Commun. Math. Phys. 282, 1–9 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0527-0
Communications in
Mathematical Physics
On the Existence of Dark Solitons in a Cubic-Quintic Nonlinear Schrödinger Equation with a Periodic Potential Pedro J. Torres1, , Vladimir V. Konotop2,3, 1 Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain.
E-mail:
[email protected]
2 Centro de Física Teórica e Computacional, Universidade de Lisboa,
Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, Lisbon 1649-003, Portugal
3 Departamento de Física, Universidade de Lisboa, Campo Grande, Ed. C8, Piso 6,
Lisboa 1749-016, Portugal Received: 5 June 2006 / Accepted: 27 January 2008 Published online: 11 June 2008 – © Springer-Verlag 2008
Abstract: A proof of the existence of stationary dark soliton solutions of a cubic-quintic nonlinear Schrödinger equation with a periodic potential is given. It is based on the interpretation of the dark soliton as a heteroclinic of the Poincaré map. 1. Introduction In the present paper we consider stationary solutions of the cubic-quintic nonlinear Schrödinger equation (CQNLS) iψt + ψx x − V (x)ψ − g1 |ψ|2 ψ − g2 |ψ|4 ψ = 0
(1)
with a real, even, and L-periodic potential: V (x) = V (−x) = V (x + L), which is also considered to be bounded. The constants g1 and g2 introduced in Eq. (1) are real. More specifically, we are interested in solutions which allow the representation ψ(t, x) = e−iωt φ(x), where ω is a real constant, through the text referred to as the frequency. Then the function φ solves the stationary equation φx x + V˜ (x)φ − g1 φ 3 − g2 φ 5 = 0
(2)
with V˜ (x) ≡ ω − V (x), subject to the nonzero boundary conditions φ(x) → φ+ (x) as x → +∞ and φ(x) → φ− (x) as x → −∞
(3)
with the functions φ± (x) being real, bounded, sign definite, and L-periodic solutions of (2). The work of PJT was supported by D.G.I. MTM2005-03483, Ministerio de Educación y Ciencia, Spain. The work of VVK was supported by the Secretaria de Estado de Universidades e Investigación (Spain) under the grant SAB2005-0195 and by the FCT (Portugal) and European program FEDER under the grant POCI/FIS/56237/2004.
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For the sake of convenience, in Fig. 1 we illustrate the concepts introduced in this paper, using the case of a particular solution considered in detail in Sect. 4. The described solutions φ(x) will be identified as dark solitons. In a general case a stationary solution of Eq. (1) ψ(t, x) from C2 ((0, ∞) × R), subjected to the boundary conditions lim x→±∞ ψ(x, t) = φ± (x)e−iωt , can be represented in the form ψ(t, x) = e−iωt+iθ(x) φ(x), where θ (x) and φ(x) are real bounded functions from C2 (R). Since the potential is real-valued, one can look for a real solution φ(x), which corresponds to θ (x) ≡const and is achieved by factoring out the x-independent phase. If however an additional constraint φ+ (x)φ− (x) < 0 is imposed, such real solutions (or more precisely solutions with phases independent on x) exhaust all possible bounded and differentiable solutions. Indeed, substituting the representation ψ(t, x) = e−iωt+iθ(x) φ(x) into Eq. (1), separating real and imaginary parts of the obtained equation, and integrating once the imaginary part of the equation with respect to x, we obtain the link φ 2 θx = C, where C is the integration constant. This relation together with the real part of the equation: φx x − θx φ + V˜ (x)φ − g1 φ 3 − g2 φ 5 = 0, allows one to conclude that if φ acquires zero at some point of the space, then C = 0, which in its turn means that the argument θ is a constant. Thus, taking into account that φ+ (x)φ− (x) < 0 for all x, a continuous function φ(x) satisfying the conditions (3) becomes zero in at least one point of the real axis, then one can consider φ(x) as pure real and satisfying Eq. (2). This fact was taken into account in the passage from (1) to (2). The model (1) having a general character, it describes weakly dispersive and weakly nonlinear wave processes, recently attracted considerable attention in connection with its application to the mean-field theory of Bose-Einstein condensates [1]. In this context,
1
φ+ 0.5
0
-0.5
φ1
φ2
φ–
-1 -5
0
x
5
Fig. 1. The dark soliton – heteroclinic (solid line), lower and upper solutions ρ1,2 (dashed-dotted line), and the hyperbolic periodic solutions φ± (x) given by (12), for the potential (11) with the parameters k = 0.5 and ρ = 1 (see below)
Dark Solitons in a Cubic-Quintic Nonlinear Schrödinger Equation with a Periodic Potential
3
ψ(x) is a macroscopic wave function, ω plays the role of the chemical potential, |ψ(x)|2 describes the linear atomic density, and V (x) is an optical lattice created by standing laser beams [2]. In particular, existence of spatially localized modes (also referred to as bright solitons) has been recently addressed in Refs. [4,5]. In Ref. [5] various families of the solutions were presented and significant differences in behavior of the stationary modes of the standard cubic nonlinear Schrödinger equation (i.e. of Eq. (2) with g1 = 0 and g2 = 0) and of the quintic nonlinear Schrödinger equation (i.e. of Eq. (2) with g1 = 0 and g2 = 0), have been found. Dark solitons of the NLS equation with a periodic potential have also been discussed in the small amplitude limit [6] and for a general case they have been studied numerically in Ref. [7] (see also [2] and references therein). The approach developed in [7] was based on the numerical study of the Poincaré map generated by Eq. (2) and considered at instants n L. In that approach dark solitons appear as heteroclinics of the map. The aim of the present paper is to extend earlier studies, providing for the first time a rigorous proof of the existence of a dark soliton solution of Eq. (1). From the mathematical point of view, the strategy of proof combines in a novel way several techniques from the classical theory of ODE’s (the upper and lower solutions [8] and the truncation arguments, as in Theorem 1 below) and the dynamics of planar homeomorphisms (the topological degree [9–11] and the free homeomorphisms [12]). We will restrict the consideration to the case g2 > 0 only, which rules out any possibility of blowing up solutions. Then without loss of generality we can set g2 = 1 through the rescaling, which is done in what follows. To conclude the Introduction, we notice that the method presented in this paper can be seen as a novel general approach working in a more general framework, including some relevant examples like the cubic Schödinger equation with inhomogeneous nonlinearity, i.e. Eq. (2) with g2 ≡ 0 and variable g1 (x). We have limited our analysis to the cubic-quintic Schrödinger equation for the sake of clarity. 2. Existence of Sign Definite Periodic Solutions We start with a proof of existence of sign definite periodic solutions. To this end we specify the condition of boundness of V (x): there exist constants Vmin , Vmax such that Vmin < V (x) < Vmax , consider ω > Vmax
(4)
and introduce the notations λ21 = ω − Vmax and λ22 = ω − Vmin . As it is clear 0 < λ21 ≤ V˜ (x) ≤ λ22 . Next we consider two stationary equations as follows ( j = 1, 2): φ j,x x + λ2j φ j − g1 φ 3j − φ 5j = 0.
(5)
Treating these equations as dynamical systems, one easily finds the (only) two nontrivial real equilibria ±ρ j , where √ g12 + 4λ2j − g1 / 2. (6) ± ρj = ± These are the hyperbolic points +ρ j and −ρ j connected by the heteroclinic orbits, which explicit forms read φj =
ρ j α j tanh(k j x) ρ 2j + α 2j − ρ 2j tanh2 (k j x)
,
(7)
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P. J. Torres, V. V. Konotop
where
αj =
3 g1 2ρ 2j + g1 , and k j = ρ j ρ 2j + , 2 2
(8)
and without loss of generality the condition φ1 (0) = φ2 (0) = 0 is imposed. Let us call ρ1,2 the positive equilibria. As it is clear φ1 (x) < φ2 (x) for x > 0. At this point some considerations about the general second order equation φx x = f (x, φ)
(9)
with f (x, φ) continuous with respect to both arguments and L-periodic in x, are required. The following definition is classical (see for instance [8] and references therein). Definition 1. A function α : [x0 , +∞) → R such that αx x (x) > f (x, α) (αx x (x) < f (x, α)) for all x > x0 is called a strict lower (upper) solution of Eq. (9). We notice that the strict inequalities are required for the proper definition of the Brouwer degree in the area between the lower and the upper solutions [8]. Now we can formulate Proposition 1. ρ1 and ρ2 are respectively strict lower and upper solutions of Eq. (2). Hence there exists an unstable L-periodic solution between them. Proof. Let us observe that for j = 1, ρ1,x x + V˜ (x)ρ1 − g1 ρ13 − ρ15 > λ21 ρ1 − g1 ρ13 − ρ15 = 0,
(10)
and similarly for j = 2. Hence, ρ1 < ρ2 are a couple of well-ordered lower and upper solutions respectively; therefore there exists a periodic solution between them [8]. Such a solution is unstable because the associated Brouwer index to the Poincaré map is −1 (see for instance [11,13]). Therefore we have a positive L-periodic solution of Eq. (2). Designating it as φ+ (x), we have that ρ1 ≤ φ+ (x) ≤ ρ2 . By the symmetry of equation we also have a negative solution φ− (x) = −φ+ (x). To give an example of a periodic solution, we consider the simplified quintic nonlinear Schrödinger model (1) with g1 = 0 and with the potential V (x) = −ρ 4 [2 − k 2 sn2 (ρ 2 x, k)]2 ,
(11)
where sn(x, k) is the Jacobi elliptic function (hereafter we use the standard notations for the Jacobi elliptic functions, which can be found e.g. in [14]), k ∈ [0, 1] is the elliptic modulus, and ρ > 0 (examples of the exact periodic solutions for the cubic nonlinear Schrödinger equation, g2 = 0, and with the potential V (x) = sn2 (x, k), were obtained in [17,18]). The respective positive definite solution reads φ+ (x) = ρ dn(ρ 2 x, k)
(12)
(dn(x, k) is the Jacobi elliptic function, where as above k is the elliptic modulus). The solution (12) corresponds to the frequency ω = ρ 4 (k 2 −3). To verify the stability of φ in ˜ the sense of the dynamical system (2) we consider a small deviation φ(x) = φ(x)−φ+ (x) at x → ∞, whose dynamics in the leading order is governed by the equation φ˜ x x − U (x, k)φ˜ = 0
(13)
Dark Solitons in a Cubic-Quintic Nonlinear Schrödinger Equation with a Periodic Potential
5
with U (x, k) = ρ 4 (4 − k 2 − 6k 2 sn(ρ 2 x, k)2 + 4k 4 sn(ρ 2 x, k)4 ) 7 ≥ ρ 4 ( − k 2 ) > 0. 4
(14)
Thus, the obtained function φ+ (x) is a hyperbolic periodic solution of Eq. (2) with the potential (11). Considering now ψ+ (x, t) = φ+ (x) exp(iρ 2 (3 − k 2 )t) as a solution of Eq. (1), performing the stability analysis as in Ref. [17], only slightly modified due to the presence of quintic nonlinearity, and taking into account that φ+ (x) > 0, one verifies that ψ+ (x, t) is linearly stable in the sense of the evolution problem (1). More sophisticated models allowing exact sign definite periodic solutions can be constructed using a kind of “reverse engineering” (i.e. by obtaining potentials starting with given periodic solutions) as it is explained in [2]. 3. Existence of a Dark Soliton In this section we prove the existence of a heteroclinic orbit connecting the periodic solutions φ− and φ+ . A battery of preparatory lemmas is necessary. Lemma 1. If φ : [x0 , +∞) → R is a bounded solution of Eq. (9), then the derivative φx is also bounded in [x0 , +∞). Proof. By the hypothesis |φ(x)| < M, where M is a constant, for all x ≥ x0 . Then, by the mean value theorem there exists xn ∈ (n L , (n+1)L) such that φ((n+1)L)−φ(n L) = φx (xn )L for n > n 0 . From here φx (xn ) < 2M/L for all n. Applying the mean value theorem one more time one obtains |φx (x) − φx (xn )| < L max | f (x, φ)|, ∀x ∈ (n L , (n + 1)L) |φ|≤M
(15)
because |φx x (x)| < max|φ|≤M | f (x, φ)|. Then |φx (x)| < L max | f (x, φ)| + |φ|≤M
2M , ∀x ≥ x0 . L
(16)
Let us denote by γ {I − P, p0 } the local index associated to the Brouwer degree of p0 as a fixed point of the homeomorphism P (see [9,10] for a rigorous definition of the Brouwer degree). The following lemma is a key ingredient in our main result. Lemma 2 ([15]). Let P : R2 → R2 be an orientation preserving homeomorphism with a unique fixed point p L such that γ {I − P, p L } = 1. Then for any p0 ∈ R2 one of the following possibilities holds i) P n ( p0 ) → p L as n → +∞, ii) P n ( p0 ) → ∞ as n → +∞. The proof relies on a basic property of free homeomorphisms exposed in [12], namely if an orientation preserving homeomorphism has a unique fixed point p L and has index different from 1, then it is a free homeomorphism. Then, by [12, Theorem 5.3] the ω-limit set of a given point has to be a connected set of the fixed point set.
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Lemma 3. If f (x, y) is strictly increasing in y, there exists atmost one L-periodic solution of (9). Proof. By contradiction, let us assume that y1 , y2 are two different L-periodic solutions of (9). First, let us suppose that y1 , y2 intersect among themselves, that is, there should be x1 , x2 such that z(x) = y1 (x) − y2 (x) verifies z(x1 ) = 0 = z(x2 ) and z(t) > 0 for t ∈ (x1 , x2 ). However, by subtracting the corresponding equations and using that f is strictly increasing, we get that z should be convex in (x1 , x2 ), which is a contradiction. Therefore, y1 , y2 do not intersect and we assume without loss of generality that y1 (x) > y2 (x) for all x. Again, z should be convex in the whole real line, but this is impossible because it is periodic. With the help of these auxiliary lemmas we are able to prove an abstract convergence result. Theorem 1. Let φ : [x0 , +∞) → R be a bounded solution of (9). Let us assume that min
x∈[0,L] y∈[inf x≥x0 φ(x),supx≥x φ(x)] 0
∂ f (x, y) > 0. ∂y
(17)
Then there exists an L-periodic solution ϕ(x) such that lim (|φ(x) − ϕ(x)| + |φx (x) − ϕx (x)|) = 0.
x→+∞
(18)
Proof. Let us define m = inf x≥x0 φ(x) and M = supx≥x0 φ(x), as well as the truncated function ⎧ ∀y ∈ [m, M] ⎨ f (x, y), ∀y > M . f˜(x, y) = f (x, M) + f y (x, M)(y − M), (19) ⎩ f (x, m) + f (x, M)(y − m), ∀y < m y f˜ is strictly increasing in y. Note also that φ is a solution of the truncated equation φx x = f˜(x, φ).
(20)
Obviously, lim y→±∞ f˜(x, y) = ±∞ uniformly in x. Hence, there exist constants α and β, such that α < β and f˜(x, α) < 0 < f˜(x, β) for all x. Such α and β is a well-ordered couple of L-periodic lower and upper solutions. Hence there exists an L-periodic solution of (20) between them [8]. Like in Proposition 1, this solution is unstable and the associated Brouwer index to the Poincaré map is −1. By the previous lemma, this solution is unique. Then, by Lemma 2, φ(x) must converge to ϕ(x) since Lemma 1 excludes the possibility ii) of Lemma 2. As ϕ(x) ∈ [m, M], it is a solution of (9). Theorem 2. Let us consider bounded functions φ1 , φ2 : [x0 , +∞) → R verifying 1) φ1 (x) < φ2 (x), ∀x > x0 , 2) φ1 x x (x) > f (x, φ1 ) and φ2 x x (x) < f (x, φ2 ), ∀x > x0 . Then there exists a solution φ(x) of (9) such that φ1 (x) < φ(x) < φ2 (x). If moreover, there exists x such that
(21)
Dark Solitons in a Cubic-Quintic Nonlinear Schrödinger Equation with a Periodic Potential
3)
min
x∈[0,L] y∈[inf x≥x0 φ1 (x),supx≥x φ2 (x)] 0
7
∂ f (x, y) > 0, ∂y
then there exists an L-periodic solution ϕ(x) such that lim (|φ(x) − ϕ(x)| + |φx (x) − ϕx (x)|) = 0.
x→+∞
(22)
Besides, ϕ(x) is the unique L-periodic solution in the interval [inf x≥x0 φ1 (x), supx≥x0 φ2 (x)]. Proof. The first assertion (21) is the classical result due to Opial [16]. The second conclusion (22) is a corollary of Theorem 1. In order to apply the above results to our model (2), we observe that now i) f (x, y) ≡ y 5 + g1 y 3 − V˜ (x)y, ii) due to parity of the potential one can consider x ≥ 0 and extend the obtained solution φ(x) as an odd function to x ≤ 0, iii) the functions φ1,2 (x) given by (7) satisfy the conditions 1) and 2) of the Theorem 2. Hence, in order to prove that there exists a solution φ(x) of (1) converging to φ± (x), found in Proposition 1, as x → ±∞, one has to verify the condition 3) of Theorem 2. As x0 can be taken arbitrarily large, this last condition is equivalent to (23) min 5y 4 + 3g1 y 2 − V˜ (x) > 0. y∈[ρ1 ,ρ2 ] x∈[0,L]
Starting with the case g1 ≥ 0 we observe that (23) is now equivalent to 5ρ14 +3g1 ρ12 −λ22 > 0. The straightforward analysis of this last inequality, which takes into account the link (6), the definition of λ1,2 , and the requirement (4) necessary for λ21 > 0, gives the following estimate for the frequency: ω − g1 g12 + 4ω − 4Vmax > −Vmin + 5Vmax . (24) Thus (24) is a sufficient condition for the existence of a dark soliton at the non-negative g1 . After some cumbersome but straightforward computations, one realizes that (24) is equivalent to the explicit bound ω > ω0 = 5Vmax − Vmin + 2g1 + g13 + 4g12 + 4g1 (4Vmax − Vmin ). (25) Considering now g1 < 0 the constraint (23) is reduced to 5ρ14 − 3|g1 |ρ22 − λ22 > 0 and subsequently to the following inequality constraint to the frequency 2 8ω + 2g1 − 10Vmax + 2Vmin − 5g1 4ω + g12 − 4Vmax +3g1 4ω + g12 − 4Vmin > 0 (26) which must be satisfied simultaneously with (4). In this case, we omit the explicit bound for ω. In both cases considered above, the conclusion is that there exists an explicitly computable ω0 such that the CQNLS has a dark soliton for any ω > ω0 .
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4. Concluding Remarks First of all we emphasize that ω is not a parameter of the original Eq. (1), but just the temporal frequency of a stationary solution. The obtained condition (26) is needed for technical reasons. Clearly, ω > ω0 implies that the origin of Eq. (2) is a center, so that the possible heteroclinics must connect non-trivial solutions. Otherwise, when the origin is a hyperbolic point one can expect the presence of homoclinics (this is well known) connecting the origin with itself, corresponding to bright solitons of Eq. (1) (for numerical examples of such homoclinics in cubic and quintic nonlinear Schrödinger equations we refer to [2,7] and to [5], respectively). The other imposed condition (4) guarantees the existence of the two nontrivial hyperbolic points (6) (we also notice that in physical terms, condition (4) means that the chemical potential is bigger than the amplitude of the periodic potential) implying the existence of sign definite periodic solutions φ± . This naturally leaves an open question about the existence of dark solitons whose asymptotics at x ± ∞ are given by sign-alternating nonlinear waves (such a possibility is suggested by the numerical simulations [2,6,7]). Finally, we consider a particular example illustrating a dark soliton, as well as other concepts introduced in the paper. To this end we recall the potential (11) and construct a dark soliton which tends to φ+ given by (12) (we thus consider now g1 = 0). This cannot be done analytically, and that is why we employ numerics. An example is shown in Fig. 1. To numerically obtain the dark soliton, we use the shooting method [7]. To this end we observe that (13) is a Hill’s equation and thus taking into account (14), from Flo˜ quet’s theorem we obtain that φ(x) → φ(x) = P(x) exp(−αx) as x → +∞, where P(x) = P(x +2K(k)/ρ 2 ) is a periodic function with the period L = 2K(k)/ρ 2 and K(k) is the complete elliptic integral of the first kind. For given parameters k and ρ one can easily compute the respective Floquet exponent α. In particular, for our choice of k = 0.5 and ρ = 1, we obtain α ≈ 2.014298. Thus, starting with a point xini = 2nK(k)/ρ 2 , where n is an integer, for which the equalities φ(xini ) = φ+ (0)−C and φx (xini ) = −Cα with a positive constant C are verified, we compute φ(0) by varying the parameter C. We start with C = 0 and increase C until we meet the condition φ(0) = 0. The respective smallest positive value of C corresponds to the dark soliton we are looking for. An example of numerical implementation of this procedure is shown in Fig. 1. References 1. Pitaevskii, L.P., Stringari, S.: Bose-Einstein condensation. Oxford: Oxford University Press, 2003 2. Brazhnyi, V.A., Konotop, V.V.: Theory of nonlinear matter waves in optical lattices. Mod. Phys. Lett. B 18, 627 (2004) 3. Brazhnyi, V.A., Konotop, V.V., Pitaevskii, L.P.: Dark solitons as quasiparticles in trapped condensates. Phys. Rev. A 73, 053601 (2006) 4. Abdullaev, F.K., Salerno, M.: Gap-Townes solitons and localized excitations in low-dimensional BoseEinstein condensates in optical lattices. Phys. Rev. A 72, 033617 (2005) 5. Alfimov, G.L., Konotop, V.V., Pacciani, P.: Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential. Phys. Rev. A 75, 023624 (2007) 6. Konotop, V.V., Salerno, M.: Modulational instability in Bose-Einstein condensates in optical lattices. Phys. Rev. A 65, 021602(R) (2002) 7. Alfimov, G.L., Konotop, V.V., Salerno, M.: Matter solitons in Bose-Einstein condensates with optical lattices. Europhys. Lett. 58, 7 (2002) 8. De Coster, C., Habets, P.: Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. In: Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations, ed. F. Zanolin, CISM-ICMS Courses and Lectures 371, New York: Springer Verlag, 1996 9. Lloyd, N.G.: Degree Theory. Cambridge: Cambridge University Press, 1978
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10. Mawhin, J.: Topological degree and boundary value problems for nonlinear differential equations. In: “Topological Methods for Ordinary Differential Equations”, (Montecatini Terme, 1991), M. Furi, P. Zecca eds., Lecture Notes in Mathematics 1537, Berlin: Springer-Verlag 1993, pp. 74–142 11. Ortega, R.: Some applications of the topological degree to stability theory. In: “Topological methods in differential equations and inclusions”, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 472, Dordrecht: Kluwer Acad. Publ., 1995, pp. 377–409 12. Brown, M.: Homeomorphisms of two-dimensional manifolds. Houston J. of Math. 11, 455–469 (1985) 13. Dancer, E.N., Ortega, R.: The index of Lyapunov stable fixed points in two dimensions. J. Dyn. Differ. Eqs. 6, 631 (1994) 14. Lawden, D.K.: Elliptic functions and applications. New York: Springer-Verlag, 1989 15. Campos, J., Torres, P.: On the structure of the set of bounded solutions on a periodic Lienard equation. Proc. Amer. Math. Soc. 127, 1453 (1999) 16. Opial, Z.: Sur les intégrales bornees de l’equation u = f (t, u, u ). Ann. Polonici Math. 4, 314–324 (1958) 17. Bronski, J.C., Carr, L.D., Deconinck, B., Kutz, J.N., Promislow, K.: Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E 63, 036612 (2001) 18. Bronski, J.C., Carr, L.D., Carratero-Gonzalez, R., Deconinck, B., Kutz, J.N., Promislow, K.: Stability of attractive Bose-Einstein condensates in a periodic potential. Phys. Rev. E 64, 056615 (2001) 19. Sulem, C., Sulem, P.: The nonlinear Schrödinger equation: Self-focusing and wave collapse. Berlin: Springer 2000 20. Rybin, A.V., Varzugin, G.G., Lindberg, M., Timonen, J., Bullough, R.K.: Similarity solutions and collapse in the attractive Gross-Pitaevskii equation. Phys. Rev. E 62, 6224 (2000) 21. Krasnosel’skii, M.A., Burd, V.Sh., Kolesov, Yu.S.: Nonlinear almost periodic oscillations. New York: John Wiley & Sons, 1973 Communicated by P. Constantin
Commun. Math. Phys. 282, 11–54 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0531-4
Communications in
Mathematical Physics
Towards a Nonperturbative Renormalization Group Analysis Haru Pinson Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA. E-mail:
[email protected];
[email protected] Received: 28 February 2007 / Accepted: 27 February 2008 Published online: 10 June 2008 – © Springer-Verlag 2008
Abstract: We prove that a certain class of convex gradient models in high dimensional spaces without the presence of a small parameter renormalizes to a free field. As a consequence we establish a certain asymptotic formula for the partition function. In some ways, this is a realization of Gawedzki and Kupiainen’s idea to use correlation inequalities to augment the rigorous renormalization group methods. We use the more particular suggestion of Spencer to use certain inequalities of Brascamp and Lieb and also the formulation of the correlation functions in terms of the solutions to some partial differential equations given by Helffer and Sjöstrand. There are interesting mathematical challenges developed through the use of the renormalization group flow. One such example is to rigorously understand the sigma models ([F,G]). Our goals are much more modest here. One issue in regard to the renormalization group flow which has not been addressed is to understand the renormalization group flow if the initial model is some distance away from the actual fixed point. In the simplest convex gradient models without the presence of a small parameter in high dimensions (d ≥ 24), we show that these models indeed flow toward the free field. These are infinite dimensional systems. Thus an abundant amount of information must be “packaged” from one scale to the next. So far various forms of cluster expansions are used to deal with this information. A small parameter is needed to control this expansion. In the situation where no such small parameter is present, we instead rely, in a word, on certain inequalities. However, the use of these inequalities, as of now, requires the Hessian of the action to be positive definite. Fortunately, just as in the case of the Ricci flow [H], where the Ricci flow preserves the positivity of the various curvature quantities, the renormalization group flow preserves the positivity of the Hessian. Thus we can use the various techniques developed by Helffer and Sjöstrand [HS]. Perhaps it is not inappropriate to recall that Hamilton also analyzed the Ricci flow for the convex case first [H]. The author is partially supported by the NSF grant DMS0301830.
12
H. Pinson
Our method is a conservative approach. One can imagine far more bold renormalization group analysis, especially in light of Hamilton’s Ricci flow theory. One can easily imagine that the various techniques, for example, the existence of the various monotonic quantities and the various Harnack inequalities, can be carried over to the renormalization group flow. Perhaps one can even contemplate φ 4 in three dimensions. The techniques involved in our analysis are not tied to the free field. Rather the convexity is the main requirement. 1. Past Works and an Outline for this Paper See the introduction in the pioneering work [WK] for the history in regard to the renormalization group. The first rigorous renormalization group analysis was done by Gawedzki and Kupiainen [GK]. Later developments occurred in the works of [FMRS,BY] and others. The renormalization group is a step by step process of integrating out different length scales. The first integration step is different. There is no scaling. This is done in Sect. 3. The key lemmas are Lemmas 3.1, 3.3. An important observation was made by Helffer and Sjöstrand in relating integrals to solutions of some partial differential equations, for example, (3.32), (3.37), (3.41). We need to invert this differential operator (3.56). This analysis is known [NS,N,G] and others. However, in our case, the Hessian becomes non local, and this causes a few minor problems. We spell out the needed modification in the Appendix (see Lemma 3.2). In doing the renormalization analysis, we need to control the effective action fˆn+1 (ψ) (see (4.19)) at the next scale for all values of ψ ∈ R|| . is the lattice on which these gradient systems are defined. The main point to this paper can be said to be contained in Lemma 4.1. This is a small field in ψ statement (the condition in Lemma 4.1 is what we mean by a small field). Of course, we need control over all values of ψ ∈ R|| . However, there is a formula (4.33) for the effective action fˆn+1 (ψ) which shows that the effective action only depends on the extracted quadratic pieces (3.71), (4.16), (4.21), which are obtained from the effective action evaluated at ψ = 0. Thus, the inductive hypothesis (the condition in Lemma 4.1) gives us control over all values of ψ. The convexity property of the integrand of (4.33) is contained in (4.35). The exponential decay or the locality of the effective action fˆn+1 (ψ) for all values of ψ ∈ R|| is then established in Lemma 4.4. In the small field region, we need to understand the one point function since all formalism of Brascamp and Lieb and Helffer and Sjöstrand are written in terms of the one point function. In the small field region, we estimate the one point function (see Lemma 4.6). We can then use the techniques of Brascamp and Lieb and Helffer and Sjöstrand to establish (4.27) and (4.28). In Section Five, we put all the arguments together to prove this theorem. 2. Models and Statement of Results Take a finite lattice ⊂ Zd with a periodic boundary condition. We can assume that the lattice has the origin as the center. Specifically, we consider lattices in 24 dimensional spaces. From now on, we use d to denote 24. Our methods can be applied to lattices in dimensions higher than 24. At least to our thinking, just fixing the dimension to 24 simplifies the exposition of our paper. We need not worry about the various quantities which otherwise would have dimensional dependence.
Towards a Nonperturbative Renormalization Group Analysis
13
Define unit vectors in Zd by e1 = (1, 0, . . . , 0), e2 = (0, 1, 0, . . . , 0), . . . , ed = (0, . . . , 0, 1).
(2.1)
Let f : R → R be a smooth, even ( f (x) = f (−x), x ∈ R), convex function whose second derivative is bounded above, 0 ≤ f (x) ≤ C.
(2.2)
We also assume that the third derivative of f is bounded (pointwise). To each lattice site x ∈ , let φx ∈ R. Define ∇x,y φ ≡ φx − φ y ,
(2.3)
M d ≡ ||.
(2.4)
With these definitions, we define the partition function 1 − dx,i=1 (∇x,x+ei φ)2 − x d+κ φx2 − dx,i=1 M Z = e
f (∇x,x+ei φ)
dφx ,
(2.5)
x∈
κ is a positive constant. Denote − dx,i=1 (∇x,x+ei φ)2 − x∈ z0 = e
1 M d+κ
φx2
dφx .
(2.6)
x∈
We prove the following. Theorem. There exist F, c(), c∞ ∈ R such that Z = e F||+c() z0
(2.7)
lim||→+∞ c() = c∞ .
(2.8)
and
The constant c∞ can depend on the particular form of the interaction f . Remark. For us, the main point to this theorem is to show that these models approach the free field under the renormalization group flow. This statement is effectively contained in Lemma 4.1. We do need the constant κ to be positive. Otherwise it looks like there will be some other contribution in || raised to a power smaller than one; the log contribution can perhaps arise too. (We knew of this possibility, but some comments at Joel Lebowitz’s conference made it clear that we should mention this aspect.) See (5.28), (5.29). Perhaps this aspect needs to be sorted out somewhere. It looks like these asymptotic formulas for the partition function pick up information very close to the zero mode. Also there will be some boundary contributions if this analysis is carried out for different boundary conditions. In an earlier version of this paper, we thought that the constant c∞ is actually zero. However, there is a zero mode contribution. Perhaps it may be worthwhile to investigate this quantity for various other models. This was done for the 2 dimensional Ising model in [PS], and the constant c∞ is zero.
14
H. Pinson
3. First Step of the Integration The first step in our renormalization group analysis is different. There is no scaling which drives the interacting term f toward zero. We can break up the domain of integration as Z = z0
e
− f 0 (φ+ψ) e
f
−(Q 0 )(φ) dφ e−(Q s0 )(ψ) dψ
z 0s
f
z0
,
(3.1)
where ei ei
f
x∈ tx φx
e−(Q 0 )(φ) dφ f z0
f −1 x,y∈ tx t y (Q 0 )x y
,
1 s −1 e−(Q 0 )(ψ) dψ = e− 4 x,y∈ tx t y (Q 0 )x y , tx ∈ R, x ∈ , s z0
(3.2)
s
x∈ tx ψx
(Q 0 )−1 xy = f
(Q s0 )−1 xy =
1
= e− 4
1 e−i p(x−y) , M d p g( p) + e−L 2 g( p) 2 2 A L
(3.3)
(3.4)
2 e−i p(x−y) e−L g( p) 1 1 −i p(x−y) β0 ( p) , (3.5) e = d 2 g( p) d −L M p A2 L 2 g( p)(g( p) + e M p h( p) ) 2 2 A L
e−L g( p) ), A2 L 2 2
β0 ( p) = e−L
2 g( p)
, h( p) = A2 L 2 g( p)(g( p) +
A x,y ≡
1 −i p(x−y) β0 ( p) e , Md p L 2 A2
g( p) = 2(d −
d
cos( pi )) +
i=1
p = ( p1 , . . . , pd ) ∈ {(
1 , M d+κ
(3.6)
(3.7)
(3.8)
2π 2π k1 , . . . , kd )|ki = 0, 1, . . . , M − 1, i = 1, . . . , d}. (3.9) M M
L is some very large positive number. A is also a large number but much smaller than L. For example, we can take A to be (log(L))2 . We denote f 0 (φ) ≡
d x,i=1
f (∇x,x+ei φ).
(3.10)
Towards a Nonperturbative Renormalization Group Analysis
15 f
The formula (3.4) defines the inverse of the quadratic form Q 0 . Thus this (3.4) defines f the quadratic form Q 0 . Similar comments apply to Q s0 . f 0 (k) eikφ dk and substitute this into both Take the Fourier transform e− f0 (φ) = e− sides of the equality (3.1) to verify (3.1). Since the resulting integrals are Gaussian, both sides can be evaluated. We define the effective action fˆ1 (ψ) by f dφ ˆ e− f1 (ψ) = e− f0 (φ+ψ)−Q 0 (φ) f . (3.11) z0 By substituting (3.11) into (3.1), (3.1) becomes s Z dψ ˆ = e− f1 (ψ) e−Q 0 (ψ) s . z0 z0
(3.12)
The quadratic form given by (3.7) can be written as
A x,y φx φ y = −
x,y
φx
1
x
2
z
L2
ˆ x,z φ + A x,x+z
− e M d+κ 2 φ , L 2 A2 x x
(3.13)
where ˆ x,z φ = 2φx − φx+z − φx−z .
(3.14)
f
The quadratic form Q 0 defined in (3.4) can then be written as f Q 0 (φ)
=
x,z
where
Bx,z =
−
L2
e M d+κ 1 ˆ x,z φ + ( Bx,z φx + d+κ ) φx2 , 2 2 L A M x
1 2
−
−
A x,x+z 2
A x,x+z 2
if z = ±ei , i = 1, . . . , d otherwise.
Our next task is to understand the following polynomial in η, γ : fˆ1,ψx ψ y (ψ)ηx γ y ,
(3.15)
(3.16)
(3.17)
x,y∈
where we have used the notation fˆ1,ψx ψ y (ψ) ≡
∂2 fˆ1 (ψ). ∂ψx ∂ψ y
(3.18)
In the subsequent discussions, we will use this notation to indicate the partial derivative. By translating φ → φ − ψ, we have f f f dφ ˆ e− f1 (ψ) = e− f0 (φ)−Q 0 (φ,φ)+2Q 0 (φ,ψ)−Q 0 (ψ,ψ) f . (3.19) z0
16
H. Pinson
By taking second derivatives of (3.19), we arrive at
fˆ1,ψx ψ y (ψ)ηx γ y = 2
x,y
x,z
−
−4
ηx γ y (
x,y
(
z2
L2
e M d+κ 1 ˆ x,z γ + 2( Bx,z ηx + d+κ ) ηx γx 2 2 L A M x z1
−
L2
e M d+κ 1 ˆ x,z 1 φ + ( Bx,z 1 + d+κ )φx ); 2 2 L A M −
L2
e M d+κ 1 ˆ y,z 2 φ + ( B y,z 2 + d+κ )φ y ) : 1 c , 2 2 L A M
(3.20)
where we denote
• =
f ; g : h c = ( f − f )(g − g )h ,
(3.21)
f ; g; h : k c = ( f − f )(g − g )(h − h )k ,
(3.22)
f
•e− f0 (φ) e−Q 0 (φ−ψ)
dφx /
f
e− f0 (φ) e−Q 0 (φ−ψ)
x
dφx .
(3.23)
x
Often, the functions h : R|| → R in (3.21) will be identically equal to one. Thus, in that case, (3.21) becomes the connected or the truncated two point function. First, we do not want to start taking the absolute values of the second derivative of fˆ1 (ψ). We must rewrite this expression (3.20) by summing by parts. The expression for (3.20) becomes 2
−
L2
e M d+κ 1 Bx,z ∇x,x+z η∇x,x+z γ + 2( 2 2 + d+κ ) ηx γx L A M x
x,z
−4{
ˆ x,z 1 ηB y,z 2 ˆ y,z 2 γ
φx ; φ y : 1 c Bx,z 1
x,z 1 ,y,z 2 L2
− e M d+κ 1 ˆ x,z 1 ηγ y +( 2 2 + d+κ )
φx ; φ y : 1 c Bx,z 1 L A M x,z ,y 1
+(
2 − Ld+κ M
e 1 ˆ y,z 2 γ + d+κ )
φx ; φ y : 1 c ηx B y,z 2 2 2 L A M x,y,z 2
+(
2 − Ld+κ M
e 1 + d+κ )2
φx ; φ y : 1 c ηx γ y }. 2 2 L A M x,y
(3.24)
We started with a system with all finite difference terms (2.5). Now there are terms ηx γ y which are not finite differences. We can remedy this situation. First note 0 = φx − φx .
(3.25)
Towards a Nonperturbative Renormalization Group Analysis
17
Perform a change of variables φx → φx + a for a ∈ R in the numerator of the exterior
• . Then, −
L2
e M d+κ 1 d φ y , (3.26) 0 = φx + a − φx |a=0 = 1 − 2( 2 2 + d+κ ) (φx − φx ) da L A M y which implies
φx ; φ y : 1 c = y
1 2(
2 − L e M d+κ 2 L A2
. +
1 M d+κ
(3.27)
)
With this equality, (3.24) becomes ˆ x,z 1 ηB y,z 2 ˆ y,z 2 γ 2 Bx,z ∇x,x+z η∇x,x+z γ − 4{
φx ; φ y : 1 c Bx,z 1 x,z 1 ,y,z 2
x,z
+(
2 − Ld+κ M
e 1 ˆ x,z 1 η∇ y,x γ + )
φx ; φ y : 1 c Bx,z 1 L 2 A2 M d+κ x,z ,y 1
1 Bx,z 1 ∇x,x+z 1 η∇x,x+z 1 γ + 2 x,z 1
−
+(
L2 M d+κ
1 e ˆ y,z 2 γ + d+κ )
φx ; φ y : 1 c ∇x,y ηB y,z 2 2 2 L A M x,y,z 2
1 B y,z 2 ∇ y,y+z 2 η∇ y,y+z 2 γ + 2 y,z 2
−
L2 M d+κ
1 e 1 − ( 2 2 + d+κ )2
φx ; φ y : 1 c ∇x,y η∇x,y γ }. 2 L A M x,y
(3.28)
We have succeeded in rewriting all terms as finite differences. We will need some estimates for the size of these coefficients in (3.28). We prove the following lemma. Lemma 3.1. a.
|x−y|
| φx ; φ y : 1 c |e C L A |x − y|l ≤ L 2+l L 30 ,
(3.29)
y
b.
|x−y|
| f 0,ψx (φ + ψ); f 0,ψ y (φ + ψ) : 1 c |e C L A |x − y|l ≤ L 2+l L 30 ,
(3.30)
y
c.
| f 0,ψx (φ + ψ); f 0,ψ y (φ + ψ);
y,z
f 0,ψz (φ + ψ) : 1 c |e
|x−y|+|x−z| CLA
(|x − y| + |x − z|)l ≤ L 6+l L 30 ,
for l = 0, 1, . . . , d and some positive small constant > 0.
(3.31)
18
H. Pinson
Remark. All the that are written in this paper are to be regarded as the same one. We will need statement a of this lemma. We do not need statements b and c of this lemma. What we need are statements (3.74) and (3.75). However, the proofs for statements b and c of this lemma carry over to (3.74) and (3.75), so we go through the proofs of statements b and c now. Later, we just say that the proofs for statements (3.74) and (3.75) can be carried out in a similar manner as statements b and c of this lemma. Proof of Lemma 3.1. These connected correlation functions, for example, (3.38), can be expressed in terms of the solutions to certain differential equations (3.39). These solutions can be estimated (3.62),(3.63) to prove our result. We show that these correlation functions, for example, φx ; φ y : 1 c decay exponentially fast as |x − y| → +∞. The exponential decay part is done through the Combes-Thomas argument, and that part of the argument occurs in (3.45)-(3.54). Also, there is a power law decay before the exponential decay sets in. To obtain the power law decay, we need to analyze the heat kernel (3.56). The perturbation theory and a derivative argument yields the estimates in Lemma 3.2. The non local piece (3.7) causes a few problems, but this is sorted out in the Appendix. By integrating the heat kernel, we obtain, for example, (3.62). Consider the differential equation 0 u = f − f ,
(3.32)
u, f : R|| → R,
(3.33)
where
0 = −
∂2 + ∇( f 0 (φ, ψ)) · ∇, ∂φx2
(3.34)
x∈
∇=(
∂ )x∈ , ∂φx
f f 0 (φ, ψ) = Q 0 (φ − ψ) + f 0 (φ).
(3.35) (3.36)
Integrating by parts in the second derivative term in 0 [HS], we arrive at the identity
( f − f )(g − g ) = ∇u · ∇g
(3.37)
R||
with g : → R. This was the important observation made by Helffer and Sjöstrand [HS]. Thus, by understanding the solution u, we can estimate the correlation functions. Applying this formalism, we can arrive at
f 0,ψx (φ + ψ); f 0,ψ y (φ + ψ) : 1 c = ∇u x · ∇ f 0,ψ y (φ + ψ) , where
ux
(3.38)
satisfies the differential equation 0 u x = f 0,ψx (φ + ψ) − f 0,ψx (φ + ψ) .
Also, doing this twice, we can arrive at
f 0,ψx (φ + ψ); f 0,ψ y (φ + ψ); f 0,ψz (φ + ψ) : 1 c = ∇u x · ∇ f 0,ψ y (φ + ψ); f 0,ψz (φ + ψ) : 1 c + ∇u x · ∇ f 0,ψz (φ + ψ); f 0,ψ y (φ + ψ) : 1 c
(3.39)
Towards a Nonperturbative Renormalization Group Analysis
19
= ∇(∇u x · ∇ f 0,ψ y (φ + ψ)) · ∇u z + ∇(∇u x · ∇ f 0,ψz (φ + ψ)) · ∇u y . (3.40) Replace x by y to define u y and so on. We need to understand the first and second derivatives of these solutions. Thus, taking the derivative of (3.39) once and twice, we arrive at 0 u φx z + (3.41) f 0,φz φz u φx z = f 0,ψx φz (φ + ψ), z ∈
0 u φx z φw +
z ,w ∈
1 x E zw,z w u φ φ + z w
= f 0,ψx φz φw (φ + ψ) −
z ∈
z ,w ∈
2 x E zw,z w u φ φ z w
f 0,φz φw φz u zx ,
(3.42)
where 1 f 0,φw φw , E zw,z w = δz,z
(3.43)
2 f 0,φz φz δw,w . E zw,z w =
(3.44)
Let us establish the exponential decay of these solutions 1
(u φx y )2 ≤ C L 4 A4 e− ALC |x−y|
(3.45)
for some large constant C. Define || by || diagonal matrix M1 , M1 = (e
u·(x−y) ˆ ALC
δz,y )z,y∈
(3.46)
for some unit vector uˆ ∈ Rd . Rewriting (3.41), we have f 0 )}U x = F x {0 + H ess(
(3.47)
U x = (u φx z )z∈ , F x = ( f 0,ψx φz (φ + ψ))z∈ .
(3.48)
f 0 )M1−1 }M1 U x = M1 F x . {0 + M1 H ess(
(3.49)
with
Thus,
Since 0 is positive definite,
M1 U x {M1 H ess( f 0 )M1−1 − H ess( f 0 ) + H ess( f 0 )}M1 U x ≤ M1 U x · M1 F x . (3.50) Also, we have |(M1 H ess( f 0 )M1−1 )z,z + (M1 H ess( f 0 )M1−1 )z ,z − 2H ess( f 0 )z,z | z
20
H. Pinson
=|
(e
)) u·((x−z)−(x−z ˆ CLA
−1+e
)−(x−z)) u·((x−z ˆ CLA
− 1)H ess( f 0 )z,z | ≤
z
1 . (3.51) C L 2 A2
See the paragraph containing (A.0) to see the estimate for (3.7). Thus, using −L 2
e M d+κ I ≤ H ess( f0 ) L 2 A2 as quadratic forms where I is the identity matrix, we have
(3.52)
1 1
M1 U x · M1 U x ≤ C M1 U x · M1 U x 2 . 2 2 2A L
(3.53)
Thus, 1
(u φx y )2 ≤ C A4 L 4 e− C L A |x−y|
(3.54)
for an appropriate choice of the unit vector u. ˆ Now we establish C C
|u φx y | ≤ + d−2 . d−2 (|x − y| + 1) L We need to invert (3.41), (3.42) (0 + H ess( f 0 ))−1 =
+∞
(3.55)
e−t (0 +H ess( f0 )) dt.
(3.56)
0
Following [NS], we use the Trotter product formula to write
t
t
e−t (0 +H ess( f0 )) = limn→+∞ e− n 0 e− n H ess( f0 ) .
(3.57)
e−t0
is positivity preserving [RSIV, p.209] and contractive on The heat kernel p || − f 0 L (R , e ) [RSIV, p.211]. Thus, we need to understand the heat equation d k(x, y, t, τ ) = − H ess( f 0 (φt , ψ))x,z k(z, y, t, τ ) dt z (H ess( f 0 (φt ))x z + x z + A x z )k(z, y, t, τ ) =−
(3.58)
z
with x z = 2dδx,z −
d
δx,z+ei + δx,z−ei +
i=1
k(x, y, τ, τ ) = δx,y ,
1 M d+κ
δx,z ,
(3.59)
(3.60)
t ≥ τ ≥ 0, and φt ∈ R|| is a continuous path for t ≥ 0. We have obtained the exponentially decaying part. We now extract the power law part.
Towards a Nonperturbative Renormalization Group Analysis
21
Lemma 3.2. |k(x, y, t, 0)| ≤
⎧ ⎪ ⎨ ⎪ ⎩
C d (t+1) 2
e
− C1
|x−y|2 t+|x−y|
+
1 C e− L |x−y| L 2 A2 L d
0 ≤ t ≤ L2 t > L 2.
C
d (t+1) 2
(3.61)
Remark. The point to this lemma is that the presence of the long range piece (3.7) in (3.4) causes a few problems in the existing heat kernel methods [N,G]. Proof. We prove this estimate in the Appendix. Thus, combining (3.56), Lemma 3.2 and the property of e−t0 as described after (3.57), we have C C + d−2 . d−2 (|x − y| + 1) L
|u φx y | ≤
(3.62)
Also, recall (3.54), 1
(u φx y )2 ≤ C L 4 A4 e− C L A |x−y| . Finally, the Schwarz inequality together with (3.62), (3.63) implies
|u φx y | ≤ L 2+ 30
(3.63)
(3.64)
y
for some small positive constant > 0. This implies statement a and b of Lemma 3.1. Similar arguments can be given to establish
|u φx y φz u φz | ≤ C L 6+ 30 , (3.65) z
y,z ,z
which implies statement c. We can go through an argument similar to the one above in obtaining (3.54) except we get the answer 1
(u φx y φz )2 ≤ C A4 L 4 e− L AC (|x−y|+|x−z|) .
(3.66)
It is also helpful to note −t (E +E 2 )
ex y,x 1y
−t H ess( f 0 ) −t H ess( f0 ) e yy .
= ex x
(3.67)
Since we have estimates for the heat kernels on the right side, we can bound (3.67). Then we can get the analogue of (3.62) and finally derive (3.65). QED Lemma 3.1. If we do not perform the change of variable φ → φ − ψ, we have e
− fˆ1 (ψ)
=
f
e
− f 0 (φ+ψ)
e−Q 0 (φ) dφ −Q f (φ) e 0 dφ
(3.68)
22
H. Pinson
and fˆ1,ψx ψ y (ψ) = f 0 (φ + ψ)ψx ψ y − f 0 (φ + ψ)ψx ; f 0 (φ + ψ)ψ y : 1 c , fˆ1,ψx ψ y ψz (ψ) = f 0 (φ + ψ)ψx ψ y ψz − ( f 0 (φ + ψ)ψx ψ y ; f 0 (φ + ψ)ψz : 1 c + f 0 (φ + ψ)ψx ψz ; f 0 (φ + ψ)ψ y : 1 c + f 0 (φ + ψ)ψ y ψz ; f 0 (φ + ψ)ψx : 1 c ) + f 0 (φ + ψ)ψx ; f 0 (φ + ψ)ψ y ; f 0 (φ + ψ)ψz : 1 c ,
(3.69)
with
f
• =
•e− f0 (φ+ψ) e−Q 0 (φ) dφ f
e− f0 (φ+ψ) e−Q 0 (φ) dφ
.
(3.70)
Letting E 1 (ψ) = E 1 ( p) =
1 ˆ f 1,ψx ψ y (0)ψx ψ y , 2 x,y
(3.71)
1 ˆ f 1,ψx ψ y (0)e−i p(x−y) , 2 y
(3.72)
we have the following lemma. Lemma 3.3. a. | b. c.
dl E 1 ( p)|β0 ( p) ≤ L l−2+ 30 , dpl |x−y|
(3.73)
| fˆ1,ψx ψ y (ψ)||x − y|l e C L A ≤ L 2+l+ 30 ,
(3.74)
y
| fˆ1,ψx ψ y ψz (ψ)|e C L A (|x−y|+|x−z|) (|x − y| + |y − z|)l ≤ L 6+l+ 30 , (3.75) 1
y,z
for l = 0, 1, . . . , d + 1. The constant is the one that appears in Lemma 3.1. Proof. a. Lemma 3.1, a, combined with (3.28) proves this statement. b and c. The integral (3.70) is slightly different from (3.23). However, the method used to prove Lemma 3.1 carries over to this situation. QED Lemma 3.3. Due to Brascamp and Lieb [BL, Theorem 4.3], note E 1 ( p) ≥ 0.
(3.76)
4. General Integration Step We have done the first integration step. We do the inductive integration step. In the first step, we have achieved
Towards a Nonperturbative Renormalization Group Analysis
Z ≡ z0
23
f dψ s dφ dψ = e− f0 (φ+ψ)−Q 0 (φ)−Q 0 (ψ) f s z0 z0 z0 s dψ ˆ = e− f1 (ψ)−Q 0 (ψ) s z0
e− f0 (ψ)−Q 0 (ψ)
ˆ
= e− f1 (0) where ˆ
e− f1 (ψ) =
e− f1 (ψ)−Q 1 (ψ)
dψ z 1 , z 1 z 0s
f
e− f0 (φ+ψ)−Q 0 (φ)
dφ
(4.1)
,
(4.2)
f 1 (ψ) = fˆ1 (ψ) − fˆ1 (0) − E 1 (ψ),
(4.3)
1 ˆ f 1,ψx ψ y (0)ψx ψ y , 2 x,y
(4.4)
E 1 (ψ) =
f
z0
Q 1 (ψ) = Q s0 (ψ) + E 1 (ψ), f z1
=
e
f
−Q 1 (φ)
dφ,
z 1s
=
e
−Q s1 (ψ)
(4.5)
dψ, z 1 =
e−Q 1 (ψ) dψ.
(4.6)
In Fourier space, (4.5) becomes Q 1 ( p) =
h( p) + E 1 ( p). β0 ( p)
(4.7)
Again, we break up the covariance into the fast and slow modes in Fourier space Q 1 ( p)−1 = (Q 1 )( p)−1 + (Q s1 )( p)−1 , f
(Q 1 )( p)−1 = Q 1 ( p)−1 (1 − β1 ( p)) = ( f
=
h( p) + E 1 ( p))−1 (1 − β1 ( p)) β0 ( p)
β0 ( p) (1 − β1 ( p)), h( p) + β0 ( p)E 1 ( p)
(Q s1 )( p)−1 = Q 1 ( p)−1 β1 ( p) = ( =
(4.9)
h( p) + E 1 ( p))−1 β1 ( p) β0 ( p)
β0 ( p) β1 ( p), h( p) + β0 ( p)E 1 ( p)
β1 ( p) = e−L
(4.8)
2(1+δ) (h( p)+β ( p)E ( p)) 0 1
(4.10)
.
(4.11)
24
H. Pinson
δ is a very small positive number but larger than that appears in Lemmas 3.1, 3.3. For example, we can take δ = 50 . Then, f s s dψ dφ dψ dψ ˆ = e− f1 (φ+ψ)−Q 1 (φ)−Q 1 (ψ) f s = e− f2 (ψ)−Q 1 (ψ) s e− f1 (ψ)−Q 1 (ψ) z1 z1 z z1 1
ˆ
= e− f2 (0) where ˆ
e− f2 (ψ) =
f
e− f2 (ψ)−Q 2 (ψ)
dψ z 2 , z 2 z 1s
f
e− f1 (φ+ψ)−Q 1 (φ)
dφ f
z1
(4.12)
,
(4.13)
Q 2 (ψ) = Q s1 (ψ) + E 2 (ψ),
(4.14)
f 2 (ψ) = fˆ2 (ψ) − fˆ2 (0) − E 2 (ψ),
(4.15)
1 ˆ f 2,ψx ψ y (0)ψx ψ y , 2 x,y
(4.16)
E 2 (ψ) =
z2 =
f
e−Q 2 (φ) dφ, z 2s =
e−Q 2 (ψ) dψ, z 2 = s
e−Q 2 (ψ) dψ.
This same procedure can be carried to the nth step, and we get Z dψ z n+1 z1 ˆ ˆ = e− fn+1 (ψ)−Q n+1 (ψ) . . . s e− f1 (0)−···− fn+1 (0) , s z0 z n+1 z n z0 where, inductively, we have e
− fˆn+1 (ψ)
=
f
e− fn (φ+ψ)−Q n (φ)
dφ
(4.18)
,
(4.19)
f n (ψ) = fˆn (ψ) − fˆn (0) − E n (ψ),
(4.20)
E n (ψ) =
f
zn
1 ˆ f n,ψx ψ y (0), 2 x,y
Q n+1 (ψ) = Q sn (ψ) + E n+1 (ψ), Q n ( p)−1 = Q n ( p)−1 + Q sn ( p)−1 in Fourier variables, f
Q n ( p)−1 = Q n ( p)−1 (1 − βn ( p)) = (E n ( p) + Q sn−1 ( p))−1 (1 − βn ( p)) f
(4.17)
(4.21)
Towards a Nonperturbative Renormalization Group Analysis
= β0 ( p) . . . βn−1 ( p)
25
(1 − βn ( p)) , h( p) + E 1 ( p)β0 ( p) + · · · + E n ( p)β0 ( p) . . . βn−1 ( p)
(4.22)
Q sn ( p)−1 = Q n ( p)−1 βn ( p) = (E n ( p) + Q sn−1 ( p))−1 βn ( p), = β0 ( p) . . . βn−1 ( p)
βn ( p) , h( p) + E 1 ( p)β0 ( p) + · · · + E n ( p)β0 ( p) . . . βn−1 ( p)
βn ( p) = e−L f zn
=
e
f
−Q n (φ)
2(1+nδ) (h( p)+E
1 ( p)β0 ( p)+···+E n ( p)β0 ( p)...βn−1 ( p))
dφ,
z ns
=
e
−Q sn (ψ)
dψ, z n =
,
(4.23)
(4.24)
e−Q n (ψ) dψ.
(4.25)
We prove the following result in the small field region. We need good estimates in this region, and the following Lemma 4.1 (the main lemma) provides that. We need statements for all regions. However, the inductive assumption on the small field field region (4.26) provides control over the large field region at the next scale through the formula (4.33). Lemma 4.1. If |ψ y | ≤
|x−y|
1 L
(1+nδ) d−2 4
e L 2(1+nδ+n )
(4.26)
for all y ∈ , then a.
1
| fˆn+1,ψx ψ y (ψ)|e 10L (1+nδ+n )
|x−y|
≤
y
b.
1
| fˆn+1,ψx ψ y ψz (ψ)|e 10L (1+nδ+n )
C L6 L
,
(4.27)
≤ C L 6.
(4.28)
d−2 2 (1+(n−1)δ)
(|x−y|+|y−z|)
y,z
Proof of Lemma 4.1. We argue by induction on n. This means that for k = 1, 2, . . . , n − 1, |ψ y | ≤
|x−y|
1 L
(1+kδ) d−2 4
e L 2(1+kδ+k )
(4.29)
implies a.
1
| fˆk+1,ψx ψ y (ψ)|e 10L (1+kδ+n )
y
b.
|x−y|
≤
C L6 L
d−2 2 (1+(k−1)δ)
,
(4.30)
1
(|x−y|+|y−z|) | fˆk+1,ψx ψ y ψz (ψ)|e 10L (1+kδ+k ) ≤ C L 6.
y,z
Then, our goal is to show that (4.29) implies (4.30) and (4.31) for k = n.
(4.31)
26
H. Pinson
As an immediate consequence of the inductive hypothesis (4.30) we have control over the extracted quadratic pieces 1 ˆ |E k+1 ( p)| = | | fˆk+1,ψx ψ y (0)||x − y|2 f k+1,ψx ψ y (0)ei p(x−y) | ≤ C| p|2 2 y y ≤
C L 6 | p|2 L 2(1+kδ+k )
(4.32) d−2 L 2 (1+(k−1)δ) for k = 1, . . . , n − 1. Note that the extracted quadratic parts E k+1 are very small. Note that the statement of this lemma involves a condition on the external field ψ. First, we need some estimates which do not depend on the size of the field ψ. To do this it is convenient to put all the integration back in and get dφ0 dφn ˆ e− fn+1 (ψ) = e−Hn (φ0 ,...,φn ,ψ) f . . . f , (4.33) z0 zn where Hn (φ0 , . . . , φn , ψ) = f 0 (φ0 + φ1 + · · · + φn + ψ) − E 1 (φ1 + · · · + φn + ψ) f
f
− · · · − E n (φn + ψ) + Q 0 (φ0 ) + · · · + Q n (φn ).
(4.34)
The consequence (4.32) of the inductive hypothesis and (3.28), (3.29) give us control over all E 1 , . . . , E n since they are evaluated when the external field is set to zero. Thus, the inductive hypothesis gives us control over the integrand of (4.33) for all ψ ∈ R|| . First let us show that these integrals (4.33) are well defined (naively, the quadratic forms E 1 (φ1 + · · · + φn + ψ) + E 2 (φ2 + · · · + φn + ψ) + · · · + E n (φn + ψ) may, for example, spoil the convergence of the integral). More importantly, the next lemma establishes the convexity property of the integrand in (4.33). Lemma 4.2. a. − E 1 (φ1 + · · · + φk + ψ) − E 2 (φ2 + · · · + φk + ψ) − · · ·
−E k (φk + ψ) +
f Q 1 (φ1 )(1 −
L − 20 ) 2
f
+ · · · + Q k (φk )(1 −
L − 20 L − 20 s ) + Q (ψ)(1 − )≥0 k 2k 2k
(4.35)
and
b. H essφ ( f k (φ + ψ) +
f Q k (φ)(1 −
L − 20 )) ≥ 0 2k
(4.36)
for k = 1, . . . , n. Remark. This inequality relies on the simple inequality (a + b)2 ≤
1 1 a 2 + b2 1−β β
for 0 < β < 1, a, b ∈ R. Statement b is an inequality as quadratic forms.
(4.37)
Towards a Nonperturbative Renormalization Group Analysis
27
It may appear strange that we are proving this statement for previous scales, meaning k being strictly less than n. We could have phrased this statement just for k = n, but the argument works by induction. Thus, it does not seem harmful that we phrased this lemma in this manner. Proof of Lemma 4.2. a. We argue by induction on k. In proving this Lemma 4.2, we will use the overall inductive assumption in regard to Lemma 4.1. Since all these quadratic forms are diagonalizable in Fourier space, we will prove these inequalities in Fourier space. We have Q 1 ( p) =
h( p) + E 1 ( p), β0 ( p)
Q l ( p) = El ( p) +
f
Q l ( p) =
(4.38)
1 Q l−1 ( p), βl−1 ( p)
(4.39)
1 Q l ( p) 1 − βl ( p)
(4.40)
for l = 1, . . . , n. In the k = 1 case, after we go into Fourier space,
− E 1 (φ1 + ψ) +
f Q 1 (φ)(1 −
L − 20 L − 20 ) + Q s1 (ψ)(1 − ) 2 2
(4.41)
becomes
−E 1 ( p)(a1 + b)2 +
1 h( p) 2 L − 20 (E 1 ( p) + )a1 (1 − ) 1 − β1 ( p) β0 ( p) 2
+
1 h( p) 2 L − 20 (E 1 ( p) + )b (1 − ) β1 ( p) β0 ( p) 2
≥ −E 1 ( p)(a1 + b)2 + (E 1 ( p) +
h( p) L − 20 )(a1 + b)2 (1 − ) β0 ( p) 2
= {−
L − 20 L − 20 h( p) E 1 ( p) + (1 − )}(a1 + b)2 ≥ 0. 2 β0 ( p) 2
(4.42)
The very last inequality is true due to h( p) ≥ O(1)| p|2 , |E 1 ( p)|β0 ( p) ≤ | p|2
(4.43)
| fˆ1,ψx ψ y (0)||x − y|2 ≤ | p|2 L 30 .
(4.44)
y
The first inequality follows from the formula (3.6). The second inequality follows from (3.28) and (3.29).
28
H. Pinson
In the inductive step, after we go into Fourier variables,
f
−E 1 (φ1 + · · · + φk + ψ) − · · · − E k (φk + ψ) + Q 1 (φ1 )(1 −
f
+ · · · + Q k (φk )(1 −
L − 20 ) 2
L − 20 L − 20 s ) + Q (ψ)(1 − ) k 2k 2k
(4.45)
becomes −E 1 ( p)(a1 + · · · + ak + b)2 − · · · − E k ( p)(ak + b)2
1 L − 20 Q 1 ( p)a12 (1 − ) + ··· + 1 − β1 ( p) 2
+
1 L − 20 1 L − 20 Q k ( p)ak2 (1 − k ) + Q k ( p)b2 (1 − k ) 1 − βk ( p) 2 βk ( p) 2 ≥ −E 1 ( p)(a1 + · · · + ak + b)2 − · · · − E k ( p)(ak + b)2
+
1 L − 20 Q 1 ( p)a12 (1 − ) + ··· 1 − β1 ( p) 2
+
Q k−1 ( p) 2 L − 20 L − 20 ak−1 (1 − k−1 ) + Q k ( p)(ak + b)2 (1 − k ) 1 − βk−1 ( p) 2 2 = −E 1 ( p)(a1 + · · · + ak + b)2 − · · · − E k ( p)(ak + b)2
+
+
1 L − 20 Q 1 ( p)a12 (1 − ) + ··· 1 − β1 ( p) 2
1 1 − βk−1 ( p)
L − 20 Q k−1 ( p) L − 20 2 )(a ) + (E ( p) + + b) (1 − ) k k 2k−1 βk−1 ( p) 2k
2 Q k−1 ( p)ak−1 (1 −
= {−E 1 ( p)(a1 + · · · + ak + b)2 − · · · − E k−1 ( p)(ak−1 + ak + b)2
+
1 1 L − 20 L − 20 2 Q 1 ( p)a12 (1 − ) + ··· + Q k−1 ( p)ak−1 (1 − k−1 ) 1 − β1 ( p) 2 1 − βk−1 ( p) 2
L − 20 L − 20 Q k−1 ( p)(ak + b) (1 − k−1 )} + {−E k ( p)(ak + b)2 k + βk−1 ( p) 2 2 1
2
+
1 βk−1 ( p)
Q k−1 ( p)(ak + b)2
L − 20 } ≥ 0. 2k
(4.46)
Towards a Nonperturbative Renormalization Group Analysis
29
The last inequality follows from Q k−1 ( p) ≥0 βk−1 ( p)
− E k ( p) +
(4.47)
and the inductive assumption. This inequality (4.47) can be seen as follows. From the inductive formula (4.38), (4.39), it can be worked out that Q k−1 ( p) =
1 {h( p) + β0 ( p)E 1 ( p) β0 ( p)β1 ( p) · · · βk−2 ( p) + · · · + β0 ( p)β1 ( p) · · · βk−2 ( p)E k−1 ( p)}.
(4.48)
From (4.43), (3.76), (4.32) (which indicates that the higher order extracted quadratic terms are small), we have Q k−1 ( p) ≥ O(1)| p|2 .
(4.49)
E k ( p) is small due to (4.32). Thus, (4.47) follows. b. For the case k = 1, we need to show
H essφ ( f 1 (φ + ψ) +
L − 20 )) = H essφ ( fˆ1 (φ + ψ) 2
f Q 1 (φ)(1 −
−E 1 (φ + ψ) +
L − 20 )) ≥ 0 2
f Q 1 (φ)(1 −
(4.50)
as quadratic forms. Since fˆ1 (φ + ψ) is convex (and thus its Hessian is semi positive definite), we need to show
H essφ (−E 1 (φ + ψ) +
f Q 1 (φ)(1 −
L − 20 )) ≥ 0. 2
(4.51)
Going into Fourier variables, this becomes
−E 1 ( p) +
f Q 1 ( p)(1 −
(Q s0 ( p) + E 1 ( p)) L − 20 L − 20 ) = −E 1 ( p) + (1 − ) 2 1 − β1 ( p) 2
=
L − 20 L − 20 1 {E 1 ( p)β1 ( p) − E 1 ( p) + Q s0 ( p)(1 − )} 1 − β1 ( p) 2 2
=
1 L − 20 L − 20 h( p) {E 1 ( p)β1 ( p) − E 1 ( p) + (1 − )} ≥ 0. 1 − β1 ( p) 2 β0 ( p) 2
(4.52)
In general,
f
H essφ ( f k (φ + ψ) + Q k (φ)(1 −
L − 20 )) 2k
L − 20 L − 20 = H essφ ( fˆk (φ + ψ) + Q sk−1 (φ)(1 − k−1 )) + H essφ (−Q sk−1 (φ)(1 − k−1 ) 2 2 L − 20 f −E k (φ + ψ) + Q k (φ)(1 − k )). (4.53) 2
30
H. Pinson
By part a and [Theorem 4.3, BL], the first term on the right side is semi positive definite. Thus, it suffices to show that the second Hessian is semi positive definite. Going into Fourier variables, we have
L − 20 L − 20 f ) − E k ( p) + Q k ( p)(1 − k ) k−1 2 2
−Q sk−1 ( p)(1 −
=
−Q sk−1 ( p)(1 −
Q sk−1 ( p) + E k ( p) L − 20 L − 20 (1 − ) − E ( p) + ) k 2k−1 1 − βk ( p) 2k
=
1 L − 20 ({E k ( p)βk ( p) + βk ( p)Q sk−1 ( p)(1 − k−1 )} 1 − βk ( p) 2
L − 20 L − 20 − E k ( p) k }). k 2 2 Since E k ( p) is small (4.32), both bracketed terms are non negative. +{Q sk−1 ( p)
(4.54) QED Lemma 4.2.
The following lemma is a necessity. Lemma 4.3. −|x−y|
1 f − |∇x,x+e1 (Q k )x y2 |
−|x−y|
C L (1+kδ) e L 1+kδ+k Clog(L 1+kδ )e L 1+kδ+k ≤ min( , ) (1 + |x − y|)d+1 (1 + |x − y|)d
(4.55)
for k = 1, . . . , n. Remark. ∇x,x+e1 is acting on the variable x in this lemma. Proof of Lemma 4.3. 1 f −
∇x,x+e1 (Q k )x y2
=
1 1 (1 − βk ( p)) }2 {β0 ( p) . . . βk−1 ( p) d M p h( p) + β0 ( p)E 1 ( p) + · · · + β0 ( p) . . . βk−1 ( p)E k ( p)
e−i p(x−y) (1 − e−i pe1 ).
(4.56)
First we can replace the sum by an integral with the error being much smaller than the right side of the inequality in the statement of Lemma 4.3. To get the exponential decay, we do a complex translation p → p−i
x−y . L 1+kδ+k |x − y|
(4.57)
All the functions in the integrand are built from the basic functions h( p), E 1 ( p),…, E k ( p). We have |
|z 1 −z 2 | x−y dl ˆq,z 1 z 2 ||z 1 − z 2 |l e L 1+kδ+k )| ≤ E ( p − i | f q dpl L 1+kδ+k |x − y| z 2
≤
L 6 L l(1+(q−1)δ+(q−1) ) L
d−2 2 (1+(q−2)δ+(q−2) )
(4.58)
Towards a Nonperturbative Renormalization Group Analysis
31
for q = 2, 3, . . . , k. We integrate by parts d or d + 1 times. To do this it is convenient 1 1 to break up the domain of integration (4.56) into two parts-| p| ≤ L 1+kδ , L 1+kδ ≤ | p|. On the first domain, we can expand in power series the exponential function in βk ( p) in (1 − βk ( p)) . h( p) + β0 ( p)E 1 ( p) + · · · + β0 ( p) . . . βk−1 ( p)E k ( p)
(4.59)
On the second domain, we do not expand this term in a power series. If we integrate by parts d + m times, we get 1 |x−y| − 1+kδ+k f −2 |∇x (Q k )x y | ≤ e L ( d d pL (d+m)(1+kδ) | p|≤
+
1 | p|> 1+kδ L
1 L 1+kδ
d d pβk−1 ( p)(
1 + βk ( p)L (d+m)(1+kδ) )). | p|d+m
(4.60)
The right side of (4.60) consists of the sum of three terms. In regard to the first term, each integration by parts produces at most L 1+kδ . In the second term, the derivative coming from the integration by parts never lands on 1 − βk ( p). Thus, the derivative lands on some power | 1p| or produces a factor of L 1+(k−1)δ . Both are bounded by | p|1d+m βk−1 ( p). In the third term, the derivative lands on 1 − βk ( p) at least once. Each integration by parts produces at most a factor L 1+kδ . Thus, if we integrate over the two regions, we get −|x−y|
1 f − |∇x,x+1 (Q k )x y2 |
C L (1+kδ) e L 1+kδ+k ≤ (1 + |x − y|)d+1
(4.61)
for m = 1 or −|x−y|
1 f − |∇x,x+1 (Q k )x y2 |
Clog(L 1+kδ )e L 1+kδ+k ≤ (1 + |x − y|)d
(4.62)
for m = 0.
QED Lemma 4.3.
In a similar manner as the derivation of (4.60), we can establish 1 f −
|∇x2 (Q k )x y2 | ≤e
−
|x−y| L 1+kδ+k
(
d
1 | p|≤ 1+kδ L
d pL
(d+m−1)(1+kδ)
+
1 | p|> 1+kδ L
d d pβk−1 ( p)(
1 | p|d+m−1
+βk ( p)L (d+m−1)(1+kδ) )).
(4.63)
∇x2 indicates a second order finite difference derivative acting on the variable x. Using similar arguments as in Lemma 4.3, we can establish −|x−y|
1 f − |∇x2 (Q k )x y2 |
−|x−y|
Clog(L (1+kδ) )e L 1+kδ+k Ce L 1+kδ+k ≤ min( , 1+(k−1)δ ) d+1 (1 + |x − y|) L (1 + |x − y|)d
for k = 1, . . . , n. We need to control the effective action over the large field regions.
(4.64)
32
H. Pinson
Lemma 4.4. |x−y|
− | fˆk+1,ψx ψ y (ψ)| ≤ L 6(1+kδ+k ) e 6L 1+kδ+k
| fˆk+1,ψx ψ y ψz (ψ)| ≤ L 10(1+kδ+k ) Ce
(4.65)
−(|x−y|+|y−z|) 6L 1+kδ+k
(4.66)
for k = 1, . . . , n. Proof of Lemma 4.4. Let us just do the k = n case. The other cases are similar. Technically it is convenient to go into the following representation: f −1 f −1 dφ0 dφn ˆ 2 2 e− fn+1 (ψ) = e−Hn ((Q 0 ) φ0 ,...,(Q n ) φn ,ψ) ··· , (4.67) || || (π ) 2 (π ) 2 where 1
1
1
1
Hn ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ) = f 0 ((Q 0 )− 2 φ0 + · · · + (Q n )− 2 φn + ψ) f
f
f
1
f
1
1
−E 1 ((Q 1 )− 2 φ1 + · · · + (Q n )− 2 φn + ψ) − · · · − E n ((Q n )− 2 φn + ψ) +I (φ0 ) + I (φ1 ) + · · · + I (φn ) (4.68) f
f
f
I (φk ) =
2 φk,x .
(4.69)
x∈
Also let
• n =
1 f −1 f 2 φ0 ,...,(Q n )− 2 φn ,ψ)
•e−Hn ((Q 0 )
dφ/
1 f −1 f 2 φ0 ,...,(Q n )− 2 φn ,ψ)
e−Hn ((Q 0 )
dφ. (4.70)
Again, we have f f f fˆn+1 (ψ)ψx ψ y = · · · Hn,ψx ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ); Hn,ψ y ((Q 0 )− 2 φ0 , . . . , 1
1
1
1
(Q n )− 2 φn , ψ) : 1 cn f
(4.71)
f f f fˆn+1 (ψ)ψx ψ y ψz = · · · Hn,ψx ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ); Hn,ψ y ((Q 0 )− 2 φ0 , . . . , 1
1
1
1
(Q n )− 2 φn , ψ); f
1
1
Hn,ψz ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ) : 1 cn . f
f
(4.72)
Since we have written similar formulas (3.69), we do not write out all the terms. Define (n + 1)|| by (n + 1)|| matrix M2 by ⎛ ⎞ 0 0 ... M0 0 ⎜ 0 M0 0 0 . . . ⎟ M2 = ⎝ , (4.73) 0 0 M0 0 . . . ⎠ ... ... ...
Towards a Nonperturbative Renormalization Group Analysis
33
where M0 is a diagonal and || by || matrix given by |x−y|
M0 = (e 3L 1+nδ+n δz,y )z,y∈ .
(4.74)
Again we set up the differential equation 1
1
{− + ∇ Hn ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ) · ∇}u x = Hn,ψx − Hn,ψx n (4.75) f
f
with ∇ = ( ∂φ∂k,x )x∈,k=0,...,n . We have 1
1
1
1
Hn,ψx ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ); Hn,ψ y ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ) : 1 cn f
f
f
1
f
1
= ∇u x · ∇ Hn,ψ y ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ) n . f
f
(4.76)
By taking a derivative of (4.75), 1
1
{− + ∇ Hn ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ) · ∇ f
f
1
1
+H ess(Hn ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ))}∇u x = ∇ Hψx . f
f
(4.77)
Thus, 1
1
{− + ∇ Hn ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ) · ∇ f
f
+M2 H ess(Hn ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ))M2−1 1
f
1
f
1
1
1
−H ess(Hn ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ)) + H ess(Hn ((Q 0 )− 2 φ0 , . . . , f
f
f
1
(Q n )− 2 φn , ψ))}W = F
(4.78)
W = M2 ∇u x , F = M2 ∇ Hψx .
(4.79)
f
Let us estimate M2 H ess(Hn ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ))M2−1 − H ess(Hn ((Q 0 )− 2 φ0 , . . . , f
1
1
f
f
1
1
(Q n )− 2 φn , ψ)). f
(4.80)
More precisely, we have terms of the form x2 ,y1 ,y2
1 f −
1 f −
|x−x1 |
(Q i )x1 2x2 E j,x2 y2 (Q k ) y2 2y1 (e 3L 1+nδ+n
−
|y1 −x| 3L 1+nδ+n
− 1)
(4.81)
for i, j, k = 1, . . . , n in the expression (4.80). We have chosen the factor 13 in the exponential in the matrix (4.74) so that the exponential growth is overridden by the 1 1 f f combination of the exponential decays in (Q i )− 2 ,E j , (Q k )− 2 in (4.81). In regard to the exponential growth term, we have the bound ||x1 − x| − |x − y1 || ≤ |x1 − y1 | ≤ |x1 − x2 | + |x2 − y2 | + |y2 − y1 |.
(4.82)
34
H. Pinson
Consider first the case j = 1 in (4.81). The extracted quadratic form E 1 is not small (see (3.28), (3.29)) unlike E k , k = 2, . . . , n (see (4.32)). The formula (3.28) for E 1 is a sum of seven terms. We consider the second and seventh terms in (3.28). The second term has two second order finite difference derivatives. Thus, using summation by parts in the variables x2 and y2 in the first line of (4.83), the second term as it appears in (3.28) can be bounded as
1 f −
x2 ,y1 ,y2
1 f −
|x−x1 |
−
|∇x22 (Q i )x1 2x2 || φx2 ; φ y2 : 1 c ||∇ y22 (Q k ) y2 2y1 ||e 3L 1+nδ+n 2|x1 −x2 |
|y1 −x| 3L 1+nδ+n
− 1|
2|y2−y1 |
− |x1 − x2 | log(L 1+nδ )e− 3L 1+nδ+n e 3L 1+nδ+n c ≤ | φ ; φ : 1 | +· · · x2 y2 L 1+nδ+n (|x1 −x2 | + 1)d+1 L 1+(n−1)δ (|y2 − y1 | + 1)d x ,y ,y 2
1
2
log(L 1+nδ+ n )3 [log(L 1+nδ+n )]3 L 2 L 30 ≤ . ≤ L n L 1+nδ+n L 1+(n−1)δ
(4.83)
In the inequalities (4.83), we have just considered the |x1 − x2 | term in (4.82). The other possibilities are indicated by . . . and can be taken care of in a similar manner. The correlation φx2 ; φ y2 : 1 c is the one in (3.23), (3.28), (3.29). Thus, (3.29) applies to this correlation. The seventh term in (3.28) as it appears in E 1 in (4.81) can be bounded as −
L2
1 e M d+κ ( 2 2 + d+κ )2 · L A M x2 ,y2 ,y1
1 f −
1 f −
|x−x1 |
|∇x2 ,y2 (Q i )x1 2x2 φx2 ; φ y2 : 1 c ∇x2 ,y2 (Q i ) y2 2y1 (e 3L 1+nδ+n −
−
|y1 −x| 3L 1+nδ+n
− 1)|
L2
e M d+κ 1 ≤ ( 2 2 + d+κ )2 · L A M 2|x1 −x2 |
2|y2 −y1 |
1+nδ e− 3L 1+nδ+n log(L 1+nδ )e− 3L 1+nδ+n 2 c |y2 − y1 |L |x − y | | φ ; φ : 1 | 2 2 x2 y2 (|x1 − x2 | + 1)d L 1+nδ+n (|y2 − y1 | + 1)d+1 x ,y ,y 2
2
1
≤
L 4 L 30 [log(L 1+nδ )]3 L 1+nδ 1 ≤ n . 4 1+nδ+n L L L2
(4.84)
Here we just consider the |y2 − y1 | term out of the three terms in (4.82) (the other terms can be handled in a similar manner). The first finite difference ∇x2 ,y2 in the second line of (4.84) acts on the variable x2 , and the second finite difference ∇x2 ,y2 acts on the variable y2 . The finite difference ∇x2 ,y2 in (4.84) is not strictly local as in Lemma 4.3. However, after extracting out the factor |x2 − y2 |, the same analysis as in Lemma 4.3 goes through, and the same bounds apply. Now we consider the j > 1 case in (4.81). The extracted quadratic terms E j , by the consequence (4.32) of the inductive hypothesis, are small, so these terms (4.81) are really small. We do not need these bounds to be so small, so we settle for crude bounds
Towards a Nonperturbative Renormalization Group Analysis
1 f −
x2 ,y1 ,y2
35 |x−x1 |
1 f −
|∇x2 (Q i )x1 2x2 ||E j,x2 y2 ||∇ y2 (Q k ) y2 2y1 ||e 3L 1+nδ+n
−
|y1 −x| 3L 1+nδ+n
2|x1 −x2 |
− 1|
2|y2 −y1 |
− |x1 − x2 | L 1+nδ e− 3L 1+nδ+n log(L 1+nδ )e 3L 1+nδ+n ≤ |E | j,x y 2 2 L 1+nδ+n (|x1 − x2 | + 1)d+1 (|y2 − y1 | + 1)d x ,y ,y 2
1
2
≤
[log(L 1+nδ+n )]3 L 1+nδ 1 ≤ n . 1+nδ+n L L2
(4.85)
We just consider the term |x1 − x2 | out of the three terms in (4.82). We have used the consequence (4.32) of the inductive hypothesis to estimate the behavior of E j . The finite differences ∇x2 and ∇ y2 arise since finite differences appear in E j . Also, from (4.35), −
1 1 L 20 f f W · W ≤ W H ess(Hn ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ))W. 2n
(4.86)
Since 1
1
− + ∇ Hn ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ) · ∇ f
f
(4.87)
is positive definite, combining (4.83) through (4.85), (4.86) and taking the dot product of (4.78) with the vector W , we have −
−
L 20 1 L 20
W · W n ≤ O( n )W · W + n W · W n ≤ W · F n n+1 2 2 L2 1
1
≤ ( W · W n ) 2 ( F · F n ) 2
(4.88)
and −
1 1 L 20 ( W · W n ) 2 ≤ ( F · F n ) 2 . n+1 2
Also,
F · F n ≤
Hn,ψx φk,w Hn,ψx φk,w n w,k
≤
f
|∇x2 (Q i )
x2 ,y2 ,w,x2 ,y2 ,i, j,k,l,m,r
≤
x2 ,y2 ,w,x2 ,y2 ,i, j,k,l,m,r
− 21 x x2
1 f −
1 f −
1 f −
E j,x2 y2 ∇ y2 (Q k ) y2 2w ∇x2 (Q l )wx2 E m,x2 y2 ∇ y2 (Q r ) y 2x | 2
2
|x−x2 | 2 −w| log(L 1+nδ ) − 1+nδ+n log(L 1+nδ ) − |y1+nδ+n L e |E | e L j,x y 2 2 d d (|x − x2 | + 1) (|y2 − w| + 1)
|y2 −x| 2| log(L 1+nδ ) − |w−x log(L 1+nδ ) − 1+nδ+n e L 1+nδ+n |E m,x2 y2 | e L d d (|w − x2 | + 1) (|y2 − x| + 1)
≤ L 4 L 15 [log(L 1+nδ+n )]8 .
(4.89)
36
H. Pinson
Here we have ignored the exponential growth factor coming from the matrix (4.73) in the definition of F (4.79) since that exponential growth factor is dominated by the other f f terms Q i , E j , . . . , Q r in (4.89). Thus, 1
(u φx k,y )2 n2 ≤ 2n+1 L 20 L 4 L 15 [log(L 1+nδ+n )]8 e ≤ L 4(1+nδ+n ) e
−
−
|x−y| 3L 1+nδ+n
|x−y| 3L 1+nδ+n
(4.90)
for k = 0, 1, . . . , n. Finally, to understand the connected two point function (4.71), we have 1
1
| Hn,ψx ; Hn,ψ y : 1 cn | ≤ | ∇u x · ∇ Hn,ψ y ((Q 0 )− 2 φ0 , . . . , (Q n )− 2 φn , ψ) n | f
f
= |
u φx k,x Hn,ψ y φk,x1 n | 1
k,x1
= |
1 f −
i, j,k,x1 ,x2 ,y2
≤
≤
1
1
i, j,k,x1 ,x2 ,y2
1 f −
1 f −
(u φx k,x )2 n2 |∇x2 (Q i )x1 2x2 E j,x2 y2 ∇ y2 (Q k ) y2 2y | 1
L 4(1+nδ+n ) e
−
|x−x1 | 3L 1+nδ+n
i, j,k,x1 ,x2 ,y2
|E j,x2 y2 |
1 f −
u φx k,x ∇x2 (Q i )x1 2x2 E j,x2 y2 ∇ y2 (Q k ) y2 2y n |
1 −x2 | log(L 1+nδ+n ) − |x1+nδ+n e L d (|x1 − x2 | + 1)
|y2 −y| log(L 1+nδ+n ) − 1+nδ+n − |x−y| e L ≤ L 6(1+nδ+n ) e 6L 1+nδ+n . d (|y2 − y| + 1)
(4.91)
The third derivative in (4.72) is similar, so we will not provide all details. Let x be such that |y − z| = min(|x − y|, |y − z|, |z − x|). Then, we have | Hn,ψx ; Hn,ψ y ; Hn,ψz : 1 cn | = | ∇u x · ∇((Hn,ψ y − Hn,ψ y n )(Hn,ψz − Hn,ψz n )) n | | u φx k,w Hn,ψ y φk,w (Hn,ψz − Hn,ψz n ) n | + | u φx k,w (Hn,ψ y − Hn,ψ y n )Hn,ψz φk,w n | ≤ k,w
≤
1
1
max(|Hn,ψ y φk,w |) (u φx k,w )2 n2 (Hn,ψz − Hn,ψz n )2 n2
k,w
≤
1
max(|Hn,ψ y φk,w |) (u φx k,w )2 n2 (2n L 20
k,w n
≤ 2 2 L 40 L 4(1+nδ+n ) L 4 L 15 [log(L 1+nδ+n )]8 e ≤ L 10(1+nδ+n ) e
−
1 6L 1+nδ+n
−
w ,l
1
Hψ2 z φl,w ) 2
1 6L 1+nδ+n
(|x−y|+|y−z|)
.
(|x−y|+|y−z|)
(4.92)
In going to the fourth line of (4.92), we have used (4.35) and Brascamp-Lieb [BL,theorem 4.1]. In going to the fifth line of (4.92), we have used (4.89) and (4.90). QED Lemma 4.4.
Towards a Nonperturbative Renormalization Group Analysis
37
We need to control the large field region, and we do this with the following lemma. Let S1 = {φ ∈ R|| ||φ y | ≤
|x−y|
1 L
d−2 4 (1+(n−1)δ)
e L 2(1+(n−1)δ+(n−1) ) for all y ∈ }.
(4.93)
The point x ∈ is fixed. We denote the characteristic functions as χ : R|| → {0, 1} which is always defined with respect to some subset of R|| . Lemma 4.5. If | φ y | ≤
3 L
(4.94)
d−2 2 (1+(n−1)δ)
for all y ∈ , then
χ S1c ≤ e−L
d−2 (1+(n−1)δ) 4
.
(4.95)
Proof. For α ∈ R, b > 0, we have eαb χ(|φz − φz | > b) ≤ 2 cosh(α(φz − φz )) − L 20 f ≤ 2 cosh(αφz )ex p(− n Q n (φ)) 2 ≤ 2ex p(
α 2 2n L
20
L d−2(1+(n−1)δ)
dφ −
f
ex p(− L220 n Q n (φ))dφ
).
(4.96)
The second inequality is true due to a theorem of Brascamp and Lieb [BL, theorem 5.1]. Letting α=
bL d−2(1+(n−1)δ)
22n L 20
we have
χ (|φz − φz | > b) ≤ e
−
b2 n 42 L 20
L d−2(1+(n−1)δ)
.
(4.97)
For each y ∈ , define B y = {φ ∈ R|| ||φ y | >
|x−y|
1 L
d−2 4 (1+(n−1)δ)
B y = {φ ∈ R|| ||φ y − φ y | >
e L 2(1+(n−1)δ+(n−1) ) },
1 2L
d−2 4 (1+(n−1)δ)
(4.98)
|x−y|
e L 2(1+(n−1)δ+(n−1) ) }.
(4.99)
The inequality in these definitions holds at one point y ∈ . Then, B y ⊂ B y .
(4.100)
Let Al = {y ∈ |L 2(1+(n−1)δ+(n−1) )l ≤ |x − y| ≤ L 2(1+(n−1)δ+(n−1) ) (l + 1)}, (4.101)
38
H. Pinson
l = 0, 1, 2, . . . ,
χ S1c ≤
∞
χ B y ≤
χ B y ≤
χ B y y
≤
∞
y
(l + 1)d L 2d(1+(n−1)δ+(n−1) ) e−e
l=0 y∈Al
d−2 2l L 4 (1+(n−1)δ)
≤ e−L
d−2 (1+(n−1)δ) 4
. (4.102)
l=0
QED Lemma 4.5. The other cases can be Proof of Lemma 4.1. Let us analyze f n,ψx ; f n,ψ y ; f n,ψz : handled in a similar manner. Consider the case diam(x, y, z) = max(|x − y|, |x − z|, |y − z|) > L 1+nδ+2n . There is an exponential decay. To connect to the exponential decay methods of Lemma 4.4, we consider the following integral representations: f f (φ)e− fn (φ+ψ)−Q n (φ) dφ
f (φ) = f e− fn (φ+ψ)−Q n (φ) dφ f (φn )e−Hn (φ0 ,...,φn ,ψ) dφ0 . . . dφn = e−Hn (φ0 ,...,φn ,ψ) dφ0 . . . dφn f −1 f −1 1 f f ((Q n )− 2 φn )e−Hn ((Q 0 ) 2 φ0 ,...,(Q 0 ) 2 φn ,ψ) dφ0 . . . dφn = . (4.103) −H ((Q f )− 21 φ ,...,(Q f )− 21 φ ,ψ) n 0 0 e n 0 dφ0 . . . dφn 1 c .
It is with respect to the last integral representation in (4.103) that we proved the exponential decay in Lemma 4.4. Thus, we set up the usual differential equation {−
∂2 1 1 f f + ∇ Hn ((Q 0 )− 2 φ0 , . . . , (Q 0 )− 2 φn , ψ) · ∇}u z 2 ∂φ k,x k,x 1
1
= f n,ψz ((Q n )− 2 φn ) − f n,ψz ((Q n )− 2 φn ) . f
f
(4.104)
1+nδ+2n
We can assume |z − x|, |z − y| ≥ L 8 , so z is far from both x, y. The definition of the gradient ∇ is given after (4.75). Then,
f n,ψx ; f n,ψ y ; f n,ψz : 1 c = ∇(( f n,ψx − f n,ψx )( f n,ψ y − f n,ψ y )) · ∇u z 1 f − =
(Q n )ww2 f n,ψx φn,w ( f n,ψ y − f n,ψ y )u φz n,w w,w
1 f −
+(Q n )ww2 ( f n,ψx − f n,ψx ) f n,ψ y φn,w u φz n,w .
(4.105)
Let us consider the first sum in (4.105), 1 1 1 f − f − | (Q n )ww2 f n,ψx φn,w ( f n,ψ y − f n,ψ y )u φz n,w | ≤
{(Q n )ww2 f n,ψx φn,w u φz n,w }2 2 w,w
( f n,ψ y − f n,ψ y )2
w,w
1 2
Towards a Nonperturbative Renormalization Group Analysis
≤
w,w
1 f −
f n,ψ y φn,w1 f n,ψ y φn,w2 ) ≤e
− |x−y|+|y−z| 1+nδ+n L
1
max(|(Q n )ww2 f n,ψx φn,w |) (u φz n,w )2 2 (
w1 ,w2
39
2n L 20 (Q n )−1 w1 w2 f
1 2
.
(4.106)
We have used Brascamp Lieb [BL, Theorem 4.1] and (4.36) in going to the second line of (4.106). The exponential decay for u φz n,w can be proved as in (4.90). Thus, n | f n,ψx ; f n,ψ y ; f n,ψz : 1 c | ≤ e−L . (4.107) y,z,diam(x,y,z)>L 1+nδ+2n
Consider the other case diam(x, y, z) ≤ L 1+nδ+2n . We decompose the external field ψ into two parts ψ = ψ1 + ψ2 .
(4.108)
{w ∈ ||w − x| ≤ L 1+nδ+3n },
(4.109)
ψ1 is the restriction of ψ to and ψ2 is the restriction of ψ to the complement. The support of the field ψ2 is far from any of the points x, y, z. We do this since the external field which is far away from the points x, y, z should have small influence on this correlation. In order to separate the effect of ψ1 and ψ2 , we have
f n,ψx ; f n,ψ y ; f n,ψz : 1 c{ψ1 +ψ2 } = f n,ψx ; f n,ψ y ; f n,ψz : 1 c{ψ1 } 1 d + dt f n,ψx ; f n,ψ y ; f n,ψz : 1 c{ψ1 +tψ2 } , (4.110) dt 0 where
f (φ) {ψ} = = =
f
f (φ)e− fn (φ+ψ)−Q n (φ) dφ f e− fn (φ+ψ)−Q n (φ) dφ f (φ)e−Hn (φ0 ,...,φn ,ψ) dφ0 . . . dφn e−Hn (φ0 ,...,φn ,ψ) dφ0 . . . dφn 1
f −1
f −1
f ((Q n )− 2 φ)e−Hn ((Q 0 ) 2 φ0 ,...,(Q n ) 2 φn ,ψ) dφ0 . . . dφn . −H ((Q f )− 21 φ ,...,(Q f )− 21 φ ,ψ) n 0 n e n 0 dφ0 . . . dφn f
We have d
f n,ψx ; f n,ψ y ; f n,ψz : 1 c{ψ1 +tψ2 } dt =−
(( f n,ψx − f n,ψx {ψ1 +tψ2 } )( f n,ψ y − f n,ψ y {ψ1 +tψ2 } ) w
·( f n,ψz − f n,ψ y {ψ1 +tψ2 } )); f n,ψw ψ2,w : 1 c{ψ1 +tψ2 } d +
(( f n,ψx − f n,ψx {ψ1 +tψ2 } )( f n,ψ y − f n,ψ y {ψ1 +tψ2 } ) dψw w ·( f n,ψz − f n,ψ y {ψ1 +tψ2 } )) {ψ1 +tψ2 } ψ2,w .
(4.111)
40
H. Pinson
Keeping in mind that ψ2 restricts to w with |w − x| > L 1+nδ+3n ,
(4.112)
the second sum in (4.111) becomes
|
w
≤
w
≤
w
d {( f n,ψx − f n,ψx {ψ1 +tψ2 } )( f n,ψ y − f n,ψ y {ψ1 +tψ2 } ) dψw ( f n,ψz − f n,ψz {ψ1 +tψ2 } )} {ψ1 +tψ2 } ψ2,w |
| ( f n,ψx ψw − f n,ψx ψw {ψ1 +tψ2 } + f n,ψx ; f n,ψw : 1 c{ψ1 +tψ2 } ) ( f n,ψz
( f n,ψ y − f n,ψ y {ψ1 +tψ2 } ) − f n,ψz {ψ1 +tψ2 } ) {ψ1 +tψ2 } ψ2,w | + · · ·
maxφ (| f n,ψx ψw − f n,ψx ψw {ψ1 +tψ2 } + f n,ψx ; f n,ψw : 1 c{ψ1 +tψ2 } |) 1
1
2 2 · ( f n,ψ y − f n,ψ y {ψ1 +tψ2 } )2 {ψ
( f n,ψz − f n,ψz {ψ1 +tψ2 } )2 {ψ |ψ2,w | + · · · 1 +tψ2 } 1 +tψ2 }
≤ e−L . n
(4.113)
In obtaining (4.113), we have pointwise control over the second derivatives f n,ψx ψw (see (4.65)). We can use the differential equation method of (4.103)-(4.106) to extract the exponential decay in f n,ψx ; f n,ψw : 1 c{ψ1 +tψ2 } . We can also use [BL, Theorem 4.1] and 1
2 . All these contributions produce a (4.65) to control ( f n,ψz − f n,ψz {ψ1 +tψ2 } )2 {ψ 1 +tψ2 } factor L 1+nδ+n raised to some power. However, there is an exponential suppression due to the large distance separation (4.112). Also note that the exponential growth (4.26) of the field ψ2 is much smaller than the exponential decay of f n,ψx ψw , for example. The dots indicate that the derivative can land other terms. The first sum in (4.111) becomes |
( f n,ψx − f n,ψx {ψ1 +tψ2 } )( f n,ψ y − f n,ψ y {ψ1 +tψ2 } )
w
×( f n,ψz − f n,ψz {ψ1 +tψ2 } ); f n,ψw ψ2,w : 1 c{ψ1 +tψ2 } | | ∇(( f n,ψx − f n,ψx {ψ1 +tψ2 } )( f n,ψ y − f n,ψ y {ψ1 +tψ2 } ) = w
×( f n,ψz − f n,ψz {ψ1 +tψ2 } ))∇u w {ψ1 +tψ2 } ψ2,w | 1 f − | (Q n )qq2 f n,ψx φn,q ( f n,ψ y − f n,ψ y {ψ1 +tψ2 } ) ≤ w,q,q
× ( f n,ψz − f n,ψz {ψ1 +tψ2 } ))u w φn,q {ψ1 +tψ2 } ψ2,w | + · · · 1 f − ≤
((Q n )qq2 f n,ψx φn,q )2 ( f n,ψ y − f n,ψ y {ψ1 +tψ2 } )2 w,q,q
1
2 ×( f n,ψz − f n,ψz {ψ1 +tψ2 } )2 {ψ 1 +tψ2 } 1
2 2 −L
(u w . φn,q ) {ψ1 +tψ2 } |ψ2,w | + · · · ≤ e n
(4.114)
Towards a Nonperturbative Renormalization Group Analysis
41
We have some comments in regard to the derivation of (4.114). In bounding 1 f −
((Q n )qq2 f n,ψx φn,q )2 ( f n,ψ y − f n,ψ y {ψ1 +tψ2 } )2 1
2 ( f n,ψz − f n,ψz {ψ1 +tψ2 } )2 {ψ 1 +tψ2 }
(4.115)
in (4.114), we need the Taylor expansion with remainder in φn about φn = φn {ψ1 +tψ2 } , f n,ψ y = f n,ψ y |φn = φn {ψ1 +tψ2 } + ∇ f n,ψ y · (φn − φn {ψ1 +tψ2 } ),
(4.116)
where f n,ψ y evaluated at φn = φn {ψ1 +tψ2 } is denoted by f n,ψ y |φn = φn {ψ1 +tψ2 } .
(4.117)
We use the following expression in (4.115): f n,ψ y − f n,ψ y {ψ1 +tψ2 } = f n,ψ y |φn = φn {ψ1 +tψ2 } + ∇ f n,ψ y · (φn − φn {ψ1 +tψ2 } ) − f n,ψ y |φn = φn {ψ1 +tψ2 } + ∇ f n,ψ y · (φn − φn {ψ1 +tψ2 } ) {ψ1 +tψ2 } = ∇ f n,ψ y · (φn − φn {ψ1 +tψ2 } )− ∇ f n,ψ y · (φn − φn {ψ1 +tψ2 } ) {ψ1 +tψ2 } .
(4.118)
The term ∇ f n,ψ y can be bounded pointwise using (4.65). The integral of powers of (φn − φn {ψ1 +tψ2 } ) can be bounded using Brascamp-Lieb [BL, Theorem 5.1]. All these contribute L 1+nδ+n raised to some power. In regard to u w φn,q , we can set up the differ1 f −
ential equation as in (4.104) and establish an analogue of (4.90). (Q n )qq2 f n,ψx φn,q has exponential decay due to (4.65). Due to the large distance separation (4.109), these two quantities provide exponential suppression to produce (4.114). Thus, (4.111) can be bounded by (4.113) and (4.114), 1 d n |
f n,ψx ; f n,ψ y ; f n,ψz : 1 c{ψ1 +tψ2 } | ≤ e−L . (4.119) 0 dt We are left to bound
f n,ψx ; f n,ψ y ; f n,ψz : 1 c{ψ1 }
(4.120)
in (4.110). Let us first prove the following lemma. Lemma 4.6. | φ y {ψ1 } | ≤
2 L
d−2 2 (1+(n−1)δ)
.
(4.121)
Proof. If ψ1 is small enough, this lemma is true. For example, if ψ1 = 0, then φ y {0} = 0. However, we have a little more room 1 d
φ y { = dt φ y { ψ1 tψ1 } dt 3d (1+nδ+4n )+ 1 (1+nδ) 3d (1+nδ+4n )+ 1 (1+nδ) } 0 L 4
=−
z
φ y ; Hn,ψz
2
L 4
ψ1,z L
3d 1 4 (1+nδ+4n )+ 2 (1+nδ)
: 1 c {
2
tψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
.
(4.122)
42
H. Pinson
Thus, | φ y { ≤|
φ y ; Hn,ψz z
ψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
ψ1,z
L
3d 1 4 (1+nδ+4n )+ 2 (1+nδ)
|
: 1 c {
tψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
L d(1+nδ+n )
1
. (4.123) L L L L In order to establish the exponential decay, we should go into the last integral representation in (4.103) as we have discussed before (see (4.104) and the discussion surrounding that equation). We have implicitly assumed that we have done this in getting to the third line of (4.123). By induction, we show that 2 | φ y { | ≤ d−2 (4.124) kψ1 } (1+(n−1)δ) 3d (1+nδ+4n )+ 1 (1+nδ) L 2 2 L 4 ≤
d−2 2 (1+(n−1)δ)
d−2 4 (1+nδ)
for all y ∈ and k = 1, 2, . . . , [L integer close to L
3d 1 4 (1+nδ+4n )+ 2 (1+nδ)
3d 1 4 (1+nδ+4n )+ 2 (1+nδ)
3d 1 4 (1+nδ+4n )+ 2 (1+nδ)
≤
|
]. [L
d−2 2 (1+(n−1)δ)
3d 1 4 (1+nδ+4n )+ 2 (1+nδ)
.
] is an
(4.125)
We do not need this to be so precise. The idea is that the inductive hypothesis on the one point function actually shows that the one point function is much smaller than what the inductive hypothesis indicates. We have shown this for k = 1. Let us do the inductive step from k to k + 1.
φ y {
φ y ; Hn,ψz : 1 c
1
−
(k+1)ψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
dt 0
{
z
= φ y {
kψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
,
1 kψ1 +tψ1
3d (1+nδ+4n )+ 1 (1+nδ) 2 L 4
}
L
3d 1 4 (1+nδ+4n )+2 (1+nδ)
ψ1,z .
(4.126)
The sum can be bounded as in (4.123), and the bound gives 1 1 | dt
φ y ; Hn,ψz : 1 c ψ1,z | kψ1 +tψ1 3d 1 { } (1+nδ+4n )+ 3d (1+nδ+4n )+ 1 (1+nδ) L 4 2 (1+nδ) 0 z 2 L 4 ≤
1 L
d−2 2 (1+(n−1)δ)
.
(4.127)
The one point function becomes
φ y {
kψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
dt 0
dt 0
{
3d 1 4 (1+nδ+4n )+ 2 (1+nδ)
tkψ1
3d (1+nδ+4n )+ 1 (1+nδ) 2 L 4
φ y ; f n,ψz : 1 c {
z,|z−y|>L 1+nδ+2n
k L
k
φ y ; f n,ψz : 1 c
z,|z−y|≤L 1+nδ+2n 1
− ×
1
−
=
ψ1,z .
}
L
3d 1 4 (1+nδ+4n )+ 2 (1+nδ)
ψ1,z
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
(4.128)
Towards a Nonperturbative Renormalization Group Analysis
43
The second sum is small e−L . The first sum in (4.128) becomes n
1
−
dt 0
×
φ y ; f n,ψz : χ E 1 + χ(E 1 )c c {
z,|z−y|≤L 1+nδ+2n
k L
3d 1 4 (1+nδ+4n )+ 2 (1+nδ)
E 1 = {φ ∈ R|| ||φ y | ≤
L
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
ψ1,z
(4.129)
1
e L 2(1+(n−1)δ+(n−1) ) for all y ∈ }. (4.130)
|x−y|
d−2 4 (1+(n−1)δ)
The first term in (4.129) with χ E 1 can be bounded as |
φ y ; f n,ψz φz1 φz2 (φz 1 − φz 1 { I1
(φz 2 − φz 2 {
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 4 2 L
· +
I2
+ φz 2 {
k L
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
· ≤
k L
L d(1+nδ+2n ) L 6 L
3(d−2) (1+(n−1)δ) 2
≤
d−2 4 (1+nδ)
1 2L
d−2 2 (1+(n−1)δ)
{
)
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
) : χ E 1 c {
)
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
ψ1,z |
1 L
) : χ E 1 c
+ φz 1 {
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
3d 1 4 (1+nδ+4n )+ 2 (1+nδ)
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 4 2 L
ψ1,z
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 2 L 4
+ φz 2 {
+ φz 1 {
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 4 2 L
3d 1 4 (1+nδ+4n )+ 2 (1+nδ)
φ y ; f n,ψz φz1 φz2 (φz 1 − φz 1 {
(φz 2 − φz 2 {
tkψ1 3d (1+nδ+4n )+ 1 (1+nδ) } 4 2 L
+ e−L
n
,
(4.131)
where I1 = {z, z 1 , z 2 ∈ ||z − y| ≤ L 1+nδ+2n , |z − z 1 | + |z 1 − z 2 | ≤ L 1+nδ+2n },
(4.132)
I2 = {z, z 1 , z 2 ∈ ||z − y| ≤ L 1+nδ+2n , |z − z 1 | + |z 1 − z 2 | > L 1+nδ+2n }.
(4.133)
By using Lemma 4.5, the second sum in (4.129) is bounded by e−L . QED Lemma 4.6. nδ
44
H. Pinson
Finally, we estimate
f n,ψx ; f n,ψ y ; f n,ψz : 1 c{ψ1 } = f n,ψx ; f n,ψ y ; f n,ψz : χ S1 + χ S1c c{ψ1 }
(4.134)
diam(x, y, z) ≤ L 1+nδ+2n
(4.135)
with
S1 = {φ ∈ R|| ||φw | ≤
|φw | ≤ ≤
L 1
L
|w−x|
1 L
d−2 4 (1+(n−1)δ)
e L 2(1+(n−1)δ+(n−1) ) ,
|w−y|
1 d−2 4 (1+(n−1)δ)
e L 2(1+(n−1)δ+(n−1) ) , |φw |
|w−z|
d−2 4 (1+(n−1)δ)
e L 2(1+(n−1)δ+(n−1) ) for all w ∈ }.
(4.136)
By Lemma 4.5 and Lemma 4.6, | f n,ψx ; f n,ψ y ; f n,ψz : χ S1c c{ψ1 } | ≤ e−L Also, | =|
d−2 (1+(n−1)δ) 4
.
(4.137)
f n,ψx ; f n,ψ y ; f n,ψz : χ S1 c{ψ1 } |
y,z,diam(x,y,z)≤L 1+nδ+2n
f n,ψx φx1 φx2 (φx1 − φx1 {ψ1 } + φx1 {ψ1 } )(φx2 − φx2 {ψ1 } + φx2 {ψ1 } );
y,z,I3
f n,ψ y φ y1 φ y2 (φ y1 − φ y1 {ψ1 } + φ y1 {ψ1 } )(φ y2 − φ y2 {ψ1 } + φ y2 {ψ1 } ); f n,ψz φz1 φz2 (φz 1 − φz 1 {ψ1 } + φz 1 {ψ1 } )(φz 2 − φz 2 {ψ1 } + φz 2 {ψ1 } ) : χ S1 c{ψ1 } +
f n,ψx φx1 φx2 (φx1 − φx1 {ψ1 } + φx1 {ψ1 } )(φx2 − φx2 {ψ1 } + φx2 {ψ1 } ); y,z,I3c
f n,ψ y φ y1 φ y2 (φ y1 − φ y1 {ψ1 } + φ y1 {ψ1 } )(φ y2 − φ y2 {ψ1 } + φ y2 {ψ1 } ); f n,ψz φz1 φz2 (φz 1 − φz 1 {ψ1 } + φz 1 {ψ1 } )(φz 2 − φz 2 {ψ1 } + φz 2 {ψ1 } ) : χ S1 c{ψ1 } | ≤ L 18
L 2d(1+nδ+2n ) L 3(d−2)(1+(n−1)δ)
+ e−L
n
≤ L 6,
(4.138)
d(x, y, z) ≤ L 1+nδ+2n ,
(4.139)
where
I3 = {x1 , x2 , y1 , y2 , z 1 , z 2 ∈ |max(|x −x1 |, |x −x2 |, |x1 − x2 |) ≤ L 1+(n−1)δ+(n−1)2 , max(|y − y1 |, |y − y2 |, |y1 − y2 |) ≤ L 1+(n−1)δ+(n−1)2 , max(|z − z 1 |, |z − z 2 |, |z 1 − z 2 |) ≤ L 1+(n−1)δ+(n−1)2 }.
(4.140)
QED Lemma 4.1.
Towards a Nonperturbative Renormalization Group Analysis
45
5. Conclusion of the Proof Now we need to vary the lattice. Assume that the origin 0 ∈ 0 ⊂ 1 is near the center 1 5 of the two lattices. Let N ∈ Z be such that M = |0 | d = L 1+ 4 N δ . Also, let N ∈ Z be 1 5 such that |1 | d = L 1+ 4 N δ . Lemma 5.1. For k = 0, 1, . . . , N + 1, a. |E k,1,0y − E k,0,0y | ≤ e f
−1
f
1 −|0 | d L 1+kδ+k
−1
b. |(Q k,1 )0y2 − (Q k,0 )0y2 | ≤ e c. |
fˆk,0 (0) fˆk,1 (0) − |≤e |1 | |0 |
,
1 −|0 | d 1+kδ+k L
1 −|0 | d 1+kδ+k L
(5.1) ,
(5.2)
.
(5.3)
Remark. E k,0,x y is the extracted quadratic part (4.21) with respect to lattice 0 . Similar definitions pertain to the other lattice 1 and so on. Proof. We argue by induction on k. We do the inductive step. It is convenient to consider the following representation: f −1 f f −1 f − 21 − 21 ˆ 2 2 e− fk,1 (ψ) = e− f0 ((Q 0 ) φ0 +···+(Q k−1 ) φk−1 +ψ)+E 1 ((Q 1 ) φ1 +···+(Q k−1 ) φk−1 +ψ) f
e+...+E k−1 ((Q k−1 )
− 21
dφ0
φk−1 +ψ)−I (φ0 )−···−I (φk−1 )
(π )
|1 | 2
...
dφk−1 (π )
|1 | 2
,
(5.4)
and let 1
1
Hk−1,1 = f 0 ((Q 0 )− 2 φ0 + · · · + (Q k−1 )− 2 φk−1 + ψ) f
1
f
1
1
−E 1 ((Q 1 )− 2 φ1 +. . . + (Q k−1 )− 2 φk−1 +ψ)−. . .− E k−1 ((Q k−1 )− 2 φk−1 +ψ). f
f
Also let
f
(5.5)
−H •e k−1,1 −I (φ0 )−···−I (φk−1 ) dφ
• 1 = −H . e k−1,1 −I (φ0 )−···−I (φk−1 ) dφ
Similar definition prevails for 1 replaced with 0. We associate the notation 1 (0) with lattice 1 (0 ). a. We have E k,1,0y =
1 ( Hk−1,1,ψ0 ψ y 1 − Hk−1,1,ψ0 ; Hk−1,1,ψ y : 1 c1 ). 2
(5.6)
This is the extracted quadratic part, so it is evaluated when the external field ψ is set to zero. Let us consider the second term on the right side of (5.6). Similar method works for 1 the first term. If |0 − y| ≥ 41 |0 | d , then statement a is automatic due to the exponential 1
decay (4.65), so we assume that |0 − y| ≤ 14 |0 | d . We deal with each factor separately in
Hk−1,1,ψ0 Hk−1,1,ψ y 1 − Hk−1,1,ψ0 1 Hk−1,1,ψ y 1 .
(5.7)
46
H. Pinson
Thus,
Hk−1,1,ψ0 Hk−1,1,ψ y 1 − Hk−1,0,ψ0 Hk−1,0,ψ y 0 = Hk−1,1,ψ0 Hk−1,1,ψ y − Hk−1,0,ψ0 Hk−1,0,ψ y 1 + Hk−1,0,ψ0 Hk−1,0,ψ y 1 − Hk−1,0,ψ0 Hk−1,0,ψ y 0 .
(5.8)
There are three correlations on the right side of (5.8). The first term is bounded as
Hk−1,1,ψ0 (Hk−1,1,ψ y − Hk−1,0,ψ y ) + (Hk−1,1,ψ0 − Hk−1,0,ψ0 )Hk−1,0,ψ y 1 . (5.9) First, since the external field ψ is set to zero, all one point functions φ y 0 , φ y 1 equal zero. Thus, correlations like φ y2 0 , φ y2 1 can be estimated using Brascamp-Lieb [BL, Theorem 5.1]. The difference Hk−1,1,ψ y − Hk−1,0,ψ y has terms like f
−1
f
−1
−1
f
f
−1
(Q i,1 ) yy2 E j,1,y y (Q k,1 ) y 2y φk,y − (Q i,0 ) yy2 E j,0,y y (Q k,0 ) y 2y φk,y . (5.10) 1 d
If max(|y |, |y |, |y |) > |20 | , then one of the points y , y , y is very far from y. There will be an exponential decay. If this condition is not satisfied, then we can f f use the induction hypothesis on the quadratic forms Q i , E j , Q k to get a bound by −
1 |0 | d 1+kδ+n L
O(e ). If a term in Hk−1,1,ψ y − Hk−1,0,ψ y cannot be paired off as in (5.10), then this means that there is a point among y , y , y (as in (5.10)) which is very far from −
1 |0 | d 1+kδ+n L
the point y. Thus, there will be some exponential decay O(e ). The second two terms on the right side of (5.8) can be rewritten as 1 dt Hk−1,0,ψ0 Hk−1,0,ψ y ;
Hk−1,0,ψ0 Hk−1,0,ψ y 1 − Hk−1,0,ψ0 Hk−1,0,ψ y 0 = 0
(H0 − H1 ) : 1 ct , where
(5.11)
−(1−t)H −t H −I (φ) k,0 k,1 •e dφ
• t = −(1−t)H −t H −I (φ) . k,0 k,1 e dφ
(5.12)
Consider the expression H0 − H1 . Replace the first letter y in (5.10) by another letter y1 . Such differences as (5.10), f
−1
f
−1
f
−1
f
−1
φi,y1 (Q i,1 ) y1 2y E j,1,y y (Q k,1 ) y 2y φk,y − φi,y1 (Q i,0 ) y1 2y E j,0,y y (Q k,0 ) y 2y φk,y , appear in H0 − H1 . If 1
|y1 |, |y |, |y |, |y | ≤ f
|0 | d , 2
(5.13)
f
then by the induction hypothesis on Q i , E j , Q k , those differences can be bounded by O(e
1 |0 | d − 1+kδ+n L
). If the condition (5.13) is violated, then that means that one of the points
Towards a Nonperturbative Renormalization Group Analysis
y1 , y , y , y is very far (at least
47
1
|0 | d 4
) from 0, y. Since this (5.11) is a connected −
1 |0 | d L 1+kδ+n
expectation, we can establish exponential decay e . f b. The quadratic form Q i , apart from some initial functions h( p) (3.6), is defined in terms of E j ’s. Thus, part a implies part b. c. We have −t H −I (φ) 1 e k,1 dφ d dt (−log( )) fˆk,1 (0) = |1 |k dt 0 (π ) 2 1 dt Hk,1 t , (5.14) = 0
where
−t H −I (φ) •e k,1 dφ
• t = −t H −I (φ) . e k,1 dφ
(5.15)
There is translation invariance present in Hk,1 . Factor out the volume |1 |, and the rest can be analyzed in a manner similar to part a. QED Lemma 5.1. 1
5
We do the integration step N + 1 times with M = |0 | d = L 1+ 4 N δ . At the end, we have Z 0 dφ z N +1 z 1 zN ˆ ˆ = e− f N +1 (φ)−Q N +1 (φ) · · · s e− f1 (0)−···− f N +1 (0) . (5.16) z0 z N +1 z 0s z 1s zN We have done this integrating out process with respect to lattice 0 . We want to take the limit as N → +∞. Take two lattices 0 ⊂ 1 with |0 | = M d ≤ |1 |. With respect to lattice 1 , we do the integrating out process and derive an expression similar to (5.16), Z 1 dφ z N +1,1 z 1,1 z N ,1 − ˆ = e− f N +1,1 (φ)−Q N +1,1 (φ) e f 1,1 (0) s s · · · zs z0 z N +1,1 z 0,1 z 1,1 N ,1 − · · · − fˆN +1,1 (0).
(5.17)
We have used the same notation z 0 in (5.16) and (5.17). z 0 is defined in (2.6). In (5.16), z 0 is defined with respect to lattice 0 . In (5.17), it is defined with respect to 1 . The actions f k,1 , k = 1, . . . , N + 1 are defined with respect to lattice 1 . N is defined by 1 5 |1 | d = L 1+ 4 N δ . The difference 1
|0 | d fˆk,1 (0) fˆk (0) − − | ≤ e L 1+kδ+k | |0 | |1 |
(5.18)
for k = 1, . . . , N + 1 is small due to Lemma 5.1, part c. For k = N + 2, . . . , N + 1, consider 1 f dφ d dt log( e−t fk−1,1 (φ)−Q k−1 (φ) f ) fˆk,1 (0) = − dt 0 z k−1,1 1 = dt < f k−1,1 (φ) >t , (5.19) 0
48
H. Pinson
where < • >t =
f
•e−t fk−1,1 (φ)−Q k−1,1 (φ) dφ f
e−t fk−1,1 (φ)−Q k−1,1 (φ) dφ
.
(5.20)
Again due to translation invariance, there is a volume |1 | factor that can be factored out from the last quantity in (5.19). It can be shown (break the integral to large and small fields; Taylor expand f k−1,1 (φ) and use (4.28) in the small field region; use Lemma 4.5 in the large field region) that |
fˆk,1 (0) C |≤ 3 (d−2)(1+(k−2)δ) |1 | L2
(5.21)
for k = N + 2, . . . , N + 1. Thus, (5.18) and (5.21) imply that ˆ
ˆ
e− f1 (0)−···− f N +1 (0) = e F1 ||+c1 ()
(5.22)
with c1 () → 0 as || → +∞. We have || β0 ( p) . . . βk−1 ( p) , z k = (π ) 2 h( p) + E ( p)β ( 1 0 p) + · · · + E k ( p)β0 ( p) . . . βk−1 ( p) p
z ks = z k
1
βk2 ( p)
(5.23)
p
for k = 0, . . . , N + 1. Thus, 1 z N +1 z N z 1 h( p) ) 2 .(5.24) ... s= ( s s z 0 z N z 1 p h( p) + E 1 ( p)β0 ( p) + · · · + E N +1 ( p)β0 ( p) . . . β N ( p)
We do not expand β terms in p as a Taylor series. However, we can Taylor expand E i terms in p. In this way, we obtain h( p) = C1 p 2 +
1 M d+κ
+ O( p 4 ),
h( p) + E 1 ( p)β0 ( p) + · · · + E N +1 ( p)β0 ( p) . . . β N ( p) 1 = h 1 ( p) p 2 + d+κ + O( p 4 ) M
(5.25)
(5.26)
with the real valued function h 1 ( p) satisfying h 1 ( p) > C > 0 for all p. Using the expansions (5.25) and (5.26), (5.24) becomes 1
e2
p =0 log(C 1 p 1
= e2
2+
1 M d+κ
p =0 log(C 1 p
+O( p4 ))−log(h 1 ( p) p2 +
2 +O( p 4 ))−log(h
1 ( p) p
1 M d+κ
+O( p4 ))
2 +O( p 4 ))+log(1+
1 1 )−log(1+ d+κ ) M d+κ (C1 p 2 +O( p 4 )) M (h 1 ( p) p 2 +O( p 4 ))
Towards a Nonperturbative Renormalization Group Analysis
=e
e
1 2
1 2
49
C1 O( p4 ) O( p4 ) p =0 log( h 1 ( p) )+log(1+ C p2 )−log(1+ h ( p) p2 ) 1 1
1 1 p =0 log(1+ M d+κ (C p 2 +O( p4 )) )−log(1+ M d+κ (h ( p) p2 +O( p4 )) ) 1 1
.
(5.27)
Note
|log(1 +
M d+κ (C
p =0
C 1 1 C )| ≤ d+κ ≤ κ. 2 4 2 M p M 1 p + O( p ))
(5.28)
p =0
In the same way, |
log(1 +
p =0
C 1 )| ≤ κ . M d+κ (h 1 ( p) p 2 + O( p 4 )) M
(5.29)
Using Fourier analysis, we have
log(
p =0
C1 C1 ) = Md ), a1,n M − log( h 1 ( p) h 1 (0) d
(5.30)
n∈Z
a1,n M =
1 (2π )d
π −π
e−i pn M log(
C1 )d d p. h 1 ( p)
(5.31)
Integration by parts d + 1 times implies Md
C . M
(5.32)
C1 ) → c∞ h 1 (0)
(5.33)
|a1,n M | ≤
n =0
Also due to Lemma 5.1, − log(
c∞ ∈ R as || → +∞. This is the source of the nonuniversality. (In the case of the 2d critical Ising model, there is no zero mode contribution to the partition function. The zero mode sector vanishes. Thus, in that case, it was shown [PS] that the constant c∞ is actually zero.) By Fourier analysis, we have
O( p 4 ) ) = Md a2,n M , 2 h 1 ( p) p d
log(1 +
a2,n M =
(5.34)
n∈Z
p
1 (2π )d
π
−π
e−i pn M log(1 +
O( p 4 ) d )d p. h 1 ( p) p 2
(5.35)
Due to the presence of p 4 in the numerator, we can integrate by parts d + 1 times. Thus, |M d
n =0
a2,n M | ≤
C . M
(5.36)
50
H. Pinson
A similar argument applies to
log(1 +
p
O( p 4 ) ). C1 p2
(5.37)
Thus, there is F2 ∈ R such that z N +1 z N z1 · · · s = e F2 ||+c2 () s s z0 z N z1 with c2 () → c∞ as || → +∞. Let
g(t) = log( with 0 ≤ t ≤ 1. Then,
e−t f N +1 (φ) e−Q N +1 (φ)
1
g(1) = −
(5.38)
dφ ) z N +1
(5.39)
dt f N +1 (φ)(χ S + χ S c ) t
(5.40)
0
with
• t =
•e−t f N +1 (φ) e−Q N +1 (φ) dφ/
e−t f N +1 (φ) e−Q N +1 (φ) dφ
(5.41)
and S = {φ ∈ R|| ||φ y | ≤
1 L
d−2 2 (1+N δ−N )
}.
(5.42)
By Lemma 4.1 (4.28), 5
| f N +1 (φ)χ S t | ≤
L d(1+ 4 N δ) L 6
→0 (5.43) 3(d−2) L 2 (1+N δ−N ) as || → +∞. f N +1 needs to be Taylor expanded to third order to apply (4.28). By a slight modification of Lemma 4.5, | f N +1 (φ)χ S c t | ≤ e−L as || → +∞. Thus,
e− f N +1 (φ)−Q N +1 (φ)
N
→0
dφ = ec3 () z N +1
(5.44)
(5.45)
with c3 () → 0
(5.46)
as || → +∞. Combining (5.22), (5.38), (5.45), we have Z = e F||+c() z0
(5.47)
c() → c∞
(5.48)
with as || → +∞. QED theorem.
Towards a Nonperturbative Renormalization Group Analysis
51
6. Appendix. The estimate in Lemma 3.2 over the interval 0 ≤ t ≤ L 2 is obtained from perturbation theory. There is the first term, and the second term results from the perturbation. On the interval t > L 2 , we can afford to get worse estimates, and a derivative argument yields the second estimate. First, we consider the case with t ∈ [τ, τ + L 2 ] and ∂ k(x, y, t, τ ) = − (H ess( f 0 (φt ))x z + x z + A x z )k(z, y, t, τ ) ∂t z k(z, y, τ, τ ) = δz,y . In regard to matrix A x z (3.7), we have the bound |x−y|
Ce− L
A x,y | ≤ d 2 2 . L L A
(A.0)
This can be seen by approximating the sum by an integral and complex translating by (x−y) p → p − i L|x−y| . By perturbation theory, k(x, y, t, τ ) = W (x, y, t, τ )+
t τ
∞ dσ W (x, z, t, σ ) (−1)m km (z, y, σ, τ ), (A.1) z
m=0
k0 (x, y, t, τ ) = −
A x z W (z, y, t, τ ),
z
km+1 (x, y, t, τ ) =
t
τ
dσ
A x z W (z, z , t, σ )km (z , y, σ, τ ),
(A.2)
z,z
where ∂ W (x, y, t, τ ) = − (H ess( f 0 (φt ))x z + x z )W (z, y, t, τ ) ∂t z W (z, y, τ, τ ) = δz,y . We have 0 ≤ W (x, y, t, τ ) ≤
C d
(t − τ + 1) 2
e
− C1
|x−y|2 t−τ +|x−y|
.
(A.3)
This result is stated in [NS]. The combination of the methods of Nash [N] and Grigor’yan [G] also yields this. The long range piece is not present, so the continuum methods can be adapted to the lattice situation. Using the estimates (A.0), (A.3) and the inductive formula (A.2), we can establish |km (x, y, t, τ )| ≤
C1m − 1 |x−y| e C1 L . A2m L 2 L d
(A.4)
52
H. Pinson
We will do the inductive step. With the condition 0 ≤ t − τ ≤ L 2 , we have |km+1 (x, y, t, τ )| ≤ ≤ ≤
t τ
t
τ
t
dσ
τ
|A x z W (z, z , t, σ )km (z , y, σ, τ )|
z,z |z−z |2
1
1
Ce− L1 |x−z| Ce− C t−σ +|z−z | C m e− C1 L |z −y| 1 dσ d 2 A2 L d L L 2 A2m L d 2 (t − σ + 1) z,z
|z−z |2
1
1
|z−z |2
1
Ce− 2L1 |x−z| Ce− 2C t−σ +|z−z | C m e− 2L |x−z|− 2C t−σ +|z−z | − C1 L |z −y| 1 dσ d 2 A2 L d L L 2 A2m L d (t − σ + 1) 2 1
z,z
≤
C1m L 2 A2(m+1) L d
e
− C 1 L |x−y|
1
τ
≤
1
t
dσ
|z−z |2
Ce− 2L1 |x−z| Ce− 2C t−σ +|z−z | L2 Ld
z,z
d
(t − σ + 1) 2
C1m+1 − C 1 L |x−y| 1 e . L 2 A2(m+1) L d
Thus, the bound (A.4) holds for τ ≤ t ≤ τ + L 2 . Since A is large, this perturbation series converges, and we get the estimate for Lemma 3.2 over the interval 0 ≤ t ≤ L 2 . Now we prove (3.61) over the interval t > L 2 . Define d
g(t) = (t + C L 2 A2 ) 2
k(x, y, t, 0)2
y
for some large constant C and t ≥ L 2 . Initially, d
g(L 2 ) ≤ (L 2 + C L 2 A2 ) 2
|k(x, y, L 2 , 0)|
y
1 ≤ C. Ld
The summability of the heat kernel follows from the estimates (A.3), (A.4) and the perturbative formula (A.1). We show that the derivative of g(t) is non-positive: d
g (t) = (t + C L 2 A2 ) 2 (2
k(x, y, t, 0)
y
d k(x, y, t, 0) dt
d + (t + C L 2 A2 )−1 k(x, y, t, 0)2 ) 2 y 2 2 d2 B y,z ∇ y,y+z k(x, y, t, 0)∇ y,y+z k(x, y, t, 0) = (t + C L A ) (−2 y,z
−2(
−L 2 M d+κ
e 1 + d+κ ) k(x, y, t, 0)2 2 2 L A M y
d + (t + C L 2 A2 )−1 k(x, y, t, 0)2 ) ≤ 0 2 y
Towards a Nonperturbative Renormalization Group Analysis
53
for sufficiently large C. Thus,
k(x, y, t, 0)2 ≤
y
C d
(t + 1) 2
over the interval t ≥ L 2 . Thus, |k(x, y, t, 0)| = |
z
≤ ≤(
z∈
t t k(x, z, t, )k(z, y, , 0)| 2 2
t t |k(x, z, t, )||k(z, y, , 0)| 2 2
1 1 t t C |k(x, z, t, )|2 ) 2 ( |k(z, y, , 0)|2 ) 2 ≤ d 2 2 (t + 1) 2 z∈ z∈
and |k(x, y, t, 0)| ≤
⎧ |x−y|2 ⎪ − ⎪ ⎨ Ce C(t+|x−y|) ⎪ ⎪ ⎩
d (t+1) 2
C
d (t+1) 2
+
− 1 |x−y|
Ce C L L 2 A2 L d
0 ≤ t ≤ L2 t ≥ L2
.
Acknowledgement. I would like to thank Professor Thomas Spencer for suggesting that we use convexity methods to do renormalization group analysis. This paper is one realization of his ideas. I would like to thank Professor Codina Cotar for her extensive comments and questions which have improved the clarity of this paper. I would like to thank Professor William Faris and Assane Lo for some comments. I would like to thank the referee for her/his insistence that I improve the clarity of this paper. In the second review, this referee provided six pages of detailed comments. I would also like to thank Professor Antti Kupiainen for his visit to the University of Arizona and his comments in regard to this paper. In the previous version of this paper, our analysis only worked in extremely high dimensions, and our arguments were unclear. Due to the comments by Professor Antti Kupiainen and the referee, we rethought the entire argument.
References [BL] [BY] [FMRS] [F] [G] [GK] [G] [H] [HS]
Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn-Minkowski and Prkopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976) Brydges, D., Yau, H.-T.: Grad φ perturbations of massless gaussian fields. Commun. Math. Phys. 129(2), 351–392 (1990) Feldman, J., Magnen, J., Rivasseau, V., Snor, R.: Construction and borel summability of infrared 44 by a phase space expansion. Commun. Math. Phys. 109(3), 437–480 (1987) Friedan, D.: Ann. Phys. 163, 318 (1985) Gawedzki, K.: Conformal field theory and strings, In: Quantum fields and strings: a course for mathematicians, Volume 2, Providence, RI: Amer. Math. Soc., 1991 Gawedzki, K., Kupiainen, A.: A rigorous block spin approach to massless lattice theories. Commun. Math. Phys. 77(1), 31–64 (1980) Grigor’yan, A.: Gaussian upper bounds for the heat kernel on arbitrary manifolds. J. Diff. Geom. 45, 33–52 (1997) Hamilton R.S.: Three-manifolds with positive ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982) Helffer, B., Sjstrand, J.: On the correlation for kac-like models in the convex case. J. Stat. Phys. 74(1-2), 349–409 (1994)
54
[NS] [N] [PS] [RSIV] [WK]
H. Pinson
Naddaf, A., Spencer, T.: On homogenization and scaling limit of some gradient perturbations of a massless free field Nash, J.: Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80(4), 931 (1958) Pinson, H., Spencer, T.: Universality in 2d Critical Ising Model, available from Haru Pinson Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Analysis of Operators, Vol. IV, New York: Academic Press, 1978 Wilson, K., Kogut, J.: The renormalization group and the expansion. Phys. Rep. 12c, 75 (1975)
Communicated by A. Kupiainen
Commun. Math. Phys. 282, 55–86 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0532-3
Communications in
Mathematical Physics
Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation Frank Merle1 , Hatem Zaag2 1 Université de Cergy Pontoise, IHES and CNRS, Département de mathématiques,
2 avenue Adolphe Chauvin, BP 222, 95302 Cergy Pontoise cedex, France. E-mail:
[email protected]
2 Université Paris 13, Institut Galilée, Laboratoire Analyse, Géométrie et Applications,
CNRS UMR 7539, 99 avenue J.B. Clément, 93430 Villetaneuse, France. E-mail:
[email protected];
[email protected] Received: 8 March 2007 / Accepted: 21 February 2008 Published online: 24 June 2008 – © Springer-Verlag 2008
Abstract: We consider here the 1 D semilinear wave equation with a power nonlinearity and with no restriction on initial data. We first prove a Liouville Theorem for that equation. Then, we consider a blow-up solution, its blow-up curve x → T (x) and I0 ⊂ R the set of non-characteristic points. We show that I0 is open and that T (x) is C 1 on I0 . All these results fundamentally use our previous result in [19] showing the convergence in selfsimilar variables for x ∈ I0 . 1. Introduction 1.1. The problem and known results. We consider the one dimensional semilinear wave equation 2 ∂tt u = ∂x2x u + |u| p−1 u, (1) u(0) = u 0 and u t (0) = u 1 , 1 2 where p > 1, u(t) : x ∈ R → u(x, t) ∈ R, u 0 ∈ Hloc,u and u 1 ∈ Lloc,u , with |v(x)|2 d x and v2H1 = v2L2 + ∇v2L2 . v2L2 = sup loc,u
a∈R |x−a|<1
loc,u
loc,u
loc,u
1 2 × Lloc,u follows from the finite The Cauchy problem for Eq. (1) in the space Hloc,u 1 2 speed of propagation and the wellposedness in H ×L (see Ginibre, Soffer and Velo [6]).
The existence of blow-up solutions for Eq. (1) follows from energy techniques (see Levine [9]). More blow-up results can be found in Caffarelli and Friedman [4,5], Alinhac [1,2] and Kichenassamy and Litman [7,8]. Most of the previous literature considered blow-up for the wave equation from the point of view of prediction. Indeed, most of the papers gave sufficient conditions to have This work was supported by a grant from the french Agence Nationale de la Recherche, project ONDENONLIN, reference ANR-06-BLAN-0185.
56
F. Merle , H. Zaag
blow-up or constructed special solutions with a prescribed behavior (see [7] and [8] for example). As we did in our earlier work ([16–19]), we adopt in this paper a different point of view and aim at describing the blow-up behavior for any blow-up solution. More precisely, this paper is dedicated to the regularity of the blow-up curve. If u is a blow-up solution of (1), we define (see for example Alinhac [1]) a continuous curve as the graph of a function x → T (x) such that u cannot be extended beyond the set Du = {(x, t) | t < T (x)}.
(2)
The set Du is called the maximal influence domain of u. From the finite speed of propagation, T is a 1-Lipschitz function. Let T¯ be the infimum of T (x) for all x ∈ R. The time T¯ and the surface are called (respectively) the blow-up time and the blow-up surface of u. A point x0 ∈ R is called a non-characteristic point if ∃δ0 = δ0 (x0 ) ∈ (0, 1) and t0 (x0 ) < T (x0 ) such that u is defined on Cx0 ,T (x0 ),δ0 ∩ {t ≥ t0 },
(3)
where ¯ ¯ Cx, ¯ ¯ t¯,δ¯ = {(x, t) | t < t − δ|x − x|}.
(4)
We denote by I0 the set of non-characteristic points. So far, it was commonly thought that I0 = R, for any blow-up solution. In a forthcoming paper [20], we prove that this is not the case. Given some (x0 , T0 ) such that 0 < T0 ≤ T (x0 ), we introduce the following self-similar change of variables: 2
wx0 ,T0 (y, s) = (T0 − t) p−1 u(x, t), y =
x − x0 , s = − log(T0 − t). T0 − t
(5)
If T0 = T (x0 ), then we simply write wx0 instead of wx0 ,T (x0 ) . This change of variables transforms the backward light cone with vertex (x0 , T0 ) into the infinite cylinder (y, s) ∈ B × [− log T0 , +∞), where B = B(0, 1). The function wx0 ,T0 (we write w for simplicity) satisfies the following equation for all y ∈ B and s ≥ − log T0 : 2 w = Lw − ∂ss
where Lw =
2( p + 1) p+3 2 ∂s w − 2y∂ y,s w + |w| p−1 w − w, 2 ( p − 1) p−1
(6)
2 1 ∂ y ρ(1 − y 2 )∂ y w and ρ(y) = (1 − y 2 ) p−1 . ρ
(7)
The Lyapunov functional for Eq. (6) E(w(s)) =
1
−1
2 1 1 ( p + 1) 2 ∂ y w (1 − y 2 ) + w (∂s w)2 + 2 2 ( p − 1)2 1 |w| p+1 ρdy − p+1
(8)
Blow-up for a Semilinear Wave Equation
57
is defined in 1 2 H = q ∈ Hloc × L loc (−1, 1) | q2H
1 2 2 2 2 q1 + q1 (1 − y ) + q2 ρdy < +∞ . ≡ −1
(9)
In [19], we find the behavior of wx0 (y, s) defined in (5) as s → ∞, where x0 is a non-characteristic or a characteristic point. More precisely, we proved this result (see Corollary 4 and Theorem 2 in [19]): Blow-up profile near a non-characteristic point. There exist positive µ0 and C0 such that if u is a solution of (1) with blow-up curve : {x → T (x)} and x0 ∈ R is non-characteristic (in the sense (3)), then there exist d(x0 ) ∈ (−1, 1), θ (x0 ) = ±1, s0 (x0 ) ≥ − log T (x0 ) such that for all s ≥ s0 (x0 ): wx (s) κ(d(x0 ), .) 0 ≤ C0 e−µ0 (s−s0 (x0 )) , − θ (x ) 0 ∂s wx0 (s) 0 H
(10)
where H and its norm are defined in (9), κ(d, y) is defined for all |d| < 1 and |y| ≤ 1 by
1
κ(d, y) = κ0
(1 − d 2 ) p−1 2
(1 + dy) p−1
where κ0 =
2( p + 1) ( p − 1)2
1 p−1
.
(11)
Moreover, we have E(wx0 (s)) → E(κ0 ) as s → ∞
(12)
wx0 (s) − θ κ(d(x0 ), y) L ∞ (−1,1) + ∂ y wx0 (s) − θ ∂ y κ(d(x0 ), y) L 2 (−1,1) +∂s wx0 (s) L 2 (−1,1) → 0 as s → ∞.
(13)
and
Blow-up behavior near a characteristic point. If x0 ∈ R is characteristic, then, there exist k(x0 ) ∈ N, θi (x0 ) = ±1 and continuous di (s) = tanh ζi (s) ∈ (−1, 1) for i = 1, . . . , k such that ⎛ k(x ) ⎞ 0 wx0 (s) ⎜ θi (x0 )κ(di (s), ·) ⎟ ∂ w (s) − ⎝ ⎠ → 0, (14) s x0 i=1 0 H
|ζi (s) − ζ j (s)| → ∞ for i = j and E(wx0 (s)) → k(x0 )E(κ0 ) as s → ∞.
Remark. When k(x0 ) = 0, the sum in (14) has to be understood as 0. In [20], we prove the existence of solutions with characteristic points. Furthermore, we greatly improve estimate (14) and give a precise description of the set of characteristic points.
58
F. Merle , H. Zaag
1.2. Statement of the results. With the result of [19], we are in a position to address the question of the regularity of the blow-up set for Eq. (1) and the notion of non-characteristic points. Note that by definition (see Alinhac [1]), is the graph of a 1-Lipschitz function T (x). In [5], Caffarelli and Friedman proved that it is a C 1 function for N ≤ 3 under restrictive conditions on initial data that ensure that for all x ∈ R N and t ≥ 0, u ≥ 0 and ∂t u ≥ (1 + δ0 )|∇u| for some δ0 > 0. Later on, they derived in [4] the same result in one dimension for p ≥ 3 and initial data in C 4 × C 3 (R). The techniques to prove these results are of elliptic type and based on the use of the maximum principle. There is no hope to generalize these techniques to the present situation. Furthermore, no results are available on the set of non-characteristic points. In this paper, we prove the following with no restrictions on the initial data: Theorem 1 (C 1 regularity of the blow-up set and continuity of the blow-up profile on I0 ). Consider u a solution of (1) with blow-up curve : {t = T (x)}. Then, the set of non-characteristic points I0 is open and T (x) is C 1 on that set. Moreover, ∀x ∈ I0 , T (x) = d(x) ∈ (−1, 1) and θ (x) is constant on connected components of I0 , where d(x) and θ (x) are such that (10) holds. Remark. From the remark after Proposition 3.5 (with δ1 = 21 ), we get the existence of a non-characteristic point. Thus, I0 is never empty and Theorem 1 is always meaningful. Unlike what was commonly thought until now, we prove in a forthcoming paper [20] the existence of blow-up solutions to (1) with R\I0 = ∅. Remark. From this theorem, the parameter d(x) related to the blow-up profile in selfsimilar variables (10), has a geometrical interpretation as the slope of T (x). In [21], Nouaili uses the results and techniques of [19] and this paper with a geometrical approach to get more regularity, namely C 1,µ0 regularity where the universal constant µ0 is introduced before (10). Remark. The techniques are based on a very good understanding of the behavior of the solution in selfsimilar variables in the energy space related to the selfsimilar variable, together with a Liouville Theorem (see Theorem 2). For a similar approach of the blowup problem, see Martel and Merle [10,11] for the critical KdV equation; see also Merle and Raphaël [12,13] for the NLS equation. Note that the main obstruction to extend the result in higher dimensions is the result of classification of all stationary solutions to (6). The proof of this theorem relies fundamentally on the convergence result (10) already obtained in [19] and on this Liouville Theorem in the u or w variable: Theorem 2 (A Liouville Theorem for Eq. (1)) Consider u(x, t) a solution to Eq. (1) defined in the cone Cx ∗ ,T ∗ ,δ ∗ (4) such that for all t < T ∗ , ∗
(T − t) ∗
2 p−1
+(T − t)
u(t) L 2 (B(0, T ∗ −t )) δ∗
(T ∗ − t)1/2 u t (t) L 2 (B(0, T ∗ −t ))
2 p−1 +1
δ∗
(T ∗ − t)1/2
+
∇u(t) L 2 (B(0, T ∗ −t )) (T ∗ − t)1/2
δ∗
≤ C∗
(15)
Blow-up for a Semilinear Wave Equation
59
for some (x∗ , T∗ ) ∈ R2 , δ∗ ∈ (0, 1) and C ∗ > 0. Then, either u ≡ 0 or u can be extended to a function (still denoted by u) defined in {(x, t) | t < T0 + d0 (x − x ∗ )} ⊃ CT ∗ ,x ∗ ,δ∗ by 1
u(x, t) = θ0 κ0
(1 − d02 ) p−1 2
(T0 − t + d0 (x − x ∗ )) p−1
,
(16)
for some T0 ≥ T ∗ , d0 ∈ [−δ∗ , δ∗ ] and θ0 = ±1, where κ0 is defined in (11). Using the selfsimilar transformation (6), we have this equivalent formulation for Theorem 2: Theorem 2’ (A Liouville Theorem for Eq. (6)). Consider w(y, s) a solution to Eq. (6) defined for all (y, s) ∈ (− δ1∗ , δ1∗ ) × R such that for all s ∈ R, w(s) H 1 (− 1 , 1 ) + ∂s w(s) L 2 (− 1 , 1 ) ≤ C ∗ δ∗ δ∗
δ∗ δ∗
(17)
for some δ∗ ∈ (0, 1) and C ∗ > 0. Then, either w ≡ 0 or w can be extended to a function (still denoted by w) defined in 1 1 ×R {(y, s) | − 1 − T0 es < d0 y} ⊃ − , δ∗ δ∗ 1
by w(y, s) = θ0 κ0
(1 − d02 ) p−1 2
(1 + T0 es + d0 y) p−1
,
(18)
for some T0 ≥ T ∗ , d0 ∈ [−δ∗ , δ∗ ] and θ0 = ±1, where κ0 is defined in (11). Remark. The limiting case δ ∗ = 1 is still open. The case δ ∗ = 0 trivially follows from the case δ ∗ > 0. We expect the result to be valid in higher dimension using the same techniques.The main obstruction comes from the classification of stationary solutions to Eq. (6), which is only proved in one dimension (see [19]) and remains open in higher dimension. Remark. From (16), we see that u is a particular solution to (1) which is defined in a half-space of equation t < T0 + d0 (x − x∗ ) (and not just the cone Cx ∗ ,T ∗ ,δ ∗ ) and blows up on a straight line of slope d0 . Note that u ≡ 0 corresponds to the limiting case T0 = +∞. Similarly, w given in (18) is a particular solution to (6). In particular, when T0 = 0, we recover the stationary solution θ0 κ(d0 , y) defined in (11). Note also that up to a Lorentz transform, the solution given in (16) is space independent and given by u(x, t) = θ0
κ0 2
(T0 − t) p−1
.
Remark. Note that deriving blow-up estimates through the proof of Liouville Theorems has been successful for different problems. For the case of the heat equation ∂t u = u + |u| p−1 u,
(19)
where u : (x, t) ∈ R N × [0, T ) → R, p > 1 and (N − 2) p < N + 2, the blow-up time T is unique for Eq. (19). The blow-up set is the subset of R N such that u(x, t) does
60
F. Merle , H. Zaag
not remain bounded as (x, t) approaches (x0 , T ). In [22,23,25] (see also the note [24]), the second author proved the C 2 regularity of the blow-up set under a non-degeneracy condition. A Liouville Theorem proved in [15] and [14] was crucially needed for the proof of the regularity result in the heat equation. In the present work, we will see that the Liouville Theorem (Theorem 2) is crucial for the regularity of the blow-up set for the wave equation (Theorem 1). Remark. The proof has a completely different structure from the proof for the heat equation (19). It is based on various energy arguments in selfsimilar variables (some of them holding even in the characteristic situation) and on a dynamical result again in selfsimilar variables obtained in [19] in the non-characteristic case. The paper is organized as follows: – In Sect. 2, we assume the Liouville Theorem and prove Theorem 1. – In Sect. 3, we prove the Liouville Theorem (Theorems 2 and 2’). The authors wish to thank the referee for his meticulous reading and valuable suggestions which undoubtedly improved the paper. 2. C 1 Regularity of the Blow-up Curve at Non-characteristic Points This section is devoted to the proof of Theorem 1 assuming Theorem 2. Proof of Theorem 1. We consider a solution u of (1) with blow-up curve : {t = T (x)}. We proceed in two parts, each making a separate subsection: – In Subsect. 2.1, we consider a non-characteristic point x0 and show that T (x) is differentiable at x = x0 with T (x0 ) = d(x0 ) where d(x0 ) is such that (10) holds. – In Subsect. 2.2, we conclude the proof of Theorem 1.
2.1. Differentiability of the blow-up curve at a given point. In this subsection, we prove the proposition: Proposition 2.1 (Differentiability of the blow-up curve at a given non-characteristic point). If x0 is a non-characteristic point, then T (x) is differentiable at x0 and T (x0 ) = d(x0 ), where d(x0 ) is such that (10) holds. Proof. We consider x0 a non-characteristic point. From (3), we have Cx0 ,T (x0 ),δ0 ∩ {t ≥ t0 } ⊂ Du
(20)
for some δ0 ∈ (0, 1) and t0 < T (x0 ). From translation invariance, we can assume that x0 = T (x0 ) = 0. Using the convergence result of [19], we see that (10) holds for some d(0) ∈ (−1, 1) and θ (0) = ±1. Up to replacing u(x, t) by −u(x, t) (also solution to Eq. (1)), we can assume that θ (0) = 1. We proceed in 2 steps.
Blow-up for a Semilinear Wave Equation
61
– In Step 1, we use the Liouville Theorem 2 to show that the convergence of w0 (s) in 1 1 1 2 (10) holds also in H × L − δ , δ , where 0
0
δ0 ∈ (δ0 , 1) is fixed.
(21)
– In Step 2, we use energy and continuation arguments in selfsimilar variables to conclude the proof by contradiction. Step 1 Convergence of w0 to κ(d(0), .) on larger sets. We claim: Lemma 2.2 (Convergence in selfsimilar variables on larger sets). It holds that w0 (s) κ(d(0), .) ∂s w0 (s) − 0 1 2 1 1 → 0 as s → ∞. H ×L − , δ0 δ0
Proof. For simplicity, we denote w0 by w. Using the uniform bound on the solution at blow-up (Theorem 2’ in [17]) and the covering technique in that paper (Proposition 3.3 in [17]), we get for all s ≥ − log T (0) + 1, w(s) (22) ∂s w(s) 1 2 1 1 ≤ K H ×L (− , ) δ0 δ0
for some constant K .We proceed by contradiction and assume that for some 0 > 0 and some sequence sn → ∞, we have w(sn ) κ(d(0), .) − (23) ∀n ∈ N, 1 2 1 1 ≥ 0 > 0. ∂s w(sn ) 0 H ×L − , δ0 δ0
Let us introduce the sequence wn (y, s) = w(y, s + sn ).
(24)
Using the uniform bound stated in (22), we can assume that 1 1 1 1 (25) wn (0) z 0 in H 1 − , and ∂s wn (0) v0 in L 2 − , δ0 δ0 δ0 δ0 as n → ∞ for some (z 0 , v0 ) ∈ H 1 × L 2 − δ1 , δ1 . Since we have from the convergence 0 0 result (10), the definitions (9) and (24) of the norm in H and wn , wn (0) κ(d(0), .) → 0 as n → ∞ 1 2 ∂s wn (0) − 0 H ×L (−1+ ,1− ) for any ∈ (0, 1), we deduce from (25) that ∀y ∈ (−1, 1), z 0 (y) = κ(d(0), y) and v0 (y) = 0 (note that we still need to determine (z 0 , v0 ) for 1 < |y| < allows us to conclude, thanks to the Liouville Theorem 2:
1 ). δ0
(26)
The following claim
62
F. Merle , H. Zaag
Claim 2.3 (Existence ofa limiting object) There exists W (y, s) a solution to (6) defined 1 1 for all (y, s) ∈ − δ , δ × R such that: 0
0
(i) W (0) = z 0 and ∂s W (0) = v0 and the convergence is strong in (25). (ii) For all s ∈ R, W (s) ∂s W (s) 1 2 1 1 ≤ K H ×L
−
(27)
, δ0 δ0
where K is defined in (22).
Proof. See Appendix A.
Indeed, from this claim, we see that W (y, s) satisfies the hypothesis of Theorem 2’. Therefore, either W ≡ 0 or there exists T0 ≥ 0, d0 ∈ [−δ0 , δ0 ] and θ0 = ±1 such that:
1 1 ∀(y, s) ∈ − , δ0 δ0
1
× R, W (y, s) = θ0 κ0
(1 − d02 ) p−1
(28)
2
(1 + T0 es + d0 y) p−1
on the one hand, where κ0 is defined in (11). On the other hand, using (26), (i) of Claim 2.3, and the definition (11) of κ(d, y), we see that 1
∀y ∈ (−1, 1), W (y, 0) = z 0 (y) = κ(d(0), y) = κ0
(1 − d(0)2 ) p−1 2
(1 + d(0)y) p−1
.
(29)
Comparing (28) and (29) when y ∈ (−1, 1) and s = 0, we see that θ0 = 1, d0 = d(0) and T0 = 0, hence, from (28), 1 1 ∀(y, s) ∈ − , × R, W (y, s) = κ(d(0), y). δ0 δ0 In particular, from (24), (25) and (i) of Claim 2.3, this implies that w(sn ) κ(d(0), .) 1 2 1 1 → 0 as n → ∞, ∂s w(sn ) − 0 H ×L − , δ0 δ0
which contradicts (23). Thus, Lemma 2.2 holds.
Step 2 Conclusion of the proof of Proposition 2.1. Our goal is to prove that T (x) is differentiable when x = 0 and that T (0) = d(0). We proceed by contradiction. From the fact that T (x) is 1-Lipschitz, we assume that there is a sequence xn such that xn → 0 and
T (xn ) → d(0) + λ with λ = 0 as n → ∞. xn
(30)
Up to extracting a subsequence and to considering u(−x, t) (also solution to (1)), we can assume that xn > 0. Since 0 is non-characteristic, we see from (20) and (21) that λ + d(0) ≥ −δ0 > −δ0 .
(31)
The following corollary transposing the convergence of w0 (s) to wxn (s) follows from Lemma 2.2:
Blow-up for a Semilinear Wave Equation
Corollary 2.4 Let δ1 =
1+δ0 2 .
63
For σn = − log
δ1 (T (xn )+δ0 xn ) δ1 −δ0
, we have
wx (σ ) w± (σ ∗ ) n n → 0 as n → ∞,
(σ ∗ ) ∂s wxn (σn ) − ∂s w± H 1 ×L 2 − 1 , 1 δ1 δ1
where ± = − sgn λ, σ ∗ = log
|λ|(δ1 − δ0 ) δ1 (λ + d(0) + δ0 )
1
and w± (y, s) = κ0
(1 − d(0)2 ) p−1 2
(1 ± es + d(0)y) p−1
(32)
is a solution to (6). Proof. We define for n large enough a time τn as the largest t such that the section of the cone Cxn ,T (xn ),δ1 at time t is included in the cone C0,0,δ0 . Note by definition of τn that we have −τn T (xn ) − τn ⊂ B 0, (33) B xn , δ1 δ0 and τn = T (xn ) − δ1 (ξn − xn ) = −δ0 ξn for some ξn ∈ R, hence τn = −
δ0 (T (xn ) + δ1 xn ). δ1 − δ0
(34)
From (30), (31) and the fact that 0 < δ0 < δ1 < 1, note that τn ≤ 0. Using the selfsimilar transformation (5), we write x , s = − log(−t), −t x − xn − 2 , u(x, t) = (T (xn ) − t) p−1 wxn (z, σ ) where z = T (xn ) − t σ = − log(T (xn ) − t). 2 − p−1
u(x, t) = (−t)
w0 (y, s) where y =
(35)
Therefore, we have 2
wxn (z, σ ) = (1 + es T (xn )) p−1 w0 (y, s), ∂ y wxn (z, σ ) = (1 + es T (xn ))
2 p−1 +1 2 p−1
∂ y w0 (y, s), (1 + es T (xn ))∂s w0 (y, s)
∂s wxn (z, σ ) = (1 + e T (xn ))
2es T (xn ) s w0 (y, s) + e (yT (xn ) + xn )∂ y w0 (y, s) , + p−1 s
(36) (37)
(38)
where y=
z + x n eσ 1 − eσ T (xn )
and
s = σ − log(1 − eσ T (xn )).
(39)
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F. Merle , H. Zaag
Using the same process as in (35) with κ(d(0), .) instead of w0 (note that κ(d(0), .) is also a solution to (6)), we define this solution to (6), 1
w¯ n (z, σ ) = (1 + e T (xn )) s
2 p−1
κ(d(0), y) =
κ0 (1 − d(0)2 ) p−1
. 2 (1 + eσ (d(0)xn − T (xn )) + d(0)z) p−1 (40)
Hence, estimates (36)-(38) hold when (wxn , w0 ) is replaced by (w¯ n , κ0 ). If we take now σ = σn ≡ − log(T (xn ) − τn ), then we see from (39) that s = σn ≡ − log(−τn ) and from (34) and (30) that as n → ∞,
eσn (d(0)xn − T (xn )) → −
λ(δ1 − δ0 ) , δ1 (λ + d(0) + δ0 )
δ1 − δ0 δ1 (δ + d(0) + λ) eσn xn → and 1 + eσn T (xn ) → 0 >0 δ0 (δ1 + d(0) + λ) δ0 (δ1 + d(0) + λ) because of (31). Therefore, if n is large enough, we get from (36)–(38), wx (σ ) w¯ n (σn ) n n ∂s wxn (σn ) − ∂s w¯ n (σn ) 1 2 1 1 H ×L − δ , δ 1 1 w0 (σn ) κ(d(0), .) ≤C , 1 2 ∂s w0 (σn ) − 0 H ×L (J )
(41)
(42)
n
where the interval Jn ⊂ − δ1 , δ1 by (33). Since we have from (32), (40) and (41) 0
0
w¯ n (σ ) w± (σ ∗ ) n → 0 as n → ∞, −
∗ ∂s w¯ n (σn ) ∂s w± (σ ) H 1 ×L 2 − 1 , 1 δ0 δ0
where w± (σ ∗ ) is defined in (32), we get the conclusion of (i) in Lemma 2.4 from Lemma 2.2 and (42). Let us conclude the proof of Lemma 2.2. We consider two cases depending on the sign of λ and reach a contradiction in both cases. Using the selfsimilar transformation (5), we introduce the following solutions to Eq. (1): 2 − p−1
vn (ξ, τ ) = (1 − τ )
wxn (y, s) with y =
ξ , s = σn − log(1 − τ ), 1−τ
(43)
and 2 − p−1
v¯± (ξ, τ ) = (1 − τ )
w±
ξ , σ ∗ − log(1 − τ ) 1−τ 1
= κ0
(1 − d(0)2 ) p−1 ∗
2
((1 − τ ) ± eσ + d(0)ξ ) p−1
.
(44)
Blow-up for a Semilinear Wave Equation
65
From Corollary 2.4, we have vn (0) v¯± (0) ∂τ vn (0) − ∂τ v¯± (0) 1 2 1 1 → 0 as n → ∞. H ×L − ,
(45)
δ1 δ1
Case where λ < 0. Here, we will reach a contradiction using Corollary 2.4 and the fact that u(x, t) cannot be extended beyond its maximal influence domain Du defined by (2). In this case, (45) holds with ± ∗= +. Since vn and v¯+ are solutions to (1) and v¯+ is defined on {(ξ, τ ), | τ < 1 + eσ + d(0)ξ } which contains the closureof D+ (τ0 ) ≡ ∗ {(ξ, τ ) | 0 ≤ τ < τ0 − |ξ |}, where τ0 is fixed in (1, min δ11 , 1 + eσ ), using (45) and the solution of the Cauchy problem for (1), we see that for some n 0 ∈ N and for all n ≥ n 0 , vn (ξ, τ ) is well defined in D+ (τ0 ). Since we have from the selfsimilar transformation (5) and (43), vn (ξ, τ ) = e
2σ
n − p−1
u(x, t) with x = xn + ξ e−σn , t = T (xn ) − e−σn (1 − τ ),
this means that for all n ≥ n 0 , u(x, t) is well defined in the set
{(x, t) | T (xn ) − e−σn ≤ t ≤ T (xn ) + (τ0 − 1)e−σn − |x − xn |}, which contains (xn , T (xn )). This is a contradiction by definition (2) of the maximal influence domain. Case where λ > 0. Here, a contradiction follows from the fact that wxn (y, s) exists for all (y, s) ∈ (−1, 1) × [− log T (xn ), +∞) and satisfies a blow-up criterion at the same time. ∗ In this case, (45) holds with ± = −. Since v¯− is defined on {(ξ, τ ) | τ < 1 − eσ + d(0)ξ } whichcontains the closure of D− (τ0 ) ≡ {(ξ, τ ), | 0 ≤ τ < min(τ0 , 1 − |ξ |)} ∗ eσ for any τ0 ∈ 0, 1 − 1−|d(0)| , using (45) and the solution of the Cauchy problem for (1), we see that for some n 0 (τ0 ) ∈ N and for all n ≥ n 0 , vn (ξ, τ ) is well defined in D+ (τ0 ). Moreover, vn (τ0 ) v¯− (τ0 ) → 0 as n → ∞. ∂τ vn (τ0 ) − ∂τ v¯− (τ0 ) 1 2 H ×L (−1+τ0 ,1−τ0 )
Using again the transformation (43) and taking s = σ ∗ − log(1 − τ0 ), we get for all s ∈ [σ ∗ , log(1 − |d(0)|)), wx (σ − σ ∗ + s) w− (s) n n → 0 as n → ∞. (46) ∂s wxn (σn − σ ∗ + s) − ∂s w− (s) 1 2 H ×L (−1,1) Since we have from (46) E wxn (σn − σ ∗ + s) → E (w− (s)) as n → ∞ and ∀s ∈ [s− , log(1 − |d(0)|)), E(w− (s)) < 0 for some s− < log(1 − |d(0)|) (47) (see Appendix B for the proof), we see that for all n large enough, we have 1 (48) E(w− (s− )) < 0. 2 Since by the definition (5), wxn is defined for all (y, s) ∈ (−1, 1) × [− log T (xn ), +∞), a contradiction follows from the following claim: E(wxn (σn − σ ∗ + s− )) ≤
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F. Merle , H. Zaag
Claim 2.5 (Blow-up criterion for equation (6), see Antonini and Merle [3]). Consider W (y, s) a solution to Eq. (6) such that W (y, s0 ) is defined for all |y| < 1 and E(W (s0 )) < 0 for some s0 . Then, W (y, s) cannot exist for all (y, s) ∈ (−1, 1)×[s0 , ∞). Proof. See Theorem 2, p. 1147 in [3]. Thus, (30) does not hold and T (x) is differentiable at x = 0 with T (0) = d(0). This concludes the proof of Proposition 2.1. 2.2. Proof of Theorem 1. Let x0 be a non-characteristic point. From (3), we know that (20) holds. One can assume that x0 = T (x0 ) = 0 from translation invariance. From [19] and Proposition 2.1, we know (up to replacing u(x, t) by −u(x, t)) that (10) holds with some d(0) ∈ (−1, 1) and θ (0) = 1, and that T (x) is differentiable at 0 with T (0) = d(0).
(49)
We proceed in 3 steps. – In Step 1, we use [19] and the fact that 0 is non-characteristic to derive that (10) holds in a small neighborhood of 0 for some d(x) ∈ (−1, 1) and θ (x) = 1 with d(x) → d(0) as x → 0. – In Step 2, using a geometrical construction and the previous step, we show that in a small open interval containing 0, the Lipschitz constant of T (x) is less than 1+d(0) 2 . – In Step 3, using Steps 1 and 2, we conclude the proof of Theorem 1. Step 1 Openness of the set of x such that (10) holds. We have from the dynamical study in selfsimilar variables (5) in [19]: Lemma 2.6 (Convergence in selfsimilar variables for x close to 0 ). For all > 0, there exists η such that if |x| < η and x is non-characteristic, then, (10) holds for wx with |d(x) − d(0)| ≤ and θ (x) = 1. Remark. Here, we don’t assume that all the points in some neighborhood of 0 are uniformly non-characteristic (that is, δ0 (x) defined in (3) may have no positive lower bound in any neighborhood of 0). We use instead the fact that in [19], we have completely understood the dynamical structure of equation (6) in H close to the stationary solution κ(d(0), y). Proof. – Since 0 is non-characteristic, we have from (22), for all s ≥ s1 for some ≤ K for some constant K , where s1 ∈ R, (w0 (s), ∂s w0 (s)) 1 1 H 1 ×L 2 −
,
δ0 δ0
δ0 ∈ (δ0 , 1) is fixed. Again from the fact that 0 is non-characteristic, note that (10) holds, hence (w0 (s), ∂s w0 (s)) converges to (κ(d(0), .), 0) as s → ∞ in the norm of H defined by (9). – Since for fixed s, we have (wx (y, s), ∂s wx (y, s)) → (w0 (y, s), ∂s w0 (y, s)) in H from the continuity of solutions to equation (6) with respect to initial data, we know that for all > 0, there exists s0 ( ) ≥ s1 and η( ) > 0 such that for all x ∈ (−η( ), η( )), wx (·, s0 ( )) κ(d(0), .) ≤ . ∂s wx (·, s0 ( )) − 0 H
Blow-up for a Semilinear Wave Equation
67
– From Theorem 3 in [19] (use in particular the first remark following Theorem 3), for a small enough fixed > 0, we have that for all x ∈ (−η( ), η( )), there exists d(x) such that wx (y, s) κ(d(x), y) → 0 as s → ∞ ∂s wx (y, s) − 0 H and |d(x) − d(0)| ≤ C . This concludes the proof of Lemma 2.6. Step 2 The Lipschitz constant of T (x) around 0 is less than (1 + |d(0)|)/2. Fix 0 small enough such that
0 > 0 and d(0) + 2 0 < 1.
(50)
Using (49) and Lemma 2.6, we see that there exists η0 > 0 such that ∀|x| ≤ η0 , |T (x) − T (0) − d(0)x| ≤ 0 |x|,
(51)
and if in addition, x is non-characteristic in the sense (3), then, (10) holds for wx with |d(x) − d(0)| ≤ 0 .
(52)
We now claim: Lemma 2.7 (The slope of T (x) around 0 is less than (1 + |d(0)|)/2). It holds that ∀x, y ∈ [−
1 + |d(0)| η0 η0 , ], |T (x) − T (y)| ≤ |x − y|. 10 10 2
Proof. We proceed by contradiction and assume that for some x0 and y0 ∈ [−
η0 η0 1 + |d(0)| , ], |T (x0 ) − T (y0 )| > |x0 − y0 |. (53) 10 10 2
Up to considering u(−x, t) (also a solution to Eq. (1)) and to renaming x0 and y0 , we can assume that y0 < x0 , T (y0 ) ≤ T (x0 ) and
1 + |d(0)| (x0 − y0 ) ≤ T (x0 ) − T (y0 ). 2
(54)
We can also assume that y0 is the minimum in [−η0 , x0 ] satisfying (54). Therefore, we have ∀x ∈ [−η0 , y0 ),
1 + |d(0)| (x0 − x) ≥ T (x0 ) − T (x). 2
(55)
Let us define x ∗ ∈ [y0 , x0 ] such that T (x ∗ ) − d(0)x ∗ =
min T (x) − d(0)x.
y0 ≤x≤x0
(56)
Note from (54) that x ∗ < x0 . Indeed, if not, then T (x0 ) − d(0)x0 ≤ T (y0 ) − d(0)y0 , hence, T (x0 ) − T (y0 ) ≤ d(0)(x0 − y0 ) which contradicts (54).
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F. Merle , H. Zaag
Considering a family of straight lines of slope d(0) + 2 0 , there exists one which is “tangent” to the blow-up curve on [x ∗ , x0 ] at some point (m ∗ , T (m ∗ )) with − η0 ≤ y0 ≤ x ∗ ≤ m ∗ ≤ x0 ,
(57)
∀x ∈ [x ∗ , x0 ], T (m ∗ ) + (d(0) + 2 0 )(x − m ∗ ) ≤ T (x).
(58)
in the sense that
We have the following: Claim 2.8 (m ∗ is a non-characteristic point). There exists η1 > 0 such that ∀x ∈ [m ∗ − η1 , m ∗ + η1 ], T (m ∗ ) −
1 + |d(0)| |x − m ∗ | ≤ T (x). 2
(59)
Proof. We claim first that −
η0 < y0 < x0 , 2
(60)
which yields by minimality in (55) (1 + |d(0)|) (x0 − y0 ) = T (x0 ) − T (y0 ) 2
(61)
Note first from (54) and (51) that 2 (T (x0 ) − T (y0 )) 1 + |d(0)| 2 ≤ [(T (x0 ) − T (0)) − (T (y0 ) − T (0))] 1 + |d(0)| 2d(0) 2 0 ≤ (x0 − y0 ) + (|x0 | + |y0 |) . 1 + |d(0)| 1 + |d(0)|
x0 − y0 ≤
Since
2|d(0)| 1+|d(0)|
(62)
< 1, this yields x0 − y0 ≤ C(d(0)) 0 (|y0 | + |x0 |) ≤ 2C(d(0)) 0 η0 ≤
η0 10
η0 by (53), (60) follows. for 0 small enough. Since |x0 | ≤ 10 Let us now prove (59). First note that m ∗ < x0 . Indeed, from (56), (61), (57) ∗ ∗ ∗ and (54), we have T (x ) ≤ T (y0 ) + d(0)(x − y0 ) = T (x0 ) + d(0)(x − x0 ) − (1+|d(0)|) − d(0) (x0 − y0 ) < T (x0 ) + (d(0) + 2 0 )(x ∗ − x0 ) since |d(0)| < 1, so 2 (58) cannot hold with m ∗ = x0 (it fails with x = x ∗ ).
– If m ∗ ∈ (y0 , x0 ), then T (m ∗ ) is a local minimum for T (x) − (d(0) + 2 0 )(x − m ∗ ), hence, since d(0) + 2 0 < 1 by (50), m ∗ is non-characteristic. – If m ∗ = y0 , then y0 = x ∗ = m ∗ by (57). From (55) and the fact that T (y0 )−d(0)y0 is the minimum of T (x) − d(0)x0 for x ∈ [y0 , x0 ], we see that m ∗ is non-characteristic, which concludes the proof of Claim 2.8.
Blow-up for a Semilinear Wave Equation
69
Thus, m ∗ is non-characteristic in the sense (3). Using (52) and Proposition 2.1, we see that T (x) is differentiable at x = m ∗ and |T (m ∗ ) − d(0)| = |d(m ∗ ) − d(0)| ≤ 0 on the one hand. On the other hand, from (57), (58) and the fact that m ∗ < x0 , we have T (m ∗ ) ≥ d(0) + 2 0 , which leads to a contradiction. This concludes the proof of Lemma 2.7. Step 3 Conclusion of the proof. η0 η0 Using Lemma 2.7, we see that for all x ∈ [− 20 , 20 ], x is non-characteristic in the sense (3). Using Proposition 2.1, we see that T is differentiable at x and T (x) = d(x), where d(x) is such that (10) holds for wx . Using Lemma 2.6, we see from (49) that T (x) = d(x) → d(0) = T (0) as x → 0 and θ (x) = 1. This concludes the proof of Theorem 1.
3. Proof of the Liouville Theorem Remark first that Theorem 2’ follows from Theorem 2. Proof of Theorem 2’ assuming Theorem 2. Consider w(y, s) a solution to Eq. (6) defined for all (y, s) ∈ (− δ1∗ , δ1∗ ) × R for some δ ∗ ∈ (0, 1) such that for all s ∈ R, (17) holds. If we introduce the function u(x, t) defined by u(x, t) = (−t)
2 − p−1
w(y, s) where y =
x and s = − log(−t), −t
(63)
then we see that u(x, t) satisfies the hypotheses of Theorem 2 with T∗ = x∗ = 0, in particular (15) holds. Therefore, the conclusion of Theorem 2 holds for u. Using again (63), we directly get the conclusion of Theorem 2’. The rest of the section is now devoted to the proof of Theorem 2. Proof of Theorem 2. Consider a solution u(x, t) to Eq. (1) defined in the backward cone Cx∗ ,T∗ ,δ ∗ (see (4)) such that (15) holds, for some (x∗ , T∗ ) ∈ R2 and δ∗ ∈ (0, 1). From the bound (15) and the resolution of the Cauchy problem of Eq. (1), we can extend the solution by a function still denoted by u(x, t) and defined in some influence domain Du of the form Du = {(x, t) ∈ R2 | t < T (x)}
(64)
for some 1-Lipschitz function T (x), where one of the following cases occurs: – Case 1. For all x ∈ R, T (x) ≡ +∞. – Case 2. For all x ∈ R, T (x) < +∞. In this case, since u(x, t) is known to be defined on C x∗ ,T∗ ,δ∗ (4), we have Cx∗ ,T∗ ,δ∗ ⊂ Du , hence from (4) and (64), ∀x ∈ R, T (x) ≥ T∗ − δ∗ |x − x∗ |. In this case, we will denote the set of non-characteristic points by I0 .
(65)
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F. Merle , H. Zaag
We proceed in 4 steps: – In Step 1, we consider wx, ¯ T¯ ) is arbitrary in D¯ u and show it ¯ T¯ defined in (5) where ( x, converges to a stationary solution of (6) as s → −∞. Using some energy estimates from [16], we conclude the proof of Theorem 2 when Case 1 holds. – From this step on, we focus on Case 2. In Step 2, we show that there exists a noncharacteristic point (in the sense (3)) with a given location. – In Step 3, we first use the convergence result (10) to show that when x0 is a noncharacteristic point, wx0 is a stationary solution to (6). It follows then that I0 is an interval. We also reduce the proof of Theorem 2 to the fact that I0 = R. – In Step 4, we proceed by contradiction and assume that I0 = R. We first show that the blow-up set is a straight line of slope ±1 outside I0 (that is, on the set of characteristic points). Then, we conclude the proof thanks to a space-time energy argument applied at a characteristic point. Step 1 Behavior for s → −∞ of wx, ¯ T¯ (s) and conclusion when Case 1 holds. We first recall some dispersion estimates from [3] and [16]: Lemma 3.1 (A Lyapunov functional for Eq. (6)) Consider w(y, s) a solution to (6) defined for all (y, s) ∈ (−1, 1) × [s0 , +∞) for some s0 ∈ R. Then: (i) For all s2 ≥ s1 ≥ s0 , we have E(w(s2 )) − E(w(s1 )) = −
4 p−1
s2 s1
1
−1
(∂s w(y, s))2
ρ(y) dyds, 1 − y2
where E is defined in (8). (ii) For all s ≥ s0 + 1,
1 2
− 21
|w(y, s)| p+1 dy ≤ C(E(w(s0 )) + 1) p .
Proof. See [3] for (i). For (ii), see Proposition 2.2 in [17] for a statement and the proof of Proposition 3.1, p. 1156 in [16] for the proof. Using energy arguments together with the finite speed of propagation similar to what we did in the blow-up situation (see [19]), we claim: Proposition 3.2 (Behavior of wx, ¯ T¯ ) ∈ D¯ u , it holds ¯ T¯ (s) as s → −∞) For any ( x, that as s → −∞, either 1 2 wx, ¯ T¯ (s) H 1 (−1,1) + ∂s wx, ¯ T¯ (s) L 2 (−1,1) → 0 in H × L (−1, 1),
or for some θ (x, ¯ T¯ ) = ±1, inf wx, ¯ T¯ )κ(d, .) H 1 (−1,1) + ∂s wx, ¯ T¯ (s) − θ ( x, ¯ T¯ (s) L 2 (−1,1) → 0, |d|<1
where κ(d, y) is defined in (11). Remark. Here, from the fact that δ ∗ < 1, we have ∗ ⊂ {t ≤ τ¯ } ∩ C x ∗ ,T ∗ ,δ ∗ ⊂ Du {t ≤ τ¯ } ∩ Cx, ¯ T¯ , 1+δ 2
for some τ¯ , hence all the point (x, ¯ T¯ ) are “non-characteristic” (in a sense adapted to s → −∞).
Blow-up for a Semilinear Wave Equation
71
Proof. The proof is similar to what we did as s → ∞ in Proposition 3 in [19] (see [19] for details). We claim first that for some s¯ ∈ R, we have ∀s ≤ s¯ , wx, ¯ T¯ (s) H 1 (−1,1) + ∂s wx, ¯ T¯ (s) L 2 (−1,1) ≤ C,
(66)
where C is related to the bound given in the hypotheses of Theorem 2. Let us prove the estimate for wx, ¯ T¯ (s) L 2 (−1,1) first. The estimate for ∂s wx, ¯ T¯ and ∂ y wx, ¯ T¯ follows in a similar way. From the selfsimilar transformation (5), we have 2s − p−1 wx, u x¯ + ye−s , T¯ − e−s . ¯ T¯ (y, s) = e Therefore, 1 4 −1 2 p−1 ¯ wx, ¯ T¯ (y, s) dy = (T − t) −1
B(x, ¯ T¯ −t)
u(x, t)2 d x where t = T¯ − e−s .
Since δ ∗ < 1, there exists s¯ (x, ¯ T¯ ) ∈ R such that for all s ≤ s¯ (x, ¯ T¯ ) (or t ≤ t¯(x, ¯ T¯ ) = T¯ − e−¯s ) T∗ − t , B(x, ¯ T¯ − t) ⊂ B x ∗ , δ∗ we see that the bound on wx, ¯ T¯ (s) L 2 (−1,1) follows from (15). Using Lemma 3.1, we see from (66) that s¯ 1 2 ρ(y) ∀s ≤ s¯ , |E(wx, (s))| ≤ C and |∂s wx, dyds ≤ C, ¯ T¯ ¯ T¯ (s)| 1 − y2 −∞ −1 and from the monotonicity of E(wx, ¯ T¯ (s)), we see that E(wx, ¯ T¯ (s)) → E − as s → −∞. We now follow exactly the steps of the proof of Proposition 3 in Sect. 6 of [19]. See [19] for more details. – From Proposition 6.2 in [19], we reduce it to prove that wx, w˜ ¯ (s) ¯ T − inf → 0 as s → −∞, w (s) 0 H 1 ×L 2 (−1,1) ∂ ¯ s x, w∈S ˜ ¯ T
(67)
where S is the set of H 1 (−1, 1) stationary solutions to (6). Recall from Proposition 2 in [19] that S = S1 ∪ S2 ∪ {0}, where S1 = {κ(d, .) | |d| < 1} and S2 = {−κ(d, .) | |d| < 1}. – Then, we proceed by contradiction to prove (67) and assume that there exists sn → −∞ as n → ∞ and 0 > 0 such that for all n ∈ N, wx, w˜ ¯ T¯ (sn ) − inf ≥ 0 > 0. 0 H 1 ×L 2 (−1,1) ∂s wx, w∈S ˜ ¯ T¯ (sn ) As in Lemma 6.4 in [19], we prove that there exists s˜n → −∞ as n → ∞ such that for some w∗ ∈ S, ∗ sn ) wx, w ¯ T¯ (˜ weakly in H 1 × L 2 (−1, 1) as n → ∞. (˜ s ) 0 ∂s wx, ¯ n ¯ T In the same way as in Lemma 6.5 in [19], we obtain a contradiction. This concludes the proof of Proposition 3.2.
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From Proposition 3.2, we derive the behavior of the Lyapunov functional E(w(s)) defined by (8) as s → −∞. Corollary 3.3 (Behavior of E(wx, ¯ T¯ (s)) as s → −∞). (i) For all d ∈ (−1, 1), E(κ(d, .)) = E(−κ(d, .)) = E(κ0 ) > 0.
(68)
(ii) For any (x, ¯ T¯ ) ∈ D¯ u , either E(wx, ¯ T¯ (s)) → 0 or E(wx, ¯ T¯ (s)) → E(κ0 ) > 0 as s → −∞. In particular, ∀s ∈ R, E(wx, ¯ T¯ (s)) ≤ E(κ0 ).
(69)
Proof. (ii) is a direct consequence of Proposition 3.2, the definition (8) of E(w(s)) and (i). Thus we only prove (i). Since κ(d, y) is a stationary solution to (6), we have Lκ(d, y) −
2( p + 1) κ(d, y) + |κ(d, y)| p−1 κ(d, y) = 0. ( p − 1)2
Multiplying this equation by κ(d, y)ρ(y) and integrating with respect to y ∈ (−1, 1), we get from the definition (7) of ρ(y), −
−1 1
+
1
−1
|∂ y κ(d, y)|2 (1 − y 2 )ρ(y) −
2( p + 1) ( p − 1)2
1
−1
κ(d, y)2 ρ(y)dy
κ(d, y) p+1 ρ(y)dy = 0.
Therefore, from the definition (8) of E(κ(d, .)) , we see that E(κ(d, .)) =
p−1 2( p + 1)
Making the change of variables Y = κ(d, y) and ρ(y) that p−1 2( p + 1)
1 −1
y+d 1+dy ,
κ(d, y) p+1 ρ(y)dy =
1
−1
κ(d, y) p+1 ρ(y)dy.
(70)
we see from the definitions (11) and (7) of
p − 1 p+1 κ 2( p + 1) 0
1
−1
ρ(Y )dY = E(κ0 ) > 0.
Thus, (68) follows from (70). This concludes the proof of Corollary 3.3. This result allows us to conclude the proof of the Liouville Theorem by energy arguments, when Case 1 holds. Indeed, Corollary 3.4 (Conclusion of the proof when Case 1 holds). If for all x ∈ R, T (x) ≡ +∞, then u ≡ 0.
Blow-up for a Semilinear Wave Equation
73
Proof. In this case, u(x, t) is defined for all (x, t) ∈ R2 . The conclusion is a consequence of the uniform bounds stated in the hypothesis of Theorem 2 and the bound for solutions of Eq. (6) in terms of the Lyapunov functional stated in (ii) of Lemma 3.1. Indeed, consider for arbitrary t ∈ R and T > t the function w0,T defined from u(x, t) by means of the transformation (5). Note that w0,T is defined for all (y, s) ∈ R2 . If s = − log(T − t), then we see from (ii) in Lemma 3.1 and (69) that
1 2
− 21
|w0,T (y, s)| p+1 dy ≤ C(E(w0,T (s0 )) + 1) p ≤ C(E(κ0 ) + 1) p ≡ C1 .
Using (5), this gives, in the original variables,
T −t 2
− T 2−t
p+1) − 2(p−1 +1
|u(x, t)| p+1 d x ≤ C1 (T − t)
.
Fix t and let T go to infinity to get u(x, t) = 0 for all x ∈ R, and then u ≡ 0, which concludes the proof of Corollary 3.4 and thus the proof of Theorem 2 in the case where T (x) ≡ +∞. Step 2 Existence of non-characteristic point with a given location. From now on, we assume that Case 2 holds. We claim the following general result on the existence of a non-characteristic point in a given cone with slope δ1 > 1. Proposition 3.5 (Existence of a non-characteristic point with a given location). For all x1 ∈ R and δ1 ∈ (δ∗ , 1), there exists x0 = x0 (x1 , δ1 ) such that (x0 , T (x0 )) ∈ C¯x1 ,T (x1 ),δ1 and Cx0 ,T (x0 ),δ1 ⊂ Du .
(71)
In particular, x0 is non-characteristic. Remark. This proposition remains valid for general solutions defined for all (x, t) such that 0 ≤ t < T (x) with T (x) ≥ T¯ > 0. Proof of Proposition 3.5. Consider x1 ∈ R and δ1 ∈ (δ∗ , 1). Note that it is enough to show the existence of x0 such that (71) holds, since this implies by the definition (3) that x0 is non-characteristic. Let us introduce E 1 = {x ∈ R | T (x) ≤ T (x1 ) − δ1 |x − x1 |}. Since δ1 > δ∗ , we have for |x| large, T (x1 ) − δ1 |x − x1 | < T∗ − δ∗ |x − x∗ | ≤ T (x), where we used (65) for the last inequality, hence the boundedness of E 1 . Since E 1 = ∅ (x1 ∈ E 1 ) and E 1 is closed, there exists x2 ∈ E 1 such that |x2 − x1 | = max |x − x1 |, if |x − x1 | > |x2 − x1 |, x∈E 1
then T (x) > T (x1 ) − δ1 |x − x1 |, and T (x2 ) = T (x1 ) − δ1 |x1 − x2 | (i.e. (x2 , T (x2 )) ∈ ∂Cx1 ,T (x1 ),δ1 ).
(72) (73)
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By continuity of T (x), there exists x0 ∈ R such that |x0 − x1 | ≤ |x2 − x1 | and T (x0 ) =
min
|x−x1 |≤|x2 −x1 |
T (x).
(74)
We claim that x0 satisfies (71). Indeed, note first from (74) and (73) that T (x0 ) ≤ T (x2 ) = T (x1 ) − δ1 |x1 − x2 | ≤ T (x1 ) − δ1 |x1 − x0 |, hence, (x0 , T (x0 )) ∈ C¯x1 ,T (x1 ),δ1 by the definition (4). For the second inequality in (71), note from (4) and (64) that it is enough to prove that for all x ∈ R, T (x0 ) − δ1 |x − x0 | ≤ T (x).
(75)
– If |x − x1 | ≤ |x2 − x1 |, then T (x0 ) − δ1 |x − x0 | ≤ T (x0 ) ≤ T (x) by (74). – If |x − x1 | ≥ |x2 − x1 |, then since we have just proved the first inequality in (71), it holds that Cx0 ,T (x0 ),δ1 ⊂ Cx1 ,T (x1 ),δ1 (use the fact that the two cones have the same slope), hence T (x0 ) − δ1 |x − x0 | ≤ T (x1 ) − δ1 |x − x1 |. Using (72), we get (75) when |x − x1 | ≥ |x2 − x1 |. Therefore, (75) holds for all x ∈ R and Cx0 ,T (x0 ),δ1 ⊂ Du . Thus, the second inequality in (71) holds. This concludes the proof of Proposition 3.5. Taking x1 = x∗ and δ1 =
1+δ∗ 2
in Proposition 3.5, we get:
Corollary 3.6 There exists a non-characteristic point x0 ∈ R (in the sense (3)). Step 4 The set of non-characteristic points is an interval. We claim the following from energy arguments (Lemma 3.1): Proposition 3.7 (Characterization of wx0 when x0 is non-characteristic). If x0 is non-characteristic, then, there exist d(x0 ) ∈ (−1, 1) and θ (x0 ) = ±1 such that for all (y, s) ∈ (−1, 1) × R, wx0 (y, s) = θ (x0 )κ(d(x0 ), y) (11). Proof. From Corollary 3.3, we know that E(wx0 (s)) → e− as s → −∞ with e− = 0 or e− = E(κ0 ) > 0.
(76)
Using the convergence result of [19] stated in (10), we see that there exists d(x0 ) ∈ (−1, 1) such that wx (s) κ(d(x0 ), y) 0 → 0 as s → ∞, − θ (x ) (77) 0 ∂s wx0 (s) 0 H where H and its norm are defined in (9). Using the definition (8) of E(w) and (68), we see that E(wx0 (s)) → e+ = E(κ(d(x0 ), .) = E(κ0 ) > 0 as s → ∞. Using Lemma 3.1, we see that 4 e+ − e− = − p−1
∞
1
−∞ −1
∂s wx0 (y, s)
2 ρ(y) dyds ≤ 0. 1 − y2
(78)
Hence, from (76), e− = E(κ0 ) and e+ − e− = 0. Therefore, from (78), we obtain ∂s wx0 (y, s) ≡ 0 for all (y, s) ∈ (−1, 1)×R, which means that wx0 is a stationary solution to (6). Using (77), we see that wx0 (y, s) = θ (x0 )κ(d(x0 ), y) for all (y, s) ∈ (−1, 1)×R. This concludes the proof of Proposition 3.7.
Blow-up for a Semilinear Wave Equation
75
We claim: Corollary 3.8 Consider x1 < x2 two non-characteristic points. Then, there exists d0 ∈ (−1, 1) and θ0 = ±1 such that: (i) for all (y, s) ∈ (−1, 1) × R, wx1 (y, s) = wx2 (y, s) = θ0 κ(d0 , y), (ii) for all x¯ ∈ [x1 , x2 ], T (x) ¯ = T (x1 ) + d0 (x¯ − x1 ) and for all (x, t) ∈ Cx,T ¯ (x),1 ¯ , 1
u(x, t) = θ0 κ0
(1 − d02 ) p−1 2
¯ p−1 ¯ − t + d0 (x − x)) (T (x)
.
(79)
Proof. Consider x1 < x2 two non-characteristic points. (i) Using Proposition 3.7, we see that there exist d(x1 ) and d(x2 ) in (−1, 1), and θ (x1 ) and θ (x2 ) in {−1, 1} such that for all (y, s) ∈ (−1, 1) × R, wxi (y, s) = θi κ(d(xi ), y), where i = 0 and 1. Going back to the original variables with (5), we see that for i = 0 and 1, 1
∀(x, t) ∈ Cxi ,T (xi ),1 , u(x, t) = θ (xi )κ0
(1 − d(xi )2 ) p−1 2
(T (xi ) − t + d(xi )(x − xi )) p−1
. (80)
Therefore, if V = Cx1 ,T (x1 ),1 ∩ Cx2 ,T (x2 ),1 , we have for all (x, t) ∈ V , 1
θ (x1 )κ0
(1 − d(x1 )2 ) p−1 2
(T (x1 ) − t + d(x1 )(x − x1 )) p−1 1
= θ (x2 )κ0
(1 − d(x2 )2 ) p−1 2
(T (x2 ) − t + d(x2 )(x − x2 )) p−1
.
Since we know from (4) that V is a non-empty open set of R2 , this yields (x1 ) θ (x1 ) = θ (x2 ) and d(x1 ) = d(x2 ) = T (xx22)−T . Thus, (i) holds with −x1 d0 = d(x1 ) = d(x2 ) =
T (x2 ) − T (x1 ) and θ0 = θ (x1 ) = θ (x2 ). x2 − x1
(81)
(ii) Consider x¯ ∈ [x1 , x2 ] and define T˜ (x) ¯ = T (x1 ) + d0 (x¯ − x1 ).
(82)
Since (x, ¯ T˜ (x)) ¯ is on the segment connecting (x1 , T (x1 )) and (x2 , T (x2 )) (use (81)), we see from (4) that for some t (x) ¯ ∈ R, we have Vx¯ ≡ Cx, ∩ {t < t (x)} ¯ ⊂ {Cx1 ,T (x1 ),1 ∪ Cx2 ,T (x2 ),1 } ∩ {t < t (x)}. ¯ ¯ T˜ (x),1 ¯ Therefore, from (80) and (81), we see that ∀(x, t) ∈ Vx¯ , 1
u(x, t) = θ0 κ0
(1 − d02 ) p−1 2
(T (x1 ) − t + d0 (x − x1 )) p−1
.
(83)
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From the uniqueness of the solution of (1), we see that u(x, t) is defined everywhere in Cx, ¯ T˜ (x),1 ¯ . In particular, the identity (83) holds in all Cx, ¯ T˜ (x),1 ¯ , which means by (82) that ˜ T (x) ¯ = T (x) ¯ = T (x1 ) + d0 (x¯ − x1 ). Hence, (79) follows from (83). This concludes the proof of Corollary 3.8. From Corollaries 3.6 and 3.8, we see that the set of non-characteristic points is some non empty interval I0 , and that for any x¯ ∈ I0 and x1 ∈ I0 , we have ¯ = d0 (x1 − x), ¯ T (x1 ) − T (x)
(84)
where d0 is the slope of the blow-up curve on I0 . Thus, using (79), we get: Corollary 3.9 The set of non-characteristic points is a non empty interval I0 and there exist d0 ∈ (−1, 1) and θ0 such that on I0 , the blow-up curve is a straight line with slope d0 . Moreover, for any x1 ∈ I0 , ∀(x, t) ∈
1
Cx,T ¯ (x),1 ¯ , u(x, t) = θ0 κ0
x∈I ¯ 0
(1 − d02 ) p−1 2
(T (x1 ) − t + d0 (x − x1 )) p−1
.
(85)
We then have the following reduction of the proof of Theorem 2: Corollary 3.10 It is enough to prove that I0 = R
(86)
in order to conclude the proof of Theorem 2. Proof. Let us conclude here the proof of Theorem 2 assuming (86). If I0 = R, then we see from Corollary 3.9 that the blow-up curve is a straight line of slope d0 whose equation is t = T (x) with ∀x ∈ R, T (x) = T (x∗ ) + d0 (x − x∗ )
(87)
and that Du = {(x, t) | t < T (x∗ ) + d0 (x − x∗ )} which contains Cx∗ ,T (x∗ ),1 by the fact that T (x∗ ) ≥ T∗ (see (65)). Using (65) and (87), Cx,T = Du , we see that |d0 | ≤ δ∗ , hence Cx∗ ,T (x∗ ),δ ∗ ⊂ Du . Moreover, since ¯ (x),1 ¯ x∈I ¯ 0
we see that (85) implies (16) with T0 = T (x∗ ) and x0 = x∗ . This concludes the proof of Theorem 2 assuming (86). Step 5 Conclusion of the proof. In this step, we proceed by contradiction to prove (86). Therefore, we assume that I0 = R. From Corollary 3.9, up to changing u(x, t) in u(−x, t) (also a solution to (1)), we can assume that I¯0 = (−∞, b] or I¯0 = [a, b]
(88)
for some a ≤ b. From Corollary 3.9, we know that the blow-up set is a straight line of slope d0 ∈ (−1, 1) on I0 . In this step, we first show that to the right of the interval I0 , the blow-up curve is a straight line of slope 1, by the fact that I0 is an interval. We then find a contradiction by energy arguments in space-time valid in the case of a non-characteristic point. More precisely:
Blow-up for a Semilinear Wave Equation
77
Lemma 3.11 It holds that ∀x ≥ b, T (x) = T (b) + x − b. Proof. Let us remark first that since the blow-up curve is 1-Lipschitz from the finite speed of propagation, we already know that for all x ≥ b, T (x) ≤ T (b) + x − b. We proceed by contradiction, and assume that for some x1 > b, we have T (x1 ) < T (b)+ x1 −b. Therefore, if λ ∈ R is the slope of the straight line connecting (b, T (b)) and (x1 , T (x1 )), we have T (x1 ) − T (b) < 1. x1 − b
λ=
Since δ∗ ∈ (0, 1) by the hypothesis of Theorem 2 and |d0 | < 1, where d0 is the slope of the blow-up set on the interval I0 , we can fix some δ1 ∈ (0, 1) such that δ1 ∈ (max(δ∗ , d0 , λ), 1). Since δ1 > λ, we have (b, T (b)) ∈ C¯x1 ,T (x1 ),δ1 , therefore, since δ1 > d0 , C¯x1 ,T (x1 ),δ1 does not contain any point from the left half-line finishing in (b, T (b)) and whose slope is d0 . Since this half-line contains the set N ≡ {(x, ¯ T (x)) ¯ | x¯ is non characteristic },
(89)
this means that for any non-characteristic point x, ¯ (x, ¯ T (x)) ¯ ∈ C¯x1 ,T (x1 ),δ1 on the one hand. On the other hand, since δ1 ∈ (δ∗ , 1), Proposition 3.5 applies and we know that there exists a non-characteristic point x0 = x0 (x1 , δ1 ) such that (x0 , T (x0 )) ∈ C¯x1 ,T (x1 ),δ1 , which is a contradiction. Thus, for all x > b, T (x) = T (b) + x − b. This concludes the proof of Lemma 3.11. From Corollary 3.9 and Lemma 3.11, we have: Claim 3.12
0
1
−1 −1
|∂s wb+1,T (b+1) (y, s )|2
ρ(y) dyds = ∞. 1 − y2
Proof. Note first that b is non-characteristic. Indeed, – if I0 = [a, b] with a = b in (88), then I0 = {b} = {x0 } by Corollary 3.6 and b is non-characteristic. – if not, then Corollary 3.9 and Lemma 3.11 imply that b is non-characteristic. We now know from Corollary 3.9 (put x1 = b) that for all (x, t) ∈ Cb,T (b),1 , 1
u(x, t) = θ0 κ0
(1 − d02 ) p−1 2
(T (b) − t + d0 (x − b))) p−1
.
Up to changing u in −u, we can assume θ0 = 1. Using the selfsimilar transformation (5) and the fact that T (b + 1) = T (b) + 1 (from Lemma 3.11), we see that when s < 0 and −1 < y < 1 − 2es , we have 1
wb+1,T (b+1) (y, s) =
κ0 (1 − d02 ) p−1 2
(1 + d0 y + es (d0 − 1)) p−1
, 1
κ0 (1 − d02 ) p−1 2es (1 − d0 ) ∂s wb+1,T (b+1) (y, s) = . p+1 p − 1 (1 + d y + es (d − 1)) p−1 0
0
(90)
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If we introduce y1 (s) = −es and y2 (s) = 1 − 2es ,
(91)
then we see that (y1 (s), y2 (s)) ⊂ (−1, 1 − 2es ) and from (90) and the definition (7) of ρ, we have for all s ∈ (−2, 0) and y ∈ (y1 (s), y2 (s)), |∂s wb+1,T (b+1) (y, s)| ≥ C
es (1 − es )
≥ C|s|
2 p−1 −1
p+1 p−1
≥
C |s|
p+1 p−1
and
2 ρ(y) −1 ≥ C(1 − es ) p−1 2 1−y
.
Since y2 (s) − y1 (s) ≥ C|s| by (91), this implies that 0 y2 (s) ρ(y) |∂s wb+1,T (b+1) (y, s)|2 dyds 1 − y2 −1 y1 (s) 0 0 2 C −1 − 2p ≥ |s| p−1 ds = +∞. (y2 (s) − y1 (s)) 2( p+1) |s| p−1 ds = C −1 −1 |s| p−1 Since (y1 (s), y2 (s)) ⊂ (−1, 1), this concludes the proof of Claim 3.12. Using energy estimates for wb+1,T (b+1) still valid in the characteristic situation (Lemma 3.1), we write 0 1 ∂s wb+1,T (b+1) (y, s)2 ρ(y) dyds (92) 1 − y2 −1 −1 p−1 E(wb+1,T (b+1) (−1)) − E(wb+1,T (b+1) (0)) . = 4 Since E(wb+1,T (b+1) (0)) ≥ 0 (use Claim 2.5 and the fact that wb+1,T (b+1) (y, s) is defined for all (y, s) ∈ (−1, 1) × R) and E(wb+1,T (b+1) (0)) ≤ E(κ0 ) (use (68)), we get from (92) 0 1 ∂s wb+1,T (b+1) (y, s)2 ρ(y) dyds < +∞, 1 − y2 −1 −1 which is a contradiction with Claim 3.12. Thus I0 = R and by Corollary 3.10, Theorem 2 is proved. A. Proof of Claim 2.3 Let us first introduce the following continuity result for solutions to Eq. (6): Proposition A.1 (Weak continuity of solutions to (6) with respect to initial data in H 1 × L 2 ). Consider a sequence of solutions Wn to Eq. (6) defined for all (y, s) ∈ (−A, A) × [0, s0 ] for some A ≥ 1 and s0 ≥ 0 such that ∀s ∈ [0, s0 ], ∀n ∈ N, Wn (s), ∂s Wn (s) H 1 ×L 2 (−A,A) ≤ M
(93)
for some M > 0. If (Wn (0), ∂s Wn (0)) weakly converges to some (z ∗ , v ∗ ) in H 1 × L 2 (−A, A) as n → ∞, then, there exists W¯ (y, s) a solution to (6) with initial data (z ∗ , v ∗ ) defined for all (y, s) ∈ (−A, A) × [0, s0 ] with the following properties:
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79
(a) For all s ∈ [0, s0 ], (Wn (s), ∂s Wn (s)) H 1 × L 2 (−A, A).
W¯ (s), ∂s W¯ (s) as n → ∞ in
There exists n 0 = n 0 (M, s0 ) ∈ N such that for all n ≥ n 0 and s ∈ [0, s0 ], −
2s
(b) Wn (s) − W¯ (s) L ∞ (−A,A) ≤ e p−1 , s − 2s (c) e 2 ∂ y Wn (s)−∂ y W¯ (s) L 2 (−A,A) +∂s Wn (s)−∂s W¯ (s) L 2 (−A,A) ≤ C(A, M)e p−1 . Proof. If we introduce 2 − p−1
u n (ξ, τ ) = (1 − τ )
Wn
ξ , − log(1 − τ ) , 1−τ
(94)
2 z ∗ (ξ ) − ξ z ∗ (ξ ), p−1
z 1∗ (ξ ) = v ∗ (ξ ) −
then we see that u n (ξ, τ ) is a solution of Eq. (1) defined in Cτ0 , where τ0 = 1 − e−s0 , Ct = {(ξ, τ ) | 0 < τ < t and |ξ | < A(1 − τ )},
(95)
and ∀τ ∈ [0, τ0 ], (u n (τ ), ∂τ u n (τ )) H 1 ×L 2 (D(τ )) ≤ M0 (1 − τ ) (u n (0), ∂τ u n (0)) (z
∗
, z 1∗ )
2 − p−1 − 21
,
as n → ∞, in H × L (−A, A), 1
2
(96) (97)
where M0 = C0 ( p)M for some C0 ( p) > 0 and D(τ ) = (−A(1 − τ ), A(1 − τ )). Note from (96) and (97) that (z ∗ , z 1∗ ) H 1 ×L 2 (−A,A) ≤ M0 .
(98)
Therefore, we can define u(ξ, τ ) as the maximal solution of (1) with initial data (z ∗ , z 1∗ ) defined in Cτ ∗ where τ ∗ ≤ 1 is maximal. Note that either τ ∗ = 1 or τ ∗ < 1 and lim sup (u(τ ), ∂τ u(τ )) H 1 ×L 2 (D(τ )) = ∞. τ →τ ∗
(99)
Defining vn = u n − u,
(100)
we see that the following lemma allows us to conclude: Lemma A.2 For all > 0, we have the following with τ = min(τ0 , τ ∗ − ): (a) supτ ∈[0,τ ] vn (τ ) L ∞ (D(τ )) → 0 as n → ∞. (b) There exists n 0 ∈ N such that for all n ≥ n 0 and τ ∈ [0, τ ], ∂ξ vn (τ ) L 2 (D(τ )) + ∂τ vn (τ ) L 2 (D(τ )) ≤ 20M0 . (c) For all τ ∈ [0, τ ], (vn (τ ), ∂τ vn (τ )) 0 weakly in H 1 × L 2 (D(τ )). Indeed, let us first show that τ = τ0 for small (in other words that τ ∗ > τ0 ) before deriving Proposition A.1 from Lemma A.2. Assume by contradiction that τ ∗ ≤ τ0 . Then,
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F. Merle , H. Zaag
using (a) and (b) of this lemma and (96), we see that for all > 0, τ = τ ∗ − and for all τ ∈ [0, τ ∗ − ], u(τ ), ∂τ u(τ ) H 1 ×L 2 (D(τ )) ≤ u n (τ ), ∂τ u n (τ ) H 1 ×L 2 (D(τ ))
+u n (τ ) − u(τ ), ∂τ u n (τ ) − ∂τ u(τ ) H 1 ×L 2 (D(τ )) ≤ C(A, M0 , τ ∗ ).
Letting → 0, we see that lim supτ →τ ∗ (u(τ ), ∂τ u(τ )) H 1 ×L 2 (D(τ )) < ∞. Since we also have τ ∗ ≤ τ0 < 1, a contradiction follows from (99). Thus, τ ∗ > τ0 and Lemma A.2 is valid for all τ ∈ [0, τ0 ]. Let us now derive Proposition A.1 from Lemma A.2. If we define s ∗ = − log(1 − τ ∗ ) and for all (y, s) ∈ (−A, A) × [0, s ∗ ), W¯ (y, s) by ξ − 2 , − log(1 − τ ) , (101) u(ξ, τ ) = (1 − τ ) p−1 W¯ 1−τ then we see from (94) that 2 − p−1 −1
∂ξ u n (ξ, τ ) = (1 − τ )
2 − p−1 −1
∂τ u n (ξ, τ ) = (1 − τ )
∂ y Wn
ξ 1−τ , − log(1 − τ )
∂s Wn + y.∂ y Wn +
2 p−1 Wn
,
(102) ,
ξ 1−τ , − log(1 − τ )
and the same holds between u and W¯ . Using (94), (101), (102) and (100) we obtain for all s ∈ [0, s0 ], 2s
− Wn (s) − W¯ (s) L ∞ (−A,A) ≤ e p−1 vn (τ ) L ∞ (D(τ )) , 2s
s
2s
s
− − ∂ y Wn (s) − ∂ y W¯ (s) L 2 (−A,A) ≤ e p−1 2 ∂ξ vn (τ )) L 2 (D(τ )) , − − ∂s Wn (s) − ∂s W¯ (s) L 2 (−A,A) ≤ e p−1 2 ∂τ vn (τ )) L 2 (D(τ )) 2 √ − 2s − s − 2s +Ae p−1 2 ∂ξ vn (τ )) L 2 (D(τ )) + 2 Ae p−1 vn (τ )) L ∞ (D(τ )) , p−1
where τ = 1 − e−s ∈ [0, τ0 ]. Therefore, the conclusion of Proposition A.1 follows from Lemma A.2. It remains to prove Lemma A.2 to conclude. Proof of Lemma A.2. (a) From the definition (100) of vn , we see that ∂τ2τ − ∂ξ2ξ vn = |u n | p−1 u n − |u| p−1 u ≡ f n (ξ, τ )
(103)
where | f n | ≤ p |u| p−1 + |u n | p−1 |vn |. Since we have from (96) and the Sobolev injection that for all τ ∈ [0, τ ], u n (τ ) L ∞ (D(τ )) ≤ C(τ0 )u n (τ ) H 1 (D(τ )) ≤ C(τ0 )M0 , we get for all τ ∈ [0, τ ] and ξ ∈ D(τ ), p−1 | f n (ξ, τ )| ≤ C(τ0 , M0 ) 1 + vn (τ ) L ∞ (D(τ )) |vn (ξ, τ )|. (104)
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Translating (97), (96) and (98) for vn , we write (vn , ∂τ vn ) (0) 0 weakly in H 1 × L 2 (−A, A) and vn (0) → 0 strongly in L q (−A, A)
(105)
for any q ∈ [2, +∞] as n → ∞, and ∀n ∈ N, vn (0) H 1 (−A,A) + ∂τ vn (0) L 2 (−A,A) ≤ 2M0 .
(106)
Using Eq. (103), we write vn (ξ, τ ) = S(τ )vn (0)(ξ ) + S1 (τ )∂τ vn (0)(ξ ) τ + (S1 (τ − s) f n (s)) (ξ )ds,
(107)
0
where S(t)h(x) =
1 1 (h(x + t) + h(x − t)) and S1 (t)h(x) = 2 2
x+t
h(x )d x , (108)
x−t
Using (108), we write from (107) αn (τ ) ≤ In (τ ) + Jn (τ ) +
τ
K n (τ, s)ds,
(109)
0
where αn (τ ) = vn (τ ) L ∞ (D(τ )) , In (τ ) = S(τ )vn (0) L ∞ (D(τ )) , (110) Jn (τ ) = S1 (τ )∂τ vn (0) L ∞ (D(τ )) and K n (τ, s) = S1 (τ − s) f n (s) L ∞ (D(τ )) . In the following, we use (105) to estimate In (τ ), Jn (τ ) and K n (τ, s). Estimate of In (τ ). From (110) and (108), we have sup In (τ ) ≤ αn (0) → 0 as n → ∞
τ ∈[0,τ ]
(111)
by (105). Estimate of Jn (τ ). Note first from (108) and (106) that for any τ ∈ [0, τ ], we have 1 ∂τ vn (. + τ, 0) − ∂τ vn (. − τ, 0) L 2 (D(τ )) 2 ≤ C∂τ vn (0) L 2 (−A,A) ≤ C M0 .
∂ξ S1 (τ )∂τ vn (0) L 2 (D(τ )) =
Similarly, for any ξ ∈ (−A, A), we have
1− |ξA| 0
1 |∂τ S1 (τ )∂τ vn (0)(ξ )| dτ = 4
2
0
1− |ξA|
|∂τ vn (ξ + τ, 0) + ∂τ vn (ξ − τ, 0)|2 dτ
≤ C∂τ vn (0)2L 2 (−A,A) ≤ C M02 ,
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which means that S1 (τ )∂τ vn (0) is 21 -Holder for (ξ, τ ) ∈ Cτ defined by (95). Then, since ∂τ vn (0) 0 as n → ∞ in L 2 (−A, A) by (105), we use 1ξ −τ <x<ξ +τ (x) as a test function and write from (108), for all (ξ, τ ) ∈ Cτ (95), 1 A ∂τ vn (x, 0)1ξ −τ <x<ξ +τ (x)d x → 0 as n → ∞. S1 (τ )∂τ vn (0)(ξ ) = 2 −A This is the pointwise convergence in Cτ . Since pointwise convergence and 1 2 -Holder imply uniform convergence, this means that sup Jn (τ ) ≤ S1 (·)∂τ vn (0) L ∞ (Cτ ) → 0 as n → ∞.
τ ∈[0,τ ]
(112)
Estimate of K n (τ, s). Fix some s and τ such that 0 ≤ s ≤ τ ≤ τ . Using (110), (108) and (104), we write, K n (τ, s) = S1 (τ − s) f n (s) L ∞ (D(τ )) ≤ f n (s) L ∞ (D(s)) ≤ C(τ0 , M0 ) 1 + αn (s) p−1 αn (s).
(113)
Using (109), (111), (112) and (113), we write for all τ ∈ [0, τ ], τ 1 + αn (s) p−1 αn (s)ds αn (τ ) − C(τ0 , M0 ) 0
≤ sup (In (τ ) + Jn (τ )) → 0 as n → ∞. τ ∈[0,τ ]
Using the Gronwall inequality with some a priori estimates, (105) and (110) , we get sup vn (τ ) L ∞ (D(τ )) = sup αn (τ ) → 0 as n → ∞,
τ ∈[0,τ ]
τ ∈[0,τ ]
which is precisely the conclusion of (a) of Lemma A.2. (b) Using (103), we can write vn = vn,1 + vn,2 , where
(114)
∂τ2τ − ∂ξ2ξ vn,1 = 0, vn,1 (0) = vn (0) and ∂τ vn,1 (0) = ∂τ vn (0), (115) ∂τ2τ − ∂ξ2ξ vn,2 = f n (ξ, τ ), vn,2 (0) = 0 and ∂τ vn,1 (0) = 0. (116)
From (107), we have vn,1 (ξ, τ ) = S(τ )vn (0)(ξ ) + S1 (τ )∂τ vn (0)(ξ ), 1 τ ξ +τ −s vn,2 (ξ, τ ) = f n (y, s)dyds. 2 0 ξ −τ +s Since the energy on slices of the light cone for the linear equation (115) 2 2 ∂ξ vn,1 (ξ, τ ) + ∂τ vn,1 (ξ, τ ) dξ D(τ )
(117) (118)
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83
is decreasing, we write from (106) ∂ξ vn,1 (τ ) L 2 (D(τ )) + ∂τ vn,1 (τ ) L 2 (D(τ )) ≤ 2 ∂ξ vn,1 (0) L 2 (|ξ |
1 τ ∂ξ vn,2 (ξ, τ ) = ( f n (ξ + τ − s, s) − f n (ξ − τ + s, s)) ds, 2 0 τ 1 ∂τ vn,2 (ξ, τ ) = ( f n (ξ + τ − s, s) + f n (ξ − τ + s, s)) ds. 2 0
Using (a) and (104), we see that for all n ∈ N, sup f n (τ ) L ∞ (D(τ )) → 0 as n → ∞.
τ ∈[0,τ ]
Therefore, we write for all τ ∈ [0, τ ], sup ∂ξ vn,2 (τ ) L 2 (D(τ )) + ∂τ vn,2 (τ ) L 2 (D(τ ))
τ ∈[0,τ ]
≤
√ 2 A sup ∂ξ vn,2 (τ ) L ∞ (D(τ )) + ∂τ vn,2 (τ ) L ∞ (D(τ )) τ ∈[0,τ ]
√ ≤ 2 2 A sup f n (τ ) L ∞ (D(τ )) → 0 τ ∈[0,τ ]
(120)
as n → ∞. Thus, (b) follows from (119) and (120) for n large. (c) Since ∀τ ∈ [0, τ ], vn (τ ) H 1 (D(τ )) ≤ C and supτ ∈[0,τ ] vn (τ ) L ∞ (D(τ )) → 0 by (a) and (b), this implies that vn 0 in H 1 (D(τ )). It remains to prove the weak convergence of ∂τ vn (τ ) to 0 in L 2 (D(τ )) as n → ∞ to conclude the proof of (c). From (114), we have ∂τ vn (ξ, τ ) = ∂τ vn,1 (ξ, τ ) + ∂τ vn,2 (ξ, τ ), Using (117) and (108), we write ∂τ vn,1 (ξ, τ ) = ∂τ S(τ )vn (0)(ξ ) + ∂τ S1 (τ )∂τ vn (0)(ξ ) 1 ∂ξ vn (ξ + τ, 0) − ∂ξ vn (ξ − τ, 0) = 2 1 + (∂τ vn (ξ + τ, 0) + ∂τ vn (ξ − τ, 0)) . 2 Since (vn , ∂τ vn ) (0) 0 in H 1 × L 2 (−A, A) by (105), we have ∂τ vn,1 (·, τ ) 0 in L 2 (D(τ )) as n → ∞. Using (120), we see that sup ∂τ vn,2 (τ ) L 2 (D(τ )) → 0 as n → ∞.
τ ∈[0,τ ]
Thus, ∂τ vn (τ ) = ∂τ vn,1 (τ ) + ∂τ vn,2 (τ ) 0 in L 2 (D(τ )) as n → ∞. This concludes the proof of (c) and Lemma A.2. We now prove Claim 2.3.
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Proof of Claim 2.3. Using the uniform bound stated in (22) and a diagonal process, we extract a subsequence (still denoted by wn ) such that for all k ∈ N, 1 1 1 1 2 wn (−k) z k in H − , and ∂s wn (−k) vk in L − , δ0 δ0 δ0 δ0 1
as n → ∞ for some (z k , vk ) ∈ H 1 × L 2 − δ1 , δ1 . From the bound (22), we can 0 0 apply Proposition A.1 on the time interval [−k, 0] and obtain the existence of Wk (y, s), a solution to (6) such that Wk (−k) = z k and ∂s Wk (−k) = vk with the following properties: – First, (wn (0), ∂s wn (0)) − (Wk (0), ∂s Wk (0))
H 1 ×L 2
− 1 , 1 δ0 δ0
≤ C(δ0 , K )e
2k − p−1
. (121)
– Second, for all s ∈ [−k, 0], (wn (s), ∂s wn (s)) (Wk (s), ∂s Wk (s)) as n → ∞ in H 1 × L 2 (−1, 1). Therefore, from (22), we obtain for all s ∈ [−k, 0], Wk (s) wn (s) ≤ lim inf 1 2 1 1 ≤ K . ∂s Wk (s) 1 2 1 1 ∂ w (s) n→∞ s n H ×L − , H ×L − , δ0 δ0
δ0 δ0
(122) Using a diagonal process, we can assume that ∀k, l ∈ N, Wk (y, s) ≡ Wl (y, s) on
−
1 1 , δ0 δ0
× [− min(k, l), 0].
Therefore, we can define W (y, s) for all (y, s) ∈ − δ1 , δ1 × (−∞, 0] by the fact that 0 0 the restriction of W to − δ1 , δ1 × [−k, 0] is Wk , for any k. Hence, from (122), we 0 0 see that (27) holds for all s ≤ 0. Since W (0) = Wk (0) for any k ∈ N, letting k go to infinity in (121) givesthe strong convergence of (wn (0), ∂s wn (0)) to (W (0), ∂s W (0)) in H 1 × L 2 − δ1 , δ1 . From the continuity of the solution to the Cauchy problem, the 0 0 same strong convergence holds for any s ≥ 0. Thus, using (22), we see that (27) holds for any s ≥ 0. This concludes the proof of Claim 2.3.
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B. Sign of E(w−(s)) for s Close to log(1 − |d(0)|) We prove (47) here. Let σ = s − log(1 − |d(0)|). In the following, we will make expansions as σ → 0− . Using (8) and (32), we write E(w− (s)) =
2 2(1 − |d(0)|)2 e2σ 2 κ0 (1 − d(0)2 ) p−1 I (σ ) 2 ( p − 1) 2 2d(0)2 2 κ0 (1 − d(0)2 ) p−1 J (σ ) + 2 ( p − 1) 2 ( p + 1) 2 κ0 (1 − d(0)2 ) p−1 K (σ ) + 2 ( p − 1) p+1 1 p+1 κ0 (1 − d(0)2 ) p−1 I (σ ), − p+1
(123)
where I (σ ) =
1 −1
2
(1 − y 2 ) p−1 g(y, σ )
, J (σ ) = 2( p+1) p−1
1 −1
p+1
(1 − y 2 ) p−1 g(y, σ )
, K (σ ) = 2( p+1) p−1
2
1
(1 − y 2 ) p−1
−1
g(y, σ ) p−1
4
,
with g(y, σ ) = 1−(1−|d(0)|)eσ +d(0)y. Since d(0)J (σ ) = d(0)o(I (σ )) and K (σ ) = p+1
κ0 p+1 from (11), 2σ 2 we see from (123) that E(w− (σ )) has the same sign as e (1 − |d(0)|) + d(0)o(1) + o(σ )−(1−d(0)2 ) which is equal to 2|d(0)|(|d(0)|−1)+d(0)o(1)+2σ (1−|d(0)|)2 +o(σ ) as σ → 0− . Since |d(0)| < 1, (47) is proved.
o(σ I (σ )) as σ → 0− from straightforward computations, and
2κ02 ( p−1)2
=
References 1. Alinhac, S.: Blowup for nonlinear hyperbolic equations. Volume 17 of Progress in Nonlinear Differential Equations and their Applications. Boston, MA: Birkhäuser Boston Inc. 1995 2. Alinhac, S.: A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations. In: Journées “Équations aux Dérivées Partielles” (Forges-les-Eaux, 2002), Nantes:Univ. Nantes, 2002 pp. Exp. No. I, 33 3. Antonini, C., Merle, F.: Optimal bounds on positive blow-up solutions for a semilinear wave equation. Internat. Math. Res. Notices. 21, 1141–1167 (2001) 4. Caffarelli, L.A., Friedman, A.: Differentiability of the blow-up curve for one-dimensional nonlinear wave equations. Arch. Rati. Mech. Anal. 91(1), 83–98 (1985) 5. Caffarelli, L.A., Friedman., A.: The blow-up boundary for nonlinear wave equations. Trans. Amer. Math. Soc. 297(1), 223–241 (1986) 6. Ginibre, J., Soffer, A., Velo, G.: The global Cauchy problem for the critical nonlinear wave equation. J. Funct. Anal. 110(1), 96–130 (1992) 7. Kichenassamy, S., Littman, W.: Blow-up surfaces for nonlinear wave equations. I. Comm. Partial Differe. Eq. 18(3-4), 431–452 (1993) 8. Kichenassamy, S., Littman, W.: Blow-up surfaces for nonlinear wave equations. II. Comm. Partial Differe. Eq. 18(11), 1869–1899 (1993) 9. Levine, H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt = −Au + F (u). Trans. Amer. Math. Soc. 192, 1–21 (1974) 10. Martel, Y., Merle, F.: A Liouville theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl. (9). 79(4), 339–425 (2000) 11. Martel, Y., Merle, F.: Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation. Ann. of Math. (2). 155(1), 235–280 (2002) 12. Merle, F., Raphael, P.: On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation. Invent. Math. 156(3), 565–672 (2004)
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13. Merle, F., Raphael, P.: The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Annals of Math (2). 161((1), 157–222 (2005) 14. Merle, F., Zaag, H.: Optimal estimates for blowup rate and behavior for nonlinear heat equations. Comm. Pure Appl. Math. 51(2), 139–196 (1998) 15. Merle, F., Zaag, H.: A Liouville theorem for vector-valued nonlinear heat equations and applications. Math. Ann. 316(1), 103–137 (2000) 16. Merle, F., Zaag, H.: Determination of the blow-up rate for the semilinear wave equation. Amer. J. Math. 125, 1147–1164 (2003) 17. Merle, F., Zaag, H.: Blow-up rate near the blow-up surface for semilinear wave equations. Internat. Math. Res. Notices. 19, 1127–1156 (2005) 18. Merle, F., Zaag, H.: Determination of the blow-up rate for a critical semilinear wave equation. Math. Annalen. 331(2), 395–416 (2005) 19. Merle, F., Zaag, H.: Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension. J. Funct. Anal. 253(1), 43–121 (2007) 20. Merle, F., Zaag, H.: Existence and characterization of characteristic points for a semilinear wave equation in one space dimension. In preparation (2008) 21. Nouaili, N.: C 1,α regularity of the blow-up curve at non-characteristic points for the one dimensional semilinear wave equation. Comm. Partial Diff. Eq. (2008) (to appear) 22. Zaag, H.: On the regularity of the blow-up set for semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. 19(5), 505–542 (2002) 23. Zaag, H.: One dimensional behavior of singular N dimensional solutions of semilinear heat equations. Comm. Math. Phys. 225(3), 523–549 (2002) 24. Zaag, H.: Regularity of the blow-up set and singular behavior for semilinear heat equations. In: Mathematics & mathematics education (Bethlehem, 2000), River Edge, NJ: World Sci. Publishing, (2002) pp 337–347 25. Zaag, H.: Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation. Duke Math. J. 133(3), 499–525 (2006) Communicated by P. Constantin
Commun. Math. Phys. 282, 87–113 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0526-1
Communications in
Mathematical Physics
p-adic Superspaces and Frobenius A. Schwarz1, , I. Shapiro2 1 Department of Mathematics, University of California, Davis, CA, USA.
E-mail: [email protected]
2 Max Planck Institute for Mathematics, Bonn, Germany.
E-mail: [email protected] Received: 16 March 2007 / Accepted: 29 January 2008 Published online: 19 June 2008 – © The Author(s) 2008
Abstract: The notion of a p-adic superspace is introduced and used to give a transparent construction of the Frobenius map on p-adic cohomology of a smooth projective variety over Zp (the ring of p-adic integers), as well as an alternative construction of the crystalline cohomology of a smooth projective variety over Fp (finite field with p elements). 1. Introduction If X is a smooth projective variety over Z or, more generally, over the ring of p-adic integers Zp , one can define the Frobenius map on the de Rham cohomology of X with coefficients in Zp [1]. This map plays an important role in arithmetic geometry (in particular it was used in the Wiles’ proof of Fermat’s Last Theorem); recently it was used to obtain interesting results in physics [8,16]. However, the construction of this map is not simple, the usual most invariant approach is based on the consideration of the crystalline site [1]. In any variation one uses the notion of a DP-ideal, that is an ideal I in a ring A with the key property that for x ∈ I , x n /n! makes sense. To be precise, one assumes the existence of operations γn : I → A, for n ≥ 0, that mimic the operations x → x n /n! and satisfy the same conditions (for instance n!γn (x) = x n ). The ring A is then called a DP-ring (DP stands for divided powers), and a DP-morphism is a ring homomorphism compatible with the DP-structure. The advantage of the crystalline cohomology ([5] is a good review) of a scheme X over Fp = Zp / pZp is that the coefficients of the theory are in Zp , though the original X was defined over Fp . Furthermore the action of the Frobenius endomorphism exists in this theory. DP neighborhoods play an essential role in defining crystalline cohomology; ˆ Iˆ), roughly speaking a DP neighborhood X of Y in X is described locally by a pair ( B, where Bˆ is the ring of functions on X and Iˆ is the ideal of the subvariety X , the important Partly supported by NSF grant No. DMS 0505735.
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requirement is that Iˆ is in fact a DP ideal. However, to construct crystalline cohomology and to analyze its relation to de Rham cohomology some technical problems must be overcome. We will show that using the ideas of supergeometry1 we can make the exposition less technical, but still completely rigorous. (Idea: Grassmann rings have divided powers naturally and at the same time there are enough of them to “feel” the entire DP neighborhood.) Thus in an appropriate (Grassmann) setting the standard notion of infinitesimal neighborhood replaces the DP envelope. To summarize, the goals of this paper are as follows. Use supergeometry to give an alternative definition of crystalline cohomology of a smooth projective variety over Fp . And secondly, to define a lifting of the action of Frobenius to the usual de Rham cohomology of a smooth projective variety over Zp . Our construction of the Grassmann neighborhood may also be applied in the case when X is not smooth over Fp , but in this case it is not likely that our cohomology coincides with the crystalline one. However, the crystalline cohomology is known not to give good answers in the non-smooth (over Fp ) case anyway, see [12] for a better cohomology. By playing with our definitions it may be possible to create a theory that gives good results in the non-smooth case. Our considerations are based on the notion of a p-adic superspace defined as a covariant functor on an appropriate subcategory of the category of Zp -algebras. It seems that this notion is interesting in itself; one can hope that it can be used to introduce and analyze “ p-adic supersymmetry” and “ p-adic superstring” making contact with the p-adic B-model of [8] and p-adic string theory (see [3] for a review). The notion of a p-adic superspace that we use is very general; it is impossible to obtain any significant results in such generality. In all of our examples however, we consider functors defining a superspace that are prorepresentable in some sense. We sketch the proof of this fact in the Appendix and show how to use it to get a more conceptual derivation of some of our statements. Let us note that a lot of what follows does not work for the even prime and so we omit that case by default. Some modifications designed to allow for the even prime are mentioned in Sect. 4.2. Finally, any functor, by default, is a covariant functor. All the rings are assumed to be unital. All varieties are of finite type over Zp . 1.1. Summary of main definitions and results. The category that serves as the source for most of our functors is described in Definition 2.1. The main notion in this paper is that of a p-adic superspace (see Definition 3.1). The replacement for a DP neighborhood (called the Grassmann neighborhood), is given in in PnZp of W ⊂ PnFp , namely W Definition 3.12. Finally, our notion of de Rham cohomology of a p-adic superspace is summarized in Definition 4.3. The Appendix contains a discussion of prorepresentable p-adic superspaces and should be considered as the correct general setting for this paper. please see Lemma 3.13. The investigation of functions For the Frobenius action on W on pt (where pt is a point in a line or P1 ) is undertaken in Sect. 4.1 and summarized in Corollary 6.6. The key points of the paper are the comparison results between our de Rham cohomology of p-adic superspaces and the usual de Rham cohomology as well as the crystalline cohomology. Namely, for a smooth V over Zp , Lemma 4.5 relates the de Rham cohomology of the completion of V with respect to p and the de Rham cohomology of the 1 A reader who is unfamiliar with supermathematics is encouraged to glance through Sect. 1.2.
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p-adic superspace associated to V . Lemma 4.11 shows that when V is also projective, the completion can be removed. The main theorem of the paper is Theorem 4.15 that establishes the isomorphism between the de Rham cohomology of a smooth projective V over Zp and the de Rham cohomology of the Grassmann neighborhood in Pn of the p-adic superspace associated to V . Corollary 4.17 compares the crystalline cohomology of W , a smooth variety over Fp , with the de Rham cohomology of its Grassmann . neighborhood W Theorem 4.15 is refined in Theorem 5.5 where it is shown that the isomorphism is in fact compatible with the Hodge filtration on one side and a very natural filtration on the other. This allows us to re-prove some divisibility estimates for the action of Frobenius on the de Rham cohomology of V . The most useful of these is stated in Lemma 5.6. The main motivation for this paper is Corollary 4.16 that shows that a smooth projective variety over Zp has an action of Frobenius on its de Rham cohomology. 1.2. Supermathematics. The present paper aims to explain crystalline cohomology from the point of view of supergeometry. However the knowledge of supergeometry itself is not essential for the reading of this text (at least formally). Our treatment of spaces as functors allows for a quick jump from the familiar commutative setting to the supercommutative one. The functorial approach to the definition of the superspace (in an essentially different form) was advocated in [15] and [7]. It is analogous to the functor approach to the theory of schemes in algebraic geometry. An excellent reference for this point of view is [6] where this theory is developed from the very beginning; groups and Lie algebras are treated as well. This treatment of algebraic geometry generalizes immediately to the setting of supergeometry by replacing commutative rings with supercommutative ones. The reader is invited to consult these papers in the case of necessity. For a more complete understanding of supermathematics we recommend [9], as well as [2] for those looking for a more abstract and conceptual picture. The very few concepts that we do need are explained below. Definition 1.1. By a supercommutative ring A we mean a Z/2Z-graded ring (i.e. A = A0 ⊕ A1 with multiplication respecting the grading) such that ai · a j = (−1)i j a j · ai where ai ∈ Ai .2 We say that a ∈ A0 is even and b ∈ A1 is odd. Remark. Any commutative ring A is also supercommutative with A1 = 0. We often use freely generated rings, i.e. B[xi , ξ j ] with B commutative, and generators xi commuting (xi x j = x j xi ) and ξ j anti-commuting (ξi ξ j = −ξ j ξi ), that is xi are even and ξi are odd, thus xi ξ j = ξ j xi . Note that specifying the parity of B, xi and ξi is sufficient to define the Z/2Z-grading. It is convenient to consider xi and ξ j as commuting and anticommuting variables. Definition 1.2. Denote by Super the category of supercommutative rings, with morphisms being algebra homomorphisms respecting the grading (i.e. a morphism preserves the Z/2Z decomposition). In other words, morphisms are parity preserving homomorphisms. 2 This is the sign rule. Whenever two odd objects are to be written in the opposite order, a minus sign is the price. This is the mildest possible modification of the usual commutativity.
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Remark. To give the reader a feeling for the morphisms in Super we point out that Hom Super (Z[x1 , . . . , xn ], A) = (A0 )n , Hom Super (Z[ξ1 , . . . , ξn ], A) = (A1 )n and Hom Super (Z[x1 , . . . , xn , ξ1 , . . . , ξm ], A) = (A0 )n × (A1 )m . We note the standard use of xi for even variables and ξi for odd. The advantage of the functorial approach to spaces is that it is deceptively simple. We can say that a superspace, in the most general sense, is a functor from Super (or from a subcategory of Super ) to Sets (the category of sets). Morphisms are natural transformations of functors. This definition ignores the most important aspect of spaces, namely that they should be local. This can be included in the definition in many ways. For the general method see [6] and compare with our approach based on the consideration of the body of a superspace (Definition 3.2) and prorepresentability (Appendix). Supergroups are functors from Super to Gr oups (the category of groups), etc. The action of a supergroup G on a superspace X can be defined very naturally: for a supercommutative ring A we should have an action of the group G(A) on the set X (A) satisfying some conditions of functoriality. We define the rest of the concepts as needed. In fact the reader is already quite familiar with some of the supermathematics that we use. For example, the de Rham complex of a smooth affine variety V is a supercommutative algebra with a differential. This algebra can be interpreted as the algebra of functions on the odd tangent space T V of V and the differential as an action of the odd line A0,1 on T V . This is thoroughly discussed in Sect. 4. 1.3. Main constructions. Let us explain our constructions in some detail. (The reader can skip this explanation. However it could be useful for some people, in particular, for readers with a background in mathematical physics.) If a (projective or affine) variety is specified by means of equations with coefficients belonging to a commutative ring R it makes sense to consider the unknowns as belonging to any R-algebra. This means that every variety of this kind (variety over R) specifies a functor on the category of R-algebras with values in the category of sets. This remark prompts a preliminary definition of a superspace over a ring R as a functor taking values in the category of sets and defined on the category of Z/2Z-graded supercommutative R-algebras or better yet on its subcategory . (Morphisms are parity preserving homomorphisms.) We define a map of superspaces as a map of functors; this definition depends on the choice of the category . In particular, in the case when R is a ring of p-adic integers Zp we take as the category of supercommutative rings of the form B ⊗ n , where B is a finitely generated commutative ring in which the multiplication by p is nilpotent and B/ p B does not contain nilpotent elements. Here n stands for the Grassmann ring (free supercommutative ring with n odd generators.) Every ring in the category can be considered as a Zp -algebra since multiplication by a series n an p n is well defined because multiplication by p is nilpotent. Considering functors on the above category we come to the definition of a p-adic superspace. The typical example of a superspace is An|k,m corresponding to a functor, that sends an R-algebra to a set of rows with n even, k even nilpotent elements and m odd elements of the algebra. A function on a superspace is a map of this superspace to A1,1 = A1|0,1 . In the p-adic case the functions on An|k,m are described in Theorem 3.8. In particular, n the functions on A0|1,0 correspond to series with divided powers an xn! , where an ∈ Zp . (Notice that in this statement it is important that the functors are defined on the category described above. We could consider the functors on the larger category of
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all supercommutative Zp -algebras; then only a series of the form an x n with an ∈ Zp corresponds to a function.) The definition of a superspace in terms of a functor is too general; one should impose some additional restrictions to develop an interesting theory. One of the possible ways is to impose some conditions on the “bosonic part” of the superspace (i.e., on the restriction of the functor to commutative R-algebras.) In particular, we notice that the category we used in the p-adic case contains the category of commutative Fp -algebras without nilpotent elements; in this case the restriction of the functor specifying a p-adic superspace Y should correspond to a variety over Fp (the body [Y ] of the superspace Y ). For any C ∈ , let us set ρ(C) = C/C nilp , where C nilp is the ideal of nilpotent elements in C. In our case C nilp is generated by p and the odd generators. From definitions we see that ρ(C) is an Fp -algebra. Thus the natural projection C → ρ(C) induces a map π : Y (C) → [Y ](ρ(C)). For every (open or closed) subvariety Z ⊂ [Y ] we can define a p-adic subsuperspace Y | Z ⊂ Y as a maximal subsuperspace of Y having Z as its body. More explicitly, Y | Z (C) = π −1 (Z (ρ(C))); we can say that Y | Z is a subsuperspace of Y over Z . We give a definition of the body only in p-adic case, but similar constructions also work in other situations. For every superspace Y over a ring R we can introduce a notion of a differential form and of the de Rham differential. (Differential forms are defined as functions on the superspace T Y that parameterizes the maps from the superspace A0,1 = A0|0,1 to Y .) One can try to define the cohomology of Y as cohomology of the differential R-module (Y ) of differential forms on Y , but this definition does not capture the whole picture (it corresponds to the consideration of the Hodge cohomology H k,0 ). The right definition of de Rham cohomology of Y can be given in terms of hypercohomology of a sheaf of differential R-modules on the body of Y . (To an open subset Z ⊂ [Y ] we assign the module (Y | Z ) of differential forms on Y | Z .) Now we are able to define our analogue of the crystalline cohomology of a projective Fp -variety X ⊂ PnFp , where PnFp stands for projective space over Fp . One can regard PnFp as a body of the p-adic projective superspace Pn ; this remark permits us to consider Pn | X (the subsuperspace of Pn over X ). We define “crystalline” cohomology of X as the de Rham cohomology of the p-adic superspace Pn | X . The Frobenius map Fr acts naturally on this cohomology. (The usual action of Fr on Pn sending every homogeneous coordinate to its p th power preserves the Fp -variety X and therefore Pn | X .) We prove that for a smooth Fp -variety X the cohomology of Pn | X coincides with the p-adic de Rham cohomology of any variety X over Zp that gives X after reduction to Fp (and therefore the Frobenius acts on the cohomology of X ); see Corollary 4.16. 2. Category Consider the local ring Zp with the maximal ideal pZp (it is a DP ideal since p n /n! which is obviously in Q p is actually in Zp because ord p n! ≤ n). Our main object is the category . Definition 2.1. Let be the category with objects B that are super commutative rings freely and finitely generated over a commutative ring B 3 (B is allowed to vary) by odd generators. More precisely, we require that B is a finitely generated commutative ring 3 A typical example of such B is Z/ p n Z.
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such that p is nilpotent, and B/ p B has no nilpotent elements. We write the Z/2Z grading of B as follows: B = 0B ⊕ 1B , where 0B is even and 1B is odd. The morphisms are parity preserving homomorphisms. The requirement that p be nilpotent is necessary to allow for evaluation of infinite series such as an x n at elements pb with b ∈ B. Also, for example, we see that B is a Zp -module (in fact it is a Z/ p N Z-module for an N large enough). This means that we can consider as a subcategory of the category of Zp -algebras. Remark. We use the notation B to denote a generic element of to emphasize the fact that it is B that is important, not the number of odd variables. The notation is meant to remind the reader of the exterior algebra (Grassmann algebra) where the field has been replaced with B. A more precise statement would be that B B[ξi ], where B[ξi ] is a polynomial ring in ξi s with coefficients in B, however in this case the variables ξi are anti-commuting, that is ξi ξ j = −ξ j ξi . The parity is determined by the total number of variables ξi . Denote by +B the ideal in B generated by ξ1 , . . . , ξn . Notice that contains the category of Fp -algebras without nilpotent elements as a full subcategory, and there is a retraction onto it that sends B to B /( p B + +B ) = B/ p B. The ideal p B + +B is to play a very important role for us. One should mention that it can be characterized by the fact that it consists exactly of the nilpotent elements of B (this follows from the lack of nilpotent elements in B/ p B). As a consequence we see that if A → B is any morphism and I ⊂ A is a nilpotent ideal, then I is carried to p B + +B by the morphism. However its most important property is explained in the following theorem. Theorem 2.2. For every B ∈ , p B + +B ⊂ B is naturally a DP ideal, i.e. there are operations γn : p B + +B → B that satisfy the axioms in [1]. Furthermore, any morphism in preserves this structure automatically. Remark. Thus every object in is in fact a DP pair ( B , p B + +B ) and any morphism preserves the DP structure. This explains our choice of . Proof. Represent B as Zp [xi ]/I , where Zp [xi ] is the polynomial algebra over Zp . Note that since Zp is torsion free Zp [xi ] ⊂ Q p [xi ] and we can define γn : Zp [xi ] → Q p [xi ] by γn (x) = x n /n! for all n. We claim that pZp [xi ] + +Zp [xi ] maps under γn to Zp [xi ] . (Thus γn define a DP structure on the pair (Zp [xi ] , pZp [xi ] + +Zp [xi ] ).) To show this it is sufficient to check that x n /n! is in Zp [xi ] for x ∈ pZp [xi ] and for x ∈ +Zp [xi ] . For x ∈ pZp [xi ], this follows from the observation that p n /n! ∈ Zp . Now suppose that e ∈ 0+ Zp [xi ] , let e = e1 + · · · + ek , where ei are even and of the form f i ξi 1 . . . ξi s , i.e. write e as the sum of monomials in ξi s. Notice that ein = 0 if n > 1. So that en = (e1 + · · · + ek )n =
n i =n
n! e1n 1 · · · ekn k = n!e1n 1 · · · ekn k . n1! · · · nk ! n =n i n i =0 or 1
Consider an element x = e + o ∈ +Zp [xi ] with e even and o odd. Then x n = (e + o)n = en + nen−1 o and we are done. Since γn satisfy the axioms for a DP structure we obtain in this way a DP structure on the pair (Zp [xi ] , pZp [xi ] + +Zp [xi ] ). Note that B = Zp [xi ] /I [ξ j ], thus it inherits
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a DP structure from Zp [xi ] if (and only if) I [ξ j ] ∩ ( pZp [xi ] + +Zp [xi ] ) is preserved by γn . But I [ξ j ] ∩ ( pZp [xi ] + +Zp [xi ] ) = I ∩ pZp [xi ] + +I , and clearly +I is preserved by γn . As for I ∩ pZp [xi ], if p f ∈ I then ( p f )n /n! = p f g with g ∈ Zp [xi ], thus ( p f )n /n! ∈ I . We conclude that B inherits a DP structure on p B + +B . Next we observe that the DP structure obtained as above does not depend on a particular representation of B as a quotient of a polynomial algebra. Namely, if we represent B as Zp [y j ]/J and obtain a DP structure on B in that way, then the identity map on B lifts to a homomorphism from Zp [xi ] to Zp [y j ] (because Zp [xi ] is free) that is automatically compatible with DP structure (since γn is just x n /n!). Because the projections to B are DP compatible by definition, the identity map is DP compatible as well. If B → C is any morphism then it lifts to a morphism of the free algebras that cover B and C as above. The lifting is again automatically compatible with DP structure, ensuring that B → C is a DP morphism.
Because of the nature of our definition of DP structure on B we use the more suggestive x n /n! instead of the more accurate γn to denote the DP operations. As we have shown above the symbol x n /n! is functorially defined for elements of the ideals p B + +B . Remark. Note that a sufficient condition on J for Zp [xi ] /J to inherit a DP structure α is that it be a ξ -homogeneous ideal, i.e. J = α J ∩ Zp [x i ]ξ , where α is a multiindex. An example used in the theorem above is J = I [ξ j ]. Perhaps one can use this observation to enlarge the category . Remark. Given any super-commutative Zp -algebra A = A0 ⊕ A1 , we may define A+ ⊂ A by A+ = p A + A1 A, generalizing the ideal p B + +B ⊂ B . This ideal is functorial, however it is not clear why it should have any DP structure. Various conditions may be imposed to ensure this. The previous remark is an example.
3. Superspaces, Neighborhoods and Frobenius We would like to base our definition of a p-adic superspace on the notion of a functor from to Sets, the category of sets. Since we are interested in studying geometric objects, we would like to impose conditions that would make the functor “local”, the easiest way to do it is through the notion of the body of a p-adic superspace. Definition 3.1. A p-adic superspace X is a functor (covariant) from to Sets, such that the restriction to the full subcategory of Fp -algebras without nilpotent elements corresponds to a variety [X ] over Fp . Definition 3.2. The body of a p-adic superspace X is the variety [X ]. Definition 3.3. A map α : X → Y of superspaces is a natural transformation from X to Y . A more familiar object, the purely even superspace, is obtained by requiring that the functor factors through 0 , the category with objects of the form 0B .
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We have the usual functors An and Pn ,4 where An ( B ) = {(r1 , . . . , rn )|ri ∈ 0B } and Pn ( B ) = {(r0 , . . . , rn )|ri ∈ 0B ,
B ri = B }/(0B )× .
Note that these are purely even. More generally we can define the superspace An,m ( B ) := {(r1 , . . . , rn , s1 , . . . , sm )|ri ∈ 0B , si ∈ 1B }. A further generalization that we will need is 1 An|k,m ( B ) := {(r1 , . . . , rn , t1 , . . . , tk , s1 , . . . , sm )|ri ∈ 0B , ti ∈ p B + 0+ B , si ∈ B },
in other words ri are even elements, ti are even nilpotent and si are odd elements. One can also define Pn,m ( B ) := {(r0 , . . . , rn , s1 , . . . , sm )|ri ∈ 0B , si ∈ 1B , B ri = B }/(0B )× but we will not need it. Remark. The most important cases from the above are A1|0,0 ( B ) = 0B , A0|1,0 ( B ) = p B + 0+ B , A0|0,1 ( B ) = 1B ; they are the main building blocks for the theory in this paper. Definition 3.4. A function on a p-adic superspace X is a natural transformation from X to the superline A1,1 . Considering all natural transformation from X to the superline A1,1 we obtain the set of functions on X . It is easily seen to be a ring by observing that the functor A1,1 takes values in the category of supercommutative rings. A very versatile notion that we will need is that of a restriction of a p-adic superspace Y to a subvariety Z (it need not be open or closed) of its body. It is the maximal subsuperspace of Y with body Z . More precisely we have the following definition. Definition 3.5. Let Y be a p-adic superspace and Z a subvariety of [Y ], then the p-adic superspace Y | Z is defined to make the following diagram cartesian: / Y ( B ) Y | Z ( B ) Z (B/ p B)
π
i
/ Y (B/ p B)
4 These functors, and others like them below, can actually be defined as usual superspaces, i.e. they can be obviously extended to the category of all supercommutative rings. Here we use their restriction to , but no extra structure of is required.
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Consider the following “local” analogue of functions on X . Definition 3.6. Define the pre-sheaf of rings O X on [X ] by setting O X (U ) to be the ring of functions on X |U , for any open U in [X ]. There is no reason to expect that the pre-sheaf O X is a sheaf in general. Thus the usual thinking about functions in terms of coordinates is not advised. However, it is a sheaf for all the superspaces that we consider in this paper. If one wants a more general setting in which O X is a sheaf, one should consider prorepresentable superspaces as defined in the Appendix. Definition 3.7. Denote by R y1 , . . . , yk the ring whose elements are formal expressions K 5 K ≥0 a K y /K !, where a K ∈ R and yi s are commuting variables. Note that K ! need not be invertible in R. We call R y1 , . . . , yk the ring of power series with divided powers. Remark. Clearly we can add such expressions, but it is also easy to see that we can n! n multiply them since ai y i /i! b j y j /j! = i+ j=n i! j! ai b j y /n!. Theorem 3.8. The functions on An|k,m are isomorphic as a ring to ⎫ ⎧ ⎬ ⎨ a I,J,T x I ξ J y T /T ! ⎭ ⎩ I,J,T ≥0
where xi and yi are even and ξi are odd, and a I,J,T ∈ Zp with a I,J,T → 0 as I → ∞. Proof. Recall that An|k,m ( B ) = {(r1 , . . . , rn , t1 , . . . , tk , s1 , . . . , sm )|ri ∈ 0B , 1 I J T ti ∈ p B + 0+ I,J,T ≥0 a I,J,T x ξ y /T ! can be evaluated at B , si ∈ B }, then every every (r, t, s) by setting x = r , ξ = s and y = t to obtain an element of B = A1,1 ( B ). Furthermore, An,m is pro-represented6 by (Zp / p N Zp )[x1 , . . . , xn , ξ1 , . . . , ξm ] ∈ (they are in because Fp [xi ] has no nilpotent elements). Here xi are even and ξi are odd. Thus7 the functions are lim (Zp / p N Zp )[x1 , . . . , xn , ξ1 , . . . , ξm ], ← − N
i.e. of the form I,J ≥0 a I,J x I ξ J with a I,J → 0 ∈ Zp as I → ∞. The case of A0|k,0 is not as trivial; the issue is that it is “pro-represented” by (Zp / p N Zp ) y1 , . . . , yk but these are not in (in Fp yi all yi are nilpotent). Thus the proof of the complete theorem is postponed until it appears as Corollary 4.13.
Definition 3.9. Given a variety V over Zp (which can be viewed as a functor from the category of commutative Zp -algebras to Sets) we can define the associated p-adic superspace X V by setting X V ( B ) = V (0B ). 5 Here and below I , J , K , T are multi-indices and T ! = t !t ! . . .. 1 2 6 The phrase F is pro-represented by C and is used here in a more narrow sense than in the Appendix. n Namely, we mean that F = lim h Cn , where h Cn = Hom(Cn , −), and Cn form an inverse system of objects in − →
the category. A good reference on pro-representable functors (in this sense) in (non-super) geometry is [13]. 7 This is an application of the Yoneda Lemma.
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Note that information is lost in passing from the variety to its associated superspace. More precisely, V and its p-adic completion Vˆ p will have the same associated superspace. (See Lemma 3.10 below; we return to this discussion in Sect. 4.) This is best illustrated by considering the functions on An . As a variety over Zp its functions are by definition Zp [x1 , . . . , xn ], however when considered as a p-adic superspace one gets the much larger ring lim (Zp / p N Zp )[x1 , . . . , xn ] ← − N
of functions8 . Observe that the functions on the purely odd affine space A0,m do not change. The crucial point for us is the metamorphosis that the functions on A0|1,0 undergo, as we pass from considering it as a variety over Zp (prorepresented by Zp [[x]]) to the associated superspace; they transform from power series to divided power series. It is this observation that motivates the present paper. Remark. Our use of An,m for a p-adic superspace is somewhat misleading as that symbol is standard for a superscheme; in particular An usually denotes (with the ground ring being implicitly Zp ) Spec(Zp [x1 , . . . , xn ]). There is no confusion however if we make explicit whether we are discussing a variety or a p-adic superspace associated to it. When we need to make the distinction explicit, we use X V for the p-adic superspace associated to a variety V . Lemma 3.10. Let V be a variety over Zp , then the body of X V is the restriction of V to Fp , i.e., [X V ] = V |Fp , and the functions on X V (as a sheaf on [X V ]) are given by the completion of the functions on V at the subvariety V |Fp , i.e., O X V = (O V )p, thus making precise the difference between V and X V . Proof. That [X V ] = V |Fp is immediate from the definition. The question of functions is local, so assume V = Spec A. Then X V is pro-represented by lim A/ p n A, so that ← − p . OXV = A
One of the most important notions of this paper is that of the infinitesimal neighborhood of one p-adic superspace inside another. It is meant to replace the DP-neighborhood. Definition 3.11. Let X ⊂ Y be p-adic superspaces, define X , the infinitesimal neighborhood of X in Y by X = Y |[X ] .9 Example. Let An → An+1 be an inclusion of p-adic superspaces, i.e. An ( B ) = 0B × · · · × 0B → 0B × · · · × 0B × {0} ⊂ An+1 ( B ), then it follows directly that n|1 n An+1 |[An ] ( B ) = 0B × · · · × 0B × ( p B + 0+ B ), so that A = A . 8 It consists of series with p-adically vanishing coefficients. 9 Note that X depends only on Y and [X ], compare with Definition 3.12.
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Suppose that W ⊂ PnFp is a possibly non-smooth Fp -variety. We want to define the notion replacing that of a DP neighborhood of W in PnZp . We do this as follows. The inclusion of varieties over Fp gives rise to a subvariety W of the body of the p-adic to denote Pn |W , this superspace Pn . Let us use the same notation as above, namely W is the infinitesimal neighborhood of W that behaves much better than W itself. as above, the DP neighborhood of W in Pn . An alternative Definition 3.12. We call W name that we sometimes use is Grassmann neighborhood. is a p-adic superspace whereas W was a variety. While not the same kind Remark. W of object, we can nevertheless define many notions for p-adic superspaces that we have for varieties. For example as we see in the next section, we may consider the de Rham cohomology of a p-adic superspace. We have the usual action of the Frobenius map Fr on the p-adic superspace Pn via raising each homogeneous coordinate to the p th power. The restriction of Fr to the body . Summarizing we get: of Pn preserves W , therefore we have an action of Fr also on W Lemma 3.13. Let W ⊂ Pn be an inclusion of varieties over F p , then any lifting of the action of Frobenius from PnFp to PnZp (i.e., a choice of homogeneous coordinates) restricts . to an action of Frobenius on W 4. De Rham Cohomology of Superspaces Let us briefly review the notion of de Rham cohomology from the point of view of superspaces. This point of view lends itself most naturally to a generalization applicable in our setting. We begin by introducing the notion of the odd tangent space to a p-adic superspace X . Remark. For a B ∈ , we denote by B [ξ ] the ring B with an adjoined extra odd variable ξ . More precisely, given a supercommutative ring R, we can form R[ξ ] by considering expressions of the form a + bξ , with multiplication defined by (a + bξ ) (c + dξ ) = ac + (ad + (−1)c¯ bc)ξ , where a, b, c, d ∈ R and c¯ is the parity of c. Definition 4.1. Let X be a p-adic superspace, define a new p-adic superspace T X , the odd tangent space of X , by T X ( B ) = X ( B [ξ ]). Functions on T X will serve as the differential forms on X . We will define the differential d and the grading on differential forms in terms of an action of a supergroup on T X . Note that there is a natural map B [ξ ] → B that sends ξ to 0. This defines a map of p-adic superspaces π : T X → X, and a corresponding map on the bodies10 [π ] : [T X ] → [X ]. 10 When X is purely even, [T X ] = [X ].
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The superspace T X carries an action11 of the supergroup A0,1 (A1 )× defined as follows. Let o ∈ 1B = A0,1 ( B ), we need to define the corresponding automorphism of T X ( B ) = X ( B [ξ ]). We accomplish that by defining a morphism in from B [ξ ] Id / to itself via B B and ξ → ξ + o; this induces the required automorphism.
Similarly we define the automorphism corresponding to e ∈ (0B )× = (A1 )× ( B ) by
Id / defining a morphism in via B B and ξ → eξ . At this point, for the sake of concreteness, let us assume that X is prorepresentable (in the sense of Definition 6.2). This is always the case in our setting. We are now ready to define the differential graded sheaf X/Zp of Zp -modules on [X ] the body of X . Its hypercohomology will be called the de Rham cohomology of X . We denote it by D RZp (X ). Let U ⊂ [X ] be an open subvariety; consider the Zp -algebra of functions on T (X |U ) (i.e. natural transformations to A1,1 ). More concisely we have the following.
Definition 4.2. Define the pre-sheaf X/Zp on [X ] by setting X/Zp = [π ]∗ OT X . Note that X/Zp carries a grading induced by the action of (A1 )× . The sections of should be thought of as differential n-forms on X . We also have a differential12 d
nX/Zp
(an operator which comes from the canonical odd element13 η ∈ A0,1 ( B [η]) for every B ). More precisely, suppose that ϕ is a function on T X . Recall that this means that for every B there is a natural map ϕ : T X ( B ) → B . Define the function dϕ by specifying that for every B , it is the following composition of maps: i
aη
ϕ
c
T X ( B ) → T X ( B [η]) → T X ( B [η]) → B [η] → B , where i is induced by the inclusion B ⊂ B [η], aη is the endomorphism of T X Id
( B [η]) = X ( B [η][ξ ]) induced by the endomorphism of B [η][ξ ] given by B → B , η → η, ξ → ξ + η; ϕ is self explanatory and c reads off the coefficient of η. One readily checks that d increases the degree by one.14 Definition 4.3. Let X be a p-adic superspace. We define the de Rham cohomology of X as the hypercohomology of •X/Zp , i.e., D RZp (X ) = H([X ], •X/Zp ). Note that it is easy to see from the definitions that D RZp (−) is a contravariant functor from the category of superspaces to the category of graded Zp -modules. Thus any endomorphism of X induces an endomorphism of D RZp (X ). 11 These definitions become more transparent when one thinks of T X as the superspace parameterizing the maps from A0,1 to X . 12 Speaking informally, it is induced by the infinitesimal action of A0,1 . 13 In fact η is the canonical element in the odd Lie algebra A0,1 (Z [η]) of A0,1 . p 14 The connection between these abstract definitions and the usual de Rham complex is made explicit in Lemma 4.4.
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Example. Let us apply the definitions in the simple example of a line. In this case X ( B ) = 0B and T X ( B ) = X ( B [ξ ]) = ( B [ξ ])0 = 0B + 1B ξ . Thus T X = A1,1 as expected. The body of X is affine, so to compute D RZp (X ) we need only compute the cohomology of the complex of Zp -modules (OA1,1 ). By Theorem 3.8 we know that as a Zp -module it is S ⊕ Sξ , where S = { ai x i |ai ∈ Zp , ai → 0}. The reader is invited to verify that the action of (A1 )× puts S in degree 0 and Sξ in degree 1, while the 1 p ). differential acts by ξ ∂x . Thus D RZp (X ) = Hd R (A More generally, by unraveling the definitions we obtain the following two lemmas that bridge the gap between the p-adic superspace approach and the usual situation. Lemma 4.4. Let Spec A be a smooth variety over Zp , then A p ). D RZp (X SpecA ) = Hd R (Spec Proof. Let X = X SpecA ; since [X ] is affine the left-hand side is computed by the complex of global functions on T X . But T X ( B ) = X ( B [ξ ]) = H om Super (A, B [ξ ]) = H om Super ( A , B ). Recall that A is the supercommutative ring generated by a (even) and da (odd) for a ∈ A subject to the usual relations (most important being the Leibniz Rule). So T X is prorepresented by A , thus OT X = ( A ) p (just like in Lemma 3.10). This is exactly the complex that computes the right-hand side, but we still need to verify that this identification is compatible with the differentials. It is sufficient to check the compatibility with the actions of A0,1 (A1 )× . Note that the action of A0,1 on A is given by the coaction (which is an algebra morphism) A → A [ξ ] a → a + daξ, da → da and the action of
(A1 )×
by A → A [x ±1 ] a → a, da → dax
while H om Super ( A , B ) = H om Super (A, B [ξ ]) {a → f (a), da → ϕ(a)} ↔ {a → f (a) + ϕ(a)ξ }. It is now straightforward to check that the action on T X is obtained in this case from the one on A .
Lemma 4.5. Let V be a smooth variety over Zp . Then D RZp (X V ) = Hd R (Vˆ p ). Proof. The left-hand side is by definition the hypercohomology of •X V /Zp on [X V ], • while the right-hand side is the hypercohomology of ( V /Zp ) on V |Fp . The two spaces p
[X V ] and V |Fp are the same (Lemma 3.10), so the question is about comparing the two complexes of sheaves locally. They are the same by the proof of Lemma 4.4.
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4.1. Functions on pt. In this section we study the most basic and at the same time the most crucial example of a DP neighborhood, namely that of a point in a line. Definition 4.6. Denote by pt the infinitesimal neighborhood of the origin in A1 . In this section we are concerned with describing explicitly the functions on pt. This is the key step in the subsequent cohomology computations. A more category theory minded reader may wish to visit the Appendix before going any further. One sees immediately from the definitions that pt( B ) = p B + 0+ B . This is our old friend A0|1,0 , and has a subfunctor that we will denote by pt 0 ; it is defined by pt 0 ( B ) = 0+ B . It is the functions on pt 0 , i.e. natural transformations to A1 that we describe first.15 Let f be one such transformation; our intention is to show that for w ∈ 0+ B , we have that ∞ f (w) = i=0 ai wi /i! with ai ∈ Zp determining f . ∞ Remark. It is clear that any expression i=0 ai wi /i! gives a function as it can be evalua0+ ted at any element of p B + B , i.e. we do have a map from such expressions to functions. However the injectivity and surjectivity of this map remains to be demonstrated below. First we need a lemma. , then Lemma 4.7. Let w = ξ j1 ξ j2 + · · · + ξ j2k−1 ξ j2k ∈ 0+ Z / pN Z p
f (w) =
k
p
ai wi /i!
i=0
and ai ∈ Zp / p N Zp depend only on f .16 Proof. We proceed by induction on k. If k = 0 then w = 0 and so by functoriality of f , f (w) ∈ Zp / p N Zp , define a0 to be f (w) and we are done. Assume that the lemma is true for k ≤ n. Let k = n + 1, w = ξ j1 ξ j2 + · · · + ξ j2k−1 ξ j2k =: x1 + · · · + xk , and setting f (w) = I a I ξ I consider ξ I = ξ j1 . . . ξ j2i . Note that by functoriality i ≤ k, and if i = k then there is only one such ξ I , denote its coefficient by ai (we have now defined an+1 ). If i < k define a map φ from Zp / p N Zp to itself by sending ξ js to ξs and the rest of ξ ’s to 0. If ξ I = xs1 . . . xsi then φ(w) = ξ1 ξ2 + · · · + ξ2i−1 ξ2i , so f (φ(w)) = · · · + ai ξ1 ξ2 . . . ξ2i−1 ξ2i by the induction hypothesis and φ( f (w)) = · · · + a I ξ1 ξ2 . . . ξ2i−1 ξ2i , so that a I = ai . 15 Here we do not need A1,1 as everything is purely even. 16 The a are defined inductively in the proof. i
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If ξ I = xs1 . . . xsi then φ(w) has fewer than i summands yet is of the form ξ ξ +· · ·+ξ ξ so that we may use the induction hypothesis to conclude that the top degree of f (φ(w)) is less than 2i, whereas φ(ξ I ) = ξ1 ξ2 . . . ξ2i−1 ξ2i has degree 2i, so that a I = 0. n+1 ai wi /i!, and we are almost done. So f (w) = I a I ξ I = i ai xs1 . . . xsi = i=0 Namely, we demonstrated that any function f , when restricted to w ∈ 0+ can be Z / pN Z p
p
written as a DP polynomial with coefficients in Zp / p N Zp of degree at most k. However, it is immediate that such an expression is unique, since wi /i! for i = 0, . . . , k form a basis of the free Zp / p N Zp -submodule of 0Z / p N Z that they span.
p
p
By functoriality of f we obtain, by considering the above lemma for all N , that the coefficients ai are given for all N by the images under the natural projection of ai ∈ Zp . ∞ i Theorem 4.8. Let e ∈ 0+ i=0 ai e /i!, with ai as above. B , then f (e) = N Proof. Let e ∈ 0+ i bi ei1 ei2 , where ei j are odd monomials in ξ ’s. Let p B = 0. B ,e = N Define a map ϕ from Zp / p N Zp to B by Zp / p Zp → B being the structure morphism, and ξ2i−1 → bi ei1 and ξ2i → ei2 , so that w = i ξ2i−1 ξ2i → e. So f (e) = f (ϕ(w)) = length(w) length(e) length(e) ϕ( f (w)) = ϕ( i=0 ai wi /i!) = i=0 ai ϕ(w)i /i! = i=0 ai ei /i! = ∞ + i i=0 ai e /i!. Here the length of an element in B is the minimal number of monomials (in the odd variables) that are needed to write it down.
Remark. A consequence of this result is that while our choice of w in Lemma 4.7 is fairly arbitrary, for instance one can reorder the coordinates, this does not in any way affect the coefficients ai . Now let us consider the functor pt( B ) = p B + 0+ B itself. We claim that the ∞ i functions are still of the form i=0 ai x /i! with coefficients in Zp .17 We reduce to the previous case to prove the following lemma, from which the claim follows easily. Lemma 4.9. Let A = Zp / p N Zp [x] and w = px + ξ1 ξ2 + · · · + ξ2k−1 ξ2k ∈ p A + 0+ A , then f (w) =
∞
ai wi /i!
i=0
and ai ∈ Zp depend only on f . Proof. First we need to define ai ∈ Zp . Recall the subfunctor pt 0 of pt that sends B N to 0+ B . If we restrict p t 0 to the subcategory of Grassmann rings with coefficients N in B, with p B = 0, then by Theorem 4.8, f | pt 0 determines (and is determined on N by) {aiN ∈ Zp / p N Zp }. We observe that by functoriality of f we may take the inverse f be a new function on limit over N to obtain {ai ∈ Zp } that determine f | pt 0 on . Let 17 Here in particular we must assume that p > 2, otherwise this expression does not define a function in general and the situation becomes more complicated.
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pt defined by f (e) = ai ei /i! for e ∈ p B + 0+ B , so that f agrees with f on p t 0 . We want to show that they agree on w also. For any N and n, let us define a map φ from A to Zp / p N Zp by ξi → ξi and x → η1 η2 + · · · + η2n−1 η2n . Note that under this map w → p(η1 η2 + · · · + η2n−1 η2n ) + ξ1 ξ2 + · · · + ξ2k−1 ξ2k ∈ 0+ Z / pN Z p
and so φ( f (w)) = f (φ(w)) =
p
ai (φ(w))i /i! = φ( f (w)).
N i Note that setting f (w) = ci x with ciN ∈ Zp / p N Zp [ξ j ], functoriality implies that we have ci ∈ Zp [ξ j ] such that f (w) = ci x i for all N . To show that f (w) = f (w) it suffices to consider the following situation. Let bi ∈ Zp [ξ j ], define g = bi x i ∈ A , suppose that 0 = φ(g) = bi (η1 η2 + · · · + η2n−1 η2n )i for all n and N . Since if i ≤ n then 0 = φ(g) implies that i!bi = 0 we see that bi = 0 in Zp [ξ j ] and g = 0. Take g = f (w) − f (w) and we are done.
∞ i Theorem 4.10. Let e ∈ p B + 0+ i=0 ai e /i!. B , then f (e) = Proof. Let e ∈ p B + 0+ i bi ei1 ei2 , where ei j are odd monomials in B , e = pb + a map ϕ from A to B by x → b, ξ ’s and b ∈ B. Suppose that p N B = 0. Define ξ2i−1 → bi ei1 and ξ2i → ei2 , so that w = px + i ξ2i−1 ξ2i → e. So that f (e) = f (ϕ(w)) = ϕ( f (w)) = ϕ ai wi /i! = ai ϕ(w)i /i! = ai ei /i!.
Remark. To summarize the above, we have a canonical map from Zp x to functions on pt. This map, as is explicitly shown in the lemmas above, is surjective. The fact that it is injective follows from the observation at the end of the proof of Lemma 4.7. 4.2. The case of p = 2. As mentioned before, the case of the even prime does not fit ∞ into the framework described. The issue is that i=0 pi /i! is convergent in Zp only for p > 2. It follows that for the case p = 2, the functions on the Grassmann neighborhood of a point in the line are not simply DP power series with coefficients in Zp , rather they form a subset of these with some conditions on the coefficients. While it is possible to describe them explicitly, one immediately sees that the homotopy of Lemma 4.14 no longer exists. Consequently one can not prove the cohomology invariance of Grassmann thickening. It seems one can introduce an alternate framework that works for all primes p. We ˆ an enlargement of our main category briefly outline it here. The idea is to introduce , which includes Grassmann rings with an infinite number of variables that allow certain infinite sums as elements.More precisely, we consider rings B = B[ξ1 , ξ2 , . . .], where elements have the form bi wi , where bi ∈ B and wi are monomials in ξ ’s of degree
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at most N , where N is fixed for each element.18 Thus ξ2i−1 ξ2i is an element, while ∞ i i=1 j=1 ξ j is not. One does not get the same functions as before for the case of the Grassmann neighborhood of a point in the line19 , but the homotopy of Lemma 4.14 now makes sense and so we can again show the cohomology invariance of Grassmann thickening by modifying all of the arguments accordingly (some of them simplify somewhat). Finally, note that the very definition of the Grassmann algebra needs modification by the addition of an extra axiom that ξ 2 = 0 for ξ odd. 4.3. De Rham cohomology in the smooth case, a comparison. Recall that to a variety V over Zp , considered as a functor from Zp -algebras to sets, we can associate a superspace X V with X V ( B ) = V (0B ). If V is smooth, then we may consider the usual de Rham cohomology of V and compare it to the D RZp (X V ). In general the two are not the same, however if V is projective then they are isomorphic. Lemma 4.11. Let V be a smooth projective variety over Zp , then Hd R (V ) D RZp (X V ). Proof. By Lemma 4.5 we need only compare Hd R (Vˆ p ) with Hd R (V ). The fact that they are isomorphic in the projective case was pointed out to us by A. Ogus, and we provide a sketch of a proof. For the relevant facts about formal schemes and the theorem on formal functions we refer to [4]. By definition, the de Rham cohomology Hd R (V ) is computed as the hypercohomology of the complex •V /Zp , which can be obtained as the cohomology of the total complex associated to the double complex of Zp -modules ⊕ I •V /Zp (U I ), where I s are finite subsets {i 1 , . . . , i s } of {0, . . . , n}, U I = Ui1 ∩ · · · ∩ Uis and U0 , . . . , Un form an open affine cover of V . The horizontal differentials are de Rham differentials and the ˇ vertical ones are Cech differentials. Clearly we have a morphism of double complexes α : ⊕ I •V /Zp (U I ) → ⊕ I (•V /Zp (U I ))
p
and the double complex on the right computes Hd R (Vˆ p ). The above morphism on the E 1 term becomes • α : H • (•V /Zp ) → H • (( V /Zp ) ) p
and it factors as follows • (•V /Zp ) → H • (( H • (•V /Zp ) → H • V /Zp ) ). p
p
j
Recall that projective morphisms preserve coherence and so H i (V /Zp ) is a finitely generated Zp -module. Because Zp is complete, by Theorem 9.7 in [4], for example, we 18 We still require that p M B = 0 for M large, thus the canonical DP ideal is still nilpotent, it need not however be DP nilpotent. 19 Instead of power series with DP one gets an extra condition that the coefficients tend to 0 in Z . p
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have that the first arrow is an isomorphism. The second arrow is an isomorphism by the theorem on formal functions. Since α is an isomorphism on E 1 , it induces an isomorphism ∼
α : Hd R (V ) → Hd R (Vˆ p ).
Given a smooth projective V over Zp we would like to define the action of Fr on its de Rham cohomology. By the above it suffices to do so for D RZp (X V ). As explained in Sect. 3, we have an action of Fr on X V (the neighborhood of X V in Pn ) and so on D RZp ( X V ). Showing that D RZp ( X V ) is isomorphic to D RZp (X V ) would accomplish our goal. Remark. Another consequence of this isomorphism is that the de Rham cohomology of V depends only on V |Fp because that is true of X V . This means in particular that for W (its Grassmann neighborhood smooth projective over Fp , the de Rham cohomology of W in the projective space over Zp ) coincides with the usual crystalline cohomology of W . That is we give a super-geometric construction of the DP envelope of W in PnZp . X V , thus also a natural map Observe that we have i : X V → X V ) → D RZp (X V ) i ∗ : D RZp ( that we will show is an isomorphism. It suffices to prove that i ∗ : X V /Zp → X V /Zp is a quasi-isomorphism of sheaves on V |Fp . Thus the question becomes local and we may assume, after induction on the codimension, that the situation is as follows. Let U ⊂ U be a pair of smooth affine varieties such that U is cut out of U by a function g on U . In this case we will show that X U = X U × pt, i.e. the infinitesimal neighborhood of U in U is a direct product of the p-adic superspaces X U (associated to U ) and our pt. Compare this with the Example following Definition 3.11 where this is discussed in the case when the function g is linear and U is an affine space. The general case is demonstrated below. We will assume that U = Spec A and U = Spec A/g, where g ∈ A. Then unraveling the definitions we see that X U ( B ) = {Hom(A, B )|g → p B + +B } Hom(A/g n , B ) = n
= Homcont ( Aˆ g , B ) = Homcont (A/g[[x]], B ) = Hom(A/g, B ) × ( p B + 0+ B ) = X U ( B ) × pt( B ),
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where Homcont denotes continuous homomorphisms ( Aˆ g is a topological ring and B is equipped with the discrete topology). The inclusion X U ⊂ X U is simply X U = X U × pt ⊂ X U × pt = XU . Next we show that the functions on X U × pt are what was expected, namely if p then: R = (A/g) Theorem 4.12. Any natural transformation f from X U × pt to A1 is given by f (φ, e) =
∞
φ(ri )ei /i!
i=0
for any φ ∈ Hom(A/g, B ) and e ∈ p B + 0+ B , where ri ∈ R depend only on f . Remark. The proof below remains valid for the case when X is given by X ( B ) = Hom(Zp xi , B ), i.e. R = Zp xi . This justifies the induction on the codimension. Proof. Begin by noting that in the proofs of Lemma 4.9 and Theorem 4.10 we can replace Zp by any p-adically complete ring without zero divisors. In particular these p. results remain valid when Zp is replaced by our R = (A/g) Thus let us define a new category (R) consisting of Grassmann rings with coefficients in R-algebras with nilpotent p-action. Denote by pt R and A1R the restrictions of similarly named functors from to (R), so that they are now functors from (R) to Sets. As before, if f is a natural transformation from pt R to A1R , then f (e) =
∞
ri ei /i!,
i=0
where e ∈ p B + 0+ B , and ri ∈ R depend only on f . Observe that any natural transformation f from X U × pt to A1 defines f : pt R → A1R 0+ by f (e) = f (φ, e), where e ∈ p B + B ∈ (R) and φ is the structure morphism. Of course the R-module structure on B provides us with a Zp -morphism φ : R → B → B , however as φ factors through R/ p N R = (A/g)/ p N (A/g) it determines a unique morphism from A/g to B and so an element of X U ( B ). Conversely, any element of X U ( B ) that factors though B makes B into an element of (R). By the above f (e) =
∞
ri ei /i!,
i=0
that is f (φ, e) =
∞
φ(ri )ei /i!
i=0
for all φ : A/g → B that factor through B. Let (φ, e) be an arbitrary element of X U × pt( B ), assume that p N B = 0 so that φ factors through R/ p N R. Let e = pb + bi ei1 ei2 and proceed as in the proof of Theorem 4.10. Define a morphism ϕ from (R/ p N R)[x] to B by
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φ : R/ p N R x ξ2i−1 ξ2i
→ → → →
B b bi ei1 ei2 .
Consider the element (π, w) ∈ X U × pt((R/ p N R)[x] ) where π : A/g → (R/ p N R)[x] is the projection onto R/ p N R ⊂ (R/ p N R)[x] and w = px +
ξ2i−1 ξ2i ,
i
then ϕ(π, w) = (φ, e).20 Thus f (φ, e) = f ϕ(π, w) = ϕ f (π, w) = ϕ
φ(ri )ei /i!. π(ri )wi /i! =
We are finally able to complete the proof of Theorem 3.8 which we restate as a corollary below. Corollary 4.13. The functions on An|k,m are isomorphic as a ring to ⎧ ⎫ ⎨ ⎬ a I,J,T x I ξ J y T /T ! , ⎩ ⎭ I,J,T ≥0
where xi and yi are even and ξi are odd, and a I,J,T ∈ Zp with a I,J,T → 0 as I → ∞. Proof. Using Theorems 4.10 and 4.12 with induction we see that the ring of functions on A0|k,0 is Zp y1 , . . . , yk . Gluing this fact with the proven part of Theorem 3.8 using Theorem 4.12 we obtain the desired result.
Denoting the functions described in Theorem 4.12 by R x and observing that the functions on X U are given by R we are done by the following lemma. Lemma 4.14. The natural map π : Rx → R is a quasi-isomorphism. Proof. In fact we show that the equally natural map ρ : R → Rx is a homotopy inverse. Note that π ◦ ρ = Id R , let F = ρ ◦ π ; we must show that there is a homotopy h such that Id Rx − F = d ◦ h + h ◦ d. It follows immediately from the abstract definitions and by using Theorem 4.12, that any w ∈ sRx can be written uniquely as w=
∞ i=0
αi x /i! + i
∞
βi x i /i!d x,
i=0
20 One should really write X × p t(ϕ)((π, w)), but that is too cumbersome. U
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where αi ∈ sR and βi ∈ s−1 R . Let h(w) = (−1)s−1
∞
βi x i+1 /(i + 1)!,
i=0
then a straightforward calculation shows that h is the desired homotopy.
At this point we have proven the main result of the section, namely: Theorem 4.15. Let V be a smooth projective variety over Zp , let X V be the associated p-adic superspace, and X V the DP neighborhood of V inside Pn , then X V ). Hd R (V ) D RZp ( Corollary 4.16. Let V be a smooth projective variety over Zp , then one has an action of Frobenius on Hd R (V ). Proof. By Lemma 3.13 we have an action of Frobenius on X V and so on its de Rham cohomology, which is isomorphic to Hd R (V ).
Corollary 4.17. Let W be a smooth projective variety over Fp , then the crystalline cohomology of W is isomorphic to the de Rham cohomology of the p-adic superspace (the Grassmann neighborhood of W in Pn ), i.e., W Zp ). Hcr ys (W ) D RZp (W Proof. Let V be any smooth projective lifting of W to Zp , i.e. W = V |Fp . Then by , and Hcr ys (W ) Hd R (V ), X V ). However XV = W Theorem 4.15, Hd R (V ) D RZp ( so we are done.
It is worth noting that the homotopy inverse ρ that was used in the proof of Lemma 4.14 can not be realized (in general) as a restriction of a global map r ∗ : D RZp (X V ) → X V ).21 Geometrically speaking we may not in general have a global retraction D RZp ( X V ) → D RZp (X V ) is an r of X V onto X V ; its existence would ensure that i ∗ : D RZp ( isomorphism of filtered modules with respect to the Hodge filtration. Consequently, the canonical lift of the Frobenius morphism Fr to D RZp (X V ) would preserve the Hodge filtration F • D RZp (X V ). Furthermore, consider the following local computation. Let x be a local function on X V , then Fr (x) = x p + py, where y is some other local function, so that p−1
Fr : f d x1 . . . d xs → p s Fr ( f )(x1
p−1
d x1 + dy1 ) . . . (xs
d xs + dys ),
i.e. under the assumption that a global retraction exists Fr (F s D RZp (X V )) ⊂ p s F s D RZp (X V ). Neither the invariance of the Hodge filtration under Fr nor the p-divisibility estimate need hold in the absence of the global retraction, in Sect. 5 we obtain some weaker p-divisibility estimates that hold in general. 21 In contrast with π which is a restriction of a global map i ∗ : D R ( Zp X V ) → D RZp (X V ).
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5. The Frobenius Map and the Hodge Filtration In this section we essentially follow B. Mazur[11] with some differences in the point of view (that is we find it more conceptual to think of DP ideals and their DP powers). We begin by recalling a definition. Definition 5.1. Let I in A be a DP ideal, then the n th DP power of I , denoted I [n] , is nk n1 the ideal generated by the products x1 /n 1 ! . . . xk /n k ! with xi ∈ I and n i ≥ n. We point out that the Hodge filtration on the de Rham cohomology of X is simply the filtration on the functions on T X given by the DP-powers of the DP ideal I X of X in T X . More precisely: Definition 5.2. For a p-adic superspace X , define a filtration, FH• on X/Zp by setting
FHi X/Zp = I X[i] . This filtration descends to D RZp (X )22 and let us still denote it by FH• there. The following lemma is immediate. Lemma 5.3. Let V be a smooth projective variety over Zp , then the isomorphism Hd R (V ) D RZp (X V ) is compatible with the Hodge filtration on the left and FH• on the right.
Remark. Because of the above lemma we will use the notation FH• to denote also the Hodge filtration on Hd R (V ). X V ) not However the Frobenius map that we are interested in is defined on D RZp ( D RZp (X V ). And while the two are isomorphic as shown previously, this isomorphism is not compatible with FH• . To fix this problem we proceed as follows: replace the FH• filtration on D RZp ( X V ) which is given by the ideal of X in T X with the one given by the DP-powers of the ideal of X itself in T X. X) Definition 5.4. Let X ⊂ X ⊂ T X be as above, define a filtration FD• P on D RZp ( as the induced filtration from X /Zp , where [i] FDi P X /Zp = I X .
In the particular case, namely the setting of Lemma 4.14 (that is the key step in proving the general case), the definition above becomes: FD• P Rx is defined by w ∈ F s Rx if w=
∞ i=0
αi x i /i! +
∞
βi x i /i!d x,
i=0
s−i s−1−i and βi ∈ . It is then not hard to show (using the observation where αi ∈ R R that the homotopy of Lemma 4.14 preserves the new filtration) that i ∗ : D RZp ( XV ) → D RZp (X V ) is an isomorphism of filtered modules, where D RZp ( X V ) is endowed with the new filtration FD• P and D RZp (X V ) has the old filtration FH• . Thus we have a refinement of Theorem 4.15. 22 The action of A0,1 on T X preserves X , thus the differential preserves I and its DP powers. X
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Theorem 5.5. Let V be a smooth projective variety over Zp , let X V be the associated p-adic superspace, and X V the DP neighborhood of V inside Pn , then XV ) Hd R (V ) D RZp ( is an isomorphism of filtered modules with the Hodge filtration on the left and FD• P on the right. Though Fr does not preserve it, FD• P is useful in the computation of divisibility 23 , where I [k] is the k th DP power of the estimates. Notice that Fr (I X[k] ) ⊂ p [k] O X X n ideal (also denoted by I X ) of X in X and [k] = minn k ord p ( pn! ) (thus ( p)[k] = ( p [k] )). Recalling the discussion at the end of Sect. 4.3, we see that Fr (FDs P D RZp ( X V )) ⊂ p [s] D RZp ( X V ). In particular if p > dim(V ) then the square brackets can be removed from s and we obtain the following lemma. Lemma 5.6. Let V be a smooth projective variety over Zp such that p > dim(V ), then Fr (FHs Hd R (V )) ⊂ p s Hd R (V ). A slightly finer statement can be derived from the above observations, one actually has j Fr (FDs P D RZp ( X V )) ⊂ p [s− j] Fr (FD P D RZp ( X V )) + p s D RZp ( XV ) j<s
⊂ p Fr (D RZp ( X V )) + p s D RZp ( X V ). The latter was sufficient for Mazur to establish a conjecture of Katz. By analogous reasoning one can introduce new filtrations on the cohomologies of X and X by considering the DP ideal of X |Fp in T X and the DP ideal of X |Fp in T X. The canonical isomorphism is now an isomorphism of filtered modules with respect to these new filtrations and they are preserved by Fr . The new filtration on D RZp (X V ) contains the Hodge filtration and satisfies the same divisibility conditions. In fact it can be easily described as follows (let us work with Hd R (V ) since it is the same as D RZp (X V )). Let FH• denote the usual Hodge filtration on Hd R (V ), then the new filtration FN• can be described thus: FNn Hd R (V ) = s+t n p [s] FHt Hd R (V ). 6. Appendix We investigate a property of a p-adic superspace that we call prorepresentability. It allows us to pass from particular examples that we considered in this paper (namely p-adic superspaces that arise in dealing with usual varieties over Zp ) to a more general class of p-adic superspaces that nevertheless share a lot of properties with our examples. 23 A very useful formula to keep in mind is ord n! = p
∞ n n − S(n) = , where S(n) is the sum of p−1 pi i=1
digits in the p-adic expansion of n.
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Recall that we have defined a map of p-adic superspaces as a map of the defining functors. It is well known (Yoneda Lemma) that the set Hom(F, G) of natural transformations from a functor F to a functor G can be easily calculated24 if F is representable, i.e. F is isomorphic to the functor h A , where h A (X ) = Hom(A, X ). Namely, in this case we have Hom(F, G) = G(A). Definition 6.1. We say that a functor F is prorepresentable if it is isomorphic to a colimit of representable functors: F = lim h A − → A∈D
for some diagram D. Thus Hom(F, G) = lim G(A), ← − A∈D
i.e. is a limit of the sets G(A), by the Yoneda Lemma and continuity of Hom(−, −). Notice that in the above definitions we use a general definition of limits and colimits; a concise reference is [10]. It is important to emphasize that the definition of a prorepresentable functor in [13] is much more restrictive. Definition 6.2. We say that a p-adic superspace X is prorepresentable, if its defining functor is locally prorepresentable, i.e. [X ] has a cover by open Ui such that the defining functors of X |Ui are prorepresentable. One can show that the property of a p-adic superspace X being prorepresentable is preserved by passing to the odd tangent space T X . Theorem 6.3. If a p-adic superspace X is prorepresentable then so is T X . Proof. This statement is local, so we may assume that X is prorepresentable as a functor. Denote by Fξ the endo-functor of that takes an object A ∈ to A[ξ ], i.e. adjoins an odd variable. Note that T X = X ◦ Fξ . Observe that Fξ extends to Super . Super is closed under limits and Fξ commutes with limits since it has a left adjoint •− . Thus we may assume that X is representable and since •− descends to we are done.
One can show that all the p-adic superspaces we consider in this paper are prorepresentable in our sense. Here we give a detailed proof of this fact for the most important functor pt, which is indeed prorepresentable (not just locally prorepresentable). In a similar way one can prove the local prorepresentability of functors corresponding to other superspaces considered in the present paper. As an application we show how to describe the functions on pt (Theorem 4.10) using the above ideas. 24 The source category does not matter and the target category is Sets.
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2n , i.e. a supercommutative ring obtained from a commutative Consider the ring C[ξi ]i=1 ring C by adjoining 2n odd anticommuting variables ξ j . Let G be the group acting on 2n generated by C[ξi ]i=1
ξ2k−1 → ξ2k , ξ2k → −ξ2k−1 and ξ2k−1 , ξ2k → ξ2k −1 , ξ2k , ξ2k −1 , ξ2k → ξ2k−1 , ξ2k . 2n is spanned by w k /k! for Lemma 6.4. The subring of G-invariants in C[ξi ]i=1 n k = 0, . . . , n, where wn = ξ1 ξ2 + · · · + ξ2n−1 ξ2n . That is 2n G ) = C y /(y)n+1 . (C[ξi ]i=1
Proof. Proceed by induction on n. For n = 0 there is nothing to prove. Let it be true for n. Note that 2(n+1)
C[ξi ]i=1
2n 2n 2n 2n = C[ξi ]i=1 ⊕ C[ξi ]i=1 ξ2n+1 ⊕ C[ξi ]i=1 ξ2n+2 ⊕ C[ξi ]i=1 ξ2n+1 ξ2n+2 .
2(n+1)
Let x ∈ C[ξi ]i=1 ξ2n+2 ensures that
be G-invariant, then the element of G that “switches” ξ2n+1 and 2n 2n ⊕ C[ξi ]i=1 ξ2n+1 ξ2n+2 . x ∈ C[ξi ]i=1
2n only and using the induction hypoConsidering the part of G that acts on C[ξi ]i=1 thesis we see that x= ak wnk /k! + bk−1 wnk−1 /(k − 1)!ξ2n+1 ξ2n+2 .
Since the action of G is degree preserving each homogeneous component of x is also G invariant, thus ak wnk /k! + bk−1 wnk−1 /(k − 1)!ξ2n+1 ξ2n+2 is G-invariant. The element of G that switches ξ1 , ξ2 and ξ2n+1 , ξ2n+2 ensures that ak = bk−1 , so that x=
k /k!. ak wn+1
Observe that if a group G acts on a set X , then the subset of fixed points X G ⊂ X can be represented as a limit of the diagram in Sets consisting of two copies of X and the arrows given by the elements of G. This is where Lemma 6.4 is used in Theorem 6.5 below.
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Theorem 6.5. The functor defining the p-adic superspace pt is prorepresentable. Proof. Consider the diagram in consisting of objects 2m 2m {(Zp / p n Zp )[ξi ]i=1 |m, n ≥ 0} {(Zp / p n Zp )[ξi ]i=1 |m, n ≥ 0} with morphisms between the copies given by elements of G, and the rest of the morphisms given by the usual projections
2m 2m (Zp / p n Zp )[ξi ]i=1 → (Zp / p n Zp )[ξi ]i=1
for n > n and
2m 2m → (Zp / p n Zp )[ξi ]i=1 (Zp / p n Zp )[ξi ]i=1
for m > m mapping the extra {ξ j } j>2m to 0. It follows from Lemma 6.4 that while the limit of the above diagram does not exist in , it exists in the category Super , where is a full subcategory, and it is equal to the ring with divided powers Zp x. Correspondingly, after passing to the category of functors from to Sets, the functor pt = Hom Super (Zp x , −) is seen to be the colimit of representable functors.
Remark. To summarize the above proof, the main ingredient is the observation that the functor pt extends to Super where it is representable. Furthermore, the representing object Zp x is a limit of a diagram of objects in . This representing diagram is not unique, however it does not prevent us from easily proving Theorem 4.10, i.e. computing the functions on pt below. Corollary 6.6. The functions on the p-adic superspace pt are Zp x. Acknowledgements. We are deeply indebted to V. Vologodsky for his help with understanding of the standard approach to the construction of the Frobenius map and to M. Kontsevich, N. Mazzari and A. Ogus for interesting discussions. The second author would also like to thank the Max Planck Institute for Mathematics in Bonn where part of the work on this paper was completed. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References 1. Berthelot, P., Ogus, A.: Notes on crystalline cohomology. Princeton, N.J.: Princeton University Press Tokyo: University of Tokyo Press, 1978 2. Deligne, P., et al: Quantum fields and strings: a course for mathematicians. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997, Providence, RI: Amer. Math. Soc. Princeton, NJ: Institute for Advanced Study (IAS), 1999 3. Freund, P.G.O.: p-adic Strings and their Applications. http://arXiv.org/list/hep-th/0510192, 2005 4. Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, No. 52. New York-Heidelberg: Springer-Verlag, 1977 5. Illusie, L.: Report on crystalline cohomology. In: Algebraic geometry (Proc. Sympos. Pure Math., vol. 29, Humboldt State Univ., Arcata, Calif., 1974), Providence, RI: Amer. Math. Soc., 1975, pp. 459–478
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6. Jantzen, J.C.: Representations of algebraic groups. Second edition. Mathematical Surveys and Monographs, 107. Providence, RI: Amer. Math. Soc., 2003 7. Konechny, A., Schwarz, A.: Theory of (k ⊕ l|q)-dimensional supermanifolds. Selecta Math. (N.S.) 6(4), 471–486 (2000) 8. Kontsevich, M., Vologodsky, V., Schwarz, A.: Integrality of instanton numbers and p-adic B-model. Phys. Lett. B 637, 97–101 (2006) 9. Manin, Y.: Gauge field theory and complex geometry. Translated from the 1984 Russian original by N. Koblitz and J. R. King. Second edition. With an appendix by Sergei Merkulov. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 289, Berlin: Springer-Verlag, 1997 10. May, J.P.: A concise course in Algebraic topology, Chicago Lectures in Mathematics, Chicago, IL: Univ of Chicago Press, 1999 11. Mazur, B.: Frobenius and the Hodge filtration (estimates). Ann. of Math. (2) 98, 58–95 (1973) 12. Ogus, A.: F-isocrystals and de Rham cohomology II – Convergent isocrystals. Duke Math. J. 51(4), 765–850 (1984) 13. Schlessinger, M.: Functors of Artin rings. Trans. Amer. Math. Soc. 130, 208–222 (1968) 14. Schwarz, A., Shapiro, I.: Supergeometry and Arithmetic Geometry. Nucl. Phys. B, 756, Issue 3, 207–218 15. Schwarz, A.: On the definition of superspace. Theoret. and Math. Phys. 60(1), 657–660 (1984) 16. Vologodsky, V.: Integrality of instanton numbers. http://arXiv.org/list/:0707.4617, 2007 Communicated by A. Connes
Commun. Math. Phys. 282, 115–160 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0528-z
Communications in
Mathematical Physics
Braided Hopf Algebras Obtained from Coquasitriangular Hopf Algebras Margaret Beattie1 , Daniel Bulacu2 1 Department of Mathematics and Computer Science, Mount Allison University,
Sackville, NB E4L 1E6, Canada. E-mail: [email protected]
2 Faculty of Mathematics and Informatics, University of Bucharest, Str. Academiei 14,
RO-010014 Bucharest 1, Romania. E-mail: [email protected] Received: 20 April 2007 / Accepted: 9 March 2008 Published online: 13 June 2008 – © Springer-Verlag 2008
Abstract: In this paper, we study the generalized quantum double construction for paired Hopf algebras with particular attention to the case when the generalized quantum double is a Hopf algebra with projection. Applying our theory to a coquasitriangular Hopf algebra (H, σ ), we see that H has an associated structure of braided Hopf algebra cop in the category of Yetter-Drinfeld modules over Hσ , where Hσ is a subHopf algebra of 0 H , the finite dual of H . Specializing to the quantum group H = SLq (N ), we find that Hσ is Uqext (sl N ), so that the duality between these quantum groups is just the evaluation map. Furthermore, we obtain explicit formulas for the braided Hopf algebra structure of SLq (N ) in the category of left Yetter-Drinfeld modules over Uqext (sl N )cop . 1. Introduction Many seemingly diverse areas of mathematics and of physics are connected by the theory of quantum groups. Classical quantum groups arose from deformations of enveloping algebras of semisimple Lie algebras and of coordinate Hopf algebras of Lie groups. An early example in the first case above was the algebra Uq (sl2 ) which appeared a quarter of a century ago in work of Kulish and Reshetikhin [13] with its Hopf algebra structure explained by Skylanin [22]. The Drinfeld-Jimbo algebras Uq (g) are the generalization to any complex semisimple Lie algebra g [6,7,10]. In the second case, the first motivating example is SLq (2). For N > 2, the quantum groups SLq (N ) are constructed thus. Given a solution to the Yang-Baxter equation on a finite dimensional vector space, the FRT construction yields a quadratic or coquasitriangular bialgebra, for example, the bialgebra Mq (N ) [9]. These yield as quotients the coquastriangular quantum groups SLq (N ). The The second author held a postdoctoral fellowship at Mount Allison University from 2005 to 2007 and would like to thank Mount Allison for their warm hospitality. Support for the first author’s research and partial support for the postdoctoral position of the second author came from an NSERC Discovery Grant. The second author now holds research support from Grant 434/1.10.2007 of CNCSIS.
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duality between these quantum groups is well-known [7,25,26]; this duality will again be recovered in a natural way from the approach in this paper. The general results in this paper lead not only to some properties of the couple (Uq (sl N ), SLq (N )) well known in quantum group theory, but also to some new ones. In addition, we will see that SLq (N ) has a braided Hopf algebra structure in the category of left Yetter-Drinfeld modules over Uqext (sl N )cop which coincides with that of the image of the transmutation object of SLq (N ) through a braided monoidal functor that will be explicitly computed, and the corresponding Radford biproduct is isomorphic to the generalized quantum double associated to this dual pair. We next outline our approach. The Drinfeld double construction plays an important role in the theory of minimal quasitriangular Hopf algebras since the Drinfeld double of a finite dimensional Hopf algebra is minimal quasitriangular. Conversely, any minimal quasitriangular Hopf algebra is a quotient of a Drinfeld double [19]. This second statement is a consequence of the following facts. If (H, R) is a quasitriangular Hopf algebra then from an expression for R ∈ H ⊗ H of minimal length, one can define two finite dimensional sub-Hopf algebras of H , denoted by R(l) and R(r ) , respectively, and there is a quasitriangular Hopf algebra morphism between the Drinfeld double of R(l) and H whose image is H R = R(l) R(r ) , the minimal quasitriangular Hopf algebra associated to (H, R). Note that there is an ∗cop isomorphism R(l) ∼ = R(r ) and this isomorphism, together with the evaluation map, gives a skew pairing from R(r ) ⊗ R(l) to k. Now suppose that (H, σ ) is a coquasitriangular Hopf algebra, not necessarily finite dimensional. The analogues of the Hopf algebras R(l) and R(r ) , denoted by Hl and Hr , are, in general, infinite dimensional sub-Hopf algebras of H 0 , the finite dual of H , and Hr Hl = Hl Hr is a sub-Hopf algebra of H 0 denoted by Hσ . In general, there is no ∗cop analogue to the isomorphism R(l) ∼ = R(r ) , but the coquasitriangular map σ induces a skew pairing on Hr ⊗ Hl . Although the classical Drinfeld double is not available in this setting, there exists a generalized quantum double, in the sense of Majid [17] or Doi and Takeuchi [5], for Hopf algebras in duality, so that the skew pairing between Hr and Hl gives a generalized quantum double, denoted D(Hr , Hl ). In general, D(Hr , Hl ) has no (co)quasitriangular structure (see Proposition 2.8 and Remark 5.4); however, there is a surjective Hopf algebra morphism from D(Hr , Hl ) to Hσ given by multiplication in H 0 . Note that, if (H, σ ) is finite dimensional then Hσ = H R∗ , the minimal quasitriangular Hopf algebra associated to (H ∗ , R), where R is the matrix of H ∗ obtained from σ . Thus, in the finite dimensional case, (Hσ , R) is quasitriangular. In the infinite dimensional case, neither Hσ nor H 0 is guaranteed a quasitriangular structure. Drinfeld [8] observed that for a finite dimensional quasitriangular Hopf algebra H , its double D(H ) is a Hopf algebra with a projection. Majid [14] proved that the converse of Drinfeld’s result also holds, and computed on H ∗ the braided Hopf algebra structure associated to this projection [20]. Here, for (H, σ ) coquasitriangular but not necessarily finite dimensional, using the evaluation pairing, we construct the generalized quantum cop cop double D(Hσ , H ). In the finite dimensional case, Hσ = (H R∗ )cop has a quasitriangucop cop lar structure and there is a Hopf algebra morphism from D(Hσ , H ) to Hσ covering cop the natural inclusion, so D(Hσ , H ) is a Hopf algebra with a projection. We show that, cop for H not necessarily finite dimensional, D(Hσ , H ) is still a Hopf algebra with projection, and we describe explicitly the induced structure on H of a braided Hopf algebra cop in the category of left Yetter-Drinfeld modules over Hσ . This paper is organized as follows. In Sect. 2, Preliminaries, we first define pairings and skew pairings on Hopf algebras and give the construction of the generalized quantum
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double. Coquasitriangular Hopf algebras give important examples of Hopf algebras with a skew pairing. As well, we describe the structure of a Hopf algebra A with projection (see [20]), so that A is isomorphic to B × K, where K is a sub-Hopf algebra of A, B is a braided Hopf algebra in the category of left Yetter-Drinfeld modules over K and × indicates the Radford biproduct. In Sect. 3, we first work in a rather general context. For U, V bialgebras such that there is an invertible pairing from U ⊗ V to k, we show that there is a projection from D(U cop , V ) to U cop covering the natural inclusion if and only if there is a bialgebra morphism γ : V → U cop satisfying a relation analogous to the coquasitriangularity condition for a skew pairing. (See also [5] and [21].) Now if U, V are Hopf algebras with an invertible pairing and γ exists as above, then V has a Hopf algebra structure in the category of left Yetter-Drinfeld modules over U cop . We describe this structure explicitly. For example, if (U, R) is quasitriangular, then such a map γ exists. Then, for H finite dimensional we prove that there is a projection π : D(H ) = D(H ∗cop , H ) → H ∗cop covering the inclusion H ∗cop → D(H ) if and only if H is coquasitriangular (Proposition 3.9). A fortiori, H has a braided Hopf algebra structure. Moreover, this structure allows us to obtain the left version of the transmutation theory for coquasitriangular Hopf algebras (Remark 4.3). The transmutation theory for coquasitriangular Hopf algebras is due to Majid [15]. Using the dual reconstruction theorem, he associated to any coquasitriangular Hopf algebra (H, σ ), a braided commutative Hopf algebra H in the category of right H -comodules, called the function algebra braided group associated to H . In Sect. 4.1 we show that H can be obtained by writing a generalized quantum double as a Radford biproduct using the projection π : D(H, H ) → H given by multiplication in H . To this projection corresponds a braided Hopf algebra H in the category of left H -comodules. By considering the co-opposite case, we obtain, after some identification, that (H cop )cop = H , as braided Hopf algebras (Proposition 4.2). cop In Sect. 4.2, we describe the “dual case”, and show that D(Hσ , H ) is always a Hopf algebra with a projection. This follows from a more general construction where we take A and X to be sub-Hopf algebras of H , Hr A and Hl X the corresponding sub-Hopf algebras of Hr and Hl respectively, and H X,A the corresponding sub-Hopf algebra of Hσ . Then cop cop there is a Hopf algebra projection from D(H X,A , X ) to H X,A which covers the inclusion. We stress the fact that this “dual case” gives rise to a new braided Hopf algebra structure cop on X (denoted by X ) in the category of left H X,A Yetter-Drinfeld modules. This new construction cannot be viewed as an example of the transmutation theory because the transmutation theory associates to an ordinary coquasitriangular Hopf algebra a braided Hopf algebra in the category of corepresentations over itself. We show that, in fact, the two constructions above are related by a non-canonical braided functor. More exactly, cop
there exists a braided functor F : M X →
H X,A cop
H X,A
Y D such that F(X ) = X (Theorem 4.10).
Nevertheless, we think it worthwhile to have the construction of X . This is firstly because cop X lies in a category of left Yetter-Drinfeld modules over H X,A which, in general, may not have a quasitriangular structure. (See the example in Sect. 6.) Secondly, this is because evaluation gives a duality between H X,A and X and the associated quantum double, cop cop D(H X,A , X ), is isomorphic to the Radford biproduct X × H X,A . In Sect. 5 we present the finite dimensional case in full detail, linking the results of this paper to those of Radford described above. Also, we note that starting with a finite dimensional quasitriangular Hopf algebra (H, R) and taking A = X = H ∗ then the corresponding braided Hopf algebra H ∗ is precisely the categorical dual of H cop , the
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associated enveloping algebra braided group of (H cop , R21 ) constructed by Majid in [16] (Remarks 5.8). Finally, in Sect. 6, we apply the constructions of Sect. 4.2 to the coquasitriangular Hopf algebra H := SLq (N ). By direct computations we show that Hl and Hr are the Borel-like Hopf algebras Uq (b+ ) and Uq (b− ), respectively, associated to Uqext (sl N ), and obtain that Hσ = Uqext (sl N ). The general theory above leads to more conceptual proofs for some well known results. Namely, there exist dual pairings of the pairs of Hopf algebras Uq (b+ )cop and Uq (b− ), Uqext (sl N ) and SLq (N ), and Uq (sl N ) and SLq (N ), respectively (Corollary 6.11 and Corollary 6.14), and Uqext (sl N ) and Uq (sl N ) are factors of generalized quantum doubles (Corollary 6.12 and Remark 6.13). Also, the Hopf algebra structure of Uq (sl N ) is achieved in a natural way (Remark 6.9 and Remark 6.13). The rest of Sect. 6 is dedicated to a description of the braided Hopf algebra structure of SLq (N ) in the category of left Uqext (sl N )cop Yetter-Drinfeld modules. The explicit formulas for this braided Hopf algebra structure can be found in Proposition 6.16 and Theorem 6.17. 2. Preliminaries Throughout, we work over a field k and maps are assumed to be k-linear. Any unexplained definitions or notation may be found in [3,11,17,18 or 23]. For B a k-bialgebra, we write the comultiplication in B as (b) = b1 ⊗ b2 for b ∈ B. For M a left B-comodule, we write the coaction as ρ(m) = m−1 ⊗ m0 . For M a right B-comodule, we will use the subscript bracket notation to differentiate subscripts from those in comultiplication expressions, i.e. we will write ρ(m) = m(0) ⊗ m(1) for m ∈ M. For k-spaces M and N, tw will denote the usual twist map from M ⊗ N to N ⊗ M.
2.1. Pairings on Hopf algebras and the generalized quantum double. We first recall the definition for two bialgebras or Hopf algebras to be in duality (see [26, Sect. 1] or [17, Sect. 1.4]); this notion was first introduced by Takeuchi and called a Hopf pairing. Throughout this section U and V will denote bialgebras over k. Recall [18, Section 9.1] that the finite dual U 0 of a bialgebra U is the subalgebra of functions f ∈ U ∗ which vanish on a right or on a left ideal of U of finite codimension, and is a bialgebra. Definition 2.1. A bilinear form , : U ⊗ V → k is called a pairing of U and V if mn, x = m, x1 n, x2 , m, x y = m 1 , xm 2 , y, 1, x = ε(x), m, 1 = ε(m),
(2.1) (2.2) (2.3)
for all m, n ∈ U and x, y ∈ V . Then U and V are said to be in duality. Remarks 2.2. (i) A bilinear form from U ⊗ V to k is called a skew pairing if it is a pairing from U cop ⊗ V to k or, equivalently, a pairing from U ⊗ V op to k. (ii) The form , is a pairing of U and V if and only if there is a bialgebra morphism φ : U → V 0 , defined by φ(u)(v) = u, v, if and only if there exists a bialgebra morphism ψ : V → U 0 defined by ψ(v)(u) = u, v. If U and V are Hopf algebras, these maps are Hopf algebra maps.
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(iii) If U and V are Hopf algebras with antipodes SU , SV respectively, and , is a pairing of U and V , then this bilinear form is invertible in the convolution algebra Hom(U ⊗ V, k) and its inverse is the bilinear form which maps u ⊗ v to SU (u), v = u, SV (v) . (iv) If , is a (skew) pairing between U and V , then there is a (skew) pairing between the sub-bialgebras φ(U ) ⊆ V 0 and ψ(V ) ⊆ U 0 defined by the bilinear form B : φ(U ) ⊗ ψ(V ) → k, B(φ(u), ψ(v)) = u, v. The form B is well-defined since φ(u) = φ(u ) if and only if u, − = u , − and ψ(v) = ψ(v ) if and only if −, v = −, v . It is straightforward to check that B is a (skew) pairing. (v) If , is a pairing of the bialgebras U and V and U ⊆ U , V ⊆ V are sub-bialgebras then the restriction of , to U⊗V is a pairing of U and V. As above, the bialgebras φ(U) and ψ(V) are also in duality. Example 2.3. For H a bialgebra, then the evaluation map provides a duality between H 0 , the finite dual of H , and H , that is, f, h = f (h) for all f ∈ H 0 and h ∈ H . Another class of examples for bialgebras in duality is provided by coquasitriangular bialgebras, also called braided bialgebras. This concept is dual to the idea of quasitriangular bialgebras (see [11 or 17] for the definition). We recall the notion of coquasitriangularity. Definition 2.4. A bialgebra H is called coquasitriangular (CQT for short) if there exists a convolution invertible k-bilinear skew pairing σ : H ⊗ H → k, i.e., for all h, h , g ∈ H, σ (hh , g) = σ (h, g1 )σ (h , g2 ), σ (g, hh ) = σ (g2 , h)σ (g1 , h ), σ (1, h) = σ (h, 1) = ε(h),
(2.4) (2.5) (2.6)
which also satisfies the coquasitriangular condition, σ (h 1 , h 1 )h 2 h 2 = h 1 h 1 σ (h 2 , h 2 ).
(2.7)
Remarks 2.5. (i) Doi [4] showed that for (H, σ ) a CQT Hopf algebra, one may define an invertible element v ∈ H ∗ by v(h) = σ (h 1 , S(h 2 )) with inverse v −1 (h) = σ (S 2 (h 1 ), h 2 ), and
S 2 (h) = v −1 (h 1 )h 2 v(h 3 ), ∀ h ∈ H. H∗
(2.8) (2.9)
Similarly, the element u ∈ defined by u(h) = σ (h 2 , S(h 1 )), is invertible with inverse u −1 (h) = σ (S 2 (h 2 ), h 1 ) and also defines the square of the antipode S of H as a co-inner automorphism of H , i.e., S 2 (h) = u(h 1 )h 2 u −1 (h 3 ), for all h ∈ H . In particular, the antipode S is bijective. (ii) Let (H, σ ) be a CQT bialgebra. Then σ is a convolution invertible pairing between H and H op and also between H cop and H . (iii) Clearly any sub-bialgebra of a CQT bialgebra (H, σ ) is CQT.
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(iv) If (H, σ ) is CQT, then so are (H op , σ −1 ) and (H cop , σ −1 ) where, if H is a Hopf −1 ) is the convolution inverse algebra, σ −1 = σ ◦ (S H ⊗ I d H ) = σ ◦ (I d H ⊗ S H −1 −1 of σ [4, 1.2]. Moreover, (H, σ21 = σ ◦ tw) is another CQT structure for H , so that (H op , σ21 := σ ◦ tw) and (H cop , σ21 ) are CQT also. Let B, H be bialgebras and ℘ an invertible skew pairing ℘ : B ⊗ H → k. We now define the bialgebra (Hopf algebra) structure on B ⊗ H to be studied in this paper. We follow the presentation in [5]. For A a bialgebra, an invertible bilinear form τ on A is called a unital 2-cocycle if for all a, b, c ∈ A, τ (a1 , b1 )τ (a2 b2 , c) = τ (b1 , c1 )τ (a, b2 c2 ) and τ (a, 1) = τ (1, a) = ε(a). If τ is a 2-cocycle on A, then we may form Aτ , the bialgebra which has coalgebra structure from A but with the new multiplication a • b = τ (a1 , b1 )a2 b2 τ −1 (a3 , b3 ). It is shown in [5] that since ℘ is an invertible skew pairing on B ⊗ H, then the bilinear form on B ⊗ H defined by τ (b ⊗ h, b ⊗ h ) = ε(b)℘ (b , h)ε(h ) is a unital 2-cocycle. Then we may form the bialgebra (B ⊗ H)τ . As a coalgebra, (B ⊗ H)τ = B ⊗ H, the unit is 1 ⊗ 1, and the multiplication is defined by (b ⊗ h)(b ⊗ h ) = ℘ (b 1 , h1 )℘ −1 (b 3 , h3 )bb 2 ⊗ h2 h .
(2.10)
For B, H Hopf algebras with bijective antipodes, the inverse of ℘ as a skew-pairing −1 on B ⊗ H is given by ℘ −1 (b, h) = ℘ (SB(b), h) = ℘ (b, SH (h)). Then (B ⊗ H)τ , also often denoted by B τ H, has antipode given by S(b ⊗ h) = (1 ⊗ SH(h)) • (SB(b) ⊗ 1) = ℘ (SB(b3 ), SH(h3 ))℘ −1 (SB(b1 ), SH(h1 ))SB(b2 ) ⊗ SH(h2 ).
(2.11)
This construction is a special case of the generalized quantum double (see [17, Chap. 7]) for matched pairs of Hopf algebras. Definition 2.6. We denote (B ⊗ H)τ by D(B, H) and will refer to this bialgebra (Hopf algebra) with multiplication as in (2.10), coalgebra structure from the tensor product B ⊗ H, and antipode as in (2.11) as a generalized quantum double. In fact, in many of our constructions, we will begin with a pairing , : U ⊗ V → k which is then a skew pairing ρ from U cop ⊗ V → k. Then, with subscripts still written in U , since SU−1 (m), SV (x) = m, x, the formulas for the multiplication, comultiplication and antipode in D(U cop , V ) are given by (m ⊗ x)(n ⊗ y) = n 3 , x1 SU−1 (n 1 ), x3 mn 2 ⊗ x2 y, (m ⊗ x) = (m 2 ⊗ x1 ) ⊗ (m 1 ⊗ x2 ), S(m ⊗ x) = m 1 , x3 m 3 , SV−1 (x1 )SU−1 (m 2 ) ⊗ SV (x2 ).
(2.12) (2.13) (2.14)
Example 2.7. (cf. [17, 7.2.5]). If H is a finite dimensional Hopf algebra, then H ∗ and H are in duality via the evaluation map as mentioned above and the double D(H ∗cop , H ) is the usual Drinfeld double, denoted in this case by D(H ).
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It is shown in [17] that if (H, σ ) is CQT, so is the double (H ⊗ H )τ = D(H, H ), where τ is the unital 2-cocycle on H ⊗ H defined by the skew-pairing σ . The next proposition shows that this generalizes to D(B, H), where B, H are CQT Hopf algebras. Proposition 2.8. Let B, H be Hopf algebras, ℘ : B ⊗ H → k a skew pairing, and D = D(B, H) = (B ⊗ H)τ the generalized quantum double of Definition 2.6. Then D is CQT if and only if B and H are. Proof. Since sub-Hopf algebras of a CQT Hopf algebra are CQT, then if D is CQT, so are B and H. Now suppose that (B, σB) and (H, σH) are CQT. Then we form the CQT Hopf algebra (A, σ ) = B ⊗ H, (σB ⊗ σH) ◦ (I dB ⊗ tw ⊗ I dH) . From the above discussion, τ : A ⊗ A → k is a unital 2-cocycle, where τ (b ⊗ h, b ⊗ h ) = ε(b)℘ (b , h)ε(h ). Then by [17, p.61 (2.24)], Aτ is CQT via the bilinear form ω defined by ω(a, a ) = τ (a 1 , a1 )σ (a2 , a 2 )τ −1 (a3 , a 3 ). Specifically, (D = Aτ , ω) is coquasitriangular where ω(b ⊗ h, b ⊗ h ) = ℘ (b1 , h 1 )σB(b2 , b 1 )σH(h1 , h 2 )℘ (SB(b 2 ), h2 ). (By using (2.7) for ℘ generously, one can even check the coquasitriangularity conditions (2.4) to (2.7) for ω directly.) Next we show how the doubles D(U cop , V ) and D(φ(U cop ), ψ(V )) are related. Lemma 2.9. Let U, V be Hopf algebras and , a pairing of U and V . Then φ(U ) ⊆ V 0 and ψ(V ) ⊆ U 0 are also Hopf algebras with a pairing B on φ(U ) ⊗ ψ(V ) defined by B(φ(u), ψ(v)) = u, v. Then φ ⊗ ψ : D(U cop , V ) → D(φ(U cop ), ψ(V )) is a surjection of Hopf algebras. Proof. From Remarks 2.2 and the fact that φ ⊗ ψ : U cop ⊗ V → φ(U cop ) ⊗ ψ(V ) is a Hopf algebra surjection, it remains only to show that φ ⊗ ψ respects the multiplication in the double. This can be checked by a straightforward computation, or by noting that for D(U cop , V ) = (U cop ⊗ V )τ and D(φ(U cop ), ψ(V )) = (φ(U cop ) ⊗ ψ(V ))τ for cocycles τ, τ as above, then τ (m ⊗ x, n ⊗ y) = τ (φ(m) ⊗ ψ(x), φ(n) ⊗ ψ(y)). 2.2. Hopf algebras with projection. Let K be a bialgebra. Recall that a left Yetter-Drinfeld module over K is a left K-module M which is also a left K-comodule, such that the following compatibility relation holds. For all κ ∈ K and m ∈ M: κ1 m−1 ⊗ κ2 · m0 = (κ1 · m)−1 κ2 ⊗ (κ1 · m)0 ,
(2.15)
where K ⊗ M κ ⊗ m → κ · m ∈ M is the left K-action. The category of left Yetter-Drinfeld modules over K and k-linear maps that preserve the K-action and K-coaction is denoted by K K Y D. The category K Y D is pre-braided. If M, N ∈ K K K Y D then M ⊗ N is a left Yetter-Drinfeld module over K via the structures defined by κ · (m ⊗ n) = κ1 · m ⊗ κ2 · n and m ⊗ n → m−1 n−1 ⊗ m0 ⊗ n0 ,
(2.16)
for all κ ∈ K, m ∈ M and n ∈ N. The pre-braiding is given by cM,N(m ⊗ n) = m−1 · n ⊗ m0 .
(2.17)
If K is a Hopf algebra then c is invertible, so K K Y D is a braided monoidal category.
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The structure of a Hopf algebra with projection was given in [20]. More precisely, if i K and A are Hopf algebras with Hopf algebra maps K A such that π ◦ i = I dK , π
then there exists a braided Hopf algebra B in the category of left Yetter-Drinfeld modules K Y D such that A ∼ B × K as Hopf algebras, where B × K denotes Radford’s biproduct = K between B and K (for more details see [20]). As k-vector space B = {a ∈ A | a1 ⊗ π(a2 ) = a ⊗ 1}. Now, B is a K-module subalgebra of A where A is a left K-module algebra via the left adjoint action induced by i, that is κ i a = i(κ1 )ai(S(κ2 )), for all κ ∈ K and a ∈ A. Moreover, B is an algebra in the braided category K K Y D, where the left coaction of K on B is given for all b ∈ B by λ B (b) = π(b1 ) ⊗ b2 . (2.18) Also, as k-vector space, B is the image of the k-linear map : A → A defined for all a ∈ A by (a) = a1 i(S(π(a2 ))). (2.19) For all a ∈ A, we define ((a)) = (a1 ) ⊗ (a2 ).
(2.20)
K This makes B into a coalgebra in K K Y D and a bialgebra in K Y D. The counit of B is K ε = ε | B . Moreover, B is a braided Hopf algebra in K Y D with antipode S given by
S(b) = i(π(b1 ))SA(b2 ),
(2.21)
where SA is the antipode of A. The Hopf algebra isomorphism χ : B × K → A is given by χ (b × κ) = bi(κ),
(2.22)
for all b ∈ B and κ ∈ K. Note that the description of a Hopf algebra with a projection in terms of a braided Hopf algebra is due to Majid [16]. 3. Generalized Quantum Doubles which are Radford Biproducts It is well-known (Majid [14]) that if H is a finite dimensional quasitriangular Hopf algebra, then the Drinfeld double D(H ) is a Radford biproduct. In this section, we give necessary and sufficient conditions for a generalized quantum double D = D(U cop , V ) to be a Radford biproduct B × U cop , and determine the structure of B as a Hopf algebra cop in U U cop YD. Suppose first that U and V are bialgebras in duality with , : U ⊗ V → k an invertible pairing, so that ρ = , is an invertible skew pairing on U cop ⊗ V . We form the generalized quantum double D = D(U cop , V ) as in Subsect. 2.1. There are bialgebra morphisms i : U cop → D(U cop , V ) given by i(m) = m ⊗ 1 and j : V → D(U cop , V ) given by j (x) = 1 ⊗ x. Proposition 3.1. Let U, V be as above.
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(i) There exists a bialgebra projection π from D = D(U cop , V ) to U cop that splits i if and only if there is a bialgebra map γ : V → U cop such that for all y ∈ V , m ∈ U , we have γ (y)m = ρ −1 (m 1 , y3 )ρ(m 3 , y1 )m 2 γ (y2 ).
(3.1)
(ii) Similarly, there exists a bialgebra projection from D to V that splits j if and only if there is a bialgebra morphism µ : U cop → V such that for all y ∈ V, m ∈ U , we have yµ(m) = ρ −1 (m 1 , y3 )ρ(m 3 , y1 )µ(m 2 )y2 . (3.2) If U, V are Hopf algebras, then these maps are Hopf algebra morphisms. Proof.
(i) Suppose that there is a bialgebra morphism γ : V → U cop satisfying (3.1). Define π to be π(m ⊗ x) = mγ (x), for all m ∈ U and x ∈ V . Then π ◦ i(m) = π(m ⊗ 1) = mγ (1) = m for m ∈ U . Furthermore, [5, 2.4] with B = J = U cop , H = V , α = I dU , β = γ , implies that π is a bialgebra morphism. Conversely, given π , define γ by γ = π ◦ j, where j : V → D(U cop , V ) is defined by j (x) = 1 ⊗ x. Then, since π, j are bialgebra maps, so is γ . To verify that (3.1) holds, we compute γ (y)m = (π ◦ j)(y)m = π(1 ⊗ y)π(m ⊗ 1) = π((1 ⊗ y)(m ⊗ 1)) = ρ −1 (m 1 , y3 )ρ(m 3 , y1 )m 2 γ (y2 ).
(ii) The proof of (ii) is analogous. Example 3.2. (cf. [5, 3.1]) Let σ be an invertible skew pairing on a bialgebra H and form D(H, H ). Then the identity map I d H satisfies (3.1) if and only if (H, σ ) is CQT if and only if the multiplication map π : D(H, H ) → H , π(h ⊗ l) = hl is a bialgebra map. Example 3.3. In the setting of Proposition 3.1, if V is quasitriangular via R = R 1 ⊗ R 2 ∈ V ⊗ V , then the map π : D(U cop , V ) → V defined by π(m ⊗ x) = m, R 1 R 2 x is a bialgebra projection. Here the map µ : U cop → V is given by µ(m) = m, R 1 R 2 . (The details can be found in [5, 2.5].) Likewise, if U is quasitriangular with the R-matrix R = R 1 ⊗ R 2 ∈ U ⊗ U , then π : D(U cop , V ) → U cop defined by π(m ⊗ x) = R 2 , xm S(R 1 ) is a Hopf algebra projection. In this case the map γ : V → U cop is given by γ (y) = R 2 , yS(R 1 ). Remarks 3.4.
(i) In later computations we will use that (3.1) is equivalent to ρ(m 1 , y2 )γ (y1 )m 2 = ρ(m 2 , y1 )m 1 γ (y2 ).
(3.3)
If m = γ (x), x ∈ V , then (3.1) becomes ρ(γ (x2 ), y2 )γ (y1 )γ (x1 ) = ρ(γ (x1 ), y1 )γ (x2 )γ (y2 ).
(3.4)
(ii) Note that if the above map γ : V → : V ⊗V → k defined by σ (x, y) = ρ(γ (x), y) = γ (x), y gives V a CQT structure. The relations (2.4), (2.5), (2.6) are easy to check and (2.7) is equivalent to U cop is injective then the map σ
γ (x1 ), y1 x2 y2 = γ (x2 ), y2 y1 x1 . This equation holds if and only if it holds when the injective map γ is applied to both sides, i.e., when (3.4) holds.
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If U, V are Hopf algebras with bijective antipodes, then for ρ a skew pairing from U cop ⊗ V to k, we have ρ −1 (m, x) = SU−1 (m), x = m, SV−1 (x). In this case, we have the identities below which are useful in the following computations and also provide generalizations of the equations describing the square of the antipode in Remarks 2.5(i). Proposition 3.5. Let U, V be Hopf algebras in duality and assume that there exists a map γ as in Proposition 3.1. Then: (i) The map ϑ ∈ V ∗ defined by ϑ(x) = γ (x1 ), SV (x2 ), for all x ∈ V , is convolution invertible with ϑ −1 (x) = γ (SV2 (x1 )), x2 . Moreover, for any x ∈ V , γ (SV2 (x)) = ϑ −1 (x1 )γ (x2 )ϑ(x3 ).
(3.5)
(ii) Similarly, the map υ ∈ V ∗ defined by υ(x) = γ (x2 ), SV (x1 ), for all x ∈ V , is convolution invertible with υ −1 (x) = γ (SV2 (x2 )), x1 . In addition, for all x ∈ V , γ (SV2 (x)) = υ(x1 )γ (x2 )υ −1 (x3 ).
(3.6)
Proof. We only sketch the proof for (i); the rest of the details are left to the reader. For all x ∈ V we have ϑ(x1 )γ (SV2 (x2 )) = γ (x3 ), SV (x4 )γ (x1 )γ (SV (x2 ))γ (SV2 (x5 )) = γ (x3 ), SV (x4 )γ (x1 )SU−1 (γ (SV (x5 ))γ (x2 )) (3.4) γ (x ), S (x )γ (x )S −1 (γ (x )γ (S (x ))) 2 V 5 1 U 3 V 4 = = γ (x2 ), SV (x3 )γ (x1 ) = γ (x1 )ϑ(x2 ), and, in a similar manner, one can prove that γ (SV2 (x1 ))ϑ −1 (x2 ) = ϑ −1 (x1 )γ (x2 ).
(3.7)
Now, for x ∈ V , using the fact that γ : V → U cop is a Hopf algebra map, we have ϑ(x1 )ϑ −1 (x2 ) = ϑ(x1 )γ (SV2 (x2 )), x3 = ϑ(x2 )γ (x1 ), x3 = γ (x2 ), SV (x3 )γ (x1 ), x4 = γ (x1 ), SV (x2 )x3 = ε(x). Similarly, using (3.7) we can show that ϑ −1 (x1 )ϑ(x2 ) = ε(x), so we are done.
Now suppose that U, V are Hopf algebras with bijective antipodes and we have a Hopf algebra projection π from D(U cop , V ) to U cop that splits i. Then there exists cop a Hopf algebra B in the category of Yetter-Drinfeld modules U U cop Y D such that D = D(U cop , V ) ∼ = B × U cop , a Radford biproduct. From Subsect. 2.2, we know that B = {a ∈ D | a1 ⊗ π(a2 ) = a ⊗ 1}. Proposition 3.6. The map θ : V → B given by θ (y) = γ (SV−1 (y2 ))⊗y1 = SU (γ (y2 ))⊗ y1 is a bijection.
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Proof. Clearly the map θ is injective since ε◦γ = ε. It remains to show that I m(θ ) = B. For y ∈ V , we have θ (y) ∈ B since θ (y)1 ⊗ π(θ (y)2 ) = (SU (γ (y4 )) ⊗ y1 ) ⊗ π(SU (γ (y3 )) ⊗ y2 ) = (SU (γ (y4 )) ⊗ y1 ) ⊗ γ (SV−1 (y3 )y2 ) = (SU (γ (y2 )) ⊗ y1 ) ⊗ 1 = θ (y) ⊗ 1. Conversely, suppose that m ⊗ y ∈ B, i.e., (m 2 ⊗ y1 ) ⊗ m 1 γ (y2 ) = (m ⊗ y) ⊗ 1 ∈ D ⊗ U cop . Then we have m ⊗ y = SU (1)m ⊗ y = SU (m 1 γ (y2 ))m 2 ⊗ y1 = SU (γ (y2 ))SU (m 1 )m 2 ⊗ y1 = εU (m)θ (y). Similarly, if z = m i ⊗ yi ∈ B, then z = θ ( ε(m i )yi ). i
i
Now we denote by V the vector space V with the structure of a Yetter-Drinfeld module induced by that of B. Proposition 3.7. The structure of the left U cop Yetter-Drinfeld module V is given by the left action and left coaction m y = m 1 , SV−1 (y1 )y2 m 2 , SV−2 (y3 ) = m, SV−1 (SV−1 (y3 )y1 )y2 ; λV (y) =
γ (SV−1 (y3 )y1 ) ⊗ y2 .
(3.8) (3.9)
Proof. For m ∈ U and y ∈ V , by Sect. 2.2, m y = θ −1 (m i θ (y)). Using (2.12), (3.1) and the fact that m, v = SU−1 (m), SV (v), we compute i(m 2 )θ (y)i(SU−1 (m 1 )) = (m 2 ⊗ 1)(γ (SV−1 (y2 )) ⊗ y1 )(SU−1 (m 1 ) ⊗ 1) = m 4 γ (SV−1 (y4 ))SU−1 (m 2 )SU−1 (m 3 ), SV−1 (y3 ) ⊗ y2 SU−1 (m 1 ), y1 (3.3) m S −1 (m )γ (S −1 (y ))S −1 (m ), S −1 (y ) ⊗ y S −1 (m ), y 4 U 3 3 2 4 2 U 1 1 V U V = −1 −1 −1 = SU (m 1 ), y1 θ (y2 )SU (m 2 ), SV (y3 ) = m, SV−1 (y1 )SV−2 (y3 )θ (y2 ), and this concludes the proof of the formula for the action. We now compute the coaction, (I dU ⊗ θ ) ◦ λV (y) = (π ⊗ I d D )(θ (y)) = (π ⊗ I d D )(γ (SV−1 (y2 )) ⊗ y1 ) = (π ⊗ I d D )((γ (SV−1 (y4 )) ⊗ y1 ) ⊗ (γ (SV−1 (y3 )) ⊗ y2 )) = γ (SV−1 (y4 )y1 ) ⊗ γ (SV−1 (y3 )) ⊗ y2 = γ (SV−1 (y3 )y1 ) ⊗ θ (y2 ), for all y ∈ V , and thus the formula for the coaction is also verified.
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We now describe the structure of V as a Hopf algebra in the category of left Yetter Drinfeld modules over U cop . cop
Proposition 3.8. The structure of V as a Hopf algebra in the category U U cop YD is given by the formulas: x · y = γ (y2 ), SV (x1 )x3 x2 y1 ;
(3.10)
(x) = γ (SV (x4 )x6 ), SV−1 (x3 )x1 x2 ⊗ x5 ; S(x) = γ (x4 ), x1 SV (x3 )SV (x2 ).
(3.11) (3.12)
The identity in V is θ −1 (1 ⊗ 1) = 1 and the counit is ε = ε. Proof. To see (3.10), we compute θ (x)θ (y)
= (2.12) = (3.4) = = =
(SU (γ (x2 )) ⊗ x1 )(SU (γ (y2 )) ⊗ y1 ) γ (y2 ), x3 SU (γ (y4 )), x1 SU (γ (y3 )γ (x4 )) ⊗ x2 y1 γ (y3 ), x4 SU (γ (y4 )), x1 SU (γ (x3 )γ (y2 )) ⊗ x2 y1 γ (y2 ), x3 γ (y3 ), SV (x1 )θ (x2 y1 ) γ (y2 ), SV (x1 )x3 θ (x2 y1 ).
Similarly, to verify (3.11), we compute (θ (x))
= = = (2.19) = (2.12, 2.1) =
( ⊗ )((θ (x))) = ( ⊗ )((SU (γ (x2 )) ⊗ x1 )) (SU (γ (x4 )) ⊗ x1 ) ⊗ (SU (γ (x3 )) ⊗ x2 )) (SU (γ (x3 )) ⊗ x1 ) ⊗ θ (x2 ) (SU (γ (x5 )) ⊗ x1 ) SU−1 (π(SU (γ (x4 )) ⊗ x2 )) ⊗ 1 ⊗ θ (x3 ) γ (x12 ), SV−1 (x4 )SU−1 (γ (x6 )), SV−1 (x5 )SU−1 (γ (x8 )), x1 ×γ (x10 ), x2 γ (SV−1 (x13 )SV (x7 )x11 ) ⊗ x3 ⊗ θ (x9 ).
Now we use the equivalent form of (3.5), ϑ −1 (SV−1 (y2 ))γ (SV−1 (y1 )) = γ (SV (y2 ))ϑ −1 (SV−1 (y1 )), to replace SU−1 (γ (x6 )), SV−1 (x5 )γ (SV (x7 )) by SU−1 (γ (x7 )), SV−1 (x6 )γ (SV−1 (x5 )), and obtain (θ (x)) = γ (x12 ), SV−1 (x4 )SU−1 (γ (x7 )), SV−1 (x6 )SU−1 (γ (x8 )), x1 ×γ (x10 ), x2 γ (SV−1 (x13 )SV−1 (x5 )x11 ) ⊗ x3 ⊗ θ (x9 ). From (3.4), γ (x12 ), SV−1 (x4 )γ (SV−1 (x5 ))γ (x11 ) = γ (x11 ), SV−1 (x5 )γ (x12 )γ (SV−1 (x4 )),
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so that (θ (x)) = γ (x11 ), SV−1 (x5 )SU−1 (γ (x7 )), SV−1 (x6 )SU−1 (γ (x8 )), x1 γ (x10 ), x2 × γ (SV−1 (x13 )x12 )γ (SV−1 (x4 )) ⊗ x3 ⊗ θ (x9 ) = γ (x10 ), SV−1 (x4 )SU−1 (γ (x6 )), SV−1 (x5 )SU−1 (γ (x7 )), x1 × γ (x9 ), x2 θ (x3 ) ⊗ θ (x8 ) = γ (SV (x4 )x8 ), SV−1 (x3 )γ (SV (x5 )x7 ), x1 θ (x2 ) ⊗ θ (x6 ) = γ (SV (x4 )x6 ), SV−1 (x3 )x1 θ (x2 ) ⊗ θ (x5 ). Finally, we verify (3.12). From (2.21), S(θ (x)) = i(π(θ (x)1 ))S(θ (x)2 ) = (SU (γ (x4 ))γ (x1 ) ⊗ 1)S(SU (γ (x3 )) ⊗ x2 ) = (SU (γ (x4 ))γ (x1 ) ⊗ 1)(1 ⊗ SV (x2 ))(γ (x3 ) ⊗ 1) = SU−1 (γ (x7 )), SV (x2 )γ (x5 ), SV (x4 )SU (γ (x8 ))γ (x1 )γ (x6 ) ⊗ SV (x3 ) = γ (x7 ), x2 γ (x5 ), SV (x4 )SU (γ (x8 ))γ (x1 )γ (x6 ) ⊗ SV (x3 ). By (3.4) we can replace γ (x7 ), x2 γ (x1 )γ (x6 ) by γ (x6 ), x1 γ (x7 )γ (x2 ) and we obtain S(θ (x)) = = = =
γ (x6 ), x1 γ (x5 ), SV (x4 )SU (γ (x8 ))γ (x7 )γ (x2 ) ⊗ SV (x3 ) γ (x6 ), x1 γ (x5 ), SV (x4 )γ (x2 ) ⊗ SV (x3 ) γ (x5 ), x1 SV (x4 )SU (γ (SV (x2 ))) ⊗ SV (x3 ) γ (x4 ), x1 SV (x3 )θ (SV (x2 )).
The final statement is clear.
We noted in Example 3.3 that if V is quasitriangular then D(U cop , V ) is a Hopf algebra with a projection. So for V = (H, R) a finite dimensional quasitriangular Hopf algebra and U = H ∗ we obtain Drinfeld’s projection [8], and by an analogue of Propositions 3.7 and 3.8, the structure of H ∗ as a braided Hopf algebra in H HYD computed by Majid in [14]. (In fact, it can be proved that this braided Hopf algebra lies in the image of a canonical braided functor from H M to H H Y D.) On the other hand, if U = H ∗ is quasitriangular, and V = H , then we have the following. Proposition 3.9. Let H be a finite dimensional Hopf algebra. Then there exists a Hopf algebra projection π : D(H ) → H ∗cop covering the natural inclusion H ∗cop → D(H ) if and only if H is CQT. Moreover, if this is the case, then π is a quasitriangular morphism. Proof. Suppose (H, σ ) is CQT and let {ei , ei } be a dual basis for H . Then H ∗ is quasitriangular with R-matrix R = σ (ei , e j )ei ⊗ e j . Since D(H ) = D(H ∗cop , H ), from i, j
Example 3.3 the map π : D(H ) → H ∗cop given by π(h ∗ ⊗ h) = σ (ei , e j )e j (h)h ∗ (ei ◦ S) = σ (ei , h)h ∗ (ei ◦ S), i, j
i
for all h ∗ ∈ H ∗ and h ∈ H , is a Hopf algebra morphism which covers the inclusion H ∗cop → D(H ).
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Conversely, if such a morphism exists, then by Proposition 3.1 there exists a Hopf algebra morphism γ : H → H ∗cop satisfying h ∗1 , x2 γ (x1 )h ∗2 = h ∗2 , x1 h ∗1 γ (x2 ), for all x ∈ H and h ∗ ∈ H ∗ , where , : H ∗ ⊗ H → k is the evaluation map. It is clear that the above condition is equivalent to γ (x1 ), y1 x2 y2 = γ (x2 ), y2 y1 x1 , for all x, y ∈ H . Now, if we define σ (x, y) = γ (x), y, for all x, y ∈ H , then one can easily check that σ defines a CQT structure on H (see also [21, Theorem 3.3.14]). Finally, the canonical R-matrix of D(H ) is R = (ε ⊗ ei ) ⊗ (ei ⊗ 1). So if (H, σ ) i
is CQT then the above morphism π is quasitriangular since (π ⊗ π )(R) = σ (e j , S −1 (ei ))e j ⊗ ei = σ −1 (e j , ei )e j ⊗ ei , i, j
i, j
and, from the dual version of Remarks 2.5 (iv), the last term defines a QT structure for H ∗cop . ∗cop
H For H finite dimensional it is not hard to see that the categories H H ∗cop Y D and H Y D are isomorphic as braided monoidal categories (the braided structures are the ones obtained from the left or right center construction, see [2]). The isomorphism is pro∗cop duced by the following functor F. If (M, ·, λ) ∈ H H ∗cop Y D then F(M) = M becomes an object in H Y D H via the structure
h • m = m−1 (h)m 0 ,
m → ei · m ⊗ ei .
F sends a morphism to itself. By the above identification, Proposition 3.9 and Propositions 3.7 and 3.8 we obtain the following which may be viewed as a left version of the transmutation theory for CQT Hopf algebras. Further details will follow in Sect. 4.1. Corollary 3.10. If H is a CQT Hopf algebra then H has a braided Hopf algebra structure, denoted by H , within H Y D H . Namely, H is a left-right Yetter-Drinfeld module over H via x → x(0) ⊗ x(1) := x2 ⊗ S −1 (x1 )S −2 (x3 ), −1 −1 (S (x3 )x1 , h)x2 , h • x = σ (h, S −1 (x1 )S −2 (x3 ))x2 = σ21
for all h, x ∈ H . H is a Hopf algebra in and
HYD
H
with the same unit and counit as H
−1 x · y = σ (S(x1 )x3 , S −1 (y2 ))x2 y1 = σ21 (y2 , S(x1 )x3 )x2 y1 , −1 (S(x4 )x6 , S −1 (x3 )x1 )x2 ⊗ x5 , (x) = σ (S −1 (x3 )x1 , S −1 (x6 )x4 )x2 ⊗ x5 = σ21 −1 (x4 , x1 S(x3 ))S(x2 ). S(x) = σ (x1 S(x3 ), S −1 (x4 ))S(x2 ) = σ21
Proof. We only note that from the proof of Proposition 3.9 the Hopf algebra morphism γ : H → H ∗cop is given in this case by γ (h) = l S −1 (h) , for all h ∈ H .
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If B is a bialgebra and M is a left Yetter-Drinfeld module over B, then the map RM : M ⊗ M → M ⊗ M, RM(m ⊗ m ) = m −1 · m ⊗ m 0 is a solution in End(M⊗3 ) of the quantum Yang-Baxter equation R12 R13 R23 = R23 R13 R12 . In the setting above, V is a left Yetter-Drinfeld module over U cop . Then a solution R V ∈ End(V ⊗ V ) to the quantum Yang-Baxter equation is given by R V (x ⊗ y) = γ (SV (y3 )y1 ) x ⊗ y2 = γ (SV−1 (y3 )y1 ), SV−1 (SV−1 (x3 )x1 )x2 ⊗ y2 . 4. Coquasitriangular Bialgebras and Generalized Quantum Doubles 4.1. Transmutation theory. As was mentioned in the introduction, the braided reconstruction theorem associates to any CQT Hopf algebra H a braided commutative Hopf algebra H in the category of right H -comodules, M H . The goal of this subsection is to show that H can be obtained from the structure of a generalized quantum double with a projection. Let X and A be sub-Hopf algebras of a CQT Hopf algebra (H, σ ). Then σ induces a skew pairing on A ⊗ X , still denoted by σ and the generalized quantum double D(A, X ) is defined. By (2.7) it follows that X A = AX , so X A is a sub-Hopf algebra of H . From [5, 3.1] the map π : D(A, X ) → AX = X A, π (a ⊗ x) = ax is a surjective Hopf algebra morphism. Although D(H, H ) is a Hopf algebra with projection, we cannot, in general, make the same claim for D(A, X ). Nevertheless, X A = AX and X are sub-Hopf algebras of (H, σ ), so σ induces a skew pairing on AX ⊗ X and π : D(AX, X ) → AX = X A, π(ax ⊗ y) = ax y is a surjective Hopf algebra morphism covering the inclusion map i : AX → D(AX, X ) with the map γ from Proposition 3.1 being the inclusion of X into AX = X A. From Propositions 3.7 and 3.8 for γ : X → AX and , = σ : AX ⊗ X → k, X has a braided Hopf algebra structure in the braided monoidal category XX AA Y D; this braided Hopf algebra is denoted, as usual, by X . The structures are given by xa y = σ (xa, S −1 (S −1 (y3 )y1 ))y2 , λ X (y) = S −1 (y3 )y1 ⊗ y2 , x · y = σ (y2 , S(x1 )x3 )x2 y1 , 1 = 1, (x) = σ (S(x4 )x6 , S −1 (x3 )x1 )x2 ⊗ x5 , ε = ε, S(x) = σ (x4 , x1 S(x3 ))S(x2 ), for all x, y ∈ X and a ∈ A. Next, we show that X lies in the image of a canonical braided functor from X M to XX AA Y D, so this general context reduces to the case X = A = H . We recall some background on CQT Hopf algebras and braided monoidal categories. For B a bialgebra, it is well-known that there is a one-to-one correspondence between the CQT-structures on B and braidings on the category of left (right) B-comodules. If
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(B, ς ) is a CQT-bialgebra, and M and N are two left (respectively right) B comodules then M ⊗ N m ⊗ n → m−1 n−1 ⊗ m0 ⊗ n0 ∈ B ⊗ M ⊗ N defines the monoidal structure of B M and cM,N : M ⊗ N → N ⊗ M, cM,N(m ⊗ n) = ς (n−1 , m−1 )n0 ⊗ m0
(4.1)
defines a braiding on B M, while M ⊗ N m ⊗ n → m(0) ⊗ n(0) ⊗ m(1) n(1) ∈ M ⊗ N ⊗ B
(4.2)
gives the monoidal structure of MB and cM,N : M ⊗ N → N ⊗ M, cM,N(m ⊗ n) = ς (m(1) , n(1) )n(0) ⊗ m(0)
(4.3)
provides a braided structure on MB (see [11,17] for terminology). We denote by (B,ς ) M the category of left B-comodules with the braiding in (4.1). Similarly, M(B,ς) is the category of right B-comodules with the braiding defined by cop (4.3). One can easily see that (B ,ς21 ) M ≡ M(B,ς ) , as braided monoidal categories, where ς21 = ς ◦ tw is the CQT structure of B cop defined in Remarks 2.5. Secondly, if (B, ς ) is a CQT bialgebra and M a left B-comodule then M is a left Yetter-Drinfeld module over B with the initial comodule structure and with the B-action defined by b · m := ς (m−1 , b)m0 . (4.4) Thus there is a well defined braided functor F(B,ς ) : (B,ς) M → B B Y D, where F(B,ς) sends a morphism to itself. Lemma 4.1. In the setting above, the braided Hopf algebra X lies in the image of the −1 −1 composite of the canonical functor (X,σ21 ) M → (X A,σ21 ) M and the functor F(X A,σ −1 ) . 21
Proof. Since λ X (x) := x−1 ⊗ x0 = S −1 (x3 )x1 ⊗ x2 , the X -action on X can be rewritten as −1 (y−1 , x)y0 , x y = σ (x, S −1 (y−1 ))y0 = σ −1 (x, y−1 )y0 = σ21
and this finishes the proof.
In general, if C is a braided monoidal category with braiding c then C in is C as a monoidal category, but with the mirror-reversed braiding c˜ M,N = c−1 N ,M . Note that, if B ∈ C is a braided Hopf algebra with comultiplication and bijective antipode S then B cop , the same object B, but with the comultiplication and antipode cop cop = c−1 = S −1 B,B ◦ and S
respectively, and with the other structure morphisms the same as for B, is a braided Hopf algebra in the category C in . Now, since the transmutation object H is a braided
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Hopf algebra in M H and not H M we must apply the above correspondence X → X to (H cop , σ21 ) rather than (H, σ ) and thus obtain a braided Hopf algebra H cop ∈ (H
cop ,σ −1 )
−1
in
M ≡ M(H,σ21 ) ≡ M(H,σ ) .
We can now prove the connection between H cop and H . The structures of H can be found in [17, Example 9.4.10]. Proposition 4.2. Let (H, σ ) be a CQT Hopf algebra. Then (H cop )cop = H as braided Hopf algebras in M(H,σ ) . Proof. One can easily see that (H cop )cop is an object of M H via the structure h → h (0) ⊗ h (1) = h 2 ⊗ S(h 1 )h 3 , and that its algebra structure within M(H,σ ) is given by h · g = σ (S −1 (h 3 )h 1 , g1 )h 2 g2 = σ (S(h 1 )h 3 , S(g1 ))h 2 g2 , for all h, g ∈ H . Clearly, the unit of (H cop )cop is the unit of H . −1 −1 Now, by (4.3), the inverse c−1 of the braiding c of M(H,σ21 ) , is defined by cM ,N(n ⊗ cop cop m) = σ (n(1) , m(1) )m(0) ⊗ n(0) . Therefore, the comultiplication of (H ) is given by h → σ (S(h 4 )h 6 , S −1 (h 3 )h 1 )c−1 (h 5 , h 2 ) = σ ((h 2 )(1) , S −1 ((h 1 )(1) ))c−1 ((h 2 )(0) ⊗ (h 1 )(0) ) = σ −1 ((h 2 )(2) , (h 1 )(2) )σ ((h 2 )(1) , (h 1 )(1) )(h 1 )(0) ⊗ (h 2 )(0) = h 1 ⊗ h 2 = (h), the comultiplication of H . The counit of (H cop )cop is ε, the counit of H . Finally, the antipode of H cop is defined by S(h) = σ (h 4 S −1 (h 2 ), h 1 )S −1 (h 3 ), so the antipode of (H cop )cop is given, for all h ∈ H , by S −1 (h) = σ (S 2 (h 3 )S(h 1 ), h 4 )S(h 2 ). Indeed, for all h ∈ H , (S ◦ S −1 )(h) = (2.4) = = (2.9) = (2.5) =
σ (h 1 S(h 6 ), S −1 (h 7 ))σ (S(h 2 )h 4 , S(h 5 ))h 3 σ (h 1 , S −1 (h 9 ))σ (S(h 7 ), S −1 (h 8 ))σ (h 2 , h 6 )σ (h 4 , S(h 5 ))h 3 σ (h 1 , S −1 (h 7 ))v(h 4 )σ (h 2 , h 5 )v −1 (h 6 )h 3 σ (h 1 , S −1 (h 5 ))σ (h 2 , S −2 (h 4 ))h 3 σ (h 1 , S −2 (h 3 )S −1 (h 4 ))h 2 = h,
as needed. In a similar way we can prove that S −1 ◦ S = I d H , the details are left to the reader. Comparing the above structures of (H cop )cop with those of H from [17] we conclude that(H cop )cop = H , as braided Hopf algebras in M(H,σ ) .
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Remark 4.3. From Corollary 3.10 we can deduce the left version of the transmutation theory for a CQT Hopf algebra (H, σ ) as follows. Observe that there is a braided functor G : (H,σ ) M → H Y D H defined by G(M) = M but now viewed as left-right H Yetter-Drinfeld module via h · m = σ (h, S −1 (m−1 ))m0 , m → ρ(m) := m0 ⊗ S −1 (m−1 ). G sends a morphism to itself. Now, one can easily see that H in Corollary 3.10 lies in the image of the functor G. In other words we can associate to (H, σ ) a Hopf algebra structure in (H,σ ) M, still denoted by H . Note that H has the left H -comodule structure given by x → λ H (x) = x−1 ⊗ x0 = S −1 (x3 )x1 ⊗ x2 , for all x ∈ H , and is a braided Hopf algebra in (H,σ ) M via the structure in Corollary 3.10. (In fact, this H is the braided Hopf algebra in Lemma 4.1 corresponding −1 to X = (H, σ21 ).) We can conclude now that H op,cop is the braided Hopf algebra (H,σ ) M associated to (H, σ ) through the left transmutation theory. By H op,cop we in denote H viewed now as a Hopf algebra in (H,σ ) M via x y =: m H ◦ c H ,H (x, y) = σ (y−1 , x−1 )y0 x0 = σ (y2 , S(x1 )x3 )x2 y1 , H op,cop = c−1 H ,H ◦ = , and the other structure morphisms equal those of H . 4.2. The “dual” case. Let (H, σ ) be a CQT bialgebra, so that σ is a pairing from H ⊗ H op to k and H cop ⊗ H to k. Then by Remarks 2.2, we have bialgebra morphisms φ : H cop → H 0 defined by φ(h)(l) = σ (h, l) and ψ : H op → H 0 defined by ψ(h)(l) = σ (l, h). We denote rh := φ(h) = σ (h, −) and Hr = I m(φ). Similarly, lh := ψ(h) = σ (−, h) and Hl = I m(ψ). More generally, if X and A are sub-bialgebras of a CQT bialgebra H then we define Hr A := φ(Acop ) = {ra = σ (a, −) | a ∈ A}, Hl X := ψ(X op ) = {l x = σ (−, x) | x ∈ X }. Proposition 4.4. Let (H, σ ) be a CQT bialgebra (Hopf algebra) and X, A ⊆ H two sub-bialgebras (sub-Hopf algebras). Then (i) Hl X and Hr A are sub-bialgebras (sub-Hopf algebras) of H 0 . The structure maps for ψ(X op ) = Hl X are given by l x y = l y l x ; l1 = ε; (l x ) = l x1 ⊗ l x2 ; ε(l x ) = ε(x); S(l x ) = l S −1 (x) , (4.5) and the structure maps for φ(Acop ) = Hr A are given by rab = ra rb ; r1 = ε; (ra ) = ra2 ⊗ra1 ; ε(ra ) = ε(a); S(ra ) = r S −1 (a) . (4.6)
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(ii) The bilinear form from Hr A ⊗ Hl X to k defined for all a ∈ A, x ∈ X , by ra ⊗ l x → σ (a, x), is a skew pairing between Hr A and Hl X . (iii) Hr A Hl X = Hl X Hr A is a sub-bialgebra of H 0 and is a sub-Hopf algebra if A, X are sub-Hopf algebras of the Hopf algebra H . (i) If X is a sub-bialgebra (sub-Hopf algebra) of H , then X op is a subbialgebra (sub-Hopf algebra) of H op . Since σ : H ⊗ H op → k is a duality, by Remarks 2.2, ψ : X op ⊆ H op → H 0 is a bialgebra morphism and thus Hl X , the image of X op under ψ, is a sub-bialgebra (sub-Hopf algebra) of H 0 with the structure maps given above. The proof for Hr A is similar, using the map φ. (ii) Follows directly from Remarks 2.2 since σ : Acop ⊗ X op → k is a skew pairing. (iii) We refer to [5, 3.2 (b)]. Here it is shown that the map from D(H cop , H op ) to H 0 defined by a ⊗ h → ra lh is a bialgebra map, and our statement follows. The proof is based on the observation that
Proof.
l y ra = σ (a1 , S(y3 ))σ (a3 , y1 )ra2 l y2 or, equivalently, ra l y =
σ (S −1 (a
3 ), y1 )σ (a1 , y3 )l y2 ra2 ,
(4.7) (4.8)
for any y ∈ X , a ∈ A. We now assume that H is a Hopf algebra. We denote by H X,A the sub-Hopf algebra of H 0 equal to Hl X Hr A = Hr A Hl X . Proposition 4.5. Let (H, σ ) be a CQT Hopf algebra, X , A and H sub-Hopf algebras of H , and Hl X and Hr A the sub-Hopf algebras of H 0 from Proposition 4.4. Let D(Hr A , Hl X ) be the generalized quantum double from the skew pairing in Proposition 4.4 (ii) induced by σ . (i) The map f : D(Hr A , Hl X ) → H X,A , f (ra ⊗ l x ) = ra l x , a ∈ A, x ∈ X , is a surjective Hopf algebra morphism. (ii) Then the evaluation map , : H X,A ⊗ H → k defined for all x ∈ X , a ∈ A and h ∈ H by l x ra , h = σ (h 1 , x)σ (a, h 2 ) = l x (h 1 )ra (h 2 ) (4.9) provides a duality between the Hopf algebras H X,A and H, and therefore between D(Hr A , Hl X ) and H via the map f in (i). In particular, , is a skew pairing on cop H X,A ⊗ H and D(Hr A , Hl X )cop ⊗ H. Proof. Statement (i) follows directly from [5, 2.4] with α = I d Hr A , β = I d Hl X and using (4.7). Statement (ii) is immediate. From now on, we assume that (H, σ ) is a CQT Hopf algebra with X, A and the cop evaluation pairing , as above. Consider the generalized quantum double D(H X,A , X ). cop From (2.12), the multiplication in D(H X,A , X ) is given by (l x ra ⊗ y)(l x rb ⊗ y ) = l x1 rb3 , S −1 (y3 )l x3 rb1 , y1 l x ra l x2 rb2 ⊗ y2 y ,
(4.10)
for all x, x , y, y ∈ X , a, b ∈ A, and since D(H X,A , X ) = H X,A ⊗ X as a coalgebra, the comultiplication is given by cop
cop
(l x ra ⊗ y) = (l x2 ra1 ⊗ y1 ) ⊗ (l x1 ra2 ⊗ y2 ). The unit is ε ⊗ 1 and the counit is defined by ε(l x ra ⊗ y) = ε(x)ε(a)ε(y).
(4.11)
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M. Beattie, D. Bulacu cop
cop
We now prove that there is a Hopf algebra projection from D(H X,A , X ) to H X,A cop cop which covers the canonical inclusion i : H X,A → D(H X,A , X ). cop
cop
Proposition 4.6. The map π : D(H X,A , X ) → H X,A defined by π(l x ra ⊗ y) = l x ra l S −1 (y) ,
(4.12)
for all x, y ∈ X , a ∈ A, is a Hopf algebra morphism such that π ◦ i = id H cop . X,A
cop Hl X
⊆ Proof. Define γ : X → morphism and that (3.1) holds. For x, y ∈ X , we have
cop H X,A
by γ (x) = l S −1 (x) . We show that γ is a bialgebra
γ (x y) = l S −1 (x y) = l S −1 (y)S −1 (x) = l S −1 (x)l S −1 (y) = γ (x)γ (y), so that γ preserves multiplication. Similarly, (γ ⊗ γ )(x1 ⊗ x2 ) = l S −1 (x1 ) ⊗ l S −1 (x2 ) = H cop (γ (x)), X,A
so that comultiplication is preserved. Clearly γ preserves the unit and counit. To verify (3.1), we compute γ (x)l y
= (2.4) = (2.7) = =
l S −1 (x)l y = l y S −1 (x) σ (S −1 (x2 ), y2 )σ (x1 , y3 )l y1 S −1 (x3 ) σ (S −1 (x3 ), y1 )σ (x1 , y3 )l S −1 (x2 )y2 l y1 , S −1 (x3 )l y3 , x1 l y2 γ (x2 ).
As well, γ (x2 )ra
= l S −1 (x2 )ra (4.7) σ (a , S(S −1 (x )))σ (a , S −1 (x ))r l −1 1 2 3 4 a2 S (x3 ) = −1 = ra1 , x2 ra3 , S (x4 )ra2 l S −1 (x3 ) .
Combining these equations, we obtain γ (x)l y ra = l y3 ra1 , x1 l y1 ra3 , S −1 (x3 )(l y2 ra2 )γ (x2 ) = (l y ra )3 , x1 (l y ra )1 , S −1 (x3 )(l y ra )2 γ (x2 ). Since π = m H 0 ◦ (I d ⊗ γ ), the statement follows from Proposition 3.1.
We now apply the results of Sect. 3 to this Radford biproduct. cop cop Proposition 4.7. Let H, X, A, γ , π be as above. Then D(H X,A , X ) ∼ = B × H X,A , where cop
B is a Hopf algebra in the category
H X,A cop
H X,A
YD. In addition,
(i) B = {l S −2 (x2 ) ⊗ x1 | x ∈ X } and is isomorphic to X as a k-space;
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cop
(ii) X is a left H X,A Yetter-Drinfeld module with the structure l x ra y = l x ra , S −1 (y1 )S −2 (y3 )y2 , λ X (y) = l S −1 (y1 )S −2 (y3 ) ⊗ y2 .
(4.13) (4.14)
Proof. Statement (i) follows directly from Proposition 3.6, while statement (ii) follows from Proposition 3.7. From now on, X will be the k-vector space X , with the structure of Hopf algebra cop
in the braided category
H X,A cop
H X,A
Y D induced from B via the above isomorphism. Next we
compute the structure maps of X in this category. H
cop
Theorem 4.8. The structure of X as a Hopf algebra in H X,A cop Y D is given by the formulas X,A
x ◦ y = σ (S(x3 )S 2 (x1 ), y2 )x2 y1 , (x) = σ (S(x1 )x3 , S(x4 )x6 )x2 ⊗ x5 ,
(4.15) (4.16)
S(x) = σ (S 2 (x3 )S(x1 ), x4 )S(x2 ),
(4.17)
for all x, y ∈ X . X has the same unit and counit as X ⊆ H . Proof. Apply the formulas in Proposition 3.8 with U = H X,A and V = X .
Remarks 4.9. (i) From Eqs. (2.8) and (2.9) in Remarks 2.5, we obtain another formula for the antipode S of X . For we note that S(x) = σ (S 2 (x3 ), x4 )σ (S(x1 ), x5 )S(x2 ) = σ (S(x1 ), x4 )v −1 (x3 )S(x2 ) = σ (S(x1 ), x4 )v −1 (x2 )S −1 (x3 ) = σ (S(x1 ), x5 )σ (S 2 (x2 ), x3 )S −1 (x4 ) = σ (S(x1 ), S −1 (x2 )x4 )S −1 (x3 ). cop
cop
(ii) The Hopf algebra isomorphism χ : X × H X,A → D(H X,A , X ) is given by χ (x × l y ra ) = (l S −2 (x2 ) ⊗ x1 )(l y ra ⊗ 1), for all x, y ∈ X and a ∈ A. (iii) One can easily check that, in general, X is neither quantum commutative nor cop
quantum cocommutative as a bialgebra in gives a triangular structure on X (that is, then X is quantum commutative, i.e.
H X,A cop
Y D. But, if σ restricted to X ⊗ X
H X,A σ −1 (x,
y) = σ (y, x), for all x, y ∈ X )
x ◦ y = (x−1 y) ◦ x0 , ∀ x, y ∈ X. (iv) Any proper sub-Hopf algebra X of a CQT Hopf algebra (H, σ ) can be viewed as a braided Hopf algebra in (at least) three different left Yetter-Drinfeld module categories by applying Proposition 4.7 and Theorem 4.8 with different A. Specicop fically, over Hl X (take A = k), over the co-opposite of Hl X Hr X = Hr X Hl X (take A = X ), and over the co-opposite of Hl X Hr = Hr Hl X (take A = H ). Note that Hl X ⊆ Hl X Hr X ⊆ Hl X Hr .
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We note that the solution to the Yang-Baxter equation from the Yetter-Drinfeld module X comes from the adjoint coaction. For if (B, σ ) is a CQT bialgebra and M a left B-comodule then we have seen that M is a left Yetter-Drinfeld module over B with the initial comodule structure and with the B-action defined by (4.4). In particular, RM ∈ End(M ⊗ M) given for all m, m ∈ M by RM(m ⊗ m ) = σ (m−1 , m −1 )m0 ⊗ m 0 , is a solution for the quantum Yang-Baxter equation. In the present setting, with (H, σ ) CQT, then X is a left X -comodule via x → S(x1 )x3 ⊗ x2 . Then R X ad ∈ End(X ⊗ X ) given for all x, y ∈ X by R X ad (x ⊗ y) = σ (S(x1 )x3 , S(y1 )y3 )x2 ⊗ y2 is a solution for the quantum Yang-Baxter equation. From the discussion at the end of Sect. 3, writing S for S X , R X (x ⊗ y) = γ (S −1 (y3 )y1 ), S −1 (S −1 (x3 )x1 )x2 ⊗ y2 = l S −1 (S −1 (y3 )y1 ) , S −1 (S −1 (x3 )x1 )x2 ⊗ y2 = σ (S −1 (S −1 (x3 )x1 ), S −1 (S −1 (y3 )y1 ))x2 ⊗ y2 = σ (S(x1 )x3 , S(y1 )y3 )x2 ⊗ y2 = R X ad (x ⊗ y). Next, we show the connection between our X and the transmutation theory. Theorem 4.10. Let (H, σ ) be a CQT Hopf algebra and A, X sub-Hopf algebras of H . cop
Then there is a braided functor F : M X →
H X,A cop
H X,A
Y D such that F(X ) = X .
Proof. We start by constructing the functor F explicitly. For M a right X -comodule, let F(M) = M as a k-vector space, endowed with the following structures: l x ra m = l x ra (S −2 (m(1) ))m(0) and λM(m) = l S −2 (m(1) ) ⊗ m(0) ,
(4.18)
cop
for any l x ra ∈ H X,A and m ∈ M. From the definitions, F(M) is both a left H X,A -module cop and comodule. Actually, F(M) is a Yetter-Drinfeld module over H X,A . Here, using the co-opposite of the structure maps in Proposition 4.4, we see that relation (2.15) is l x1 ra2 , S −2 (m(1) )l x2 ra1 l S −2 (m(2) ) ⊗ m(0) = l x2 ra1 , S −2 (m(2) )l S −2 (m(1) )l x1 ra2 ⊗ m(0) , and this holds since l x1 ra2 , y1 l x2 ra1 l y2
= (4.8) = (2.5) = (2.7) = =
σ (y1 , x1 )σ (a2 , y2 )l x2 ra1 l y3 σ (y1 , x1 )σ (a4 , y2 )σ (S −1 (a3 ), y3 )σ (a1 , y5 )l x2 l y4 ra2 σ (y1 , x1 )σ (a1 , y3 )l y2 x2 ra2 σ (y2 , x2 )σ (a1 , y3 )l x1 y1 ra2 l x2 ra1 , y2 l y1 l x1 ra2 ,
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137 cop
for all x, y ∈ X and a ∈ A. So F is a well defined functor from M X to
H X,A cop
H X,A
Y D.
(F sends a morphism to itself.) We claim that F has a monoidal structure defined by ϕ0 = I d : k → F(k) and ϕ2 (M, N) : F(M) ⊗ F(N) → F(M ⊗ N), the family of natural isomorphisms in cop
H X,A cop
H X,A
Y D given by ϕ2 (M, N)(m ⊗ n) = σ −1 (m(1) , n(1) )m(0) ⊗ n(0) ,
for all m ∈ M ∈ M X and n ∈ N ∈ M X (for more details see for example [11, XI.4]). cop To this end, observe that ϕ2 (M, N) is left H X,A -linear since ϕ2 (M, N)(l x ra (m ⊗ n)) (2.16) ϕ (M, N)(l r m ⊗ l r n) 2 x 2 a1 x 1 a2 = (4.18) l r , S −2 (n m )ϕ (M, N)(m ⊗ n ) x a 2 (1) (1) (0) (0) = −2 −1 = l x ra , S (n(2) m(2) )σ (m(1) , n(1) )m(0) ⊗ n(0) (2.7) l x ra , S −2 (m(1) n(1) )σ −1 (m(2) , n(2) )m(0) ⊗ n(0) = (4.2, 4.18) σ −1 (m , n )l r (m ⊗ n ) (1) (1) x a (0) (0) = = l x ra ϕ2 (M, N)(m ⊗ n), cop
for all x ∈ X , a ∈ A, m ∈ M and n ∈ N. The fact that ϕ2 (M, N) is left H X,A -colinear can be proved in a similar manner, the details are left to the reader. Clearly, ϕ2 (M, N) is bijective and ϕ0 makes the left and right unit constraints in cop
M X and
H X,A cop
H X,A
Y D compatible. So it remains to prove that ϕ2 respects the associativity
constraints of the two categories above. It is not hard to see that this fact is equivalent to σ (h 1 , g1 )σ (h 2 g2 , h ) = σ (g1 , h 1 )σ (h, h 2 g2 ), for all h, h , g ∈ H , i.e., σ is a 2-cocycle, a well known fact which follows by applying the properties of coquasitriangularity or see [4]. Moreover, we have F a braided functor, this means F(cM,N) ◦ ϕ2 (M, N) = ϕ2 (N, M) ◦ cF(M),F(N) , for all M, N ∈ M X . Indeed, on one hand, by (4.3) we have F(cM,N) ◦ ϕ2 (M, N)(m ⊗ n) = n ⊗ m. On the other hand, by (2.17) and (4.18) we compute: ϕ2 (N, M) ◦ cF(M),F(N) (m ⊗ n) = ϕ2 (N, M)(l S −2 (m(1) ) n ⊗ m(0) ) = σ (S −2 (n(1) ), S −2 (m(1) )ϕ2 (N, M)(n(0) ⊗ m(0) ) = n ⊗ m, as needed.
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Finally, by Proposition 4.2 and Proposition 4.7 it follows that F(X ) = X , as objects cop
in
H X,A cop
H X,A
Y D. Furthermore, since F is a braided functor and X is a braided Hopf algebra cop
in M X it follows that F(X ) is a braided Hopf algebra in
H X,A cop
H X,A
Y D with multiplication
given by m F(X ) : F(X ) ⊗ F(X )
ϕ2 (X ,X )
- F(X ⊗ X )
F(m X )
- F(X ),
the comultiplication defined by F(X ) : F(X )
F( X )
- F(X ⊗ X )
ϕ2−1 (X ,X )
- F(X ) ⊗ F(X ),
and the same antipode as those of X . More precisely, according to the proof of Proposition 4.2 the multiplication • is given by x•y
= σ −1 (x(1) , y(1) )x(0) · y(0) = σ −1 (x(1) , S(y1 )y3 )x(0) · y2
= σ (x(2) , S −1 (y4 )y1 )σ (x(1) , S(y2 ))x(0) y3 (2.5) σ (x , S −1 (y ))x y = σ (S(x )S 2 (x ), y )x y , 2 3 1 2 2 1 (1) (0) 1 = for all x, y ∈ X . Similarly, we have F(X ) (x) = σ ((x1 )(1) , (x2 )(1) )(x1 )(0) ⊗ (x2 )(0) = σ (S(x1 )x3 , S(x4 )x6 )x2 ⊗ x5 , for all x ∈ X . Comparing to the structures in Theorem 4.8 we conclude that F(X ) = X cop
as braided Hopf algebras in
H X,A cop
H X,A
Y D, and this finishes the proof.
We end this section with a few comments. Remarks 4.11. Let (H, σ ), X and A be as above. cop cop (i) One can easily see that the map γ : A → Hr A ⊆ H X,A given by γ (a) = ra is a Hopf algebra morphism. Moreover, for this map γ , (3.1) reduces to ra rb = σ −1 (b3 , a3 )σ (b1 , a1 )rb2 ra2 which holds by (2.7). So on the k-vector space A we have a braided Hopf algebra cop
structure, denoted by A, in cop H X,A
H X,A cop
H X,A
Y D. Namely, A is a left Yetter-Drinfeld module over
via l x ra b = l x ra , S −1 (b1 )S −2 (b3 )b2 , λ A (a) = r S −1 (a3 )a1 ⊗ a2 ,
and a Hopf algebra with the same unit and counit as A and, −1 (S(a3 )S 2 (a1 ), b2 )a2 b1 , a · b = σ (b2 , S(a1 )a3 )a2 b1 = σ21 −1 (S(a1 )a3 , S(a4 )a6 )a2 ⊗ a5 , (a) = σ (S(a4 )a6 , S −1 (a3 )a1 )a2 ⊗ a5 = σ21 −1 2 (S (a3 )S(a1 ), a4 )S(a2 ). S(a) = σ (a4 , a1 S(a3 ))S(a2 ) = σ21
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Comparing with the structure of X we conclude that A can be obtained from X by −1 replacing (H, σ ) with (H, σ21 ) and interchanging X and A. For this, observe that if l˜a −1 0 and r˜x are the elements of H corresponding to H˜ = (H, σ21 ) then r˜x = l S −1 (x) and l˜a = r S(a) , so H˜ A,X := H˜ l˜A H˜ r˜X = Hr A Hl X = H X,A , as Hopf algebras. The remaining details are now trivial. (ii) Although it might seem more natural to try to obtain a braided Hopf algebra cop
structure in
H X,A cop
H X,A cop H X,A
Y D on X A = AX , for this we would need a Hopf algebra map
π : AX → satisfying condition (3.1) in Proposition 3.1. A candidate is π(ax) = ra l S −1 (x) but, in general, it is not well defined. (Take for example X = A = H = (SLq (N ), σ ), the CQT Hopf algebra defined in the last section of this paper.) 5. The Finite Dimensional Case In this section we discuss how our results thus far relate to those of Radford [19] for finite dimensional Hopf algebras. Throughout this section, (H, R) is a finite dimensional quasitriangular (QT for short) Hopf algebra, so its dual linear space H ∗ has a CQT structure given by σ : H ∗ ⊗H ∗ → k, σ ( p, q) = p(R 1 )q(R 2 ), for all p, q ∈ H ∗ , where R := R 1 ⊗ R 2 ∈ H ⊗ H . Thus, we can consider the sub-Hopf algebras Hl∗ and Hr∗ of H ∗∗ defined in the previous section. Identifying H ∗∗ and H via the canonical Hopf algebra isomorphism : H → H ∗∗ , (h)(h ∗ ) = h ∗ (h), for all h ∈ H and h ∗ ∈ H ∗ , we will prove that the sub-Hopf algebras Hl∗ and Hr∗ of H ∗∗ can be identified with the sub-Hopf algebras R(l) and R(r ) of H constructed in [19]. Recall that R(l) := {q(R 2 )R 1 | q ∈ H ∗ } and R(r ) := { p(R 1 )R 2 | p ∈ H ∗ }. Also, note that, if we write R =
m
u i ⊗ vi ∈ H ⊗ H , with m as small as possible, then
i=1
{u 1 , . . . , u m } is a basis for R(l) and {v1 , . . . , vm } is a basis for R(r ) , respectively. We extend {u 1 , . . . , u m } to a basis {u 1 , . . . , u m , . . . , u n } of H and denote by {u i }i=1,n the dual basis of H ∗ corresponding to {u i }i=1,n . Similarly, we extend {v1 , . . . , vm } to a basis {v1 , . . . , vm , . . . , vn } of H and denote by {v i }i=1,n the dual basis of H ∗ corresponding to {vi }i=1,n . Lemma 5.1. For the above context the following statements hold: i) The map µl : Hl∗ → R(l) defined by µl (lq ) = q(R 2 )R 1 , for all q ∈ H ∗ , is a well defined Hopf algebra isomorphism. Its inverse is given by µl−1 (u i ) = lvi , for all i = 1, m. In particular, {lvi }i=1,m is a basis for Hl∗ . ii) The map µr : Hr∗ → R(r ) given by µr (r p ) = p(R 1 )R 2 , for all p ∈ H ∗ , is a well defined Hopf algebra isomorphism. Its inverse is defined by µr−1 (vi ) = ru i , for all i = 1, m. In particular, {ru i }i=1,m is a basis for Hr∗ .
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Proof. We show only i), the proof of ii) being similar. First observe that lq (h ∗ ) = σ (h ∗ , q) = h ∗ (R 1 )q(R 2 ) = (q(R 2 )R 1 )(h ∗ ), for all h ∗ ∈ H ∗ . Thus Hl∗ = {(q(R 2 )R 1 ) | q ∈ H ∗ } = (R(l) ) ∼ = R(l) . Clearly µl is precisely the restriction and corestriction of −1 at (R(l) ) and R(l) , respectively. Hence µl is well defined and a Hopf algebra isomorphism. Finally, for all h ∗ ∈ H ∗ we have lvi (h ∗ ) = σ (h ∗ , v i ) = h ∗ (R 1 )v i (R 2 ) =
m
h ∗ (u j )v i (v j ),
j=1
and therefore lvi = 0, for all i ∈ {m + 1, · · · , n}, and lvi (h ∗ ) = h ∗ (u i ) = (u i )(h ∗ ), i.e. lvi = (u i ), for all i = 1, m. This shows that µl−1 (u i ) = lvi , for all i = 1, m, so the proof is complete. ∗cop
From Proposition 4.4, σ gives a pairing on Hr ⊗ Hl∗ , and we consider the generalized quantum double D(Hr∗ , Hl∗ ). On the other hand, Hr∗ and Hl∗ are finite dimensional Hopf algebras, so we can construct the usual Drinfeld quantum doubles D(Hl∗ ) and D(Hr∗ ). To demonstrate the connections between these Hopf algebras, we first need the following lemma. Lemma 5.2. Suppose that , : U ⊗ V → k defines a duality between the finite dimensional Hopf algebras U and V such that there exists a Hopf algebra isomorphism : V ∗ → U with the property (v ∗ ), v = v ∗ (v), ∀ v ∗ ∈ V ∗ , v ∈ V.
(5.1)
Then D(U cop , V ) ∼ = D(V ) ∼ = D(U op,cop )op as Hopf algebras. Proof. As coalgebras D(U cop , V ) = U cop ⊗ V
−1 ⊗I d ∼ =
V ∗cop ⊗ V = D(V ).
Thus it will be enough to check that ⊗ I d : D(V ) → D(U cop , V ) is an algebra morphism. For all x ∗ , y ∗ ∈ V ∗ and x, y ∈ V we compute ( ⊗ I d) (x ∗ ⊗ x)(y ∗ ⊗ y) = y1∗ (S −1 (x3 ))y3∗ (x1 )(x ∗ y2∗ ) ⊗ x2 y
= (y1∗ ), S −1 (x3 )(y3∗ ), x1 (x ∗ )(y2∗ ) ⊗ x2 y = (x ∗ ) ⊗ x (y ∗ ) ⊗ y = ( ⊗ I d) x ∗ ⊗ x ( ⊗ I d) y ∗ ⊗ y , as needed. Clearly, ⊗ I d respects the units, so ⊗ I d : D(U cop , V ) → D(V ) is a Hopf algebra isomorphism. From [19, Theorem 3] we know that D(U op,cop )op ∼ = D(U ∗ ) as Hopf algebras. But ∗ ∗∗ ∗ ∼ ∼ U = V = V as Hopf algebras, and therefore D(U ) ∼ = D(V ) as Hopf algebras. We conclude that D(U op,cop )op ∼ = D(V ) as Hopf algebras, and this finishes the proof.
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Proposition 5.3. Let (H, R) be a finite dimensional QT Hopf algebra with an R-matrix m R= u i ⊗ vi ∈ H ⊗ H , with m as small as possible. If Hl∗ and Hr∗ are the sub-Hopf i=1
algebras of H ∗∗ ∼ = H defined above then D(Hr∗ , Hl∗ ) ∼ = D(Hl∗ ) ∼ = D((Hr∗ )op )op , as Hopf algebras. Proof. We apply Lemma 5.2 for U = (Hr∗ )cop and V = Hl∗ . So we shall prove that there exists a Hopf algebra morphism : (Hl∗ )∗ → (Hr∗ )cop (or, equivalently, from (Hl∗ )∗cop to Hr∗ ) satisfying (5.1), this means (η), lq = η(q), for all η ∈ (Hl∗ )∗ and q ∈ H ∗ . ∗cop We will use the Hopf algebra isomorphism ξ : R(l) → R(r ) from [19, Prop. 2(c)] ∗ . Note that ξ −1 (v ) = u˜ i , for all i = 1, m, defined by ξ(ζ ) = ζ (R 1 )R 2 , for all ζ ∈ R(l) i where u˜ i is the restriction of u i at R(l) . Now, define : (Hl∗ )∗cop → Hr∗ as the composition of the following Hopf algebra isomorphisms: (µl−1 )∗
Explicitly, we have (η) = m
η(lvi )ru i , lv s =
m
i=1
m i=1
compute (η), lv s to be
−1
µr ξ - R ∗cop R(r ) Hr∗ . (l)
: (Hl∗ )∗cop
η(lvi )ru i , for all η ∈ (Hl∗ )∗ , and this allows us to
η(lvi )σ (u i , v s ) =
i
m
η(lvi )u i (u j )v s (v j ) = η(lv s ),
i, j=1
for all s = 1, m. Since {lv s | s = 1, m} is a basis for Hl∗ it follows that (η), lq = η(q), for all η ∈ (Hl∗ )∗ and q ∈ H ∗ , so the proof is finished. Remark 5.4. Let U and V be two Hopf algebras in duality via the bilinear form , . If t u i ⊗ vi ∈ U ⊗ V such that, for all u ∈ U and v ∈ V , there is an element i=1 t
u i , vvi = v and
i=1
then R =
t
t u, vi u i = u,
(5.2)
i=1
(1 ⊗ vi ) ⊗ (u i ⊗ 1) is an R-matrix for D(U cop , V ).
i=1
Let now (H, R) be a finite dimensional QT Hopf algebra with R = m as small as possible. Then
m i=1
m
u i ⊗ vi , with
i=1
ru i ⊗ lvi ∈ Hr∗ ⊗ Hl∗ satisfies the conditions in (5.2).
Indeed, for all p ∈ H ∗ we have
m m m r p , lvi ru i = p(u j )v i (v j )ru i = p(u i )(vi ) = ( p(R 1 )R 2 ) = r p , i=1
i, j=1
i=1
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and in a similar manner one can verify that R=
m i=1
m i=1
ru i , lq lvi = lq , for all q ∈ H ∗ . Therefore,
(1 ⊗ lvi ) ⊗ (ru i ⊗ 1) is an R-matrix for D(Hr∗ , Hl∗ ).
On the other hand, a basis on (Hl∗ )∗ can be obtained by using the inverse of the Hopf algebra isomorphism constructed in Proposition 5.3. Since −1 (ru i ) = u˜ i ◦ µl it follows that {u˜ i ◦ µl | i = 1, m} is a basis of (Hl∗ )∗ . Moreover, we can easily see that m it is the basis of (Hl∗ )∗ dual to the basis {lvi | i = 1, m} of Hl∗ , so R = (1 ⊗ lvi ) ⊗ i=1
(u˜ i ◦ µl ⊗ 1) is an R-matrix for D(Hl∗ ). Thus, we can conclude that the Hopf algebra isomorphism ⊗ I d : D(Hl∗ ) → D(Hr∗ , Hl∗ ) constructed in Proposition 5.3 is actually a QT Hopf algebra isomorphism, that is, in addition, (( ⊗ I d) ⊗ ( ⊗ I d)) (R) = R. The verification of the details is left to the reader. We are now able to prove that, for our context, the Hopf algebra surjection from Proposition 4.5 is in fact a QT morphism, and that it can be deduced from the surjective QT morphism F : D(R(l) ) → H R := R(l) R(r ) = R(r ) R(l) considered in [19, Theorem ∗ and ∈ R , where ξ(ζ ) = ζ (R 1 )R 2 2]. Namely, F(ζ ⊗ ) = ξ(ζ ), for all ζ ∈ R(l) (l) ∗cop is the Hopf algebra isomorphism from R(l) to R(r ) considered in Proposition 5.3. Proposition 5.5. With the above conditions and notations, the following diagram is commutative:
f
D(Hr∗ , Hl∗ ) −1 ⊗ I d ∼ = ? D(Hl∗ )
- H ∗ := H ∗ H ∗ = H ∗ H ∗ σ r l r l 6 ∼ = µr−1 µl−1
(µl−1 )∗ ⊗ µ F -l D(R(l) ) H R := R(r ) R(l) = R(l) R(r ) ∼ =
Here (µl−1 )∗ is the transpose of µl−1 and µr−1 µl−1 is defined by µr−1 µl−1 ( p(r 1 )r 2 q(R 2 )R 1 ) = r p lq , ∀ p, q ∈ H ∗ , where R = R 1 ⊗ R 2 = r 1 ⊗ r 2 is the R matrix of H . Proof. Straightforward. We only note that p(r 1 )r 2 q(R 2 )R 1 = p (r 1 )r 2 q (R 2 )R 1 iff −1 (r p )−1 (lq ) = −1 (r p )−1 (lq ), iff −1 (r p lq ) = −1 (r p lq ), iff r p lq = r p lq , so µr−1 µl−1 is well defined. Corollary 5.6. Hσ∗ is QT with =
m i=1
lvi ⊗ ru i and in this way f becomes a surjective
QT Hopf algebra morphism. Finally, we are able to show that in the finite dimensional case the morphism π in Proposition 4.6 arises from the particular situation described above.
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Corollary 5.7. Let (H, R) be a finite dimensional QT Hopf algbera with an R matrix m R= u i ⊗vi , m as small as possible. For any sub-Hopf algebra X of H ∗ there exists a i=1
∗cop
surjective Hopf algebra morphism π : D(Hσ ∗cop ∗cop inclusion i Hσ∗ : Hσ → D(Hσ , X ).
∗cop
, X ) → Hσ
covering the canonical
Proof. From Corollary 5.6 and Example 3.3, such a morphism π exists. Moreover, using the definition of π in Example 3.3, for all p, q ∈ H ∗ and h ∗ ∈ X , we have π(l p rq ⊗ h ∗ ) =
m
lu i , h ∗ l p rq l S ∗−1 (vi ) =
i=1
=
m
m
u i (u j )h ∗ (v j )l p rq l S ∗−1 (vi )
i, j=1
h ∗ (vi )l p rq l S ∗−1 (vi ) =
i=1
n
h ∗ (vi )l p rq l S ∗−1 (vi )
i=1
= l p rq l S ∗−1 (h ∗ ) , where S ∗ is the antipode of H ∗ as usual.
Remarks 5.8. (i) Here, the sub-Hopf algebra Hσ∗ of H ∗∗ ∼ = H is the smallest sub-Hopf algebra C of H ∗∗ such that R := ( ⊗ )(R) ∈ C ⊗ C and R played a crucial role in the definition of π . Now, for H CQT not necessarily finite dimensional with sub-Hopf algebras X and A as in Proposition 4.6, we cannot ensure that H X,A has a quasitriangular structure but we can define the projection π in the same way. (ii) Applying Corollary 5.7 with X = H ∗ we see that H ∗ has a braided Hopf algebra ∗cop structure in the braided category of left Yetter-Drinfeld modules over Hσ . Iden∗cop
tifying Hσ
cop
denoted in what follows by H ∗ . From Proposition 4.7 the structure of cop H R Yetter-Drinfeld module is given by
cop
HR cop Y D, HR ∗ H as left
with H R we obtain that H ∗ is a braided Hopf algebra in
· ϕ = (S ∗−1 (ϕ1 )S ∗−2 (ϕ3 ))()ϕ2 = S −2 (2 ) ϕ S −1 (1 ), λ H ∗ (ϕ) = (S ∗−1 (ϕ1 )S ∗−2 (ϕ3 ))(R 2 )R 1 ⊗ ϕ2 = R 1 ⊗ R 2 · ϕ, for all ∈ H R and ϕ ∈ H ∗ , and from Theorem 4.8 the structure of H ∗ as a cop
bialgebra in
HR cop Y D HR
is the following. The multiplication is defined by
ϕ◦ψ = ϕ3 (S(R11 ))ϕ1 (S 2 (R21 ))ψ2 (R 2 )ϕ2 ψ1 = S(R11 ) ϕ S 2 (R21 ) ⊗ R 2 ψ, for all ϕ, ψ ∈ H ∗ , the unit is ε, the comultiplication is given by H ∗ (ϕ) = ϕ1 (S(R11 ))ϕ3 (R21 )ϕ4 (S(R12 ))ϕ6 (R22 )ϕ2 ⊗ ϕ5 = R21 ϕ1 S(R11 ) ⊗ R22 ϕ2 S(R12 ), for all ϕ ∈ H ∗ , and the counit is (1). Finally, according to part 1) of Remarks 4.9, H ∗ is a braided Hopf algbera with antipode S H ∗ (ϕ) = ϕ1 (S(R 1 ))ϕ4 (R12 )ϕ2 (S −1 (R22 ))ϕ3 ◦ S −1 = (R12 ϕ S(R 1 )S −1 (R22 )) ◦ S −1 .
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Using similar arguments to those in Subsect. 4.1 one can easily see that H ∗ coincides with (H cop ) , the categorical right dual of H cop in H cop M, viewed as a braided Hopf algecop
HR cop Y D through HR cop H , the associated
bra in
a composite of two canonical braided functors. (The structures
of enveloping algebra braided group of (H cop , R21 := R 2 ⊗ R 1 ), can be obtained from [17, Example 9.4.9], and then the braided structure of (H cop ) can be deduced from [1] or [2].) More precisely, H ∗ = F(H cop ,R21 ) G(H cop ,R21 ) ((H cop ) ), as braided Hopf algebras in cop
HR cop Y D, HR
where, in general, if (H, R) is a QT Hopf algebra then
(i) F(H, R) : H M → H H M is the braided functor which acts as identity on objects and morphisms and for M ∈ H M, F H (M) = M as H -module, and has the left H -coaction given by λ M (m) = R 2 ⊗ R 1 · m, for all m ∈ M; (ii) G(H,R) : (H,R) M → (H R ,R) M is the functor of restriction of scalars which is braided because the inclusion H R → H is a QT Hopf algebra morphism. The verification of all these details is left to the reader. 6. The Hopf Algebras SLq (N) and Uqext (sl N ) In this section we apply the results in Sect. 4 to the CQT Hopf algebra H = SLq (N ) introduced in [26]. We will show by explicit computation that Hσ ∼ = Uqext (sl N ). The computation should be compared to results of [5, 3.2] and the description of Uqext (sl N ) from [12, Ch. 8, Theorem 33]. Our approach is to study Hσ , a sub-Hopf algebra of H 0 , which is the image of the map ω in [5, 3.2(b)]. The image of the map θ in [5, 3.2(a)] is a sub-bialgebra of (H 0 )cop . Constructing the isomorphism between the images of ω and θ seems to be no simpler computationally than the direct calculations we supply below. To make this section as self-contained as possible, we first recall the definition of the CQT bialgebra Mq (N ), and outline the construction of SLq (N ). For V a k-vector space with finite basis {e1 , . . . , e N } and any 0 = q ∈ k, we associate a solution c of the Yang-Baxter equation, c : V ⊗ V → V ⊗ V , by c(ei ⊗ e j ) = q δi, j e j ⊗ ei + [i > j](q − q −1 )ei ⊗ e j , for all 1 ≤ i, j ≤ N , where δi, j is Kronecker’s symbol and [i > j] is the Heaviside symbol, that is, [i > j] = 0 if i ≤ j and [i > j] = 1 if i > j (see for instance [11, Prop. VIII 1.4]). By the FRT construction, to any solution c of the quantum Yang-Baxter equation we can associate a CQT bialgebra, denoted by A(c), (see [9,11]), obtained by taking a quotient of the free algebra generated by the family of indeterminates (xi j )1≤i, j≤N . For the map c above, A(c) is denoted by Mq (N ) and has the following structure, cf. [11, Exercise 10, p. 197]. As an algebra Mq (N ) is generated by 1 and by (xi, j )1≤i, j≤N . (Note that we write xi, j as xi j if the meaning is clear.) Multiplication in Mq (N ) is defined by the following relations: xim xin = q xin xim , ∀ n < m, x jm xim = q xim x jm , ∀ i < j, x jn xim = xim x jn , ∀ i < j and n < m,
(6.1) (6.2) (6.3)
x jm xin − xin x jm = (q − q −1 )xim x jn , ∀ i < j and n < m.
(6.4)
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The coalgebra structure on the xi j is that of a comatrix coalgebra, that is (xi j ) =
N
xik ⊗ xk j , ε(xi j ) = δi, j ,
(6.5)
k=1
for all 1 ≤ i, j ≤ N . A CQT structure on Mq (N ) is given by the skew pairing σ : Mq (N ) ⊗ Mq (N ) → k defined on generators xim , x jn by σ (xim , x jn ) = q δi, j δm,i δn, j + [ j > i](q − q −1 )δm, j δn,i ,
(6.6)
and satisfying (2.4) to (2.7). Complete details of this construction can be found in [11, Theorem VIII 6.4] or see [5, (3.3)] and [4]. This skew pairing is invertible with inverse obtained in the formulas above by replacing q by q −1 . For z ∈ k such that z N = q −1 , we define another skew pairing σ on Mq (N )⊗ Mq (N ) by defining σ (1, −) = σ (−, 1) = ε and σ (xim , x jn ) = zσ (xim , x jn ) and extending. For example, σ (xim xkl , x jn ) = z 2
N r =1
(6.7)
σ (xim , x jr )σ (xkl , xr n ). Since σ
satisfies (2.7) when h, h are generators xi j , then by [11, Lemma VIII 6.8], σ satisfies (2.7), and σ gives Mq (N ) a CQT structure. Explicitly, σ (xii , xii ) = zq; σ (xii , x j j ) = z for i = j; σ (xi j , x ji ) = z(q − q −1 ) if i < j; σ (xi j , xst ) = 0 otherwise.
(6.8) (6.9) (6.10) (6.11)
The inverse σ −1 is obtained by replacing q by q −1 and z by z −1 . The bialgebra Mq (N ) does not have a Hopf algebra structure, but it possesses a remarkable grouplike central element detq = (−q)−l( p) x1 p(1) · · · x N p(N ) , p∈S N
which allows us to construct SLq (N ) := Mq (N )/(detq − 1). Here S N denotes the group of permutations of order N , and l( p) the number of the inversions of p ∈ S N . As an algebra, S L q (N ) is generated by (xi j )1≤i, j≤N with relations (6.1)-(6.4) and (−q)−l( p) x1 p(1) · · · x N p(N ) = 1. (6.12) p∈S N
The coalgebra structure is the comatrix coalgebra structure. To define the antipode S of SLq (N ), denote X = (xi j )1≤i, j≤N and then define Yi j as the generic (N − 1) × (N − 1) matrix obtained by deleting the i th row and the j th column of X . Then S(xi j ) = (−q) j−i detq (Y ji ) = (−q) j−i (−q)−l( p) x1 p(1) · · · x j−1 p( j−1) x j+1 p( j+1) · · · x N p(N ) , p∈S j,i
(6.13)
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where S j,i is the set of bijective maps from {1, . . . , j − 1, j + 1, . . . , N } to {1, . . . , i − 1, i + 1, . . . , N }. Moreover, for all 1 ≤ i, j ≤ N , we have S 2 (xi j ) = q 2( j−i) xi j .
(6.14)
We include the next lemma to provide complete details of the construction. Lemma 6.1. For σ the skew pairing defined by (6.8) to (6.11), for all x ∈ Mq (N ), σ (detq , x) = σ (x, detq ) = ε(x),
(6.15)
and thus σ is well defined on S L q (N ) ⊗ S L q (N ). Proof. Since detq is a grouplike element, from (2.4) and (2.5), it suffices to check that (6.15) holds on generators. From (6.6) it follows that σ (xim , x jn ) = 0 unless i ≤ m, and if i = m, then σ (xim , x jn ) = q δi, j δn, j . Now we compute σ (detq , xi j ) = (−q)−l( p) σ (x1 p(1) · · · x N p(N ) , xi j ) p∈S N
=
(−q)−l( p)
p∈S N
(−q)
p∈S N , p(N )=N
=q
δN , j
σ (x1 p(1) · · · x N −1 p(N −1) , xik )σ (x N p(N ) , xk j )
k=1
=
N
−l( p)
N
σ (x1 p(1) · · · x N −1 p(N −1) , xik )σ (x N N , xk j )
k=1
(−q)
−l( p)
σ (x1 p(1) · · · x N −1 p(N −1) , xi j ).
p∈S N −1
By induction, it follows that N
σ (detq , xi j ) = q k=1
δk, j
δi, j = qε(xi j ),
and hence, σ (detq , xi j ) = z N qε(xi j ) = ε(xi j ), for all 1 ≤ i, j ≤ N . In a similar manner, using the fact that σ (xim , x jn ) = 0 unless j ≥ n, and if j = n, then σ (xim , x jn ) = q δi, j δm,i , one can prove that N
σ (xi j , detq ) = z q N
k=1
δk, j
δi, j = ε(xi j ).
Thus σ is well defined on the quotient Mq (N )/(detq − 1) and S L q (N ) has a CQT structure. We now apply the results of Sect. 4 to the CQT Hopf algebra H = (S L q (N ), σ ). For all 1 ≤ i, j ≤ N , let us denote ri j := r xi j and li j := l xi j . (Note that we will insert commas in these subscripts only for more complicated expressions.)
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Lemma 6.2. Let H = (S L q (N ), σ ). ri j (xmn ) = z(q − q −1 )δi,n δ j,m , ∀ i < j; li j (xmn ) = z(q − q
−1
)δi,n δ j,m , ∀ i > j;
lii (xmn ) = rii (xmn ) = zq li j = r ji = 0, ∀ i < j.
δi,m
δm,n ;
(6.16) (6.17) (6.18) (6.19)
Proof. Equations (6.16)-(6.18) follow directly from (6.8) to (6.10). Now suppose that i < j. From (6.11), li j and r ji are 0 on generators xmn . Let a, b ∈ {xmn | 1 ≤ m, n ≤ N }. Since σ is a skew pairing, li j (ab) = σ (ab, xi j ) =
N
lik (a)lk j (b).
k=1
Since i < j we cannot have both i ≥ k and k ≥ j, so it follows that the product lik (a)lk j (b) = 0, for all k = 1, N . By mathematical induction it follows that li j = 0, for any i < j. The statement for r ji is proved similarly. Corollary 6.3. The maps lii and rii are equal grouplike elements of H 0 . Also, denoting lii−1 = l S −1 (xii ) = S ∗ (lii ), we have lii−1 (xmn ) = rii−1 (xmn ) = z −1 q −δi,m δm,n .
(6.20)
Proof. The fact that lii and rii are grouplike follows directly from (6.19). Then, since these are algebra maps equal on generators, they are equal. The rest is immediate. The next lemma describes the commutation relations for the generators of Hr and Hl . Lemma 6.4. In Hr the commutation relations for the generators ri j are given by: rim rin = qrin rim , ∀ n < m; r jm rim = qrim r jm , ∀ i < j; r jn rim = rim r jn ; r jm rin − rin r jm = (q − q
−1
(6.21)
)rim r jn , ∀ i < j, n < m. (6.22)
In Hl the commutation relations for the generators are given by: lin lim = qlim lin , ∀ n < m ; lim l jm = ql jm lim , ∀ i < j; lim l jn = l jn lim ; lin l jm − l jm lin = (q − q
−1
(6.23)
)l jn lim , ∀ i < j and n < m. (6.24)
As well, for all i, j, l11 l22 · · · l N N = r11 r22 · · · r N N = ε and rii r j j = lii l j j = l j j lii = r j j rii .
(6.25)
Proof. Relations (6.21) to (6.24) follow from (6.1) to (6.4) together with (4.5), (4.6). The first relation in (6.25) follows from (6.12), (6.19) and Corollary 6.3. The second relation in (6.24) implies that lii l j j − l j j lii = (q − q −1 )l ji li j if i < j and by (6.19) this is 0. Therefore, lii l j j = l j j lii , for all 1 ≤ i, j ≤ N .
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Note that (6.22) and (6.24) imply that r jm rin = rin r jm if i < j, n < m and either i > m or j > n; lin l jm = l jm lin if i < j, n < m and either i < m or j < n.
(6.26)
We now describe another set of algebra generators for Hl and Hr . Proposition 6.5. Let 1 ≤ i ≤ N and 1 ≤ s, t ≤ N − 1 and define: i := lii , K i−1 := l−1 ; K ii −1 E s := l−1 s+1,s+1 ls+1,s = qls+1,s ls+1,s+1 ; −1 −2 −1 Fs := (q − q −1 )−2 r−1 ss rs,s+1 = (q − q ) qrs,s+1 rss .
i , K i−1 , and the E s , with the following relations: As an algebra Hl is generated by K i K j K i K i−1 K 1 K N = ε; j = K i , K i−1 = K i = ε, K 2 · · · K K
(6.27)
i−1 = q δi,t −δi,t+1 E t ; i E t K K E t E s = E s E t , if | s − t |> 1; E s2 E t − (q + q −1 )E s E t E s + E t E s2 = 0, if | s − t |= 1.
(6.28) (6.29) (6.30)
i , K i−1 , and the Fs , with relations Similarly, Hr is generated as an algebra by the K (6.27) and i−1 = q δi,t+1 −δi,t Ft ; i Ft K K Ft Fs = Fs Ft , if | s − t |> 1; Fs2 Ft − (q + q −1 )Fs Ft Fs + Ft Fs2 = 0, if | s − t |= 1.
(6.31) (6.32) (6.33)
Proof. We prove only the statements for Hl ; the proofs for Hr are similar. Note that the i → K i−1 , E s → Fs , is a well defined algebra isomorphism from Hl to Hr . map K Since {li j | i ≥ j} is a set of algebra generators for Hl , it suffices to prove that for any 1 ≤ i ≤ N and 1 ≤ j ≤ N − i the element li+ j,i can be written as a linear ±1 ±1 combination of products of the elements K 1 , . . . , K N , E 1 , . . . , E N −1 to see that these i+1 E i = q −1 E i K i+1 . elements generate. For 1 ≤ i ≤ N − 1, we have li+1,i = K Suppose that li+ j,i can be written as a linear combination of products of the elements ±1 ±1 K 1 , . . . , K N , E 1 , . . . , E N −1 . By the second relation in (6.24), li+ j,i li+ j+1,i+ j − li+ j+1,i+ j li+ j,i = (q − q −1 )li+ j+1,i li+ j,i+ j for all 1 ≤ j ≤ N − i − 1. Now we compute li+ j+1,i =
−1 −1 (q − q −1 )−1 [li+ j,i li+ j+1,i+ j li+ j,i+ j − li+ j+1,i+ j li+ j,i li+ j,i+ j ]
(6.23, 6.24)(q − q −1 )−1 q −1 [l −1 −1 i+ j,i E i+ j li+ j+1,i+ j+1 li+ j,i+ j − li+ j+1,i+ j li+ j,i+ j li+ j,i ] = −1 −1 (6.23) (q 2 − 1)−1 [l −1 i+ j+1 K i+ i+ j+1 K i+ E i+ j K i+ j,i E i+ j K j −q j li+ j,i ], = and the statement follows by mathematical induction.
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Now we show that (6.27) to (6.30) hold. Relations (6.27) are immediate from (6.25). In order to prove (6.28) observe that −1 −1 −1 i E t K i−1 = lii l−1 K t+1,t+1 lt+1,t lii = lt+1,t+1 lii lt+1,t lii .
We now verify (6.28) case by case. Case I. If i < t or i > t + 1, by (6.24) we have lii lt+1,t = lt+1,t lii , and therefore δi,t −δi,t+1 i−1 = l−1 i E t K Et . K t+1,t+1 lt+1,t = E t = q
Case II. If i = t, by (6.23) we have ltt lt+1,t = qlt+1,t ltt , so that −1 δt,t −δt,t+1 −1 t E t K Et . K t = qlt+1,t+1 lt+1,t = q E t = q
Case III. If i = t + 1, again using (6.23) we have lt+1,t lt+1,t+1 = qlt+1,t+1 lt+1,t , and −1 −1 −1 t+1 E t K E t = q δt+1,t −δt+1,t+1 E t . K t+1 = lt+1,t lt+1,t+1 = q
Now to verify (6.29), we first note that if t < s − 1, (6.26) implies that the pairs lt+1,t and ls+1,s+1 , lt+1,t and ls+1,s , and ls+1,s and lt+1,t+1 commute and thus −1 E t E s = l−1 t+1,t+1 lt+1,t ls+1,s+1 ls+1,s = E s E t .
If t > s + 1 then s < t − 1, so interchanging s and t in the argument above we obtain Es Et = Et Es . It remains to prove the relation in (6.30). We first compute E s E s+1
= (6.26) = (6.24) = (6.23, 6.24) =
−1 l−1 s+1,s+1 ls+1,s ls+2,s+2 ls+2,s+1 −1 l−1 s+1,s+1 ls+2,s+2 ls+1,s ls+2,s+1 −1 −1 l−1 s+2,s+2 ls+1,s+1 [ls+2,s+1 ls+1,s + (q − q )ls+2,s ls+1,s+1 ] −1 −1 −1 q −1 l−1 s+2,s+2 ls+2,s+1 ls+1,s+1 ls+1,s + (q − q )ls+2,s+2 ls+2,s .
In other words, we have proved that E s E s+1 = q −1 E s+1 E s + (q − q −1 )l−1 s+2,s+2 ls+2,s ,
(6.34)
for all 1 ≤ s ≤ N − 2. Clearly, this is equivalent to E s+1 E s = q E s E s+1 − q(q − q −1 )l−1 s+2,s+2 ls+2,s , for all 1 ≤ s ≤ N − 2. Using the two relations above we obtain E s2 E s+1 = q −1 E s E s+1 E s + (q − q −1 )E s l−1 s+2,s+2 ls+2,s , E s+1 E s2 = q E s E s+1 E s − q(q − q −1 )l−1 s+2,s+2 ls+2,s E s , from which we compute E s2 E s+1 − (q −1 + q)E s E s+1 E s + E s+1 E s2 −1 = (q − q −1 )[E s l−1 s+2,s+2 ls+2,s − qls+2,s+2 ls+2,s E s ].
(6.35)
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Now, since E s l−1 s+2,s+2 ls+2,s
= (6.26) = (6.23) = (6.24) =
−1 l−1 s+1,s+1 ls+1,s ls+2,s+2 ls+2,s −1 l−1 s+1,s+1 ls+2,s+2 ls+1,s ls+2,s −1 ql−1 s+2,s+2 ls+1,s+1 ls+2,s ls+1,s −1 −1 ql−1 s+2,s+2 ls+2,s ls+1,s+1 ls+1,s = qls+2,s+2 ls+2,s E s ,
it follows that E s2 E s+1 − (q −1 + q)E s E s+1 E s + E s+1 E s2 = 0, for all 1 ≤ s ≤ N − 2. Similarly, again from (6.34) and (6.35), 2 E s = q E s+1 E s E s+1 − q(q − q −1 )E s+1 l−1 E s+1 s+2,s+2 ls+2,s , 2 = q −1 E s+1 E s E s+1 + (q − q −1 )l−1 E s E s+1 s+2,s+2 ls+2,s E s+1 ,
and therefore 2 2 E s+1 E s − (q + q −1 )E s+1 E s E s+1 + E s E s+1 −1 = (q − q −1 )[l−1 s+2,s+2 ls+2,s E s+1 − q E s+1 ls+2,s+2 ls+2,s ].
Now, (6.23) q 2 l−1 −1 q E s+1 l−1 s+2,s+2 ls+2,s s+2,s+2 ls+2,s+1 ls+2,s ls+2,s+2 = (6.23) ql−1 −1 s+2,s+2 ls+2,s ls+2,s+1 ls+2,s+2 = (6.23) l−1 −1 −1 s+2,s+2 ls+2,s ls+2,s+2 ls+2,s+1 = ls+2,s+2 ls+2,s E s+1 , = 2 E − (q + q −1 )E 2 and therefore E s+1 s s+1 E s E s+1 + E s E s+1 = 0, for all 1 ≤ s ≤ N − 2, and this finishes the proof.
Next we fix some notation: Definition 6.6. For any x and y, let [x, y]q := q x y − yx. (i) For 1 ≤ i ≤ j ≤ N , define Fi,i := 1, Fi,i+1 := q(1 − q −2 )2 Fi , and Fi, j := q(1 − q −2 ) j−i+1 [F j−1 , [F j−2 , · · · [Fi+1 , Fi ]q · · ·]q ]q , for j ≥ i + 2, i.e., the Fi, j are defined inductively by Fi, j+1 = (1 − q −2 )[F j , Fi, j ]q . (ii) Similarly, define E j, j := 1, E j+1, j := q −1 E j and, Ei, j := q −1 (q 2 − 1) j−i [· · · [[E j , E j+1 ]q , E j+2 ]q , . . . , E i−1 ]q , for i ≥ j + 2, i.e., the Ei, j are defined inductively by Ei+1, j = (q 2 − 1)−1 [Ei, j , E i ]q . Now, using Proposition 6.5, li j and ri j can be expressed in terms of the new generators ±1 K i , E s and Fs in the spirit of Serre relations. j , and ri j = Fi, j K i . Corollary 6.7. For 1 ≤ i ≤ j ≤ N , l ji = E j,i K
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Proof. We prove the statement for ri j by mathematical induction on j; the similar proof for l ji is left to the reader. If j = i, the statement is clear. If j = i + 1, then the statement i for ri j with follows from Lemma 6.4 and the definitions. Now assume that ri j = Fi j K j ≥ i + 1 and we show that it holds for ri, j+1 . By (6.22), we have r j, j+1 ri, j − ri, j r j, j+1 = (q − q −1 )ri, j+1 r j, j , and thus ri, j+1
=
−1 (q − q −1 )−1 [r j, j+1 ri, j r−1 j, j − ri, j r j, j+1 r j, j ]
(6.21) (q − q −1 )−1 [qr −1 −1 −1 j, j+1 r j, j ri, j − q ri, j r j, j r j, j+1 ] = (6.21) (q − q −1 )[F r − q −1 r F ] j i, j i, j j = = (1 − q −2 )[F j , ri, j ]q i ]q by the induction assumption = (1 − q −1 )[F j , Fi j K (6.31) (1 − q −2 )[F , F ] K j i j q i = i , = (1 − q −2 )Fi, j+1 K and the proof is complete.
The next proposition describes the comultiplication, counit and antipode for Hl and Hr . Proposition 6.8. The coalgebra structure , ε and the antipode S for Hl are determined by i±1 ) = K i±1 ⊗ K i±1 , ε( K i±1 ) = 1, ( K (E s ) = ε ⊗ E s + E s ⊗ i±1 ) S( K
=
i∓1 , K
(6.36)
−1 K s+1 K s ,
S(E s ) =
ε(E s ) = −1 s K s+1 , −E s K
0,
(6.37) (6.38)
for 1 ≤ i ≤ N and 1 ≤ s ≤ N − 1. Similarly, the coalgebra structure , ε and the antipode S for Hr are determined by (6.36), the first equality in (6.38) and −1
s+1 K s ⊗ Fs , ε(Fs ) = 0, (Fs ) = Fs ⊗ ε + K S(Fs ) =
s K −1 −K s+1 Fs .
(6.39) (6.40)
Proof. We give the details for E s and leave those for Fs to the reader. Also, we note that i → K i−1 , E s → Fs , defined in the proof of Proposition 6.5, the algebra isomorphism K cop is actually a Hopf algebra isomorphism between Hl and Hr . Now, we compute: −1 −1 (E s ) = (l−1 s+1,s+1 ls+1,s ) = (ls+1,s+1 ⊗ ls+1,s+1 )
s+1
ls+1,k ⊗ lks
k=s −1 = (l−1 s+1,s+1 ⊗ ls+1,s+1 )(ls+1,s ⊗ lss + ls+1,s+1 ⊗ ls+1,s )
−1 = Es ⊗ K s+1 K s + ε ⊗ E s , as needed. Also, ε(E s ) = ε(l−1 s+1,s+1 ls+1,s ) = δs+1,s = 0, for all 1 ≤ s ≤ N − 1. The formula for S(E s ) is then immediate.
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−1 −1 i+1 i K i+1 Remark 6.9. For 1 ≤ i ≤ N − 1 we now define K i := K , and then Ki = K ±1 define Hl to be the subalgebra of Hl generated by {K i , E i | 1 ≤ i ≤ N − 1}. From Proposition 6.5 one can easily see that the relations between the algebra generators of Hl are (6.29), (6.30) and
K i±1 K i∓1 = ε, K i K j = K j K i , K i E j K i−1 = q ai j E j ,
(6.41)
for all 1 ≤ i, j ≤ N − 1, where aii = 2, ai j = −1 if | i − j |= 1, and ai j = 0 if | i − j |> 1. Actually, Hl is a sub-Hopf algebra of Hl . The induced Hopf algebra structure is given by (K i±1 ) = K i±1 ⊗ K i±1 , ε(K i±1 ) = 1, (E i ) = ε ⊗ E i + E i ⊗ K i , ε(E i ) = 0, S(K i±1 ) = K i∓1 , S(E i ) = −E i K i−1 , for all 1 ≤ i ≤ N − 1. Similarly, define Hr to be the subalgebra of Hr generated by {K i± , Fi | 1 ≤ i ≤ N − 1}. Then its algebra generators satisfy the relations in (6.32), (6.33), the first two relations in (6.41) and K i F j K i−1 = q −ai j F j , ∀ 1 ≤ i, j ≤ N − 1,
(6.42)
where ai j are the scalars defined above. Moreover, Hr is a sub-Hopf algebra of Hr . The elements K i± are grouplike elements and (Fi ) = K i−1 ⊗ Fi + Fi ⊗ ε, ε(Fi ) = 0, S(Fi ) = −K i Fi , for all 1 ≤ i ≤ N − 1. Also, the map K i → cop algebra isomorphism from Hl to Hr .
K i−1 ,
(6.43)
E i → Fi , is well defined and a Hopf
We can now describe the Hopf algebra structure of Hσ := Hl Hr . Proposition 6.10. The Hopf algebra Hσ := Hl Hr can be presented as follows. It is the i±1 | 1 ≤ i ≤ N } ∪ {E s , Fs | 1 ≤ s ≤ N − 1} with relations algebra generated by { K (6.27)–(6.33) and s K −1 −1 E s Ft − Ft E s = δs,t (q − q −1 )−1 [ K s+1 − K s K s+1 ],
(6.44)
for all 1 ≤ s, t ≤ N − 1. The Hopf algebra structure of Hσ is given by the relations in Proposition 6.8. In other words, Hσ ≡ Uqext (sl N ), the extended Hopf algebra of Uq (sl N ) defined in [12, Sect. 8.5.3]. Proof. It remains only to prove (6.44). It suffices to show first that (6.44) holds when applied to xmn for all 1 ≤ m, n ≤ N , and then to show that if the maps in (6.44) are equal on a and b then they are also equal on ab. The relations from Lemma 6.2 will be key in the following computations. Also we s K −1 will sometimes use the notation K s = K s+1 from Remark 6.9. i (xmn ) = lii (xmn ) = zq δi,m δm,n by (6.18) and K i−1 (xmn ) = l−1 (xmn ) = Recall that K ii z −1 q −δi,m δm,n by (6.20). Thus, E s (xmn ) =
l−1 s+1,s+1 ls+1,s (x mn )
=
N
l−1 s+1,s+1 (x mk )ls+1,s (x kn )
k=1
= q −δm,s+1 (q − q −1 )δs+1,n δs,m = (q − q −1 )δs+1,n δs,m .
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Similarly, we compute Ft (xmn ) = (q − q −1 )−1 δt,n δt+1,m , and therefore E s Ft (xmn ) =
N
E s (xmk )Ft (xkn ) = (q − q −1 )δs,m Ft (xs+1,n )
k=1
= δs,m δt,n δs,t = δs,m δs,t δm,n , and Ft E s (xmn ) =
n
Ft (xmk )E s (xkn ) = (q − q −1 )−1 δt+1,m E s (xtn )
k=1
= δt+1,m δs+1,n δs,t = δs+1,m δs,t δm,n . We conclude that (E s Ft − Ft E s )(xmn ) = δs,t [δs,m − δs+1,m ]δm,n .
(6.45)
On the other hand, s K −1 K s+1 (x mn ) =
N
−1 s (xmk ) K K s+1 (x kn )
k=1
= zq δs,m δm,k z −1 q −δs+1,m δk,n = q δs,m −δs+1,m δm,n . δs+1,m −δs,m δ −1 Similarly, K m,n . Thus s K s+1 (x mn ) = q δs,m −δs+1,m −1 s K −1 − q δs+1,m −δs,m ]δm,n . [K s+1 − K s K s+1 ](x mn ) = [q
Since q δs,m −δs+1,m − q δs+1,m −δs,m
⎧ , if m ∈ {s, s + 1} ⎨0 , if m = s = q − q −1 , ⎩ −(q − q −1 ) , if m = s + 1
it follows that δs,t (q − q −1 )−1 [K s − K s−1 ](xmn ) = δs,t [δs,m − δs+1,m ]δm,n . Together with (6.45) this proves that (6.44) holds on generators. Since the K s are grouplike, the right hand side of (6.44) is (K s , K s−1 )-primitive. Also the left hand side is (K s , K t−1 )-primitive. This follows from the fact that the matrix (ai j )1≤i, j≤N defined in Remark 6.9 is symmetric, and thus K t−1 E s ⊗ Ft K s = q −ats E s K t−1 ⊗ q ast K s Ft = E s K t−1 ⊗ K s Ft . If s = t, then both sides of (6.44) are 0 on generators a, b ∈ S L q (N ), and thus, being skew-primitive, on ab. Otherwise, both sides are (K s , K s−1 )-primitive and thus equal on the product ab. The statement then follows by induction. From Proposition 6.8 and Proposition 6.10 we conclude that the Hopf algebras Hl and Hr are precisely Uq (b− ) and Uq (b+ ), the Borel-like Hopf algebras associated to Uqext (sl N ). The fact that Uq (b+ )cop and Uq (b− ) are Hopf algebras in duality is well known but using the general theory above we are now able to present a more conceptual proof.
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Corollary 6.11. The pairing , : Uq (b+ ) ⊗ Uq (b− ) → k defined by −1
−1
−1
−1
i , K j = K i , K j = zq δi, j , K i , K j = K i , K j = z −1 q −δi, j , K i , E s = K i−1 , E s = 0, Fs , K j = Fs , K −1 K j = 0, Fs , E t = (q − q −1 )−1 δs,t , for all 1 ≤ i, j ≤ N and 1 ≤ s, t ≤ N − 1, defines a duality between Uq (b+ )cop and Uq (b− ) as in Proposition 4.4. Also, Uq (b− ) is self dual, in the sense that there is a duality between Uq (b− ) and itself given for all 1 ≤ i, j ≤ N and 1 ≤ s, t ≤ N − 1 by −1 −δi, j δi, j j = K i−1 , K −1 i , K −1 −1 i , K , K , K j =z q j = K i , K j = zq −1
−1
i , E s = K i , E s = E s , K i = E s , K i = 0, K E s , E t = (q − q −1 )−1 δs,t . Proof. The first set of equations follows from Proposition 4.4 and from (6.8)- (6.11). We verify only the third equation of the first set. Note first that rss , li j = 0 unless i = j by (6.8)- (6.11) so that r−1 ss , E t = 0. Now, Fs , E t
= (6.37) = = =
(q − q −1 )−2 r−1 ss rs,s+1 , E t −1
−1 (q − q −1 )−2 [r−1 ss , εrs,s+1 , E t + rss , E t rs,s+1 , K t+1 K t ]
(q − q −1 )−2 rs,s+1 , l−1 t+1,t+1 lt+1,t (q − q −1 )−2 [rss , l−1 t+1,t+1 rs,s+1 , lt+1,t +rs,s+1 , l−1 t+1,t+1 rs+1,s+1 , lt+1,t ] −1
t+1 (xss )lt+1,t (xs,s+1 ) = (q − q −1 )−2 K (6.17, 6.18) (q − q −1 )−2 z −1 q −δs,t+1 z(q − q −1 )δ = (q − q −1 )−1 δ , s,t s,t = cop
as we claimed. The second set of equations follows from the isomorphism between Hr and Hl described at the beginning of the proof of Proposition 6.8.
Corollary 6.12. The Hopf algebra Uqext (sl N ) is isomorphic to a factor of the generalized quantum double D(Uq (b+ ), Uq (b− )) ≡ D(Uq (b− )cop , Uq (b− )). Proof. It is an immediate consequence of Corollary 6.11 and Proposition 4.5.
Remark 6.13. Further to Remark 6.9, define Hσ = Hl Hr = Hσ = Hr Hl , a sub-Hopf algebra of Hσ ≡ Uqext (sl N ). As expected, Hσ ≡ Uq (sl N ), the Hopf algebra with algebra generators {K i±1 , E i , Fi | 1 ≤ i ≤ N − 1} and relations from Remark 6.9 as well as E i F j − F j E i = δi j
K i − K i−1 , ∀ 1 ≤ i, j ≤ N − 1, q − q −1
from Proposition 6.10. The comultiplication, the counit and the antipode are defined in Remark 6.9. (For more details see [11, VI.7& VII.9].)
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155
Note that Hr and Hl are exactly the Borel-like Hopf algebras associated to Uq (sl N ) cop and the situation is similar to that of Corollary 6.11, i.e., there is a duality on Hr ⊗ Hl given by −1 −1 ai j −ai j K i , K j = K i−1 , K −1 , j = q , K i , K j = K i , K j = q
K i , E s = K i−1 , E s = Fs , K i = Fs , K i−1 = 0, Fs , E t = (q − q −1 )−1 δs,t , cop
for all 1 ≤ i, j ≤ N and 1 ≤ s, t ≤ N − 1. Moreover, since Hr is isomorphic to Hl , then Hl is self dual and the generalized quantum double D(Hr , Hl ) ≡ D((Hl )cop , Hl ). Furthermore, one can easily see that the associated generalized quantum double D(Hr , Hl ) can be identified as a sub-Hopf algebra of D(Hr , Hl ); hence we have a Hopf algebra morphism from D(Hr , Hl ) to Hσ . The image of this morphism is Hσ , so Uq (sl N ) is also a factor of a generalized quantum double. Corollary 6.14. The evaluation pairings on Uqext (sl N )⊗SLq (N ) and Uq (sl N )⊗SLq (N ), are given by −1
i , xmn = zq δi,m δm,n , K i , xmn = z −1 q −δi,m δm,n , K E s , xmn = (q − q −1 )δs+1,n δs,m , Fs , xmn = (q − q −1 )−1 δs,n δs+1,m ; and K i , xmn = q δi,m −δi+1,m δm,n , K i−1 , xmn = q δi+1,m −δi,m δm,n , E i , xmn = (q − q −1 )δi+1,n δi,m , Fi , xmn = (q − q −1 )−1 δi,n δi+1,m , respectively. Proof. Both Uqext (sl N ) and Uq (sl N ) are sub-Hopf algebras of SLq (N )0 , so the evaluation map gives dual pairings. The formulas for the pairings come directly from Lemma 6.2 or from the proof of Proposition 6.10. From Theorem 4.8, SLq (N ) is a braided Hopf algebra in the braided category of left Yetter-Drinfeld modules over Uqext (sl N )cop . We end this paper by computing the structures of the braided Hopf algebra SLq (N ). First we need the following. Lemma 6.15. For any 1 ≤ s ≤ N − 1 and 1 ≤ m, n ≤ N we have ls+1,s (S(xmn )) = −z −1 (q − q −1 )δs+1,n δs,m ,
(6.46)
rs,s+1 (S(xmn )) = −z −1 q −2 (q − q −1 )δs,n δs+1,m .
(6.47)
Proof. As usual, we only prove (6.46); (6.47) can be proved similarly. Recall that σ −1 is obtained by replacing q by q −1 and z by z −1 in (6.6) and (6.7). We compute ls+1,s (S(xmn )) = σ (S(xmn ), xs+1,s ) = σ −1 (xmn , xs+1,s ) = z −1 [q −δm,s+1 δm,n δs,s+1 + [s + 1 > m](q −1 − q)δs+1,n δs,m ] = −z −1 (q − q −1 )δs+1,n δs,m , since [s + 1 > m]δs,m = δs,m , so the proof is complete.
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Now we can describe concretely the left Yetter-Drinfeld module structure of SLq (N ) over Uqext (sl N )cop . Proposition 6.16. SLq (N ) is a left Yetter-Drinfeld module over Uqext (sl N )cop via the structure i xmn = q δi,n −δi,m xmn , K i−1 xmn = q δi,m −δi,n xmn , K E s xmn = (1 − q −2 )[q −1 δs+1,n xms − q δs,n −δs+1,n δs,m xs+1n ], Fs xmn = q(1 − q −2 )−1 [q δs,m −δs+1,m δs,n xms+1 − q −1 δs+1,m xsn ], N −1 j K xmn −→ q 2( j−n) E j,n K m ⊗ xm j j=n
+
−1
i (−q)−l( p)+(m−i) Em, p(m) · · · Ei+1, p(i+1) K
⊗ xi j ,
1≤i≤m−1 p∈Bi,m
where the Ei, j are from Definition 6.6. Also for i ≤ m −1, Bi,m denotes the set of bijective maps p : {i + 1, · · · , m} → {i, · · · , m − 1} such that p(k) ≤ k, for all i + 1 ≤ k ≤ m. Proof. We apply Proposition 4.7 to this setting. By (4.13), and recalling again from i±1 (xmn ) = (zq δi,m )±1 δm,n for 1 ≤ i, m, n ≤ N , we have (6.18) and (6.20) that K i±1 xmn = K
N
i±1 (S −1 (xm j )S −2 (xkn ))x jk = K
j,k=1
=q
N
i±1 (xkn )x jk i∓1 (xm j ) K K
j,k=1
±(δi,n −δi,m )
xmn .
From (6.46), E s (S(xmn )) =
l−1 s+1s+1 ls+1s , S(x mn )
=
N
s+1 (xin )ls+1s (S(xmi )) K
i=1
= zq δs+1,n ls+1s (S(xmn )) = −q δs+1,n (q − q −1 )δs,m δs+1,n = −q(q − q −1 )δs,m δs+1,n . Now, we use the above computation together with the fact that E s (xmn ) = (q − q −1 )δs+1,n δs,m and K s (xmn ) = q δs+1,m −δs,m δs,m from the proof of Proposition 6.10, to compute the action of E s on xmn , E s xmn (4.13) =
N
E s (S −1 (xmi )S −2 (x jn ))xi j
i, j=1 −1 (6.37, 6.14) [q 2( j−n) δ E (x ) + q 2(m−i)+2( j−n) E (S(x )) K m,i s jn s mi s+1 K s (x jn )]x i j = N
i, j=1
=
q −1 (q − q −1 )[q −1 δs+1,n xms − δs,m
N
q 2( j−n) K s (x jn )xs+1 j ]
j=1
=
(1 − q
−2
)[q
−1
δs+1,n xms − q
δs,n −δs+1,n
δs,m xs+1n ],
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157
as claimed. Similarly, one can compute Fs xmn ; the verification of the details is left to the reader. Finally, from (4.14) and (6.14), we see that the coaction of Uqext (sl N )cop on SLq (N ) is defined by N
xmn →
l S −1 (xmi )S −2 (x jn ) ⊗ xi j =
i, j=1
q 2( j−n)+2(m−i) l jn l S(xmi ) ⊗ xi j .
1≤i, j≤N
From (6.13), l S(xmi ) = (−q)i−m
(−q)−l( p) lx1, p(1) ···xi−1, p(i−1) xi+1, p(i+1) ···x N , p(N )
p∈Si,m
= (−q)
i−m
(−q)−l( p) l N , p(N ) · · · li+1, p(i+1) li−1, p(i−1) · · · l1, p(1) ,
p∈Si,m
and this forces k ≥ p(k), for any k = i. Since p is bijective, non-zero summands occur only when (i) i = m and p(k) = k, for all k = i; (ii) i ≤ m − 1 and p(1) = 1, · · · , p(i − 1) = i − 1, { p(i + 1), · · · , p(m)} = {i, · · · , m − 1}, p(m + 1) = m + 1, · · · , p(N ) = N , and p(k) ≤ k, for any k ∈ {i + 1, · · · , m}. In other words, we have proved that l S(xmm ) = l−1 mm , and that for 1 ≤ i ≤ m − 1, l S(xmi ) = (−q)i−m
(−q)−l( p) l N N · · · lm+1,m+1 lm, p(m) · · · li+1, p(i+1) li−1,i−1 · · · l11 .
p∈Bi,m
From (6.26), for i + 1 ≤ k ≤ m, m + 1 ≤ t ≤ N and 1 ≤ s ≤ i − 1 we have lkp(k) ltt = ltt lkp(k) and lss lkp(k) = lkp(k) lss , so that, using (6.25), l S(xmi ) = (−q)i−m
(6.48)
(−q)−l( p) lmp(m) · · · li+1 p(i+1) lii−1 · · · l−1 mm ,
p∈Bi,m
k , for all for 1 ≤ i ≤ m − 1. Now, by Corollary 6.7 we have lkp(k) = Ek, p(k) K k ∈ {i + 1, . . . , m}. In particular, if i = m − 1 we then have −1 −1 −1 −1 l S(xmi ) = −q −1 lmp(m) K m−1 K m = −q Em, p(m) K i .
s E t = E t K s , for all s > t + 1, Now let 1 ≤ i ≤ m − 2. From (6.28) it follows that K m−t , for all so from the definition of Ei, j we conclude that K m−t Ek, p(k) = Ek, p(k) K 0 ≤ t ≤ m − i − 2 and i + 1 ≤ k ≤ m − t − 1. Consequently, i−1 , l S(xmi ) = (−q)i−m (−q)−l( p) Em, p(m) · · · Ei+1, p(i+1) K p∈Bi,m
for all 1 ≤ i ≤ m − 1. Again using Corollary 6.7 for l jn and the formulas above for l S(xmi ) we obtain the coaction in the statement of the theorem.
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We now apply the results of Theorem 4.8 to obtain the multiplication, comultiplication, unit, counit and antipode for SLq (N ). Theorem 6.17. The structure of SLq (N ) as a braided Hopf algebra in the category of left Uqext (sl N )cop Yetter-Drinfeld modules is the following. The unit and counit are as in SLq (N ). The multiplication is defined by xim ◦ x jn = q δi,n −δm,n xim x jn + (q − q −1 )δi,n
q 2(s−i)−δs,m xsm x js
s>n
−[m > n]q δi,n (q − q −1 )x
−1 2 in x jm − [m > n + 1](q − q ) δi,n
q 2(s−i) xss x jm.
n<s<m
Comultiplication is given by q 2s+δs,i +δs,m xis ⊗ xsm + (q − q −1 )[xim ⊗ x+>m (xim ) = q −2N −1 {q −δi,m − δi,m
s
q
2(s+δs,i )
s;t>i
− [i > m]q
2i+1
xts ⊗ xst + x+>i ⊗ xim ] + (q − q −1 )2 [ xsm ⊗ xis − [m > i]q
2m+1
q 2s−1 xsm ⊗ xis
s>i,m
xsm ⊗ xis ]
s>m
s>i
− (q − q −1 )3
q 2s xtm ⊗ xit },
t>s>i,m
where we denoted
x+>k
:=
q 2s xss . The antipode is determined by
s>k
>m S(xim ) = q [q −2m−δi,m S(xim ) − (q − q −1 )δi,m S(x− )], −2s := q xss and S is the antipode of SLq (N ). 2N +1
>m where x−
s>m
Proof. From Lemma 6.2, li j (xmn ) = zq δm,i δm,n δi, j + [i > j]z(q − q −1 )δn,i δm, j , li−j (xmn ) = z −1 q −δm,i δm,n δi, j − [i > j]z −1 (q − q −1 )δn,i δm, j , where li−j denotes the map σ −1 (−, xi j ) = l S −1 (xi j ) . Now, the multiplication of SLq (N ) from Theorem 4.8 can be rewritten as x ◦ y = σ −1 (x3 , y2 )σ (S 2 (x1 ), y3 )x2 y1 , and together with (6.14) this allows us to compute xim ◦ x jn =
N
q 2(s−i) σ −1 (xtm , xuv )σ (xis , xvn )xst x ju
s,t,u,v=1
=
N
q 2(s−i) l− uv (x tm )lvn (x is )x st x ju
s,t,u,v=1
=
N s,t,u,v=1
q 2(s−i) q −δt,u δt,m δu,v − [u > v](q − q −1 )δm,u δv,t
× q δi,v δi,s δv,n + [v > n](q − q −1 )δs,v δi,n xst x ju .
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Splitting the sum above into four separate sums, we obtain the formula in the statement of the theorem. The computation of (xim ) is much more complicated. Firstly, observe that the comultiplication in Theorem 4.8 can be rewritten as (x) = σ (x1 , x6 )σ −1 (x2 , x8 )σ (x4 , x9 )v(x5 )x3 ⊗ x7 ,
(6.49)
where v is the map v(y) = σ (y1 , S(y2 )) as in (2.8). Next, we compute v(xim ) =
N
σ (xik , S(xkm )) =
k=1
=
N
N
q 2(m−k) l− km (x ik )
k=1
z −1 q 2(m−k) (q −δi,k δi,k δk,m − [k > m](q − q −1 )δi,m )
k=1
= z −1 [q −δi,m − q(1 − q −2 )
(q −2 )k−m ]δi,m
k>m
= z −1 [q −1 − q −1 (1 − (q −2 ) N −m )]δi,m = z −1 q −2(N −m)−1 δi,m , for all 1 ≤ i, m ≤ N . Therefore, by (6.49) we have (xim ) =
N
le f (xia )l− gh (x ab )lhm (x cd )v(x de )x bc ⊗ x f g
a,···,h=1
=
N
q −2(N −d)−1 q δi,d δi,a δd, f + [d > f ](q − q −1 )δd,a δi, f
a,b,c,d, f,g,h=1
× q −δa,g δa,b δg,h − [g > h](q − q −1 )δg,b δh,a × q δc,h δc,d δh,m + [h > m](q − q −1 )δh,d δc,m xbc ⊗ x f g . Again, splitting the above sum into eight separate sums, a tedious but straightforward computation yields the formula for (xim ) in the statement of the theorem. Finally, the equation for S(xi j ) follows easily by the general formula in Theorem 4.8 and by computations similar to the ones above. Acknowledgement. Thanks to the referees for their helpful comments.
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Commun. Math. Phys. 282, 161–198 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0530-5
Communications in
Mathematical Physics
On Adiabatic Pair Creation Peter Pickl1,2 , Detlef Dürr2 1 Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090 Vienna, Austria 2 Mathematisches Institut der Universität München, Theresienstraße 39, 80333 München, Germany.
E-mail: [email protected]; [email protected] Received: 29 May 2007 / Accepted: 4 February 2008 Published online: 17 June 2008 – © Springer-Verlag 2008
Abstract: We give here a rigorous proof of the well known prediction of pair creation as it arises from the Dirac equation with an external time dependent potential. Pair creation happens with probability one if the potential changes adiabatically in time and becomes overcritical, which means that an eigenvalue curve (as a function of time) bridges the gap between the negative and positive spectral continuum. The potential can be thought of as being zero at large negative and large positive times. The rigorous treatment of this effect has been lacking since the pioneering work of Beck, Steinwedel and Süßmann [1] in 1963 and Gershtein and Zeldovich [8] in 1970. Contents 1. 2. 3. 4. 5. 6. 7. 8.
Introduction . . . . . . . . . . . . . . . . . . The Result . . . . . . . . . . . . . . . . . . . Skeleton of the Proof and Content of Sections Generalized Eigenfunctions . . . . . . . . . . Propagation Estimates . . . . . . . . . . . . . µ-Convergence of the Eigenspaces . . . . . . Proof of Theorem 2.4 . . . . . . . . . . . . . Appendix: Proof of Lemma 5.1 (24) . . . . . .
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161 165 167 168 170 176 183 194
1. Introduction Adiabatic pair creation (APC) has been called—unfortunately misleading—spontaneous pair creation (see e.g. [9] and references therein). The creation of electron positron pairs in very strong external classical electromagnetic fields arises straightforwardly from the Dirac sea interpretation of negative energy states. After Dirac [5] it has been discussed as an academic problem by Klein [15], Sauter [33] , Heisenberg and Euler [11], Schwinger
162
P. Pickl, D. Dürr
[36] and Brezin and Itzykson [3]. A more realistic setting was considered by Beck, Steinwedel and Süßmann [1] and worked out by Gershtein and Zeldovich [8] as APC. In the common physics language using the so called Dirac sea picture APC may be shortly described as follows (we explain this below in much greater detail): An adiabatically increasing electric potential lifts a particle from the sea to the positive energy subspace where it scatters and when the potential is gently switched off one has one free electron and one unoccupied state—a hole—in the sea. The experimental verification needs very strong classical fields [9] and is discussed elsewhere [23]. In this respect we would like to remark that a coherent analysis of the existence of APC has been lacking until recently [21]. Earlier quantitative results based on an ad hoc and incoherent analysis (see for example [16,17,26,29,35,37]) give an incorrect rate and distribution of outgoing momenta of the spontaneously created pairs (see [23] for a short discussion). For the case of a potential well and for the long range Coulomb potential correct rates have been obtained in [24,42]. There are also results and discussions in the mathematical physics literature related to APC, notably [18,19,25,34]. But those results are for models which avoid the adiabatic transition of the potential from undercritical to overcritical values and thus do not come to grips with the heart of the problem of APC, which is the control of the wavefunction evolution within the neighborhood of the spectral edge mc2 . In APC one considers the so called external field problem, where interactions between the charges are neglected. Vacuum polarization will in general perturb the external field and - using mean field approximation -(see [10]) one may think of the external field as an effective field. The existence of APC in second quantized external field Dirac theory (if the latter exists1 ) is equivalent to the existence of certain types of solutions of the Dirac equation (see e.g. [18] and [21]) which we describe below. The existence of APC in terms of the second quantised S-matrix theory of the Dirac equation with external field is “by definition” equivalent to the existence of these types of solutions of the Dirac equation. We shall in fact formulate our result in terms of the solutions of the Dirac equation and use the Dirac sea picture for the interpretation of the particular solution the existence of which we prove in this paper. Consider the Dirac equation with an external electric field. Then the potential A can be chosen as a real valued multiple of the 4 × 4 unit matrix. (We wish to note that the results can be extended to general four potentials. Concerning strong magnetic fields we wish to call attention to the recent work of Dolbeault et.al. [6] as well as [23]). Amc2 gives the potential in the units eV. We assume that the potential A varies slowly with time, expressed by Aετ , where ε is a dimensionless small parameter (given by the physics, see [23] for some examples) which in this work will eventually be sent to zero 2 mc to obtain limit results. Here τ = mc t and x = r are the dimensionless microscopic time- and space-scales and the Dirac equation in the standard representation reads with the notation x = (x1 , x2 , x3 ) ∈ R3 and ∂l := ∂∂x : l
∂ψτ (x) = −i αl ∂l ψτ (x) + Aετ (x)ψτ (x) + βψτ (x) ∂τ 3
i
l=1
=: (D0 + Aετ (x))ψτ (x) .
(1)
1 It is well known that the lifting of the Dirac evolution (with a smooth field of compact support) to Fock space (second quantisation) is possible if and only if the Shale-Stinespring condition is satisfied [40], which is the case if and only if the magnetic field vanishes. On the other hand, the S-matrix can always be lifted.
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We introduce in (1) the macroscopic time scale s = ετ . We wish to restrict ourselves to potentials As which can be factorized into a space- and a time dependent factor As (x) := A(x)µ(s), a restriction of a technical nature which eases notations and computations and which furthermore helps to picture a spatial potential well, which changes its depth with time. With the new time scale we have i
1 1 ∂ψs (x) = (D0 + A(x)µ(s))ψs (x) =: Dµ(s) ψs (x). ∂s ε ε
(2)
Furthermore we wish to restrict ourselves to potentials As (x) which are smooth, bounded, compactly supported in x and s, and positive. The spectrum of the free Dirac operator D0 is absolutely continuous and given by (−∞, −1] ∩ [1, ∞), defining “negative and positive energy” subspaces. In the Dirac sea interpretation wavefunctions which lie in the positive energy subspace of the free Dirac operator are interpreted as wavefunctions of electrons. The “vacuum” of the second quantized Dirac equation corresponds in the Dirac sea picture to all “states of the negative energy subspace being occupied by particles”–the Dirac sea. “Holes” in the Dirac sea are unoccupied negative energy states which are interpreted as anti-electrons, i.e. positrons. The goal of our paper is to assert that there exist solutions of (2) which describe pair creation. We explain what that means looking at Fig. 1. Consider first the spectrum of the time dependent Dirac operator Dµ(s) . At large negative and large positive times when As = 0, Dµ(s) = D0 and the spectrum is that of D0 . At times at which As = 0 there may be eigenvalues in the gap [−1, 1], while the continuous spectrum remains unchanged. The eigenvalues change with the strength of Energy (mc 2 ) Scattering state
1
φ 0
overcritical
undercritical
-1
s
sc
E(s) of bound state
Hole
Fig. 1. Schematic presentation of the adiabatic pair creation. It depicts the spectrum of the Dirac operator Dµ(s) as function of s. Depending on the strength of the potential there may exist bound state energy curves E(s), one or more of which may bridge the spectral gap (overcritical case). Also schematically drawn are bound states Φ at various undercritical times. No bound states exist in the lower and upper spectral continua (−∞, −1) and (1, ∞). For more information see the text
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the potential, i.e. with time s (bound state energy curve E(s) in Fig. 1). Suppose first that no eigenvalue reaches 1, i.e. no bound state energy curve bridges the gap (undercritical case). The adiabatic theorem (see e.g. [38]) ensures that there is no tunneling across spectral gaps meaning that the bound states stay more or less intact when the potentials change adiabatically. In terms of solutions of the Dirac equation (2) this means the following: When ε goes to zero, there exists no solution which for large negative times lies completely within the negative continuous energy subspace and for large positive times has parts in the positive continuous energy subspace. That means in the Dirac sea interpretation that the probability of creating a pair is zero. However, when the external field becomes overcritical (at time sc in Fig. 1), i.e. when the largest eigenvalue curve reaches the positive continuum, the bound state ceases to exist and becomes a continuum state (a “resonance”) in the positive continuum subspace. Then there exists a solution of the Dirac equation which follows adiabatically the path of this bound state, which for large negative times must develop into a wavefunction which lies entirely in the negative continuum energy subspace and for positive times may have a part in the positive continuum energy subspace. As indicated in Fig. 1 when the potential decreases with increasing time there is again a bound state energy curve bridging the gap. In principle the solution of the Dirac equation could have a part which “follows” the bound state back into the negative continuous energy subspace and remains there when the potential is switched off. In the Dirac sea interpretation such a solution of the Dirac equation would correspond to pair creation with a probability determined by the absolute squares of the two parts of the wave function. We show however, that no such “back sliding” is adiabatically possible, i.e. no such solution of the Dirac equation exists. The former bound state scatters in the positive continuum energy subspace, i.e. it stays there for all later times (“Scattering state” in Fig. 1). In the Dirac sea picture the “returning” bound state remains for sure empty and upon becoming a state within the negative continuum energy subspace there is now an unoccupied state in the sea (“Hole” in Fig. 1): APC is accomplished with probability one. One pictorial way to describe APC is to imagine the potential acting as an elevator, lifting a particle from the sea to the “upper”(positive) continuum. The scenario is symmetric under change of sign of the potential: It then transports an unoccupied state (a hole) from the positive continuum to the sea and catches a particle from the sea when it is switched off. The hole (positron) then scatters. We understand now the type of solution of (2) we wish to study, namely a solution which at some moment of time at which an overcritical bound state exists is equal to that bound state. The scenario described above translates mathematically into the task to establish scattering for such solutions of the time-inhomogeneous Dirac equation (2). To show to what extent the scenario of APC holds one must control first that the bound states stay on the adiabatic time scale intact until the time sc the eigenvalues reach the positive continuum. That is the content of an adiabatic lemma without a gap and “relatively easy” to establish. The solution of (2) is thus adiabatically essentially represented by the “time dependent bound states” until sc . Then we must control the propagation of the wavefunction (the resonance) emerging from the bound state during over-criticality. We wish to show that it scatters. This task is difficult, since the Dirac operator changes with time. The time evolution will be controlled by generalized eigenfunctions, i.e. by a stationary phase argument, which is not standard because the generalized eigenfunctions themselves depend now on time. But we must also take into account the bad (resonant) behavior of the generalized eigenfunctions near criticality. (We wish to note that also [32] is concerned with the wavefunction propagation for time dependent Hamiltonians but under generic smallness assumptions on the potential, assumptions which are not
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fulfilled in our problem). They become unbounded for critical k-values (which are small) and hence the situation is very much different from the usual scattering situation governed by “plane waves” (see [23] for a heuristic argument giving some intuition). As we shall find out, the decay time of a wave, say from a bounded spatial region (i.e. the time in which roughly half of the mass left the region), is now of the order of ε−2/3 in units of the microscopic time as compared to O(1) in the common plane wave scattering situation. This means that the resonance lingers around the range of the potential for a much longer time than in the usual scattering of wavefunctions. Such a metastable state decay has already been suggested by [1]. We shall give in the next section the result: Theorem 2.4 and Corollary 2.5. The rest of the paper is devoted to the proof of the theorem. The proof is technically very involved. Instead of describing here what is in the sections to follow we first give the result and then give in Sect. 3 a skeleton of the proof with a description of the contents of the sections. 2. The Result We begin with Notation 2.1. For any a ∈ R3 we write its norm as |a| = a. The functions we mainly consider are spinors in the space L n (R3 , C4 ), n = 1, 2, ∞. We shall denote this space if no ambiguity arises simply by L n . We shall have two scalar products: (i) For a, b ∈ C4 : ab := 4α=1 aα∗ bα , where ∗ denotes complex conjugation. √ For a ∈ C4 : |a| = aa (ii) For ψ, χ ∈ L 2 : ψ, χ := ψ(x)χ (x)d 3 x, ψ = √ ψ, ψ. We shall also consider the operator norm · op of operators D from L 2 to 2 L given by D op = sup Φ =1 DΦ . Warning: Constants appearing in estimates will generically be denoted by C. We shall not distinguish constants appearing in a sequence of estimates, i.e. in X ≤ CY ≤ C Z the constants may differ. In the following we will only consider potentials which are bounded, compactly supported, positive and purely electrostatic. The latter implies that A will be a multiple of the unit matrix (since we stick to one inertial frame throughout the paper). Thus A can be written as a scalar function. To have the possibility of pair creation the external (scalar) field µ(s)A has to become critical for some time s and the first such time will be set s = 0 and we choose µ(0) = 1, i.e. A is critical. Criticality means for us that the (generalized) eigenvalue equation for the energy-eigenvalue 1, (D0 + A)Φ = Φ ,
(3)
has only bound states solutions (i.e. L 2 -solutions). In general there could be also resonances [14], but generically this is not the case for the critical Dirac operator (in our setting). We shall now collect the conditions in a form most convenient for our considerations. Condition 2.2. For A : R3 → R+ and µ : R → R we shall require that (i) A has compact support S A ; A, ∇ A are bounded and A is critical. Furthermore D1 = D0 + A has no resonances for E = 1 and E = 1 is an n-fold degenerate eigenvalue for some n ∈ N with eigenspace denoted by N : N := {Φ ∈ L 2 : (D0 + A − 1)Φ = 0}.
(4)
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(ii) There exists a δ > 0 such that for any µ ∈ [0, 1 + δ] there exist not more than one eigenvalue E µ of the operator Dµ = D0 + µA. Warning: We shall use the symbol µ as a fixed parameter and as a function µ(s). (iii) µ : R → R is continuously differentiable, its derivative µ is bounded, µ(0) = 1 and µ (0) > 0. There exists si < 0 and s f > 0 such that µ(s) = 0 if s < si or s > sf. Remark 2.3. The condition above is fulfilled by a large class of critical potentials A. (i) The absence of resonances is generically fulfilled (see [14]). Assumption (ii) is not essential but makes the proof less heavy. Under this condition (see e.g [40]) the operator of interest defining (2) namely 1ε Dµ(s) generates a unitary time evolution denoted by U ε (s, s0 ) obeying i∂s U ε (s, s0 ) =
1 Dµ(s) U ε (s, s0 ) , ε
(5)
generating solutions of (2). The following theorem and its corollary assert that there exists a scattering solution of the Dirac equation (2) which at large negative times is an element of the negative energy spectral subspace of the free Dirac operator (A = 0) and at large positive times it is an element of the positive energy spectral subspace of the free Dirac operator. Theorem 2.4. (Decay of Bound States). Assume Condition 2.2. Let Φµ(s0 ) be a normalized bound state of Dµ(s0 ) for some s0 ∈ (si , 0]. Let U ε (s, s0 ) be given by (5), i.e. ψsε = U ε (s, s0 )Φµ(s0 ) is the solution of the Dirac equation (2) with ψsε0 = Φµ(s0 ) . Then for all χ ∈ L 2 , lim ψsε , χ = 0
ε→0
(6)
for any s > 0. As a corollary of Theorem 2.4 and the adiabatic theorem (see e.g. [39]) we obtain that the solution starts in the negative energy spectral subspace of the free Dirac-Hamiltonian D0 and ends in the positive energy spectral subspace. Denoting by P0+ , P0− the corresponding spectral projectors we formulate Corollary 2.5. (Adiabatic Pair Creation). Let ψsε be as in Theorem 2.4. Then for all s < si , lim ψsε , P0− ψsε = 1,
(7)
lim ψsε , P0+ ψsε = 1.
(8)
ε→0
and for all s > s f ε→0
The proof of this consists in observing that (7) follows directly from the initial condition of ψsε (bound state in the gap) and the adiabatic theorem [38]. For (8) one must apply Theorem 2.4, which ensures that the scattering state is orthogonal to any bound state in the gap, then the statement follows again using the adiabatic theorem.
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3. Skeleton of the Proof and Content of Sections The proof of Theorem 2.4 consists in controlling the propagation of ψsε . For sufficiently small ε, ψsε follows more or less the bound states Φs . Reaching the critical time s = 0 the bound state “vanishes” in the positive energy subspace of the free Dirac Hamiltonian. One needs to show that ψsε will stay there for all later times. Note that we need to control the wavefunction evolution for a time dependent hamiltonian. The proof has thus naturally two parts: (1) Show that a bound state with energy in the energy gap between −1 and 1 reaches adiabatically the upper spectral edge without “injuries” and ending up in N (Lemma 7.1). (2) Show that any state in N scatters during the time in which the potential stays overcritical (Corollary 7.7). The proof of (1) is done in several sections. Contrary to what one might expect at first sight it is quite involved. (1.1) The problem is the possible degeneracy of the bound states. We must show that the eigenspaces Nµ of E µ when µ goes to one converge to N (Corollary 6.5). That is done in the first part of Sect. 6. The proof is long and tricky and aims at the definition of an invertible operator Q µ which maps Nµ to N . This map will be used in the next step. (1.2) We need good control of “how a bound state converges”. We show that to every bound state Φ ∈ N the sequence of bound states (Φµ )µ ⊂ Nµ given by Φµ = Q −1 µ Φ is “good” in the sense that it is differentiable with respect to µ with relatively soft divergent behavior of the derivative when µ tends to one (Lemma 6.6). This is done in Sect. 6.1. (1.3) The “good sequences” (Φµ )µ will be used in Sect. 7.1 to establish an adiabatic lemma without a gap, Lemma 7.1. The proof is a two scale argument. We first propagate adiabatically to times s0 very close to 0 and then by the uniform (in ε) estimate (70) we can close the gap. This establishes the first part of the proof. The proof of (2) is naturally much more involved than that of (1). (2.1) We wish to show that any bound state at the spectral edge scatters. What we need to establish is that the wavefunction leaves the range of the scattering potential sufficiently fast, faster than ε−1 , the time after which the potential is undercritical again. Such control is rather easy when the potential does not depend on time, but here it depends on time: How does one control the evolution of wavefunctions for time dependent hamiltonians? The most direct and physical way is using generalized eigenfunctions. The point is however that the generalized eigenfunctions are singular near the spectral edge! The essential question is: How bad? We recall here and rely heavily on a result of [21,22] on generalized eigenfunctions near criticality (Theorem 4.2). That is done in Sect. 4. (2.2) The eigenfunctions allow us to control the wavefunction evolution for potentials constant in time. Section 5 gives preliminary estimates with Lemma 5.1. Equation (24) shows what we need to have when dealing with generalized eigenfunctions, namely an estimate in the sup-norm. The proof of Lemma 5.1 involves the tricky use of momentum cut-offs. Corollary 5.2 formulates then three estimates for the L 2 − norms. At first sight the third estimate ((iii) of Corollary 5.2) seems the only relevant one, but the first two are in fact needed for technical
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reasons later. (The technical reason is that we must bring the “static potential estimates” in contact with the true time evolution, i.e. with the non-static potential). The proofs here are “straightforward” applications of stationary phase methods, taking however the singular eigenfunctions behavior into account by tricky cutoffs of small momenta. The stationary phase application leading to good decay is unfortunately lengthy, while not difficult or tricky. Therefore we decided to shift that calculation to the Appendix. (2.3) In sect. 7.2 contact with the true time evolution is made. First we consider the wavefunction evolution for “short times” (Lemma 7.3). Short means macroscopic times of order one. We introduce the time σ which may roughly be thought of as being the first time at which µ reaches a maximum. Here we use the estimate (iii) of Corollary 7.2, which is the translation of Corollary 5.2 to the macroscopic time scale. It is shown that most of the wavefunction will have left the range of the potential by a macroscopic time of order ε1/3 , i.e. ε−2/3 on the microscopic time scale. That proof uses Cook’s argument in combination with physical insights: We need to compare the true evolution until time s ≤ σ with a “static potential” evolution. The potential will be “frozen” to the value it has at a time s. There is a big error between the true and the auxiliary time evolution. But in terms of the evolution of the relevant part of the wavefunction the error is not so big, since most of the wavefunction will have left the range of the potential. So the error is transported to a region in space which we do not care so much about. That idea is behind this part of the proof. Of course, we must insure that for very long times, this error does not come back! But that is done in the next step. (2.4) In Sect. 7.3 we extend our result to all times (Lemma 7.3). Most of the wavefunction has left the range of the potential, we must insure that it stays like that. Again we use Cook’s method, but now we freeze the potential at the value it has at time σ . In fact we can choose here any value s > 0 for which the potential is overcritical. The physical idea is clear: Since most of the wavefunction is outside of the range of the potential it moves freely and the critical potential is roughly the zero potential. To avoid problems arising from small k values, we use density arguments and cut off small momenta. It is here where Corollary 7.2 (i) and (ii) come into play. (2.5) Section 7.4 collects simply the results to establish the second part of Theorem 2.4, namely that the bound state which reached the upper edge scatters. 4. Generalized Eigenfunctions Consider for µ ∈ R, k ∈ R3 , j = 1, 2, 3, 4 the solutions of the Lippmann Schwinger equation (9) ϕµ (k, j, x) = G +k (x )µA(x − x )ϕµ (k, j, x − x )d 3 x + ϕ0 (k, j, x), where the G +k is the kernel of (E k − D 0 )−1 = limδ→0 (E k − D 0 + iδ)−1 [40] and the ϕ0 (k, j, x) are the solutions of the free Dirac eigenvalue equation D0 ϕ0 (k, j, ·) = ±E k ϕ0 (k, j, ·) (with “+” for j = 1, 2 and “-” for j = 3, 4) observing the boundary condition ϕ0 (k, j, 0) = 1. Note, that (9) holds trivially for µ = 0.
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Lemma 4.1. (GEF Properties). Let A satisfy Condition 2.2 (i) and δ > 0 be such that µ = 1 is the only critical value in [1, 1 + δ]. Then (a) Any solution ϕµ (k, j, ·) of (9) satisfies Dµ ϕµ (k, j, ·) = ±E k ϕµ (k, j, ·),
(10)
with “+” for j = 1, 2 and “-” for j = 3, 4. (b) For all x the functions ϕµ (k, j, x) are infinitely often continuously differentiable with respect to k for k = 0 . (c) The scattering system (D0 , Dµ = D0 + µA) is asymptotically complete for any µ ∈ [1, 1 + δ]. In particular the wave operator Ωµ+ defined via Ωµ+ ψ ≡ lim ei Dµ t e−i D0 t ψ for all ψ ∈ L 2 t→∞
exists, is isometric and ac Ran Ωµ+ = Hµ , ac is the spectral subspace of the absolutely continuous spectrum of D . where Hµ µ (d) For µ = 1 the solutions ϕµ (k, j, ·) define a generalized Fourier transform, i.e. an ac ⊂ L 2 (R3 , C4 ) → L 2 ((R3 × {1, 2, 3, 4}), C) by isometry F : Hµ 3
Fµ (ψ)(k, j) := (2π )− 2
ϕ µ (k, j, x)ψ(x)d 3 x
(11)
and ψ(x) =
4
3
(2π )− 2 ϕµ (k, j, x)Fµ (ψ)(k, j)d 3 k,
(12)
j=1
where the integrals are in the l.i.m.-sense (see e.g. [27]). Furthermore Plancherel holds ψ, χ =
4
Fµ (ψ)∗ (k, j)Fµ (χ )(k, j)d 3 k
(13)
|Fµ (ψ)(k, j)|2 d 3 k =: Fµ (ψ) .
(14)
j=1
as well as Parseval ψ =
4 j=1
Proof. (a) and (b) have been proven in [7] (see Lemma 3.4. therein). Also (c) is not new, it is known to hold for short range potentials (see for example Theorems 8.2, 8.3 and 8.20 in [40]). (d) follows also from Lemma 3.4. in [7], where it is shown that µout where the Fµ (ψ)(k, j) = ψ · stands for the (ordinary) Fourier transform and ψµout + out is defined by Ωµ ψµ = ψ.
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Since Ωµ+ is isometric we have ψ, χ = Ωµ+ ψ, Ωµ+ χ = ψµout , χµout =
4
µout∗ (k, j) ψ χµout (k, j)d 3 k
j=1
=
4
Fµ (ψ)∗ (k, j)Fµ (χ )(k, j)d 3 k,
j=1
i.e. (13) and (14) hold.
As we deal with a time dependent external field which grows from under-criticality to over-criticality in the course of which we need very good control on the evolution of the wavefunction, we need uniform estimates on the generalized eigenfunctions of the operator D0 + µA. Uniform estimates have not been given before. It is known that for critical potentials the generalized eigenfunctions diverge for k → 0 [13], but that is not sufficient to control the propagation. What is sufficient are estimates on the L ∞ -norm of the generalized eigenfunctions of D0 + µA and their k-derivatives uniform in k and uniform in µ ∈ [1 − δ, 1 + δ] for some δ > 0. The uniform estimates we need are provided in [22]. We cite the crucial Corollaries 3.7. and 6.3. in [22], using it (following Remark 3.6. in [22]) both for positive and negative energies, i.e. for spin indices j = 1, 2 and j = 3, 4. Assuming that there is no bound state nor resonance with energy −1 we could give a much better bound for negative energies, but that is not necessary for our estimates below where the bounds on j = 1, 2 are essential. Theorem 4.2. (Upper Bound for the Sup-Norm of the GEF of Lemma 4.1). There exist δ > 0, constants νl , 1 ≤ l ≤ n and a constant c > 0 so that the following holds: For (m) the m th derivative ϕµ := ∂km ϕµ , m ∈ N0 , there exist constants Cm so that for every k ∈ R3 , j = 1, 2, 3, 4 and for every µ ∈ [1 − δ, 1 + δ], ⎛ m+1 ⎞ n k ⎠ . (15) (1 + | · |)−m ϕµ(m) (k, j, ·) ∞ < Cm ⎝k −m + |µ − 1 − νl k 2 | + ck 3 l=1
Furthermore there exist µ (k, j, ·) ∈ N and C uniform in k ∈ R3 and µ ∈ [1−δ, 1+δ] so that ϕµ (k, j, ·) − µ (k, j, ·) ∞ < C.
(16)
5. Propagation Estimates In this section we want to apply Theorem 4.2 to get estimates on the time propagation of wave-functions for the static Dirac Hamiltonian Dµ uniform in µ ∈ [1 − δ, 1 + δ]. Under Condition 2.2 (see e.g. [40]) Dµ generates a unitary time evolution denoted by Vµ (t, 0), i.e. i∂t Vµ (t, 0) = Dµ Vµ (t, 0) ,
(17)
which applied to eigenfunctions reads Vµ (t, 0)ϕµ (k, j, x) = e−i E k t ϕµ (k, j, x) .
(18)
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This formula explains the role of the generalized eigenfunctions and it gives us the most direct control on the evolution of wavefunctions by expanding the wavefunction into generalized eigenfunctions. The estimates we are after are such that we can control the wavefunction evolution of the bound states in N during overcriticality. The bound states must decay fast enough (i.e. scatter fast enough) so that they are outside of the range S A of the potential before the potential becomes undercritical again. The naive picture of scattering theory suggests that the Fourier transform (given by plane waves) of the state governs the spreading. But we are here in a delicate situation analogous to resonant behavior. The generalized eigenfunctions are not at all like plane waves as we see from (15) and we must control the spreading given by such badly behaved eigenfunctions. We need to separate very very slow spreadings of the wavefunction (whose contribution will be hopefully negligible because of small probability) from the moderately fast spreading which make the state scatter. The borderline will be given by the k-value of the “resonance”, i.e., where µ − 1 ≈ νl k 2 (cf (15)). µ should be thought of as being only slightly bigger than 1, because that is the dangerous regime, the regime where k is small with large probability. For technical reasons we also separate very large momenta. Thus we will give propagation estimates for a wavefunction separating large, intermediate and small momenta, using the mollifier ρ κ ∈ C ∞ given by 0, for k ≤ 1, ρ (k) := (19) 1, for k ≥ 2; and for κ > 0 we define ρ κ (k) := ρ
k κ
(20)
and the operator ρκ,µ := Fµ−1 ρ κ Fµ .
(21)
Note that [Dµ , ρκ,µ ] = 0. Lemma 5.1. (Cutoff and propagation estimates - stationary case. Let δ > 0 be such that there is no bound state of Dµ for µ ∈ (1, 1 + δ], i.e. Hac (Dµ ) = L 2 . Let S ⊂ R3 be compact. Then there exists a C < ∞ such that for any χ ∈ L 2 with suppχ ⊂ S and any 0 < κ < 1 we have that for all µ ∈ (1, 1 + δ] and all 0 ≤ t < ∞ , n 3 k χ (22) (1 − ρκ,µ )χ ≤ Cκ 2 sup 1 + 2 | + ck 3 |µ − 1 − ν k l k<2κ l=1
and
2 n k 1+ χ . (23) 2 3 |µ − 1 − νl k | + ck
Vµ (t, 0)(1 − ρκ,µ )χ ∞ ≤ Cκ 3 sup
k<2κ
l=1
Furthermore let κ < ∞. For all m ∈ N0 there exists Cm < ∞ such that 1S Vµ (t, 0)ρκ,µ (1 − ρκ,µ )χ ∞ < κ 3 t −m Cm κ −4
n m 1 −2 χ κ + sup |µ − 1 − νl k 2 | + ck 3 2κ>k>κ l=1
(24)
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and Dµ χ . (25) κ Proof. Using linearity we can assume without loss of generality that χ = 1. We start with (22). Let χ be as in the lemma. Then by (21) and (14), ρκ,µ ρκ,µ χ ≤
(1 − ρκ,µ )χ = (1 − ρ κ )Fµ (χ ) ≤ 4 sup {| Fµ (χ )(k, j) |} 1 − ρ κ .
(26)
j,k<2κ
By (20) and (19) 3
ρκ − 1 = κ 2
1 |ρ ( p) − 1 |2 d 3 p
2
≤ Cκ 3/2 .
(27)
Furthermore
3 sup {| Fµ (χ )(k, j) |} ≤ sup { (2π )− 2 | ϕ µ (k, j, x)χ (x) | d 3 x}
j,k<2κ
j,k<2κ
3
≤ sup { ϕµ (k, j, ·) ∞ }(2π )− 2 χ 1 ,
(28)
j,k<2κ
where by Schwarz
χ 1 = | χ (x) | d 3 x = 1S | χ (x) | d 3 x ≤ χ (x) | S | ≤ C .
(29)
For sup j,k<2κ { ϕµ (k, j, ·) ∞ } we have by Theorem 4.2 (15) for m = 0 that n k . sup { ϕµ (k, j, ·) ∞ } ≤ C sup 1 + 2 | + ck 3 |µ − 1 − ν k l j,k<2κ k<2κ
(30)
l=1
Hence for (28) with (29) sup {| Fµ (χ )(k, j) |} ≤ C sup
j,k<2κ
n k 1+ . |µ − 1 − νl k 2 | + ck 3
k<2κ
(31)
l=1
This with (27) in (26) yields (22). Next we establish (23). Observing the definitions Vµ (t, 0)(1 − ρκ,µ )χ ∞ 4 − 23 3 ≤ κ (k)) | d k (2π ) | Vµ (t, 0)ϕµ (k, j, x)Fµ (χ )(k, j)(1 − ρ
∞
j=1
4 − 23 −i E k t 3 = ϕµ (k, j, x)Fµ (χ )(k, j)(1 − ρ κ (k)) | d k (2π ) | e
∞
j=1
≤ 4 sup { ϕµ (k, j, ·) ∞ } sup {| Fµ (χ )(k, j) |} 1 − ρ κ 1 . j,k<2κ
j,k<2κ
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Similarly as in (27) we find that 1 − ρ κ 1 < Cκ 3 and with (30) and (31)) we get (23). We now turn to (24). As above Vµ (t, 0)ρκ,µ (1 − ρκ,µ )χ (x) 4 3 ρκ (1 − ρ κ )Fµ (χ )(k, j)d 3 k (2π )− 2 Vµ (t, 0)ϕµ (k, j, x) = j=1
=
4
3
ρκ (1 − ρ κ )Fµ (χ )(k, j)d 3 k. (2π )− 2 exp (−it E k ) ϕµ (k, j, x)
(32)
j=1
It is for this term that we need the behavior of the derivatives of the eigenfunctions Ek ∂k exp (−it E k ) = exp (−it E k ). (cf (15)). We shall use a stationary phase method, using i kt The rigorous estimate of this formula is based on a simple straightforward computation which is done in the Appendix. We shall only describe here in a heuristic manner how the estimate comes about. First we recall (15), ⎛ m+1 ⎞ n k ⎠ (1 + x)−m ϕµ(m) (k, j, ·) ∞ ≤ Cm ⎝k −m + . (33) |µ − 1 − νl k 2 | + ck 3 l=1
This enters also in the k-derivatives of Fµ (χ ). Since χ has compact support in S we obtain |∂km Fµ (χ )(k, j)| = |∂km ϕµ (k, j, ·), χ | ≤ Cm (1 + x)−m ϕµ(m) (k, j, ·) ∞ (1 + x)m χ 1 ≤ CCm (1 + x)−m ϕµ(m) (k, j, ·) ∞ ,
(34)
where in the last step we followed (29). Doing the partial integration we need to apply the operator ∂k Ekk = k12 + Ekk ∂k . Relevant to us is only the “small” k-behavior (k ≥ κ), i.e. we need to count the inverse powers of k. In that sense ∂k Ekk ∼ ∂k Ekk
1 k2
+ k1 ∂k . Fur-
ther observe that the relevant term in (32) to which is applied is the product 1 1 ϕµ (k, j, x) ρκ (1 − ρ κ )Fµ (χ )(k, j). But ∂k ρ κ ∼ κ ∼ k , while ∂k ϕµ(m) (k, j, x) = ϕµ(m+1) (k, j, x)
n k ≈ (1 + x)ϕµ(m) (k, j, x) k −1 + . 2 3 |µ − 1 − νl k | + ck
l=1
Likewise for Fµ (χ ). Since we are interested in the supremum over the compact set S the factor (1 + x) can be estimated by a constant. The upshot is then that the contribution of the terms is essentially the m th power of n 1 1 −1 Ek k ≈ 2+ ∂k k + k k k |µ − 1 − νl k 2 | + ck 3 l=1 n 1 1 ≈ 2 + k |µ − 1 − νl k 2 | + ck 3 l=1
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multiplied by the product ϕµ Fµ (χ ), yielding roughly
m 2 n n 1 1 k + 1 + . 2 2 3 2 3 k |µ − 1 − νl k | + ck |µ − 1 − νl k | + ck l=1
l=1
The second factor can be bounded by c−2 k −4 ≤ Cκ −4 . This gives with the volume factor κ 3 the right-hand side of (24). Finally we establish (25). By (14) and using E k2 ≥ k 2 , ρκ,µ ρκ,µ χ ≤
2
|Fµ (χ )| d k ≤ 2 3
k>κ
1 ≤ 2 κ
k>κ
k>κ
E k−2 Fµ (Dµ χ )∗ Fµ (Dµ χ )d 3 k
Fµ (Dµ χ )∗ Fµ (Dµ χ )d 3 k =
Dµ χ 2 . κ2
The results of Lemma 5.1 can now be used to estimate the decay behavior of any compactly supported wavefunction χ ∈ L 2 Corollary 5.2. (Propagation Estimate - stationary case). Let S ⊂ R3 be compact, 0 < ξ < 1. There exists a δ > 0 (possibly smaller than the δ of Lemma 5.1) such that there exists a C < ∞ and for all m ∈ N a constant Cm such that for all µ ∈ (1, 1 + δ], all 3 − 2(1−ξ )
t > (µ − 1)
and all χ ∈ L 2 with suppχ ⊂ S the following holds: 1
(i) Let Vµ be defined by (17), then for κ = t − 2 (1−ξ ) and for all w ≥ t, − m 1S Vµ (w, 0)ρκ,µ χ ≤ Cm ( Dµ χ ), w 1
(ii) for κ = t − 2 (1−ξ ) , 3
1
(1 − ρκ,µ )χ ≤ Ct − 4 (1−ξ ) (µ − 1)− 4 χ , (iii) 1
3
1S Vµ (t, 0)χ ≤ C|µ − 1|− 2 t − 2 (1−ξ ) ( χ + Dµ χ ). We note that we want to have good decay estimates, so ξ should be thought of as being small. We also wish to remark that we shall need both estimates (i) and (ii) as well as (iii). (iii) is better than (i) and (ii) together but it is not suitable for “density arguments” which we shall use later on when we compare the true time evolution with the Vµ -evolution. Proof. Using linearity we can assume without loss of generality that χ = 1. We start 1 with (ii). With our choice κ = t − 2 (1−ξ ) we obtain in view of (22), (1 − ρκ,µ )χ ≤ Ct
− 34 (1−ξ )
sup k<2κ
1+
n l=1
k . |µ − 1 − νl k 2 | + ck 3
On Adiabatic Pair Creation
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3
Since by assumption t > (µ − 1) 2(1−ξ ) we have κ 2 < (µ − 1)3/2 (µ − 1) and hence for δ small enough we are below the resonant k-values, i.e. inf l |µ−1−νl k 2| ≥ (µ−1)/2, and thus the supremum of the bracket term is less than 1 + Cκ/(µ − 1) . Thus
3 Cκ (1 − ρκ,µ )χ ≤ Ct − 4 (1−ξ ) 1 + µ−1
3 C(µ − 1)3/4 ≤ Ct − 4 (1−ξ ) 1 + µ−1 3
≤ Ct − 4 (1−ξ ) (µ − 1)−1/4 , which establishes (ii). We now prove (i). Using 1S ψ ≤ 1S 1S ψ ∞ ≤ C 1S ψ ∞ ,
(35)
we have with the high momentum cutoff κ to be specified below 1S Vµ (w, 0)ρκ,µ χ ≤ 1S Vµ (w, 0)ρκ,µ (1 − ρκ,µ )χ + 1S Vµ (w, 0)ρκ,µ ρκ,µ χ ≤ C 1S Vµ (w, 0)ρκ,µ (1 − ρκ,µ )χ ∞ + ρκ,µ ρκ,µ χ . −
(36)
3
Let w ≥ t > (µ − 1) 2(1−ξ ) . κ will be chosen large enough, so that the first term encompasses the resonant regime (νl k 2 ≈ µ − 1). We obtain with Lemma 5.1 formula (24) using w ≥ t > (µ − 1)
3 − 2(1−ξ )
1
and κ = t − 2 (1−ξ ) ,
3 m 1S Vµ (w, 0)ρκ,µ (1 − ρκ,µ )χ ∞ ≤ CCm κ 3 w −m t 2−2ξ t 1−ξ + C(µ − 1)− 2 m+2 ≤ CCm κ 3 w −m Ct 1−ξ m κ 3 w −mξ +2−2ξ . ≤C
For the second term in (36) we get with Lemma 5.1 formula (25) ρκ,µ ρκ,µ χ ≤
Dµ χ . κ
Hence for (36) we obtain m κ 3 w −mξ +2−2ξ + Dµ χ . 1S Vµ (w, 0)ρκ,µ χ ≤ C κ Choosing κ := w mξ/4 we find the bound m Dµ χ w − 1S Vµ (w, 0)ρκ,µ χ ≤ C
mξ 4
+2−2ξ
.
Choosing m such that mξ/4 − 2 + 2ξ > m (i) follows. Next we prove (iii). By (35) 1S Vµ (t, 0)χ ≤ 1S Vµ (t, 0)ρκ,µ χ + 1S Vµ (t, 0)(1 − ρκ,µ )χ ∞ ≤ 1S Vµ (t, 0)ρκ,µ χ + Vµ (t, 0)(1 − ρκ,µ )χ ∞ .
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For the first summand use (i) with w = t and m = 2. For the second summand use − 21 (1−ξ ) Lemma 5.1 formula (23) with κ = t to obtain 2 n k − 23 (1−ξ ) Vµ (t, 0)(1 − ρκ,µ )χ ∞ ≤ Ct sup 1 + |µ − 1 − νl k 2 | + ck 3 k<2κ l=1
≤ C|µ − 1|
− 21 − 23 (1−ξ )
t
,
where the bound comes as in the proof of (ii) with the only difference being that we now have the square. 6. µ-Convergence of the Eigenspaces Let [µ B , 1] be an interval of parameters for which bound states for Dµ Φµ := (D0 + µA)Φµ = E µ Φµ exist. In the course of this paper we shall adjust µ B < 1 according to our needs. Note that E µ ∈ [−1, 1]. Let Nµ denote the eigenspace of E µ . In [14] it is shown that Condition 2.2 (i) implies that there exist constants µ B < 1, C > 0 and C > 0 such that C < ∂µ E µ < C
(37)
for all µ B ≤ µ ≤ 1. Lemma 6.1. Let A satisfy Condition 2.2. Then for µ B close enough to 1, dim N = dim Nµ for all µ ∈ [µ B , 1). Proof. We shall present a proof which prepares notation and results which we shall need later on for the control of the µ-derivative of the bound states: Given a critical degenerate bound state Φ there is a special sequence of undercritical bound states converging to Φ in a “nicely” differentiable manner. This special sequence will be called a “good sequence corresponding to Φ”. This notion is essential for the proof of the Adiabatic Lemma without a gap. The good sequence will be constructed via a “good operator” Q µ , which will be a bijection mapping Nµ to N . The inverse of Q µ will be constructed from the resolvent of an operator Rµ which we construct first. So before we prove the lemma we shall be concerned with Rµ and Q µ the upshot of which is Corollary 6.3. Having that, the actual proof of the lemma is very short. Let PN , PNµ be the projections on the eigen-spaces N , Nµ . With (Dµ − E µ )PNµ = 0
(38)
Dµ − Dν = (µ − ν)A,
(39)
D1 − E µ PNµ = (1 − µ)A PNµ .
(40)
and
we have
On Adiabatic Pair Creation
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Since for µ ∈ [µ B , 1) E µ is in the resolvent set of D1 −1 PNµ = (1 − µ) D1 − E µ A PNµ .
(41)
Multiplying with PN − 1 = −PN ⊥ and observing that [D1 , PN ⊥ ] = 0 yields −1 PN ⊥ A PNµ , (PN − 1) PNµ = −(1 − µ) D1 − E µ thus
−1 PN PNµ = 1 − (1 − µ) D1 − E µ PN ⊥ A PNµ .
(42)
Note that (D1 − E µ )−1 is now applied to vectors orthogonal to N . Therefore the term has good chances of being controllable. Again observing [D1 , PN ⊥ ] = 0, define Rµ : L 2 → N ⊥ by −1 PN ⊥ A. (43) Rµ := D1 − E µ Introducing the operator Q µ := 1 − (1 − µ)Rµ
(44)
PN PNµ = Q µ PNµ .
(45)
we get
With the following lemma we shall establish that Q µ is invertible. Lemma 6.2. There exists C < ∞ such that 13
Rµ op < C (1 − µ)− 16 .
(46)
Proof. Let χ ∈ L 2 with χ = 1. Set ξ = PN ⊥ (Aχ ) ,
(47)
and note for later that ξ ≤ A ∞ . Also for later use we define and note that Φ := PN (Aχ ),
Φ ≤ A ∞ .
(48)
With (43) −1 R µ χ = D1 − E µ ξ.
(49)
Set 3
rµ = (1 − µ)− 8
(50)
and let B0 (rµ ) = {x ∈ R3 : x < rµ }. Choose µ B < 1 so that S A ⊂ B0 (rµ ) for all µ ∈ [µ B , 1]. Let {Φ l , 1 ≤ l ≤ n} be an orthonormal basis in N . For large enough rµ (µ B close enough to one) we have that the vectors PN (1B(rµ ) Φ l ), 1 ≤ l ≤ n are
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linearly independent, thus {PN (1B(rµ ) Φ l ), 1 ≤ l ≤ n} is a basis of N . Hence there µ ∈ N such that exists a Φ µ ) = PN (1B(rµ ) ξ ). PN (1B(rµ ) Φ
(51)
We define now the spacial cutoff parts µ, ξµa := 1B(rµ ) ξ − 1B(rµ ) Φ ξµb
:= ξ
− ξµa ,
(52) (53)
which are orthogonal to N (cf. (47)). By Schwarz inequality ξµa 1
≤
1B(rµ ) ξµa
≤
4 3 πr 3 µ
1 2
µ . ξ + 1B(rµ ) Φ
µ = Φ µ we have with (51), Since PN Φ µ 2 = 1B(rµ ) Φ µ , Φ µ , PN Φ µ µ = 1B(rµ ) Φ 1B(rµ ) Φ µ , Φ µ = 1B(rµ ) ξ, Φ µ = PN 1B(rµ ) Φ µ ≤ 1B(rµ ) ξ 1B(rµ ) Φ µ , = 1B(rµ ) ξ, 1B(rµ ) Φ hence µ ≤ 1B(rµ ) ξ ≤ ξ ≤ A ∞ . 1B(rµ ) Φ Hence there exists a constant C < ∞ so that 1/2 4 a π ξµ 1 ≤ 2 A ∞rµ3/2 . 3
(54)
For (49) we obtain −1 a −1 b R µ χ = D1 − E µ ξµ + D 1 − E µ ξµ , and we wish to show that for some C < ∞, −1 a,b 13 D1 − E µ ξµ < C (1 − µ)− 16 .
(55)
“(55) a”: We introduce 1 (k, j, ·) ∈ N from Theorem 4.2 (cf. (16)). Since ξµa , 1 (k, j, ·) = 0 we have F1 ξµa (k, j) := (2π )−3/2 ϕ1 (k, j, x), ξµa = (2π )−3/2 ϕ1 (k, j, x) − 1 (k, j, x), ξµa . Thus
a F1 ξ (k, j) ≤ (2π )−3/2 ϕ1 (k, j, ·) − 1 (k, j, ·) ∞ ξ a 1 . µ µ
By using Theorem 4.2 and (54) we get a F1 ξ (k, j) ≤ Cr 3/2 . µ µ
(56)
On Adiabatic Pair Creation
179
√ Note that D1 ϕ1 (k, j, ·) = E k ϕ1 (k, j, ·) = k 2 + 1ϕ1 (k, j, ·). Observing (14) we get, reading in the following E k as the for the function of k which multiplies the symbol generalized Fourier transform F1 ξµa (k, j), −1 a D1 − E µ ξµ =
1 F1 ξµa . Ek − Eµ
Using E k ≥ 1 > E µ and hence E k − E µ ≥ 1 − E µ > 0, E k − E µ > E k − 1 ≥ 0 we compute −1 a ξµ ≤ D1 − E µ
1 1|k|<(1−E µ )1/2 F1 ξµa Ek − Eµ 1 + 11>|k|>(1−E µ )1/2 F1 ξµa Ek − Eµ 1 + 1|k|>1 F1 ξµa Ek − Eµ 1 ≤ 1|k|<(1−E µ )1/2 F1 ξµa 1 − Eµ 1 + 11>|k|>(1−E µ )1/2 F1 ξµa Ek − 1 1 + 1|k|>1 F1 ξµa . Ek − 1
By (56) we obtain with appropriate constants C1 , C2 , C3 , −1 a D1 − E µ ξµ ≤
C1 r 3/2 (1 − E µ )3/4 1 − Eµ µ
1/2 +C2 rµ3/2 11>|k|>(1−E µ )1/2 (E k − 1)−2 d 3 k +C3 .
Noting that E k − 1 ≥ k 2 /2 for |k| < 1, we obtain ⎛
≤ rµ3/2 ⎝C1 (1 − E µ )−1/4 + C2 4π
1 (1−E µ )1/2
1/2 ⎞ 4k −2 dk
⎠ + C3
1/2 −1/2 8π + C3 . = rµ3/2 C1 (1 − E µ )−1/4 + √ C2 1 − E µ −1 3 Because E µ → 1 for µ → 1 and rµ is given by (50) there exists an appropriate constant C < ∞ such that −1 a D1 − E µ ξµ ≤ Crµ3/2 (1 − E µ )−1/4 . (57) By (37) 1 − E µ ≥ C(1 − µ). This and (50) yield (55) a.
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“(55) b”: By (52) and (53) µ . ξµb = ξ − ξµa ≤ ξ − 1B(rµ ) ξ + 1B(rµ ) Φ µ ∈ N we have with (51), Since Φ µ | = |PN 1B(rµ ) ξ, Φ µ | = |1B(rµ ) ξ, PN Φ µ | |1B(rµ ) ξ, Φ µ , Φ µ 2 . µ | = 1B(rµ ) Φ = |1B(rµ ) Φ On the other hand recalling that ξ ⊥N we obtain by Schwarz inequality µ | = |ξ − 1B(rµ ) ξ, Φ µ | ≤ ξ − 1B(rµ ) ξ Φ µ , |1B(rµ ) ξ, Φ hence µ ≤ C ξ − 1B(rµ ) ξ 1B(rµ ) Φ
µ Φ . µ 1B(rµ ) Φ
Clearly µ Φ = 1, µ µ→1 1B(r ) Φ µ lim
thus for µ B close enough to 1 there exists a C < ∞ so that µ ξµb ≤ ξ − 1B(rµ ) ξ + 1B(rµ ) Φ ≤ C ξ − 1B(rµ ) ξ .
(58)
By (47) and the fact that A has compact support and by (48) with (47) we have for rµ large enough that ξ is outside the ball B(rµ ) a multiple of Φ. Hence ξ − 1B(rµ ) ξ is outside the ball B(rµ ) a multiple of Φ. Since Φ ∈ N with norm bounded in (48) its decay properties are known from the Greens function of D1 − 1 (see e.g. [14,22]), namely | Φ |≤ C x −2 , we obtain ξµb
1
≤C
x x>rµ
−4 3
d x
2
−1
≤ Crµ 2
with appropriate constants C < ∞. It follows, again using E k − E µ ≥ 1 − E µ , that −1 b D1 − E µ ξµ =
1 1 F1 ξµb ≤ ξ b Ek − Eµ 1 − Eµ µ −1
≤ | 1 − E µ |−1 Crµ 2 . As before, (37) and (50) yield (55) b.
(59)
On Adiabatic Pair Creation
181
Corollary 6.3. For µ B close enough to 1, 3
(1 − µ)Rµ op ≤ (1 − µ) 16 , ∀ µ ∈ [µ B , 1).
(60)
Furthermore lim Q µ op = 1
µ→1
and the inverse of the operator (44) Q −1 µ =
∞
(1 − µ) j Rµj
(61)
j=0
exists as a bounded operator on L 2 . Proof. The corollary follows immediately from (44) and Lemma 6.2.
With this we establish now Lemma 6.1. Let n µ := dim Nµ . Since Q µ is invertible for µ B close enough to 1 it follows by (45) that n ≥ n µ . It is left to show that for µ B close enough to 1 n ≤ n µ for all µ ∈ [µ B , 1). Assume that for any 0 < µ B < 1 there exists a µ ∈ [µ B , 1) such that n > n µ . Then the n µ dimensional vectors {v j , j = 1, . . . , n} defined by their coordinates vlj := Φµl , Φ j are linearly dependent, i.e. there exist nontrivial α j such that nj=1 α j v j = 0. Then n Φ := j=1 α j Φ j = 0, Φ ∈ N and Φ satisfies PNµ Φ = 0. But for any ∆E ∈ R we have (Dµ − 1 + ∆E)Φ 2 = (D0 + (µ − 1)A − 1 + ∆E)Φ 2 = (∆E − (1 − µ)A)2 Φ, Φ = (∆E)2 + (−2∆E(1 − µ)A + (1 − µ)2 A2 )Φ, Φ. Choosing ∆E = (1 − µ) A ∞ < 1 for µ B close enough to 1, it follows that (Dµ − 1 + ∆E)Φ 2 < (∆E)2 .
(62)
On the other hand PNµ Φ = 0 implies by virtue of Condition 2.2 (ii) that Φ lies in the absolutely continuous spectrum. Writing Pµ+ and Pµ− for the spectral projections onto the positive and negative absolutely continuous spectral subspaces of Dµ and using that ∆E < 1 we obtain (Dµ − 1 + ∆E)Φ 2 = (Dµ − 1 + ∆E)Pµ+ Φ 2 + (Dµ − 1 + ∆E)Pµ− Φ 2 ≥ (1 − 1 + ∆E)Pµ+ Φ 2 + (−1 − 1 + ∆E)Pµ− Φ 2 ≥ (∆E)2 Pµ+ Φ 2 + Pµ− Φ 2 ≥ (∆E)2 Pµ+ Φ 2 + (∆E)2 Pµ− Φ 2 = (∆E)2 . This contradicts (62) and hence n = n µ .
Corollary 6.4. Q µ is a bijection from Nµ to N for µ B close enough to 1. Proof. By (45) Q µ maps Nµ into N , thus “onto” by Lemma 6.1 and Corollary 6.3.
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6.1. µ-Derivative of the Bound States. Using Q µ we can construct now for each element Φ ∈ N a sequence of elements (ζµ )µ by ζµ := Q −1 µ Φ ∈ Nµ which is “good” in several respects: Corollary 6.5. For µ B close enough to 1 holds: For each Φ ∈ N , Φ = 1, the corresponding sequence (ζµ )µ∈[µ B ,1) := (Q −1 µ Φ)µ∈[µ B ,1) satisfies (i) ζµ ∈ Nµ for all µ ∈ [1, µ B ], (ii) PN ζµ = Φ (iii) limµ→0 ζµ = 1. We call the sequence (ζµ )µ∈[µ B ,1) the good sequence corresponding to Φ. Proof. (i) follows from Corollary 6.4. (ii) follows from (61) using that Rµ maps into N ⊥ (iii) is immediate using (60) in (61). Below we shall need the µ-Derivative of Φµ given by Lemma 6.6. (µ-Derivative of the Bound States). Let Φ ∈ N be normalized and let ζµ , µ ∈ [µ B , 1] be the good sequence corresponding to Φ. Then2 13
∂µ ζµ ≤ C(1 − µ)− 16 , µ ∈ [µ B , 1]
(63)
with some constant C < ∞. Since N is finite dimensional the constant can be chosen uniformly on N . Proof. We shall use the formula (61) to differentiate ζµ = Q −1 µ Φ (cf. (61)). Formally ∂µ ζ µ = ∂ µ
∞ j (µ − 1)Rµ Φ j=0
= (∂µ (µ − 1)Rµ )
∞ j−1 j (µ − 1)Rµ Φ j=1
= Rµ
∞
j−1 j (µ − 1)Rµ Φ
j=1
+(µ − 1)(∂µ Rµ )
∞ j−1 j (µ − 1)Rµ Φ. j=1
Hence by Corollary 6.3 we obtain rigorously ∂µ ζµ ≤ C Rµ op + (µ − 1)∂µ Rµ op .
(64)
For the second term observe that formally −1 ∂µ Rµ χ = ∂µ D1 − E µ (Aχ − Aχ , ΦΦ) −2 = ∂µ E µ D1 − E µ (Aχ − Aχ , ΦΦ) −1 = ∂µ E µ D1 − E µ Rµ χ , 2 This estimate is not optimal, but sufficient for what is needed later. It seems reasonable to conjecture that the correct exponent is − 21 .
On Adiabatic Pair Creation
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which can be justified using for example the spectral decomposition of D1 . From this we get ∂µ Rµ op ≤ C(1 − µ)−1 Rµ op , hence ∂µ ζµ ≤ C Rµ op and (63) follows with Lemma 6.2.
(65)
7. Proof of Theorem 2.4 We now study the true time evolution U ε (0, s) as given by (5). To prove Theorem 2.4 we have to control the time propagation of ψsε . This propagation is naturally qualitatively different for s < 0 (“adiabatic bound state evolution”) and s > 0 (“scattering”). Hence we control the propagation for s < 0 and s > 0 separately. 7.1. Control of ψsε for s ≤ 0. Usually adiabatic theory assumes a spectral gap. Here we are in a situation where the eigenvalue E µ(s) will close the gap to the upper continuum. We need to control the adiabatic change of the bound states without a gap condition. Lemma 7.1. (Adiabatic Lemma without a gap). Let s < 0 be such that a normalized µ(s) of Dµ(s) with energy E µ(s) > −1 exists. Then bound state Φ µ(s) = 1. lim PN U ε (0, s)Φ
ε→0
(66)
µ(s) be a normalized bound state. By the Adiabatic Theorem [38] Proof. Let s < 0, Φ we have that for any s < s0 < 0, µ(s) = 1. lim PNµ(s0 ) U ε (s0 , s)Φ
ε→0
(67)
ε ε Setting Φ µ(s0 ) := PNµ(s0 ) U (s0 , s)Φµ(s) we note that (67) is equivalent to
µ(s) − Φ ε = 0. lim U ε (s0 , s)Φ µ(s0 )
ε→0
We now define an approximate time evolution
i s ε Φsε0 ,µ(s) := exp − E µ(v) dv Q −1 µ(s) Q µ(s0 ) Φµ(s0 ) , ε s0
(68)
(69)
and note that ε . Φsε0 ,µ(s0 ) = Φ µ(s0 ) ε / Q µ(s0 ) Φ ε is normalized and in N and hence ζ ε Now Φsε0 := Q µ(s0 ) Φ µ(s0 ) µ(s0 ) s0 ,µ(s) := ε ε Q −1 µ(s) Φs0 is the good sequence corresponding to Φs0 . Equation (69) becomes
i s ε ε Φs0 ,µ(s) = Q µ(s0 ) Φµ(s0 ) exp − E µ(v) dv ζsε0 ,µ(s) . ε s0
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P. Pickl, D. Dürr
We compare the approximate time evolution with the true one 0 ε ε ε U = ∂ (0, u)Φ Φsε0 ,µ(0) − U ε (0, s0 )Φ u µ(s0 ) s0 ,µ(u) du s0
i u Du − i∂u exp − E µ(v) dv ζsε0 ,µ(u) du ε ε s0 s0
0 i u i ε ε U (0, u) Du − E µ(u) exp − E µ(v) dv ζsε0 ,µ(u) du = Q µ(s0 ) Φµ(s0 ) ε ε s0 s0
0 i u ε ε U (0, u) exp − E dv ∂u ζsε0 ,µ(u) du. + Q µ(s0 ) Φ µ(v) µ(s0 ) ε s0 s0
ε = i Q µ(s0 ) Φ µ(s0 )
0
U ε (0, u)
Since (Du − E µ(u) )Φµ(u) = 0, ε Φsε0 ,µ(0) − U ε (0, s0 )Φ µ(s0 )
0 i u ε ε = Q µ(s0 ) Φ U (0, u) exp − E dv ∂u ζsε0 ,µ(u) du. µ(v) µ(s0 ) ε s0 s0 Hence by unitarity of U ε , Lemma 6.6 and Condition 2.2 (ii), Φsε0 ,µ(0)
ε C ε ≤ Q µ(s0 ) Φ − U (0, s0 )Φ µ(s0 ) µ(s0 ) ε
0
13
u − 16 du
s0
3 16 3 ≤ Q µ(s0 ) op C s016 ≤ Cs016 3
by Corollary 6.3. Thus uniformly in ε, ε = 0. lim Φsε0 ,µ(0) − U ε (0, s0 )Φ µ(s0 )
s0 →0
(70)
Furthermore we obtain from (70) that uniformly in ε, ε = lim | Φ ε lim Φsε0 ,µ(0) − U ε (0, s0 )Φ µ(s0 ) s0 ,µ(0) − 1| = 0 , s0 →0
s0 →0
so that lim lim Φsε0 ,µ(0) = 1 .
s0 →0 ε→0
Since Φsε0 ,µ(0) ∈ N , ε ε µ(s) − Φ ε PN U ε (0, s)Φ s0 ,µ(0) ≤ PN U (0, s)Φµ(s) − PN Φs0 ,µ(0) µ(s) − U ε (0, s0 )Φ ε ≤ U ε (0, s)Φ µ(s0 ) ε ε + U ε (0, s0 )Φ µ(s0 ) − Φs0 ,µ(0)
µ(s) − Φ ε = U ε (s0 , s)Φ µ(s0 )
ε ε + U ε (0, s0 )Φ µ(s0 ) − Φs0 ,µ(0) .
It follows with (68), (70) and (71), that µ(s) = lim lim PN U ε (0, s)Φ µ(s) lim PN U ε (0, s)Φ ε→0
s0 →0 ε→0
= lim lim Φsε0 ,µ(0) = 1. s0 →0 ε→0
(71)
On Adiabatic Pair Creation
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7.2. Propagation Estimates for the time dependent Case: “Short” Times s > 0. We shall introduce a time σ > 0 which is a time of order one. For example the time at which the switching factor µ(s) is halfway between 1 and its maximum. Our estimates will be valid until this time. It is in fact the crucial time after which the critical bound state has already left the range of the potential. We shall in the next section consider “long” times, i.e. the times bigger than σ . We shall now consider the auxiliary time evolution (17) on ε (s, 0) where the macroscopic time scale s = tε. That evolution will be denoted by Vµ(v) v is fixed! It is defined by ε i∂s Vµ(v) (s, 0) =
1 ε Dµ(v) Vµ(v) (s, 0) . ε
(72)
ε given by (72). Instead of µ ∈ (1, 1+δ] We first reformulate our Corollary 5.2 for Vµ(v) we have now v ∈ (0, σ ]. For the chosen σ we can replace in view of (37) the factor µ(v) − 1 corresponding to µ − 1 in Corollary 5.2 by v at little extra costs. We formulate first this adjustment as
Corollary 7.2. (Propagation Estimate - stationary case). Let S ⊂ R3 be compact, 0 < ξ < 1. There exists a σ > 0 such that there exists a C < ∞ and for all m ∈ N 3 − 2(1−ξ )
constants Cm < ∞ such that for all v ∈ (0, σ ], all u > ε(µ(v) − 1) χ ∈ L 2 with suppχ ⊂ S the following holds: 1
, and all
1
(i) for κ = ε 2 (1−ξ ) u − 2 (1−ξ ) and for all s ≥ u, ε 1S Vµ(v) (s, 0)ρκ,µ(v) χ ≤ Cm εm s −m ( Dµ(v) χ ), 1
1
(ii) for κ = ε 2 (1−ξ ) u − 2 (1−ξ ) , 3
3
3
3
(1 − ρκ,µ(v) )χ ≤ Cε 4 (1−ξ ) u − 4 (1−ξ ) v −1/4 χ , (iii) ε 1S Vµ(v) (u, 0)χ ≤ Cv −1/2 ε 2 (1−ξ ) u − 2 (1−ξ ) ( χ + Dµ(v) χ ).
We shall use that to control the time evolution of a wavefunction under the influence of the time dependent Dirac operator. We wish to note, that in contrast to Corollary 7.2 the estimates in the following Lemmata (i.e. Lemma 7.3 - 7.6 are done for a fixed wave function and not for a whole set of wavefunctions. Hence the constants C and Cm below depend in fact on more criteria of χ (in particular the energy Dµ(0) χ )) than the C and Cm in Corollary 7.2. Lemma 7.3. (Propagation Estimates - time dependent Case: “Short” times). Let U ε (s, u) be given by (5). Let S ⊃ S A be compact and let 0 < ξ < 1/3. Let χ be normalized with suppχ ⊂ S and Dµ(0) χ < ∞. Let σ > 0 be a possible choice of σ for Corollary 7.2 and such that C ≥ ∂s µ(s) ≥ C > 0 on (0, σ ]. Then there exists C < ∞ such that for all s ∈ (0, σ ], 1 3 3 (73) 1S U ε (s, 0)χ ≤ C ε 2 − 2 ξ s − 2 .
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Remark 7.4. This estimate gives the decay time of the critical bound state. It is of the order of ε1/3 , i.e. ε−2/3 on the microscopic time scale. One should compare this with the decay of an L 2 -function in a non-critical situation which is of order one on the microscopic time scale. Proof. Using that χ is normalized the lemma follows trivially for s ≤ ε1/3−ξ by choosing C > 1. Let s > ε1/3−ξ and ψsε := U ε (s, 0)χ .
(74)
ε Now Vµ(s) is controllable with help of Corollary 7.2. We shall “replace” the propagator ε . Then ε U by Vµ(v) ε ε ∂s U ε (s, 0) = ε−1 Ds U ε (s, 0) ∂s Vµ(v) (s, 0) = ε−1 Dµ(v) Vµ(v) (s, 0),
and in view of (39)
ε ∂u Vµ(v) (s, u)U ε (u, 0) du (75) 0 s i =− V ε (s, u) Dµ(v) − Dµ(u) U ε (u, 0)du ε 0 µ(v) i s ε =− V (s, u) (µ(v) − µ(u)) AU ε (u, 0)du. ε 0 µ(v)
ε U ε (s, 0) − Vµ(v) (s, 0) =
s
Hence ψsε = U ε (s, 0)χ ε = Vµ(v) (s, 0)χ +
i ε
s 0
ε (s, u)Aψuε du. (µ(u) − µ(v)) Vµ(v)
(76)
We shall now choose a “good” v. The good choice is v = s. We shall explain why: The “error” coming from (µ(s) − µ(u)) Aψuε for u close to s is very small. The “error” coming from earlier times is large in L 2 , but the propagation time s − u is also large and hence most of the wavefunction will have left the region S (cf. Corollary 7.2)). So our strategy is not to show that the “error” is small in L 2 (which would not work) but to show that the “error” which is not small in L 2 leaves the region S and what is left of the error in the relevant region is small and thus in fact deserves to be called an error. The estimates in Corollary 7.2 are only valid from a small time on. This is an inheritance of the singular behavior of the generalized eigenfunctions and must be taken into account. This will lead to a slight complication which makes another splitting necessary. In detail: choosing u = s we obtain, splitting the time according to the idea above introducing another cutoff σ which will be specified below (and which takes care of the applicability of Corollary 7.2), i s ε ε ψsε = Vµ(s) (s, 0)χ + (s, u)Aψuε du (µ(u) − µ(s)) Vµ(s) ε s− σ σ i s− ε + (s, u)Aψuε du. (µ(u) − µ(s)) Vµ(s) ε 0
On Adiabatic Pair Creation
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Hence 1S ψsε
≤
ε 1S Vµ(s) (s, 0)χ +
+
1 ε
s− σ 0
1 ε
s
s− σ
(µ(s) − µ(u)) Aψuε du
ε (s, u)Aψuε du. (µ(s) − µ(u)) 1S Vµ(s)
(77)
The second summand is bounded by C ε
s
s− σ
(s − u) A ∞ 1S A ψuε du ≤
C 2 σ . ε
(78)
For (77) we wish to apply now Corollary 7.2 (iii) to the first and third term. For the first term ε 1S Vµ(s) (s, 0)χ
we have the replacement of variables u =s and v =s, hence we need that s > ε(µ(s) − 1)
3 − 2(1−ξ )
≥ ε(Cs)
3 − 2(1−ξ )
, i.e. that s
3 1+ 2(1−ξ )
But since s > ε1/3−ξ we have that s
>C
3 − 2(1−ξ )
3 1+ 2(1−ξ )
ε = Cε.
>ε
(79)
3 1+ 2(1−ξ ) (1/3−ξ )
. Since 1 +
3 2(1−ξ )
1−3ξ 5 (1/3 − ξ ) = 5−2ξ 6 1−ξ ≤ 6 < 1 the condition for the corollary is fulfilled provided that ε is small enough (Cε1/6 < 1, with the C of formula (80)). Hence 3
3
3
3
3
3
3
ε 1S Vµ(s) (s, 0)χ ≤ Cs −1/2 ε 2 − 2 ξ s − 2 + 2 ξ ( χ + Dµ(s) χ ) ≤ Cε 2 − 2 ξ s −2+ 2 ξ , (80)
since χ = 1 and Dµ(s) χ ≤ |µ(s) − µ(0)| Aχ + D0 χ ≤ C A ∞ + Dµ(0) χ ≤ C . Next we use Corollary 7.2 on the third term of (77), that is on the term ε (s, u)Aψ ε with the replacement of variables u 1S Vµ(s) =s − u and v =s and χ = Aψuε , u which has support S A ⊂ S. Therefore we need that 3 − 2(1−ξ )
s − u > ε(µ(s) − 1)
> Cεs
3 − 2(1−ξ )
.
(81)
Choosing σ = Cεs
3 − 2(1−ξ )
(82)
with the C of formula (81), this is satisfied for all u < s − σ , i.e. for the integrand of the third summand. Hence we have for the third term that 3 3 3 3 ε (s, u)Aψuε ≤ s −1/2 ε 2 − 2 ξ (s − u)− 2 + 2 ξ Aψuε + Dµ(0) Aψuε . (83) 1S Vµ(s)
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To control this term we need to find an upper bound on Dµ(s) Aψuε uniform in 0 ≤ s, u ≤ σ , which we will do next. We have that ∂s ψ ε , Dµ(s) ψ ε s s = ψsε , (∂s Dµ(s) )ψsε + (∂s ψsε ), Dµ(s) ψsε + ψsε , Dµ(s) ∂s ψsε i i = ψsε , A(∂s µ(s))ψsε + Dµ(s) ψsε , Dµ(s) ψsε + ψsε , Dµ(s) Dµ(s) ψsε ε ε ≤ (∂s µ(s)) A ∞ 1S A ψsε . Integrating and observing that Dµ(0) χ < ∞ implies |χ , Dµ(0) χ | < ∞, we obtain |ψsε , Dµ(s) ψsε | < C with a constant C uniform in 0 ≤ s ≤ σ . Using this we get 2 ψ ε | < C with another constant C uniform in 0 ≤ s ≤ σ . similarly that |ψsε , Dµ(s) s Hence with (39) Dµ(s) Aψuε ≤ (µ(s) − µ(u))A2 ψuε + ADµ(u) ψuε + 3j=1 α j (∂ j A)ψuε ≤ (µ(s) − µ(u)) A 2∞ + A ∞ Dµ(u) ψuε + ∇ A ∞ ≤ C
(84)
with a constant C uniform in 0 ≤ s ≤ σ . Using (83), (84) and that ψuε is normalized we get σ 1 s− ε (s, u)Aψuε du (µ(s) − µ(u)) 1S Vµ(s) ε 0 σ C s− ≤ (µ(s) − µ(u)) ε 0 3 3 3 3 ×s −1/2 ε 2 − 2 ξ (s − u)− 2 + 2 ξ Aψuε + Dµ(s) Aψuε du 1 3 CC s −1/2 3 − 3 ξ s ε 2 2 (s − u)− 2 + 2 ξ ( A ∞ + C) du ≤ ε 0 1
3
3
≤ Cε 2 − 2 ξ s 2 ξ . This, (78) (with σ˜ replaced by (82)) and (80) introduced in (77) yields 3
1S ψsε ≤ Cε 2 − 2 ξ s −2+ 2 ξ + Cεs − 1−ξ + Cε 2 − 2 ξ s 2 ξ
−3−3ξ 1 3 3 1 3 1 3 3 3 ≤ Cξ ε 2 − 2 ξ s − 2 εs − 2 + 2 ξ + ε 2 + 2 ξ s 2−2ξ + s 2 + 2 ξ . 3
3
3
1
3
3
Since σ > s > ε1/3−ξ it follows that for ε small enough, 1
3
1
3
εs − 2 + 2 ξ ≤ εε− 6 σ 2 ξ < 1 1
3
ε2+2ξs
−3−3ξ 2−2ξ 3
3
< ε 2 + 2 ξ ε− 1
3
3
1+ξ 1−3ξ 2 1−ξ
1
3
≤ ε 2 + 2 ξ ε−
3
s 2 + 2 ξ < σ 2 + 2 ξ < C. Hence 1
3
3
1S ψsε ≤ Cε 2 − 2 ξ s − 2 .
1+ξ 2
= εξ < 1
On Adiabatic Pair Creation
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7.3. Propagation Estimates for the Time Dependent Case: “Long” Times s > 0. Lemma 7.3 gives estimates on the decay behavior for times smaller than σ . In principle the Lemma can be extended also for larger times for a very large class of potentials Aµ(s) . ε This seems alright as long as the propagator Vµ(s) leads to fast enough decay, i.e. as long as µ(s) is bounded away from one. But we are especially interested in the case that µ(s) attains the critical value µ(s) = 1 again after time σ , since the potential will be switched off again. We shall need a different technique to estimate the decay behavior in this situation for times s > σ (cf. Lemma 7.5). This techniques will be based on the fact that by time σ most of the wavefunction has already left the area S A of the potential. This allows us to choose in the comparison of U ε (s, 0)χ with Vvε χ a fixed value of v, in fact we shall use v = σ , in contrast to Lemma 7.3 where we chose v = s. This has the advantage, that we can use fixed cutoffs in Fourier space, i.e. we can use Corollary 7.2 (i) and (ii). Lemma 7.5. (Propagation Estimates - Time Dependent Case: “Long” times). Let S, σ , ξ and χ satisfy the conditions of Lemma 7.3. Then there exists a C < ∞ such that for all s > σ , 1
3
1S U ε (s, 0)χ ≤ Cε 12 − 4 ξ .
(85)
L 2 -function
Proof. Despite the fact that an has mostly left any compact region by time σ , to show that it scatters is still not easy. The reason is that we deal with a time evolution which is generated by a time dependent Hamiltonian. We shall use again a freezing of the potential defining an auxiliary time evolution. We start with an auxiliary lemma about the auxiliary time evolution with which we shall later compare the true evolution: Lemma 7.6. (Auxiliary Lemma). Let S, σ , ξ and χ satisfy the conditions of Lemma 7.3. ε (s) be the unitary defined by U ε (s, 0) = U ε (s, 0) for s ≤ σ and U ε (s, σ ) = Let U ε Vµ(σ ) (s, σ ) for s > σ . Let ε (s, 0)χ . χsε := U
(86)
sε , a C < ∞ and for any m ∈ N a Cm such that for all s > σ , Then there exists a ψ sε ≤ Cε 12 − 4 ξ χsε − ψ 1
3
(87)
and sε ≤ Cm εm/3−1 s −m . 1S ψ χsε
ψsε
χsε
(88) ε (s, σ )χ ε Vµ(σ σ )
Proof. With the notation (74) = for s ≤ σ and = for s > σ . Using (76) with s = v = σ we obtain i σ ε ε ε χσε = Vµ(σ (σ, 0)χ + (µ(v) − µ(σ )) Vµ(σ ) ) (σ, v)A(x)ψv dv. ε 0 ε (s, σ ) yields Hence applying Vµ(σ )
χsε
=
ε Vµ(σ ) (s, 0)χ
i + ε
σ 0
ε ε (µ(v) − µ(σ )) Vµ(σ ) (s, v)A(x)ψv dv
2/3 i σ −ε ε ε ε (s, 0)χ + = Vµ(σ (µ(v) − µ(σ )) Vµ(σ ) ) (s, v)A(x)ψv dv ε 0 i σ ε ε + (µ(v) − µ(σ )) Vµ(σ ) (s, v)A(x)ψv dv. ε σ −ε2/3
(89)
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The splitting of the integrals are done for application of Corollary 7.2 (i) and (ii) to control (89) and will become clearer in a moment. We must process in various steps. We sε of Lemma 7.6. define (in view of Corollary 7.2) now the function ψ sε := V ε (s, 0)ρκ,µ(σ ) χ ψ µ(σ ) 2/3 i σ −ε ε ε + (µ(v) − µ(σ )) Vµ(σ ) (s, v)ρκ,µ(σ ) Aψv dv. ε 0
(90)
We note that by definition ε (s, σ )ψ sε = U σε . ψ
(91)
Now sε ≤ 1S V ε (s, 0)ρκ,µ(σ ) χ 1S ψ µ(σ ) σ −ε2/3 1 ε ε + (µ(v) − µ(σ )) 1S Vµ(σ ) (s, v)ρκ,µ(σ ) Aψv dv . ε 0
(92)
We subtract now (90) from (89), we take the norms, use the triangle inequality and use ε , unitarity of Vµ(σ ) sε ≤ χ − ρκ,µ(σ ) χ χsε − ψ 2/3 1 σ −ε |µ(v) − µ(σ )| Aψvε − ρκ,µ(σ ) Aψvε dv + ε 0 1 σ |µ(v) − µ(σ )| Aψvε dv. +C ε σ −ε2/3 Using that Aψvε ≤ A ∞ one gets after trivial reordering sε ≤ 1 − ρκ,µ(σ ) χ + Cε1/3 χsε − ψ 2/3 CC σ −ε + (σ − v) 1 − ρκ,µ(σ ) Aψvε dv. ε 0
(93)
We shall now estimate the terms in (92) and (93) using Corollary 7.2. The terms are ε (s, 0)ρ ε ε 1S Vµ(σ κ,µ(σ ) χ , (1 − ρκ,µ(σ ) )χ , 1S Vµ(σ ) (s, v)ρκ,µ(σ ) Aψv and (1 − ) ε ρκ,µ(σ ) )Aψv . Note that χ and Aψvε are compactly supported and have finite energy (by (84) and the assumptions of the lemma). For application of Corollary 7.2 we must check whether −
3
the inequality for the propagation time (i.e. s ≥ u > ε(µ(v) − 1) 2(1−ξ ) ) is satisfied. We first want to use Corollary 7.2 (i) and (ii) on χ with the following replacements of variables: s =s, v =σ and u =σ . Hence the condition of the Corollary reads now −
3
s ≥ σ > ε(µ(σ ) − 1) 2(1−ξ ) . The first inequality is satisfied by assumption of the lemma. Since ∂v µ(v) ≥ C > 0 for all 0 < v < σ (by assumption of the lemma) and µ(0) = 1 we have that µ(σ ) − 1 > 0. Hence for small enough ε we have that 3 − 2(1−ξ )
σ > ε(µ(σ ) − 1)
and Corollary 7.2 (i) and (ii) yields, observing the replacements
1S Vµ(σ ) (s, 0)ρκ,µ(σ ) χ ≤ Cm ( Dµ(σ ) χ )εm s −m
(94)
On Adiabatic Pair Creation
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and 3
3
3
3
(1 − ρκ,µ(σ ) )χ ≤ Cσ − 4 (1−ξ ) ε 4 (1−ξ ) σ −1/4 χ = Cσ −1+ 4 ξ ε 4 (1−ξ ) .
(95)
Next we want to use Corollary 7.2 (i) and (ii) on 1S Vµ(σ ) (s, v)ρκ,µ(σ ) Aψvε with v ≤ σ − ε2/3 . Thus we must make the following replacements of variables in the corollary: χ = A(x)ψvε , v =σ , u =σ − v and s =s − v. Then the condition of the Corollary −
3
becomes s − v ≥ σ − v > ε(µ(σ ) − 1) 2(1−ξ ) , which is why we did the splitting of the integrals in (89) in the first place, namely we have that v ≤ σ − ε2/3 , so that the condition is satisfied for small enough ε. Hence we can use the corollary on A(x)ψvε , making the correct replacements to obtain 1S Vµ(σ ) (s, v)ρκ,µ(σ ) Aψvε ≤ Cm ( Dµ(σ ) A(x)ψvε )(s − v)−m εm
(96)
and 3
3
1
(1 − ρκ,µ(σ ) )Aψvε ≤ Cε 4 (1−ξ ) (σ − v)− 4 (1−ξ ) σ − 4 Aψvε .
(97)
Equations (94)–(97) can now be used to control (92) and (93). Inserting (95) and (97) into (93) yields sε ≤ Cσ −1+ 4 ξ ε 4 (1−ξ ) + Cε1/3 χsε − ψ 2/3 3 1 3 C σ −ε + (σ − v)(σ − v)− 4 (1−ξ ) σ − 4 ε 4 (1−ξ ) Aψvε dv . ε 0 3
3
Now Lemma 7.3 comes into play. Without the control of Aψvε which the lemma provides us with, the last summand would be of order ε−1/4 and thus would explode as ε → 0. But the estimates of Lemma 7.3 are only good for times larger than ε1/3 . For smaller times the trivial estimate Aψvε ≤ C is better. Thus we split the v integral accordingly and arrive at sε ≤ Cσ −1+ 4 ξ ε 4 (1−ξ ) + Cε1/3 χsε − ψ 1/3 1 3 3 C ε + (σ − v) 4 + 4 ξ ε 4 (1−ξ ) σ −1/4 Aψvε dv ε 0 1 3 3 C σ + (σ − v) 4 + 4 ξ ε 4 (1−ξ ) σ −1/4 Aψvε dv . ε ε1/3 3
3
Now use Lemma 7.3 on the last summand. Observe first that by (74), Aψvε ≤ A ∞ 1S A U ε (v, 0)χ ≤ A ∞ 1S A ∪S U ε (v, 0)χ . We apply to 1S A ∪S U ε (v, 0)χ Lemma 7.3 and get, estimating σ − v ≤ σ ≤ C, that for all s ≥ σ , 3 3 C 3 3 sε ≤ Cσ −1+ 4 ξ ε 4 (1−ξ ) + Cε1/3 + σ 4 ξ ε 4 (1−ξ ) ε1/3 dv χsε − ψ ε 1 3 1 3 3 3 C σ + (σ − v) 4 + 4 ξ ε 4 (1−ξ ) σ −1/4 ε 2 − 2 ξ v − 2 dv ε ε1/3 3
3
1
3
1
9
≤ Cε 4 − 4 ξ + Cε1/3 + Cε 12 − 4 ξ + Cε 12 − 4 ξ 1
3
≤ Cε 12 − 4 ξ , which is (87).
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Next we estimate (92). Introducing (94) and (96) yields sε ≤ Cm ( Dµ(σ ) χ )εm s −m 1S ψ 2/3 C σ −ε + (σ − v) Cm ( Dµ(σ ) Aψvε )εm (s − v)−m dv. ε 0
(98)
Recall that s ≥ σ , so for ε small enough we have s(1 − ε2/3 σ −1 ) ≥ σ (1 − ε2/3 σ −1 ), hence s − σ + ε2/3 ≥ ε2/3 sσ −1 , so for v ≤ σ − ε2/3 , 2
(s − v)−m ≤ (s − σ + ε2/3 )−m ≤ ε− 3 m s −m σ m . Using this and the fact that Dµ(σ ) Aψvε is bounded (cf. (84)) we get for (98), sε ≤ Cm εm s −m + Cm εm/3−1 s −m ≤ 2Cm εm/3−1 s −m , 1S ψ which is (88).
We shall now prove Lemma 7.5. Using (75) we have for s > σ that ε ε i s ε (s, σ ) ψ ε (v, σ )ψ σ = − σε dv, U (s, v) (µ(σ ) − µ(v)) A(x)U U (s, σ ) − U ε σ and therefore by (91), sε + σε ≤ 1S ψ 1S U (s, σ )ψ ε
1 ε
s σ
vε dv . (µ(v) − µ(σ )) A ∞ 1S ψ
Using (88), Cm s (µ(σ ) − µ(v)) v −m εm/3−1 dv ε σ Cm s −m+1 m/3−1 + v ε dv ε σ
σε ≤ Cm εm/3−1 s −m + 1S U ε (s, σ )ψ ≤ Cm εm/3−1 s −m
≤ Cm εm/3−1 s −m + Cm (s −m+2 − σ −m+2 )εm/3−2 ≤ Cm εm/3−1 s −m + Cm s −m+2 εm/3−1−1 ≤ Cm s −m εm/3−2 .
ε (σ, 0)χ = U ε (σ, 0)χ , thus We turn now to 1S U ε (s, 0)χ . Recall that χσε = U 1S U ε (s, 0)χ = 1S U ε (s, σ )χ ε σ ε σ − χσε + 1S U ε (s, σ )ψ σε ≤ 1S U ε (s, σ ) ψ ε σ − χσε + 1S U ε (s, σ )ψ σε ≤ ψ 1
3
≤ Cε 12 − 4 ξ + Cm s −m εm/3−2 , where we used (87) and (99). Choosing m large enough the lemma follows.
(99)
On Adiabatic Pair Creation
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7.4. Proof of APC. We come now to the proof of Theorem 2.4. We wish to establish that for s > 0 and χ ∈ L 2 : limε→0 ψsε , χ = 0. From Lemma 7.1 we have that limε→0 (1 − PN )ψ0ε = 0. Therefore by lim ψsε , χ = lim U ε (s, 0)PN ψ0ε , χ + lim U ε (s, 0)(1 − PN )ψ0ε , χ ,
ε→0
ε→0
ε→0
and lim U ε (s, 0)(1 − PN )ψ0ε , χ ≤ lim (1 − PN )ψ0ε = 0
ε→0
ε→0
Theorem 2.4 follows from Corollary 7.7. (Decay of the Critical Bound State). Let s > 0 and χ ∈ L 2 . Then lim |U ε (s, 0)PN ψ0ε , χ | = 0.
(100)
ε→0
Proof. Note that PN projects on the subspace with energy 1, hence D0 PN ψ0ε ≤ 1. For the proof it is very convenient to use a two scale argument. Let j (x) ∈ C ∞ be a mollifier with j (x) = 1 for x ≤ 1 and j (x) = 0 for x ≥ 2, define for any δ > 0 1 := j P ψ ε and χ 2 := j χ . jδ (x) := j (δx), χδ,ε δ N 0 δ δ Clearly limδ→0 (1 − jδ ) = 0 for ∈ L 2 . Since N is a finite dimensional subset of L 2 , we thus have lim χ − χδ2 = lim (1 − jδ )χ = 0 and lim (1 − jδ )PN op = 0
δ→0
δ→0
(101)
δ→0
and 1 D0 χδ,ε = D0 jδ PN ψ0ε ≤ C sup ∂k jδ ∞ PN ψ0ε + jδ D0 PN ψ0ε k=1,2,3
= Cδ + jδ PN ψ0ε ≤ Cδ + ψ0ε < ∞. We shall now use Lemma 7.3 and Lemma 7.5. Let s > 0 and δ > 0. Set ξ = 1/12 and let Sδ be the support of jδ . Let {Φ l } be a orthonormal basis of N . For each Φ l there exists a Cδ,l < ∞ such that 1
1Sδ U ε (s, 0) jδ Φ l ≤ Cδ,l ε 48 . Let Cδ := maxl=1...,n {Cδ,l }, then the operator 1Sδ U ε (s, 0) jδ PN is a matrix on N with norm bounded by 1
1Sδ U ε (s, 0) jδ PN op ≤ nCδ ε 48 . Thus 1
1
1 1Sδ U ε (s, 0)χδ,ε = 1Sδ U ε (s, 0) jδ PN ψ0ε ≤ nCδ ε 48 PN ψ0ε ≤ nCδ ε 48 .
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Hence |U ε (s, 0)PN ψ0ε , χ | ≤ |U ε (s, 0)PN ψ0ε , χδ2 | + |U ε (s, 0)PN ψ0ε , χ − χδ2 | ≤ | jδ U ε (s, 0)PN ψ0ε , χ | + PN ψ0ε χ − χδ2 ≤ 1Sδ U ε (s, 0)PN ψ0ε , χ | + χ − χδ2
≤ 1Sδ U ε (s, 0)PN ψ0ε χ + χ − χδ2 1 1 ≤ 1Sδ U ε (s, 0)χδ,ε + 1Sδ U ε (s, 0) PN ψ0ε − χδ,ε + χ − χδ2 1
1 + χ − χδ2 = Cδ ε 48 + PN ψ0ε − χδ,ε 1
= Cδ ε 48 + (1 − jδ )PN ψ0ε + (1 − jδ )χ 1
≤ Cδ ε 48 + (1 − jδ )PN op + (1 − jδ )χ . Taking the limits ε → 0 and δ → 0 the corollary follows in view of (101).
Acknowledgements. The paper has greatly profited from discussions with Stefan Teufel and Herbert Spohn. In particular Sect. 6 has been much simplified by very helpful remarks made by Stefan Teufel on an earlier version of the manuscript. We are grateful for his very careful reading. We thank ESI (Vienna) for hospitality and funds. Work was partly funded by DFG and by FWF Projekt P17176-N02.
8. Appendix: Proof of Lemma 5.1 (24) Recall (32) Vµ (t, 0)ρκ,µ (1 − ρκ,µ )χ (x) 4 3 = ρκ (1 − ρ κ )Fµ (χ )(k, j)d 3 k. (2π )− 2 exp (−it E k ) ϕµ (k, j, x) j=1
We estimate the right hand side via the stationary phase method, i.e. we integrate by Ek parts. Using i kt ∂k exp (−it E k ) = exp (−it E k ) m partial integrations yield - writing
Ek m Ek Ek ..., ∂k := ∂k ∂k k k k where ∂k acts on everything to the right Vµ (t, 0)ρκ,µ (1 − ρκ,µ )χ (x) m 4 ∞ 3 i (2π )− 2 exp (−it E k ) = − t j=1 0
Ek m 2 × ∂k ϕµ (k, j, x) ρκ (1 − ρ κ )Fµ (χ )(k, j)k dΩdk k m 4 3 i = − k −2 (2π )− 2 exp (−it E k ) t j=1
Ek m ϕµ (k, j, x) ρκ (1 − ρ κ )Fµ (χ )(k, j)k 2 d 3 k. × ∂k k
On Adiabatic Pair Creation
195
Since ρ κ (k) = 0 for k ≤ κ and k ≥ κ, 4 1S Vµ (u, 0)χ ∞ ≤ t −m π κ 3 4 3 m
−2 Ek 2 k ∂ ϕ (k, j, x) ρ (1 − ρ )F (χ )(k, j)k sup k µ κ κ µ . k 2κ>k>κ,x∈S , j ∞
(102)
We next show that for any j, l, r ∈ N0 there exist C j,l,r so that ∂k
Ek k
m
k 2 f (k) =
C j,l,r E km−2r k −m−l+r +2 ∂k f (k). j
(103)
j+l+r =m
We prove this equation by induction over m. For m = 0 (103) follows trivially. Assume that (103) holds for some m ∈ N. It follows that
E k m+1 2 Ek Ek m 2 ∂k ∂k k f (k) = ∂k k f (k) k k k Ek j C j,l,r E km−2r k −m+2−l+r ∂k f (k) = ∂k k j+l+r =m j = ∂k C j,l,r E km−2r +1 k (−m−1+2)−l+r ∂k f (k) j+l+r =m
j C j,l,r ∂k E km−2r +1 k (−m−1+2)−l+r ∂k f (k)
=
j+l+r =m
+
j C j,l,r E km−2r +1 ∂k k (−m−1+2)−l+r ∂k f (k)
j+l+r =m
+
C j,l,r E km−2r +1 k −m+3−l+r ∂k
j+1
f (k).
j+l+r =m
Using that E k =
√ k 2 + 1 we have that ∂k E km = m E km−1 ∂k k 2 + 1 = m E km−2 k.
Setting m = m + 1, j = j + 1, l = l + 1 and r = r + 1 yields
E k m+1 2 ∂k k f (k) = k
−2 r − C j,l,r E km k m +2−l+r ∂k f (k) j
j+l+ r = m
+
−2r − C j,l,r E km k m +2−l+r ∂k f (k) j
j+ l+r = m
+
j+l+r = m
−2r − Cj,l,r E km k m +2−l+r ∂k
j+1
f (k)
196
P. Pickl, D. Dürr
for appropriate Cj,l,r < ∞, C j,l,r < ∞ and C j,l,r < ∞, and (103) follows for m = m + 1. Induction yields that (103) holds for all m ∈ N0 . Note that for k → 0, k −2 E km−2r k −m+2−l+r is of order k −m−l+r . For k → ∞ E k is of order k, hence k −2 E km−2r k −m+2−l+r is of order k −l−r (hence bounded for large k). Since we only observe κ → 0, it follows with (103) that for any m, j ∈ N0 , there exist Cm, j < ∞ such that
m Ek m 2 j |k ∂k k f (k) | ≤ Cm, j k −2m+ j | ∂k f (k) | . k 2
(104)
j=0
κ (1 − ρ κ )Fµ (χ ). Using the product rule of In our case (cf. (102)) we have f = ϕµ ρ differentiation it follows that j
∂k ϕµ ρ κ (1 − ρ κ )Fµ (χ ) j j j j = C j1 , j2 , j3 , j4 ∂k 1 ϕµ ∂k 2 ρ κ 1 − ∂k 3 ρ κ ∂k 4 Fµ (χ ) , j1 + j2 + j3 + j4 = j
where C j1 , j2 , j3 , j4 is a combinatorial factor. With (20), (33) and (34) we get using that κ < κ, j | ∂k ϕµ ρ κ (1 − ρ κ )Fµ (χ )| < C j1 , j2 , j3 , j4 C j1 C j4 κ − j2 − j3 (1 + x) j1 j1 + j2 + j3 + j4 = j
j1 + j4 +2 n k × κ + . |µ − 1 − νl k 2 | + ck 3 l=1 n k Collecting the worst terms (i.e. handling the two cases κ −1 < l=1 |µ−1−νl k 2 |+ck 3 and “ ≥ ” separately) we get with an appropriate constant C j that
−1
j
κ (1 − ρ κ )Fµ (χ )| |∂k ϕµ ρ ⎛
j+2 ⎞ n k ⎠ < C j (1 + x) j ⎝κ − j−2 + . |µ − 1 − νl k 2 | + ck 3 l=1
With (104), again collecting the worst terms, it follows that
Ek m 2 k ϕµ ρ κ (1 − ρ κ )Fµ (χ ) |< (1 + x)m Cm κ −m−2 k −m + k −2m | k 2 ∂k k n m+2 1 m 2 +(1 + x) Cm k . |µ − 1 − νl k 2 | + ck 3 l=1
Thus (recall that κ < 1 and that S is compactly supported, hence (1 + x)m is bounded by some constant)
−2 Ek m 2 k ∂ sup ϕ (k, j, x) ρ (1 − ρ )F (χ )(k, j)k k µ κ µ κ k 2κ>k>κ,x∈S , j ∞ ⎛ ⎛ m+2 ⎞⎞ n 1 ⎠⎠ < Cm ⎝κ −2m−2 + sup ⎝k 2 . (105) 2 | + ck 3 |µ − 1 − ν k l 2κ>k>κ l=1
On Adiabatic Pair Creation
Since κ < 1 and thus
197
⎧ ⎨
n 2 ⎫ ⎬ 1 1 sup k2 < 2 4, 2 3 |µ − 1 − νl k | + ck ⎭ c κ 2κ>k>κ ⎩ l=1
(105) is bounded from above by Cm κ
−4
κ
−2m
n m 1 + sup . |µ − 1 − νl k 2 | + ck 3 2κ>k>κ l=1
With (102) (and using that for positive a, b and m ∈ N we have (a + b)m ≥ a m + bm ) Eq. (24) follows.
References 1. Beck, F., Steinwedel, H., Süssmann, G.: Bemerkungen zum Klein’schen Paradoxon. Z. Phys 171, 189–198 (1963) 2. Bhabba, H.J.: The Creation of Electron Pairs by Fast Charged Particles. Proc. R. Soc. London Ser. A 152, 559–586 (1935) 3. Brezin, E., Itzykson, C.: Pair Production in Vacuum by an Alternating Field. Phys. Rev. D. 2, 1191–1199 (1970) 4. Cowan, T. et al.: Observation of correlated narrow-peak structures in positron and electron spectra from superheavy collision systems. Phys. Rev. Lett. 56, 444–447 (1986) 5. Dirac, P.: The Principles of Quantum Mechanics. Oxford: Oxford University Press, 1930 6. Dolbeault, J., Esteban, M.J., Loss, M.: Relativistic hydrogenic atoms in strong magnetic fields. http:// arxiv.org/list/math/0607027v1, 2006 7. Dürr, D., Pickl, P.: Flux-across-surfaces Theorem for a Dirac-particle. J. Math. Phys. 44, 423–465 (2003) 8. Gershtein, S., Zeldovich, Y.: Positron Production During the Mutual Approach of Heavy Nuclei and the Polarization of the Vacuum. Sov. Phys. JETP 30, 358–361 (1970) 9. Greiner, W., Müller, B., Rafelski, J.: Quantum Electrodynamics of Strong Fields. Berlin: Springer Verlag, 1985 10. Hainzl, C., Lewin, M., Solovej, J.P.: Mean-field approximation in Quantum Electrodynamics. The no-Photon Case. Comm. Pure Appl. Math. 60, 546–596 (2007) 11. Heisenberg, W., Euler, H.: Consequences of Dirac’s Theory of the Positron. Z. Phys. 98, 714 (1936) 12. Ikebe, T.: Eigenfunction expansions associated with the Schrödinger operators and their application to scattering theory. Arch. Rat. Mech. Anal. 5, 1–34 (1960) 13. Jensen, A., Kato, T.: Spectral Properties of Schrödinger operators and time-decay of the wavefunctions. Duke Math. J. 46(3), 583–611 (1979) 14. Klaus, M.: On coupling constant thresholds and related eigenvalue properties of Dirac operators. J. Reine Angew.Math. 362, 197–212 (1985) 15. Klein, O.: Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac. Z. Phys. 53, 157 (1929) 16. Müller, B.: Positron creation in superheavy quasimolecules. Ann. Rev. Nucl. Science 26, 351–383 (1976) 17. Müller, B., Peitz, H., Rafelski, J., Greiner, W.: Solutions of the Dirac Equation for Strong External Fields. Phys. Rev. Lett. 28, 1235–1238 (1972) 18. Nenciu, G.: On the adiabatic limit for Dirac particles in external fields. Commun. Math. Phys. 76, 117–128 (1980) 19. Nenciu, G.: Existence of spontaneous pair creation in the external field approximation of Q.E.D. Commun. Math. Phys. 109, 303–312 (1987) 20. O’Connell, R.F.: Effect of the Anomalous Magnetic Moment of the Electron on Spontaneous Pair Production in a Strong Magnetic Field. Phys. Rev. Lett. 21, 397–398 (1968) 21. Pickl, P.: Existence of Spontaneous Pair Creation, Dissertation, 2005 22. Pickl, P.: Generalized Eigenfunctions for Dirac Operators Near Criticality. J. Math. Phys. 48, 1 (2007) 23. Pickl, P., Dürr, D.: Adiabatic Pair Creation in Heavy Ion and Laser Fields. Eur. Phys. Lett. 81, 40001 (2008) 24. Popov, V.S.: Positron Production in a Coulomb Field with Z > 137. Zh. Eksp. Teor. Fiz. 59, 965–84 (1970)
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25. Prodan, E.: Spontaneous transitions in quantum mechanics. J. Phys. A: Math. Gen. 32, 4877–4881 (1999) 26. Rafelski, J., Fulcher, L.P., Greiner, W.: Superheavy Elements and an Upper Limit to the Electric Field Strength. Phys. Rev. Lett. 27, 958–961 (1971) 27. Reed, M., Simon, B.: Functional Analysis. San Diego: Academic Press, 1980 28. Rein, D.: Über den Grundzustand überschwerer Atome. Z. Phys 221, 423–430 (1969) 29. Reinhardt, J., Müller, U., Müller, B., Greiner, W: The decay of the vacuum in the field of superheavy nuclear systems. Z. F. Physik A 303, 173–188 (1981) 30. Riesz, F., von Sz.-Nagy, B.: Functional Analysis. New York: F. Ungar. Publ. Co., 1955 31. Roberts, C.D., Schmidt, S.M., Vinnik, D.V.: Quantum Effects with an X-Ray Free-Electron Laser. Phys. Rev. Lett. 89, 153901 (2002) 32. Rodnianski, I., Schlag, W.: Time decay for solutions of Schrödinger equations with rough and time dependent potentials. Invent. Math. 155, 451–513 (2004) 33. Sauter, F.: Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs. Z. Phys. 69, 742 (1931) 34. Scharf, G., Seipp, H.P.: Charged Vacuum, Spontaneous Positron Production and all that. Phys. Lett. 108B, 196–198 (1982) 35. Schweppe, J., et al.: Observation of a Peak Structure in Positron Spectra from U+Cm Collisions. Phys. Rev. Lett. 51, 2261–2264 (1983) 36. Schwinger, J.: On Gauge Invariance and Vacuum Polarization. Phys. Rev. 82, 664–679 (1951) 37. Smith, K., Peitz, H., Müller, B., Greiner, W.: Induced Decay of the Neutral Vaccum in Overcritical Fields Occurring in Heavy-Ion Collisions. Phys. Rev. Lett. 32, 554–556 (1974) 38. Teufel, S.: Adiabatic Perturbation Theory in Quantum Dynamics. Berlin: Springer Verlag, 2003 39. Teufel, S.: The flux-across-surfaces theorem and its implications for scattering theory. Dissertiation an der Ludwig-Maximilians-Universität, München, 1999 40. Thaller, B.: The Dirac equation, Springer Verlag, Berlin, 1992 41. Yamada, O.: Eigenfunction expansions and scattering theory for Dirac operators. Publ. RIMS. Kyoto Univ. 11, 651–689 (1976) 42. Zeldovich, Ya.B., Popov, V.S.: Electronic Structure of Superheavy Atoms. Sov. Phys. Usp. 14(6), 673–694 (1972) Communicated by H. Spohn
Commun. Math. Phys. 282, 199–208 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0453-1
Communications in
Mathematical Physics
Eigenvalue Bounds for Perturbations of Schrödinger Operators and Jacobi Matrices With Regular Ground States Rupert L. Frank1 , Barry Simon2, , Timo Weidl3, 1 Department of Mathematics, Princeton University, Princeton, NJ 08544, USA.
E-mail: [email protected]
2 Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA.
E-mail: [email protected]
3 Stuttgart University, Department of Mathematics and Physics, Pfaffenwaldring 57,
70569 Stuttgart, Germany. E-mail: [email protected] Received: 28 June 2007 / Accepted: 19 July 2007 Published online: 14 March 2008 – © Springer-Verlag 2008
Abstract: We prove general comparison theorems for eigenvalues of perturbed Schrödinger operators that allow proof of Lieb–Thirring bounds for suitable non-free Schrödinger operators and Jacobi matrices. 1. Introduction Consider a Schrödinger operator, H0 = − + V0
(1.1)
on L 2 (Rν ) which we suppose obeys inf spec(− + V0 ) = 0.
(1.2)
(By subtracting a constant, we can always arrange this, and by assuming this, the notation simplifies.) We are interested in controlling the negative eigenvalues of H = H0 + V.
(1.3)
E 1 (V0 ; V ) ≤ E 2 (V0 ; V ) ≤ · · · ≤ E n (V0 ; V ) ≤ · · ·
(1.4)
We let
be either the negative eigenvalues or 0, that is, E j (V0 ; V ) = min(0; inf{λ, dim P(−∞,λ] (H ) ≥ j}).
(1.5)
Supported in part by NSF Grant DMS-0140592 and U.S.–Israel Binational Science Foundation (BSF) Grant No. 2002068. Supported in part by DFG grant WE-1964 2/1.
200
R. L. Frank, B. Simon, T. Weidl
We say that H0 has a regular ground state if and only if there exists a function, u 0 , on Rν obeying (− + V0 )u 0 = 0, 0 < c1 ≤ u(x) ≤ c2 < ∞,
(1.6) (1.7)
for some c1 , c2 . We take c1 = inf u, c2 = sup u, and let 2 c2 β(V0 ) = . c1
(1.8)
Our main result is that any bound on the number or sums of eigenvalues for the operator − + V can be carried (with a change in the coupling constant) to the operator H0 + V. This is based on the following observation: Theorem 1.1. For any V ≤ 0 and V0 obeying (1.2) with regular ground state, we have for all j, |E j (0; β −1 V )| ≤ |E j (V0 ; V )| ≤ |E j (0; βV )|.
(1.9)
This result is remarkable for its generality and also for the simplicity of its proof. We will see in Sect. 2 that it can be used to compare not only V0 and 0 but two arbitrary V0 ’s with relatively bounded ground states. Of course, (1.9) immediately implies bounds on moments of bound states: Sγ (V0 ; V ) =
∞
|E j (V0 ; V )|γ ,
(1.10)
j=1
where we look at this only for γ ≥ 0 and interpret 00 = 0 so S0 is the number of strictly negative eigenvalues. Clearly, Theorem 1.1 implies: Corollary 1.2. For any γ > 0 and β = β(V0 ) given by (1.8), Sγ (0; β −1 V ) ≤ Sγ (V0 ; V ) ≤ Sγ (0; βV ). The standard Lieb–Thirring inequalities (reviewed in [8,14]) assert ν Sγ (0; βV ) ≤ L γ ,ν |V (x)|γ + 2 d ν x
(1.11)
(1.12)
for γ ≥ 21 in ν = 1, γ > 0 in ν = 2, and γ ≥ 0 in ν ≥ 3. In some cases, the optimal constants are known, for example, L 1 ,1 = 21 . (These yield good constants in 2 our perturbed estimates but we do not claim optimal constants for our situation!) Clearly, (1.11) implies: Corollary 1.3. ν
Sγ (V0 ; V ) ≤ L γ ,ν β γ + 2
ν
|V (x)|γ + 2 d ν x
(1.13)
and, in particular, ∞ j=1
in ν = 1 dimension.
1
|E j (V0 ; V )| 2 ≤
1 2
β
|V (x)| d x
(1.14)
Eigenvalue Bounds
201
One can also obtain logarithmic estimates as in [13] and Hardy–Lieb–Thirring bounds as in [5]. Since it is known [20,17,14] that for ν = 1 and V ≤ 0, S 1 (0; V ) ≥ 41 |V (x)| d x, (1.15) 2
we conclude that 1 S 1 (V0 ; V ) ≥ 2 4β
|V (x)| d x.
(1.16)
These results are of interest because there are many cases which are known to have regular ground states. Example 1.4. If V0 is periodic, then there is a positive periodic ground state. If V0 is ν locally L 2 (if ν ≥ 3, locally L 1 if ν = 1, and locally L p with p > 1 if ν = 2), then it is known that eigenfunctions are continuous (see [18]) and thus, H0 has a regular ground state. Example 1.5. We will discuss Jacobi matrices in Sects. 3 and 4. It is known (see [3, 15,19]) that elements in the isospectral torus of finite gap Jacobi matrices have regular ground states. Example 1.6. If u is any function obeying (1.7), then V0 = state.
u u
has a regular ground
In Sect. 2, we will review the ground state representation and prove a stronger theorem than Theorem 1.1. As hinted, it is the ground state representation that is critical. In this regard, we should emphasize that the variational argument we use in Sect. 2 has appeared earlier in work of the Birman school—we would mention, in particular, Lemma 6.1 of Birman, Laptev, and Suslina [2], although it may have appeared earlier in their work. Our novelty here is the wide applicability, the use in CLR and Lieb–Thirring bounds, and the applicability to the discrete case and Szeg˝o estimates. As we will explain in Sect. 4, an initial motivation for this work was critical Lieb–Thirring bounds for finite gap almost periodic Jacobi matrices in connection with Szeg˝o’s theorem for such situations. Ground state representations do not seem to be in the literature for Jacobi matrices, so we do this first in Sect. 3, and then prove an analog of Theorem 1.1 for Jacobi matrices in Sect. 4. Section 5 discusses some other cases. 2. Comparison for Schrödinger Operators Fundamental to our results is the ground state representation that if (1.6) holds for u 0 , continuous and strictly positive on Rν , then gu 0 , H0 gu 0 = |∇g|2 u 20 d ν x. (2.1) Ground state representations go back to Jacobi [11]. For Schrödinger operators, it appears at least as far back as Birman [1] and it was used extensively in constructive quantum field theory (especially by Segal, Nelson, Gross, and Glimm–Jaffe; see Glimm–Jaffe [7]). As a basis for comparison theorems, it was used by Kirsch–Simon [12] and, as noted above, in a similar context by Birman–Laptev–Suslina [2].
202
R. L. Frank, B. Simon, T. Weidl
We will be cavalier about technical assumptions needed for (2.1). From one point of view, we can use (2.1) as a definition of H0 ! Namely, the right side of (2.1) defined for g’s with distributional derivative making the right side finite is easily seen to be a closed 0 quadratic form on Hu 0 ≡ L 2 (Rν , u 20 d ν x) defining a positive selfadjoint operator H −1 2 ν ν on Hu 0 . The unitary operator W : L (R , d x) → Hu 0 by W g = u 0 g lets us define 0 W and our results hold for perturbations of that. H0 = W −1 H It is not hard to prove that if V0 = V0,+ + V0,− with V0,+ ∈ L 1loc (Rν , d ν x) and V0,− ∈ K ν , the Kato class, then the selfadjoint operator H0 defined as the form closure of − + V0 on C0∞ obeys (2.1) if u 0 is a positive distributional solution of (1.6). Notice that we do not need (1.2), but only inf spec(H0 ) ≥ 0 for this to work, and Theorem 2.1 below holds in that case (although inf spec(H0 ) = inf spec(H1 )) under the hypothesis of the theorem. Here is our main result: Theorem 2.1. Let H0 , H1 have the form (2.1) for positive continuous functions u 0 , u 1 . Suppose u0 u0 ≤ sup <∞ (2.2) 0 < inf u1 u1 and let β≡
sup( uu 01 )
2
inf( uu 01 )
.
(2.3)
For any V ≤ 0, let E j be given by (1.5). Then |E j (V0 ; V )| ≤ |E j (V1 ; βV )|.
(2.4)
Remark. By interchanging V0 and V1 and replacing V by β −1 V, we get the complementary inequality |E j (V1 ; β −1 V )| ≤ |E j (V0 ; V )|.
(2.5)
Lemma 2.2. Let V ≤ 0. Let τ > 0. If for some g, gu 0 , (H0 + V )gu 0 ≤ −τ gu 0 , gu 0 ,
(2.6)
gu 1 , (H1 + βV )gu 1 ≤ −τ gu 1 , gu 1 .
(2.7)
then
Proof. Let −2 u0 2 u1 = inf , β+ = sup u1 u0 2 −2 u0 u1 β− = inf = sup , u1 u0
(2.8) (2.9)
so −1 β+ = ββ− ⇒ ββ+−1 = β− .
(2.10)
Eigenvalue Bounds
203
Since −V g 2 ≥ 0 and |∇g|2 ≥ 0, we have −1 2 2 (∇g) u 1 d x ≤ β− (∇g)2 u 20 d x − V g 2 u 21 d x ≥ −β+−1 V g 2 u 20 d x
(2.11) (2.12)
so, by (2.10) and (2.1), −1 gu 0 , (H0 + V )gu 1 gu 1 , (H1 + βV )gu 1 ≤ β− −1 gu 0 , gu 0 . ≤ −τβ−
(2.13)
But gu 0 , gu 0 ≥ β− gu 1 , gu 1
(2.14)
and τ > 0, so RHS of (2.13) ≤ −τ gu 1 , gu 1 , proving (2.7). Proof of Theorem 2.1. If E j (V0 ; V ) = 0, there is nothing to prove. If τ ≡ |E j (V0 ; V )| > 0, there is a space, H j , of dimension at least j so ψ, (H0 + V )ψ ≤ −τ ψ, ψ for ψ ∈ H j . By the lemma, if ϕ =
u1 u 0 ψ,
(2.15)
we have
ϕ, (H1 + βV )ϕ ≤ −τ ϕ, ϕ.
(2.16)
Thus, there is a space of dimension at least j where (2.16) holds. By the min-max principle (see [16]), E j (V1 ; βV ) ≤ −τ,
(2.17)
which is (2.4). 3. Ground State Representation for Jacobi Matrices While we are interested mainly in semi-infinite one-dimensional Jacobi matrices, that is, tridiagonal semi-infinite matrices, we can consider the higher-dimensional case as well, so we will. So far as we know, there is no prior literature on the ground state representation for discrete operators, so we start with that in this section. In Zν , we let δ j , j = 1, . . . , ν, be the ν component vectors with 1 in the j th place and 0 elsewhere. So k ± δ j are the 2ν neighbors of k ∈ Zν . A Jacobi operator is parametrized by a symmetric a j > 0 for each j, ∈ Zν with | j − | = 1 and a real number bk for each k ∈ Zν . We will suppose sup |bk | + sup |ak | < ∞. k
The Jacobi operator associated with these parameters is the operator J on 2 (Zν ) with (J ϕ) = a ±δ j ϕ ±δ j + b ϕ . (3.1) ±, j=1,...,ν
We will use J ({a m }, {b }) if we want to make the dependence on a and b explicit.
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R. L. Frank, B. Simon, T. Weidl
Lemma 3.1. Let f be a bounded real-valued function on Zν and M f the diagonal matrix on 2 (Zν ) which is multiplication by f . Then [M f , J ({a m }, {b })] = J ({a m ( f − f m ), b ≡ 0}), [M f , [M f , J ({a m }, {b })]] = J ({a m ( f − f m ) , b ≡ 0}). 2
(3.2) (3.3)
Proof. Equation (3.2) is an elementary calculation and it implies (3.3). Theorem 3.2. Let J be a Jacobi operator on 2 (Zν ) with parameters {a m }, {b }. Suppose u is a positive “sequence” parametrized by Zν so that a m u m + b u = 0 (3.4) |m− |=1
for all ∈ Zν . Then for any f with f u ∈ 2 (Zν ), we have a m u u m ( f − f m )2 . f u, (−J ) f u =
(3.5)
m, |m− |=1
Remark. In particular, this shows −J ≥ 0. Proof. It suffices to prove (3.5) for f of finite support and take limits. For the left side converges since −J is a bounded operator, and by positivity, the right side converges (a priori perhaps to ∞, but by the equality to a finite limit; it is only here that positivity of u is used). For finite sequences, f , use (3.3), taking expectations in a vector u˜ which equals u on { | f (m) is non-zero for some m with |m − | ≤ 1}. Then M f J u˜ = 0, so u, ˜ [M f , [M f , J ]]u ˜ = −2 f u, J f u. 4. Comparison for Jacobi Operators In this section, we will prove an analog of Theorem 1.1 for Jacobi operators. One difference that we have to expect is that of sup spec(J ) = 0; the same is true of λJ for any λ > 0, but the ground states are the same. Thus, comparison of the two J ’s cannot involve only the ground state ratio but also a setting of scales which will enter as a ratio of a’s. In the Schrödinger case, the scale is set by the − rather than −λ. For notation, we let E j ({a m , b }) be the max of zero and the j th eigenvalue of J ({a m , b }) counting from the top. Here is our main result: (0)
(0)
(1)
(1)
Theorem 4.1. Let {a m , b } and {a m , b } be two sets of bounded Jacobi parameters with positive sequences u (0) , u (1) obeying (3.4) for {a (0) , b(0) }, {a (1) , b(1) }, respectively. Let
(0) 2 u β+ = sup , (4.1) (1) u
(0) 2 u β− = inf , (4.2) (1) u
Eigenvalue Bounds
205
γ− = inf
(0) (0) (0)
a j u j u
(1) (1) a (1) j u j u
.
(4.3) (0)
Suppose β+ < ∞ and β− > 0. Then for perturbations {δa m , δb } with a m + δa m > 0, (0) (0) E j ({a m +δa m , b +δb })
⎡ ⎤⎫⎞ ⎛⎧ ⎬ ⎨ (1) (1) ≤ E j ⎝ ηa m , ηb + β ⎣|δb | + |δa m |⎦ ⎠ , ⎭ ⎩ |m− |=1
(4.4) where η=
γ− β−
β=
β+ . β−
(4.5)
Remark. 1. We only have a one-sided inequality as we would in the Schrödinger case if we did not demand V ≤ 0. Since δa terms are never of a definite sign, we cannot have them in a two-sided comparison. But there is clearly a two-sided comparison if δa = 0 and δb > 0. 2. Note that rescaling u (0) or u (1) which changes β+ , β− , γ− does not change η or β. Similarly, η scales properly under changes of the scale of a. Following Hundertmark–Simon [9], we begin with a reduction to the case δa = 0, δb ≥ 0: Lemma 4.2. One has ⎫⎞ ⎛⎧ ⎬ ⎨ (0) (0) (0) (0) E j ({a m + δa m , b + δb }) ≤ E j ⎝ a m , b + |δb | + |δa m | ⎠ . (4.6) ⎭ ⎩ |m− |=1
Thus, it suffices to prove (4.4) when δa m = 0 and δb ≥ 0. As in [9], this follows from
0 a
a a ≤ 0 0
0 a
and b ≤ |b|. Proof of Theorem 4.1. The proof is identical to the proof in Sect. 2, the sole change being that (2.11) needs to be replaced by ,m | −m|=1
So instead of a
(1) (1)
−1 2 a m u u (1) m | f − f m | ≤ γ−
β− β−
= 1, we get η =
γ− β− .
,m | −m|=1
(0) (0)
2 a m u u (0) m | f − fm | .
(4.7)
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Let W : 2 (Zν ) → 2 (Zν ) by (W f )n = (−1)|n| f n
(4.8)
with |n| = |n 1 | + · · · + |n ν |. Then WJ ({a m , b })W −1 = −J ({a m , −b }).
(4.9)
This allows one to control eigenvalues below inf spec(J ) in the same way. We define th E− j to be the min of 0 and the j eigenvalue of J counting from the bottom. We call a sequence u on Zν alternating positive if (−1)|m| u m > 0 for all m. The result is: (0)
(0)
(1)
(1)
Theorem 4.3. Let {a m , b } and {a m , b } be two sets of bounded Jacobi parameters (0) (1) with alternating positive sequences u m , u m obeying (3.4) for {a (0) , b(0) }, {a (1) , b(1) }, respectively. Define β+ , β− , γ− , η, β as in (4.1)–(4.3) and (4.5). Then (4.4) holds with E j replaced by |E − j |. (0)
(0)
Corollary 4.4. Let {a m , b } be a one-dimensional periodic set of Jacobi parameters or the almost periodic parameters associated with a finite gap spectrum (see [19]). Let J0 be the associated half-line Jacobi matrix. Let J be the Jacobi matrix associated with (0) (0) {a m + δa m , b + δb }, where
|δa m | +
| −m|=1 =1,2,3,...
∞
|δb | < ∞.
(4.10)
=1
Let E 1− < E 2− < · · · < inf spec(J0 ) < sup spec(J0 ) < · · · < E 2+ < E 1+ be the eigenvalues of J outside the convex hull of spec(J0 ). Then
dist(E k± , σ (J0 )) 2 ≤ C 1
(|δa m | + |δb |)
(4.11)
k,±
for a constant C depending only on J0 . Proof. We only need that J0 and WJ0 W −1 have regular ground states. This follows from Floquet theory for the periodic case and from the detailed analysis of Jost solutions for the almost periodic case; see [3,15,19]. Then compare to the free J0 (a m = 1 if | − m| = 1, b = 0) and use the bound of [9]. For the periodic case, this is proven by Damanik, Killip, and Simon [4], who also prove this where the sum in (4.11) is over all eigenvalues including the ones in gaps. It remains an interesting question relevant to the study of the Szeg˝o condition (see [3,10]) to get the bound in the almost periodic case. In [10], Hundertmark and Simon prove the weaker bounds where the 21 power is replaced by any p > 21 or where p = 21 , but there is an ε added to the sum.
Eigenvalue Bounds
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5. Some Final Remarks All we needed for our arguments is some kind of ground state representation. That means we can replace − by L0 = −
ν
∂ j A jk (x) ∂k
(5.1)
j,k=1
with {A jk (x)}1≤ j,k≤ν a strictly positive matrix. Hence, if (L 0 + V )u 0 = 0, then f u 0 , (L 0 + V ) f u 0 = ∇ f, A∇ f |u 0 (x)|2 d ν x, and we can still compare to − although sup A and inf A will enter. We can compare magnetic field operators where the magnetic field is fixed but V0 , V vary. For if (− + V0 )u 0 = 0 for some positive u 0 , then for Ha ≡ −(∇ − ia)2 + V0 we have that f u 0 , Ha f u 0 = |(∇ − ia) f |2 |u 0 |2 d ν x. For the discrete case, the key was not tridiagonal matrices, but ones with a ground state representation. For example, non-negative off-diagonal will do. Using the representation of Frank, Lieb, and Seiringer [6], one can treat some perturbations of (−)α , 0 < α < 1. Acknowledgements. It is a pleasure to thank Fritz Gesztesy, Yehuda Pinchover, Robert Seiringer, and Simone Warzel for useful comments, and Michael Aizenman for being a sensitive editor.
References 1. Birman, M.Sh.: On the spectrum of singular boundary-value problems. Mat. Sb. (N.S.) 55 (97), 125–174 (1961) (Russian) 2. Birman, M.Sh., Laptev, A., Suslina, T.A.: The discrete spectrum of a two-dimensional second-order periodic elliptic operator perturbed by a decreasing potential. I. A semi-infinite gap. St. Petersburg Math. J. 12, 535–567 (2001); Russian original in Algebra i Analiz 12, 36–78 (2000) 3. Christiansen, J., Simon, B., Zinchenko, M.: In preparation 4. Damanik, D., Killip, R., Simon, B.: Perturbations of orthogonal polynomials with periodic recursion coefficients. Preprint 5. Ekholm, T., Frank, R.: On Lieb–Thirring inequalities for Schrödinger operators with virtual level. Commun. Math. Phys. 264, 725–740 (2006) 6. Frank, R.L., Lieb, E.H., Seiringer, R.: Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators. JAMS, in press, DOI: 10.1090/s0894-0347-07-00582-6, 2007 7. Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View. New York-Berlin: Springer-Verlag, 1981 8. Hundertmark, D.: Some bound state problems in quantum mechanics. In: “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday”, F. Gesztesy et al., eds., Proc. Symp. Pure Math. 76.1, Providence, RI: Amer. Math. Soc., 2007, pp. 463–496 9. Hundertmark, D., Simon, B.: Lieb–Thirring inequalities for Jacobi matrices. J. Approx. Theory 118, 106–130 (2002) 10. Hundertmark, D., Simon, B.: Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices. J. Math. Anal. Appl. 340, 892–900 (2008) 11. Jacobi, C.G.J.: Zur Theorie der Variationsrechnung und der Differentialgleichungen. J. für Math. von Crelle 17, 68–82 (1837)
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12. Kirsch, W., Simon, B.: Comparison theorems for the gap of Schrödinger operators. J. Funct. Anal. 75, 396–410 (1987) 13. Kovarik, H., Vugalter, S., Weidl, T.: Spectral estimates for two-dimensional Schrödinger operators with application to quantum layers. Commun. Math. Phys. 275(3), 827–838 (2007) 14. Laptev, A., Weidl, T.: Recent results on Lieb–Thirring inequalities. In: Journées “ Équations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000), Exp. No. XX, Nantes: Univ. Nantes, 2000 15. Peherstorfer, F., Yuditskii, P.: Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points. Proc. Amer. Math. Soc. 129, 3213–3220 (2001) 16. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, IV: Analysis of Operators. New York: Academic Press, 1978 17. Schmincke, U.-W.: On Schrödinger’s factorization method for Sturm–Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A 80, 67–84 (1978) 18. Simon, B.: Schrödinger semigroups. Bull. Amer. Math. Soc. 7, 447–526 (1982) 19. Sodin, M., Yuditskii, P.: Almost periodic Jacobi matrices with homogeneous spectrum, infinitedimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal. 7, 387–435 (1997) 20. Zaharov, V.E., Faddeev, L.D.: The Korteweg–de Vries equation is a fully integrable Hamiltonian system. Funk. Anal. i Pril. 5, 18–27 (1971) (Russian) Communicated by M. Aizenman
Commun. Math. Phys. 282, 209–238 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0533-2
Communications in
Mathematical Physics
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models O. Khorunzhiy Département de Mathématiques, Université de Versailles – Saint-Quentin, Versailles, 78035, France. E-mail: [email protected] Received: 10 July 2007 / Accepted: 31 January 2008 Published online: 6 June 2008 – © Springer-Verlag 2008
Abstract: Regarding the adjacency matrices of n-vertex graphs and related graph Laplacian, we introduce two families of discrete matrix models constructed both with the help of the Erd˝os-Rényi ensemble of random graphs. Corresponding matrix sums represent the characteristic functions of the average number of walks and closed walks over the random graph. These sums can be considered as discrete analogues of the matrix integrals of random matrix theory. We study the diagram structure of the cumulant expansions of logarithms of these matrix sums and analyze the limiting expressions as n → ∞ in the cases of constant and vanishing edge probabilities. 1. Introduction It has been about thirty years since map enumeration problems of theoretical physics were related with the diagram representation of the formal cumulant expansions of the form gk 1 1 gVN = log E e Cum k (VN ), (1.1) N2 k! N 2 k≥1
where the average is taken with respect to the Gaussian measure over the space of herq mitian N -dimensional matrices {H N } and VN is given by Tr H N , or more generally, by a linear combination of such traces with certain degrees q ≥ 3 [5,12]. This relation between maps, diagrams and random matrices is still a source of numerous results that reveal deep links between various branches of mathematics, mathematical physics and theoretical physics (see e.g. papers [2,7,21] for references and reviews of results). In these studies, the nature of the large-N limit of the right-hand side of (1.1) and existence of its asymptotic expansion in degrees of 1/N 2 are problems of the primary importance [3,8,11].
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In the present paper we examine the diagram structure of a discrete analog of (1.1), where the role of H N is played by the n × n adjacency matrix An of the Erd˝os-Rényi random graphs with the edge probability pn . In this setting, VN of (1.1) can be replaced by random variables (q) Xn
= Tr
q An
=
n i=1
q (An )ii
and
(q) Yn
=
n
q
(An )i j , q ≥ 2,
(1.2)
i, j=1
and this gives two different discrete matrix models related to the powers of the adjacency matrix and of the Laplace operator on graphs. Variables X and Y represent the numbers of close walks of q steps and walks of q steps over the graphs, respectively. (q) (q) We study the limiting behavior of the cumulants of X n and Yn in the following 1 three asymptotic regimes, when pn = O(1), n pn 1, or pn = O(1/n) as n → ∞. Our main result is that in all of these three regimes, the following limits exist (cf. (1.1)): 1 (q) (q) g → Fk (ω), n → ∞, Cum Y (1.3) k n n pn n 2 with the appropriate choice of the normalizing factors gn . These limiting expressions depend on the asymptotic regime indicated by ω ∈ {1, 2, 3}. The same statement as (q) (1.3) holds for random variables X n with corresponding changes of gn and the limiting expressions. To prove convergence (1.3), we develop a diagram technique similar to that commonly accepted in the random matrix theory. We show that the diagram structure of the cumulants of (1.3) is closely related with the trees with k labeled edges. In the simplest case of q = 2 and ω = 2, we derive explicit recurrent relations that determine the limits (1.3) and prove that the corresponding exponential generating function verifies the Lagrange (or Pólya) equation. Regarding other values of q and ω, we obtain several generalizations of this equation. Using (1.3), we show that the Central Limit Theorem is valid for the centered and (q) (q) renormalized variables X n and Yn . This describes asymptotic properties of the averaged numbers of walks and closed walks over the random graphs. At the same time, our results imply the CLT for the normalized spectral measure of the adjacency matrices of the Erd˝os-Rényi random graphs. This improves known results about the convergence of the corresponding normalized spectral measures [1,16]. Finally, we indicate an asymp(q) (q) totic regime when the CLT does not hold for variables X n and Yn . The paper is organized as follows. In Sect. 2 we introduce the families of matrix sums that can be called the Erd˝os-Rényi matrix models. The key observation here is that the graph Laplacian generates the Erd˝os-Rényi measure on graphs. This allows us to (2) identify Yn of (1.2) as a natural analog of the quartic potential VN = Tr H N4 of (1.1). In Sect. 3, we develop a general diagram technique to study the cumulant expansions of the form (1.1) for discrete Erd˝os-Rényi matrix models of the first and the second type (1.2). In Sect. 4, we prove convergence of normalized cumulants in three main asymptotic regimes. The issue of the Central Limit Theorem is discussed at the end of Sect. 4. In Sect. 5, we study the classes of connected diagrams and derive recurrent relations for (q) their numbers. In Sect. 6, we complete the study of limiting expressions Fk (ω) (1.3) and consider the formal limiting transition for the free energy. This free energy takes different forms in dependence on the edge probability as n → ∞. Section 7 contains a summary of our results.
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2. Discrete Erd˝os-Rényi Matrix Models Regarding (1.1), one can rewrite the mathematical expectation as follows: g 1 g q q E exp Tr H N Tr H N d H N , β > 0, (2.1) = exp −β Tr H N2 + N C N HN N where H N is N × N hermitian matrix, C N = C N (β) is a normalization constant, and the matrix integral runs over the space H N of all hermitian matrices with respect to the Lebesgue measure d H N . The ensemble of random matrices {H N } distributed according to the Gaussian measure with the density exp{−β Tr H N2 } is known as the Gaussian Unitary invariant Ensemble, abbreviated as GUE . This ensemble plays a fundamental role in the random matrix theory (see monograph [19] and references therein). The matrix integral of (2.1) is known as the partition function of the matrix model with the potential q VN = Tr H N . Regarding Tr H N2 of (2.1) as a formal kinetic energy term, one can speculate about the trace of the Laplace operator; being restricted to the space of functions on graphs, it leads immediately to a discrete analog of the matrix integral of (2.1). In this case the integral H N of (2.1) is replaced by the sum over the set of all graphs with n vertices, or equivalently, over the set of all n-dimensional adjacency matrices of graphs. This point of view is fairly natural and one benefits from two important counterparts of it. On one hand, the use of a graph Laplacian indicates a way to find natural analogs of the matrix models with quartic potentials and clarify relations between the weights generated by Tr H 2 and Tr H 4 . This helps to distinguish two matrix models (1.2), where (q) (q) X n resembles the traditional ones and Yn represents a class of new matrix models. On the other hand, a simple but important observation relates the measure generated by the trace of the graph Laplacian with the Erd˝os-Rényi ensemble of random graphs. This connection reveals the invariance of the Erd˝os-Rényi probability distribution with respect to the space transformations. We see that the Gaussian measure of GUE (2.1) is replaced in the discrete case by another measure with nice properties we are going to describe.
2.1. Graph Laplacian and Erd˝os-Rényi random graphs. Given a finite graph with the set of n labeled vertices Vn = {v1 , . . . , vn } and the set of simple non-oriented edges E m = {e(1) , . . . , e(m) }, the discrete analog of the Laplace operator (γ ) on the graph γ is defined by relation (γ ) = ∂ ∗ ∂,
(2.2)
where ∂ is the difference operator determined on the space of complex functions on vertices, Vn → C and ∂ ∗ is its conjugate determined on the space of complex functions on edges, E m → C. It can be easily shown that in the canonical basis, the linear operator (γ ) = n has an n × n real symmetric matrix with the elements deg(v j ), if i = j, i j = −1, (2.3) if i = j and (vi , v j ) ∈ E, 0, otherwise, where deg(v) is the vertex degree (see [6,20] for more details).
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If one considers the n × n adjacency matrix A = A(γ ) of the graph γ ,
Ai j =
1, if (vi , v j ) ∈ E, i = j, 0, otherwise,
then one can rewrite the definition of (2.2) in the form i j = Bi j − Ai j with Bi j = δi, j
n
Ail ,
(2.4)
l=1
where δi, j is the Kronecker δ-symbol δi, j =
1, if i = j, 0, if i = j.
It follows from (2.2) that (γn ) has positive eigenvalues. Let us consider the set n of all possible simple non-oriented graphs γn with the set V = Vn of n labeled vertices. Obviously, |n | = 2n(n−1)/2 . Given an element γ ∈ n , it is natural to consider the trace Tr (γ ) as the “kinetic energy” of the graph γ . Then we can assign to each graph γn the Gibbs weight exp{−β Tr (γn )}, β > 0 and introduce the discrete analog of the integral (2.1) by relation
Z n (β, Q) =
exp{−β Tr n + Q(γn )},
(2.5)
γn ∈n
where n = (γn ) and Q is an application: n → R that we specify later. Let us note that one should normalize the sum (2.5) by |n |, but this does not play any role with respect to further results. In what follows, we omit the subscript n in n . Relation (2.4) implies that Tr =
n
n
ii =
i=1
Ai j = 2
i, j=1
Ai j .
(2.6)
1≤i< j≤n
Then we can rewrite (2.5) in the form Z n (β, Q) =
γn ∈n
e Q(γn )
e−2β Ai j .
(2.7)
1≤i< j≤n
It is easy to see that n(n−1)/2 Z n (β, 0) = 1 + e−2β .
(2.8)
Then the normalized partition function can be represented as Zˆ n (β, Q) = Z n (β, Q)/Z n (β, 0) = Eβ e Q(γ ) ,
(2.9)
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
213
where Eβ {·} denotes the mathematical expectation with respect to the measure supported on the set n . This measure assigns to each element γ ∈ n probability e−2β|E(γ )| Pn (γ ) = n(n−1)/2 , 1 + e−2β where E(γ ) denotes the set of edges of the graph γ . Given a couple (i, j), i, j ∈ {1, . . . , n}, one can determine a random variable ai j on the probability space (n , Pn ) that is the indicator function of the edge (vi , v j ), 1, if (vi , v j ) ∈ E(γ ), ai j (γ ) = 0, otherwise. It is easy to show that the random variables {ai j }1≤i≤ j≤n are jointly independent and are of the same Bernoulli distribution depending on β such that e−2β = p, (β) 1 − δ , with probability i j ai j = (2.10) 1 + e−2β 0, with probability 1 − p. The term 1 − δi j reflects the property that graphs γ have no loops. The probability space (n , Pn ) is known as the Erd˝os-Rényi (or Bernoulli) ensemble of random graphs with the edge probability p [13]. Since the series of pioneering papers by Erd˝os and Rényi, the asymptotic properties of graphs (n , Pn ), such as the size and the number of connected components, the maximal and minimal vertex degree and many others, are extensively studied (see [4,13]). Spectral properties of corresponding random matrices A (2.3) and (2.4) are considered in a series of papers (in particular, see [1,10,14–16]). In the present paper we study the random graph ensemble (n , Pn ) from another point of view motivated by the asymptotic behavior of partition functions (2.9). 2.2. Quartic potential and Erd˝os-Rényi matrix models. Let us determine the discrete analog of the integral (2.1) with quartic potential Tr H N4 . Once Tr H 2 replaced by Tr (∂ ∗ ∂) = Tr (2.2), it is natural to consider Tr (∂ ∗ ∂∂ ∗ ∂) = Tr 2 as the analog of Tr H 4 . Then the partition function (2.5) reads as Z n (β, g) = exp{−β Tr n + gn Tr 2 },
(2.11)
γn ∈n
where gn is to be specified. It follows from (2.3) and (2.4) that Tr 2 = Tr B 2 + Tr A2 =
n i, j=1
(A2 )i j +
n
Ai j .
i, j=1
Then, using (2.6) and repeating computations of (2.7) and (2.8), we obtain the representation
n(n−1)/2 −2β 1 + e Eβ {e gn Yn }. (2.12) Zˆ n (β, gn ) = Z n (β, gn )/Z n (β, 0) = 1 + e−2β
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In this relation, we have denoted β = β − gn and introduced the variable Yn =
n
ail al j ,
(2.13)
i, j,l=1
where ai j are jointly independent random variables of the law (2.10) with β replaced by β . The average Eβ denotes the corresponding mathematical expectation. In what follows, we omit the subscripts β and β when they are not necessary. One can generalize (2.11), (2.13) and consider the mathematical expectation n (q) (q) E exp(gn Yn ) , with Yn = (Aq )i j ,
(2.14)
i, j=1
as the normalized partition function of the discrete Erd˝os-Rényi matrix model that we will refer to as the q-step walks model. Then it is natural to say that the average (q) (q) E exp(gn X n ) , with X n = Tr (Aq ), (2.15) represents the discrete Erd˝os-Rényi model for q-step closed walks. The models (2.14) and (2.15) represent the main subject of our studies. Let us note qthat one more analog of the matrix integral (2.1) is given by the average E exp(gn Tr n ) . This matrix sum is of less interest for us because it can be considered as a certain combination of the X - and Y -models (see the end of Sect. 4 for more details). Finally, let us discuss one more analogy between the discrete model (2.11) and the gaussian matrix integrals (2.1). We mean the invariance property of the probability measure with respect to the group of space transformations [19]. Let An denote the set of of all n-dimensional symmetric matrices whose elements are equal to 0 or 1 and the diagonal elements are zeros. It is not hard to see that if the n × n orthogonal matrix ϒ is such that ϒ Aϒ −1 ∈ An for all A ∈ An , then ϒ verifies the following properties: a) each element of ϒ takes values 0 or 1, b) any given line of ϒ contains only one non-zero element, and c) any given column of ϒ contains only one non-zero element. Clearly, the set of all such orthogonal matrices Yn is in one-to-one correspondence with the symmetric group of permutations Sn . It is easy to see that ϒ ∈ Yn determines a basis change in R N because ϒ re-enumerates the vectors of the canonical basis associated with the graph. Equality Tr (ϒϒ T ) = Tr shows that the probability measure P on An , P(A) = C −1 exp{−β Tr } is invariant with respect to the group of transformations Yn , P(ϒ Aϒ T ) = P(A).
(2.16)
It is not hard to prove the inverse statement: if the probability measure P on the set An verifies (2.16) for all ϒ ∈ Yn , then the random variables given by the matrix elements of A are jointly independent.
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
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This is in complete analogy with the well-known fact of random matrix theory that the invariant Gaussian distribution on the space of hermitian (or real symmetric) matrices generates independent random variables [19]. This means that the Erd˝os-Rényi measure plays a very special role in the discrete matrix models similar to that played by the Gaussian measure in the continuous case. We do not pursue this topic here and return to asymptotic behavior of the sums (2.11) in the limit of infinite n.
3. Cumulant Expansions and Connected Diagrams In this section we develop a diagram technique to study the terms of the expansion gk Cum k (Vn ), log E e gVn = k!
(3.1)
k≥1
where Vn represents X n (2.15) or Yn (2.14). Let us stress that the random variables ai j are bounded and therefore the series (3.1) is absolutely convergent for any finite n and sufficiently small g. Given a vector α = (i 1 , . . . , ir ), we introduce random variables Uα , Uα =
for X -model ai1 i2 ai2 i3 · · · aiq i1 , r = q ai1 i2 ai2 i3 · · · aiq iq+1 , r = q + 1 for Y -model
(3.2)
such that Vn = {α}n Uα , where the sum runs over all possible values of i s , s = 1, . . . , r . 1 Then one can write the relation Cum k (Vn ) =
{α1 }n1
···
Cum{Uα1 , . . . , Uαk },
(3.3)
{αk }n1
where by definition Cum{Uα1 , . . . , Uαk } =
dk log E{exp(z 1 Uα1 + · · · + z k Uαk )}|zl =0 . dz 1 · · · dz k
αk )}, α k = (α1 , . . . , αk ) are known as the Coefficients Cum{Uα1 , . . . , Uαk } = Cum{U(
semi-invariants of the family U(
αk ) = {Uα j }kj=1 [18]. To simplify (3.3), we separate the set {1, . . . , n}⊗(r k) into the classes of equivalence according to the properties of the family U(
αk ). The rule is that given α k , we pay major attention not to the values of variables i j but rather to the presence of copies of the same random variable a in U(
αk ). This approach is fairly common in random matrix theory [10,24]. It leads to the diagram representation of the classes of equivalence (see e.g. [7] for a review and for a mathematical description). This method is used to study random matrices with Gaussian or centered random variables. We modify it to the study of cumulants (3.3) of the Erd˝os-Rényi models.
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3.1. Connected diagrams. The diagrams we construct for the X -model and Y -model are very similar and we describe them in common. Let us consider a graph λ with r labeled vertices {θ1 , . . . , θr } with r determined by (3.2). The graph λ contains q edges: these are ε j = (θ j , θ j+1 ), j = 1, . . . , q; for the X -model we have θq+1 = θ1 . Given α = (i 1 , . . . , ir ), one can assign to each θ j the value of i j , j = 1, . . . , r and then each edge ε j denotes a random variable ai j ,i j+1 . To study cum{U(
αk )} with given α k , we consider the set k = {λ1 , . . . , λk } of k labeled graphs λ; we will say that λl is the element number l of the diagram we construct (l) and denote by {ε j }rj=1 the edges of this element (see Fig. 1). The diagram consists of k elements and a number of arcs σ that join some of the edges of λ’s. We draw an arc ( j ,l ) (l) (l ) that joins ε j and ε j when in α k σ j,l (l )
(l)
(l )
(l)
(l)
(l )
(l )
(l)
i j = i j , i j+1 = i j +1 or i j = i j +1 , i j+1 = i j .
(3.4)
Here we assume that l < l . It follows from (3.4) that the arc can have one of the two orientations that we call the direct and the inverse one, respectively. If l = l , we assume j < j and again consider the direct and the inverse arcs. ( j ,l ) (l ) and denote We say that the edges ε(l) j and ε j represent the feet of the arc σ j,l (l )
(l)
this by relation ε j ε j . Clearly, this relation separates the set of edges (l) E(k, r ) = ε j , j = 1, . . . , r ; l = 1, . . . , k into classes of equivalence that we call the color groups of edges. It is possible that a class contains one edge only. In this case we say that there is a simple color group. Certainly, we color different classes in different colors. An important remark is that it is sufficient to consider the reduced diagrams; this (l) (l ) (l ) means that if there are three edges that belong to the same color group, ε j ε j ε j , ( j ,l )
with l < l < l , then we draw the arcs σ j,l
(l) (l ) ε j , ε j
θj
εj
and
(l ) (l ) ε j , ε j
( j ,l )
and σ j ,l
between the nearest neighbors ( j ,l)
only. The same concerns the arcs σ j,l
whose feet belong both
θ j+1
θq
θ2 θ1
λ
(1)
θ1
λ1
(2)
(3)
θ1
θ1
λ2
λ3
Fig. 1. A graph λ and a diagram δ3 for X -model with q = 7
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
217
to λl . Regarding a reduced diagram, each color group has its minimal edge determined in the obvious manner. To summarize, we consider the set k and define the application α k → δ(
αk ) of the set of all α k to the set of diagrams by drawing all arcs prescribed by α k and reducing of set of arcs obtained to the set of arcs between nearest neighbors (
αk ). As a result, we get a diagram δk = δ(
αk ) = ( k , (
αk )) (see Fig. 1). We say that a diagram δk is non-connected when there exist at least two disjoint subsets τ1 and τ2 of {1, . . . , k} such that τ1 ∪ τ2 = {1, . . . , k} and there is no arc with one foot in (τ1 ) = {λ j , j ∈ τ1 } and another in (τ2 ) = {λ j , j ∈ τ2 }. If there is no such subsets, then we say that the diagram is connected. The following statement is a well-known fact from probability theory. Lemma 3.1. If α k is such that the corresponding diagram δk = δk (
αk ) is non-connected, then Cum{U(
αk )} = 0. Proof. It is clear if there exist two subsets τ1 and τ2 as described before, then the σ -algebras generated by random variables {Uµ , µ ∈ 1 } and {Uν , ν ∈ 2 } are independent. The fundamental property of semi-invariants is that in this case Cum{U(
αk )} vanishes [18]. 3.2. Cumulants and sums over connected diagrams. It is clear that for any given set α k = {α1 , . . . , αk } there exists only one diagram δk = δ(
αk ). We agree that two diagrams δk = ( k , k ) and δk = ( k , k ) are not equal, δk = δk if in obvious bijection J ( k ) = k preserving orderings, we have J (k ) = k . Let us say that two vectors α k and α k are equivalent, α k ∼ α k if δ(
αk ) = δ(
αk ). The relation ∼ separates the set {1, . . . , n}⊗(r k) into the classes of equivalence that we denote by C(δk ), δk = δ(
αk ). Let N (δk ) = |C(δk )| be the cardinality of the equivalence class C(δk ). Relation (3.4) means that two edges ε and ε belong to one color group if and only if the random variables a and a assigned to these edges by α k are equal. This is true for any given α k ∈ C(δk ). Let us denote by m(δk ) the number of color groups of δk . Clearly, random variables that belong to different groups are jointly independent. Lemma 3.2. The right-hand side of relation (3.3) can be represented as ··· Cum{Uα1 , . . . , Uαk } = N (δk )W (δk ), {α1 }n1
(3.5)
δk ∈Dk
{αk }n1
where Dk is the set of all possible connected reduced diagrams of the form ( k , ) and W (δk ) = (−1)s−1 (s − 1)! (Ea)m(δk ) (Ea)χ (πs ,δk ) . (3.6) πs ∈k
In this relation πs denotes a partition πs = (τ1 , . . . τs ) of the set {1, 2, . . . , k} into s subsets; k is the set of all possible partitions, and χ (πs , δk ) is the number of additional color groups generated by πs . Proof. By definition of the semi-invariant [18], we have that Cum{Uα1 , . . . , Uαk } = (−1)s (s − 1)! E{U(τ1 )} · · · E{U(τs )}, πs ∈k
(3.7)
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O. Khorunzhiy
where we denoted U(τ j ) = µ∈τ j Uµ . It is easy to see that the right-hand side of (3.7) depends on δk only, because it does not change when α k is replaced by another α k from the same equivalence class C(δk ). Denoting by W (δk ) the right-hand side of (3.7), we get (3.5). To prove (3.6), let us choose an element α k(0) of the equivalence class C(δk ) and consider first the trivial partition π0 = (τ0 ) with τ0 = {1, . . . , k}. Then χ (πs , δk ) = 0 and E Vα (0) · · · Vα (0) = (Ea)m(δk ) , 1
k
where we have used the independence of random variables {ai j } that belong to different color groups of δk and the fact that a ∈ {0, 1}. Now let us consider a partition πs = π0 . It is clear that there exist at least one color group of two or more edges such that its elements belong to two or more different subsets (τ j ) = {λt , t ∈ τ j }. In this case we say that this color group is separated into subgroups; each subgroup contains edges of this color that belong to one subset T j . We color the edges of each new subgroup in the same color, the edges of the different subgroups - into different colors. It follows from (3.7) that the color group separated by πs into v + 1 new subgroups provides the factor (Ea)v+1 to the right-hand side of (3.6). We say that v is the number of additional color groups generated by πs . Regarding all initial color groups, we get the factor (Ea)m(δk ) (Ea)v with v = χ (πs , δk ). Lemma 3.2 is proved. To study N (δk ), let us consider the set of vertices of k , (l)
k = {θ j , l = 1, . . . , k; j = 1, . . . , r }. αk ). Let us say Given α k , we draw the arcs according to the rule (3.4) and get δk = δk (
(l) (l ) (l) that equality i j = i j of (3.4) identifies corresponding vertices θ j and θ (lj ) and denote this by
(l) (l ) θj ∼ = θ j .
(3.8)
It is easy to see that relation (3.8) separates k into classes of equivalence. We determine the minimal element of the class as the vertex whose numbers j and l take minimal values among those of vertices that belong to this class. If there is no θ (lj ) such that (l) ∼ (l ) (l) θ = θ , then we say that θ belongs to the class of equivalence consisting of one j
j
j
element. Let us denote the total number of the classes of equivalence by ν(δk ).
Lemma 3.3. Given δk ∈ Dk the cardinality N (δk ) = |C(δk )| verifies asymptotic relation N (δk ) = n ν(δk ) (1 + o(1)),
(3.9)
in the limit n → ∞. Proof. It is clear that two equivalent vectors α k ∼ α k generate the same partition of k into classes of equivalence. Inversely, given a diagram δk and regarding the corresponding partition of k , we get a separation of the set of variables (l)
Ik = {i j , j = 1, . . . , r ; l = 1, . . . , k}
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
219
into ν(δk ) groups. Variables that belong to the same group are equal between them. To get all possible α k from the same equivalence class, we allow variables i (l) j to take all possible values from 1 to n with the obvious restriction that variables from different groups take different values. Then obviously N (δk ) = n(n − 1) · · · (n − ν(δk ) + 1) = n ν(δk ) (1 + o(1)), as n → ∞. (3.10) Lemma 3.3 is proved. Let us complete this subsection with the following useful remark. The set of diagrams Dk we constructed gives a graphical representation of vectors α k and describes the classes of equivalence of such vectors. One can push forward this representation and consider the set of graphs of diagrams G k = G(δk ) generated in a natural way by diagrams δk by gluing the edges of elements λ in the way prescribed by the arcs of δk . Actually, this is what is usually done in the standard diagram approach of random matrix theory. ˆ has ν(δk ) labeled vertices θˆ that correspond to the minˆ E) The graph G(δk ) = (, imal elements of the classes of equivalence of k . The vertices θˆ and θˆ are joined by an edge (θˆ , θˆ ) ∈ Eˆ if there is an edge ε ∈ E(k, r ) that joins corresponding classes of equivalence of k . Certainly, the number of edges of G(δk ) is equal to the number of all color groups m(δk ). The graph representation is very useful when δk contains k − 1 arcs only. We say that δk are tree-like because in this case, the graphs G(δk ) of Y -model are trees with color edges (see Fig. 2). Remembering that the arcs σ ∈ have the direct and the inverse orientation (3.4), we agree to consider the graphs G(δk ) with the edges of δk glued in the inverse sense only. In what follows, we will use representations of α k by both diagrams δk and graphs G(δk ). Note that G(δk ) has no loops. 4. Diagrams and Limits of the Cumulants In this section we characterize the classes of connected diagrams that provide the leading contribution to the cumulants of X and Y models. The following three asymptotic regimes known from the spectral theory of random matrices are also distinguished in the present context. We refer to them as:
(4)
θ1
(3)
θ1 (1)
θ1
λ1
(2)
θ1
λ2
(3)
θ1
λ3
(4)
θ1
λ4
(1)
θ1
λ Fig. 2. Tree-like diagram and corresponding tree for Y -model with q = 3
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O. Khorunzhiy
• the “full random graphs” regime, when p = Const as n → ∞; • the “dilute” regime, when p = cn /n and 1 cn n; • the “sparse” regime, when p = c/n and c = const as n → ∞. One can consider also the fourth asymptotic regime of the “very sparse” random graphs, when p = cn /n and cn → 0 as n → ∞. In this regime, the diagrams of the leading contribution to cumulants are degenerated, so we formulate the corresponding results as remarks to the proofs of the main theorems. We omit subscripts in cn when they are not necessary. 4.1. Asymptotic behavior of cumulants. In this subsection we formulate the main results on the asymptotic behavior of the cumulants Cum k (Un ). We start with the Y -model. In what follows, we denote by u n vn the asymptotic relation u n = vn (1 + o(1)) as n → ∞. (q)
Theorem 4.1. Asymptotic behavior of Cum k (Yn ) is determined by the following classes of diagrams: (A) in the full random graphs regime, the leading contribution to the sum (3.5) is given by the diagrams Dk(1) (Y ) that have exactly k − 1 arcs; the graph G(δk ), δk ∈ Dk(1) (Y ) is a tree; then (q) W (δk ) n (q−1)k+2 , (4.1) Cum k (Yn ) (1)
δk ∈Dk (Y ) (1)
where W (δk ) is given by (3.6) considered with δk ∈ Dk (Y (q) ); (B) in the dilute random graphs regime, the leading contribution to (3.5) is given by the same diagrams as in (A), δk ∈ Dk(2) (Y (q) ) = Dk(1) (Y (q) ), and (q) Cum k (Yn ) (Ea)(q−1)k+1 n (q−1)k+2 ; (4.2) (1)
δk ∈Dk (Y )
(C) in the sparse random graphs regime, the leading contribution to (3.5) is given by the diagrams D(3) (Y (q) ) such that the graph G(δ˜k ), δ˜k ∈ D(3) (Y (q) ) is a tree Tl with l edges, 1 ≤ l ≤ (q − 1)k + 1; then (q) Cum k (Yn )
(q−1)k+1 l=1
δk
∈D (3) (Y ),G(δ
(Ea)l nl+1 .
(4.3)
k )=Tl
Corollary of Theorem 4.1. Remembering that Ea = p = pn , we can reformulate the results of Theorem 4.1 in the following form: 1 1 (q) (q) lim = Fk (ω), ω = 1, 2, 3, Cum k Yn (4.4) n→∞ pn n 2 (npn )q−1 (q)
where the numbers Fk (i) represent the contributions of corresponding families of diagrams (weighted by W (δk ) in the full random graphs regime). We discuss relation (4.4) (q) in more details in Subsect. 4.3. Explicit expressions for some of Fk (ω) will be obtained in Sect. 6.
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
221
Let us consider the X -model. Now the difference between the cases of the even and odd numbers q becomes crucial. Theorem 4.2. Asymptotic behavior of Cum k (X n ) is determined by the following classes of diagrams: (A) in the full random graphs regime, the leading contribution to the sum (3.5) is given by the diagrams Dk(1) (X (q) ) that have exactly k − 1 arcs; in this case
(q)
Cum k (X n )
W (δk ) n (q−2)k+2 ,
(4.5)
(1) δk ∈Dk (X )
(q)
(1)
where expressions for W (δk ) is given by (3.6) with δk ∈ Dk (X n ); (B) in the dilute random graphs regime, the leading contribution to (3.5) in the case of the X -model with even q = 2q is given by the diagrams Dk(2) (X ) such that the graph G(δk ), δk ∈ Dk(2) (X ) is a tree with the maximal possible number of edges; then (2q )
Cum k (X n
)
(Ea)(q −1)k+1 n (q −1)k+2 ;
(4.6)
(2) δk ∈Dk (X (2q ) )
in the case of the X -model with odd q the leading contribution to (3.5) is given by the (2) (2) diagrams D˜ k (X ) such that the graph G(δ˜k ), δ˜k ∈ D˜ k (X ) is a cycle with the maximal possible number of edges; then (2q +1)
Cum k (X n
)
(Ea)2q +1 n 2q +1 ;
(4.7)
(2) δk ∈D˜ k (X (2q +1) )
(C) in the sparse random graphs regime, the leading contribution to (3.5) in the case of even q = 2q + 1 is given by the diagrams D(3) (X ) such that the graph G(δk ), δk ∈ D(3) (X ) is a tree Tl with l edges, 1 ≤ l ≤ (q − 1)k + 1; then (2q ) Cum k (X n )
(q −1)k+1 l=1
δk
∈D (3) (X ),G(δ
(Ea)l nl+1 ;
(4.8)
k )=Tl
in the case of X -model with odd q = 2q + 1, the leading contribution to (3.5) is given (1) by the diagrams D˜ (3) (X ) such that G(δk ) is a graph Tl with l edges and one cycle only; then (2q +1) ) Cum k (X n
(q −1)k+1 l=1
δk
∈D (3) (X ),G(δ
(Ea)l nl .
(4.9)
(1) k )=Tl
Corollary of Theorem 4.2. In the full random graphs regime, 1 1 (q) (q) lim = k (1). Cum X k n n→∞ pn 2 pq−1 n q−2
(4.10)
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O. Khorunzhiy
In the dilute and sparse regimes we have convergence of the even powers 1 1 (2q ) (2q ) = k (ω), ω = 2, 3, lim X Cum k n −1 q n→∞ cn c
(4.11)
and the odd powers converge with different normalizations; lim
1cn
1 (2q +1) (2q +1) Cum k (X n ) = k (2) c2q +1
(4.12)
and lim
n→∞,c=const
(2q +1)
Cum k (X n
(2q +1)
) = k
(3).
(4.13)
We discuss relations (4.10)–(4.13) in Subsect. 4.3. Explicit expressions for some of (q) k (ω) are obtained in Sect. 6. 4.2. Proof of Theorems 4.1 and 4.2. Proof of Theorem 4.1. In the full random graphs regime, Ea is a constant and all the terms of the sum (3.6) are of the same order of magnitude. The leading contribution to (3.5) comes from the diagrams δk such that the graph G(δk ) has the maximally possible number of vertices. In this case the number of arcs in δk is minimal, i.e. is equal to k − 1 and G(δk ) is a tree. This proves (4.1). In the case of dilute and sparse graphs, relation Ea 2 = o(Ea) shows that the leading contribution to the sum (3.6) is obtained from those partitions πs that χ (πs ; δk ) = 0. Since δk is connected, then only the trivial partition π0 verifies this condition. Let us denote by l the number of edges in G(δk ). If G(δk ) is given by a tree, then ν(δk ) = l + 1 and (Ea)l |N (δk )| = ncl (1 + o(1)). If G(δk ) is not a tree, then by the Euler theorem, ν(δk ) < l and such diagrams provide a contribution to (3.5) of the order o(ncl ). In the case of dilute random graphs, c → ∞ and the leading contribution is obtained from the graphs with l = (q − 1)k + 1; other trees do not contribute. This proves (4.2). To show (4.3), it remains to note that in the sparse random graphs regime all trees with 1 ≤ l ≤ (q − 1)k + 1 provide contributions of the same order of magnitude. Theorem 4.1 is proved. Proof of Theorem 4.2. In the full random graph regime, the leading contribution to (3.5) comes from those δk that have k − 1 arcs. This implies relation (4.5). It is easy to see that in the dilute and sparse regimes, the only trivial partition π0 con tributes to (3.6). In the case of X (2q ) , we repeat arguments of the proof of Theorem 4.1 and get relations (4.6) and (4.8). Let us pass to the case of odd q = 2q + 1 and consider diagrams δ1 with one element λ1 . It is clear that graphs G(δ1 ) always contain at least one cycle. Then G(δk ) also contain at least one cycle. If the number of edges of G(δk ) is equal to l, then the contribution of such a diagram is of the order (Ea)l nl+1−w , where w is the number of cycles in G(δk ). The graphs with one cycle only provide the leading contribution of the order (Ea)l nl . In the dilute random graphs regime the leading contribution comes from the graphs with a maximal number of edges. Then we conclude that in this case G(δk ) represents a cycle with 2q + 1 edges and 2q + 1 vertices. This proves (4.7). In the case of the sparse regime, we consider δk such that G(δk ) has exactly one cycle. This implies (4.9). Theorem 4.2 is proved.
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
223
Remark. It follows from the proof of Theorems 4.1 and 4.2 that if pn = c/n and c → 0 (q) (q) as n → ∞, then the leading contribution to Cum k (Yn ) and Cum k (Yn ) is given by the connected diagrams δk such that G(δk ) have a minimal number of edges. It is easy to see that in this case (q)
Cum k (Yn ) = 2k−1 pn n 2 (1 + o(1)) = 2k−1 cn(1 + o(1)),
(4.14)
and (2q )
Cum k (X n
) = 2k−1 pn n 2 (1 + o(1)) = 2k−1 cn(1 + o(1)).
(2q +1)
Also we have Cum k (X n
(4.15)
) = O( pn3 n 3 ) = O(c3 ) as n → ∞, c → 0.
4.3. Central Limit Theorem for variables X and Y . Regarding the definition of the cumulants, it is easy to see that for any constant C, Cum k (Vn + C) = Cum k (Vn ) + Cδk,1 . Also, if there exists such a sequence bn that bn → ∞ and 1 Cum k (Vn ) → φk , k ≥ 1, bn then
Cum k
Vn − EVn √ bn
→
(4.16)
φk , if k = 2; 0, if k = 2.
(4.17)
Relation (4.17) means √ that the probability law of the centered random variable V˜n = (Vn − EVn )/ bn converges to the Gaussian distribution N (0, φ2 ). Regarding (q) (q) centered and normalized variables X n and Yn , we formulate the Central Limit Theorem. Theorem 4.3. If n → ∞, and pn determines one of the three asymptotic regimes indicated by ω, the following random variables converge in law to the Gaussian distribution:
(q)
1 pn
n2
·
(q)
Yn − EYn ( pn n)q−1
(q)
→ N (0, F2 (ω)), ω = 1, 2, 3,
(4.18)
and
1 pn n 2
(2q )
·
Xn
(2q )
− EX n ( pn n)q −1
(2q )
→ N (0, 2
(ω)), ω = 2, 3.
(4.19)
Also, convergence to the standard normal distribution holds in the case of the X model in the full random graphs asymptotic regime;
(q)
1 pn
n2
·
(q)
X n − EX n ( pn n)q−2
(q)
→ N (0, 2 ),
(4.20)
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O. Khorunzhiy
and in the regime of dilute random graphs for X (q) with odd q = 2q + 1:
1 c
q +1/2
(2q +1)
Xn
(2q +1)
− EX n
(2q +1)
→ N (0, 2
(2q +1)
There is no convergence to the normal distribution of X n random graphs.
(B)).
(4.21)
in the regime of sparse
Proof. The proof follows immediately from relations (4.4) and (4.11) with properties (4.16) and (4.17) taken into account. Let us discuss relations of Theorem 4.3 with the spectral theory of random matrices. Remembering that pn n = c and introducing the normalized adjacency matrices 1 Aˆ (n,c) = √ An , c (n,c)
we can introduce variables M2q
=
1 n
2q
Tr Aˆ (n,c) and rewrite (4.19) in the form
√ (n,c) (2q ) (n,c) → N (0, 2 (ω)), ω = 2, 3. nc M2q − EM2q
(4.22)
Also, we deduce from relations (4.11)–(4.13) that (n,c)
lim EM2q
n→∞
(2q )
= 1
(ω) = m 2q (ω) and
(n,c)
lim EM2q +1 = 0.
n→∞
(4.23)
(n,c)
Variables Mq represent the moments of the normalized eigenvalue counting measure of random matrices Aˆ (n,c) , (n,c)
σn (λ) = #{ j : λ j
≤ λ}n −1 .
Convergence (4.23) implies the weak convergence in the average of the measures σn . This convergence is established in the present setting and in more general cases in papers [1] and [16]. In particular, the semi-circle law is proved for Aˆ (n,c) in the case of sparse random graphs [16]. The limiting numbers m 2q = (2q + 1)!/q !(q + 1)! represent the even moments of the semi-circle (or Wigner) distribution [24]. The Central Limit The(n,c) orem (4.22) improves results of [1,16] and implies convergence of the moments M2q with probability 1 in both of the asymptotic regimes of dilute and sparse random graphs. Regarding random variables Tr q , it is not difficult to reformulate the diagram technique of the previous section and establish the analogs of the Theorems 4.1 and 4.2. Indeed, the diagram representation of the term Tr (Au 1 B v1 · · · Au s B vs ) is given by a closed chain of u 1 + · · · + u s edges, with the off-spreads of v1 , v2 , and vs edges placed at the properly chosen vertices of the chain. It is easy to show that in the asymptotic regime of dilute random graphs, the term Tr B q dominates the other ones and the leading contribution to Cum k (Tr B q ) coincides with the one obtained for the Y -model. The case of the sparse random graphs regime is more complicated and requires more analysis. (q) Returning to variables Yn , we deduce from (4.4) that lim
n→∞
1 (q) (q) E Yn = F1 (ω) = 1, ncq
(4.24)
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
225 (q)
because there exists only one connected diagram of one element. Variable Yn counts the number of q-step walks over the graph, and relation (4.24) shows that this number is asymptotically proportional to ncq . It is natural to expect this result because the average degree of a vertex in the Erd˝os-Rényi random graph converges to c, as n → ∞ [13]. It follows from (4.18) that 1 (q) γn 1 (q) ∼√ , Yn − E Yn q q nc nc nc where γn converges in law to a Gaussian random variable. Then convergence with probability 1 holds 1 (q) Yn → 1, n → ∞ ncq in the asymptotic regimes of dilute and sparse random graphs. The limiting variance of γn depends on the asymptotic regime. We return to this question in Sect. 6. Finally, let us note that relations (4.14) and (4.15) imply that 1 (q) Cum k (Yn ) → 2k−1 as n → ∞, c → 0. cn (2q )
The same is true for the cumulants of X n . This means that the Central Limit Theorem does not hold for these variables in the asymptotic regime of very sparse random graphs. 5. Number of Tree Diagrams In this section we derive recurrent relations that determine the number of connected diagrams δk = ( k , ) with a minimal number of arcs, (1)
Dk = {( k , ) : || = k − 1}. In the case of the Y -model, the graphs G(δk ) have the tree structure and we refer to δk as tree diagrams. In the case of the X model we refer to such diagrams as tree-like diagrams. It is clear that (1)
(1)
(q)
|Dk (X (q) )| = |Dk (Y (q) )| = dk , so we consider the case of the Y -model only. 5.1. Recurrent relations and Lagrange equation. First let us recall that the arc in δk is interpreted by G(δk ) as gluing between edges that make a color group. Let us say that the edges that are not glued are not colored and stay grey. So, we forget the color of the simple color groups consisting of one edge only. The different color groups and grey edges correspond to the edges of G(δk ). It is convenient to color the arcs that join the edges of one group in the same color as the edges of this group. We start with the following simple statement. (1)
Lemma 5.1. Let a diagram δk ∈ Dk (Y (q) ) has arcs of s different colors. Then there are (q − 1)k − s + 1 grey edges in δk .
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O. Khorunzhiy
Proof. Let us assume that there are µ j arcs of each color. Obviously, µ1 +· · ·+µs = s −1 and the total number of colored edges is k − s + 1. Taking into account that the total number of edges in δk is qk, we obtain the result. The lemma is proved. (q)
Lemma 5.2. Given q ≥ 2, the numbers dk equalities (q)
dk
=
(1)
= |Dk (Y (q) )| with k ≥ 1 are given by
k! (q) h , (q − 1)k + 1 k
(5.1)
(q)
where the sequence {h k }k≥0 is determined by the recurrent relation (q)
hk =
(q − 1)k + 1 · k
(q) (q) h j1 · · · h jq , k ≥ 1
(5.2)
j1 +...+ jq =k−1 ji ≥0
(q)
with initial condition h 0 = 1. Proof. Let us recall that the diagrams under consideration are reduced, and then each color group has a maximal edge. Let us count first the number of the diagrams δ¯k such that the last element λk contains one and only one edge that is the maximal edge of a color group. This means that only one arc ends at λk by its right foot (or leg). Let us denote this arc by σ¯ . Obviously, one can choose one of the q edges of λk to put this right leg. The left leg of σ¯ (1) joins λk with the connected tree diagram δk−1 ∈ Dk−1 (Y (q) ). By Lemma 5.1, there are (q − 1)(k − 1) − s + 1 grey edges and s maximal edges of s color groups, where one can put the left foot of σ¯ (see Fig. 3). Then we obtain the relation (q)
(1)
|D¯ k | = ((q − 1)(k − 1) + 1) dk−1 . Now let us consider the case when there are l arcs σ¯ 1 , . . . , σ¯ l that end at λk with 2 ≤ l ≤ min(q, k − 1). There are ql possibilities to choose the emplacements for the right legs of these arcs. To put the left legs of these arcs, one has to choose the l subsets
(1) , . . . , (l) of ji = | (i) | elements such that j1 + · · · + jl = k − 1 and ji ≥ 1,
....
λ1
λ2
λ3
λ4
Fig. 3. Example of δ¯k with possible positions for the left foot of σ¯
λ5
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
227
then to create l connected sub-diagrams ( (i) , (i) ), i = 1, . . . , l and to choose the emplacements for the left legs of σ¯ i in ( (i) , S (i) ). This produces q l
j1 +···+ jl =k−1 ji ≥1
l (k − 1)!
(q) ((q − 1) ji + 1)d ji j1 ! · · · jl ! i=1
(q)
diagrams. Summing up, we derive for numbers {dk , k ≥ 1} with given q ≥ 2 the following recurrent relation: (q)
dk
=
q
I[1,k−1] (l) ×
l=1
q l
j1 +···+ jl =k−1 ji ≥1
l (k − 1)!
(q) ((q − 1) ji + 1)d ji , j1 ! · · · jl !
(5.3)
i=1
(q)
with initial condition d1 = 1. Here we denoted by I[1,k−1] (·) the indicator function of the interval [1, k − 1]. Introducing the auxiliary numbers h j , such that hj =
(q − 1) j + 1 (q) dj , j!
j ≥ 1,
we reduce (5.3) to relation q q k I[1,k−1] (l) × hk = l (q − 1)k + 1 l=1
h j1 h j2 · · · h jl , k ≥ 1. (5.4)
j1 +···+ jl =k−1 ji ≥1
If we introduce an auxiliary number h 0 = 1, then we can rewrite (5.4) in more (k) compact form. Indeed, we can make an agreement that each labeled edge εl of the element λk serves as the right foot of the corresponding arc σ¯ l but some of these arcs can have the right leg “empty”, i.e. with no elements attached to their right feet. Then the variable l of the sum of (5.4) represents the number of non-empty arcs, and ql stands for the choice of these non-empty arcs from the set σ¯ 1 , . . . , σ¯ q . This corresponds to the choice of the variables ji in the product h j1 · · · h jq that take zero value. Finally, we set (q)
h j = h j , j ≥ 0 and conclude that relation (5.4) is equivalent to the recurrent relation (5.3). Lemma 3.5 is proved. Lemma 5.3. The generating function Hq (z) =
+∞
(q)
hk zk
(5.5)
k=0
is regular in the domain {z : |z| ≤ 1/(q 2 e)} and verifies the equation q−1 Hq (z) = exp{qz Hq (z) }. (q)
It follows from (5.6) that the numbers h k can be found from recurrent relations
(5.6)
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O. Khorunzhiy
(q)
(q)
h j1 · · · h jq−1 = q k (q − 1)k
j1 +···+ jq−1 =k ji ≥0
(k + 1)k−1 (q) , h 0 = 1. k!
(5.7)
Remark. Relation (5.6) with q = 2 is known as the Lagrange (or Pólya) equation [17,22]. In the next subsection we give another derivation of (5.6) by using the notion of color trees. In Sect. 6 we obtain one more generalization of the Lagrange-Pólya equation. Proof of Lemma 5.3. It is easy to deduce from (5.2) that Hq (z) verifies the equality z q Hq (z) − 1 = z(q − 1)(Hq (z)) + (Hq (ζ ))q dζ 0
that is equivalent to the differential equation q q Hq (z) d Hq (z) = q−1 , dz 1 − (q 2 − q)z Hq (z) Substitution
Hq (0) = 1.
q−1 ψ(z) = z Hq (z)
(5.8)
(5.9)
transforms (5.8) to an elementary equation ψ (z) =
ψ(z) 1 · . z 1 − (q 2 − q)ψ(z)
Resolving this equation with the obvious initial condition, we conclude that ψ(z) verifies the Pólya equation [22] ψ(z) = zeq(q−1)ψ(z) .
(5.10)
Then (5.6) easily follows from (5.9) and (5.10). Using the standard method of the contour integration, we deduce from (5.10) explicit expressions for the coefficients ψk of the generating function ψ(z) = k≥0 ψk z k . The first observation is that the inverse function ψ ∗ (w) = we−q(q−1)w is regular in the vicinity of the origin. By the Cauchy formula, we have 1 ψ(z) ψk = dz. 2π i z k+1 By changing variables by z = ψ ∗ (w), we get dz = (1 − q(q − 1)w)e−q(q−1)w dw and find that 1 1 − q(q − 1)w q(q−1)wk ψk = e dw. 2π i wk Then ψk = (q(q − 1))k−1 and (5.7) follows. Lemma 5.3 is proved.
k k−2 , (k − 1)!
(5.11)
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
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5.2. Color trees. In this subsection we give another derivation of the Lagrange equation (5.6) based on the graph representation G(δk ) of the diagrams δk . For simplicity, we consider the case of the Y -model with q = 2 only. If δk has k − 1 arcs, then G(δk ) is a tree. We say that the arcs that indicate the groups of edges to be glued are all of the same color. Then the corresponding edges of G(δk ) are of this color. We say that the edges that are not glued are grey. Let us derive expressions for the number of the elements of the set Tk = |Gk |, Gk = {G(δk ), δk ∈ D(1) (Y (2) )}. We denote by Tˆm the number of rooted trees of the form G(δm ). Let us consider the left element of λ1 as the root edge ρ for the trees we construct (see Fig. 4). We can attach to this root r elements ν1 , . . . νr that we choose by mr ways. We glue these elements by their right edges. Then we separate m − r elements into the groups of l1 , l2 , . . . , lr elements and construct rooted trees and attach them by their roots to the free elements of ν1 , . . . , νr . There )! are (m−r l1 !···lr ! possibilities to separate m − r elements into these groups. Then we get Tˆm =
m
2r
r =1
m r
l1 +···+lr =m−r
(m − r )! ˆ Tl · · · Tˆlr , m ≥ 1, l 1 ! · · · lr ! 1
(5.12)
where the sum runs over all li ≥ 0 and we accept that Tˆ0 = 1. Simplifying (5.12) and taking into account that Tˆ0 = 1, we derive from (5.12) the following relation for the generating function t (z) = k≥0 tk z k , tk = Tˆk /k! (cf. (5.6)): t (z) = exp{2zt (z)}.
(5.13)
Then Tˆm = 2m (m + 1)m−1 and the numbers tm are given by recurrent relations tm =
m−1 m+1 t j tm−1− j , t0 = 1. m
(5.14)
j=0
^
ν1
Tl
1
ν2 νr ^ Tl r
^
^
Tm
.. . (1) θ1
λ1
Tm
1 (1)
θ1
Fig. 4. Rooted color tree and color tree
λ1
2
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O. Khorunzhiy
Now let us return to the construction of the tree G(δk ). Now the left edge of λ1 can serves as the root for the left rooted tree of m 1 elements and the right element serves as the root for the rooted tree of m 2 = k − 1 − m 1 elements. Then
Tk =
m 1 +m 2 =k−1 m i ≥0
(k − 1)! ˆ ˆ Tm 1 Tm 2 , m 1 !m 2 !
where Tˆ0 = 1. Then Tk k 2k (k + 1)k−2 t j tk−1− j = = tk = . (k − 1)! k+1 (k − 1)! k−1
j=0
Then equality Tk = 2k (k + 1)k−2 follows. In the general case q ≥ 2, one can easily repeat this procedure of tree construction (q) and derive (5.6) directly, without use of the numbers h k .
6. Limits of the Cumulants (q)
In this section we specify the limits of the cumulants of the Y -model Fk (i) (4.4) and (q) of the X -model k (i) (4.10)–(4.13). We start with the case of the full random graphs regime, but the most attention is paid to the dilute and sparse random graphs regimes.
6.1. Cumulants of full X and Y models. In Sect. 4 we have shown that the leading contribution to the cumulants of (4.1) and (4.5) is given by those diagrams δk that have exactly k − 1 arcs and the graph G(δk ) with the maximal number of vertices. The weight of the diagram W (δk ) is determined by relation (3.6), where all terms (Ea)l are of the same order of magnitude (see the proof of Lemma 3.2). Regarding the Y -model and taking into account that in this case m(δk ) = (q − 1)k + 1 (see Lemma 5.1), we combine (3.6) with (4.1) and conclude that (4.4) is true with (q)
Fk (1) =
(−1)s−1 (s − 1)! p χ (πs ;δk ) ,
(6.1)
(1) δk ∈Dk (Y ) πs ∈k
where χ (πs ; δk ) is determined in Lemma 3.2. Considering the X -model, it is easy to see that (3.6) gives the same expression for W (δk ) as for the Y -model. Using (4.5), we arrive at the conclusion that (4.10) is true (q) (q) with k (A) = Fk (A) (6.1). Regarding the first two cumulants, it is easy to compute that (q)
(q)
(q)
(q)
F1 (1) = 1 (1) = 1 and F2 (1) = 2 (1) = 2q 2 (1 − p).
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
231
6.2. Cumulants of the dilute Y -model. Regarding (4.2) with Ea = c/n and using results of Sect. 5, we conclude that (4.4) takes the form of 1 1 (q) (q) (q) (1) = 2k−1 |Dk (Y )| = 2k−1 dk , (6.2) Cum k Fk (2) = lim Yn q−1 n,c→∞ nc c (q)
where dk are determined by relations (5.1) and (5.2). In (6.2), we have taken into account that the k − 1 arc of the diagrams δk can be drawn in the direct and inverse sense. This produces the factor 2k−1 . Let us consider in more detail the case of q = 2 that corresponds to the continuous matrix model (2.1) with the quartic potential. In this case relation (5.2) takes the form (2)
hk =
k+1 k
j1 + j2 =k1 , ji ≥0
(2) (2)
(2)
h j1 h j2 , h 0 = 1.
(2)
It follows from (5.7) that in this case h k = 2k (k + 1)k−1 /k! and (2)
dk
= 2k (k + 1)k−2 .
(6.3)
(2)
The combinatorial meaning of dk can be explained with the help of the notion of the dual diagrams that we briefly describe below. Regarding δk ∈ D(1) (Y (2) ), we construct the dual diagram dˆk by transforming the edges of elements λ j into vertices joined by an edge; the arcs remain without changes and indicate the vertices of δˆk to be glued (see Fig. 5). It is easy to see that the graph (2) G(δˆk ) is a tree. Then dk is equal to the number of all non-rooted trees constructed with the help of k labeled edges. Since the edges of δk are labeled to be the left and the right ˆ k ) as the oriented one. one, we can think about the edges of G(δ Regarding the dual diagrams in the general case of q ≥ 2, we see that these are trees when q = 2 and trees constructed from oriented chains of q edges when q > 2. It is not (q) clear whether it is possible to derive from (5.7) explicit expressions for dk with q > 2. 6.3. Cumulants of the sparse Y -model. The diagrams of D(3) (Y ) are of more complicated structure than those of D(1) (Y ) and we can not obtain recurrent relations to (q) determine Fk (3) in the general case of q ≥ 2 and k ≥ 1. We present the results for (q) (2) Fk (3) and F1 (3) only.
2 1
2
3
4
4 1
Fig. 5. Dual diagrams and dual trees in the case of q = 2, k = 4
3
232
O. Khorunzhiy (2)
Lemma 6.1. The coefficients Fk (3), k ≥ 1 are determined by relations 2k−1 (k − 1)! wk , ck
(6.4)
k−s 1 wˆ j wˆ k−s− j , k ≥ 1, (s − 1)!
(6.5)
Fk(2) (3) = where wk are determined by relation wk =
k s=1
j=0
and the numbers wˆ j , j ≥ 0 are such that the generating function Wˆ (z) = verifies the equation Wˆ (z) = 1 − ce2z + c exp 2z[e2z Wˆ (z) − 1] .
ˆkz k≥0 w
k
(6.6)
Proof. Following the lines of Subsect. 5.2, we consider first the set of the rooted trees of the type we are interested in and denote their contribution by Tˆm , where m is the number of elements λ. In the present case the trees are constructed with the help of elements λ, but there can be double elements that are glued to one of the edges, and there can be simple elements, and there are elements glued to simple elements all along (see Fig. 6). Let us consider the root edge ρ. We denote by u the number of double elements attached to ρ and by r the number of simple elements attached to ρ by one edge. Also we denote by vi , i = 1, . . . , r the numbers of elements glued all along to these r simple elements. Taking into account possibilities to choose u, r , and vi elements from the set of m labeled elements, we obtain the recurrent formula Tˆm = c
m u=0
×
2u
m−u m−u−r m − u m m −u −r · 2r 2v1 · · · 2vr u r v1 , . . . , vr r =1 v1 +···+vr =V, V =1 m −u −r −V ˆ (6.7) Tl1 · · · Tˆlr , l 1 , . . . , lr
l1 +···+lr =m−u−r −V
v1 r
u
(1)
θ1
λ1
Fig. 6. Elements to construct a rooted tree and example of the tree
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
233
where we denoted the multinomial coefficients m −u −r (m − u − r )! = . v1 , . . . , vr v1 ! · · · vr ! (m − u − r − V )! Factor c in the right-hand side of (6.7) takes into account the contribution (or weight) of the root ρ. Simplifying (6.7) and denoting wˆ m = Tˆm /m! for m ≥ 1, we get relation wˆ m = c
m 2u u=0
u!
·
m−u r =1
2r · r!
m−u−r V =v1 +...+vr ,V =1
2v1 · · · 2vr v1 ! · · · vr !
wˆ l1 · · · wˆ lr .
l1 +···+lr =m−u−r −V
Passing to the generating function Wˆ (z), we derive the equality Wˆ (z) − 1 = ce2z exp 2ze2z Wˆ (z) − 1 ,
(6.8)
that is equivalent to (6.6). Let us note here that (6.8) generalizes the Pólya equation (5.13). (3) Returning to the set Dk (Y ), we consider λ1 as the simple root element. Assuming that there are s − 1 elements glued all along this root element, we dispose of k − s elements to construct the rooted trees using the left and the right edge of λ1 as the roots (3) ρ1 and ρ2 . Denoting the number Tk = |Dk (Y )|, we get relation Tk =
k k−1 s=1
s−1
m 1 +m 2 =k−s m i ≥0
k−s ˆ ˆ Tm 1 Tm 2 . m1, m2
Simplifying this relation and passing to variables wˆ m = Tˆm /(m −1)!, we obtain equality (6.5). This completes the proof of Lemma 6.1. In the general case of q ≥ 2 we determine the first cumulant only, (q) F1 (3)
=
lim n→∞, p=c/n
c=Const
n
E Ai1 i2 Ai2 i3 · · · Aiq iq+1 .
(6.9)
i 1 ,...,i q+1 =1
The vertices θ of λ are ordered and we denote θ1 = ρ and θ2 = ν. It follows from (q) Theorem 4.1 that F1 (3) is determined by the number of diagrams δ1 consisting of one element λ such that the corresponding graph G(δk ) is a tree of l vertices, 2 ≤ l ≤ q. This problem is closely related with the studies done in [1,16]. These studies are related to the X model and we present the results in the next subsection. In what follows, we describe briefly a corresponding approach and modify it to suit our situation. To study the right-hand side of (6.8), it is useful to introduce the notion of a walk ξ . A walk ξq of q steps is an ordered sequence of q letters, starting with ρ and followed by ν always. Given a sequence Iq+1 = (i 1 , i 2 , . . . , i q+1 ), we construct the corresponding ξq = ξ(Iq+1 ) by the following recurrent rule: regarding the value i s+1 , s ≥ 1, we compare it with the values of {i 1 , . . . , i s } and write a letter number s + 1 in ξ . If there is no i j , 1 ≤ j ≤ s such that i s+1 = i j , then we write a new letter that is not present in ξs . If there is such j, 1 ≤ j ≤ s such that Is+1 = i j , then we write on the s + 1th place the letter number j of ξs . We can represent the walks graphically, by drawing the root vertex ρ, the next vertex ν and the corresponding edge, and then passing along ξq and creating the new vertices
234
O. Khorunzhiy
at the instants, when the new letters occur in ξq . We also draw the new edges in this case (see Fig. 7). It is clear that there is one-to-one correspondence between the diagrams δ1 and walks ξq . Also, the walk ξq generates in a natural way the graph G(ξq ) isomorphic to G(δk ). (q)
Lemma 6.2. The number F1 (3) = Fq c1−q is given by the sum Fq = cFq−1 +
q
Fq (r ),
(6.10)
r =2
where the numbers Fq (r ), r ≥ 2 are determined by the following recurrent relations: Fq (r ) = c
r q−v q−v−u [ v−1 ] + [ s ][ r ] − 1 2
v=2 u=0
s=0
[ v−1 2 ]
2
2
[ v2 ] − 1
Fq−u−v (s)Fu (r − v). (6.11)
In (6.11), [x] denotes the largest integer less than or equal to x and the following initial conditions are assumed: Fl (0) = δl,0 , Fr − j (r ) = 0 for j > 0 and F2 (2) = 1. Remark. Relations (6.10) and (6.11) are similar to those derived in [1,16] for the number of closed walks of an even number of steps such that their graph is a tree. Proof. Let us consider the set q (r ) of walks such that there are r steps that start or end in ρ. We denote by f q (r ) = | q (r )| the cardinality of this set. Let us consider the subset q (r, v, s) ⊂ q (r ) consisting of the walks that pass the edge e1 = (ρ, ν) v times and that have s steps attached to ν that do not pass e1 (see Fig. 8). We denote the set of such steps ν → π with π = ρ by Sν→ρ¯ . Certainly, such a walk has r − v steps that belong to the set Sρ→ν¯ . Any walk of q (r, v, s) can be separated into three parts or in other words into the following three subwalks: the first one contains the steps ρ → ν and ν → ρ only, the second one is attached to ν with the elements of Sν→ρ¯ (the left part of the walk) and the third one is attached to ρ by the elements of Sρ→ν¯ (the right part of the walk). We denote the left and the right parts by q−u−v (s) and by u (r − v), respectively, where u denotes the number of steps in the right-hand part of the walk. We denote by q (r, v, s, u) the set of elements of q (r, v, s) such that the right-hand part of the walk consists of u steps.
6
ν
5
3
2
ν
1
θ1
θ2
ρ
4
Fig. 7. A diagram δ1 , the walk ξq and corresponding graph G(δ1 ) = G(ξq )
ρ
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
′ Ξq−u− (µ) q−u−
235
10
2
s
u
ν
r−
ρ
Ξ′′(r− )
3
u
1
9
5
4
6
7 8
Fig. 8. The first-passage decomposition and example of the walk
Then we can write the equality v−1 [ 2 ] + [ 2s ] [ r2 ] − 1 f q−u−v (s) f u (r − v), | q (r, v, s, u)| = [ v2 ] − 1 [ v−1 2 ]
(6.12)
where we have taken into account the obvious equalities | q−u−v (s)| = | q−u−v (s)| and | u (r − v)| = | u (r − v)|. Let us describe the form of the combinatorial coefficients in the right-hand side of (6.12). If r and v are both even, then there are v/2 steps in Sν→ρ and r/2 − v/2 steps in Sρ→π¯ . Therefore there is a choice to perform each of the r/2 − v/2 steps after one of v/2 steps ν → ρ. This produces the second binomial coefficient of (6.12). If r is odd and v is even, then the last step of the form ρ → π with π = ρ is to be performed after all the steps ν → ρ are done. Then we get again the combinatorial coefficient as that given by (6.12). If v is odd, then there are still [v/2] steps ν → ρ. Finally, let us explain the first binomial coefficient of (6.12). If s and v are both even, then there are v/2+s/2−1 possibilities to perform s/2 steps of Sν→ρ¯ after v/2 steps of v/2−1 Sρ→ν . If s is odd, then the last step is performed after that all v/2 steps are performed. If v is odd, then there are [v/2] + 1 steps ρ → ν. Summing (6.12) with respect to parameters α, u, and s, and taking into account that Ea = c/n, we get relation (6.11) for the contribution to Fq determined by the walks with r, r ≥ 2 steps attached to ρ. If r = 1, then the corresponding contribution is given by cFq−1 . Summing all contributions, we obtain formula (6.10). The lemma is proved.
Remark. It is easy to see from (6.12) that f 3 (2) = 0. Moreover, it is not hard to show that f 2l+1 (2s) = 0 for all l, s ≥ 1. This is in agreement with the obvious observation that the set of walks 2l+1 (2, s) is empty. 6.4. Cumulants of the dilute and sparse X -model. Analysis of cumulants of the X -model in the dilute random graphs regime is very similar to that of the Y -model in the same regime. Lemma 6.3. Relations (4.11) and (4.12) are true with k 2q 1 (2q +1) (2q ) (q ) k (2) = (4q + 2)k−1 and k (2) = 2k−1 dk , (6.13) q +1 q (q )
where dk
is determined by relations (5.1) and (5.2) with q replaced by q .
236
O. Khorunzhiy
Proof. According to Theorem 4.2, the leading contribution to (4.7) in the case of odd q = 2q + 1 is given by the diagrams δk such that the graph G(δk ) is of maximal number of edges and contains one cycle only. Taking into account that each element λ j is represented by a cyclic graph with 2q + 1 edges, the only possibility to get rid of the “extra” cycles in δk is to glue the elements λ j along each other. To get the maximal number of edges in δk , we keep the cyclic structure of λ j . When doing this, we have to choose the direction and the edge of λ1 to glue the first elements of λ j , j = 2, . . . , k. The number of possibilities is given by 2k−1 × (2q + 1)k−1 and we get the first equality of (6.13). Regarding the case of even q = 2q , we conclude that to construct a tree with the maximal number of edges we have first to construct a tree from each element λ j and then to draw k − 1 arcs between these k trees obtained. It is easy to see that the maximal size of the tree obtained by gluing the edges of λ j is q and the number of different trees is given by the Catalan number (q + 1)−1 2q q [22]. Then we produce the connected diagrams exactly as in the case of the dilute Y model. This gives the second relation of (6.13). The lemma is proved. Similarly to the Y -model, the sparse random graphs regime of the X -model is more complicated than the dilute random graphs regime. The average value of the random vari (2q ) (2q ) (2q ) able X n = Tr A2q is studied in [1,16]. The limit 1 (3) = limn→∞ 1q EX n is nc determined by a system of recurrent relations that have the form similar to that described in Lemma 6.2. We do not present these results. (2q ) is studied in [23] in the more general setting of weighted The variance of X n graphs. The resulting expression is also determined by a system of recurrent relations. This system is much more complicated than that for the first moment of X n . We refer the reader to the paper [23] for corresponding expressions and statements. 6.5. Formal limit of the partition function. Returning to the normalized partition function (2.12) that describes the Y -model with q = 2, one can see that
−2β 1 + e 1 n − 1 1 tg Y ˆ n (β, tgn ) = e n n , (6.14) log log Z log E + β pn n 2 2 pn n 1 + e−2β pn n 2 where β = β − tgn and gn = ( pn n)−1 . It is clear that the three asymptotic regimes introduced in Sect. 4 are characterized by the corresponding behavior of the inverse “temperature” β and elementary analysis shows that
⎧ 0, if β = Const; ⎨ 1 + e−2β n−1 0, if β = 21 log nc , 1 c n; log lim = n→∞ 2 pn n ⎩ exp{2tc−1 }−1 1 + e−2β , if β = 21 log nc , c = Const, 2 where pn = e−2β (1 + e−2β )−1 . Then, assuming that the limit of the last term of (6.14) exists, we conclude that (4.4) implies the relation lim
n→∞
1 δω,3 log Zˆ n (β, tgn ) = 2 pn n 2
∞ k (2) t Fk (ω) 2t exp −1 + , c k!
(6.15)
k=1
where ω indicates the full, the dilute, and the sparse random graphs regimes, and δω,3 denotes the Kronecker δ-symbol.
On Connected Diagrams and Cumulants of Erd˝os-Rényi Matrix Models
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(2)
Comparing expressions (6.1), (6.2), and (6.4) for Fk with ω = 1, 2, 3, we see the (2) difference between the limiting rate functions of (6.15). Indeed, Fk does not depend on c in the dilute random graphs regime, ω = 2, and is given by a polynomial in degrees of 1/c in the sparse random graph regime, ω = 3. In particular, regarding the small-t expansion of the right hand-side of (6.15) in the sparse random graphs limit and taking into account that F1(2) (3) = 1+c−1 , we see that this approximation to the rate function is t (1 + 2c )(1 + o(1)), t → 0. The same difference between the rate functions is observed in 2t the general case of q, with the exponential term of (6.15) replaced by 21 (exp{ cq−1 } − 1). Let us repeat that (6.15) relies strongly on the existence of the limit 1 tg V e n n = f q (t) lim log E (6.16) β n→∞ pn n 2 (q)
(q)
with Vn = X n or Yn . This problem is difficult to solve by using the cumulant expansion (3.1). As it is pointed out many times (see for example [9]), the terms of the formal relation (3.1) with positive g are used to enumerate combinatorial structures related to the matrix integrals. In the rigorous sense, the series (3.1) never converges. Existence and analyticity in g of the terms of asymptotic expansion (3.1) with respect to n is proved for the matrix integrals just recently (see e.g. [3,8]) by using powerful techniques of the integrable models and the Riemann-Hilbert problem. 7. Summary The Gibbs weight determined by the trace of the graph Laplacian generates a measure µn on the ensemble of n-dimensional adjacency matrices of simple non-oriented graphs. This measure is invariant with respect to the symmetric group of permutations and determines the Erd˝os-Rényi ensemble of random graphs with the edge probability pn . Regarding the sum over the set of weighted adjacency matrices as the analog of the matrix integrals, we determine the discrete analog of matrix models with the quartic potential. In the general case of the q-power potential, we distinguish two different families of discrete Erd˝os-Rényi matrix models. They are related with the numbers of (q) (q) q-step walks and q-step closed walks over the random graphs denoted by Yn and X n , respectively. The logarithm of the standard partition function of these models is determined by the (q) (q) cumulants of random variables X n and Yn . We develop a diagram technique to study the cumulants in three different asymptotic regimes of full, dilute and sparse random graphs; these are determined by the limiting behavior of the edge probability pn as n → ∞. We prove that the limits of these cumulants, when properly normalized, exist in all of the three asymptotic regimes. As a consequence, the Central Limit Theorem is shown to be true for centered and normalized variables X and Y . This implies CLT for the moments of the normalized spectral measure of the adjacency matrix of random Erd˝os-Rényi graphs. We show that the limiting expressions of the cumulants are related with the number of non-rooted trees constructed with the help of labeled edges. In the simplest case of the dilute random graphs regime, the exponential generating function of these numbers H (z) verifies the Lagrange-Pólya equation. Passing to the sparse random graphs regime, we derive more general equations that determine H (z) in this case.
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These results imply a conclusion that the asymptotic regimes we consider are different not only with respect to the normalization factors of the cumulants, but also with respect to the rate functions of large deviations formally determined as the limit f q (t) (6.16). It should be noted that we did not prove rigorously the existence of this limit because our results concern the leading terms of the cumulants only. References 1. Bauer, M., Golinelli, O.: Random incidence matrices: moments of the spectral density. J. Stat. Phys. 103, 301–337 (2001) 2. Bessis, D., Itzykson, C., Zuber, J.-B.: Quantum field theory techniques in graphical enumeration. Adv. Appl. Mathem. 1, 127–172 (1980) 3. Bleher, P., Its, A.: Asymptotics of the partition function of a random matrix model. Ann. Inst. Fourier (Grenoble) 55, 1943–2000 (2005) 4. Bollobás, B.: Random Graphs. Cambridge: Cambridge University Press, 2001 5. Brézin, E., Itzykson, C., Parisi, G., Zuber, J.-B.: Planar diagrams. Commun. Math. Phys. 59, 35–51 (1978) 6. Chung, F.R.K.: Spectral Graph Theory. Providence, RI: Amer. Math. Soc., 1997 7. Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2D gravity and random matrices. Phys. Rep. 254, 1–133 (1995) 8. Ercolani, N.M., McLaughlin, K.D.T-R.: Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration. Int. Mathem. research Notices 14, 755–820 (2003) 9. Eynard, B.: Formal matrix integrals and combinatorics of maps (in press) In: Random Matrices, Random Processes and Integrable Systems, Montreal. Proceedings, CRM Series in Mathematical Physics, to appear, available at http://arxiv.org/list/math-ph/0611087, 2006 10. Füredi, Z., Komlós, J.: The eigenvalues of random symmetric matrices. Combinatorica 1, 233–241 (1981) 11. Guionnet, A.: Large deviations and stochastic calculus for large random matrices. Probab. Surv. 1, 72–172 (2004) 12. ’t Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461–473 (1974) 13. Janson, S., Luczak, T., Ru´cinski, A.: Random Graphs. New York: Wiley, 2000 14. Krivelevich, M., Sudakov, B.: The largest eigenvalue of sparse random graphs. Comb. Probab. Comput. 12, 61–72 (2003) 15. Khorunzhiy, O., Kirsch, W., Müller, P.: Lifshits tails for spectra of Erd˝os-Rényi random graphs. Ann. Appl. Probab. 16, 295–309 (2006) 16. Khorunzhiy, O., Shcherbina, M., Vengerovsky, V.: Eigenvalue distribution of large weighted random graphs. J. Math. Phys. 45, 1648–1672 (2004) 17. Lando, S.K.: Lectures on Generating Functions . Student Mathematical Library, Vol. 23, Amer. Math. Soc. Providence, RI, 2003 18. Malyshev, V.A., Minlos, R.A.: Gibbs random fields. Cluster expansions. Mathematics and its Applications (Soviet Series), Dordrecht: Kluwer Academic Publishers Group, 1991 19. Mehta, M.L.: Random Matrices. New York: Academic Press, 1991 20. Mohar, B.: The Laplacian spectrum of graph, graph theory, combinatorics, and applications. New York: Wiley, 1991 21. Okounkov, A.: Random matrices and random permutations. Intern. Math. Res. Not. 2000, 1043–1095 (2000) 22. Stanley, R.P.: Enumerative Combinatorics, Vol. II. Cambridge: Cambridge University Press, 1999 23. Vengerovsky, V.V.: Asymptotics of the correlator of an ensemble of sparse random matrices. Mat. Fiz. Anal. Geom. 11, 135–160 (2004) 24. Wigner, E.P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548–564 (1955) Communicated by M. Aizenman
Commun. Math. Phys. 282, 239–286 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0540-3
Communications in
Mathematical Physics
(Co)cyclic (Co)homology of Bialgebroids: An Approach via (Co)monads 2 Gabriella Böhm1 , Drago¸s Stefan ¸ 1 Research Institute for Particle and Nuclear Physics, P.O.B.49, 1525 Budapest 114,
Hungary. E-mail: [email protected]
2 Faculty of Mathematics and Informatics, University of Bucharest, 14 Academiei Street,
010014 Bucharest, Romania Received: 14 July 2007 / Accepted: 7 March 2008 Published online: 17 June 2008 – © Springer-Verlag 2008
Abstract: For a (co)monad Tl on a category M, an object X in M, and a functor : M → C, there is a (co)simplex Z ∗ := Tl ∗+1 X in C. The aim of this paper is to find criteria for para-(co)cyclicity of Z ∗ . Our construction is built on a distributive law of Tl with a second (co)monad Tr on M, a natural transformation i : Tl → Tr , and a morphism w : Tr X → Tl X in M. The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads Tl = T ⊗ R (−) and Tr = (−) ⊗ R T on the category of R-bimodules. R · · · ⊗ RT ⊗ R X is the cyclic R-module The functor can be chosen such that Z n = T ⊗ R (−) → (−)⊗ R T is given by the flip tensor product. A natural transformation i : T ⊗ map and a morphism w : X ⊗ R T → T ⊗ R X is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. The notion of a stable anti-Yetter-Drinfel’d module over certain bialgebroids, the so-called × R -Hopf algebras, is introduced. In the particular example when T is a module coring of a × R -Hopf algebra B and X is a stable anti-YetterDrinfel’d B-module, the para-cyclic object Z ∗ is shown to project to a cyclic structure on T ⊗ R ∗+1 ⊗B X . For a B-Galois extension S ⊆ T , a stable anti-Yetter-Drinfel’d S ∗+1 B-module TS is constructed, such that the cyclic objects B ⊗ R ∗+1 ⊗B TS and T ⊗ are isomorphic. This extends a theorem by Jara and Stefan ¸ for Hopf Galois extensions. As an application, we compute Hochschild and cyclic homologies of a groupoid with coefficients in a stable anti-Yetter-Drinfel’d module, by tracing it back to the group case. In particular, we obtain explicit expressions for (coinciding relative and ordinary) Hochschild and cyclic homologies of a groupoid. The latter extends results of Burghelea on cyclic homology of groups. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 1. The (Co)cyclic Object Associated to a Transposition Map . . . . . . . . . . 242
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1.A Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . 1.B Monads and distributive laws . . . . . . . . . . . . . . . . . . . . . . . 1.C Admissible septuples and transposition maps. The main result . . . . . 1.D Examples of admissible septuples and transposition maps. Applications 1.E The dual construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Cyclic (Co)homology of Bialgebroids . . . . . . . . . . . . . . . . . . . . 2.A (Co)module algebras of bialgebroids . . . . . . . . . . . . . . . . . . . 2.B (Co)module corings of bialgebroids . . . . . . . . . . . . . . . . . . . 2.C Stable anti-Yetter-Drinfel’d modules of × R -Hopf algebras . . . . . . . 2.D Galois extensions of × R -Hopf algebras . . . . . . . . . . . . . . . . . 3. Cyclic Homology of Groupoids . . . . . . . . . . . . . . . . . . . . . . . . 3.A Anti-Yetter-Drinfel’d modules for groupoids . . . . . . . . . . . . . . . 3.B Hochschild and cyclic homology with coefficients . . . . . . . . . . . . 3.C Cyclic homology of groupoids . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction Cyclic cohomology of Hopf algebras originates from the work [CM98] of Connes and Moscovici on the index theory of transversally elliptic operators. Their local index formula in [CM95] gives a generalization of the Chern character to non-commutative geometry. In order to give a geometrical interpretation of the non-commutative Chern character in terms of non-commutative foliations, in [CM98] a cocyclic structure was constructed on a cosimplex Z C∗ M = H ⊗∗ , associated to the coalgebra underlying a Hopf algebra H over a field K. The cocyclic operator was given in terms of a so-called modular pair in involution. In the subsequent years the Connes-Moscovici cocyclic module was placed in a broader and broader context. In [KR03] (see also [HKRS2]) to any (co)module algebra T of a Hopf algebra H , and any H -(co)module X , there was associated a para-cyclic module with components T ⊗∗+1 ⊗ X . Dually, for any (co)module coalgebra T of a Hopf algebra H , and any H -(co)module X , there is a para-cocyclic module with components T ⊗∗+1 ⊗ X . The Connes-Moscovici cosimplex Z C∗ M turns out to be isomorphic to a quotient of the para-cocyclic module associated to the regular module coalgebra T := H and an H -comodule defined on K. For bialgebras, the Connes-Moscovici construction was generalized in [Kay05]. In the papers [HKRS1] and [JS], ¸ a modular pair in involution was proven to be equivalent to a stable anti-Yetter-Drinfel’d module structure on the ground field K. In [HKRS2], the para-cocyclic module T ⊗∗+1 ⊗ X , associated to an H -module coalgebra T and a stable anti-Yetter-Drinfel’d H -module X , was shown to project to a cocyclic object whose components are the H -module tensor products T ⊗∗+1 ⊗ H X . The way in which the para-cocyclic object H ⊗∗+1 projects to the Connes-Moscovici cosimplex Z C∗ M , is an example of this scenario. Dually, the para-cyclic module, associated to an H -comodule algebra, was proven to have a cyclic submodule. In the spirit of [JS], ¸ one can follow a dual approach. That is, para-cocyclic modules can be constructed for (co)module algebras of Hopf algebras, and para-cyclic modules for (co)module coalgebras, in both cases with coefficients in H -(co)modules. Taking coefficients in a stable anti-Yetter-Drinfel’d module, it was shown in [JS] ¸ that in this case the para-cyclic object associated to a module coalgebra possesses a cyclic quotient.
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In [KR05] an isomorphism was proven between the cyclic quotient of the para-cyclic object in [JS] ¸ of H as an H -module coalgebra on one hand, and the cyclic subobject of the para-cyclic object in [HKRS2] of H as an H -comodule algebra on the other. Constructions in Sect. 2 of the current paper follow the root in [JS]. ¸ Since this framework is dual to that suggested in [HKRS2] (cf. also [Kay06]), some might like to call it a dual Hopf (co)cyclic theory. However, we do not use this somewhat involved terminology in the paper, but remind the reader of the difference between the two possible dual approaches. In [Kay06] Kaygun proposed a unifying approach to the para-(co)cyclic objects corresponding to a (co)module (co)algebra of a Hopf algebra. Starting with a (co)algebra T and an object X in a symmetric monoidal category S, he introduced the notion of a transposition map. It is a morphism w : X ⊗ T → T ⊗ X in S, satisfying conditions reminiscent to half of the axioms of a distributive law in [Be]. Any transposition map w was shown to determine a para-(co)cyclic structure on the (co)simplex T ⊗∗+1 ⊗ X in S. In particular, canonical transposition maps were constructed for (co)module (co)algebras T and (co)modules X of a bialgebra. Connes and Moscovici’s index theory of transversally elliptic operators lead beyond cyclic homology of Hopf algebras. In dealing with the general, non-flat case, in [CM01] certain bialgebroids (in fact × R -Hopf algebras) arose naturally. Bialgebroids can be thought of as a generalization of bialgebras to a non-commutative base algebra R, while × R -Hopf algebras generalize Hopf algebras. There are a few papers in the literature, e.g. [KR04] and [Ra], attempting to extend Hopf cyclic theory to non-commutative base algebras. However, an understanding of the subject, comparable to that in the classical case of a commutative base ring (or field), is missing yet. The aim of the current paper is to give a universal construction of para-(co)cyclic (co)simplices, including examples coming from (co)module algebras and (co)module corings for bialgebroids. When replacing bialgebras over a commutative ring K by bialgebroids over a noncommutative K-algebra R, the monoidal category of K-modules becomes replaced by the monoidal category of R-bimodules. Indeed, (co)module algebras of an R-bialgebroid are in particular algebras, and (co)module corings are coalgebras, in the category of R-bimodules. The main difference is that the category of K-modules is symmetric. In contrast, the category of R-bimodules is not even braided in general. Hence Kaygun’s elegant theory [Kay06], formulated in a symmetric monoidal category S, is not applicable. Our key observation is that the role the symmetry plays in Kaygun’s work is that it defines a compatible natural transformation i between the two (co)monads T ⊗ (−) and (−) ⊗ T on the symmetric monoidal category S, induced by a (co)algebra T in S. Note that these (co)monads on S are connected by a trivial distributive law. Guided by this observation, in Sect. 1 we start with a distributive law of two (co)monads Tl and Tr on any category M. In addition, we allow for the presence of a functor : M → C (it is the identity functor on S in [Kay06]). Then, for any object X in M, there is a (co)simplex in C, given at degree n by Tl n+1 X . In Sects. 1.C and 1.E we show that it is para-(co)cyclic provided that there exist a natural transformation i : Tl → Tr and a morphism Tr X → Tl X in M, satisfying symmetrical conditions generalizing Kaygun’s axioms of a transposition map. Examples of this situation are collected in Sect. 1.D. Among other (classical) examples, we show that Škoda’s functorial construction in [Šk] of a para-cyclic object in the category of endofunctors, Majid and Akrami’s para-cyclic modules associated to a ribbon algebra in [AM], and Rangipour’s cyclic module in [Ra] determined by a coring, fit our framework. It is discussed in Sects. 2.A and 2.B how the general results in Sects. 1.C and 1.E cover the particular cases when the two
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(co)monads Tl = T ⊗ R (−) and Tr = (−) ⊗ R T are induced by a (co)module algebra or (co)module coring T of an R-bialgebroid B, the functor is defined via the coequalizer of the R-actions in a bimodule, and X is a B-(co)module. The components of the resulting R · · · ⊗ RT ⊗ R X . In this para-(co)cyclic module are cyclic R-module tensor products T ⊗ way we obtain examples which extend both some para-(co)cyclic objects in [Kay06] and [HKRS2] and their cyclic duals. By the above procedure, we associate a para-cyclic object in particular to a module coring C and a comodule X of an R-bialgebroid B. Following [JS], ¸ in Sect. 2.C we look for situations in which it projects to the B-module tensor product C ⊗ R · · ·⊗ R C ⊗B X . Restricting at this point our study to × R -Hopf algebras B, we define stable anti-YetterDrinfel’d modules for B. In parallel to the case of Hopf algebras [JS, ¸ Theorem 4.13], [HKRS2, Theorem 2.1], we prove cyclicity of the simplex C ⊗ R · · · ⊗ R C ⊗B X , whenever X is a stable anti-Yetter-Drinfel’d module. The simplest example of a cyclic simplex is associated to an algebra extension S ⊆ T . S n+1 ; face and Its components are given by the n + 1 fold cyclic tensor product T ⊗ degeneracy maps are determined by the algebra structure of T and the cyclic map is given by the cyclic permutation of the tensor factors. In Sect. 2.D, for a Galois extension S ⊆ T by a × R -Hopf algebra B, we construct a stable anti-Yetter-Drinfel’d module S n+1 and TS := T /{ s · t − t · s | s ∈ S, t ∈ T }. We prove that the cyclic simplices T ⊗ ⊗ n+1 B R ⊗B TS are isomorphic. This extends [JS, ¸ Theorem 3.7]. A most fundamental class of examples of bialgebroids (in fact × R Hopf algebras) is given by algebras (over fields), generated by a groupoid of finitely many objects. As an application of our abstract theory, we compute explicitly the relative Hochschild and cyclic homologies of such a groupoid, with coefficients in a stable anti-YetterDrinfel’d module. By our results, any Galois extension by the groupoid provides us with a stable anti-Yetter-Drinfel’d module. In particular, the groupoid algebra B is a Galois R n+1 extension of its base algebra R. Applying the isomorphism of the simplices B ⊗ and ⊗ n+1 R B ⊗B B R , we obtain the R-relative cyclic homology of B. Since R is a separable algebra, it is equal to ordinary cyclic homology of B, hence our results extend those by Burghelea on the cyclic homology of groups [Burg]. Similar formulae were obtained by Crainic for cyclic homology of étale groupoids [Cra]. Observe that any groupoid (with an arbitrary set of objects) can be obtained as a direct limit of groupoids with finite sets of objects. Certainly, the algebra generated by a groupoid with infinitely many objects is no longer unital. However, one can still consider its cyclic homology, as a homology of Connes’ complex, associated to a presimplicial object. Since the homology functor commutes with direct limits, we can extend our formula of cyclic homology to arbitrary groupoids. Throughout the paper K denotes a commutative and associative unital ring. The term K-algebra means an associative and unital algebra over K. 1. The (Co)cyclic Object Associated to a Transposition Map In this first section we establish a general categorical framework – in terms of admissible septuples and their transposition maps – to produce para-cocyclic, and dually, para-cyclic objects. 1.A. Notation and conventions. In the 2-category CAT we denote horizontal composition (of functors) by juxtaposition, while ◦ is used for vertical composition (of natural
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transformations). That is, for two functors F : C → C , G : C → C and an object X in C, instead of G(F(X )) we write G F X . For two natural transformations µ : F → F and ν : G → G we write G µX ◦ν F X : G F X → G F X instead of G (µ X )◦ν F (X ) . In equalities of natural transformations we shall omit the object X in our formulae. Inspired by the diagrammatic computation in a 2-category (in particular CAT), we shall use a graphical representation of morphisms in a category. For functors F 1 , . . . , F n , G 1 , . . . , G m , which can be composed to F 1 F 2 · · · F n : D1 → C and G 1 G 2 · · · G m : D2 → C, and objects X in D1 and Y in D2 , a morphism f : F 1 F 2 · · · F n X → G 1 G 2 · · · G m Y will be represented vertically, with the domain up, as in Fig. 1(a). Furthermore, for a functor T : C → C , the morphism T f will be drawn as in (b). Keeping the notation from the first paragraph of this section, the picture representing µG X is shown in diagram (c). The composition g ◦ f of the morphisms f : X → Y and g : Y → Z will be represented as in diagram (d). For the multiplication m T and the unit u T of a monad T on C (see Definition 1.1), and an object X in C, to draw m T X and u T X we shall use the diagrams (e) and (f), while for a distributive law t : RT → T R (see Definition 1.3) t X will be drawn as in the picture (g). If t is invertible, the representation of t −1 X is shown in diagram (h). For simplifying diagrams containing only natural transformations, we shall always omit the last string that corresponds to an object in the category. That is, we work with diagrams in CAT whenever it is possible. We shall use the following method to perform computations with such diagrams. In view of associativity of composition, any diagram, representing a well-defined composition of morphisms, can be thought of as a tower with several layers. Any part of the diagram, corresponding to a layer, can be substituted with any other equivalent representation of it. Usually, equivalent representations are obtained from formulas that define the notions that we deal with, or equations that have been previously proved. 1.B. Monads and distributive laws. Monads represent the main ingredient in our approach to cyclic (co)homology. The definition of monads traces back to Godement’s book [Go], where they are called “standard constructions”. In the literature they are also called “triples”, see for example [EM]. Definition 1.1. A monad on a category C is a triple (T , m T , u T ), where T : C → C is a functor, m T : T 2 → T and u T : I d C → T are natural transformations such that the first two diagrams in Fig. 2, expressing associativity and unitality, are commutative. We call m T and u T the multiplication and the unit of the monad T , respectively. For two monads (T , m T , u T ) and (T , m T , u T ) on C, we say that a natural transformation ϕ : T → T is a morphism of monads if the last two diagrams in Fig. 2 are commutative.
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T3
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Example 1.2. Let (C, ⊗, a, l, r, 1) be a monoidal category with unit object 1, associativity constraint a and unit constraints l, r. For details about monoidal categories the reader is referred to [Kass, Chap. XI]. An algebra in C is a triple (T, m T , u T ) such that m T : T ⊗ T → T defines an associative multiplication on T with unit u T : 1 → T . To such an algebra one can associate two monads Tl := T ⊗ (−) and Tr := (−) ⊗ T on C. The multiplication m Tl and the unit u Tl of Tl are given by m Tl X := (m T ⊗ X ) ◦ a−1 T,T,X
and
u Tl X := (u T ⊗ X ) ◦ l −1 X ,
for every X in C. Analogously, for X in C, m Tr X and u Tr X are defined by m Tr X := (X ⊗ m T ) ◦ a X,T,T
and
u Tr X := (X ⊗ u T ) ◦ r −1 X .
A homomorphism ϕ : T → T of algebras in C induces monad morphisms ϕl : Tl → Tl and ϕ r : Tr → Tr . For example, ϕl X := ϕ ⊗ id X , for any object X in C. A particular case of these constructions, which is very important for our work, is obtained when we take C to be the category R-Mod-R of bimodules over an ordinary K-algebra R (i.e. R is an algebra in the category of K-modules, where K is a commutative ring). The category R-Mod-R is monoidal with respect to the R-module tensor product ⊗ R . The unit object is R. An algebra in R-Mod-R is called an R-ring. R-rings (T, m T , ϕ) are in bijective correspondence with K-algebra maps ϕ : R → T . Indeed, for an algebra (T, m T , ϕ) in R-Mod-R, composition of the canonical epimorphism T ⊗K T → T ⊗ R T with m T : T ⊗ R T → T defines a K-algebra structure on T such that ϕ is a K-algebra homomorphism. Conversely, via a K-algebra homomorphism ϕ : R → T , T becomes an R-bimodule. Multiplication of T induces a morphism m T from T ⊗ R T to T , which makes T an associative algebra in R-Mod-R. The unit of T is ϕ. (With a slight abuse of notation, we denote both multiplication maps T ⊗ R T → T and T ⊗K T → T by the same symbol m T .) Consequently, a K-algebra homomorphism ϕ : R → T defines two monads T ⊗ R (−) and (−) ⊗ R T on R-Mod-R. Distributive laws were introduced by J. Beck [Be]. They give a way to compose two monads in order to obtain a monad. Definition 1.3. A distributive law between two monads (R, m R , u R ) and (T , m T , u T ) is a natural transformation t : RT → T R satisfying the four conditions in Fig. 3. Remark 1.4. Since we are using for the first time the diagrammatic representation of morphisms, let us write out explicitly the first and the third relations in Fig. 3. They read as t ◦ m R T = T m R ◦ t R ◦ Rt,
t ◦ u R T = T u R.
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Fig. 3. The definition of distributive laws
Example 1.5. Let T be an algebra in a monoidal category as in Example 1.2. Keeping the notation from Example 1.2, the natural transformation t : Tr Tl → Tl Tr , given by t X := a T,X,T
(1.1)
for any object X in C, is a distributive law. Note that the first equality in Fig. 3 follows by the Pentagon Axiom [Kass, p. 282, Diagram (2.6)], applied to the quadruple (T, X, T, T ). Similarly, by applying the Pentagon Axiom for (T, T, X, T ) we deduce the second equality in the definition of distributive laws. The fourth equality in Fig. 3 is a consequence of l X ⊗T ◦ a1,X,T = l X ⊗ T , see [Kass, Lemma XI.2.2]. The other relation in the above cited lemma can be used to prove that the third equality in Fig. 3 holds too. Example 1.6. Let C be a braided monoidal category with braiding c X,Y : X ⊗Y → Y ⊗X . For the definition and properties of braided monoidal categories see [Kass, Chap. XIII]. If R and T are algebras in C then t X := a T,R,X ◦ (c R,T ⊗ X ) ◦ a−1 R,T,X defines a distributive law t : Rl Tl → Tl Rl , where Rl and Tl are constructed as in Example 1.2. Obviously, t −1 : Tl Rl → Rl Tl is also a distributive law. 1.C. Admissible septuples and transposition maps. The main result. In this section we introduce admissible septuples and transposition morphisms of them. We show that to these data one associates functorially para-cocyclic objects. Our aim is twofold. On one hand, in this way we obtain a very general but at the same time technically very simple framework. In particular, it can be used to associate para-cocyclic objects to (co)module algebras of bialgebroids, cf. Sect. 2.A. On the other hand, the resulting setting will be easily dualized to describe in Sect. 2.B the situation dual to that in Sect. 2.A, i.e. the (para-)cyclic objects associated to (co)module corings of bialgebroids. Definition 1.7. An admissible septuple S := (M, C, Tl , Tr , , t, i) is defined by the following data: • • • • •
Two categories M and C; Two monads Tl and Tr on M; A functor : M → C; A distributive law t : Tr Tl → Tl Tr ; A natural transformation i : Tl → Tr .
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These data are assumed to satisfy the relations i ◦ u Tl = u Tr
i ◦ m Tl = m Tr ◦ i Tr ◦ t ◦ i Tl .
and
(1.2)
Examples of admissible septuples will be given in Sect. 1.D, where also several applications of the main result of this section, Theorem 1.10, will be indicated. By [We, p. 281], to every monad Tl : M → M and object X in M one can associate a cosimplicial object of components Tl n+1 X in M. Thus in particular an admissible septuple S in Definition 1.7 determines a cosimplicial object in M. It can be transported to C via the functor : M → C in Definition 1.7. The resulting cosimplex in C will be denoted by Z ∗ (S, X ). By construction, Z n (S, X ) = Tl n+1 X and, for every k ∈ {0, . . . , n}, the coface maps dk : Tl n X → Tl n+1 X and the codegeneracy maps sk : Tl n+2 X → Tl n+1 X are given by dk := Tl k u Tl Tl n−k X,
sk := Tl k m Tl Tl n−k X.
(1.3)
Our aim is to construct a category WS such that Z ∗ (S, −) can be regarded as a functor from WS to the category of para-cocyclic objects in C. Observe that, for an admissible septuple S in Definition 1.7, the distributive law t is lifted to a natural transformation t n : Tr Tl n → Tl n Tr , t n := Tl n−1 t ◦ Tl n−2 t Tl ◦ · · · ◦ Tl t Tl n−2 ◦ t Tl n−1 .
(1.4)
Definition 1.8. Let S := (M, C, Tl , Tr , , t, i) be an admissible septuple. We say that an arrow w : Tr X → Tl X in M is a transposition morphism with respect to S if w ◦ u Tr X = u Tl X
and
w ◦ m Tr X = m Tl X ◦ Tl w ◦ t X ◦ Tr w.
(1.5)
The category of pairs (X, w), with w : Tr X → Tl X a transposition morphism of S, will be denoted by WS . A morphism from (X, w) to (X , w ) is an arrow f : X → X in M such that Tl f ◦ w = w ◦ Tr f . Morphisms w : Tr X → Tl X satisfying (1.5), for a distributive law t : Tr Tl → Tl Tr , were termed t-algebras in [Burr]. Based on [Burr, Prop. I.1.1], transposition morphisms can be characterized as in Proposition 1.9 below. Recall that a module of a monad (T, m T , u T ) on a category M is a pair (Y, ), consisting of an object Y and a morphism : T Y → Y in M, such that ◦ T = ◦ m T Y and ◦ u T Y = IdY (i.e. is associative and unital). A morphism of T -modules (Y, ) → (Y , ) is a morphism f : Y → Y in M, such that f ◦ = ◦ T f . Proposition 1.9. Consider an admissible septuple S := (M, C, Tl , Tr , , t, i). There is a bijective correspondence between objects (X, w) in the category WS and Tr -modules of the form (Tl X, ), satisfying m Tl X ◦ Tl ◦ t Tl X = ◦ Tr m Tl X.
(1.6)
Moreover, a morphism f : X → X in M is a morphism in WS if and only if Tl f is a Tr -module morphism. Proof. Similarly to the proof of [Burr, Prop. I.1.1] one checks that, for an object (X, w) in WS , an associative and unital Tr -action on Tl X satisfying (1.6) is given by w := m Tl X ◦ Tl w ◦ t X : Tr Tl X → Tl X . Conversely, note that for a Tr -module (Tl X, ), (1.6) is equivalent to = m Tl X ◦ Tl ◦ Tl Tr u Tl X ◦ t X . With this identity at hand,
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the pair (X, w := ◦ Tr u Tl X ) is checked to be an object in WS . A straightforward computation shows that the two constructions are mutual inverses. A morphism Tl f is a morphism of Tr -modules (Tl X, w ) → (Tl X , w ) if m Tl X ◦ Tl w ◦ Tl Tr f ◦ t X = m Tl X ◦ Tl Tl f ◦ Tl w ◦ t X.
(1.7)
If f is a morphism in WS then (1.7) obviously holds. In order to prove the converse implication, compose both sides of (1.7) with Tr u Tl X on the right. Theorem 1.10. Consider an admissible septuple S and a transposition map w : Tr X → Tl X in WS . The cosimplicial object Z ∗ (S, X ) is para-cocyclic with respect to wn := Tl n w ◦ t n X ◦ i Tl n X.
(1.8)
We shall denote this para-cocyclic object by Z ∗ (S, w). For a morphism f : (X, w) → (X , w ) in WS , the morphisms Tl n+1 f : Z n (S, w) → Z n (S, w ) determine a morphism of para-cocyclic objects. Proof. In Fig. 4 we show that the morphism (1.8) is compatible with the coface maps, that is wn ◦ d0 = dn
and
wn ◦ dk = dk−1 ◦ wn−1
(1.9)
for any k ∈ {1, . . . , n}. The proof of the first equation is given in three steps in the left picture. To simplify the diagrams, we draw the n strings representing Tl n as a black stripe. For the first equality we used the compatibility between i and the unit of Tl , that is the first equation in (1.2). Next we applied n times the compatibility relation between the distributive law t and the unit of Tr , i.e. the third equality in Fig. 3. The first relation in (1.5) implies the third equality. The second relation in (1.9) follows in a similar way, as it is shown in the right picture in Fig. 4. Note that the leftmost black stripe represents Tl k−1 and the other one represents Tl n−k . Since u Tl is a natural transformation, the box representing it can be pushed down along the string until it meets the crossing t. By the fourth identity in Fig. 3, one can push u Tl under the string in the crossing. To conclude the proof of this equality, we use once again that u Tl is a natural transformation to move it to the bottom of the diagram.
Fig. 4. The proof of the relations (1.9)
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Fig. 5. The proof of the relations (1.10)
Next we prove that the morphism (1.8) and the codegeneracy maps are compatible too, that is wn ◦ s0 = sn ◦ (wn+1 )2
and
wn ◦ sk = sk−1 ◦ wn+1
(1.10)
for any k ∈ {1, . . . , n}. The proof of the first relation can be found in the left picture in Fig. 5. As before, the black stripe represents Tl n . The morphisms corresponding to the first two diagrams are equal in view of the second equation in (1.2). By applying n times the first identity in Fig. 3, it follows that the second and the third diagrams represent the same morphism. Taking into account the second relation in (1.5) we got the penultimate equality, while for the last one we used that i is a natural transformation. The other relation in (1.10) immediately follows by the second identity in Fig. 3 and the fact m Tl is natural (see the second picture in Fig. 5). Since the coface and codegeneracy morphisms (1.3) are defined in terms of natural transformations, the morphisms Tl n+1 f : Z n (S, X ) → Z n (S, X ) determine a morphism Z ∗ (S, f ) : Z ∗ (S, X ) → Z ∗ (S, X ) of cosimplicial objects, for any morphism f : X → X in M. It follows immediately from the definition of a morphism in WS that if f : (X, w) → (X , w ) is a morphism in WS then Z ∗ (S, f ) is a morphism of para-cocyclic objects Z ∗ (S, w) → Z ∗ (S, w ). Corollary 1.11. Let S be an admissible septuple as in Definition 1.7 and let w : Tr X → Tl X be a transposition morphism in WS . Consider the corresponding para-cocyclic morphism wn in (1.8). If the coequalizer Z n (S, w)
(wn )n+1 Id Z n (S,w)
// Z n (S, w)
/ Z n (S, w)
exists in C, for every non-negative integer n, then it defines a cocyclic cosimplex Z ∗ (S, w). Proof. Let k ∈ {0, . . . , n}. It follows by (1.9) that dk satisfies dk ◦(wn )n+1 = (wn+1 )n+2 ◦ dk . Similarly, by (1.10), the codegeneracy morphism sk satisfies sk ◦(wn )n+1 = (wn−1 )n ◦ sk . Hence dk and sk determine coface morphisms dk X and codegeneracy morphisms sk on n n Z (S, X ). Together with the projection w n of wn onto Z (S, X ) they define a cocyclic object ( Z ∗ (S, w), d∗ , s ∗, w ∗ ).
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1.D. Examples of admissible septuples and transposition maps. Applications. In this section we shall apply Theorem 1.10 to several examples of admissible septuples. In this way we shall show that the most known (co)cyclic objects in the literature can be obtained as direct applications of the result obtained in Sect. 1.C. A functorial construction of a para-cyclic object in a category of endofunctors, of a somewhat similar flavour to that in Theorem 1.10, was proposed in [Šk]. The following example is its dual version. Example 1.12. Let T = (T, m, u) be a monad on a category M and t : T T → T T be a distributive law. Assume that t satisfies the Yang-Baxter relation tT ◦ T t ◦ tT = T t ◦ tT ◦ T t
(1.11)
of natural transformations T T T → T T T , and m ◦ t ◦ t = m. As a consequence of (1.11), also T 0 = (T, m ◦ t, u) is a monad, and t can be regarded as a distributive law T 0 T → T T 0 . Furthermore, the datum S := (M, M, T, T 0 , IdM , t, Id T ) (where IdM denotes the identity functor M → M and Id T is the identity natural transformation T → T ) is an admissible septuple. For any object X in M, the identity morphism Id T X is a transposition morphism. The corresponding para-cocyclic morphism is t n in (1.4). The simplest example of an admissible septuple can be obtained by starting with a morphism ϕ : R → T of K-algebras. As in Example 1.2, we define two monads on the category M := R-Mod-R by Tl := T ⊗ R (−) and Tr := (−) ⊗ R T. The category C is, by definition, the category Mod-K of K-modules. The functor is constructed below. Definition 1.13. On the objects X ∈ R-Mod-R, the functor : R-Mod-R → Mod-K is defined as a coequalizer X ⊗K R
x⊗r →x·r x⊗r →r ·x
//
X
pX
/ X .
For a morphism f : X → Y of R-bimodules, f is the unique K-linear map such that pY ◦ f = f ◦ p X. Hence p can be interpreted as a natural epimorphism from the forgetful functor U : R-Mod-R → Mod-K to . Remark 1.14. An R-bimodule X can be considered as a left or right module for the enveloping algebra R e := R ⊗K R op of R. In terms of the functor in Definition 1.13, R Y of two R-bimodules X and Y is defined by X ⊗ R Y := the cyclic tensor product X ⊗ (X ⊗ R Y ) ∼ = X ⊗ R e Y . With this interpretation in mind, the K-module X ∼ = R ⊗ Re X can be seen as the cyclic tensor product of R and X . For R-bimodules X 1 , . . . , X n , the n-fold cyclic module tensor product is defined as R · · · ⊗ R X n := (X 1 ⊗ R · · · ⊗ R X n ) = (X 1 ⊗ R · · · ⊗ R X k ) X 1⊗ R (X k+1 ⊗ R · · · ⊗ R X n ), ⊗ R · · · ⊗ R xn . for k ∈ {1, . . . , n − 1}. It is generated by the cyclic tensor monomials x1 ⊗ It is well known that the symmetry c X,Y : X ⊗K Y → Y ⊗K X induces a natuR Y ∼ R X . In particular, there is a natural isomorral isomorphism i X,Y : X ⊗ = Y⊗ R X n → X 2⊗ R Xn⊗ R · · · ⊗ R · · · ⊗ R X 1 , that maps a generator phism i X 1 ,...,X n : X 1 ⊗ R · · · ⊗ R · · · ⊗ R x1 . R xn to x2 ⊗ R xn ⊗ x1 ⊗
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As we have noticed in Example 1.5, the associativity constraint of the monoidal category R-Mod-R defines a distributive law t : Tr Tl → Tl Tr . Thus, in this particular case that we are investigating, t X is the canonical isomorphism (T ⊗ R X ) ⊗ R T ∼ = T ⊗ R (X ⊗ R T ), for any X in R-Mod-R. Let us define i X : Tl X → Tr X by i X := i T,X , as in Remark 1.14. In terms of these natural transformations we can give one of the main examples of admissible septuples. Proposition 1.15. Let ϕ : R → T be a morphism of K-algebras. The following data: • • • •
the categories M := R-Mod-R and C := Mod-K, the monads Tl := T ⊗ R (−) and Tr := (−) ⊗ R T , the functor : M → C, X := R ⊗ R e X , the natural transformation t X : (T ⊗ R X ) ⊗ R T → T ⊗ R (X ⊗ R T ) defined by the canonical isomorphism; R X → X⊗ R T , t⊗ R x → x ⊗ R t, • the natural transformation i X : T ⊗ define an admissible septuple ST .
Proof. Let X be an R-bimodule. By definition, R X Tl n X = R ⊗ R e (T ⊗ R n ⊗ R X ) ∼ = T ⊗R n ⊗
and
R n R T ⊗ Tr n X ∼ . = X⊗
Via these identifications, i Tl X = i T,T,X and i Tr X = i T,X,T . So the conditions (1.2) take the form R X) = X⊗ R ϕ i T,X ◦ (ϕ ⊗ identities which are obvious.
and
R X ) = (X ⊗ R m T ) ◦ i T ⊗ R T,X , i T,X ◦ (m T ⊗
Let ST be the admissible septuple associated to an algebra morphism ϕ : R → T . A morphism of R-bimodules w : X ⊗ R T → T ⊗ R X is a transposition map in WST if, and only if, it satisfies the conditions and w ◦ (X ⊗ R m T ) w ◦ (X ⊗ R ϕ) = ϕ ⊗ R X = (m T ⊗ R X ) ◦ (T ⊗ R w) ◦ (w ⊗ R T ),
(1.12)
where m T : T ⊗ R T → T denotes the multiplication map t ⊗ R t → tt . Note in passing the similarity of conditions (1.12) to some of those defining an entwining structure over R. (For the definition of entwining structures see [BMa, Def. 2.1], and for a reformulation over an arbitrary base algebra R see [BB, Sect. 2.3].) By Proposition 1.9, there is a bijective correspondence between transposition maps w : X ⊗ R T → T ⊗ R X on one hand, and right T -actions on T ⊗ R X , which are left T -module maps with respect to the left T -action t · (t ⊗ R x) = t t ⊗ R x, on the other hand. Theorem 1.16. Let ST be the admissible septuple associated to an algebra morphism ϕ : R → T . Let w : X ⊗ R T → T ⊗ R X be a transposition map in WST , that is, a morphism of R-bimodules satisfying (1.12). Then there is a cocylic quotient Z ∗ (ST , w) of R ∗+1 ⊗ T ⊗ R X such that its cocyclic structure is induced by the para-cocyclic morphisms R n+1 R n+1 ⊗R X → T ⊗ ⊗R X , wn : T ⊗ R w ◦ i T,...,T,X , wn := T ⊗ R n ⊗ (1.13) where i T,...,T,X is the K-linear map defined in Remark 1.14.
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Proof. Apply Theorem 1.10 for S = ST . It yields a para-cocyclic object Z ∗ (ST , w) whose para-cocyclic operator is given in formula (1.8). For ST , the map t n is the identity R n+1 morphism of T ⊗ ⊗ R X , cf. (1.4). Hence wn satisfies (1.13). We conclude the proof by applying Corollary 1.11. Corollary 1.17. Let ST be the admissible septuple associated to a K-algebra homomorphism ϕ : R → T as in Proposition 1.15. Then the canonical isomorphism wT : R ⊗ R T → T ⊗ R R is a transposition map in WST . The corresponding cocy R n+1 . The clic cosimplex Z ∗ (ST , wT ) has in degree n the K-module Z n (ST , wT ) = T ⊗ coface and codegeneracy maps are R t1 ⊗ R · · · ⊗ R t1 ⊗ R · · · ⊗ R 1T ⊗ R tk ⊗ R · · · ⊗ R tn−1 ) = t0 ⊗ R tk−1 ⊗ R tn−1 , dk (t0 ⊗ R t1 ⊗ R · · · ⊗ R t1 ⊗ R · · · ⊗ R tk tk+1 ⊗ R tk+2 ⊗ R · · · ⊗ R tn+1 ) = t0 ⊗ R tk−1 ⊗ R tn+1 , sk (t0 ⊗ where k ∈ {0, . . . , n}. The cocyclic operator is given by R tn ) = t1 ⊗ R tn ⊗ R t1 ⊗ R · · · ⊗ R t2 ⊗ R · · · ⊗ R t0 . wn (t0 ⊗ Remark 1.18. As u T : k → T , the unit of a K-algebra T is an algebra map, we can apply Proposition 1.15 to get an admissible septuple Su T . Corresponding transposition maps were also considered by Kaygun in [Kay06] to construct cocyclic K-modules. His approach should be considered, however, dual to ours (see related remarks in the Introduction). It follows by an observation in [Burr, p. 11] that for the admissible septuple ST , associated to an algebra morphism ϕ : R → T in Proposition 1.15, any R-T bimodule Y admits a transposition morphism wY : Y ⊗ R T → T ⊗ R Y , y ⊗ R t → 1T ⊗ R y · t. In particular, for any R-bimodule X , the pair (X ⊗ R T, (u T ⊗ R X ⊗ R T ) ◦ (X ⊗ R m T )) is an object in WST . Corresponding para-cocyclic objects are given in the following Example 1.19. Let ST be the admissible septuple associated to a K-algebra homomorphism ϕ : R → T as in Proposition 1.15. For any R-T bimodule Y , there is a para-cocyclic cosimplex Z ∗ (ST , wY ), given in degree n by the K-module Z n (ST , wY ) = R n+1 ⊗ R Y . The coface and codegeneracy maps are T⊗ R t1 ⊗ R · · · ⊗ R y) = t0 ⊗ R t1 ⊗ R · · · ⊗ R 1T R tn−1 ⊗ R tk−1 ⊗ dk (t0 ⊗ R tn−1 ⊗ R tk ⊗ R · · · ⊗ R y, ⊗ R t1 ⊗ R · · · ⊗ R y) = t0 ⊗ R t1 ⊗ R · · · ⊗ R tk tk+1 R tn+1 ⊗ R tk−1 ⊗ sk (t0 ⊗ ⊗ R tk+2 ⊗ R · · · ⊗ R tn+1 ⊗ R y, where k ∈ {0, . . . , n}. The para-cocyclic operator is given by R tn ⊗ R tn ⊗ R t1 ⊗ R · · · ⊗ R y) = t1 ⊗ R · · · ⊗ R 1T ⊗ R y · t0 . wn (t0 ⊗ Note that wn is degenerate in the sense that the cocyclic quotient of Z ∗ (ST , wY ) (cf. R Y , in every degree n. Corollary 1.11) is given by Z n (ST , wY ) = R ⊗ Next we are going to associate an admissible septuple to every ribbon algebra. Recall that a ribbon algebra is an algebra (T, m T , u T ) in a braided monoidal category M together with an automorphism σ : T → T in M such that σ ◦ uT = uT
and
σ ◦ m T = m T ◦ (σ ⊗ σ ) ◦ c2T,T .
(1.14)
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Ribbon algebras appeared in [AM], where they are used to define cyclic homology of quasialgebras (non-associative algebras that are obtained by a cochain twist). We shall show that the ribbon automorphism σ can be used to define a certain admissible septuple. For, we start with an algebra (T, m T , u T ) and an automorphism σ : T → T in a braided monoidal category. It is easy to see that T is also an associative and unital algebra with respect to m T := m T ◦ cT,T and u T := u T . To make a distinction between T and the new algebra, the latter one will be denoted by T . Consider the monads Tl and Tl on M, defined as in Example 1.2. In the following, Tl will play the role of Tr in the definition of an admissible septuple. We have seen in Example 1.6 that cT,T : T ⊗ T → T ⊗ T induces a distributive law t : Tl Tl → Tl Tl , t X := a T,T,X ◦ (cT,T ⊗ X ) ◦ a−1 T,T,X , where X is an arbitrary object in M. Furthermore, we take C = M and = IdM . By definition, the natural transformation i : Tl → Tl is i X := (σ ⊗ X ). It is not difficult to prove that the relations in (1.2) hold if, and only if, the identities in (1.14) are satisfied. Thus, we have the following Proposition 1.20. The algebra (T, m T , u T ) in a braided monoidal category M is a ribbon algebra with ribbon automorphism σ if and only if ST,σ := (M, M, Tl , Tl , IdM , t, i), the septuple constructed above, is admissible. Let (T, m T , u T ) be a ribbon algebra with ribbon automorphism σ . In view of Proposition 1.20, we can speak about transposition morphisms with respect to ST,σ . A morphism w : T ⊗ X → T ⊗ X in M is a transposition map in WST,σ if, and only if (1.15) w ◦ (u T ⊗ X ) = u T ⊗ X, w◦(m T ⊗ X ) ◦ (cT,T ⊗ X ) = (m T ⊗ X ) ◦ (T ⊗ w) ◦ (cT,T ⊗ X ) ◦ (T ⊗ w). (1.16) Note that, in the second equation, we omitted the associativity constraints, to make the formula as short as possible. In fact, in view of the Coherence Theorem, we can omit bracketing in any equality involving morphisms in an arbitrary monoidal category M. (For MacLane’s Coherence Theorem consult e.g. [MacL, Theorem 1, p. 162] or [Mj, pp. 420–421]). Relations (1.15) and (1.16) already appeared in the definition of braided twistors, structures that are used to construct new associative and unitary multiplications on T ⊗T. For details the reader is referred to [LPvO]. As an application of Theorem 1.10 we get Proposition 1.21 below. Note that, for simplifying the formulae of coface, codegeneracy and para-cocyclic morphisms, we omitted the associativity and unit constraints. Proposition 1.21. Let (T, m T , u T , σ ) be a ribbon algebra in a braided monoidal category M. For every object X in M, the sequence Z n (ST,σ , X ) := T ⊗n+1 ⊗ X defines a cosimplicial object, with respect to the coface and codegeneracy morphisms dk := T ⊗k ⊗ u T ⊗ T ⊗n−k ⊗ X
and
sk := T ⊗k ⊗ m T ⊗ T ⊗n−k ⊗ X,
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where k ∈ {0, . . . , n}. Moreover, if w : T ⊗ X → T ⊗ X is a transposition map in WST,σ , then Z ∗ (ST,σ , X ) admits a para-cocyclic structure with respect to the operator wn := (T ⊗n ⊗ w) ◦ (T ⊗n−1 ⊗ cT,T ⊗ X ) ◦ · · · ◦ ×(cT,T ⊗ T ⊗n−1 ⊗ X ) ◦ (σ ⊗ T ⊗n ⊗ X ). We shall denote this para-cocyclic object by Z ∗ (ST,σ , w). Remarks 1.22. (i) Every ribbon algebra (T, σ ) in a braided category (M, ⊗, a, l, r, c, 1) can be seen as an algebra with ribbon element σ −1 in the opposite braided category of M. Recall that the opposite of the braided category M is (M, ⊗, a, l, r, c, 1), where −1 c X,Y = cY,X .
(ii) To every (para)-cocyclic object with invertible para-cocyclic morphism, there corresponds a (para)-cyclic object, namely its cyclic dual. Roughly speaking, the cyclic dual is obtained by interchanging the coface and codegeneracy morphisms and inverting the para-cocyclic operator. The interested reader can find the definition of the cyclic dual in [KR05]. The cyclic dual of Z ∗ (ST,σ −1 , Id T ) is, modulo a sign in the formula of wn , the cyclic object in [AM, Theorem 4]. Note that, via the identification T ⊗ 1 ∼ = T , the identity morphism Id T can be regarded as a transposition map in WST,σ −1 . Thus, for an arbitrary w in WST,σ −1 , the cyclic dual of Z ∗ (ST,σ −1 , w) may be interpreted as a generalization of cyclic homology introduced in [AM]. Other examples of para-cocyclic objects, obtained as applications of Theorem 1.10, will be discussed in Theorems 2.4 and Theorem 2.7. 1.E. The dual construction. In this section we turn to studying the situation dual to that in Sect. 1.C, i.e. application of Theorem 1.10 to the opposite categories C op and Mop . By Mop we mean the category with the same classes of objects and morphisms in M, with composition opposite to that in M. Note that any diagram expressing an identity of morphisms in M, yields a diagram in Mop , by interchanging the top and the bottom. In particular, a comonad on M is a monad on Mop . That is, a triple (Tl , Tl , ε Tl ), consisting of a functor Tl : M → M and natural transformations Tl : Tl → Tl 2 and ε Tl : Tl → I d M . Their compatibility axioms are obtained by reversing the arrows in the first two diagrams in Fig. 2. For two comonads (Tl , Tl , ε Tl ) and (Tr , Tr , ε Tr ) on a category M, a dual distributive law is a distributive law for the monads Tl and Tr on Mop . That is, a natural transformation t : Tl Tr → Tr Tl such that the relations encoded in the up-down mirror images of the diagrams in Fig. 3 hold. To dualize admissible septuples we need two comonads Tl and Tr on a category M, a dual distributive law t : Tl Tr → Tr Tl , a covariant functor : M → C and a natural transformation i : Tr → Tl that satisfy the identities εTl ◦ i = εTr
and
Tl ◦ i = i Tl ◦ t ◦ i Tr ◦ Tr .
(1.17)
Such a dual admissible septuple S 0 = (M, C, Tl , Tr , , t, i) determines a simplicial object Z ∗ (S 0 , X ) in C, which in degree n is given by Z n (S 0 , X ) = Tl n+1 X . Its face maps dk : Tl n+1 X → Tl n X and degeneracy maps sk : Tl n+1 X → Tl n+2 X are dk := Tl k εTl Tl n−k X,
sk := Tl k Tl Tl n−k X,
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for any k ∈ {0, . . . , n}. An arrow w : Tl X → Tr X in M is a transposition morphism with respect to the dual admissible septuple S 0 if, and only if εTr X ◦ w = εTl X
and
Tr X ◦ w = Tr w ◦ t X ◦ Tl w ◦ Tl X.
(1.18)
Morphisms between transpositions maps can be easily defined by duality. The category of transposition maps with respect to S 0 will be denoted by WS 0 . Note that t can be lifted to a natural transformation t n : Tl n Tr → Tr Tl n , t 0n := t Tl n−1 ◦ Tl t Tl n−2 ◦ · · · ◦ Tl n−2 t Tl ◦ Tl n−1 t.
(1.19)
Now we can state, for future reference, the dual of Theorem 1.10. Theorem 1.23. Consider a dual admissible septuple S 0 as above and a transposition morphism w : Tl X → Tr X in WS 0 . The simplex Z ∗ (S 0 , X ) is para-cyclic with paracyclic morphism wn := i Tl n X ◦ t 0n X ◦ Tl n w.
(1.20)
We shall denote this cyclic object by Z ∗ (S 0 , w). For a morphism f : (X, w) → (X , w ) in WS 0 , the morphisms Tl n+1 f : Z n (S 0 , w) → Z n (S 0 , w ) determine a morphism of para-cyclic objects. Dually to Example 1.12, we have the following Example 1.24. Let T = (T, , ε) be a comonad on a category M and t : T T → T T a dual distributive law. Assume that t satisfies the Yang-Baxter relation 1.11 and t ◦t ◦ = . Then T 0 = (T, t ◦ , ε) is a comonad, and t can be regarded as a distributive law T 0 T → T T 0 . Furthermore, the datum S := (M, M, T 0 , T , IdM , t, Id T ) is a dual admissible septuple. For any object X in M, the identity morphism Id T X is a transposition morphism. The corresponding para-cyclic morphism is tn0 in (1.19). Note that if in addition t is an invertible morphism in M then its properties assumed above are equivalent to the premises in [Šk, Theorem 1]. Let R be an algebra over a commutative ring K. It was recalled in Example 1.2 that R-Mod-R, the category of R-bimodules, is monoidal with respect to the tensor product ⊗ R and unit object R. By definition, an R-coring (C, , ) is a coalgebra in (R-Mod-R, ⊗ R , R). Proposition 1.25. Let (C, C , C ) be an R-coring. The following data: • the category C := Mod-K of K-modules and the category M := R-Mod-R of R-bimodules, • the comonads Tl := C ⊗ R (−) and Tr := (−) ⊗ R C on R-Mod-R, • the functor : R-Mod-R → Mod-K, M → R ⊗ R e M in Definition 1.13, • the trivial dual distributive law t X : C ⊗ R (X ⊗ R C) → (C ⊗ R X ) ⊗ R C, RC → C⊗ R X , given by the flip map x ⊗ R c → • the natural morphism i X : X ⊗ R x, c⊗ define a dual admissible septuple SC .
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Proof. We have to check the identities in (1.17). Recall that, for any R-bimodule X , the cyclic tensor product (X ) := R ⊗ R e X is isomorphic to the quotient of X modulo the K-submodule [X, R] generated by all commutators [x, r ], where x ∈ X and r ∈ R. Hence rx = xr , where z denotes the class of z in the quotient module, for any z ∈ X . To prove the first relation in (1.17), note that R c) = ε R c). εTl ◦ i (x ⊗ C (c)x = xε C (c) = ε Tr (x ⊗ For the coproduct in the coring C we use a Sweedler type notation, namely we write C (c) = c(1) ⊗ R c(2) , with implicit summation understood. A straightforward computation yields R c), R c) = c(1) ⊗ R c(2) ⊗ R x = i Tl ◦ t ◦ i Tr ◦ Tr (x ⊗ Tl ◦ i (x ⊗ for any x ∈ X and c ∈ C. Thus the second relation in (1.17) is also proven.
Let SC be the dual admissible septuple associated to an R-coring (C, C , C ). In this particular case, a map of R-bimodules w : C ⊗ R X → X ⊗ R C is a transposition map in WSC if, and only if, (X ⊗ R εC ) ◦ w = εC ⊗ R X and (X ⊗ R C ) ◦ w = (w ⊗ R C) ◦ (C ⊗ R w) ◦ (C ⊗ R X ).
(1.21)
Theorem 1.26. Let SC be the dual admissible septuple associated to an R-coring, as in Proposition 1.25. Let w : C ⊗ R X → X ⊗ R C be a transposition map in WSC , that is, a morphism of R-bimodules satisfying (1.21). Then there is a cyclic subobject R ∗+1 ⊗ R X whose cyclic structure is a restriction of the para-cyclic Z ∗ (SC , w) of C ⊗ R n+1 R n+1 morphism wn : C ⊗ ⊗R X → C ⊗ ⊗R X , −1 R w , wn := i C,...,C,X ◦ C ⊗R n ⊗ where i C,...,C,X is the isomorphism constructed in Remark 1.14. Proof. Proceed as in the proof of Theorem 1.16.
Dually to Corollary 1.17, the following holds. Corollary 1.27. Let SC be the dual admissible septuple associated to an R-coring, as in Proposition 1.25. Then the canonical isomorphism wC : C ⊗ R R → R ⊗ R C is a transposition map in WSC . The corresponding cyclic object Z ∗ (SC , wC ) has in degree R n+1 n the K-module Z n (SC , wC ) = C ⊗ . The face and degeneracy maps are for 0 ≤ k < n, R c1 ⊗ R · · · ⊗ R cn ) = c0 ⊗ R · · · ⊗ R C (ck )ck+1 ⊗ R · · · ⊗ R cn , dk (c0 ⊗ R c1 ⊗ R · · · ⊗ R cn−1 C (cn ), for k = n, R cn−2 ⊗ c0 ⊗ R c1 ⊗ R · · · ⊗ R · · · ⊗ R C (ck ) ⊗ R R cn ) = c0 ⊗ R ck−1 ⊗ R ck+1 ⊗ sk (c0 ⊗ · · · ⊗ R cn , for 0 ≤ k ≤ n. The cyclic operator is defined by R · · · ⊗ R c0 ⊗ R · · · ⊗ R cn ) = cn ⊗ R cn−1 . wn (c0 ⊗ A symmetrical version of the construction in Corollary 1.27 is described in [Ra, Prop. 3.1].
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Example 1.28. Let ϕ : S → T be a homomorphism of algebras over a commutative ring K. It determines Sweedler’s T -coring T ⊗ S T , where on T ⊗ S T we take the obvious T -bimodule structure. The coproduct T ⊗ S T and the counit T ⊗ S T are respectively defined by T ⊗ S T : T ⊗ S T → (T ⊗ S T ) ⊗T (T ⊗ S T ), T ⊗ S T : T ⊗ S T → T,
T ⊗ S T (t ⊗ S t ) = t ⊗ S 1T ⊗ S t , T ⊗ S T (t ⊗ S t ) = tt ,
where in the definition of T ⊗ R T we identified (T ⊗ S T )⊗T (T ⊗ S T ) and T ⊗ S T ⊗ S T . For S-bimodules T and X , let v : T ⊗ S X → X ⊗ S T be an S-bimodule map. For t ∈ T and x ∈ X we shall use the notation v(t ⊗ S x) = xv ⊗ S tv , where in the right hand side implicit summation is understood. Corollary 1.29. Let ϕ : S → T be a homomorphism of algebras over a commutative ring K, X be an S -bimodule and v : T ⊗ S X → X ⊗ S T be an S -bimodule map satisfying v ◦ (ϕ ⊗ S X ) = X ⊗ S ϕ and v ◦ (m T ⊗ S X ) = (X ⊗ S m T ) ◦ (v ⊗ S T ) ◦ (T ⊗ S v). ˆ S X whose face and There is a cyclic object Z ∗ (T /S, v), with Z n (T /S, v) = T ⊗ˆ S n+1 ⊗ degeneracy maps are S tn ⊗ t ⊗ ···⊗ S x) dk (t0 ⊗ S 1 S t ⊗ ···⊗ ···⊗ x, for 0 ≤ k < n, t t ⊗ t ⊗ t ⊗ = 0 S 1 S S kk+1 S S n S (tn )v t0 ⊗ S t1 ⊗ S · · · ⊗ S tn−1 ⊗ S (x)v , for k = n, S tn ⊗ S t1 ⊗ S · · · ⊗ S x) sk (t0 ⊗ S tk ⊗ S tn ⊗ S t1 ⊗ S · · · ⊗ S 1T ⊗ S · · · ⊗ S x, for 0 ≤ k ≤ n. = t0 ⊗ The para-cyclic map is given by S · · · ⊗ S x) = (tn )v ⊗ S t0 ⊗ S · · · ⊗ S (x)v . S tn ⊗ S tn−1 ⊗ vn (t0 ⊗ Proof. In terms of the map v, we can equip X ⊗ S T with a T -bimodule structure by t1 (x ⊗ S t)t2 = v(t1 ⊗ S x)tt2 . Moreover, v ⊗S T : T ⊗S X ⊗S T ∼ = (T ⊗ S T ) ⊗T (X ⊗ S T ) → (X ⊗ S T ) ⊗T (T ⊗ S T ) ∼ = X ⊗S T ⊗S T is a transposition map for Sweedler’s T -coring T ⊗ S T and the T -bimodule X ⊗ S T , in the sense of (1.18). Consequently, we can apply Corollary 1.27 to Sweedler’s coring in Example 1.28. One proves that the corresponding para-cyclic object has Z ∗ (T /S, v) as an underlying simplicial structure and v∗ as a para-cyclic map. Remarks 1.30. (i) Let us take X = S. The canonical isomorphism v : T ⊗ S S → S ⊗ S T satisfies the hypothesis of Corollary 1.29. The corresponding para-cyclic (in fact cyclic) object was used in [JS] ¸ to define relative cyclic homology. Moreover, this cyclic object and the cocyclic object Z ∗ (ST , wT ) in Corollary 1.17 are (cyclic) dual to each other. In the particular case when R = k and ϕ = u T , we rediscover the cyclic object introduced
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by A. Connes in order to define the cyclic homology of an algebra, cf. [We, p. 330]. Thus Z ∗ (ST , wT ) is the cyclic dual of Connes’ cyclic object. (ii) Note that the construction of Z ∗ (T /S, v) can be performed for any algebra T S with ⊗, the tensor in a symmetric monoidal category M, by replacing everywhere ⊗ product in M. Therefore, para-cyclic objects in [Kay06] and [HKRS2] are examples of this type. (iii) If w : X ⊗ S T → T ⊗ S X is an invertible S -bimodule map satisfying (1.12), then v = w −1 satisfies the relations in Corollary 1.29. Conversely, in the case when the morphism v in the above construction is invertible, then its inverse is a transposition map in the sense of (1.12). As a matter of fact, the corresponding para-cocyclic object in Theorem 1.16 is a cyclic dual of the para-cyclic object in Corollary 1.29. This suggests a categorical approach to cyclic duality, details of which will be studied elsewhere. Other examples of para-cyclic objects, obtained as applications of Theorem 1.23, will be discussed in Theorems 2.9 and Theorem 2.11. 2. Cyclic (Co)homology of Bialgebroids In this section we apply the categorical framework, obtained in Sect. 1, to examples provided by (co)module algebras and (co)module corings of bialgebroids, and analyze the structure of the resulting para-(co)cyclic objects.
2.A. (Co)module algebras of bialgebroids. In this section we consider admissible septuples ST , coming from a K-algebra homomorphism ϕ : R → T as in Proposition 1.15. As we have seen in Proposition 1.15, ST determines a cosimplex
R X, Z n (ST , X ) = T ⊗ R n+1 ⊗
(2.1)
for any R-bimodule X . Coface and codegeneracy maps are given by R n−k R ϕ ⊗ R X and R R T ⊗ ⊗ dk = T ⊗ R k ⊗ sk = T ⊗ R k ⊗ R T ⊗ R n−k ⊗ R X, ×m T ⊗
(2.2)
where k = 0, . . . , n and m T denotes the multiplication map T ⊗ R T → T . Furthermore, by Theorem 1.16, the cosimplex Z ∗ (ST , X ) is para-cocyclic provided that there is a transposition map w : X ⊗ R T → T ⊗ R X . Conditions (1.12) characterizing a transposition map are reminiscent of some of the axioms of an entwining structure (over an algebra R), cf. [BB, Sect. 2.3]. Main examples of entwining structures over non-commutative algebras arise from Doi-Koppinen data of bialgebroids (in the sense of [BCM]). In a similar manner, the aim of this section is to construct canonical transposition maps in the case when T is a (co)module algebra of an R-bialgebroid B and X is a B-(co)module. Bialgebroids can be thought of as a generalization of bialgebras to arbitrary, noncommutative base algebras. The first form of the structure that is known today as a left bialgebroid was introduced by Takeuchi in [Ta] under the name × R -bialgebra. Another definition and the name ‘bialgebroid’ was proposed by Lu in [Lu]. The two definitions were proven to be equivalent in [BMi]. ‘Left’ and ‘right’ versions of bialgebroids were defined in [KSz].
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Definition 2.1. Consider an algebra R over a commutative ring K. A left bialgebroid B over R consists of the data (B, ξ, ζ, , ). Here B is a K-algebra and ξ and ζ are K-algebra homomorphisms R → B and R op → B, respectively, such that their ranges are commuting subalgebras in B. In terms of the maps ξ and ζ , B can be equipped with an R-bimodule structure as r1 · b · r2 := ξ(r1 )ζ (r2 )b,
for r1 , r2 ∈ R and b ∈ B.
By definition, the coproduct : B → B ⊗ R B and the counit : B → R equip this bimodule with an R-coring structure. Between the algebra and coring structures of B the following compatibility axioms are required. For the coproduct we introduce the index notation (b) = b(1) ⊗ R b(2) , where implicit summation is understood: (i) b(1) ζ (r ) ⊗ R b(2) = b(1) ⊗ R b(2) ξ(r ), for r ∈ R and b ∈ B. ⊗ b b , for b, b ∈ B. (ii) (1 B ) = 1 B ⊗ R 1 B and (bb ) = b(1) b(1) R (2) (2) (iii) (1 B ) = 1 R and (bb ) = bξ((b )) , for b, b ∈ B. Axiom (i) in Definition 2.1 needs to be imposed in order for the second condition in axiom (ii) to make sense. Axiom (iii) implies that also (bb ) = bζ ((b )) , for b, b ∈ B. It follows by the R-module map properties, unitality and multiplicativity of the coproduct that ξ(r1 )ζ (r2 )bξ(r3 )ζ (r4 ) = ξ(r1 )b(1) ξ(r3 ) ⊗ R ζ (r2 )b(2) ζ (r4 ), (2.3) for r1 , r2 , r3 , r4 ∈ R and b ∈ B. Since the coproduct is coassociative, the SweedlerHeynemann index notation can be used. That is, for the iterated power of the coproduct we write (⊗ R B ⊗ R · · ·⊗ R B)◦· · ·◦(⊗ R B)◦(b) = b(1) ⊗ R · · ·⊗ R b(n−1) ⊗ R b(n) , for any positive integer n and b ∈ B. Note that the axioms in Definition 2.1 are not invariant under changing the multiplication in B to the opposite multiplication. Definition 2.1 has a symmetrical counterpart, known as a right bialgebroid. For the details we refer to [KSz]. Definition 2.1 is motivated by the following result of Schauenburg. Consider two algebras R and B over a commutative ring K and two K-algebra homomorphisms ξ : R → B and ζ : R op → B, whose ranges are commuting subalgebras of B. Clearly, in this setting any (left or right) B module can be equipped with an R bimodule structure using the maps ξ and ζ . For example, for a left B-module V with action : B⊗K V → V , one can define an R-bimodule structure as r1 ⊗K v ⊗K r2 → ξ(r1 )ζ (r2 ) v. With respect to the resulting R-actions, B-module maps are R-bimodule maps. That is, there exists a forgetful functor from the category of (left or right) B-modules to the category of R-bimodules. Theorem 2.2. [Sch98, Theorem 5.1] Consider two algebras R and B over a commutative ring K and two K-algebra homomorphisms ξ : R → B and ζ : R op → B, whose ranges are commuting subalgebras of B. There exists a right (resp. left) bialgebroid (B, ξ, ζ, , ) if and only if the forgetful functor from the category of right (resp. left) B-modules to the category of R-bimodules is strict monoidal. That is, R is a right (resp. left) B-module and the R-module tensor product of two right (resp. left) B-modules is a right (resp. left) B-module.
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Similarly to the case of a bialgebra, in Theorem 2.2 for a left R-bialgebroid (B, ξ, ζ, , ) the following B-actions are used on R, and on the R-module tensor product of two left B-modules V and W , respectively. b r := bξ(r )
and
b (v ⊗ R w) := b(1) v ⊗ R b(2) w,
(2.4)
for r ∈ R, v ⊗ R w ∈ V ⊗ R W and b ∈ B. It was proven in [Sch98, Theorem 5.1] that the diagonal action in the second equation in (2.4) is meaningful by axiom (i) in Definition 2.1. In light of Theorem 2.2, one can speak about right (resp. left) module algebras of a right (resp. left) bialgebroid B, i.e. about algebras in the monoidal category of right (resp. left) B-modules. Definition 2.3. Consider an algebra R over a commutative ring K and a left R-bialgebroid B. A left B-module algebra is a K-algebra and left B-module T , with B-action : B ⊗K T → T , such that the multiplication in T is R-balanced and b 1T = ξ (b) 1T
and
b (tt ) = (b(1) t)(b(2) t ),
(2.5)
for b ∈ B and t, t ∈ T . Note that for a left module algebra T of a left R-bialgebroid B = (B, ξ, ζ, , ), there is a canonical K-algebra homomorphism R → T , r → ξ(r ) 1T . Hence there is a corresponding admissible septuple as in Proposition 1.15. A left comodule of a left R-bialgebroid B = (B, ξ, ζ, , ) is defined as a left comodule of the underlying R-coring (B, , ). That is, a left R-module X , together with a left R-module map X → B ⊗ R X , x → x[−1] ⊗ R x[0] (where implicit summation is understood), satisfying coassociativity and counitality axioms. Note that a left B-comodule X can be equipped with an R-bimodule structure by introducing a right R-action x · r := x[−1] ξ(r ) · x[0] ,
for r ∈ R and x ∈ X.
(2.6)
With respect to the resulting bimodule structure, B-comodule maps are R-bimodule maps. In particular, the left B-coaction on X is an R-bimodule map in the sense that, for r, r ∈ R and x ∈ X , (r · x · r )[−1] ⊗ R (r · x · r )[0] = ξ(r )x[−1] ξ(r ) ⊗ R x[0] .
(2.7)
Furthermore, for any x ∈ X and r ∈ R, x[−1] ⊗ R x[0] · r = x[−1] ζ (r ) ⊗ R x[0] .
(2.8)
Theorem 2.4. Let R be an algebra over a commutative ring K and let B be a left bialgebroid over R. Consider a left B-module algebra T with B-action and a left B-comodule X with coaction x → x[−1] ⊗ R x[0] (where implicit summation is understood). Then a transposition map for the admissible septuple ST , associated via Proposition 1.15 to the K-algebra map R → T , r → ξ(r ) 1T , is given by
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w : X ⊗ R T → T ⊗ R X,
x ⊗ R t → x[−1] t ⊗ R x[0] .
(2.9)
Hence the cosimplex (2.1) admits a para-cocyclic structure R tn ⊗ R tn ⊗ R · · · ⊗ R x) = t1 ⊗ R · · · ⊗ R x[−1] t0 ⊗ R x[0] . wn (t0 ⊗
(2.10)
Proof. The map (2.9) is a well defined left R-module homomorphism by (2.7). Its right R-module map property follows by (2.8). Conditions (1.12) follow by definition (2.5) of a module algebra as it is shown below. Denote by ϕ the algebra homomorphism R → T , r → ξ(r ) 1T = ζ (r ) 1T . Omitting canonical isomorphisms R ⊗ R X ∼ =X∼ = X ⊗ R R, we have w ◦ (X ⊗ R ϕ) (x) = x[−1] 1T ⊗ R x[0] = ζ (x[−1] ) 1T ⊗ R x[0] = 1T ⊗ R (x[−1] ) · x[0] = 1T ⊗ R x = (ϕ ⊗ R X ) (x), (m T ⊗ R X ) ◦ (T ⊗ R w) ◦ (w ⊗ R T ) (x ⊗ R t ⊗ R t ) = (x[−1] t)(x[0][−1] t ) ⊗ R x[0][0] = (x[−1](1) t)(x[−1](2) t ) ⊗ R x[0] = x[−1] tt ⊗ R x[0] = w ◦ (X ⊗ R m T ) (x ⊗ R t ⊗ R t ). Analogously to (2.6), also a right comodule V of a right R-bialgebroid B = (B, ξ, ζ, , ) can be equipped with an R-bimodule structure by introducing a left R-action, for r ∈ R and v ∈ V, (2.11) r · v := v [0] · ξ(r )v [1] , where v → v [0] ⊗ R v [1] denotes the right B-coaction on V , with implicit summation understood. Hence there exists a forgetful functor from the category of right B-comodules to the category of R-bimodules. With this observation in mind, the next theorem follows by a symmetrical form of [Sch98, Prop. 5.6]. Theorem 2.5. Consider an algebra R over a commutative ring K and a right R-bialgebroid B. Then the forgetful functor from the category of right B-comodules to the category of R-bimodules is strict monoidal. That is, R is a right B-comodule and the R-module tensor product of two right B-comodules is a right B-comodule. Similarly to a bialgebra, in Theorem 2.5 for a right R-bialgebroid B = (B, ξ, ζ, , ) the following B-coactions on R, and on the R-module tensor product V ⊗ R W of two right B-comodules V and W , are used. R → R ⊗R B ∼ = B, r → ξ(r ) and V ⊗ R W → V ⊗ R W ⊗ R B, v ⊗ R w → v [0] ⊗ R w [0] ⊗ R v [1] w [1] . The coaction on V ⊗ R W is well defined by the right bialgebroid versions of properties (2.7) and (2.8), i.e. the identities (r · v · r )[0] ⊗ R (r · v · r )[1] = v [0] ⊗ R ξ(r )v [1] ξ(r ) r · v [0] ⊗ R v [1] = v [0] ⊗ R ζ (r )v [1] ,
and (2.12)
for r, r ∈ R and v ∈ V . Symmetrically, the forgetful functors from the category of left comodules of a right R-bialgebroid, and from the categories of right or left comodules of a left R-bialgebroid, to the category of R-bimodules are strict (anti-)monoidal. In light of Theorem 2.5, one can speak about right comodule algebras of a right bialgebroid B, i.e. about algebras in the monoidal category of right B-comodules.
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Definition 2.6. Consider an algebra R over a commutative ring K and a right R-bialgebroid B. A right B-comodule algebra is a K-algebra and right B-comodule T , with coaction t → t [0] ⊗ R t [1] , such that the multiplication in T is R-balanced and, for t, t ∈ T , [1] [0] 1[0] ⊗ R (tt )[1] = t [0] t [0] ⊗ R t [1] t [1] . T ⊗ R 1T = 1T ⊗ R 1T and (tt )
(2.13)
For example, the constituent algebra in a right bialgebroid B is itself a (so-called right regular) right B-comodule algebra via the coaction given by the coproduct in B. Note that the second condition in (2.13) is meaningful since the multiplication in T is R-balanced and the second condition in (2.12) holds. For a right comodule algebra T of a right R-bialgebroid B, there is a canonical K-algebra homomorphism R → T , r → r · 1T = 1T · r , in terms of which r · t = (r · 1T )t and t · r = t (r · 1T ). Hence there is a corresponding admissible septuple as in Proposition 1.15. Theorem 2.7. Let R be an algebra over a commutative ring K and let B be a right bialgebroid over R. Consider a right B-comodule algebra T , with coaction t → t [0] ⊗ R t [1] (where implicit summation is understood) and a right B-module X with action . Then a transposition map for the admissible septuple ST , associated via Proposition 1.15 to the K-algebra map R → T , r → r · 1T = 1T · r , is given by w : X ⊗ R T → T ⊗ R X,
x ⊗ R t → t [0] ⊗ R x t [1] .
Hence the cosimplex (2.1) admits a para-cocyclic structure R tn ⊗ R tn ⊗ R · · · ⊗ R x) = t1 ⊗ R · · · ⊗ R t0[0] ⊗ R x t0[1] . wn (t0 ⊗ Proof. The map w is a well defined R-bimodule homomorphism by (2.12). Conditions (1.12) follow by definition (2.13) of a comodule algebra. 2.B. (Co)module corings of bialgebroids. In this section we consider comonads Tl := C ⊗ R (−) on the category of R-bimodules as in Proposition 1.25, determined by a coring (C, , ) over an algebra R. Let be the functor in Definition 1.13. By Proposition 1.25, for any R-bimodule X , there is an associated simplicial module Z ∗ (SC , X ), which in degree n is given by
R X. Z n (SC , X ) = C ⊗ R n+1 ⊗
(2.14)
Face and degeneracy maps in degree n are R n−k R C ⊗ R C⊗ R X and dk = C ⊗ R k ⊗ ⊗ R k R n−k ⊗ ⊗ R C ⊗ R C R X, ⊗ ⊗ sk = C
(2.15)
for k ∈ {0, . . . , n}. As we have seen in Proposition 1.25, we can choose a second comonad Tr := (−) ⊗ R C on the category of R-bimodules, and a natural transformation i : R C → C ⊗ R (−), given by the flip map. Taking the trivial natural transformation (−) ⊗ t : C ⊗ R (−) ⊗ R C → C ⊗ R (−) ⊗ R C, the conditions in (1.17) hold. By Theorem 1.26, the simplex Z ∗ (SC , X ) is para-cyclic provided that there exists a morphism w : C ⊗ R X → X ⊗ R C, satisfying (1.21). Note the similarity of conditions (1.21) to some of the axioms of an entwining structure over R. Similarly to the way Doi-Koppinen data
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determine entwining structures, in this section we construct dual transposition maps in the case when C is a (co)module coring of a bialgebroid B and X is a B-(co)module. In light of (a symmetrical version of) Theorem 2.5, one can speak about left comodule corings of a left bialgebroid B, i.e. about coalgebras in the monoidal category of left B-comodules. Definition 2.8. Consider an algebra R over a commutative ring K and a left R-bialgebroid B = (B, ξ, ζ, , ). A left B-comodule coring is an R-coring and left B-comodule C, with one and the same underlying R-bimodule structure, such that for c ∈ C, ξ C (c) = ζ C (c[0] ) c[−1] and c[−1] ⊗ c[0](1) ⊗ c[0](2) R
R
= c(1)[−1] c(2)[−1] ⊗ c(1)[0] ⊗ c(2)[0] , R
R
(2.16)
where C is the counit and C : c → c(1) ⊗ R c(2) is the coproduct of C and c → c[−1] ⊗ R c[0] is the B-coaction on C. The second condition in (2.16) is meaningful by (2.7) and (2.8). Theorem 2.9. Let R be an algebra over a commutative ring K and let B be a left bialgebroid over R. Consider a left B-comodule coring C with coaction c → c[−1] ⊗ R c[0] , and a left B-module X with action . Then a transposition map w : C ⊗ R X → X ⊗ R C for the dual admissible septuple SC , associated via Proposition 1.25 to the R-coring C, is given by w(c ⊗ R x) := c[−1] x ⊗ R c[0] . Hence the simplex (2.14) admits a para-cyclic structure R cn ⊗ R cn−1 ⊗ R · · · ⊗ R x) = cn[0] ⊗ R c0 ⊗ R · · · ⊗ R cn[−1] x. wn (c0 ⊗ Proof. The map w is a well defined R-bimodule homomorphism by (2.7) and (2.8). Conditions (1.21) follow by definition (2.16) of a comodule coring. In light of Theorem 2.2, one can speak about right (resp. left) module corings of a right (resp. left) bialgebroid B, i.e. about coalgebras in the monoidal category of right (resp. left) B-modules. Definition 2.10. Consider an algebra R over a commutative ring K and a right R-bialgebroid B = (B, ξ, ζ, , ). A right B-module coring is an R-coring and right B-module C, with one and the same underlying R-bimodule structure, such that for c ∈ C and b ∈ B, C (c b) = ξ(C (c))b (c b)
(1)
⊗ R (c b)
(2)
=c
and (1)
b(1) ⊗ R c(2) b(2) ,
(2.17)
where C is the counit and C : c → c(1) ⊗ R c(2) is the coproduct in C, the symbol denotes the B-action on C and for the coproduct in B the index notation : b → b(1) ⊗ R b(2) is used, with implicit summation understood. For example, the constituent R-coring in a right R-bialgebroid B is itself a (so called right regular) right B-module coring via the action given by the product in B.
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Theorem 2.11. Let R be an algebra over a commutative ring K and let B be a right bialgebroid over R. Consider a right B-module coring C with action , and a right B-comodule X with coaction x → x [0] ⊗ R x [1] . Then a transposition map w : C ⊗ R X → X ⊗ R C, for the dual admissible septuple SC , associated via Proposition 1.25 to the R-coring C, is given by w(c ⊗ R x) := x [0] ⊗ R c x [1] . Hence the simplex (2.14) admits a para-cyclic structure, R cn ⊗ R cn−1 ⊗ R · · · ⊗ R x) = cn x [1] ⊗ R c0 ⊗ R · · · ⊗ R x [0] . (2.18) wn (c0 ⊗ Proof. The map w is a well defined R-bimodule homomorphism by (2.12). Conditions (1.21) follow by definition (2.17) of a module coring. 2.C. Stable anti-Yetter-Drinfel’d modules of × R -Hopf algebras. For a right module coring C and right comodule X of a right bialgebroid B = (B, ξ, ζ, , ) over a K-algebra R, there is a K-module isomorphism R X ∼ C ⊗ R n+1 ⊗ = C ⊗ R n+1 ⊗ R e X.
Assume that X has an additional left B-module structure. In this case, corresponding to the K-algebra homomorphism R e → B, r ⊗K r → ξ(r )ζ (r ) = ζ (r )ξ(r ), there is a R n+1 ⊗ R X → C ⊗ R n+1 ⊗B X , where C ⊗ R n+1 is understood canonical epimorphism C ⊗ to be a right B-module via the diagonal action (c1 ⊗ R · · · ⊗ R cn+1 ) b := c1 b(1) ⊗ R · · · ⊗ R cn+1 b(n+1) , given by the iterated coproduct in B. It is a well defined action by (a symmetrical version of) [Sch98, Theorem 5.1]. Lemma 2.12. Let R be an algebra over a commutative ring K and let B be a right bialgebroid over R. Consider a right B-module coring C and a left B-module right B-comodule X . Then the simplex in Theorem 2.11 projects to a simplex C ⊗ R n+1 ⊗B X . Proof. Since the coproduct C and the counit C of C are right B-module maps by definition, face and degeneracy maps of the simplex in Example 1.27 are right B-module maps with respect to the diagonal action. Hence we can take their tensor product with the identity map on X over the algebra B, yielding a simplex as stated. The task of this section is to find criteria for the cyclicity of the quotient simplex in Lemma 2.12. In order to do so, some restriction on the involved bialgebroid is needed. Definition 2.13. [Sch00, Theorem and Definition 3.5]. Let R be an algebra over a commutative ring K. A right R-bialgebroid B = (B, ξ, ζ, , ) is said to be a right × R -Hopf algebra provided that the map ϑ : B ⊗ R op B → B ⊗ R B,
b ⊗ R op b → bb(1) ⊗ R b(2)
(2.19)
is bijective. In the domain of the map in (2.19), R op -module structures are given by right and left multiplication by ζ (r ), for r ∈ R. In the codomain of the map in (2.19), R-module structures are given by right multiplication by ξ(r ) and ζ (r ), for r ∈ R. The notion of a × R -Hopf algebra extends that of a Hopf algebra. Indeed, if B is a bialgebra over a commutative ring R, with coproduct b → b(1) ⊗ R b(2) , then the map (2.19) is bijective if and only if B is a Hopf algebra. In this case the inverse is given in terms of the antipode S as ϑ −1 (b ⊗ R b ) := bS(b(1) ) ⊗ R b(2) .
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For an algebra R, consider a right × R -Hopf algebra B. Since the map ϑ in (2.19) is a left B-module map, its inverse is determined by the restriction ϑ −1 (1B ⊗ R b) =: b− ⊗ R op b+ , where implicit summation is understood. Lemma 2.14, which is essentially a symmetrical version of [Sch00, Prop. 3.7], collects properties of the map b → b− ⊗ R op b+ . Lemma 2.14. For an algebra R, consider a right × R -Hopf algebra B = (B, ξ, ζ, , ). Write (b) =: b(1) ⊗ R b(2) for the coproduct and in terms of the map (2.19) put b− ⊗ R op b+ := ϑ −1 (1B ⊗ R b). The following identities hold, for b, b ∈ B and r ∈ R. (i) b− b+ (1) ⊗ R b+ (2) = 1B ⊗ R b, (ii) b(1) b(2) − ⊗ R op b(2) + = 1B ⊗ R op b , b ⊗ op b b , (iii) (bb )− ⊗ R op (bb )+ = b− − R + + (iv) 1B − ⊗ R op 1B + = 1B ⊗ R op 1B , (v) b− ⊗ R op b+ (1) ⊗ R b+ (2) = b(1) − ⊗ R op b(1) + ⊗ R b(2) , (2) (vi) b− (1) ⊗ R b− ⊗ R op b+ = b+− ⊗ R b− ⊗ R op b++ , (vii) b = ζ (b− ) b+, (viii) b− b+ = ξ (b) , (x) ζ (r )b− ⊗ R op b+ = b− ⊗ R op b+ ζ (r ). Next Definition 2.15 extends [JS, ¸ Definition 4.1] or [HKRS1, Definition 2.1]. Definition 2.15. For an algebra R, consider a right × R -Hopf algebra B = (B, ξ, ζ, , ). Let X be a right B-comodule and left B-module. Denote the right B-coaction on X by x → x [0] ⊗ R x [1] , for x ∈ X (where implicit summation is understood) and denote the left B-action by b x, for b ∈ B and x ∈ X . We say that X is an anti-Yetter-Drinfel’d module provided that the following conditions hold. (i) The R-bimodule structures of X , underlying its module and comodule structures, coincide. That is, for x ∈ X and r ∈ R, x · r = ζ (r ) x
and
r · x = ξ(r ) x,
(2.20)
where x ·r denotes the right R-action on the right B-comodule X and r ·x is the canonical left R-action (2.11) coming from the right B-coaction. (ii) For b ∈ B and x ∈ X , (b x)[0] ⊗ R (b x)[1] = b(1) + x [0] ⊗ R b(2) x [1] b(1) − ,
(2.21)
where for the coproduct and the inverse of the map (2.19) the respective index notations (with implicit summation), (b) = b(1) ⊗ R b(2) , and ϑ −1 (1B ⊗ R b) = b− ⊗ R op b+ are used, for b ∈ B. The anti-Yetter-Drinfel’d module X is said to be stable if in addition, for any x ∈ X , x [1] x [0] = x.
(2.22)
We need to show that condition (ii) in Definition 2.15 is meaningful, i.e. the expression on the right hand side of (2.21) is well defined. This follows by the following Lemma 2.16. For an algebra R, consider a right × R -Hopf algebra B = (B, ξ, ζ, , ). Let X be a right B-comodule and left B-module. Keeping the notations in Definition 2.15, assume that axiom (i) in Definition 2.15 holds. Then the following hold:
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(1) Considering B as a left R-module via ζ , the R-module tensor product X ⊗ R B is a left B-module via the action b (x ⊗ R b) := b+ x ⊗ R bb− .
(2) X ⊗ R B is a B-B op bimodule, via the left B-action in part (i) and the right B op -action (x ⊗ R b) b := x ⊗ R b b. (3) For any elements x ∈ X and r ∈ R, ξ(r ) x [0] ⊗ R x [1] = x [0] ⊗ R x [1] ζ (r ). Proof. By the first condition in (2.20) and Lemma 2.14 (ix), there is a well defined map B ⊗K (X ⊗ R B) → X ⊗ R B,
b ⊗K (x ⊗ R b) → b+ x ⊗ R b b− .
It is an associative left B-action by part (iii) of Lemma 2.14 and it is unital by part (iv). This proves claim (1). Claim (2) is obvious. Claim (3) follows by the following computation, for r ∈ R and x ∈ X : ξ(r ) x [0] ⊗ R x [1] = ξ(r ) x [0] ⊗ R x [1] = x [0][0] · (ζ (r )x [0][1] ) ⊗ R x [1] = x [0] ⊗ R x [1](2) ζ (ζ (r )x [1](1) ) = x [0] ⊗ R ζ (r )x [1] = x [0] ⊗ R x [1] ζ (r ). In the first equality we used that, by unitality and the right R-module map property of the coproduct, the map (2.19) satisfies ϑ(1B ⊗ R op ξ(r )) = 1B ⊗ R ξ(r ). Hence ξ(r )− ⊗ R op ξ(r )+ = 1B ⊗ R op ξ(r ). The second equality follows by the second condition in (2.20). The third equality follows by coassociativity of the B-coaction on X . The fourth equality follows by counitality of and the right bialgebroid version of (2.3), i.e. the identity ξ(r1 )ζ (r2 )bξ(r3 )ζ (r4 ) = ζ (r2 )b(1) ζ (r4 ) ⊗ R ξ(r1 )b(2) ξ(r3 ), (2.23) for r1 , r2 , r3 , r4 ∈ R and b ∈ B.
Using the notations in Lemma 2.16, the right-hand side of (2.21) is equal to the well defined expression b(1) x [0] ⊗ R x [1] b(2) . Remark 2.17. Note that a Hopf algebra H over a commutative ring K is a ×K -Hopf algebra. The bialgebroid structure is given by the equal source and target maps K → H , κ → κ1 H , and the coproduct and counit in H . The canonical map (2.19) has an inverse ϑ −1 (h ⊗K h) = h S(h (1) )⊗K h (2) , where h → h (1) ⊗K h (2) is the usual Sweedler index notation for the coproduct, with implicit summation understood. That is, h − ⊗K h + = S(h (1) ) ⊗K h (2) . Clearly, in this case (2.20) becomes the trivial condition xκ = (κ1 H ) x = κ x, for κ ∈ K and any element x of a left H -module right H -comodule X . This condition simply expresses the requirement that the left and right K-actions on X are equal, and are induced by the H -module structure. The second condition (2.21) in Definition 2.15 reduces to (b x)[0] ⊗ (b x)[1] = b(2) x [0] ⊗ b(3) x [1] S(b(1) ), for b ∈ H and x ∈ X , which is the defining property of a (left-right) anti-Yetter-Drinfel’d module X of a Hopf algebra H in [JS, ¸ Def. 4.1] or [HKRS1, Def. 2.1]. So we conclude that Def. 2.15 generalizes these definitions.
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Example 2.18. For an algebra R, consider a right × R -Hopf algebra B = (B, ξ, ζ, , ). Note that left B-actions on R, satisfying ζ (r ) r = rr , are in bijective correspondence with maps χ : B → R, obeying the following properties, for r ∈ R, b, b ∈ B: χ (1 B ) = 1 R . χ (ζ (r )b) = χ (b)r χ (bb ) = χ bζ (χ (b )) , Indeed, in terms of such a map χ , one can put b r := χ (bζ (r )). Furthermore, right B-coactions on R, with underlying right regular R-module structure, are in bijective correspondence with grouplike elements in B, i.e. g ∈ B such that (g) = g ⊗ R g and (g) = 1 R . Indeed, in terms of a grouplike element g, a right B-coaction on R is given by r → 1 R ⊗ R gξ(r ). One checks that the left B-module determined by χ and the right B-comodule determined by g combine into an anti-Yetter-Drinfel’d module on R if and only if, for r ∈ R and b ∈ B, ξ(r )g = χ ξ(r ) , and gξ χ (b) = b(2) gb(1) − ζ χ (b(1) + ) . The anti-Yetter-Drinfel’d module R is stable if in addition χ gξ(r ) = r , for all r ∈ R. The pair (χ , g) with these properties generalizes the notion of a modular pair in involution for a Hopf algebra in [CM01] or a weak Hopf algebra in [V]. Proposition 2.19. Let B be a right × R -Hopf algebra over an algebra R. Consider a right B-module coring C with B-action : (1) For an anti-Yetter-Drinfel’d module X of B, the para-cyclic object in Theorem 2.11 projects to a para-cyclic structure on C ⊗ R n+1 ⊗B X . (2) For a stable anti-Yetter-Drinfel’d module X of B, the para-cyclic object C ⊗ R n+1 ⊗B X in part (1) is cyclic, in which case it will be denoted by Z ∗ (C, M). Proof. We need to show that the composite map R n+1 C⊗ ⊗R X
wn
R n+1 / C⊗ ⊗R X
/ / C ⊗ R n+1 ⊗ X B
is B-balanced, i.e. that cn b(n+1) x [1] ⊗ c0 b(1) ⊗ · · · ⊗ cn−1 b(n) ⊗ x [0] R R R B = cn (b x)[1] ⊗ c0 ⊗ · · · ⊗ cn−1 ⊗ (b x)[0] , R
R
R
B
for b ∈ B, x ∈ X and c0 ⊗ R · · · ⊗ R cn ∈ C ⊗ R n+1 . By counitality of the coproduct in B, the left hand side is equal to cn b(n+2) x [1] ⊗ R c0 b(2) ζ (b(1) ) ⊗ R c1 b(3) ⊗ R · · · ⊗ R cn−1 b(n+1) ⊗B x [0] = cn b(n+2) x [1] ξ (b(1) ) ⊗ R c0 b(2) ⊗ R c1 b(3) ⊗ R · · · ⊗ R cn−1 b(n+1) ⊗B x [0] = cn b(n+2) x [1] b(1) − b(1) + ⊗ R c0 b(2) ⊗ R c1 b(3) ⊗ R · · · ⊗ R cn−1 b(n+1) ⊗B x [0] = cn b+ (n+2) x [1] b− b+ (1) ⊗ R c0 b+ (2) ⊗ R · · · ⊗ R cn−1 b+ (n+1) ⊗B x [0] = cn b+ (2) x [1] b− ⊗ R c0 ⊗ R · · · ⊗ R cn−1 b+ (1) ⊗B x [0] = cn b+ (2) x [1] b− ⊗ R c0 ⊗ R · · · ⊗ R cn−1 ⊗B b+ (1) x [0] = cn (b x)[1] ⊗ R c0 ⊗ R · · · ⊗ R cn−1 ⊗B (b x)[0] .
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The second equality follows by part (viii) in Lemma 2.14 and the third one follows by part (v). The last equality is a consequence of Lemma 2.14 (v) and (2.21). This proves that the para-cyclic map wn in Theorem 2.11 factors to a map w n : C ⊗ R n+1 ⊗B X → ⊗ n+1 R C ⊗B X , hence it defines a para-cyclic structure on the simplex in Lemma 2.12. This completes the proof of part (1). Furthermore, ( wn )n+1 takes an element c0 ⊗ R · · · ⊗ R cn ⊗B x ∈ C ⊗ R n+1 ⊗B X to c0 x [1](1) ⊗ · · · ⊗ cn x [1](n+1) ⊗ x [0] = c0 ⊗ · · · ⊗ cn x [1] ⊗ x [0] R R B R R B = c0 ⊗ · · · ⊗ cn ⊗ x [1] x [0] . R
R
B
Hence if X is a stable anti-Yetter-Drinfel’d module, i.e. condition (2.22) holds, then ( wn )n+1 is the identity map. Thus we have claim (2) proven. Remark 2.20. Let B = (B, ξ, ζ, , ) be a right bialgebroid over an algebra R. Consider B as an R op -bimodule via right multiplications by ξ and ζ . This bimodule has an R op coring structure with counit and coproduct cop : b → b(2) ⊗ R op b(1) , co-opposite to . Together with the opposite algebra B op , they constitute a left R op -bialgebroid op op Bcop = (B op , ξ, ζ, cop , ). A right B-module algebra T determines a left Bcop -module algebra T op , canonically. Furthermore, a right B-comodule X can be looked at as a op op left Bcop -comodule. Application of Theorem 2.4 to the left R op -bialgebroid Bcop , the op op left Bcop -module algebra T op and the left Bcop -comodule X , yields a para-cocyclic cosimplex that has in degree n, op ⊗ R op n+1 R op X ∼ R X, ⊗ T = T ⊗ R n+1 ⊗ where the isomorphism is given by reversing the order of the factors, i.e. R op tn ⊗ R op t1 ⊗ R op · · · ⊗ R op x t0 ⊗
R t0 ⊗ R tn−1 ⊗ R · · · ⊗ R x. tn ⊗
→
RT ⊗ R X are the maps in R · · · ⊗ The resulting coface and codegeneracy maps on T ⊗ (2.2) and the para-cocyclic map comes out as R · · · ⊗ R x R tn ⊗ t0 ⊗
→
R t0 ⊗ R · · · ⊗ R x [0] . R tn−1 ⊗ tn x [1] ⊗
(2.24)
By the right B-module map property of the maps ϕ : R → T , r → 1T ξ(r ), and the multiplication map m T : T ⊗ R T → T , the coface and codegeneracy maps (2.2) project to T ⊗ R n+1 ⊗B X . Note moreover that the para-cocyclic map (2.24) and the para-cyclic map (2.18) are of the same form. Hence it follows by the computation in the proof of Proposition 2.19 that also the para-cocyclic map (2.24) projects to T ⊗ R n+1 ⊗B X , whenever X is an anti-Yetter-Drinfel’d module of B. That is, in this case T ⊗ R ∗+1 ⊗B X is a para-cocyclic object, which is cocyclic if the anti-Yetter-Drinfel’d module X is stable. 2.D. Galois extensions of × R -Hopf algebras. For a right × R -Hopf algebra B over an algebra R, consider a right comodule algebra T with coaction t → t [0] ⊗ R t [1] (where implicit summation is understood). The subalgebra S of coinvariants in T consists of those elements s ∈ T for which s [0] ⊗ R s [1] = s ⊗ R 1B . To the inclusion map S → T one can associate a cyclic simplex
Z ∗ (T /S) = T ⊗ S ∗+1
(2.25)
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as in Corollary 1.29. On the other hand, it follows by Proposition 2.19 that, regarding B as a right B-module coring, for any stable anti-Yetter-Drinfel’d module X of B there is a cyclic simplex Z ∗ (B, X ) = B ⊗ R ∗+1 ⊗B X,
(2.26)
where B ⊗ R n+1 is understood to be a right B-module via the diagonal action. In this section, under the additional assumption that T is a B-Galois extension of S, we construct a stable anti-Yetter-Drinfel’d module X := T /{ st − ts | s ∈ S, t ∈ T } ∼ = S ⊗Se T of B, such that the cyclic simplices (2.25) and (2.26) are isomorphic. This extends [JS, ¸ Theorem 3.7]. In a right comodule algebra T of a right bialgebroid B over an algebra R, we denote the coaction by : t → t [0] ⊗ R t [1] , where implicit summation is understood. For the iterated power of the coaction we write ( ⊗ R B ⊗ R n−1 ) ◦ · · · ◦ ( ⊗ R B) ◦ (t) =: t [0] ⊗ R · · · ⊗ R t [n−1] ⊗ R t [n] . Definition 2.21. Let B be a right bialgebroid over an algebra R. A right B-comodule algebra T is said to be a B-Galois extension of its coinvariant subalgebra S if the canonical map can : T ⊗ S T → T ⊗ R B,
t ⊗ S t → t t [0] ⊗ R t [1]
(2.27)
is bijective. For example, if B = (B, ξ, ζ, , ) is a right × R -Hopf algebra, then the right regular B-comodule algebra is a B-Galois extension of the coinvariant subalgebra ζ (R) ∼ = R op . Let B be a right R-bialgebroid and T a right B-comodule algebra. It follows by the right R-module map property of a right B-coaction and (2.13) that, for a coinvariant s ∈ S and r ∈ R, ((1T · r )s)[0] ⊗ R ((1T · r )s)[1] = s ⊗ R ξ(r ) = (s(1T · r ))[0] ⊗ R (s(1T · r ))[1] . Hence, applying the counit of B to the second factor on both sides, we conclude that the elements s ∈ S commute with 1T · r , for all r ∈ R. Hence T ⊗ R B is a right S-module, with action (t ⊗ R b) · s := ts ⊗ R b. Consider a right × R -Hopf algebra B over an algebra R, and a B-Galois extension S ⊆ T . As in Lemma 2.14, in terms of the maps (2.19) and (2.27), introduce the index notations can−1 (1T ⊗ R b) =: b{−} ⊗ S b{+}
and
ϑ −1 (1B ⊗ R b) =: b− ⊗ R op b+ ,(2.28)
for b ∈ B, where in both cases implicit summation is understood. The following lemma is a right bialgebroid version of [H, Lemma 4.1.21]. It extends Lemma 2.14 (vi). Lemma 2.22. Consider a right × R -Hopf algebra B = (B, ξ, ζ, , ) over an algebra R, and a B-Galois extension S ⊆ T . Using the notations in (2.28), for any b ∈ B the following pentagonal equation holds in (T ⊗ R B) ⊗ S T : b{−}[0] ⊗ R b{−}[1] ⊗ S b{+} = b+ {−} ⊗ R b− ⊗ S b+ {+} .
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Proof. The second condition in (2.12) implies that there is a well defined bijection can13 : (T ⊗ R B) ⊗ S T → T ⊗ R (B ⊗ R op B), (t ⊗ R b) ⊗ S t → t t [0] ⊗ R (b ⊗ R op t [1] ), where in the R op -module tensor product B ⊗ R op B the R op -actions R op ⊗K B ⊗K R op → B, r1 ⊗K b ⊗K r2 → ζ (r2 )bζ (r1 ) are used, and B ⊗ R op B is meant to be a left R-module via the action r ·(b⊗ R op b ) = b⊗ R op b ζ (r ). A straightforward computation using (2.13) shows that the S-bimodule map and right R-module map (2.27), and the left R-module map (2.19) satisfy (can ⊗ R B) ◦ (T ⊗ S can) = (T ⊗ R ϑ) ◦ can13 ◦ (can ⊗ S T ). Hence, by bijectivity of all involved maps, −1 (can ⊗ S T ) ◦ (T ⊗ S can−1 ) = can−1 13 ◦ (T ⊗ R ϑ ) ◦ (can ⊗ R B).
Application of this identity to 1T ⊗ S 1T ⊗ R b yields the claim.
For an algebra extension S ⊆ T , T has a canonical S-bimodule structure. Hence application of the functor : S-Mod-S → Mod-K in Definition 1.13 to T yields a S T . K-module T ∼ = S⊗ Proposition 2.23. Consider a right × R -Hopf algebra B over an algebra R, and a B-Galois extension S ⊆ T . Then the quotient S T TS := S ⊗
(2.29)
is a stable anti-Yetter-Drinfel’d module. Proof. Since the S-, and R-actions on T commute (cf. second paragraph following Definition 2.21), there is a unique R-bimodule structure on TS such that the epimorphism pT : T → TS is an R-bimodule map. Furthermore, by the S-bimodule map property of the coaction : t → t [0] ⊗ R t [1] in T , the map ( pT ⊗ R B)◦ : T → TS ⊗ R B coequalizes the left and right S-actions on T . Hence there exists a unique right B-comodule structure on TS such that pT : T → TS is a right B-comodule map. The algebra map S → T equips T with an S-bimodule structure. The center S of the S-bimodule T ⊗ T is an algebra, with multiplication ( (T ⊗ T ) S S i ui ⊗S u i )( j v j ⊗ S v j ) = i, j v j u i ⊗ S u i v j . Recall from [JS, ¸ Sect. 2.2] that for any any S M ∼ T -bimodule M, the quotient S ⊗ = M/{ s · m − m · s | s ∈ S, m ∈ M } is a left S (T ⊗ S T ) -module via the action ( u i ⊗ S u i ) ⊗K p M (m) → p M (u i mu i ), (T ⊗ S T ) S ⊗K M S → M S , i
i
S M denotes the canonical epimorphism. In particular, TS is a left where p M : M → S ⊗ (T ⊗ S T ) S -module. On the other hand, for a Galois extension S ⊆ T by a right R-bialgebroid B, using the notation in (2.28), the map B → (T ⊗ S T ) S ,
b → b{−} ⊗ S b{+}
(2.30)
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is an algebra homomorphism. Indeed, by the S-bimodule map property of the coaction on T it follows that can(sb{−} ⊗ S b{+} ) = s ⊗ R b = can(b{−} ⊗ S b{+} s). Hence, by bijectivity of can, b{−} ⊗ S b{+} ∈ (T ⊗ S T ) S , for all b ∈ B. The map (2.30) is unital and multiplicative since by (2.13) for all b, b ∈ B, can(1T ⊗ S 1T ) = 1T ⊗ S 1B
and
can(b{−} b{−} ⊗ S b{+} b{+} ) = 1T ⊗ R bb ,
and can is bijective. This proves that TS is a left B-module, with a so-called MiyashitaUlbrich type action b pT (t) = pT (b{+} tb{−} ).
(2.31)
It remains to check the compatibility conditions in Definition 2.15 between the B-module and B-comodule structures on TS . It follows by the R-bimodule map property of (2.27) that ζ (r ){−} ⊗ S ζ (r ){+} = r · 1T ⊗ S 1T and ξ(r ){−} ⊗ S ξ(r ){+} = 1T ⊗ S 1T · r.
(2.32)
Hence ζ (r ) pT (t) = pT t (r · 1T ) = pT (t · r ) = pT (t) · r and ξ(r ) pT (t) = pT (1T · r )t = pT (r · t) = r · pT (t). Furthermore, for b ∈ B and t ∈ T ,
(b pT (t))[0] ⊗ R (b pT (t))[1] = pT b{+}[0] t [0] b{−}[0] ⊗ R b{+}[1] t [1] b{−}[1] = pT b(1){+} t [0] b(1){−}[0] ⊗ R b(2) t [1] b(1){−}[1] {+} {−} ⊗ R b(2) t [1] b(1) − = pT b(1) + t [0] b(1) + = b(1) + pT (t)[0] ⊗ R b(2) pT (t)[1] b(1) − .
The first equality follows by (2.31), the second condition in (2.13) and the comodule map property of pT . The second equality is a consequence of the right B-comodule map property of (2.27), hence of the map (2.30). The third equality results from the application of Lemma 2.22. The last equality follows by (2.31) and the comodule map property of pT again. Thus we proved that TS is an anti-Yetter-Drinfel’d module. Finally, by the comodule map property of pT and the identity for t ∈ T, (2.33) t [0] t [1]{−} ⊗ S t [1]{+} = can−1 can(1T ⊗ S t) = 1T ⊗ S t, it follows that pT (t)[1] pT (t)[0] = t [1] pT (t)[0] = pT t [1]{+} t [0] t [1]{−} = pT (t). That is, the anti-Yetter-Drinfel’d module TS is stable.
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Lemma 2.24. For an algebra R over a commutative ring K, consider a right R-bialgebroid B and a B-Galois extension S ⊆ T . Using the notation in (2.28), for any non-negative integer n there exist S-bimodule isomorphisms T ⊗ S n+1 ∼ = T ⊗ R B⊗ R n , αn (t0 ⊗ · · · ⊗ tn ) = t0 t1[0] t2[0] · · · tn[0] ⊗ t1[1] t2[1] · · · tn[1] ⊗ t2[2] t3[2] · · · tn[2] ⊗ S
S
R
R
R
[n−1] [n−1] ⊗ tn[n] , · · · ⊗ tn−1 tn R
R
αn−1 (t ⊗ b1 ⊗ · · · ⊗ bn ) = t b1 {−} ⊗ b1 {+} b2 {−} ⊗ b2 {+} b3 {−} ⊗ R
R
R
S
S
S
· · · ⊗ bn−1 {+} bn {−} ⊗ bn {+} . S
S
S n+1 Projections of the above isomorphisms yield K-module isomorphisms αn : T ⊗ → ⊗ n TS ⊗ R B R , where TS is the R-bimodule (2.29).
Proof. It follows by the S-bimodule map property of the right B-coaction on T that αn is a well defined S-bimodule map. The to-be-inverse αn−1 is well defined by (2.32) and Lemma 2.14 (iii). We prove by induction that the maps αn and αn−1 are mutual inverses. For n = 0, both α0 and α0−1 are equal to the identity map on T , hence they are mutual inverses. It follows by the second condition in (2.13) that, for all values of n, −1 = (T ⊗ S n ⊗ S can−1 ) ◦ (αn−1 ⊗ R B). αn+1 = (αn ⊗ R B) ◦ (T ⊗ S n ⊗ S can) and αn+1
Hence if αn−1 is the inverse of αn then αn+1 is also an S-bimodule isomorphism with −1 inverse αn+1 . Applying the functor in Definition 1.13, from the category of S-bimodules to the category of K-modules, it takes αn to the required K-module isomorphism αn : S n+1 → TS ⊗ R B ⊗ R n . T⊗ Lemma 2.25. Let B be a right × R -Hopf algebra over an algebra R. For the inverse of the canonical map (2.19) use the index notation in (2.28). Then, for any non-negative integer n, there exist right B-module isomorphisms B ⊗ R n ⊗ R op B ∼ = B ⊗ R n+1 , βn b1 ⊗ R · · · ⊗ R bn ⊗ R op b = b1 b(1) ⊗ R · · · ⊗ R bn b(n) ⊗ R b(n+1) , (1) (n) βn−1 b1 ⊗ R · · · ⊗ R bn ⊗ R b = b1 b− ⊗ R · · · ⊗ R bn b− ⊗ R op b+ . Proof. For any right B-module N , the B-bimodule isomorphism (2.19) induces a right B-module isomorphism N ⊗B ϑ : N ⊗B B⊗ R op B ∼ = N ⊗ R op B → N ⊗B B⊗ R B ∼ = N ⊗R B. Consider B ⊗ R n as a right B-module via the diagonal action. Then βn = B ⊗ R n ⊗B ϑ is a right B-module isomorphism as stated. Theorem 2.26. Let B be a right × R -Hopf algebra over an algebra R and let S ⊆ T be a B-Galois extension. Consider the right regular B-module coring and the stable antiYetter-Drinfel’d module TS in Prop. 2.23. Then the associated cyclic simplex Z ∗ (B, TS ) in Proposition 2.19 is isomorphic to the cyclic simplex Z ∗ (T /S) in Corollary 1.29. Proof. In terms of the maps in Lemmas 2.24 and 2.25, for any non-negative integer n, one constructs a K-module isomorphism ωn as the composition of the following morphisms S n+1 T⊗
αn
/ TS ⊗ B ⊗ R n R
/ B⊗ R n ⊗op TS R
∼ =
/ B⊗ R n ⊗op B ⊗ TS R
B
βn ⊗B TS
/ B⊗ R n+1 ⊗ TS . B
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Explicitly, t1 ⊗ ··· ⊗ tn = t1[1] t2[1] · · · tn[1] ⊗ t2[2] · · · tn[2] ⊗ · · · ⊗ tn[n] ⊗ 1B ωn t0 ⊗ S
S
S
R
R
R
R
⊗ pT (t0 t1[0] t2[0] · · · tn[0] ),
(2.34)
B
S T denotes the canonical epimorphism. We show that (2.34) where pT : T → TS ∼ = S⊗ is a homomorphism of cyclic simplices. Denote the counit in B by . By multiplicativity of the coproduct in B, for any integer 0 ≤ k < n, ⊗ k ··· ⊗ tn ) = (B R ⊗ ⊗ B ⊗ R n−k ) ⊗ TS ◦ ωn (to ⊗ R
=
B
R
(t1[1] · · · tn[1] ⊗ R
S
S
[k+1] ⊗ tk[k] · · · tn[k] ⊗ tk+2 · · · tn[k+1] ⊗ R R R
···
· · · ⊗ tn[n−1] ⊗ 1B ) ⊗ pT (t0 t1[0] · · · tn[0] ) R
B
R
··· ⊗ tk tk+1 ⊗ tk+2 ⊗ ···⊗ tk−1 ⊗ tn ). = ωn−1 (to ⊗ S
S
S
S
S
S
Furthermore, using the form of the (diagonal) right B-action on B ⊗ R n (in the second equality) and the form of the left B-action (2.31) on TS (in the third equality), one computes ⊗ n ··· ⊗ tn ) (B R ⊗ ) ⊗ TS ◦ ωn (to ⊗ R B S S [1] [1] ⊗ [2] [2] ⊗ t · · · tn · · · ⊗ tn[n] ⊗ pT (t0 t1[0] · · · tn[0] ) = t1 · · · tn R 2 R R B [1] [2] [n−1] ⊗ t2[2] · · · tn−1 ⊗ · · · ⊗ tn−1 ⊗ 1B tn[1] ⊗ pT (t0 t1[0] · · · tn[0] ) = t1[1] · · · tn−1 R R R R B [1] [2] [n−1] ⊗ t2[2] · · · tn−1 ⊗ · · · ⊗ tn−1 ⊗ 1B ⊗ = t1[1] · · · tn−1 R
R
R
B
R
pT (tn[1]{+} t0 t1[0] · · · tn[0] tn[1]{−} ) [1] [2] [n−1] [0] ⊗ t2[2] · · · tn−1 ⊗ · · · ⊗ tn−1 ⊗ 1B ⊗ pT (tn t0 t1[0] · · · tn−1 ) = t1[1] · · · tn−1 R
R
R
B
R
t1 ⊗ ··· ⊗ tn−1 ), = ωn−1 (tn to ⊗ S
S
(2.35)
S
where the penultimate equality follows by (2.33). This proves that the map ωn is compatible with the face maps. Denote the coproduct in B by . It follows by its multiplicativity that, for any integer 0 ≤ k < n, ⊗ k ··· ⊗ tn ) (B R ⊗ ⊗ B ⊗ R n−k ) ⊗ TS ◦ ωn (to ⊗ R
B
R
S
S
[k+1] [k+2] · · · tn[k+1] ⊗ tk+1 · · · tn[k+2] ⊗ = (t1[1] · · · tn[1] ⊗ · · · ⊗ tk+1 R
R
R
R
· · · ⊗ tn[n+1] ⊗ 1B ) ⊗ pT (t0 t1[0] · · · tn[0] ) R
B
R
··· ⊗ 1T ⊗ tk+1 ⊗ ···⊗ tk ⊗ tn ). = ωn+1 (to ⊗ S
S
S
S
S
S
Furthermore, by unitality of and of the B-coaction on T , ⊗ n ··· ⊗ tn ) (B R ⊗ ) ⊗ TS ◦ ωn (to ⊗ R B S S [1] [1] = t1 t2 · · · tn[1] ⊗ t2[2] · · · tn[2] ⊗ · · · ⊗ tn[n] ⊗ 1B ⊗ 1B ⊗ pT (t0 t1[0] t2[0] · · · tn[0] ) R
R
··· ⊗ 1T ). tn ⊗ = ωn+1 (to ⊗ S
S
S
R
R
R
B
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This proves that the map ωn is compatible with the degeneracy maps. Finally, by similar steps as in (2.35), the cyclic map w n in Proposition 2.19 is checked to satisfy ··· ⊗ tn ) w n ◦ ωn (to ⊗ S S [1] [1] [1] [n] [n] tn ⊗ tn[n+1] = t0 t1 t2 · · · tn[1] ⊗ t1[2] · · · tn[2] ⊗ · · · ⊗ tn−1 R
R
R
R
⊗ pT (t0[0] t1[0] t2[0] · · · tn[0] ) B [1] [2] [n] ⊗ t1[2] · · · tn−1 ⊗ · · · ⊗ tn−1 ⊗ 1B = t0[1] t1[1] t2[1] · · · tn−1 R
R
R
R
[0] ⊗ pT (tn t0[0] t1[0] t2[0] · · · tn−1 ) B
t0 ⊗ ··· ⊗ tn−1 ). = ωn (tn ⊗ S
S
S
Hence ωn is compatible with the cyclic maps as well, which proves the claim.
3. Cyclic Homology of Groupoids Consider a groupoid G (i.e. a small category in which all morphisms are invertible) with a finite set G 0 of objects and an arbitrary set G 1 of morphisms. Via the map associating to x ∈ G 0 the identity morphism x → x, we consider G 0 as a subset of G 1 . Composition in G is denoted by ◦ while the source and target maps G 1 → G 0 are denoted by s and t, respectively. For any field K, the K-vector space B := KG 1 , spanned by the elements of G 1 , has a right × R -Hopf algebra structure over the commutative base algebra R := KG 0 . Structure maps are the following. Multiplication in the K-algebra B is given on basis elements g, g ∈ G 1 by g ◦ g , if g and g are composable, i.e. s(g) = t (g ), and zero otherwise. We denote by juxtaposition this product, linearly extended to all elements of B. The unit element is 1 B = x∈G 0 x. Similarly, R is a commutative K-algebra with minimal orthogonal idempotents { x ∈ G 0 }. Both algebra maps ξ and ζ : R → B are induced by the inclusion map G 0 → G 1 . That is, B is an R-module (or R-bimodule, with coinciding left and right actions) via multiplication on the right. The coproduct is diagonal on the basis elements g ∈ G 1 , i.e. (g) := g ⊗ R g. The counit maps g ∈ G 1 to s(g) ∈ G 0 . The canonical map (2.19) has the explicit form on the generating set {g ⊗ R g | g, g ∈ G 1 }, ϑ : B ⊗ R B → B ⊗ R B,
g ⊗ R g → gg ⊗ R g .
(Note that R-module structures in the domain and codomain are different, cf. Definition 2.13.) It obviously has an inverse ϑ −1 (g ⊗ R g ) = gg −1 ⊗ R g . In this final section we apply the theory developed in the earlier sections to the groupoid bialgebroid B and its stable anti-Yetter-Drinfel’d modules. In this way, we obtain expressions for Hochschild and cyclic homologies of a groupoid with finitely many objects. Describing then any groupoid as a direct limit of groupoids with finitely many objects, we extend the computation of cyclic homology to arbitrary groupoids. Similar arguments don’t seem to apply in the case of Hochschild homology. 3.A. Anti-Yetter-Drinfel’d modules for groupoids. The subject of this section is a complete characterization of (stable) anti-Yetter-Drinfel’d modules of a groupoid bialgebroid. As a first step, we study comodules.
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Proposition 3.1. Let G be a small groupoid with a finite set of objects. Let B be the groupoid R-bialgebroid associated to G. Then any right B-comodule M has a direct sum decomposition M ∼ = ⊕g∈G 1 Mg (as an R-module) such that the R-action · and the B-coaction satisfy the conditions (i) m · x = δx,s(g) m and (ii) (m) = m ⊗ R g, ∼ ⊕ 1 Mg , which for x ∈ G 0 , g ∈ G 1 and m ∈ Mg . Conversely, on an R-module M = g∈G is subject to condition (i), there is a unique right B-coaction satisfying (ii). Proof. Recall that by definition R acts on B by right multiplication. For any g ∈ G 1 there is an R-module map χg : B → R, h → δg,h s(g). Introduce the map πg := (M ⊗ R χg ) ◦ : M → M. We claim that M is isomorphic to a direct sum of the R-modules Mg := Im(πg ). Since 1 B is a free K-module, there exist (non-unique) elements {m g | g ∈ G } in M, in terms of which (m) = g∈G 1 m g ⊗ R g, hence πg (m) = m g · s(g). By construction, for a given element m ∈ M there are only finitely many elements g ∈ G 1 such that m g = 0, hence πg (m) = 0. By coassociativity of , g,h∈G 1
(m h )g ⊗ R g ⊗ R h =
g∈G 1
m g ⊗ R g ⊗ R g.
Hence applying M ⊗ R χg ⊗ R χh , for some g , h ∈ G 1 , we conclude that (m h )g · s(g )s(h ) = δg ,h m g · s(g ), i.e. πg ◦ πh = δg ,h πg . By counitality of , for all m ∈ M, π (m) = m g · s(g) = m. g 1 1 g∈G
g∈G
This proves the direct sum decomposition of M as an R-module. Condition (i) follows by the computation, for m ∈ M, g ∈ G 1 and x ∈ G 0 , πg (m) · x = m g · s(g)x = δs(g),x m g · s(g) = δs(g),x πg (m). Moreover, for m ∈ M, (m) = mg ⊗R g = m g ⊗ R gs(g) = m g · s(g) ⊗ R g 1 1 g∈G g∈G g∈G 1 = π (m) ⊗ R g. 1 g g∈G
Hence, by orthogonality of the projections πg , (πh (m)) =
g∈G 1
πg (πh (m)) ⊗ R g = πh (m) ⊗ R h,
which proves condition (ii). Conversely, assume that for an R-module M ∼ = ⊕g∈G 1 Mg condition (i) holds. Put g : Mg → Mg ⊗ R B, m → m ⊗ R g. One can check that it makes Mg to a right B-comodule. By universality of a direct sum, this defines a unique right B-coaction on M, such that condition (ii) holds.
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Note that property (ii) in Proposition 3.1 characterizes uniquely the elements m of a component Mg . Indeed, if for m ∈ M, (m) = m ⊗ R g, then by counitality of the coaction we obtain πg (m) = m · s(g) = m. We are ready to characterize (stable) anti-Yetter-Drinfel’d modules of groupoids. Theorem 3.2. Let G be a small groupoid with a finite set of objects. Let B be the groupoid bialgebroid associated to G. A left B-module M ∼ = ⊕g∈G 1 Mg , with action , is an antiYetter-Drinfel’d B-module if and only if the following conditions hold. (i) For x ∈ G 0 , g ∈ G 1 and m ∈ Mg , δx,s(g) m = x m = δx,t (g) m. (ii) For g, h ∈ G 1 and m ∈ Mg , the element h m is zero if hgh −1 = 0 in B and it belongs to Mhgh −1 if hgh −1 = 0 in B. The anti-Yetter-Drinfel’d B-module M is stable if and only if in addition g m = m, for all g ∈ G 1 and m ∈ Mg . Proof. By Proposition 3.1 there is a unique B-coaction on M corresponding to the given direct sum decomposition. For g ∈ G 1 , it takes m ∈ Mg to m ⊗ R g. The R-bimodule structure corresponding to this coaction comes out, for x, y ∈ G 0 , g ∈ G 1 and m ∈ Mg , as x · m · y = m · (xgy) = δs(g),y δt (g),x m · s(g) = δs(g),y δt (g),x m. Hence axiom (2.20) of an anti-Yetter-Drinfel’d module translates to condition (i) in the theorem. A straightforward computation shows that for a groupoid bialgebroid, axiom (2.21) of an anti-Yetter-Drinfel’d module takes the form (h m) = h m ⊗ R hgh −1 ,
(3.1)
for h, g ∈ G 1 and m ∈ Mg . If hgh −1 = 0 in B, then the right-hand side of (3.1) vanishes. Since is a monomorphism of R-modules (split by M ⊗ R ), this is equivalent to h m = 0. If hgh −1 = 0 in B, then (3.1) is equivalent to h m ∈ Mhgh −1 (cf. Proposition 3.1, and discussions following it). This completes the proof. As a consequence of Theorem 3.2 (condition (i)), in the direct sum decomposition of an anti-Yetter-Drinfel’d module M of a groupoid bialgebroid, Mg is non-zero for only those elements g ∈ G 1 for which s(g) = t (g), i.e. which are loops. That is, introducing the notation L(G) := { g ∈ G 1 | s(g) = t (g) }, one can write M ∼ = ⊕l∈L(G ) Ml . Our next aim is to decompose an anti-Yetter-Drinfel’d module M of a groupoid bialgebroid B as a direct sum of anti-Yetter-Drinfel’d modules. For a loop l ∈ L(G), denote by [l] the orbit of l in L(G) for the adjoint action, that is, the set of different non-zero elements of the form glg −1 , as g runs through G 1 . This gives a (G-invariant) partition T (G) of L(G). Using Theorem 3.2 one concludes that M[l] := ⊕l ∈[l] Ml is an anti-Yetter-Drinfel’d B-module, and
M[l] , (3.2) M∼ = [l]∈T (G )
as anti-Yetter-Drinfel’d modules. Let us give an alternative description of the anti-YetterDrinfel’d module M[l] . Introduce the following subalgebras of the groupoid algebra B. For l ∈ L(G), let Bl be the group algebra of the centralizer Gl1 of l in the group { l ∈ L(G) | s(l ) = s(l) }. That is, Bl := KGl1 ≡ K{ l ∈ L(G) | l ll −1 = l }.
(3.3)
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The algebra Bl possesses a unit (the element s(l) = t (l)), different from the unit element of B. For x ∈ G 0 , let B(x) denote the vector space spanned by the elements g ∈ G 1 , such that s(g) = x. Consider the group bialgebra structure of Bl . Clearly, for any l ∈ L(G), the component Ml in the direct sum decomposition of an anti-Yetter-Drinfel’d B-module M is an anti-Yetter-Drinfel’d module of the group bialgebra Bl (in the sense of [JS], ¸ cf. Theorem 3.2). Moreover, B(s(l)) is a B-Bl bimodule and a right Bl -module coalgebra (with coproduct induced by the map g → g ⊗K g, for g ∈ G 1 such that s(g) = s(l).) Lemma 3.3. Let G be a small groupoid with a finite set of objects. Let B be the groupoid bialgebroid over a field K, associated to G. Keeping the notation introduced above, B(s(l)) is free as a right Bl -module, for any l ∈ L(G). Proof. Since any morphism in a groupoid is invertible, the right action of the group Gl1 on the set { g ∈ G 1 | s(g) = s(l) } is faithful in the sense that gl = g implies l = s(l). Hence the claim follows by the fact that KX is a free module for a group algebra KG whenever G acts faithfully on the set X . Indeed, fix a K-basis { ex | x ∈ X } of KX , such that ex · g = ex·g . Fix a section f of the canonical epimorphism from X to the set of G-orbits X/G. By construction, { e f (O) | O ∈ X/G } is a generating set of the KG-module KX . It is also linearly independent over KG, by the following reasoning. Assume that, for some coefficients aO = g∈G αO,g g ∈ KG, 0= e f (O ) · a O = αO,g e f (O)·g . O∈X/G
O∈X/G g∈G
Since G acts on X faithfully, in the above sum each element e f (O) appears exactly once. Hence αO,g = 0, for all O ∈ X/G and g ∈ G. Thus we have the claim proven. Proposition 3.4. Let G be a small groupoid with a finite set of objects. Let B be the groupoid bialgebroid associated to G. Let M be an anti-Yetter-Drinfel’d B-module. Keeping the notation introduced above, there is an isomorphism of anti-Yetter-Drinfel’d B-modules M[l] ∼ = B(s(l)) ⊗ Bl Ml ,
for all l ∈ L(G).
Proof. First we construct a left B-module isomorphism ϕl : B(s(l)) ⊗ Bl Ml → M[l] ,
g ⊗ Bl m → g m.
(3.4)
For any g ∈ G 1 such that s(g) = s(l), consider the map ψg : Mglg−1 → B(s(l)) ⊗ Bl Ml ,
g → g ⊗ Bl g −1 m.
In order to see that the map ψg does not depend on g, only on glg −1 , choose another element h ∈ G 1 , such that glg −1 = hlh −1 . Note that this implies in particular t (g) = t (h). Then, for m ∈ Mglg−1 , ψh (m) = h ⊗ Bl h −1 m = h ⊗ Bl h −1 t (h) m = h ⊗ Bl h −1 t (g) m = h ⊗ Bl h −1 gg −1 m = hh −1 g ⊗ Bl g −1 m = t (h)g ⊗ Bl g −1 m = t (g)g ⊗ Bl g −1 m = g ⊗ Bl g −1 m = ψg (m),
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where in the fifth equality we used that h −1 g is an element of Bl . Thus we conclude by universality of a direct sum on the existence of a map ψ : M[l] ∼ = ⊕l ∈[l] Ml → B(s(l)) ⊗ Bl Ml , mapping m ∈ Mglg−1 to ψg (m) = g ⊗ Bl g −1 m. A straightforward computation shows that ψ is the inverse of ϕl in (3.4). Next we show that B(s(l)) ⊗ Bl Ml is an anti-Yetter-Drinfel’d module with respect to the direct sum decomposition B(s(l)) ⊗ Bl Ml ∼ = ⊕l ∈[l] ψ(Ml ), hence (3.4) is an isomorphism of anti-Yetter-Drinfel’d modules, as stated. For m ∈ Ml , l ∈ L(G), h, h ∈ G 1 and y ∈ G 0 , such that hlh −1 = l and s(h ) = t (h), y (h ⊗ Bl m) = yh ⊗ Bl m = δt (h),y h ⊗ Bl m = δs(l ),y h ⊗ Bl m,
h (h ⊗ Bl m) = h h ⊗ Bl m ∈ ψ(Mh l h −1 ).
In view of Theorem 3.2 this implies that B(s(l)) ⊗ Bl Ml is an anti-Yetter-Drinfel’d module with respect to the given decomposition, hence it completes the proof. 3.B. Hochschild and cyclic homology with coefficients. In view of Proposition 2.19, there is a cyclic simplex associated to a groupoid bialgebroid B, the right regular B-module coring and any stable anti-Yetter-Drinfel’d B-module M. At degree n, it is given by Z n (B, M) = B ⊗ R n+1 ⊗ B M (where B acts on B ⊗ R n+1 via the diagonal right action). In this section we compute its Hochschild and cyclic homologies. With an eye on the decomposition of M in Sect. 3.A, computations start with the following Lemma 3.5. Let G be a small groupoid with a finite set of objects. Let B be the groupoid bialgebroid over a field K, associated to G. Let M be an anti-Yetter-Drinfel’d B-module. Using notations introduced in Sect. 3.A, there is an isomorphism of right Bl -modules, B ⊗ R n+1 ⊗ B B(x) ∼ = B(x)⊗K n+1 , for all l ∈ L(G) and x := s(l). Here B ⊗ R n+1 ⊗ B B(x) is understood to be a right Bl -module via the last factor and the group algebra Bl acts on B(x)⊗K n+1 via the diagonal action. Proof. Since R = KG 0 is a separable K-algebra, B ⊗ R n+1 is isomorphic to the subspace B × n+1 of B ⊗K n+1 , spanned by those elements g0 ⊗K · · ·⊗K gn for which s(gi ) = s(gi+1 ), for all i = 0 · · · n. Thus it suffices to prove B × n+1 ⊗ B B(x) ∼ = B(x)⊗K n+1 . We claim that the right Bl -module map B(x)⊗K n+1 → B × n+1 ⊗ B B(x), g0 ⊗K · · ·⊗K gn → (g0 ⊗K · · ·⊗K gn ) ⊗ B x (3.5) is an isomorphism. Since the map B × n+1 ⊗K B(x) → B(x)⊗K n+1 , (g0 ⊗K · · · ⊗K gn ) ⊗K h → g0 h ⊗K · · · ⊗K gn h factorizes through B × n+1 ⊗ B B(x), it defines a unique map B × n+1 ⊗ B B(x) → B(x)⊗K n+1 , (g0 ⊗K · · ·⊗K gn ) ⊗ B h → g0 h ⊗K · · ·⊗K gn h. (3.6) Obviously, (3.5) and (3.6) are mutual inverses.
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Proposition 3.6. Let G be a small groupoid with a finite set of objects. Let B be the groupoid bialgebroid over a field K, associated to G. Let M be a stable anti-YetterDrinfel’d B-module, with decomposition (3.2). For all l ∈ L(G), the cyclic simplex Z ∗ (B, M[l] ) = B ⊗ R ∗+1 ⊗ B M[l] is isomorphic to Z ∗ (B(s(l)), Ml ) = B(s(l))⊗K ∗+1 ⊗ Bl Ml , corresponding (as in [J¸S, Remark 4.16]) to the module coalgebra B(s(l)), and stable anti-Yetter-Drinfel’d module Ml , of the group bialgebra Bl . Proof. Combining the isomorphisms in Proposition 3.4 and Lemma 3.5, we obtain an isomorphism of vector spaces B(s(l))⊗K n+1 ⊗ Bl Ml ∼ = B ⊗ R n+1 ⊗ B B(s(l)) ⊗ Bl Ml ∼ = B ⊗ R n+1 ⊗ B M[l] (g0 ⊗K · · · ⊗K gn ) ⊗ Bl m
−→ (g0 ⊗ R · · · ⊗ R gn ) ⊗ B m, with inverse B ⊗ R n+1 ⊗ B Mhlh −1 (g0 ⊗ R · · ·⊗ R gn )⊗ B m → (g0 h ⊗K · · ·⊗K gn h)⊗ Bl h −1 m. It is left to the reader to check that it is an isomorphism of cyclic objects. Hochschild and cyclic homologies of a group, with coefficients in a stable antiYetter-Drinfel’d module, were computed in [JS, ¸ Corollary 5.13]. Hence in what follows we relate Hochschild and cyclic homologies of the cyclic object Z ∗ (B(s(l)), Ml ) ∼ = Z ∗ (B, M[l] ) in Proposition 3.6 to the known respective homology of Z ∗ (Bl , Ml ). Lemma 3.7. Let (C, , ) be a coalgebra over a field K. Consider the corresponding simplex C ⊗K ∗+1 (with face maps ∂i induced by and degeneracy maps µi induced by ∗ (C) is acyclic, i.e. Hn (C ∗ (C)) = δn,0 K, for any ). Then the associated complex C non-negative integer n. Moreover, if (C, , ) is a right module coalgebra of a K-Hopf ∗ (C) is a free resolution of K in the algebra H and free as a right H -module, then C category of right H -modules. Proof. We need to show that the chain complex ···
δn+1
/ C ⊗K n+1
δn
/ C ⊗K n
/ ···
δ1
/C
δ0 =
/K
/0
n (−1)i ∂i . Indeed, the map C ⊗K is surjective, having a is exact, where δn = i=0 section . Since C is a faithfully flat module of the field K, this implies surjectivity of . Take an element g ∈ C such that (g) = 1K . One easily checks that, for z ∈ C ⊗K n+1 such that δn (z) = 0, δn+1 (g ⊗K z) = z − (g ⊗K δn (z)) = z. Assume now that (C, , ) is a right module coalgebra of a K-Hopf algebra H . ∗ (C) is an acyclic complex in the category of right H-modules. It remains to Then C show that C ⊗K n+1 is a free H -module whenever C is. Indeed, in this case C ⊗K n+1 is free as a right module of H ⊗K n+1 via factorwise action. By [JS, ¸ Lemma 2.10] (cf. Lemma 2.25), H ⊗K n+1 is free as a right H -module via the diagonal action. Hence C ⊗K n+1 ∼ = C ⊗K n+1 ⊗ H ⊗K n+1 H ⊗K n+1 is a free right H -module. Lemma 3.8. Let H be a Hopf algebra over a field K and let i : C → C be a morphism of right H -module coalgebras. Let M be a stable anti-Yetter-Drinfel’d H -module. If both C and C are free as right H -modules, then the induced morphism i ∗ : Z ∗ (C, M) → Z ∗ (C , M) of cyclic objects gives rise to isomorphisms both of Hochschild and cyclic homologies. That is, HH∗ (C, M) ∼ = HH∗ (C , M)
and
HC∗ (C, M) ∼ = HC∗ (C , M).
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∗ (C) and C ∗ (C ) are free resolutions of K over H . Thus Proof. By Lemma 3.7, both C ⊗ ∗+1 K i : C∗ (C) → C∗ (C ) is a quasi-isomorphism. Then so is i ⊗K ∗+1 ⊗ H M : ∗ (C ) ⊗ H M, yielding an isomorphism C∗ (C) ⊗ H M → C ∗ (C) ⊗ H M) ∼ ∗ (C ) ⊗ H M) ≡ HHn (C , M), HHn (C, M) ≡ Hn (C = Hn (C for all non-negative integers n. The isomorphism of Hochschild homologies implies the isomorphism of cyclic homologies, see e.g. (a dual form of) [Lo, 2.2.3]. Theorem 3.9. Let G be a small groupoid with a finite set of objects. Let B be the groupoid bialgebroid over a field K, associated to G. Let M be a stable anti-Yetter-Drinfel’d B-module, with decomposition (3.2). For all l ∈ L(G), HH∗ (B, M[l] ) ∼ = HH∗ (Bl , Ml )
and
HC∗ (B, M[l] ) ∼ = HC∗ (Bl , Ml ).
Proof. By Proposition 3.6, the cyclic objects Z ∗ (B, M[l] ) and Z ∗ (B(s(l)), Ml ) are isomorphic. Hence HH∗ (B, M[l] ) ∼ = HH∗ (B(s(l)), Ml ). Furthermore, the right Bl -module coalgebra B(s(l)) is a free right Bl -module by Lemma 3.3. Hence we conclude by Lemma 3.8 that the inclusion map Bl → B(s(l)) induces an isomorphism HH∗ (B(s(l)), Ml ) ∼ = HH∗ (Bl , Ml ). Combination of these isomorphisms proves the theorem. Let us assume now that K is a field of characteristic zero. In view of Theorem 3.9, one can compute also
HH∗ (B, M) ∼ HH∗ (Bl , Ml ) HC∗ (Bl , Ml ) . and HC∗ (B, M) ∼ = = [l]∈T (G )
[l]∈T (G )
Since Bl is a group algebra of Gl1 (cf. (3.3)), HH∗ (Bl , Ml ) = H∗ (Gl1 , Ml ) is the group ¸ Corollary 5.13], for the cyclic homology of Gl1 with coefficients in Ml . Applying [JS, homology we get 1 i≥0 1H∗−2i Gl /l, Ml , if l has finite order HC∗ (Bl , Ml ) = H∗ Gl /l, Ml , if l has infinite order. 3.C. Cyclic homology of groupoids. The results in Sect. 3.B can be specialized further to stable anti-Yetter-Drinfel’d modules provided by groupoid Galois extensions, cf. Proposition 2.23. This enables us, in particular, to compute ordinary (i.e. non-relative) Hochschild and cyclic homologies of a groupoid. Let G be a small groupoid with a finite set of objects. Let B be the groupoid bialgebroid over a field K, associated to G. Recall from [CaDGr, Sect. 3] that a Galois extension S ⊆ T by B has an equivalent description as follows. T is a strongly G-graded K-algebra, that is, T ∼ = ⊕g∈G 1 Tg (as a vector space), Tg◦g if s(g) = t (g ) Tg Tg = 0 if s(g) = t (g ), 1T = x∈G 0 1Tx , and S is equal to the subalgebra ⊕x∈G 0 Tx . Note that each direct summand Tg is an R = KG 0 -module via v · x := v1Tx = δs(g),x v, for v ∈ Tg and x ∈ G0. S T is a stable anti-Yetter-Drinfel’d module. In the By Proposition 2.23, TS := S ⊗ following lemma its structure is investigated.
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Lemma 3.10. Let G be a small groupoid with a finite set of objects. Let B be the groupoid bialgebroid over a field K, associated to G. Let S ⊆ T be a Galois extension by B. Using the notations introduced in Sect. 3.A, there is an isomorphism of anti-Yetter-Drinfel’d modules
S Tl . TS ∼ S⊗ = l∈L(G )
The (Miyashita-Ulbrich) action is given, for g ∈ G 1 , l ∈ L(G) and v ∈ Tl , by g pTl (v) =
n
pTl (bi vai ),
i=1
S Tl denotes the canonical where pTl : Tl → S ⊗ n epimorphism and the elements {a1 , . . . , an } ⊆ Tg−1 and {b1 , · · · , bn } ⊆ Tg satisfy i=1 ai bi = 1Ts(g) . ∼ ⊕ 1 Tg is G-graded, each direct summand Tg is an S-bimodule. Proof. Since T = g∈G Hence
∼ S Tg . S T = S⊗ S⊗ g∈G 1
We claim that only those elements g ∈ G 1 give a non-zero contribution to this direct S Tg is isomorphic to the quotient of Tg sum, for which s(g) = t (g). Recall that S ⊗ with respect to the commutator subspace [S, Tg ] = {qv − vq | q ∈ S, v ∈ Tg }. Take an element g ∈ G 1 such that s(g) = t (g) and an element v ∈ Tg . By strong grading of T , there exist elements {a1 , · · · , an } ⊆ Tg−1 and {b1 , · · · , bn } ⊆ Tg such n ai bi = 1Ts(g) . Then that i=1 n
[vai , bi ] =
i=1
n
vai bi − bi vai = v1Ts(g) = v,
i=1
where the penultimate equality follows by the fact that, for all values of i, bi vai ∈ Tg Tg Tg−1 is zero by the assumption that s(g) = t (g). Since for all values of i, vai ∈ Tg−1 Tg = Ts(g) ⊆ S, we conclude that Tg ⊆ [S, Tg ]. Since the converse inclusion is S Tg ∼ obvious, we have S ⊗ = Tg /[S, Tg ] = 0 proven. 1 In order to write down the Miyashita-Ulbrich action, consider n again g ∈ G and elements {a1 , · · · , an } ⊆ Tg−1 and {b1 , . . . , bn } ⊆ Tg such that i=1 ai bi = 1Ts(g) . The canonical map can : T ⊗ S T → T ⊗ R B satisfies n n ai ⊗ S bi ) = ai bi ⊗ R g = 1Ts(g) ⊗ R g = 1T · s(g) ⊗ R g = 1T ⊗ R g. can( i=1
i=1
Hence the stated form of the Miyashita-Ulbrich action follows by (2.31).
Corollary 3.11. Let G be a small groupoid with a finite set of objects. Let B be the groupoid bialgebroid over a field K, associated to G. Let S ⊆ T be a Galois extension by B. Using the notations introduced in Sect. 3.A, the S-relative Hochschild and cyclic homologies of T are given, respectively, by
S Tl ) and HC∗ (T /S) = S Tl ). HH∗ (T /S) = HH∗ (Bl , S ⊗ HC∗ (Bl , S ⊗ [l]∈T (G )
[l]∈T (G )
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Proof. By Theorem 2.26, HH∗ (T /S) ∼ = HH∗ (B, TS ) and HC∗ (T /S) ∼ = HC∗ (B, TS ). Hence the claim follows by Theorem 3.9 and considerations following it, together with Lemma 3.10. A particular example R ⊆ B of a Galois extension by a groupoid G is provided by the inclusion of the base algebra R = KG 0 in the groupoid algebra B = KG 1 . Applying Corollary 3.11 to it, we obtain formulae for the R-relative Hochschild and cyclic homologies of the groupoid G. Since the base algebra R = KG 0 is separable over K, relative homologies coincide with ordinary ones, cf. [Kad]. Therefore we obtain the following corollary, extending results in [Burg] on cyclic homology of groups. Similar expressions were derived also by Crainic in [Cra] for étale groupoids. Corollary 3.12. Let B be the groupoid algebra of a small groupoid G with a finite set of objects over a field K of characteristic zero. Using the notations introduced in Sect. 3.A we have
HH∗ (B) = H∗ (Gl1 , K), [l]∈T (G )
⎛
⎜
HC∗ (B) = ⎜ ⎝
⎞
[l]∈T (G ) i≥0 ord(l)<∞
⎛
⎞
⎟ ⎜
⎜ H∗−2i (Gl1 / l , K)⎟ ⎠ ⎝
[l]∈T (G ) ord(l)=∞
⎟ H∗ (Gl1 / l , K)⎟ ⎠.
Recall that a groupoid G is connected if, for any x, y ∈ G 0 , there exists at least one g ∈ G 1 such that s(g) = x and t (g) = y. For a connected groupoid G we fix an object x ∈ G 0 and we denote by G the group of loops l ∈ L(G) such that s(l) = t (l) = x. (Since G is connected by assumption, different choices of x lead to isomorphic subgroups G of G.) Let T (G) denote the set of conjugacy classes in G and let {gσ | σ ∈ T (G)} be a transversal of T (G). Lemma 3.13. Let G be a connected groupoid. Keeping the above notation, there is an one-to-one correspondence between T (G) and T (G). Proof. We have to show that any orbit in L(G) for the adjoint action contains precisely one element of the set {gσ | σ ∈ T (G)}. That is, any loop l in G is equivalent to a certain gσ and that two elements gσ and gτ are equivalent if, and only if σ = τ . First, let us take l ∈ L(G). Since G is connected, there is a morphism g such that s(g) = x and t (g) = s(l). Let l := g −1 ◦ l ◦ g. By construction, l ∈ G, so there are h ∈ G and σ ∈ T (G) such that h ◦ l ◦ h −1 = gσ . Since l = (g ◦ h −1 ) ◦ gσ ◦ (g ◦ h −1 )−1 it follows that l and gσ are conjugated in L(G). Let us take σ and τ in T (G) and assume that gσ and gτ define the same element in T (G). Then there is g ∈ G 1 such that gσ = g ◦ gτ ◦ g −1 . Since both loops gσ and gτ have the same source, we deduce that the source and the target of g must be x. Hence g ∈ G and the conjugacy classes of gσ and gτ in T (G) are equal. In conclusion, σ = τ. Remark 3.14. Lemma 3.13 tells us, in particular, that {gσ | σ ∈ T (G)} is also a transversal of T (G). Explicitly, a given element gσ represents the following orbit σ ∈ T (G). Recall that we defined the group G in terms of a fixed object x in G 0 . For every y ∈ G 0 we pick up a fixed g y ∈ HomG (x, y). It defines a group isomorphism G → HomG (y, y), h → g y ◦ h ◦ g −1 g −1 y . It maps σ ∈ T (G) to the conjugacy class σ y = {g y ◦ h ◦ y | h ∈ σ} σ = y∈G 0 σ y . in the group HomG (y, y). The orbit of gσ for the adjoint G-action is
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Corollary 3.15. Let G be a connected groupoid with finitely many objects. Let M be a stable anti-Yetter-Drinfel’d module over B, where B is the groupoid algebra of G over a field of characteristic zero. (i) MG := g∈G M g is a stable anti-Yetter-Drinfel’d module over KG and the inclusions KG ⊆ B and MG ⊆ M induce isomorphisms HH∗ (B, M) ∼ = HH∗ (KG, MG ) and HC∗ (B, M) ∼ = HC∗ (KG, MG ). (ii) The inclusion KG ⊆ B induces isomorphisms ∼ HH∗ (KG) HH∗ (B) = and HC∗ (B) ∼ = HC∗ (KG). Proof. (i) Obviously MG is a stable anti-Yetter-Drinfel’d module over KG. Consider the following diagram. / σ ∈T (G) Z ∗ (KG, Mσ ) / σ ∈T (G) Z ∗ (KBgσ , Mgσ ) Z ∗ (KG, MG ) Z ∗ (B, M) /
σ ∈T (G)
Z ∗ (B, M σ) /
σ ∈T (G)
Z ∗ (B(x), Mgσ )
The leftmost morphism in the bottom row comes from the decomposition (3.2), while the other one is the direct sum of the arrows that were constructed in Proposition 3.6. Note that both maps are induced by appropriate inclusions and they are isomorphisms. The morphisms in the top row have the same properties, as any group can be regarded as a groupoid with one object. By definition, the vertical arrows are the canonical morphisms induced by inclusions, so they make the squares commutative. In view of the proof of Theorem 3.9, the rightmost vertical arrow gives isomorphisms both of Hochschild and cyclic homologies. Then also the leftmost vertical morphism does so. (ii) The subalgebra R := KX of B is a separable K-algebra. Hence HH∗ (B) ∼ = R B is a stable anti-Yetter-Drinfel’d HH∗ (B, R). By Lemma 3.10 we know that M := R ⊗ module over B. Since for any g ∈ G, either [R, Kg] = Kg or [R, Kg] = 0, depending on the fact that g is a loop or not, we get M := g∈L(G ) Kg. Hence MG := KG. Obviously the action of B on KG induced from the Ulbrich-Miyashita action is the adjoint action of KG on itself. We conclude by applying the first part of the corollary and the isomorphisms HH∗ (KG) ∼ ¸ = HH∗ (KG, (KG)ad ) and HC∗ (KG) ∼ = HC∗ (KG, (KG)ad ), cf. [JS]. Our final aim is to compute HC∗ (B), the ordinary cyclic homology of the groupoid algebra B of a groupoid G that may have an infinite number of objects. For such a groupoid, its groupoid algebra B is not unital anymore. Nevertheless, to define cyclic homology of B one can proceed as for unital algebras, cf. [Lo, Chap. 2, §2.1]. The point is that Connes’ complex C λ (B) still exists, although it is now associated to a precyclic object, that is to a presimplicial structure endowed with cyclic operators. Here, by presimplicial object we mean a sequence of objects together only with face maps. The defining properties of face maps and cyclic operators are the same as in the definition of cyclic objects, neglecting of course the relations that involve the degeneracy maps. The key ingredient of our computation is a description of any groupoid as a direct limit of groupoids with finitely many objects. Since cyclic homology is defined as homology of Connes’ complex and the homology functor commutes with direct limits, we obtain cyclic homology of an arbitrary groupoid as a direct limit. Note however that, for nonunital algebras, Hochschild homology is constructed in a different way. For the definition, see for example [Lo, Chap. 1, §1.2]. Therefore, we can not apply the same arguments to compute Hochschild homology of an arbitrary groupoid.
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Theorem 3.16. Let G be a connected groupoid. If B is the groupoid algebra over a field K of characteristic zero, then HC∗ (B) ∼ = HC∗ (KG). Proof. Let x be a given object in G 0 . Let G denote the group of loops l ∈ L(G) such that s(l) = x. We order the set X := {X ⊂ G 0 | x ∈ X and X is finite} with respect to inclusion. Trivially X is a direct system. For X ∈ X we define G X to be the full subgroupoid of G such that G X0 = X and we denote its groupoid algebra by B X . Note that B = X ∈X B X , so C∗λ (B) = X ∈X C∗λ (B X ). As the direct limit in the category of vector spaces is exact, it follows that the homology functor and direct limit commute. Thus HC∗ (B) ∼ C λ (B X ) ∼ H C∗λ (B X ) ∼ HC∗ (B X ), = H∗ (C∗λ (B)) ∼ = H∗ −lim = −lim = −lim −→ ∗ −→ ∗ −→ X ∈X
X ∈X
X ∈X
where the latter direct system is defined by the canonical maps HC∗ (B X ) → HC∗ (BY ), with X, Y in X such that X ⊂ Y. We claim that these maps are isomorphisms. Indeed, let us consider the following commutative diagram Z ∗ (KG) KKK rr KKK r r r KKK r r r K% r yr / Z ∗ (BY ) Z ∗ (B X ) By the second part of Corollary 3.15, the oblique arrows induce isomorphisms in cyclic homology. Then, passing to cyclic homology, also the horizontal map yields an isomorphism. We can now conclude the proof of the theorem by remarking that, for any X ∈ X, HC∗ (B X ) ∼ HC∗ (B X ). = −lim −→ X ∈X
Thus, taking X = {x} we get the required isomorphism.
The computation performed in Theorem 3.16 can be extended to an arbitrary (discrete) groupoid G. Let (Gi )i∈I be the connected components of G. For each i we pick up xi ∈ Gi0 and we denote the set of loops l with s(l) = xi by G i . We have the following result. Corollary 3.17. Let G be a discrete groupoid. If B denotes the groupoid algebra of G over a field of characteristic zero, then
HC∗ (KG i ). (3.7) HC∗ (B) ∼ = i∈I
In addition, if Gi0 is a finite set for every i ∈ I , then a similar isomorphism holds in Hochschild homology.
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Proof. For i ∈ I, let Bi be the groupoid algebra of Gi . As a vector space, B is isomorphic to i∈I Bi . Via this identification, the multiplication of B can be extended to the direct sum. It is easy to see that, for two families (bi )i∈I and (bi )i∈I in i∈I Bi , we have (bi )i∈I · (bi )i∈I = (bi bi )i∈I . The corollary now follows by the isomorphism HC∗ ( i∈I Bi ) ∼ = i∈I HC∗ (Bi ). If I is a finite set, then this isomorphism can be found in [Lo, Exercise 2.2.1]. Since cyclic homology and direct limits commute, the isomorphism can be extended for an arbitrary set I. Since for non-unital algebras Hochschild homology is constructed in a different way (cf. [Lo, Chap. 1, §1.2]), the above arguments can not be applied to deduce an isomorphism for Hochschild homology, analogous to (3.7). Nevertheless, in the case when G 0 is finite, B is an unital algebra. Thus HH∗ (B) can be computed as the Hochschild homology of Connes’ cyclic object. The required isomorphism now follows by [We, Theorem 9.1.8], proceeding as for cyclic homology. As an application of Corollary 3.17 we shall compute HH∗ (B) and HC∗ (B), where B is the groupoid algebra of the groupoid associated to a G-set X . Throughout the remaining part of the paper we fix an arbitrary (discrete) group G that acts to the left on an arbitrary set X . The action of G on X maps a pair (x, g) ∈ X × G to g x ∈ X. For a G-set X as above, one constructs a groupoid G as follows. By definition, the set of objects in G is G 0 := X while, for x, y ∈ X, we set HomG (x, y) = (x, g) ∈ X × G | g x = y . Note that the source of (x, g) is x and its target is g x. Thus the composition (x, g)◦(x , g ) is defined if, and only if x = g x and, in this case (x, g) ◦ (x , g ) := (x , gg ). The set of morphisms in G is G 1 = X × G. Therefore, the groupoid algebra B of G has X × G as a basis. To describe the multiplication on this basis let us recall some wellknown facts about twisted semigroup algebras by a 2-cocycle. Let S be a semigroup. A function ω : S × S → K is called a 2-cocycle if, for any p1 , p2 , p3 in S, ω( p1 , p2 )ω( p1 p2 , p3 ) = ω( p2 , p3 )ω( p1 , p2 p3 ). The semigroup algebra of S is defined as in the group case: as a vector space it has S as a basis and the multiplication on this basis is given by the multiplication in S. We shall denote this algebra by KS. The cocycle ω can be used to deform the multiplication of KS such that we get another associative algebra structure on the vector space KS. Its multiplication is defined by p1 · p2 = ω( p1 , p2 ) p1 p2 . The resulting algebra will be denoted by Kω S. Certainly, it is not unital in general. Still, even if S has no neutral element, the algebra Kω S may have a unit e := x∈X x, where X is an appropriate finite subset of S. In fact, it is easy to see that e is a unit element in the algebra Kω if, and only if ω is X -normalized, i.e. for any p, q ∈ S, ω(x, p) = δ p,q = ω( p, x). {x∈X |x p=q}
{x∈X | px=q}
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Let us turn back to the groupoid algebra of G, where X is a G-set. We define the semigroup S := X × G with the multiplication (x, g)(y, h) = (y, gh). It is not difficult to see that ω X : S × S → K, given by 0, if h y = x ω X (x, g), (y, h) := 1, if h y = x
(3.8)
is a 2-cocycle, which is X -normalized if, and only if X is finite. Obviously, B = Kω S as non-unital algebras, in general. For a finite G-set X , this is an equality of unital algebras. We denote the set of G-orbits in X by X. Let us remark that there is an one-to-one correspondence between X and the set of connected components of G. This bijection maps an orbit 0 ∈ X to the full subgroupoid Go defined uniquely such that Go0 = o. We choose a transversal {xo ∈ X | o ∈ X} for X. Thus, any x ∈ X is in the orbit of a certain xo and xo and xo are in the same orbit if, and only if o = o . Moreover, the set G o of loops l such that s(l) = xo is the stabilizer of xo, G o := {g ∈ G | g xo = xo}. Hence, a direct application of Corollary 3.17 yields the following. Theorem 3.18. Let X be a G-set. If ω X denotes the 2-cocycle in (3.8) and K is a field of characteristic zero, then
HC∗ (KG o). HC∗ (Kω X G) = o∈X
If in addition X is finite, then HH∗ (Kω X G)=
HH∗ (KG o).
o∈X
Acknowledgements. The first author was financially supported by the Hungarian Scientific Research Fund OTKA K 68195 and the Bolyai János Scholarship, while the second author was supported by Contract 2-CEx06-11-20 of the Romanian Ministry of Education and Research. Both authors are grateful to Bachuki Mesablishvili, Zoran Škoda and the referee for their helpful comments.
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Akrami, S.E., Majid, S.: Braided cyclic cocycles and non-associative geometry. J. Math. Phys. 45, 3883–3911 (2004) Beck, J.: Distributive laws, Lecture Notes in Mathematics 80, Berlin-Heidelberg-New York: Springer Verlag, 1969, pp. 119–140 Böhm, G., Brzezi´nski, T.: Strong connections and the relative chern-galois character for corings. Int. Math. Res. Not. 42, 2579–2625 (2005) Brzezi´nski, T., Caenepeel, S., Militaru, G.: Doi-koppinen modules for quantum groupoids. J. Pure Appl. Algebra 175, 45–62 (2002) Brzezi´nski, T., Majid, S.: Coalgebra bundles. Commun. Math. Phys. 191, 467–492 (1998) Brzezi´nski, T., Militaru, G.: Bialgebroids, ×a -bialgebras and duality. J. Algebra 251, 279–294 (2002) Burghelea, D.: The cyclic homology of the group rings. Comment. Math. Helv. 60, 354–365 (1985)
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Burroni, E.: Algébres non déterministiques et D-catégories. Cahiers de Topologie et Géometrie Différentielle 14, 417–475, 480–481 (1973) [CaDGr] Caenepeel, S., De Groot, E.: Galois theory for weak Hopf algebras. Rev. Roumaine Math. Pures Appl. 52, 51–76 (2007) [CM95] Connes, A., Moscovici, H.: Local index formula in noncommutative geometry. Geom. Funct. Anal. 5, 174–243 (1995) [CM98] Connes, A., Moscovici, H.: Hopf algebras, cyclic cohomology and the transverse index theorem. Commun. Math. Phys. 198, 199–246 (1998) [CM01] Connes, A., Moscovici, H.: Differential cyclic cohomology and Hopf algebraic structures in transverse geometry. In: Essays on geometry and related topics. Vol. 1-2, Monogr. Enseign. Math. 38, Geneva: Enseignement Math., 2001, pp. 217–255 [Cra] Crainic, M.: Cyclic cohomology of étale groupoids: the general case. K -Theory 17, 319–362 (1999) [EM] Eilenberg, S., Moore, J.C.: Adjoint functors and triples. Ill. J. Math. 9, 381–398 (1965) [Go] Godement, R.: Théorie des faisceaux. Paris: Hermann, 1957 [HKRS1] Hajac, P.M., Khalkhali, M., Rangipour, B., Sommerhäuser, Y.: Stable anti-Yetter-Drinfeld modules. C. R. Math. Acad. Sci. Paris 338, 587–590 (2004) [HKRS2] Hajac, P.M., Khalkhali, M., Rangipour, B., Sommerhäuser, Y.: Hopf-cyclic homology and cohomology with coefficients. C. R. Math. Acad. Sci. Paris 338, 667–672 (2004) [H] Hobst, D.: Antipodes in the theory of noncommutative torsors. PhD thesis, Ludwig-Maximilians Universität München, 2004, Berlin: Logos Verlag, 2004 [JS] ¸ Jara, P., Stefan, ¸ D.: Hopf-cyclic homology and relative cyclic homology of Hopf-Galois extensions. Proc. London Math. Soc. (3) 93, 138–174 (2006) [Kad] Kadison, L.: Cyclic homology of triangular matrix algebras, In: Topology Hawaii (Honolulu, HI, 1990), Singapore: World Sci., 1992 [KSz] Kadison, L., Szlachányi, K.: Bialgebroid actions on depth two extensions and duality. Adv. Math. 179, 75–121 (2003) [KLV] Kasangian, S., Lack, S., Vitale, E.: Coalgebras, braidings, and distributive laws. Theory and Applications of Categories 13, 129–146 (2004) [Kass] Kassel, C.: Quantum groups. Graduate Text in Mathematics 155. Berlin-Heidelberg-New York: Springer, 1995 [Kay05] Kaygun, A.: Bialgebra cyclic homology with coefficients. K -Theory 34, 151–194 (2005) [Kay06] Kaygun, A.: The universal Hopf cyclic theory. To appear in J. Noncommut. Geom., available at http://arxiv.org/list/math/0609311 [KR03] Khalkhali, M., Rangipour, B.: Invariant cyclic homology. K -Theory 28, 183–205 (2003) [KR04] Khalkhali, M., Rangipour, B.: Para-Hopf algebroids and their cyclic cohomology. Lett. Math. Phys. 70, 259–272 (2004) [KR05] Khalkhali, M., Rangipour, B.: A note on cyclic duality and Hopf algebras. Comm. Algebra 33, 763–773 (2005) [LPvO] Lopez, J., Panaite, F., van Oystaeyen, F.: General twisting of algebras. Adv. Mathematics 212, 315–337 (2007) [Lo] Loday, J.-L.: Cyclic Homology. Berlin-Heidelberg-New York: Springer-Verlag, 1992 [Lu] Lu, J.H.: Hopf algebroids and quantum groupoids. Int. J. Math. 7, 47–70 (1996) [MacL] Mac Lane, S.: Categories for the working mathematician. Berlin-Heidelberg-New York: SpringerVerlag, 1974 [Mj] Majid, S.: Foundations of quantum group theory. Cambridge: Cambridge University Press, 1995 [Ra] Rangipour, B.: Cyclic cohomology of corings. http://arxiv.org/list/math/0607248, 2006 [Sch98] Schauenburg, P.: Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules. Appl. Categorical Str. 6, 193–222 (1998) [Sch00] Schauenburg, P.: Duals and doubles of quantum groupoids (× R -Hopf algebras). In: Andruskiewitsch, N., Ferrer-Santos, W.R., Schneider, H.-J. (eds.), AMS Contemp. Math. 267, RI: Amer. Math. Soc., Providence 2000, pp. 273–293 [Šk] Škoda, Z.: Cyclic structures for simplicial objects from comonads. http://arxiv.org/list/math/ 0412001, 2004 [Ta] Takeuchi, M.: Groups of algebras over A ⊗ A. J. Math. Soc. Japan 90, 459–492 (1977) [V] Vecsernyés, P.: Larson-Sweedler theorem and the role of grouplike elements in weak Hopf algebras. J. Algebra 270, 471–520 (2003) [We] Weibel, C.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics 38. Cambridge: Cambridge, University Press, 1994 Communicated by A. Connes
Commun. Math. Phys. 282, 287–322 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0552-z
Communications in
Mathematical Physics
The Pauli Principle Revisited Murat Altunbulak, Alexander Klyachko Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey. E-mail: [email protected]; [email protected] Received: 22 November 2006 / Accepted: 25 February 2008 Published online: 4 July 2008 – © Springer-Verlag 2008
Abstract: By the Pauli exclusion principle, no quantum state can be occupied by more than one electron. One can state this as a constraint on the one electron density matrix that bounds its eigenvalues by 1. Shortly after its discovery, the Pauli principle was replaced by anti-symmetry of the multi-electron wave function. In this paper we solve a longstanding problem about the impact of this replacement on the one electron density matrix, that goes far beyond the original Pauli principle. Our approach uses Berenstein and Sjamaar’s theorem on the restriction of an adjoint orbit onto a subgroup, and allows us to treat any type of permutational symmetry. Electronic Supplementary Material: The online version of this article (doi:10.1007/s00220-008-0552-z) contains supplementary material, which is available to authorized users. Contents 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . . . A Review of the Berenstein-Sjamaar Paper One Point ν-Representability . . . . . . . Beyond the Basic Constraints . . . . . . . Connection with Representation Theory . . Analysis of Some Small Systems . . . . .
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1. Introduction The Pauli exclusion principle, discovered in 1925, claims that no quantum state can be occupied by more than one electron. In terms of the electron density matrix1 ρ 1 There is no agreement on a proper normalization of the one-electron matrix. To avoid confusion we call it electron density matrix for Dirac’s normalization to the number of particles Tr ρ = N , and reserve the term reduced state for the probability normalization Tr ρ = 1.
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this amounts to the inequality ψ|ρ|ψ ≤ 1, that bounds its eigenvalues by one. The following year Heisenberg and Dirac replaced the Pauli principle by skew symmetry of a multi-electron wave function [11, Ch. 4]. The subject of this study is the impact of this replacement on the electron density matrix. The latter determines the light scattering and therefore quite literally represents a visible state of the electron system. The impact goes far beyond the original Pauli principle. As an example, consider a three electron system ∧3 H6 with one-electron space H6 of dimension 6. Then the spectrum λ of the electron density matrix, arranged in non-increasing order, is bounded by the following (in)equalities discovered by Borland and Dennis [3]: λ1 + λ6 = λ2 + λ5 = λ3 + λ4 = 1, λ4 ≤ λ5 + λ6 .
(1)
The authors established the sufficiency of these constraints and refer for a complete proof to M.B. Ruskai and R.L. Kingsley.2 It worth reading their comment: We have no apology for consideration of such a special case. The general N -representability problem is so difficult and yet so fundamental for many branches of science that each concrete result is useful in shedding light on the nature of general solution. In spite of some bogus claims [29], refuted in [32], this result had stood for more than three decades as the only known solution of the N -representability problem beyond two electrons ∧2 Hr and two holes ∧r −2 Hr . For the latter systems the problem is easy and the constraints amount to double degeneracy of the spectrum, starting from the head λ2i−1 = λ2i for two electrons and from the tail λr −2i = λr −2i−1 for two holes [5], where we set λi = 0 for i > r , and λi = 1 for i < 1. Here we solve this longstanding problem. The content of the paper is as follows. In Sect. 2 we recast the Berenstein-Sjamaar theorem [1, Thm 3.2.1] into a usable form (Theorem 1). This provides a theoretical basis for our study. We start Sect. 3 by a variation of the above problem, called ν-representability, that takes into account both spin and orbital occupation numbers. Mathematically this amounts to replacing the exterior power ∧ N H by a representation Hν defined by a Young diagram ν of order N . Theorem 2 gives a formal solution of the ν-representability problem. We derive from it the majorization inequality λ ν, that plays the rôle of the Pauli principle. This inequality is necessary and sufficient for λ to be occupation numbers of an unspecified mixed state (Theorem 3). Theorem 4 deals with a class of systems where the majorization inequality alone provides a criterion for pure ν-representability. This includes the so-called closed shell, meaning a system of electrons of total spin zero. The corresponding Young diagram ν consists of two columns of equal length. For this system all constraints on the occupation numbers are given by the Pauli type inequality v (a) that governed λ ≤ 2. Next in Theorem 5 we calculate the topological coefficients cw the constraints on the occupation numbers in Theorem 2. This gives it the full strength we need in the next section. Section 4 starts with analysis of pure ν-representability for a toy example of two-row diagrams, that allows us to illustrate the basic technique (Theorem 6). These are exceptional systems where the constraints on the occupation numbers are given by a finite set 2 Recently M.B. Ruskai published the proof [33] derived from known constraints on the spectra of Hermitian matrices A, B, and C = A + B. Conceptually the N -representability problem is close to the Hermitian spectral problem [15,16], but a direct connection between them, beyond sporadic coincidences, is unlikely. R.L. Kingsley’s independent solution apparently has never been published.
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of inequalities independent of the rank. Then we return to the original N -representability problem, that appears to be the most difficult one. For example, in contrast to Theorem 6, no finite system of inequalities can describe N -representability for a fixed N > 1 and arbitrary big rank (Corollary 3 to Proposition 5). This forces us to restrict either the rank, as we do in the last section, or the type of the inequalities. Here we focus on the inequalities with 0/1 coefficients. It turns out that under some natural conditions such an inequality should be either of the form λi1 + λi2 + · · · + λi N −1 ≤ N − 2, with
k (i k
(2)
− k) = r − N + 1, or of the form λi1 + λi2 + · · · + λi p ≤ N − 1,
(3)
with p ≥ N and k (i k − k) = Np . We call them Grassmann inequalities of the first and second kind respectively. A surprising result is that these inequalities actually hold true with very few exceptions (Theorems 7 and 8). In the simplest case N = 3 we get from (2) inequalities λk+1 + λr −k ≤ 1, 0 ≤ k < (r − 1)/2 that hold for any even rank r ≥ 6. This constraint prohibits more than one electron to occupy two symmetric orbitals and supersedes the original Pauli principle. For r = 6, due to the normalization i λi = 3, the inequalities degenerate into Borland-Dennis equalities (1). For odd rank the first inequality k = 0 should be either skipped or replaced 2 by the weaker one λ1 + λr ≤ 1 + r −1 . We treat Grassmann inequalities of the second kind (3) only for lowest levels p = N , N + 1. For N = 3 and p = N + 1 they amount to four inequalities: λ2 + λ3 + λ4 + λ5 ≤ 2, λ1 + λ2 + λ5 + λ6 ≤ 2,
λ1 + λ3 + λ4 + λ6 ≤ 2, λ1 + λ2 + λ4 + λ7 ≤ 2,
(4)
that hold for arbitrary rank r and give all the constraints for r ≤ 7. For r = 6 they turn into Borland-Dennis conditions (1). In Sect. 5 we briefly discuss a connection of the ν-representability with representation theory, that provides information complementary to Theorem 2. A combination of the two approaches leads to an algorithm for solution of the problem for any fixed rank. The algorithm, along with other tools, has been used in calculations reported in the last Sect. 6. Eventually this led to a complete solution of the N -representability problem for rank r ≤ 10. However, we provide a rigorous justification only for r ≤ 8. We also give an example of constraints on the spin and orbital occupation numbers for a system of three electrons of total spin 1/2. The first sections may be mathematically more demanding than the rest of the paper. We recommend books [7–9] as general references on Schubert calculus, Lie algebra, and representation theory. The theoretical results of the paper belong to the second author. They were often inspired by calculations, that at this stage couldn’t be accomplished by a computer without intelligent human assistance and insight.
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2. A Review of the Berenstein-Sjamaar Paper Let M be a compact connected Lie group with the Lie algebra m and its dual coadjoint representation m∗ . For coadjoint orbit O ⊂ m∗ of group M and a Cartan subalgebra t ⊂ m consider the composition : O → m∗ → t∗ known as the moment map. By Kostant’s theorem its image is a convex polytope spanned by the W -orbit of some weight µ ∈ t∗ which can be taken from a fixed positive Weyl chamber t∗+ . Here W = N (t)/Z (t) is the Weyl group of M. This gives a parameterization of the coadjoint orbits Oµ by the dominant weights µ ∈ t∗+ . Example 1. In this paper we will mostly deal with the unitary group U(n) whose Lie algebra u(n) consists of all Hermitian3 n × n matrices. Let us identify u(n) with its dual via the invariant trace form (A, B) = Tr(AB). Then the (co)adjoint orbit Oµ consists of all Hermitian matrices A of spectrum µ : µ1 ≥ µ2 ≥ · · · ≥ µn and the moment map : Oµ → t is given by orthogonal projection into the Cartan subalgebra of diagonal matrices t. Kostant’s theorem in this case amounts to Horn’s observation that the diagonal entries of Hermitian matrices of spectrum µ form a convex polytope with vertices wµ obtained from µ by permutations of the coordinates µi . This is equivalent to the majorization inequalities d1 d1 + d2 d1 + d2 + d3 ··· d1 + d2 + · · · + dn
≤ ≤ ≤ ··· =
µ1 , µ1 + µ2 , µ1 + µ2 + µ3 , ··· µ1 + µ2 + · · · + µn
(5)
for the diagonal entries d : d1 ≥ d2 ≥ · · · ≥ dn of matrix A. We will use for them a shortcut d µ. Consider now an immersion f : L → M of another compact Lie group L and the induced morphisms f ∗ : l → m and f ∗ : m∗ → l∗ of the Lie algebras and their duals. In the paper [1] Berenstein and Sjamaar found a decomposition of the projection f ∗ (Oµ ) ⊂ l∗ of an M-orbit Oµ ⊂ m∗ into L-orbits Oλ ⊂ f ∗ (Oµ ). Here we paraphrase their main result in a form suitable for the intended applications. Fix Cartan subalgebras t L → t M , and for every test spectrum a ∈ t L consider the inclusion of the adjoint orbits of groups L and M, ϕa : Oa → O f∗ (a) ,
(6)
through a and f ∗ (a) respectively. Topologically the orbits are (generalized) flag varieties. They carry a hidden complex structure coming from the representation Oa = L/Z L (a) = L C /Pa ,
(7)
where Pa ⊂ L C is a parabolic subgroup of the complexified group L C whose Lie algebra pa is spanned by t L and the root vectors X α such that α, a ≥ 0. One can say this in another way: Pa = {g ∈ L C | lim eta ge−ta exists}, t→−∞
which makes it clear that f : Pa → P f∗ (a) . 3 Hereafter we treat u(n) as the algebra of Hermitian, rather than skew-Hermitian, operators at the expense of a modified Lie bracket [X, Y ] = i(X Y − Y X ).
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We will use the parabolic subgroups to construct canonical bases in cohomologies H ∗ (Oa ) and H ∗ (O f∗ (a) ). Let TL ⊂ B ⊂ Pa be a Borel subgroup containing a maximal torus TL with Lie algebra t L . The flag variety Oa = L C /Pa splits into a disjoint union of Schubert cells Bv Pa /Pa , parameterized by the left cosets v ∈ W L /W Z L (a) or in practice by representatives of minimal length = (v) in these cosets. We actually prefer to deal with shifted cells v −1 Bv Pa /Pa = B v Pa /Pa depending on the Borel subgroups B v ⊃ TL modulo conjugation by the Weyl group of the centralizer W (Z L (a)). The closure of B v Pa /Pa is known as the Schubert variety, and its cohomology class σv ∈ H 2 (v) (Oa ) is called the Schubert cocycle. These cocycles form the canonical basis of the cohomology ring H ∗ (Oa ). Inclusion (6) induces a morphism of the cohomologies ϕa∗ : H ∗ (O f∗ (a) ) → H ∗ (Oa ), v (a) of the decomposition given in the canonical bases by the coefficients cw v cw (a)σv . ϕa∗ : σw →
(8)
(9)
v
They play a crucial rôle in the next theorem. We extend them by zeros if either v ∈ W L or w ∈ W M is not the minimal representative of a coset in W L /W Z L (a) or W M /W Z M ( f∗ (a)) respectively. Theorem 1. In the above notations the inclusion Oλ ⊂ f ∗ (Oµ ) is equivalent to the following system of linear inequalities λ, va ≤ µ, w f ∗ (a)
(a, v, w)
v (a) = 0. for all a ∈ t L , v ∈ W L , w ∈ W M such that cw
Proof. This is not the way Berenstein and Sjamaar stated their result. Instead, for some generic a0 ∈ t L they fix positive Weyl chambers t+L a0 and t+M f ∗ (a0 ) and use them to define Schubert cocycles σv ∈ H ∗ (Oa ) and σw ∈ H ∗ (O f∗ (a) ) for all other a ∈ t+L . Hence their Schubert cocycles σw are canonical in the above sense iff f ∗ (a) and f ∗ (a0 ) are in the same Weyl chamber. The set of such a ∈ t+L form a convex polyhedral cone called the principle cubicle. It is determined by a0 , and different choices of a0 produce a polyhedral decomposition of the positive Weyl chamber t+L into cubicles. For every cubicle Berenstein and Sjamaar gave a system of linear constraints on the dominant weights λ, µ, so that all together they provide a criterion for the inclusion Oλ ⊂ f ∗ (Oµ ). For the principal cubicle the constraints are simplest and are as follows [1, Thm 3.2.1]: v −1 λ ∈ f ∗ (w −1 µ − C),
for
v cw (a0 ) = 0,
(10)
where C is a cone spanned by the positive roots in t∗M . Note that f ∗ (C) is the cone dual to the principal cubicle and therefore the above condition can be recast into the inequalities v −1 λ, a ≤ f ∗ (w −1 µ), a ⇐⇒ λ, va ≤ µ, w f ∗ (a),
(11)
v (a ) = 0. The coefficients that hold for all a from the principle cubicle provided that cw 0 v cw (a) are actually constant inside the cubicle, and therefore the last condition can be v (a) = 0. Thus we arrive at the inequalities (a, v, w) for the principle changed to cw cubicle. Other inequalities (a, v, w) follow by choosing another cubicle as the principle
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one. They are equivalent to the remaining more complicated inequalities in [1, Thm 3.2.1], but look different since Berenstein and Sjamaar use other non-canonical Schubert cocycles. Example 2. Quantum marginal problem [17]. Let’s illustrate the above theorem with immersion of unitary groups f : U(H A ) × U(H B ) → U(H AB ),
g A × g B → g A ⊗ g B ,
where H AB = H A ⊗ H B . As we have seen in Example 1 the coadjoint orbit of U(H AB ) consists of the isospectral Hermitian operators ρ AB : H AB understood here as mixed states. The projection f ∗ (ρ AB ) = ρ A ⊗ 1 + 1 ⊗ ρ B amounts to reduced operators ρ A : H A and ρ B : H B implicitly defined by the equations Tr H A (ρ A X A ) = Tr H AB (ρ AB X A ),
Tr H B (ρ A X B ) = Tr H AB (ρ AB X B )
(12)
for all Hermitian operators X A : H A and X B : H B . This means that ρ A , ρ B are just the visible states of the subsystems H A , H B . In this setting Theorem 1 tells us that all constraints on the decreasing spectra λ AB = Spec(ρ AB ), λ A = Spec(ρ A ), and λ B = Spec(ρ B ) are given by the inequalities ↓ AB A B ai λu(i) + b j λv( (a + b)k λw(k) , (13) j) ≤ i
j
k
for all test spectra a : a1 ≥ a2 ≥ · · · ≥ an , b : b1 ≥ b2 ≥ · · · ≥ bm from the Cartan uv (a, b) = 0. Here (a + b)↓ subalgebras t A , t B and permutations u, v, w such that cw denotes the sequence ai + b j arranged in decreasing order. The order determines the canonical Weyl chamber containing f ∗ (a, b). The pairs (a, b) with fixed order of terms ai + b j in (a + b)↓ form a cubicle. The adjoint orbit Oa ⊂ u(H A ) is a classical flag variety understood as the set of Hermitian operators X A : H A of spectrum a = Spec X A . Denote it by Fa (H A ). Then the morphism (6) is given by the equation ϕab : Fa (H A ) × Fb (H B ) → Fa+b (H AB ), (X A , X B ) → X A ⊗ 1 + 1 ⊗ X B , (14) uv (a, b) are determined by the induced morphism of the cohomoand the coefficients cw logies ∗ ϕab : H ∗ (Fa+b (H AB )) → H ∗ (Fa (H A )) ⊗ H ∗ (Fb (H B )) uv σw
→ cw (a, b) · σu ⊗ σv .
(15)
u,v uv (a, b) = 1 for identical One can find the details of their calculation in [17]. Note that cw permutations u, v, w. Hence we get for free the following basic inequality: ↓ ai λiA + b j λ Bj ≤ (a + b)k λkAB , (16) i
valid for all test spectra a, b.
j
k
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3. One Point ν-Representability In this section we apply the above results to the morphism f : U(H) → U(Hν ) given by an irreducible representation Hν of group U(H) with a Young diagram ν of order N = |ν|. For a column diagram we return to the N -fermion system ∧ N H, while a row diagram corresponds to the N -boson space S N H. However, the main reason to consider the general para-statistical representations Hν is not a uniform treatment of fermions and bosons, but taking into account spin. Observe that the state space of a single particle with spin splits into the tensor product H = Hr ⊗ Hs of the orbital Hr and the spin Hs degrees of freedom. The total N -fermion space decomposes into spin-orbital components as follows [35]: t ∧ N (Hr ⊗ Hs ) = Hrν ⊗ Hsν , (17) |ν|=N
where ν t stands for the transpose diagram. In many physical systems, like electrons in an atom or a molecule, the total spin is a well defined quantity that singles out a specific component of this decomposition. Theorem 1 applied to the component gives all constraints on the possible spin and orbital occupation numbers, see the details in n ◦ 3.1.1 below. 3.1. Physical interpretation. Let’s now relate Theorem 1 to the N -representability problem and its ramifications indicated above. We’ll refer to the latter as the ν-representability problem. It is instructive to think about X ∈ u(H) as an observable and treat ρ ∈ u(H)∗ as a mixed state with the duality pairing given by the expectation value of X in state ρ, X, ρ = Tr H Xρ
(18)
(forget for a while about the positivity ρ ≥ 0 and normalization Tr ρ = 1). We want to elucidate the physical meaning of the projection f ∗ : u(Hν )∗ → u(H)∗ uniquely determined by the equation f ∗ (X ), ρ ν = X, f ∗ (ρ ν ),
X ∈ u(H),
ρ ν ∈ u(Hν )∗ .
In the above setting (18) it reads as follows: Tr Hν (Xρ ν ) = Tr H (X f ∗ (ρ ν )),
∀X ∈ u(H).
(19)
A good point to start with is Schur’s duality between irreducible representations of the unitary U(H) and the symmetric S N groups, H⊗N = Hν ⊗ S ν . (20) |ν|=N
The latter group acts on H⊗N by permutations of the tensor factors, and its irreducible representations S ν show up in the right-hand side. One can treat H⊗N as a state space of N -particles, and for identical particles all physical quantities should commute with S N . Looking into the right-hand side of (20) we see that such quantities are linear combinations of operators ρ ν ⊗ 1 acting in the component Hν ⊗ S ν and equal to zero elsewhere. In the case of a genuine mixed state ρ ν , i.e. a nonnegative operator of trace
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1, one can treat (ρ ν ⊗ 1)/ dim S ν as a mixed state of N identical particles obeying some para-statistics of type ν. Let ρi : H be its i th reduced state. Since ρ ν ⊗ 1 commutes with S N , the reduced state ρ = ρi is actually independent of i. However, occasionally we retain the index i just to indicate the tensor component where it operates. Proposition 1. In the above notations f ∗ (ρ ν ) = Nρ.
(21)
Proof. We have to check that (21) fits Eq. (19): Tr Hν (Xρ ν ) = Tr Hν ⊗S ν X
ρν ⊗ 1 ρν ⊗ 1 = Tr = Tr H X i ρi = N Tr H Xρ, ⊗N X H dim S ν dim S ν i
where X i is a copy of X acting in the Tr H⊗N X i
i th
component of
H⊗N ,
so that
ρν ⊗ 1 = Tr H X i ρi dim S ν
by definition (12) of the reduced state.
A general ν-representability problem concerns the relationship between the spectrum µ of a mixed state ρ ν and spectrum λ of its particle density matrix Nρ. The latter spectrum is known as the occupation numbers4 of the system in state ρ ν . Formally the constraints on the spectra are given by Theorem 1. Remark 1. The above construction allows for a given mixed state ρ ν to define the higher order reduced matrices. Their characterization would have almost unlimited applications. Indeed, behavior of most systems of physical interest is governed by two-particle interaction. As a result, the energy of a state becomes a linear functional of its two-point reduced matrix. To minimize the energy and to find the correlation matrix of the ground state one has to elucidate all the constraints that a two-point reduced matrix should satisfy. This problem and the whole program are known as the Coulson challenge5 [6]. In the form just described it may be unfeasible even for quantum computers [23]. For other approaches and the current state of the art see [26]. This problem is far beyond the scope of our paper. Nevertheless, the characterization of one point reduced matrices given below imposes also new constraints on the higher reduced states. 3.1.1. Constraints on spin and orbital occupation numbers Let’s return to a system of N fermions, this time of smallest possible spin s = 1/2, dim Hs = 2. In this case spin-orbital decomposition (17) involves only terms Hrν ⊗ Hsν
t
(22)
with at most a two-column diagram ν. The sizes of the columns α ≥ β are determined by equations α + β = N,
α − β = 2J,
(23)
4 More precisely, the occupation numbers of natural orbitals. The latter are defined as eigenvectors of the particle density matrix. 5 Also known as two-particle N -representability or, following D. Herschbach, a holy grail of theoretical chemistry.
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where J is the total spin of the system, so that Hsν = H J is just the spin J representation of the group SU(Hs ) = SU(2). Consider now a pure N -fermion state of total spin J t
ψ ∈ Hrν ⊗ H J , where the diagram ν is determined by Eqs. (23). Let ρ ν and ρ J be its reduced states in the orbital and spin components respectively. The basic fact is that the reduced states are isospectral Spec ρ ν = Spec ρ J . Hence Spec ρ ν can be identified with the spin occupation numbers. On the other hand Theorem 1, in view of Proposition 1, relates Spec ρ ν with the orbital occupation numbers given by the spectrum of the particle density matrix Nρ. In this way one can produce all constraints on allowed spin and orbital occupation numbers, provided that a solution of the ν-representability problem is known for twocolumn diagrams. We address this issue in Sects. 3.2 and 3.3. See also Corollary 1 in Sect. 3.2. 3.2. Formal solution of the ν-representability problem. Henceforth we treat the lower index r as the rank of the Hilbert space Hr . Recall that the character of the representation Hrν , i.e. the trace of a diagonal operator z = diag(z 1 , z 2 , . . . , zr ) ∈ U(Hr ),
(24)
in some orthonormal basis e of Hr , is given by Schur’s function Sν (z 1 , z 2 , . . . , zr ). It has a purely combinatorial description in terms of the so called semistandard tableaux T of shape ν. The latter are obtained from the diagram ν by filling it with numbers 1, 2, . . . , r strictly increasing in columns andweakly in rows. Then the Schur function can be written as a sum of monomials z T = i∈T z i , Sν (z) = zT , T
corresponding to all semistandard tableaux T of shape ν. The monomials are actually the weights of representation Hrν , meaning that z · eT = z T eT
(25)
for some basis eT of Hrν parameterized by the semistandard tableaux. Denote by t ⊂ u(Hr ) and tν ⊂ u(Hrν ) the Cartan subalgebras of real diagonal operators in the bases e and eT respectively, so that the differential of the above group action z : eT → z T eT gives the morphism
f ∗ : t → tν ,
f ∗ (a) : eT → aT eT ,
(26)
where aT := i∈T ai . As in Example 2 we treat the orbits Oa and O f ∗ (a) as flag varieties Fa (Hr ) and Fa ν (Hrν ) consisting of Hermitian operators of spectra a : a1 ≥ a2 ≥ · · · ≥ ar and a ν respectively. Here a ν consists of the quantities aT arranged in the non-increasing order a ν := {aT | T = semistandard tableau of shape ν}↓ .
(27)
Finally, we need the morphism ϕa : Fa (Hr ) → Fa ν (Hrν ),
X → f ∗ (X ),
(28)
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together with its cohomological version ϕa∗ : H ∗ (Fa ν (Hrν )) → H ∗ (Fa (Hr )), v (a): given in the canonical bases by coefficients cw v cw (a)σv . ϕa∗ : σw →
(29)
(30)
v
Theorem 2. In the above notations all constraints on the occupation numbers λ of the system Hrν in a state ρ ν of spectrum µ are given by the inequalities ai λv(i) ≤ akν µw(k) (31) i
k
v (a) = 0. for all test spectra a and permutations v, w such that cw
Proof. In view of Proposition 1, this is what Theorem 1 tells. One has to remember that the left action of a permutation on “places” is inverse to its right action on indices. That is why the permutations v and w, acting on a and f ∗ (a) = a ν in Theorem 1, move to the indices of λ and µ in the inequality (31). v (a) depends only on the order in which quantities a appear in the The coefficient cw T ν spectrum a . The order changes when the test spectrum a crosses a hyperplane ai = aj. HT |T : i∈T
j∈T
The hyperplanes cut the set of all test spectra into a finite number of polyhedral cones called cubicles. For each cubicle one has to check the inequality (31) only for its extremal edges. As a result, the ν-representability amounts to a finite system of linear inequalities. Remark 2. Let’s emphasize once again the difference between the Berenstein-Sjamaar Theorem [1, Thm 3.2.1] and its version used in this paper. In the settings of Theorem 2 it manifests itself in the way the quantities aT are ordered in the spectrum a ν , or which parabolic subgroup is used for definition of Schubert cocycles. Berenstein and Sjamaar choose a specific order of tableaux T , while we rely on the natural order of the quantities aT = i∈T ai . The latter choice allows to treat the inequalities uniformly, and to avoid a rather cumbersome transformation every time the test spectrum passes from one cubicle to another. Recall from Sect. 3.1.1 that the theorem also describes a relationship between the spin and orbital occupation numbers. We keep for them the above notations µ and λ respectively. Corollary 1. All constraints on spin and orbital occupation numbers of N -electron system in a pure state of total spin J are given by the inequalities (31), applied to two column diagram ν determined by Eqs. (23), and bounded to mixed states ρ ν of rank not exceeding dimensionality 2J + 1 of the spin space. v (a) to Sect. 3.3 and focus instead We postpone the calculation of the coefficients cw on some general results that can be deduced from the theorem as it stands.
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3.2.1. Basic inequalities Being a ring homomorphism, ϕa∗ maps unit into unit ϕa∗ (1) = 1, v (a) = 1 for identical permutations v, w. Hence the following basic inequality that is cw ai λi ≤ akν µk i
k
holds for all test spectra a. Let’s look at it more closely for a pure state ρ ν = |ψψ| in which case the right-hand side is maximal and the inequality takes the form ai λi ≤ a1ν = max ai = ai νi , (32) T
i
i∈T
i
where ν1 ≥ ν2 ≥ · · · ≥ 0 are rows of ν. The maximum in the right-hand side is attained for the tableau T of shape ν whose i-row is filled by i. The normalization i λi = N = j ν j allows to shift the test spectra into the positive domain a1 ≥ a2 ≥ · · · ≥ 0, so that they become nonnegative linear combinations of the fundamental weights ωk = (1, 1, . . . , 1, 0, 0, . . . , 0).
(33)
k
Hence it is enough to check (32) for a = ωk , that gives the majorization inequality λ ν, cf. Example 1. Thus we arrive at the first claim of the following result that characterizes occupation numbers of system Hν in an unspecified mixed state. Theorem 3. The occupation numbers of the system Hν in an arbitrary mixed state satisfy the majorization inequality λ ν,
(34)
and any such λ can be realized as the occupation numbers of some mixed state. Proof. The second claim follows from two observations: 1. The occupation numbers of a coherent state ψ ∈ Hν , that is a highest vector of the representation, are equal to ν. 2. The set of allowed occupation numbers, written in any order, form a convex set. Indeed, the polytope given by the majorization inequality (34) is just a convex hull of vectors obtained from ν by permutations of coordinates, cf. Example 1. Hence by 1 and 2 it consists of legitimate occupation numbers. Proof of 1. Consider a decomposition of the complexified Lie algebra u(H) ⊗ C = gl(H) = n− + h + n+ , into a diagonal Cartan subalgebra h = t ⊗ C accompanied with lower- and uppertriangular nilpotent subalgebras n∓ . By definition n+ annihilates the highest vector ψ ∈ Hν of weight ν. Hence ψ|X ± |ψ = X ∓ ψ|ψ = 0 for all X ± ∈ n± . Then by Eq. (19) ψ|X ± |ψ = Tr Hν (X ± |ψψ|) = Tr H (X ± f ∗ (|ψψ|)) = 0,
∀X ± ∈ n± .
This means that ρ = f ∗ (|ψψ|) is a diagonal matrix. On the other hand tψ = t, νψ for t ∈ t, hence as above t, ν = ψ|t|ψ = Tr Hν (t|ψψ|) = Tr H (t f ∗ (|ψψ|)) = Tr H (tρ) = t, ρ, that is Spec ρ = ν.
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Proof of 2. Let ρ1ν , ρ2ν be mixed states, with the particle densities ρ1 , ρ2 , and the occupation numbers λ1 , λ2 . We apply to ρ1 , ρ1ν a unitary rotation ρ1 → Uρ1 U ∗ , ρ1ν → Uρ1ν U ∗ that transforms orthonormal eigenvectors of ρ1 into that of ρ2 in a prescribed order. The resulting new operators ρ1 , ρ2 commute and have the original spectra λ1 , λ2 . Then the particle density matrix ρ = p1 ρ1 + p2 ρ2 of the convex combination ρ ν = p1 ρ1ν + p2 ρ2ν has spectrum λ = p1 λ1 + p2 λ2 . For a column diagram ν the majorization inequality λ ν amounts to the Pauli exclusion principle λi ≤ 1. In general, we refer to it as the Pauli constraint. Note that the above proof shows that equality in (34) is attained for coherent states only. The second part of Theorem 3 extends Coleman’s result [5] for ∧ N H. Recall, that the theorem solves the ν-representability problem for unspecified mixed states. We will see later that for pure states the answer in general is much more complicated. Nevertheless, there are surprisingly many systems for which the majorization inequality alone is sufficient for pure ν-representability. We address them in the next item. 3.2.2. Pure moment polytope One of the most striking features of Theorem 2 is the linearity of the constraints (31). As a result, the allowed spectra (λ, µ) form a convex polytope, called (noncommutative) moment polytope. The convexity still holds for any fixed µ = Spec ρ ν , and in particular for the occupation numbers λ of all pure states. We refer to the latter case as the pure moment polytope. It sits inside the positive Weyl chamber, and its multiple kaleidoscopic reflections in the walls of the chamber generally form a nonconvex rosette, consisting of all legitimate occupation numbers written in an arbitrary order. It can be convex only if all constraints on the occupation numbers are given by the majorization inequality λ ν alone. Here we describe a class of representations Hν with this property. This happens, for example, for a system of N ≥ 2 bosons. In this case ν is a row diagram and the majorization inequality imposes no constraints on λ. By Theorem 3 this means that every nonnegative spectrum λ of trace N represents occupation numbers of some mixed state. However for bosons one can easily find a pure state that does the job:
ψ= λi ei⊗N ∈ S N H, i
where ei is an orthonormal basis of H. This makes the bosonic N -representability problem trivial. A more interesting physical example constitutes the so-called closed shell, meaning a system of electrons of total spin zero. The corresponding diagram ν consists of two columns of equal length. We will see shortly that in this case the Pauli constraint λ ≤ 2 shapes the pure moment polytope. Observe that it is enough to construct pure states whose occupation numbers are generators of the cone cut out of the Weyl chamber by the majorization inequality λ ν. Then the convexity does the rest. Recall that in the proof of Theorem 3 we have already identified ν with the occupation numbers of a coherent state. Due to the majorization inequality λ ν, the entropy of its reduced state is minimal possible. For that reason coherent states are generally considered closest to classical ones [30]. At the other extreme one finds the so-called completely entangled states ψ ∈ Hν whose particle density matrix ρ = f ∗ (|ψψ|) is scalar and the reduced entropy is maximal [19]. By definition (19) we have
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TrH (Xρ) = TrHν (X |ψψ|) = ψ|X |ψ, so that the completely entangled states can be described by the equation ψ|X |ψ = 0,
∀ X ∈ su(H).
(35)
Let’s call a system Hrν exceptional if the SU(Hr )-representation Hrν is equivalent to one of the following: Hr , its dual Hr∗ , and, for odd rank r , ∧2 Hr , ∧2 Hr∗ . The Young diagram ν of an exceptional system can be obtained from an r × m rectangle by adding an extra column of length 1, r − 1, 2, r − 2 respectively. One readily realizes that the exceptional systems contain no completely entangled states, say because the reduced matrix of ψ ∈ ∧2 Hr has an even rank. Proposition 2. In every non-exceptional system Hν there exists a completely entangled state. Proof. The result is actually well known, but in a different context. The entanglement equation (35) is nothing but the stationarity condition for the length of vector ψ|ψ with respect to the action of the complexified group SL(H). It is known [34] that every stationary point is actually a minimum, and an SL(H)-orbit contains a minimal vector if and only if the orbit is closed. As a result, we end up with the problem of existence of a nonzero closed orbit, or, what is the same, the existence of a nonconstant polynomial invariant. The proposition just reproduces a known answer to the latter question [34]. By admitting other simple Lie groups we find only two more exceptional representations: the standard representation of the symplectic group Sp(n) and a halfspinor representation of Spin(10). Now we can solve the pure ν-representability problem for a wide class of systems, including the above mentioned closed shell. Theorem 4. Suppose that all columns of Young diagram ν are multiple, meaning that every number in the sequence of column lengths ν1t ≥ ν2t ≥ ν3t ≥ · · · appears at least twice. Then all constraints on the occupation numbers of the system Hν in a pure state are given by the majorization inequality λ ν alone. Proof. We’ll proceed by induction on the height of the diagram ν. The triviality of the bosonic N -representability problem provides a starting point for the induction. Let now λ be a vertex of the polytope cut out of the positive Weyl chamber by the majorization inequality λ ν. Note that the latter includes the equation Tr λ = Tr ν. Then the following alternative holds: 1. Either all nonzero components of λ are equal, 2. Or one can split λ and ν into two parts λ = λ |λ , ν = ν |ν containing the first p components and the remaining ones, both satisfying the inequalities λ ν , λ ν . Indeed, the second claim states that the p th majorization inequality in (5) turns into an equation. On the other hand, if all the majorization inequalities are strict, and λ contains different nonzero entries, then one can linearly vary these entries preserving the nonincreasing order of λ and the majorization λ ν. As result we get a line segment in the polytope containing λ, which is impossible for a vertex. We have to prove that every vertex λ represents occupation numbers of some pure state. Consider the above two cases separately.
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Case 1. Let λ contain r equal nonzero entries and Hr ⊂ H be a subspace of dimension r . The conditions of the theorem ensure that the system Hrν is non-exceptional, hence by Proposition 2 it contains a state ψ ∈ Hrν with occupation numbers equal to the nonzero part of λ. In the bigger system Hν ⊃ Hrν its occupation numbers will be extended by zeros. Case 2. Let the system have rank r = p + q. Choose a decomposition Hr = H p ⊕ Hq and consider a restriction of the representation Hrν onto subgroup U(H p ) × U(Hq ) Hrν =
µ,π
ν cµπ Hµp ⊗ Hqπ ,
(36)
ν where cµπ are the omnipresent Littlewood-Richardson coefficients. Observe that ν cν ν = 1, and therefore Hνp ⊗ Hqν ⊂ Hrν . By the induction hypothesis there exist states ψ ∈ Hνp and ψ ∈ Hqν with occupation numbers λ , λ and particle densities ρ , ρ respectively. Then decomposable state ψ = ψ ⊗ ψ has particle density ρ ⊕ ρ , and its occupation numbers are equal to λ = λ |λ .
Let’s extract for reference a useful corollary from the last part of the proof. ν is nonzero. Then Corollary 2. Suppose that the Littlewood-Richardson coefficient cµπ µ π merging the occupation numbers λ , λ of the systems H p , Hq form legitimate occupation numbers of the system Hνp+q .
Remark 3. The restriction on the column’s multiplicities of diagram ν is needed only to ensure that the components of any splitting ν = ν |ν |ν | . . . are non-exceptional. The latter condition holds for any two-row diagram [α, β], β = 1 for dim H ≥ 3. This gives examples of systems beyond Theorem 4, say for ν = [3, 2], whose pure moment polytope is given by the majorization inequality alone. More such diagrams can be produced as follows: take ν as in Theorem 4 and remove one cell from its last row. This works when the last row contains at least three cells and the rank of the system is bigger than the height of ν. A complete classification of all such systems is still missing. 3.2.3. Dadok-Kac construction In the last two theorems we encounter the problem of constructing a pure state with given occupation numbers. The problem lies at the very heart of the ν-representability and one shouldn’t expect an easy solution. Nevertheless, there is a combinatorial construction that produces a state with diagonal density matrix, whose spectrum can be easily controlled. It has been used first by Borland and Dennis [3] to forecast the structure of the moment polytope for small fermionic systems. Later on Müller [27] formalized and advanced their approach to the limit. It fits into a general Dadok-Kac construction [10] that works for any representation. Below we follow the notations introduced at the beginning of Sect. 3.2. Let x = diag(x1 , x2 , . . . , xr ) be a typical element from the Cartan subalgebra t ⊂u(Hr ). For a given semi-standard tableau T call the linear form ωT : x → x T = i∈T xi the weight of the basic vector eT ∈ Hrν . We also need nonzero weights of the adjoint representation αi j : x → xi − x j , i = j called roots. Let’s turn the set of semi-standard tableaux of shape ν into a graph by connecting T and T each time ωT − ωT is a root, i.e. the contents of T and T , considered as multi-sets, differ by exactly one element.
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Proposition 3. Let T be a set of semi-standard tableaux of shape ν containing no connec ted pairs. Then every state ψ = c e ∈ Hν with support T has a diagonal T T T ∈T particle density matrix with entries λi = |cT |2 , (37) T i
where every tableau T is counted as many times as the index i appears in it. Proof. The proof refines the arguments used in Claim 1 of Theorem 3, from which we borrow the notation. As in the above theorem we have to prove ψ|X |ψ = 0 for every X ∈ n+ + n− . It is enough to consider root vectors X α that form a basis of n+ + n− . Then ψ|X α |ψ = c T cT eT |X α |eT . T,T ∈T
Since X α eT has weight α + ωT , it is orthogonal to eT , except for ωT = ωT + α. The latter is impossible for T, T ∈ T, and therefore the reduced state of ψ is diagonal. A straightforward calculation gives the diagonal entries (37). We’ll have a chance to use this construction in Sect. 4.1. Note that for a fixed support T the set of unordered spectra (37) form a convex polytope. It is not known when this approach exhausts the whole moment polytope. The smallest fermionic system where it fails is ∧3 H8 , see Sect. 6. v (a). To progress further and to give Theorem 1 3.3. Calculation of the coefficients cw v (a). Berenstein and Sjamaar left this full strength one has to calculate the coefficients cw problem mostly untouched. However, in the ν-representability settings, highlighted in Theorem 2, this can be done very explicitly.
3.3.1. Canonical generators To proceed we first need an alternative description of the cohomology of the flag variety Fa (Hr ) [2]. Recall that the latter is understood here as the set of Hermitian operators in Hr of given spectrum a. To avoid technicalities, we assume the spectrum to be simple, a1 > a2 > · · · > ar . Let Ei be the eigenbundle on Fa (Hr ) whose fiber at X ∈ Fa (Hr ) is the eigenspace of operator X with eigenvalue ai . Their Chern classes xi = c1 (Ei ) generate the cohomology ring H ∗ (Fa (Hr )) and we refer to them as the canonical generators. The elementary symmetric functions σi (x) of the canonical generators are the characteristic classes of the trivial bundle Hr and thus vanish. This identifies the cohomology with the ring of coinvariants H ∗ (Fa (Hr )) = Z[x1 , x2 , . . . , xr ]/(σ1 , σ2 , . . . , σr ).
(38)
This approach to the cohomology is more functorial and for that reason leads to an easy calculation of the morphism (29), ϕa∗ : H ∗ (Fa ν (Hν )) → H ∗ (Fa (H)).
Recall that the spectrum a ν consists of the quantities aT = i∈T ai arranged in decreasing order, where T runs over all semi-standard tableaux of shape ν. We define x T = i∈T xi in a similar way.
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Proposition 4. Let xi and xkν be the canonical generators of H ∗ (Fa (H)) and H ∗ (Fa ν (Hν )) respectively. Then ϕa∗ (xkν ) = x T ,
when akν = aT .
(39)
In other words, ϕa∗ (xkν ) is obtained from akν by the substitution ai → xi . Proof. The eigenbundle Ei is equivariant with respect to the adjoint action X → u X u ∗ of the unitary group U(H). Therefore it is uniquely determined by the linear representation of the centralizer D = Z (X ) in a fixed fiber Ei (X ) or by its character εi : D → S1 = {z ∈ C∗ | |z| = 1}. In the eigenbasis e of the operator X the centralizer becomes a diagonal torus with typical element z = diag(z 1 , z 2 , . . . , zr ) and the character εi : z → z i . Let now X ν = ϕa (X ), D ν = Z (X ν ), and eT be the weight basis of Hν , introduced in Sect. 3.2, parameterized by the semi-standard tableaux T of shape ν and arranged in the order of eigenvalues a ν . Then the character of the pull-back ϕa−1 (Ekν ) is just the weight i∈T εi of the k th vector eT , where the tableau T is determined from the equation akν = aT , cf. (25). Thus ϕa−1 (Ekν ) = i∈T Ei , and we finally get ϕa∗ (xkν ) = ϕa∗ (c1 (Ekν )) = c1 (ϕa−1 (Ekν )) = c1 (
Ei ) =
i∈T
xi = x T .
i∈T
Remark 4. Formula (39) may look ambiguous for a degenerate spectrum a, while in fact it is perfectly self-consistent. Indeed, consider a small perturbation a, ˜ resolving multiple components of a, and the natural projection π : Fa˜ (H) → Fa (H) that maps X = i a˜ i |ei ei | into X = i ai |ei ei |, where ei is an orthonormal eigenbasis of X . It is known [2] that π induces the isomorphism π ∗ : H ∗ (Fa (H)) H ∗ (Fa˜ (H))W (D) ,
(40)
where the right-hand side denotes the algebra of invariants with respect to permutations of the canonical generators x˜i with the same unperturbed eigenvalue ai = α. Such
= Z ( permutations form the Weyl group W (D) of the maximal torus D X ) in D = Z (X ). For example, characteristic classes of the eigenbundle Eα with the multiple eigenvalue α = ai correspond to elementary symmetric functions of the respective variables x˜i . Equation (39), as it stands, depends on a specific ordering of the unresolved spectral values ai and akν . However, when ϕa∗ is applied to invariant elements with respect to the above Weyl group, the ambiguity vanishes. Note also that the Schubert cocycle σw ∈ H ∗ (Fa˜ (H)) is invariant with respect to W (D) if and only if w is the shortest representative in its left coset modulo W (D). Such cocycles form the canonical basis of cohomology H ∗ (Fa (H)).
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v (a) we have to return to 3.3.2. Schubert polynomials To calculate the coefficients cw the Schubert cocycles σw and express them via the canonical generators xi . This can be accomplished by the divided difference operators
f (. . . , xi , xi+1 , . . .) − f (. . . , xi+1 , xi , . . .) xi − xi+1
∂i : f (x1 , x2 , . . . , xn ) →
(41)
as follows. Write a permutation w ∈ Sn as a product of the minimal number of transpositions si = (i, i + 1), w = si1 si2 · · · si .
(42)
The number of factors (w) = #{i < j | w(i) > w( j)} is called the length of the permutation w. The product ∂w := ∂i1 ∂i2 · · · ∂i is independent of the reduced decomposition and in terms of these operators the Schubert cocycle σw is given by the equation σw = ∂w−1 w0 (x1n−1 x2n−2 · · · xn−1 ),
(43)
where w0 = (n, n − 1, . . . , 2, 1) is the unique permutation of maximal length. The right-hand side of Eq. (43) makes sense for independent variables xi and in this setting it is called the Schubert polynomial Sw (x1 , x2 , . . . , xn ), deg Sw = (w). They were first introduced by Lascoux and Schützenberger [21,22] who studied them in a series of papers. See [24] for further references and a concise exposition of the theory. We borrow from [21] the following table, in which x, y, z stand for x1 , x2 , x3 : w 3210 2310 3120 3201 1320 2130
Sw x 3 y2 z x 2 y2 z x 3 yz x 3 y2 x 2 yz + x y 2 z x 2 yz
w 2301 3021 3102 1230 0321 1302
Sw x 2 y2 x3 y + x3z x3 y x yz x 2 y + x 2 z + x y2 x 2 y + x y2
w 2031 2103 3012 0231 0312 1032
Sw x2 y + x2z x2 y x3 x y + yz + zx x 2 + x y + y2 x2 + xy + xz
w 1203 2013 0132 0213 1023 0123
Sw xy x2 x+y+z x+y x 1
Extra variables xn+1 , xn+2 , . . . added to (43) leave the Schubert polynomials unaltered. For that reason they are usually treated as polynomials in an infinite ordered alphabet x = (x1 , x2 , . . .). With this understanding every homogeneous polynomial can be decomposed into Schubert components as follows: f (x) =
∂w f · Sw (x).
(w)=deg( f )
Applying this to the polynomial ϕa∗ (Sw (x ν )) = Sw (ϕa∗ (x ν )) =
(v)= (w)
v cw (a) · Sv (x),
and using Proposition 4 we finally arrive at the following result.
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Theorem 5. Forv the ν-representability problem the coefficients of the decomposition ϕa∗ (σw ) = v cw (a)σv are given by the formula v (a) = ∂v Sw (x ν ) |xkν →x T , cw
(44)
where the tableau T is derived from equation akν = aT , and the operator ∂v acts on the variables xi , replacing xkν via specialization xkν → x T = i∈T xi . Note that this equation is independent of an ordering of the unresolved spectral values akν . Indeed, the Schubert polynomial Sw (x ν ) is symmetric in the respective variables xkν , provided that w is the minimal representative in its left coset modulo the centralizer of the spectrum a ν in the symmetric group. Only such permutations correspond to the Schubert cocycles σw ∈ H ∗ (Fa ν (Hν )), cf. Remark 4. 4. Beyond the Basic Constraints Here we use the above results to derive some general inequalities for the pure ν-representability problem beyond the Pauli constraint λ ν. We start with a complete solution of the problem for two-row diagrams, and then turn to the initial N -representability problem that appears to be the most difficult one. 4.1. Two-row diagrams. For the two-row diagram ν = [α, β] the majorization inequality λ ν just tells us that λ1 ≤ α. As we know, for β = 1 it shapes the whole moment polytope, see Remark 3 to Theorem 4. Here we elucidate the remaining case ν = [N − 1, 1], and thus solve the pure ν-representability problem for all two-row diagrams. The result can not be extended to three-row diagrams, nor even to three fermion systems, where the number of independent inequalities increases with the rank, see Corollary 3 below. For convenience and future reference we collect all known facts in the next theorem. Theorem 6. For a system Hrν of rank r ≥ 3 with the two-row diagram ν = [α, β], α + β = N , all constraints on the occupation numbers of a pure state are given by the following conditions: 1. 2. 3. 4.
Basic inequality: λ1 ≤ α for β = 1. Inequality: λ1 − λ2 ≤ N − 2 for ν = [N − 1, 1], N > 3. Inequalities: λ1 − λ2 ≤ 1, λ2 − λ3 ≤ 1 for ν = [2, 1]. Even degeneracy: λ2i−1 = λ2i for ν = [1, 1].
Proof. We have already addressed Cases 1 and 4 in Remark 3 and the Introduction respectively. Case 2. Necessity. To prove the inequality λ1 − λ2 ≤ N − 2 we have to put it into the form of Theorem 2: ai λv(i) ≤ akν µw(k) . (45) i
k
This suggests the test spectrum a = (1, 0, 0, . . . , 0, −1) and the shortest permutation v that transforms it into (1, −1, 0, 0, . . . , 0), which is the cyclic one v = (2, 3, 4, . . . , r ).
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Thus we get the left-hand side of the inequality. To interpret its right-hand side N − 2, notice that the spectrum a ν starts with the terms a ν = (N − 1, N − 1, . . . , N − 1, N − 2, . . .), r −2
corresponding to semi-standard tableaux T with first row of ones and the indices 2, 3, . . . , r filling a unique place in the second row. Since for pure state µ = (1, 0, 0, . . . , 0), then the shortest permutation w that produces N − 2 in the right-hand side of (45) is also cyclic, w = (1, 2, 3, . . . , r − 1). The corresponding Schubert polynomial is just the monomial Sw (x ν ) = x1ν x2ν · · · xrν−2 . This is a special case of Grassmann permutations discussed in the next Sect. 4.2. Specialization xkν → x T of Theorem 5 transforms it into the product P(x) =
r −1
[(N − 1)x1 + xi ].
i=2
Taking the reduced decomposition v = s2 s3 · · · sr −1 we infer v cw (a) = ∂v P(x) = ∂2 ∂3 · · · ∂r −1 P(x).
The right-hand side is a constant, and the operators ∂i do not touch x1 . Hence we can put x1 = 0, that gives v cw (a) = ∂2 ∂3 · · · ∂r −1 (x2 x3 · · · xr −1 ) = 1. v (a) = 0, the inequality follows from Theorem 2. Since cw
Case 2. Sufficiency. By the convexity it is enough to construct extremal states whose occupation numbers are vertices of the polytope cut out of the Weyl chamber by the inequality λ1 − λ2 ≤ N − 2 and the normalization Tr λ = N . The vertices are given first of all by the fundamental weights normalized to trace N ωk = (N /k, N /k, . . . , N /k , 0, 0, . . . , 0) k
that generate the edges of the Weyl chamber, except for ω1 forbidden by the constraint λ1 − λ2 ≤ N − 2. The latter is replaced by the intersections τk of segments [ω1 , ωk ] with the hyperplane λ1 − λ2 = N − 2, τk = (N − 2 + 2/k, 2/k, . . . , 2/k , 0, 0, . . . , 0). k
Here we tacitly assume that N > 3, since otherwise ω2 would be also forbidden. The same condition ensures that the system Hkν is non-exceptional for k ≥ 2, hence ωk are occupation numbers of some pure states by Proposition 2. To deal with the remaining vertices τk we invoke the Dadok-Kac construction, Sect. 3.2.3 and observe that the state 1 i i k · · · k ψk = k1 k k · · · k + √ 2 2≤i
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has disconnected support and the occupation numbers τk , k ≥ 2. Here for clarity we write tableau T instead of the weight vector eT and skip an overall normalization factor. Case 3. Here we only briefly sketch the proof which follows a similar scheme. The second inequality in the form λ2 − λ3 ≤ N − 2 holds for all N , but it becomes redundant v (a) for N > 3. It can be deduced from Theorem 2 by calculation of the coefficient cw for the same a and w as above, but with another permutation v = (1, 2)(3, 4, . . . , r ). Then, keeping the notations of Case 2, we get v (a) = ∂3 ∂4 · · · ∂r −1 ∂1 P(x1 , x2 , . . . , xr −1 ) cw P(x1 , x2 , . . . , xr −1 ) − P(x2 , x1 , . . . , xr −1 ) = ∂3 ∂4 · · · ∂r −1 . x1 − x2
The operators ∂k , k ≥ 3 do not affect the variables x1 , x2 . Therefore in the fraction we can pass to the limit x1 , x2 → 0 equal to (N − 2)x3 x4 · · · xr −1 , which gives v (a) = N − 2 = 0. cw To prove sufficiency of the above inequalities we again have to look at the vertices of a polytope cut out of the Weyl chamber by the constraints λ1 − λ2 ≤ 1, λ2 − λ3 ≤ 1, Tr λ = 3. This time, along with ωk , k ≥ 3 and τk , k ≥ 2, there are vertices of another type ηk = (1 + 1/k, 1 + 1/k, 1/k, 1/k, . . . , 1/k , 0, 0, . . . , 0) k
for k ≥ 3. They represent occupation numbers of the following states with disconnected support: √ √ 2 i . ψk = k + 1 21 1 + 2 23 2 + i 3
Remark 5. Two-row diagrams naturally appear in the description of bosonic systems, like photons where polarization plays the rôle of spin. Representation with diagram can be applied both for bosons and fermions. In this case we calculated all constraints on the spin and orbital occupation numbers for small ranks, see Sect. 6.1. It appears that the constraints are stable and independent of the rank. 4.2. Grassmann inequalities. Let’s return to the initial pure N -representability problem for system ∧ N Hr and consider a constraint on its occupation numbers with 0/1 coefficients, λi1 + λi2 + · · · + λi p ≤ b,
(46)
called the Grassmann inequality. For example, all constraints (4) for the system ∧3 H7 are Grassmannian. We assume that the Grassmann inequality is essential, meaning that it defines a facet of the moment polytope. Then it should fit into the form of Theorem 2 with a = (1, 1, . . . , 1, 0, 0, . . . , 0) p
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and the Grassmann permutation or shuffle v = [i 1 , i 2 , . . . , i p , j1 , j2 , . . . , jq ] := [I, J ],
(47)
where I and J are increasing sequences of lengths p and q, p + q = r . This is the shortest permutation that produces the left-hand side of inequality (46). Our terminology stems from the observation that for the test spectrum a the flag variety Fa (H) reduces q to the Grassmannian Gr p (H) consisting of all subspaces in H of dimension p and codimension q. It is instructive to think about the Grassmann permutation v = [I, J ] geometrically as a path connecting the SW and N E corners of the p × q rectangle, with the k th unit step running to the North for k ∈ I and to the East for k ∈ J . The path cuts out of the rectangle a Young diagram γ at its N W corner. We’ll refer to I and J as the vertical and horizontal sequences of the diagram γ ⊂ p × q and denote the corresponding shuffle by vγ = [I, J ]. The length of the shuffle vγ is equal to the size |γ | of the diagram γ and its Schubert polynomial reduces to the much better understood Schur function Svγ (x) = Sγ (x1 , x2 , . . . , x p ). Observe that γ p−k+1 = i k − k, and the size of the Young diagram γ is related to its vertical sequence by the equation |γ | = (i k − k). (48) 1≤k≤ p
To get the strongest inequality (46) we choose w to be a cyclic6 permutation w = (1, 2, . . . , + 1) = [2, 3, . . . , + 1, 1, + 2, + 3 . . . , r ] of length = (v) = |γ | for which the right-hand side b = (∧ N a) +1 of (45) is minimal and equal to the ( + 1)st term of the non-increasing sequence ∧ N a = {a K := ak1 + ak2 + · · · + ak N | 1 ≤ k1 < k2 < · · · < k N ≤ r }↓ . The sequence consists of nonnegative numbers m, each taken with multiplicity p q . m N −m Recall that w also should be the minimal representative in its left coset modulo the stabilizer of ∧ N a. For the cyclic permutation this amounts to the inequality (∧ N a) > (∧ N a) +1 = b, which tells us that the first terms of ∧ N a contain all the components bigger than b. The number of such terms is bounded by the inequality p q = = |γ | ≤ pq. (49) m N −m m>b
To avoid sporadic constraints, assume that the inequality we are looking for is stable, i.e. remains valid for an arbitrary big rank r . Then the left-hand side should be linear in q = r − p and the sum contains at most two terms: m = N and m = N − 1. Thus we end up with two possibilities: 6 Actually w is always cyclic for an essential pure ν-representability inequality. We’ll address this issue elsewhere.
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1. b = N − 2, p = N − 1, = r − p, that gives the inequality λi1 + λi2 + · · · + λi N −1 ≤ N − 2,
(50)
with k (i k − k) = r − p. 2. b = N − 1, p ≥ N , = Np , that gives the inequality
with
p k (i k − k) = N .
λi1 + λi2 + · · · + λi p ≤ N − 1,
(51)
We will refer to them as the Grassmann inequalities of the first and second kind respectively. For the inequalities of the first kind the sum k (i k − k) = r − p increases with the rank, and therefore some of the involved occupation numbers should move away from the head of the spectrum. In contrast, the constraints of the second kind deal only with a few leading occupation numbers that are independent of the rank. We analyze them below for p = N + 1 and postpone the more peculiar first kind to the next section. The final result is that these inequalities actually hold true with very few exceptions. The cyclic permutation w is a special type of shuffle with the column Young diagram of height . The corresponding Schur function is just the monomial Sw (y) = y1 y2 . . . y . Applying to Sw the specialization of Theorem 5 we arrive at the product P(x) = (xk1 + xk2 + · · · + xk N ) = cγ Sγ (x1 , x2 , . . . , x p ). (52) 1≤k1
γ
Being symmetric, it can be expressed via Schur functions and, by Theorem 2, each time Sγ (x) enters into the decomposition with nonzero coefficient cγ = 0 we get inequality λi1 + λi2 + · · · + λi p ≤ N − 1,
(53)
where i 1 < i 2 < · · · < i p is the vertical sequence of Young diagram γ ⊂ p × q, |γ | = Np . The product P(x) represents the top Chern class of the exterior power ∧ N E p of the q tautological bundle E p on the Grassmannian Gr p and the decomposition (52) has been discussed in this context [20]. However, known results are very limited. Example 3. For N = 2 and any p ≥ N the product (xi + x j ) = Sδ (x1 , x2 , . . . , x p ) P(x) = 1≤i< j≤ p
is just the Schur function with a triangular Young diagram δ = [ p − 1, p − 2, . . . , 0], see [25]. This gives for the two fermion system ∧2 H the inequality (54) λ1 + λ3 + λ5 + λ7 · · · ≤ 1, that, due to the normalization i λi = 2, degenerates into equality and implies even the degeneracy λ2i−1 = λ2i of the occupation numbers. On the other hand, for arbitrary N and minimal value p = N we get P(x) = x1 + x2 + · · · + x N = S(x).
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The vertical sequence of the one-box diagram
gives a nontrivial inequality
λ1 + λ2 + · · · + λ N −1 + λ N +1 ≤ N − 1
(55)
that forces the N th electron into the N th orbital, when the preceding orbitals are fully occupied. We improve it below. In the rest of this section we focus upon the next case p = N + 1 that provides an infinite series of inequalities. Observe that in this setting a row diagram γ of length N + 1 = Np produces a false inequality λ1 + λ2 + · · · + λ N + λ2N +2 ≤ N − 1,
(56)
that fails for a coherent state given by one Slater determinant e1 ∧e2 ∧. . .∧e N . Similarly, the column inequality λ2 + λ3 + . . . + λ N +2 ≤ N − 1
(57)
fails for even N . Indeed, in this case the system ∧ N H N +2 ⊂ ∧ N Hr is non-exceptional and hence, by Proposition 2, the spectrum λ=
1 (N , N , . . . , N , 0, 0 . . . , 0) N +2 N +2
represents legitimate occupation numbers violating the inequality. Quite unexpectedly, all the other diagrams produce a valid constraint. In plain language the result can be stated as follows: Theorem 7. The occupation numbers of the N -fermion system ∧ N H in a pure state satisfy the following constraint:
each time (57).
λi1 + λi2 + · · · + λi N +1 ≤ N − 1 k (i k
− k) = N + 1, except for inequality (56) and, for even N , inequality
Proof. For p = N + 1 the decomposition (52) takes the form P(x) = (x1 + x2 + · · · + xi + · · · + x N +1 ) = (σ1 − xi ) 1≤i≤N +1
=
(−1)
k
σ1N +1−k σk
0≤k≤N +1
=
1≤i≤N +1
cγ Sγ (x1 , x2 , . . . , x N +1 ),
γ
where σk (x) = S[1k ] (x) are elementary symmetric functions, or what is the same Schur functions for the column diagram [1k ]. For Young diagrams τ ⊂ γ denote by t (γ /τ ) the number of standard tableaux of skew shape γ /τ . Then cγ = (−1)k t (γ /[1k ]). (58) k≥0 N +1−k Indeed, the coefficient at Sγ in σ1N +1−k σk = S[1] S[1k ] is equal to the number of k ways to build γ from the column diagram [1 ] by adding cells one at a time. Numbering
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the cells in the order of their appearance gives a standard tableaux of shape γ /[1k ] that encodes the whole building process. Thus the coefficient is t (γ /[1k ]) and Eq. (58) follows. For a column diagram γ we infer from the last equation cγ =
N +1
(−1)k =
k=0
0, 1,
N ≡ 0 mod 2, N ≡ 1 mod 2.
Henceforth we assume that γ is not a column. Let’s combine successive even and odd terms of the sum (58) cγ = [t (γ /[12i ]) − t (γ /[12i+1 ])]. (59) i≥0
We claim that t (γ /[1k ]) − t (γ /[1k+1 ]) = t (γ /[2, 1k−1 ]),
(60)
where meaningless terms understood as zeros, e.g. the right-hand side for k = 0. Indeed, the building process can be described as an extension of the partially filled tableau 1 2 · · · k
to a full standard tableau of shape γ . One can put the number k + 1 either just below k or next to 1. For the first choice the number of ways to complete the tableau is t (γ /[1k+1 ]), while for another one the number is t (γ /[2, 1k−1 ]). Hence t (γ /[1k ]) = t (γ /[1k+1 ]) + t (γ /[2, 1k−1 ]). Combining the last two equations we arrive at the following representation of the coefficient cγ as a sum of nonnegative terms cγ =
t (γ /[2, 12i−1 ]).
(61)
i>0
For a row diagram all terms vanish, while otherwise t (γ /[2, 1]) = 0. Hence cγ > 0 if the diagram is neither a row nor a column. The result now follows from Theorem 2. Example 4. For N = 3 the theorem gives four inequalities listed below together with the corresponding diagrams :
λ2 + λ3 + λ4 + λ5 ≤ 2,
:
λ1 + λ3 + λ4 + λ6 ≤ 2, (62)
:
λ1 + λ2 + λ5 + λ6 ≤ 2,
:
λ1 + λ2 + λ4 + λ7 ≤ 2.
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They are valid for arbitrary rank r and give all constraints on the occupation numbers for r ≤ 7. Observe also an improved version of the inequality (55) λ1 + λ2 + · · · + λ N −1 + λ N +1 + λ2N +1 ≤ N − 1,
(63)
coming from the diagram [N , 1], and another inequality λ2 + λ3 + · · · + λ N +2 ≤ N − 1, originated from a column diagram and valid only for odd N . Remark 6. We have considered above only Grassmann inequalities of the lowest levels p = N , N + 1. The higher levels provide further improvements. For example, the inequalities (55) and (63) are just the first terms of an infinite series corresponding to increasing values of p, λi1 + λi2 + λi3 + · · · + λi p ≤ N − 1,
(64)
where i k = k + Nk−1 −1 . For N = 2 this gives the inequality (54) and the double degeneracy of the occupation numbers, while for N = 3 we get the inequality λ1 + λ2 + λ4 + λ7 + λ11 + λ16 + · · · ≤ 2, where the differences between the successive indices are natural numbers 1, 2, 3, 4, . . .. The details will be given elsewhere. 4.3. Grassmann inequalities of the first kind. Formally we have such an inequality λi1 + λi2 + · · · + λi N −1 ≤ N − 2 each time the Schur function Sγ = Svγ enters into the decomposition P(x) = (x1 + x2 + · · · + x N −1 + x j ) = cv Sv (x). N ≤ j≤r
(65)
(66)
(v)=
Here γ is a Young diagram of size = r − N + 1 with the vertical sequence formed by the indices in the above inequality, and vγ is the corresponding shuffle. In contrast to the previous case, the product is not a symmetric function and its decomposition into Schubert polynomials is a challenge. Let’s try a simple case of a row diagram that produces the inequality λ1 + λ2 + · · · + λ N −2 + λr ≤ N − 2.
(67)
A close look shows that it fails for odd = r − N + 1 = 2m − 1 for the spectrum λ = (1, 1, . . . , 1, 1/m, 1/m, . . . , 1/m ) N −2
2m
obtained by merging the occupation numbers of the systems ∧ N −2 H N −2 and ∧2 H2m , see Corollary 2 of Theorem 4. Nevertheless Proposition 5. The inequality (67) holds for even = r − N + 1. In this case the Schur function with a row diagram enters into the decomposition (66) with unit coefficient.
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Proof. The row diagram γ corresponds to the cyclic permutation v = vγ = (r, r − 1, . . . , N , N − 1) = sr −1 sr −2 · · · s N −1 , where si = (i, i + 1) are transpositions. We have to calculate the coefficient cv of the decomposition (66) given by the equation cv = ∂v P(x) = ∂r −1 ∂r −2 · · · ∂ N −1 P(x). The operator ∂v does not affect the variables xi , i < N − 1, so we can set them to zero and deal with the polynomial P0 (x) = (x N −1 + xi ) = x N −k −1 x i 1 x i 2 · · · x i k . N ≤i≤r
N ≤i 1
We claim that
∂v x N −k −1 x i 1 x i 2
· · · xik =
(−1)k 0
for i s = r − k + s, otherwise.
(68)
Let’s start with the second case i 1 ≤ r − k = + N − k − 1. In the following calculation we set to zero all variables that are not affected by the subsequent operators ∂ j . With this convention we get +N −k−i 1 ∂i1 −2 ∂i1 −3 · · · ∂ N −1 x N −k xi1 xi2 · · · xik . −1 x i 1 x i 2 · · · x i k = x i 1 −1
(69)
The resulting monomial is divisible by an si1 −1 -invariant factor xi1 −1 xi1 that commutes with the operator ∂i1 −1 . Hence everything vanishes in the next step as a result of the action ∂i1 −1 and setting xi1 −1 = 0. In the case i 1 = r − k + 1 = + N − k the right-hand side of (69) is just the product of the last k variables xr −k+1 xr −k+2 · · · xr and application of the remaining operators ∂ j , r − k ≤ j ≤ r − 1 gives (−1)k . Finally, from Eq. (68) we infer 1, is even, k (70) cv = (−1) = 0, is odd, 0≤k≤
and the result follows from Theorem 2.
Remark 7. The inequality (67) is most appealing for N = 3, λ1 + λr ≤ 1,
(71)
where it supersedes the Pauli principle λ1 ≤ 1 for even r . Note that for the three electron system one- and two-point density matrices are isospectral and therefore the above inequality holds for both of them. We first came across this result reading paper [14], where the authors observed that if the 2-point density matrix of a three fermion system in state ψ ∈ ∧3 Hr has an eigenvalue equal to one, then the corresponding eigenform ω ∈ ∧2 Hr can’t have the full rank r . This is trivial for odd r , since rank of ω is always even. For even rank this follows from (71). Moreover, in the latter case the state ψ ∈ ∧3 Hr itself has rank less than r . M.B. Ruskai also conjectured inequality (71) in her analysis of three fermion and three hole systems [33].
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Observe the following result, anticipated by many experts. It may appear not so trivial if compared with Theorems 4 and 6. Corollary 3. No finite set of inequalities gives all constraints on occupation numbers of N -fermion system ∧ N H, N > 1 of arbitrary big rank. Proof. Indeed, a finite set Q of linear inequalities L α (λ) ≤ bα includes only finitely many occupation numbers λi , i < M. Every inequality that follows from Q is a nonnegative combination of the inequalities from Q, the ordering conditions λi − λi−1 ≤ 0, and a multiple of the normalization equation ri=1 λi = N . Suppose now that the inequality of Proposition 5, λ1 + λ2 + · · · + λ N −2 + λr ≤ N − 2,
(72)
can be deduced from the system Q for some r M and even = r − N + 1. The coefficients at λi in the left side for i ≥ M should come from the following linear combination with non-negative coefficients ai , a1 (λ2 − λ1 ) + a2 (λ3 − λ2 ) + · · · + ar −1 (λr − λr −1 ) − ar λr = −λ1 a1 + λ2 (a1 − a2 ) + · · · + λr −1 (ar −2 − ar −1 ) + λr (ar −1 − ar ), amended with a multiple of the normalization equation. The Abel transformation shown in the second line implies that the coefficients ai should form an arithmetical progression ai = ai + b for M ≤ i < r , while ar = ar + b − 1 ≥ 0. Suppose now that a ≥ 0. Then the same combination of inequalities from Q that produces (72) and the same coefficients ai for i < r together with ar = ar + b ≥ 0, ar +1 = a(r + 1) + b − 1 ≥ 0 would give a false inequality of rank r + 1 obtained from (72) by replacing r → r + 1. Recall that the inequality (72) fails for odd = r − N + 1. For a ≤ 0 a similar consideration gives a false inequality of rank r − 1. Proposition 5 can be extended to two-row diagrams γ = [ −k, k]. For three fermions this leads to the constraints λk+1 + λr −k ≤ 1,
for
k + 1 < r − k,
(73)
that prohibit more than one electron to occupy two complementary orbitals. It holds both for even and odd r for k > 0. The corresponding coefficients cγ = c( , k) of the decomposition (66) satisfy the recurrence relation c( , k) = c( − 1, k) + c( − 1, k − 1) and form the left half of the Pascal triangle 0 1 0 1 0 1 0 1 0 1
3
1
2 2
3
1
4
0 2
6 12
0 6
0 0
0
2
6 9
0
1
−1
0 −1 −2 −6
−1 −2 −6
−1 −1 −4 −12
0 −2 −6
−1 −2 −9
0 −3
−1 −3
0
−1
with apex at = −1, and the 0/1 boundary condition for k = 0 set by Eq. (70). We return to the Pascal recurrence relation in a more general framework below, see Eq. (79).
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Observe a zero in the forth line of the Pascal triangle, corresponding to diagram . In general, a column diagram should have zero coefficient, because it produces inequality λ1 + λ2 + · · · + λ N − + · · · + λ N ≤ N − 2
(74)
that fails for a coherent state given by one Slater determinant. It turns out that the Grassmann inequality of the first kind (65) holds for all diagrams, except for a column and an odd row. To wit Theorem 8. The occupation numbers of N -fermion system ∧ N Hr in a pure state satisfy the following constraint: λi1 + λi2 + · · · + λi N −1 ≤ N − 2
(75)
each time k (i k − k) = r − N + 1, except for inequality (74) and, for odd = r − N + 1, inequality (67). Proof. We have to show that the Schur function Sγ (x) = Svγ (x) enters into the decomposition (x1 + x2 + · · · + x N −1 + x j ) = cv Sv (x), (76) Pr (x) = N ≤ j≤r
(v)=
provided that γ ⊂ p × q is neither a column nor an odd row. Here p = N − 1, q = = |γ | = r − p. Note first of all, that the coefficients of this decomposition are nonnegative for v ∈ Sr and can be positive only for shuffles v = vγ . The first claim holds in general for the coefficients cvw (a) of Theorem 2, ϕa∗ (σw ) =
v
v cw (a)σv ,
since the cycle ϕa−1 (σw ) ⊂ Fa (Hr ) is effective. Here v runs over representatives of minimal length in left coset modulo stabilizer of a. To include all permutations v ∈ Sr one has to deal with a small perturbation a˜ that resolves multiple entries of a. However, −1 since ϕa−1 ˜ (σw ) ⊂ Fa˜ (Hr ) is pull back of ϕa (σw ) ⊂ Fa (Hr ) via natural projection π : Fa˜ (Hr ) → Fa (Hr ) defined in Remark 4, then decomposition of ϕa−1 ˜ (σw ) and ϕa−1 (σw ) involve the same Schubert cycles σv . This prove the second claim. Let’s add as a warning, that the decomposition (76) actually contains Schubert polynomials Sv with permutations v ∈ / Sr . The rest of the proof is purely algebraic. We’ll proceed by induction on r keeping N fixed. For the first meaningful case r = N + 1, = 2, as we know, only the row diagram appears in the decomposition. Suppose now the induction hypothesis holds for Pr (x), and consider the next polynomial, Pr +1 (x) = (x1 + x2 + · · · + x N −1 + xr +1 )Pr (x) = (x1 + x2 + · · · + x N −1 + xr +1 ) cv Sv (x). (v)=
(77)
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We can find its Schubert components using a version of Monk’s formula, (αi − α j )Svti j , (α1 x1 + α2 x2 + α3 x3 · · · )Sv (x) = (vti j )= (v)+1
where ti j = (i, j), i < j < ∞ is a transposition, see [24, p. 86]. For a typical term of (77) this gives (x1 + x2 + · · · +x N −1 + xr +1 )Sv = Svti j − 1≤i
sgn(r + 1 − j)Svt j,r +1 ,
(78)
N ≤ j=r +1
where the sums include only those transpositions t for which (vt) = (v) + 1. We are interested in the terms u γ = vt ∈ Sr +1 that are shuffles coming from a Young diagram γ ⊂ p × ( + 1) of size + 1. Let’s single out the row diagram for which Proposition 5 gives the coefficient cγ . The remaining shuffles u γ do not move the last index r + 1, and therefore permutation v = u γ ti, j has a bigger length than u γ for j ≥ r + 1. Hence a non-row Schur component Sγ in (78) comes from the sum Svti j 1≤i
for v = u γ ti j , (v) = (u γ ) − 1 = |γ | − 1. Then v ∈ Sr , and Sv (x) enters into decomposition (76) only for a shuffle v = vτ . In this case the relation vτ = u γ ti j just means that τ is obtained from γ by removing a cell. As a result, we arrive at the recurrence relation cτ , (79) cγ = γ /τ =cell
that holds for all non-row diagrams γ . This implies that cγ > 0 if one can obtain an even row from γ by removing cells one at a time from a non-row diagram. This can be done for any diagram different from a column or an odd row. The inequality (75) now follows from Theorem 2. Example 5. For a four fermion system ∧4 Hr the theorem gives inequality λi + λ j + λk ≤ 2, that holds for odd rank r ≥ 7 and pairwise distinct indices satisfying equation i + j + k = r + 3. For even r one has to exclude the row inequality λ1 + λ2 + λr ≤ 2. For two-row diagrams Eq. (79) amounts to the Pascal recurrence relation discussed in Remark 7. In general, it allows to get an explicit formula for the coefficient cγ that is surprisingly similar to the one given in the proof of Theorem 8, where we borrow the notations. Corollary 4. cγ =
(−1)k t (γ /[k]) = t (γ /[2i, 1]), k≥0
i>0
where the second equality holds for diagrams γ different from rows and columns.
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Proof. Applying Eq. (79) recurrently in conjunction with Proposition 5 we find out that cγ is equal to the number of ways to obtain an even row from γ by removing cells one at a time from a non-row diagram. If γ is not a row or a column, then the last step in the process will be [2i, 1] → [2i]. Encoding the process by the standard tableaux, we arrived at the second formula. The first one follows from the identity t (γ /[2i, 1]) = t (γ /[2i]) − t (γ /[2i + 1]), cf. the proof of Theorem 8, and holds for all diagrams. 5. Connection with Representation Theory The solution of the ν-representability problem suggested by Theorem 2 is not feasible, except for very small systems. For example, for four fermions ∧4 H8 we confront an 8 immense symmetric group of degree 4 = 70. Besides, listing of the extremal edges for systems of this size is all but impossible. A representation theoretical interpretation of the ν-representability discussed below often allows to mollify or circumvent these difficulties. Let’s consider a composition of the Schur functors H → Hν called a plethysm µ [Hν ]µ = m λ Hλ . (81) |λ|=|ν|·|µ|
µ
It splits into U(H) irreducible components Hλ of multiplicity m λ . It is instructive to treat the diagrams λ and µ as spectra. We are interested in their asymptotic behavior µ for m λ = 0 and |µ| → ∞. Therefore we normalize them to a fixed size µ = µ/|µ|,
λ = λ/|µ|, so that Tr µ = 1 and Tr λ = N = |ν|. µ Theorem 9. Every time m λ = 0 the couple ( λ, µ) belongs to the moment polytope of the µ, with occupation numbers λ. system Hν , i.e. there exists its mixed state ρ ν of spectrum Moreover every point of the moment polytope is a convex combination of such spectra ( λ, µ) of a bounded size |µ| ≤ M < ∞.
The theorem is a special case of Mumford’s description of the moment polytope, see his Appendix in [28]. It also holds in more general Berenstein-Sjamaar settings [1]. 5.1. Practical algorithm. For a fixed M the convex hull of the spectra ( λ, µ) from Theorem 9 gives an inner approximation to the moment polytope, while any set of inequalities of Theorem 2 amounts to its outer approximation. This suggests the following approach to the mixed ν-representability problem, which combines both theorems. 1. Find all irreducible components Hλ ⊂ [Hν ]µ for |µ| ≤ M. 2. Calculate the convex hull of the corresponding spectra ( λ, µ) that gives an in ⊂ P for the moment polytope P. inner approximation P M in that are given by the inequalities of Theorem 2. 3. Identify the facets of P M out ⊃ P. They cut out an outer approximation P M in out . 4. Increase M and continue until P M = P M The algorithm became practical by generosity of the authors of LiE package [4], who made it publicly available. It allows to handle plethysms efficiently. We also benefit from
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Convex package by Franz [13], who apply a similar approach to the quantum marginal problem for three qutrits [12,17]. One can incorporate in the algorithm additional constraints on the spectrum of the mixed state ρ ν . In many problems this is just a restriction on the rank rk ρ ν ≤ p, that bounds the number of rows of µ. For example, a pure state ρ ν = |ψψ| has rank one, the corresponding diagram µ = [m] reduces to a row, and the plethysm amounts to the symmetric power S m (Hν ). More generally, for spin-orbital occupation numbers of a system of electrons of total spin J , we have to deal with mixed states of rank 2J + 1, see Corollary 1 to Theorem 2, and respectively with the diagrams µ of at most that height. 5.2. Particle-hole duality. Here is another application of Theorem 9. Recall, that we arrived at the ν-representability problem from the spin-orbital decompositions (17) of Sect. 3. In this setting the Young diagram ν comes together with a rectangular frame r × s ⊃ ν, where r and s are dimensions of the orbital and spin spaces respectively. Let ν ∗ be the complementary diagram to ν in the frame r × s, that is νi∗ = s − νr +1−i . ∗ One can think about the representation Hrν as describing the holes of the system Hrν . ∗ These are dual systems with a natural pairing Hrν ⊗ Hrν → Hrr ×s = det(Hr )⊗s , that ∗ can be extended to a pairing of the plethysms [Hrν ]µ ⊗ [Hrν ]µ → det(Hr )⊗sm , where ∗ m = |µ|. The latter duality means that if Hrλ is a component of [Hrν ]µ , then Hrλ is a ∗ component of [Hrν ]µ of the same multiplicity. Here λ∗ is the complementary diagram to λ ⊂ r × sm. In view of Theorem 9 this implies ∗
Corollary 5. The moment polytope of the hole system Hrν is obtained from the moment polytope of Hrν by the transformation (λ, µ) → (λ∗ , µ), where λi∗ = s − λr +1−i . 6. Analysis of Some Small Systems Here we take the challenge to explore all the constraints on the occupation numbers. This is clearly a mission impossible. It moves us from a garden of the carefully selected species we dealt with in the preceding sections, into the midst of a wild jungle with no order or end in sight. To succeed in this environment we try the algorithm of Sect. 5.1 first. However, due to computer limitation, it can be accomplished only for very small systems. For the pure N -representability problem these are the systems for which Borland and Dennis made their prophesy 35 yeas ago [3]. To move further we use any tool available, from a clever guess to numerical optimization. The final outcome of this endeavour are all the constraints for the systems of rank not exceeding 10. For r ≤ 8 we provide a rigorous proof below. We also have a proof for the system ∧3 H9 based on other ideas, not discussed here. For the remaining cases the constraints are complete only beyond a reasonable doubt. To resolve the doubt one has to verify independently that the vertices of the constructed polytope are legitimate occupation numbers. We did this using a variety of methods for most of the vertices, but some still evaded all efforts. For the latter we resort to numerical optimization to check that they indeed can be approached very closely within the moment polytope. The biggest system we treated ∧5 H10 is bounded by 161 inequalities. We are ready to bet a bottle of decent wine for every additional essential constraint found.
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M. Altunbulak, A. Klyachko Table 1. N -representability inequalities for system ∧3 H6 v ∈ S6 (2 6 5 4 3) (1 2 5 4 3) (1 3)(2 4) (1 4 3 2)
Inequalities λ1 + λ6 ≤ 1 λ2 + λ5 ≤ 1 λ3 + λ4 ≤ 1 λ4 ≤ λ5 + λ6
w ∈ S20 (1 2 3 4 5) (1 2 3 4)
v (a) cw 1 1 1 1
Table 2. N -representability inequalities for system ∧3 H7 Inequalities λ2 + λ3 + λ4 + λ5 λ1 + λ3 + λ4 + λ6 λ1 + λ2 + λ4 + λ7 λ1 + λ2 + λ5 + λ6
≤2 ≤2 ≤2 ≤2
v ∈ S7 (1 2 3 4 5) (2 3 4 6 5) (3 4 7 6 5) (3 5)(4 6)
w ∈ S35 (1 2 3 4 5)
v (a) cw 1 1 1 1
6.1. Spin and orbital occupation numbers. Let’s start with a simple example of constraints on spin µ and orbital λ occupation numbers for a system of three electrons of the total spin J = 1/2. By Corollary 1 to Theorem 2 the problem is equivalent to mixed ν-representability for ν = and Spec ρ ν = (µ1 , µ2 ). A calculation based on the algorithm of Sect. 5.1 shows that the constraints amount to 5 inequalities λ1 − λ2 ≤ 1 + µ2 , λ2 − λ3 ≤ 1 + µ2 , λ1 − λ3 ≤ 2 − µ2 , λ1 − λ2 − λ3 ≤ 1, 2λ1 − λ2 + λ4 ≤ 4 − µ2 , that apparently are independent of the rank. We test them for r = 4, 5. Recall that λ and µ are arranged in non-increasing order and are normalized to the traces 3 and 1 respectively. 6.2. Pure N -representability. The known solution for two fermions, together with the particle-hole duality of Sect. 5.2, bound the pure N −representability problem to the range 3 ≤ N ≤ r/2. For rank r ≤ 8 this leaves us with systems ∧3 H6 , ∧3 H7 , ∧3 H8 , and ∧4 H8 . For three of them ∧3 H6 , ∧3 H7 and ∧4 H8 the algorithm of Sect. 5.1 runs flawlessly and terminates at M = 4, 8, 10, respectively. The independent constraints grouped by v (a), and cycle decomposition of the the test spectra a, together with the coefficients cw permutations v, w are given in Tables 1–3. The remaining system ∧3 H8 is much harder to resolve. 6.2.1. System ∧3 H8 We managed to decompose plethysm S m (∧3 H8 ) up to degree m = 24, but still have had a discrepancy between the inner and the outer approximations in , except for one, fit Theorem 2. For to the moment polytope. Actually all facets of P24 the remaining facet λ1 + λ5 + λ6 ≥ 1 we use a numerical minimization of the linear form L(λ) = λ1 + λ5 + λ6 over all particle density matrices. It turns out that the form attains its minimum, equal to 27 28 , at the vertex 1 (15, 15, 15, 15, 6, 6, 6, 6). 28
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Table 3. N -representability inequalities for system ∧4 H8 Inequalities
v ∈ S8
w ∈ S70
v (a) cw
λ1 ≤ 1 λ5 − λ6 − λ7 − λ8 ≤ 0 λ1 − λ2 − λ7 − λ8 ≤ 0 λ1 − λ3 − λ6 − λ8 ≤ 0 λ1 − λ4 − λ6 − λ7 ≤ 0 λ1 − λ4 − λ5 − λ8 ≤ 0 λ3 − λ4 − λ7 − λ8 ≤ 0 λ2 − λ4 − λ6 − λ8 ≤ 0 λ2 + λ3 + λ5 − λ8 ≤ 2 λ1 + λ3 + λ6 − λ8 ≤ 2 λ1 + λ2 + λ7 − λ8 ≤ 2 λ1 + λ2 + λ3 − λ4 ≤ 2 λ1 + λ4 + λ5 − λ8 ≤ 2 λ1 + λ2 + λ5 − λ6 ≤ 2 λ1 + λ3 + λ5 − λ7 ≤ 2
(1) (1 5 4 3 2) (2 3 4 5 6) (3 4 5 7 6) (4 5 8 7 6) (4 6)(5 7) (1 3 2)(4 5 6) (1 2)(4 5 7 6) (1 2 3 5 4) (2 3 6 5 4) (3 7 6 5 4) (4 5 6 7 8) (2 4)(3 5) (3 5 4)(6 7 8) (2 3 5 4)(7 8)
(1)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
(1 2 3 4 5)
(1 2 3 4 5)
Table 4. N -representability inequalities for system ∧3 H8 Inequalities
v ∈ S8
λ2 + λ3 + λ4 + λ5 ≤ 2 λ1 + λ2 + λ4 + λ7 ≤ 2 λ1 + λ3 + λ4 + λ6 ≤ 2 λ1 + λ2 + λ5 + λ6 ≤ 2 λ1 + λ2 − λ3 ≤ 1 λ2 + λ5 − λ7 ≤ 1 λ1 + λ6 − λ7 ≤ 1 λ2 + λ4 − λ6 ≤ 1 λ1 + λ4 − λ5 ≤ 1 λ3 + λ4 − λ7 ≤ 1 λ1 + λ8 ≤ 1 λ2 − λ3 − λ6 − λ7 ≤ 0 λ4 − λ5 − λ6 − λ7 ≤ 0 λ1 − λ3 − λ5 − λ7 ≤ 0 λ2 + λ3 + 2λ4 − λ5 − λ7 + λ8 ≤ 2 λ1 + λ3 + 2λ4 − λ5 − λ6 + λ8 ≤ 2 λ1 + 2λ2 − λ3 + λ4 − λ5 + λ8 ≤ 2 λ1 + 2λ2 − λ3 + λ5 − λ6 + λ8 ≤ 2 λ1 + λ2 − 2λ3 − λ4 − λ5 ≤ 0 λ1 − λ2 − λ3 + λ6 − 2λ7 ≤ 0 λ1 − λ3 − λ4 − λ5 + λ8 ≤ 0 λ1 − λ2 − λ3 − λ7 + λ8 ≤ 0 2λ1 − λ2 + λ4 − 2λ5 − λ6 + λ8 ≤ 1 λ3 + 2λ4 − 2λ5 − λ6 − λ7 + λ8 ≤ 1 2λ1 − λ2 − λ4 + λ6 − 2λ7 + λ8 ≤ 1 2λ1 + λ2 − 2λ3 − λ4 − λ6 + λ8 ≤ 1 λ1 + 2λ2 − 2λ3 − λ5 − λ6 + λ8 ≤ 1 2λ1 − 2λ2 − λ3 − λ4 + λ6 − 3λ7 + λ8 ≤ 0 −λ1 + λ3 + 2λ4 − 3λ5 − 2λ6 − λ7 + λ8 ≤ 0 2λ1 + λ2 − 3λ3 − 2λ4 − λ5 − λ6 + λ8 ≤ 0 λ1 + 2λ2 − 3λ3 − λ4 − 2λ5 − λ6 + λ8 ≤ 0
(1 2 3 4 5) (3 4 7 6 5) (2 3 4 6 5) (3 5)(4 6) (3 4 5 6 7 8) (1 2 5 4 3)(7 8) (2 6 5 4 3)(7 8) (1 2 4 3)(6 7 8) (2 4 3)(5 6 7 8) (1 3)(2 4)(7 8) (2 8 7 6 5 4 3) (1 2)(3 4 5 8 7 6) (1 4 3 2)(5 8 7 6) (3 4 6)(5 8 7) (1 4 8 7 5) (1 4 8 6 7 5 2) (1 2)(3 4 8 5 6 7) (1 2)(3 5 4 8 6 7) (3 6 4 7 5 8) (2 6)(3 4 5 8 7) (2 8 5 7 4 6 3) (2 8 7 3 4 5 6) (2 4 3 8 5 7 6) (1 4)(2 3 8 5) (2 6)(3 8 7 4) (3 8)(4 5 7 6) (1 2)(3 8)(5 7 6) (2 6 4 5 3 8 7) (1 4 2 3 8 5)(6 7) (3 8)(4 7) (1 2)(3 8)(4 7 5)
w ∈ S56 (1 2 3 4 5)
(1 2 3 4 5 6)
(1 2 3 4 5 6 7) (1 2 3 4 5 6 7) (1 2 3 . . . 10 11) (1 2 3 . . . 11 12) (1 2 3 . . . 12 13)
(1 2 3 . . . 12 13)
(1 2 3 . . . 14 15)
v (a) cw
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Adding this vertex gives a polytope P where all facets are covered by Theorem 2. Thus P is the genuine moment polytope for ∧3 H8 given by 31 independent inequalities listed in Table 4.
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M. Altunbulak, A. Klyachko Table 5. Vertices of the moment polytope of ∧4 H8 and the corresponding extremal states Extremal states
Vertices
[1234] [1234] + [1256] + [3456] [1234] + [1256] [1234] + [1256] + [1357] + [1467] + [2367] + [2457] + [3456] [1234] + [1256] + [1357] + [1467] √ √2[1234] + [1256] + [1357] + [2367] [1256] + [1357] + [2457] + [3456] √2[1234] + √ 3[1234] + √2[1256] + [1357] + [2457] √ √2[1234] + 2[1256] + [1357] + [1467] + [2367] + [2457] 2[1234] + [1256] + [1357] [1234] + [5678] √ 2[1234] + [1256] + [1278] + [1357] + [1368] [1234] + [1256] + [1278] √ + [1357] + [1458] + [2358] + [2457] + [3456] √ √3[1234] + [1256] 2[1234] √ + 2[1256] + [1357] + [1368] + [1458] + [1467] 2[1234] + √2[1256] + [1357] + [1458] + [2358] + [2457] 2[1234] + 2[1256] + [1357] + [1368] + [2358] + [2367] √ √2[1234] + [1256] + [1357] + [1458] 3[1234] + [1256] + [1357] + [2358] [1234] + [1256] √ + [1278] + [1357] + [1368] + [1458] + [1467] √ 3[1234] + 2[1256] + [1357] + [1368] √ 3[1234] + [1256] + [1278] + [1357] + [1368] + [2358] + [2367]
(1 : 1 : 1 : 1 : 0 : 0 : 0 : 0) (1 : 1 : 1 : 1 : 1 : 1 : 0 : 0) (2 : 2 : 1 : 1 : 1 : 1 : 0 : 0) (1 : 1 : 1 : 1 : 1 : 1 : 1 : 0) (2 : 1 : 1 : 1 : 1 : 1 : 1 : 0) (2 : 2 : 2 : 1 : 1 : 1 : 1 : 0) (2 : 2 : 2 : 2 : 2 : 1 : 1 : 0) (3 : 3 : 2 : 2 : 2 : 1 : 1 : 0) (3 : 3 : 2 : 2 : 2 : 2 : 2 : 0) (4 : 3 : 3 : 2 : 2 : 1 : 1 : 0) (1 : 1 : 1 : 1 : 1 : 1 : 1 : 1) (3 : 2 : 2 : 1 : 1 : 1 : 1 : 1) (3 : 3 : 1 : 1 : 1 : 1 : 1 : 1) (3 : 3 : 3 : 3 : 3 : 1 : 1 : 1) (4 : 2 : 2 : 2 : 2 : 2 : 1 : 1) (4 : 4 : 3 : 3 : 3 : 1 : 1 : 1) (4 : 4 : 4 : 2 : 2 : 2 : 1 : 1) (5 : 3 : 3 : 3 : 3 : 1 : 1 : 1) (5 : 5 : 5 : 3 : 3 : 1 : 1 : 1) (7 : 3 : 3 : 3 : 3 : 3 : 3 : 3) (7 : 5 : 5 : 3 : 3 : 3 : 1 : 1) (7 : 7 : 7 : 3 : 3 : 3 : 3 : 3)
We are actually unhappy with employing of numerical optimization, that can produce no rigorous result. Nevertheless, it provides a helpful hint about missed vertices. After some guesses and trials we found the state ψ = 2[123] +
√ √ √ √ 10[145] + 5[347] + 2[356] + 2[258] + 2[368] + [178],
whose occupation numbers give the vertex (82). This provides a rigorous proof of the completeness the above constraints. Here [i jk] = ei ∧ e j ∧ ek is the Slater determinant or, in our general notations, weight vector eT corresponding to the semi-standard tableau T transpose to [i jk]. Six triplets [i jk] in the support of ψ, excluding one [356] typeset in boldface, form a disconnected set. They are remnants of our failed attempt to produce the missed vertex by the Dadok-Kac construction in Sect. 3.2.3. Extra tableau [356] in the support increases the number of adjustable parameters, but makes the problem nonlinear. For those people who don’t trust a computer assisted proof we give an extremal state for every vertex of the moment polytope for the systems ∧3 H7 , ∧3 H8 , and ∧4 H8 listed in Tables 5-6. They are sufficient for a computer independent proof, provided that one v (a) in Tables 2–4. takes for granted the values of the coefficients cw 6.2.2. Systems of rank 9 and 10 The results here are less definite. Only for the smallest system ∧3 H9 do we have a rigorous justification of completeness for the system of 52 independent inequalities. For the next one, ∧4 H9 , we found 60 constraints, that give a polytope with 103 vertices. For all of them, except for two: [16, 16, 16, 6, 6, 6, 6, 6, 6]/21,
[20, 14, 14, 14, 14, 4, 4, 4, 4]/23,
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Table 6. Vertices of the moment polytope of ∧3 H8 and the corresponding extremal states. The first ten lines give the same data for ∧3 H7 Extremal states [123] [123]+[145] [123]+[145]+[246]+[356] √ 2[123]+[145]+[246] [123]+[145]+[167]+[246]+[257]+[347]+[356] √ √2[123]+[167]+[246]+[257]+[145] √ 2[123]+ 2[145]+[246]+[257]+[347]+[356] [123]+[145]+[167] √ √ √2[123]+[145]+[246]+[347] 3[123]+ 2[145]+[246]+[257] [178]+[368]+[258]+[567]+[347]+[246]+[145]+[123] √ √ √ 2[145]+ √2[123] √ √2[178]+[368]+[567]+[246]+ 2[246]+[145]+ 3[123] √ √ √ √2[178]+[258]+[567]+ √ 2[258] √3[123]+ √3[145]+[246]+ 2[347]+[356]+ 3[178]+ 2[567]+[347]+[246]+2[145]+ 5[123] √ [178]+[246]+[145]+ 2[123] √ [178]+[258]+[246]+[145]+ 2[123] √ [258]+[567]+[145]+ 3[123] √ √ 2[368] √ √2[145]+[246]+[347]+[356]+ 2[178]+[246]+[145]+ √ √2[123] [368]+[347]+ 2[145]+ 3[123] √ √ √ √ 2[356]+ 2[258]+2[368]+[178] 5[347]+ 2[123]+ 10[145]+ √ √ 2[145]+ 3[123] [178]+[567]+ √ √ √ 2[246]+√ 3[356]+ 5[567]+2[258] 2[123]+ √ √ 2[178]+[258]+ 2[246]+[145]+ √ √ √ √ 3[123]√ 2√2[145]+ √2[246]+ √2[347]+ 3[356]+ √ 3[368]√ 3[258]+ 3[178] 2√ 3[123]+ 6[145]+ 2[356]+2[567]+ √ 2[368]√ √ √ √2[145]+2[246]+[347]+[356]+ 2[178]+[124] √ 6[258]+2[368]+2 √ √3[246]+ 2[347]+ 3[258]+[567]+ 2[347]+ 2[246]+2[123] √ √ √ 3[145]+ 6[246]+3[347]+2[356]+ 3[258]+ 14[368] √ √ √ √ 2[178]+[258]+ 3[246]+ 2[145]+ 5[123] √ √ √ √ √ 2[123]+ 2[246]+ 2[356]+ 3[567]+ 3[258]+ 2[368] √ √ √ √ √ 6[246]+ 6[347]+ 2[145]+ 5[356]+ 2[258]+3[368] 2 √ √ √ √ 2[145]+2[123] √ √5[178]+[347]+ 2[246]+ 2[145]+ √6[123] √ √3[178]+[258]+2[246]+ 3[178]+ √ 2[567]+[246]+2[145]+ √ 5[123]√ [123]+ 3[145]+2[347]+2[356]+ 3[258]+ 3[368]
Vertices (1 : 1 : 1 : 0 : 0 : 0 : 0 : 0) (2 : 1 : 1 : 1 : 1 : 0 : 0 : 0) (1 : 1 : 1 : 1 : 1 : 1 : 0 : 0) (3 : 3 : 2 : 2 : 1 : 1 : 0 : 0) (1 : 1 : 1 : 1 : 1 : 1 : 1 : 0) (2 : 2 : 1 : 1 : 1 : 1 : 1 : 0) (2 : 2 : 2 : 2 : 2 : 1 : 1 : 0) (3 : 1 : 1 : 1 : 1 : 1 : 1 : 0) (3 : 3 : 3 : 3 : 1 : 1 : 1 : 0) (5 : 5 : 3 : 3 : 3 : 1 : 1 : 0) (1 : 1 : 1 : 1 : 1 : 1 : 1 : 1) (2 : 1 : 1 : 1 : 1 : 1 : 1 : 1) (2 : 2 : 1 : 1 : 1 : 1 : 1 : 1) (3 : 3 : 3 : 3 : 3 : 1 : 1 : 1) (4 : 2 : 2 : 2 : 2 : 2 : 1 : 1) (4 : 3 : 2 : 2 : 1 : 1 : 1 : 1) (4 : 4 : 2 : 2 : 2 : 2 : 1 : 1) (4 : 4 : 3 : 3 : 1 : 1 : 1 : 1) (4 : 4 : 4 : 4 : 2 : 1 : 1 : 1] (5 : 3 : 2 : 2 : 2 : 2 : 1 : 1) (5 : 5 : 3 : 3 : 2 : 1 : 1 : 1) (5 : 5 : 5 : 5 : 2 : 2 : 2 : 2) (6 : 3 : 3 : 3 : 2 : 2 : 1 : 1) (6 : 5 : 5 : 5 : 2 : 2 : 1 : 1) (6 : 6 : 3 : 3 : 3 : 2 : 2 : 2) (6 : 6 : 4 : 4 : 4 : 1 : 1 : 1) (7 : 5 : 5 : 5 : 2 : 2 : 2 : 2) (7 : 7 : 4 : 4 : 4 : 2 : 1 : 1) (9 : 5 : 5 : 5 : 3 : 3 : 3 : 3) (9 : 6 : 4 : 4 : 4 : 3 : 3 : 3) (9 : 8 : 8 : 8 : 3 : 3 : 3 : 3) (9 : 9 : 5 : 5 : 3 : 3 : 3 : 2) (9 : 9 : 9 : 9 : 4 : 4 : 2 : 2) (10 : 10 : 10 : 10 : 4 : 4 : 3 : 3) (11 : 6 : 6 : 5 : 5 : 5 : 2 : 2) (11 : 11 : 6 : 6 : 4 : 4 : 3 : 3) (12 : 6 : 6 : 5 : 5 : 5 : 3 : 3) (12 : 12 : 7 : 7 : 4 : 4 : 4 : 4)
we have proved rigorously that they belong to the moment polytope. The remaining two vertices were checked only numerically. It turns out that the same two vertices would provide the completeness of 125 constraints for ∧4 H10 . The occupation numbers of the remaining systems ∧3 H10 and ∧5 H10 are bounded by 93 and 161 inequalities, but many vertices are still waiting a confirmation by non-numerical methods. The facets and vertices of the moment polytopes for all systems of rank ≤ 10 are available as electronic supplementary material in the online version of this article doi:10.1007/s00220-008-0552-z.
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References 1. Berenstein, A., Sjamaar, R.: Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion. J. Amer. Math. Soc. 13(2), 433–466 (2000) 2. Bernstein, I., Gelfand, I., Gelfand, S.: Schubert cells and cohomology of the space G/P. Russ. Math. Survey 28(3), 1–26 (1973) 3. Borland, R.E., Dennis, K.: The conditions on the one-matrix for three-body fermion wavefunctions with one-rank equal to six. J. Phys. B: Atom Molec. Phys. 5, 7–15 (1972) 4. Cohen, A.M., van Leeuwen, M., Lisser, B.: LiE, a software package for Lie group theoretical computations, available at http://www-mathlabo.univ-poitiers.fr/~maavl/LiE/ 5. Coleman, A.J.: Structure of Fermion Density Matrices. Rev. Mod. Phys. 35, 668–686 (1963) 6. Coleman, A.J., Yukalov, V.I.: Reduced density matrices: Coulson’s challenge. Berlin: Springer, 2000 7. Fulton, W., Harris, J.: Representation theory. New York: Springer, 1991 8. Fulton, W.: Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997 9. Fulton, W.: Schubert varieties and degeneracy loci. Berlin: Springer, 1998 10. Dadok, J., Kac, V.: Polar representations. J. Algebra 92(2), 504–524 (1985) 11. Duck, T., Sudarshan, E.C.G.: Pauli and the spin-statistics theorem. Singapore: World Scientific, 1997 12. Franz, M.: Moment polytopes of projective G-varieties and tensor products of symmetric group representations. J. Lie Theory 12, 539–549 (2002) 13. Franz, M.: Convex, a Maple package for convex geometry, available at http://www-fourier.ujf-grenoble. fr/~franz/convex/ 14. Grudzi´nski, H., Hirsch, J.: Search for new conditions for fermion N -representability. http://arXiv.org/ list/math-ph/0311026, 2003 15. Klyachko, A.: Stable bundles, representation theory, and Hermitian operators. Selecta Math. 4, 419–445 (1998) 16. Klyachko, A.: Vector bundles, Linear representations, and Spectral problems. Proc. Int. Congress of Math. Beijing 2002, Invited Lectures, Vol. II, Beijing: Higher Edication Press, 2003, pp. 599–614 17. Klyachko, A.: Quantum marginal problem and representations of the symmetric group. http://arXiv.org/ list/quant-ph/0409113, 2004 18. Klyachko, A.: Quantum marginal problem and N -representability. J. Phys. Conf. Series 36, 72–86 (2006) 19. Klyachko, A.: Dynamic symmetry approach to entnglement. In: Proc. NATO Advanced Study Inst., Cargese, Corsica, France, 2005, J.-P. Gazeau et. al. eds., Amsterdam: IOS Press, 2007, pp. 25–54 20. Lascoux, A.: Classes de Chern d’un produit tensoriel. C. R. Acad. Sci. Paris 286, 385–387 (1978) 21. Lascoux, A., Schützenberger, M.-P.: Symmetry and flag manifolds. Lecture Notes in Mathematics 25, 159–198 (1974) 22. Lascoux, A., Schützenberger, M.-P.: Polyôme de Schubert. C. R. Acad. Sci. Paris 294, 447–450 (1982) 23. Liu, Y.-K., Christandl, M., Verstraete, V.: N-representability is QMA-complete. Phys. Rev. Lett. 98, 110503 (2007) 24. Macdonald, I.G.: Schubert polynomials. London Math. Soc. Lecture Notes 166, 73–99 (1991) 25. Macdonald, I.G.: Symmetric functions and Hall polynomials. Oxford: Clarendon Press, 1995 26. Mazziotti, D.A. (ed): Reduced density matrix mechanics with application to many electron atoms and molecules. New York: John Wiley and Sons, 2007 27. Müller, C.W.: Sufficient conditions for pure state N -representability. J. Phys. A: Math. Gen. 32, 4139–4148 (1999) 28. Ness, L.: A stratification of the null cone via moment map. Amer. J. Math. 106, 1281–1329 (1984) 29. Peltzer, C.P., Brandstatter, J.J.: Studies in the theory of generalized density operators III. J. Math. Anal. Appl. 33, 263–277 (1971) 30. Perelomov, A.M.: Generalized coherent states and their applications, Berlin: Springer, 1986 31. Ruskai, M.B.: N -representability problem: Particle-hole equivalence. J. Math. Phys. 11, 3218–3224 (1970) 32. Ruskai, M.B.: Comments on Peltzer–Brandstatter papers: Two counterexamples. J. Math. Anal. Appl. 44, 131–135 (1973) 33. Ruskai, M.B.: Connecting N -representability to Weyl’s problem: The one particle density matrix for N = 3 and R = 6. J. Phys. A: Math. Theor. 40, F961–F967 (2007) 34. Vinberg, E., Popov, V.: Invariant theory. In: “Algebraic Geometry IV”, A.N. Parshin, I. Shafarevich, eds., Berlin: Springer, 1992 35. Weyl, H.: The theory of groups and quantum mechanics. New York: Dover, 1931 Communicated by M.B. Ruskai
Commun. Math. Phys. 282, 323–337 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0551-0
Communications in
Mathematical Physics
Root Asymptotics of Spectral Polynomials for the Lamé Operator Julius Borcea, Boris Shapiro Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden. E-mail: [email protected]; [email protected] Received: 1 February 2007 / Accepted: 31 January 2008 Published online: 28 June 2008 – © Springer-Verlag 2008
Abstract: The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schrödinger equation with finite gap potential given by the Weierstrass ℘-function on the real line. In this paper we establish several natural (and equivalent) formulas in terms of hypergeometric and elliptic type integrals for the density of the appropriately scaled asymptotic distribution of these eigenvalues when the integervalued spectral parameter tends to infinity. We also show that this density satisfies a Heun differential equation with four singularities. 1. Introduction and Main Results The algebraic form of the classical Lamé equation [21, Chap. 23] was introduced by Lamé in the 1830’s in connection with the separation of variables in the Laplace equation by means of elliptic coordinates in Rl . Lamé’s equation is given by d2 1 d Q(z) 2 + Q (z) + V (z) S(z) = 0, (1) dz 2 dz where Q l (z) is a real degree l polynomial with all real and distinct roots and V (z) is a polynomial of degree at most l − 2 whose choice depends on the type of solution to (1) one is looking for. In the second half of the nineteenth century several famous mathematicians including Bôcher, Heine, Klein and Stieltjes studied the number and various properties of the so-called Lamé solutions of the first kind (of given degree and type) to Eq. (1). These are also known as Lamé polynomials of a certain type. Such solutions exist for special choices of V (z) and are characterized by the property that their logarithmic derivative is a rational function. For a given Q(z) of degree l ≥ 2
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with simple roots there exist 2l different possibilities for Lamé solutions depending on whether these solutions are smooth at a given root of Q(z) or have a square root singularity, see [14,21] for more details. A generalized Lamé equation [21] is a second order differential equation of the form d2 d Q(z) 2 + P(z) + V (z) S(z) = 0, (2) dz dz where Q(z) is a complex polynomial of degree l and P(z) is a complex polynomial of degree at most l − 1. As it was first shown by Heine [10] for a generic equation of either of the forms (1) or (2) and an arbitrary positive integer n there are exactly n+l−2 n polynomials V (z) such that S(z) is a polynomial of degree n. Below we concentrate on the most classical case of a cubic polynomial Q(z) with all real roots as treated in e.g. [21]. Already Lamé and Liouville knew that in this case the (unique) root of V (z) is real and located between the minimal and the maximal roots of Q(z). The latter property was further generalized by Heine, Van Vleck and other authors. Note that if deg Q(z) = 3 then V (z) is at most linear and that for a given value of the positive integer n there are at most n + 1 such polynomials. Before formulating our main results let us briefly review – following mostly [13, §3] and [14] – some necessary background on the version of the Lamé equation used below and its spectral polynomials. Setting, as one traditionally does, Q(z) = 4(z − e1 ) (z − e2 )(z − e3 ) with e1 > e2 > e3 we can rewrite Eq. (1) as 3 d2 n(n + 1)z + E 1 1 d − 3 + S(z) = 0. (3) dz 2 2 z − ei dz 4 i=1 (z − ei ) i=1
(The chosen representation for the linear polynomial V (z) will shortly become clear.) Notice that several equivalent forms of the Lamé equation are classically known. Among those one should mention two algebraic forms, the Jacobian form and the Weierstrassian form, respectively, see [21, §23.4] and Remark 1 below. Equation (3) presents the most commonly used real algebraic form of the Lamé equation which is smooth while the other (real) algebraic form is singular.1 A Lamé solution of the first kind to Eq. (3) is a solution of the form S(z) = (z − ˜ ˜ is a polynomial. where each κi is either 0 or 21 and S(z) e1 )κ1 (z − e2 )κ2 (z − e3 )κ3 S(z), Lamé solutions which are pure polynomials – i.e., for which κ1 = κ2 = κ3 = 0 – are said to be of type 1, those with a single square root are of type 2, those with two square roots are of type 3 and, finally, those involving three square roots are said to be of type 4. (“Types” are sometimes called “species”, see, e.g., [21].) One can easily check that (3) has a Lamé solution if and only if n is a nonnegative integer. Moreover, if n is even then only solutions of types 1 and 3 exist. Namely, for appropriate choices of the 3n energy constant E one gets exactly n+2 2 distinct independent solutions of type 1 and 2 independent solutions of type 3. If n is odd then only solutions of types 2 and 4 exist. In this case, for appropriate choices of the energy constant E one gets exactly 3(n+1) 2 distinct independent solutions of type 2 and n−1 2 independent solutions of type 4. In both cases the total number of Lamé solutions equals 2n + 1, which coincides with the number of independent spherical harmonics of order n. Let Rn (E) = 2n+1 j=0 (E − E j ), n ∈ Z+ , denote the monic polynomial of degree 2n +1 whose roots are exactly the values of the energy E at which Eq. (3) has a Lamé solution. 1 The authors thank the anonymous referee for this observation.
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These polynomials are often referred to as spectral polynomials in the literature; their study goes back to Hermite and Halphen. The most recent results in this direction can be found in [3,13,16,17,20] and [8,9]. In particular, [13] contains an excellent survey of this topic as well as a comprehensive table with these polynomials (and their modified versions) correcting several mistakes that occurred in previous publications. Article [9] is apparently the first attempt to give a (somewhat) closed formula for Rn (E); for this the authors use yet another family of polynomials which they call elliptic Bernoulli polynomials. Since explicit formulas for Rn (E) seem to be rather complicated and difficult to handle, in this paper we study the asymptotics of the root distribution of appropriately scaled versions of Rn (E) (Corollary 1). In spite of a more than 150 years long history of the Lamé equation the only source discussing questions similar to ours that we were able to locate is [6]. We should also mention that the results below are actually much more precise and therefore supersede those of loc. cit. Let us now introduce eight spectral polynomials Rnκ1 ,κ2 ,κ3 (E) related to the eight types of Lamé solutions mentioned above, namely ⎧ ⎨ R 0,0,0 (E)R 21 , 12 ,0 (E)R 21 ,0, 21 (E)R 0, 21 , 21 (E) when n is even, n n n n (4) Rn (E) = 1 1 1 1 1 1 ⎩ R 2 ,0,0 (E)R 0, 2 ,0 (E)R 0,0, 2 (E)R 2 , 2 , 2 (E) when n is odd. n
n
n
n
Note for example that if n is even then Rn0,0,0 (E) is the (unique) monic polynomial of degree n+2 2 whose roots are precisely the values of E for which Eq. (3) has a pure polynomial solution. The results of the present paper actually hold not only for Eq. (3) but also for generalized Lamé equations (cf. (2)) of the form 3 d 2 αi d V (z) + 3 + S(z) = 0, (5) dz 2 z − ei dz i=1 (z − ei ) i=1
where αi > 0, i = 1, 2, 3, and V (z) is an undetermined linear polynomial. This case was thoroughly treated by Stieltjes in [15]. In particular, he proved that for any positive integer n there exist exactly n + 1 polynomials Vn, j (z), 1 ≤ j ≤ n + 1, such that (5) has a polynomial solution S(z) of degree n. Moreover, the unique root tn, j of Vn, j (z) lies in the interval (e3 , e1 ) and these n + 1 roots are pairwise distinct. Consider now the polynomial Spn (t) =
n+1
(t − tn, j ),
j=1
which is the scaled version of the spectral polynomial Rn0,0,0 (E). More exactly, for any even n one has that Spn (t) =
Rn0,0,0 (n(n+1)t)
n + 1} is ordered so that tn,1 <
n+2
(n(n+1)) 2 tn,2 < . . .
. Assume further that the set {tn, j : 1 ≤ j ≤
< tn,n+1 and associate to it the finite measure
1 δ(z − tn, j ), n+1 n+1
µn =
j=1
where δ(z − a) is the Dirac measure supported at a. The measure µn thus obtained is clearly a real probability measure that one usually refers to as the root-counting measure of Spn (t).
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Below we shall make use of some well-known notions from the theory of special functions that can be found in e.g. [1]. In fact several of our formulas rely on various integral representations and transformation properties of the Gauss hypergeometric series F(a, b, c; z) =2 F1 (a, b, c; z) =
∞ (c) (a + m)(b + m) z m , (a)(b) (c + m) m! m=0
where denotes Euler’s Gamma-function, as well as the complete elliptic integral of the first kind
1 dt . (6) K(ζ ) = 2 0 (1 − t )(1 − ζ 2 t 2 ) Recall from [1, §15.3.1] that if (c) > (b) > 0 then a convenient way of rewriting F(a, b, c; z) is given by F(a, b, c; z) =
(c) (b)(c − b)
1
t b−1 (1 − t)c−b−1 (1 − t z)−a dt.
(7)
0
The above integral represents a one-valued analytic function in the z-plane cut along the real axis from 1 to ∞ and thus it gives the analytic continuation of F(a, b, c; z). To simplify our formulas let us also define a real-valued function of three real variables (x1 , x2 , x3 ) → ω(x1 , x2 , x3 ) by setting ω(x1 , x2 , x3 ) =
(|x1 | + |x2 |)|x3 | . |x1 ||x2 − x3 | + |x2 ||x1 − x3 |
(8)
Note that ω(x1 , x2 , x3 ) is symmetric in its first two arguments and that 0 ≤ ω(x1 , x2 , x3 ) ≤ 1 whenever x1 < 0, x2 > 0 and x1 ≤ x3 ≤ x2 .
(9)
We should emphasize that the latter conditions on x1 , x2 , x3 – hence also the bounds for ω(x1 , x2 , x3 ) given in (9) – always hold in the present context. We are now ready to state our main results. Theorem 1. For any choice of real numbers e1 > e2 > e3 and positive numbers α1 , α2 , α3 the sequence of measures {µn }n∈Z+ strongly converges to the probability measure µ Q supported on the interval [e3 , e1 ] with density ρ Q given by any of the following equivalent expressions: (i)
1 + 1 − ω(e1 − e2 , e3 − e2 , s − e2 )2 (e1 − e3 )|s − e2 |ω(e1 − e2 , e3 − e2 , s − e2 ) ⎛ ⎞ 2 1 − ω(e1 − e2 , e3 − e2 , s − e2 )2 ⎠, ×K⎝i 1 − 1 − ω(e1 − e2 , e3 − e2 , s − e2 )2
1 ρ Q (s) = π
where ω is the function defined in (8).
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(ii)
327
1 + 1 − ω(e1 − e2 , e3 − e2 , s − e2 )2 (e1 − e3 )|s − e2 |ω(e1 − e2 , e3 − e2 , s − e2 ) 1 1 2 1 − ω(e1 − e2 , e3 − e2 , s − e2 )2 , , 1; − ×F . 2 2 1 − 1 − ω(e1 − e2 , e3 − e2 , s − e2 )2
1 ρ Q (s) = 2
(iii)
ρ Q (s) =
ω(e1 −e2 , e3 −e2 , s −e2 ) 2(e1 −e3 )|s −e2 |(1+ω(e1 −e2 , e3 −e2 , s −e2 )) 1 − ω(e1 − e2 , e3 − e2 , s − e2 ) 1 1 , , 1; . ×F 2 2 1 + ω(e1 − e2 , e3 − e2 , s − e2 )
(iv) 1 2 [(e1 +e3 −2e2 )(s −e2 )+2(e1 −e2 )(e2 −e3 )+(e1 −e3 )|s −e2 |] 1 1 (e1 +e3 −2e2 )(s −e2 )+2(e1 −e2 )(e2 −e3 )−(e1 −e3 )|s −e2 | ×F , , 1; , 2 2 (e1 +e3 −2e2 )(s −e2 )+2(e1 −e2 )(e2 −e3 ) + (e1 − e3 )|s − e2 |
ρ Q (s) = √
that is,
e1 dx 1 when e3 < s < e2 , ρ Q (s) = √ 2π e2 (e1 − x)(x − e2 )(x − 1e3 )(x − s)
e2 dx 1 when e2 < s < e1 . ρ Q (s) = √ 2π e3 (e1 − x)(e2 − x)(x − e3 )(s − x)
Theorem 2. The density function ρ Q defined in Theorem 1 satisfies the following Heun differential equation: 8Q(s)ρ Q (s) + 8Q (s)ρ Q (s) + Q (s)ρ Q (s) = 0,
(10)
where Q(s) = (s − e1 )(s − e2 )(s − e3 ). Both indices of this equation at the finite regular singularities e1 , e2 , e3 vanish while its indices at ∞ equal 21 and 23 . Corollary 1. The root-counting measures for each of the eight normalized spectral polynomials Rnκ1 ,κ2 ,κ3 (n(n + 1)t) as well as that of the normalized spectral polynomial Rn (n(n + 1)t) converge to the measure µ Q defined in Theorem 1, see Fig. 1 below. Remark 1. Note that Eq. (3) is usually lifted to the elliptic curve defined by y 2 = Q(z) and that on this equation takes the so-called Weierstrassian form 2 d − [n(n + 1)℘ (u) + E] S(u) = 0, du 2 where u is the canonical coordinate on the universal covering of given by
∞ dw u= . √ Q(w) z
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Fig. 1. Comparison of the theoretical density with the numerical density of the measure µ Q for Q(z) = z 3 − z
If Q(z) has all real roots then n(n + 1)℘ (u) becomes a finite gap real periodic potential on R, see [11]. Depending on the parity on n four of the eight spectral polynomials Rnκ1 ,κ2 ,κ3 (E) determine the ends of bands of the spectrum of the corresponding Schrödinger operator on R. Hopefully our results will find applications to the spectral theory of finite gap potentials. To end this introduction let us mention that in the present context one can actually “guess” the last two formulas in Theorem 1 (iv) by WKB-type considerations, as we were kindly informed by K. Takemura in the final stages of this work. However, such arguments fail to provide an accurate mathematical proof. By contrast, our methods use solely rigorous results involving orthogonal polynomials, elliptic integrals and hypergeometric functions, and we are currently unaware of any possible “shortcuts” in this set-up. Finally, we should also note that so far neither our methods nor WKB-type “guesses” could provide an answer to these questions in more general situations, see §3 below. 2. Proofs Proof of Theorem 1. Theorem 1 essentially follows from the main result of [7] after some amount of work. First we express the polynomial Spn (t) as the characteristic polynomial of a certain matrix. In order to make this matrix tridiagonal so as to simplify the calculations we assume wlog that Q(z) = (z − e3 )z(z − e1 ) = z 3 + vz 2 + wz with e3 < e2 = 0 < e1 . Set T = (z 3 + vz 2 + wz)
d2 d + (αz 2 + βz + γ ) − θn (z − t), 2 dz dz
where v, w, α, β, γ are fixed constants and θn , t are variables. Assuming that S(z) = a0 z n + a1 z n−1 + · · · + an with undetermined coefficients ai , 0 ≤ i ≤ n, we are looking
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for the values of θn , t and ai , 0 ≤ i ≤ n, such that T (S(z)) = 0. Note that T (S(z)) is in general a polynomial of degree n + 1 whose leading coefficient equals a0 [n(n − 1) + αn − θn ]. To get a non-trivial solution we therefore set θn = n(n − 1 + α). Straightforward computations show that the coefficients of the successive powers z n , z n−1 , . . . , z 0 in T (S(z)) can be expressed in the form of a matrix product Mn A, where A = (a0 , a1 , . . . , an )T and Mn is the following tridiagonal (n + 1) × (n + 1) matrix: ⎛ ⎞ t − ξn,1 αn,2 0 0 ··· 0 0 0 ··· 0 0 ⎜ γn,2 t − ξn,2 αn,3 ⎟ ⎜ 0 ⎟ γ t − ξ α · · · 0 0 ⎜ ⎟ n,3 n,3 n,4 ⎜ . ⎟ .. .. .. .. .. .. ⎜ . ⎟ . . . . . . Mn := ⎜ . ⎟ ⎜ ⎟ .. .. ⎜ ⎟ . . αn,n 0 0 0 ⎜ 0 ⎟ ⎝ 0 0 0 · · · γn,n t − ξn,n αn,n+1 ⎠ 0 0 0 · · · 0 γn,n+1 t − ξn,n+1 with v(n − i)(n − i + 1) + β(n − i + 1) , i ∈ {1, . . . , n + 1}, θn (n − i)(n − i + 1) + α(n − i + 1) = − 1, i ∈ {2, . . . , n + 1}, θn w(n − i + 1)(n − i + 2) + γ (n − i + 2) = , i ∈ {2, . . . , n + 1}. θn
ξn,i = − αn,i γn,i
(11)
A similar matrix can be found in [10] and also in [19]. The matrix Mn depends linearly on the indeterminate t which appears only on its main diagonal. If the linear system Mn A = 0 is to have a nontrivial solution A = (a0 , a1 , . . . , an )T the determinant of Mn has to vanish. This gives the polynomial equation Spn (t) = det(Mn ) = 0. The sequence of polynomials {Spn (t)}n∈Z+ does not seem to satisfy any reasonable recurrence relation. In order to overcome this difficulty and to be able to use the technique of 3-term recurrence relations with variable coefficients (which is applicable since Mn is tridiagonal) we extend the above polynomial sequence by introducing an additional parameter. Namely, define Spn,i (t) = det Mn,i , i ∈ {1, . . . , n + 1}, where Mn,i is the upper i × i principal submatrix of Mn . One can easily check (see, e.g., [2, p. 20]) that the following 3-term relation holds: Spn,i (t) = (t − ξn,i )Spn,i−1 (t) − ψn,i Spn,i−2 (t), i ∈ {1, . . . , n + 1},
(12)
where ξn,i is as in (11) and ψn,i = αn,i γn,i , i ∈ {2, . . . , n + 1}.
(13)
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Here we used the (standard) initial conditions Spn,0 (t) = 1, Spn,−1 (t) = 0. It is wellknown that if all ξn,i ’s are real and all ψn,i ’s are positive then the polynomials Spn,i (t), i ∈ {0, . . . , n + 1}, form a (finite) sequence of orthogonal polynomials. In particular, all their roots are real. Under the assumptions of Theorem 1 the reality of ξn,i is obvious. Assuming that the positivity of ψn,i is also settled (see Lemma 1 below) let us complete the proof of Theorem 1. For this we invoke [7, Theorem 1.4] which translated in our notation claims that if there exist two continuous functions ξ(τ ) and ψ(τ ), τ ∈ [0, 1], such that lim
i/(n+1)→τ
ξi,n = ξ(τ ),
lim
i/(n+1)→τ
ψi,n = ψ(τ ), ψ(τ ) ≥ 0 ∀τ ∈ [0, 1],
then the density of the asymptotic root-counting measure of the polynomial sequence {Spn (t)}n∈Z+ = {Spn,n+1 (t)}n∈Z+ is given by
1 ω[ξ(τ )−2√ψ(τ ),ξ(τ )+2√ψ(τ )] dτ, 0
where for any x < y one has
ω[x,y] (s) =
1 √ π (y−s)(s−x)
if s ∈ [x, y],
0
otherwise.
From the explicit formulas for ξn,i and ψn,i (see (11) and (13)) one easily gets ξ(τ ) = ψ(τ ) =
lim
ξi,n = −v(1 − τ )2 ,
lim
ψi,n = −w(1 − (1 − τ )2 )(1 − τ )2 .
i/(n+1)→τ i/(n+1)→τ
Notice that the above limits are independent of the coefficients α, β, γ and that by the assumption on Q(z) made at the beginning of this section one has w = e3 e1 < 0, which in its turn implies that ψ(τ ) ≥ 0 for τ ∈ [0, 1]. The required density ρ Q (s) is therefore given by
1 dτ ρ Q (s) = + √ √ 0 π ξ(τ ) + 2 ψ(τ ) − s s − ξ(τ ) + 2 ψ(τ )
1 dτ = , 2 + 0 π −4w 1 − (1 − τ )2 (1 − τ )2 − v(1 − τ )2 + s √+ where · is meant to remind that the integrand√ vanishes whenever the expression under the square root becomes negative. Introducing ν = 1 − τ we get
1 dν 1 ρ Q (s) = +. 2π 0 2 2 ν[(4w − v )ν − (4w + 2vs)ν − s 2 ] √+ In order to get rid of · we rewrite
νmax (s) dν 1 , (14) ρ Q (s) = √ √ 2 ν(ν − νmin (s))(νmax (s) − ν) 2π v − 4w νmin (s)
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where νmin (s) and νmax (s) are the minimal and maximal roots of of the equation (4w − v 2 )ν 2 − (4w + 2vs)ν − s 2 = 0 with respect to ν, that is, νmin (s) = −(v 2 − 4w)−1 [2w + vs + 2 w(w + vs + s 2 )], νmax (s) = −(v 2 − 4w)−1 [2w + vs − 2 w(w + vs + s 2 )]. A few remarks are in order at this stage. First, by the real-rootedness of Q(z) one has v 2 − 4w = (e1 − e3 )2 > 0. Second, we claim that ξ(τ ) − 2 ψ(τ ), ξ(τ ) + 2 ψ(τ ) ⊆ [e3 , e1 ] for τ ∈ [0, 1]. Indeed, e1 − ξ(τ ) − 2 ψ(τ ) = e1 + v(1 − τ )2 − 2 −w(1 − (1 − τ )2 )(1 − τ )2 = e1 (1 − (1 − τ )2 ) − 2 e1 (1 − (1 − τ )2 ) −e3 (1 − τ )2 − e3 (1 − τ )2 2 = e1 (1 − (1 − τ )2 ) − −e3 (1 − τ )2 ≥ 0 and similarly ξ(τ ) − 2 ψ(τ ) − e3 = −v(1 − τ )2 − 2 −w(1 − (1 − τ )2 )(1 − τ )2 − e3 = e1 (1 − τ )2 − 2 e1 (1 − τ )2 −e3 (1 − (1 − τ )2 ) − e3 (1 − (1 − τ )2 ) 2 e1 (1 − τ )2 − −e3 (1 − (1 − τ )2 ) ≥ 0. = It follows that whenever s is such that the integrand in the above formulas is nonvanishing one has e3 ≤ s ≤ e1 hence w(w + vs + s 2 ) = −e3 e1 (s − e3 )(e1 − s) ≥ 0 and 2w + vs = e1 (e3 − s) + e3 (e1 − s) < 0 for all s as above. Therefore νmin (s) + νmax (s) = −2(2w + vs)(v 2 − 4w) > 0 and νmin (s)νmax (s) = s 2 (v 2 − 4w) ≥ 0 from which we conclude that νmax (s) > νmin (s) ≥ 0 for e3 < s < e1 . Recall the definition and properties of the function ω from (8)–(9) and note that √ |s| v 2 − 4w (e1 − e3 )|s| =− (15) ω(e1 , e3 , s) = e1 (s − e3 ) − e3 (e1 − s) 2w + vs so that νmin (s) and νmax (s) may actually be rewritten as follows: ! ! |s| 1 − 1 − ω(e1 , e3 , s)2 |s| 1 + 1 − ω(e1 , e3 , s)2 , νmax (s) = . νmin (s) = (e1 − e3 )ω(e1 , e3 , s) (e1 − e3 )ω(e1 , e3 , s)
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Using these expressions combined with the fact that for any x < y one has "
y dν x−y 2 =√ K √ x x ν(ν − x)(y − ν) x (which readily follows from (6)) we deduce from (14) that ⎛ ⎞ 2 2 1 − ω(e , e , s) 1 1 + 1 − ω(e1 , e3 , s)2 ⎝ 1 3 ⎠. K i ρ Q (s) = π (e1 − e3 )|s|ω(e1 , e3 , s) 1 − 1 − ω(e1 , e3 , s)2 Now from (6)–(7) and the well-known identities (1) = 1, ( 21 ) = 1 1 π 2 , , 1; ζ K(ζ ) = F 2 2 2 hence
1 ρ Q (s) = 2
√
(16)
π one easily gets
1 1 2 1 − ω(e1 , e3 , s)2 1 + 1 − ω(e1 , e3 , s)2 F , , 1; − (17) (e1 − e3 )|s|ω(e1 , e3 , s) 2 2 1 − 1 − ω(e1 , e3 , s)2
by (16). It is a remarkable fact due to Kummer and Goursat that for special choices of the numbers a, b, c the hypergeometric series F(a, b, c; z) obeys certain quadratic transformation laws. To complete the proof we need precisely such transformation properties, namely formulas 15.3.19-20 in [1] with a = 41 , b = 21 and c = 1: √ √ 2 z 1− 1−z 2 1 1 1 1 1 , , 1; − = , , 1; F . √ √ √ √ F 2 2 1− z 2 2 1+ 1−z 1+ 1−z 1− z The above identity for z = 1 − ω(e1 , e3 , s)2 together with (17) then yields 1 1 1 − ω(e1 , e3 , s) ω(e1 , e3 , s) F , , 1; ρ Q (s) = 2(e1 − e3 )|s|(1 + ω(e1 , e3 , s)) 2 2 1 + ω(e1 , e3 , s)
(18)
which by (15) amounts to 1 ρ Q (s) = √ 2 [(e1 + e3 )s − 2e1 e3 + (e1 − e3 )|s|] 1 1 (e1 + e3 )s − 2e1 e3 − (e1 − e3 )|s| ×F , , 1; . 2 2 (e1 + e3 )s − 2e1 e3 + (e1 − e3 )|s|
(19)
To prove the last two formulas of Theorem 1 (iv) we use (19) and (7) with a = b = 21 and c = 1 in order to get an integral representation of ρ Q (s) in which one makes the following variable substitutions: t =−
e3 (e1 − x) e1 (x − e3 ) if e3 < s < 0, t = − if 0 < s < e1 . e1 (x − e3 ) e3 (e1 − x)
The desired expressions are then obtained by straightforward computations.
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We thus established all formulas stated in Theorem 1 in the special case when e3 < e2 = 0 < e1 . The general case reduces to this one by noticing that if e3 < e2 < e1 and Q(z) = (z − e3 )(z − e2 )(z − e1 ) then all the above arguments may be used for the polynomial Q(z + e2 ). Hence the expressions for ρ Q (s) given in Theorem 1 in the general case are obtained from (16)–(19) simply by replacing e1 , e3 and s with e1 − e2 , e3 − e2 and s − e2 , respectively.
Lemma 1. Let Q(z) = z 3 + vz 2 + wz and P(z) = αz 2 + βz + γ be two polynomials such that Q(z) has three real distinct roots e1 > e2 = 0 > e3 and α1 P(z) α2 α3 = + + Q(z) z − e1 z − e2 z − e3 with α1 , α2 , α3 > 0. Then the coefficients ψn,i defined in (13) are all positive. Proof. One immediately gets α > 0, which implies that θn > 0 for any positive integer n. Recall from (11) and (13) that ψn,i = αn,i γn,i . Now αn,i = θn−1 {[(n − i)(n − i + 1) − n(n − 1)] + α[(n − i + 1) − n]} and since both α and θn are positive it follows that αn,i < 0. Let us show that under our assumptions one also has γn,i < 0. For this it is clearly enough to show that both w and γ are negative. Obviously, w = e3 e1 < 0 and since α1 , α2 , α3 > 0 the quadratic polynomial P(z) has two real roots interlacing with e3 , e2 , e1 . Therefore P(z) has one positive and one negative root and positive leading coefficient, hence γ = P(0) < 0.
Proof of Theorem 2. In order to deduce the differential equation satisfied by ρ Q (s) we note first that the restrictions of ρ Q (s) to (e3 , e2 ) and (e2 , e1 ), respectively, are two branches of the same multi-valued analytic function (this can be seen e.g. from the last two expressions in Theorem 1). Thus it suffices to derive the linear differential equation satisfied by ρ Q (s) restricted to, say, (e2 , e1 ). Specializing formula (iv) of Theorem 1 to this case we get ρ Q (s) = where
I Q (s) = 0
1
I Q (s) , √ 2π (e1 − e2 )(s − e3 )
dw " ! 2 −e3 )(e1 −s) w(1 − w) 1 − (e w (e1 −e2 )(s−e3 )
factor I Q (s) is the same as the and s ∈ (e2 , e1 ). By (7) we see that up to a constant ! 2 −e3 )(e1 −s) hypergeometric series F 21 , 21 , 1; (e , which is known to satisfy the following (e1 −e2 )(s−e3 ) Riemann differential equation (see, e.g., [1, §15.6]): 1 1 (e3 − e2 )(e3 − e1 ) I Q (s) + I Q (s) = 0. + I Q (s) + s − e1 s − e2 4(s − e3 )Q(s) √ √ Substituting I Q (s) = K s − e3 ρ Q (s), where K := 2π e1 − e2 is but a constant, in the latter equation we get after some straightforward algebraic manipulations 3 #3 1 3s − i=1 ei ρ Q (s) = 0, ρ Q (s) + (20) ρ Q (s) + 3 s − ei 4 i=1 (s − ei ) i=1
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which after multiplication by 8Q(s) coincides with the required equation (10). To calculate the indices recall from e.g. [12] that for a second order linear differential equation ρ (s) + a1 (s)ρ (s) + a2 (s)ρ(s) = 0 its indicial equation at a finite regular or regular singular point s˜ has the form ζ (ζ − 1) + α1 ζ + α2 = 0, where α1 = lims→˜s (s − s˜ )a1 (s) and α2 = lims→˜s (s − s˜ )2 a2 (s). Thus for (20) the indicial equation at each ei , i = 1, 2, 3, has the form ζ 2 = 0, which implies that both corresponding indices vanish and that a solution to (20) might have a logarithmic singularity at any of these points, see Remark 2 below. The indicial equation of ρ (s) + a1 (s)ρ (s) + a2 (s)ρ(s) = 0 at ∞ has the form ζ (ζ + 1) − α˜ 1 ζ + α˜ 2 = 0, where α˜ 1 = lims→∞ sa1 (s) and α˜ 2 = lims→∞ s 2 a2 (s). Therefore the indicial equation at ∞ for (20) is ζ (ζ + 1) − 3ζ + 43 = 0 whose roots are 21 and 23 .
Remark 2. It is not difficult to show that
! 16(e1 −e2 )(e2 −e3 ) log (e1 −e3 )|s−e2 | 1 as s → e2 , ρ Q (s) ≈ √ 2π (e1 − e2 )(e2 − e3 )
so that ρ Q (s) always has a logarithmic singularity at e2 . Proof of Corollary 1. The main idea of the proof of Corollary 1 is that the polynomial part of any Lamé solution to (3) itself satisfies a very similar differential equation. Let us illustrate this in the case of Lamé solutions of type 2. (The other cases can be ˜ dealt with in the same way.) Assume that S(z) is a polynomial of degree n such that 1 1 ˜ solves the equation S(z) := (z − e1 ) 2 (z − e2 ) 2 S(z) % $ 3 1 1 4Q(z) S (z) + S (z) + V (z)S(z) = 0 2 z − ei i=1
for some linear polynomial V (z). Recall that the (unique) root of each such V (z) is 1 1
, ,0
also a root of the scaled spectral polynomial Rn2 2 (n(n + 1)t). Substituting S(z) = 1 1 ˜ one gets after some straightforward calculations that S(z) ˜ satisfies (z−e1 ) 2 (z−e2 ) 2 S(z) the equation & ' 3 1 3 1 ˜ 4Q(z) S˜ (z) + + + = 0, (21) S˜ (z) + V˜ (z) S(z) 2 z − e1 z − e2 z − e3 where V˜ (z) = V (z) + (z − e1 ) − (z − e2 ) + 4(z − e3 ). Notice that in the coefficient in front of S˜ (z) the numerators of the first two simple fractions are increased by 1 while the remaining one is unchanged. Thus we are practically in the situation covered by Theorem 1. The only (slight) difference is that we are not considering the asymptotic distribution of the roots of polynomials V˜ (z) but that of V (z). However, since V˜ (z) = V (z) + (z − e1 ) − (z − e2 ) + 4(z − e3 ) and the leading coefficient θn of V (z) tends to ∞ as n → ∞ it follows that both these families of polynomials actually have the same asymptotic root distribution. Finally, note that by (4) the scaled spectral polynomial
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1
0.8
0.6
0.4
0.2
0 -0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Fig. 2. The roots of the scaled spectral polynomial Sp50 (t) for Q(z) = z(z − 1)(z + 21 − i)
Rn (n(n + 1)t) equals the product of four corresponding polynomials Rnκ1 ,κ2 ,κ3 (n(n + 1)t) and that all eight polynomial families {Rnκ1 ,κ2 ,κ3 (n(n + 1)t)}n∈Z+ have the same asymptotic root distribution when n → ∞. This implies that the family {Rn (n(n + 1)t)}n∈Z+ itself has the same limiting root distribution.
3. Remarks and Conjectures 1. So far we were unable to extend our method of proving Theorem 1 to the case of a cubic polynomial Q(z) with complex roots. The main difficulty comes from the fact that Theorem 1.4 of [7] seems to fail in this case. Indeed, if it were true then the support of the resulting root-counting measure would be two-dimensional. However, numerical experiments strongly suggest that this support is one-dimensional, see Fig. 2. Problem 1. Generalize Theorem 1 to the case of a complex cubic polynomial Q(z). Based on our previous experience (comp. [4,5]) we conjecture that: (i) The asymptotic root distribution of the Van Vleck polynomials for Eq. (5) with complex coefficients is independent of the αi ’s and depends only on the leading polynomial Q(z); (ii) Denoting the above limiting distribution by µ Q we claim its support is straigh( z that dt tened out in the canonical local coordinate w(z) = z 0 √ Q(t) .
2. Even more difficulties occur when dealing the more general Lamé equation (2) since in this case Van Vleck polynomials and their roots cannot be found by means of a determinantal equation. They are in fact related to a more complicated situation when the rank of a certain non-square matrix is less than the maximal one, see [5]. Problem 2. Describe the asymptotic distribution of the roots of all Van Vleck polynomials for Eq. (2) when n → ∞. An illustration of this asymptotic distribution is given in the next picture. (Fig. 3).
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1
0
-1
-2
-3 0
0.5
1
1.5
2
2.5
3
Fig. 3. The union of the roots of 861 quadratic Van Vleck polynomials corresponding to Stieltjes polynomials
of degree 40 for the classical Lamé equation Q(z)S (z) + Q 2(z) S (z) + V (z)S(z) = 0 with Q(z) = (z 2 + 1)(z − 3i − 2)(z + 2i − 3)
Acknowledgements. We are grateful to M. Tater from the Institute of Nuclear Physics, Czech Academy of Sciences, for interesting discussions on this subject during his visit to Stockholm in November 2006. We would especially like to thank K. Takemura from Yokohama City University for numerous illuminating discussions prior to and during his visit to Stockholm in September 2007. We would also like to thank the anonymous referee for his insightful comments and suggestions.
References 1. Abramowitz, M., Stegun, I. A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Appl. Math. Ser. 55, New York: Dover Publications, 1964 2. Arscott, F.M.: Periodic differential equations. New York: Pergamon Press, 1964 3. Belokolos, D., Enolskii V., Z.: Reduction of Abelian functions and algebraically integrable systems. II. Complex analysis and representation theory. J. Math. Sci. (New York) 108, 295–374 (2002) 4. Borcea, J.: Choquet order for spectra of higher Lamé operators and orthogonal polynomials. J. Approx. Theory 151, 164–180 (2008) 5. Borcea, J., Brändén, P., Shapiro B.: Higher Lamé equations, Heine-Stieltjes and Van Vleck polynomials. In preparation 6. Bourget, A.: Nodal statistics for the Van Vleck polynomials. Commun. Math. Phys. 230, 503–516 (2002) 7. Kuijlaars, A.B.J., Van Assche, W: The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients. J. Approx. Theory 99, 167–197 (1999) 8. Grosset, M.P., Veselov, A.P.: Elliptic Faulhaber polynomials and Lamé densities of states. Int. Math. Res. Not. 2006, Art. ID 62120, 31 pp 9. Grosset, M.P., Veselov, A.P.: Lamé’s equation, quantum top and elliptic Bernoulli polynomials. http:// arXiv.org/list/math-ph/0508068, 2005 10. Heine, E.: Handbuch der Kugelfunctionen, Vol. 1, Berlin: G. Reimer Verlag, 1878, pp. 472–479 11. Ince, E.L.: Further investigations into periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, 83–99 (1940) 12. Ince, E.L.: Ordinary differential equations. New York: Dover Publications, 1944 13. Maier, R.S.: Lamé polynomials, hyperelliptic reductions and Lamé band structure. Philos. Trans. Roy. Soc. London Ser. A 366, 1115–1153 (2008) 14. Poole, E.G.C.: Introduction to the theory of linear differential equations. New York: Dover Publications, 1960 15. Stieltjes, T.: Sur certains polynômes qui vérifient une équation différentielle linéaire du second ordre et sur la théorie des fonctions de Lamé. Acta Math. 8, 321–326 (1885) 16. Takemura, K: Analytic continuation of eigenvalues of the Lamé operator. J. Differ. Eqs. 228, 1–16 (2006) 17. Takemura, K.: On eigenvalues of the Lamé operator. http://arXiv.org.list/math.CA/0409247, 2007 18. Takemura, K.: Private communication, December 2006
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19. Turbiner, A.: Quasi-exactly solvable differential equations. In: CRC Handbook of Lie group analysis of differential equations, Vol. 3, ed. N. H. Ibragimov, Boca Raton, FL: CRC Press, 1996 20. Volkmer, H.: Four remarks on eigenvalues of the Lamé equation. Analysis and Applic. 2(2), 161–175 (2004) 21. Whittaker, E.T., Watson, G.: A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions with an account of the principal transcendental functions. Reprint of the 4th (1927) edition, Cambridge Mathematical Library, Cambridge: Cambridge Univ. Press, 1996 Communicated by N.A. Nekrasov
Commun. Math. Phys. 282, 339–355 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0560-z
Communications in
Mathematical Physics
Precise Constants in Bosonization Formulas on Riemann Surfaces. I Richard A. Wentworth Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA. E-mail: [email protected] Received: 2 March 2007 / Accepted: 13 March 2008 Published online: 3 July 2008 – © Springer-Verlag 2008
Abstract: A computation of the constant appearing in the spin-1 bosonization formula is given. This constant relates Faltings’ delta invariant to the zeta-regularized determinant of the Laplace operator with respect to the Arakelov metric. 1. Introduction The bosonization formulas on Riemann surfaces relate zeta-regularized determinants of Laplace operators acting on sections of line bundles and on scalars [AMV,VV,ABMNV, Sn,DS,F2]. They play an important role in conformal field theory and perturbative string theory (for a survey of the subject, see [DP3]). Proofs of these identities in the mathematical literature generally proceed by computing either first or second variations of certain combinations of Green’s and theta functions with respect to the Riemann moduli. As a consequence, all current formulations leave undetermined constants of integration, depending only on the genus and spin, which must be evaluated by other means. There has been renewed interest recently in the precise values of these constants (cf. [DGP]). In [W1], using ideas of Belavin-Knizhnik [BK] and D’Hoker-Phong [DP1], along with the results in [W2] on the behavior of the Arakelov metric on degenerating surfaces, values for the constants cg associated to genus = g and spin = 1 were obtained. The argument was heuristic, however, since it involved techniques from functional integration where normalizations can be somewhat arbitrary. The goal of this note is to give a rigorous proof of the result in [W1]. For the purposes of this paper, we may define cg in terms of the relationship between Faltings’ delta invariant δ(M) of a compact Riemann surface M of genus g ≥ 1 and the determinant of the Laplace operator on functions with respect to the Arakelov metric (cf. [Fl]), δ(M) = cg − 6 log
det (M,Arak.) . area(M, Arak.)
Research supported in part by NSF grant DMS-0505512.
(1.1)
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The spin-1 bosonization formula states that 3/4 ϑ( p1 +· · · + pg − z − ) i< j G( pi , p j ) det (M,Arak.) cg /8 =e , g area(M, Arak.) det Im det ωi ( p j ) i=1 G( pi , z) (1.2) where ϑ is the theta function associated to M, is the Riemann divisor, ωi is a basis of abelian differentials, is the period matrix, G(z, w) is the Arakelov-Green’s function, and { pi , z} ⊂ M are generic points. The main result is Theorem 1.3. The value of cg in (1.1) is cg = (1 − g)c0 + gc1 , c1 = −8 log(2π ), c0 = −24ζ (−1) + 1 − 6 log(2π ) − 2 log(2), where ζ (s) is the Riemann zeta function. Remark 1.4. In [W1] one of the terms in the computation of the conformal anomaly was mistakenly neglected (see [W3] and the proof of Lemma 5.2 below). With this correction, the expression in Theorem 1.3 agrees with the result in [W1]. It is noteworthy that the path integral approach to factorization computes this constant exactly. Remark 1.5. The constant: ag = (1 − g)a0 , a0 = c0 − c1 = −24ζ (−1) + 1 + 2 log(2π ) − 2 log(2), is called “Deligne’s constant” and is the normalization in another formulation of the bosonization formulas (see [D]). We note that this value of cg implies that formally: δ(P1 ) = 0 (see [J2, §7]). Remark 1.6. In addition to [W1], expressions for cg have previously appeared in the mathematical literature (cf. [GSo,So,J2]). These disagree slightly with the result obtained in Theorem 1.3. The proof of Theorem 1.3 is obtained via degeneration. Let Mt denote a family of genus g Riemann surfaces degenerating as t → 0 to a semistable nodal curve M0 with irreducible components M + and M − of genus g + and g − , respectively. We will use the following Theorem 1.7 ([W2], Main Theorem; see also [J1]). 4g + g − lim δ(Mt ) + log |t| = δ(M + ) + δ(M − ). t→0 g To evaluate cg we compare this result with the degeneration of the determinant of the Laplace operator for the Arakelov metric, and then we use (1.1). The asymptotic behavior of determinants of laplacians is a widely studied subject (cf. [DP2,Sa,K,V,GIJR,H, Wo1]). The new ingredient that is useful here is the result of [BFK]. Let γ denote a simple closed separating curve on M, with R + ∪ R − = M − γ the connected components. Given a Riemannian metric h on M, let R ± have the induced metrics.
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Theorem 1.8 ([BFK], Theorem B*). det N(M,γ ,h) det (M,h) = det(R + ,h) det(R − ,h) . area(M, h)
h (γ ) Here, the determinants on the right-hand-side are evaluated with respect to the spectra for the Dirichlet problems on the surfaces with boundary R ± , h (γ ) denotes the length of γ with respect to the metric h, and N(M,γ ,h) denotes the Neumann jump operator for functions on γ ; in this case, it is the sum of the Dirichlet-to-Neumann operators for R ± . In Sect. 2 we show how Theorem 1.8 can be used to factorize det M in terms of det M ± (see Proposition 2.5). This argument is closely parallel to the one in [W1], where the sewing property of path integrals was used in lieu of Theorem 1.8. In Sect. 3 we make a simple observation concerning the asymptotic behavior of the Neumann jump operator N(M,γ ,h) as the surface degenerates along γ . The key result, Proposition 3.5, is that N(M,γ ,h) is equal to a universal operator on L 2 -functions on the circle, modulo trace-class operators with asymptotically vanishing norm (for more detailed treatments, see [GG,PW,L,MM]). In Sect. 4 we briefly review the results needed on the Arakelov metric, and in Sect. 5 we complete the proof of Theorem 1.3. As a final remark, we emphasize that the method described above gives the asymptotic behavior of determinants up to the zeroth order term for any family of conformal metrics, provided one has sufficient information on the degeneration (essentially, a C 0 estimate on the metric and its curvature, with bounded growth of derivatives; see Proposition 4.6). This paper illustrates the case of the Arakelov metric, but the technique applies equally well to the hyperbolic metric, for example. Indeed, using the expansion in [Wo2] one can quickly recover the asymptotic behavior for the case of pinching along a separating curve, log det (Mt ,hyp.) = (1/6) log |t| + O(1) that is a consequence of the (more precise) expression given in [Wo1, Theorem 5.3], without passing through the explicit evaluation of determinants in terms of Selberg zeta functions. This also explains why evaluating the next order term is more difficult in this case, since the procedure in [Wo2] relates the hyperbolic metric on Mt to the complete hyperbolic metrics on the punctured surfaces M ± − { p ± }, and not to the hyperbolic metrics on the closed surfaces M ± . The conformal factors relating these two are, of course, quite complicated (cf. [JL]). 2. Factorization of Determinants Let M + and M − be a pair of closed Riemann surfaces with genera g ± ≥ 1. For a complex parameter t, 0 < |t| ≤ 1 we construct a degenerating family Mt of closed surfaces of genus g = g + + g − using the “plumbing construction”: z + z − = t, for local coordinates z ± on M ± centered at points p ± (for more details, see [W2]). Let γt denote the curve in Mt given by |z + | = |z − | = |t|1/2 . If we set Bt = {z ∈ C : |z| ≤ |t|1/2 }, identify z with z ± , and let Rt± = M ± − Bt , then Mt − γt is conformally equivalent to the disjoint union Rt+ ∪ Rt− . Fix conformal metrics h ± on M ± . For simplicity, we will assume in advance that z ± are normalized in the sense that with respect to these coordinates h ± (0) = 1. Let h t , t = 0 be a family of conformal metrics on Mt . Then on Rt± there are conformal factors ± h t M −γ = h ± e2σt . (2.1) t
t
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Set Q(Mt ,h t ) =
det (Mt ,h t ) , area(Mt , h t )
(2.2)
where det denotes the zeta-regularized determinant of the (positive) Laplace-Beltrami operator over the nonzero spectrum {λ j }∞ j=1 : log det = −ζ (0) ,
ζ (s) =
∞
λ−s j .
j=1
Similarly, we define Q(M ± ,h ± ) =
det (M ± ,h ± ) . area(M ± , h ± )
We wish to derive an expression for Q Mt in terms of Q M ± . To do this, we pass through the relative determinants. Let Q(R ± ,h t ) = det(R ± ,h t ) , t
t
Q(R ± ,h ± ) = det(R ± ,h ± ) t
t
the Laplacian on Rt± with respect and the metrics h ± on the other.
denote the determinants for to the restriction of the metric h t on the one hand, In this note we always choose Dirichlet boundary conditions on a manifold with boundary. Since h t and h ± are conformally related as in (2.1), we have the Polyakov-Alvarez formula (cf. [A]): S
± ±
(σ ± )
Q(R ± ,h t ) = Q(R ± ,h ± ) e (Rt ,h ) t , (2.3) t t 1 1 S(R,h) (σ ) = − d Ah 2K h σ + |∇σ |2 − dsh {2kh σ + 3∂n σ } , (2.4) 12π R 12π ∂ R where K h and kh are the Gauss and geodesic curvatures, respectively, for the metric h. Let Nt denote the Neumann jump operator for (Mt , γt , h t ), and Nt± the Neumann jump operators for the surfaces (M ± , γt± , h ± ), where γt± = ∂ Rt± . Finally, we set Q Bt = det Bt , Q(Bt ,h) = det(Bt ,h) , where Bt has the Euclidean metric, or a general metric h, respectively. Then we have the following Proposition 2.5. (γ ) det N + det N −
ht t t t 2 + − lim Q(Mt ,h t ) (Q Bt ) exp −S(Rt+ ,h + ) (σt ) − S(R − ,h − ) (σt ) t t→0
h + (γt ) h − (γt ) det Nt = Q(M + ,h + ) Q(M − ,h − ) . Proof. Applying Theorem 1.8 and (2.3) we have det N t Q(Mt ,h t ) = Q(Rt+ ,h + ) Q(R − ,h − ) exp S(Rt+ ,h + ) (σt+ ) + S(R − ,h − ) (σt− ) . t t
h t (γt )
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Using Theorem 1.8 again for (M ± , h ± ): Q(M ± ,h ± ) = Q(R ± ,h ± ) Q(Bt ,h ± ) t
det Nt± .
h ± (γt± )
Now Q(Bt ,h ± ) and Q Bt are related by the Liouville action (2.4) for a smooth conformal factor. Hence, Q(Bt ,h ± ) /Q Bt → 1 as t → 0 (recall the normalization h ± (0) = 1). The result follows.
3. Asymptotics of the Neumann Jump Operator Let M be a compact Riemann surface and z = r eiθ a local coordinate. For a real parameter ε, 0 < ε ≤ 1, set B(ε) = {x ∈ M : |z(x)| < ε}, and R(ε) = M − B(ε) (n.b. in the notation of the previous section, B(ε) = Bt and R ± (ε) = Rt± , where |t| = ε2 ). Assume M is equipped with a conformal metric that is euclidean |dz|2 in a neighborhood of B(1). Let P R(ε) : L 2 (S 1 ) → L 2 (S 1 ) (resp. P B(ε) ) denote the Dirichlet-to-Neumann operator for R(ε) (resp. B(ε)). Normals are always taken to be outward pointing. Note that ker P R(ε) consists of the constants, and by Stokes’ theorem, P R(ε) : L 20 (S 1 ) → L 20 (S 1 ), where L 20 (S 1 ) is the subspace of L 2 (S 1 ) orthogonal to the constants. Define the following auxiliary operators on L 2 (S 1 ):
Tε :
an einθ −→
n∈Z
Uε± : V:
n∈Z
1 n (ε ± ε−n )an einθ , 2 n∈Z −→ nan einθ ,
an einθ −→
n∈Z
an einθ
n∈Z
|V| :
n∈Z
εn − ε−n an einθ , εn + ε−n
n∈Z
an einθ −→
|n|an einθ .
n∈Z
Remark 3.1. On L 20 (S 1 ), the operators V(Uε− )−1 and V(Tε )−1 + |V| are trace-class with norm tending to zero as ε → 0. Remark 3.2. By direct computation: εP B(ε) = |V|. We also define the (unbounded) operator E R(ε) : L 2 (S 1 ) → L 2 (S 1 ) as follows. For f . Extend f ∈ L 2 (S 1 ), let u be the harmonic function on R(1) with boundary values u to a harmonic function on R(ε) (also denoted u). Then E R(ε) ( f ) = u ∂ R(ε) . We have the following simple Lemma 3.3. E R(ε) ( f ) ≥ f for all 0 < ε ≤ 1 and all f ∈ L 2 (S 1 ). In particular, −1 E R(ε) is uniformly bounded as ε → 0.
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Proof. It suffices to prove the estimate for f smooth. Let u be harmonic on R(ε) with u ∂ R(1) = f . Then
2π
f 2 − E R(ε) ( f )2 =
dθ u 2 (1, θ ) − u 2 (ε, θ )
0
2π d dθ u 2 (r, θ ) dr ε 0 1 dr 2π ∂u r dθ u(r, θ ) (r, θ ) =2 r ∂r ε 0 1 dr = −2 ds u∂n u ∂ R(r ) ε r 1 dr = −2 d A |∇u|2 R(r ) ε r ≤ 0. =
1
dr
We also have Lemma 3.4. E R(ε) preserves the orthogonal splitting L 2 (S 1 ) = C ⊕ L 20 (S 1 ). Proof. Let u be a harmonic function on R(ε) with u ∂ R(1) = f , u ∂ R(ε) = g. Then applying Green’s theorem with the harmonic function v = log r on the annulus B(1) − B(ε), 0= (u∂n v − v∂n u) ds ∂(R(ε)−R(1)) f dθ − gdθ − P R(ε) (g) log ε ds. = S1
Since P R(ε) (g) ∈ L 20 (S 1 ), we have f dθ = S1
∂ R(ε)
S1
S1
gdθ =
S1
E R(ε) ( f )dθ.
The following result shows that the operators εP R(ε) and εP B(ε) are asymptotically close as ε → 0. −1 . Proposition 3.5. On L 20 (S 1 ), εP R(ε) = −V(Tε )−1 + V(Uε− )−1 E R(ε) −1 is well-defined on Proof. First, note that by Lemma 3.4 the composition (Uε− )−1 E R(ε) 2 1 L 0 (S ). With this understood, the result follows by a direct calculation. Namely, by expanding functions in Fourier modes, one computes
E R(ε) = Uε+ − V −1 Uε− P R(1) ,
−εP R(ε) E R(ε) = VUε− − Uε+ P R(1) . The proposition is then a consequence of these two facts.
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Corollary 3.6. Let t be any of det Nt / h t (γt ), or det Nt± / h ± (γt± ) from Sect. 2. Then lim t = 1/2.
t→0
Proof. Consider Nt+ , the argument being similar for the other two cases. By conformal invariance of det Nt+ / h + (γt+ ) (cf. [GG], the method in [EW], or note that this follows from (2.3) and Theorem 1.8) it suffices to consider the locally euclidean case. Also, notice that ζNt+ (0) = −1
(3.7)
for all t. Indeed, if we scale the metric by c2 ,
h + (γt+ ) → c h + (γt+ ) ,
Nt+ → c−1 Nt+ .
Then by conformal invariance
log
det Nt+ det Nt+ det(c−1 Nt+ ) + (0) + 1) log c + log = log = −(ζ , N t
h + (γt+ ) c h + (γt+ )
h + (γt+ )
and since c is arbitrary, (3.7) holds. As a consequence, log det(|t|1/2 Nt+ ) = ζNt+ (0) log |t|1/2 + log det Nt+ = − log |t|1/2 + log det Nt+ , det Nt+ log det(|t|1/2 Nt+ ) = log + log(2π ).
h + (γt+ )
(3.8)
On the other hand, by Proposition 3.5, Remarks 3.1 and 3.2, and Lemma 3.3, it follows that |t|1/2 Nt+ = 2|V| + {trace-class}, where the trace norm of the remainder tends to zero as t → 0. Hence by (3.8) and [L, Lemma 4.1],
lim log t = lim log det(|t|1/2 Nt+ ) − log(2π ) = log det(2|V|) − log(2π ).
t→0
t→0
Now spec(2|V|) = {2n}∞ n=0 , and each of the nonzero eigenvalues has multiplicity 2. Hence, ζ2|V | (s) = 21−s ζ (s). The result then follows from the special values of the Riemann zeta function: ζ (0) = −1/2, ζ (0) = −(1/2) log(2π ).
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4. The Arakelov Metric Recall the definition of the Arakelov metric (cf. [Ar,Fl,J1,W2,F2]). Given a compact g Riemann surface M of genus g ≥ 1, let {Ai , Bi }i=1 be a symplectic set of generators g of H1 (M) and choose {ωi }i=1 to be a basis of abelian differentials normalized such that Ai ω j = δi j . Let i j = Bi ω j be the associated period matrix with theta function ϑ. Set √ g −1 µ= (Im )i−1 j ωi ∧ ω j . 2g
(4.1)
i, j=1
Then M µ = 1. The Arakelov-Green’s function G(z, w) is symmetric with a zero of order one along the diagonal satisfying
√ ∂z ∂z¯ log G(z, w) = π −1µz z¯ , z = w ;
(4.2)
µ(z) log G(z, w) = 0,
(4.3)
M
and the Arakelov metric h = h z z¯ |dz|2 is defined by log h z z¯ = 2 lim {log G(z, w) − log |z − w|} . w→z
(4.4)
The Arakelov metric is “admissible” in the sense of [Fl]; hence, √ Ric(h) = 4π −1(g − 1)µ.
(4.5)
For more details we refer to the papers cited above. Consider now the situation in Sect. 2, where h t and h ± denote the Arakelov metrics on Mt and M ± , respectively (see [Ar,Fl]). The purpose of this section is to prove ±
Proposition 4.6 (cf. [W2], Eq. (8.1)). Let h t = e2σt h ± . Then for z ∈ Rt± , σt± (z)
=
g∓ g
2
g∓ log |t| − 2 g
log G ± (z, p ± ) + r ± (t, z),
(4.7)
where lim sup |r ± (t, z)| = 0. Moreover, if z ± are the plumbing coordinates then there t→0
z∈Rt±
is a constant C > 0 independent of t such that sup |z ± |=|t|1/2
|∂z ± σt± (z ± )| ≤ C|t|−1/2 .
(4.8)
In the statement above, G ± (z, w) denote the Arakelov-Green’s functions on M ± . The uniformity of the estimate is an improvement on the result of [W2] and is made possible by the explicit expression for the Arakelov metric in [F2] (Eq. (4.28) below). We require two preliminary technical results.
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347
Lemma 4.9. Let ωt be a holomorphic family of abelian differentials on Mt such that (pointwise) ωt (z) → ω0 (z) for z ∈ M + − { p + }, and ωt (z) → 0 for z ∈ M − − { p − }, where ω0 is an abelian differential on M + . Let Wt± , W0+ be the local expressions for ωt and ω0 in the plumbing coordinates z ± . Then there are functions f (t, x), g(t, x), analytic in a neighborhood of (0, 0), and h(t) analytic in a neighborhood of 0, such that t t2 + + h(t) + f (t, z + t/z ) + g(t, z + + t/z + ), z+ (z + )2 t t Wt− (z − ) = − − h(t) − − 2 f (t, z − + t/z − ) − tg(t, z − + t/z − ). z (z ) Wt+ (z + ) =
(4.10) (4.11)
In particular, if we assume f (0, 0) = 0, then for |z ± | sufficiently small, log |Wt+ (z + )| = log |W0+ (z + )| + r + (t, z + ), log |Wt− (z − )| = log |t| + log |W˙ − (z − )| + r − (t, z − ), 0
(4.12) (4.13)
where W0+ (z + ) = f (0, z + ), h(0) f (0, z − ) − g(0, z − ), W˙ 0− (z − ) = − − − z (z − )2 and lim
sup
t→0 |z ± |≥|t|1/2
|r ± (t, z ± )| = 0. Moreover, there is a constant C independent of t such
that sup |z ± |=|t|1/2
∂z + W ± (z ± ) ≤ C|t|−1/2 . t
Proof. By [F1, p. 40] we have an expansion dX , am (t)X m dX + bn (t)X n ωt = Y m≥0
(4.14)
(4.15)
n≥0
where X = (1/2)(z + + z − ) = (1/2)(z + + t/z + ), Y = (1/2)(z + − z − ) = −(1/2)(z − − t/z − ).
(4.16) (4.17)
The am (t), bn (t) are analytic near t = 0, and the series are convergent for (X , t) in a neighborhood of (0, 0). Moreover, using the assumption that ωt (z) → 0 on M − − { p − }, we have b0 (0) = 0, and bn+1 (0) = an (0) for all m ≥ 0. Now, substituting the expressions for X , dX , and Y, we find am (t) t t m bn (t) 1 t n + + 1 − z z Wt+ (z + ) = + + + 2m+1 (z + )2 z+ 2n z + z+ m≥0 n≥0 1 b0 (t) t t m + = + (a (t) + b (t)) z + m m+1 t z+ 2m+1 z+ m≥0 1 (an (t) − bn+1 (t)) t 2 t n + z + + . − 2n+1 t (z + )2 z n≥0
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R. A. Wentworth
Since b0 (t)/t and (an (t) − bn+1 (t))/t are regular at t = 0, this gives the expression in (4.10) with b0 (t) , t 1 f (t, x) = (am (t) + bm+1 (t))x m , 2m+1 h(t) =
m≥0
g(t, x) = −
1 (an (t) − bn+1 (t)) xn. 2n+1 t n≥0
Equation (4.11) follows from (4.10) and the fact that dz + = −tdz − /(z − )2 . For fixed z + write t t2 h(t) + g(t, z + + t/z + ), z+ (z + )2 1 1 t 2t F (t) = ∂1 f + + ∂2 f + + h + + h + + 2 g z z z (z ) 2 2 t t + + 2 ∂1 g(t, z + + t/z + ) + + 3 ∂2 g(t, z + + t/z + ). (z ) (z ) F(t) = f (t, z + + t/z + ) +
Hence, for |z + | ≥ |t|1/2 , we have F(t) = f (0, 0) + O(|t|1/2 ), uniformly. This implies (4.12). For fixed z − write G(t) = f (t, z − + t/z − ) + z − h(t) + (z − )2 g(t, z − + t/z − ), 1 G (t) = ∂1 f + − ∂2 f + z − h + (z − )2 ∂1 g(t, z − + t/z − ) + z − ∂2 g(t, z − + t/z − ). z It follows that G(t) = G(0) + O(|t|1/2 ), uniformly for |z − | ≥ |t|1/2 . By assumption, G(0) = 0 for |z − | sufficiently small, and hence Wt− (z − ) = −
t G(0)(1 + O(|t|1/2 )), (z − )2
so the estimate (4.13) follows. The estimate (4.14) follows immediately from (4.10) and (4.11).
For the second result, let E t (z, w), E ± (z, w) denote the Schottky prime forms on Mt , M ± (cf. [F1,F2] and Eq. (4.22) below). When evaluated at points on the Riemann surface, we will assume that lifts to a fixed fundamental domain have been chosen. Also, when points are in the pinching region we will assume the expressions for the prime forms are given with respect to the coordinates z ± . The following asymptotics are well-known. The point again is the uniformity of the estimates. Lemma 4.18. For z, w ∈ Rt+ , log |E t (z, w)| = log |E + (z, w)| + r1+ (z, w, t),
(4.19)
where for fixed w, lim sup |r1+ (z, w, t)| = 0 (and similarly for fixed z). A similar result t→0 z∈R + t
holds for points in Rt− . For z ∈ Rt+ and w ∈ Rt− , log |E t (z, w)|= − log |t|1/2 + log |E + (z, p + )|+log |E − (w, p − )|+r2 (z, w, t),
(4.20)
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349
where for fixed w, lim sup |r2 (z, w, t)| = 0 (and similarly for fixed z). Moreover, if z ± t→0 z∈R + t
are the plumbing coordinates and w is fixed, then there is a constant C > 0 independent of t such that sup |z ± |=|t|1/2
|∂z ± E t± (z ± , w)| ≤ C|t|−1/2 .
(4.21)
Proof. Recall that the prime form may be defined as follows. Let ϑ(Z ), Z ∈ Cg , denote the theta function for the period matrix . Choose a nonsingular δ ∈ , where ⊂ J ac(M) is the theta divisor in the Jacobian of M. Then by [F1, Corollary 2.3], − E 2 (z, w) =
ϑ(z − w + δ)ϑ(z − w − δ) , Hδ (z)Hδ (w)
(4.22)
where Hδ (z) =
g
∂ Z i ϑ(δ)ωi (z).
(4.23)
i=1 g
Now consider the degenerating family Mt . The collection {Ai , Bi }i=1 may be chosen
g+ so that {Ai , Bi }i=1
g
is a symplectic homology basis for M + and {Ai , Bi }i=g+ +1 is a symplectic homology basis for M − . Let ωi,t , ωi± be abelian differentials on Mt and M ± , respectively, normalized with respect to {Ai }, and let t and ± be the associated period matrices. Then as t → 0, ⎧ + + + + ⎪ ⎪ωi (z) i ≤ g , z ∈ M − { p } ⎪ ⎨0 + − i ≤ g , z ∈ M − { p− } (4.24) ωi,t (z) −→ ⎪ωi− (z) i > g + , z ∈ M − − { p − } ⎪ ⎪ ⎩0 i > g+ , z ∈ M + − { p+ } and t becomes block diagonal (+ , − ) (cf. [F1, p. 38]). Choose δt so that lim δt = (δ + , δ − ) ∈ + × J ac(M − ).
t→0
If z, w ∈ Rt+ , we also have ϑt (z − w + δt ) → ϑ + (z − w + δ + )ϑ − (δ − ), and this is uniform in z, w (cf. [W2, Sect. 3 and Prop. A.1]). For simplicity and without loss of generality, assume ϑ − (δ − ) = 0 and Hδ++ ( p + ) = 0. Then from (4.23), Hδt (z) → Hδ++ (z)ϑ − (δ − ),
(4.25)
and by Lemma 4.9 this is uniform for z ∈ Rt+ . Hence, the expression (4.22) immediately implies (4.19). Now suppose z, w ∈ Rt− . From [W2, Prop. 3.8] we have ϑt (z−w ± δt ) = t
ϑ − (z− p − ± δ − )ϑ − (w− p − ∓ δ − )E − (z, w) + + Hδ + ( p ) + O(|t|2 ), ϑ − (δ − )E − (z, p − )E − (w, p − ) (4.26)
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R. A. Wentworth
where the O(|t|2 ) term depends on two derivatives of ϑ(Z , ) with respect to i j and is smooth in Z . Hence, again this is uniform. We then also have, Hδt (z) = t
ϑ − (z − p − + δ − )ϑ − (z − p − − δ − ) + + Hδ + ( p ) + O(|t|2 ). ϑ − (δ − )(E − (z, p − ))2
(4.27)
Again, (4.19) follows. The uniformity for the remainder term in log |Hδt (z)| comes from (4.13) as applied to Hδt (z) and Hδt (w), and as mentioned, the expansion (4.26) is also uniform. The expression (4.20) follows. For (4.21), note that from (4.22), g i=1 ∂ Z i ϑt (z − w + δt )ωi (z) 2∂z E t (z, w) = E t (z, w) ϑt (z − w + δt ) g ∂z Hδt (z) i=1 ∂ Z i ϑt (z − w − δt )ωi (z) − + . ϑt (z − w − δt ) Hδt (z) Now fix w ∈ Rt− . By (4.19), |E t (z, w)| is uniformly bounded from above for |z + | = |t|1/2 , and |Hδt (z + )| is uniformly bounded from below (see Lemma 4.9). Similarly, |ϑt (z − w ± δt )| ∼ |t|1/2 (see (4.26)). Hence, sup |z ± |=|t|1/2
|∂z + E t (z + , w)| ≤ C1 |t|−1/2 + C2
sup |z ± |=|t|1/2
|∂z + Hδt (z + )|
and the desired bound in (4.21) follows from (4.14) applied to Hδt (z). The case w ∈ Rt+ is similar. Finally, in case z ∈ Rt+ , w ∈ Rt− , use ϑt (z − w + δt ) → ϑ + (z − p + + δ + )ϑ − (w − p − + δ − ), (4.25), and (4.27) to obtain (4.20). The uniformity and statement about derivatives follows as above. This completes the proof.
Proof of Proposition 4.6. From [F2, Eq. (1.31)], the Arakelov metric may be expressed in local conformal coordinates as h(z) = C(M)|s(z)|4/g exp{(4/g(g − 1))B[k z , k z ]} |dz|2 , where C(M) is an explicit constant depending on M but independent of z, vector of Riemann constants associated to z, z 1 + jj + ωi (x) ωj, k zj = 2 Ai x
(4.28) kz
is the
i = j
and B[k z , k z ] = π
g z z (Im )i−1 j (Im k )i (Im k ) j . i=1
The function s(z) is given in [F2, Prop. 1.2]. Important here is the relation g g ϑ( i=1 pi − z − ) E( pi , z) s(z) = g , s(z 0 ) ϑ( i=1 pi − z 0 − ) i=1 E( pi , z 0 )
(4.29)
where p1 , . . . , pg are generic points. Using Theorem 1.7 one can in fact recover the asymptotics in [W2, Eq. (8.1)] from (4.28). Relevant to the proof of Proposition 4.6, however, is the uniformity. From (4.28) and (4.29) this amounts to uniformity and bound on derivatives of the prime form, since the uniformity of the theta function and Riemann constants follows from the results in [W2]. Hence, Proposition 4.6 is a consequence of Lemma 4.18.
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351
5. Proof of the Main Theorem We now apply the formulation of Sects. 2 and 3 to Q(Mt ,h t ) and Q(M ± ,h ± ) , where h t and h ± are the Arakelov metrics on Mt and M ± , respectively. The main result is Proposition 5.1. 2g + g − c0 lim log Q(Mt ,h t ) − log |t| = log(Q(M + ,h + ) Q(M − ,h − ) ) − , t→0 3g 6 where c0 is defined in Theorem 1.3. For this we require Lemma 5.2. In the notation of Sect. 2, lim
t→0
S(Rt+ ,h + ) (σt+ ) + S(R − ,h − ) (σt− ) + t
1 2g + g − − 3 3g
log |t| + 1 = 0.
Proof. This is essentially [W1, Eq. (3.5)]. However, there is an error in the computation due to the omission of the last term in (2.4). See [W3] or (5.8) below for the correction. Moreover, the uniformity of the asymptotics of the Arakelov metric, established in the previous section, was assumed in that paper. Hence, for the sake of completeness, we include the full computation here. Let µt , µ± be the forms corresponding to (4.1) on Mt and M ± . It follows from (4.24) that µt −→ (g ± /g)µ± ,
(5.3)
and from Lemma 4.9, µt is uniformly bounded in the plumbing coordinates as t → 0. By (4.5) we have K h ± d Ah ± = −4π(g ± − 1)µ± . Using this and Proposition 4.6, 1 − 12π
Rt±
d Ah ± 2K h ± σt±
2 = (g ± − 1) 3
Rt±
µ± (g ∓ /g)2 log |t|
−2(g ∓ /g) log G ± (z, p ± ) + o(1)
=
2 ± (g − 1)(g ∓ /g)2 log |t| + o(1), 3
(5.4)
where we have also used the normalization (4.3). For the next term in (2.4), use the estimates in Proposition 4.6 and the fact, again from (4.5) and also (5.3), that d Ah ± σt± = 4π(g ∓ /g)µ± + o(1),
(5.5)
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R. A. Wentworth
where o(1) converges pointwise to zero and is uniformly bounded on Rt± with respect the euclidean measure in the plumbing coordinates. By (4.7), (4.8), and (5.5), we have 1 − d Ah ± |∇σt± |2 12π Rt± 1 1 = d Ah ± σt± σt± − dsh ± σt± ∂n σt± 12π Rt± 12π ∂ Rt± 1 = d Ah ± σt± σt± 12π Rt± 1 + dsh ± (g + g − /g 2 ) log |t| + o(1) ∂n σt± 12π ∂ Rt± 1 = d Ah ± σt± (g ∓ /g)2 log |t| − 2(g ∓ /g) log G ± (z, p ± ) 12π Rt± + (g + g − /g 2 ) log |t| + o(1) 2 1 g∓ = log |t| + o(1). (5.6) 3 g Since the metrics h ± are normalized at p ± , 1 1 − dsh ± kh ± σt± = − (g + g − /g 2 ) log |t| + o(1) 6π ∂ Rt± 3
(by (4.7)).
(5.7)
Finally, −
1 4π
∂ Rt±
dsh ± ∂n σt± = −
1 4π
Rt±
d Ah ± σt± = −
g∓ + o(1) g
(by (5.5)). (5.8)
Combining the results (5.4) and (5.6)-(5.8) for Rt+ and Rt− , we obtain the statement of the lemma.
Proof of Proposition 5.1. It suffices to consider the factors in Proposition 2.5. By Corollary 3.6,
h t (γt ) det Nt+ det Nt− 1 −→ .
h + (γt ) h − (γt ) det Nt 2 By [Wb, Eq. (28)], (Q Bt )2 = 22/3 (2π )−1 |t|−1/3 exp(−4ζ (−1) − 5/6). The result now follows from Lemma 5.2.
Proof of Theorem 1.3. Comparing Proposition 5.1 with Theorem 1.7 and the definition (1.1) of cg , we find: cg+1 − cg = c1 − c0 , where c1 is the constant for g = 1 surfaces and c0 is as in Proposition 5.1. The g = 1 constant c1 has been evaluated (cf. [Fl,P]): c1 = −8 log(2π ). The expression for cg now follows.
Precise Constants in Bosonization Formulas
353
Appendix At the suggestion of the referee, in this appendix we clarify the relationship between the expansions for holomorphic abelian differentials on degenerating surfaces found in [F1] and [Y]. Let ωt (z) → ω0 (z) be as in the statement of Lemma 4.9. We assume moreover that the periods A j ωt are fixed independent of t. From (4.10) and (4.11) we have W0+ (z + ) = f (0, z + ), (5.9) + h(0) ∂2 f (0, z ) , (5.10) W˙ 0+ (z + ) = + + ∂1 f (0, z + ) + z z+ h (0) ∂1 ∂2 f (0, z + ) ∂22 f (0, z + ) 2 W¨ 0+ (z + ) = 2 + +∂12 f (0, z + )+2 + + + 2 g(0, z + ), z z+ (z + )2 (z ) (5.11) − h(0) f (0, z ) W˙ 0− (z − ) = − − − − g(0, z − ), (5.12) z (z − )2 where the dots indicate derivatives with respect to t. From (5.10), the restriction of ω˙ 0 (z) = lim
t→0
ωt (z) − ω0 (t) t
to M + is an abelian differential with a pole of order at most one at p + . Hence, it is holomorphic. On the other hand, by the normalization it must have zero A j periods. It therefore vanishes identically. From (5.12), the restriction of ω˙ 0 to M − is an abelian differential with a pole of order two at p − , and the coefficient of the 1/(z − )2 term is − f (0, 0) = −W0+ (0), from (5.9). Hence, for z ∈ M − we have the t-expansion ωt (z) = −t W0+ (0)ω− (z, p − ) + O(t 2 ),
(5.13)
where ω− (z, w) is the abelian differential of the second kind on M − , normalized to have zero A j -periods and an expansion 1 1 − ω− (z, w) = S + ( p) + · · · dzdw (z − w)2 6 in local coordinates about a point p, where S − is a holomorphic projective connection on M − (see [F1, Cor. 2.6]). Examining the constant term in the t-expansion (5.12) we find 1 2 1 ∂2 f (0, 0) + g(0, 0) = W0+ (0)S − ( p − ). 2 6 From (5.11), the restriction of ω¨ 0 to M + is an abelian differential with a pole of order two at p + , and the coefficient of the 1/(z + )2 term is ∂22 f (0, 0) + 2g(0, 0) =
1 + W (0)S − ( p − ). 3 0
Hence, for z ∈ M + we have the t-expansion ωt (z) = ω0 (z) +
t2 + W (0)S − ( p − )ω− (z, p − ) + O(t 3 ). 6 0
(5.14)
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The expressions (5.13) and (5.14) agree with [Y, Eq. (36)]. In [F1, Eq. 47], the t-expansion was carried out with respect to a local expression of the differentials in the “pinching coordinate” X . Since the restrictions of dX to M ± − { p ± } are differentials that themselves depend on t (see (4.16)), the O(t) terms calculated there are incorrect. Acknowledgement. The author is grateful to J. Jorgenson, D.H. Phong, and S. Zelditch for very useful discussions. Many thanks also to Alexey Kokotov for numerous suggestions during the revision of an earlier version of this paper.
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Alvarez, O.: Theory of strings with boundaries: fluctuations, topology, and quantum geometry. Nucl. Phys. B 216, 125–184 (1983) Alvarez-Gaumé, L., Moore, G., Vafa, C.: Theta functions, modular invariance, and strings. Commun. Math. Phys. 106(1), 1–40 (1986) Arakelov, S.: An intersection theory for divisors on an arithmetic surface. Math. USSR Izv. 8, 1179–1192 (1974) Belavin, A., Knizhnik, V.: Complex geometry and the theory of quantum strings. Soviet Phys. JETP 64 no. 2, 214–228 (1986), translated from Zh.Eksper. Teoret. Fiz. 91(2), 364–390 (1986) (Russian) Alvarez-Gaumé, L., Bost, J.-B., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys. 112(3), 503–552 (1987) Burghelea, D., Friedlander, L., Kappeler, T.: Meyer-Vietoris type formula for determinants of elliptic differential operators. J. Funct. Anal. 107(1), 34–65 (1992) Deligne, P.: Le déterminant de la cohomologie. Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., 67, Providence, RI: Amer. Math. Soc., 1987, pp. 93–177 D’Hoker, E., Phong, D.H.: Length-twist parameters in string path integrals. Phys. Rev. Lett. 56(9), 912–915 (1986) D’Hoker, E., Phong, D.H.: On determinants of laplacians on Riemann surfaces. Commun. Math. Phys. 104(4), 537–545 (1986) D’Hoker, E., Phong, D.H.: The geometry of string perturbation theory. Rev. Mod. Phys. 60(4), 917–1065 (1988) D’Hoker, E., Gutperle, M., Phong, D.H.: Two-loop superstrings and s-duality. Nucl. Phys. B 722(1-2), 81–118 (2005) Dugan, M., Sonoda, H.: Functional determinants on Riemann surfaces. Nucl. Phys. B 289(1), 227–252 (1987) Edward, J., Wu, S.: Determinant of the Neumann operator on smooth jordan curves. Proc. Amer. Math. Soc. 111(2), 357–363 (1991) Faltings, G.: Calculus on arithmetic surfaces. Ann. of Math. (2) 119(2), 387–424 (1984) Fay, J.: Theta functions on Riemann surfaces. Lecture Notes in Mathematics, Vol. 352. Berlin-New York:Springer-Verlag, 1973 Fay, J.: Kernel functions, analytic torsion, and moduli spaces. Mem. Amer. Math. Soc. 96, no. 464 (1992) Gava, E., Iengo, R., Jayaraman, T., Ramachandran, R.: Multiloop divergences in the closed bosonic string theory. Phys. Lett. B 168(3), 207–211 (1986) Gillet, H., Soulé, C.: Analytic torsion and the arithmetic Todd genus. With an Appendix by D. Zagier. Topology 30(1), 21–54 (1991) Guillarmou, C., Guillopé, L.: The determinant of the Dirichlet-to-Neumann map for surfaces with boundary. http://arXiv.org/list/math/0701727, 2007 Hejhal, D.: Regular b-groups, degenerating Riemann surfaces, and spectral theory. Mem. Amer. Math. Soc. 88, no. 437 (1990) Jorgenson, J.: Asymptotic behavior of Faltings’s delta function. Duke Math. J. 61(1), 221–254 (1990) Jorgenson, J.: Degenerating hyperbolic Riemann surfaces and an evaluation of the constant in Deligne’s arithmetic Riemann-Roch theorem. Preprint, 1991 Jorgenson, J., Lundelius, R.: Continuity of relative hyperbolic spectral theory through metric degeneration. Duke Math. J. 84(1), 47–81 (1996)
Precise Constants in Bosonization Formulas
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Kierlanczyk, M.: Determinants of Laplacians. MIT thesis, 1986 Lee, Y.: Burghelea-Friedlander-Kappeler’s gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion. Trans. Amer. Math. Soc. 355(10), 4093–4110 (2003) Müller, J., Müller, W.: Regularized determinants of laplace-type operators, analytic surgery, and relative determinants. Duke Math. J. 133(2), 259–312 (2006) Polchinski, J.: Evaluation of the one loop string path integral. Commun. Math. Phys. 104(1), 37–47 (1986) Park, J., Wojciechowski, K.: Adiabatic decomposition of the ζ -determinant and Dirichlet to Neumann operator. J. Geom. Phys. 55(3), 241–266 (2005) Sarnak, P.: Determinants of laplacians. Commun. Math. Phys. 110(1), 113–120 (1987) Smit, D.-J.: String theory and algebraic geometry of moduli spaces. Commun. Math. Phys. 114, 645–685 (1988) Sonoda, H.: Calculation of a propagator on a Riemann surface. Phys. Lett. B 178(4), 390–394 (1986) Soulé, C.: Géométrie d’Arakelov des surfaces arithmétiques. Séminaire Bourbaki, Vol. 1988/89. Astérisque No. 177-178, Exp. No. 713, 327–343 (1989) Verlinde, E., Verlinde, H.: Chiral bosonization, determinants and the string partition function. Nucl. Phys. B 288(2), 357–396 (1987) Voros, A.: Spectral functions, special functions and the selberg zeta function. Commun. Math. Phys. 110(3), 439–465 (1987) Weisberger, W.: Conformal invariants for determinants of laplacians on Riemann surfaces. Commun. Math. Phys. 112(4), 633–638 (1987) Wentworth, R.: Asymptotics of determinants from functional integration. J. Math. Phys. 32(7), 1767–1773 (1991) Wentworth, R.: The asymptotics of the Arakelov-Green’s function and Faltings’ delta invariant. Commun. Math. Phys. 137, 427–459 (1991) Wentworth, R.: Erratum to “Asymptotics of determinants from functional integration”. Available at:http://math.jhu.edu/~raw/papers/ Wolpert, S.: Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces. Commun. Math. Phys. 112(2), 283–315 (1987) Wolpert, S.: The hyperbolic metric and the geometry of the universal curve. J. Diff. Geom. 31, 417–472 (1990) Yamada, A.: Precise variational formulas for abelian differentials. Kodai Math. J. 3(1), 114–143 (1980)
Communicated by L. Takhtajan
Commun. Math. Phys. 282, 357–393 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0558-6
Communications in
Mathematical Physics
Numerical Kähler-Einstein Metric on the Third del Pezzo Charles Doran1 , Matthew Headrick2 , Christopher P. Herzog3 , Joshua Kantor1 , Toby Wiseman4 1 Department of Mathematics, University of Washington, Seattle, WA 98195-1560, USA 2 Stanford Institute for Theoretical Physics, Stanford, CA 94305-4060, USA 3 Department of Physics, Princeton University, Princeton, NJ 08544, USA.
E-mail: [email protected]
4 Blackett Laboratory, Imperial College London, London SW7 2AZ, UK
Received: 19 March 2007 / Accepted: 3 April 2008 Published online: 2 July 2008 – © Springer-Verlag 2008
Abstract: The third del Pezzo surface admits a unique Kähler-Einstein metric, which is not known in closed form. The manifold’s toric structure reduces the Einstein equation to a single Monge-Ampère equation in two real dimensions. We numerically solve this nonlinear PDE using three different algorithms, and describe the resulting metric. The first two algorithms involve simulation of Ricci flow, in complex and symplectic coordinates respectively. The third algorithm involves turning the PDE into an optimization problem on a certain space of metrics, which are symplectic analogues of the “algebraic” metrics used in numerical work on Calabi-Yau manifolds. Our algorithms should be applicable to general toric manifolds. Using our metric, we compute various geometric quantities of interest, including Laplacian eigenvalues and a harmonic (1, 1)-form. The metric and (1, 1)-form can be used to construct a Klebanov-Tseytlin-like supergravity solution. 1. Introduction Kähler metrics on manifolds play an important role in mathematics and physics. As Yau demonstrated [1], in the Kähler case it is often possible to prove the existence (or nonexistence) of metrics which solve the Einstein equation. While it is extremely valuable to know whether they exist, for many purposes one also wants to know their specific form. The existence theorems, however, are generally non-constructive, and explicit examples of Kähler-Einstein metrics are rare. This state of affairs naturally leads to the following question: Is it possible, in practice, to find accurate numerical approximations to these metrics using computers? The last two years have seen significant success, with a variety of different algorithms providing numerical solutions to the Einstein equation on the Calabi-Yau surface K3 [2–4] and on a three-fold [5]. The Kähler property proved to be as crucial for this numerical work as it was for the existence theorems. In this paper we extend this success to a non-Calabi-Yau manifold, namely CP2 2 blown up at three points (CP2 #3CP ), also known as the third del Pezzo surface (dP3 ). This manifold is known by work of Siu [7] and Tian-Yau [6] to admit a Kähler-Einstein
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metric with positive cosmological constant, but as in the Calabi-Yau case that metric is not known explicitly. An important property of dP3 that differentiates it from CalabiYau manifolds is that it is toric. Toric manifolds are a special class of Kähler manifolds whose U (1)n isometry group (where n is the manifold’s complex dimension) allows even greater analytical control, and we develop algorithms for solving the Einstein equation that specifically exploit this structure. We should also note that the Kähler-Einstein metric on dP3 is unique (up to rescaling); therefore, rather than having a moduli space of metrics as we have in the case of Calabi-Yau manifolds, there is only one metric to compute. On the physics side, the Kähler-Einstein metric on dP3 is important because it can be used to construct an example of a gauge/gravity duality. These dualities provide a bridge between physical theories of radically different character, allowing computation in one theory using the methods of its dual. Given the Kähler-Einstein metric on dP3 , one can construct a five-dimensional Sasaki-Einstein metric. Compactifying type IIB supergravity on this manifold one obtains an AdS5 supergravity solution, which has a known superconformal gauge theory dual [8–10]. An interesting generalization includes a 3-form flux from wrapped D5-branes. This flux can be written in terms of a harmonic (1, 1)-form on dP3 , which we also compute numerically in this paper. The resulting supergravity solution (the analogue of the Klebanov-Tseytlin solution on the conifold [11]) is nakedly singular, but is the first step toward finding the full supergravity solution on the smoothed-out cone (the analogue of the Klebanov-Strassler warped deformed conifold [12]). This smoothed-out cone is dual to a cascading gauge theory, and knowing the explicit form of the supergravity solution would be useful for both gauge theory and cosmology applications. This physics background is explained in detail in Sect. 2. In Sect. 3, we review the mathematical background necessary for understanding the rest of the paper. Here we closely follow the review article on toric geometry by Abreu [13]. We explain the two natural coordinate systems on a toric manifold, namely complex and symplectic coordinates. In complex coordinates, the metric is encoded in the Kähler potential, and in symplectic coordinates in the symplectic potential; these two functions are related by a Legendre transform. In either coordinate system, the Einstein equation reduces to a single nonlinear partial differential equation, of Monge-Ampère type, for the corresponding potential. Thanks to the U (1)n symmetry, this PDE is in half the number of dimensions of the original manifold (two real dimensions for dP3 ). In this section we also derive the equation we need for the (1, 1)-form, and give all the necessary details about dP3 . In this paper we describe three different methods to solve the Monge-Ampère equation. In Sect. 4 we explain the first two methods, which involve numerically simulating Ricci flow in complex and symplectic coordinates respectively. Specifically, we use a variant of Ricci flow (normalized) that includes a term, ∂gµν = −2Rµν + 2gµν , ∂t
(1)
whose fixed points are clearly Einstein metrics with cosmological constant . According to a recent result of Tian-Zhu [14], on a manifold which admits a Kähler-Einstein metric, the flow (60) converges to it starting from any metric in the same Kähler class. Our simulations behaved accordingly, yielding Kähler-Einstein metrics accurate (at the highest resolutions we employed) to a few parts in 106 (and which agree with each other to within that error). Numerical simulations of Ricci flow have been studied before in a variety of contexts [15–17], but as far as we know this is the first time they have been
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used to find a new solution to the Einstein equation (aside from a limited exploration of its use on K3 [2]). In this section we also explore the geometry of this solution, and discuss a method for computing eigenvalues and eigenfunctions of the Laplacian in that background. In Sect. 5 we introduce a different way to represent the metric, based on certain polynomials in the symplectic coordinates. This non-local representation is a symplectic analogue of the “algebraic” metrics on Calabi-Yau manifolds employed in numerical work by Donaldson [3] and Douglas et al. [4,5]. To demonstrate the utility of this representation we give a low order polynomial fit to the numerical solutions found by Ricci flow, that can be written on one line and yet agrees with the true solution to one part in 103 , and everywhere satisfies the Einstein condition to better than 10%. In Sect. 6 we discuss our third method to compute the Kähler-Einstein metric. As above we represent the metric using polynomials in the symplectic coordinates, but now we constrain the polynomial coefficients by solving the Monge-Ampère equation order by order in the coordinates. This leaves a small number of undetermined coefficients, which we compute by minimizing an error function. Using this method we obtain numerical metrics of similar accuracy to those found by Ricci flow. By a similar method we also calculate eigenvalues and eigenfunctions of the Laplacian, and the harmonic (1, 1)-form. We conclude the paper with a brief discussion of the three methods and their relative merits in Sect. 7. At the two websites [21], we have made available for download the full numerical data representing our metrics, as well as Mathematica notebooks that input the data and allow the user to work with those metrics. The codes used to generate the data are also available on those websites. We believe that all of the methods we present here can be applied to a general toric manifold. We will report elsewhere on an application to dP2 [18], which does not admit a Kähler-Einstein metric but does admit a Kähler-Ricci soliton [19]. For future work, there is also a natural analogue of dP3 to study in three complex dimensions. From Batyrev’s classification of toric Fano threefolds, it follows that there are precisely two of them that admit Kähler-Einstein metrics which are not themselves products of lower dimensional manifolds [20, Sect. 4]. One of these is CP3 . The other is the total space of the projectivization of the rank two bundle O ⊕ O(1, −1) over CP1 × CP1 . Its Delzant polytope (see Sect. 3 for the definition) possesses a D4 symmetry, and a fundamental region is simply a tetrahedron. While this work was in progress we have learned that Kähler-Einstein metrics on dP3 have also been computed in [22] using the methods of [3]. 2. Gauge/Gravity Duality A prototypical example of a gauge/gravity duality which provides the physics motivation for studying dP3 is the Klebanov-Strassler (KS) supergravity solution [12]. (Mathematicians may wish to skip this section.) The KS solution is a solution of the type IIB supergravity equations of motion. The space-time is a warped product of Minkowski space R1,3 and the deformed conifold X . The affine variety X has an embedding in C4 defined by 4 i=1
z i2 = ,
(2)
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where z i ∈ C. There are also a variety of nontrivial fluxes in this solution which we will return to later. One important aspect of the KS solution is its conjectured duality to a non-abelian gauge theory, namely the cascading SU (N )× SU (N + M)N = 1 supersymmetric gauge theory with bifundamental fields Ai and Bi , i = 1 or 2, and superpotential W = i j kl Ai Bk A j Bl .
(3)
This theory is similar in a number of respects to QCD; it exhibits renormalization group flow, chiral symmetry breaking, and confinement. Moreover, all of these properties can be understood from the dual gravitational perspective. In addition to its gauge theory applications, the KS solution is important for cosmology. Treating the deformed conifold as a local feature of a compact Calabi-Yau manifold, the KS solution provides a string compactification with a natural hierarchy of scales in which all the complex moduli are fixed [23]. Stabilizing the Kähler moduli as well [24], the KS solution can become a metastable string vacuum and thus a model of the real world. In this context, inflation might correspond to the motion of D-branes [25] and cosmic strings might be the fundamental and D-strings of type IIB string theory [26]. One naturally wonders to what extent the cosmological and gauge theoretic applications depend upon the choice of the deformed conifold. A natural way to generalize X is to consider smoothings of other Calabi-Yau singularities. One such family of singularities involves a Calabi-Yau where a del Pezzo surface dPn shrinks to zero size. (Here dPn is CP2 blown up at n points.) Note we are distinguishing here between resolutions—or Kähler structure deformations—where even dimensional cycles are made to be of finite size, and smoothings—complex structure deformations—where a three dimensional cycle is made finite. Using toric geometry techniques, Altmann [27] has shown that the total space of the canonical bundle over dP1 admits no smoothings, while dP2 admits one and dP3 two.1 The higher dPn are not toric. Thus two relatively simple candidates for generalizing the KS solution are smoothed cones over the complex surfaces dP2 and dP3 . Without knowing the details of the metric on the smoothed cone X , one can show that a generalization of the KS solution exists for such warped products [32]. The solution will have a ten dimensional line element of the form ds 2 = h( p)−1/2 ηµν d x µ d x ν + h( p)1/2 ds X2 ,
(4)
where ds X2 is the line element on X , p ∈ X , the map h : X → R+ is called the “warp factor”, and ηµν is the Minkowski tensor for R1,3 with signature (− + ++). There are also a variety of nontrivial fluxes turned on in this solution. There is a five form flux F5 = dC4 + dC4 , where C4 =
1 d x 0 ∧ d x 1 ∧ d x 2 ∧ d x 3, gs h
(5)
and where gs is the string coupling constant. The finite smoothing indicates the presence of a harmonic (2, 1)-form ω2,1 which we take to be imaginary self-dual: X ω2,1 = iω2,1 . From ω2,1 we construct a three-form flux G 3 = Cω2,1 , where C is a constant related to the rank of the gauge group in the dual theory. The warp factor satisfies the relation X h = −
gs2 G abc (G ∗ )abc , 12
(6)
1 The physical relevance of this fact for supersymmetry breaking was pointed out in [28–30] (see also [31] for a more recent account).
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where the indices on G 3 are raised and the Laplacian X is constructed using the line element ds X2 without the warp factor. Although this solution holds for general X , clearly in order to know detailed behavior of the fluxes and warp factor as a function of p, we need to know a metric on X . From this perspective, we have chosen to study dP3 and not dP2 in this paper because dP3 is known to have a Kähler-Einstein metric [6] while dP2 does not [33]. Given that dP3 is Kähler-Einstein, we can construct a singular Calabi-Yau cone over dP3 in a straightforward manner: ds X2 = dr 2 + r 2 (dψ + σ )2 + dsV2 ,
(7)
where σ = −2i(∂ f − ∂¯ f ) and f is half the Kähler potential on V = dP3 . Here dsV2 is a Kähler-Einstein line element on dP3 . In a hopefully obvious notation, r is the radius of the cone and ψ an angle. Although such a cone over dP2 probably exists as well, it will involve an irregular Sasaki-Einstein manifold as an intermediate step; the metric on the Sasaki-Einstein manifold over dP2 is not yet known. Given that dP3 is Kähler-Einstein, the problem of finding ω2,1 on the singular cone reduces to finding a harmonic (1, 1)-form θ on dP3 such that θ ∧ ω = 0 (where ω is the Kähler form on V ) and V θ = −θ , as pointed out in [34]. The relation between ω2,1 and θ is ω2,1 = (−idr/r + dψ + σ ) ∧ θ . In addition to finding a numerical Kähler-Einstein metric on dP3 , we will also find a numerical (1, 1)-form θ , thus yielding a singular generalization of the KS solution for dP3 . Historically, before the KS solution, Klebanov and Tseytlin [11] derived exactly such a singular solution for the singular conifold. Although we have not produced a numerical solution for h( p), with the explicit metric and numerical (1, 1)-form for dP3 in hand, we have all the necessary ingredients to calculate h( p). The KT solution for dP1 is known [35]. The next step would be to find a Ricci flat metric and imaginary self-dual (2,1)-form on the smoothed cone over dP3 , thus providing a generalization of the KS solution. Such a solution would open up many future directions of study, both in gauge theory and cosmology. To name a handful of possibilities, one could compute k-string tensions of the confining low energy gauge theory dual to this dP3 background, generalizing work of [36]. Alternatively, treating the SUGRA solution as a cosmology, one could compute annihilation cross sections of cosmic strings [26] or slow roll parameters for D-brane inflation [25].
3. Mathematical Background 3.1. Complex and symplectic coordinates. The formalism we use to construct our numerical Kähler-Einstein metric on dP3 is based on work by Guillemin [37] and later developed by Abreu [13]. Here we summarize this formalism. We consider an n (complex) dimensional compact Kähler manifold M. The manifold is equipped with the following three tensors: – A complex structure J µ ν satisfying J µ ν J ν λ = −δ µ λ . – A symplectic form (also called in this context a Kähler form) ω, which is a nondegenerate closed two-form. – A positive-definite metric ds 2 = gµν d x µ d x ν .
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These tensors are related to each other by: gµν = ωµλ J λ ν .
(8)
Now let M also be a toric manifold. Tensors which are invariant under its U (1)n = T n group of diffeomorphisms we call toric. In particular, we will restrict our attention to toric metrics. Let M ◦ be the subset of M which is acted on freely by that group. There is a natural set of n complex coordinates z = u + iθ on M ◦ , where u ∈ Rn and θ ∈ T n (θi ∼ θi + 2π ); the U (1)n acts on θ and leaves u fixed. Given that the manifold is Kähler, the metric may locally be expressed in terms of the Hessian of a Kähler potential f (z).2 Because gµν is invariant under the action of U (1)n , the potential can be chosen to be a function of u. The line element is ds 2 = gi j¯ dz i d z¯ j + gı¯ j d z¯ i dz j = Fi j (du i du j + dθi dθ j ),
(9)
where we have introduced Fi j (u): gi j¯ = 2
∂2 f 1 ∂2 f 1 = = Fi j . ∂z i ∂ z¯ j 2 ∂u i ∂u j 2
(10)
The Kähler form is ω = 2i∂ ∂¯ f = igi j¯ dz i ∧ d z¯ j = Fi j du i ∧ dθ j .
(11)
The complex structure in these coordinates is trivial: −J u i θ j = J θi u j = δ ij ,
J u u = J θ θ = 0.
(12)
It is often convenient to work with symplectic coordinates w = x + iθ , which are related to the complex coordinates by: x≡
∂f . ∂u
(13)
Under this map (also known as the moment map), Rn is mapped to the interior P ◦ of a convex polytope P ⊂ Rn which is given by the intersection of a set of linear inequalities, P = {x : la (x) ≥ 0 ∀a},
la (x) = va · x + λa ;
(14)
the index a labels the faces, and the normal vector va to each face is a primitive element of Zn . These va define the toric fan in the complex coordinates z. We will see that the λa determine the Kähler class of the metric. In terms of these new coordinates, the Kähler form becomes trivial: ω = d xi ∧ dθi . (15) Introducing the symplectic potential, which is the Legendre transform of f , g(x) = u · x − f (u),
(16)
ds 2 = G i j d xi d x j + G i j dθi dθ j ,
(17)
the line element can be written
2 We follow the conventions of Abreu [13]. Note that f is one-half the usual definition of the Kähler potential.
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where Gi j =
363
∂2g , ∂ xi ∂ x j
(18)
and G i j is the inverse of G i j . Note that, while Fi j is regarded as a function of u and G i j as a function of x, under the mapping (13) the two matrices are equal to each other. The complex structure in symplectic coordinates is given by: J xi θ j = −G i j ,
J θi x j = G i j ,
J x x = J θ θ = 0.
(19)
To summarize, the complex and symplectic coordinate systems are related to each other by a Legendre transform: x=
∂f , ∂u
u=
∂g , ∂x
f (u) + g(x) = u · x,
Fi j (u) = G i j (x).
(20)
Note that the whole system (complex and symplectic coordinate systems) has four gauge invariances under which the metric is invariant and the relations (20) are preserved: 1. f (u) → f (u) + c, g(x) → g(x) − c for any constant c; 2. f (u) → f (u) + k · u, x → x + k for any vector k; 3. g(x) → g(x) + k · x, u → u + k for any vector k; 4. x → M · x, u t → u t · M −1 for any element M ∈ G L(n, Z) (the ring Z is necessary to preserve the integrality of the boundary vectors va ). However, gauge invariances (2) and (4) are broken by the polytope down to the subgroup of Rn G L(n, Z) under which P is invariant. 3.2. Boundary conditions and the canonical metric. The complement of M ◦ in M consists of points where one or more circles in the T n fiber degenerate. In order to have a smooth metric on all of M, there are two boundary conditions that must be imposed on f (u), or equivalently g(x). These are somewhat easier to express in the symplectic coordinate system. The first condition is that, as we approach a face of the polytope, the part of the metric parallel to the face, should not degenerate (or become infinite). Technically the requirement is that the function det G i j la , (21) a
which is positive in the interior of the polytope, should extend to a smooth positive function on the entire polytope. The second condition is that the shrinking circle(s) should go to zero size at the correct rate in order to avoid having a conical singularity. To express this boundary condition in terms of g, Guillemin [37] and Abreu [13] introduced the canonical symplectic potential: 1 gcan ≡ la ln la . (22) 2 a This canonical potential leads to a metric on M that is free of conical singularities. Furthermore, every smooth metric corresponds to a g that differs from gcan by a function that is smooth on the entire polytope; we will call this function h: g = gcan + h.
(23)
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Fig. 1. The polytopes for the five compact toric manifolds with positive first Chern class: from left to right, CP1 × CP1 , CP2 , dP1 , dP2 , dP3 . Each polytope is drawn such that the λa for all its faces are equal, corresponding to the Kähler class being proportional to the first Chern class
So far, we considered a toric manifold with a fixed metric. In general, a given toric manifold will admit many different toric metrics, i.e. Kähler metrics invariant under the given U (1)n diffeomorphism group. These will be described by functions g(x) (or f (u)) that differ by more than the gauge transformations listed above. Metrics with the same polytope (modulo the gauge transformations 2 and 4, which act on the polytope) are in the same Kähler class. In general, the topology of the manifold is determined by the number and angles of the polytope’s faces, i.e. the vectors va (modulo gauge transformation 4), while the Kähler class is determined by their positions, i.e. the numbers λa (modulo gauge transformation 2). If the λa are all equal, then the Kähler class is proportional to the manifold’s first Chern class. More specifically, if λa = −1 for all a, then [ω] = 2π c1 (M). The case of interest, dP3 , has c1 (M) > 0, so we must take > 0. 3.3. Examples. In one complex dimension, there is only one compact toric manifold, CP1 . The corresponding polytope P is simply the interval, whose length determines the Kähler modulus. We take P = [−λ, λ]. The canonical symplectic potential 1 ((λ + x) ln(λ + x) + (λ − x) ln(λ − x)) 2 √ yields the round metric of radius λ: gcan (x) =
2 = dscan
λ2
λ λ2 − x 2 2 dθ . dx2 + 2 −x λ
(24)
(25)
The Kähler coordinate u is related to x by x = λ tanh u,
(26)
f can (u) = λ ln cosh u,
(27)
2 dscan = λ sech2 u(du 2 + dθ 2 ).
(28)
and the Kähler potential is giving the metric in the form
There are five compact toric surfaces with positive first Chern class (i.e. toric Fano surfaces); their polytopes are shown in Fig. 1. CP1 × CP1 has two moduli, the sizes of the two CP1 factors; when these are equal the canonical metric is Einstein. CP2 has only a size modulus, and again the canonical metric is Einstein, as shown in Appendix A. The del Pezzo surfaces dP1 , dP2 , and dP3 have two, three, and four moduli respectively. Their
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canonical metrics are never Einstein. Indeed, dP1 and dP2 do not admit Kähler-Einstein metrics at all [33], essentially because their Lie algebra of holomorphic one-forms is not reductive. The case of dP3 is special. On the one hand, it is known to admit a toric Kähler-Einstein metric [6]. On the other hand, that metric is not the canonical one; indeed it is not known in closed form—hence the necessity of computing it numerically. dP3 is discussed in more detail in Sect. 3.8 below. 3.4. The Monge-Ampère equation. As usual for a Kähler manifold, the Ricci curvature tensor Ri j¯ can be written in complex coordinates z in a simple way, namely Ri j¯ = −
∂2 ln det gkl¯. ∂z i ∂ z¯ j
(29)
Recall that there is a two-form R = i Ri j¯ dz i ∧ d z¯ j associated with Ri j¯ where the class [R] = 2π c1 (M). We are interested in Kähler-Einstein metrics on M, that is metrics which satisfy the following relation: Ri j¯ = gi j¯
(30)
for some fixed . As explained above, this implies that λa = −1 for all a. The sign of is not arbitrary but is fixed by the first Chern class of M, which we are now assuming to be positive. Given (29), we may integrate (30) twice, yielding ln det Fi j = −2 f + γ · u − c,
(31)
where γ and c are integration constants. In symplectic coordinates, this becomes ln det G i j = −2g +
∂g · (2x − γ ) + c. ∂x
(32)
Both (31) and (32) are examples of Monge-Ampère type equations, which must be solved with the boundary conditions discussed in Sect. 3.2 above. The values of the constants γ and c in these equations are arbitrary; given a solution with one set of values, a solution with any other set can be obtained using gauge transformations (1) and (2). However, as discussed above, gauge transformation (2) acts on the polytope P by a translation. Therefore, if we fix the position of P, there will be a unique γ such that (32) admits a solution. Since the boundary conditions are non-standard, it is worth exploring them in more detail. Again, we work in the symplectic coordinate system. Recall that the condition for a smooth metric was that h(x), defined by g = gcan + h,
(33)
be smooth on P, including on its boundary. We can re-write (32) in terms of h as follows: ∂h ∂ 2h ik · (2x − γ ) − ρcan + c, = −2h + (34) ln det δi j + G can ∂ xk ∂ x j ∂x where ρcan ≡ ln det G ican j + 2gcan −
∂gcan · (2x − γ ). ∂x
(35)
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Normally, for a second-order PDE, we would expect to have to impose, for example, Dirichlet or Neumann boundary conditions in order to obtain a unique solution. However, in the case of (32), the coefficient of the normal second derivative goes to zero (linearly) k on the boundary (near the face a of the polytope, G ik can va ∼ la ). Therefore, under the assumption that h and its derivatives remain finite on the boundary, the equation itself imposes a certain (mixed Dirichlet-Neumann) boundary condition. If we were to try to impose an extra one, we would fail to find a solution. This is illustrated by the case of CP1 . Setting = λ−1 , we have ρcan = 0, so that (34) becomes 2 λ2 − x 2 h = (−h + xh ) + c; ln 1 + (36) λ λ we see that the coefficient of h vanishes on the polytope boundary. For future reference we record here the formulas for the Ricci and Riemann tensors in symplectic coordinates. Their non-zero components are ∂G i j ∂ ∂2 1 ln det G i j , Rxi x j = R θi θ j = − G kl (37) 2 ∂ xi ∂ x j ∂ x k ∂ xl 1 Rxi x j xk xl = Rxi x j θk θl = R θi θ j xk xl = R θi θ j θk θl = G mn G ml[i G j]nk , 2 (38) R θi x j θk xl = −R xi θ j θk xl = −R θi x j xk θl = R xi θ j xk θl 1 G i jkl − G mn G i jm G kln − G mn G ml(i G j)nk , (39) = 2 where we’ve defined G i jk ≡
∂G jk , ∂ xi
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∂G jkl . ∂ xi
(40)
3.5. Volumes. Here follows a short discussion about volumes useful for understanding the relation between λa and above. We know that the volume of a complex surface M is
1 Vol(M) = ω2 , (41) 2 M while the volume of a curve C is
ω.
Vol(C) =
(42)
C
From the Kähler-Einstein condition (30), (11), and the fact that the class of the Ricci form is related to the first Chern class, [R] = 2π c1 (M), it follows that [ω] = 2π c1 (M)/ and that 2π 2 Vol(M) = 2 c1 (M)2 . (43) For CP2 blown up at k points, c12 = 9 − k. Meanwhile, for our curve, Vol(C) =
2π c1 (M) · C.
(44)
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In symplectic coordinates, it is easy to compute these volumes. From (15) or (17), the volume (41) reduces to 4π 2 times the area of P. Setting = 1 corresponds to setting the λa = 1. For curves, the computation is similarly easy. For example, some simple torus invariant curves correspond to edges of P, and the volume computation reduces to measuring the length of an edge of P. 3.6. The Laplacian. The Laplacian acting on a scalar function ψ is, ψ =
1 ∂µ det gαβ g µν ∂ν ψ . det gαβ
(45)
In symplectic coordinates, the determinant of the metric det gαβ = 1. We consider a function ψ invariant under the torus action, implying no dependence on the two angular coordinates so that ψ = ψ(x1 , x2 ). Thus, the Laplacian can be written in a simpler fashion: ∂ψ ∂ Gi j . (46) ψ = ∂ xi ∂x j The Laplacian thus depends on a third derivative of g. However since we are interested in Einstein metrics, we may take (32), differentiate it, and use it to eliminate this derivative. One then finds the form ij
E ψ = G E
∂ 2ψ ∂ψ − 2xi . ∂ xi ∂ x j ∂ xi
(47)
From this equation one deduces the interesting fact that the symplectic coordinates are eigenfunctions of −E with eigenvalue 2—the symplectic coordinates are close to being harmonic coordinates. 3.7. Harmonic (1, 1)-forms. A compact toric variety M of dimension n defined by a fan {va } of m rays has Betti number b2 = m − n [38]. Moreover, dP3 has no holomorphic (nor antiholomorphic) 2-forms and thus must have m − n harmonic (1,1)-forms. These (1,1)-forms are in one-to-one correspondence with torus invariant Weil divisors Da modulo linear equivalence. The equivalence relations are vai Da ∼ 0. (48) a
From their connection with the divisors Da , perhaps it is not surprising that these (1,1)-forms θa can be constructed from functions µa which have a singularity of the form ln(va · x + 1) along the boundaries of the polytope [13]. Moreover, they should satisfy Maxwell’s equations, dθ = 0 and d θ = 0.
(49)
The first equation dθ = 0 is immediate from the local description of θi j¯ as ∂i ∂j¯ µ. The other equation is ¯
0 = D i θi j¯ = g ki Dk¯ θi j¯ .
(50)
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Using the Bianchi identity, we find then
¯ ¯ 0 = g ki Dj¯ θi k¯ = ∂j¯ g i k θi k¯ , or
¯
g i k θi k¯ = constant.
(51) (52)
Now this last equation is rather interesting. The left hand side is the Laplacian operator acting on µ. In symplectic coordinates, the left hand side can be rewritten to yield ∂ i j ∂µ G = constant. (53) ∂ xi ∂x j Using the Kähler-Einstein condition, the left hand side becomes (47). The constant is easy to establish. We have that
Vol(M)g i j¯ θi j¯ = θ ∧ ω.
(54)
M
The Kähler form is self-dual under the Hodge star, ω = ω. Conventionally, we may write [θ ] = 2π c1 (D), assuming θ is the curvature of a line bundle O(D). Using finally that [ω] = 2π c1 (M) (for = 1), we find that g i j¯ θi j¯ = 2
D·K , K2
(55)
where K is the canonical class of M. This constant is often referred to as the slope of O(D). 3.8. More on dP3 . For dP3 , the toric fan is described by the six rays spanned by v1 = (1, 0) ; v2 = (1, 1) ; v3 = (0, 1) ; v4 = (−1, 0) ; v5 = (−1, −1) ; v6 = (0, −1).
(56)
This fan leads to a dual polytope P which is a hexagon. As discussed above, when all six λa are equal, the Kähler class is proportional to the first Chern class; we will choose λa = 1. In this case the polytope has a dihedral symmetry group D6 , generated by the following Z2 reflection and Z6 rotation: 01 1 1 R1 = , R2 = . (57) 10 −1 0 This discrete symmetry will be shared by the Kähler-Einstein metric on dP3 , and will therefore play an important role in our computations. Note that the element R23 acts as x → −x, which sets γ = 0 in (31) and (32). Note also that the D6 acts naturally on the group of Cartier divisors on dP3 (and hence on the harmonic (1,1)-forms), as can easily be seen by thinking of the set of va as divisors on the manifold. The hexagon P has a natural interpretation as the intersection of the polytope for a symmetric unit (CP1 )3 , which is a cube with coordinates x1 , x2 , x3 satisfying |xi | ≤ 1, with the plane x1 + x2 + x3 = 0. The intersection defines a natural embedding of dP3 into (CP1 )3 . Furthermore, the canonical metric on dP3 is simply the one induced from the Fubini-Study metric on (CP1 )3 .
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While the action of the symmetries in symplectic coordinates is geometrically clear, we want to make explicit their description in complex coordinates. Note that the action of R2 on the complex coordinates u i is given by the inverse transpose of R2 since u = ∂g/∂ x. Thus R2 acts as (u 1 , u 2 ) → (u 2 , u 2 − u 1 ). Similarly, R1 which is its own inverse transpose, acts by (u 1 , u 2 ) → (u 2 , u 1 ). More explicitly, the relation between the complex and symplectic coordinates for the canonical potentials is given by X 2 = e2u 1 =
1 + x1 1 + x1 + x2 , 1 − x1 1 − x1 − x2
(58)
1 + x2 1 + x1 + x2 . 1 − x2 1 − x1 − x2 To invert these relations involves solving a cubic equation in one of the xi . There are six affine coordinate patches associated to the cones formed by pairs of neighboring rays va . We denote the cone formed by the ray va and va+1 to be σa and the coordinate system on σa by (ξa , ηa ). The complex coordinates on the patches a = 1, . . . , 6 are σ1 : (ξ1 , η1 ) = (Y, X Y −1 ), σ2 : (ξ2 , η2 ) = (X −1 Y, X ), σ3 : (ξ3 , η3 ) = (X −1 , Y ), (59) σ4 : (ξ4 , η4 ) = (Y −1 , X −1 Y ), σ5 : (ξ5 , η5 ) = (X Y −1 , X −1 ), σ6 : (ξ6 , η6 ) = (X, Y −1 ). Y 2 = e2u 2 =
Note that ηa = 1/ξa+1 , and that when ξa = ηa+1 , then ξa+1 = ηa . Because of these relations, we can consider an atlas on dP3 where for each σa , we restrict ξa ≤ 1 and ηa ≤ 1. These six polydisks Pa tile dP3 . In the symplectic coordinate system (x1 , x2 ), our atlas divides the hexagon up into six pieces. The boundaries are given by the conditions eu 1 = 1, eu 2 = 1, and eu 1 −u 2 = 1 or in symplectic coordinates, the three lines 2x1 + x2 = 0, x1 + 2x2 = 0, and x1 − x2 = 0. The action of D6 maps one Pa into another. For example in patch 3 since X and Y are the exponentials of u 1 and u 2 , R2 acts on our coordinates by sending (X −1 , Y ) → (Y −1 , X −1 Y ), mapping patch 3 into patch 4. 4. Ricci Flow In this section we consider the flow, on the space of Kähler metrics on M, defined by (1): ∂gµν = −2Rµν + 2gµν , ∂t
(60)
where we have set = 1. Note that, since the Ricci form is a closed two-form, a Kähler metric remains Kähler along the flow. Furthermore, if [ω] = [R] for the initial metric, then the flow will stay within that Kähler class. If the flow converges, the limiting metric will clearly be Einstein. A recent result of Tian and Zhu [14] implies that, on dP3 , starting from any initial Kähler metric obeying [ω] = [R], the flow will indeed converge to the Kähler-Einstein metric. Numerical simulation of the flow can therefore be used as an algorithm for finding the Kähler-Einstein metric. Thanks to the toric symmetry, (60) can be written as a parabolic partial differential equation for a single function in 2 space and 1 time dimensions, and therefore represents
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a simple problem in numerical analysis. Specifically, according to (29), we can write (60) in terms of the Kähler potential f (u): ∂f = ln det Fi j + 2 f + c. ∂t
(61)
Here c, which is independent of u but may depend on t, can be chosen arbitrarily. In particular, the zero mode of f (which is pure gauge) is clearly unstable according to (61), and c is useful for controlling that instability. In principle we could also add a term γ · u to the right-hand side. As explained in Sect. 3.8, however, due to the symmetry of the polytope for dP3 , we know that γ = 0 in the solution to the Monge-Ampère Eq. (31). In the symplectic coordinates, (61) takes the form ∂g ∂g = ln det G i j + 2 −x · + g − c. (62) ∂t ∂x Note that, because the Kähler potential is t-dependent, the mapping relating u to x is as well. To derive (62), we used the fact that, since f and g are related by a Legendre transform, ∂g/∂t|x = −∂ f /∂t|u . Note also that (62) does not quite describe Ricci flow in the symplectic coordinate system; rather it describes Ricci flow supplemented with a t-dependent diffeomorphism (described by the flow equation ∂gµν /∂t = −2Rµν + 2gµν + 2∇(µ ξν) ). Under pure Ricci flow, symplectic coordinates do not stay symplectic, since the Ricci tensor (37) is not the Hessian of a function, hence the necessity of supplementing the flow with a diffeomorphism. The two flows (61) and (62) were simulated by two independent computer programs, which yielded consistent results. In both cases we represented the (Kähler or symplectic) potential using standard real space finite differencing, and therefore obtain the resulting approximation to the geometry in the form of the potential at an array of points. (In the next section we discuss how to present this information more compactly using polynomial approximations.) We relegate technical details of these implementations to Appendix B. When using finite difference methods it is important to show that one obtains
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a suitable convergence to a continuum limit upon refinement of the discrete equations, and we present tests that confirm this, and estimate errors in the same appendix. 4.1. Complex coordinates. We now discuss in more detail the implementation of Ricci flow using complex coordinates. Our canonical Kähler potential f can is defined in P ◦ , the interior of the hexagon. To work on our atlas (59), we use Kähler transformations to modify the Kähler potential so that f is smooth on each coordinate patch, including the two relevant edges of the hexagon. This Kähler transformation must also respect the U (1) isometries so must only depend on the u i . The only such Kähler transformations are affine linear combinations of the u i . (For a function h(u 1 + iθ1 , u 2 + iθ2 ) depending ¯ ≡ 0 reduces to ∂ 2 h ≡ 0.) only on the u i , the condition on Kähler transformations ∂ ∂h ∂u i ∂u j In the interior of the hexagon, we can thus modify f without modifying the metric by adding a linear combination of u 1 and u 2 . In symplectic coordinates 1 va · κ ln(va · x + 1), 2 6
u·κ =
a=1
where κ ∈ R2 . We find that for any Kähler potential f , regardless of coordinate patch, 2 2 2 f 1 ∂f 2f 1 ∂ f ∂ f ∂ ∂ + + det Fi j = ξ 2 η2 . − ξ ∂ξ ∂ξ 2 η ∂η ∂η2 ∂ξ ∂η
(63)
We can rewrite (61), yielding 2 2 1 ∂ f ∂2 f 1 ∂ f ∂2 f ∂ f ∂f − = ln + + + 2( f + ln ξ + ln η) + c. 2 2 ∂t ξ ∂ξ ∂ξ η ∂η ∂η ∂ξ ∂η (64) This shift of f by logarithms is a Kähler transformation. In each coordinate patch, we can write ln ξa + ln ηa = u · κa , where κ1 = (1, 0), κ2 = (0, 1), κ3 = (−1, 1), κ4 = (−1, 0), κ5 = (0, −1), and κ6 = (1, −1). Moreover, for these choices of κa , f a = f + u · κa is well behaved everywhere inside Pa including the edges. We simulate Ricci flow using a Kähler-Einstein potential f a defined on a neighborhood Ua of Pa , Ua = {(ξa , ηa ) : ξa , ηa < L , L > 1} which satisfies (64). As an initial condition, we take f a (t = 0) = f can + ln ξa + ln ηa . Given two patches, a and b, then for a point in the overlap, p ∈ Ua ∩ Ub , f b ( p) = f a ( p) + u · (κb − κa ).
(65)
These Kähler transformations are consistent with the canonical potential and thus fix the same Kähler class. There is then an element R ∈ D6 such that R(Pb ) = Pa which relates f a and f b . In particular, if p is in Ub and Rp is in Ua : f b ( p) = f a (Rp).
(66)
Thus, with these quasiperiodic boundary conditions along the interior edges ξa = L and ηa = L, we can determine f by working solely on the patch Ua . In addition to quasiperiodic boundary conditions along the interior edges of Ua , close to the exterior boundary of Ua , we use the condition that the normal derivative to the
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boundary must vanish. These Neumann boundary conditions arise because in complex coordinates, approaching the exterior boundary should be like approaching the center of the complex plane. The Ricci flow in this domain was represented using second order accurate finite differencing with various resolutions up to a grid F I J of 250 × 250 points. This discretization, its convergence to the continuum and error estimates at this resolution are discussed in Appendix B, and data presented in this section is given either from extrapolating to the continuum or using this highest resolution. In particular we estimate that the Kähler potential computed at this resolution is accurate pointwise to one part in 105 . We noted earlier that while Ricci flow may converge, the zero mode of the potential f is not guaranteed to converge since it does not appear in the metric. For convenience we wish to obtain a flow of f that does converge, so we promote the constant c to be time dependent along the flow, but still a constant on the geometry (i.e. we introduce a flow dependent constant Kähler transformation), and choose c such that the value of F N /2,N /2 , where N is the grid size, does not change. At large times, c tends to a constant. At the end of the flow, we found it convenient to set c = 0 by adding an appropriate constant value to the grid F. 4.2. Symplectic coordinates. We move on to discuss the implementation of Ricci flow based on symplectic coordinates. As discussed at the beginning of the section, although we work in symplectic coordinates, we use Ricci flow defined by the complex coordinates; since the relation between the two coordinate systems moves around as the metric changes, in the symplectic coordinates this is Ricci flow plus diffeomorphism. In order to deal with the boundary conditions on the edge of the polytope, it is useful to work with h rather than g (recall that g = gcan + h and gcan is the canonical symplectic potential). In terms of h, (62) becomes ∂h ∂h ∂ 2h + 2 −x · = ln det δi j + G ik + h + ρcan − c, (67) can ∂t ∂ xk ∂ x j ∂x where
∂gcan + g + 2 −x · ρcan ≡ ln det G ican can = ln(L 1 + L 2 + L 3 ), j ∂x 1 + LL 23 −1 L1 L2 ik G can = , −1 1 + LL 31 L1 + L2 + L3
L 1 = l1l4 = 1 − x12 ,
L 2 = l3l6 = 1 − x22 ,
(68)
(69)
L 3 = l2 l5 = 1 − (x1 + x2 )2 . (70)
Due to the hexagon’s D6 symmetry, it is sufficient to simulate the flow within the square domain 0 ≤ x1 ≤ 1, −1 ≤ x2 ≤ 0. Ricci flow in this domain was represented using second order accurate finite differencing with various resolutions up to a grid of 400 × 400 points. The differencing, continuum convergence and errors at this resolution are discussed in Appendix B with data presented in this section given either from extrapolating to the continuum or using this highest resolution. In that appendix we estimate that the symplectic potential computed at this resolution is accurate to one part in 106 at a given point. As for the complex coordinates, the zero mode of the symplectic potential is pure gauge and does not converge. The constant c was promoted to depend on flow time and chosen to keep h(0, 0, t) = 0.
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Fig. 3. The Ricci flow using symplectic coordinates starting with (top) the canonical metric (h 0 = 0), (middle) h 0 = − 21 (x12 + x22 + x1 x2 ), (bottom) h 0 = (x12 + x22 + x1 x2 )2 . Left side: g = gcan + h versus x1 along the line x2 = 0, plotted at intervals of 0.1 units of Ricci flow time (t = 0 is the top curve). Right side: g versus t at the (arbitarily chosen) point (x1 , x2 ) = (0.3, −0.2). We see that all three initial conditions converge to the same fixed point
4.3. Results. As predicted by the Tian-Zhu theorem, the metric converges smoothly and uneventfully to the Kähler-Einstein one. In the symplectic implementation, the Ricci flow was simulated starting with a variety of initial functions h 0 (x) = h(x, t = 0) (always corresponding, of course, to positive-definite initial metrics). Three examples are shown in Fig. 3. For every initial function investigated, the flow converged to the same fixed point h E (x) = h(x, t = ∞), which necessarily represents the Kähler-Einstein metric. The exponential approach to the fixed point is controlled by the scalar Laplacian; this will be discussed in detail in the next subsection. The final complex and symplectic potentials found by the independent implementations are plotted in Figs. 4 and 5, along with the respective canonical potentials. The results are plotted in the fundamental domain actually simulated, and one should use the D6 symmetry to picture the potentials extended over the whole domain. To compare the Kähler potential f E (u) computed in complex coordinates with the symplectic potential gE (x) computed in symplectic coordinates, we numerically performed a Legendre transform to obtain a Kähler potential f E (u) from gE (x). Plotting f E − f E in Fig. 6, it can be seen that they differ by less than 5 × 10−6 . (In fact the agreement may be slightly better as there is likely some error introduced in doing the numerical
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Legendre transform.) Thus our results appear to agree to around the same order that we believe they are accurate. In order to get some feeling for the form of the Kähler-Einstein metric, and how it compares to the canonical one, it is helpful to plot some curvature invariants. Of course, any invariant depending solely on the Ricci tensor, such as the Ricci scalar, will be trivial, so we need to go to invariants constructed from the Riemann tensor. (Expressions for the Riemann tensor are given in Sect. 3.4 above.) For example, the sectional curvature of the x1 -x2 plane (at fixed θi ), which is Rx1 x2 x1 x2 / det(G i j ), is plotted for the canonical and Einstein metrics in Fig. 7. Also of interest is the Euler density
1 2 µν µνρλ e= R , (71) − 4R R + R R µν µνρλ 32π 2 which integrates to the Euler character of the manifold, which is 6 for dP3 . The first two terms inside the parentheses cancel in the case of an Einstein metric. Figure 8 shows 2 4π 2 e for the canonical and Einstein metrics. The √ factor of 4π takes account of the coordinate volume of the fiber. Recalling that g = 1 in symplectic coordinates, the plotted quantity should integrate to 2 over the plotted region, which covers one-third of the polytope. This can easily be checked in both cases by numerical integration. 4.4. Laplacian eigenvalues. An important geometric quantity is the spectrum of the scalar Laplacian. Here we illustrate a simple method to compute low-lying eigenfunctions. The natural flow associated with the scalar Laplacian is diffusion, and the late
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Fig. 5. Top left: The canonical symplectic potential gcan . Top right: The Einstein symplectic potential gE = gcan + h E . Bottom: h E . These are plotted in the range 0 ≤ x1 ≤ 1, −1 ≤ x2 ≤ 0, which is onethird of the hexagon; the values on the rest of the hexagon are determined from these by its D6 symmetry
0 2 10 4 10
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Fig. 6. The difference between the final Kähler potential computed in complex coordinates and the Legendre transform of the final symplectic potential computed in symplectic coordinates
time asymptotic behavior of the diffusion flow is dominated by the eigenfunction with lowest eigenvalue. Hence simulating diffusive flow on the dP3 geometry and extracting the asymptotics of this flow allow the lowest eigenfunction to be studied. We may classify the eigenfunctions under the action of the D6 and U (1)2 isometries. Since the flow
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Fig. 7. The sectional curvature of the x1 − x2 plane at fixed angle θi , for the canonical (left) and Einstein (right) metric. The two are quantitatively different but qualitatively similar
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Fig. 8. 4π 2 e, where e is the Euler density, for the canonical (left) and Einstein (right) metrics. The Euler density is distributed differently but integrates to the same value—2 over this region, and 6 over the whole polytope—for any two metrics. It is more evenly distributed for the Einstein metric than for the canonical metric, but still peaked at the vertices of the polytope
equation is invariant under these symmetries, if we start with initial data that transforms in a particular representation, the function at any later time in the flow will remain in this representation. For simplicity we will focus on eigenfunctions which transform trivially, but obviously the method straightforwardly generalizes to compute the low-lying eigenfunctions in other sectors. As for the Ricci flow, the flow does not depend on second normal derivatives of ψ at the boundaries of the hexagon domain, and hence we do not require boundary conditions for ψ there, except to require it to be smooth. The lowest eigenfunction of −E in the symmetry sector we study is ψ = constant which has zero eigenvalue. We are interested in the next lowest mode which has positive
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0
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Fig. 9. Decay of ψ under diffusion towards a constant as a function of diffusion time t. The log of ψ(t, 0, 0) − ψ0 is plotted, the slope giving the eigenvalue for the lowest (non-constant) symmetric eigenfunction. The 3 curves correspond to 3 different initial profiles for ψ although we see that the decay quickly becomes dominated by the lowest eigenmode
eigenvalue and non-trivial eigenfunction, denoted ψ1 (x) with eigenvalue λ1 . Then we consider the diffusion flow on our del Pezzo solution, ∂ ψ(t, x) = E ψ(t, x), (72) ∂t and start with initial data for ψ that is symmetric and will hence remain symmetric. At late times, the flow will generically behave as, ψ(t, x) = ψ0 + ψ1 (x)e−λ1 t + O(e−λ2 t ),
(73)
where ψ0 is a constant, corresponding to the trivial zero eigenmode, and λ2 is the next lowest eigenvalue λ2 > λ1 . Waiting long enough and subtracting out the trivial constant, the late flow is given by ψ1 , the eigenfunction we wish to compute. In Fig. 9 we plot the log of ψ(t, 0, 0) − ψ0 as a function of the flow time t for 3 different initial data. Once the higher eigenmodes have decayed away, we clearly see the flows tend to the same exponential behaviour. We estimate this eigenvalue by fitting the exponential decay as λ1 = 6.32. In Fig. 10 we plot the eigenfunction ψ1 (x), normalized so that ψ1 (0, 0) = 1. Note that, as expected, for different initial data we consistently obtain the same function. This lowest eigenvalue and eigenfunction can also be obtained from the approach to the fixed point of the Ricci flow. For concreteness let us work in symplectic coordinates; corresponding expressions will hold in complex coordinates. Expanding h about its fixed-point value, h = h E + δh, (74) the flow equation (67) becomes, to first order in δh, ∂δh = (E + 2)δh − δc, (75) ∂t where δc depends on how c is chosen. Since we used initial conditions for the Ricci flow that respected the D6 symmetry, we should find that the potentials approach their fixed point values the same way as ψ above, with a shift of 2 in the exponent: h(t, x) = h E (x) + (ψ1 (x) − ψ1 (0)) e−(λ1 −2)t + · · · ;
(76)
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1 0.5 0 -0.5 -1
0 -0.2 -0.4
0 0.2
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Fig. 10. The lowest eigenfunction of the scalar Laplacian which transforms trivially under the D6 and U (1)2 symmetries
the eigenfunction is shifted by a constant because c was chosen to keep h(t, 0) = 0 along the flow. Our numerical flows confirm this expectation. The corrections in (76) involve both higher eigenvalues of the Laplacian and higher-order effects in δh. 5. Symplectic Polynomials In the Ricci flow simulation in symplectic coordinates discussed in Sect. 4.2, the function h(x)—which encodes the symplectic potential and therefore the metric—was represented by its values on a lattice of points in x1 , x2 . In this section we will discuss a different way to represent the same function, namely as a polynomial in x1 , x2 . Since the solution h E to the Monge-Ampère equation is a smooth function, it can be represented to good accuracy with a vastly smaller amount of data in this way: a few polynomial coefficients as compared to values on thousands of lattice points. Furthermore, quite independent of the solutions found in the previous section, the problem of finding an approximate solution to the Monge-Ampère equation can be expressed as an optimization problem for the polynomial coefficients; we will use this fact to develop a third algorithm in the following section that is quite different in character from Ricci flow. It is interesting to note that the metrics obtained from polynomial expressions for h(x) are the symplectic analogues of the so-called “algebraic” metrics on Calabi-Yau manifolds that have been used for numerical work by Donaldson [3] and Douglas et al. [4,5]. The algebraic metrics, which are defined for a Calabi-Yau embedded in a projective space, have a Kähler potential that differs from the induced Fubini-Study one by (the logarithm of) a finite linear combination of a certain basis of functions, namely the pull-backs of the Laplacian eigenfunctions on the embedding projective space. This is a generalization of the usual strategy of representing a function by expanding it in a basis of Laplacian eigenfunctions (such as Fourier modes); since the eigenfunctions on the Calabi-Yau depend on the metric that one is trying to find, one instead uses the eigenfunctions on the embedding space, which are known in closed form (and are indeed very simple). In our case, we consider the embedding of dP3 in (CP1 )3 , which, as discussed in Sect. 3.8, is described in symplectic coordinates by the equation x1 + x2 + x3 = 0 (where the xi are the symplectic coordinates on the respective CP1 factors). The first
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n eigenspaces of the Laplacian on (CP1 )3 (with respect to the Fubini-Study metric), restricted functions that are invariant under the U (1)3 isometry group, are spanned by the monomials in x1 , x2 , x3 up to order n. This is precisely the basis of functions we use to expand the difference h between the symplectic potential g and the induced one gcan . We now describe some fits to the numerical solutions of the last section in terms of polynomials up to sixth order in x1 and x2 , and quantify how well those polynomials do in solving the Einstein equation. These small polynomials likely provide sufficiently accurate approximations to the Einstein symplectic potential for most purposes, while at the same time being more tractable than the full numerical data for analytic calculations. We begin by noting that, since h E is invariant under the hexagon’s D6 symmetry group, it is sufficient to consider invariant polynomials. As shown in Appendix C, every invariant polynomial can be expressed in terms of the two basic invariant polynomials, U = x12 + x1 x2 + x22 ,
V = x12 x22 (x1 + x2 )2 .
(77)
We simply do a least-squares fit of the polynomial coefficients to the lattice values of h E obtained in the Ricci flow in symplectic coordinates, that is, we minimize 2
1 1 √ √ g (h fit − h E )2 − g (h fit − h E ) . (78) α2 = VdP3 dP3 VdP3 dP3 (Any constant difference between h fit and h E is irrelevant). At successive orders in x we find the following fits: h fit 0 −0.24U −0.2214U − 0.0215U 2 −0.22412U − 0.01450U 2 − 0.00521U 3 + 0.00734V
α 0.06 10−3 10−4 10−5
β 0.5 0.1 0.03 0.007
(79)
In each case we have written only the significant digits of the coefficients.3 Independently of our numerical result h E , it is useful to know how far the metric corresponding to h fit deviates from being Einstein. In Fig. 11, the pointwise rms deviation of the eigenvalues of the Ricci tensor from 1, 1 (Rµν − gµν )2 , (80) D≡ 4 is plotted for these four functions. The global rms deviation from being Einstein, β, where
1 √ 2 β2 = gD , (81) VdP3 dP3 is also shown in the table above. As expected, each successive order gives a substantially better approximation, and a metric that is substantially closer to being Einstein. For eigenfunctions of the Laplacian, we may perform the same invariant polynomial fits as we did for h; at quadratic order we find ψ1 ≈ 0.985 − 2.37U . We regard Table 79 as a key result of this paper. The last line of the table provides in an extremely compact, analytic form, an approximation to the true symplectic potential 3 These digits do not change between the run with 200 lattice points and the run with 400 lattice points (except the last digit of each coefficient in the sixth-order approximation), and are therefore presumably equal to their continuum values.
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1 2 4 (Rµν − gµν ) versus x 1 , x 2 for the four polynomial functions listed in (79). The deviation
Fig. 11. D ≡ from being Einstein is most significant along the edges of the hexagon in the first two cases, and at the corners in the second two
on dP3 . It deviates from the true potential pointwise by at most ∼ 0.1%, and satisfies the Einstein condition well within the hexagon, giving at most a 10% error near the hexagon corners as measured by the pointwise rms deviation of the Ricci tensor eigenvalues, defined above. 6. Constrained Optimization In the previous section we used the results of Ricci flow to find a polynomial approximation to h(x), the smooth part of the Kähler-Einstein symplectic potential (recall that h ≡ g − gcan ). Here we instead search for a polynomial approximation to h using the Monge-Ampère equation directly. A simple approach would be to consider the space of polynomials of some given order, and minimize an error function built from the MongeAmpère equation on that space. However, if one wishes to obtain high accuracies, one needs to go to high orders, and then this brute force approach rapidly becomes intractable due to the large number of polynomial coefficients and the difficulty of searching in a high dimensional space. We therefore take advantage of the analytic properties of the Monge-Ampère equation to constrain the polynomial coefficients, by requiring the polynomial to solve it order by order in xi . As we will see, this leaves only a small number of undetermined parameters, dramatically simplifying the error function minimization. We now explain the details of the method. We begin by noting that the exact solution h E (x) to the Monge-Ampère equation is an analytic function of the xi , which can be seen in a couple of ways. As noted, the
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symplectic coordinates are eigenfunctions of the Laplacian. Hence, in these coordinates the Ricci curvature operator has the same character as it does in harmonic cooordinates: it is actually an elliptic operator. Because the Einstein equations are analytic in the metric, they will have analytic solutions. Somewhat more directly, the Monge-Ampère equation we are solving is elliptic at the Kähler-Einstein potential and analytic, and so the solution will be analytic. Before we constrain the polynomial approximation to h(x) using the Monge-Ampère equation, we constrain it by imposing the hexagon’s D6 discrete symmetry group. As discussed in Sect. 5, any invariant polynomial in xi can be written in the form h= ci, j U i V j , (82) i, j
where U and V are given in Eq. (77). To eighteenth order in x1 and x2 we write the series as follows h = A0 + A1 U + A2 U 2 + · · · + A8 U 8 + A9 U 9 + · · · + V (B0 + B1 U + B2 U 2 + B3 U 3 + B4 U 4 + B5 U 5 + B6 U 6 + · · · ) + V 2 (C0 + C1 U + C2 U 2 + C3 U 3 + · · · ) + V 3 (D0 + · · · ) + · · · .
(83)
Plugging this series into the Monge-Ampère equation (34) (with γ = 0 and = 1), yields constraints on the ci, j that relate the ci, j with j > 0 to the ci,0 . To make the expressions a little simpler, we introduce a new constant α: 1 A0 = − ln 3 − ln α. 2
(84)
We worked out the relations up to order 18 in x1 and x2 . The first relation is that A1 = −1 ± α. The numerical Ricci flow results are consistent only with the plus sign. The next few relations are 1 α2 , A2 = − + 6 4
A3 = −
5α 25 2 2B0 11α 3 1 145α 4 − + ; A4 = − − − α B0 + ; 27 27 72 28 378 189 1152
25 1 85 2 1 α B0 + (−4 + 5α) ; B2 = α B0 + (−32 + 51α 2 ). 14 28 32 192 The power of this method is that since the Monge-Ampère equation determines many of the coefficients, when we determine the remaining ones by minimizing an error function, then at a given order there are far fewer parameters to solve for. Truncating at eighteenth order in x1 and x2 , we need fit only 4 parameters A0 , B0 , C0 , and D0 to find an approximation to h. In contrast, minimizing an error function using the most general unconstrained eighteenth order polynomial given in Eq. (83) involves searching a 22 dimensional space. In principle if we took an arbitrarily high order expansion, then about the origin of the hexagon, in the region where the series for h converges, we would solve the MongeAmpère equation precisely. However we see that we still have undetermined constants in the series, and these correspond to the fact that we must provide boundary conditions to determine a solution fully. A posteriori, our numerics strongly suggest that h converges everywhere in the interior of the Delzant polytope. B1 =
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Constraining h to have the correct behavior on the boundary of the polytope should completely determine the remaining constants in the power series expansion. Hence we fit the remaining parameters by requiring Eq. (34) be satisfied at the boundary of the hexagon. As emphasized in the discussion around Eq. (36), the boundary conditions do not need to be supplied separately—they are enforced by Eq. (32) itself. For dP3 , along x1 = 1, Eq. (32) reduces to the boundary condition BC ≡ 1−2x2 −2x22 +(1−x22 )(1−(1+x2 )2 ))h x2 x2 −exp 2(h x1 + x2 h x2 − h) = 0. (85) To find this expression, we have assumed that h is smooth at the boundary x1 = 1. At the corners x2 = 0 and x2 = −1, Eq. (85) reduces further to h = h x1 and h = −h x2 + h x1 respectively. Explicitly, we determine the remaining parameters (A0 through D0 in the eighteenth order truncation) by minimizing M= |BC( p)|2 , (86) p
summed over twenty equally spaced points along the boundary x1 = 1. We find, as we include more terms in the series, order 2 6 10 14 18
M 0.02 9 × 10−5 3 × 10−6 5 × 10−8 2 × 10−9
α B0 C0 D0 0.757 0.7753 0.011 0.77616 0.00508 0.776226 0.00480 −0.00055 0.776235 0.004781 −0.00015 0.004
(87)
The full expression for the 18th-order polynomial is given in a Mathematica notebook available for download at the websites [21]. In Appendix D we show the dependence of h at various locations in the polytope as a function of the number of terms taken in the expansion. We see that convergence for h is fast—apparently faster than polynomial—in the number of terms, and in particular for everywhere tested within the hexagon, and also on its boundary, we see convergence. In particular we see no sign of poor behaviour near or on the boundaries of the hexagon. We also see that the values the series converges to are in excellent agreement with the continuum extrapolated values of h found from the Ricci flow method and detailed in Appendix B. From these data we estimate that the potential given by the eighteenth order expansion differs from the true solution by approximately one part in 106 , and hence is comparable in this respect to the 400 × 400 Ricci flow result. The figure of merit M does not give a very good indication of the degree of accuracy of our fit globally. To understand how well we are doing globally, we use the same
local estimate of error as in the previous section, D = 41 (Rµν − gµν )2 . We find that the maximum value D attains in the domain decreases with each increase in order of the expansion. The maximum is found on the lines connecting the origin to the hexagon vertices, and hence in Fig. 12, we plot this error estimate along one of these lines, D(x1 , 0), for the 6th , 12th and 18th order polynomial expansions. As expected the error is smallest at the origin, and most error is localized near the boundaries. Since the error is rather localized near the hexagon vertex, we have plotted this error against ln (1 − x1 ) to demonstrate that it is indeed finite at the vertex. We see that while the symplectic potential taken pointwise may be accurate to 1 part in 106 as stated above, since the
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D x1 ,0 0.012 0.01 0.008 0.006 0.004 0.002
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D x1 ,0 0.00175 0.0015 0.00125 0.001 0.00075 0.0005 0.00025 2
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Fig. 12. D, the error in the Einstein condition, along the line x2 = 0 from the origin x1 = 0 to the hexagon corner x1 = 1, where the maximum of D in the hexagon domain occurs. The top left plot is for a 6th order expansion, the top right is 12th order, and the bottom is 18th order. We see the error in the Einstein condition is quite localized near the hexagon corner (hence we plot against ln (1 − x1 )), but remains finite there, and decreases everywhere with increasing numbers of terms in the expansion
error is localized in the hexagon corners, the quality of the solution is a little worse in these regions. We see for the 18th order approximation that the error in the Einstein condition, estimated by D, is about one part in 103 at the vertex. 6.1. Laplacian eigenvalues. We can use this 18th order fit to extract the lowest eigenvalues (of eigenfunctions invariant under the D6 action) on our manifold. Note that in coordinates where the metric is analytic, eigenfunctions of the Laplacian are analytic: elliptic equations with analytic coefficients have analytic solutions. Thus we can play a very similar game, expressing the eigenvectors as a local power series near the origin of the hexagon in the U and V variables: ψ(U, V ) =
1 + X 1 U + X 2 U 2 + X 3 U 3 + Y1 V + · · · . 10
(88)
We have chosen to normalize ψ(0, 0) = 0.1. We could in principle have used the differential equation to constrain some of the X i and Yi , but we did not, aiming for a fit whose errors are more evenly distributed over the hexagon. We fit a hundred equally spaced points in a square domain 0 < x1 < 0.9 and −0.9 < x2 < 0 and minimize |ψ + λψ|2 , (89) Mψ = p
as a function of λ and the X i and Y1 . By searching for successive local minima of Mψ , we can extract successively higher eigenvalues. We find order Mψ X1 X2 X3 Y1 2 0.006 −0.239 4 0.002 −0.246 0.011 6 3 × 10−5 −0.245 0.006 0.006 −0.024
λ1 6.27 6.325 6.322
(90)
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order Mψ X 1 X 2 X 3 Y1 λ2 4 0.5 −0.69 0.79 17.4 6 0.008 −0.67 0.70 0.16 −1.28 17.2
(91)
For the lower eigenvalue 6.32, note that the ratio of the first two coefficients −2.39 is in reasonably good agreement with the corresponding ratio −2.37/0.985 = −2.41 determined in Sect. 5. The choice to minimize Mψ in the domain 0 < |xi | < 0.9 was a compromise that requires some justification. First, since (32) enforces its own boundary conditions, minimizing Mψ close to the boundary x1 = 1 is enforcing the boundary condition to first order in x1 − 1. Second, the power series approximation for ψ experimentally does not appear to have good convergence properties near the boundary. Experimentally, the best values for the coefficients of the truncated power series (in the sense of agreeing with the coefficients of the power series itself) are obtained by making a compromise between minimizing over a set of points that extends to the boundary and minimizing over a set of points for which the power series has good convergence properties. 6.2. Harmonic (1,1)-forms. As noted in Sect. 3.7, harmonic (1, 1)-forms and eigenfunctions of the Laplacian must satisfy a very similar equation. We end this section with a computation of the harmonic (1, 1)-form θa . Unlike the eigenfunctions computed above, θa does not transform trivially under D6 . Thus, we assume that µa has a more general expansion of the form cnm x1n x2m . (92) µa = ln(1 + va · x) + n,m
We use the same least squares approach as above, minimizing Mθ = |µa − const|2 .
(93)
p
Because of the explicit x1 ↔ x2 symmetry, we start with a = 2 and set cnm = cmn . Fitting to sixth order in x1 and x2 , we find c10 c20 c30 c40 c50 c60
= −0.2250 = 0.0638 = −0.0301 = 0.0126 = −0.0150 = 0.0088
c11 c21 c31 c41 c51
= 0.0311 = 0.0059 = 0.0005 c22 = −0.0073 = −0.0245 c32 = −0.0240 = 0.0223 c42 = 0.0281 c33 = 0.0196
(94)
The value of Mθ ∼ 10−4 at the minimum implies an average error of 10−3 at each of the 100 points. Note the error gets much worse outside the fitting domain |xi | > 0.9. The philosophy in this section is similar to that in the discussion of eigenfunctions: we are attempting to find more accurate values of the ci j rather than attempting to minimize the global error. The fit also yields F i j θi j = 0.6672, consistent with our expectations. We know that 1 θa . 2 6
ω=
a=1
(95)
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8.04 8.02 8 7.98 7.96 0
0 0.2 0.4 0.2
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Fig. 13. The value of 2 for our 6th order fit
Clearly F i j¯ Fi j¯ = 2. By the dihedral symmetry group, the value of F i j¯ θi j¯ should be independent of a. We conclude that (θa )i j¯ F i j¯ =
2 . 3
(96)
From θ2 , we can reconstruct the other θa by applying the D6 group action. To test how good our approximation to θ2 was, we computed 2 = i j i j where i j = a (θa )i j , using our best fit for θ2 . Since i j should be 2Fi j , 2 should be approximately eight. A plot of 2 is shown in Fig. 13. For the purposes of the KT solution described in the Introduction, we need a θ such ij that θ Fi j = 0. From the preceding discussion, any linear combination of the form a ca θa such that a ca = 0 will have this property. We also require that θ = −θ . In fact, the condition a ca = 0 enforces anti-self-duality. The reason is that the Hodge star treated as a linear operator acting on the space of harmonic (1,1)-forms has signature (+ − −−). We know that ω = ω; thus any (1,1)-form orthogonal to ω must be antiself-dual. In general, the numerics suggest that for such a θ , θ i j θi j will be a nontrivial function of both xi and thus that solving for h( p) requires solving a PDE in three real variables.
7. Discussion In this paper we have described three different methods to find the Kähler-Einstein metric on dP3 . All three methods exploit the Kähler and toric structures of the manifold, allowing us, using modest computing resources, to compute the metric in both Kähler and symplectic coordinates to an accuracy of one part in 106 . The results of the different methods are consistent to within that error. We expect that this accuracy is sufficient if one wishes to compute geometric quantities for either physical or mathematical applications, and we have made available the data along with Mathematica notebooks to allow manipulation of these results [21]. We noted that, for a lesser accuracy of one part in 103 , a simple expression for the smooth part h of the symplectic potential, g = gcan + h, already provides such an
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approximation, and we repeat it here: h(x1 , x2 ) = −0.22412U − 0.01450U 2 − 0.00521U 3 + 0.00734V, U = x12 + x1 x2 + x22 , V = x12 x22 (x1 + x2 )2 , where gcan is given in Eq. 22. The resulting metric satisfies the Einstein condition everywhere to better than 10% as discussed in Sect. 5. Simulation of Ricci flow has proven to be an effective way to solve the Einstein equation. We have found that implementing the flow is a little simpler in symplectic coordinates: the domain is naturally compact; the symmetries are more manifest; and the boundary conditions are simpler. Our codes (which were not optimized for speed) converged in a few hours for the highest resolutions. For higher accuracy than attained here, one could optimize the flow simulation, for example by taking more advantage of the discrete symmetries than we have done. More generally, Ricci flow simulation, using an explicit finite differencing method as we have done, can be thought of as a particular iterative scheme for solving the Monge-Ampère equation. If one is interested only in solving that equation, and not in accurately simulating Ricci flow, then this scheme could be modified to improve speed. For example, by replacing the Jacobi-type updating method by a Gauss-Seidel method, one obtains a faster algorithm (experimentally, 50% faster in complex coordinates). To obtain a parametric improvement in speed would likely require a non-local modification such as multi-grid. The constrained optimization approach we have demonstrated uses the symplectic polynomials, reducing the size of the search space by solving the Monge-Ampère equation order by order in xi , and has proven very powerful. It is as accurate as the Ricci flow results, but is quicker. One drawback is that the hexagon origin is singled out as the point where the solution is best, and the error in the solution becomes tightly localized at the corners of the hexagon. Most computational time is invested in determining the constraints in the series expansion, and this algebraic problem gets worse the more terms that are included in the expansion. However, once one has this solution, the numerical minimization of the error function is simple. The two methods are complementary in the sense that for Ricci flow the time is spent in numerically computing the flow, whereas for the optimization the time is spent algebraically computing the expansion of the potential. The principle advantage of Ricci flow is that the method is very general, and while it benefits from the Kähler and toric structures it certainly applies to more general problems which do not possess them. It is not clear how widely applicable the constrained optimization approach is, as it likely works due to the special properties resulting from those mathematical structures. However, it would be interesting to investigate its application to other situations. It would also be interesting to compare these approaches, particularly the constrained optimization, with Donaldson’s method [3]. Acknowledgements We would like to thank Ken Bube, Simon Donaldson, Richard Hamilton, Julien Keller, Liam McAllister, Gang Tian, and Ursula Whitcher for discussion. C.H. would like to acknowledge the support of the String Phenomenology Workshop at the KITP, UCSB and the Physics Department at the University of Texas, Austin where part of this work was done. C.D. is supported in part by a Royalty Research Fund Scholar Award from the Office of Research, University of Washington. M.H. is supported by the Stanford Institute for Theoretical Physics and by NSF grant PHY 9870115. C.H. is supported in part by U.S. Department of Energy under Grant No. DE-FG02-96ER40956 and by the National Science Foundation under Grants No. PHY9907949 and No. PHY-0455649. J.K. is supported in part by a VIGRE graduate fellowship. T.W. is supported by a PPARC advanced fellowship and the Halliday award.
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A. Canonical Metric on CP2 Consider CP2 which has the fan, unique up to S L(2, Z) transformations, v1 = (1, 0), v2 = (0, 1), and v3 = (−1, −1). We choose all the λa = 1 in gcan to assure that the class of the resulting Kähler form is proportional to the first Chern class, 1 [(1 + x1 ) ln(1 + x1 ) + (1 + x2 ) ln(1 + x2 ) + (1 − x1 − x2 ) ln(1 − x1 − x2 )] . 2 (97) From gcan , we can reconstruct the complex coordinates
gcan =
1 + x1 =
3e2u 1 3e2u 2 ; 1 + x = ; 2 1 + e2u 1 + e2u 2 1 + e2u 1 + e2u 2 1 − x1 − x2 =
3 1 + e2u 1
+ e2u 2
(98)
.
The canonical Kähler potential is then f can =
3 3 ln(1 + e2u 1 + e2u 2 ) − ln 3 − u 1 − u 2 . 2 2
(99)
The expression gcan satisfies (32) provided c = ln(4/3) and γ = 0. For CP2 , the canonical metric is the Fubini-Study metric. In terms of the traditional homogenous coordinates (X 1 , X 2 , X 3 ) on CP2 , the Kähler potential is traditionally written, with a different choice of normalization of the volume, in the patch X 3 = 0, ln(1 + |X 1 / X 3 |2 + |X 2 / X 3 |2 ).
(100)
Thus, we identify |X 1 / X 3 | = exp(u 1 ) and |X 2 / X 3 | = exp(u 2 ). The canonical metric is always Kähler-Einstein for Cartesian products of projective spaces [37]. B. Numerical Ricci Flow Implementation and Error Estimates In this appendix we give technical details of our two finite difference implementations of Ricci flow, and also discuss the continuum convergence and estimate errors for the resolutions used.
B.1. Implementation in complex coordinates. We work on a square domain slightly bigger than a unit polydisk, 0 ≤ ξ ≤ L and 0 ≤ η ≤ L (L > 1), where f is assumed to have Neumann boundary conditions along ξ = 0 and η = 0. Along the internal boundaries, ξ = L and η = L, we employ a kind of periodic boundary condition enforced by the Z6 symmetry of the hexagon. We take f (ξ, L) = f (1/L , ξ L) + ln L ,
(101)
to map the boundary η = L for 0 ≤ ξ < 1/L back into our square domain. For 1/L < ξ < 1, we take a composition of the above map: f (ξ, L) = f (1/(Lξ ), ξ ) + ln L 2 ξ.
(102)
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For the ξ = L boundary, we take the preceding rules with η and ξ switched. For the points along the boundary with ξ > 1 or η > 1, we use the rule f (ξ, η) = f (1/ξ, 1/η) + 2 ln ξ + 2 ln η.
(103)
Note that as indicated in Sect. 4.1, due to the Z6 symmetry, for any point p = (ξ, η), with ξ > 1 or η > 1, the value of f ( p) should be related by a Kähler transformation to the value of f at a point inside the unit polydisk. Although we are only enforcing the Z6 symmetry along the boundary of our coordinate patch, the symmetry will hold globally. First, our initial potential f can respects the symmetry. Second, Ricci flow preserves the symmetry. To discretize (61), we approximated f (ξ, η) by its values F I J on an N × N grid with lattice spacing = L/(N − 1) and used standard second order finite differencing for derivatives. In evaluating the first order derivatives at the boundary, we took 1 ∂ f ∂ 2 f 1 ∂ f ∂ 2 f = and = . (104) ξ ∂ξ ξ =0 ∂ξ 2 ξ =0 η ∂η η=0 ∂η2 η=0 To impose a discrete version of the boundary conditions, we added extra rows and columns along the grid. To impose Neumann boundary conditions, we imposed that F0,J = F2,J and F I,0 = F I,2 . To impose the periodic boundary conditions along the (I, N ) and (N , J ) boundaries, we mapped the point (I, N + 1) or (N + 1, J ) back inside the grid using the symmetries and used a bicubic interpolation to compute a best value for f . B.2. Implementation in symplectic coordinates. We represented h on a uniform square lattice in the variables x1 , x2 . Such a lattice has several advantages, aside from simplicity. First, according to (17), these points are also spread uniformly according to the measure of any symplectic metric. Second, the lattice is itself invariant under the symmetries of the hexagon (since these are elements of G L(2, Z)). In fact, in view of this symmetry group, the lattice should in a sense be considered triangular, with “edges” running not just horizontally and vertically, but also along the diagonals with slope −1. In other words, a given lattice point (x1 , x2 ) has six nearest neighbors: (x1 ± , x2 ), (x1 , x2 ± ), and (x1 ± , x2 ∓ ) (where is the lattice spacing). First and second derivatives were calculated using these nearest neighbor points in a way that was accurate to second order in the lattice spacing and respected the hexagon symmetries. Generally speaking, the lattice spacing was chosen to be one over an integer, so that the polytope boundaries passed through lattice points. In view of the free boundary conditions for h, on these boundaries the necessary derivatives were computed by extrapolation, using next-to-nearest neighbor points to give third order accuracy. B.3. Simulation of Ricci flow. The Ricci flow was simulated by an explicit method, with first-order accurate time derivatives and a time step of ∝ 2 . The constant of proportionality is of order one, but depends on the initial metric since the equation is non-linear. For the canonical choices of initial potentials in the complex coordinate case we required a time step 17 2 , while in the symplectic case we required 21 2 . However modifying the initial potential may require a smaller initial timestep. The diffusive nature of the flow requires the time step to be the square of the spatial lattice interval. The errors in the time derivatives are therefore of the same order as in the
Numerical Kähler-Einstein Metric on the Third del Pezzo hE 1, 1 0.2438
389 hE 0.5, 0.5
a)
0.05697
0.24382
b)
0.056975
0.24384 0.24386
0.05698
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0.24392
0.01
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0.02
0.03
0.04
c 0.592175
c)
0.59215 0.592125 0.592075
0.04
0.59205 0.592025
Fig. 14. (a) h E (1, −1) and (b) h E (0.5, 0.5) as functions of the number of lattice points, compared to a fit assuming second order scaling to the continuum, fitted using the highest two resolution points. Then (c) shows the value of c such that h(0, 0) vanishes, the extrapolated continuum value of which is 0.592016
spatial derivatives. In order to proceed to much higher resolutions implicit differencing, such as the Crank-Nicholson scheme should be used, or the time steps would become prohibitively small. However, for the resolutions we have used here, the explicit method is quite manageable.
B.4. Convergence tests. In both the complex and symplectic implementations various resolutions were used to compute the Kähler-Einstein metric, both to check convergence to the continuum and estimate error. Taking the example of the symplectic implementation, we uniformly covered the coordinate square 0 ≤ x1 ≤ 1, −1 ≤ x2 ≤ 0. Various lattice sizes were used, including 25 × 25, 50 × 50, 100 × 100, 200 × 200, 400 × 400. The calculations were performed on a desktop computer, with the lowest resolution 25 × 25 taking seconds to run, and the highest resolution 400 × 400 taking many hours. We estimate h E as h for sufficient flow time that the update of h in a time step is of order the machine precision. Since second order differencing was used to implement the flow locally, we expect that any quantity measured, say O, should scale to the continuum as Ocontinuum + 2 Ocorrection + O(3 ). Values of the relaxed function h at different coordinate locations were used to check this scaling, and indeed give this consistent second order scaling behaviour. In Fig. 14 we give an example, plotting the value of the boundary point h E (1, −1), and also the value of an interior point h E (0.5, −0.5), where we note that in the flow, h(0, 0) is fixed to zero by the appropriate choice of c, which is shown in Fig. 14c. Using the two highest resolution points we fit the second order scaling behaviour above, and see a very good fit to the lower resolution points, with h E (1, −1) = −0.2439 + 0.0772 and h E (0.5, −0.5) = −0.0570 + 0.0142 . These fits indicate the error in the value of h E at a point is about 10−6 for the highest resolution 400 ×400 grid calculated in the symplectic implementation.
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f 1,1 0.296175 0.29615 0.296125 0.296075 0.29605 0.296025
0.005 0.01 0.015 0.02 0.025 0.03
Fig. 15. A plot of the value of the Kähler potential at the center of the hexagon as a function of . The four data points correspond to our 40 × 40, 80 × 80, 160 × 160, and 250 × 250 grids. The linear fit was made with the two data points with smallest , giving 0.296008 + 0.18982 λ 6.32 6.322 6.324 6.326 6.328 6.332
0.01
0.02
0.03
0.04
6.334
Fig. 16. The eigenvalue as a function of number of lattice points for the lowest non-constant eigenfunction which transforms trivially under the hexagon and U (1) × U (1) symmetry. Again we see good agreement with a second order scaling fit function
Likewise in the complex coordinate implementation, various resolutions were computed, up to 250 × 250. The value of the potential at the center of the hexagon, (ξ = 1, η = 1) is plotted in Fig. 15 and we see that the convergence to the continuum value is quadratic in , again consistent with the second order spatial finite differencing. This and other such tests suggest our best 250 × 250 grid in the complex case is accurate to about a part in 105 . Note that in both the symplectic and complex coordinates, the value of 2 f + c at the center of the hexagon was found to be 0.592016. The diffusion flow used to study the eigenfunctions of the scalar Laplacian was also differenced to second order accuracy. The flow was simulated using the same explicit method, with time step 21 2 , and for the same resolutions as above. Using the lowest non-constant eigenfunction which transforms trivially under the action of the U (1)2 and D6 isometries, we plot in Fig. 16 the eigenvalue extracted for different resolutions, using the same initial data for the diffusion, and again fitting second order scaling to the two highest resolutions. We see consistent second order scaling behaviour. C. D6 -Invariant Polynomials In this section, we describe the set of polynomials bn,m x1n x2m P= n,m
invariant under the dihedral group acting on the hexagon.
(105)
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Our dihedral group is generated by two elements R1 and R2 (see (57)). If we take a linear combination of the xi , v · x, then Ri (v · x) = v · Ri · x = (Rit v) · x, i.e the Ri act on v by their transpose. Now, R2 has eigenvalues eπi/3 and e−πi/3 . A convenient basis of eigenvectors is v1 = (eπi/6 , e−πi/6 ), v2 = (e−πi/6 , eπi/6 ). The basis is convenient because a polynomial that is symmetric in a1 ≡ v1 · x and a2 ≡ v2 · x is symmetric in x1 and x2 . Consider the polynomial P= cn,m a1n a2m . (106) n,m
Symmetry under interchange of a1 and a2 requires cn,m = cm,n . Moreover, we have πi (n − m) a1n a2m , R2 (a1n a2m ) = exp (107) 3 which implies that n − m ≡ 0 mod 6. We conclude that the most general polynomial invariant under the group action can be decomposed into a sum of polynomials of the form a1n a2m + a1m a2n , where n − m ≡ 0 mod 6. For example, two important invariant polynomials are a1 a2 = x12 + x1 x2 + x22 ≡ U, 1 3 (a + a23 )2 = x12 x22 (x1 + x2 )2 ≡ V. 27 1 We now argue that any invariant polynomial can be decomposed into sums and products of U and V . Assume a1n a2m + a1m a2n is left invariant by R2 and assume m is the minimum of m and n. Then a1n a2m + a1m a2n = (a1 a2 )m (a1n−m + a2n−m ) = U m (a1n−m + a2n−m ). One can prove inductively that a1n−m +a2n−m can be written in terms of U and V whenever (n − m) is a multiple of 6. So U and V generate the polynomials invariant under R1 and R2 . D. Convergence and Error Estimates of the Constrained Optimization Method In this appendix we give convergence results for the smooth part of the symplectic potential, h(x1 , x2 ) determined in Sect. 6. At all points in the hexagon domain the value of h was observed to converge quickly with increasing numbers of terms taken in the series expansion at the origin. In Fig. 17 we plot the value of h at 3 points. Note that one of these points lies on the boundary of the hexagon. All other points checked, both in the hexagon interior and on the boundary, gave qualitatively similar convergence. In the figure we also compare the data with the extrapolated continuum results for the Ricci flow given in the previous appendix (suitably adjusting for the different value of c). We see excellent agreement. We may also estimate that the eighteenth order results and the estimated infinite order result differ at the 1 part in 106 level.
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h 0,0 0.2948 0.295 0.2952 0.2954 0.2956 0.2958 0.296 0.2962
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0.5396 0.5397 0.5398 0.5399 0.54
n
Fig. 17. (a) h(0, 0), (b) h(0.5, −0.5) and (c) h(1, −1) as functions of the number terms in the expansion. The dashed lines show the extrapolated continuum from the Ricci flow results
References 1. Yau, S.-T.: Calabi’s Conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. 74, 1798 (1977) Yau, S.-T.: On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equations. Comm. Pure App. Math. 31, 339–411 (1978) 2. Headrick, M., Wiseman, T.: Numerical Ricci-flat metrics on K3. Class. Quant. Grav. 22, 4931 (2005) 3. Donaldson, S.: Some numerical results in complex differential geometry. http://arXiv.org/list/math.DG/ 0512625, 2005 4. Douglas, M.R., Karp, R.L., Lukic, S., Reinbacher, R.: Numerical Calabi-Yau metrics. http://arXiv.org/ list/hep-th/0612075, 2006 5. Douglas, M.R., Karp, R.L., Lukic, S., Reinbacher, R.: Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic. http://arXiv.org/list/hep-th/0606261, 2006 6. Tian, G.: On Kähler-Einstein metrics on certain Kähler manifolds with c1 (M) > 0. Invent. Math. 89, 225–246 (1987) Tian, G., Yau, S.T.: On Kähler-Einstein metrics on complex surfaces with C1 > 0. Commun. Math. Phys. 112, 175–203 (1987) 7. Siu, Y.T.: The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group. Ann. of Math. (2) 127(3), 585–627 (1988) 8. Feng, B., Hanany, A., He, Y.H.: Phase structure of D-brane gauge theories and toric duality. JHEP 0108, 040 (2001) 9. Hanany, A., Iqbal, A.: Quiver theories from D6-branes via mirror symmetry. JHEP 0204, 009 (2002) 10. Beasley, C.E., Plesser, M.R.: Toric duality is Seiberg duality. JHEP 0112, 001 (2001) 11. Klebanov, I.R., Tseytlin, A.A.: Gravity duals of supersymmetric SU(N) x SU(N+M) gauge theories. Nucl. Phys. B 578, 123 (2000) 12. Klebanov, I.R., Strassler, M.J.: Supergravity and a confining gauge theory: Duality cascades and chiSBresolution of naked singularities. JHEP 0008, 052 (2000) 13. Abreu, M.: Kähler geometry of toric manifolds in symplectic coordinates. In: Symplectic and Contact Topology, Y. Eliashberg et al, eds. Fields Inst. Common., Providence, RI: Amer. Math. Soc., 2003, pp. 1–24 14. Tian, G., Zhu, X.: Convergence of Kähler-Ricci Flow. J. Amer. Math. Soc., 20, 675–699 (2006) 15. Hori, K., Kapustin, A.: Duality of the fermionic 2d black hole and N = 2 Liouville theory as mirror symmetry, JHEP 0108, 045 (2001) http://arXiv.org/list/hep-th/0104202, 2001
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16. Garfinkle, D., Isenberg, J.: Critical behavior in Ricci flow, http://arXiv.org/list/arXiv:math.DG/0306129, 2003; Garfinkle, D., Isenberg, J.: Numerical studies of the behavior of Ricci flow. In: Geometric evolution equations, Vol. 367 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2005, pp. 103–114 17. Headrick, M., Wiseman, T.: Ricci flow and black holes. Class. Quant. Grav. 23, 6683 (2006) 18. Headrick, M., Wiseman, T.: Numerical Kähler-Ricci soliton on the second del Pezzo. http://arXiv.org/ abs/0706.2329, 2007 19. Wang, X., Zhu, X.: Kähler-Ricci solitons on toric manifolds with positive first Chern Class. Adv. Math. 188(1), 87–103 (2004) 20. Baytrev, V., Selivanova, E.: Einstein-Kähler metrics on symmetric toric Fano manifolds. J. Reine. Agnew. Math. 512, 225–236 (1999) 21. www.stanford.edu/~headrick/dP3/ www.phy.princeton.edu/~cpherzog/dP3/ 22. Keller, J.: Ricci iterations on Kähler classes, http://arXiv.org/abs/0709.1490, 2007 23. Giddings, S.B., Kachru, S., Polchinski, J.: “Hierarchies from fluxes in string compactifications,” Phys. Rev. D 66, 106006 (2002) 24. Kachru, S., Kallosh, R., Linde, A., Trivedi, S.P.: De Sitter vacua in string theory. Phys. Rev. D 68, 046005 (2003) 25. Kachru, S., Kallosh, R., Linde, A., Maldacena, J.M., McAllister, L., Trivedi, S.P.: Towards inflation in string theory. JCAP 0310, 013 (2003) 26. Copeland, E.J., Myers, R.C., Polchinski, J.: Cosmic F- and D-strings. JHEP 0406, 013 (2004) 27. Altmann, K.: The versal deformation of an isolated toric Gorenstein singularity. http://arXiv.org/abs/ alg-geom/9403004, 1994 28. Berenstein, D., Herzog, C.P., Ouyang, P., Pinansky, S.: Supersymmetry breaking from a Calabi-Yau singularity. JHEP 0509, 084 (2005) 29. Franco, S., Hanany, A., Saad, F., Uranga, A.M.: Fractional branes and dynamical supersymmetry breaking. JHEP 0601, 011 (2006) 30. Bertolini, M., Bigazzi, F., Cotrone, A.L.: Supersymmetry breaking at the end of a cascade of Seiberg dualities. Phys. Rev. D 72, 061902 (2005) 31. Brini, A., Forcella, D.: Comments on the non-conformal gauge theories dual to Y(p,q) manifolds. JHEP 0606, 050 (2006) 32. Grana, M., Polchinski, J.: Supersymmetric three-form flux perturbations on AdS(5). Phys. Rev. D 63, 026001 (2001) 33. Matsushima, Y.: Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kaehlérienne. Nagoya Math. J. 11, 145–150 (1957) 34. Franco, S., He, Y.H., Herzog, C., Walcher, J.: Chaotic duality in string theory. Phys. Rev. D 70, 046006 (2004); Franco, S., He, Y.H., Herzog, C., Walcher, J.: Chaotic cascades for D-branes on singularities. In: String theory: From Gange. Interactions to Cosuology (Cargese, 7–19 June 2004), NATO Science Servies II. Math. Phys. and Chem., Berlin-Heidelberge-New York: Springer 2006 35. Herzog, C.P., Ejaz, Q.J., Klebanov, I.R.: Cascading RG flows from new Sasaki-Einstein manifolds. JHEP 0502, 009 (2005) 36. Herzog, C.P., Klebanov, I.R.: On string tensions in supersymmetric SU(M) gauge theory. Phys. Lett. B 526, 388 (2002) 37. Guillemin, V.: Kähler structures on toric varieties. J. Diff. Geom. 40, 285–309 (1994) 38. Fulton, W.: Introduction to Toric Varieties. Princeton NJ: Princeton University Press, 1993, p 92 Communicated by M.R. Douglas
Commun. Math. Phys. 282, 395–418 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0557-7
Communications in
Mathematical Physics
Quantisations of Piecewise Parabolic Maps on the Torus and their Quantum Limits Cheng-Hung Chang1,2 , Tyll Krüger3 , Roman Schubert4 , Serge Troubetzkoy5 1 Institute of Physics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC 2 Taiwan National Center for Theoretical Sciences, 101 Section 2 Kuang Fu Road,
Hsinchu 300, Taiwan, ROC
3 Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136,
D-10623 Berlin, Germany
4 School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK 5 Centre de physique théorique, Fédération de Recherches des Unités de Mathématiques
de Marseille, Institut de mathématiques de Luminy and Université de la Méditerranée, Luminy, Case 907, F-13288 Marseille Cedex 9, France Received: 20 April 2007 / Accepted: 29 January 2008 Published online: 27 June 2008 – © Springer-Verlag 2008
Abstract: For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so-called quantum limits, and one would like to understand which invariant measures can occur that way, thereby classifying the semiclassical behaviour of eigenfunctions. We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail. In particular we can construct examples of ergodic maps which have singular ergodic measures as quantum limits, and examples of non-ergodic maps where arbitrary convex combinations of absolutely continuous ergodic measures can occur as quantum limits. The maps we quantise are obtained by cutting and stacking. 1. Introduction The correspondence principle in quantum mechanics states that in the semiclassical limit → 0 classical mechanics emerges and governs quantum mechanical quantities for small de Broglie wavelength. One manifestation of this principle is that the Wignerfunctions of eigenfunctions converge weakly to invariant probability measures on phase space, the so-called quantum limits. It is one of the big open problems in the field to classify the set of quantum limits, and it is in general not known which invariant measures can occur as quantum limits. In particular the case that the classical system is ergodic has attracted a lot of attention. In this case the celebrated quantum ergodicity theorem, [Šni74,Zel87,CdV85], states that almost all eigenfunctions have the ergodic Liouville measure as quantum limit, and one would like to know if in fact all eigenfunctions converge to the Liouville measure, i.e., if quantum unique ergodicity holds or if there are exceptions. Possible candidates for exceptions would be quantum limits concentrated on periodic orbits, a phenomenon called strong scarring. Another
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very interesting case is when the classical system is of mixed type, i.e., the phase space has several invariant components of positive measure, or, if there exist several invariant measures which are continuous relative to Liouville measure. Here the question is to what extent the quantum mechanical system respects the splitting of the classical system into invariant components, i.e., is a typical quantum limit ergodic, or can every convex combination of invariant measures appear as a quantum limit. There has been recently considerable progress in some of these questions. For the quantized cat map it was shown that quantum unique ergodicity does not hold, in [FNDB03] a sequence of eigenfunctions was constructed whose quantum limit is a convex combination of the Liouville measure and an atomic measure supported on a periodic orbit. It was furthermore shown that the orbit can carry at most 1/2 of the total mass of the measure [BDB03,FN04]. The eigenvalues of the quantized cat map have large multiplicities and this behaviour depends on the choice of the basis of eigenfunctions, in [KR00] it was shown that for a so called Hecke basis of eigenfunctions quantum unique ergodicity actually holds. Cat maps on a higher dimensional torus were considered in [Kel07,Kel08], quantum unique ergodicity for Hecke basis was shown in [Kel08], and in [Kel07] the existence of quantum limits concentrated on certain submanifolds was derived, this latter result is even valid for a certain class of perturbed cat maps. Maps which are not hyperbolic provide as well very interesting examples. In [MR00] a uniquely ergodic map was studied, which we will describe below in more detail, and in [Ros06] quantizations of perturbed Kroneker maps which remain uniquely ergodic were investigated and bounds on their rate of quantum ergodicity obtained. Quantum maps provide as well model examples for non-ergodic maps, in [MO05] the localisation of eigenfunctions or quasimodes on invariant subsets in phase space was studied for a class of maps on the torus which were constructed from two twist maps. The more detailed knowledge on the classical dynamics in the case of maps allows to derive stronger results than in the case of quantized classical flows, [Sch01]. On compact Riemannian manifolds of negative curvature quantum unique ergodicity was conjectured in [RS94] and for arithmetic manifolds it was recently proved by Lindenstrauss for Hecke bases of eigenfunctions, [Lin06]. The non-arithmetic case is still open, but in [Ana06,AN06] the authors succeeded in proving lower bounds for the entropy of quantum limits on manifolds of negative curvature. Quantum limits concentrated on submanifolds have been described in [Kel08a]: In the case that the universal cover of a manifold is a product of hyperbolic planes the system has constants of motion, but one has classical and quantum ergodicity on certain submanifolds in phase space. In this paper we introduce a class of model systems for which the set of quantum limits can be determined very precisely. This work was motivated by a paper of Marklof and Rudnick where they gave an example of a quantum ergodic map which one can prove to be quantum uniquely ergodic [MR00].They mention that there are no examples known where a quantum ergodic map is not quantum uniquely ergodic. The purpose of our work was to provide such examples, in fact examples which are quite close in nature to those considered by Marklof and Rudnick. The map they considered was a skew product map of the torus T2 = R2 /Z2 of the form: p + 2q p mod 1, → F: f (q) q where f ( p) is an irrational rotation of the circle R/Z. In this article we consider skew products of the same form for other functions f ( p). We will prove results for any f ( p) which can be obtained by cutting and stacking. The cutting and stacking technique is an
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ergodic theoretic tool which gives a isomorphic representation of any aperiodic measure preserving transformation as a piecewise isometry on the interval (with at most countable number of pieces) with Lebesgue measure as an invariant measure. For any such F we give a new proof of the quantum ergodicity theorem, i.e., Lebesgue measure is a full density quantum limit. Furthermore we prove that certain nonatomic singular ergodic measures appear as quantum limits, but these quantum limits must have 0 density. We also have results about non-ergodic measures. Since a circle rotation is an interval exchange transformation on two intervals, the main example we will consider is when f ( p) is an interval exchange transformation on more intervals. These occur for instance in Poincaré maps of polygonal billiards, see [BS04] for the special properties of eigenfunctions in these systems and [GMO04] for a related quantized map and its eigenvalue statistics. An interval exchange transformation (IET for short) is an invertible map of the interval I := [0, 1) such that we can partition I into a finite set of intervals Ii := [ai , bi ) such that the map f restricted to each piece Ii is a translation. Lebesgue measure is always invariant for IETs but there are examples known of IETs which are not uniquely ergodic. A consequence of our result is that if f ( p) is an aperiodic IET then each invariant measure of F which is the product of Lebesgue measure with an invariant measure of f ( p) is a quantum limit. The plan of the paper is as follows. In Sect. 2 we give a quick review of quantisation of maps on the torus, and introduce the maps we study and their quantisation. In particular we prove Egorov’s theorem for these maps. In Sect. 3 we turn our attention to quantum limits; we first give a general proof of quantum ergodicity for maps with singularities, and then show that for our particular class of maps the quantum limits can be understood purely in terms of the orbits of discretisations of the classical map. Then, in Sect. 4, we finally come to our main result. We first review the cutting and stacking construction to obtain maps and then show how it can be combined with discretisations to get a detailed understanding of quantum limits. In Theorem 4 we summarise our main findings. Finally in the last two sections we discuss two examples and give some conclusions. 2. Quantisation We give a short summary of the quantisation of maps on the torus, for more details and background we refer to [DEG03,DB01]. The Hilbert space. For ( p, q) ∈ R2 we introduce the phase space translation operator i
ˆ pq) ˆ , T( p, q) := e− (q p−
where pψ(x) ˆ := i ψ (x) and qψ(x) ˆ := xψ(x) for ψ ∈ S(R), are the momentum and position operators, respectively. These operators are unitary on L 2 (R) and satisfy for ( p, q), ( p , q ) ∈ R2 , i
T( p + p , q + q ) = e− 2 (q p − pq ) T( p, q)T( p , q ), and they provide therefore a unitary irreducible representation of the Heisenberg group on L 2 (R). The state space of the classical map is obtained from R2 by identifying integer translates which gives the two torus T2 = R2 /Z2 .
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By mimicking this procedure the quantum mechanical state space is defined to be the space of distributions on R which satisfy T(1, 0)ψ = ψ,
T (0, 1)ψ = ψ.
One finds that these two conditions can only be fulfilled (for ψ = const.) if Planck’s constant meets the condition 1 = N, 2π where N is a positive integer. The allowed states then turn out to be distributions of the form 1 Q ψ(x) = √ (Q)δ x − N N Q∈Z
with (Q) a complex number satisfying (Q + N ) = (Q). So the (Q) are functions on Z N = Z/N Z and the space of these functions will be denoted by H N , it is N -dimensional and through the coefficients (Q), Q = 0, 1, . . . , N −1 it can be identified with C N . There is a map S N : S(R) → H N defined by S N ψ := (−1) N nm T(n, m)ψ n,m∈Z
which is onto. If we equip H N furthermore with the inner product ψ, φ N :=
1 ∗ (Q)(Q) N Q∈Z N
then H N is a Hilbert space and S N is an isometry. Observables. In classical mechanics observables on the torus are given by functions on T2 ; these can be expanded into Fourier series a= aˆ n e(−ω(z, n)), n∈Z2
where z = (q, p) ∈ T2 , ω(z, n) = qn 2 − pn 1 and aˆ n := T2 a(z)e(ω(z, n)) dz denotes the n th Fourier coefficient. We use here and in the following the notation e(x) = e2π ix 2π i and e N (x) := e N x . These observables can be quantised by replacing e(−ω(z, n)) by the translation operator n n 1 2 , T N (n) := T N N which acts on H N . This is called Weyl quantisation, to a classical observable a ∈ C ∞ (T2 ) a corresponding quantum observable is defined by OpN [a] := aˆ n T N (n), (1) n∈Z2
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which is an operator on H N . For example, if a depends only on q then the corresponding operator is just multiplication with a, OpN [a]ψ(q) = a(q)ψ(q),
(2)
and in terms of the coefficients (Q) the action of OpN [a] is given by (Q) → a (Q/N ) (Q). Let us recall a few well known properties of the operators defined by (1), see [DEG03, DB01]. They form an algebra and for a, b ∈ C ∞ (T2 ) one has OpN [a] OpN [b] = OpN [ab] + O(1/N ).
(3)
If a and b have disjoint support, then ab = 0 and one can obtain the stronger result OpN [a] OpN [b] = O(N −∞ ),
(4)
furthermore, if χ = 1 on a neighbourhood of the support of a then OpN [χ ] OpN [a] = OpN [a] + O(N −∞ ), we will use these two relations below. Since the T N (n) are unitary the norm of OpN [a] can be estimated from (1) by |a(n)| ˆ ≤C |∂ α a|, (5) || OpN [a]|| ≤ n∈Z2
|α|≤3
and the trace of a Weyl operator can as well be expressed in terms of the symbol 1 a dµT2 , lim Tr OpN [a] = N →∞ N T2
(6)
where µT2 is the normalized Lebesgue measure on T2 . Quantisation of a map. Let F : T2 → T2 be a volume preserving map. One calls a sequence of unitary operators U N : H N → H N , N ∈ N, a quantisation of the map F if the correspondence principle holds, i.e., if for sufficiently nice functions a one has U N∗ OpN [a]U N ∼ OpN [a ◦ F],
(7)
for N → ∞. If this relation holds, it is often called Egorov’s theorem and it means that in the semiclassical limit, i.e., for N → ∞, quantum evolution of the observable approaches the classical time evolution. Let us now turn to the specific class of maps we want to quantise. They are given by p + 2q p mod 1, (8) → F: f (q) q where f : [0, 1] → [0, 1] is a piecewise affine map given by a cutting and stacking construction which we will describe in detail in Sect. 4. For the construction of the quantisation we only need the property that the singularity set S ⊂ [0, 1] is nowhere dense. In order to quantise this map we proceed similar to the construction in [MR00], i.e., use a sequence of approximations to f . Consider the discretized interval D N := {Q/N ; Q ∈ {0, 1, 2, . . . , N − 1}},
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C.-H. Chang, T. Krüger, R. Schubert, S. Troubetzkoy
i.e., the support of the Hilbert space elements (2). For each N ∈ N we will call a map f N : D N → D N an approximation of f if it is close to f in a certain sense which we will now explain. Since f is not assumed to be continuous we do not approximate it uniformly in the supremum norm. Let f (S) := {q : ∃q0 ∈ S such that q = limq →q0 f (q )}. We measure the difference between f and an approximation f N only away from the set f (S). Let us call the relevant set Iε := {q ∈ [0, 1]; dist(q, f (S)) ≥ ε}. In the construction of f N in Sect. 4 we will choose a sequence ε N with lim N →∞ ε N = 0. For any fixed ε N the relevant measure for the quality of the approximation will be δ N := δ N (ε N ) :=
sup
Q/N ∈Iε N
| f N (Q/N ) − f (Q/N )|.
Any approximation f N then defines via (8) an approximation FN of F. The quantisation of F is now defined to be the sequence of unitary operators U N (Q) = e N −( fˆN−1 (Q))2 ( fˆN−1 (Q)),
(9)
(10)
where fˆN (Q) := N f N (Q/N ) denotes the map induced by f N on Z N = Z/N Z. This is indeed a unitary operator on H N , with its adjoint given by U N∗ (Q) = e N (Q 2 )( fˆN (Q)).
(11)
That this sequence of operators U N is really a quantisation of the map F is the content of the Egorov theorem (7) which we will now prove. In our case we have to be careful at the singularities of the map. The singularities of f and F can be naturally identified, thus without confusion we can denote by S the set of singularities of F as well. By CS∞ (T2 ) we denote the space of functions in C ∞ (T2 ) which vanish in a neighbourhood of F(S) (here F(S) is the set of all limit points of F(z) as z aproaches S). We then find Theorem 1. For any a ∈ CS∞ (T2 ) we have U N∗ OpN [a]U N = OpN [a ◦ FN ] + Oa (N −∞ ), and there are constants C(a), ε0 (a) > 0 such that for ε N < ε0 (a), ||U N∗ OpN [a]U N − OpN [a ◦ F]|| ≤ C(a) δ N . Proof. The map F and its quantisation U N can be decomposed into a product of two simpler maps and operators. Namely, with F (1) ( p, q) = ( p + 2q, q),
and F (2) ( p, q) = ( p, f (q))
we have F = F (2) ◦ F (1) . These maps can be quantised separately as (1)
U N (Q) := e N (−Q 2 )(Q),
(2)
and U N (Q) := ( fˆN−1 (Q)),
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where f N denotes a discretisation of f on the Heisenberg lattice. We then have (2)
(1)
UN = UN UN , and therefore it is sufficient to study the conjugation of an Weyl operator for the two (1) operators separately. In the case of U N it is well known that Egorov’s theorem is exactly fulfilled (1) ∗
UN
(1)
OpN [a]U N = OpN [a ◦ F (1) ],
see [MR00]. For the study of the second operator U N(2) we will use a partition of unity associated with the symbol a. Since a vanishes near the images of the singularities of f the support of a consists of a finite number of connected components, let us call this number Ja and let I j ⊂ [0, 1], j = 1, . . . , Ja , be the projections of these connected components to the q variable. For each j we choose a smooth function χ j (q) which is supported in a neighbourhood of I j which contains no singularities of f and satisfies f = 1 on a smaller a a neighbourhood of I j . If we define χ0 := 1 − Jj=1 χ j then we have 1 = Jj=0 χj. We observe now that for the Q ∈ {0, . . . , N } such that Q/N ∈ I j there is a t N , j ∈ Z such that fˆN (Q) = Q + t N , j mod N ; let us denote the extension of this map to all (2) Q ∈ Z/N Z by fˆN , j . Denote the corresponding unitary operator by U N , j , it is defined by U (2) (Q) := ( fˆ−1 (Q)) and it was shown in [MR00] that it satisfies N, j
N
(2) ∗
(2)
(2)
U N , j OpN [a]U N , j = OpN [a ◦ FN , j ],
(12)
(2)
where FN , j ( p, q) = ( p, fˆN (Q)/N ) for q = Q/N . We now have (2) (2) χ j U N (Q) = χ j (Q/N )( fˆN−1 (Q)) = χ j (Q/N )( fˆN−1 (Q)) = χ U j N , j (Q) ,j and so (2) UN
=
Ja
(2)
(2)
χ j U N , j + χ0 U N ,
j=1
which gives (2) ∗ (2) U N OpN [a]U N
=
Ja
(2) ∗
(2)
U N , j χ j OpN [a]χ j U N , j
j=1
⎛ ∗
+ U N(2) ⎝
⎞ χi OpN [a]χ j + χ0 OpN [a]χ0 ⎠ U N(2) .
i, j≥0,i= j
Since χi and χ j have disjoint support for i = j and i, j ≥ 1, and χ0 and a have disjoint support, too, we find using (4), and the remark afterwards, that the second term is Oa (N −∞ ). For the first term we observe that χ j OpN [a]χ j = OpN [χ 2j a] + Oa (N −∞ )
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since χ j is 1 in a neighbourhood of one connected component of the support of a and 0 on all other components, again by (4); this gives us with (12) Ja
(2) ∗
(2)
U N , j χ j OpN [a]χ j U N , j =
j=1
Ja
(2) ∗
(2)
U N , j OpN [χ 2j a]U N , j + Oa (N −∞ )
j=1
=
Ja
(2)
OpN [(χ 2j a) ◦ FN , j ] + Oa (N −∞ ).
j=1
But
Ja
2 j=1 (χ j a) ◦
FN(2), j = a ◦ FN(2) , and so (2) ∗
UN
(2)
(2)
OpN [a]U N = OpN [a ◦ FN ] + Oa (N −∞ ).
This proves the first part of the theorem. For the second part we have to estimate || OpN [a ◦ F] − OpN [a ◦ FN ]|| = || OpN [b N ]||, the images of the where b N := a ◦ F − a ◦ FN , and, since a is supported away from singularities of F, b N is a smooth function. By (5) || OpN [b N ]|| ≤ C |α|≤3 |∂ α b N | for some C which does not depend on b N , and on the support of a both F and FN have the same differential and their difference is δ N for N large enough, so we obtain || OpN [b N ]|| ≤ C
|∂ α a|δ N .
|α|≤k
So if we can choose our approximations f N in a way that ε N → 0 and δ N → 0 for N → ∞, then the sequence of unitary operators U N reproduces the classical map F in the semiclassical limit N → ∞, and so the correspondence principle holds. Definition 1. A sequence of operators U N for which δ N (ε N ) and ε N tend to 0 for N → ∞ will be called a proper quantisation of F. The restriction on the support of the classical observables is necessary in order that a ◦ FN and a ◦ F are smooth for N large enough. For a general a the composition a ◦ F is discontinuous which causes problems with the Weyl quantisation. Theorem 1 is not valid without the assumption on the singularities. This is shown by the following counter-example. Proposition 1. Let s ∈ S, a(q) ∈ C ∞ (T2 ) depend only on q with a(s) = 0 and let √ 2 gs (q) := N e−(q−s) /N be a Gaussian centred at s and ψs := S N gs be its projection to H N . Then there exists a constant C such that
lim inf || U N∗ OpN [a]U N − OpN [a ◦ F] ψs || ≥ C|a(s)|||ψs ||. N →∞
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Proof. We first observe that since a depends only on q, OpN [a] is just multiplication with a and it follows then from the definition of U N , (10) and (11), that U N∗ OpN [a]U N is given by multiplication with a ◦ f N . And since by (8) a ◦ F = a ◦ f if a depends only on q, we find that the operator U N∗ OpN [a]U N − OpN [a ◦ F] is given by multiplication with a ◦ f N − a ◦ f , and so N
1 |(a ◦ f N −a ◦ f )(n/N )|2 |s (n/N )|2 , || U N∗ OpN [a]U N −OpN [a ◦ F] ψs ||2 = N n=1
where s (q) = m∈Z gs (q − m). Because f is discontinuous at s there exists an ε > 0, a C > 0 and a N0 such that |(a ◦ f N − a ◦ f )(q)| ≥ C|a(s)|, for q ∈ [s − ε, s + ε] and N ≥ N0 . So if we split N 1 |(a ◦ f N − a ◦ f )(n/N )|2 |s (n/N )|2 N n=1 1 |(a ◦ f N − a ◦ f )(n/N )|2 |s (n/N )|2 = N |n/N −s|≤ε
1 + N
|(a ◦ f N − a ◦ f )(n/N )|2 |s (n/N )|2 ,
|n/N −s|>ε
the first term can be estimated from below as 1 N
|(a ◦ f N −a ◦ f )(n/N )|2 |s (n/N )|2 ≥ C 2 |a(s)|2
|n/N −s|≤ε
1 N
|s (n/N )|2 .
|n/N −s|≤ε
Now s (q) is strongly localised near s; in particular we have |s (n/N )|2 ≤ Ce−ε for |n/N − s| ≥ ε, and this implies 1 N
|s (n/N )|2 =
|n/N −s|≤ε
N 1 |s (n/N )|2 + O(e−c/N ) = ||ψs ||2 + O(e−c/N ) N n=1
and 1 N
|(a ◦ f N − a ◦ f )(n/N )|2 |s (n/N )|2 = O(e−c/N )
|n/N −s|>ε
for any c < ε2 . Putting these estimates together yields then finally N 1 |(a ◦ f N − a ◦ f )(n/N )|2 |s (n/N )|2 ≥ C 2 |a(s)|2 ||ψs ||2 + O(e−c/N ). N n=1
2 /N
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C.-H. Chang, T. Krüger, R. Schubert, S. Troubetzkoy
We want to close this section with some comments about the underlying motivation for the specific quantisation assumptions on the neighbourhood of the singularities. Classically the singularities act like points with infinite local expansion rate respectively Lyapunov exponent. Therefore any perturbation in a small neighbourhood of the singularity set gives rise to an error which becomes unbounded if the perturbation approaches the singularity set. Since the quantised maps are a specific kind of perturbation it is natural to leave the allowed error big for points close to the singularity set. 3. Quantum Limits and Orbits We will now discuss the implications of Theorem 1 for the eigenfunctions of the quantised map. We will denote a orthonormal basis of eigenfunctions of U N by ψkN , k = 1, . . . , N , U N ψkN = e N (θkN )ψkN , 2π i
where we use the notation e N (x) = e N x and θkN are the eigenphases. Each eigenfunction defines a linear map on the algebra of observables OpN [a] → ψkN , OpN [a]ψkN and the leading term for N → ∞ depends only on the principal symbol σ (a). The limit points of the sequence of all these maps defined by the eigenfunctions define measures on the set of classical observables and are called quantum limits (see, e.g., [MR00]). To put it more explicitly, a measure ν on T2 is called a quantum limit of the system defined N by the U N if there exist a sequence of eigenfunctions {ψk j j } j∈N such that N N a(z) dν = lim ψk j j , OpN [a]ψk j j . j→∞
T2
One of the major goals in quantum chaos, and more generally in semiclassical analysis, is to determine all quantum limits that can occur and the relative density of the corresponding subsequences of eigenfunctions. For a subsequence of eigenfunctions N F = {ψk j j } j∈N we denote by F j := {ψkNi i ; ψkNi i ∈ H N j } the subset of eigenfunctions lying in H N j . (Notice that since j labels both the Hilbert spaces via N j and particular elements in them, via k j , we will often have N j = Ni for j N
and i different.) We say now that a subsequence of eigenfunctions F = {ψk j j } j∈N has density α(F) ∈ [0, 1] if lim
j→∞
|F j | = α(F), Nj
(13)
provided that the limit exists. Egorov’s theorem usually implies that all quantum limits are invariant measures for the classical map. In our case the same is true, but we have to be careful at the singularities. If the set of singularities S is nowhere dense then the space CS∞ (T2 ) in Theorem 1 is large enough so that as an immediate consequence we have: Corollary 1. Let us denote by Minv (F) the convex set of F-invariant probability measures on T2 , and by Mqlim (U N ) the set of quantum limits µ of U N with µ(S) = 0, then
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Mqlim (U N ) ⊂ Minv (F) So we only have to look at invariant measures as candidates for quantum limits. In the simplest case that there is only one invariant probability measure, i.e., that the system is uniquely ergodic, all eigenfunctions must converge to this measure, and we have the so called unique quantum ergodicity. This was the situation in the example of Marklof and Rudnick, [MR00]. We will study now the relationship between properties of quantum limits and the density of subsequences of eigenfunctions converging to them more closely. Our first result gives an upper bound on the density. Theorem 2. Let U N be a proper quantisation of F and let µ be a quantum limit of U N with support ⊂ T2 . Then any sequence of eigenfunctions which converge to µ has at most density µT2 (), where µT2 is the Lebesgue measure on T2 . Proof. Let aε ∈ C ∞ (T2 ), ε ∈ (0, 1], be a sequence satisfying aε | = 1 and limε→0 aε (z) = 0 for all z ∈ T2 \, i.e., a sequence approximating the characteristic function N of . If F = {ψk j j } j∈N is a sequence of eigenfunctions with µ as quantum limit then N
N
lim ψk j j , OpN [aε ]∗ OpN [aε ]ψk j j = 1,
j→∞
ψ∈F j ψ, OpN [aε ]
and therefore (13)
lim
j→∞
∗ Op [a ]ψ N ε
1 ψ, OpN [aε ]∗ OpN [aε ]ψ = α(F). Nj ψ∈F j
ψ, OpN [aε ]∗ OpN [aε ]ψ
But since of U N j we have
will approach |F j | for large j, so by
∗
≥ 0 and F j is a subset of a basis of eigenfunctions
ψ, OpN [aε ] OpN [aε ]ψ ≤
ψ∈F j
Nj
N
N
ψk j , OpN [aε ]∗ OpN [aε ]ψk j
k=1
and by (6) and (3) Nj 1 Nj Nj ∗ |aε |2 dµT2 , lim ψk , OpN [aε ] OpN [aε ]ψk = 2 j→∞ N j T k=1 2 so α(F) ≤ T2 |aε | dµT2 . We now take the limit ε → 0 and use that T2 |aε |2 dµT2 → µT2 (), then α(F) ≤ µT2 ().
Since µT2 (S) = 0 it follows in particular that a possible sequence of eigenfunctions converging to a quantum limit concentrated on S must have density 0. This result is as well interesting for non-ergodic maps, because it gives an upper bound on the number of eigenfunctions whose quantum limits are supported on an invariant subset of T2 by the volume of . In case the system is ergodic, we can actually determine the quantum limit of most (N ) eigenfunctions. Let us say that a subsequence F = {ψk j j } has full density one if (N j )
lim N →∞
|{ψk
j
∈H N }|
N
= 1.
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C.-H. Chang, T. Krüger, R. Schubert, S. Troubetzkoy
Theorem 3. Let U N be a proper quantisation of F and assume that µT2 is ergodic. Then there exists a subsequence of eigenfunctions of full density one which converges to µT2 . This is the usual quantum ergodicity result, but our proof differs from the standard one (see e.g. [DEG03]) in that we rely on the convexity definition of ergodicity; this is more convenient when dealing with maps with singularities as has been observed in [GL93]. Recall that µT2 is ergodic if it is extremal in the convex set of invariant probability measures, i.e., if µT2 = αµ1 + (1 − α)µ2 with µ2 = µT2 then α = 1 and µ1 = µT2 . N
Proof. The existence of a subsequence F = {ψk j j } j∈N of full density one of eigenfunctions with quantum limit µT2 is equivalent to lim
N →∞
where a =
N 1 |ψkN , OpN [a]ψkN − a| = 0, N k=1
adµT2 , see [DEG03].
N
We will first show that by ergodicity every subsequence F = {ψk j j } j∈N of positive density satisfies 1 lim ψ, OpN [a]ψ = a. (14) j→∞ |F j | ψ∈F j
To see this we consider the sequence a j :=
1 ψ, OpN [a]ψ, |F j | ψ∈F j
this is a bounded sequence since OpN [a] is bounded, and therefore there exists a convergent subsequence {ai }i∈S , where i runs through a subset S of N. Now using (6) we have with a convergent subsequence Ni 1 a = lim ψkNi , OpN [a]ψkNi i∈S, i→∞ Ni k=1 |Fi | 1 = lim ψ, OpN [a]ψ i∈S,i→∞ Ni |Fi | ψ∈Fi
+
lim
Ni − |Fi | 1 Ni Ni − |Fi |
i∈S,i→∞
= α(F)
a dµ1 + (1 − α(F))
(15) ψkNi , OpN [a]ψkNi
N
ψk i ∈H Ni \Fi
a dµ2
where µ1 and µ2 are invariant measures defined by 1 lim ψ, OpN [a]ψ = a dµ1 i∈S,i→∞ |Fi | lim
i∈S,i→∞
1 Ni − |Fi |
N ψk i ∈H Ni \F Ni
ψ∈Fi
ψkNi , OpN [a]ψkNi =
a dµ2 .
(16) (17)
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These two measures exist by the assumption that the subsequence {ai }i∈S was convergent, and they are invariant by Theorem 1. But Eq. (15) can be rewritten as µ = αµ1 + (1 − α)µ2 and if µ is ergodic and α = 0 this is only possible if µ1 = µ, and this proves that lim a N j = a.
j→∞
Since this holds for every convergent subsequence of {a j } j∈N a is the only limit point and (14) follows. Now assume that lim
N →∞
N 1 |ψkN , OpN [a]ψkN − a| = C > 0, N k=1
then there must either exist a subsequence {k j } j∈N of positive density with N
N
ψk j j , OpN [a]ψk j j − a ≥ C/2 or one with N
N
ψk j j , OpN [a]ψk j j − a ≤ −C/2. N
N
But the mean value of the sequence ψk j j , OpN [a]ψk j j − a must tend to 0 by (14) and so we have a contradiction if C = 0. The previous results, Corollary 1, Theorem 2 and Theorem 3, are quite general, they are valid for all quantised maps which satisfy Egorov’s theorem. We now turn to a more concrete study of the eigenfunctions for the specific quantum maps (10). Our aim is to show that the quantum limits are determined by the spatial distribution of the periodic orbits of the discretisation of the classical map. The eigenvalue equation U N ψ = e N (θ )ψ leads to the following recursion equation for ψ; ( fˆN (Q)) = e N (θ − Q 2 )(Q).
(18)
From this recursion relation we obtain |( fˆN (Q))|2 = |(Q)|2 , and this implies that the probability densities in position space defined by the eigenfunctions are invariant under the map f N . In order to determine these densities it is therefore sufficient to determine the spatial distribution of the orbits of f N . For further investigation we note that each periodic orbit of f N carries at least one eigenfunction. And we can determine the eigenfunctions and eigenvalues more explicitly, let O be a periodic orbit of period |O| = K , then the recursion relation (18) gives K −1 2 k (Q) = e N K θ − fˆ (Q) (Q). N
k=0
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C.-H. Chang, T. Krüger, R. Schubert, S. Troubetzkoy
So if ψ should be an eigenfunction with eigenvalue e N (θ ) we get the condition Kθ −
K −1
fˆNk (Q)
2
= Nm
k=0
with m ∈ {0, 1, . . . , K − 1}. This determines the eigenvalues, and then the corresponding eigenfunctions follow from the recursion relation and the normalisation condition. Summarising we get: Proposition 2. Let O be an orbit of period K = |O| of f N , then there exists K orthogonal eigenfunctions of U N with support O. The eigenphases are given by N k K
θk = SO + with k ∈ {0, 1, . . . , K − 1} and SO :=
K −1 2 1 ˆk f N (Q) , K k =0
and a normalised eigenfunction corresponding to θk is given by ⎛ ⎞ 1/2 k N k m 2 ˆ ˆ ⎝ ⎠ [ f N (Q 0 )] , k ( f N (Q 0 )) = e N k θk − K m=0
where Q 0 ∈ O is an arbitrary point on the orbit and k ∈ {0, 1, . . . , K − 1}. The quantum lattice D N of N points is a disjoint union of all periodic orbits of f N , and on each of these orbits are as many eigenfunctions concentrated as the orbit is long. But that means that the orbits determine the quantum limits and the relative density of the corresponding sequence of eigenfunctions. To each periodic orbit O we can associate a probability measure on [0, 1] δO (q) :=
1 Q δ q− |O| N
(19)
Q∈O
which is invariant under f N . ) (N ) Corollary 2. Let O(N be the j , j = 1, . . . , J N be the periodic orbits of f N and let δ j corresponding probability measures (19). Assume that there is an invariant measure ν k) of f and a sequence of periodic orbits {O(N jk }k∈N such that (N ) lim δ k k→∞ jk
then ν is a quantum limit of U N .
= ν,
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409
In all our examples the sequence (Nk ) contains all natural numbers. Thus there are k) two possible definitions of the density of a sequence of periodic orbits, G = {O(N jk }k∈N , (N )
α(G) = lim
N →∞
|O j N | N
or
β(G) = lim
N →∞
1 N
k:Nk =N
(N )
|O jk k |,
whenever the limit exists, which we will call the α-density or β-density of G, respectively. This corollary suggests that the set of quantum limits coincides with the set of limits (N ) points of the sequence of orbit measures δ j and that moreover the relative densities of the convergent subsequences coincide too. This is true if there are no multiplicities in the eigenvalues. If there are eigenvalues of multiplicity larger than one, then the eigenspace can mix the contribution of the different orbits. But even in this case there always exists (N ) a choice of a basis of eigenfunctions corresponding to the orbit measures δ j . Notice that in this case the β-density of the sequence of orbits coincides with the density of the corresponding sequence of eigenfunctions defined in (13).
4. Cutting and Stacking Constructions Cutting and stacking is a popular method in ergodic theory to construct maps on the interval which are isomorphic models of arbitrary measure preserving dynamical systems. The construction gives a piecewise isometric mapping on the interval with Lebesgue measure as an invariant measure. One can also think of this transformation as a countable interval exchange transformation. The final mapping will be defined only Lebesgue almost everywhere. None the less we can use this model to study certain other invariant measures which are well behaved with respect to the cutting and stacking construction. For a very readable introduction into cutting and stacking see the recent book by Shields [Shi96] or survey article by Friedman [Fri92]. We will now give a short description of the basic construction scheme and the relevant definitions. A stack (or column) S is a finite family of enumerated disjoint intervals {I j }h(S) j=1 , where h(S) is called the height of S. The I j are subintervals of [0, 1] of equal length which is called the width of S. There are two possible conventions: either one can take all the intervals to be open, or all the intervals to be closed on the left and open on the right. Which convention we choose is not important for this paper. The intervals I1 and Ih(s) are called the bottom and top of S respectively. We define a transformation f S as follows, if the point x ∈ I j is not on the top of S and not on the boundary of I j then it gets mapped to the point directly above it (see Fig. 1(a)). Since I j+1 and I j have equal width, f S is simply the canonical identification map between I j and I j+1 . Interpreted in [0, 1] this means that f S : I j → I j+1 such that x → x + ∂ − I j+1 − ∂ − I j , where ∂ − denotes the left boundary point of an interval. The construction clearly defines f −1 on all I j except at the bottom. h(S )
A stack family S is a finite or countable set of stacks {Si } = {{I ij } j=1i } such that all I ij are disjoint and ∪I ij = [0, 1]. In this paper we will work only with finite stack families. On S one defines a transformation f S by f S | Si = f Si except on the collection of top intervals.
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I 3k cut
fS
Ik
1
2
Ik
(a) a stack
f Sí
cut
3
Ik
Ik
(b) cutting a stack
I 1k
I 2k
(c) stacking the third stack on top of the second stack
Fig. 1. The cutting and stacking construction. In a given stack (a) the mapping f S is defined, except at the top interval. In (b) the stack is cut into three substacks, and in (c) the third substack is stacked onto the second one. This gives an extension f S of the map f S which was not defined on the top of substack two before
Given two stacks Si and S j , i = j of equal width one can define a new stack S by stacking S j on Si that is h(S )+h(S j )
S = {Ik }k=1i Ik
=
Iki
,
for k ≤ h(Si ) and Ik = Ik−h(Si ) for k > h(Si ). j
Correspondingly one gets a new transformation f S which agrees with f Si and f S j i except on Ih(S , where f Si was not defined before. i) It remains to define the cutting of stacks. A cutting of a stack S = {Ik } is a splitting of S into two (or more) disjoint stacks S1 and S2 with intervals {Ik1 } and {Ik2 } such that Ik1 ∪ Ik2 = Ik and ∂ − Ik1 < ∂ + Ik1 = ∂ − Ik2 < ∂ + Ik2 ∀k that is Ik1 is always the left component of the partition of Ik into Ik1 and Ik2 (Fig. 1(b)). The definition of f {S1 ,S2 } is as above. Multiple cutting of S is defined analogously. A stack family S(n) is obtained from a stack family S(n −1) by cutting and stacking, if each Si (n) from S(n) can be obtained by successive cuttings and stackings of stacks from S(n − 1). By construction f S (n) is an extension of f S (n−1) . If one has a sequence {S(n)}n≥1 of stack families such that each S(n) is obtained from S(n − 1) by cutting and stacking and furthermore lim width(Sk (n)) = 0, (20) n→∞
Sk (n)∈S (n)
then lim f S (n) = f is an invertible transformation on [0, 1] defined everywhere except at a set of zero Lebesgue measure. Note that f is always aperiodic. The “partition” of [0, 1] into the intervals of S(1), 1 the starting object of the construction, gives a natural symbolic dynamics for f . The coding is unique for all points whose infinite orbit is defined. 1 Here we can ignore the boundary points of the interval since f is not defined on them.
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1
1/2
1/4
1/8
1/2
3/4
7/8
1
Fig. 2. The von Neumann-Kakutani transformation
For convenience, we denote the intervals of S(n) by I ij (n), where i indexes the stack and j the interval in the stack. We consider the set i (n) ∪ ∪i,n ∂ I1i (n) . S = ∪i,n ∂ Ih(S i) The set S = S ( f ) is called the singularity set of the map f . It consists of all the points of discontinuity of f and all the points where the map f or f −1 is not defined. The boundary points of the intervals I ij which are not top or bottom intervals are not included in this set, the map is defined and continuous on such points! Furthermore ∪i, j,n ∂ I ij (n) is included in the set ∪k∈Z f k S, where f k S := {x : f −k x ∈ S}. We give an simple example of a map f constructed by cutting and stacking due J. von Neumann and S. Kakutani (unpublished, see [Fri92]). In this example there is exactly one stack at each stage in the construction. The first stack S1 is of height two, the bottom interval is I1 = [0, 1/2) and the top is I2 := [1/2, 1). Suppose the n − 1st stack Sn−1 has been defined. Now inductively define Sn by cutting Sn−1 in half and stack the right half above the left half to form Sn . The stack Sn is of height 2n with base [0, 2−n ) and top [1 − 2−n , 1). The map of the interval defined is shown in Fig. 2. Historically the cutting and stacking construction was invented to represent the dynamics with respect to a single invariant measure as a countable interval exchange transformation with the canonical invariant Lebesgue measure. The construction is universal in the sense that for every measurable dynamical system (M, g, µ) one can explicitly give a cutting and stacking representation ([0, 1] , f (g) , µ L ) [Shi96]. The following proposition which will be needed for the application of Theorem 3 seemed to be unknown.
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Proposition 3. Let (, σ ) be a symbolic dynamical system over a finite or countable alphabet and let µ be a shift-invariant measure such that µ ([w]) > 0 for all cylindersets corresponding to finite words w appearing in . Then the associated cutting and stacking construction yields a representation ([0, 1] , f, µ L ) of (, σ, µ) such that all nonatomic invariant measures ν ∗ on (, σ ) have a corresponding isomorphic invariant measure ν on ([0, 1] , f ) with the property ν (S ( f )) = 0. Proof. By definition all finite symbolic words have a representation as orbit segments of the cutting and stacking construction. Furthermore to each nonperiodic symbol sequence corresponds a unique point in [0, 1] whose orbit under f is well defined and commutes with the shift. By the Poincaré recurrence theorem every f − invariant measure is supported on recurrent points. Since the singular orbits, respectively symbolic sequences, are the ones which eventually or asymptotically fall onto the singularity set they do not intersect with the recurrent nonperiodic symbol sequences. Therefore all invariant measures- except the finitely supported ones- give zero measure to the singularity set. 4.1. The approximating family. Each f defined by cutting and stacking provides us with a natural approximation family { f S (n) } which we will use now to define the approQ ximation mappings on the rational points D N = { N : Q ∈ {0, . . . , N − 1}} for the i quantisation. Let the points in G i, j (n, N ) := D N ∩ I j (n) be enumerated from left to right and let K (N , I ij (n)) := G i, j (n, N ) and K (N , Si (n)) := min{K (N , I ij (n))}. j
K (N , Si (n)) is just the smallest number of points from the discretisation in an interval in the stack Si (n). Let Gˆ i, j (n, N ) be the set of the first K (N , Si (n)) points from j,i K (N ,S (n)) G i, j (n, N ) denoted by {xe }e=1 i . We define f N ,n first on Dˆ N ,n = ∪i, j Gˆ i, j (n, N ) by setting j,i
j+1,i
f N ,n xe = xe
for j < h i (n) := h (Si (n)) . (21)
j,i j,i We call these the internal orbit segments. Clearly f N ,n xe − f xe = O N1 . We call each approximation mapping f N ,n on D N whose restriction to Dˆ N ,n is given by the above construction an ergodic approximation. i (n). Let Dˇ N ,n be the set of points not in Dˆ N ,n and not in any of the top intervals Ih(S i) For x ∈ Dˇ N ,n let f N ,n x be the closest point to f x. Note that f N ,n is not necessarily
an invertible map, thus the construction implies that max | f N ,n x − f x| = O N1 . It is x∈ Dˇ N ,n
clear that for fixed n we have Dˆ N ,n = 1. N →∞ D N lim
(22)
To complete the definition of f N ,n , it remains to define the mapping of the points Gˆ i,h i (n) (n, N ) on the tops of the stacks to points Gˆ i,1 (n, N ) on the bottoms of the stacks. This will be done in a way to produce periodic orbits which approximately mimic
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413 µ 2 typical orbit segment
µ 1 typical orbit segment
*
µ typical orbit segment
x5 x4 x3 x2 x1 f(x) fN(x) x0
Fig. 3. The approximate mapping f N on the discrete set D N (denoted by the full dots), and the way the orbit segments from different stacks are concatenated in order to produce a quantum limit of the form α1 µ1 + α2 µ2 . A singular limit µ∗ is obtained by a stack of small width and where the orbits are concatenated from top to the bottom
a given invariant measure. Furthermore it remains to link N to a given n to get a good approximation. In essence we have to require that for each fixed stage n construction we have enough discretisation points in each stack. That means that with the increase of N we pass from n to n+1 only as we pass a critical threshold value Nn . This can be done already without a precise description of the gluing between the top and bottom of the stacks. We need that the approximation family f N is good enough to apply Theorem 1 for a sequence ε N → 0 such that δ N (ε N ) → 0. Note that Theorem 1 does not impose any requirements on the rate of convergence. The basic idea is to keep N large enough compared with n such that all intervals I ij (n) of S(n) contain sufficiently many points from D N . Let bn = min width(Si (n)). Choose a function n(N ) going to infinity such that i
lim min K (N , Si (n(N ))) = ∞,
N →∞
which is equivalent to require
(23)
i
1 bn
= o(N ). Furthermore let ε N = max width(Si (n)). i
Equation (20) implies that ε N tends to 0 since n(N ) tends to infinity. With this choice of ε N the points where f N ,n(N ) is not yet defined do not contribute to δ N (ε N ), hence one obtains δ N (ε N ) = O( N1 ). An approximation family f N := f N ,n(N ) for any function n(N ) satisfying the above requirements is called proper. We say that a measure µ appears as a quantum limit if one can find a proper approximating family f N and associated quantisation U N such that µ is a quantum limit of U N . The notion of quantum limit as well as the notion of density were introduced in Corollary 2. Theorem 4. a) If µ is an absolute continuous ergodic measure for f , then µ appears as a positive α and β− density quantum limit. Furthermore the quantum limit has full α and β− density if µ is the Lebesgue measure. b) If µ is a nonatomic, singular ergodic measure for f then µ appears as a quantum limit. Furthermore the quantum limit must have zero α and β− density. c) If µ1 and µ2 are two absolute continuous ergodic invariant measures, then α1 µ1 + α2 µ2 appears as a positive α and β− density quantum limit for any α1 , α2 ∈ (0, 1) with α1 + α2 = 1.
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d) If µ1 and µ2 are two ergodic measures at least one of which is singular, then α1 µ1 + α2 µ2 appears as a quantum limit for any α1 , α2 ∈ (0, 1) with α1 + α2 = 1. Furthermore the quantum limit must have zero α and β− density. A transformation f : [0, 1] → [0, 1] is called a finite rank transformation if one can construct it via cutting and stacking such that the number of stacks #{Si (n)} in the n th stack family S(n) is bounded (independent of n). Corollary 3. If f is a finite rank transformation, then every ergodic non atomic invariant measure appears as the quantum limit of any proper ergodic approximation family. Proof. For a finite rank transformation, the singularity set S is a countable set which has only a finite number of points of density. Thus any non-atomic invariant measure can not be supported on S. 1 A point x is called µ-typical if limm,u→∞ m+u+1
m
δ f i x = µ. For the proof
i=−u
of the theorem we need the following simple fact whose proof is omitted since it is immediate from the definition of weak convergence of measures. Proposition 4. Fix xµ-typical. Let j (m, u) and ε( j) be functions such that j (m, u) → ( j) ∞ for m, u → ∞ and ε( j) → 0 as j → ∞. Let {yk }k∈Z be a family of sequences with m ( j) 1 the property that yk − f k x ≤ ε( j) for −u ≤ k ≤ m. Then limm,u→∞ m+u+1 k=−u ( j (m,u)) = µ. δ yk Proof of Theorem 3. For the proof of part b) and part a) for α− density we complete the definition of f N as follows. Let us complete each internal orbit segment into a periodic h (n),i orbit by setting f N xe i = xe1,i (compare (21)). For points x ∈ D N \ S where the map is not yet defined we have the freedom to map x anywhere; for preciseness define f N x to be the closest point to f x. By Corollary 2 it is enough to show that there is a sequence of periodic orbits on D N whose point-mass average converge for N → ∞ to the considered measure µ. (N ) Fix an arbitrary enumeration O j of the periodic orbits on D N . For µ absolute m
1 continuous the set of points x ∈ [0, 1] with limm,u→∞ m+u+1 δ f i x = µ has i=−u
positive Lebesgue measure and in the case µ is the Lebesgue measure it has full measure. Let x be µ-typical and consider for each N the stack Si (n(N )) in which x is placed, (N ) where n(N ) is a function satisfying the requirements of Eq. (23). Let O jl be the set of periodic orbits in ∪ j Gˆ i, j (n(N ), N ). They stay width(Si (n(N )))-close to the orbit segment { f −m 0 x, . . . , f n 0 x}, where m 0 and n 0 are the smallest and largest iterates such that f Ski (n(N )) x is still defined. Note that m 0 + n 0 + 1 = h (Si (n(N ))). Since width(Si (n(N ))) ≤ ε N := max width(Si (n(N ))) → 0 for N → ∞ i
(N )
we can apply the above proposition to the family O jl (with fixed l) to conclude that µ is a quantum limit, however this construction has not yet proved the positive density.
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To prove the positive density we need a quantified version of the above. Let Jq be the set of subintervals of [0, 1] with boundary points of the form qp . For the convergence of a sequence to a measure it is clearly enough to check the characteristic function averages with respect to the elements of ∪q Jq . A stack Si (n(N )) is called ε − q−good with respect to µ if for all x ∈ I1i (n(N )) (i.e. x in the base of the stack Si (n(N ))), µ(J ) − ε ≤
h i (n) 1 1 J f |iSi (n(N )) (x) ≤ µ(J ) + ε f or ∀J ∈ Jq , h i (n(N )) i=0
where h i (n(N )) denotes the height of the stack Si (n(N )). Denote the family of such stacks by G(n, q, ε, µ) and by G(n, q, ε, µ) the set of points contained in G(n, q, ε, µ). Clearly one has ∀q, ε > 0, lim µ L (G(n, q, ε, µ)) = µ L (x : x is µ − typical) and
n→∞
lim µ(G(n, q, ε, µ)) = 1.
n→∞
Let ε(n) be a sufficiently slowly decreasing function and q(ε(n)) be a sufficiently slowly increasing function that lim µ L (G(n, q(n), ε(n), µ)) = µ L (x : x is µ − typical) .
n→∞
(24)
With this new notation we are ready to prove that µ is a quantum limit of positive (N ) density. We define the sequence O jl of periodic orbits which give rise to the desired quantum limit as follows. For fixed N the set G(n(N ), q(n(N )), ε(n(N )), µ) ∩ Dˆ N ,n(N ) (N )
consists of a collection of points of periodic orbits. The sequence O jl consists of the set of these orbits. The positive β− density and full density in the case of Lebesgue measure then follows from Eqs. (24) and (22). To prove part a) for the α− density we only need to modify the map f N on top of the stacks. This will be done such that the collection (N ) of periodic orbits O jl becomes just one periodic orbit for each N . This completes the proof of part a). To prove part b) observe that due to Proposition 3 we have µ (S) = 0 and hence we can apply Theorem 1 to get an invariant measure out of the quantum-limit. One considers the set of µ -typical points. From the proof of Proposition 3 it follows that the orbit of every µ-typical point does not intersect nor converge to the singularity set S. Since the set of µ−typical points has zero Lebesgue measure we can obtain only a zero density quantum limit just as in the proof of part a). To prove part c) one has to modify the construction of the approximating mapping f N in the following way. Instead of making f N periodic within each stack Si (n(N )) we want to connect two stacks say Si (n(N )) and S j (n(N )), where the orbit segments in the i th stack, respectively j th stack, are approximately typical for µ1 , respectively µ2 , to get an average of µ1 and µ2 . For l = 1, 2 let Al (N ) := {i : Si (n(N )) ∈ G(n (N ) , q(n), ε(n), µl ). On Dˆ N ,n(N ) j,i j+1,i j,i define f N as before by f N xe = xe for j < h i (n). For xe ∈ G(n (N ) , q(n), ε(n), µl ) ∩ Dˆ N ,n(N ) one has for ∀J ∈ Jq(n) , 1 1 1 µl (J ) − ε(n) + O( ) ≤ 1 J f Nk xe1,i ≤ µl (J ) + ε(n) + O( ). N h i (n) N 0≤k≤h i (n)−1
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Let θl := µ L (x : x is µl − typical) and note that # A1 (N ) ∩ Dˆ N ,n(N ) θ1 → for N → ∞. θ2 # A2 (N ) ∩ Dˆ N ,n(N ) Thus by gluing all the orbit segments of f N in the sets A1 (N ) ∩ Dˆ N ,n(N ) and A2 (N ) ∩ Dˆ N ,n(N ) in such a way that they form one periodic orbit we obtain a family of periodic
l orbits with quantum limit α1 µ1 + α2 µ2 , where αl = θ1θ+θ . The α and β− densities are 2 just θ1 + θ2 . It is easy to construct in the same spirit approximation families f N for any values 1 2 and hence α2 > θ1θ+θ . Take in α1 and α2 = 1− α1 . Suppose first that α1 < θ1θ+θ 2 2 α1 each stack Si (n(N )) with i ∈ A1 (N ) approximately α2 of the internal orbit segments. The function n (N ) is sufficiently slowly growing (23) to ensure that there are enough points in the discretisation set Dˆ N ,n(N ) ; we can guarantee the convergence to αα21 . Gluing these segments together with all the internal orbit segments of A2 (N ) yields a single periodic orbit O(N ) . The family of periodic orbits O(N ) N defines a quantum limit for 1 the measure α1 µ1 + α2 µ2 with α and β− density αα21 θ1 + θ2 > 0. The case α1 > θ1θ+θ 2 is analogous. The proof of d) follows immediately by combining the arguments from parts b) and c).
5. Examples 5.1. Interval exchange maps. Consider a permutation πof {1, 2, . . . , n} and a vecn tor v = (v1 , . . . , vn ) such that vi > 0 for all i and i=1 vi = 1. Let u 0 = 0, u i = v1 + · · · + vi and i = (u i−1 , u i ). The interval exchange transformation T = Tπ,v , T : [0, 1] → [0, 1] is the map that is an isometry of each interval i which rearranges these intervals according to the permutation π . The Lebesgue measure is always an invariant measure for an IET. A typical IET is uniquely ergodic, however there exist minimal, non-uniquely ergodic IETs. The first example of a minimal, non-uniquely ergodic IET was given by Keynes and Newton [KN76] and Keane [Kea77]. The number of ergodic invariant measures for a minimal IET on m intervals is bounded by the m/2 [Kat73,Vee78]. The set of invariant measures always includes absolutely continuous measures but can also include singular measures. It is known that an interval exchange transformation on m intervals is at most of rank m; in particular it is a finite rank transformation (see for example [Fer97]). In fact the typical IET is of rank 1 [Vee84], although we will not use this fact. Thus we can apply Corollary 3 to conclude: 1. any uniquely ergodic IET is quantum uniquely ergodic, 2. any minimal, non-uniquely ergodic IET is not quantum uniquely ergodic, 3. any absolutely continuous invariant measures appear as a positive density quantum limit, 4. any singular ergodic invariant measure appears as a zero density quantum limit. 5.2. The full shift. Another example of a cutting and stacking transformation f B that has µ L as an ergodic invariant measure and admits further singular measures µ such
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that µ(S) = 0 is given by the full shift. Take any cutting and stacking model of the full two-sided shift on two symbols with Bernoulli-measure p0 = p1 = 21 (for details of such models we refer to the book [Shi96]). Note that although the full shift has many periodic orbits the cutting and stacking model has none. We remark that one could introduce some periodic orbits at the boundaries of the subintervals but they would all sit or fall at singularity points and hence do not appear as quantum limits, in other words there are no scars in quantised cutting and stacking skew product mappings. By Proposition 3 all other invariant measures of the full shift have no support on the singularity set. Hence we can apply Theorem 3. It is interesting to note that the fractal-dimensions (box or Hausdorff dimension) of the singularity set are rather large and that the upper and lower dimensions do not coincide. A straightforward counting argument for instance
shows that the upper and lower box dimensions are in the open interval 21 , 1 . 6. Comments and Conclusions We have shown in this paper that for a rather general class of dynamical systems on the torus the variety of different invariant measures can be recovered as quantum limits of the corresponding proper families of quantised maps. The quantisation scheme here used is based on the one introduced by [MR00]. For a discussion of alternative quantisation procedures and a critical comparison we refer to the recent work [Zel05]. One of the main features in our systems is the presence of singularities. In the quantisation procedure this provides enough freedom to obtain eigenfunctions reflecting the typical orbit structure with respect to any non-atomic ergodic measure. It is an interesting question whether our results are still valid in case the classical dynamical system has no singularities. We conjecture that similar statements can be obtained. For this it seems natural to replace the top-bottom gluing scheme in the interval exchange approximating family by cutting and “crossover-concatenation” of touching period orbits. Concerning the quantisation of flows one might hope that a good understanding of the associated quantised Poincaré maps can guide one to a deeper understanding of concrete features of eigenfunctions and spectrum. A natural class of examples to study these questions are polygonal billiards. In the case of rational polygons the associated Poincare maps for the directional flow are interval exchange transformations which can be quantised similar to the quantisation used in this paper. It would be interesting to compare the results obtained that way with the semiclassical properties of the direct flow quantisation via the billiard Hamiltonian. Acknowledgements. R.S. was supported by the EPSRC under grant GR/T28058/01.
References [AN06] [Ana06] [BS04] [BDB03] [CdV85]
Anantharaman, N., Nonnenmacher, S.: Half–delocalization of eigenfunctions for the Laplacian on an Anosov manifold. Ann. Inst. Fourier 57(7), 2465–2523 (2007) Anantharaman, N.: Entropy and the localization of eigenfunctions. To appear in Ann. Math, accepted in 2006 Bogomolny, E., Schmit, C.: Structure of wave functions of pseudointegrable billiards. Phys. Rev. Let. 92(24), 244102 (2004) Bonechi, F., De Bièvre, S.: Controlling strong scarring for quantized ergodic toral automorphisms. Duke Math. J. 117(3), 571–587 (2003) Colin de Verdière, Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102(3), 497–502 (1985)
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[DB01] [DEG03] [Fer97] [FN04] [FNDB03] [Fri92] [GL93] [GMO04] [Kat73] [Kea77] [Kel07] [Kel08] [Kel08a] [KN76] [KR00] [Lin06] [MO05] [MR00] [Ros06] [RS94] [Sch01] [Shi96] [Šni74] [Vee78] [Vee84] [Zel87] [Zel05]
C.-H. Chang, T. Krüger, R. Schubert, S. Troubetzkoy
De Bièvre, S.: Quantum chaos: a brief first visit, Second Summer School in Analysis and Mathematical Physics (Cuernavaca, 2000), Contemp. Math., Vol. 289, Providence, RI: Amer. Math. Soc., 2001, pp. 161–218 Degli Esposti, M., Graffi, S. (eds.): The mathematical aspects of quantum maps. Lecture Notes in Physics, Vol. 618, Berlin-Heidelberg-New York: Springer, 2003 Ferenczi, S.: Systems of finite rank. Colloq. Math. 73(1), 35–65 (1997) Faure, F., Nonnenmacher, S.: On the maximal scarring for quantum cat map eigenstates. Commun. Math. Phys. 245(1), 201–214 (2004) Faure, F., Nonnenmacher, S., De Bièvre, S.: Scarred eigenstates for quantum cat maps of minimal periods. Commun. Math. Phys. 239(3), 449–492 (2003) Friedman, N.A.: Replication and stacking in ergodic theory. Amer. Math. Monthly 99, 31–41 (1992) Gérard, P., Leichtnam, É.: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71(2), 559–607 (1993) Giraud, O., Marklof, J., O’Keefe, S.: Intermediate statistics in quantum maps. J. Phys. A 37(28), L303–L311 (2004) Katok, A.B.: Invariant measures of flows on orientable surfaces. Dokl. Akad. Nauk SSSR 211, 775–778 (1973) Keane, M.: Non-ergodic interval exchange transformations. Israel J. Math. 26(2), 188–196 (1977) Kelmer, D.: Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms. Commun. Math. Phys. 276(2), 381–395 (2007) Kelmer, D.: Arithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the Multidimensional Torus. To appear in Ann. of Math, 2008 Kelmer, D.: Quantum ergodicity for products of hyperbolic planes. J. Mod. Dyn. 2(2), 287–313 (2008) Keynes, H.B., Newton, D.: A “minimal”, non-uniquely ergodic interval exchange transformation. Math. Z. 148(2), 101–105 (1976) Kurlberg, P., Rudnick, Z.: Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J. 103(1), 47–77 (2000) Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math. (2) 163(1), 165–219 (2006) Marklof, J., O’Keefe, S.: Weyl’s law and quantum ergodicity for maps with divided phase space, with an appendix “Converse quantum ergodicity” by Steve Zelditch. Nonlinearity 18(1), 277– 304 (2005) Marklof, J., Rudnick, Z.: Quantum unique ergodicity for parabolic maps. Geom. Funct. Anal. 10(6), 1554–1578 (2000) Rosenzweig, L.: Quantum unique ergodicity for maps on the torus. Ann. Henri Poincaré 7, 447– 469 (2006) Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161(1), 195–213 (1994) Schubert, R.: Semiclassical localization in phase space. PhD thesis, Ulm, 2001 Shields, P.C.: The ergodic theory of discrete sample paths, Graduate Studies in Mathematics, Vol. 13, Providence, RI: Amer. Math. Soc., 1996 Šnirel’man, A.I.: Ergodic properties of eigenfunctions. Uspehi Mat. Nauk 29, no. 6(180), 181–182 (1974) Veech, W.A.: Interval exchange transformations. J. Analyse Math. 33, 222–272 (1978) Veech, W.A.: The metric theory of interval exchange transformations. i. generic spectral properties. Amer. J. Math. 106(6), 1331–1359 (1984) Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987) Zelditch, S.: Quantum maps and automorphisms, The breadth of symplectic and Poisson geometry, Progr. Math., Vol. 232, Boston, MA: Birkhäuser Boston, 2005, pp. 623–654
Communicated by P. Sarnak
Commun. Math. Phys. 282, 419–433 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0550-1
Communications in
Mathematical Physics
Asymptotics and Dimensional Dependence of the Number of Critical Points of Random Holomorphic Sections Benjamin Baugher Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA. E-mail: [email protected] Received: 25 April 2007 / Accepted: 8 February 2008 Published online: 2 July 2008 – © Springer-Verlag 2008
Abstract: We prove two conjectures from [DSZ2,DSZ3] concerning the expected number of critical points of random holomorphic sections of a positive line bundle. We show that, on average, the critical points of minimal Morse index are the most plentiful for holomorphic sections of O(N ) → CPm and, in an asymptotic sense, for those of line bundles over general Kähler manifolds. We calculate the expected number of these critical points for the respective cases and use these to obtain growth rates and asymptotic bounds for the total expected number of critical points in these cases. This line of research was motivated by landscape problems in string theory and spin glasses. 1. Introduction In the series of articles [DSZ1,DSZ2,DSZ3], the authors have been studying the statistics of critical points of Gaussian random holomorphic sections and their application to the vacuum selection problem in string theory. The purpose of this article is to prove two conjectures from these papers. m In [DSZ2,DSZ3] it was informally conjectured that the expected number N Ncrit ,h (CP ) m of critical points of random holomorphic sections of O(N ) → CP grows exponentially with the dimension. This was based on a conjectured formula for the expected number of critical points of minimal Morse index and the evidence from calculations in small m dimensions that the expected number N Ncrit ,q,h (CP ) of critical points of Morse index q decreased as q increased. In [DSZ3, Sect. 7.3] this conjectured growth rate was used as a basis for the heuristic estimate of the growth rate for the expected density of vacua in string/M theory. It was also noted that it is consistent with the analogous estimates of the growth rate of the number of metastable states of spin glasses [F]. In this paper we show that this conjecture is indeed true by proving the conjectured formula for the case q = m and verifying that the observed behavior as q increases holds in all dimensions. On more general Kähler manifolds the formula for the expected number N Ncrit ,h of critical points is much more difficult to evaluate. Because things simplify as the degree
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of the bundle gets large, an asymptotic expansion of N Ncrit ,h and an integral formula for its leading coefficient b0 were derived in [DSZ2]. The leading coefficient was shown to be universal and therefore, based on calculations on CPm , it was conjectured that the critical points of minimal Morse index were the most plentiful as N → ∞, and upper and lower asymptotic bounds for N Ncrit ,h were estimated in Conjecture 4.4 of [DSZ2]. We are able to apply our methods to b0 and work out a proof of this conjecture as well, with an improvement on the upper bound estimate. 1.1. Background. The setting for this paper is the N th tensor power of a positive Hermitian line bundle (L N , h N ) → (M m , ωh ) over a compact Kähler manifold of dimension m. Here ωh is the Kähler form and is given by ωh = 2i h , where h = −∂ ∂¯ log h is the curvature form of the metric. The connection on the bundle is taken to be the Chern connection ∇ of h N . Relative to this connection, the critical points of a holomorphic section s ∈ H 0 (M, L N ) are given by ∇s(z) = 0 and the set of critical points of s will be denoted by Crit (s, h N ). We note that in general, the critical point equation is not holomorphic and thus the cardinality of Crit (s, h N ) is a non-constant random variable on the space H 0 (M, L N ). Indeed we see that in a local frame e we can write s = f e and then ∇s = (∂ f − f ∂ K )⊗ e L , where K = − log e2h N is the Kähler potential. Thus the critical point equation in the local frame is ∂ f − f ∂ K = 0, which is holomorphic only when K is. The space H 0 (M, L N ) is endowed with the Gaussian measure γ N given by dγ N (s) =
1 −c2 e dc , πd
s=
d
cjej,
j=1
where dc is Lebesgue measure and {e j } is an orthonormal basis of H 0 (M, L N ) relative to the inner product 1 s1 , s2 = h N (s1 (z), s2 (z)) ωhm m! M induced by h N on H 0 (M, L N ). The expected distribution of critical points of s ∈ H 0 (M, L N ) with respect to γ N is defined to be ⎡ ⎤ ⎣ Kcrit δz ⎦ dγ N (s), N ,h = H 0 (M,L)
z∈Crit (s, h N )
where δz is the Dirac point mass at z, and the expected total number of critical points is then given by N Ncrit = Kcrit ,h N ,h (z). M
The critical points of s with respect to ∇ are the same as those of log s2h N , and therefore as an aid in the analysis of the statistics of the critical points we consider their Morse indices. Recall that the Morse index q of a critical point of a real-valued function is given by the number of negative eigenvalues of its Hessian. For a positive line bundle it is well-known that m ≤ q ≤ 2m [Bo]. We let Kcrit N ,q,h denote the expected distribution
Asymptotics and Dimensional Dependence of Critical Points
421
of critical points of Morse index q, and N Ncrit ,q,h denote the expected number of these critical points. It follows that Kcrit N ,h (z) =
2m
Kcrit N ,q,h (z) ,
N Ncrit ,h =
q=m
2m
N Ncrit ,q,h .
q=m
We now recall the relevant results from [DSZ2]. First, we have the integral formula m for N Ncrit ,q,h (CP ). Theorem 1. The expected number of critical points of Morse index q for random sections s ∈ H 0 (CPm , O(N )) is given by m 2 +m+2
m N Ncrit ,q,h (CP )
2 2 = m
j=1
×
(N − 1)m+1 j! (m + 2)N − 2
e(m+2−2/N )λm 1
Y2m−q
m dλ mj=1 λ j (λ) e− j=1 λ j
for q > m for q = m
for N ≥2, where Y p = {λ ∈ Rm : λ1 > · · · > λ p > 0 > λ p+1 > · · · > λm } and (λ) = i< j (λi − λ j ) is the Vandermonde determinant. crit Next we have the complete asymptotic expansions of Kcrit N ,q,h (z) and N N ,q,h on a general Kähler manifold.
Theorem 2. For any positive Hermitian line bundle (L , h) → (M, ωh ) over any compact Kähler manifold with ωh = 2i h , the expected distribution of critical points of Morse index q of random sections in H 0 (M, L N ) relative to the Hermitian Gaussian measure induced by h and ωh has an asymptotic expansion of the form −1 N −m Kcrit + b2q (z)N −2 + · · · } N ,q,h (z) ∼ {b0q + b1q (z)N
ωhm , m!
m ≤ q ≤ 2m ,
where the b jq = b jq (m) are curvature invariants of order j of ωh . In particular, b0q is the universal constant −(m+2 ) 2 b0q = π (1) det(2H H ∗ − |x|2 I ) e−(H,x),(H,x) d H d x , Sm,q−m
where Sm,k := {(H, x) ∈ Sym(m, C) × C : index(2H H ∗ − |x|2 I ) = k} . Corollary 1. Let (L , h) → (M, ωh ) be a positive holomorphic line bundle on a compact Kähler manifold, with ωh = 2i h . Then the expected number of critical points of Morse index q (m ≤ q ≤ 2m) of random sections in H 0 (M, L N ) has the asymptotic expansion m m π b0q π β1q crit m m m−1 N m−1 c1 (L) N + c1 (M) · c1 (L) N N ,q,h ∼ m! (m − 1)!
+ β2q ρ 2 dVolh + β2q c1 (M)2 · c1 (L)m−2 M
+β2q c2 (M) · c1 (L)m−2 N m−2 + · · · ,
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B. Baugher
, β
are universal constants depending only on the dimension where b0q , β1q , β2q , β2q 2q m.
We ask the interested reader to refer to [DSZ1,DSZ2,DSZ3] for additional background information. m 1.2. Results. Our first result gives the exact formula for N Ncrit ,q,h (CP ) when q = m, shows that this number decreases as q increases, and gives an upper and lower bound m for the total expected number N Ncrit ,h (CP ) which holds for all N and m. m Theorem 3. Let N Ncrit ,q,h (CP ) denote the expected number of critical points of Morse index q for random sections s ∈ H 0 (CPm , O(N )), then m N Ncrit ,m,h (CP ) =
2(m + 1)(N − 1)m+1 , (m + 2)N − 2
and when N > 2 m crit m N Ncrit ,q+1,h (CP ) < N N ,q,h (CP ).
Therefore, 2(m + 1)(N − 1)m+1 2(m + 1)2 (N − 1)m+1 m < N Ncrit . ,h (CP ) < (m + 2)N − 2 (m + 2)N − 2 m In order to obtain the exact formula for N Ncrit ,m,h (CP ) we apply a modification of Selberg’s integral formula to the integral in Theorem 1. The second part of the theorem then follows from a change of variable argument. These arguments are presented in §2. m From this theorem we see that N Ncrit ,q,h (CP ) grows exponentially with the dimenm sion. This verifies the conjectured growth rate for N Ncrit ,q,h (CP ) that was used in [DSZ3, Sect. 7.3] as a basis for their heuristic estimate of the growth rate for the expected density of vacua in string/M theory. The modulus of the spectral determinant shows up in the various integral formulas for the expected number of critical points ([AD,BM,DSZ2,F]). As the modulus presents a serious technical challenge in evaluating the integral, it is often dropped from the calculation (see [AD] and [BM]), which results in counting the critical points with signs. In string theory this is known as computing the “supergravity index”, while in spin glass theory there is some debate over the validity and implications of the calculation (see [ABM] and references therein). In our case, Morse theory tells us that the number of critical points of each s ∈ H 0 (CPm , O(N )) counted with signs is a topological invariant and is given by ∗1,0 (−1)q = cm (TCP m ⊗ O(N )) z:∇s(z)=0
=
m (N − 1)m+1 + (−1)m , N m− j = (−1) j m+1 j N j=0
where q is the Morse index of z. We see that this “index counting” provides a good estimate of the total expected number of critical points, giving the correct growth rate except for the coefficient.
Asymptotics and Dimensional Dependence of Critical Points
423
Next we turn our attention to b0q and note that the absolute value sign in (1) prevents the direct application of Wick methods. Therefore in §3 we utilize a variant of the Itzykson-Zuber formula in random matrix integrals, as was done in the simplification of the formula for β2q in [DSZ2], to derive the following alternative formula for b0q . Theorem 4. In all dimensions, b0q (m) =
2
m 2 +m+2 2
π m (m + 2) m−1 j=1 j!
e(m+2)λm for q > m m − mj=1 λ j . × dλ j=1 λ j (λ) e × 1 for q = m Y2m−q
Here Y p= {λ ∈ Rm : λ1 > · · · > λ p > 0 > λ p+1 > · · · > λm } (λ) = i< j (λi − λ j ) is the Vandermonde determinant.
and
We see that the integral in the above theorem is almost identical to the one in Theorem 1, so we apply the methods of §2 to this formula to obtain: Theorem 5. Let n q (m) := πm! b0q (m) denote the leading coefficient in the expansion of 2m N Ncrit q=m n q (m): ,q,h , and let n(m) = m
m m N Ncrit ,q,h ∼ n q (m) c1 (L) N ,
m m N Ncrit ,h ∼ n(m) c1 (L) N .
Then
m+1 n m (m) = 2 m+2
and
n q+1 (m) <
2m − q 2m − q + 1
2 n q (m) ,
and hence the expected total number of critical points m m N Ncrit ,h ∼ n(m) c1 (L) N
with
2
2m + 3 m+1 < n(m) < . m+2 3
This theorem proves Conjecture 4.4 in [DSZ2] which was made based on calculations in small dimensions for the leading coefficient in the CPm case. These results are part of the author’s ongoing thesis research at the Johns Hopkins University which is being advised by S. Zelditch. 2. Proof of Theorem 3 m In this proof we first work out the formula for the minimal Morse index case N Ncrit ,m,h (CP ) crit crit m m and then proceed to show that N N ,q,h (CP ) > N N ,q+1,h (CP ) for m ≤ q ≤ 2m. The proofs of the intermediate lemmas will be given in the subsections below. From Theorem 1 we have m 2 +m+2
m N Ncrit ,m,h (CP )
We then use
2 2 = m
j=1
(N − 1)m+1 j! (m + 2)N − 2
Ym
m − mj=1 λ j dλ. j=1 λ j (λ) e
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B. Baugher
Lemma 1. m 2 +m+2
2 2 m
j=1
j!
m
j=1 λ j
0<λm <···<λ1 <∞
(λ) e−
m
j=1 λ j
dλ = 2(m + 1)
(2)
to obtain m N Ncrit ,m,h (CP ) =
2(m + 1)(N − 1)m+1 . (m + 2)N − 2
(3)
For the general case, we recall that m 2 +m+2
m N Ncrit ,q,h (CP )
2 2 = m
j=1
×
(N − 1)m+1 j! (m + 2)N − 2
m dλ mj=1 λ j (λ) e− j=1 λ j
Y2m−q
e(m+2−2/N )λm for q > m . 1 for q = m
(4)
We will show in §2.2 that: Lemma 2. For m ≥ 1 and 0 ≤ p ≤ m, let
m e(m+c)λm for p < m − m−1 λj j=1 Pc, p (m) = dλ λ j (λ) e × −λm , for p = m e Yp j=1 where Y p is as in Theorem 1. Then P0, r (m) = P0, s (m) for 0 ≤ r, s ≤ m, and for c > 0, Pc, p−1 (m) <
p p+c
2 Pc, p (m).
From this we see first that for N = 2 we can apply the above lemma with p = 2m − q crit (CPm ) = N crit (CPm ) for m ≤ r, s ≤ 2m. and c = 0 to the integral in (4). Thus N2,r,h 2,s,h crit (CPm ) = 1 and therefore From (3) we see that N2,m,h crit N2,h (CPm ) ≡
2m
crit N2,q,h (CPm ) = m + 1
q=m
for m ≥ 1. Then, when N > 2, we apply Lemma 2 with p = 2m − q and c = 1 − m N Ncrit ,q+1,h (CP )
<
2
2m − q 2m − q + 1 −
2 N
m N Ncrit ,q,h (CP ).
Therefore, 2(m + 1)(N − 1)m+1 2(m + 1)2 (N − 1)m+1 m (CP ) < < N Ncrit . ,h (m + 2)N − 2 (m + 2)N − 2
2 N
to obtain
Asymptotics and Dimensional Dependence of Critical Points
425
2.1. Proof of Lemma 1. First, we need the following well-known theorem (see [Se]). Theorem 6 (Selberg’s Integral Formula). For any positive integer n, let (λ) ≡ (λ1 , · · · , λn ) = |(λ)|2γ
n
λα−1 (1 − λ j )β−1 . j
j=1
Then
1
1
···
0
(λ)dλ =
0
n−1 j=0
(1 + γ + jγ ) (α + jγ ) (β + jγ ) ,
(1 + γ ) (α + β + γ (n + j − 1))
when α, β, γ ∈ C with Re α > 0, Re β > 0, Re γ > -min
Reβ 1 Reα n , (n−1) , (n−1)
(5)
.
As a corollary, we have a special limiting case of the above formula (see [As]). Corollary 2. For any positive integer n, let (λ) ≡ (λ1 , · · · , λn ) = |(λ)|2γ
n
λα−1 e−λ j . j
j=1
Then
∞
∞
···
0
(λ)dλ =
0
n−1 j=0
(1 + γ + jγ ) (α + jγ ) ,
(1 + γ )
valid for complex α, γ with Re α > 0, Re γ > -min
1 Re α n , (n−1)
(6)
. x
This formula is obtained by setting β = m and making the change of variables x j → mj in (5) and then letting m → ∞. In order to simplify the notation we will use P(m) to denote the integral on the LHS of (2). We see that we can rewrite this integral as m − mj=1 λ j P(m) = dλ. j=1 λ j |(λ)| e 0<λm <···<λ1 <∞
We then note that the integrand on the RHS of the above equation is symmetric under permutations of λ. Therefore, m m 1 λ j |(λ)| e− j=1 λ j dλ. (7) P(m) = j=1 m! Rm+ It is easy to see that the integrals in (6) and (7) are equal when α = 2, γ = n = m. Consequently, P(m) =
1 2,
and
m−1 m j j 1 ( 23 + 2 ) (2 + 2 ) = (m + 1) 2− j j! , 3 m!
( ) 2 j=0 j=1
where the last equality follows from an application of Gauss’s multiplication formula. The desired formula is then obtained by substituting P(m) back into (2).
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B. Baugher
2.2. Proof of Lemma 2 . In order to simplify the discussion we will examine the case where p = m separately from the others. In this case ⎞ ⎛ ⎞⎛ m m−1 m m ⎝ Pc, m (m) = λj⎠ ⎝ (λi − λ j )⎠ e− j=1 λ j dλ. 0<λm <···<λ1 <∞
j=1
i=1 j=i+1
We make the change of variables
{λ1 , . . . , λm } →
m
λi ,
i=1
to obtain
⎛
Pc, m (m) =
Rm +
⎝
m m
m
λi , . . . , λm
i=2
⎞⎛
j m−1 m−1
λj⎠ ⎝
i=1 j=i
⎞ λk ⎠ e −
m j=1
j λj
dλ .
i=1 j=i k=i
Next we see that ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ j j m m m m−1 m−1 m−1 m−1 m−1 ⎝ λj⎠ ⎝ λk ⎠ = λm ⎝ λj⎠ ⎝ λk ⎠ i=1 j=i
i=1 j=i k=i
⎛
i=1 j=i
j m−1 m
= λm ⎝
i=1 j=i k=i
⎞ λk ⎠
(8)
i=1 j=i k=i
=
j m m
λk ,
i=1 j=i k=i
and so Pc, m (m) =
Rm +
⎛ ⎞ j m m m ⎝ λk ⎠ e− j=1
j λj
dλ .
i=1 j=i k=i
Now we consider Pc, p (m) when 0 ≤ p < m. For these cases ⎛ ⎞ m−1 m m−1 m Pc, p (m) = λ j ⎝ (λi − λ j )⎠ e− j=1 λ j +(m+c)λm dλ, Y p j=1 i=1 j=i+1 and we make the change of variables
p p λ1 , . . . , λ p → λi , λi , . . . , λ p , i=1
i=2
⎧ ⎫ m ⎨ ⎬ λ p+1 , . . . , λm → −λ p+1 , −(λ p+1 + λ p+2 ), . . . , − λi ⎩ ⎭ i= p+1
(9)
Asymptotics and Dimensional Dependence of Critical Points
to obtain
⎛
Pc, p (m) =
⎝
Rm +
⎛
×⎝
⎞⎛
p p
λj⎠ ⎝
i=1 j=i p
m
m
i
427
⎞⎛ λj⎠ ⎝
i= p+1 j= p+1
j
⎞⎛ λk ⎠ ⎝
i=1 j= p+1 k=i
m
j p−1 p−1
⎞ λk ⎠
i=1 j=i k=i
⎞
j m
λk ⎠ e −
p j=1
j λ j − mj= p+1 ( j+c) λ j
dλ .
i= p+2 j=i k=i
We can combine the first quantity with the third, and the second with the fifth, as we did in (8), and thus ⎛ ⎞⎛ ⎞ p j j p m m ⎝ Pc, p (m) = λk ⎠⎝ λk ⎠ Rm +
⎛
×⎝
i=1 j=i k=i p
m
j
⎞
i= p+1 j=i k=i
λk ⎠e −
p j=1
j λ j − mj= p+1 ( j+c) λ j
dλ .
i=1 j= p+1 k=i
Now it is clear that ⎞⎛ ⎞ ⎛ p j j j p p p m m ⎝ λk ⎠ ⎝ λk ⎠ = λk , i=1 j=i k=i
and then
⎛ ⎝
j p m
i=1 j= p+1 k=i
⎞⎛ λk ⎠ ⎝
i=1 j=i k=i
i=1 j=i k=i
j m m
⎞ λk ⎠ =
i= p+1 j=i k=i
j m m
λk .
i=1 j=i k=i
Therefore, ⎛
Pc, p (m) =
Rm +
⎝
j m m
⎞ λk ⎠ e −
p j=1
j λ j − mj= p+1 ( j+c) λ j
dλ.
(10)
i=1 j=i k=i
We note that in this formula the only dependence on p is in the exponential, and we see from (9) that this formula also holds for the p = m case as well. When c = 0, the formula does not depend on p at all, so we see that P0, r (m) = P0, s (m) for 0 ≤ r, s ≤ m. Next we let c > 0 and rewrite (10) as follows, m p m Pc, p (m) = I(λ1 , . . . , λm ) λi e− j=1 j λ j − j= p+1 ( j+c) λ j dλ , Rm +
i=1
where I(λ1 , . . . , λm ) =
j m m i=1 j=i+1 k=i
λk .
428
B. Baugher
Then we make the change of variable λ p → Pc, p−1 (m) =
p p+c λ p
in the formula for Pc, p−1 to obtain
2 p p I(λ1 , . . . , p+c λ p , . . . , λm ) m p+c R m + p m × λi e− i=1 i λi − j= p+1 ( j+c) λ j dλ i=1
<
p p+c
2 Pc, p (m).
3. Proof of Theorem 4 We begin with an intermediate lemma. This formula follows with only slight modifications from the derivation given in [DSZ2, Sect. 6.3] of a similar formula for the constant β2q . For the sake of completeness we present the entire proof below. Lemma 3. b0q (m) =
(−i)m(m−1)/2 π 2m m−1 j=1 j! × ··· Y2m−q
R
(λ) (ξ ) mj=1 |λ j | eiλ,ξ " # dξ1 · · · dξm dλ . i R 1− i j ξj j≤k 1 + 2 (ξ j + ξk ) 2
Here, (λ) and Y2m−q are as in Theorem 4, and the iterated dξ j integrals are defined in the distribution sense. Proof. First, we let Iε,ε =
1 ∗ 1 2 |det(2P)| ei,P−H H + 2 |x| I d m π Hm Hm (m−q) Sym(m,C)×C × e−Tr H H
∗ −|x|2
e−Tr
∗ −ε Tr P P ∗
d H d x d P d ,
(11)
where Hm is the space of m × m Hermitian matrices, Hm (m − q) = {P ∈ Hm : index P = m − q}, and dm = dimC (Sym(m, C) × C) = 21 (m 2 + m + 2). We note that absolute convergence in the above integral is guaranteed by the Gaussian factors in each of the variables (H, x, P, ). Then we have that b0q (m) =
1 π m (2π )m
2
lim lim Iε,ε .
ε →0 ε→0
(12)
$ ∗ 1 2 ∗ To verify this, first evaluate ei,P−H H + 2 |x| e−εTr d to obtain a dual Gaussian, which approximates the delta function δ H H ∗ − 1 |x|2 (P). As → 0, the d P integral then 2
yields the integrand at P = H H ∗ − 21 |x|2 I ; then we let ε → 0 to obtain the original integral.
Asymptotics and Dimensional Dependence of Critical Points
429
Next we conjugate P to a diagonal matrix D(λ) with λ = (λ1 , . . . , λm ) by an element h ∈ U(m). We recall that m (2π )( 2 )
, (13) φ(P) d P = cm φ(h D(λ)h ∗ )(λ)2 dh dλ , cm = m Hm Rm U(m) j=1 j! where dh is unit mass Haar measure on U(m) (see for example [ZZ, (1.9)]), and use this to obtain m 2m c ∗ 2 Iε,ε = d m (λ)2 |λ j | e−Tr H H −|x|
π m U(m) Hm Y2m−q Sym(m,C)×C ×e Here Y p
%
∗ λ2 i, h D(λ)h ∗ + 21 |x|2 I −H H ∗ − Tr + j
e
j=1
d H d x dλ d dh .
denotes the set of points in Rm
with exactly p coordinates positive. Again using (13) applied this time to , we obtain m
)2 2m (cm 2 2 Iε,ε = (λ) (ξ ) |λ j |
π dm U(m) U(m) Rm Y2m−q Sym(m,C)×C j=1
×
∗ ∗ 1 2 ∗ eig D(ξ )g , h D(λ)h + 2 |x| I −H H −Tr H H ∗ −|x|2 − (εξ 2j +ε λ2j )
×e
d H d x dλ dξ dh dg .
We then transfer the conjugation by g to the right side of the , in the first exponent and make the change of variables h → gh, H → g H g t to eliminate g from the integrand: m
)2 2m (cm 2 2
Iε,ε = (λ) (ξ ) |λ j |
π dm U(m) Rm Y2m−q Sym(m,C)×C ×e
j=1 iD(ξ ), h D(λ)h ∗ + 21 |x|2 I −H H ∗ −Tr H H ∗ −|x|2 − (εξ 2j +ε λ2j )
$
e
d H d x dλ dξ dh .
iD(ξ ),h D(λ)h ∗
Next we recognize the integral U(m) e dh as the well-known ItzyksonZuber-Harish-Chandra integral [Ha] (cf., [ZZ]): det[eiλ j ξk ] j,k m−1 J (D(λ), D(ξ )) = (−i)m(m−1)/2 . (14) j=1 j! (λ)(ξ ) We substitute (14) into the above integral and expand det[eiξ j λk ] jk = (−1)σ eiξ,σ (λ) , σ ∈Sm
obtaining a sum of m! integrals. However, by making the change of variables σ (λ) → λ and noting that (λ ) = (−1)σ (λ), we see that the integrals of all these terms are equal, and so we obtain m c
Iε,ε = (−i)m(m−1)/2 dm (λ) (ξ ) |λ j | eiλ,ξ
π m Rm Y2m−q Sym(m,C)×C j=1 ( ) & ' 1 2 ∗ ∗ 2 × exp i D(ξ ), 2 |x| I − H H − Tr H H − |x| × exp −ε ξ 2j − ε λ2j d H d x dλ dξ ,
430
B. Baugher
where 2
cm =
2m π m(m−1) m . j=1 j!
The phase
+ * 1 (H, x; ξ ) := i D(ξ ), |x|2 I − H H ∗ − Tr H H ∗ − |x|2 2 ⎞ ⎤ ⎡ ⎛ m i 2 2 2 = − ⎣H HS + i ξ j |H jk | + ⎝1 − ξ j ⎠ |x| ⎦ 2 j,k=1 j ⎡ ⎞ ⎤ ⎛ i ,jk |2 + ⎝1 − i = −⎣ 1 + (ξ j + ξk ) | H ξ j ⎠ |x|2 ⎦ , 2 2 j≤k
where
j
√ 2 H jk for j < k ,jk = H . for j = k H jk
Thus,
Iε,ε = (−i)m(m−1)/2 cm (15) m 2 2 (λ) (ξ ) |λ j | eiλ,ξ I(λ, ξ ) e−ε ξ j −ε λ j dξ dλ , ×
Y2m−q
where
Rm
j=1
1 e (H,x;ξ ) d H d x π dm C Sym(m,C) 2 1 − 1− 2i j ξ j |x| = e dx & ) i C j≤k 1 + 2 (ξ j + ξk ) π = & ). i i 1 − 2 j ξj j≤k 1 + 2 (ξ j + ξk )
I(λ, ξ ) =
To evaluate limε,ε →0+ Iε,ε , we first observe that the map 2 (ξ ) eiλ,ξ I(λ, ξ ) e− ε j ξ j dξ (ε1 , . . . , εm ) →
Rm is a continuous map from [0, +∞)m to the tempered distributions. In addition, since the
integrand in (15) is invariant under identical simultaneous permutations of the ξ j and the λ j , it follows that the integral equals m! times the corresponding integral over Ym−k . Hence, by (12) and (15), we have (−i)m(m−1)/2 lim b0q (m) = lim dλ ε →0+ ε1 ,...,εm →0+ Y2m−q π 2m m−1 j=1 j! m 2
2 (λ) (ξ ) |λ j | eiλ,ξ I(λ, ξ ) e− ε j ξ j −ε λ j dξ . × Rm
j=1
Asymptotics and Dimensional Dependence of Critical Points
431
Letting ε1 → 0, . . . , εm → 0, ε → 0 sequentially, produces the desired result.
3.1. Evaluating the inner integral. The last step is to evaluate the inner integral. We begin by writing (−i)m(m−1)/2 b0q (m) = π 2m m−1 j=1 j!
m Y2m−q j=1
|λ j | (λ) Iλ dλ ,
(16)
where Iλ =
& Rm 1 −
i 2
(ξ ) eiλ,ξ dξ #. ) " i ξj j≤k 1 + 2 (ξ j + ξk )
In order to simplify the formula, we make the change of variables ξ j → t j + i to obtain Iλ = −(−2i)
m 2 +m+2 2
e−
λj
Iλ,m+2 ,
where Iλ,c =
(R−i)m
&
(t) eiλ,t ) dt . t j + ic 1≤ j≤k≤m (t j + tk )
Putting this together we have
b0q (m) =
(−i)m π
2 −1
2
m 2 +m+2 2
m−1 2m j=1
j!
dλ Y2m−q
m
|λ j | (λ) e−
λj
Iλ,m+2 .
(17)
j=1
Now we need the following lemma from [DSZ2] where the authors evaluated the integral using iterated residues to derive the result. Lemma 4. Let 0 ≤ p ≤ m and let c > 0. Then for λ1 > · · · > λ p > 0 > λ p+1 > · · · > λm , we have
⎧ π m cλm 2 ⎪ ⎨ i m −1 e cm ) & dt = ⎪ t j + ic ⎩ i m 2 −1 π 1≤ j≤k≤m (t j + tk ) c (t) eiλ,t
(R−i)m
for p < m for p = m
.
By setting p = 2m − q and c = m + 2 in the above lemma and substituting this formula into (17) we obtain the desired result.
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B. Baugher
4. Proof of Theorem 5 From Theorem 4 and the definition of n q we obtain m m 2 +m+2 m 2 2 m dλ λ j (λ) e− j=1 λ j n q (m) = (m + 2) j=1 j! Y2m−q j=1
e(m+2)λm for q > m . × 1 for q = m
(18)
When q = m, we can apply Lemma 1 directly to the above integral and simplify to obtain n m (m) = 2
m+1 . m+2
(19)
For m ≤ q ≤ 2m, we can apply Lemma 2 with p = 2m − q and c = 1 to the integral in (18) and therefore 2 2m − q n q (m). (20) n q+1 (m) < 2m − q + 1 2m By definition n(m) = q=m n q (m), thus it follows from (19) and (20) that 2 2m−1 i 2m − j m+1 m+1 +2 n(m) < 2 m+2 m+2 2m − j + 1 i=m j=m 2m−1 2m − i 2 m+1 =2 1+ m+2 m+1 i=m m(2m + 1) m+1 1+ =2 m+2 6(m + 1) 2m + 3 . = 3 Acknowledgements. I would like to thank my advisor, S. Zelditch, for his guidance and helpful suggestions. I would also like to thank H. Hezari for reviewing the manuscript and providing useful comments.
References [Ao] [AD] [As] [ABM] [Bo] [BM] [DSZ1]
Aomoto, K.: Jacobi polynomials associated with selberg integrals. SIAM J. Math. Anal. 18, 545–549 (1987) Ashok, S., Douglas, M.: Counting flux vacua. J. High Energy Phys. 0401, 060 (2004) Askey, R.: Some basic hypergeometric extensions of integrals of selberg and andrews. SIAM J. Math. Anal. 11, 938–951 (1980) Aspelmeier, T., Bray, A.J., Moore, M.A.: Complexity of ising spin glasses. Phys. Rev. Lett. 92(8), 087203 (2004) Bott, R.: On a theorem of lefschetz. Michigan Math. J. 6, 211–216 (1959) Bray, A.J., Moore, M.A.: Metastable states in spin glasses. J. Phys. C 13, L469–L476 (1980) Douglas, M.R., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua. Commun. Math. Phys. 252(1-3), 325–358 (2004)
Asymptotics and Dimensional Dependence of Critical Points
[DSZ2] [DSZ3] [F] [Ha] [Se] [ZZ]
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Douglas, M.R., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua, ii: asymptotics and extremal metrics. J. Differ. Geom. 72(3), 381–427 (2006) Douglas, M.R., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua, iii: string/m models. Commun. Math. Phys. 265(3), 617–671 (2006) Fyodorov, Y.V.: Complexity of random energy landscapes, glass transition, and absolute value of the spectral determinant of random matrices. Phys. Rev. Lett. 92(24), 240601 (2004) Harish-Chandra: Differential operators on a semisimple lie algebra. Amer. J. Math. 79, 87–120 (1957) Selberg, A.: Bemerkninger om et multipelt integral. Norske Mat. Tidsskr. 26, 71–78 (1944) Zinn-Justin, P., Zuber, J.-B.: On some integrals over the u(n) unitary group and their large n limit. Random Matrix Theory. J. Phys. A 36, 3173–3193 (2003)
Communicated by M.R. Douglas
Commun. Math. Phys. 282, 435–467 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0529-y
Communications in
Mathematical Physics
A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation Justin Holmer1 , Svetlana Roudenko2 1 Department of Mathematics, University of California, Berkeley, CA 94720, USA.
E-mail: [email protected]
2 Department of Mathematics and Statistics, Arizona State University,
Tempe, AZ 85287, USA. E-mail: [email protected] Received: 18 May 2007 / Accepted: 10 February 2008 Published online: 27 June 2008 – © Springer-Verlag 2008
Abstract: We consider the problem of identifying sharp criteria under which radial H 1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂t u + u + |u|2 u = 0 scatter, i.e., approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the scale-invariant quantities u 0 L 2 ∇u 0 L 2 and M[u]E[u], where u 0 denotes the initial data, and M[u] and E[u] denote the (conserved in time) mass and energy of the corresponding solution u(t). The focusing NLS possesses a soliton solution eit Q(x), where Q is the ground-state solution to a nonlinear elliptic equation, and we prove that if M[u]E[u] < M[Q]E[Q] and u 0 L 2 ∇u 0 L 2 < Q L 2 ∇ Q L 2 , then the solution u(t) is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution eit Q(x), for which equality in these conditions is obtained, is global but does not scatter. We further show that if M[u]E[u] < M[Q]E[Q] and u 0 L 2 ∇u 0 L 2 > Q L 2 ∇ Q L 2 , then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle [17] in their study of the energy-critical NLS. 1. Introduction Consider the cubic focusing nonlinear Schrödinger (NLS) equation on R3 : i∂t u + u + |u|2 u = 0,
(1.1)
where u = u(x, t) is complex-valued and (x, t) ∈ R3 × R. The initial-value problem posed with initial-data u(x, 0) = u 0 (x) is locally well-posed in H 1 (see Ginibre-Velo [10]; standard reference texts are Cazenave [3], Linares-Ponce [21], and Tao [30]). Such solutions, during their lifespan [0, T ∗ ) (where T ∗ = +∞ or T ∗ < +∞), satisfy mass conservation M[u](t) = M[u 0 ], where M[u](t) = |u(x, t)|2 d x,
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and energy conservation E[u](t) = E[u 0 ], where 1 1 2 E[u](t) = |∇u(x, t)| d x − |u(x, t)|4 d x 2 4 (and we thus henceforth denote these quantities M[u] and E[u] respectively, with no reference to the time t). The equation also has several invariances, among them (in each of the following cases, u˜ is a solution to (1.1) if and only if u is a solution to (1.1)): • Spatial translation. For a fixed x0 ∈ R3 , let u(x, ˜ t) = u(x + x0 , t). • Scaling. For a fixed λ ∈ (0, +∞), let u(x, ˜ t) = λu(λx, λ2 t). 2 ˜ t) = ei xξ0 e−itξ0 u(x − 2ξ0 t, t). • Galilean phase shift. For a fixed ξ0 ∈ R3 , let u(x, The scale-invariant Sobolev norm is H˙ 1/2 , although we find it more useful, as described below, to focus on the scale invariant quantities u(t) L 2 ∇u(t) L 2 and M[u]E[u]. The Galilean invariance leaves only the L 2 norm invariant, while translation leaves all Sobolev norms invariant. We note that these two symmetries do not preserve radiality, while the scaling symmetry does. The nonlinear elliptic equation − Q + Q + |Q|2 Q = 0,
Q = Q(x), x ∈ R3 ,
(1.2)
has an infinite number of solutions in H 1 . Among these there is exactly one solution of minimal mass1 , called the ground-state solution, and it is positive (real-valued), radial, smooth, and exponentially decaying (see Appendix B of Tao’s text [30] for exposition). We henceforth denote by Q this ground-state solution. If we let u(x, t) = eit Q(x), then u is a solution to (1.1), and is called the standard soliton. A whole family of soliton solutions to (1.1) can be built from the standard soliton via the invariances of the NLS equation (1.1): u(x, t) = eit ei x·ξ0 e−it|ξ0 | λ u(λ(x − (x0 + 2ξ0 t)), λ2 t). 2
(1.3)
The standard soliton has the property that the quantities u 0 L 2 ∇u 0 L 2 and M[u]E[u] are minimal among all solitons (1.3). Indeed, these quantities are independent of translation and scaling, and the introduction of a Galilean phase shift only increases their values. Since solutions to the linear Schrödinger equation completely disperse (spread out, and shrink in a variety of spatial norms) as t → ±∞, the soliton solutions by their definition do not scatter (approach a solution of the linear Schrödinger equation). Indeed, soliton solutions represent a perfect balance between the focusing forces of the nonlinearity and the dispersive forces of the linear component. The basic line of thought in the subject, motivated by heuristics (Soffer [25]), rigorous partial results (Tao [28,29]), numerical simulation (Sulem-Sulem [27]), and analogy with the completely integrable one-dimensional case, is that a solution of (1.1) either completely disperses as t → ∞ (linear effects dominate), blows-up in finite time (nonlinear effects dominate) or the solution resolves into a sum of solitons propagating in 1 In view of the connection between solutions Q to (1.2) and solutions u(t) = eit Q to (1.1), and the fact that u(t) L 2 ∇u(t) L 2 is a scale invariant quantity for solutions u(t) to (1.1), it might be more natural to classify the family of solutions Q to (1.2) in terms of the quantity Q L 2 ∇ Q L 2 rather than the mass. √ However, any solution Q to (1.2) must satisfy the Pohozhaev identity Q L 2 ∇ Q L 2 = 3Q2 2 , and L thus, the two classifications are equivalent.
Scattering of 3D Cubic NLS
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different directions or at different speeds plus dispersive radiation as t → ∞ (nonlinear effects and linear effects balance). Since the smallest value of u 0 L 2 ∇u 0 L 2 among all soliton solutions is Q L 2 ∇ Q L 2 , it seems reasonable to conjecture, even for nonradial data, that if u 0 L 2 ∇u 0 L 2 < Q L 2 ∇ Q L 2 , then the solution scatters provided we can rule out blow-up. Ruling out blow-up in this situation is straightforward provided M[u]E[u] < M[Q]E[Q] using the conservation of mass and energy and a result of M. Weinstein stating that an appropriate Gagliardo-Nirenberg inequality is optimized at Q. The main result of this paper is the resolution of this conjecture under the assumption of radial data, which appears below as Theorem 1.1(1)(b). Theorem 1.1. Let u 0 ∈ H 1 be radial and let u be the corresponding solution to (1.1) in H 1 with maximal forward time interval of existence [0, T ). Suppose M[u]E[u] < M[Q]E[Q]. (1) If u 0 L 2 ∇u 0 L 2 < Q L 2 ∇ Q L 2 , then (a) T = +∞ (the solution is globally well-posed in H 1 ), and (b) u scatters in H 1 . This means that there exists φ+ ∈ H 1 such that lim u(t) − eit φ+ H 1 = 0 .
t→+∞
(2) If u 0 L 2 ∇u 0 L 2 > Q L 2 ∇ Q L 2 , then T < +∞ (the solution blows-up in finite time). It is straightforward to establish as a corollary the same result for negative times: take the complex conjugate of the equation and replace t by −t. Since the hypotheses in Theorem 1.1 (1),(2) apply to u 0 if and only if they apply to u¯ 0 , we obtain that the hypotheses of (1) imply that u scatters both as t → +∞ and t → −∞ and the hypotheses of (2) imply that u blows-up both in finite positive time and in finite negative time. An interesting open question is whether or not there exist solutions u with M[u]E[u] ≥ M[Q]E[Q] that exhibit different behavior in the positive and negative directions.2 The proof of Theorem 1.1(1)(b) is based upon ideas in Kenig-Merle [17], who proved an analogous statement for the energy-critical NLS. The key dynamical quantity in the proof of Theorem 1.1 is a localized variance xu(t) L 2 (|x|≤R) . The virial identity states that if xu 0 L 2 < ∞, then u satisfies ∂t2 |x|2 |u(x, t)|2 d x = 24E[u] − 4∇u(t)2L 2 . (1.4) x
We use a localized version of this identity in both the proof of Theorem 1.1(2) and the rigidity lemma (see §6) giving Theorem 1.1(1)(b). On a heuristic level (keeping in mind that u(t) L 2 is conserved), under the hypotheses of Theorem 1.1(1), the right side of (1.4) is strictly positive, which pushes the variance xu(t) L 2 to +∞ as t → +∞, which says roughly that the mass of u is being redistributed to large radii, meaning that it “disperses”, and we expect the effect of the nonlinearity to diminish and scattering to occur. On the other hand, under the hypotheses of Theorem 1.1(2), the right side of (1.4) is strictly negative, which pushes the variance xu(t) L 2 to 0 in finite time, meaning that all the mass of u concentrates at the origin and we expect blow-up. We do not use (1.4) directly, however, for two reasons. First, it requires the additional hypothesis that the initial data has finite variance—an assumption we would like to avoid. Secondly, in 2 Solutions with this type of behavior have recently been constructed by Duyckaerts–Roudenko [7] at the threshold M[u]E[u] = M[Q]E[Q].
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the case of the scattering argument, we don’t see a method for proving scattering given only the strict convexity (in time) of the variance and its divergence to +∞, although it is heuristically consistent with scattering. The problem is that large variance can be produced by a very small amount of mass moving to very, very large radii, while still leaving a significant amount of mass at small radii. Therefore, to prove the scattering claim in Theorem 1.1(1)(b), we instead use a localized virial identity, as Kenig–Merle [17] did, involving a localized variance. If a very small amount of mass moved to very, very large radii, it would not affect the localized variance dramatically. For the 3d cubic defocusing NLS i∂t u + u − |u|2 u = 0 scattering has been established for all H 1 solutions (regardless of “size”) even for nonradial data by Ginibre-Velo [11] using a Morawetz inequality. This proof was simplified by Colliander–Keel–Staffilani–Takaoka–Tao [4] using a new interaction Morawetz inequality they discovered. These Morawetz estimates, however, are not positive definite for solutions to the focusing equation (1.1), and thus, cannot be applied directly to our problem. It remains open whether or not one could prove suitable bounds on the nonpositive terms to recover the results of this paper. For (1.1), Tao [28] proved a few results in the direction of the soliton resolution conjecture, assuming the solution is radial and global (has globally bounded H 1 norm). It is shown that for large data, radial solutions asymptotically split into (i) a (smooth) function localized near the origin (which is either zero or has a non-zero mass and energy and obeys an asymptotic Pohozhaev identity), (ii) a radiation term evolving by the linear Schrödinger flow, and (iii) an error term (approaching zero in the H˙ 1 norm). Further results for mass supercritical, energy subcritical NLS equations in higher dimensions (N ≥ 5) were established by Tao in [29]. Equation (1.1) frequently arises, often in more complex forms, as a model equation in physics. In 2d, it appears as a model in nonlinear optics — see Fibich [8] for a review. When coupled with a nonlinear wave equation, it arises as the Zakharov system [33] in plasma physics. According to [19] p. 7, in the mass supercritical case “the most important partial case p = 3, d = 3 corresponds to the subsonic collapse of Langmuir waves in plasma”. Furthermore, (1.1) arises as a model for the Bose–Einstein condensate (BEC) in condensed matter physics. There, it appears as the Gross–Pitaevskii (GP) equation (in 1d, 2d, and 3d), which is (1.1) with a (real) potential V = V (x): i∂t u + u − V u + a|u|2 u = 0 .
(1.5)
It is derived by mean-field theory approximation (see Schlein [26]), and |u(x, t)|2 represents the density of the condensate at time t and position x. The coefficient a in the nonlinearity is governed by a quantity called the s-scattering length. Some elements used in recent experiments (7 Li, 85 Rb, 133 Cs) possess a negative s-scattering length in the ground state and are modeled by (1.5) with a < 0. V (x) is an external trapping potential imposed by a system of laser beams and is typically taken to be harmonic V (x) = β|x|2 . These “unstable BECs” (where a < 0) have been investigated experimentally recently (see the JILA experiments [6]) and a number of theoretical predictions have been confirmed, including the observation of “collapse events” (corresponding to blow-up of solutions to (1.5)). A few articles have appeared (for example [1]) in the physics literature discussing the critical number of atoms required to initiate collapse. The “critical number of atoms” corresponds to “threshold mass M[u]” in our terminology, and connects well with the mathematical investigations in this paper.
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The format of this paper is as follows. In §2, we give a review of the Strichartz estimates, the small data theory, and the long-time perturbation theory. We review properties of the ground state profile Q in §3 and recall its connection to the sharp GagliardoNirenberg estimate of M. Weinstein [32]. In §4, we introduce the local virial identity and prove Theorem 1.1 except for the scattering claim in part (1)(b). In §5–6, we prove Theorem 1.1(1)(b). This is done in two stages, assuming that the threshold for scattering is strictly below the one claimed. First, in §5, we construct a solution u c (a “critical element”) that stands exactly at the boundary between scattering and nonscattering. This is done using a profile decomposition lemma in H˙ 1/2 , obtained by extending the H˙ 1 methods of Keraani [15]. We then show that time slices of u c (t), as a collection of functions in H 1 , form a precompact set in H 1 (and thus, u c has something in common with the soliton eit Q(x)). This enables us to prove that u c remains localized uniformly in time. In §6, this localization is shown to give a strict convexity (in time) of a localized variance which leads to a contradiction with the conservation of mass at large times. In §7, we explain how Theorem 4.2 should carry over to more general nonlinearities and general dimensions (mass supercritical and energy subcritical cases) of NLS equations. 2. Local Theory and Strichartz Estimates We begin by recalling the relevant Strichartz estimates (e.g., see Cazenave [3], Keel–Tao [16]). We say that (q, r ) is H˙ s Strichartz admissible (in 3d) if 2 3 3 + = − s. q r 2 Let u S(L 2 ) =
sup (q,r ) L 2 admissible 2≤r ≤6, 2≤q≤∞
u L qt L r . x
10 3 In particular, we are interested in (q, r ) equal to ( 10 3 , 3 ) and (∞, 2). Define
u S( H˙ 1/2 ) =
sup
(q,r ) H˙ 1/2 admissible 3≤r ≤6− , 4+ ≤q≤∞
u L qt L r , x
where 6− is an arbitrarily preselected and fixed number < 6; similarly for 4+ . We will, in particular, use (q, r ) equal to (5, 5), (20, 10 3 ), and (∞, 3). Now we consider dual Strichartz norms. Let u S (L 2 ) =
inf
(q,r ) L 2 admissible 2≤q≤∞, 2≤r ≤6
u
q
L t L rx
,
where (q , r ) is the Hölder dual to (q, r ). Also define u S ( H˙ −1/2 ) =
inf
(q,r ) H˙ −1/2 admissible 4+ − + − 3 ≤q≤2 , 3 ≤r ≤6
u
q
L t L rx
.
3 For some inequalities, the range of valid exponents (q, r ) can be extended. The Kato inequality (2.2) imposes the most restrictive assumptions that we incorporate into our definitions of S( H˙ 1/2 ) and S( H˙ −1/2 ).
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The Strichartz estimates are eit φ S(L 2 ) ≤ cφ L 2 and
t ei(t−t ) f (·, t )dt 0
≤ c f S (L 2 ) .
S(L 2 )
By combining Sobolev embedding with the Strichartz estimates, we obtain eit φ S( H˙ 1/2 ) ≤ cφ H˙ 1/2 and
t ei(t−t ) f (·, t )dt 0
S( H˙ 1/2 )
≤ cD 1/2 f S (L 2 ) .
(2.1)
We shall also need the Kato inhomogeneous Strichartz estimate [14] (for further extensions see [9] and [31]) t ei(t−t ) f (·, t ) dt ≤ c f S ( H˙ −1/2 ) . (2.2) S( H˙ 1/2 )
0
10
10/3 5/4
3 L x on the right In particular, we will use L 5t L 5x and L 20 t L x on the left side, and L t side. We extend our notation S( H˙ s ), S ( H˙ s ) as follows: If a time interval is not specified (that is, if we just write S( H˙ s ), S ( H˙ s )), then the t-norm is evaluated over (−∞, +∞). To indicate a restriction to a time subinterval I ⊂ (−∞, +∞), we will write S( H˙ s ; I ) or S ( H˙ s ; I ).
Proposition 2.1. (Small data) Suppose u 0 H˙ 1/2 ≤ A. There is δsd = δsd (A) > 0 such that if eit u 0 S( H˙ 1/2 ) ≤ δsd , then u solving (1.1) is global (in H˙ 1/2 ) and u S( H˙ 1/2 ) ≤ 2 eit u 0 S( H˙ 1/2 ) , D 1/2 u S(L 2 ) ≤ 2 c u 0 H˙ 1/2 . (Note that by the Strichartz estimates, the hypotheses are satisfied if u 0 H˙ 1/2 ≤ cδsd .) Proof. Define u 0 (v) = e
it
t
u0 + i
ei(t−t ) |v|2 v(t )dt .
0
Applying the Strichartz estimates, we obtain D 1/2 u 0 (v) S(L 2 ) ≤ cu 0 H˙ 1/2 + cD 1/2 (|v|2 v) L 5/2 L 10/9 x
t
and u 0 (v) S( H˙ 1/2 ) ≤ eit u 0 S( H˙ 1/2 ) + cD 1/2 (|v|2 v) L 5/2 L 10/9 . t
x
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Applying the fractional Leibnitz [18] and Hölder inequalities, 2 D 1/2 (|v|2 v) L 5/2 L 10/9 ≤ v2L 5 L 5 D 1/2 v L ∞ D 1/2 v S(L 2 ) . 2 ≤ v S( H˙ 1/2 ) t Lx t
x
t
Let
x
1 . ≤ min √ , 24c 24c A
δsd
1
Then u 0 : B → B, where B = v v S( H˙ 1/2 ) ≤ 2eit u 0 S( H˙ 1/2 ) , D 1/2 v S(L 2 ) ≤ 2cu 0 H˙ 1/2 , and u 0 is a contraction on B.
Proposition 2.2. (H 1 scattering) If u 0 ∈ H 1 , u(t) is global with globally finite H˙ 1/2 Strichartz norm u S( H˙ 1/2 ) < +∞ and a uniformly bounded H 1 norm supt∈[0,+∞) u(t) H 1 ≤ B, then u(t) scatters in H 1 as t → +∞. This means that there exists φ + ∈ H 1 such that lim u(t) − eit φ + H 1 = 0.
t→+∞
Proof. Since u(t) solves the integral equation t
ei(t−t ) (|u|2 u)(t ) dt , u(t) = eit u 0 + i 0
we have
u(t) − eit φ + = −i
+∞
ei(t−t ) (|u|2 u)(t )dt ,
(2.3)
t
where
+∞
φ+ = u0 + i
e−it (|u|2 u)(t )dt .
0
Applying the Strichartz estimates to (2.3), we have u(t) − eit φ + H 1 ≤ c|u|2 (1 + |∇|)u L 5/2
10/9 [t,+∞) L x
≤ cu2L 5
5 [t,+∞) L x
≤ cBu2L 5
u L ∞ 1 t Hx
5 [t,+∞) L x
Send t → +∞ in this inequality to obtain the claim.
.
The following long-time perturbation result is similar in spirit to Lemma 3.10 in Colliander–Keel–Staffilani–Takaoka–Tao [5], although more refined than a direct analogous version since the smallness condition (2.4) is expressed in terms of S( H˙ 1/2 ) rather than D −1/2 S(L 2 ). This refinement is achieved by employing the Kato inhomogeneous Strichartz estimates [14].
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Proposition 2.3. (Long time perturbation theory) For each A 1, there exists 0 = 0 (A) 1 and c = c(A) 1 such that the following holds. Let u = u(x, t) ∈ Hx1 for all t and solve i∂t u + u + |u|2 u = 0 . Let u˜ = u(x, ˜ t) ∈ Hx1 for all t and define def
˜ 2 u˜ . e = i∂t u˜ + u˜ + |u| If u ˜ S( H˙ 1/2 ) ≤ A , e S ( H˙ −1/2 ) ≤ 0 , and ei(t−t0 ) (u(t0 ) − u(t ˜ 0 )) S( H˙ 1/2 ) ≤ 0 ,
(2.4)
then u S( H˙ 1/2 ) ≤ c = c(A) < ∞ . Proof. Let w be defined by u = u˜ + w. Then w solves the equation ˜ 2 w) + (2 u˜ |w|2 + u¯˜ w 2 ) + |w|2 w − e = 0. i∂t w + w + (u˜ 2 w¯ + 2|u|
(2.5)
Since u ˜ S( H˙ 1/2 ) ≤ A, we can partition [t0 , +∞) into N = N (A) intervals4 I j = ˜ S( H˙ 1/2 ;I j ) ≤ δ is suitably small (δ to be [t j , t j+1 ] such that for each j, the quantity u chosen below). The integral equation version of (2.5) with initial time t j is t i(t−t j ) w(t) = e w(t j ) + i ei(t−s) W (·, s) ds, (2.6) tj
where W = (u˜ 2 w¯ + 2|u| ˜ 2 w) + (2 u˜ |w|2 + u¯˜ w 2 ) + |w|2 w − e. By applying the Kato Strichartz estimate (2.2) on I j , we obtain w S( H˙ 1/2 ;I j ) ≤ ei(t−t j ) w(t j ) S( H˙ 1/2 ;I j ) + c u˜ 2 w L 10/3 L 5/4 Ij
(2.7)
x
+ c uw ˜ 2 L 10/3 L 5/4 + c w 3 L 10/3 L 5/4 + e S ( H˙ −1/2 ;I j ) . Ij
x
10/3
w L 5
Ij
x
Observe u˜ 2 w L 10/3 L 5/4 ≤ u ˜ 2 20 x
Ij
L I Lx j
Ij
L 5x
≤ u ˜ 2S( H˙ 1/2 ;I ) w S( H˙ 1/2 ;I j ) j
≤ δ 2 w S( H˙ 1/2 ;I j ) . Similarly, uw ˜ 2 L 10/3 L 5/4 ≤ δw2S( H˙ 1/2 ;I ) , and w 3 L 10/3 L 5/4 ≤ w3S( H˙ 1/2 ;I ) . Ij
x
j
Ij
x
j
4 The number of intervals depends only on A, but the intervals themselves depend upon the function u. ˜
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Substituting the above estimates in (2.7), w S( H˙ 1/2 ;I j ) ≤ ei(t−t j ) w(t j ) S( H˙ 1/2 ;I j ) + cδ 2 w S( H˙ 1/2 ;I j )
(2.8)
+ cδw2S( H˙ 1/2 ;I ) + cw3S( H˙ 1/2 ;I ) + c 0 . j
j
Provided 1 1 , and ei(t−t j ) w(t j ) S( H˙ 1/2 ;I j ) + c 0 ≤ min 1, √ δ ≤ min 1, 6c 2 6c (2.9) we obtain w S( H˙ 1/2 ;I j ) ≤ 2ei(t−t j ) w(t j ) S( H˙ 1/2 ;I j ) + 2c 0 .
(2.10)
Now take t = t j+1 in (2.6), and apply ei(t−t j+1 ) to both sides to obtain ei(t−t j+1 ) w(t j+1 ) = ei(t−t j ) w(t j ) + i
t j+1
ei(t−s) W (·, s) ds.
(2.11)
tj
Since the Duhamel integral is confined to I j = [t j , t j+1 ], by again applying the Kato estimate, similarly to (2.8) we obtain the estimate ei(t−t j+1 ) w(t j+1 ) S( H˙ 1/2 ) ≤ ei(t−t j ) w(t j ) S( H˙ 1/2 ) + cδ 2 w S( H˙ 1/2 ;I j ) + cδw2S( H˙ 1/2 ;I ) + cw3S( H˙ 1/2 ;I ) + c 0 . j
j
By (2.10) and (2.11), we bound the previous expression to obtain ei(t−t j+1 ) w(t j+1 ) S( H˙ 1/2 ) ≤ 2ei(t−t j ) w(t j ) S( H˙ 1/2 ) + 2c 0 . Iterating beginning with j = 0, we obtain ei(t−t j ) w(t j ) S( H˙ 1/2 ) ≤ 2 j ei(t−t0 ) w(t0 ) S( H˙ 1/2 ) + (2 j − 1)2c 0 ≤ 2 j+2 c 0 . To accommodate the second part of (2.9) for all intervals I j , 0 ≤ j ≤ N − 1, we require that 1 N +2 . (2.12) 2 c 0 ≤ min 1, √ 2 6c We review the dependence of parameters: δ is an absolute constant selected to meet the first part of (2.9). We were given A, which then determined N (the number of time subintervals). The inequality (2.12) specifies how small 0 needs to be taken in terms of N (and thus, in terms of A).
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3. Properties of the Ground State Weinstein [32] proved that the sharp constant cGN in the Gagliardo–Nirenberg estimate f 4L 4 ≤ cGN f L 2 ∇ f 3L 2
(3.1)
is attained at the function Q (the ground state described in the introduction), i.e., cGN = Q4L 4 /(Q L 2 ∇ Q3L 2 ). By multiplying (1.2) by Q, integrating, and applying integration by parts, we obtain −Q2L 2 − ∇ Q2L 2 + Q4L 4 = 0 . By multiplying (1.2) by x · ∇ Q, integrating, and applying integration by parts, we obtain the Pohozhaev identity 3 1 3 Q2L 2 + ∇ Q2L 2 − Q4L 4 = 0 . 2 2 4 These two identities enable us to obtain the relations ∇ Q2L 2 = 3Q2L 2 , Q4L 4 = 4Q2L 2 ,
(3.2)
4 4 = √ .5 3Q L 2 ∇ Q L 2 3 3Q2L 2
(3.3)
and thus, reexpress cGN = We also calculate
1 1 ∇ Q2L 2 − Q4L 4 2 4
1 1 Q2L 2 ∇ Q2L 2 = Q4L 2 . 6 2 (3.4) For later purposes we recall a version of the Gagliardo–Nirenberg inequality valid only for radial functions, due to Strauss [26]. In R3 , for any R > 0, we have M[Q]E[Q] =
Q2L 2
f 4L 4 (|x|>R) ≤
=
c f 3L 2 (|x|>R) ∇ f L 2 (|x|>R) . R2
(3.5)
4. Global versus Blow-up Dichotomy In this section we show how to obtain Theorem 1.1 part (1)(a) and part (2). This was proved in Holmer–Roudenko [13] for general mass supercritical and energy subcritical NLS equations with H 1 initial data, but for self-containment of this exposition we outline the main ideas here. Before giving the proof, we observe that the following quantities are scaling invariant: ∇u L 2 · u L 2 and E[u] · M[u]. Next, we quote a localized version of the virial identity as in Kenig–Merle [17]. We refer, for example, to Merle–Raphaël [22] or Ozawa–Tsutsumi [23] for a proof. 5 Numerical calculations show Q2 ∼ = 18.94, which gives cGN ∼ = 0.0406 (in R3 ). L 2 (R 3 )
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Lemma 4.1. (Local virial identity) Let χ ∈ C0∞ (R N ), radially symmetric and u solve i∂t u + u + |u| p−1 u = 0. Then 1 1 2 2
2 2 2 ∂t χ (x) |u(x, t)| d x = 4 χ |∇u| − χ |u| − 4 − χ |u| p+1 . 2 p+1 (4.1) We prove a slightly stronger version of Theorem 1.1 parts (1)(a) and (2) that is valid for the nonradial initial condition. The generalization of this theorem to all mass supercritical and energy critical cases of NLS can be found in §7 as well as in [13]. A different type of condition for global existence, phrased as u 0 L 2 ≤ γ∗ (∇u 0 L 2 ) for a certain monotonic function γ : R+ → R+ , is given by Bégout [2]. Theorem 4.2. (Global versus blow-up dichotomy) Let u 0 ∈ H 1 (R3 ) (possibly nonradial), and let I = (−T∗ , T ∗ ) be the maximal time interval of existence of u(t) solving (1.1). Suppose that M[u 0 ] E[u 0 ] < M[Q] E[Q]. (4.2) If (4.2) holds and
∇u 0 L 2 u 0 L 2 < ∇ Q L 2 Q L 2 ,
(4.3)
then I = (−∞, +∞), i.e., the solution exists globally in time, and for all time t ∈ R,
If (4.2) holds and then for t ∈ I ,
∇u(t) L 2 u 0 L 2 < ∇ Q L 2 Q L 2 .
(4.4)
∇u 0 L 2 u 0 L 2 > ∇ Q L 2 Q L 2 ,
(4.5)
∇u(t) L 2 u 0 L 2 > ∇ Q L 2 Q L 2 .
(4.6)
Furthermore, if (a) |x|u 0 ∈ solution blows up in finite time.
L 2 (R3 ),
or (b) u 0 is radial, then I is finite, and thus, the
We recently became aware that the global existence assertion and the blow-up assertion under the hypothesis |x|u 0 ∈ L 2 (R3 ) in this theorem previously appeared in the literature in Kuznetsov–Rasmussen–Rypdal–Turitsyn [20]. We have decided to keep the proof below since it is short and for the convenience of the reader (there are significant notational differences between our paper and theirs). Remark 4.3. 6 Since this theorem applies to the nonradial case, we remark that one should exploit the Galilean invariance to extend the class of solutions u to which it applies. Since u is global [respectively, blows up in finite time] if and only if a Galilean transformation of it is global [respectively, blows up in finite time], given u consider for some ξ0 ∈ R3 the transformed solution w(x, t) = ei x·ξ0 e−it|ξ0 | u(x − 2ξ0 t, t). 2
We compute ∇w2L 2 = |ξ0 |2 M[u] + 2ξ0 · P[u] + ∇u2L 2 , 6 We thank J. Colliander for supplying this comment.
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where the vector P[u] = Im M[w] = M[u] and E[w] =
u∇u ¯ d x is the conserved momentum. Therefore,
1 |ξ0 |2 M[u] + ξ0 · P[u] + E[u]. 2
To minimize E[w] and ∇w L 2 , we take ξ0 = −P[u]/M[u]. Then we test the condition (4.2), and (4.3) or (4.5) for w, rather than u. This means that for P = 0, the hypothesis (4.2) can be sharpened to P[u]2 M[u] − + E[u] < M[Q]E[Q], 2M[u] and the hypothesis (4.3) can be sharpened to P[u]2 2 + ∇u 0 L 2 u 0 2L 2 < Q2L 2 ∇ Q2L 2 , − M[u] and similarly for (4.5). Proof. Multiplying the definition of energy by M[u] and using (3.1), we have 1 u4L 4 u 0 2L 2 4 1 cGN ∇u3L 2 u 0 3L 2 . 4 3 3 3
2 Define f (x) = 21 x 2 − cGN 4 x . Then f (x) = x − 4 cGN x = x 1 − 4 cGN x , and 1 = ∇ Q L 2 Q L 2 by (3.3). Note that thus, f (x) = 0 when x0 = 0 and x1 = 43 cGN 1 2 2 f (0) = 0 and f (x1 ) = 6 ∇ Q L 2 Q L 2 . Thus, the graph of f has a local minimum at x0 and a local maximum at x1 . The condition (4.2) together with (3.4) imply that M[u 0 ]E[u 0 ] < f (x1 ). Combining this with energy conservation, we have 1 ∇u2L 2 u 0 2L 2 − 2 1 ≥ ∇u2L 2 u 0 2L 2 − 2
M[u]E[u] =
f (∇u(t) L 2 u 0 L 2 ) ≤ M[u 0 ] E[u(t)] = M[u 0 ] E[u 0 ] < f (x1 ).
(4.7)
If initially u 0 L 2 ∇u 0 L 2 < x1 , i.e., the condition (4.3) holds, then by (4.7) and the continuity of ∇u(t) L 2 in t, we have u 0 L 2 ∇u(t) L 2 < x1 for all time t ∈ I which gives (4.4). In particular, the H˙ 1 norm of the solution u is bounded, which proves global existence (and thus, global wellposedness) in this case. If initially u 0 L 2 ∇u 0 L 2 > x1 , i.e., the condition (4.5) holds, then by (4.7) and the continuity of ∇u(t) L 2 in t, we have u 0 L 2 ∇u(t) L 2 > x1 for all time t ∈ I which gives (4.6). We can refine this analysis to obtain the following: if the condition (4.5) (together with (4.2)) holds, then there exists δ1 > 0 such that M[u 0 ] E[u 0 ] < (1 − δ1 )M[Q] E[Q], and thus, there exists δ2 = δ2 (δ1 ) > 0 such that u 0 2L 2 ∇u(t)2L 2 > (1 + δ2 ) ∇ Q2L 2 Q2L 2 for all t ∈ I . Now if u has a finite variance, we recall the virial identity ∂t2 |x|2 |u(x, t)|2 d x = 24E[u 0 ] − 4∇u(t)2L 2 .
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Multiplying both sides by M[u 0 ] and applying the refinement of inequalities (4.2) and (4.6) mentioned above as well as (3.4), we get M[u 0 ] ∂t2 |x|2 |u(x, t)|2 d x = 24 M[u 0 ] E[u 0 ] − 4∇u(t)2L 2 u 0 2L 2 < 24 ·
1 6
(1 − δ1 ) ∇ Q2L 2 Q2L 2
− 4(1 + δ2 )∇ Q2L 2 Q2L 2 = −4(δ1 + δ2 )∇ Q2L 2 Q2L 2 < 0, and thus, I must be finite, which implies that blow up occurs in finite time. If u 0 is radial, we use a localized version of the virial identity (4.1). Choose χ (r ) (radial) such that ∂r2 χ (r ) ≤ 2 for all r ≥ 0, χ (r ) = r 2 for 0 ≤ r ≤ 1, and χ (r ) is constant for r ≥ 3. Let χm (r ) = m 2 χ (r/m). The rest of the argument follows the proof of the main theorem in Ogawa–Tsutsumi [23], although we include the details here for the convenience of the reader. We bound each of the terms in the local virial identity (4.1) as follows, using that χm (r ) = 6 for r ≤ m and 2 χm (r ) = 0 for r ≤ m:
2 4 χm |∇u| ≤ 8 |∇u|2 , 3 R c − 2 χm |u|2 ≤ 2 |u|2 , m m≤|x|≤3m |u|4 + c |u|4 − χm |u|4 ≤ −6 |x|≤m m≤|x|≤3m 4
≤ −6 |u| + c |u|4 . R3
|x|≥m
Adding these three bounds and applying the radial Gagliardo-Nirenberg estimate (3.5), we obtain that for any large m > 0, we have 2 2 ∂t χm (|x|) |u(x, t)| d x ≤ 24E[u 0 ] − 4 |∇u|2 c1 c2 + 2 u 0 3L 2 ∇u L 2 + 2 |u|2 . m m m<|x| Let > 0 be a small constant to be chosen below. Use Young’s inequality in the third term on the right side to separate the L 2 -norm and gradient term and then absorb the gradient term into the second term with the chosen . Multiplying the above expression by M[u 0 ], we get M[u 0 ] ∂t2 χm (|x|) |u(x, t)|2 d x ≤ 24 E[u 0 ]M[u 0 ] − (4 − )∇u2L 2 u 0 2L 2 + ≤ −c3 ∇ Q2L 2 Q2L 2 +
c( ) c2 u 0 8L 2 + 2 u 0 4L 2 m4 m
c( ) c2 u 0 8L 2 + 2 u 0 4L 2 , 4 m m
where c3 ≡ −4(1 − δ1 ) + (4 − )(1 + δ2 ) = +4(δ1 + δ2 ) − (1 + δ2 ).
(4.8)
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Select = (δ1 , δ2 ) > 0 so that c3 > 0 and then take m = m(c3 , , M[u 0 ]) large enough so that the right side of (4.8) is bounded by a strictly negative constant. This implies that the maximal interval of existence I is finite.
The next two lemmas provide some additional estimates that hold under the hypotheses (4.2) and (4.3) of Theorem 4.2. These estimates will be needed for the compactness and rigidity results in §5–6. Lemma 4.4. (Lower bound on the convexity of the variance) Let u 0 ∈ H 1 (R3 ) satisfy (4.2) and (4.3). Furthermore, take δ > 0 such that M[u 0 ]E[u 0 ] < (1 − δ)M[Q]E[Q]. If u is the solution of the Cauchy problem (1.1) with initial data u 0 , then there exists cδ > 0 such that for all t ∈ R, 24E[u] − 4∇u(t)2L 2 = 8∇u(t)2L 2 − 6u(t)4L 4 ≥ cδ ∇u(t)2L 2 .
(4.9)
Proof. By the analysis in the proof of Theorem 4.2, there exists δ2 = δ2 (δ) > 0 such that for all t ∈ R, u 0 2L 2 ∇u(t)2L 2 ≤ (1 − δ2 )2 ∇ Q2L 2 Q2L 2 .
(4.10)
Let h(t) =
1 2 2 2 4 8u , ∇u(t) − 6u u(t) 0 L2 0 L2 L2 L4 Q2L 2 ∇ Q2L 2
and set g(y) = y 2 − y 3 . By the Gagliardo-Nirenberg estimate (3.1) with sharp constant cGN and (3.3), ∇u(t) L 2 u 0 L 2 . h(t) ≥ 8 g ∇ Q L 2 Q L 2 By (4.10), we restrict attention to 0 ≤ y ≤ 1 − δ2 . By an elementary argument, there exists c = c(δ2 ) such that g(y) ≥ c y 2 if 0 ≤ y ≤ 1 − δ2 , which completes the proof.
Lemma 4.5. (Comparability of gradient and energy) Let u 0 ∈ H 1 (R3 ) satisfy (4.2) and (4.3). Then 1 1 ∇u(t)2L 2 ≤ E[u] ≤ ∇u(t)2L 2 . 6 2 Proof. The second inequality is immediate from the definition of energy. The first one is obtained by observing that 1 1 1 1 ∇u2L 2 − u4L 4 ≥ ∇u2L 2 1 − 21 cGN ∇u L 2 u L 2 ≥ ∇u2L 2 , 2 4 2 6 where we used (3.1), (3.3) and (4.4).
In the proofs of Proposition 5.4 and 5.5, we will need the following result called existence of wave operators since the map + : ψ + → v0 is called the wave operator (see the proposition for the meaning of ψ + and v0 ).
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Proposition 4.6. (Existence of wave operators) Suppose ψ + ∈ H 1 and 1 + 2 ψ L 2 ∇ψ + 2L 2 < M[Q]E[Q]. 2
(4.11)
Then there exists v0 ∈ H 1 such that v solving (1.1) with initial data v0 is global in H 1 with 1 ∇v(t) L 2 v0 L 2 ≤ Q L 2 ∇ Q L 2 , M[v] = ψ + 2L 2 , E[v] = ∇ψ + 2L 2 , 2 and lim v(t) − eit ψ + H 1 = 0.
t→+∞
Moreover, if eit ψ + S( H˙ 1/2 ) ≤ δsd , then v0 H˙ 1/2 ≤ 2 ψ + H˙ 1/2 and v S( H˙ 1/2 ) ≤ 2 eit ψ + S( H˙ 1/2 ) . Proof. We want to solve the integral equation +∞
ei(t−t ) (|v|2 v)(t ) dt
v(t) = eit ψ + − i
(4.12)
t
first for t ≥ T with T large. This is achieved as in the proof of the small data scattering theory (Proposition 2.1), since there exists T 0 such that eit ψ + S( H˙ 1/2 ;[T,+∞)) ≤ δsd . By estimating (4.12), we obtain ∇v S(L 2 ;[T,+∞)) ≤ cψ + H˙ 1 + c∇(v 3 ) S (L 2 ) ≤ cψ + H˙ 1 + c∇v S(L 2 ;[T,+∞)) v2S( H˙ 1/2 ;[T,+∞)) , where in the last step, we used · S (L 2 ) ≤ · L 10/7 L 10/7 and the Hölder partition 7 10
=
3 1 1 10 + 5 + 5 . Thus, ∇v S(L 2 ;[T,+∞))
t
x
≤ 2 c ψ + H˙ 1 . Using this, we obtain similarly,
∇(v(t) − eit ψ + ) S(L 2 ;[T,+∞)) → 0 as T → +∞. Since v(t) − eit ψ + → 0 in H 1 as t → +∞, eit ψ + → 0 in L 4 as t → +∞, and ∇eit ψ + L 2 is conserved, we have 1 1 1 E[v] = lim ∇eit ψ + 2L 2 − eit ψ + 4L 4 = ψ + 2L 2 . t→+∞ 2 4 2 Immediately, we obtain M[v] = ψ + 2L 2 . Note that we now have M[v]E[v] < M[Q]E[Q] by (4.11). Observe that lim ∇v(t)2L 2 v2L 2 = lim ∇eit ψ + 2L 2 eit ψ + 2L 2
t→+∞
t→+∞
= ∇ψ + 2L 2 ψ + 2L 2 ≤ 2M[Q]E[Q] 1 = ∇ Q2L 2 Q2L 2 , 3 where in the last two steps we used (4.11) and (3.4). Take T sufficiently large so that ∇v(T ) L 2 v L 2 ≤ ∇ Q L 2 Q L 2 . By Theorem 4.2, we can evolve v(t) from T back to time 0.
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5. Compactness Definition 5.1. Suppose u 0 ∈ H 1 and u is the corresponding H 1 solution to (1.1) and [0, T ∗ ) the maximal forward time interval of existence. We shall say that SC(u 0 ) holds if T ∗ = +∞ and u S( H˙ 1/2 ) < ∞. To prove Theorem 1.1(1)(b), we must show that if u 0 L 2 ∇u 0 L 2 <Q L 2 ∇ Q L 2 , and M[u]E[u] < M[Q]E[Q], then SC(u 0 ) holds. We already know that under these hypotheses, we have an a priori bound on ∇u(t) L 2 , and thus, the maximal forward time of existence is T ∗ = +∞ (this is the energy subcritical case). The goal is, therefore, to show that the global-in-time H˙ 1/2 Strichartz norm is finite. 4 and u ∇u By Lemma 4.5, if M[u]E[u] < 16 δsd 0 L2 0 L 2 < Q L 2 ∇ Q L 2 , then 4 . u 0 4H˙ 1/2 ≤ u 0 2L 2 ∇u 0 2L 2 ≤ 6M[u]E[u] ≤ δsd
Thus, by the small data theory (Proposition 2.1), SC(u 0 ) holds. Let (M E)c be the number defined as the supremum over all δ for which the following statement holds true: “If u 0 is radial with u 0 L 2 ∇u 0 L 2 < Q L 2 ∇ Q L 2 and M[u]E[u] < δ, then SC(u 0 ) 4 ≤ (M E) . If (M E) ≥ M[Q]E[Q], then holds.” We then clearly have 0 < 16 δsd c c Theorem 1.1(1)(b) is true. We, therefore, proceed with the proof of Theorem 1.1(1)(b) by assuming that (M E)c < M[Q]E[Q] and ultimately deduce a contradiction (much later, in §6). By definition of (M E)c , we have (C.1) If u 0 is radial and u 0 L 2 ∇u 0 L 2 < Q L 2 ∇ Q L 2 and M[u]E[u] < (M E)c , then SC(u 0 ) holds. (C.2) There exists a sequence of radial solutions u n to (1.1) with corresponding initial data u n,0 such that u n,0 L 2 ∇u n,0 L 2 < Q L 2 ∇ Q L 2 and M[u n ]E[u n ] (M E)c as n → +∞, for which SC(u n,0 ) does not hold for any n. The goal of this section is to use the above sequence u n,0 (rescaled so that u n,0 L 2 = 1 for all n) to prove the existence of an H 1 radial solution u c to (1.1) with initial data u c,0 such that u c,0 L 2 ∇u c,0 L 2 < Q L 2 ∇ Q L 2 and M[u c ]E[u c ] = (M E)c for which SC(u c,0 ) does not hold (Proposition 5.4). Moreover, we will show that K = { u c (t) | 0 ≤ t < +∞ } is precompact in H 1 (Proposition 5.5), which will enable us to show that for each > 0, there is an R > 0 such that, uniformly in t, we have |∇u c (t, x)|2 d x ≤ |x|>R
(Lemma 5.6). This will then play into the rigidity theorem of the next section that will ultimately lead to a contradiction. Before stating and proving Proposition 5.4, we introduce some preliminaries in the spirit of the results of Keraani [15], since we are not able to directly apply his results as was possible in Kenig-Merle [17]. Note in the following lemma that φn , ψ j and WnM are functions of x alone, in notational contrast to the analogous lemma in Keraani ([15] Prop. 2.6). Lemma 5.2. (Profile expansion) Let φn (x) be a radial uniformly bounded sequence in H 1 . Then for each M there exists a subsequence of φn , also denoted φn , and (1) For each 1 ≤ j ≤ M, there exists a (fixed in n) radial profile ψ j (x) in H 1 . j (2) For each 1 ≤ j ≤ M, there exists a sequence (in n) of time shifts tn .
Scattering of 3D Cubic NLS
451
(3) There exists a sequence (in n) of remainders WnM (x) in H 1 , such that φn =
M
j
e−itn ψ j + WnM .
j=1
The time sequences have a pairwise divergence property: For 1 ≤ i = j ≤ M, we have lim |t i n→+∞ n
j
− tn | = +∞.
(5.1)
The remainder sequence has the following asymptotic smallness property7 : lim eit WnM S( H˙ 1/2 ) = 0. lim M→+∞ n→+∞
(5.2)
For fixed M and any 0 ≤ s ≤ 1, we have the asymptotic Pythagorean expansion, φn 2H˙ s =
M
ψ j 2H˙ s + WnM 2H˙ s + on (1).
(5.3)
j=1
Note that we do not claim that the remainder WnM is small in any Sobolev norm, i.e. for all we know it might be true that for some s, 0 ≤ s ≤ 1, we have lim inf lim WnM H˙ s > 0 . M→+∞ n→+∞
Fortunately, the Strichartz norm smallness (5.2) will suffice in our application. Proof. Since φn is assumed uniformly bounded in H 1 , let c1 be such that φn H 1 ≤ c1 . Note the interpolation inequality θ v L q L r ≤ v1−θ q˜ r˜ v L ∞ L 3 , t
x
Lt Lx
t
x
where (q, r ) is any H˙ 1/2 Strichartz admissible pair (so
= 1 and 3 ≤ r < +∞), Observe that (q, ˜ r˜ ) is also H˙ 1/2 2 q
+
3 r
(so 0 < θ ≤ 1), r˜ = 2r , and q˜ = θ = Strichartz admissible. By this inequality and the Strichartz estimates (for 0 < θ < 35 )8 , we get 3 2r −3
4r 2r −3 .
eit WnM θL ∞ L 3 . eit WnM L qt L r ≤ c WnM 1−θ H˙ 1/2 x
t
x
Since we will have WnM H˙ 1/2 ≤ c1 , it will suffice for us to show that = 0. lim lim sup eit WnM L ∞ 3 t Lx M→+∞
n→+∞
7 We can always pass to a subsequence in n with the property that eit W M n S( H˙ 1/2 ) converges. Therefore, we use lim and not lim sup or lim inf. Similar remarks apply for the limits that appear in the Pythagorean expansion. 8 This restriction is for consistency with our definition of S( H˙ 1/2 ) in §2.
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J. Holmer, S. Roudenko
Let A1 ≡ lim supn→+∞ eit φn L ∞ 3 . If A 1 = 0, the proof is complete with t Lx j ψ = 0 for all 1 ≤ j ≤ M. Suppose A1 > 0. Pass to a subsequence so that 1 limn→+∞ eit φn L ∞ 3 = A 1 . We will show that there is a time sequence tn and a t Lx profile ψ 1 ∈ H 1 such that eitn φn ψ 1 and ψ 1 H˙ 1/2 ≥ 1
A51 . 210 c14
For r > 1 yet to be
chosen, let χ (x) be a radial Schwartz function such that χˆ (ξ ) = 1 for r1 ≤ |ξ | ≤ r and χ(ξ ˆ ) is supported in 2r1 ≤ |ξ | ≤ 2r . By Sobolev embedding, eit φn − χ ∗ eit φn 2L ∞ L 3 ≤ |ξ |(1 − χˆ (ξ ))2 |φˆ n (ξ )|2 dξ t x 2 ˆ |ξ ||φn (ξ )| dξ + |ξ ||φˆ n (ξ )|2 dξ ≤ |ξ |≤ r1
|ξ |≥r
1 1 ≤ φn 2L 2 + φn 2H˙ 1 r r c12 . ≤ r Take r =
16c12 A21
so that
c12 r
=
A21 16 ,
and then we have for n large, 1 A1 . 2
χ ∗ eit φn L ∞ 3 ≥ t Lx Note that
∞ χ ∗ eit φn 3L ∞ L 3 ≤ χ ∗ eit φn 2L ∞ L 2 χ ∗ eit φn L ∞ t Lx t
t
x
≤
φn 2L 2 χ
∗e
x
it
∞, φn L ∞ t Lx
and thus, we have ∞ ≥ χ ∗ eit φn L ∞ t Lx
A31 8c12
.
Since φn are radial functions, so are χ ∗ eit φn , and by the radial Gagliardo–Nirenberg inequality, we obtain ∞ ≤ χ ∗ eit φn L ∞ t L {|x|≥R}
1 c1 1/2 1/2 χ ∗ eit φn L 2 ∇χ ∗ eit φn L 2 ≤ . x x R R
Therefore, by selecting R large enough, ∞ χ ∗ eit φn L ∞ ≥ t L {|x|≤R}
A31 16c12
.
Let tn1 and xn1 (with |xn1 | ≤ R) be sequences such that for each n, |χ ∗ eitn φn (xn1 )| ≥ 1
A31 32c12
,
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453
or, written out,
R3
χ (xn1
− y) e
itn1
A3 φn (y) dy ≥ 12 . 32c1
Pass to a subsequence such that xn1 → x 1 (possible since |xn1 | ≤ R). Then since χ (x 1 −·) − χ (xn1 − ·) L 3/2 → 0 as n → +∞, we have 3 1 itn1 ≥ A1 . χ (x − y) e φ (y) dy n 64c2 3 R 1 Consider the sequence eitn φn , which is uniformly bounded in H 1 . Pass to a subsequence 1 so that eitn φn ψ 1 , with ψ 1 ∈ H 1 radial and ψ 1 H 1 ≤ lim sup φn H 1 ≤ c1 . By the above, we have 3 1 1 ≥ A1 . χ (x − y) ψ (y) dy 64c2 3 R 1 1
By Plancherel and Cauchy-Schwarz applied to the left side of the above inequality, we obtain χ H˙ −1/2 ψ 1 H˙ 1/2 ≥
A31 64c12
.
By converting to radial coordinates, we can estimate χ H˙ −1/2 ≤ r . Thus, ψ 1 H˙ 1/2 ≥
A31 64c12
·
A5 1 = 101 4 . r 2 c1
Let Wn1 = φn − e−itn ψ 1 . Since eitn φn ψ 1 , we have that for any 0 ≤ s ≤ 1, 1
1
φn , e−itn ψ 1 H˙ s = eitn φn , ψ 1 H˙ s → ψ 1 2H˙ s , 1
1
(5.4)
and, by expanding Wn1 2H˙ s , we obtain lim Wn1 2H˙ s = lim φn 2H˙ s − ψ 1 2H˙ s .
n→+∞
n→+∞
From this with s = 1 and s = 0 we deduce that Wn1 H 1 ≤ c1 . Let A2 = lim supn→+∞ eit Wn1 L ∞ 3 . If A 2 = 0, then we are done. If A 2 > 0, t Lx then repeat the above argument, with φn replaced by Wn1 to obtain a sequence of time 2 shifts tn2 and a profile ψ 2 ∈ H 1 such that eitn Wn1 ψ 2 and ψ 2 H˙ 1/2 ≥
A52 210 c14
.
We claim that |tn2 − tn1 | → +∞. Indeed, suppose we pass to a subsequence such that tn2 − tn1 → t 21 finite. Then ei(tn −tn ) [eitn φn − ψ 1 ] = eitn [φn − e−itn ψ 1 ] = eitn Wn1 ψ 2 . 2
1
1
2
1
2
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J. Holmer, S. Roudenko
Since tn2 − tn1 → t 21 and eitn φn − ψ 1 0, the left side of the above expression 1 2 converges weakly to 0, so ψ 2 = 0, a contradiction. Let Wn2 = φn − eitn ψ 1 − eitn ψ 2 . Note that 1
φn , eitn ψ 2 H˙ s = e−itn φn , ψ 2 H˙ s 2
2
= e−itn (φn − eitn ψ 1 ), ψ 2 H˙ s + on (1) 2
1
= e−itn Wn1 , ψ 2 H˙ s + on (1) 2
→ ψ 2 2H˙ s , where the second line follows from the fact that |tn1 − tn2 | → ∞. Using this and (5.4), we compute lim Wn2 2H˙ s = lim φn 2H˙ s − ψ 1 2H˙ s − ψ 2 2H˙ s ,
n→+∞
n→+∞
and thus, Wn2 H 1 ≤ c1 . We continue inductively, constructing a sequence tnM and a profile ψ M such that M it e n WnM−1 ψ M and A5 (5.5) ψ M H˙ 1/2 ≥ 10M4 . 2 c1 j
Suppose 1 ≤ j < M. We shall show that |tnM − tn | → +∞ inductively by assuming j+1 that |tnM − tn | → +∞, . . . , |tnM − tnM−1 | → +∞. Suppose, passing to a subsequence j that tnM − tn → t M j finite. Note that M −t j ) n
ei(tn
M
= eitn
j
(eitn Wn
j−1
M −t j+1 ) n
− ψ j ) − ei(tn
M −t M−1 ) n
ψ j+1 − · · · − ei(tn
ψ M−1
WnM−1 .
The left side converges weakly to 0, while the right side converges weakly to ψ M , which is 1 M nonzero; contradiction. This proves (5.1). Let WnM = φn −e−itn ψ 1 −· · ·−e−itn ψ M . Note that φn , e−itn
M
M
φn , ψ M H˙ s
M
(φn − eitn ψ 1 − · · · − eitn
M
WnM−1 , ψ M H˙ s + on (1),
ψ M = eitn = eitn = eitn
M−1
1
ψ M−1 ), ψ M H˙ s + on (1)
where the middle line follows from the pairwise divergence property (5.1). Thus, M φn , e−itn ψ M → ψ M 2H˙ s . The expansion (5.3) is then shown to hold by expanding WnM 2H˙ s . By (5.5) and (5.3) with s = 21 , we have +∞
M=1
A5M
2
210 c14
and hence, A M → 0 as M → +∞.
≤ lim φn 2H˙ 1/2 ≤ c12 , n→+∞
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455
Corollary 5.3. (Energy Pythagorean expansion) In the situation of Lemma 5.2, we have E[φn ] =
M
j
E[e−itn ψ j ] + E[WnM ] + on (1).
(5.6)
j=1 1 → L 4 (which follows from the radial Proof. We will use the compact embedding Hrad rad j Gagliardo–Nirenberg estimate of Strauss [26]) to address a j for which tn converges to a finite number (if one exists). We will also use the decay of linear Schrödinger solutions in the L 4 norm as time → ∞. There are two cases to consider. j
Case 1. There exists some j for which tn converges to a finite number, which without loss we assume is 0. In this case we will show that lim WnM L 4x = 0, for M > j,
n→+∞
lim e−itn ψ i L 4x = 0, for all i = j, i
n→+∞
and lim φn L 4 = ψ j L 4 ,
n→+∞
which, combined with (5.3) for s = 1, gives (5.6). j
Case 2. For all j, |tn | → ∞. In this case we will show that j
lim e−itn ψ j L 4x = 0, for all j
n→+∞
and lim φn L 4 = lim WnM L 4 ,
n→+∞
n→+∞
which, combined with (5.3) for s = 1, gives (5.6). j−1
Proof of Case 1. In this situation, we have, from the proof of Lemma 5.2 that Wn 1 → L 4 , it follows that W j−1 → ψ j ψ j . By the compactness of the embedding Hrad n rad i strongly in L 4 . Let i = j. Then we claim that eitn ψ i L 4 → 0 as n → ∞. Indeed, j since tn = 0, by (5.1), we have |tni | → +∞. For a function ψ˜ i ∈ H˙ 3/4 ∩ L 4/3 , from Sobolev embedding and the L p spacetime decay estimate of the linear flow, we obtain j c eitn ψ i L 4 ≤ cψ i − ψ˜ i H˙ 3/4 + i 1/4 ψ˜ i L 4/3 . |tn | By approximating ψ i by ψ˜ i ∈ Cc∞ in H˙ 3/4 and sending n → +∞, we obtain the claim. Recalling that j−1
= φn − e−itn ψ 1 − · · · − e−itn 1
j−1
Wn
ψ j−1 ,
we conclude that φn → ψ j strongly in L 4 . Recalling that j−1
WnM = (Wn
j+1
− ψ j ) − e−itn
ψ j+1 − · · · − e−itn
we also conclude that WnM → 0 strongly in L 4 for M > j.
M
ψM,
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J. Holmer, S. Roudenko
Proof of Case 2. Similar to the proof of Case 1.
Proposition 5.4. (Existence of a critical solution) There exists a global (T ∗ = +∞) solution u c in H 1 with initial data u c,0 such that u c,0 L 2 = 1, E[u c ] = (M E)c < M[Q]E[Q], ∇u c (t) L 2 < Q L 2 ∇ Q L 2 for all 0 ≤ t < +∞, and u c S( H˙ 1/2 ) = +∞. Proof. We consider the sequence u n,0 described in the introduction to this section. Rescale it so that u n,0 L 2 = 1; this rescaling does not affect the quantities M[u n ]E[u n ] and u n,0 L 2 ∇u n,0 L 2 . After this rescaling, we have ∇u n,0 L 2 < Q L 2 ∇ Q L 2 and E[u n ] (M E)c . Each u n is global and non-scattering, i.e. u n S( H˙ 1/2 ) = +∞. Apply the profile expansion lemma (Lemma 5.2) to u n,0 (which is now uniformly bounded in H 1 ) to obtain M
j e−itn ψ j + WnM , (5.7) u n,0 = j=1
where M will be taken large later. By the energy Pythagorean expansion (Corollary 5.3), we have M
j=1
j
lim E[e−itn ψ j ] + lim E[WnM ] = lim E[u n,0 ] = (M E)c ,
n→+∞
n→+∞
n→+∞
and thus (recalling that each energy is ≥ 0 — see Lemma 4.5), j
lim E[e−itn ψ j ] ≤ (M E)c ∀ j.
(5.8)
n→+∞
Also by s = 0 of (5.3), we have M
j=1
M[ψ j ] + lim M[WnM ] = lim M[u n,0 ] = 1. n→+∞
n→+∞
(5.9)
Now we consider two cases; we will show that Case 1 leads to a contradiction and thus does not occur; Case 2 will manufacture the desired critical solution u c . Case 1. More than one ψ j = 0. By (5.9), we necessarily have M[ψ j ] < 1 for each j, which by (5.8) implies that for n sufficiently large, j
j
M[e−itn ψ j ]E[e−itn ψ j ] < (M E)c . j
For a given j, there are two cases to consider: Case (a). If |tn | → +∞ (passing to j j j a subsequence we have tn → +∞ or tn → −∞), we have e−itn ψ j L 4 → 0 (as discussed in the proof of Corollary 5.3), and thus, 1 ψ j 2L 2 ∇ψ j 2L 2 < (M E)c 2
Scattering of 3D Cubic NLS
457
j
(we have used ∇e−itn ψ j L 2 = ∇ψ j L 2 ). Let NLS(t)ψ denote the solution to (1.1) with initial data ψ. By the existence of wave operators (Proposition 4.6), there exists ψ˜ j such that j
j NLS(−tn )ψ˜ j − e−itn ψ j H 1 → 0, as n → +∞
with ψ˜ j L 2 ∇ NLS(t)ψ˜ j L 2 < Q L 2 ∇ Q L 2 , 1 M[ψ˜ j ] = ψ j 2L 2 , E[ψ˜ j ] = ∇ψ j 2L 2 , 2 and thus, M[ψ˜ j ]E[ψ˜ j ] < (M E)c , NLS(t)ψ˜ j S( H˙ 1/2 ) < +∞. j
Case (b). On the other hand, if for a given j we have tn → t∗ finite (and there can be at most one such j by (5.1)), we note that by continuity of the linear flow in H 1 , j
e−itn ψ j → e−it∗ ψ j strongly in H 1 , and we let ψ˜ j = NLS(t∗ )[e−it∗ ψ j ] so that NLS(−t∗ )ψ˜ j = e−it∗ ψ j . In either case, associated to each original profile ψ j we now have a new profile ψ˜ j such that j
j NLS(−tn )ψ˜ j − e−itn ψ j H 1 → 0 as n → +∞. j
j It now follows that we can replace e−itn ψ j by NLS(−tn )ψ˜ j in (5.7) to obtain
u n,0 =
M
NLS(−tn )ψ˜ j + W˜ nM , j
j=1
where
lim
lim eit W˜ nM S( H˙ 1/2 ) = 0.
M→+∞ n→+∞
The idea of what follows is that we approximate NLS(t)u n,0 ≈
M
NLS(t − tn )ψ˜ j j
j=1
via a perturbation theory argument, and since the right side has bounded S( H˙ 1/2 ) norm, so must the left-side, which is a contradiction. To carry out this argument, we introduce the notation v j (t) = NLS(t)ψ˜ j , u n (t) = NLS(t)u n,0 , and9 u˜ n (t) =
M
j
v j (t − tn ).
j=1 9 u˜ , and e also depend on M, but we have suppressed the notation. n n
458
J. Holmer, S. Roudenko
Then i∂t u˜ n + u˜ n + |u˜ n |2 u˜ n = en , where en = |u˜ n |2 u˜ n −
M
j
j
|v j (t − tn )|2 v j (t − tn ).
j=1
We claim that there is a (large) constant A (independent of M) with the property that for any M, there exists n 0 = n 0 (M) such that for n > n 0 , u˜ n S( H˙ 1/2 ) ≤ A. Moreover, we claim that for each M and > 0 there exists n 1 = n 1 (M, ) such that for n > n1, en L 10/3 L 5/4 ≤ . x
t
Note that since u˜ n (0) − u n (0) = W˜ nM , there exists M1 = M1 ( ) sufficiently large such that for each M > M1 there exists n 2 = n 2 (M) such that n > n 2 implies eit (u˜ n (0) − u n (0)) S( H˙ 1/2 ) ≤ . Thus, we may apply Proposition 2.3 (long-time perturbation theory) to obtain that for n and M sufficiently large, u n S( H˙ 1/2 ) < ∞, a contradiction.10 Therefore, it remains to establish the above claims, and we begin with showing that u˜ n S( H˙ 1/2 ) ≤ A for n > n 0 = n 0 (M), where A is some large constant independent of M. Let M0 be large enough so that eit W˜ nM0 S( H˙ 1/2 ) ≤ δsd . Then for each j > M0 , we have eit ψ j S( H˙ 1/2 ) ≤ δsd , and by the second part of Proposition 4.6 we obtain v j S( H˙ 1/2 ) ≤ 2eit ψ j S( H˙ 1/2 ) for j > M0 .
(5.10)
By the elementary inequality: for a j ≥ 0, ⎛ ⎞5/2 M M
⎝ 5/2 aj⎠ − a j ≤ cM |a j ||ak |3/2 , j=1 j=k j=1 we have u˜ n 5L 5 L 5 = t
x
M0
j=1
≤
M0
j=1
M
v j 5L 5 L 5 + t
x
v j 5L 5 L 5 t x
v j 5L 5 L 5 + cross terms t
j=M0 +1
+2
5
M
j=M0 +1
x
(5.11) e
it
ψ j 5L 5 L 5 t x
+ cross terms ,
10 The order of logic here is: The constant A, which is independent of M, is put into Prop. 2.3, which gives a suitable . We then take M1 = M1 ( ) as above, and then take n = max(n 0 , n 1 , n 2 ).
Scattering of 3D Cubic NLS
459
where we used (5.10) to bound middle terms. On the other hand, by (5.7), eit u n,0 5L 5 L 5 = t
x
M0
M
eit ψ j 5L 5 L 5 + t
j=1
x
eit ψ j 5L 5 L 5 + cross terms. (5.12) t
j=M0 +1
x
The “cross terms” are made ≤ 1 by taking n 0 = n 0 (M) large enough and appealing to (5.1). We observe that since eit u n,0 L 5 L 5 ≤ cu n,0 H˙ 1/2 ≤ c , (5.12) shows that t x it ψ j 5 e is bounded independently of M provided n > n 0 . the quantity M 5 5 j=M0 +1 Lt Lx
Then, (5.11) gives that u˜ n L 5 L 5 is bounded independently of M provided n > n 0 . A t x similar argument establishes that u˜ n L ∞ 3 is bounded independently of M for n > n 0 . t Lx Interpolation between these exponents gives that u˜ n L 20 L 10/3 is bounded independently x t of M for n > n 0 . Finally, by applying the Kato estimate (2.2) to the integral equation for i∂t u˜ n + u˜ n + |u˜ n |2 u˜ n = en and using that en S( H˙ −1/2 ) ≤ 1 (proved next), we obtain that u˜ n S( H˙ 1/2 ) is bounded independently of M for n > n 0 . We now address the next claim, that for each M and > 0, there exists n 1 = n 1 (M, ) such that for n > n 1 , en L 10/3 L 5/4 ≤ . The expansion of en consists of ∼ M 3 cross x t terms of the form v j (t − tn )v k (t − tnk )v (t − tn ), j
where not all three of j, k, and are the same. Assume, without loss, that j = k, and j thus, |tn − tnk | → ∞ as n → +∞. We estimate v j (t − tn )v k (t − tnk )v (t − tn ) L 10/3 L 5/4 j
x
t
≤ v (t j
j − tn )v k (t
− tnk ) L 10 L 5/3 v (t x t
− tn ) L 5 L 5 . t
x
Now observe that j
v j (t − (tn − tnk )) · v k (t) L 10 L 5/3 → 0, t
10/3
since v j and v k belong to L 20 t Lx
x
j
and |tn − tnk | → ∞.
Case 2. ψ 1 = 0, and ψ j = 0 for all j ≥ 2. By (5.9), we have M[ψ 1 ] ≤ 1 and by (5.8), we have limn→+∞ E[e−itn ψ 1 ] ≤ (M E)c . If tn1 converges (to 0 without loss of generality), we take ψ˜ 1 = ψ 1 and then we 1 have NLS(−tn1 )ψ˜ 1 − e−itn ψ 1 H 1 → 0 as n → +∞. If, on the other hand, tn1 → +∞, 1 then since eitn ψ 1 L 4 → 0, 1
1 1 ∇ψ 1 2L 2 = lim E[e−itn ψ 1 ] ≤ (M E)c . n→+∞ 2 Thus, by the existence of wave operators (Proposition 4.6), there exists ψ˜ 1 such that M[ψ˜ 1 ] = M[ψ 1 ] ≤ 1, E[ψ˜ 1 ] = 21 ∇ψ 1 2L 2 ≤ (M E)c , and NLS(−tn1 )ψ˜ 1 − e−itn ψ 1 H 1 → 0 as n → +∞. 1
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In either case, let W˜ nM = WnM + (e−itn ψ 1 − NLS(−tn1 )ψ˜ 1 ). Then, by the Strichartz estimates, 1
1 e−it W˜ nM S( H˙ 1/2 ) ≤ e−it WnM S( H˙ 1/2 ) + ce−itn ψ 1 − NLS(−tn1 )ψ˜ 1 H˙ 1/2 ,
and therefore, limn→+∞ e−it W˜ nM S( H˙ 1/2 ) = limn→+∞ e−it WnM S( H˙ 1/2 ) . Hence, we now have u n,0 = NLS(−tn1 )ψ˜ 1 + W˜ nM with M[ψ˜ 1 ] ≤ 1, E[ψ˜ 1 ] ≤ (M E)c , and lim sup lim W˜ nM S( H˙ 1/2 ) = 0 . M→+∞ n→+∞
Let u c be the solution to (1.1) with initial data u c,0 = ψ˜ 1 . Now we claim that u c S( H˙ 1/2 ) = ∞, and thus, M[u c ] = 1 and E[u c ] = (M E)c , which will complete the proof. To establish this claim, we use a perturbation argument similar to that in Case 1. Suppose A := NLS(t − tn1 )ψ˜ 1 S( H˙ 1/2 ) = NLS(t)ψ˜ 1 S( H˙ 1/2 ) = u c S( H˙ 1/2 ) < ∞. Obtain 0 = 0 (A) from the long-time perturbation theory (Proposition 2.3), and then take M sufficiently large and n 2 = n 2 (M) sufficiently large so that n > n 2 implies W˜ nM S( H˙ 1/2 ) ≤ 0 . We then repeat the argument in Case 1 using Proposition 2.3 to obtain that there exists n large for which u n S( H˙ 1/2 ) < ∞, a contradiction.
Proposition 5.5. (Precompactness of the flow of the critical solution) With u c as in Proposition 5.4, let K = { u c (t) | t ∈ [0, +∞) } ⊂ H 1 . Then K is precompact in H 1 (i.e. K¯ is compact in H 1 ). Proof. Take a sequence tn → +∞; we shall argue that u c (tn ) has a subsequence converging in H 1 .11 Take φn = u c (tn ) (a uniformly bounded sequence in H 1 ) in the profile expansion lemma (Lemma 5.2) to obtain profiles ψ j and an error WnM such that u c (tn ) =
M
j
e−itn ψ j + WnM
j=1 j
with |tn −tnk | → +∞ as n → +∞ for fixed j = k. By the energy Pythagorean expansion (Corollary 5.3), we have M
j=1
j
lim E[e−itn ψ j ] + lim E[WnM ] = E[u c ] = (M E)c ,
n→+∞
n→+∞
11 By time continuity of the solution in H 1 , we of course do not need to consider the case when t is bounded n and thus has a subsequence convergent to some finite time.
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and thus (recalling that each energy is ≥ 0 – see Lemma 4.5), j
lim E[e−itn ψ j ] ≤ (M E)c ∀ j .
n→+∞
Also by s = 0 of (5.3), we have M
M[ψ j ] + lim M[WnM ] = lim M[u n,0 ] = 1. n→+∞
j=1
n→+∞
We now consider two cases, just as in the proof of Proposition 5.4; both Case 1 and Case 2 will lead to a contradiction. Case 1. More than one ψ j = 0. The proof that this leads to a contradiction is identical to that in Proposition 5.4, so we omit it. Case 2. Only ψ 1 = 0 and ψ j = 0 for all 2 ≤ j ≤ M, so that u c (tn ) = e−itn ψ 1 + WnM . 1
(5.13)
Just as in the proof of Proposition 5.4, Case 2, we obtain that M[ψ 1 ] = 1,
lim E[e−itn ψ 1 ] = (M E)c , 1
n→+∞
lim M[WnM ] = 0, and
n→+∞
lim E[WnM ] = 0 .
n→+∞
By the comparability lemma (Lemma 4.5), lim WnM H 1 = 0 .
(5.14)
n→+∞
Next, we show that (a subsequence of) tn1 converges.12 Suppose that tn1 → −∞. Then eit u c (tn ) S( H˙ 1/2 ;[0,+∞)) ≤ ei(t−tn ) ψ 1 S( H˙ 1/2 ;[0,+∞)) + eit WnM S( H˙ 1/2 ;[0,+∞)) . 1
Since lim ei(t−tn ) ψ 1 S( H˙ 1/2 ;[0,+∞)) = lim eit ψ 1 S( H˙ 1/2 ;[−t 1 ,+∞)) = 0 1
n→+∞
n→+∞
n
and eit WnM S( H˙ 1/2 ) ≤ 21 δsd , we obtain a contradiction to the small data scattering theory (Proposition 2.1) by taking n sufficiently large. On the other hand, suppose that tn1 → +∞. Then we can similarly argue that for n large, 1 δsd , 2 and thus, the small data scattering theory (Proposition 2.1) shows that eit u c (tn ) S( H˙ 1/2 ;(−∞,0]) ≤
u c S( H˙ 1/2 ;(−∞,tn ]) ≤ δsd . Since tn → +∞, by sending n → +∞ in the above, we obtain u c S( H˙ 1/2 ;(−∞,+∞)) ≤ δsd , a contradiction. Thus, we have shown that tn1 converges to some finite t 1 . 1 1 Since e−itn ψ 1 → e−it ψ 1 in H 1 and (5.14) holds, (5.13) shows that u c (tn ) converges in H 1 .
12 In the rest of the argument, take care not to confuse t 1 (associated with ψ 1 ) with t . n n
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Lemma 5.6. (Precompactness of the flow implies uniform localization) Let u be a solution to (1.1) such that K = { u(t) | t ∈ [0, +∞) } is precompact in H 1 . Then for each > 0, there exists R > 0 so that |∇u(x, t)|2 ≤ , for all 0 ≤ t < +∞. |x|>R
Proof. If not, then there exists > 0 and a sequence of times tn such that |∇u(x, tn )|2 d x ≥ . |x|>n
Since K is precompact, there exists φ ∈ H 1 such that, passing to a subsequence of tn , we have u(tn ) → φ in H 1 . By taking n large, we have both 1 |∇φ(x)|2 ≤ 4 |x|>n and
R3
which is a contradiction.
|∇(u(x, tn ) − φ(x))|2 d x ≤
1 , 4
6. Rigidity Theorem We now prove the rigidity theorem. Theorem 6.1. (Rigidity) Suppose u 0 ∈ H 1 satisfies
and
M[u 0 ]E[u 0 ] < M[Q]E[Q]
(6.1)
u 0 L 2 ∇u 0 L 2 < Q L 2 ∇ Q L 2 .
(6.2)
Let u be the global H 1 solution of (1.1) with initial data u 0 and suppose that K = { u(t) | t ∈ [0, +∞) } is precompact in H 1 . Then u 0 = 0. Proof. Let φ ∈ C0∞ , radial, with φ(x) =
|x|2 0
for |x| ≤ 1 . for |x| ≥ 2
For R > 0 define z R (t) = R 2 φ( Rx ) |u(x, t)|2 d x. Then x
¯ ∇u(t) (∇φ) |z R (t)| ≤ 2R u(t) dx ≤ c R |∇u(t)| |u(t)| d x. (6.3) R 0<|x|<2R
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Using Hölder’s inequality and Theorem 1.1(1)(a), we bound the previous expression by c R ∇u(t) L 2 u L 2 ≤ c R ∇ Q L 2 Q L 2 = c˜ R. Thus, we obtain
|z R (t) − z R (0)| ≤ 2 c˜ R for t > 0. Next we estimate using the localized virial identity (4.1): x x |x| 1 |∇u|2 − 2 (2 φ) |u|2 − (φ) |u|4 z
R (t) = 4 φ
R R R R |x| c |∇u|2 − 2 ≥8 |∇u|2 + 4 φ
|u|2 R R R<|x|<2R |x|≤R R<|x|<2R −6 |u|4 − c |u|4
(6.4)
z
R (t)
≥ 8
|x|≤R
R<|x|<2R
|x|≤R
|∇u| − 6 2
|x|≤R
|u|
4
− c1
|u|2 2 4 |∇u| + 2 + |u| . R R<|x|<2R
Since (6.1)
holds, take δ > 0 such that M[u 0 ]E[u 0 ] ≤ (1 − δ)M[Q]E[Q]. Let = c1−1 cδ |∇u 0 |2 , where cδ is as in (4.9). 1 Since
{u(t)|t ∈ [0, ∞)} is precompact in H , by Lemma 5.6 there exists R1 > 0 such that |x|>R1 |∇u(t)|2 ≤ 19 . Next, because of mass conservation, there exists R2 > 0
such that R12 |u|2 < 19 . Finally, the radial Gagliardo-Nirenberg inequality (3.5) yields 2
the existence of R3 > 0 such that c c 1 |u(t)|4 ≤ 2 ∇u(t) L 2 (|x|>R3 ) u 0 3L 2 ≤ 2 ∇u 0 L 2 u 0 3L 2 ≤ , 9 R3 R3 |x|>R3
with R32 > 9 c ∇u 0 L 2 u 0 3L 2 ; in the above chain we used the gradient-energy comparability (Lemma 4.5) with t = 0 on the left side. Take R = max{R1 , R2 , R3 } to obtain 1 |u|2 |∇u|2 + 2 + |u|4 ≤ cδ |∇u 0 |2 . c1 (6.5) R 3 |x|>R By (4.9) and Lemma 4.5, we also have 2 4 8 |∇u| − 6 |u| ≥ cδ (6.6) |∇u 0 |2 . Splitting the integrals on the left side of the above expression into the regions {|x| > R} and {|x| < R} and applying (6.5), we get 2 8 |∇u|2 − 6 |u|4 ≥ cδ |∇u 0 |2 . 3 |x|≤R |x|≤R Hence, we obtain z
R (t) ≥ 13 cδ ∇u 0 2L 2 , which implies by integration from 0 to t that z R (t) − z R (0) ≥ 13 cδ ∇u 0 2L 2 t. Taking t large, we obtain a contradiction with (6.4), which can be resolved only if ∇u 0 L 2 = 0.
To complete the proof of Theorem 1.1(1)(b), we just apply Theorem 6.1 to u c constructed in Proposition 5.4, which by Proposition 5.5, meets the hypotheses in Theorem 6.1. Thus u c,0 = 0, which contradicts the fact that u c S( H˙ 1/2 ) = ∞. We have thus obtained that if u 0 L 2 ∇u 0 L 2 < Q L 2 ∇ Q L 2 and M[u]E[u] < M[Q]E[Q], then SC(u 0 ) holds, i.e. u S( H˙ 1/2 ) < ∞. By Proposition 2.2, H 1 scattering holds.
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7. Extensions to General Mass Supercritical, Energy Subcritical NLS Equations Consider the focusing mass supercritical, energy subcritical nonlinear Schrödinger equation NLS p (R N ):
i∂t u + u + |u| p−1 u = 0, (x, t) ∈ R N × R, u(x, 0) = u 0 (x) ∈ H 1 (R N ),
(7.1)
with the choice of nonlinear exponent p and the dimension N such that 0 < sc < 1, where sc =
2 N − . 2 p−1
The initial value problem with u 0 ∈ H 1 (R N ) is locally well-posed, see [10]. Denote by I = (−T∗ , T ∗ ) the maximal interval of existence of the solution u (e.g., see [3]). This implies that either T ∗ = +∞ or T ∗ < +∞ and ∇u(t) L 2 → ∞ as t → T ∗ (similar properties for T∗ ). The solutions to this problem satisfy mass and energy conservation laws, in particular, 1 1 2 E[u(t)] = |∇u(x, t)| − |u(x, t)| p+1 d x = E[u 0 ]. 2 p+1 The Sobolev H˙ sc norm is invariant under the scaling u → u λ (x, t) = λ2/( p−1) u(λx, λ2 t) (u λ is a solution of NLS p (R N ), if u is). The general Gagliardo–Nirenberg inequality (see [32]) is valid for values of p and N such that 0 ≤ sc < 113 : N ( p−1)
p+1
2− (N −2)( p−1)
u L p+1 (R N ) ≤ cGN ∇u L 2 (2R N ) u L 2 (R N )2
,
(7.2)
where p+1
cGN =
Q L p+1 (R N ) N ( p−1)
2− (N −2)( p−1)
∇ Q L 2 (2R N ) Q L 2 (R N )2
and Q is the ground state solution (positive solution of minimal L 2 norm) of the equation − (1 − sc )Q + Q + |Q| p−1 Q = 0.
(7.3)
(See [32] and references therein for discussion on the existence of positive solutions of class H 1 (R N ) to this equation.)14 The corresponding soliton solution to (7.1) is u(x, t) = ei(1−sc )t Q(x). The generalization of Theorem 4.2 (or Theorem 1.1 without scattering) to all c 0 < sc < 1 is based on using the scaling invariant quantity ∇usLc2 (R N ) · u1−s . L 2 (R N ) 13 It is also valid for s = 1 becoming the Sobolev embedding, see Remark 7.3. c 14 In the case p = 3, N = 3, we have s = 1 , and thus, the normalization for Q chosen here is different c 2
from that in the main part of this paper. The normalization of Q taken here was chosen since it enables us to draw a comparison with the sc = 1 endpoint result of Kenig-Merle [17].
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Theorem 7.1. Consider NLS p (R N ) with (possibly non-radial) u 0 ∈ H 1 (R N ) and 0 < sc < 1. Suppose that E[u 0 ]sc M[u 0 ]1−sc < E[Q]sc M[Q]1−sc , E[u 0 ] ≥ 0.
(7.4)
If (7.4) holds and c c ∇u 0 sLc2 (R N ) u 0 1−s < ∇ QsLc2 (R N ) Q1−s , L 2 (R N ) L 2 (R N )
(7.5)
then for any t ∈ I , c c < ∇ QsLc2 (R N ) Q1−s , ∇u(t)sLc2 (R N ) u 0 1−s L 2 (R N ) L 2 (R N )
(7.6)
and thus I = (−∞, +∞), i.e., the solution exists globally in time. If (7.4) holds and c c > ∇ QsLc2 (R N ) Q1−s , ∇u 0 sLc2 (R N ) u 0 1−s L 2 (R N ) L 2 (R N )
(7.7)
c c > ∇ QsLc2 (R N ) Q1−s . ∇u(t)sLc2 (R N ) u 0 1−s L 2 (R N ) L 2 (R N )
(7.8)
then for t ∈ I ,
Furthermore, if (a) |x|u 0 ∈ L 2 (R N ), or (b) u 0 is radial with N > 1 and 1 + N4 < p < min{1 + N 4−2 , 5}, then I is finite, and thus, the solution blows up in finite time. The finite-time blowup conclusion and (7.8) also hold if, in place of (7.4) and (7.7), we assume E[u 0 ] < 0. The proof of this theorem is similar to Theorem 4.2 and can be found in [13]. Remark 7.2. A finite-time T blow-up solution to a mass-supercritical energy subcritical NLS equation satisfies a lower bound on the blow-up rate: ∇u(t) L 2 ≥ c(T − t)−α , where α = α( p, d). This is obtained by scaling the local-theory, and it implies that the c quantity u 0 sLc2 ∇u(t)1−s → ∞, thus strengthening the conclusion (7.8). A stronger L2 result in this direction was recently obtained by Merle-Raphaël [22]: if u(t) blows-up in finite time T ∗ < ∞, then limt→T ∗ u(t) H˙ sc = ∞ (in fact, it diverges to ∞ with a logarithmic lower bound). Remark 7.3. This theorem provides a link between the mass critical NLS and energy critical NLS equations: Consider sc = 1; the theorem holds true by the work of Kenig-Merle [17, Sect. 3]. In this case there is no mass involved, the Gagliardo-Nirenberg inequality (7.2) becomes the Sobolev inequality, the condition (7.4) is E[u 0 ] < E[Q], where Q is the radial positive decreasing (class H˙ 1 (R N )) solution of Q + |Q| p−1 Q = 0, and the conditions (7.5) – (7.8) involve only the size of ∇u 0 L 2 in relation to ∇ Q L 2 (R N ) . In regard to the case sc = 0, (7.4) should be replaced by M[u] < M[Q] and (7.5) becomes the same statement. Under these hypotheses, the result of M. Weinstein [32] states that ∇u(t)2L 2
≤2
1−
u 0 2L 2 Q2L 2
−1 E[u],
E[u] > 0,
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and thus, global existence holds. We do not recover this estimate as a formal limit in (7.6),15 however, the conclusion about the global existence in this case does hold true. Our intention here is not to reprove the sc = 0 endpoint result – only to draw a connection to it. The hypothesis (7.7) should be replaced by its formal limit u 0 L 2 > Q L 2 , which is the complement of (7.4). Thus, the only surviving claim in Theorem 7.1 regarding blow-up in the sc = 0 limit is that it should hold under the hypothesis E[u 0 ] < 0. Blow-up under this hypothesis is the classical result of Glassey [12] in the case of finite variance, and in the radial case it is the result of Ogawa-Tsutsumi [23]. We expect that the proof of scattering for NLS p (R N ) with u 0 ∈ H 1 (R N ) and 0 < sc < 1 when (7.4) and (7.5) hold will carry over analogously to the N = 3, p = 3 case, provided (i) N > 1 (the radial assumption in 1D does not help to eliminate the translation defect of compactness); (ii) the Kato estimate (as in (2.2)) or the more refined Strichartz estimates by Foschi [9] are sufficient to complete the long term perturbation argument16 . Acknowledgement. J.H. is partially supported by an NSF postdoctoral fellowship. S.R. would like to thank Mary and Frosty Waitz for their great hospitality during her visits to Berkeley. We both thank Guixiang Xu for pointing out a few misprints and the referee for helpful suggestions.
References 1. Bergé, L., Alexander, T., Kivshar, Y.: Stability criterion for attractive Bose-Einstein condensates. Phys. Rev. A 62(2), 023607 (2000) 2. Bégout, P.: Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation. Adv. Math. Sci. Appl. 12(2), 817–827 (2002) 3. Cazenave, T.: Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, 10. New York: New York University, Courant Institute of Mathematical Sciences, Providence, RI: Amer. Math. Soc. 2003 4. Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R3 . Comm. Pure Appl. Math. 57(8), 987–1014 (2004) 5. Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3 . http://arxiv.org/PS_cache/math/pdf/0402/ 0402129v7.pdf, 2006 6. Donley, E., Claussen, N., Cornish, S., Roberts, J., Cornell, E., Wieman, C.: Dynamics of collapsing and exploding Bose-Einstein condensates. Nature 412, 295–299 (2001) 7. Duyckaerts, T., Roudenko, S.: Threshold solutions for the focusing 3d cubic Schrödinger equation, arxiv.org (preprint). arxiv:0806.1752[math.AP] 8. Fibich, G.: Some modern aspects of self-focusing theory. In: Self-Focusing: Past and Present, R.W. Boyd, S.G. Lukishova, Y.R. Shen, eds., to be published by Springer, in August 2008, available at http://www. math.tau.ac.il/%7Efibich/Manuscripts/Fibich_G_SF_Springer.pdf 9. Foschi, D.: Inhomogeneous Strichartz estimates. J. Hyper. Diff. Eq. 2(1), 1–24 (2005) 10. Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equation. I. The Cauchy problems; II. Scattering Theory, General Case. J. Funct. Anal. 32(1–32), 33–71 (1979) 11. Ginibre, J., Velo, G.: Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pures Appl. (9) 64(4), 363–401 (1985) 12. Glassey, R.T.: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equation. J. Math. Phys. 18(9), 1794–1797 (1977) 13. Holmer, J., Roudenko, S.: On blow-up solutions to the 3D cubic nonlinear Schrödinger equation. Appl. Math. Res. Express, Vol. 2007, article ID abm004, doi:10.1093/amrx/abm004, 2007 15 It might appear as a formal limit if one were to refine the estimate (7.6) to account for the gain resulting from the strict inequality in (7.4) (as we did in the proof of Theorem 4.2) before passing to the sc → 0 limit. 16 It may be necessary, for example, to express the estimates in terms of the norm D sc −α (· · · ) S( H˙ α ) for some 0 < α < sc , rather than · S( H˙ sc ) .
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14. Kato, T.: An L q,r -theory for nonlinear Schrödinger equations. In: Spectral and scattering theory and applications, Adv. Stud. Pure Math., 23, Tokyo: Math. Soc. Japan, 1994, pp. 223–238 15. Keraani, S.: On the defect of compactness for the Strichartz estimates of the Schrödinger equation. J. Diff. Eq. 175, 353–392 (2001) 16. Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955–980 (1998) 17. Kenig, C.E., Merle, F.: Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case. Invent. Math. 166(3), 645–675 (2006) 18. Kenig, C., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46(4), 527–620 (1993) 19. Kosmatova, N.E., Shvets, V.F., Zakharov, V.E.: Computer simulation of wave collapses in the nonlinear Schrödinger equation. Physica D 52, 16–35 (1991) 20. Kuznetsov, E.A., Juul Rasmussen, J., Rypdal, K., Turitsyn, S.K.: Sharper criteria for the wave collapse. Phys. D 87(1-4), 273–284 (1995) 21. Linares, F., Ponce, G.: Introduction to nonlinear dispersive equations. Rio de Janeiro: IMPA, 2004 22. Merle, F., Raphaël, P.: Blow-up of the critical norm for some radial L 2 supercritical nonlinear Schrödinger equations. http://arxiv.org/list/math/0605378v2, 2006 23. Ogawa, T., Tsutsumi, Y.: Blow-Up of H 1 solution for the Nonlinear Schrödinger Equation. J. Diff. Eq. 92, 317–330 (1991) 24. Schlein, B.: Derivation of the Gross-Pitaevskii hierarchy. In: Mathematical physics of quantum mechanics, Lecture Notes in Phys., 690, Berlin: Springer, 2006, pp. 279–293, see also arxiv.org http://arXiv.org/ list/math-ph/0504078, 2005 25. Soffer, A.: Soliton dynamics and scattering. In: Proceedings of the International Congress of Mathematicians, (Madrid, Spain, 2006), Zurich: European Math. Soc., 2006, pp. 459–471 26. Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55(2), 149–162 (1977) 27. Sulem, C., Sulem, P-L.: The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. New York: Springer-Verlag, 1999 28. Tao, T.: On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation. Dyn. Partial Differ. Equ. 1(1), 1–48 (2004) 29. Tao, T.: A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations. Dynamics of PDE 4, 1–53 (2007) 30. Tao, T.: Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC, Providence, RI: Amer. Math. Soc. 2006 31. Vilela, M.: Regularity of solutions to the free Schrödinger equation with radial initial data. Illinois J. Math. 45(2), 361–370 (2001) 32. Weinstein, M.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87(4), 567–576 (1982) 33. Zakharov, V.E.: Collapse of Langmuir waves, Zh. Eksp. Teor. Fiz. 62, 1745–1751 (1972), (in Russian); Sov. Phys. JETP, 35 908–914 (1972), (English) Communicated by P. Constantin
Commun. Math. Phys. 282, 469–518 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0502-9
Communications in
Mathematical Physics
Fluctuation Relations for Diffusion Processes Raphaël Chetrite, Krzysztof Gaw¸edzki Université de Lyon, C.N.R.S., ENS-Lyon, Laboratoire de Physique, 46 Allée d’Italie, 69364 Lyon, France. E-mail: [email protected] Received: 18 July 2007 / Accepted: 29 September 2007 Published online: 22 May 2008 – © Springer-Verlag 2008
Abstract: The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the time-reversed process. The origin of a variety of fluctuation relations is traced to the use of different time reversals. It is also shown how the application of the presented approach to the tangent process describing the joint evolution of infinitesimally close trajectories of the original process leads to a multiplicative extension of the fluctuation relations.
1. Introduction Nonequilibrium statistical mechanics attempts a statistical description of closed and open systems evolving under the action of time-dependent conservative forces or under time-independent or time dependent non-conservative ones. Fluctuation relations are robust identities concerning the statistics of entropy production or performed work in such systems. They hold arbitrarily far from thermal equilibrium. Close to equilibrium, they reduce to Green-Kubo or fluctuation-dissipation relations, usually obtained in the scope of linear response theory [87,44]. Historically, the study of fluctuation relations originated in the numerical observation of Evans, Cohen and Morriss [23] of a symmetry in the distribution of fluctuations of microscopic pressure in a thermostatted particle system driven by external shear. The symmetry related the probability of occurrence of positive and negative time averages of pressure over sufficiently long time intervals and predicted that the former is exponentially suppressed with respect to the latter. Ref. [23] attempted to explain this observation by a symmetry, induced by the time-reversibility, of the statistics of partial sums of finite-time Lyapunov exponents in dissipative dynamical systems. This was further elaborated in [25] where an argument was given explaining Member of C.N.R.S.
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such a symmetry in a transient situation when one starts with a simple state which evolves under dynamics, see also [26]. In refs. [35,36], Gallavotti and Cohen provided a theoretical explanation of the symmetry observed numerically in [23] employing the theory of uniformly hyperbolic dynamical systems. In this theory, the stationary states correspond to invariant measures of the SRB type [82] and the entropy production is described by the phase-space contraction [74]. The authors of [35,36] established a fluctuation theorem about the rate function describing the statistics of large deviations of the phase-space contraction in a time-reversible dynamics. To relate to the behavior of realistic systems, they formulated the chaotic hypothesis postulating that many such systems behave, for practical purposes, as the uniformly hyperbolic ones. They interpreted the numerical observations of ref. [23] as a confirmation of this hypothesis. The difference between the fluctuation relations for a transient situation analyzed in [25,26] and the stationary one discussed in [35,36] was subsequently stressed in [16]. The debate about the connection between the transient and stationary fluctuation relations still continues, see e.g. [77] and [33]. In another early development, Jarzynski established in [48] a simple transient relation for the statistics of fluctuations of work performed on a system driven by conservative time-dependent forces. This relation is now known under the name of the Jarzynski equality. A similar observation, but with more limited scope, was contained in earlier work [4–6], see [52] for a recent comparison. The simplicity of the Jarzynski equality and its possible applications to measurements of free-energy landscape for small systems attracted a lot of attention, see [72,73] and the references therein. The first studies of fluctuation relations dealt with the deterministic dynamics of finitely-many degrees of freedom. Such dynamics may be also used to model systems interacting with environment or with heat reservoirs. To this end, one employs simplified finite-dimensional models of reservoirs forced to keep their energy constant [24]. This type of models was often used in numerical simulations and in discussing fluctuation relations, see e.g. [33]. A more realistic treatment of reservoirs would describe them as infinite systems prepared in the thermal equilibrium state. Up to now, only infinite systems of non-interacting particles could be treated effectively, see [21,22]. A less realistic description of interaction with environment or with reservoirs consists of replacing them by a random noise, usually shortly correlated in time. This leads to Markovian stochastic evolution equations. Stochastic models are often easier to control than deterministic ones and they became popular in modeling nonequilibrium dynamics. In [49], Jarzynski generalized his relation to time-dependent Markov processes with the instantaneous generators satisfying the detailed balance relation. At almost the same time, Kurchan has shown in [56] that the stationary fluctuation relations hold for the stochastic Langevin-Kramers evolution. His result was extended to more general diffusion processes by Lebowitz and Spohn in [59]. In [63], Maes has traced the origin of fluctuation relations to the Gibbsian nature of the statistics of the dynamical histories, see a recent discussion of the fluctuation relations from this point of view in [64]. Searles and Evans generalized their transient fluctuation relation to the stochastic setup in [76]. Finally, within the stochastic approach, the scope of the transient fluctuation relations was further extended due to the works of Crooks [18,19], Jarzynski [51], Hatano and Sasa [46], Speck and Seifert [78] and Chernyak, Chertkov and Jarzynski [10], just to cite only the papers that most influenced the present authors. It is worth stressing that the general transient fluctuation relations do not impose the time reversibility of the dynamics but compare the fluctuation statistics of the original process and of its time reversal. Such an extension of the scope of fluctuation relations is a possibility in the
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stationary case as well, but it becomes a necessity in many transient situations. Within the theory of the hyperbolic dynamical systems, the stationary fluctuation theorem of [35] was recently generalized to the random dynamics in [8]. In [1], Balkovsky, Falkovich and Fouxon noticed another robust relation concerning the large deviations of finite-time Lyapunov exponents in the context of homogeneous hydrodynamic flows. It was remarked in [29], that this observation, which we shall call, following [40], the multiplicative fluctuation relation, provides an extension of the previously known fluctuation relations for the phase-space contraction. The simple argument presented in [1] dealt with a transient situation. It was very similar to the original Evans-Searles argument as formulated later in [26]. The multiplicative fluctuation relation was explicitly checked in the Kraichnan model of hydrodynamic flows [1,29,11]. The theoretical work on fluctuation relations has established most of them as mathematical identities holding within precisely defined models, but concerning statistics of events that are rare, especially for macroscopic systems. The relevance of such identities to numerical simulations and, even more, to real experiments, required a confirmation. Numerical (see e.g. [7,45,83,38,86]) and experimental testing of the fluctuation relations (see e.g. [15,37,17,2,54,47]) has attracted over the years a lot of attention, inspiring further developments. It will probably remain an active field in the future. It is not, however, the topic of the present paper. The growing number of different fluctuation relations made urgent a development of a unifying approach. Several recent reviews partially provided such a unification from different points of view, see ref. [26,64,57,10]. In the present paper, we attempt another synthesis, with the aim of supplying a uniform derivation of most of the known fluctuation relations, including the multiplicative ones. We shall work in the setup of (possibly non-autonomous) diffusion processes in finite-dimensional spaces, somewhat similar, but more general than the one adopted in [59]. The systems considered include, as special cases, the deterministic dynamics, the Langevin stochastic equation, and the Kraichnan model of hydrodynamic flow. This is certainly not the most general setup possible for discussing fluctuation relations (for example, the discrete-time dynamics, the stochastic dynamics with jumps, or non-Markovian evolutions are not covered), but it is general enough for a unified discussion of a variety of aspects of fluctuation relations. Most of our considerations are simple extensions of arguments that appeared earlier in usually more constrained contexts. There are two basic ideas that we try to exploit to obtain a larger flexibility than in the previous discussions of fluctuation relations. The first one concerns the possible time-reversed processes that we admit. This idea appeared already in [10], where two different time inversions were used for the Langevin dynamics with non-conservative forces, leading to two different backward processes and two different fluctuation relations. We try to exploit the freedom of choice of the time-inversion in a more systematic way. The second idea, which seems original to us, although it is similar in spirit to the first one, is to obtain new fluctuation relations by considering new diffusion processes derived from the original one. In particular, we show that the multiplicative fluctuation relations for general diffusion processes may be obtained by writing a more standard relation for the tangent diffusion process describing a simultaneous evolution of infinitesimally close trajectories of the original process. The same idea may be used [13] to explain additional fluctuation relations, like the one for the rate function of the difference of finite-time Lyapunov exponents “along unstable flag” that was observed in [11] for the anisotropic Kraichnan model.
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The present paper is organized as follows. In Sect. 2, we define the class of diffusion processes that will be discussed and list four special cases. Section 3 recalls the notions of transition probabilities and generators of a diffusion process, as well as the detailed balance relation. In Sect. 4, we introduce the tangent diffusion process induced from the original one and define the phase-space contraction. Time inversions leading to different backward processes are discussed in Sect. 5, with few important examples listed in Sect. 6. A formal relation between the expectations in the forward and in the backward process is introduced in Sect. 7. As examples, we discuss the case of tangent process in the homogeneous Kraichnan flow, a simple generalization of the detailed balance relation and the 1st law of thermodynamics for the Langevin dynamics. Section 8 is devoted to a general version of the Jarzynski equality, whose different special cases are reviewed, and Sect. 9 to a related equality established by Speck and Seifert in [78]. We formulate the Jarzynski equality as a statement that for a certain functional W of the diffusion process, the expectation value of e−W is normalized. In Sect. 10, the functional W is related to the entropy production and the positivity of its expectation value is interpreted as the 2nd law of thermodynamics for the diffusive processes. In Sect. 11, we show how the general Jarzynski equality reduces in the linear response regime to the Green-Kubo and Onsager relations for the transport coefficients and to the fluctuation-dissipation theorem. In Sect. 12, we discuss briefly a peculiar one-dimensional Langevin process in which the equilibrium is spontaneously broken and replaced by a state with a constant flux, leading to a modification of the fluctuation-dissipation relation. The model is well known from the theory of one-dimensional Anderson localization and describes also the separation of infinitesimally close particles with inertia carried by a one-dimensional Kraichnan flow. Section 13.3 formulates in the general setup of diffusion processes what is sometimes termed a detailed fluctuation relation [51,19], an extension of the Crooks fluctuation relations [18]. Few special cases are retraced in Sect. 14. Up to this point of the paper, the discussion is centered on the transient evolution where the system is initially prepared in a state that changes under the dynamics. In Sect. 15, we discuss the relation of the transient fluctuation relations to the stationary ones which pertain to the situation where the initial state is preserved by the evolution. The stationary relations are usually written for the rate function of large deviations of entropy production observed in the long-time regime. In our case, they describe the long time asymptotics of the statistics of W. The Gallavotti-Cohen relation was the first example of such identities. We show how the fluctuation relation for the tangent process in the homogeneous Kraichnan flow discussed in Sect. 7 leads to a generalization of the Gallavotti-Cohen relation that involves the large-deviations rate function of the so called stretching exponents whose sum describes the phase-space contraction. In Sect. 16, we extend such a multiplicative fluctuation relation to the case of general diffusion processes. Section 17 contains speculation about possible versions of fluctuation relations for multi-point motions and Sect. 18 collects our conclusions. Few simple but more technical arguments are deferred to Appendices in order not to overburden the main text, admittedly already much more technical than most of the work on the subject. Some of the technicalities are due to a rather careful treatment of the intricacies related to the conventions for the stochastic differential equations that are usually omitted in physical literature. The aim at generality, even without pretension of mathematical rigor, places the stress on the formal aspects and makes this exposition rather distant from physical discourse, although we make an effort to include many examples that illustrate general relations in more specific situations. The physical content is, however, more transparent in examples to such examples which are scarce in the present text but which abound in
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the existing literature to which we often refer. Certainly, the paper will be too formal for many tastes, and we take precautions to warn the potential reader who can safely omit the more technical passages. After submission of the first version of the paper, we received the article [66] which influenced our revision of Sect. 10. 2. Forward Process As mentioned in the Introduction, the present paper deals with non-equilibrium systems modeled by diffusion processes of a rather general type. More concretely, the main objects of our study are the stochastic processes xt in Rd (or, more generally, on a d-dimensional manifold), described by the differential equation x˙ = u t (x) + vt (x) ,
(2.1)
where x˙ ≡ ddtx and, on the right hand side, u t (x) is a time-dependent deterministic vector field (a drift), and vt (x) is a Gaussian random vector field with mean zero and covariance i j ij (2.2) vt (x) vs (y) = δ(t − s) Dt (x, y) . Due to the white-noise nature of the temporal dependence of vt (typical vt are distributional in time), Eq. (2.1) is a stochastic differential equation (SDE). We shall consider it with the Stratonovich convention1 [71,67], keeping for the Stratonovich SDEs the notation of the ordinary differential equations (ODEs). Examples of systems described by Eq. (2.1) include four special cases that we shall keep in mind. ij
Example 1. Deterministic dynamics. Here vt (x) ≡ 0 and Dt (x, y) ≡ 0 so that Eq. (2.1) reduces to the ODE x˙ = u t (x) .
(2.3)
Example 2. Lagrangian flow in the Kraichnan model. This is a process used in modeling turbulent transport. The SDE (2.1), where one usually takes u t (x) ≡ 0, describes the motion of tracer particles in a stationary Gaussian ensemble of velocities vt (x) white in time. Such an ensemble, with an appropriate time-independent spatial covariance D i j (x, y), was designed by Kraichnan [62] to mimic turbulent velocities. In particular, homogeneous flows are modeled by imposing the translation invariance D i j (x, y) = D i j (x − y) and isotropic ones by assuming that D i j (x, y) is rotation-covariant. In this paper, we shall consider only the case when D i j (x, y) is smooth. A discussion of the case with D i j (x, y) non-smooth around the diagonal, pertaining to the fully developed turbulence, may be found in [29], or, on a mathematical level, in [60]. Example 3. Langevin dynamics. Here Eq. (2.1) takes the form2 x˙ i = − i j ∂ j Ht (x) + i j ∂ j Ht (x) + G it (x) + ζti ,
(2.4)
1 The choice of the Stratonovich convention guarantees that u and v transform as vector fields under a t t change of coordinates. 2 We use throughout the paper the summation convention.
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where is a constant non-negative matrix and an antisymmetric one, the Hamiltonian Ht is a, possibly time dependent, function, G t is an additional force, and ζt is the d-dimensional white noise with the covariance i j ζt ζs = 2 δ(t − s) β −1 i j . (2.5) In this example, the noise ζt plays the role of the (space-independent) random vector ij field vt so that Dt (x, y) = 2β −1 i j . For G t ≡ 0 and a time independent Hamiltonian Ht ≡ H , the Langevin dynamics is used to model the approach to thermal equilibrium at inverse temperature β [46]. The deterministic vector field − i j ∂ j H drives the solution towards the minimum of H (if it exists) whereas the Hamiltonian vector field i j ∂ j H preserves H . The noise ζt generates thermal fluctuations of the solution. Note that its spatial covariance is aligned with the matrix appearing in the dissipative force − i j ∂ j H (such an alignment, known from Einstein’s theory of Brownian motion, is often called the Einstein relation). Inclusion of the Hamiltonian vector field permits to model systems where the noise acts only on some degrees of freedom, e.g. the ones at the ends of a coupled chain, with the rest of the degrees of freedom undergoing a Hamiltonian dynamics. The introduction of a time-dependence and/or of the force G t permits to model nonequilibrium systems. In the particular case of vanishing , the SDE (2.4) reduces to the ODE x˙ i = i j ∂ j Ht (x) + G it (x)
(2.6)
describing a deterministic Hamiltonian dynamics in the presence of an additional force Gt . Example 4. Langevin-Kramers equation. This is a special case of the Langevin dynamics that takes place in the phase space of n degrees of freedom with x = (q, p) and 1 0 1 −1 = 00 γ0 , = −1 p + Vt (q), G t = (0, f t (q)), 0 , Ht = 2 p · m where γ = 0 is a non-negative n × n matrix, m −1 a positive one, and 1 the unit one. Here, Eq. (2.4) reduces to the standard relation pi = m i j q˙ j between momenta and velocities, where m is the mass matrix, and to the second order SDE, m i j q¨ j = −γik q˙ k − ∂i Vt (q) + f ti (q) + ζi ,
(2.7)
that we shall call Langevin-Kramers equation, with the n-dimensional white noise ζ such that ζit ζ jt = 2β −1 γi j δ(t − t ) . The Langevin-Kramers equation has the form of the Newton equation with the friction −γ q˙ and white-noise ζt forces supplementing the conservative one −∇Vt and the additional one f t . It was discussed in [57] in a very similar context. In the limit of a strongly overdamped system when the friction term becomes much larger than the second order one, the Langevin-Kramers equation (2.7) reduces to the first order SDE, γik q˙ k = − ∂i Vt (q) + f ti (q) + ζi , which, if γ > 0, may be cast again into the form (2.4) but with = γ −1 , = 0 and Ht = Vt . One should keep in mind this change when applying the results described below for the Langevin dynamics (2.4) to the overdamped Langevin-Kramers dynamics.
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3. Transition Probabilities and Detailed Balance Let us recall some basic facts about the diffusion processes in order to set the notations. We shall denote by E tx0 the expectation of functionals of the Markov process xt solving the SDE (2.1) with the initial condition xt0 = x. For t ≥ t0 , the relation t0 Pt0 ,t (x, dy) g(y) ≡ (Pt0 ,t g)(x) (3.1) E x g(xt ) = defines the transition probabilities Pt0 ,t (x, dy) of the process xt and the operator Pt0 ,t . The transition probabilities satisfy the normalization condition Pt0 ,t (x, dy) = 1 and the Chapman-Kolmogorov chain rule Pt0 ,t (x, dy) Pt,t (y, dz) = Pt0 ,t (x, dz) . The evolution of the expectation values is governed by the second-order differential operators L t defined by the relation d t0 E g(xt ) = E tx0 (L t g)(xt ) . dt x
(3.2)
The explicit form of L t is found by a standard argument that involves the passage from the Stratonovich to the Itô convention. For reader’s convenience, we give the details in Appendix A. The result is: 1
ij
L t = uˆ it ∂i + 2 ∂ j dt ∂i ,
(3.3)
where ij
ij
dt (x) = Dt (x, x)
and
1
ij
uˆ it (x) = u it (x) − 2 ∂ y j Dt (x, y)| y=x .
(3.4)
Due to the relation (3.1), Eq. (3.2) may be rewritten as the operator identity ∂t Pt0 ,t = Pt0 ,t L t . Together with the initial condition Pt0 ,t0 = 1, it implies that Pt0 ,t is given by the time-ordered exponential Pt0 ,t
− → = T exp
t
L s ds
t0
=
∞
L s1 L s2 ....L sn n=0 t0 ≤s1 ≤s2 ≤....≤t
ds1 ds2 ...dsn . (3.5)
In particular, Pt0 ,t = e(t−t0 )L ≡ Pt−t0 in the stationary case with u t ≡ u and Dt ≡ D. The operator L t ≡ L is then called the generator of the process. The stochastic process xt may be used to evolve measures. Under the stochastic dynamics, the initial measure µt0 (d x) evolves at time t to the measure µt0 (d x) Pt0 ,t (x, dy) . (3.6) µt (dy) = We shall use below the shorthand notation: µt = µ0 P0,t . For measures with densities µt (d x) = ρt (x) d x with respect to the Lebesgue measure d x, Eq. (3.6) is equivalent to the evolution equation 1 ij ∂t ρt = ∂i − uˆ it + 2 dt ∂ j ρt ≡ L †t ρt , (3.7)
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where L †t is the (formal) adjoint of the operator L t . The latter relation may be rewritten as the continuity equation 1 ij
∂t ρt + ∇ · j = 0 with jti = (uˆ it − 2 dt ∂ j )ρt ,
(3.8)
where ∇ · j ≡ ∂i jti is the divergence of the density current jt corresponding to the measure µt (the probability current, if µt is normalized). In the case with no explicit time dependence when L t ≡ L, an invariant density ρ, corresponding to an invariant measure µ(d x) = ρ(x) d x of the process, satisfies the equation L † ρ = 0 which may be rewritten in the form of the current conservation condition ∇ · j = 0. We shall often write the invariant density ρ(x) in the exponential form as e−ϕ(x) . One says that the process satisfies the detailed balance relation with respect to ϕ if the density current j related to the measure µ(d x) = e−ϕ(x) d x vanishes itself, i.e. if 1
uˆ i = − 2 d i j ∂ j ϕ . Equivalently, this condition may be written as the relation L † = e−ϕ L eϕ , for the generator of the process or as the identity µ(d x) Pt (x, dy) = µ(dy) Pt (y, d x)
(3.9)
for the transition probabilities. In all these three forms, it implies directly that µ is an invariant measure. The converse, however, is not true: there exist stationary diffusion processes with invariant measures that do not satisfy the detailed balance relation. The generator of the stationary Langevin equation with = 0 and G = 0 satisfies the detailed balance relation with respect to ϕ = β H so that the Gibbs density ρ(x) = e−β H (x) , and, if the latter is normalizable, the Gibbs probability measure µG (d x) = Z −1 e−β H (x) d x, are invariant under such dynamics. The invariance still holds when = 0 but, in this case, the detailed balance relation fails. We shall see below how to generalize the latter to catch also the case with conservative forces when = 0. 4. Tangent Process and Phase-Space Contraction One may generate other processes of a similar nature from the diffusive process (2.1). Such constructions will play an important role in studying fluctuation relations. As the first example, let us consider the separation δxt between the solution xt of Eq. (2.1) with the initial value x0 = x and another solution infinitesimally close to xt . Such a separation evolves according to the law δxt = Xt (x) δx0 , where the matrix Xt (x) with the entries Xti j (x) =
∂xti j
∂x0
(x)
(4.1)
solves the (Stratonovich) SDE
X˙ i j = ∂k u it + ∂k vti (xt ) X kj
(4.2)
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with the initial condition X0 (x) = 1. Together with Eq. (2.1), the SDE (4.2) defines a diffusion process (xt , Xt ) that we shall call the tangent process. In particular, the quantity − ln det Xt that represents the accumulated phase-space contraction along the trajectory xt , solves the SDE d − ln det X = −(∇ · u t + ∇ · vt )(xt ) . (4.3) dt The right hand side of Eq. (4.3) is the phase-space contraction rate. We infer that T − ln det Xt = −
T (∇ · u t )(xt ) dt −
0
(∇ · vt )(xt ) dt .
(4.4)
0
The second integral on the right hand side should be interpreted with the Stratonovich convention. The phase-space contraction is an important quantity in the study of nonequilibrium dynamics and it will reappear in the sequel. 5. Backward Processes Among the diffusion processes that may be generated from the original process (2.1) are the ones which may be interpreted as its time reversals. The action of time inversion on space-time will be given by the transformation (t, x) −→ (T − t, x ∗ ) ≡ (t ∗ , x ∗ )
(5.1)
for an involution x → x ∗ . It may be lifted to the level of process trajectories by defining the transformed trajectory x˜ t by the relation x˜ t = xt∗∗ .
(5.2)
In general, however, we shall not define the time-reversed process as x˜ t because, in the presence of dissipative deterministic forces like friction, such time inversion would lead to an anti-dissipative dynamics. We shall then allow for more flexibility. In order to define the time-reversed process, we shall divide the deterministic vector field u t into two parts u t = u t,+ + u t,− ,
(5.3)
that we shall loosely term dissipative and conservative, choosing different time-inversion rules for them. The time-reversed process xt will be given by the SDE x˙ = u t (x ) + vt (x )
(5.4)
with the deterministic vector field u t = u t,+ + u t,− and the random one vt defined by the equations u t,± (x) = ±(∂k x ∗ i )(x ∗ ) u tk∗ ,± (x ∗ ) i
and
v t (x) = ±(∂k x ∗ i )(x ∗ ) vtk∗ (x ∗ ) . i
(5.5)
Note that u t,+ transforms as a vector field under the involution x → x ∗ and u t,− as a pseudo-vector field. For vt we may use whichever rule since vt and −vt have the same distribution. The SDE (5.4) for the time-reversed process xt coincides with the one for the process x˜ t defined by Eq. (5.2) if and only if u t,+ vanishes and vt is transformed
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according to the pseudo-vector rule. We shall call xt the backward process referring to xt as the forward one. The random vector field vt of the backward process is again Gaussian with mean zero and white-noise behavior in time. Its covariance is i j i j vt (x) vs (y) = δ(t − s) Dt (x, y) , where i j
Dt (x, y) = (∂k x ∗ i )(x ∗ ) Dtkl∗ (x ∗ , y ∗ ) (∂l x ∗ j )(y ∗ ) .
(5.6)
As before, see Eqs. (3.4), we shall denote i j
i j
i j
dt (x) = Dt (x, x), uˆ it (x) = u it (x) − 2 ∂ y j Dt (x, y)| y=x . 1
(5.7)
Remark 1. Using the chain rule (∂ j x ∗ i )(x ∗ )(∂k x ∗ j )(x) = δki , it is easy to see that the time-inversion transformations (5.5) are involutive. Let us emphasize that the choice of a time inversion consists of the choice of the involution (5.1) and of the splitting (5.3) of u t . We shall call the process time-reversible (for a given choice of time inversion) if the deterministic vector fields u and u of the forward and of the backward processes coincide and if the respective random vector fields vt and vt have the same distribution, i.e. if u it,+ (x) + u it,− (x) = (∂k x ∗ i )(x ∗ ) u kt∗ ,+ (x ∗ ) − u kt∗ ,− (x ∗ ) and if Dt (x, y) = (∂k x ∗ i )(x ∗ ) Dtkl∗ (x ∗ , y ∗ ) (∂l x ∗ j )(x ∗ ) . ij
Note that the first identity is equivalent to the relations 1 u it,± (x) = 2 u it (x) ± (∂k x ∗ i )(x ∗ ) u kt∗ (x ∗ )
(5.8)
(5.9)
and can be always achieved by taking such a splitting of u t . It may be not easy, however, to realize physically the backward process corresponding to the splitting (5.9). The second condition (5.8) is a non-trivial constraint on the distribution of the the white-noise velocity vt . Nevertheless, if Dt is time-independent, it may be satisfied by choosing the trivial involution x ∗ ≡ x. Parallelly to the splitting (5.3) of the drifts u t and u t , we shall divide the operators generating the forward and the backward evolution into two parts: L t = L t,+ + L t,−
L t = L t,+ + L t,− , according to the formulae: 1
ij
L t,+ = uˆ it,+ ∂i + 2 ∂ j dt ∂i , i 1 i j L t,+ = uˆ t,+ ∂i + 2 ∂ j dt ∂i ,
L t,− = u it,− ∂i , L t,− = u t,− ∂i . i
The time-inversion rules become even more transparent when expressed in terms of the split generators. Let R denote the involution operator acting on the functions by (R f )(x) = f (x ∗ ) .
(5.10)
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Lemma 1. L t,± = ± R L t ∗ ,± R .
(5.11)
Proof of Lemma 1, involving a straightforward although somewhat tedious check, is given in Appendix B. Below, similarly as for the forward process, we shall denote by E tx 0 the expectation of functionals of the backward process satisfying the initial condition xt0 = x. For t ≥ t0 , the relations E tx 0
g(xt )
=
(Pt0 ,t g)(x)
with
Pt0 ,t
− → = T exp
t
L s ds
t0
define the operators whose kernels give the transition probabilities of the time-reversed process xt . 6. Examples of Time-Inversion Rules The preceding considerations were very general. Physically, not all time-inversion rules for the diffusive processes (2.1) described above are on equal footing. In particular situations, some rules may be more natural or easier to implement than other ones. Let us list here a few cases of special time inversions that were discussed in the literature and/or will be used below. 6.1. Natural time inversion. Taking the trivial splitting u t,+ = 0, u t,− = u t combined with an involution x → x ∗ leads to the time-inversion rules that produce the backward process with trajectories related by the transformation (5.2) to the ones of the forward process if the pseudo-vector field rule is used when transforming vt . This is the time inversion usually employed for the deterministic systems but it may be used more generally. 6.2. Time inversion with uˆ t,+ = 0. Consider the time inversion corresponding to an arbitrary involution x → x ∗ and the choice uˆ t,+ = 0, u t,− = uˆ t .
(6.1)
of the splitting of u t . Such a time inversion is a slight modification of the natural one to which it reduces in the case of deterministic dynamics (2.3) with vt ≡ 0. As we show in Appendix C, the backward dynamics corresponding to the splitting (6.1) is given by the relations uˆ it,+ (x) =
1 i j d (x) (∂ j 2 t
ln σ )(x), u it,− (x) = −(∂k x ∗ i )(x ∗ ) uˆ kt∗ (x ∗ ) ,
(6.2)
where σ (x) = σ (x ∗ )−1 denotes the absolute value | det(∂ j x ∗ i )(x)| of the Jacobian of the involution x → x ∗ . The time inversion considered here will be used to obtain ij fluctuation relations in the limiting case of deterministic dynamics (2.3) when Dt is set to zero and the backward dynamics is given by the ODE x˙ i = u it (x )
for
u it (x) = −(∂k x ∗ i )(x ∗ ) u kt∗ (x ∗ ) ,
obtained from the ODE (2.3) by the natural time inversion.
(6.3)
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6.3. Time inversion in the Langevin dynamics. To explain why the rules of time inversion with non-vanishing u t,+ are more generally needed, we consider the case of the Langevin dynamics that involves the dissipative force −∇ Ht . Let us arbitrarily split the corresponding drift u t into two parts: u t = −∇ Ht + ∇ Ht + G t = u t,+ + u t,− ,
(6.4)
see Eq. (2.4). Recall the relation (2.5) that aligns the matrix with the covariance of the white-noise vt = ζt . It is natural to require the backward dynamics to be also of the Langevin type but for the time-reversed Hamiltonian Ht (x) = Ht ∗ (x ∗ ). This requires that u t = − ∇ Ht + ∇ Ht + G t = u t,+ + u t,− ,
(6.5)
and that vt (x) = ζt with the covariance of the white noise ζt aligned with matrix as in Eq. (2.5). Upon restriction to linear involutions x ∗ = r x with the matrix r squaring to 1, the transformation rules (5.5) become u t,± (x) = ±r u t ∗ ,± (r x), ζt = ±r ζt ∗ . The condition on the covariance of ζt imposes the relation = r r T . Applying r to both sides of Eq. (6.5) taken at time t ∗ and at point r x, we infer that − r r T ∇ Ht (x) + r r T ∇ Ht (x) + r G t ∗ (r x) = u t,+ (x) − u t,− (x) . The latter identity, together with Eq. (6.4), result in the relations u t,+ (x) = −∇ Ht (x) + 2 ( + r r T )∇ Ht (x) + 1
u t,− (x) = − 2 ( − r r T )∇ Ht (x) + 1
1 (G t (x) + r G t ∗ (r x)) , 2 1 (G t (x) − r G t ∗ (r x)) . 2
At least when is strictly positive, Ht is not a constant, and the extra force G t is absent, one infers that the component u t,+ cannot vanish identically by considering the contraction (∇ Ht ) · u t,+ . We shall call canonical a choice of the time inversion for the Langevin dynamics for which = r r T = , = −r r T = , u t,+ = −∇ Ht , u t,− = ∇ Ht + G t .
(6.6) (6.7)
Note that such a time inversion treats the force G t as a part of u t,− even when this force is of the non-conservative type. The Langevin dynamics is time-reversible under a canonical time inversion if Ht = Ht and G t = G t . For the Langevin-Kramers equation, the standard phase-space involution (q, p)∗ = r (q, p) = (q, − p) verifies Eqs. (6.6) and it leads to the particularly simple canonical time-inversion rules with Vt = Vt ∗ ,
f t = f t ∗
and to the time-reversibility if Vt = Vt ∗ and f t = f t ∗ .
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481
6.4. Reversed protocol. The time inversion corresponding to the choice u t,+ = u t , u t,− = 0
(6.8)
and the trivial involution x ∗ ≡ x was termed in [10] a reversed protocol. It may be viewed as consisting of the inversion of the time-parametrization in the vector fields in the SDE (2.1), if the vector-field rule is used to reverse vt . In the stationary case, where it results in time-reversibility, such a time inversion was employed already in [59]. Here, we shall admit also a possibility of a non-trivial involution x → x ∗ . The reversed protocol leads then to the backward process with u it,+ = (∂k x ∗ i )(x ∗ ) u kt∗ (x ∗ ), u t,− = 0, vti = (∂k x ∗ i )(x ∗ ) vtk∗ (x ∗ ) . i
(6.9)
6.5. Current reversal. Suppose that e−ϕt are densities satisfying L †t e−ϕt = 0. Such densities would be preserved by the evolution if the generator of the process were frozen to L t . The density current corresponding to e−ϕt has the form 1 ij jti = uˆ it + 2 dt (∂ j ϕt ) e−ϕt , see Eq. (3.8). It is conserved due to the relation L †t e−ϕt = 0. The time inversion defined by the choice 1 ij
uˆ it,+ = − 2 dt ∂ j ϕt , u it,− = uˆ it +
1 ij d ∂ ϕ, 2 t j t
(6.10)
and an arbitrary involution x → x ∗ leads, after an easy calculation using the results of Appendix C, to the backward process with 1 i j
uˆ it,+ = − 2 dt ∂ j ϕt , u it,− (x) = −(∂k x ∗ i )(x ∗ ) u kt∗ ,− (x ∗ ) vti = ±(∂k x ∗ i )(x ∗ ) vti ∗ (x ∗ )
(6.11)
for ϕt (x) = (ϕt ∗ + ln σ )(x ∗ ). The density current for the backward process correspon ding to the densities e−ϕt is 1 i j jti = uˆ it + 2 dt (∂ j ϕt ) e−ϕt
∗
= u it,− (x) e−ϕt (x) = −(∂k x ∗ i )(x ∗ ) u kt∗ ,− (x ∗ ) e−ϕt ∗ (x ) σ (x) 1 ij = −(∂k x ∗ i )(x ∗ ) uˆ kt∗ (x ∗ ) + 2 dt ∗ (x ∗ )(∂ j ϕt ∗ )(x ∗ ) e−ϕt σ (x) = − (∂k x ∗ i )(x ∗ ) j k (x ∗ ) σ (x)
(6.12)
−ϕt = 0. We shall term the and is also conserved, as is easy to check. It follows that L † t e time inversion corresponding to the choices (6.10) the current reversal. For x ∗ ≡ x when it just reverses the sign of the density current, it was already employed in an implicit way in [43], and was introduced explicitly (under a different name) in [10]. The latter reference discussed also a simple two-dimensional model for which the inverse protocol and the current reversal led to different backward processes.
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6.6. Complete reversal. Finally, modifying slightly the last scheme, let us suppose the densities ρt = e−ϕt evolve under the dynamics solving Eq. (3.7). With the same splitting (6.10) as for the current reversal, we obtain the backward process for which Eqs. (6.11) and (6.12) still hold for ϕt (x) = (ϕt ∗ + ln σ )(x ∗ ). We shall call the corresponding time inversion the complete reversal. Unlike in the other examples, it depends also on the choice of the initial density ρ0 and may be difficult to realize physically. The time reflected densities ρt = e−ϕt evolve now according to the backward-process version of Eq. (3.7). The current reversal and the complete reversal coincide in the case without explicit time dependence and with the choice of ϕt ≡ ϕ such that e−ϕ d x is an invariant measure.
7. Relation Between Forward and Backward Processes A comparison between the forward and the backward processes will be at the core of fluctuation relations that we shall discuss. To put the processes in the two time directions back-to-back, we shall adapt to the present setup the arguments developed in Sect. 5 of [59]. Let us introduce a perturbed version of the generator L t of the forward process, L 1t = L t − 2 uˆ it,+ ∂i − (∂i uˆ it,+ ) + (∂i u it,− ) .
(7.1)
Operator L 1t is related in a simple way to the generator of the backward process: † 1 ij R L 1t R = R ∂i uˆ it,+ − ∂i u it,− + 2 ∂i dt ∂ j − (∂i uˆ it,+ ) + (∂i u it,− ) R = R L t,+ R − R L t,− R = L t ∗ ,
(7.2)
where R is defined by Eq. (5.10) and the last equality is a consequence of the relations (5.11). Let us consider the time-ordered exponential of the integral of L 1t . Using the relation L 1t = (R L t ∗ R)† that follows from Eq. (7.2), we infer that
Pt10 ,t
− → ≡ T exp
t L 1s ds t0
− → = R T exp
∗ t0
← − = T exp
∗
t0
(R L s R)† ds
t∗
† † L s ds R = R Pt∗ ,t ∗ R . 0
(7.3)
t∗
− → ← − Above, the first inversion of the time order from T to T was due to the change of integration variables s → s ∗ = T − s, and the second one, to the fact that the hermitian x) conjugation reverses the order in the product of operators. Let us remark that A(y,d d x dy † ∗ ∗ is the kernel of the operator A and A(x , dy ) of the operator R A R if A(x, dy) is the kernel of a real operator A. Rewriting Eq. (7.3) in terms of the kernels, with these comments in mind, we obtain the identity d x Pt10 ,t (x, dy) = dy Pt∗ ,t ∗ (y ∗ , d x ∗ ) . 0
(7.4)
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Remark 2. The transition probability of the backward process on the right hand side may be replaced by the one of the forward process in the time-reversible case. Note that the 2nd order differential operator L 1t differs from L t only by lower order terms, see Eq. (7.1). A combination of the Cameron-Martin-Girsanov and the FeynmanKac formulae [79] permits to express the kernel Pt10 ,t (x, dy) as a perturbed expectation for the forward process. ij Lemma 2. If the matrix dt (x) is invertible for all t and x then Pt10 ,t (x, dy) = E tx0 e
−
t
Js ds
t0
δ(xt − y) dy ,
(7.5)
where Jt = 2 uˆ t,+ (xt ) · dt−1 (xt ) x˙ t − 2 uˆ t,+ (xt ) · dt−1 (xt ) u t,− (xt ) − (∇ · u t,− )(xt ) (7.6) is a (local) functional of the solution xt of the SDE (2.1). The right hand side of Eq. (7.6) uses the vector notation. The first term in the expression for Jt has to be interpreted with the Stratonovich convention. Proof of Lemma 2 is deferred to Appendix D. A combination of the relations (7.5) and (7.4) gives immediately Proposition 1. d x E tx0 e
−
t t0
Js ds
δ(xt − y) dy = dy Pt∗ ,t ∗ (y ∗ , d x ∗ ) . 0
(7.7)
This is the first fluctuation relation of a series to be considered. It connects the transition probability of the backward process to an expectation in the forward process weighted with an exponential factor. Let us illustrate this relation in a few particular situations related to the examples of the diffusion processes considered in Sect. 2. Example 5. Tangent process in the stationary homogeneous Kraichnan model. Recall Sect. 4 devoted to the definition of a tangent process. Let us consider the tangent process (xt , Xt ) with fixed initial data x0 = x and X0 = 1 for the homogeneous Kraichnan model. As was discussed in detail in [40], in this case, the distribution of the process Xt may be obtained by solving, instead of the SDE (4.2) with u t ≡ 0, a simpler linear Itô SDE d X = St dt X
(7.8)
with a matrix-valued white-noise St such that
j Sti k Ss l = −δ(t − s) ∂k ∂l D i j (0) .
In other words, in Eq. (4.2), we may replace ∂k v i (xt ) by ∂k vti (0) ≡ Sti k , if we change the SDE convention to the Itô one at the same time. Consequently, in the homogeneous Kraichnan model, the process Xt may be decoupled from the original process xt . Let
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R. Chetrite, K. Gaw¸edzki ij
ij
ij
ji
ij
ji
us abbreviate: −∂k ∂l Dt (0) = Ckl . Remark the symmetries Ckl = Clk = Clk = Ckl . The Itô SDE (7.8) may be rewritten as the equation 1 ik l X˙ i j = − 2 Ckl X j + Sti l X l j
(7.9)
that employs the Stratonovich convention. Upon the use of the notations: 1
ik l X j , Vti j (X ) = Sti l X l j , U i j (X ) = − 2 Ckl
it may be cast into the form X˙ = U (X ) + Vt (X ) ,
(7.10)
falling within the scope of (stationary) diffusion SDEs (2.1) and defining a Markov process Xt . The covariance of the white-noise “velocity” Vt (X ) is
j ij ij ij Vtik (X ) Vs l (Y ) = δ(t − s) Dkl (X, Y ) with Dkl (X, Y ) = Cnm X nk Y ml . As in the general case (3.4), we shall denote: ij ij dkl (X ) = Dkl (X, X ) Uˆ i j (X ) = U i j (X ) −
1 2
d + 1 ni k Cnk X j . ∂ X˜ k D ik (X, Y )Y =X = − jl l 2
Let us apply the reversed-protocol time inversion discussed in Sect. 6.4 to the forward SDE (7.10). It corresponds to the trivial splitting of U with U+ = U and U− = 0 and to an involution X → X ∗ that we shall also take trivial: X ∗ ≡ X . The backward evolution is then given by the same equation (7.9) with St replaced by St = St ∗ , a matrix-valued white noise with the same distribution as St . The time-reversibility follows. Suppose that the covariance C of the white noise S(t) is invertible3 , i.e. that there exists a matrix ij −1 ln i n (C −1 )ln jm such that C kl (C ) jm = δm δk . Then the matrix −1 l −1 n −1 (d −1 )ln jm (X ) = (X ) p (X ) r (C ) jm pr
ij
provides the inverse of dkl (X ). Substituting these data into Eq. (7.6), we obtain d j −1 n ˙ m ˙m ln | det Xt |. Jt =2(Uˆ ) l (Xt ) (d −1 )ln jm (Xt ) Xt n = − (d + 1) (Xt ) m Xt n = − (d + 1) dt The relation (7.7) applied to the case at hand leads to the identity d X 0 Pt (X 0 , d X ) | det X 0 |−(d+1) | det X |d+1 = d X Pt (X, d X 0 ) ,
(7.11)
where Pt (X 0 , d X ) denotes the transition probability of the forward process Xt solving the SDEs (7.8) or (7.9) and d X 0 on the left hand side and d X on the right hand side stand for the Lebesgue measures on the space of d × d matrices. We made use of the fact that the backward process has the same law as the forward one. Equation (7.11) is nothing else but the detailed balance relation with respect to ϕ(X ) = (d + 1) ln | det X |. 3 The assumption about inversibility of C may be dropped at the end by a limiting argument.
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485
Indeed, note that the density current corresponding to the density ρ(X ) = | det X |−d+1 1 j (X ) = Uˆ (X ) + 2 d(X )∇ϕ | det X |−d+1 , see Eq. (3.8), vanishes. Integrating the left hand side of the identity (7.11) against a function f (X 0 , X ) and using the relation Pt (X 0 , d X ) = Pt (1, d(X X 0−1 )) that follows from the invariance of the corresponding SDE under the right multiplication of X by invertible matrices, we obtain the equalities f (X 0 , X ) det(X X 0−1 )d+1 d X 0 Pt (X 0 , d X ) = f (X 0 , X ) det(X X 0−1 )d+1 d X 0 Pt (1, d(X X 0−1 )) = f (X 0 , X X 0 ) (det X )d+1 d X 0 Pt (1, d X ) = f (X −1 X 0 , X 0 ) (det X ) d X 0 Pt (1, d X ) , where we twice changed variables in the iterated integrals. On the other hand, the integration of the right hand side of Eq. (7.11) against f (X 0 , X ) gives f (X 0 , X ) d X Pt (X, d X 0 ) = f (X, X 0 ) d X 0 Pt (X 0 , d X ) = f (X X 0 , X 0 ) d X 0 Pt (1, d X ) = f (X −1 X 0 , X 0 ) d X 0 Pt (1, d X −1 ) . Comparing the two expressions, we infer that Pt (1, d X ) (det X ) = Pt (1, d X −1 ) .
(7.12)
This is a version of the Evans-Searles [25] fluctuation relation for the stationary homogeneous Kraichnan model. In the context of general hydrodynamic flows, it was formulated and proven by a change-of-integration-variables argument in [1], see also [40]. We shall return in Sect. 15 to the relation (7.12) in order to examine some of its consequences. Subsequently, we shall generalize it in Sect. 16 to arbitrary diffusion processes of the type (2.1). Example 6. Generalized detailed balance relation. Consider the complete-reversal rules discussed in Sect. 6.6 and corresponding to the choice (6.10). Since, by virtue of the assumption that the densities e−ϕt evolve under the dynamics, see Eq. (3.7), L †t e−ϕt = −∂i u it,− e−ϕt = e−ϕt (u it,− ∂i ϕt − ∂i u it,− ) = −e−ϕt ∂t ϕt ,
(7.13)
the last two terms in the definition (7.6) reduce to − (∂t ϕt )(xt ) in this case so that Jt = −(∇ϕt )(xt ) · x˙ t − (∂t ϕt )(xt ) = −
d ϕt (xt ) . dt
(7.14)
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R. Chetrite, K. Gaw¸edzki
Upon integration over time, this produces boundary terms and Eq. (7.7) implies the generalized detailed balance relation: µ0 (d x) P0,T (x, dy) = µT (dy) P0,T (y ∗ , d x ∗ ) ,
(7.15)
for µt (d x) = e−ϕt (x) d x. Note that Eq. (7.15) holds for any choice of the involution x → x ∗ . Upon integration over x, it assures that the measures µt stay invariant under the dynamics, what was assumed from the very beginning. In the case with no explicit time dependence, i.e. when L t ≡ L, Eq. (7.15) holds, in particular, for ϕt ≡ ϕ such that µ = e−ϕ d x is an invariant measure. In that case, the generalized detailed balance relation reduces to the detailed balance one (3.9) if u − in the splitting (6.10) vanishes and x ∗ ≡ x. This was the case in Example 5. Below, we shall see examples where the invariant measure µ is known and the generalized detailed balance relation holds but where the detailed balance itself fails. Some of those cases fall under the scope of the Langevin dynamics. Let us discuss them first. Example 7. 1st law of thermodynamics and generalized detailed balance for Langevin dynamics. For Langevin dynamics with the splitting (6.7) of the drift, a direct substitution yields Jt = −β(∇ Ht )(xt ) · x˙ t + β(∇ H )(xt ) · G t (xt ) − (∇ · G t )(xt ) ≡ JtLan . Upon the use of the dynamical equation (2.4), T
T Jt
Lan
0
dt =
β (∇ Ht )(xt ) · (∇ Ht )(xt ) − β (∇ Ht )(xt ) · ζt
0
−(∇ · G t )(xt ) dt ≡ βQ ,
(7.16)
where Q may be identified with the heat transferred to the environment modeled by the thermal noise. On the other hand, using the original expression for JtLan together d with the (Stranonovich convention) identity dt Ht (xt ) = (∇ Ht )(xt ) · x˙ t + (∂t Ht )(xt ), we obtain the relation T JtLan dt = −β U + βW ,
(7.17)
0
where U = HT (xT ) − H0 (x0 ) is the change of the internal energy of the system and T W =
(∂t Ht )(xt ) + (∇ H )(xt ) · G t (xt ) − β −1 (∇ · G t )(xt ) dt
(7.18)
0
may be interpreted as the work performed on the system. With these interpretations, a comparison of the two expressions for the integral of JtLan leads to the 1st law of thermodynamics: U = −Q + W .
(7.19)
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487
This was discussed in a simple example of the forced and damped oscillator in [53]. In the absence of the extra force G t , the expression for the work reduces to T W =
(∂t Ht )(xt ) dt
(7.20)
0
and represents the so called Jarzynski work introduced in [48] for deterministic Hamiltonian dynamics. In the stochastic Langevin-Kramers dynamics, the expressions for the heat and the work become: T Q =
q˙t · γ q˙t − q˙t · ζt dt, W =
0
T
(∂t Vt )(qt ) + q˙t · f t (qt ) dt . (7.21)
0
The second quantity is equal to the sum of the Jarzynski work and of the work of the external force f t . It was introduced and discussed in [57]. In the stationary case, it reduces to the injected work [56] and, up to the β-factor, coincides with the “action functional” (for uniform temperature) given by Eq. (6.3) of [59]. Note that the general expression (7.18) for work also makes sense in the case of deterministic dynamics (2.6) obtained from the SDE (2.4) by setting = 0, in particular for the deterministic Hamiltonian evolution with G t ≡ 0. If G t ≡ 0, the splitting (6.7) is a special case of the splitting used for the current reversal for ϕt = β Ht , see Eq. (6.10). In particular, if Ht ≡ H then the transition probabilities of the Langevin process satisfy the generalized detailed balance relation (7.15) that takes the form µ(d x) PT (x, dy) = µ(dy) PT (y ∗ , d x ∗ )
(7.22)
for µ(d x) = e−β H (x) d x and any involution x → x ∗ = r x. The latter identity replaces in the presence of the conservative force ∇ H the detailed balance relation (3.9) and still assures that the Gibbs density e−β H is invariant under such Langevin dynamics. If the involution r satisfies additionally the relations (6.6) and H (r x) = H (x), resulting in the time-reversibility, then one may replace PT by PT in Eq. (7.15). Example 8. Linear Langevin equation. Consider the linear SDE x˙ = M x + ζt ,
(7.23)
where M is a d × d matrix and ζt is the white noise with the covariance (2.5) and matrix strictly positive. We shall be interested in cases when the matrix −1 M is non-symmetric. For an elementary discussion of mathematical aspects of such SDEs see e.g. [42]. In the context of nonequilibrium statistical mechanics, examples of such linear equations were considered in [58] as models of a harmonic chain of oscillators interacting with environment of variable temperature or, quite recently, in [81] for modeling coiled polymers in a shearing flow. The diffusion process xt that solves Eq. (7.23) with the initial value x0 = x is given by the formula t xt = e
tM
x + 0
e (t−s)M ζs ds .
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R. Chetrite, K. Gaw¸edzki
The transition probabilities of this process are Gaussian and have the explicit form
β Pt (x, dy) = det(2πβ −1 Ct )−1/2 exp − y − e t M x · Ct−1 y − e t M x dy , 2 (7.24) where t Ct = 2
T
es M es M ds
(7.25)
0
is a strictly positive matrix. Suppose that all the eigenvalues λ of M have negative real parts. Under this condition, et M tends to zero exponentially fast when t → ∞ so that C∞ ≡ C is finite and Pt (x, dy)
−→
t→∞
β det(2πβ −1 C)−1/2 exp − y · C −1 y dy , 2
with the right hand side defining the unique invariant probability measure of the process. This Gaussian measure has the form of the Gibbs measure for the quadratic Hamiltonian 1 2
H (x) =
x · C −1 x .
(7.26)
Introducing the matrix = + MC
(7.27)
that is antisymmetric: ∞ + = 2 + 2
T
e s M (M + M T )e s M ds
T
0
∞ = 2 + 2
d sM sMT ds = 0, e e ds
0
the linear SDE (7.23) may be rewritten in the Langevin form (2.4) as x˙ = −∇ H (x) + ∇ H (x) + ζt .
(7.28)
Conversely, the last SDE with H as in Eq. (7.26) for some C > 0 is turned into the form (7.23) upon setting M = −( − )C −1 .
(7.29)
Note that the last equation implies the relation (7.27) for . In Appendix E, we show that M given by Eq. (7.29) has necessarily all eigenvalues with negative real part and that C may be recovered from M as C∞ given by Eq. (7.25) with t = ∞. This establishes the equivalence between the SDEs (7.23) and (7.28).
Fluctuation Relations for Diffusion Processes
489
The probability current associated by the formula (3.8) to the Gaussian invariant Gibbs measure µG (d x) = Z −1 e−β H (x) d x is j (x) = Z −1 C −1 x e−β H (x) . It vanishes only when = 0. In the latter case, the transition probabilities (7.24) satisfy the detailed balance relation (3.9) for ϕ = β H + ln Z . If = 0 then only a generalized detailed balance relation (7.22) holds for any choice of the linear involution x → x ∗ = r x. If moreover r r T = , r r T = − and rCr T = C , then PT on the right hand side of Eq. (7.15) may be replaced by PT . 8. Jarzynski Equality We shall exploit further consequences of the relation (7.7) between the forward and the backward processes. In this section we shall derive an identity that generalizes the celebrated Jarzynski equality [48,49] and shall prepare the ground for obtaining more refined fluctuation relations following the ideas of [31,63] and [19]. Let ϕ0 and ϕT be two functions generating measures µ0 (d x) = e−ϕ0 (x) d x, µT (d x) = e−ϕT (x) d x ,
(8.1)
respectively. In particular, we could take e−ϕT (x) such that the measure µT is related to µ0 by the dynamical evolution (3.6), i.e. µT = µ0 P0,T , but we shall not assume such a choice unless explicitly stated. In general, the measures (8.1) may be not normalizable but we shall impose the normalization condition later on. We shall associate to µ0 and µT the time-reflected measures
∗
∗
µ0 (d x) = e−ϕ0 (x) d x = e−ϕT (x ) d x ∗ , µT (d x) = e−ϕT (x) d x = e−ϕ0 (x ) d x ∗ . Let us modify the functional
T
Jt dt introduced in the last section by boundary terms
0
ϕ ≡ ϕT (xT ) − ϕ0 (x0 ) by setting T W = ϕ +
Jt dt .
(8.2)
0
The functional W will be the basic quantity in what follows. Its physical interpretation in terms of the entropy production will be discussed in Sect. 10 below. For any functional F on the space of trajectories xt parametrized by time in the ˜ interval [0, T ], we shall denote by F˜ the functional defined by F(x) = F(x˜ ), where x˜ is given by Eq. (5.2). We shall also introduce the shorthand notation 0 E 0,T x,y F(x) = E x F(x) δ(xT − y)
for the (unnormalized) expectation of the process xt with fixed initial and final points, and similarly for the backward process. The following refinement of the relation (7.7) of Proposition 1 holds:
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R. Chetrite, K. Gaw¸edzki
Proposition 2. −W (x) ∗ ˜ dy = µ0 (dy ∗ ) E 0,T µ0 (d x) E 0,T x,y F(x) e y ∗,x ∗ F(x ) d x .
(8.3)
Proof of Proposition 2 is contained in Appendix F. Note that the explicit dependence on the choice of measures µ0 and µ0 trivially cancels the one buried in W. In particular, for F ≡ 1, Proposition 2 reduces to Proposition 1 with t0 = 0 and t = T . As before, the backward-process expectation E may be replaced by the forward-process one E for the time-reversible process. If the measures µ0 and µ0 are normalized then we may use them as the probability distributions of the initial points of the forward and of the backward process, respectively. The corresponding probability measures M(dx) and M (dx ) on the space of trajectories on the time-interval [0, T ] are given by the relations F(x) M(dx) = E 0x F(x) µ0 (d x) ≡ F , (8.4) F(x ) M (dx ) = E 0 (8.5) x F(x ) µ0 (d x) ≡ F . Upon integration over x and y, the identity (8.3) induces the following equality between the expectations with respect to the trajectory measures M and M : Corollary 1.
F e−W
=
F˜ .
(8.6)
It was stressed in [63], and even more explicitly in [19], that the identity of the type of (8.6), comparing the expectations in the forward and the backward processes, is a source of fluctuation relations. An important special case of Eq. (8.6) is obtained by setting F ≡ 1. It was derived in [48] in the context of the Hamiltonian dynamics and in [49] in the one of Markov processes: Corollary 2. (Jarzynski equality).
e−W
= 1.
(8.7)
Let us illustrate the meaning of the above relation by considering a few special cases. Example 9. The case of Langevin dynamics. With the splitting (6.7) used forthe canonical time inversion, upon taking ϕt = β(Ht − Ft ), where Ft = −β −1 ln e−β Ht (x) d x denotes the free energy, we infer from Eq. (7.17) that W = β(W − F) ,
(8.8)
where F = FT − F0 is the free energy change and W is the work given by Eq. (7.18). The difference W − F is often called the dissipative work. The Jarzynski equality (8.7) may be rewritten in this case in the original form
e−βW = e−β F , (8.9) in which it has become a tool to compute the differences between free energies of equilibrium states from nonequilibrium processes [45,17,72,73].
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Example 10. The case of deterministic dynamics. Upon splitting the drift u t with uˆ t,+ ≡ 0 related to the modified natural time inversion described in Sect. 6.2, the expression (7.6) reduces to at . Jt = −(∇ · uˆ t )(xt ) ≡ Jtn
(8.10)
ij
For the deterministic dynamics where Dt (x, y) ≡ 0, the difference between the vector at reduces to fields uˆ t and u t disappears and Jtn Jtdet = −(∇ · u t )(xt ) .
(8.11)
The right hand side represents the phase-space contraction rate along the trajectory xt , see Eq. (4.3). In this case, T W = ϕ −
T (∇ · u t )(xt ) dt =
0
d
ϕ (x ) − (∇ · u )(x ) dt ≡ W det . t t t t dt
(8.12)
0
For ϕT = ϕ0 = ϕ, the last integral in Eq. (8.12) was termed “the integral of the dissipation function” in [26]. In the case of the deterministic dynamics (2.6) obtained from the Langevin equation by setting = 0, the expression (8.12) for W reduces to the one of Eq. (8.8) if we take ϕt = β(Ht − Ft ). In the deterministic case, the Jarzynski equality (8.7) reads
T
e0
(∇·u t )(xt ) dt
e−ϕT (xT ) dx0 = 1
(8.13)
and may be easily proven directly. To this end recall Eq. (4.4) which implies for the T deterministic case that (∇ · u t )(xt ) dt = ln det XT (x0 ), where the matrices Xt (x) of 0
the tangent process are given by Eq. (4.1). The equality (8.13) is then obtained by the change of integration variables x0 → xT whose Jacobian is equal to det XT (x0 ). Example 11. The reversed protocol case. In the setup of Sect. 6.4 with u t,− = 0, Jt = 2 uˆ t (xt ) · dt−1 (xt ) x˙ t ≡ Jttot . In the stationary case, the integral
T
Jttot dt, rewritten with use of the Itô convention, was
0
termed an “action” in [59], see Eq. (5.3) therein. In [43], it was considered in the context of the Langevin equation with the extra force G t (but without the Hamiltonian term ∇ Ht ). It was then identified as βQtot with the quantity Qtot interpreted, following [68], as the total heat produced in the environment. The functional W of the forward process is given here by the formula T W = ϕ + 2 0
uˆ t (xt ) · dt−1 (xt ) x˙ t dt ≡ W tot .
(8.14)
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In particular, for the Langevin dynamics (2.4), one obtains: T W
tot
= ϕ + β
(−∇ Ht + ∇ Ht + G t )(xt ) · −1 x˙ t dt .
(8.15)
0
The Jarzynski equality (8.7) was discussed for this case in [43,78,10]. Note that W tot is not well defined for the Langevin-Kramers dynamics. On the other hand, for the linear Langevin equation of Example 8 and for ϕt = β(H − F), T W
tot
= β
xt · (C
−1
+M T
−1
) x˙ t dt = −β
0
T
xt · C −1 −1 x˙ t dt (8.16)
0
and it vanishes if = 0. A long time asymptotics of the probability distribution of a quantity differing from the last one by a boundary term was studied in [81]. Example 12. Hatano-Sasa equality [43]. In the current-reversal setup of Sect. 6.5, with the splitting (6.10) of the drift u t induced by the normalized densities e−ϕt such that L †t e−ϕt = 0, Jt = −(∇ϕt )(xt ) · x˙ t ≡ Jtex ,
(8.17)
since now the last two terms on the right hand side of Eq. (7.6) vanish, compare to Eq. (7.13). Upon integration, this gives: T
T Jtex
dt = − ϕ +
0
(∂t ϕt )(xt ) dt.
(8.18)
0
In [43], the integral given by Eq. (8.17) was identified in the context of the Langevin equation with the force G t as equal to βQex , where Qex was termed the excess heat, following [68]. The difference Qtot − Qex = Qhk was called, in turn, the housekeeping heat and was interpreted as the heat production needed to keep the system in a nonequilibrium stationary state, see again [68,43,78,10]. Using in the definition (8.2) the functions ϕ0 and ϕT from the same family, we infer from Eq. (8.18) that T W =
(∂t ϕt )(xt ) dt ≡ W ex .
(8.19)
0
The equality (8.7) for this case was proven by Hatano-Saso [43], see also [57]. Note that in the stationary case, W ex = 0. The Langevin dynamics discussed in Example 9 provides a special instance of the situation considered here if G t ≡ 0. Consequently, in that case, W ex is equal to the dissipative Jarzynski work (in the β −1 units) β(W − F) with W given by Eq. (7.20). Example 13. The case of complete reversal. Recall that for the complete reversal rule of Sect. 6.6 based on the choice of densities e−ϕt evolving dynamically, Jt is the total time derivative, see Eq. (7.14). The use in the definition (8.2) of the functions from the same family annihilates the functional W: W ≡ 0.
(8.20)
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9. Speck-Seifert Equality Let us consider the two functionals W tot and W ex of the process xt introduced in Examples 11 and 12. We shall take them with ϕt satisfying L †t e−ϕt =
the same functions
= 1 = e−W hold simultaneously. In [78] 0. The two Jarzynski equalities e−W a third equality of the same type, this time involving the quantity tot
T W
hk
= W
tot
−W
=
ex
ex
∇ϕt (xt ) + 2 uˆ t (xt ) · dt−1 (xt ) x˙ t dt
0
was established in the context of the Langevin equation where W hk = βQhk = βQtot − βQex is the housekeeping heat (in the β −1 units). We shall prove here a generalization of the result of [78]. To this end, let us consider, besides the original process xt satisfying the SDE (2.1), the Markov process xt satisfying the same equation but with the drift uˆ t replaced by uˆ t = −uˆ t − dt ∇ϕt .
(9.1)
We shall denote by · the expectation defined by Eq. (8.4) but referring to the process −ϕt = 0, where the operators L are given by xt . Note in passing the relations L † t e t Eq. (3.3) with uˆ t replacing uˆ t . In particular, in the stationary case, the processes xt and xt have the same invariant measure. Proposition 3.
F e−W
hk
=
F .
(9.2)
Proof. The above identity may be proven directly with the use of the Cameron-MartinGirsanov formula, see Appendix D, by comparing the measures of the processes xt and xt corresponding to SDEs differing by a drift term. Here we shall give another proof based on applying twice the relation (8.6). First, we use this relation with the functional tot ex F replaced by F e−W +2W for the current-reversal time inversion with the trivial involution x ∗ ≡ x and the vector-field rule for vt . This results in the equality
where the expectation
·
F e−W
hk
=
˜ tot ˜ ex F˜ e−W +2W ,
(9.3)
pertains to the backward dynamics with
uˆ it = −uˆ it ∗ − dt ∗ ∂ j ϕt ∗ , vti = vti ∗ , ij
see Eqs. (6.11). Now, we observe that the same backward process may be obtained by the reversed-protocol time inversion, again for x ∗ ≡ x, from the process xt introduced above. The identity (8.6) applied for the processes xt and xt reads:
F e−W
=
F˜ ,
(9.4)
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where T
W = ϕ + 2
uˆ t (xt ) · dt−1 (xt ) x˙ t dt
0
is the functional W referring to the dynamics with uˆ t = uˆ t,+ given by Eq. (9.1). The tot ex application of Eq. (9.4) to F = F e−W +2W reduces the right hand side of Eq. (9.3) −( tot ex to the expectation F e W −2W +W ) . The equality (9.2) follows by checking that W
tot
− 2W
ex
+W
T = ϕ + 2
uˆ t (xt ) · dt−1 (xt ) x˙ t
0
T dt − 2
(∂t ϕt )(xt ) dt 0
T + ϕ + 2 0 T
= 2 ϕ − 2
− uˆ t (xt ) − dt (xt )∇ϕt (xt ) · dt−1 (xt ) x˙ t dt
∂t ϕt (xt ) + ∇ϕt (xt ) · xt dt = 0 .
0
Setting F ≡ 1 in the identity (9.2), we obtain the result that was established by a different argument in [78] in the context of the Langevin equation: Corollary 3. (Speck-Seifert equality).
hk e−W = 1.
(9.5)
10. Entropy Production An immediate consequence of the Jarzynski equality (8.7) and of the Jensen inequality (i.e. of convexity of the exponential function) is Corollary 4. (2nd law of thermodynamics for diffusion processes). W ≥ 0.
(10.1)
To explain the relation of the latter inequality to the 2nd law of thermodynamics, let us first remark that the quantity on the left hand side has the interpretation of a relative entropy. Recall, that for two probability measures µ(dx) and ν(dx) = e−w(x) µ(dx), the relative entropy of ν with respect to µ is defined by the formula S(µ|ν) = w(x) µ(dx) and is always non-negative. Now, the identity (8.6) may be read as the relation M˜ (dx) = e−W M(dx)
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between the measures M(dx) and M˜ (dx) ≡ M (d x˜ ). In other words, e−W is the relative (Radon-Nikodym) density of the trajectory measure M˜ with respect to the measure M. It follows that W = S(M| M˜ ) so that the inequality (10.1) expresses the positivity of the relative entropy. Up to now, the measures µ0 and µT were unrelated. Let us consider the particular case when µT is obtained by the dynamical evolution (3.6) from µ0 so that µT = µ0 P0,T . In this case, the relative entropy S(M| M˜ ) may be interpreted as the overall entropy production in the forward process between times 0 and T , relative to the backward process, see [27,65,39]. Let for a measure ν(d x) = ρ(x)d x, S(ν) = − ln ρ(x) ν(d x) denotes its entropy. Using the definition (8.2), we may rewrite W = S(µT ) − S(µ0 ) + Senv , (10.2) where the difference S(µT ) − S(µ0 ) is the change of entropy of the fixed-time distribution of the process during the time T and T Senv =
Jt dt .
(10.3)
0
The latter quantity will be interpreted as the mean entropy production in the environment modeled by the stochastic noise, measured relative to the backward process. The inequality (10.1) states that the overall entropy production cannot be negative in mean. In this sense, it is a version of the 2nd law of thermodynamics for the diffusion processes under consideration. In the stationary case, where µT = µ0 , the overall mean entropy production reduces to the one in the environment Senv . The rate of change of the fixed-time entropy S(µt ) for µt = µ0 P0,t is easily calculated with the use of Eq. (3.7) to be d S(µt ) 1 = uˆ t · ∇ϕt + 2 (∇ϕt ) · dt (∇ϕt ) (x) µt (d x) (10.4) dt for µt (d x) = e−ϕt (x) d x. Rewriting the expression (7.6) for Jt in terms of the Itô convention, it is also easy to show that Jt = 2 uˆ t,+ · dt−1 uˆ t,+ + (∇ · uˆ t,+ ) − (∇ · u t,− ) (x) µt (d x) . (10.5) The average (10.5) represents the instantaneous mean rate of the entropy production in the environement. Combining the last two expressions, we obtain the relation d S(µt ) 1 1 + Jt = 2 (uˆ t,+ + 2 dt (∇ϕt )) · dt−1 (uˆ t,+ + 2 dt (∇ϕt )) (x) µt (x) (10.6) dt which is explicitly positive. This provides still another proof of the positivity of the expectation W which is the time integral of the latter expression if the measure µT is obtained by evolving dynamically µ0 . See Eq. (13) in [66] for the special case of the latter relation. If µT = µ0 P0,T then one has to distinguish between those two measures and the relation (10.2) is modified to W = S(µ0 P0,T ) − S(µ0 ) + Senv + S(µ0 P0,T )|µT ) ,
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i.e. the right hand side is increased by the relative entropy of the measure µT with respect to the measure obtained from µ0 by the dynamical evolution. Consequently, the average W is minimal when µT = µ0 P0,T . Note that Senv as defined by Eq. (10.3) depends on the time inversion employed (more precisely, on the splitting of u t ), and the quantities obtained by employing different time inversions are, in general, different. They may have different physical relevance. We may talk about the total mean entropy production in the environment T tot Senv
Jttot
=
T dt =
0
dt
2 uˆ t · dt−1 uˆ t + (∇ · uˆ t ) (x) µt (d x) ,
0
if the reversed protocol of Sect. 6.4 and Example 11 is used or about the excess mean entropy production
T ex Senv
=
dt
1 ∇ · uˆ t − 2uˆ t · (∇ϕt ) − 2 (∇ϕt ) · dt (∇ϕt ) (x) µt (d x)
0
in the environment for the current reversal of Sect. 6.5 and Example 12 (in the latter formula, e−ϕt satisfies L †t e−ϕt = 0 and is, in general different from the density of µt = µ0 P0,t ). The Speck-Seifert equality (9.5) combined with the Jensen inequality ex does not exceed S tot which may be also seen directly since imply that Senv env tot ex Senv − Senv = W hk
T =2
dt
1 1 (uˆ t + 2 dt · (∇ϕt )) · dt−1 (uˆ t + 2 dt · (∇ϕt )) (x) µt (d x) ≥ 0 .
0
As an illustration, consider the stationary Langevin equation with the vanishing additioex = 0 although S tot may be non-zero if = 0. In particular, nal force where Senv env in the linear case studied in Example 8, T tot Senv
Jttot
=
T dt =
0
β xt · M T −1 Mxt + tr M dt = −T tr −1 M
0
by Eq. (10.5). For the general diffusion process and the drift splitting corresponding to the modified natural time inversion of Sect. 6.2, i.e. for uˆ t,+ ≡ 0, n at Senv
T = 0
at Jtn dt
T = −
dt 0
(∇ · uˆ t )(x) µt (d x) ,
(10.7)
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see Eq. (8.10). The difference tot Senv
−
n at Senv
T = 4
T (µt ) dt,
(10.8)
0
where the functional 1 2
T (µt ) =
uˆ t · dt−1 uˆ t + (∇ · uˆ t ) (x) µt (d x)
(10.9)
was called the traffic in [66]. For the Langevin equation with the splitting (6.7) corresponding to the canonical time inversion, the entropy production in the environment is proportional to the mean heat transferred to the environment as given by Eq. (7.16): T Lan Senv
= βQ =
dt
β (∇ Ht )(x) · (∇ Ht )(x) − i j ∂i ∂ j H (x)
0
−(∇ · G t )(x) µt (d x) .
In the deterministic case when Jt is given by Eq. (8.11), the mean rate of entropy production in the environment is det Jt = − (∇ · u t )(x) µt (d x) for µt = µ0 P0,t obtained by the dynamical evolution from µ0 with P0,t (x0 , dy) = δ(y − xt )dy. For uniformly hyperbolic dynamical systems without explicit time dependence, the measures µt tend for large t to the invariant SRB measure µ∞ and the mean rate of entropy production in the environment converges to the expectation of the phase-space contraction rate −∇ · u with respect to µ∞ [74]. A discussion of the relation between of the phase-space contraction and the production of thermodynamic entropy in deterministic dynamics employing models of finite-dimensional thermostats may be found in [33]. Finally, let us remark that if the complete reversal of Sect. 6.6 is employed to define the backward process then the overall entropy production vanishes because W ≡ 0 in this case, see Eq. (8.20). With our flexibility of the choice of backward processes, there are always ones with respect to which there is no overall entropy production! 11. Linear Response for the Langevin Dynamics 11.1. Green-Kubo formula and Onsager reciprocity. As noted in [23,32,59], fluctuation relations may be viewed as extensions to the non-perturbative regime of the Green-Kubo and Onsager relations for the nonequilibrium transport coefficients valid within the linear response description of the vicinity of the equilibrium. Here, for the sake of completeness, we shall show how such relations follow formally from the Jarzynski equality (8.7) for the Langevin dynamics. To this end, we shall consider the latter with a time independent Hamiltonian Ht ≡ H and the additional time-dependent force G t (x) = gta G a (x) ,
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R. Chetrite, K. Gaw¸edzki
where the couplings gta , a = 1, 2, are arbitrary (regular) functions of time (and the summation over the index a is understood). In the case at hand, we infer from Eq. (8.8) that T W = gta J a (xt ) dt for J a = β(∇ H ) · G a − ∇ · G a . 0
In particular, for the Langevin-Kramers equation (2.7), J a = β f ia (q) q˙ i is the power injected by the external force f a (in the β −1 units). The quantities J a are often called fluxes associated to the forces G a . Let us denote by F the expectation defined by Eq. (8.4) with µ0 standing for the Gibbs measure Z −1 e−β H d x and by F 0 the same expectation taken for gta ≡ 0, i.e. in the equilibrium system. Expanding Eq. (8.7) up to the second order in gta and abbreviating J a (xt ) ≡ Jta , we obtain the identity T − 0
gta Jta 0 dt −
T T 0
gta gtb Jta Rbt 0 dt dt +
0
1 2
T T 0
gta gtb Jta Jtb 0 dt dt = 0 ,
0
(11.1) where the insertion of the response field Rat is defined by the relation δ F . F Rat 0 = δgta g≡0 Note that Jta Rbt 0 = 0 for t > t because of the causal nature of the stochastic evolution. The vanishing of the term linear in gta in Eq. (11.1) implies that the equilibrium expectation of the fluxes J a vanishes a Jt 0 = Z −1 J a (x) e−β H (x) d x = 0 , which is easy to check directly. Stripping the quadratic term in Eq. (11.1) of arbitrary functions gta , we infer that a b Jt Rt 0 = θ (t − t ) Jta Jtb 0 . The integration of the latter equation over t ≥ 0 results in the relation ∂ ∂gb
g=0
Jta
t =
Jta Jtb 0 dt ,
(11.2)
0
where on the left hand side we consider the derivative with respect to the coupling gb constant in time. In the limit t → ∞, we may expect the convergence of the expectation a a Jt in the presence of the time-independent force a ga G (and of its derivatives over gb ) to the nonequilibrium stationary expectation Jt st (and its derivatives). Let us also assume that the temporal decay of the stationary equilibrium correlation function of the fluxes is sufficiently fast, e.g. exponential. These may be often established for the dynamics governed by the Langevin equation by studying the properties of its generator. With these assumptions, Eq. (11.2) implies
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Proposition 4. (Green-Kubo formula). ∂ ∂gb
g=0
Jta st
t =
Jta Jtb 0 dt .
−∞
The stationary equilibrium correlation function Jta Jtb 0 depends only on the difference t − t of times. Besides, if the system is time-reversible, then Jta Jtb 0 = Jtb Jta 0 and the Green-Kubo formula may be rewritten in the form a b b a ∂ 1 1 a = J dt = J J Jt Jt 0 dt 0 t t t st ∂gb 2 2 g=0
which implies Corollary 5. (Onsager reciprocity). ∂ Jta st = ∂g b g=0
∂ ∂ga
g=0
Jtb st .
11.2. Fluctuation-dissipation theorem. Let us consider again the Jarzynski equality for the Langevin dynamics, this time in the absence of the additional force G t but with a time dependent Hamiltonian Ht (x) = H (x) − h ta O a (x) ,
(11.3)
where h ta , a = 1, 2, vanish at t = 0 and O a (x) are functions of x (“observables”). In this case, Eq. (8.8) reduces to the relation T W = −β
h˙ ta O a (xt ) − β F ,
0
where
β F = − ln
e
−β H (x) − h T a O a (x)
d x + ln
e−β H (x) d x .
Expanding the left hand side of the Jarzynski equality (8.7) up to the second order in h ta and abbreviating O a (xt ) ≡ Ota , we infer that β
T 0
h˙ ta Ota 0 dt − β h T a O0a 0 = 0
(11.4)
and that 1 2
β2
T T 0 0
T T h˙ ta h˙ t b Ota Otb 0 dt dt + β h˙ ta h t b Ota Rtb 0 dt dt 0 0 − 21 β 2 h T a h T b O0a O0b 0 = 0 ,
(11.5)
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where the insertion of the response field Rta is defined similarly as that of Rat before by δ F . F Rta 0 = δh ta h≡0 Again, similarly as before, Ota Rtb 0 = 0 for t > t because of causality. The first order equality (11.4) is equivalent to the time-independence of the equilibrium expectation of Ota . As for the second order relation (11.5), upon expressing h ta as the integral of h˙ ta , it is turned into the equality β
T T
h˙ ta h˙ t b
0 0
Ota
Otb
−
Ota
Otb
0
dt
dt
= 2
T T t 0 0 0
h˙ ta h˙ t b Ota Rtb 0 dt dt dt .
After the change of the order of integration over t and t followed by the interchange of those symbols, the right hand side becomes 2
T T T 0 0 t
T T t h˙ ta h˙ t b Ota Rtb 0 dt dt dt = 2 h˙ ta h˙ t b θ (t − t ) Ota Rtb 0 dt dt dt 0 0 t
with the use of causality. Stripping the resulting identity of the integrals against arbitrary functions h˙ ta , we obtain the identity t t a b a b a b a b β Ot Ot 0 − β Ot Ot 0 = θ (t − t ) Ot Rt 0 dt + θ (t − t) Ot Rt 0 dt t
t
which is the integrated version of the differential relation between the dynamical 2-point correlation function and the response function: Proposition 5. (Fluctuation-dissipation theorem). For t > t , − ∂t Ota Otb 0 = β −1 Ota Rtb 0 .
(11.6)
Note the explicit factor β in this identity. Relations between the dynamical correlation functions and the response functions were used in recent years to extend the concept of temperature to nonequilibrium systems [20,14]. 12. One-Dimensional Langevin Equation with Flux Solution Let us consider, as an illustration, the one-dimensional Langevin equation of the form
x˙ = −∂x Ht (x) + ζt
(12.1)
with ζt ζt = 2β −1 δ(t − t ) (any force is a gradient in one dimension). As before, xt will represent the Markov process solving the SDE (12.1). First, let us consider the time-independent case with a polynomial Hamiltonian H (x) = ax k + . . . with a = 0 and the dots representing lower order terms. • If k = 0 then, up to a linear change of variables, xt is a Brownian motion and does not have an invariant probability measure. • If k = 1 then xt + at is, up to a linear change of variables, a Brownian motion and xt still does not have an invariant probability measure.
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• If k ≥ 2 and is even then for a > 0 the Gibbs measure µ0 (d x) = Z −1 e−β H (x) d x provides the unique invariant probability measure of the process xt . It satisfies the detailed balance condition j (x) = 0, where j (x) is the probability current defined by Eq. (3.8). If a < 0, however, then the Gibbs density e−β H (x) is not normalizable4 . In this case, the process xt escapes to ±∞ in finite time with probability one and it has no invariant probability measure. • If k ≥ 3 and is odd then the Gibbs density e−β H (x) is not normalizable. The process xt escapes in finite time to −∞ if a > 0 and to +∞ if a < 0, but it has a realization with the trajectories that reappear immediately from ±∞. Such a resuscitating process has a unique invariant probability measure µ0 (d x) = ±N
−1
e
−β H (x)
x
eβ H (y) dy d x ≡ e−ϕ0 (x) d x
(12.2)
∓∞
with the density e−ϕ0 (x) = O(x −k+1 ) when x → ±∞ and N the (positive) normalization constant. The measure µ0 corresponds to a constant probability current j (x) = ∓(β N )−1 and the model provides the simplest example on a nonequilibrium steady state with a constant flux. Let us look closer at the last case. Adding the time-dependence and taking ϕt as in Eq. (12.2) but with Ht replacing H , we obtain the Hatano-Sasa version of the Jarzynski equality (8.7) with W = W ex given by Eq. (8.19). Suppose, in particular, that the time dependence of Ht has the form (11.3) with functions O a having compact support. Let us introduce also the deformed observables x a (x) = O
∓∞
O a (y) eβ H (y) dy x ∓∞
. eβ H (y) dy
Expanding the Jarzynski identity (8.7) to the second order in h at as in Sect. 11.2, one obtains: Proposition 6. (Deformed fluctuation-dissipation relation). For t > t , − ∂t Aat Otb 0 = β −1 Aat Rtb 0 ∓(β N )−1 (∂x O b )(x) d x Pt−t (x, dy) Aa (y) ,
(12.3)
where Pt (x, dy) is the transition probability in the stationary process and a . Aa = O a − O Remark 3. It is easy to show directly, that Eq. (12.3) still holds if Aa is replaced by O a . Note that the term on the right hand side of (12.3) violating the standard fluctuationdissipation theorem (11.6) contains the constant flux of the probability current j (x) as a factor. Proof of Proposition 6 and of its version with Aa replaced by O a will be given in [12]. 4 This leads to the breaking of the quantum-mechanical supersymmetry underlying the Fokker-Planck formulation of the Langevin dynamics [85,69,80].
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The Langevin equation (12.1) with the flux solution arises when one studies the tangent process for particles with inertia moving in the one-dimensional homogeneous Kraichnan ensemble of velocities vt (y) with the covariance vt (y) vs (y ) = δ(t − s) D(y − y ) , see Example 2. The position y and the velocity w of such particles satisfy the SDE [3] y˙ = w, w˙ =
1 (−w τ
+ vt (y)) ,
where τ is the so called Stokes time measuring the time-delay of particles with inertia as compared to the Lagrangian particles that follow the flow. The separation between two infinitesimally close trajectories of particles satisfies the equations [84] d dt
δy = δw,
d dt
δw =
1 (−δw τ
+ (∂ y vt )(y) δy)
(12.4)
and, similarly as in Example 5, we may replace τ1 ∂ y vt (y) on the right hand side by a white noise ζ (t) with the covariance ζt ζs = −δ(t − s) τ −2 D (0) , where the primes denote the spatial derivatives. The ratio x = x˙ = −x 2 −
1 x τ
+ ζt
δw δy
satisfies then the SDE (12.5)
1 2 x , a third order polynomial. The which has the form (12.1) with H (x) = 13 x 3 + 2τ solution with the trajectories appearing at +∞ after disappearing at −∞ corresponds to the solution for (δy, δw) with δy passing through zero with positive speed. The top Langevin exponent for the random dynamical system (12.4) is obtained as the mean value of x (which is the temporal logarithmic derivative of |δy|) in the invariant probability measure (12.2) with constant flux [84]. A very similar SDE arose earlier [41] in the one-dimensional Anderson localization in white-noise potential V (y) , where one studies the stationary Schrödinger equation
− ψ (y) + V (y) ψ(y) = E ψ(y) . By setting x = ψ /ψ, one obtains then the evolution SDE x = −x 2 − E + V (y)
(12.6)
that has an invariant probability measure with constant flux, as already noticed in [41]. The expectation value of x in that measure may be expressed by the Airy functions [61]. It gives the (top) Lyapunov exponent which is always positive, reflecting the permanent localization in one dimension. The SDE (12.5) may be obtained from (12.6) but taking in the latter E = − 4τ1 2 and by the substitutions x − 2τt → x, V → ζ and y → t. 1 This shifts the Lyapunov exponent down by − 2τ and the top exponent for the inertial particles may have both signs [84].
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13. Detailed Fluctuation Relation For a general pair of forward and backward diffusion processes (2.1) and (5.4), it is still possible to obtain identities resembling the generalized detailed balance relation (7.15) at the price of adding constraints on the process trajectories. Let us introduce a functional W of the backward process by mimicking the definition (8.2) of W for the forward process: W = ϕ +
T
Jt dt
0
with Jt = 2 uˆ t,+ (xt ) · dt−1 (xt ) x˙ t − 2 uˆ t,+ (xt ) · dt−1 (xt ) u t,− (xt ) − (∇ · u t,− )(xt ) , see Eq. (7.6). Since the time inversion is involutive, the mirror version of the identity (8.3), −W (x ) ∗ ∗ ˜ µ0 (d x ) E 0,T dy = µ0 (dy ∗ ) E 0,T x ,y F (x ) e y ∗,x ∗ F (x) d x d x , (13.1) ˜ must also hold. Taking x = y ∗ , y = x ∗ and F = F˜ e−W , we infer that the compatibility of identities (8.3) and (13.1) imposes the equality
˜ , W = −W
(13.2)
which may be also checked directly. We infer that, whatever the time inversion used in their definition, the entropy-production functionals W for the forward and the backward processes are related by the natural time inversion. The replacement in Eq. (8.3) of the functional F(x) by the functional F(x) δ(W(x) − W ) including the constraint fixing the value of W, leads then to Proposition 7. (Detailed fluctuation relation). ∗ W µ0 (d x) E 0,T x,y F(x) δ W(x) − W dy = µ0 (dy ) e
∗ ˜ ×E 0,T y ∗,x ∗ F(x ) δ W (x ) + W d x . (13.3)
The primes on the right hand side may be dropped in the time-reversible case if, additionally, ϕ0 = ϕ0 and ϕT = ϕT .
A relation of this type, named the ”detailed fluctuation theorem”, was established in [51] in the setup of Hamiltonian dynamics. It is close in spirit to the earlier observation made for the long-time asymptotics of deterministic dynamical systems in [31]. We shall view Proposition 7 as a source of fluctuation relations that hold for the diffusion processes (2.1), including the Jarzynski equality (8.7) already discussed and various identities that appeared in the literature in different contexts, see [51,18,19,57]. Taking, in particular, F ≡ 1 in Eq. (13.3) and introducing the joint probability distributions of the end-point of the process and of the entropy production functional W, E 0,T x,y δ(W(x) − W ) dy dW = P0,T (x, dy, dW ) , E 0,T x,y δ(W (x ) − W ) = P0,T (x, dy, dW ) ,
we obtain
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Corollary 6. µ0 (d x) P0,T (x, dy, dW ) = µT (dy) e W P0,T (y ∗ , d x ∗ , d(−W )) .
(13.4)
This may be viewed as an extension to a general diffusive SDE (2.1) of the detailed balance relation (3.9), or of its generalization (7.15). In particular, when the backward process is obtained by the complete reversal of Sect. (6.6) with W ≡ 0, the latter relation reduces to Eq. (7.15) with both sides multiplied by δ(W )dW . In the case when the measures µ0 and µ0 are normalized, Proposition 7 gives rise, upon integration over x and y, to a detailed fluctuation relation between the forward and the backward processes with the initial points sampled with measures µ0 and µ0 , respectively: Corollary 7.
F δ W − W = eW F˜ δ W + W .
Finally, taking F = 1 in the latter identity and denoting p0,T (dW ) = δ W − W dW, p0,T (dW ) = δ W − W dW , we obtain Corollary 8. (Crooks relation) [18,19]. p0,T (dW ) = e W p0,T (d(−W )) .
(13.5)
Note that p0,T (dW ) is the distribution of the random variable W if the time-zero values of the forward process xt are distributed with the measure µ0 and, similarly, (dW ) is the distribution of the random variable W if x is distributed with p0,T 0 (dW ) = p the measure µ0 . In particular, in the time-reversible case, p0,T 0,T (dW ) if ϕ0 = ϕ0 and ϕT = ϕT . Finally, note that integrating the Crooks relation (13.5) multiplied by e−W over W , one recovers the Jarzynski equality (8.7). 14. Special Cases 14.1. Deterministic case. As already explained in Sect. 6.2 and 13, taking uˆ t,+ = 0 and u t,− = uˆ t leads in the limit of the deterministic dynamics (2.3) to the expression (8.12) for W. The time-reversed dynamics corresponds to the vector fields of Eqs. (6.2). It reduces in the deterministic case to the ODE (6.3). The functional W of the backward process, that could be also found from the relation (13.2), takes the form
T
W = ϕ +
(∇ ln σ )(xt ) · (x˙t − u t,− ) − (∇ · u t,− )(xt ) dt .
0
In the deterministic limit, this simplifies to the expression
T
W = ϕ − 0
(∇ · u t )(xt ) dt
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which is of the same form as Eq. (8.12) for W. Proposition 7 and Corollaries 6,7 and 8 still hold in the deterministic limit. In particular, in the time-reversible deterministic case with u = u and ϕT = ϕ0 = ϕ0 , the fluctuation relation (13.5) reduces to Corollary 9. (Evans-Searles transient fluctuation theorem) [25,26]. p0,T (dW ) = eW p0,T (d(−W )) . The latter relation may also be proven directly by a change of the integration variables x0 → xt [26]. 14.2. Reversed protocol case. For the reversed protocol time inversion of Sect. 6.4 and Example 11 that corresponds to the choice (6.8), the backward process is given by Eq. (6.9) and
T
W = ϕ + 2
uˆ t (xt ) · dt−1 (xt ) x˙ t dt
0
and has the same form as W, see Eq. (8.14). For such a time inversion with x ∗ ≡ x, employed already in the stationary context in [59], the fluctuation relation (13.5) for the choice of ϕt such that L †t e−ϕt = 0 was established in [10]. 14.3. Current reversal case. For the time inversion (6.10) discussed in Sect. 6.5 and Example 12, the functional W of the backward process is given by the expression of the same form as Eq. (8.19):
T
W =
(∂t ϕt )(xt ) dt
0
for ϕt (x) = (ϕt ∗ +ln σ )(x ∗ ). The fluctuation relation (13.5) for this type of time inversion (with x ∗ ≡ x) was proven in [10]. Integrated against e−W , Eq. (13.5) reduces to the Hatano-Sasa case of the Jarzynski equality (8.7) that we discussed in Example 12. 14.4. Langevin dynamics case. Recall that for the Langevin dynamics (2.4), the backward process obtained by using a canonical time inversion defined by Eqs. (6.6) and (6.7) is also of the Langevin type with u t = −∇ Ht + ∇ Ht + G t ,
(14.1)
where Ht (x) = Ht ∗ (r x), G t (x) = −r G t ∗ (r x). The white noise ζt = ±r ζt ∗ has the same distribution as ζt . Consequently, for ϕt = β(Ht − Ft ), the functional W is given by the primed version of Eq. (8.8) and is equal to the dissipative work (in the β −1 units). If, instead of the canonical time inversion, we use the reversed protocol with x ∗ ≡ x , then the backward process is again the Langevin dynamics with u t given by Eq. (14.1), except that this time Ht (x) = Ht ∗ (x) and G t (x) = G t ∗ (x). The white noise ζt = ζt ∗
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has again the same distribution as ζt . The functional W is given in that case by the primed version of Eq. (8.15). The two time inversions lead to the equivalent backward processes for the Langevin-Kramers equation but, as already mentioned, W tot is not well defined in the case of the reversed protocol. Finally, if we apply the current-reversal time inversion (6.10) with x ∗ ≡ x to the Langevin dynamics (2.4) with G t ≡ 0 by setting ϕt = β(Ht − Ft ) = ϕt∗ = β(Ht∗ − Ft∗ ) for Ht (x) = Ht ∗ (x), the drift of the backward dynamics becomes u t = − ∇ Ht − ∇ Ht and has the changed sign of the antisymmetric matrix with respect to the forward process. The white noise ζt = ±ζt ∗ . Here both W and W have the form of the dissipative work.
15. Transient Versus Stationary Fluctuation Relations The fluctuation relations considered up to now dealt with the quantities related to finitetime evolution in a random process that, in general, was not stationary. Such simple relations, whose prototypes were the Evans-Searles fluctuation relation [25] or the Jarzynski equality [48] are called transient fluctuation relations. On the other hand, as was recalled in Introduction, Gallavotti and Cohen have established in [35] a fluctuation relation for quantities pertaining to the long-time evolution in stationary deterministic dynamical systems of chaotic type and similar relations were subsequently obtained for the Langevin dynamics and Markov processes in [57] and [59]. Such fluctuation relations, that are commonly termed stationary, are usually more difficult to establish than the transient ones and require some non-trivial work that involves the existence and the properties of the stationary regime of the dynamics. Such properties are in general harder to establish in the non-random case than in the random one. Also, in the random case, the invariant measure of the process, if it exists, is usually smooth. It could be used as the measure µ0 (d x) = e−ϕ0 (x) d x = µT (d x) = µ0 (d x ∗ ) in the definition (8.2), leading to the exact detailed fluctuation relation (13.3) pertaining to the stationary evolution. On the other hand, in the dissipative deterministic systems, the invariant (SRB) measures are not smooth, so that they may not be used this way and the exact stationary fluctuation relations may be obtained only in the asymptotic long-time regime. Let us discuss briefly a formal relation between such asymptotic fluctuation relations and the transient ones, sweeping under the rug the hard points. We shall consider the stationary case of the SDE (2.1), with u t ≡ u and Dt (x, y) ≡ D(x, y). Under precise conditions, the Markov process xt that has decaying dynamical correlations and attains at long times the steady state independent of the initial (or/and final) position [42,55]. In such a situation, the distribution of the functional W is expected (and may often be proven with some work) to take for long time T and for W/T = O(1) the large deviation form P0,T (x, dy, dW ) ∝ e−T ζ (W /T ) dy dW
(15.1)
independent of x and y. The function ζ is called the large deviations rate function. It has vanishing minimum. More exactly, the relation (15.1) means that
Fluctuation Relations for Diffusion Processes
− sup ζ (w) ≤ lim w∈I
1 ≤ lim ln T →∞ T
T →∞
507
1 ln T
P0,T (x, y|W) dW
TI
P0,T (x, y|W) dW ≤ − inf ζ (w)
TI
w∈I
for any interval I in the real line. In particular, in the limit T → ∞, the distribution of W/T concentrates at the non-random value w0 , where the rate function ζ attains its minimum. With similar assumptions about the inverse process, we shall denote by ζ the large deviation rate function of the functional W . The detailed fluctuation relation (13.4) implies then immediately, if the boundary term ϕ0 (y ∗ )/T = (ϕT (y)+ln σ (y))/T converges to zero when T → ∞, a relation between the rate functions ζ and ζ : Corollary 10. (Stationary fluctuation relation). ζ (w) = ζ (−w) − w.
(15.2)
Equation (15.2) connects the statistics of large deviations of W for the forward and for the backward stationary stochastic processes. Note that the equality ζ ≥ 0 implies that the asymptotic value w0 of W/T is non-negative. This conclusion may be also drawn from the 2nd law (10.1). In the special case of a stationary time-reversible dynamics, the inverse process coincides with the direct one so that ζ = ζ . Equation (15.2) compares then the large deviations of W of opposite signs in the forward process. In particular, it states that the probability that W/T takes values opposite to the most probable ones around w0 is suppressed by the exponential factor e−T w0 for large times T . T Recall from the definition (8.2) that W differs from the extensive quantity Jt dt 0
by a boundary term which should not contribute to the large deviations if ϕT stays bounded, although the presence of such terms may change the time-scales on which the large deviation regime is effectively visible. On the contrary, unbounded ϕT may give contributions to the large deviations statistics [83,9,86,70]. For the deterministic T T dynamics where Jt dt = − (∇ · u)(xt ) dt is the phase-space contraction along the 0
0
trajectory, see Eq. (8.11), the identity (15.2) with ζ = ζ is essentially the original Gallavotti-Cohen fluctuation relation [35,31] established rigorously by the authors for the reversible Anosov dynamical systems with discrete time. For such systems, the thermodynamic formalism [75,34] may be used to prove the existence of the stationary (SRB) measure and of the large deviations regime for the phase-space contraction, see also [74] for a somewhat different approach. In [56], the fluctuation relation (15.2) was discussed for the Langevin-Kramers dynamics, see also [59,57]. Its version considered here for a general stationary diffusion process is equivalent in the case of vanishing time-inversion-odd drift u − to the fluctuation relation discussed in [59], see Eq. (5.8) therein. As another (although related) example of how the transient fluctuation relations yield stationary ones involving large deviations, let us recall the case of the tangent process in the homogeneous Kraichnan model leading to the Itô multiplicative SDE (7.8) (or the Stratonovich SDE (7.10) equivalent to it) and defining the matrix-valued process Xt .
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We have established for it the transient fluctuation relation (7.12) that may be rewritten as the identity E 01 det(XT ) f (XT ) = E 01 f (X−1 T )
(15.3)
for functions f of real d × d matrices with positive determinant. Such matrices X may be cast into the form X = O diag(eρ1 , . . . , eρd ) O −1
(15.4)
with a diagonal matrix of non-increasingpositive entries sandwiched between two orthogonal ones. Note that ln det X = ρi . The so called stretching exponents ρ1 ≥ · · · ≥ ρd are uniquely defined by Eq. (15.4). Consider functions f (X ) that are left- and right-invariant under the action of the orthogonal group O(d). They may be viewed as functions of the vector ρ of the stretching exponents. The distribution PT (d ρ) of such exponents is defined by the relation E 01 f (XT ) = f (ρ) PT (d ρ) . ρ1 ≥..≥ρd
The identity (15.3) implies then that PT (d ρ) e
ρi
= PT (d(−ρ)) ,
(15.5)
where −ρ = (−ρd , . . . , −ρ1 ) is the vector of the stretching exponents of the matrix X −1 . In a few particular situations (e.g. in the isotropic case), it has been established that for long times and ρ/T = O(1), the distribution of the stretching exponents takes the large deviation form ) ∝ e−T Z (ρ/T d ρ PT (d ρ)
and the identity (15.5) implies then the stationary fluctuation relation Z ( σ) −
d
σi = Z (−σ) ,
(15.6)
i=1
see [11]. Since − ρi represents the phase-space contraction − ln det Xt in the Kraichnan model, the relation (15.6) may be viewed as a modified Gallavotti-Cohen identity (15.2) for the homogeneous Kraichnan model. The modification goes in two directions. On one hand, the original Gallavotti-Cohen relation involved the deterministic dynamics, whereas the relation (15.6) pertains to random Kraichnan dynamics. On the other hand, it refers to the “multiplicative” large deviations for the vector ρ of the stretching exponents containing more detailed information than the phase-space contraction represented by − ρi . For example, the most probable values of the stretching rates σi = ρi /T for which Z ( σ ) = 0 define the Lyapunov exponents λi whereas the most probable phase-space contraction rate is equal to the negative of their sum. We shall see in the next section how to extend such multiplicative fluctuation relations to the general diffusive processes. The source of such an extension resides in transient relations that may be proven for general random or deterministic dynamical systems by a simple change-of-variables argument à la Evans-Searles [26], as first indicated in [1].
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16. Multiplicative Fluctuation Relations As we have mentioned above, the SDE (2.1) defining the diffusive process xt may be used to induce other diffusive processes, the simplest example being the tangent process (xt , Xt ) introduced in Sect. 4 and satisfying the SDEs x˙ = u t (x) + vt (x),
X˙ = Ut (x, X ) + Vt (x, X )
with Uti j (x, X ) = (∂k u it )(x) X k j , Vti j (x, X ) = (∂k vti )(x) X k j , see Eq. (4.2). The covariance of the white noise vector field (vt , Vt ) is given by the relations (2.2) and vti (x) Vsk l (y, Y ) = δ(t − s) ∂ y m Dtik (x, y) Y ml , p
j
pj
Vt r (x, X ) vs (y) = δ(t − s) ∂x n Dt (x, y) X nr , p
pk
Vt r (x, X ) Vsk l (y, Y ) = δ(t − s) ∂x n ∂ y m Dt (x, y) X nr Y ml . One may now apply the theory developed above for general diffusion processes to the case of the tangent process. As an example, let us consider the natural time inversion of Sect. 6.1 corresponding to the trivial splitting (u t,+ , Ut,+ ) = 0, (u t,− , Ut,− ) = (u t , Ut ) and to the involution (x, X )∗ = (x ∗ , X ∗ ) with (X ∗ )i j = (∂k x ∗ i )(x) X k j . The backward process (xt , Xt ) satisfies in this case the SDE x˙ = u t (x ) + vt (x ),
X˙ = Ut (x , X ) + Vt (x , X )
with u it (x) = −(∂k x ∗ i )(x ∗ ) u kt∗ (x ∗ ) , Ut (x, X ) = −(∂k ∂m x ∗ i )(x ∗ ) (X ∗ )mj u kt∗ (x ∗ ) − (∂m x ∗ i )(x ∗ ) Utm∗ j (x ∗ , X ∗ ) = (∂n u it )(x) X n j and, similarly, vti (x) = ±(∂k x ∗ i )(x ∗ ) vtk∗ (x ∗ ), Vt (x, X ) = (∂n vti )(x) X n j . Note that the backward process (xt , Xt ) defined this way coincides with the tangent process of xt . Equations. (3.4) applied to case at hand give: i d+1 uˆ t,+ (x) , (Uˆ t,+ )kl (x, X ) = − 2 ∂ y n D in (x, y)| y=x , ∂x n ∂ y m D km (x, y)| y=x X nl ij dt (x) ∂ y m Dtik (x, y)| y=x X ml d+1 −1 r = − 2 0 , (X ) p pj pk ∂x n Dt (x, y)| y=x X nr ∂x n ∂ y m Dt (x, y)| y=x X nr X ml
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in the matrix notation, where the matrix on the right hand side is the counterpart of ij dt (x) for the tangent process. Substituting the above expression to the definition (7.6), we infer that p p Jt = −(d + 1)(Xt−1 )r p X˙ t r + (d + 1)(Xt−1 )r p ∂n u t (xt ) Xtn r − (d + 1) ∂n u nt (xt ) d ln det Xt . = −(d + 1) dt
The relation (7.7) of Sect. 7 gives then for the case of the tangent process the identity d x d X 0 P0,T (x, X 0 ; dy, d X ) (det X 0 )−(d+1) (det X )d+1 = dy d X P0,T (y ∗ , X ∗ ; d x ∗ , d X 0∗ ) that may be viewed as an extension of the relation (7.11) obtained in Example 5 for the homogeneous Kraichnan process to a general diffusive process. Similarly as in Example 5, we infer from the above equation the multiplicative fluctuation relation d x P0,T (x, 1; dy, d X ) (det X ) = dy P0,T (y ∗ , 1∗ ; d x ∗ , d(X −1 )∗ ) .
(16.1)
Suppose that we are given a Riemannian metric γ on Rd (for example the usual flat one). Since the matrix X = XT maps the tangent space at x = x0 to the one at y = xT , see Eq. (4.2), it is natural to define the stretching exponents ρ of X by the relation (15.4) with O and O mapping the canonical basis of Rd into a basis orthonormal with respect to the metric γ (x) and γ (y), respectively. The joint probability distribution P0,T (x, dy, d ρ) of the end-point of the process xt and of the stretching exponents of Xt is then given by the relation f (ρ) P0,T (x, dy, d ρ) f (X ) P0,T (x, 1, dy, d X ) = ρ1 ≥..≥ρd
for functions f (X ) left- and right-invariant under the action of the orthogonal groups preserving, respectively, the metric γ (x) and γ (y). Similarly we introduce the kernels (x , dy , d ρ P0,T ) using the transition probabilities of the backward process and the metric γ obtained from γ by the involution x → x ∗ . Equation (16.1) implies then the identity
vγ (d x) P0,T (x, dy; d ρ) e
ρi
i
= vγ (dy) P0,T (y, d x ∗ ; d(−ρ)) ,
where vγ (d x) is the metric volume measure. For the stationary dynamics, we may expect the emergence of the large deviations regime for the stretching rates with ) ∼ dy d ρ P0,T (x, dy; d ρ) = e−T Z (ρ/T
for large T and ρ/T = O(1), and similarly for the backward process. One obtains then the identity Z ( σ) −
d i=1
σi = Z (−σ) .
(16.2)
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As usually, the rate function Z for the backward process may be replaced by Z for a time-reversible dynamics. The relation (16.2) generalizes the fluctuation relation (15.6) obtained for the Lagrangian flow in the homogeneous Kraichnan model that was timereversible. The multiplicative fluctuation relations were studied recently in [30] also for particles with inertia carried by the homogeneous Kraichnan flow. Due to the Stokes friction force, the standard time-reversibility is broken in such a system, leading to a modification of the relation between the rate functions Z and Z . 17. Towards N-Point Hierarchy of Fluctuation Relations Another way to induce new diffusive processes from the original one described by the SDE (2.1) is to consider simultaneously its N solutions starting at different initial points. They may be viewed as a solution of the SDE x˙ = ut (x) + vt (x)
(17.1)
with x = (x1 , . . . , x N ), ut (x) = (u t (x1 ), . . . , u t (x N )), and vt (x) = (vt (x1 ), . . . , vt (x N )). The covariance of the white noise vector field vt ≡ (vt,1 , . . . , vt,N ) appearing on the right hand side is j
ij
i vt,m (x) vs,n ( y) = δ(t − s) Dt (xm , yn ) .
The spatial part of the covariance restricted to the diagonal is ij
ij
dt,mn (x) = Dt (xm , xn ) . The machinery producing the fluctuation relations described in this paper may be applied to the N -point diffusion process governed by the SDE (17.1), at least if the matrix ij ij dt,mn (x) is invertible, recall that the inverse of the matrix dt (x) appears in the expression (7.6) for Jt . We postpone a closer examination of the possible hierarchy of fluctuation relations obtained this way to the future. Here, let us only remark that the tangent process (xt , Xt ), which was studied in the preceding section and led to the multiplicative fluctuation relation (16.1), could be viewed as a limiting case of the (d + 1)-point process where the last d points are infinitesimally close to the first one. 18. Conclusions We have developed a unified approach to fluctuation relations for finite-dimensional diffusion processes. The setup of the paper covered the cases of deterministic dissipative continuous-time dynamical systems, of the Langevin dynamics with non-conservative forces, and of the Kraichnan model of hydrodynamic flows. The fluctuation relations were obtained by comparing the forward diffusion process to the backward one produced by a time inversion. We have admitted different time inversions that treated differently two parts of the deterministic drift in the diffusion equations. This was physically motivated in situations when one part of the drift was assimilated with a dissipative and another one with a conservative force, but was used in other situations as well, leading to a greater flexibility. As particular cases, we discussed the natural time inversion used for deterministic systems, its slight modification for stochastic dynamics that permitted to take easily the deterministic limit of fluctuation relations, as well as the reverse
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protocol and the current reversal discussed in a similar context in [10], and the complete reversal. We showed that any of the allowed time inversions leads to a detailed fluctuation relation (13.3) of Proposition 7 that may be viewed as a constrained version of the generalized detailed balance relation to which the identity (13.3) reduces in the case of the complete reversal. The constraint fixes the value of the entropy production measured relative to the corresponding backward process. We obtained various transient fluctuation relations as corollaries of the detailed one. Among examples were the Evans-Searles fluctuation relation (14.1), the Crooks one (13.5), and various versions of the Jarzynski equality (8.7), including the original ones for the deterministic Hamiltonian dynamics and for the Langevin dynamics with local detailed balance (8.9), the one for reversed protocol, and the Hatano-Sasa one. By comparing the detailed fluctuation relations for two different time inversions, we obtained also a generalization (9.2) of the Speck-Seifert equality (9.5). For the sake of completeness, we included into the paper a derivation from the Jarzynski equality of the Green-Kubo and the Onsager relations, and of the fluctuation-dissipation theorem. On a simple example of a one-dimensional Langevin equation with spontaneously broken equilibrium, we indicated how in such a situation the Hatano-Sasa version of the Jarzynski equality induced corrections to the fluctuation-dissipation theorem proportional to the flux of the probability current. In the case of stationary diffusion processes, we pointed out that the transient fluctuation relations may give rise to the asymptotic symmetries of the large-deviations rate function of the entropy production which were established first by Gallavotti-Cohen for the uniformly hyperbolic dynamical systems and were extended later to (some) diffusion processes by Kurchan and Lebowitz-Spohn. Finally, we wrote explicitly a detailed fluctuation relation for the induced tangent diffusion process obtained from the original one. This produced a multiplicative transient fluctuation relation that led for long times to a Gallavotti-Cohen-type symmetry of the large-deviations rate function for the stretching exponents governing the behavior of infinitesimally close trajectories of the diffusion process. We speculated that considering distant multi-point trajectories of the process should give rise to a hierarchy of fluctuation relations. It could also provide a way to produce fluctuation relations for flow processes describing the simultaneous evolution of all trajectories of the process [55]. A similar extension should also permit to formulate fluctuation relations for hydrodynamic flows modeling fully developed turbulence [40,60]. We postpone such questions to further studies. Appendix A. The Stratonovich SDE (2.1) defining the process xt is equivalent to the Itô SDE d x i = u it (x) + u˜ it (x) dt + vti (x) dt , with the correction term u˜ it (x) =
1 ∂ j 2 x
ij
Dt (x, y)| y=x .
By the Itô calculus, g(xt ) satisfies the Itô SDE dg(x) = u it (x) + u˜ it (x) ∂i g(x) dt + vti (x)∂i g(x) dt +
1 ij d (x)∂i ∂ j g(x) dt 2 t
with the second order Itô term. For the expectation of g(xt ), this gives the ODE d t0 E g(xt ) = E tx0 u it (xt ) + u˜ it (xt ) ∂i g(xt ) + dt x
1 ij d (xt )∂i ∂ j g(xt ) 2 t
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from which the formula L t = u it + u˜ it (xt ) ∂i +
1 ij d ∂∂ 2 t i j
,
easily seen to be equivalent to Eq. (3.3), follows. Appendix B. Proof of Lemma 1. (L t,− Rg)(x) = u it,− (x)∂i (Rg)(x) = u it,− (x) (∂i x ∗ k )(x)(∂k g)(x ∗ ) ∗ = −(u k t ∗ ∂k g)(x ) = −(R L t ∗ ,− g)(x) , 1 ij ∂ d (x)∂i (Rg)(x) 2 j t 1 u it,+ (x)(∂i x ∗ k )(x)(∂k g)(x ∗ ) − 2 ∂ y j D i j (x, y)| y=x 1 ij + 2 (∂ j x ∗ l )(x)∂x ∗l dt (x)(∂i x ∗ k )(x)(∂k g)(x ∗ )
(L t,+ Rg)(x) = uˆ it,+ (x)∂i (Rg)(x) + =
(∂i x ∗ k )(x)(∂k g)(x ∗ )
∗ ∗l ij ∗k ∗ = (u k t ∗ ,+ ∂k g)(x ) − 2 (∂ j x )(y)∂ y ∗l D (x, y)| y=x (∂i x )(x)(∂k g)(x ) ij 1 − 2 ∂x ∗l ∂ j x ∗ l (x) dt (x)(∂i x ∗ k )(x)(∂k g)(x ∗ ) 1
+ 2 ∂x ∗l (∂ j x ∗ l )(x)dt (x)(∂i x ∗ k )(x)(∂k g)(x ∗ ) 1
ij
∗ = (u k t ∗ ,+ ∂k g)(x ) −
1 ij ∂ (∂ j x ∗ l )(y)Dt (x, 2 y ∗l ∗
y)(∂i x ∗ k )(x)| y=x (∂k g)(x ∗ )
∗ + 2 ∂x ∗l dtkl ∗ (x )(∂k g)(x ) 1
∗ = (uˆ k t ∗ ,+ ∂k g)(x ) +
1 (∂ d kl ∂ g)(x ∗ ) 2 l t∗ k
= (R L t ∗ ,+ g)(x) ,
where we have used the relations (3.4), (5.5, (5.6) and (5.7).
Appendix C. In order to prove the first of the equalities (6.2), let us note that the condition uˆ t,+ = 0 means that u it,+ (x) =
1 ∂ j 2 y
ij
Dt (x, y)| y=x
so that, according to Eqs. (5.5) and (5.6), u t ∗ ,+ (x ∗ ) = (∂k x ∗ i )(x) 2 ∂ yl Dtkl∗ (x, y)| y=x 1
i
= 2 ∂ y n (∂ j x ∗ n )(y ∗ )(∂l x ∗ j )(y)(∂k x ∗ i )(x)Dtkl (x, y)| y=x 1
= 2 (∂ j x ∗ n )(y ∗ )∂ y n (∂k x ∗ i )(x)Dtkl (x, y)(∂l x ∗ j )(y)| y=x 1 + 2 ∂x n (∂ j x ∗ n )(x ∗ ) (∂k x ∗ i )(x)dtkl (x)(∂l x ∗ j )(x) 1
i j
= 2 ∂ y j Dt ∗ (x ∗ , y)| y=x ∗ + 1
i j
= 2 ∂ y j Dt ∗ (x ∗ , y)| y=x ∗ + 1
1 i j (∂ x ∗ k )(x)(∂k ∂ j x ∗ n )(x ∗ ) dt ∗ (x ∗ ) 2 n 1 i j ∗ d (x )(∂ j ln σ )(x ∗ ) , 2 t∗
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where we used the identity (∂ j ln σ )(x ∗ ) = (∂n x ∗ k )(x)(∂ j ∂k x ∗ n )(x ∗ ) to obtain the last equality. The first of the relations in Eqs. (6.2) follows. The second one is an immediate consequence of the transformation rule in Eqs. (5.5). Appendix D. Proof of Lemma 2. The Cameron-Martin-Girsanov formula5 says that if yt is the diffusion process solving the SDE y˙ = wt (y) + u t (y) + vt (y) ,
(D.1)
then E tx0 F(y) = E tx0 F(x) e−U (x) for xt solving the SDE (2.1) and U (x) =
t
− ws (xs ) · ds−1 (xs ) x˙ s + ws (xs ) · ds−1 (xs ) uˆ s (xs )
t0
+ 2 ws (xs ) · ds−1 (xs ) ws (xs ) + 1
1 (∇ 2
· ws )(xs ) ds
if the functional F depends on the process restricted to the time interval [t0 , t]. The first term under the integral in the expression for U (x) has to be interpreted with the Stratonovich rule. Denoting by L˜ t the generator of the process yt solving the SDE (D.1), 1 ij L˜ t = (wti + uˆ it )∂i + 2 ∂ j dt ∂i ,
we obtain this way the relation
− → P˜t0 ,t (x, dy) ≡ E tx0 δ(yt − y) dy = T exp
t
L˜ s ds (x, dy)
t0
=
E tx0 e−U (x) δ(xt
− y) dy .
Next, if f t (x) is a time-dependent function then, by the Feynman-Kac formula,
− → T exp
t
( L˜ s − f s ) ds (x, dy) = E tx0 e
−U (x)−
t
t0
f s (xs ) ds
δ(xt − y) dy .
t0
The application of the latter formula for wt = −2uˆ t,+ and f t = −∇ · uˆ t,+ + ∇ · u t,− gives Eq. (7.5) in view of the relation (7.1).
5 We have transformed the formula usually written in the Itô convention [79] to the Stratonovich one.
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Appendix E. Here we show that the matrix M given by Eq. (7.29), where and C are strictly positive and is antisymmetric, has eigenvalues with negative real parts and that the matrix C may be recovered from Eq. (7.25) by setting t = ∞. If λ is an eigenvalue of M, i.e. if − ( − )C −1 xλ = λxλ for some xλ = 0 then λ =
− xλ · C −1 ( − )C −1 xλ − xλ · C −1 C −1 xλ = < 0. xλ · C −1 xλ xλ · C −1 xλ
Equation (7.29) implies that MC + C M T = −2 which is solved by C∞ given by Eq. (7.25) with t = ∞. Besides, this is the unique solution because if M D + D M T = 0 then d tM T e De t M = 0 dt and D = lim e t M De t M t→∞
T
= 0.
Appendix F. Proof of Proposition 2. It is enough to check the last identity for the so called cylindrical functionals F(x) = f (xt1 , . . . , xtn ) for 0 ≤ t1 ≤ · · · ≤ tn ≤ T . Since e
−W
= e
ϕ0 (x0 )
e
−
t1 0
t2 Js ds − Js ds
e
t1
··· e
−
T tn
Js ds
e−ϕT (xT ) ,
then, by virtue of Eq. (7.5), the left hand side of Eq. (8.3) is equal to 1 dx f (x1 . . . , xn ) P0,t (x, d x1 ) Pt11 ,t2 (x1 , d x2 ) · · · Pt1n ,T (xn , dy) e−ϕT (y) 1 with the integral over x1 , · · · , xn . With the use of relation (7.4), this may be rewritten as ∗ ∗ ∗ ∗ ∗ ∗ e−ϕT (y) dy f (x1 . . . , xn ) P0,t ∗ (y , d x n ) · · · Pt ∗ ,t ∗ (x 2 , d x 1 ) Pt ∗ ,T (x 1 , d x ) n 2 1
1
∗ → x , as and, after the change of variables xi+1 n−i ∗ ∗ e−ϕT (y) dy f (xn∗ . . . , x1∗ ) P0,t ∗ (y , d x 1 ) · · · Pt ∗ ,t ∗ (x n−1 , d x n ) Pt ∗ ,T (x n , d x ). n 2 1
1
∗
This is equal to the left hand side of the identity (8.3) since e−ϕT (y) dy = e−ϕ0 (y ) dy ∗ .
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Acknowledgements. The authors are grateful to S. Ciliberto, G. Falkovich, I. Fouxon, G. Gallavotti and P. Horvai for discussions.
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Commun. Math. Phys. 282, 519–573 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0548-8
Communications in
Mathematical Physics
Einstein Supergravity and New Twistor String Theories Mohab Abou-Zeid1 , Christopher M. Hull2,3 , Lionel J. Mason4 1 Theoretische Natuurkunde, Vrije Universiteit Brussel & The International Solvay Institutes,
Pleinlaan 2, 1050 Brussels, Belgium. E-mail: [email protected]
2 Theoretical Physics Group, The Blackett Laboratory, Imperial College London,
Prince Consort Road, London SW7 2BW, United Kingdom. E-mail: [email protected] 3 The Institute for Mathematical Sciences, Imperial College London, 53 Prince’s Gate, London SW7 2PG, United Kingdom 4 The Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, United Kingdom. E-mail: [email protected] Received: 1 November 2007 / Accepted: 23 November 2007 Published online: 27 June 2008 – © Springer-Verlag 2008
Abstract: A family of new twistor string theories is constructed and shown to be free from world-sheet anomalies. The spectra in space-time are calculated and shown to give Einstein supergravities with second order field equations instead of the higher derivative conformal supergravities that arose from earlier twistor strings. The theories include one with the spectrum of N = 8 supergravity, another with the spectrum of N = 4 supergravity coupled to N = 4 super-Yang-Mills, and a family with N ≥ 0 supersymmetries with the spectra of self-dual supergravity coupled to self-dual superYang-Mills. The non-supersymmetric string with N = 0 gives self-dual gravity coupled to self-dual Yang-Mills and a scalar. A three-graviton amplitude is calculated for the N = 8 and N = 4 theories and shown to give a result consistent with the cubic interaction of Einstein supergravity. 1. Introduction The string theories in twistor space proposed by Witten and by Berkovits [1–3] give a formulation of N = 4 supersymmetric Yang-Mills theory coupled to conformal supergravity. They provide an elegant derivation of a number of remarkable properties exhibited by the scattering amplitudes of these theories, giving important results for super-Yang-Mills tree amplitudes in particular [4,5]. However, in these theories the conformal supergravity is inextricably mixed in with the gauge theory so that, in computations of gauge theory loop amplitudes, conformal supergravity modes propagate on internal lines [6]. There appears to be no decoupling limit giving pure super-YangMills amplitudes, and although there has been considerable progress in studying the twistor-space Yang-Mills amplitudes at loops (see e.g. [7] and references therein), the results do not follow from the known twistor strings. A twistor string that gave Einstein supergravity coupled to super-Yang-Mills would be much more useful, and might be expected to have a limit in which the gravity could be decoupled to give pure gauge theory amplitudes. (By Einstein supergravity, we mean a supergravity with 2nd order
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field equations for the graviton, in contrast to conformal supergravity which has 4th order field equations.) Indeed, it is known that MHV amplitudes for Einstein (super) gravity [8] have an elegant formulation in twistor space [1,9–11], and it is natural to ask whether these can have a twistor string origin. In this paper, we propose new twistor string models which give Einstein (super) gravity coupled to Yang-Mills. The new theories are constructed by gauging certain symmetries of the Berkovits twistor string. The structure of the theory is very similar to that of the Berkovits model, but the gauging adds new terms to the BRST operator so that the vertex operators have new constraints and gauge invariances. In this paper we construct a family of theories for which the world-sheet anomalies cancel, and find their spectra. We postpone a detailed discussion of the interactions and scattering amplitudes to a subsequent paper, but do show that there is a non-trivial cubic graviton interaction for two of the theories, so that at least these theories are non-trivial. The theories of [1–3] give target space theories that are anomalous in general, with the anomalies canceling only for 4-dimensional gauge groups. It is to be expected that these anomalies should arise from inconsistencies in the corresponding twistor string model, but the mechanism for this is as yet unknown [6]. If there are such inconsistencies in the Berkovits twistor string that only cancel in special cases, there should be similar problems for our theories, and this may rule out some of the models we construct, or restrict the choice of gauge group. We find two classes of anomaly-free theories. The first is formulated in N = 4 super-twistor space. Gauging a symmetry of the string theory generated by one bosonic and four fermionic currents gives a theory with the spectrum of N = 4 Einstein supergravity coupled to N = 4 super-Yang-Mills with arbitrary gauge group, while gauging a single bosonic current gives a theory with the spectrum of N = 8 Einstein supergravity, provided the number of N = 4 vector multiplets is six. In the Yang-Mills sector, the string theory is identical to that of Berkovits, so that it gives the same tree level Yang-Mills amplitudes. Both theories have the MHV 3-graviton interaction (with two positive helicity gravitons and one negative helicity one) of Einstein gravity. The gauging introduces new ghost sectors into our twistor string theories, and in the second family of string theories, gauging different numbers of bosonic and fermionic symmetries allows anomalies to be cancelled against ghost contributions for strings in twistor spaces with 3 complex bosonic dimensions and any number N of complex fermionic dimensions, corresponding to theories in four-dimensional space-time with N supersymmetries. We then find the spectrum of states arising from ghost-independent vertex operators. For N = 0, we find a theory with the bosonic spectrum of selfdual gravity together with self-dual Yang-Mills and a scalar, and for N < 4 we find supersymmetric versions of this self-dual theory. As twistor theory has been particularly successful in formulating self-dual gravity [12] and self-dual Yang-Mills [13], it seems fitting that these theories should emerge from twistor string theory. With N = 4, we find a theory whose spectrum is that of N = 4 Einstein supergravity coupled to N = 4 super-Yang-Mills with an arbitrary gauge group. It is intriguing that some of the theories we find have similar structure to N = 2 string theories [14]. One of the achievements of twistor theory was to give a general solution of the self-dual and conformally self-dual Einstein equations. Penrose’s non-linear graviton construction [12] provides an equivalence between 4-dimensional space-times M with self-dual Weyl curvature and certain complex 3-folds, the curved projective twistor spaces PT , providing an implicit construction of general conformally self-dual spacetimes. For flat space-time, the corresponding twistor space PT is CP3 . In Euclidean signature, there is an elegant realisation of the twistor space PT corresponding to a
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space M with signature + + ++ as the projective primed spin-bundle over M, the bundle of primed spinors π A on M identified under complex scalings π A ∼ tπ A , so that it is a CP1 bundle over M [15]. For other signatures, the construction of curved twistor space PT is not quite so straightforward, and will be reviewed in Sect. 3. New twistor spaces, and hence new conformally self-dual space-times, can be constructed by deforming the complex structure of a suitable region of a given twistor space PT0 (such as a neighbourhood PT0 of a projective line in CP3 ). The complex structure of a space can be specified by a (1,1) tensor field J satisfying J 2 = −1 that is integrable, so that the Nijenhuis tensor N (J ) vanishes. Given the complex structure J0 of PT0 , one can construct a new complex structure J = J0 + λJ1 + λ2 J2 + · · ·
(1.1)
as a power series in a parameter λ, imposing the conditions J 2 = −1 and N (J ) = 0. In holomorphic coordinates for J0 , J 2 = −1 implies that J1 decomposes into a section j of (0,1) ⊗ T (1,0) and its complex conjugate on PT0 . The linearised condition N (J ) = 0 is equivalent to ∂¯ j = 0. Furthermore, j represents an infinitesimal diffeomorphism if ¯ for some section α of T (1,0) . Thus a deformation corresponds to an element j = ∂α of the first Dolbeault cohomology group on twistor space with values in the holomorphic tangent bundle. Moreover, the linearised deformations J1 are unobstructed to all orders and determine the tangent space to the moduli space of complex structures if certain second cohomology groups vanish, which they do when PT0 is a small enough neighbourhood of a line. Witten’s twistor string [1] is a topological string theory on (super-)twistor space and has physical states corresponding to deformations of the complex structure of the target space PT0 . The corresponding vertex operator constructed from J1 is physical precisely when j represents an element of H∂¯1 (PT0 ). The twistor space string field theory action for Witten’s theory has a term with a Lagrange multiplier imposing N (J ) = 0 [6] and the corresponding term in the space-time action is √ d 4 x gU ABC D W ABC D , (1.2) where W ABC D is the anti-self-dual part of the Weyl tensor. If this were the complete gravity action, then U ABC D would be a Lagrange multiplier imposing the vanishing of W ABC D , so that the Weyl tensor would be self-dual. However, in addition there is a term U 2 , which arises from D-instantons in Witten’s 2topological B-model [6,30]. Integrating out U gives the conformal gravity action W . In split + + −− space-time signature, there is a three real dimensional submanifold PTR of complex twistor space PT . In the flat case, PTR ⊂ PT is the standard embedding of RP3 ⊂ CP3 , and the information about deformations of the complex structure is encoded in an analytic vector field f on PTR . It was shown in [16] that conformally selfdual space-times in split signature can also be constructed by deforming the embedding of PTR to some PTR in PT instead of deforming the complex structure of some region in PT to give PT . The deformations of the anti-self-dual conformal structure correspond to deformations of the embedding of PTR in CP3 and are determined at first order by a vector field f on PTR , or more precisely by a section of the normal bundle to PTR ⊂ CP3 . Berkovits’ twistor string [2,3] has open strings with boundaries on the real twistor space PTR , and (conformal) supergravity physical states are created by an open string
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vertex operator constructed from a vector field f defined on PTR , corresponding to deformations of the embedding of PTR in PT. There is an important variant of the Penrose construction that applies to the Ricci-flat case (in fact, this is the original non-linear graviton construction). A special case of the conformally self-dual spaces are those that are Ricci-flat, so that the full Riemann tensor is self-dual. The corresponding twistor spaces PT then have extra structure, as will be discussed in Sect. 3. In particular, they have a fibration PT → CP1 . The holomorphic one-form on CP1 pulls back to give a holomorphic one-form on PT which takes the form Iαβ Z α d Z β in homogeneous coordinates Z α , for some Iαβ (Z ) = −Iβα (Z) (which are the components of a closed 2-form on the non-projective twistor space T ). The dual bi-vector I αβ = 21 αβγ δ Iγ δ defines a Poisson structure and is called the infinity twistor. Consider for example flat space-time M = R4 in signature + + ++, which has conformal compactification S 4 . The twistor space is CP3 , which is a CP1 bundle over S 4 : it is the projective primed spin bundle over the conformal compactification of M. If conformal invariance is broken, then there is a distinguished point at infinity. Removing the point at infinity from S 4 to leave R4 amounts to removing the fibre over this point in the twistor space, leaving PT = CP3 − CP1 , the projective primed spin bundle over R4 . However, PT is also a bundle over CP1 with fibres C2 , the planes through the missing CP1 . A projective line joining two points X α and Y β in twistor space can be represented by a bivector X [α Y β] , and the infinity twistor is the bivector corresponding to the projective line over the point at infinity in S 4 . Choosing a point at infinity, or an infinity twistor, breaks the conformal group down to the Poincaré group. For Minkowski space, the infinity twistor determines the light-cone at infinity in the conformal compactification. A similar situation obtains more generally: the infinity twistor breaks conformal invariance. Self-dual space-times are obtained by seeking deformations of the complex structure of twistor space as before, but now Ricci-flatness in space-time places further restrictions on the deformations allowed. In the split signature picture, the vector field f on RP3 is required to be a Hamiltonian vector field with respect to the infinity twistor, so that in homogeneous coordinates we can write f α = I αβ
∂h ∂ Zβ
(1.3)
for some function h of homogeneity degree 2 on RP3 . In the linearised theory, such a function h corresponds to a positive-helicity graviton in space-time via the Penrose transform, and the non-linear graviton construction gives the generalisation of this to the non-linear theory. In the Dolbeault picture, the tensor J1 is given by a (0, 1)-form j α of the form j α = I αβ
∂h , ∂ Zβ
(1.4)
where h is a (0, 1)-form representing an element of H 1 (PT , O(2)). This suggests seeking a twistor string that is a modification of either the Berkovits or the Witten string theories which introduces explicit dependence on the infinity twistor, such that there are extra constraints on the vertex operators imposing that the deformation of the complex structure be of the form (1.3) or (1.4). Then the leading term in the action analogous to (1.2) should have a multiplier imposing self-duality, not just conformal self-duality, and further terms quadratic in the multiplier (from instantons in Witten’s
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approach) could then give Einstein gravity. A formulation of Einstein gravity of just this form was discussed in [17]. We will present such a modification of the Berkovits twistor string here. The key ingredient is that the one-form corresponding to the infinity twistor is used to construct a current, and the corresponding symmetry is gauged. The resulting gauge-fixed theory is given by the Berkovits twistor string theory plus some extra ghosts, and there are extra terms in the BRST operator involving these ghosts. The dynamics and vertex operators are of the same form as for the Berkovits twistor string, but the extra terms in the BRST charge give extra constraints and gauge invariances for the vertex operators, including the constraint (1.3) that takes us from conformal gravity to Einstein gravity. Variants of the theory are obtained by also gauging some fermionic currents. The case of N = 4 is particularly interesting as in that case the spectrum is parity invariant and is that of N = 4 Einstein supergravity (together with N = 4 Yang-Mills). We expect that similar refinements of Witten’s twistor string should also be possible. A key difference between our models and the twistor strings of refs. [1–3] is that space-time conformal invariance is broken. The magnitude of the infinity twistor defines a length scale in space-time, and so determines the gravitational coupling κ. The theory has two independent coupling constants: the gravitational coupling κ, determined by the magnitude of the infinity twistor, and the Yang-Mills coupling gY M , arising as in [6]. Then for the N = 4 theory there is a limit in which κ → 0 and supergravity decouples from the super-Yang-Mills, so that, if the twistor string theory is consistent at loops, it will have a decoupling limit that gives N = 4 super-Yang-Mills loop amplitudes. The plan of the paper is as follows. In Sect. 2, relevant aspects of twistor theory are reviewed, including special features of different space-time signatures, super-twistor space, the Penrose transform and the infinity twistor. In Sect. 3, the non-linear graviton construction of Penrose is reviewed, and its generalisations to bosonic spaces of split signature and to super-twistor spaces are given. In particular, we adapt [16] to the Ricci-flat case. In Sect. 4, the Berkovits twistor string theory is reviewed. In Sect. 5, the gauging of symmetries of so-called beta-gamma systems is studied. In Sect. 6, this analysis is applied to the Berkovits twistor string, gauging various symmetry groups of the theory and calculating the world-sheet anomalies. In Sect. 7, the conditions for anomaly cancellation are solved, and a number of anomaly-free bosonic and supersymmetric models is found. The spectra of these models are found in Sect. 8, where they are compared to known (super)gravity theories. In Sect. 9, we give a sample calculation of a nontrivial three point function in the theory giving N = 4 supergravity coupled to N = 4 super-Yang-Mills. Finally, in Sect. 10 we discuss our results and the space-time theories that might emerge from our twistor strings. Our conventions are those of Penrose, see for example [18], apart from our choice of sign of the helicity, which is opposite to that of Penrose.
2. Twistor Space and the Infinity Twistor 2.1. Twistor space for flat complex space-time. We start by considering complexified flat space-time C4 , and postpone the discussion of the real slices giving space-times of signature (4, 0), (3, 1) or (2, 2) to the next subsection. The twistor space T corresponding to flat complex space-time is also C4 , with coordinates Z α , α = 0, 1, 2, 3. We also use Z α as homogeneous coordinates on projective twistor space PT = CP3 , which is obtained by identifying Z α ∼ λZ α for complex λ = 0. The Z α transform as a 4 under the
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complexified conformal group1 S L(4, C) and decompose into two-component spinors under the complexified Lorentz group S L(2, C) × S L(2, C): Z α = (ω A , π A ), where A = 0, 1 and A = 0 , 1 are spinor indices for the two S L(2, C) factors. Spinor indices are raised and lowered with AB = [AB] , 01 = 1, and its dual and primed counterparts. Complex flat space-time CM is C4 with complex coordinates x A A and complexvalued metric
ds 2 = AB A B d x A A d x B B .
(2.1)
A point x A A in CM corresponds to a two dimensional linear subspace of T given by the incidence relation
ω A = x A A π A .
(2.2)
In the projective twistor space PT, these two-dimensional subspaces determine projective lines (i.e. CP1 ’s), so that each point x A A in CM corresponds to a CP1 in PT. However, some two-dimensional subspaces in T cannot be expressed in this way, and these correspond to ‘points at infinity’ in the conformal compactification CM of CM. The conformal compactification is obtained by adding a light cone at infinity I to CM [18]. The vertex i of the lightcone I at infinity corresponds to the subspace π A = 0, and other points of I correspond to two-dimensional subspaces lying in the three-spaces α A π A = 0 in which one linear combination of the two components of π vanishes. There is then a one-to-one correspondence between points in compactified space-time CM and two dimensional linear subspaces of T, or projective lines in CP3 . A two dimensional linear subspace of T is determined by two vectors X α , Y α that lie in it, or equivalently by a simple bi-vector, that is a bi-vector P αβ = −P βα satisfying the simplicity condition P [αβ P γ δ] = 0
(2.3)
which implies P αβ = X [α Y β] for some X, Y . Then a point in compactified space-time corresponds to the linear subspace in T determined by a simple bi-vector P αβ . As P αβ and λP αβ (λ = 0) determine the same linear space, we are only interested in equivalence classes under scaling, so that the 6-dimensional space of bivectors P αβ is reduced to the space CP5 of scaling equivalence classes, and the simplicity condition selects a quadric in CP5 . In this way, the conformal compactification CM is represented as a complex 4-quadric in CP5 [18]. Instead of using a simple bi-vector, one can equivalently use the simple 2-form Pαβ = 21 αβγ δ P γ δ in T (where a simple 2-form is one satisfying P[αβ Pγ δ] = 0). A point Z α in twistor space corresponds to an ‘α-plane’ in CM, which is a totally null self-dual 2-plane. This can be seen by regarding the incidence relation (2.2) as a condition on x A A for fixed Z α , the general solution of which is x A A = x0A A + λ A π A ; this describes a 2-plane parametrised by λ A . The two-form orthogonal to the two-plane 1 Strictly speaking, the complexified conformal group is P G L(4, C) = S L(4, C)/Z , as the centre Z 4 4 acts trivially, but this Z4 will not play a role in this paper.
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is given by the symmetric bi-spinor π A π B , and is null and self-dual. In this way, the twistor space PT can be defined as the space of α-planes in CM, and this formulation is useful as it generalises to curved space-times. A standard tool for studying twistor correspondences is the double fibration of the bundle of primed spinors S over space-time and over twistor space
CM
q
S
r
T
.
(2.4)
Using coordinates (x, π A ) on the spin bundle, q is the projection q(x A A , π B ) = x A A , whose fibre at x A A is the spin space at x A A . The other projection r takes (x A A , π A ) ∈ S to the point (ω A , π B ) = (x A A π A , π B ) ∈ T. The fibre at Z α = (x A A π A , π B ) is the set of all (x, π A ) ∈ S with Z α = (x A A π A , π B ), which is the 2-surface (x A A + λ A π A , π A ) parameterised by λ A ; this surface is the lift to the spin bundle of the α-plane corresponding to Z α with tangent spinor π A . There is clearly a corresponding double fibration of the projective spin bundle PS, but now over projective twistor space PT. The Penrose transform can be understood in terms of this double fibration as pulling back objects from twistor space using r ∗ and then pushing them down to space-time using q∗ . The space T has various canonical structures. The space T − 0 has a natural fibration over PT and the Euler homogeneity operator ϒ = Zα
∂ ∂ Zα
(2.5)
is a vector field which points up the fibres of the line bundle {T − 0} → PT. We will represent objects on PT by their pull-backs to T. Thus functions on PT are given by functions on T that are annihilated by ϒ. The line bundle O(n) over PT has sections that are functions on T that are homogeneous of degree n, i.e. ϒ f = n f . Similarly, a form α on PT with values in O(n) pulls back to a form on T (which we will also denote by α) satisfying ¯ ι(ϒ)α = ι(ϒ)α = 0, Lϒ¯ α = 0, Lϒ α = nα,
(2.6)
where ι(ϒ) denotes the interior product (i.e. contraction) with ϒ. We will denote the space of p-forms on PT with values in O(n) as p (n). We define the 3-form =
1 αβγ δ Z α dZ β ∧ dZ γ ∧ dZ δ , αβγ δ = [αβγ δ] , 0123 = 1. 6
(2.7)
This annihilates ϒ (i.e. ι(ϒ) = 0), but it does not descend to PT, since it has homogeneity degree 4. However, it does so descend when multiplied by functions that are of homogeneity degree −4, and gives an isomorphism (3,0) (PT) O(−4) (or alternatively defines a holomorphic section of (3,0) (4)). This also determines the holomorphic volume form d on T: d =
1 αβγ δ dZ α ∧ dZ β ∧ dZ γ ∧ dZ δ . 6
(2.8)
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2.2. The infinity twistor. The conformal compactification CM of space-time is invariant under the full conformal group. In order to break conformal invariance to conformal Poincaré invariance (i.e. the Poincaré group together with dilations), we choose a point in CM to be the point i at infinity, and the complexified conformal Poincaré group is the subgroup of S L(4, C) preserving this point. In particular, with a further choice of an origin 0 in CM, this chooses a Lorentz subgroup S L(2, C) × S L(2, C) ⊂ S L(4, C), and different choices of i, 0 lead to different conjugate Lorentz subgroups. The point i at infinity in CM corresponds to a bi-vector I αβ up to scale which is simple, I [αβ I γ δ] = 0,
(2.9)
and which is called the infinity twistor. The infinity twistor can also be represented by the 2-form τ on T defined by τ=
1 Iαβ dZ α ∧ dZ β , 2
where I αβ = 21 εαβγ δ Iγ δ . Choosing a point 0 in CM to be the origin x µ = 0 corresponds to choosing a second two-form µ (dual to a simple bi-vector), and this can be chosen so that2 d = 4µ ∧ τ.
(2.10)
The choice of i, 0 in CM selects an S L(2, C) × S L(2, C) subgroup of S L(4, C) that preserves µ and τ separately, and this is the double cover of the rotation group S O(4, C) preserving the origin x = 0 and the point at infinity in CM. It is natural to use 2-component spinor notation for this S L(2, C) × S L(2, C) subgroup, with Z α = (ω A , π A ). Then τ=
R A B 1 AB dω A ∧ dω B dπ A ∧ dπ B , µ = 2 2R
(2.11)
for some R. The corresponding space-time metric is
ds 2 = R 2 AB A B d x A A d x B B ,
(2.12)
so that scaling the infinity twistor by R leads to a conformal scaling of the metric by R 2 , and the scale of the infinity twistor determines the length scale in space-time. For the rest of the paper, we will set R = 1. The infinity twistor determines the projective line I in PT corresponding to i by Z α Iαβ = 0, which in adapted coordinates is the line π A = 0, while the origin x = 0 corresponds to the line µ A = 0. Removing the light-cone at infinity I from CM leaves complex space-time CM while removing the line I in PT corresponding to the infinity twistor gives the twistor space PT = PT − I . As I is the CP1 ⊂ PT given by π A = 0, PT consists of points Z α = (ω A , π A ) in which at least one component of π is non-zero. For 2 If no choice of origin is made, the two-form µ is defined by (2.10) up to the addition of multiples of dπ . A
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non-conformal theories, it is natural to use PT , and this (and its curved generalisations) is the twistor space that will be used in our constructions. The infinity twistor determines a projection T → S A to S A , the dual primed spinor space, given by Z α = (ω A , π A ) → π A . Projectively, this projection determines a fibration PT → CP1 . The infinity twistor I αβ defines a Poisson structure of homogeneity −2 by { f, g} I := I αβ
∂ f ∂g ∂ f ∂g = AB . ∂ Zα ∂ Zβ ∂ω A ω B
We further define the one-form
k = Iαβ Z α d Z β = A B π A dπ B ,
(2.13)
for which τ = 21 dk = 21 A B dπ A ∧ dπ B ; k is the pull-back of a holomorphic one-form on CP1 with weight 2 and will play a central role in our construction. 2.3. Twistor spaces for real space-times. We can choose a real slice M ⊂ CM in such a way that the metric has signature ( p, 4 − p) for p = 0, 1, 2, and the subgroup of the complexified conformal group that preserves the real slice is a real form of S L(4, C). For Euclidean signature, Lorentzian signature, or split signature (2, 2), the real conformal groups are SU ∗ (4) = S L(2, H) = Spin(5, 1), SU (2, 2) = Spin(4, 2) and S L(4, R) = Spin(3, 3) respectively, where H denotes the quaternions.3 The conformal group acts on the twistor space T = C4 , with Z α transforming as a complex Weyl spinor for S O(6, C). For split signature, this representation is reducible: it decomposes into the direct sum of two copies of the real Majorana-Weyl representations of Spin(3, 3), and it is possible to impose a reality condition on the twistors, giving the real twistor space RP3 . However, for the other two signatures, the Weyl representation is irreducible so that twistors are necessarily complex. We can characterise the real slices M of CM as fixed points of a complex conjugation τ : CM → CM which, in local coordinates that are real on the appropriate real slice, are given by standard complex conjugation, τ (x µ ) = (x µ )∗ . A point x µ in CM is represented by a complex matrix x AB . The different conjugations can be expressed on this matrix as follows. For space-time of split signature, τ (x AB ) = (x AB )∗ is the entry-by-entry complex conjugate, for Lorentzian signature τ (x AB ) is the hermitian conjugate τ (x) = x † , while for Euclidean signature τ (x AB ) = xˆ AB , where xˆ = x ∗ with the real anti-symmetric 2 × 2 matrix (given in terms of the Pauli matrix σ2 by = iσ2 ).4 Complex conjugation x → τ x in CM leads to a map on twistor space. In split signature and in Euclidean signature, τ sends α-planes to α-planes, but in Lorentz signature it sends α-planes to β-planes where β-planes are totally null 2-planes in CM that are anti-self-dual. The space of such β-planes together with tangent spinor λ A , is dual twis tor space T∗ with coordinates Wα = (λ A , µ A ); a point in T∗ corresponds to the β-plane in CM defined by the dual incidence relation µ A = x A A λ A . The complex conjugation τ on CM therefore induces a complex conjugation τ : T → T in split signature and 3 Again, we are ignoring factors of Z here. 4 4 Note that in this definition, neither the map x → x¯ nor x → x are invariant under the S O(4) rotation
group, only the composition x → x ¯ is.
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Euclidean signature, but in Lorentz signature, it determines an anti-holomorphic map τ : T → T∗ . We have the complex conjugate twistor space T¯ (i.e. T with the opposite complex structure) with coordinates Z¯ α¯ = (Z α )∗ on twistor space, and their counterparts on dual twistor space T† with coordinates W¯ α¯ = (Wα )∗ . For the real and split signature complex structure, τ is an isomorphism from T¯ to T and in the Lorentzian case it is a natural map from T¯ to T∗ , and this can be used to express conjugate twistors in T¯ in terms of twistors in T or T∗ , so that conjugate twistor indices are never needed explicitly. We now describe features of twistor geometry appropriate to each signature in more detail. 2.3.1. Lorentzian signature. In the case of Lorentzian signature, the conformal group SU (2, 2) preserves a Hermitian metric α β¯ , and this defines the map τ : T¯ → T∗ ¯ under which Z¯ α¯ = (Z α )∗ → α β¯ Z¯ β , so that each conjugate twistor can be identified with a dual twistor. Complex conjugation on CM leads to an anti-holomorphic map ¯ Z α → Z¯ α = α β¯ Z¯ β from T → T∗ . The real Minkowski space-time M is the subspace
of CM in which x AB is Hermitian and is preserved by this conjugation. This is the standard case, discussed in detail in e.g. [18].
2.3.2. Split signature. For extensive discussions of the twistor correspondences in split signature see [16,19]. Here we give a summary of the main ideas. For split signature, the real space-time M is the subspace of CM with x AB real. The ordinary complex conjugation on CM that preserves M is represented by the ordinary component-by-component complex conjugation on T, viz. Z α → (Z α )∗ , that fixes the real slice TR = R4 ⊂ C4 = T and hence PTR = RP3 ⊂ PT. Points of this real slice correspond to totally real α-planes in M and there is a totally real version of the twistor correspond to real projective lines (i.e. RP1 s) in correspondence in which points in M PTR via the incidence relation ω A = x A A π A , where now ω A , π A and x A A are all real. 2 2 is the conformal compactification of M, which is M = S × S /Z2 . Here M In order to use deformed twistor correspondences in split signature, we will also need to use the correspondence between M and the complex twistor space PT. Each point x ∈ M corresponds to a complex line L x = CP1 in PT that intersects the real slice PTR in a real line L Rx = RP1 . This real line divides L x into two discs Dx± , each with of boundary L Rx ⊂ PTR . The space of such discs naturally defines a double cover M conformally compactified Minkowski space M (which is the space of all L Rx ⊂ PTR ). = S 2 × S 2 with the conformal structure that is determined by the split signature In fact M product metric g = π1∗ h − π2∗ h, where h is the standard round metric on S 2 and π1 , π2 : S 2 × S 2 → S 2 are the two = S 2 × S 2 /Z2 is obtained from factor projections. The conformal compactification M the double cover M by identifying under the Z2 that acts as the joint antipodal map on both S 2 factors. can be thought of as two copies M± of M glued together across the double cover of M the lightcone at infinity I . With the choice of the infinity twistor, we have the fibration PT = PT − I → CP1 as above. The condition that iπ A π¯ A be positive, negative or zero defines PT± and PT0 . The holomorphic discs in PT± project to ±iπ A π¯ A > 0
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in CP1 and correspond respectively to points of M± , whereas the holomorphic discs in PT0 correspond to points of the double cover I of I . This will be important later for the Berkovits string, where the open string world-sheets are holomorphic discs. The with two copies of space-time M, and moduli space of discs in twistor space gives M to get just one copy, the theory must be restricted to one in which the world-sheets are discs in one half of twistor space, say in PT+ . 2.3.3. Euclidean signature. The anti-linear map τ : T → T is given by the conjugation Z α → Zˆ α where, if Z α = (ω A , π A ), then Zˆ α = (ωˆ A , πˆ A ), with ωˆ A = (ω¯ 1 , −ω¯ 0 ) and πˆ A = (π¯ 1 , −π¯ 0 ). The conjugation extends to multi-spinors and the real Euclidean space-time M is the subspace of CM preserved by this, x AB = xˆ AB . The conjugation Z α → Zˆ α is then the lift of the complex conjugation x µ → (x µ )∗ on CM preserving real Euclidean slices. The conjugation Z α → Zˆ α is quaternionic in the sense that Zˆˆ α = −Z α so that it defines a complex structure that anticommutes with the standard one. It therefore has no fixed points (as Z α = Zˆ α implies Z α = −Z α ), and it is induced by the standard quaternionic conjugation on spinors: πˆ A = (π¯ 1 , −π¯ 0 ) and similarly for ω A . of Euclidean R4 is given by adding a single The conformal compactification M 4 point i at infinity to give S . The Euclidean signature correspondence is particularly straightforward since we have a fibration PT = CP3 → S 4 given by sending Z α to the point in Euclidean space corresponding to the projective line through Z α and Zˆ α (this includes a line at infinity corresponding to π A = 0). The fibre over any point x A A in S 4 is a CP1 with projective coordinates π A , and the corresponding point in PT is
(ω A , π A ) = (x A A π A , π A ).
(2.14) (ω A , π
Conversely, a point in PT with holomorphic coordinates local non-holomorphic coordinates (x A A , π A ) by ω A πˆ A − ωˆ A π A A A , π A . (x , π A ) = π A πˆ A
A )
is represented in
(2.15)
The CP1 fibre at each point is the space of primed spinors π A , identified under scaling, so that PT is the projective primed spin bundle over S 4 . Similarly, T − 0 is the bundle of primed spinors minus the zero section, and we can again use the formulae (2.14),(2.15). To obtain M = R4 , we choose a point i on S 4 to be the point at infinity, and this corresponds to an infinity twistor I , specifying the CP1 fibre over i. Then the twistor space for R4 is given by removing this CP1 , so that PT = PT − CP1 is the projective spin bundle over R4 . Choosing an infinity twistor and an origin chooses a subgroup SU (2) × SU (2) ⊂ SU ∗ (4) and a decomposition of Z α into holomorphic coordinates (ω A , π A ) transforming under this SU (2)× SU (2); in this frame, the twistor correspondence is given by (2.14),(2.15) on T = T − {π A = 0} so that the point at infinity is x A A = ∞, corresponding to the 2-plane in T (or CP1 in PT) given by {π A = 0}. 2.4. The Penrose transform. The Penrose transform identifies fields of helicity −n/2 satisfying the massless wave equation on a suitable region U ⊂ CM with the cohomology group H 1 (PT(U ), O(n − 2)) for PT(U ) the corresponding subset of PT. A
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Dolbeault representative of this group is a (0, 1)-form α with values in O(n − 2) such ¯ = 0, where α is defined modulo ∂¯ g with g a smooth section of O(n − 2). that ∂α The corresponding massless space-time field of helicity |n|/2 for n ≤ 0 is given by the integral formula φ A1 ...A−n (x) = π A1 · · · π A−n α ∧ πC dπ C . (2.16)
ω A =x A A π A
For n ≥ 0, the massless space-time field of helicity −n/2 is given by ∂ ∂ φ A1 ... An (x) = · · · A α ∧ πC dπ C . A n 1 ω ω A =x A A π A ∂ω
(2.17)
ˇ Alternatively, a Cech representative can be chosen for the cohomology class, and the space-time fields are then given by a contour integral formula. This can be implemented simply when it is possible to cover PT(U ) by two open sets, V0 and V1 (this is the case ˇ for PT , for which we can take V0 = {π0 = 0} and V1 = {π1 = 0}). Then the Cech cohomology class can be represented by a holomorphic function f of homogeneity n −2 on V0 ∩ V1 . The analogues of the above formulae are then, for n ≤ 0, φ A1 ...A−n (x) = π A1 · · · π A−n f πC dπ C (2.18)
and, for n ≥ 0,
φ A1 ...An (x) =
∂ ∂ · · · A f πC dπ C . A n 1 ∂ω ω
(2.19)
In both (2.18) and (2.19) the contour is a suitable circle in V0 ∩ V1 ∩ {ω A = x A A π A }. In split signature, instead of considering cohomology classes, we can consider smooth functions defined on PTR that are homogeneous of degree n − 2 and apply the integral formulae (2.18) and (2.19), where now is taken to be the real line {ω A = x A A π A } in PTR for x A A a point in real split signature Minkowski space. In the case of n = 0 this is known as the X-ray transform, and it is a classic theorem that these formulae define an isomorphism from functions on PTR to solutions of the ultrahyperbolic wave equation on M [20]. The close relationship between the Penrose transform and the X-ray transform was observed by Atiyah [21]. The connection between the X-ray transform and the Penrose transform can be understood naively by requiring f to be analytic, ˇ extending it to some complex neighbourhood of PTR and reinterpreting it as a Cech cohomology class. However there are a number of issues that this approach does not deal with; a full treatment of the relationship between the X-ray and Penrose transforms is given in [22,23]. For the most part, it is this X-ray transform version of the Penrose transform that is used by Witten and Berkovits in [1,2].
2.5. Super-twistor space. The superspace with N supersymmetries has space-time coor dinates x A A and anti-commuting coordinates θaA , θ˜ a A , where a, b = 1, . . . , N . The latter are space-time spinors and transform as an N -dimensional representation of an R-symmetry group, which is U (N ) or SU (N ) for Lorentzian signature, G L(N , R) or S L(N , R) for split signature and U ∗ (N ) or SU ∗ (N ) for Euclidean signature.
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The complexified superconformal group is S L(4|N ; C) and its real forms are SU (2, 2|N ) for Lorentzian signature, S L(4|N ; R) for split signature and SU ∗ (4|N ) for Euclidean signature. The group S L(4|N ; C) is realised on the space C4|N with coordinates Z I = (Z α , ψ a ) ∈ C4|N , consisting of the usual commuting coordinates Z α as before and anti-commuting coordinates ψ a , a = 1, . . . , N . Super-twistor space T[N ] is the subset C4|N − C0|N on which Z α = 0, and the projective super-twistor space PT[N ] = CP3|N is the space of equivalence classes under complex scalings [24]: PT[N ] = CP3|N = {Z I = (Z α , ψ a ) ∈ C4|N − C0|N }/{Z I ∼ λZ I , λ ∈ C× }. Note that in this definition we have a fibration PT[N ] → PT given by (Z α , ψ a ) → Z α . However, this fibration is not preserved by the action of the superconformal group S L(4|N ; C). The N = 4 superspace is special for twistor theory because in that case there is a global holomorphic volume form on the projective super-twistor space, s = dψ1 dψ2 dψ3 dψ4 , with the bosonic 3-form defined in (2.7). This has weight zero, since each dψa has weight −1 according to the Berezinian integration rule ψ1 dψ1 = 1. A A ˜ a A arises as the Anti-chiral super-Minkowski space CM− [N ] with coordinates x + , θ space of CP1|0 s in PT[N ] via the incidence relations
(ω A , π A , ψ a ) = (x+A A π A , π A , θ˜ a A π A ),
(2.20)
where we have used π A as homogeneous coordinates on CP1|0 . Chiral super-Minkowski space CM+[N ] with coordinates x−A A , θaA arises as the space of CP1|N s in PT[N ] via the incidence relations
(ω A , π A , ψ a ) = (x−A A π A + ψ a θaA , π A , ψ a ),
(2.21)
where now we have used (π A , ψ a ) as homogeneous coordinates on the CP1|N s. A point of full super-Minkowski space CM[N ] with coordinates x A A , θaA , θ˜ a A arises from a choice of CP1|N in PT[N ] together with a choice of CP1|0 ⊂ CP1|N , so that full super-Minkowski space is the space of ‘flags’ CP1|0 ⊂ CP1|N in PT[N ] [24]. Taking (2.20) and (2.21) together we have x+A A = x−A A + θ˜ a A θaA and it is usual to define x A A = 21 (x+A A + x−A A ).5 The massless field formulae generalising (2.16) and (2.17) now give rise to superfields encoding supermultiplets. The easiest way to see this is to expand out an element Fn ∈ H 1 (PT[N ] (U ), O(n)) as follows: Fn = f (n) + f (n−1)a ψ a + f (n−2)a1 a2 ψ a1 ψ a2 + f (n−3)a1 a2 a3 ψ a1 ψ a2 ψ a3 + · · · . Here f (n−k)... has homogeneity degree n − k so that its Penrose transform is a massless field of helicity −(n − k − 2) on space-time with skew-symmetric indices a1 , . . . , ak , and it transforms as a k th rank anti-symmetric tensor under the R-symmetry group. It is possible to perform the transform on Fn to obtain a superfield directly on CM± , the ± depending on whether we integrate over CP1|0 s or CP1|N fibres. Particularly 5 To obtain standard conventions in Lorentz signature we must take x A A = i y A A for real y A A ; our conventions are adapted to split and Euclidean signature.
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interesting examples are furnished by the cases of n = ±2 in the context of linearised N = 4 Einstein supergravity. We can define H + (x− , θaA ) = F2 (x−A A π A + ψ a θaB , π A , ψ b )π A dπ A d4 ψ (2.22) CP1|4
and H (x+ , θ˜ a A ) =
−
CP
1|0
F−2 (x+A A π A , π A , θ˜ a A π B )π A dπ A .
(2.23)
The integrand of (2.22) can be expanded in ψ a using Taylor series in the anti-commuting coordinates and the variables ψ a can be integrated out to yield a power series in θaB ; the standard Penrose transform in the form (2.18) can then be applied to the coefficients to yield a superfield on chiral super Minkowski space. Equation (2.23) can be expanded as a Taylor series in θ˜ a A to obtain a series whose coefficients can be integrated using (2.19) to obtain a superfield on anti-chiral super-Minkowski space CM− [N ] . This directly gives formulae for the full chiral and anti-chiral superfields for N = 4 supergravity in terms of the component fields. In order to obtain an anti-chiral or a chiral superfield for other values of n or N , we need to either repeatedly differentiate Fn with respect to ω A , or to multiply it by enough factors of π A . In the first case, this will reduce the homogeneity to −2 and enable us to apply (2.23) to obtain an anti-chiral superfield; in the second case, we arrange for homogeneity N − 2 and obtain a chiral superfield by applying (2.22). As before, the space of CP1|0 s (resp. CP1|N s or flags CP1|0 ⊂ CP1|N ) in PT[N ] is a conformal compactification of chiral (resp. anti-chiral or full) super Minkowski space on which the superconformal group acts. We will wish to break conformal invariance on super-twistor space by choosing points at infinity and a scale. There are three ways in which we can break superconformal invariance; we can choose points at infinity in either the chiral, anti-chiral or full Minkowski space, and these lead to different structures. A choice of a point at infinity in chiral super-Minkowski space corresponds to a choice of a line I , a CP1|0 , in PT[N ] and coordinates (ω A , π A , ψ a ) can be chosen so that I is given by π A = 0 = ψ a . This determines a projection p1 : PT[N ] − I → CP1|N given in homogeneous coordinates by p1 : (ω A , π A , ψ a ) → (π A , ψ a ). The fibres of the projection are the CP2|0 s through I . If we choose a point in anti-chiral Minkowski space, then this gives a choice of a superline I[N ] = CP1|N and we can then choose coordinates (ω A , π A , ψ a ) so that I[N ] is the set π A = 0. This, as before, leads to a fibration p : PT[N ] − I[N ] → CP1|0 given by p1 : (ω A , π A , ψ a ) → π A with fibres the CP2|N s through I[N ] . The richest structure is obtained by choosing a vertex i of a super-light-cone at infinity I in the full conformally compactified super-Minkowski space (as opposed to one of its chiral versions). This is equivalent to the choice of a ‘flag’ CP1|0 ⊂ CP1|N ⊂ PT[N ] , i.e. the pair I ⊂ I[N ] . These lead to corresponding projections of PT[N ] = PT − I[N ] , p1
p0
PT[N ] −→ CP1|N −→ CP1|0 ,
Z I = (ω A , π A , ψ a ) → (π A , ψ a ) → π A .
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We will also be interested in the case in which there is only the projection p : PT[N ] → CP1|0 . We will see that this is a weaker structure and there will correspondingly be a larger class of deformations. We can define the Poisson structure { f, g} I := I I J
∂ f ∂g ∂ f ∂g = AB ∂ZI ∂ZJ ∂ω A ω B
as in the bosonic case, and p0 can then be used to pull back the 1-form
I I J Z I dZ J = A B π A dπ B from CP1|0 . These are special cases of more general correspondences between points of chiral Minkowski space and rank two bi-vectors X I J = X [I J ] up to scale, and between points of anti-chiral Minkowski space and simple (rank two) two-forms X I J up to scale. Alternative representations can be obtained by use of the volume form I1 ...I4+N and its inverse on T[N ] . 3. The Non-Linear Graviton 3.1. The conformally anti-self-dual case. Penrose’s non-linear graviton construction provides a correspondence between curved twistor spaces and conformally anti-self-dual space-times, and so gives a general construction of such space-times. This arises from nontrivial deformations of the flat twistor correspondence in which, on the one hand, the space-time is deformed from flat space to one with a curved conformal structure with anti-self-dual Weyl curvature, and, on the other, the complex structure of a region in twistor space is deformed away from that of a region in projective space. One cannot deform the complex structure of the whole of flat twistor space as PT = CP3 is rigid and has no continuous deformations, so we instead consider deformations of PT , which is CP3 with a line removed. This has topology R4 × S 2 . We will find it convenient to start by describing the non-projective twistor space. A curved twistor space T will be taken to be a 4-dimensional complex manifold equipped with a vector field ϒ and a non-vanishing holomorphic 3-form such that • ϒ gives T the structure of a line bundle over the space PT = T /{ϒ} of orbits of ϒ, for which ϒ is the Euler vector field (in local coordinates (z, z 1 , z 2 , z 3 ) where (z 1 , z 2 , z 3 ) are coordinates on PT and z is a linear coordinate up the fibre, ϒ = z∂/∂z). • ϒ and satisfy Lϒ = 4, ι(ϒ) = 0.
(3.1)
• PT contains a holomorphically embedded Riemann sphere that has the same normal bundle as a complex projective line in CP3 . The last condition is in fact rather mild and holds automatically not only for any twistor space that is constructed as described below from a conformally anti-self dual spacetime, but also for any twistor space that is an arbitrary small deformation of such a twistor space. The space-time is reconstructed as the moduli space of such Riemann spheres; given one such sphere, Kodaira theory implies the existence of a full four-dimensional family [55].
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The existence of the holomorphic volume form d implies that T is a non-compact Calabi-Yau space.6 The global existence of ϒ and allows us to introduce local complex coordinates Z α on T such that ∂ 1 ϒ = Z α α , = αβγ δ Z α dZ β dZ γ dZ δ ∂Z 6 as in the flat case, with αβγ δ = [αβγ δ] , 0123 = 1. We now turn to the relation between curved twistor space and space-time. For complexified Minkowski space, a twistor corresponds to an α-plane, i.e. a totally null selfdual two-plane. In a curved complex space-time CM, which is a complex 4 manifold with a holomorphic metric g (so that locally the metric is gµν (x)d x µ d x ν , depending on the complex coordinates x µ but not their complex conjugates), α–plane elements in the tangent space are not generally integrable, i.e. one cannot in general find a two surface whose tangent planes are α-planes. A two-surface whose tangent plane is an α-plane at every point is called an α-surface. The nececessary and sufficient condition for there to exist α-surfaces through each α-plane element at every point is that the self-dual part of the Weyl curvature should vanish, ψ˜ A B C D = 0.
(3.2)
If (3.2) holds, then the 3 complex dimensional curved twistor space PT is the space of such α–surfaces. An α-surface through x is specified by an α-plane in the tangent space at x, and this in turn is fixed by a choice of primed ‘tangent’ spinor π A at x, up to complex scalings, so that the space of tangent vectors is given by π A λ A as λ A varies. A point in the non-projective twistor space T is determined by an α-surface in CM and a tangent spinor π A that is parallelly propagated over the α-surface using the LeviCivita connection of any metric in the conformal class. It is a non-trivial fact that the parallel propagation of such a ‘tangent’ spinor over its α-surface is independent of the choice of conformal factor for the metric in the conformal class. A point in the projective twistor space PT is given by the α-plane together with π A up to complex scalings of π A . For Euclidean signature, we saw that in the flat case the twistor space PT = CP3 is the projective spin bundle over compactified space-time S 4 . This generalises, and for Euclidean signature, the curved twistor space PT for a conformally anti-self-dual space M is the projective spin bundle over M, where the fibre at a point x is a CP1 with homogeneous coordinates given by the primed spinors π A at x, while T is the corresponding non-projective spin bundle. In terms of coordinates (x, π A ), ϒ = π A ∂/∂π A and = π A Dπ A ∧π B πC BC e B B ∧eCC , where D is the covariant exterior derivative with the Levi-Civita connection of some metric in the conformal class, and e A A are the A A µ pull-backs from space-time to the spin bundle of the ‘solder forms’ eµ d x constructed from a vielbein eµA A .7 6 The second condition allows us to give a construction of T in terms of PT as the total space of the line bundle T = ((3,0) )1/4 over PT . This definition arises by analogy with the flat case, where (3,0) is O(−4)
because the holomorphic (3, 0)-form has weight 4 and so it needs to be multiplied by a weight −4 function to define a (3, 0)-form. Since T − {0} is the total space of the line bundle O(−1) minus its zero-section, it is therefore the fourth root of (3,0) . With this definition of T , the existence of on T is tautological as T is a covering of the bundle of 3-forms and so is the pull-back to T of the corresponding 3-form at that point. As the (3, 0)-form has weight 4, it is not a (3, 0)-form on PT, so that PT is not a Calabi-Yau space. 7 In this form, the construction makes sense for compact space-times of Euclidean signature with complicated topology: a celebrated result of Taubes is that Euclidean signature anti-self-dual conformal structures can be found on arbitrary compact 4-manifolds, possibly after performing a connected sum with a finite number of CP2 s, and so there are many nontrivial compact examples of twistor spaces.
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The famous result of Penrose [12] is that the space-time CM together with its anti-self-dual conformal structure can be reconstructed from the complex structure of T together with (ϒ, ) as described above, or from PT and its complex structure. The existence of the correspondence is preserved under small deformations, either of the complex structure on PT , or of the anti-self dual conformal structure on CM. Thus one can attempt to construct anti-self-dual space-times by deforming, say, PT . The key idea is that a point x ∈ CM corresponds to a Riemann sphere CP1x (the Riemann sphere with homogeneous coordinates π A ) in PT consisting of those α-surfaces through x. It follows from Kodaira theory that the moduli space of deformations of CP1x in PT is necessarily four dimensional, and naturally contains CM as an open set (in general it is some analytic continuation of CM). Furthermore, this family of CP1x s still survives after deformations of the complex structure of PT . If CM arises as such a moduli space, an anti-self-dual conformal structure can be defined on CM by declaring points x and y to be null separated if CP1x and CP1y intersect. The fact that the existence of such a correspondence survives deformations of the complex structure on PT means that, given one conformally anti-self-dual spacetime, a family of new conformally self-dual space-times can be constructed by deforming the complex structure of the corresponding curved twistor space PT , and so the equations governing the deformation of the complex structure correspond to the field equations for conformal anti-self-dual gravity. The data of the conformal structure on CM is then encoded in the complex structure of PT . There are two standard ways to represent the complex structure. The Dolbeault approach (cf. the introduction) is to regard PT as a real 6-manifold with an almost complex structure, i.e. a (1, 1)-tensor J subject to the integrability condition that its Nijenhuis tensor N (J ) vanishes. We can equivalently encode J into a ∂¯ operator, the restriction of the exterior derivative to the 1-forms (0,1) in the −i eigenspace of J . With ˇ this restriction, N (J ) = 0 is equivalent to ∂¯ 2 = 0. The Cech approach is to consider PT as a 3 complex dimensional manifold formed by choosing a suitable open cover Vi , picking holomorphic coordinates on each Vi and then encoding the data of the manifold in the biholomorphic patching functions defined on the overlaps Vi V j . Both these points of view lead to a cohomological understanding of the deformation theory, the first ˇ via Dolbeault cohomology and the second via Cech cohomology. In either approach, the deformations of the complex structure are parametrised by H 1 (PT , T (1,0) ). If we consider linearised deformations of PT, we obtain the following description of linearised conformal gravity. ¯ We represent f ∈ H 1 (PT , T (1,0) ) by a (0, 1)-form f α (Z ) = f α β¯ (Z )d Z¯ β taking values in the bundle of holomorphic vector fields on T , with the condition that f α has homogeneity degree 1 and is defined up to the gauge freedom f α → f α + a(Z )Z α for some (0, 1)-form a(Z ) of homogeneity zero. This freedom can be fixed by the requirement that ∂ f α /∂ Z α = 0, which is the condition that the measure d is holomorphic for the deformed complex structure ∂¯ + f (Z )α ∂/∂ Z α . This implies that f (Z )α ∂/∂ Z α is a deformation of T that preserves both and d. The Penrose transform of f α gives a helicity +2 field ψ ABC D in space-time satisfying the field equation of linearised conformal gravity, which is the linearised Bach equation [25]: ∇ CA ∇ BD ψ ABC D = 0; see [26,27] for details.
(3.3)
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Following [6] and [30], the negative helicity conformal graviton can be represented by an element g ∈ H 1 (PT(U ), 1 (−4)). The pull-back of g to T gives a 1-form gα (Z )dZ α on T, where g(ϒ) = Z α gα = 0 and the components gα have weight −5. The Penrose transform of gα gives a Weyl spinor ψ˜ A B C D , now of helicity −2, satisfying ∇ BC ∇ AD ψ˜ A B C D = 0.
(3.4)
The Penrose transform in this case is the opposite helicity to that of f α , and can be derived using the methods of [27,30]; it is discussed from a different point of view in [6], where g appears as the component ψ 1 ψ 2 ψ 3 ψ 4 g of the cohomology class b in H 1 (PT[4] , T ∗ ) on super-twistor space, where T ∗ is the cotangent bundle. 3.1.1. Real space-times. The non-linear graviton construction cannot be applied to conformally curved Lorentzian space-times, as a real Lorentzian space-time satisfying (3.2) is conformally flat; the self-dual part of the Weyl curvature is the complex conjugate of the anti-self-dual part. However, it can be applied to the other two signatures by constructing a complex space-time and seeking a suitable real submanifold. The specialisation to Euclidean space-times gives the construction of general conformally anti-self-dual spaces. In this case, the twistor space is a CP1 bundle over space-time, so that the space-time is obtained from the twistor space by projection [15]. In split signature the non-linear graviton construction changes character, and there are two ways of constructing self-dual spaces [16,28]; see also [19]. For flat space in this signature, there is a complex twistor space PT = CP3 and a real subspace PTR = RP3 fixed by the complex conjugation τ : Z → Z ∗ inherited by twistor space from that on complex space-time, x µ → (x µ )∗ . There are two routes to the curved space generalisation. In the first, one deforms the complex structure of a region of the complex twistor space PT = CP3 to obtain a curved twistor space PT as before, but in such a way as to preserve the complex conjugation. The fixed point set PTR of the conjugation defines an analogue of PTR in the deformed case and induces a complex conjugation on space-time that fixes a real slice of split signature. In the second, the complex twistor space PT = CP3 is kept fixed but the real subspace is deformed from PTR to a subspace PTR . Both approaches lead to considering deformations of the real twistor space from PTR to PTR , but this is embedded in different complex spaces in the two cases. The two kinds of deformations are both locally encoded in the same cohomology classes on the real twistor space, but the second approach is better behaved globally and does not require analyticity of the space-time, so it is more powerful. However, it is the first approach that has been used to give a non-linear interpretation of the Berkovits string theory, in which open strings move in PT with boundaries lying in PTR . In §4, we will propose a modification of the Berkovits string theory that corresponds to the second approach, in which there is a natural geometric interpretation of the vertex operators. In the first approach, points in space-time correspond to CP1 ’s in PT that are invariant under the conjugation, while in the second they correspond to discs in PT with boundary on PTR . We now describe the two constructions in more detail. In the first, the twistor space PT was the deformation of a region in flat twistor space in such a way that the complex conjugation τ : PT → PT is preserved. We can construct such a twistor space starting with a real split signature space-time M that is real analytic.8 The real analyticity 8 This assumption is nontrivial as generic solutions will be non-analytic (this can be seen to follow from the second construction). Nevertheless, such non-analytic solutions can be approximated arbitrarily closely by analytic ones, and the construction captures the full functional freedom of these solutions.
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can be used to find a complexification CM of the real split-signature space M. This can be found locally by allowing the coordinates to take complex values, and using the analyticity of the transition functions for the coordinates we can extend the charts and transition functions to construct a complex manifold CM which contains M as a real slice (i.e. a slice fixed by complex conjugation of the coordinates we have just constructed). The analyticity of the metric implies that it can be extended to a holomorphic metric on CM. The complex non-linear graviton construction of §3.1 can be used locally on any suitable open set U ⊂ CM to define a twistor space PTU corresponding to U . The complex conjugation on space-time again sends α-planes to α-planes, inducing a complex conjugation on PTU that fixes a real slice PTU R which is a totally real 3-dimensional submanifold of the complex twistor space. A point x in the real spacetime M corresponds to a holomorphic Riemann sphere in the complex twistor space that intersects PTU R in a circle and cuts the Riemann sphere into two discs Dx± . In the reverse direction, the complex twistor space can be used to reconstruct a complex conformally anti-self-dual space as before. This naturally has a complex conjugation that determines a real slice, on which the complex conformal structure restricts to give a real conformally anti-self-dual structure. In order to construct the global complex twistor space PT , we first need to choose a suitable open cover {Ui } of CM and construct the twistor space PTUi for each open set; we then glue these twistor spaces together, identifying points in PTUi with those in PTU j whose corresponding α-surfaces coincide in Ui ∩ U j . However, this natural extension gives a PT which is a non-Hausdorff manifold [28]; see the Appendix for a brief description of this space. In the second approach, we consider general anti-self-dual conformal structures on S 2 × S 2 . Recall that the conformal compactification of split signature flat space R2,2 is S 2 × S 2 /Z2 , with double cover S 2 × S 2 . It turns out that there is only the conformally flat anti-self-dual conformal structure on S 2 × S 2 /Z2 , while there is an infinite dimensional family of nontrivial such conformal structures on the double cover S 2 × S 2 [16]. Real points in S 2 × S 2 correspond to Riemann spheres that intersect the real subspace PTR , dividing each sphere into two discs Dx± . The best way to understand the twistor theory in this case is to focus on one of the two discs, say Dx+ , rather than the Riemann spheres. In Euclidean space we were able to represent the twistor space T as the bundle of primed spinors S because we could solve the incidence relation ω A = x A A π A with x A A = (ω A πˆ A − ωˆ A π A )/(πˆ B π B ) when x A A was real. Thus the coordinate trans formation between (ω A , π A ) and (x A A , π A ) is locally invertible and in fact globally invertible if x A A = ∞ is allowed. In the context of the double fibration (2.4), when the spin bundle S is restricted to the real slice M, the projection r from S to T is one-to-one and identifies the spin bundle with the twistor space. In split signature, with π A complex, x A A = (ω A π¯ A − ω¯ A π A )/(π¯ B π B ) solves the incidence relation so that there is locally a one-to-one correspondence between the points in the bundle of complex spinors on M and twistor space. However, this fails where π¯ B π B vanishes, i.e. when π A is a complex multiple of a real spinor. This is because at real values of x and π A there are real α-planes, and such planes correspond to points of PTR . Indeed, the bundle SR of real spinors is foliated by the lifts of real α-planes to SR , with the lifted α-plane through (x, π A ) given by the α-plane through x with tangent spinor π A , i.e. the 2-surface in SR of the form (x A A + λ A π A , π A ) parameterised by λ A . Thus, there is a one-to-one identification between PS − {π¯ A π A = 0} and points in PT − PTR , but PTR itself is a quotient of PSR by its foliation by α-planes.
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The set S0 = {(x, π A ) ∈ S : π¯ A π A = 0} is a co-dimension-1 hypersurface in S and divides S into two halves S± on which ±i π¯ A π A ≥ 0 with common boundary S0 . We define the corresponding bundles of projective primed spinors PS± and PS0 by the same conditions on π¯ A π A . Working now on S 2 × S 2 with a general anti-self-dual conformal structure, it is still possible to distinguish between PS+ and PS− globally and we focus on one half, say PS+ .9 This is a bundle of discs over M with boundary PS0 . It turns out that PS+ has an integrable complex structure and is naturally a complex manifold—in the conformally flat case, PS+ is PT − PTR . The boundary, PS0 , is naturally foliated by the lifts of real α-surfaces in M as in the conformally flat case and the quotient is PTR , the space of real α-planes. There is a natural way to glue PTR to the boundary of PS+ to obtain a smooth compact complex manifold which is a copy of CP3 topologically.10 If the original space-time is smooth, it can be shown that this gluing can be performed in such a way that the twistor space has a smooth complex structure. If our anti-self-dual conformal structure on S 2 × S 2 is a continuous deformation of the standard conformal structure, then this twistor space must be the standard PT because the complex structure on CP3 is rigid. However, the embedding of PTR into PT will be a deformation of the standard embedding of the real slice PTR inside PT. The original space-time together with its anti-self-dual conformal structure can be reconstructed as the moduli space of holomorphically embedded discs in PT, with boundary in PTR in the appropriate topological class [16]. The central role played by discs in this approach makes open string theory seem rather natural. Linearised deformations of the embedding of PTR in PT correspond to sections of the normal bundle to PTR over PTR . These can be naturally represented as purely imaginary tangent vector fields on PTR ; they can be represented as vector fields on TR of the form i f α ∂/∂ Z α , where f α is real with homogeneity degree 1, defined up to f α → f α + Z α for of weight 0. This freedom can be fixed with the gauge choice ∂α f α = 0. The only such vector fields that give trivial deformations are the generators of SL(4, C). The non-linear version of this is to define a submanifold TR in T by the constraint Z α = X α + i F α (X α ),
(3.5)
where X α = Z β + Z¯ β is real and F α is a real function of four real variables of homogeneity degree one. Given PTR ⊂ PT , there is some freedom in the choice of TR corresponding to the shift
Z α → Z α = eiθ(X ) X α + i F α , (3.6) where θ is an arbitrary function of X α of weight 0; this changes the non-projective real slice, but not the projective one. Infinitesimally, (3.6) induces F α → F α + θ (X )X α + · · · .
(3.7)
This freedom can be fixed by imposing that det (δβα + i∂α F β ) be real. This implies that ∂α F α = ∂α F [α ∂β F β ∂γ F γ ] ,
(3.8)
which is an analogue of the Calabi-Yau condition on T . Clearly, this is a non-linear generalisation of the ∂α f α = 0 condition above. 9 On S 2 × S 2 /Z , it is not possible to distinguish between PS and PS ; the space-time is not simply + − 2 connected and, as one traverses a non-contractible loop, PS± interchange. 10 This is done by considering the manifold with boundary PS ∪ PS and compressing each horizontal lift + 0 of an α-plane to a point.
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Our primary interest in this paper will be in the second construction described above, but for completeness we give a discussion of the connection between the two approaches in an appendix. 3.2. The Ricci-flat case. We now return to complex space-time and suppose that the Ricci tensor vanishes in addition to ψ˜ A B C D = 0. This is the case if and only if the full Riemann curvature is anti-self-dual, and this is equivalent to the condition that the primed spin connection is flat, so that there exists a two complex dimensional vector space C2 of covariantly constant primed spinor fields. We saw in §3.1 that each point in T corresponds to an α-surface in space-time with a non-vanishing parallelly propagated tangent spinor field π A (x) defined over it. If the full Riemann curvature is self-dual, then a tangent spinor π A (x) on an α-surface is naturally the restriction of a covariantly constant spinor field on the whole space-time and determined by a constant spinor π A ∈ C2 , e.g. the value of the covariantly constant spinor field π A (x0 ) at some point x0 . Thus we have a projection p : T → C2 − {0} that takes an α-plane with tangent spinor π A (x) to π A (x0 ). We can use this projection to characterise the twistor space for a Ricci-flat space-time. A non-projective twistor space is a complex 4-manifold T satisfying the three conditions given in §3.1. Such a twistor space corresponds to a conformally anti-self-dual spacetime, and for this to be Ricci-flat, the twistor space T must in addition have • a projection p : T → C2 − {0} such that p∗ ϒ = π A ∂/∂π A . This condition arises because ϒ generates scalings of the tangent spinors to α-planes. The compatibility of ϒ with the Euler vector field on C2 means that the projection descends to p : PT → CP1 , giving a fibration over CP1 of the projective twistor space.11 The fibres are two-dimensional complex manifolds (but have no linear structure in the curved case, although, as we will see, they do have certain symplectic and Poisson structures). In order to clarify these conditions, we can introduce global coordinates π A on the base C2 − 0 of the fibration p : T → C2 − 0 and use them to build local coordinates (ω A , π A ) on T . These coordinates will be homogeneous coordinates for PT . As T is fibred over C2 − 0, the pull-back of the volume form gives a globally-defined two-form τ on T given by 1 1 Iαβ dZ α ∧ dZ β = A B dπ A ∧ dπ B , 2 2 and a holomorphic 1-form τ=
k = Iαβ Z α dZ β = π A dπ A
(3.9)
on PT (and T ) given by the pull-back of the holomorphic 1-form on CP1 . We can now restrict our choice of coordinates ω A so that 1 (3.10) d = αβγ δ dZ α ∧ dZ β ∧ dZ γ ∧ dZ δ = 2 AB dω A ∧ dω B ∧ τ. 6 11 Note that the existence of a projective twistor space with a projection to CP1 is not sufficient to reconstruct the projection p : T → C2 as, thinking of C2 − 0 as the total space of the C∗ bundle O(−1) over CP1 , p ∗ O(−1) will not in general be equivalent as a line bundle over PT to T → PT . Given p : PT → CP1 , in order to guarantee that there is a Ricci-flat metric in the conformal equivalence class, we need to require that p ∗ O(−1) is an equivalent line bundle to T as an independent condition.
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This can be expressed as the condition that we have a holomorphic (2, 0) form µ on the fibres given in local coordinates by µ=
1 AB dω A ∧ dω B , 2
(3.11)
where AB is the constant alternating symbol (note that only contractions of this form with vertical vectors up the fibres are defined). Then d = 4µ ∧ τ, = 2µ ∧ k.
(3.12)
Dually, there is a Poisson structure determined by a bi-vector I αβ and this is in turn given by AB , the inverse of AB , by { f, g} I := I αβ
∂ f ∂g ∂ f ∂g := AB . ∂ Zα ∂ Zβ ∂ω A ∂ω B
Since d and τ are globally defined by construction, Eq. (3.12) implies that µ is globally defined up to the addition of multiples of dπ A . The Poisson structure I αβ is globally and unambiguously defined, as the relation I αβ = 21 αβγ δ Iγ δ determines it uniquely. We now consider the implications of the condition that these structures be globally defined. We introduce two coordinate patches: U0 on which π0 does not vanish, and U1 on which π1 does not vanish. We then introduce local coordinates ‘up the fibres’ of p, w0A on U0 and w1A on U1 . These can be elevated to homogeneous coordinates on the respective patches by defining ω0A = π0 w0A and ω1A = π1 w1A . The coordinates are related in the overlap by the patching relations ω0A = F A (ω1A , π A ) for some transition function F A , and these are required to be homogeneous: F A (λω1A , λπ A ) = λF A (ω1A , π A ). This means that, as in the flat case, we can define the homogeneity operator ϒ = Z 0α ∂/∂ Z 0α = Z 1α ∂/∂ Z 1α . The requirement that the Poisson structure be expressed in its normal form on each patch is that { f, g} I = I αβ
∂ f ∂g ∂ f ∂g ∂ f ∂g ∂ f ∂g = AB = I αβ α = AB . α β β A B ∂ Z0 ∂ Z ∂ Z1 ∂ Z ∂ω0 ∂ω0 ∂ω1A ∂ω1B 0 1
A similar condition arises for the µ and in both cases the condition amounts to the requirement AB = C D
∂FA ∂FB ∂ω1C ∂ω1D
(3.13)
that the patching conditions preserve AB . Given a global I αβ , the equation 1 αβ I αβγ δ = Iγ δ 2 determines globally the scale of αβγ δ , and vice versa. Thus, the condition for Ricci flatness can be expressed as the condition that we have a global holomorphically defined
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simple bi-vector I αβ that determines a Poisson structure, and we will refer to this as the infinity twistor, as in the flat case.12 An infinitesimal deformation f α of the complex structure is an element of 1 ˇ H (PT , T (1,0) ), represented either as a Cech cocycle or as a Dolbeault form. The condition that it preserves the Poisson structure I αβ is that it is a Hamiltonian vector field that can be expressed as f α = I αβ
∂h ∂ Zβ
for some h ∈ H 1 (PT , O(2)). This is the linearised form of (3.13). Whereas the Penrose transform of a general f α subject to the gauge equivalence under f α → f α + a(Z )Z α gives a spin-2 field ψ ABC D satisfying the higher derivative equation (3.3), the Penrose transform of h gives a spin-2 field ψ ABC D satisfying the usual spin-2 equations
∇ A A ψ ABC D = 0.
(3.14)
3.2.1. Ricci-flat case in split signature. In the second of the two approaches to the split signature non-linear graviton construction, the complex twistor space is taken to be PT = CP3 , and conformally anti-self-dual space-times are constructed from deformations of a real slice PTR , which is itself an arbitrary small deformation of the real subspace RP3 . However, in the Ricci flat case, PTR is no longer an arbitrary deformation; instead it is subject to certain conditions as will now be explained. Again we take T to have an infinity twistor I αβ defined on it, and this determines a projection from T = T − {π A = 0} to C2 − 0 given by Z α → π A together with the corresponding projection p : PT → CP1 . This should be compatible with the real slice in the sense that PTR should project to RP1 ⊂ CP1 . Equivalently, PTR should lie inside the real codimension-1 hypersurface := p −1 (RP1 ) ⊂ PT , which can also be defined by the equation π A π¯ A = 0 with π¯ A = (π¯ 0 , π¯ 1 ) the standard complex conjugation. This is the analogue of the existence of the projection p : PT → CP1 and we need to express the second part of the condition for Ricci flatness in this context. On PT the line bundles O(n) of homogeneous functions of degree n are equal to the pull-backs of the corresponding line bundles from CP1 . Thus, on , the complex line bundles O(n) naturally have a fibrewise complex conjugation fixing the real sub-bundles OR (n), which are the pull-backs of the corresponding real sub-bundles of O(n) on RP1 (i.e. these real line sub-bundles are spanned by homogeneous polynomials of degree n in π A with real coefficients). The second condition necessary in order that PTR ⊂ PT corresponds to a Ricci-flat anti-self-dual conformal structure is that the O(4)-valued 3-form , when restricted to PTR , lies in OR (4), or equivalently that the restriction to PTR of the O(2)-valued 2-form µ = 21 dω A ∧ dω A up the fibres is real. This can be stated geometrically by observing first that, on each 4 real-dimensional fibre of p over RP1 , the form µ defines a complex symplectic form with values in O(2), and its imaginary part defines a real symplectic form with values in OR (2). Our requirement is then that on each fibre p −1 (π A ) of p over RP1 , the intersection of PTR with p −1 (π A ) should be Lagrangian with respect to , i.e., | P TR ∩ p−1 (π A ) = 0 for each π A . This will guarantee that µ is real on restriction to PTR , since we have required that the restriction of its imaginary part vanishes; it then follows from Eq. (3.12) that is real. 12 In fact, if we relax the simplicity condition, we obtain the condition that the space-time admits an Einstein metric for which the Ricci scalar can be non-zero.
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An infinitesimal deformation of PTR preserving this condition is therefore generated by a Hamiltonian vector field preserving µ, and so it is determined by a Hamiltonian function h which will be a global section of OR (2) defined over PTR (a finite deformation can then be obtained from a generating function). To be more explicit, we can decompose ω A into its real and imaginary parts, ω A = A ω R +iω IA , where ω RA and ω IA are real; then = 2dω RA ∧dω I A . Assuming the deformation to be transverse to ∂/∂ω IA , we can express PTR in , on which π A is real, as the graph ω IA = F A (ω RA , π A ), where F A has homogeneity degree one. Then the Lagrangian condition is ∂ F A = 0. ∂ω RA These conditions can be solved by introducing a smooth real function H (ω RA , π A ) on TR of homogeneity degree two and defining F A (ω RA , π A ) = AB
∂H . ∂ω RB
It can be seen that this automatically incorporates the condition (3.8). Infinitesimally, a deformation of PTR to PTR is given by pushing PTR along the vector field β
i f α (Z R )
∂h ∂ ∂h ∂ ∂ = i I αβ α = i AB , B B ∂ ZR ∂ ZI ∂ ZI ∂ω RA ∂ω IB
where we have written Z α = Z αR + i Z αI for Z αR and Z αI real, and h = h(Z αR ) is the infinitesimal analogue of H . The vector field is understood to be a normal vector field to the real slice, so it can be taken to be imaginary. As a final note, we observe that the hypersurface divides PT into two halves PT± according to ±iπ A π¯ A > 0. The holomorphic discs in PT with boundary on RP3 divide into those that lie entirely in , and those that lie in one of PT± . Those in PT± correspond to two distinct copies M± of space-time R4 , whereas those in correspond to points at (null) infinity. We will wish to work with just one copy of space-time, so we discard PT− and work only with the holomorphic discs in PT+ and hence just the one copy M+ of space-time. 3.2.2. Superspace, super-twistor space and anti-self-dual supergravity. We can consider deformations of super-twistor space PT[N ] to obtain anti-self-dual solutions to the conformal supergravity equations. The formal definition of such a deformed complex supermanifold has been studied in the mathematics literature [51,52]. Here we use the more general physics formulation in which both fermionic coordinates and fermionic constants are allowed. A supermanifold is constructed by patching together coordinate charts {Ui } with coordinates Z iI = (Z iα , ψia ) on each patch, where the Z iα are bosonic and the ψia fermionic. On the overlaps, the coordinates are related by patching functions Z iI := (Z iα , ψia ) = PiIj (Z Jj ) := (Piαj (Z Jj ), Piaj (Z Jj )),
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where Piαj is an even function, and Piaj is odd.13 We also require that the matrices ∂ PiIj /∂ Z Jj have non-zero super-determinant (in fact, it must be possible to choose coordinates so that it is equal to 1 in the N = 4 case for which the super-twistor spaces are super-Calabi-Yau; note that our projective twistor spaces are not Calabi-Yau for general N ). A complex supermanifold, e.g. PT[N ] , is composed of an underlying ordinary comβ plex manifold, PT (the ‘body’) with patching functions Piαj (Z j , 0) with all anticommuting coordinates and parameters set to zero, and a rank N vector bundle E → PT (the ‘soul’) whose patching functions are ∂ Piaj /∂ψ bj |ψ b =0 , again with all odd parameters j set to zero. It is an important feature of generic complex supermanifolds that they are not in general obtained by simply reversing the Grassmann parity of the coordinates up the fibres of the vector bundle E → PT (whereas this is the case for real supermanifolds). The higher derivatives of the patching functions with respect to odd variables encode information that cannot be gauged away. One necessary restriction for a complex supermanifold to be a super-twistor space is the requirement that the ψ a have homogeneity degree 1. One way of expressing this is to say that the bundle E should have degree −N (i.e. first Chern class −N ). As discussed earlier, the space CM of rational curves in PT in the appropriate topological class will be a space-time with anti-self-dual conformal structure. These rational curves will have deformations away from the body, and their moduli space CM+[N ] will be chiral superspace with body CM. The full superspace is obtained as the space of flags CP1|0 ⊂ CP1|N in PT[N ] , with the chiral and anti-chiral superspaces arising as the space of CP1|0 s and CP1|N s respectively. We are not aware of a full presentation of this construction in the literature, and to give one here would take us too far afield. An infinitesimal deformation of PT[N ] can be obtained by varying the patching functions, and such an infinitesimal variation is given in local coordinates on the overlap of two coordinate charts by a tangent vector f = f α ∂/∂ Z iα + f a ∂/∂ψia , where f α is even and f a is odd. To deform the complex structure, we use such a vector field on each overlap and a nontrivial deformation is defined modulo infinitesimal coordinate transformations on the open sets; thus the nontrivial deformations are parametrised by the cohomology group H 1 (PT[N ] , T (1,0) ), where T (1,0) is (the sheaf of sections of) the holomorphic tangent bundle of the supermanifold. This group was studied in the case of N = 4 in [6] and the spectrum of N = 4 conformal supergravity was obtained (see the end of Sect. 4). A similar analysis can be carried out for other values of N . In order to obtain an anti-self-dual version of Einstein supergravity, we need to impose the supersymmetric analogues of the constraints imposed on a twistor space to obtain Ricci-flat anti-self-dual four-manifolds as described in §3.2. There is now some ambiguity because, in the supersymmetric case, the restriction to Poincaré invariance gives a projection to CP1|N and hence also to CP1|0 . In order to obtain a straightforward supermultiplet starting from helicity −2 and increasing to helicity (N − 4)/2 in the linearised theory, we require that we have a projection p1 : PT[N ] → CP1|N
(3.15)
(and thence a further projection p : PT[N ] → CP1|0 ) and a global holomorphic volume form s with values in the pull-back of O(4 − N ) from CP1|0 . 13 Here fermionic parameters are allowed in these functions.
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To make this more explicit, we introduce the non-projective super-twistor space T[N ] , which as before can be defined as the total space of the pull-back of the line bundle O(−1) from CP1 using p. The projection p1 then determines a projection p : T[N ] → C1|N . We can introduce coordinates (π A , ψ a ), A = 0 , 1 , a = 1 . . . , N on C1|N and complete these to a local coordinate system Z I on T[N ] by adjoining local coordinates ω A (A = 0, 1) of homogeneity degree 1. In this case we can define ‘infinity twistors’ I I J and I I J on the non-projective twistor space T[N ] by setting
I I J dZ I ∧ dZ J = dπ A ∧ dπ A , IJ N I (ds ) I J K 1 ...K N +2 dZ K 1 . . . dZ K N +2 = I I J dZ I ∧ dZ J a=1 dψ a . It is now straightforward to see that deformations of super-twistor space preserving these structures must be of the form ∂ ∂h ∂ fI = IIJ , h ∈ H 1 (PT[N ] , O(2)). I ∂Z ∂ZI ∂ZJ Such an h precisely describes an anti-self-dual supergravity multiplet, starting with helicity 2 and going down to helicity (4 − N )/2; this will be discussed in more detail in Sect. 8.3. It is also possible to consider deformations of PT[N ] that preserve less structure. For example, later we will consider the case where we only preserve the projection p : PT[N ] → CP1 . In such cases, the space of possible deformations will be larger and correspond to more fields on space-time. 4. The Berkovits Twistor String 4.1. The Berkovits open string theory. The Berkovits string is a theory of maps from the world-sheet to a curved super-twistor space with coordinates Z I = (ω A , π A , ψ a ), Z˜ I = (ω˜ A , π˜ A , ψ˜ a ). In the following, we will find it useful to use a notation that can handle different signatures and different reality properties in a unified way. There are three different cases that we will consider: (i) Z I are complex coordinates on a complex super-twistor space T and Z˜ I are the complex conjugate coordinates Z˜ = (Z )∗ , (ii) Z I , Z˜ I are independent real coordinates on a space TR × TR for some real twistor space TR , (iii) Z I , Z˜ I are independent complex coordinates on a space T × T for some complex twistor space T . For space-times of signature ++++ or +++−, the twistors are necessarily complex, while for signature + + −− either complex or real twistors can be used. In the flat case, Z I , Z˜ I are complex conjugate coordinates on C4|4 , real coordinates on R4|4 × R4|4 , or complex coordinates on C4|4 ×C4|4 ; then we write Z I = (ω A , π A , ψ a ), Z˜ I = (ω˜ A , π˜ A , ψ˜ a ). For open strings in any of the three cases, the boundary of the world-sheet ∂ is constrained to map to the submanifold defined by Z = Z˜ . For case (i) with complex Z , this is the real submanifold PTR that arose in §3.1.1. We use world-sheet coordinates σ, σ˜ with world-sheet metric ds 2 = 2dσ d σ˜ . For Euclidean world-sheet signature, σ, σ˜ are complex conjugate coordinates σ˜ = σ ∗ while for Lorentzian world-sheet signature, σ, σ˜ are independent real null coordinates.
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The fields include maps Z I (σ, σ˜ ), Z˜ I (σ, σ˜ ) from the world-sheet to super-twistor space and these are world-sheet scalar fields. The action is
S = d 2 σ Y I ∂˜ Z I + Y˜ J ∂ Z˜ J − A˜ J − A J˜ + SC , (4.1) where Y I , Y˜ I are conjugate momenta of conformal dimensions (1, 0) and (0, 1) respectively and ∂ = ∂/∂σ , ∂˜ = ∂/∂ σ˜ . The world-sheet gauge fields A, A˜ couple to currents J = YI Z I ,
J˜ = Y˜ I Z˜ I ,
(4.2)
so that there is a local symmetry 1 Z˜ I → t˜ Z˜ I , Y˜ I → Y˜ I , t˜ 1 A → A + ∂ t˜. t˜
Z I → t Z I , YI → 1˜ A˜ → A˜ + ∂t, t
1 YI , t
(4.3)
This symmetry ensures that the theory projects to one defined on a projective twistor space PT, PTR × PTR or PT × PT . The action is real for Euclidean world-sheets if one chooses case (i) above, all variables are complex, and the tilde operation is complex conjugation, so that for any ˜ = ∗ . For Lorentzian world-sheets the action is real if all variables are real, field , ˜ are independent real variables. For Euclidean requiring signature + + −−, and , world-sheets the parameter t is complex and the gauge symmetry (4.3) is G L(1, C) while for Lorentzian world-sheets t, t˜ are independent real parameters and the gauge ˜ group is G L(1, R) × G L(1, R). For the case of Lorentzian world-sheets in which , are independent real variables, ‘Wick rotation’ gives a theory on Euclidean world-sheets ˜ become independent complex variables, leading to case (iii) above, and in which , it is the action of this theory that is used in the Euclidean path integral. The term SC in (4.1) is the action for an additional matter system which is a conformal field theory with Virasoro central charges cC = c˜C and currents j r and j˜r , for r = 1, . . . dim G. Here G is some group whose Kaˇc-Moody algebra is generated by the currents. The Kaˇc-Moody central charges are denoted by k = k˜ and the group G becomes a Yang-Mills gauge group in space-time. Open strings are included in the model with the boundary conditions Z I = Z˜ I , Y I = Y˜ I ,
j r = j˜r
(4.4)
on ∂. For complex Z with Z˜ = Z ∗ , the string endpoints lie in a real subspace TR of T , which projects onto a real subspace PTR of PT. In the flat case, this is RP3|4 ⊂ CP3|4 and (4.4) breaks the S L(4|4; C) symmetry to S L(4|4; R). This boundary condition is natural for the case of split space-time signature + + −−, where the real subspace plays a natural and important role, as was discussed in §2.3 and §3.1.1. As the interpretation of the results for other signatures is less clear, we will restrict ourselves to the split spacetime signature + + −− in what follows. For independent real Z , Z˜ and split space-time signature, the ends of the strings lie in the diagonal PTR in PT = PTR × PTR . For the flat twistor space PT = RP3|4 × RP3|4 , the endpoints lie in the diagonal RP3|4 , breaking the conformal symmetry from S L(4|4; R) × S L(4|4; R) to the diagonal subgroup. In either case, the boundary theory lives on a real twistor space PTR (which is RP3|4 in the flat case) and the scaling symmetry is broken to G L(1, R) by the boundary conditions.
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˜ c) Quantisation gives the usual conformal gauge ghosts (b, c) and (b, ˜ together with G L(1) ghosts (u, v) and (u, ˜ v) ˜ (v and v˜ have conformal dimensions (0, 0), while u and u˜ have dimensions (1, 0) and (0, 1)). Variables φ˜ with a tilde are right-moving (∂ φ˜ = 0), ˜ = 0). The matter stress-energy tensor is while those without are left-moving (∂φ T m = YI ∂ Z I + T C , T˜ m = Y˜ I ∂˜ Z˜ I + T˜ C ,
(4.5)
T˜ C
TC
and are the left and right-moving stress-energy tensors for the current where algebra. The stress-energy tensor for the ghosts is T gh = b∂c + ∂ (bc) + u∂v, ˜ b˜ c) T˜ gh = b˜ ∂˜ c˜ + ∂( ˜ + u˜ ∂˜ v. ˜
(4.6)
The open string theory is defined by the boundary conditions (4.4) on the twistor variables, together with additional boundary conditions on the ghosts: ˜ v = v, c = c, ˜ b = b, ˜ u = u. ˜ The BRST charges are Q= Q˜ =
(4.7)
dσ (cT + v J + cu∂v + cb∂c) ,
d σ˜ c˜ T˜ + v˜ J˜ + c˜u˜ ∂˜ v˜ + c˜b˜ ∂˜ c˜ ,
(4.8)
and they are nilpotent provided the additional matter system has Virasoro central charge cC = 28; this value cancels the contributions c = −26 of the (b, c) system and c = −2 of the (u, v) system to the conformal anomaly. There is no G L(1) × G L(1) anomaly because of cancellation between bosons and fermions. The physical open string states are BRST cohomology classes represented by vertex operators that are G L(1) neutral and are dimension one primary fields with respect to the Virasoro and Kaˇc-Moody generators (4.5), (4.6) and (4.2). The super-Yang-Mills vertex operators are the dimension one operators constructed with Kaˇc-Moody currents of the auxiliary matter system [2]: Vφ = jr φ r (Z ),
(4.9)
φ r (Z )
are any Lie-algebra-valued functions that are invariant under scalings where the of Z I (i.e. any Lie-algebra-valued functions on RP3|4 ) and have conformal weight zero. The dimension one vertex operators [6] V f = Y I f I (Z ), Vg = g I (Z )∂ Z I
(4.10)
are G L(1)-invariant provided the functions f I carries G L(1) charge +1 (i.e. it is in O(1)) and g I carries G L(1) charge −1 (i.e. it is in O(−1)). They will be physical if the f I and g I satisfy ∂ I f I = 0, Changing f
I,g
I
Z I g I = 0.
(4.11)
by δ f I = Z I , δg I = ∂ I χ ,
(4.12)
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gives operators in the same BRST cohomology class as those given in (4.10), so that (4.12) are gauge invariances giving physically equivalent states [2,6]. The vertex operators (4.10) give the states of conformal supergravity [6]. Since f I has G L(1) charge 1, the vector field f = fI
∂ ∂ZI
(4.13)
on T is invariant under scaling, and the first equivalence relation in (4.12) means that f can be interpreted as a vector field on PT [6]. The first constraint in (4.11) means that f is a volume-preserving vector field. The second constraint in (4.11) means that the one-form g = gI d Z I
(4.14)
is well-defined on PT [6]. The second gauge equivalence in (4.12) means that g is an abelian gauge field. The functions φ r (Z ) in (4.9) are superfields which can be expanded in terms of ordinary functions on twistor space with values in the line bundles O(0), O(−1), O(−2), O(−3), O(−4). By the Penrose transform, these represent fields of helicities (1, 21 , 0, − 21 , −1) with the correct R-symmetry representations to describe N = 4 super-Yang-Mills states [1,2]. Likewise, the spectrum of Minkowski space helicity states associated with the vertex operators (4.10) follows from the expansions of the superfields f I (Z ) and g I (Z ) in powers of ψ [6]. The analysis of [6] shows that, taking (4.11,4.12) into account, f A (Z ) and f A (Z ) each describe the helicity states (+2, + 23 , +1, + 21 , 0) of an N = 4 supergravity multiplet (with the correct R-symmetry representations) while f a (Z ) describe the helicity states (+ 23 , +1, + 21 , 0, − 21 ) of (four) gravitino multiplets. Similarly, g A , g A give two supergravity multiplets with negative helicities (0, − 21 , −1, − 23 , −2) and ga give (four) gravitino multiplets (+ 21 , 0, − 21 , −1, − 23 ). Taken together, the spacetime fields described by the vertex operators V f and Vg given in (4.10) can be identified with the physical states of N = 4 conformal supergravity. 4.2. Generalised boundary conditions. In split signature, the non-linear graviton can be constructed from deformations of a real subspace PTR in a fixed flat twistor space PT, as was reviewed in §3.1.1. This suggests a modification of the Berkovits string model in which, for the case (i) of complex Z , the strings live in PT and the open string boundaries are constrained to lie in the general subspace PTR defined in terms of functions F α by (3.5) instead of the real subspace defined by the condition Z = Z ∗ . We then consider a string theory in which the boundary condition Z I = Z˜ I is replaced with Z I − Z˜ I = Fˆ I (Z J + Z˜ J )
(4.15)
for some function Fˆ I of homogeneity degree one. There is a gauge freedom in the definition of F, which can be multiplied by a function of homogeneity degree 0 (see also the discussion following Eq. (3.5)). This can be fixed by imposing the condition that sdet(δ JI + ∂ J Fˆ I ) = sdet(δ JI − ∂ J Fˆ I ), where sdet denotes the super-determinant. This is the condition that the Calabi-Yau forms d in Z α and in Z˜ α agree. The corresponding boundary conditions for Y are found by requiring the surface term in the variation of the
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action to vanish. Varying the action (4.1) gives terms proportional to the field equations together with a surface term
1 (Y I δ Z − Y˜ I δ Z˜ I ) = 2 ∂
I
∂
(Y I − Y˜ I )(δ Z I + δ Z˜ I ) + (Y I + Y˜ I )(δ Z I − δ Z˜ I ) , (4.16)
where the boundary ∂ is specified by σ + σ˜ = 0. Using Eq. (4.15), this will vanish if the boundary conditions for Y are modified to become Y J − Y˜ J = − Fˆ I ,J (Y I + Y˜ I ).
(4.17)
In the cases (i) or (iii) above in which Z˜ α and Z α are independent quantities, the deformation of the boundary condition amounts to a deformation of the location of the diagonal subspace inside PTR × PTR or PT × PT, where the world-sheet boundary is constrained to lie. In the complex case (ii) in which Z is complex and Z˜ = (Z )∗ and the boundary is the real axis σ = σ ∗ , it is useful to write Fˆ = i F so that (4.15) becomes Z I − Z¯ I = i F I (Z J + Z¯ J ),
(4.18)
where sdet(δ JI + i∂ J F I ) is constrained to be real (in order to fix the gauge freedom). This is a supersymmetric version of (3.5), and the boundary condition (4.17) becomes Y J − Y¯ J = −i F I ,J (Y I + Y¯ I ).
(4.19)
With these boundary conditions, the worldsheets of degree 1 correspond to points of the compactified space-time S 2 × S 2 , and this has the non-trivial split signature anti-selfdual conformal structure determined by F I . The construction of §3.1.1 then suggests that the geometric interpretation of the vertex operator V f = Y I f I should be that f I determines an infinitesimal variation in F I , and so deforms the boundary conditions. Next we turn to the interpretation of the vertex operator Vg = g I ∂ Z I . If one adds a boundary term ∂
G I (Z J + Z˜ J )∂(Z I + Z˜ I )
(4.20)
to the action (4.1), for some G I = G I (Z J + Z˜ J ), then the condition that the surface term in the variation of the action vanishes is now Y J − Y˜ J = − Fˆ I ,J (Y I + Y˜ I ) + 2G [I,J ] ∂(Z J + Z˜ J ),
(4.21)
so that the surface term leads to a modification of the boundary conditions for Y . Then the vertex operator g I ∂ Z I corresponds to a deformation of G I . The quantisation of the string models based on the generalised boundary conditions (4.15) and (4.21) will be discussed elsewhere.
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5. Gauged β-γ Systems 5.1. 1-form symmetries. The system (sometimes referred to as a β-γ system) S = d 2 σ Y I ∂˜ Z I ,
(5.1)
where the Z I are coordinates on some manifold (or supermanifold) M, has recently been discussed in [32,33]. The Berkovits twistor string has kinetic terms of this form, with super-twistor space as the target space. If k i = k iI d Z I are 1-forms on M labeled by an index i, i = 1, . . . p, then the chiral currents K i = k iI ∂ Z I
(5.2)
∂˜ K i = 0
(5.3)
are conserved:
˜ i = 0, and generate a symmetry with parameters αi (σ ) satisfying ∂α i J δ Z I = 0, δY I = k iI ∂αi + 2αi k[I,J ]∂ Z .
(5.4)
The rigid symmetry with constant parameters was discussed in [32]. Both bosonic and fermionic local symmetries can be considered, and below we consider models with d bosonic currents and n fermionic currents and p = d + n. The currents K i commute, so they satisfy an abelian Kaˇc-Moody algebra with vanishing central charge: [K i (σ ), K j (σ )] = 0.
(5.5)
This can be promoted to a local symmetry by coupling to gauge fields B˜ i to give the action
(5.6) S = d 2 σ Y I ∂˜ Z I − B˜ i K i , which is invariant under (5.4) and ˜ i δ B˜ i = ∂α
(5.7)
for general local parameters αi (σ, σ˜ ). Gauge-fixing and introducing ghosts si and antighosts r i gives the action
˜ i , (5.8) S = d 2 σ Y I ∂˜ Z I + r i ∂s and the BRST charge Q=
dσ si K i
(5.9)
is nilpotent. For the vertex operator V f = f I Y I , i J I i i I [Q, V f ] = (∂si ) f I k iI + 2si f I k[I,J ] ∂ Z = ∂(si f k I ) − si [L f k ] I ∂ Z , (5.10)
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and so f I Y I is BRST invariant provided i (5.11) f I k iI = 0, f I k[I,J ] = 0, while the integrated vertex operator V f is invariant (up to a surface term) provided the Lie derivative of k i with respect to the vector field f vanishes,
L f k i = 0.
(5.12)
Changing the vertex operator g I ∂ Z I by a BRST exact term leads to the symmetry δg I = ηi k iI
(5.13)
for any ηi (Z ), since ηi k iI ∂ Z I = {Q, ηi r i }. This can be generalised to the case in which the one-forms k i are not globallydefined14 but are local sections of a bundle [34]. For example, the k i might be a local section of the co-frame bundle, i.e. a local basis for the cotangent bundle T ∗ M. If M is a bundle over some E, the ki could be a local section of the co-frame bundle of E (or rather the pull-back of this co-frame bundle). We will be interested mainly in the case in which M is projective (super-)twistor space, and is a bundle over E where E is CP1 or CP1|N . Given an open cover {Ur } of M, suppose there is a set of 1-forms kri in each patch Ur , with j
kri = (L r s )i j ks
(5.14)
in the overlaps Ur ∩ Us , and transition functions L r s in G L(d|n) if the kri consist of d bosonic one-forms and n fermionic ones. The kri are then sections of a bundle X over M, and we can introduce a connection one-form ( Bˆ r )i = ( Bˆ r )i I d Z I with transition functions j ˆ ˆi ( Bˆ r )i I = (L r−1 s )i ( Bs ) j I + ∂ I α
(5.15)
for the bundle Xˆ whose structure group is the group of fibre translations (with parameters αˆ i ). Then the gauged theory is well-defined provided the gauge fields B˜ i are taken to be connections on the pull-back of Xˆ to a bundle over the world-sheet, by a similar construction to that given in [34]. The theory is locally the same as that described above. 5.2. 1-form symmetries and scale symmetry. A natural generalisation of the construction of the last section would be to consider a set of vector fields V j = V jI (Z )∂/∂ Z I on M, and construct the currents V jI Y I . A necessary condition for the current algebra to close is that the V j are closed under the Lie bracket, so that they generate the action of a group L on M. In certain circumstances, the corresponding symmetries can be gauged, resulting in a theory on the quotient space M/L. Thus the gauging leads to replacing M with M/L, and gauging symmetries from vectors and 1-forms on M is equivalent to gauging symmetries from 1-forms only on M/L. There is then no loss of generality in 14 As emphasised by E. Witten, a geometrically clearer formulation of the construction and of its generalisation can be given in terms of the distribution (i.e. the sub-bundle of the cotangent bundle T ∗ M of M) generated by the k i . In particular, the distribution does not depend on the choice of basis for the one-forms k i .
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considering general M without gauging the symmetries generated by vector fields on M. However, it will be useful to consider the case of the Euler vector field ϒ = ZI
∂ ∂ZI
(5.16)
generating the one-dimensional group L S of scale transformations. Gauging the symmetries from 1-forms and ϒ on M is then the same as gauging 1-forms alone on the projective space P M = M/L S , but using the formulation on M will be useful for the Berkovits twistor string. Suppose the one-forms k i have scaling weights h i under the action of (5.16), so that for each i, Lϒ k i = h i k i ,
(5.17)
where Lϒ is the Lie derivative with respect to ϒ, and have constant vertical projections, so that ι(ϒ)k i = ei for some constants ei , i.e. Z I k iI = ei .
(5.18)
If h i = 0, ei = 0, then k i is horizontal and is the pull-back of a form on P M, the projective space given by taking the quotient by the action of the scalings generated by ϒ. Then the current J = Y I Z I has the commutation relations [J (σ ), K i (σ )] = h i K i (σ )δ(σ − σ ) + ei δ (σ − σ )
(5.19)
for each i. If Z I = (Z α , Z a ) and Y I = (Yα , Ya ), where Yα , Z α with α = 1, . . . , D are bosonic β-γ systems and Ya , Z a with a = 1, . . . , N are fermionic b-c systems, then [J (σ ), J (σ )] = δ (σ − σ )(D − N ).
(5.20)
J, K i
generate a Kaˇc-Moody algebra which is non-abelian if the Then the currents weights h i are not all zero and which has central charges ei , D − N . If the ei were not constant, the algebra would not close and one would need to introduce the ei as extra generators. ˜ B˜ i only if ei = 0, so This symmetry can be gauged by introducing gauge fields A, i that the k are all horizontal; it will now be assumed that this is the case. The gauged action is
(5.21) S = d 2 σ Y I ∂˜ Z I − A˜ J − B˜ i K i , which is invariant under the gauge transformations given by (5.4) together with δ A˜ = 0
(5.22)
˜ i − h i Aα ˜ i. δ B˜ i = ∂α
(5.23)
and
It is also invariant under the scaling symmetry Z I → t Z I , YI →
1 YI , t
1˜ A˜ → A˜ + ∂t, t
B˜ i → t −h i B˜ i . (5.24)
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Introducing ghosts v, si and anti-ghosts u, r i , the BRST charge is now i i [si K − vh i si r ] . Q = dσ v J +
(5.25)
i
The ghost si is a world-sheet scalar with scaling weight −h i (transforming as si → t −h i si under G L(1)) while the antighost r i has world-sheet conformal dimension one and scaling weight h i . Then Q 2 is proportional to κv∂v, where i (h i )2 (5.26) κ = D−N− i
with i = 1 for bosonic symmetries (with αi a bosonic parameter) and i = −1 for fermionic symmetries (with αi a fermionic parameter). The constant κ is the central charge for the Kaˇc-Moody algebra generated by the currents h i si r i (5.27) Jg f = J − i
which generate scalings of the gauge-fixed action, and quantum consistency (cancellation of the anomaly in the scaling symmetry) requires κ = 0.15 6. Gauging the Berkovits Twistor String The formalism of the previous section will now be applied to the Berkovits twistor string, generalised to a target space T that is a supermanifold with D bosonic dimensions and N fermionic ones; the flat twistor space is C D|N , R D|N × R D|N or C D|N × C D|N . The case of physical interest is D = 4, and we will see that, remarkably, this value is selected by anomaly cancellation in some of the models. We saw in §3.2 that the twistor space T for a Ricci-flat space-time is fibred over C2 − 0, so that PT is fibred over CP1 , and this in particular implies the existence of the 1-form k given by (3.9), corresponding to an infinity twistor. In the flat case, this requires working with PT = CP3 − CP1 , which has such a fibration, whereas the full twistor space CP3 does not. In the supersymmetric case, PT is fibred over CP1|0 or CP1|N , and in the latter case a local basis of N fermionic 1-forms on CP1|N pull back to N locally defined fermionic 1-forms k a on super-twistor space. In this section we will assume that the target space T is equipped with a set of 1-forms k i and gauge the corresponding symmetries. In the following sections, we will suppose that these 1-forms arise from a fibration of the super-twistor space that follows from the condition for a Ricci-flat space-time, and find that the gauging restricts the physical states of the string theory so that they can be associated with deformations of the super-twistor space preserving the fibration structure, and hence the Ricci-flatness. Given a set of 1-forms k i = k iI (Z )d Z I and k˜ i = k˜ iI ( Z˜ )d Z˜ I of weights h i , h˜ i there are currents K i = k iI ∂ Z I ,
K˜ i = k˜ iI ∂˜ Z˜ I .
(6.1)
15 It was pointed out to us by E. Witten that, if a global and everywhere nonzero function w exists on M then the last term (involving the scaling weights h i ) in the anomaly (5.26) can be eliminated by adding to the BRST operator Q a term proportional to ∂v log w. This is natural in the formulation in terms of the distribution generated by the one-forms k i rather than that using a specific choice of k i adopted here.
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These are conserved Kaˇc-Moody currents for the free theory given by (4.1) with A = A˜ = 0. For the case of Euclidean world-sheets, in which σ˜ = σ ∗ and Z˜ = Z ∗ , the currents K˜ i are the complex conjugates of the K i . For the other cases, the K˜ i and the K i are independent currents satisfying K i = K˜ i on the boundary as a result of the boundary conditions (4.4). We assume that the 1-forms satisfy Z I k iI = 0,
Z˜ I k˜ iI = 0,
(6.2)
so that the central charges ei , e˜i vanish and gauging is possible. Then gauging the symmetries generated by K i , K˜ i gives the action
(6.3) S = d 2 σ Y I ∂˜ Z I + Y˜ J ∂ Z˜ J − A˜ J − A J˜ − Bi K˜ i − B˜ i K i + SC , and this is invariant under (5.4), (5.22), (5.23) together with the corresponding symmetries with parameter α, ˜ t˜. For open strings, the boundary conditions (4.4) are imposed as before. Under the symmetries with parameter α, α, ˜ the action changes by a total derivative term
˜ α K − α˜ K˜ , δS = d 2 σ (∂ − ∂) (6.4) and with the boundary conditions (4.4), this vanishes for gauge transformations in which the parameters satisfy α = α˜
(6.5)
on the boundary. Gauge-fixing by choosing conformal gauge and requiring all gauge fields to vanish introduces the ghosts (u, v) and (u, ˜ v) ˜ of the Berkovits string, together with the ghost system (r i , si ) of the last section and its conjugate system (˜r i , s˜i ). The open string theory is defined by the boundary conditions (4.4) on the twistor variables and ˜ v = v, c = c, ˜ b = b, ˜ u = u, ˜ r i = r˜ i , si = s˜i (6.6) on the ghosts. The BRST operators are i i i Q = dσ cT + v J + si K + cu∂v + cb∂c + cr ∂si − vh i si r , Q˜ =
i
d σ˜ c˜ T˜ + v˜ J˜ + s˜i K + c˜u˜ ∂˜ v˜ + c˜b˜ ∂˜ c˜ + c˜r˜ ∂ s˜i − ˜i
i˜
v˜ h˜ i s˜i r˜
i
.
i
(6.7) In Q 2, there are two potentially non-zero terms: a conformal anomaly term proportional to C c∂ 3c, where C is the Virasoro central charge, and a gauge anomaly term proportional to k v∂v, where k is the Kaˇc-Moody central charge. The Virasoro central charge is C = D − N + cC − 28 − 2(d − n),
(6.8)
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where D − N comes from the Y Z system, cC is the central charge of the auxiliary matter system SC , the contribution −28 = −26 − 2 comes from the bc and uv systems, and −2(d −n) comes from the (r i , si ) system consisting of d fermionic ghosts and n bosonic ones. The Kaˇc-Moody central charge is k= D−N− i (h i )2 , (6.9) i
where i = 1 for bosonic symmetries (with αi bosonic) and i = −1 for fermionic symmetries (with αi fermionic). The gauge anomaly cancels if κ = 0. If κ = 0, one might attempt to cancel the anomaly against a contribution from the matter system SC . If the matter system SC has a current JC generating a G L(1) Kaˇc-Moody symmetry with central charge κC , and SC is chosen to contain the coupling A˜ JC , then k = D − N + κC − i (h i )2 . (6.10) i
However, this is likely to lead to problems from mixing between the auxiliary matter system and the twistor space sector, and its most natural interpretation would be as a change in the definition of the twistor space. We therefore restrict ourselves to solutions with D−N− i (h i )2 = 0, (6.11) i
so that no resort to such a compensating coupling is needed. ˜ k˜ from Q. ˜ Quantum consistency There will be similar anomalies with coefficients C, ˜ ˜ requires C = C = 0 and k = k = 0. In the next section, some string theories in which these anomalies cancel will be considered. 7. World-Sheet Anomaly Cancellation in Twistor Strings 7.1. No supersymmetry. Consider first the bosonic case in which N = 0, n = 0, so that the twistor space PT is an ordinary (bosonic) complex manifold of dimension D − 1. The Penrose construction of the non-linear graviton for D = 4 requires the projective twistor space PT to be fibred over CP1 . We then restrict ourselves to twistor spaces in which PT is fibred over CP1 (or in the real case, to spaces PTR × PTR with PTR fibred over RP1 ). Then there is a holomorphic 1-form on CP1 , given by A B π A ∧dπ B , 1 where π A are homogeneous coordinates on CP , and its pull-back to PT is k = Iαβ Z α d Z β
(7.1)
with Iαβ the dual of the infinity twistor. This in turn pulls back to a 1-form on (nonprojective) twistor space T , again given by (7.1). This 1-form has weight h = 2. Gauging the symmetry generated by this 1-form then gives the Kaˇc-Moody central charge k = D − h 2 = D − 4, which vanishes precisely when D takes the value D = 4 needed for Penrose’s twistor space, and no κC is needed. Then from (6.8) with D = 4, d = 1, we find C = cC − 26 so the matter system can be taken to be a critical bosonic string with cC = 26.
(7.2)
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7.2. N supersymmetries, PT fibred over CP1|N . Suppose now that there are N fermionic dimensions, and the projective twistor space is fibred over CP1|N (or RP1|N × RP1|N ). On CP1|N , a section of the co-frame bundle gives one bosonic one-form and N fermionic ones. The bosonic 1-form is the globally-defined k given in (7.1), while the N locally-defined fermionic one-forms k a are of the form k a = dψ a + eaA dπ A and are of weight h a = 1. Here
eaA
(7.3)
satisfies
π A eaA = −ψ a ,
(7.4)
so that the k a satisfy ι(ϒ)k a = 0. In a patch where π A ρ A = 0 for some fixed spinor ρ A , this can be solved by eaA = −
ψ a ρ A π B ρB
so that
k a = π A ρ A d
ψa π B ρB
(7.5) .
(7.6)
These forms pull back to one-forms (k, k a ) on PT and T , so they can be used in the construction of the last section. The k a are only locally-defined, but the gauging is still defined globally, as discussed at the end of §5.1. Now from (6.9), the Kaˇc-Moody central charge k is independent of N and κ = D − 4,
(7.7)
so that anomaly cancellation again selects D = 4. Then (6.8) gives C = cC − (26 − N ),
(7.8)
so that the matter system should be chosen to have cC = 26 − N . 7.3. General weights. The form (7.1) is of weight h = 2, but a 1-form of general weight h can be made by multiplying by a function w(Z ) of weight h − 2 (so that w is a section of O(h − 2)) to give kˆ = w(Z )I I J Z I d Z J . Similarly, multiplying (7.3) by a
wa (Z )
(7.9)
that is a section of O(h a − 1) gives for each a
kˆ a = wa (Z )(dψ a − eaA dπ A )
(7.10)
which is of weight h a . Introducing such factors gives many formal anomaly-free solutions for which the central charges (6.8) and (6.9) vanish. For example, choosing all kˆ a to be of equal weights h , the conditions are 0 = D − N + cC − 30, 0 = D − N − h 2 + N (h )2 .
(7.11)
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In the bosonic case N = 0, the only solution with D = 4 is the model with h = 2 and matter central charge cC = 26 discussed in §7.1; however, formally there are higher dimensional solutions of (7.11) with h 2 = D, cC = 30 − D.
(7.12)
For the case D = 4 with N fermionic currents, cC = 26 + N , h 2 − N (h )2 = 4 − N .
(7.13)
For h = 1, there are solutions with h = 2 and cC = 26 + N (including an N = 4 model which is distinct from the N = 4 model with cC = 22 discussed in §7.2), and there are additional solutions of (7.13) with h > 1. It is straightforward to find further anomaly-free solutions corresponding to currents of general weights h, h a . 7.4. Weightless forms. An important special case of the construction with general w, wa ˆ kˆ a consists in choosing w of weight −2 and all the wa of weight −1, which gives forms k, all with weights 0. Then (6.9) gives the same constraint D = N as for the Berkovits string, and with D = 4 this selects N = 4. If one gauges kˆ and n of the kˆ a with 0 ≤ n ≤ N , then the central charge is C = cC − 30 + 2n.
(7.14)
There are two models of particular interest with D = N = 4, that with n = 0 and that with n = 4. If w is chosen to depend on π A only, then the one-form kˆ is closed, d kˆ = 0. In a patch where π A ρ A = 0 for some fixed spinor ρ A with k a given by (7.6), choosing wa = (π A ρ A )−1 for each a gives ψa ˆk a = d (7.15) π B ρB which automatically satisfies d kˆ a = 0. More generally, for any wa (π ) on CP1 of weight −1, we can choose kˆ a = d(ψ a wa ) (with no sum over a). A potential problem with this construction is that functions w(Z ), wa (Z ) of negative weights can have singularities. For example, for weight −1, w = (π A ρ A )−1 is singular A A A on the surface π ρ A = 0 on which π = λρ for arbitrary parameter λ. A function w(Z ) of weight h on CP1 will have −h singularities if h < 0, and it is not clear how to define the construction at these singularities. For the case of real twistor space with Z , Z˜ independent and real, there are nonsingular functions of negative weights. For example, a function of weight −2 on RP1 is given by w(π ) =
1 MA B π
A π B
,
(7.16)
where π A are real homogeneous coordinates for RP1 , and this is non-singular if the constant symmetric real matrix M A B is positive definite, since the point π A = 0 is
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excluded. This can then be pulled back to a non-singular function of weight −2 on any space that is fibred over RP1 . For a real twistor space given by a region of RP3|4 ×RP3|4 , or more generally one that is of the form PTR × PTR for some real PTR that is fibred over RP1 ×RP1 , non-singular functions w(π ), w( ˜ π˜ ) can be constructed in this way, and ˆ˜ Z˜ ) of weight h = h˜ = 0. ˆ ), k( they can be used to construct well-defined one-forms√k(Z A function w of weight −1 can be defined as w = w as w is positive. For the complex case, w(Z ) can be chosen to be non-singular in a holomorphic disc with boundary on the real subspace, so that it is non-singular on the embedding of the open string world-sheet in super-twistor space. For a twistor space PT fibred over CP1 , w can be chosen as w=
1
(7.17)
(ρ1A π A )(ρ2B π B )
for some fixed complex spinors ρ1A , ρ2A . Then each singularity lies in a plane ρ A π A = 0. Recall that twistor space divides into two parts PT± with ±iπ A π¯ A ≥ 0 and that these two parts correspond to two copies of space-time. To obtain just one copy of space-time, we choose PT+ , say, as the twistor space, and the space of holomorphic discs in this part of twistor space with boundary on PTR gives a complete copy of space-time. If we take both ρ1A , ρ2A to lie in PT− , then w(Z ) is non-singular on PT+ and the gauging of the twistor string is well-defined for world-sheets that are discs in PT+ . In the complex case with Z˜ = Z ∗ , the cancellation of the surface term in the variation (6.4) requires that wα = w˜ α˜ = (wα)∗ on the boundary. If w(Z ) is real on the real axis Z = Z ∗ , this gives the boundary condition α = α˜ as before, but if w is a complex function on the real axis, then the boundary conditions of α and hence of the ghosts s are modified. However, in the case of Euclidean world-sheet, in which Z and Z˜ are independent complex variables, the boundary condition is Z = Z˜ and it is possible that w(Z ), w( ˜ Z˜ ) can be chosen so that w(Z ) = w( ˜ Z˜ ) on the boundary with w(Z ) non-singular on the holomorphic disc, and the boundary condition on α is α = α. ˜ The models in which the zero-weight one-form (7.9) or the one-forms (7.9), (7.10) are gauged are then well-defined both for the real case, and for the complex case with independent complex coordinates Z , Z˜ . The models depend on an arbitrary function w, or on the functions w and wa , but these only enter into the BRST charge. It will be seen in the next section that the spectrum is independent of w, wa , provided these functions are chosen to have no zeroes or poles; tree-level amplitudes at degree zero are also independent of the choice of w, wa , as will be checked explicitly in an example in §9. 8. Spectra of the Twistor String Theories 8.1. Physical vertex operators. In this section, we will investigate the constraints and gauge invariances for the vertex operators V f , Vg , Vφ for each of the anomaly-free theories of the last section, and obtain the ghost-independent part of the BRST cohomology. We will discuss the ghost-dependent vertex operators elsewhere. The gauged twistor string is constructed on a twistor space with a set of 1-forms k i = k iI d Z I with weights h i defined by (5.17) and satisfying Z I k iI = 0.
(8.1)
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The vertex operator V f = Y I f I (Z ) is physical provided ∂ I f I = 0,
f I k iI = 0,
i f I k[I,J ] =0
for each i. However, the gauge invariance (4.12) is now modified, as {Q, u} = J + h i r i si .
(8.2)
(8.3)
i
If all the weights h i vanish, then J is BRST trivial for any (Z ) of zero weight, and δf I = ZI
(8.4)
changes V f by a BRST trivial term. However, if any of the weights h i are non-zero, then the extra ghost terms in (8.3) mean that (8.4) is not a symmetry. This is just as well, as the constraints (8.2) are only invariant under (8.4) if all the h i are zero. The vertex operator Vg = g I (Z )∂ Z I is physical provided Z I g I = 0,
(8.5)
δg I = ∂ I χ , δg I = ηi k iI
(8.6)
and it has the gauge invariances
for any χ (Z ) and any ηi (Z ) of weights −h i . The Yang-Mills vertex operator Vφ = jr φ r (Z ) receives no further constraints from the gauging. In the following the spectrum will be analysed for the anomaly-free strings of the last section in the flat case. The twistor space is PT[N ] = PT[N ] − I and results from removing the appropriate (super)line I (which is I = CP1|0 or I[N ] = CP1|N in the complex case, and RP1|0 × RP1|0 or RP1|N × RP1|N in the real case) from CP3|N or RP3|N × RP3|N . The vertex operators live on the boundary of the world-sheet, which in turn lies in RP3|N . 8.2. Self-dual gravity without supersymmetry. Consider first the bosonic N = 0 theory of §7.1 with the one-form k = Iαβ Z α d Z β
(8.7)
on the twistor space PT = CP3 − CP1 (or PTR = RP3 − RP1 in the real case), so that kα = −Iαβ Z β , k[α,β] = −Iαβ .
(8.8)
The coordinates on twistor space are Z α = (ω A , π A ) and
k = A B π A dπ B .
(8.9)
Then f I = ( f A , f A ) are of degree one and the constraints (8.2) imply ∂f A = 0, ∂ω A
f A = 0,
(8.10)
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which in turn imply f A = AB
∂h ∂ω B
(8.11)
for some twistor function h(Z ) homogeneous of degree 2. Via the twistor transform, this corresponds to a space-time field of helicity 2 satisfying the field equations of linearised Einstein gravity [26]. The 1-form g = gα dZ α in the vertex operator gα ∂ Z α satisfies Z α gα = 0, which means that gα is defined on the projective twistor space, and moreover it follows from (8.6) that it is defined up to two gauge freedoms: gα → gα + ∂α χ , gα → gα + Iαβ Z β η.
(8.12)
The four components of gα are subject to one constraint and two gauge invariances, and the remaining degree of freedom is conveniently represented by a function f˜ of homogeneity degree −2 defined by h˜ = I αβ ∂α gβ = AB ∂ A g B ,
(8.13)
which is invariant under the two gauge transformations given in (8.12). This function of degree −2 corresponds to a space-time scalar field. Finally, the Yang-Mills vertex operator with functions φr (Z ) of degree zero gives states of helicity +1 in the adjoint of the gauge group G. Thus the spectrum of this theory consists of a state of helicity +2, a scalar state of spin 0 and dim(G) states of helicity +1. Note that the state of spin zero could come from a scalar field or a 2-form gauge field. An interacting theory with this spectrum is self-dual gravity coupled to self-dual Yang-Mills and a scalar (or 2-form gauge field), and this has covariant field equations but no covariant action. In the absence of the scalar, the field equations would be R = ∗R,
F = ∗F,
(8.14)
where R is the curvature 2-form, F is the Yang-Mills field strength and ∗ denotes the Hodge duality operation. Finding out whether this interacting theory arises, and finding the form of the scalar coupling, requires investigating the interactions arising from string amplitudes. This will be discussed elsewhere.
8.3. Supergravity with N supersymmetries. Consider next the case of §7.2, with projective twistor space PT[N ] of dimension 3|N (given by CP3|N − CP1|0 , or RP3|N − RP1|0 in the real case) that is fibred over CP1|N , and the gauging associated with the bosonic one-form (8.9) and the N fermionic one-forms
k a = dψ a − eaA dπ A .
(8.15)
The vector field f I decomposes as f I = ( f α , f a ) = ( f A , f A , f a ) and the conditions (8.2) imply ∂f A = 0, ∂ω A
f A = 0,
f a = 0,
(8.16)
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and again f A = AB
∂h ∂ω B
(8.17)
for some super-twistor function h(Z ) homogeneous of degree 2. Consider first the case N = 4. Then h(Z ) has an expansion h(Z I ) = g(Z α ) + χa (Z α )ψ a + Aab (Z α )ψ a ψ b + d (Z α )abcd ψ a ψ b ψ c + ϕ(Z α )abcd ψ a ψ b ψ c ψ d , (8.18) where Z α = (ω A , π A ) are the coordinates on bosonic twistor space. This gives twistor fields g, χa , Aab , abc , ϕ in O(2), O(1), O(0), O(−1), O(−2) respectively. Via the twistor transform, these correspond to space-time fields of helicities 2, 3/2, 1, 1/2, 0 in the S L(4, R) representations (1, 4, 6, 4 , 1) respectively. We then obtain the following µ positive helicity fields in space-time: a graviton gµν , four gravitini χa , six helicity one µ fields Aab , four helicity half fields abc and a scalar ϕ. These satisfy the field equations of linearised N = 4 supergravity. For general N , one again has an expansion h(Z I ) = g(Z α ) + χa (Z α )ψ a + Aab (Z α )ψ a ψ b + · · ·
(8.19)
terminating with a term of order ψ N , giving twistor fields in O(2), O(1), . . . , O(2 − N ) corresponding to space-time fields of helicities 2, 3/2 . . . , 2 − (N /2) in the S L(N , R) representations (1, N, N(N − 1)/2, . . . , N , 1) respectively. For the vertex operator g I ∂ Z I , g I = (g A , g A , ga ) and the symmetry (8.6) with the one-forms k a can be used to set ga = 0. Then (8.13) again defines a function of homogeneity degree −2 that is invariant under the remaining symmetries, and gives rise to the conjugate multiplet to the one obtained from f . For N = 4, this is ˜ I ) = g(Z h(Z ˜ α )abcd ψ a ψ b ψ c ψ d + χ˜ d (Z α )abcd ψ a ψ b ψ c ˜ a (Z α )ψ a + ϕ(Z + A˜ ab (Z α )ψ a ψ b + ˜ α ),
(8.20)
˜ abc , ϕ˜ in O(−6), O(−5), O(−4), O(−3), O(−2) giving twistor functions g, ˜ χ˜ a , A˜ ab , corresponding to helicities −2, −3/2, −1, −1/2, 0 with multiplicities 1, 4, 6, 4, 1 respectively. For general N , this gives twistor fields in O(−2 − N ), . . . , O(−3), O(−2) corresponding to helicities −N /2, . . . , −1/2, 0. Finally, the Yang-Mills sector is represented by a function of degree zero in supertwistor space, corresponding to helicities 1, 1/2, . . . , −N /2 in the S L(N , R) representations (1, N, N(N − 1)/2, . . . , N , 1), and for N > 4, there are higher-spin fields with helicities less than −1. For N = 4 this is the spectrum of N = 4 supergravity coupled to N = 4 superYang-Mills. For N < 4, this is a self-dual supergravity theory coupled to self-dual Yang-Mills. Interacting self-dual supergravity theories in 2 + 2 dimensions have been discussed in [35–40]. For N > 4, we find multiplets with spins greater than two, and with more than one state of helicity −2. Free theories can be written down for all these spectra, but the possibilities for interactions are more limited. However, there is the intriguing possibility of self-dual interactions for these theories, as the usual higher-spin inconsistencies are absent for certain self-dual theories. The possibility of interactions will be discussed in Sect. 10.
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8.4. N = 8 supergravity. Consider the theory of §7.4 formulated in N = 4 supertwistor space with the gauging for the single weightless 1-form kˆ = w(Z )I I J Z I d Z J ,
(8.21)
where w is of degree −2. We need only assume a fibration over CP1|0 , so that the flat twistor space can be taken to be PT[4] = CP3|4 − CP1|4 (or the real analogue thereof). We choose w = w(π ) so that kˆ is closed, d kˆ = 0. For real twistors Z , Z˜ , the function w could be chosen as in (7.16), and for complex ones as in (7.17). Starting with the vector field f I , we work through the various conditions and gauge equivalences as follows. In this case, the constraints (8.2) are weaker than in §8.3 as d kˆ = 0, but there is now a gauge invariance of the type (8.4) since the form has weight h = 0. We set f I = ( f α , f a ) = ( f A , f A , f a ). We fix the gauge freedom f I → f I + Z I from Eq. (8.4) by requiring that ∂f A = 0, ∂ω A
(8.22)
which in turn implies f A = AB
∂h ∂ω B
(8.23)
for some twistor function h(Z ) homogeneous of degree 2. This has the expansion (8.18) and gives the space-time fields of helicities 2, 3/2, 1, 1/2, 0 of the positive helicity N = 4 supergravity multiplet. For w = w(π ), (8.21) implies kˆ[I,J ] = 0, so that the constraints (8.2) give
∂ I f I = 0 w(π ) f A π A = 0,
(8.24)
implying that f A = π A λ for some λ of homogeneity degree −1. The function λ can be understood to be determined in terms of the f a by the condition ∂ I f I = 0 (cf. Eq. (8.2)) and so λ does not represent any independent degrees of freedom. We expand the f a to obtain e f e = χ e (Z α ) + Aae (Z α )ψ a + ab (Z α )ψ a ψ b + ϕ ea (Z α )abcd ψ b ψ c ψ d ˜ e (Z α )abcd ψ a ψ b ψ c ψ d . + (8.25)
We have used the same symbol as in Eq. (8.18) to denote fields of the same helicity. Equation (8.25) gives four gravitino multiplets, each with states of helicities 3/2, 1, 1/2, 0, −1/2, and so leads to a further four gravitini, sixteen helicity one fields, twenty four helicity one half fields, sixteen scalars and four helicity minus one half fields. The 1-form g = g I dZ I in the vertex operator g I ∂ Z I satisfies Z I g I = 0, which means that g I is defined on the projective twistor space; moreover g I is defined up to two gauge freedoms: g I → g I + ∂ I χ , g I → g I + w I I J Z J η.
(8.26)
We define a gauge-invariant function h˜ of homogeneity degree −2 by (8.13) and this again gives rise to the conjugate supergravity multiplet with helicities −2,−3/2,−1, −1/2,0 and multiplicities 1, 4, 6, 4, 1 respectively.
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The fermionic components ga contribute further states to the spectrum. In order to see this and find the full spectrum, we write g I = (gα , ga ) = (g A , g A , ga ). The gauge freedom g I → g I + ∂ I χ can be fixed by imposing the gauge condition g A π A = 0. This implies g A = π A ξ for some ξ which can then be set to zero by use of the gauge freedom δg I = I I J Z J η. Consider next the two degrees of freedom in g A . One is the component ω A g A , which is determined in terms of the ga by the final constraint Z I g I = 0 (cf. (8.5)) and so is not an independent degree of freedom. This leaves one degree of freedom represented by the gauge-invariant function h˜ given by (8.13), corresponding to the negative helicity N = 4 supergravity multiplet. ˜ determine the The remaining components ga are unconstrained and, together with h, gauge fixed conponents of g I . The ga can be expanded as ge = χ˜ e (Z α )abcd ψ a ψ b ψ c ψ d + A˜ de (Z α )abcd ψ a ψ b ψ c ˜ eab (Z α )ψ a ψ b + ϕ(Z + ˜ α )ea ψ a + (Z α )e .
(8.27) f a.
This gives four negative helicity gravitino multiplets, conjugate to those from Note that the spectrum is independent of the choice of w(π ). Combining all the positive and negative helicity states, we obtain a spectrum consisting of a graviton h µν , 8 gravitini, 22 vector fields, 32 spin-half fields abc and 34 scalars. This is six N = 4 vector multiplets short of the full N = 8 supergravity spectrum. In addition, the YangMills vertex operator gives vector multiplets in the adjoint of some group G. If G is six-dimensional, then the spectrum of N = 8 supergravity is obtained. 8.5. N = 4 supergravity coupled to super-Yang-Mills. Consider the theory of §7.4 formulated in N = 4 super-twistor space with the gauging for the weightless 1-form kˆ = w(Z )I I J Z I d Z J ,
(8.28)
where w is of degree −2, and the four weightless 1-forms kˆ a = w (Z )(dψ a − eaA dπ A ),
(8.29)
where w is of degree −1. We assume a fibration over CP1|4 , so that the flat twistor space can be taken to be PT[4] = CP3|4 − CP1|0 (or the real analogue thereof). It will be ˆ kˆ a are closed, and that they have no zeroes or assumed that w, w are chosen so that k, poles on the boundary space defined by the boundary condition Z = Z˜ (which is RP3|4 for the Lorentzian world-sheet theory). It was shown in the previous subsection that the constraints from kˆ imply that the vertex operator V f is determined by a function h(Z ) of degree 2 and four functions f a of degree 1, while Vg is given in terms of a function ˜ ) of degree −2 and four functions ga of degree −1. The constraints f I kˆ a = 0 from h(Z I the fermionic 1-forms give w f a = 0;
(8.30)
= Z = Z˜ , while the symmetry this implies that δg I = ηa kˆ aI can be used to set ga = 0. In this way the gravitino multiplets are eliminated, ˜ ) of degree −2, and this gives the leaving the twistor functions h(Z ) of degree 2 and h(Z spectrum of N = 4 supergravity. In addition, the vertex operators Vφ give the spectrum of N = 4 super-Yang-Mills with gauge group G, so the spectrum of N = 4 supergravity coupled to N = 4 super-Yang-Mills is obtained. fa
0 as w is chosen to have no zeroes on
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8 and N
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4 Supergravity
The scattering amplitudes for the Berkovits string, calculated from open string correlation functions with vertex operators V f , Vg , Vφ inserted on the world-sheet boundary, give rise to nontrivial scattering amplitudes and hence to interactions for the space-time fields [2,3,6]. The n-point tree-level amplitude is given by the formula [2,3] cV1 (σ1 )cV2 (σ2 )cV3 (σ3 ) dσ4 V4 (σ4 ) . . . dσn Vn (σn )R , (9.1) d
d
where Vi are any of the vertex operators V f , Vg , Vφ and . . . d is the correlation function on a disc of degree d, corresponding to a gauge instanton on the disc with a topologically non-trivial configuration for the gauge field A characterised by the integer d [3]. The coordinates are written as Z I = ρ Zˆ I , where ρ is a scale factor (which is complex for complex Z ), and a BRST-invariant operator R is R = δ(ρ − 1)v + · · · .
(9.2)
This has the property that it gives an insertion of the zero-mode of the ghost v, so that the integration over v is non-zero, and regulates the integral over ρ. (Changing the insertion point σ0 changes R(σ0 ) by a BRST exact term, so that the amplitude is independent of σ0 .) Integrating out ρ, v leaves an amplitude defined on a ‘small Hilbert space’ of G L(1)neutral states independent of the v zero-mode, giving results defined on the projective twistor space [3]. Consider now the new theories based on weightless forms of §7.4, §8.4, §8.5, corresponding to N = 8 supergravity or N = 4 supergravity coupled to super-Yang-Mills. These new string theories are similar to the Berkovits string, and the twistor fields Y, Z have the same world-sheet dynamics and the same vertex operators. However, there is an additional ghost sector and the extra terms in the BRST operator give extra constraints and extra gauge invariances for the twistor wave-functions f I , g I , while there are no further constraints or invariances for the Yang-Mills wave-functions φr . In the N = 8 theory, there is an extra anti-commuting ghost s of conformal weight zero, which has one zero mode on the disc, so that one insertion of the s zero-mode is needed to obtain a non-zero amplitude. For any BRST-invariant vertex operator cV , scV is also BRSTinvariant, so that a non-zero amplitude is given by replacing e.g. cV1 (σ1 ) with scV1 (σ1 ) in (9.1). Upon integrating over the s zero-mode, the amplitude (9.1) is recovered. For the N = 4 theories of Sect. 8.5, there is in addition one zero-mode for each of the four commuting ghosts s a , and the integral over these can be handled by choosing appropriate pictures for the vertex operators Vi . A convenient choice is to replace cV1 (σ1 ) with sδ 4 (s a )cV1 (σ1 ) in (9.1). Again, on integrating out the ghost zero modes s, s a , the formula (9.1) is recovered. As a result, after integrating out the zero-modes of the new ghosts, the tree-level correlation functions for the N = 4 and N = 8 theories of §8.4 and §8.5 have the same form as for the Berkovits string in [2,3,6] when written in terms of f I , g I , φr . However, in our case these wave-functions are subject to further constraints and have further gauge invariances. As we have seen, these can be used to write f I , g I in terms of the unconstrained wave-functions h, h˜ (defined by (8.17),(8.13)) for the N = 4 theory, or ˜ f a , ga for the N = 8 theory. These are wave-functions for supergravity and matter h, h, systems whose field equations are of 2nd order in space-time derivatives for bosons (1st order for fermions), not those for conformal supergravity with 4th order equations
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˜ f a , ga , the scattering amplitudes of for bosons. When written in terms of h, h˜ or h, h, the new twistor strings should then give interactions for Einstein gravitons and matter. These will be systematically investigated and compared with known gravity amplitudes elsewhere, but it is straightforward to see that non-vanishing amplitudes are obtained in certain examples, confirming that these theories have non-trivial interactions, and moreover we can compare these with the known MHV gravity amplitudes. We now check this for tree-level amplitudes at degree zero by first calculating amplitudes in terms of f I , g I using the procedure described in [3,6], and then writing these in terms of the h, h˜ defined by (8.17) and (8.13). The Yang-Mills amplitudes are the same as for the Berkovits string. At degree zero, the amplitudes Vg Vg Vg , V f Vg Vg vanish automatically. Now consider the amplitude V f1 V f2 Vg3 . Following the procedure given in [6], we obtain the formula V f1 V f2 Vg3 = s f 1I f 2J ∂[I g3J ] , (9.3) RP3|4
where s is the volume form on RP3|4 . Briefly, this formula follows upon identifying the open string worldsheet with the upper-half complex plane, inserting open string vertex operators on the real axis, and evaluating the correlation function V f1 (σ1 )V f2 (σ2 ) Vg3 (σ3 ) of three vertex operators given in terms of the f I and g I by V f = Y I f I (Z ) and Vg = ∂ Z I g I (Z ). This correlation function is computed by taking contractions and using the OPE, δ JI . (9.4) σ1 − σ2 The contractions give rise to a factor of (σ1 −σ2 )(σ2 −σ3 )(σ3 −σ1 ) in the denominator that cancels an identical factor in the numerator coming from the integral over zero-modes of the conformal ghost c. The result is then integrated over the space of zero-modes of the fields Z I (σ ), which are just constant maps from the disc to twistor space, giving an integral over RP3|4 . To obtain the formula (9.3), one also needs to integrate certain terms by parts and use the fact that ∂ I f I = 0. Furthermore, it can be checked that, for our vertex V f with ∂h i f iI = AB B , 0, 0 , i = 1, 2, 3, (9.5) ∂ω Z I (σ1 )Y J (σ2 ) ∼
the formula for the remaining amplitude V f1 V f2 V f3 given in [6] (Eq. (5.10) of that paper) yields 1 × V f1 V f2 V f3 = (σ1 − σ2 )(σ2 − σ3 )(σ3 − σ1 )
∂h ∂h 2 ∂h 3 1 s AB C D E F − C B E D AF . E B A D 3|4 ∂ω ∂ω ∂ω ∂ω ∂ωC ∂ω F RP (9.6) We now focus on the amplitudes between two positive helicity and one negative helicity graviton states so we consider the case in which the wave functions are given in ˜ We choose terms of functions h, h.
∂h 1 ∂h 2 4 f 1I = AB B , 0, 0 , f 2I = AB B , 0, 0 , g3I = g3A a=1 ψ a , 0, 0 , ∂ω ∂ω (9.7)
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where h 1 , h 2 and g3A are functions of the bosonic twistor coordinates Z α alone, g3A has weight −5 and AB
∂ g3B = h˜ 3 , ∂ω A
(9.8)
where h˜ 3 has homogeneity degree −6. Performing the integrals over the odd variables, the integral (9.3) now becomes ∂ ∂ AB V f1 V f2 Vg3 = s h1 h 2 h˜ 3 , (9.9) ∂ω A ∂ω B RP3 where is the volume form on RP3 . We now take h 1 , h 2 , and h˜ 3 to be momentum A A A A eigenstates with momenta Pi = pi pi , i = 1, 2, 3:
ω A Pi A A α A h i = exp πB α B
π A α A p1B α B
3
δ(π A p1A )
(9.10)
for i = 1, 2 and
ω A Pi A A α A h˜ 3 = exp πB α B
π A α A p1B α B
−5
δ(π A p1A ).
(9.11)
Here α A is a fixed spinor on which the representatives (9.10) and (9.11) in fact do not depend (see e.g. [1,41]). The integral (9.9) can now be done; after some delta-function manipulations, this yields the standard formula for the three point MHV amplitude for gravity in split signature (or in Lorentz signature with complex momenta) [9–11]: 6
p1A p2A 4 V f1 V f2 Vg3 = δ (P1 + P2 + P3 )
(9.12) 2
2 . p3B p1B p2C p3C Thus the new N = 4 and N = 8 twistor string theories each have at least one non-trivial interaction, and this gives precisely the helicity (++−) 3-graviton interaction of Einstein gravity. Under scaling the infinity twistor I I J → R I I J , AB → R AB , so that if f I , g I are ˜ Then the amplitude scales as R −1 , so that kept fixed, then h → R −1 h and h˜ → R h. −1 R sets the strength of the gravitational coupling. 10. Discussion In this paper, a number of new twistor string theories have been constructed. They were shown to be free from perturbative world-sheet anomalies, and the ghost-independent part of the spectra in space-time have been found. The full BRST cohomology including ghost-dependent vertex operators will be discussed elsewhere. The key questions that remain are whether these give fully consistent quantum theories, and whether they have non-trivial interactions. We have seen in Sect. 9 that non-vanishing 3-point supergravity amplitudes are obtained in the N = 4 and N = 8 cases, so these theories have non-trivial interactions. Other amplitudes for these theories, and those for the other theories, will be discussed elsewhere.
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The string theories giving the N = 4 and N = 8 theories involve arbitrary functions w, w of homogeneity −2 and −1 respectively. These can be chosen to be non-singular for the theory with Lorentzian world-sheet and independent real coordinates Z , Z˜ (with target space RP3|4 × RP3|4 in the flat case) and for the Wick-rotated version of this with Euclidean world-sheet and independent complex coordinates Z , Z˜ (with target space CP3|4 × CP3|4 in the flat case). There is also a theory with Euclidean world-sheet obtained from this by setting Z˜ = Z ∗ (with target space CP3|4 in the flat case); in this case, we can choose w, w to be non-singular on the disc but complex on the boundary, resulting in a modification of the boundary conditions for the ghosts, or we can choose w, w to be real on the boundary but singular on the disc. With the latter choice, however, the gauging of the weightless one-forms may be problematic. The N = 4 and N = 8 theories then arise from the real theory with Lorentzian world-sheet and real Z , Z˜ , while the amplitudes are calculated using the Euclidean version of this. The Berkovits twistor string gives a theory of N = 4 superconformal gravity coupled to N = 4 super-Yang-Mills for any gauge group that can arise as a current algebra of a c = 28 conformal field theory. However, it is known that N = 4 superconformal gravity coupled to N = 4 super-Yang-Mills has an SU (4) (or S L(4, R) in split signature) R-symmetry anomaly that cancels only if G is 4-dimensional [42,43], so G = SU (2) × U (1) or U (1)4 . This is so for the theory with minimal kinetic term W 2 , but a similar result is expected to apply for the theory with non-minimal kinetic term e−2 e2W arising from the twistor string [6]. This suggests that the Berkovits string may only be consistent at loops for special gauge groups, and that there are constraints and potential inconsistencies that have not yet been found. In [6], it was suggested that these may come from open string tadpole cancellation. At loops, there may be interactions with a closed string sector, and further issues could arise from closed strings. (Closed string vertex operators are constructed from products of left-moving and right-moving vertex operators, so that one might expect the closed string spectrum to be related to the tensor product of the open string spectrum with itself. The twistor space spectrum appears to be the tensor product of that for open strings, but it is not clear what this means for the space-time spectrum, as the conventional Penrose transform does not apply to non-holomorphic fields (Z , Z˜ ).) The new string theories described here have the same form as the Berkovits string, but with extra terms in the BRST operator. It is therefore to be expected that for these theories, too, there will be further constraints that will eliminate some models. We do not understand these constraints from the string theory perspective, but some clues might be obtained from the corresponding space-time theories. The new theories have different symmetries from those of conformal supergravity (for example, they do not have a gauged R-symmetry or a conformal symmetry) and so they will have different anomalies, and different constraints from anomaly cancellation. Interestingly, there are supersymmetric theories which can be defined in 2 + 2 dimensions that have no analogue in 3 + 1 dimensional space-time, and the spectra of some of these arise here. First, the theory of Sect. 8.2 has the spectrum of self-dual gravity coupled to self-dual Yang-Mills and a scalar (or 2-form gauge field). Consistent non-linear interactions are possible classically for this theory, with field equations given by some scalar-dependent modification of (8.14). There is no covariant action for such field equations, but there are non-covariant actions of the type proposed by Plebanski [44]. The theory is a chiral one in 2 + 2 dimensional space-time, and so it is prone to potential anomalies. An interacting theory of self-dual gravity coupled to self-dual Yang-Mills in 2 + 2 dimensions arises from the N = 2 string [14], and this is believed to be a consistent quantum theory
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(however, see [45,46]). This suggests the intriguing possibility that the N = 0 twistor string found here could be dual to an N = 2 string theory. A string theory with the spectrum of self-dual gravity coupled to self-dual Yang-Mills and a 2-form gauge field is given by the N = 2 string whose target space is generalised Kähler [53]; this is obtained by coupling the (2, 2) supersymmetric sigma-model with torsion [54] to N = 2 worldsheet supergravity. The theories of Sect. 8.3 with N < 4 give supersymmetric extensions of this bosonic theory with self-dual supergravity coupled to self-dual super-Yang-Mills and N supersymmetries, and these could be consistent non-trivial theories if the N = 0 theory is. For N = 4, we have two twistor theories, both of which have the spectrum of N = 4 supergravity coupled to N = 4 super-Yang-Mills. One is the theory of §8.3 with N = 4 (for any gauge group that can arise as a current algebra of a c = 22 conformal field theory) and the other is the theory of §8.5. However, there are a number of different supersymmetric theories with this spectrum, and the question we now turn to is which of these arises in the twistor string. Consider first the Yang-Mills sector, for which there is the free theory and two possible interacting supersymmetric theories. For N = 4 Yang-Mills, there is the standard non-chiral theory, which can be rewritten in the Chalmers-Siegel form [47] with Yang-Mills kinetic term E F + E 2 , where E is a self-dual 2-form and F = d A + A2 is the usual Yang-Mills field strength. There is also Siegel’s chiral theory with Yang-Mills kinetic term E F [48]. This is sometimes called a self-dual theory, but it has the same spectrum as the usual super-Yang-Mills theory. It differs from the usual theory in that the interactions are chiral, i.e. they are not symmetric under the parity transformation interchanging positive and negative helicities, and the action is linear in the negative helicity fields (such as E). The full non-chiral N = 4 super-Yang-Mills theory is obtained in the Berkovits string, and the same is true for our N = 4 theory as it is the same as that of Berkovits in the Yang-Mills sector. The supergravity sector has the spectrum of N = 4 Einstein supergravity, and we have seen that it has at least one non-trivial interaction. Just as for Yang-Mills, there is the possibility of either the standard non-chiral theory or of one with chiral interactions. A formulation of Einstein gravity with chiral interactions was discussed in [17,40]. The fields consist of a vierbein eµ a (the analogue of the Yang-Mills connection A) and an ab which is anti-self-dual in the Lorentz indices independent Lagrange multiplier field ωµ ab ab (the analogue of the anti-self-dual Lagrange multiplier field E). The multiplier ωµ imposes the constraint that the anti-self-dual part of the Levi-Civita spin-connection (e) constructed from e vanishes, so that the corresponding curvature is self-dual. An N = 4 supersymmetric version of this theory was given by Siegel [40], with component action given by truncating the N = 8 component action of ref. [40]. To determine whether the free, chiral or the non-chiral interacting N = 4 supergravity arises from the two N = 4 string theories requires further analysis of the scattering amplitudes, and we will return to this elsewhere. However, the theory of §8.5 has the usual non-chiral Yang-Mills interactions and has a non-trivial cubic gravitational coupling, so it is presumably the full Yang-Mills theory coupled to either chiral or non-chiral N = 4 supergravity. The usual non-chiral interacting N = 4 supergravity coupled to Yang-Mills theory has no anomalies, but it is expected to have ultra-violet divergences. Nonetheless, it has a limit in which gravity decouples to leave N = 4 super-Yang-Mills, and this is believed to be a consistent ultra-violet finite field theory. The theory of chiral N = 4 supergravity coupled to N = 4 super-Yang-Mills is likely to have better ultraviolet behaviour than the full supergravity (and might conceivably be finite) and it has a similar decoupling limit so that, whichever supergravity theory arises, there should
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be a decoupling limit giving pure N = 4 super-Yang-Mills amplitudes. This limit in the twistor theory is given by scaling the infinity twistor so that I I J → 0. Then from (1.3), for any supergravity wave-function h, the corresponding f α will vanish and so any amplitude involving h will vanish. It will be interesting to check that this leads to a full decoupling of gravity at all orders in perturbation theory. There is then the intriguing possibility that this twistor string can give N = 4 super-Yang-Mills in this limit. For the N = 4 supergravity and Yang-Mills theories, a relation with N = 2 strings has also been suggested in [48,49], and again there is the possibility of a link between our twistor strings and an N = 2 string theory. A relation between Siegel’s N = 4 supersymmetric N = 2 string and a different twistor string theory was suggested in [50]. Next, consider the theory of Sect. 8.4, giving the spectrum of N = 4 supergravity plus four N = 4 gravitino multiplets, together with super-Yang-Mills (for any gauge group that can arise as a current algebra of a c = 26 conformal field theory). There are then 8 gravitini of helicity +3/2 and 8 gravitini of helicity −3/2, so that the theory should be an N = 8 supergravity theory. Again, there is the possibility of either a theory with chiral interactions, or a non-chiral one. (There is also the possibility of a free theory.) If it is a standard non-chiral N = 8 supergravity, the total number of vector fields should be 28 and this requires the number of Yang-Mills multiplets to be six. This suggests that, if the twistor string gives a consistent non-chiral theory, there must be a constraint fixing the number of vector multiplets to be 6. The Berkovits string is expected to have a constraint fixing the number of vector multiplets to be 4, to cancel the anomalies of conformal supergravity, and both constraints could arise in the same, as yet unknown, way. Alternatively, the theory arising could be Siegel’s chiral N = 8 supergravity [40], in which the negative helicity fields appear linearly. In [40], Siegel argued that the N = 2 string gives N = 4 chiral Yang-Mills from the open string sector and N = 8 chiral supergravity from the closed string sector, and that the chirality of the interactions implied that the supergravity and super-Yang-Mills fields do not couple, so that one can consistently have N = 8 chiral supergravity and an arbitrary number of N = 4 chiral Yang-Mills multiplets. It will be interesting to see whether either of these interacting N = 8 supergravity theories arise here. If the space-time theories arising from the perturbative string theory are chiral supergravities, then it is possible that non-perturbative effects could give rise to the non-chiral interactions, as they do for Yang-Mills in Witten’s topological twistor string [1]. Finally, for the models of Sect. 8.3 with N > 4, the spectrum is chiral with states of spin greater than 2, and with more than one state of spin 2. It is believed that there are no chirally-symmetric theories with spins higher than 2 or with more than one graviton which have non-trivial interactions, but the no-go theorems do not apply to chiral theories. Consider first the N > 4 Yang-Mills theories, with helicities 1, 1/2, . . . , −N /2 in the S L(N , R) representations (1, N, N(N − 1)/2, . . . , N , 1), and all in the adjoint of the Yang-Mills gauge group, so that for N > 4 there are negative helicity states of spin greater than one. The field equation for a free massless field A1 A2 ...An of helicity −n/2 is
∇ B A1 A1 A2 ...An = 0.
(10.1)
For a field in a representation of the gauge group, the corresponding field equation is (10.1), where ∇ is the Yang-Mills covariant derivative. For n ≥ 2 this is consistent only if the Yang-Mills connection is self-dual, FA B = 0.
(10.2)
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The chiral N = 4 theory is of this type, with self-dual Yang-Mills coupled to a field E A B with field equation of the form (10.1). There are then consistent chiral interactions for the N > 4 Yang-Mills multiplets of this type provided the Yang-Mills equation is the self-duality condition (10.2). It remains to investigate whether such interactions can be supersymmetric, and we will return to this elsewhere. For N > 4, the chirality of the spectrum will mean that it is unlikely that there will be a covariant action. Similar considerations apply to the N > 4 supergravities arising from the twistor strings, in which there are negative helicity states of spin greater than two. The field equation for a free massless field of helicity −n/2 is again (10.1), but with ∇ denoting the gravitational covariant derivative. In curved space, this has an integrability condition for n > 2 (the Buchdahl constraint) given by ψ˜ A B C D = 0,
(10.3)
where ψ˜ A B C D is the anti-self-dual part of the Weyl-curvature. For Lorentzian signature, this would imply that space-time is conformally flat, but for Euclidean or split signatures, non-trivial conformally self-dual spaces are possible. A free field of helicity −n/2 can then be consistently coupled to conformally self-dual gravity. Self-dual supergravities for N ≤ 8 have been given in [40], and it is to be expected that these can be coupled to the free supermultiplet with helicities 0, −1/2, . . . , −N /2. Such theories could provide consistent interactions for the space-time theory arising from the N ≤ 8 twistor strings, with the self-dual supergravity fields arising from the twistor field f and the negative helicity multiplet from the twistor field g. For N > 8 supergravity, just as for N > 4 Yang-Mills, there are consistent interactions that can be written down and it remains to be seen whether these can be supersymmetric. Much remains to be done to investigate the interactions of the theories presented in this paper. It would be interesting to find and analyse super-twistor space actions, following [30,31], and to seek corresponding modifications of Witten’s topological twistor string that gave similar results. It is conceivable that some of the strings found here give free theories, and that others may be inconsistent. However, it is encouraging that suitable interacting supersymmetric space-time theories exist for many of the cases, and interesting that the interactions are typically chiral for N = 4, 8. However, the most promising theories are the N = 4 theory giving an interacting theory of supergravity coupled to super-Yang-Mills, and the one giving N = 8 supergravity. The N = 4 theory has a decoupling limit giving pure Yang-Mills, opening the prospect of a twistor string formulation of super-Yang-Mills loop amplitudes. Acknowledgements. We would like to thank Nathan Berkovits, Edward Witten, Parameswaran Nair, Roger Penrose, Simon Salomon, David Skinner, Kellogg Stelle and Arkady Tseytlin for helpful discussions; we are especially grateful to Edward Witten for his advice and remarks. MA thanks the Theory Division at CERN for hospitality and financial support. MA and LJM also thank the Institute for Mathematical Sciences at Imperial College London for hospitality and financial support. The work of MA was also supported in part by a PPARC Postdoctoral Research Fellowship with grant reference PPA/P/S/2000/00402, by the ‘FWOVlaandere’ through projects G.0034.02 and G.0428.06, by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P5/27, and by the European Union FP6 RTN programme MRTN-CT-2004005104. The work of LJM was also partially supported by the European Union through the FP6 Marie Curie RTN Enigma (contract number MRTN-CT-2004-5652) and by a Leverhulme Senior Research Fellowship.
A. Appendix: Relation Between Split Signature Constructions In this appendix, we continue our discussion in §3.1.1 of two distinct twistor constructions for space-times of split signature. In the first construction, we obtained a deformed
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twistor space PT with a complex conjugation τ : PT → PT whose fixed point set defined a real slice PTR , whereas in the second we considered a deformation PTR of the real slice PTR inside PT. Although the first construction is perhaps more intuitive, the second is more powerful and has a better conceptual fit with the Berkovits open twistor string model, so we will derive the first construction from the second. We will assume that we have obtained a twistor space PT by suitably gluing together the twistor spaces for small open sets in space-time, with the assumption that the space-time is S 2 × S 2 globally and admits an analytic conformal structure. This space is non-Hausdorff, and we give a brief description of it here. The second construction starts from the data of PTR ⊂ PT determined by Eq. (3.5): Z α − Z¯ α = i F α (Z β + Z¯ β ).
(A.1)
With the assumption of analyticity, F α can be analytically continued to become a holomorphic function F α (Z β ) on a neighbourhood containing TR (initially, F α (Z β ) was defined only for real values of Z α ). Thus Eq. (A.1) will make sense when Z¯ α is replaced by Z˜ α , where Z˜ α is close to, but not necessarily equal to Z¯ α . This gives the equation Z α − Z˜ α = i F α (Z β + Z˜ β ),
(A.2)
where now Z˜ α is an independent variable that is no longer the complex conjugate of Z α . For F α sufficiently small, this equation can be solved for Z˜ α in terms of Z β as Z˜ α = P α (Z β ) Pα.
(A.3) Zα
Z˜ α
for some invertible functions Since (3.5) was defined for ∈ TR and ∈ TR , the analytic continuation (A.2) will be defined for Z α in some neighbourhood V of TR ⊂ PT+ and, from the reality properties of (3.5), the P α will map V holomorphically onto the complex conjugate set V¯ ⊂ PT− . It follows from this definition that the real slice PTR is given by the subset of V on which Z˜ α = Z¯ α , since (A.2) then reduces to (3.5). We will construct PT by gluing together two copies of CP3 using P α (Z β ). We now take Z α to be holomorphic coordinates on one copy of CP3 , denoted PT+ , and Z˜ α to be coordinates on another copy denoted PT− . We construct PT by interpreting Eq. (A.3) as a patching relation for constructing a complex manifold by gluing the neighbourhood V ⊂ PT+ to V¯ ⊂ PT− . We note, however, that this global description is not Hausdorff. Furthermore, the full space PT admits a complex conjugation τ which interchanges PT+ and PT− so that τ maps the point Z α ∈ PT+ to the point Z˜ α = Z¯ α ∈ PT− and vice-versa. In order to see that this is well defined, we need to check that it is compatible with the patching (A.3); if Z α ∈ V then τ (Z α ) is the point in PT− with Z˜ α = Z¯ α , but Z α is identified with Z˜ α = P α (Z β ) in PT− whose conjugate point is Z α = P α (Z β ) in PT+ . For τ to give the same point in each case, we need to see that Z¯ α = P α (P β (Z γ )). This follows from the fact that (A.3) is equivalent to (A.2) and F α is a real function for real values of its argument, so that its analytic continuation satisfies F β (Z α + Z˜ α ) = F β ( Z¯ α + Z¯˜ α ). Thus (A.2) implies Z¯˜ α − Z¯ α = i F α ( Z¯ β + Z¯˜ β ), and this equation is the same as (A.2) except that the role of Z α has been taken by Z¯˜ α and that of Z˜ α by Z¯ α . Thus we have Z¯ α = P α ( Z¯˜ β ) = P α (P β (Z γ )) as desired.
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Given a holomorphic disc Dx in PT+ with boundary on PTR , we can define the Riemann sphere CP1x = Dx ∪ τ (Dx ) in PT since τ fixes PTR and hence glues the boundary of Dx to that of τ (Dx ). It is a standard theorem in complex analysis that this embedding will actually be holomorphic along ∂ Dx as well as over the interiors of Dx and τ (Dx ). We can carry out the non-linear graviton construction on PT and construct the space CM of Riemann spheres in PT in the same family as CP1x . This will be four complex dimensional as before, and admit a holomorphic conformal structure that is anti-selfdual. The anti-holomorphic involution τ on PT takes Riemann spheres to Riemann spheres, and so it induces a complex conjugation on CM that preserves the conformal structure; thus it fixes a real slice M ⊂ CM on which the conformal structure is real. The points of the real slice correspond to Riemann spheres in PT that are mapped to themselves by the anti-holomorphic involution. Such Riemann spheres contain an equatorial circle that is fixed by the involution, and which must lie in the fixed points PTR in PT . Thus such a Riemann sphere corresponds to a pair of holomorphic discs in PT with common boundary on PTR and conversely a disc D gives rise to the Riemann sphere D ∪ τ (D) as described above. References 1. Witten, E.: Perturbative gauge theory as a string theory in twistor space. Commun. Math. Phys. 252, 189258 (2004) 2. Berkovits, N.: An alternative string theory in twistor space for N = 4 super-Yang-Mills. Phys. Rev. Lett. B93, 011601 (2004) 3. Berkovits, N., Motl, L.: Cubic twistorial string field theory. JHEP 0404, 056 (2004) 4. Parke, S., Taylor, T.: An amplitude for N gluon scattering. Phys. Rev. Lett. 56, 2459 (1986) 5. Berends, F.A., Giele, W.T.: Recursive calculations for processes with N gluons. Nucl. Phys. B306, 759 (1988) 6. Berkovits, N., Witten, E.: Conformal supergravity in twistor-string theory. JHEP 0408, 009 (2004) 7. Cachazo, F., Svrˇcek, P.: Lectures on twistor strings and perturbative Yang-Mills theory. Proc. Sci. RTN2005, 005 (2005) 8. Berends, F.A., Giele, W.T., Kuijf, H.: On relations between multi-gluon and multigraviton scattering. Phys. Lett. B 211, 91 (1988) 9. Giombi, S., Ricci, R., Robles-Llana, D., Trancanelli, D.: A note on twistor gravity amplitudes. JHEP 0407, 059 (2004) 10. Nair, V.P.: A note on MHV amplitudes for gravitons. Phys. Rev. D 71, 121701 (2005) 11. Bjerrum-Bohr, N.E.J., Dunbar, D.C., Ita, H., Perkins, W.B., Risager, K.: MHV-vertices for gravity amplitudes. JHEP 0601, 009 (2006) 12. Penrose, R.: Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav. 7, 31 (1976) 13. Ward, R.S.: On selfdual gauge fields. Phys. Lett. A 61, 81 (1977) 14. Ooguri, H., Vafa, C.: Geometry of N = 2 strings. Nucl. Phys. B 361, 469 (1991) 15. Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Rimannian geometry. Proc. Roy. Soc. A362, 425 (1978) 16. LeBrun, C., Mason, L.J.: Nonlinear gravitons, null geodesics and holomorphic discs. http://arXiv.org/ list/math.DG/0504582, 2005 17. Abou-Zeid, M., Hull, C.M.: A chiral perturbation expansion for gravity. JHEP 0602, 057 (2006) 18. Huggett, S., Tod, K.: An introduction to twistor theory. Second Edition, Cambridge: Cambridge University Press, 1994 19. Mason, L.J.: Global anti-self-dual Yang-Mills fields in split signature and their scattering. To appear in Crelle’s journal, http://arXiv.org/list/math-ph/0505039, 2005 20. John, F.: The ultrahyperbolic differential equation with four independent variables. Duke Math. J., 4, 300–322, (1938) reprinted in 75 years of the Radon transform (Vienna 1992), Conf. Proc. Lecture Notes Math. Phys., IV, Cambridge, MA: International Press, 1994, pp. 301–323 21. Atiyah, M.F.: Geometry of Yang-Mills fields. Accademia Nazionale dei Lincei Scuola Normale Superiore, Pisa: Lezione Fermiane, 1979 22. Bailey, T.N., Eastwood, M.G., Gover, R., Mason, L.J.: The Funk transform as a Penrose transform. Math. Proc. Camb. Phil. Soc. 125(1), 67–81 (1999)
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Commun. Math. Phys. 282, 575 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0567-5
Communications in
Mathematical Physics
Erratum
The Paraboson Fock Space and Unitary Irreducible Representations of the Lie Superalgebra osp(1|2n) S. Lievens, N. I. Stoilova, , J. Van der Jeugt Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium. E-mail: [email protected]; [email protected]; [email protected] Published online: 4 July 2008 – © Springer-Verlag 2008
Commun. Math. Phys. 281, 805–826 (2008)
The original version of this article unfortunately contained a mistake. The spelling of N. I. Stoilova’s name was incorrect in the HTML version.
The online version of the original article can be found under doi:10.1007/s00220-008-0503-8. Permanent address: Institute for Nuclear Research and Nuclear Energy, Boul. Tsarigradsko Chaussee 72,
1784 Sofia, Bulgaria
NIS was supported by a project from the Fund for Scientific Research – Flanders (Belgium) and by project
P6/02 of the Interuniversity Attraction Poles Programme (Belgian State – Belgian Science Policy).
Commun. Math. Phys. 282, 577–623 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0561-y
Communications in
Mathematical Physics
A Spinor Approach to Walker Geometry Peter R. Law1 , Yasuo Matsushita2 1 4 Mack Place, Monroe, NY 10950, USA. E-mail: [email protected]; [email protected] 2 Section of Mathematics, School of Engineering, University of Shiga Prefecture,
Hikone 522-8533, Japan. E-mail: [email protected] Received: 11 January 2007 / Accepted: 12 February 2008 Published online: 15 July 2008 – © Springer-Verlag 2008
Abstract: A four-dimensional Walker geometry is a four-dimensional manifold M with a neutral metric g and a parallel distribution of totally null two-planes. This distribution has a natural characterization as a projective spinor field subject to a certain constraint. Spinors therefore provide a natural tool for studying Walker geometry, which we exploit to draw together several themes in recent explicit studies of Walker geometry and in other work of Dunajski [11] and Pleba˜nski [30] in which Walker geometry is implicit. In addition to studying local Walker geometry, we address a global question raised by the use of spinors.
1. Introduction Our conventions and notation for the tensor and exterior algebras and curvature are stipulated in Appendix One. Let (M, g) be a pseudo-Riemannian manifold of dimension n and D a parallel distribution of q-planes on M, i.e., D is invariant under paralleltransla- tion. Let Dm be the plane at m ∈ M. Write Dm = Nm ⊕ Q m , where Nm := ker ξg |Dm (where ξg is defined just after (A1.2)) and Q m is a nondegenerate linear complement of Nm . If Nm = 0R , then Dm is itself nondegenerate and possesses a unique orthogonal ⊥ , the latter forming a parallel (n − q)-distribution. The almost product complement Dm structure P corresponding to these complementary distributions is therefore orthogonal and parallel. Since ∇g is torsion free, P must be integrable and (M, g, P) is locally decomposable pseudo-Riemannian, i.e., M is locally product and each point m has a neighbourhood U with local coordinates (u α , x β ) which are simultaneously Frobenius for each distribution and with respect to which (U, g) is a pseudo-Riemannian product; see, for example, [47]. Walker [41] studied the case when Nm is nontrivial, showed one can choose local coordinates with respect to which the metric assumes a canonical form in [43], and treated the special case when Dm = Nm in [42]. When (M, g) is 2n-dimensional, g is
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of neutral signature, and there exists a parallel distribution of totally null n-planes, we shall call this very natural geometry (M, g, D) a Walker geometry. Walker geometry has proven of utility in several topics of independent interest: the holonomy Lie algebras of four-dimensional neutral metrics, [13]; isotropic Kähler metrics, [10] and [9]; the Osserman condition on the Jacobi operator, [2] and [8]; the Osserman condition on the conformal Jacobi operator, [4]; while Matsushita et al. [25] present a Walker geometry on an 8-torus which is a counterexample to the neutral analogue of the Goldberg conjecture, which asserts the integrability of the almost complex structure underlying an almost Kähler-Einstein Riemannian geometry on a compact manifold. Of course, the structure of a Walker geometry is very natural in the context of neutral geometry, and has been investigated for its own interest: [22,23] and [5]. In this paper, we show that four-dimensional Walker geometry has an intimate connexion with spinors and relate this fact to the local geometry of Walker four-manifolds. In the remainder of this introduction, we first make a few simple observations regarding arbitrary Walker geometry and then outline the subsequent sections of this paper. We first quote a result from [41]. 1.1. Lemma. A distribution D on a pseudo-Riemannian manifold is parallel iff ∇Y X ∈ D for all (local) sections X of D and arbitrary (local) vector fields Y . Consequently, a parallel distribution is integrable and the integral surfaces are totally geodesic. Now let (M, g, D) be a Walker geometry of dimension 2n. Walker [42] showed that one can find local coordinates (x 1 , . . . , x n , y 1 , . . . , y n ) =: (x α , y α ), α = 1, . . . , n, and hence an atlas of such, so that D = ∂x α : α = 1, . . . , n R and with respect to which the metric takes the canonical form −W 1n 0 1 g ab = , (1.1) (gab ) = n n , 1n W 1n 0n where W is an unspecified symmetric matrix (lower case concrete indices take values 1, . . . , 2n and upper case concrete indices 1, . . . , n). We call any coordinates, with respect to which the metric takes Walker’s canonical form (1.1), Walker coordinates. Consider two charts of Walker coordinates (x α , y α ) and (u α , v α ) with nontrivial intersection. Since D = ∂x α : α = 1, . . . , n R = ∂u α : α = 1, . . . , n R , the Jacobian for the transformation between the two coordinate systems is of the form ⎛ α α⎞ ∂u ∂u B C ∂ x β ∂ yβ . J := ⎝ ∂v α ∂v α ⎠ = 0n D ∂ x β ∂ yβ
Since both coordinates systems are Walker, the metric has components of the form (1.1) with respect to each, i.e., on the intersection, g = τ J g J , where g and g are each of the form (1.1), from which one deduces that ⎛ α α⎞ ∂u ∂u B C ∂ x β ∂ yβ whence det(J ) = 1. (1.2) J := ⎝ ∂v α ∂v α ⎠ = 0n τ B −1 ∂ x β ∂ yβ
Note that
⎛
∂xα β ⎝ ∂uα ∂y ∂u β
⎞
∂xα ∂v β ⎠ ∂ yα ∂v β
= J −1 =
B −1 −B −1 C τ B . τB 0n
(1.3)
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1.2. Lemma. The atlas {(x α , y α )} of Walker coordinates defines a canonical orientation for M, given by the orientation class [∂x 1 , . . . , ∂x n , ∂ y 1 , . . . , ∂ y n ] of the coordinate basis, and may be represented inthe customary fashion0 by the equivalence class [∂x 1 ∧ . . . ∧ ∂x n ∧ ∂ y 1 ∧ . . . ∧ ∂ y n ] in 2n (Tm M) /R+ ∼ = S or the equivalence class [d x 1 ∧ . . . ∧ d x n ∧ dy 1 ∧ . . . ∧ dy n ] in 2n (Tm M)• /R+ ∼ = S0 . Indeed, the globally defined 2n-form := d x 1 ∧ . . . d x n ∧ dy 1 ∧ . . . dy n
(1.4)
is in fact the volume form for (M, g) and the following 2n-vector is also globally defined and equals the volume element V : ∂x 1 ∧ . . . ∧ ∂x n ∧ ∂ y 1 ∧ . . . ∧ ∂ y n . Thus, only orientable even-dimensional manifolds can admit a Walker geometry. Proof. These assertions follow immediately from (1.2–3) and 0n 1n = (−1)n 1n 0n = (−1)n . 1n W W 1n
(1.5)
1.3. Observation. Observe that ξg : ∂x α → dy α while ξg : ∂ y α → d x α + terms in dy’s, whence one computes ξg : V = ∂x 1 ∧ . . . ∧ ∂x n ∧ ∂ y 1 ∧ . . . ∧ ∂ y n → (−1)n d x 1 ∧ . . . ∧ d x n ∧dy 1 ∧ . . . ∧ dy n , which concurs with (A1.3). Now D = ∂x 1 , . . . , ∂x n R = ker (dy 1 ∧ . . . ∧ dy n ) . Thus, D can be characterized by its image under the Plücker mapping, viz., [∂x 1 ∧ . . . ∧ ∂x n ] ∈ P n (T M) or by [dy 1 ∧ . . . ∧ dy n ] ∈ P n (T M)• . One computes that dy 1 ∧ . . . ∧ dy n is SD as an element of n (T M• ) with respect to the Hodge star operator defined by the canonical orientation and g. By (A1.6), ξg−1 (dy 1 ∧ . . . ∧ dy n ) = ξg−1 (∗dy 1 ∧ . . . ∧ dy n ) = (−1)n ∗ ξg−1 (dy 1 ∧ . . . ∧ dy n ) i.e., ∗ ∂x 1 ∧ . . . ∧ ∂x n = (−1)n ∂x 1 ∧ . . . ∧ ∂x n .
(1.6)
Thus, while dy 1 ∧ . . . ∧ dy n is always SD, ∂x 1 ∧ . . . ∧ ∂x n is SD/ASD according to whether n is even/odd. 1.4. Observation. From (1.2–3), ∂x 1 ∧ . . . ∧ ∂x n = det(B)∂u 1 ∧ . . . ∧ ∂u n , dy 1 ∧ . . . ∧ dy n = det(τ B)dv 1 ∧ . . . ∧ dv n ,
and det(B) > 0 is the condition for the induced orientations on D to agree. Interchanging any x α with any distinct x β and y α with y β yields another set of Walker coordinates with the opposite orientation for D (but of course the same orientation for M). Thus, supposing D is indeed orientable, given a specified orientation for D, one can choose a
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subatlas of Walker coordinates whose induced orientation on D agrees with the specified one. For four-dimensional Walker geometries, Walker coordinates will typically be denoted (u, v, x, y) and W in (1.1) is written ac . (1.7) W = cb In Appendix One, in addition to stipulating our conventions and some notation, we have collected together the expressions of standard objects with respect to Walker coordinates. We recast this information into spinorial form in §2, which focuses on the local geometry of four-dimensional Walker geometry. Appendix Two contains the essential background in spinors for four-dimensional neutral geometry we require. In §3, we impose a natural condition on four-dimensional Walker geometry and thereby refine a result of Dunajski [11]. In §4, we consider another natural restriction on Walker geometry; namely, the existence of a complementary parallel totally null distribution and demonstrate that some previously known formalisms arise naturally as special cases of Walker geometry. Finally, in §5, we address a global issue that arises naturally in the spinor approach to four-dimensional Walker geometry. 2. Local Four-Dimensional Walker Geometry Let P be a totally null two-pane in the four-dimensional, pseudo-Euclidean linear space R2,2 of neutral signature. Under the identification (A2.1), any two linearly independent elements of P can be written in the form κ A λ A and µ A ν A . Orthogonality requires at least one of κ D µ D = 0 and λ D ν D = 0, i.e., either κ A ∝ µ A or λ A ∝ ν A . In the latter { κ A η A : η A ∈ S }. case, P takes the form { η A λ A : η A ∈ S }, in the former case As is well known, the quadric Grassmannian Q 2 R2,2 of totally null two-planes in R2,2 is homeomorphic to O(2), and can be described as the planes of the form { x, L(x) : x ∈ R2 }, parametrized by L ∈ O(2). Under the identification (A2.1), each element, called an α-plane, of the component Q +2 R2,2 ↔ SO(2) takes the form Z [π ] := { λ A π A : λ A ∈ S }, for some [π A ] ∈ PS ∼ = S1 , while each ele− 2,2 ment, called a β-plane, of the component Q 2 R ↔ ASO(2) takes the form A λ A : λ A ∈ S }, for some [σ A ] ∈ PS ∼ S1 . Under the Plücker embedding { σ W[σ ] := = of G 2 R4 into P 2 R4 ∼ = RP5 , the element Z [π ] of Q +2 R2,2 is mapped to the 2,2 is projective class of the SD bivector AB π A π B while the element W[σ ] of Q − 2 R mapped to the projective class of the ASD bivector σ A σ B A B , see (A2.2–3). A SD bivector F ab is simple iff null, F ab Fab = 0, which, with F ab = AB ψ A B , in turn is equivalent to ψ A B null, i.e., ψ A B = π A π B , for some π A (see [28], §3.5). 2,2 is precisely the intersection of Thus, the Plücker embedding of Q ± 2 R 2 2,2 P ± R with the projectivized subset of simple bivectors, which is a quadric sur face in RP5 . The standard basis of R2,2 yields, via (A1.17), a -ON basis for 2 R2,2 which provides an explicit isomorphism 2 R2,2 = 2+ R2,2 ⊥ 2− R2,2 ∼ = R1,2 ⊥ R1,2 ( ⊥ denotes the orthogonal direct sum) in which the space of simple SD/ASD bivectors is precisely the null cone in 2± R2,2 , whence the Plücker embed 2,2 is identified with the space of generators of that null cone, i.e., with ding of Q ± 2 R 1 S .
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Thus, the image of an α/β-plane under the Plücker mapping is the projective class of a SD/ASD bivector, whence they are also called SD/ASD planes. Moreover, if P = V1 , V2 R , with φ i := ξg (Vi ), then ker( (φ 1 ∧ φ 2 ) = P and, by (A1.6), φ 1 ∧ φ 2 is SD/ASD according as V1 ∧ V2 is SD/ASD. Note that the employment of spinors to obtain this correspondence between the components of Q 2 R2,2 and SD/ASD simple bivectors is of course not necessary. Now let (M, g) be a connected, neutral four-dimensional manifold. Since any Walker manifold has a canonical orientation, we shall assume, without loss of generality, that M is orientable. Let D be a totally null two-dimensional distribution on (M, g). Thus, each Dm is either an α-plane or β-plane in Tm M. In fact: 2.1. Lemma. D is a field of α-planes or a field of β-planes. Proof. Since Mis orientable, there is a globally consistent notion of duality in 2 (T M) and 2 (T M)• . Let U := { m ∈ M : Dm is SD }. For p ∈ U , choose a neighbourhood V of p on which there are smooth vector fields v and w spanning D. Then ψ := v ∧ w is SD at p whence the continuous bivector ψ + ∗ψ (which equals either 2ψ or zero) is nonzero at p and thus near p. It follows that U is open. The complement of U is open by a similar argument. By (1.6), the distribution of a four-dimensional Walker geometry is SD with respect to the canonical orientation and, as we shall always adopt the canonical orientation, thus a distribution of α-planes. (For any set of Walker coordinates (u, v, x, y), (A1.18–20) confirms that ∂u ∧ ∂v = s1+ − s3+ is indeed SD with respect to the canonical orientation.) Let, therefore, D be a distribution of α-planes, i.e., every element of Dm is of the form η A π A , with [π A ] fixed at m and η A arbitrary. This statement only makes use of spinors locally. Supposing that (M, g) is SO+ -orientable, i.e., admits a reduction to SO+ (2, 2), then one can construct the bundle B of SO+ -oriented frames. Locally, one can construct a bundle of spin frames as a two-fold cover of the restriction of the bundle of SO+ -oriented frames and, equally locally, associated bundles of spinors. The obstruction to gluing these local bundles together to obtain a two-fold covering of B and associated bundles over M of spinors arises from the sign ambiguity in the two-fold covering of SO+ (2, 2) by Spin+ (2, 2) when nontrivial topology (specifically, the second StiefelWhitney class) of M can obstruct a consistent choice of signs. But this problem does not arise when one employs the local lifts of transition functions for B to glue together local (trivial) bundles of projective classes of spinors. Thus, PS M is well defined, provided (M, g) is SO+ -orientable, and unique (distinct spin structures arise from different ways of choosing signs, but the ambiguity at a point is always one of sign). Hence, when (M, g) is SO+ -orientable, the distribution D is equivalent to a global . In the absence of this orientability condition, this characterization is section of PS M purely local. The section [π A ], whether understood locally or globally, will be called the projective spinor field defining D (locally or globally).
2.2. Local lifts of [π A ]. Let U be a neighbourhood over which all bundles are triv ial. Construct a trivial bundle of projective spinors on U (either by restriction if PS M exists or otherwise by direct construction) and also construct trivial spinor bundles SU , with fibre S and SU with fibre S. Using the isomorphism (A2.1), one can construct an isomorphism T M|U ∼ = SU ⊗SU . Applying this isomorphism to a local smooth element of
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D|U yields a spinor representation of the vector field in the form λ A π A , where these spinors are defined up to scaling freedom π A → f π A and λ A → f −1 λ A , for any nonvanishing smooth function f on U . The spinor π A projects onto [π A ]. Any local spinor field which projects onto [π A ] will be called a a local scaled representative (LSR) of [π A ]. Employing any such LSR π A to describe D, one easily checks that integrability of D is equivalent to
π A π B ∇b π A = 0,
(2.1)
noting that if this equation holds for some LSR π A then it holds for any. When integrable, the integral manifolds of D will be called α-surfaces. Thus, when D is integrable, M is foliated by α-surfaces. By 1.1, a distribution D of α-planes is parallel iff
π A ∇b π A = 0,
(2.2)
which, like (2.1), if true for some LSR of [π A ] is true for any. Equation (2.2) is equivalent to
∇b π A = Pb π A ,
(2.3)
where the one-form Pb depends on the LSR π A as follows: under
π A → f π A
Pb → Pb + f −1 ∇b f.
(2.4)
Equation (2.3) expresses the fact that the LSR π A is recurrent, see [41]. Indeed, Walker, [41] §5, characterized the condition for a distribution to be parallel in terms of recurrence, which, in the present circumstances, amounts to (2.3). The condition (2.1) in Lorentzian signature characterizes shear-free geodetic null congruences, see [29], §7.3. For complex spacetimes, condition (2.1) is most usefully interpreted as describing distributions of totally null complex two-planes, see [29](7.3.18) and, for example, [31] and [3]. The geometrical interpretation of (2.1) in neutral geometry is thus a natural real analogue of that complex spacetime geometry. Walker geometry is therefore a specialization of the real neutral analogue of this complex geometry and those familiar with complex general relativity will recognize the parallels. 2.3. Walker’s canonical form [42]. Suppose that D is a parallel distribution of α-planes with projective spinor field [π A ]. Let ( p, q, x, y) be Frobenius coordinates for (the integrable) D. Since d x and dy are zero when restricted to the distribution, one can write, for any LSR π A , d x = µ A π A , dy = ν A π A with ν D µ D = 0.
The vector fields U a := µ A π A and V a := ν A π A span D. Noting that ∇b Vc = ∇b ∇c y = ∇c ∇b y = ∇c Vb , then
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U b ∇b V a = g ac U b ∇b Vc = g ac U b ∇c Vb = g ac U b π B (∇c ν B )+g ac U b ν B (∇c π B ) (the first summand of which is zero) = U b ν B π B P a = 0. Similarly, V b ∇b U a = 0 and it follows that [U, V ] = U b ∇b V a − V b ∇b U a = 0. The pair of equations U f = 1, V f = 0 have trivial integrability conditions (see [42]), as do the equations U g = 0, V g = 1. Let u and v be solutions of these systems respectively. Let B be the nonsingular matrix expressing {∂ p , ∂q } in terms of {U, V } (both are frames of D). Then, by the definition of u and v, ∂(u, v, x, y) B C = 02 12 ∂( p, q, x, y) is nonsingular, whence (u, v, x, y) are legitimate local coordinates. Computing the metric g ab : it vanishes on d x, dyR ; g(du, d x) = du(U ) = U (u) = 1; g(du, dy) = du(V ) = V (u) = 0; g(dv, d x) = dv(U ) = U (v) = 0; and g(dv, dy) = dv(V ) = V (v) = 1; which together give the form (1.1) of Walker’s canonical form. From Walker’s canonical form for these coordinates, one observes that U a := ab g (d x)b = ∂u and V a := g ab (dy)b = ∂v , i.e., the Walker coordinates are Frobenius coordinates for D satisfying ∂u = µ A π A
∂v = ν A π A
d x = µ A π A dy = ν A π A
with ν D µ D = 0.
(2.5)
2.4. Remarks. From (2.5) d x ∧ dy = (ν D µ D ) AB π A π B =: (ν D µ D )σπ ,
∂u ∧ ∂v = (ν D µ D ) AB π A π B =: (ν D µ D )π .
(2.6)
If one replaces π A by f π A , then one must replace µ by µ/ f , ν by ν/ f , whence (ν D µ D ) by (ν D µ D )/ f 2 . Thus, a suitable scaling yields an LSR π A so that ν D µ D = ±1, whence d x ∧ dy = ±σπ and ∂u ∧ ∂v = ±π . Thus, given any Walker coordinates (u, v, x, y), there is an LSR π A such that d x ∧ dy = ±σπ , ∂u ∧ ∂v = ±π ,
(2.7)
and this LSR is unique up to sign. If D is orientable, then one can choose an atlas of Walker coordinates so that ∂u ∧ ∂v defines a consistent orientation on D, see 1.4. Thus, for an orientable distribution, one can choose an atlas of Walker coordinates so that (2.7) holds with constant sign for all charts of the atlas. We will have more to say about orientability of D in §5. If (u, v, x, y) are Walker coordinates, by (A1.7) so are (v, u, y, x). Hence, by breaking the Walker symmetry (A1.7), it is always possible to choose Walker coordinates (u, v, x, y) for which the plus sign occurs in (2.7). For such Walker coordinates, and the LSR π A determined by (2.7), we will write ∂u = α A π A
∂v = β A π A
d x = α A π A dy = β A π A ,
where {α A , β A } is an (unprimed) spin frame.
(2.8)
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2.5. Proposition. The projective spinor [π A ] of a Walker geometry (M, g, D, [π A ]) is a Weyl Principal Spinor (WPS, see [19]) of multiplicity at least two. Indeed, any LSR π A satisfies ˜ A B C D π C π D + 2π A π B = 0.
(2.9)
Thus, [π A ] is a WPS of multiplicity at least three in scalar-flat Walker geometries and W + = 0 ⇒ S = 0. Moreover, [π A ] is also a principal spinor of the Ricci spinor AB A B , whence the Einstein endomorphism G a b = 2 A A B B maps D to itself (see A1.8).
Proof. Working with some LSR π A of [π A ], 0 = ∇ b π A ∇b π A = π A π A + ∇ b π A ∇b π A . Since the second term vanishes by (2.3), then
0 = π A
πA .
(2.10)
Furthermore, 0 = ∇CB π A ∇b π A = π A ∇CB ∇b π A + ∇CB π A (∇b π A ) = π A ∇CB ∇b π A , since the second term in the sum again vanishes by (2.3). Using (A2.8),
1 . + ∇CB ∇b = −∇ BC ∇ BB = − C B + ∇ B[C ∇ BB ] = − C B C B 2 Thus, one obtains from the previous calculation 1 A 0=π C B π A + C B π A 2 1 = π A C B π A + C B π A π A 2
˜ C B A D π D − πC B A − C A π B = πA
by (A2.9) and (2.10)
˜ B C A D π A π D + 2π B πC . =
Transvecting by π C shows that [π A ] is a WPS of multiplicity at least two. Similarly,
0 = ∇CB (π A ∇b π A )
= π A ∇CB ∇b π A + ∇CB π A ∇b π A
= −π A ∇ B C ∇ B B π A 1 = −π A π + π CB A CB A 2 =
C B π A −C B A D π A π D .
= −π A
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(One can write AB A B = A AB π A π B + B AB π(A ξ B ) + C AB ξ A ξ B ,
where ξ D π D = 1 and A AB , B AB , and C AB are symmetric. AB A B π A π B = 0 entails C AB = 0.)
2.6. Classification of the SD Weyl curvature endomorphism. The classification of the Weyl curvature endomorphisms of four-dimensional neutral metrics according to their Jordan canonical form (JCF) was given in [17] and according to the algebraic structure of the corresponding Weyl spinors in [19]. By 2.5, the SD Weyl spinor of any four-dimensional Walker geometry is algebraically special. From (A1.27), the eigenvalues of the SD Weyl curvature endomorphism W + of any four-dimensional Walker geometry are −S/6, S/12, and S/12 and Díaz-Ramos et al. [8] determined the Jordan canonical forms of W + . Here we refine their result. For any eigenvalue, let m(M) denote the algebraic (geometric) multiplicity. There are four cases; see (A1.23–24) for notation: i) S = 0 but S 2 + AS + 3B 2 = 0; W + can be diagonalized and has JCF ⎛ ⎞ S ⎝2 0 0 ⎠ 0 −1 0 , − 12 0 0 −1
−S/6 has m = M = 1; S/12 has m = M = 2. Since [π A ] is a real WPS of multiplicity at least two, from [19], p. 2106, this information indicates W + must be of type {22}Ia, and the multiplicity of [π A ] is exactly two. ii) S = 0 and S 2 + AS + 3B 2 = 0; W + has JCF ⎛ S ⎞ −6 0 0 ⎝ 0 S 1 ⎠, 12 S 0 0 12 −S/6 has m = M = 1; S/12 has M = 1, m = 2. It follows from [19], p. 2106, that W + is of type {211}II or {112}II; in each case the double WPS must be [π A ]. iii) S = 0, B = 0, A = 0; from (A1.25), the matrix representation of W + is ⎛ ⎞ 1 0 1 A + W = − ⎝ 0 0 0 ⎠, 12 −1 0 −1
⎛ with JCF
⎞ 000 ⎝0 0 1 ⎠ , 000
and 0 is the only eigenvalue, with M = 2. From [19], p. 2106, W + must be type ˜ A B C D is null with four-fold WPS [π A ]. {4}II, i.e., iv) S = 0 but B = 0; W + has JCF J3 (0), i.e., 0 is the only eigenvalue and M = 1. From [19], p. 2106, W + is of type {31}III and [π A ] is a WPS of multiplicity three.
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2.7. Remark. Since W + is algebraically special, it is never, in particular, of type Ib, see [17] and [19]. If, therefore, (M, g) is compact Einstein, then χ (M) ≤ 3τ (M)/2, see [17,26]. If further, W − is not of type Ib, then in fact χ (M) ≤ −3|τ (M)|/2 ≤ 0. This conclusion holds if (M, g) is a compact Einstein Walker geometry which is, for example: SD (W − = 0); or algebraically special in W − ; or Kähler with the orientation induced by the complex structure agreeing with the Walker canonical orientation, in which case W − is of type Ia, see [15]. Examples of four-dimensional Walker geometries with ASD Weyl curvature of type Ib are presented in [4].
In 2.6 we relied upon the computations recorded in Appendix One and the result of Díaz-Ramos et al. [8] which is also based on these computations. A systematic development of spinor analysis of Walker geometry would proceed by computing the neutral analogues of spin coefficients, [28], §4.5, and thence the spinor equivalents of the curvature. The notation for the spin coefficients for neutral signature, however, requires modification of that employed in Lorentz signature; e.g., the priming operation of [28](4.5.17) is not appropriate for neutral signature. Spin coefficients for neutral signature will be presented elsewhere; here we shall follow expediency and further exploit the known results of Appendix One to deduce the spinor equivalents of curvature. To this end, Walker’s canonical form for the metric ds 2 = 2d x(du +
c c b a d x + dy) − 2dy(−dv − d x − dy), 2 2 2 2
suggests utilizing the following null tetrad: a := d x m˜ a := dy n a := du +
c b a c d x + dy m a := −(dv + d x + dy), 2 2 2 2
which we shall call the Walker null tetrad associated to a set of Walker coordinates. The null tetrad determines unique (up to an overall sign) spin frames. In particular, assuming we have chosen Walker coordinates (u, v, x, y) and a LSR π A so that (2.8) is satisfied, write n a = ν A ζ A , where ζ D π D = 0. Then n a m˜ a = 0 implies ν D β D = 0. a A A D So n = β ξ , with ξ π D = 0. But then n a a = 1 implies ξ D π D = 1. Writing m a = γ A κ A , then m a m˜ a = −1, but m a a = 0 implies γ D α D = 0. Then, m a n a = 0 implies κ D ξ D = 0, whence m a = λα A ξ A , for some λ ∈ R. But then m a m˜ a = −1 implies λ = 1. Thus, {α A , β A } and {π A , ξ A } are the unique (up to an overall sign) spin frames associated to the null tetrad, and a = d x = α A π A , c a n a = du + d x + dy = β A ξ A , 2 2 a = ∂u = α A π A , a c n a = − ∂u − ∂ v + ∂ x = β A ξ A , 2 2
m˜ a = dy = β A π A , b c m a = −(dv + d x + dy) = α A ξ A , 2 2 (2.11) a A A m˜ = ∂v = β π , c b m a = ∂u + ∂v − ∂ y = α A ξ A . 2 2
Note that [∂u , ∂v , ∂x , ∂ y ] = [a , m˜ a , n a , −m a ], the latter being a Witt frame which 4 when a = b = c = 0 (R4 being the reduces to the standard Witt frame for R2,2 ∼ = Rhb hb standard hyperbolic four-dimensional pseudo-Euclidean space as in [32]).
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From the null tetrad, one constructs a -ON basis as follows: 2−a 1 1 c U a := √ (a + n a ) = √ ∂u − ∂v + ∂ x , 2 2 2 2 c 1 1 2−b a a a V := √ (m˜ − m ) = √ − ∂u + ∂v + ∂ y , 2 2 2 2 2+a 1 a 1 c a a X := √ ( − n ) = √ ∂u + ∂v − ∂ x , 2 2 2 2 c 1 1 2+b a a a Y := √ (m˜ + m ) = √ ∂u + ∂v − ∂ y . 2 2 2 2
(2.12)
We note that [U a , V a , X a , Y a ] = [∂u , ∂v , ∂x , ∂ y ], i.e., the -ON frame (2.12) has the canonical orientation; moreover it is well behaved under the Walker symmetry (A1.7): U a ↔ V a and X a ↔ Y a . From (2.12) and (A1.17), one obtains the following -ON bases for the spaces of SD and ASD bivectors: Ua ∧ V a + Xa ∧ Y a √ 2 4 + ab − c2 1 = √ ∂u ∧ ∂v + c∂u ∧ ∂x − a∂u ∧ ∂ y + b∂v ∧ ∂x 2 2 2 −c∂v ∧ ∂ y + 2∂x ∧ ∂ y ;
S1+ :=
S2+ := S3+ := =
S1− := S2− := S3− :=
Ua ∧ Xa + V a ∧ Y a 1 = − √ (∂u ∧ ∂x + ∂v ∧ ∂ y ); √ 2 2 Ua ∧ Y a − V a ∧ Xa (2.13) √ 2 4 − ab + c2 1 ∂u ∧ ∂v − c∂u ∧ ∂x + a∂u ∧ ∂ y − b∂v ∧ ∂x √ 2 2 2 +c∂v ∧ ∂ y − 2∂x ∧ ∂ y ; (a + b) Ua ∧ V a − Xa ∧ Y a 1 ∂u ∧ ∂v + ∂u ∧ ∂ y − ∂v ∧ ∂x ; =√ − √ 2 2 2 Ua ∧ Xa − V a ∧ Y a 1 = √ (c∂u ∧ ∂v − ∂u ∧ ∂x + ∂v ∧ ∂ y ); √ 2 2 b−a Ua ∧ Y a + V a ∧ Xa 1 ∂u ∧ ∂v − ∂u ∧ ∂ y − ∂v ∧ ∂x . =√ √ 2 2 2
Writing these (A)SD bivectors in terms of spinors:
a ∧ m˜ a − n a ∧ m a AB (π A π B + ξ A ξ B ) = = =: AB 1A B ; √ √ 2 2 a ∧m a − a ∧ m a A α B + β A β B ) A B ˜ n (α S1− = =− =: −1AB A B ; √ √ 2 2 S1+
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S2± =
na
∧ a
± m˜ a
√ 2
∧ ma
=
⎧ √ AB (A B ) =: − AB A B ; ⎪ ⎨− 2 π ξ 3 ⎪ ⎩−√2α (A β B) A B =: − AB A B ; 3
(2.14)
S3+ =
a ∧ m˜ a + n a ∧ m a AB (π A π B − ξ A ξ B ) =: AB 2A B ; = √ √ 2 2
S3− =
n a ∧ m˜ a + a ∧ m a (α A α B − β A β B ) A B = =: 2AB A B . √ √ 2 2
Noting the conventions of Appendix One, the SD Weyl curvature endomorphism W + satisfies: Si+ + C
i
j
:= W + (S +j ) :=
1 AB ˜ A B C D C D Cj D C D 2
˜AB = AB
C D C D j
˜ i j, =: AB iA B
(2.15)
i.e., the matrix representation + C of W + (acting on 2+ , the space of SD bivectors) with ˜ of the endorespect to the basis {S1+ , S2+ , S3+ } coincides with the matrix representation B A ˜ morphism C D (acting on the space S S of symmetric rank two primed spinors) with respect to the basis {1A B , 2A B , 3A B }. Similarly, the matrix representation − C of the ASD Weyl curvature endomorphism W − (acting on 2− ) with respect to the basis {S1− , S2− , S3− } coincides with the matrix representation of the endomorphism AB C D (acting on S S) with respect to the basis {1AB , 2AB , 3AB }. From the matrix representations of W + and W − given in (A1.21–25) with respect to the bases (A1.19–20) of 2+ and 2− , one can calculate + C and − C. From the expression ˜ one can then compute ABC D and for given in [19], Eq. (20), and its analogue for , ˜ A B C D themselves. From (2.13) and (A1.19–20), one finds 1 S1+ = √ (c2 + 5)s1+ − 2cs2+ − (c2 + 3)s3+ , 4 2 1 S3+ = √ (3 − c2 )s1+ + 2cs2+ + (c − 52 )s3+ , 4 2 1 S2+ = √ (cs1+ − s2+ − cs3+ ), 2 s− S1− = √1 , 2
s− S2− = − √2 , 2
Writing + Z iab := AB iA B , (2.14) and (2.16) entail:
s− S3− = − √3 . 2
(2.16)
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1 Z 1 = √ (c2 + 5)s1+ − 2cs2+ − (c + 32 )s3+ , 4 2 1 + Z 2 = √ (3 − c2 )s1+ + 2cs2+ + (c − 52 )s3+ , 4 2 1 + Z 3 = √ (−cs1+ + s2+ + cs3+ ). 2 +
(2.17)
The matrix J expressing the + Z i in terms of the si+ is therefore ⎞ ⎛ 2 √ ⎛ c2 + 5 2 −4c + 5 3 − c c 2⎝ 2 1 2c 4 ⎠ whence J −1= J= √ ⎝ −2c c −3 4 4 2 −(3 + c2 ) c2 − 5 4c 4c
2c 2c 4
⎞ c2 + 3 c 2 − 5⎠ . 4c (2.18)
Hence, from (A1.25), +
C = J −1 + W J ⎛ A − 6Bc − 3S(c2 + 1) 1 ⎜ A − 6Bc − S(3c2 − 1) =− ⎝ −6(B + Sc) 48 .
−A + 6Bc + S(3c2 − 1) −A + 6Bc + S(3c2 − 5) 6(B + Sc)
⎞ 6(B + Sc) 6(B + Sc)⎟ ⎠ (2.19) 8S
Equating this expression to the tilde version of [19](20) one finds: ˜0 = 0 = ˜2 = ˜1
S (B + Sc) 6Bc − A + S(3c2 − 1) ˜3 = − ˜4 = , (2.20) 12 8 24
whence ˜ A B C D =
S B + Sc π(A π B ξC ξ D ) + π(A π B πC ξ D ) 2 2 6Bc − A + S(3c2 − 1) π A π B πC π D . + 24
(2.21)
From (2.20–21), one can obtain the results of 2.6 directly. Referring to [19](22–24), one computes I = S 2 /24, J = −S 3 /288, whence I 3 = 6J 2 , and 0 = (λ + S/6)(λ − S/12)2 ˜ A B C D . The expression (2.21) confirms 2.5: [π A ] is is the eigenvalue equation for a real WPS of multiplicity at least two, and of multiplicity at least three when S = 0. If [π A ] is of multiplicity exactly two, then S = 0. Given the eigenvalues just deduced, from the diagram in [19], p. 2106, if the geometric multiplicities coincide with the ˜ A B C D is type {22}Ia (in particular, W + is diagonalizalgebraic multiplicities, then ˜ A B C D = 6(S/12)π(A π B ηC η D ) , for some able). It follows from [19], §5.6, that spinor η A = pπ A + qξ A . Equating this expression with (2.21) yields the condition ˜ A B C D cannot be of type S 2 + AS + 3B 2 = 0 of 2.6(i). When S 2 + AS + 3B 2 = 0, {22}Ia; with S = 0 still, one sees from [19], p. 2106, that the only possible types with [π A ] a real WPS of multiplicity two, are types {211}II or {112}II, in which cases the geometric multiplicity of S/12 is one, rather than two (in particular, W + is not diagonalizable). If now S = 0 but B = 0, then [π A ] is a WPS of multiplicity three, whence ˜ A B C D has type {31}III, and there is a single eigenvalue (namely zero) of algebraic multiplicity three and geometric multiplicity one. Finally, if S = 0, B = 0, but A = 0, ˜ A B C D is of type {4}II, and the zero then [π A ] is a WPS of multiplicity four, whence
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eigenvalue now has geometric multiplicity two. Thus, one may derive the results of 2.6 directly from (2.21) (bearing in mind that this expression could be computed directly by first computing spin coefficients without exploiting the results of Appendix One). Turning now to the ASD Weyl curvature, and with − Z iab := AB A B , one finds −
s− Z 1 = −S1− = − √1 , 2
−
s− Z 2 = S3− = − √3 , 2
−
s− Z 3 = −S2− = √2 , 2 (2.22)
whence the analogue of (2.18) is ⎛ ⎞ 1 ⎝−1 0 0⎠ 0 0 1 K =√ 2 0 −1 0
and
K −1
⎛ ⎞ √ −1 0 0 = 2 ⎝ 0 0 −1⎠ . 0 1 0
Hence, referring to (A1.22), one computes ⎛ ⎞ P + 3Q −3Y 3(T + X ) 1 − ⎝ 3Y P − 3Q 3(T − X )⎠ . C = K −1 − W K = 12 −3(T + X ) 3(T − X ) −2P
(2.23)
(2.24)
Equating this expression to [19](20) and, as in Appendix One, using the numerals 1, 2, 3 ,4 to denote u, v, x, y, yields: b11 b12 − c11 a11 + b22 − 4c12 , 1 = , 2 = , 2 4 12 a12 − c22 a22 , 4 = , 3 = 4 2 0 =
(2.25)
whence ABC D =
b11 a11 + b22 − 4c12 β A β B βC β D − (b12 − c11 )α(A β B βC β D) + 2 2 a22 α A α B αC α D . ×α(A α B βC β D) − (a12 − c22 )α(A α B αC β D) + 2
(2.26)
It is clear from the dyad components in (2.25) that, generically, there will be no relation between the I and J of [19](24), which therefore impose no constraints on ABC D for the general Walker metric. We note that the -ON frame (A1.18) determines, as in (2.12), a null tetrad {L a , N a , M a , M˜ a }, which one computes to be √ A 1 c 1 1 L a = √ (−a∂u + 2∂x ) = β A 2ξ + √ π A , N a = √ a = √ α A π A , 2 2 2 2 √ A 1 c M˜ a = √ −2c∂u − b∂v + 2∂ y = −α A 2ξ + √ π A , 2 2 1 1 M a = − √ m˜ a = − √ β A π A . (2.27) 2 2 The frames are (up to an overall sign) {β A , −α A } and √ spin √ Aassociated √ A A { 2ξ + (c/ 2)π , −π / 2}. One can therefore obtain the components of the Weyl
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spinors with respect to these spin frames either in the manner followed above or simply by re-expressing (2.21) and (2.26) in terms of the relevant spin frames. Putting o A := β A ,
ι A := −α A ,
o A :=
√ A c 2ξ + √ π A , 2
πA ι A := − √ , 2
(2.28)
one finds (A + S) S ι A ι B ιC ι D − Bo(A ι B ιC ι D ) + o(A o B ιC ι D ) , 6 2 the analysis of which is similar to that of (2.21), and ˜ A B C D = −
(2.29)
a22 a11 + b22 − 4c12 ι A ι B ιC ι D − (c22 − a12 )o(A ι B ιC ι D) + o(A o B ιC ι D) 2 2 b11 o A o B oC o D . −(c11 − b12 )o(A o B oC ι D) + (2.30) 2 Returning to the Walker null tetrad, consider the Ricci spinor: 1 1 S Rab − gab = E ab . AB A B = 2 4 2 ABC D =
From (A1.26–27), one can compute the components of E ab with respect to the null tetrad (2.11) and thence the dyad components of AB A B . (Note that by 2.5 AB A B π A π B = 0, i.e., AB0 0 = 0.) One finds: 1 a E a b b = 0; 2 1 ϒ = ma E a bmb = ; 2 2 1 θ = a E a b n b = ; 2 2 1 ν a b + m˜ a E b n = ; 2 2
1 µ a E a b m b = − ; 2 2 1 = a E a b m˜ b = 0; 2 (2.31) 1 = m˜ a E a b m˜ b = 0; 2 1 ζ = na E a b nb = ; 2 2
00 = 000 0 =
01 = 000 1 =
02 = 001 1
10 = 010 0
11 = 010 1 21 = 110 1
12 = 011 1 =
20 = 110 0 22 = 111 1
1 1 m a E a b n b = − (2η + 2cθ + bν − aµ). 2 4
It follows, see also 2.5, that AB A B = A AB π A π B + B AB π(A ξ B ) ,
(2.32)
with 1 ϒβ A β B + (2η + 2cθ + bν − aµ)α(A β B) + ζ α A α B , 2 = µβ A β B + 2θ α(A β B) − να A α B .
A AB = B AB
(2.33)
With respect to the basis induced by {α A , β B } for S ⊗ S, A AB and B AB have components 1 ϒ −(2cθ + 2η + bν − aµ)/2 , AAB = ζ 2 −(2cθ + 2η + bν − aµ)/2 µ −θ BAB = . −θ −ν
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3. Walker Geometry with Parallel LSRs Dunajski [11] considered a four manifold M with a neutral metric g and a global parallel spinor π A and called such a null-Kähler four-manifold. Since a global parallel spinor π A satisfies (2.2), [π A ] defines a parallel distribution of α-planes and hence a Walker geometry. In fact, the considerations in [11] are essentially local, and his main result is naturally subsumed as a feature of Walker geometry, as we explain in this section. We first note the following restrictions imposed on the curvature by the presence of a parallel spinor ([11]). Note that these restrictions are local, rather than global, in nature, in the sense that they hold on the domain of any parallel spinor.
3.1. Proposition. If (M, g) is a neutral four-manifold and π A a parallel spinor on some domain (with nonempty interior) in M, then on that domain: g is Ricci-scalar flat; [π A ] is a principal spinor of the Ricci spinor of multiplicity two, i.e., AB A B = FAB π A π B for ˜ A B C D some symmetric FAB ; and [π A ] is a WPS of multiplicity four, i.e., if nonzero, is of type {4}II and case (iii) of 2.6 pertains.
Proof. Work locally on the interior of the domain of π A ; the curvature conditions then extend to the closure of that domain by continuity. The proof follows the argument of 2.5 but beginning with ∇b π A = 0 rather than (2.2), which eliminates a factor of π A from the computations and therefore increases the multiplicity of [π A ] as a principal spinor ˜ A B C D by one. From (A2.9), A B π A = −3π A , whence of both AB A B and A π parallel entails = 0, which fact entails [π A ] is a WPS of multiplicity four. ˜ A B C D = cπ A π B πC π D . Substituting the expressions obtained for the Thus, Weyl and Ricci curvature spinors into (A2.10) yields, since π A is parallel,
π A π B πC π D ∇ BA c = π(C π D ∇ BA ) FAB , π A π B ∇ C A FC D = 0.
In fact, transvecting the first, by say π B , yields the second. Transvecting the first by η B ηC η D , where η D π D = 1, yields π A ∇ BA c = η B ∇ BA FAB . Transvecting the second equation above by η B yields π A ∇ A C FC D = 0. It follows that
π A ∇ BA c = 0 ⇔ ∇ CA FC D = 0.
(3.1)
3.2. Proposition. A Walker geometry (M, g, D, [π A ]) admits, on a neighbourhood of each point, a parallel LSR π A iff the curvature conditions of 3.1 pertain on M. Proof. The necessity follows immediately from 3.1. For sufficiency, choose any LSR π A on a neighbourhood of some point and consider f π A , where f is a smooth positive function. Observe that 0 = ∇b ( f π A ) = (∇b f )π A + f ∇b (π A ) is equivalent to (∇b f )π A = − f ∇b π A = − f Pb π A , by (2.3), i.e., equivalent to ∇b ln( f ) = −Pb . One can there fore find an f to rescale π A to be parallel iff −Pb is a gradient. By the Poincaré lemma, this is so iff Pb is closed, at least locally, i.e., iff ∇[c Pb] = 0. Now
∇c ∇b π A = (∇c Pb )π A + Pb ∇c π A
= (∇c Pb + Pb Pc ) π A .
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Hence, ∇[c ∇b] π A = ∇[c Pb] π A , i.e., Pb is closed iff ∇[c ∇b] π A = 0. This condition is not true in general, so one cannot always find an LSR of [π A ] that is parallel. From (A2.7) and (A2.9), however, the curvature conditions of 3.1 do entail ∇[c ∇b] π A = 0.
3.3. Lemma. For any Walker geometry (M, g, D, [π A ]), suppose given a parallel LSR π A of [π A ] on a neighbourhood of some point m ∈ M. Then one can choose Walker coordinates (u, v, x, y) on a (smaller) neighbourhood of m for which (2.8) is valid with this parallel LSR. Proof. Put L a := µ A π A and M˜ a := ν A π A , where {µ A , ν A } is a spin frame, i.e., ν D µ D = 1. Then L a ∧ M˜ b = AB π A π B =: σπ is parallel and therefore a closed two-form. Thus, locally, σπ = dφ, for some one-form φ. Now, σπ is of rank one, i.e., σπ ∧ σπ = 0. Since φ is defined only up to the addition of an exact one-form, one can exploit this freedom to ensure that φ ∧ dφ is nonvanishing. It follows from Darboux’s theorem (e.g., Theorem 6.2 in [38]) that there exist local coordinates (x, p, q, y) such that φ = xdy + dp, whence σπ = dφ = d x ∧ dy. Now D = L a , M˜ a R and is the kernel of the endomorphism of the tangent space: v → v σπ . In particular, 0 = La
σπ = L a
(d x ∧ dy) = d x(L)dy − dy(L)d x.
Since d x and dy are linearly independent one-forms,then d x(L) = dy(L) = 0, whence L ∈ ∂/∂ p, ∂/∂qR . Together with a similar computation for M˜ a , one deduces that ker(d x) ∩ ker(dy) = ∂/∂ p, ∂/∂qR = L a , M˜ a R = D. Thus, ( p, q, x, y) are Frobenius coordinates for D. It follows, as in 2.3, that d x = α A π A , dy = β A π A , for some spinors α A and β A satisfying β D α D = 0. But σπ = d x ∧ dy = (β D α D ) AB π A π B , whence β D α D = 1. Putting U a = α A π A and V a = β A π A , Walker’s argument 2.3 constructs functions u and v so that (u, v, x, y) are Frobenius coordinates, with ∂u = α A π A and ∂v = β A π A , which is the desired result.
3.4. Lemma. Given any Walker coordinates (u, v, x, y), the LSR π A defined by (2.7) is parallel iff d x ∧ dy, equivalently, ∂u ∧ ∂v , is parallel.
Proof. All that needs to be verified here is that π A π B parallel entails π A is parallel. Supposing π A π B is parallel, then by (2.3) 0 = ∇c (π A π B ) = 2Pc π A π B , which entails that Pc is zero, i.e., π A is indeed parallel. In a slightly different formulation, for any LSR π A , by (2.6) d x∧dy = (ν D µ D )σπ =: ± f σπ , with f positive. Hence, ∇b (d x ∧ dy) = ±(∇b f + 2 f Pb )π . Thus, d x ∧ dy is √ parallel iff −Pb = ∇b (ln f ). From the proof of 3.2, one notes that this last equation √ is precisely the condition for f π A to be parallel; and d x ∧ dy = ±√ f π .
3.5. Lemma. The LSR π A determined up to sign by the condition (2.7) is parallel iff a1 + c2 = 0, b2 + c1 = 0.
(3.2)
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Proof. From (A1.8), a direct computation yields ∇1 (∂u ∧ ∂v ) = 0, ∇2 (∂u ∧ ∂v ) = 0, ∇3 (∂u ∧ ∂v ) =
a1 +c2 2 ∂u
∧ ∂v , ∇4 (∂u ∧ ∂v ) =
b2 +c1 2 ∂u
∧ ∂v .
3.6. Remarks. The conditions (3.2) imply a12 + c22 = 0 = b12 + c11 , a11 = b22 = −c12 ,
(3.3)
which entail simplifications of the curvature. Obviously one expects to recover the results of 3.1. Indeed, from A1.6 one sees immediately that S = 0 and from A1.5 that Rij = 0 unless i, j are 3 or 4. More geometrically, in A1.8 θ = µ = ν = 0, whence the Einstein endomorphism (here equal to the Ricci endomorphism since S = 0) maps the tangent space to D with kernel D. Since the contraction of both ∂u and ∂v (which have linearly independent unprimed spinor parts) on the Ricci tensor is zero, then (since S = 0) one deduces that the Ricci spinor AB A B is null of the form FAB π A π B . By (3.2), the B of (A1.24) vanishes. Since S = 0, assuming W + = 0, it follows ˜ A B C D is null. that case (iii) of 2.6 pertains, i.e., π A is WPS of multiplicity four and Substituting S = B = 0 in (2.21) yields ˜ A B C D = −
A π A π B πC π D . 24
(3.4)
Equation (3.3) entails a simplification in the formulae (2.25–26) for the ASD Weyl curvature: c11 b22 c12 b11 b12 a11 , 1 = = − , 2 = = =− , 2 2 2 2 2 2 c22 a12 a22 3 = = − , 4 = . 2 2 2 0 =
(3.5)
Hence, in the present circumstances, the ASD Weyl curvature is zero iff a, b, and c are affine functions of u and v, whose coefficients are functions of x and y. Imposing (3.2) then yields a = Eu + Fv + G, b = Mu + N v + P, c = −N u − Ev + T,
(3.6)
i.e., for a Walker geometry (M, g, D, [π A ]), if some LSR π A is parallel and (u, v, x, y) are Walker coordinates for which (2.7) holds, then the ASD curvature vanishes iff a, b, and c are of the form as in (3.6), leaving 7 arbitrary functions of x and y.
3.7. Observation. Equations (3.2) can be simply solved as follows. Take c to be any smooth function of (u, v, x, y) such that cv is integrable wrt u and cu is integrable wrt v; then one directly solves for a and b. Presume in fact that c is at least C 4 . Assuming (3.2) hold, then a is an antiderivative of −cv wrt u, and b is an antiderivative of −cu wrt v. Let g :=: c du denote a specific antiderivative of c wrt u and h :=: c dv a specific antiderivative of c with respect to
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v. Since gu = h v there exists a function k :=: c dudv such that ku = h and kv = g. Put 1 ϑ := c dudv + m(u, x, y) + n(v, x, y), (3.7) 2 where m and n are arbitrary (suitably) smooth functions. Then, ϑuv = c/2, 2ϑvv = cv du + 2n vv = − au du + 2n vv , and 2ϑuu = cu dv + 2m uu = − bv + 2m uu , where the remaining integrals are specific antiderivatives of their integrands. Use the freedom in the choice of m and n to ensure that in fact ac −ϑvv ϑuv −ϑ22 ϑ12 W = =2 =2 . (3.8) cb ϑuv −ϑuu ϑ12 −ϑ11 ˜ A B C D is equivalent to A = 0. It is therefore natural to By (3.4), the vanishing of re-express A in terms of ϑ. With the advantage of hindsight obtained from [11] and the results of §4, we wrote (A1.23) in the form we require here. Observing that under (3.2), in (A1.23) each of the first and second pairs of terms in the final line cancel, while the last three terms together equal −S, which is zero under (3.2), one may ignore the final line. Using A1.1, for an arbitrary function f : k k ∂k f = g ij ∂i ∂j f f = g ab ∇a ∇b f = g ij ∂i ∂j f − 2 13 + 24 −(a1 + c2 )∂1 f − (b2 + c1 )∂2 f.
(3.9)
Thus, when (3.2) hold, f = g ij ∂i ∂j f = −a f 11 − 2c f 12 − b f 22 + 2 f 13 + 2 f 24 .
(3.10)
Using (3.2) to rewrite certain terms, it is now routine to verify that: the first line of (A1.23) equals 12 (ϑ13 ); the second line equals 12 (ϑ24 ), the third through fifth lines together equal 3 (ab) = 12 (ϑ11 ϑ22 ) and the sixth and seventh lines together equal −3 (c2 ) = −12 (ϑ12 )2 . Hence, A = 12 ϑ13 + ϑ24 + ϑ11 ϑ22 − (ϑ12 )2 . (3.11) Thus, when (3.2) hold, one may write the metric in the form (1.1) with W determined by a function ϑ as in (3.8), the curvature conditions of 3.1 hold and the vanishing of ˜ A B C D is equivalent to the vanishing of (3.11). This result refines that of Dunajski [11], who studied ASD, four-dimensional, neutral metrics admitting a parallel spinor and also wrote the condition for the vanishing of the SD Weyl curvature in the form: ϑ13 + ϑ24 + ϑ11 ϑ22 − (ϑ12 )2 =: P,
P = 0.
(3.12)
The homogeneous form of this second order PDE is Pleba˜nski’s [30] second heavenly equation characterizing, locally, the metrics of (in the real case) neutral four-manifolds whose only nontrivial curvature is the ASD Weyl curvature. We elucidate this connexion in §4. Of course, any Walker metric with W of the form (3.8) trivially satisfies (3.2). Now consider the form that the ASD Weyl curvature (2.26) and (3.5) takes under (3.2). Substituting (3.8) into (2.25) or (3.5) yields: 0 = −ϑ1111 , 1 = −ϑ1112 , 2 = −ϑ1122 , 3 = −ϑ1222 , 4 = −ϑ2222 . (3.13)
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With
δ A := α A ∂v − β A ∂u = α A m˜ b ∇b − β A b ∇b = π A ∇ A A ,
(3.14)
one finds that ABC D = −ϑ1111 β A β B βC β D + 4ϑ1112 α(A β B βC β D) − 6ϑ1122 α(A α B βC β D) +4ϑ1222 α(A α B αC β D) − ϑ2222 α A α B αC α D = −δ A δ B δC δ D ϑ. (3.15) Equation (3.15) is the form of the ASD Weyl spinor obtained by Pleba˜nski [30] under the assumption that all the other curvature vanishes, and by Dunajski [11] assuming 3.1 and the vanishing of the SD Weyl curvature. We have found this form is valid wherever a Walker metric has a parallel LSR, i.e., under the constraints imposed on the curvature in ˜ A B C D is not required. In short, Walker 3.1; note, in particular, that the vanishing of geometry provides the natural context for this generalization of Dunajski’s result. In accordance with 3.1, under (3.2) θ = µ = ν = 0 in A1.8, whence B AB = 0 in (2.33). Hence, AB A B =A AB π A π B where A AB =
1 ϒβ A β B +2ηα(A β B) +ζ α A α B . 2
(3.16)
Under (3.2), one further finds that ϒ/2 = P11 , η/2 = −P12 , and ζ /2 = P22 , whence one can express (3.16) in the form AB A B = (δ A δ B P) π A π B .
(3.17)
3.8. Theorem. In summary, a Walker geometry (M, g, [π A ]) with a parallel LSR on some coordinate domain, is determined on that domain by a single function ϑ of Walker ˜ A B C D of type {4} with WPS [π A ] or coordinates with: the metric given by (3.8); vanishes when (3.12) holds; ABC D = −δ D δC δ B δ A ϑ and thus vanishes iff ϑ is a cubic polynomial in u and v of the form −12ϑ = Mu 3 + 3Euv 2 + 3N u 2 v + Fv 3 + 3Gv 2 − 6T uv + 3Pu 2 + K 1 u + K 2 v + K 3 , consistent with (3.6), where the coefficients are arbitrary functions of (x, y) and K 1 , K 2 and K 3 express the residual freedom in ϑ not constrained by (3.8); S = 0; the Ricci spinor is given by (3.17), whence the Walker geometry is Einstein iff P is affine in u and v. 4. Complementary Distributions We begin this section by considering an orientable four-dimensional neutral manifold (M, g) which admits a pair of complementary distributions of totally null two-planes. By 2.1, each distribution is a distribution of either α-planes or of β-planes. Since at any point, any α-plane intersects any β-plane in a one-dimensional subspace, a complementary pair of totally null distributions must both be distributions of α-planes or distributions of β-planes. If one of the distributions is parallel, then one has a Walker geometry, and with respect to the canonical orientation the two distributions are each distributions of α-planes.
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Hence, we consider two complementary distributions of α-planes on (M, g). Denote these distributions by Dπ and Dχ , where the distributions have associated projec tive spinor fields [π A ] and [χ A ] respectively. The complementarity is equivalent to D χ π D = 0 for any LSRs. We now suppose both distributions are integrable. In a suitable neighbourhood of any point m ∈ M, one can choose Frobenius coordinates ( p, q, x, y) for Dπ with Dπ = ∂ p , ∂q R and Frobenius coordinates (w, z, r, s) for Dχ with Dχ = ∂w , ∂z R . Since d x and dy separately annihilate Dπ , for any LSR π A of [π A ] they are of the form γ A π A , for suitable γ A . One can rescale π A , and interchange the roles of x and y if necessary, so that one may suppose the coordinates ( p, q, x, y) and LSR π A are chosen so that d x = α A π A ,
dy = β A π A ,
β D α D = 1,
d x ∧ dy = σπ . (4.1)
whence
Similarly, one may suppose the coordinates (w, z, r, s) and an LSR χ A are chosen so that dr = µ A χ A , ds = ν A χ A , ν D µ D = 1,
whence
dr ∧ ds = σχ , (4.2)
where σχ = AB χ A χ B . Since the distributions are complementary, the functions (r, s, x, y) constitute local coordinates which are simultaneously Frobenius with respect to both distributions, specifically Dπ = ∂r , ∂s R , Dχ = ∂x , ∂ y R ,
(4.3)
(see for example [16], p. 182) and (4.1) and (4.2) are of course still valid, being coordinate-independent statements. With respect to the coordinates (r, s, x, y), the metric must take the form 02 D 02 τ D −1 ab (gab ) = τ , whence g (4.4) = D 02 D −1 02 2 for some (2 × 2)-matrix D. Note that det(gab ) = det(D) . Writing the metric as 2d x(D11 dr + D21 ds) − 2dy(−D12 dr − D22 ds),
(4.5)
one can extract the null tetrad: a := d x, m˜ a , := dy,
n a := D11 dr + D21 ds, m a := −(D12 dr + D22 ds).
(4.6)
One computes a = (D −1 )11 ∂r + (D −1 )12 ∂s , m˜ a = (D −1 )21 ∂r + (D −1 )22 ∂v ,
n a = ∂x , m a = −∂ y .
(4.7)
From (4.1–2) and (4.4), one computes (D −1 )11 (D −1 )21 (D −1 )12 (D −1 )22
:= g ab (dr, d x) = (µ D α D )(χ D π D ), := g ab (dr, dy) = (µ D β D )(χ D π D ); := g ab (ds, d x) = (ν D α D )(χ D π D ), := g ab (ds, dy) = (ν D β D )(χ D π D ),
(4.8)
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whence
−1 −1 D = (µ α D )(ν E β E ) − (µ D β D )(ν E α E ) (χ D π D )−2 . det(D) = det(D −1 )
But, {α A , β A } and {µ A , ν A } are both spin frames, and therefore related by an element of SL(2;R); indeed, µ A = −(µ D β D )α A + (µ D α D )β A , ν A = −(ν D β D )α A + (ν D α D )β A , whence 1 =
−(µ D β
D
)(ν E α
E
) + (ν D β
D
)(µ E α
det(D) = (χ Putting ξ
A
n a = (χ
A
:= χ /(χ D
D
π D ) so that ξ
D
D
E ),
(4.9)
and therefore
π D )−2 > 0.
(4.10)
π D = 1, then, from (4.6),
π D )(D11 µ A + D21 ν A )ξ A , m a = −(χ D π D )(D12 µ A + D22 ν A )ξ A .
With
κ A := −(χ D π D )(D12 µ A + D22 ν A ), λ A = (χ D π D )(D11 µ A + D21 ν A ), the matrix
F = (χ D π D )
−D12 D11 −D22 D21
has determinant (χ D π D )2 det(D) = 1, by (4.10). Thus, F is an element of SL(2;R) and {κ A , λ A } is a spin frame. From (4.8–10), one checks that F is inverse to the element of SL(2;R) in (4.9) expressing {µ A , ν A } in terms of {α A , β A }, whence κ A = α A and λA = β A. In summary, the null tetrad (4.6) is a = α A π A = d x, m a = α A ξ A = −(D12 dr + D22 ds),
m˜ a = β A π A = dy, (4.11) n a = β A ξ A = D11 dr + D21 ds.
The complementary distributions are equivalent to an almost product structure P, an endomorphism of T M satisfying P 2 = 1. Here, P = 12 ⊕ −12 with respect to the decomposition T M = Dπ ⊕ Dχ . It is straightforward to check that P is in fact an anti-orthogonal automorphism of (T p M, g). By analogy with the Kähler form, define ω := g(P , ), which is indeed a two-form. In fact, one has: P 2 = 1, gab P a c P b d = −gcd , ωab := gcb P c a = −ωba ,
(4.12)
any two of which entails the third. The integrability of the two complementary distributions is equivalent to the integrability of the almost product structure (i.e., the vanishing of the Nijenhuis tensor of P) and equivalent to the integrability of the induced GL(2; R) × GL(2; R)-structure on the bundle of frames. It is well known that an integrable almost product structure (in any of these equivalent forms) is in turn equivalent to a locally product structure on M (see, for example, [47], Ch. XI); indeed, an atlas of coordinates of the form (r, s, x, y) as constructed above constitute an atlas for the locally product structure in the present circumstances; the Jacobian of the transformations between such coordinates systems M 0 must be of the form 0 N with M, N ∈ GL(2; R). The following result is also well known, e.g., [47], Ch. X.
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4.1. Lemma. The almost product structure is parallel iff the two complementary distributions are each parallel. So, now suppose that Dπ and Dχ are each parallel; in particular, (M, g) is a Walker geometry, in two ways. Now P is obviously parallel iff the associated two-form ω is parallel, whence ω is a symplectic form naturally associated with the complementary parallel totally null distributions. One computes ω = D11 dr ∧ d x + D12 dr ∧ dy + D21 ds ∧ d x + D22 ds ∧ dy.
(4.13)
In the more usual theory of (almost) product spaces in which the almost product structure is an orthogonal automorphism of the tangent space with respect to a metric which restricts to nondegenerate metrics on the complementary distributions, the condition ∇ P = 0 is equivalent to (M, g) being locally a Riemannian product ([47], Ch. X). In the present circumstances, we should therefore expect that ∇ P = 0 is equivalent to some condition on the metric. To investigate, it will prove convenient to write the (locally product) coordinates (r, s, x, y) derived above in the form (x a ) = (x α , x α ), where Greek indices range over 1 and 2 here. Then, c c ∇a ∂r = a1 ∂c , ∇a ∂s = a2 ∂c ; c c ∇a ∂x = a3 ∂c , ∇a ∂ y = a4 ∂c ;
from which it follows, from 1.1 and (4.3), that γ
Dπ is parallel iff aβ = 0
γ
Dχ is parallel iff aβ = 0
and
(4.14)
(see also [47], Ch. X). From (4.4), one directly computes that: γ
aβ = 21 g γ
m
γ µ
γ
aβ = 21 g γ m (gam,β + gβ m,a − gaβ ,m )
(gam,β + gβm,a − gaβ,m )
= 21 g (gaµ,β + gβµ,a − gaβ,µ ) = 21 g γ µ (gα µ,β − gα β,µ ) = 21 (D −1 )γ µ (τ D)α µ,β − (τ D)α β,µ ,
= 21 g γ µ (gaµ ,β + gβ µ ,a − gaβ ,µ ) = 21 g γ µ (gα µ ,β − gα β ,µ ) = 21 (τ D −1 )γ µ Dα µ ,β − Dα β ,µ ,
whence Dπ is parallel iff Dµα ,β = Dβα ,µ and Dχ is parallel iff Dα µ ,β = Dα β ,µ , i.e., Dπ is parallel iff
∂ Dµα ∂ Dβα ∂ Dα µ ∂ Dα β = and Dχ is parallel iff = . µ β ∂x ∂x ∂xµ ∂xβ
(4.15)
(The analogues of (4.15) for nondegenerate complementary distributions may be found in [47], Ch. X.) The nontrivial components of the second condition of(4.15) (β = µ ) are the integrability conditions for functions φα satisfying ∂φα /∂ x β = Dα β . Substituting this expression into the first condition of (4.15), the nontrivial components (β = µ) are now the integrability conditions for a function (context should preclude confusion with the Walker volume form; the reason for the choice of this symbol will become apparent shortly) satisfying r = ∂φ0 /∂ x α and s = ∂φ1 /∂ x α , whence Dα β =
∂φα ∂x
β
=
∂ 2
∂xα∂xβ
.
(4.16)
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Thus, the metric in locally product coordinates (x α , x α ), see (4.4), is determined by a single function according to (4.16). Note that spinors do not, in fact, play a necessary role in this derivation (so far in this section, spinors have played an essentially descriptive role) nor does the four dimensionality; stripping away all reference to spinors and the null tetrad (4.6), one obtains an argument valid for a neutral manifold of arbitrary dimension 2n with parallel complementary totally null distributions (now α and α each range over 1, . . . , n and the null tetrad is replaced by a Witt frame): 4.2. Theorem. Let (M, g) be a neutral manifold of dimension 2n admitting complementary totally null distributions. This structure is equivalent to an almost product structure P which is an anti-orthogonal automorphism of each (T p M, g). Integrability of P (vanishing of its Nijenhuis tensor) is equivalent to integrability of the two distributions, and coordinates (x α , x α ) which are simultaneously Frobenius for both distributions provide a locally product structure for (M, g). When P is parallel, equivalently the two distributions are parallel, with respect to the local product coordinates the metric takes the form (gab ) =
0n D τD 0 n
with D as in (4.16), and the associated two-form ω is a symplectic form with coordinate expression ω=
∂ 2 ∂ x α∂ x β
dxα ∧ dxβ .
Note that the canonical orientation induced by the one distribution is (−1)n times that of the other. We will refer to this geometry as double Walker geometry. It is already known, however, under other names. As an instance of locally product geometry, there is an obvious strong analogy with Kähler geometry. Indeed, an almost product structure P on a 2n-dimensional manifold whose eigenspaces are both of dimension n is also known as a paracomplex structure. When, in addition, M carries a (necessarily) neutral metric g compatible with P in the sense that P is an anti-orthogonal automorphism of each (T p M, g), then the triple (M, g, P) is called almost paraHermitian, paraHermitian when P is integrable, and paraKähler when ∇ P = 0 (that a parallel almost product structure P is indeed integrable is again analogous to the case for almost complex structures, deduced, in particular, by Walker [45–46]; see also [47]). The results above indicate this analogy is far reaching. Indeed, there is an extensive body of literature on paracomplex geometry, see the reviews [6,7]. (Almost) paraHermitian geometry has also been called biLagrangian geometry because the eigenspaces of P are totally null with respect to g, see [14]. These various names reflect different emphases upon g, P, and ω; and perhaps because the topic is not widely known. Indeed, Theorem 4.2 has been presented independently on several occasions, e.g., in [39] and [1], though it dates back to the earliest work in paracomplex geometry. Indeed, Rashevskij [33] studied the properties of a metric of the form (4.16) on a locally product 2n-dimensional manifold and then Rozenfeld [34] explicitly
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drew the parallel with Kähler geometry. Paracomplex geometry has multiple independent origins, [20–21] being notable; a short, but informative, history of the origins of the subject may be found in [7]. We suggest that (almost) paraHermitian geometry is most naturally construed as a special kind of neutral geometry. ParaKähler geometry is then a special kind of Walker geometry which we have called double Walker. Our intention here is not to simply make matters worse by adding yet further terminology. Both paraKähler and double Walker are useful terms which stress different aspects of this interesting geometry. But it is the underlying neutral geometry which is, in our opinion, the natural setting. It may be argued, however, that the root of these geometries is paracomplex geometry, which involves no metric. We would suggest that the significance of (almost) paracomplex geometry within (almost) product geometries stems from its algebraic origins in the paracomplex (also called Lorentz) numbers. Though one can treat this algebra purely algebraically, it is most naturally regarded as a ‘normed’ (-Euclidean) algebra, i.e., as an algebraic structure on R1,1 , and thus naturally a feature of neutral geometry (it is argued in [18] that R1,1 is best understood as a neutral geometry rather than two-dimensional Lorentzian geometry due to the role of anti-isometries). All this geometry also has a natural description as G-structures on the frame bundle F(M) of M. Paracomplex geometry is a GL(n; R) × GL(n; R)-structure. Let P(M) denote the bundle of almost-product frames for the given reduction. Adding the compatible neutral metric g allows one to construct from any almost-product frame, an almost-product frame that is also a Witt frame with respect to g and thus a further reduction to W (M), say. The group of this bundle of admissible frames is the intersection of GL(n; R) × GL(n; R) with O(n,n), but where the latter must be expressed in the form appropriate to Witt bases rather than -ON bases. The result is that the symmetry group of W (M) takes the form B 0n : B ∈ GL(n; R) , 0n τ B −1 and thus is isomorphic to GL(n;R). Thus, almost paraHermitian geometry and its refinements may be viewed as certain GL(n;R)-structures on 2n-dimensional manifolds. Returning now to the four-dimensional case of our primary concern, in which a double Walker geometry has an unambiguous canonical orientation, we encounter another co-incidence. Pleba˜nski [30] derived two canonical coordinate forms for a complex space-time (i.e., a four-dimensional complex manifold carrying a holomorphic Riemannian metric) whose only nontrivial curvature is the ASD Weyl curvature. Pleba˜nski’s results carry over to the real category to apply to neutral metrics in four dimensions. With respect to the coordinates (r, s, x, y), (4.16) takes the form r x r y D= sx sy which, given our choice of ordering of the coordinates, is of Pleba˜nski’s first heavenly form. Pleba˜nski’s [30] first result is that, when det(D) = 1 (Pleba˜nski’s first heavenly equation), this form is a canonical form for a neutral metric in four dimensions whose only nontrivial curvature is the ASD Weyl curvature. Such metrics are thereby given, locally, by a single function satisfying the first heavenly equation. What we have derived is the first heavenly form, without the constraint of the first heavenly equation, as a canonical form for a neutral metric with parallel complementary totally null distributions, and we recognize that this form is nothing but the paraKähler form of the
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metric for a paraKähler geometry. We note that (4.11) is a slight variation on Pleba˜nski’s heavenly null tetrad. To appreciate the significance of the first heavenly equation we state some definitions and facts well known to those familiar with complex general relativity and twistor theory, adapted to the context of neutral signature. But first we note in passing that Pleba˜nski’s first heavenly equation actually appeared many years earlier in [35]; see also [12]. 4.3. Definitions & facts. Let (M, g) be a four-dimensional neutral manifold. One can apply the Hodge star operator to the fully covariant Riemann tensor by applying it to either of the pair of (abstract) indices in which the Riemann tensor is skew to obtain two closely related ‘duals’, see [28], §4.6. It then turns out that the Riemann tensor is SD/ASD with respect to one of these notions of duality iff it is also SD/ASD with respect to the other, and that when SD/ASD, the Ricci curvature vanishes and the Riemann tensor equals the SD/ASD Weyl tensor. One says (M, g) is half-flat when the Riemann tensor is SD/ASD; specifically, right flat when the Riemann tensor is ASD, i.e., the only nontrivial curvature is the ASD Weyl curvature, and left flat when the Riemann tensor is SD, i.e., the only nontrivial curvature is the SD Weyl curvature. (M, g) itself is said to be SD/ASD according as the Weyl curvature is SD/ASD, a weaker condition. In (M, g), one can construct, locally, parallel primed/unprimed spin frames iff (M, g) is right/left flat. The forward implication follows by applying 3.1 to the two elements of the spin frame; the converse can be proved by a straightforward adaptation of the classical result that one can construct, locally, parallel ON frame fields iff the Riemann tensor vanishes (e.g., [37], pp. 261–263). In more modern terms, (M, g) is right/left flat iff the induced connexion on the bundle of primed/unprimed spinors is flat, see. e.g., [16], §II.9, or [37], pp. 402–403.
Now suppose that Dπ and Dχ admit parallel local scaled representatives π A and χ , at least on some common domain. On that domain, χ D π D is also constant, whence by a constant scaling of χ one can suppose, without loss of generality, that χ D π D = 1. For these choices of π A and χ A , one can choose, by 3.3, coordinates ( p, q, x, y) and (w, z, r, s) as in (4.1–2) and proceed with the previous construction up to (4.16). Of course, ξ A = χ A . By (4.10), det(D) = 1, i.e., Pleba˜nski’s first heavenly equation now holds. Moreover, the results of 3.1 must hold with respect to both π A and ξ A . But these ˜ A B C D , AB A B , and S must vanish, whence (M, g) is right flat. facts entail that What if we assume det(D) = 1, equivalently ξ A = χ A , in the construction leading up to (4.16) but not assume that parallel local scaled representatives are available? Pleba˜nski’s [30] result assures one that (M, g) is indeed right flat and that π A and ξ A , characterized through the heavenly tetrad (4.11), are indeed parallel. Rather than repeat this analysis, however, we return to the assumption that Dπ and ξ D , where from now on we employ the LSR ξ A of [χ A ] defined just after (4.10), are parallel complementary distributions without any presumption that either has parallel LSRs, i.e., to the double Walker (paraKähler) case. The next result follows immediately from 2.5 and demonstrates the utility of viewing the geometry as Walker geometry. A
4.4. Corollary. Let (M, g, Dπ , [π A ], Dξ , [ξ A ]) be a four-dimensional double Walker (paraKähler) geometry. Then [π A ] and [ξ A ] are each WPSs of multiplicity at least two, ˜ A B C D ∝ π(A π B ξC ξ D ) , i.e., ˜ A B C D is of type {22}Ia, or zero; S = 0 whence
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˜ A B C D is zero; and the Ricci spinor is of the form AB A B = B AB π(A ξ B ) . This iff geometry is therefore Einstein iff B AB = 0. We now note that, when π A can be chosen parallel, (3.8) substituted into (1.1) yields Pleba˜nski’s second heavenly form for the metric of a right-flat space, though without the constraint of Pleba˜nski’s second heavenly equation; rather (3.11) pertains. Hence, for a double Walker geometry, we follow Pleba˜nski [30] and define functions u := x , v := y .
(4.17)
Considering (u, v, x, y) as functions of (r, s, x, y), the Jacobian is ⎞ ⎛ xr xs x x x y τ D K ⎜ yr ys yx yy ⎟ , = ⎝ 0 0 1 0 ⎠ 02 12 0 0 0 1
(4.18)
which therefore has nonzero determinant. Thus, one may employ (u, v, x, y) as local coordinates. Now (4.1) is still valid. One computes du = xr dr + xs ds + x x d x + x y dy, dv = yr dr + ys ds + yx d x + yy dy,
(4.19)
whence one can rewrite (4.11) as: a = d x, m˜ a = dy, n a = D11 dr + D21 ds = r x dr + sx ds = du − x x d x − x y dy, (4.20) m a = −D12 dr − D22 ds = −r y dr − sy ds = −dv + yx d x + yy dy, and du = n a + x x a + x y m˜ a , dv = −m a + x y a + yy m˜ a .
(4.21)
One can now compute g ab with respect to the coordinates (u, v, x, y): g ab (du, du) = 2x x , g ab (du, dv) = 2x y , g ab (dv, dv) = 2 yy , g ab (du, d x) = 1, g ab (du, dy) = 0, g ab (dv, d x) = 0, g ab (dv, dy) = 1, g ab (d x, d x) = g ab (d x, dy) = g ab (dy, dy) = 0. Hence, one obtains, with respect to (u, v, x, y), Walker’s canonical form (1.1) for the metric with x x x y . (4.22) W = −2 x y yy
Thus, (u, v, x, y) are Walker coordinates for (M, g, Dπ , [π A ]) and the heavenly tetrad (4.11), equivalently (4.21), is exactly the Walker null tetrad (2.11) for these Walker coordinates. Hence, (2.21), (2.26), and (2.32–33) give the curvature spinors with respect to the spin frames associated to the heavenly tetrad but with 4.4 in force, so in fact S ˜ A B C D = π(A π B ξC ξ D ) , 2 AB A B = B AB π(A ξ B )
(4.23)
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with B AB as in (2.33). Moreover, the following equations must hold: B + Sc = 0,
0 = 6Bc − A + S(3c2 − 1) = 3Bc − A − S,
A AB = 0,
(4.24)
with A AB as in (2.33). Note that the first two equations entail S 2 + AS + 3B 2 = 0 in accord with 2.6(i). In particular, every four-dimensional paraKähler metric must locally be of this form. Note also, from (4.10) and (4.18) that [∂r , ∂s , ∂x , ∂v ] = [∂u , ∂v , ∂x , ∂v ], i.e., the locally product coordinates have the canonical orientation of the double Walker geometry. Suppose now that [π A ] has a parallel LSR. One can employ 3.3 to choose the original set of coordinates ( p, q, x, y) satisfying (4.1) with that parallel LSR π A . Then one can introduce ϑ as in (3.7) and W takes the form (3.8), where ϑ is subject to (3.12). The ˜ A B C D = 0 by (4.23) ASD Weyl curvature is given by (3.15). By 3.1, S = 0 whence A A (equivalently, [π ] is a WPS of multiplicity four but ξ is also a WPS of multiplicity two; impossible). It also follows from 3.1, and 2.5 applied to ξ A , that AB A B = 0 (equivalently, combine (3.16) and (4.23)). From (3.17), δ A δ B f = 0. Thus, if only Dπ has a parallel LSR, then a double Walker geometry (M, g, Dπ , [π A ], Dξ , [ξ A ]) is right flat. It then follows from 3.2 that ξ A can be rescaled so as to be parallel, say f ξ A is parallel. Then f = f ξ D π D is constant, whence ξ A itself must be parallel. Although the description of the double Walker geometry relative to Walker coordinates is asymmetrical in the roles played by Dπ and Dξ , the geometry itself is not and one can of course reverse the roles. Hence: 4.5. Theorem. Given a four-dimensional double Walker (paraKähler) geometry, if either distribution admits a parallel LSR, the geometry is right flat on that domain and the other distribution also admits a parallel LSR on that domain.
Consequently, suppose now that both π A and ξ A in (4.11 & 20) are parallel. Then a m ∧ n b = AB ξ A ξ B is parallel. One computes ma ∧ nb =
ab − c2 c a b ∂u ∧ ∂v + ∂u ∧ ∂x − ∂u ∧ ∂ y + ∂v ∧ ∂x 4 2 2 2 c − ∂v ∧ ∂ y + ∂ x ∧ ∂ y . 2
(4.25)
Using (A1.8), one computes (4.26) ∇1 (m a ∧ n b ) = 0 = ∇2 (m a ∧ n b ), 2 (c − ab) b a 1 ∂u ∧ ∂v − ∂v ∧ ∂x + ∂u ∧ ∂ y − ∂x ∧ ∂ y ∇3 (m a ∧ n b ) = (au + cv ) 8 4 4 2 2cx − 2a y − (c2 )v + bav − acu ∂u ∧ ∂ x 4 2cx − 2a y + bav − acu + 2cau ∂v ∧ ∂ y , + 4 +
(4.27)
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(c2 − ab) b a 1 ∇4 (m ∧ n ) = (bv + cu ) ∂u ∧ ∂v − ∂v ∧ ∂x + ∂u ∧ ∂ y − ∂x ∧ ∂ y 8 4 4 2 2bx − 2c y − 2cbv + bcv − abu ∂u ∧ ∂ x + 4 2bx − 2c y + (c2 )u + bcv − abu ∂v ∧ ∂ y . + (4.28) 4 a
b
Hence, under (3.2), and with P as in (3.12), (4.27–28) become ∇3 (m a ∧ n b ) = Pv (∂u ∧ ∂x + ∂v ∧ ∂ y ),
∇4 (m a ∧ n b ) = −Pu (∂u ∧ ∂x + ∂v ∧ ∂ y ),
from which one deduces that ξ A is parallel iff Pu = Pv = 0, i.e., P depends only on x and y. Following Pleba˜nski [30] again, let F be an antiderivative of P with respect to x: Fx := P. Define := ϑ − u F. Then, uu = ϑuu , uv = ϑuv , vv = ϑvv , i.e., (3.8) holds with replacing ϑ, as do (3.13 & 15). Of course, δ A δ B P = 0. Finally, ux + vy − ( uv )2 + uu vv = ϑux − P + ϑvy − (ϑuv )2 + ϑuu ϑvv = 0, (4.29) which is Pleba˜nski’s second heavenly equation. Thus, Pleba˜nski’s second heavenly form is the special case of Walker’s canonical form for a double Walker (paraKähler) geometry with parallel LSRs for one, hence both, distributions. Of course, every right-flat neutral four-fold is, locally, such a double Walker geometry in many ways.
5. Global Lifts of [π A ] Our considerations so far have been essentially local in nature. In this section we con sider a four-dimensional Walker geometry (M, g, D, [π A ]) for which the frame bundle admits a reduction with structure group SO+ (2, 2). As noted just prior to 2.2, with is well defined globally, respect to this reduction the bundle of projective spinors PS M A so the spinor field [π ] corresponding to the Walker distribution D is a global section . The question we address here is whether this section can be lifted to a section of PS M ? Obviously, we must assume the bundle S exists which, given the of the bundle S M M SO+ (2, 2)-reduction, requires that the second Stiefel-Whitney class of M vanishes. We will also assume M is connected and paracompact. Before proceeding, we say a few words about our assumptions. One can reformulate the existence of an SO+ (2, 2)-reduction in various ways, most obviously as the existence of a distribution of oriented two-planes, for which, when M is compact, there are well known necessary and sufficient topological conditions. Matsushita has studied these issues and we direct the reader to [24] for a recent review. In general, the assumption of a Walker geometry admitting an SO+ (2, 2)-reduction cannot have a purely topological characterization as evidenced by the explicit examples in the literature cited in the Introduction of such Walker geometries on R4 . It would, however, be of interest to determine in the compact case any topological conditions imposed by the existence of the Walker geometry beyond those equivalent to an O(2, 2)- or SO+ (2, 2)-reduction. Returning to the question of a global lifting of [π A ], we first note that if π A is such a global lift, then AB π A π B is globally defined and defines an orientation for D, see 2.4. Thus, orientability of D is a necessary condition. We elaborate on this point at the end of the section. By an open covering of the Walker geometry, we shall mean an open
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covering U = { Ui : i ∈ I } of M such that each Ui carries an LSR πiA of [π A ]. We can frame our question as: given an open covering of the Walker geometry, can one scale the LSRs so that they agree on nontrivial intersections Ui j := Ui ∩ U j , i, j ∈ I? 5.1. Lemma. With notation and assumptions as in the previous paragraphs, the obstruc is a specific element π in H 1 (M, C ∗ ), where C ∗ is tion to a global lifting of [π A ] to S M the sheaf of germs of nowhere-vanishing smooth functions on M. Thus, whether a global lift of [π A ] exists or not is determined by the topology of M and the sheaf C ∗ . Proof. For any open covering of the Walker geometry, on a nontrivial intersection Ui j one has πiA = f i j π jA , where f i j is a nowhere-vanishing, smooth function on Ui j . One ˇ easily checks that Ui j → f i j defines a Cech 1-cocycle with coefficients in C ∗ on U. This assignment is well defined under restriction to refining coverings, i.e., on passing to a refinement, the induced LSRs define the 1-cocycle obtained by restricting the original 1-cocycle to the refinement. It is routine to confirm that given two open coverings of the Walker geometry, with open coverings U and V of M respectively, the 1-cocycles induced on a common refinement of U and V are cohomologous. Hence, the possible open coverings of the Walker geometry define a certain element π of H 1 (M, C ∗ ). Since H 1 (U, S) → H 1 (M, S) is injective for any sheaf S, if π is zero then for any open covering of the Walker geometry, f i j is a coboundary, f i j = (δh)i j = h j h i−1 . Then h i πiA = h j π jA , which thereby defines a global lift of [π A ].
Conversely if π A is a global lift of [π A ] then one can construct open coverings of the Walker geometry with f i j = 1, whence the 1-cocycle is a coboundary: f i j = (δh)i j = h j / h i , where h i ≡ 1, for all i, i.e., π = 0. Thus, π ∈ H 1 (M, C ∗ ) is the obstruction to a global lifting of [π A ].
If a Walker geometry (M, g, D, [π A ]) satisfies the curvature conditions of 3.1, then 3.2 says one can construct an open covering of the Walker geometry with parallel LSRs. Call such a parallel open covering of the Walker geometry. One can repeat the argument of 5.1, the difference being that the 1-cocycle takes coefficients in a different sheaf. 5.2. Lemma. If a Walker geometry admits parallel open coverings, the obstruction to constructing a global parallel lifting of [π A ] is a certain cohomology class ρ in H 1 (M, R∗ ), where R∗ is the constant (multiplicative) sheaf of nonzero real numbers (the sheaf of germs of locally constant R∗ -valued functions). To explore these obstructions, consider the commutative diagram of short exact sequences: j
ψ
0 −→ S⏐0 !→ R⏐∗ −→ ⏐p ⏐ 1 i φ 0 0 −→ S !→ C ∗ −→
R⏐+ −→ 0 ⏐q C + −→ 0
(5.1)
where S0 is the constant multiplicative sheaf with fibres isomorphic as groups to Z2 , R+ is the constant (multiplicative) sheaf of positive real numbers, C + is the sheaf of germs
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of positive R-valued smooth functions, both φ and ψ are f → | f | for the appropriate domain, and the remaining mappings are the obvious inclusions. If C denotes the sheaf germs of smooth functions, exp : C → C + is a sheaf isomorphism, whence these two sheaves have the same cohomology. Since C is a fine sheaf, H p (M, C + ) = 0 for p ≥ 1. Also exp : R → R+ is a sheaf isomorphism, whence p H p (M, R+ ) ∼ = H p (M, R) ∼ = HDR (M), where the last is the de Rham cohomology of M. One obtains from the long exact cohomology sequences the commutative diagram: j∗
ψ∗
0
−→
H 0 (M,S0 )
!→
H 0 (M,R∗ )
−→
0
−→
H 0 (M,S0 )
!→
i∗
H 0 (M,C ∗ )
−→
⏐ ⏐ 1
⏐ ⏐p ∗
φ∗
δ∗
H 0 (M,R+ )
−→
H 0 (M,C + )
−→
⏐ ⏐q ∗
δ∗
j∗
ψ∗
H 1 (M,S0 )
−→
H 1 (M,R∗ )
−→
H 1 (M,S0 )
−→
i∗
H 1 (M,C ∗ )
−→
⏐ ⏐ 1
⏐ ⏐p ∗
φ∗
δ∗
H 1 (M,R+ )
−→
H 1 (M,C + )=0
−→
⏐ ⏐q ∗
δ∗
(5.2)
Beyond the portion shown, the bottom row is just δ∗
φ∗
i∗
0 −→ H p (M, S0 ) −→ H p (M, C ∗ ) −→ 0 for each p > 1, whence i ∗ is clearly an isomorphism for p > 1. Since C + ⊂ C ∗ and R+ ⊂ R∗ , a cohomology class c of H p (M, C + ) also defines an element d of H p (M, C ∗ ) which is mapped by φ∗ to c, i.e., φ∗ is always onto. Similarly, ψ∗ is always onto. It follows by exactness, equivalently by their very definition, that all the connecting homomorphisms δ∗ are trivial mappings. Since φ∗ : H 1 (M, C ∗ ) → H 1 (M, C + ) is a surjection onto a trivial space and δ∗ a trivial mapping, it follows that i ∗ : H 1 (M, S0 ) → H 1 (M, C ∗ ) is an isomorphism. Hence, for p > 0, H p (M, C ∗ ) ∼ = H p (M, S0 ) ∼ = H p (M, Z2 ).
(5.3)
Since the δ∗ ’s are trivial, one can isolate the following commutative diagram, with exact rows, from (5.2): j∗
ψ∗
0 1 ∗ 1 + 0 −→ H 1 (M, ⏐ R ) −→ H (M, ⏐ R ) −→ 0 ⏐ S ) −→ H (M, ⏐p ⏐q ⏐ ∗ ∗ 1 φ∗ i∗ 1 0 1 ∗ 1 0 −→ H (M, S ) −→ H (M, C ) −→ H (M, C + ) = 0 −→ 0
(5.4)
H 1 (M, R∗ ) 1 HDR . (M) ∼ = H 1 (M, R) ∼ = H 1 (M, R+ ) ∼ = 1 H (M, S0 )
(5.5)
Thus
5.3. Theorem. Suppose M is connected and paracompact. Let (M, g, D, [π A ]) be a four-dimensional Walker geometry. Suppose M admits an SO+ (2, 2)-reduction of the . Supframe bundle, whence [π A ] is a global section of the projective spinor bundle PS M pose further that w2 (M) = 0, whence (M, g) admits spinor structures. The obstruction is an element of to a global lifting of [π A ] to a section of S M H 1 (M, C ∗ ) ∼ = H 1 (M, Z2 ) ∼ = Hom (H1 (M, Z), Z2 ) ∼ = Hom (π1 (M), Z2 ) . (5.6) The nontrivial elements of Hom (π1 (M), Z2 ) are in bijective correspondence with the subgroups of π1 (M) of index two, which are, in turn, in bijective correspondence with the isomorphism classes of two-fold covering spaces of M.
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If the curvature conditions of 3.1 pertain, the open coverings of the Walker geometry define π ∈ H 1 (M, C ∗ ) and the parallel open coverings define ρ ∈ H 1 (M, R∗ ); 1 (M) vanishes. From the latter group is isomorphic to H 1 (M, Z2 ) precisely when HDR (5.4), π = i ∗ (β) = ( p∗ ◦ j∗ )(β) = p∗ (ρ), for a unique β ∈ H 1 (M, S0 ), whence ρ − j∗ (β) ∈ ker p∗ . If π = 0, then β = 0 and ρ ∈ ker p∗ but is not necessarily zero, i.e., [π A ] may have a global lifting but not a parallel global lifting, even when the con1 (M) = 0, however, then j , whence p , are isomorphisms, ditions of 3.1 pertain. If HDR ∗ ∗ ρ = j∗ (β), p∗ (ρ) = π , and π = 0 iff β = 0 iff ρ = 0. Remarks. The isomorphisms in (5.6) and thereafter are standard interpretations of H 1 (M, Z2 ) following, in the first instance, from the Universal Coefficient Theorem. When the curvature conditions of 3.1 pertain, one can construct a parallel open covering using a simple covering in the sense of [16], pp. 167–168. The cohomology of M ˇ is then given by the Cech cohomology with respect to this covering. Let { f i j } be the 1-cocycle obtained from the parallel open covering, which has coefficients in R∗ ⊂ C ∗ , i.e., it represents both ρ and π . Now | f i j | defines a 1-cocycle with coefficients in C + . As H 1 (M, C + ) = 0, then | f i j | = h j / h i , for some 0-cochain {h i } with coefficients in C + . But { f i j /| f i j |} is also a 1-cocycle with coefficients in S0 ⊂ C ∗ , whence { f i j } is cohomologous to { f i j /| f i j |} = { f i j h j / h i } as 1-cocycles with coefficients in C ∗ and { f i j /| f i j |} represents β. If π = 0, then f i j = f j / f i , for some 0-cochain { f j } with coefficients in C ∗ . Then | f i j | = | f j |/| f i | and f i j /| f i j | = ( f j /| f j |)( f i /| f i |)−1 , i.e., β is trivial. But { f i j } may not be a coboundary with coefficients in R∗ . Now | f i j | also 1 (M) ∼ H 1 (M, R+ ) = 0, then defines a 1-cocycle with coefficients in R+ , so if HDR = | f i j | = g j /gi , where {gi } is a 0-cochain with coefficients in R+ . Now { f i j } is cohomologous to { f i j /| f i j |} = { f i j g j /gi } as 1-cocycles with coefficients in R∗ , i.e., j∗ (β) = ρ. Now π = 0 iff β = 0 iff ρ = 0. Finally, if { f i j } is the 1-cocycle defined by an open covering of the Walker geometry, so πiA = f i j π jA , then i := AB πiA πiB = ( f i j )2 AB π jA π jB =: ( f i j )2 j . Now ( f i j )2 is a smooth, positive function, which indicates D must be orientable. Indeed, ( f i j )2 /(| f i j |)2 is a one-cocycle with coefficients in S0 which everywhere takes the trivial value and thus belongs to the trivial cohomology class. In other words, {( f i j /| f i j |)2 } must represent the first Stiefel-Whitney class of D viewed as a bundle. We thus see that D must be orientable merely because we have assumed the existence of a global spinor bundle, since this fact allows one to compare LSRs on overlaps and deduce that the i s are positive multiples of each other. On the other hand, the cohomology class π = { f i j /| f i j |} need not be trivial, so orientability of D is merely a necessary condition of the context in which the question of global lifts arises. Acknowledgements. We thank Mike Eastwood for a useful tip in connexion with §5, Pedro Gadea for kindly and expeditiously providing us with copies of his papers on paracomplex geometry, and an anonymous referee for remarks that improved the paper.
Appendix One: Local Geometry with Respect to Walker Coordinates In this appendix we record, for ease of reference in this and other papers, local coordinate expressions for standard geometric objects. We also specify our choice of conventions and in this sense standardize the coordinate expressions. We employ the abstract index notation of [28]; italic indices will be ‘abstract’, i.e., serve to denote the tensor space to which the object they are attached to belongs, while
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bold upright indices are ‘concrete’, i.e., they take numerical values and typically serve to label components of geometric objects with respect to some basis or the elements of a basis themselves. We also employ the standard summation convention for concrete indices when convenient; while repeated abstract indices in a formula, one as superscript and one as subscript, indicates the standard pairing between a linear space and its dual. For exterior algebras, of the two conventions commonly found in the literature, we employ those of [40] (and many other texts such as [36]). This choice simplifies the expression for volume forms and elements. We shall, however, retain the definition of symmetrization and skew symmetrization, denoted by round and square brackets respectively, of abstract and concrete indices employed in [28]. Thus, if va and wb are two vectors, then v a ∧ w b = v a ⊗ w b − wa ⊗ v b = 2v [a w b] .
(A1.1)
Given a real linear space V equipped with a scalar product g (of arbitrary signature), the induced scalar product on the exterior algebra p (V) of p-vectors is given by (U, W ) :=
1 a1 ...a p b1 ...b p U W ga1 b1 . . . ga p b p , p!
(A1.2)
where U a1 ...a p and W b1 ...b p denote the multivectors U and W as tensors. The scalar product gab induces an identification of V with its linear dual V• by v → g(v, ), which we shall call the correlation ξg : V → V• with inverse ξg−1 : V• → V. These identifications extend to arbitrary tensor spaces and are conveniently represented by index lowering and raising via gab and g ab using abstract indices, where g ab is the scalar product induced on V• in the standard way. The induced scalar product on the exterior algebra p (V• ) is given by a formula analogous to (A1.2) with forms replacing multivectors and g ab replacing gab . The definition (A1.2) ensures that if {v1 , . . . , vn } is a pseudo-orthonormal (-ON) basis for V then the multivectors vi1 ∧ . . . ∧ vi p , i 1 < · · · < i p , form a -ON basis for p (V). Now suppose g is of signature (r, s), r +s = n, that V is oriented, and that {v1 , . . . , vn } is an oriented -ON basis with dual basis {φ 1 , . . . , φ n }. Putting ν j := ξg (v j ), then ν j = j φ j , where j = ± according as j ≤ r or j > r . The orientation class [v1 , . . . , vn ] can also be represented by the equivalence class of v1 ∧. . .∧vn in n (V)/R+ or, equivalently, by the equivalence class of φ 1 ∧. . .∧φ n in n (V• )/R+ (as those n-forms which are positive when evaluated on oriented bases). The n-fold wedge product of the elements of any oriented -ON basis yield one and the same n-vector in n (V). This element we call the volume element of (V, g) and represent as a tensor by V a1 ...an . Similarly, the dual bases of oriented -ON bases all generate the same n-form φ 1 ∧ . . . φ n , which we call the volume form of (V, g) and denote by ea1 ...en as a tensor. Note that Va1 ...an := ga1 b1 . . . gan bn V b1 ...bn = ξg (v1 ∧ . . . ∧ vn ) = ν 1 ∧ . . . ∧ ν n = (−1)s ea1 ...an . (A1.3) The Hodge star operator is defined in the usual way on p (V• ) by ∗ α :=
1 ei ...i α j ... j g i1 j1 . . . g i p j p , p! 1 n i p
(A1.4)
where αa1 ...a p denotes the p-form α as a tensor. The well known formula α ∧ ∗β = (α, β)ea1 ...an pertains, where (α, β) is the induced scalar product on p (V• ). Similarly,
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one defines the Hodge star operator on p (V) by ∗ U :=
1 i1 ...in ji ... j p V U gi1 j1 . . . gi p j p , p!
(A1.5)
and U ∧ ∗W = (U, W )V a1 ...an . These two Hodge star operators are related as follows: ξg (∗U ) = (−1)s ∗ ξg (U ).
(A1.6)
When the context is clear, we may write either p (V) or p (V• ) simply as p . In the four-dimensional case, 2± will denote the subspaces of self dual(SD)/anti-self dual (ASD) multivectors or forms. Now let (M, g, D) be a Walker four-manifold. We typically denote a set of Walker coordinates by (u, v, x, y), but it is preferable to write coordinate expressions for geometrical objects in a form independent of the choice of letters used; to this end the Walker coordinates will be designated by the numerals 1, 2, 3, 4 respectively, when convenient; in particular, ∂1 := ∂u , ∂2 := ∂v , ∂3 := ∂x , and ∂4 := ∂ y . The canonical form of the metric is given in (1.1) and (1.7).
A1.1. The Christoffel symbols. The Christoffel symbols in a Walker coordinate system are: i i i = 12 = 22 = 0, 11 ⎧1 ⎧1 c1 , i = 1; ⎪ ⎪ ⎪ 2 a1 , i = 1; ⎪ ⎨ ⎨2 i i 1 1 14 = 2 b1 , i = 2; 13 = 2 c1 , i = 2; ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 0, otherwise; 0, otherwise; ⎧1 ⎧1 a2 , i = 1; c2 , i = 1; ⎪ ⎪ ⎪ ⎪ ⎨2 ⎨2 i i 1 1 24 = 2 b2 , i = 2; 23 = 2 c2 , i = 2; ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 0, otherwise; 0, otherwise; ⎧ ⎧ 1 1 ⎪ ⎪ (aa + ca + a ), i = 1; 1 2 3 ⎪ ⎪ 2 2 (a4 + ac1 + cc2 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 (2c3 + ca1 + ba2 − a4 ), i = 2; ⎨ 1 (b3 + cc1 + bc2 ), 2 2 i i 33 = 34 = 1 ⎪ ⎪ ⎪ ⎪ i = 3; − 2 a1 , − 21 c1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩− 1 a , ⎩− 1 c , i = 4; 2 2 2 2 ⎧ 1 ⎪ ⎪ 2 (2c4 + ab1 + cb2 − b3 ), i = 1; ⎪ ⎪ ⎪ ⎪ ⎨ 1 (b4 + cb1 + bb2 ), i=2 2 i 44 = ⎪ ⎪ − 21 b1 , i = 3; ⎪ ⎪ ⎪ ⎪ ⎩− 1 b , i = 4. 2 2
i = 1; i = 2; i = 3; i = 4;
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A1.2. The geodesic equations. With X a = (u, v, x, y), the Lagrangian is L := (1/2)gab X˙ a X˙ b = u˙ x˙ + v˙ y˙ +
a 2 b 2 (x) ˙ + ( y˙ ) + c x˙ y˙ , 2 2
which is invariant under the interchange u ↔ v, x ↔ y, a ↔ b.
(A1.7)
Note that this interchange constitutes a fundamental symmetry of Walker coordinates which all Walker coordinate expressions must manifest. The Euler equations are:
∂L ∂ u˙ ∂L ∂ v˙ ∂L ∂ x˙
∂L ∂ y˙
· · ·
·
−
au ∂L bu = x¨ − (x) ˙ 2 − ( y˙ )2 − cu x˙ y˙ = 0, ∂u 2 2
−
av ∂L bv = y¨ − (x) ˙ 2 − ( y˙ )2 − cv x˙ y˙ = 0, ∂v 2 2
−
∂L = u¨ + au u˙ x˙ + av v˙ x˙ + cu u˙ y˙ + cv v˙ y˙ ∂x 1 + (aau + cav + ax )(x) ˙ 2 2 1 + (abu + cbv − bx + 2c y )( y˙ )2 + (a y + acu + ccv )x˙ y˙ = 0, 2
−
∂L = v¨ + cu u˙ x˙ + cv v˙ x˙ + bu u˙ y˙ + bv v˙ y˙ ∂y 1 + (cau + bav + 2cx − a y )(x) ˙ 2 2 1 + (cbu + bbv + b y )( y˙ )2 + (bx + ccu + bcv )x˙ y˙ = 0, 2
from which in fact the Christoffel symbols may be directly read off. The geodesic equations were previously published in [13]. In particular, putting x = constant, y = constant reduces these four equations to the pair u¨ = 0, v¨ = 0, i.e., (u, v, x, y) = (αs + β, γ s + δ, µ, ν), for any constants α, β, γ , δ, µ and ν, is a null geodesic lying in the integral surface of D through (β, δ, µ, ν). As the ratio α/γ varies, one obtains a one-parameter family of null geodesics lying in, and sweeping out, this α-surface. Writing ∇i := ∇∂i , and noting that of course ∇i ∂j = ijk ∂k = ∇j ∂i , one computes the covariant derivatives of the coordinate basis:
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∇1 ∂1 = 0, ∇2 ∂2 = 0, ∇2 ∂1 = 0 = ∇1 ∂2 , 1 1 1 ∇3 ∂3 = (a3 + ca2 + aa1 )∂1 + (2c3 − a4 + ba2 + ca1 )∂2 − (a1 ∂3 + a2 ∂4 ), 2 2 2 1 1 1 ∇4 ∂4 = (ab1 + cb2 − b3 + 2c4 )∂1 + (b4 + cb1 + bb2 )∂2 − (b1 ∂3 + b2 ∂4 ), 2 2 2 1 1 ∇1 ∂3 = (a1 ∂1 + c1 ∂2 ) = ∇3 ∂1 , ∇1 ∂4 = (c1 ∂1 + b1 ∂2 ) = ∇4 ∂1 , 2 2 (A1.8) 1 1 ∇2 ∂3 = (a2 ∂1 + c2 ∂2 ) = ∇3 ∂2 , ∇2 ∂4 = (c2 ∂1 + b2 ∂2 ) = ∇4 ∂2 , 2 2 1 1 1 ∇3 ∂4 = (a4 + ac1 + cc2 )∂1 + (b3 + cc1 + bc2 )∂2 − (c1 ∂3 + c2 ∂4 ) = ∇4 ∂3 . 2 2 2 One confirms that D = ∂u , ∂v R is indeed parallel. If ∂u and ∂v are actually parallel, then a, b & c depend only on x and y. In fact, Walker [42,44] asserted that in this case, one can actually choose the coordinates (u, v, x, y) so that a = c = 0 and b is a function of x & y only.
A1.3. Riemann Curvature. We use the curvature conventions of [27], which for the Riemann curvature are R(X, Y )Z = ∇[X,Y ] Z − [∇ X , ∇Y ]Z , Rijkl := R(X i , X j , X k , X l ) := g R(X k , X l )X j , X i ,
(A1.9) (A1.10)
agreeing with those for the Riemann curvature in [28]. The coordinate expression for R i jkl is then: i i m i i R i jkl = kj,l − lj,k + kj lm − ljm km .
Direct computation yields: R i j12 = 0, ⎧ 1 ⎧ 1 ⎪ ⎪ ⎨− 2 a11 , i = 1; ⎨− 2 a12 , i = 1; i i R 113 = − 1 c11 , i = 2; R 213 = − 1 c12 , i = 2; ⎪ ⎪ ⎩ 2 ⎩ 2 0, otherwise; 0, otherwise; ⎧ 1 ⎪ − 2 (aa11 + ca12 ), i = 1; ⎪ ⎪ ⎪ ⎪ ⎪ ⎨− 1 (2c13 + 2ca11 + 2ba12 − 2a14 + b1 a2 − c1 c2 ), i = 2; 4 R i 313 = 1 ⎪ ⎪ i = 3; ⎪ 2 a11 , ⎪ ⎪ ⎪ ⎩1a , i = 4; 2 12
(A1.11)
Spinor Approach to Walker Geometry
R i 413 =
R i 114
R i 314
R i 323
i = 1; i = 2; i = 3;
i = 4; 2 12 ⎧ ⎪ − 41 2c14 + 2ab11 + 2cb12 − 2b13 + a1 b1 + b2 c1 − b1 c2 − (c1 )2 , i = 1; ⎪ ⎪ ⎪ ⎪ ⎪ ⎨− 1 (cb11 + bb12 ), i = 2; 2
1 ⎪ ⎪ ⎪ 2 b11 , ⎪ ⎪ ⎪ ⎩1b , 2 12 ⎧ 1 ⎧ 1 ⎪ ⎪ ⎨− 2 a12 , i = 1; ⎨− 2 a22 , i = 1; i R 223 = − 1 c22 , i = 2; = − 1 c12 , i = 2; ⎪ ⎪ ⎩ 2 ⎩ 2 0, otherwise; 0, otherwise; ⎧ 1 ⎪ ⎪− 2 (aa12 + ca22 ), ⎪ ⎪ ⎪ ⎪− 1 2c + 2ca + 2ba − 2a + a c + a b − a c − (c )2 , ⎨ 23 12 22 24 1 2 2 2 2 1 2 4 = 1 ⎪ ⎪ ⎪ 2 a12 , ⎪ ⎪ ⎪ ⎩1a ,
R i 423 =
R i 124
⎧ 1 ⎪ ⎪ 4 (2c13 − a14 − ac11 − 2cc12 + a2 b1 − c1 c2 ), i = 1; ⎪ ⎪ ⎪ ⎪ ⎨− 1 (cc11 + bc12 ), i = 2; 2
⎪ ⎪ − 21 c11 , i = 3; ⎪ ⎪ ⎪ ⎪ ⎩− 1 c , i = 4; 2 12 ⎧ 1 ⎧ 1 ⎪ ⎪ ⎨− 2 c11 , i = 1; ⎨− 2 c12 , i = 1; i = − 1 b11 , i = 2 R 214 = − 1 b12 , i = 2 ⎪ 2 ⎪ ⎩ ⎩ 2 0, otherwise; 0, otherwise; ⎧ ⎪ − 21 (ac11 + cc12 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 2c14 − 2b13 − 2cc11 − 2bc12 + a1 b1 − b1 c2 + b2 c1 − (c1 )2 , 4 = 1 ⎪ ⎪ ⎪ 2 c11 , ⎪ ⎪ ⎪ ⎩1c ,
R i 414 =
R i 123
613
i=3 i = 4;
i = 1; i = 2; i = 3;
i = 4; 2 22 ⎧ 1 2 ⎪ ⎪ 4 2c23 − 2a24 − 2ac11 − 2cc22 − a2 c1 + a1 c2 + a2 b2 − (c2 ) , i = 1; ⎪ ⎪ ⎪ ⎪ ⎨− 1 (cc12 + bc22 ), i = 2; 2
1 ⎪ ⎪ ⎪ 2 c12 , ⎪ ⎪ ⎪1 ⎩ 2 c22 , ⎧ 1 ⎪ ⎨− 2 c12 , i = 1; = − 1 b12 , i = 2; ⎪ 2 ⎩ 0, otherwise;
i = 3; i = 4; R i 224
⎧ 1 ⎪ ⎨− 2 c22 , i = 1; = − 1 b22 , i = 2; ⎪ 2 ⎩ 0, otherwise;
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R i 324 =
R i 424 =
R i 134
R
i
234
R i 334
⎧ 1 ⎪ ⎪− 2 (ac12 + cc22 ), ⎪ ⎪ ⎪ ⎪ ⎨ 1 (2c − 2b − 2cc 24
4
32
12
i = 1; − 2bc22 + a2 b1 − c1 c2 ), i = 2;
1 ⎪ ⎪ i = 3; ⎪ 2 c12 , ⎪ ⎪ ⎪ ⎩1c , i = 4; 2 22 ⎧ 1 ⎪ ⎪− 4 (2c24 + 2ab12 + 2cb22 − 2b23 + a2 b1 − c1 c2 ), i = 1; ⎪ ⎪ ⎪ ⎪ ⎨− 1 (cb12 + bb22 ), i = 2; 2
1 ⎪ ⎪ ⎪ 2 b12 , ⎪ ⎪ ⎪1 ⎩ 2 b22 , ⎧ 1 ⎪ − 2c13 + c1 c2 − a2 b1 ), ⎨ 4 (2a 14 = 41 2c14 − 2b13 + a1 b1 + b2 c1 − b1 c2 − (c1 )2 , ⎪ ⎩0, ⎧ 1 2 ⎪ ⎨ 4 2a24 − 2c23 + a2 c1 − a1 c2 − b2 c2 + (c2 ) , = 41 (2c24 − 2b23 + a2 b1 − c1 c2 ), ⎪ ⎩0,
i = 3; i = 4; i = 1; i = 2; otherwise; i = 1; i = 2; otherwise;
⎧ 1 2 ⎪ ⎪ 4 2aa14 +2ca24 −2ac13 −2cc23 +ca2 c1 +ac1 c2 −ca1 c2 −aa2 b1 −ca2 b2 +c(c2 ) , ⎪ ⎪ 1 ⎪ ⎪ ⎪ 2 (2c34 + a1 c4 − b3 c2 + b2 c3 − a4 c1 + ca14 + ba24 − cc13 − bc23 − a44 − b33 ) ⎨ = + 41 a2 b4 +a3 b1 −a4 b2 −a1 b3 +aa1 b1 +ca1 b2 −cc1 c2 + ba2 c1 −ba1 c2 −a(c1 )2 , ⎪ ⎪ ⎪ 1 ⎪ ⎪ 14 − 2c13 − a2 b1 + c1 c2 ), ⎪− 4 (2a ⎪ ⎩ 1 − 4 2a24 − 2c23 − a1 c2 + a2 c1 − a2 b2 + (c2 )2 ,
⎧ ⎪ − 1 (2c +a c −a c +b c −b c −ac14 − cc24 + ab13 + cb23 − a44 − b33 ) ⎪ ⎪ 2 34 1 4 4 1 2 3 3 2 ⎪ ⎪ 1 ⎪ 2 ⎪ ⎨ − 4 a3 b1 −a1 b3 +a2 b4 −a4 b2 −cc1 c2 +ab1 c2 +ca1 b2 +ba2 b2 −ac1 b2 −b(c2 ) , R i 434 = − 1 2bb23 +2cb13 −2bc24 −2cc14 +cb1 c2 +bc1 c2 −cb2 c1 −bb1 a2 −cb1 a1 +c(c1 )2 , ⎪ ⎪ 1 4 ⎪ 2 ⎪ ⎪ ⎪ 4 2b13 − 2c14 + b1 c2 − b2 c1 − a1 b1 + (c1 ) , ⎪ ⎩1 4 (2b23 − 2c24 − a2 b1 + c1 c2 ),
i = 1; i = 2; i = 3; i = 4; i = 1; i = 2; i = 3; i = 4.
A1.4. Fully covariant Riemann curvature. Straightforward computation from A1.3 yields: R12jk = 0, 1 1 R1313 = a11 , R1314 = c11 , R1323 = 2 2 1 R1334 = − (2a14 − 2c13 − a2 b1 + c1 c2 ), 4 1 1 R1414 = b11 , R1423 = c12 , R1424 = 2 2
1 a12 , 2
1 b12 , 2
R1324 =
1 c12 , 2
Spinor Approach to Walker Geometry
R1434 = R2323 = R2334 = R2424 = R3434 =
615
1 2b13 − 2c14 − a1 b1 + b1 c2 − b2 c1 + (c1 )2 , 4 1 1 a22 , R2324 = c22 , 2 2 1 − 2a24 − 2c23 − a2 b2 + a2 c1 − a1 c2 + (c2 )2 , 4 1 1 b22 , R2434 = (2b23 − 2c24 − a2 b1 + c1 c2 ), 2 4 1 − (2c34 + a1 c4 − a4 c1 + b2 c3 − b3 c2 − cc1 c2 − a44 − b33 ) 2 1 − a3 b1 − a1 b3 + a2 b4 − a4 b2 + aa1 b1 + ba2 b2 + ca1 b2 4 +ca2 b1 − a(c1 )2 − b(c2 )2 .
A1.5. The Ricci tensor. The definition of the Ricci tensor in [27] is: c Rab := R c bac = Rac cb = −Racb = R c abc .
(A1.12)
The Ricci curvature is defined in [28] as the negative of (A1.12) so we must modify the equations [28](4.6.20–23) and [28](4.6.25) by removing a minus sign so as to preserve the definitions of AB A B and , see Appendix Two. R11 = 0, R22 = 0, R33 R34 R44
1 1 (a11 + c12 ), R14 = (b12 + c11 ), 2 2 1 1 = (a12 + c22 ), R24 = (b22 + c12 ), 2 2
R12 = 0,
R13 =
R23 1 2ca12 − 2a24 + 2c23 + aa11 + ba22 + a2 b2 + a1 c2 − a2 c1 − (c2 )2 , = 2 1 = (2cc12 + ac11 + bc22 + a14 + b23 − c13 − c24 − a2 b1 + c1 c2 ), 2 1 2c14 − 2b13 + 2cb12 + ab11 + bb22 + a1 b1 + b2 c1 − b1 c2 − (c1 )2 . = 2
A1.6. The scalar curvature. S = a11 + b22 + 2c12 . A1.7. The Ricci endomorphism.
Ri1
⎧1 (a11 + c12 ), i = 1; ⎪ ⎪ ⎨2 = 21 (b12 + c11 ), i = 2; ⎪ ⎪ ⎩ 0, otherwise;
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⎧1 (a12 + c22 ), i = 1; ⎪ ⎪ ⎨2 R i 2 = 21 (b22 + c12 ), i = 2; ⎪ ⎪ ⎩ 0, otherwise; ⎧ 1 2 ⎪ ⎪ 2 2c23 −2a24 +ba22 +ca12 −ac12 −cc22 +a2 b2 +a1 c2 −a2 c1 −(c2 ) =: ζ, ⎪ ⎪ ⎪ ⎪ ⎨ 1 (ac11 +cc12 −ca11 −ba12 +a14 +b23 −c13 −c24 −a2 b1 +c1 c2 ) =: η, 2 i R 3= 1 ⎪ ⎪ ⎪ 2 (a11 + c12 ), ⎪ ⎪ ⎪ ⎩ 1 (a + c ), 2
12
22
⎧ 1 ⎪ ⎪ 2 (bc22 +cc12 −ab12 −cb22 +a14 +b23 −c13 − c24 − a2 b1 + c1 c2 ) =: ", ⎪ ⎪ ⎪ ⎪ ⎨1 2c14 −2b13 +ab11 +cb12 −cc11 −bc12 +a1 b1 +b2 c1 −b1 c2 −(c1 )2 =: ϒ, 2 i R 4= 1 ⎪ ⎪ ⎪ 2 (b12 + c11 ), ⎪ ⎪ ⎪ ⎩1 (b + c ), 2
22
12
i = 1; i = 2; i = 3; i = 4; i = 1; i = 2; i = 3; i = 4.
A1.8. The Einstein endomorphism. The diagonal elements of E a b := R a b − (S/4)δ a b are: E 1 1 = E 3 3 = −E 2 2 = −E 4 4 =
a11 − b22 . 4
The off-diagonal elements are as in A1.7. Putting θ :=
a11 − b22 b12 + c11 a12 + c22 , µ := , ν := , 4 2 2
one computes E(∂1 ) E(∂2 ) E(∂3 ) E(∂4 )
= θ ∂1 + µ∂2 , = ν∂1 − θ ∂2 , = ζ ∂1 + η∂2 + θ ∂3 + ν∂4 , = "∂1 + ϒ∂2 + µ∂3 − θ ∂4 ,
where ζ , η, " and ϒ are defined in A1.7. A1.4 was reported by Ghanam & Thompson [13], though with a typographical error. Matsushita [23] reported A1.4–6 and the covariant form of the Einstein endomorphism. Chaichi et al. [5] computed curvature properties under the perhaps ad hoc assumption c = 0. Díaz-Ramos et al. [8] also reported A1.4–6. Generally, these authors employed distinct curvature conventions to us; we have been motivated by choices which maintain a close correspondence with the conventions of [28] so as to facilitate the employment of spinors.
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A1.9. Conformal curvature. With the conventions of [27], the Weyl conformal curvature is given by: S 1 (gad gbc − gac gbd ) + (gad Rbc − gac Rbd + gbc Rad − gbd Rac ) 6 2 S 1 + (gad gbc − gac gbd ) + (gad E bc − gac E bd + gbc E ad − gbd E ac ), 12 2 (A1.13)
Rabcd = Cabcd − = Cabcd
and the Weyl curvature components are: C12jk = 0 C1313 = C1323 = C1334 = C1414 = C1434 = C2323 = C2334 = C2424 = C2434 = C3434 =
for
(j, k) = (3, 4); C1234 =
S , 12
1 1 (a11 − c12 + b22 ), C1314 = (c11 − b12 ), 6 4 1 1 (a12 − c22 ), C1324 = c12 , 4 2 1 − (3a14 − 3c13 − 5cc12 − 3bc22 + 3c24 − 3b23 − ca11 + 3ab12 + 2cb22 ), 12 1 1 1 C1423 = − (a11 − 4c12 + b22 ), C1424 = (b12 − c11 ), b11 , 2 12 4 1 (b a11 + 3ab11 + 3cb12 + bb22 − bc12 − 3cc11 ), 12 1 1 a22 , C2324 = − (a12 − c22 ), 2 4 1 − (aa11 + 3ca12 + 3ba22 − 3cc22 − ac12 + ab22 ), 12 1 (a11 − c12 + b22 ) 6 1 (2ca11 + 3ba12 − 3a14 + 3b23 − cb22 − 3c24 − 3ac11 − 5cc12 + 3c13 ), 12 1 3ba1 c2 + 3cb1 a2 + 6bca12 + baa11 − 3ca1 b2 − 4abc12 − 6cac11 12 −3bc1 a2 + abb22 + 6acb12 − 6cbc22 + 3ac1 b2 + 3a 2 b11 + 6c1 a4 − 3ab1 c2 −6a1 c4 + 3a1 b3 + 6c2 b3 − 3a2 b4 − 3b1 a3 − 6b2 c3 + 3b2 a4 − 12c34 +6a44 + 6b33 + 6ac14 − 6ab13 − 8c2 c12 − 6ca14 + 6cc13 + 6cc24 −6cb23 + 3b2 a22 − 6ba24 + 6bc23 + 2c2 a11 + 2c2 b22 .
A1.10. The curvature endomorphism. Define the curvature endomorphism of the space 2 (T p M) of bivectors as the tensor contraction: R(F) :=
1 S ab 1 a bc 1 ab R cd F cd = C ab cd F cd + F + (E c F − E b c F ac ), 2 2 12 2 (A1.14)
ab ij where the second expression follows from (A1.13). If R cd has components R kl with respect to a frame {e1 , . . . , e4 }, then R has matrix R ij kl with respect to the induced
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basis { eia ∧ ejb : i < j } for 2 (T p M). The trace of this endomorphism is tr(R) =
R ij ij =
i<j
1 ij S R ij = . 2 2
(A1.15)
ij
This definition ensures that R is the identity on S4 . 2 (T p M) equipped with the induced scalar product is isomorphic to R2,4 and the Hodge star operator ∗ induces, via its eigenspaces, the orthogonal decomposition 2 = 2+ ⊕ 2− , under which + S W Z R= ∗ + 16 , (A1.16) Z W− 12 with R self adjoint. W + and W − are induced by the self-dual and anti-self-dual parts of the Weyl conformal tensor respectively and, with 2+ (T p M) ∼ = 2− (T p M) ∼ = R1,2 , W + and W − are self adjoint with respect to the induced scalar products, while Z is induced by the Einstein endomorphism E a b , and ∗Z is the adjoint of Z ∈ Hom(2− , 2+ ). To compute the curvature endomorphism, one requires a basis of 2 compatible with the decomposition 2 = 2+ ⊕ 2− . If {e1 , e2 , e3 , e4 } is an oriented -ON frame, then the following is a -ON frame of 2 = 2+ ⊕ 2− : e1 ∧ e2 + e3 ∧ e4 , √ 2 e1 ∧ e3 + e2 ∧ e4 , s2+ := √ 2 e1 ∧ e4 − e2 ∧ e3 , s3+ := √ 2 s1+ :=
e1 ∧ e2 − e3 ∧ e4 , √ 2 e1 ∧ e3 − e2 ∧ e4 s2− := , √ 2 e1 ∧ e4 + e2 ∧ e3 s3− := . √ 2 s1− :=
(A1.17)
A simple choice of -ON frame constructed from Walker coordinates (u, v, x, y) is provided by 1 (1 − a)∂1 + ∂3 , 2 1 e2 := −c∂1 + (1 − b)∂2 + ∂4 , 2 e1 :=
1 e3 := − (1 + a)∂1 + ∂3 , 2 (A1.18) 1 e4 := −c∂1 − (1 + b)∂2 + ∂4 . 2
The matrix relating these frames has determinant one so [e1 , e2 , e3 , e4 ] = [∂1 , ∂2 , ∂3 , ∂4 ], i.e., {e1 , e2 , e3 , e4 } is a -ON frame with the canonical orientation (see §1) and thus suitable for employment in (A1.17). If one takes the Walker coordinates (v, u, y, x) derived from the symmetry (A1.7), the -ON frame obtained from them via (A1.18) also possesses the canonical orientation but opposite SO+ -orientation (i.e., opposite ‘time’ and ‘space’ orientations). With the choice (A1.18), one obtains 1 + ab ∂1 ∧ ∂2 + 2c ∂1 ∧ ∂3 − a ∂1 ∧ ∂4 + b ∂2 ∧ ∂3 + 2 ∂3 ∧ ∂4 , 2 (A1.19) s2+ = c ∂1 ∧ ∂2 + ∂1 ∧ ∂3 + ∂2 ∧ ∂4 , ab − 1 s3+ = ∂1 ∧ ∂2 + 2c ∂1 ∧ ∂3 − a ∂1 ∧ ∂4 + b ∂2 ∧ ∂3 + 2 ∂3 ∧ ∂4 , 2
s1+ =
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619
and a+b ∂1 ∧ ∂2 + ∂1 ∧ ∂4 − ∂2 ∧ ∂3 , 2 = −c ∂1 ∧ ∂2 + ∂1 ∧ ∂3 − ∂2 ∧ ∂4 , a−b ∂1 ∧ ∂2 + ∂1 ∧ ∂4 + ∂2 ∧ ∂3 . = 2
s1− = − s2− s3−
(A1.20)
The matrix representations of the Weyl curvature endomorphisms have been reported in [8] and [10]. Putting P := a11 + b22 − 4c12 , Q := a22 + b11 , T := a12 − c22 , X := b12 − c11 , Y := a22 − b11 , then the matrix representation of W − is ⎛ ⎞ −(P + 3Q) 3(T + X ) 3Y 1 − 2P 3(T − X ) ⎠ . W := − ⎝−3(T + X ) 12 −3Y 3(T − X ) −(P − 3Q)
(A1.21)
(A1.22)
With A = 6ab13 − 6bc23 − 12cc13 − 12b33 + 12c34 − 6ac14 + 6ba24 + 12ca14 + 12c34 − 12a44 − 3a(−bc12 − 2c2 b1 + ab11 ) − 3b(ba22 − 2a2 c1 − ac12 ) − 6c(ba12 + a1 b2 + a2 b1 − ac11 ) + 6(−bc23 − c2 b3 + a3 b1 + ab13 ) + 6(ba24 + a2 b4 − a4 c1 − ac14 ) + −6ac1 b2 + 6acc11 − 6bc2 a1 − 6bca12 + 12ca1 b2 − 12c2 a11 + 12b2 c3 − 12cc13 + 12a1 c4 + 12ca14 − 6a4 c1 − 6a4 b2 − 6a1 b3 − 6b3 c2 − a11 − b22 − 2c12 , (A1.23) where we note that for use in 3.7 the right-hand side is expressed intentionally in a redundant form in that we have not grouped all like terms together, in particular the second term in the fourth line (−6ca1 b2 ) cancels half of the fifth term in the sixth line (12ca1 b2 ) to yield the term 6ca1 b2 , and B:=2(a14 − b23 −c13 +c24 )−2ca11 − ba12 + ab12 + ac11 − bc22 − 2cc12 , (A1.24) the matrix representation of W + is ⎛ ⎞ A 3B A+S 1 + 2S −3B ⎠ . W := − ⎝ −3B 12 −(A + S) −3B −(A + 2S)
(A1.25)
One computes det(W + − λ13 ) = (S/6 + λ)(λ − S/12)2 ,
(A1.26)
whence the eigenvalues of W + are −
S S S , , . 6 12 12
(A1.27)
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P. R. Law, Y. Matsushita
Díaz-Ramos et al. [8] showed that W + possesses only certain possible Jordan canonical forms, see 2.6, indicating that generic four-dimensional Walker geometry manifests certain self duality properties, as explained by our spinor analysis of Walker geometry. The Einstein endomorphism (i.e., traceless Ricci tensor) determines 0 Z . (A1.28) E := ∗ Z 0 From (A1.14) E(F) =
1 a bc 1 (E c F − E b c F ac ) = − (E a c F cb − E b c F ca ). 2 2
(A1.29)
1 E(X ∧ Y ) = − (X ∧ E(Y ) + E(X ) ∧ Y ). 2
(A1.30)
In particular,
Using A1.8 and (A1.30), Davidov & Muškarov [10] computed the matrix representation of Z with respect to the bases (A1.19–20): ⎞ ⎛ ϒ + ζ + c(ν − µ) η + " − 2θ c ϒ − ζ − c(ν + µ) 1 ⎠ , (A1.31) µ−ν 2θ µ+ν Z =− ⎝ 2 − ϒ + ζ + c(ν − µ) −η + " − 2θ c −ϒ − ζ − c(ν + µ) which completes the description of the curvature endomorphism. Both Díaz-Ramos et al. [8] and Davidov & Muškarov [10] used these results to characterize the vanishing of the ASD Weyl curvature of a four-dimensional Walker geometry; the latter authors also obtained characterizations of some other curvature conditions while the former authors studied the Osserman condition on the Jacobi operator. Appendix Two: Spinors for Four-Dimensional Neutral Metrics The two-component spinor formalism for R2,2 is more directly analogous to that for C4 equipped with the standard C-bilinear scalar product than that for R1,3 . That said, it is mostly straightforward to adapt the results in [28] to the context of neutral signature, though one must be aware of a few features peculiar to neutral geometry. The two-component spinor formalism is based on an isomorphism, via Clifford algebras, of R4 with S ⊗ S , where S ∼ = S ∼ = R2 , but S and S are independent spaces. Each of S and S is more appropriately viewed as isomorphic to the symplectic plane 2 , with symplectic forms and . (Objects constructed from the tensor algebra of S Rsp are indicated by abstract indices bearing a prime, in which case primes on the symbol denoting the object itself are dropped, whence A B for . The actual isomorphism of interest is then R2,2 = (R4 , η) ∼ = (S ⊗ S , ⊗ ). See [19] for a brief sketch of this isomorphism and some basic spinor algebra and geometry. In particular, the isomorphism R2,2 ∼ = S ⊗ S may be taken to be 1 v1 + v3 v4 − v2 1 2 3 4 =: v AA , (A2.1) v = (v , v , v , v ) ↔ √ 4 2 1 3 2 v +v v −v whence s(v, v) = 2 det v AA ; in particular, v is null iff v AA is singular, equivalently
v A A is decomposable as an element of S ⊗ S .
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When employing results from [28], the main fact to bear in mind is that there is no natural identification between S and S (which for neutral signature are real linear spaces whereas in the case of Lorentz signature they are complex conjugate (linear) spaces of each other). In this appendix we will merely record a few results which we require in the main body of the paper. For R2,2 , (A1.3) indicates that Vabcd = eabcd and, by (A1.6) ξg (∗U ) = ∗ξg (U ) for any multivector U , i.e., the Hodge star operators on multivectors and forms coincide under the identification of multivectors and forms via the metric. The volume form of the standard orientation of R2,2 is (A2.2) eabcd = AC B D A D B C − AD BC A C B D . Representing an element of 2 R2,2 as a skew tensor F ab , the decomposition into SD and ASD summands is
F ab = AB ψ A B + φ AB A B ,
(A2.3)
A B
∈ S S and φ AB ∈ S S. Note that ∗2 = 1. where ψ The spinorial representation of the curvature may be obtained exactly as in [28], §4.6, the only difference being that all curvature spinors are real objects whence the SD and ˜ A B C D is the spinorial repreASD Weyl spinors are independent objects: AB C D sentation of the SD Weyl tensor and ABC D A B C D that of the ASD Weyl curvature tensor. The fully covariant Riemann tensor is given by ˜ A B C D + ABC D A B C D Rabcd = AB C D + ABC D A B C D + C D A B AB C D +2( AC A C B D B D − AD A D BC B C ),
(A2.4)
˜ A B C D and ABC D are fully symmetric while the Ricci spinor where the Weyl spinors satisfies AB A B = (AB)(A B ) . Because our definition A1.5 of the Ricci tensor is the negative of that employed in [28], we obtain: Rab = 2 AB A B − 6 AB A B ,
(A2.5)
whence S gab = E ab , (A2.6) 4 where E ab is the fully covariant version of the Einstein endomorphism, i.e., the trace-free Ricci tensor (not to be confused with the ‘Einstein tensor’ G ab of [28]). As in [28], §4.9, S = −24, 2ab := 2 AB A B = Rab −
ab := 2∇[a ∇b] = A B
AB
+ AB
A B ,
(A2.7)
where
AB
:= ∇ X (A ∇ B) X ,
A B
:= ∇ X (A ∇ B ) X .
(A2.8)
The spinor Ricci identities for arbitrary spinors κ A and τ A are: AB κC
= ABC E κ E − (κ A BC + AC κ B ),
A B τC
˜ A B C E τ E − (τ A B C + A C τ B ), =
AB τC
= ABC E τ E , (A2.9) E A B κC = A B C E κ .
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The Bianchi equation may be written, see [28], §4.10,
A A ˜ = ∇A ∇ BA ABC D = ∇(B C D)A B , ∇ B ABC D (B C D )AB , ∇ C A C D A B
= −3∇
D B
(A2.10)
.
References 1. Bérard Bergery, L., Ikemakhen, A.: Sur l’holonomie des variétés pseudo-riemanniennes de signature (n, n). Bull. Soc. Math. France 125, 93–114 (1997) 2. Blaži´c, N., Bokan, N., Raki´c, Z.: Osserman Pseudo-Riemannian Manifolds of Signature (2, 2). J. Austral. Math. Soc. 71, 367–395 (2001) 3. Boyer, C.P., Finley III, J.D., Pleba˜nski, J.F.: Complex General Relativity, H and HH Spaces-A Survey of One Approach. In: Held, A. (ed.) General Relativity and Gravitation: One Hundred Years After the Birth of Albert Einstein, Vol. 2. New York: Plenum Press, 1980, pp. 241–281 4. Brozos-Vázquez, M., García-Río, E., Vázquez-Lorenzo, R.: Conformally Osserman four-dimensional manifolds whose conformal Jacobi operators have complex eigenvalues. Proc. R. Soc. London A 462, 1425–1441 (2006) 5. Chaichi, M., García-Río, E., Matsushita, Y.: Curvature properties of four-dimensional Walker metrics. Class. Quantum Grav. 22, 559–577 (2005) 6. Cruceanu, V., Gadea, P.M., Mu˜noz Masqué, J.: Para-Hermitian and Para-Kähler Manifolds. Quad. Inst. Mat. Univ. Messina 1, 1–72 (1995) 7. Cruceanu, V., Fortuny, P., Gadea, P.M.: A survey on paracomplex geometry. Rocky Mountain J. Math. 26, 83–115 (1996) 8. Díaz-Ramos, J.C., García-Río, E., Vázquez-Lorenzo, R.: Four-dimensional Osserman metrics with nondiagonalizable Jacobi operators. J. Geom. Anal. 16, 39–52 (2006) 9. Davidov, J., Díaz-Ramos, J.C., García-Río, E., Matsushita, Y., Muškarov, O., Vázquez-Lorenzo, R.: Almost Kähler Walker four-manifolds. J. Geom. Phys. 57, 1075–1088 (2007) 10. Davidov, J., Muškarov, O.: Self-dual Walker metrics with two-step nilpotent Ricci operator. J. Geom. Phys. 57, 157–165 (2006) 11. Dunajski, M.: Anti-self-dual four-manifolds with a parallel spinor. Proc. R. Soc. London A 458, 1205–1222 (2002) 12. Dunajski, M., West, S.: Anti-self-dual conformal structures in neutral signature. http://arxiv.org.math. DG/0610280 2006 13. Ghanam, R., Thompson, G.: The holonomy Lie algebras of neutral metrics in dimension four. J. Math. Phys. 42, 2266–2284 (2001) 14. Jensen, G.R., Rigoli, M.: Neutral Surfaces in Neutral Four-Spaces. Le Matematiche XLV, 407–443 (1990) 15. Kamada, H.: Self-duality of neutral metrics on four-dimensional manifolds. In: The Third Pacific Rim Geometry Conference (Seoul, 1996), Monographs Geom. Topology 25. Cambridge, MA: International Press, 1998, pp. 79–98 16. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, Volume I. New York: Wiley-Interscience, 1963 17. Law, P.R.: Neutral Einstein metrics in four dimensions. J. Math. Phys. 32, 3039–3042 (1991) 18. Law, P.R.: Neutral Geometry and the Gauss-Bonnet Theorem for Two-Dimensional Pseudo-Riemannian Manifolds. Rocky Mountain J. Math. 22, 1365–1383 (1992) 19. Law, P.R.: Classification of the Weyl curvature spinors of neutral metrics in four dimensions. J. Geom. Phys. 56, 2093–2108 (2006) 20. Libermann, P.: Sur les structures presque paracomplexes. C. R. Acad. Sci. Paris Sér. I Math. 234, 2517–2519 (1952) 21. Libermann, P.: Sur le problème d’équivalence de certaines structures infinitésimales. Ann. Mat. Pura Appl. (4) 36, 27–120 (1954) 22. Matsushita, Y.: Four-Dimensional Walker metrics and symplectic structures. J. Geom. Phys. 52, 89–99 (2004) 23. Matsushita, Y.: Walker 4-manifolds with proper almost complex structures. J. Geom. Phys. 55, 385–398 (2005) 24. Matsushita, Y.: The existence of indefinite metrics of signature (++−−) and two kinds of almost complex structures in dimension four. In: Contemporary Aspects of Complex Analysis, Differential Geometry and Mathematical Physics (Proceedings of the 7th International Workshop on Complex Structures and Vector Fields, 2004, Plovdiv, Bulgaria), Hackensack, NJ: World Scientific Publishing Co., 2005, pp. 210–226
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25. Matsushita, Y., Haze, S., Law, P.R.: Almost Kähler-Einstein Structures on 8-Dimensional Walker Manifolds. Monatsh. Math. 150, 41–48 (2007) 26. Matsushita, Y., Law, P.R.: Hitchin-Thorpe-Type Inequalities for Pseudo-Riemannian 4-Manifolds of Metric Signature (+ + −−). Geo. Ded. 87, 65–89 (2001) 27. O’Neill, B.: Semi-Riemannian Geometry: With Applications to Relativity. Orlando: Academic Press, 1983 28. Penrose R., Rindler, W.: Spinors and Space-time, Volume 1: Two-Spinor Calculus and Relativistic Fields. Cambridge: Cambridge University Press, 1984 29. Penrose, R., Rindler, W.: Spinors and Space-time, Volume 2: Spinor and Twistor Methods in Space-Time Geometry. Cambridge: Cambridge University Press, 1986 30. Pleba˜nski, J.F.: Some Solutions of Complex Einstein Equations. J. Math. Phys. 16, 2395–2402 (1975) 31. Pleba˜nski, J.F., Hacyan, S.: Null geodesic surfaces and Goldberg-Sachs theorem in complex Riemannian spaces. J. Math. Phys. 16, 2403–2407 (1975) 32. Porteous, I.R.: Topological Geometry. Second Edition. Cambridge: Cambridge University Press, 1981 33. Rashevskij, P.K.: The scalar field in a stratified space. Trudy Sem. Vektor. Tenzor. Anal. 6, 225–248 (1948) 34. Rozenfeld, B.A.: On unitary and stratified spaces. Trudy Sem. Vektor. Tenzor. Anal. 7, 260–275 (1949) 35. Sibata, T., Morinaga, K.: Complete and Simpler Treatment of Wave Geometry. J. Sci. Hiroshima University, Series A. 6, 173–189 (1936) 36. Spivak, M.: A Comprehensive Introduction to Differential Geometry, Volume One. Second Edition. Berkeley: Publish or Perish Inc., 1979 37. Spivak, M.: A Comprehensive Introduction to Differential Geometry, Volume Two. Second Edition. Berkeley: Publish or Perish Inc., 1979 38. Sternberg, S.: Lectures on Differential Geometry. Second Edition. New York: Chelsea Publishing Company, 1983 39. Thompson, G.: Normal form of a metric admitting a parallel field of planes. J. Math. Phys. 33, 4008–4010 (1992) 40. Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics 94. New York: Springer-Verlag, 1983 41. Walker, A.G.: On parallel fields of partially null vector spaces. Quart. J. Math. (Oxford) 20, 135–145 (1949) 42. Walker, A.G.: Canonical form for a Riemannian space with a parallel field of null planes. Quart. J. Math. Oxford (2) 1, 69–79 (1950) 43. Walker, A.G.: Canonical forms (II): Parallel partially null planes. Quart. J. Math. Oxford (2) 1, 147–152 (1950) 44. Walker, A.G.: On Ruse’s spaces of recurrent curvature. Proc. London Math. Soc. (2) 52, 36–64 (1950) 45. Walker, A.G.: Connexions for parallel distributions in the large. Quart. J. Math. Oxford (2) 6, 301–308 (1955) 46. Walker, A.G.: Connexions for parallel distributions in the large II. Quart. J. Math. Oxford (2) 9, 221–231 (1958) 47. Yano, K.: Differential Geometry on Complex and Almost Complex Spaces. New York: MacMillan, 1965 Communicated by G.W. Gibbons
Commun. Math. Phys. 282, 625–662 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0554-x
Communications in
Mathematical Physics
On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization S. M. Khoroshkin1 , I. I. Pop2 , M. E. Samsonov3 , A. A. Stolin2 , V. N. Tolstoy4 1 Institute for Theoretical and Experimental Physics, Moscow, Russia 2 Department of Mathematics, Gothenburg University, Gothenburg, Sweden.
E-mail: [email protected]
3 Dublin Institute for Advanced Studies, Dublin, Ireland 4 Institute of Nuclear Physics, Moscow State University, Moscow, Russia
Received: 5 June 2007 / Accepted: 4 April 2008 Published online: 4 July 2008 – © Springer-Verlag 2008
Abstract: We study classical twists of Lie bialgebra structures on the polynomial current algebra g[u], where g is a simple complex finite-dimensional Lie algebra. We focus on the structures induced by the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. It turns out that quasi-trigonometric r -matrices fall into classes labelled by the vertices of the extended Dynkin diagram of g. We give the complete classification of quasi-trigonometric r -matrices belonging to multiplicity free simple roots (which have coefficient 1 in the decomposition of the maximal root). We quantize solutions corresponding to the first root of sl(n). 1. Introduction Recall that, given a Lie algebra g, the classical Yang-Baxter equation (CYBE) with one spectral parameter is the equation [X 12 (u), X 13 (u + v)] + [X 12 (u), X 23 (v)] + [X 13 (u + v), X 23 (v)] = 0,
(1.1)
where X (u) is a meromorphic function of one complex variable u, defined in a neighbourhood of 0, taking values in g ⊗ g. In their outstanding paper [2], A. Belavin and V. Drinfeld investigated solutions of the CYBE for a simple complex Lie algebra g. They considered so-called nondegenerate solutions (i.e. X (u) has maximal rank for generic u). It was proved in [2] that nondegenerate solutions are of three types: rational, trigonometric and elliptic. Moreover the authors completely classified trigonometric and elliptic solutions, the last ones exist only in the case g = sl(n). One can see that any rational solution of CYBE provides the Lie bialgebra structure on the polynomial Lie algebra g[u] for a simple Lie algebra g. On the contrary, there are no clear Lie bialgebra structures related to elliptic solutions of CYBE. For trigonometric solutions of CYBE, the situation is as follows. Any trigonometric solution has the form Y (ek(u−v) ), where Y is a g ⊗ g-valued rational function and k is
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some constant. After setting ek(u−v) = z this solution provides a Lie bialgebra structure either on Lie algebra g[z, z −1 ] or on its twisted version but does not induce, generally speaking, a Lie bialgebra structure on the polynomial Lie algebra g[z]. Therefore we are motivated to introduce a class of solutions of ‘trigonometric’ type that will induce Lie bialgebra structures on g[u]. Let Ω denote the quadratic Casimir element of g. We say that a solution X of the CYBE is quasi-trigonometric if it is of the form: X (u, v) =
vΩ + p(u, v), u−v
(1.2)
where p(u, v) is a polynomial with coefficients in g ⊗ g. We will prove that by applying a certain holomorphic transformation and a change of variables, any quasi-trigonometric solution becomes trigonometric, in the sense of Belavin-Drinfeld classification. In the works [8,9] to any Lie bialgebra V. Drinfeld assigned another Lie bialgebra, the so-called classical double. F. Montaner and E. Zelmanov [27] proved that for any Lie bialgebra structure on g[u] its classical double is isomorphic as a Lie algebra to one of four Lie algebras. We will consider two of them: g((u −1 )) and g((u −1 )) ⊕ g. The study of the Lie bialgebra structures given by quasi-trigonometric solutions will be based on the description of the classical double. We show that all quasi-trigonometric solutions induce the same classical double g((u −1 )) ⊕ g. Moreover we construct a one-to-one correspondence between this type of solution and a special class of Lagrangian subalgebras of the g((u −1 )) ⊕ g. It turns out that such Lagrangian subalgebras can be embedded into some maximal orders of g((u −1 )) ⊕ g, which correspond to vertices of the extended Dynkin diagram of g. This embedding enables us to classify quasi-trigonometric solutions of CYBE which correspond to multiplicity free roots. We also use the classification of Manin triples for reductive Lie algebras in terms of the generalized Belavin-Drinfeld data obtained by P. Delorme [6]. In particular, we get a complete combinatorial description of quasi-trigonometric solutions of CYBE, related to Lie algebra sl(n). The goal of the second part of the paper is to propose a quantization scheme for some of the Lie bialgebra structures on g[u] for g = sl(n) described in the first part of the paper. In all these cases the quantization is given by an explicit construction of the corresponding twist. More precisely, the corresponding Hopf algebra is isomorphic to Uq (g[u]) with twisted comultiplication, where Uq (g[u]) is defined as a certain subalgebra of the quantum affine algebra Uq ( g). This result confirms the natural conjecture made in [21] and recently proved in [17]: any classical twist can be quantized. For the construction of twist quantizations of quasi-trigonometric solutions of CYBE, we use nontrivial embeddings of certain Hopf subalgebras of the quantized universal n ). This enables us enveloping algebra Uq (sln+1 ), called seaweed algebras [7] into Uq (sl to ‘affinize’ the finite dimensional twists constructed in [15] and [19]. 2. Lie Bialgebra Structures and Classical Twists Let g denote an arbitrary complex Lie algebra. We recall that a Lie bialgebra structure on g is a 1-cocycle δ : g −→ ∧2 g which satisfies the co-Jacobi identity. In other words, δ provides a Lie algebra structure for g∗ compatible with the structure of g. To any Lie bialgebra (g, δ) one associates the so-called classical double D(g, δ). It is defined as the unique Lie algebra structure on the vector space g ⊕ g∗ such that: a) it induces the given Lie algebra structures on g and g∗ ,
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b) the bilinear form Q defined by Q(x1 + l1 , x2 + l2 ) = l1 (x2 ) + l2 (x1 )
(2.1)
is invariant with respect to the adjoint representation of g ⊕ g∗ . Let δ1 be a Lie bialgebra structure on g. Suppose s ∈ ∧2 g satisfies [s 12 , s 13 ] + [s 12 , s 23 ] + [s 13 , s 23 ] = Alt(δ1 ⊗ id)(s),
(2.2)
where Alt(x) := x 123 + x 231 + x 312 for any x ∈ g⊗3 . Then δ2 (a) := δ1 (a) + [a ⊗ 1 + 1 ⊗ a, s]
(2.3)
defines a Lie bialgebra structure on g. We call s a classical twist and say that the bialgebra structures (g, δ1 ) and (g, δ2 ) are related by a classical twist. The construction of the double suggests another notion of equivalence between bialgebras. Namely, we say that Lie bialgebra structures δ1 and δ2 on g are in the same twisting class if there is a Lie algebra isomorphism f : D(g, δ1 ) −→ D(g, δ2 ) satisfying the properties: 1) Q 1 (x, y) = Q 2 ( f (x), f (y)) for any x, y ∈ D(g, δ1 ), where Q i denotes the canonical form on D(g, δi ), i = 1, 2. 2) f ◦ j1 = j2 , where ji is the canonical embedding of g in D(g, δi ). For a finite-dimensional g, it was shown in [20] that two Lie bialgebra structures are in the same twisting class if and only if they are related by a classical twist. Example 1. Let g be finite-dimensional. All Lie bialgebra structures induced by triangular r -matrices are related by classical twists. The classical double corresponding to any triangular r -matrix is isomorphic to the semidirect sum g g∗ such that g∗ is a commutative ideal and [a, l] = ad∗ (a)(l) for any a ∈ g and l ∈ g∗ . Another example of twisting is the following: Example 2. Suppose g is simple and let δ0 be the Lie bialgebra structure induced by the standard Drinfeld-Jimbo r -matrix. Then the entire Belavin-Drinfeld list [1] is obtained by twisting the standard structure δ0 . The classical double corresponding to any r -matrix from this list is isomorphic to g ⊕ g. Now, if we pass to the case of infinite-dimensional Lie bialgebra structures, we encounter more examples of twisting. Let us recall several facts from the theory of rational solutions as it was developed in [33]. We let again g denote a simple Lie algebra. Denote by K the Killing form and let Ω be the corresponding Casimir element of g. We look for functions X : C2 −→ g ⊗ g which satisfy [X 12 (u 1 , u 2 ), X 13 (u 1 , u 3 )] + [X 12 (u 1 , u 2 ), X 23 (u 2 , u 3 )] + +[X 13 (u 1 , u 3 ), X 23 (u 2 , u 3 )] = 0, X 12 (u, v) = −X 21 (v, u).
(2.4) (2.5)
Remark 1. We will call these two equations the classical Yang-Baxter equation (CYBE). In the case of rational and quasi-trigonometric solutions, the unitarity condition (2.5) can actually be dropped. We will prove in the Appendix that (2.5) is a consequence of (2.4).
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Ω Definition 1. Let X (u, v) = u−v + p(u, v) be a function from C2 to g ⊗ g, where p(u, v) is a polynomial with coefficients in g ⊗ g. If X satisfies the CYBE, we say that X is a rational solution.
Let R be a commutative ring and let L be a Lie algebra over R. Let us denote by Aut R (L) the group of automorphisms of L over R. In other words we consider such automorphisms of L, which satisfy the condition f (rl) = r f (l), where r ∈ R, l ∈ L . Let f : R → S be a homomorphism of commutative rings. Then L ⊗ R S is a Lie algebra over S and the formula f ∗ (l ⊗ s) = f (l) ⊗ s defines a group homomorphism f ∗ : Aut R (L) → Aut S (L ⊗ R S). Let C[[u −1 ]] be the ring of formal power series in u −1 and C((u −1 )) its field of quotients. Consider the Lie algebras g[u] = g ⊗ C[u], g[[u −1 ]] = g ⊗ C[[u −1 ]] and g((u −1 )) = g ⊗ C((u −1 )). Let us consider the following natural embedding i : C[u] → C((u −1 )), which defines a group homomorphism i ∗ : AutC[u] (g[u]) → AutC((u −1 )) (g((u −1 ))). Clearly it is an embedding. If σ (u) ∈ Aut C[u] (g[u]), we denote i ∗ (σ (u)) by σˆ (u). In what follows we will need some other similar constructions: 1. Let ev(α) : C[u] → C be the evaluation at α ∈ C. Then we denote ev(α)∗ (σ (u)) ∈ Aut C (g) by σ (α). 2. If inc : C → C[u] is the natural embedding of rings and σ ∈ Aut C (g), then inc∗ (σ ) is a constant automorphism of Aut C[u] (g[u]), i.e. inc∗ (σ )(α) = σ for any α ∈ C. Definition 2. Two rational solutions X 1 and X 2 are called gauge equivalent if there exists σ (u) ∈ Aut C[u] (g[u]) such that X 2 (u, v) = (σ (u) ⊗ σˆ (v))X 1 (u, v). At this point we note that X 1 (u, v) and X 2 (u, v) can be considered as elements of a completed tensor product of g[u] and g((v −1 )). Therefore, (σ (u) ⊗ σˆ (v))X 1 (u, v) is well-defined and also belongs to the completed tensor product of g[u] and g((v −1 )). Later we will show that (σ (u) ⊗ σˆ (v))X 1 (u, v) is a rational solution if X 1 (u, v) is. Remark 2. It was proved in [33] that any rational solution can be brought by means of a gauge transformation to the form: X (u, v) =
Ω + p00 + p10 u + p01 v + p11 uv, u−v
where p00 , p10 , p01 , p11 ∈ g ⊗ g. We recall that any rational solution induces a Lie bialgebra structure on the polynomial current algebra g[u]. Let us consider a rational solution X and define the map δ X : g[u] −→ g[u] ∧ g[u] by δ X (a(u)) = [X (u, v), a(u) ⊗ 1 + 1 ⊗ a(v)],
(2.6)
for any a(u) ∈ g[u]. Obviously δ X is a 1-cocycle and therefore induces a Lie bialgebra structure on g[u]. The following result, proved in [33], shows that all Lie bialgebra structures corresponding to rational solutions have the same classical double.
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Let D X (g[u]) be the classical double corresponding to a rational solution X of the CYBE. Then D X (g[u]) and g((u −1 )) are isomorphic as Lie algebras, with an inner product which has the following form on g((u −1 )): Q( f (u), g(u)) = Resu=0 K ( f (u), g(u)),
(2.7)
where f (u), g(u) ∈ g((u −1 )). Remark 3. This result means in fact that g((u −1 )) can be represented as a Manin triple g((u −1 )) = g[u] ⊕ W , where W is a Lagrangian subalgebra with respect to the invariant form Q. In the case g = sl(n) all rational solutions were described in the following way: Let dk = diag(1, ..., 1, u, ..., u) (k many 1’s), 0 ≤ k ≤ [ n2 ]. Then it was proved in [33] that every rational solution of the CYBE defines some Lagrangian subalgebra W contained in dk−1 sl(n)[[u −1 ]]dk for some k. These subalgebras are in one-to-one correspondence with pairs (L , B) verifying: (1) L is a subalgebra of sl(n) such that L + Pk = sl(n), where Pk denotes the maximal parabolic subalgebra of sl(n) not containing the root vector eαk of the simple root αk ; (2) B is a 2-cocycle on L which is nondegenerate on L ∩ Pk . In case g = sl(2) one has just two non-standard rational r -matrices, up to gauge equivalence: Ω + h α ∧ e−α u−v
(2.8)
Ω + ue−α ⊗ h α − vh α ⊗ e−α , u−v
(2.9)
X 1 (u, v) = and X 2 (u, v) =
where eα = e12 , e−α = e21 and h α = e11 − e22 is the usual basis of sl(2). 3. Quasi-Trigonometric Solutions of the CYBE Another interesting case of infinite-dimensional Lie bialgebra structures on g[u] is provided by a class of trigonometric type solutions of the CYBE, called quasi-trigonometric solutions. These solutions were first introduced in [21]. Definition 3. We say that a solution X of the CYBE is quasi-trigonometric if it is of the form: X (u, v) =
vΩ + p(u, v), u−v
(3.1)
where p(u, v) is a polynomial with coefficients in g ⊗ g. The term quasi-trigonometric is motivated by the relationship between this type of solution of CYBE and trigonometric solutions in the Belavin-Drinfeld meaning. The following result, whose proof we give in the Appendix, illustrates this fact.
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Theorem 1. Let X (u, v) be a quasi-trigonometric solution of the CYBE. There exists a holomorphic transformation and a change of variables such that X (u, v) becomes a trigonometric solution, in the sense of Belavin-Drinfeld classification. vΩ Example 3. A function X (u, v) = u−v + r , where r ∈ g ⊗ g, satisfies the CYBE if and only if r is a solution of the modified classical Yang-Baxter equation, i.e.
r + r 21 = Ω, [r 12 , r 13 ] + [r 12 , r 23 ] + [r 13 , r 23 ] = 0.
(3.2) (3.3)
Let r0 denote the standard Drinfeld-Jimbo r -matrix. We fix a Cartan subalgebra h and the associated root system. We choose a system of generators eα , e−α and h α , where α is a positive root, such that K (eα , e−α ) = 1. Then r0 =
1 ( eα ∧ e−α + Ω). 2
(3.4)
α>0
Correspondingly we have a quasi-trigonometric solution X 0 (u, v) =
vΩ + r0 . u−v
Definition 4. A quasi-trigonometric solution X (u, v) = constant if p(u, v) is a constant polynomial.
(3.5) vΩ u−v
+ p(u, v) is called quasi-
We remind the reader that if σ (u) ∈ AutC[u] (g[u]), then σ (u) naturally defines an automorphism σˆ (u) of AutC((u −1 )) (g((u −1 ))). We also note that a quasi-trigonometric solution X (u, v) is an element of the completed tensor product of g[u] and g((v −1 )). Hence, the element (σ (u) ⊗ σˆ (v))X (u, v) is well-defined and belongs to the completed tensor product of g[u] and g((v −1 )). Proposition 1. Let X 1 be a quasi-trigonometric solution and σ (u) ∈ AutC[u] (g[u]). Then X 2 (u, v) = (σ (u) ⊗ σˆ (v))X 1 (u, v) is also a quasi-trigonometric solution. vΩ Proof. Let X 1 (u, v) = u−v + p(u, v). Since X 2 obviously satisfies the CYBE, one has to check that X 2 is quasi-trigonometric. It is enough to show the following: vΩ vΩ = + p(u, v) , (3.6) σ (u) ⊗ σˆ (v) u−v u−v
where p(u, v) ∈ g[u] ⊗ g[v] is a polynomial. Let i : g[u] −→ g((u −1 )) be the natural embedding and ev(α) : g[u] −→ g the evaluation at α ∈ C (we use the same notations as for the corresponding ring homorphisms). We first notice that v vΩ = (i ⊗ i)(σ (u) ⊗ σ (v))(Ω). σ (u) ⊗ σˆ (v) u−v u−v
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Thus, in order to prove (3.6), we have to check that σ (u) ⊗ σ (v)(Ω) = Ω + p(u, ˜ v), where p(u, ˜ v) ∈ g[u] ⊗ g[v] is a polynomial, which vanishes at u = v. The latter fact can be proved as follows: Let us consider p(u, ˜ v) := (σ (u) ⊗ σ (v))(Ω) − Ω. It is easy to prove that if q(u) ∈ g[u], then ev(α)(σ (u)(q(u))) = σ (α)(q(α)). Thus, we have the following: ev(α) ⊗ ev(α)(σ (u) ⊗ σ (v)(Ω) = (σ (α) ⊗ σ (α))(Ω) = Ω because Ω is invariant, which implies that ev(α) ⊗ ev(α)( p(u, ˜ v)) = p(α, ˜ α) = 0. Since this holds for all α ∈ C, we deduce that p(u, ˜ v) vanishes at u = v. This ends the proof of the statement for the quasi-trigonometric solutions and it is clear that the same proof works in the rational case.
Definition 5. Two quasi-trigonometric solutions X 1 and X 2 are called gauge equivalent if there exists σ (u) ∈ AutC[u] (g[u]) such that X 2 (u, v) = (σ (u) ⊗ σˆ (v))X 1 (u, v).
(3.7)
Any quasi-trigonometric solution X of the CYBE induces a Lie bialgebra structure on g[u]. Let δ X be the 1-cocycle defined by δ X (a(u)) = [X (u, v), a(u) ⊗ 1 + 1 ⊗ a(v)],
(3.8)
for any a(u) ∈ g[u]. It is expected that all Lie bialgebra structures corresponding to quasi-trigonometric solutions induce the same classical double. Let us consider the direct sum of Lie algebras g((u −1 ))⊕g, together with the following invariant bilinear form: Q(( f (u), a), (g(u), b)) = K ( f (u), g(u))0 − K (a, b).
(3.9)
Here the index zero means that we have taken the free term in the series expansion. Remark 4. The Lie algebra g[u] is embedded into g((u −1 ))⊕g via a(u) −→ (a(u), a(0)) and is naturally identified with V0 := {(a(u), a(0)); a(u) ∈ g[u]}.
(3.10)
Consider the following Lie subalgebra of g((u −1 )) ⊕ g: W0 = {(a + f (z), b) : f ∈ z −1 g[[z −1 ]], a ∈ b+ , b ∈ b− , ah + bh = 0}.
(3.11)
Here h is the fixed Cartan subalgebra of g, b± are the positive (negative) Borel subalgebras and ah denotes the Cartan part of a. We make the remark that V0 ⊕ W0 = g((u −1 )) ⊕ g and both V0 and W0 are isotropic with respect to the form Q. In order to show that all quasi-trigonometric solutions induce the same classical double, we will first prove the following result:
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Theorem 2. There exists a natural one-to-one correspondence between quasitrigonometric solutions of the CYBE and linear subspaces W of g((u −1 )) ⊕ g such that 1) W is a Lie subalgebra in g((u −1 ))⊕g such that W ⊇ u −N g[[u −1 ]] for some N > 0; 2) W ⊕ V0 = g((u −1 )) ⊕ g; 3) W is a Lagrangian subspace with respect to the inner product of g((u −1 )) ⊕ g. Proof. Let V0 and W0 be the Lie algebras given in Remark 4. We choose dual bases in V0 and W0 respectively. Let {k j } be an orthonormal basis in h. The canonical basis of V0 is formed by eα u k , e−α u k , k j u k for any α > 0, k > 0 and all j; (e−α , e−α ), (eα , eα ) for any α > 0, and (k j , k j ), for all j. The dual basis of W0 is the following: e−α u −k , eα u −k , k j u −k for any α > 0, k > 0 and all j; (eα , 0) and (0, −e−α ) for all α > 0, and 21 (k j , −k j ), for all j. Let us simply denote these dual bases by {vi } and {w0i } respectively. We notice that the quasi-trigonometric solution X 0 can be written as X 0 (u, v) = (τ ⊗ τ )( w0i ⊗ vi ), (3.12) i
where τ denotes the projection of g((u −1 )) ⊕ g onto g((u −1 )). We denote by H om c (W0 , V0 ) the space of those linear maps F : W0 −→ V0 such that K er F ⊇ u −N g[[u −1 ]] for some N > 0. It is the space of linear maps F which are continuous with respect to the “u −1 - adic” topology. Let us contruct a linear isomorphism Φ : V0 ⊗ V0 −→ H om c (W0 , V0 ) in the following way: Φ(x ⊗ y)(w0 ) = Q(w0 , y) · x,
(3.13)
for any x, y ∈ V0 and any w0 ∈ W0 . It is easy to check that Φ is indeed an isomorphism. The inverse mapping is Ψ : H om c (W0 , V0 ) −→ V0 ⊗ V0 defined by Ψ (F) = F(w0i ) ⊗ vi . (3.14) i
We make the remark that this sum is finite since F(w0i ) = 0 only for a finite number of indices i. The next step is to construct a bijection between H om c (W0 , V0 ) and the set L of linear subspaces W of g((u −1 )) ⊕ g such that W ⊕ V0 = g((u −1 )) ⊕ g and W ⊇ u −N g[[u −1 ]] for some N > 0. This can be done in a very natural way. For any F ∈ H om c (W0 , V0 ) we take W (F) = {w0 + F(w0 ); w0 ∈ W0 }.
(3.15)
The inverse mapping associates to any W the linear function FW such that for any w0 ∈ W0 , FW (w0 ) = −v, uniquely defined by the decomposition w0 = w + v0 with w ∈ W and v0 ∈ V0 . Therefore we have a bijection between V0 ⊗ V0 and L. By a straightforward computation, one can show that a tensor r (u, v) ∈ V0 ⊗ V0 satisfies the condition r (u, v) = −r 21 (v, u) if and only if the linear subspace W (Φ(r )) is Lagrangian with respect to Q. Let us suppose now that X (u, v) = X 0 (u, v) + r (u, v) and r (u, v) = −r 21 (v, u). Then X (u, v) satisfies (2.4) if and only if W (Φ(r )) is a Lie subalgebra of g((u −1 )) ⊕ g.
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Indeed, since r is unitary, we have that W (Φ(r )) is a Lagrangian subspace with respect to Q. It is enough to check that X (u, v) satisfies (2.4) if and only if Q([w1 + Φ(r )(w1 ), w2 + Φ(r )(w2 )], w3 + Φ(r )(w3 )) = 0
(3.16)
for any elements w1 , w2 and w3 of W0 . This follows by direct computations. vΩ In conclusion, we see that a function X (u, v) = u−v + p(u, v) is a quasi-trigonometric solution if and only if W (Φ( p − r0 )) is a Lagrangian subalgebra of g((u −1 )) ⊕ g. This ends the proof.
Remark 5. If W is a Lagrangian subalgebra of g((u −1 )) ⊕ g satisfying the conditions of Theorem 2, then the corresponding solution X (u, v) is constructed in the following way: take a basis {wi } in W which is dual to the canonical basis {vi } of V0 and construct the tensor r (u, v) = wi ⊗ vi . (3.17) i
Let π denote the projection of g((u −1 )) ⊕ g onto g[u] which is induced by the decomposition g((u −1 )) ⊕ g = V0 ⊕ W0 . Explicitly, π(an z n + · · · + a0 + a−1 u −1 + · · · , b) = an u n + · · · + a1 u 1 + (a0h + bh) + a0− + b+ . 2
(3.18)
Here a0 = a0h + a0+ + a0− and b = bh + b+ + b− are the decompositions with respect to g = h ⊕ n+ ⊕ n− . Then X (u, v) = X 0 (u, v) + (π ⊗ π )( r (u, v)).
(3.19)
At this point we note the following fact that we will prove in the Appendix: Proposition 2. Let W be a Lie subalgebra satisfying Conditions 2) and 3) of Theorem 2. Let r be constructed as in (3.17). Assume r induces a Lie bialgebra structure on g[u] by δr (a(u)) = [ r (u, v), a(u) ⊗ 1 + 1 ⊗ a(v)]. Then W ⊇ u −N g[[u −1 ]] for some positive N . For any quasi-trigonometric solution X of the CYBE denote by D X (g[u]) the classical double of g[u] corresponding to X . Theorem 3. For any quasi-trigonometric solution X of the CYBE there exists an isomorphism of Lie algebra D X (g[u]) and g((u −1 )) ⊕ g identical on g[u], which transforms the canonical bilinear form on D X (g[u]) to the form Q. Proof. One can easily check that if W is a Lagrangian subalgebra of g((u −1 )) ⊕ g, vΩ corresponding to a quasi-trigonometric solution X (u, v) = u−v + p(u, v), then W is ∗ isomorphic to (g[u]) with the Lie algebra structure induced by X . Indeed, with the notation introduced in Theorem 2, let F := FW . It is enough to check that for any v0 ∈ V0 and w1 , w2 ∈ W0 the following equality is satisfied: Q(v0 , [w1 + F(w1 ), w2 + F(w2 )]) =< δ X (v0 ), w1 ⊗ w2 >,
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where <, > denotes the pairing between V0⊗2 and W0⊗2 induced by Q. This equality is implied by the following identities: Q(v0 , [w1 , w2 ]) = < δ X 0 (v0 ), w1 ⊗ w2 >, Q(v0 , [F(w1 ), w2 ]) = < [ p − r0 , 1 ⊗ v0 ], w1 ⊗ w2 >, Q(v0 , [w1 , F(w2 )]) = < [ p − r0 , v0 ⊗ 1], w1 ⊗ w2 > .
Remark 6. Theorem 3 states in particular that all quasi-trigonometric solutions of CYBE are in the same twisting class. This can be seen directly, since by the definition of quasitrigonometric solutions they are related to the solution (3.5) by classical twists. Let σ (u) ∈ Aut C[u] (g[u]). Let σ (u) = σ (u)⊕σ (0) be an automorphism of g((u −1 ))⊕g. Theorem 4. Let X 1 and X 2 be quasi-trigonometric solutions of the CYBE. Suppose that W1 and W2 are the corresponding Lagrangian subalgebras of g((u −1 )) ⊕ g. The following conditions are equivalent: 1) X 1 (u, v) = (σ (u) ⊗ σˆ (v))X 2 (u, v); 2) W1 = σ (u)W2 . Proof. 1) ⇒ 2). Let us begin by proving this for the particular case X 1 = X 0 and X 2 = (σ (u) ⊗ σˆ (v))X 0 (u, v). The Lagrangian subalgebra corresponding to X 0 is W0 given by (3.11). On the other hand, one can check the Lagrangian subalgebra W2 , corresponding to the solution X 2 , consists of elements f := ( f, σ (vi )) · σ (w0i ) = ( σ −1 ( f ), vi ) · σ (w0i ), i
i
for any f ∈ W0 . Here {vi } and are the dual bases of V0 and W0 introduced in the proof of Theorem 2. We show that W2 = σ (W0 ). Let g denote the projection of σ −1 ( f ) onto W0 induced by the decomposition V0 ⊕ W0 = g((u −1 )) ⊕ g. Then g= ( σ −1 ( f ), vi ) · w0i , {w0i }
i
which implies that f = σ (g). Therefore W2 ⊆ σ (W0 ). The other inclusion is similar. Let us pass to the general case. If X (u, v) = X 1 0 (u, v)+r (u, v) is a quasi-trigonometric solution with r (u, v) = ak u k ⊗ b j v j , then the corresponding W1 consists of ele ments of the form f + ( f, b j u j )ak u k , for any f in W0 . Now let X 2 (u, v) = (σ (u) ⊗ σˆ (v))X 1 (u, v). The corresponding subalgebra W2 is formed by elements of the form ( f, σ (vi )) · σ (w0i ) + ( f, σ (b j u j )) σ (ak u k ). fr := i
σ (h), where h := g + (g, b j u j )ak u k and g is the projection It is easy to see that fr = of σ −1 ( f ) onto W0 . These considerations prove that σ (W1 ) = W2 . 2) ⇒ 1). Suppose that W2 = σ (W1 ). Let X 2 := (σ (u) ⊗ σˆ (v))X 1 (u, v). It is a quasi2 . Because trigonometric solution which has a corresponding Lagrangian subalgebra W 2 = 2 . Since the correspondence 1) ⇒ 2) we obtain that W σ (W1 ) and thus W2 = W between solutions and subalgebras is one-to-one, we get that X 2 = X 2.
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Definition 6. We will say that W1 and W2 are gauge equivalent (with respect to Aut C[u] (g[u])) if Condition 2) of Theorem 4 is satisfied. vΩ Theorem 5. Let X (u, v) = u−v + p(u, v) be a quasi-trigonometric solution of the CYBE and W the corresponding Lagrangian subalgebra of g((u −1 )) ⊕ g. Then the following are equivalent:
1) X is quasi-constant. 2) W is contained in g[[u −1 ]] ⊕ g. Proof. We keep the notations from the proof of Theorem 2 and also those from Remark 5. Let r (u, v) = p(u, v) − r0 and F = Φ(r (u, v)). If p(u, v) is constant, then F(w0 ) ∈ g ⊗ g for any w0 ∈ W0 . Therefore W (F) ⊆ g[[u −1 ]] ⊕ g. Conversely, let us suppose that W is included in g[[u −1 ]] ⊕ g. The orthogonal of g[[u −1 ]] ⊕ g with respect to Q is obviously u −1 g[[u −1 ]]. Since W is a Lagrangian subalgebra, it follows that W contains u −1 g[[u −1 ]]. According to the previous remark, r (u, v) = (π ⊗ π )( r (u, v)), where π −1 is the projection ionto g[u] induced by the decomposition g((u )) ⊕ g = V0 ⊕ W0 and r (u, v) = i w ⊗ vi . Now it is clear that r is a constant. This ends the proof.
4. Classification of Quasi-Trigonometric Solutions We have seen that gauge equivalent solutions correspond to gauge equivalent subalgebras W . Thus, the classification of quasi-trigonometric solutions is equivalent to the classification of W satisfying the conditions of Theorem 2. In order to classify such W we will use a method from [35] which allows us to embed W in a suitable C-subalgebra of g((u −1 )) ⊕ g. Having fixed a Cartan subalgebra h of g, let R be the correspondig set of roots and Γ the set of simple roots. Denote by gα the root space corresponding to a root α. Let h(R) be the set of all h ∈ hsuch that α(h) ∈ R for all α ∈ R. Consider the valuation on C((u −1 )) defined by v( k≥n ak u −k ) = n. For any root α and any h ∈ h(R), set Mα (h):={ f ∈ C((u −1 )) : v( f ) ≥ α(h)}. Consider Oh := h[[u −1 ]] ⊕ (⊕α∈R Mα (h) ⊗ gα ).
(4.1)
As a direct corollary of Theorem 4 from [35], the following result can be deduced: Theorem 6. Up to a gauge equivalence, any subalgebra W which corresponds to a quasi-trigonometric solution can be embedded into Oh × g, where h is a vertex of the following standard simplex {h ∈ h(R) : α(h) ≥ 0 for all α ∈ Γ and αmax ≤ 1}. Vertices of the above simplex correspond to vertices of the extended Dynkin diagram of g, the correspondence being given by the following rule: 0 ↔ αmax h i ↔ αi ,
where αi (h j ) = δi j /k j and k j are given by the relation k j α j = αmax . We will write Oα instead of Oh if α is the root which corresponds to the vertex h. Remark 7. We have Oαmax = g[[u −1 ]]. We have already seen that a quasi-trigonometric solution is quasi-constant if and only if its corresponding W is embedded into Oαmax × g.
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Remark 8. It might happen that two Lagrangian subalgebras W1 and W2 are gauge equivalent even though they are embedded into different Oα1 × g and Oα2 × g. If there exists an automorphism of the Dynkin diagram of g taking α1 into α2 , then W1 and W2 are gauge equivalent and the corresponding quasi-trigonometric solutions as well. Let us suppose now that α is a simple root which can be sent to −αmax by means of an automorphism. Such a root has coefficient one in the decomposition of αmax and will be called a multiplicity free root. Let us denote by pα the standard parabolic subalgebra of g generated by all root vectors corresponding to simple roots and their opposite except −α. Let lα denote the set of all pairs in pα × pα with equal Levi components. This is a Lagrangian subalgebra of g × g, where g × g has been endowed with the following invariant bilinear form Q ((a, b), (c, d)) := K (a, c) − K (b, d).
(4.2)
Theorem 7. The set of subalgebras W ⊆ Oα × g, corresponding to quasi-trigonometric solutions, is in a one-to-one correspondence with the set of Lagrangian subalgebras l of g × g, with respect to the bilinear form Q , which satisfy the condition l ⊕ lα = g × g. Proof. The proof is based on the following result from [34]: Let G be the simply connected Lie group with Lie algebra g. Denote by G ad the Lie group G/Z (G). Let H be the Cartan subgroup with Lie algebra h and Had its image in G ad . If α is a multiplicity free root, then Oα and Oαmax are conjugate by an element of Had (C((u −1 ))). Suppose now that W ⊆ Oα × g, then (Oα × g)⊥ ⊆ W ⊥ = W . It follows that W Oα × g ∼ Oαmax × g ∼ ⊆ = = g × g. (Oα × g)⊥ (Oα × g)⊥ (Oαmax × g)⊥
(4.3)
Denote by l the image of the quotient (O W ⊥ in g × g. One can check that l is a α ×g) Lagrangian subalgebra of g × g with respect to Q . Moreover the image of g[u]∩(Oα ×g) in g×g, after passing to the quotient as above, is precisely lα . Since W is transversal to g[u], it follows that l should be transversal to lα . Conversely, if l is a Lagrangian subalgebra transversal to lα in g × g, then its corresponding lift, W , is transversal to g[u].
We see that in the case of multiplicity free roots, the classification of quasi-trigonometric solutions reduces to the following Problem. Given a multiplicity free root α, find all subalgebras l of g × g which build a Manin triple (Q , lα , l), with respect to the invariant bilinear form Q on g × g. This problem was solved in [30] by using the classification of Manin triples for complex reductive Lie algebras which had been obtained by P. Delorme in [6]. The classification of Manin triples was expressed in terms of so-called generalized BelavinDrinfeld data. Let us recall Delorme’s construction. Let r be a finite-dimensional complex, reductive, Lie algebra and B a symmetric, invariant, nondegenerate bilinear form on r. The goal in [6] is to classify all Manin triples of r up to conjugacy under the action on r of the simply connected Lie group R whose Lie algebra is r. One denotes by r+ and r− respectively the sum of the simple ideals of r for which the restriction of B is equal to a positive (negative) multiple of the Killing form. Then the derived ideal of r is the sum of r+ and r− .
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Let j0 be a Cartan subalgebra of r, b0 a Borel subalgebra containing j0 and b0 be its opposite. Choose b0 ∩ r+ as a Borel subalgebra of r+ and b0 ∩ r− as Borel subalgebra of r− . Denote by Σ+ (resp., Σ− ) the set of simple roots of r+ (resp., r− ) with respect to the above Borel subalgebras. Let Σ = Σ+ ∪ Σ− and denote by W = (Hα , X α , Yα )α∈Σ a Weyl system of generators of [r, r]. Definition 7 (Delorme, [6]). One calls (A, A , ia, ia ) generalized Belavin-Drinfeld data with respect to B when the following five conditions are satisfied: (1) A is a bijection from a subset Γ+ of Σ+ on a subset Γ− of Σ− such that B(H Aα , H Aβ ) = −B(Hα , Hβ ), α, β ∈ Γ+ .
(4.4)
(2) A is a bijection from a subset Γ+ of Σ+ on a subset Γ− of Σ− such that B(H A α , H A β ) = −B(Hα , Hβ ), α, β ∈ Γ+ .
(4.5)
(3) If C = A−1 A is the map defined on dom(C) = {α ∈ Γ+ : A α ∈ Γ− } by Cα = A−1 A α, then C satisfies: For all α ∈ dom(C), there exists a positive integer n such that α, . . . , C n−1 α ∈ dom(C) and C n α ∈ / dom(C). (4) ia (resp., ia ) is a complex vector subspace of j0 , included and Lagrangian in the orthogonal a (resp., a ) to the subspace generated by Hα , α ∈ Γ+ ∪ Γ− (resp., Γ+ ∪ Γ− ). (5) If f is the subspace of j0 generated by the family Hα + H Aα , α ∈ Γ+ , and f is defined similarly, then (f ⊕ ia) ∩ (f ⊕ ia ) = 0.
(4.6)
Let R+ be the set of roots of j0 in r which are linear combinations of elements of Γ+ . . The bijections A and A can then be extended One defines similarly R− , R+ and R− ). If A satisfies Condition (1), by linearity to bijections from R+ to R− (resp., R+ to R− then there exists a unique isomorphism τ between the subalgebra m+ of r spanned by X α , Hα and Yα , α ∈ Γ+ , and the subalgebra m− spanned by X α , Hα and Yα , α ∈ Γ− , such that τ (Hα ) = H Aα , τ (X α ) = X Aα , τ (Yα ) = Y Aα for all α ∈ Γ+ . If A satisfies (2), then one defines similarly an isomorphism τ between m + and m − . Theorem 8 (Delorme, [6]). (i) Let BD = (A, A , ia, ia ) be generalized Belavin-Drinfeld data, with respect to B. Let n be the sum of the root spaces relative to roots α of j0 in b0 , which are not in R+ ∪ R− . Let i := k ⊕ ia ⊕ n, where k := {X + τ (X ) : X ∈ m+ }. R+
Let n be the sum of the root spaces relative to roots α of j0 in b0 , which are not in . Let i := k ⊕ i ⊕ n , where k := {X + τ (X ) : X ∈ m }. ∪ R− + a Then (B, i, i ) is a Manin triple. (ii) Every Manin triple is conjugate by an element of R to a unique Manin triple of this type.
Remark 9. One says that the Manin triple constructed in (i) is associated to the generalized Belavin-Drinfeld data BD and the system of Weyl generators W. Such a Manin triple will be denoted by TBD,W .
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Recall that Γ denotes the set of simple roots relative to a Cartan subalgebra h of g. For a subset S of Γ , let [S] be the set of roots in the linear span of S. Let m S := h + α∈[S] gα , gα , p S := m S + n S . We also consider g S := [m S , m S ], h S := h ∩ g S n S := α>0,α ∈[S] / and ζ S := {x ∈ h : α(x) = 0, ∀α ∈ S}. Consider the Lagrangian subalgebra l S of g × g which consists of all pairs from p S × p S with equal components in m S . In [30], the general result of Delorme was used in order to determine Manin triples of the form (Q , l S , l). This enables one to give the classification of all Lagrangian subalgebras l of g × g which are transversal to a given l S . We devote the rest of this section to presenting the main results of [30]. We refer to [30] for the proofs. First of all, let us choose a suitable system of Weyl generators for g × g. Let b be a Borel subalgebra of g containing the Cartan subalgebra h. Then b0 := b × b is a Borel subalgebra of g × g containing the Cartan subalgebra j0 := h × h. One denotes by Σ+ the set of pairs (α, 0), and by Σ− the set of pairs (0, −α), where α ∈ Γ . Let Σ := Σ+ ∪ Σ− . Let (X α , Yα , Hα )α∈Γ be an arbitrary Weyl system of generators for g. A system of Weyl generators of g × g with respect to this choice of simple roots can be chosen as follows: X (α,0) = (X α , 0), H(α,0) = (Hα , 0), Y(α,0) = (Yα , 0), X (0,−α) = (0, Yα ), H(0,−α) = (0, −Hα ), Y(0,−α) = (0, X α ). A Manin triple associated to some generalized Belavin-Drinfeld data BD for g × g with respect to this Weyl system will simply be noted by TBD . Let θ¯S be the automorphism of g S uniquely defined by the properties θ¯S (X α ) = Yα , ¯θ S (Yα ) = X α and θ¯S (Hα ) = −Hα for all α ∈ S. Recall that there is a short exact sequence 1 −→ G S −→ Aut(g S ) −→ Aut S −→ 1,
(4.7)
where Aut S denotes the group of automorphisms of the Dynkin diagram of g S . Let θ S be the image of θ¯S in Aut S . Therefore θ¯S can be written uniquely as θ¯S = ψ S Ad g0 ,
(4.8)
where g0 ∈ G S and ψ S is the unique automorphism of g S satisfying the properties: ψ S (X α ) = X θ S (α) , ψ S (Yα ) = Yθ S (α) , ψ S (Hα ) = Hθ S (α) for all α ∈ S. Theorem 9. For any Manin triple (Q , l S , l), there exists a unique generalized BelavinDrinfeld data BD = (A, A , ia, ia ) where A : S × {0} −→ {0} × (−S), A(α, 0) = (0, −θ S (α)) and ia = diag(ζS ), such that (Q , l S , l) is conjugate to the Manin triple TBD = (Q , i, i ). Moreover, up to a conjugation which preserves l S , the Lagrangian subalgebra l is of the form l = (id × Ad g0 )(i ), where g0 ∈ G S is the unique element from the decomposition (4.8). Lemma 1. Let A : S × {0} −→ {0} × (−S), A(α, 0) = (0, −θ S (α)) and ia = diag(ζS ). A quadruple (A, A , ia, ia ) is generalized Belavin-Drinfeld data if and only if the pair (A , ia ) satisfies the following conditions: (1) A : Γ1 × {0} −→ {0} × (−Γ2 ) is given by an isometry A˜ between two subsets Γ1 and Γ2 of Γ : A (α, 0) = (0, − A˜ (α)). (2) Let dom ( A˜ , S) := {α ∈ Γ1 : A˜ (α) ∈ S ∩ Γ2 }. Then for any α ∈ dom ( A˜ , S) there exists a positive integer n such that α, (θ S A˜ )(α),..., (θ S A˜ )n−1 (α) ∈ dom ( A˜ , S) but (θ S A˜ )n (α) ∈ / dom ( A˜ , S).
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(3) Consider ∆ S := {(h + h , −ψ S (h) + h ) : h ∈ h S , h ∈ ζ S }. Let f be the subspace of h × h spanned by pairs (Hα , −H A˜ (α) ) for all α ∈ Γ1 . Let ia be a Lagrangian subspace of a := {(h 1 , h 2 ) ∈ h × h : α(h 1 ) = 0, β(h 2 ) = 0, ∀α ∈ Γ1 , ∀β ∈ Γ2 }. Then (f ⊕ ia ) ∩ ∆ S = 0.
(4.9)
Definition 8. A triple (Γ1 , Γ2 , A˜ ) is called S-admissible if condition (2) of Lemma 1 is satisfied. Let Ω0 denote the Cartan component of the Casimir element Ω. Let π1 (resp., π2 ) be the projection of h onto h S (resp., ζ S ). Let K 0 be the restriction of the Killing form K of g to h, which permits an identification between h and h∗ . If R is an endomorphism of h, denote by R ∗ the adjoint of R regarded as an endomorphism of h. Lemma 2. (i) Suppose that (f ⊕ ia ) ∩ ∆ S = 0. Then there exists a unique linear endomorphism R of h such that f ⊕ ia = {(Rh, R h) : h ∈ h},
(4.10)
where R h := π1 (h) − π2 (h) − ψ S π1 (Rh) + π2 (Rh), and (ψ S π1 + π2 )R + R ∗ (ψ S π1 + π2 ) = idh.
(4.11)
(ii) There exists a bijection between the Lagrangian subspaces ia of a satisfying the condition (f ⊕ ia ) ∩ ∆ S = 0, and the endomorphisms R of h verifying (4.11) and the additional condition: R((ψ S π1 + π2 )(Hγ ) + (π2 − π1 )(H A˜ (γ ) )) = Hγ , ∀γ ∈ Γ1 .
(4.12)
(iii) There exists a bijection between endomorphisms R of h verifying (4.11) and (4.12) and tensors r ∈ h ⊗ h satisfying the following conditions: (4.13) (id ⊗ (ψ S π1 + π2 ))(r ) + ((ψ S π1 + π2 ) ⊗ id)(r 21 ) = Ω0 , (( A˜ (γ )(π2 − π1 ) ⊗ id)(r ) = ((ψ S π1 + π2 ) ⊗ γ )(r ), ∀γ ∈ Γ1 . (4.14) Corollary 1. Let A : S×{0} −→ {0}×(−S), A(α, 0) = (0, −θ S (α)) and ia = diag(ζ S ). There exists a one-to-one correspondence between generalized Belavin-Drinfeld data (A, A , ia, ia ) and pairs formed by an S-admissible triple (Γ1 , Γ2 , A˜ ) and a tensor r ∈ h ⊗ h satisfying conditions (4.13), (4.14). Theorem 10. Suppose that l is a Lagrangian subalgebra of g × g transversal to l S . Then, up to a conjugation which preserves l S , one has l = (id × Ad g0 )(i ), where i is constructed from an S-admissible triple (Γ1 , Γ2 , A˜ ) and a tensor r ∈ h ⊗ h satisfying conditions (4.13), (4.14). Let α be a multiplicity free root of g. Set S = Γ \ {α}. We write θα instead of θ S and ψα instead of ψ S . We make the remark that a triple (Γ1 , Γ2 , A˜ ) is S-admissible if and only if it is in one of two situations: I. If α ∈ / Γ2 , then (Γ1 , θα (Γ2 ), θα A˜ ) is an admissible triple in the sense of [1].
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II. If α ∈ Γ2 and A˜ (β) = α, for some β ∈ Γ1 , then (Γ1 \ {β}, θα (Γ2 \ {α}), θα A˜ ) is an admissible triple in the sense of [1]. By applying Theorem 10 in the particular case S = Γ \ {α} and working with the tensor r˜ := ((ψα π1 + π2 ) ⊗ id)(r ) instead of r , one obtains Theorem 11. Let α be a multiplicity free root. Suppose that l is a Lagrangian subalgebra of g × g transversal to lα . Then, up to a conjugation which preserves lα , one has l = (id × Ad g0 )(i ), where i is constructed from a pair formed by (Γ1 , Γ2 , A˜ ) and a tensor r˜ ∈ h ⊗ h satisfying the following conditions: (1) (Γ1 , Γ2 , A˜ ) is of type I or II from above. (2) r˜ satisfies r˜ + r˜ 21 = Ω0 ,
(4.15)
(3) If (Γ1 , Γ2 , A˜ ) is of type I, then r˜ satisfies (θα A˜ (γ ) ⊗ id)(˜r ) + (id ⊗ γ )(˜r) = 0, ∀γ ∈ Γ1 .
(4.16)
(4) If (Γ1 , Γ2 , A˜ ) is of type II and A˜ (β) = α, then r˜ satisfies (4.16) for all γ ∈ Γ1 \{β} and (α(π2 − ψα π1 ) ⊗ id)(˜r ) = (id ⊗ β)(˜r ).
(4.17)
The construction of the quasi-trigonometric solutions can be summed up as follows. Suppose that (Γ1 , Γ2 , A˜ ) is of type I or II from above. Then one finds the tensor r˜ and consequently r . This induces a unique endomorphism R of h, which in turn enables one to construct the subspace ia , according to (4.10). This is enough to reconstruct i . Then l := (id × Ad g0 )(i ) is a Lagrangian subalgebra of g × g which is transversal to lα . Moreover, l can be lifted to a Lagrangian subalgebra W of g((u −1 )) ⊕ g which is transversal to g[u]. By choosing two appropriate dual bases in g[u] and W respectively, we reconstruct the quasi-trigonometric solution X (u, v). We will illustrate this procedure by several examples. Example 4. Quasi-trigonometric solutions for sl(2). Let e, f, h be the canonical basis of sl(2) and α be the simple root with root vector e. Then Γ = {α}. We have two cases: I. Γ1 = Γ2 = ∅ and r˜ = 41 h ⊗ h. Correspondingly we get one quasi-trigonometric solution: X 0 (u, v) =
vΩ + r0 , u−v
(4.18)
where Ω = e ⊗ f + f ⊗ e + 21 h ⊗ h and r0 = e ⊗ f + 41 h ⊗ h is the Drinfeld-Jimbo r -matrix for sl(2). II. Γ1 = Γ2 = {α}, A˜ = id and r˜ = 41 h ⊗ h. The corresponding quasi-trigonometric solution is X 1 (u, v) = X 0 (u, v) + (u − v)e ⊗ e.
(4.19)
Example 5. Quasi-trigonometric solutions for sl(3). Denote by α the simple root with root vector e12 and by β the one with root vector e13 . Then Γ = {α, β} and both roots are singular. We will present the quasi-trigonometric solutions corresponding to the root α.
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I. Γ1 = Γ2 = ∅. Then r˜ = a(e11 − e33 ) ⊗ (e22 − e33 ) + b(e22 − e33 ) ⊗ (e11 − e33 ) + 1 1 1 3 (e11 − e33 ) ⊗ (e11 − e33 ) + 3 (e22 − e33 ) ⊗ (e22 − e33 ), where a + b = − 3 . The corresponding quasi-trigonometric solution is quasi-constant: X 0 (u, v) =
vΩ + r0 , u−v
(4.20)
where r0 is the Drinfeld-Jimbo non-skewsymmetric r -matrix in sl(3). II. Γ1 = {α}, Γ2 = {α}, A˜ (α) = α. Then r˜ = − 13 (e11 − e33 ) ⊗ (e22 − e33 ) + 13 (e11 − e33 ) ⊗ (e11 − e33 ) + 13 (e22 − e33 ) ⊗ (e22 − e33 ). It follows that the corresponding solution of the CYBE is again quasi-constant: X 1 (u, v) =
vΩ + r1 , u−v
(4.21)
where r1 is another non-skewsymmetric r -matrix in sl(3). III. Γ1 = {α}, Γ2 = {β}, A˜ (α) = β. Then r˜ = − 13 (e22 − e33 ) ⊗ (e11 − e33 ) + 13 (e11 − e33 ) ⊗ (e11 − e33 ) + 13 (e22 − e33 ) ⊗ (e22 − e33 ). This data allows one to construct the following Lagrangian subalgebra which is transversal to lα : ⎧⎛⎛ ⎫ ⎞ ⎛ ⎞⎞ ab0 −a − d 0 0 ⎨ ⎬ ⎠,⎝∗ a b ⎠⎠ : a, b, c, d ∈ C . (4.22) l = ⎝⎝ c d 0 ⎩ ⎭ ∗ ∗ −a − d ∗ c d Correspondingly, one obtains the following solution: X 2 (u, v) = X 0 (u, v) − u(e12 ⊗ e32 ) + v(e32 ⊗ e12 ) 1 − (e11 − e22 ) ⊗ (e22 − e33 ). 6
(4.23)
III’. Γ1 = {β}, Γ2 = {α}, A˜ (β) = α and the same r˜ as in III. We have a quasitrigonometric solution which is gauge equivalent to (4.23). IV. Γ1 = Γ2 = {α, β}, A˜ (α) = β, A˜ (β) = α. Then r˜ = − 13 (e22 − e33 ) ⊗ (e11 − e33 ) + 13 (e11 − e33 ) ⊗ (e11 − e33 ) + 13 (e22 − e33 ) ⊗ (e22 − e33 ). This data induces the following Lagrangian subalgebra: l = {(X, T X T −1 ) : X ∈ sl(3)}, where T = e13 + e21 + e32 . This is a subalgebra transversal to lα . The corresponding quasi-trigonometric solution is X 3 (u, v) = X 0 (u, v) − u(e12 ⊗ e32 + e13 ⊗ e12 + e12 ⊗ e13 ) +v(e32 ⊗ e12 + e12 ⊗ e13 + e13 ⊗ e12 ) + (e13 + e23 ) ∧ e23 1 + (e11 − e33 ) ∧ (e11 − e22 ). (4.24) 6
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Remark 10. Solutions corresponding to the simple root β are gauge equivalent to the solutions corresponding to α. The solutions with non-constant polynomial part are the following: 1 Y2 (u, v)=X 0 (u, v)+v(e21 ⊗ e23 )−(ue23 ⊗ e21 )− (e11 −e22 ) ⊗ (e22 −e33 ), 6
(4.25)
which is equivalent to X 2 (u, v), and Y3 (u, v) = X 0 (u, v) − u(e13 ⊗ e23 + e23 ⊗ e13 + e23 ⊗ e21 ) +v(e13 ⊗ e23 + e23 ⊗ e13 + e21 ⊗ e23 ) + (e13 + e21 ) ∧ e12 1 + (e11 − e22 ) ∧ (e22 − e33 ), (4.26) 6 which is equivalent to X 3 (u, v). Example 6. Two examples of quasi-trigonometric solutions for sl(N ) of CremmerGervais type. We reconstruct two Lagrangian subalgebras which provide two quasitrigonometric solutions for g = sl(N ). These Lagrangian subalgebras are transversal to lα1 and are related to the Cremmer-Gervais Lie bialgebra structure on g. Let us consider the set of simple roots Γ = {α1 , ..., α N −1 } and take S = Γ \ {α1 }. Let us denote by (X α , Yα , Hα )α∈Γ the standard Weyl system. In order to construct an S-admissible triple (Γ1 , Γ2 , A˜ ), let us first determine the map θ S : S −→ S. One can easily check that θ S is the following involution: θ S (αi ) = α N +1−i , for all i = 2, ..., N −1, and that g0 = TN −1 is the (N − 1) × (N − 1) matrix with 1 on the second diagonal and 0 elsewhere. I. Consider Γ1 = {α1 , ..., α N −2 }, Γ2 = {α2 , ..., α N −1 } and A˜ (αi ) = α N −i . This is an S-admissible triple. Indeed, θ S A˜ (αi ) = αi+1 and (Γ1 , θ S (Γ2 ), θ S A˜ ) is an admissible triple in the sense of [1], which is known to be related to the CremmerGervais Lie bialgebra structure on g (see [5]). The tensor r˜ satisfying (4.15), (4.16) is the Cartan part of the Cremmer-Gervais non-skewsymmetric constant r -matrix. Let n− α N −1 denote the sum of all eigenspaces of negative roots which contain α N −1 in their decomposition. Let n− α1 be the sum of all eigenspaces of negative roots which contain α1 in their decomposition. One can easily check that the Lagrangian subalgebra i − constructed from this data is the following: i = n ⊕ia ⊕k , where n = n− α N −1 ×nα1 , ia = span((diag(1, 1, ..., 1, −N +1), diag(−N +1, 1, ..., 1)) and k is spanned by (X αi , Yα N −i ), (Yαi , X α N −i ), (Hαi , −Hα N −i ), i = 1, ..., N − 2. Let us consider g0 as an element of S L(N ) and take − l1 = (id × Ad g0 )(i ) = n− α N −1 × nα1 ⊕ {(x, τ (x)) : x ∈ mα N −1 },
(4.27)
where τ (ei j ) = ei+1, j+1 and mα N −1 denotes the reductive part of pα N −1 . This Lagrangian subalgebra is transversal to lα1 and therefore induces a quasi-trigonometric solution corresponding to α1 . Denote this solution by X 1 (u, v). II. We consider Γ1 = Γ2 = Γ and A˜ (αi ) = α N −i . This is indeed an S-admissible triple since (Γ1 \ {α N −1 }, θ S (Γ2 \ {α1 }), θ S A˜ ) is an admissible triple in the sense of [1]. We have Γ1 \ {α N −1 } = {α1 , ..., α N −2 }, θ S (Γ2 \ {α1 }) = {α2 , ..., α N −1 } and θ S A˜ (αi ) = αi+1 . The tensor r˜ , which satisfies the system (4.15), (4.16) for γ = αi , i = 1, ..., N − 2 and (4.17), is as in Case I.
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We obtain i = {(X, AdU (X )) : X ∈ sl(N )}, where U ∈ S L(N ) is the matrix with 1 on the second diagonal and zero elsewhere. Finally take l2 = (id × Ad g0 )(i ) = {(X, Ad T (X )) : X ∈ sl(N )},
(4.28)
where T = g0 U = e1N + e21 + e32 + · · · + e N ,N −1 . This Lagrangian subalgebra is transversal to lα1 and consequently provides a quasi-trigonometric solution X 2 (u, v), corresponding again to the root α1 . Solutions X 1 (u, v) and X 2 (u, v) will be called quasi-trigonometric solutions of Cremmer-Gervais type. Regarding their quantization, we will quantize instead the following solutions: X 1 (u, v) = (σ ⊗ σ )X 1 (u, v), X 2 (u, v) = (σ ⊗ σ )X 2 (u, v),
(4.29) (4.30)
where σ (A) = −At . 5. Quantum Twists and Their Affinization The aim of the second part of our paper is to quantize certain quasi-trigonometric solutions of the CYBE and the corresponding Lie bialgebra stuctures on g[u] in case g = sl(N ). We already know that all of them are in the same twisting class and therefore the corresponding quantum groups are isomorphic as algebras but with different comultiplications. However, these comultiplications can be obtained from each other via quantum twisting (see [17,21]). So, we would like to outline some basic elements of quantum twisting of Hopf algebras (see [14], pp. 84-85). Suppose given a Hopf algebra A := A(m, ∆, , S) with a multiplication m : A ⊗ A → A, a coproduct ∆ : A → A ⊗ A, a counit : A → C, and an antipode S : A → A. (1) (2) An invertible element F ∈ A ⊗ A, F = i f i ⊗ f i is called a quantum twist if it satisfies the cocycle equation F 12 (∆ ⊗ id)(F) = F 23 (id ⊗ ∆)(F) ,
(5.1)
and the “unital” normalization condition ( ⊗ id)(F) = (id ⊗ )(F) = 1 .
(5.2)
Now we can define a twisted Hopf algebra A(F) := A(F) (m, ∆(F) , , S (F) ) which has the same multiplication m and the counit mapping but the twisted coproduct and antipode (1) (2) f i S( f i ) (a ∈ A). ∆(F) (a) = F∆(a)F −1 , S (F) (a) = u S(a)u −1 , u = i
(5.3) The Hopf algebra A is called quasitriangular if it has an additional invertible element (universal R-matrix) R, which relates the coproduct ∆ with its opposite coproduct ∆˜ by the transformation ˜ ∆(a) = R ∆(a)R −1
(a ∈ A),
(5.4)
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with R satisfying the quasitriangularity conditions (∆ ⊗ id)(R) = R 13 R 23 ,
(id ⊗ ∆)(R) = R 13 R 12 .
(5.5)
The twisted (“quantized”) Hopf algebra A(F) is also quasitriangular with the universal R-matrix R (F) defined as follows
R (F) = F 21 R F −1 , (2)
(5.6)
(1)
where F 21 = ⊗ f i . So, the first step is to find a quantization of the Lie i fi bialgebra structure on g[u] defined by X 0 (u, v), which was described in Sect. 3. It is well-known that the corresponding quantum group is the so-called Uq (g[u]) which is a parabolic subalgebra of the quantum affine algebra Uq (ˆg). In case sl(N ) this algebra will be defined below. However, it turns out that it is more convenient to work with its extended version Uq (gl N [u]). The second step is to find explicit formulas for the quantum twists. We use two methods. A) Affinization by Hopf isomorphism. Let F be a quantum twist and let Sup(F) be a minimal Hopf subalgebra, whose tensor square contains F, which we call the support of F. Similarly we define the support of a classical twist as the minimal Lie sub-bialgebra, whose tensor square contains the given classical twist. It turns out that for certain quasi-trigonometric solutions for sl(N ), the corresponding support (in sl(N )[u]!) is isomorphic to the support of a certain classical twist in sl(N + 1), which is however constant! This observation enables us to apply results of [15, 19], where constant twists from the Belavin–Drinfeld list were quantized. Of course, the corresponding quantum twists, one in Uq (sl(N )[u]) and the second in Uq (sl(N + 1)), have isomorphic quantum supports. We will call this method affinization by Hopf isomorphism. B) Affinization by automorphism. Let F be some constant twist of Uq (g) and ω be some automorphism of Uq (g[u]) such that ω(Uq (g)) Uq (g). Then, under some conditions the element Fπ ω := (ω−1 π(ω) ⊗ 1)F(π ⊗ id)(∆(ω−1 ))∆(ω) will be also a quantum twist, i.e. it satisfies the cocycle equation (5.1). Here π : Uq (g[u]) → Uq (g) is the canonical projection (the images of the affine roots are zero). The method is interesting on its own but what is more important is that it leads to quantization of rational solutions of the CYBE. We consider these two methods on examples for the quantum algebra Uq (gl N [u]). 6. A Quantum Seaweed Algebra and its Affine Realization As we already mentioned it is more convenient to use instead of the simple Lie algebra sl N its central extension gl N . The polynomial affine Lie algebra gl N [u] is generated by (n) Cartan–Weyl basis ei j := ei j u n (i, j = 1, 2, . . . N , n = 0, 1, 2, . . .) with the defining relations (m) (n+m) [ei(n) − δil ek(n+m) . j , ekl ] = δ jk eil j
(6.1)
The total root system Σ of the Lie algebra gl N [u] with respect to an extended Cartan subalgebra generated by the Cartan elements eii (i = 1, 2, . . . , N ) and d = u(∂/∂u) is given by Σ (gl N [u]) = {i − j , nδ + i − j , nδ | i = j; i, j = 1, 2, . . . , N ; n = 1, 2, . . .}, (6.2)
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where i (i = 1, 2, . . . , N ) is the orthonormal basis of a N -dimensional Euclidean space R N dual to the Cartan subalgebra of gl N . (n) We have the following correspondence: ei j = enδ+i − j for i = j, n = 0, 1, 2, . . . . We choose the following system of positive simple roots: Π (gl N [u]) = {αi := i − i+1 , α0 := δ + N − 1 | i = 1, 2, . . . N − 1}.
(6.3)
Now we would like to introduce the seaweed algebra, which is important for our purposes. Let sw N +1 be a subalgebra of gl N +1 generated by the root vectors: e21 , ei,i+1 , ei+1,i for i = 2, 3, . . . , N and e N ,N +1 , and also by the Cartan elements: e11 + e N +1N +1 , eii for i = 2, 3, . . . , N . It is easy to check that sw N +1 has the structure of a seaweed Lie algebra (see [7]). (0) (0) (0) ˆ N be a subalgebra of gl N [u] generated by the root vectors: e21 , ei,i+1 , ei+1,i for Let sw (1)
(0)
i = 2, 3, . . . , N and e N ,1 , and also by the Cartan elements: eii for i = 1, 2, 3, . . . , N . It ˆ N and sw N +1 are isomorphic. This isomorphism is easy to check that the Lie algebras sw (0) for i = 1, 2, . . . , N − 1, is described by the following correspondence: ei+1,i ↔ ei+1,i (0)
(1)
ei,i+1 ↔ ei,i+1 for i = 2, 3, . . . , N − 1, e N ,N +1 ↔ e N ,1 for i = 2, 3, . . . , N − 1, and (0) (0) ˆ N an affine (e11 + e N +1,N +1 ) ↔ e11 , eii ↔ eii for i = 2, 3, . . . , N . We shall call sw realization of sw N +1 . Now let us consider the q-analogs of the previous Lie algebras. The quantum algebra Uq (gl N ) is generated by the Chevalley elements1 ei,i+1 , ei+1,i (i = 1, 2, . . . , N − 1), q ±eii (i = 1, 2, . . . , N ) with the defining relations: q eii q −eii = q −eii q eii = 1, q eii q e j j = q e j j q eii , q eii e jk q −eii = q δi j −δik e jk (| j − k| = 1), [ei,i+1 , e j+1, j ] = δi j
q eii −ei+1,i+1 − q ei+1,i+1 −eii , q − q −1
[ei,i+1 , e j, j+1 ] = 0 for |i − j| ≥ 2,
(6.4)
[ei+1,i , e j+1, j ] = 0 for |i − j| ≥ 2, [[ei,i+1 , e j, j+1 ]q , e j, j+1 ]q = 0 for |i − j| = 1, [[ei+1,i , e j+1, j ]q , e j+1, j ]q = 0 for |i − j| = 1. where [eβ , eγ ]q denotes the q-commutator: [eβ , eγ ]q := eβ eγ − q (β,γ ) eγ eβ .
(6.5)
The Hopf structure on Uq (gl N ) is given by the following formulas for a comultiplication ∆q , an antipode Sq , and a co-unit εq : 1 We denote the generators in the classical and quantum cases by the same letter “e”. It should not cause any misunderstanding.
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∆q (q ±eii ) = q ±eii ⊗ q ±eii , ∆q (ei,i+1 = ei,i+1 ⊗ 1 + q ei+1,i+1 −eii ⊗ ei,i+1 , ∆q (ei+1,i ) = ei+1,i ⊗ q Sq (q
±eii
)=q
∓eii
Sq (ei,i+1 ) = −q εq (q
(6.6)
+ 1 ⊗ ei+1,i ;
,
eii −ei+1,i+1
Sq (ei+1,i ) = −ei+1,i q ±eii
eii −ei+1,i+1
ei,i+1 , ei+1,i+1 −ei,i
(6.7) ;
) = 1, εq (ei j ) = 0 for |i − j| = 1.
(6.8)
In order to construct composite root vectors ei j for |i − j| ≥ 2 we fix the following normal ordering of the positive root system ∆+ (see [24,25,36]) 1 − 2 ≺ 1 − 3 ≺ 2 − 3 ≺ 1 − 4 ≺ 2 − 4 ≺ 3 − 4 ≺ · · · ≺ 1 − k ≺ 2 − k ≺ · · · ≺ k−1 − k ≺ . . . ≺ 1 − N ≺ 2 − N ≺ · · · ≺ N −1 − N . (6.9) According to this ordering we set ei j := [eik , ek j ]q −1 ,
e ji := [e jk , eki ]q ,
(6.10)
where 1 ≤ i < k < j ≤ N . It should be stressed that the structure of the composite root vectors does not dependent on the choice of the index k in the r.h.s. of the definition (6.10). In particular, we have ei j := [ei,i+1 , ei+1, j ]q −1 = [ei, j−1 , e j−1, j ]q −1 , e ji := [ e j,i+1 , ei+1,i ]q = [e j, j−1 , e j−1,i ]q ,
(6.11)
where 2 ≤ i + 1 < j ≤ N . Using these explicit constructions and the defining relations (6.4) for the Chevalley basis it is not hard to calculate the following relations between the Cartan–Weyl generators ei j (i, j = 1, 2, . . . , N ): q ekk ei j q −ekk = q δki −δk j ei j [ei j , e ji ] =
(1 ≤ i, j, k ≤ N ),
q eii −e j j − q e j j −eii q − q −1
[ei j , ekl ]q −1 = δ jk eil
(1 ≤ i < j ≤ N ),
(1 ≤ i < j ≤ k < l ≤ N ),
[eik , e jl ]q −1 = (q − q −1 ) e jk eil
(1 ≤ i < j < k < l ≤ N ),
(6.12) (6.13) (6.14) (6.15)
[e jk , eil ]q −1 = 0
(1 ≤ i ≤ j < k ≤ l ≤ N ),
(6.16)
[ekl , e ji ] = 0
(1 ≤ i < j ≤ k < l ≤ N ),
(6.17)
[eil , ek j ] = 0
(1 ≤ i < j < k < l ≤ N ),
(6.18)
[e ji , eil ] = e jl q eii −e j j
(1 ≤ i < j < l ≤ N ),
(6.19)
[ekl , eli ] = eki q ekk −ell
(1 ≤ i < k < l ≤ N ),
(6.20)
[e jl , eki ] = (q −1 − q) ekl e ji q e j j −ekk
(1 ≤ i < j < k < l ≤ N ).
(6.21)
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These formulas can also be obtained from the relations between the elements of the Cartan–Weyl basis for the quantum superalgebra Uq (gl(N |M) (see [39]). If we apply the Cartan involution (ei∗j = e ji ) to the formulas above, we will get all relations between elements of the Cartan–Weyl basis. The quantum algebra Uq (gl N [u]) (N ≥ 3) is generated (as a unital associative algebra) by the algebra Uq (gl N ) and the additional element e(1) N 1 with the relations: (0)
(1)
(0)
(1)
q ±eii e N 1 = q ∓(δi1 −δi N ) e N 1 q ±eii , (0) [ei,i+1 , e(1) N 1 ] = 0 for i = 2, 3, . . . , N − 2, (0)
(1)
[ei+1,i , e N 1 ] = 0 for i = 1, 2, . . . , N − 1, (0)
(0)
(1)
[e12 , [e12 , e N 1 ]q ]q = 0, (0)
(0)
(6.22)
(1)
[e N −1,N , [e N −1,N , e N 1 ]q ]q = 0, (0)
(1)
(1)
(1)
(1)
[[e12 , e N 1 ]q , e N 1 ]q = 0, (0)
[[e N −1,N , e N 1 ]q , e N 1 ]q = 0. The Hopf structure of Uq (gl N [u]) is defined by the formulas (6.6)-(6.8) for Uq (gl(0) N ) (1) and the following formulas for the comultiplication and the antipode of e N 1 : (1)
(1)
(0)
(0)
(1)
∆q (e N 1 ) = e N 1 ⊗ 1 + q e11 −e N N ⊗ e N 1 , (1)
Sq (e N 1 ) = −q
(0) (0) e N N −e11
(1)
eN 1.
(6.23) (6.24)
ˆ N are Quantum analogs of the seaweed algebra sw N +1 and its affine realization sw inherited from the quantum algebras Uq (gl N +1 ) and Uq (gl N [u]). Namely, the quantum algebra Uq (sw N +1 ) is generated by the root vectors: e21 , ei,i+1 , ei+1,i for i = 2, 3, . . . , N and e N ,N +1 , and also by the q-Cartan elements: q e11 +e N +1,N +1 , q eii for i = 2, 3, . . . , N ˆ N ) is generated with the relations satisfying (6.4). Similarly, the quantum algebra Uq (sw (0) (0) (0) (1) by the root vectors: e21 , ei,i+1 , ei+1,i for i = 2, 3, . . . , N and e N ,1 , and also by the (0)
q-Cartan elements: q eii for i = 1, 2, 3, . . . , N with the relations satisfying (6.4) and ˆ N ) are isomorphic as associative (6.22). It is clear that the algebras Uq (gl N +1 ) and Uq (sw algebras but they are not isomorphic as Hopf algebras. However if we introduce a new coproduct in the Hopf algebra Uq (gl N +1 ) (F
)
∆ q 1,N +1 (x) = F1,N +1 ∆q (x)F−1 1,N +1 (∀x ∈ Uq (gl N +1 )),
(6.25)
F1,N +1 := q −e11 ⊗e N +1,N +1 ,
(6.26)
where we obtain the isomorphism of Hopf algebras (F1,N +1 )
Uq (F
)
ˆ N ). (sw N +1 ) Uq (sw
(6.27)
Here Uq 1,N +1 (sw N +1 ) denotes the quantum seaweed algebra Uq (sw N +1 ) with the twisted coproduct (6.25).
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7. Cartan Part of Cremmer-Gervais r-Matrix First of all we recall the classification of quasi-triangular r -matrices for a simple Lie algebra g. The quasi-triangular r -matrices are solutions of the system r 12 + r 21 = Ω, [r 12 , r 13 ] + [r 12 , r 23 ] + [r 13 , r 23 ] = 0,
(7.1)
where Ω is the quadratic Casimir two-tensor in g ⊗ g. Belavin and Drinfeld proved that any solution of this system is defined by a triple (Γ1 , Γ2 , τ ), where Γ1 , Γ2 are subdiagrams of the Dynkin diagram of g and τ is an isometry between these two subdiagrams. Further, each Γi defines a reductive subalgebra of g, and τ is extended to an isometry (with respect to the corresponding restrictions of the Killing form) between the corresponding reductive subalgebras of g. The following property of τ should be satisfied: τ k (α) ∈ Γ1 for any α ∈ Γ1 and some k. Let Ω0 be the Cartan part of Ω. Then one can construct a quasi-triangular r -matrix according to the following Theorem 12 (Belavin–Drinfeld [1]). Let r0 ∈ h ⊗ h satisfy the system r012 + r021 = Ω0 ,
(7.2)
(α ⊗ 1 + 1 ⊗ α)(r0 ) = h α ,
(7.3)
(τ (α) ⊗ 1 + 1 ⊗ α)(r0 ) = 0 for any α ∈ Γ1 . Then the tensor r = r0 + e−α ⊗ eα + α>0
α>0;k≥1
e−α ∧ eτ k (α)
(7.4)
(7.5)
satisfies (7.1). Moreover, any solution of the system (7.1) is of the above form, for a suitable triangular decomposition of g and suitable choice of a basis {eα }. In what follows, aiming the quantization of algebra structures on the polynomial Lie algebra gl N [u]) we shall use the twisted two-tensor q r0 (N ) , where r0 (N ) is the Cartan part of the Cremmer–Gervais r -matrix for the Lie algebra gl N when Γ1 = {α1 , α2 , . . . , α N −2 } Γ2 = {α2 , α3 , . . . , α N −1 } and τ (αi ) = αi+1 . An explicit form of r0 (N ) is given by the following proposition (see [16]). Proposition 3. The Cartan part of the Cremmer–Gervais r -matrix for gl N is given by the following expression r0 (gl N ) =
N 1 eii ⊗ eii + 2 i=1
1≤i< j≤N
N + 2(i − j) eii ∧ e j j . 2N
(7.6)
It is easy to check that the Cartan part (7.6), r0 (N ) := r0 (gl N ), satisfies the conditions (7.7) k ⊗ id + id ⊗ k r0 (N ) = ekk for k = 1, 2, . . . , N , k−1 k ⊗ id+id ⊗ k r0 (N ) = (k − k ) C1 (N ) − eii for 1 ≤ k < k ≤ N , i=k +1
(7.8)
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where C1 (N ) is the normalized central element (the Casimir element of first order): C1 (N ) :=
N 1 eii . N
(7.9)
i=1
In particular (7.7) and (7.8) imply the Belavin–Drinfeld conditions (7.3) and (7.4), i.e. (7.10) αk ⊗ id + id ⊗ αk r0 (N ) = h αk := ekk − ek+1,k+1 , τ (αk ) ⊗ id + id ⊗ αk r0 (N ) = αk +1 ⊗ id + id ⊗ αk r0 (N ) = 0 (7.11) for k = 1, 2 . . . , N −1 , k = 1, 2 . . . , N −2 , where αk = k −k+1 and αk = k −k +1 are the simple roots of the system Π (gl N ) (see (6.3)). Now we consider some properties of the two-tensor q r0 (N ) . First of all it is obvious that this two-tensor satisfies the cocycle equation. Furthermore, in order to construct a quantum twist corresponding to the Cremmer–Gervais r -matrix (7.6) we introduce new Cartan–Weyl basis elements eij (i = j) for the quantum algebra Uq (gl N ) as follows: eij
= ei j q ((i − j )⊗id)(r0 (N )) = ei j q
e ji
= q (id⊗( j −i ))(r0 (N )) e
j−1 k=i
j
ji
=q
k=i+1
ekk −( j−i)C1 (N )
ekk −( j−i)C1 (N )
,
(7.12)
e ji ,
(7.13)
for 1 ≤ i < j ≤ N . Permutation relations for these elements can be easily obtained from relations (6.12)–(6.21). For example, we have [ei j , eji ] = [ei j , e ji ] q ((i − j )⊗id+id⊗( j −i ))(r0 (N )) 2
=
q
j−1 k=i
ekk −2( j−i)C1 (N )
2
j
− q k=i+1 q − q −1
(7.14)
ekk −2( j−i)C1 (N )
.
It is not hard to check that the Chevalley elements ei,i+1 and ei+1,i have the following r (N ) 0 coproducts after twisting by the two-tensor q : q r0 (N ) ∆q (ei,i+1 )q −r0 (N ) = ei,i+1 ⊗ q 2((i −i+1 )⊗id)(r0 (N )) + 1 ⊗ ei,i+1 = ei,i+1 ⊗ q 2eii −2C1 (N ) + 1 ⊗ ei,i+1 ,
q
r0 (N )
∆q (ei+1,i )q −r0 (N )
= =
ei+1,i ei+1,i
⊗1+q
−2(id⊗(i+1 −i ))(r0 (N ))
(7.15) ⊗ ei+1,i
⊗ 1 + q 2ei+1,i+1 −2C1 (N ) ⊗ ei+1,i ,
(7.16)
for 1 ≤ i < N . Since the quantum algebra Uq (gl N ) is a subalgebra of the quantum (1) affine algebra Uq (gl N [u]) let us introduce the new affine root vector e N 1 in accordance with (7.12): e N 1 = e N 1 q (( N −1 )⊗id)(r0 (N )) = e N 1 q e N N −C1 (N ) . (1)
(7.17)
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The coproduct of this element after twisting by the two-tensor q r0 (N ) has the form q r0 (N ) ∆q (e N 1 )q −r0 (N ) = e N 1 ⊗ q 2((1 − N )⊗id)(r0 (N )) + 1 ⊗ e N 1 (1)
(1)
(1)
(1)
(1)
= e N 1 ⊗ q 2e N N −2C1 (N ) + 1 ⊗ e N 1 .
(7.18)
Consider the quantum seaweed algebra Uq (sw N +1 ) after twisting by the two-tensor q r0 (N +1) . Its new Cartan–Weyl basis and the coproduct for the Chevalley generators are given by formulas (7.12), (7.13) and (7.15), (7.16), where N should be replaced by N +1, and where i = 1 in (7.12) and (7.15), and j = N in (7.13), and i = N in (7.16). In particular, for the element eN ,N +1 we have eN ,N +1 = e N ,N +1 q e N N −C1 (N +1) , q
r0 (N +1)
∆q (eN ,N +1 )q −r0 (N +1) = eN ,N +1
⊗q
2e N N −2C1 (N +1)
(7.19) +1 ⊗ eN ,N +1 .
(7.20)
Comparing the Hopf structures of the quantum seaweed algebra Uq (sw N +1 ) after twiˆ N ), after twisting by the sting by the two-tensor q r0 (N +1) and its affine realization Uq (sw two-tensor q r0 (N ) we see that these algebras are isomorphic as Hopf algebras: ˆ N ))q −r0 (N ) . q r0 (N +1) ∆q (Uq (sw N +1 ))q −r0 (N +1) q r0 (N ) ∆q (Uq (sw
(7.21)
In terms of new Cartan–Weyl bases this isomorphism, “ı”, is arranged as follows: ı(eij ) = ei(0) j ı(e ji ) = ı(eii − C1 (N + 1)) =
for 2 ≤ i < j ≤ N ,
(0) e ji for 1 ≤ (0) eii − C1 (N ),
(7.22)
i < j ≤ N − 1,
(7.23)
for 2 ≤ i ≤ N , N
ı(eiN +1 ) = ei1 = ei1 q ((i −1 )⊗id)(r0 (N )) = ei1 q k=i (1)
(1)
(1)
(7.24) ekk −(N +1−i)C1 (N )
(7.25)
(1)
for 2 ≤ i ≤ N , where the affine root vectors ei1 (2 ≤ i < N ) are defined by the formula (cf. 6.10): (1)
(0)
(1)
ei1 = [ei N , e N 1 ]q −1 .
(7.26)
8. Affine Realization of Cremmer-Gervais Twist In order to construct a twist corresponding to the Cremmer-Gervais r -matrix (7.6) we will follow the papers [15,19]. Let R be a universal R-matrix of the quantum algebra Uq (gl N +1 ). According to [24] it has the following form: R = R · K,
(8.1)
where the factor K is a q-power of Cartan elements (see [24]) and we do not need its explicit form. The factor R depends on the root vectors and it is given by the following formula: R = R12 (R13 R23 )(R14 R24 R34 ) · · · (R1,N +1 R2,N +1 · · · R N ,N +1 ) ⎛ ⎞ j−1 N +1 ⎝↑ =↑ Ri j ⎠ , j=2
i=1
(8.2)
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where (8.3) Ri j = expq −2 ((q − q −1 )ei j ⊗ e ji ), xn , (n)q ! ≡ (1)q (2)q . . . (n)q , (k)q ≡ (1 − q k )/(1 − q). (8.4) expq (x) := (n)q ! n≥0
It should be noted that the product of factors Ri j in (8.2) corresponds to the normal ordering (6.9) where N is replaced by N + 1. Let R := q r0 (N +1) R q −r0 (N +1) . It is evident that ⎛ ⎞ j−1 N +1 ⎝↑ R =↑ Rij ⎠ , (8.5) j=2
i=1
where Rij = expq −2 ((q − q −1 )eij ⊗ e ji ).
(8.6)
Here eij and e ji are the root vectors (7.12) and (7.13) where N should be replaced by N + 1. Let T be a homomorphism which acts on the elements eij (1 ≤ i < j ≤ N + 1) by formulas T (eij ) = eτ (i j) = ei+1, j+1 for 1 ≤ i < j ≤ N , and T (ei,N +1 ) = 0 for all i = 1, 2, . . . , N . We set ⎛ ⎞ j−1 N +1−k (k) ⎝↑ R (k) := (T k ⊗ id)(R ) =↑ Ri j ⎠ , (8.7) j=2
i=1
where (k)
Ri j
= expq −2 (q − q −1 )T k (eij ) ⊗ e ji = expq −2 (q − q −1 )ei+k, j+k ⊗ e ji
(8.8)
for k ≤ N − j. According to [15,19], the Cremmer-Gervais twist FC G in Uq (gl N +1 ) is given as follows: FC G = F · q r0 (N +1) ,
(8.9)
F = R (N −1) R (N −2) · · · R (1) .
(8.10)
where
It is easy to see that the support of the twist (8.10) is the quantum seaweed algebra Uq (sw N +1 ) with the coproducts (7.15) and (7.16), where N should be replaced by N + 1. From the results of the previous section it follows that we can immediately obtain an affine realization Fˆ C G which twists the quantum affine algebra Uq (gl N [u]): FˆC G = Fˆ · q r0 (N ) , Fˆ := (ı ⊗ ı)(F) = Rˆ (N −1) Rˆ (N −2) · · · Rˆ (1) , ⎛ ⎞ j−1 N +1−k (k) ⎝↑ Rˆ (k) = ↑ Rˆ i j ⎠ , j=2
i=1
(8.11) (8.12) (8.13)
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where
(k) (0) (0) for 1 ≤ i < j ≤ N − k, Rˆ i j = expq −2 (q −q −1 )ei+k, j+k ⊗ e ji (k) (1) (0) Rˆ i,N +1−k = expq −2 (q −q −1 )e N ,i+k ⊗ e N +1−k,i for 1 ≤ i < N + 1 − k.
(8.14) (8.15)
Finally, using the isomorphism (6.27) from Sect. 6 we obtain the following two results: ˆ N ), and let R ˆ be the uniTheorem 13. Let Fˆ C G be the twist Fˆ C G reduced to Uq (sl 21 −1 ˆ N ). Then the R-matrix Fˆ R ˆ ˆ versal R-matrix for Uq (sl C G F C G quantizes the quasitrigonometric solution (4.30). Now we turn to quantization of the quasi-trigonometric solution given by (4.29). The ˆ N are isomorphic as Lie isomorphism (6.27) shows that classical limits sw N +1 and sw bialgebras. Computations show that the support of solution (4.29) is contained in the support of the solution (4.30). So, we can push the twist related to the solution (4.29) to sl(N + 1). It is not difficult to see that such an obtained twist will be defined by the following Belavin-Drinfeld triple for sl(N + 1): {α2 , ..., α N −1 } → {α3 , ..., α N }. In fact, this is exactly the Cremmer-Gervais twist for sl(N ), embedded into sl(N + 1) as the N × N block in the right low corner. The corresponding constant twist for the above Belavin-Drinfeld triple can be quantized by means of [15]. Using once again (6.27), we get a quantum twist which we denote s . by Fˆ CG ˆ N ). Then the R-matrix Theorem 14. Let Rˆ be the universal R-matrix for Uq (sl s21 s −1 ˆ ˆ ˆ FCG R (FCG ) quantizes the quasi-trigonometric solution (4.29). Remark 11. In fact, using the isomorphism (6.27) we can quantize all the quasi-trigonometric solutions of the CYBE corresponding to the first simple root α1 of sl(N ). Let W be the Lagrangian subalgebra of sl(N )((u −1 )) ⊕ sl(N ) contained in Oα1 ⊕ sl(N ). Then it is not difficult to show that the support of the corresponding classical ˆ N . Therefore, it provides twist is contained in Oα1 ∩ sl(N )[u], which is isomorphic to sw ˆ N and sw N +1 are isomorphic as Lie bialgebras. a classical twist in sw N +1 since sw Now we can again use results of [15,19] to get the corresponding quantum affine twist. 9. Affinization by Automorphism and Quantization of Rational r-Matrices The aim of this section is to quantize certain rational r -matrices. We begin with the following result: Theorem 15. Let π : Uq (g[u]) −→ Uq (g) be the canonical projection sending all the affine generators to 0. Let F ∈ Uq (g)⊗Uq (g) be a twist. Let ω ∈ Uq (g[u]) be invertible. Consider the following elements: Fπ ω = (ω−1 π(ω) ⊗ 1)F(π ⊗ id)(∆(ω−1 ))∆(ω) ∈ Uq (g[u]) ⊗ Uq (g) and F = Fπ ω F −1 . Then Fπ ω is a twist if the following condition holds: F12 (π ⊗ id ⊗ id)(∆ ⊗ id)(F ) = F23 F12 .
(9.1)
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Proof. We will verify the cocycle equation for the equivalent element Fπ ω := (ω ⊗ ω)Fπ ω ∆(ω−1 ) = (π ⊗ id) (ω ⊗ ω)F∆(ω−1 ) .
Note that Fπ ω ∈ Uq (g) ⊗ Uq (g[u]). It follows that Assoc(Fπ ω ) := (Fπ ω )12 (∆ ⊗ id)(Fπ ω )(id ⊗ ∆)((Fπ ω )−1 )(Fπ ω )−1 23 ∈ Uq (g) ⊗ Uq⊗2 (g[u]).
On the other hand
Assoc(Fπ ω ) = (π ⊗ id ⊗ id) ω⊗3 Assoc(Fπ ω )(ω−1 )⊗3 .
If we take into account (π ⊗ id)(F ) = 1 ⊗ 1 and property (9.1), then we get Assoc(Fπ ω ) =
−1 = Ad(π(ω) ⊗ ω⊗2 ) F12 F23 (∆ ⊗ id)(F)(id ⊗ ∆)(F −1 )(F23 )−1 (F23 ) = Ad(π(ω) ⊗ ω⊗2 ) F23 Assoc(F)(F23 )−1 . (9.2)
Since F is a twist we deduce that Assoc(Fπ ω ) = 1 ⊗ 1 ⊗ 1. This ends the proof.
Remark 12. A kind of converse statement is true: Assume that J ∈ Uq (g[u]) ⊗ Uq (g) and (π ⊗ id)(J ) are twists. Denote F := (π ⊗ id)(J ). Then there exists an invertible element ω ∈ Uq (g[u]) such that J = (ω−1 π(ω) ⊗ 1)F(π ⊗ id)(∆(ω−1 ))∆(ω). −1 Indeed, observe that the twists (S ⊗ S)(J21 ) and J have one and the same classical limit. By results [17] and [13], there exists ω such that −1 ) = (ω ⊗ ω)J ∆(ω−1 ) ∈ Uq (g) ⊗ Uq (g[u]). (S ⊗ S)(J21
It is not difficult to see that J = (ω−1 π(ω) ⊗ 1)F(π ⊗ id)(∆(ω−1 )) · ∆(ω). If an element ω satisfies the conditions of Theorem 15 we will call it an affinizator as it allows to construct an affine extension for a non-affine twist Ψ . Such element of course is not unique but some affinizators allow to construct Ψω which are compatible with the Yangian degeneration. Consider as an example the affinization of the coboundary twist F = (expq 2 (λ e−α ) ⊗ expq 2 (λ e−α ))∆(expq −2 (−λ e−α )) with ω = expq 2 (µ q −h α eδ−α ). In this case we obtain (− 1 1⊗h α )
Fπ ω = (1 ⊗ 1 + (q 2 − 1)λ q −h α eδ−α ⊗ q −h α + (q 2 − 1)µ e−α ⊗ 1)q 2 2
.
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Let us form the equation for which the different ω are the solutions. In order to find such ω, we consider the following equation: (1) (1) (2) µ(id ⊗ S)(Fπ ω ) = ω−1 π(ω)Fi π(ω j )S(ω j )S(Fi2 ), i, j
(1) (1) (2) (2) where F = i Fi ⊗ Fi and ∆(ω) = i ωi ⊗ ωi . Now we would like to explain how ω-affinization can be used to find a Yangian degeneration of the affine Cremmer–Gervais twists. Let us consider the case sl3 . We set (τ ) (0) (0) F = FC G 3 := expq 2 (−(q − q −1 )ζ eˆ12 ⊗ eˆ32 ) · Kˆ3 ,
(9.3)
4 2 5 7 Kˆ 3 = q 9 h 12 ⊗h 12 + 9 h 12 ⊗h 23 + 9 h 23 ⊗h 12 + 9 h 23 ⊗h 23
(9.4)
where
with h i j := eii − e j j . The twist (9.3) belongs to Uq (sl3 ) ⊗ Uq (sl3 )[[ζ ]]. long The following affinizator ω3 was constructed in [31]. It is given by the following formula: ζ qζ 2 long (1) (1) 2h ⊥ 2h ⊥ β α ω3 = expq 2 q eˆ21 expq 2 − q eˆ31 1 − q2 (1 − q 2 )2 2 ζ ζ ζ2 (0) (0) (0) × expq −2 expq −2 expq −2 , (9.5) eˆ eˆ eˆ 1 − q 2 32 1 − q 2 21 1 − q 2 32 1 2 2 1 ⊥ where h ⊥ α = 3 (e11 + e22 ) − 3 e33 and h α = 3 e11 − 3 (e22 + e33 ). For convenience sake we remind the reader that ⊥
⊥
(0) 0 q h β −h α eˆ12 = e12 ⊥
(0)
(0)
eˆ32 = q −h β e32 ,
(1) eˆ31
=
⊥ (1) ⊥ q h α −h β e31 ,
long
Theorem 16. The elements ω3 consequently Fπ ω is a twist.
⊥
(0) (0) eˆ21 = q h β e21 , (0)
(0) (0)
0 = e0 e −1 eˆ31 32 21 − q e21 e32 ,
(1) eˆ32
=
(0) (1) e12 e31
(9.6)
(1) (0) − qe31 e12 .
(τ )
, F = FC G 3 satisfy the conditions of Theorem 15 and
Proof. Straigtforward.
It turns out that Fπ ω has a rational degeneration. To define this rational degeneration we have to introduce the so-called f -generators: ⊥
(0) (1) (0) , f 1 = q 2h β eˆ31 + q −1 ζ eˆ31 , f 0 = (q − q −1 ) eˆ31 (0)
⊥
(1)
(0)
f 2 = (1 − q −2 ) eˆ32 , f 3 = q h α eˆ32 − ζ eˆ32 . ˆ ˆ 3 ) generated by Let us consider the Hopf subalgebra of UqK3 (sl (0)
(0)
{h 12 , h 23 , f 0 , f 1 , f 2 , f 3 , eˆ12 , eˆ21 }.
(9.7)
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When q → 1 we obtain the following Yangian twist (see [31]): ⊥
(0)
⊥
(−h β ⊗1) F π ω = (1 ⊗ 1 − ζ 1 ⊗ f 3 − ζ 2 h ⊥ (1 ⊗ 1 + ζ 1 ⊗ e21 )(−h β ⊗1) β ⊗ f 2) (0) (0) (0) × exp(ζ 2 e12 h 13 ⊗ f 0 ) exp(−ζ e12 ⊗ f 1 ) · exp(−ζ e12 ⊗ f 2) ⊥
⊥
((h β −h α )⊗1) , × (1 ⊗ 1 − ζ 1 ⊗ f 3 − ζ 2 h ⊥ α ⊗ f 2)
(9.8)
where the overlined generators are the generators of Y (sl3 ). In the evaluation representation we have: f 0 → e31 , f 1 → u e31 , f 2 → e32 , f 3 → u e32 , e21 → e21 , e12 → e12 .
(9.9)
Therefore we have obtained the following result:
Theorem 17. The Yangian twist F π ω quantizes the following classical rational r −matrix: r (u, v) =
Ω ⊥ ⊥ + h⊥ α ⊗ ve32 − ue32 ⊗ h α + h β ∧ e21 u−v + e21 ⊗ ve31 − ue31 ⊗ e21 + e12 ∧ e32 .
(9.10)
To obtain a quantization of the second non-constant rational r -matrix for sl3 we take the following affinizator ω3short and apply it to F = q r0 (3) , where the Cartan part of the Cremmer-Gervais constant r -matrix for sl3 has the form: r0 (3) =
1 1 2 h α1 ⊗ h α1 +h α2 ⊗ h α2 + h α1 ⊗ h α2 +h α2 ⊗ h α1 + h α1 ∧ h α2 . (9.11) 3 3 6
We have
(0) ω3short = expq −2 ζ eˆ21 expq 2 −
ζ ⊥ (1) q 2h α eˆ31 1 − q2
expq −2
ζ (0) , (9.12) e ˆ 1 − q 2 32
where (0)
1
(0)
(0)
⊥
(0)
(1)
⊥
(1)
eˆ21 = q − 3 (h 12 −h 23 ) e21 , eˆ32 = q −h α e32 , eˆ31 = q −h α e31 . We have to calculate
Aff ω short (q r0 (3) ) := (π ⊗ id) ◦ (ω3short ⊗ ω3short )q r0 (3) ∆(ω3short )−1 . 3
(9.13)
Using standard commutation relations between q-exponents, Fπ ω can be brought to the following form: ⊥ ⊥ (1) ⊥ (0) (0) (−h α ⊗1) 1 ⊗ 1 + ζ 1 ⊗ q 2h α eˆ31 + ζ q −2h α ⊗ Ad expq 2 (ζ eˆ21 ) (eˆ32 ) 2 1 1 (0) (− 3 (h 12 −h 23 )⊗1) r0 (3) × 1 ⊗ 1 + ζ (1 − q 2 ) 1 ⊗ eˆ21 −2 q . q
q
(9.14)
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The q-Hadamard formula allows us to calculate the Ad-term explicitly: ⊥ (0) (0) (0) (0) Ad expq −2 (ζ eˆ21 ) (eˆ21 ) = eˆ21 + ζ q −h β e31 , (0)
(0) (0)
(9.15)
(0) (0)
where e31 := e21 e32 − q e32 e21 . To define a rational degeneration we introduce g-generators, which satisfy the Yangian relations as q → 1: ⊥
(0)
(1)
⊥
(0)
(0)
g0 = (q − q −1 )q −h β e31 , g1 = q 2h α eˆ31 + ζ q −h β e31 , g2 = (q 2 − 1) eˆ21 . ⊥
Using g-generators we can calculate the rational degeneration of the twist Fπ ω : (−h ⊥α ⊗1) (0) F π ω = 1 ⊗ 1 + ζ 1 ⊗ (g 1 + e32 ) − ζ 2 h ⊥ ⊗ g 0 α (− 1 (h −h )⊗1) × 1 ⊗ 1 − ζ 1 ⊗ g 2 3 12 23 .
(9.16)
Theorem 18. The Yangian twist F π ω quantizes the following rational r -matrix: r (u, v) =
1 Ω ⊥ ⊥ −u e31 ⊗ h ⊥ α +v h α ⊗ e31 + h α ∧ e32 − (h 12 −h 23 ) ∧ e21 . (9.17) u−v 3
Therefore we have quantized all non-trivial rational r -matrices for sl3 classified in [33]. 10. Solutions for sl(2) and Deformed Hamiltonians We consider the case sl(2). Let σ + = e12 , σ − = e21 and σ z = e11 − e22 . Recall that in sl(2) we have two quasi-trigonometric solutions, modulo gauge equivalence. The nonquasi-constant solution is X 1 (z 1 , z 2 ) = X 0 (z 1 , z 2 ) + (z 1 − z 2 )(σ + ⊗ σ + ). This solution is gauge equivalent to the following: X a,b (z 1 , z 2 ) =
z2 Ω 1 + σ− ⊗ σ+ + σz ⊗ σz z1 − z2 4 − z z + a(z 1 σ ⊗ σ − z 2 σ ⊗ σ − ) + b(σ − ⊗ σ z − σ z ⊗ σ − ). (10.1)
The above quasi-trigonometric solution was quantized in [23]. Let π1/2 (z) be the 2 ). In this representation, the generator two-dimensional vector representation of Uq (sl e−α acts as a matrix unit e21 , eδ−α as ze21 and h α as e11 − e22 . The quantum R-matrix 2 ) in the tensor product π1/2 (z 1 ) ⊗ π1/2 (z 2 ) is the following: of Uq (sl R0 (z 1 , z 2 ) = e11 ⊗ e11 + e22 ⊗ e22 + +
z1 − z2 (e11 −1 q z 1 − qz 2
⊗ e22 + e22 ⊗ e11 )
q −1 − q (z 2 e12 ⊗ e21 + z 1 e21 ⊗ e12 ). q −1 z 1 − qz 2
(10.2)
Proposition 4. The R-matrix given by R : = R0 (z 1 , z 2 ) +
z1 − z2 ((b −1 q z 1 − qz 2 − z
+ az 2 )σ z ⊗ σ −
+ (q −1 az 1 + qb)σ ⊗ σ + (b + az 2 )(q −1 az 1 + qb)σ − ⊗ σ − ) is a quantization of the quasi-trigonometric solution X a,b .
(10.3)
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Corollary 2. The rational degeneration R F (u 1 , u 2 ) =
P12 u1 − u2 (1 − η − ξ u2σ z ⊗ σ − u1 − u2 − η u1 − u2
(10.4)
+ξ(u 1 − η)σ − ⊗ σ z + ξ 2 u 2 (u 1 − η)σ − ⊗ σ − ), where P12 denotes the permutation of factors in C2 ⊗ C2 , is a quantization of the following rational solution of the CYBE: Ω + ξ(u 1 σ − ⊗ σ z − u 2 σ z ⊗ σ − ). u1 − u2
r (u 1 , u 2 ) =
(10.5)
The Hamiltonians of the periodic chains related to the twisted R-matrix were computed in [23]. We recall this result. We consider t (z) = T r0 R0N (z, z 2 )R0N −1 (z, z 2 )...R01 (z, z 2 )
(10.6)
a family of commuting transfer matrices for the corresponding homogeneous periodic chain, [t (z ), t (z")] = 0, where we treat z 2 as a parameter of the theory and z = z 1 as a spectral parameter. Then the Hamiltonian Ha,b,z 2 = (q −1 − q)z
d t (z) |z=z 2 t −1 (z 2 ) dz
(10.7)
can be computed by a standard procedure: − − z Ha,b,z 2 = H X X Z + (C(σkz σk+1 + σk− σk+1 ) + Dσk− σk+1 .
(10.8)
k
Here C = ((q − 1)/2)(b − az 2 q −1 ), D = (az 2 + b)(q −1 az 2 + qb), σ + = e12 , σ − = e21 , σ z = e11 − e22 and HX X Z =
q + q −1 z z − + σk σk+1 ). (σk+ σk+1 + σk− σk+1 + 2
(10.9)
k
We see that, by a suitable choice of parameters a, b and z 2 , we can add to the XXZ − z Hamiltonian an arbitrary linear combination of the terms k σkz σk+1 + σk− σk+1 and − − σ σ and the model will remain integrable. k k k+1 Moreover, it was proved in [23] that the Hamiltonian Hη,ξ,u 2 = ((q −1 − q)u − q −1 η)
d t (u) |u=u 2 t −1 (u 2 ) du
(10.10)
for t (u) = T r0 R0N (u, u 2 )R0N −1 (u, u 2 )...R01 (u, u 2 ), ξ((q −1
(10.11) (q −1 ξ η)/2
is given by the same formula (10.7), where C = − 1)/2)u 2 − D = ξ 2 u 2 (q −1 u 2 − qη). Now it also makes sense in the XXX limit q = 1: − − z (C(σkz σk+1 + σk− σk+1 ) + Dσk− σk+1 ), Hη,ξ,u 2 = H X X X + k
where C = −ξ η/2 and D =
ξ 2u
2 (u 2
− η).
and
(10.12)
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Appendix In this appendix we give the proofs of the following results mentioned in the text: Proposition 5. Let X be a rational or quasi-trigonometric solution of (2.4). Then X satisfies the unitarity condition (2.5). Proof. The proof is almost a word by word transcription of the proof of [3], Prop. 4.1. Interchanging u 1 and u 2 and also the first and second factors in g⊗3 in Eq. (2.4), we obtain [X 21 (u 2 , u 1 ), X 23 (u 2 , u 3 )] + [X 21 (u 2 , u 1 ), X 13 (u 1 , u 3 )] + + [X 23 (u 2 , u 3 ), X 13 (u 1 , u 3 )] = 0.
(10.13)
Adding (10.13) and (2.4), we get [X 12 (u 1 , u 2 ) + X 21 (u 2 , u 1 ), X 13 (u 1 , u 3 ) + X 23 (u 2 , u 3 )] = 0.
(10.14)
Ω a) Suppose X is rational, i.e. X (u, v) = u−v + p(u, v), where p is a polynomial. For u 1 and u 2 fixed, let us multiply (10.14) by u 2 − u 3 and let u 3 → u 2 . It follows that
[X 12 (u 1 , u 2 ) + X 21 (u 2 , u 1 ), Ω 23 ] = 0.
(10.15)
It is known that if a tensor r ∈ g ⊗ g satisfies [r ⊗ 1, Ω 23 ] = 0, then r = 0. It follows that X 12 (u 1 , u 2 ) + X 21 (u 2 , u 1 ) = 0. vΩ b) Suppose X is quasi-trigonometric, i.e. X (u, v) = u−v + q(u, v), where q is a polynomial function. By the same procedure we get [X 12 (u 1 , u 2 ) + X 21 (u 2 , u 1 ), u 2 Ω 23 ] = 0
(10.16)
which also implies the unitarity condition.
Proposition 6. Let W be a Lie subalgebra satisfying Conditions 2) and 3) of Theorem 2. Let r be constructed as in (3.17). Assume r induces a Lie bialgebra structure on g[u] by δr (a(u)) = [ r (u, v), a(u) ⊗ 1 + 1 ⊗ a(v)]. Then W ⊇ u −N g[[u −1 ]] for some positive N. Proof. Since W is a Lagrangian subalgebra, it is enough to prove that W is bounded. Let us write r (u, v) = X 0 (u, v) + Γm , (10.17) m
where Γm is the homogeneous polynomial of degree m with coefficients in g ⊗ g: Γm = amnk u n v k . (10.18) n+k=m
It is enough to prove that there exists a positive integer N such that Γm = 0 for m ≥ N . We know that δr (a) should belong to g[u] ⊗ g[v] for any element a of g. On the other hand, one can see that [Γm , a ⊗ 1 + 1 ⊗ a] is either 0 or has degree m. This implies that [Γm , a ⊗ 1 + 1 ⊗ a] = 0 for m large enough. Therefore Γm = Pm (u, v)Ω
(10.19)
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with Pm (u, v) ∈ C[[u, v]]. Let us compute the following: [Γm , au ⊗ 1 + 1 ⊗ av] = Pm (u, v)(u − v)[Ω, a ⊗ 1] + Pm (u, v)v[Ω, a ⊗ 1 + 1 ⊗ a] = Pm (u, v)(u − v)[Ω, a ⊗ 1]. We choose an element a such that [Ω, a ⊗ 1] = 0. We obtain that if Pm (u, v) is not identically zero then Pm (u, v)(u − v)[Ω, a ⊗ 1] is a homogeneous polynomial of degree m + 1. Consequently, δr (au) = Pm (u, v)(u − v)[Ω, a ⊗ 1] (10.20) m
cannot belong to g[u] ⊗ g[v] unless Pm (u, v) = 0 for m large enough.
Theorem 19. Let X (z 1 , z 2 ) be a quasi-trigonometric solution of the CYBE. 1. Then there is a transformation Ψ (z), holomorphic around z = 1, such that z1 Ψ (z 1 )−1 ⊗ Ψ (z 2 )−1 X (z 1 , z 2 ) = Y ( ), z2 where Y (z) =
Ω z−1
+ s(z), with s(z) holomorphic around z = 1.
2. Y (eλ ) is a trigonometric solution of the CYBE in the sense of Belavin–Drinfeld. Ω Proof. Let us consider X (z 1 , z 2 ) = zz12−z + p(z 1 , z 2 ), where p(z 1 , z 2 ) is a polynomial. 2 Let {Ii } be an orthonormal basis in g with respect to the Killing form and {cikj } denote the structure constants of g with respect to {Ii }. Let us write p(z 1 , z 2 ) = pi j (z 1 , z 2 )Ii ⊗ I j . i, j
We set h(z) =
pi j (z, z)[Ii , I j ] =
i, j
pi j (z, z)cikj Ik .
i, j,k
Repeating the arguments of [3], one can prove that h(z) and X (z 1 , z 2 ) satisfy z1
∂ X (z 1 , z 2 ) ∂ X (z 1 , z 2 ) + z2 = [h(z 1 ) ⊗ 1 + 1 ⊗ h(z 2 ), X (z 1 , z 2 )]. ∂z 1 ∂z 2
Suppose Ψ (z) is a function with values in Aut(g), which satisfies the differential equation z
dΨ (z) = (adh(z))Ψ (z), dz
(10.21)
and the initial condition Ψ (1) = id. Then the function Y (z 1 , z 2 ) defined as Y (z 1 , z 2 ) = Ψ (z 1 )−1 ⊗ Ψ (z 2 )−1 X (z 1 , z 2 ), satisfies the CYBE and depends on z 1 /z 2 only. By construction, Ψ (z) is holomorphic in a neighborhood of z = 1 and clearly Y (z) = Ω z−1 + s(z), where s(z) is a holomorphic function in the same neighborhood.
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Now we turn to the proof of the second statement. Apply the change of variables z 1 = eu , z 2 = ev . Then let us prove that Ψ (eu ) is holomorphic in the entire complex plane and Y (u, v) is a trigonometric solution of the CYBE. Let Ψ1 (u) := Ψ (eu ). Clearly this operator satisfies the equation dΨ1 (u) = (adh 1 (u))Ψ1 (u), du
(10.22)
where h 1 (u) = h(eu ). Let U (u) be the matrix of the operator Ψ1 (u) in the basis {Ii }. Let a i j (u, v) := i j p (eu , ev ) (the decomposition of h 1 (u) in the basis {Ii }). Equation (10.22) is equivalent to dU (u) = H (u)U (u), du where H (u) is the matrix with elements h i j (u) = csi j crs t a r t (u, u).
(10.23)
(10.24)
s,r,t
Since the matrix function H (u) is holomorphic in C, the matrix equation (10.23) admits a unique solution u satisfying U (0) = E. This solution is holomorphic in C because U (u) = Pexp( 0 H (v)dv) (ordered exponential) and u u v1 U (u) = 1 + H (v)dv + ( H (v1 )H (v2 )dv2 )dv1 + · · · ≤ 0 0 0 u ≤ exp( H (v) dv. 0
Moreover, according to [3], the linear operator Ψ (u), corresponding to U (u), is an automorphism of g. Clearly Y (eu , ev ) depends only on u − v and, as a function in one variable, has poles when eu−v = 1. Hence it is trigonometric in the sense of Belavin–Drinfeld. This ends the proof.
Acknowledgements. The paper has been partially supported by the Royal Swedish Academy of Sciences under the program “Cooperation between Sweden and former USSR”, the grant for the support of scientific schools NSch-3036.2008.2, and the grants ANR 05-BLAN-0029-01, RFBR-08-01-00392 (S.M.Kh. and V.N.T.). The authors are thankful to the referee of the first version of the paper for pointing out the importance of the papers [6] and [17].
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5. Cremmer, E., Gervais, J.L.: The quantum group structure associated to nonlineary extended Virasoro algebras. Commun. Math. Phys. 134(3), 619–632 (1990) 6. Delorme, P.: Classification des triples de Manin pour les algebres de Lie reductives complexes. J. Algebra 246, 97–174 (2001) 7. Dergachev, V., Kirillov, A.A.: Index of Lie algebras of seaweed type. J. Lie Theory 10(2), 331–343 (2000) 8. Drinfeld, V.G.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations. Soviet Math. Dokl. 27, 68–71 (1983) 9. Drinfeld, V.G.: Quantum groups. In: Proceedings ICM (Berkeley 1986) 1, Providence, RI: Amer. Math. Soc., 1987, pp. 798–820 10. Drinfeld, V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1, 1419–1457 (1990) 11. Drinfeld, V.G.: On constant, quasiclassical solutions of the Yang-Baxter quantum equation. Soviet Math. Dokl. 28(3), 667–671 (1983) 12. Drinfeld, V.G.: On some unsolved problems in quantum group theory. In: Quantum groups, Lecture Notes in Math., 1510, Berlin: Springer, 1992, pp. 1–8 13. Etingof, P., Kazhdan, D.: Quantization of Lie bialgebras I. Selecta Math. 2(1), 1–41 (1986) 14. Etingof, P., Schiffmann, O.: Lectures on Quantum Groups. Somerville, MA: International Press, 1998 15. Etingof, P., Schedler, T., Schiffmann, O.: Explicit quantization of dynamical R-matrices for finitedimensional semisimple Lie algebras. J. AMS 13, 595–609 (2000) 16. Gerstenhaber, M., Giaquinto, A.: Boundary solutions of the classical Yang-Baxter equation. Lett. Math. Phys. 40(4), 337–353 (1997) 17. Halbout, G.: Formality theorem for Lie bialgebras and quantization of twists and coboundary r -matrices. Adv. Math. 207, 617–633 (2006) 18. Hodges, T.J.: Nonstandard quantum groups associated to certain Belavin–Drinfeld triples. In: Perspectives on quantization (South Hadley, MA, 1996), Contemp. Math., 214, Providence, RI: Amer. Math. Soc., 1998, pp. 63–70 19. Isaev, A.P., Ogievetsky, O.V.: On quantization of r -matrices for Belavin-Drinfeld triples. Phys. Atomic Nuclei 64(12), 2126–2130 (2001) 20. Karolinsky, E., Stolin, A.A.: Classical dynamical r -matrices, Poisson homogeneous spaces and Lagrangian subalgebras. Lett. Math. Phys. 60, 257–274 (2002) 21. Khoroshkin, S.M., Pop, I.I., Stolin, A.A., Tolstoy, V.N.: On some Lie bialgebra structures on polynomial algebras and their quantization. Preprint no. 21, 2003/2004, Mittag-Leffler Institute, Sweden (2004) 22. Khoroshkin, S.M., Stolin, A.A., Tolstoy, V.N.: Deformation of the Yangian Y (sl2 ). Commun. Alg. 26(3), 1041–1055 (1998) 23. Khoroshkin, S.M., Stolin, A.A., Tolstoy, V.N.: q-Power function over q-commuting variables and deformed XXX and XXZ chains. Phys. Atomic Nuclei 64(12), 2173–2178 (2001) 24. Khoroshkin, S.M., Tolstoy, V.N.: Universal R-matrix for quantized superalgebras. Commun. Math. Phys. 141(3), 599–617 (1991) 25. Khoroshkin, S.M., Tolstoy, V.N.: Twisting of quantum (super)algebras. Connection of Drinfeld’s and Cartan–Weyl realizations for quantum affine algebras. MPIM preprint, MPI/94-23, Bonn (1994); http://arxiv.org/list/hep-th/9404036, 1994 26. Kulish, P.P., Mudrov, A.I.: Universal R-matrix for esoteric quantum groups. Lett. Math. Phys. 47(2), 139–148 (1999) 27. Montaner, F., Zelmanov, E.: Bialgebra structures on current Lie algebras. Preprint, University of Wisconsin, Madison, 1993 28. Panyushev, D.I.: Inductive formulas for the index of seaweed Lie algebras. Mosc. Math. J. 1(2), 221–241 (2001) 29. Pop, I.: Lie bialgebra structures and their quantization. Doctoral thesis, Department of Mathematical Sciences, Göteborg University, Sweden, 2005 30. Pop, I.: On quasi-trigonometric solutions of CYBE and generalized Belavin-Drinfeld data. Preprint, Department of Mathematical Sciences, Göteborg University, 2008 31. Samsonov, M.: Semi-classical Twists for sl3 and sl4 Boundary r -matrices of Cremmer-Gervais type. Lett. Math. Phys., 72(3), 197–210 (2005) 32. Samsonov, M.: Quantization of semi-classical twists and noncommutative geometry. Lett. Math. Phys., 75(1), 63–77 (2006) 33. Stolin, A.A.: On rational solutions of Yang-Baxter equation for sl(n). Math. Scand. 69, 57–80 (1991) 34. Stolin, A.A.: On rational solutions of Yang-Baxter equation. Maximal Orders in Loop Algebra. Commun. Math. Phys. 141, 533–548 (1991) 35. Stolin, A.A.: A geometrical approach to rational solutions of the classical Yang-Baxter equation. Part I. Symposia Gaussiana, Conf.A, Berlin, New York: Walter de Gruyter, 1995, pp. 347–357 36. Tolstoy, V.N.: Extremal projectors for quantized Kac–Moody superalgebras and some of their applications. Lecture Notes in Phys., 370, Berlin: Springer, 1990, pp. 118–125
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Commun. Math. Phys. 282, 663–695 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0565-7
Communications in
Mathematical Physics
Universality of the REM for Dynamics of Mean-Field Spin Glasses 4 ˇ Gérard Ben Arous1 , Anton Bovier2,3 , Jiˇrí Cerný 1 Courant Institute of the Mathematical Sciences, New York University, 251 Mercer Street,
New York, NY 10012, USA. E-mail: [email protected]
2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39,
10117 Berlin, Germany. E-mail: [email protected]
3 Mathematics Institute, Berlin University of Technology, Strasse des 17. Juni 136,
10269 Berlin, Germany
4 Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland.
E-mail: [email protected] Received: 15 June 2007 / Accepted: 28 April 2008 Published online: 9 July 2008 – © Springer-Verlag 2008
Abstract: We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N -dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γβ, p > 0, such that for all exponential time scales, exp(γ N ), with γ < γβ, p , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ /β 2 < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REMlike trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class. 1. Introduction and Results Aging has become one of the main paradigms to describe the long-time behavior of complex and/or disordered systems. Systems that have strongly motivated this research are spin glasses, where aging was first observed experimentally in the anomalous relaxation patterns of the magnetization [LSNB83,Cha84]. To capture the features of activated dynamics, early on people introduced effective dynamics where the state space is reduced to the configurations with lowest energy [DDOL85,KH89]. The theoretical modeling of aging phenomena took a major leap with the introduction of so-called trap models by Bouchaud and Dean in the early 1990’ies [Bou92,BD95] (see [BCKM98] for a review). These models reproduce the characteristic power law behavior seen experimentally, while being sufficiently simple to allow for a detailed analytical treatment. While trap models are heuristically motivated to capture the behavior of the dynamics of spin glass models, there is no clear theoretical, let alone mathematical derivation of these from an
ˇ G. Ben Arous, A. Bovier, J. Cerný
664
underlying spin-glass dynamics. The first attempt to establish such a connection was made in [BBG02,BBG03a,BBG03b] where it was shown that, starting from a particular Glauber dynamics of the Random Energy Model (REM), at low temperatures and at a time scale slightly shorter than the equilibration time of the dynamics, an appropriate time-time correlation function of the dynamics converges to that given by Bouchaud’s REM-like trap model. ˇ ˇ ˇ ˇ On the other hand, in a series of papers [BC05,B CM06,B C08,B C07] a systematic investigation of a variety of trap models was initiated. In this process, it emerged that there appears to be an almost universal aging mechanism based on α-stable subordinators that governs aging in most trap models. It was also shown that the same feature holds for the dynamics of the REM at shorter time scales than those considered in [BBG03a,BBG03b], and that this also happens at high temperatures, provided appropriate time scales are ˇ ˇ considered [BC08]. For a general review on trap models see [BC06]. However, both in the REM and in the trap models that were analyzed so far, the random variables describing the quenched disorder were considered to be independent. Aging in correlated spin glass models was investigated rigorously only in some cases of spherical SK models and at very short time scales [BDG01]. In the present paper we show for the first time that the same type of aging mechanism is also relevant in correlated spin glasses, at least on time scales that are short compared to the equilibration time (but exponentially large in the volume of the system). Let us first describe the class of models we are considering. Our state spaces will be the N -dimensional hypercube, S N ≡ {−1, 1} N . R N : S N × S N → [−1, 1] denotes N σi τi . The Hamiltonian of the as usual the normalized overlap,√R N (σ, τ ) ≡ N −1 i=1 p-spin SK-model is defined as N H N , where H N : S N → R is a centered Gaussian process indexed by S N with covariance E[H N (σ )H N (τ )] = R N (σ, τ ) p ,
(1.1)
for 3 ≤ p ∈ N. We will denote by H the σ -algebra generated by the random variables {H N (σ ), σ ∈ S N , N ∈ N}. The corresponding Gibbs measure is given by −1 β e µβ,N (σ ) ≡ Z β,N
√
N H N (σ )
,
(1.2)
where Z β,N denotes the normalizing partition function. We define the dynamics as a nearest neighbor continuous time Markov chain σ N (·) on S N with transition rates
N −1 e−β w N (σ, τ ) = 0,
√
N H N (σ ) ,
if dist(σ, τ ) = 1, otherwise;
(1.3)
here dist(·, ·) is the graph distance on the hypercube, 1 |σi − τi |. 2 N
dist(σ, τ ) =
(1.4)
i=1
A simple way to construct this dynamics is as a time change of a simple random walk on S N : We denote by Y N (k) ∈ S N , k ∈ N, the simple unbiased random walk (SRW)
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on S N started at some fixed point of S N , say at (1, . . . , 1). For β > 0 we define the clock-process by S N (k) =
k−1
√ ei exp β N H N (Y N (i)) ,
(1.5)
i=0
where {ei , i ∈ N} is a sequence of mean-one i.i.d. exponential random variables. We denote by Y the σ -algebra generated by the SRW random variables {Y N (k), k ∈ N, N ∈ N}. The σ -algebra generated by the random variables {ei , i ∈ N}, will be denoted by E. For non-integer t ≥ 0 we define S N (t) = S N (t) and we write S N−1 for the generalized right-continuous inverse of S N . Then the process σ N (·) can be written as σ N (t) ≡ Y N (S N−1 (t)).
(1.6)
Obviously, σ N is reversible with respect to the Gibbs measure µβ,N , and S N (k) is the instant of the k th jump of σ N . We will consider all random processes to be defined on an abstract probability space (, F, P). Note that the three σ -algebras, H, Y, and E, are all independent under P. We will systematically exploit the construction of the dynamics given by (1.3) or (1.6). The same dynamics was used in the analysis of the REM and in most work on trap models. It differs substantially from more popular dynamics such as the Metropolis or the heat-bath algorithm. The main difference is that in these dynamics the trajectories are not independent of the environment and are biased against going up in energy. This may have a substantial effect, and we do not know whether our results will apply, at least qualitatively, in these cases. The fact is that we currently do not have the tools to analyze these dynamics even in the case of the REM! Let Vα (t) be an α-stable subordinator with the Laplace transform given by E[e−λVα (t) ] = exp(−tλα ).
(1.7)
Our main technical result is the following theorem that describes the asymptotic behavior of the clock process. Theorem 1.1. There exists a function, ζ ( p), such that, for all p ≥ 3, and γ satisfying 0 < γ < min β 2 , ζ ( p)β , (1.8) under the conditional distribution P[·|Y] the law of the stochastic process 2 2 S¯ N (t) = e−γ N S N t N 1/2 e N γ /2β , t ≥ 0,
(1.9)
defined on the space of càdlàg functions equipped with the Skorokhod M1 -topology, converges, Y-a.s., to the law of the γ /β 2 -stable subordinator, Vγ /β 2 (K t), t ≥ 0, where K is a positive constant depending on γ , β, and p. Moreover, the function ζ ( p) is increasing and satisfies √ (1.10) ζ (3) 1.0291 and lim ζ ( p) = 2 ln 2. p→∞
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We will explain in Sect. 5 what the M1 -topology is. Roughly, it is a weak topology that does not convey much information at the jumps of the limiting process: e.g., it allows for several jumps of the approximating processes at rather short distances in time to merge to one big jump of the limit process. This will actually occur in our models for p < ∞, while it does not happen in the REM. Therefore, we cannot replace the M1 -topology with the stronger J1 -topology in Theorem 1.1. To control the behavior of spin-spin correlation functions that are commonly used to characterize aging, we need to know more on how these jumps occur at finite N . What we will show is that if we slightly coarse-grain the process S¯ N over blocks of size o(N ), the rescaled process does converge in the J1 -topology. What this says, is that the jumps of the limiting process are compounded by smaller jumps that are made over ≤ o(N ) steps of the SRW. In other words, the jumps of the limiting process come from waiting times accumulated in one slightly extended trap, and during this entire time only a negligible fraction of the spins are flipped. This will imply the following aging result. Theorem 1.2. Let AεN (t, s) be the event defined by ≥1−ε . AεN (t, s) = R N σ N teγ N , σ N (t + s)eγ N
(1.11)
Then, under the hypothesis of Theorem 1.1, for all ε ∈ (0, 1), t > 0, s > 0, and α = γ /β 2 ,
sin απ t/(t+s) α−1 ε u (1 − u)−α du. (1.12) lim P[A N (t, s)] = N →∞ π 0 Remark. We will in fact prove the stronger statement that aging in the above sense occurs along almost every random walk trajectory, that is
sin απ t/(t+s) α−1 lim P[AεN (t, s)|Y] = u (1 − u)−α du, Y-a.s. (1.13) N →∞ π 0 Let us discuss the meaning of these results. eγ N is the time-scale at which we want to observe the process. According to Theorem 1.1, at this time the random walk will make 2 2 about N 1/2 e N γ /2β eγ N steps. Since this number is also much smaller than 2 N (as follows from (1.10)), the random walk will essentially visit that number of sites. If the random process H N were i.i.d., then the maximum of H N along the trajectory 1/2 2 2 2 2 ∼ N 1/2 γ /β, and the time up to time N 1/2 e N γ /β would be 2 ln(N 1/2 e N γ /2β ) spent in the site with maximal H N would be of order eγ N . Since Theorem 1.1 holds ˇ also in the i.i.d. case, that is in the REM (see [BC08]), the time spent in the maximum is comparable to the total time and the convergence to the α-stable subordinator implies that the total accumulated time is composed of pieces of order eγ N that are collected along the trajectory. In fact, each jump of the subordinator corresponds to one visit to a site that has waiting time of that order. In a common metaphor, the sites are referred to as traps and the mean waiting times as their depths. The theorem in the general case states that the same is essentially true in the p-spin model. The difference is that now the traps do not consist of a single site, but of a deep valley (along the trajectory) whose bottom has approximately the same energy as in the i.i.d. case and whose shape and width we will describe quite precisely. Remarkably, the number of sites contributing significantly to the residence time in the valley is essentially finite, and different valleys are statistically independent.
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The fact that traps are finite may appear quite surprising to those familiar with the statics of p-spin models. From the results there (see [Tal03,Bov06]), it is known that the Gibbs measure concentrates on “lumps” whose diameter is of order N ε p , with ε p > 0. The mystery is however solved easily: the process H N (σ ) does indeed decrease essentially linearly with speed N −1/2 from a local maximum. Thus, the residence times in such sites decrease geometrically, so that the contributions of a neighborhood of size K of a local maximum amounts to a fraction of (1 − c−K ) of the total time spent in that valley; for the support of the Gibbs measure, one needs however to take into account the entropy, that is the fact that the volume of the balls of radius r increases like N r . For the dynamics, at least at our time-scales, this is, however, irrelevant, since the SRW leaves a local minimum essentially ballistically. The proof of Theorem 1.1 relies on the combination of detailed information on the properties of the SRW on the hypercube, which is provided in Sect. 4 (but see also ˇ [Mat89,BG06, CG08]), and comparison of the process H N on the trajectory of the SRW to a simpler Gaussian process using interpolation techniques à la Slepian, familiar from extreme value theory of Gaussian processes. Let us explain this in more detail. On the time scales we are considering, the SRW makes t N 1/2 exp(N γ 2 /2β 2 ) t N 1/2 exp(N ζ ( p)2 /2) 2 N steps. In this regime the SRW is extremely “transient”, in the sense that (i) starting from a given point x, for times t ≤ ν ∼ N ω , ω < 1, the distance from x grows essentially linearly with speed one, that is there are no backtrackings with high probability; (ii) the SRW will never return to a neighborhood of size ν of the starting point x, with high probability. The upshot is that we can think of the trajectory of the SRW essentially as of a straight line. Next we consider the Gaussian process restricted to the SRW trajectory. We expect that the main contributions to the sums S N (k) come from places where H N is maximal (on the trajectory). We expect that the distribution of these extremes does not feel the correlations between points farther than ν apart. On the other hand, for points closer than ν, the correlation function R N (Y N (i), Y N ( j)) p can be well approximated by a linear function 1 − 2 pN −1 |i − j| (using that R N (Y N (i), Y N ( j)) ∼ 1 − 2N −1 |i − j|). This is convenient since this process has an explicit representation in terms of i.i.d. random variables, which allows for explicit computations (in fact, this is one of the famous processes for which the extremal distribution can be computed explicitly [Sle61,She71]). Thus the idea is to cut the SRW trajectory into blocks of length ν and to replace the original process H N (Y N (i)) by a new one Ui , where Ui and U j are independent, if i, j are not in the same block, and E[Ui U j ] = 1 − 2 pN −1 |i − j| if they are. For the new process, Theorem 1.1 is relatively straightforward. The main step is the computation of Laplace transforms in Sect. 2. Comparing the real process with the auxiliary one is the bulk of the work and is done in Sect. 3. The properties of SRW needed are established in Sect. 4. In Sect. 5 we present the proofs of the main theorems. Our results exhibit considerable universality of the REM for dynamics of p-spin models with p ≥ 3. This dynamic universality is close to the static universality of the REM, which shows that various features of the landscape of energies (that is of the Hamiltonian H N ) are insensitive to correlations. This static universality in a microcanonical context has been introduced by [BM04] (see [BK06a,BK06b] for rigorous results in the context of spin-glasses). The static results closest to our dynamics question are given in [BGK06,BK07], where it is shown that the statistics of extreme values for the restriction of H N to a random set X N ⊂ S N are universal, for p ≥ 3 and |X N | = ecN , for c small enough.
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2. Behavior of the One-Block Sums √ ν ei eβ N Ui , where In this section we analyze the distribution of the block-sums, i=1 {ei , i ∈ N} are mean-one i.i.d. exponential random variables, and {Ui , i = 1, . . . , ν} is a centered Gaussian process with the covariance EUi U j = 1 − 2 pN −1 |i − j|; ν = ν N is a function of N of the form
ν = N ω ,
with ω ∈ (1/2, 1).
(2.1)
As explained in the introduction, this process will serve as a local approximation of the corresponding block sums along a SRW trajectory. We characterize the distribution of the block-sums in terms of its Laplace transform ν √ −γ N β N Ui F N (u) = E exp −ue ei e . (2.2) i=1
Proposition 2.1. For all γ such that γ /β 2 ∈ (0, 1) there exists a constant, K = K (γ , β, p), such that, uniformly for u in compact subsets of [0, ∞), lim N 1/2 ν −1 e N γ
N →∞
2 /2β 2
[1 − F N (u)] = K u γ /β . 2
(2.3)
Proof. For all N the argument of the limit on the left-hand side of (2.3) is continuous and increasing in u ∈ [0, ∞). The same is true for the right-hand side of (2.3). Therefore, the uniform convergence claimed in the proposition is a direct consequence of the point-wise convergence for u ∈ (0, ∞), which we will prove in the following. We first compute the conditional expectation in (2.2) given the σ -algebra, U, generated by the Gaussian process U , ν ν √ 1 −γ N β N Ui √ E exp −ue ei e U = −γ N eβ N Ui i=1 i=1 1 + ue ν (2.4) √ −γ N β N Ui = exp − g ue e , i=1
where g(x) ≡ ln(1 + x).
(2.5)
Note that g(x) is monotone increasing and non-negative for x ∈ R+ . We use the wellknown fact (see e.g. [Sle61]) that the random variables Ui can be expressed using a sequence of i.i.d. standard normal variables, Z i , as follows. Set Z 1 = (U1 + Uν )/ (4 − 4 p(ν − 1)/N )1/2 and Z k = (Uk − Uk−1 )/(4 p/N )1/2 , k = 2, . . . , ν. Then Z i are i.i.d. standard normal variables and Ui = 1 Z 1 + · · · + i Z i − i+1 Z i+1 − · · · − ν Z ν , where
1 =
1−
p (ν − 1) N
and
2 = · · · = ν =
(2.6)
p . N
(2.7)
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Observe that
ν
2 i=1 i
669
= 1. Let us define G i (z) = G i (z 1 , . . . , z ν ) as
G i (z) = 1 z 1 + · · · + i z i − i+1 z i+1 − · · · − ν z ν .
(2.8)
Using this notation we get ν
√ ν dz − 12 i=1 z i2 −γ N β N G i (z) 1 − F N (u) = e g ue e 1 − exp − . ν/2 Rν (2π ) i=1
(2.9) We divide the domain of integration into several parts according to which of the G i (z) is maximal. Define Dk = {z ∈ Rν : G k (z) ≥ G i (z)∀i = k}. On Dk we use the substitution √ if i ≤ k, z i = bi + i (γ N − ln u)/(β N ), (2.10) √ z i = bi − i (γ N − ln u)/(β N ), if i > k. It will be useful to define kj=i+1 a j as kj=1 a j − ij=1 a j , which is meaningful also for k < i + 1. Using this definition G k (b) − G i (b) = 2
k
ν b j .
(2.11)
j=i+1
Set θ = − ln(u)/(γ N ) and define ⎫ ⎧ √ k ⎬ ⎨ γ p |k − i|(1 + θ ) ≥ 0 ∀i = k . bj + Dk = b ∈ Rν : ⎭ ⎩ β
(2.12)
j=i+1
After a straightforward computation we find that (2.9) equals (up to a multiplicative correction converging to one as N → ∞) ν
ν γ√ db 2 2 2 − 12 i=1 bi2 − β N G k (b)(1+θ) e−N γ /2β u γ /β e e (2π )ν/2 k=1 Dk ν β √ N G (b)−2β √ p k k j=i+1 b j −2 pγ |k−i|(1+θ) × 1 − exp − g e . i=1
(2.13) To finish the proof we have to show that asymptotically the only dependence in 2 (2.13) on u (or θ ) is through the factor u γ /β , and that the sum is of order ν N −1/2 . We change variables once more to a j = b j /(1 + θ ) in order to remove the dependence of the integration domains Dk on u. Then the sum (without the prefactor) in (2.13) can be expressed as ν
(1 + θ )ν da − 1 (1+θ)2 ν a 2 − γβ √ N G k (a)(1+θ)2 i=1 i e e 2 (2π )ν/2 k=1 Dk ν (β √ N G (a)−2β √ p k a j −2 pγ |k−i|)(1+θ) k j=i+1 × 1 − exp − g e , (2.14) i=1
where Dk = a ∈ Rν : kj=i+1 a j +
γ
√ β
p
|k − i| ≥ 0 ∀i = k .
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Let δ > 0 be such that (1 + δ)γ /β 2 < 1, and let N > | ln u|/(γ δ), so that |θ | ≤ δ. We first examine the square bracket in the above expression for a fixed k. On Dk ν (β √ N G (a)−2β √ p k a −2 pγ |k−i|)(1+θ) k j=i+1 j exp − g e i=1
√ ≥ exp −νg eβ N G k (a)(1+θ) .
(2.15)
Write G k (a) as (recall (2.1)) G k (a) =
ξ − ω ln N √ . (1 + θ )β N
(2.16)
The square bracket of (2.14) is then smaller than − γ (ξ −ω ln N )(1+θ) 1 − exp −νg eξ −ω ln N e β2 ≤N
γ ω(1+θ) β2
e
. 1 − exp −νg eξ /ν
− γ 2ξ (1+θ) β
− γ ξ (1+θ)
If (1 + θ )γ /β 2 < 1, then the function e β 2 bounded in ξ ∈ R and ν. Indeed, if ξ ≥ 0, then e
− γ 2ξ (1+θ) β
(2.17)
1 − exp −νg eξ /ν is uniformly
− γ ξ (1+θ) 1 − exp −νg eξ /ν ≤ e β2 ≤ 1.
If ξ < 0, then, since g(x) ≤ x, ≤ 1 − exp −eξ , 1 − exp −νg eξ /ν
(2.18)
(2.19)
which behaves like eξ , as ξ → −∞. This compensates the exponentially growing prefactor if (1 + θ )γ /β 2 < 1. Thus, under this condition, the bracket of (2.14) increases at most polynomially with N . Therefore, there exists δ > 0 small, such that the domain of integration in (2.14) may be restricted to ai ’s satisfying ν −1
ν
ν
ai2 ∈ (1 − δ, 1 + δ), |a1 | ≤ N 1/4 ,
i=1
|ai | ≤ ν 1+δ .
(2.20)
i=1 δ
The integral over the remaining ai ’s decays at least as e−N , for some δ > 0 (by a simple large deviation argument). For all a satisfying (2.20), |G k (a)| ≤ N 1/4 + N −1/2 ν 1+δ 1/2 N and thus, for any fixed u, uniformly in such a’s, we have e
− γβ
e
√
N G k (a)(1+θ)2
− γβ
√
N G k (a)
N →∞
N →∞
−−−−→ 1, and
1
e− 2 (1+θ) e
− 12
2 ν a 2 i=1 i
ν
2 i=1 ai
N →∞
−−−−→ 1.
(2.21)
Also, (1 + θ )ν −−−−→ 1. Hence, up to a small error, we can remove all but the last occurrence of θ in (2.14). Finally, taking xi = ai for i ≥ 2, x1 = N 1/2 G k (a), and thus √ x1 − p(x2 + · · · + xk − xk+1 − · · · − xν ) a1 = , (2.22) √
1 N
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671
(2.14) is, up to a small error, equal to 1 ν 2 ν
x12 x12 a12 dx e− 2 i=2 xi γ exp − x1 − exp − + N 1/2 (2π )ν/2 β 2 2 12 N 2 12 N k=1 Dk 1 ν √ (2.23) k (1+θ)βx1 − 2β p j=i+1 x j +2 pγ |k−i| (1+θ) g e e × 1 − exp − . i=1
The last exponential term on the first line can be omitted. Indeed, −
x12 a12 1 √ N →∞ + = x1 p(x2 + · · · − xν ) − 2 p(x2 + · · · − xν )2 −−−−→ 0 2 2 2 2 1 N
1 N (2.24)
uniformly for all |x1 | ≤ N (1+δ)/2 and |x2 + · · · − xν | ≤ ν (1+δ)/2 if δ > 0 is sufficiently δ small. The integral over the remaining x is again at most e−N , again by a large-deviation argument. Now we estimate the integral over x2 , . . . , xν in (2.23). Namely, ν √ 1 ν 2
dxe− 2 i=2 xi (1+θ)βx1 − 2β p kj=i+1 x j +2 pγ |k−i| (1+θ) g e 1 − exp − , (ν−1)/2 D¯ k (2π ) i=1
(2.25) where D¯ k is the restriction of Dk to the last ν − 1 coordinates (which does not depend on the value of the first one). Let V = (V2 , . . . , Vν ) be a sequence of i.i.d. standard normal random variables and let Rik = kj=i+1 V j . Using this notation we rewrite (2.25) as
ν √ (1+θ)βx1 − 2β p Rik +2 pγ |k−i| (1+θ) g e e E 1 − exp − ; V ∈ D¯ k . (2.26) i=1
Restricting the summation to i = k, we get a lower bound E 1 − exp −g e(1+θ)βx1 ; V ∈ D¯ k ∼ e(1+θ)βx1 P[V ∈ D¯ k ] as x1 → −∞. (2.27) The probability P[V ∈ D¯ k ] is bounded from below by the probability that a two-sided random walk √ (R(i), i ∈ Z) with standard normal increments and R(0) = 0 satisfies R(i) ≥ −γ p|i|/β for all i ∈ Z. This probability is positive and does not depend on N . This implies that there exists c > 0, independent of k, N , and u, such that, for all x1 < 0, (2.25) ≥ ce(1+θ)βx1 .
(2.28)
Using g(x) ≤ x and 1 − e−x ≤ x we get an upper bound for (2.25), namely (2.25) ≤ e(1+θ)βx1
ν √ k E e− 2β p Ri +2 pγ |k−i| (1+θ) ; V ∈ D¯ k . i=1
(2.29)
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672 γ
√
Relaxing the condition V ∈ D¯ k to Rik ≥ − β |k − i| in the i th term of the summation and using the fact that Rik is a centered normal random variable with variance |k − i|, we get, by a straightforward Gaussian computation, that (2.25) ≤ e(1+θ)βx1
ν i=1
p
C 2 2 e−γ p|k−i|/(2β ) ≤ Ce(1+θ)βx1 , √ |k − i|
(2.30)
where C depends only on β, γ , and p. The bounds (2.28) and (2.30) imply that (2.23) is bounded from above and from below (with different constants) by
x12 γ −1/2 C νN dx1 exp − x1 − (2.31) (1 ∧ ce(1+θ)βx1 ) = C ν N −1/2 . 2N β 2 R 1 This proves that (2.23), and thus (2.14), are of the right order. Moreover, these bounds imply that we can restrict the domain of integration over x1 in (2.23) to a large compact ¯ C] ¯ without losing precision. interval [−C, Finally, observe that (2.25) is an increasing function of min(k, ν − k). Therefore, there exists a bounded function, c¯ : R → [0, ∞), such that, as this minimum increases, (2.25) converges to c(x ¯ 1 )(e(1+θ)βx1 ∧1). Using this fact and the bounds (2.28) and (2.30), it is easy to see that there exists C > 0 such that (2.23) behaves as Cν N −1/2 (1 + o(1)), ¯ C] ¯ then the derivative of the integrand in as N → ∞. Finally, observe that if x1 ∈ [−C, (2.25) with respect to θ is bounded uniformly in N and k. This can be used to show that the constant C is independent of u. This completes the proof of Proposition 2.1. We close this section with a short description of the shape of the valleys mentioned in the introduction. First, it follows from (2.10) and the following computations that the most important contribution√to the Laplace transform comes from realizations for which max{Ui : 1 ≤ i ≤ ν} ∼ γ N /β with an error of order N −1/2 . It is the “geometrical” sequence in (2.30) which shows that only finitely many neighbors of the maximum actually contribute to the Laplace transform. The same can be seen, at least heuristically, from a simple calculation √ ! " |i| γ√ γ N E Uk+i Uk = − Cβ,γ , p √ , N = (2.32) β β N which means that, disregarding the fluctuations, the energy decreases linearly with the distance from the local maximum and thus the mean waiting times decrease exponentially. 3. Comparison Between the Real and the Clock Process We now come to the main task, the comparison of the clock-process sums with those in which the real Gaussian process is replaced by a simplified process. For a given realization, Y N , of the SRW, we set X 0N (i) = H N (Y N (i)) (the dependence on Y N will be suppressed in the notation). Then X 0N (i) is a centered Gaussian process indexed by N with covariance matrix i0j = E[X 0N (i)X 0N ( j)] = R N (Y N (i), Y N ( j)) p .
(3.1)
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We further define the comparison process, X 1N (i), as a centered Gaussian process with covariance matrix 1 − 2 pN −1 |i − j|, if i/ν = j/ν, 1 1 1 i j = E[X N (i)X N ( j)] = (3.2) 0, otherwise. √ √ For h ∈ (0, 1) we define the interpolating process X hN (i) ≡ 1 − h X 0N (i) + h X 1N (i). Let ∈ N, 0 = t0 < · · · < t = T , and u 1 , . . . , u ∈ R+ be fixed. For any Gaussian process X = (X (i), i ∈ N) we define a function, FN (X ) = FN (X ; {ti }, {u i }), as ⎞ ⎤ ⎡ ⎛ (N )−1 tk r √ u k β N X (i) ⎠ ⎦ FN (X ; {ti }, {u i }) ≡ E ⎣exp ⎝− e e X (X ) i eγ N k=1 i=tk−1 r (N ) ⎛ ⎞ (3.3) (N )−1 tk r u √ k = exp ⎝− g γ N eβ N X (i) ⎠ , e k=1 i=tk−1 r (N )
where
+ , 2 2 r (N ) = N 1/2 e N γ /2β .
(3.4)
Observe that E[F(X 0 ; {ti }, {u i })|Y] is a joint Laplace transform of the distribution of the properly rescaled clock process at times ti . The following approximation is the crucial step of the proof. Proposition 3.1. If the assumptions of Theorem 1.1 are satisfied, then, for all sequences {ti } and {u i }, lim E FN X 0N ; {ti }, {u i } |Y] − E FN X 1N ; {ti }, {u i } = 0, Y-a.s. (3.5) N →∞
Proof. The well-known interpolation formula for functionals of two Gaussian processes due (probably) to Slepian and Kahane (see e.g. [LT91]) reads in our context
r (N )T ∂ 2 FN (X hN ) 1 1 0 1 0 1 E[FN (X N ) − FN (X N )|Y] = dh (i j − i j )E Y . 2 0 ∂ X (i)∂ X ( j) i, j=1 i= j
(3.6) We will show that the integral in (3.6) converges to 0. Let k(i) be defined by tk(i)−1 r (N ) ≤ i < tk(i) r (N ). The second derivative in (3.6) is equal to u k(i) u k( j) β 2 N β √ N (X h (i)+X h ( j)) N N e e2γ N u √ h u k(i) √ h k( j) × g γ N eβ N X N (i) g γ N eβ N X N ( j) FN (X hN ) e e u k(i) u k( j) β 2 N β √ N (X h (i)+X h ( j)) N N ≤ e e2γN u u √ √ h h k( j) k(i) × exp −2g γ N eβ N X N (i) − 2g γ N eβ N X N ( j) , e e
(3.7)
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674
where we used that g (x) = (1 + x)−1 = exp(−g(x)) (recall (2.5)), and we omitted in the summation of FN (X hN ) all terms different from i and j. To estimate the expected value of this expression we need the following technical lemma. Lemma 3.2. Let c ∈ [−1, 1] and let U1 , U2 be two standard normal variables with covariance E[U1 U2 ] = c, λ a small constant, 0 < λ < min{1 − γβ −2 , γβ −2 } ¯ N (c) = (which will stay fixed), and u, v > 0. Define N (c) = N (c, β, γ , u, v) and ¯ N (c, β, γ , u, v, λ) by N (c)=
√ √ uvβ 2 N √ E exp β N (U1 + U2 ) − 2g ueβ N U1 −γ N − 2g veβ N U2−γ N 2γ N e (3.8)
and
√ 2 −1/2 ∧ N exp − γ N C (1 − c) , if 1 ≥ c > γβ −2 + λ − 1, 2 β (1+c) ¯ N (c) = if c ≤ γβ −2 + λ − 1, C N exp N (β 2 (1 + c) − 2γ ) , (3.9)
where C ≡ C(γ , β, u, v, λ) is a suitably chosen constant independent of N and c. Then ¯ N (c). N (c) ≤
(3.10)
√
Proof. Set κ± = 2(1 ± c). Let U¯ 1 , U¯ 2 be two independent standard normal variables. Then U1 and U2 can be written as U1 =
1 (κ+ U¯ 1 + κ− U¯ 2 ), 2
U2 =
1 (κ+ U¯ 1 − κ− U¯ 2 ). 2
(3.11)
Hence, U1 + U2 = κ+ U¯ 1 . For x ≥ 0 and y ≥ 0 the function g satisfies g(x) + g(y) = g(x + y + x y) ≥ g(x + y). Moreover, ue x + ve−x ≥ min(u, v)e|x| . Hence, √ √ g ueβ N U1 −γ N + g veβ NU2 −γ N √ √ (3.12) κ+ β N U¯ 1 κ− β N U¯ 2 . + ≥ g min(u, v) exp −γN 2 2 Setting min(u, v) = u, ¯ we can bound N (c) from above by
√ 1 κ β √ N y + 1 κ β √ N |y |−γ N y12 + y22 uvβ 2 N dy + − 1 2 2 2 . exp − + β N κ+ y1 − 2g ue ¯ 2 e2γ N R2 2π (3.13)
√
Substituting z 1 = y1 − β N κ+ and z 2 = y2 we get
z 12 + z 22 uvβ 2 N β 2 κ+2 N /2 dz e exp − e2γ N 2 R2 2π ! 2 2 ". (3.14) √ √ β κ+ βκ+ βκ− −γ z1 + |z 2 | N N+ . × exp −2g u¯ exp 2 2 2
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The second line of the last expression is always smaller than one. Therefore, N (c) ≤
uvβ 2 N β 2 κ+2 N /2 2 e = C(γ , β, u, v)N exp N (β (1 + c) − 2γ ) , (3.15) e2γ N
¯ This which is the same expression as appears on the second line of definition (3.9) of . √ estimate is however not always optimal. Indeed, note that the function exp(−2g(ue ¯ N x )) converges to the indicator function 1x<0 as N → ∞. The role of x will be played by the square bracket in the expression (3.14). If β 2 κ+2 /2 − γ > 0, this bracket is positive for z close to 0 and the integrand in (3.14) is typically very small. We thus fix λ satisfying the assumptions of Lemma 3.2, and consider c such that c > γβ −2 + λ − 1. This is equivalent to β 2 κ+2 /2 − γ > λ for some λ = λ (λ) > 0. In this case we need another substitution, namely ! " 1 κ− 2γ v1 − , z1 = √ |v2 | − N βκ+ − κ+ βκ+ N (3.16) v2 z2 = √ . N This substitution transforms the domain where the square bracket of (3.14) is negative into the half plane v1 < 0: The expression inside of the braces in (3.14) equals βκ+ v1 /2. Substituting (3.16) into (z 12 + z 22 )/2 produces an additional exponential prefac (β 2 κ+2 −2γ )2 N tor exp − 2β . Another prefactor N −1 comes from the Jacobian. The remai2κ2 + ning terms can be bounded from above by
v22 dv 2γ κ− βκ+ v1 /2 v1 − exp − + βκ+ − |v2 | − 2g(ue ¯ ) , (3.17) 2N βκ+ κ+ R2 2π which can be separated into a product of two integrals. The integral over v2 contains two terms: one with v22 and the second with |v2 |. If we ignore the quadratic one (which can be done only if κ− = 0, that is c = 1), then the integral over v2 can be bounded from above by 2γ κ− −1 −1 βκ+ − ≤ C(λ)κ− ≤ C(λ)(1 − c)−1/2 , (3.18) βκ+ κ+ where C(λ) is a constant depending only on λ. If κ−√ = 0, then the term with |v2 | disappears and the integration over v2 gives a factor C N . To bound the integral over v1 in (3.17) observe that the integrand behaves as exp{−2v1 γ /βκ+ } as v1 → ∞, and as exp{(βκ+ − (2γ /βκ+ ))v1 } as v1 → −∞. Therefore, the integral over v1 is bounded uniformly by some λ-dependent constant for all values of c ≥ −1 + (γ /β 2 ) + λ. Putting everything together we get √ uvβ 2 N β 2 κ 2 N /2 1 (β 2 κ+2 − 2γ )2 N + N (c) ≤ C (1 − c)−1/2 ∧ N exp − e e2γ N N 2β 2 κ+2 (3.19) . √ γ 2N −1/2 ¯ = N (c). = C(γ , β, u, v, λ) (1 − c) ∧ N exp − 2 β (1 + c) This finishes the proof of Lemma 3.2.
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We can now return to the proof of Proposition 3.1, that is to formula (3.6). Let Di j = dist(Y N (i), Y N ( j)). Observe that Di j is always smaller than |i − j|. Hence, for i/ν = j/ν, p p i0j = 1 − 2N −1 Di j ≥ 1 − 2N −1 |i − j| ≥ i1j . (3.20) Since i1j = 0 for (i, j) with i/ν = j/ν, i0j − i1j < 0 if and only if i0j < 0. The absolute value of (3.6) is thus bounded from above by
r (N )T ∂ 2 FN (X hN ) ∂ 2 FN (X hN ) 1 1 0 1 0 |Y] + (i j )− E dh (i j − i j )+ E Y . 2 0 ∂ X (i)∂ X ( j) ∂ X (i)∂ X ( j) i, j=1 i= j
(3.21) Given the sequence {u i } that was fixed at the beginning of this section, we define ˜ N (c) = max{ ¯ N (c, β, γ , u i , u j , λ) : 1 ≤ i, j ≤ }.
(3.22)
˜ N (c) can be written as in (3.9) with a new large constant C. Observe that Clearly, ˜ N (c) is an increasing function of c ∈ [−1, 1]. Lemma 3.2 and the computation just preceding it then imply that (3.21) is bounded from above by 1 2
1
dh 0
r (N )T
˜ N (1 − h)i0j + hi1j + (i0j )− ˜ N (1 − h)i0j . (i0j − i1j )+
i, j=1 i= j
(3.23) 0d
2d N −1 ) p .
We define, with a slight abuse of notation, = (1 − That is to say, 0d is 0 0 the covariance of X N (i) and X N ( j) if Di j = d. Using this notation, (3.23) is smaller than ⎧ ⎪ r (N )T
1 N ⎨ ⎪ 0 ˜ N (1 − h)0d dh 1{Di j = d}(d )+ ⎪ 0 d=0 ⎪ ⎩ i, j=1 i/ν= j/ν
+
r (N )T
˜ N 0d 1{Di j = d}(0d − i1j )
i, j=1,i= j i/ν= j/ν
+
r (N )T i, j:|i− j|≥N /2
(3.24)
⎫ ⎬
˜ N (0) . 1{Di j = d}(0d )− ⎭
In the last two lines we used the fact that the integral over h is bounded by the supremum ˜ N implies ˜ N is increasing. Finally, the definition of of its integrand, and the fact that that, for c ≥ 0,
1
1 γ2N √ ˜ N ((1 − h)c)dh ≤ Ce− β 2 (1+c) (1 − (1 − h)c)−1/2 ∧ N dh ≤ 2. 0
0
(3.25)
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677
To control (3.24) we need to count the pairs (i, j) with Di j = d. The following proposition, that is proved in the next section, provides sufficiently good estimates for our purposes. Proposition 3.3. Let γ and β satisfy the hypothesis of Theorem 1.1, let T > 0, and let ν be as in (2.1). Then, for any η > 0, there exists a constant, C = C(β, γ , ν, η), such that, Y-a.s., for all but finitely many values of N , for all d ∈ {0, . . . , N }, !
r (N )T
2 −N
1{Di j = d} ≤ C T r (N ) 2 2
i, j=1 i/ν= j/ν
" N −1 ηd + r (N )T ν e , d
(3.26)
where d = min(d, N − d). Moreover, we can choose ε < 1 such that, Y-a.s., for all but finitely many values of N , r (N )T Cνr (N )T {d ∨ 1}N −2 , if d ≤ εN ν −1 , 0 1 1{Di j = d}(d − i j ) ≤ if ν ≥ d ≥ εN ν −1 . Cr (N )T ν 2 N −1 , i, j=1,i= j i/ν= j/ν
(3.27) This proposition and the estimate (3.25) imply that the first line of (3.24) is smaller than the sum of the following two terms: N 2N N γ C (0d )+ exp − 2 T 2 r (N )2 2−N (3.28) d β (1 + 0d ) d=0 and C
N r (N )T eηd d=0
ν
(0d )+ exp
γ 2N − 2 . β (1 + 0d )
(3.29)
By (3.27), the second line of (3.24) is bounded by C
εN /ν d=0
ν r (N )T ν{d ∨ 1} r (N )T ν 2 ˜ N (0d ). ˜ N (0d ) + C N2 N
(3.30)
d=εN /ν
The third line is non-zero only if p is odd. By (3.26) it is smaller than N d=N /2
" ! N 2d p ˜ N (0). + r (N )T ν −1 eηd 1 − C T 2 r (N )2 2−N d N −
(3.31)
We estimate (3.28) first. Let I (u) be defined by I (u) = u ln u + (1 − u) ln(1 − u) + ln 2, and let J N (u) = 2
−N
N N u
π N N I (u) e . 2
(3.32)
(3.33)
ˇ G. Ben Arous, A. Bovier, J. Cerný
678 N →∞
By Stirling’s formula we have that J N (u) −−−−→ (4u(1 − u))−1 , uniformly on compact subsets of (0, 1). Moreover, J N (u) ≤ C N 1/2 for all u ∈ [0, 1]. From the definitions of ˜ N , we find that r (N ) and (3.28) ≤ C
N
2
T N
1/2
d=0
. 2d p d d 1− JN , exp N ϒ p,β,γ N + N N
(3.34)
where ϒ p,β,γ (u) =
γ2 γ2 − I (u) − 2 . 2 β β (1 + |1 − 2u| p )
(3.35)
Lemma 3.4. There exists a function, ζ ( p), such that, for all p ≥ 2 and γ , β satisfying γ < min{ζ ( p)β, β 2 }, there exist positive constants δ, δ , and c such that ϒ p,β,γ (u) ≤ −δ,
for all u ∈ [0, 1] \ (1/2 − δ , 1/2 + δ ),
(3.36)
and ϒ p,β,γ (u) ≤ −c(u − 1/2)2 ,
for all u ∈ (1/2 − δ , 1/2 + δ ).
Moreover, ζ ( p) is increasing and satisfies (1.10), that is, ζ (2) = 2−1/2 , ζ (3) 1.0291,
and
lim ζ ( p) =
p→∞
√
2 ln 2.
(3.37)
(3.38)
(1/2) = 0 Proof. The function ϒ p,β,γ and its derivatives satisfy ϒ p,β,γ (1/2) = ϒ p,β,γ and 2 2γ 4 − 1 , if p = 2, β2 ϒ p,β,γ (1/2) = (3.39) −4 otherwise.
The second derivative is always negative for β, γ , p satisfying the assumptions of the lemma. Therefore (3.37) holds. For any δ > 0 and |u − 1/2| ≥ δ , the function I (u) is strictly positive and the function (u) ≡ 1 − 1/(1 + |1 − 2u| p ) is bounded. Therefore, if γ /β is sufficiently small (how small defines the function ζ ( p)), then ϒ p,β,γ (u) < −δ. This proves (3.36). The monotonicity of ζ ( p) follows from the monotonicity of in p. The function ϒ2,β,γ (u) is increasing in γ 2 /β 2 and I (u) ≥ (1 − 2u)2 /2. Thus, for all γ < 2−1/2 β, 1 1 1 − (1 − 2u)2 . 1− (3.40) ϒ2,β,γ (u) < 2 1 + (1 − 2u)2 2 The right-hand side of the last inequality is equal to 0 for u = 1/2 and its derivative, 1 2(1 − 2u) 1 − ≷ 0, for all u ≶ 1/2, (3.41) (1 + (1 − 2u)2 )2 which implies that (3.36) is true for all γ < 2−1/2 β, and so the first part of (3.38) holds. Obviously, (0) = 1/2, (0) = −2 p, I (0) = ln 2 and I (0) = −∞. Hence, for √ γ /β = 2 ln 2, there exists u > 0 small enough such that ϒ p,β,γ (u) is positive. This
Universality of the REM for Dynamics of Mean-Field Spin Glasses
0.2
0.2
0.1
0.1
679
0.2 0.1
u 0.1
0.2
0.4
0.6
0.8
u
u
1
0.2
0.1
0.4
0.6
0.8
1
0.1
0.2
0.4
0.6
0.8
1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
Fig. 1. Function ϒ p,γ ,β for p = 2, 3, 4 and various values of γ /β
√ implies that ζ ( p) < 2 ln 2. If u ∈ (0, 1/2) then lim p→∞ (u) = 0. This yields the third claim of (3.38). The value of ζ (3) was obtained numerically. For illustration the graphs of the function ϒ p,β,γ are plotted in Fig. 1 for p = 2, 3, 4, √ β = 1, and √ γ = 0 (solid lines), γ = 1/2 (dashed lines), γ = 1 (dash-dotted lines) and γ = 2 ln 2 (dotted lines). We can now finish the bound of (3.28), resp. of (3.34). Lemma 3.4 and the bounds on the function J N yield that, for d/N ∈ / (1/2 − δ , 1/2 + δ ), the summands decrease exponentially in N . Therefore they can be neglected. The remaining part can be bounded by C
(1/2+δ )N
d=(1/2−δ )N
2d p T 2 N 1/2 1 − exp −cN (d/N − 1/2)2 N +
δ
|x| p e−c N x dx
∞ 2 N →∞ ≤ C T 2 N 3/2 N −( p+1)/2 |u| p e−c u du −−−−→ 0,
≤ C T 2 N 3/2
2
(3.42)
−δ
−∞
however only if p ≥ 3. Similarly, for (3.29) we have (3.29) ≤ C
N
TN
1/2 −1
ν
d=0
2d p ˜ 1− exp(N ϒ(d/N )), N +
(3.43)
where, setting u = min(u, 1 − u), ϒ˜ p,β,γ (u) =
γ2 γ2 + ηu. − 2 2 2β β (1 + |1 − 2u| p )
(3.44)
It is easy to check that there are positive values of δ, δ , and η, such that ϒ˜ p,β,γ (u) < −δ, for all u ≥ δ . Therefore all such d can be neglected. Around d = 0 the function ϒ˜ p,β,γ (x) can be approximated by a linear function −cx, c > 0, and the summation by an integration. As an upper bound we get, using (2.1), CT N
3/2 −1
δ
ν
N →∞
e−cN x dx ≤ C T N 1/2 ν −1 −−−−→ 0.
0
An analogous bound works for d close to N and p even.
(3.45)
ˇ G. Ben Arous, A. Bovier, J. Cerný
680
For (3.30) we have ! "−1/2 εN /ν T νd C T ν N ϒ˜ p,β,γ (0) 2d p ˜ +C e N ϒ p,β,γ (d/N ) 1 − 1 − e N N 3/2 N d=1 (3.46) "−1/2 ! ν 2 Tν 2d p N ϒ˜ p,β,γ (d/N ) 1− 1− +C e . N 1/2 N
(3.30) ≤
d=εN /ν
The first term converges to zero. The linear approximation of ϒ˜ p,β,γ and of the bracket in the second term yields an upper bound
ε N →∞ 1/2 CT N ν x 1/2 e−c N x dx ≤ C T N −1 ν −−−−→ 0. (3.47) 0
The third term is smaller than ν 3 e−c N /ν , which is also negligible. ˜ N (0) = Ce−N γ 2 /β 2 , it is easy to see that the contribution of the Finally, since second term in the bracket of (3.31) tends to 0. The contribution of the first term is equal (up to a constant) to p N 2d N − 1 T 2 N 2−N d N d=N /2 ⎧ ⎫ (3.48) 3/5 p 2N ⎨ ⎬ N +i N 2 −N 1/2 −i 2 /2N ≤ CT −1 N e N2 , + ⎩ ⎭ d N 3/5 d≥N /2+N
i=1
2 where we used that Nd ≤ C N −1/2 2 N e−i /2N for d = (N + i)/2 and i N 2/3 . The first term in (3.48) tends to 0 by a standard moderate deviation argument. The second one can be approximated by
∞ 2 N →∞ C T 2 N 1−( p/2) x p e−x /2 dx −−−−→ 0 (3.49) 0
for p ≥ 3. This completes the proof of Proposition 3.1.
4. Random Walk Properties In this section we prove Proposition 3.3. We write Px for the law of the simple random walk Y N conditioned to start in x. Let Q = {Q i , i ∈ N} be the Ehrenfest Urn Markov chain, that is a birth-death process on {0, . . . , N } with transition probabilities pi,i−1 = 1 − pi,i+1 = i/N . We use Pk and E k to denote the law of (the expectation with respect to) Q conditioned on Q 0 = k. Under P0 , Q i has the same law as dist(Y N (0), Y N (i)). We define Tk as the hitting time of k by Q, Tk = min{i ≥ 1 : Q i = k}. It is a well-known fact that, for k < l < m, l−1 N −1 −1 Pl [Tm < Tk ] = i=k i −1 . m−1 N −1 i=k
(4.1)
i
Finally, let pk (d) = P0 (Q k = d). We need the following lemma to estimate pk (d) for large k.
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681
Lemma 4.1. There exists K large enough such that, for all k ≥ K N 2 ln N ≡ K(N ) and x, y ∈ S N , P y [Y N (k) = x ∪ Y N (k + 1) = x] −N −8N , (4.2) − 2 ≤2 2 and thus
pk (d) + pk+1 (d) −N N ≤ 2−4N . − 2 d 2
(4.3)
Proof. The beginning of the argument is the same as in [Mat87]. We construct a coupling between Y N (which by definition starts at site 1 = (1, . . . , 1) ∈ S N ) and another process Y N . This process is a simple random walk on S N , with initial distribution µN being uniform on those x ∈ S N with dist(x, 1) even. The coupling is the same as in [Mat87]. This coupling provides a certain random time, T N , which can be used to bound the variational distance d∞ between µ and the distribution µkN of Y N (k): for k even d∞ (µN , µkN ) ≡ max |µN (A) − µkN (A)| ≤ P[T N > k]. A⊂S N
(4.4)
The law of T N is as follows. Let U = dist(Y N (0), 1). That is U is a binomial random variable with parameters N and 1/2 conditioned on being even. Consider another simple random walk, Y˜U , on SU , started from 1. The distribution of T N is then the same as the distribution of the hitting time of the set {x ∈ SU : dist(1, x) = U/2} by Y˜U . It is proved in [Mat87] that P(T N > N ln N ) → c < 1. It is then easy to see that, P[T N ≥ K(N )] ≤ c K N ≤ 2−8N ,
(4.5)
if K is large enough. Thus, for even k ≥ K(N ), d∞ (µN , µkN ) ≤ 2−8N , and thus |µN (x) − µkN (x)| ≤ 2−8N , for all x ∈ S N . A similar claim for k odd is then not difficult to prove. The second part of the lemma is a direct consequence of the first part. Lemma 4.2. Let γ , β, ν satisfy the hypothesis of Proposition 3.3. Then, there exists a constant, C = C(β, γ , ν), such that, Y-a.s., for all but finitely many values of N , for all d ∈ {0, . . . , N }, r (N )T
1{Di j = d} ≤ Cr (N )T 1{d ≤ ν}.
(4.6)
i, j=1,i= j i/ν= j/ν
Proof. The lemma is trivially true for d > ν. For d ≤ ν, observe first that (4.6) is bounded from above by an i.i.d. sum of m = r (N )T /ν “block” random variables which have the same distribution as i,ν j=1,i= j 1{Di j = d}. To control these block variables we first compute ρ(d) = E 0
ν
1{Q i = d}.
(4.7)
i=1
We have ρ(0) ≥ N −1 and ρ(d) ≥ P0 [Td ≤ ν]. This probability is decreasing in d and P0 [Tν ≤ ν] =
N −ν+1 N N −1 2 · ... ≥ e−ν /N . N N N
(4.8)
ˇ G. Ben Arous, A. Bovier, J. Cerný
682
Thus ρ(d) ≥ e−ν /N for all d ≤ ν. Using Tν ≤ ν, the decomposition on the first visit of d, and the standard relation between the Green’s function and the escape probability, we get
T T ν ν 1 ρ(d) ≤ E 0 1{Q i = d} = 1 + E d 1{Q i = d} = 1 + . Pd [Tν < Td ] 2
i=1
i=1
(4.9) However, using (4.1), N −1 −1 N −d N −d d Pd+1 [Tν < Td ] = Pd [Tν < Td ] = = 1 − O(ν N −1 ). N N ν−1 N −1 −1 i=d
i
(4.10) Since ν N , ρ(d) ≤ 2. Consider now the one-block contribution to (4.6), ν
1{Di j = d} ≡ ν 2 Z˜ .
(4.11)
i, j=1
Of course, Z˜ ∈ [0, 1] and, using the results of the previous paragraph, e−ν
2 /N
(2ν)−2 ≤ E[ Z˜ ] ≤ 4ν −1 .
(4.12)
Denoting by Z˜ k a sequence of i.i.d. copies of Z˜ , we obtain from Hoeffding’s inequality [Hoe63]: m
2 P (4.13) Z˜ k ≥ 2mE[ Z˜ k ] ≤ exp{−2mE[ Z˜ k ]2 } ≤ exp{−me−2ν /N (2ν)−4 }, i=1
where we used the lower bound from (4.12). Since ν 2 /N N , by the Borel-Cantelli lemma, the left-hand side of (4.6) is a.s. bounded by ν 2 2mE[ Z˜ ] ≤ Cr (N )T, for all N large enough and all d ≤ ν. This completes the proof of Lemma 4.2.
(4.14)
Proof of Proposition 3.3. We prove (3.27) first. Since it is trivially verified for d ∈ {ν − 1, ν}, we will assume that d ≤ ν − 2. Observe that, for i, j in the same block, 2 2 p(|i − j| − d) d 2d p 2 p|i − j| = +O . 0d − i1j = 1 − − 1− N N N N2 (4.15) The contribution of the error term is smaller than the right-hand side of (3.27), as follows from Lemma 4.2.
Universality of the REM for Dynamics of Mean-Field Spin Glasses
683
To compute the contribution of the main term, let
ν ν ρ(d) ˜ = E0 (i − d)1{Q i = d} = (i − d) pi (d). i=1
(4.16)
i=1
We need upper and lower bounds on ρ(d) ˜ to proceed with a Hoeffding-type argument. The lower bound is easy: by considering the path with Q 2 = 0 and Q d+2 = d, we get 2 using (4.8) that ρ(d) ˜ ≥ N −1 e−ν /N . The upper bound is slightly more complicated, (ν−d)/2 (ν−d)/2 d + 2k d +k k ρ(d) ˜ = 2kpd+2k (d) ≤ 2k N k k=1
≤C
(ν−d)/2
k
k=1
≤C
(ν−d)/2
(d
+ 2k)k
√ k k e−k k
k=1
d +k N
k (4.17)
(2e)k (dk −1 + 2)k (ν N −1 )k ≤ C
k=1
(ν−d)/2
c(d + 2)ν N −1
k
k=1
≤ C(d ∨ 1)ν N
−1
if d ≤ εN /ν for some small ε. Otherwise, trivially, ρ(d) ˜ ≤ ν 2 . The one-block contribution of the first term of (4.15) to (3.27) is then given by ν 2p 2p 3 ˜ ν Z, (|i − j| − d)1{Di j = d} ≡ N N
(4.18)
i, j=1
2 with Z˜ ∈ [0, 1], E[ Z˜ ] ≥ cN −1 e−ν /N ν −3 , and C{d ∨ 1}N −1 ν −1 , E[ Z˜ ] ≤ 1,
if d ≤ εN /ν, if ν ≥ d ≥ εN /ν.
(4.19)
Then, as in the proof of Lemma 4.2, Hoeffding’s inequality and (4.19) imply that the contribution of the first term of (4.15) to (3.27) is smaller than Cr (N )T {d ∨ 1}ν N −2 or Cr (N )T ν 2 N −1 , respectively, which was to be shown. Finally, we prove (3.26). We restrict the summation to i < j, since the terms with j > i give the same contribution. We first consider the contribution of pairs (i, j) such that j − i ≥ K(N ), so that in particular, i/ν = j/ν. With R = r (N )T , Lemma 4.1 yields ⎡ ⎤ R R 2 −N N ⎣ ⎦ . (4.20) E 1{Di j = d} = p j−i (d) ≤ C R 2 d j−i≥K(N )
Moreover, ⎡ R Var ⎣
j−i≥K(N )
⎤ 1{Di j = d}⎦
j−i≥K(N )
=
R
R
j1 −i 1 ≥K(N ) j2 −i 2 ≥K(N )
0
1
0
P Di1 , j1 = Di2 , j2 = d − P Di1 , j1
(4.21) 1 0 1 = d P Di2 , j2 = d .
ˇ G. Ben Arous, A. Bovier, J. Cerný
684
We can again suppose that i 1 ≤ i 2 . The right-hand side of (4.21) is non-zero only if i 1 ≤ i 2 ≤ j1 < j2 or i 1 ≤ i 2 < j2 ≤ j1 . We will consider only the first case. The second one can be treated analogously. In is not difficult to see, using Lemma 4.1, that if i 2 − i 1 ≥ K(N ) or j2 − j1 ≥ K(N ) then the difference of probabilities in the above summation is at most 2−4N . Therefore, the contribution of such (i 1 , i 2 , j1 , j2 ) to the variance is at most R 4 2−4N . If i 2 − i 1 < K(N ) and j2 − j1 < K(N ) then, using again Lemma 4.1, 1 0 1 0 −N N . (4.22) P Di1 , j1 = Di2 , j2 = d ≤ P Di1 , j1 = d ≤ C2 d We choose ε > 0. For d ≤ (1 − ε)N /2 we have 0 1 P Di1 , j1 = Di2 , j2 = d j1 −i 1 ≥K(N ) j2 −i 2 ≥K(N ) i 2 −i 1
(4.23) N ≤ CK(N )2 R 2 e−N I ((1−ε/2)/2) N −3 R 2 ν −2 , ≤ CK(N )2 R 2 2−N d
say. For d ≥ (1 − ε)N /2, that is |d − N /2| ≤ εN /2, we have, for ε small enough (how small depend on γ and β), that 2−N Nd N 7 R −2 . Then, 0 1 P Di1 , j1 = Di2 , j2 = d j1 −i 1 ≥K(N ) j2 −i 2 ≥K(N ) i 2 −i 1
2 4 2 −N N −3 4 −2N N N R 2 . ≤ CN R 2 d d
(4.24)
We have thus found that the expectation of the summation over j − i > K(N ) is smaller than the right-hand side of (3.26) and the variance of the same summation is much smaller than N −3 times the right-hand side of (3.26) squared. A straightforward application of the Chebyshev inequality and the Borel-Cantelli Lemma then gives the desired a.s. bound for pairs j − i ≥ K(N ) and all d ∈ {0, . . . , N }. Choose again ε > 0. For j −i < K(N ), observe first that if d ≥ (ln N )1+ε ln N then the summation over such pairs (i, j) in (3.26) is always smaller than K(N )R Rν −1 eηd , for all η > 0. For the remaining d’s, that is d < (ln N )1+ε , let K N ≥ K be the smallest constant such that K N N 2 ln N is a multiple of ν. Since ν N 2 , K N − K 1. As the difference between K and K N is negligible, we will use the same notation K(N ) for K N N 2 ln N and we will simply suppose that K(N ) is a multiple of ν. The summation in (3.26) for j − i ≤ K(N ) can be bounded from above by r (N )T 0< j−i
1{Di j = d} ≤
K(N ) K K(N )−1 R/ (N ) k=0
=0
1{DK(N )+k,K(N )+k+m = d},
m= jk
(4.25) where jk is the smallest integer such that (K(N ) + k)/ν = (K(N ) + k + jk )/ν, which does not depend on . We define random variables, Z ( j, d), by K(N ) 1 1{DK(N )+k,K(N )+k+m = d}. Z ( j, d) = K(N ) m= j
(4.26)
Universality of the REM for Dynamics of Mean-Field Spin Glasses
685
The sequence {Z ( j, d) : ≥ 0}, for fixed j and d, is a sequence of i.i.d. variables with values in [0, 1]. Let E N = {d : d < (ln N )1+ε , d ≥ N /2}. For d ∈ E N , we have N P[Z (k, d) > 0] ≤ P0 [Td ≤ K(N )] ≤ Pz d [T1 < K(N )] d (4.27) N λK 2 ≤ e Ez d e−λT1 /N ln N , d where z d is any point on the hypercube with dist(1, z d ) = d and, with a slight abuse of notation, T1 is the hitting time of 1 by the simple random walk Y N . According to ˇ Lemma 3.4 of [CG08], Ez d exp(−λT1 m(N )−1 ) ≤ (2−N m(N )λ−1 + ξ N (d))(1 + o(1)), (4.28) −1 n−k n 1 for N ln N m(N ) 2 N , with ξn (k) = 2−n n2 nk j=1 k+ j j . Taking 2 m(N ) = N ln N and d ∈ E N it is not difficult to check that, for ε small enough, 2 Ez d e−λT1 /N ln N ≤ 2−N (1−ε) . (4.29) Hence, ⎡ P⎣
⎧ K(N ) )−1 R/ 2 ⎨K(N d∈E N
⎩
k=0
=0
⎫⎤ ⎬ Z ( jk , d) > 0 ⎦ ⎭
N R(ln N )1+ε 2−N (1−ε) ≤ C2−ε N , ≤C 1+ε (ln N )
(4.30)
for some ε small. Hence, d ∈ E N do not pose any problem, by the Borel-Cantelli lemma again. To treat d ≤ (ln N )1+ε ν we will distinguish two cases: jk ≤ d + 6 and jk > d + 6. For the first case, observe that, for any d < ν, there are at most (d + 6)K(N )/ν values of k ∈ {0, . . . , K(N ) − 1} such that jk ≤ d + 6. Clearly, Z ( jk , d) ≤ Z (0, d). Moreover, by similar arguments as in Lemma 4.2, E[Z (0, d)] ≥ 1/(N K(N )), and E[Z (0, d)] ≤ C/K(N ). Hence, by Hoeffding’s inequality, the probability ⎡ ⎤ R/ K(N ) C R ⎦ P ⎣K(N ) Z (0, d) ≥ (4.31) K(N ) =0
decreases, for C large, at least exponentially with N . Hence, for jk ≤ d + 6, Y-a.s, K(N )
R/ K(N ) =0
Z ( jk , d) ≤ K(N )
R/ K(N )
Z (0, d) ≤
=0
CR . K(N )
(4.32)
For j > d + 6 and N large enough, Z ( j, d) ≤ Z (d + 6, d). We claim that cN −6 ≤ K(N )E[Z (d + 6, d)] ν −1 .
(4.33)
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Indeed, the lower bound is trivially obtained by considering a path that returns 6 times to its starting point in the first 12 steps and then continues without backtracking to a distance d. To get the upper bound in (4.33), we first bound the probability that the chain Q started at 0 hits d between times d + 6 and K(N ). This probability is bounded by P0 [Td+6 = d + 6] + P0 [Td+6 = d + 6]Pd+6 [Td < K(N )].
(4.34)
The first term is smaller than c(d + 6)2 N −1 ν −1 . For the second term, Pd+6 [Td < Td+6] ≤ C N −5 (ln(N ))5(1+ε) N −4 . Moreover, before time K(N ), there are at most K(N ) trials to reach d, so Pd+6 [Td < K(N )] ≤ K(N )N −4 ν −1 . So (4.34) ν −1 . If Q hits d after d + 6 it spends there on average a time less than 2. This gives the upper bound in (4.33). From (4.33), it follows by another Hoeffding’s type argument that, for j > d + 6, ⎡ ⎤ R/ K(N ) R ⎦ P ⎣K(N ) Z ( j, d) ≥ (4.35) νK(N ) =0
decreases at least exponentially in N and thus the inequality question fails Y-a.s. for all but finitely many values of N . Putting together all arguments of the last three paragraphs and summing over k we get, Y-a.s., for all but finitely many N , K(N ) K(N )−1 R/ k=0
K(N )Z ( jk , d) ≤ dK(N )ν −1
=0
CR R + K(N ) ≤ C Rν −1 eηd . K(N ) νK(N ) (4.36)
This completes the proof.
5. Convergence of Clock Process We will prove the convergence of the rescaled clock process to the stable subordinator on the space D([0, T ], R) equipped with the Skorokhod M1 -topology. This topology is not commonly used in the literature, therefore we shortly recall some of its properties and compare it with the more standard Skorokhod J1 -topology, which we will need later, too. The reader is referred to [Whi02] for more details on both topologies, and to [Bil68] for a thorough account on the J1 -topology. 5.1. Topologies on the Skorokhod space. Consider the space D = D([0, T ], R) of càdlàg functions. The J1 -topology is the topology given by the J1 -metric d J1 , d J1 ( f, g) = inf { f ◦ λ − g∞ ∨ λ − e∞ }, λ∈
f, g ∈ D,
(5.1)
where is the set of strictly increasing functions mapping [0, T ] onto itself such that both λ and its inverse are continuous, and e is the identity map on [0, T ]. Also the M1 -topology is given by a metric. For f ∈ D let f be its completed graph, namely
f = {(t, z) ∈ [0, T ] × R : z = α f (t−) + (1 − α) f (t), α ∈ [0, 1]}.
(5.2)
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A parametric representation of the completed graph f (or of f ) is a continuous bijective mapping φ(s) = (φ1 (s), φ2 (s)), [0, 1] → f , whose first coordinate φ1 is increasing. If ( f ) is the set of all parametric representation of f , then the M1 -metric, d M1 , is defined by d M1 ( f, g) = inf{φ1 − ψ1 ∞ ∨ φ2 − ψ2 ∞ : φ ∈ ( f ), ψ ∈ (g)}.
(5.3)
The space D equipped with both M1 - and J1 -topologies is Polish. The M1 -topology is weaker than the J1 -topology: As an example, consider the sequence f n = 1{[1 − 1/n, 1)} + 2 · 1{[1, T ]},
(5.4)
which converges to f = 2 · 1{[1, T ]} in the M1 -topology but not in the J1 -topology. One often says that the M1 -topology allows “intermediate jumps”. We will need a criterion for tightness of probability measures on D. To this end we define several moduli of continuity, w f (δ) = sup min | f (t) − f (t )|, | f (t ) − f (t)| : t ≤ t ≤ t ≤ T, t − t ≤ δ , . w f (δ) = sup inf | f (t) − α f (t ) − (1 − α) f (t )| : t ≤ t ≤ t ≤ T, t − t ≤ δ , α∈[0,1] v f (t, δ) = sup | f (t ) − f (t )| : t , t ∈ [0, T ] ∩ (t − δ, t + δ) . (5.5) The following result is a restatement of Theorem 12.12.3 of [Whi02] and Theorem 15.3 of [Bil68]. Lemma 5.1. The sequence of probability measures {Pn } on D([0, T ], R) is tight in the J1 -topology if (i) For each positive ε there exists c such that Pn [ f : f ∞ > c] ≤ ε,
n ≥ 1.
(5.6)
(ii) For each ε > 0 and η > 0, there exist a δ, 0 < δ < T , and an integer n 0 such that Pn [ f : w f (δ) ≥ η] ≤ ε,
n ≥ n0,
(5.7)
and Pn [ f : v f (0, δ) ≥ η] ≤ ε and Pn [ f : v f (T, δ) ≥ η] ≤ ε,
n ≥ n 0 . (5.8)
The same claim holds for the M1 -topology with w f (δ) in (5.7) replaced by w f (δ).
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5.2. Proof of Theorem 1.1. To prove the convergence of the rescaled clock process S¯ N (·) = e−γ N S N (·r (N )) to the stable subordinator Vγ /β 2 , we check first the convergence of finite-dimensional marginals. Let ∈ N, 0 = t0 < · · · < t = T and u 1 , . . . , u ∈ R+ be fixed. Then, ¯ ¯ E exp − u i S N (tk ) − S N (tk−1 ) |Y] (5.9) i=1 0 1 = E FN (X N ; {ti }, {u i }) |Y] = E FN (X N ; {ti }, {u i }) + o(1), as follows from Proposition 3.1. 1 0 The value of E FN (X 1N ; {ti }, {u i }) is not difficult to calculate. Define j N (i) = ti r (N )/ν. Then ⎞⎤ ⎡ ⎛ tk r (N )−1 √ u 1 (i) k 1 β N X ⎠⎦ N E FN (X N ; {ti }, {u i }) = E ⎣exp ⎝− ei e eγ N k=1 i=tk−1 r (N ) ⎡ (5.10) ⎤ j (k) ν−1 N √ uk 1 ≥ E⎣ exp − γ N e jν+i eβ N X N ( jν+i) ⎦ . e k=1 j= j N (k−1)+1
i=0
X 1N
is a piecewise independent process, the product in (5.10) is a Since the process product of independent random variables. The expectations of all of them can be then bounded using Proposition 2.1. We get, for δ > 0 fixed and N large enough, E FN (X 1N ; {ti }, {u i }) ≥
j N (k)
F N (u k )
k=1 j= j N (k−1)+1
≥
1 − (1 + δ)ν N −1/2 e−N γ
2 /2β 2
γ /β 2
K uk
j N (k)− j N (k−1)−1
(5.11)
k=1
≥
γ /β 2 , exp −(1 + 2δ)(tk − tk−1 )K u k
k=1
which is (up to 1 + 2δ term) the Laplace transform of Vγ /β 2 (K ·). A corresponding upper bound can be constructed analogously. To check the tightness for S¯ N in D([0, T ], R) equipped with the Skorokhod M1 -topology we use Lemma 5.1. Since the processes S¯ N are increasing, it is easy to see that condition (i) is equivalent to the tightness of the distribution of S¯ N (T ), which can be checked easily from the convergence of the Laplace transform of the marginal at time T (the limiting Laplace transform tends to 1 as u → 0). Since S¯ N are increasing, the oscillation function wS¯ (δ) is always equal to zero. So N checking (ii) boils down to controlling the boundary oscillations v S¯ N (0, δ) and v S¯ N (T, δ). For the first quantity (using again the monotonicity of S¯ N ) this amounts to check that P[ S¯ N (δ) ≥ η] < ε if δ is small enough and N large enough. Using the convergence of the marginal at time δ, it is sufficient to take δ such that P[Vγ /β 2 (K δ) ≥ η] ≤ ε/2, and to take N0 such that, for all N ≥ N0 , P[ S¯ N (δ) ≥ η] − P[Vγ /β 2 (K δ) ≥ η] ≤ ε/2. (5.12) The reasoning for v S¯ N (T, δ) is analogous.
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5.3. Coarse-grained clock process. To prove our aging result, that is Theorem 1.2, we need to modify the result of Theorem 1.1 slightly. Let S˜ N be the “coarse-grained” clock processes, 1 S˜ N (t) = γ N S N (νtr (N )ν −1 ). e
(5.13)
For these processes we can strengthen the topology used in Theorem 1.1, that is we can replace the M1 - by the J1 -topology. Theorem 5.2. If the hypothesis of Theorem 1.1 is satisfied, then N →∞ S˜ N (t) −−−−→ Vγ /β 2 (K t),
Y − a.s.,
(5.14)
weakly in the J1 -topology on the space of càdlàg functions D([0, T ], R). Unfortunately, we cannot prove the theorem with the estimates we have already at our disposition. We should return and√improve some of them. First we show that traps with energies “much √ smaller” than √ γ N /β almost do not contribute to the clock process. Let Bm = γ N /β − m/(β N ) and let S¯ Nm (t) = e−γ N
tr (N )
√ ei exp β N X 0N (i) 1{X 0N (i) ≤ Bm }.
(5.15)
i=0
Lemma 5.3. For every T and η, ε > 0 there exists m large enough such that P[ S¯ Nm (T ) ≥ η|Y] ≤ ε,
Y-a.s.
(5.16)
Proof. To prove this lemma we should improve/modify slightly the calculations of Sects. 2 and 3. With the notation of Sect. 2 define ν √ m −γ N β N Ui F N = E exp −e ei e 1{Ui ≤ Bm } (5.17) i=1
(comparing with (2.2) observe that we set u = 1). We will show that lim N 1/2 ν −1 e N γ
N →∞
2 /2β 2
[1 − F Nm ] = K m ,
(5.18)
with K m → 0 as m → ∞. The proof of this claim is completely analogous to the proof of Proposition 2.1. One should only modify the domains of integrations. More precisely, the definition of Dk which appears after (2.9) should be replaced by Dkm = Dk ∩ {z : √ G k (z) ≤ Bm }. Hence, Dk becomes Dkm = Dk ∩ {b : G k (b) ≤ −m/(β/ N )}, which then restricts the domain of integration in (2.31) to (−∞, −m/β]. Hence, the constant K m can be made arbitrarily small by choosing m large. We define, similarly as in Sect. 3, ⎛ ⎞ T r (N )−1 √ FNm (X ) = exp ⎝− g e−γ N eβ N X (i) 1{X (i) ≤ Bm } ⎠ . (5.19) i=0
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As in Proposition 3.1 we show lim E FNm (X 0N ) |Y] − E FNm (X 1N ) = 0, N →∞
Y-a.s.
(5.20)
We use again (3.6) to show this claim. Although the indicator function is not differentiable, we will proceed as if it was, setting (1{x ≤ B}) = −δ(x − M), where δ denotes the Dirac delta function. As usual, this can be justified e.g. by using smooth approximations of the indicator function. The second derivative of FNm (X ) equals u 2 β 2 N β √ N (X (i)+X ( j)) β √ N X (i)−γ N β √ N X ( j)−γ N m g ue FN (X ) e g ue e2γ N δ B (X (i)) δ B (X ( j)) 1{X ( j) ≤ Bm } − m √ × 1{X (i) ≤ Bm } − m √ β N β N 2 2 √ √ √ u β N ≤ 2γ N eβ N (X (i)+X ( j)) exp −2g ueβ N X (i)−γ N − 2g ueβ N X ( j)−γ N e δ B (X (i)) δ B (X ( j)) × 1{X (i) ≤ Bm } − m √ 1{X ( j) ≤ Bm } − m √ . (5.21) β N β N We should now bound the contributions of four terms. The one with the product of two indicator functions is easy, because we can use directly the result of Lemma 3.2. For the remaining three terms, those with the product of one indicator and one delta function, and the one with two delta functions, the calculation should be repeated. However, in the ¯ end we find that (5.21) is bounded by (Cov(X (i), X ( j))) as before. The presence of the delta functions actually simplifies the calculations slightly. The proof then proceeds as in Sect. 3. We can now finish the proof of Lemma 5.3. By (5.17) and (5.20), 0 1 E exp(− S¯ Nm (T ))|Y = E FNm (X 0N )|Y = E FNm (X 1N )|Y + o(1) (5.22) 2 2 = (1 − K m f (N )−1 e−N γ /2β )T r (N )/ν + o(1) = e−K m T + o(1). Since K m → 0 as m → ∞, P[ S¯ Nm (T ) ≥ η|Y] ≤
1 0 1 − E exp(− S¯ Nm (T )) |Y 1 − e−η
can be made arbitrarily small by taking m large enough.
(5.23)
We study now how the blocks where the process visits sites with energies larger than Bm are distributed along the trajectory. To this end we set for any Gaussian process X , s Nm (i; X ) = 1{∃ j : iν < j ≤ (i + 1)ν, X ( j) > Bm }.
(5.24)
We define the point process H Nm (X ) on [0, T ] by H Nm (X ; dx)
=
T r (N )/ν
s Nm (i; X )δiν/r (N ) (dx).
(5.25)
i=0
Lemma 5.4. For every m ∈ R the point processes H Nm (X 0N ) converge to a homogeneous Poisson point process on [0, T ] with intensity ρm ∈ (0, ∞), Y-a.s.
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Proof. To show this lemma we use Proposition 16.17 of Kallenberg [Kal02]. According to it, to prove the convergence of H Nm (X 0N ) to a Poisson point process with intensity ρm it is sufficient to check that, for any interval I ⊂ [0, T ], lim P[H Nm (X 0N ; I ) = 0|Y] = e−ρm |I |
(5.26)
lim sup E[H Nm (X 0N ; I )|Y] ≤ ρm |I |,
(5.27)
N →∞
and N →∞
where |I | denotes the Lebesgue measure of I . The proof of the first claim is completely similar to the previous ones. We start with a one-block estimate for (5.26): lim N 1/2 ν −1 e N γ
2 /2β 2
N →∞
Using the notation of Sect. 2, we get E[s Nm (0, U )] =
Am
E[s Nm (0, U )] = ρm .
1 ν dz 2 e− 2 i=1 zi , ν/2 (2π )
(5.28)
(5.29)
where Am = {z : ∃k ∈ {1, . . . , ν}G k (z) > Bm }. Dividing the domain of integration according to the maximal G k (z), this is equal to ν
1 ν dz 2 e− 2 i=1 zi , (5.30) ν/2 Dk (2π ) k=1
where Dk = {z : G k (z) > Bm , G i (z) ≤ G k (z)∀i = k}. Using the substitution z i = bi ± i Bm on Dk (where + sign is used for i ≤ k and − sign for i > k) we get ν
ν db −N γ 2 /2β 2 mγ /β 2 − 12 i=1 bi2 −Bm G k (b) e e e e , (5.31) ν/2 (2π ) Dk k=1
where
Dk
= {b : G k (b) > 0,
k
j=i+1 b j +|k −i| ν Bm
≥ 0∀i = k}. The same reasoning
as before then allows to show that the last expression behaves like ρm ν N −1/2 e−γ as N → ∞. To compare the real process with the block-independent process, let FN (I ; X ) = 1{max{X (i) : iν/r (N ) ∈ I } ≤ Bm }.
2 N /2β 2
,
(5.32)
The difference between E[FN (I ; X 0N )|Y] and E[FN (I ; X 1N )] is again given by the Gaussian comparison formula (3.6). This time the second derivative equals δ(X (i) − Bm )δ(X ( j) − Bm ) 1{X (k) ≤ Bm } ≤ δ(X (i) − Bm )δ(X ( j) − Bm ). k=i, j
(5.33) If the covariance of X (i) and X ( j) is equal to c, then the expectation of the last expression is given by the value of the joint density of X (i), X ( j) at the point (Bm , Bm ), which is . γ 2N 2 (2π(1 − c2 ))−1 e−Bm /(1+c) ≤ C(1 − c2 )−1 exp − 2 . (5.34) β (1 + c)
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¯ The exponential term is the same as in (c). The polynomial prefactor is however ˜ replaced different, it diverges faster as c → 1. We should thus return to (3.24) with by the right-hand side of (5.34). First
1 −1 1 (5.35) 1 − (1 − h)2 c2 dh = c−1 arg tanh(c) ≈ − ln(1 − c) 2 0 as c → 1, which is not bounded for all c as before. The estimates (3.28) and (3.29) are influenced by this change. For (3.28) we can actually neglect this change, because the main contribution to this term came from the neighborhood of d = N /2 (or c = 0) and was exponentially small in the neighborhood of d = 1 (or c ∼ 1/N ). In the treatment of (3.29), the change has a stronger effect and after some computations (3.45) turns into
δ N →∞ Ct N 3/2 ν −1 ln(c/x)e−cN x dx ≤ Ct N 1/2 ν −1 ln N −−−−→ 0. (5.36) 0
¯ implies a change in the control of Finally, the change of the polynomial prefactor of (3.30). Equation (3.46) becomes (3.30) ≤ C
ν
t N −3/2 d 2 [1 − (1 − 2d N −1 )2 p]−1 exp(N ϒ˜ p,β,γ (d/N )), (5.37)
d=0
and the linearization of ϒ˜ p,β,γ gives a new form of (3.47), namely
ε N →∞ 3/2 Ct N xe−c N x dx ≤ Ct N −1/2 −−−−→ 0.
(5.38)
0
Therefore, using (5.28) P[H Nm (X 0N ; I ) = 0|Y] = E[FN (I ; X 0N )|Y] = E[FN (I ; X 1N )] + o(1) = (1 − E[s Nm (0, U )])|I |r (N )/ν → e−ρm |I | . This completes the proof of (5.26). It is easy to check Eq. (5.27). By definition, E[H Nm (X 0N ; I )|Y] =
E[s Nm (i, X 0N )|Y].
(5.39)
(5.40)
i:iν/r (N )∈I
Since i0j ≥ i1j for i, j in the same block, E[s Nm (i, X 0N )|Y] ≤ E[s Nm (i, X 1N )]. Therefore, (5.40) ≤ |I |r (N )/νE[s Nm (0, U )] = ρm |I |. This completes the proof of Lemma 5.4.
(5.41)
Proof of Theorem 5.2. Checking the convergence of finite-dimensional marginals as well as condition (i) and the second part of (ii) of Lemma 5.1 is analogous to the case of the original clock process S¯ N . We should thus only prove the first part of condition (ii). Namely, for any η and ε there exist δ such that P[w S¯ N (δ) ≥ η] ≤ ε, for all N large enough.
(5.42)
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Let w f ([τ, τ + δ]) = sup{min(| f (t2 ) − f (t)|, | f (t) − f (t1 )|) : τ ≤ t1 ≤ t ≤ t2 ≤ τ + δ}. (5.43) Fix m such that P[ S¯ Nm (T ) ≥ η/2] ≤ ε/2, which is possible according to Lemma 5.3. If H Nm (X n0 ; [τ, τ + δ]) ≤ 1 then w S¯ N ([τ, τ + δ]) ≤ S¯ Nm (τ + δ) − S¯ Nm (τ ) ≤ S¯ Nm (T ).
(5.44)
Hence, P[w S¯ N ([τ, τ + δ]) ≥ η|i S¯ Nm (T ) ≤ η/2] ≤ P[H Nm (X 0N ; [τ, τ + δ]) ≥ 2] ≤ Cρm δ 2 . (5.45) We can now show (5.42). The estimate w S˜ N (δ) ≤ max{w S˜ N ([τ, τ + 2δ]) : 0 ≤ τ ≤ T, τ = kδ, k ∈ N}
(5.46)
yields P[w S˜ N (δ) ≥ η|Y] ≤
T δ −1
P[w S˜ N ([kδ, (k + 2)δ]) ≥ ε|Y]
k=0
≤ P[ S¯ Nm (T ) ≥ η/2] +
T δ −1
(5.47) P[H Nm (X 0N ; [kδ, (k + 2)δ]) ≥ 2]
k=0
≤ ε/2 + C T δ −1 ρm δ 2 ≤ ε if δ is chosen small enough. This completes the proof.
Proof of Theorem 1.2. Let R N be the range of the coarse grained process S˜ N . Obviously, for any 1 > ε > 0, AεN (t, s) ⊃ {R N ∩ (t, s) = ∅},
(5.48)
because if the above intersection is empty, then σ N makes less than ν steps in the time interval [teγ N , seγ N ], and thus the overlap of σ N (teγ N ) and σ N (seγ N ) is O(ν/N ). If R N ∩ (t, s) = ∅, then there exist u such that S˜ N (u) ∈ (t, s). Moreover, it follows from Theorem 5.2 that, for any δ, there exist η such than P[ S˜ N (u + η) ∈ (s, t)] ≥ 1 − δ.
(5.49)
This means that the process σ N makes at least ηr (N ) steps between times t and s and thus the overlap between σ N (teγ N ) and σ N (seγ N ) is with high probability close to 0. Hence P[AεN (t, s)|Y] is very well approximated by P[R N ∩ (t, s) = ∅|Y]. Since the stable subordinators do not hit points, that is P[∃u : Vγ /β 2 (u) = t] = 0, and S˜ N converge in the J1 -topology, N →∞
P[R N ∩ (t, s) = ∅|Y] −−−−→ P[{Vγ /β 2 (u) : u ≥ 0} ∩ (s, t) = ∅].
(5.50)
The right-hand side of this equation is given by the formula (1.13), as follows from the arc-sine law for stable subordinators.
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Acknowledgements. This work was initiated during a concentration period on metastability and aging at the Max-Planck Institute for Mathematics in the Sciences in Leipzig. GBA and AB thank the MIP-MIS and Stefan Müller for their kind hospitality during this event. AB’s research is supported in part by DFG in the DutchGerman Bilateral Research Group “Mathematics of Random Spatial Models from Physics and Biology”. Finally, we thank the referees of this paper for their careful reading and helpful suggestions.
References [BBG02] [BBG03a] [BBG03b] ˇ [BC05] ˇ [BC06] ˇ [BC07] ˇ [BC08] [BCKM98] ˇ [BCM06] [BD95] [BDG01] [BG06] [BGK06] [Bil68] [BK06a] [BK06b] [BK07] [BM04] [Bou92] [Bov06] ˇ [CG08] [Cha84] [DDOL85] [Hoe63] [Kal02] [KH89] [LSNB83] [LT91]
Ben Arous, G., Bovier, A., Gayrard, V.: Aging in the random energy model. Phys. Rev. Lett. 88(8), 087201 (2002) Ben Arous, G., Bovier, A., Gayrard, V.: Glauber dynamics of the random energy model. I. Metastable motion on the extreme states. Commun. Math. Phys. 235(3), 379–425 (2003) Ben Arous, G., Bovier, A., Gayrard, V.: Glauber dynamics of the random energy model. II. Aging below the critical temperature. Commun. Math. Phys. 236(1), 1–54 (2003) ˇ Ben Arous, G., Cerný, J.: Bouchaud’s model exhibits two aging regimes in dimension one. Ann. Appl. Probab. 15(2), 1161–1192 (2005) ˇ Ben Arous, G., Cerný, J.: Dynamics of trap models. In: École d’Été de Physique des Houches, Session LXXXIII “Mathematical Statistical Physics”, Amsterdam: Elsevier, 2006, pp. 331–394 ˇ Ben Arous, G., Cerný, J.: Scaling limit for trap models on Zd . Ann. Probab. 35(6), 2356– 2384 (2007) ˇ Ben Arous, G., Cerný, J.: The arcsine law as a universal aging scheme for trap models. Commun. Pure Appl. Math. 61(3), 289–329 (2008) Bouchaud, J.-P., Cugliandolo, L., Kurchan, J., Mézard, M.: Out of equilibrium dynamics in spinglasses and other glassy systems. In: A.P. Young, ed. Spin glasses and random fields. Singapore: World Scientific, 1998 ˇ Ben Arous, G., Cerný, J., Mountford, T.: Aging in two-dimensional Bouchaud’s model. Probab. Theory Related Fields 134(1), 1–43 (2006) Bouchaud, J.-P., Dean, D.S.: Aging on Parisi’s tree. J. Phys I(France) 5, 265 (1995) Ben Arous, G., Dembo, A., Guionnet, A.: Aging of spherical spin glasses. Probab. Theory Related Fields 120(1), 1–67 (2001) Ben Arous, G., Gayrard, V.: Elementary potential theory on the hypercube. http://arXiv.org/list/ math.PR/0611178, 2006 Ben Arous, G., Gayrard, V., Kuptsov, A.: A new REM conjecture. http://arXiv.org/list/math.PR/ 0612373, 2006 Billingsley, P.: Convergence of probability measures. New York: John Wiley & Sons Inc., 1968 Bovier, A., Kurkova, I.: Local energy statistics in disordered systems: A proof of the local REM conjecture. Commun. Math. Phys. 263(2), 513–533 (2006) Bovier, A., Kurkova, I.: A tomography of the GREM: beyond the REM conjecture. Commun. Math. Phys. 263(2), 535–552 (2006) Ben Arous, G., Kuptsov, A.: The limits of REM universality. In preparation, 2007 Bauke, H., Mertens, S.: Universality in the level statistics of disordered systems. Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 70(2), 025102 (2004) Bouchaud, J.-P.: Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France) 2, 1705–1713 (1992) Bovier, A.: Statistical mechanics of disordered systems. Cambridge: Cambridge University Press, 2006 ˇ Cerný, J., Gayrard, V.: Hitting time of large subsets of the hypercube. to appear in Random Structures and Algorithms, 2008, doi:10.1002/rsa.20217, 2008 Chamberlin, R.V.: Time decay of the thermoremanent magnetization in spin-glasses as a function of the time spent in the field-cooled state. Phys. Rev. B 30(9), 5393–5395 (1984) De Dominicis, C., Orland, H., Lainée, F.: Stretched exponential relaxation in systems with random free energies. J- Phys.(Paris). 46, L463–L466 (1985) Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13–30 (1963) Kallenberg, O.: Foundations of modern probability. In: Probability and its Applications. New York: Springer-Verlag, 2002 Koper, G.J.M., Hilhorst, H.J.: Nonexponential relaxation in the random energy model. Physica A 160, 1–23 (1989) Lundgren, L., Svedlindh, P., Nordblad, P., Beckman, O.: Dynamics of the relaxation-time spectrum in a CuMn spin-glass. Phys. Rev. Lett. 51(10), 911–914 (1983) Ledoux, M., Talagrand, M.: Probability in Banach spaces. Berlin: Springer-Verlag, 1991
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Communicated by F. Toninelli
Commun. Math. Phys. 282, 697–719 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0553-y
Communications in
Mathematical Physics
Decay and Non-Decay of the Local Energy for the Wave Equation on the De Sitter–Schwarzschild Metric Jean-François Bony, Dietrich Häfner Institut de Mathématiques de Bordeaux, UMR 5251 du CNRS, Université de Bordeaux I, 351 cours de la Libération, 33 405 Talence cedex, France. E-mail: [email protected]; [email protected] Received: 20 June 2007 / Accepted: 6 March 2008 Published online: 15 July 2008 – © Springer-Verlag 2008
Abstract: We describe an expansion of the solution of the wave equation on the De Sitter–Schwarzschild metric in terms of resonances. The principal term in the expansion is due to a resonance at 0. The error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an ε derivative loss in the angular directions.
1. Introduction There has been important progress in the question of local energy decay for the solution of the wave equation in black hole type space-times over the last years. The best results are now known in Schwarzschild space-time. We refer to the papers of Blue–Soffer [4], Blue–Sterbenz [5], Dafermos–Rodnianski [12] and references therein for an overview. See also the paper of Finster–Kamran–Smoller–Yau for the Kerr space-time [13]. Results on the decay of local energy are believed to be a prerequisite for a possible proof of the global nonlinear stability of these space-times. Today global nonlinear stability is only known for the Minkowski space-time (see [11]). From our point of view one of the most efficient approaches to the question of local energy decay is the theory of resonances. Resonances correspond to the frequencies and rates of dumping of signals emitted by the black hole in the presence of perturbations (see [9, Chap. 4.35]). On the one hand these resonances are today an important hope of effectively detecting the presence of a black hole as we are theoretically able to measure the corresponding gravitational waves. On the other hand, the distance of the resonances to the real axis reflects the stability of the system under the perturbation: larger distances correspond to more stability. In particular the knowledge of the localization of resonances gives precise information about the decay of the local energy and its rate. The aim of the present paper is to show how this method applies to the simplest model of a black hole: the De Sitter–Schwarzschild black hole.
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In Euclidean space, such results are already known, especially for non trapping geometries. The first result is due to Lax and Phillips (see their book [16, Theorem III.5.4]). They have proved that the cut-off propagator associated to the wave equation outside an obstacle in odd dimension ≥ 3 (more precisely the Lax–Phillips semi-group Z (t)) has an expansion in terms of resonances if Z (T ) is compact for a given T . In particular, there is a uniform exponential decay of the local energy. From Melrose–Sjöstrand [19], this assumption is true for non trapping obstacles. Va˘ınberg [33] has obtained such results for general, non trapping, differential operators using different techniques. In the trapping case, we know, by the work of Ralston [22], that it is not possible to obtain uniform decay estimates without loss of derivatives. In the exterior domain of two strictly convex obstacles, the local energy decays exponentially with a loss of derivatives, by the work of Ikawa [15]. This situation is close to the one treated in this paper. We also mention the works Tang–Zworski [31] and Burq–Zworski [8] concerning the resonances close to the real line and the work of Christiansen–Zworski [10] for the wave equation on the modular surface and on the hyperbolic cylinder. Thanks to the work of Sá Barreto and Zworski [25], we have a very good knowledge of the localization of resonances for the wave equation on the De Sitter–Schwarzschild metric. Using their results we can describe an expansion of the solution of the wave equation on the De Sitter–Schwarzschild metric in terms of resonances. The main term in the expansion is due to a resonance at 0. The error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an ε derivative loss in the angular directions. For initial data in the complement of a one-dimensional space the local energy is integrable if we permit a (ln−ω )α derivative loss with α > 1. This estimate is almost optimal in the sense that it becomes false for α < 21 . The method presented in this paper does not directly apply to the Schwarzschild case. This is not linked to the difficulty of the photon sphere which we treat in this paper, but to the possible accumulation of resonances at the origin in the Schwarzschild case. The exterior of the De Sitter–Schwarzschild black hole is given by (M, g), M = Rt × X with X =]r− , r+ [r ×S2ω , 1 2 1/2 2M 2 2 −2 2 2 2 − r g = α dt − α dr − r dω , α = 1 − , r 3
(1.1) (1.2)
where M > 0 is the mass of the black holes and > 0 with 9M 2 < 1 is the cosmological constant. r− and r+ are the two positive roots of α = 0. We also denoted by dω2 the standard metric on S2 . The corresponding d’Alembertian is (1.3) g = α −2 Dt2 − α 2 r −2 Dr (r 2 α 2 )Dr + α 2 r −2 ω , where D• = 1i ∂• and −ω is the positive Laplacian on S2 . We also denote = α 2 r −2 Dr (r 2 α 2 )Dr − α 2 r −2 ω , P the operator on X which governs the situation on L 2 (X, r 2 α −2 dr dω). We define −1 , P = r Pr
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Fig. 1. The resonances of P near the real axis
on L 2 (X, α −2 dr dω), and, in the coordinates (r, ω), we have P = α 2 Dr (α 2 Dr ) − α 2 r −2 ω + r −1 α 2 (∂r α 2 ). We introduce the Regge–Wheeler coordinate given by x (r ) = α −2 .
(1.4)
In the coordinates (x, ω), the operator P is given by P = Dx2 − α 2 r −2 ω + α 2 r −1 (∂r α 2 ),
(1.5)
on L 2 (R × S2 , d x dω). Let V = α 2 r −2 and W = α 2 r −1 (∂r α 2 ) be the potentials appearing in P. As stated in Proposition 2.1 of [25], the work of Mazzeo–Melrose [18] (see also Guillarmou [14]) implies that for χ ∈ C0∞ (R), Rχ (λ) = χ (P − λ2 )−1 χ , has a meromorphic extension from the upper half plane to C, whose poles λ are called resonances. The set of the resonances is denoted by Res P. We recall the main result of [25]: Theorem 1.1 (Sá Barreto–Zworski). There exist K > 0 and θ > 0 such that for any ˜ from the set of pseudo-poles C > 0 there exists an injective map, b, 1
(1 − 9M 2 ) 2 3
32 M
1 1 1 ±N ± − i N0 + , 2 2 2
into the set of poles of the meromorphic continuation of (P − λ2 )−1 : L 2comp → L 2loc such that all the poles in
C = {λ; Im λ > −C, |λ| > K , Im λ > −θ | Re λ|}, ˜ are in the image of b˜ and for b(µ) ∈ C , ˜ b(µ) − µ → 0 as |µ| → ∞. 1 − 23 M −1 (1 − 9M 2 ) 2 (± ± 21 ) − i 21 ( j + 21 ) , ∈ N, j ∈ N0 , then If µ = µ± , j = 3 ˜ the corresponding pole, b(µ), has multiplicity 2 + 1.
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The natural energy space E for the wave equation is given by the completion of C0∞ (R × S2 ) × C0∞ (R × S2 ) in the norm (u 0 , u 1 )2E = u 1 2 + Pu 0 , u 0 .
(1.6)
It turns out that this is not a space of distributions. The problem is very similar to the problem for the wave equation in dimension 1. We therefore introduce another energy space mod (−∞ < a < b < ∞) defined as the completion of C ∞ (R × S2 ) × C ∞ (R × S2 ) Ea,b 0 0 in the norm b 2 2 (u 0 , u 1 )E mod = u 1 + Pu 0 , u 0 + |u 0 (s, ω)|2 ds dω. a,b
a
S2
mod and E mod are equivalent. We will Note that for any −∞ < a < b < ∞ the norms Ea,b 0,1 mod therefore only work with the space E0,1 in the future and note it from now on as E mod . Let us write the wave equation g u = 0 as a first order system in the following way: i∂t v = Lv 0 i . with L = −i P 0 v(0) = v0
Let Hk be the scale of Sobolev spaces associated to P. We denote by Hc2 the completion of H2 in the norm u22 := Pu, u + Pu2 . Then (L , D(L) = Hc2 ⊕ H1 ) is selfadjoint on E. We denote by E k the scale of Sobolev spaces associated to L. Note that because of λ i , (1.7) (L − λ)−1 = (P − λ2 )−1 −i P λ the meromorphic extension of the cut-off resolvent of P entails a meromorphic extension of the cut-off resolvent of L and the resonances of L coincide with the resonances of P. Recall that (−ω , H 2 (S2 )) is a selfadjoint operator with compact resolvent. Its eigenvalues are ( + 1), ≥ 0 with multiplicity 2 + 1. We denote P = r −1 Dx r 2 Dx r −1 + α 2 r −2 ( + 1)
(1.8)
the operator restricted to H = L 2 (R)×Y , where Y is the eigenspace of the eigenvalue ( + 1). In the following, P will be identified with the operator on L 2 (R) given by (1.8). The operators L and the spaces E , E mod , E k are defined similarly to the operator L and the spaces E, E mod , E k . Let be the orthogonal projector on E mod . For ≥ 1, the space E mod and E are the same and the norms are equivalent uniformly with respect to . Using Proposition II.2 of Bachelot and Motet-Bachelot [3], the group e−it L preserves the space E mod and there exist C, k > 0 such that e−it L uE mod ≤ Cek|t| uE mod . From the previous discussion, the same estimate holds for L with k = 0 uniformly in ≥ 1. In particular, (L − z)−1 is bounded on E mod for Im z > k, and we denote E mod,−j = (L − z) j E mod ⊂ D (R × S2 ) × D (R × S2 ) for j ∈ N0 . We first need a result on P:
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Proposition 1.2. For ≥ 1, the operator P has no resonance and no eigenvalue on the real axis. For = 0, P0 has no eigenvalue in R and no resonance in R \ {0}. But, 0 is a simple resonance of P0 , and, for z close to 0, we have (P0 − z 2 )−1 =
iγ r r | · + H (z), z
(1.9)
where γ ∈]0, +∞[ and H (z) is a holomorphic (bounded) operator near 0. Equation (1.9) is an equality between operators from L 2comp to L 2loc . The proof of Proposition 1.2 is given in Sect. 2.1. For χ ∈ C0∞ (R) we denote henceforth: χ (λ) = χ (L − λ)−1 χ . R For a resonance λ j we define m(λ j ) by the Laurent expansion of the cut-off resolvent near λ j : ∞
χ (λ) = R
Ak (λ − λ j )k .
k=−(m(λ j )+1) χ
We also define π j,k by χ
π j,k =
−1 2πi
(−i)k Rχ (λ)(λ − λ j )k dλ. k!
(1.10)
The main result of this paper is the following: Theorem 1.3. Let χ ∈ C0∞ (R). 2 )1/2 1 1 (i) Let 0 < µ ∈ / (1−9M 2 N0 + 2 such that there is no resonance with Im z =−µ. 31/2 M Then there exists M > 0 with the following property. Let u ∈ E mod such that −ω M u ∈ E mod . Then we have χ e−it L χ u =
m(λ j )
χ
e−iλ j t t k π j,k u + E 1 (t)u,
(1.11)
λ j ∈Res P k = 0 Im λ j >−µ
with
E 1 (t)uE mod e−µt −ω M uE mod ,
(1.12)
and the sum is absolutely convergent in the sense that
m(λ j )
λ j ∈Res P k = 1 Im λ j >−µ
χ
π j,k −ω −M L(E mod ) 1.
(1.13)
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(ii) There exists ε > 0 with the following property. Let g ∈ C([0, +∞[), lim|x|→∞ g(x) = 0, positive, strictly decreasing with x −1 ≤ g(x) for x large. Let u = (u 1 , u 2 ) ∈ E mod be such that (g(−ω ))−1 u ∈ E mod . Then we have r χ r, χ u 2 −it L + E 2 (t)u, χu = γ (1.14) χe 0 with
E 2 (t)uE mod g(eεt ) (g(−ω ))−1 u E mod .
(1.15)
Remark 1.4. a) By the results of Sá Barreto and Zworski we know that there exists µ > 0 such that 0 is the only resonance in Im z > −µ. Choosing this µ in (i) the sum on the right-hand side contains a single element which is r χ r, χ u 2 . γ 0 b) Again by the paper of Sá Barreto and Zworski we know that λ j = b(µε , j ) for all the λ j ’s outside a compact set (see Theorem 1.1). For such λ j , we have m j (λ j ) = 0 χ χ χ and π j,k = π j,k = π j,k is an operator of rank 2 + 1. c) Let E mod,⊥ = {u ∈ E mod ; r, χ u 2 = 0}. By part (ii) of the theorem, for u ∈ E mod,⊥ , the local energy is integrable if (ln−ω )α u ∈ E mod , for some α > 1, and decays exponentially if −ω ε u ∈ E mod for some ε > 0. d) We can replace −ω M by P2M in the first part of the theorem. And, by an interpolation argument, we can obtain the following estimate: for all ε > 0, there exists δ > 0 such that r χ r, χ u 2 + E 3 (t)u, χ e−it L χ u = γ (1.16) 0 with
E 3 (t)uE mod e−δt Pε u E mod .
(1.17)
Remark 1.5. In the Schwarzschild case the potential V (x) is only polynomially decreasing at infinity and we cannot apply the result of Mazzeo–Melrose. Therefore we cannot exclude a possible accumulation of resonances at 0. This difficulty has nothing to do with the presence of the photon sphere which is treated by the method presented in this paper. Remark 1.6. Let u ∈ E mod,⊥ be such that (ln−ω )α u ∈ E mod for some α > 1. Then we have from part (ii) of the theorem, for λ ∈ R, ∞ χ e−it (L−λ) χ u dt mod (ln−ω )α uE mod . (1.18) E
0
This estimate is almost optimal since it becomes false for α < (λ ∈ R): ∞ χ (λ)u = i χ e−it (L−λ) χ u dt. R 0
1 2.
Indeed we have
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Thus from (1.18) we obtain the resolvent estimate χ (λ)(ln−ω )−α L(E mod,⊥ ,E mod ) 1. R It is easy to see that this entails the resolvent estimate χ (P − λ2 )−1 χ (ln ( + 1))−α
1 , |λ|
for ≥ 1. We introduce the semi-classical parameter h 2 = ( ( + 1))−1 and
= h 2 Dx2 + V (x) + h 2 W (x) as in Sect. 2.3. Then, for R > 0, the above estimate P gives the semi-classical estimate: α
− z)−1 χ | ln h| , χ ( P h
for 1/R ≤ z ≤ R (see (2.25) and (2.26)). Such an estimate is known to be false for α < 21 and z = z 0 , the maximum value of the potential V (x) (see [1, Prop. 2.2]). Remark 1.7. Let P1 be the projection on the first variable, P1 (u 1 , u 2 ) = u 1 . If u ∈ E mod is such that (g(−ω ))−1 (L + i)u ∈ E mod , then P1 χ e−it L χ u ∈ C 0 (R × S2 ) and the remainder term in (1.14) satisfies (1.19) P1 E 2 (t)u L ∞ (R×S2 ) g(eεt ) (g(−ω ))−1 (L + i)u E mod . Moreover, if u ∈ E mod is such that (g(−ω ))−1 (L + i)2 u ∈ E mod , then χ e−it L χ u ∈ C 0 ((R × S2 ) × (R × S2 )) and the remainder term in (1.14) satisfies (1.20) E 2 (t)u L ∞ ((R×S2 )×(R×S2 )) g(eεt ) (g(−ω ))−1 (L + i)2 u E mod . The proof of the theorem is based on resolvent estimates. Using (1.7) we see that it is sufficient to prove resolvent estimates for χ (P − λ2 )−1 χ . This is the purpose of the next section.
2. Estimate for the Cut-Off Resolvent In this section, we obtain estimates for the cut-off resolvent of P , the operator P restricted to the spherical harmonic . We will use the description of the resonances given in Sá Barreto–Zworski [25]. Recall that Rχ (λ) = χ (P − λ2 )−1 χ ,
(2.1)
has a meromorphic extension from the upper half plane to C. The resonances of P are defined as the poles of this extension. We treat only the case Re ∗ λ > −1 since we can obtain the same type of estimates for Re λ < 1 using Rχ (−λ) = Rχ (λ). Theorem 2.1. Let C0 > 0 be fixed. The operators χ (P − λ2 )−1 χ satisfy the following estimates uniformly in :
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i) For all R > 0, the number of resonances of P is bounded in B(0, R). Moreover, there exists C > 0 such that 1 χ (P − λ2 )−1 χ ≤ χ (P − λ2 )−1 χ ≤ C , (2.2) |λ − λ j | λ j ∈Res P |λ j |<2R
for all λ ∈ B(0, R). As usual, the resonances are counted with their multiplicity. ii) For R large enough, P has no resonance in [R, /R] + i[−C0 , 0]. Moreover, there exists C > 0 such that C , (2.3) χ (P − λ2 )−1 χ ≤ λ2 for λ ∈ [R, /R] + i[−C0 , C0 ]. iii) Let R be fixed. For large enough, the resonances of P in [ /R, R ]+i[−C0 , 0] are the b(µ+ , j ) given in Theorem 1.1 (in particular their number is bounded uniformly in ). Moreover, there exists C > 0 such that 1 , (2.4) χ (P − λ2 )−1 χ ≤ CλC |λ − λ j | λ j ∈Res P |λ−λ j |<1
for λ ∈ [ /R, R ] + i[−C0 , C0 ]. Furthermore, P has no resonance in [ /R, R ] + i[−ε, 0], for some ε > 0, and we have lnλ C| Im λ| lnλ e , (2.5) χ (P − λ2 )−1 χ ≤ C λ for λ ∈ [ /R, R ] + i[−ε, 0]. iv) Let C1 > 0 be fixed. For R large enough, P has no resonance in {λ ∈ C; Re λ > R , and 0 ≥ Im λ ≥ −C0 − C1 lnλ}. Moreover, there exists C > 0 such that χ (P − λ2 )−1 χ ≤
C C| Im λ| e , λ
(2.6)
for Re λ > R and C0 ≥ Im λ ≥ −C0 − C1 lnλ. The results concerning the localization of the resonances in this theorem are proved in [3] and [25], Fig. 2 summarizes the different estimates of the resolvent. In zone I which is compact, the result of Mazzeo–Melrose [18] gives a bound uniform with respect to (away from the possible resonances). In particular, part i) of Theorem 2.1 is a direct consequence of this work. In zone II, the result of Zworski [34] gives us a good (uniform with respect to ) estimate of the resolvent. Here, we use the exponential decay of the potential at +∞ and −∞. By comparison, the corresponding potential for the Schwarzschild metric does not decay exponentially, and our present work cannot be extended to this setting. Note that this problem concerns only zones I and II, but zones III and IV can be treated in the same way. In zone III, we have to deal with the so called “photon sphere”. The estimate (2.4) follows from a general polynomial bound of the resolvent in dimension 1 (see [6]). In zone IV, the potentials ( + 1)V and W are very small in comparison to λ2 . So they do not play any role, and we obtain the same estimate as in the free case of − (or as for non trapping geometries).
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Fig. 2. The different zones in Theorem 2.1
2.1. Estimate close to 0. This part is devoted to the proof of Proposition 1.2 and of part i) of Theorem 2.1. Since χ (P − λ2 )−1 χ has a meromorphic extension to C, the number of resonances in B(0, R) is always bounded and point i) of Theorem 2.1 is clear. It is a classic result (see Theorem XIII.58 in [23]) that P has no eigenvalue in R \ {0}. On the other hand, from Proposition II.1 of the work of Bachelot and Motet-Bachelot [3], 0 is not an eigenvalue of the operators P . Moreover, by the limiting absorption principle [20], x−α (P − (z + i0))−1 x−α < ∞, (2.7) for z ∈ R \ {0} and any α > 1, we know that P has no resonance in R \ {0}. We now study the resonance 0 using a technique specific to the one dimensional case. We start by recalling some facts about outgoing Jost solutions. Let
(x), Q = − + V
(2.8)
∈ C ∞ (R) decaying exponentially at infinity. For be a Schrödinger operator with V Im λ > 0, there exists a unique pair of functions e± (x, λ) such that ⎧ ⎨ (Q − λ2 )e± (x, λ) = 0, ⎩ lim e± (x, λ) − e±iλx = 0. x→±∞
∈ C ∞ (R) decays The function e± is called the outgoing Jost solution at ±∞. Since V ∞ exponentially at infinity, the functions e± can be extended, as C (R) functions of x, analytically in a strip {λ ∈ C; Im λ > −ε}, for some ε > 0. Moreover, in such a strip, they satisfy |e± (x, λ) − e±iλx | = O(e−|x|(Im λ+δ) ) for ± x > 0, |∂x e± (x, λ) ∓ iλe
±iλx
| = O(e
−|x|(Im λ+δ)
) for ± x > 0,
(2.9) (2.10)
for some δ > 0. All these properties can be found in Theorem XI.57 of [24]. Using these Jost solutions, the kernel of (Q − λ2 )−1 , for Im λ > 0 takes the form R(x, y, λ) =
1 (e+ (x, λ)e− (y, λ)H (x − y) + e− (x, λ)e+ (y, λ)H (y − x)) , w(λ) (2.11)
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where H (x) is the Heaviside function
H (x) = and
1 for x > 0, 0 for x ≤ 0,
w(λ) = (∂x e− )e+ − (∂x e+ )e− ,
(2.12)
is the wronskian between e− and e+ (the right-hand side of (2.12) does not depend on x). In particular, w(λ) is an analytic function on {λ ∈ C; Im λ > −ε}. Since the e± are always non-zero thanks to (2.9), the resonances are the zeros of w(λ). Such a discussion can be found in the preprint of Tang–Zworski [32]. Remark that P is of the form (2.8). If 0 is a resonance of one of the P ’s with ≥ 1, the Jost solutions e± (x, 0) are collinear. In particular, from (2.9) and (2.10), the C ∞ function e+ (x, 0) converges to two non zero limits at ±∞ and ∂x e+ (x, 0) goes to 0 as x → ±∞. Since P = r −1 Dx r 2 Dx r −1 + α 2 r −2 ( + 1), (2.13) we get, by an integration by parts, R 0= (P e+ )e+ d x −R
= ( + 1)
R −R
|αr −1 e+ |2 d x +
R −R
R |r Dx (r −1 e+ )|2 d x − ir −1 e+ Dx (r −1 e+ ) . −R
(2.14) Since ∂x (r −1 e+ ) = r −1 ∂x e+ − r −2 α 2 e+ , the last term in (2.14) goes to 0 as R goes to +∞. Thus, if ≥ 1, (2.14) gives e+ = 0 and 0 is not a resonance of P . We now study the case = 0. If u ∈ C 2 (R) satisfies P0 u = 0, we get from (2.13), r 2 Dx r −1 u = −iβ, where β ∈ C is a constant. Then
x
u(x) = αr (x) + βr (x) 0
1 r 2 (t)
dt,
where α, β ∈ C are constants. Note that x x 1
r (x) := r (x) dt = + O(1), 2 (t) r r ± 0 as x → ±∞. Since e± (x, 0) are C ∞ functions bounded at ±∞ from (2.9) which satisfy P0 u = 0, the two functions e± (x, 0) are collinear to r and then w(0) = 0 which means that 0 is a resonance of P0 . The resolvent of P0 thus has the form (P0 − λ2 )−1 =
J 1 + H (λ), + ··· + λJ λ
where H (λ) is an analytic family of bounded operators near 0 and J = 0. For all λ = iε with ε > 0, we have λ2 (P0 − λ2 )−1 L 2 →L 2 = ε2 (P0 + ε2 )−1 L 2 →L 2 ≤ 1,
Decay and Non-Decay on the De Sitter–Schwarzschild Metric
707
from the functional calculus. This inequality implies that J ≤ 2 and 2 L 2 →L 2 ≤ 1. If f (x) ∈ L 2loc is in the range of 2 , we have f ∈ L 2 and P0 f = 0. Then, f ∈ H s for all s and f is an eigenvector of P0 for the eigenvalue 0. This is impossible because P0 has no eigenvalue. Thus 2 = 0 and J = 1. So w(λ) has a zero of order 1 at λ = 0. Since e± (x, 0) = r (x)/r± , (2.11) implies that the kernel of 1 is given by 1 (x, y) =
1 w (0)r
+ r−
r (x)r (y) = iγ r (x)r (y).
(2.15)
Finally, since iε(P0 + ε2 )−1 → 1 as ε → 0 and since P0 + ε2 is a strictly positive operator, we get −i 1 u, u ≥ 0 for all u ∈ L 2comp . In particular, −iiγ > 0 and then γ ∈]0, +∞[. 2.2. Estimate for λ small in comparison to . In this section, we give an upper bound for the cut-off resolvent for λ ∈ [R, /R] + i[−C0 , C0 ]. We assume that λ ∈ [N , 2N ] + i[−C0 , C0 ] with N ∈ [R, /R], and define a new semi-classical parameter h = N −1 , a new spectral parameter z = h 2 λ2 ∈ [1/4, 4] + i[−4C0 h, 4C0 h] and
= −h 2 + h 2 ( + 1)V (x) + h 2 W (x). P
(2.16)
With these notations, we have
− z)−1 . (P − λ2 )−1 = h 2 ( P
(2.17)
We remark that β 2 := h 2 ( + 1) 1 in our window of parameters. The potentials V and W have a holomorphic extension in a sector = {x ∈ C; | Im x| ≤ θ0 | Re x| and | Re x| ≥ C},
(2.18)
for some C, θ0 > 0. From the form of α 2 (see (1.2)), there exist κ± > 0 and functions f ± ∈ C ∞ (R± ; [1/C, C]), C > 0, analytic in such that V (x) = e∓κ± x f ± (x),
(2.19)
for x ∈ and ± Re x > 0. Moreover, f ± have a (non zero) limit for x → ±∞, x ∈ . Under these hypotheses, and following Proposition 4.4 of [25], we can use the specific estimate developed by Zworski in [34] for operators like (2.16) with V satisfying (2.19). In the beginning of Sect. 4 of [34], Zworski defines a subtle contour θ briefly described in Fig 3.
θ = P
| is defined by Recall that the distorted operator P θ
θ u = ( Pu)
| P θ
(2.20)
for all u analytic in and then extended as a differential operator on L 2 (θ ) by means
in the sector Sθ = {e−2is r ; 0 < s < of almost analytic functions. The resonances of P
θ in that set. For the θ and r ∈]0, +∞[} = e2i]−θ,0] ]0, +∞[ are then the eigenvalues of P general theory of resonances, see the paper of Sjöstrand [27] or his book [28].
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J.-F. Bony, D. Häfner
Fig. 3. The set and the contour θ
has no resonance in [1/4, 4] For θ large enough, Proposition 4.1 of [34] proves that P +i[−4C0 h, 4C0 h]. Moreover, for z in that set, this proposition gives the uniform estimate
θ − z)−1 ≤ C. ( P
(2.21)
Since θ coincides with R for x ∈ supp χ , we have
θ − z)−1 χ ,
− z)−1 χ = χ ( P χ(P
(2.22)
from Lemma 3.5 of [29]. Using (2.16), we immediately obtain χ (P − λ2 )−1 χ ≤
C , λ2
(2.23)
which is exactly (2.3).
2.3. Estimate for λ of order . In this part, we study the cut-off resolvent for the energy λ ∈ [ /R, R ] + i[−C0 , C0 ]. In this zone, we have to deal with the photon sphere. We define the new semi-classical parameter h = ( ( + 1))−1/2 and
= −h 2 + V (x) + h 2 W (x). P
(2.24)
− z)−1 , (P − λ2 )−1 = h 2 ( P
(2.25)
As previously, we have
where z = h 2 λ2 ∈ [1/2R 2 , R 2 ] + i[−3RC0 h, 0] ⊂ [a, b] + i[−ch, ch],
(2.26)
with 0 < a < b and 0 < c. The form of V is given in Fig. 4. In particular, V admits at x0 a non-degenerate maximum at energy z 0 > 0. On the other hand, for z = z 0 , z > 0, the energy level z is non trapping for p0 (x, ξ ) = ξ 2 + V (x), the principal semi-classical
We define P
θ by standard distortion (see Sjöstrand [27]) and can apply the symbol of P. following general upper bound on the cut-off resolvent in dimension one.
Decay and Non-Decay on the De Sitter–Schwarzschild Metric
709
Fig. 4. The potential V (x)
Lemma 2.2 (Lemma 6.5 of [6]). We assume that n = 1 and that the critical points of
p0 (x, ξ ) on the energy level E 0 are non-degenerate (i.e. the points (x, ξ ) ∈ p0−1 ({E 0 }) such that ∇ p0 (x, ξ ) = 0 satisfy Hess p0 (x, ξ ) is invertible). Then, there exists ε > 0 such that, for E ∈ [E 0 − ε, E 0 + ε] and θ = N h with N > 0 large enough,
θ − z)−1 = O(h −M ) ( P
z∈Res P |z−z j |<εθ
h |z − z j |
(2.27)
for |z − E| < εθ/2 and some M > 0 which depends on N . Note that there is a slight error in the statement of the lemma in [6]. Indeed, M depends on N , and in the proof of this lemma, the right hand side of (6.18), O(ln(1/θ )), must be replaced by O(θ h −1 ln(1/θ )). Recall that, from Proposition 4.3 [25], which is close to the work of Sjöstrand [26] on the the resonances associated to a critical point, there exists an injective map b(h) from (2.28) 0 (h) = µ j = z 0 − i h |V (x0 )|/2( j + 1/2); j ∈ N0 ,
such that into the set of resonances of P b(h)(µ) − µ = o(h), µ ∈ 0 (h),
(2.29)
and such that all the resonances in [a/2, 2b] + i[−ch, ch] are in the range of b(h). In par is bounded in [a/2, 2b]+i[−ch, ch]. Furthermore, ticular, the number of resonances of P
the operator P has no resonance in
(h) = [a/2, 2b] + i[−εh, ch], for any ε > 0 and h small enough. Using a compactness argument, we get (2.27) for all z ∈ [a, b] + i[−ch, ch]. −1 −1 −1
Thus, √ from (2.25), (2.26), χ ( P − z) χ = χ ( Pθ − z) χ , the estimate λ h = ( + 1) λ for λ ∈ [ /R, R ] + i[−C0 , 0], Lemma 2.2 and the previous discussion, we get 1 χ (P − λ2 )−1 χ ≤ CλC , (2.30) |λ − λ j | z j ∈Res P |λ−λ j |<1
for λ ∈ [ /R, R ] + i[−C0 , C0 ] and (2.4) follows.
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J.-F. Bony, D. Häfner
has no resonance in (h) and in this set On the other hand, P ⎧ −M on (h), ⎨h
− z)−1 χ χ ( P 1 ⎩ on (h) ∩ {Im z > 0}. | Im z|
(2.31)
We can now apply the following version, due to Burq [7], of the so-called “semi-classical maximum principle” introduced by Tang–Zworski [30]. Lemma 2.3 (Burq). Suppose that f (z, h) is a family of holomorphic functions defined for 0 < h < 1 in a neighborhood of
(h) = [a/2, 2b] + i[−ch, ch], with 0 < a < b and 0 < c, such that ⎧ −M ⎨h | f (z, h)| 1 ⎩ | Im z|
on (h), on (h) ∩ {Im z > 0}.
Then, there exists h 0 , C > 0 such that, for any 0 < h < h 0 , | f (z, h)| ≤ C
| ln h| C| Im z|| ln h|/ h e , h
(2.32)
for z ∈ [a, b] + i[−ch, 0]. This lemma is strictly analogous to Lemma 4.7 of [7]. Combining (2.25), (2.26), λ h −1 λ with this lemma, we obtain χ (P − λ2 )−1 χ ≤ C
lnλ C| Im λ| lnλ e , λ
(2.33)
for λ ∈ [ /R, R ] + i[−ε, 0], for some ε > 0. 2.4. Estimate for the very large values of λ. Here, we study the resolvent for |λ| . More precisely, we assume that λ ∈ [N , 2N ] + i[−C ln N , C0 ], for some C > 0 fixed and N . We define the new semi-classical parameter h = N −1 and z = h 2 λ2 ∈ h 2 [N 2 /2, 4N 2 ] + i h 2 [−4C N ln N , 4C0 N −1 ] ⊂ [a, b] + i[−ch| ln h|, ch], for some 0 < a < b and 0 < c. Then, P can be written
− z), P − λ2 = h −2 ( P where
= −h 2 + µV (x) + νW (x), P with µ = ( + 1)h 2 , ν = h 2 . For N , the coefficients µ, ν are small, and the
is uniformly non trapping for z ∈ [a, b]. We can expect a uniform bound of operator P the cut-off resolvent in [a, b] + i[−ch| ln h|, ch]. Such a result is proved in the following lemma.
Decay and Non-Decay on the De Sitter–Schwarzschild Metric
711
Lemma 2.4. For all χ ∈ C0∞ (R), there exist µ0 , ν0 , h 0 , C > 0 such that, for all µ < µ0 ,
has no resonance in [a, b] + i[−ch| ln h|, ch]. Moreover ν < ν0 and h < h 0 , P
− z)−1 χ ≤ C eC| Im z|/ h , χ ( P h
(2.34)
for all z ∈ [a, b] + i[−ch| ln h|, ch]. Assume first Lemma 2.4. For λ ∈ [N , 2N ] + i[−C ln N , C0 ], we have
− z)−1 χ χ (P − λ2 )−1 χ = h 2 χ ( P C| Im z|/ h ≤ Che C 4C| Im λ| e ≤ , |λ| and the estimate (2.6) follows. Proof of Lemma 2.4. For µ and ν small and fixed, the estimate (2.34) is already known. The proof can be found in the book of Va˘ınberg [33] in the classical case and in the paper of Nakamura–Stefanov–Zworski [21] in our semi-classical setting. To obtain Lemma 2.4, we only have to check the uniformity (with respect to µ and ν) in the proof of [21, Prop. 3.1]. • Limiting absorption principle. The point is to note that A = xh Dx + h Dx x,
(2.35)
is a conjugate operator for all µ, ν 1. Let g ∈ C0∞ ([a/3, 3b]; [0, 1]) be equal to 1
is well defined on D(A), and its closure, A, is near [a/2, 2b]. The operator g( P)Ag( P)
is of class C r (A) if there self-adjoint. The operator P is of class C 2 (A). Recall that P
exists z ∈ C \ σ ( P) such that
− z)−1 e−it A , R t → eit A ( P is C r for the strong topology of L 2 (see [2, Sect. 6.2] for more details). We have
A] = 4 P
− 4µV − 4νW − 2µx V − 2νx W . i h −1 [ P,
(2.36)
In particular, for µ and ν small enough, we easily obtain
P,
A]1[a/2,2b] ( P)
≥ ah1[a/2,2b] ( P).
1[a/2,2b] ( P)i[
(2.37)
Note that this Mourre estimate is uniform with respect to µ, ν. It is also easy to check that x−1 A ≤ C,
+ i)−1 [ P,
A] ≤ Ch, ( P −1
+ i) [[ P,
A], A] ≤ Ch 2, ( P
+ i)−1 [ P,
[ P,
A]] ≤ Ch 2, ( P −1
+ i) A[ P,
[ P,
A]] ≤ Ch 2, ( P uniformly in µ, ν.
(2.38)
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J.-F. Bony, D. Häfner
∈ C 2 (A), the Mourre estimate (2.37) and the upper bound (2.38) are The regularity P the key assumptions for the limiting absorption principle. In particular, from, for instance, the proof of Proposition 3.2 in [1] which is an adaptation of the theorem of Mourre [20], we obtain the following estimate: For α > 1/2, there exist µ0 , ν0 , h 0 , C > 0, such that
− z)−1 x−α ≤ Ch −1 , x−α ( P
(2.39)
for all µ < µ0 , ν < ν0 , h < h 0 and z ∈ [a/2, 2b] + i]0, ch]. In particular,
− z)−1 χ ≤ Ch −1 χ ( P
(2.40)
for z ∈ [a/2, 2b] + i[0, ch]. • Polynomial estimate in the complex. The second point of the proof is to obtain a polynomial bound of the distorted resolvent. To obtain such bounds, we use the paper
(x), of Martinez [17]. In this article, the author studies the resonances of Q = −h + V
is a C ∞ (Rn ) function which can be extended analytically in a domain like where V (see (2.18)) and decays in this domain. If the energy level z 0 is non trapped for the symbol
(x), the operator Q has no resonance in [z 0 − δ, z 0 + δ] + i[Ah ln h, 0] q(x, ξ ) = ξ 2 + V for a δ small enough and any A > 0. Moreover, (Q θ − z)−1 ≤ Ch −C
(2.41)
for z ∈ [z 0 − δ, z 0 + δ] + i[Ah ln h, 0]. Here Q θ denotes the distorted operator outside of a large ball of angle θ = Bh| ln h|, with B A.
satisfies the previous assumption on Q, for µ and ν fixed small enough. Of course, P But, following line by line the proof of (2.41) in [17, Sect. 4], one can prove that (2.41) is uniformly true for µ, ν 1. This means that there exist µ0 , ν0 , h 0 , C > 0 such that
− z)−1 χ = χ ( P
θ − z)−1 χ ≤ Ch −C , χ ( P
(2.42)
for all µ < µ0 , ν < ν0 , h < h 0 and z ∈ [a/2, 2b] + i[ch ln h, 0]. • Semi-classical maximum principle. To finish the proof, we use a version of the semiclassical maximum principle. This argument can be found in [21, Prop. 3.1], but we give it here for the convenience of the reader. We can construct a holomorphic function f (z, h) with the following properties: | f | ≤ C for z ∈ [a/2, 2b] + i[ch ln h, 0], | f | ≥ 1 for z ∈ [a, b] + i[ch ln h, 0], | f | ≤ h M for z ∈ [a/2, 2b] \ [2a/3, 3b/2] + i[ch ln h, 0], where M is the constant C given in (2.42). We can then apply the maximum principle in [a/2, 2b] + i[ch ln h, 0] to the subharmonic function
− z)−1 χ + ln | f (z, h)| + M Im z , ln χ ( P c h proving the lemma with (2.40) and (2.42).
Decay and Non-Decay on the De Sitter–Schwarzschild Metric
713
3. Proof of the Main Theorem 3.1. Resolvent estimates for L . The cut-off resolvent estimates for P give immediately cut-off resolvent estimates for L . Proposition 3.1. Let χ ∈ C0∞ (R). Then the operator χ (L − λ)−1 χ sends E mod into itself and we have uniformly in : χ (L − z)−1 χ L(E mod ) zχ (P − z 2 )−1 χ .
(3.1)
mod as well as the fact Proof. Using Theorem 2.1, (1.7), the equivalence of the norms Ea,b ∞
χ = χ we see that it is sufficient that we can always replace u by χ
u, χ
∈ C0 (R), χ to show:
χ (P − z 2 )−1 χ
u H 1 , χ (P − z 2 )−1 χ u H 1 2 −1
χ (P − z )
2 −1
χ u H 1 z χ (P − z )
(3.2)
χ
u L 2 ,
(3.3)
χ (P − z 2 )−1 P χ u L 2 z χ (P − z 2 )−1 χ
u H 1 .
(3.4)
Using complex interpolation we see that it is sufficient to show: χ (P − z 2 )−1 χ
u H 2 , χ (P − z 2 )−1 χ u H 2 2 −1
χ (P − z )
2 −1
χ (P − z )
2 −1
χ u H 2 z χ (P − z ) 2
2 −1
P χ u L 2 χ (P − z )
χ
u L 2 ,
χ
u H 2 .
(3.5) (3.6) (3.7)
We start with (3.7) which follows from χ (P − z 2 )−1 P χ = χ (P − z 2 )−1 χ P + χ (P − z 2 )−1 [P , χ ]u. Let us now observe that P χ (P − z 2 )−1 χ u = [P , χ ](P − z 2 )−1 χ + χ (P − z 2 )−1 P χ u =χ
(P + i)−1 [P , [P , χ ]](P − z 2 )−1 χ u + χ (P + i)−1 [P , χ ](P − z 2 )−1 (P + i)χ u +χ (P − z 2 )−1 P χ u. From this identity we obtain (3.5) and (3.6) using (3.7) (for (3.5)) and the uniform boundedness of (P + i)−1 [P , [P , χ ]]. 3.2. Resonance expansion for the wave equation. For the proof of the main theorem we follow closely the ideas of Va˘ınberg [33, Chap. X.3]. If N is a Hilbert space we will denote by L 2ν (R; N ) the space of all functions v(t) with values in N such that e−νt v(t) ∈ L 2 (R; N ). Let u ∈ E mod and −it L u t ≥ 0, e v(t) = 0 t < 0. Then v ∈ L 2ν (R; E ) for all ν > 0. We can define ∞ v(t)eikt dt v(k) ˜ = 0
714
J.-F. Bony, D. Häfner
as an element of E for all k with Im k > 0. The function v˜ depends analytically on k when Im k > 0. Also, on the line Im k = ν the function belongs to L 2 (R; E ). We have the inversion formula: ∞+iν 1 v(t) = e−ikt v(k) ˜ dk 2π −∞+iν and the integral converges in L 2ν (R; E ) for all ν > 0. From the functional calculus we know that v(k) ˜ = −i(L − k)−1 u for all k with Im k > 0. We therefore obtain for all t ≥ 0: ∞+iν 1 e−it L u = (L − k)−1 e−ikt u dk, 2πi −∞+iν
(3.8)
χ (k) the where the integral is convergent in L 2ν (R; E ). In the following, we denote by R −1 meromorphic extension of χ (L − k) χ . Lemma 3.2. Let χ ∈ C0∞ (R), N ≥ 0. Then, there exist bounded operators B j ∈ mod,−q mod,− j−q mod,−q mod,−N −1−q L(E ; E ), j = 0, . . . , N , q ∈ N0 and B ∈ L(E ; E ), q ∈ N0 such that χ (k) = R
N
j=0
1 1 χ Bj + BR (k)χ , j+1 (k − i(ν + 1)) (k − i(ν + 1)) N +1
(3.9)
for some χ
∈ C0∞ (R) with χ χ
= χ. Proof. We proceed by induction over N . For N = 0, we write (L − k)−1 +
1 1 = (L − i(ν + 1))(L − k)−1 , k − i(ν + 1) k − i(ν + 1)
and choose B0 = −χ 2 . Then χ (k) − R
1 1 χ
B0 = (k)χ , Bχ , χR k − i(ν + 1) k − i(ν + 1)
where Bχ , χ , with χ = χ χ
, is in the space L(E χ = χ (L − i(ν + 1)) Let us suppose that the lemma is proved for N ≥ 0. We put
m,−q
B N +1 =
(3.10) m,−1−q
; E
1 B χ 2χ . (k − i(ν + 1)) N +1
).
(3.11)
Using (3.10), we get χ (k) = R
N
j=0
=
N +1
j=0
1 1 χ Bj + BR χ j+1 (k − i(ν + 1)) (k − i(ν + 1)) N +1 1 1 χ , Bj + B Bχ , χ R χ (k − i(ν + 1)) j+1 (k − i(ν + 1)) N +2
with χ ∈ C0∞ (R) with χχ
= χ
. This proves the lemma.
(3.12)
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715
Fig. 5. The paths j
Let us define
χ (k) = R χ (k) − R
1
j=0
1 Bj. (k − i(ν + 1)) j+1
Then, Lemma 3.2 implies, for Im k ≤ ν, 1 R (k)L(E mod ;E mod ) . k2 χ
χ (k) mod mod,−2 R L(E ;E )
Now observe that
∞+iν
−∞+iν
Bj e−ikt dk = 0. (k − i(ν + 1))− j−1
(3.13)
(3.14)
Therefore (3.8) becomes: 1 2πi
χ e−it L χ u =
∞+iν
−∞+iν
χ (k)e−ikt u dk, R
where the previous integral is absolutely convergent in L(E mod ; E mod,−2 ). We first show part (i) of the theorem. Integrating along the path indicated in Fig. 5 we obtain by the Cauchy theorem: 1 2πi
X +iν
−X +iν
Let I j = enough,
e
−ikt
1 2πi
χ (k)u dk= R
m(λ j )
χ
e−iλ j t t k π j,k u+
λ j ∈Res P k=0 Im λ j >−µ
j
5
1
χ (λ)u dλ. e−itλ R 2πi j j=1
(3.15)
χ (λ)u dλ. Using (2.6), (3.1) and (3.13), we have, for t large e−itλ R
I3 E mod,−2
X +iν
χ (s)u mod,−2 ds e−ist R E
X −i lnX ν 1 − lnX
X 2
ets+C|s| ds uE mod
etν −2 X uE mod . t
(3.16)
We now take the limit X goes to +∞ in the L(E mod ; E mod,−2 ) sense in (3.15). The integrals I3 and I5 go to 0 thanks to (3.16) and, in the integrals I2 and I4 , the paths •
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J.-F. Bony, D. Häfner
are replaced by paths which extend • in a natural way and which go to ∞. We denote them again by • . We remark that
Bj e−ikt dk = 0, (k − i(ν + 1))− j−1
4 ∪1 ∪2
(3.17)
where the integral is absolutely convergent in L(E mod ; E mod,−2 ). On the other hand, we have the estimate, for t large enough, I1 E mod
I2 E mod
R
−R
χ (s − iµ)u mod ds e−µt R E
R
sC ds uE mod e−µt C+1 uE mod , (3.18) e−µt −R +∞ −i(R +s−i(µ+lns))t Rχ (R + s − i(µ + lns))u mod ds e E 0 ∞ e−µt e− lnst eC(lns+µ) ds uE mod e−µt uE mod , (3.19)
0
and a similar estimate holds for I4 . Since all these estimates hold in L(E mod ), (3.18) and (3.19) give the estimate of the rest (1.12) with M = (C + 1)/2. The estimate (1.13) follows from (1.10), Theorem 2.1 iii) and Proposition 3.1. Let us now show part (ii) of the theorem. We choose 0 > −µ > sup{Im λ; λ ∈ (Res P)\{0}} and the integration path as in part (i) of the theorem. We first suppose eε t > for some ε > 0 to be chosen later. Then the estimate for I1 can be replaced by
I1 E mod e((C+1)ε −µ)t uE mod .
Let us now suppose ≥ eε t . On the one hand we have the inequality: χ e−it L χ L(E mod ) 1,
since the norms on E mod and on E are uniformly equivalent for ≥ 1. On the other hand by the hypotheses on g we have
g(e2ε t ) . 1≤ g( ( + 1)) It follows: χ e
−it L
χ L(E mod )
g(e2ε t ) . g( ( + 1))
This concludes the proof of the theorem if we choose ε sufficiently small and put ε := min{2ε , µ − (C + 1)ε }.
Decay and Non-Decay on the De Sitter–Schwarzschild Metric
717
Proof of Remark 1.4 d). We note that for u ∈ D(P ), we have P u , u = r −1 Dx r 2 Dx r −1 + V ( + 1) u , u ≥ V ( + 1) u , u , √ V u 2 ≤ (P + 1)u 2 .
and then
(3.20) (3.21)
Estimate (1.12) can be written E 1 (t)E mod e−µt −ω M χ0 uE mod , with χ0 ∈ C0∞ (R) and χ0 χ = χ . Let χ j ∈ C0∞ (R), j = 1, . . . , 2M with χ j+1 χ j = χ j √ for j = 0, . . . , 2M − 1. Remark that there exists C > 0 such that V > 1/C on the support of χ2M . Using the radial decomposition u = u , we get −ω M χ0 uE mod sup 2M χ0 u E mod
sup 2M−1 (P + 1)χ0 u E mod = sup 2M−1 χ1 (P +1)χ0 u E mod
sup χ2M (P + 1)χ2M−1 (P + 1) · · · χ1 (P + 1)χ0 u E mod
(P + 1)2M uE mod .
(3.22)
Finally, for the interpolation argument, we use the fact that e−it L L(E mod ;E mod ) e−it L L(E ;E ) = 1,
for ≥ 1.
(3.23)
Proof of Remark 1.7. We only prove the first part since the proof of the second part is 2 3 analogous. As we are localized in space, we can use the Sobolev embedding H (R ) → u1 −it L χ . C 0 (R3 ). Using Pu 1 ≤ L u 2 mod , it is sufficient to consider (L + i)χ e E For this purpose, we write Lχ e−it L χ = (L + i)−1 [L , [L , χ ]]e−it L χ + (L + i)−1 [L , χ ]e−it L [L , χ ] +(L + i)−1 [L , χ ]e−it L χ (L + i) + χ e−it L [L , χ ] + χ e−it L χ L . (3.24) It is easy to check that the operators (L + i)−1 [L , [L , χ ]] and (L + i)−1 [L , χ ] can be extended to bounded operators on E mod . Note also that all commutators can be multiplied on the left and on the right by a cut-off function χ
∈ C0∞ (R) with χ
χ = χ without changing them. By Theorem 1.3, the left hand side of (3.24) is equal to r χ r, χ u 2 + L E 2 (t), Lγ 0
2 (t), in E mod,−1 . By the same theorem, the right hand side of (3.24) is equal to α + E where α is a constant in time and
2 (t)E mod g(eεt ) (g(−ω ))−1 (L + i)u mod . E E
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It follows α = Lγ
r χ r, χ u 2 , and thus the first part of the remark. 0
Acknowledgements. We would like to thank A. Bachelot for fruitful discussions during the preparation of this article. This work was partially supported by the ANR project JC0546063 “Equations hyperboliques dans les espaces temps de la relativité générale : diffusion et résonances”.
References 1. Alexandrova, I., Bony, J.-F., Ramond, T.: Semiclassical scattering amplitude at the maximum point of the potential. Asymptotic Analysis 58(1–2), 57–125 (2008) 2. Amrein, W., Boutet de Monvel, A., Georgescu, V.: C0 -groups, commutator methods and spectral theory of N -body Hamiltonians, Progress in Mathematics, Vol. 135, Basel-Boston: Birkhäuser Verlag, 1996 3. Bachelot, A., Motet-Bachelot, A.: Les résonances d’un trou noir de Schwarzschild. Ann. Inst. H. Poincaré Phys. Théor. 59(1), 3–68 (1993) 4. Blue, P., Soffer, A.: Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole. http://arXiv.org/list/math/0612168, 2006 5. Blue, P., Sterbenz, J.: Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space. Commun. Math. Phys. 268(2), 481–504 (2006) 6. Bony, J.-F., Michel, L.: Microlocalization of resonant states and estimates of the residue of the scattering amplitude. Commun. Math. Phys. 246(2), 375–402 (2004) 7. Burq, N.: Smoothing effect for Schrödinger boundary value problems. Duke Math. J. 123(2), 403–427 (2004) 8. Burq, N., Zworski, M.: Resonance expansions in semi-classical propagation. Commun. Math. Phys. 223(1), 1–12 (2001) 9. Chandrasekhar, S.: The mathematical theory of black holes. International Series of Monographs on Physics, Vol. 69, The Clarendon Press, Oxford: Oxford University Press, 1992 10. Christiansen, T., Zworski, M.: Resonance wave expansions: two hyperbolic examples. Commun. Math. Phys. 212(2), 323–336 (2000) 11. Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space. Princeton Mathematical Series, Vol. 41, Princeton, NJ: Princeton University Press, 1993 12. Dafermos, M., Rodnianski, I.: A proof of Price’s law for the collapse of a self-gravitating scalar field. Invent. Math. 162(2), 381–457 (2005) 13. Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: Decay of solutions of the wave equation in the Kerr geometry. Commun. Math. Phys. 264(2), 465–503 (2006) 14. Guillarmou, C.: Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds. Duke Math. J. 129(1), 1–37 (2005) 15. Ikawa, M.: Decay of solutions of the wave equation in the exterior of two convex obstacles. Osaka J. Math. 19(3), 459–509 (1982) 16. Lax, P., Phillips, R.: Scattering theory, Second ed. With appendices by C. Morawetz and G. Schmidt Pure and Applied Mathematics, Vol. 26, London-New York: Academic Press Inc., 1989 17. Martinez, A.: Resonance free domains for non globally analytic potentials. Ann. Henri Poincaré 3(4), 739–756 (2002) 18. Mazzeo, R., Melrose, R.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 75(2), 260–310 (1987) 19. Melrose, R., Sjöstrand, J.: Singularities of boundary value problems. I. Commun. Pure Appl. Math. 31(5), 593–617 (1978) 20. Mourre, E.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys. 78(3), 391–408 (1980) 21. Nakamura, S., Stefanov, P., Zworski, M.: Resonance expansions of propagators in the presence of potential barriers. J. Funct. Anal. 205(1), 180–205 (2003) 22. Ralston, J.: Solutions of the wave equation with localized energy. Commun. Pure Appl. Math. 22, 807–823 (1969) 23. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. New York: Academic Press, 1978 24. Reed, M., Simon, B.: Methods of modern mathematical physics. III. New York: Academic Press, 1979 25. Sá Barreto, A., Zworski, M.: Distribution of resonances for spherical black holes. Math. Res. Lett. 4(1), 103–121 (1997) 26. Sjöstrand, J.: Semiclassical resonances generated by nondegenerate critical points. In: Pseudodifferential operators (Oberwolfach, 1986), Lecture Notes in Math., Vol. 1256, Berlin-Heidelberg-New York: Springer, 1987, pp. 402–429
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27. Sjöstrand, J.: A trace formula and review of some estimates for resonances, Microlocal analysis and spectral theory (Lucca, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 490, Dordrecht: Kluwer Acad. Publ., 1997, pp. 377–437 28. Sjöstrand, J.: Lectures on resonances. Preprint available on http://www.math.polytechnique.fr/ ~sjoestrand, 2007, pp. 1–169 29. Sjöstrand, J., Zworski, M.: Complex scaling and the distribution of scattering poles. J. Amer. Math. Soc. 4(4), 729–769 (1991) 30. Tang, S.-H., Zworski, M.: From quasimodes to resonances. Math. Res. Lett. 5(3), 261–272 (1998) 31. Tang, S.-H., Zworski, M.: Resonance expansions of scattered waves. Comm. Pure Appl. Math. 53(10), 1305–1334 (2000) 32. Tang, S.-H., Zworski, M.: Potential scattering on the real line. Preprint available on http://math.berkeley. edu/~zworski/, 2007, pp. 1–46 33. Va˘ınberg, B.: Asymptotic methods in equations of mathematical physics, Gordon & Breach Science Publishers, 1989 34. Zworski, M.: Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces. Invent. Math. 136(2), 353–409 (1999) Communicated by G.W. Gibbons
Commun. Math. Phys. 282, 721–731 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0546-x
Communications in
Mathematical Physics
Multiple Bound States of Nonlinear Schrödinger Systems Zhaoli Liu1 , Zhi-Qiang Wang2 1 School of Mathematical Sciences, Capital Normal University, Beijing 100037, P.R. China 2 Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322, USA.
E-mail: [email protected] Received: 5 July 2007 / Accepted: 5 February 2008 Published online: 27 June 2008 – © Springer-Verlag 2008
Abstract: This paper is concerned with existence of bound states for Schrödinger systems which have appeared as several models from mathematical physics. We establish multiplicity results of bound states for both small and large interactions. This is done by different approaches depending upon the sizes of the interaction parameters in the systems. For small interactions we give a new approach to deal with multiple bound states. The novelty of our approach lies in establishing a certain type of invariant sets of the associated gradient flows. For large interactions we use a minimax procedure to distinguish solutions by analyzing their Morse indices. 1. Introduction In this paper we study standing wave solutions (1 , . . . , N ) : Rn → C N of the time-dependent system of N coupled nonlinear Schrödinger equations given by ⎧ N ⎪ ⎪ ⎨ − i ∂ = + βi j |i |2 j for x ∈ Rn , t > 0, j j ∂t (1) i=1 ⎪ ⎪ ⎩ j (x, t) → 0 as |x| → +∞, t > 0, j = 1, . . . , N , where βi j are constants satisfying βi j = β ji , n = 2, 3, N ≥ 2. A solitary wave of this system is a solution with j (x, t) = eiλ j t u j (x), j = 1, . . . , N . This ansatz leads to the elliptic system − u j + λ j u j =
N
βi j u i2 u j in Rn , u j (x) → 0 as |x| → ∞, j = 1, . . . , N .
(2)
i=1
Throughout the paper, λ j and β j j are positive constants for all j. The system (1) appears in many physical problems, especially in nonlinear optics. Physically, the solution j
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denotes the j th component of the beam in Kerr-like photorefractive media ([1]). The positive constant β j j is for self-focusing in the j th component of the beam. The coupling constant βi j (i = j) is the interaction between the i th and the j th components of the beam. When N = 2, problem (1) also arises in the Hartree-Fock theory for a double condensate, i. e., a binary mixture of Bose-Einstein condensates in two different hyperfine states |1 and |2 ([13]). Physically, j are the corresponding condensate amplitudes, β j j and β12 are the intraspecies and interspecies scattering lengths. The sign of the scattering length β12 determines whether the interactions of states |1 and |2 are repulsive or attractive. For more references we refer the reader to [1,10,13–17,23,27]. The goal of this paper is to establish multiplicity results of the above nonlinear Schrödinger system (2) for small interaction constants (both positive and negative) and for large positive interaction constants. The existence theory of solutions has received great interest recently, see [2,3,7,8,18,21,26] for the existence of a ground state or bound state solution, [19,20,22,24] for semiclassical states or singularly perturbed settings, and [11,28,29] for existence of multiple solutions of two coupled equations with negative interactions. Our paper is devoted to establishing multiplicity of nontrivial bound state solutions (see Theorems 2.1 and 3.1). This is done for the parameter regime of both small interaction constants (both positive and negative) and large positive interaction constants (i.e., βi j , i = j), a topic which has not been studied in the above works. The difficulty in obtaining multiplicity results of nontrivial bound state solutions lies in distinguishing nontrivial solutions (we will call a solution nontrivial if all components of the solution are non-zero) from semi-nontrivial solutions (i.e., solutions where at least one component is zero). We give existence results of nontrivial solutions both for small interaction parameters βi j , i = j for a general N system with N ≥ 2 and for large interactions for the case N = 2. In the case of small interactions we make use of invariant sets methods for gradient flows to distinguish nontrivial and semi-trivial solutions. For the case of large interactions, we use estimates of Morse indices of solutions to separate solutions. The reason for the interactions to be small or large may be justifiable in light of the nonexistence results of nontrivial positive solutions in [8] in which it is proved that there exist two positive numbers β < β such that the two-system (system (2) with N = 2) has nontrivial positive solutions if and only if β12 is less than β or greater than β . Thus it seems to be natural to require βi j to be small or large to obtain existence of nontrivial solutions. However, we do not know whether for some parameters the system (1) has no nontrivial solutions at all. The paper is organized as follows. Section 2 is devoted to the case of small interactions and Sect. 3 to the case of large interactions. We finish each section with a few concluding remarks. 2. Small Interaction Constants In this section we consider small interactions (both positive and negative) between the components. We assume N ≥ 2. Theorem 2.1. Let n = 2, 3 and let λ j and β j j be fixed positive constants. Then for any k ∈ N, there exists βk > 0 such that for |βi j | ≤ βk , i = j, the system N −u j + λ j u j = i=1 βi j u i2 u j , in Rn , (3) u j (x) → 0, as |x| → ∞, j = 1, 2, · · · , N has as least k pairs of nontrivial spherically symmetric solutions.
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Proof. Let E = Hr1 (Rn ) consist of spherically symmetric functions in H 1 (Rn ) in which we use the equivalent norms
1/2 2 2 u j = |∇u| + λ j u , j = 1, 2, · · · , N . Rn
N
The product space
EN
= E × E × · · · × E is endowed with the norm
N u = ( u j 2j )1/2 , u = (u 1 , u 2 , · · · , u N ) ∈ E N . j=1
Spherically symmetric solutions of (3) correspond to critical points of the functional J ( u) =
N 1 1 u 2 − βi j u i2 u 2j , u = (u 1 , u 2 , · · · , u N ) ∈ E N . n 2 4 R i, j=1
Note that J ∈ C 2 (E N ) and ∇ J ( u ) = u − A( u ), u = (u 1 , u 2 , · · · , u N ) ∈ E N , where A( u ) = ((A( u ))1 , (A( u ))2 , · · · , (A( u )) N ) and (A( u )) j = (− + λ j I )−1 (
N
βi j u i2 u j ).
i=1
Let ϕ t ( u ) with the maximal interval of existence [0, η( u )) be the solution of the initial value problem d t t dt ϕ = −∇ J (ϕ ), for t ≥ 0, 0 ϕ = u . We say ϕ : {(t, u )| u ∈ E N , t ∈ [0, η( u ))} → E N is the gradient flow of J . A subset N F of E is said to be an invariant set for the flow if ϕ(t, u ) ∈ F for all u ∈ F and t ∈ [0, η( u )). For two invariant sets F ⊂ G, we say F is strictly invariant with respect to G if ϕ(t, u ) ∈ int G F for all u ∈ F and t ∈ (0, η( u )), where intG F is the interior of F in G. Define A0 = { u ∈ EN |
lim
t→η( u )−0
ϕ t ( u ) = 0}.
Since 0 is a strict local minimizer of J , A0 is an open subset of E N . We claim that ∂A0 is an invariant set. Otherwise, there would be u 0 ∈ ∂A0 and t0 > 0 such that ϕ t0 ( u 0 ) ∈ ∂A0 . Then either ϕ t0 ( u 0 ) ∈ A0 or ϕ t0 ( u 0 ) ∈ E N \ A0 . If ϕ t0 ( u 0 ) ∈ A0 then t limt→η( u 0 )−0 ϕ ( u 0 ) = 0 and u 0 ∈ A0 , a contradiction. If ϕ t0 ( u 0 ) ∈ E N \ A0 , then the continuous dependence of ϕ on initial data would imply that ϕ t0 ( u 1 ) ∈ E N \ A0 for any u 1 ∈ A0 sufficiently close to u 0 , which is also a contradiction. Of course, A0 depends on βi j , i = j. Assume |βi j | ≤ β ∗ (i = j) and β ∗ is small enough such that b1 I ≤ (βi j ) ≤ b2 I , where b1 = 21 min{β11 , · · · , β N N }, b2 = 2 max{β11 , · · · , β N N }, and I is the N × N identity matrix. Then a careful inspection
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shows that there exist r > 0 and α > 0 independent of βi j (i = j) such that Br (0) ⊂ A0 and inf ∂ A0 J ≥ α, and for any finitely dimensional subspace F of E N , there exists R > 0 independent of βi j (i = j) such that A0 ∩ F ⊂ { u | u ≤ R}. Now fix k ∈ N and let E 1 be a k-dimensional subspace of E. Define ck = sup J˜, E 1N
where J˜ is the functional J in which (βi j ) is replaced with b1 I . Clearly, ck < ∞, ck is independent of βi j (i = j), and for |βi j | ≤ β ∗ (i = j), sup J ≤ ck . E 1N
For any u ∈ E N , if J ( u ) ≤ ck and ∇ J ( u ) ≤ 1 then u ≥ 4J ( u ) − (∇ J ( u ), u ) ≥ u 2 , 4ck + which implies u < 2 + ck . Therefore, ∇ J ( u ) > 1,
u ≥ 2 + ck . if J ( u ) ≤ ck and
(4)
We show that u ) ≥ 2 + ck , ϕ t (
if 0 < J ( u ) ≤ ck and u ≥ 2 + 2ck and J (ϕ t ( u )) ≥ 0. (5)
If not then there would be u and t0 > 0 such that 0 < J ( u ) ≤ ck , u ≥ 2 + 2ck , and J (ϕ t0 ( u )) ≥ 0, but ϕ t0 ( u ) < 2 + ck . Let t1 ∈ (0, t0 ) be such that ϕ t1 ( u ) = 2 + ck and ϕ t ( u ) > 2 + ck for t ∈ (0, t1 ). Then a contradiction can be obtained as u ) ≥ u − u − ϕ t1 ( u ) 2 + ck = ϕ t1 ( t1 ∇ J (ϕ t ( u )) dt ≥ 2 + 2ck − 0 t1 ≥ 2 + 2ck − ∇ J (ϕ t ( u )) 2 dt 0
u ) − J (ϕ t1 ( u ))) = 2 + 2ck − (J ( t0 > 2 + 2ck − (J ( u ) − J (ϕ ( u ))) ≥ 2 + ck . u ≥ 2 + 2ck Using (4) and (5), a standard argument shows that if 0 < J ( u ) ≤ ck and then there exists t > 0 such that J (ϕ t ( u )) ≤ 0. Therefore, since ∂A0 is an invariant set and inf ∂ A0 J ≥ α > 0, if u ∈ ∂A0 and J ( u ) ≤ ck then u < 2 + 2ck . Or, in other word, ∂A0 ∩ J ck ⊂ int(B2+2ck (0)),
(6)
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where J ck = { u | u ∈ E N , J ( u ) ≤ ck }. For u ≤ 4 + 4ck , we have (A( u )) j j = C1 (− + λ j I )
−1
(
N
βi j u i2 u j ) W 2,2 (Rn )
i=1
≤ C2
N
βi j u i2 u j L 2 (Rn )
i=1
≤ C3
N
|βi j | u i i2 u j j
i=1
≤ C3 β j j u j 3j + 16C3 max{|βi j | : i = j}(1 + ck )2 u j j , 1 where Ci are constants independent of u , k, and βi j . Let βk = min{β ∗ , 64C (1+c 2 } and 3 k) √ u ≤ 4 + 4ck fix βi j with |βi j | ≤ βk for i = j. There exists εk ∈ (0, r/ N ) such that if and 0 < u j j ≤ εk then
(A( u )) j j < u j j . Define, for ε > 0 and j = 1, 2, · · · , N , Dεj = { u | u = (u 1 , u 2 , · · · , u N ) ∈ E N , u j j ≤ ε}. Then for 0 < ε ≤ εk , A(Dεj ∩ B4+4ck (0)) ⊂ int(Dεj ),
j = 1, 2, · · · , N .
According to [12, §4], for any u ∈ Dεj ∩ B3+3ck (0) and any 0 < ε ≤ εk there exists t0 > 0 such that ϕ t ( u ) ∈ int(Dεj ) ∩ B4+4ck (0) for t ∈ (0, t0 ). This observation together with (6) implies that ϕ t ( u ) ∈ ∂A0 ∩ J ck ∩ int(Dεj ) if u ∈ ∂A0 ∩ J ck ∩ Dεj and t ∈ (0, η( u )) ε c k for any j = 1, 2, · · · , N , 0 < ε ≤ εk . Therefore, ∂A0 ∩ J ∩ D j , j = 1, 2, · · · , N , 0 < ε ≤ εk , are strictly invariant sets with respect to ∂A0 ∩ J ck . Define u ∈ ∂A0 ∩ J ck | ∃ t > 0 such that ϕ t ( u ) ∈ ∪ Nj=1 int(Dεj k )}. A1 = { Then A1 is an open subset of ∂A0 ∩ J ck , and ∂A0 ∩ J ck \ A1 is closed and invariant for the flow. We want to prove gen(∂A0 ∩ J ck \ A1 ) ≥ k, where gen(·) is the genus of a closed symmetric subset of E N . Note that E 1N ∩ ∂A0 \ A1 ⊂ J ck ∩ ∂A0 \ A1 . For u ∈ A1 , since ∂A0 ∩ J ck ∩ Dεj , j = 1, 2, · · · , N , 0 < ε ≤ εk , are strictly invariant ε /2
sets with respect to ∂A0 ∩ J ck , there exists t > 0 such that ϕ t ( u ) ∈ ∪ Nj=1 D j k + function τ : A1 → R defined by ε /2
τ ( u ) = inf{t ≥ 0 : ϕ t ( u ) ∈ ∪ Nj=1 D j k } is even and continuous. Define a map h : E 1N ∩ A1 → E 1N as h( u ) = (γ1 ( u )u 1 , γ2 ( u )u 2 , · · · , γ N ( u )u N ),
and the
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where γi ( u) =
⎧ ⎪ ⎨ 1, ⎪ ⎩
τ ( u)
τ ( u) 2 ( u ) i εk ϕi
if |ϕi ( u ) i ≥ εk , τ ( u) − 1, if εk /2 ≤ ϕi ( u ) i < εk , τ ( u)
if ϕi
0,
( u ) i < εk /2,
and ϕit is the i th component of ϕ t . Then h : E 1N ∩ A1 → E 1N is odd and continuous. For u ) is zero. any u ∈ E 1N ∩ A1 , the definition of h implies that at least one component of h( √ On the other hand, since Br (0) ⊂ A0 and εk ∈ (0, r/ N ) and ϕ τ ( u ) ( u ) ∈ A1 ⊂ ∂A0 , at least one component of h( u ) is nonzero. Define W = { u ∈ E 1N : u 1 = u 2 = · · · = u N } and let V be the orthogonal complement of W in E 1N . Thus any u ∈ E 1N can be uniquely u ) = v . Since decomposed as u = w + v , w ∈ W, v ∈ V . Define g : E 1N → V as g( N W ∩ h(E 1 ∩ A1 ) = ∅, g ◦ h : E 1N ∩ A1 → V \ {0} is odd and continuous. Therefore gen(E 1N ∩ A1 ) ≤ dim V = (N − 1)k, which implies gen(J ck ∩ ∂A0 \ A1 ) ≥ gen(E 1N ∩ ∂A0 ) − gen(E 1N ∩ A1 ) ≥ N k − (N − 1)k = k. Define for i = 1, 2, · · · , k, u ), di = inf sup J ( A∈ i u ∈A
where i = {A | A ⊂ J ck ∩ ∂A0 \ A1 , gen(A) ≥ i}. Now standard arguments (see, for example, [25]) can be used to obtain the conclusion. Remark 2.2. a) The result holds for general potential V j (x) in place of λ j for j = 1, . . . , N . We only need inf Rn V j > 0 and some sort of compactness condition such that 1 n the embedding from E j := {u ∈ H (R )| V j u 2 < ∞} into L q (Rn ) is compact for all q ∈ [2, 2∗ ). This is assured for example by assuming V j (x) → ∞ as |x| → ∞. A more general one (e.g., [6]) is that there is r > 0 such that for each b > 0, lim|y|→∞ µ({x ∈ Rn |V j (x) ≤ b} ∩ Br (y)) = 0, where µ(·) is the Lebesgue measure in Rn . b) Our methods allow extensions of the results to more general nonlinearities. We consider, for example, the system ∂F −u j + λ j u j = ∂u (u 1 , u 2 , · · · , u N ), in Rn , j (7) u j (x) → 0, as |x| → ∞, j = 1, 2, · · · , N , where n = 2, 3, λ j > 0, and F(u) =
mi j N i, j=1 l=1
βi jl |u i |ri jl |u j |si jl
Multiple Bound States of Nonlinear Schrödinger Systems
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with m i j ∈ N, ri jl ≥ 2, si jl ≥ 2, and ri jl + si jl < 6 in case n = 3. We define S = {(i, j, l) | l = 1, 2, · · · , m i j and i, j = 1, 2, · · · , N } and T = {(i, j, l) ∈ S | i = j} ∪ {(i, j, l) ∈ S | ri jl > 2, si jl > 2}. The proof of Theorem 2.1 can be adapted to prove the following result. Assume that (ri jl +si jl ) is a constant independent of (i, j, l). Let βi jl > 0 be fixed for all (i, j, l) ∈ T . Then for any k ∈ N, there exists βk > 0 such that if |βi jl | ≤ βk for all (i, j, l) ∈ S \ T , (7) has as least k pairs of nontrivial spherically symmetric solutions. Similar results hold for a system in which the potential takes the form in a) and the nonlinearity takes the form above. c) We do not know how small βk ’s are, but it seems βk is non-increasing. 3. Large Interaction Constants In this section we consider large interaction parameters. We will deal with only the case of two coupled equations, i.e., N = 2. Rewriting µi = βii and β = β12 we have a system ⎧ ⎨ −u + λ1 u = µ1 u 3 + βv 2 u, in Rn , (8) −v + λ2 v = µ2 v 3 + βu 2 v, in Rn , ⎩ u(x) → 0, v(x) → 0, as |x| → ∞. Theorem 3.1. Let n = 2, 3 and for i = 1, 2 let λi and µi be fixed positive constants. Then for any k ∈ N, there exists β k > 0 such that for β > β k the system (8) has at least k pairs of nontrivial spherically symmetric solutions. For the proof of this theorem we need some preliminaries and preparation. First let us recall that solutions of (8) correspond to critical points of the C 2 functional on E 2 with E = Hr1 (Rn ) 1 1 J ((u, v)) = (|∇u|2 + |∇v|2 ) + (λ1 u 2 + λ2 v 2 ) 2 Rn 2 Rn (9) 1 1 β 4 4 2 2 − µ1 u − µ2 v − u v . 4 Rn 4 Rn 2 Rn Proposition 3.2. For any β > 0, (8) has a sequence of solutions (u k , vk ) such that J ((u k , vk )) → ∞ as k → ∞ and the Morse index of (u k , vk ) is no more than k. Proof. The existence of a sequence of solutions follows from standard methods. By the symmetric mountain pass theorem or by a linking argument ([4]) we get the existence of a sequence of critical points of J having critical values tending to infinity. This sequence of solutions can also be constructed as critical points of a constraint problem, namely, critical points of I ((u, v)) = |∇u|2 + λ1 u 2 + |∇v|2 + λ2 v 2 Rn
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subject to M = {(u, v) ∈ E 2 | (u, v) = (0, 0), (∇ J ((u, v)), (u, v)) = 0}. Let (ck ) be the sequence of critical values of I , which tends to infinity, characterized as ck = inf
sup I ((u, v)),
A∈ k (u,v)∈A
where k = {A ⊂ M | A = −A, closed, gen(A) ≥ k }. By the arguments in Chang’s book [9], Chap. 2, Sect. 2.2, if K ck = {(u, v) ∈ M | I (u, v) = ck , ∇ I (u, v) = 0} is isolated then there exists a critical point (u k , vk ) such that its Morse index is no more than k − 1. +1 In general we use a perturbation argument. Let A := K cckk−1 = {(u, v) ∈ M | ck − 1 ≤ I (u, v) ≤ ck + 1, ∇ I (u, v) = 0} then A is compact. If the Morse indices of all critical points of A are greater than k, for each (u, v) ∈ A there is a > 0 such that the linearized eigenvalue problem at (u, v) has at least k + 1 eigenvalues less than or equal to −a. Due to compactness of A, there exist δ > 0 and a0 > 0 such that for all (u, v) ∈ Aδ , the δ-neighborhood of A, the linearized eigenvalue problem at (u, v) has at least k + 1 eigenvalues less than or equal to −a0 . Choose ε > 0 such that ε < 14 max{a0 , 1}. Fix these δ > 0 and ε > 0. Then we perturb I to an even functional I˜ which agrees with I outside Aδ , I − I˜ C 2 ≤ ε and I˜ has only non-degenerate critical points in Aδ . For this perturbed functional we apply the minimax procedure again to define c˜k . Then ck − ε ≤ c˜k ≤ ck + ε by the property of I˜. Thus we may assume for I˜, K c˜k is isolated. Then there exists a critical point of I˜ at this level whose Morse index is no more than k. This is a contradiction since this critical point of I˜ is in Aδ and its Morse index has to be at least k + 1. Proposition 3.3. Consider − U = c0 |U | p−1 U, U ∈ Cb2 (Rn ),
(10)
where 1 < p < 2∗ − 1, c0 > 0 and Cb2 (Rn ) consists of bounded C 2 functions. Then any radial solution of (10) with finite Morse index (with respect to the class of radial functions) must be zero. I.e., if i(U ) < ∞, then U = 0, where i(U ) is the number of negative eigenvalues of − − c0 p|U | p−1 in Hr2 (Rn ). Proof. This was proved in [5] for the equation considered without radial symmetry. Checking the proofs there one sees that all the arguments are still valid when confined in the class of radially symmetric functions. Proposition 3.4. Consider − u + λ1 u = µ1 u 3 , and define for each integer k, Sk1 := {u ∈ E | u is a solution of (11) with Morse index no more than k}. Then for any k, Sk1 is bounded in L ∞ (Rn ).
(11)
Multiple Bound States of Nonlinear Schrödinger Systems
729
There is a similar statement by defining Sk2 for the second equation, −u + λ2 u = µ2 u 3 . Proof. First, for any u ∈ Sk1 , |u(0)| is a global maximum of |u(x)|. In fact, if s ≥ 0 is a zero of u then multiplying the equation −u − (n√− 1)u /r + λ1 u = µ1 u 3 with u and taking the integral from s to ∞ yields |u(s)| > 2λ1 /µ1 . Let s2 > s1 ≥ 0 be two local maximizers of |u|. By multiplying the equation with u and taking the integral from s1 to s2 we√have (u 2 (s2 ) − u 2 (s1 ))(2λ1 − µ1 (u 2 (s2 ) + u 2 (s1 ))) > 0 which together with |u(si )| > 2λ1 /µ1 implies |u(s1 )| > |u(s2 )|. We prove the proposition by contradiction. Let us assume there is a sequence u m ∈ Sk1 such that Mm := |u m (0)| = maxRn |u m | → ∞ as m → ∞. Then we define wm (x) = Mm−1 u m (Mm−1 x), which satisfies the following equation 3 −wm + Mm−2 λ1 wm = µ1 wm . 2 (Rn ) as m → ∞ and By elliptic estimates, for a subsequence we have wm → U in Cloc U is a non-zero solution in Cb2 (Rn ) of (10) in the case p = 3. Since i(u m ) ≤ k we have i(U ) ≤ k. Thus U = 0, which is a contradiction with |U (0)| = 1.
Let i be fixed. For w ∈ Si1 and for l = 1, 2, . . ., let νl (w) denote the l th eigenvalue of the following weighted eigenvalue problem in E: − ϕ + λ2 ϕ = νl (w)w 2 ϕ.
(12)
1 > 0 such that ν (w) ≤ C 1 for all Proposition 3.5. For each i and l, there exists Ci,l l i,l w ∈ Si1 .
Again a similar statement can be proved for the eigenvalue problem −ϕ + λ1 2 can be obtained such that ϕ = νl (w)w 2 ϕ, where w ∈ Si2 , and, accordingly, a bound Ci,l 2 νl (w) ≤ Ci,l . √ Proof. Note that |w(0)| > 2λ1 /µ1 for all w ∈ Si1 . From Proposition 3.4, the elliptic estimate shows that Sk1 is bounded in C 1 (Rn ). Therefore, there exist r > 0 and a0 > 0 such that |w(x)| ≥ a0 for x ∈ Br (0) and all w ∈ Si1 . Now since νl (w) can be characterized by |∇ϕ|2 + λ2 ϕ 2 Rn inf sup , νl (w) = 2 2 F⊂E, dim F=l ϕ∈F, ϕ=0 Rn w ϕ we see νl (w) ≤
Ci,1 l
1 := 2 a0
inf
1 (B (0)), dim F=l F⊂H0, r r
Finally we complete the proof of Theorem 3.1.
sup
ϕ∈F, ϕ=0
Rn
|∇ϕ|2 + λ2 ϕ 2 . 2 Rn ϕ
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Proof of Theorem 3.1. Let k be fixed. We show that there is a β k > 0 such that for β > β k , (u i , vi ) for i = 1, . . . , k in Proposition 3.2 are all nontrivial, i.e., all components of (u i , vi ) are nonzero for i = 1, 2, . . . k. We need to rule out the possibility that (u i , vi ) is of the form (w1 , 0) or (0, w2 ) with w j ( j = 1, 2) being a nonzero solution of −w j + λ j w j = µ j w 3j . We will analyze the Morse indices of these semi-trivial solutions and compare them with the Morse index of (u i , vi ) which is no more than k for i = 1, 2, . . . , k. First, if w1 is a nonzero solution to −u + λ1 u = µ1 u 3 having Morse index m such that m > k, it is easy to see that (w1 , 0) as a solution of the system (8) has Morse index at least m for functional J and this is independent of the size of β. Next we claim that there is β k such that for β > β k , any critical point of J of the form (w1 , 0) or (0, w2 ) with w1 ∈ Sk1 and w2 ∈ Sk2 have Morse index greater than k. This follows from Proposition 3.5. In fact, when we linearize the system (8) at a solution of the form (w1 , 0) with w1 ∈ Sk1 we have ϕ1 ∈ E, ϕ1 − λ1 ϕ1 + 3µ1 w12 ϕ1 = 0, ϕ2 − λ2 ϕ2 + βw12 ϕ2 = 0, ϕ2 ∈ E. The first equation has an eigenvector corresponding to a negative eigenvalue. 1 the Morse index of (w , 0) as According to Proposition 3.5 we see that for β ≥ Ck,k 1 2 the Morse index of a solution of the system (8) is at least k + 1. Similarly, for β ≥ Ck, k 1 , C 2 }, (0, w2 ) as a solution of the system (8) is at least k + 1. If we set β k = max{Ck, k k, k k then for β ≥ β , the critical point (u i , vi ) for i = 1, 2, . . . , k provided by Proposition 3.2 must be nontrivial since we know the Morse index of (u i , vi ) is no more than k for i = 1, 2, . . . , k. Remark 3.6. a) We do not know how large β k ’s are. b) The solutions given by our results likely are not one-sign solutions. But we do not have a more precise nodal property of the solutions. Acknowledgement. The authors would like to thank the referee for suggestions of improving the writing of the paper, in particular the extensions in Remark 2.2. The first author is supported by NSFC(10571123), PHR(IHLB), and NSFB(KZ200610028015).
References 1. Akhmediev, N., Ankiewicz, A.: Partially coherent solitons on a finite background. Phys. Rev. Lett. 82, 2661–2664 (1999) 2. Ambrosetti, A., Colorado, E.: Bound and ground states of coupled nonlinear Schrödinger equations. C. R. Math. Acad. Sci. Paris 342, 453–458 (2006) 3. Ambrosetti, A., Colorado, E.: Standing waves of some coupled nonlinear Schrödinger equations. J. London Math. Soc. 75, 67–82 (2007) 4. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973) 5. Bahri, A., Lions, P.L.: Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math. 45, 1205–1215 (1992) 6. Bartsch, T., Pankov, A., Wang, Z.-Q.: Nonlinear Schrödinger equations with steep potential well. Comm. Contem. Math. 3, 1–21 (2001) 7. Bartsch, T., Wang, Z.-Q.: Note on ground states of nonlinear Schrödinger systems. J. Partial Differ. Eqs. 19, 200–207 (2006) 8. Bartsch, T., Wang, Z.-Q., Wei, J.C.: Bound states for a coupled Schrödinger system. J. Fixed Point Th. Appl. 2, 353–367 (2007)
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9. Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and Their Applications, 6, Boston: Birkhäuser (1993) 10. Christodoulides, D.N., Coskun, T.H., Mitchell, M., Segev, M.: Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett. 78, 646–649 (1997) 11. Dancer, E.N., Wei, J.C., Weth, T.: A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Preprint 12. Deimling, K.: Ordinary Differential Equations in Banach Spaces, Lecture Notes in Mathematics, vol. 596. Berlin: Springer-Verlag, 1977 13. Esry, B.D., Greene, C.H., Burke, J.P. Jr.., Bohn, J.L.: Hartree-Fock theory for double condensates. Phys. Rev. Lett. 78, 3594–3597 (1997) 14. Genkin, G.M.: Modification of superfluidity in a resonantly strongly driven Bose-Einstein condensate. Phys. Rev. A, 65 (2002), No.035604 15. Hioe, F.T.: Solitary waves for N coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 82, 1152– 1155 (1999) 16. Hioe, F.T., Salter, T.S.: Special set and solutions of coupled nonlinear Schrödinger equations. J. Phys. A: Math. Gen. 35, 8913–8928 (2002) 17. Kanna, T., Lakshmanan, M.: Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. Phys. Rev. Lett., 86, 5043(1–4) (2001) 18. Lin, T.-C., Wei, J.C.: Ground state of N coupled nonlinear Schrödinger equations in R n , n ≤ 3. Commun. Math. Phys. 255, 629–653 (2005) 19. Lin, T.-C., Wei, J.C.: Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 403–439 (2005) 20. Lin, T.-C., Wei, J.C.: Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials. J. Differ. Eq. 299(2), 538–569 (2006) 21. Maia, L.A., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Eq. 229, 743–767 (2006) 22. Montefusco, E., Pellacci, B., Squassina, M.: Semiclassical states for weakly coupled nonlinear Schrödinger systems. Preprint 23. Mitchell, M., Chen, Z., Shih, M., Segev, M.: Self-Trapping of partially spatially incoherent light. Phys. Rev. Lett. 77, 490–493 (1996) 24. Pomponio, A.: Coupled nonlinear Schrödinger systems with potentials. J. Differ. Eq. 227, 258–281 (2006) 25. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math. 65, Providence, R.I.: Amer. Math. Soc., 1986 26. Sirakov, B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in Rn . Commun. Math. Phys. 271, 199–221 (2007) 27. Timmermans, E.: Phase separation of Bose-Einstein condensates. Phys. Rev. Lett. 81, 5718–5721 (1998) 28. Wei, J.C., Weth, T.: Nonradial symmetric bound states for a system of two coupled Schrödinger equations. Rend. Lincei Mat. Appl. 18, 279–294 (2007) 29. Wei, J.C., Weth, T.: Radial solutions and phase seperation in a system of two coupled Schrödinger equations. Arch. Rational Mech. Anal., On-line First, doi:10:1007/s00205-008-0121-9, 2008 Communicated by I.M. Sigal
Commun. Math. Phys. 282, 733–796 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0505-6
Communications in
Mathematical Physics
Unstable Surface Waves in Running Water Vera Mikyoung Hur1 , Zhiwu Lin2 1 Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue,
Cambridge, MA 02139-4307, USA. E-mail: [email protected]
2 University of Missouri-Columbia, Department of Mathematics, Columbia, MO 65203-4100, USA.
E-mail: [email protected] Received: 16 July 2007 / Accepted: 18 October 2007 Published online: 15 May 2008 – © Springer-Verlag 2008
Abstract: We consider the stability of periodic gravity free-surface water waves traveling downstream at a constant speed over a shear flow of finite depth. In case the free surface is flat, a sharp criterion of linear instability is established for a general class of shear flows with inflection points and the maximal unstable wave number is found. Comparison to the rigid-wall setting testifies that the free surface has a destabilizing effect. For a class of unstable shear flows, the bifurcation of nontrivial periodic traveling waves is demonstrated at all wave numbers. We show the linear instability of small nontrivial waves that appear after bifurcation at an unstable wave number of the background shear flow. The proof uses a new formulation of the linearized water-wave problem and a perturbation argument. An example of the background shear flow of unstable smallamplitude periodic traveling waves is constructed for an arbitrary vorticity strength and for an arbitrary depth, illustrating that vorticity has a subtle influence on the stability of free-surface water waves. 1. Introduction The water-wave problem in its simplest form concerns two-dimensional motion of an incompressible inviscid liquid with a free surface, acted on only by gravity. Suppose, for definiteness, that in the (x, y)-Cartesian coordinates gravity acts in the negative y-direction and that the liquid at time t occupies the region bounded from above by the free surface y = η(t; x) and from below by the flat bottom y = 0. In the fluid region {(x, y) : 0 < y < η(t; x)}, the velocity field (u(t; x, y), v(t; x, y)) satisfies the incompressibility condition and the Euler equation
∂x u + ∂ y v = 0
(1.1)
∂t u + u∂x u + v∂ y u = −∂x P ∂t v + u∂x v + v∂ y v = −∂ y P − g,
(1.2)
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where P(t; x, y) is the pressure and g > 0 denotes the gravitational constant of acceleration. The flow is allowed to have rotational motions and characterized by the vorticity ω = vx − u y . The kinematic and dynamic boundary conditions at the free surface {y = η(t; x)} v = ∂t η + u∂x η and P = Patm
(1.3)
express, respectively, that the boundary moves with the velocity of the fluid particles at the boundary and that the pressure at the surface equals the constant atmospheric pressure Patm . The impermeability condition at the flat bottom states that v=0
at {y = 0}.
(1.4)
It is a matter of common experience that waves which may be observed on the surface of the sea or on the river are approximately periodic and propagating of permanent form at a constant speed. In case ω ≡ 0, namely in the irrotational setting, waves of this kind are referred to as Stokes waves, whose mathematical treatment was initiated by formal but far-reaching considerations of Stokes [53] himself. The existence theory of Stokes waves dates back to the construction due to Levi-Civita [36] and Nekrasov [50] in the infinite-depth case and due to Struik [54] in the finite-depth case of small-amplitude waves, and it includes the global theory due to Krasovskii [35] and Keady and Norbury [34]. Stokes waves of greatest height exist [58,46] and are shown to have stagnation at wave crests [5]. A nice survey on the existence of Stokes waves include [59] and [14, Chaps. 10, 11]. In the finite-depth case, it is recently shown by Constantin [16] that there are no closed paths in the Stokes waves and each particle experiences a slight forward drift. While the irrotational assumption may serve as an approximation under certain circumstances and has been used in a majority of the existing research, surface water-waves typically carry vorticity, e.g. shear currents on a shallow channel and wind-drift boundary layers. Moreover, the governing equations for water waves allow for rotational steady motions. Gerstner [27] early in 1802 found an explicit formula for a family of periodic traveling waves on deep waters with a particular nonzero vorticity. An extensive existence theory of periodic traveling water waves with vorticity appeared in the construction due to Dubreil-Jacotin [25] of small-amplitude waves. Recently, for a general class of vorticity distributions, Constantin and Strauss [18] in the finite-depth case and Hur [32] in the infinite-depth case accomplished the bifurcation analysis for periodic traveling waves of large amplitude. Partial results on the location of possible stagnation are found in [20,60]. Waves of Stokes’ kind is one of the few exact solutions of the free-surface water-wave problem, and as such it is important to understand the stability of these solutions. In this paper, we investigate the linear instability of periodic gravity water-waves with vorticity. The stability of water waves in case of zero vorticity has been under research much by means of numerical computations and formal analysis, especially in the works of Longuet-Higgins and his coworkers. Numerical studies of stability of Stokes waves under perturbations of the same period, namely the superharmonic perturbations, indicate that [48,55] instability sets in only when the wave amplitude is large enough to link with wave breaking ([49]) and small-amplitude Stokes waves are found to be linearly stable under the same-period perturbations. The instability of large Stokes waves and solitary waves is recently proved by Lin ([44,43]), under the assumption of no secondary bifurcation which is confirmed numerically. MacKay and Saffman [45] considered linear stability of small-amplitude Stokes waves by the general results of the Hamiltonian system. The
Unstable Surface Waves in Running Water
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Hamiltonian formulation in terms of the velocity potential, however, does not avail in the presence of vorticity except when the vorticity is constant [61]. The analysis of Benjamin and Feir [11] showed that there is a “sideband” instability for small Stokes waves, meaning that the perturbation has a different period than the steady wave. The Benjamin-Feir instability was made mathematically rigorous by Bridges and Mielke [12]. Analytical works on the stability of water waves with vorticity, on the other hand, are quite sparse. A recent contribution due to Constantin and Strauss [19] concerns two different kinds of formal stability of periodic traveling water-waves with vorticity under perturbations of the same period. First, in the case when the vorticity decreases with depth an energy-Casimir functional H is constructed as a temporal invariant of the nonlinear water-wave problem, whose first variation gives the exact equations for steady waves [17]. In [19], the second variation of H is shown to be positive for some special perturbations and the water-wave system is called H-formally stable under these perturbations. The use of such an energy-Casimir functional in studying stability of ideal fluids is pioneered by Arnold [3] for the fixed boundary case. The second approach of [19] uses another functional J , which is essentially the dual of H in the transformed variables but not an invariant. Its first variation gives the exact equations for steady waves in the transformed variables ([17]), which served as the basis in [18] for the existence theory of traveling waves. The J -formal stability then means the positivity of its second variation. The “exchange of stability” theorem due to Crandall and Rabinowitz [21] applies to conclude that the J -formal stability of the trivial solutions switches exactly at the bifurcation point and that steady waves along the curve of local bifurcation are J -formally stable provided that both the depth and the vorticity strength are sufficiently small. The main results. As a preliminary step toward the stability and instability of nontrivial periodic waves, we examine the linear stability and instability of flat-surface shear flows. The linear stability of shear flows in the rigid-wall setting is a classical problem, whose theories date back to the necessary condition for linear instability due to Rayleigh [52]. We refer to [23,24,26,38,39] and references therein for historic and recent results on this problem. In [39], Lin obtained linear instability criteria for several classes of shear flows in a channel with rigid walls, and in this paper we generalize these to the free-surface setting. More specifically, our conclusions include: (1) The linear stability of shear flows with no inflection points (Theorem 6.4), which generalizes Rayleigh’s criterion in the rigid-wall setting [52] to the free-surface setting; (2) A sharp criterion of linear instability for a class of shear flows with one inflection value (Theorem 4.2); and (3) A sufficient condition of linear instability for a class of shear flows with multiple inflection values (Theorem 6.1) including any monotone flows. Our result testifies that free surface has a destabilizing effect compared to rigid walls. Our next step is to understand the local bifurcation of small-amplitude periodic traveling waves in the physical space. While our setting is similar to [19] in that it hinges on the existence results of periodic waves in [18] via the local bifurcation, the choice of the bifurcation parameter and the dependence of other parameters on the bifurcation parameter and free parameters in the description of the background shear flow are different. In our setting, it is natural to consider that the shear profile and the channel depth are given and that the speed of wave propagation is chosen to ensure the local bifurcation. The relative flux and the vorticity-stream function relation are then computed. In contrast, in the bifurcation analysis [18] in the transformed variables, the wave speed, as well as the relative flux and the vorticity-stream function relation are held fixed. In turn, the shear
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profile and the channel depth vary along the bifurcation curve. Lemma 2.3 establishes the equivalence between the bifurcation equation (2.7) (equivalently [18, (3.8)]) in the transformed variables and the Rayleigh system (2.9)–(2.10) to obtain the bifurcation results for a large class of shear flows. In addition, our result helps to clarify the nature of the local bifurcation of periodic traveling water-waves that it does not necessarily involve the exchange of stability of trivial solutions (Remark 4.14). Our third step is to show under some technical assumptions that the linear instability of the background shear flow persists along the local curve of bifurcation of small-amplitude periodic traveling waves (Theorem 5.1). An example of such an unstable shear flow is U (y) = a sin b(y − h/2)
for y ∈ [0, h],
where h, b > 0 satisfy hb π and a > 0 is arbitrary (Remark 5.2). In particular, by choosing a and h to be arbitrarily small, we can construct linearly unstable small periodic traveling water-waves with an arbitrarily small vorticity strength and an arbitrarily small channel depth. This indicates that the formal stability of the second kind in [19] (see discussions above) is quite different from the linear stability of the physical water wave problem. Our example also shows that adding an arbitrarily small vorticity to the water-wave system may affect the superharmonic stability of small-amplitude periodic irrotational waves in a water of arbitrary depth. Thus, it is important to take into account the effects of vorticity in the study of the stability of water waves. Temporal invariants of the linearized water-wave problem are derived and their implications for the stability of the water-wave system are discussed (Sect. 3.3). In the case when the vorticity-stream function relation is monotone, the energy functional ∂ 2 H in [19] is indeed an invariant of the linearized water-wave problem. Other invariants are also derived. However, even with these additional invariants as constraints, the quadratic form ∂ 2 H is in general indefinite, indicating that a steady (pure gravity) water-wave may be an energy saddle. A similar observation was made by Bona and Sachs [8] in the irrotational case. Therefore, a successful proof of the stability for the full water-wave problem would require to use the full equations instead of just a few invariants. Ideas of the proofs. Our approach in the proof of the linear instability of free-surface shear flows uses the Rayleigh system (4.1)–(4.2), which is related to that in the rigid-wall setting [39]. The main difference from [39] lies in the complicated boundary condition (4.2) on the free surface, which renders the analysis more involved. The instability property depends on the wave number, which is considered as a parameter. As in the rigid-wall setting [39], the key to a successful instability analysis is to locate the neutral limiting modes, which are a neutrally stable solution of the Rayleigh system and contiguous to unstable modes. For certain classes of flows, neutral limiting modes in the free-surface setting are characterized by the inflection values. This together with the local bifurcation of unstable modes from each neutral limiting wave number gives complete knowledge on the instability at all wave numbers. The instability analysis of small-amplitude nontrivial waves taken here is based on a new formulation which directly linearizes the Euler equation and the kinematic and dynamic boundary conditions on the free surface around a periodic traveling wave. Its growing-mode problem then is written as an operator equation for the stream function perturbation restricted on the steady free-surface. The mapping by the action-angle variables is employed to prove the continuity of the operator with respect to the amplitude parameter. In addition, in the action-angle variables, the equation of the particle trajectory takes a very simple form. The persistence of instability along the local curve
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of bifurcation is established by means of Steinberg’s eigenvalue perturbation theorem [51]. In addition, growing-mode solutions is proved to acquire regularity up to that of the steady profiles. This paper is organized as follows. Section 2 is the discussion on the local bifurcation of periodic traveling water-waves when a background shear flow in the physical space is given. Section 3 includes the formulation of the linearized periodic water-wave problem and the derivation of its invariants. Section 4 is devoted to the linear instability of shear flows with one inflection value, and subsequently, Sect. 5 is to the linear instability of small-amplitude periodic waves over an unstable shear flow. Section 6 revisits the linear instability of shear flows for a more general class. 2. Existence of Small-Amplitude Periodic Traveling Water-Waves We consider a traveling-wave solution of (1.1)–(1.4), that is, a solution for which the velocity field, the wave profile and the pressure have space-time dependence (x − ct, y), where c > 0 is the speed of wave propagation. With respect to a frame of reference moving with the speed c, the wave profile appears to be stationary and the flow is steady. The traveling-wave problem for (1.1)–(1.4) is further supplemented with the periodicity condition that the velocity field, the wave profile and the pressure are 2π/α-periodic in the x-variable, where α > 0 is the wave number. It is traditional in the traveling-wave problem to introduce the relative stream function ψ(x, y) such that ψx = −v,
ψy = u − c
(2.1)
and ψ(0, η(0)) = 0. This reduces the traveling-wave problem for (1.1)–(1.4) to a stationary elliptic boundary value problem [18, Sect. 2]: For a real parameter B and a function γ ∈ C 1+β ([0, | p0 |]), β ∈ (0, 1), find η(x) and ψ(x, y) which are 2π/α -periodic in the x-variable, ψ y (x, y) < 0 in {(x, y) : 0 < y < η(x)} 1 , and − ψ ψ 2 |∇ψ| + 2gy ψ
= = = =
γ (ψ) in 0 < y < η(x), 0 on y = η(x), B on y = η(x), − p0 on y = 0,
(2.2a) (2.2b) (2.2c) (2.2d)
where
η(x)
p0 =
ψ y (x, y)dy
(2.3)
0
is the relative total flux2 . The vorticity function γ gives the vorticity-stream function relation, that is, ω = γ (ψ). The assumption of no stagnation, i.e. ψ y (x, y) < 0 in the fluid region {(x, y) : 0 < y < η(x)}, guarantees that such a function is well-defined globally, see [18]. 1 In other words, there is no stagnation in the fluid region. Field observations [37] as well as laboratory experiments [57] indicate that for wave patterns which are not near the spilling or breaking state, the speed of wave propagation is in general considerably larger than the horizontal velocity of any water particle. 2 p < 0 is independent of x. 0
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V. M. Hur, Z. Lin
Furthermore, under this physically motivated stipulation, interchanging the roles of the y-coordinates and ψ offers an alternative formulation to (2.2) in a fixed strip, which serves as the basis of the existence theories in [25,18,32]. The nonlinear boundary condition (2.2c) at the free surface y = η(x) expresses Bernoulli’s law. The steady hydrostatic pressure in the fluid region is given by −ψ(x,y) 2 1 P(x, y) = B − 2 |∇ψ(x, y)| − gy − γ (− p)dp. (2.4) 0
In this setting, α and B are considered as parameters whose values form part of the solution. The wave number α in the existence theory is independent of other physical parameters and hence is held fixed, while in the stability analysis in Sect. 5 it serves as parameter. The Bernoulli constant B measures the total mechanical energy of the flow and varies along a solution branch. 2.1. The local bifurcation theorem in [18]. This subsection contains a summary of the existence result in [18] via the local bifurcation theorem of small-amplitude travellingwave solutions to (2.2), provided that the total flux p0 and the vorticity-stream function relation γ are given. A preliminary result for the local bifurcation is to find a curve of trivial solutions, which correspond to horizontal shear flows under a flat surface. As in [18, Sect. 3.1], let p
( p) = γ (− p )dp , min = min ( p) 0. 0
[ p0 ,0]
Lemma 2.1 ([18], Lemma 3.2). Given p0 < 0 and γ ∈ C 1+β ([0, | p0 |]), β ∈ (0, 1), for each µ ∈ (−2 min , ∞) the system (2.2) has a solution p dp y( p) = , µ + 2 ( p ) p0 which corresponds to a parallel shear flow in the horizontal direction u(x, y) = U (y; µ) = c − µ + 2 ( p(y))
(2.5)
and v(x, y) ≡ 0 in the channel {(x, y) : 0 < y < h(µ)}, where 0 dp . h(µ) = √ µ + 2 ( p) p0 The hydrostatic pressure is P(y) = −gy for y ∈ [0, h(µ)]. Here, p(y) is the inverse of y = y( p) and determines the stream function ψ(y; µ) = − p(y; µ); c > 0 is arbitrary. In the statement of Theorem 2.2 below, instead of B the squared (relative) upstream flow speed µ = (U (h) − c)2 of a trivial shear flow (2.5) serves as the bifurcation parameter. For each µ ∈ (−2 min , ∞) the Bernoulli constant B is determined uniquely in terms of µ by 0 dp B = µ + 2g . (2.6) √ µ + 2 ( p) p0 The following theorem [18, Theorem 3.1] states the existence result of a one-parameter curve of small-amplitude periodic water-waves for a general class of vorticities and their properties, in a form convenient for our purposes.
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Theorem 2.2 (Existence of small-amplitude periodic water-waves). Let the speed of wave propagation c > 0, the flux p0 < 0, the vorticity function γ ∈ C 1+β ([0, | p0 |]), β ∈ (0, 1), and the wave number α > 0 be given such that the system ⎧ 3 2 ⎪ ⎨(a (µ)M p ) p = α a(µ)M for p ∈ ( p0 , 0) (2.7) µ3/2 M p (0) = g M(0) ⎪ ⎩ M( p ) = 0 0 admits a nontrivial solution for some µ0 ∈ (−2 min , ∞), where a(µ) = a(µ; p) = √ µ + 2 ( p). Then, for ε 0 sufficiently small there exists a one-parameter curve of steady solution-pair µε of (2.6) and (ηε (x), ψε (x, y)) of (2.2) such that ηε (x) and ψε (x, y) are 2π/α -periodic in the x-variable, of C 3+β class, where β ∈ (0, 1), and ψεy (x, y) < 0 throughout the fluid region. At ε = 0 the solution corresponds to a trivial shear flow under a flat surface: (i0) The flat surface is given by η0 (x) ≡ h(µ0 ) =: h 0 and the velocity field is (ψ0y (x, y), −ψ0x (x, y)) = (U (y) − c, 0), where U (y) is determined in (2.5); (ii0) The pressure is given by the hydrostatic law P0 (x, y) = −gy for y ∈ [0, h 0 ]. At each ε > 0 the corresponding nontrivial solution enjoys the following properties: (iε) The bifurcation parameter has the asymptotic expansion µε = µ0 + O(ε)
as ε → 0
and the wave profile is given by ηε (x) = h ε + α −1 δγ ε cos αx + O(ε2 )
as ε → 0,
where h ε = h(µε ) is given in Lemma 2.1 and δγ depends only on γ and p0 ; the mean height satisfies h ε = h 0 + O(ε)
as ε → 0.
Furthermore, the wave profile is of mean-zero; that is, 2π/α (ηε (x) − h ε )d x = 0.
(2.8)
0
(iiε) The velocity field (ψεy (x, y), −ψεx (x, y)) in the steady fluid region {(x, y) : 0 < x < 2π/α, 0 < y < ηε (x)} is given by ψεx (x, y) = εψ∗x (y) sin αx + O(ε2 ), ψεy (x, y) = U (y) − c + εψ∗y (y) cos αx + O(ε2 ) as ε → 0, where ψ∗x and ψ∗y are determined from the linear theory. (iiiε) The hydrostatic pressure has the asymptotic expansion Pε (x, y) = −gy + O(ε)
as ε → 0.
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The condition that the system (2.7) admits a nontrivial solution for some µ0 ∈ (−2 min , ∞) is necessary and sufficient for the local bifurcation [18, Sect. 3]. A sufficient condition ([18]) for the solvability of (2.7) and therefore the local bifurcation is 0 α 2 ( p − p0 )2 (2 ( p) − 2 min )1/2 + (2 ( p) − 2 min )3/2 dp < gp02 , p0
which is satisfied when p0 is sufficiently small.
2.2. The bifurcation condition for a given shear flow. Our instability analyses in Sect. 4 and Sect. 5 are carried out in the physical space, where a shear-flow profile and the water depth are held fixed. The bifurcation analysis in the proof of Theorem 2.2, on the other hand, is carried out in the space of transformed variables, where the travel speed c, as well as the relative flux p0 and the vorticity-stream function relation γ are held fixed, and the shear flow U (y) − c and the water depth h vary along the curve of local bifurcation. In this subsection, we study the local bifurcation in the physical space, with a given shear flow U (y) and the water depth h, which is relevant to the later instability analyses. The natural choice for parameters is the speed of wave propagation c > max U and the wave number k. Our first task is to relate the bifurcation equation (2.7) in transformed variables with the Rayleigh system in the physical variables. Lemma 2.3. For the shear flow U (y) with y ∈ [0, h] which is defined via Lemma 2.1, the bifurcation equation (2.7) is equivalent to the following Rayleigh equation: (U − c)(φ − k 2 φ) − U φ = 0
for y ∈ (0, h)
(2.9)
with the boundary conditions
φ (h) =
g U (h) φ(h) and φ(0) = 0, + (U (h) − c)2 U (h) − c
(2.10)
where (c, k) = (c, α), c > max U , and φ(y) = (c − U (y))M( p(y)). Here and in the sequel, the prime denotes the differentiation in the y-variable. Proof. Notice that c > max U . Indeed, a(µ; p) =
µ + 2 ( p) = −(U (y( p)) − c) > 0.
Since ∂∂ py = −ψ (y) = −(U (y) − c), it follows that ∂ p = ∂ y ∂∂ py = − U 1−c ∂ y . Let M( p(y)) = (y), then (2.7) is written as ((U − c)2 ) − α 2 (U − c)2 = 0 for y ∈ (0, h), (U − c)2 (h) = g(h) and (0) = 0. Let φ(y) = (c − U (y))(y), and the above system becomes (2.9)–(2.10).
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Remark 2.4. We illustrate how to construct downstream-traveling periodic waves of small-amplitude bifurcating from a fixed background shear-flow U (y) for y ∈ [0, h]. First, one finds the parameter values (c, k) = (c, α) with c > max U and α > 0 such that the Rayleigh system (2.9)–(2.10) admits a nontrivial solution. The wave speed then determines the bifurcation parameter via µ0 = (U (h)−c)2 , and the flux and the vorticity function are determined by h p0 = (U (y) − c)dy and γ ( p) = −U (y( p)), (2.11) 0
respectively. By Lemma 2.3, the bifurcation equation (2.7) with µ0 , p0 and γ as above has a nontrivial solution. Moreover, each shear flow U (h)−c for c > max U corresponds to a trivial solution in Lemma 2.1. Indeed, µ and c has a one-to-one correspondence via µ = (U (h) − c)2 ; The (relative) stream function defined as h ψ(y) = − (U (h) − c)dy y
is monotone and its inverse y = y(−ψ) is well-defined. Then, Theorem 2.2 applies in the setting above to give a local curve of bifurcation of periodic waves. The lemma below obtains for a large class of shear flows the local bifurcation by showing that the Rayleigh system (2.9)–(2.10) has a nontrivial solution. Lemma 2.5. If U ∈ C 2 ([0, h]), U (h) < 0 and U (h) > U (y) for y = h,
(2.12)
then for any wave number k > 0 there exists c(k) > U (h) = max U such that the system (2.9)–(2.10) has a nontrivial solution φ with φ > 0 in (0, h]. Proof. For c ∈ (U (h), ∞) and k > 0, let φc be the solution of (2.9), or equivalently,
(U − c)φc − U φc − k 2 (U − c)φc = 0 for y ∈ (0, h) (2.13) with φc (0) = 0 and φc (0) = 1. An integration of the above equation on the interval [0, h] yields that h (U (h) − c)φc (h) − (U (0) − c) − φc (h)U (h) − k 2 (U − c)φc dy = 0. 0
Note that the bifurcation condition (2.10) is fulfilled if and only if the function h g f (c) = c − U (0) + k 2 φc (h) (c − U )φc dy − (2.14) c − U (h) 0 has a zero at some c(k) > U (h). It is easy to see that f is a continuous function of c for c > U (h). First, we claim that φc (y) > 0 for y ∈ (0, h]. Suppose, on the contrary, that φc (y0 ) = 0 for some y0 ∈ (0, h]. Note that (2.9) can be written as a Sturm-Liouville equation φc − k 2 φc −
U φc = 0. U −c
(2.15)
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Since (c − U ) +
U (c − U ) = 0 for y ∈ (0, h), c−U
by Sturm’s first comparison theorem the function c − U (y) must have a zero on the interval (0, y0 ). A contradiction then proves the claim. Our goal is to show that f (c) > 0 for c large enough and f (c) < 0 as c → U (h)+. Then by continuity, f vanishes at some c > U (h). First, when c → ∞ the sequence of solutions φc of (2.9), or equivalently (2.15) converges in C 2 , that is, φc → φ∞ . The limit φ∞ satisfies the boundary value problem φ∞ − k 2 φ∞ = 0 for y ∈ (0, h) (0) = 1. Therefore, φ is bounded, continuous, and positive with φ∞ (0) = 0 and φ∞ ∞ on (0, h]. By the definition (2.14), then it follows that f (c) → ∞ as c → ∞. Next is to examine f (c) as c → U (h)+. Denote ε = c − U (h) > 0. We claim that:
φc (h) C1 > 0 for ε > 0 sufficiently small,
(2.16)
where C1 > 0 is independent of ε. To see this, it is convenient to write (2.13) as
(c − U )φc − (c − U ) φc = k 2 (c − U )φc > 0 for y ∈ (0, h), whence (c − U (y))φc (y) − (c − U (y)) φc (y) > c − U (0) > 0 for y ∈ (0, h). (2.17) Since (c − U )φc − (c − U ) φc = (c − U )2
φc c−U
,
(2.18)
it follows from (2.17) that
φc c−U
>
c − U (0) , (c − U )2
and an integration of the above on [0, h] yields that h c − U (0) φc (h) > (c − U (h)) dy. (c − U (y))2 0 Our assumption on U (y) asserts that 0 U (h) − U (y) β(h − y) for y ∈ [h − δ, h] and δ > 0 sufficiently small, where β > 0 is a constant. Thus, h 1 dy φc (h) > ε(c − U (0)) 2 h−δ (ε + β(h − y)) (h−δ)/ε 1 = (c − U (0)) d x C1 > 0, (1 + βx)2 0 where C1 > 0 is independent of ε > 0. This proves the claim (2.16). To prove that f (c) < 0 as c → U (h)+, we consider the following two cases.
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Case 1. max φc (y) = φc (h). It follows that
h f (c) c − U (0) + C1 k 2 (c − U )dy − 0
g c − U (h)
<0
provided that c − U (h) = ε > 0 is small enough. Case 2. max φc = φc (yc ), where yc ∈ (0, h). Since φc (yc ) 0 and φc (yc ) > 0, by (2.15) we have U (yc ) > 0. On the other hand, U (h) 0 and hence yc ∈ [0, h − δ] for some δ > 0. Note that c−U (y) is bounded away from zero on the interval y ∈ [0, h −δ]. Since the coefficients of (2.15) are uniformly bounded for c on y ∈ [0, h−δ], the solution φc is uniformly bounded on y ∈ [0, h −δ]. In particular, 0 < φc (yc ) C2 independently for c. Therefore, f (c) c − U (0) + k 2 hC2 max(c − U (y)) − [0,h]
gC1 < 0, c − U (h)
when ε = c − U (h) is small enough. This completes the proof.
Lemma 2.3 and Remark 2.4 ensure the local bifurcation from a shear flow satisfying (2.12) at any wave number k > 0, as stated below. Theorem 2.6. If U ∈ C 2 ([0, h]), U (h) < 0 and U (h) > U (y) for y = h then for an arbitrary wavelength 2π/k, where k > 0, there exist small-amplitude periodic waves bifurcating in the sense as in Theorem 2.2 from the flat-surface shear flow U (y), where c(k) > max U . In the irrotational setting, i.e. U ≡ 0, the parameter values c and k for which the Rayleigh system (2.9)–(2.10) is solvable give the dispersion relation (i.e. [22]) c2 =
g tanh(kh) . k
In the case of a nonzero background shear flow, such an explicit algebraic relation is in general unavailable. Still, the solvability of the Rayleigh problem (2.9)–(2.10) may be considered to give a generalized dispersion relation. Moreover, we have the following quantitative information about c(k). Lemma 2.7. Given a shear flow U (y) in [0, h], let k and c(k) > max U be such that (2.9)–(2.10) has a nontrivial solution. Then, (a) c(k) is bounded for k > 0; (b) If k1 = k2 then c(k1 ) = c(k2 ); (c) In the long wave limit k → 0+, the limit of the wave speed c(0) satisfies the Burns condition [15] h dy 1 (2.19) = . 2 g 0 (U − c(0)) Proof. As in the proof of Lemma 2.5, the solvability of (2.9)–(2.10) is equivalent to the vanishing of the function f (c) defined by (2.14). (a) The proof of Lemma 2.5 implies that for each A > 0 there exists C A such that f (c) is positive when c > C A , k < A. In interpretation, c(k) C A for k < A. Thus, it
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suffices to show that f (c) 0 when c and k are large enough. Indeed, let c > max U > U be large enough so that c−U 1 and let us denote by φ1 and φ2 the solutions of φ1 + 1 − k 2 φ1 = 0 and φ2 + −1 − k 2 φ2 = 0, respectively, with φ j (0) = 0 and φ j (0) = 1, where j = 1, 2. It is straightforward to see that 1 1 φ1 (y) = √ sin h( k 2 + 1)y. sin h( k 2 − 1)y and φ2 (y) = √ k2 − 1 k2 + 1 Sturm’s second comparison theorem [31] then asserts that the solution φc,k of (2.15) and φ1 , φ2 satisfy that φc,k φ1 φ 2, φ1 φc,k φ2
and thus φ1 φc,k φ2 . It is then easy to see that f (c) > 0 when k is big enough. (b) Suppose on the contrary that c(k1 ) = c(k2 ) = c for k1 < k2 . Let us denote by φc,k1 and φc,k2 the corresponding nontrivial solutions of (2.9)–(2.10). By Sturm’s second comparison theorem [31], it follows that (h) φc,k 1
φc,k1 (h)
<
(h) φc,k 2
φc,k2 (h)
.
This contradicts the boundary condition (2.10). (c) The Rayleigh equation (2.13) for k = 0 implies that (c − U (y))φc (y) + U (y)φc (y) = m, where m is a constant. So by (2.18),
φc m = , c−U (c − U )2 and an integration of above from 0 to h gives h 1 φc (h) =m dy. 2 c − U (h) 0 (c − U (y)) On the other hand, by (2.10), m = (c − U (h))φc (h) + U (h)φc (h) = A combination of above gives (2.19).
g φc (h). (c − U (h))
The limiting parameter value µ which corresponds to the limiting wave speed c(0) gives the lowest hydraulic head B defined in (2.6); see [18, Sect. 3] for detail. The limiting wave speed c(0) is the critical parameter near which small solitary waves of elevation exist [56,33].
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3. Linearization of the Periodic Gravity Water-Wave Problem This section includes the detailed account of the linearization of the water-wave problem (1.1)–(1.4) around a periodic traveling wave which solves (2.2). The growing-mode problem is formulated as a set of operator equations. Invariants of the linearized problem are derived, and their implications in the stability of water waves are discussed.
3.1. Derivation of the linearized problem of periodic water waves. A periodic traveling-wave solution of (2.2) is held fixed, and it serves as the undisturbed state about which the system (1.1)–(1.4) is linearized. The derivation is performed in the moving frame of references, in which the wave profile appears to be stationary and the flow is steady. Let us denote the undisturbed wave profile and (relative) stream function by ηe (x) and ψe (x, y), respectively, which satisfy the system ( 2.2). The steady (relative) velocity field (u e (x, y) − c, ve (x, y)) = (ψey (x, y), −ψex (x, y)) is given by (2.1), and the hydrostatic pressure Pe (x, y) is determined by (2.4). Let De = {(x, y) : 0 < x < 2π/α, 0 < y < ηe (x)} and Se = {(x, ηe (x)) : 0 < x < 2π/α} denote, respectively, the undisturbed fluid domain of one period and the steady wave profile. The steady vorticity ωe (x, y) is given by ωe = − ψe = γ (ψe ). The linearization concerns a slightly-perturbed time-dependent solution of the nonlinear problem (1.1)–(1.4) near the steady state (ηe (x), ψe (x, y)). Let us denote the small perturbation of the wave profile, the velocity field and the pressure by ηe (x) + η(t; x), (u e (x, y) − c + u(t; x, y), ve (x, y) + v(t; x, y)) and Pe (x, y) + P(t; x, y), respectively. We expand the nonlinear equations (1.1)–(1.4) around the steady state in the order of small perturbations and restrict the first-order terms to the steady domain and the boundary to obtain linearized equations for the deviation η(t; x), (u(t; x, y), v(t; x, y)), and P(t; x, y) in the wave profile, the velocity field and the pressure from those of the undisturbed state. In the steady fluid domain De , the velocity deviation (u, v) satisfies the incompressibility condition ∂x u + ∂ y v = 0 and the linearized Euler equation ∂t u + (u e − c)∂x u + u ex u + ve ∂ y u + vey v = −∂x P ∂t v + (u e − c)∂x v + vex u + ve ∂ y v + vey v = −∂ y P,
(3.1)
(3.2)
where P is the pressure deviation. Equation (3.1) allows us to introduce the stream function ψ(t; x, y) for the velocity deviation (u(t; x, y), v(t; x, y)): ∂x ψ = −v and ∂ y ψ = u. Let us denote by ω(t; x, y) the deviation of vorticity. By definition, ω = − ψ. The linearized vorticity equation is ∂t ω + (ψey ∂x ω − ψex ∂ y ω) + (ωex ∂ y ψ − ωey ∂x ψ) = 0. Since ωe = γ (ψe ), the last term can be written as −γ (ψe )(ψey ∂x ψ − ψex ∂ y ψ).
(3.3)
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The linearized kinematic and dynamic boundary conditions restricted to the steady free boundary Se are v + vey η = ∂t η + (u e − c)∂x η + (u + u ey η)ηex
(3.4)
and P + Pey η = 0, respectively. In terms of the stream function, (3.4) is further written as 0 = ∂t η + ψey ∂x η + (∂x ψ + ηex ∂ y ψ) + (ψex y + ηex ψeyy )η = ∂t η + ψey ∂x η + ∂τ ψ + ∂τ ψey η = ∂t η + ∂τ (ψey η) + ∂τ ψ, where ∂τ f = ∂x f + ηex ∂ y f denotes the tangential derivative of a function f defined on the curve {y = ηe (x)}. Alternatively, ∂τ f (x) = ∂x f (x, ηe (x)). The bottom boundary condition of the linearized motion is ∂x ψ = 0 on {y = 0}. Our next task is to examine the time-evolution of ψ on the steady free surface Se . This links the tangential derivative of the pressure deviation P on the steady free surface Se with ψ and η on Se . For a function f defined on Se , let us denote by ∂n f = ∂ y f − ηex ∂x f the normal derivative of f on the curve {y = ηe (x)}. Lemma 3.1. On the steady free surface Se , the normal derivative ∂n ψ satisfies ∂t ∂n ψ + ∂τ (ψey ∂n ψ) + ∂τ ψ + ∂τ P = 0, where = ωe (x, ηe (x)) is the (constant) value of steady vorticity on Se . Proof. The linearized Euler equation (3.2) can be rewritten in the form
ve u v + (u e − c)u + ve v − ωe −ω , − P = ∂t v −u −(u e − c)
(3.5)
by linearizing the nonlinear convection term according to the identity
1 2 u v 2 = u +v −ω . (u, v) · v −u 2 Taking dot product of (3.5) with the vector (1, ηex ) and restricting the result to Se , we have
−∂τ P = ∂t (u + vηex ) + ∂τ (u e − c)u + ve v − (v − uηex )
= ∂t ψ y − ηex ψx + ∂τ (u e − c)ψ y − (u e − c)ηex ψx − −ψx − ηex ψ y = ∂t ∂n ψ + ∂τ ((u e − c)∂n ψ) + ∂τ ψ, where in the above derivation we use the steady kinematic equation ve = (u e − c)ηex on Se .
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In summary, there results in the linearized water-wave problem: ∂t ω + (ψey ∂x ω − ψex ∂ y ω) = γ (ψe )(ψey ∂x ψ − ψex ∂ y ψ) in De , ∂t η + ∂τ (ψey η) + ∂τ ψ = 0 on Se ; P + Pey η = 0 on Se ; ∂t ∂n ψ + ∂τ (ψey ∂n ψ) + ∂τ ψ + ∂τ P = 0 on Se ; ∂x ψ = 0 on {y = 0}.
(3.6a) (3.6b) (3.6c) (3.6d) (3.6e)
Note that the above linearized system may be viewed as one for ψ(t; x, y) and η(t; x). Indeed, P(t; x, ηe (x)) is determined through (3.6c) in terms of η(t; x) and other physical quantities are similarly determined in terms of ψ(t; x, y) and η(t; x). 3.2. The growing-mode problem. A growing mode refers to as an exponentially growing solution to the linearized water-wave problem (3.6) of the form (η(t; x), ψ(t; x, y)) = (eλt η(x), eλt ψ(x, y)) and P(t; x, ηe (x)) = eλt P(x, ηe (x)) with Re λ > 0. For such a solution, the linearized vorticity equation (3.6a) further reduces to λω + (ψey ∂x ω − ψex ∂ y ω) − γ (ψe )(ψey ∂x ψ − ψex ∂ y ψ) = 0 in De ,
(3.7)
where ω = − ψ. Let (X e (s; x, y), Ye (s; x, y)) be the particle trajectory of the steady flow X˙e = ψey (X e , Ye ) (3.8) Y˙e = −ψex (X e , Ye ) with the initial position (X e (0), Ye (0)) = (x, y). Here, the dot above a variable denotes the differentiation in the s-variable. Integration of (3.7) along the particle trajectory (X e (s; x, y), Ye (s, x, y)) for s ∈ (−∞, 0) yields [41, Lemma 3.1] 0 λeλs ψ(X e (s), Ye (s))ds = 0 in De . (3.9a) ψ + γ (ψe )ψ − γ (ψe ) −∞
For a growing mode the boundary conditions (3.6b), (3.6c) and (3.6d) on Se become
d d ψey (x, ηe (x))η(x) = − ψ(x, ηe (x)), (3.9b) dx dx (3.9c) P(x, ηe (x)) + Pey (x, ηe (x))η(x) = 0,
d d d ψey (x, ηe (x))ψn (x) = − P(x, ηe (x)) − ψ(x, ηe (x)). λψn (x) + dx dx dx (3.9d) λη(x) +
The kinematic boundary condition (3.6e) at the flat bottom {y = 0} implies that ψ(x, 0) is a constant. Observe that (3.9a)–(3.9d) remain unchanged by adding a constant to ψ. So we can impose the bottom boundary condition by ψ(x, 0) = 0.
(3.9e)
In summary, the growing-mode problem for periodic traveling water-waves is to find a nontrivial solution of (3.9a)–(3.9e) with Re λ > 0.
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3.3. Invariants of the linearized water-wave problem. In this subsection, the invariants of the linearized water-wave problem (3.6a)–(3.6e) are derived, and their implications in the stability of water waves are discussed. With the introduction of the Poisson bracket, defined as [ f 1 , f 2 ] = ∂x f 1 ∂ y f 2 − ∂ y f 1 ∂x f 2 , the linearized vorticity equation (3.6a) is further written as ∂t ω − [ψe , ω] + γ (ψe )[ψe , ψ] = 0 in De ,
(3.10)
where ω = − ψ. Recall that ∂τ f = ∂x f + ηex ∂ y f and ∂n f = ∂ y f − ηex ∂x f, where f is a function defined on Se . For the case when the vorticity-stream relation is monotone, we show that the following energy functional is an invariant of the linearized system. Lemma 3.2. Provided that either γ ( p) < 0 or γ ( p) > 0 on [0, | p0 |], then for any d solution (η, ψ) of the linearized water-wave problem (3.6), we have dt E(η, ψ) = 0, where 2 −1 2 1 1 |∇ψ| d yd x − E(η, ψ) = 2 2 γ (ψe ) |ω| d yd x De De (3.11) 2 ∗ 2 1 1 − P |η| d x + Re ψ ∂ ψη d x − ψ |η| d x. ey ey n ey 2 2 Se
Se
Se
In the irrotational setting, i.e. γ ≡ 0, the invariant functional becomes 2 2 1 1 E(η, ψ) = ψey ∂n ψη∗ d x. 2 |∇ψ| d yd x − 2 Pey |η| d x + Re De
Se
Se
Proof. By integration by parts, d 2 1 ∂t (∇ψ) · ∇ψ ∗ d yd x 2 |∇ψ| d yd x = Re dt De De ∗ ∗ = Re ∂t ωψ d yd x + ∂t (∂n ψ)ψ d x + De
Se
(3.12)
∂t ∂ y ψψ ∗ d x
{y=0}
:= (I ) + (I I ) + (I I I ). A substitution of ∂t ω by the linearized vorticity equation (3.10) yields that
−γ (ψe )[ψe , ψ] + [ψe , ω] ψ ∗ d yd x (I ) = Re D e − 21 [ψe , γ (ψe )|ψ|2 ] + [ψe , ωψ ∗ ] − [ψe , ψ ∗ ]ω d xd y. = Re De
The second equality in the above uses that 2 1 2 [ψe , γ (ψe )|ψ| ]
= 21 [ψe , γ (ψe )]|ψ|2 + 21 γ (ψe )([ψe , ψ]ψ ∗ + [ψe , ψ ∗ ]ψ) = γ (ψe ) Re[ψe , ψ]ψ ∗ ,
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which follows since [ψe , g(ψe )] = 0 for any g. With the observation that
De
[ψe , f ]d yd x =
De
=
Se
∇ · ((ψey , −ψex ) f )d yd x
(ψey , −ψex ) · (−ηex , 1) f d x +
f ψex d x = 0 y=0
for any function f , the integral (I ) further reduces to (I ) = Re
De
−[ψe , ψ]∗ ωd yd x.
A simple substitution of [ψe , ψ] by the linearized vorticity equation (3.10) then yields that (I ) = Re γ (ψe )−1 (∂t ω∗ − [ψe , ω]∗ )ωd yd x De = Re ( 21 γ (ψe )−1 ∂t |ω|2 − 21 [ψe , γ (ψe )−1 |ω|2 ])d yd x (3.13) De d −1 2 1 = 2 γ (ψe ) |ω| d yd x. dt De This uses that −1 2 1 2 [ψe , γ (ψe ) |ω| ]
= γ (ψe )−1 Re[ψe , ω]∗ ω.
Next is to examine the surface integral (I I ). Simple substitutions by the linearized boundary conditions (3.6b), (3.6c) and ( 3.6d) into (I I ) and an integration by parts yield that (I I ) = Re ∂τ (Pey η − ψey ∂n ψ − ψ)ψ ∗ d x Se = Re (Pey η − ψey ∂n ψ)(−∂τ ψ ∗ )d x Se = Re (Pey η − ψey ∂n ψ)(∂t η∗ + ∂τ (ψey η∗ ))d x Se d 2 ∗ 1 = Pey |η| d x −Re ψey ∂n ψ∂t η d x +Re ∂τ (P +ψey ∂n ψ)ψey η∗ d x. dt Se 2 Se Se The second equality uses that Re
Se
(∂τ ψ)ψ ∗ d x =
d 1 |ψ(x, ηe (x))|2 d x = 0. 2 Se d x
More generally, Re Se (∂τ f ) f ∗ d x = 0 for any function f defined on Se . With another simple substitution by the boundary condition (3.6d), the last term in the computation
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of (I I ) is written as Re ∂τ (P + ψey ∂n ψ)ψey η∗ d x Se = − Re (∂t ∂n ψ + ∂τ ψ)ψey η∗ d x Se ∗ = − Re (∂t (∂n ψ)ψey η d x + Re (∂t η + ∂τ (ψey η))ψey η∗ )d x Se Se d ∗ 1 = − Re ψey ∂t (∂n ψ)η d x + ψey |η|2 d x. dt Se 2 Se The last equality uses that Re Se ∂τ (ψey η)(ψey η)∗ d x = 0. Therefore, d d d 2 ∗ 1 1 Pey |η| d x − Re ψey ∂n ψη d x + ψey |η|2 d x. (3.14) (I I ) = dt Se 2 dt Se dt Se 2 Finally, it is straightforward to see that (I I I ) = ψ ∗ (x, 0)
{y=0}
∂t (∂ y ψ)d x = 0.
This, together with (3.13) and (3.14) proves that E(η, ψ) is an invariant. In the irrotational setting, i.e. γ = 0, the area integral (I ) is zero, = 0, and the other terms remain the same. This completes the proof.
Remark 3.3. Our energy functional E agrees with the second variation ∂ 2 H of the energyCasimir functional in [19]. Recall that the hydrostatic pressure of the steady solution is given as in (2.4) by −ψe (x,y) 2 1 Pe (x, y) = B − 2 |∇ψe (x, y)| − gy + γ (− p)dp, 0
where B is the Bernoulli constant. Differentiation of the above and restriction on Se then yield that −Pey − ψey = 21 ∂ y |∇ψe |2 + g, where = γ (ψ = 0) = ωe (x, ηe (x)). Thus, when γ is monotone, it follows that 2 |∇ψ| d yd x − γ (ψe )−1 |ω|2 d yd x 2E(η, ψ) = De De 2 2 1 + ψey ∂n ψη∗ d x. 2 ∂ y |∇ψe | + g |η| d x + 2 Re Se
Se
∂ 2H
This is exactly the expression of the second variation in [19] of the following energy-Casimir functional (in our notations)
|∇(ψ − cy)|2 + gy − B − F(ω) d yd x, (3.15) H(η, ψ) = 2 Dη around the steady state (ηe , ψe ). Here, Dη = {(x, y) : 0 < y < η(t; x)} and (F )−1 = γ . The quadratic form ∂ 2 H is used in [19] to study a formal stability.
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Our next invariant is the linearized horizontal momentum. The result is free from restrictions on γ . Lemma 3.4. For any solution (η, ψ) of the linearized problem of (3.6), the identity d (ψ + ψey η)d x = 0 dt Se holds true. Proof. We integrate over the steady fluid region De of the linearized equation for the horizontal velocity ∂t ∂ y ψ + (ψey , −ψex ) · ∇(∂ y ψ) + (∂ y ψ, −∂x ψ) · ∇ψey = −Px and apply the divergence theorem to arrive at d ∂ y ψ d yd x + (∂ y ψ, −∂x ψ) · (−ηex , 1)ψey d x = − Px d yd x. (3.16) dt De Se De It is straightforward to see that De
∂ y ψd yd x =
Se
ψ d x.
In view of (3.6b), the second term on the left hand side of (3.16) is written as (∂τ ψ)ψey d x = (∂t η + ∂τ (ψey η))ψey d x Se Se d = ψey η d x − ψey ∂τ (ψey )ηd x. dt Se Se With the use of Stokes’ theorem and the dynamic boundary condition (3.5), the right side of (3.16) becomes Px d yd x = Pηex d x = Pey ηηex d x. De
Se
Se
On the other hand, the steady Euler equation restricted to the steady free-surface Se yields that −Pex = ψey ψex y − ψex ψeyy = ψey (ψex y + ηex ψeyy ) = ψey ∂τ ψey . Since Pex + Pey ηex = 0 on Se , a simple substitution then proves the assertion.
Next, the integration of (3.6b) on Se shows that Se η d x is an invariant. Finally, multiplication on the linearized vorticity equation (3.10) by ξ and then integration yield that ωξ d yd x De
is an invariant for any function ξ ∈ ker(ψey ∂x − ψex ∂ y ) ⊂ L 2 (De ). We summarize our results.
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Proposition 3.5. The linearized problem (3.6) has the energy invariant: 2 −1 2 1 1 E(η, ψ) = |∇ψ| d yd x − 2 2 γ (ψe ) |ω| d yd x De De 2 2 1 1 ∂ + ( |∇ψ | ) + g |η| d x + Re ψey ∂n ψη∗ d x y e 2 2 Se
Se
when γ is monotone and 2 1 E(η, ψ) = 2 |∇ψ| d yd x D e 2 2 1 1 ∂ ( |∇ψ | ) + g |η| d x + Re + y 2 e 2 Se
Se
ψey ∂n ψη∗ d x
when γ ≡ 0 (irrotational). In addition, (3.6) has the following invariants: M(η, ψ) = (ψ + ψey η)d x, S e m(η, ψ) = ηd x, S e F(η, ψ) = ωξ d yd x, for any ξ ∈ ker(ψey ∂x − ψex ∂ y ). De
The nonlinear water-wave problem (1.1)–(1.4) has the following invariants [19, Sect. 2]: 2 1 2 E= 2 (u + v ) + gy d yd x (energy), D(t) M= u dyd x (horizontal momentum), D(t) m= d yd x (mass), D(t) F= f (ω)d yd x (Casimir invariant), D(t)
where D(t) = {(x, y) : 0 < x < 2π/α , 0 < y < η(t; x)} is the fluid domain at time t of one wave length, and the function f is arbitrary such that the integral F exists. The invariants of the linearized problem E, M, m, and F in Proposition 3.5 can be obtained by expanding the invariants of the nonlinear problem E, M, m and F, respectively, around the steady state (ηe (x), ψe (x, y)). The quadratic form E(η, ψ) is the second variation of the energy functional H defined in (3.15) (see Remark 3.3), which is a combination of E, M, m and F. The invariants M and m of the linear problem are the first variations of M and m, respectively. We note that F is the first variation of F, since by the assumption of no stagnation and the monotonicity of γ it follows that ker(ψey ∂x − ψex ∂ y ) = { f (ψe ) : f is arbitrary} = { f (ωe ) : f is arbitrary}.
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We now make some comments on the implications of these invariants on the stability of water waves. A traditional approach of proving stability for conservative systems is the energy method, for which one tries to show that a steady state is an energy minimizer under the constraints of other invariants such as momentum, mass, etc. This method has been widely used in the stability analysis of various approximate equations such as the KdV equation [6,10] and water waves with a large surface tension [47]. However, steady waves of the full pure gravity water-wave problem in general are not (constrained) energy minimizers. This was first observed ([8]) in the irrotational case. By our discussions below, even with the favorable vorticity, the situation remains the same. Note that if a steady gravity water-wave is an energy minimizer under the constraints of fixed M, m and F, then at the linearized level the second variation E should be positive under the constraints that the variations M, m and F are zero. In the irrotational setting, the first two terms in the expression (3.12) of E(η, ψ) yield a positive norm De
|∇ψ|2 d yd x +
Se
|η|2 d x.
(3.17)
2 −ψ 2 −ψ 2 0 and P (x, 0) = −g by the maximum (Indeed, since Pe = −2ψex ey y ex x eyy principle Pe attains its minimum at the free surface Se . Furthermore,since Pe takes a constant on Se by the Hopf lemma Pey < 0 on Se .) The last term Re Se ψey ∂n ψη∗ d x of the right side of (3.12), however, does not have a definite sign and contains a 1/2 -higher derivative than that of (3.17). Thus, it cannot be bounded by the norm (3.17), even with the constraints M = m = 0. Consequently, the quadratic form E(η, ψ) is highly indefinite unless ψey ≡ 0, that is, for a trivial flow. In other words, steady gravity water waves in the irrotational case are in general not (constrained) energy minimizers. Indeed, in [29,13], steady water-waves were constructed as energy saddles by variational methods. With a nonzero vorticity, control of the mixed-type term in the energy functional by other positive terms fails for the same reason as in the irrotational setting. Let us consider the case of a favorable vorticity with γ < 0. Under some additional assumptions, it is shown in [19] that the first three terms in the expression (3.11) of E(η, ψ) gives a positive norm
De
(|∇ψ|2 + |ω|2 )d yd x +
Se
|η|2 d x.
(3.18)
However, due to the lack of control of the boundary value of ψ on Se , one could not use the elliptic regularity to get a better control for ψ, such as ψ H 2 . Thus, the mixed-type term Re Se ψey ∂n ψη∗ d x is still not controllable by (3.18). In [19], several classes of perturbations are introduced to make this mixed term controllable by (3.18), and therefore get the positivity of E (η, ψ) and a formal stability for these special perturbations. However, it is difficult to see that these special classes are invariant during the evolution of the water-wave problem. So, it remains unclear how to pass from such formal stability to genuine stability, even for these special perturbations. It is not hard to see that by adding other invariants as constraints, one can relax the assumptions to get the positive term (3.18) such as in [41] for the fixed boundary case, but the mixed-type term remains uncontrollable. In conclusion, the quadratic form E(η, ψ) is in general indefinite also in the rotational setting, and steady water waves with vorticity are expected to be energy saddles. The above discussions of steady water waves as energy saddles do not imply that steady water-waves are necessarily unstable. Indeed, as mentioned in the Introduction,
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the small Stokes waves are believed to be stable [48,55] under perturbations of the same period. For the rotational case, under the assumption of a monotone γ , the corresponding trivial solutions with shear flows defined in Lemma 2.1 can not have an inflection point, since γ (ψe (y)) = −U (y)/U (y) = 0 and U < 0 by the no-stagnation assumption. Thus by Theorem 6.4, such shear flows are linearly stable to perturbations of any period. Since small-amplitude waves with any monotone vorticity relation γ bifurcate from these strongly stable shear flows, they are likely to be stable. But, a successful stability analysis of any nontrivial steady gravity water waves would require to use the full set of equations instead of a few invariants. 4. Linear Instability of Shear Flows with Free Surface This section is devoted to the study of the linear instability of a free-surface shear flow (U (y), 0) with y ∈ [0, h], which is a steady solution of the water-wave problem (1.1)– (1.4) with P(x, y) = −gy. In the traveling frame of reference with the speed c > max U , this may be recognized as a trivial solution of the traveling-wave problem (2.2): ηe (x) ≡ h and (ψey (x, y), −ψex (x, y)) = (U (y) − c, 0). Throughout this section, we write U for U − c to simplify notations. We seek for normal mode solutions of the form η(t; x) = ηh eiα(x−ct) , ψ(t; x, y) = φ(y)eiα(x−ct) and P(t; x, y) = P(y)eiα(x−ct) to the linearized water-wave problem (3.6). Here, α > 0 is the wave number and c is the complex phase speed. It is equivalent to find solutions to the growing-mode problem (3.9) of the form λ = −iαc and η(x) = ηh eiαx , ψ(x, y) = φ(y)eiαx and P(x, ηe (x)) = Ph eiαx . Note that Re λ > 0 if and only if Im c > 0. Since γ (ψe (y)) = ωey (y)/ψey (y) = −U (y)/U (y) and (X e (s), Ye (s)) = (x + U (y)s, y), the linearized vorticity equation (3.9a) translates into the Rayleigh equation (U − c)(φ − α 2 φ) − U φ = 0 for y ∈ (0, h).
(4.1)
Here and elsewhere the prime denote the differentiation in the y-variable. The boundary conditions (3.9b), (3.9c) and (3.9d) on the free surface are simplified to be (c − U (h))ηh = φ(h),
Ph − gηh = 0
and (c − U (h))φ (h) = Ph + U (h)φ(h), respectively. Eliminating ηh and Ph from the above yields that
(U (h) − c)2 φ (h) = g + U (h)(U (h) − c) φ(h). The bottom boundary condition (3.9e) becomes φ(0) = 0. In summary, the growingmode problem is reduced to study the Rayleigh equation (4.1) with the boundary condition
(U (h) − c)2 φ (h) = g + U (h)(U (h) − c) φ(h) (4.2) φ(0) = 0.
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A shear profile U is said to be linearly unstable if there exists a nontrivial solution of (4.1)–(4.2) with Im c > 0. The Rayleigh system (4.1)–(4.2) is alternatively derived in [62] by linearizing directly the water-wave problem (1.1)–(1.4) around (U (y), 0) in y ∈ [0, h]. Note that for a real number c > max U the Rayleigh system (4.1)–(4.2) becomes the bifurcation equation (2.9)–(2.10). In the case of rigid walls at y = h and y = 0, that is, the Dirichlet boundary conditions φ(h) = 0 = φ(0) in place of (4.2), the instability of a shear flow is a classical problem, which has been under extensive research since Lord Rayleigh [52]. Recently, by a novel analysis of neutral modes, Lin [39] established a sharp criterion for linear instability in the rigid-wall setting for a general class of shear flows. Our objective in this section is to obtain analogous results in the free-surface setting. Below is the definition of the class of shear flows studied in this section. By an inflection value we mean the value of U at an inflection point. Definition 4.1. A function U ∈ C 2 ([0, h]) is said to be in the class K+ if U has an unique inflection value Us and K (y) = −
U (y) U (y) − Us
(4.3)
is bounded and positive on [0, h]. An example of class K+ flows is U (y) = sin my, for which K (y) = m 2 . For U (y) in the class K+ , consider the Sturm-Liouville equation φ − α 2 φ + K (y)φ = 0 for y ∈ (0, h)
(4.4)
with the boundary conditions
φ (h) = gr (Us )φ(h) if U (h) = Us φ(h) = 0 if U (h) = Us , φ(0) = 0.
(4.5) (4.6)
Here, gr (c) =
g U (h) . + 2 (U (h) − c) U (h) − c
(4.7)
The following theorem gives a sharp instability criterion for shear flows in class K+ , for the free surface case. Theorem 4.2 (Linear instability of free-surface shear flows in K+ ). Consider a flow d2 2 the lowest eigenvalue of − dy U (y) in class K+ . Denote by −αmax 2 − K (y) with the boundary condition (4.5)–(4.6), which is assumed to be negative. Then for each α ∈ (0, αmax ) , there exists an unstable solution-triple (φ, α, c) (with Im c > 0) of (4.1)–(4.2). The interval of unstable wave numbers (0, αmax ) is maximal in the sense d2 that the flow is linearly stable if either the operator − dy 2 − K (y) on y ∈ (0, h) with (4.5)–(4.6) is nonnegative or α αmax .
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2 is characterized From the usual variational consideration, the lowest eigenvalue −αmax as h 2 − K (y)|φ|2 )dy + gr (Us )|φ(h)|2 0( φ 2 (4.8a) − αmax = inf h 2 φ∈H 1 (0,h) 0 |φ| dy φ(0)=0
in case U (h) = Us , and 2 − αmax
=
inf
φ∈H 1 (0,h) φ(0)=0=φ(h)
h 2 − K (y)|φ|2 )dy 0( φ h 2 0 |φ| dy
(4.8b)
in case U (h) = Us . 4.1. Neutral limiting modes. The proof of Theorem 4.2 makes use of neutral limiting modes, as in the rigid-wall setting [39]. Definition 4.3 (Neutral limiting modes). A triple (φs , αs , cs ) with αs positive and cs real is called a neutral limiting mode if it is the limit of a sequence of unstable solutions {(φk , αk , ck )}∞ k=1 of (4.1)–(4.2) as k → ∞. The convergence of φk to φs is in the almost-everywhere sense. For a neutral limiting mode, αs is called the neutral limiting wave number and cs is called the neutral limiting wave speed. Lemma 4.8 below establishes that neutral limiting wave numbers form the boundary points of the interval of unstable wave numbers, and thus the stability investigation of a shear flow is reduced to find all neutral limiting modes and then study the stability properties near them. In general, it is difficult to locate all neutral limiting modes. For flows in class K+ , nonetheless, neutral limiting modes are characterized by the inflection value. Proposition 4.4. For U ∈ K+ a neutral limiting mode (φs , αs , cs ) must solve (4.4)–(4.6) with cs = Us . For the proof of Proposition 4.4, we need several properties of unstable solutions. First, Howard’s semicircle theorem holds true in the free-surface setting [62, Theorem 1]. That is, any unstable eigenvalue c = cr + ici (ci > 0) of the Rayleigh equation (4.1)–(4.2) must lie in the semicircle (cr − 21 (Umin + Umax ))2 + ci2 ( 21 (Umin − Umax ))2 ,
(4.9)
where Umin = min[0,h] U (y) and Umax = max[0,h] U (y). The identities below are important for future considerations. Lemma 4.5. If φ is a solution (4.1)–(4.2) with c = cr + ici and ci = 0 then for any q real the identities
h
2 φ + α 2 |φ|2 + U (U − q) |φ|2 dy = Re gr (c) + cr − q Im gr (c) |φ(h)|2 , |U − c|2 ci 0 0
h
(4.10)
2 φ + α 2 |φ|2 + U (U − q) |φ|2 dy = Re gs (c) − cr − q Im gs (c) φ (h)2 |U − c|2 ci (4.11)
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hold true, where gr is defined in (4.7) and gs (c) =
(U (h) − c)2 . g + U (h)(U (h) − c)
(4.12)
Proof. We rewrite the Rayleigh equation (4.1) as − φ + α 2 φ +
U φ = 0. U −c
(4.13)
Multiplication of the above by φ ∗ and integration by parts with the boundary condition (4.2) yield h
2 φ + α 2 |φ|2 + U |φ|2 dy = gr (c)|φ(h)|2 . U −c 0 Its real and imaginary parts read h
2 φ + α 2 |φ|2 + U (U − cr ) |φ|2 dy = Re gr (c) |φ (h)|2 |U − c|2 0 and
U |φ|2 dy = Im gr (c)|φ(h)|2 , 2 0 |U − c| respectively. Combining (4.14) and (4.15) then establishes (4.10). Similarly, combining the real and the imaginary parts of h
2 φ + α 2 |φ|2 + U |φ|2 dy = g ∗ (c)|φ (h)|2 s U −c 0 ci
leads to (4.11).
(4.14)
h
(4.15)
(4.16)
Our next preliminary result is an a priori H 2 -estimate of unstable solutions near a neutral limiting mode. Lemma 4.6. For U ∈ K+ , let {(φk , αk , ck )}∞ k=1 be a sequence of unstable solutions to (4.1)–(4.2) such that φk L 2 = 1. If αk → αs > 0 and Im ck → 0+ as k → ∞, then φk H 2 C, where C > 0 is independent of k. Proof. By the semicircle theorem (4.9), ck ∈ [Umin , Umax ] for any k > 1. Thus ck → cs ∈ [Umin , Umax ], as k → ∞. The proof is divided into the following two cases. Case 1. U (h) = cs . The proof is similar to that of [39, Lemma 3.7]. It is straightforward to see that Im gr (ck ) C0 , | Re gr (ck )| + (4.17) Im ck where C0 > 0 is independent of k. For simplicity of notations, the subscript k is suppressed in the estimates below and C is used to denote generic constants which are independent of k. Let c = cr + ici . We write (4.10) as h 2 )(U − q) (U − U s φ + α 2 |φ|2 + K (y) |φ|2 dy |U − cr |2 + ci2 0
cr − q Im gr (c) |φ(h)|2 . = Re gr (c) + ci
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Note that gr (cs ) is well defined. Setting q = Us − 2 (Us − cr ) in the above identity leads to h h 2 (U −Us )2 + 2(U −Us )(Us −cr ) 2 2 2 ( φ + α |φ| )dy K (y) |φ| dy + C|φ(h)|2 |U − cr |2 + ci2 0 0 h (U − cr )2 − (Us − cr )2 2 = K (y) |φ| dy + C|φ(h)|2 2 + c2 |U − c | 0 r i h 1 2 K (y)|φ|2 dy + C(ε φ L 2 + φ2L 2 ). ε 0 One chooses ε sufficiently small to conclude that φ H 1 C. Next is the H 2 -estimate. Multiplication of (4.13) by −(φ ∗ ) and integration over the interval [0, h] yield h h 2 2 U φ dy. (4.18) (φ + α 2 φ )dy − α 2 gr∗ (c)|φ(h)|2 = (φ ∗ ) U −c 0 0 In view of the Rayleigh equation for φ ∗ , the right side is written as ∗
h
h U U ∗ U 2 ∗ α φ + φ dy = φ φ dy (φ ) U −c U −c U −c 0 0 h 2 h U U = α2 |φ|2 dy + |φ|2 dy. 2 U − c |U − c| 0 0 The real part of (4.18) then reads h 2 2 (φ + α 2 φ )dy = α 2 Re gr (c) |φ(h)|2 0
h 2 h U U (U − cr ) 2 + α2 |φ| dy + |φ|2 dy 2 2 |U − c| |U − c| 0 0
h 2 2 2 2 2 = α 2 Re gr (c)|φ(h)| − ( φ + α |φ| )dy + 0
h
2 U |φ|2 dy, |U − c|2
0
where in the last equality we use (4.14). So h 2 2 (φ + 2α 2 φ + α 4 |φ|2 )dy = 2α 2 Re gr (c)|φ(h)|2 + 0
0
h
2 U |φ|2 dy. |U − c|2 (4.19)
By (4.10) with q = Us , h h 2 U −U (U −Us ) 2 2 ∞ |φ| dy K |φ| dy (since − U (U − Us ) 0) L 2 2 |U −c| |U −c| 0 0
h
2 Im gr (c) 2 2 2 ∞ |φ(h)| = K L ( φ + α |φ| )dy − Re gr (c) + (cr − Us ) ci 0 C φ2H 1 ,
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where we use the bound (4.17) and that ||φ(h)| C0 φ H 1 . Therefore, by (4.19) and the φ H 1 bound, we get φ H 2 C as desired. Case 2. U (h) = cs . The proof is similar to that of Case 1 except that one uses φ(h) = gs (c)φ (h) in place of φ (h) = gr (c)φ(h). It can be checked that Im gs (ck ) C0 d(ck , U (h)), | Re gs (ck )| + (4.20) Im ck where C0 > 0 is independent of k and d(ck , U (h)) = | Re ck − U (h)| + (Im ck )2 . Since U (h) = cs , it follows that d(ck , U (h)) → 0 as k → ∞. The same computations as in Case 1 establish that h h 2 2 (φk + αk2 |φk |2 )dy K (y)|φk |2 dy + C0 d(ck , U (h)) φk (h) 0
and
h 0
0
2 2 (φk + 2αk2 φk + αk4 |φk |2 )dy
2 U + |φ |2 dy = 2 k 0 |U − c|
h 2 2 (φk + αk2 |φk |2 )dy + d(ck , U (h)) φk (h) C 2αk2 Re
0
h
C 0
gs (ck )|φk (h)|2
h
2 2 1 2 2 ε φk + , + αk |φk | dy + d(ck , U (h)) φk (h) ε
(4.21)
where C > 0 is independent of k. Consequently, by choosing ε small, 2 φk 2H 2 C1 (φk 2L 2 + d(ck , U (h)) φk (h) ) C2 (φk 2L 2 + d(ck , U (h))φk 2H 2 ), where C1 , C2 > 0 are independent of k. Since d(ck , U (h)) → 0 as k → ∞ it follows from above that φk 2H 2 C.
For U ∈ K+ , if c is in the range of U then U (y) = c holds at a finite number of points [39, Remark 3.2], which are denoted by y1 < y2 < · · · < ym c . Let y0 = 0 and ym c +1 = h . We state our last preliminary result. Lemma 4.7 ([39], Lemma 3.5). Let φ satisfy (4.1) with α positive and c in the range of U and let U (y) = c for y ∈ y1 , y2 , . . . , ym c . If φ is sectionally continuous on the open intervals (y j , y j+1 ), j = 0, 1, . . . , m c , then φ cannot vanish at both endpoints of any intervals (y j , y j+1 ) unless it vanishes identically on that interval. Proof of Proposition 4.4. Let (φs , αs , cs ) be a neutral limiting mode with αs > 0 and cs ∈ [Umin , Umax ], and let {(φk , αk , ck )}∞ k=1 be a sequence of unstable solutions of (4.1)–(4.2) such that (φk , αk , ck ) converges to (φs , αs , cs ) as k → ∞. We normalize the sequence by setting φk L 2 = 1. First, the result of Lemma 4.6 says that φk H 2 C, where C > 0 is independent of k. Consequently, φk converges to φs weakly in H 2 and strongly in H 1 . Then
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φs H 2 C and φs L 2 = 1. Let y1 , y2 , . . . , ym s be the roots of U (y) − cs and let S0 be the complement of the set of points {y1 , y2 , . . . , ym s } in the interval [0, h]. Since φk converges to φs uniformly in C 2 on any compact subset of S0 , it follows that φs exists on S0 . Since (U − ck )−1 , φk and their derivatives up to second order are uniformly bounded on any compact subset of S0 , it follows that φs satisfies φs − αs2 φs −
U φs = 0 U − cs
almost everywhere on [0, h]. Moreover,
g U (h) φs (h) and φs (0) = 0 + φs (h) = (U (h) − cs )2 U (h) − cs
(4.22)
(4.23)
in case U (h) = cs , and φs (h) = 0 and φs (0) = 0
(4.24)
in case U (h) = cs . Next, we claim that cs is the inflection value Us . By Definition 4.1, U (y j ) = −K (y j )(cs − Us ) has the same sign for j = 1, . . . , m s , say positive. Let ms E δ = ∪i=1 {y ∈ [0, h] : y − y j < δ}. Clearly, E δc ⊂ S0 and U (y) > 0 for y ∈ E δ when δ > 0 sufficiently small. The proof is again divided into two cases. Case 1. U (h) = cs . Since φs is not identically zero, Lemma 4.7 asserts that φs (y j ) = 0 for some y j . If cs were not an inflection value then near such a y j it must hold that U (U − Umin + 1) U 2 |φ | dy |φ |2 dy = ∞. (4.25) s 2 s |U − cs |2 Eδ |y−y j |<δ |U − cs | Since
2 φ + α 2 |φk |2 + U (U − Umin + 1) |φk |2 dy k k |U − ck |2 0 U (U − Umin + 1) |U (U − q)| |φk |2 dy − sup , 2 |U − ck | |U − ck |2 Eδ Ec
h
δ
by Fatou’s Lemma and (4.25) it follows that h
2 φ + α 2 |φk |2 + U (U − Umin + 1) |φk |2 dy = ∞. lim inf k k k→∞ 0 |U − ck |2 On the other hand, (4.10) with q = Umin − 1 yields h
2 φ + α 2 |φk |2 + U (U − Umin + 1) |φk |2 dy k k |U − ck |2 0
Re ck − Umin + 1 |φk (h)|2 Cφk H 2 C1 , = Re gr (ck ) + Im ck where C1 > 0 is independent of k. A contradiction proves the claim. Case 2. U (h) = cs . The proof is identical to that of Case 1 except that we use (4.11) in place of (4.10) and hence is omitted.
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The following lemma allows us to continuate unstable modes until a neutral limiting wave number is reached, analogous to [39, Theorem 3.9]. Lemma 4.8. For U ∈ K+ , the set of unstable wave numbers is open, whose any boundd2 ary point α satisfies that −α 2 is an eigenvalue of the operator − dy 2 − K (y) on y ∈ (0, h) with the boundary conditions (4.5)–(4.6). Proof. An unstable solution (φ0 , α0 , c0 ) of (4.1)–(4.2) is held fixed, with α0 > 0 and Im c0 > 0. For r1 , r2 > 0, let us define Ir1 = {α > 0 : |α − α0 | < r1 } and Br2 = {c ∈ C+ : |c − c0 | < r2 }, where C+ denotes the set of complex numbers with positive imaginary part. Our goal is to show that for each α ∈ Ir1 , there exists c(α) ∈ Br2 for some r2 > 0 and φα ∈ H 2 such that (φα , α, c(α)) satisfies (4.1)–(4.2). For α ∈ (0, ∞) and c ∈ C+ , let φ1 (x; α, c) and φ2 (x; α, c) be the solutions of the differential equation φ − α 2 φ −
U φ=0 U −c
for y ∈ (0, h)
normalized at h, that is, φ1 (h) = 1, φ2 (h) = 0, φ1 (h) = 0, φ2 (h) = 1. It is standard that φ1 and φ2 are analytic as functions of c in C+ . Let us consider a function on (0, ∞) × C+ , defined as (α, c) = φ1 (0; α, c) + gr (c)φ2 (0; α, c),
(4.26)
where gr is defined in (4.7). Clearly, is analytic in c and continuous in α. Note that (α, c) = 0 if and only if the system (4.1)–(4.2) has an unstable solution φα (y) = φ1 (y; α, c) + gr (c)φ2 (y; α, c). Since (α0 , c0 ) = 0 and the zeros of an analytic function are isolated, (α0 , c) = 0 on {|c − c0 | = r2 }, for r2 > 0 sufficiently small. Then, by the continuity of (α, c) in α, we have (α, c) = 0 on {|c − c0 | = r2 }, when α ∈ Ir1 and r1 is sufficiently small. Let us consider the function 1 ∂/∂c(α, c) dc, N (α) = 2πi |c−c0 |=r2 (α, c) where α ∈ Ir1 . Observe that N (α) counts the number of zeros of (α, c) in Br2 . Since N (α0 ) > 0 and N (α) is continuous as a function of α in Ir1 it follows that N (α) > 0 for any α ∈ Ir1 . This proves that the set of unstable wave numbers is open. By definition, the boundary points of the set of unstable wave numbers must be neutral limiting wave numbers, say αs . Proposition 4.4 asserts that −αs2 must be a negative d2 eigenvalue of the operator − dy 2 − K (y) on (0, h) with (4.5)–(4.6). This completes the proof.
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4.2. Proof of Theorem 4.2. The proof of Theorem 4.2 is to examine the bifurcation of unstable modes from each neutral limiting mode. This reduces to the study of the bifurcation of zeros of the algebraic equation (α, c) = 0 from (αs , cs ) as is pointed out in the proof of Lemma 4.8. However, the differentiability of (α, c) in c at a neutral limiting mode (αs , cs ) can only be established on half of the c−neighborhood, and thus the standard implicit function theorem does not apply. To overcome this difficulty, we construct a contraction mapping in such a half neighborhood and prove the existence of an unstable mode near (αs , cs ) as is done in [39] for the rigid-wall case. Moreover, the existence of an unstable mode can only be established for wave numbers slightly to the left of αs . Therefore, unstable modes with wave numbers slightly to the left of αs can be continuated to the zero wave number. Our method is almost the same as in the rigid-wall setting ([39, Sect. 4]). Below we construct unstable solutions for wave numbers slightly to the left of a neutral limiting wave number. Proposition 4.9. For U (y) in the class K+ , let Us be the inflection value and y1 , y2 , . . . , ym s be the inflection points, as such U (y j ) = Us for i = 1, 2, . . . , m s . If (φs , αs , Us ) is a neutral limiting mode with αs positive, then for ε ∈ (ε0 , 0), where |ε0 | is sufficiently small, there exist φε and c(ε) such that U φε − αs2 + ε φε − (4.27) φε = 0 for y ∈ (0, h), U − Us − c(ε) (4.28) φε (h) = gr (Us + c(ε))φε (h) and φε (0) = 0 with Im c(ε) > 0. Moreover, lim c(ε) = 0,
dc (ε) = lim ε→0− dε
0
h
(4.29)
ε→0−
⎞−1 h ms K (y ) K j φ 2 (y j ) + p.v. φs2 dy ⎝iπ φs2 dy + A⎠ ; |U (y j )| s 0 U −Us
⎛
j=1
(4.30) here p.v. means the Cauchy principal part and 2g + U (h) A = (U (h)−Us )3 (U (h)−Us )2 0
if U (h) = Us if U (h) = Us .
(4.31)
Proof. As in the proof of Lemma 4.8, for c ∈ C+ and ε < 0, let φ1 (y; ε, c) and φ2 (y; ε, c) be the solutions of φ − (αs2 + ε)φ − normalized at h, that is
U φ = 0 for y ∈ (0, h) U − Us − c
(4.32)
φ1 (h) = 1, φ2 (h) = 0, φ1 (h) = 0, φ2 (h) = 1.
It is standard that φ1 and φ2 are analytic as a function of c in C+ and that φ1 and φ2 are linearly independent with Wronskian 1. The neutral limiting mode is normalized so that φs (h) = 1 and φs (h) = gr (Us ). The proof is again divided into two cases.
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Case 1. U (h) = Us . Let us define φ0 (y; ε, c) = φ1 (y; ε, c) + gr (Us + c)φ2 (y; ε, c), (ε, c) = φ1 (0; ε, c) + gr (Us + c)φ2 (0; ε, c), where gr is given in (4.7). It is readily seen that φ0 solves (4.32) with φ0 (h) = 1 and φ0 (h) = gr (Us + c). It is easy to see that (ε, c) is analytic in c ∈ C+ and differentiable in ε. Note that an unstable solution to (4.27)–(4.28) exists if and only if (ε, c) = 0 for some Im c > 0. The Green’s function of (4.32) is written as G(y, y ; ε, c) = φ¯ 1 (y; ε, c)φ2 (y ; ε, c) − φ2 (y; ε, c)φ¯ 1 (y ; ε, c), where φ¯ 1 (y; ε, c) is the solution of (4.32) with φ¯ 1 (h) = 1 and φ¯ 1 (h) = gr (Us ). A similar computation as in [39, pp. 336] for φ j (y; ε, c) ( j = 1, 2) yields that h ∂ (ε, c) = − G(y, 0; ε, c)φ0 (y; ε, c)dy (4.33) ∂ε 0 and
−U d gr (Us + c)φ2 (0; ε, c). φ (y; ε, c)dy + 2 0 (U − U − c) dc s 0 (4.34) Let us define the triangle in C+ as
∂ (ε, c) = ∂c
h
G(y, 0; ε, c)
(R,b) = {cr + ici : |cr | < Rci , 0 < ci < b} and the Cartesian product in (0, ∞) × C+ as E (R,b1 ,b2 ) = (−b2 , 0) × (R,b1 ) , where R, b1 , b2 > 0 are to be determined later. We claim that: (a) For fixed R, both φ¯ 1 (·; ε, c) and φ0 (·; ε, c) uniformly converge to φs in C 1 [0, h] as (ε, c) → (0, 0) in E (R,b1 ,b2 ) . That is, for any δ > 0 there exists some b0 > 0 such that whenever b1 , b2 < b0 and (ε, c) ∈ E (R,b1 ,b2 ) the inequalities φ¯ 1 (·; ε, c) − φs 1 , φ0 (·; ε, c) − φs C 1 δ C hold. (b) φ2 (·; ε, c) converges uniformly to φ2 (y; 0, 0) in the sense of (a). We denote φ2 (y; 0, 0) = φz (y) , then φz (0) = − φ 1(0) , since the Wronskian of φ¯ 1 , φ2 and its s limit (φs , φz ) is 1. The proof of (a) and (b) is very similar to [39, pp. 337] and we skip it. In the Appendix, we prove that h 1 ∂ (ε, c) → φ 2 dy, (4.35) ∂ε φs (0) 0 s ⎞ ⎛ h ms K (y j ) 2 ∂ 1 ⎝ K (ε, c) → − φ (y j ) + p.v. φs2 dy + A⎠ iπ ∂c φs (0) |U (y j )| s 0 U − Us j=1
(4.36)
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uniformly as ε → 0− and c → 0 in E (R,b1 ,b2 ) , where A is defined by (4.31) and y j ( j = 1, . . . , m s ) are the inflection points for Us . Let us denote B=
1 φs (0)
C=−
h
φs2 dy,
0
1 φs (0)
h
p.v. 0
K (y) 2 φs dy + A , U − Us
(4.37)
s K (y j ) 2 π φ (y j ). φs (0) |U (y j )| s
m
D=−
j=1
Lemma 4.7 asserts that φs is nonzero at one of the inflection points, say φs (y j ) = 0. Note that by [39, Remark 4.2], for class K+ flows, U (y j ) = 0 for j = 1, 2, . . . , m s . Consequently, D < 0. The remainder of the proof is identically the same as that of [39, Theorem 4.1] and hence we only sketch it. Define f (ε, c) = (ε, c) − Bε − (C + Di)c, B f (ε, c) F(ε, c) = − ε− . C +iD C +iD
(4.38) (4.39)
Note that for each ε < 0 fixed, a zero of (ε, ·) corresponds to a fixed point of the mapping F(ε, ·). It is shown in [39, pp. 338-339] that for ε ∈ (ε0 , 0), where |ε0 | is sufficiently small, the mapping F(ε, ·) is contracting on (R,b(ε)) for some R > 0 and b(ε) = −2D B(C 2 + D 2 )−1 ε. So for each ε ∈ (ε0 , 0) there exists a unique c(ε) ∈ (R,b(ε)) such that F(ε, c(ε)) = c(ε) and thus (ε, c(ε)) = 0. It can also be shown that c(ε) is differentiable in ε in the interval (ε0 , 0). Since c(ε) ∈ (R,b(ε)) , it is immediate that (4.29) holds. Finally, differentiation of (ε, c(ε)) = 0 yields c (ε) = −
∂/∂ε , ∂/∂c
which, in view of (4.35) and (4.36), implies (4.30). Case 2. U (h) = Us . The proof is almost identical to that of Case 1, and thus we only indicate some differences. Let us define (ε, c) = gs (Us + c)φ1 (0; ε, c) + φ2 (0; ε, c), where gs is defined in (4.12). Note that φs (h) = φs (0) = 0. Thereby, the Green’s function is written as G(y, y ; ε, c) = φ1 (y; ε, c)φ2 (y ; ε, c) − φ2 (y; ε, c)φ1 (y ; ε, c). The same computations as in Case 1 yield that 1 ∂ →− ∂ε φs (0)
0
h
φs2 dy
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and ⎞ ⎛ h ms K (y j ) 2 1 ⎝ K ∂ (ε, c) → − φ (y j ) + p.v. φs2 dy ⎠ iπ ∂c φs (0) |U (y j )| s 0 U − Us j=1
uniformly as ε → 0− and c → 0 in E (R,b1 ,b2 ) . This completes the proof.
Proof of Theorem 4.2. Let −α 2N < −α 2N −1 < · · · < −α12 < 0 be negative eigenvalues 2
d of the operator − dy 2 − K (y) on (0, h) with boundary conditions (4.5)–(4.6). That is, 2 α N = αmax , where −αmax is defined either in (4.8a) or (4.8b). We deduce from Lemma 4.8 and Proposition 4.9 that to each α ∈ (0, α N ) with α = α j ( j = 1, . . . , N ) an unstable solution is associated. Our goal is to show the instability at α = α j for each j = 1, . . . , N − 1. Case 1. U (h) = Us . Let {(φk , αk , ck )}∞ k=1 be a sequence of unstable solutions such that αk → α j + as k → ∞. After normalization, we may assume φk (h) = 1. Note that φk satisfies U φk − αk2 φk − φk = 0 for y ∈ (0, h) (4.40) U − ck
and φk (h) = gr (ck ), φk (0) = 0. Below we will prove that Im ck δ > 0, where δ is independent of k. Since the coefficients of (4.40) and gr (ck ) are bounded uniformly for k, the solutions φk of the above Rayleigh equations are uniformly bounded in C 2 , and subsequently, φk converges in C 2 as k → ∞, say to φ∞ . The semicircle theorem (4.9) ensures that ck → c∞ as k → ∞. Note that Im c∞ δ > 0. By continuity, φ∞ satisfies φ∞ − α 2j φ∞ −
U φ∞ = 0 for y ∈ (0, h), U − c∞
(h) = g (c ) and φ (0) = 0. That is, (φ , α , c ) is an unstable with φ∞ (h) = 1, φ∞ r ∞ ∞ ∞ j ∞ solution of (4.1)–(4.2). It remains to show that {Im ck } has a positive lower bound. Suppose on the contrary that Im ck → 0 as k → ∞. We claim that φk L 2 C, where C > 0 is independent of k. Otherwise, φk L 2 → ∞ as k → ∞. Let ϕk = φk /φk L 2 , then ϕk L 2 = 1. Lemma 4.6 then dictates that ϕk H 2 C independently of k. Subsequently, Proposition 4.4 ensures that (ϕk , αk , ck ) converges to a neutral limiting mode (ϕs , αs , Us ). By continuity, ϕs L 2 = 1 and
ϕs − α 2j ϕs + K (y)ϕs = 0 for y ∈ (0, h). On the other hand, ϕs (h) = ϕs (h) = 0 since ϕk (h) = 1/φk L 2 → 0 and ϕk (h) = gr (ck )/φk L 2 → 0 as k → ∞. Correspondingly, ϕs ≡ 0 on [0, h] . A contradiction proves the claim. Since φk L 2 is bounded uniformly for k, Lemma 4.6 and Proposition 4.4 apply and φk H 2 C, ck → Us and φk → φs in C 1 , where φs satisfies φs − α 2j φs −
U φs = 0 for y ∈ (0, h) U − Us
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with φs (h) = 1, φs (h) = gr (Us ) and φs (0) = 0. An integration by parts yields that h
U U 2 2 φs (φk − αk φk − 0= φk ) − φk (φs − α j φs − φs ) dy U − ck U − Us 0 h h U = (α 2j −αk2 ) φs φk dy − (ck −Us ) φs φk dy + gr (ck )−gr (Us ). 0 0 (U −ck )(U −Us ) Let us denote
Bk = 0
h
φs φk dy,
U gr (ck ) − gr (Us ) φs φk dy + . (U − c )(U − U ) ck − Us k s 0 h It is immediate that limk→∞ Bk = 0 |φs |2 dy. We shall show in the Appendix that Dk = −
h
lim Dk = A + iπ
k→∞
ms K (y j ) 2 φ (y j ), |U (y j )| s
(4.41)
j=1
where A is defined by (4.31) and a j ( j = 1, 2, . . . , m s ) are inflection points corresponding to the inflection value Us . Since Im (limk→∞ Dk ) > 0 (see the proof of Proposition 4.9) it follows that Im ck = (αk2 − α 2j ) Im(Bk /Dk ) < 0 for k large. A contradiction proves that {Im ck } has a positive lower bound, uniformly for k. Case 2. U (h) = Us . We normalize φk so that φk (h) = 1 and φk (h) = gs (ck ). The proof is identically the same as that of Case 1 except that h U gs (ck ) φs φk dy + . Dk = − ck − Us 0 (U − ck )(U − Us ) We shall show in the Appendix that lim Dk = iπ
k→∞
h ms K (y j ) 2 K φ (y ) + p.v. φs2 dy. j |U (y j )| s U − U s 0
(4.42)
j=1
This proves that there exists an unstable solution for each α ∈ (0, αmax ). d2 It remains to prove linear stability in case either the operator − dy 2 − K (y) on y ∈ (0, h) with (4.5)–(4.6 ) is nonnegative or α αmax . Suppose otherwise, there exists an unstable mode at a wave number α αmax . By Lemma 4.8, we can continuate this unstable mode for wave numbers larger than α until the growth rate becomes zero. By Lemma 6.3, this continuation must stop at a wave number αs > α, where there is a neutral limiting mode. Then by Proposition 4.4, −αs2 is a negative eigenvalue of d2 2 2 − dy 2 − K (y) with (4.5)-(4.6). But −αs < −αmax which is a contradiction to the fact 2
d 2 that −αmax is the lowest eigenvalue of − dy 2 − K (y) with (4.5)–(4.6). This completes the proof.
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d Remark 4.10. For U ∈ K+ , let −αd2 be the lowest eigenvalue of − dy 2 − K (y) on y ∈ (0, h) with the Dirichlet boundary conditions φ(h) = 0 = φ(0). If U (h) = Us then αmax = αd . We claim that if U (h) = Us then αmax > αd . To see this, let φd be d2 the eigenfunction of − dy 2 − K (y) on y ∈ (0, h) with the Dirichlet boundary conditions 2
d corresponding to −αd2 and let φm be the eigenfunction of − dy 2 − K (y) with the bound2 . By Sturm’s theory, we can assume ary conditions (4.5)–(4.6) corresponding to −αmax φd , φm > 0 on y ∈ (0, h). An integration by parts yields that h 2 φd (φm − αmax φm + K (y)φm ) − φm (φd − αd2 φd + K (y)φd ) dy 0= 0
2 = −φm (h)φd (h) + (αd2 − αmax )
h
φd φm dy.
0
Since φd (h) < 0 the claim follows. For flows in class K+ , the lowest eigenvalue of (4.4) measures the range of instability, for both the free surface and the rigid wall cases. That means, (0, αmax ) is the interval of unstable wave numbers in the free-surface setting (Theorem 4.2) and (0, αd ) is the interval of unstable wave numbers in the rigid-wall setting [39, Theorem 1.2]. The fact that αmax > αd thus indicates that the free surface has a destabilizing effect. 4.3. Monotone unstable shear flows. In general, for a given shear flow profile in the class K+ , one might show the existence of a neutral limiting mode and thus the existence of growing modes, by numerically computing the negativity of (4.8). For monotone flows with one inflection point, however, the existence of neutral limiting modes can be shown ([62]) by a comparison argument. Lemma 4.11. For any monotone shear flow U (y) with exactly one inflection point ys in the interior, there exists a neutral limiting mode. That is, (4.1)–(4.2) has a nontrivial solution for which c = U (ys ) and α > 0. Proof. This result is given in Theorem 4 of [62]. Here we present a detailed proof for completeness and also for clarification of some arguments in [62]. We consider an increasing flow U (y) only. A decreasing flow can be treated in the same way. Let Us = U (ys ) be the inflection value. Denoted by φα the solution of the Rayleigh equation
U 2 − α φα = 0 for y ∈ (0, h) φα + Us − U with φα (0) = 0 and φα (0) = 1. As in the proof of Lemma 2.5, an integration of the above over (0, h) yields that h (U − Us )φα dy = 0, (U (h) − Us )φα (h) − (U (0) − Us ) − φα (h)U (h) − α 2 0
and thus φα (h) α2 U (0) − Us U (h) + = + φα (h) (U (h) − Us )φα (h) U (h) − Us (U (h) − Us )φα (h)
0
h
(U − Us )φα dy.
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It is straightforward to see that the boundary condition (4.2) is satisfied if and only if the function h α2 U (0) − Us g + f (α) = (U − Us )φα dy − (U (h) − Us )φα (h) (U (h) − Us )φα (h) 0 (Us − U (h))2 vanishes at some α > 0. We claim that φα (y) > 0 on y ∈ (0, h] for any α 0 . Suppose otherwise, let yα ∈ (0, h] be the first zero of φα other than 0, that is, φα (yα ) = 0 and φα (y) > 0 for y ∈ (0, yα ). Then, yα > ys must hold. Indeed, if yα ys were to be true, then Sturm’s first comparison theorem would apply to φα and U − Us on [0, yα ] to assert that U − Us must vanish somewhere in (0, yα ) ⊂ (0, ys ). This uses that (U − Us ) +
U (U − Us ) = 0. Us − U
A contradiction then asserts that yα > ys . Correspondingly, φα and U − Us have exactly one zero in [0, yα ]. On the other hand, by Sturm’s second comparison theorem [31], it follows that U (yα ) φα (yα ) . φα (yα ) U (yα ) − Us This contradicts since φα (yα ) = 0 and the left hand side is −∞. Therefore, φα (y) > 0 for y ∈ (0, h] and for any α 0. In particular, φα (h) > 0 for any α 0. Consequently, f is a continuous function of α and f (0) < 0. It remains to show that f (α) > 0 for α > 0 big enough. Thereby, by continuity f vanishes at some α > 0. Let UU M and α 2 > M. Let us denote by φ1 and φ2 the s −U solutions of φ1 + (M − α 2 )φ1 = 0 and φ2 + (−M − α 2 )φ2 = 0
for y ∈ (0, h),
respectively, with φi (0) = 0 and φi (0) = 1. It is straightforward that φ1 (y) = √
1 α2 − M
sinh
1 sinh α 2 + M y. α 2 − M y and φ2 (y) = √ α2 + M
As in the proof of Lemma 2.7 (a), Sturm’s second comparison theorem [31] implies that 1 1 sinh α 2 + M y. sinh α 2 − M y φα (y) √ √ α2 − M α2 + M This together with the monotone property of U establishes that f (α) C1 α − C2 for some constants C1 , C2 > 0. For details we refer to [62, Theorem 4]. Thus, f (α) > 0 if α > 0 is sufficiently large. This completes the proof .
Since a monotone flow with one inflection value is in class K+ , the above lemma combined with Theorem 4.2 asserts its instability. Corollary 4.12. Any monotone shear flow with exactly one inflection point in the interior is unstable in the free-surface setting, for wave numbers in an interval (0, αmax ) with αmax > 0.
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Remark 4.13. In the free-surface setting, there are two different kinds of neutral modes, which are solutions to the Rayleigh system (4.1)–(4.2) with Im c = 0. Neutral limiting modes have their phase speed in the range of the shear profile. For flows in class K+ , moreover, the phase speed of a neutral limiting mode must be the inflection value of the shear profile (Proposition 4.4) and it is contiguous to unstable modes (Proposition 4.9). On the other hand, Lemma 2.5 shows that under the conditions U (h) < 0 and U (h) > U (y) for h = y, a neutral mode exists with the phase speed c > max U . Such a neutral mode is used in the local bifurcation of nontrivial periodic waves in Theorems 2.2 and 2.6. In view of the semicircle theorem (4.9), however, such a neutral mode is not contiguous to unstable modes. This implies that neutral modes governing the stability property are different from those governing the bifurcation of nontrivial waves. This is an important difference in the free-surface setting. Since in the rigid wall setting, for any possible neutral modes the phase speed must lie in the range of U , which follows easily from Sturm’s first comparison theorem. For class K+ flows, such neutral modes are contiguous to unstable modes ([39]) and the bifurcation of nontrivial waves from these neutral modes can also be shown ([9]). Thus, in the rigid-wall setting, the same neutral modes govern both stability and bifurcation. Remark 4.14. In [19], the J -formal stability was introduced via a quadratic form, which is related to the local bifurcation of nontrivial waves in the transformed variables (see also the Introduction), and it was concluded that this formal stability of the trivial solutions switches exactly at the bifurcation point. Below we discuss two examples for which the linear stability property does not change along the line of trivial solutions passing the bifurcation point, which indicates that the J -formal stability in [19] is unrelated to linear stability of the physical water wave problem. However, the J -formal stability results [19] do give more information about the structure of the periodic water wave branch. Let us consider a monotone increasing flow U (y) on y ∈ [0, h] with one inflection point ys ∈ (0, h), for example, U (y) = a sin b(y − h/2) on y ∈ [0, h] for which ys = h/2. By Lemma 4.11 and Corollary 4.12, such a shear flow is equipped with a neutral limiting mode with c = U (ys ) and α = αmax > 0, and it is linearly unstable for any wave number α ∈ (0, αmax ). In addition, by Lemma 2.5 and Theorem 2.6 for any wave number α ∈ (0, αmax ) this shear flow has a neutral mode with c(α) > U (h) = max U . Moreover, such a neutral mode is a nontrivial solution to the bifurcation equation (2.9)– (2.10), and thus there exists a local curve of bifurcation of nontrivial waves with a wave speed c(α) and period 2π/α. Let p0 and γ be the flux and vorticity relation determined by U (y) , c(α) and h via (2.11). Consider the trivial solutions with shear flows U (y; µ) defined in Lemma 2.1, with the above p0 , γ and the parameter µ. The bifurcation point U (y; µ0 ) = U (y) − c(α) corresponds to µ0 = (U (h) − c(α))2 . The instability of U (y; µ0 ) at the wave number α is continuated to shear flows U (y; µ) with µ near µ0 , which can be shown by a similar argument as in the proof of Lemma 4.8. So at the bifurcation point µ0 , there is NO switch of stability of trivial solutions. Let us consider U ∈ C 2 ([0, h]) satisfying that U (y) > 0 and U (y) < 0 in y ∈ [0, h]. For such a shear flow Lemma 2.5 and Theorem 2.6 apply as well and there exists a local curve of bifurcation of nontrivial waves for any wave number α which travels at the speed c(α) > max U , where c(α) is chosen so that the bifurcation equation (2.9)–(2.10) is solvable. For this bifurcation flow U (y) − c(α), the vorticity relation γ determined via (2.11) is monotone since U does not change sign. Consequently, any shear flows U (y; µ) defined in Lemma 2.1 with the same γ has no inflection points, as is also remarked at the end of Sect. 3. Therefore, by Theorem 6.4 all trivial solutions
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corresponding to these shear flows U (y; µ) are stable. This again shows that the bifurcation of nontrivial periodic waves does not involve the switch of stability of trivial solutions. 5. Linear Instability of Periodic Water Waves with Free Surface We now turn to investigating the linear instability of periodic traveling waves near an unstable background shear flow. Suppose that a shear flow (U (y), 0) with U ∈ C 2+β ([0, h 0 ]) and U ∈ K+ has an unstable wave number α > 0, that is, for such a wave number α > 0 the Rayleigh system (4.1)–(4.2) has a nontrivial solution with Im c > 0. Suppose moreover that for the unstable wave number α > 0 the bifurcation equation (2.9)–(2.10) is solvable with some c(α) > max U . Then Remark 2.4 and Theorem 2.2 apply to state that there exists a one-parameter curve of small-amplitude traveling-wave solutions (ηε (x), ψε (x, y)) satisfying (2.2) with the period 2π/α and the wave speed c(α), where ε 0 is the amplitude parameter. A natural question is: are these small-amplitude nontrivial periodic waves generated over the unstable shear flow also unstable? The answer is YES under some technical assumptions, which is the subject of the forthcoming investigation.
5.1. The main theorem and examples. We prove the linear instability of the steady periodic water-waves (ηε (x), ψε (x, y)) by finding a growing-mode solution to the linearized water-wave problem. As in Sect. 4, let Dε = {(x, y) : 0 < x < 2π/α, 0 < y < ηε (x)} and Sε = {(x, ηε (x)) : 0 < x < 2π/α} denote, respectively, the fluid domain of the steady wave (ηε (x), ψε (x, y)) of one period and the steady surface. The growing-mode problem (3.9) of the linearized periodic waterwave problem around (ηε (x), ψε (x, y)) reduces to
ψ + γ (ψε )ψ − γ (ψε )
0 −∞
λeλs ψ(X ε (s), Yε (s))ds = 0 in Dε ;
(5.1a)
d d ψεy (x, ηε (x))η(x) = − ψ(x, ηε (x)); (5.1b) dx dx P(x, ηε (x)) + Pεy (x)η(x) = 0; (5.1c)
d d d ψεy (x, ηε (x))ψn (x) = − P(x, ηε (x)) − ψ(x, ηε (x)); (5.1d) λψn (x) + dx dx dx ψ(x, 0) = 0. (5.1e) λη(x) +
Here and in the sequel, let us abuse notation and denote that Pεy (x) = Pεy (x, ηε (x)), that is, the restriction of Pεy (x, y) on the steady wave-profile y = ηε (x). By Theorem 2.2, Pεy (x) = Pεy (x, ηε (x)) = −g + O(ε). Recall that ψn (x) = ∂ y ψ(x, ηε (x)) − ηεx (x)∂x ψ(x, ηε (x)) is the derivative of ψ(x, ηε (x)) in the direction normal to the free surface (x, ηε (x)) and that = γ (0) is the vorticity of the steady flow of ψε (x, y) on the steady wave-profile y = ηε (x). Note that is a constant independent of ε.
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Theorem 5.1 (Linear instability of small-amplitude periodic water-waves). Let the shear flow U (y) ∈ C 2+β ([0, h 0 ]), β ∈ (0, 1), be in class K+ . Suppose that U (h 0 ) = Us , where Us is the inflection value of U , and that αmax defined by (4.8a) is positive, as such Theorem 4.2 applies to find the interval of unstable wave numbers (0, αmax ). Suppose moreover that for some α ∈ (αmax /2, αmax ) there exists c(α) > max U such that the bifurcation equation (2.9)–(2.10) has a nontrivial solution. Let us denote by (ηε (x), ψε (x, y)) the family of nontrivial waves with the period 2π/α and the wave speed c(α), bifurcating from the trivial solution η0 (x) ≡ h 0 and (ψ0y (y), −ψ0x (y)) = U (y) − c(α), 0), where ε 0 is the amplitude parameter. Provided that g + U (h 0 )(U (h 0 ) − Us ) > 0,
(5.2)
then for each ε > 0 sufficiently small, there exists an exponentially growing solution (eλt η(x), eλt ψ(x, y)) of the linearized system (3.6), where Re λ > 0, with the regularity property (η(x), ψ(x, y)) ∈ C 2+β ([0, 2π/α]) × C 2+β (D¯ ε ). Remark 5.2 (Examples). As discussed in Remark 4.14, any increasing shear flow U ∈ C 2+β ([0, h 0 ]), β ∈ (0, 1), with exactly one inflection point in y ∈ (0, h 0 ) satisfies αmax > 0. Moreover, Lemma 2.5 applies and small-amplitude periodic waves bifurcate at any wave number α > 0. Since (5.2) holds true, therefore, by Theorem 5.1 small-amplitude periodic waves bifurcating from such a shear flow at any wave number α ∈ (αmax /2, αmax ) are unstable. Below, we discuss in detail such an example: U (y) = a sin b(y − h 0 /2)
for y ∈ [0, h 0 ],
(5.3)
where h 0 , b > 0 satisfy h 0 b π and a > 0 is arbitrary. (1) The shear flow in (5.3) is unstable under periodic perturbations of a wave number α ∈ (0, αmax ), where αmax > 0. Note that in the rigid-wall setting [39], the same shear flow is stable under perturbations of any wave number. This indicates that the free surface has a destabilizing effect. This serves as an example of Remark 4.10 since αmax > 0 and αd = 0. (2) The amplitude a and the depth h 0 in (5.3) may be chosen arbitrarily small, and the shear flow as well as the nontrivial periodic waves near the shear flow are unstable for any wave number α ∈ (0, αmax ), which contrasts with the result in [19] that small-amplitude rotational periodic water-waves are J -formally stable if the vorticity strength and the depth are sufficiently small. Thus, as also commented in Remark (4.14), the J -formal stability in [19] is not directly related to the linear stability of water waves. Indeed, while ∂J (η, ψ) = 0 gives the equations for steady steady waves, the linearized water-wave problem is not in the form ∂t (η, ψ) = (∂ 2 J )(η, ψ), which is implicitly required in [19] in order to apply the Crandall-Rabanowitz theory [21] of the exchange of stability. (3) Our example (5.3) also indicates that adding an arbitrarily small vorticity to the irrotational water wave system of an arbitrary depth may induce instability. That means, although small irrotational periodic waves are found to be stable under perturbations of the same period [48,55], they are not structurally stable; vorticity has a subtle influence on the stability of water waves.
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The proof of Theorem 5.1 uses a perturbation argument. At ε = 0 the trivial solution (η0 (x), ψ0 (x, y)) corresponds to the shear flow (U (y) − c(α), 0) under the flat surface {y = h 0 }. To simplify notations, in the remainder of this section, we write U (y) for U (y) − c(α), as is done in Sect. 5. Thereby, U (y) < 0. Since α is an unstable wave number of U (y), there exist an unstable solution φα to the Rayleigh system (4.1)–(4.2) and an unstable phase speed cα . That is, φα ≡ 0, Im cα > 0 and U φα − α 2 φα + φα = 0 for y ∈ (0, h 0 ), U − cα
g U (h 0 ) φα (h 0 ), φα (h 0 ) = + (U (h 0 ) − cα )2 U (h 0 ) − cα φα (0) = 0.
(5.4)
This corresponds to a growing mode solution satisfying (5.1) at ε = 0, where λ0 = −iαcα has a positive real part. Our goal is to show that for ε > 0 sufficiently small, there exists λε near λ0 such that the growing-mode problem (5.1) at (ηε (x), ψε (x, y)) is solvable. First, the system (5.1) is reduced to an operator equation defined in a function space independent of ε. Then, by showing the continuity of this operator with respect to the small-amplitude parameter ε, the continuation of the unstable mode follows from the eigenvalue perturbation theory of operators. 5.2. Reduction to an operator equation. The purpose of this subsection is to reduce the growing mode system (5.1) to an operator equation on L 2per (Sε ). Here and elsewhere the subscript per denotes the periodicity in the x-variable. The idea is to express η(x) on Sε and ψ(x, y) in Dε (and hence P(x, ηε (x))) in terms of ψ(x, ηε (x)). Our first task is to relate η(x) with ψ(x, ηε (x)). Lemma 5.3. For |λ − λ0 | (Re λ0 )/2 , where λ0 = −iαcα , let us define the operator C λ : L 2per (Sε ) → L 2per (Sε ) by x 1 1 −1 φ(x) + λeλa(x ) ψεy (x )φ(x )d x C λ φ(x) = − λa(x) ψεy (x) ψεy (x)e 0 (5.5) 2π/α λ λa(x ) −1
− e ψεy (x )φ(x )d x , ψεy (x)eλa(x) 1 − eλa(2π/α) 0 x −1 where a(x) = 0 ψεy (x , ηε (x ))d x . For simplicity, here and in the sequel we identify ψey (x) with ψey (x, ηε (x)) and φ(x) with φ(x, ηε (x)), etc. Then, (a) The operator C λ is analytic in λ for |λ − λ0 | (Re λ0 )/2, and the estimate λ C 2 K L (S )→L 2 (S ) per
ε
per
ε
holds, where K > 0 is independent of λ and ε. (b) For any φ ∈ L 2per (Sε ), the function ϕ = C λ φ is the unique L 2per (Sε )- weak solution of the first-order ordinary differential equation λϕ +
d d ψεy (x)ϕ = − φ. dx dx
(5.6)
1 (S ) then ϕ ∈ C 1 (S ) is the unique classical solution of If, in addition, φ ∈ Cper ε per ε (5.6).
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Proof. Assertions of (a) follow immediately since ψεy (x) < 0 and thus a(x) < 0 and since Re λ Re λ0 /2 > 0. 1 (S ) to motivate the definition of C λ . (b) First, we consider the case φ ∈ Cper ε Let us write (5.6) as the first-order ordinary differential equation
d 1 d 1 d λ+ ϕ+ ψεy ϕ = − φ, dx ψεy dx ψεy d x which has an unique 2π/α-periodic solution x 1 d ϕ(x) = − eλa(x ) φ(x )d x λa(x) dx ψεy (x)e 0 2π/α 1 d − eλa(x ) φ(x )d x . λa(2π/α) dx 1−e 0 An integration by parts of the above formula yields that ϕ(x) = C λ φ is as defined in (5.5). In case φ ∈ L 2per (Sε ), the integral representation (5.5) makes sense and solves (5.6) in the weak sense. Indeed, an integration by parts shows that ϕ defined by (5.5) satisfies the weak form of Eq. (5.6), 2π/α
d d λϕ(x)h(x) − ψεy (x)ϕ(x) h(x) − φ(x) h(x) d x = 0 dx dx 0 1 ([0, 2π/α]). In order to show the uniqueness, for any 2π/α-periodic function h ∈ Hper 2 suppose that ϕ˜ ∈ L per (Sε ) is another weak solution of (5.6). Let ϕ1 = ϕ − ϕ. ˜ Then, 2 ϕ1 ∈ L per (Sε ) is a weak solution of the homogeneous differential equation
λϕ1 +
d ψεy (x)ϕ1 = 0. dx
x 2π/α ϕ1 (x)d x = 0. Note that h 1 (x) = 0 ϕ1 (x )d x defines a It is readily seen that 0 1 ([0, 2π/α]). Multiplication of the above homogeneous 2π/α-periodic function in Hper ∗ equation by (λh 1 ) and an integration by parts then yield that
∗ 2π/α
d |λ|2 ϕ1 (x)h ∗1 (x) − ψεy (x)λ∗ ϕ1 (x) dx 0 = Re h 1 (x) dx 0 2π/α 2π/α d 1 2 2 |h 1 | d x − Re λ = |λ| ψεy (x) |ϕ1 |2 d x dx 2 0 0 2π/α = − Re λ ψεy (x) |ϕ1 |2 d x. 0
Here and elsewhere, the asterisk denotes the complex conjugation. Since Re λ > 0 and ψεy (x) < 0, it follows that ϕ1 ≡ 0, and in turn, ϕ ≡ ϕ. ˜ This completes the proof.
Formally, the operator C λ can be written as
−1 d d ψεy (x)φ(x) φ(x). C λ φ(x) = − λ + dx dx
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Let us denote f (x) = ψ(x, ηε (x)). With the use of C λ then the boundary conditions (5.1b)–(5.1 d) are written in terms of f as η(x) = C λ f (x), P(x, ηε (x)) = −Pεy (x)C λ f (x), ψn (x) = −C λ (Pεy (x)C λ + id) f (x), where id : L 2 (Sε ) → L 2 (Sε ) is the identity operator. Our next task is to relate ψ(x, y) in Dε with f (x) = ψ(x, ηε (x)). Given b ∈ L 2per (Sε ), let ψb ∈ H 1 (Dε ) be a weak solution of the elliptic partial differential equation 0 ψ + γ (ψε )ψ − γ (ψε ) λeλs ψ(X ε (s), Yε (s))ds = 0 in Dε , (5.7a) −∞
ψn (x) := ∂ y ψ(x, ηε (x)) − ηεx (x)∂x ψ(x, ηε (x)) = b(x) on Sε , ψ(x, 0) = 0,
(5.7b) (5.7c)
such that ψb is 2π/α-periodic in the x-variable. Lemma 5.6 below proves that the boundary value problem (5.7) is uniquely solvable and ψb ∈ H 1 (Dε ) provided that |λ − λ0 | (Re λ0 )/2 and (ηε (x), ψε (x, y)) is near the trivial solution with the flat surface y = h 0 and the unstable shear flow (U (y), 0) given in Theorem 5.1. This, together with the trace theorem, allows us to define an operator Tε : L 2per (Sε ) → L 2per (Sε ) by Tε b(x) = ψb (x, ηε (x)),
(5.8)
which is the unique solution ψb of (5.7) restricted on the steady surface Sε . Be definition, it follows that f (x) = ψ(x, ηε (x)) = Tε ψn (x). This, together with the boundary conditions written in terms of f as above yields that f = −Tε C λ (Pεy (x)C λ + id) f.
(5.9)
The growing-mode problem (5.1) is thus reduced to find a nontrivial solution f (x) = ψ(x, ηε (x)) ∈ L 2per (Sε ) of Eq. (5.9), or equivalently, to show that the operator id + Tε C λ (Pεy (x)C λ + id) has a nontrivial kernel for some λ ∈ C with Re λ > 0. The remainder of this subsection concerns the unique solvability of ( 5.7). Our first task is to compare (5.7) at ε = 0 with the Rayleigh system, which is useful in later consideration. 2 Lemma 5.4. For U (y) in class K+ , y ∈ [0, h 0 ] and U (h 0 ) = Us , let −αmax be the 2 d lowest eigenvalue of − dy 2 − K (y) for y ∈ (0, h 0 ) subject to the boundary conditions
U (h 0 ) g φ (h 0 ) = φ(h 0 ) and φ(0) = 0, + (5.10) (U (h 0 ) − Us )2 U (h 0 ) − Us 2
d where K is defined in (4.3). Let −αn2 be the lowest eigenvalue of − dy 2 − K (y) for y ∈ (0, h 0 ) subject to the boundary conditions
φ (h 0 ) = 0 and φ(0) = 0. If g
+ U (h
0 )(U (h 0 ) − Us )
> 0, then αmax > αn .
(5.11)
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Proof. The argument is nearly identical to that in Remark 4.10. Let us denote by φm d2 the eigenfunction of − dy 2 − K (y) on y ∈ (0, h 0 ) with (5.10) corresponding to the 2
d 2 and by φn the eigenfunction of − dy eigenvalue −αmax 2 − K (y) on y ∈ (0, h 0 ) with
(5.11) corresponding to the eigenvalue −αn2 . By the standard theory of Sturm-Liouville operators we can assume φm > 0 and φn > 0 on y ∈ (0, h 0 ). An integration by parts yields that h0 2 φn (φm − αmax φm + K (y)φm ) − φm (φn − αn2 φn + K (y)φn ) dy 0= 0
2 = (αn2 − αmax )
The assumption g
h0
φn φm dy +
0
+ U (h
g + U (h 0 )(U (h 0 ) − Us ) φm (h 0 )φn (h 0 ). (U (h 0 ) − Us )2
0 )(U (h 0 ) − Us )
> 0 then proves the assertion.
Our next task is the unique solvability of the homogeneous problem of (5.7). Lemma 5.5. Assume that U (h 0 ) = Us and g + U (h 0 )(U (h 0 ) − Us ) > 0. For ε > 0 sufficiently small and |λ − λ0 | (Re λ0 )/2, the following elliptic partial differential equation: 0 ψ + γ (ψε )ψ − γ (ψε ) λeλs ψ(X ε (s), Yε (s))ds = 0 in Dε (5.12a) −∞
subject to
ψ(x, 0) = 0
(5.12b)
and the Neumann boundary condition ψn = 0
on Sε
(5.12c)
admits only the trivial solution ψ ≡ 0. The main difficulty in the proof of Lemma 5.5 is that the domain Dε depends on the small amplitude parameter ε > 0 whereas the statement of Lemma 5.5 calls for an estimate of solutions of the system (5.12) uniform for ε > 0. In order to compare (5.12) for different values of ε, we employ the action-angle variables, which map the domain Dε into a common domain independent of ε. For any (x, y) ∈ Dε , let us denote by {(x , y ) : ψε (x , y ) = ψε (x, y) = p} the streamline containing (x, y) , by σ the arc-length variable on the streamline {ψε (x , y ) = p}, and by σ (x, y) the value of σ corresponding to the point (x, y) along the streamline. Let us define the normalized action-angle variables as σ (x,y) α h0 1 I = dy d x , and θ = υε (I ) dσ , 2π h ε |∇ψε | {ψε (x y )= p} {ψε (x ,y )< p} 0 (5.13) where h 0 and h ε are the mean water depth at the parameter values 0 and ε, respectively, and
−1 1 2π υε (I ) = . α {ψε (x ,y )= p} |∇ψε |
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The action variable I represents the (normalized) area in the phase space under the streamline {(x , y ) : ψε (x , y ) = ψε (x, y) = p} and the angle variable θ represents the position along the streamline of ψε (x, y). The assumption of no stagnation, i.e. ψεy (x, y) < 0 throughout Dε , implies that all streamlines are non-closed. For, otherwise, the horizontal velocity ψεy must change signs on a closed streamline. Moreover, for ε > 0 sufficiently small, all streamlines are close to those of the trivial flow, that is, they are almost horizontal. Therefore, the action-angle variable (θ, I ) is defined globally in Dε . The mean-zero property (2.8) of the wave profile ηε (x) implies that the area of the (steady) fluid region Dε is (2π/α)h ε . Accordingly, by the definition in (5.13) it follows that 0 < I < h 0 and 0 < θ < 2π/α, independently of ε. Let us define the mapping by the action-angle variables by Aε (x, y) = (θ, I ). From the above discussions follows that Aε is bijective and maps Dε to D = {(θ, I ) : 0 < θ < 2π/α, 0 < I < h 0 }. At ε = 0, the action-angle mapping reduces to the identity mapping on D0 = D. For ε > 0, the mapping has a scaling effect. More precisely, h0 f (A−1 (θ, I ))dθ d I = f (x, y)d yd x ε hε D Dε for any function f defined in Dε . Another motivation to employ the action-angle variables comes from the fact that they simplify the equation on the particle trajectory3 . In the action-angle variables (θ, I ), the characteristic equation (3.8) becomes ([4, Sect. 50], [41, p. 94]) θ˙ = −υε (I ) I˙ = 0, where the dot above a variable denotes the differentiation in the σ -variable. This observation is very useful for future considerations. Proof of Lemma 5.5. Suppose on the contrary that there would exist sequences εk → 0+, λk → λ0 as k → ∞ and ψk ∈ H 1 (Dεk ) such that ψk ≡ 0 is a solution of (5.12) with ε = εk . After normalization, ψk L 2 (Dε ) = 1. We claim that k
ψk H 2 (Dε
k)
C,
where C > 0 is independent of k. Indeed, by Minkovski’s inequality it follows that 0 λs γ (ψε )ψk − γ (ψε ) λe ψ (X (s), Y (s))ds k εk εk k k
−∞
(5.14)
L 2 (Dεk )
0 |λ| eRe λs ψk (X εk (s), Yεk (s)) L 2 (D ) ds γ (ψεk ) L ∞ ψk L 2 (Dε ) + εk k −∞
|λ| |λ0 | + δ γ (ψεk ) L ∞ 1 + . (5.15) = γ (ψεk ) L ∞ ψk L 2 (Dε ) 1 + k Re λ δ
3 In the irrotational setting, i.e., γ ≡ 0, a qualitative description of particle trajectories is obtained [16] by studying a specific nonlinear boundary value problem for harmonic functions.
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This uses the fact that the mapping (x, y) → (X εk (s), Yεk (s)) is measure preserving and that |λ − λ0 | < Re λ0 /2. The standard elliptic regularity theory [2] for a Neumann problem adapted for (5.12) then proves the estimate (5.14). This proves the claim. In order to study the convergence of {ψk }, we perform the mapping by the actionangle variables (5.13) and write Aεk (x, y) = (θ, I ). Note that the image of Dεk under the mapping Aεk is D = {(θ, I ) : 0 < θ < 2π/α , 0 < I < h 0 }, which is independent of k. Let us denote Aψk (θ, I ) = ψk (A−1 εk (θ, I )). It is immediate 2 to see that Aψk ∈ H (D). Since h ε = h 0 + O(ε) it follows that Aψk L 2 (D) = (h 0 / h ε )ψk L 2 (Dε
k)
= 1 + O(ε).
This, together with (5.14), implies that Aψk → ψ∞ weakly in H 2 (D)
as k → ∞,
Aψk → ψ∞ strongly in L 2 (D)
as k → ∞
for some ψ∞ . By continuity, ψ∞ L 2 (D0 ) = 1. Our goal is to show that ψ∞ ≡ 0 and thus prove the assertion by contradiction. At the limit as εk → 0, the limiting mapping A0 is the identity mapping on D = D0 and the limit function ψ∞ satisfies
ψ∞ + γ (ψ0 )ψ∞ − γ (ψ0 ) ∂ y ψ∞ (x, h 0 ) = 0; ψ∞ (x, 0) = 0.
0 −∞
λ0 eλ0 s ψ∞ (X 0 (s), Y0 (s))ds = 0
in D; (5.16)
Here, λ0 = −iαcα and (X 0 (s), Y0 (s)) = (x + U (y)s, y) . Since the above equation and the boundary conditions are separable in the x and y variables, ψ∞ can be written as ψ∞ (x, y) =
∞
eilαx φl (y).
l=0
In case l = 0, the boundary value problem (5.16) reduces to φ0 = 0 for y ∈ (0, h 0 ) φ0 (h 0 ) = 0, φ0 (0) = 0, and thus, φ0 ≡ 0. Next, consider the solution φ1 of (5.16) when l = 1: φ1 − α 2 φ1 − UU−cα φ1 = 0 for y ∈ (0, h 0 ) φ1 (h 0 ) = 0, φ1 (0) = 0.
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This uses that γ (ψ0 ) = −U /U . Recall that for the unstable wave number α and the unstable wave speed cα , there exists an unstable solution φα of the Rayleigh system (5.4). As is done in the proof of Lemma 5.4, an integration by parts yields that
U U dy φα − φα φ1 − α 2 φ1 − φ1 U − cα U − cα 0
g U (h 0 ) = φα (h 0 )φ1 (h 0 ). + (U (h 0 ) − cα )2 U (h 0 ) − cα
0=
h0
φ1 φα − α 2 φα −
Since φα (h 0 ) = 0, we have φ1 (h 0 ) = 0, which together with φ1 (h 0 ) = 0, implies that φ1 ≡ 0. For l 2, the solution φl of (5.16) ought to satisfy U φl − l 2 α 2 φl − U −c φl = 0 for y ∈ (0, h 0 ) α /l φl (h 0 ) = 0, φl (0) = 0. An integration by parts yields that
h0 0
φ 2 + l 2 α 2 |φl |2 + l
U |φl |2 dy = 0, U − cα /l
and subsequently, for any q real it follows that (see the proof of Lemma 4.6)
φ 2 + l 2 α 2 |φl |2 + U (U − q) |φl |2 dy = 0. l 2 |U − cα /l|
h0
0
The same argument as in proving Lemma 4.6 applies to assert that
h0
0
2 (φl + l 2 α 2 |φl |2 )dy
h0
K (y)|φl |2 dy.
0
−U (y)/(U (y) − U
Recall that K (y) = s ) > 0. d2 2 Let −αn be as in Lemma 5.4 the lowest eigenvalue of dy 2 − K (y) on y ∈ (0, h 0 ) with the boundary conditions φ (h 0 ) = 0 and φ(0) = 0. By Lemma 5.4, αmax > αn . On the other hand, the variational characterization of −αn2 asserts that 0
h
(|φl2
− K (y)|φl | )dy 2
−αn2
h
|φl |2 dy.
0
Accordingly,
h0 0
(l 2 α 2 − αn2 )|φl |2 dy 0
2 must hold. Since α > αmax /2 and l 2, it follows that l 2 α 2 − αn > 2αmax − αn2 > 0. Consequently, φl ≡ 0. Therefore, ψ∞ ≡ 0, which contradicts since φ∞ L 2 = 1. This completes the proof.
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779
Lemma 5.6. Under the assumption of Lemma 5.5, for any b ∈ L 2 (Sε ), there exists an unique solution ψb to (5.7). Moreover, the estimate ψb H 1 (Dε ) Cb L 2 (Sε )
(5.17)
holds, where C > 0 is independent of ε, b. Proof. The proof uses the theory of Fredholm alternative as adapted to usual elliptic problems [28, Sect. 6.2]. Let us introduce the Hilbert space H (Dε ) = {ψ ∈ H 1 (Dε ) : ψ(x, 0) = 0}
(5.18)
and a bilinear form Bz : H × H → R, defined as
Bz [φ, ψ] = ∇φ · ∇ψ ∗ + φ(K λ ψ)∗ d yd x + z (φ, ψ) . Dε
Here, λ
K ψ = −γ (ψε )ψ + γ (ψε )
0 −∞
λeλs ψ(X ε (s), Yε (s))ds,
z ∈ R and (·, ·) denotes the L 2 (Dε ) inner product. By the estimate (5.15) it follows that |Bz [φ, ψ]| (C(γ , δ) + z)φ H 1 ψ H 1
(5.19)
Bz [φ, φ] c0 φ2H 1
(5.20)
and for z > 2C(γ , δ), where
|λ0 | + δ C(γ , δ) = γ (ψε ) L ∞ 1 + δ
and c0 = min(1, C(γ , δ)) > 0. Then, by the Lax-Milgram theorem there exists a bounded operator L z : H ∗ → H such that Bz [L z f, φ] = f, φ for any f ∈ H ∗ and φ ∈ H , where H ∗ denotes the dual space of H and ·, · is the duality pairing. For b ∈ L 2 (Sε ), the trace theorem [28, Sect. 5.5] permits us to define b∗ ∈ H ∗ by ∗ b(x)ψ ∗ (x, ηε (x))d x for ψ ∈ H. < b , ψ >= Sε
Note that ψb ∈ H is a weak solution of (5.7) if and only if Bz (ψb , φ) = < zψb + b∗ , φ >
for all φ ∈ H.
That is to say, ψb = L z (zψb + b∗ ), or equivalently (id − z L z )ψb = L z b∗ . The operator L z : L 2 (Dε ) → L 2 (Dε ) is compact. Indeed, L z φ H 1 (Dε ) L z H ∗ →H φ H ∗ C φ L 2 (Dε )
(5.21)
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V. M. Hur, Z. Lin
for any φ ∈ L 2 (Dε ). Moreover, the result of Lemma 5.5 states that ker(I − z L z ) = {0}. Thus, by the Fredholm alternative theory for compact operators, Eq. (5.21) is uniquely solvable for any b ∈ L 2 (Sε ) and ψb = (id − z L z )−1 L z b∗ .
(5.22)
Next is the proof of (5.17). From (5.22) it follows that ψb L 2 (Dε ) (id − z L z )−1 2 2 L z H ∗ →H b∗ H ∗ Cb L 2 (Sε ) , L →L
where C > 0 is independent of b and ε. Then, by (5.21) it follows that ψb H 1 (Dε ) = L z (zψb ) + L z b∗ H C(ψb L 2 (Dε ) + b L 2 (Sε ) ) C b L 2 (Sε ) , where C, C > 0 are independent of b and ε. This completes the proof.
Similar considerations to the above proves the unique solvability of the inhomogeneous problem. Corollary 5.7. Under the assumption of Lemma 5.5, for any f ∈ H ∗ (Dε ) there exists an unique weak solution ψ f ∈ H 1 (Dε ) to the elliptic problem 0 ψ + γ (ψε )ψ − γ (ψε ) λeλs ψ(X ε (s), Yε (s))ds = f in Dε , ψn = 0 on Sε , ψ(x, 0) = 0. The estimate
−∞
ψ f H 1 (Dε ) C f H ∗ (Dε )
holds, where C > 0 is independent of f and ε. Moreover, if f ∈ improved estimate ψ f H 2 (Dε ) C f L 2 (Dε )
(5.23) L 2 (D
ε)
then an (5.24)
holds, where C > 0 is independent of f and ε. Proof. As in the proof of Lemma 5.6, the function ψ f is a solution to the above boundary value problem if and only if (id − z L z )ψ f = L z f. The unique solvability and the H 1 -estimate (5.23) are identically the same as those in Lemma 5.6. When f ∈ L 2 (Dε ) the elliptic regularity theory [2] for the Neumann boundary condition implies (5.24).
For any b ∈ L 2per (Sε ) let us define the operator Tε b = ψb |Sε = ψb (x, ηε (x)) ,
(5.25)
where ψb is the unique solution of (5.7) in Lemma 5.6 with periodicity. Then, the elliptic estimate (5.17) and the trace theorem [28, Sect. 5.5] implies that Tε b H 1/2 (Sε ) Cψb H 1 (Dε ) C b L 2 . Therefore, Tε : L 2per (Sε ) → L 2per (Sε ) is a compact operator.
(5.26)
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5.3. Proof of Theorem 5.1. This subsection is devoted to the proof of Theorem 5.1 pertaining to the linear instability of small-amplitude periodic traveling waves over an unstable shear flow. For ε 0 sufficiently small and |λ − λ0 | (Re λ0 )/2, let us denote F(λ, ε) = T (λ, ε) C (λ, ε) (Pεy C (λ, ε) + id) : L 2per (Sε ) → L 2per (Sε ),
(5.27)
where C(λ, ε) = C λ and T (λ, ε) = Tε are defined in (5.5) and (5.25), respectively. In the light of the discussion in the previous subsection, it suffices to show that for each small parameter ε 0 there exists λ(ε) with |λ(ε) − λ0 | (Re λ0 )/2 such that the operator id + F(λ(ε), ε) has a nontrivial null space. The result of Theorem 4.2 states that there exists λ0 = −iαcα with Im cα > 0 such that id + F(λ0 , 0) has a nontrivial null space. The proof for ε > 0 uses a perturbation argument, based on the following lemma due to Steinberg [51]. Lemma 5.8. Let F(λ, ε) be a family of compact operators on a Banach space, analytic in λ in a region in the complex plane and jointly continuous in (λ, ε) for each (λ, ε) ∈ × R. Suppose that id − F(λ0 , ε) is invertible for some λ0 ∈ and all ε ∈ R. Then, R(λ, ε) = (id − F(λ, ε))−1 is meromorphic in for each ε ∈ R and jointly continuous at (z 0 , ε0 ) if λ0 is not a pole of R(λ, ε); its poles depend continuously on ε and can appear or disappear only at the boundary of . In order to apply Lemma 5.8 to our situation, we need to transform the operator (5.27) to one on a function space independent of the parameter ε. This calls for the employment of the action-angle mapping Aε , as is done in the proof of Lemma 5.5: Aε : Dε → D and Aε (x, y) = (θ, I ), where the action-angle variables (θ, I ) are defined in (5.13) and D = {(θ, I ) : 0 < θ < 2π/α , 0 < I < h 0 }. Note that Aε maps Sε bijectively to {(θ, h 0 ) : 0 < θ < 2π/α}. The latter may be identified with (2π/α, h 0 ). This naturally induces an homeomorphism Bε : L 2per (Sε ) → L 2per ([0, 2π/α]) by (Bε f )(θ ) = f (A−1 ε (θ, h 0 )). Let us denote the following operators from L 2per ([0, 2π/α]) to itself: T˜ (λ, ε) = Bε T ((λ, ε)) (Bε )−1 , C˜ (λ, ε) = Bε C (λ, ε) (Bε )−1 , P˜ ε = Bε Pεy (Bε )−1 ,
and ˜ ε)(P˜ ε C˜ (λ, ε) + id). ˜ F(λ, ε) = Bε F(λ, ε)(Bε )−1 = T˜ (λ, ε) C(λ,
(5.28)
Since T (λ, ε) is compact and T (λ, ε) and C(λ, ε) are analytic in λ with |λ − λ0 | ˜ (Re λ0 )/2, the operator F(λ, ε) is compact and analytic in λ. Subsequently, F(λ, ε) is compact and analytic in λ. Clearly, Pεy (x), C(λ, ε) and Bε are continuous in ε, and ˜ ε) and P˜ ε are continuous in ε. The key technical lemma is to show the in turn, C(λ, ˜ continuity of T˜ (λ, ε) in ε and thus obtain the continuity of F(λ, ε) in ε.
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V. M. Hur, Z. Lin
Lemma 5.9. For ε 0 sufficiently small and |λ−λ0 | (Re λ0 )/2, the operator T˜ (λ, ε) satisfies the estimate T˜ (λ, ε1 ) − T˜ (λ, ε2 ) L 2per ([0,2π/α]) → L 2per ([0,2π/α]) C|ε1 − ε2 |,
(5.29)
where C > 0 is independent of λ and ε. Proof. For a given b ∈ L 2per ([0, 2π/α]), let us denote bε = Bε−1 b ∈ L 2 (Sε ) . By definition T˜ (λ, ε) b = Bε Tε bε = Bε (ψb,ε |Sε ), where ψb,ε ∈ H 1 (Dε ) is the unique weak solution of 0 λeλs ψb,ε (X ε (s), Yε (s))ds = 0 in Dε ; ψb,ε + γ (ψε )ψb,ε − γ (ψε ) −∞
(5.30a) (5.30b) (5.30c)
(ψb,ε )n = bε on Sε ; ψb,ε (x, 0) = 0.
Our goal is to estimate the L 2 -operator norm of T˜ (λ, ε1 ) − T˜ (λ, ε2 ) in terms of |ε1 − ε2 |. Since the domain of the boundary value problem (5.30) depends on ε, we use the action-angle mapping Aε j ( j = 1, 2) to transform functions and the Laplacian operator in Dε j ( j = 1, 2) to those in the fixed domain D. To simplify notations, we use Aε j ( j = 1, 2) to denote the induced transformations for functions and operators. Let ψ j = Aε j (ψb,ε j ), which are H 1 (D) -functions, and let γ j (I ) = Aε j (γ (ψε j )),
j = Aε j ( ),
where j = 1, 2. By definition, Aε j (bε j ) = Bε j (bε j ) = b. Note that the characteristic equation (3.8) in the action-angle variables (θ, I ) becomes θ˙ = υ j (I ) I˙ = 0 for j = 1, 2, where υ j (I ) = υε j (I ). That means, the trajectory (X ε j (s), Yε j (s)) in the phase space transforms under the mapping Aε j into (θ + υ j (I )s, I ). Since ψ j ( j = 1, 2) are 2π/α-periodic in θ , we have the Fourier expansions ψ j (θ, I ) = eilθ ψ j,l (I ), j = 1, 2. l
Under the action-angle mapping Aε j ( j = 1, 2) the left side of (5.30) becomes
0 λeλs ψb,ε (X ε (s), Yε (s))ds Aε j γ (ψε )ψb,ε − γ (ψε ) −∞ 0 ilθ λs il(θ+υ j (I )s) e ψ j,l (I ) − λe e ψ j,l (I )ds (5.31) = γ j (I ) l
= γ j (I )
l
−∞
ilυ j (I ) ψ j,l (I )eilθ , λ + ilυ j (I )
l
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783
and thus the system (5.30) becomes ilυ j (I ) j ψ j + γ j (I ) ψ j,l (I )eilθ = 0 in D, λ + ilυ j (I ) l
∂ I ψ j (θ, h 0 ) = b(θ ), ψ j (θ, 0) = 0. Accordingly, the difference ψ1 −ψ2 is a weak solution of the partial differential equation ilυ1 1 (ψ1 − ψ2 ) + ( 1 − 2 )ψ2 + γ1 (ψ1,l (I ) − ψ2,l (I ))eilθ λ + ilυ1 l ilυ2 ilυ1 ilυ2 ψ2,l (I )eilθ + (γ1 − γ2 ) + γ1 − ψ2,l (I )eilθ = 0 λ + ilυ1 λ + ilυ2 λ + ilυ2 l
l
(5.32) with the boundary conditions ∂ I (ψ1 − ψ2 )(θ, h 0 ) = 0, (ψ1 − ψ2 )(θ, 0) = 0.
(5.33a) (5.33b)
Let us write (5.32) as 1 (ψ1 − ψ2 ) + γ1
l
ilυ1 (I ) (ψ1,l (I ) − ψ2,l (I ))eilθ = f, λ − ikυ1 (I )
(5.34)
where f = f 1 + f 2 + f 3 with f 1 = −( 1 − 2 )ψ2 , iλl(υ1 − υ2 ) ψ2,l (I )eilθ , f 2 = −γ1 (λ + ilυ1 )(λ + ilυ2 ) l ilυ2 f 3 = −(γ1 − γ2 ) ψ2,l (I )eilθ . λ + ilυ2 l
We estimate f 1 , f 2 , f 3 separately. To simplify notations, C > 0 in the estimates below denotes a generic constant independent of ε and λ. First, we claim that f 1 ∈ H ∗ (D) with the estimates f 1 H ∗ (D0 ) C|ε1 − ε2 |b L 2 ([0,2π/α]) , where H ∗ (D) is the dual space of H (D) = {ψ ∈ H 1 (D) : ψ(θ, I ) = ψ(θ + 2π/α, I ), ψ(θ, 0) = 0}. Let us write j
j
j
j
j
j = a I I ∂ I I + a I θ ∂ I θ + aθθ ∂θθ + b I ∂ I + bθ ∂θ
for j = 1, 2,
and the difference of the coefficients as 1 2 a¯ I I = a 1I I − a 2I I , a¯ I θ = a 1I θ − a 2I θ , a¯ θθ = aθθ − aθθ , 1 2 1 2 b¯ I = b I − b I , b¯θ = bθ − bθ .
(5.35)
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V. M. Hur, Z. Lin
¯ it follows that Then formally, for any φ ∈ H (D) ∩ C 2 ( D) f 1 φ d I dθ D 0
−φ a¯ I I ∂ I I + a¯ I θ ∂ I θ + a¯ θθ ∂θθ + b¯θ ∂θ + b¯ I ∂ I ψ2 d I dθ = D 0 = [∂ I ψ2 ∂ I (a¯ I I φ) + ∂ I ψ2 ∂θ (a¯ I θ φ) + ∂θ ψ2 ∂θ (a¯ θθ φ)] d I dθ D0
φ b¯ I ∂ I ψ2 + b¯θ ∂θ ψ2 d I dθ − a¯ I I φb (θ ) dθ. −
(5.36)
{I =h 0 }
D0
This uses that ψ2 and φ are periodic in the θ -variable and that ∂ I ψ2 (θ, h 0 ) = b(θ ),
φ(θ, 0) = 0.
Note that the elliptic estimate (5.17) and the equivalence of norms under the transformation A−1 ε2 assert that ψ2 H 1 (D) C A−1 ψ2 = C ψb,ε 1 ε2
C bε 2
H 1 (Dε2 )
L 2 (Sε2 )
2
H (Dε2 )
C b L 2 ([0,2π/α]) .
Since |a¯ I I |C 1 + |a¯ I θ |C 1 + |a¯ θθ |C 1 + b¯θ C 1 + b¯ I C 1 = O(|ε1 − ε2 |), by using the trace theorem from (5.36) the estimate follows
f 1 φ d I dθ C |ε1 − ε2 | ψ2 H 1 (D) + b L 2 ([0,2π/α]) φ H 1 (D) D
C |ε1 − ε2 | b L 2 ([0,2π/α]) φ H 1 (D) . This proves the estimate (5.35). We claim that if b is smooth then the formal manipulations in (5.36) are valid and ¯ Note that Theorem 2.2 ensures that the steady state (ηε2 (x), ψε2 (x, y)) ψ2 ∈ C 2 ( D). is in C 3+β class, where β ∈ (0, 1). Since b is smooth it follows that bε2 = Bε−1 b is at 2 2 least in H Sε2 . Then, the similar argument as in the regularity proof of Theorem 5.1 below asserts that ψb,ε2 ∈ H 7/2 (Dε2 ) ⊂ C 2 (D¯ ε2 ). Since the definition of the actionangle variables guarantees that the mapping Aε2 is at least of C 2 , subsequently, ψ2 = ¯ This proves the claim. If b ∈ L 2 , an approximation of b by smooth Aε2 ψb,ε2 ∈ C 2 ( D). functions establishes (5.35). Next, since 1 1 2 1 λ + ilυ = (| Re λ|2 + |Im λ − lυ |2 )1/2 | Re λ| Re λ , j j 0 by the estimate (5.17) it follows that f 2 L 2 (D) C|ε1 − ε2 |ψ2 L 2 (D) C|ε1 − ε2 |b L 2 ([0,2π/α]) .
(5.37)
Unstable Surface Waves in Running Water
Similarly,
785
f 3 L 2 (D0 ) C|ε1 − ε2 |b L 2 ([0,2π/α]) .
Combining the estimates (5.35), (5.37) and (5.38) asserts that f ∈
(5.38) H ∗ (D)
and
f H ∗ (D) C|ε1 − ε2 |b L 2 ([0,2π/α]) .
1 Let ψ = ψ1 − ψ2 ∈ H 1 (D) and φ = A−1 ε1 ψ ∈ H Dε1 . It remains to transform back to the physical space of the boundary value problem for ψ and to compute the operator norm of T˜ (λ, ε1 ) − T˜ (λ, ε2 ). Under the transformation A−1 ε1 , Eqs. (5.34), (5.33a), (5.33b) become 0 λeλs φ(X ε1 (s), Yε1 (s))ds = A−1 φ + γ (ψε1 )φ − γ (ψε1 ) ε1 f in Dε1 ; −∞
(φ ε1 )n = 0 on Sε1 ; φ ε1 (x, 0) = 0.
∗ Then, A−1 ε1 f ∈ H (Dε1 ) and −1 C f H ∗ (D) C|ε1 − ε2 |b L 2 ([0,2π/α]) . Aε1 f ∗ H (Dε1 )
Corollary 5.7 thus applies to assert that ψ H 1 (D) φ H 1 (Dε
1)
C A−1 f ε1
C|ε1 − ε2 |b L 2 ([0,2π/α]) .
H ∗ (Dε1 )
Finally, by the trace theorem it follows T˜ (λ, ε1 ) b − T˜ (λ, ε2 ) b L 2per ([0,2π/α]) = (ψ1 − ψ2 ) (θ, h 0 ) L 2per ([0,2π/α]) C ψ H 1 (D) C|ε1 − ε2 |b L 2per ([0,2π/α]) . This completes the proof.
We are now in a position to prove our main theorem. Proof of Theorem 5.1. For |λ − λ0 | (Re λ0 )/2, where λ0 = −iαcα , and ε 0 small, ˜ consider the family of operators F(λ, ε) on L 2per ([0, 2π/α]), defined by (5.28). The ˜ discussions following (5.28) and Lemma 5.9 assert that F(λ, ε) is compact, analytic in λ and continuous in ε. By our assumption, Im cα > 0 and cα is an unstable eigenvalue of the Rayleigh system (4.1)–(4.2) which corresponds to ε = 0. In other words, λ0 is a pole of (id + F(λ, 0))−1 . Subsequently, it is a pole of (id + F˜ (λ, 0))−1 . Since λ0 is an isolated pole, we may choose ˜ δ > 0 small enough so that the operator id + F(λ, 0) is invertible on |λ − λ0 | = δ. By ˜ the continuity of F(λ, ε) in ε, the following estimate ˜ ˜ F(λ, ε) − F(λ, 0) L 2per ([0,2π/α]) → L 2per ([0,2π/α]) Cε ˜ holds. Hence, id + F(λ, ε) is invertible on |λ − λ0 | = δ for ε 0 sufficiently small. ˜ Then by Lemma 5.8, the poles of (id + F(λ, ε))−1 are continuous in ε and can only
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V. M. Hur, Z. Lin
appear or disappear in the boundary of {λ : |λ − λ0 < δ}. Therefore, for each ε 0, −1 ˜ in |λ(ε) − λ0 | < δ. Thus, Re λ(ε) > 0 and there exists a pole λ(ε) of id + F(λ, ε) there exists a nonzero function f˜ ∈ L 2per ([0, 2π/α]) such that ˜ (id + F(λ, ε)) f˜ = 0.
(5.39)
Below we construct an exponentially growing solution to the linearized system (3.6a)–(3.6e). Define f = (Bε )−1 f˜ ∈ L 2 (Sε ), then
(I + F(λ, ε)) f = 0
˜ Let ψ (x, y) ∈ by (5.39) and the definition of F. (5.7) with λ = λ(ε) and
H 1 (D
(5.40) ε)
to be the unique solution of
ψn (x) = b = −C λ (Pεy (x)C λ + I ) f (x).
(5.41)
By (5.40), (5.41) and the definition of F, we have f = Tε b = ψ(x, ηε (x)), and thus ψn (x) = −C λ (Pεy (x)C λ + I )ψ(x, ηε (x)). Define
η(x) = C λ [ψ(x, ηε (x))] ∈ L 2 (Sε ),
(5.42)
and P(x, ηε (x)) = −Pεy (x)η(x), then (5.41) becomes ψn (x) = −C λ (P(x, ηε (x)) + ψ(x, ηε (x))) . (5.43) λ(ε)t Now we show that e ψ (x, y) , eλ(ε)t η(x) satisfies the linearized system (3.6a)(3.6e). The bottom boundary condition (3.6e) is satisfied since ψ (x, 0) = 0. Equation (3.6c) is automatic. By Lemma 5.3 and Eqs. (5.42), (5.43), η (x) and ψn (x) satisfy Eqs. (5.1b) and (5.1d) weakly. Equivalently, Eqs. (3.6b) and (3.6d) are satisfied weakly. Since ψ (x, y) satisfies Eq. (5.7a), we have 0 ω = − ψ = γ (ψε )ψ − γ (ψε ) λeλs ψ(X ε (s), Yε (s))ds. (5.44) −∞
As shown in [42], the above equation implies that the vorticity ω satisfies Eq. (3.7) weakly. Equivalently, Eq. (3.6a) is satisfied weakly. In summary, [eλ(ε)t ψ (x, y) , eλ(ε)t η(x)] is a weak solution of the linearized system (3.6a)–(3.6e). Our last step of the proof is to get the regularity of the growing-mode (eλ(ε)t η(x), eλ(ε)t ψ(x, y)) and thus show that it is a classical solution of (3.6a)-(3.6e). By (5.41) and Lemma 5.6, it follows that ψ ∈ H 1 (Dε ). We claim that ψ ∈ H 2 (Dε ). Indeed, by the
Unstable Surface Waves in Running Water
787 1
trace theorem, ψ ∈ H 1 (Dε ) implies that ψ(x, ηε (x)) ∈ H 2 (Sε ). Since the operator C λ 1 is regularity preserving, by (5.43) ψn (x) ∈ H 2 (Sε ). This, together with the facts that ω = − ψ ∈ L 2 (Dε ) and that the steady state (ηε (x), ψε (x, y)) ∈ C 3+α , α ∈ (0, 1) (see [18] or Theorem 2.2), implies that ψ ∈ H 2 (Dε ) by the regularity theory ([2]) of elliptic boundary problems. Then, by using the trace theorem and (5.43) again, we get 3 ψ(x, ηε (x)) and ψn (x) ∈ H 2 (Sε ). In order to obtain the higher regularity for ψ, we need to show that ω ∈ H 1 (Dε ). The argument presented below is a simpler version of that in [40]. Taking the gradient of (5.44) yields that 0 ∇ω =∇(γ (ψε ))ψ + γ (ψε )∇ψ − ∇(γ (ψε )) λeλs ψ(X ε (s), Yε (s))ds −∞ (5.45) 0 ∂(X ε (s), Yε (s)) λs ds. λe ∇ψ(X ε (s), Yε (s)) − γ (ψε ) ∂(x, y) −∞ Note that the particle trajectory is written in the action-angle variables (θ, I ) = Aε (x, y) as (X ε (s; x, y), Yε (s; x, y)) = A−1 ε ((θ + υε (I )s, I )). This relies on that the action-angle mapping Aε is globally defined, as a consequence of the fact that the steady flow has no stagnation. With the use of the above description of the trajectory the estimate of the Jacobi matrix ∂ (X ε (s; x, y), Yε (s; x, y)) C1 |s| + C2 (5.46) ∂ (x, y) follows, where C1 , C2 > 0 are independent of s. It is straightforward to see that by calculations as in proving (5.15), the L 2 -norm of the first three terms of (5.45) is bounded by the H 1 norm of ψ. The last term in (5.45) is treated as 0 ∂(X ε (s), Yε (s)) γ ds 2 (ψ ) λeλs ∇ψ(X ε (s), Yε (s)) ε L ∂(x, y) −∞ 0 ||γ (ψε )|| L ∞ |λ|eRe λs (C1 |s| + C2 )||∇ψ(X ε (s), Yε (s))|| L 2 (Dε ) ds C ψ H 1 (Dε ) .
−∞
This uses (5.46), Re λ δ > 0 and the fact that the mapping (x, y) → (X ε (s), Yε (s)) is measure-preserving. Therefore, ∇ω L 2 (Dε ) C ψ H 1 (Dε ) . In turn, ω ∈ H 1 (Dε ). Since ψn (x) ∈ H 3/2 (Sε ), by the elliptic regularity theorem [2] it follows that that ψ ∈ H 3 (Dε ). In view of the trace theorem this implies ψn ∈ H 5/2 (Sε ). We repeat the process again. Taking the gradient of (5.45) and using the linear stretching property (5.46) of the trajectory, it follows that ω ∈ H 2 (Dε ). The elliptic regularity applies to assert that ψ ∈ H 4 (Dε ) ⊂ C 2+β (D¯ ε ) , where β ∈ (0, 1). By the trace theorem then it follows that ψ(x, ηε (x)) ∈ H 7/2 (Sε ). On account of (5.42) this implies that η ∈ H 7/2 (Sε ) ⊂ C 2+β ([0, 2π/α]). Therefore, (eλ(ε)t η(x), eλ(ε)t ψ(x, y)) is a classical solution of (3.6). This completes the proof.
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6. Instability of General Shear Flows Linear instability of free-surface shear flows is of independent interest. This section extends our instability result in Theorem 4.2 to a more general class of shear flows. The following class of flows was introduced in [39] and [42] in the rigid-wall setting. Definition 6.1. A function U ∈ C 2 ([0, h]) is said to be in the class F if U takes the same sign at all points such that U (y) = c, where c is in the range of U but not an inflection value of U . Examples of the class-F flows include all monotone flows and symmetric flows with a monotone half. Moreover, if U (y) = f (U (y))k(y) for f continuous and k(y) > 0, then U is in class F. All flows in class K+ are in class F . The lemma below shows that for a flow in class F a neutral limiting wave speed must be an inflection value. The main difference of the proof from that in the class K+ case (Proposition 4.4) is the lack of an uniform H 2 -bound for the unstable mode sequence. Lemma 6.2. For U ∈ F, let {(φk , αk , ck )}∞ k=1 with Im ck > 0 be a sequence of unstable solutions satisfying (4.1)–(4.2). If (αk , ck ) converges to (αs , cs ) as k → ∞ with αs > 0 and cs is in the range of U , then cs must be an inflection value of U . Proof. Suppose on the contrary that cs is not an inflection value. Let y1 , y2 , . . . , ym be in the pre-image of cs so that U (y j ) = cs , and let S0 be the complement of the set of points {y1 , y2 , . . . , ym } in the interval [0, h]. Since cs is not an inflection value, Definition 6.1 asserts that U (y j ) takes the same sign for j = 1, 2, . . . , m, say positive. As in the proof of Proposition 4.4, let E δ = {y ∈ [0, h] : |y − y j | < δ for some j, where j = 1, 2, · · · , m}. It is readily seen that E δc ⊂ S0 . Note that U (y) > 0 for y ∈ E δ if δ > 0 small enough. We normalize the sequence by setting φk L 2 = 1. The result of Lemma 3.6 in [39] implies that φk converges uniformly to φs on any compact subset of S0 . Moreover, φs exists on S0 and φs satisfies φs − αs2 φs −
U φs = 0 U − cs
for y ∈ (0, h).
Our first task is to show that φs is not identically zero. Suppose otherwise. The proof is again divided into two cases. Case 1. U (h) = cs . In this case, [h − δ1 , h] ⊂ S0 for some δ1 > 0. As is done in the proof of Proposition 4.4, for any q real, it follows that 0
h
2 φ + α 2 |φk |2 + U (U − q) |φk |2 dy k k |U − ck |2 h 2 U (U − q) U (U − q) 2 φ dy + α 2 + |φ | dy + |φk |2 dy k k k 2 2 c |U − c | |U − c | k k 0 Eδ Eδ h (U − q)| 2 |U φ dy + α 2 − sup |φk |2 dy. k k 2 c c |U − c | k 0 E E δ δ
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We choose q = Umin − 1, then by (4.10), h
2 φ + α 2 |φk |2 + U (U − Umin + 1) |φk |2 dy k k |U − ck |2 0
Im gr (ck ) = Re gr (ck ) + (Re ck − Umin + 1) |φk (h)|2 Im ck
h 2 1 h 2 2 φk dy + C|φk (h)| C1 ε |φk | dy . ε h−δ1 h−δ1 If ε is chosen to be small then the above two inequalities lead to h |U (U − Umin + 1)| 2 2 |φk | dy − Cε |φk |2 dy. 0 αk − sup |U − ck |2 h−δ1 E δc Ec
(6.1)
(6.2)
δ
Since φk converges to φs ≡ 0 uniformly on E δc and [h − δ1 , h], this implies 0 αs2 /2 when k is large enough. A contradiction proves that φs is not identically zero. Case 2. U (h) = cs . From (4.21), we have h h 2 2 |φk |2 U φ + 2α 2 φ 2 + α 4 |φk |2 dy = 2α 2 Re gs (ck ) φ (h)2 + dy. k k k k k k |U − ck |2 0 0 The imaginary part of (4.16) yields h 2 U |φk |2 Im ck dy = − Im gs (ck ) φk (h) . 2 0 |U − ck | = max Denote Umax [0,h] U (y). Combining the above two identities, we have
h h 2 2 U Umax + 1 − U |φk |2 2 4 2 φ + 2α φ + α |φk | dy + dy k k k k |U − ck |2 0 0
2
Im gs (ck ) Umax + 1 φk (h) = 2αk2 Re gs (ck ) − Im ck 2 C d (ck , U (h)) φ (h) Cd (ck , U (h)) φk 2 2 , k
(6.3)
(6.4)
H
where we use (4.20). Since d(ck , U (h)) = | Re ck − U (h)| + (Im ck )2 → 0, so for k large enough we have
U U αk4 αs4 max + 1 − U 2 |φ | − sup , 0 dy k 2 4 |U − ck |2 E δc Ec δ
which is a contradiction. This proves that φs is not identically zero. Subsequently, Lemma 4.7 asserts that φs (y j ) = 0 for some y j . Below, we get a contradiction from the assumption that cs is not an inflection value. In Case 1 when U (h) = cs , it is straightforward to see that U (U − Umin + 1) U 2 |φ | dy |φ |2 dy = ∞, s 2 s |U − cs |2 Eδ |y−y j |<δ |U − cs |
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since φs (y j ) = 0 and by our assumption U > 0 on |y − y j | < δ . Fatou’s lemma then states that U (U − Umin + 1) lim inf |φk |2 dy = ∞. k→∞ E δ |U − ck |2 Then similar to the estimate (6.1) above, we have h
2 U (U − Umin + 1) 2 2 2 φk + αk |φk | + 0= |φk | dy |U − c|2 0
Im gr (ck ) − Re gr (ck ) + (Re ck − Umin + 1) |φk (h)|2 Im ck U (U − Umin + 1) |U (U − Umin + 1)| 2 |φ | dy − sup − Cε > 0, k |U − ck |2 |U − ck |2 Eδ Ec δ
for k large. A contradiction asserts that cs is an inflection value. For Case 2 when U (h) = cs , similarly we have
U Umax + 1 − U |φk |2 dy = +∞, lim inf k→∞ |U − ck |2 Eδ and from (6.4)
U U U Umax + 1 − U max + 1 − U 2 |φk | dy − sup 0 > 0, |U − ck |2 |U − ck |2 Eδ E δc when k is large. Another contradiction completes the proof.
The proof of the above lemma indicates that a flow in class F is linearly stable when the wave number is large. Lemma 6.3. Assume g = 1. Then for any flow U (y) in class F, there exists αmax > 0 such that when α αmax there are no unstable solutions to (4.1)–(4.2). Proof. Suppose otherwise. Then, there would exist a sequence of unstable solutions {(φk , αk , ck )}∞ k=1 of (4.1)–(4.2) such that αk → ∞ as k → ∞. After normalization, let φk L 2 = 1. First we show that limk→∞ Im ck = 0. If Im ck δ > 0 for some δ, then 1/ |U − ck | and |gr (ck )| are uniformly bounded. Accordingly, with q = Re ck in (4.10) it follows that h
2 φ + α 2 |φk |2 + U (U − Re ck ) |φk |2 dy − Re gr (ck ) |φk (h)|2 0= k k |U − c |2 0 k
h h U (U − Re ck ) 2 1 h 2 2 2 |φk | dy φk dy − C ε φk dy + αk − sup + ε 0 |U − ck |2 0 0 U (U − Re ck ) C − > 0, αk2 − sup ε |U − ck |2 when k is big enough. This contradiction shows that ck → cs ∈ [Umin , Umax ] when k → ∞. The remainder of the proof is nearly identical to that of Lemma 6.2 and hence is omitted.
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The following theorem gives a necessary condition for the free surface instability that the flow profile should have an inflection point, which generalizes the classical result of Lord Rayleigh [52] in the rigid wall case. Theorem 6.4. A shear flow U (y) without an inflection point is linearly stable in the free surface setting. Proof. Suppose otherwise; then, there would exist an unstable solution (φ, α, c) to (4.1)– (4.2) with α > 0 and Im c > 0. Lemma 4.8 allows us to continue this unstable mode for wave numbers to the right of α until the growth rate becomes zero. Note that a flow without an inflection point is trivially in class F . So by Lemma 6.3, this continuation must end at a finite wave number αmax and a neutral limiting mode therein. On the other hand, Lemma 6.2 asserts that the neutral limiting wave speed cs corresponding to this neutral limiting mode must be an inflection value. A contradiction proves the assertion.
Remark 6.5. Our proof of the above no-inflection stability theorem is very different from the rigid wall case. In the rigid-wall setting, where φ(h) = 0, the identity (4.15) reduces to
h
ci 0
U |φ|2 dy = 0, |U − c|2
which immediately shows that if U is unstable (ci > 0) then U (y) = 0 at some point y ∈ (0, h). The same argument was adapted in [62, Sect. 5] for the free-surface setting, however, it does not give linear stability for general flows with no inflection points. More specifically, in the free-surface setting, (4.15) becomes
h
ci 0
U |φ|2 dy = |U − c|2
2g(U (h) − cr ) U (h) + 4 |U (h) − c| |U (h) − c|2
|φ(h)|2 ,
which only implies linear stability ([62, Sect. 5]) for special flows satisfying U (y) < 0, U (y) 0 or U (y) > 0, U (y) 0. In the proof of Theorem 6.4, we use the characterization of neutral limiting modes and remove above additional assumptions. Let us now consider a shear flow U ∈ F with multiple inflection values U1 , U2 , . . . , Un . Lemma 6.2 states that a neutral limiting wave speed cs must be one of the inflection values U1 , U2 , . . . , Un , say cs = U j . By localizing the estimates in the proof of Lemma 4.6 around inflection points with the inflection values Ui , we can get an uniform H 2 bound for the unstable mode sequence. We skip the details, which are similar to the case of rigid walls treated in [42]. Thus, neutral limiting modes for flows in class F are also characterized by inflection values. Proposition 6.6. If U ∈ F has inflection values U1 , U2 , . . . , Un , then for a neutral limiting mode (φs , αs , cs ) with αs > 0 the neutral limiting wave speed must be one of the inflection values, that is, cs = U j for some j. Moreover, φs must solve φs − αs2 φs + K j (y)φs = 0 for y ∈ (0, h),
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with boundary conditions φs (h) = gr (U j ), φs (0) = 0 φs (h) = 0, φs (0) = 0
if U (h) = U j , if U (h) = U j
(6.5)
where K j (y) = −U (y)/(U (y) − U j ). One may exploit the instability analysis of Theorem 4.9 for a flow in class F with possibly multiple inflection values. The main difference of the analysis in class F from that in class K+ is that unstable wave numbers in class F may bifurcate to the left and to the right of a neutral limiting wave number, whereas unstable wave numbers in class K+ bifurcate only to the left of a neutral limiting wave number. In the rigid-wall setting, with an extension of the proof of [39, Theorem 1.1], Lin [42, Theorem 2.7] analyzed this more complicated structure of the set of unstable wave numbers. The remainder of this section establishes an analogous result in the free-surface setting. In order to study the structure of unstable wave numbers in class F with possibly multiple inflection values, we need several notations to describe. A flow U ∈ F is said to be in class F + if each K j (y) = −U (y)/(U (y) − U j ) is nonzero, where U j for j = 1, · · · , n are inflection values of U . It is readily seen that for such a flow K j takes the same sign at all inflection points of U j . A neutral limiting mode (φ j , α j , U j ) is said to be positive if the sign of K j is positive at inflection points of U j , and negative if the sign of K j is negative. Proposition 6.6 asserts that −αs2 is a negative eigenvalue of d2 − dy 2 − K j (y) on y ∈ (0, h) with boundary conditions (6.5). We employ the argument in the proof of Theorem 4.9 to conclude that an unstable solution exists near a positive (negative) neutral limiting mode if and only if the perturbed wave number is slightly to the left (right) of the neutral limiting wave number. Thus, the structure of the set of unstable wave numbers with multiple inflection values is more intricate. We remark that a class-K+ flow has a unique positive neutral limiting mode and hence unstable solutions bifurcate to the left of a neutral limiting wave number. Let us list all neutral limiting wave numbers in the increasing order. If the sequence contains more than one successive negative neutral limiting wave numbers, then we pick the smallest (and discard others). If the sequence contains more than one successive positive neutral limiting wave numbers, then we pick the largest (and discard others). If the smallest member in this sequence is a positive neutral limiting wave number, then we add zero into the sequence. Thus, we obtain a new sequence of neutral limiting wave numbers. Let us denote the resulting sequence by α0− < a0+ < · · · < − − + α− N < α N , where, α0 (might be 0),. . . , α N are negative neutral limiting wave numbers + + and α0 , . . . , α N are positive neutral limiting wave numbers. The largest member of the sequence must be a positive neutral wave number since no unstable modes exist to its right. Theorem 6.7. For U ∈ F + with inflection values U1 , U2 , . . . , Un , let α0− < a0+ < · · · < − N + + α− N < α N be defined as above. For each α ∈ ∪ j=0 (α j , α j ), there exists an unstable solution of (4.1)–(4.2). Moreover, the flow is linear stable if either α α +N or all 2
d operators − dy 2 − K j (y) ( j = 1, 2, . . . , n) on y ∈ (0, h) with (6.5) are nonnegative.
Theorem 6.7 indicates that there might exist a gap in (0, α +N ) of stable wave numbers. Indeed, in the rigid-wall setting, a numerical computation [7] demonstrates that for a certain shear-flow profile the onset of the unstable wave numbers is away from zero, that is α0− > 0.
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Appendix A: Proofs of (4.35), (4.36), ( 4.41) and (4.42) Our first task is to show that the limit (4.35) holds as ε → 0− uniformly in E (R,b1 ,b2 ) . In the proof of Theorem 4.9, we have already established that both φ¯ 1 (y; ε, c) and φ0 (y; ε, c) uniformly converge to φs in C 1 , as (ε, c) → (0, 0) in E (R,b1 ,b2 ) . Moreover, φ2 (0; ε, c) → − φ 1(0) uniformly as (ε, c) → (0, 0) in E (R,b1 ,b2 ) . So the function s
G(y, 0; ε, c)φ0 (y; ε, c) = φ¯ 1 (y; ε, c) φ2 (0; ε, c) −φ2 (y; ε, c) φ¯ 1 (0; ε, c) φ0 (y; ε, c) converges uniformly to −φs2 (y) /φs (0) in C 1 [0, d],as (ε, c) → (0, 0) in E (R,b1 ,b2 ) .Then the uniform convergence of (4.35) follows from (4.33). The proof of (4.36) uses the following lemma. Lemma A.1 ([39], Lemma 7.3). Assume that a sequence of differentiable functions ∞ 1 { k }∞ k=1 converges to ∞ in C and that {ck }k=1 converges to zero, where Im ck > 0 and | Re ck | R Im ck for some R > 0 . Then, h h ms K (a j ) U K (y) φs (a j ),
dy = p.v.
dy + iπ lim − k ∞ 2 k→∞ |U (a j )| 0 (U − Us − ck ) 0 U − Us i=1 (A.1) provided that U (y) = 0 at each a j . Here a1 , . . . , am s are roots of U − Us . We now prove (4.36). That is, we shall show that (4.36 ) holds uniformly in E (R,b1 ,b2 ) . k k Suppose for some δ > 0 and a sequence {(εk , ck )}∞ k=1 in E (R,b1 ,b2 ) with max(b1 , b2 ) tending to zero ∂ (εk , ck ) − (C + i D)| > δ ∂c holds, where C and D are defined in (4.37). Let us write h ∂ U d (εk , ck ) = − gr (Us + c)φ2 (0; εk , ck ) = I + I I,
dy + 2 k ∂c (U − U − c ) dc s k 0 |
where
k (y) = G(y, 0; εk , ck )φ0 (y; εk , ck ) → −
1 φ 2 in C 1 . φs (0) s
By Lemma A.1, it follows that h ms K (a j ) 2 K (y) 2 1 φ (a j ) . lim I = − φs dy + iπ p.v. k→∞ φs (0) |U (a j ) s 0 U − Us k=1
A straightforward calculation yields that lim I I = −
k→∞
1 2g A d 1 U (h) gr (Us + c) =− =− , + dc φs (0) (U (h) − Us )3 (U (h) − Us )2 φs (0) φs (0)
where A is given in (4.31). Therefore, ∂ (εk , ck ) = C + i D. k→∞ ∂c A contradiction then proves the uniform convergence. The proofs of (4.41) and (4.42) use the following lemma. lim
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1 Lemma A.2 ([39], Lemma 7.1). Assume that {ψk }∞ k=1 converges to ψ∞ in C ([0, h]) ∞ and that {ck }k=1 with Im ck > 0 converges to zero. Let us denote Wk (y) = U (y) − Us − Re ck . Then, the limits h h Wk ψ∞ lim ψ dy = p.v. dy, (A.2) k k→∞ 0 W 2 + Im c2 U − Us 0 k k h ms ψ∞ (a j ) Im ck (A.3) lim ψk dy = π k→∞ 0 W 2 + Im c2 |U (a j )| k k j=1
hold provided that U (y) = 0 at each a j . Here, a1 , . . . , am s satisfy U (a j ) = Us . We now prove (4.41). Since K (y)φs φk → K (y)φs2 in C 1 ([0, h]), by (A.2) and (A.3) it follows that h U − φs φk dy (U − ck )(U − Us ) 0 h h Wk Im ck = K (y)φ φ dy + i K (y)φs φk dy (A.4) s k 2 2 2 2 0 Wk + Im ck 0 Wk + Im ck h ms K (a j ) 2 K φs2 dy + iπ φ (a j ) → p.v. |U (a j )| s 0 (U − Us ) j=1
as k → ∞. Since ck → Us as k → ∞ in the proof of Theorem 4.2, it follows that lim
k→∞
gr (ck ) − gr (Us ) 2g U (h) = gr (Us ) = + . ck − Us (U (h) − Us )3 (U (h) − Us )2
(A.5)
Addition of (A.4) and (A.5) proves (4.41). In case U (h) = Us the same computations as above prove (4.42). This uses that gs (ck ) U (h) − ck = 0. = − lim k→∞ ck − Us k→∞ g + U (h)(U (h) − ck ) lim
Acknowledgement. The work of Vera Mikyoung Hur is supported partly by the NSF grant DMS-0707647. The work of Zhiwu Lin is supported partly by the NSF grants DMS-0505460 and DMS-0707397. The authors thank the anonymous referee for valuable comments and suggestions.
References 1. Adams, R.A.: Sobolev spaces, New York: Academic Press, 1975 2. Agranovich, M.S.: Elliptic boundary problems, Partial Differential equations IX, Encyclopaedia of Math. Sci., 79, Berlin-Heidelberg-New York: Springer-Verlag, 1997, pp. 1–132 3. Arnold, V.I.: On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid. Dokl. Akad. Nauk SSSR 162, 975–978 (1965) 4. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Second Edition, New York: Springer-Verlag, 1989 5. Amick, C., Fraenkel, L.E., Toland, J.F.: On the Stokes conjecture for the wave of extreme form. Acta Math. 148, 193–214 (1982) 6. Angulo Pava, J., Bona, J.L., Scialom, M.: Stability of cnoidal waves. Adv. Differ. Eqs. 12, 1321–1374 (2006)
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7. Balmforth, N.J., Morrison, P.J.: A necessary and sufficient instability condition for inviscid shear flow. Studies in Appl. Math. 102, 309–344 (1999) 8. Bona, J.L., Sachs, R.L.: The existence of internal solitary waves in a two-fluid system near the KdV limit. Geophys. Astrophys. Fluid Dyn. 48(1–3), 25–51 (1989) 9. Barcilon, A., Drazin, P.G.: Nonlinear waves of vorticity. Stud. Appl. Math. 106(4), 437–479 (2001) 10. Benjamin, T.B.: The stability of solitary waves. Proc. Royal Soc. London Ser. A 338, 153–183 (1972) 11. Benjamin, T.B., Feir, J.E.: Disintegration of wave trains on deep water. J. Fluid. Mech. 27, 417–437 (1967) 12. Bridges, T.J., Mielke, A.: A proof of the Benjamin-Feir instabillity. Arch. Rational Mech. Anal. 133, 145–198 (1995) 13. Buffoni, B, Séré, É., Toland, J.F.: Surface water waves as saddle points of the energy. Calculus of Variations and Partial Differ. Eqs. 17, 199–220 (2003) 14. Buffoni, B., Toland, J.F.: Analytic theory of global bifurcation: an introduction. Princeton NJ: Princeton University Press, 2003 15. Burns, J.C.: Long waves in running water. Proc. Camb. Phil. Soc. 49, 695–706 (1953) 16. Constantin, A.: The trajectories of particle in Stokes waves. Invent. Math. 166, 523–535 (2006) 17. Constantin, A., Sattinger, D., Strauss, W.: Variational formulations for steady water waves with vorticity. J. Fluid Mech. 548, 151–163 (2006) 18. Constantin, A., Strauss, W.: Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57, 481–527 (2004) 19. Constantin, A., Strauss, W.: Stability properties of steady water waves with vorticity. Comm. Pure Appl. Math. 60, 911–950 (2007) 20. Constantin, A., Strauss, W.: Rotational steady water waves near stagnation. Phil. Trans. Roy. Soc. London A 365, 2227–2239 (2007) 21. Crandall, M.G., Rabinowitz, P.H.: Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rat. Mech. Anal. 52, 161–180 (1973) 22. Crapper, G.D.: Introduction to water waves. Ellis Horwood Series: Mathematics and Its Applications. Chichester: Ellis Horwood/New York: Halsted Press, 1984 23. Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge Monographs on Mechanics and Appl. Math. Cambridge: Cambridge University Press, 1981 24. Drazin, P.G., Howard, L.N.: Hydrodynamic stability of parallel fluid of an inviscid fluid. Adv. Appl. Math. 9, 1–89 (1966) 25. Dubreil-Jacotin, M.-L.: Sur la dérmination rigoureuse des ondes permanentes péiodiquesd’ampleur finie. J. Math. Pures Appl. 13, 217–291 (1934) 26. Friedlander, S., Howard, L.N.: Instability in paralel flow revisited. Stud. Appl. Math. 101, 1–21 (1998) 27. Gerstner, F.: Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 2, 412–445 (1809) 28. Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics 19, Providence RI: Amer. Math. Soc., 1998 29. Garabedian, P.R.: Surface waves of finite depth. J. d’Anal. Math. 14, 161–169 (1965) 30. Guo, Y., Strauss, W.: Instability of periodic BGK equilibria. Comm. Pure. Appl. Math. 48, 861–894 (1995) 31. Hartman, P.: Ordinary differential equations. Reprint of the second edition. Boston, MA: Birkhäser, 1982 32. Hur, V.M.: Global bifurcation of deep-water waves. SIAM J. Math. Anal. 37, 1482–1521 (2006) 33. Hur, V.M.: Exact solitary water waves with vorticity. Arch. Rat. Mech. Anal. 188, 213–244 (2008) 34. Keady, G., Norbury, J.: On the existence theory for irrotational water waves. Math. Proc. Cambridge Philos. Soc. 83, 137–157 (1978) 35. Krasovskii, J.: On the theory of steady-state waves of finite amplitude. USSR Comput. Math. and Math. Phys. 1, 996–1018 (1962) 36. Levi-Civita, T.: Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda. Rendus Accad. Lincei 33, 141–150 (1924) 37. Lighthill, J.: Waves in Fluids. Cambridge: Cambridge University Press, 1978 38. Lin, C.C.: The theory of hydrodynamic stability. Cambridge: Cambridge University Press, 1955 39. Lin, Z.-W.: Instability of some ideal plane flows. SIAM J. Math. Anal. 35, 318–356 (2003) 40. Lin, Z.-W.: Nonlinear instability of ideal plane flows. Int. Math. Res. Not. 41, 2147–2178 (2004) 41. Lin, Z.-W.: Some stability and instability criteria for ideal plane flows. Commun. Math. Phys. 246, 87–112 (2004) 42. Lin, Z.-W.: Some recent results on instability of ideal plane flows. In: Nonlinear partial differential equations and related analysis, Contemp. Math., 371, Providence, RI: Amer. Math. Soc. 2005, pp. 217–229 43. Lin, Z.-W.: Instability of large solitary water waves. http://arxiv,org/list/math-plus/0803.0339, 2008 44. Lin, Z.-W.: Instability of large Stokes waves. In preparation 45. Mackay, R.S., Saffman, P.G.: Stability of water waves. Proc. Roy. Soc. London Ser. A 496, 115–125 (1986) 46. McLeod, J.B.: The Stokes and Krasovskii conjectures for the wave of greatest height. University of Wisconsin Mathematics Research Center Report 2041, 1979
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47. Mielk, A.: On the energetic stability of solitary water waves. In: Recent developments in the mathematical theory of water waves (Oberwolfach, 2001). R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 360 (1799), 2337–2358 (2002) 48. Longuet-Higgins, M.S.: The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. Roy. Soc. London Ser. A 360(1703), 471–488 (1978) 49. Longuet-Higgins, M.S., Dommermuth, D.G.: Crest instabilities of gravity waves. III. Nonlinear Development and Breaking. J. Fluid Mech. 336, 33–50 (1997) 50. Nekrasov, A.I.: On steady waves. Izv. Ivanovo-Voznesenk. Politekhn. 3 (1921) 51. Steinberg, S.: Meromorphic families of compact operators. Arch. Rat. Mech. Anal. 31 pp. 372–379 (1968/1969) 52. Rayleigh, L.: On the stability or instability of certain fluid motions. Proc. London Math. Soc. 9, 57–70 (1880) 53. Stokes, G.G.: Considerations relative to the greatest height of oscillatory irrotatioanl waves which can be propagated without change of form, Mathematical and physical papers 1, 1880, Cambridge, pp. 225–228 54. Struik, D.: Détermination rigoureuse des ondes irrotationelles périodiques dans un canal à profondeur finie. Math. Ann. 95, 595–634 (1926) 55. Tanaka, M.: The stability of steep gravity waves. J. Phys. Soc. Japan 52, 3047–3055 (1983) 56. Ter-Krikorov, A.M.: A solitary wave on the surface of a turbulent liquid. (Russian) Zh. Vydisl. Mat. Fiz. 1, 1077–1088 1961. Translated in U.S.S.R. Comput. Math. and Math. Phys. 1, 1253–1264 (1962) 57. Thomas, G., Klopman, G.: Wave-current interactions in the nearshore region. In: Gravity waves in water of finite depth, Advances in fluid mechanics, 10, Southampton, United Kingdom, 1997, pp. 215–319 58. Toland, J.F.: On the existence of a wave of greatest height and Stokes’s conjecture. Proc. Roy. Soc. London Ser. A 363(1715), 469–485 (1978) 59. Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996) 60. Varvaruca, E.: On some properties of traveling water waves with vorticity. Siam. J. Math. Anal. 39, 1686–1692 61. Wahlen, E.: A Hamiltonian formulation of water waves with constant vorticity. Lett. Math. Phys. 79, 303–315 (2007) 62. Yih, C.-S.: Surface waves in flowing water. J. Fluid. Mech. 51, 209–220 (1972) Communicated by P. Constantin
Commun. Math. Phys. 282, 797–818 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0463-z
Communications in
Mathematical Physics
Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems Walter A. de S. Pedra1,2,3 , Manfred Salmhofer1,2 1 Theoretical Physics, University of Leipzig, Postfach 100920, 04009 Leipzig, Germany.
E-mail: [email protected]
2 Max–Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany 3 Institut für Mathematik, Universität Mainz, D-55128 Mainz, Germany
Received: 18 July 2007 / Accepted: 17 December 2007 Published online: 6 May 2008 – © Springer-Verlag 2008
Dedicated to Jürg Fröhlich in celebration of his 61st birthday Abstract: It is known that perturbation theory converges in fermionic field theory at weak coupling if the interaction and the covariance are summable and if certain determinants arising in the expansion can be bounded efficiently, e.g. if the covariance admits a Gram representation with a finite Gram constant. The covariances of the standard many– fermion systems do not fall into this class due to the slow decay of the covariance at large Matsubara frequency, giving rise to a UV problem in the integration over degrees of freedom with Matsubara frequencies larger than some (usually the first step in a multiscale analysis). We show that these covariances do not have Gram representations on any separable Hilbert space. We then prove a general bound for determinants associated to chronological products which is stronger than the usual Gram bound and which applies to the many–fermion case. This allows us to prove convergence of the first integration step in a rather easy way, for a short–range interaction which can be arbitrarily strong, provided is chosen large enough. Moreover, we give – for the first time – nonperturbative bounds on all scales for the case of scale decompositions of the propagator which do not impose cutoffs on the Matsubara frequency. 1. Gram Representations and Determinant Bounds Let X be a set and M : X2 → C, (x, y) → M(x, y). We call M an (X × X)-matrix and use the notation M = (Mx y )x,y∈X (if X = {1, . . . , n}, we call it as usual an (n × n)–matrix). Definition 1.1. Let M be an (X × X)-matrix. A triple (H, v, w), where H is a Hilbert space and v and w are maps from X to H, is called a Gram representation of M if ∀ x, x ∈ X :
Mx x = vx , wx ,
(1)
and if there is a finite constant γ M > 0 such that 2 . sup vx wx ≤ γ M
x,x ∈X
(2)
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γ M is called the Gram constant of M associated to the Gram representation (H, v, w). If M has a Gram representation, then the Gram estimate (see, e.g., Lemma B.30 of [S98b]) implies that for all n ∈ N and all x1 , . . . , xn , y1 , . . . , yn ∈ X, n det (Mx y )k,l ≤ vx w y ≤ γ M 2n . k l k k
(3)
k=1
Every (n × n)–matrix A has a Gram representation – the equation A = 1 · A (where 1 denotes the unit matrix) means that Akl = ek , al Cn , where ek is the k th row of 1 th and al is the l column of A. The associated Gram estimate | det A| ≤ l al 2 , the 2 Hadamard bound, has associated Gram constant γHad = maxl al 2 . Although considering diagonal matrices shows that the Hadamard bound is optimal, the way it was derived here is basis–dependent, and its application in an arbitrary basis can lead to a significant overestimate of the determinant. For instance, the matrix P = v ⊗ v, where v = (1, √ . . . , 1)T ∈ Cn has Pkl = 1 for all k, l, so the above Gram representation gives 2 γHad = n, thus the bound | det P| ≤ n n/2 . On the other hand, P has the Gram representation Pi j = w · w T with w = n −1/2 (1, . . . , 1), which gives the bound det P ≤ 1. Thus the main issue about Gram bounds for a given class of matrices is not the existence of some bound, but its size, and its dependence on n. Specifically, what is really needed in the proof of convergence of fermionic perturbation theory given in [SW] are bounds of the following type: there is a finite constant δ such that for all n ∈ N and all x1 , . . . , xn , y1 , . . . , yn ∈ X, (4) sup det(C xi y j Pi j )i, j ≤ δ 2n . P∈Pn,1
Here Pn,1 denotesthe set of complex hermitian (n × n)–matrices P = (Pi j ) that are nonnegative, i.e. i,n j=1 Pi j c¯i c j ≥ 0 for all c1 , . . . , cn ∈ C, and that have diagonal elements Pii ≤ 1. Such matrices P arise naturally in interpolation constructions of the tree expansion for the connected functions; they are positive if the tree expansion is chosen well [AR,SW]. We briefly recall Lemmas 7 and 8 of [SW]: The positivity of P implies that P = Q 2 = Q ∗ Q with Q ≥ 0, i.e. Pi j = qi , q j ,
(5)
where qi is the i th column of Q. Because qi , qi = Pii ≤ 1 the Gram constant of P is bounded by 1. If C has a Gram representation (H, v, w), then the matrix with elements Mi j = C xi y j Pi j has a Gram representation Mi j = vxi ⊗ qi , w y j ⊗ q j ,
(6)
and M has the same Gram constant as C because qi ≤ 1 for all i. Definition 1.2. Let C be an (X × X)-matrix. A finite constant δC > 0 is called a determinant bound of C if for all n ∈ N and all x1 , . . . , xn , y1 , . . . , yn ∈ X, det( pi , q j C x y )i, j ≤ δC 2n . (7) sup i j (n)
p1 ,..., pn ,q1 ,...,qn ∈B1 (n)
Here B1 = {ξ ∈ Cn : ξ 2 ≤ 1} denotes the closed n–dimensional unit ball.
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We have replaced the supremum over P ∈ Pn,1 by that over a larger set in Definition 1.2 because this makes the definition robust under the operation of taking arbitrary submatrices (positivity is spoiled by that operation). If C has a Gram representation with Gram constant γC , then C also has a determinant bound δC = γC by the same argument as above, i.e. writing pi , q j C xi y j = pi ⊗ vxi , q j ⊗ w y j . However, the Gram representation is not necessary for a useful determinant bound, and in this paper, we prove optimal determinant bounds for a class of covariance matrices for which no Gram representation with a good Gram constant is known. As will be discussed in Sect. 2, these matrices arise naturally in time–ordered perturbation theory and standard functional integral representations of many–fermion systems. The constructions we give are motivated by similar ones in [FKT04], and we shall discuss this relation in more detail in Sect. 2. Theorem 1.3. Let K , k ∈ N0 , k + K ≥ 1, and C0 , . . . , Ck+K be (X × X)–matrices. Assume that for all l ∈ {0, . . . , k + K }, Cl has a Gram representation with Gram constant γl . Let (J , ) be a totally ordered set, and for all l ∈ {1, . . . , k + K } let ϕl and ϕl be functions from X to J . Denote 1 A = 1 if A is true and 1 A = 0 otherwise. Then the (X × X)–matrix M given by Mx y = (C0 )x y +
k k+K (Cl )x y 1ϕl (x)ϕl (y) + (Cl )x y 1ϕl (x)ϕl (y) l=1
has determinant bound δ M =
k+K
(8)
l=k+1
γl .
l=0
Theorem 1.3 is proven in Sect. 3. The bound given in Theorem 1.3 is optimal in the following sense. Let us assume that for each l, the Gram representation for the Cl is optimal in that the Gram constant γCl satisfies γC2l = supx,y∈X (Cl )x y , and that the decomposition (8) is nonredundant in the sense that for any choice of x and y, only one of the summands is nonzero (in particular, C0 = 0). Then the determinant bound given in Theorem 1.3 is optimal up to a factor k + K because
1 2
δ M ≤ (k + K )
sup |Mx y |
x,y∈X
(9)
and because, by Definition 1.2, the determinant bound δ M of a matrix M satisfies (10) δ M 2n ≥ sup det(Mxi y j δi j ) = ( sup |Mx y |)n . x1 ,...,xn ∈X y1 ,...,yn ∈X
x,y∈X
2. The Matsubara UV Problem for Fermion Systems In this section, we specify the covariances for the many–fermion models, and then briefly review the well–known problem with the standard Gram representation due to the slow decay at large frequencies which is caused by the indicator functions from time ordering, which are special cases of the ones appearing in (8) (the Matsubara UV problem). We show that, if a Gram representation of these covariances exists, it has rather unusual properties. Then we state our main results for these models which follow directly
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from our new determinant bounds. A detailed analysis of these models will be given in [PS]. We consider the standard many–fermion model, as formulated for instance in [BR] or in [S98b], Chapter 4. The Hamiltonian of this model is of the form H = H0 + V . The free part H0 is given by a hopping term (if a lattice model is considered) or a differential operator (if a continuum model is considered). In either case, the relevant data for the present discussion are a momentum space B dual to configuration space X and an energy function E : B → R, p → E(p), which assigns an energy E(p) to a particle with (quasi)momentum p ∈ B. The interaction part V of H describes the interaction of two or more particles (see below). To be specific, we review briefly how E arises in some relevant cases. For a continuum system in d spatial dimensions without a crystal potential, X = Rd , B = Rd , and E(p) = p2 − µ, where the parameter µ > 0, the chemical potential, is a Lagrange parameter used to adjust the particle density. Particles in a crystal are modelled by a periodic Schrödinger operator containing a potential that is –periodic, where ⊂ Rd is a lattice of maximal rank. In this case, B is the torus B = Rd / # , where # is the dual lattice to . The operator has a band spectrum p → (eν (p))ν∈N , with the index ν labelling the bands. The case of a single E is obtained by restricting to a single band ν = ν0 and setting E(p) = eν0 (p) − µ. For a (one–band) lattice model on a spatial lattice , B = Rd /# is again a torus, and E(p) is the Fourier transform of the hopping matrix (see [S98b], Chapter 4). The motivation for restricting to a single band is that the interesting case is the one where E(p) has a nontrivial zero set, and that in many interesting cases, the bands do not overlap, so that for this zero set, only a single band matters. In field theoretic constructions, one often considers configuration spaces X = /L that have sidelengths L ∈ N, hence finite volume, in which case momentum space is discrete: B = B L = L −1 # / # . We shall consider the cases of finite and infinite volume in parallel and use the conventions of [S98b], Appendix A, for the Fourier transform. We denote by µ(dp) the natural invariant Haar measure on the torus B; specifically, for the continuous torus corresponding to infinite volume it is given by (2π )−d times Lebesgue measure, for the discrete torus B L corresponding to a finite volume it is given by the inverse of the volume times the counting measure. We shall drop the subscript L on B L when no confusion can arise. The interaction part of the Hamiltonian is assumed to be given by a two–body potential v, where v(x −y) is the interaction energy of a configuration with one particle at x and one particle at y. Most of the present paper is concerned with properties of the covariance, in which the interaction plays no role. However, the decay properties of the interaction are important for convergence of expansions, see below. The correct treatment of the interaction is difficult, but some progress has been made by multiscale expansion methods. One of the purposes of the present paper is to simplify and extend parts of this analysis, namely the ultraviolet (UV) integration, which is quite different from the analysis of the infrared singularity which arises in the limit of zero temperature. We briefly discuss the UV problems arising in such models. There is a spatial UV problem associated to continuum interactions that have a singularity at coinciding points, such as, for instance, a Yukawa potential e−α|x| /|x|, but this is not the issue we address here. There are also different UV problems associated to the covariances. The first one is related to the noncompactness of momentum space in the first example mentioned above. A similar problem arises for the periodic Schrödinger operator, namely there is
Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems
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an infinite number of bands. For the lattice system, the lattice spacing provides a natural spatial ultraviolet cutoff. The UV problem we are concerned with here is the discontinuity of the covariance as a function of the time variable, and the corresponding slow decay of its Fourier transform in the dual variable, the Matsubara frequency. In the continuum case X = Rd , we shall therefore impose a cutoff on the spatial part of momentum. We do this by using the measure µa where, for a > 0, µa (dp) = χ (ap)µ(dp), with χ a nonnegative function on Rd of compact support chosen such that χ (p)µ(dp) = 1, hence µa (B) = a −d . The UV cutoff parameter a scales similarly to a lattice spacing: if X = aZd , µ(B) = a −d . For a general lattice , which may have different spacings in the different directions, we define a by µ(B) = a −d , so that a is a geometric mean of the lattice spacings, and set µa = µ. Let β > 0, f β (E) = (1 + eβ E )−1 , and for (τ, E) ∈ (−β, β] × R let −τ E (1 − f β (E)) for 0 < τ ≤ β −e (11) C(τ, E) = for −β < τ ≤ 0. e−τ E f β (E) Extend the function C to a function on R × R that is 2β – periodic in τ . Note that C(τ + β, E) = −C(τ, E).
(12)
In the application, the parameter β is the inverse temperature, and the Fermi function f β is the expected occupation number for free fermions. Definition 2.1. The free covariance (free one–particle Green function) for a many– fermion system is the inverse Fourier transform of the map p → C(τ, E(p)) :
C(τ,x),(τ ,x ) = µa (dp) eip·(x−x ) C(τ − τ , E(p)). (13) B
More generally, let h ∈ L 1 (B, µa ) and define
(h) µa (dp) h( p) eip·(x−x ) C(τ − τ , E(p)). C(τ,x),(τ ,x ) = B
(14)
The function (13) arises in time–ordered expansions relative to a quasifree state corresponding to a quadratic Hamiltonian H0 with dispersion relation E, as discussed above. If we denote the fermionic field operators in a second–quantized formulation by ax and (+) (−) set aτ,x = eτ H0 ax∗ e−τ H0 and aτ,x = eτ H0 ax e−τ H0 , (−) (+) C(τ,x),(τ ,x ) = −ω0 T[aτ,x aτ ,x ] , (15) where ω0 denotes the quasifree state corresponding to H0 and to the inverse temperature β, around which we expand, and T denotes time [AGD]. As ω0 is a KMS state, ordering (15) makes sense for all τ, τ ∈ R with 0 ≤ τ − τ ≤ β. Because the field operators obey the canonical anticommutation relations, the time ordering, which avoids commutator terms (keeping only the fermionic antisymmetry), leads to discontinuities in the function, which are explicit in (11). Thus the discontinuity of C reflects the microscopic structure of the physical system, as encoded in the anticommutation relations of the field operators that generate the observable algebra. In the above definitions, we have assumed for simplicity that C x y and v(x − y) depend only on space coordinates x, y ∈ X , with X as above. It is straightforward to generalize
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our arguments to the case with spin or additional indices on which the fields depend (e.g. for the usual models with SU (N ) symmetry, this just amounts to replacing C by C ⊗ 1 N , where 1 N denotes the N –dimensional unit matrix, and the representations by inner products used below can be adapted in the obvious way by tensoring with a factor C N and using that δi, j = ei , e j for any orthonormal basis of C N ). Obviously, (13) can be regarded as defining an (Xd × Xd )–matrix, where Xd = [0, β) × X .
(16)
ˆ d = M F × B, X
(17)
Let where M F =
π β (2Z + 1).
The Fourier transform of C is
ˆ C(ω, p) =
1 ˆ d. , (ω, p) ∈ X i ω − E(p)
(18)
The standard way to obtain a Gram representation for (regularized) covariances in quanˆ d ), tum field theory is via their Fourier representation. In our present setting, if Dˆ ∈ L 1 (X 2 ˆ then a Gram representation for D is obtained simply by setting H = L (Xd ), and for (τ, x) ∈ Xd , 1/2 ˆ p) , vτ,x (ω, p) = e−iτ ω+ip·x D(ω, −1/2 ˆ ˆ p) wτ,x (ω, p) = e−iτ ω+ip·x D(ω, D(ω, p). (19) ˆ 1 , and the dominated convergence theorem implies The Gram constant is γ D = D continuity of the maps (τ, x) → vτ,x and (τ, x) → wτ,x . However, the Cˆ in (18) decays so slowly as a function of the Matsubara frequency ω ˆ d ) (this must be so because C itself has a discontinuity, so its Fourier transthat Cˆ ∈ L 1 (X form cannot be integrable). Thus the standard procedure to obtain a Gram representation fails. Lemma 2.2. Let U be the (R × R)-matrix given by 1, s≥t Ust = . 0, s
(20)
If (H, v, w) is a Gram representation of U , then H is non-separable and the maps t → vt and t → wt are discontinuous at all t ∈ R. Proof. For all s, t ∈ R, Ust = vs , wt , so for t > t, vt , wt − wt = 1 and for t < t, vt , wt − wt = −1. Thus, by the Schwarz inequality and the bound supt vt ≤ γU , ∀t, t : t = t ⇒ wt − wt ≥
1 . γU
(21)
Thus the map t → wt is discontinuous everywhere. Reversing the roles of vt and wt in the above argument implies the same for the map t → vt . An obvious variant of this argument implies discontinuity in the weak topology as well. Set W = {wt : t ∈ R}. Let A ⊂ H be countable. For all x ∈ A, Eq. (21) and the triangle inequality imply that {y ∈ H : y − x < 4γ1U } contains at most one element of W . Thus the 4γ1U – neighbourhood of A contains only countably many elements of W , hence A is not dense in H.
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Corollary 2.3. The covariance matrix of the many–fermion system given by (13) has no Gram representation on a separable Hilbert space. Proof. The function τ → D(τ, E) = C(τ, E) − C(τ, 0) is continuous in τ . Its Fourier transform, ω → − is in 1 . Thus
D(τ,x),(τ ,x ) =
E , iω(iω − E)
µa (d p) eip·(x−x ) D(τ − τ , E(p))
(22)
(23)
has the Gram representation given in (19). An elementary argument involving direct sums of Hilbert spaces shows that C = D + D has a Gram representation if and only if D has a Gram representation. Assume that C, given by (13), has a Gram representation on a separable Hilbert space H. Then C − D has a Gram representation on a direct sum of separable Hilbert spaces, which is itself separable. But C − D is 1 a δx,x ) (Uτ,τ − 2
(24)
a a with δx,x µ (d p) eip·(x−x ) and U as in Lemma 2.2, which has no Gram represen = tation on any separable Hilbert space. Our main use of Gram representations is, of course, to bound determinants of the type occurring in (4). Lemma 2.2 does not exclude that a useful Gram representation, i.e. one with a good Gram constant, can be found, but it shows that the representation will be very different from the ones used so far in fermion models, which all involve separable Hilbert spaces and where continuity of the maps v and w holds. One can attempt to circumvent the above problem by introducing a UV cutoff > 0, which restricts the sum over frequencies ω to a finite set (for instance by regularizing ˆ χ (ω, p) = C(ω, ˆ to C p) χ (ω/ ), where χ is a smooth function of compact support). This obviously makes the standard Gram constant finite, Of course, a UV cutoff cannot simply be imposed, because it implies that the time-ordered imaginary-time correlation functions are continuous and therefore not physical. The Gram constant γCχ ∼ log diverges for → ∞. One can attempt to perform the limit → ∞ by multiscale and renormalization techniques. The approach via determinant bounds developed in the next sections is, however, much simpler and more natural that such a multiscale approach, and it makes the latter unnecessary. Recall that momentum space is B = Rd for a continuous system and B = Rd / # for a system on a lattice , that in the continuum case, µa contains an ultraviolet cutoff, and that B L = L −1 # / # is the corresponding momentum space for the finite–volume system. The main result about the determinant bound of many–fermion covariances is as follows. Theorem 2.4. Let E : B → R be bounded and measurable. Then the fermionic covariance matrix C (h) given in (14) has determinant bound 1/2 a δC (h) = 2 . (25) µ (dp) |h(p)| In particular, the covariance C defined in (13) has δC = 2µa (B)1/2 .
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Theorem 2.4 is proven after Corollary 4.2. As mentioned after Theorem 1.3, this bound is optimal up to the prefactor 2. In Sect. 4, we discuss the decay constant of these covariances and prove a convergence theorem for the expansion for the fermionic effective action. In Sect. 5, we discuss the properties of covariances obtained by a splitting into small and large frequencies and prove that the integration over fields with large frequencies, which usually is the first step in a multiscale treatment, is given by convergent expansions, for arbitrarily large initial interaction strength. When rewriting traces using Trotter–type formulas, to obtain functional integral representations, one typically obtains time–discretized covariances. The bounds given here apply to them as well, uniformly in the parameter n that defines the discretization [PS]. 3. Determinants and Chronological Products In this section we show that determinants corresponding to a general chronological ordering have good determinant bounds and prove Theorem 1.3. We first recall some standard facts and fix notation. Definition 3.1. Let V be a finite–dimensional vector space over C. 1. Let k ∈ N. A totally antisymmetric k–linear map α : V k → C is called k–form. The vector space of all k–forms is identified with k V ∗ . We also set 0 V ∗ = C. k ∗ l ∗ 2. Let k, l ∈ N. The exterior product of α ∈ V and β ∈ V , α ∧ β ∈ k+l V ∗ , acts on v1 , . . . , vk+l ∈ V as (α ∧ β) (v1 , . . . , vk+l ) 1 = sgn (σ ) α(vσ (1) , . . . , vσ (k) ) β(vσ (k+1) , . . . , vσ (k+l) ). k!l!
(26)
σ ∈Sk+l
Here Sn denotes the setof permutations of {1, . . . , n}. The exterior algebra V ∗ over the vector space V is
V∗ =
∞ k
V∗
(27)
k=0
We identify
V with
V ∗∗ , the exterior algebra over V ∗ .
The following condition defines a duality between the spaces α = α1 ∧ . . . ∧ αk ∈ k V ∗ and v = v1 ∧ . . . ∧ vk ∈ k V ,
k
V ∗ and
α, v = det(αi (v j ))i, j . This duality defines a vector space isomorphism
k
V∗ → (
k
V : for (28)
k
V )∗ :
α1 ∧ . . . ∧ αk , v1 ∧ . . . ∧ vk = α1 ∧ . . . ∧ αk (v1 , . . . , vk )
(29)
(this isomorphism is unique only up to a multiplicative factor, and different conventions are used in the literature). the isomorphisms (29), k ∈ N, canonically induce an ∗ Finally, isomorphism between V ∗ and V .
Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems
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Definition 3.2. Let End V ∗ denote the set of endomorphisms of V ∗ . 1. For w ∈ V define w ∈ End V ∗ by the condition ∀v ∈
2. For α ∈ V ∗ let (α∧) ∈ End
∀β ∈
V : wα, v = α, w ∧ v.
(30)
V ∗ be defined by
V ∗ : (α∧) : β → α ∧ β.
(31)
Lemma 3.3. These endomorphisms obey canonical anticommutation relations: 1. (α1 ∧)(α2 ∧) + (α2 ∧)(α1 ∧) = 0, for all α1 , α2 ∈ V ∗ . 2. u 1 u 2 + u 2 u 1 = 0, for all u 1 , u 2 ∈ V . 3. (α∧)u + u(α∧) = α(u), for all α ∈ V ∗ and all u ∈ V . Proof. Items 1 and 2 are clear. Item 3 holds because for all u ∈ V , u : k V ∗ → k−1 ∗ k ∗ ∗ V is an antiderivation of degree -1: for all α ∈ V and all β ∈ V , u(α ∧ β) = (uα) ∧ β + (−1)k α ∧ (uβ). Lemma 3.4. Let n ∈ N, α1 , . . . , αn ∈ V ∗ and v1 , . . . , vn ∈ V . Then n(n−1) = (−1) 2 v1 . . . vn (α1 ∧ . . . ∧ αn ). det αi (v j ) 1≤i, j≤n
(32)
Proof. Observe that (32) makes sense because the right hand side of this equation is an element of 0 V ∗ = C. Equation (30) implies by induction that v1 . . . vn (α1 ∧ . . . ∧ αn ) = α1 ∧ . . . ∧ αn , vn ∧ . . . ∧ v1 Inverting the order of the vi and using (28) gives the claim.
(33)
Definition 3.5. Let (J , ) be a totally ordered set. For j, j ∈ J , j = j denote 1 if j j 1 j j = (34) 0 if j j. 1. For J, J ⊂ J define ρ(J, J ) = (−1) N J,J , where N J,J is the number of pairs ( j, j ) ∈ J × J with j j . 2. Let K ∈ N and J = ( j1 , . . . , j K ) be a finite sequence in J , such that k = l ⇒ jk = jl . Let π ∈ S K denote the unique permutation chosen such that for all k ∈ {1, . . . , K − 1}, jπ(k) ≺ jπ(k+1) . Let ε1 , . . . , ε K ∈ End V ∗ . The J–chronological product of ε1 , . . . , ε K is TJ [ε1 , . . . , ε K ] = sgn (π )
K
επ(ν) .
(35)
ν=1
3. Let J = { j1 , . . . , jn } , J = { j1 , . . . , jn } with j1 ≺ . . . ≺ jn , j1 ≺ . . . ≺ jn and J ∩ J = ∅. Let ε1 , . . . , ε2n ∈ End V ∗ and J = ( j1 , . . . , jn , j1 , . . . , jn ). For this
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special choice we denote T J,J [ε1 , . . . , ε2n ] = TJ [ε1 , . . . , ε2n ] and call it the
(J, J )–chronological
(36)
product of ε1 , . . . , ε2n .
An obvious consequence is Lemma 3.6. LetJ and J be chosen as in item 3 and π as in item 2 of Definition 3.5. Then sgn (π ) = ρ(J, J ).
(37)
This sign is chosen in the definition (35) of the chronological product because in our application the εi will be odd elements of the graded algebra End V ∗ . In general, the sign involved in the chronological product is well–defined only if each εi is either even or odd, and the sign includes only the permutations of odd elements. The main result of this section is the following generalization of Lemma 3.4. Theorem 3.7. Let (J , ) be a totally ordered set and J and J be chosen as in Definition 3.5. For α1 , . . . , αn ∈ V ∗ and v1 , . . . , vn ∈ V define the (n × n)–matrix M by Mkl = αk (vl ) 1 jk jl .
(38)
det M = (−1)n(n−1)/2 T J,J [v1 , . . . , vn , (α1 ∧), . . . , (αn ∧)]1.
(39)
Then Proof. Induction on n. The case n = 1 is obvious. Let n ≥ 2 and assume (39) to hold for matrices of size n − 1. By definition and by Lemma 3.6, the chronological product T J,J [· · ·] on the right-hand side of (38) is ρ(J, J )A1 . . . A2n , with Ai ∈ {v1 , . . . , (αn ∧)}. Suppose that A1 = (αm ∧) for some m. Then A2 . . . A2n 1 = 0, so the right hand side of (39) vanishes. The indicator function in the definition of M implies that the m th row of M is zero, so that the left-hand side of (39) vanishes, too. Thus we may assume that A1 ∈ {v1 , . . . , vn }. Because J is ordered, A1 = v1 . Use A1 A2 . . . A2n =
2n
(−1)k A2 . . . Ak−1 (A1 Ak + Ak A1 )Ak+1 . . . A2n
k=2
0
−A2 . . . A2n A1 .
(40)
When applied to 1 ∈ V ∗ , the last term vanishes because A1 1 = 0. By Lemma 3.3, A1 Ak + Ak A1 = αm (v1 ) if Ak = αm ∧ for some m ∈ {1, . . . , n}, and zero otherwise. The position k where αm ∧ appears in the product is k = 1 + |{ j ∈ J ∪ J : j ≺ jm }| = 1 + m − 1 + |{ j ∈ J : j ≺ jm }|.
(41)
Thus (−1)k = (−1)m ρ({ jm }, J ). Let I = J \ {1} and Im = J \ { jm }. The remaining product A2 . . . Ak−1 Ak+1 . . . A2n times the sign factor ρ(I, Im ) equals the (I, Im )–chronological product, so T J,J [v1 , . . . , vn , (α1 ∧), . . . , (αn ∧)]1 (42) n σm (J, J ) T I,Im [v2 , . . . , vn , (α1 ∧), . . . , (αm−1 ∧), (αm+1 ∧), . . . , (αn ∧)]1 = m=1
Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems
807
with σm (J, J ) = ρ(J, J ) (−1)m ρ({ jm }, J ) ρ(I, Im ).
(43)
ρ(J, J ) = ρ(I, Im ) ρ(J, { jm }) ρ({1}, Im ) ,
(44)
ρ({ jm }, J ) ρ(J, { jm }) = (−1)|J | = (−1)n .
(45)
σm (J, J ) = (−1)m+n .
(46)
By definition,
ρ({1}, Im ) = 1, and
Thus
The inductive hypothesis applies to the chronological product on the right hand side of (42). Combine (−1)n(n−1)/2+m+n = (−1)(n−1)(n−2)/2 (−1)m−1 . The statement of the theorem follows by identifying the right hand side of (42) as the Laplace expansion for the determinant. In the remainder of this section, we prepare and give the proof of Theorem 1.3. Lemma 3.8. Assume that the space V is a Hilbert space with scalar product ·, ·V . In this case we identify V with its dual V ∗ (v ∈ V → v, ·V ∈ V ∗ ) and consequently k k V with k V ∗ ∼ = ( V )∗ (see (28) and (29)). 1. The scalar product ·, ·V of V induces, for each k ∈ N, through the identifica tion of elements of k V with elements of its dual ( k V )∗ a norm · on k V : u2 = u, u. This norm fulfills the parallelogram identity u + v2 + u − v2 = 2u2 + 2v2 , ∀u, v ∈ hence it defines a compatible scalar product on are Hilbert spaces. 2. (u)† = (u∧) and (u∧)† = u, for all u ∈ V . 3. max{u, (u∧)} ≤ u, for all u ∈ V .
k
V . Thus
k k
V ,
(47)
V and hence
V
Proof. 1. To see that · is nondegenerate, use the defining identity (28). The other propertiesare clear. Item 2 follows directly from Definition 3.2.1. To see 3, let u ∈ V and w ∈ V . Then by Lemma 3.3, w, (u(u∧) + (u∧)u)w = w2 u2 . Thus u2 = sup w, (u(u∧) + (u∧)u)w ≥ max{u2 , (u∧)2 }. w∈ V w=1
(48)
In Definition 3.1, we required the space V to be finite–dimensional, to avoid a discussion of subtleties in the relation between V and its dual. In our applications, we can always achieve that V is a finite–dimensional subspace of a Hilbert space or a reflexive Banach space, by taking V as a space spanned by finitely many vectors. For Hilbert spaces, we could alternatively also have dropped the condition of finite dimensionality in the above.
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Lemma 3.9. Let ϕ, ϕ : N → J be functions into a totally ordered set (J , ). Let H be a Hilbert space. For all n ∈ N and all v1 , . . . , vn , w1 , . . . , wn ∈ H, n ≤ det vk , wl H 1ϕ (k)ϕ(l) vk wk . k,l
(49)
k=1
The same inequality holds with 1ϕ (k)ϕ(l) replaced by 1ϕ (k)ϕ(l) . Proof. For n ≥ 1 let Nn = {1, . . . , n}. Define Gn = { j ∈ J : ∃k, l ∈ Nn : ϕ (k) = ϕ(l) = j}.
(50)
Obviously, |Gn | ≤ n. Let m = max{(ϕ )−1 ({ j}) ∩ Nn , ϕ −1 ({ j}) ∩ Nn } j∈J
(51)
and set J˜n = J × {0, 1} × {1, . . . , m}. Extend the ordering lexicographically, i.e. ( j, µ, ν) ( j , µ , ν ) ⇔ j j or [ j = j and µ µ ] or [ j = j and µ = µ and ν > ν ]. Then (J˜n , ) is totally ordered. For j ∈ Gn , there are r ≤ m and k1 , . . . , kr ∈ Nn such that for all ρ ≤ r , ϕ (kρ ) = j, and there are s ≤ m, l1 , . . . , ls ∈ Nn such that for all σ ≤ s, ϕ(lσ ) = j. We now extend ϕ to ϕ˜ and ϕ to ϕ˜ as follows. Case of the matrix with 1ϕ (k)ϕ(l) . In this case, 1ϕ (k)ϕ(l) = 0 if ϕ (k) = ϕ(l). To obtain 1ϕ˜ (k)ϕ(l) = 0, we make ϕ˜ (k) smaller by setting ϕ˜ (kρ ) = (ϕ (kρ ), 0, ρ) and ˜ ϕ(l ˜ σ ) = (ϕ(lσ ), 1, σ ). Case of the matrix with 1ϕ (k)ϕ(l) . In this case, 1ϕ (k)ϕ(l) = 1 if ϕ (k) = ϕ(l). To obtain 1ϕ˜ (k)ϕ(l) = 1, we make ϕ˜ (k) bigger by setting ϕ˜ (kρ ) = (ϕ (kρ ), 1, ρ) and ˜ ϕ(l ˜ σ ) = (ϕ(lσ ), 0, σ ). For j ∈ J \ Gn , j = ϕ (k), we set ϕ˜ (k) = (ϕ (k), 0, ρ) etc. By definition of the lexicographical ordering on J˜ , it does not matter which convention one chooses on J \ Gn . By construction, ϕ˜ (Nn ) = J and ϕ(N ˜ n ) = J are disjoint, and |J | = |J | = n. We may permute the rows and columns of the matrix such that ϕ(m ˜ 1 ) ≺ ϕ(m ˜ 2 ) if m 1 < m 2 and similarly for ϕ˜ . This does not change the absolute value of the determinant. We can now apply Theorem 3.7, to represent the determinant as a chronological product. The norm estimate in Lemma 3.8 implies the statement. Definition 3.10. Let n ∈ N and A be a complex (n × n)–matrix. We say that (A, γ ) holds iff for all p ∈ {1, . . . n} and all sequences a1 < · · · < a p and b1 < · · · < b p in {1, . . . , n}, sup (n)
v1 ,...,v p ,w1 ,...,w p ∈B1
det vq , wr Aaq ,br 1≤q,r ≤ p ≤ γ 2 p .
(52)
Lemma 3.11. Let n and k ∈ N and A(1) , . . . , A(k) be complex (n ×n)–matrices. Assume that for all l ∈ {1, . . . , k} there are γl > 0 such that the property (A(l) , γl ) holds. Then A(1) + · · · + A(k) , γ1 + · · · + γk holds.
Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems
809
Proof. Inductionon k. For k = 1, the statement is obvious. In the induction step, let k ≥ 2, and assume A(2) + · · · + A(k) , γ2 + · · · + γk to hold. Let p ∈ {1, . . . n}, a1 < · · · < (n) a p , and b1 < . . . < b p in {1, . . . , n}, and v1 , . . . , v p , w1 , . . . , w p ∈ B1 . Let B and C be k and Cq,r = vq , wr i=2 Aa(i)q ,br the matrices with elements Bq,r = vq , wr Aa(1) q ,br k Also, set γ1 = l=2 γl . Then by the generalized Laplace expansion for determinants det(B + C) = ε p (S, T ) det B S,T det C S c ,T c , (53) S,T ⊂{1,..., p} |S|=|T |
where S c = {1, . . . , p} \ S and ε p (S, T ) ∈ {−1, 1}, and the subscripts denote the submatrices of B and C defined by the sets. Let s = |S| = |T |. By hypothesis of the lemma, for all S, T , det B S,T ≤ γ 2s (54) 1 and by the inductive hypothesis, det C S c ,T c ≤ γ 2( p−s) . 1 p2 2 p Thus, using s ≤ 2s , |det(B + C)| ≤
p 2 p s=0
s
γ1
2s
2( p−s) γ1
(55)
k
2 p ≤ γl .
(56)
l=1
Proof of Theorem 1.3.. Call the n × n submatrices of the summands in (8) Ml . By Lemma 3.11, it suffices to show that for all l ∈ {0, . . . , k + K }, (Ml , γl ) holds. The matrix Cl has a Gram representation (H, g, h) with Gram constant γl . Then v , wCn (Cl )x y = v ⊗ gx , w ⊗ h y Cn ⊗H
(57)
and, if v ≤ 1, v ⊗ gx = v gx ≤ gx , similarly for w ⊗ h y . Ml is obtained (for l > 0) by multiplying this with an indicator function. Every submatrix of Ml is of the same form as Ml and satisfies the hypotheses of Lemma 3.9. Thus (Ml , γl ) holds. That all submatrices are involved in property , as necessary for the inductive argument in the proof of Lemma 3.11, is the reason for taking the supremum over the larger set in Definition 1.2, instead of taking a supremum over P ∈ Pn,1 . Submatrices of a P ∈ Pn,1 are in general not positive. By contrast, the property of having a Gram representation on Cn with Gram constant 1 is stable under taking submatrices. 4. Convergent Expansions without UV Cutoffs In this section we apply the results of Sect. 3 to the many–fermion covariances introduced in Sect. 2. We give explicit determinant and decay bounds, and prove Theorem 2.4. Moreover, we show that, for a multiscale expansion with the standard Fermi surface cutoff functions and sectorization, our results yield all standard power counting bounds without requiring a cutoff on the Matsubara frequencies, so that the analytic structure as a function of the frequencies can be preserved in such a multiscale analysis.
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4.1. Determinant bound. In the following, we apply Theorem 1.3 to the covariance (14), of which (13) is the special case h = 1. Before stating the details of the representation we briefly motivate it. By definition, C(τ, E) = −1τ >0 e−τ E f β (−E) + 1τ ≤0 e−τ E f β (E).
(58)
Let ε > 0 and 1 (s, ε) = √ π
ε f β (−ε) . is − ε
(59)
Then, since ε > 0, s → (s, ε) ∈ L 2 (R), (·, ε)2 ≤ 1, and, ∀τ ≥ 0, ε > 0 :
e
−ετ
f β (−ε) =
R
ds eisτ |(s, ε)|2 .
(60)
Thus, if τ = t − t > 0, e−ετ f β (−ε) = vt , vt with vt (s) = e−ist (s, ε). To use this for C we need to respect the signs in (58), hence rewrite, for τ ∈ [−β, β], ⎧ −τ E −e f β (−E) ⎪ ⎪ ⎨ (β−τ )E −e f β (E) C(τ, E) = −(β+τ )E f (−E) ⎪ e β ⎪ ⎩ −τ E f β (E) e
if τ if τ if τ if τ
> 0 and E > 0 and E ≤ 0 and E ≤ 0 and E
>0 <0 >0 <0
(61)
using f β (−E) = eβ E f β (E). By Tonelli’s theorem and an obvious decomposition of the remaining factors in the integrand, we can represent C(t,x),(t ,x ) by integration over p. Note that the vt defined above vanishes at E = 0, but that C(τ, 0) = 21 − 1τ >0 = 0, so it is necessary to restrict to functions E(p) whose zero level set has measure zero. Lemma 4.1. Let E : B → R be measurable and assume that µa ({p ∈ B : E(p) = 0}) = 0 .
(62)
Let h ∈ L 1 (B, µa ) with h(p) ≥ 0 for all p ∈ B. For x = (t, x) ∈ Xd and σ ∈ {−1, 1} define h(p)1σ E(p)>0 , (s, |E(p)|) h(p)1 E(p)<0 .
gxσ (s, p) = e−ip·x−ist (s, |E(p)|) h x (s, p) = e−ip·x+ist
(63)
Then for all x ∈ Xd , gx+ , gx− and h x are in H = L 2 (R × B, ds ⊗ dµa ), with norms 1/2 bounded by h1 , and the covariance (14) has the representation (h)
− + − gβ−t,x , gt+ ,x + h t ,x C(t,x),(t ,x ) = 1t>t −gt,x + +1t≤t gt,x + h t,x , gt+ −β,x + h t ,x .
(64)
Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems
811
Proof. The integrand in (14) is bounded, so we can remove the set of measure zero {p ∈ B : E(p) = 0} from the integral. On its complement, the Gram representation given in the lemma converges absolutely as an iterated integral first over s, then over p, hence by Tonelli’s theorem in any order of integration, and the L 2 –norms are finite by the same argument. The bound for the norms is obvious from the properties of . By the support properties of the functions, − − + + −gt,x − gβ−t,x , gt+ ,x + h t ,x = −gt,x , gt+ ,x + −gβ−t,x , h t ,x
(65)
+ + gt,x + h t,x , gt+ −β,x + h t ,x = gt,x , gt+ −β,x + h t,x , h t ,x .
(66)
and
Decomposing the integration domain into B± = {p ∈ B : ±E(p) > 0}, (64) follows from (60) and (61). The condition that h ≥ 0 in Lemma 4.1 was just for convenience in stating the result in a simple form. With an obvious generalization, replacing h(p) by h(p) |h(p)|−1/2 , and defining a few more functions g˜ to take care of the necessary complex conjugations, a representation with the same properties as (64) can be obtained for general h ∈ L 1 (B, µa ). In the applications below, h will be a scaling function, hence nonnegative. Corollary 4.2. Under the hypotheses of Lemma 4.1, the many–fermion covariance (14) has a determinant bound δC (h) with 1 1/2 1/2 √ h1 ≤ δC (h) ≤ 2h1 2
(67)
(for h = 1, corresponding to the covariance (13), h1 = µa (B) = a −d ). Proof. The indicator functions in the times t and t correspond to the choices (J , ) = ([−β, β], >), ϕ1 (t, x) = ϕ1 (t, x) = t and ϕ2 (t, x) = ϕ2 (t, x) = −t. The upper bound follows from the explicit representation given in Lemma 4.1 by applying Theorem 1.3. Let
ρ± = µa (dp) f β (±E(p)) h(p), (68) B
then ρ− = h1 − ρ+ . Set x = x . Then considering the cases t = t and t ↑ t gives 1 h1 . sup C x(h) (69) ≥ max{ρ+ , ρ− } ≥ x 2 x,x ∈Xd The lower bound for δC (h) now follows from (10).
Proof of Theorem 2.4. To apply Lemma 4.1, we need to satisfy the zero measure condition. For ε > 0, define E ε : B → R by E ε (p) = ε/2 if |E(p)| ≤ ε/2 and E ε (p) = E(p) otherwise. Obviously, E − E ε ∞ ≤ ε, and {p ∈ B : E ε (p) = 0} = ∅. Because β < ∞, the covariance C (h) is a continuous function of E in ·∞ , so C (h) is the limit ε → 0 of the covariance C (h,ε) given by E ε . By construction, E ε satisfies the conditions of Lemma 4.1 so Corollary 4.2 implies the bound (25) for C (h,ε) . That bound is uniform in ε.
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The representation of C (h) given in Lemma 4.1 generalizes one found in [FKT04], where determinants of matrices of the form 0, tk − tl ≤ 0 Mkl = vk , wl −(tk −tl ) , , tk − tl > 0 e for vectors vk , wl in a Hilbert space H and real numbers tk , tl , were considered. The result of [FKT04] corresponds to the special case of the function ˜ ) = −e−τ 1τ >0 , C(τ
(70)
which is the limit β → ∞ of (58) at E = 1. Thus our method applies to that case, with ˜ (s) = (is − 1)−1 . 4.2. Decay constant. Under very mild conditions on E, the determinant bounds we have proven are uniform in β (see Corollary 4.2). One must of course not jump to the conclusion that this implies convergence of perturbation series uniformly in the temperature because a finite determinant bound is only one condition for convergence of the perturbation expansion. The second is the finiteness of the decay constants
β
(k ,k) αC 0 = dτ dx |C(τ, x)| |τ |k0 |x|k (71) −β
X
for k0 ≥ 0 and k ≥ 0. In this paper, we only discuss the case k0 = k = 0, and denote αC(0,0) = αC because the simplest convergence theorem requires only this data, and because the generalization is straightforward. For our many–fermion covariance, the existence of a nonempty Fermi surface that is not degenerated to a point implies that the decay constant grows polynomially in β and diverges in the zero–temperature limit. Only for special situations, such as a model for an insulator, for which |E(p)| ≥ E min > 0, the decay constant is uniform in β. For simplicity we assume here the case of a continuous torus B. The case of a discrete torus corresponding to a finite volume is similar, and treated in [PS]. For z ∈ C and ε ≥ 0 set ||| z |||ε = max{|z|, ε}. Lemma 4.3. Let E ∈ C d+2 (B, R). Let 0 < < 1 and assume that h(p) = f ( E(p) )g(p), where f ∈ C ∞ (R, R+0 ) and g ∈ C ∞ (B, R+0 ). Let b ∈ Nd0 be a multiindex and b = |b|. 1. There is a constant K d > 0 such that for b ≤ d + 1,
β
b (h) dτ xb C(τ,x),(0,0) ≤ K d m−b −β
m=0
supp h
µa (dp) . ||| E(p) |||m+1 1
(72)
β
2. If there is κ0 > 0 such that for all E for which Sˆ E,g = {p ∈ supp g : E(p) = E} is nonempty, inf p∈ Sˆ E,g |∇ E(p)| ≥ η > 0, and the submanifold Sˆ E,g of B has Gauss curvature bounded below pointwise by κ0 , then there is a constant K˜ d > 0 such that for b ≤ $ d+2 2 %,
β −β
(h) dτ xb C(τ,x),(0,0) ≤
K˜ d |x|
d−1 2
b m=0
m−b
dE
1 E ∈supp f
||| E |||m+1 1 β
.
(73)
Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems
Proof. We have
(h)
813
b ∂ µa (dp) C(τ, E(p)) h(p) −i ∂p eip·x .
xb C(τ,x),(0,0) =
(74)
Upon integration by parts, the derivative can act in four places — on C, on either of the factors f and g in h, or (for the continuum system) on the spatial ultraviolet cutoff function χ in µa (dp) = χ (ap)dp. Thus (h) xb C(τ,x),(0,0) =
b b−m
−n
µa (dp) m (τ, E(p)) f (n)
E(p)
ip·x G (b) , m,n (p) e
m=0 n=0 (b)
(b)
where G m,n ∈ C d+2−b (B, R) is independent of and satisfies supp G m,n ⊂ supp g, and m (τ, E) =
dm C(τ, E). dE m
(75)
Taking the absolute value inside all sums and integrals and using that
β dτ |m (τ, E)| ≤ const ||| E ||| 1 −m−1 ,
(76)
β
−β
we obtain (72). To prove (73), we rewrite (h)
xb C(τ,x),(0,0) =
b b−m
−n
dE m (τ, E) f (n)
E
S E,G (b) (x), m,n
(77)
m=0 n=0
where
S E,G (b) (x) = m,n
ip·x µa (dp) δ(E − E(p)) G (b) . m,n (p) e
By standard theorems about the Fourier transform of surfaces [St], d−1 S E,G (b) (x) ≤ const |x|− 2 m,n
(78)
(79)
with a constant that depends on κ0 and E, and which is finite under our regularity assumption on E. Finally, we use again (76). The regularity assumptions on E in Lemma 4.3 are not optimized. For improved bounds using smoothing techniques, see [PS]. The scaling function h can be chosen C ∞ in our applications, so that the assumptions of Lemma 4.3 on h are not restrictive. This lemma allows us to bound decay constants as follows. Corollary 4.4. Let E ∈ C d+2 (B, R). 1. αC ≤ const β d+1 . 2. If the system is an insulator, i.e. if there is E 0 > 0 such that for all p ∈ B, |E(p)| ≥ E 0 , then αC (h) ≤ const E 0−d−1 .
(80)
The constant is proportional to the volume of the support of h. For h = 1, it is proportional to µa (B).
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If there is E 1 such that for all energies E with |E| ≤ E 1 the level sets satisfy the hypotheses of Lemma 4.3, item 2, then we also have: 3. αC ≤ const
E 1−d−1 + β
d+3 2
.
(81)
4. If f (x) = 0 unless 1 ≤ |x| ≤ 2, then d+1
αC (h) ≤ const − 2 . (82) √ p 5. For a sector of angular radius , i.e. g( p) = γ ( √ ), with γ supported near 0, αC (h) ≤ const −1 .
Proof. The first bound follows by the standard summation argument from ||| E(p) ||| 1 ≥ β
1 β . The case of an insulator follows immediately from ||| E(p) ||| β1 ≥ E 0 . To prove (81), we E(p) insert a partition of unity χ< ( E(p) ) + χ> ( ) = 1, where χ< (x) vanishes for |x| ≥ 1.
The support condition on f in item 4 implies ||| E(p) ||| 1 ≥ . Again, summation implies β the result. The sector estimate is similar.
4.3. Convergence theorem. In the following we state a theorem about convergence of expansions for the effective action which generalizes the main theorem of [SW]. As in [SW], we define an interaction V by its interaction vertices vn,m : Xn × Xm → C as (83) dn X dm X vn,m (X , X )ψ¯ n (X )ψ m (X ), V () = m,n≥0
where X = (X 1 , . . . X m ) and ψ m (X ) = ψ(X 1 ) . . . ψ(X m ). For h > 0, let V h = |vn,m |h n+m ,
(84)
m,n≥0 m+n≥1
where |vn,m | = max sup i∈Nn+m
Xi
dX j |vn,m (X 1 , . . . , X n+m )|.
(85)
j=i
Theorem 4.5. Let C be an (X × X)–matrix, considered as a covariance for a fermionic Gaussian integral, with finite determinant bound δC and decay bound αC . Denote ωC = 2αC δC−2 . Let h > 0, h = h + 2δC , and let V be an interaction with V h < ∞. Then the effective action W (V, C), defined as
W (V, C) = log dµC ( ) e V ( +) , (86) exists and is analytic in V : let in powers of V . Then for all P P W (V ) − p=1
W (V, C) = ≥ 1,
1 p≥1 p! W p (V, C)
be the expansion of W
V h P+1 1 W p (V, C) ≤ ωC P . p! 1 − ωC V h h
(87)
Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems
815
Proof. Same as in [SW], except that in the bound for the determinants, Lemma 6 of [SW], the Gram constant is replaced by the determinant bound δC . The coefficients in the expansion of W (V, C)() in the fields are the amputated connected Green functions, so the above theorem implies their analyticity in the interaction. In particular, analyticity holds for all cases listed in Corollary 4.4, with the appropriate constants. In case of an insulator, the convergence radius is uniform in the temperature. In case of scaled propagators, one obtains power counting bounds that are on all scales operationally equivalent to those with a frequency cutoff. That no ω-cutoff is needed implies that the analytic structure as a function of ω need not be mutilated in a multiscale construction. 5. Bounds for the Integration over Large Frequencies In a multiscale analysis of many–fermion systems, the integration over fields with large Matsubara frequency is often the first integration step in the analysis. In the following we give bounds for the effective action obtained by this integration step. We first decompose the covariance C (h) given in (14) in an ultraviolet and an infrared part. Let χ< and χ> ∈ C ∞ (R, [0, 1]) with χ< + χ> = 1, χ< (0) = 1, with constants κ > 0 and α > 0 such that χ< (x) ≤ κ|x|−α for all |x| ≥ 1. Abbreviate the covariance (h) C(τ,x),(τ ,x ) = C (h) (τ − τ, x − x). The covariance (h,<) C (τ, x)
ω h(p) 1 = µa (dp) e−iωτ +ip·x χ< β ω iω − E(p)
(88)
is the infrared part of C (h) , and (h,>)
C
(h,<)
(τ, x) = C (h) (τ, x) − C
(τ, x)
(89)
is the ultraviolet part of C (h) . An obvious variant of this decomposition is one where the argument of the function χ< is −2 (ω2 + E(p)2 ). Our bounds adapt to this choice in an obvious way, so we will not discuss it further here. By standard properties of Grassmann Gaussian integration, the convolution with the (h,>) (h,<) Gaussian measure C (h) = C + C becomes an iterated convolution, first with (h,>) (h,<) (see, e.g. [S98b]). C , then with C 5.1. Determinant bound. Lemma 5.1. Let χ< be chosen as above, ≥ 1, and β > π . Let E be continuous. Then (h,>) the determinant bound of C satisfies δ 2 (h,>) ≤ h1 (K + 2 ln ) + C
where K = 10 + 2κ(α −1 + (β)−1 ).
|E(p)|≤1
µa (dp) |h(p)| ln
1 , ||| E(p) ||| πβ
(90)
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Proof. By our hypothesis on the decay of χ< , the Fourier transform of the covariance (h,<) (h,<) C is 1 in the Matsubara frequency. Thus C has a Gram representation of type (19), with finite Gram constant γ< . By (89) and Theorem 1.3, a determinant bound for (h,>) C is given by δC (h) +γ< . δC (h) was bounded in Corollary 4.2, so it suffices to estimate γ< . By definition, γ<
2
|h(p)| 1 ω µa (dp)χ< ( = ) . |iω − E(p)| β
(91)
ω∈M F
The contribution from |ω| ≥ 1 is bounded by κ 1 1 κ ω 1 h1 . χ< ( ) ≤ 2h1 π + ln + + |ω| β α β ω∈M F
Forthe contribution from |ω| < 1, we will repeatedly use the elementary bound 1 2u −1 ≤ 1, so the contribution from ω∈M F 1|ω|
(92)
ω∈M F
and, bounding the sum by an integral, 1 β
ω∈M F |E(p)|≤|ω|≤1
1 1 ≤ |iω − E(p)| β
ω∈M F |E(p)|≤|ω|≤1
2 1 1 ≤ + ln . |ω| π ||| E(p) ||| πβ
(93)
5.2. Decay constant. In this section we show that for a strict cutoff function χ< , and (h,>) is bounded under natural assumptions on the function E, the decay constant of C −1 by a multiple of . Thus the extra factor log from the determinant bound can be avoided in this bound. Lemma 5.2. Assume that χ< satisfies χ< (x) = 1 for |x| ≤ 1 and χ< (x) = 0 for |x| ≥ 2. Let ≥ 1. Assume that the dispersion function E is the Fourier transform E = Fˆ of some F ∈ L 1 (, C), and that the inverse Fourier transform g of h satifies g ∈ L 1 (, C). There is a constant K > 0, depending only on χ< , such that if 2K F1 < 1 and (h,>) −1 E∞ < 1, the decay constant of C satisfies αC (h,>) ≤
g1 K . 1 − 2K −1 F1
In particular, if K F1 < 41 , then αC (h,>) ≤
2K g1 .
(94)
Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems
817
Proof. Let u(τ ) =
ω 1 −iωτ , e χ< β π
(95)
ω∈ β Z
then (h,>)
C
= C (h) − u ∗ C (h) ,
where the convolution is in τ . By summation by parts, n ω π 1 −iωτ n i τ , e (δ χ< ) e β − 1 u(τ ) = β ω
(96)
(97)
where δ is the difference operator (δ f )(ω) = f (ω + πβ ) − f (ω). Using that for all τ π i τ |τ | ≥ 2 |τβ| and that χ< is smooth, it follows that with |τ | ≤ β, e β − 1 = 2 sin π2β |u(τ )| ≤
1 K , 4 (1 + |τ |)3
(98)
where K depends on the sup norms of the first three derivatives of χ< . Let a(τ ) = C(τ, 0). By definition, a(s) = θ + (−s) − 21 , where θ + (t) = 1 for t ≥ 1 β and zero otherwise. Because −β u(s)ds = χ< (0) = 1,
β a(τ ) − (u ∗ a)(τ ) =
ds u(s) [a(τ ) − a(τ − s)].
(99)
−β
The
1 2
drops out, and a(τ ) − (u ∗ a)(τ ) = sgn(τ ) I (τ ) u(s)ds, where [−β, −β + τ ] ∪ [τ, β] for τ > 0 I(τ ) = [−β, τ ] ∪ [β + τ, β] for τ ≤ 0.
(100)
Our hypothesis on g and (98) imply that (h,>)
A satisfies
(τ, x) = g(x) [a(τ ) − (u ∗ a)(τ )]
(h,>) A (τ, x) ≤ K g1 −1 . 1
(101)
(102)
The same bound holds with (g, h) replaced by (F, E). For all (τ, x), ω 1 e−iωτ E(p)h(p) ipx (h,>) (h,>) χ µa (dp) e . C (τ, x) = A (τ, x) + > β ω (iω)2 B 1 − E(p) iω (103) ω Because χ> ( ) = 0 for |ω| ≤ , the condition −1 E∞ < 1 implies that the geometric series for (1− E(p)/iω)−1 converges uniformly in p. By dominated convergence,
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the summation can be exchanged with the integral over p and the summation over ω. n th Moreover, by the support properties of χ< , we may insert a factor χ> ( 2ω ) in the n order term in this expansion, to get ∞ (h,>) (h,>) (h,>) (E,>) (E,>) C (τ, x) = A (τ, x) + A ∗ A/2 ∗ . . . ∗ A/2 (τ, x), (104) n=1 (E,>)
where the convolution is in τ and x and n factors A/2 appear in the product. The standard L 1 bound for the convolution implies
∞ (h,>) (h,>) (E,>) n (105) C ≤ A 1 + A/2 1
1
n=1
1
which converges by the hypotheses on g, F, and by (102), and yields the bound (94). Theorem 4.5 directly applies and implies convergence of the effective action obtained from the integration over large frequencies. Note that because of the way the constants depend on , the initial interaction can be taken arbitrarily strong (as long as it is summable): if U denotes the coupling constant of a quartic interaction, convergence of the 2 expansion for the effective action holds for all U with U (ln ) small enough, which can always be achieved by taking large enough. Thus, for arbitrarily strong coupling, the initial integration step is given by a convergent expansion. The consequences and some possible extensions of this are discussed in [S07]. Acknowledgement. This work was supported by DFG grant Sa 1362/2, by the Max–Planck society, and by DFG-Forschergruppe FOR718.
References [AGD] [AR] [BR] [FKT98] [FKT02] [FKT04] [S98a] [S98b] [S07] [SW] [PS] [St]
Abrikosov, A.A., Gorkov, L.P., Dzyaloshinski, I.E.: Methods of Quantum Field Theory in Statistical Physics. New York: Dover, 1963 Abdesselam, A., Rivasseau, V.: Explicit fermionic tree expansions. Lett. Math. Phys. 44, 77–88 (1998) Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics, Vol. 1,2, Berlin-Heidelberg-New York: Springer, 2002 Feldman, J., Knörrer, H., Trubowitz, E.: A representation for fermionic correlation functions. Commun. Math. Phys. 195, 465–493 (1998) Feldman, J., Knörrer, H., Trubowitz, E.: Fermionic Functional Integrals and the Renormalization Group. CRM Monograph Series, Vol. 16, Providence, RI: Amer. Math. Soc., 2002 Feldman, J., Knörrer, H., Trubowitz, E.: Convergence of pertubation expansions in fermionic models: part 1. Commun. Math. Phys. 247, 195–242 (2004) Salmhofer, M.: Continuous renormalization for fermions and fermi liquid theory. Commun. Math. Phys. 194, 249–295 (1998) Salmhofer, M.: Renormalization. Heidelberg: Springer, 1998 Salmhofer, M.: Dynamical adjustment of propagators in renormalization group flows. Ann. Phys. (Leipzig) 16(3), 171–206 (2007) Salmhofer, M., Wieczerkowski, C.: Positivity and convergence in fermionic quantum field theory. J. Stat. Phys. 99, 557–586 (2000) Pedra, W., Salmhofer, M.: On the Mathematical Theory of Fermi Liquids in Two Dimensions. To appear Stein, E.: Harmonic Analysis. Chapter 3; Princeton, NJ: Princeton University Press, 1993
Communicated by H.-T. Yau
Commun. Math. Phys. 282, 819–864 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0534-1
Communications in
Mathematical Physics
Modules-at-Infinity for Quantum Vertex Algebras Haisheng Li Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA. E-mail: [email protected] Received: 19 July 2007 / Accepted: 30 January 2008 Published online: 10 June 2008 – © Springer-Verlag 2008
Abstract: This is a sequel to [Li4] and [Li5] in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian DY(sl2 ), denoted by DYq (sl2 ) and DYq∞ (sl2 ) with q a nonzero complex number. For each nonzero complex number q, we construct a quantum vertex algebra Vq and prove that every DYq (sl2 )-module is naturally a Vq -module. We also show that DYq∞ (sl2 )-modules are what we call Vq -modules-at-infinity. To achieve this goal, we study what we call S-local subsets and quasi-local subsets of Hom(W, W ((x −1 ))) for any vector space W , and we prove that any S-local subset generates a (weak) quantum vertex algebra and that any quasilocal subset generates a vertex algebra with W as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity. 1. Introduction This is a sequel to [Li4] and [Li5] in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In [Li4] and [Li5], partially motivated by Etingof-Kazhdan’s notion of quantum vertex operator algebra over C[[]] (see [EK]), we formulated and studied a notion of quantum vertex algebra over C and we established general constructions of (weak) quantum vertex algebras and modules. The general constructions were illustrated by examples in which quantum vertex algebras were constructed from certain Zamolodchikov-Faddeev-type algebras. The main goal of this paper is to establish a natural connection of (centerless) double Yangians with quantum vertex algebras over C. For each finite-dimensional simple Lie algebra g, Drinfeld introduced a Hopf algebra Y (g), called a Yangian (see [D]), as a deformation of the universal enveloping algebra Partially supported by NSA grant H98230-05-1-0018 and NSF grant DMS-0600189.
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of the Lie algebra g ⊗ C[t]. The Yangian double DY(g) is a deformation of the algebra U (g ⊗ C[t, t −1 ]). In the simplest case with g = sl2 (cf. [S,Kh]), the following is one of the defining relations in terms of generating functions: e(x1 )e(x2 ) =
x1 − x2 + e(x2 )e(x1 ). x1 − x2 −
(1.1)
In this paper we study two versions of DY(sl2 ) with the formal parameter being evaluated at a nonzero complex number q. With a direct substitution, the defining relation (1.1) becomes e(x1 )e(x2 ) =
x1 − x2 + q e(x2 )e(x1 ), x1 − x2 − q
(1.2)
where x1 − x2 + q = (x1 − x2 + q) q i (x1 − x2 )−i−1 ∈ C(((x1 − x2 )−1 )) ⊂ C[x2 ]((x1−1 )). x1 − x2 − q i≥0
In this way we get a version of DY(sl2 ), which we denote by DYq∞ (sl2 ). Notice that for a module W of highest weight type where the generating functions are elements of Hom(W, W ((x))), the expression x1 − x2 + q e(x2 )e(x1 ) x1 − x2 − q does not exist in general. Because of this, DYq∞ (sl2 ) admits only modules W of lowest weight type where the generating functions such as e(x) are elements of Hom(W, W ((x −1 ))). Note that (quantum) vertex algebras and their modules are modules of highest weight type in nature and so far we have used only modules of highest weight type for various algebras to construct (quantum) vertex algebras and their modules. Motivated by this, we then consider another version of DY(sl2 ), which we denote by DYq (sl2 ), by expanding −x2 +q the same rational function xx11−x as follows: 2 −q x1 − x2 + q = −(x1 − x2 + q) q −i−1 (x1 − x2 )i ∈ C[[(x1 − x2 )]] ⊂ C[[x1 , x2 ]]. −q + x1 − x2 i≥0
Contrary to the situation with DYq∞ (sl2 ), the algebra DYq (sl2 ) admits only modules of highest weight type, including vacuum modules. For every nonzero complex number q, we construct a universal vacuum DYq (sl2 )-module Vq and by applying the general construction theorems of [Li4] and [Li5] we show that there exists a canonical quantum vertex algebra structure on Vq and that on any DYq (sl2 )-module there exists a canonical Vq -module structure. As it was mentioned before, the algebra DYq∞ (sl2 ) admits only lowest-weight-type modules. This is also the case for the Lie algebra of pseudo-differential operators. Nevertheless, we hope to associate such algebras with (quantum) vertex algebras in some natural way. Having this in mind, we systematically study how to construct quantum vertex algebras from suitable subsets of the space E o (W ) = Hom(W, W ((x −1 )))
Modules-at-Infinity for Quantum Vertex Algebras
821
for a general vector space W , developing a general theory analogous to that of [Li4]. For a vector space W , a subset T of E o (W ) is said to be S-local if for any a(x), b(x) ∈ T , there exist ai (x), bi (x) ∈ T, f i (x) ∈ C(x), i = 1, . . . , r, and a nonnegative integer k such that (x1 − x2 )k a(x1 )b(x2 ) = (x1 − x2 )k
r
ιx,∞ ( f i )(x1 − x2 )bi (x2 )ai (x1 ),
i=1
where ιx,∞ f i (x) denotes the formal Laurent series expansion of f i (x) at infinity. A subset T of E o (W ) is said to be quasi-local if for any a(x), b(x) ∈ T , there exists a nonzero polynomial p(x1 , x2 ) such that p(x1 , x2 )a(x1 )b(x2 ) = p(x1 , x2 )b(x2 )a(x1 ). We prove that any S-local subset generates a weak quantum vertex algebra and that any quasi-local subset of E o (W ) generates a vertex algebra in a certain natural way. To describe the structure on W we formulate a notion of (left) quasi module-at-infinity for a vertex algebra and for a weak quantum vertex algebra. For a vertex algebra V , a (left) quasi V -module-at-infinity is a vector space W equipped with a linear map YW from V to Hom(W, W ((x −1 ))), satisfying the condition that YW (1, x) = 1W (the identity operator on W ) and that for u, v ∈ V , there exists a nonzero polynomial p(x1 , x2 ) such that p(x1 , x2 )YW (u, x1 )YW (v, x2 ) = p(x1 , x2 )YW (v, x2 )YW (u, x1 ), p(x2 + x0 , x2 )YW (Y (u, x0 )v, x2 ) = ( p(x1 , x2 )YW (u, x1 )YW (v, x2 )) |x1 =x2 +x0 . For a module-at-infinity, the following opposite Jacobi identity holds for u, v ∈ V : x1 − x2 x2 − x1 YW (v, x2 )YW (u, x1 ) − x0−1 δ YW (u, x1 )YW (v, x2 ) x0−1 δ x0 −x0 x1 − x0 YW (Y (u, x0 )v, x2 ). = x2−1 δ x2 This notion of a left module-at-infinity for a vertex algebra V coincides with the notion of a right module, which was suggested in [HL]. As an application, we show that DYq∞ (sl2 )-modules are canonical modules-at-infinity for the quantum vertex algebra Vq . We also show that lowest-weight type modules for the Lie algebras of pseudo-differential operators on the circle are quasi modules-at-infinity for some vertex algebras associated with the affine Lie algebra of a certain infinitedimensional Lie algebra. For the double Yangians DY(g) associated with a general simple Lie algebra g, it is straightforward to show that the universal vacuum module has a unique weak quantum vertex algebra structure. As with the simplest case, the hard part is to establish the nondegeneracy so that we have a quantum vertex algebra. We believe that the same method will work essentially, but it will require much more work. We shall study general double Yangians DY(g) and their central extensions DY (g) (see [Kh]) in terms of quantum vertex algebras in a separate paper. This paper is organized as follows: In Sect. 2, we introduce a version of the double Yangian DY(sl2 ) and we associate it with quantum vertex algebras and modules. In
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H. Li
Sect. 3, we study quasi-compatible subsets and prove that any quasi-compatible subset canonically generates a nonlocal vertex algebra. In Sect. 4, we study S-local subsets and modules-at-infinity for quantum vertex algebras. In Sect. 5, we study quasi local subsets and quasi modules-at-infinity for vertex algebras. 2. Associative Algebra DYq (sl 2 ) and Quantum Vertex Algebras In this section, we first recall the notions of weak quantum vertex algebra and quantum vertex algebra and we then define an associative algebra DYq (sl2 ) over C with q an arbitrary nonzero complex number, which is a version of the (centerless) double Yangian DY(sl2 ), and we associate a quantum vertex algebra to the algebra DYq (sl2 ). Throughout this paper, we use the standard formal variable notions and conventions, including the formal delta functions with multi-variables and the formal binomial expansion convention, as established in [FLM] and [FHL] (cf. [LL]). In this paper, N denotes the set of nonnegative integers and letters x, y, z, xi , yi , z i for i ∈ N are mutually commuting independent formal variables. We begin with the notion of nonlocal vertex algebra ([K,BK,Li2]). A nonlocal vertex algebra is a vector space V , equipped with a linear map Y : V → Hom(V, V ((x))) ⊂ (End V )[[x, x −1 ]] vn x −n−1 (vn ∈ End V ) v → Y (v, x) =
(2.1)
n∈Z
and equipped with a distinguished vector 1, such that for v ∈ V , Y (1, x)v = v, Y (v, x)1 ∈ V [[x]] and
lim Y (v, x)1 = v,
x→0
(2.2) (2.3)
and such that for u, v, w ∈ V , there exists a nonnegative integer l such that (x0 + x2 )l Y (u, x0 + x2 )Y (v, x2 )w = (x0 + x2 )l Y (Y (u, x0 )v, x2 )w.
(2.4)
For a nonlocal vertex algebra V , we have d Y (v, x) dx where D is the linear operator on V , defined by d D(v) = Y (v, x)1 |x=0 (= v−2 1) dx [D, Y (v, x)] = Y (Dv, x) =
for v ∈ V,
(2.5)
for v ∈ V.
Furthermore, for v ∈ V , e x D Y (v, x1 )e−x D = Y (e x D v, x1 ) = Y (v, x1 + x), Y (v, x)1 = e
xD
v.
(2.6) (2.7)
A weak quantum vertex algebra (see [Li4,Li5]) is a vector space V (over C) equipped with a distinguished vector 1 and a linear map Y : V → Hom(V, V ((x))) ⊂ (End V )[[x, x −1 ]], vn x −n−1 (where vn ∈ End V ) v → Y (v, x) = n∈Z
Modules-at-Infinity for Quantum Vertex Algebras
823
satisfying the condition that Y (1, x)v = v, Y (v, x)1 ∈ V [[x]] and
lim Y (v, x)1 = v
x→0
for v ∈ V,
(2.8) (2.9)
and that for any u, v ∈ V , there exist u (i) , v (i) ∈ V, f i (x) ∈ C((x)), i = 1, . . . , r , such that r x1 − x2 x2 − x1 Y (u, x1 )Y (v, x2 ) − x0−1 δ x0−1 δ f i (−x0 ) x0 −x0 i=1
×Y (v (i) , x2 )Y (u (i) , x1 ) x1 − x0 Y (Y (u, x0 )v, x2 ). = x2−1 δ x2
(2.10)
In terms of the notion of nonlocal vertex algebra, a weak quantum vertex algebra is simply a nonlocal vertex algebra that satisfies the S-locality (cf. [EK]) in the sense that for any u, v ∈ V , there exist u (i) , v (i) ∈ V, f i (x) ∈ C((x)), i = 1, . . . , r , such that (x1 − x2 )k Y (u, x1 )Y (v, x2 ) = (x1 − x2 )k
r
f i (x2 − x1 )Y (v (i) , x2 )Y (u (i) , x1 )
i=1
(2.11) for some nonnegative integer k depending on u and v. The notion of quantum vertex algebra involves the notion of unitary rational quantum Yang-Baxter operator, which here we recall: A unitary rational quantum Yang-Baxter operator on a vector space H is a linear map S(x) : H ⊗ H → H ⊗ H ⊗ C((x)) such that S21 (−x)S(x) = 1, S12 (x)S13 (x + z)S23 (z) = S23 (z)S13 (x + z)S12 (x). In this definition, S21 (x) = σ12 S(x)σ12 , where σ12 is the flip map on H ⊗ H (u ⊗ v → v ⊗ u), S12 (x) = S(x) ⊗ 1 : H ⊗ H ⊗ H → H ⊗ H ⊗ H ⊗ C((x)), and S13 (x), S23 (x) are defined accordingly. A quantum vertex algebra ([Li4], cf. [EK]) is a weak quantum vertex algebra V equipped with a unitary rational quantum Yang-Baxter operator S(x) : V ⊗V → V ⊗ V ⊗ C((x)) such that for u, v ∈ V , (2.10) holds with S(x)(v ⊗ u) = ri=1 v (i) ⊗ u (i) ⊗ f i (x). Remark 2.1. Recall from [EK] that a nonlocal vertex algebra V is nondegenerate if for every positive integer n, the linear map Z n : V ⊗n ⊗ C((x1 )) · · · ((xn )) → V ((x1 )) · · · ((xn )) defined by Z n (v(1) ⊗ · · · ⊗ v(n) ⊗ f ) = f Y (v(1) , x1 ) · · · Y (v(n) , xn )1
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is injective. It was proved ([Li4], Theorem 4.8, cf. [EK], Proposition 1.11) that if V is a nondegenerate weak quantum vertex algebra, then the S-locality of vertex operators Y (v, x) for v ∈ V uniquely defines a linear map S(x) : V ⊗ V → V ⊗ V ⊗ C((x)) such that V equipped with S(x) is a quantum vertex algebra and S(x) is the unique rational quantum Yang-Baxter operator making V a quantum vertex algebra. In view of this, we shall use the term “a nondegenerate quantum vertex algebra” for a nondegenerate weak quantum vertex algebra which is a quantum vertex algebra with the canonical rational quantum Yang-Baxter operator. Let V be a nonlocal vertex algebra. A V -module is a vector space W equipped with a linear map YW : V → Hom(W, W ((x))) satisfying the condition that YW (1, x) = 1 (the identity operator on W ) and for any u, v ∈ V, w ∈ W , there exists a nonnegative integer l such that (x0 + x2 )l YW (u, x0 + x2 )YW (v, x2 )w = (x0 + x2 )l YW (Y (u, x0 )v, x2 )w. (2.12) Now, assume that V is a weak quantum vertex algebra and let (W, YW ) be a module for V viewed as a nonlocal vertex algebra. It was proved ([Li4], Lemma 5.7) that for any u, v ∈ V , x1 − x2 −1 x0 δ YW (u, x1 )YW (v, x2 ) x0 r x2 − x1 f i (−x0 )YW (v (i) , x2 )YW (u (i) , x1 ) −x0−1 δ −x0 i=1 x1 − x0 −1 YW (Y (u, x0 )v, x2 ), = x2 δ (2.13) x2 where u (i) , v (i) ∈ V, f i (x) ∈ C((x)) are the same as those in (2.10). Remark 2.2. Here, we recall a general construction of weak quantum vertex algebras from [Li4]. Let W be any vector space (over C) and set E(W ) = Hom(W, W ((x))). A subset T of E(W ) is S-local if for any a(x), b(x) ∈ T , there exist f i (x) ∈ C((x)), a (i) (x), b(i) (x) ∈ T (i = 1, . . . , r ) (finitely many) such that (x1 − x2 )k a(x1 )b(x2 ) =
r
(x1 − x2 )k f i (x2 − x1 )b(i) (x2 )a (i) (x1 )
i=1
for some nonnegative integer k. In this case, for any n ∈ Z, we define (see [Li5]) r n n (i) (i) a(x)n b(x) = Resx1 (x1 − x) a(x1 )b(x)−(−x + x1 ) f i (x − x1 )b (x)a (x1 ) . i=1
(Note that it is well defined.) It was proved ([Li4], Theorem 5.8) that every S-local subset of E(W ) generates a weak quantum vertex algebra with W as a module with YW (a(x), x0 ) = a(x0 ).
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Next, we introduce a version of the double Yangian DY(sl2 ) (associated to the three-dimensional simple Lie algebra sl2 ). Let T (sl2 ⊗ C[t, t −1 ]) denote the tensor algebra over the vector space sl2 ⊗ C[t, t −1 ]. From now on we shall simply use T for this algebra. We equip T with the Z-grading which is uniquely defined by deg(u ⊗ t n ) = n
for u ∈ sl2 , n ∈ Z,
making T a Z-graded algebra T = ⊕n∈Z Tn . For n ∈ Z, set I [n] = ⊕m≥n Tm ⊂ T . This defines a decreasing filtration of T = ∪n∈Z I [n] with ∩n∈Z I [n] = 0. Denote by T the completion of T associated with this filtration. For u ∈ sl2 , set u(n)x −n−1 , u(x) = n∈Z
where u(n) = u ⊗ t n . We also write sl2 (n) = sl2 ⊗ t n for n ∈ Z. Let e, f, h be the standard Chevalley generators of sl2 . Definition 2.3. Let q be a nonzero complex number. We define DYq (sl2 ) to be the quotient algebra of T modulo the following relations: q + x1 − x2 e(x2 )e(x1 ), −q + x1 − x2 −q + x1 − x2 f (x2 ) f (x1 ), f (x1 ) f (x2 ) = q + x1 − x2 x2 h(x2 ), [e(x1 ), f (x2 )] = x1−1 δ x1 q + x1 − x2 h(x1 )e(x2 ) = e(x2 )h(x1 ), −q + x1 − x2 h(x1 )h(x2 ) = h(x2 )h(x1 ), −q + x1 − x2 h(x1 ) f (x2 ) = f (x2 )h(x1 ), q + x1 − x2
e(x1 )e(x2 ) =
where it is understood that (±q + x1 − x2 )−1 =
(±q)−i−1 (x2 − x1 )i ∈ C[[x1 , x2 ]]. i∈N
It is straightforward to see that DYq (sl2 ) admits a (unique) derivation d such that [d, u(x)] =
d u(x) dx
for u ∈ sl2 .
(2.14)
That is, [d, u(n)] = −nu(n − 1)
for u ∈ sl2 , n ∈ Z.
(2.15)
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H. Li
Definition 2.4. As a convention, we define a DYq (sl2 )-module to be a T -module W such that for any w ∈ W , sl2 (n)w = 0
for n sufficiently large
and such that all the defining relations of DYq (sl2 ) in Definition 2.3 hold. A vector w0 of a DYq (sl2 )-module W is called a vacuum vector if u(n)w0 = 0 for u ∈ sl2 , n ≥ 0. A vacuum DYq (sl2 )-module is a module W equipped with a vacuum vector that generates W . The following are some basic properties of a general vacuum DYq (sl2 )-module: Lemma 2.5. Let W be a vacuum DYq (sl2 )-module with a vacuum vector w0 as a generator. Set F0 = Cw0 and Fk = 0 for k < 0. For any positive integer k, we define Fk to be the linear span of the vectors a1 (−m 1 ) · · · ar (−m r )w0 for r ≥ 1, a1 , . . . , ar ∈ sl2 , m 1 , . . . , m r ≥ 1 with m 1 +· · ·+m r ≤ k. Then the subspaces Fk for k ∈ Z form an increasing filtration of W and for any a ∈ sl2 , m, k ∈ Z, a(m)Fk ⊂ Fk−m .
(2.16)
Furthermore, Tm w0 = 0 for m ≥ 1. Proof. We first prove (2.16). It is true for k < 0 as Fk = 0 by definition. With F0 = Cw0 , we see that (2.16) holds for k = 0. Assume k ≥ 1. From definition, (2.16) always holds for m < 0. Let a, b ∈ {e, f, h}. From the defining relations of DYq (sl2 ) we have λi j b(n + i)a(m + j) + αh(m + n) (2.17) a(m)b(n) = ±b(n)a(m) + i, j≥0, i+ j≥1
for all m, n ∈ Z, where λi j , α ∈ C, depending on a, b. Using this fact and induction on k we obtain (2.16), noticing that a(m)F0 = 0 for m ≥ 0. From (2.16) we get Tm w0 = Tm F0 ⊂ F−m = 0
for m ≥ 1.
It also follows from (2.16) that ∪k≥0 Fk is a submodule of W . Since w0 generates W , we must have W = ∪k≥0 Fk . This proves that the subspaces Fk for k ∈ Z form an increasing filtration of W .
Lemma 2.6. Let W be a vacuum DYq (sl2 )-module with a vacuum vector w0 as a generator. For n ∈ N, define E n to be the linear span of the vectors u (1) (m 1 ) · · · u (r ) (m r )w0 for 0 ≤ r ≤ n, u (i) ∈ {e, f, h}, m i ∈ Z. Then the subspaces E n for n ∈ N form an increasing filtration of W and for each n ∈ N, E n is linearly spanned by the vectors e(−m 1 ) · · · e(−m r ) f (−n 1 ) · · · f (−n s )h(−k1 ) · · · h(−kl )w0 , where r, s, t ≥ 0 and m i , n j , kt are positive integers such that m 1 > · · · > m r , n 1 > · · · > n s , k1 ≥ · · · ≥ kl , r + s + l ≤ n.
(2.18)
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Proof. As w0 generates W , the subspaces E n for n ∈ N form an increasing filtration for W . It remains to prove the spanning property. For any nonnegative integer n, let E n be the span of the vectors a1 (−m 1 ) · · · ar (−m r )w0 for 0 ≤ r ≤ n, a1 , . . . , ar ∈ sl2 , m 1 , . . . , m r ≥ 1. By definition, E n ⊂ E n for n ≥ 0. Using induction (on k) and (2.17) we get a(m)E k ⊂ E k+1 if m < 0, a(m)E k ⊂ E k if m ≥ 0
for any a ∈ sl2 , m ∈ Z, k ∈ N. Using this and induction we have E n ⊂ E n for n ≥ 0. Thus E n = E n for all n ≥ 0. For every nonnegative integer n, from Lemma 2.5, the subspaces E n ∩ Fm for m ∈ N form an increasing filtration of E n . The spanning property of E n follows from this filtration and (2.17).
The following is a tautological construction of a vacuum module. Let d be the derivation of T such that d(a ⊗ t n ) = −n(a ⊗ t n−1 ) for a ∈ sl2 , n ∈ Z. Set T+ =
Tn
and J = T C[d]T+ .
n≥1
With J a left ideal of T , T /J is a (left) T -module and for any v ∈ T , sl2 (n)(v + J ) = 0 for n sufficiently large. Definition 2.7. We define Vq to be the quotient T -module of T /J , modulo all the defining relations of DYq (sl2 ). Denote by 1 the image of 1 in Vq . From the construction, (Vq , 1) is a vacuum DYq (sl2 )-module. As d J ⊂ J , Vq admits an action of d such that d1 = 0,
[d, u(x)] =
d u(x) dx
for u ∈ sl2 .
(2.19)
It is clear that for any vacuum DYq (sl2 )-module (W, w0 ) on which d acts such that dw0 = 0 and [d, u(x)] =
d u(x) dx
for u ∈ sl2 ,
there exists a unique DYq (sl2 )-module homomorphism from Vq to W , sending 1 to w0 . We are going to show that Vq has a certain normal basis and there is a canonical quantum vertex algebra structure on Vq . To show that Vq has a certain normal basis, we shall construct a vacuum DYq (sl2 )-module with this property.
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Remark 2.8. Here, we construct a vertex superalgebra. Let g = g0 ⊕ g1 be a three-dimensional Lie superalgebra with g0 = Ch¯ (the even subspace) and g1 = Ce¯ ⊕ C f¯ (the odd subspace), where ¯ [h, ¯ e] ¯ f¯] = 0, [h, ¯ h] ¯ = 0. [e, ¯ e] ¯ + = [ f¯, f¯]+ = 0, [e, ¯ f¯]+ = h, ¯ = 0, [h,
(2.20)
(One can show that this is indeed a Lie superalgebra by embedding g into the Clifford ¯ associated with the vector space g1 equipped with a symmetric algebra over the ring C[h] bilinear form.) Form the loop Lie superalgebra L(g) = g ⊗ C[t, t −1 ]. Viewing C as a trivial g ⊗ C[t]-module, we form the induced L(g)-module VL(g) = U (L(g)) ⊗U (g⊗C[t]) C. Set 1 = 1 ⊗ 1 ∈ VL(g) . ¯ consisting of the vectors It follows from the P-B-W theorem that VL(g) has a basis B, ¯ ¯ e(−m ¯ ¯ 1 ) · · · e(−m r ) f¯(−n 1 ) · · · f¯(−n s )h(−k 1 ) · · · h(−k l )1,
(2.21)
where r, s, t ≥ 0 and m i , n j , kt are positive integers such that m 1 > · · · > m r , n 1 > · · · > n s , k 1 ≥ · · · ≥ kl . Identify g as a subspace of VL(g) through the map u → u(−1)1 for u ∈ g. Then there exists a (unique) vertex superalgebra structure on VL(g) with 1 as the vacuum vector and ¯ We define a Z-grading on g ⊗ C[t, t −1 ] by with Y (u, x) = u(x) for u = e, ¯ f¯, h. wt(g ⊗ t n ) = −n
for n ∈ Z,
making g ⊗ C[t, t −1 ] a Z-graded Lie superalgebra. Then V is N-graded with V(0) = C1 such that u(m)V(n) ⊂ V(n−m)
for u ∈ g, m, n ∈ Z.
We are going to define a vacuum DYq (sl2 )-module structure on the vertex superalgebra V = VL(g) . Lemma 2.9. There exists a unique element (t) ∈ Hom(V, V ⊗ C[t]) such that (t)1 = 1, (t)e¯ = e¯ ⊗ t, (t) f¯ = f¯ ⊗ t, (t)h¯ = h¯ ⊗ t 2 , (t)Y (v, x) = Y ((t − x)v, x)(t) for v ∈ V.
(2.22) (2.23)
Furthermore, we have (t)e(x) ¯ = (t − x)e(x)(t), ¯ (t) f¯(x) = (t − x) f¯(x)(t), 2¯ ¯ (t)h(x) = (t − x) h(x)(t), and [D, (t)] =
d dt (t),
(x)(t) = (t)(x).
(2.24) (2.25)
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829
Proof. Let us equip C[t] with the vertex algebra structure for which 1 is the vacuum vector and Y ( p(t), x)q(t) = e−x(d/dt) p(t) q(t) = p(t − x)q(t) for p(t), q(t) ∈ C[t]. Then equip V ⊗ C[t] with the tensor product vertex superalgebra structure where we denote the vertex operator map by Yten . Thus Yten (u ⊗ t n , x) = Y (u, x) ⊗ (t − x)n
for u ∈ V, n ∈ Z.
We have ¯ x1 ), Y (e, ¯ x2 )]+ ⊗ (t − x1 )(t − x2 ) = 0, [Yten (e¯ ⊗ t, x1 ), Yten (e¯ ⊗ t, x2 )]+ = [Y (e, ¯ ¯ ¯ [Yten ( f ⊗ t, x1 ), Yten ( f ⊗ t, x2 )]+ = [Y ( f , x1 ), Y ( f¯, x2 )]+ ⊗ (t − x1 )(t − x2 ) = 0, ¯ x1 ), Y (e, [Yten (h¯ ⊗ t 2 , x1 ), Yten (e¯ ⊗ t, x2 )] = [Y (h, ¯ x2 )] ⊗ (t − x1 )2 (t − x2 ) = 0, ¯ x1 ), Y ( f¯, x2 )] ⊗ (t − x1 )2 (t − x2 ) = 0, [Yten (h¯ ⊗ t 2 , x1 ), Yten ( f¯ ⊗ t, x2 )] = [Y (h, [Yten (e¯ ⊗ t, x1 ), Yten ( f¯ ⊗ t, x2 )]+ = [Y (e, ¯ x1 ), Y ( f¯, x2 )]+ ⊗ (t − x1 )(t − x2 ) x2 ¯ x2 ) ⊗ (t − x1 )(t − x2 ) = x −1 δ x2 Y (h, ¯ x2 ) ⊗ (t − x2 )2 , Y (h, = x1−1 δ 1 x1 x1 ¯ x1 ), Y (h, ¯ x2 )] ⊗ (t − x1 )2 (t − x2 )2 = 0. [Yten (h¯ ⊗ t 2 , x1 ), Yten (h¯ ⊗ t 2 , x2 )] = [Y (h, It follows that there exists a (unique) vertex-superalgebra homomorphism θ from V to V ⊗ C[t] such that θ (e) ¯ = e¯ ⊗ t,
θ ( f¯) = f¯ ⊗ t,
¯ = h¯ ⊗ t 2 . θ (h)
(2.26)
Let us alternatively denote by (t) the vertex superalgebra homomorphism θ (from V ¯ = h¯ ⊗ t 2 . to V ⊗ C[t]). Then (t)1 = 1, (t)(e) ¯ = e¯ ⊗ t, (t)( f¯) = f¯ ⊗ t, (t)(h) Furthermore, for u, v ∈ V , we have (t)Y (u, x)v = θ (Y (u, x)v) = Yten (θ (u), x)θ (v) = Y ((t − x)u, x)(t)v, where Y is viewed as a C[t]-map. The rest follows immediately.
Proposition 2.10. Let q be any nonzero complex number and let V = VL(g) be the vertex superalgebra as in Lemma 2.9. The assignment e(x) = e(x)(q ¯ + x),
¯ f (x) = f¯(x)(−q + x), h(x) = q h(x)(q + x)(−q + x)
uniquely defines a vacuum DYq (sl2 )-module structure on V with 1 as the generating vacuum vector and d d d e(x), [D, f (x)] = f (x), [D, h(x)] = h(x). (2.27) [D, e(x)] = dx dx dx Furthermore, for n ∈ N, define E n to be the linear span of the vectors u (1) (m 1 ) · · · u (r ) (m r )1 for 0 ≤ r ≤ n, u (i) ∈ {e, f, h}, m i ∈ Z. Then E n has a basis consisting of the vectors e(−m 1 ) · · · e(−m r ) f (−n 1 ) · · · f (−n s )h(−k1 ) · · · h(−kl )1, where r, s, t ≥ 0 and m i , n j , kt are positive integers such that m 1 > · · · > m r , n 1 > · · · > n s , k1 ≥ · · · ≥ kl , r + s + l ≤ n.
(2.28)
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H. Li
Proof. Using Lemma 2.9 we have e(x1 )e(x2 ) = = = = = f (x1 ) f (x2 ) = = = = =
e(x ¯ 1 )(q + x1 )e(x ¯ 2 )(q + x2 ) e(x ¯ 1 )e(x ¯ 2 )(q + x1 )(q + x2 )(q + x1 − x2 ) −e(x ¯ 2 )e(x ¯ 1 )(q + x2 )(q + x1 )(q + x1 − x2 ) −e(x ¯ 2 )(q + x2 )e(x ¯ 1 )(q + x1 )(q + x1 − x2 )(q + x2 − x1 )−1 q + x1 − x2 e(x2 )e(x1 ), −q − x2 + x1 f¯(x1 )(−q + x1 ) f¯(x2 )(−q + x2 ) f¯(x1 ) f¯(x2 )(−q + x1 )(−q + x2 )(−q + x1 − x2 ) − f¯(x2 ) f¯(x1 )(−q + x2 )(−q + x1 )(−q + x1 − x2 ) − f¯(x2 )(−q + x2 ) f¯(x1 )(−q + x1 )(−q + x1 − x2 )(−q + x2 − x1 )−1 −q + x1 − x2 f (x2 ) f (x1 ), q − x2 + x1
[e(x1 ), f (x2 )] = e(x ¯ 1 )(q + x1 ) f¯(x2 )(−q + x2 ) − f¯(x2 )(−q + x2 )e(x ¯ 1 )(q + x1 ) = (q + x1 − x2 )e(x ¯ 1 ) f¯(x2 )(q + x1 )(−q + x2 ) −(−q + x2 − x1 ) f¯(x2 )e(x ¯ 1 )(q + x1 )(−q + x2 ) = (q + x1 − x2 )(e(x ¯ 1 ) f¯(x2 ) + f¯(x2 )e(x ¯ 1 ))(q + x1 )(−q + x2 ) x2 ¯ = (q + x1 − x2 )x1−1 δ h(x2 )(q + x1 )(−q + x2 ) x1 x2 ¯ = q x1−1 δ h(x2 )(q + x2 )(−q + x2 ) x1 x2 h(x2 ), = x1−1 δ x1 [h(x1 ), h(x2 )] ¯ 2 )(q + x2 )(−q + x2 ) ¯ 1 )(q + x1 )(−q + x1 )h(x = q 2 h(x 2¯ ¯ 1 )(q + x1 )(−q + x1 ) −q h(x2 )(q + x2 )(−q + x2 )h(x 2 ¯ 1 )h(x ¯ 2 )(q + x1 )(−q + x1 ) = q (q + x1 − x2 )(−q + x1 − x2 )h(x 2 ¯ 2 )h(x ¯ 1) ×(q + x2 )(−q + x2 ) − q (q + x2 − x1 )(−q + x2 − x1 )h(x ×(q + x1 )(−q + x1 )(q + x2 )(−q + x2 ) ¯ 1 ), h(x ¯ 2 )](q + x1 )(−q + x1 ) = q 2 (xq +1 −x2 )(−q + x1 − x2 )[h(x ×(q + x2 )(−q + x2 ) = 0, h(x1 )e(x2 ) ¯ 1 )(q + x1 )(−q + x1 )e(x = q h(x ¯ 2 )(q + x2 ) ¯ = q h(x1 )e(x ¯ 2 )(q + x1 )(−q + x1 )(q + x2 )(q + x1 − x2 )(−q + x1 − x2 ) ¯ 1 )(q + x2 )(q + x1 )(−q + x1 )(q + x1 − x2 )(−q + x1 − x2 ) = q e(x ¯ 2 )h(x
Modules-at-Infinity for Quantum Vertex Algebras
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q(q + x1 − x2 )(−q + x1 − x2 ) ¯ 1 )(q + x1 )(−q + x1 ) e(x ¯ 2 )(q + x2 )h(x (q + x2 − x1 )2 q + x1 − x2 e(x2 )h(x1 ), = −q − x2 + x1 h(x1 ) f (x2 ) ¯ 1 )(q + x1 )(−q + x1 ) f¯(x2 )(−q + x2 ) = q h(x ¯ 1 ) f¯(x2 )(q + x1 )(−q + x1 )(−q + x2 )(q + x1 − x2 )(−q + x1 − x2 ) = q h(x ¯ 1 )(−q + x2 )(q + x1 )(−q + x1 )(q + x1 − x2 )(−q + x1 − x2 ) = q f¯(x2 )h(x q(q + x1 − x2 )(−q + x1 − x2 ) ¯ ¯ 1 )(q + x1 )(−q + x1 ) = f (x2 )(−q + x2 )h(x (−q + x2 − x1 )2 −q + x1 − x2 = f (x2 )h(x1 ). q − x2 + x1 =
This proves that V becomes a DYq (sl2 )-module. As (x)1 = 1, it is clear that 1 is a vacuum vector for DYq (sl2 ). Now it remains to prove that 1 generates V as a DYq (sl2 )module. Let W be the DYq (sl2 )-submodule of V generated by 1. Using Lemma 2.9 we have (x1 )e(x) = (x1 )(e(x)(q ¯ + x)) = e(x)(q ¯ + x)(x1 − x)(x1 ) = (x1 − x)e(x)(x1 ). ¯ Similar relations also hold for f¯(x) and h(x). As (x)1 = 1, by induction we have ¯ (x)W ⊂ W ((x)). Then it follows that W is stable under the actions of e(n), ¯ f¯(n), h(n) for n ∈ Z. Thus W = V . This proves that 1 generates V as a DYq (sl2 )-module and then proves that V is a vacuum DYq (sl2 )-module. Now we prove the last assertion. With the spanning property having been established in Lemma 2.6 we only need to prove the independence. Recall that V = n∈N V(n) is N-graded with V(0) = C1. For n ∈ N, set F¯n = V(0) ⊕ V(1) ⊕ · · · ⊕ V(n) ⊂ V. We know that F¯n has a basis consisting of the vectors ¯ ¯ e(−m ¯ ¯ 1 ) · · · e(−m r ) f¯(−n 1 ) · · · f¯(−n s )h(−k 1 ) · · · h(−k l )1, where r, s, t ≥ 0 and m i , n j , kt are positive integers such that m 1 > · · · > m r , n 1 > · · · > n s , k 1 ≥ · · · ≥ kl ,
mi +
nj +
kt ≤ n.
From the commutation relations in Lemma 2.9, we have (t)e(m) ¯ = (t e(m) ¯ − e(m ¯ + 1))(t), (t) f¯(m) = (t f¯(m) − f¯(m + 1))(t), 2 ¯ ¯ ¯ + 1) + h(m ¯ + 2))(t) (t)h(m) = (t h(m) − 2t h(m for m ∈ Z. With (t)1 = 1, using induction we get (t)w ≡ t m w mod F¯n−1 [t] for w ∈ F¯n , n ≥ 0,
832
H. Li
where m is a nonnegative integer depending on w. As e(x) = e(x)(q ¯ + x) =
1 ( j) x j e(x) ¯ (q), j! j≥0
for any m ∈ Z we have e(m) =
1 e(m ¯ + i)(i) (q). i! i≥0
For u ∈ {e, f, h} and for m ∈ Z, w ∈ F¯k , we have u(m)w ≡ α u(m)w ¯ mod F¯k−m−1 for some nonzero complex number α. It follows immediately that E n has a basis as claimed.
With Vq being universal, from Proposition 2.10 we immediately have: Corollary 2.11. For n ∈ N, let E n be the subspace of Vq , linearly spanned by the vectors u (1) (m 1 ) · · · u (r ) (m r )1 for 0 ≤ r ≤ n, u (i) ∈ {e, f, h}, m i ∈ Z. Then the subspaces E n for n ≥ 0 form an increasing filtration of Vq and for each n ≥ 0, E n has a basis consisting of the vectors e(−m 1 ) · · · e(−m r ) f (−n 1 ) · · · f (−n s )h(−k1 ) · · · h(−kl )1,
(2.29)
where r, s, t ≥ 0 and m i , n j , kt are positive integers such that m 1 > · · · > m r , n 1 > · · · > n s , k1 ≥ · · · ≥ kl , r + s + l ≤ n. In view of Corollary 2.11, we can and we should consider sl2 as a subspace of Vq through the map u → u(−1)1 for u ∈ sl2 . The following is our main result: Theorem 2.12. Let q be any nonzero complex number and let (Vq , 1) be the universal vacuum DYq (sl2 )-module. There exists one and only one weak quantum vertex algebra structure on Vq with 1 as the vacuum vector such that Y (e, x) = e(x), Y ( f, x) = f (x), Y (h, x) = h(x), and the weak quantum vertex algebra Vq is nondegenerate. Furthermore, for any DYq (sl2 )-module W , there exists one and only one Vq -module structure YW on W such that YW (e, x) = e(x), YW ( f, x) = f (x), YW (h, x) = h(x).
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Proof. We shall follow the procedure outlined in [Li4] and [Li5]. Let W be any DYq (sl2 )module and let W = Vq ⊕ W be the direct sum module. Set U = {e(x), f (x), h(x)} ⊂ E(W ). With (x1 − x2 )x1−1 δ(x2 /x1 ) = 0, we have (x1 − x2 )e(x1 ) f (x2 ) = (x1 − x2 ) f (x2 )e(x1 ). Combining this with other defining relations, we see that U is an S-local subset. By ([Li4], Theorem 5.8), U generates a weak quantum vertex algebra VW where the identity operator 1W is the vacuum vector and YE denotes the vertex operator map. Furthermore, the vector space W is a faithful VW -module with YW (a(x), x0 ) = a(x0 ) for a(x) ∈ VW . It follows from ([Li4], Prop 6.7) and the defining relations of DYq (sl2 ) that VW is a vacuum DYq (sl2 )-module with e(x0 ), f (x0 ), h(x0 ) acting as YE (e(x), x0 ), YE ( f (x), x0 ), YE (h(x), x0 ). As Vq is universal, there exists a DYq (sl2 )-module homomorphism ψ from Vq to VW , sending 1 to 1W . Since Vq is a DYq (sl2 )-submodule of W , it follows that ψ maps Vq into Vq ⊂ W . Notice that Vq as a DYq (sl2 )-module is generated by 1 and that we have the operator d on Vq with the property (2.19). Now we can apply Theorem 6.3 of [Li4], asserting that there exists one and only one weak quantum vertex algebra structure on Vq with the required properties. It follows from Theorem 6.5 of [Li4] that W is a Vq -module with W as a submodule. Now it remains to prove that Vq is nondegenerate. For n ∈ N, define E n to be the linear span of the vectors u (1) (m 1 ) · · · u (r ) (m r )1 for 0 ≤ r ≤ n, u (i) ∈ {e, f, h}, m i ∈ Z. By Proposition 3.15 of [Li5], the subspaces E n (n ∈ N) form an increasing filtration of Vq with E 0 = C1 such that ak E n ⊂ E m+n for a ∈ E m , m, n ∈ N, k ∈ Z. Denote by Gr E (Vq ) the associated graded nonlocal ˆ fˆ, hˆ denote the images of e, f, h in vertex algebra. Notice that e, f, h ∈ E 1 . Let e, ˆ ˆ ˆ f , h} is a generating subset of Gr E (Vq ) and we have E 1 /E 0 ⊂ Gr E (Vq ). Then {e, q + x1 − x2 e(x ˆ 2 )e(x ˆ 1 ), −q + x1 − x2 −q + x1 − x2 ˆ fˆ(x1 ) fˆ(x2 ) = f (x2 ) fˆ(x1 ), q + x1 − x2 ˆ 1 ), e(x ˆ 1 ) fˆ(x2 ) = fˆ(x2 )e(x ˆ 2) = e(x ˆ 1 )e(x
ˆ 1 )h(x ˆ 2 ) = h(x ˆ 2 )h(x ˆ 1 ), h(x q + x1 − x2 ˆ 1 ), ˆ 1 )e(x ˆ 2) = e(x ˆ 2 )h(x h(x −q + x1 − x2 ˆ 1 ) fˆ(x2 ) = −q + x1 − x2 fˆ(x2 )h(x ˆ 1 ). h(x q + x1 − x2 From Corollary 2.11, for each n ≥ 0, E n+1 /E n has a basis consisting of the vectors ˆ ˆ e(−m ˆ ˆ 1 ) · · · e(−m r ) fˆ(−n 1 ) · · · fˆ(−n s )h(−k 1 ) · · · h(−k l )1, where r, s, t ≥ 0 and m i , n j , kt are positive integers such that m 1 > · · · > m r , n 1 > · · · > n s , k1 ≥ · · · ≥ kl , r + s + l = n + 1. It will be proved in [KL] that Gr E (Vq ) is nondegenerate. Then by ([Li5], Prop. 3.14), Vq is nondegenerate.
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Remark 2.13. In the defining relations of the algebra DYq (sl2 ), let us use the following expansion: 1 (∓q)i (x1 − x2 )−1−i ∈ C[x2 ][[x1−1 ]]. = x1 − x2 ± q i≥0
By doing this one gets a new algebra which we denote by DYq∞ (sl2 ). Unlike DYq (sl2 ), the algebra DYq∞ (sl2 ) only admits modules of lowest weight type. To relate DYq∞ (sl2 ) with quantum vertex algebras we shall need a new theory.
3. Quasi-Compatibility and Quasi-Modules-at-Infinity for Nonlocal Vertex Algebras In this section we study quasi-compatible subsets of Hom(W, W ((x −1 ))) for a general vector space W and we show that from any quasi-compatible subset, one can construct a canonical nonlocal vertex algebra. We formulate a notion of quasi module-at-infinity for a nonlocal vertex algebra and we show that the starting vector space W is naturally a quasi module-at-infinity for the nonlocal vertex algebra generated by a quasi-compatible subset. The theory and the results of this section are analogous to those in [Li4]. Let W be any vector space over C, which is fixed throughout this section. Set E o (W ) = Hom(W, W ((x −1 ))) ⊂ (End W )[[x, x −1 ]].
(3.1)
Denote by 1W the identity operator on W , a distinguished element of E o (W ). Note that E o (W ) is naturally a vector space over the field C((x −1 )). Let G denote the group of linear transformations on C: G = {g(z) = c0 z + c1 | c0 ∈ C× , c1 ∈ C}.
(3.2)
Group G acts on E o (W ) with g ∈ G acting as Rg defined by Rg a(x) = a(g(x))
for a(x) ∈ E o (W ),
(3.3)
where as a convention a(g(x)) =
an (c0 x + c1 )
n∈Z
for a(x) =
n∈Z an x
−n−1
−n − 1 c0−n−1−i c1i an x −n−1−i = i n∈Z i∈N
−n−1 .
Definition 3.1. An ordered sequence a1 (x), . . . , ar (x) in E o (W ) is said to be quasicompatible if there exists a nonzero polynomial p(x1 , x2 ) such that ⎞ ⎛ ⎝ p(xi , x j )⎠ a1 (x1 )a2 (x2 ) · · · ar (xr ) ∈ Hom(W, W ((x1−1 , x2−1 , . . . , xr−1 ))). 1≤i< j≤r
(3.4)
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A subset S of E o (W ) is said to be quasi-compatible if every finite sequence in S is quasi-compatible. A sequence a1 (x), . . . , ar (x) is said to be compatible if there exists a nonnegative integer k such that ⎛ ⎞ ⎝ (xi − x j )k ⎠ a1 (x1 )a2 (x2 ) · · · ar (xr ) ∈ Hom(W, W ((x1−1 , x2−1 , . . . , xr−1 ))). 1≤i< j≤r
(3.5) A subset S
of E o (W ) is said to be compatible if every finite sequence in
S is compatible.
Hom(W, W ((x1−1 , x2−1 ))),
A(x2 , x2 ) exists
Remark 3.2. Note that for any A(x1 , x2 ) ∈ in Hom(W, W ((x2−1 ))). Furthermore, x0 ∂ ∂x A(x2 + x0 , x2 ) = e 1 A(x1 , x2 ) |x1 =x2
exists in (Hom(W, W ((x2−1 ))))[[x0 ]] and A(x2 + x0 , x2 ) = 0 if A(x1 , x2 ) = 0. We also have x1 x1 A(x1 , x2 ) = x2−1 δ A(x2 , x2 ), x2−1 δ x2 x2 x1 A(x1 , x2 ) = A(x2 , x2 ), Resx1 x2−1 δ x2 x2 + x0 Resx1 x1−1 δ A(x1 , x2 ) = A(x2 + x0 , x2 ). x1 Let C(x1 , x2 ) denote the field of rational functions. We have fields C((x1−1 ))((x2−1 )) and C((x1−1 ))((x2 )) of formal series. As these fields contain C[x1 , x2 ] as a subring, there exist unique field-embeddings ιx1 ,∞;x2 ,∞ : C(x1 , x2 ) → C((x1−1 ))((x2−1 )), ιx1 ,∞;x2 ,0 : C(x1 , x2 ) →
C((x1−1 ))((x2 )).
(3.6) (3.7)
Let C(x) denote the field of rational functions. Define field-embeddings ιx,0 : C(x) → C((x)), sending f (x) to the formal Laurent series expansion of f (x) at x = 0, and ιx,∞ : C(x) → C((x −1 )), sending f (x) to the formal Laurent series expansion of f (x) at x = ∞. Remark 3.3. We shall often use the following simple fact. Suppose that U is a vector space over C and F, G ∈ U ((x −1 ))((x0 )) satisfy the relation q(x, x0 )F(x, x0 ) = q(x, x0 )G(x, x0 ) for some nonzero polynomial q(x, x0 ). Then F = G. This is simply because U ((x −1 ))((x0 )) is a vector space over the field C((x −1 ))((x0 )) and 0 = q(x, x0 ) ∈ C[x, x0 ] ⊂ C((x −1 ))((x0 )). Note that for a quasi-compatible pair (a(x), b(x)) in E o (W ), by definition there exists a nonzero polynomial p(x1 , x2 ) such that
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p(x1 , x2 )a(x1 )b(x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))).
(3.8)
As ιx,∞;x0 ,0 (1/ p(x0 + x, x)) ∈ C((x −1 ))((x0 )), ( p(x1 , x)a(x1 )b(x)) |x1 =x+x0 ∈ (Hom(W, W ((x −1 ))))[[x0 ]], we have ιx,∞;x0 ,0 (1/ p(x0 + x, x)) ( p(x1 , x)a(x1 )b(x)) |x1 =x+x0 ∈ (Hom(W, W ((x −1 ))))((x0 )). Definition 3.4. Let (a(x), b(x)) be a quasi-compatible pair in E o (W ). We define YE o (a(x), x0 )b(x) = ιx,∞;x0 ,0 (1/ p(x0 + x, x)) ( p(x1 , x)a(x1 )b(x)) |x1 =x+x0 ∈ E o (W )((x0 )), where p(x1 , x2 ) is any nonzero polynomial such that (3.8) holds. It is easy to show that YE o (a(x), x0 )b(x) is well defined, i.e., the expression on the right hand side does not depend on the choice of polynomial p(x1 , x2 ). Write YE o (a(x), x0 )b(x) = a(x)n b(x)x0−n−1 . (3.9) n∈Z
The following is an immediate consequence: Lemma 3.5. Let (a(x), b(x)) be a quasi-compatible pair in E o (W ). Then a(x)n b(x) ∈ E o (W ) for n ∈ Z. Furthermore, let p(x1 , x2 ) be a nonzero polynomial such that p(x1 , x2 )a(x1 )b(x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))) and let k be an integer such that x0k ιx,∞;x0 ,0 (1/ p(x0 + x, x)) ∈ C((x −1 ))[[x0 ]]. Then a(x)n b(x) = 0
for n ≥ k.
(3.10)
We shall need the following result: Lemma 3.6. Let (ai (x), bi (x)) (i = 1, . . . , n) be quasi-compatible ordered pairs in E o (W ). Suppose that n
gi (z, x)ai (z)bi (x) ∈ Hom(W, W ((z −1 , x −1 )))
(3.11)
i=1
for some polynomials g1 (z, x), . . . , gn (z, x). Then n n gi (x + x0 , x)YE o (ai (x), x0 )bi (x) = gi (z, x)ai (z)bi (x) |z=x+x0 . i=1
i=1
(3.12)
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Proof. Let g(z, x) be a nonzero polynomial such that g(z, x)ai (z)bi (x) ∈ Hom(W, W ((z −1 , x −1 )))
for i = 1, . . . , n.
From Definition 3.4, we have g(x + x0 , x)YE o (ai (x), x0 )bi (x) = (g(z, x)ai (z)bi (x)) |z=x+x0
for i = 1, . . . , n.
Then using (3.11) we have g(x + x0 , x)
n
gi (x + x0 , x)YE o (ai (x), x0 )bi (x)
i=1
=
n
gi (x + x0 , x) (g(z, x)ai (z)bi (x)) |z=x+x0
i=1
=
g(z, x)
n i=1
= g(x + x0 , x)
gi (z, x)ai (z)bi (x) |z=x+x0
n
gi (z, x)ai (z)bi (x) |z=x+x0 .
(3.13)
i=1
As both
n
i=1 gi (x
+ x0 , x)YE o (ai (x), x0 )bi (x) and
n
gi (z, x)ai (z)bi (x) |z=x+x0
i=1
lie in (Hom(W, W ((x −1 ))))((x0 )), from Remark 3.3, (3.12) follows immediately.
A quasi-compatible subspace U of E o (W ) is said to be closed if a(x)n b(x) ∈ U
for a(x), b(x) ∈ U, n ∈ Z.
(3.14)
We are going to prove that any closed quasi-compatible subspace containing 1W of E o (W ) is a nonlocal vertex algebra. First we prove the following result: Lemma 3.7. Let V be a closed quasi-compatible subspace of E o (W ). Let ψ(x), φ(x), θ (x) ∈ V and let f (x, y) be a nonzero polynomial such that f (x, y)φ(x)θ (y) ∈ Hom(W, W ((x −1 , y −1 ))), f (x, y) f (x, z) f (y, z)ψ(x)φ(y)θ (z) ∈ Hom(W, W ((x −1 , y −1 , z −1 ))).
(3.15) (3.16)
Then f (x + x1 , x) f (x + x2 , x) f (x + x1 , x + x2 )YE o (ψ(x), x1 )YE o (φ(x), x2 )θ (x) = ( f (y, x) f (z, x) f (y, z)ψ(y)φ(z)θ (x)) | y=x+x1 ,z=x+x2 . (3.17)
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Proof. With (3.15), from Definition 3.4 we have f (x + x2 , x)YE o (φ(x), x2 )θ (x) = ( f (z, x)φ(z)θ (x)) |z=x+x2 ,
(3.18)
f (y, x) f (y, x + x2 ) f (x + x2 , x)ψ(y)YE o (φ(x), x2 )θ (x) = ( f (y, x) f (y, z) f (z, x)ψ(y)φ(z)θ (x)) |z=x+x2 .
(3.19)
which gives
From (3.16) the expression on the right-hand side lies in (Hom(W, W ((y −1 , x −1 )))) [[x2 ]], so does the expression on the left-hand side. That is, f (y, x) f (y, x + x2 ) f (x + x2 , x)ψ(y)YE o (φ(x), x2 )θ (x) ∈ (Hom(W, W ((y −1 , x −1 ))))[[x2 ]]. Multiplying by ιx,∞;x2 ,0 ( f (x + x2 , x)−1 ), which lies in C((x −1 ))((x2 )), we have f (y, x) f (y, x + x2 )ψ(y)YE o (φ(x), x2 )θ (x) ∈ (Hom(W, W ((y −1 , x −1 ))))((x2 )).
(3.20)
In view of Lemma 3.6, by considering the coefficient of each power of x2 , we have f (x + x1 , x) f (x + x1 , x + x2 )YE o (ψ(x), x1 )YE o (φ(x), x2 )θ (x) = ( f (y, x) f (y, x + x2 )ψ(y)(YE o (φ(x), x2 )θ (x)) | y=x+x1 .
(3.21)
Using this and (3.18) we have f (x + x1 , x) f (x + x2 , x) f (x + x1 , x + x2 )YE o (ψ(x), x1 )YE o (φ(x), x2 )θ (x) = ( f (y, x) f (x + x2 , x) f (y, x + x2 )ψ(y)YE o (φ(x), x2 )θ (x)) | y=x+x1 = ( f (y, x) f (z, x) f (y, z)ψ(y)φ(z)θ (x)) | y=x+x1 ,z=x+x2 , as desired.
To state our first result we shall need a new notion. Definition 3.8. Let V be a nonlocal vertex algebra. A (left) quasi V -module-at-infinity is a vector space W equipped with a linear map YW : V → Hom(W, W ((x −1 ))) ⊂ (End W )[[x, x −1 ]], satisfying the condition that YW (1, x) = 1W and that for any u, v ∈ V , there exists a nonzero polynomial p(x1 , x2 ) such that p(x1 , x2 )YW (u, x1 )YW (v, x2 ) ∈ Hom(W, W ((x1−1 , x2−1 )))
(3.22)
and p(x0 + x2 , x2 )YW (Y (u, x0 )v, x2 ) = ( p(x1 , x2 )YW (u, x1 )YW (v, x2 )) |x1=x2 +x0 .
(3.23)
A quasi V -module at infinity (W, YW ) is called a (left) V -module-at-infinity if for any u, v ∈ V , there exists a nonnegative integer k such that
and
(x1 − x2 )k YW (u, x1 )YW (v, x2 ) ∈ Hom(W, W ((x1−1 , x2−1 )))
(3.24)
x0k YW (Y (u, x0 )v, x2 ) = (x1 − x2 )k YW (u, x1 )YW (v, x2 ) |x1 =x2 +x0 .
(3.25)
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Here we make a new notation for convenience. Let a(x) = n∈Z an x −n−1 be any formal series (with coefficients an in some vector space). For any m ∈ Z, we set an x −n−1 . (3.26) a(x)≥m = n≥m
Then for any polynomial q(x) we have Resx x m q(x)a(x) = Resx x m q(x)a(x)≥m .
(3.27)
Now we are in a position to prove our first key result: Theorem 3.9. Let V be a closed quasi-compatible subspace of E o (W ), containing 1W . Then (V, YE o , 1W ) carries the structure of a nonlocal vertex algebra with W as a faithful (left) quasi module-at-infinity where the vertex operator map YW is given by YW (α(x), x0 ) = α(x0 ). Furthermore, if V is compatible, W is a V -module-at-infinity. Proof. For any a(x) ∈ E o (W ), as 1W a(x2 ) = a(x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))), by definition we have YE o (1W , x0 )a(x) = 1W a(x) = a(x), d
YE o (a(x), x0 )1W = (a(x1 )1W )|x1 =x+x0 = a(x + x0 ) = e x0 d x a(x). For the assertion on the nonlocal vertex algebra structure, it remains to prove the weak associativity, i.e., for ψ, φ, θ ∈ V , there exists a nonnegative integer k such that (x0 + x2 )k YE o (ψ, x0 + x2 )YE o (φ, x2 )θ = (x0 + x2 )k YE o (YE o (ψ, x0 )φ, x2 )θ.
(3.28)
Let f (x, y) be a nonzero polynomial such that f (x, y)ψ(x)φ(y) ∈ Hom(W, W ((x −1 , y −1 ))), f (x, y)φ(x)θ (y) ∈ Hom(W, W ((x −1 , y −1 ))), f (x, y) f (x, z) f (y, z)ψ(x)φ(y)θ (z) ∈ Hom(W, W ((x −1 , y −1 , z −1 ))). By Lemma 3.7, we have f (x + x2 , x) f (x + x0 + x2 , x) f (x + x0 + x2 , x + x2 )YE o (ψ(x), x0 + x2 ) YE o (φ(x), x2 )θ (x) = ( f (z, x) f (y, x) f (y, z)ψ(y)φ(z)θ (x)) | y=x+x0 +x2 ,z=x+x2 . (3.29) On the other hand, let n ∈ Z be arbitrarily fixed. Since ψ(x)m φ(x) = 0 for m sufficiently large, there exists a nonzero polynomial p(x, y), depending on n, such that p(x + x2 , x)(YE o (ψ(x)m φ(x), x2 )θ (x) = ( p(z, x)(ψ(z)m φ(z))θ (x)) |z=x+x2 (3.30) for all m ≥ n. With f (x, y)ψ(x)φ(y) ∈ Hom(W, W ((x −1 , y −1 ))), from Definition 3.4 we have f (x2 + x0 , x2 )(YE o (ψ(x2 ), x0 )φ(x2 ))θ (x) = ( f (y, x2 )ψ(y)φ(x2 )θ (x)) | y=x2 +x0 . (3.31)
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Using (3.27), (3.30) and (3.31) we get Resx0 x0n f (x + x0 + x2 , x) f (x + x0 + x2 , x + x2 ) p(x + x2 , x) ·YE o (YE o (ψ(x), x0 )φ(x), x2 )θ (x) = Resx0 x0n f (x + x0 + x2 , x) f (x + x0 + x2 , x + x2 ) p(x + x2 , x) ·YE (YE o (ψ(x), x0 )≥n φ(x), x2 )θ (x) = Resx0 x0n f (x + x0 + x2 , x) f (x + x0 + x2 , x + x2 ) · p(z, x)YE o (ψ(z), x0 )≥n φ(z))θ (x) |z=x+x2 = Resx0 x0n f (x + x0 + x2 , x) f (x + x0 + x2 , x + x2 ) · ( p(z, x)YE o (ψ(z), x0 )φ(z))θ (x)) |z=x+x2 = Resx0 x0n ( f (z + x0 , x) f (z + x0 , z) p(z, x)(YE o (ψ(z), x0 )φ(z))θ (x)) |z=x+x2 = Resx0 x0n ( f (y, x) f (y, z) p(z, x)ψ(y)φ(z)θ (x)) | y=z+x0 ,z=x+x2 . (3.32) Combining (3.32) with (3.29) we get Resx0 x0n f (x + x2 , x) f (x + x0 + x2 , x) f (x + x0 + x2 , x + x2 ) · p(x + x2 , x)YE o (ψ(x), x0 + x2 )YE o (φ(x), x2 )θ (x) = Resx0 x0n f (x2 + x, x) f (x + x0 + x2 , x) f (x + x0 + x2 , x + x2 ) · p(x + x2 , x)YE o (YE o (ψ(x), x0 )φ(x), x2 )θ (x).
(3.33)
Notice that both sides of (3.33) involve only finitely many negative powers of x2 . In view of Remark 3.3 we can multiply both sides by ιx,∞;x2 ,0 ( p(x + x2 , x)−1 f (x + x2 , x)−1 ) (in C((x −1 ))((x2 )) to get Resx0 x0n f (x + x0 + x2 , x) f (x + x0 + x2 , x + x2 )YE o (ψ(x), x0 + x2 )YE o (φ(x), x2 )θ (x) = Resx0 x0n f (x + x0 + x2 , x) f (x + x0 + x2 , x + x2 )YE o (YE o (ψ(x), x0 )φ(x), x2 )θ (x). Since f (x, y) does not depend on n and since n is arbitrary, we have f (x + x0 + x2 , x) f (x + x0 + x2 , x + x2 )YE o (ψ(x), x0 + x2 )YE o (φ(x), x2 )θ (x) = f (x + x0 + x2 , x) f (x + x0 + x2 , x + x2 )YE o (YE o (ψ(x), x0 )φ(x), x2 )θ (x). (3.34) Write f (x, y) = (x − y)k g(x, y) for some k ∈ N, g(x, y) ∈ C[x, y] with g(x, x) = 0. Then f (x + x0 + x2 , x) = (x0 + x2 )k g(x + x0 + x2 , x), f (x + x0 + x2 , x + x2 ) = x0k g(x + x0 + x2 , x + x2 ). Since g(x, x) = 0, we have ιx,∞;z,0 g(x + z, x)−1 ∈ C((x −1 ))[[z]], so that ιx,∞;z,0 g(x + z, x)−1 |z=x0 +x2 , ιz,∞;x0 ,0 g(z + x0 , z)−1 |z=x+x2 ∈ C((x −1 ))[[x0 , x2 ]]. By cancellation, from (3.34) we obtain (x0 + x2 )k YE o (ψ(x), x0 + x2 )YE o (φ(x), x2 )θ (x) = (x0 + x2 )k YE o (YE o (ψ(x), x0 )φ(x), x2 )θ (x), as desired. This proves that (V, YE o , 1W ) carries the structure of a nonlocal vertex algebra.
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841
Next, we prove that W is a quasi module-at-infinity. For a(x), b(x) ∈ V , there exists a nonzero polynomial h(x, y) such that h(x, y)a(x)b(y) ∈ Hom(W, W ((x −1 , y −1 ))). Then h(x1 , x2 )YW (a(x), x1 )YW (b(x), x2 ) = h(x1 , x2 )a(x1 )b(x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))) and h(x0 + x2 , x2 )YW (YE o (a(x), x0 )b(x), x2 ) = h(x0 + x2 , x2 )(YE o (a(x), x0 )b(x))|x=x2 = (h(x1 , x2 )a(x1 )b(x2 )) |x1 =x2 +x0 = (h(x1 , x2 )YW (a(x), x1 )YW (b(x), x2 )) |x1 =x2 +x0 . Therefore W is a (left) quasi V -module-at-infinity with YW (α(x), x0 ) = α(x0 ) for α(x) ∈ V . Finally, if V is compatible, the polynomial h(x, y) is of the form (x − y)k with k ∈ N. Then W is a V -module-at-infinity, instead of a quasi V -module-at-infinity.
In practice, we are often given an unnecessarily closed quasi-compatible subspace. Next, we are going to show that every quasi-compatible subset is contained in some closed quasi-compatible subspace. The following is an analogue of a result in [Li2] and [Li4]: Proposition 3.10. Let ψ1 (x), . . . , ψr (x), a(x), b(x), φ1 (x), . . . , φs (x) ∈ E o (W ). Assume that the ordered sequences (a(x), b(x)) and (ψ1 (x), . . . , ψr (x), a(x), b(x), φ1 (x), . . . , φs (x)) are quasi-compatible (compatible). Then for any n ∈ Z, the ordered sequence (ψ1 (x), . . . , ψr (x), a(x)n b(x), φ1 (x), . . . , φs (x)) is quasi-compatible (compatible). Proof. Let f (x, y) be a nonzero polynomial such that f (x, y)a(x)b(y) ∈ Hom(W, W ((x −1 , y −1 ))) and
⎛ ⎝
⎞⎛ f (yi , y j )⎠ ⎝
1≤i< j≤r
· f (x1 , x2 )
⎞⎛
f (yi , z j )⎠ ⎝
1≤i≤r,1≤ j≤s r
f (x1 , yi ) f (x2 , yi )
i=1
s
⎞
f (z i , z j )⎠
1≤i< j≤s
f (x1 , z i ) f (x2 , z i )
i=1
·ψ1 (y1 ) · · · ψr (yr )a(x1 )b(x2 )φ1 (z 1 ) · · · φs (z s ) ∈ Hom(W, W ((y1−1 , . . . , yr−1 , x1−1 , x2−1 , z 1−1 , . . . , z s−1 ))). Set P=
1≤i< j≤r
f (yi , y j ),
Q=
1≤i< j≤s
f (z i , z j ),
R=
1≤i≤r, 1≤ j≤s
(3.35)
f (yi , z j ).
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H. Li
Let n ∈ Z be arbitrarily fixed. There exists a nonnegative integer k such that x0k+n ιx2 ,∞;x0 ,0 f (x0 + x2 , x2 )−1 ∈ C((x2−1 ))[[x0 ]].
(3.36)
Using (3.36) and Definition 3.4 we obtain r
f (x2 , yi )k
i=1
s
f (x2 , z j )k
j=1
·ψ1 (y1 ) · · · ψr (yr )(a(x2 )n b(x2 ))φ1 (z 1 ) · · · φs (z s ) r s f (x2 , yi )k f (x2 , z j )k = Resx0 x0n i=1
j=1
·ψ1 (y1 ) · · · ψr (yr )(YE (a, x0 )b)(x2 )φ1 (z 1 ) · · · φs (z s ) r s f (x2 , yi )k f (x2 , z j )k = Resx1 Resx0 x0n i=1
j=1
x2 + x0 ·ιx2 ,∞;x0 ,0 ( f (x2 + x0 , x2 ) x1 · ( f (x1 , x2 )ψ1 (y1 ) · · · ψr (yr )a(x1 )b(x2 )φ1 (z 1 ) · · · φs (z s )) r s n k = Resx1 Resx0 x0 f (x1 − x0 , yi ) f (x1 − x0 , z j )k −1
)x1−1 δ
i=1
j=1
x2 + x0 ·ιx2 ,∞;x0 ,0 ( f (x2 + x0 , x2 ) x1 · ( f (x1 , x2 )ψ1 (y1 ) · · · ψr (yr )a(x1 )b(x2 )φ1 (z 1 ) · · · φs (z s )) ⎞k ⎛ r s ∂ −x = Resx1 Resx0 x0n e 0 ∂ x1 ⎝ f (x1 , yi ) f (x1 , z j )⎠ −1
)x1−1 δ
i=1
j=1
x2 + x0 ·ιx2 ,∞;x0 ,0 ( f (x2 + x0 , x2 ) x1 · ( f (x1 , x2 )ψ1 (y1 ) · · · ψr (yr )a(x1 )b(x2 )φ1 (z 1 ) · · · φs (z s )) ⎞k ⎛ k−1 r s ∂ t ⎝ (−1)t n+t x = Resx1 Resx0 f (x1 , yi ) f (x1 , z j )⎠ t! 0 ∂ x1 t=0 i=1 j=1 x2 + x0 ·ιx2 ,∞;x0 ,0 ( f (x2 + x0 , x2 )−1 )x1−1 δ x1 (3.37) · ( f (x1 , x2 )ψ1 (y1 ) · · · ψr (yr )a(x1 )b(x2 )φ1 (z 1 ) · · · φs (z s )) . t Notice that for any polynomial B and for 0 ≤ t ≤ k − 1, ∂∂x1 B k is a multiple of B. Using (3.35) we have −1
PQR
r i=1
f (x2 , yi )
s j=1
)x1−1 δ
f (x2 , z j )
k−1 (−1)t t=0
t!
x0n+t
∂ ∂ x1
t
Modules-at-Infinity for Quantum Vertex Algebras
⎛ ×⎝
r
f (x1 , yi )
i=1
s
843
⎞k f (x1 , z j )⎠
j=1
x2 + x0 ·ιx2 ,∞;x0 ,0 ( f (x2 + x0 , x2 ) x1 · ( f (x1 , x2 )ψ1 (y1 ) · · · ψr (yr )a(x1 )b(x2 )φ1 (z 1 ) · · · φs (z s )) ∈ Hom(W, W ((y1−1 , . . . , yr−1 , x2−1 , z 1−1 , . . . , z s−1 ))) ((x0 ))[[x1±1 ]]. −1
)x1−1 δ
Then PQR
r
f (x2 , yi )k+1
i=1
s
f (x2 , z j )k+1 ψ1 (y1 ) · · · ψr (yr )(a(x)n b(x))
j=1
×(x2 )φ1 (z 1 ) · · · φs (z s ) ∈ Hom(W, W ((y1−1 , . . . , yr−1 , x2−1 , z 1−1 , . . . , z s−1 ))).
(3.38)
This proves that (ψ1 (x), . . . , ψr (x), a(x)n b(x), φ1 (x), . . . , φs (x)) is quasi-compatible. It is clear that the assertion with compatibility holds.
The following is our second key result: Theorem 3.11. Every maximal quasi-compatible subspace of E o (W ) is closed and contains 1W . Furthermore, for any quasi-compatible subset S, there exists a (unique) smallest closed quasi-compatible subspace S containing S and 1W , and (S, YE o , 1W ) carries the structure of a nonlocal vertex algebra with W as a faithful (left) quasi moduleat-infinity where the vertex operator map YW is given by YW (ψ(x), x0 ) = ψ(x0 ). If S is compatible, then W is a module-at-infinity for S. Proof. Let K be any maximal quasi-compatible subspace of E o (W ). Clearly, K +C1W is quasi-compatible. With K maximal we must have 1W ∈ K . Let a(x), b(x) ∈ K , n ∈ Z. It follows from Proposition 3.10 and an induction that any finite sequence in K ∩ {a(x)n b(x)} is quasi-compatible. Again, with K maximal we must have a(x)n b(x) ∈ K . This proves that K is closed. The rest of the assertions follow immediately from Theorem 3.9.
Recall that C(x) denotes the field of rational functions and ιx,0 and ιx,∞ are the field embeddings of C(x) into C((x)) and C((x −1 )), respectively. Proposition 3.12. Let V be a nonlocal vertex algebra generated by a quasi-compatible subset of E o (W ). Suppose that the following relation holds: (x1 − x2 )k p(x1 , x2 )a(x1 )b(x2 ) r ιx,∞ (qi )(x1 − x2 )u i (x2 )vi (x1 ), = (x1 − x2 )k p(x1 , x2 )
(3.39)
i=1
where a(x), b(x), u i (x), vi (x) ∈ V and p(x, y) ∈ C[x, y], qi (x) ∈ C(x), k ∈ N with p(x, x) = 0. Then there exists a nonnegative integer k such that
(x1 − x2 )k YE o (a(x), x1 )YE o (b(x), x2 ) r ιx,0 (qi )(−x2 + x1 )YE o (u i (x), x2 )YE o (vi (x), x1 ). (3.40) = (x1 − x2 )k i=1
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Proof. Let θ (x) ∈ V . By Lemma 3.7, there exists a nonzero polynomial f (x, y) such that f (x + x1 , x) f (x + x1 , x + x2 ) f (x + x2 , x)YE o (a(x), x1 )YE o (b(x), x2 )θ (x) = ( f (y, x) f (y, z) f (z, x)a(y)b(z)θ (x)) | y=x+x1 ,z=x+x2 (3.41) and such that f (x + x1 , x) f (x + x1 , x + x2 ) f (x + x2 , x)YE o (u i (x), x1 )YE o (vi (x), x2 )θ (x) = ( f (y, x) f (y, z) f (z, x)u i (y)vi (z)θ (x)) | y=x+x1 ,z=x+x2 (3.42) for i = 1, . . . , r . Let 0 = g(x) ∈ C[x] such that g(x)qi (x) ∈ C[x] for i = 1, . . . , r . Then f (x + x1 , x) f (x + x1 , x + x2 ) f (x + x2 , x)(x1 − x2 )k p(x + x1 , x + x2 ) ·g(x1 − x2 )YE o (a(x), x1 )YE o (b(x), x2 )θ (x) = (x1 − x2 )k ( f (y, x) f (y, z) f (z, x)g(y − z) p(y, z)a(y)b(z)θ (x)) | y=x+x1 ,z=x+x2 = (x1 − x2 )k
r f (y, x) f (y, z) f (z, x) p(y, z) (gqi )(y − z)u i (y)vi (z)θ (x) | y=x+x1 ,z=x+x2
×
i=1
= f (x + x1 , x) f (x + x1 , x + x2 ) f (x + x2 , x)(x1 − x2 )k p(x + x1 , x + x2 ) r (gqi )(x1 − x2 )YE o (u i (x), x2 )YE o (vi (x), x1 )θ (x) · i=1
= f (x + x1 , x) f (x + x1 , x + x2 ) f (x + x2 , x)(x1 − x2 )k p(x + x1 , x + x2 ) r ιx,0 (qi )(−x2 + x1 ))YE o (u i (x), x2 )YE o (vi (x), x1 )θ. g(x1 − x2 ) ·
(3.43)
i=1
Notice that we can multiply both sides by ιx,∞;x1 ,0 f (x + x1 , x)−1 ιx,∞;x2 ,0 f (x + x2 , x)−1 to cancel the factors f (x +x1 , x) and f (x +x2 , x) (recall Remark 3.3). Since p(x, x) = 0, we can also cancel the factor p(x + x1 , x + x2 ). By cancelation we get (x1 − x2 )k f (x + x1 , x + x2 )g(x1 − x2 )YE o (a(x), x1 )YE o (b(x), x2 )θ (x) = (x1 − x2 )k f (x + x1 , x + x2 )g(x1 − x2 ) r ιx,0 (qi )(−x2 + x1 ))YE o (u i (x), x2 )YE o (vi (x), x1 )θ (x). · i=1
Write g(x) = x l g(x), ¯ where l ≥ 0, g(x) ¯ ∈ C[x] with g(0) ¯ = 0. Similarly, write f (x, z) = (x − z)t f¯(x, z), where t ≥ 0, f¯(x, z) ∈ C[x, z] with f¯(x, x) = 0. By a
further cancelation we get (3.40) with k = k + l + t. Recall that G denotes the group of linear transformations on C. Lemma 3.13. Let be a group of linear transformations and let V be a vertex algebra generated by a quasi-compatible subset of E o (W ). Assume that Rg a(x) (= a(g(x))) ∈ V
for g ∈ , a(x) ∈ V.
Modules-at-Infinity for Quantum Vertex Algebras
845
Then YE o (Rg a(x), x0 )Rg b(x) = Rg YE o (a(x), g0 x0 )b(x)
(3.44)
for g ∈ , a(x), b(x) ∈ V , where g(x) = g0 x + g1 . Proof. Let g ∈ , a(x), b(x) ∈ V . There exists a nonzero polynomial p(x1 , x2 ) such that p(x1 , x2 )a(x1 )b(x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))). Then p(x + g0 x0 , x)YE o (a(x), g0 x0 )b(x) = ( p(x1 , x)a(x1 )b(x)) |x1 =x+g0 x0 . Substituting x with g(x) we get p(g(x) + g0 x0 , g(x))Rg(x) (YE o (a(x), g0 x0 )b(x)) = ( p(x1 , g(x))a(x1 )b(g(x))) |x1 =g(x)+g0 x0 =g(x+x0 ) = ( p(g(x1 ), g(x))a(g(x1 )b(x)) |x1 =x+x0 . We also have p(g(x1 ), g(x2 ))a(g(x1 ))b(g(x2 )) ∈ Hom(W, W ((x1−1 , x2−1 ))), so that p(g(x) + g0 x0 , g(x))YE o (a(g(x)), x0 )b(g(x)) = ( p(g(x1 ), g(x))a(g(x1 ))b(g(x))) |x1 =x+x0 . Consequently, p(g(x) + g0 x0 , g(x))YE o (a(g(x)), x0 )b(g(x)) = p(g(x) + g0 x0 , g(x))Rg(x) YEo (a(x), g0 x0 )b(x). By cancelation, we obtain (3.44).
The following is an analogue of ([Li3], Prop. 4.3): Proposition 3.14. Let a(x), b(x), c(x) ∈ E o (W ). Assume that f (x1 , x2 )a(x1 )b(x2 ) = f (x1 , x2 )b(x2 )a(x1 ), g(x1 , x2 )a(x1 )c(x2 ) = g(x ˜ 1 , x2 )c(x2 )a(x1 ), ˜ 1 , x2 )c(x2 )b(x1 ), h(x1 , x2 )b(x1 )c(x2 ) = h(x ˜ where f (x, y), g(x, y), g(x, ˜ y), h(x, y), h(x, y) are nonzero polynomials. Then for any n ∈ Z, there exists k ∈ N, depending on n, such that f (x3 , x)k g(x3 , x)a(x3 )(b(x)n c(x)) = f (x3 , x)k g(x ˜ 3 , x)(b(x)n c(x))a(x3 ).
(3.45)
Proof. Let n ∈ Z be arbitrarily fixed. Let k be a nonnegative integer such that x0k+n ιx,∞;x0 ,0 (h(x + x0 , x)−1 ) ∈ C((x −1 ))[[x0 ]]. In the proof of Proposition 4.3 of [Li3], take α = 1 and replace ιx,x0 with ιx,∞ x0 ,0 . Then the same arguments prove (3.45).
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H. Li
4. Associative Algebra DYq∞(sl 2 ) and Modules-at-Infinity for Quantum Vertex Algebras In this section we continue to study S-local subsets of E o (W ) for a vector space W and we prove that any S-local subset generates a weak quantum vertex algebra with W as a canonical module-at-infinity. We introduce another version DYq∞ (sl2 ) of the double Yangian and we prove that every DYq∞ (sl2 )-module W is naturally a module-at-infinity for the quantum vertex algebra Vq which was constructed in Sect. 2. First we prove a simple result that we shall need later: Lemma 4.1. Let V be a nonlocal vertex algebra and let (W, YW ) be a (left) quasi V -module-at-infinity. Then d YW (v, x) for v ∈ V. (4.1) dx Proof. For any v ∈ V , by definition there exists a nonzero polynomial p(x1 , x2 ) such that YW (Dv, x) =
p(x1 , x2 )YW (v, x1 )YW (1, x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))), p(x2 + x0 , x2 )YW (Y (v, x0 )1, x2 ) = ( p(x1 , x2 )YW (v, x1 )YW (1, x2 )) |x1 =x2 +x0 . With Y (v, x0 )1 = e x0 D v and YW (1, x2 ) = 1W , we get p(x2 + x0 , x2 )YW (e x0 D v, x2 ) = ( p(x1 , x2 )YW (v, x1 )) |x1 =x2 +x0 = p(x2 + x0 , x2 )YW (v, x2 + x0 ). As both YW (e x0 D v, x2 ) and YW (v, x2 + x0 ) lie in (Hom(W, W ((x2−1 ))))[[x0 ]], in view of Remark 3.3 we have YW (e x0 D v, x2 ) = YW (v, x2 + x0 ) = e
x0 d dx
2
YW (v, x2 ),
which implies (4.1).
The following is straightforward to prove: Lemma 4.2. Let W be any vector space and let A(x1 , x2 ) ∈ Hom(W, W ((x2−1 ))((x1−1 ))), B(x1 , x2 ) ∈ Hom(W, W ((x1−1 ))((x2−1 ))), C(x2 , x0 ) ∈ (Hom(W, W ((x2−1 )))((x0 )). Then
x1 − x2 x2 − x1 −1 A(x1 , x2 ) − x0 δ B(x1 , x2 ) x0 −x0 x1 − x0 C(x2 , x0 ) = x2−1 δ x2
x0−1 δ
(4.2)
if and only if there exist a nonnegative integer k and F(x1 , x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))) such that (x1 − x2 )k A(x1 , x2 ) = F(x1 , x2 ) = (x1 − x2 )k B(x1 , x2 ), x0k C(x2 , x0 ) = F(x2 + x0 , x2 ).
(4.3) (4.4)
Modules-at-Infinity for Quantum Vertex Algebras
847
gn x −n ∈ C((x −1 )) with m ∈ Z. We have −n −n (−1)i gn x1−n−i x2i ∈ C[x2 ]((x1−1 )). g(x1 − x2 ) = gn (x1 − x2 ) = i n≥m n≥m
Remark 4.3. Let g(x) =
n≥m
i∈N
Furthermore, for any ψ(x), φ(x) ∈ Hom(W, W ((x −1 ))) with W a vector space, the product g(x1 − x2 )ψ(x2 )φ(x1 ) exists in Hom(W, W ((x2−1 ))((x1−1 ))), and we have x1 − x2 x1 − x2 −1 −1 g(−x0 )ψ(x2 )φ(x1 ) = x0 δ g(x1 − x2 )ψ(x2 )φ(x1 ). x0 δ −x0 −x0 Using Lemma 4.2 we immediately have: Lemma 4.4. Let V be a nonlocal vertex algebra, let (W, YW ) be a V -module-at-infinity, and let u, v, u (i) , v (i) ∈ V, f i (x) ∈ C(x) (i = 1, . . . , r ). Then
x2 − x1 YW (u, x1 )YW (v, x2 ) −x0 r x1 − x2 ιx,∞ ( f i )(−x0 )YW (v(i) , x2 )YW (u (i) , x1 ) +x0−1 δ x0 i=1 x1 − x0 −1 YW (Y (u, x0 )v, x2 ) = x2 δ x2
−x0−1 δ
if and only if there exists a nonnegative integer k such that (x1 − x2 )k YW (u, x1 )YW (v, x2 ) r ιx,∞ ( f i )(−x1 + x2 )YW (v (i) , x2 )YW (u (i) , x1 ). = (x1 − x2 )k i=1
The following is an analogue of ([Li4], Prop. 6.7): Proposition 4.5. Let V be a nonlocal vertex algebra, let (W, YW ) be a V -module-atinfinity, and let n ∈ Z, u, v, u (i) , v (i) ∈ V, f i (x) ∈ C(x) (i = 1, . . . , r ), c(0) , . . . , c(s) ∈ V. If (x1 − x2 )n Y (u, x1 )Y (v, x2 ) − (−x2 + x1 )n
r
ιx,0 ( f i )(x2 − x1 )
i=1
×Y (v (i) , x2 )Y (u (i) , x1 ) s ∂ j −1 1 x2 ( j) Y (c , x2 ) x1 δ = j! ∂ x2 x1 j=0
(4.5)
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H. Li
on V , then (x1 − x2 )n
r
ιx,∞ ( f i )(−x1 + x2 )YW (v (i) , x2 )YW (u (i) , x1 )
i=1
−(−x2 + x1 )n YW (u, x1 )YW (v, x2 ) s ∂ j −1 1 x2 YW (c( j) , x2 ) x1 δ = j! ∂ x2 x1
(4.6)
j=0
on W . If (W, YW ) is faithful, the converse is also true. Proof. Let k be a nonnegative integer such that k > s and n + k ≥ 0. From (4.5) we get (x1 −x2 )k+n Y (u, x1 )Y (v, x2 ) =
r (x1 −x2 )k+n ιx,0 ( f i )(x2 −x1 )Y (v (i) , x2 )Y (u (i) , x1 ). i=1
By Corollary 5.3 in [Li5] (cf. [EK]) we have the S-skew symmetry Y (u, x)v =
r
ιx,0 ( f i (−x))e x D Y (v (i) , −x)u (i) .
(4.7)
i=1
We also have the following S-Jacobi identity: x1 − x2 −1 Y (u, x1 )Y (v, x2 ) x0 δ x0 r x2 − x1 −1 ιx0 ,0 ( f i (−x0 ))Y (v (i) , x2 )Y (u (i) , x1 ) −x0 δ −x0 i=1 x1 − x0 −1 Y (Y (u, x0 )v, x2 ). = x2 δ x2
(4.8)
By taking Resx0 x0n we get (x1 − x2 ) Y (u, x1 )Y (v, x2 ) − (−x2 + x1 ) n
n
r
ιx,0 ( f i )(x2 − x1 )
i=1
×Y (v (i) , x2 )Y (u (i) , x1 ) ∂ j −1 1 x2 . Y (u n+ j v, x2 ) x1 δ = j! ∂ x2 x1
(4.9)
j≥0
Combining this with (4.5) we obtain u n+ j v = c( j)
for j = 0, . . . , s, and u n+ j v = 0
for j > s.
Let l be a sufficiently large nonnegative integer such that (x1 − x2 )l YW (u, x1 )YW (v, x2 ) = F(x1 , x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))), x0l YW (Y (u, x0 )v, x2 ) = F(x2 + x0 , x2 )
(4.10)
Modules-at-Infinity for Quantum Vertex Algebras
849
and (x1 − x2 )l YW (v (i) , x2 )YW (u (i) , x1 ) = G i (x1 , x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))), x0l YW (Y (v (i) , −x0 )u (i) , x1 ) = G i (x1 , x1 − x0 ) for all i = 1, . . . , r . Let p(x) be a nonzero polynomial such that p(x) f i (x) ∈ C[x]
for all i = 1, . . . , r.
Note that by Lemma 4.1, we have YW (Dv, x) = (d/d x)YW (v, x) for v ∈ V . Using all of these and the S-skew symmetry we get p(−x0 )F(x2 + x0 , x2 ) = p(−x0 )x0l YW (Y (u, x0 )v, x2 ) r p(−x0 ) f i (−x0 )x0l YW (e x0 D Y (v (i) , −x0 )u (i) , x2 ) = =
i=1 r
p(−x0 ) f i (−x0 )x0l YW (Y (v (i) , −x0 )u (i) , x2 + x0 )
i=1
= =
r i=1 r
p(−x0 ) f i (−x0 ) (G i (x1 , x1 − x0 )) |x1 =x2 +x0 p(−x0 ) f i (−x0 )G i (x2 + x0 , x2 ),
i=1
which implies p(x2 − x1 )F(x1 , x2 ) =
r ( p f i )(x2 − x1 )G i (x1 , x2 ). i=1
Then p(x2 − x1 )(x1 − x2 )l YW (u, x1 )YW (v, x2 ) r (x1 − x2 )l ιx,∞ ( f i )(−x1 + x2 )YW (v (i) , x2 )YW (u (i) , x1 ). = p(x2 − x1 ) i=1
As both (x1 − x2 )l YW (u, x1 )YW (v, x2 ) and r (x1 − x2 )l ιx,∞ ( f i )(−x1 + x2 )YW (v (i) , x2 )YW (u (i) , x1 ) i=1
lie in Hom(W, W ((x2−1 ))((x1−1 ))), by Remark 3.3 we get (x1 − x2 )l YW (u, x1 )YW (v, x2 ) r ιx,∞ ( f i )(−x1 + x2 )YW (v (i) , x2 )YW (u (i) , x1 ). = (x1 − x2 )l i=1
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Now by Lemma 4.2 we have x2 − x1 −1 −x0 δ YW (u, x1 )YW (v, x2 ) −x0 r x1 − x2 ιx0 ,∞ ( f i (−x0 ))YW (v (i) , x2 )YW (u (i) , x1 ) +x0−1 δ x0 i=1 − x x 1 0 YW (Y (u, x0 )v, x2 ). = x2−1 δ x2
(4.11)
Using this and (4.10) we obtain (4.6). For the converse, we trace back, assuming that W is faithful and (4.6) holds. Let k be a nonnegative integer such that k + n ≥ 0 and k > s. Then (x1 − x2 )k+n YW (u, x1 )YW (v, x2 ) r ιx,∞ ( f i )(−x1 + x2 )YW (v (i) , x2 )YW (u (i) , x1 ). = (x1 − x2 )k+n
(4.12) (4.13)
i=1
Using Lemma 4.2 we get (4.11). Combining (4.11) with (4.6) we obtain (4.10), using the assumption that YW is injective. Using (4.12) we also have x2 − x1 YW (u, x1 )YW (v, x2 ) −x0−1 δ −x0 r x1 − x2 ιx0 ,∞ ( f i (−x0 ))YW (v (i) , x2 )YW (u (i) , x1 ) +x0−1 δ x0 i=1 r x2 + x0 = x1−1 δ ιx0 ,∞ ( f i (−x0 ))YW (Y (v (i) , −x0 )u (i) , x1 ) x1 i=1 r x2 + x0 = x1−1 δ ιx0 ,∞ ( f i (−x0 ))YW (Y (v (i) , −x0 )u (i) , x2 + x0 ) x1 i=1 r x1 − x0 ιx0 ,∞ ( f i (−x0 ))YW (e x0 D Y (v (i) , −x0 )u (i) , x2 ). = x2−1 δ x2 i=1
Combining this with (4.11) we get the S-skew symmetry, with which we obtain the S-Jacobi identity (4.8). Then using (4.10) we obtain (4.5).
Definition 4.6. Let W be a vector space. A subset U of E o (W ) is said to be S-local if for any a(x), b(x) ∈ U , there exist u (1) (x), v (1) (x), . . . , u (r ) (x), v (r ) (x) ∈ U and f 1 (x), . . . , fr (x) ∈ C(x) such that (x1 − x2 )k a(x1 )b(x2 ) = (x1 − x2 )k
r i=1
for some nonnegative integer k.
ιx,∞ ( f i )(−x1 + x2 )u (i) (x2 )v (i) (x1 )
(4.14)
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Theorem 4.7. Let W be a vector space and let U be an S-local subset of E o (W ). Then U is compatible and the nonlocal vertex algebra U generated by U is a weak quantum vertex algebra with W as a module-at-infinity. Proof. Notice that the relation (4.14) implies that (x1 − x2 )k a(x1 )b(x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))). Thus any ordered pair in U is compatible. As in the proof of Lemma 3.2 of [Li4], using induction we see that any finite sequence in U is compatible. That is, U is compatible. By Theorem 3.11, U generates a nonlocal vertex algebra U inside E o (W ) with W as a module-at-infinity. As U is S-local, from Proposition 3.12, the vertex operators YE o (a(x), x0 ) for a(x) ∈ U form an S-local subset of U . Because U generates U , by Lemma 2.7 of [Li5], U is a weak quantum vertex algebra.
Lemma 4.8. Let a(x), b(x) ∈ E o (W ). Suppose that there exist u (i) (x), v (i) (x) ∈ E o (W ), f i (x) ∈ C(x) (i = 1, . . . , r ) such that (x1 − x2 )k a(x1 )b(x2 ) = (x1 − x2 )k
r
ιx,∞ ( f i )(x1 − x2 )u (i) (x2 )v (i) (x1 ) (4.15)
i=1
for some nonnegative integer k. Then (a(x), b(x)) is compatible and YE o (a(x), x0 )b(x) r x − x1 x1 − x a(x1 )b(x) − x0−1 δ ιx,∞ ( f i ) = Resx1 x0−1 δ x0 −x0 i=1
(i)
(i)
×(−x0 )u (x)v (x1 ).
(4.16)
Proof. We have (x1 − x2 )k a(x1 )b(x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))), so that x0k YE o (W ) (a(x), x0 )b(x) = (x1 − x2 )k a(x1 )b(x2 ) |x1 =x2 +x0 . Then we have the Jacobi identity, then the iterate formula.
Now we come back to double Yangians. Recall that T is the tensor algebra over the space sl2 ⊗ C[t, t −1 ] and T = n∈Z Tn is Z-graded with deg(sl2 ⊗ t n ) = n for n ∈ Z. For n ∈ Z, set J [n] = m≤−n Tm . This gives a decreasing filtration. Denote by T˜ the completion of T associated with this filtration.
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Definition 4.9. Let q be a nonzero complex number as before. We define DYq∞ (sl2 ) to be the quotient algebra of T˜ modulo the following relations: x1 − x2 + q e(x2 )e(x1 ), x1 − x2 − q x1 − x2 − q = f (x2 ) f (x1 ), x1 − x2 + q x2 h(x2 ), = −x1−1 δ x1 x1 − x2 + q = e(x2 )h(x1 ), x1 − x2 − q = h(x2 )h(x1 ), x1 − x2 − q f (x2 )h(x1 ), = x1 − x2 + q
e(x1 )e(x2 ) = f (x1 ) f (x2 )
e(x1 ), f (x2 )
h(x1 )e(x2 ) h(x1 )h(x2 ) h(x1 ) f (x2 ) where it is understood that (x1 − x2 ± q)−1 =
(±q)i (x1 − x2 )−i−1 ∈ C[[x1−1 , x2 ]]. i∈N
We define a DYq∞ (sl2 )-module to be a T (sl2 ⊗ C[t, t −1 ])-module W such that for every w ∈ W , sl(n)w = 0
for n sufficiently small
(4.17)
and such that all the defining relations for DYq∞ (sl2 ) hold. Then for any DYq∞ (sl2 )module W , the generating functions e(x), f (x), h(x) are elements of E o (W ). Recall from Sect. 2 the quantum vertex algebra Vq . Then we have: Theorem 4.10. Let q be any nonzero complex number and let W be any DYq∞ (sl2 )module. There exists one and only one structure of a Vq -module-at-infinity on W with YW (e, x) = e(x), YW ( f, x) = f (x), YW (h, x) = h(x). Proof. The uniqueness is clear as e, f, h generate Vq . The proof for the existence is similar to the proof of Theorem 2.12. Set U = {e(x), f (x), h(x)} ⊂ E o (W ). From the defining relations of DYq∞ (sl2 ), U is an S-local subset. Then, by Theorem 4.7, U generates a weak quantum vertex algebra VW with W as a faithful module-at-infinity where YW (a(x), x0 ) = a(x0 ) for a(x) ∈ VW . Using Proposition 4.5, we see that VW is a DYq (sl2 )-module with e(x0 ), f (x0 ), h(x0 ) acting as YE o (e(x), x0 ), YE o ( f (x), x0 ), YE o (h(x), x0 ). Clearly, 1W is a vacuum vector of VW viewed as a DYq (sl2 )-module. Then (VW , 1W ) is a vacuum DYq (sl2 )-module with an operator D such that D(1W ) = 0 and [D, u(x)] =
d u(x) dx
for u ∈ sl2 .
By the universal property of Vq , there exists a DYq (sl2 )-module homomorphism θ from Vq to VW , sending 1 to 1W . Since sl2 generates Vq as a nonlocal vertex algebra, it follows that θ is a homomorphism of nonlocal vertex algebras. Using θ we obtain a structure of a Vq -module-at-infinity on W with the desired property.
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5. Quasi Modules-at-Infinity for Vertex Algebras In this section we study quasi-local subsets of E o (W ) for a vector space W and we prove that every quasi-local subset generates a vertex algebra with W as a quasi module-atinfinity. We give a family of examples related to infinite-dimensional Lie algebras of a certain type, including the Lie algebra of pseudo-differential operators on the circle. First we prove: Lemma 5.1. Let V be a vertex algebra and let (W, YW ) be a quasi module-at-infinity for V viewed as a nonlocal vertex algebra. Then for u, v ∈ V , there exists a nonzero polynomial p(x1 , x2 ) such that p(x1 , x2 )YW (v, x2 )YW (u, x1 ) = p(x1 , x2 )YW (u, x1 )YW (v, x2 ).
(5.1)
Furthermore, if (W, YW ) is a module-at-infinity, then for u, v ∈ V , x1 − x2 x2 − x1 −1 −1 YW (v, x2 )YW (u, x1 ) − x0 δ YW (u, x1 )YW (v, x2 ) x0 δ x0 −x0 x1 − x0 YW (Y (u, x0 )v, x2 ). (5.2) = x2−1 δ x2 Proof. It basically follows from the arguments of [LL] (Theorem 3.6.3). Let u, v ∈ V . From definition, there exist F(x1 , x2 ), G(x1 , x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))), 0 = p(x1 , x2 ) ∈ C[x1 , x2 ] such that p(x1 , x2 )YW (u, x1 )YW (v, x2 ) = F(x1 , x2 ), p(x1 , x2 )YW (v, x2 )YW (u, x1 ) = G(x1 , x2 ) and p(x2 + x0 , x2 )YW (Y (u, x0 )v, x2 ) = F(x2 + x0 , x2 ), p(x1 , x1 − x0 )YW (Y (v, −x0 )u, x1 ) = G(x1 , x1 − x0 ).
(5.3) (5.4)
Using the skew symmetry of the vertex algebra V and Lemma 4.1, we have YW (Y (v, −x0 )u, x1 ) = YW (e−x0 D Y (u, x0 )v, x1 ) = YW (Y (u, x0 )v, x1 − x0 ). Now (5.4) is rewritten as p(x1 , x1 − x0 )YW (Y (u, x0 )v, x1 − x0 ) = G(x1 , x1 − x0 ), which gives p(x2 + x0 , x2 )YW (Y (u, x0 )v, x2 ) = G(x2 + x0 , x2 ). Combining this with (5.3) we get F(x1 , x2 ) = G(x1 , x2 ), proving (5.1). For the second assertion, the polynomial p(x1 , x2 ) in the above argument is of the form (x1 − x2 )k for k ∈ N. Then it follows from Lemma 4.2.
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Remark 5.2. Let V be a vertex algebra. A right V -module (see [HL,Li1]) is a vector space W equipped with a linear map YW : V → Hom(W, W ((x −1 ))) ⊂ (End W )[[x, x −1 ]] satisfying the following conditions: YW (1, x) = 1W ,
(5.5)
and for u, v ∈ V , x1 − x2 x2 − x1 −1 −1 YW (v, x2 )YW (u, x1 ) − x0 δ YW (u, x1 )YW (v, x2 ) x0 δ x0 −x0 x1 − x0 YW (Y (u, x0 )v, x2 ). (5.6) = x2−1 δ x2 In view of Lemma 5.1, this notion of right V -module is equivalent to the notion of (left) V -module-at-infinity. The following is a counterpart of the notion of quasi-local subset in [Li3]: Definition 5.3. Let W be a vector space. A subset S of E o (W ) is said to be quasi-local if for any a(x), b(x) ∈ S, there exists a nonzero polynomial p(x1 , x2 ) such that p(x1 , x2 )a(x1 )b(x2 ) = p(x1 , x2 )b(x2 )a(x1 ). In case that for any a(x), b(x) ∈ S, there exists a nonnegative integer k such that (x1 − x2 )k a(x1 )b(x2 ) = (x1 − x2 )k b(x2 )a(x1 ), we say S is local. Specializing Theorem 3.11 we have: Theorem 5.4. Let W be a vector space. Every quasi-local subset U of E o (W ) is quasicompatible and the nonlocal vertex algebra U generated by U inside E o (W ) is a vertex algebra with W as a (left) quasi-module-at-infinity, where YW (a(x), x0 ) = a(x0 ) for a(x) ∈ U . If U is local, W is a module-at-infinity. Proof. It is clear that any quasi-local subset U is quasi-compatible. By Theorem 3.11, U generates a nonlocal vertex algebra U with W as a quasi module-at-infinity. From Proposition 3.12, U is a local subspace of the nonlocal vertex algebra U in the sense that the adjoint vertex operators associated to the vectors of U are mutually local. As U generates U as a nonlocal vertex algebra, from [Li2] (Proposition 2.17) U is a vertex algebra. As locality implies compatibility, the last assertion follows from Theorem 3.11.
The following notion was due to [GKK]: Definition 5.5. Let be a subgroup of G (the group of linear transformations on C). A subset S of E o (W ) is said to be -local if for any a(x), b(x) ∈ S, there exist finitely many g1 (x), . . . , gr (x) ∈ such that (x1 − g1 (x2 )) · · · (x1 − gr (x2 ))[a(x1 ), b(x2 )] = 0.
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Recall the following notion from [Li6] (cf. [Li3]): Definition 5.6. Let be an abstract group. A -vertex algebra is a vertex algebra V equipped with two group homomorphisms R : → G L(V ), φ : → C×
(5.7)
Rg Y (u, x)v = Y (Rg u, φ(g)−1 x)Rg v
(5.8)
such Rg 1 = 1 and
for g ∈ , u, v ∈ V . Example 5.7. Let V = n∈Z V(n) be a Z-graded vertex algebra in the sense that V is a vertex algebra equipped with a Z-grading such that 1 ∈ V(0) and u m V(n) ⊂ V(k+n−m−1)
for u ∈ V(k) , k, m, n ∈ Z.
(5.9)
Let be a group acting on V by automorphisms that preserve the grading and let φ : → C× be any group homomorphism. Then V becomes a -vertex algebra with Rg = φ(g)−L(0) g
for g ∈ ,
where L(0) denotes the degree operator on V (see [Li6]). Note that the projection ddx : G → C× is a group homomorphism. Then any group homomorphism : → G gives rise to a group homomorphism 0 = ddx ◦ : → C× . Definition 5.8. Let V be a -vertex algebra. A quasi V -module-at-infinity is a quasimodule-at-infinity (W, YW ) for V viewed as a nonlocal vertex algebra, equipped with a group homomorphism : → G, such that φ = 0 (=
d dx
◦ ) and
YW (Rg v, x) = YW (v, (g)(x))
for g ∈ , v ∈ V,
(5.10)
and {YW (v, x) | v ∈ V } is ( )-local. We shall need the following technical result: Lemma 5.9. Let V be a -vertex algebra, let : → G be a group homomorphism with 0 = φ, and let (W, YW ) be a quasi-module-at-infinity for V viewed as a nonlocal vertex algebra. Assume that {YW (u, x) | u ∈ U } is ( )-local and YW (Rg u, x) = YW (u, (g)(x))
for g ∈ , u ∈ U,
where U is a -submodule and a generating subspace of V . Then (W, YW ) is a quasiV -module-at-infinity.
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Proof. First we prove that for u, v ∈ V , if YW (u, x) and YW (v, x) are quasi compatible, then YW (u n v, x) = YW (u, x)n YW (v, x)
for n ∈ Z.
(5.11)
Let p(x1 , x2 ) be a nonzero polynomial such that p(x1 , x2 )YW (u, x1 )YW (v, x2 ) ∈ Hom(W, W ((x1−1 , x2−1 ))). We have p(x0 + x, x)YE o (YW (u, x), x0 )YW (v, x) = ( p(x1 , x)YW (u, x1 )YW (v, x)) |x1 =x+x0 . On the other hand, there exists a nonzero polynomial q(x1 , x2 ) such that q(x0 + x, x)YW (Y (u, x0 )v, x) = (q(x1 , x)YW (u, x1 )YW (v, x)) |x1 =x+x0 . Then p(x0 + x)q(x0 + x, x)YW (Y (u, x0 )v, x) = p(x0 + x)q(x0 + x, x)YE o (YW (u, x), x0 )YW (v, x). Consequently, we get YW (Y (u, x0 )v, x) = YE o (YW (u, x), x0 )YW (v, x), proving (5.11). It follows from Proposition 3.14 and induction that {YW (v, x) | v ∈ V } is ( )-local. Suppose that YW (Rg u, x) = YW (u, (g)(x)) and YW (Rg v, x) = YW (v, (g)(x)) for some g ∈ , u, v ∈ V . By suitably choosing a nonzero polynomial p(x1 , x2 ) we have p(φ(g)−1 x0 + x, x)YW (Rg Y (u, x0 )v, x) = p(φ(g)−1 x0 + x, x)YW (Y (Rg u, φ(g)−1 x0 )Rg v, x) = p(x1 , x)YW (Rg u, x1 )YW (Rg v, x) |x1 =x+φ(g)−1 x0 = ( p(x1 , x)YW (u, (g)(x1 ))YW (v, (g)(x))) |x1 =x+φ(g)−1 x0 = p(φ(g)−1 x0 + x, x)YW (Y (u, x0 )v, (g)(x)), which implies YW (Rg Y (u, x0 )v, x) = YW (Y (u, x0 )v, (g)(x)), noticing that (g)(x + φ(g)−1 x0 ) = (x) + x0 . Then it follows from induction that YW (Rg v, x) = YW (v, (g)(x)) Thus W is a quasi V -module-at-infinity.
Now we have:
for all g ∈ , v ∈ V.
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Theorem 5.10. Let W be a vector space, let be a subgroup of G (the group of linear transformations on C), and let S be any -local subset of E o (W ). Set · S = span{Rg a(x) = a(g(x)) | g ∈ , a(x) ∈ S}. Then · S is -local and · S is a -vertex algebra with group homomorphisms R : → G L( · S) and φ : → C× defined by Rg (α(x)) = α(g(x)) for g ∈ , α(x) ∈ · S, φ(g(x)) = g0 for g(x) = g0 x + g1 ∈ .
(5.12) (5.13)
Furthermore, W is a quasi- · S-module-at-infinity with being the identity map. Proof. For g(x), h(x) ∈ , we have g(x1 ) − h(x2 ) = g0 x1 − (g0−1 h 0 x2 + g0−1 (h 1 − g1 )) = g0 (x1 − (g −1 h)(x2 )). With this, it is clear that · S is -local. By Theorem 5.4, · S generates a vertex algebra · S inside E o (W ) and W is a quasi module-at-infinity for · S. It follows from Lemma 3.13 and induction that · S is -stable. In view of Lemma 3.13, · S equipped with the action of and with the group homomorphism φ is a -vertex algebra. Furthermore, for α(x), β(x) ∈ · S, as YW (α(x), x1 ) = α(x1 ) and YW (β(x), x2 ) = β(x2 ), YW (α(x), x1 ) and YW (β(x), x2 ) are -local. For g ∈ , α(x) ∈ · S, we have YW (Rg α(x), x0 ) = YW (α(g(x)), x0 ) = α(g(x0 )) = YW (α(x), g(x0 )). It follows from Lemma 5.9 that W is a quasi-module-at-infinity for · S viewed as a -vertex algebra.
We shall need the following result: Lemma 5.11. Let W be a vector space and let a(x), b(x) ∈ E o (W ). Suppose that k r ∂ j −1 βi x2 1 , (5.14) i, j (x2 ) x1 δ − [a(x1 ), b(x2 )] = j! ∂ x2 x1 i=1 j=0
where β1 , . . . , βk are distinct nonzero complex numbers with β1 = 1 and i, j (x) ∈ E o (W ). Then (a(x), b(x)) is quasi local and a(x)n b(x) = 0 for n > r and a(x)n b(x) = 1,n (x) for 0 ≤ n ≤ r . Proof. Set p(x, z) = (x − β1 z)r +1 · · · (x − βk z)r +1 and q(x, z) = (x − β2 )r +1 · · · (x − βk z)r +1 . From (5.14) we have p(x1 , x2 )[a(x1 ), b(x2 )] = 0. Thus (a(x), b(x)) is quasi-local. Furthermore, we have p(x + x0 , x)YE o (a(x), x0 )b(x) = ( p(x1 , x)a(x1 )b(x)) |x1 =x+x0 . In view of Lemma 4.2 we have x1 − x x − x1 p(x1 , x)b(x)a(x1 ) − x0−1 δ p(x1 , x)a(x1 )b(x) x0−1 δ x0 −x0 x + x0 p(x1 , x)YE o (a(x), x0 )b(x). = x1−1 δ x1
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Multiplying both sides by x0−r −1 and using delta-function substitution we get x1 − x x − x1 q(x1 , x)b(x)a(x1 ) − x0−1 δ q(x1 , x)a(x1 )b(x) x0−1 δ x0 −x0 x + x0 q(x1 , x)YE o (a(x), x0 )b(x). = x1−1 δ x1 Taking Resx0 we get − q(x1 , x)[a(x1 ), b(x)] = =
Resx0 x1−1 δ
x + x0 x1
q(x1 , x)YE o (a(x), x0 )b(x)
q(x1 , x)a(x) j b(x)
j≥0
1 j!
∂ ∂x
j
x1−1 δ
x x1
. (5.15)
On the other hand, from (5.14) we have − q(x1 , x)[a(x1 ), b(x)] =
r j=0
1 q(x1 , x)1, j (x) j!
∂ ∂x
j
x1−1 δ
x x1
. (5.16)
Assume s j=0
1 q(x1 , x)A j (x) j!
∂ ∂x
j
x1−1 δ
x x1
= 0,
where A j (x) ∈ E o (W ) for 0 ≤ j ≤ s. As q(x, 1) and (x − 1) are relatively prime, we have 1 = q(x, 1) f (x) + (x − 1)s+1 g(x) for some f (x), g(x) ∈ C[x]. Then s
=
j=0 s j=0
A j (x)
1 j!
∂ ∂x
j
x1−1 δ
x x1
1 ∂ j x A j (x) q(x1 /x, 1) f (x1 /x) + (x1 /x − 1)s+1 g(x1 /x) x1−1 δ j! ∂ x x1
= 0, which implies A j (x) = 0 for 0 ≤ j ≤ s. Using this fact, combining (5.15) with (5.16) we obtain the desired relations.
Next, we study certain infinite-dimensional Lie algebras including the Lie algebra of pseudo-differential operators on the circle. Let g be a (possibly infinite-dimensional) Lie algebra equipped with a nondegenerate symmetric invariant bilinear form ·, ·. Associated with the pair (g, ·, ·), one has an (untwisted) affine Lie algebra gˆ = g ⊗ C[t, t −1 ] ⊕ Ck, where k is central and [a ⊗ t m , b ⊗ t n ] = [a, b] ⊗ t m+n + mδm+n,0 a, bk
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for a, b ∈ g, m, n ∈ Z. Defining deg(g ⊗ t m ) = −m for m ∈ Z and deg k = 0 makes gˆ a Z-graded Lie algebra. For a ∈ g, form the generating function a(x) = (a ⊗ t n )x −n−1 . n∈Z
We say that a gˆ -module W is of level ∈ C if k acts on W as scalar . A vacuum vector in a gˆ -module is a nonzero vector v such that (g ⊗ C[t])v = 0 and a vacuum gˆ -module is a gˆ -module W equipped with a vacuum vector which generates W . Let be any complex number. Denote by C the 1-dimensional (g ⊗ C[t] ⊕ Ck)-module with g ⊗ C[t] acting trivially and with k acting as scalar . Form the induced gˆ -module Vgˆ (, 0) = U (ˆg) ⊗U (g⊗C[t]⊕Ck) C . Set 1 = 1 ⊗ 1, which is a vacuum vector. From definition, Vgˆ (, 0) is a universal vacuum gˆ -module of level . Identify g as a subspace of Vgˆ (, 0) through the linear map a → a(−1)1. It is now well known (cf. [FZ]) that there exists one and only one vertexalgebra structure on Vgˆ (, 0) with 1 as the vacuum vector and with Y (a, x) = a(x) for a ∈ g. Defining deg 1 = 0 makes Vgˆ (, 0) a Z-graded gˆ -module and the vertex algebra Vgˆ (, 0) equipped with this Z-grading is a Z-graded vertex algebra. Let be a subgroup of Aut(g), preserving the bilinear form ·, ·. Each g ∈ canonically lifts to an automorphism of the Z-graded Lie algebra gˆ . Then acts on the vertex algebra Vgˆ (, 0) by automorphisms preserving the Z-grading. Let φ : → C× be a group homomorphism. For g ∈ , set Rg = φ(g)−L(0) g ∈ G L(Vgˆ (, 0)), where L(0) denotes the Z-grading operator. This defines a -vertex-algebra structure on Vgˆ (, 0). Consider the following completion of the Z-graded affine Lie algebra gˆ : gˆ (∞) = g ⊗ C((t −1 )) ⊕ Ck,
(5.17)
[a ⊗ p(t), b ⊗ q(t)] = [a, b] ⊗ p(t)q(t) + Rest p (t)q(t)a, bk
(5.18)
where
for a, b ∈ g, p(t), q(t) ∈ C((t −1 )). Proposition 5.12. Let g be a Lie algebra equipped with a nondegenerate symmetric invariant bilinear form ·, · and let be a group acting on g by automorphisms preserving the bilinear form ·, · and satisfying the condition that for any u, v ∈ g, [gu, v] = 0 and gu, v = 0
for all but finitely many g ∈ .
Let : → G be a group homomorphism. Define a new bilinear multiplicative operation [·, ·] on the vector space gˆ (∞) = g ⊗ C((t −1 )) ⊕ Ck by [a ⊗ p(t), k] = 0 = [k, a ⊗ p(t)] , [ga, b] ⊗ p(g(t))q(t) + Rest p (g(t))g (t)q(t)ga, bk [a ⊗ p(t), b ⊗ q(t)] = g∈
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for a, b ∈ g, p(t), q(t) ∈ C((t −1 )). Then the subspace, linearly spanned by the elements ga ⊗ p(g(t)) − a ⊗ p(t) for g ∈ , a ∈ g, p(t) ∈ C((t −1 )), is a two-sided ideal of the nonassociative algebra and the quotient algebra, which we denote by gˆ (∞)[ ], is a Lie algebra. Proof. Let act on the Lie algebra gˆ (∞) by for g ∈ , a ∈ g, p(t) ∈ C((t −1 )), λ ∈ C.
g(a ⊗ p(t) + λk) = ga ⊗ p(g(t)) + λk
It is straightforward to see that acts on gˆ (∞) by automorphisms. We have
[g(a ⊗ p(t)), b ⊗ q(t)] =
g∈
[ga ⊗ p(g(t)), b ⊗ q(t)]
g∈
=
[ga, b]⊗ p(g(t))q(t)+Rest q(t) p (g(t))g (t)ga, bk
g∈
for a, b ∈ g, p(t), q(t) ∈ C((t −1 )) and g(a ⊗ p(t) + λk) − (a ⊗ p(t) + λk) = ga ⊗ p(g(t)) − a ⊗ p(t). Now the assertions follow immediately from ([Li6], Lemma 4.1).
Let π : gˆ (∞) → gˆ (∞)[ ] be the natural map. For a ∈ g, set a (x) =
π(a ⊗ t n )x −n−1 ∈ gˆ (∞)[ ] [[x, x −1 ]].
n∈Z
Lemma 5.13. For g ∈ , a ∈ g, we have (ga) (x) = φ(g)a (g(x)).
(5.19)
For a, b ∈ g, we have g(x2 ) [a (x1 ), b (x2 )] = x1 g∈ ∂ −1 g(x2 ) + ga, bk , x1 δ ∂ x2 x1
where g(x) = (g)(x) ∈ G.
[ga, b] (x2 )x1−1 δ
(5.20)
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Proof. Let g(x) = g0 x +g1 ∈ G, where g0 ∈ C× , g1 ∈ C. Then g −1 (x) = g0−1 (x −g1 ) and −1 g x + g g (t) t − g1 g(x) 0 1 x −1 δ = x −1 δ = g0 t −1 δ = g0 t −1 δ . (5.21) x g0 x t t From the definition we have π(ga ⊗ t n ) = π(a ⊗ (g −1 (t))n ) for g ∈ , a ∈ g, n ∈ Z. Then we get (ga) (x) = π(ga ⊗ t n )x −n−1 = π(a ⊗ (g −1 (t))n )x −n−1 n∈Z
n∈Z
−1 g (t) −1 =π a⊗x δ x g(x) −1 = g0 π a ⊗ t δ t = g0 a (g(x)),
proving (5.19). As for (5.20), notice that g(t) t g(x2 ) t x2−1 δ = x1−1 δ x2−1 δ g(t)m t n x1−m−1 x2−n−1 = x1−1 δ x1 x2 x1 x2
m,n∈Z
and Rest
mg(t)m−1 g (t)t n x1−m−1 x2−n−1 = −Rest
m,n∈Z
ng(t)m t n−1 x1−m−1 x2−n−1
m,n∈Z
∂ −1 g(t) −1 x2 t δ = Rest x δ ∂ x2 1 x1 t ∂ −1 g(x2 ) . = x δ ∂ x2 1 x1
Now (5.20) follows.
Now we are in a position to present our main result of this section, which is an analogue of a theorem of [Li6]: Theorem 5.14. Let g, ·, ·, , be given as in Proposition 5.12 and let W be any gˆ (∞)[ ]-module of level ∈ C such that a (x) ∈ E o (W ) for a ∈ g. Then on W there exists one and only one structure of a quasi-module-at-infinity for Vgˆo (−, 0) viewed as a -vertex algebra with YW (a, x) = a (x) for a ∈ g, where go denotes the opposite Lie algebra of g. Proof. Set U = {a (x) | a ∈ g} ⊂ E o (W ). For a, b ∈ g, let g1 , . . . , gr ∈ such that [ga, b] = 0 and ga, b = 0 for g ∈ / {g1 , . . . , gr }. It follows from (5.20) that (x1 − g1 (x2 ))2 · · · (x1 − gr (x2 ))2 [a (x1 ), b (x2 )] = 0.
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Thus U is a -local subspace of E o (W ). From (5.19), · U = U . By Theorem 5.10, U generates a -vertex algebra U with W as a quasi-module-at-infinity with YW (α(x), x0 ) = α(x0 ). Combining (5.20) with Lemma 5.11 we get a (x)0 b (x) = −[a, b] (x), a (x)1 b (x) = −a, b1W , and a (x)n b (x) = 0
(5.22)
for n ≥ 2. In view of the universal property (cf. [P,Li6]) of Vgˆo (−, 0), there exists a (unique) vertex-algebra homomorphism from Vgˆo (−, 0) to U , sending a to a (x) for a ∈ g. Consequently, W is a quasi-module-at-infinity for Vgˆo (−, 0) viewed as a vertex algebra. Furthermore, for g ∈ , a ∈ g we have YW (Rg a, x) = φ(g)−1 (ga) (x) = a (g(x)) = YW (a, g(x)). As g generates Vgo (−, 0) as a vertex algebra, it follows from Lemma 5.9 that W is a quasi-module-at-infinity for Vgˆo (−, 0) viewed as a -vertex algebra.
Now, let be any abstract group. As in [Li6], we define an associative algebra gl with a C-basis consisting of symbols E α,β for α, β ∈ and with E α,β E µ,ν = δβ,µ E α,ν
for α, β, µ, ν ∈ ,
and we equip gl with a nondegenerate symmetric associative bilinear form defined by E α,β , E µ,ν = δα,ν δβ,µ
for α, β, µ, ν ∈ .
Defining Tα ∈ G L(gl ) for α ∈ by Tα E µ,ν = E αµ,αν
for α, β, µ, ν ∈
(cf. [GKK]) we have a group action of on gl by automorphisms preserving the bilinear form. We can also view gl as a Lie algebra with ·, · an invariant bilinear form. Furthermore, for any α, β, µ, ν ∈ , we have [Tg E α,β , E µ,ν ] = 0
and
Tg E α,β , E µ,ν = 0
for all but finitely many g ∈ . Associated with the pair (gl , ·, ·), we have an (unt and its completion gl (∞). wisted) affine Lie algebra gl Let : → G be a group homomorphism. For α ∈ , we set α(x) = (α) = α0 x + α1 ∈ G, (∞)[ ]. where α0 , α1 ∈ C with α0 = 0. From Proposition 5.12 we have a Lie algebra gl For α, β ∈ , we have [E α,e (x1 ), E β,e (x2 )] g(x2 ) ∂ −1 g(x2 ) −1 + E gα,g , E β,e k = [E gα,g , E β,e ](x2 )x1 δ x δ x1 ∂ x2 1 x1 g∈ −1 β(x2 ) α (x2 ) − E β,α −1 (x2 )x1−1 δ = E βα,e (x2 )x1−1 δ x1 x1 ∂ −1 β(x2 ) . (5.23) +δαβ,e k x δ ∂ x2 1 x1
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(∞)[ ] [[x, x −1 ]]. Note that For α ∈ , denote by Aα (x) the image of E α,e (x) in gl E β,α −1 (x) = (Tα −1 E αβ,e )(x) = α0 E αβ,e (α −1 (x)). By (5.23) we have −1 β(x2 ) α (x2 ) − α0 Aαβ (α −1 (x2 ))x1−1 δ x1 x1 ∂ −1 β(x2 ) +δαβ,e k x δ . (5.24) ∂ x2 1 x1
[Aα (x1 ), Aβ (x2 )] = Aβα (x2 )x1−1 δ
(∞)[ ]-module of level ∈ C such that E α,β (x) ∈ E o (W ) for all Let W be any gl α, β ∈ . In view of Theorem 5.14, there exists one and only one quasi-module-atinfinity structure on W for the -vertex algebra V gl o (−, 0).
Example 5.15. Consider the special case with = Z. (The associative algebra glZ is simply gl∞ , the associative algebra of doubly infinite complex matrices with only finite many nonzero entries.) Let φ be the group embedding of Z into G defined by φ(n)(x) = x + n for n ∈ Z. We have x2 + n x2 − m −1 −1 [Am (x1 ), An (x2 )] = Am+n (x2 )x1 δ − Am+n (x2 − m)x1 δ x1 x1 ∂ −1 x2 + n +δm+n,0 k x δ ∂ x2 1 x1 x2 + n x1 + m −1 −1 − Am+n (x1 )x2 δ = Am+n (x2 )x1 δ x1 x2 ∂ −1 x2 + n (5.25) +δm+n,0 k x δ ∂ x2 1 x1 for m, n ∈ Z. This is exactly the relation of the Lie algebra of pseudo-differential operators on the circle. This Lie algebra (without central extension) was studied in [GKK] in terms of a notion called -conformal algebra. Note that this Lie algebra does not admit nontrivial modules of highest weight type. References [BK] [D] [EK] [FHL] [FLM] [FZ] [HL] [GKK]
Bakalov, B., Kac, V.: Field algebras. Internat. Math. Res. Notices 3, 123–159 (2003) Drinfeld, V.G.: Hopf algebras and quantum yang-baxter equation. Soviet Math. Dokl. 32, 254–258 (1985) Etingof, P., Kazhdan, D.: Quantization of lie bialgebras, V. Selecta Mathematica (N. S.) 6, 105–130 (2000) Frenkel, I., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104, 1993 Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Boston: Academic Press, 1988 Frenkel, I., Zhu, Y.-C.: Vertex operator algebras associated to representations of affine and virasoro algebras. Duke Math. J. 66, 123–168 (1992) Huang, Y.-Z., Lepowsky, J.: A theory of tensor products for modules categories for a vertex operator algebra, I. Selecta Mathematica (N. S.) 1, 699–756 (1995) Golenishcheva-Kutuzova, M., Kac, V.G.: -conformal algebras. J. Math. Phys. 39, 2290–2305 (1998)
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[K]
Kac V.G.: Vertex Algebras for Beginners. University Lecture Series, vol. 10, Providence, RI: Amer. Math. Soc., 1997 Karel, M., Li, H.-S.: Some quantum vertex algebras of Zamolodchikov-Faddeev type. Preprint, arXiv:0801.2901 Khoroshkin, S.: Central extension of the Yangian Double. arXiv: q-alg/9602031 Khoroshkin, S., Tolstoy, V.: Yangian double. Lett. Math. Phys. 36, 373–402 (1996) Lepowsky, J., Li, H.-S.: Introduction to Vertex Operator Algebras and Their Representations. Progress in Math. 227, Boston: Birkhäuser, 2004 Li, H.-S.: Regular representations of vertex operator algebras. Commun. Contemp. Math. 4, 639–683 (2002) Li, H.-S.: Axiomatic G 1 -vertex algebras. Commun. Contemp. Math. 5, 281–327 (2003) Li, H.-S.: A new construction of vertex algebras and quasi modules for vertex algebras. Adv. Math. 202, 232–286 (2006) Li, H.-S.: Nonlocal vertex algebras generated by formal vertex operators. Selecta Mathematica (N. S.) 11, 349–397 (2005) Li, H.-S.: Constructing quantum vertex algebras. Int. J. Math. 17, 441–476 (2006) Li, H.-S.: On certain generalizations of twisted affine lie algebras and quasi-modules for -vertex algebras. J. Pure Appl. Alg. 209, 853–871 (2007) Primc, M.: Vertex algebras generated by lie algebras. J. Pure Appl. Alg. 135, 253–293 (1999) Smirnov, F.: Dynamical symmetries of massive integrable models. J. Modl. Phys. A 7(Suppl. 1B), 813–838 (1992)
[KL] [Kh] [KT] [LL] [Li1] [Li2] [Li3] [Li4] [Li5] [Li6] [P] [S]
Communicated by Y. Kawahigashi
