Commun. Math. Phys. 271, 1–53 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0158-2
Communications in
Mathematical Physics
Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams László Erd˝os1, , Manfred Salmhofer2,3 , Horng-Tzer Yau4, 1 Institute of Mathematics, University of Munich, Theresienstr. 39, 80333 Munich, Germany.
E-mail:
[email protected]
2 Max–Planck Institute for Mathematics, Inselstr. 22, 04103 Leipzig, Germany 3 Theoretical Physics, University of Leipzig, Postfach 100920, 04009 Leipzig, Germany 4 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
Received: 6 December 2005 / Accepted: 13 June 2006 Published online: 31 January 2007 – © Springer-Verlag 2007
Abstract: We consider random Schrödinger equations on Rd for d ≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0 . The space and time variables scale as x ∼ λ−2−κ/2 , t ∼ λ−2−κ with 0 < κ < κ0 (d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψt converges weakly to the solution of a heat equation in the space variable x for arbitrary L 2 initial data. The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper [10] the analysis of the non-repetition diagrams was presented. In this paper we complete the proof by estimating the recollision diagrams and showing that the main terms, i.e. the ladder diagrams with renormalized propagator, converge to the heat equation.
1. Introduction The Schrödinger equation is time reversible and has no dissipation. The long time dynamics of a quantum particle in a small random environment nevertheless exhibits a stochastic behavior that can be described by a diffusion equation. Our goal is to establish this fact rigorously in the weak coupling regime. We have announced in [10] that under a scaling of space and time with inverse powers of the coupling constant λ, the Wigner distribution of the solution to the random Schrödinger equation converges to the solution of a heat equation as λ → 0. Our approach is based on graphical expansion methods coupled with a certain truncation scheme. The first part of the proof was given in [10]; the current paper contains the second and final part. To help the orientation of the reader, we now summarize the important notations and the main result below. Partially supported by NSF grant DMS-0200235 and EU-IHP Network “Analysis and Quantum” HPRNCT-2002-0027. Partially supported by NSF grant DMS-0602038.
2
L. Erd˝os, M. Salmhofer, H.-T. Yau
The quantum dynamics of a single particle in a random potential is given by the Schrödinger equation i∂t ψt = H ψt ,
ψt ∈ L 2 (Rd ), t ∈ R.
(1.1)
The Hamiltonian is a Schrödinger operator, 1 H := − + λV, 2
(1.2)
acting on L 2 (Rd ) with a random potential V = Vω (x) and a small positive coupling constant λ. The potential is given by Vω (x) := B(x − y)dμω (y), (1.3) Rd
where B is a single site potential profile and μω is a Poisson point process on Rd with homogeneous unit density and with independent, identically distributed random coupling constants. More precisely, for almost all realizations ω consists of a countable, locally finite collection of points, {yγ (ω) : γ = 1, 2, . . .}, and random charges {vγ (ω) : γ = 1, 2, . . .} such that the random measure is given by μω =
∞
vγ (ω)δ yγ (ω) ,
(1.4)
γ =1
where δ y denotes the Dirac mass at y ∈ Rd . The Poisson process {yγ (ω)} is independent of the charges {vγ (ω)}. The charges are real i.i.d. random variables with distribution Pv and with moments m k := Ev vγk satisfying m 2 = 1, m 2d < ∞,
m 1 = m 3 = m 5 = 0.
(1.5)
The expectation with respect to the random process {yγ , vγ } is denoted by E. For the single-site potential, we assume that B is a spherically symmetric Schwarz function with 0 in the support of its Fourier transform. A quantum wave ψ ∈ L 2 (Rd ) function can be represented on the phase space by its Wigner transform η η (1.6) Wψ (x, v) := e2πiv·η ψ(x + )ψ(x − )dη, 2 2 with the convention that integrals without explicit domains denote integration over Rd with respect to the Lebesgue measure. Define the rescaled Wigner distribution as X ,V . (1.7) Wψε (X, V ) := ε−d Wψ ε The Fourier transform of the kinetic energy (dispersion relation) is e( p) := 21 p 2 , the 1 1 ∇e( p) = 2π p. velocity is given by 2π d For any function h : R → C and energy value e ≥ 0 we introduce the notation dν(q) , (1.8) [h](e) := h(v)δ(e − e(v))dv := h(q) |∇e(q)| e
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
3
where dν(q) is the restriction of the Lebesgue measure onto the energy surface e := √ {q : e(q) = e} that is the ball of radius 2e. The normalization of the measure [·]e is given by d
[1](e) := cd−1 (2e) 2 −1 , where cd−1 is the volume of the unit sphere S d−1 . We consider the scaling t = λ−κ λ−2 T , x = λ−κ/2 λ−2 X = X/ε, ε = λκ/2+2
(1.9)
(1.10)
with some κ ≥ 0. On the kinetic scale, κ = 0, the limiting equation is the linear Boltzmann equation with a collision kernel σ (u, v) := 2π | B(u − v)|2 δ(e(u) − e(v)).
(1.11)
The generator of the corresponding momentum jump process v(t) on the energy surface e , e > 0, is L e f (v) := du σ (u, v)[ f (u) − f (v)], e(v) = e. (1.12) The choice κ > 0 corresponds to the long time limit of the Boltzmann equation with diffusive scaling. The Markov process {v(t)}t≥0 with generator L e is exponentially mixing (see Lemma A.1 in the Appendix). Let ∞
1 Di j (e) := Ee v (i) (t)v ( j) (0) dt, v = (v (1) , . . . , v (d) ), i, j = 1, 2, . . . d, 2 (2π ) 0 be the velocity autocorrelation matrix, where Ee denotes the expectation with respect to this Markov process in equilibrium. By the spherical symmetry of B and e(U ), the autocorrelation matrix is constant times the identity: ∞
1 Di j (e) = De δi j , De := Ee v(t) · v(0) dt . (1.13) 2 (2π ) d 0 The main result is the following theorem. Theorem 1.1. Let d ≥ 3 and ψ0 ∈ L 2 (Rd ) be a normalized initial wave function. Let ψ(t) := exp(−it H )ψ0 solve the Schrödinger equation (1.1). Let O(x, v) be a Schwarz function on Rd × Rd . For any e > 0, let f be the solution to the heat equation ∂T f (T, X, e) = De X f (T, X, e) with the initial condition
(1.14)
0 (v)|2 (e). f (0, X, e) := δ(X ) |ψ
Then there exist 0 < κ0 (d) ≤ 2 such that for 0 < κ < κ0 (d) and for ε and λ related by (1.10), the rescaled Wigner distribution satisfies ε lim dX dv O(X, v)EWψ(λ (X, v) = dX dv O (X, v) f (T, X, e(v)), −κ−2 T ) λ→0
(1.15) and the limit is uniform on T ∈ [0, T0 ] with any fixed T0 . In d = 3 one can choose κ0 (3) = 1/500.
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L. Erd˝os, M. Salmhofer, H.-T. Yau
Our result shows that the quantum dynamics on the time scale t ∼ λ−2−κ is given by a heat equation and the diffusion coefficient can be computed from the long time behavior of the underlying Boltzmann dynamics. Heuristically, this statement can be understood from two facts. First, the Boltzmann equation correctly describes the limit of the quantum evolution on the kinetic time scale t ∼ λ−2 , λ → 0 (see [18, 7, 5 and 15]). Second, the long time limit of the linear Boltzmann evolution is a diffusion. This two step limiting argument is, however, misleading (e.g. in d = 2, localization is expected to occur at all values of λ, so that no diffusion occurs). In higher dimensions, quantum correlations that are small on the kinetic scale and are neglected in the first limit may become important on the longer time scale, too. To prove Theorem 1.1, we need to control the full quantum dynamics up to the time scale t ∼ λ−2−κ and prove that the quantum correlations are not sufficiently strong to destroy the heuristic picture. The approach of this paper applies also to lattice models and yields a derivation of Brownian motion from the Anderson model [8, 9], where the Hamiltonian (1.2) is defined on 2 (Zd ). The random potential is given by vγ δ(x − γ ), x ∈ Zd , (1.16) Vω (x) = γ ∈Zd
where δ(·) is the lattice delta function. The dynamics of the Anderson model was postulated by Anderson [3] to be localized for large coupling constant λ and extended for small coupling constant (away from the band edges and in dimension d ≥ 3). The localization conjecture was first established rigorously by Goldsheid, Molchanov and Pastur [13] in one dimension, by Fröhlich-Spencer [12], and later by Aizenman-Molchanov [1] in several dimensions, and many other works have since contributed to this field. The progress for the extended state regime, however, has been limited. It was proved by Klein [14] that all eigenfunctions are extended on the Bethe lattice (see also [2, 11]). In Euclidean space, Schlag, Shubin and Wolff [17] proved that the eigenfunctions cannot be localized in a region smaller than λ−2+δ for some δ > 0 in d = 2. Chen [5] extended this result to all dimensions d ≥ 2 and δ = 0 (with logarithmic corrections). Extended states for a Schrödinger equation with a sufficiently decaying random potential were proven by Rodnianski and Schlag [16] and Bourgain [4] (see also [6]). However, all known results for Anderson-type models in Euclidean space are in regions where the dynamics have typically finitely many effective collisions. Under the diffusive scaling of this paper, see (1.10), the number of effective scatterings is a negative fractional power of the scaling parameter. In particular, it goes to infinity in the scaling limit, as it should be the case if we aim to obtain a Brownian motion. 2. Summary of Part I 2.1. Notations. We introduce a few notations. The letters x, y, z will denote configuration space variables, while p, q, r, u, v, w will be used for d-dimensional momentum variables. The norm · denotes the standard L 2 (Rd ) norm and we set
x m ∂ α f (x) ∞
f m,n := |α|≤n
with x := (2 + x 2 )1/2 (here α is a multiindex). The bracket (· , ·) denotes the standard scalar product on L 2 (Rd ) and · , · will denote the pairing between the Schwarz space and its dual on the phase space Rd × Rd . The Fourier transform is denoted by hat. For
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
5
functions on the phase space, f (x, v), x, v ∈ Rd , Fourier transform will always be understood in the first variable only: f (ξ, v) := e−2πiξ ·x f (x, v)dv. By the regularization argument in Sect. 3.2 of [10], we can cutoff the high momentum regime of ψ0 and we can multiply B by a λ-dependent cutoff function in momentum space at a negligible error. When dealing with estimates for any fixed λ > 0, we can thus assume that supp B( p) ⊂ {| p| ≤ λ−δ }
0 is compact, supp ψ
(2.1)
for any fixed δ > 0. Universal constants and constants that depend only on the dimension d, on the final time T0 and on ψ0 and B will be denoted by C. The same applies to the hidden constants in the O(·) and o(·) notations. In [10] the self-energy operator was defined as the multiplication operator in momentum space θ ( p) := (e( p)),
(α) := lim ε (α) , ε→0+
for any r with e(r ) = α, where
ε (α, r ) :=
ε (α) := ε (α, r )
| B(q − r )|2 dq . α − e(q) + iε
(2.2)
(2.3)
The function (α) is Hölder continuous with exponent 21 , and it decays as α −1/2 (Lemma 3.1 and 3.2 in [10]). If we write (α) = R(α) − iI(α), where R(α) and I(α) are real functions, and recall I m(x + i0)−1 = −π δ(x), we have I(α) = −Im (α) = π δ(e(q) − α)| B(q − r )|2 dq (2.4) for any r satisfying α = e(r ). The dispersion relation was renormalized by adding the self-energy term: ω( p) := e( p) + λ2 θ ( p), and the Hamiltonian was rewritten as
, H = H0 + V
H0 := ω( p),
:= λV − λ2 θ ( p). V
(2.5)
The renormalization compensates for the immediate recollisions in the Duhamel expansion (see Sect. 3). The rate of the immediate recollisions is of order λ2 , thus their total effect is λ2 t 1. The renormalization removes this instability. We note that ω( p) is not the self-consistently renormalized dispersion relation, but only its approximation up to O(λ4 ). Since this error is negligible on our time scale due to λ4 t 1, we use ω to simplify the technicalities associated with the analysis of the self-consistent dispersion relation. We define the renormalized propagator (with η-regularization): Rη (α, v) :=
1 . α − ω(v) + iη
In Appendix B.1 we prove the following estimates on the renormalized propagator.
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L. Erd˝os, M. Salmhofer, H.-T. Yau
Lemma 2.1. Suppose that λ2 ≥ η ≥ λ2+4κ with κ ≤ 1/12. Then we have, |h( p − q)|d p C h 2d,0 | log λ| log α , ≤ √ |α − ω( p) + iη| α 1/2 |q| − 2|α| and for 0 ≤ a < 1,
(2.6)
Ca h 2d,0 λ−2(1−a) |h( p − q)|d p , ≤ √ |α − ω( p) + iη|2−a α a/2 |q| − 2|α|
(2.7)
|h( p − q)|d p Ca h 2d,0 η−2(1−a) . ≤ √ 2−a |α − e( p) + iη| α a/2 |q| − 2|α|
(2.8)
For a = 0 and with h := B, the following more precise estimate holds. There exists a constant C0 , depending only on finitely many B k,k norms, such that 2
λ | B( p − q)|2 d p (2.9) ≤ 1 + C0 λ−12κ λ + |α − ω(q)|1/2 . 2 |α − ω( p) − iη|
(see (2.5)) by the 2.2. The expansion. We expand the unitary kernel of H = H0 + V Duhamel formula and after taking the expectation, we organize the expansion into sums of Feynman diagrams. In order to avoid the infinite summations (1.4) in the expansion, we have reduced the problem to a large finite box, L = [−L/2, L/2]d ⊂ Rd with periodic boundary conditions (see Sect. 3.3 of [10]). The infinite volume Poisson process μω was replaced with μω =
M γ =1
vγ δ yγ ,
where M is a Poisson random variable with mean | L |, the points {yγ }γM=1 are uniformly distributed on L and the real charges vγ have distribution Pv . All random variables are independent. Lemma 3.4 of [10] guarantees that these modifications have no effect on the final result if L → ∞ is taken before any other limit. After the Duhamel expansion, taking the expectation and letting L → ∞, we regain the infinite volume formulas for the Feynman graphs. In [10] we used primes to denote the restricted quantities, but to avoid the heavy notation here we will omit them, except when stating the theorems. All quantities throughout this section are understood on L with M random points. We recall the Duhamel formula from Sect. 4 of [10]. For any fixed integer N ≥ 1, ψt := e−it H ψ0 =
N −1
ψn (t) + N (t),
(2.10)
n=0
with
t
e−isn H0 V
... V
e−is1 H0 ψ0 , ψn (t) := (−i)n [ds j ]n+1 e−isn+1 H0 V 1 0 t
ψ N −1 (s) N (t) := (−i) ds e−i(t−s)H V 0
(2.11) (2.12)
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
with the notation 0
t
[ds j ]n1 :=
t
...
0
⎛ t
⎝
0
n
⎞ ⎛ ds j ⎠ δ ⎝t −
j=1
n
7
⎞ sj⎠ .
j=1
M
= −λ2 θ ( p) + γ =1 λVγ with Vγ (x) := vγ B(x − yγ ), the terms in Substituting V (2.11) and (2.12) are summations over collision history. Denote by ˜ n , n ≤ ∞, the set of sequences γ˜ = (γ˜1 , γ˜2 , . . . , γ˜n ), and by Wγ˜ the associated potential λVγ˜ Wγ˜ := −λ2 θ ( p)
γ˜ j ∈ {1, 2, . . . , M} ∪ {ϑ},
(2.13)
if γ˜ ∈ {1, . . . , M} if γ˜ = ϑ.
The tilde refers to the fact that the additional {ϑ} symbol is also allowed. An element γ˜ j of the sequence γ˜ ∈ ˜ is called potential label and j is called potential index if γ˜ j ∈ {1, 2, . . . , M}, otherwise they are called ϑ-label and ϑ-index, respectively. A potential label carries a factor λ, a ϑ-label carries λ2 . For any γ˜ ∈ ˜ n we define the following fully expanded wave function with truncation t ψ∗t,γ˜ := (−i)n−1 [ds j ]n1 Wγ˜n e−isn H0 Wγ˜n−1 . . . e−is2 H0 Wγ˜1 e−is1 H0 ψ0 (2.14) 0
and without truncation t −isn+1 H0 ψt,γ˜ := (−i)n [ds j ]n+1 Wγ˜n e−isn H0 Wγ˜n−1 . . . e−is2 H0 Wγ˜1 e−is1 H0 ψ0 . 1 e 0
(2.15) In the notation the star (∗) will always refer to truncated functions. Each term ψt,γ˜ along the expansion procedure is characterized by its order n and by a sequence γ˜ ∈ ˜ n . The main terms are given by non-repetitive sequences that contain only potential labels, i.e. we define knr := γ = (γ1 , . . . , γk ) : γ j ∈ {1, . . . , M}, γi = γ j if i = j ⊂ ˜ n . (2.16) The sum of the corresponding elementary wave functions is denoted by nr ψt,k := ψt,γ .
(2.17)
γ ∈knr
The rate of collisions is O(λ2 ), thus the total number of collisions is typically of order k ∼ λ2 t. We thus set K := [λ−δ (λ2 t)]
(2.18)
([ · ] denotes integer part) to be an upper threshold for the number of collisions in the expanded terms. Here δ = δ(κ) > 0 is a small positive number to be fixed later on.
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L. Erd˝os, M. Salmhofer, H.-T. Yau
2.3. Structure of the proof. In Sect. 5 of [10] the Main Theorem was proved from three key theorems. For completeness, we repeat these three statements here. Recall that the prime indicates restriction to L and hence dependence on L , M. Theorem 2.2 (L 2 -estimate of the error terms). Let t = O(λ−2−κ ) and K be given by (2.18). If κ < κ0 (d) and δ is sufficiently small (depending only on κ), then K −1 2 nr lim lim E ψt − ψt,k = 0.
λ→0 L→∞
L
k=0
In d = 3 dimensions, one can choose κ0 (3) =
1 500 .
2 Theorem 2.3 (Only the ladder diagram contributes). Let κ < 34d+39 , ε = λ2+κ/2 , 2+κ −2−κ ), and K be given by (2.18). For a sufficiently small positive δ η = λ , t = O(λ and for any 1 ≤ k ≤ K we have 1 17 13 nr 2 (2.19)
L = Vλ (t, k) + O λ 3 −( 3 d+ 2 )κ−O(δ) , lim E ψt,k L→∞ 1 17 13 L , E W ε nr L = Wλ (t, k, O) + O λ 3 −( 3 d+ 2 )κ−O(δ) (2.20) lim O ψ L→∞
t,k
as λ 1. Here λ2k e2tη Vλ (t, k) := (2π )2 ×
k+1
∞
−∞
dαdβ ei(α−β)t
Rη (α, p j )Rη (β, p j )
j=1
⎛ ⎝
k+1
⎞ 0 ( p1 )|2 d p j ⎠ |ψ
j=1 k
| B( p j+1 − p j )|2 ,
(2.21)
j=1
∞ λ2k e2tη i(α−β)t dαdβ e Wλ (t, k, O) := dξ (2π )2 −∞ ⎛ ⎞ k+1 vk+1 )W ε (ξ, v1 ) × ⎝ dv j ⎠ O(ξ, ψ0 j=1
×
k+1 j=1
k εξ εξ Rη β, v j − Rη α, v j + | B(v j − v j+1 )|2 . (2.22) 2 2 j=1
The definition (2.22) does not apply literally to the free evolution term k = 0; this term is defined separately: 2 v)W 0 (εξ, v). Wλ (t, k = 0, O) := dξ dv eitεv·ξ e2tλ Im θ(v) O(ξ, (2.23) Theorem 2.4 (The ladder diagram converges to the heat equation). Under the conditions of Theorem 2.3 and setting t = λ−2−κ T , we have K −1 lim Wλ (t, k, O) = dX dv O(X, v) f (T, X, e(v)), (2.24) λ→0
k=0
where f is the solution to the heat equation (1.14).
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
9
The main result of [10] was the proof of Theorem 2.3. In Sects. 3–5 of this paper we prove Theorem 2.2 and in Sect. 6 we prove Theorem 2.4. We now explain the ideas behind these theorems. The actual estimates are somewhat weaker than the heuristics predicts. The proof of Theorem 2.2 consists in controlling the wave functions of collision histories that contain ϑ-labels or repeated potential labels. The repeated potential labels correspond to recollisions with the same obstacle. Immediate recollision with the same obstacle occurs with an amplitude O(λ2 ). Over the total history of the evolution, this would yield a large contribution of order λ2 t 1. However, the wave functions of these collision histories will be resummed with those containing ϑ-labels. Thanks to the choice of the renormalization counterterm, λ2 θ ( p), the contributions of the immediate recollisions and the ϑ-labels cancel each other up to leading order. Each resummed term thus has an amplitude O(λ4 t). The full propagator in the error term, N (t) (2.12), however, will be estimated by unitarity (4.4). This estimate effectively loses an extra t factor. To compensate for it, we have to continue the expansion up to two immediate recollisions in the error term. The amplitudes of the non-immediate recollisions are much smaller and they can be estimated individually. Heuristically, the probability of such recollisions can be understood in classical mechanics. Since the mean free path is λ−2 , returning to an already visited obstacle after visiting another obstacle at distance λ−2 has probability O(λ2d ). Another scenario is when the particle collides with obstacle γ1 , then it bounces back from a nearby obstacle γ2 , |γ1 − γ2 | = O(1), and then it recollides with γ1 . This situation is atypical and it is penalized by O(λ4 ) because the time elapsed between these collisions is O(1) while the collision cross-section is O(λ2 ). In conclusion, the probability of a non-immediate recollision among the total k ∼ λ2 t collisions is at most O(kλ4 ), thus the total effect of these recollisions on our time scale is negligible, even when multiplied with the additional factor t from the unitarity estimates for N (t). This outline neglects the key analytic difficulty originated from the growth of the combinatorics of Feynman diagrams. The amplitude of non-repetition wave functions can be written as a sum of k! Feynman diagrams. Only one of them, the ladder diagram, contributes to the heat equation. All other diagrams can be estimated by O(λ2 ) due to phase cancellations. This estimate is not sufficient to sum up all diagrams since their number, k! ∼ exp(λ−const ), is exponentially large. The size of a few combinatorially simple diagrams is indeed O(λ2 ), but much stronger estimates were obtained in [10] as the combinatorial complexity of the diagram increases. This improvement balances the increased combinatorial factor for more complicated diagrams and it allows us to control the expansion for non-repetition wave functions for time scale t ∼ λ−2−κ . In this paper, we extend the classification scheme to include all diagrams arising from the Duhamel expansion. Thanks to the stopping rules, this part involves only a few extra collisions. The main idea of this paper is to design a surgery of Feynman diagrams so that a general diagram can be decomposed into a repetition and a non-repetition part: the repetition part involves only a few variables and the integration can be estimated accurately; the non-repetition part is reduced to Theorem 2.3. The errors from the surgery are controlled by the small factors from the repetition part. This renders all repetition diagrams negligible. Thus we prove that among all diagrams only the ladder diagrams without repetition contribute to the final heat equation. In practice, the scheme used in this paper is much more complicated than is stated here. But this description gives a good first idea. Several estimates in Sects. 4 and 5 are also used in the proof for the lattice case [9]. We thus present a unified setup in these sections that is directly applicable for both cases.
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L. Erd˝os, M. Salmhofer, H.-T. Yau
Finally, the proof of Theorem 2.4 is a fairly explicit but delicate calculation involving singular integrals. The proof shows how the long time evolution of the Boltzmann equation emerges from the ladder diagrams. 3. The Stopping Rules We use the Duhamel expansion to identify the non-repetition error terms to be estimated in Theorem 2.2. This method allows for the flexibility that at every new term of the expansion we perform the separation into elementary waves, ψ∗s, γ , and we can decide whether we want to stop (keeping the full propagator as in (2.12)) or we continue to expand that term further. This decision will depend on the collision history, γ , and it will be given by a precise algorithm, the stopping rules. The key idea is that once the collision history γ˜ is “sufficiently” atypical, i.e. it contains either atypical recollision or too many collisions, we stop the expansion for that elementary wave function immediately to reduce the number and the complexity of the expanded terms. Not every recollision is atypical. An immediate second collision with the same obstacle contributes to the main term; this is actually the reason why the dispersion relation had to be corrected with the self-energy λ2 θ ( p). In a sequence γ˜ we thus identify the immediate recollisions inductively starting from γ˜1 (due to their graphical picture, they are also called gates). The gates must involve potential labels and not ϑ. For example, the sequence γ˜ = (a, ϑ, a, b, b, c, d, d, ϑ, ϑ, e, e, f ) has three gates (see Fig. 1). In the sequence (a, b, b, c, c, c) there are two gates. Any potential label which does not belong to a gate will be called a skeleton label. The index j of a skeleton label γ j in γ˜ is called a skeleton index. The set of skeleton indices is S(γ˜ ). Similar terminology is used for the gates. In the first example 1, 3, 6, 13 are skeleton indices and a, a, c, f are skeleton labels, in the second example 1, 6 are skeleton indices and a, c are skeleton labels. The ϑ terms are never part of the skeleton. This definition is recursive so we can identify skeleton indices successively along the expansion procedure. Notice that a skeleton index may become a gate index at a later stage of the expansion, but never the other way around. The formal definition is as follows. Let In := {1, 2, . . . , n}. Definition 3.1 (Skeleton labels and indices). Let γ˜ = (γ˜1 , γ˜2 , . . . , γ˜n ) ∈ ˜ n and let γ˜ ∗ := (γ˜1 , γ˜2 , . . . , γ˜n−1 ) be its truncation. The set of skeleton indices of γ˜ , S(γ˜ ) ⊂ In , is defined inductively (on the length of γ˜ ) as follows. If γ˜ ∈ ˜ 1 , then S(γ˜ ) := {1} if γ˜1 = ϑ and S(γ˜ ) := ∅ otherwise. Furthermore, for any γ˜ ∈ ˜ n , n ≥ 2, let ⎧ if γ˜n = ϑ S(γ˜ ∗ ) ⎨ if n − 1 ∈ S(γ˜ ∗ ) and γ˜n = γ˜n−1 S(γ˜ ) := S(γ˜ ∗ ) \ {n − 1} ⎩ S(γ˜ ∗ ) ∪ {n} if γ˜n = ϑ and [γ˜n = γ˜n−1 or n − 1 ∈ S(γ˜ ∗ )]. a 1
θ
a
2
3
b 4
b
c 5 6
d 7
d
θ 8 9
Non-skeleton index Skeleton index Fig. 1. Gates and skeletons
θ
e
10 11
e
f 12 13
Gate
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
11
Finally, γn is called a skeleton label if n ∈ S(γ ). For any γ˜ ∈ ˜ n , let k(γ˜ ) := |S(γ˜ )|, be the number of skeleton indices in γ˜ , let Inθ (γ˜ ) := { j : γ˜ j = ϑ} be the set of θ -indices and t (γ˜ ) := |Inθ (γ˜ )|. Denote the total λ2 -power collected from non-skeleton indices by r (γ˜ ) :=
1 [n − k(γ˜ )] + t (γ˜ ). 2
(3.25)
Notice that r (γ˜ ) is integer. The exact stopping rule requires somewhat tedious definitions of different types of elementary wave functions. First we give these definitions intuitively, state our final representation formula for ψt using these concepts, then we give the precise definitions and prove the formula. Sequences where the only repetitions in potential labels occur within the gates are called non-repetitive sequences. A special case is the set of non-repetitive sequences, knr , without gates and θ -labels. The repetitive sequences are divided into the following categories (Fig. 2). If two non-neighboring skeleton labels coincide, then the collision history includes a genuine (non-immediate) recollision. If a skeleton label coincides with a gate label, then we have a triple collision of the same obstacle. If two neighboring skeleton labels coincide and are not immediate recollisions because there are gates or ϑ’s in between, then we have a nest. We stop the expansion at an elementary truncated wave function (2.14) characterized by γ˜ , if any of the following happens (precise definitions will be given in Definition 3.3). •
The number of skeleton indices in γ˜ reaches K (see (2.18)). We denote the sum of the truncated elementary non-repetitive wave functions up to time s with at most one λ2 power from the non-skeleton labels or ϑ’s and with K skeleton indices by (≤1),nr ψ∗s,K . The superscript (≤1) refers to the number of collected λ2 powers from non-skeleton labels. a
a
a
Genuine recollision
Triple collision with a gate a
Skeleton index
a
b
b
θ
Nest
Non-skeleton index Fig. 2. Repetition patterns
a
a
12
L. Erd˝os, M. Salmhofer, H.-T. Yau (2),last
•
We have collected λ4 from non-skeleton labels. We denoted by ψ∗s,k the sum of 2 the truncated elementary wave functions up to time s with two λ power from the non-skeleton indices (the word last indicates that the last λ power was collected at the last collision). We observe a repeated skeleton label, i.e., a recollision or a nest. The corresponding (≤1),r ec (≤1),nest wave functions are denoted by ψ∗s,k , ψ∗s,k . We observe three identical potential labels, i.e., a triple collision. The corresponding (≤1),tri wave functions are denoted by ψ∗s,k .
• •
(≤1),nr Finally, ψt,k denotes the sum of non-repetitive elementary wave functions without truncation (i.e. up to time t) with at most one λ2 power from the non-skeleton indices or ϑ’s and with k skeleton indices. In particular, the non-repetition wave funcnr above) contribute to this sum. For notational tions without gates and ϑ’s (denoted by ψt,k (0),nr nr to explicitly indicate the number of λ2 -powconsistency, we will rename ψt,k := ψt,k ers collected from gates or ϑ’s. This stopping rule gives rise to the following representation.
Proposition 3.2. [Duhamel formula]. For any K ≥ 1 we have ψt = e−it H ψ0 =
K −1
(≤1),nr
ψt,k
k=0
+
K
t
−i
(2),last ψ∗s,k
0
(≤1),nr ds e−i(t−s)H ψ∗s,K
(≤1),r ec + ψ∗s,k
(1),nest + ψ∗s,k
(1),tri + ψ∗s,k
.
(3.26)
k=0
Proof of Proposition 3.2. We start with the precise definitions. For γ˜ ∈ ˜ n and < n we introduce the notation γ˜[1,] := (γ˜1 , . . . , γ˜ ) to denote the beginning segment, or truncation, of length of the sequence γ˜ . Definition 3.3 (Sets of sequences). For 0 ≤ r ≤ 2, k ≤ n we let (r ) ˜ n,k := {γ˜ ∈ ˜ n : k(γ˜ ) = k, r (γ˜ ) = r }
be the set of sequences with k skeleton indices and λ2r collected from non-skeleton indices. Let (r ),nr (r ) ˜ n,k := {γ˜ ∈ ˜ n,k : [γ˜ j = γ˜ j = θ ] =⇒ [| j − j | = 1, j, j ∈ S(γ˜ )]}
be the set of non-repetitive sequences. For r = 0 we have n = k and we set knr := (0),nr . Let ˜ k,k
(r ),last (r ),nr ˜ n,k := {γ˜ ∈ ˜ n,k : γ˜n ∈ S(γ˜ )}
be the set of non-repetitive sequences whose last element is non-skeleton. Let (r ),nr ˜ n,k ˜ nnr := k≤n r ≤2
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
13
be the set of all non-repetitive sequences of length n and let nr ˜ n∗ := {γ˜ ∈ ˜ n \ ˜ nnr : γ˜[1,n−1] ∈ ˜ n−1 }
be the set of sequences that are repetitive, but their proper truncations are non-repetitive. We let (r ),tri (r ) ˜ n,k := ˜ n∗ ∩ {γ˜ ∈ ˜ n,k : γ˜n ∈ S(γ˜ ), ∃ j ≤ n − 2,
γ˜ j , γ˜ j+1 ∈ S(γ˜ ), γ˜n = γ˜ j = γ˜ j+1 } be the set of triple-collision sequences, i.e. sequences whose last entry γ˜n is a part of a triple collision with a gate. Let (r ),r ec (r ) (r ),tri ˜ n,k := ˜ n∗ ∩ {γ˜ ∈ ˜ n,k \ ˜ n,k : ∃ j ∈ S(γ˜ ), j ≤ n − 2,
γ˜ j = γ˜n , S(γ˜ ) ∩ { j + 1, . . . , n − 1} = ∅} be the set of recollision sequences. Finally, let (r ),nest (r ) (r ),r ec (r ),tri : ∃ j ∈ S(γ˜ ), j ≤ n − 2, γ˜ j = γ˜n := ˜ n∗ ∩ γ˜ ∈ ˜ n,k \ ˜ n,k ∪ ˜ n,k ˜ n,k be the set of nested sequences. In all cases we introduce the notation (≤R),# ˜ n,k :=
R
(r ),# ˜ n,k ,
r =0
where # = tri, r ec, nest, nr, last refers to the structure of the wave function. (r ),last
∩ Notice that the last index n of any γ˜ ∈ ˜ n∗ is a skeleton index, in particular ˜ n,k ∗ ˜ n = ∅. This index can create a repetition in three different ways: triple collision, (r ),tri (r ),r ec recollision or nest. It is therefore clear from the definition, that the sets ˜ n,k , ˜ n,k , (r ),nest (r ),nr ˜ n,k and ˜ n,k for 0 ≤ r ≤ 2 are disjoint. Moreover, for triple collision and nested sequences we have r ≥ 1. The next lemma shows that these sets include the appropriate beginning segment of any infinite sequence. Lemma 3.4. Let a positive integer K be given. Let γ˜ = (γ˜1 , γ˜2 , . . .) ∈ ˜ ∞ be an infinite sequence. Then there exist a unique k ≤ K and n ∈ [k, k + 4] such that the truncation of length n of γ˜ , γ˜[1,n] belongs to the (disjoint) union (≤1),nr ˜ (n, K ) := ˜ n,K ∪
K
(1),tri (≤1),r ec (1),nest (2),last ˜ n,k . (3.27) ∪ ˜ n,k ∪ ˜ n,k ∪ ˜ n,k
k=0
Proof. We look at the increasing family of truncated sequences γ˜[1,2] , γ˜[1,3] , . . . induc(r ),nr tively. If γ˜[1,n] ∈ ˜ n,K for some n and r ≤ 1, then it falls into the first set of (3.27). Otherwise there is an n ≤ K + 4 such that γ˜[1,n−1] is non-repetitive with r ≤ 1, but γ˜[1,n] is either repetitive or r (γ˜[1,n] ) = 2. If it is repetitive, then γ˜n is a skeleton index, so r (γ˜[1,n] ) = r (γ˜[1,n−1] ) ≤ 1, and the repetition can be a triple collision, recollision or nest with k ≤ K . If γ˜[1,n] is non-repetitive, then the r has increased from r (γ˜[1,n−1] ) = 1 to r (γ˜[1,n] ) = 2 and γ˜n is non-skeleton. These four possibilities correspond to the remaining sets in (3.27). The disjointness of these sets follows from their definition.
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L. Erd˝os, M. Salmhofer, H.-T. Yau
For 0 ≤ r ≤ 2 and # = r ec, nest, tri, last, nr , let (r ),#
ψ(∗)t,k :=
k+2r
ψ(∗)t,γ˜
n=k+r γ˜ ∈˜ (r ),# n,k
be the wave function with k skeleton labels, with λ2r total power collected from nonskeleton terms and with recollision, nest, triple collision or no repetition (with the last collision being skeleton or not) specified by #. The notation (∗) indicates that the same definition is used for the wave functions with or without truncation. Finally we set (≤1),# (0),# (1),# ψ(∗)t,k := ψ(∗)t,k + ψ(∗)t,k .
We stop the expansion at the elementary truncated wave function (2.14) characterized by γ˜ ∈ ˜ n , if γ˜ falls into one of the sets in (3.27), but none of its proper truncations γ˜[1,n ] fell into the appropriate sets (3.27) with n replaced with n . Lemma 3.4 shows that the expansion is stopped for every term for a unique reason. This procedure proves Proposition 3.2. 4. Error Terms The content of Theorem 2.2 is that the main contribution to the wave function ψt in (3.26) (0),nr comes from the fully expanded non-recollision terms with r = 0, i.e ψt,k . Here we show that the contribution of all other terms are negligible. Each error term in (3.26) has a specific reason to be small. The result can be summarized in the following Theorem which is proven in Sects. 4 and 5. We recall that prime indicates restriction to L . Theorem 4.1. We assume t = λ−2−κ T , T ∈ [0, T0 ], and 1 ≤ k ≤ K . If κ < then (r ),# 2 lim E ψ∗t,k = o(λ4+2κ+2δ ), λ → 0, L→∞
2 34d+39 ,
(4.1)
for the following choices of parameters: {# = r ec, r = 0, 1}, {# = nest, tri, r = 1} or {# = last, r = 2}. Furthermore, for k = K and r = 0, 1, (r ),nr 2 lim E ψ∗t,K = o(λ4+2κ+2δ ),
(4.2)
(1),nr 2 lim E ψt,k
= o(λ2κ+2δ ).
(4.3)
L→∞
and for k < K , L→∞
Proof of Theorem 2.2 using Theorem 4.1. By (3.26) and the unitarity of the operator e−i(t−s)H , we have t K 2 (r ),# 2 −i(t−s)H 2 (r ),# ds e ψ∗s,k ≤ t K sup E E ψ ∗s,k ,
0
k≤K
k=0 0≤s≤t
(4.4)
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
15
where the value of r is determined by # according to the terms on the right-hand side of (3.26). The non-repetition terms with r = 1 (first term on the right-hand side of (3.26)) are fully expanded and there is no need for unitarity. After a Schwarz inequality, K K −1 2 −1 (1),nr 2 (1),nr ≤K ψt,k E E ψ . t,k
k=0
k=0
Given Theorem 4.1, all these error terms are negligible if we first take L → ∞ and then λ → 0. The natural way to prove Theorem 4.1 would be to rewrite E ψ 2 into a sum of Feynman graphs, similarly to Proposition 7.2 in [10], then to identify a repetition subgraph of a few vertices (recollisions, nests, etc.) that renders the graphs small and remove them by graph surgeries after extracting a small factor. We then should sum up the remaining graphs on the core indices for all possible permutations and lumps as in Sect. 9 of [10]. For the sake of technical simplifications, however, at the price of a smaller κ we follow a somewhat different path. We first identify the vertices in the graph (called core indices) that carry no complication whatsoever (no repetition, no gate, no θ ). We then ¯ by a Schwarz inequality to reduce the symmetrize the non-core indices in ψ and ψ, number of repetition patterns. For graphs with sufficient high combinatorial complexity (with large joint degree, see Definition 4.4 below) we simply neglect the possible gain from the repetition patterns by removing them from the graph with a crude estimate. The necessary small factor will come from Proposition 9.2 of [10] with q being large. For graphs with low complexity, the gain comes from analyzing the repetition patterns case by case. Since there are not too many low-complexity graphs, we can neglect the possible gain from the complexity and still sum up for all combinatorial patterns of the core indices. 4.1. Feynman graphs and their values. In this section we collect the necessary definitions from [10] to estimate the values of Feynman graphs. More details can be found in Sects. 7.1 and 7.2 of [10]. The Feynman graph is an oriented circle graph on N ≥ 2 vertices and with two distinguished vertices, denoted by 0, 0∗ . The number of vertices between 0 and 0∗ are n and n , in particular N = n + n + 2. The vertex set can thus be identified with the set − 1, . . . , 1} ˜ equipped with the circular ordering. V = Vn,n := {0, 1, 2, . . . , n, 0∗ , n˜ , n In := { 1, 2, . . . , n }. The set of oriented edges, L(V), We set In := {1, 2, . . . , n} and
can be partitioned into L(V) = L ∪ L so that L contains the edges between In ∪ {0, 0∗ }
contains the edges between and L In ∪ {0, 0∗ }. For v ∈ V we use the notation v − 1 and v + 1 for the vertex right before and after v in the circular ordering. We also denote ev− = (v − 1, v) and ev+ = (v, v + 1) the edge right before and after the vertex v, respectively. For each e ∈ L(V), we introduce a momentum we and a real number αe associated to this edge. The collection of all momenta is denoted by w = {we : e ∈ L(V)} and dw = ⊗e dwe is the Lebesgue measure. The notation v ∼ e will indicate that an edge e is adjacent to a vertex v. Let P = {Pμ : μ ∈ I } be a partition of the set V \ {0, 0∗ } = In ∪ In into nonempty, pairwise disjoint sets, where I = I (P) is the index set to label the sets in the partition. Let m(P) := |I (P)|. The sets Pμ are called P-lumps or just lumps. We assign an auxiliary variable, u μ ∈ Rd , μ ∈ I (P), to each lump. The vector of auxiliary momenta is denoted
16
L. Erd˝os, M. Salmhofer, H.-T. Yau
by u := {u μ : μ ∈ I (P)}. We will always assume that they add up to zero uμ = 0
(4.5)
μ∈I (P)
and that they satisfy |u μ | ≤ O(λ−2κ−4δ ). The set of all partitions of the vertex set V \ {0, 0∗ } is denoted by PV . When we wish to indicate the n, n dependence and identify V \ {0, 0∗ } with In ∪ In , then the set of all partitions on In ∪ In will be denoted by Pn,n instead of PV . For any P ⊂ V, we let L + (P) := {(v, v + 1) ∈ L(V) : v + 1 ∈ P, v ∈ P} denote the set of edges that go out of P, with respect to the orientation of the circle graph, and similarly L − (P) denote the set of edges that go into P. We set L(P) := L + (P) ∪ L − (P). For any ξ ∈ Rd we define the following product of delta functions: (4.6) ±we δ ±we − u μ . (P, w, u) := δ ξ + e∈L ± ({0∗ })
μ∈I (P)
e∈L ± (Pμ )
The sign ± indicates that momenta we is added or subtracted depending on whether the edge e is outgoing or incoming, respectively. For each subset G ⊂ V \ {0, 0∗ }, we define 0 (we )| NG (w) := |ψ | B(wev− − wev+ )| wev− − wev+ −2d . (4.7) e∼0
v∈V \{0,0∗ }\G
v∈G
We also define the restricted Lebesgue measure dμ(w) := 1(|w| ≤ ζ )dw, ζ := λ−κ−3δ ,
dμ(w) := ⊗e dμ(we ).
(4.8)
On the support of this restriction will not substantially influence our integrals (see (7.9)–(7.10) of [10]). With these notations, we define, for any P ∈ PV and g = 0, 1, 2, . . ., the E-value of the partition 1 N −2 sup E g (P, u, α) := λ dμ(w) |αe − ω(we ) + iη| G : |G |≤g e∈L(V )
×(P, w, u)NG (w).
(4.9)
The E-value depends also on the parameters λ, η, but we will not specify them in the notation. We will also need a truncated version of this definition: 1 N −2 sup dμ(w) E ∗g (P, u, α) := λ |αe − ω(we ) + iη| G : |G |≤g e∈L(V) e ∈ L({0∗ })
×(P, w, u)NG (w).
(4.10)
In . The lumps of a partition containing Let P ∈ Pn,n be a partition on the set In ∪ only one vertex will be called single lumps. The vertices 0 and 0∗ will not be considered
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
17
single lumps. Let G = G(P) be the set of edges that go into a single lump and let g(P) := |G(P)| be its cardinality. In case of n = n , we will use the shorter notation Vn = Vn,n , Pn = Pn,n , etc. We will always have |n − n | ≤ g(P) ≤ 4, n, n ≤ K .
(4.11)
We also introduce a function Q that will represent the momentum dependence of the observable. We can assume, for convenience, that Q ∞ ≤ 1. We define [−θ (we )] [−θ (we )] B(we − we+1 ) B(we − we+1 ) M(w) := e∈L∩G
∩G e∈L
(we0+ )ψ (we0− )Q ×ψ
1 2
e∈L\G e ∼0
e∈L\G e ∼0∗
(we0∗ − + we0∗ + ) ,
(4.12)
where e + 1 denotes the edge succeeding e ∈ L(V) in the circular ordering. Let α, β ∈ R, P ∈ Pn,n , and define 1 1 n+n +g(P) dμ(w) V (P, α, β) := λ α − ω(we ) − iη β − ω(we ) + iη e∈L
×(P, w, u ≡ 0)M(w).
e∈L
(4.13)
The truncated version, V∗ (P, α, β), is defined analogously but the α and β denominators that correspond to e ∈ L({0∗ }) are removed. We set Y := λ−100 and define Y e2tη dαdβ eit (α−β) V(∗) (P, α, β) (4.14) V(∗) (P) := (2π )2 −Y and e2tη E (∗)g (P, u) := (2π )2
Y −Y
dαdβ E (∗)g (P, u, α),
(4.15)
The where α in E (∗)g (P, u, α) is defined as αe = α for e ∈ L and αe := β for e ∈ L. notation (∗) indicates the same formulas with and without truncation. We will call these numbers the V -value and E-value of the partition P, or sometimes, of the corresponding Feynman graph. Strictly speaking, the V - and E-values depend on ξ through and the V -value depends on the choice of Q as well. When necessary, we will make these dependencies explicit in the notation, e.g. E ξ or Vξ (P; Q). The E-value is a convenient estimate for the V -value of the graph (see (7.14) of [10]) V(∗) (P) ≤ (Cλ)g E (∗)g (P, u ≡ 0) (4.16) with g = g(P). We will use the notation E (∗)g (P) := E (∗)g (P, u ≡ 0). For the graphical representation of the Duhamel expansion we will really need e2tη ◦ V(∗) (P) := dαdβ V(∗) (P, α, β), (4.17) (2π )2 R i.e. a version of V(∗) (P) with unrestricted dα dβ integrations. (The circle superscript in V ◦ will refer to the unrestricted version of V .) The difference between the restricted
18
L. Erd˝os, M. Salmhofer, H.-T. Yau
and unrestricted V -values are negligible even when we sum them up for all partitions (Lemma 7.1 of [10]). Sometimes we will use the numerical labelling of the edges. We will label the edge between ( j − 1, j) by e j , the edge between ( j, j − 1) by e j˜ and we set en+1 := (n, 0∗ ), e := (0∗ , n˜ ), e1 = (0, 1) and e 1 := ( 1, 0). Therefore the edge set L = L(Vn,n ) is n +1 identified with the index set In+1 ∪ In +1 and we set p j := we j ,
p˜ j := we j .
(4.18)
4.2. Resummation for core indices. We need to identify the non-repetitive potential labels in a sequence. Definition 4.2 (Core of a sequence). Let γ˜ ∈ ˜ n , then the set of core indices of γ˜ is defined as Incor e (γ˜ ) := j ∈ S(γ˜ ) : γ˜ j = γ˜i , ∀ i = j and we set c(γ˜ ) = |Incor e (γ˜ )|. The corresponding γ˜ j labels are called core labels. The subsequence of core labels form an element in cnr , i.e. a sequence of different potential labels. The elements of
Innc (γ˜ ) := In \ Incor e (γ˜ ) ∪ Inθ (γ˜ ) are called non-core potential indices. In other words, the core indices are those skeleton indices (Definition 3.1) that do not participate in any recollision, gate, triple collision or nest. Given the stopping rules (Sect. 3), the number of non-core potential indices and θ -indices together is at most 4. The number of core indices c = c(γ˜ ) is related to the number of skeleton indices k = k(γ˜ ) as follows: ⎧ if # = nr, last ⎨ k c := k − 1 if # = tri ple (4.19) ⎩ k − 2 if # = nest, r ec. For any γ˜ ∈ ˜ n , the index set In is partitioned as In = Incor e ∪ Innc ∪ Inθ into core indices, non-core potential indices and θ -indices. Let τ = τ (γ˜ ) := (τ1 , τ2 , . . . , τc ) ∈ cnr denote the core labels of the sequence γ˜ . We also introduce the notation τ[a,b] := (τa , τa+1 , . . . , τb ) if a ≤ b, and τ[a,b] = ∅ otherwise. (1),nr ˜ Now let γ˜ ∈ (n, K ) ∪ ˜ n,k (see (3.27) and Definition 3.3 for the notation). We recall that the total number of gates and θ ’s is given by r . The possible values of r are determined by # = r ec, nest, tri ple, last according to (3.27) or r = 1 if # = nr and k < K . We will refer to a gate or θ index as a gate/θ -index in short. We rewrite each error term in (3.26) by first summing over core labels τ . For fixed number of core indices c we sum over all possible locations of non-core indices. If a non-core index is inserted between the (w − 1)th and w th core indices, we characterize its location by w. The locations of non-core indices within the sequence are given by a location code w. For example, if # = last, then w ∈ Ic+1 encodes that the first gate/θ -index is located
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
19
between the (w −1)th and w th core indices. The location of the second gate/θ -index need not be encoded because it is fixed to be after the last core index. If # = r ec and r = 0, then w ∈ Ic encodes that the first recollision label is between the (w − 1)th and w th core indices. The most complicated case is # = r ec, r = 1, when the code w consists of two numbers, w = (w1 , w2 ) ∈ Ic × Ic+1 , where w1 and w2 encode the location of the recollision and gate/θ , respectively. If w1 = w2 , an extra binary code determines whether the gate/θ is immediately before or after the recollision. The set of possible (r ),# location codes therefore depends on #, c and r , and it will be denoted by W = Wc . The detailed description of the set W in the other cases is obvious but lengthy and we omit the formal details. The precise structure of W is not important, but we remark that its cardinality satisfies |W | ≤ (c + 1)2 in all cases. We thus have the following resummation formula: (r ),# (r ),# ψ(∗)t,k = ψ(∗)t,τ,w , (4.20) τ ∈cnr w∈W
(r ),#
where ψ(∗)t,τ,w is the wave function with core labels τ , with structure #, with r gates/θ indices and with location of non-core indices given by w. We note that the wave function (r ),# ψ(∗)t,τ,w includes a summation over the non-core labels with the restriction that they are distinct from the core labels τ . Having specified the locations of the r gates/θ -indices within the sequence of core indices, we introduce another code h ∈ {g, θ }r , called gate-code, to specify whether there is a gate or a θ at the given location. This gives the decomposition (r ),# h,# ψ(∗)t,τ,w = ψ(∗)t,τ,w (4.21) h∈{θ,g}r
h,# . with the obvious definition of ψ(∗)t,τ,w
4.3. Symmetrization of the non-core indices. Starting from (4.20), we can use the Schwarz inequality (r ),# (r ),# ! (r ),# 2 E ψ(∗)t,k
= E ψ(∗)t,τ,w , ψ(∗)t,τ ,w w,w ∈W
≤ |W |
τ ∈cnr
w∈W τ,τ ∈cnr
τ ∈cnr
! (r ),# (r ),# E ψ(∗)t,τ,w , ψ(∗)t,τ ,w ,
where c is given by k and # according to (4.19), and recall that W depends on #, c, r . Notice that any non-core potential label appears in a pair (in a gate, nest or recollision) and none of them appear more than six times by the stopping rules. Since the first, third and fifth moments of the random variables vγ are zero, the expectation in (4.22) is nonzero only if τ and τ are paired, i.e. if there is a permutation σ ∈ Sc such that τ = σ (τ ), meaning τi = τσ (i) . Therefore ! (r ),# 2 (r ),# (r ),# E ψ(∗)t,k
≤ |W | E ψ(∗)t,τ,w , ψ(∗)t,σ (τ ),w w∈W σ ∈Sc τ ∈cnr
≤ |W |
w∈W σ ∈Sc h,h ∈{g,θ}r τ ∈cnr
! h,# h ,# E ψ(∗)t,τ,w , ψ(∗)t,σ (τ ),w .
(4.22)
20
L. Erd˝os, M. Salmhofer, H.-T. Yau (r ),#
Note that each wave function ψ(∗)t,τ,w has been further decomposed into a sum over h-codes according to (4.21). However, we did not estimate the h = h cross terms by an additional Schwarz inequality. The term with a gate must cancel the term with a θ exactly at the same location, i.e. ψ g and ψ θ would not individually be negligible, but their sum is of smaller order. h,# Notice also that each wave function ψ(∗)t,τ,w may involve summations over one or two further non-core potential labels. We use the convention that the recollision or nest label is denoted by ν, the label of the gate or triple is denoted by μ. In case of a second gate, # = last, its label will be μ. ˚ According to the non-repetition rules, (ν, μ, μ) ˚ may not h,# h ,# coincide with each other or with any element of τ . In the products ψ (∗)t,τ,w ψ(∗)t,σ (τ ),w there is no repetition among τ and σ (τ ) indices other than the ones prescribed by σ . h ,# ˚ , However, if the additional non-core labels within ψ(∗)t,σ (τ ),w are denoted by ν , μ or μ then there may be a few coincidences between primed and non-primed non-core labels. Those coincidences are allowed that do not violate the non-repetition rules requiring that ν, μ, μ˚ are distinct and their primed counterparts are also distinct among themselves. Once the number of core indices c, a location-code w and a gate-code h are fixed, this defines a unique insertion of the non-core indices into the sequence of core indices Ic . The core and non-core indices in the given order can be identified with In , where n, the total number of collisions, is given by n = k + r + |{ j : h j = g}| and k is given by (4.19). This naturally defines an embedding map s = swh : Ic → In . Simi Ic → In can be defined. The precise definition depends on #, r and h in a larly, swh : natural way. For illustration, we describe the most complicated case; # = r ec, r = 1 and h = g. In this case n = c + 4. For definitiveness, we consider the case when the recollision precedes the gate w1 ≤ w2 . In this case the complete collision sequence is (τ[1,w1 −1] , ν, τ[w1 ,w2 −1] , μ, μ, τ[w2 ,c] , ν). Let ⎧ if j < w1 ⎨ j swh=g ( j) := j + 1 if w1 ≤ j < w2 ⎩ j + 3 if w ≤ j ≤ c. 2 With this notation, the pairing of core indices, originally determined by σ ∈ Sc , is given swh (σ ( j))} as subsets of the full index set In ∪ In . by the pairs {swh ( j), Given (#, c, σ, w, h, h ), we define the partition D0 = D0 (#, c, σ, w, h, h ) of In ∪ In by lumping only those indices that are required to carry the same potential label by the prescribed structure # and the gates. The vertices with θ always remain a single lump, the remaining vertices are paired. For example, if # = r ec, r = 1, w = (w1 , w2 ), h = h = g, we obtain D0 := {s( j), s(σ ( j))} j∈Ic , {w1 , n}, {w2 + 1, w2 + 2}, { w1 , n }, {w 2 + 1, w 2 + 2} (see Fig. 3) and all other cases are similar. The elements of D0 consisting of pairs of core indices, {s( j), s(σ ( j))} j∈Ic , are called core elements of D0 and those elements of D0 that contain non-core indices will be called non-core elements. In the example above, the last four elements are the non-core elements of D0 describing the two gates and two recollisions. The non-core potential labels ν, μ, ν , μ will correspond to these non-core elements, respectively. Some of the non-core elements may be lumped together according to the possible coincidence between the {ν, μ} and {ν , μ } since the non-repetition rule does not prevent it. This procedure defines new partitions D that we will call derived partitions,
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
1
τ1
2
τ2
3
w 1-1
τ3
τw
0
1
-1
w 1 w 1+1 w 2
ν
τw
1
τw
ν’
w 2+1 w 2+2 2
-1
μ
μ
μ’
μ’
21
w 2+3
τw
2
n-1
τc
n
ν ν’
0*
τ σ(1) Fig. 3. Symmetrized recollision with a gate. D0 consists of the paired vertices
denoted by D D0 . In this particular case, there are seven possible lumpings of the four non-core elements without violating the non-repetition rule within the sets (τ, ν, μ) and (τ, ν , μ ). Fig. 4 shows the partition D derived from the above partition D0 when μ = μ but ν = ν . In general D is defined by lumping together a few non-core elements of D0 under the constraint of the non-repetition rule (Fig. 4). The single elements of D0 that correspond to θ are never lumped. Note that each of these pairings gives rise to the appearance of exactly four or six identical potential labels; higher moments do not appear. The number of such quartets and sextets is denoted by 4 (D) and 6 (D). Clearly 4 (D) ≤ 2 and 6 (D) ≤ 1. Let D∗ ⊂ D denote the collection of non-single elements of D. Note that for each element of D∗ one selects a distinct potential label. The quantity (4.22) contains a summation over all such potential labels. We will use Lemma 6.1 from [10] for the index set D∗ to evaluate this sum. Let A ∈ A(D∗ ) be a partition of the set D∗ . We define P(A, D) ∈ Pn,n to be the In whose lumps are given by the equivalence relation that two elements partition of In ∪ of In ∪ In are P(A, D)-equivalent if their D-lump(s) are A-equivalent. The single lumps of D remain single in P (these are the θ indices). ◦ (P) from (4.14) and (4.17). Since we will We recall the definition of V(∗) (P) and V(∗) 2 compute the L -norm, the momentum shift at 0∗ is chosen to be ξ = 0 (see (4.6)) and the Q function in (4.12), representing the observable, will be Q ≡ 1. Furthermore, when defining Feynman graphs, we will always assume the following range of parameters (see (7.25) of [10]) unless stated otherwise: η = λ2+κ , t = λ−2−κ T, T ∈ [0, T0 ],
K = [λ−δ (λ2 t)],
k ≤ K , ζ = λ−κ−3δ , g ≤ 8,
Fig. 4. Partition D lumps some non-core elements of D0
(4.23)
22
L. Erd˝os, M. Salmhofer, H.-T. Yau
with a sufficiently small δ > 0 that is independent of λ but depends on κ. We recall that η is the regularization of the propagator, K is the upper threshold for the number of skeleton indices, k, in the expansion, ζ is the momentum cutoff (see (4.8)) and g is the number of exceptional vertices where the standard | B(win − wout )| potential decay is not present (this happens for the single lumps). All estimates will be uniform in ξ and in T ∈ [0, T0 ]. The rules of taking the expectation in (4.22) are different for the continuum and the lattice random potentials, see (1.3) and (1.16). Lemma 6.1 of [10] presents a formulation that includes both cases. For each fixed (#, c, σ, w, h, h ), by using (1.5) we obtain, similarly to Proposition 7.2 of [10], ! h,# h ,# ◦ lim E ψ(∗)t,τ,w , ψ(∗)t,σ m (D) c(A)V(∗) (P(A, D)) (τ ),w = L→∞
τ ∈cnr
D D0
A∈A(D∗ )
(4.24) (D)
with m (D) := m 4 4
(D)
m 66
(we recall m k = E vγk and (1.5)). Here c(|Aν |), c(A) := ν∈I (A)
where c(n) is defined differently for the continuum and the lattice case, see Lemma 6.1 of [10]. However, for the argument in the sequel we need only the estimate |c(n)| ≤ n n−2 , which is valid for both cases. The summary of the results in this section is Proposition 4.3. Let k ≤ K , let # = r ec, nest, tri ple, last and r be one of the possible values allowed by (3.27) or r = 1 if # = nr , k < K . Let c be given by (4.19), and let (r ),# be the set of location codes. Then W = Wc (r ),# 2 ◦
≤ |W | m (D) c(A)V(∗) (P(A, D)). lim E ψ(∗)t,k L→∞
w∈W σ ∈Sc h,h ∈{g,θ}r D D0
A∈A(D∗ )
(4.25) 4.4. Splitting into high and low complexity regimes. Given (#, c, σ, w, h, h ), we consider the partition D0 of In ∪ In as defined above and let D D0 . Note that the collection D∗ contains all core elements of D0 , i.e. all pairs of core indices {s( j), s(σ ( j))} j∈Ic . The restriction of a partition A ∈ A(D∗ ) onto these core elements can therefore be naturally identified with a partition of Ic using the map {s( j), s(σ ( j))} j∈Ic → j ∈ Ic . We denote this restricted partition by A. In the sequel we shall therefore view A ∈ Ac , i.e. as a partition on Ic . The restricted partition A together with σ also generates a partition P( A, σ ) on the μ ∪ σ ( A μ ), where A μ are the lumps set Ic ∪ Ic . The lumps Pμ of P( A, σ ) are given by A of A. Notice that the (s, s)-image of the restriction of P(A, D) ∈ Pn,n onto the set of e is exactly P( Incor A, σ ). Since the cardinality of non-core elements core indices Incor e ∪ ∗ of D is at most 4, for any given D and A there can exist at most (c + 4)4 partitions, A ∈ A(D∗ ), whose restriction onto the core elements is A. We recall the definition of joint degree from [10]:
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
23
Definition 4.4. (i) Let A ∈ Ak be a partition of Ik = {1, 2 . . . , k}. Set aν := |Aν |, ν ∈ I (A), to be the size of the ν th lump. Let S(A) :=
Aν
ν∈I (A) aν ≥2
be the union of nontrivial lumps. The cardinality of this set, s(A) := |S(A)|, is called the degree of the partition A. (ii) Let A ∈ Ak and σ ∈ Sk . The number 1 q(A, σ ) := max deg(σ ), s(A) 2
(4.26)
is called the joint degree of the pair (σ, A) of the permutation σ and partition A. The sum (4.25) will be split into two parts and estimated differently. In the regime of high combinatorial complexity, i.e., when the joint degree q( A, σ ) of σ and A is bigger than a threshold q ≥ 1 (to be determined later), then we can use the method of Sect. 9 (especially Proposition 9.2) from [10] robustly. This will be explained in Sect. 4.5. For low combinatorial complexity we use the special structure given by the recollisions, nests, triple collisions or gates (Sect. 4.6). The threshold q will be chosen differently for the estimates (4.1)–(4.2) and for (4.3). The precise estimate is the following: (r ),#
lim E ψ(∗)t,k 2 ≤ (I ) + (I I ) + O(λ5 )
L→∞
(4.27)
with (I ) := |W |(c + 4)4 ×
A ∈Ac q(A ,σ )≥q
m (D)
w∈W h,h σ ∈Sc D D0
sup |V(∗) (P(A, D))c(A)| : A = A ,
(4.28)
A
where the supremum is over all possible A ∈ A(D∗ ) whose restriction A is the given partition A ; and (I I ) := |W |
w∈W σ ∈Sc h,h ∈{g,θ}r D D0
m (D)
A∈A(D∗ ) q( A,σ )
V(∗) (P(A, D))c(A) . (4.29)
We recall that D0 is determined by (#, c, σ, w, h, h ). The error term O(λ5 ) comes from ◦ (· · · ) with V (· · · ); see Lemma 7.1 of [10]. replacing V(∗) (∗)
24
L. Erd˝os, M. Salmhofer, H.-T. Yau
4.5. Case of high combinatorial complexity. Here we estimate the term (I) in (4.28). Clearly m (D) ≤ m 4 2 m 6 ≤ C. We estimate V(∗) (P(A, D)) by using (4.16). Then, by applying Operation I from Appendix C, we break up all the lumps Pμ in the partition P(A, D) that involve elements from non-core indices, Innc ∪ Innc , in such a way that all ∗ non-core indices must form single lumps. Let P (A, D) denote this new partition. Note that the projection of P∗ (A, D) onto the core indices is unchanged. The number of applications of Operation I is at most 6. Using Lemma C.1, we can estimate E (∗)g (P(A, D)) in terms of supu E (∗)g (P∗ (A, D), u) with an additional factor of at most 6 , where := [C K ζ ]d = O(λ−2dκ−O(δ) ). Then we apply Operation II (Appendix C) to remove all single lumps with non-core indices and use (C.1) from Lemma C.2. After removing the non-core indices, the remaining vertex set can naturally be identified with Ic ∪ Ic by using the (s, s) maps, and the e is identified with P( partition P∗ (A, D) restricted to core indices Incor e ∪ Incor A, σ ). Lemma C.2 is applied at most 8 times, therefore we obtain that for any σ , D and A in the sum (4.28): |V(∗) (P(A, D))| ≤ C6 (λη−1 )8 sup E (∗)g (A , σ, u).
(4.30)
u,g≤8
The application of Operation II is schematically shown on Fig. 5. The summations over h, h ∈ {g, θ }r and D D0 in (4.28) contribute with at most a constant factor since r ≤ 2 and the number of different D’s is at most 7. The cardinality of W can be bounded by (c + 1)2 and c ≤ K ≤ Cλ−κ−δ . Therefore we obtain (I ) ≤ Cλ−8−κ(16+2d)−O(δ) sup E (∗)g (A , σ, u)|c(A )|. σ ∈Sc
A ∈Ac q(A ,σ )≥q
u,g≤8
Using Proposition 9.2 from [10], we have 1
17
13
(I ) ≤ Cλq[ 3 −( 3 d+ 2 )κ−O(δ)]−8−(16+2d)κ . We immediately see that the contribution of (I) to the error term in (4.3) in Theorem 4.1 satisfies the announced bound with a sufficiently small δ if κ<
2q − 48 . (34d + 39)q + 112 + 12d
(4.31)
The bounds (4.1)–(4.2) are satisfied if κ<
V
2q − 72 . (34d + 39)q + 112 + 12d
8
< C (λ/η) Eg
Fig. 5. Estimate after removing all non-core indices
(4.32)
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
25
4.6. Case of small combinatorial complexity. Here we control the term (II) in (4.29). First we estimate the combinatorics. Lemma 4.5. For any q ∈ N, c ≤ K and structure type #, we have sup |c(A)| ≤ (Cq K )3q+3 , sup w,h,h σ ∈S D D0 c
A∈A(D∗ ) q( A,σ )
where we recall that D0 depends on (#, c, σ, w, h, h ). Proof. The bound #{σ ∈ Sk : (σ ) = } ≤ (Ck)k−+1
(4.33)
(see (8.14) from [10]) shows that the number of permutations σ ∈ Sc with deg(σ ) < q is bounded by (C K )q using c ≤ K . The number of A’s whose restriction yields the same A is at most (c + 3)4 ≤ (C K )4 . The number of A ∈ Ac with s( A) < 2q is bounded "∗ a j −2 2q−1 2q−1 by c ≤ (C K ) . Finally, |c(A)| ≤ j a j ≤ (2q)2q−2 . The individual terms in (4.29) are estimated in the following proposition whose proof will be given in Sect. 5. (r ),#
2 Proposition 4.6. We assume (4.23) and κ < 34d+39 . Let σ ∈ Sc , w ∈ Wc , h, h ∈ r {g, θ } , where # and r vary in the different cases and let D D0 (#, c, σ, w, h, h ).
1) Let A ∈ A(D∗ ) such that q( A, σ ) < q, where q is a fixed number. Then the following individual estimates hold: (1a) [Many collisions]. Let # = nr , r = 0, 1 and c = K , then δ
|V∗ (P(A, D))| ≤ C q λ 2 K .
(4.34)
(1b) [Recollision]. Let # = r ec, r = 0, 1, then |V∗ (P(A, D))| ≤ C q λ6−4dκ(q+3) .
(4.35)
(1c) [Triple collision]. Let # = tri ple, r = 1, then |V∗ (P(A, D))| ≤ C q λ6−4dκ(q+3) . 1) Now let A ∈ Ac be given. Then the following estimates hold: (2a) [Non-repetition with a gate]. Let # = nr , r = 1, then 1 17 sup V (P(A, D))c(A) ≤ Cλ 3 −( 3 d+8)κ−O(δ) . σ,w
h,h ∈{g,θ}r D D0
(4.37)
A∈A(D∗ ) A=A
(2b) [Last]. Let # = last, r = 2, then sup V∗ (P(A, D))c(A) ≤ Cλ6−(14d+6)κ−O(δ) . σ,w h,h ∈{g,θ}r D D0
(4.36)
A∈A(D∗ ) A=A
(4.38)
26
L. Erd˝os, M. Salmhofer, H.-T. Yau
(2c) [Nest]. Let # = nest, r = 1, then sup σ,w
h,h ∈{g,θ} D D0
A∈A(D∗ ) A=A
V∗ (P(A, D))c(A) ≤ Cλ6−(10d+8)κ−O(δ) .
(4.39)
Combining Lemma 4.5 with these estimates, and using |W | ≤ K 2 , we see that 1
17
(I I ) ≤ (Cq K )3q+7 λ 3 −( 3 d+8)κ−O(δ) for the case # = nr , r = 1 (Case (2a) above), and (I I ) ≤ (Cq K )3q+7 λ6−4dκ(q+3)−O(δ) for all other cases (with q ≥ 2). So the contributions of the error terms from (II) to (r ),# E ψ(∗)t,k 2 (see (4.27)) satisfy the bound (4.3) if 1 , 9q + 17d + 51
(4.40)
2 (4d + 3)q + (12d + 9)
(4.41)
κ< and they satisfy (4.1)–(4.2) if κ<
and δ is sufficiently small. Combining this with (4.31)–(4.32) and optimizing, we obtain that there exists κ0 (d) > 0 such that for any κ < κ0 , the systems of inequalities (4.31)– (4.40) and (4.32)–(4.41) have solutions for q. For d = 3, the optimal κ0 (d) is a bit above 1 500 . This finishes the proof of Theorem 4.1. 5. Proof of Proposition 4.6 In each case except (1a), the corresponding Feynman graph has a specific subgraph of a few vertices (recollision, nest, etc.) that renders the value small. We shall prove that this subgraph gains at least a factor λ4+2κ+2δ required in Theorem 4.1. Then we remove all repetition patterns from the graph, we use the robust bounds sup sup E(σ, u) ≤ C| log λ|2 ,
(5.1)
sup sup E ∗ (σ, u) ≤ Cλ2 | log λ|2
(5.2)
σ ∈Sk u σ ∈Sk u
from Lemma 10.2 of [10] to conclude the estimate.
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
27
5.1. Many collisions. The estimate in Case (1a) will come from the fact that any graph can be robustly estimated by the ladder graph and the value of the ladder of length L always carries a factor 1/L!. This effect is the best seen in the time integral form. We first change V∗ (P) back to V∗◦ (P) with an error smaller than O(λ10K −O(1) ) by Lemma 7.1 from [10]. We then apply the K -identity (formula (6.2) in [10]) to the definition of V∗◦ (P) given in (4.17) to obtain ◦ n+n +g ˜ n )(P, w, u ≡ 0)M(w) V∗ (P) = λ dpdp˜ K (t, p, n)K (t, p, with P = P(A, D) and
t
K (t, p, n) := (−i)
n−1 0
[ds j ]n1
n
e−is j ω( p j ) .
j=1
We recall that g = g(P(A, D)) denotes the number of single lumps, or, equivalently, the number of θ labels in h and h . Note that we use the labelling w and p, p˜ in parallel, keeping in mind the relabelling convention from (4.18) (see also Sect. 7.2 in [10] for more details). We use a Schwarz inequality: ◦ n+n +g ˜ n )|2 (P, w, u ≡ 0)|M(w)|. |V∗ (P)| ≤ λ dpdp˜ |K (t, p, n)|2 + |K (t, p, By using Operation I, we can break up A into single lumps. Since s( A) ≤ 2q, we have s(A) ≤ 2q + 4, thus Operation I will be applied at most 2q + 3 times. If r = 0, i.e. the original graph was a non-repetition graph, and thus n = n = k, then the trivial partition A0 corresponds to a partition P with a complete pairing. Thus both p and p˜ momenta can be used as independent variables and |V∗◦ (P)|
≤
2q+3
(Cλ)
2k+g
0 ( p1 )|2 dp |K (t, p, k)|2 |ψ
k
| B( p j − p j+1 )|2 .
j=1
The estimate (4.34) is then completed by the bound (5.3) from the following Lemma with any 1/2 < a < 1. The proof will be given below. Lemma 5.1. For any 0 ≤ a < 1, t = T λ−2−κ , there exists a constant Ca such that I (k) :=
0 ( p1 )|2 dp|K (t, p, k)|2 |ψ
k
| B( p j − p j+1 )|2
j=1
≤
T λ−2−κa )k−1
(Ca [(k − 1)!]a
| log λ|2 .
In particular, I (k) ≤ (Ca λ−2+δa )k−1 if k ≥ T λ−κ−δ and λ 1.
(5.3)
28
L. Erd˝os, M. Salmhofer, H.-T. Yau
Proof. This lemma is essentially the same as Lemma 3.1 in [7]. The only differences are that here we estimate the truncated value, so K has one less time integration and the individual integrals are performed by using Lemma 2.1. The details are left to the reader. (1),nr
is the location of the gate/θ -index among the core Finally, if r = 1 and w ∈ Wc indices, then pw = pw+1 or pw = pw+2 (depending whether we have a θ or a gate) is forced by and similarly for p˜ w . In this case, the estimates in Lemma 5.1 are worse by a factor of t. This factor can be absorbed into the main term λδa K . This completes the proof of (4.34). 5.2. Recollision and triple collision. For the proof of (4.35), we break up the partition A into the trivial partition A0 using Operation I. Since s( A) ≤ 2q, we have s(A) ≤ 2q + 4, thus Operation I will be applied at most 2q + 3 times. Clearly |V∗ (P(A, D))| ≤ 2q+3 λg sup E ∗ (P(A0 , D0 ), u)
(5.4)
u
by using (4.16) and Lemma 9.5 from [10]. The single lumps are removed by Operation II from E ∗ (P(A0 , D0 ), u), at the price λ/η; the total contribution of one θ -removal is (λ2 /η) ∼ λ−κ . The possible gates are eliminated by Operation IV at the expense of λ2 η−1 | log η| ∼ λ−κ | log λ| each. Since r ≤ 1, we lose at most a factor λ−2κ | log η|2 in this way (see Fig. 6; the double line denotes truncated propagators). The vertex set of the remaining graph is naturally identified with Ic+2 ∪ Ic+2 ∪ {0, 0∗ }, the permutation σ provides a pairing between the elements Ic+2 \ {w1 , c + 2} and Ic+2 \ {# w1 , c + 2}, furthermore {w1 , c + 2} and {# w1 , c + 2} each form a lump (see picture). We denote this partition by P∗ and by using Proposition 5.2 below, we will obtain (4.35). Later we need to estimate asymmetric recollision graphs as well, so we formulate the following proposition in a more general setup: Proposition 5.2. Consider the Feynman graph on the vertex set Vk , k ≥ 3, choose numbers a, b, a , b ∈ Ik such that b − a ≥ 2, b − a ≥ 2. Let σ be a bijection between Ik \ {a, b} and Ik \ {a , b }. Let P∗ be the partition on the set Ik ∪ Ik consisting of the lumps { j, σ ( j)}, j ∈ Ik \ {a, b} and {a, b}, {a , b } (Fig. 7). Then sup E ∗g (P∗ , u) ≤ Cλ6−3κ ζ 4d .
(5.5)
u,g≤8
We also need a “one-sided” version of this estimate (Fig. 8). Proposition 5.3. Consider the Feynman graph on the vertex set Vk,k−2 , k ≥ 3, choose numbers a, b ∈ Ik such that b − a ≥ 2. Let σ be a bijection between Ik \ {a, b} and
2
E
−2κ
*g
<λ
2
| logλ | E g+2 *
1
Fig. 6. Removal of gates from a recollision
3
w
w+1
c+2 c+3
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
m+1
a
a+1
b
b+1
6−3κ
m
E
29
*g a’
a’+1
b’
ζ
4d
b’+1
Fig. 7. Estimate of a two-sided recollision graph
a
E
a+1
b
b+1
4−κ
*g
ζ 4d
Fig. 8. Estimate of a one-sided recollision graph
Ik−2 consisting of the lumps { j, σ ( j)}, Ik−2 . Let P∗ be the partition on the set Ik ∪ j ∈ Ik \ {a, b} and {a, b}. Then sup E g (P∗ , u) ≤ Cλ2−κ ζ 4d ,
(5.6)
u,g≤8
and for the truncated version sup E ∗g (P∗ , u) ≤ Cλ4−κ ζ 4d .
(5.7)
u,g≤8
Proof of Proposition 5.2. We use p = ( p1 , . . . , pk+1 ) and their tilde-counterparts to denote the momenta to express E ∗g (P∗ , u) =
sup
Y
G : |G |≤g −Y
dαdβ G (α, β)
with
k
1 1 |α − ω( p j ) − iη| |β − ω( p˜ j ) + iη| j=1 ×δ( pk+1 − p˜ k+1 )δ pa+1 − pa + ( pb+1 − pb ) − u a ×δ − p˜ a +1 + p˜ a − ( p˜ b +1 − p˜ b ) − u˜ a
G (α, β) := λ
2k
×
˜ dμ(p)dμ(p)
k δ p j+1 − p j − ( p˜ σ ( j)+1 − p˜ σ ( j) ) − u j NG (w), j=1 j =a,b
where the u-momenta are labelled as u = (u 1 , . . . , u b−1 , u b+1 , u k , u˜ a ) and we used the identification from (4.18) between the w and p, p˜ notations.
30
L. Erd˝os, M. Salmhofer, H.-T. Yau
Without recollision, the momenta p formed a spanning set of all momenta, and simi˜ Since now there is a delta function among the p momenta, we need to exchange larly for p. one tilde-momentum (out of p˜ a , p˜ a +1 , p˜ b , p˜ b +1 ) with a non-tilde momentum (out of pa , pa+1 , pb , pb+1 ). We will call them exchange momenta. For the moment, we choose p˜ b and pb to be the exchange momenta and we partition the set of all p, p˜ momenta into two subsets of size k + 1 each: A := { p1 , p2 , . . . , pb−1 , pb+1 , . . . , pk+1 , p˜ b }, B := { p˜ 1 , p˜ 2 , . . . , p˜ b −1 , p˜ b +1 , . . . , p˜ k+1 , pb }. It is straightforward to check that all A-momenta can be uniquely expressed in terms of linear combinations of the B-momenta (plus the u-momenta) and conversely. In particular pb−1 = pb − ( p˜ σ (b−1)+1 − p˜ σ (b−1) ) − u b , p˜ b −1 = p˜ b − ( pm+1 − pm ) + u m with
m := σ −1 (b − 1).
The letters on the pictures indicate the indices of the corresponding p or p˜ momenta. We perform a Schwarz estimate to separate A and B-momenta at the expense of squaring the propagators, but we keep the denominators with p1 , p˜ 1 , pb−1 , p˜ b −1 pb , p˜ b common and only on the first power: k j=1
1 1 1 ≤ (a) + (b) , |α − ω( p j ) − iη| |β − ω( p˜ j ) + iη| 2
(a) :=
j=1,b−1,b
(b) :=
j=1,b−1,b
(5.8)
k 1 1 1 , |α − ω( p j ) − iη| |β − ω( p˜ j ) + iη| j=2 |α − ω( p j ) − iη|2 j =b−1,b
1 1 |α − ω( p j ) − iη| |β − ω( p˜ j ) + iη|
k j=2 j =b −1,b
1 . |β − ω( p˜ j ) + iη|2
(with a little abuse of notations we used j for 1, b − 1 and b when j = 1, b − 1 and b, respectively). Since common factors can be explicitly expressed both in terms of A and B-momenta, we can compute the integral of (a) by first integrating all B-momenta that removes all delta functions, then estimating the A-momentum integrals. A similar procedure works for (b). The result is (with m := σ −1 (b − 1)) Y k+1 1 ∗ 6 E∗g (P , u) ≤ λ sup dαdβ dμ( p˜ b ) j=1 dμ( p j ) NG (w) |α − ω( p1 ) − iη| j =b −Y |G |≤g 1 1 × |β − ω( p1 ) + iη| |α − ω( pb−1 ) − iη| 1 × |β − ω( p˜ b − ( pm+1 − pm ) + u m ) + iη| 1 1 × |α − ω( pb+1 − pa + pa+1 − u a ) − iη| |β − ω( p˜ b ) + iη| k λ2 . (5.9) × |α − ω( p j ) − iη|2 j=2 j =b−1,b
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
31
We recall the key technical bound Lemma 10.8 from [10] to estimate integrals with shifted denominators. We also recall the notation |||q||| := η + min{|q|, 1}. We start with estimating λ2 λ2 η−1 ≤ |α − ω( pb+1 ) − iη|2 |α − ω( pb+1 ) − iη| and integrating out pb+1 by using (10.25) from [10]. We collect a point singularity ||| pa − pa+1 + u a |||−1 and a factor Cλ2 η−1 ζ d−3 | log η|2 . This argument works if b < k; the b = k case is even easier since the denominator with pk+1 is not present. Then we perform the p˜ b integration again by (10.25) from [10], and we collect a new point singularity ||| pm+1 − pm + u m |||−1 and a factor Cζ d−3 | log λ|2 . Note that these two point singularities are not identical, because m, as an inverse image of σ , is not equal to a. Next we integrate out pb−1 yielding C| log λ| from (2.6). If pb−1 appears in one of the point singularities, then we use the bound 1 dμ( p) ≤ Cζ d−2 | log η| sup (5.10) |α − ω( p) + iη| ||| p − r ||| α,r (see (A.3) and (A.7) from [10]) to collect Cζ d−2 | log λ| and the point singularity disappears. If pb−1 appears in both point singularities, then we separate them by the telescopic estimate (A.1) of [10] before applying (5.10). Now we integrate out all p j ’s with j = 1, b − 1, b, b + 1 in decreasing order with the successive integration scheme (10.7)–(10.9) from Sect. 10.1.2 of [10]. The factor NG (w) provides the necessary | B( p j − p j−1 )|2 terms with at most eight exceptions, namely when j − 1 ∈ G or σ ( j − 1) ∈ G (recall |G| ≤ 8). At each exceptional index | B( p j − p j−1 )|2 is replaced with p j − p j−1 −2d and we use (2.7) instead of (10.8) of [10]. Thus the successive production of the factors (1 + Cλ1−12κ ) breaks at these indices and we obtain a uniform constant C instead. The successive scheme also breaks at the indices j = b − 1, b, b + 1 that have already been integrated out. Furthermore, it also may break at j = m + 1, a + 1, i.e., at the indices where the point singularities are first affected (unless b − 1 ∈ {m + 1, a + 1} and the point singularity has already been integrated out). At each of these indices we use (5.10) and collect Cλ2 η−1 ζ d−2 | log η|2 instead of the constant factor from (10.7)–(10.9) of [10]. Since there are at most 13 exceptional indices, so we collect at most C 13 [λ2 η−1 ζ d−2 | log η|2 ]2 (1 + Cλ1−12κ ) K . The other factors of NG , that are not explicitly used in the successive integration, are 0 ( p1 )|2 . Finally the dα, dβ integrals contribestimated by supremum norm, except |ψ 2 0 ( p1 )|2 . ute with an additional C| log λ| . The last p1 -integral is finite by the factor |ψ Collecting these estimates and recalling that (5.10) has been used twice, the result is (5.5). Proof of Proposition 5.3. This proof is very similar to the previous one but the estimate is weaker since (10.25) from [10] can be used only once. We just indicate that the set of A and B momenta are as follows: A := { p1 , p2 , . . . , pb−1 , pb+1 , . . . , pk+1 } , and we leave the details to the reader.
B := { p˜ 1 , p˜ 2 , . . . , p˜ k−1 , pb },
32
L. Erd˝os, M. Salmhofer, H.-T. Yau
2
< C λ | log λ |
E*g
2
E*g+2
6
4
< C λ | log λ |
E*g
Eg+2
Fig. 9. Estimate of triple collisions for w < c + 1 and w = c + 1
The case of the triple collision, (4.36), can be easily reduced to the case of a recollision (Fig. 9). We first use the analogue of the estimate (5.4). Clearly g = 0 in the case of a triple collision. Then we remove half of each of the two gates (Lemma C.3) and we collect a factor λ2 | log λ|2 . The resulting Feynman graph has either a recollision or a gate at the end. In the first case we apply Proposition 5.2. In the second case, we remove half of each gate by a second application of Lemma C.3, then the estimate (C.2) from Lemma C.2 together with (5.1) can be applied. 5.3. Cancellation with a gate. The key mechanism behind the estimates (4.37)–(4.39) is the cancellation between a gate and a θ -label, We first present estimates on general graphs. 5.3.1. Cancellation between a gate and θ . Fix n, n integers and consider a partition P ∈ Pn,n with no single lump on the set In ∪ In within the vertex set V = Vn,n . Let 1 ≤ m ≤ n + 1 be an integer. We define two new cyclically ordered sets: − 1, . . . , 1}, ˜ V := {0, 1, 2, . . . , m − 1, ♣, m, . . . , n, 0∗ , n˜ , n − 1, . . . , 1} ˜ V := {0, 1, 2, . . . , m − 1, ♦, ♥, m, . . . , n, 0∗ , n˜ , n
(5.11)
with additional elements ♣, ♦, ♥. For the result of this section it would make no difference if the extra elements were inserted into the sequence of tilde-variables. These sets can be naturally identified with Vn+1,n and Vn+2,n and we will use this identification with the obvious choice of the relabelling map. We define two partitions on these sets, P ∈ Pn+1,n and P ∈ Pn+2,n , simply by adding the single lump {♣} to P in the first case and the double lump {♦, ♥} in the second case. This will correspond to adding a ϑ label or a gate whose potential labels have been paired to the original partition P, respectively. The following lemma shows that the V -value of these two partitions cancel each other up to the lowest order (Fig. 10).
Vη
pm–1 pm
q
pm pm+1
+ Vη
pm–1 pm θ pm
pm+1
2 –1/2
< Cλ η
Eη
pm–1
Fig. 10. Cancellation of a gate and θ (partition is the same elsewhere)
pm pm+1
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
33
Lemma 5.4. With the notations above and assuming λ3 η λ2 , we have V(∗) (P ) + V(∗) (P ) ≤ Cλ2 η−1/2 E (∗)g=0 (P).
(5.12)
Proof of Lemma 5.4. Introduce the notations p := ( p1 , . . . , pn+1 ), p˜ := ( p˜ 1 , . . . , p˜ n +1 ), dp := d p1 d p2 . . . d pn+1 and similarly for
$
˜ Then we have dp.
Y e2tη dαdβ ei(α−β)t V(∗) (P ) + V(∗) (P ) = λn+n +g(P) (2π )2 −Y × dp dp˜ (P, w, u ≡ 0)M(w) × (α, pm )
n+1, (n) j=1
1 α − ω( p j ) − iη
n +1, (n )
j=1
1 , β − ω( p˜ j ) + iη (5.13)
with
% (α, pm ) :=
& λ2 | B(q − pm )|2 dq − θ ( pm ) . α − ω(q) − iη α − ω( pm ) − iη
(5.14)
The expression n + 1, (n) on the product sign indicates that for the truncated values (*) the last fraction is not present, i.e. j runs up to n. Notice that the sum of the contributions of a gate and a ϑ inserted between m and m − 1 yields an additional factor (α, pm ) in the V -value of the P partition. Using (B.1) and (B.9) with ε = η, ε → 0 + 0, we have | B(q − pm )|2 dq − θ( pm ) ≤ C η1/2 + η−1/2 |α − λ2 (α) − e( pm )| . α − ω(q) − iη Therefore, using (B.5) and λ3 η λ2 , we have λ2 |(α) − θ ( pm )| 2 −1/2 1+ ≤ Cλ2 η−1/2 | (α, pm )| ≤ Cλ η |α − ω( pm ) + iη|
(5.15)
uniformly in α and pm . The proof shows that the cancellation between the gate and the ϑ is completely local in the graph. In particular, if we fix L locations, maybe with multiplicity, between the elements of Vn,n , and we consider all possible 2 L combinations of insertions of gates and ϑ’s at these locations, then we gain a factor λ2 η−1/2 from each location. More precisely, let υ ∈ NVn,n be a given sequence of integers, υ0 , υ1 , . . . , υ0∗ , labelled by the elements of Vn,n . The number υ j indicates how many gates or ϑ’s are inserted between the j th and ( j − 1)th vertex. Let |υ| := j υ j be the total number of insertions. A sequence S ∈ {g, ϑ}|υ| encodes whether the insertion is gate or ϑ. Fix a sequence υ and for any S ∈ {g, ϑ}|υ| we define the extended set V S consisting of Vn,n and we insert extra single or double symbols for ϑ and gate indices (determined by
34
L. Erd˝os, M. Salmhofer, H.-T. Yau
S) at the locations given by υ. In the example above, we have υ = (0, 0, . . . , 0, 1, 0 . . . 0) (i.e. qm = 1, the rest is zero), |υ| = 1, and V corresponds to S = {ϑ}, while V corresponds to S = {g}. Given a partition P ∈ Pn,n , we also define the extended partition P S on V S by simply adding the single symbols as single lumps and the double symbols as paired lumps to P. Lemma 5.5. With the notations above, for any fixed P ∈ Pn,n and υ ∈ NVn,n , |υ| 2 −1/2 V (P ) E (∗) (P). (5.16) (∗) S ≤ Cλ η S∈{g,ϑ}|υ| Proof. Notice that the sum of the contributions of a gate and a ϑ inserted at the same place between m and m −1 yields a factor of (α, pm ), while the same insertion between of each other, m
and m − 1 yields a factor (β, p˜ m ). These insertions are independent thanks to the summation over all possible S-combinations. So S V (P S ) is represented by an expression similar to (5.13), where a total factor [ (α, pm )]qm [ (β, p˜ m )]qm m=1,...,n,0∗
m= n ,..., 1,0
is inserted. The uniform estimate (5.15) for each factor gives (5.16). Next we will apply Lemmas 5.4–5.5 to prove (4.37)–(4.39). The difficulty is that these estimates hold only if the gate remains isolated even after the lumping procedure, otherwise the gain comes from the artificial recollision introduced by the lump. Recall that the lumping procedure has two steps. The original partition D0 may lump nontrivially, yielding the derived partition D, due to a few possible coincidences among the gate or recollision labels of ψ and ψ. Then the non-single elements of D (denoted by D∗ ) lump into a coarser partition imposed by A ∈ A(D∗ ) due to the connected graph formula. We will estimate the value of individual graphs only. The number of terms in the summations in (4.37)–(4.39) is bounded by O((c + 4)4 ) ≤ C K 4 . This extra factor C K 4 will be added to the factors gained in the cases discussed below to obtain (4.37)–(4.39). 5.3.2. Non-repetition graphs with a gate. To prove (4.37), we first note that the non(1),nr repetition rules in ψt,k force D to be identical with D0 unless h, h = g. In this latter case there is a gate both in the expansion of ψ and ψ, and either D = D0 or D lumps the two gate-lumps in D0 together. We first consider the case D = D0 . Then D has a lump consisting of all four gate indices, {w, w + 1, w
, w + 1}. The corresponding Feynman graphs can be identified with certain nontrivial lumpings of non-repetition graphs on Ic+2 ∪ Ic+2 , where the indices w and w + 1 are lumped together. More precisely, for a given σ ∈ Sc , w ∈ Ic , h = h = g, A ∈ Ac , V (P(A, D))c(A) = V (A, σ )c(A), (5.17) D=D0
A∈A(D∗ ) A=A
A∈Ac+2 , A=A w≡w+1(mod A)
where σ ∈ Sc+2 is the natural extension of σ where two new elements, w, w + 1, are added to the base set Ic , the indices are shifted by the embedding map swh (Sect. 4.3) and
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
35
σ (w) = w, σ (w + 1) = w + 1. The summation has at most c + 1 terms and it expresses the choice of joining w, w + 1 to one of the existing lumps in A or keeping the lump {w, w + 1} separate in A. Because of lumping w and w + 1, the partition A is non-trivial, and q(A, σ ) ≥ 1 (see (4.26)). After estimating |V (· · · )| by E(· · · ), each term on the right-hand side is estimated by the bound 1 17 3 q(A,σ ) sup E (∗)g (A, σ, u) ≤ C| log λ|2 λ 3 −( 3 d+ 2 )κ−O(δ) (5.18) u
2 (see (9.4) from [10]). that holds whenever σ ∈ Sk , A ∈ Ak and κ ≤ 34d+9 The estimate (5.18), combined with the combinatorial bound on c(A) and on the summation (Lemma 4.5), gives (4.37) in the case when D = D0 . Now we focus on the case D = D0 . If at least one of the gate-lumps, {w, w + 1} or { w, w + 1}, do not remain isolated in A, then we can repeat the argument above since P(A, D) has a non-trivial lump of size at least 4. Finally, we can assume that the gate lumps remain isolated in A. We fix A and consider the sum of four terms corresponding to h, h ∈ {g, θ }. The partition A is defined by adding the gate lump(s) to A . Note that c(A) is the same for all these four cases since replacing a θ index with an isolated gate lump adds only a single lump to A. For these partitions, we apply Lemma 5.5 (with the choice υw = υw
, all other υ’s are zero) to κ obtain a factor (Cλ1− 2 )2 . The remaining non-repetition graphs bounded by O(| log λ|2 ) by using Proposition 9.2 from [10] with q = 0. This completes the proof of (4.37).
5.3.3. Last gate. First we consider the case when the lumps {w, w + 1} ⊂ In and { w, w + 1} ⊂ In of the two first gates remain isolated in P = P(A, D). As in the previous section, by applying Lemma 5.5, we can sum up the two times two possibilities for the first components of the code h and h , i.e. sum up four Feynman diagrams that differ only by the choice of gate or θ at the w or w
position. We collect (λ1−κ/2 )2 . The remaining two gates will be isolated from the rest in P by Operation I, then half of each gate is removed by Operation III, collecting 2 λ2 | log λ|2 . Finally, (C.2) can be used to remove the remaining two halves of the gates, collecting λ2 . By (5.1), the untruncated E-values of the remaining graph are bounded by O(| log λ|2 ). The total estimate is Cλ6−(4d+1)κ−O(δ) . If only one of the lumps, {w, w + 1} or { w, w + 1}, remains isolated in P, we can still apply Lemma 5.4 to obtain a cancellation of order λ1−κ/2 from adding up the V -values of those pairs of graphs that differ only by changing this gate to θ . Note that the value c(A) is again the same for these two graphs. Next we consider those lumps among {w, w + 1} and { w, w + 1} that do not remain isolated in P. By breaking up lumps via Operation I, we can ensure that every such gate is either lumped exactly with one other gate or with a core index pair. For definiteness, let {w, w + 1} from In be such a gate. We also isolate all other gates from the rest. To do that, Operation I is used at most four times at the total expense of 4 . We distinguish three cases: Case 1. The gate {w, w + 1} is lumped with a core index-pair ( j, σ ( j)), where σ is the natural extension of σ from Ic to In . If j is next to w or w+1, say j = w+2 (Fig. 11), then both vertices of the gate can be removed by Operation III since the momenta between (w − 1, w) and (w, w + 1) do not appear in any delta function, and we gain λ2 | log η|2 from this gate. We can now remove the two gates at the end (using Operation III and (C.2) as above), collect an additional λ4 | log λ|2 . The remaining gate at { w, w + 1} can
36
L. Erd˝os, M. Salmhofer, H.-T. Yau w w+1
j
j
E* g
<
2
2
λ |logη | E * g+2
σ( j )
σ( j )
Fig. 11. Case 1. Removal of a gate lumped to an adjacent core index
be removed by Operation IV at the expense of λ−κ | log λ|. By Proposition 9.2 from [10], the remaining graphs are bounded by O(| log λ|2 ). We thus collect Cλ6−(8d+1)κ−O(δ) . If j is not next to w or w + 1 (Fig. 12), then we remove w + 1 by Operation III, break up the lump into { j, w} and { σ ( j)} by Operation I and remove the single lump { σ ( j)} by Operation II. The total price for these steps is λ2 η−1 | log η|2 . We can now again remove the two gates at the end, collect λ4 | log λ|2 and we end up with a graph with a one sided recollision, so (5.6) from Proposition 5.3 applies. We collect Cλ6−(14d+2)κ−O(δ) in this case. Case 2. The gate {w, w+1} is lumped with the other gate in In , i.e. with {n −1, n}. If w+1 and n−1 are neighbors (Fig. 13), then we can remove three vertices n−2 = w+1, n−1, n by Operation III. We collect λ3 | log η|3 . On the In side, we remove the gate that is not adjacent to 0∗ by Operation IV at the expense of λ−κ | log λ| as above. The other gate is adjacent to 0∗ ; first we remove its leg not adjacent to 0∗ by Operation III, collecting λ| log λ|. Finally, we remove the two remaining vertices, originally with indices n − 3 and n that now form single lumps and they are both adjacent to 0∗ . By using (C.2), we gain a factor λ2 . Altogether we thus collect Cλ6−(8d+1)κ−O(δ) . If w + 1 and n − 1 are not neighbors (Fig. 14), then we remove w + 1 and n, gaining λ2 | log η|2 and the remaining partition again has a one-sided recollision. The other gate adjacent to 0∗ can be removed by collecting λ2 | log λ|2 , the remaining gate collects λ−κ | log λ|, and finally we use (5.6). The result is again λ6−(12d+1)κ−O(δ) . Case 3. Finally, if neither {w, w + 1} nor { w, w + 1} falls into Case 1 or 2, then each of them either remains isolated and collects λ1−κ/2 from Lemma 5.4 or is lumped with another gate on the “opposite side” (Fig. 15). For definiteness, assume {w, w + 1} is w w+1 j
E *g
< λ |log η |2 E *g+2 η 2
σ( j )
Fig. 12. Case 1. Removal of a gate lumped to a non-adjacent core index w w+1
4
3
< C λ |log η| E *g+4
E *g
Fig. 13. Case 2. Removal of two adjacent lumped gates w w+1
E *g
2
2
< C λ |log η| E
*g+2
Fig. 14. Case 2. Removal of half of each gate lumped on the same side
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II g
E*g
37
g+1
< Cλ |log λ| E*g+2 2
2
Fig. 15. Case 3. Removal of two opposite lumped gates − 1. Then we simply remove lumped with { , + 1} in In , where is either w
or n half of each gate, say w + 1 and + 1 using Operation III, gain λ2 | log η|2 and we extend the set of core indices to include w and extend the permutation σ by adding σ (w) = . Therefore we effectively gained λ| log η| from each such gate. Finally, after having gained either λ1−κ/2 or λ| log λ| from each gate, we gain λ2 | log λ|2 from the remaining truncated graph (5.1)–(5.2) and we thus collect at least Cλ6−(8d+1)κ−O(δ) . This completes the proof of (4.38).
5.3.4. Nest. The procedure is very similar to the analysis of the last gate, so we just outline the steps. We point out that the main reason why the nested graphs are small is the cancellation between the gate and θ inside the nest. This is a different mechanism than the one used in [7]. First we consider the cases when both nests are independent, i.e. no part of the nest in In is lumped with any part of the nest in In . We will show that one can gain at least λ3−(2d+2)κ−O(δ) from each such nest. The total gain from both nests is then λ6−(4d+4)κ−O(δ) . Consider the gate {n − 2, n − 1} inside the nest. If this gate remains an isolated lump in P, then it cancels the same graph with θ up to order λ1−κ/2 by Lemma 5.4. After this cancellation, the outer shell of the nest, {n − 3, n}, becomes a gate in the reduced graph that is adjacent with 0∗ . After separating it from the rest by Operation I, if necessary, its removal yields λ2 because of the truncation. Therefore we gain at least 1 λ3−κ/2 ≤ λ3−(2d+ 2 )κ−O(δ) from this nest. If the gate {n − 2, n − 1} is not isolated in P, then we again distinguish two cases. If {n − 2, n − 1} is lumped with a core index j < n − 3 but not with its outer shell, {n − 3, n}, then we can always create a one-sided recollision in such a way that n − 2 will be paired with σ ( j) in the extended permutation, while n − 1 is removed (Operation III, gain λ| log λ|) and j is removed (Operation II, lose λ−1−κ ). The net result is λ−κ | log λ| and the outer shell of the nest, {n − 3, n}, becomes a genuine recollision. We may have to separate this from the rest of the graph by an Operation I before applying (5.7). This will effectively give λ3−2κ ζ 4d | log λ| O(1) ≤ λ3−(6d+2)κ−O(δ) for this nest. In this calculation we used only λ3−κ from (5.7), the additional λ is due to the truncation on the “other side” and will be counted there. If the gate {n − 2, n − 1} is lumped with its outer shell, then the momenta between (n − 3, n − 2), (n − 2, n − 1) and (n − 1, n) can be freely integrated, we can remove the nest completely and collect λ4 ≤ Cλ4−2dκ from this nest. We may have to use Operation I once to separate the nest from the rest of the graph. So far we have treated the cases when the two nests are independent. In the remaining cases some parts of the nests are lumped with each other. If the gate {n − 2, n − 1} is lumped with the other gate {n − 2, n − 1}, then half 2 of each gate is removed, say n − 1, n − 1 by Operation III (gain λ | log λ|2 ), the other halves are separated from the rest of the graph (Operation I) and then connected by extending the permutation σ (n − 2) = n − 2 (i.e. we include n − 2 among the core
38
L. Erd˝os, M. Salmhofer, H.-T. Yau
indices). The resulting graph has two recollisions, so (5.5) applies and the total size is λ8−3κ ζ 4d | log λ| O(1) . Finally, if {n−2, n−1} is lumped with the outer shell {n − 3, n } of the other nest, then we isolate and remove the gate {n − 2, n − 1} inside the other nest (price: λ−κ | log λ|), remove n and n − 1 each as one half of a gate, gaining λ2 and the remaining graph is a one-sided truncated recollision graph after possibly isolating the recollision lump {n − 3, n} from the rest. Thus (5.7) gives λ4−κ ζ 4d | log λ| O(1) , after a possible application of Operation I. The total size is O(λ6−(8d+2)κ−O(δ) ). Collecting the various cases, we obtain (4.39). 6. Convergence of the Ladder Diagrams to the Heat Equation We start the proof of Theorem 2.4 by noticing that ◦ ·) dξ, Wλ (t, k, O) = Vεξ A0 , O(ξ,
k≥1
with A0 being the trivial partition on Ik , where we chose the function Q(v) in the defi v) (see (4.17) for the nition of V ◦ to be ξ -dependent, namely Q(v) = Q ξ (v) := O(ξ, definition of V ◦ ). First we note that the dξ integration can be restricted to the regime {|ξ | ≤ λ−δ } with a negligible error (even after summation over k): ∗ ◦ ·) dξ, (6.1) A0 , O(ξ, Wλ (t, k, O) = ◦k + o(1) , ◦k := Vεξ 1≤k
1≤k
where use the notation
∗
· · · dξ :=
· · · 1(|ξ | ≤ λ−δ )dξ.
To see (6.1), we first recall that replacing V ◦ (· · · ) with V (· · · ) yields a negligible error even after summing up for all k (Lemma 7.1 of [10]). The linearity of the estimate in v)| guarantees the integrability in ξ since O is a Schwarz function.
Q ξ ∞ = supv |O(ξ, We then use the estimate ·) ≤ Q ξ ∞ sup E(σ = id, u) Vεξ A0 , O(ξ, ξ,u
and the uniform bound (5.1) and finally we conclude (6.1) by the arbitrarily fast decay of Q ξ ∞ in ξ . From the definition of ◦k , we have ◦k
k+1 dαdβ it (α−β)+2tη ∗ vk+1 )W 0 (εξ, v1 ) =λ e dξ dμ(v j ) O(ξ, 2 R (2π ) j=1 ' ( k+1 εξ εξ 2 × Rη β, v j − | B(v j − v j+1 )| . Rη α, v j + (6.2) 2 2
2k
j=1
To simplify the notation, in (6.2) we followed the convention that | B(v j − v j+1 )| = 1 for j = k + 1 because of the non-existence of vk+2 . Similar convention will be followed
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
39
later, also for | B(v j−1 − v j )|2 = 1 if j = 1. Note also that the measure dv j has been changed to dμ(v j ) by using the support properties of B and ψ0 (2.1). The estimates of the error terms were performed with the choice η = λ2+κ . However, ◦k , given by (6.2), is clearly independent of η; this follows from the K -identity (formula (6.2) in [10]). Therefore we can change the value of η to η := λ2+4κ for the rest of this calculation and we define R(α, v) := Rη (α, v) , with η := λ2+4κ . We also recall, that the restriction of the dαdβ integration in (6.2) to any set that contains {α, β : |α|, |β| ≤ Y = λ−100 } results in negligible errors, even after the summation over k (Lemma 7.1 of [10]). We will consider the set D := {(α, β) : |α + β| ≤ 2Y, |α − β| ≤ 2Y }. We denote by k the version of ◦k given by formula (6.2) with the dαdβ integrals restricted to D, k := λ2k D
dαdβ Integrand from (6.2) , 2 (2π )
then
| ◦k − k | = o(1).
1≤k≤K
We also remind the reader that this argument literally does not apply to the trivial k = 0 case, when the dα dβ integral in (6.2) gives free evolutions and this term should be computed directly: 0 :=
∗
2 v)W 0 (εξ, v) + o(1), dξ dv eitεv·ξ e2tλ Im θ(v) O(ξ,
(6.3)
where the error term comes from the error term in θ (v + εξ/2) − θ (v − εξ/2) = 2iI(v) + O(εξ ). By using tλ2 → ∞, the bound Im θ (v) ≤ −c1 min{| p|d−2 , | p|−1 }, (from Lemma 3.2 of [10]) and the decay of the observable, one easily obtains that | 0 | = o(1) anyway. To evaluate the integral (6.2), we need the following crucial technical lemma which is proven in the Appendix. Lemma 6.1. Let κ < 1/8, define γ := (α + β)/2 and let η satisfy λ2+4κ ≤ η ≤ λ2+κ . Then for |r | ≤ λ2+κ/4 we have,
λ2 f ( p) dp α − ω( ¯ p − r ) − iη β − ω( p + r ) + iη f ( p) δ(e( p) − γ ) d p + O(λ1/2−4κ ) f 4d,1 . (6.4) = −2πiλ2 (α − β) + 2 p · r − 2i[λ2 I(γ ) + η]
:=
40
L. Erd˝os, M. Salmhofer, H.-T. Yau
Now we compute k by applying Lemma 6.1. Denote a := (α + β)/2 and b := λ−2 (α − β). By using ε = λ2+κ/2 , η = λ2+4κ , we have
εξ εξ R β, v − dv ϒ(ξ, v) R α, v + 2 2 −2πiϒ(ξ, v) δ(e(v) − a) = dv + O(λ1/2−4κ ) ϒ 4d,1 . b + λκ/2 v · ξ − 2i[I(a) + λ4κ ]
λ2
(6.5)
In the applications, ϒ(ξ, v) will always be supported on |v| ≤ ζ , therefore the measure dv can be freely changed to dμ(v). We now replace the product of k+1 factors in the restricted version of (6.2) one by one. We need a λ2 factor for each application of (6.5), thus we need λ2k+2 in (6.2). But (6.2) contains only λ2k , the missing λ2 comes from the change of variables dαdβ = λ2 dadb. The domain of integration, (α, β) ∈ D, is replaced by the domain D ∗ := {(a, b) : |a| ≤ Y, |b| ≤ 2λ−2 Y }. For any = 1, 2, . . . , k + 1, we introduce the notation ⎛ ⎞ ∗ k+1 dadb ⎜ ⎟ Fk, := dξ dμ(v j )⎠ 2 ⎝ D ∗ (2π ) j=1 j =
2πi F j (ξ, v j )| B(v j − v j+1 )|2 δ(e(v j ) − a) × b + λκ/2 v j · ξ − 2i[I(a) + λ4κ ] j=1 2πi F (ξ, v−1 ) δ(e(v−1 ) − a) × b + λκ/2 v−1 · ξ − 2i[I(a) + λ4κ ] % εξ εξ × dμ(v )ϒ (v−1 , v , v+1 ) λ2 R α, v + R β, v − 2 2 & −2πi δ(e(v ) − a) − b + λκ/2 v · ξ − 2i[I(a) + λ4κ ] k εξ εξ 2 2 × λ R α, v j + R β, v j − F j (ξ, v j ) | B(v j − v j+1 )| 2 2 j=+1 εξ εξ ×λ2 R α, vk+1 + (6.6) R β, vk+1 − Fk+1 (ξ, vk+1 ) 2 2 −2
with
⎧ 0 (εξ, v1 ) | B(v1 − v2 )|2 for = 1 ⎨W ϒ (v−1 , v , v+1 ) := | B(v−1 − v )|2 | B(v − v+1 )|2 for 2 ≤ ≤ k ⎩ B(v1 − v2 )|2 for = k + 1 |O(ξ, vk+1 )| |
and ⎧ 0 (εξ, v) for j = 1 ⎨W F j (ξ, v) := 1 for 2 ≤ j ≤ k ⎩ O(ξ, v) for j = k + 1.
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
41
The formula (6.6) is literally valid for 2 ≤ ≤ k. For = 1 the first product and the factor in the second line are absent, for = k + 1 the factors in the last two lines are absent. We also recall the convention made after (6.2) about the intepretation of | B(v j − v j+1 )| for j = k + 1. With these notations and by introducing τ := λ2 t = λ−κ T , and W (k) := Wλ (t, k, O), we obtain the following telescopic estimate from (6.2): ∗ dadb W (k) − dξ eiτ b+2tη 2 ∗ (2π ) D k
j=1
≤
k+1
Fk, + o(1) .
(6.7)
k
Now we explain how to estimate Fk, for the general case (2 ≤ ≤ k), the modifications for the two extrema are straightforward. 0 by supremum norm and estimate all denominators in the first First we estimate W two lines by their imaginary part: 1 1 . (6.8) ≤ κ/2 4κ b + λ v j · ξ − 2i[I(a) + λ ] 2I(a) Then the v1 , v2 , . . . , v−2 variables are integrated out in this order, by using (2.4), yielding a total factor 1 from the product in the first line of (6.6). By recalling (1.9) and the estimate d
I(a) = −Im (a) ≥ c1 min{|e| 2 −1 , e−1/2 } from Lemma 3.2 of [10], the integral of v−1 is estimated trivially by 1 Ca 1/2 ≤ a . δ(e(v−1 ) − a) dμ(v−1 ) ≤ 2I(a) I(a)
(6.9)
This estimate is used if a ≤ ζ 2 /2, otherwise the integral is zero by the support of dμ, so we obtain a factor O(λ−2κ−O(δ) ). In the regime |b| ≥ λ−κ , we have |b − λκ/2 v−1 · ξ | ≥ |b|/2 using |v−1 | ≤ ζ and |ξ | ≤ λ−δ . The estimate (6.8) can thus be changed to 2|b|−1 , improving estimate (6.9) to ≤ Ca 1/2 /|b| ≤ Cζ /|b|. The integral dμ(v ) in (6.6) is estimated by O(λ1/2−4κ ) sup ϒ (v−1 , · , v+1 )4d,1 v−1 ,v+1
≤ O(λ
ζ ) = O(λ1/2−(4d+4)κ−O(δ) )
1/2−4κ 4d
(6.10)
by using (6.5) and the fact that all v j variables satisfy |v j | ≤ ζ . For = 1 and = k + 1 we also used that the initial data and the observable are Schwarz functions. For the dμ(v j ), j = + 1, + 2, . . . , k, integrals we separate the resolvents by Schwarz inequality,
42
L. Erd˝os, M. Salmhofer, H.-T. Yau k R(α, v j + · · · )R(β, v j − · · · ) j=+1
≤
k k 2 R(α, v j + · · · )|2 + |R(β, v j − · · · ) , j=+1
j=+1
and we use the successive integration scheme (see Sect. 10.1.2 of [10]) to collect a constant factor. Before we integrate out the last momentum variable, vk+1 , we perform the da db integration. We can change back the a, b variables to α, β, we perform dα dβ integrals to collect a C| log λ|2 factor since D ⊂ {|α|, |β| ≤ 2Y }. This argument applies unless = k + 1 and the last line in (6.6) is absent. In this case, however, ≥ 2 (the k = 0 case is treated separately, see (2.23)), and then the denominator with v−1 in the second line of (6.6) is present. We use the a, b variables. Recall that δ(e(v−1 ) − a) restricts the domain of the da integration to |a| ≤ ζ 2 /2, giving a contribution O(ζ 2 ). The domain of the b integral is larger, |b| ≤ 2λ−2 Y , but in the regime |b| ≥ λ−κ we have collected an additional |b|−1 factor in (6.9), thus the db integration contributes at most with a factor O(λ−κ ). Finally we integrate vk+1 by using the integrability of Fk+1 = O and the ξ integral gives a factor O(λ−3δ ). By collecting these estimates, we arrive at k+1
Fk, ≤ Cλ1/2−(4d+9)κ−O(δ)
1≤k
and that is negligible, since κ < 1/(8d + 18). Now we focus on the main term on the left-hand side of (6.7). First we extend the db integration from |b| ≤ 2λ−2 Y to R. It is easy to see that the error is negligible; all denominators can be bounded by |b|/2 and the result from the db integral in the region |b| ≥ 2λ−2 Y , db ≤ λ2k , k+1 |b|≥2λ−2 Y |b| is negligible even after multiplying the C k from the dv j integrals. We can also extend the da integration from [−Y, Y ] to R, since, due to the factor δ(e(v j ) − a) and the cutoff in v j , the integrand is zero for |a| ≥ Y . Now we write ∞ −i κ/2 = e−iτ j (b+λ ...) dτ j b + λκ/2 v j · ξ − 2i[I(a) + λ4κ ] 0 and perform the db integration. We obtain
⎛ ⎞ ⎛ ⎞ k+1 k+1 da −2τ I (a)−τ λ4κ ⎝ ∞ ⎠ ⎝ e W (k) = dξ dt j δ τ − τj⎠ 2π R 0 k
∗
j=1
vk+1 )W 0 (εξ, v1 ) + o(1). × O(ξ,
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
43
We now replace dμ(v j ) with dv j and remove the cutoff in ξ to perform the Fourier transform. We also replace W0 with F0 and remove the k < K cutoff from the summation: ⎛ ⎞ ⎛ ⎞ ∞ k+1 ∞ k+1 W (k) = da e−2τ I (a) ⎝ dτ j ⎠ δ ⎝τ − τj⎠ dX k
R
k=0
×
⎛
dv1 δ(e(v1 ) − a) ⎝
j=1 0
k+1
j=1
⎞
B(v j − v j−1 )|2 δ(e(v j ) − a)⎠ dv j 2π |
j=2
× O(X, vk+1 )F0 X − (2π )−1 λκ/2 (
τ j v j ), v1 + o(1)
(6.11)
j
with initial data 0 (v)|2 . F0 (X, v) := δ(X )|ψ 0 (v1 ) is compactly supported The replacement of dμ(v j ) with dv j is justified since ψ and thus all other v j ’s are restricted to a compact energy range by the delta functions " j δ(e(v j ) − a). The removal of the ξ -cutoff is allowed since the integrand of the dξ -integral can be majorized by
k da −2t I (a) 2I(a)t 0 (εξ, v1 ) sup |O(ξ, v)| e dv1 δ(e(v1 ) − a)W k! v R 2π k≤K
v)| ≤ sup |O(ξ,
(6.12)
v
Here we whose integral vanishes in the regime |ξ | ≥ λ−δ due to the assumptions on O. used (2.4) to perform the v j integrations successively and the time integration yielded 0 (εξ, v) with |ψ 0 (v)|2 comes from the uniformly integraτ k /k!. The replacement of W 0 (· ± εξ ) − ψ 0 (·) → 0 as ξ ble bound (6.12) and from the uniformity of the limit ψ runs over any compact set. Finally, the removal of the k ≤ K cutoff in the summation follows from the same majorization as (6.12) together with
2I(a)τ k (Cτ ) K ≤ ≤ (Cλδ ) K → 0. k! K! k>K
For a fixed energy e > 0 we consider the continuous time Markov process {v(t)}t≥0 on the energy surface e with generator (1.12). This process is exponentially mixing with the uniform measure on e being the unique invariant measure (see Lemma A.1 in ψ the Appendix). Let Ee denote the expectation value with respect to this process starting from the initial state ψ = ψ0 given by the normalized measure (v)|2 δ(e(v1 ) − e) dv |ψ
|2 (e) |ψ on e (for the notations, see Sect. 1). Let Ee denote the expectation with respect to the |2 (e)de be the energy distribution of ψ. The co-area equilibrium. Let dμψ (e) = |ψ formula, ∞ 2
| (e) de = |ψ (v)|2 dv |ψ 0
and ψ ∈ L 2 guarantee that dμψ is absolutely continuous.
44
L. Erd˝os, M. Salmhofer, H.-T. Yau
From (6.11) and τ = λ−κ T we have ∞ W (k) = Eeψ O λκ/2 x(τ ), v(τ ) dμψ (e) + o(1) k
0
with x(τ ) := 0
τ
1 v(s)ds. 2π
Due to the exponential mixing and the continuity of O, the replacement of Eψ with the equilibrium measure Ee gives a negligible error since τ → ∞. By the central limit theorem for additive functionals of exponentially mixing Markov chains, λκ/2 x(τ ) converges to a centered Gaussian random variable with covariance matrix τ
λκ Ee λκ/2 x(τ ) ⊗ λκ/2 x(τ ) = v(s) ⊗ v(s E ) dsds → 2T D(e). e (2π )2 0 Since the equilibrium measure is uniform, the covariance matrix is diagonal, Di j (e) = De δi j . The diffusion coefficient, De , is finite and positive. This proves Theorem 2.4. A. Mixing Properties of the Boltzmann Generator The Boltzmann velocity process with generator L e introduced in (1.12) enjoys very good statistical properties. The proof uses standard arguments which we only indicate below. Lemma A.1. For each e > 0 the Markov process {v(t)}t≥0 with generator L e is uniformly exponentially mixing. The unique invariant measure is the uniform distribution, [ · ](e)/[1](e), on the energy surface e . Sketch of the proof. Let P t (u, A) be the transition kernel for any u ∈ e , A ⊂ e . Since the transition rate σ (u, v) is continuous on e and 0 ∈ supp ( B) holds, there exists an √ d open set S ⊂ R , diam(S) ≤ 2e and there exists a δ = δ(e) > 0 such that σ (u, v) ≥ δ whenever u − v ∈ S, u, v ∈ e . Since the state space e is compact it follows that the transition kernel P t satisfies a uniform Doeblin-type condition, 1 P t (u, A) dt ≥ C(e)|A| A ⊂ e , inf u∈e 0
with some e-dependent positive constant, where |A| is the restriction of the Lebesgue measure (on e ) of the set A. It is clear that the Markov process {v(t)}t≥0 is irreducible and aperiodic, therefore it is uniformly exponentially mixing. Moreover, the rate of the mixing is uniform as e runs through a compact energy interval since in this case C(e) is uniformly separated away from zero. It is easy to see that the uniform measure on e is invariant and by exponential mixing it is the only invariant measure. B. Estimates on Propagators B.1. Proof of Lemma 2.1. The following lemma proves (2.6) and (2.7). The proof of (2.8) is analogous but easier and will be omitted. Lemma B.1. Let κ < 1/6 and η satisfying λ2+4κ ≤ η ≤ λ2+κ . For any 0 ≤ a ≤ 1 we have the following approximation result: 2−a 1 1 − |h( p − q)|d p α − ω( p) + iη α − e( p) − λ2 (α) + iη |h( p − q)| ≤ Cλ1−6κ d p. (B.1) |α − e( p) − λ2 (α) + iη|2−a
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
45
Moreover, for any 0 ≤ a < 1 we have
and
|h( p − q)| d p Ca h 2d,0 λ−2(1−a) ≤ √ 2 2−a |α − e( p) − λ (α) + iη| α a/2 |q| − 2|α|
(B.2)
Ca h 2d,0 log λ log α |h( p − q)| d p . ≤ √ |α − e( p) − λ2 (α) + iη| α 1/2 |q| − 2|α|
(B.3)
Proof. To prove (B.1), we rewrite it as 2−a 1 1 |h( p − q)| d p α − ω( p) + iη − α − e( p) − λ2 (α) + iη 2−a λ2 ((e( p)) − (α)) = |h( p − q)| d p . (B.4) (α − ω( p) + iη)(α − e( p) − λ2 (α) + iη) From the Hölder continuity, |(α) − (α )| ≤ C|α − α |1/2
(B.5)
(Lemma 3.1 from [10]), we can bound (B.4) by 2−a 1/2 λ2 |e( p) − α| |h( p − q)|d p. (α − ω( p) + iη) (α − e( p) − λ2 (α) + iη) Since |e( p) − α|1/2 ≤ |ω( p) − α|1/2 + O(λ), this integral is bounded by (λ2 η−1/2 + λ3 η−1 )2−a
|h( p − q)| d p . |α − e( p) − λ2 (α) + iη|2−a
Notice that in these estimates we used that the imaginary part of ω( p) is negative. To prove the estimates (B.2) and (B.3), we rewrite the integrals by the co-area formula (0 ≤ a ≤ 1): ∞ |h( p − q)|d p ds = (s) (B.6) 2 2−a 2 |α − e( p) − λ (α) + iη| |α − s − λ (α) + iη|2−a 0 with
(s) :=
√ | p|= 2s
1 |h( p − q)|d p =√ |∇e( p)| 2s
√ | p|= 2s
Using the decay properties of h, we have √ s
h 2d,0 , · (s) ≤ √ |q| − 2s s
|h( p − q)|d p.
46
so
L. Erd˝os, M. Salmhofer, H.-T. Yau
|h( p − q)|d p |α − e( p) − λ2 (α) + iη|2−a √ ∞ s ds ≤ C h 2d,0 . √ s |q| − 2s |α − s − λ2 (α) + iη|2−a 0 √ For a = 1 the last integral can be directly estimated by C log η α −1/2 |q| − 2|α| log α , yielding (B.3). To prove (B.2), i.e. for a < 1, we recall that = R − iI with non-negative real I and R. The last integral is estimated by √ ∞ s ds I := √
1−a/2 0 s |q| − 2s ( α − s)2 + (λ2 I( α ) + η)2 with α := α − λ2 R(α). We used the Hölder continuity of I, λ2 I(α) = λ2 I( α ) + O(λ3 ) and the fact that the error can be absorbed into η. First we assume that | α | ≤ 1, then the estimate on (α) from Lemma 3.2. of [10] yields I( α ) ≤ c1 | α |1/2 . The I integral can be estimated √ √ ∞ 2 1 s ds s ds I ≤ . √
1−a/2 + |q| 0 ( s |q| − 2s s 2−a 2 α − s)2 + (λ2 | α |1/2 + η)2 √ The second term is bounded by |q| −1 ∼ |q| − 2|α| −1 . In the first term we consider
:= λ2 | two cases. If | α| ≤ β α |1/2 + η, then √ 2 s ds ≤ +
1−a/2
| α −s|≤2β | α −s|≥2β 0 α |1/2 + η)2 ( α − s)2 + (λ2 | √ ds −(2−a)
sds + ≤β α − s|3/2−a
| |s|≤2β | α −s|≥β
a−1/2 ) ≤ O(λ−2(1−a) ), = O(β
≥ η ≥ λ3 in by using that in the second regime s and |s − α | are comparable and that β the last step. This proves (B.2) for | α | ≤ 1. Next we consider the regime | α | ≥ 1, then I( α ) ≤ c1 | α |−1/2 and we have 1(| α − s| ≥ 1/2)ds I ≤ √ 1/2 s |q| − 2s | α − s|2−a 1(| α − s| ≤ 1/2)ds + √
1−a/2 . 1/2 α − s)2 + (λ2 | s |q| − 2s ( α |−1/2 + η)2 √ The first integral is bounded by Ca α −1/2 |q| − 2|α| −1 , by using that α ∼ α . The second integral is bounded by 1/2 ds 1 Ca λ−2(1−a) ≤ , √ √
1−a/2 α 1/2 |q| − 2α −1/2 [s 2 + (λ2 | α a/2 |q| − 2α α |−1/2 + η)2 and this completes the proof of (B.2).
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
47
We now prove the more accurate estimate (2.9). We have λ2 1 λ2 Im . = 2 2 2 |α − ω( p) − iη| λ I(e( p)) + η α − e( p) − λ R(e( p)) − i(λ2 I(e( p)) + η) From the resolvent identity and with the notations e = e( p), α = α − λ2 R(α), the last term equals to (I ) + (I I ) + (I I I ) with λ2
1 , (B.7)
α − e − i(λ2 I( α ) + η) λ2 (I(e) − I( 1 α )) λ2 Im , (I I ) := − 2 λ I( α) + η λ2 I(e) + η α − e − λ2 (e) − iη & % λ2 λ2 ((α) − (e)) 1 (I I I ) := − 2 . Im λ I( α) + η
α − e − i(λ2 I(α) + η) α − e − λ2 (e) − iη $
Our goal is to estimate | B( p − q)|2 (I ) + (I I ) + (I I I ) d p. We recall two continuity properties of ε (α, r ) from Lemma 3.1 of [10]: (B.8) |ε (α, r ) − e (α, r )| ≤ C |r | − |r | , (I ) :=
λ2 I( α) + η
Im
|ε (α, r ) − ε (α , r )| ≤ C(|ε − ε | + |α − α |)ε−1/2
(B.9)
if ε ≥ ε > 0. In estimating the integral of term (I), we first use (B.8) to change q to
q with e( q) = α , then we use (B.9) with ε → 0 + 0 and ε = λ2 I( α ) + η = O(λ2 ) to obtain λ2 I( α ) + O(λ) + O(| α − e(q)|1/2 ) | B( p − q)|2 (I ) d p = 2 λ I( α) + η ≤ 1 + O(λ2 η−1 [λ + |α − ω(q)|1/2 ]). In the second term of (B.7) we use (B.5) and we drop the positive I( α ) and I(e) terms in the denominators λ2 2 |λ2 I(e( p)) + η| | α − e( p)|1/2 |(I I )| ≤ C η | α − e( p) + λ2 (R(α) − R(e( p)))|2 + |λ2 I(e( p)) + η|2 λ2 2 | α − e( p)|1/2 ≤ Cλ2 , η | α − e( p) + λ2 (R( α ) − R(e( p)))|2 + η2
2 α ) − R(α)) = O(λ6 ) η2 based upon (B.5). where we used that λ2 (R( To perform the d p integration, we distinguish two regimes depending on whether | α − e( p)| is bigger or smaller than K λ4 for a sufficiently large fixed K . When | α − e( p)| ≥ K λ4 , then λ2 |R( α ) − R(e( p))| < 21 | α − e( p)|, hence |(I I )| ≤ Cλ2
λ2 η
2
η−1/2 , | α − e( p)| + η
and, by using a bound analogous to (B.3), the corresponding integral is bounded by | B( p − q)|2 d p Cλ6 η−5/2 ≤ O(λ6 η−5/2 | log η|). | α − e( p)| + η
48
L. Erd˝os, M. Salmhofer, H.-T. Yau
When | α − e( p)| ≤ K λ4 , then we can trivially estimate |(I I )| ≤ C(λ2 η−1 )4 and after the co-area formula, the volume factor is given by ∞ √ 1(| α − s| ≤ K λ4 ) s S(s)ds = O(λ4 ), 0
with
S(e) :=
S d−1
√ | B( 2e(φr − φ))|2 dφ,
where φr ∈ S d−1 is fixed. Recalling the properties of$ S(e) from the proof of Lemma 3.2 in [10], we see that the contribution to the integral | B( p − q)|2 (I I ) d p is of order 3 −1 4 O((λ η ) ). Finally, the last term in (B.7) is estimated as 2 λ 1 |α − e|1/2 . |(I I I )| ≤ Cλ2 η | α − e| + η |α − e + λ2 R(e)| + η In the regime where |α − e( p)| ≥ K λ2 (with some large K ) we obtain 2 2 λ |α − e|1/2 1 2 −1/2 λ ≤ Cλ η |(I I I )| ≤ Cλ2 2 η (|α − e| + η) η |α − e| + η and after integration we collect O(λ4 η−3/2 | log η|). In the regime where |α − e( p)| ≤ K λ2 we have |(I I I )| ≤ O(λ5 η−3 ) and the volume factor is O(λ2 ), therefore the integral is O(λ7 η−3 ). Collecting the error terms we arrive at the proof of Lemma 2.1. B.2. Proof of Lemma 6.1. We can assume that f is a real function and write f ( p) = p −2d g( p) with g 2d,0 < C f 4d,1 . We can restrict the integration regime in (6.4) to | p| ≤ λ−1 since the contribution of the outside regime is O(λ2d ) by a Schwarz inequality (to separate the two denominators) and a trivial application of Lemma B.1 with a = 0. This large momentum cutoff will be done with the insertion of a function χ (λ p ) with a smooth, compactly supported χ , χ ≡ 1 on [−1, 1]. We can also assume that |α − e( p − r )| ≤ λ, |β − e( p + r )| ≤ λ, otherwise at least one of the denominators can be estimated by O(λ−1 ) and the other one integrated out by (2.6) to give O(λ| log λ|). Since |α − e( p − r )| ≥ |α − e( p)| − C(| p| + |r |)|r | ≥ |α − e( p)| − O(λ1+κ/4 ), we obtain that |α − e( p)| ≤ 2λ and similarly |β − e( p)| ≤ 2λ, in particular |α − β| ≤ 4λ and |γ − e( p)| ≤ 2λ. ¯ ) − iη. The We replace the first denominator of (6.4) by α − e( p) + p · r − λ2 (γ error term of this replacement, by the resolvent expansion, is bounded by λ9/2 χ (λ p )| f ( p)|d p , ¯ ) − iη| |β − ω( p + r ) + iη| |α − ω( ¯ p − r ) − iη| |α − e( p) + p · r − λ2 (γ (B.10) where we have used the estimate ¯ ) − iη + O(λ5/2 ), α − e( p − r ) − λ2 θ( p − r ) − iη = α − e( p) + p · r − λ2 (γ which follows from the above restrictions on the integration domain |r | ≤ λ2+κ/4 and the Hölder continuity (B.5). To estimate the error term (B.10), we bound the β denom-
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
49
inator trivially by η−1 and use the Schwarz inequality to separate the remaining two denominators 1 1 1 + . ≤ 2 |α − ω( ¯ p − r ) − iη| |α − e( p) + · · · | |α − ω( ¯ p − r ) − iη| |α − e( p) + · · · |2 The integral of the first term can be bounded by Lemma B.1 with a = 0; for the second term we rewrite e( p) + p · r = e( p + r ) − 21 r 2 and use (2.8) after a shift in α and p. We arrive at = 0 + O(λ1/2−4κ ) with λ2 χ (λ p ) f ( p)d p . 0 = ¯ ) − iη β − e( p) − p · r − λ2 (γ ) + iη α − e( p) + p · r − λ2 (γ To compute 0 we can choose a coordinate system where the vector r points in the n th direction: r = |r |(0, . . . , 0, 1). We can write d p
0 = λ2 2 R α − β + 2|r | p − 2i[λ I(γ ) + η] % 1 × 2 n−1 β − e( p) − p |r
| − λ (γ ) + iη R & 1 χ (λ p ) f ( p)d p⊥ . (B.11) − ¯ ) − iη α − e( p) + p |r | − λ2 (γ Lemma B.2. Let F be a C 1 -function on R with |F(Q)| ≤ C Q −2 and let ∞ F(Q) dQ Y (z) := z −Q 0 for any z = α + iε with 0 < ε ≤ 1/2. Then |Y (z) − Y (z )| ≤ C|F(0)| | log z − log z | + |z − z || log ε| F 2d,1 , where
z
=
α
+ iε
and ε ≥
ε
(B.12)
> 0.
Proof. This lemma is essentially Lemma 3.10 in [7]. For completeness we recall the proof. Choose a branch of the complex logarithm on the upper half plane and use integration by parts: ∞
|Y (z) − Y (z )| ≤ |F(0)|| log z − log z | + F (Q) log(z − Q) − log(z − Q) dQ . 0
The second term is estimated by (z,z )
d|ξ | 0
∞
|F (Q)| dQ, |ξ − Q|
where (z, z ) is any path in the upper half plane that connects z and z and d|ξ | is the arclength measure. A simple exercise shows ∞ |F (Q)| dQ ≤ C F 2d,1 | log(Im ξ )|. |ξ − Q| 0 Choose a path from z = α + iε to α + iε then to α + iε along straight line segments. After integration we obtain (B.12).
50
L. Erd˝os, M. Salmhofer, H.-T. Yau
We now change the denominators in the square bracket in (B.11) to γ − e( p) ± iη. This requires a change of order O(λ) in the denominators using the estimates on |α − γ | and p |r |. With the help of Lemma B.2 such change yields an error of order λ3 | log λ|η−1 f 2d,1 in 0 . After these changes, we can remove the cutoff χ (λ p ) at a price of O(λ2d ) as before, and we have d p
0 = λ2 2 R α − β + 2|r | p − 2i[λ I(γ ) + η] % & 1 1 − f ( p)d p⊥ (B.13) × γ − e( p) − iη Rd−1 γ − e( p) + iη modulo negligible errors involving f 4d,1 . The inner integral is evaluated as
∞
I := i Im 0
f ∗ (u, p )u
d−3 2
du
∗
with f (u, p ) :=
γ − 21 p 2 − 21 u + iη
S d−2
f (u 1/2 θ, p )dθ.
For p in the range γ − 21 p 2 ≤ −λ, we have |I | ≤
Cη f 4d,0 p 2d
∞
0
u
d−3 2
du
|λ + 21 u|2 u 2d
= O(ηλ−1 ),
f∗
by using the decay of inherited from f . The contribution of this regime to 0 is therefore of order λ| log λ|, and hence negligible. In the regime |γ − 21 p 2 | ≤ λ one can estimate |I | ≤ C p −2d . The d p -volume is at most O(λ1/2 ), so the contribution of this regime to 0 is at most of order λ5/2 η−1 , and hence also negligible. Finally, we can concentrate on the regime γ − 21 p 2 ≥ λ. We can use the estimate (for ε > ε > 0) ∞ g(x) Im dx = −πg(0) + O(ε /ε) + O(ε | log ε |) −ε x + iε if g ∈ C 1 with a bounded derivative. We obtain, with ε = γ − 21 p 2 , ε = η, that I = −2πi(2γ − p 2 )
d−3 2
f ∗ (2γ − p 2 , p ) + O(λ1+4κ ),
where the error is integrable in p . Therefore it is negligible in 0 . Substituting the main term into (B.13), we obtain the main term in (6.4). C. General Estimates on Circle Graphs We define four operations on a partition given on the vertex set of a circle graph on N vertices and we estimate how the E-value of the partition changes. Operation I was already defined in Sect. 9 of [10], here we repeat the definition and the corresponding estimate for convenience. Operation I: Breaking up lumps. Consider a Feynman graph on N vertices (Sect. 4.1). Given a partition of the set V \ {0, 0∗ }, P = {Pμ : μ ∈ I (P)} ∈ PV , we define a new partition P∗ by breaking up one
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
51
of the lumps into two smaller nonempty lumps. Let Pν = Pν ∪ Pν with Pν ∩ Pν = ∅ and P∗ = {Pν , Pν , Pμ : μ ∈ I (P) \ {ν}}. In particular I (P∗ ) = I (P) ∪ {ν , ν } \ {ν} and m(P∗ ) = m(P) + 1. The following estimate was proven in Lemma 9.5 of [10]. Lemma C.1. With the notation above, we have E (∗)g (P, u, α) ≤ dr E (∗)g (P∗ , u∗ (r, ν), α), |r |≤N ζ
where the new set of momenta u∗ = u∗ (r, ν) is given by u ∗μ := u μ , μ ∈ I (P) \ {ν} and u ∗ν = u ν − r , u ∗ν = r . In our estimates we will always have N ≤ 2K and then sup E (∗)g (P, u, α) ≤ sup E (∗)g (P∗ , u, α) u
with :=
[C K ζ ]d
=
u
O(λ−2dκ−O(δ) )
(see (2.18) and (4.8)).
Operation II: Removing the lump of a single vertex. Let v ∈ V \ {0, 0∗ } be a vertex and let P ∈ PV such that Pσ = {v} for some σ ∈ I (P), i.e. the single element set {v} is a lump. Define V ∗ := V \ {v}, L(V ∗ ) := L(V) ∪ {(v − 1, v + 1)} \ {(v − 1, v), (v, v + 1)}, i.e. we simply remove the vertex v from the circle graph and connect the vertices v − 1, v + 1. Let P∗ ∈ PV ∗ , P∗ := P \ { {v} } be P after simply removing the lump {v}. In particular, I (P∗ ) = I (P) \ {σ }. Lemma C.2. With the notations above sup E (∗)g (P, u, α) ≤ Cλη−1 sup E (∗)g+1 (P∗ , u∗ , α). u∗
u
(C.1)
If both neighbors of 0∗ , v = v , form single lumps in P, then both of these lumps can be simultaneously removed to obtain a partition P∗ := P \ {{v}, {v }} with the estimate sup E ∗g (P, u, α) ≤ Cλ2 sup E g+2 (P∗ , u∗ , α). u∗
u
(C.2)
Proof. The factor λ in the estimate (C.1) is due to the fact that each vertex (apart from 0 and 0∗ ) carries a factor λ and |V ∗ | = |V| − 1. Let Pν be the lump of the vertex v − 1 right before to v in the circular ordering and assume v − 1 = 0, 0∗ (otherwise we consider v + 1 and the proof is slightly modified). We use the trivial bound |αev−
1 ≤ η−1 − ω(wev− ) + iη|
(C.3)
(recall that Im ω ≤ 0 from Lemma 3.2 of [10]) and the bound | B(wev+ − wev− )| | B(wev− − we(v−1)− )| ≤
C wev+ − we(v−1)− 2d
(C.4)
(uniformly in wev− ) to obtain the necessary decay between the two newly consecutive momenta. The same bound holds if some of the B(·) on the left-hand side is replaced with · −2d due to the set G. Now we integrate wev− to obtain a new delta function from ±we − u ν dμ(wev− )δ wev+ − wev− − u σ δ wev− + ⎛ ≤ δ ⎝wev+ +
e∈L ± (Pν ) : e=ev−
e∈L ± (Pν ) : e=ev−
⎞
±we − (u σ + u ν )⎠
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L. Erd˝os, M. Salmhofer, H.-T. Yau
and clearly wev+ +
e∈L ± (Pν ) : e=ev−
±we =
±we .
e∈L ± (Pν∗ )
The new auxiliary momentum associated to Pν is u ν +u σ and Pσ disappeared, so the sum of the auxiliary momenta remain unchanged, and (4.5) continues to hold. This proves (C.1). For the proof of (C.2), if v and v are the vertices on both sides of 0∗ , then they can be removed and their neighbours can be connected directly to 0∗ yielding a non-truncated value of a graph with two vertices less. This gives a factor λ2 . The appropriate redefinition of the auxiliary momenta is straightforward. Operation III: Removing half of a gate. Let v, v + 1 ∈ V \ {0, 0∗ } be two subsequent vertices and let P ∈ PV be such that v ≡ v+1 (mod P). In the main application this will arise when v, v+1 are connected and form a gate. Define V ∗ := V \{v +1}, L(V ∗ ) := L(V)∪{(v, v +2)}\{(v, v +1), (v +1, v +2)}, i.e. we simply remove the vertex v + 1 from the circle graph with the adjacent edges and add a new edge between the vertices v, v + 2. Let P∗ ∈ PV ∗ be identical to the partition P except that v + 1 is simply removed from its lump. In particular, I (P) = I (P∗ ). Lemma C.3. With the notations above E (∗)g (P, u, α) ≤ Cλ| log η| E (∗)g+1 (P∗ , u, α). Proof. Note that the momentum wev+ of the edge between v and v + 1 does not appear in the delta functions in the definition of E (∗)g (P, u, α) (see (4.9)). Before integrating out this momentum in (4.9), we use the bound | B(wev+ − wev− )| | B(wev− − we(v−1)− )| ' ( 1 1 C + ≤ wev+ − we(v−1)− 2d wev+ − wev− 2d wev− − we(v−1)− 2d to ensure the decay between the momenta wev− and we(v+2)− , that are consecutive in the new graph. The same bound holds if some of the B(·) is already replaced with · −2d . The integration of wev− yields C| log η| by using (2.6). Operation IV: Removing an isolated gate. Let v, v + 1 ∈ V \ {0, 0∗ } be two subsequent vertices and let a partition P ∈ PV be such that v ≡ v + 1 (mod P). Define V ∗ := V \ {v, v + 1}, L(V ∗ ) := L(V) ∪ {(v − 1, v + 2)} \ {(v − 1, v), (v, v + 1), (v + 1, v + 2)}, i.e. we simply remove the gate. Let P∗ ∈ PV ∗ be P after removing the lump {v, v + 1}. Combining Operations III and II, we immediately obtain: Lemma C.4. With the notations above sup E (∗)g (P, u, α) ≤ Cλ2 η−1 | log η| sup E (∗)g+2 (P∗ , u∗ , α). u
u∗
Quantum Diffusion of Random Schrödinger Evolution in the Scaling Limit II
53
Note that this bound is not optimal. The removal of a gate affects the value of the graph only by a constant factor, but the corresponding estimate is more complicated and we do not aim at optimizing the value of κ. Acknowledgement. The authors are grateful for the financial support and hospitality of the Erwin Schrödinger Institut, Vienna, Max Planck Institut, Leipzig, Stanford University and Harvard University, where part of this work has been done.
References 1. Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993) 2. Aizenman, M., Sims, R., Warzel, S.: Absolutely continuous spectra of quantum tree graphs with weak disorder. Commun. Math. Phys. 264, 371–389 (2006) 3. Anderson, P.: Absences of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958) 4. Bourgain, J.: Random lattice Schrödinger operators with decaying potential: some higher dimensional phenomena. Lecture Notes in Mathematics, Vol. 1807, Berlin-Heidelberg: Springer, 2003, pp. 70–99 5. Chen, T.: Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3. J. Stat. Phys. 120(1–2), 279–337 (2005) 6. Denisov, S.A.: Absolutely continuous spectrum of multidimensional Schrödinger operator. Int. Math. Res. Not. 2004(74), 3963–3982 (2004) 7. Erd˝os, L., Yau, H.-T.: Linear Boltzmann equation as the weak coupling limit of the random Schrödinger equation. Commun. Pure Appl. Math. LIII, 667–735 (2000) 8. Erd˝os, L., Salmhofer, M., Yau, H.-T.: Towards the quantum Brownian motion. Lecture Notes in Physics, 690, In: Mathematical Physics of Quantum Mechanics, Selected and Refereed Lectures from QMath9, Asch, J., Joye, A. (eds) Berlin-Heidelberg: Springer, 2006, pp. 233–258 9. Erd˝os, L., Salmhofer, M., Yau, H.-T.: Quantum diffusion for the Anderson model in scaling limit. Submitted to Ann. Inst. H. Poincaré (2006), available at http://xxx.lanl.gov/abs/math-ph/0502025, 2005 10. Erd˝os, L., Salmhofer, M., Yau, H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit I. The non-recollision diagrams. Available at http://xxx.lanl.gov/abs/math-ph/0512014, 2005 11. Froese, R., Hasler, D., Spitzer, W.: Absolutely continuous spectrum for the Anderson model on a tree: a geometric proof of Klein’s theorem. Preprint http://xxx.lanl.gov/math-ph/051150, 2005 12. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983) 13. Goldsheid, I.Ya., Molchanov, S.A., Pastur, L.A.: A pure point spectrum of the one dimensional Schrödinger operator. Funct. Anal. Appl. 11, 1–10 (1997) 14. Klein, A.: Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res. Lett. 1, 399–407 (1994) 15. Lukkarinen, J., Spohn, H.: Kinetic limit for wave propagation in a random medium. Preprint. http://xxx.lanl.gov/math-ph/0505075, 2005 16. Rodnianski, I., Schlag, W.: Classical and quantum scattering for a class of long range random potentials. Int. Math. Res. Not. 5, 243–300 (2003) 17. Schlag, W., Shubin, C., Wolff, T.: Frequency concentration and location lengths for the Anderson model at small disorders. J. Anal. Math. 88, 173–220 (2002) 18. Spohn, H.: Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17(6), 385–412 (1977) Communicated by M. Aizenman
Commun. Math. Phys. 271, 55–91 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0175-1
Communications in
Mathematical Physics
Pattern Densities in Non-Frozen Planar Dimer Models Cédric Boutillier Centrum voor Wiskunde en Informatica, P.O. Box 94079, NL-1090 GB Amsterdam, The Netherlands. E-mail:
[email protected] Received: 1 February 2006 / Accepted: 23 June 2006 Published online: 23 January 2007 – © Springer-Verlag 2007
Abstract: In this paper, we introduce a family of observables for the dimer model on a bi-periodic bipartite planar graph, called pattern density fields. We study the scaling limit of these objects for non-frozen Gibbs measures of the dimer model, and prove that they converge to a linear combination of a derivative of the Gaussian massless free field and an independent white noise.
1. Introduction A dimer configuration C of a graph G is a subset of edges of G such that every vertex of G is incident to exactly one edge of C. The dimer model is a system from statistical mechanics, obtained by endowing the set of all possible dimer configurations with a probability measure. It was introduced in the 1930s [5] to give a model for adsorption of diatomic molecules (dimers) on the surface of a crystal, represented by a planar periodic graph. This model of statistical mechanics is one of the rare models that can be solved exactly. For an introduction to the dimer model, see for example [16]. In this paper, we study the scaling limit of pattern density fields in the dimer model, when the mesh of the graph goes to zero. The random density field of a pattern (made of a finite number of molecules) associates to every domain D of the plane, the number of copies of this pattern seen in D. We prove that the fluctuations of these random fields are Gaussian when the mesh size goes to zero. But before describing our results in detail, we give some background on the dimer model.
1.1. The dimer model. Let G be a graph with at least one dimer configuration. Suppose for the moment that G is finite. One can define a Boltzmann probability measure on the dimer configurations of G as follows: positive weights we are assigned to the edges e of G and the probability of a dimer configuration C is chosen to be proportional to the
56
C. Boutillier
product of the weights of the edges it contains: 1 P [C] = we , (1) Z e∈C where the normalizing factor Z = C e∈C we is called the partition function. Kasteleyn showed [10] that if G is planar, the partition function Z can be expressed as the Pfaffian of a weighted adjacency matrix for a well-chosen orientation of the graph. If moreover the graph is bipartite, Z reduces to the determinant of a certain matrix K, called the Kasteleyn operator, a cousin of the adjacency matrix of G, whose rows are indexed by white vertices, and columns by black vertices. In particular, it has the property that if e = (w, b), |K(w, b)| = we . See [18] to see how to define K. Because of the correspondence between determinants of submatrices of a matrix and those of its inverse, the correlations are given by determinants of submatrices of K−1 [11]. When the graph G is infinite, it may have an infinite number of dimer configurations, and it is not possible anymore to define directly a Boltzmann measure. For planar bipartite Z2 -periodic graphs, endowed with periodic weights on edges, this notion is replaced by that of Gibbs probability measure. A probability measure µ on the space of dimer configurations of a Z2 -periodic graph (endowed with the usual product σ -algebra) is called a Gibbs measure if it has the following properties: – it is ergodic under the action of Z2 by translation, – if the dimer configuration is fixed in an annular region, then the random dimer configuration inside and outside the annulus are independent, and the induced probability measure inside the annulus is the Boltzmann measure defined above. We will assume from now on that G is a planar bipartite Z2 -periodic graph, with periodic weights on edges, and that the quotient G/Z2 is a finite graph. A dimer configuration M can be interpreted as a discrete surface via the height function [25] defined on the faces of G. The slope of a Gibbs measure µ is the average slope of the corresponding random surface model, which is well defined, since µ is ergodic. The set of possible slopes for Gibbs measures, called the Newton Polygon of the dimer model on G, is a polygonal convex set of R2 with vertices in Z2 . Sheffield proved [23] that for any (s, t) in the interior of the Newton polygon, there exists a unique ergodic Gibbs measure µ(s,t) on dimer configurations of G with slope (s, t). It is sometimes more convenient to parametrize the set of Gibbs measures with an external magnetic field B = (Bx , B y ), instead of the slope (s, t), defined as the conjugate variable of the slope for the Fenchel-Legendre transform of the surface tension σ (s, t) of the corresponding random surface model. We will only consider here the case when B = (0, 0), since the other measures can be obtained as measures without external field for different weights. We refer the reader to [18] for more details about the magnetic field and its relation with weights. The fundamental domain G 1 of G is supposed to have the same number of white and black vertices. The white (resp. black) vertices are labeled, say, from 1 to n: w1 , . . . , wn (resp. b1 , . . . , bn ). If v is a vertex of G, then vx = v + (x, y) denotes the translate of v by the lattice vector x = (x, y) ∈ Z2 . If (z, w) is in the unit torus T2 = S1 × S1 , a function f is said to be (z, w)-periodic if ∀ v ∈ G, ∀ (x, y) ∈ Z2 ,
f (vx,y ) = z −y w x f (v).
The space of (z, w)-periodic functions supported on black (resp. on white) vertices is an n-dimensional vector space.
Pattern Densities in Non-Frozen Planar Dimer Models
57
As in the case of a planar finite region, an infinite Kasteleyn matrix K is defined. Its action on functions defined on black vertices returns a function on white vertices K(w, b) f b . (K f )w = w∼b
If e = (w, b), we will often write Ke instead of K(w, b). K is a periodic operator. We can thus define its Fourier transform K (z, w) as the restriction of K to the (z, w)-periodic functions, which is a n × n matrix. The rows of K (z, w) are indexed by the white vertices of the fundamental domain G 1 (more precisely, by the (z, w)-periodic function taking value 1 at a given white vertex of the fundamental domain, and 0 elsewhere). Its columns are indexed by the black vertices of G 1 . The correlations between dimers under the Gibbs measure are expressed in terms of determinants whose entries are those of an infinite matrix, the inverse Kasteleyn operator K−1 . The entries of this infinite matrix are the limits of those of the inverse of K , defined on the quotient G/(N Z2 ), when N goes to ∞. These entries are defined by the following inverse Fourier Transform formula: if b and w are respectively a black and a white vertex in the fundamental domain, and x = (x, y), x = (x , y ) are lattice points, the coefficient K−1 (bx , wx ) is given by dz dw −1 K−1 (bx , wx ) = K−1 (bx−x , w) = z −(y−y ) w (x−x ) K bw (z, w) 2iπ z 2iπ w T2
= T2
z −(y−y ) w (x−x ) Q bw (z, w) dz dw , P(z, w) 2iπ z 2iπ w
where K −1 (z, w) is the inverse of the n × n matrix K (z, w), P(z, w) its determinant and Q b,w (z, w) is the cofactor associated to K w,b (z, w). The probability that edges e1 = (w1 , b1 ), . . . ek = (wk , bk ) appear in the random dimer configuration is given by k
P [e1 , . . . ek ] = K(w j , b j ) det K−1 (bi , w j ) . (2) j=1
1≤i, j≤k
The way the correlation between edges decays with the distance depends on the number of zeros of P(z, w) on the unit torus [18]. Three behaviours are possible for a Gibbs measure. The measure can be: – frozen (or solid), when there are bi-infinite dual paths on which edges appear with probability 0 or 1, – in a massless (or liquid) phase, when K−1 (bx , w) decays linearly with |x|, and thus correlations decay polynomially. A more precise statement is given in Lemma 2, – in a massive (or gaseous) phase, when K−1 (bx , w) and thus correlations, decay exponentially. The terms in backets are those introduced by Kenyon, Okounkov and Sheffield in [18] to qualify the different phases of the model. We prefer to use here the terms frozen, massless, and massive, coming from field theory. In the random interface interpretation of this model, the terms rough and smooth are used for massless and massive respectively. In this paper, we will deal with non-frozen Gibbs measures.
58
C. Boutillier
1.2. Scaling limits of pattern densities. A dimer configuration on a bipartite planar graph can be interpreted via the height function as a discrete surface [25]. From this point of view, scaling limits of dimer models on planar bipartite graphs have already been the object of several studies: a law of large number has been established [2, 17] showing that this discrete surface approaches a limit shape when the mesh size goes to zero. The fluctuations of the height function around the limit shape have also been studied in the case of a graph embedded in a bounded region [13, 15], as well as in the case of an isoradial infinite graph with critical weights [3]. The continuous limiting object for these fluctuations is the massless free field [22, 7]. For the convergence of other models from statistical physics to the massless free field, see for example [19, 6]. In this paper, we are interested in the scaling limit of dimer models on Z2 -periodic planar graphs but from a different standpoint. Instead of looking at the height function, we consider other observables, called pattern density fields. A geometric realization of a Z2 -periodic planar bipartite graph G is a map from G to R2 preserving the Z2 -periodicity of G: vertices of G are mapped to points of R2 , edges to segments, and Z2 acts on the image of G by translation. We do not require to be an embedding. Let be a geometric realization of G such that the quotient of R2 by the action of 2 Z has area 1. For each scaling factor ε > 0, we define the scaled geometric realization ε = ε and G ε = ε (G) the image of G by the map ε . A pattern P is a finite set of edges {e1 , . . . , ek }, together with a marked vertex. The position of the pattern in G ε is given by the coordinates of the image by ε of this marked vertex. If e j goes from white vertex w j to black vertex b j , then the probability of seeing such a pattern in a random dimer configuration is given by formula (2) k
P [P] = K(w j , b j ) det K−1 (bi , w j ) . 1≤i, j≤k
j=1
In order to get some information about spatial distribution of patterns, and the way they interact with each other, we define for every pattern P a family of (discrete) random fields NPε , called pattern density fields. For a given ε, NPε is a random distribution, associating to every domain D the number of copies of P seen in D in a random dimer configuration of G ε . More precisely, if u ε is the image by ε of the marked vertex attached to P, the action of NPε on a smooth test function ϕ is given by ϕ(u εx )IPx , (3) NPε (ϕ) = x∈Z2
where Px and u εx are the translates by x of P and u ε , and is the indicator function IPx of the pattern Px , equal to 1 or 0 whether Px is in the random dimer configuration C or not. To simplify notations, we will use the symbol Px to represent also the indicator function IPx . With this new notation,NPε (ϕ) becomes ϕ(u εx )Px . (4) NPε (ϕ) = x∈Z2
ε2 E NPε (ϕ) is a Riemann sum of P[P]ϕ, and thus converges to P[P] ϕ(u)du when ε goes to 0. The aim of the paper is to prove the following convergence of the fluctuations of this field around its mean value:
Pattern Densities in Non-Frozen Planar Dimer Models
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Theorem 1. When ε goes to zero, the normalized fluctuation field
N˜ Pε = ε NPε − E NPε converges to a Gaussian field, which is the linear combination of a derivative of the massless free field and an independent white noise. See Sect. 1.4 for a more precise statement. Central limit theorems for Gibbsian fields and convergence to white noise have been the subject of many studies in the literature (see for example [1, 4, 9, 20, 21]), where usually fast mixing properties are required. Such central limit theorems exist also for linear statistics of a broad class of determinantal random fields [24] with Hermitian kernels. However, since the kernel K−1 appearing in dimer models is not Hermitian, those results do not apply. Before stating precisely the main results, we recall some basic facts we will need about Gaussian fields.
1.3. Gaussian Fields. A Gaussian field is somehow an infinite dimensional generalization of the notion of Gaussian vector. See [8] for an introduction. As in the classical situation with a Gaussian vector, all the moments can be expressed in terms of the second moment, by Wick’s formula: Proposition 1 (Wick’s formula). Let W be a centered Gaussian field. All the moments of W are determined by the covariance: Let ϕ1 , . . . , ϕn be smooth test functions. Then
E [W (ϕ1 ) · · · W (ϕn )] =
0
if n is odd,
n/2
E W (ϕik )W (ϕ jk ) if n is even.
τ pairings k=1 τ ={{i 1 , j1 },...,{i n/2 , jn/2 }}
(5) In particular, if all the test functions are equal and E W (ϕ)2 = σ 2 , then we recover the usual formula for the moments of a Gaussian random variable:
0 if n is odd, n E W (ϕ) = (6) n (n − 1)!! σ if n is even,
where (n − 1)!! = (n − 1)(n − 3) · · · 3 · 1 is the number of pairings of n elements. We say that a sequence of random fields (Wn ) converges weakly in distribution to a random field W if for all smooth test functions ϕ, the real random variable Wn (ϕ) converges in distribution to W (ϕ). When W is Gaussian, the distribution of W (ϕ) is determined by its moments, and one has just to check that the moments of Wn (ϕ) converge to that of W (ϕ), i.e. that Wick’s formula is asymptotically satisfied by the sequence (Wn ). 1.4. Statement of the result and outline of the paper. The main result of the paper is the following Central Limit Theorem when the measure on dimer configurations is non-frozen.
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C. Boutillier
Theorem 2. Consider the dimer model on a planar bipartite Z2 -periodic graph G with a generic non-frozen Gibbs measure µ. Let P be a pattern of G and N˜ Pε be the random field of density fluctuations of pattern P. Then when ε goes to 0, N˜ Pε converges weakly in distribution to a Gaussian random field NP . – In the massless phase, NP is a linear combination of a directional derivative of the massless free field and an independent white noise. There exist a vector a = (ax , a y ) ∈ R2 and a constant A depending on P such that covariance structure has the following form:
1 E NP (ϕ1 )NP (ϕ2 ) = π
G(u, v)∂a ϕ1 (u)∂a ϕ2 (v)|du||dv| + A
R 2 ×R 2
ϕ1 (u)ϕ2 (u)|du|,
R2
(7) 1 where G(u, v) = − 2π log |u − v| is the Green function on the plane, and ∂a ϕ = ∂ϕ ∂ϕ ax ∂ x + a y ∂ y . – In the massive phase, NP is a white noise and there exists a constant A depending on P such that E [NP (ϕ1 )NP (ϕ2 )] = A ϕ1 (u)ϕ2 (u)|du|. (8)
R2
In other words, for any choice of ϕ1 , . . . , ϕn ∈ Cc∞ (R2 ), the distribution of the random vector ( N˜ Pε ϕ1 , . . . , N˜ Pε ϕn ) converges to that of the Gaussian vector (NP ϕ1 , . . . , NP ϕn ) whose covariance structure is mentioned in the theorem. As the distribution of a Gaussian vector is characterized by its moments, it is sufficient to prove the convergence of the moments of ( N˜ Pε ϕ1 , . . . , N˜ Pε ϕn ) to those of (NP ϕ1 , . . . , NP ϕn ), given by Wick’s formula (5). In the massive phase, the intensity A has a simple probabilistic interpretation: it is the sum of the covariances between the indicator function of the pattern P0 at the origin and all its translates Px , A=
P [P0 , Px ] − P [P0 ] P [Px ] .
(9)
x∈Z2
In the massless phase, the sum above is not absolutely convergent, and A is more difficult to interpret. The direction a in which the derivative is taken has an explicit expression in terms of the matrices K and Q. When the pattern is made of one single edge e, the direction is that of the dual edge e∗ , but has not a simple geometric description in general. The proof goes in two steps: first we prove the convergence of the second moment of the fluctuation field N˜ Pε to the covariance of NP , and then we prove that Wick’s formula is satisfied asymptotically. As the arguments are different for the two kinds of phases, the proof of Theorem 2 is decomposed into three cases. In Sect. 3, we give the proof for a pattern made of a single edge in the generic massless case (when the two zeros of P on the unit torus are distinct) and discuss briefly what happens in the non-generic case. In Sect. 4, the
Pattern Densities in Non-Frozen Planar Dimer Models
61
proof is extended to any admissible pattern ( i.e. a pattern appearing with positive probability) in the generic massless case. The situation when the measure is in a massive phase is discussed in Sect. 5. The correlations between different pattern density fields are presented in Sect. 6, providing a generalization of Theorem 2. In the last section two explicit computations are given: one on the square lattice, and the other on the so-called square-octagon graph. Before entering into the details of the proof, we present in Sect. 2 some properties of the different non-frozen phases of the dimer model. 2. The Non-Frozen Phases of the Dimer Model We present here some properties of massless and massive phases of the dimer model on a planar bipartite periodic graph G. See [18] for more details.
2.1. Massless phase of dimer models. Recall that this phase is called liquid in [18]. Map from G ∗ to R2 . When given a measure on dimer configurations of G in the massless phase, there is a natural function ∗ from the dual graph G ∗ to R2 , described by the following lemma. This map gives the appropriate geometry to study the massless phase of planar dimer models. In particular, when dimer weights are critical, ∗ coincides with the isoradial embedding of G ∗ described in [14]. In the massless phase, the characteristic polynomial P has two zeros on the unit torus, that are complex conjugate of each other, and generically distinct. Let (z 0 , w0 ) be one of them. Lemma 1. The 1-form e = (w, b) → i K wb (z 0 , w0 )Q bw (z 0 , w0 )
(10)
is a divergence-free flow. Its dual is therefore the gradient of a mapping from G ∗ to R2 C. This function ∗ is Z2 -periodic and the symmetries of its range are generated by the two vectors xˆ = i z 0 ∂∂zP (z 0 , w0 ) and yˆ = iw0 ∂∂wP (z 0 , w0 ). Proof. Recall that Q(z, w) is the transposed comatrix of K (z, w). The divergence of the 1-form ω : e → i K wb (z 0 , w0 )Q bw (z 0 , w0 ) at some black vertex b is given by (divω)(b) = i K w b (z 0 , w0 )Q bw (z 0 , w0 ) w ∼b
= i Q(z 0 , w0 ) · K (z 0 , w0 ) b,b = i P(z 0 , w0 ) = 0. Similarly, one can check that the divergence of this flow is also 0 at every white vertex w. Thus, since G is planar, there exists a function ∗ : G ∗ → C such that ω = d. The fact that ∗ is periodic is a consequence of the fact that G and the Kasteleyn operator are both periodic. There exist two complex numbers xˆ and yˆ such that for every f ∈ G ∗ and every (x, y) ∈ Z2 , the difference between the image of f x,y and that of f itself is given by ∗ ( f x,y ) − ∗ ( f ) = x xˆ + y yˆ . (11)
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C. Boutillier
γy
γx
Fig. 1. A piece of a planar bipartite periodic graph G. The shaded zone represents its fundamental domain G 1 , delimited by the two paths γx and γ y .
Define α and β to be the partial derivatives of P at the root (z 0 , w0 ) with respect to the first and the second variable respectively α=
∂P (z 0 , w0 ), ∂z
β=
∂P (z 0 , w0 ). ∂w
(12)
Let us prove now that the numbers xˆ and yˆ are given by i z 0 α and iw0 β respectively. Let γx and γ y be the paths delimiting respectively the lower horizontal boundary and the leftmost vertical boundary of the fundamental domain G 1 (Fig. 1). The complex number xˆ equals the sum of the complex numbers ±ω(e) over all edges e crossing γx . These are exactly the edges of the fundamental domain whose weights have been multiplied by z ±1 in the Fourier transform of the Kasteleyn operator K (z, w), and the sign of the power of z is the same as that in front of ω(e). Noticing that if m = 1 z ∂ m m −1 (13) z z = m z = −z if m = −1 ∂z 0 if m = 0 we can write
∂ K wb (z 0 , w0 ) Q bw (z 0 , w0 ) i z0 xˆ = ∂z w,b∈G 1 ∂K (z 0 , w0 ) · Q(z 0 , w0 ) . = i z 0 tr ∂z
On the other hand, since the characteristic polynomial P(z, w) is the determinant of K (z, w), then ∂P (z 0 , w0 ) ∂z n ∂ Kb j (z 0 , w0 ), . . . , K bn (z 0 , w0 )), det(K b1 (z 0 , w0 ), . . . , = i z0 ∂z
i z0 α = i z0
j=1
(14)
Pattern Densities in Non-Frozen Planar Dimer Models
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where K b j (z, w) is the j th column of the matrix K (z, w). Expanding each determinant with respect to the column containing derivatives, we get i z0 α = i z0
n n ∂ K wk b j j=1 k=1
∂z
(z 0 , w0 ) ×
∂ Kb j Cof wk b j K b1 (z 0 , w0 ), . . . , (z 0 , w0 ), . . . , K bn (z 0 , w0 ) . ∂z
(15)
∂ Kb
Since the cofactor Cof wk b j (K b1 , . . . , ∂z j (z 0 , w0 ), . . . , K bn (z 0 , w0 )), obtained by removing the k th row and the j th column, does not depend on the j th column, we can replace it by K b j (z 0 , w0 ). This cofactor then by definition is Q b j wk (z 0 , w0 ). Thus, i z0 α = i z0
n n ∂ K wk b j j=1 k=1
∂z
(z 0 , w0 )Q b j wk (z 0 , w0 )
∂K (z 0 , w0 ) · Q(z 0 , w0 ) = xˆ . = i z 0 tr ∂z
(16)
The same argument applied to γ y gives the formula iw0 β = yˆ .
(17) G∗
R2 ,
to we will choose the root In what follows, to construct the function from (z 0 , w0 ) on the unit torus for which the frame (ˆx, yˆ ) is direct. This is equivalent to requiring that w0 β > 0. (18) Im z0 α To get a geometric realization of G from ∗ , just pick a point in each dual face of ∗ (G ∗ ) in a periodic way. Note that sometimes, as in the case of isoradial embeddings [14] the obtained realization is not a plane, but can be thought of as a globally flat manifold with conic singularities, obtained by gluing together the images of faces of G ∗ . Asymptotics of K−1 (b, w). The coefficients of K−1 decay linearly. More precisely, if b and w are in the same fundamental domain, and bx,y is a translate of b by (x, y), then we have the following asymptotics for K−1 (bx,y , w): Lemma 2. Let (z 0 , w0 ) be the root of P on the unit torus satisfying (18). Then the asymptotic expression for the coefficients of K−1 is given by z 0 −y w0 x Q b,w (z 0 , w0 ) 1 −1
K (bx,y , w) = −Re +O |x|2 + |y|2 iπ αz 0 x + βw0 y −y z w x Q b,w (z 0 , w0 ) 1 = −Re 0 0 (19) +O |x|2 + |y|2 π x xˆ + y yˆ with α, β, xˆ and yˆ as defined in Lemma 1. The proof of this lemma is given in [18].
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C. Boutillier
Note that the geometry of the map from G ∗ to R2 appears in this analytical result. The denominator up to a factor π is the vector separating the image of the fundamental domains of bx,y and w under this function. Moreover, if b and w are the ends of an edge e then (19) can be rewritten as −y z 0 w0x ie∗ 1 −1
+O K(w, b) K (bx,y , w) = Re , (20) |x|2 + |y|2 π x xˆ + y yˆ where e∗ denotes the complex number representing the dual edge of e in the geometrical representation ∗ of G ∗ . Thus correlations between edges decay polynomially with the distance. 2.2. Massive phases. Recall that these phases are called gaseous in [18]. In these phases, the characteristic polynomial P has no zeros on the unit torus. If b and w are a black and a white vertex of G 1 , the fraction Q bw /P is analytic on the unit torus and its Fourier coefficients K−1 (bx , w) decay exponentially with x: ∃ C1 , C2 > 0, ∀ x ∈ Z2 , ∀ b, w ∈ G 1 , K−1 (bx , w) ≤ C1 e−C2 |x| . Hence the correlations also decay exponentially. 2.3. Density fields and partition function. In this paragraph, the calculations are purely formal and try to give some heuristics on the information one could get from these pattern density fields. Consider a dimer model for which we assign to a configuration C the weight w(C). Let Z 0 be the partition function of this model w(C). (21) Z0 = C
We now perturb the partition function by modifying locally the configuration weights. Let ϕ be a smooth test function. Fix a pattern P and a ε > 0. We multiply every weight ε w(C) by a factor etεϕ(u x ) whenever there is a copy of a given pattern P located at x. Up to a multiplicative constant exp(tεP [P] x ϕ(u εx )), the new partition function for the model with these new weights is ε ˜ε Zt = w(C)etε x ϕ(u x )(IPx −P[P ]) = w(C)et NP (ϕ) . C
C
This can be generalized to a perturbation involving several patterns. Formally, the successive derivatives of Z t /Z 0 at t = 0 are the moments of the random variable N˜ Pε (ϕ) with respect to the unperturbed probability measure. ∂ k Z t ˜ ε ϕ)k . ( N = E 0 P ∂t k Z 0 t=0
Statistics of patterns for a finite ε describe how the partition function transforms under an infinitesimal modification of the weights. Understanding these statistics in the limit when ε goes to 0 is a first step in the comprehension how the partition function is modified and might give insights on how to construct probability measures corresponding to the modified weights.
Pattern Densities in Non-Frozen Planar Dimer Models
65
3. The Massless Case: Edge Densities We first concentrate on the proof of a simple particular case of Theorem 2. We suppose that the probability measure on dimer configurations of G is a generic point in the massless phase, and consider the fluctuations of the density random field N˜ eε for a pattern consisting in a single edge e = (w, b). The precise statement we prove in this section is the following: Theorem 3. The random field N˜ eε converges weakly in distribution, as ε goes to 0, to a Gaussian random field Ne with covariance 1 E [Ne (ϕ1 )Ne (ϕ2 )] = ∂e∗ ϕ1 (u)G(u, v)∂e∗ ϕ2 (v)|du||dv| + A ϕ1 (u)ϕ2 (u)|du| π (22) for a certain A ≥ 0, and where e∗ is the vector representing the dual edge of e in the geometric realization ∗ of G ∗ presented in Lemma 1. Note that the particular geometry introduced by the map is particularly well-adapted to the problem. As we discussed in the previous section, we first prove the convergence of the second moment, and then that of higher moments. In this section, as we are interested in copies of an edge e = (w, b), the only vertices we will deal with are most of the time translates of w and b. To simplify notations, we will write K−1 (x − x ) instead of K−1 (bx , wx ) and Ke will stand for K(w, b). 3.1. Convergence of the second moment. The second moment (ϕ1 , ϕ2 ) → E N˜ eε (ϕ1 ) N˜ eε (ϕ2 ) of N˜ eε is a continuous bilinear positive form on Cc∞ (R2 ). We prove that this bilinear form converges to a non-degenerate bilinear form, that will define the covariance structure for the limit Gaussian field Ne . Proposition 2. There exists a non-negative constant A such that ∀ϕ1 , ϕ2 ∈ Cc∞ (R2 ) 1 lim E N˜ eε (ϕ1 ) N˜ eε (ϕ2 ) = ∂e∗ ϕ1 (u 1 )G(u 1 , u 2 )∂e∗ ϕ2 (u 2 )|du 1 ||du 2 | ε→0 π R 2 ×R 2
+A
ϕ1 (u)ϕ2 (u 2 )|du|,
(23)
R2 1 log |u − v| is the Green function on the plane. where G(u, v) = − 2π
Before going into the proof of this proposition, we give some interpretation of the expression for the covariance. The right-hand side can be physically interpreted as the energy of interaction between two magnetic dipoles with moment density ϕ1 e∗ and ϕ2 e∗ , plus a term of interaction at very short range. Suppose that there is an excess of edges e in the random dimer configuration, in some region D1 . These dimers behave collectively as a magnetic dipole: their presence influences the rest of the dimer configuration as if a magnetic field created by a dipole with a density e∗ over D1 was applied to the system: an edge whose dual edge has an orientation closed to that of the magnetic field at that point is more likely to appear.
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C. Boutillier
Proof. Using the invariance by translation of the Kasteleyn operator and hence of the correlations, we rewrite the second moment as a convolution of two distributions, applied to a test function: ϕ1 (u x )ϕ2 (u x )E [(ex − e)(ex − e)] E N˜ eε (ϕ1 ) N˜ eε (ϕ2 ) = ε2 x,x ∈Z2
= ε2
(24)
ϕ1 (u x )ϕ2 (u x )E (e − e)(ex−x − e)
(25)
x,x ∈Z2
= ϕ1ε ∗ F ε , ϕ2 ,
(26)
where e represents the indicator function of the event {the edge e belongs to the random dimer configuration} and e = P [e]. The two distributions ϕ1ε and F ε are defined by ϕ1 (u x )δu εx , Fε = E [(e − e)(ex − e)] δu εx . ϕ1ε = ε2 xy
x
The distribution ϕ1ε converges weakly to ϕ1 when ε goes to zero. We will now prove the convergence of F ε to some distribution F, which will ensure that ϕ1ε ∗ F ε converges weakly to ϕ1ε ∗ F ε , since the support of ϕ1ε is contained in the fixed compact supp(ϕ1 ), and hence that ϕ1ε ∗ F ε , ϕ2 converges. Let ψ be a smooth test function with compact support. Let us prove the convergence of F ε , ψ = E [(e − e)(ex − e)] ψ(u εx ) = Cov(e, ex )ψ(u εx ). (27) x ε F , ψ
x
looks vaguely like a Riemann sum of a particular function. At first sight, The problem is that, due to the asymptotics of K−1 , the function would behave as 1/u 2 , which is not integrable in the vicinity of 0. Therefore, we decompose the sum over x in the definition of F ε depending on whether the norm of x is larger than M = 1/ε or not, that is whether u εx is in B = −i(αz 0 s + βw0 t) ; (s, t) ∈ [−1, 1]2 or not, ψ(u εx )Cov(e, ex ) + ψ(u εx )Cov(e, ex ) F ε , ψ = |x|>M
=
|x|≤M
ψ(u εx )Cov(e, ex ) +
|x|>M
+ψ(0)
ψ(u εx ) − ψ(0) Cov(e, ex )
|x|≤M
Cov(e, ex ).
(28)
|x|≤M
The fact that we subtracted and added ψ(0) in the second sum removed the nonintegrable singularity at 0. The following two lemmas state the convergence of the three sums. Lemma 3. ∗ 2 1 e ε lim ψ(u x )Cov(e, ex ) = ψ(z)Re |du|, (29) 2 ε→0 2π u |x|>1/ε
lim
ε→0
|x|≤1/ε
1 ψ(u εx ) − ψ(0) Cov(e, ex ) = 2π 2
R2 \B
B
e∗ (ψ(z) − ψ(0))Re u
2 |du|. (30)
Pattern Densities in Non-Frozen Planar Dimer Models
67
Moreover the sum of the two previous limits can be rewritten as 1 ∂e∗ ψ(u)∂e∗ G(u, 0)|du| + A1 ψ(0) π R2
for some constant A1 . Proof. If x = (0, 0), the covariance between edges e = e(0,0) and ex = e(x,y) is given by
2 K−1 (0) K−1 (x) − Ke K−1 (0) K−1 (−x) K−1 (0)
Cov(e, ex ) = P [e and ex ] − P [e] = 2
Ke2 det
= −Ke2 K−1 (x)K−1 (−x). Using asymptotics of K−1 for large x, we get the following asymptotic expression for the covariance between two distinct edges −y y 3 z 0 w0x ie∗ z 0 w0−x ie∗ ε2 ε Cov(e, ex ) = − 2 Re Re + O π u εx −u εx u εx ∗ 2 ∗ 2 3 ε2 e ε2 ε 2y −2x e = − 2 Re + Re(z 0 w0 ) ε + O . 2π u εx 2π 2 ux u εx Since the second term is oscillating, it will not contribute to the limit. The sum in the left-hand side of (29), modulo the oscillating terms, can be interpreted as the integral of a piecewise constant function approximating e∗ 2 1 − 2 ψ(u)Re IR2 \B (u). 2π u2 As the approximating functions are bounded uniformly in ε by an integrable function, and converge almost everywhere, then by Lebesgue theorem, the first sum converges to e∗ 2 1 − Re ψ(z)|du|. (31) 2π 2 u2 R2 \B
In the same way, the sum in the left-hand side of (30) is the integral of a piecewise constant function approximating e∗ 2 1 IB (u), − 2 (ψ(u) − ψ(0))Re 2π u2 and for the same reasons it converges to e∗ 2 1 − Re ψ(z) − ψ(0) |du|. 2 2 2π u B
(32)
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C. Boutillier
We rewrite the sum of the limit using Green’s formula inside and outside of B, noticing ∗ 2 1 1 Re eu is the second derivative of the Green function G(u, 0) = − 2π log |u| that 2π ∗ along the vector e : ! 1 2 1 ∂e∗ G(u, 0)ψ(z)|du| = ψ(u)∂e∗ G(u, 0)nin , e∗ dσ π π ∂B
R2 \B
− B
1 2 1 ∂ ∗ G(u, 0)(ψ(u) − ψ(0))|du| = π e π −
1 π !
∂e∗ ψ(u)∂e∗ G(u, 0)|du|,
(33)
R2 \B
∂B
1 π
(ψ(u) − ψ(0))∂e∗ G(u, 0)nout , e∗ dσ ∂e∗ ψ(u)∂e∗ G(u, 0)|du|,
(34)
B
where nin and nout are the unit normal vector fields on ∂B pointing respectively inward and outward. The two integrals on B and R2 \ B combine to give an integral over R2 . The two integrals on ∂B cancel out partially. It remains only ! −ψ(0) ∂e∗ G(u, 0)next , e∗ dσ = A1 ψ(0). ∂B
This completes the proof of the lemma.
We now prove the convergence of the third sum of (28). Lemma 4. |x|≤M Cov(e, ex ) converges when M goes to infinity to a limit A2 . Proof. The sum of the covariances is given in terms of K and K−1 by Cov(e, ex ) = P [e] (1 − P [e]) + Cov(e, ex ) |x|≤M
0<|x|≤M
= Ke K
−1
(0) − Ke2
K−1 (x)K−1 (−x).
(35)
|x|≤M
K−1 (x) is by definition the xth Fourier coefficient of the function f = Q/P defined on the unit torus T2 . As P has simple zeros, f is in L1 (T2 ). The convolution dξ dζ (36) f (ξ, ζ ) f (zξ, wζ ) ( f ∗ f )(z, w) = 2iπ ξ 2iπ ζ T2
is also in L1 (T2 ) and its xth Fourier coefficient is exactly K−1 (x)K−1 (−x). Establishing the convergence of the sum is now a problem of pointwise convergence of a Fourier series. If f had been continuous at (z, w) = (1, 1), then the Fourier series would have converged to f (1, 1). The problem is that f (ξ, ζ )2 is not integrable and thus, the function f ∗ f is not defined when z and w are both equal to 1. However, f ∗ f is smooth in a punctured neighborhood of (1, 1), has directional limits when (z, w) converges to
Pattern Densities in Non-Frozen Planar Dimer Models
69
(1, 1), varying continuously with the direction. We can then prove an analogue in two dimensions of Dirichlet’s theorem for f ∗ f to show that the Fourier series at (z, w) = (1, 1) when M goes to infinity, converges to a mixture of the directional limits of f ∗ f . More precisely, if t = arg(w)/ arg(z) and (t) the limit of f ∗ f when (z, w) goes to (1, 1) with t fixed, then
lim
M→+∞
And thus
|x|≤M
|x|≤M
1 K (x)K (−x) = 2 π −1
+∞
−1
−∞
1 + t dt . (t) log 1−t t
(37)
Cov(e − e)(ex − e) converges to a limit that we denote by A2 .
We now come back to the proof of the convergence of the distribution F ε . The three sums in (28) defining F ε , ψ converge and the sum of the limits is 1 ∂e∗ ψ(u)∂e∗ G(u, 0)|du| + Aψ(0), (38) F, ψ = π R2
where A = A1 + A2 . Thus, when ε goes to 0, F ε converges to the distribution F defined by the formula above, and hence ϕ1ε ∗ F ε to ϕ1 ∗ F. Denoting by ∂ (u) and ∂ (v) respectively the operator of partial differentiation with respect to the variable u and v, and noticing that, since G(u, v) = G(u − v), we have: ∂ (u) G(u, v) = −∂ (v) G(u, v), we get the following expression for the limit covariance structure: lim E ( N˜ eε ϕ1 )( N˜ eε ϕ2 ) ε→0
= ϕ1 ∗ F, ϕ2 1 ∂e∗ ϕ1 (u)∂e(u) ϕ1 (u)ϕ2 (u)|du| = ∗ G(u, v)ϕ2 (v)|du||dv| + A π R 2 ×R 2
1 =− π
R2
∂e∗ ϕ1 (u)∂e(v)∗ G(u, v)ϕ2 (v)|du||dv| + A
R 2 ×R 2
1 = π
∂e∗ ϕ1 (u)G(u, v)∂e∗ ϕ2 (v)|du||dv| + A
R 2 ×R 2
ϕ1 (u)ϕ2 (u)|du|
R2
ϕ1 (u)ϕ2 (u)|du|.
(39)
R2
Thus the covariance of N˜ eε converges to the expression given in Proposition 2.
3.2. Convergence of higher moments. We now prove the convergence of the moments of order ≥ 3 of N˜ eε to those of the Gaussian field Ne . Proposition 3. For every n ≥ 3, the n th moment of N˜ eε converges to that of Ne when ε goes to zero. In other words, for every ϕ1 , . . . , ϕn ∈ Cc∞ (R2 ), lim E N˜ eε ϕ1 · · · N˜ eε ϕn = E [Ne ϕ1 · · · Ne ϕn ] . ε→0
70
C. Boutillier
Since Ne is Gaussian, it is sufficient to show that in the of N˜ eε sat limit, the moments isfy Wick’s formula. Moreover, as (ϕ1 , . . . , ϕn ) → E N˜ eε ϕ1 · · · N˜ eε ϕn is a symmetric n-linear form, we just have to prove Proposition 3 when all the ϕi are equal to some test function ψ, the general case being obtained by polarization. The previous proposition reduces then to showing that Proposition 4. "
lim E ( N˜ eε ψ)n = E (Ne ψ)n =
ε→0
0 if n is odd n/2
. 2 if n is even (n − 1)!! E (Ne ψ)
In this section, we are thus interested in the limit of
E ( N˜ eε ψ)n = εn ψ(u ε1 ) · · · ψ(u εn )E (ex1 − e) · · · (exn − e) .
(40)
x1 ,··· ,xn
A first step in the proof is to study the convergence of a related quantity εn (ψ), defined by a sum of the same general term as for E ( N˜ eε ψ)n , but with a set of indices (x j ) restricted to distinct points: εn (ψ) = εn
x1 ···xn distinct
ψ(u εx1 ) · · · ψ(u εxn )E (ex1 − e) · · · (exn − e) .
(41)
3.2.1. Convergence of εn (ψ) To prove the convergence of εn (ψ), we have to understand the asymptotic behavior of the correlations between distinct edges, when they are far from each other. A simple expression is given by Kenyon in [12] to compute these correlations using a unique determinant. Lemma 5 ([12]). Let e1 = (w1 , b1 ), · · · , en = (wn , bn ) be distinct edges. Their correlation is given by E [(e1 − e¯ 1 ) · · · (en − e¯ n )] =
n j=1
K(w j , b j )
K−1 (bi , w j ) .. det . . 1≤i, j≤n −1 K (b j , wi ) 0
0
This formula allows us to give an explicit expression for εn (ψ) in terms of the operators K and K−1 . Since the matrix in Lemma 5 has zeros on the diagonal, only permutations with no fixed point will contribute to the expansion of the determinant as a sum over the ˆ n be the set of such permutations. Every permutation σ ∈ S ˆ n is symmetric group. Let S decomposed as a product of disjoint cycles γ1 · · · γ p . The supports of these cycles form p a partition (l )l=1 of {1, . . . , n}, whose parts l have cardinality at least 2. The terms
Pattern Densities in Non-Frozen Planar Dimer Models
71
coming from permutations leading to the same partition are put together and we get: n εn E ψ(u εx j )(ex j − e¯ ) x1 ···xn distinct
= εn Ken
= εn Ken
=
j=1 n x1 ···xn j=1 distinct
ψ(u εx j ) det
sgn(σ )
x1 ···xn ˆ distinct σ ∈Sn p
0
K−1 (bi ,w j )
K−1 (b j ,wi )
0
n
ψ(u εx j )K−1 (xσ ( j) − x j )
γ cycle supp(γ )=l
p
(42)
j=1
sgn(γ )ε|γ |
{}l=1 l=1
|γ |
x j1 ,··· ,x j|γ | k=1 distinct
ψ(u εx j )Ke K−1 (xγ ( jk ) − x jk ) + o(1) . k
The error term o(1) comes from the fact that in the second line, we allow two x j whose indices are in different components of (l ) to be equal. We now examine the convergence of a term in brackets, associated to a cycle γ ,
sgn(γ )ε|γ |
|γ |
x j1 ,··· ,x j|γ | k=1 distinct
ψ(u εx j )Ke K−1 (xγ ( jk ) − x jk ). k
According to Subsect. 3.1, we know that when γ is a transposition the corresponding term converges. When the length of γ is at least 3, we have the following lemma. Lemma 6. For any cycle γ of length m ≥ 3, and any ψ ∈ Cc∞ (R2 ), we have m
lim εm
ε→0
=
x1 ,···xm k=1 distinct
ψ(u εxk )Ke K−1 (xγ (k) − xk )
2m−1 π m
(ie∗ )m ψ(u 1 ) · · · ψ(u m ) |du 1 | · · · |du m |. Re (u γ (1) − u 1 ) · · · (u γ (m) − u m )
1
(R 2 )m
(43)
Proof. From the behavior of K−1 at long range, we get an asymptotic expression for the product m
Ke K−1 (xγ ( j) − x j )
j=1
=
m j=1
Re
−(yγ ( j) −y j )
εz 0
+(x
−x j )
w0 γ ( j) π(u xγ ( j) − u εx j )
ie∗
+
small terms
(ie∗ )m εm = m−1 m Re 2 π (u εxγ (1) − u εx1 ) · · · (u εxγ (m) − u εxm )
(44)
+
oscillating . terms
(45)
72
C. Boutillier
The oscillating part of this asymptotic expansion once summed over x1 , . . . , xm will not contribute to the limit. The sum of the leading term multiplied by εm ψ(u εx1 )· · ·ψ(u εxm ) can be interpreted as the integral of a piecewise constant function, converging almost everywhere to m ∗ ie 1 . (46) ψ(u 1 ) · · · ψ(u m )Re 2m−1 π m u γ ( j) − u j j=1
As all the functions are dominated by a constant times the integrable function |ψ(u 1 ) · · · ψ(u m )| , |(u γ (1) − u 1 ) · · · (u γ (m) − u m )| the convergence follows from Lebesgue theorem. Once the convergence of all these terms is proven we can combine their limit to get the limit of εn (ψ). When summing over all cycles with a given support = { j1 , . . . , jm } of cardinality m ≥ 3, we get the following limit: lim
ε→0
=
γ cycle supp(γ )=
1 2m−1 π m
m ε (R 2 )m
m
x j1 ,··· ,x jm k=1 distinct
j∈
ψ(u εx j )Ke K−1 (xγ ( jk ) − x jk ) + o(1) k
ψ(u j )Re
γ cycle j∈ suppγ =
1 |du 1 | · · · |du m | u γ ( j) − u j
which equals zero according to the following lemma: Lemma 7. Let m ≥ 3, and u 1 , . . . , u n be distinct complex numbers. Then
m
γ ∈Sm i=1 m−cycles
1 = 0. u γ (i) − u i
(47)
Proof. Denote by f the function of u 1 , . . . , u m defined by the left-hand side of (47). When m is odd, the result is obvious, since γ and γ −1 give opposite contributions. For a general m, since m-cycles form a conjugation class in the group of permutations Sm , the function f is a rational fraction invariant under permutation of the variables u 1 , . . . , u m . The denominator of this fraction is the Vandermonde V = i< j (u i − u j ), and the numerator is of lower degree than V . Since V is antisymmetric under permutation, the denominator has to be antisymmetric as well. But the only antisymmetric polynomial of lower degree than V is 0. The limit of Eq. (42) will contribute only the partitions all of whose components have cardinality 2, i.e. a pairing. If n is odd, {1, . . . , n} cannot be partitioned into parts of
Pattern Densities in Non-Frozen Planar Dimer Models
73
two elements, and the limit n (ψ) of εn (ψ) is zero. When n is even, there are (n − 1)!! pairings and the limit n (ψ) of εn (ψ) is then
n (ψ) =
n/2
lim ε2 ε→0
pairings k=1 {i 1 , j1 },...,{i n/2 , jn/2 }
xi k =x jk
ψ(u εxi )ψ(u εx j )Cov(exik , ex jk ) k
k
= (n − 1)!! (2 (ψ))n/2 . For the moment, we discussed the limit of a restricted sum on distinct edges, which is not exactly the expression for the n th moment. We will now deal with the case when some edges can coincide. 3.2.2. Proof of Proposition 4. In the expression of the n th moment
εn
x1 ,··· ,xn
ψ(u ε1 ) · · · ψ(u εn )E (ex1 − e) · · · (exn − e)
the correlations are not given by Lemma 5 as soon as at least two edges coincide. To understand the behavior of this expression, we must be able to express in terms of K and K−1 correlations of the form E (ex1 − e)k1 · · · (ex p − e)k p (48) when e1 , . . . , e p are distinct, and k1 , . . . , k p ≥ 1. Using the fact that the indicator function of an edge e, also denoted by e, satisfies e j = e for j ≥ 1, Newton’s is formula yields (e − e) = k
k k j=0
j
e (−e) j
k− j
=e
k k j=1
j
(−e)k− j + (−e)k = αke (e − e) + βke ,
where αke and βke are deterministic, depending only on e and k. Since all the edges we consider are translates one from another, and thus have the same probability, we will simply denote these coefficients by αk and βk . Here are some particular values of αk and βk that will be useful later α1 = 1,
β1 = 0,
β2 = e(1 − e).
Correlation (48) can be rewritten with these notations as
E (ex1 − e)k1 · · · (ex p − e)k p = E (αk1 (e1 − e) + βk1 ) · · · (αk p (e p − e) + βk p ) α˜ J β˜ J¯ E (e j − e) , (49) = J ⊂{1,··· , p}
where α˜ J =
j∈J
αk j and β˜ J¯ =
j ∈J /
βk j .
j∈J
74
C. Boutillier
In Eq. (40), E ( N˜ eε ψ)n is expressed as a sum over all edges. We want now to rewrite this sum as a sum over distinct edges, using partitions of {1, . . . , n}. To every n-tuple of p lattice points (x1 , . . . , xn ) is naturally associated a partition {l }l=1 : each component l of this partition is an equivalence class for the relation i ∼ j ⇔ xi = x j . Denoting by nl the cardinal of l , we rewrite Eq. (40) as p E ( N˜ eε ψ)n = εn E ψ n j (u εx j )(ex j − e¯ )n j {l }l=1 x1 ,··· ,x p distinct p
= εn
j=1
α˜ J β˜ J¯
p {l }l=1 J ⊂{1,..., p}
×
(x j ) j∈J distinct
E
ψ n j (u εx j )(ex j − e¯ )
j∈J
ψ nl (u εxl ),
(50)
l ∈J / (xl )l ∈J /
where means that the xl , l ∈ J indexing the sum are not only disctinct from each other, but also from values of any x j , j ∈ J . Denote by q the number of l reduced to a single element. As β1 = 0, β˜ J¯ is zero unless J contains the indices of these l . Thus, the cardinality of a subset J giving a non-zero contribution must be at least q. For such a J , the last sum over (xl )l ∈J / in (50) is a Riemann sum without its renormalizing factor, and therefore is O(ε−2( p−|J |) ). Furthermore, the sum over (x J ) j∈J can be expressed by polarization in terms of ε|J | , and is therefore O(ε−|J | ). Since |J | ≥ q and n=
p
nl ≥ q + 2( p − q) = 2 p − q,
l=1
the contribution of J to (50) is at most O(εn−2 p+|J | ) which will be negligible in the limit except when |J | = q and nl = 2 for all l ∈ / J . For such J and (l ), we have α˜ J = 1,
β˜ J¯ = (¯e(1 − e¯ ))(n−q)/2 .
(51)
Thus the only partitions that will contribute to the limit are “partial pairings”, matching by pairs 2m = (n − q) elements of {1, . . . , n}. For a fixed m, there are n! n (2m − 1)!! = m 2m 2 m!(n − 2m)! such partitions, all giving the same contribution. Summing over m we get m n/2 n ε n ε 2 2 ε ˜ E ( Ne ψ) = (2m − 1)!! n−2m (ψ) ε e¯ (1 − e¯ ) ψ (u x ) + O(ε). 2m x m=0
Pattern Densities in Non-Frozen Planar Dimer Models
75
The Riemann sum ε2 e¯ (1 − e¯ ) ψ 2 (u εx ) converges to e¯ (1 − e¯ ) ψ 2 (u)|du| = E Ne (ψ)2 − 2 (ψ). R2
If n is odd, so is n − 2m. In this case, lim εn−2m (ψ) = 0, and therefore lim E ( N˜ eε ψ)n = 0. ε→0
(52)
If n is even, lim εn−2m (ψ) = n−2m (ψ) = (n − 2m − 1)!! 2 (ψ)n/2−m
ε→0
(n − 2m)! 2 (ψ)n/2−m . 2n/2−m (n/2 − m)! Therefore, the limit of E ( N˜ eε ψ)n is given by =
(53) (54)
n/2 ε n ˜ lim E ( Ne ψ) =
ε→0
(n − 2m)! 2 (ψ)n/2−m n! 2m m!(n − 2m)! 2n/2−m (n/2 − m)! m=0 m 2 × e¯ (1 − e¯ ) ψ (u)|du|
= (n − 1)!!
n/2 n/2
m 2 (ψ)n/2−m e¯ (1 − e¯ ) ψ 2 (u)|du|
m m=0 n/2 2 = (n − 1)!! 2 (ψ) + e¯ (1 − e¯ ) ψ (u)|du| = (n − 1)!! E Ne (ψ)2
n/2
,
which is exactly what we wanted to prove. This therefore ends the proof of Theorem 2 for a pattern made of one edge and a generic Gibbs measure in the massless phase. 3.3. A word about the non-generic case. When the two roots of the characteristic polynomial P(z, w) on the unit torus coincide, the measure is still in the massless phase, and the correlations between edges at distance r still decay like r −2 . However, since z 0 and w0 are real, the leading term in the asymptotics of K−1 is not oscillating anymore, which will induce a “resonance phenomenon” in the system. The first two sums in Eq. (28) defining the distribution F ε appearing in the study of the convergence of the second moment still have a finite limit when ε goes to zero. On the contrary, due to the resonance, the third sum Cov(e, ex ) |x|≤1/ε
in this case diverges. More precisely, this sum is O(log(1/ε)). Therefore the second moment diverges. However, one can prove that (log(1/ε))−1 N˜ eε converges weakly in distribution to a white noise. We will not show it here.
76
C. Boutillier
4. The Massless Case: General Pattern Densities This section is devoted to the proof of an analogue of Theorem 3 for multi-edged patterns. Let N˜ Pε be the density fluctuation field of a pattern P for a generic Gibbs measure in the massless phase. Theorem 4. When ε goes to zero, N˜ Pε converges weakly in distribution to the Gaussian field NP whose covariant structure is given by 1 ∂P ∗ ϕ1 (u 1 )G(u 1 , u 2 )∂P ∗ ϕ2 (u 2 )|du 1 ||du 2 | E [NP (ϕ1 )NP (ϕ2 )] = π R 2 ×R 2
+A
ϕ1 (u 1 )ϕ2 (u)|du|,
R2
where the vector P ∗ ∈ R2 and the nonnegative constant A depend only on the Gibbs measure and the pattern P. The stategy of the proof is very similar to that of Theorem 3 for edge density fluctuations. The problem is that there is no simple analogue of Lemma 5 for correlations between non-overlapping patterns, which is why the proof needs a little more combinatorial work in this case. After having introduced the different notations required to deal easily with these patterns, we prove the theorem, following the structure of the proof given in the last section, and explaining in detail only parts that are specific to patterns made of more than one edge. 4.1. Notations. Let P be a pattern containing k distinct edges e1 = (w1 , b1 ), . . . , ek = (wk , bk ). The probability of such a pattern to appear in a random dimer configuration is −1 1 1 K (b , w ) · · · K−1 (b1 , wk ) .. .. .. (55) P = P [P] = Ke1 · · · Kek det . . . . K−1 (bk , w1 ) · · · K−1 (bk , wk )
More generally, the probability to see n non-overlapping copies P1 , . . . , Pn of P obtained respectively by translation of a lattice vector x1 , . . . , xn is given up to a constant by a determinant of matrix nk × nk defined by blocks A11 · · · A1n n
P [P1 · · · Pn ] = Ke1 · · · Kek det ... . . . ... , (56) An1 · · · Ann where the entries of the block A I J are coefficients of K−1 between black vertices of P I and white vertices of P J , −1 1 K (bx I , wx1 J ) · · · K−1 (b1x I , wxk J ) .. .. .. AI J = (57) . . . . −1 k 1 −1 k k K (bx I , wx J ) · · · K (bx I , wx J )
Pattern Densities in Non-Frozen Planar Dimer Models
77
The matrix A I I does not depend on I . We denote by E this matrix, whose determinant is used to compute P¯ = P [P]. We suppose that the pattern appears with positive probability P¯ > 0, and in particular, that E is invertible. Defining B I J as the product E −1 A I J and B as the whole block matrix (B I J ), we can rewrite the joint probability of P1 , . . . , Pn as Ik B1,2 · · · B1,n .. B2,1 Ik . (58) P [P1 · · · Pn ] = (P)n det . .. . .. .. . . Bn,1 · · · Bn,n−1 Ik Instead of using a single integer i to denote the row (resp. the column) of an entry in such a matrix defined by blocks, it will be more convenient to use a couple of integers (I, α), where I is the index of the block row (resp. of the block column) and α is the relative position in the I th block row (resp. block column). The relation between the two sets of indices is simply i = k(I − 1) + α. If the coordinates x j are all distinct but some patterns partially overlap, define P˜ j = + j−1 P j \ i=1 Pi for all j ∈ {1, . . . , n}. We have then P [P1 · · · Pn ] = P P˜ 1 · · · P˜ n . Up to a relabeling of the patterns, we can assume that none of the P˜ j is empty. Thus the joint probability of these patterns is also given by the determinant of a matrix defined by blocks of size |P˜ 1 | + · · · + |P˜ n |. 4.2. Asymptotics of correlations. The following lemma gives asymptotic correlations between distant patterns. Lemma 8. Let (x j ) = x1 , . . . , xn be n distinct lattice points. The correlations between the patterns Px1 , . . . , Pxn can be rewritten as
small ¯ · · · (Pxn − P) ¯ = Hγ (x j ) + , E (Px1 − P) terms ˆn S∈S
γ cycle of S
where the functions Hγ have the following asymptotic behaviour:
tr((E −1 Q)|γ | ) ε|γ | oscillating Hγ (x j ) = sgn(γ ) |γ |−1 |γ | Re + ε ε terms 2 π j∈suppγ u xγ ( j) − u x j
k with Q = Q bα wβ (z 0 , w0 ) α,β=1 , and satisfy Hγ (x j ) ≤ ε|γ | j∈suppγ
C |u γ ( j) − u j |
78
C. Boutillier
for every u 1 , . . . , u n in a ε-neighborhood of u εx1 , . . . , u εxn . The error term o(1) is uniformly bounded in x1 , . . . , xn and goes to zero when the distance between the patterns goes to infinity. Proof. We first derive the asymptotic expression for the correlations when the patterns are far from each other. When |xi − x j | is large enough for every i = j, the patterns are disjoint and expression (58) for correlations can be used. Expanding the products in the expectation, we get
¯ · · · (Pxn − P) ¯ = ¯ n− p E Px j . . . Px j E (Px1 − P) (−P) p 1 j1 ,..., j p
= P [P]n
(−1)n−|C|
C⊂{1,...,n}
In δ12 B12 · · · δ1n B1n δ21 B21 In det . . .. .. δn1 Bn1 In
(59)
where the non-diagonal block δ I J B I J is either B I J or 0 depending on whether (I, J ) belongs to C × C or not. Expressing each determinant as a sum over the symmetric group Snk and gathering the terms coming from the same permutation, one can notice that the contributions of permutations fixing a whole block are vanishing, due to the alternating sign in the sum over C. The permutations contributing to the correlations are those whose support intersects each block. Therefore, we have
¯ · · · (Pxn − P) ¯ = P [P]n E (Px1 − P) sgn(σ ) B(I,α),σ ((I,α)) . σ ∈Snk fixing no block
I,α
The main contribution to this sum is given by the “special” permutations, the support of which intersects each block exactly once. Let σ be such a permutation. The n nonfixed elements are (1, α1 ), . . . , (n, αn ). For every I ∈ {1, . . . , n}, the non-fixed element ˆ n is a fixed-point free permu(I, α I ) is sent to σ ((I, α I )) = (S(I ), α S(I ) , where S ∈ S n tation, having the same signature as σ . There are k special permutations σ leading to the same S, corresponding to the different possible choices of the non-fixed points in each block α1 , . . . , αn . The contribution of the other permutations will be negligible because of the extra ε coming from additional entries of K−1 between non-fixed points: n ¯ = P [P]n E (Px j − P) sgn(σ ) B(I,α),σ ((I,α)) σ ∈Snk fixing no block
j=1
= P [P]n
ˆn S∈S
I,α
sgn(S)
n k I =1 α I =1
(B I,S(I ) )α I ,α S(I ) +
small . terms (60)
Pattern Densities in Non-Frozen Planar Dimer Models
79
The main term in Eq. (60) has an expression in terms of traces of products of block matrices B I J , n k
B I,S(I )
I =1 α I =1
α I α S(I )
=
tr B I1 ,I2 · · · B I p ,I1 .
(61)
γ =(I1 ,...,I p ) cycle of S
Let us have a look to a particular trace tr B I1 ,I2 · · · B I p ,I1 . Recall that B I J is the product of E −1 whose entries will be denoted by eαβ and A I J whose entries (A I J )αβ are the coefficient of K−1 taken between a black vertex ‘α’ of pattern I and a white vertex ‘β’ of pattern J . The asymptotics of (A I J )αβ when patterns are far away from each other are given by Lemma 2: −y +y +x −x z 0 J I w0 J I Q αβ (z 0 , w0 ) ε small β −1 α (A I J )αβ = K (bx I , wx J ) = − Re + . (62) ε ε terms π ux J − uxI To simplify notations, we introduce −y J +y I
ζI J =
z0
u εx J
w0+x J −x I − u εx I
(63)
and write Q αβ instead of Q bα wβ (z 0 , w0 ). The trace tr B I1 ,I2 · · · B I p ,I1 can therefore be rewritten as
tr B I1 ,I2 · · · B I p ,I1 = (E −1 A I1 I2 )α1 α2 · · · (E −1 A I p I1 )α p α1 α1 ,...,αk
−ε p = eα1 β1 · · · eα p β p π α1 ,...,αk β1 ,··· ,βk
small
. × Re Q β1 α1 ζ I1 I2 · · · Re Q β p α1 ζ I p I1 + terms As in the case of edge densities, there will be only two non-oscillating terms in the expansion of this product of real parts: those for which the phases contained in the ζ I J compensate exactly. With the convention that p + 1 = 1, one has
tr B I1 I2 · · · B I p I1 = p −ε p
small eα j β j Q β j α j+1 ζ I j I j+1 + Q β j α j+1 ζ I j I j+1 + = terms α1 ,...,αk 2π β1 ,··· ,βk
j=1
p p (−ε) p oscillating = . eα j β j Q β j α j+1 ζ I j I j+1 + eα j β j Q β j α j+1 ζ I j I j+1 + pπ p terms 2 α1 ,...,αk β1 ,··· ,βk
j=1
j=1
The product of the ζ Ik Ik+1 is equal to p j=1
ζ I j ,I j+1 =
p j=1
1 u εI j+1 − u εI j
(64)
80
C. Boutillier
and we can rewrite the trace of B I1 ,I2 · · · B I p ,I1 as
(−ε) p tr((E −1 Q) p ) oscillating tr B I1 ,I2 · · · B I p ,I1 = p−1 p Re ε + terms 2 π (u I2 − u εI1 ) . . . (u εI1 − u εI p )
(65)
giving the asymptotics for Hγ . When the patterns are not disjoint anymore, then a similar analysis can be done, in defining new patterns as the connected components of P1 ∪ · · · ∪ Pn . The bound can be extended to this case. 4.3. Convergence of the second moment. Proposition 5. ∀ ϕ1 , ϕ2 ∈ Cc∞ (R2 ) lim E N˜ Pε (ϕ1 ) N˜ Pε (ϕ2 ) ε→0 1 ∂P ∗ ϕ1 (u 1 )G(u 1 , u 2 )∂P ∗ ϕ2 (u 2 )|du 1 ||du 2 | + A ϕ1 (u)ϕ2 (u)|du|. = π R 2 ×R 2
R2
(66) The proof of the convergence of the second moment goes exactly as that of Sect. 3. The second moment of N˜ Pε can be expressed as a convolution of two distributions applied to a test function (67) E ( N˜ Pε ϕ1 )( N˜ Pε ϕ2 ) = ϕ ε ∗ F ε , ϕ2 with the same definitions as before for ϕ1ε and F ε , ϕ1 (u εx )δ(· − u εx ) Fε = Cov(P, Px )δ(· − u εx ), ϕ1ε = ε2 x
(68)
x
ϕ1ε converges weakly to ϕ1 . The convergence of F ε to a distribution F is proven exactly in the same way as in Sect. 3. The only difficulty that could appear is the analogue of Lemma 4 proving the convergence of x Cov(P, Px ). Cov(P, Px ) is a linear combination of products of diverse values of K−1 . If we interpret these products as Fourier coefficients of a convolution of functions, then the convergence becomes more obvious: we saw in Sect. 3 that the function whose Fourier coefficients are the product of two K−1 is not defined at (1, 1), but has directional limits when (z, w) converges to (1, 1), and the sum of the Fourier coefficient was an average of these directional limits. When more than two K−1 are involved, the function is even continuous thanks to the multiple convolutions, and the Fourier series converges at (z, w) = 1. F ε converges then to the distribution F acting on a test function ψ as
tr (E −1 Q)2 1 F, ψ = ψ(u)Re du + Aψ(0). (69) π 2π u 2 R2
The complex number representing the vector P ∗ along which are taken the derivatives is a square root of tr(E −1 Q E −1 Q), and after application of Green’s formula, we get the expression of Proposition 5 for ϕ1 ∗ F, ϕ2 .
Pattern Densities in Non-Frozen Planar Dimer Models
81
4.4. Convergence of higher moments. Proposition 6. Let n ≥ 3, and ϕ1 , . . . , ϕn ∈ Cc∞ (R2 ),
0 if n is odd n/2
lim E NP (ϕ1 ) · · · NP (ϕn ) = . (70) E NP (ϕil )NP (ϕ jl ) if n is even ε→0
˜ε
˜ε
pairings l=1
Proof. As before, it is sufficient to study the case where all the ϕi are equal to some fixed smooth test function ψ. The nth moment is the given by
˜ε
E ( NP ψ)
n
=ε
,
n
E
x1 ,...,xn
n
ψ(u lε )(Pxl
¯ . − P)
(71)
l=1
We know from Lemma 8 the asymptotics of the general term of the sum , ε E n
n
ψ(u lε )(Pxl
¯ − P)
l=1
=
sgn γ
ˆ n γ cycle of S S∈S
+
=
(−2)
p (l )1 l=1
+
(2π )
oscillating small + terms terms p
tr (E −1 Q)|γ | Re |γ |
2ε2|γ |
|l | −ε2 2π
ψ(u lε )
l∈suppγ
u εxγ (l) − u εxl
−1 | | l Re tr (E Q)
γ cycle suppγ =l
ψ(u εj )
uε j∈l xγ ( j)
− u εx j
small oscillating + , terms terms
where the (l ) are partitions of {1, . . . , n} whose components have size at least 2. The expression we obtained is very close to that of Eq. (44). From this point, the same arguments as for edges yield the proof of the proposition. 5. The Massive Case In a massive (gaseous) phase, K−1 decays exponentially. There exist two constants C1 and C2 such that j ∀ x, x ∈ Z2 , K−1 (bix , w00 ) ≤ C1 · e−C2 |x| . The precise statement of Theorem 2 in this particular context for a pattern consisting in a single edge e is the following:
82
C. Boutillier
˜ε Theorem 5. . The random field Ne converges weakly in distribution to a white noise of amplitude
∂2F , ∂ 2 log Ke
where log P(z, w)
F= T2
dz dw 2iπ z 2iπ w
(72)
is the free energy per fundamental domain of the dimer model. The proof is exposed in the first two subsections, and the case of a more complex pattern is briefly discussed in Subsect. 5.3. 5.1. Convergence of the second moment. Proposition 7. ∀ϕ1 , ϕ2 ∈
Cc∞ (R2 )
lim E N˜ eε (ϕ1 ) N˜ eε (ϕ2 ) =
ε→0
∂ 2F ∂ log Ke2
ϕ1 (u)ϕ2 (u)|du|. R2
Proof. As in Sect. 3, the covariance E N˜ eε (ϕ1 ) N˜ eε S(ϕ2 ) is a convolution of distributions ϕ1ε ∗ F ε applied to the test function ϕ2 , where ϕ1ε = ε2 ϕ1 (u x )δ(· − u εx ), Fε = Cov(e, ex )δ(· − u εx ), x
x
ϕ1ε
converges weakly to ϕ1 . We have now to prove that the distribution F ε converges 2 toward the distribution F = ∂ ∂logFK2 δ. If it is the case, then the limit of the second moment e will be ∂ 2F ∂ 2F ϕ1 ∗ F, ϕ2 = ϕ ∗ δ, ϕ = ϕ1 (u)ϕ2 (u)|du|. (73) 1 2 ∂ log Ke2 ∂ log Ke 2 R2
If x = (0, 0), the covariance between edges e and ex is given by
Cov(e, ex ) = E (e0,0 − e)(ex − e) = −Ke2 K−1 (x)K−1 (−x).
(74)
Let ψ be a smooth test function with compact support, and N a large integer. We decompose the sum over x in the expression of F ε , ψ depending on whether the norm of x is larger than N or not. F ε , ψ = ψ(u εx ) − ψ(0) Cov(e, ex ) ψ(u εx )Cov(e, ex ) + |x|>N
+ψ(0)
|x|≤N
Cov(e, ex ).
|x|≤N
In the first sum, there are at most O(ε−2 ) terms since the support of ψ is bounded. As |x| > N , each term in this sum is bounded by some constant times e−2C2 N . Therefore the whole sum is O(ε−2 e−2C2 N ). In the second sum, since |x| ≤ N , the distance between
Pattern Densities in Non-Frozen Planar Dimer Models
83
u εx and 0 is less than εN . As ψ is smooth, ψ(u εx ) − ψ(0) is O(εN ). Since there are O(N 2 ) terms in the second sum, it is O(N 3 ε). If we choose for instance N of order ε−1/4 , these two sums converge to zero, when ε goes to zero. The third sum is absolutely convergent, the limit of F ε , ψ is ψ(0) Cov(e, ex ), x∈Z2
meaning that F ε converges weakly to x∈Z2 Cov(e, ex )δ0 and the limit of the second moment is proportional to the L2 scalar product. The coefficient of proportionality Cov(e, ex ) x∈Z2
can be rewritten in terms of the polynomials P(z, w) and Q bw (z, w) = Q e (z, w):
Cov(e, ex ) = E [(e − e)(e − e)] + E (e0,0 − e)(ex − e) x=(0,0)
x∈Z2
= P [e] (1 − P [e]) − Ke2 = Ke K = T2
Since Q e (z, w) =
−1
(0) − Ke2
K−1 (x)K−1 (−x)
x=(0,0) −1
K (x)K−1 (−x)
x
Ke Q e (z, w) Ke2 Q e (z, w)2 dz dw − . P(z, w) P(z, w)2 2πi z 2πiw
∂ ∂Ke
P(z, w), we have finally ∂ 2F ∂ ∂ dz dw = Cov(e, ex ) = Ke Ke log(P(z, w)) . ∂Ke ∂Ke 2πi z 2πiw ∂ log Ke2 2
x∈Z
T2
5.2. Higher moments. Proposition 8. The Wick formula is verified in the limit. 0 if n is odd, n/2 ε n lim E ( N˜ e ψ) ) = ε 2 (n − 1)!!E ( N˜ e ) if n is even. ε→0 Proof. As in the massless case, the proof begins with the study of the restricted n th moment εn (ψ) defined by Eq. (41). Lemma 5 yields an asymptotic expression with the same structure as in the generic massless case (42): p εn (ψ) = sgn(γ )(εKe )|γ | × p
{}l=1 l=1
γ cycle supp(γ )=l |γ |
x j1 ,··· ,x j|γ | k=1 distinct
ψ(u εx j )K−1 (xγ ( jk ) k
− x jk ) + o(1) .
(75)
84
C. Boutillier
The contributions of cycles of length greater than 3 vanish in the limit, as an application of the following lemma: Lemma 9. ∀m ≥ 3,
lim εm
ε→0
m x1 ,··· ,xm j=1
ψ(u εx j )K−1 (x j+1 − x j ) = 0.
Proof. Define x = x1 and for j ≥ 2 xj = x j − x j−1 . The sum can be rewritten using these new notations and invariance by translation of operator K−1 as m εm ψ(u εx ) ψ(u εx +x +···+x )K−1 (xj ) K−1 (−x − x2 · · · − xm ). x
x2 ···xm
j=2
2
j
As ψ is a continuous function on a compact set, it is bounded. The sum on x has O(ε−2 ) terms. K−1 (b−x −x2 ···−xn , w) is bounded independently from the xj ’s. As K−1 decays exponentially, the sum on x2 , . . . , xn is bounded. The whole sum is thus a O(εm−2 ), which goes to zero when ε goes to zero as soon as m ≥ 3. Thus εn (ψ) converges to n (ψ) = (n −1)!! 2 (ψ). The end of the proof deals with collisions between edges, and is identical to what has been done for Proposition 4, leading to the result. 5.3. Patterns in a massive phase. Combining the notations and the techniques introduced in Sect. 4 to deal with correlations between patterns, and following the steps of the proof of theorem 5, one can prove the following Theorem 6. Let P be a pattern in a dimer model endowed with a Gibbs measure in a massive phase. The random field N˜ Pε of density fluctuation of pattern P converges weakly in distribution to a white noise. The proof is omitted here. 6. Correlations between Density Fields In the previous sections, only fluctuations of the density field associated to one fixed pattern was considered. One can ask what happens for correlations between density fields associated to different patterns. To what extent does a high density of some pattern in a given region of the plane have an influence on the density of another pattern in another region? This question is answered by the following theorem generalizing the results of the previous sections. Theorem 7. Consider a dimer model with a generic non-frozen Gibbs measure µ. – Let P1 and P2 be two patterns. The bilinear form E N˜ Pε 1 (·) N˜ Pε 2 (·) on Cc∞ (R2 )
converges when ε goes to zero to a bilinear form E NP1 (·)NP2 (·) .
Pattern Densities in Non-Frozen Planar Dimer Models
85
– If µ is generic in the massless phase, there exists a constant AP1 P2 such that for every test function ϕ1 and ϕ2 ,
1 ∂P1∗ ϕ1 (u)G(u, v)∂P2∗ ϕ2 (v)|du||dv| E NP1 (ϕ1 )NP2 (ϕ2 ) = π R 2 ×R 2
ϕ1 (u)ϕ2 (u)|du|.
+AP1 P2
(76)
R2
– If µ is in a massive phase, there exists a constant AP1 P2 such that for every test function ϕ1 and ϕ2 ,
E NP1 (ϕ1 )NP2 (ϕ2 ) = AP1 P2 ϕ1 (u)ϕ2 (u)|du|. R2
distinct). When ε goes to zero, the mul– Let P1 , . . . , Pn be patterns (not necessarily
ε ε ˜ ˜ tilinear form E NP1 (·) · · · NPn (·) converges. The limit E NP1 (·) · · · NPn (·) is given by the Wick formula: for every test functions ϕ1 , . . . , ϕn , if n is odd 0 n/2
E NP1 (ϕ1 ) · · · NPn (ϕn ) = E NPi (ϕi k )NP j (ϕ jk ) if n is even . k k pairings k=1 {{i 1 , j1 },{i n/2 , jn/2 }}
The coefficients AP1 P2 are in general not known in a closed form. However some relations between them can be found. For example, let v be a vertex of the graph, and ∗ sum to zero e1 , . . . , em the edges incident with v. The complex numbers e1∗ , . . . , em ∗ since they represent the edges of the dual face v . Even more, for every ε > 0, the linear combination N˜ eε1 + · · · + N˜ eεm is identically zero. Indeed, for every x ∈ Z2 , there is exactly one edge incident with vx in the random dimer configuration. Therefore the sum of indicator functions (e1 )x + · · · + (em )x is always equal to 1. This relation between the random fields N˜ eεj at a microscopic level yields relations between the different coefficients Aei e j . Precisely, m m Aei e j = 0. (77) i=1 j=1
7. Examples The theorems in the previous sections state a convergence of density fluctuation in the scaling limit to a linear combination of a derivative of the massless free field and a white noise. However, they do not give an explicit form for the white noise amplitude. In this section we present some cases for which a closed expression for the white noise amplitude can be provided in terms of the weights on edges. The first case is the dimer model on the graph Z2 with periodic weights a, b, c, d around white vertices. The second case is the dimer model on the square-octagon graph.
86
C. Boutillier
_ w0 b* y^
−z0 a*
b
x^
c*
a
c d
d*
Fig. 2. On the left, a piece of the geometric realization (isoradial embedding in this case) of Z2 fixed by the weights a, b, c, d assigned to edges. The region delimited by the tick dotted contour is a fundamental domain, and the coloring-preserving symmetries are generated by xˆ and yˆ . On the right, a white face and the quantities (angles and side lengths) related to it.
7.1. Dimer densities on Z2 . The graph we consider here is the graph Z2 with a bipartite coloring of its vertices. Weights are assigned to edges according to their directions: a, b, c, d counterclockwise around white vertices and clockwise around black vertices. If none of the weights is greater than the sum of the others, the corresponding dimer model is critical [14]: the graph can be embedded in the plane such that all the faces of the graph as well as those of the dual graph are inscribed in circles of a given radius (see Fig. 2). The Gibbs measure with no magnetic field on dimer configurations is in the massless phase. The dual faces are similar to the cyclic quadrilateral with sides a, b, c and d. The area of such a quadrilateral is Area =
1/ (−a + b + c + d)(a − b + c + d)(a + b − c + d)(a + b + c − d) 4
and the radius R of its circumscribed circle is defined by the relation R2 =
(ab + cd)(ac + bd)(ad + cb) . (−a + b + c + d)(a − b + c + d)(a + b − c + d)(a + b + c − d)
The fact we chose the fundamental domain to have area 1 leads to the following expression for the complex numbers representing the dual edges in the embedding of Z2 : ia ib −icz 0 idz 0 a∗ = √ b∗ = c∗ = d∗ = √ , √ √ 2 Area w0 2 Area w0 2 Area 2 Area where (z 0 , w0 ) is the root of the characteristic polynomial P(z, w) = a +
cz b − + dz w w
on the unit torus, with the additional constraint that Im(z 0 ) > 0.
Pattern Densities in Non-Frozen Planar Dimer Models
87
The following theorem is a particular case of Theorem 7. Theorem 8. Let G F F a massless free field in the plane and W an independent white noise with unit variance. The vector-valued random field N˜ aε N˜ ε b N˜ ε = N˜ ε c N˜ ε
d
converges weakly in distribution to the vector-valued Gaussian Field
Na +1 ∂a∗ 0 abcd 1 ∂b ∗ Nb −1 N = √ ∂ ∗ G F F + W. c c 8π R 2 Area +1 π −1 Nd ∂d ∗ Proof. The only point we have to explain is the computation of the white noise amplitude. This coefficient can be identified from the limit of the second moment of N˜ aε . The proof of the convergence of the second moment, as before, goes through the proof of the convergence of the distribution F ε defined by (27). The sum over x in the definition of F ε is decomposed into two parts depending on whether u εx belongs to some case, we considered the neighborhood B = neighborhood of 0 or not. In the 2general i(αz 0 s + βz 0 t) ; (s, t) ∈ [−1, 1] . However, here we will take an infinite strip S = {i(αz 0 s + βw0 t) ; (s, t) ∈ R × [−1, 1]} . The condition on x = (x, y) corresponding to u εx ∈ S is x ∈ Z and |y| ≤ M, where M = 1/ε. We have F ε , ψ = ψ(u εx ) − ψ(0) Cov(e, ex ) ψ(u εx )Cov(e, ex ) + x∈Z |y|>M
+ψ(0)
x∈Z |y|≤M
Cov(e, ex ).
x∈Z |y|≤M
The inverse Kasteleyn operator in this case is given by −1
−1
K (x, y) = K (bx , w0 ) = T2
dz dw z −y w x . a + b/w − cz/w + dz 2iπ z 2iπ w
The fact we have an infinite strip allows us to make use of one dimensional Fourier series in the x-direction to compute the third sum. Indeed, K−1 (x, y) is the x th Fourier coefficient of the function f y (w) defined by f y (w) = S1
dz z −y . a + b/w − cz/w + dz 2iπ z
88
C. Boutillier
Hence, for a fixed y,
−1
−1
K (x, y)K (−x, −y) =
x∈Z
f y (w) f −y (w) S1
dw . 2iπ w
For y = 0, the functions f y and f −y have disjoint support, therefore the sum above is zero. When y = 0, the sum is equal to
K−1 (x, 0)K−1 (−x, 0) =
x∈Z
f 02 (w) S1
dw = 2iπ w
a+bw cw−d >1
dw . 2iπ(a + b/w)2 w
The third sum equals
ψ(0)
Cov(e, ex ) = ψ(0) aK−1 (0) − a 2
K−1 (x, y)K−1 (−x, −y)
|y|≤M x∈Z
x∈Z |y|≤M
= ψ(0)
a+bw cw−d >1
= ψ(0)
adz − (a + b/w)2iπ w
a+bw cw−d >1
a 2 dz (a + b/w)2 2iπ w
−abIm(w0 ) . π |a + b/w0 |2
The other term coming from the application of the Green formula can also be computed and turns out to be equal to 1 −ψ(0) 2Area π
−abIm(w0 ) |a +
b w0 |
2 .
Hence the variance of the white noise appearing in the limit of N˜ aε is given, after some calculations, by 1 −abIm(w0 ) abIm(w0 ) abcd 1 + = . π |a + b/w0 |2 2Area 8π R 2 Area A similar computation for E N˜ aε (ϕ1 ) N˜ bε (ϕ2 ) leads to the same expression, with a negative sign. As the expression of the coefficient is invariant under cyclic permutation of a, b, c, d, the amplitudes for the other pairs are easily deduced. When d = 0, the dimer model on Z2 is equivalent to the dimer model on the honeycomb lattice with periodic weights a, b, c. One can notice that in this case, the amplitude
Pattern Densities in Non-Frozen Planar Dimer Models
89
w4 b4 w1
b1
w3
b3
b2 w2
Fig. 3. A portion of the square-octagon graph. Edges are oriented from white end to black end, except those whose orientation is represented on the figure.
of the white noise vanishes. The interaction between dimers on the honeycomb lattice is purely magnetostatic. We conjecture that it is true only for that particular model. 7.2. Dimer densities on the square-octagon graph. The square-octagon graph is a Z2 periodic graph whose fundamental domain is presented in Fig. 3. It contains four white and four black vertices. When every edge is assigned a weight equal to 1, the characteristic polynomial is given by
1 w1 0 − 1z 1 1 1 0 1 1 P(z, w) = det 0 z 1 w = 5 − z − z − w − w. −1 0 1 1 When the magnetic field is zero, or weak enough, the dimer model is in a massive phase. The fluctuations of the density field of an edge can be therefore computed by taking derivatives of the free energy of the system with respect to the weights, as explained in Sect. 5. To compute for instance the amplitude of the white noise in the limit density of the edges (w1 , b1 ), we assign to these edges a weight ea and to the others a weight equal to 1, and compute the second derivative of the free energy F associated to this model with respect to a. For these new weights, the characteristic polynomial is now ea w1 0 − 1z 1 1 1 0 1 1 a a a Pa (z, w) = det 0 z 1 w = 4 + e − e z − z − e w − w. −1 0 1 1
90
C. Boutillier
If a is small enough, Pa (z, w) has no zeros on the unit torus, and the dimer model is still in a massive phase in the absence of magnetic field. In every point of this massive phase, the free energy is constant and given by dz dw F = log Pa (z, w) 2iπ z 2iπ w T2
log 1 −
= log(4 + ea ) +
dθ dφ 1 a+iθ −iθ a+iφ −iφ . e + e + e + e 4 + ea 2π 2π
[0,2π ]2 θ−φ Performing the change of variables α = θ+φ 2 , β = 2 and moving the contour of integration over α from [0, 2π ] to [−ia/2, −ia/2 + 2π ] using analyticity and periodicity in α, we finally get an expression of F in terms of an absolutely convergent series: 4ea/2 dα dβ (78) Fa = log(4 + ea ) + log 1 − cos(α) cos(β) a 4+e 2π 2π [0,2π ]2
= log(4 + ea ) −
k ∞ 1 4ea/2 k=1
k
4 + ea
2π 2π dα dβ cosk (α) cosk (β) (79) 2π 2π 0
0
a/2 2k ∞ (2k)! 2 e 1 a . = log(4 + e ) − 2k 4 + ea (k!)2
(80)
k=1
Fa can be expressed as the value of a certain generalized hypergeometric function. The Taylor expansion up to order 2, involving the complete elliptic integrals K and E,
16 K 16 1 3K 16 a2 25 25 − E 25 − + O(a 3 ), Fa = F + a+ 2 5π 2π 2 gives information on the statistics of the copies of edge (w1 , b1 ). The constant coefficient is the free energy of the initial model, the coefficient of a is the probability of (w1 , b1 ),
1 3K 16 25 , P [(w1 , b1 )] = − 2 5π and the coefficient of a 2 /2 gives the amplitude of the white noise describing the scaling limit of the fluctuations of the number of edges (w1 , b1 ),
16 K 16 ε 2 25 − E 25 ˜ ϕ(u)2 du. lim E ( N(w1 ,b1 ) ϕ) = ε→0 2π R2
Similarly, one can compute the probability of seeing an edge of a square, for example (w2 , b1 ), and the amplitude of the white noise:
1 3K 16 25 , P [(w2 , b1 )] = + 4 10π 2K 16 ε 2 25 ˜ lim E ( N(w1 ,b1 ) ϕ) = ϕ(u)2 du. ε→0 5π R2
Pattern Densities in Non-Frozen Planar Dimer Models
91
The fact we see elliptic functions showing up is related to the fact that the spectral curve {Pa (z, w) = 0} in this case is a genus-1 algebraic curve. Acknowledgements. We warmly thank Richard Kenyon for proposing to study pattern densities in dimer models. We are grateful to him for the many fruitful discussions. This work has been done when the author was at Unversité Paris-XI. The last part of writing this paper was done in a project at CWI, financially supported by the Netherlands Organization for Scientific Research (NWO).
References 1. Bertini, L., Cirillo, E.N.M., Olivieri, E.: Renormalization-group transformations under strong mixing conditions: Gibbsianness and convergence of renormalized interactions. J. Stat. Phys. 97, 831–915 (1999) 2. Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J.Amer. Math. Soc. 14, 297– 346 (2001) 3. de Tilière, B.: Conformal invariance of isoradial dimer models & the case of triangular quadrititlings. http://arxiv.org/list/ math.PR/0512395, 2005 4. Dobrushin, R.L., Tirozzi, B.: The central limit theorem and the problem of equivalence of ensembles. Commun. Math. Phys. 54, 173–192 (1977) 5. Fowler, R.H., Rushbrooke, G.S.: Statistical theory of perfect solutions. Trans. Faraday Soc. 33, 1272– 1294 (1937) 6. Giacomin, G., Olla, S., Spohn, H.: Equilibrium fluctuations for ∇φ interface model. Ann. Probab. 29, 1138–1172 (2001) 7. Glimm, J., Jaffe, A.: Quantum physics. New York: Springer-Verlag, 1981 8. Guelfand, I.M., Vilenkin, N.Y.: Les distributions. Tome 4: Applications de l’analyse harmonique, Traduit du russe par G. Rideau. Collection Universitaire de Mathématiques, No. 23, Paris: Dunod, 1967 9. Iagolnitzer, D., Souillard, B.: Lee-Yang theory and normal fluctuations. Phys. Rev. B (3), 19, 1515–1518 (1979) 10. Kasteleyn, P.W.: Graph theory and crystal physics. In: Graph Theory and Theoretical Physics, London: Academic Press, 1967, pp. 43–110 11. Kenyon, R.: Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Statist. 33, 591–618 (1997) 12. Kenyon, R.: Conformal invariance of domino tiling. Ann. Probab. 28, 759–795 (2000) 13. Kenyon, R.: Dominos and the Gaussian free field. Ann. Probab. 29, 1128–1137 (2001) 14. Kenyon, R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150, 409– 439 (2002) 15. Kenyon, R.: Height fluctuations in the honeycomb dimer model. http://arxiv.org/list/ math-ph/0405052, 2004 16. Kenyon, R.: An introduction to the dimer model. In: School and Conference on Probability Theory, ICTP Lect. Notes, XVII, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 267–304 17. Kenyon, R., Okounkov, A.: Limit shapes and the complex Burgers equation. http://arxiv.org/list/mathph/0507007,2005 18. Kenyon R., Okounkov A., Sheffield, S.: Dimers and amoebae. Ann. of Math. (2), 163, 1019–1056 (2006) 19. Naddaf, A., Spencer, T.: On homogenization and scaling limit of some gradient perturbations of a massless free field. Comm. Math. Phys. 183, 55–84 (1997) 20. Neaderhouser, C.C.: Some limit theorems for random fields. Commun. Math. Phys. 61, 293–305 (1978) 21. Newman, C.M.: Normal fluctuations and the FKG inequalities. Commun. Math. Phys. 74, 119–128 (1980) 22. Sheffield, S.: Gaussian Free Field for mathematicians. http://arxiv.org/abs/math/0312099, 2003 23. Sheffield, S.: Random Surfaces: Large Deviations Principles and Gradient Gibbs Measure Classifications, PhD thesis, Stanford University, 2004, and Asterisque 304, (2005) 24. Soshnikov, A.: Gaussian limit for determinantal random point fields. Ann. Probab. 30, 171–187 (2002) 25. Thurston, W.P.: Conway’s tiling groups. Amer. Math. Monthly 97, 757–773 (1990) Communicated by H. Spohn
Commun. Math. Phys. 271, 93–101 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0160-8
Communications in
Mathematical Physics
The Singular Set for the Composite Membrane Problem Henrik Shahgholian Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden. E-mail:
[email protected] Received: 2 March 2006 / Accepted: 6 June 2006 Published online: 9 January 2007 – © Springer-Verlag 2006
Abstract: In this paper we study the behavior of the singular set {u = |∇u| = 0}, for solutions u to the free boundary problem u = f χ{u≥0} − gχ{u<0} , with f > 0, f (x) + g(x) < 0, and f, g ∈ C α . Such problems arise in an eigenvalue optimization for composite membranes. Here we show that if for a singular point z ∈ {u = ∇u = 0}, there are r0 > 0, and c0 > 0 such that the density assumption |{u < 0} ∩ Br (z)| ≥ c0 r 2 ,
∀ r < r0 ,
holds, then z is isolated. The density assumption can be motivated by the following example: u = x12 ,
f = 2, g < −2, and {u < 0} = ∅.
1. Introduction and Background In this paper we analyze properties of singular sets of solutions to a certain eigenvalue optimization problem, which has been very much in focus lately (see [5, 6]). The problem, in physical terms, amounts to building a body of a prescribed shape out of given materials of varying densities, in such a way that the body has a prescribed mass and with the property that the fundamental frequency of the resulting membrane (with fixed boundary) is lowest possible. The reformulated and slightly more general mathematical problem is as follows. A bounded Lipschitz domain ⊂ R2 is given. Also given are Supported in part by the Swedish Research Council. This work is part of the ESF program Global.
94
H. Shahgholian
two numbers α > 0, and A ∈ [0, ||], (|| denotes the Lebesgue measure of ). Let λ be the lowest eigenvalue of the problem, −v + αχ D v = λv in
v = 0 on ∂,
(1.1)
and set (α, A) =
inf
D⊂,|D|=A
λ (α, D).
Then one is interested in the optimal pair (v, D), solving the above problem. The existence of such an optimal pair is shown using minimization of the corresponding functional. Moreover it is known that any optimal pair has the property that D = {v ≤ t}, for some t ≥ 0. Now, after rewriting the equation u := t − v, and taking into consideration that D = {v ≤ t}, one arrives at u = (αχ{u≥0} − λ)(u − t).
(1.2)
One of the main questions that has puzzled several mathematicians is whether the singular set Su := {u = |∇u| = 0} is isolated, or small enough. In this paper we consider a more general problem of the type u = f χ{u≥0} − gχ{u<0} ,
(1.3)
where f, g are C α -functions. We also assume, throughout this paper, f > 0,
f +g <0
on the singular set Su .
(1.4)
The assumption f > 0 is needed to avoid degeneracy in blow-up arguments. It is however, compatible with the applications in the composite membrane problem. In terms of the original problem it means that t > 0 and that α < λ. Also the case f + g ≥ 0, can be handled easier, due to strong tools such as Alt-Caffarelli-Friedman monotonicity formula [1], and that of Caffarelli-Jerison-Kenig [4]; see [9, 10] for a general treatment of this case. We also refer the reader to a recent paper by J. Andersson, and G.S. Weiss [2], where they treat the case f ≡ 0. Our main result is the following. Theorem 1.1. Let D ⊂ R2 be a bounded domain and f, g ∈ C α (D). Suppose u solves problem (1.3) in D (with given boundary condition), and that condition (1.4) is satisfied at z ∈ Su := {u = |∇u| = 0}, i.e. f (z) > 0,
f (z) + g(z) < 0.
(1.5)
Suppose further that for z ∈ Su , we have positive constants c0 , r0 such that |{u < 0} ∩ Br (z)| ≥ c0 r 2 , for all 0 < r < r0 . Then z is an isolated point of Su .
(1.6)
Singular Set Composite Membrane
95
The proof uses a simple blow-up argument in combination with monotonicity of certain energy functionals, due to G.S. Weiss. Define u r (x) :=
u(r x + z) , r2
fr (x) = f (r x + z).
Then the following function (due to Weiss [8]) W (r ) = W (r, u, z) := |∇u r |2 + 2 fr u r+ + 2gr u r− − 2 B1 (0)
∂ B1 (0)
u r2 ,
has the property that W (r ) + C0 r α/2
is monotone increasing in r .
(1.7)
Here C0 depends on the C α -norms of f, g. If f, g are constants then we may take C0 = 0 and also claim that the function W (r ) is strictly monotone unless u is homogeneous of degree two.
(1.8)
For a proof of a much more general version of this see Appendix. We also refer to [8], and [7] for variations of this monotonicity formula. Let us now state the following lemma. Lemma 1.2. If u is a non-trivial degree two homogeneous solution to our problem (1.3), with f = f 0 , g = g0 constants, and f 0 + g0 < 0, then either Su = {0} or (after rotation) Su = {x1 = 0}. The reader may find a proof of this in [3], where the author tacitly assumes {u < 0} = ∅. The case {u < 0} = ∅ was unfortunately forgotten to be considered in [3]. The proof of this, however, follows by straightforward computations, or simply by the fact that due to homogeneity, if z ∈ Su and z = 0, then the ray L z := {t z, t > 0} ∈ Su . We may rotate and assume L z is the positive x1 -axis. It is not hard to realize that by Hopf’s boundary lemma, for each y ∈ L z we have u ≥ 0 in Br (y) for some r > 0. In particular, f 0 x12 /2 is a solution in Br (y). Now by uniqueness of Cauchy-Kowalevskaya theorem we must have u = f 0 x12 /2. Lemma 1.3. Let u be a solution to our problem (1.3), with f, g ∈ W 1, p ( p > 1), and suppose that for some sequence r j 0, u(r j x) r 2j converges to a non-trivial function u 0 . Then u 0 is a degree two homogeneous function, solving our problem. The proof of this follows from the Weiss monotonicity function. Indeed, W (sr j , u) = W (s, u j ) converges to W (0+ , u) = C(u) as r j tend to zero, and we obtain W (s, u 0 ) = C(u). In particular u 0 solves our problem with f, g constants, and the Weiss function is constant. Hence the monotonicity theorem tells us that u 0 is homogeneous of degree two.
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Lemma 1.4. The solution to our problem in Theorem 1.1 has the non-degeneracy proprty sup u ≥
Br (z)
bz 2 r , 2n
for some constant bz > 0, and all r < 1 − |z|. Proof. Let w(x) = u(x) − bz |x − z|2 /2n, with 0 < bz ≤ inf Br (z) f (x), r < r0 . Then w ≥ 0 due to the assumption f + g < 0. Hence w takes its non-negative maximum (w(0) = 0) in Br (z) on the boundary. Therefore sup u ≥ bz r 2 /2n.
Br (z)
2. Proof of Theorem 1.1 Let us assume z is the origin, and that is is not an isolated point of Su , i.e. there exists x j ∈ Su , with r j := |x j | → 0. We have two possibilities. (A) There exists a constant M such that M j ≤ Mr 2j ,
for j = 1, 2, · · · .
(B) There exists a sequence α j , tending to infinity, such that M j ≥ α j r 2j ,
for j = 1, 2, · · · .
(2.1)
Now if (A) above is true then u j (x) :=
u(r j x) r 2j
is bounded. In particular, by compactness and by Lemma 1.4, for a subsequence u j converges to a non-trivial limit function u 0 , solving our problem u 0 = f (0)χ{u 0 ≥0} − g(0)χ{u 0 <0} . Using the monotonicity function of Weiss W (0+ , u) = lim W (sr j , u) = lim W (s, u j ) = W (s, u 0 ) j
j
for any constant s < 1. By the Weiss monotonicity argument (1.7)-(1.8), u 0 is a degree two homogeneous global solution to our problem. Now at the same time x˜ j = x j /r j ∈ Su j converges (for yet another subsequence) to a point x 0 ∈ Su 0 and with |x 0 | = 1. Since condition (1.6) is stable under scaling |{u j < 0} ∩ Br (0)| ≥ c0 r 2 , we conclude by Lemma 1.2 that Su 0 = {0}. A contradiction.
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In Case (B) we use a homogeneous scaling at r j = |x j |, u j (x) = u(r j x)/M j . The idea is to use the argument in [8] (Proof of Proposition 4.1). Obviously u j are bounded in L 2∂ B1 . Let now f j :=
r 2j Mj
f (r j x), g j :=
r 2j Mj
g(r j x).
Then by the monotonicity function 2 2 2 2 rj rj W (1, u j , f j , g j ) = W (r j , u, f, g) ≤ (W (1, u, f, g) + C) → 0 Mj Mj as j tends to infinity. In particular 2 2 2 rj rj 2 2 |∇u j | ≤ 2 uj + |u j | . ( f +g) (W (1, u, f, g)+C)+ Mj Mj B1 ∂ B1 B1 (2.2) Since |u j | ≥ −C (for some constant C), it is not hard to see (using monotonicity of the integral ∂ Bt h for subharmonic functions h) that |u j | ≤ |u j | + C. ∂ B1
B1
Putting this into estimate (2.2), and using Hölder’s inequality, we conclude u j ∈ W 1,2 (B1 ). Hence there is a subsequence of u j converging weakly in W 1,2 to a limit function u 0 . Now the compact embedding on the boundary (i.e. the trace theorem) implies that u 0 L 2 (∂ B1 ) = 1. Moreover |u 0 (0)| = |∇u 0 (0)| = 0,
(2.3)
and due to the assumption in (B), Mk j ≥ αk 4−k j , which leads to |u j | ≤
4−k j 1 ≤ , → Mk j αj
0,
i.e., u 0 is harmonic. It also follows from inequality (2.2) that 2 |∇u 0 | ≤ 2 u 20 . B1
∂ B1
(2.4)
(2.5)
Using (2.3)-(2.4)-(2.5) in conjunction with Lemma 4.1 in [8] we conclude that u 0 is a degree two homogeneous harmonic function. Now on the other hand we have that the sequence x˜ j = x j/r j ∈ Su j (where |x˜ 0 | = 1) and hence there is limit point x 0 ∈ Su 0 , with |x 0 | = 1. This of course is a contradiction, as for any degree two homogeneous harmonic function h we must have Sh = {0}.
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3. Appendix In this Appendix we establish a generalization of Weiss’s monotonicity formula. The proof of such a result follows the same lines of that of the proof of Theorem M in [7], that was done for the one-phase case. We assume that ω is a modulus of continuity satisfying ω(ρ) log 1 ρ ρ
0
and that
ω
F (t) := 0
t
1 ω(τ ) log τ
dρ < ∞,
n+3 n+2
τ
−
(3.1)
1 n+1 n+2 τ t n+2
dτ.
(3.2)
Theorem 3.1 [Monotonicity formula]. Let u be a solution to our problem (1.3), in B1 (0) with f, g ∈ C 0 (B1 (0)), having a modulus of continuity ω(r ), satisfying (3.1). Then there exists a constant C M = C(M, ω, n) such that W (u r , fr , gr ) + C M F ω (r )
for 0 < r < 1/2,
with F ω as in (3.2). Remark 3.2. In the above, when f, g = 1, we can take C M = 0 and therefore W (r ) itself is monotone. This is the original case of Weiss’s monotonicity formula; see [8]. Moreover, in this case something more can be shown: if W (r1 ) = W (r2 ) for r1
F(u r , fr , gr ) =
B1 (0)
Then
W (u s , f s , gs ) − W (u r , fr , gr ) =
B1 (0)
fr u r+ + gr u r− , G(u r ) =
s
r
= I1 + 2I2 , d dt ,
and I1 := r
s
E (u t ) + 2F(u t , f t , gt ) − 2G (u t ) dt,
I2 := F(u s , f s , gs ) − F(u r , fr , gr ) − r
Using integration by parts one can show s t I1 = 2 r
∂ B1 (0)
u r2 .
E (u t )dt + 2F(u s , f s , gs )−2F(u r , fr , gr ) s −2 G (u t )dt r
where =
∂ B1 (0)
s
F(u t , f t , gt )dt.
2 u t d xdt ≥ 0.
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Now to estimate I2 , we see that I2 (u, f, g) = I2 (u, f, 0) + I2 (u, 0, g), and it is enough to estimate one of these terms, as the estimate for the second one follows in the same way. Let now σ = s/r , and approximate f with a smooth function f with the same modulus of continuity ω(r ); for instance by taking a convolution with a mollifier. Next we rewrite σ d −n−2 f (σ x) u σ − f (x) u − I2 (u s , f s , 0) = r f (t x) u t dt d x dt B 1 r σ d = r −n−2 f (t x) u t dt d x dt B 1 σr = r −n−2 x · ∇ f (t x) u t d x dt. 1
Br
By changing to polar coordinates and integrating by parts, we’ll have r x · ∇ f (t x) u t d x = ∂ρ f (tρθ )u t (ρθ )ρ n dρ d Hθn−1 Br ∂ B1 0 r n n [ f (tr θ ) − 1]u t (r θ )r − = [ f (tρθ ) − 1]∂ρ (u t (ρθ )ρ )dρ d Hθn−1 . ∂ B1
0
Now, recall that in general we have the Zygmund class estimate |x|2 log |u| ≤ C M
1 1 1 , |∇u| ≤ C M , |x| ≤ |x| log |x| |x| 2
for solutions of our problem. Then we obtain x · ∇ f (t x) u t d x Br r 1 1 ω(tr )r n+2 log + ≥ −C M ω(tρ)ρ n+1 log dρ . tr tρ 0 Since this inequality is purely in terms of the modulus of continuity ω, we can let → 0 to obtain I2 (u s , f s , 0) − I2 (u r , fr , 0) σ r 1 1 1 ω(tr ) log + n+2 ω(tρ)ρ n+1 log dρ dt. ≥ −C M tr r tρ 1 0 Changing the variables, we can rewrite the above inequality as I2 (u s , f s , 0) − I2 (u r , fr , 0) s t s CM 1 τ n+1 1 ≥− ω(t) log dt + ω(τ ) n+2 log dτ dt r t t τ r r 0 1 s ω(t) log s t τ n+1 1 t s dt + ≥ −C M ω(τ ) n+3 log dτ dt . r t t τ r r 0
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We also have s
s s τ n+1 1 τ n+1 1 ω(τ ) n+3 log dτ dt = ω(τ ) n+3 log dt dτ t τ t τ 0 0 0 τ s 1 1 1 1 n+1 ω(τ )τ − = log dτ. n+2 0 τ n+2 s n+2 τ t
Thus, introducing, ω
F (t) := 0
t
1 ω(τ ) log τ
n+3 n+2
τ
−
1 n+1 n+2 τ t n+2
dτ,
we obtain s [F(s) − F(r )] I2 (u s , f s , 0) − I2 (u r , fr , 0) ≥ −C M r
for any 0 < r ≤ s ≤ 1/2. Finally, we drop the factor s/r in the right-hand side, by taking a partition r = s0 < s1 < . . . < s N = s and applying the inequality for I2 (u si+1 , f si+1 , 0) − I2 (u si , f si , 0), and then summing up, and letting the size of the partition tend to 0. Thus we arrive at I2 (u s , f s , 0) − I2 (u r , fr , 0) ≥ −C M [F(s) − F(r )],
for any 0 < r ≤ s ≤ 1/2. Arguing similarly for g, and putting all the above together we have W (u r , fr , gr ) + C M F ω (r ) . The theorem is proved. where C M = 4C M
as r ,
In the above theorem, it is not hard to see that once ω(t) ≤ Ct α , then F ω (r ) can be replaced by Cr α/2 . References 1. Alt, H.W., Caffarelli, L.A., Friedman, A.: Variational problems with two phases and their free boundaries. Trans. Amer. Math. Soc. 282, 431–461 (1984) 2. Andersson, J., Weiss, G.S.: Cross-shaped and degenarte singularities in an unstable elliptic free boundary problem. J. Differ. Eqs. 2005. In press. 3. Blank, I.: Eliminating mixed asymptotics in obstacle type free boundary problems. Commun. Partial Differ. Eq. 29(7–8) 1167–1186 (2004) 4. Caffarelli, L.A., Jerison, D., Kenig, C.E.: Some new monotonicity theorems with applications to free boundary problems. Ann. Math. (2) 155(2) 369–404 (2002) 5. Chanillo, S., Grieser, D., Kurata, K.: The free boundary problem in the optimization of composite membranes. In: Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999), Contemp. Math. 268, Providence, RI: Amer. Math. Soc. 2000, pp. 61–81 6. Chanillo, S., Grieser, D., Imai, M., Kurata, K., Ohnishi, I.: Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. Commun. Math. Phys. 214(2), 315–337 (2000) 7. Petrosyan, A., Shahgholian, H.: Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem. Submitted, available at http://www.math.purdue.edu/ arshak/pdf/gecm-energyfinal.pdf 8. Weiss, G.S.: An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary. Interfaces Free Bound. 3(2), 121–128 (2001)
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9. Shahgholian, H., Uraltseva, N., Weiss, G.S.: Global solutions of an obstacle-problem-like equation with two phases. Monatsh Math 142(12), 27–34 (2004) 10. Shahgholian, H., Uraltseva, N., Weiss, G.S.: The Two-Phase Membrane Problem – Regularity of the Free Boundaries in Higher Dimensions. Submitted, available at http://www.ms.u-tokyo.ac.jp/ gw/url.pdf Communicated by P. Constantin
Commun. Math. Phys. 271, 103–178 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0173-3
Communications in
Mathematical Physics
Supersymmetric Vertex Algebras Reimundo Heluani, Victor G. Kac Department of Mathematics, MIT, Cambridge, MA 02139, USA. E-mail:
[email protected];
[email protected] Received: 27 March 2006 / Accepted: 9 August 2006 Published online: 20 January 2007 – © Springer-Verlag 2007
Abstract: We define and study the structure of SUSY Lie conformal and vertex algebras. This leads to effective rules for computations with superfields. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . 3. Structure Theory of N W = N SUSY Vertex Algebras 3.1 Formal distribution calculus . . . . . . . . . . . 3.2 N W = N SUSY Lie conformal algebras . . . . 3.3 Identities and existence theorem . . . . . . . . 3.4 The universal enveloping SUSY vertex algebra 4. Structure Theory of N K = N SUSY Vertex Algebras 5. Examples . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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103 110 118 118 125 135 148 149 166 177
1. Introduction 1.1. Vertex algebras were introduced about 20 years ago by Borcherds [Bor86]. They provide a rigorous definition of the chiral part of 2-dimensional conformal field theory, intensively studied by physicists. Since then, they have had important applications to string theory and conformal field theory, and to mathematics, by providing tools to study the most interesting representations of infinite dimensional Lie algebras. Since their appearance, they have been extensively studied in many papers and books (for the latter we refer to [FLM88, FHL93, Kac96, Hua97, FBZ01, BD04]). Supported in part by NSF grants DMS-0201017 and DMS-0501395.
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1.2. The purpose of the present paper is to define and study the structure of supersymmetric (SUSY) vertex algebras. This theory encompasses the formalism of superfields extensively used by physicists (see eg. [Coh87, DRS90] and references therein). For earlier discussions of SUSY vertex algebras see [Kac96, § 5.9] and [Bar00]. Recall [Kac96] that a vertex algebra (V, |0, T, Y ) is a vector superspace V (space of states) with an even vector |0 (vacuum vector), an even endomorphism T (translation operator), and a parity preserving bilinear product with values in Laurent series in an indeterminate z over V: V ⊗ V → V ((z)), a ⊗ b → Y (a, z)b = (a(n) b)z −1−n , n∈Z
subject to the following axioms (a, b ∈ V ) : • (vacuum axioms) Y (a, z)|0|z=0 = a, T |0 = 0, • (translation invariance) [T, Y (a, z)] = ∂z Y (a, z), • (locality) (z − w) N [Y (a, z), Y (b, w)] = 0 for some N ∈ Z+ . The vertex algebras which arise in conformal field theory carry a conformal vector ν ∈ V , so that the coefficients of the corresponding field Y (ν, z) ≡ L(z) = L n z −2−n (1.2.1) n∈Z
satisfy the Virasoro commutation relations [L m , L n ] = (m − n)L m+n + δm,−n
m3 − m c, 12
(1.2.2)
where c ∈ C is the central charge, and also the following two properties hold: • L −1 = T, (1.2.3) • L 0 is diagonalizable on V with eigenvalues bounded below. Sometimes, such a vertex algebra V (called conformal) admits a “supersymmetry”, namely, V carries a superconformal vector τ , such that the corresponding field Y (τ, z) ≡ G(z) = G n z −3/2−n (1.2.4) n∈ 21 +Z
satisfies the following properties: (1) ν = 21 G −1/2 τ is a conformal vector with central charge c; (2) the operators G n (n ∈ 1/2 + Z) along with the Virasoro operators L n (n ∈ Z), appearing in L(z) = Y (ν, z), form the Neveu-Schwarz algebra with central charge c, namely, along with the Virasoro relations (1.2.2), the following commutation relations hold: n c 1 2 [G m , L n ] = m − m − δm,−n . G m+n , [G m , G n ] = 2L m+n + 2 3 4 (1.2.5)
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Then V is called an N = 1 superconformal vertex algebra. In particular, we then can obtain an enhanced translation invariance property as follows. Let S = G −1/2 , let θ be an odd indeterminate, commuting with z : θ 2 = 0, θ z = zθ , and anticommuting with S. To each a ∈ V associate a superfield: Y (a, z, θ ) = Y (a, z) + θ Y (Sa, z).
(1.2.6)
Then one can show, using the so-called commutator formula ([Kac96, (4.6.4)]), that [S, Y (a, z, θ )] = (∂θ − θ ∂z )Y (a, z, θ ).
(1.2.7)
Since, by the second formula in (1.2.5) and (1.2.3), we have S 2 = T,
(1.2.8)
formula (1.2.7) implies the usual translation invariance [T, Y (a, z, θ )] = ∂z Y (a, z, θ ). This leads to the following definition of an N K = 1 SUSY vertex algebra (V , |0, S, Y ), where S is an odd endomorphism of the space of states V , and Y is a parity preserving bilinear product with values in V ((z))[θ ] : V ⊗ V → V ((z))[θ ], a ⊗ b → Y (a, z, θ )b = θ 1−i z −1−n a(n|i) b, n∈Z
i=0,1
subject to the following axioms: • (vacuum axioms) Y (a, z, θ )|0|z=0,θ=0 = a, S|0 = 0, • (translation invariance) [S, Y (a, z, θ )] = (∂θ − θ ∂z )Y (a, z, θ ), • (locality) (z − w) N [Y (a, z, θ ), Y (b, w, ζ )] = 0 for some N ∈ Z+ . This SUSY vertex algebra is called conformal if there exists an odd vector τ ∈ V , such that Y (τ, z, θ ) = G(z) + 2θ L(z), and such that the coefficients of the expansions of these fields as in (1.2.1) and (1.2.4) satisfy the commutation relations (1.2.2) and (1.2.5) of the Neveu-Schwarz algebra, S = G −1/2 and L 0 is diagonalizable with eigenvalues bounded below. Of course, a SUSY vertex algebra (V, |0, S, Y (a, z, θ )) gives rise to an ordinary vertex algebra (V, |0, T = S 2 , Y (a, z) = Y (a, z, 0)), and a superconformal vector τ gives rise to a conformal vector ν = 21 Sτ . However, computations with superfields are much more effective than with the ordinary fields. 1.3. Now, the Neveu-Schwarz is the simplest, after Virasoro, among the superconformal Lie algebras. A superconformal Lie algebra is a pair (L , F ), where L is a Lie superalgebra and i i −1−n F = a (z) = an z n∈Z
i∈I
is a finite family of formal distributions whose coefficients ani span L , satisfying the following properties: −2−n , such that (1) F contains a Virasoro formal distribution L(z) = n∈Z L n z i i [L −1 , a (z)] = ∂z a (z);
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(2) the formal distributions a i (z) are pairwise local: (z − w) N [a i (z), a j (w)] = 0 for some N ∈ Z+ , or, equivalently: [a i (z), a j (w)] =
N −1
ij
cs (w)∂ws δ(z, w)
s=0 ij
for some formal distributions cs (z); ij (3) cs (z) ∈ C[∂z ]F ; (4) L is a non-split central extension of an almost simple Lie superalgebra L˜ (i.e. all non-zero ideals of L˜ contain its derived subalgebra L˜ ). 1.4. A complete list of centerless superconformal Lie algebras consists of four series ˜ +2), K (1|N ), and two exceptions K (1|4) and (N ∈ Z+ ) : W (1|N ), S(1|N +2; a), S(1|N C K (1|6) (see [FK02]). Here W (1|N ) denotes the Lie superalgebra of all derivations of C[z, z −1 , θ 1 , . . . , θ N ], where z is an even indeterminate and θ i ’s are odd anticommuting indeterminates, commuting with z (in particular, W (1|0) is the centerless Virasoro ˜ algebra). The Lie superalgebras S(1|N ; a) and S(1|N ) (resp. K (1|N )) are subalgebras of W (1|N ) consisting of vector fields, annihilating certain supervolume forms (resp. N i i preserving the super-contact form1 dz + i=1 θ dθ , up to multiplication by a function), K (1|4) is the derived algebra of K (1|4), and C K (1|6) is a certain subalgebra of K (1|6). The only algebras on the list that admit a central extension are W (1|N ) with ˜ N ≤ 1, S(1|2; a), S(1|2), K (1|N ) with N ≤ 4 and K (1|4) (see [KvdL89] and [FK02]). Note that the Neveu-Schwarz algebra is a central extension of K (1|1). The subalgebra L≤ = span{ani |i ∈ I, n ≥ 0} is called the annihilation subalgebra of L . The reason for this name comes from the fact that the space V (L ) = U (L )/U (L )L≤ ,
(1.4.1)
where U (L ) denotes the universal enveloping superalgebra of L , carries a canonical structure of a vertex algebra, called the universal enveloping vertex algebra of L , where |0 is the image of 1, T is induced by L −1 , and the image of F is contained in the space of fields of V (L ), so that L≤ |0 = 0. In all cases the subalgebra L≤ consists of all vector fields of L which are regular at the origin. Denote by L< the subalgebra of L≤ , consisting of vector fields, which vanish at the origin. Then in all cases we can find a complementary subalgebra Ltr (the “translation” subalgebra) L≤ = Ltr ⊕ L< . (1.4.2) In the cases W and S one can choose Ltr = spanC {∂z , ∂θ 1 , . . . , ∂θ N }. However, in the remaining cases the simplest choice is Ltr = spanC {∂z , ∂θ 1 − θ 1 ∂z , . . . , ∂θ N − θ N ∂z }. Denote these Lie superalgebras by Wtr and K tr respectively, and by HW (resp. H K ) the universal enveloping superalgebra of Wtr (resp. K tr ). It is an associative algebra on one even generator T and N odd generators S i , subject to the relations: [T, S i ] = 0, [S i , S j ] = 0(resp. = 2δi, j T ). 1 Here and further we use the sign convention of [DM99],which differs from the one used in [KvdL89] and [FK02]. The main difference is that the de Rham differential d is an even derivation.
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Note that the operators T and S of an N K = 1 SUSY vertex algebra define a representation of the associative superalgebra H K for N = 1 (see 1.2.8). This leads to the definition of an N W (resp. N K ) = N SUSY vertex algebra as a HW (resp. H K )-module V with the vacuum vector |0 and a parity preserving bilinear product with values in V ((z))[θ i , . . . , θ N ], c a ⊗ b → Y (a, z, θ 1 , . . . , θ N )b = θ J z −1−n a(n|J ) b, n∈Z,J
where J runs over all ordered subsets of {1, . . . , N }, J c denotes the ordered complement of J in {1, . . . , N } and θ J = θ j1 . . . θ jk for J = { j1 , . . . , jk }, subject to the vacuum, translation invariance and locality axioms. The vacuum and locality axioms generalize in the obvious way; translation invariance means the following: [T, Y (a, z, θ )] = ∂z Y (a, z, θ ), [S i , Y (a, z, θ )] = ∂θ i Y (a, z, θ )
in the N W = N case,
[S , Y (a, z, θ )] = (∂θ i − θ ∂z )Y (a, z, θ ) in the N K = N case. i
i
Of course, in general (V, |0, T, Y (a, z) = Y (a, z, 0)) is an ordinary vertex algebra. 1.5. Given a superconformal Lie algebra L , and a decomposition (1.4.2) of its annihilation subalgebra, such that Ltr Wtr (resp. K tr ), we define an L -conformal N W (resp. N K ) = N SUSY vertex algebra V by the property that V carries a representation of L such that the representation of U (Ltr ) coincides with that of HW (resp. H K ), the formal distributions of L are represented by fields of V , and L 0 is diagonalizable with eigenvalues bounded below. The “minimal” example of an L -superconformal vertex algebra is V (L ), given by (1.4.1), or V c (L ) = V (L )/(C − c1)V (L ) in the case L has a central element C, called the SUSY Virasoro vertex algebra associated to L . 1.6. We develop the structure theory of SUSY vertex algebras along the lines of [Kac96]. First, we develop the calculus of formal superdistributions. As usual, the key role is played by the formal super delta-function δ(Z , W ) = (θ 1 − ζ 1 ) . . . (θ N − ζ N )(i z,w − i w,z )
1 , z−w
(1.6.1)
where Z = (z, θ 1 , . . . , θ N ), W = (w, ζ 1 , . . . , ζ N ), and i z,w signifies the expansion in the domain |z| > |w|. The key result is the decomposition formula of a local formal superdistribution: ( j|J ) a(Z , W ) = ∂W δ(Z , W )c j|J (W ), (1.6.2) j≥0,J
where
c j|J (W ) = res Z (Z − W ) j|J a(Z , W ),
(1.6.3)
and res Z stands for the coefficient of θ 1 . . . θ N z −1 . There are actually two cases, W and j|J K , of this formula, depending on the choice of ∂W in (1.6.2) and the respective choice j|J of (Z − W ) in (1.6.3). We let for J = { j1 < · · · < js } : j|J
∂W = ∂wj ∂θ j1 . . . ∂θ js
in case W,
j|J ∂W
in case K ,
= ∂wj (∂θ j1 + θ j1 ∂w ) . . . (∂θ js + θ js ∂w )
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and in both cases we let ∂W = (−1)|J |(|J |+1)/2 ∂ j|J /j!. (The second choice will be j|J denoted by DW .) Respectively, we let (Z − W ) j|J = (z − w) j (θ j1 − ζ j1 ) . . . (θ js − ζ js )
j N j|J i i (Z − W ) = z − w − θ ζ (θ j1 − ζ j1 ) . . . (θ js − ζ js )
in case W, in case K .
i=1
1.7. Next, the formal Fourier transform is defined by F Z ,W a(Z , W ) = res Z exp((Z − W ) )a(Z , W ), where = (λ, χ 1 , . . . , χ N ), λ is an even indeterminate, χ i ’s are odd indeterminates, commuting with λ and satisfying the following commutation relations: χiχ j + χ j χi = 0
in case W,
χ χ + χ χ = −2δi, j λ
in case K .
i
j
j
i
Defining the -bracket [a(W ) b(W )] = F Z ,W [a(Z ), b(W )] of formal superdistributions, we arrive at the notion of the N W (resp. N K )= N SUSY Lie conformal algebra R, which, as usual, encodes the singular part of the operator product expansion (OPE) of a local pair of superfields. The case N = 0 is that of an ordinary Lie conformal superalgebra [Kac96]. The new phenomenon for N > 0 is that the -bracket has parity N mod 2. Consequently, the bracket [a b]| =0 induces on R/(T R + i S i R) a structure of a Lie superalgebra of parity N mod 2. On the other hand, the structure of a SUSY Lie conformal algebra is an important part of the structure of a SUSY vertex algebra. As in the case of vertex algebras, the only missing ingredient is the normally ordered product of superfields, which is defined as usual: : a(Z )b(Z ) := a+ (Z )b(Z ) + (−1) p(a) p(b) b(Z )a− (Z ), c where for a superfield a(Z ) = n∈Z,J θ J z −1−n a(n|J ) we let a− (Z ) =
θ J z −1−n a(n|J ) , a+ (Z ) = a(Z ) − a− (Z ). c
n≥0
We prove that the non-commutative Wick formula, which allows to compute the singular part of the OPE for normally ordered products, generalizes to the SUSY N W = N and N K = N cases almost verbatim: ( p(a)+N ) p(b) : b[a c] : + [[a b] c]d . (1.7.1) [a : bc :] =: [a b]c : + (−1) 0
Furthermore, we show that the uniqueness and existence theorems, as well as all basic identities for vertex algebras, extend to the SUSY case with minor modification of signs.
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In particular, we show that a N W (resp. N K ) = N SUSY vertex algebra is a N W (resp. N K )= N SUSY Lie conformal algebra with the bracket: [a b] = res Z e Z Y (a, Z )b, together with a unital, “quasicommutative” and “quasiassociative” differential superalgebra structure ::, which are related by formula (1.7.1). This is an equivalent definition of a SUSY vertex algebra, analogous to the one given in [BK03] for vertex algebras. Removing “quantum corrections”, we obtain the definition of a SUSY Poisson vertex algebra. 1.8. As in the vertex algebra case, most of the basic examples of SUSY vertex algebras are constructed starting with a SUSY Lie conformal algebra R, and taking its universal enveloping vertex algebra V (R) or V c (R). This can be defined as the universal enveloping vertex algebra V (L ) of the corresponding formal superdistribution Lie algebra V (L ) (resp. V c (L )), defined in the same way as in (1.4.1). By abuse of terminology, we often say that V (R) is a SUSY vertex algebra generated by R. The simplest example of an N K = N SUSY vertex algebra is the well-known boson-fermion system, generated by one superfield of parity N mod 2, subject to the following -bracket: [ ] = 1|N
for even N , [ ] = 0|N
for odd N .
Viewed as an ordinary vertex algebra, this SUSY vertex algebra in the case N = 1 is generated by one boson and one fermion (hence the name “boson-fermion” system). Another example is the SUSY vertex algebra, generated by the current SUSY Lie conformal algebra, associated to a Lie superalgebra g with an invariant bilinear form ( , ). For N even, the corresponding SUSY N W = N or N K = N -bracket is defined as [a b] = [a, b] + λ(a, b), a, b ∈ g. For N odd, we consider the vector superspace g with reversed parity, and define for a, ¯ b¯ ∈ g :
N p(a) i ¯ = (−1) [a¯ b] χ . [a, b] + (a, b) i=1
In the case N K = 1, using the normally ordered products of the above superfields, one reproduces the construction of the N K = 1 super Virasoro field from [KT85]. Unfortunately, we do not know how to construct a SUSY lattice vertex algebra. A less routine example is the following. Let g be a Lie algebra and F a g-module. We associate to this data an N K = 1 SUSY Lie conformal algebra R(g, F), which is a free H K -module over the vector superspace: (F ⊕ g∗ ) ⊕ (g ⊕ g∗ ), where (F ⊕ g∗ ) and (g ⊕ g∗ ) are the even and odd parts respectively is the change of parity functor, with the following non-zero -brackets (X, Y ∈ g, f ∈ F, α ∈ g∗ , α¯ ∈ g∗ ): [X Y ] = [X, Y ],
[X f ] = X f,
[X α] = X α + λ < α, X >,
[X α] ¯ = X α + χ < α, X > .
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The corresponding SUSY universal enveloping vertex algebra is denoted by V (g, F). The SUSY N K = 1 vertex algebra V (g, F) in the case when g is the Lie algebra of vector fields on a manifold M, F is the space of functions on M and g∗ is the space of differential 1−forms on M, is used in [BZHS06] to construct the chiral de Rham complex [MSV99], as a sheaf of N K = 1 SUSY vertex algebras on M, and study its SUSY properties. This approach makes calculations in the chiral de Rham complex more natural and simple. 1.9. In the subsequent paper [Hel06] the formalism of SUSY vertex algebras is applied to associate to any (strongly) conformal SUSY N W (resp. N K )= N vertex algebra V a vector bundle V X on any supercurve (resp. superconformal curve) X , along the lines of [FBZ01], where this is done for ordinary curves. 1.10. Throughout the paper we use the usual conventions of the superalgebra theory [Kac77, DM99]. For example, by a commutator of elements a and b of an associative superalgebra, we always mean [a, b] = ab − (−1) p(a) p(b) ba, where p(a) stands for the parity of a. In particular, by a commutative superalgebra we mean that ab = (−1) p(a) p(b) ba. Likewise, an odd operator and an odd indeterminate always anticommute. 2. Preliminaries In this section we recall some notation and basic results on vertex algebras. We also give the first examples of SUSY vertex algebras constructed via ordinary vertex algebras. The reader is referred to [Kac96] for an introduction to the vertex algebra theory. Definition 2.1. Let A be a Lie superalgebra. An A -valued formal distribution is a formal expression of the form: B(z) = B(n) z −1−n , n∈Z
where B(n) ∈ A have the same parity for all n ∈ Z; this parity is called the parity of B(z). The coefficients B(n) are called the Fourier modes of B(z), and z is an indeterminate. A pair of formal distributions B(z), C(w) is local if (z − w) N [B(z), C(w)] = 0
for some N ∈ Z+ .
If A = End(V ), where V is a vector superspace, with the usual superbracket, we say that B(z) is a field if, for every v ∈ V , B(n) v = 0 for large enough n. 2.1. Let V be a vertex algebra2 (see 1). The map Y is called the state-field correspondence and we will use this map to identify a vector a ∈ V with its corresponding field Y (a, z). 2.2. Given a vertex algebra V we denote
a(n) b = a(n) b ,
[aλ b] =
λk k≥0
k!
a(k) b,
: ab := a(−1) b.
(2.2.1)
The first operation is called the n th product, the second is called the λ-bracket and the third the normally ordered product. 2 We will denote a vertex algebra by its space of states V when there is no possible confusion.
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2.3. For each n ∈ Z, define the n th product of End(V )-valued fields A(z) and B(z) as follows. Denote by i z,w the expansion in the domain |z| > |w|: i z,w z w (z −w) = z m
n
k
m+k
w k j k w i z,w 1 − = (−1) z m+k− j w n+ j . j z n
(2.3.1)
j≥0
Define
A(w)(n) B(w) = resz i z,w (z − w)n A(z)B(w)
−i w,z (z − w)n (−1) p(A) p(B) B(w)A(z) ,
(2.3.2)
where p(A) denotes the parity of A(w). It can be shown that the following n th product identity holds (cf. [Kac96, Prop. 4.4]) Y (a(n) b, z) = Y (a, z)(n) Y (b, z) ∀n ∈ Z,
(2.3.3)
Y (T a, z) = ∂z Y (a, z).
(2.3.4)
hence,
2.4. In a vertex algebra V we have the following commutator formulas [Kac96, p. 112]: m [a(m) , b(n) ] = a( j) b (m+n− j) , j j≥0
∂wj w m [a(m) , Y (b, w)] = Y (a( j) b, w). j! j≥0
This formula shows that the space of Fourier modes of all fields of a vertex algebra is closed under the Lie bracket, and, moreover, the commutation relations are expressed in terms of j th products. Definition 2.2. A Lie conformal algebra is a super C[∂]-module R equipped with a parity preserving bilinear map [ λ ] : R ⊗ R → C[λ] ⊗ R,
a ⊗ b → [aλ b],
satisfying the following axioms: • Sesquilinearity: [∂aλ b] = −λ[aλ b], [aλ ∂b] = (∂ + λ)[aλ b]. • Skew-commutativity:
[bλ a] = −(−1) p(a) p(b) [a−∂−λ b].
• Jacobi identity: [aλ [bμ c]] = [[aλ b]λ+μ c] + (−1) p(a) p(b) [bμ [aλ c]], for all a, b, c ∈ R. Here λ and μ are commuting indeterminates.
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Given a Lie conformal algebra R, we can associate to it a vertex algebra V (R) (cf. [Kac96, BK03]) called the universal enveloping vertex algebra of R, as defined in the introduction. If R is generated by some vectors {ai } as a C[∂]-module, we say that V (R) is generated by the same vectors. If C ∈ R is a central element such that ∂C = 0, given any complex number c, we denote by V c (R) the quotient of V (R) by the ideal (C − c)V (R). One can show [Kac96 ] that a vertex algebra V is canonically a Lie conformal algebra with the λ-bracket defined in (2.2.1) and ∂ = T . Example 2.3. The Virasoro vertex algebra Vir c is generated by an even field L satisfying: [L λ L] = (∂ + 2λ)L +
c 3 λ . 12
(2.4.1)
The complex number c is called the central charge. Expanding the corresponding field as in (1.2.1) we obtain the familiar commutation relations of the Virasoro algebra (1.2.2). 2.5. Let ν ∈ V be a conformal vector (see 1.2) and let L(z) be the corresponding Virasoro field. A vector a ∈ V satisfying [L λ a] = (T + λ)a + O(λ2 ) is said to have conformal weight . If, moreover, a satisfies [L λ a] = (T + λ)a we say that a is primary. Example 2.4. The Neveu Schwarz (NS) vertex algebra is generated by an even Virasoro field L (satisfying (2.4.1)) and an odd primary field G of conformal weight 3/2, satisfying the commutation relation: [G λ G] = 2L +
λ2 c. 3
If we expand the corresponding fields as in (1.2.1) and (1.2.4) we obtain the commutation relations (1.2.5) and (1.2.2) of the Neveu-Schwarz algebra. As we have seen in the introduction, given a vertex algebra with an N = 1 superconformal vector τ , we obtain an N K = 1 SUSY vertex algebra (see also 4.13 for a definition) by defining the superfields (1.2.6). In particular, the Neveu-Schwarz algebra gives rise to such an N K = 1 SUSY vertex algebra. Below we give some examples of vertex algebras with an N = 1 superconformal vector. By the construction in 1, they are automatically N K = 1 SUSY vertex algebras. Example 2.5 [Kac96, Ex. 5.9a]. Let V be the universal enveloping vertex algebra of the Lie conformal algebra generated by an even vector (free boson) α and an odd vector (free fermion) ϕ, namely [αλ α] = λ,
[ϕλ ϕ] = 1,
[αλ ϕ] = 0.
Then V is a (simple) vertex algebra with a family of N = 1 superconformal vectors τ = (α(−1) ϕ(−1) + mϕ(−2) )|0, of central charge c =
3 2
− 3m 2 .
m ∈ C,
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Example 2.6 [KT85, Kac96, Thm 5.9]. Let g be a finite dimensional Lie algebra with a non-degenerate invariant symmetric bilinear form ( , ), normalized by the condition (α, α) = 2 for a long root α, and let h ∨ be the dual Coxeter number. We construct a vertex algebra V k (gsuper ) generated by the usual currents a, b ∈ g, satisfying: [aλ b] = [a, b] + (k + h ∨ )λ(a, b), and the odd super currents a¯ ∈ g (as before stands for reversal of parity), satisfying: ¯ = [a, b], [aλ b]
¯ = (k + h ∨ )(a, b). [a¯ λ b]
Let a i and bi be dual bases of g. Provided that k = −h ∨ the vertex algebra V k (gsuper ) admits an N = 1 superconformal vector ⎛ ⎞ 1 ⎝ i ¯ i 1 j i r ⎠ |0, τ= a(−1) b(−1) + ([a i , a j ], a r )b¯(−1) b¯(−1) b¯(−1) k + h∨ 3(k + h ∨ ) i
i, j,r
of central charge kdimg 1 + dimg. k + h∨ 2 This is known as the Kac-Todorov construction. The formulas in [Kac96] should be corrected as above. ck =
Example 2.7 [Kac96, Thm 5.10]. The N = 2 vertex algebra is generated by a Virasoro field L of central charge c, an even field J , primary of conformal weight 1, and two odd fields G ± , primary of conformal weight 3/2. The remaining commutation relations are: c ± ± ± λ, [G ± λ G ] = 0, [Jλ G ] = ±G , 3 c 1 [G +λ G − ] = L + ∂ J + λJ + λ2 . 2 6
[Jλ J ] =
This vertex algebra contains an N = 1 superconformal vector: τ = G +(−1) |0 + G − (−1) |0. Also, this vertex algebra admits a Z/2Z × C∗ family of automorphisms. The generator of Z/2Z is given by L → L, J → −J and G ± → G ∓ . The C∗ family is given by G + → μG + and G − → μ−1 G − . Applying these automorphisms, we get a family of N = 1 superconformal vectors. By expanding the corresponding fields −3/2−n L(z) = L n z −2−n , G ± (z) = G± , J (z) = Jn z −1−n , nz n∈Z
n∈1/2+Z
n∈Z
we get the commutation relations of the Virasoro operators L n , and the following remaining commutation relations: m n ± ± [Jm , Jn ] = δm,−n c, G [Jm , G ± [G ± , n ] = ±G m+n , m, Ln] = m − 3 2 m+n m−n c 1 Jm+n + m2 − δm,−n . [G +m , G − [L m , Jn ] = −n Jm+n , n ] = L m+n + 2 6 4
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Sometimes it is convenient to introduce a different set of generating fields for this vertex algebra. We define L˜ = L − 1/2∂ J . This is a Virasoro field with central charge ˜ = (∂ +2λ) L. ˜ With respect to this Virasoro element, G + is primary of zero, namely [ L˜ λ L] − conformal weight 2 and G is primary of conformal weight 1; J has conformal weight 1 but is no longer a primary field. To summarize the commutation relations, we write Q(z) = G + (z) = Q n z −2−n , H (z) = G − (z) = Hn z −1−n , n∈Z
˜ L(z) =
n∈Z
Tn z
−2−n
.
(2.5.1a)
n∈Z
The corresponding λ-brackets of these fields are given by: λ2 c, 6
˜ = (∂ + 2λ) L, ˜ [ L˜ λ L]
[ L˜ λ J ] = (∂ + λ)J −
[ L˜ λ H ] = (∂ + λ)H,
c [Hλ Q] = L˜ − λJ + λ2 . 6
[ L˜ λ Q] = (∂ + 2λ)Q,
(2.5.1b)
The commutation relations between the Fourier coefficients are: [Tm , Tn ] = (m − n)Tm+n , [Tm , Hn ] = −n Hm+n , [Tm , Q n ] = (m − n)Q m+n ,
[Q m , Q n ] = [Hm , Hn ] = 0, c δm,−n , 12 c + m(m − 1) δm,−n . 6 (2.5.1c)
[Tm , Jn ] = −n Jm+n − m(m + 1) [Hm , Q n ] = Tm+n − m Jm+n
2.6. In the subsequent sections, we will study in detail the structure theory of SUSY vertex algebras. Here we introduce some basic notation, used in the examples further on. Let V be a vector superspace over C. Let z be an even indeterminate and θ 1 , . . . , θ N be odd anticommuting indeterminates which commute with z. For an ordered subset I = (i 1 , . . . , i k ) ⊂ {1, . . . , N }, we will write θ I = θ i1 . . . θ ik and let N \ I be the ordered complement of I in {1, . . . , N }. An End(V )-valued superfield is an expression of the form: A(z, θ 1 , . . . , θ N ) = θ N \I A(n|I ) z −1−n , (2.6.1) (n|I ):n∈Z
where I runs over all ordered subsets of the set {1, . . . , N }, A(n|I ) ∈ End(V ), and for each I and v ∈ V we have A(n|I ) v = 0 for n large enough. We will usually write A(z, θ ) or simply A(Z ) for this field, where Z = (z, θ 1 , . . . , θ N ). Remark 2.8. Define an N = 2 superconformal vertex algebra as a vertex algebra with a vector τ and two operators S 1 , S 2 satisfying [T , S i ] = 0,
[S i , S j ] = 2δi, j T,
such that the corresponding fields J (z) = −iY (τ, z), L(z) = 21 Y (S 2 S 1 τ, z) and G (1) (z) ≡ G + (z) + G − (z) = −Y (S 2 τ, z),
G (2) (z) ≡ i G + (z) − G − (z) = Y (S 1 τ, z),
(2.6.2)
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115 (i)
satisfy the λ-brackets of Example 2.7, L −1 = T , G −1/2 = S i , and L 0 is diagonalizable with eigenvalues bounded below. Then we obtain an N K = 2 SUSY vertex algebra (see 4.13 for a definition) by letting Y (a, Z ) = Y (a, z) + θ 1 Y (S 1 a, z) + θ 2 Y (S 2 a, z) + θ 2 θ 1 Y (S 1 S 2 a, z). Similarly, given a vertex algebra with two vectors ν, τ and an odd operator S such that [T, S] = 0, S 2 = 0 and the associated fields: J (z) = −Y (τ, z), Q(z) = Y (Sτ, z),
H (z) = Y (ν, z), ˜ L(z) = Y (Sν, z) − ∂z J (z),
satisfy the commutation relations (2.5.1b), T−1 = T , Q −1 = S, T0 is diagonalizable with eigenvalues bounded below, and J0 is diagonalizable, we obtain an N W = 1 SUSY vertex algebra (see 3.3.1 for a definition) by letting: Y (a, Z ) = Y (a, z) + θ Y (Sa, z). Example 2.9. Following the previous remark, we can give the N = 2 vertex algebra, as defined in Example 2.7, the structure of an N K = 2 SUSY vertex algebra by letting − + 2 S 1 = (G +(0) + G − (0) ) and S = i(G (0) − G (0) ). Also we check directly that letting √
−1J(−1) |0,
(2.6.3)
√ −1J (z) + θ 1 G (2) (z) − θ 2 G (1) (z) + 2θ 1 θ 2 L(z),
(2.6.4)
τ= we get: Y (τ, z, θ i ) =
where G (1) (z) = G + (z) + G − (z) and G (2) (z) = i(G + (z) − G − (z)). It follows that [S i , S j ] = 2δi j T , τ(0|0) = 2T , τ(0|1) = −S 1 and τ(0|2) = −S 2 (cf. 5.6 below). We note that G (i) are primary of conformal weight 3/2, and J is primary of conformal weight 1. The other commutation relations between the generating fields L , J, G (i) (i = 1, 2) are cλ2 , [G (1) λ G (2) ] = −i (∂ + 2λ) J, 3 [Jλ G (1) ] = −i G (2) , [Jλ G (2) ] = i G (1) ,
[G (i) λ G (i) ] = 2L +
or, equivalently, (i) [G (i) m , Gn ]
= 2L m+n (2)
1 c 2 (2) δm,−n , [G (1) + m − m , G n ] = i (n − m) Jm+n , 4 3
[Jm , G (1) n ] = −i G m+n ,
(1)
[Jm , G (2) n ] = i G m+n .
Similarly, we can view the N = 2 vertex algebra of Example 2.7 as an N W = 1 ˜ Q, H, and J with SUSY vertex algebra as follows. We will use the generating fields L, the commutation relations (2.5.1c). Define the superfields: Y (a, z, θ ) = Y (a, z) + θ Y (Q −1 a, z), and let T = T−1 , S = S 1 = Q −1 , so that T and S commute and S 2 = 0.
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Note that defining the vectors ν = H(−1) |0 and τ = −J(−1) |0 we have in particular ˜ Y (ν, z, θ ) = H (z) + θ ( L(z) + ∂z J (z)), Y (τ, z, θ ) = −J (z) + θ Q(z). Therefore, if we consider the Fourier modes as defined in (2.6.1), we have ν(0,0) = T, τ(0,0) = S. ˜ Moreover, it is easy to see that the field L(z) + ∂z J (z) is also a Virasoro field and the ˜ H, Q, J are positive with respect to this conformal weights of the generating fields L, Virasoro field as well. It follows that the operator ν(1,0) acts diagonally with non-negative eigenvalues (cf. Definition 5.2 below). Example 2.10 [Kac96, Ex. 5.9d]. Consider the vertex algebra generated by a pair of free charged bosons α ± and a pair of free charged fermions ϕ ± , where the only non-trivial commutation relations are: [α ± λ α ∓ ] = λ,
[ϕ ± λ ϕ ∓ ] = 1.
This vertex algebra contains the following family of N = 2 vertex subalgebras (m ∈ C): G ± =: α ± ϕ ± : ±m∂ϕ ± , J =: ϕ + ϕ − : −m(α + + α − ), 1 1 m L =: α + α − : + : ∂ϕ + ϕ − : + : ∂ϕ − ϕ + : − ∂(α + − α − ). 2 2 2 The vector τ given by (2.6.3) provides this vertex algebra with the structure of an N K = 2 SUSY vertex algebra, by letting T = L −1 and S i = G (i) −1/2 (see (2.6.2)). As in Example 2.9, we can view this vertex algebra as an N W = 1 SUSY vertex algebra. Example 2.11. An example of an N W = N SUSY vertex algebra for each N can be constructed as follows. Denote by A the set of all ordered monomials θ i1 . . . θ is and consider the superalgebra C[t, t −1 , θ 1 , . . . , θ N ], where t is even and θ i are odd indeterminates. Let W (1|N ) be the Lie superalgebra of derivations of this superalgebra, and define the following collection of W (1|N )-valued formal distributions: j n −1−n F = a (z) = (t a∂ j )z a ∈ A, j = 0, 1, . . . , N , n∈Z
where ∂ j = ∂θ j if j > 0 and ∂0 = ∂t . The pair (W (1|N ), F ) is a formal distribution Lie superalgebra. The corresponding Lie conformal superalgebra is the free C[∂]-module W N generated by the vectors a j , with a ∈ A and j = 0, . . . , N , and the following λ-brackets (cf. [Kac96, FK02]): [a i λ b j ] = (a∂i b) j + (−1) p(a) ((∂ j a)b)i , i, j ≥ 1, [a i λ b0 ] = (a∂i b)0 − (−1) p(b) (ab)i λ, [a 0 λ b0 ] = −∂(ab)0 − 2(ab)0 λ. Let V (W N ) be the associated universal enveloping vertex algebra. The field L(z) = −10 (z) +
n i=1
∂z (θ i )i (z),
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is a Virasoro field, and the elements (θ i ) j are primary of conformal weight 1, while the elements −1i are primary of conformal weight 2. We will need later its Fourier modes, which are given by: L n = −t n+1 ∂t − (n + 1) t n θ i ∂θ i . We define T = L −1 = −∂t . In order to be consistent with previous notation we define the fields Q i (z) = −1i (z) and write down their Fourier modes which are Q in = −t n+1 ∂θ i . In particular, we define S i = Q i−1 for i ≥ 1 and note that (S i )2 = 0 and [T, S i ] = 0. In order to construct an N W = N SUSY vertex algebra from the vertex algebra V (W N ) we proceed as before, defining the superfields I (I −1) (−1) 2 θ I Y (S i1 . . . S is a, z), (2.6.5) Y (a, z, θ 1 , . . . , θ N ) = I
where the summation is taken over all ordered subsets I = (i 1 , . . . , i s ) of {1, . . . , N }. (In this and the next examples the same θ is used both in the definition of Lie superalgebras and of the superfields, but we hope that this will cause no confusion.) It is straightforward to check that the data (V (W N ), T, S i , |0, Y ) is indeed an N W = N SUSY vertex algebra. Moreover, this is a W (1|N )-conformal N W = N SUSY vertex algebra, as defined in 1. We shall return to this example in 5.1. Example 2.12. We can similarly construct an N K = N SUSY vertex algebra V (K N ) for any N . For this we define a subalgebra ) of W (1|N ), of those differential Ki(1|N operators preserving the form ω = dt + θ dθ i up to multiplication by a function (recall that we consider d to be an even derivation, as in [DM99], but not in [KvdL89] and [FK02]). It consists of differential operators of the form: 1 D = f ∂0 + (−1) p( f ) (θ i ∂0 + ∂i )( f )(θi ∂0 + ∂i ), 2 N
f
(2.6.6)
i=1
for f ∈ C[t, t −1 , θ 1 , . . . , θ N ]. These operators satisfy [D f , D g ] = D { f,g} , where
{ f, g} =
1 i f − θ ∂i f 2 N
∂ 0 g − ∂0 f
i=1
N N 1 1 i g−1 θ ∂i g + (−1) f ∂i f ∂i g. 2 2 i=1
i=1
In particular K (1|N ) contains the operators n+1 n i t θ ∂θ i , n ∈ Z, 2 1 n−1/2 i j 1 n+1/2 i θ ∂θ j , n ∈ + Z. t = −t (∂θ i − θ ∂t ) + n + θ 2 2
L n = −t n+1 ∂t − G (i) n
It is easy to see that the operators L n span a centerless Virasoro Lie algebra.
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As in the W (1|N ) case, we construct the corresponding Lie conformal superalgebra as follows. It is the free C[∂]-module K N generated by vectors a ∈ A, with the following λ-brackets [FK02]:
n r r + s 1 − 1 ∂(ab) + (−1)r − 2 ab, [aλ b] = ∂i a∂i b + λ 2 2 2 i=1
where a = θ ii . . . θ ir , b = θ j1 . . . θ js . We denote by V (K N ) its universal enveloping vertex algebra, and we define the (i) operators T = L −1 and S i = G −1/2 . Now we define the state-field correspondence as in (2.6.5): I (I −1) Y (a, z, θ ) = (−1) 2 θ I Y (S I a, z). I
All the properties of an N K = N SUSY vertex algebra are straightforward to check as in the previous cases. Moreover, this is a K (1|N )-conformal N K = N SUSY vertex algebra. We will return to this example in 5.5. 3. Structure Theory of NW = N SUSY Vertex Algebras In this section we develop the structure theory of SUSY Lie conformal algebras and SUSY vertex algebras along the lines of [Kac96] (see also [DSK05] for a better exposition). Proofs are rather straightforward adaptations of those in the vertex algebra case, the only difficulty being the problem of signs. 3.1. Formal distribution calculus 3.1.1. In what follows we fix the ground field to be the complex numbers C and N to be a non-negative integer. Let θ 1 , . . . , θ N be Grassmann variables and I = {i 1 , . . . , i k } be an ordered k-tuple: 1 ≤ i 1 < · · · < i k ≤ N . We will denote θ I = θ i1 . . . θ ik ,
θ N = θ1 . . . θ N .
For an element a in a vector superspace we will denote (−1)a = (−1) p(a) , where p(a) ∈ Z/2Z is the parity of a, and, given a k-tuple I as above, we will let (−1) I = (−1)k . Given two disjoint ordered tuples I and J , we define σ (I, J ) = ±1 by θ I θ J = σ (I, J )θ I ∪J , and we define σ (I, J ) to be zero if I ∩ J = ∅. Also, unless noted otherwise, all “union” symbols “∪” will denote disjoint unions3 . It follows easily, by looking at θ I θ J θ K , that for three mutually disjoint tuples, I, J and K we have: σ (I, J )σ (I ∪ J, K ) = σ (I, J ∪ K )σ (J, K ), σ (I, J ) = (−1) I J σ (J, I ).
(3.1.1.1)
Here and further (−1) I J stands for (−1)(I )(J ) . We will denote by N \ I the ordered complement of I in {1, . . . , N } and define σ (I ) := σ (I, N \ I ). It follows from the definitions that θ I θ N \I = σ (I )θ N . 3 This will not be true in Section 4 where we analyze N = n SUSY vertex algebras. K
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3.1.2. Let Z = (z, θ 1 , . . . , θ N ) and W = (w, ζ 1 , . . . , ζ N ) denote two sets of coordinates in the formal superdisk D = D 1|N . As before, all θ i and ζ j anticommute. Let C[[z]] be the algebra of formal power series in z; its elements are series n≥0 an z n with an ∈ C. The superalgebra of regular functions in D is defined as C[[Z ]] := C[[z]]⊗C[θ 1 , . . . , θ N ]. Similarly, we define the superalgebra C[[Z , W ]] := C[[z, w]]⊗ C[θ 1 , . . . , θ N , ζ 1 , . . . , ζ N ]. For any C-algebra R, we denote by R((z)) the algebra of R-valued formal Laurent series, its elements are series of the form n∈Z an z n such that an ∈ R and there exists N0 ∈ Z such that an = 0 for all n ≤ N0 . If R is a field, so is R((z)). We denote R((Z )) := R((z)) ⊗C C[θ 1 , . . . , θ N ]. Denote also by C((Z ))((W )) the superalgebra R((W )), where R = C((Z )); its elements are Laurent series in W whose coefficients are Laurent series in Z . Similarly we have the superalgebra C((W ))((Z )). Denote by C((z, w)) the field of fractions of C[[z, w]] and put C((Z , W )) := C((z, w)) ⊗C C[θ 1 , . . . , θ N , ζ 1 , . . . , ζ N ]. One may think of this superalgebra as the algebra of meromorphic functions in the formal superdisk D 2|2N . Given such a meromorphic function, we can “expand it near the w axis”, to obtain an element of C((Z ))((W )). Indeed, C[[z, w]] embeds naturally in C((z))((w)) and C((w))((z)) respectively. Since C((z, w)) is the ring of fractions of C[[z, w]] and C((z))((w)) and C((w))((z)) are fields, these embeddings induce respective algebra embeddings i z,w
i w,z
C((z))((w)) ← C((z, w)) → C((w))((z)). (A concrete example is given by (2.3.1).) Tensoring with the corresponding Grassmann superalgebras, we obtain superalgebra embeddings i z,w
i w,z
C((Z ))((W )) ← C((Z , W )) → C((W ))((Z )). Let U be a vector superspace. An U -valued formal distribution is an expression of the form Z n|I an|I , an|I ∈ U . a(Z ) = (n|I ):n∈Z
The space of such distributions will be denoted U [[Z , Z −1 ]]. We denote by C[Z , Z −1 ] := C[z, z −1 ] ⊗ C[θ 1 , . . . , θ N ] the superalgebra of Laurent polynomials. A U -valued formal distribution is canonically a linear functional C[Z , Z −1 ] → U . To see this, we define the super residue as the coefficient of Z −1|N : res Z a(Z ) = a−1|N . This clearly satisfies res Z ∂z a(Z ) = res Z ∂θ a(Z ) = 0.
(3.1.2.1)
Given a U -valued formal distribution a(Z ) we obtain a linear map C[Z , Z −1 ] → U given by f (Z ) → res Z a(Z ) f (Z ). Conversely, every formal distribution arises in this way. Indeed we have: res Z Z n|I a(Z ) = σ (I )a−1−n|N \I .
(3.1.2.2)
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Therefore the formal distribution a(Z ) can be written as a(Z ) = Z −1−n|N \I a(n|I ) , (n|I ):n∈Z
where
a(n|I ) = σ (I ) res Z Z n|I a(Z ).
We can similarly define U -valued formal distributions in two variables, as expressions of the form a(Z , W ) = Z j|J W k|K a j|J,k|K , a j|J,k|K ∈ U . ( j|J ),(k|K )
The space of such formal distributions will be denoted U [[Z , Z −1 , W, W −1 ]]. Note that in the case U = C, both algebras C((Z ))((W )) and C((W ))((Z )) are embedded in C[[Z , Z −1 , W, W −1 ]]. We will denote by i z,w and i w,z the corresponding embeddings of C((Z , W )) in C[[Z , Z −1 , W, W −1 ]]. When f (Z , W ) is a Laurent polynomial (that is a polynomial in z, z −1 , w, w−1 and the odd variables) then the embeddings i z,w f and i w,z f coincide on C[[Z , Z −1 , W, W −1 ]]. Indeed, it is immediate to see that C((Z ))((W )) ∩ C((W ))((Z )) = C[[Z , W ]][z −1 , w −1 ],
(3.1.2.3)
where the intersection is taken in C[[Z , Z −1 , W, W −1 ]]. The images under these embeddings are different for other functions, as we will see below (cf. 3.1.2). A U -valued formal distribution in two variables is called local if there exists a non-negative integer n such that (z − w)n a(Z , W ) = 0. 3.1.3. Note that the differential operators ∂z , ∂θ i and ∂w , ∂ζ i act in the usual way on the spaces C((Z , W )), C[[Z , Z −1 , W, W −1 ]]. For j ∈ Z+ and J = ( j1 , . . . , jk ) we will denote j|J
∂Z We define
j
= ∂z ∂θ j1 . . . ∂θ jk .
J (J +1)
J (J +1)
(−1) 2 (−1) 2 j|J ∂Z , Z j|J . Z ( j|J ) := j! j! One checks easily that the embeddings i z,w and i w,z defined above commute with the j|J j|J action of the differential operators ∂ Z and ∂W . We will denote ( j|J )
∂Z
:=
Z − W = (z − w, θ 1 − ζ 1 , . . . , θ N − ζ N ), Z n|I = z n θ I , (θ i − ζ i ), ∂W = (∂w , ∂ζ 1 , . . . , ∂ζ N ), (Z − W ) j|J = (z − w) j
(3.1.3.1)
i∈J
and for any formal power series f (Z ) ∈ C[[Z ]] we have its Taylor expansion: f (Z ) = e(Z −W )∂W f (W ),
(3.1.3.2)
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where (Z − W )∂W = (z − w)∂w +
(θ i − ζ i )∂ζ i .
i
Expanding the exponential in (3.1.3.2) we obtain: ( j|J ) f (Z ) = (−1) J (Z − W ) j|J ∂W f (W ). ( j|J ): j≥0
Remark 3.1.1. In the definition of formal distributions and super residues, we can replace C by any commutative superalgebra A , and U by any A -module. We see immediately that the residue map is of parity N mod 2, that is, for χ ∈ A , and u(Z ) an U -valued distribution, we have: res Z χ u(Z ) = (−1)χ N res Z u(Z ). On the other hand, this residue map is a morphism of right A -modules, namely: res Z u(Z )χ = res Z u(Z ) χ .
Proposition 3.1.2. There exists a unique C-valued formal distribution δ(Z , W ) such that for every function f ∈ U [Z , Z −1 ] we have res Z δ(Z , W ) f (Z ) = f (W ). Proof. For uniqueness, let δ and δ be two such distributions, then β = δ − δ satisfies res Z β(Z , W ) f (Z ) = 0 for all functions f (Z ). Decomposing β(Z , W ) = βn|I,m|J W m|J Z n|I , and multiplying by Z k|L we see that β−1−k|N −L ,m|J = 0 for all m|J , hence β = 0. Existence will be proved below. 3.1.4. We define the formal δ-function as the C-valued formal distribution in two variables, given by δ(Z , W ) = (i z,w − i w,z )(Z − W )−1|N = (i z,w − i w,z )
(θ − ζ ) N . z−w
(3.1.4.1)
It follows that ∂w(n) δ(Z , W ) :=
1 n ∂ δ(Z , W ) = (i z,w − i w,z )(Z − W )−1−n|N . n! w
This distribution has the following properties: (1) (2) (3) (4) (5) (6)
n|I
(Z − W )m|J ∂W δ(Z , W ) = 0 whenever m > n or J I , (n− j|I \J ) (n|I ) (Z − W ) j|J ∂W δ(Z , W ) = σ (I \ J, J )∂W δ(Z , W ) if n ≥ j and I ⊃ J , N δ(Z , W ) = (−1) δ(W, Z ), j|J j|J ∂ Z δ(Z , W ) = (−1) j+N +J ∂W δ(W, Z ), δ(Z , W )a(Z ) = δ(Z , W )a(W ), where a(Z ) is any formal distribution, res Z δ(Z , W )a(Z ) = a(W ), where a(Z ) is any formal distribution,
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R. Heluani, V. G. Kac n|I
(7) exp ((Z − W ) ) ∂W δ(Z , W ) = ( +∂W )n|I δ(Z , W ), where = (λ, χ 1 , . . . , χ N ), χ i are odd anticommuting variables, λ is even, λ commutes with χ i , and we write (Z − W ) = (z − w)λ + (θ i − ζ i )χ i , ( + ∂W ) = (λ + ∂w , χi + ∂θ i ). i
Proof. Writing ∂ζI = ∂ζ i1 . . . ∂ζ ik we have (Z − W )m|J ∂W δ(Z , W ) = (z − w)m n!(i z,w − i w,z )(z − w)−1−n (θ − ζ ) J ∂ζI (θ − ζ ) N . n|I
Now this clearly vanishes if m ≥ 1 + n since then the two embeddings i z,w and i w,z coincide on the regular function (z − w)m−n−1 . The other factor is clearly zero if J I since for every j ∈ J \ I we have a factor (θ j − ζ j ) in ∂ζI (θ − ζ ) N . This proves (1). In order to prove (2) we write: n|I
(Z − W ) j|J ∂W δ(Z , W ) = n!(i z,w − i w,z )(z − w) j−1−n (θ − ζ ) J ∂ζI (θ − ζ ) N n! I \J (n− j)!(i z,w −i w,z )(z−w)−1−(n− j) (−1) J σ (J, I \J )(θ −ζ ) J ∂θJ ∂ζ (θ −ζ ) N (n − j)! J (J +1) n! I \J (−1) 2 σ (J, I \ J )(n − j)!(i z,w − i w,z )(z − w)−1−(n− j) ∂ζ (θ − ζ ) N = (n − j)! J (J +1) n! n− j|I \J ∂ = (−1) 2 σ (J, I \ J ) δ(Z , W ). (3.1.4.2) (n − j)! W
=
(3) is obvious and (4) follows from (3) easily. In order to prove (5) we see that from (1) we have δz = δw, therefore we get δ(Z , W )z k = δ(Z , W )w k . On the other hand, also from (1) it follows that δ(Z , W )θ i = δ(Z , W )ζ i . Hence δ(Z , W )θ I = δ(Z , W )ζ I and we have proved that δ(Z , W )Z n|I = δ(Z , W )W n|I . The result follows by linearity now. (6) follows by taking residue in (5). To prove (7) we first expand the exponential in power series: n|I n|I exp ((Z − W ) ) ∂W δ(Z , W ) = (−1) J (Z − W )( j|J ) j|J ∂W δ(Z , W ). ( j|J ): j≥0
(3.1.4.3) Now using (2) we see that this is: n n− j|I \J j|J σ (J, I \ J )∂W δ(Z , W ). j
(3.1.4.4)
( j|J ): j≥0
On the other hand we can expand the right-hand side of (7) as: n I \J λ j ∂wn− j σ (J, I \ J )χ J ∂ζ ( + ∂W )n|I = (λ + ∂w )n (χ + ∂ζ ) I = j ( j|J ): j≥0 n n− j|I \J j|J σ (J, I \ J )∂W = . j ( j|J ): j≥0
Comparing with (3.1.4.4) we get the result.
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Lemma 3.1.3. Let a(Z , W ) be a local formal distribution in two variables. Then a(Z , W ) can be uniquely decomposed as a(Z , W ) =
( j|J ) ∂W δ(Z , W ) c j|J (W ),
(3.1.4.5)
( j|J ): j≥0
where the sum is finite. The coefficients c j|J are given by c j|J (W ) = res Z (Z − W ) j|J a(Z , W ).
(3.1.4.6)
Proof. First we note that if a(Z , W ) is local then the sum on the right-hand side is finite. Let b(Z , W ) be the difference between the right hand side and the left-hand side of (3.1.4.5). We find: res Z (Z − W )k|K b(Z , W ) = res Z (Z − W )k|K a(Z , W ) ( j|J ) (Z − W )k|K ∂W δ(Z , W ) c j|J (W ) − res Z ( j|J ): j≥0
( j−k|J \K ) δ(Z , W ) c( j|J ) (W ) = ck|K (W ) − res Z ∂W = ck|K (W ) − res Z δ(Z , W )ck|K (W ) = 0, where in the second line we have used (2) of 3.1.4. It follows that b(Z , W ) has no negative powers of z. Moreover, b(Z , W ) is local, since a(Z , W ) is, and the right-hand side of (3.1.4.5) is local by (1) of 3.1.4. We can write then b(Z , W ) =
Z j|J b j|J (W ),
( j|J ): j≥0
and since (z − w)n b(Z , W ) = 0 we obtain: n Z j|J w n−k b j−k|J (W ) = 0, k ( j|J )
j≥k≥0
which easily shows that b(Z , W ) = 0. Uniqueness is clear by taking residues on both sides of (3.1.4.5). 3.1.5. Let a(Z , W ) be a formal distribution in two variables. We define its formal Fourier transform by: F Z ,W a(Z , W ) = res Z exp ((Z − W ) ) a(Z , W ),
(3.1.5.1)
where = (λ, χ 1 , . . . , χ N ), λ is an even variable, and χ i are odd anticommuting variables, commuting with λ.
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Expanding this exponential we have (recall (3.1.4.3)): j|J (Z − W )( j|J ) a(Z , W ) F Z ,W a(Z , W ) = res Z =
( j|J ): j≥0
(−1)
JN
j|J
res Z (Z − W )( j|J ) a(Z , W ) =
( j|J ): j≥0
(−1) J N ( j|J ) c j|J (W ),
( j|J ): j≥0
(3.1.5.2) where c j|J are defined by (3.1.4.6) and we write, as before
j|J
( j|J )
= λ χ ...χ , j
j1
jk
J (J +1) 2
(−1) := j!
j|J .
Proposition 3.1.4. The formal Fourier transform satisfies the following properties: (1) sesquilinearity: F Z ,W ∂z a(Z , W ) = −λF Z ,W a(Z , W ) = [∂w , F Z ,W ]a(Z , W ), F Z ,W ∂θ i a(Z , W ) = −(−1) N χ i F Z ,W a(Z , W ) = (−1) N [∂ζ i , F Z ,W ]a(Z , W ). (2) For any local formal distribution a(Z , W ) we have: W a(Z , W ), (−1) N F Z ,W a(W, Z ) = F Z− −∂ ,W
= F Z ,W a(Z , W )| =− −∂W ,
(3.1.5.3)
where − − ∂W = (−λ − ∂w , −χ i − ∂ζ i ). (3) For any formal distribution in three variables a(Z , X, W ) we have + F Z ,W F X,W a(Z , X, W ) = (−1) N F X,W F Z ,X a(Z , X, W ),
where = (γ , η1 , . . . , η N ), with ηi odd anticommutative variables and γ is even and commutes with ηi , + is the sum (λ + γ , χ i + ηi ), and the superalgebra C[ , ] is commutative. Proof. The proof of the first equality of both lines of (1) follows from (3.1.2.1). For the first equality of the second line we have F Z ,W ∂θ i a(Z , W ) = res Z exp ((Z − W ) ) ∂θ i a(Z , W )
= − res Z ∂θ i exp ((Z − W ) ) a(Z , W ) = − res Z χ i exp ((Z − W ) ) a(Z , W ) = −(−1) N χ i F Z ,W a(Z , W ). For the second equality of the second line of (1) we have:
[∂ζ i , F Z ,W ]a(Z , W ) = (−1) N res Z ∂ζ i exp ((Z − W ) ) a(Z , W )
− exp ((Z − W ) ) ∂ζ i a(Z , W ) == (−1) N res Z ∂ζ i exp ((Z − W ) ) a(Z , W ) = −χ i F Z ,W a(Z , W ).
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To prove (2) it is enough, by Lemma 3.1.3, to prove the statement when a(Z , W ) = j|J ∂W δ(Z , W ) c(W ). In this case we have: j|J j|J F Z ,W a(W, Z ) = F Z ,W ∂ Z δ(W, Z ) c(Z ) = F Z ,W (−1) j+J +N ∂W δ(Z , W )c(Z ). Now using (7) in 3.1.4 we can express the last expression above as: (−1) j+J +N res Z ( + ∂W ) j|J δ(Z , W )c(Z ) = (−1) j+J +J N +N ( + ∂W ) j|J res Z δ(Z , W )c(Z ) = (−1) j+J +J N +N ( + ∂W ) j|J c(W ). On the other hand we have F Z ,W a(Z , W )| =− −∂W = (−1) J N (− − ∂W ) j|J c(W ) = (−1) j+J +J N ( + ∂W ) j|J c(W ). The proof of (3) is straightforward: F Z ,W F X,W = res Z exp ((Z − W ) ) res X exp ((X − W ) ) = res Z res X exp ((Z − W ) + (X − W ) )
= (−1) N res X res Z exp ((Z − X ) + (X − W )( + )) = (−1) N res X exp ((X − W )( + )) res Z exp ((Z − X ) ) + = (−1) N F X,W F Z ,X .
(3.1.5.4)
The sign (−1) N appears when we commute the residue maps (recall that they have parity N mod 2). 3.2. N W = N SUSY Lie conformal algebras 3.2.1. Let g be a Lie superalgebra. A pair of g-valued formal distributions a(Z ), b(Z ) is called local if the distribution [a(Z ), b(W )] is local. By the decomposition Lemma 3.1.3 we have for such a pair: ( j|J ) ∂W δ(Z , W ) c j|J (W ), [a(Z ), b(W )] = ( j|J ): j≥0
where
c j|J (W ) = res Z (Z − W ) j|J [a(Z ), b(W )].
We define a(W )( j|J ) b(W ) = c j|J (W ) and we call this operation the ( j|J )-product. Let us also define the -bracket of two g-valued formal distributions by [a b](W ) = F Z ,W [a(Z ), b(W )],
(3.2.1.1)
where F Z ,W is the formal Fourier transform defined in 3.1.5. It follows from the definitions and from (3.1.5.2) that [a b] = (−1) J N ( j|J ) a( j|J ) b. ( j|J ): j≥0
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Note also that the -bracket has parity N mod 2 (this follows from the fact that the residue map has parity N mod 2). A pair (g, R) consisting of a Lie superalgebra g and a family R of pairwise local g-valued formal distributions a(Z ), whose coefficients span g, stable under all j|J th products and under the derivations ∂z and ∂θ i is called an N W = N formal distribution Lie superalgebra. Proposition 3.2.1. The -bracket defined in (3.2.1.1) satisfies the following properties: (1) Sesquilinearity for a pair (a(Z ), b(W )): [∂z a b] = −λ[a b],
[a ∂w b] = (∂w + λ)[a b],
(3.2.1.2)
[∂θ i a b] = −(−1) N χ i [a b], [a ∂ζ i b] = (−1)a+N (∂ζ i + χ i )[a b]. (3.2.1.3) (2) Skew-symmetry for a local pair (a(Z ), b(W )): [b a] = −(−1)ab+N [a− −∂W b]. (3) Jacobi identity for a triple (a(Z ), b(X ), c(W )) : [a [b c]] = (−1)a N +N [[a b] + c] + (−1)(a+N )(b+N ) [b [a c]], where = (γ , η1 , . . . , η N ) and the superalgebra C[ , ] is commutative. Proof. In order to prove the first equation in (3.2.1.3) we use Proposition 3.1.4 (1): [∂θ i a b] = F Z ,W [∂θ i a(Z ), b(W )] = F Z ,W ∂θ i [a(Z ), b(W )] = −(−1) N χ i F Z ,W [a(Z ), b(W )] = −(−1) N χ i [a b]. For the second equation we have by Proposition 3.1.4 (1): [a ∂ζ i b] = F Z ,W [a(Z ), ∂ζ i b(W )] = F Z ,W (−1)a ∂ζ i [a(Z ), b(W )] = (−1)a [F Z ,W , ∂ζ i ] + (−1) N ∂ζ i F Z ,W [a(Z ), b(W )] = (−1)a+N (χ i + ∂ζ i )F Z ,W [a(Z ), b(W )] = (−1)a+N χ i + ∂ζ i [a b]. Skew-symmetry follows from the skew-symmetry property of the Fourier transform (2) as follows: [b a] = F Z ,W [b(Z ), a(W )] = −(−1)ab F Z ,W [a(W ), b(Z )] W = −(−1)ab+N F Z− −∂ [a(Z ), b(W )] ,W
= −(−1)ab+N [a− −∂W b].
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Finally, to prove the Jacobi identity we use Proposition 3.1.4 (3) [a [b c]] = F Z ,W [a(Z ), F X,W [b(X ), c(W )]] [a(Z ), [b(X ), c(W )]] = (−1)a N F Z ,W F X,W [[a(Z ), b(X )], c(W )] = (−1)a N F Z ,W F X,W [b(X ), [a(Z ), c(W )]] + (−1)ab+a N F Z ,W F X,W + F Z ,X [[a(Z ), b(X )], c(W )] = (−1)a N +N F X,W [b(X ), F Z ,W [a(Z ), c(W )]] + (−1)ab+a N +bN +N F X,W
= (−1)a N +N [[a b] + c] + (−1)(a+N )(b+N ) [b [a c]]. Definition 3.2.2. Let C[T, S] := C[T, S 1 , . . . , S N ] be the commutative superalgebra freely generated by an even element T and N odd elements S i . A N W = N SUSY Lie conformal algebra is a Z/2Z-graded C[T, S]-module R with a C-bilinear operation [ ] : R ⊗C R → C[ ] ⊗C R of parity N mod 2 satisfying the following three axioms: (1) Sesquilinearity: [T a b] = −λ[a b], [S i a b] = −(−1) N χ i [a b],
[a T b] = (T + λ)[a b], [a S i b] = (−1)a+N S i + χ i [a b].
(2) Skew-symmetry: [b a] = −(−1)ab+N [b− −∇ a], where ∇ = (T, S 1 , . . . , S N ), the -bracket in the RHS means compute first the bracket and then let = − − ∇. (3) Jacobi identity: [a [b c]] = (−1)a N +N [[a b] + c] + (−1)(a+N )(b+N ) [b [a c]].
(3.2.1.4)
We will drop the adjective SUSY when no confusion may arise. Remark 3.2.3. Even though in this case the situation is simple, it is instructive to realize the bracket as a morphism of C[ ]-modules. Consider the co-commutative Hopf superalgebra H = C[ ] with comultiplication λ = λ⊗1+1⊗λ, χ i = χ i ⊗1+1⊗χ i . Note that C[∇] H . Consider H as a H -module with the adjoint action (which is trivial in this case, given that H is super-commutative). Then we may think of H ⊗ R as an H module; the action is given by h → h. Similarly R ⊗ R is an H -module. The -bracket is then a H -module homomorphism of degree (−1) N . Namely, let φ denote the morphism R ⊗ R → H ⊗ R which is given by the -bracket. Then for every h ∈ H we have φh − (−1)h N hφ = 0, as elements in Hom(R ⊗ R, H ⊗ R). Similarly, the Jacobi identity is an identity in Hom(R ⊗ R ⊗ R, H ⊗ H ⊗ R). We will expand on this in Remark 4.11.
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Remark 3.2.4. According to Proposition 3.2.1, given any N W = N SUSY formal distribution Lie superalgebra (g, R), the space R is a SUSY Lie conformal algebra where T = ∂w and S i = ∂ζ i , and the -bracket is defined by (3.2.1.1). Definition 3.2.5. A Lie superalgebra of degree p ∈ Z/2Z is a vector superspace h with a bilinear operation { , } : h ⊗ h → h of parity p satisfying: (1) Skew-symmetry:{a, b} = −(−1)ab+ p {b, a}. (2) Jacobi identity: a, {b, c} = (−1)ap+ p {a, b}, c + (−1)(a+ p)(b+ p) b, {a, c} . Lemma 3.2.6. Let h be a Lie superalgebra of degree p ∈ Z/2Z. Define g as a vector superspace to be h if p = 0 mod 2 or h with the reversed parity if p = 1 mod 2. Define the bilinear operation [ , ] : g ⊗ g → g by: [a, b] = (−1)ap+ p {a, b}, ¯ Then where the right-hand side is computed in h and then we reverse the parity if p = 1. (g, [ , ]) is a Lie superalgebra which we will denote as Lie(h). Proof. We have: [b, a] = (−1)bp+ p {b, a} = −(−1)bp+ab {a, b} = −(−1)(a+ p)(b+ p) [a, b], which is skew-symmetry for the Lie algebra provided the parity in g is shifted by p. To check the Jacobi identity we have: [a, [b, c]] = (−1) pb+ap {a, {b, c}} = (−1) pb+ p {{a, b}, c} + (−1)ab+ p {b, {a, c}} = = (−1) pb+ p+(a+b+ p) p+ap [[a, b], c] + (−1)ab+ p+ap+bp [b, [a, c]] = = [[a, b], c] + (−1)(a+ p)(b+ p) [b, [a, c]]. Lemma 3.2.7. Let R be a N W = N SUSY Lie conformal algebra. Then R/∇R is naturally a Lie superalgebra of degree N mod 2 with bracket {a + ∇R, b + ∇R} = [a b] =0 + ∇R. Proof. The fact that the bilinear map { , } is well defined follows from sesquilinearity. Skew-symmetry and the Jacobi identity follow from the corresponding axioms for the SUSY Lie conformal algebra R. Lemma 3.2.8. Let R be an N W = N SUSY Lie conformal algebra. Then R˜ := R ⊗ C[W, W −1 ] is an N W = N SUSY Lie conformal algebra with -bracket: [a ⊗ f b ⊗ g] = (−1) f b [a +∂W b] ⊗ f (W )g(W )|W =W ,
(3.2.1.5)
and with T˜ = T ⊗ id + id ⊗∂w and S˜i = S i ⊗ id + id ⊗∂ζ i . Proof. We prove here skew-symmetry; the other axioms are checked in a similar way: [a ⊗ f b ⊗ g] = (−1) f b [a +∂W b] ⊗ f (W )g(W )|W =W = −(−1)ab+N + f b [b− −∂W −∇ a] ⊗ f (W )g(W )|W =W = −(−1)(a+ f )(b+g)+N +ga [b− −∂W −∇−∂W +∂W a] ⊗ g(W ) f (W )|W =W
= −(−1)(a+ f )(b+g)+N [b ⊗ g− −∇˜ a ⊗ f ].
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˜ ∇˜ R˜ 3.2.2. For any N W = N SUSY Lie conformal algebra R, we put L(R) = R/ and Lie(R) := Lie(L(R)) (see Lemmas 3.2.6 and 3.2.7). For each a ∈ R, let a
∈ L(R) be the image of a ⊗ W n|I . Similarly define a(n|I ) ∈ Lie(R) as the image of the following element of L(R) : (−1)a I σ (I )a , and define, for each a ∈ R, the following Lie(R)-valued formal distribution a(Z ) = Z −1− j|N \J a( j|J ) ∈ Lie(R)[[Z , Z −1 ]]. (3.2.2.1) j∈Z,J
Using (3.2.1.5) with f = W n|I and g = W k|K and putting = 0 we compute explicitly the Lie bracket (of parity N mod 2) in L(R) : n {a , b } = (−1)a J +b(I −J ) × j j≥0,J
×σ (J, I \ J )σ (I \ J, K )(a( j|J ) b) . (3.2.2.2) It is straightforward to check using Lemma 3.2.6, that the Lie bracket in Lie(R) is given by: n [a(n|I ) , b(k|K ) ] = (−1)(a+N −I )(N −K ) × (−1)(I −J )(N −J ) j ( j|J ): j≥0
×σ (I )σ (J, I \ J )σ (I \ J, (N \ K )\(I \ J )) a( j|J ) b (n+k− j|K ∪(I\J )) . (3.2.2.3) Proposition 3.2.9. Let R be an N W = N SUSY Lie conformal algebra, a, b two vectors in R, and a(Z ), b(W ) the corresponding Lie(R)-valued formal distributions defined by (3.2.2.1). Then ( j|J ) ∂W δ(Z , W ) a( j|J ) b (W ). (3.2.2.4) [a(Z ), b(W )] = j≥0,J
Proof. First we expand ( j|J )
∂W
δ(Z , W ) =
n (−1) I −J σ (J )σ (N \ I, I \ J )Z −1−n|N \I W n− j|I \J . j
n∈Z,I
(3.2.2.5) Now using (3.2.2.3) we have: n (−1)(I −J )(N −J ) σ (I )σ (J, I \ J )× [a(Z ), b(W )] = j n∈Z,I
k∈Z,K
× σ (I \ J, (N \ K ) \ (I \ J ))Z −1−n|N \I W −1−k|N \K a( j|J ) b (n+k− j|K ∪(I \J ) . (3.2.2.6)
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On the other hand we have W −1−k|N \K = σ (I \ J, (N \K )\(I \ J ))W n− j|I \J W −1−k−n+ j|(N \K )\(I \J ) ,
(3.2.2.7)
and, due to (3.1.1.1), σ (I )σ (J, I \ J ) = (−1)(I −J )(N −I ) σ (N \ I, I \ J )σ (J ). Now substituting (3.2.2.7) in (3.2.2.6) and using (3.2.2.8) we obtain (3.2.2.4).
(3.2.2.8)
Proposition 3.2.10. Let R be an N W = N SUSY Lie conformal algebra, then the pair (Lie(R), R) is an N W = N SUSY formal distribution Lie superalgebra. Proof. The fact that the family of distributions (3.2.2.1) is closed under ( j|J )th products and that they are pairwise local follows from Proposition 3.2.9 since a( j|J ) b = 0 for j 0 in R. The fact that this family is closed under the derivations ∂z , ∂θ i follows from the following identities which are straightforward to check (T a)( j|J ) = − ja( j−1|J ) , (S i a)( j|J ) = σ (ei , N \ J )a( j|J \ei ) .
(3.2.2.9)
3.2.3. Note from (3.2.1.5) that (−∂w , −∂ζ i ) are derivations of the (0|0)th product of ˜ Since these operators supercommute with (T˜ , S˜ i ), they induce derivations (T, S i ) R. of the Lie superalgebra Lie(R), given by the formulas: T (a( j|J ) ) = − ja( j−1|J ) , σ (N \ J, ei )a( j|J \ei ) i S (a( j|J ) ) = 0
if i ∈ J, if i ∈ / J.
(3.2.3.1)
Note that Lie(R) contains a subalgebra Lie(R)≤ spanned by vectors a( j|J ) with j ≥ 0. This subalgebra, called the annihilation subalgebra, is stable under the action of ∇ = (T, S i ). Moreover, it is straightforward to check, using (3.2.3.1), that the formal distributions (3.2.2.1) satisfy: T a(Z ) = ∂z a(Z ), S i a(Z ) = ∂θ i a(Z ) (3.2.3.2) namely, the N W = N formal distribution Lie superalgebra (Lie(R), R) is regular. 3.2.4. Recall that we have defined ( j|J )th products of formal distributions for j ≥ 0 in 3.2.1. In order to define these products for j < 0 we let for a formal distribution a(Z ) = Z j|J a j|J : a+ (Z ) = Z j|J a j|J , a− (Z ) = Z j|J a j|J . ( j|J ): j≥0
( j|J ): j<0
It follows easily from the definitions that a+ (W ) = res Z i z,w (Z − W )−1|N a(Z ),
a− (W ) = − res Z i w,z (Z − W )−1|N a(Z ). (3.2.4.1)
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Indeed, we have
i z,w (Z − W )−1|N =
(−1) J σ (J )W m|J Z −1−m|N \J .
(m|J ):m≥0
Hence:
res Z i z,w (Z − W )−1|N a(Z ) = res Z
(−1) J σ (J )W m|J Z −1−m|N \J Z n|I an|I
(m|J ):m≥0
(n|I ):n∈Z
= res Z
(−1) J σ (J )σ (N \ J, J )W m|J Z −1|N am|J
(m|J ):m≥0
=
W m|J am|J = a+ (W ).
(m|J ):m≥0
The second equation in (3.2.4.1) follows similarly, or by noting that it is a consequence of the first equation in (3.2.4.1), the definition of the δ function (3.1.4.1) and property (5) in 3.1.4. Differentiating (3.2.4.1) we find: ( j|J )
(−1) J N ∂W
( j|J )
(−1) J N ∂W
a(W )+ = σ (J ) res Z i z,w (Z − W )−1− j|N \J a(Z ),
a(W )− = −σ (J ) res Z i w,z (Z − W )−1− j|N \J a(Z ).
(3.2.4.2)
These equations (3.2.4.2) are called the super Cauchy formulae. Definition 3.2.11. Let V be a vector superspace. An End(V )-valued formal distribution a(Z ) is called a field if for every vector v ∈ V we have a(Z )v ∈ V ((Z )), i.e. there are finitely many negative powers of z in a(Z )v. For two such fields we define their normally ordered product to be : a(Z )b(Z ) : := a+ (Z )b(Z ) + (−1)ab b(Z )a− (Z ).
(3.2.4.3)
3.2.5. The normally ordered product of fields is again a well defined field. Indeed, when applied to any vector v ∈ V the first summand in (3.2.4.3) clearly has finitely many negative powers of z since b(Z )v ∈ V ((Z )) and a+ (Z ) has only non-negative powers of z. For the second summand we see that a− (Z )v ∈ V [Z , Z −1 ], namely it is a Laurent polynomial with values in V , therefore b(Z )a− (Z )v ∈ V ((Z )) as we wanted. Lemma 3.2.12. : a(W )b(W ) : = res Z i z,w (Z − W )−1|N a(Z )b(W ) −
−(−1)ab i w,z (Z − W )−1|N b(W )a(Z ) .
Proof. This is immediate by (3.2.4.1).
(3.2.5.1)
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3.2.6. Given the last lemma and the Cauchy formulae (3.2.4.2) it is natural to define ( j|J ) a(W )(−1− j|N \J ) b(W ) = σ (J )(−1) J N : ∂W a(W ) b(W ) : . (3.2.6.1) Differentiating (3.2.5.1) we find:
i z,w (Z − W )−1− j|N \J a(Z )b(W )− −(−1)ab i w,z (Z − W )−1− j|N \J b(W )a(Z ) .
a(W )(−1− j|N \J ) b(W ) = res Z
Similarly, from the definition of the j|J th products for j ≥ 0 in 3.2.1 we have: res Z i z,w (Z − W ) j|J a(Z )b(W ) − (−1)ab i w,z (Z − W ) j|J b(W )a(Z ) = = res Z (Z − W ) j|J a(Z )b(W ) − (−1)ab b(W )a(Z ) = = res Z (Z − W ) j|J [a(Z ), b(W )] = a(W )( j|J ) b(W ). Therefore we have proved that for every j ∈ Z and every tuple J we have: a(W )( j|J ) b(W ) = res Z i z,w (Z − W ) j|J a(Z )b(W )− −(−1)ab i w,z (Z − W ) j|J b(W )a(Z ) .
(3.2.6.2)
(3.2.6.3)
Proposition 3.2.13. The following identities analogous to sesquilinearity for all pairs j|J are true: (∂w a(W ))( j|J ) b(W )
∂w a(W )( j|J ) b(W )
∂ζ i a(W ) ( j|J ) b(W )
∂ζ i a(W )( j|J ) b(W )
= − ja(W )(− j−1|J ) b(W ), = (∂w a(W ))( j|J ) b(W ) + a(W )( j|J ) ∂w b(W ), = σ (J \ ei , ei )a(W )( j|J \ei ) b(W ), = (−1) N −J ∂ζ i a(W ) ( j|J ) b(W )+
+ (−1)a a(W )( j|J ) ∂ζ i b(W ) ,
(3.2.6.4)
where ei is the tuple consisting of only one element {i} and we recall that we are defining σ (ei , J \ ei ) to be zero if i ∈ J . Proof. The first two equations are standard and their proof is similar to the last two. We will prove the last two equations by using (3.2.6.3). If i ∈ J the result is obvious: res Z i z,w (Z − W ) j|J ∂θ i a(Z )b(W ) = −(−1) J res Z ∂θ i i z,w (Z − W ) j|J a(Z )b(W ) = −(−1) J σ (ei , J \ ei ) res Z i z,w (Z − W ) j|J \ei a(Z )b(W ).
(3.2.6.5)
Similarly we have: − (−1)(a+1)b res Z i w,z (Z − W ) j|J b(W )∂θ i a(Z ) = = (−1)ab+J res Z ∂θ i i w,z (Z − W ) j|J b(W )a(Z ) = = (−1)ab+J σ (ei , J \ ei ) res Z i w,z (Z − W ) j|J \ei b(W )a(Z ).
(3.2.6.6)
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Adding (3.2.6.5) and (3.2.6.6) and using (3.1.1.1), we obtain the third equation in (3.2.6.4). Finally, to prove the last relation in (3.2.6.4) we expand:
∂ζ i a(W )( j|J ) b(W ) = ∂ζ i res Z i z,w (Z − W ) j|J a(Z )b(W )− − (−1)ab i w,z (Z − W ) j|J b(W )a(Z ) = = (−1) N res Z −σ (ei , J \ ei )i z,w (Z − W ) j|J \ei a(Z )b(W )+ +(−1) J +a i z,w (Z − W ) j|J a(Z )∂ζ i b(W )− −(−1)ab σ (ei , J \ ei )i w,z (Z − W ) j|J \ei b(W )a(Z )− −(−1)ab+J i w,z (Z − W ) j|J ∂ζ i b(W )a(Z ) = = −(−1) N σ (ei , J \ ei )a(W )( j|J \ei ) b(W ) + (−1) N +J +a a(W )( j|J ) ∂ζ i b(W ) = (3.2.6.7) = (−1) N −J ∂ζ i a(W ) ( j|J ) b(W ) + (−1)a a(W )( j|J ) ∂ζ i b(W ) . Proposition 3.2.14. The following identity holds for any ( j|J ) and any three fields a = a(W ), b = b(W ), c = c(W ):
[a (b( j|J ) c)] = (−1)(a+K +N )(J +N ) σ (J, K ) (k|K ) [a b]( j+k|J ∪K ) c+ = (k|K ):k≥0
+ (−1)(a+N )(b+N −J ) b( j|J ) [a c].
(3.2.6.8)
Proof. The left-hand side is
res Z exp ((Z − W ) ) [a(Z ), (b(W )( j|J ) c(W ))] = = res Z exp ((Z − W ) ) [a(Z ), res X i x,w (X − W ) j|J b(X )c(W )]− −(−1)bc [a(Z ), res X i w,x (X − W ) j|J c(W )b(X )] = = (−1)a(N −J ) res Z res X exp ((Z − W ) ) i x,w (X − W ) j|J [a(Z ), b(X )c(W )]− − (−1)bc+a(N −J ) res Z res X exp ((Z − W ) ) i w,x (X − W ) j|J [a(Z ), c(W )b(X )]. (3.2.6.9)
Using the identity [a, bc] = [a, b]c + (−1)ab b[a, c] we can write the first term of the RHS of the last equality as:
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(−1)a(N −J ) res Z res X exp ((Z − X + X − W ) ) × × i x,w (X − W ) j|J [a(Z ), b(X )]c(W ) + (−1)a(N −J +b) res Z res X exp ((Z − W ) ) × × i x,w (X − W ) j|J b(X )[a(Z ), c(W )] = = (−1)a(N −J )+N +J N res X exp ((X − W ) ) i x,w (X − W ) j|J [a b](X )c(W )+ + (−1)a(N −J +b)+N +J N +bN res X i x,w (X − W ) j|J b(X )[a c](W ) = = (−1)(a+N )(N −J ) res X
K (K +1) 2
(k|K ):k≥0
× i x,w (X − W )
k|K
(−1) k!
k|K ×
(X − W ) j|J [a b](X )c(W )+
+ (−1)(a+N )(N −J +b) res X i x,w (X − W ) j|J b(X )[a c](W ) = = (−1)
(a+N )(N −J )
K (K +1) 2
res X
(k|K ):k≥0
(−1) k!
σ (K , J )×
× k|K i x,w (X − W )k+ j|K ∪J [a b](X )c(W )+ + (−1)(a+N )(N −J +b) res X i x,w (X − W ) j|J b(X )[a c](W ).
(3.2.6.10)
Similarly the second term in the RHS of the last equality of (3.2.6.9) can be written as: − (−1)bc+a(N −J ) res Z res X exp ((Z − W ) ) × × i w,x (X − W ) j|J [a(Z ), c(W )]b(X ) − (−1)bc+a(N −J +c) res Z res X × × exp ((Z − W ) ) i w,x (X − W ) j|J c(W )[a(Z ), b(X )] = = −(−1)bc+a(N −J )+N +J N res X i w,x (X − W ) j|J [a c](W )b(X )− − (−1)bc+(a+N )(N −J +c) res X exp ((X − W ) ) i w,x (X − W ) j|J c(W )[a b](X ) = −(−1)bc+(a+N )(N −J ) res X i w,x (X − W ) j|J [a c](W )b(X )− − (−1)
bc+(a+N )(N −J +c)
res X
(k|K ):k≥0
K (K +1) 2
(−1) k!
×
× σ (K , J ) k|K i w,x (X − W )k+ j,K ∪J c(W )[a b](X ).
(3.2.6.11)
Now adding (3.2.6.10) and (3.2.6.11) we get (3.2.6.8) (recall that the -bracket has parity N mod2). Remark 3.2.15. If we multiply both sides of (3.2.6.8) by (−1)
J (J +1+2a) 2
j!
j|J ,
and sum over all pairs ( j|J ) with j ≥ 0 we obtain the Jacobi identity for the -bracket that we have already proved in Proposition 3.2.1. Therefore, the identities (3.2.6.8) for j ≥ 0 are equivalent to the Jacobi identity (3.2.1.4). Next we note that if we replace b by ∂w b in (3.2.6.8) we obtain the same identity with j replaced by j − 1 whenever j ≤ −1. Similarly, replacing b by ∂ζ i b we obtain the same identity with J replaced by J \ ei . It follows the identity (3.2.6.8) is equivalent to
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the Jacobi identity (3.2.1.4) and (3.2.6.8) with ( j|J ) = (−1|N ). In this case the formula (3.2.6.8) looks as follows: [a : bc :] =
λk k≥0
k!
[a b](k−1|N ) c + (−1)(a+N )b : b[a c] : .
Rewriting the sum as the sum of the k = 0 term and the rest, this becomes: (a+N )b [a : bc :] =: [a b]c : +(−1) : b[a c] : + [[a b] c]d . (3.2.6.12) 0
Here the integral 0 is computed by taking the indefinite integral in the even variable γ of ∂ηN of the integrand, and then taking the difference of the values at the limits. This is the super analogue of the non-commutative Wick formula [Kac96]. Thus, the identity (3.2.6.8) is equivalent to the Jacobi identity plus this non-commutative Wick formula. The following lemma is proved as in the ordinary vertex algebra case [Kac96, Lem. 3.2]. Lemma 3.2.16 (Dong’s Lemma). Given three pairwise local formal distributions a, b, c, the pair (a, b( j|J ) c) is local for any ( j|J ). 3.3. Identities and existence theorem In this section we define N W = N SUSY vertex algebras, derive their identities, and prove an existence theorem as in the non-super case [Kac96, Thm. 4.5]. Definition 3.3.1. An N W = N SUSY vertex algebra consists of a vector superspace V , an even vector |0 ∈ V , N odd operators S i (the odd translation operators), an even operator T (the even translation operator), and a parity preserving linear map Y from V to the space of End(V )-valued superfields a → Y (a, Z ). The following axioms must be satisfied: • Vacuum axioms: Y (a, Z )|0 = a + O(Z ), T |0 = S i |0 = 0,
i = 1, . . . , N .
• Translation invariance [S i , Y (a, Z )] = ∂θ i Y (A, Z ), [T, Y (a, Z )] = ∂z Y (a, Z ). • Locality
(3.3.0.13)
(z − w)n [Y (a, Z ), Y (b, W )] = 0 for some n ∈ Z+ .
As before, O(Z ) is an element of V [[Z ]] which vanishes at Z = 0. Morphisms between N W = N SUSY vertex algebras are linear maps f : V1 → V2 such that: f ◦ T1 = T2 ◦ f,
f (Y1 (a, Z )b) = Y ( f (a), Z ) f (b), ∀ a, b ∈ V1 .
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3.3.1. Given a N W = n SUSY vertex algebra V , we can define the ( j|J ) product of two vectors of V as follows. Expand the field Y (a, Z ) for a ∈ V : Z −1− j|N \J a( j|J ) , (3.3.1.1) Y (a, Z ) = ( j|J ): j∈Z
and define the j|J -product of two vectors in V as:
a( j|J ) b := a( j|J ) b .
(3.3.1.2)
This is a C-bilinear product on V of parity N − J mod 2. We can rewrite the axioms of the vertex algebra in terms of these products. For example, the vacuum axioms are equivalent to: a(−1|N ) |0 = a,
a( j|J ) |0 = 0 if j ≥ 0,
T |0 = S |0 = 0, i
(3.3.1.3) (3.3.1.4)
and translation invariance is equivalent to: [T, a( j|J ) ] = − ja( j−1|J ) , σ (N \ J, ei )a( j|J \ei ) i [S , a( j|J ) ] = 0
if i ∈ J, if i ∈ / J.
(3.3.1.5)
Of course the fact that Y (a, Z ) is a field is equivalent to a( j|J ) b = 0 for j 0, given a, b ∈ V . Theorem 3.3.2. Let U be a vector superspace and V be a space of pairwise local End(U )-valued fields such that V contains the constant field Id, it is invariant under the derivations ∂z , ∂θ i and closed under all ( j|J )th products. Then V is a N W = N SUSY vertex algebra with vacuum vector Id, translation operators T a(Z ) = ∂z a(Z ) and S i a(Z ) = ∂θ i a(Z ), and the ( j|J ) products are given by the RHS of (3.2.6.3) multiplied by σ (J )4 . Proof. To check the vacuum axioms we have: a(Z )( j|J ) 1 = σ (J ) res Z (Z − W ) j|J [a(Z ), 1] = 0 if j ≥ 0, a(Z )(−1|N ) 1 =: a(Z )1 := a(Z ), ∂z 1 = ∂θ i 1 = 0. To check translation invariance we have: ∂z (a(Z )( j|J ) b(Z )) − a(Z )( j|J ) ∂z b(Z ) = (∂z a(Z ))( j|J ) b(Z ), but this is − ja(Z )( j−1|J ) b(Z ), according to (3.2.6.4). Therefore we see that the first equation in (3.3.1.5) holds. For the odd translation operators we write (note that the parity of a( j|J ) is a + N − J since Y is parity preserving and our choice of decomposing the field in (3.3.1.1)): σ (J ) ∂θ i (a(Z )( j|J ) b(Z )) − (−1)a+N −J a(Z )( j|J ) ∂θ i b(Z ) = = (−1) N −J σ (J )(∂θ i a(Z ))( j|J ) b(Z ), 4 This normalization becomes is necessary because of our choice in (3.2.6.1), see also Theorem 3.3.8.
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and again by (3.2.6.4) we see that this is −(−1) N σ (J )σ (ei , J \ei )a(Z )( j|J \ei ) b(Z ) = σ (N \J, ei )σ (J \ei )a(Z )( j|J \ei ) b(Z ), proving the second identity in (3.3.1.5). In order to check locality, we expand σ (J )X −1− j|N \J a(W )( j|J ) b(W ) Y (a(W ), X )b(W ) = ( j|J ): j∈Z
= res Z
(−1)(N −J )N σ (J )X −1− j|N \J ×
( j|J ): j∈Z
× i z,w (Z −W ) j|J a(Z )b(W )−(−1)ab i w,z (Z −W ) j|J b(W )a(Z ) = res Z (−1) N −J σ (J ) i z,w (Z − W ) j|J X −1− j|N \J × ( j|J ): j∈Z
× a(Z )b(W ) − (−1)ab i w,z (Z − W ) j|J X −1− j|N \J b(W )a(Z ) . (3.3.1.6) We note that (−1)(N −J ) σ (J )(Z − W ) j|J X −1− j|N \J = i z,w δ(Z − W, X ). (3.3.1.7) i z,w ( j|J ): j∈Z
Therefore the RHS of (3.3.1.6) reads: res Z i z,w δ(Z − W, X )a(Z )b(W ) − (−1)ab i w,z δ(Z − W, X )b(W )a(Z ) . With this last equation we can compute then the commutator [Y (a(W )), Y (b(W ))]c(W ). Indeed, the product Y (a(W ), X )Y (b(W ), Y )c(W ) is given by: res Z resU i u,w i z,w δ(U − W, X )δ(Z − W, Y )a(U )b(Z )c(W )− − (−1)bc i u,w i w,z δ(U − W, X )δ(Z − W, Y )a(U )c(W )b(Z )− − (−1)a(b+c) i w,u i z,w δ(U − W, X )δ(Z − W, Y )b(Z )c(W )a(U )+ + (−1)a(b+c)+bc i w,u i w,z δ(U − W, X )δ(Z − W, Y )c(W )b(Z )a(U ) , (3.3.1.8) and we get a similar expression for the product Y (b(W ), Y )Y (a(W ), X )c(W ). Subtracting we obtain: [Y (a(W ), X ), Y (b(W ), Y )]c(W ) = = res Z resU i u,w i z,w δ(U − W, X )δ(Z − W, Y )[a(U ), b(Z )]c(W )− − (−1)(a+b)c i w,u i w,z δ(U − W, X )δ(Z − W, Y )c(W )[a(U ), b(Z )] . (3.3.1.9) Let n ∈ Z+ be such that (u − z)n [a(U ), b(Z )] = 0. Multiplying (3.3.1.9) by (x − y)n we obtain that the RHS vanishes. Indeed, using (x − y) = (z − u) − ((z − w) − x) + ((u − w) − y), we see that all terms in the expansion of (x − y)n vanish when multiplied by δ functions, with the exception of (z − u)n . But this term vanishes when multiplied by the factors [a(U ), b(Z )] in (3.3.1.9). Therefore we have proved locality and the theorem.
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Corollary 3.3.3. An identity that holds for elements of any N W = N SUSY vertex algebra, holds for any collection of pairwise local fields. Lemma 3.3.4. Let V be a vector superspace and let |0 be an even vector of V . Let a(Z ), b(Z ) be two End(V )-valued fields such that a(Z )|0 ∈ V [[Z ]] and b(Z )|0 ∈ V [[Z ]]. Then for all ( j|J ), a(W )( j|J ) b(W )|0 ∈ V [[W ]] and the constant term is σ (J )a( j|J ) b(−1|N ) |0.
(3.3.1.10)
Proof. Applying both sides of (3.2.6.3) to the vacuum, we see that the second term on the RHS of (3.2.6.3) vanishes since it contains only positive powers of z. The first term in the RHS contains only positive powers of w since i z,w (Z − W ) j|J does and b(W )|0 ∈ C[[W ]]. Letting W = 0 we get
a(W )( j|J ) b(W )|0|W =0 = res Z Z j|J a(Z ) b(−1|N ) |0 . (3.3.1.11) It follows from (3.1.2.2) that the RHS of (3.3.1.11) is (3.3.1.10).
The following lemma is straightforward Lemma 3.3.5. Let A and B1 , . . . , B N be linear operators on a vector superspace U . Suppose that A is even and Bi are odd and they pairwise (super) commute, i.e. ABi = Bi A, Bi B j = −B j Bi . Then there exists a unique solution f (Z ) ∈ U [[Z ]] to the system of differential equations: ∂z f (Z ) = A f (Z ),
∂θ i f (Z ) = Bi f (Z ) (i = 1, . . . , N ),
(3.3.1.12)
for any initial condition f (0) = f 0 . Proof. Using (3.3.1.12), the coefficients of f (Z ) can be computed by induction, given f0 . Proposition 3.3.6. Let V be a N W = N SUSY vertex algebra. Then for every a, b ∈ V : (a) Y (a, Z )|0 = exp(Z ∇)a, (b) exp(Z ∇)Y (a, W ) exp(−Z ∇) = i w,z Y (a, Z + W ), (c) Y (a, Z )( j|J ) Y (b, Z )|0 = σ (J )Y (a( j|J ) b, Z )|0, where ∇ = (T, S 1 , . . . , S N ) and Z ∇ = zT + i θ i S i . Proof. We note that both sides in (a) and (c) are elements of V [[Z ]] whereas both sides of (b) are elements of End(V )[[W, W −1 ]][[Z ]]. Note that by evaluating at Z = 0 we get equalities in all three cases, the only non-trivial case is (c), but it follows from Lemma 3.3.4. Let us denote the right-hand side in each case by X (Z ). It is easy to show that it satisfies the following systems of equations respectively: (1) ∂z X (Z ) = T X (Z ), and ∂θ i X (Z ) = S i X (Z ). (2) ∂z X (Z ) = [T, X (Z )] and ∂θ i X (Z ) = [S i , X (Z )] by the translation axioms. (3) ∂z X (Z ) = T X (Z ) and ∂θ i X (Z ) = S i X (Z ) by the translation axioms (recall that T |0 = S i |0 = 0). In order to apply Lemma 3.3.5, we have to show that the left-hand side of (a), (b) and (c) satisfies the same differential equations (1), (2) and (3) respectively;
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(1) It is immediate by the translation invariance and the second of the vacuum axioms. (2) Denoting Y (Z ) = e Z ∇ Y (a, W )e−Z ∇ , we have: ∂z Y (Z ) = T Y (Z ) − Y (Z )T = [T, Y (Z )], and similarly: ∂θ i Y (Z ) = S i Y (Z ) + (−1)a e Z ∇ Y (a, W )(−S i )e−Z ∇ = S i Y (Z ) − (−1)a Y (Z )S i = [S i , Y (Z )]. (3) Denote Y (Z ) = Y (a, Z )( j|J ) Y (b, Z )|0 and recall that from Proposition 3.2.13, ∂z and ∂θ i are derivations of the ( j|J ) products. To simplify notation, we will denote a(Z ) = Y (a, Z ) and b(Z ) = Y (b, Z ). We have: S i Y (W ) = S i res Z i z,w (Z − W ) j|J a(Z )b(W )|0−
− (−1)ab i w,z (Z − W ) j|J b(W )a(Z )|0 = = (−1) N +J res Z i z,w (Z − W ) j|J [S i , a(Z )]b(W )|0 + (−1)a i z,w (Z − W ) j|J a(Z )[S i , b(W )]|0 − (−1)ab i w,z (Z − W ) j|J [S i , b(W )]a(Z )|0−
− (−1)ab+b i w,z (Z − W ) j|J b(W )[S i , a(Z )]|0 ,
and, using S i |0 = 0, = (−1) N +J res Z i z,w (Z − W ) j|J (∂θ i a(Z ))b(W )|0+ + (−1)a i z,w (Z − W ) j|J a(Z )(∂ζ i b(W ))|0− − (−1)ab i w,z (Z − W ) j|J (∂ζ i b(W ))a(Z )|0
− (−1)ab+b i w,z (Z − W ) j|J b(W )(∂θ i a(Z ))|0 = (−1) N +J (∂ζ i a(W ))( j|J ) b(W ) + (−1)a a(W )( j|J ) (∂ζ i b(W )) |0 =
= ∂ζ i a(W )( j|J ) b(W )|0 . The proof for T is similar.
Proposition 3.3.7. (Uniqueness) Let V be a N W = N SUSY vertex algebra and let a(Z ) be an End(V )-valued field such that the pair (a(Z ), Y (b, Z )) is local for every b ∈ V , and a(Z )|0 = 0, then a(Z ) = 0. Proof. By locality there exists n ∈ Z+ such that (z − w)n a(Z )Y (b, W )|0 = (−1)ab (z − w)n Y (b, W )a(Z )|0 = 0. By Proposition 3.3.6 (1), the left-hand side is (z − w)n a(Z )e W ∇ b. Letting W = 0, we get z n a(Z )b = 0, and this holds for all b, therefore a(Z ) = 0.
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As a simple corollary of the previous proposition and Proposition 3.3.6 we obtain the following Theorem 3.3.8. In an N W = N SUSY vertex algebra the following identities hold (1) Y (a( j|J ) b, Z ) = σ (J )Y (a, Z )( j|J ) Y (b, Z ) ( ( j|J )-th product identity). (2) Y (a(−1|N ) b, Z ) =: Y (a, Z )Y (b, Z ) :. (3) Y (T a, Z ) = ∂z Y (a, Z ). (4) Y (S i a, Z ) = ∂θ i Y (a, Z ). (5) We have the following OPE formula (the sums are finite): ( j|J ) σ (J )(∂W δ(Z , W ))Y (a( j|J ) b, W ) [Y (a, Z ), Y (b, W )] = ( j|J ): j≥0
=
(i z,w − i w,z )(Z − W )−1− j|N \J Y (a( j|J ) b, W ).
( j|J ): j≥0
(3.3.1.13) Proof. (1) is the combined statement of Dong’s Lemma 3.2.16, and Propositions 3.3.7 and 3.3.6 (c). (2) follows from (1) by letting j|J = −1|N . To prove (3) we write, using (3.3.1.5), the (−2|N )-product identity, (3.2.6.1) and the vacuum axiom: Y (T a, Z ) = Y (a(−2,N ) |0, Z ) = Y (a, Z )(−2|N ) Id =: ∂z Y (a, Z ) Id := ∂z Y (a, Z ). (4) follows similarly: Y (S i a, Z ) = Y (a(−1,N \ei ) |0, Z ) = = −σ (N \ ei , ei )σ (ei , N \ ei )(−1) N : ∂θ i Y (a, Z ) Id := ∂θ i Y (a, Z ). Finally (5) follows from (1) and the decomposition Lemma 3.1.3.
Corollary 3.3.9. Let ei = {i}. One has (cf. (3.2.6.4)): (S i a)( j|J ) = σ (ei , N \ J )a( j|J \ei ) , (T a)( j|J ) = − ja( j−1|J ) , T (a( j|J ) b) = (T a)( j|J ) b + a( j|J ) T (b), S i (a( j|J ) b) = (−1) N −J (S i a)( j|J ) b + (−1)a a( j|J ) S i b . Lemma 3.3.10.
i x,z δ(X − Z , W ) = i w,z δ(X, W + Z ).
Proof. For simplicity let us assume N = 0, the general result follows easily. Denote: ψ = i x,z i x−z,w (x − w − z)−1 ∈ C[[x, x −1 , z, z −1 , w, w −1 ]], ϕ = i w,z i x,w+z (x − w − z)−1 ∈ C[[x, x −1 , z, z −1 , w, w −1 ]]. It is straightforward to check that both ψ and ϕ are elements of K [[z, w]] where K = C((x)). On the other hand, since both compositions i x,z i x−z,w and i w,z i x,w+z commute with multiplication by x, z and w, we have (x − w − z)(ψ − ϕ) = 0, hence ψ = ϕ, since K [[z, w]] has no zero divisors. Similarly, we have: (i x,z i w,x−z − i w,z i w+z,x )(x − w − z)−1 = 0, and the lemma follows.
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3.3.2. Taking the generating series in Theorem 3.3.8(1) we obtain for the left-hand side: W −1− j|N \J Y (a( j|J ) b, Z ) = Y Y (a, W )b, Z . ( j|J ): j∈Z
On the right-hand side we obtain
W −1− j|N \J σ (J ) res X i x,z (X − Z ) j|J Y (a, X )Y (b, Z )−
( j|J ): j∈Z
= res X
− (−1)ab i z,x (X − Z ) j|J b(Z )a(X ) = (−1) N \J σ (J ) i x,z (X − Z ) j|J W −1− j|N \J Y (a, X )Y (b, Z )− ( j|J ): j∈Z
− (−1)ab i z,x (X − Z ) j|J W −1− j|N \J Y (b, Z )Y (a, X ) .
But, according to (3.3.1.7), this is res X i x,z δ(X − Z , W )Y (a, X )Y (b, Z ) − (−1)ab i z,x δ(X − Z , W )Y (b, Z )Y (a, X ) . (3.3.2.1) Using Lemma 3.3.10, the first term gives i w,z Y (a, W + Z )Y (b, Z ).
(3.3.2.2)
In order to compute the second term we expand in Taylor series (cf. 3.1.3.2) i z,x δ(X − Z , W ) =
(k|K ):k≥0
(k|K )
(−1) K X k|K ∂−Z δ(−Z , W ).
Hence the second term in (3.3.2.1) reads: − (−1)ab res X
(k|K )
(−1) K X k|K ∂ Z
(k|K ):k≥0
= − res X
δ(−Z , W )Y (b, Z )X −1−n|N \I a(n|I ) =
(−1)ab+(N −I )(b+N −K )+K σ (K , N \ I )×
(k|K ):k≥0 (k|K ) × X k−1−n|K ∪(N \I ) ∂−Z δ(−Z , W )Y (b, Z )a(n|I ) = (k|K ) (−1)(a+N −K )b+N σ (K )∂−Z δ(−Z , W )Y (b, =− (k|K ):k≥0
Z )a(k|K ) .
Adding this to (3.3.2.2) and changing Z by −Z we obtain the important formula Y Y (a, W )b, −Z = i w,z Y (a, W − Z )Y (b, −Z )− (k|K ) − (−1)(a+N −K )b+N σ (K )∂ Z δ(Z , W )Y (b, −Z )a(k|K ) . (3.3.2.3) (k|K ):k≥0
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Note now that by acting on any vector c ∈ V and multiplying this last equation by a sufficiently high power of (z −w) the second term vanishes, therefore we obtain associativity for the vertex operators, namely: (z − w)n Y Y (a, W )b, −Z c = (z − w)n Y (a, W − Z )Y (b, −Z )c, n 0. (3.3.2.4) As in [FBZ01, 3.2.3] we obtain an equivalent formulation which is called the Cousin property. Recall the embedding: i z,w : C((Z , W )) → C((Z ))((W )). Given f ∈ C((Z , W )), i z,w f is called the expansion of f in the domain |z| > |w|. Corollary 3.3.11. (Cousin property) For any N W = n SUSY vertex algebra V and vectors a, b, c ∈ V , the three expressions: Y (a, Z )Y (b, W )c ∈ V ((Z ))((W )), (−1) Y (b, W )Y (a, Z )c ∈ V ((W ))((Z )), Y (Y (a, Z − W )b, W ) c ∈ V ((W ))((Z − W )) ab
are the expansions, in the domains |z| > |w|, |w| > |z| and |w| > |w − z| respectively, of the same element of V [[Z , W ]][z −1 , w −1 , (z − w)−1 ]. Proof. By the locality axiom, there exists n ∈ Z+ such that: (z − w)n Y (a, Z )Y (b, W )c = (−1)ab (z − w)n Y (b, W )Y (a, Z )c. Since the LHS is an element of V ((Z ))((W )) and the RHS is an element of V ((W ))((Z )), it follows that they are both equal to some ϕ ∈ V [[Z , W ]][z −1 , w −1 ] (cf. (3.1.2.3)). Since i z,w and i w,z are algebra morphisms, we get ϕ ϕ , (−1)ab Y (b, W )Y (a, Z )c = i w,z . Y (a, Z )Y (b, W )c = i z,w n (z − w) (z − w)n The rest of the corollary is proved in a similar way, using (3.3.2.4). Theorem 3.3.12. (Skew-symmetry) In an N W = N SUSY vertex algebra the following identity, called skew-symmetry, holds Y (a, Z )b = (−1)ab e Z ∇ Y (b, −Z )a.
(3.3.2.5)
Proof. By the locality axiom we have for n 0, (z − w)n Y (a, Z )Y (b, W )|0 = (z − w)n (−1)ab Y (b, W )Y (a, Z )|0. Now by (1) in Proposition 3.3.6 we can write this as: (z − w)n Y (a, Z )e W ∇ b = (z − w)(−1)ab Y (b, W )e Z ∇ a = (z−w)n (−1)ab e Z ∇ e−Z ∇ Y (b, W )e Z ∇ a = (z−w)n (−1)ab e Z ∇ i w,z Y (b, W − Z )a, (3.3.2.6) where in the last line we used (2) of Proposition 3.3.6. Now both sides in (3.3.2.6) are formal power series in W . Indeed, since b( j|J ) a = 0 for j 0 we see that by making n large enough we may assume that there are no negative powers of w in the RHS. We can then let W = 0 in (3.3.2.6) and multiply by z −n to obtain (3.3.2.5).
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3.3.3. Expanding both sides in (3.3.2.5) we have: Z −1− j|N \J a( j|J ) b = (−1)ab ×
( j|J ): j∈Z
(−Z )
−1−k|N \K
∇ ( j|J ) Z j|J ×
( j|J ): j≥0
b(k|K ) a = (−1)
ab
(k|K ):k∈Z
(−1)1+k+N −K ∇ ( j|J ) ×
( j|J ): j≥0
(k|K ):k∈Z
×σ (J, N \ K )Z j−1−k|J ∪(N \K ) b(k|K ) a. Taking the coefficient of Z −1−n|N \I on both sides we get: a(n|I ) b = (−1)ab (−1)1−n+N +J −I × j≥0,J ∩I =∅
× (−∇)( j|J ) σ (N \ (I ∪ J ), J )b(n+ j|I ∪J ) a. (3.3.3.1) In particular, when (n|I ) = (−1|N ) in (3.3.3.1), we get: : ab : −(−1)ab : ba := (−1)ab
(−T ) j b(−1+ j|N ) a , j! j≥1
or, equivalently, after exchanging a and b : : ab : −(−1)
ab
: ba :=
0 −∇
[a b]d .
(3.3.3.2)
Identity (3.3.3.2) is called the quasi-commutativity of the normally ordered product. 3.3.4. Define the following formal Fourier transform by FZ a(Z ) = res Z e Z a(Z ). It is a linear map from the space of U -valued formal distributions in Z to U [[ ]]. It has the following properties which are immediate to check: FZ ∂z a(Z ) = −λFZ a(Z ),
FZ
(3.3.4.1)
FZ ∂θ i a(Z ) = −(−1) N χ i FZ a(Z ), FZ e Z ∇ a(Z ) = FZ +∇ a(Z ) if a(Z ) ∈ U ((Z )),
(3.3.4.2)
FZ a(−Z ) = −FZ− a(Z ), ( j|J ) ∂W δ(Z , W ) = (−1) J N e W ( j|J ) .
(3.3.4.4)
(3.3.4.3)
(3.3.4.5)
Theorem 3.3.13. Let V be a N W = N SUSY vertex algebra. Then V is a N W = N SUSY Lie conformal algebra with -bracket: (−1) J N σ (J ) ( j|J ) (a( j|J ) b). (3.3.4.6) [a b] = FZ Y (a, Z )b = ( j|J ): j≥0
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Proof. The sesquilinearity relations follow from Corollary 3.3.9 for j ≥ 0. Applying FZ to both sides of (3.3.2.5) and using (3.3.4.3) and (3.3.4.4) we get the skew-symmetry relation. In order to prove the Jacobi identity, apply FZ to the OPE formula (3.3.1.13) applied to c, and use (3.3.4.5) to obtain (cf. (3.2.6.8)): [a Y (b, W )c] = (−1)ab+bN Y (b, W )[a c] + e W Y ([a b], W )c. to both sides of this formula we get the Jacobi identity. Applying FW
Theorem 3.3.14. Let V be a N W = N SUSY vertex algebra. The following identity called “quasi-associativity” of the normally ordered product holds for every a, b, c ∈ V :
:: ab : c : − : a : bc ::= a(−2− j|N ) b( j|N ) c + (−1)ab b(−2− j|N ) a( j|N ) c . j≥0
j≥0
Equivalently
∇
:: ab : c : − : a : bc ::= 0
d a [b c] + (−1)ab
∇ 0
d b [a c],
where the integral is computed as follows: expand the -bracket, put the powers of on the left, under the sign of integral, then take the definite integral by the usual rules inside the parenthesis. Proof. Applying both sides of Theorem 3.3.8 (2) to c and taking the constant coefficient, the LHS is :: ab : c :. By (3.2.4.3), the RHS of Theorem 3.3.8 (2) applied to c is
(−1)(N −K )(a+N −J ) σ (N \ J, N \ K )Z −2− j−k|N \(J ∩K ) a( j|J ) b(k|K ) c + j<0,J
k,K ∪J =N
+
(−1)(N −J )(b+N −K )+ab σ (N \ K , N \ J )Z −2− j−k|N \(J ∩K ) b(k|K ) a( j|J ) c .
j≥0,J
k,K ∪J =N
(3.3.4.7) To compute the constant coefficient in the last formula, we let K = J = N , and k = −2 − j, to get
a(−2− j|N ) b( j|N ) c + (−1)ab b(−2− j|N ) a( j|N ) c . j≥−1
j≥0
Noting that the term with j = −1 in the first summand in the last formula is :a : bc ::, the theorem follows. We thus arrive to the following equivalent definition of an N W = N SUSY vertex algebra (cf. [BK03]): Definition 3.3.15. An N W = N SUSY vertex algebra is a tuple (V , T , S i , [· ·], |0, ::), where • (V, T, S i , [· ·]) is an N W = N SUSY Lie conformal algebra,
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• (V, |0, T, S i , ::) is a unital quasicommutative quasiassociative differential superalgebra (i.e. T is an even derivation of :: and S i (i = 1, . . . , N ) are odd derivations of ::), • the -bracket and the product :: are related by the non-commutative Wick formula (3.2.6.12). Proof. We have shown that this definition follows from Definition 3.3.1. For the converse, we refer the reader to [BK03]. The proof carries over to the SUSY case with minor modifications. Removing the “quantum corrections” we arrive to the following definition: Definition 3.3.16. An N W = N Poisson SUSY vertex algebra is tuple (V , |0, T , S i , {· ·}, ·), where • (V, T, S i , {· ·}) is an N W = N SUSY Lie conformal algebra, • (V, |0, T, S i , ·) is an unital commutative associative differential superalgebra, • the following Leibniz rule is satisfied: {a bc} = {a b}c + (−1)(a+N )b b{a c}. Theorem 3.3.17. Let V be an N W = N SUSY vertex algebra. For each a, b ∈ V , k ∈ Z and K ⊂ {1, . . . , N }, the following identity, called the Borcherds identity, holds: i z,w (Z − W )k|K Y (a, Z )Y (b, W )−(−1)ab i w,z (Z − W )k|K Y (b, W )Y (a, Z ) = ( j|J ) = σ (J, K )σ (J ∪ K ) ∂W δ(Z , W ) Y (a(k+ j|K ∪J ) b, W ). (3.3.4.8) j≥0,J
Proof. The LHS of (3.3.4.8) is local since multiplied by (z − w)n for n 0 it is equal to (Z − W )n+k|K [Y (a, Z ), Y (b, W )] = 0, by the locality axiom. Therefore we can apply the decomposition Lemma 3.1.3 to the LHS of (3.3.4.8). We have c j|J (W ) = σ (J, K ) res Z i z,w (Z − W )k+J |K ∪J Y (a, Z )Y (b, W )− −(−1)ab σ (J, K ) i w,z (Z − W )k+ j|K ∪J Y (b, W )Y (a, Z ) , therefore the theorem follows from (3.2.6.3) and Theorem 3.3.8 (1).
Proposition 3.3.18. Let V be a N W = N SUSY vertex algebra. Then [a(n|I ) , Y (b, W )] = (−1) J N +I N +I J σ (J )σ (I ) × ( j|J ): j≥0
( j|J ) × ∂W W n|I Y a( j|J ) b, W .
(3.3.4.9)
[a(n|I ) , Y (b, W )] = Y (e−W ∇ a(n|I ) e W ∇ b, W ).
(3.3.4.10)
If, moreover, n ≥ 0, this becomes:
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Proof. Multiplying the OPE formula (3.3.1.13) by Z n|I and taking residues we obtain in the left-hand side σ (I )[a(n|I ) , Y (b, W )], while the right-hand side is ( j|J ) res Z (−1) I (N −J ) σ (J )(∂W δ(Z , W )Z n|I )Y (a( j|J ) b, W ) = ( j|J ): j≥0
= res Z
( j|J )
(−1) I (N −J ) σ (J )(∂W
δ(Z , W )W n|I )Y (a( j|J ) b, W ) =
( j|J ): j≥0
=
( j|J )
(−1) I (N −J )+J N σ (J )(∂W
W n|I )Y (a( j|J ) b, W )
( j|J ): j≥0
hence (3.3.4.9) follows. Note that when n ≥ 0, the RHS of (3.3.4.9) is a finite sum of fields of V times monomials W k|K with k ≥ 0. This easily implies that this field is local with respect to all fields of V , hence the LHS of (3.3.4.10) is local with respect to all fields of V . On the other hand, the adjoint action of ∇ on a(n|I ) either decreases n or I (cf. (3.3.1.5)). Using the formula Ade X = eadX for an even element X of a Lie superalgebra, we get that e−W ∇ a(n|I ) e W ∇ is a finite sum, involving only positive powers of w, hence the RHS of (3.3.4.10) is also local with respect to all fields of V . To apply the uniqueness theorem, we need to check that both sides agree when valuated at the vacuum vector. The left-hand side is given by [a(n|I ) , Y (b, W )]|0 = a(n|I ) e W ∇ b, where we used the fact that a(n|I ) |0 = 0 and Proposition 3.3.6 (1). On the other hand, by the same proposition the left-hand side is Y (e−W ∇ a(n|I ) e W ∇ , W )|0 = e W ∇ e−W ∇ a(n|I ) e W ∇ |0, and (3.3.4.10) follows.
Remark 3.3.19. As a consequence of (3.3.4.9) we see that by taking the coefficient of W −1−k|N \K we obtain the commutator [a(n|I ) , b(k|K ) ] as a linear combination of Fourier modes of fields in V . This rather complicated formula says that the linear span of Fourier modes of End(V )-valued fields is a Lie superalgebra. In order to compute explicitly the Lie bracket, we compute the coefficient of W −1−k|N \K on the left-hand side of (3.3.4.9) to obtain: (−1)(a+N −I )(N −K ) [a(n|I ) , b(k|K ) ]. (3.3.4.11) To compute this coefficient on the right-hand side we first expand: J (J −1)
(∂W W n|I )W −1−l|N \L = j|J
(−1) 2 n! × (n − j)!
× σ (J, I \ J )σ (I \ J, N \ L)W n− j−1−l|(I \J )∪(N \L) .
(3.3.4.12)
Note that in order for the corresponding term in (3.3.4.9) not to vanish, we must have J ⊂ I and in order for the coefficient of W −1−k|N \K not to be zero in (3.3.4.12) we must have (K ∩ I ) ⊂ J . Now we set n − j − l − 1 = −1 − k and (I \ J ) ∪ (N \ L) = N \ K to obtain l = n + k − j and L = K ∪ (I \ J ). We get then for the right-hand side (J +I )(N −J ) n σ (J )σ (I )× (−1) j ( j|J ): j≥0
× σ (J, I \ J )σ (I \ J, (N \ K ) \ (I \ J )) a( j|J ) b (n+k− j|K ∪(I \J )) .
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Combining with (3.3.4.11), we obtain: [a(n|I ) , b(k|K ) ] = (−1)(a+N −I )(N −K )
(−1)(I −J )(N −J )
( j|J ): j≥0
n σ (J )× j
× σ (I )σ (J, I \ J )σ (I \ J, (N \ K ) \ (I \ J )) a( j|J ) b (n+k− j|K ∪(I \J )) . (3.3.4.13)
3.3.5. We can define the tensor product of two N W = N SUSY vertex algebras in the usual way, namely, let V and W be two N W = N SUSY vertex algebras. The space of states is the vector superspace V ⊗ W . The vacuum vector is |0V ⊗ |0W . Let us denote YV and YW the corresponding state-field correspondences. We define the state field correspondence Y for V ⊗ W as Y (a ⊗ b, Z ) = YV (a, Z ) ⊗ YW (b, Z ) = (−1)a(N −K ) σ (N \ K , N \ J )Z −2− j−k|(N \(J ∩K )) a( j|J ) ⊗ b(k|K ) , = ( j|J ),(k|K )
(3.3.5.1) where the endomorphism a( j|J ) ⊗ b(k|K ) is defined to be (a( j|J ) ⊗ b(k|K ) )(v ⊗ w) = (−1)(b+N −K )v a( j|J ) v ⊗ b(k|K ) w. Note that in order for σ not to vanish in (3.3.5.1) we must have J ∪ K = N . Finally, we i . let the translation operators be T = TV ⊗ I d + I d ⊗ TW and S i = SVi ⊗ I d + I d ⊗ SW All the axioms of SUSY vertex algebra are straightforward to check. Theorem 3.3.20. (Existence). Let V be a vector superspace, |0 ∈ V an even vector, T an even endomorphism of V and S i , i = 1, . . . , N . odd endomorphisms of V , pairwise anticommuting between themselves and commuting with T . Suppose moreover that T |0 = S i |0 = 0. Let F be a family of End(V )-valued fields a α (Z ) = Z −1− j|N \J a(αj|J ) j∈Z,J
indexed by α ∈ A, such that (1) (2) (3) (4)
a α (Z )|0| Z =0 = a α ∈ V, [T, a α (Z )] = ∂z a α (Z ) and [S i , a α (Z )] = ∂θ i a α (Z ), all pairs (a α (Z ), a β (Z )) are local, the vectors a(αjss |Js ) . . . a(αj11 |J1 ) |0 span V .
Then the formula Y (a(αjss |Js ) . . . a(αj11 |J1 ) |0, Z ) =
= σ (Ji )a αs (Z )( js |Js ) . . . a(αj22 |J2 ) a(αj11 |J1 ) Id . . . (3.3.5.2) gives a well defined structure of an N W = N SUSY vertex algebra on V , with vacuum vector |0, translation operators T, S i , and such that Y (a α , Z ) = a α (Z ). Such a structure is unique.
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Proof. Let F¯ be the minimal family of End(V )-valued fields containing F , closed under all ( j|J )-products and under the derivations ∂z and ∂θ i , and subject to the conditions (1)–(3) of the theorem. By Theorem 3.3.2, F¯ is an N W = N SUSY vertex algebra. Define a map ϕ : F¯ → V by a(Z ) → a(Z )|0 Z =0 . This map is surjective by (4). Let a(Z ) ∈ ker ϕ. It follows easily by (2) and the fact that T |0 = S i |0 = 0 that a(Z )|0 = 0. Since ϕ is surjective, for each b ∈ V , there exists b(W ) ∈ ϕ −1 (b). By (3) there exists j ∈ Z+ such that (z − w) j a(Z )b(W )|0 = (−1)ab (z − w) j b(W )a(Z )|0 = 0. Letting W = 0 and canceling z j we obtain a(Z )b = 0 for every b ∈ V , hence a(Z ) = 0 and ϕ is also injective. We obtain thus a state-field correspondence Y : a → Y (a, Z ). Formula (3.3.5.2) follows from the ( j|J )-product identity in Theorem 3.3.8 (1). 3.4. The universal enveloping SUSY vertex algebra In this section we construct maps ϕ and ϕ , used in [Hel06] to define the conformal blocks, and we construct an N W = N SUSY vertex algebra attached to each N W = N SUSY Lie conformal algebra. 3.4.1. Let V be an N W = N SUSY vertex algebra. According to Theorem 3.3.13 it is a SUSY Lie conformal algebra. It follows by Proposition 3.2.10 that the pair (Lie(V ), V ) is an N W = N formal distribution Lie superalgebra. Recall from Lemma 3.2.8 and 3.2.2 the construction of the Lie superalgebra Lie(V ) = V˜ /∇˜ V˜ , where V˜ = V ⊗C C[X, X −1 ] and ∇˜ V˜ is the space spanned by vectors of the form: T a ⊗ f (X ) + a ⊗ ∂x f (X ), S i a ⊗ f (X ) + (−1)a N a ⊗ ∂ηi f (X ),
(3.4.1.1)
for a ∈ V , f (X ) ∈ C[X, X −1 ], and we change the parity if N is odd. Let ϕ : Lie(V ) → End(V ) be the linear map defined by a = a ⊗ X n|I → (−1)a I σ (I )a(n|I ) ,
a ∈ V.
(3.4.1.2)
Lie (V )
by the vector Similarly, we construct V ⊗C C((X )) and consider its quotient space generated by vectors of the form (3.4.1.1), with reversed parity if N is odd. Then (3.4.1.2) defines a map ϕ : Lie (V ) → End(V ). Comparing (3.2.2.3) and (3.3.4.13) and noting the extra factor σ (J ) in (3.3.4.6) we obtain the following Theorem 3.4.1. The maps ϕ, and ϕ are Lie superalgebra homomorphisms. 3.4.2. Let R be an N W = N SUSY Lie conformal algebra, and let (Lie(R), R) be the corresponding N W = N formal distribution Lie superalgebra (cf. Proposition 3.2.10). The Lie bracket in Lie(R) is given by (3.2.2.3). Recall from 3.2.3 that Lie(R) is a regular N W = N formal distribution Lie superalgebra. In particular, it carries an even derivation T and N odd derivations S i , i = 1, . . . , N defined by (3.2.3.1). Moreover, the annihilation subalgebra Lie(R)≤ is invariant by these derivations. Theorem 3.4.2. Let R be an N W = N SUSY Lie conformal algebra. Let V = V (R) be the quotient of U (Lie(R)) by the left ideal generated by Lie(R)≤ . Then V admits an N W = N SUSY vertex algebra structure whose vacuum vector |0 is the image of 1 in V , and the translation operators T , S i (i = 1, . . . , N ), are obtained by extending the corresponding derivations on Lie(R) by the Leibniz rule. This vertex algebra is called the universal enveloping vertex algebra of R.
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Proof. Let F be the family of Lie(R)-valued formal distributions F = a(Z )a ∈ R , where a(Z ) was defined in (3.2.2.1). Note that this family defines a family of End(V )-valued formal distributions, where the action is by left multiplication. This family satisfies (1)–(4) of Theorem 3.3.20. Indeed, (2) follows from (3.2.3.2), (3) follows from Proposition 3.2.9 and (4) follows since the vectors a( j|J ) with a ∈ R, j ∈ Z and J ⊂ {1, . . . , N } span Lie(R). The theorem will follow from the existence Theorem 3.3.20 if we show that these distributions are in fact End(V )-valued fields. For that, given a α1 , . . . , a αs ∈ R, we have to prove that for any a ∈ R and J ⊂ {1, . . . , N }, we have: a(n|J ) a(αj11 |J1 ) . . . a(αjss |Js ) |0 = 0 for n 0. This is proved by induction on s, using (3.2.2.3). 4. Structure Theory of N K = N SUSY Vertex Algebras 4.1. In this section we develop the structure theory of N K = N SUSY vertex algebras, where N is a positive integer. This kind of structure has been studied, in some particular cases, in the physics literature. Roughly speaking an N K = N SUSY vertex algebra is an N W = N SUSY vertex algebra, where the differential operators ∂θ i are replaced by the differential operators D iZ = D eZi = ∂θ i + θ i ∂z . To describe the corresponding SUSY Lie conformal superalgebras, perhaps the language of H -pseudoalgebras is more convenient [BDK01]. On the other hand, we are interested in their universal enveloping vertex algebras and in particular we want a description along the lines of the previous sections. In order to have a uniform notation between this section and the previous ones, given two sets of coordinates Z = (z, θ i ) and W = (w, ζ i ) we will denote
N i i j j θ ζ ,θ − ζ , Z −W = z−w− i=1
(θ − ζ ) J =
(θ i − ζ i ), (Z − W ) j|J = z − w −
i∈J
N
j θiζ i
(4.1.1) (θ − ζ ) J ,
i=1
where j ∈ Z and J is an ordered subset of {1, . . . , N }. As before, we define Z j|J = z j θ J . Note that (Z − W )−1|0 =
N k=0
N
i=1 θ
iζi
k
(z − w)k+1
,
(4.1.2)
therefore (Z − W )−1|N coincides with that in the N W = N case: (Z − W )−1|N =
(θ − ζ ) N . z−w
(4.1.3)
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The differential operators D iZ satisfy the commutation relations j
[D iZ , D Z ] = 2δi, j ∂z ,
(4.1.4)
and, as before, we denote for J = ( j1 , . . . , jk ) : D Z = (∂z , D 1Z , . . . , D ZN ),
j|J
DZ
j
j
( j|J )
j
= ∂z D Z1 . . . D Zk ,
DZ
=
J (J +1) 2
(−1) j!
j|J
DZ . (4.1.5)
0|J DZ
Occasionally, when j = 0, we will write = Finally, in this section we will consider not necessarily disjoint subsets I, J ⊂ {1, . . . , N } as in the N W = N case. Given I and J , ordered subsets of {1, . . . , N }, we will write I J = (I \ J ) ∪ (J \ I ). Note, however, that still (Z − W ) j|J (Z − W )k|K = 0 if J ∩ K = ∅. We will use the same formal δ-function δ(Z , W ) as before. Remarkably, the new binomial (Z − W ) j|J , given by (4.1.1) “behaves” with respect to the operators j|J j|J DW , in the same way as the old binomials (3.1.3.1) with respect to ∂W . D ZJ .
Lemma 4.1. The following identity is true: ( j|J )
DW
δ(Z , W ) = σ (J )(i z,w − i w,z )(Z − W )−1− j|N \J .
(4.1.6)
Proof. Let us assume for simplicity that j = 0, the general case follows easily from this, differentiating by w. We will prove the lemma by induction on J . When J = ∅, it follows from (4.1.3) that (4.1.6) coincides with the formula (1.6.1) for δ(Z , W ). When J = ei = {i}, the left-hand side of (4.1.6) is given by (θ − ζ ) N z−w
(θ − ζ ) N (θ − ζ ) N \ei + ζi . = −(i z,w − i w,z ) −σ (ei ) z−w (z − w)2
i i −DW δ(Z , W ) = −DW (i z,w − i w,z )
On the other hand, using (4.1.2), we get: −1|N \ei
(Z − W )
=
k≥0
k θi ζ i (θ − ζ ) N \ei (z − w)k+1
θi ζ i (θ − ζ ) N \ei + (θ − ζ ) N \ei z−w (z − w)2 (θ − ζ ) N \ei ζi − σ (ei , N \ ei ) = (θ − ζ ) N , z−w (z − w)2 =
hence (4.1.6) follows when J = ei . To prove the general case, assume that the lemma I = σ (e , I \ e )D i D I \ei , we have by the induction is valid for J = I \ ei . Since DW i i W W hypothesis, I DW δ(Z , W ) = σ (ei , I \ ei )σ (I \ ei , N \ (I \ ei ))(−1) i (i z,w ×DW
−1|N \(I \ei )
− i w,z )(Z − W )
.
(I −1)I 2
× (4.1.7)
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We expand the last factor as: k−1 θ jζ j D (Z − W ) =− kζ (θ − ζ ) N \(I \ei ) − (z − w)k+1 k≥1
j j k
j j k θ ζ θ ζ N \I i − σ (ei , N \ I ) (θ − ζ ) + (k + 1)ζ (θ − ζ ) N \(I \ei ) . k+1 (z − w) (z − w)k+2 i
−1|N \(I \ei )
i
k≥0
k≥0
(4.1.8) Relabeling the indexes we see that the first and last term cancel. Finally we note that, by (3.1.1.1): σ (ei , I \ ei )σ (ei , N \ I )σ (I \ ei ) = (−1) I σ (I ). (4.1.9) Combining (4.1.9), (4.1.8) and (4.1.7) we obtain the lemma for j = 0. 4.2. Most of the results proved in Sect. 3 for the N W = N situation carry over to the N K = N setting with the following modifications: • replace ∂ Z = (∂z , ∂θ 1 , . . . , ∂θ N ) by D Z = (∂z , D 1Z, . . . , D ZN ), N i i j • replace Z − W = (z − w, θ i − ζ i ) by Z − W = z − w − i=1 θ ζ ,θ − ζ j , • replace (Z − W ) j|J = (z − w) j i∈J (θ i − ζ i ) by
j N j|J i i θ ζ (θ i − ζ i ), (Z − W ) = z − w − i=1
i∈J
• replace the commutative associative “translation” superalgebra C[T, S i ] by the noncommutative associative “translation” superalgebra H generated by the set ∇ = (T, S 1 , . . . , S N ), where T is an even generator and S i are odd generators, subject to the relations: [T, S i ] = 0, [S i , S j ] = 2δi j T ; (4.2.1) • replace the commutative associative “parameter” superalgebra C[λ, χ i ] by the noncommutative associative “parameter” superalgebra L , generated by the set = (λ, χ 1 , . . . , χ N ), where λ is an even generator and χ i are odd generators, subject to the relations: [λ, χ i ] = 0, [χ i , χ j ] = −2δi j λ. (4.2.2) Note that we have an isomorphism H → L given by ∇ → − . Lemma 4.2. The formal δ-function satisfies the properties (1)–(7) of 3.1.4 after replacing ∂W by DW and writing + DW = (λ + ∂w , χ i + D i ). Proof. (1) is clear from Lemma 4.1. In order to prove (2) we use Lemma 4.1 to write: (n|I )
(Z − W ) j|J DW
δ(Z , W ) = σ (I )σ (J, N \ I )× × (i z,w − i w,z )(Z − W )−1−n+ j|N \(I \J ) . (n− j|I \J )
δ(Z , W ) the result follows from the following propApplying Lemma 4.1 to DW erty of σ , which follows from (3.1.1.1): σ (J, N \ I )σ (I \ J ) = σ (I )σ (I \ J, J ). Properties (3)–(7) are proved as in 3.1.4.
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Lemma 4.3. D iZ (Z − W ) j|J = σ (ei , J \ ei )(Z − W ) j|J \ei + jσ (ei , J )(Z − W ) j−1|J ∪ei . Proof. We prove the lemma by direct computation when j ≥ 0 : j D iZ (Z − W ) j|J = (∂θ i + θ i ∂z ) z − w − θ i ζ i (θ − ζ ) J = − jζ i (Z − W ) j−1|J + σ (ei , J \ ei )(Z − W ) j|J \ei + θ i j (Z − W ) j−1|J = σ (ei , J \ ei )(Z − W ) j|J \ei + jσ (ei , J )(Z − W ) j−1|J ∪ei . When j = −1 we have
θ i ζ i k (θ − ζ ) J i i − W) = ∂θ i + θ ∂ z (z − w)k+1 k≥0
i i k−1 (θ − ζ ) J iθ ζ i = kζ + σ (ei , J \ ei )× k+1 (z − w) k≥0
i i k
i i k (θ − ζ ) J \ei (θ − ζ ) J iθ ζ iθ ζ i × − θ (k + 1) (z − w)k+1 (z − w)k+2 k≥0 k≥0
i i k (θ − ζ ) J ∪ei iθ ζ −1|J \ei = σ (ei , J \ ei )(Z − W ) − σ (ei , J ) (k + 1) (z − w)k+2 D iZ (Z
−1|J
k≥0
= σ (ei , J \ ei )(Z − W )
−1|J \ei
− σ (ei , J )(Z − W )−2|J ∪ei . 1|0
The general case follows from these by noting that D Z = (D iZ )2 = ∂z , hence (Z − W )− j−1|J =
1 i 2j (D ) (Z − W )−1|J . j! Z
Therefore we get for j ≥ 0 : D iZ (Z − W )− j−1|J = =
1 i 2 j+1 (D ) (Z − W )−1|J j! Z
1 i 2j (D Z ) σ (ei , J \ ei )(Z − W )−1|J − −σ (ei , J )(Z − W )−2|J ∪ei j!
= σ (ei , J \ ei )(Z − W )−1− j|J \ei − ( j + 1)σ (ei , J )(Z − W )− j−2|J ∪ei . The following decomposition lemma is now proved in the same manner as Lemma 3.1.3: Lemma 4.4. Let a(Z , W ) be a local distribution in two variables. Then a(Z , W ) can be uniquely decomposed in the following finite sum: ( j|J ) a(Z , W ) = DW δ(Z , W ) c j|J (W ). ( j|J ): j≥0
The coefficients c j|J are given by c j|J (W ) = res Z (Z − W ) j|J a(Z , W ).
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N N i i Remark 4.5. Let (Z − W ) = z − w − i−1 θ ζ λ + i=1 (θ i − ζ i )χ i . Note that i satisfy the same commutation relations (4.2.2) as λ, χ i , therefore −∂ , −∂w and −DW w i −DW generate an associative superalgebra isomorphic to L . This allows us to consider i and ∂ act as the following derivations of the L as a module over H , by letting DW w superalgebra L : i i [DW , χ j ] = 2δi j λ, [∂w , χ i ] = [∂w , λ] = [DW , λ] = 0.
(4.2.3)
Lemma 4.6. i , exp ((Z − W ) )]. D iZ exp ((Z − W ) ) = χ i exp ((Z − W ) ) = −[DW
Proof. Since the exponent is a sum of non-commuting terms, the derivative of the expo N j j j nential is not as obvious as in the N W = N case. Let A = j=1 (θ − ζ )χ . We have: ⎛⎛ ⎞ ⎞ N exp ((Z − W ) ) = exp ⎝⎝z − w − θ j ζ j ⎠ λ⎠ exp(A), j=1
∂θ i Ak =
k−1
A j χ i Ak− j−1 .
j=0
Since
[A, χ i ]
∂θ i Ak =
k−1
=
−2(θ i
− ζ i )λ
we obtain
χ i Ak−1 − 2 jλ(θ i − ζ i )Ak−2 = kχ i Ak−1 − k(k − 1)λ(θ i − ζ i )Ak−2 ,
j=0
therefore
∂θ i exp(A) = χ i − λ(θ i − ζ i ) exp(A)
(4.2.4)
from which the first equality of the lemma follows easily. The proof of the second equality of the lemma is similar. Note that from (4.2.3) we have: i , A] = −χ i − 2λ(θ i − ζ i ), [DW
from where it follows as in (4.2.4) that i , exp(A)] = −(χ i + λ(θ i − ζ i )) exp(A), [DW
and the lemma follows by a straightforward computation.
Now we are in position to define the formal Fourier transform and N K = N SUSY Lie conformal algebras as we did in 3.1.5. We put F Z ,W a(Z , W ) = res Z exp ((Z − W ) ) a(Z , W ), which formally looks exactly like (3.1.5.1) but in this expression the variables χ i in = (λ, χ 1 , . . . , χ N ) do not commute, but rather satisfy (4.2.2), and (Z − W ) is given by (4.1.1) instead of (3.1.3.1). Using this formal Fourier transform, we define the bracket of two formal distributions a(W ) and b(W ) as in (3.2.1.1). We can now state the N K = N version of Proposition 3.1.4, on the properties of the Fourier transform:
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Proposition 4.7. The formal Fourier transform satisfies the following properties: (1) sesquilinearity: F Z ,W ∂z a(Z , W ) = −λF Z ,W a(Z , W ) = [∂w , F Z ,W ]a(Z , W ), i , F Z ,W ]a(Z , W ). F Z ,W D iZ a(Z , W ) = −(−1) N χ i F Z ,W a(Z , W ) = (−1) N [DW
(2) For any local formal distribution a(Z , W ) we have: W a(Z , W ), (−1) N F Z ,W a(W, Z ) = F Z− −∇ ,W
= F Z ,W a(Z , W )| =− −∇W ,
(4.2.5)
i ). where − − ∇W = (−λ − ∂w , −χ i − DW (3) For any formal distribution in three variables a(Z , W, X ) we have
a(Z , W, X ) = (−1) N F X,W F Z ,X a(Z , W, X )| = + , F Z ,W F X,W
F a(Z , W, X ), where the RHS is computed as follows. First compute F X,W Z ,X where denotes another set of indeterminates = (ψ, υ 1 , . . . , υ N ), where ψ is an even indeterminate and υ i are odd indeterminates, subject to the relations:
[ψ, υ i ] = 0, [υ i , υ j ] = −2δi j ψ. Then commute to the right of , using [λ, ψ] = [λ, υ i ] = 0, [χ i , ψ] = [χ i , υ j ] = 0.
(4.2.6)
Finally, replace = + . Proof. (1) and (2) are proved in the same way as in the N W = N case, with the aid of Lemma 4.6. Note that the exponentials involved in (3) do not commute, hence the argument in (3.1.5.4) is no longer valid. We need to check: exp((Z − W ) ) exp((X − W ) ) = exp((X − W ) ) exp((Z − X ) )| = + . (4.2.7) Note that the RHS of (4.2.7) can be also computed as exp((X − W )( + )) exp((Z − X ) ),
(4.2.8)
where we have to use the following commutation relations between and : [ηi , χ j ] = 2λδi, j , [γ , χ i ] = [γ , λ] = [λ, ηi ] = 0,
(4.2.9)
which follow from (4.2.6) after replacing by + . In order to check (4.2.7), recall that, given two operators A, B, such that their commutator [A, B] = C commutes with both A and B, we have e A e B = eC e B e A .
(4.2.10)
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Let Z = (z, θ i ), W = (w, ζ i ) and X = (x, π i ). Now we expand: exp ((Z − W ) ) exp ((X − W ) ) = exp (z − w − θ i ζ i )λ × π i ζ i )γ exp × exp (θ i − ζ i )χ i exp (x − w − (π i − ζ i )ηi . (4.2.11) Note also that we have exp (θ i − ζ i )χ i = exp (θ i − ζ i )χ i = (1 + (θ i − ζ i )χ i ) i i i i i i (1 + θ i χ i )(1 − ζ i χ i )(1 + θ i ζ i λ) = = eθ χ e−ζ χ eθ ζ λ ζ i χ i exp = exp θi ζ iλ , θ i χ i exp −
(4.2.12)
therefore (4.2.11) reads: exp ((Z − W ) ) exp ((X − W ) ) = exp ((z − w)λ) × ζ i χ i exp ((x − w)γ ) exp ζ i ηi . ×exp π i ηi exp − θ i χ i exp − (4.2.13) Commuting the exponentials using (4.2.10) and (4.2.9), (4.2.13) can be expressed as: exp ((z − w)λ + (x − w)γ ) exp − ζ i χ i exp π i ηi × × exp − ζ i ηi exp θ i χ i exp −2 θ i π i λ . (4.2.14) Multiplying and dividing by exp( π i χ i ) and using (4.2.12) we can express (4.2.14) as exp ((z − w)λ + (x − w)γ ) exp − ζ i χ i exp π i ηi × × exp − ζ i ηi exp θiπi λ . π i χ i exp (θ i − π i )χ i exp − Combining again the exponentials it is easy to express this as exp ((z − w)λ + (x − w)γ ) exp (π i − ζ i )(χ i + ηi ) × × exp − π i ζ i (λ + γ ) exp θi πiλ , (θ i − π i )χ i exp − which is equal to (4.2.8).
The definition of ( j|J )th products for j ≥ 0 and the definition of an N K = N formal distribution Lie superalgebra generalizes in a straightforward way the corresponding definitions in 3.2.1. We have now the N K = N version of Proposition 3.2.1, on the properties of the -bracket.
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Proposition 4.8. The -bracket defined as in (3.2.1.1) satisfies the following properties: (1) Sesquilinearity for a pair (a(Z ), b(W )) : [∂z a b] = −λ[a b], [D iZ a b]
[a ∂w b] = (∂w + λ)[a b],
= −(−1) χ [a b], N
i
i [a DW b]
= (−1)
a+N
i (DW
(4.2.15)
+ χ )[a b]. (4.2.16) i
(2) Skew-symmetry for a local pair (a(Z ), b(W )) : [b a] = −(−1)ab+N [a− −∇W b]. (3) Jacobi identity for a triple (a(Z ), b(X ), c(W )) : [a [b c]] = (−1)a N +N [[a b] + c] + (−1)(a+N )(b+N ) [b [a c]],
(4.2.17)
where = (γ , η1 , . . . , η N ), with ηi odd, γ even, subject to the relations [γ , ηi ] = [γ , γ ] = 0, [ηi , η j ] = −2δi j γ . In order to compare both sides of (4.2.17) the first term in the RHS of (4.2.17) is computed as follows. First compute [[a b] c]. Then commute
to the right of using (4.2.6) and finally replace = + . We also need to commute to the right of in the second term in the RHS of (4.2.17) using: [γ , λ] = [γ , χ i ] = 0,
[ηi , λ] = [ηi , χ j ] = 0.
(4.2.18)
Proof. (1) and (2) are proved in the same way as in the N W = N case, with the aid of (4.2.7). The proof of (3) is formally the same as in the N W = N , namely: [b(X ), c(W )]] [a [b c]] = F Z ,W [a(Z ), F X,W [a(Z ), [b(X ), c(W )]] = (−1)a N F Z ,W F X,W = (−1)a N F Z ,W F X,W [[a(Z ), b(X )], c(W )]+ [b(X ), [a(Z ), c(W )]] + (−1)ab+a N F Z ,W F X,W + F Z ,X [[a(Z ), b(X )], c(W )]+ = (−1)a N +N F X,W + (−1)ab+a N +bN +N F X,W [b(X ), F Z ,W [a(Z ), c(W )]]
= (−1)a N +N [[a b] + c] + (−1)(a+N )(b+N ) [b [a c]], where in the fifth line we used (3) in Proposition 4.7. The fact that we need to commute
and as explained in the statement, follows as in the considerations in Proposition 4.7. Note also that in the sixth line we used = (−1) N F X,W F Z ,W , F Z ,W F X,W
and in order to compare both sides of this equation we need to use (4.2.18).
Remark 4.9. In practice, we write the LHS of (4.2.17) as a linear combination of terms of the form [a(Z ) q( )d(W )], where q( ) is a polynomial in of parity p and d(W ) is a formal distribution, which we replace by (−1) p(a+N ) q( )[a(Z ) d(W )]. This procedure permutes and in the LHS of (4.2.17). The same procedure is applied to the RHS of (4.2.17). In order to compare both sides of (4.2.17) we permute to the right of in the LHS. The first term in the RHS is computed by first evaluating [[a b] c] and then replacing by + (without permuting and ). Similarly, we do not need to permute and in the second term of the RHS of (4.2.17).
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Definition 4.10. An N K = N SUSY Lie conformal algebra is a Z/2Z-graded H -module R, endowed with a parity N mod 2 C-bilinear map R ⊗C R → L ⊗C R denoted (as before, we omit the symbol ⊗ in the -bracket) a ⊗ b → [a b] =
(−1) J N ( j|J ) a( j|J ) b,
j≥0,J
finite
where a( j|J ) b ∈ R. This data should satisfy the following axioms: (1) sesquilinearity (this is an equality in L ⊗ R): [S i a b] = −(−1) N χ i [a b],
[a S i b] = (−1)a+N S i + χ i [a b], (4.2.19)
where in the RHS of the second equation, to obtain an element of L ⊗ R, we first compute the -bracket, and then we commute S i to the right using the relations [S i , χ j ] = 2δi j λ (cf. (4.2.3)). (2) skew-symmetry (this is an equality in L ⊗ R): [a b] = −(−1)ab+N [b− −∇ a],
(4.2.20)
where the commutator on the right-hand side is computed as follows: first compute [b a] = j≥0,J j|J c j|J ∈ L ⊗R, where L is another copy of L generated by the set = (γ , η1 , . . . , η N ), where γ is an even generator, ηi are odd generators, subject to the relations [γ , ηi ] = 0, [ηi , η j ] = −2δi j γ . Then replace by −∇ − = (−T − λ, −S 1 − χ 1 , . . . , −S N − χ N ) and apply T and S i to c j|J ∈ R. (3) Jacobi identity (this is an equality in L ⊗ L ⊗ R): [a [b c]] = (−1)a N +N [[a b] + c] + (−1)(a+N )(b+N ) [b [a c]],
(4.2.21)
where [[a b] + c] is computed as follows, first compute [[a b] c] ∈ L ⊗L ⊗R, where L is another copy of L generated by the set = (ψ, υ 1 , . . . , υ N ), where ψ is an even generator, υ i are odd generators, subject to the relations [ψ, υ i ] = 0, [υ i , υ j ] = −2δi j ψ. Then replace by + = (λ + γ , θ 1 + η1 , . . . , θ N + η N ) to obtain an element of L ⊗ L ⊗ R. Similarly we compute the LHS as follows. First compute [a [b c]] to obtain an element of L ⊗ L ⊗ R. Then commute to the right of using (4.2.18) to obtain an element of L ⊗ L ⊗ R (cf. Remark 4.9). It follows that given an N K = N formal distribution Lie superalgebra (g, R), the space i . R is an N K = N SUSY Lie conformal algebra with T = ∂w and S i = DW
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Remark 4.11. We want to give an explanation for the commutation relations [S i , χ j ] = 2δi j λ appearing in sesquilinearity. For this, we give an abstract description of the axioms of a Lie conformal algebra as follows. Let H be a co-commutative Hopf superalgebra with comultiplication : H → H ⊗ H and antipode S (note that this is the case in Definition 4.10). Let R be a (left) H -module. The spaces R ⊗R and H ⊗R are canonically H modules with h → h and we consider H as a H -module with the adjoint action. An H -Lie conformal algebra structure in R is a linear map φ : R⊗R → H ⊗R satisfying the following axioms (see [BDK01]): • φ is an homomorphism of H -modules, namely, the following diagram is commutative for any h ∈ H : R⊗R
φ
h
R⊗R
/ H ⊗R h
φ
/ H ⊗R
• (Sesquilinearity) Let L h be the operator of left multiplication by h in H . The following diagram is commutative: R⊗R
φ
L h ⊗1
h⊗1
R⊗R
/ H ⊗R
φ
/ H ⊗R
• (Skew-symmetry) Let A and B be two H -modules. Let σ12 be the permutation isomorphism A ⊗ B B ⊗ A. Let μ : H ⊗ R → R be the natural multiplication coming from the H -module structure in R. The following diagram is commutative: R⊗R
φ
σ12
R⊗R
/ H ⊗R (S⊗μ)◦(⊗1)
−φ
/ H ⊗R
• (Jacobi identity) Define three morphisms R ⊗3 → H ⊗2 ⊗ R corresponding to the three terms in the Jacobi identity. First, let μ1{23} be the composition σ12 (1⊗φ)σ12
1⊗φ
R ⊗3 −−→ R ⊗ H ⊗ R −−−−−−−→ H ⊗2 ⊗ R. Similarly, we define μ2{13} to be the composition: σ12 (1⊗φ)σ12
(1⊗φ)
R ⊗3 −−−−−−−→ H ⊗ R ⊗2 −−−→ H ⊗2 ⊗ R. Finally, let ν : H ⊗ H → H be the multiplication map. We define μ{12}3 to be the composition: φ⊗1
1⊗φ
(ν⊗1⊗1)(1⊗⊗1)
R ⊗3 −−→ H ⊗ R ⊗2 −−→ H ⊗2 ⊗ R −−−−−−−−−−−→ H ⊗2 ⊗ R.
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The Jacobi identity is the following axiom: μ1{23} = μ{12}3 + μ2{13} . Note the last factor σ12 in the definition of μ1{23} . This factor is analogous to commuting to the right of in the LHS of the Jacobi identity (4.2.21) as explained in Definition 4.10. In the N K = N case, identifying: S i → −χ i ,
T → −λ,
γ → λ,
ηi → χ i ,
and, as in 3.2.6, changing the parity of R if N is odd and defining φ(a ⊗ b) = (−1)a N +N [a b], it is straightforward to check that the axioms of an N K = N SUSY Lie conformal algebra, as in Definition 4.10, get transformed into the axioms of an H -Lie conformal algebra. 4.3. Lemmas 3.2.7 and 3.2.8 hold in the N K = N setting replacing ∂W with DW in (3.2.1.5). This allows us to construct a Lie superalgebra of degree N mod 2 L(R) = ˜ ∇˜ R, ˜ and the corresponding Lie superalgebra Lie(R), from any N K = N SUSY Lie R/ conformal algebra R. In order to prove the N K = N versions of Propositions 3.2.9 and 3.2.10, we note that for J = ( j1 , . . . , jk ), we have j|J
DW = ∂wj (∂ζ j1 + ζ j1 ∂w ) . . . (∂ζ jk + ζ jk ∂w ) j+K |J \K = σ (K , J \ K )ζ K ∂W .
(4.3.1)
K ⊂J
Let now a = a ⊗ W n|I ∈ L(R) for each a ∈ R. Using (4.3.1) and (3.2.1.5) with f = W n|I , g = W k|K and letting = 0, we compute the Lie bracket (of parity N mod 2) in L(R) :
(n) j+(J \I ) × j! j≥0,J
×σ (J \I, J ∩I )σ (J ∩I, I \J )σ (J \I, I \J )σ (I J, K ) a( j|J ) b , (4.3.2) {a , b } =
(−1)a J +b(I −J )+
(J ∩I )(J ∩I −1) J (J −1) + 2 2
where (n)k = n(n −1) . . . (n −k +1). Defining a(n|I ) as the image of (−1)a I σ (I )a in Lie(R) and using (4.3.2) and Lemma 3.2.6 we compute the Lie bracket in Lie(R) : [a(n|I ) , b(k|K ) ] = (−1)(a+N −I )(N −K )
(−1) J (N −I )+I N +
(J ∩I )(J ∩I −1) J (J +1) + 2 2
×
j≥0,K
×
(n) j+(J \I ) σ (I J, N \ (K ∪ I J ))σ (I )σ (J \ I, J ∩ I )σ (J ∩ I, I \ J )× j!
× σ (J \ I, I \ J ) a( j|J ) b (n+k− j−(J \I )|K ∪(I J )) . (4.3.3)
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Substituting (3.2.2.5) in (4.3.1) we find: (J ∩I )(J ∩I −1) J (J +1) ( j|J ) + 2 +I +(N −I )(J −I ) 2 DW δ(Z , W ) = (−1) × n∈Z,I
×
(n) j+(J \I ) σ (J \ I, J ∩ I )σ (J ∩ I )× j!
× σ (I \ J, N \ I )σ (J \ I, I \ J )Z −1−n|N \I W n− j−(J \I )|I J . For each a ∈ R define the following Lie(R)-valued formal distribution: a(Z ) = Z −1−n|N \I a(n|I ) .
(4.3.4)
(4.3.5)
n∈Z,I
Using (4.3.3) and (4.3.4) we obtain the N K = N analog of Proposition 3.2.9: ( j|J ) DW δ(Z , W ) a( j|J ) b (W ). [a(Z ), b(W )] = j≥0,J
To prove the N K = N analog of Proposition 3.2.10 we need the following identity which is straightforward to check: σ (ei , N \ J )a( j|J \ei ) for ei ∈ J, (S i a)( j|J ) = for ei ∈ / J. − jσ (ei , N \ (J ∪ ei ))a( j−1|J ∪ei ) 4.4. As in 3.2.3, the N K = N formal distribution Lie superalgebra Lie(R) carries an even derivation T and N odd derivations S i (i = 1, . . . , N ), given by:
T a( j|J ) = − ja( j−1|J ) ,
σ (N \ J, ei )a( j,J \ei ) ei ∈ J, i S a( j,J ) = jσ (N \ (J ∪ ei ), ei )a( j−1|J ∪ei ) ei ∈ / J. It follows easily that the formal distributions (4.3.5) satisfy: T a(Z ) = ∂z a(Z ),
S i a(Z ) = (∂θ i − θ i ∂z )a(Z ), i = 1, . . . , N ,
and therefore (Lie(R), R) is a regular N K = N formal distribution Lie superalgebra. We define the normally ordered product of fields by the same formula (3.2.4.3) as in the N W = N case, and all the other products by using the derivations DW instead of ∂W . It is easy to check that the N K = N analog of the Cauchy formulae (3.2.4.2) hold with DW instead of ∂W . Lemma 3.2.12 is still valid in the N K = N setting (recall that (Z − W )−1|N is the same in both situations) therefore Eq. (3.2.6.3) is still valid in this context. The N K = N version of Proposition 3.2.13 is: Proposition 4.12. The following identities analogous to sesquilinearity for all pairs ( j|J ) are true:
i DW a(W ) b(W ) = −(−1) J σ (ei , J )a(W )( j|J \ei ) b(W )+ ( j|J ) + jσ (ei , J )a(W )( j−1|J ∪ei ) b(W ) , (4.4.1)
i N −J i DW DW a(W )( j|J ) b(W ) = (−1) a(W ) b(W )+ ( j|J ) a i +(−1) a(W )( j|J ) DW b(W ) .
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Proof. According to Lemma 4.3 we have: res Z i z,w (Z − W ) j|J D iZ a(Z )b(W ) = = −(−1) J res Z D iZ i z,w (Z − W ) j|J a(Z )b(W ) = = −(−1) J res Z σ (ei , J \ ei )i z,w (Z − W ) j|J \ei + + jσ (ei , J )i z,w (Z − W ) j−1|J ∪ei a(Z )b(W ).
(4.4.2)
Similarly we have:
−(−1)(a+1)b res Z i w,z (Z − W ) j|J b(W ) D iZ a(Z ) = = (−1)ab+J res Z D iZ i w,z (Z − W ) j|J b(W )a(Z ) = = (−1)ab+J res Z σ (ei , J \ ei )i w,z (Z − W ) j|J \ei + + jσ (ei , J )i w,z (Z − W ) j−1|J ∪ei b(W )a(Z ).
(4.4.3)
Adding (4.4.2) and (4.4.3) we obtain:
i DW a(W ) b(W ) = −(−1) J σ (ei , J \ ei )a(W )( j|J \ei ) b(W )+ ( j|J )
+ jσ (ei , J )a(W )( j−1|J ∪ei ) b(W ) .
i is a derivation of all ( j|J )-products is proved in the same way as in The fact that DW (3.2.6.7):
i i DW a(W )( j|J ) b(W ) = DW res Z i z,w (Z − W ) j|J a(Z )b(W )− − (−1)ab i w,z (Z − W ) j|J b(W )a(Z ) = (−1) N res Z −σ (ei , J \ ei )i z,w (Z − W ) j|J \ei − jσ (ei , J )i z,w (Z − W ) j−1|J ∪ei ×
i ×a(Z )b(W ) + (−1) J +a i z,w (Z − W ) j|J a(Z )DW b(W )+
+(−1)ab σ (ei , J \ ei )i w,z (Z − W ) j|J \ei + jσ (ei , J )i w,z (Z − W ) j−1|J ∪ei b(W )a(Z )− i −(−1)ab+J i w,z (Z − W ) j|J DW b(W )a(Z ) = = −(−1) N σ (ei , J \ ei )a(W )( j|J \ei ) b(W ) − (−1) N jσ (ei , J )a(W )( j−1|J ∪ei ) b(W )+ i + (−1) N +J +a a(W )( j|J ) DW b(W ) = N −J i a i DW a(W ) b(W ) + (−1) a(W )( j|J ) DW b(W ) . = (−1) ( j|J )
4.5. Even though the general Proposition 3.2.14 is no longer valid in the N K = N setting, we easily check that its proof works in the particular case ( j|J ) = (−1|N ). Therefore, the non-commutative Wick formula (3.2.6.12) is still valid in the N K = N case. The N K = N version of Dong’s Lemma 3.2.16 is proved as in the usual vertex algebra case.
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Definition 4.13. An N K = N SUSY vertex algebra is the data consisting of a super vector space V , an even vector |0 ∈ V , N odd endomorphisms S i and a parity preserving linear map Y from V to End(V )-valued superfields a → Y (a, Z ), satisfying the following axioms: • vacuum axioms: Y (a, Z )|0 = a + O(Z ), S i |0 = 0, • translation invariance:
i = 1, . . . , N ,
[S i , Y (a, Z )] = D¯ iZ Y (a, Z ),
where D¯ iZ = ∂θ i − θ i ∂z , • locality: (z − w)n [Y (a, Z ), Y (b, W )] = 0
for some n ∈ Z+ .
4.6. We define the ( j|J )-products for a N K = N SUSY vertex algebra, as in the N W = N case, by (3.3.1.2). As in 3.3.1 we see easily that the vacuum axioms may be formulated as (3.3.1.4) and translation invariance is equivalent to: σ (N \ J, ei )a( j,J \ei ) ei ∈ J, [S i , a( j,J ) ] = (4.6.1) jσ (N \ (J ∪ ei ), ei )a( j−1|J ∪ei ) ei ∈ / J. 4.7. It follows easily from (4.6.1) and the vacuum axioms that [S i , S j ] = 2δi, j T,
[S i , T ] = 0,
where T is an even operator satisfying: [T, a( j|J ) ] = − ja( j−1|J ) or equivalently:
∀a, ( j|J ),
[T, Y (a, Z )] = ∂z Y (a, Z ).
With these results we can prove the N K = N version of Theorem 3.3.2. Theorem 4.14. Let U be a vector superspace and V a space of pairwise local End(U )valued fields such that V contains the constant field Id, it is invariant under the derivations D iZ = ∂θ i + θ i ∂z and closed under all ( j|J )th products. Then V is an N K = N SUSY vertex algebra with vacuum vector Id, the translation operators are S i a(Z ) = D iZ a(Z ), the ( j|J ) product is the one for distributions multiplied by σ (J ). Proof. The proof goes like the proof of 3.3.2. To check translation invariance we see that D iZ 1 = 0 and that
σ (J )D iZ a(Z )( j|J ) b(Z ) − (−1)a+N −J a(Z )( j|J ) D iZ b(Z ) = b(Z ). = (−1) N −J σ (J ) D iZ a(Z ) ( j|J )
But in view of (4.4.1) this is:
− (−1) N σ (J ) σ (ei , J )a(Z )( j|J \ei ) b(Z ) + jσ (ei , J )a(W )( j−1|J ∪ei ) b(Z ) = = σ (N \J, ei )σ (J \ei )a(Z )( j|J \ei ) b(Z )+ jσ (N \(J ∪ei ), ei )σ (J ∪ei )a(Z )( j−1|J ∪ei ) b(Z ), proving Eq. (4.6.1). Locality is proved in the same way as in 3.3.2.
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Lemma 3.3.4 is still valid for N K = N SUSY vertex algebras. Its proof parallels the proof for N W = N SUSY vertex algebras. Lemma 3.3.5, on the existence and uniqueness of solutions to a system of differential equations, is straightforward to generalize to the N K = N setting. The proof of Proposition 3.3.6 in this context is more subtle: Proposition 4.15. Let V be a N K = N SUSY vertex algebra. Then for every a, b ∈ V we have: (1) Y (a, Z )|0 = exp(Z ∇)a, (2) exp(Z ∇)Y (a, W ) exp(−Z ∇) = i w,z Y (a, W + Z ), (3) Y (a, Z )( j|J ) Y (b, Z )|0 = σ (J )Y (a( j|J ) b, Z )|0, where ∇ = (T, S 1 , . . . , S N ), Z ∇ = zT + θ i S i , and we define W + Z = W − (−Z ) = (z + w + ζ i θ i , θ j + ζ j )5 . Proof. As in the proof of Proposition 3.3.6 we note that both sides of (1) and (3) are elements of V [[Z ]], whereas both sides of (2) are elements of End(V )[[W, W −1 ]][[Z ]]. By evaluating at Z = 0 we get equalities in all three cases. Indeed (1) and (2) are trivial, and (3) follows from the N K = N version of Lemma 3.3.4. We need to show that both sides in each equation satisfy the same system of differential equations. (1) Similarly to the proof of Lemma 4.6, we expand: i i k ( iθ S ) i i ¯ D Z exp(Z ∇) = ∂θ i − θ ∂z exp(zT ) k! k≥0 i
= (S + T θ ) exp(Z ∇) − θ T exp(Z ∇) = S i exp(Z ∇), (4.7.1) i
i
from where the RHS X (Z ) of (1) satisfies the system of differential equations: D¯ iZ X (Z ) = S i X (Z ). Similarly by translation invariance we have for the LHS of (1): D¯ iZ Y (a, Z )|0 = [S i , Y (a, Z )]|0 = S i Y (a, Z )|0. We also point out that a computation similar to (4.7.1) shows that D¯ iZ exp(−Z ∇) = − exp(−Z ∇)S i ,
(4.7.2)
which is not entirely obvious since S i does not commute with the exponential. (2) By translation invariance we have: D¯ iZ Y (a, W + Z ) = −ζ i ∂w+z+ ζ i θ i + ∂ζ i +θ i − θ i ∂w+z+ ζ i θ i Y (a, W + Z ) = i i = D¯ W +Z Y (a, W + Z ) = [S , Y (a, W + Z )].
On the other hand, letting Y (Z ) = e Z ∇ Y (a, W )e−Z ∇ we have (cf. (4.7.2)): D¯ iZ Y (Z ) = S i Y (Z ) − (−1)a Y (Z )S i = [S i , Y (Z )]. 5 Note that Z + W = W + Z
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(3) For the RHS we have by translation invariance and the vacuum axioms: S i Y (a( j|J ) b, Z )|0 = [S i , Y (a( j|J ) b, Z )]|0 = D¯ iZ Y (a( j|J ) b, Z )|0. To prove that the LHS satisfies the same differential equation, we proceed exactly in the same way as in the proof of Proposition 3.3.6. We only need the fact that D¯ iZ is a derivation of all ( j|J )-products. But ∂z = (D iZ )2 is a derivation since: ∂z a(Z )( j|J ) b(Z ) = (−1) N −J D iZ (D iZ a(Z ))( j|J ) b(Z ) + (−1)a a( j|J ) D iZ b(Z ) = (∂z a(Z ))( j|J ) b(Z ) + (−1)a+1 (D iZ a(Z ))( j|J ) D iZ b(Z ) + +(−1)a (D iZ a(Z ))( j|J ) D iZ b(Z ) + a(Z )( j|J ) ∂z b(Z ), therefore D¯ iZ = D iZ − 2θ i ∂z is a derivation of all ( j|J )-products.
The uniqueness Proposition 3.3.7 is still valid in the N K = N setting, As its corollary, we obtain an analogous version of Theorem 3.3.8, namely Theorem 4.16. On an N K = N SUSY vertex algebra the following identities hold (1) (2) (3) (4)
Y (a( j|J ) b, Z ) = σ (J )Y (a, Z )( j|J ) Y (b, Z ) . Y (a(−1|N ) b, Z ) =: Y (a, Z )Y (b, Z ) :. Y (S i a, Z ) = D iZ Y (a, Z ). We have the following OPE formula: ( j|J ) σ (J )(DW δ(Z , W ))Y (a( j|J ) b, W ). [Y (a, Z ), Y (b, W )] = ( j|J ): j≥0
Remark 4.17. Note that as a consequence of (3) we obtain [S i , Y (a, Z )] = Y (S i a, Z ), in contrast to the N W = N and, in particular, the ordinary vertex algebra case. Corollary 4.18. (S i a)( j|J ) = σ (ei , N \ J )a( j|J \ei ) − jσ (ei , N \ (J ∪ ei ))a( j−1|J ∪ei ) σ (ei , N \ J )a( j|J \ei ) for ei ∈ J = − jσ (ei , N \ (J ∪ ei ))a( j−1|J ∪ei ) for ei ∈ / J,
S i a( j|J ) b = (−1) N −J (S i a)( j|J ) b + (−1)a a( j|J ) S i b . 4.8. In order to prove the N K = N version of the associativity formulas (3.3.2.3) and (3.3.2.4), we proceed as in 3.3.2, by taking the generating series of 4.16 (1) and using the following N K = N version of the Taylor expansion. For a formal power series a(Z ) ∈ C[[Z ]] we have: a(W + Z ) =
( j|J ): j≥0
(−1)
J (J −1) 2
W j|J j|J D Z a(Z ) = e W D Z a(Z ). j!
(4.8.1)
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Indeed, the usual Taylor expansion is: j|J J (J −1) ∂ W (−1) 2 a(W + Z ) = a w + z + ζ i θi, ζ j + θ j = a(W + Z )|W =0 . j!
In this case:
1|0
1|0
∂W a(W + Z )|W =0 = D Z a(Z ), 0|i
∂W a(W + Z )|W =0 = (θ i ∂z + ∂θ i )a(Z ) = D iZ a(Z ), proving (4.8.1). Also, according to our prescription to add coordinates we see that (X − Z ) − W = X − (W + Z ) = X − (W − (−Z )), and note that Eq. (3.3.1.7) is still valid in the N K = N setting. The proof in 3.3.2 generalizes now easily. Similarly, we obtain as a corollary, the N K = N version of the Cousin property 3.3.11. The proofs for skew-symmetry in Theorem 3.3.12 and quasi-commutativity for the normally ordered product as in 3.3.3 carry over verbatim to the N K = N case. 4.9. Defining the Fourier Transform as in 3.3.4 we obtain an analogous result to Theorem 3.3.13, namely an N K = N SUSY vertex algebra gives rise to an N K = N SUSY Lie conformal algebra. The N K = N version of quasi-associativity for the normally ordered product is the same and is proved in the same way as Theorem 3.3.14. As in the N W = N case, we have the following equivalent definition of N K = N SUSY vertex algebras: Definition 4.19. An N K = N SUSY vertex algebra is a tuple (V, T, S i , [· ·], |0, ::), i = 1, . . . , N , where • (V, T, S i , [· ·]) is an N K = N SUSY Lie conformal algebra, • (V, |0, T, S i , ::) is a unital quasicommutative quasiassociative differential superalgebra (i.e. T is an even derivation of :: and S i are odd derivations of ::), • the -bracket and the product :: are related by the non-commutative Wick formula (3.2.6.12). 4.10. The definition of an N K = N Poisson SUSY vertex algebra is straightforward to generalize. Similarly, the N K = N version of Borcherds identity in Theorem 3.3.17 and the N K = N version of the commutator formula in Proposition 3.3.18, are the same with ∂W replaced by DW and are proved in the same way as for N W = N SUSY vertex algebras. Following the argument in Remark 3.3.19 and using (4.3.1), we obtain the formula for the commutator of the Fourier coefficients of the End(V )-valued fields of the N K = N SUSY vertex algebra: it is equal to the RHS of (4.3.3), multiplied by σ (J ). The rest of Sect. 3.3 carries over to the N K = N case with minor modifications, in particular we define tensor products of N K = N SUSY vertex algebras as in 3.3.5 and we have an existence theorem as in 3.3.20 that we restate here: Theorem 4.20 (Existence of N K = N SUSY vertex algebras). Let V be a vector space, |0 ∈ V an even vector, T an even endomorphism of V and S i , i = 1, . . . , N odd endomorphisms of V , satisfying [S i , S j ] = 2δi, j T . Suppose moreover that S i |0 = 0. Let F be a family of fields a α (Z ) = ( j|J ) Z −1− j|N \J a(αj|J ) , indexed by α ∈ A, and such that
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a α (Z )|0| Z =0 = a α ∈ V , [S i , a α (Z )] = D¯ iZ a α (Z ), all pairs (a α (Z ), a β (Z )) are local, the vectors a(αjss |Js ) . . . a(αj11 |J1 ) |0 span V .
Then the formula Y (a(αjss |Js ) . . . a(αj11 |J1 ) |0, Z ) =
= σ (Ji )a αs (Z )( js |Js ) . . . a(αj22 |J2 ) (Z ) a(αj11 |J1 ) (Z ) Id . . . defines a structure of an N K = N SUSY vertex algebra on V , with vacuum vector |0, translation operators S i and such that Y (a α , Z ) = a α (Z ), α ∈ A. Such a structure is unique. 4.11. The results in Sect. 3.4 generalize to this context without difficulty. In particular, we obtain the universal enveloping N K = N SUSY vertex algebra of an N K = N SUSY Lie conformal algebra. In the same way as in Theorem 3.4.1, we obtain: Theorem 4.21. Let V be an N K = N SUSY vertex algebra. Let A = C[X, X −1 ], define L(V ) to be the quotient of V˜ = A ⊗C V by the span of vectors of the form: S i a ⊗ f (X ) + (−1)a a ⊗ D iX f (X ), and let L (V ) be its completion with respect to the natural topology in A . Then L(V ) (resp. L (V )) carries a natural Lie superalgebra of degree N mod 2 structure. Let Lie(V ) (resp. Lie (V )) be the corresponding Lie superalgebra. The map ϕ : Lie(V ) → End(V ) (resp. ϕ : Lie (V ) → End(V )), given by formula (3.4.1.2), is a Lie superalgebra homomorphism. 5. Examples Example 5.1 (V (W N ) series). Let X = (x, ξ 1 , . . . , ξ N ), where x is even and ξ i are odd anticommuting variables commuting with x. Consider the Lie algebra g = W (1|N ) of derivations of C[X, X −1 ]. It is spanned by elements of the form X j|J ∂x and X j|J ∂ξ i (cf. Example 2.11). Define the following g-valued formal distributions: L(Z ) = −δ(Z , X )∂x ,
Q i (Z ) = −δ(Z , X )∂ξ i , i = 1, . . . , N .
(5.1)
A straightforward computation shows that these distributions satisfy the following commutation relations: [L(Z ), L(W )] = δ(Z , W )∂w L(W ) + 2 (∂w δ(Z , W )) L(W ),
[L(Z ), Q i (W )] = δ(Z , W )∂w Q i (W ) + ∂ζ i δ(Z , W ) L(W )+ + (∂w δ(Z , W )) Q i (W ),
[Q (Z ), Q (W )] = δ(Z , W )∂ζ i Q (W ) + (−1) ∂ζ i δ(Z , W ) Q j (W )−
− (−1) N ∂ζ j δ(Z , W ) Q i (W ). i
j
j
N
(5.2)
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In particular, the distributions (5.1) are pairwise local. Let F be the family of g-valued formal distributions j|J j|J F = ∂ Z L(Z ), ∂ Z Q i (Z ) j ≥ 0, J ⊂ {1, . . . , N }, i = 1, . . . , N . Then (g, F ) is an N W = N SUSY formal distribution Lie superalgebra. Let W N be the corresponding N W = N SUSY Lie conformal algebra. It is generated as a C[T, S i ]-module by a vector L of parity N mod 2 and N -vectors Q i , i = 1, . . . , N , of parity N + 1 mod 2 satisfying the following -brackets (which follow from (5.2)) [L L] = (T + 2λ)L , [Q i Q j ] = (S i + χ i )Q j − χ j Q i , [L Q i ] = (T + λ)Q i + (−1) N χ i L .
(5.3)
When N = 0, this is the centerless Virasoro conformal algebra, which admits a central extension defined by: λ3 [L λ L] = (T + 2λ)L + C, 12 where C is even, central, and satisfies T C = 0. Translating the formulas in [FK02], it follows that when N = 1, W1 admits a central extension of the form: λχ C, [L L] = (T + 2λ)L , [Q Q] = S Q + 3 (5.4) λ2 [L Q] = (T + λ)Q − χ L + C, 6 where C is even, central, and satisfies T C = SC = 0. When N = 2, W2 admits a central extension given by: [L L] = (T + 2λ)L ,
[L Q i ] = (T + λ)Q i + χ i L ,
λ C, 6 where C is as above. It follows from [KvdL89, FK02] that these algebras do not admit central extensions for N ≥ 3. If N ≤ 2, we let V (W N ) be the universal enveloping N W = N SUSY vertex algebra of the central extension of W N as given by Theorem 3.4.2, and let V (W N )c be the quotient of V (W N ) by the ideal (C − c|0)(−1|N ) V (W N ), where c ∈ C is called the central charge. When N = 1, expanding the superfields as 1 Q(Z ) = −J (z) + θ G + (z), L(Z ) = G − (z) + θ L(z) + ∂z J (z) , 2 [Q i Q i ] = S i Q i ,
[Q 1 Q 2 ] = (S 1 + χ 1 )Q 2 − χ 2 Q 1 +
we check that the fields L , J, G ± satisfy the commutation relations of the N = 2 vertex algebra as defined in Example 2.7. When N ≥ 3, we let V (W N ) be the universal enveloping N W = N SUSY vertex algebra of W N . It follows from the definitions that Lie(W N ) = W (1|N ), the Lie superalgebra of derivations of C[X, X −1 ]. Also, Lie(W N )≤ = W (1|N )≤ is the Lie superalgebra of derivations of C[X ]. Denote by W (1|N )< = Lie(W N )< ⊂ Lie(W N )≤ the Lie superalgebra of vector fields vanishing at the origin; it is spanned by vectors of the form X j|J ∂x and X j|J ∂ξ i , with j + J > 0.
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Definition 5.2. An N W = N SUSY vertex algebra V is called conformal if there exists N + 1 vectors ν, τ 1 , . . . , τ N in V such that their associated superfields L(Z ) = Y (ν, Z ) and Q i (Z ) = Y (τ i , Z ) satisfy (5.3) (or possibly a central extension) and moreover: i = Si , • ν(0|0) = T , τ(0|0) • the operator ν(1|0) acts diagonally with eigenvalues bounded below and with finite dimensional eigenspaces.
If moreover, the action of Lie(W N )< on V can be exponentiated to the group of automorphisms of the disk D 1|N , we will say that V is strongly conformal. This amounts to the following extra condition: N i • the operators ν(1|0) and i=1 σ (ei )τ(0|e have integer eigenvalues. i) If a ∈ V is an eigenvector of the operator ν(1|0) of eigenvalue , we say that a has conformal weight . This happens if a satisfies [L a] = (T + λ)a + O(λ2 ) + O(χ 1 , . . . , χ N ). If, moreover, a satisfies [L a] = (T + λ)a we say that a is primary. As in the ordinary vertex algebra case, the conformal weight (a) is an important book-keeping device: (T a) = (a) + 1, (S i a) = (a), (a( j|J ) b) = (a) + (b) − j − 1. (Note that (: ab :) = (a) + (b).) Furthermore, letting (χ i ) = 0 and (λ) = 1, all terms in [a b] have conformal weight (a) + (b) − 1. Remark 5.3. It is clear from this definition that the N W = N SUSY vertex algebra V (W N )c defined in Example 5.1 is strongly conformal. Example 5.4 (Free Fields). As an example of a strongly conformal N W = N SUSY vertex algebra, we will compute explicitly the free fields case, namely, let α, C be even vectors and let ϕ be an odd vector. Consider the N W = N SUSY Lie conformal algebra generated by these three vectors, where C is central and annihilated by ∇, and the other commutation relations are: [α ϕ] = C. Let F˜ N be its universal enveloping N W = N SUSY vertex algebra and FN its quotient by the ideal (C − |0)(−1|N ) F˜ N . Expanding the superfields α(Z ) = a(z) + θ ψ(z),
ϕ(Z ) = φ(z) + θ b(z),
we find that the fields a, b, ψ and φ generate the well known bc − βγ -system, namely, the non-trivial λ-brackets are (up to skew-symmetry): [bλ a] = [ψλ φ] = 1. When N = 1, this SUSY vertex algebra admits a N W = 1 strongly conformal structure with: ν = α(−2|1) ϕ(−1|1) |0, τ = −α(−1|0) ϕ(−1|1) |0, and central charge c = 3. The associated fields L = Y (ν, Z ) and Q = Y (τ, Z ) are: L =: (T α)ϕ :
Q = − : (Sα)ϕ : .
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In order to check the commutation relations (5.4) we use the non-commutative Wick formula (3.2.6.12) to find: [α L] = T α, [ϕ L] = λϕ, [α Q] = Sα, [ϕ Q] = −χ ϕ. And now by skew-symmetry and sesquilinearity we obtain: [L α] = T α, [L T α] = (T + λ)T α, [Q α] = Sα, [Q Sα] = −χ Sα.
[L ϕ] = (λ + T )ϕ, [L Sα] = (S + χ )T α, [Q ϕ] = (S + χ )ϕ,
(5.5) (5.6) (5.7) (5.8)
Formula (5.5) says that α and ϕ are primary fields of conformal weight 0 and 1 respectively. With these formulas and using again the Wick formula (3.2.6.12) we obtain [L L] = [L : (T α)ϕ :] =: ((T + λ)T α)ϕ : + : T α(λ + T )ϕ := = 2λL+ : (T (T α))ϕ : + : T αT ϕ := (T + 2λ)L , since the integral term obviously vanishes. For the other commutation relations we compute: [Q Q] = −[Q : (Sα)ϕ :] = χ : (Sα)ϕ : + : Sα(χ + S)ϕ : + [χ Sα ϕ]d = 0
= : SαSϕ : +
(η − χ )ηd = S Q + λχ .
0
Finally for the last commutator we find: [L Q] = −[L : (Sα)ϕ :] = − : ((S + χ )T α)ϕ : − : Sα(λ + T )ϕ : − − [(S + χ )T α ϕ]d = T Q − χ : (T α)ϕ : +λQ + (η − χ )γ d = 0
0
λ2 = (T + λ)Q − χ L + . 2 According to (5.4) this is a conformal N W = 1 SUSY vertex algebra with central charge 3. It is easy to check that this SUSY vertex algebra is indeed strongly conformal. Example 5.5 (V (K N ) series). Consider now the Lie subsuperalgebra K (1|N ) ⊂ W (1|N ) consisting of those derivations of C[X, X −1 ] preserving the form ω = dx +
N
ξ i dξ i ,
i=1
up to multiplication by a function (cf. Example 2.12). Define the following g-valued formal distribution: G(Z ) = −2δ(Z , X )∂x − (−1) N
N i=1
D iX δ(Z , X ) D iX .
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It follows from (2.6.6) that its Z -coefficients form a basis of K (1|N ). A straightforward computation shows that this formal distribution satisfies the following commutation relation: [G(Z ), G(W )] = 2δ(Z , W )∂w G(W ) + (4 − N ) (∂w δ(Z , W )) G(W ) + N i i DW +(−1) N δ(Z , W ) DW G(W ), i=1
in particular, the pair of g-valued formal distributions (G(Z ), G(Z )) is local. Letting j|J F = ∂ Z G(Z ), j ≥ 0, J ⊂ {1, . . . , N } , we obtain an N K = N formal distribution Lie superalgebra (g, F ). Let K N be the associated N K = N SUSY Lie conformal algebra. It is generated as a free H -module by a vector G of parity N mod 2 satisfying the following -bracket (for the definition of the algebra H , see 4.2)
N i i [G G] = 2T + (4 − N )λ + χ S G. (5.9) i=1
When N ≤ 3, K N admits a non-trivial central extension, obtained by adding the term λ3−N χ N C to the RHS of (5.9), where C is even, central and satisfies T C = S i C = 0, 3 cf. [FK02]. When N = 4, K4 admits a central extension, obtained by adding the term λC to the RHS of (5.9). It follows from [KvdL89, FK02] that K N does not admit central extensions when N > 4. When N ≤ 4, we let V (K N ) be the universal enveloping N K = N SUSY vertex algebra of the central extension of K N and define V (K N )c to be its quotient by the ideal (C − c|0)(−1|N ) V (K N ), where c ∈ C is called the central charge. When N ≥ 5, we let V (K N ) be the universal enveloping N K = N SUSY vertex algebra of K N . In the case N = 1, if we expand the corresponding superfield as G(z, θ ) = G(z) + 2θ L(z), we find that the fields G(z) and L(z) generate a Neveu Schwarz vertex algebra of central charge c as in Example 2.4. When N = 2 expanding the corresponding superfield as (cf. 2.6.4) √ G(z, θ 1 , θ 2 ) = −1J (z) + θ 1 G (2) (z) − θ 2 G (1) (z) + 2θ 1 θ 2 L(z). We find that the corresponding fields J, L , G ± satisfy the commutation relations of the N = 2 vertex algebra as in Example 2.7. When N = 4 the corresponding N K = 4 SUSY vertex algebra is not simple. Indeed the SUSY Lie conformal superalgebra K4 ⊂ K4 generated by S i G, i = 1, . . . , 4 is an ideal. The central extension of K4 described above restricts to a central extension of K4 whose cocycle is given by: α1 (S i G, S j G) = −χ i χ j C1 .
(5.10)
This SUSY Lie conformal algebra admits another central extension given by (cf. [FK02]): α2 (S i G, S j G) = χ 1 χ 2 χ 3 χ 4 C2 . We let
V (K4 )c1 ,c2
(5.11)
be the corresponding N K = 4 SUSY vertex algebra with Ci = ci .
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Note that Lie(K N ) = K (1|N ) by definition, while Lie(K N )≤ = K (1|N )≤ is the Lie superalgebra of regular vector fields preserving ω up to multiplication by a function. We will denote by K (1|N )< = Lie(K N )< ⊂ Lie(K N )≤ the Lie superalgebra of regular vector fields, preserving ω up to multiplication by a function, and vanishing at the origin. A field G satisfying the commutation relations (5.9) with a central extension, will be called a super Virasoro field. Definition 5.6. Let N ≤ 4, an N K = N SUSY vertex algebra V is called conformal if there exists a vector τ ∈ V (called the conformal vector) such that the corresponding field G(Z ) = Y (τ, Z ) satisfies (5.9) with a central extension, and moreover • τ(0|0) = 2T , τ(0|ei ) = σ (N \ ei , ei )S i , • the operator τ(1|0) acts diagonally with eigenvalues bounded below and finite dimensional eigenspaces. If moreover, the representation of Lie(K N )< can be exponentiated to the group of automorphisms of the disk D 1|N preserving the SUSY structure ω = dx +
N
ξ i dξ i ,
i=1
we will say that V is strongly conformal. This amounts to the extra condition √ • the operator τ(1|0) has integer eigenvalues and, if N = 2, the operator −1τ(0|N ) has integer eigenvalues. If a vector a in a conformal N K = N SUSY vertex algebra V is an eigenvector of τ(1|0) with eigenvalue 2, we say that a has conformal weight . This happens iff a satisfies
N i i χ S a + O( 2 ), [G a] = 2T + 2λ + i=1
O( 2 )
denotes a polynomial in with vanishing constant and linear terms. If, where moreover, a satisfies
N i i [G a] = 2T + 2λ + χ S a, i=1
we say that a is primary. For example, formula (5.9) with a central extension, says that G has conformal weight 2 − N /2, and it is primary if the central extension is trivial. As in the N W = N case, the conformal weight is an important book-keeping device 1 (T a) = (a) + 1, (S i a) = (a) + , (a( j|J ) b) 2 N − J . = (a) + (b) − j − 1 + 2 (Note that (: ab :) = (a) + (b).) Furthermore, letting (λ) = 1 and (χ i ) = 1/2, all terms in [a b] have conformal weight (a) + (b) − 1 + N /2. Remark 5.7. The N K = N SUSY vertex algebra V (K N )c defined in Example 5.5 is strongly conformal when N < 4, and V (K4 )c1 ,c2 is strongly conformal as well.
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Example 5.8 (Free fields). The well-known boson-fermion system is an N K = N vertex algebra generated by one superfield. Let be a vector of parity (−1) N , C an even vector, and define a N K = N SUSY Lie conformal algebra generated by and C where C is central, satisfies T C = S i C = 0 and the remaining commutation relations are: [ ] = 1|N C, when N is even, and [ ] = 0|N C, when N is odd. Skew symmetry is clear and the Jacobi identity is obvious since all triple brackets vanish. We let B˜ N be the corresponding universal enveloping N K = N SUSY vertex algebra, and let B N be its quotient by the ideal (C − |0)(−1|N ) B˜ N . To show an application of the above formalism, as well as the subtleties involved in calculations, we will show explicitly that the N K = 1 SUSY vertex algebra B1 is conformal, the corresponding super Virasoro field being G =: (S ) : +mT ,
m ∈ C.
Indeed, from sesquilinearity (4.2.19) and skew-symmetry (4.2.20) we find [ S ] = (S + χ )χ = λ,
[S ] = −λ,
where we used [S, χ ] = 2λ and χ 2 = −λ. Using sesquilinearity once more we get: [S S ] = χ λ,
[ T ] = λχ .
Now we can use the N K = 1 version of the non-commutative Wick formula (3.2.6.12) to find: [ G] = (λ + χ S) + mλχ , [S G] = λ(χ − S) − mλ2 , [T G] = −λ(λ + χ S) − mλ2 χ , where we note that all the integral terms vanish. Using skew-symmetry again we get: [G ] = (λ + 2T + χ S) − mλχ , [G S ] = (λ + T )(χ + 2S) − mλ2 [G T ] = (T + λ)(λ + 2T + χ S) − mλ2 χ .
(5.12)
With these formulas we can use again (3.2.6.12) to get: [G G] =: (λ + T )(χ + 2S) − mλ2 : + + : S (λ+2T +χ S) −mλχ ) : +m(T +λ)(λ+2T +χ S) +(1/2−m 2 )λ2 χ , where again the integral term is easily seen to be 1/2λ2 χ . Note that from quasi-commutativity of the normally ordered product we find :
:= 0, from where the expression above reduces to: 2λ : (S ) : +χ : (T ) : +2 : (S 3 ) : −mλ2 + : S ((λ + 2T + χ S) − mλχ ) : +m(T +λ)(λ+2T +χ S) +(1/2−m 2 )λ2 χ .
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Expanding this expression and after a simple cancellation we find [G G] = (2T + 3λ + χ S) G + (1/2 − m 2 )λ2 χ . Therefore B1 is a strongly conformal N K = 1 SUSY vertex algebra with central charge 3/2 − 3m 2 . Note that, by (5.12), has conformal weight 1/2 (but it is not primary if m = 0). Expanding the superfield
(Z ) = ϕ(z) + θ α(z), we find easily that [ϕλ ϕ] = 1,
[αλ α] = λ,
hence the name boson-fermion system. Example 5.9 (Super Currents). Let g be a simple or abelian Lie superalgebra with a nondegenerate invariant supersymmetric bilinear form ( , ). If N is even, then we define a SUSY Lie conformal algebra (either N K = N or N W = N ) generated by g with commutation relations: [a b] = [a, b] + (k + h ∨ )(a, b)λ
∀a, b ∈ g,
where 2h ∨ is the eigenvalue of the Casimir operator on g. When N is odd we let g be g with reversed parity, and for each element a ∈ g we let a¯ be the same element though in g. In this case we define a SUSY Lie conformal algebra generated by g with commutation relations:
¯ = (−1) [a¯ b]
a
∨
[a, b] + (k + h )(a, b)
N
χ
i
.
i=1
We let V k (g) be the associated universal enveloping SUSY vertex algebra6 , either the N K = N or the N W = N vertex algebra; the choice will be clear in each context, as well as the value of N . When N = 1, the corresponding N K = 1 SUSY vertex algebra is strongly conformal, the corresponding conformal vector is 1 1 ai i ¯i i j r i ¯ j ¯r ¯ (−1) : (S a¯ )b : + τ= ([a , a ], a ) : b : b b :: , k + h∨ 3(k + h ∨ ) where {a i } and {bi } are dual bases for g with respect to ( , ). This is known as the Kac-Todorov construction [KT85]. The super Virasoro field Y (τ, Z ) has central charge c=
ksdimg sdimg + , k + h∨ 2
and the fields a¯ ∈ g¯ have conformal weight 1/2. 6 Here as before, we are considering a central extension of a SUSY Lie conformal algebra, and then we identify the central element with a multiple of the vacuum vector in the universal enveloping SUSY vertex algebra.
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Example 5.10 (N = 2 vertex algebra). As a vertex algebra it is generated by 4 fields (cf. Example 2.7). As we have seen in Example 5.5, this is a N K = 2 SUSY vertex algebra generated by one field G. On the other hand, the N = 2 vertex algebra admits an embedding of the N = 1 vertex algebra. Therefore we can view the N = 2 vertex algebra as an N K = 1 SUSY vertex algebra. As such, this algebra is generated by two superfields G and J , where G is a super Virasoro field of central charge c and J is even, primary of conformal weight 1. The remaining -bracket is given by: c [J J ] = G + λχ . 3 This is computed using the decomposition Lemma 4.4. Example 5.11 (N = 4 vertex algebra). As a vertex algebra it is generated by 8 fields: a Virasoro field, three currents (for the Lie algebra sl2 ) and four fermions [Kac96, p. 187]. This vertex algebra admits an embedding of the Neveu Schwarz vertex algebra, therefore we can consider a N K = 1 SUSY vertex algebra structure on it. As a N K = 1 vertex algebra, it is of rank 3|1, generated by an N = 1 conformal vector G with central charge c and three even vectors J i , i = 1, 2, 3. Each pair (G, J i ) generates an N = 2 vertex algebra, viewed as an N K = 1 SUSY vertex algebra, as in the previous example. The remaining commutation relations are: √ [J i J j ] = −1εi jk (S + 2χ )J k , i = j, where ε is the totally antisymmetric tensor. This vertex algebra is the universal enveloping vertex algebra of the central extension of the superconformal Lie algebra S(1|2; 0) (cf. [FK02]). Example 5.12 (bc − βγ system). This is a N K = 1 SUSY vertex algebra generated by n even fields B i and n odd fields i . The only non-vanishing -brackets (up to skew-symmetry) are: [B i j ] = δi j . This SUSY vertex algebra is strongly conformal with super Virasoro field n G= : (S B i )(S i ) : + : (T B i ) i : , i=1
and central charge 3n. The fields B i (resp. i ) are primary of conformal weight 0 (resp. 1/2). Let σisj , s = 1, 2, 3, be three n × n matrices satisfying √ (σ s )2 = Id . σ i σ j = −1εi jk σ k , The fields Ji =
n
σ ijk : S B j k :, i = 1, 2, 3,
j,k=1
together with G generate an N = 4 vertex algebra as in the previous example7 (cf. [BZHS06]). 7 Note however that these fields J i differ from those used in [BZHS06] by a factor of
√ −1.
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Example 5.13. Here we explain briefly the construction of the chiral de Rham complex of a smooth manifold introduced in [MSV99], using the above formalism of N K = 1 SUSY vertex algebras [BZHS06]. Let U be a differentiable manifold. Let T be the tangent bundle of U and T ∗ be its cotangent bundle. We let T = (U, T ) be the space of vector fields on U and A = (U, T ∗ ) be the space of differentiable 1-forms on U . We let C = C ∞ (U ) be the space of differentiable functions on U . Denote by < , >: A ⊗ T → C the natural pairing, and, as before, by the functor of change of parity. Consider now an N K = 1 SUSY Lie conformal algebra R generated by the vector superspace C ⊕ T ⊕ A ⊕ A. That is, we consider differentiable functions (to be denoted f, g, . . . ) as even elements, vector fields X, Y, . . . as odd elements, and finally we have two copies of the space of differential forms. For differential forms, which we consider to be even elements, ¯ . . . . The α, β, · · · ∈ A, we will denote the corresponding elements of A by α, ¯ β, nonvanishing -brackets in R are given by (up to skew-symmetry): [X f ] = X ( f ),
[X Y ] = [X, Y ]Lie ,
[X α] = Lie X α + λ < α, X >,
[X α] ¯ = Lie X α + χ < α, X >,
where [, ]Lie is the Lie bracket of vector fields and Lie X is the action of X on the space of differential forms by the Lie derivative. The fact that (5.13) satisfies the Jacobi identity is a straightforward computation (cf. 1.8). We let V (U ) be the corresponding universal enveloping N K = 1 SUSY vertex algebra of R. This vertex algebra is too big. We impose some relations in V (U ) as follows. Let 1U denote the constant function 1 in U . Let d : C → A be the de Rham differential. Define I (U ) ⊂ V (U ) to be the ideal generated by elements of the form: : f g : −( f g), : f X : −( f X ), 1U − |0,
: f α : −( f α),
T f − d f,
: f α¯ : −( f α),
Sf − df .
Define the N K = 1 SUSY vertex algebra ch
(U ) := V (U )/I (U ).
The following theorem is a reformulation of the corresponding result in [LL05]: Theorem 5.14. (1) Let M ⊂ Rn be an open submanifold. The assignment U → sheaf of SUSY vertex algebras ch M on M.
ch (U )
defines a
ϕ
(2) For any diffeomorphism of open sets M − → M we obtain a canonical isomorphism of SUSY vertex algebras phisms
M
ϕ
− →
ϕ M − →
ch (M)
M, we have
ch (ϕ)
−−−−→ ch (ϕ
ch (M ).
◦ ϕ)
=
Moreover, given diffeomor-
ch (ϕ ) ◦
ch (ϕ).
This theorem allows one to construct a sheaf of SUSY vertex algebras in the Grothendieck topology on Rn (generated by open embeddings). This in turn allows one to attach to any smooth manifold M, a sheaf of SUSY vertex algebras ch M , called the chiral de Rham complex of M.
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Example 5.15 (Free Fields). We can generalize Examples 5.8, 5.12, and 5.4 as follows. Let A = A0¯ ⊕ A1¯ be a vector superspace, and let ( , ) be a non-degenerate bilinear form in A. Recall that the bilinear form ( , ) is said to be of parity p ∈ Z/2Z if (a, b) = 0 unless p(a) + p(b) = p, and it is supersymmetric (resp. skew-supersymmetric) if (a, b) = (−1)ab (b, a) (resp (a, b) = −(−1)ab (b, a)). Let H = C[T, S i ] in the N W = N case, and let H be defined as in 4.2 in the N K = N case. Let R = H ⊗ A ⊕ CC, where C is an even element such that T C = S i C = 0. Given a non-zero homogeneous polynomial Q( ) of degree s (in PBW basis) and parity p, define the following -bracket on A ⊕ CC : [a b] = Q( )(a, b)C, a, b ∈ A, and C central,
(5.13)
and extend it to R by sesquilinearity. Then the Jacobi identity automatically holds since all triple brackets are zero. Skewsymmetry holds if and only if (a, b) = −(−1)ab (−1) N +s (b, a).
(5.14)
Thus, (5.13) defines a structure of a SUSY Lie conformal algebra, provided that (5.14) holds together with the following parity condition: p + p(( , )) = N
mod 2.
Thus, R is a SUSY Lie conformal algebra if and only if N + s is even (resp. odd) and the bilinear form ( , ) is supersymmetric (resp. skew-supersymmetric) of parity (N − p) mod 2. The corresponding free field SUSY vertex algebra F(A, Q) is the quotient of the universal enveloping vertex algebra V (R) by the ideal (C − |0)(−1|N ) V (R). Example 5.16 (Spin 7 vertex algebra). In [SV95], Shatashvili and Vafa constructed a vertex algebra associated to any manifold with Spin7 holonomy. This vertex algebra is generated by four fields and comes equipped with an N = 1 superconformal vector, therefore we can view it as an N K = 1 SUSY vertex algebra. As such, it is generated by a super Virasoro field G of central charge 1/2, and a (non-primary) even field X of conformal weight 2. The corresponding -brackets, derived from the OPE in [SV95], using Lemma 4.4, are: 2 χλ [G X ] = (2T + χ S + 4λ) X + G + λ3 , 2 3 5 2 5 T SX + T G + 6 : GX : [X X ] = 2 4 8 15 + 8 (χ T + λS + 2λχ ) X + λ(T + λ)G + λ3 χ . 4 3 Note that this -bracket is quadratic in the generating fields. Thus, this SUSY vertex algebra is not the universal enveloping SUSY vertex algebra of a SUSY Lie conformal algebra. Expanding these superfields as: G(Z ) = G(z) + 2θ T (z), we obtain the generating fields as in [SV95].
˜ X (Z ) = X˜ (z) + θ M(z),
Supersymmetric Vertex Algebras
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Example 5.17 (Odake’s vertex algebra). In [Oda89], Odake constructed a vertex algebra, generated by eight fields, as an extension of the N = 2 vertex superalgebra, associated to manifolds with SU3 holonomy. It carries therefore an N = 1 superconformal vector, so we can view this algebra as an N K = 1 SUSY vertex algebra. As such, it is generated by two superfields G, J forming an N = 2 vertex algebra of central charge 9, as in Example 5.10, and two odd superfields X ± , primary of conformal weight 3/2. The remaining -brackets are as follows: [J X ± ] = ±(S + 3χ )X ± ,
[X ± X ± ] = 0,
[X + X − ] = (: J G : + : J S J : +T G + T S J ) +
+ χ (: J J : +T J ) + λ(G + S J ) + 2λχ J + λ2 χ . Note that these relations are also quadratic in the generators as in the previous example. Expanding the generating superfieds as 1 ¯ J (Z ) = I (z) + θ √ (G(z) − G(z)), 2 √ X + (Z ) = X (z) + θ 2Y (z),
1 ¯ G(Z ) = √ (G(z) + G(z)) + 2θ T (z), 2 √ X − (Z ) = X¯ (z) + θ 2Y¯ (z),
we obtain the generating fields as in [Oda89]. Acknowledgement. We would like to thank Namhoom Kim for providing his notes and for many discussions on the subject of the paper. We are grateful to the referees for corrections and useful suggestions.
References [Bar00]
[BD04] [BDK01] [BK03] [Bor86] [BZHS06] [Coh87] [DM99] [DRS90] [DSK05] [FBZ01] [FHL93] [FK02] [FLM88]
Barron, K.: N = 1 Neveu Schwarz vertex operator superalgebras over Grassmann algebras with odd formal variables. In: Representations and Quantizations: Proceedings of the International Conference on Representation Theory 1998, Berlin Heidelberg New York: China Higher Ed. Press and Springer Verlag, 2000, pp 9–36 Beilinson, A., Drinfeld, V.: Chiral Algebras. AMS Colloquium Publications Vol. 51, Providence, RI: Amer. Math. Soc., 2004 Bakalov, B., D’Andrea, A., Kac, V.G.: Theory of finite pseudoalgebras. Adv. Math. 162(1), 1– 140 (2001) Bakalov, B., Kac, V.G.: Field algebras. Int. Math. Res. Not. 3, 123–159 (2003) Borcherds, R.: Vertex algebras, kac-moody algebras and the monster. Proc. Nat. Acad. Sci. USA 83(10), 3068–3071 (1986) Ben-Zvi, D., Heluani, R., Szczesny, M.: Supersymmetry of the chiral de Rham complex. http:// arxiv.org/list/math.QA/0601532, 2006 Cohn, J.D.: N = 2 super-riemann surfaces. Nucl. Phys. B284, 349–364 (1987) Deligne, P., Morgan, J.W.: Notes on supersymmetry. In: Quantum fields and strings: A course for mathematicians, Vol. 1. Providence, RI:Amer. Math. Soc., 1999 Dolgikh, S.N., Rosly, A.A., Schwarz, A.S.: Supermoduli spaces. Commun. Math. Phys. 135(1), 91–100 (1990) De Sole, A., Kac, V.G.: Finite vs affine W -algebras. Japanese J. Math. (to appear), available at http://arxiv. math-ph/0511055, 2005 Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves. Mathematical surveys and monographs, Vol. 88, Providence, RI: Amer. Math. Soc. 2001 Frenkel, I.B., Huang, Y., Lepowsky, J.Vol. 104 : On axiomatic approaches to vertex operators algebras and modules. Mem. Amer. Math. Soc. , 494 (1993) Fattori, D., Kac, V.G.: Classification of finite simple lie conformal superalgebras. J. Algebra 258(1), 23–59 (2002) Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster. Pure and Applied Mathematics, Vol. 134. New York London: Academic Press Inc., 1988
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[Hel06] [Hua97] [Kac77] [Kac96] [KT85] [KvdL89] [LL05] [MSV99] [Oda89] [SV95]
R. Heluani, V. G. Kac
Heluani, R.: SUSY vertex algebras and supercurves. Preprint, http://arxiv.org/list/math.QA/ 0603591, 2006 Huang, Y.Z.: Two dimensional conformal geometry and vertex operator algebras. Progress in Mathematics, Vol. 148. Boston, MA: Birkhäuser Boston Inc., 1997 Kac, V.G.: Lie superalgebras. Advances in Mathematics, 26(1), 8–96 (1977) Kac, V.G.: Vertex algebras for beginners. University Lecture series, Vol. 10. Providence, RI:Amer. Math. Soc., 1996, Second edition 1998 Kac, V.G., Todorov, I.T.: Superconformal current algebras and their unitary representations. Commun. Math. Phys. 102(2), 337–347 (1985) Kac, V.G., van de Leur, J.: On classification of superconformal algebras. In: Strings-88, Singapore: World Scientific, 1989, pp. 77–106 Lian, B., Linshaw, A.: Chiral equivariant cohomology I. Preprint. http://arxiv.org/list/math.DG/0501084, 2005 Malikov, A., Shechtman, V., Vaintrob, A.: Chiral de Rham complex. Commun. Math. Phys 204(2), 439–473 (1999) Odake, S.: Extension of n = 2 superconformal algebra and Calabi-Yau compactification. Mod. Phys. Lett. A 4(6), 557–568 (1989) Shatashvili, S.L., Vafa, C.: Superstrings and manifolds of exceptional holonomy. Selecta Mathematica 1(2), 347–381 (1995)
Communicated by Y. Kawahigashi
Commun. Math. Phys. 271, 179–198 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0178-y
Communications in
Mathematical Physics
Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators Krzysztof Bogdan , Tomasz Jakubowski Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wrocław, Poland. E-mail: [email protected]; [email protected] Received: 5 May 2006 / Accepted: 8 September 2006 Published online: 9 January 2007 – © Springer-Verlag 2007
Abstract: We construct a continuous transition density of the semigroup generated by α/2 + b(x) · ∇ for 1 < α < 2, d ≥ 1 and b in the Kato class Kdα−1 on Rd . For small time the transition density is comparable with that of the fractional Laplacian. 1. Introduction Let d be a natural number, α ∈ (1, 2), and let b = (b j )dj=1 : Rd → Rd be a function in
a Kato class Kdα−1 defined below. Our aim is to construct and estimate the semigroup with (weak) generator α/2 f (x) + dj=1 b j (x)∂ j f (x).
Theorem 1. There is a continuous transition density p (t, x, y) such that Pt f (x) − f (x) g(x) d x = α/2 f (x) + b(x) · ∇ f (x) g(x) d x, (1) lim+ t→0 Rd t Rd where f, g ∈ Cc∞ (Rd ), and Pt f (x) = Rd p (t, x, y) f (y) dy. Similar perturbations were studied by other authors for the Laplacian, which corresponds to the limiting case α = 2. In particular, Cranston and Zhao ([16]) proved for b ∈ Kd1 that the harmonic measure and the Green function for the Lipschitz domain and the perturbed operator + b(x) · ∇ are comparable with those of . Zhang ([43, 44]) showed later that the corresponding transition density has Gaussian bounds. Kim and Song ([32]) generalized these results to perturbations + μ · ∇ given by a vector of signed measures μ ∈ Kd1 . The potential theory of the fractional Laplacian α/2 is less developed and it is intensely studied at present because it may be considered representative for a class of Partially supported by KBN and MEN.
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non-local pseudo differential operators ([22–24]). So far the primary focus in the theory of the additive perturbations of α/2 was on the Schrödinger-type perturbations and the properties of the resulting (perturbed) Green function. We refer the interested reader to the work of Chen and Song [36, 37, 12, 13], Bogdan and Byczkowski [3, 4], and Takeda and Uemura [38, 40, 39]. We also remark that the Green function of other generators, e.g. of the relativistic operator ([10]), may be studied by a similar perturbation technique ([33, 31, 19]). The developments hinge on the various versions of the so-called 3G Theorem for the (unperturbed) Green function of α/2 , which are obtained from the so-called boundary Harnack principle ([15, 28, 20], see, e.g., [14, 2] for the results and references in case of the Laplacian). We refer the reader to [7] for a general perspective, more references and a definitive version of the boundary Harnack principle for α/2 . We would also like to mention a forthcoming paper [25] on gradient perturbations of α/2 in terms of the Green functions. Generally, in the present paper we study the asymptotics of the corresponding (perturbed) semigroup instead of that of the Green functions. As a natural continuation of [6, 27, 26] we focus here on gradient perturbations, however similar methods apply in some other settings even more easily. In particular, in the forthcoming paper [5] we give the estimates for the transition density of time-dependent Schrödinger perturbations of α/2 . In passing, we refer the interested reader to [9, 41] for information on the various asymptotics of the unperturbed semigroup generated by more general homogeneous operators satisfying a positive maximum principle. Lemma 3 below is an example of such an estimate, see also [11]. Needless to say, the estimates distinctly differ from the asymptotics of the Gaussian kernel, and our methods do not directly apply to perturbations of . We remark that analogous perturbations of the fractional Laplacian were recently studied within the framework of the theory of pseudo differential equations in [18] and [30]. The authors of these papers consider equations u t = α/2 u + F(t, x, u, ∇u), where F is, among other assumptions, a continuous time dependent function. We like to note that our restriction to α > 1 is also adopted in [18] and [30] (see [6], too). It renders ∇ and F(t, x, ·, ∇·) a small perturbation of α/2 . Within our present setting we note that even the existence of the semigroup generated by α/2 + b(x) · ∇ was not studied before in this generality. Our construction and estimates of the semigroup are based on a new technique, the estimates for the transition density and resolvent kernel of α/2 which we call the 3P Theorem and 3U Theorem. A certain role is also played by a delicate “transference” property (12). These facilitate the construction and lead to sharp estimates of the perturbed transition density p (t, x, y). To state the estimates, we denote p(t, x, y) = pt (y − x), where t > 0 and pt is the smooth function determined by α pt (z)ei z·ξ dz = e−t|ξ | , ξ ∈ Rd . Rd
(2)
(3)
The semigroup Pt f (x) = Rd p(t, x, y) f (y)dy has the fractional Laplacian as generator ([1, 42, 3]). In particular, for f ∈ Cc∞ (Rd ), and x ∈ Rd we have 1 f (x + y) − f (x) α/2 f (x) = lim+ (Pt f (x) − f (x)) = lim+ c dy. t→0 t ε→0 |y|d+α |y|>ε
Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators
181
Theorem 2. For every 0 < t0 < ∞ there is C = C(d, α, b, t0 ) such that C −1 p(t, x, y) ≤ p (t, x, y) ≤ C p(t, x, y), 0 < t < t0 , x, y ∈ Rd ,
(4)
and C → 1 as t0 → 0. The perturbed semigroup {Pt } may be considered in the context of the α-stable OrnsteinUhlenbeck semigroup, for which b(x) = −x, see [35, 34, 29, 27] and [26]. However, b(x) = −x does not belong to Kdα−1 . By inspecting available explicit formulas (see [35, 27]) it can be seen that the transition density of the Ornstein-Uhlenbeck semigroup does not satisfy the inequality in (4) for large x, y. Thus the assumption b ∈ Kdα−1 is not superfluous. Our confinement to α ∈ (1, 2), rather than α ∈ (0, 2), can be explained by the fact that even the semigroup generated by α/2 +∇ on the line (d = 1) is not comparable with Pt if α ∈ (0, 1). Indeed, the corresponding transition density is p (t, x, y) = pt (y − (x + t)), for all α ∈ (0, 2). For x = y and t → 0+ by (5) and (6) below we have p(t, x, y) ≈ t −1/α , whereas p (t, x, y) = pt (t) ≈ t −1/α for α ∈ [1, 2), and p (t, x, y) ≈ t −α for α ∈ (0, 1). Thus in the latter case p (t, x, y) and p(t, x, y) are not comparable for small t. The paper is organized as follows. In Sect. 2 we give basic estimates for p(t, x, y) and its resolvent kernel, and we define and characterize the Kato class Kdα−1 . In Sect. 3 we consider the series of kernels kn (t, x, y) suggested by the perturbation formula for semigroups and we construct the kernel p (t, x, y). In Sect. 4 we prove Theorem 1 and 2. To give an informal idea of the construction of p (t, x, y) let us denote by P the inverse, or fundamental solution, of ∂/∂t − α/2 and let P be the inverse, if it exists, of α/2 − A , where A = b(x) · ∇ . It can be expected that P = P + P A P , and ∂/∂t − b b x b n n so P = ∞ n (t − s, x, y), the kernel of P(Ab P) , n=0 P(Ab P) . We inductively define k and we study the kernel of P given by means of ∞ k (t, x, y). n=0 n Similar reasoning was used, e.g., in [43] and [44] but here we simultaneously prove the estimates and the semigroup property of k(t, x, y), which makes the arguments quite delicate. We like to note that the procedure was inspired by the elementary approach of [21] to perturbations of the Green function. All the functions considered in the sequel are Borel measurable. When we write f (x) ≈ g(x), we mean that there is a number 0 < C < ∞ independent of x, i.e. a constant, such that for every x we have C −1 f (x) ≤ g(x) ≤ C f (x). The notation C = C(a, b, . . . , c) means that C is a constant which depends only on a, b, . . . , c. 2. Preliminaries Throughout the paper d ∈ {1, 2, . . .} and, unless stated otherwise, α ∈ (1, 2). Lemma 3. There exists a constant C1 = C1 (d, α) such that t t −1 −d/α −d/α ≤ p(t, x, y) ≤ C1 . C1 ∧t ∧t |y − x|d+α |y − x|d+α
(5)
The estimate is well-known, and proved, e.g., in [8]. Noteworthy, t ≤ t −d/α |y − x|d+α
iff |y − x|α ≥ t.
(6)
The following estimates (3P) and (3U) (see Theorem 8 below) may be considered analogues of the 3G Theorem mentioned in Sect. 1.
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Theorem 4 (3P). There exists a constant C2 = C2 (d, α) such that x, y, z ∈ Rd , t, s > 0.
p(t, x, z) ∧ p(s, z, y) ≤ C2 p(t + s, x, y),
(7)
Proof. For η > 0 and r ≥ 0 we define p˜ η (r ) = η−d/α ∧(ηr −d−α ). Note that r → p˜ η (r ) is non-increasing and p(t, x, y) ≈ p˜ t (|x − y|). Hence it suffices to verify that p˜ t (r ) ∧ p˜ s (ρ) ≤ c1 p˜ t+s (r + ρ),
(8)
where r = |x − z| and ρ = |y − z|. We have p˜ t (r ) ∧ p˜ s (ρ) = t −d/α ∧ (tr −d−α ) ∧ s −d/α ∧ (sρ −d−α ) r + ρ −d−α t + s −d/α ∧ (t + s) ≤ 2 2 ≤ 26d/α (t + s)−d/α ∧ (t + s)(r + ρ)−d−α = 26d/α p˜ t+s (r + ρ).
We remark that such (3P) holds for all α ∈ (0, 2) but fails to hold for the Gaussian kernel (α = 2). Since for any a, b > 0, we have ab = (a ∨ b)(a ∧ b) and (a ∨ b) ≤ a + b, the inequality (7) implies that p(t, x, z) p(s, z, y) ≤ C2 ( p(t, x, z) + p(s, z, y)), p(t + s, x, y)
x, y, z ∈ Rd , t, s > 0.
(9)
Lemma 5. For all x, y ∈ Rd and t > 0 it holds that |∇x p(t, x, y)| ≈ |y − x|
t −(d+2)/α . ∧ t |y − x|d+2+α
(10)
Proof. Let ηt (u) be the density function of the α/2-stable subordinator at time t and 2 let gt (x) = (4π t)−d/2 e−|x| /4t be the Gaussian kernel. There is a constant c1 such −1−α/2 that η1 (u) ≤ c1 u for all u > 0, see [17, Theorem 37.1], so by scaling property ηt (u) ≤ c1 tu −1−α/2 . Note that pt is a C ∞ (Rd ) function because its Fourier transform is fast decreasing, see (3). Furthermore, pt attains its maximum at 0, hence ∇x pt (0) = 0. Let x ∈ Rd \ {0}. By the subordination formula ([42, 8])
∞
∇x pt (x) = ∇x
gu (x)ηt (u) du.
0
Let e j = (0, . . . , 0, 1, 0, . . . , 0) where 1 is on jth place, and let −|x|/2 < s < |x|/2. By the mean value theorem there exists ξ ∈ (−s, s) such that gu (x + se j ) − gu (x) ∂ = gu (x + ξ e j ) s ∂x j xj + ξ |x|gu (x/2) = gu (x + ξ e j ) ≤ . 2u u
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∞ Since 0 |x|gu (x/2)u −2−α/2 du < ∞, by dominated convergence theorem we may differentiate under the integral sign ∞ x ∞ gu (x) ∇x pt (x) = ηt (u)du ∇x gu (x)ηt (u) du = − 2 0 u 0 ˜ (11) = −2π x p(d+2) (t, x), where x˜ ∈ Rd+2 is such that |x| ˜ = |x| and p(d+2) (t, x) ˜ is the transition density of the semigroup of the fractional Laplacian in dimension d + 2. By (5) t −(d+2)/α |∇x pt (x)| = 2π |x| p(d+2) (t, x) . (12) ˜ ≈ |x| ∧ t |x|d+2+α
Remark 6. Lemma 3, Theorem 4, and Lemma 5 actually hold for all α ∈ (0, 2) by the same proofs with respective changes in constants. However, some of the following estimates of the resolvent kernel would take on a slightly different form for α = d = 1. In view of our applications in what follows we stick to the assumption α ∈ (1, 2). For λ > 0 and x ∈ Rd we define
u λ (x) =
∞
e−λt pt (x) dt.
0
Lemma 7. We have u λ (x) ≈ (λ(d−α)/α ∨ |x|α−d ) ∧ (λ−2 |x|−d−α ). Proof. By the scaling property ∞ u λ (x) = e−s p(sλ−1 , x) ds = λd/α−1 u 1 (λ1/α x),
(13)
(14)
0
hence it suffices to show
u 1 (x) ≈ (1 ∨ |x|
α−d
) ∧ |x|
−d−α
=
1 ∨ |x|α−d for |x| ≤ 1, for |x| > 1. |x|−d−α
(15)
By (5) and (6) we have
|x|α
u 1 (x) ≈ 0
e−t t dt + |x|d+α
∞
|x|α
e−t t −d/α dt.
For |x| > 1 we have 1 −t |x|α −t ∞ −t 1 −d−α e t e t e t |x| < dt ≤ dt ≤ dt = |x|−d−α . d+α d+α d+α 2e |x| |x| |x| 0 0 0 Furthermore, ∞ |x|α
−t −d/α
e t
dt ≤ |x|
−d
∞ |x|α
α
e−t dt = |x|−d e−|x| ≤ |x|−d−α .
This gives (15) for |x| > 1. For |x| ≤ 1 we consider two cases: d = 1 < α and d > α.
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For d > α we have |x|α −t |x|α α |x|−d−α |x| |x|α−d e t |x|α−d −d−α ≤ , t dt ≤ dt ≤ |x| t dt ≤ 2e e |x|d+α 2 0 0 0 and
∞ |x|α
e−t t −d/α dt ≤
∞
|x|α
α |x|α−d , d −α
t −d/α dt =
which gives (15) for |x| ≤ 1 and d > α. For d = 1 < α we simply have ∞ ∞ e−t t −d/α dt ≤ e−t t −d/α dt ≤ Since proof.
|x|α 0
|x|α
1
∞
e−t t −d/α dt.
0
e−t t |x|−1−α dt < |x|α−1 < 1 we obtain u 1 (x) ≈ 1. This completes the
When α < d the formula (13) simplifies: u λ (x) ≈ |x|α−d ∧ (λ−2 |x|−d−α ).
(16)
We denote u λ (x, y) = u λ (y − x). Theorem 8 (3U). There exists a constant C8 such that u λ (x, z) ∧ u λ (z, y) ≤ C8 u λ (x, y),
x, y, z ∈ Rd , λ > 0.
(17)
Proof. According to (14) it suffices to show (17) only for λ = 1. We note that u λ (x) is decreasing in |x|. Since |x − y| ≤ 2(|x − z| ∨ |z − y|) by (13) we have u 1 (x, z) ∧ u 1 (z, y) ≤ u 1 ((x − y)/2) ≈ 1 ∨ (|x − y|/2)α−d ∧ (|x − y|/2)−d−α ≤ 2d+α (1 ∨ |x − y|α−d ) ∧ |x − y|−d−α ≈ u 1 (x, y).
Lemma 9. For λ > 0, u λ is continuously differentiable off the origin, and |∇z u λ (z)| ≈ |z|α−d−1 ∧ (λ−2 |z|−α−d−1 ), z = 0.
(18)
Proof. By the dominated convergence theorem and (12) u λ is differentiable (hence continuous) off the origin and ∞ ∞ −λt ∇z e pt (z) dt = e−λt ∇z pt (z) dt. (19) 0
Let z˜ ∈
0
be such that |˜z | = |z|. By (11), ∞ e−λt ∇z p(t, z) dt = −2π z ∇z u λ (z) =
Rd+2
0
∞ 0
e−λt p(d+2) (t, z˜ ) dt.
(20)
By (16) for dimension d + 2, we obtain (18). The continuity of the integral on the right-hand side as a function of z = 0 is tantamount to the continuity of u λ in dimension d + 2.
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Definition 10. We say that b belongs to the Kato class Kdα−1 if lim sup |b(y)||y − x|α−1−d dy = 0. ε→0 x∈Rd
|y−x|<ε
This uniform local integrability yields the following global integrability: Lemma 11. Let |b| ∈ Kdα−1 and β > (α − 1)/α. We have that tβ sup |b(y)| dy → 0 as t → 0+ . d+1+(β−1)α α |y − x| d x∈R |y−x| ≥t
(21)
The supremum in (21) is finite for every t > 0. Proof. For z ∈ Rd and r > 0 we let
√ C z,r = {y ∈ Rd : |yi − z i | ≤ r/(2 d), i = 1 . . . , d}.
Let k = (k1 , . . . kd ) ∈ Zd and K n = {k ∈ Zd :
max |ki | = n}. Let x ∈ Rd , t ≥ 0 and
i=1,...,d
r = t 1/α . We have |b(y)|t β r αβ dy = |b(y)| dy d+1+(β−1)α |y − x|d+1+(β−1)α |y−x|α ≥t |y − x| |y−x|≥r √
|b(y)| r αβ |y − (x + kr/ d)|d−(α−1) ≤ dy √ |y − x|d+1+(β−1)α C x+kr/√d,r |y − (x + kr/ d)|d−(α−1) d ≤
k∈Z \{0} ∞
√ n=1 k∈K n C x+kr/ d,r ∞
≤ c1
n=1 k∈K n
|b(y)| r d+1+(β−1)α dy √ |y − (x + kr/ d)|d−(α−1) |y − x|d+1+(β−1)α
C x+kr/√d,r
|b(y)| 1 dy. √ d+1+(β−1)α d−(α−1) n |y − (x + kr/ d)|
We have #K n = (2n + 1)d − (2n − 1)d < d2d (2n + 1)d−1 , hence c2 = ∞ n=1 k∈K n −2−(β−1)α < ∞. Since |b| ∈ Kα−1 , for every ε > 0 there n −d−1−(β−1)α < ∞ n=1 c3 n d exists t0 > 0 such that C z,r |b(y)||y − z|−d−1+α dy < ε/(c1 c2 ), z ∈ Rd , provided 1/α
r < t0 . Therefore |y−x|α ≥t
|b(y)|
tβ |y −
x|d+1+(1−β)α
dy ≤ c1
∞
n=1 k∈K n
ε/(c1 c2 ) d+1+(1−β)α n
= ε.
The finiteness of the supremum in (21) for every t > 0 follows from (21) and comparability of the integrand in (21) for different t > 0.
Using Lemma 11 we obtain the following characterization. α−1 if and only if Corollary 12. Let β > α−1 α . A function b belongs to Kd 1 tβ |b(y)| dy = 0. ∧ lim sup d+1−α t→0 x∈Rd Rd |y − x| |y − x|d+1−α+αβ
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K. Bogdan, T. Jakubowski
Proof. We note that t β /|y − x|d+1−α+αβ ≤ 1/|y − x|d+1−α iff t ≤ |y − x|α . On {y : |y − x|α < t} we use the definition of Kdα−1 . On {y : |y − x|α ≥ t} we use Lemma 11.
The singularity of the above integrand at y = x is the same for each β, but its decay at infinity may be manipulated; in what follows we will use β = 1, β = (2α − 1)/α, and β = 2. 3. Construction Recall that α > 1. The following lemma, and Lemma 17 below are related to the smallness of b(x) · ∇ as a perturbation of α/2 . Lemma 13. Let b ∈ Kdα−1 . For every t > 0 there is C4 = C4 (d, α, b, t) with the following properties. For all x, y ∈ Rd we have t p(t − s, x, z)|b(z)| |∇z p(s, z, y)| ds dz ≤ C4 p(t, x, y), (22) Rd
0
C4 is nondecreasing in t, C4 → 0 as t → 0, and C4 /t → 0 as t → ∞. Proof. By (10) and (3P) we have p(t − s, x, z)|∇z p(s, z, y)| p(t, x, y) 1 p(t − s, x, z) p(s, z, y) ≤ c1 |y − z|−1 ∧ s − α p(t, x, y) 1 ≤ c2 ( p(t − s, x, z) + p(s, z, y)) |y − z|−1 ∧ s − α .
W =
By (5),
1 1 − αd − α1 ∧ s ∧ (t − s) |z − x|d |y − z| d 1 1 −α − α1 ∧ s + c4 ∧ s |y − z|d |y − z| d+1 −d−1 ∧ ((t − s) ∧ s)− α ≤ 2c4 (|z − x| ∧ |y − z|) 1 1 − d+1 − d+1 α α + 2c ∧ (t − s) ∧ (t − s) ≤ 2c4 4 |z − x|d+1 |y − z|d+1 1 1 − d+1 − d+1 α α + 2c . + 2c4 ∧ s ∧ s 4 |z − x|d+1 |y − z|d+1
W ≤ c4
All four terms may be dealt with similarly. For example, t 1 − d+1 −d−1 −d−1+α α ds ≤ c (t|z − x| . (23) ∧ s ) ∧ |z − x| 5 |z − x|d+1 0 t d+1 By Corollary 12 with β = 1, supv∈Rd Rd 0 |b(z)| |v − z|−d−1 ∧ s − α ds dz is finite and tends to 0 as t → 0. Clearly, it is nondecreasing in t. By a choice, the same is true of C4 . As the right-hand side of (23) grows sublinearly with t, by the monotone convergence theorem we have C4 /t → 0 as t → ∞.
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We will inductively define functions |k|n : R × Rd × Rd → R. Namely, for t > 0 and x, y ∈ Rd , we let |k|0 (t, x, y) = p(t, x, y), and we define t |k|n (t, x, y) = |k|n−1 (t − s, x, z)|b(z)||∇z p(s, z, y)| dz ds, (24) 0
Rd
for n ≥ 1. When t ≤ 0, we let |k|n (t, x, y) = 0 for all n ≥ 0, x, y ∈ Rd . Let C4 = C4 (d, α, b, t) be the constant of Lemma 13. If we suppose that |k|n−1 (t, x, y) ≤ C4n−1 p(t, x, y), which is satisfied for n = 1, then by Lemma 13, t |k|n (t, x, y) ≤ C4n−1 p(t − s, x, z)|b(z)||∇z p(s, z, y)| ds dz Rd
0
≤ C4n p(t, x, y) < ∞.
(25)
Thus, (25) holds for all n ≥ 0, t ∈ R, and x, y ∈ Rd . We will now define functions kn : R × Rd × Rd → R. For t > 0 and x, y ∈ Rd , we let k0 (t, x, y) = p(t, x, y), and we inductively define t kn−1 (t − s, x, z)b(z) · ∇z p(s, z, y) dz ds, (26) kn (t, x, y) = 0
Rd
for n ≥ 1. When t ≤ 0 we let kn (t, x, y) = 0 for all n ≥ 0, x, y ∈ Rd . By induction and (25) every kn (t, x, y) is well defined and |kn (t, x, y)| ≤ |k|n (t, x, y) ≤ C4n p(t, x, y) < ∞.
(27)
Lemma 14. For every n, kn (t, x, y) is jointly continuous where t = 0. Proof. Clearly, k0 (t, x, y) is continuous at t = 0. For n = 1, 2, . . . we will use induction and the following version of (26): kn−1 (t − s, x, z)b(z) · ∇z k0 (s, z, y) dz ds. (28) kn (t, x, y) = R Rd
The integrand is a continuous function of (t, x, y) except for (s, z) = (0, y) and (s, z) = (t, x). We claim uniform integrability of kn−1 (t − s, x, z)|∇z k0 (s, z, y)| with respect to |b(y)|dyds. Indeed, since the poles of the factors are separated, we may consider them individually. The first factor is comparable to k0 (t − s, x, y) by (27), and so it is bounded off s = t. We have t sup k0 (t − s, x, z)ds|b(z)|dz ≤ sup e u 1/ε (x, z)|b(z)|dz → 0, x∈Rd
Rd
t−ε
Rd
x∈Rd
as ε → 0 by (13) and Corollary 12 with β = (2α − 1)/α. We also have ε sup |∇x k0 (s, x, z)|ds|b(z)|dz ≤ sup e |∇x u 1/ε (x, z)||b(z)|dz → 0, x∈Rd
Rd
0
x∈Rd
Rd
as ε → 0 by (11), (20), (18), and Corollary 12 with β = 2. We return to (28). By uniform integrability we have continuity of the integral as a function of (t, x, y) for t = 0 provided we integrate over a compact subset of R × Rd . For large z we simply use the dominated convergence theorem.
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Lemma 15. For small t > 0 there is C5 = C5 (d, α, b, t) such that C5−1 p(t, x, y) ≤
∞
kn (t, x, y) ≤
n=0
∞
|kn (t, x, y)| ≤ C5 p(t, x, y),
(29)
n=0
for all x, y ∈ Rd , C5 is nondecreasing in t, and C5 → 1 as t → 0. Proof. By Lemma 13 and (27) if t > 0 is small enough then C4 < 1/2, ∞
|k|n (t, x, y) ≤
n=1
C4 p(t, x, y), 1 − C4
(30)
and ∞
|kn (t, x, y)| ≤
n=0 ∞
kn (t, x, y) ≥ p(t, x, y) −
n=0
∞
1 p(t, x, y), 1 − C4 |kn (t, x, y)| ≥
n=1
1 − 2C4 p(t, x, y). 1 − C4
(31)
We see that (29) holds with C5 = (1 − C4 )/(1 − 2C4 ). Lemma 13 yields that C5 is nondecreasing in t, and C5 → 1 as t → 0.
∞ To analyze the series n=0 kn (t, x, y) for large t > 0 we will utilize the Laplace transform. For λ > 0 and f ∈ Kdα−1 we define the resolvent operator u λ (x, y) f (y) dy. (32) Uλ f (x) = Rd
Lemma 16. If f ∈ Kdα−1 then Uλ f and ∇Uλ f are bounded and continuous. Proof. Let λ > 0. By Corollary 12 with β = (2α−1)/α we have that supx∈Rd Rd | f (y)| (1 + |y − x|)−d−α dy < ∞. By(13), continuity of u λ off zero, and the dominated convergence theorem, the integral |y−x|>r u λ (x, y) f (y) dy is bounded and continuous in x for every r > 0. We let r → 0. By the estimate (13) at the origin, Corollary 12, and uniform convergence we see that Uλ f is bounded and continuous. We will next verify that for every x ∈ Rd , ∇x Uλ f (x) = ∇x u λ (x, y) f (y) dy = ∇x u λ (x, y) f (y) dy. (33) Rd
Rd
To this end for y, z ∈ Rd we denote v = v(y, z) =
|u λ (z) − u λ (y)| , |z − y|
if y, z = 0, y = z,
and we let v = 0 if y = 0 or z = 0 or y = z. We will estimate v. Case 1. If z = t y with t > 1, y = 0, then the mean value theorem yields u λ (z)−u λ (y) = (z−y)·∇u λ (y+ξ(z−y)), with some ξ ∈ (0, 1). By (18) there is a constant c = c(d, α, λ) such that v ≤ c(|y|α−d−1 ∧ |y|−α−d−1 ).
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Case 2. If |z| > |y| > 0, we let z ∗ = (|z|/|y|)y, so that |z − y| ≥ |z|−|y| = |z ∗ |−|y| = |z ∗ − y|. By this and Case 1, we have v=
|u λ (z ∗ ) − u λ (y)| |u λ (z ∗ ) − u λ (y)| ≤ ≤ c(|y|α−d−1 ∧ |y|−α−d−1 ). |z − y| |z ∗ − y|
By Case 2, analogous arguments for |y| > |z| > 0, and our definition of v, v ≤ c (|y| ∧ |z|)α−d−1 ∧ (|y| ∧ |z|)−α−d−1 , y, z ∈ Rd . For η ∈ Rd \ {0} we let vη (y) =
|u λ (y + η) − u λ (y)| , |η|
y ∈ Rd .
We claim that {vη } are uniformly integrable with respect to the measure f (y)dy on Rd . Indeed, for y ∈ Rd , vη (y) ≤ c (|y| ∧ |y + η|)α−d−1 ∧ (|y| ∧ |y + η|)−α−d−1 = g0 (y) ∨ gη (y), (34) where gη (y) = c(|y + η|α−d−1 ∧ |y + η|−α−d−1 ). Let K > 0. By Corollary 12, vη (y)| f (y)| dy ≤ g0 (y)| f (y)| dy + gη (y)| f (y)| dy → 0 {y : vη (y)>K }
{g0 >K }
{gη >K }
(uniformly in η) as K → ∞. We finally let x ∈ Rd , η ∈ Rd \ {0}. For B = B(x, 2|η|) ⊂ Rd we have u λ (y − x + sη) − u λ (y − x) ∂ f (y) dy, u λ (x, y) f (y) dy = lim s→0 ∂η B s B by differentiability of u λ off the origin and uniform integrability. By (34), Corollary 12, and dominated convergence theorem analogous equality holds for B c . This proves (33). The boundedness and the continuity of the right-hand side of (33) follows from Lemma 9 and Corollary 12 (with β = 2) similarly as those of Uλ f , see the beginning of the proof.
The following is an analogue of Lemma 13. Lemma 17. For every λ > 0 there is C6 = C6 (d, α, b, λ) such that u λ (x, z)|b(z)| |∇z u λ (z, y)| dz ≤ C6 u λ (x, y), x, y ∈ Rd , Rd
C6 is non-increasing in λ, and C6 → 0 as λ → ∞. Proof. We denote
wλ (z) =
|z|−1 for d ≥ 2, λ(α−1)/α |z|α−2 ∧ |z|−1 for d = 1 < α.
(35)
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K. Bogdan, T. Jakubowski
In view of Lemma 9 and Lemma 7 we have |∇z u λ (z)| ≈ wλ (z)u λ (z). By (3U), W = u λ (x, z)|b(z)||∇z u λ |(z, y) dz Rd u λ (x, z)u λ (z, y) ≤ c1 u λ (x, y) |b(z)|wλ (y − z) dz d u λ (x, y) R |b(z)| wλ (z − x) ∨ wλ (y − z) u λ (x, z) ∨ u λ (z, y) dz. ≤ c2 u λ (x, y) Rd
The functions wλ (y − z) and u λ (z, y) are decreasing in |y − z|, hence |b(z)| wλ (z − x)u λ (x, z) ∨ wλ (y − z)u λ (z, y) dz W ≤ c2 u λ (x, y) d R ≤ c3 u λ (x, y) sup |b(z)| |z − x|α−d−1 ∧ (λ−2 |z − x|−α−d−1 ) dz. x∈Rd
Rd
By Corollary 12 with β = 2 and t = 1/λ we get the assertion of the lemma.
For a differentiable function f : Rd → R we let Ab f (x) = b(x) · ∇ f (x). We note that multiplication by a bounded function preserves the condition Kdα−1 . Thus, by Lemma 16, Ab Uλ : Kdα−1 → Kdα−1 . For λ > 0, n = 0, 1, . . ., and x, y ∈ Rd we define ∞ (n) |u|λ (x, y) = e−λt |k|n (t, x, y) dt. 0
Of course, |u|(0) λ = u λ . By (11) and (19), for n ≥ 1 we have |u|(n) λ (x, y) =
=
e−λt ∞
0
= =
∞
Rd
R
d
Rd
t 0 ∞
Rd
|k|n−1 (t − s, x, z) |b(z)| |∇z p(s, z, y)| dz ds dt
e−λ(t−s) |k|n−1 (t −s, x, z) |b(z)| e−λs |∇z p(s, z, y)| dt ds dz ∞ (n−1) |u|λ (x, z)|b(z)| e−λs |∇z p(s, z, y)| ds dz 0
s
0
(n−1) |u|λ (x, z)|b(z)||∇z u λ (z,
y)|dz.
(36) (n−1)
Let C6 = C6 (d, α, b, λ) be the constant of Lemma 17. If we suppose that |u|λ C6n−1 u λ , which is satisfied for n = 0, then by (36) and Lemma 17, (n) |u|λ (x, y) ≤ C6n−1 u λ (x, z)|b(z)||∇z u λ (z, y)| dz ≤ C6n u λ (x, y). Rd
Thus, (37) holds for all n, λ > 0, and x, y ∈ Rd .
≤
(37)
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If x = y, then u λ (x, y) < ∞, and by (37), ∞ (n) e−λt kn (t, x, y) dt u λ (x, y) = 0
is absolutely convergent, and (n) n |u (n) λ (x, y)| ≤ |u|λ (x, y) ≤ C 6 u λ (x, y).
(38)
(n)
Lemma 18. Uλ (Ab Uλ )n has kernel u λ (x, y), which is continuous for x = y. Proof. By Fubini’s theorem (see also 36), and (38), for n ≥ 1 and x = y we have (n) (n−1) u λ (x, y) = uλ (x, y) b(z) · ∇z u λ (z, y) dz. (39) Rd
By the definition (32), the statement of the lemma is true for n = 0. By induction, (33) and (39), for every f ∈ Kdα−1 (in particular, for bounded f ) (n−1) n uλ (x, y) b(y) · ∇Uλ f (y) dy Uλ (Ab Uλ ) f (x) = d R (n−1) = uλ (x, y) b(y) · ∇ y u λ (y, z) f (z) dz dy Rd Rd (n−1) uλ (x, y)b(y) · ∇ y u λ (y, z) dy f (z) dz = d Rd R (n) = u λ (x, z) f (z) dz, Rd
for all n ≥ 1. The use of Fubini’s theorem is justified here by (36), (38), (13), and Corollary 12 with β = (2α − 1)/α. The continuity of u (n) λ at x = y follows from Lemma 9 when n = 0, and from (38), uniform integrability, and induction for n = 1, 2, . . . .
By Lemma 17 and (38), there is λ0 such that C6 < 1 for λ > λ0 . Thus ∞
∞
e−λt |k|n (t, x, y) dt ≤
n=0 0
u λ (x, y) . 1 − C6
(40)
In particular, for all x = y the series k(t, x, y) =
∞
kn (t, x, y)
(41)
n=0
absolutely converges for almost every t > 0. By the proof of Lemma 15 and (27), the series converges almost uniformly on (0, t0 ]×Rd ×Rd , where t0 > 0 is a small constant. Lemma 14 yields the following result. Corollary 19. k(t, x, y) is jointly continuous for small t > 0.
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In view of (41) and (40), for λ > λ0 and x = y we can define u λ (x, y) =
∞
(n)
u λ (x, y) =
n=0
∞
∞
e−λt kn (t, x, y) dt.
(42)
n=0 0
For f ∈ Kdα−1 , λ > λ0 , and x ∈ Rd we let Uλ f (x) = Rd u λ (x, y) f (y)dy, see Lemma 16 and (40). We will prove the following resolvent equation. Lemma 20. Let λ, η > λ0 . Then Uλ Uη =
Uλ − Uη η−λ
.
(43)
Proof. By (42) and (40) for f ∈ Kdα−1 and λ > λ0 we have Uλ f (x) =
∞
Uλ (Ab Uλ )n f (x).
(44)
n=0
In what follows we will drop f from the notation. Using (44) and the resolvent equation for Uλ we obtain ∞ ∞
(η − λ)Uλ Uη = (η − λ) Uλ + Uλ (AUλ )n (AUη )m Uη n=1
= (η − λ)Uλ Uη = (Uλ − Uη )
∞
= Uλ
∞
(AUη )m + (η − λ)Uλ
m=0 ∞
∞ ∞
= Uλ
(AUλ )n−1 A(η − λ)Uλ Uη (AUη )m
n=1 m=0 ∞ ∞
(AUη )m − Uη + Uλ
m=0 ∞
− Uλ
(AUλ )n A(Uλ − Uη )(AUη )m
n=0 m=0
(AUη )m − Uη + Uλ
m=0 ∞ ∞
(AUλ )n Uη (AUη )m
n=1 m=0
(AUη )m + Uλ
m=0
m=0 ∞ ∞
∞ ∞
(AUλ )n+1 (AUη )m
n=0 m=0
(AUλ )n (AUη )m+1
n=0 m=0
= Uλ
∞
(AUη )m − Uη + Uλ
m=0
−
Uλ
∞ ∞
n=0 m=0
∞ ∞
(AUλ )n (AUη )m − Uλ
n=0 m=0 ∞
n m (AUλ ) (AUη ) − Uλ (AUλ )n
∞
(AUη )m
m=0
n=0
= Uλ − Uη . We note that the calculations may also be understood as compositions of integral kernels (by considering bounded f ).
Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators
Lemma 21. For 0 < s, t ≤ t0 /2 and x, y ∈ Rd we have k(t + s, x, y) = k(t, x, z)k(s, z, y) dz. Rd
Proof. Let η > λ > λ0 . By Lemma 20 for x ∈ Rd the equality u η (x, y) − u λ (x, y) , u λ (x, z)u η (z, y) dz = λ−η Rd
193
(45)
(46)
holds for a.e. y ∈ Rd . By Lemma 18 and (37) the function u λ (x, y) is continuous at x = y. By (37), Lemma 7, and uniform integrability, the left-hand side of (46) is continuous at x = 0, and so (46) holds for all y = x. We will calculate the Laplace transform of k(t + s, x, y) for x = y: ∞ ∞ e−λt e−ηs k(t + s, x, y) ds dt 0 0 ∞ ∞ e(η−λ)t e−η(t+s) k(t + s, x, y) ds dt = 0
−t ∞ 0
e−λt e−ηs k(t + s, x, y) ds dt − 0 −t ∞ t ∞ e−(λ−η)t u η (x, y) dt − e−λt e−η(r −t) k(r, x, y) dr dt = 0
= =
u η (x, y) λ−η u η (x, y) λ−η
−
0
0
e−(λ−η)t e−ηr k(r, x, y) dt dr
r
∞ 0
Rd
0
0
−
Of course, ∞ ∞ 0
∞ ∞
u η (x, y) − u λ (x, y) 1 −(λ−η)r −ηr e . e k(r, x, y) dt dr = λ−η λ−η
e−λt e−ηs k(t, x, z)k(s, z, y) dz ds dt =
Rd
u λ (x, z)u η (z, y) dz.
By (46) and the uniqueness of the Laplace transform ([42]) we get ∞ ∞ e−ηs k(t + s, x, y) ds = e−ηs k(t, x, z)k(s, z, y) dz ds, 0
0
Rd
(47)
for almost all t > 0. Note that the left-hand side of (47), ∞ t e−ηs k(t + s, x, y) ds = eηt u η (x, y) − e−ηs k(s, x, y) ds 0
0
is continuous in t > 0. By (40), (29) and the dominated convergence theorem the righthand side of (47) is also continuous in t ∈ (0, t0 /2], hence (47) holds for all t ∈ (0, t0 /2]. By the uniqueness of the Laplace transform, (45) holds for every t ∈ (0, t0 /2] and almost every s > 0. By Corollary 19 and (40) both sides of the equation are continuous in (s, t) ∈ (0, t0 /2] × (0, t0 /2], which proves (45). Finally, both sides of (45) are continuous in x and y, for s, t ∈ (0, t0 /2], which extends (45) to x = y.
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Remark 22. It is possible to prove Lemma 21 directly from the definition of k(t, x, y) ([5]). The proof presented in this paper gives, however, additional information on the resolvent operators. For t > 0 and x, y ∈ Rd , we define p (t/2, x, z) p (t/2, z, y) dz, p (t, x, y) = Rd
p (t, x, y) = k(t, x, y), For bounded f , t > 0 and x ∈
for t > t0 /2,
for t ≤ t0 /2.
Rd ,
we define p (t, x, y) f (y)dy. Pt f (x) =
Lemma 23. For all s, t > 0 we have
Rd Pt Ps
. = Pt+s
for t, s ≤ t /2. By definition of p (t, x, y), Proof. By Lemma 21, Pt Ps = Pt+s 0
P2t = Pt Pt
for all t > 0.
(48)
Let t, s > 0 and let n ∈ N be such that t/2n , s/2n ≤ t0 /2. We note that Pt/2 n and Ps/2n commute, so by (48), n n n n Pt Ps = (Pt/2 n ) (Ps/2n ) = (Pt/2n Ps/2n ) = (P(t+s)/2n ) = Pt+s .
4. Proofs of Theorems 1 and 2 Proof of Theorem 1. By Lemma 23 we have p (t, x, y) = p (t/2, x, z) p (t/2, z, y) dz, Rd
(49)
for all s, t > 0 and x, y ∈ Rd . By (11) Rd ∇x p(t, x, y) dy = 0 for all t > 0, x ∈ Rd , hence for all n ≥ 1 we have t kn (t, x, y) dy = kn−1 (t − s, x, z)b(z) · ∇z p(s, z, y) ds dz dy Rd Rd Rd 0 t = kn−1 (t − s, x, z)b(z) · ∇z p(s, z, y) dy ds dz Rd
= 0.
0
Rd
By dominated convergence and (30) Rd p (t, x, y) dy = Rd p(t, x, y) dy = 1 for small t > 0, which extends to all t > 0 by (45). Continuity of p (t, x, y) for all t > 0 follows from Corollary 19, (49), and dominated convergence. It remains to prove that α/2 − b(x) · ∇ is the weak generator of the Markov semigroup Pt with the transition density p (t, x, y). Let f, g ∈ Cc∞ (Rd ). We will calculate the limit Pt f (x) − f (x) g(x) d x lim t→0 Rd t ∞
1 = lim kn (t, x, y) f (y) dy − f (x) g(x) d x. t→0 t Rd Rd n=0
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Since k0 (t, x, y) = p(t, x, y), for t > 0 we have 1 α/2 f (x) g(x) d x. k0 (t, x, y) f (y) dy − f (x) g(x) d x = lim t→0 t Rd Rd Rd We will verify that Rd k1 (t, x, y)t −1 f (y) dy converges in the weak sense to b(x) · ∇ f (x). Let It = Rd Rd k1 (t, x, y)t −1 f (y)g(x) d y d x. By (2), Fubini’s theorem and integration by parts we have t 1 It = p(t − s, x, z)b(z) · ∇z p(s, z, y) dz ds f (y)g(x) d y d x t Rd Rd 0 Rd t 1 p(t − s, x, z)g(x)b(z) · ∇ y p(s, z, y) f (y) d y dz ds d x =− t Rd 0 Rd Rd t 1 p(t − s, x, z) p(s, z, y) ds b(z) · ∇ y f (y)g(x) dz dy d x. = Rd Rd Rd 0 t We now estimate It − Rd b(z) · ∇z f (z)g(z) dz. Let K 1 be the support of ∇ f and let K 2 be the support of g. Denote K = {z ∈ Rd : dist(z, K 1 ∪ K 2 ) ≤ 1}. The function h(x, y) = ∇ f (y)g(x) is (uniformly) continuous and has compact support K 1 × K 2 . Let ε > 0. There exists δ > 0 such that for every z ∈ Rd and (x, y) ∈ B(z, δ) × B(z, δ) we have |∇ f (y)g(x) − ∇ f (z)g(z)| < ε. Then It − b(z) · ∇ f (z)g(z) dz Rd
≤
≤
Rd
Rd
Rd
t 0
p(t − s, x, z) p(s, z, y) ds |b(z)||h(x, y) − h(z, z)| d x d y dz t
p(t − s, x, z) p(s, z, y) ds |b(z)||h(x, y)| d x d y dz t K c K2 K1 0 t p(t − s, x, z) p(s, z, y) + 2||h|| ds |b(z)| d x d y dz t K (B(z,δ)×B(z,δ))c 0 t p(t − s, x, z) p(s, z, y) ds |b(z)| d x d y dz +ε t K B(z,δ) B(z,δ) 0 t
= J1 + J2 + J3 . Here ||h|| = supx,y |h(x, y)| < ∞. We estimate J1 first. By (5) J1 ≤ c1
Kc
K1
t|b(z)| ||h|| d x dz ≤ c1 |K 1 | sup |x − z|d+α x∈K 1
|x−z|>1
t|b(z)| ||h|| dz −→ 0, |x − z|d+α
as t → 0. To estimate J2 we note that if (x, y) ∈ (B(z, δ) × B(z, δ))c then |x − z| > δ or |y − z| > δ. Thus, for t → 0 we have t|b(z)| −α J2 ≤ c1 4||h|| d x dz ≤ 2tc2 δ ||h|| |b(z)| dz −→ 0. d+α K |x−z|>δ |x − z| K
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Finally, p(t − s, x, z) p(s, z, y) ds |b(z)| d x d y dz ≤ ε J3 ≤ ε |b(z)| dz. t K Rd Rd 0 K Since the last expression is arbitrary small, It → Rd b(z) · ∇z f (z)g(z) dz. ∞ By (30), n=1 kn (t, x, y) ≤ ct p(t, x, y) with ct → 0, as t → 0. Therefore for all n ≥ 2 and g ∈ Cc∞ (Rd ) we have 1 | p (t, x, y) − p(t, x, y) − k1 (t, x, y)|| f (y)g(x)| d y d x t Rd t ct ≤ p(t − s, x, z)|b(z)||∇z p(s, z, y)| dz ds | f (y)g(x)| dy −→ 0, t Rd Rd 0 Rd
t
as t → 0, see the estimates of It above. This completes the proof.
Proof of Theorem 2. By Lemma 15 we have C5−1 p(t, x, y) ≤ p (t, x, y) ≤ C5 p(t, x, y) for 0 < t ≤ t0 and x, y ∈ Rd . Here C5 = C5 (d, α, b, t0 ). Let t > 0 and n be the smallest natural number such that t0 n ≥ t. We have p (t, x, y) = ... p ( nt , x, x1 ) p ( nt , x1 , x2 ) . . . p ( nt , xn−1 , y) d x1 . . . d xn−1 Rd Rd n ≤ C5 ... p( nt , x, x1 ) p( nt , x1 , x2 ) . . . p( nt , xn−1 , y) d x1 . . . d xn−1 Rd
=
Rd
C5n p(t, x,
t/t0
y) ≤ C5 C5
p(t, x, y) ≤ C5 ec1 t p(t, x, y).
Similarly we prove p (t, x, y) ≥ C5−1 e−c2 t p(t, x, y).
Acknowledgement. We thank the referee for corrections and helpful comments.
References 1. Berg, C., Forst, G.: Potential theory on locally compact abelian groups. New York: SpringerVerlag 1975 2. Bogdan, K.: Sharp estimates for the Green function in Lipschitz domains. J. Math. Anal. Appl. 243(2), 326–337 (2000) 3. Bogdan, K., Byczkowski, T.: Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133(1), 53–92 (1999) 4. Bogdan, K., Byczkowski, T.: Potential theory of Schrödinger operator based on fractional Laplacian. Probab. Math. Statist. 20(2, Acta Univ. Wratislav. No. 2256), 293–335 (2000) 5. Bogdan, K., Hansen, W., Jakubowski, T.: On time-dependent Schrödinger perturbations. (2006) preprint. 6. Bogdan, K., Kulczycki, T., Nowak, A.: Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes. Illinois J. Math. 46(2), 541–556 (2002) 7. Bogdan, K., Kwa´snicki, M., Kulczycki, T.: Estimates and structure of α-harmonic functions. 2006 submitted (available at http://front.math.ucdavis.edu/math.PR/0607561). 8. Bogdan, K., Stós, A., Sztonyk, P.: Harnack inequality for stable processes on d-sets. Studia Math. 158(2), 163–198 (2003) 9. Bogdan, K., Sztonyk, P.: Estimates of potential kernel and Harnack’s inequality for anisotropic fractional Laplacian. 2006 submitted (available at http://front.math.ucdavis.edu/math.PR/0507579) 10. Carmona, R., Masters, W.C., Simon, B.: Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91(1), 117–142 (1990)
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11. Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on d-sets. Stochastic Process. Appl. 108(1), 27–62 (2003) 12. Chen, Z.-Q., Song, R.: Intrinsic ultracontractivity and conditional gauge for symmetric stable processes. J. Funct. Anal. 150(1), 204–239 (1997) 13. Chen, Z.-Q., Song, R.: Drift transforms and Green function estimates for discontinuous processes. J. Funct. Anal. 201(1), 262–281 (2003) 14. Chung, K.L., Zhao, Z.X.: From Brownian motion to Schrödinger’s equation. Volume 312 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag, 1995 15. Cranston, M., Fabes, E., Zhao, Z.: Conditional gauge and potential theory for the Schrödinger operator. Trans. Amer. Math. Soc. 307(1), 171–194 (1988) 16. Cranston, M., Zhao, Z.: Conditional transformation of drift formula and potential theory for 21 + b(·) · ∇. Commun. Math. Phys. 112(4), 613–625 (1987) 17. Doetsch, G.: Introduction to the theory and application of the Laplace transformation. New York: Springer-Verlag, 1974, translated from the second German edition by Walter Nader 18. Droniou, J., Imbert, C.: Fractal first order partial differential equations. Arch. Ration. Mech. Anal. 182(2), 299–331 (2006) 19. Grzywny, T., Ryznar, M.: Estimates for perturbations of fractional Laplacian. 2006, submitted 20. Hansen, W.: Uniform boundary Harnack principle and generalized triangle property. J. Funct. Anal. 226(2), 452–484 (2005) 21. Hansen, W.: Global comparison of perturbed Green functions. Math. Ann. 334(3), 643–678 (2006) 22. Jacob, N.: Pseudo differential operators and Markov processes. Vol. I. London: Imperial College Press, 2001 23. Jacob, N.: Pseudo differential operators and Markov processes. Vol. II. London: Imperial College Press, 2002 24. Jacob, N.: Pseudo differential operators and Markov processes. Vol. III. London: Imperial College Press, 2005 25. Jakubowski, T.: Estimates of Green function for fractional Laplacian perturbed by gradient. Preprint, 2006 26. Jakubowski, T.: The estimates of the mean first exit time from the ball for the α–stable Ornstein Uhlenbeck processes. (2006) submitted, available at http://math.univ-angers.fr/publications/prepub/fichiers/00225.pdf 27. Jakubowski, T.: On Harnack inequality for α-stable Ornstein-Uhlenbeck processes. (2006) submitted, available at http://math.univ-angers.fr/publications/prepub/fichiers/00224.pdf 28. Jakubowski, T.: The estimates for the Green function in Lipschitz domains for the symmetric stable processes. Probab. Math. Statist. 22(2, Acta Univ. Wratislav. No.2470), 419–441 (2002) 29. Janicki, A., Weron, A.: Simulation and chaotic behavior of α-stable stochastic processes. Volume 178 of Monographs and Textbooks in Pure and Applied Mathematics. New York: Marcel Dekker Inc., 1994 30. Karch, G., Woyczy´nski, W.: Fractal Hamilton-Jacobi-KPZ equations. Trans. Amer. Math. Soc., to appear 31. Kim, P., Lee, Y.-R.: Generalized 3G theorem and application to relativistic stable process on non-smooth open sets. 2006, submitted 32. Kim, P., Song, R.: Two-sided estimates on the density of Brownian motion with singular drift. To appear in Ill. J. Math. 33. Ryznar, M.: Estimates of Green function for relativistic α-stable process. Potential Anal. 17(1), 1– 23 (2002) 34. Samorodnitsky, G., Taqqu, M.S.: Stable non-Gaussian random processes. Stochastic Modeling. New York: Chapman & Hall, 1994 35. Sato, K.-i.: Lévy processes and infinitely divisible distributions. Volume 68 of Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 1999, translated from the 1990 Japanese original, revised by the author. 36. Song, R.: Probabilistic approach to the Dirichlet problem of perturbed stable processes. Probab. Theory Related Fields, 95(3), 371–389 (1993) 37. Song, R.: Feynman-Kac semigroup with discontinuous additive functionals. J. Theoret. Probab. 8(4), 727– 762 (1995) 38. Takeda, M.: Conditional gaugeability and subcriticality of generalized Schrödinger operators. J. Funct. Anal. 191(2), 343–376 (2002) 39. Takeda, M.: Gaugeability for Feynman-Kac functionals with applications to symmetric α-stable processes. Proc. Amer. Math. Soc. 134(9), 2729–2738 (electronic) (2006 ) 40. Takeda, M., Uemura, T.: Subcriticality and gaugeability for symmetric α-stable processes. Forum Math. 16(4), 505–517 (2004) 41. Watanabe, T.: Asymptotic estimates of multi-dimensional stable densities and their applications. (2004), Trans. Amer. Math. Soc, in press
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Commun. Math. Phys. 271, 199–221 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0179-x
Communications in
Mathematical Physics
Least Energy Solitary Waves for a System of Nonlinear Schrödinger Equations in Rn Boyan Sirakov1,2 1 Modalx, Ufr Segmi, Université Paris 10, 92001 Nanterre Cedex, France 2 Cams, Ehess, 54 bd Raspail, 75270 Paris Cedex 06, France. E-mail: [email protected]
Received: 15 May 2006 / Accepted: 5 July 2006 Published online: 12 January 2007 – © Springer-Verlag 2007
Abstract: In this paper we consider systems of coupled Schrödinger equations which appear in nonlinear optics. The problem has been considered mostly in the onedimensional case. Here we make a rigorous study of the existence of least energy standing waves (solitons) in higher dimensions. We give: conditions on the parameters of the system under which it possesses a solution with least energy among all multi-component solutions; conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie. 1. Introduction The concept of incoherent solitons in nonlinear optics has attracted considerable attention in the last ten years, both from experimental and theoretical point of view. The two experimental studies [20] and [21] demonstrated the existence of solitons made from both spatially and temporally incoherent light. These papers were followed by a large amount of theoretical work on incoherent solitons. We shall quote here [13, 12, 1, 4], where a comprehensive list of references on this subject can be found. It is shown for instance in the recent works [7, 4] (see also the references there) that, for photorefractive Kerr media, a good approximation describing the propagation of self-trapped mutually incoherent wave packets is the following system of coupled Schrödinger equations: j
2ik j
∂ψm j j + x ψm + αk 2j I (x, t)ψm = 0, ∂t
where I (x, t) =
Nf Nj j=1 m=1
j
j
λm |ψm (x, t)|2 .
(1)
200
B. Sirakov j
Here ψm (the (m, j)-component of the beam) is a complex function defined on Rn × R+ , n ≤ 3, is the Laplace operator, N f is the number of frequencies, N j is the number of waves at a particular frequency ω j , k j is a constant multiple of the frequency ω j , j and λm are the so-called time averaged mode-occupancy coefficients (we refer to [4] for more precision on the meaning of the constants in this equation). We note that for this problem all coefficients in (1) are positive. We will search for soliton (stationary wave) solutions of (1) in the form j
j
j
ψm (t, x) = eiκm t u m (x), j
(2) j
where u m : Rn → R is the spatial profile of the m th wave at frequency ω j , and κm is the propagation speed of this wave. Substituting (2) into (1) and renaming indices and constants leads us to the following real elliptic system for the vector function u = (u 1 , . . . , u d ) : Rd → Rn , u = (0, . . . , 0), ⎛ ⎞ d −u i + λi u i = ⎝ μi j |u j |2 ⎠ u i , i = 1, . . . , d. (3) j=1
In this paper we consider the case when u can be scaled, i.e. u i can be replaced by si u i , si > 0, in such a way that s 2j μi j = si2 μ ji , for all i = j. Note this is always possible for systems of two equations (d = 2). So we can suppose μi j = μ ji , and (3) is the Euler-Lagrange system for the energy functional E(u) =
d d 1 1 |∇u i (x)|2 + λi |u i (x)|2 d x − μi j u i2 (x)u 2j (x) d x. 2 Rn 4 Rn i=1 i, j=1
This functional is well defined if u i are in the Sobolev space H 1 (Rn ), by virtue of the embeddings H 1 (Rn ) → L 4 (Rn ), valid for n ≤ 3. The following essential remark has to be done immediately: for the solution to be of the type we are interested in, at least two of its components u i have to be different from the trivial zero wave. Note that solutions with one nontrivial and d − 1 zero components always exist (see the next section). To fix vocabulary, we shall refer to all other eventual solutions as nonstandard solutions. We will always consider solitons with finite energy. We will be in particular interested in existence of least energy nonstandard solutions of (3), that is, solutions with minimal energy on the set of solutions u = (u i )i of (3), such that u i ≡ 0 for at least two different indices i. It is unnecessary to stress the importance of least energy in physics - in general, the most often observed phenomena tend to minimize some underlying energy. For instance, for photorefractive Kerr media, in cases when we are able to compute explicitly the least energy solutions - Theorem 1 below - these solutions turn out to represent the profiles of the so-called bright photorefractive screening solitons, known in physics since the founding works [24, 19] (see also [13], and the references in that paper). To explain in one phrase the essence of the results we obtain, we will show that, somewhat surprisingly, there always exist ranges of positive parameters in (3), for which this system has a least energy solution, and ranges of positive parameters for which the functional cannot be minimized on the natural set where the eventual solutions lie. We will see that E, which looks quite “scalar” with respect to the vector u, has richer structure,
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and differs in its behaviour from its scalar counterpart, when nonstandard solutions are searched for (this remark will become clear later). In all the papers that we quoted above the authors considered the one-dimensional case, n = 1. The motivation for our work comes from a recent paper by Lin and Wei [16], which seems to be the first attempt to make a rigorous study of the higher dimensional case. The main existence result in [16] states that a nonstandard least energy solution of (3) exists provided the off-diagonal entries of the matrix (μi j ) are sufficiently small. It is our goal here to consider arbitrary coefficients and to give explicit ranges for existence and non-existence of least-energy solutions. In order to simplify the presentation, we shall concentrate on a system of two equations ⎧ ⎨ u 1 − u 1 + μ1 u 31 + βu 1 u 22 = 0 (4) u 1 , u 2 ∈ H 1 (Rn ) ⎩ 3 u 2 − λu 2 + μ2 u 2 + βu 21 u 2 = 0, where λ, μ1 , μ2 > 0. We stress however that all our results extend straightforwardly to systems with an arbitrary number of equations, see Sect. 4. Note that the constant 1 in the u 1 term does not introduce a restriction, since this can always be obtained by using renumbering of u 1 , u 2 and a scaling of x. This also permits to suppose λ ≥ 1, without restricting the generality. A solution u = (u 1 , u 2 ) of (4) which has a zero component (u 1 ≡ 0 or u 2 ≡ 0) will be called a standard solution. The vector (0, 0) will be referred to as the trivial solution. We shall search for nonstandard solutions of (4), or, equivalently, for nonstandard critical points of the functional: 1 1 |∇u 1 |2 + u 21 + |∇u 2 |2 + λu 22 − E(u) = μ1 u 41 + 2βu 21 u 22 + μ2 u 42 2 Rn 4 Rn on the energy space H := H 1 (Rn ) × H 1 (Rn ). We denote with Hr the set of couples in H who are radially symmetric with respect to a fixed point in Rn . As in [16] we consider the set N = u ∈ H, u 1 ≡ 0, u 2 ≡ 0, Rn |∇u 1 |2 + u 21 = Rn μ1 u 41 + βu 21 u 22 ,
Rn
|∇u 2 |2 + λu 22 = Rn βu 21 u 22 + μ2 u 42 .
Note that any nonstandard solution of (4) has to belong to N (multiply the equations in (4) by u 1 , u 2 , and integrate over Rn ). We set A = inf E(u), u∈N
Ar =
inf
u∈N ∩Hr
E(u).
(5)
The following proposition shows the role of A and Ar . Proposition 1.1. If A or Ar is √ attained by a couple u ∈ N then this couple is a solution of (4), provided −∞ < β < μ1 μ2 . We now state our main results. We show that there exist (explicitly given) intervals I1 , I2 , I3 ⊂ R such that I1 contains zero, I3 is a neighbourhood of infinity, I2 is between I1 and I3 , and A (or Ar ) is attained for β ∈ I1 ∪ I3 , while it is not attained for β ∈ I2 . Let w1 (x) = w1 (|x|) be the unique positive solution of the scalar equation −w + w = w 3
in Rn .
(6)
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The function w1 is well studied (see the next section) and will play an important role in our analysis. Our first result concerns the case λ = 1. Going back to (1)–(3), we see that this is the case when the propagation speeds are adjusted to the frequencies. Theorem 1. Suppose λ = 1 in system (4). (i) If 0 ≤ β < min{μ1 , μ2} then A = Ar is attained by the couple √ √ μ1 k + βl = 1 ( kw1 , lw1 ), where βk + μ2 l = 1.
(7)
(ii) If min{μ1 , μ2} ≤ β ≤ max{μ1 , μ2} and μ1 = μ2 then system (4) does not have a nonstandard solution with √ nonnegative components. (iii) If min{μ1 , μ2} ≤ β < μ1 μ2 then A and Ar are not attained. (iv) If β > max{μ1 , μ2} then A = Ar is attained by the same couple as in (i), which is a solution of (4). Remark 1. It was shown in [16] that A is attained if 0 < β < β0 (λ, μ1 , μ2 , n), where β0 is a (unknown) small constant. Remark 2. The couple considered in (i) and (iv) is obviously a solution of (4) with λ = 1, whenever the solution of the linear system in (7) is such that k > 0, l > 0. The result here states that this couple is actually a least energy solution – this was an open problem, see for example 1.4. We conjecture that under the hypoth√[17], Remark √ eses of (i) or (iv) the couple ( kw1 , lw1 ) is the unique positive solution to (4). Note that when λ = μ1 = μ2 = β = 1 system (4) has an infinity of positive solutions (cosθ w1 , sinθ w1 ), θ ∈ (0, π/2). Remark 3. If β ≥ 0 then any nonstandard nonnegative solution of (4) is strictly positive and radial, by the strong maximum principle and the results in [5], see Sect. 3.5. On the other hand, if A is attained by a couple (u 1 , u 2 ) ∈ H then it is attained by (|u 1 |, |u 2 |) ∈ H , so whenever minimizers for A exist and are solutions of (4), we have A = Ar . The next theorem deals with the general case λ ≥ 1 and β ∈ R. n
n
Theorem 2. Suppose λ ≥ 1 in (4) and set ν1 = μ1 λ1− 4 , ν2 = μ2 λ 4 −1 . (i) Let ν0 be the smaller root of the equation λ−n/4 x 2 − (ν1 + ν2 )x + ν1 ν2 = 0. If −∞ < β < ν0 then Ar is attained by a solution of (4). In addition, if 0 ≤ β < ν0 then A = Ar . (ii) If μ2 ≤ β ≤ μ1 and μ2 < μ1 then system (4) does not have a nonstandard solution with nonnegative √ components. (iii) If μ2 ≤ β < μ1 μ2 then A and Ar are not attained. n (iv) If β > λ 4 max{ν1 , ν2} then A = Ar is attained by a solution of (4). Remark 4. The conditions in (i) and (iv) in Theorem 2 reduce to those from Theorem 1, when λ = 1. The ranges in Theorem 2 (i) and (iv) are not the best we can get, we have given them in this form to avoid introducing heavy notations at this stage. We will see in the course of the proof how (i) and (iv) can be improved (with the help of the function h(λ), defined in Sect. 3), we refer to Sect. 3.3, Proposition 3.4 and Sect. 3.4, Proposition 3.7 for precise statements. When λ = 1 it is open, and quite interesting, to find out what the optimal ranges for existence are.
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Remark 5. The statement (i) above includes cases when β < 0. In many applications this is known as “repulsive interaction”. Note it was shown in [16] that A is not attained when β < 0. Theorem 2 above shows that, nevertheless, nonstandard solutions of (3) exist in this case. Remark 6. It will be shown that in cases (iv) we obtain a minimizer even with respect to the standard solutions, see Sect. 3.4. The next section contains further comments on this problem, and some preliminary results. The proofs of the main theorems can be found in Sect. 3. In Sect. 4 we state more general results for systems of d equations. The results in Sect. 4 also have the advantage to underline the symmetries and the relations between the coefficients of the system in the conditions for existence of nonstandard solutions. Note. After this work was submitted, the author learned that existence of solutions for Schrödinger systems was studied, by different methods, in several recent (independent) works - [2, 3, 18]. All these papers contain statements of type (iv) in our theorems (existence of minimizers for β > β1 ), with existence ranges different from ours. The papers [2, 3] contain also results on existence of solutions of (3) in ranges of the type β < β0 , without results on the energy of these solutions.
2. Preliminaries and Further Comments In this section we comment on our problem more extensively, and recall some known facts in the theory of elliptic equations and systems. Existence and properties of standard solutions of (4) are very well studied. Let us recall some facts. For each u ∈ H 1 (Rn ) we denote |∇u|2 + λu 2 . u2λ := Rn
Proposition 2.1. Consider the minimization problems Sλ,μ =
inf
u∈H 1 (Rn )\{0}
u2λ
Rn
μu 4
1/2 ,
Tλ,μ = inf
u∈M0
1 1 μu 4 , u2λ − 2 4 Rn
where M0 = u ∈ H 1 (Rn ), u ≡ 0 : u2λ = Rn μu 4 . Then the function 1
wλ,μ (x) = μ− 2
√
√ λ w1 ( λx)
is a minimizer for Tλ,μ and the unique positive solution of the equation −w + λw = μw 3
in Rn .
In addition, we have Tλ,μ =
1 2 S , 4 λ,μ
1
n
Sλ,μ = μ− 2 λ1− 4 S1,1 .
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This proposition is easily proved by scaling and by using known results for (6) (see for example [23], we will give a brief proof in Sect. 3.4, for the case of a system). By [10] any positive solution of (6) is radially symmetric and strictly decreasing in the radial variable. The uniqueness of radial solutions of (6) goes back to Coffman [8], see also Kwong [14]. By Proposition 2.1 system (4) has exactly two nonnegative standard solutions : (u 1 , 0) and (0, u 2 ), where u 1 (x) = w1,μ1 (x), u 2 (x) = wλ,μ2 (x). (8) Further, it is known that (6) has an infinity of radial and nonradial solutions, which give an infinity of standard solutions of (4). We go back to the case of a system. Let us immediately note that the functional E has a sort of “scalar” geometry on H , in the following sense: it can be written as 1 1 E(u 1 , u 2 ) = u2H − (Mu 2 , u 2 ), 2 4 Rn where u := (u 1 , u 2 ), u 2 := (u 21 , u 22 ), u2H := u 1 21 + u 2 2λ is a norm on H , μ1 β is such that (Mu 2 , u 2 ) = μ1 u 41 + 2βu 21 u 22 + μ2 u 42 , and M = β μ2 c0 (u 41 + u 42 ) ≤ (Mu 2 , u 2 ) ≤ C0 (u 41 + u 42 ), √ for some positive constants c0 , C0 , as long as − μ1 μ2 < β. This basically means that all Critical Point Theory (see for example [22, 23]) for scalar functionals can be applied to E(u 1 , u 2 ). For instance, E satisfies the hypotheses of the Symmetric Mountain Pass Lemma [22] (or the Fountain Theorem, [23]), which immediately yields the existence of an infinity of solutions of (4), such that (u 1 , u 2 ) = (0, 0). However, a priori nothing prevents these from being standard. So, in general, it is unavoidable to distinguish between restricting the solutions (u 1 , u 2 ) to being different from the couple (0, 0) or to being such that u 1 ≡ 0, u 2 ≡ 0. If only the former is done, we will need extra information in order to conclude that we have a nonstandard solution. Borrowing from the scalar theory, one may envision several ways to prove existence of nonstandard solutions of (4). First, one may try to directly search for critical points of E on H , through use of the Mountain Pass Lemma, for example. The drawback of this otherwise very powerful method is that it does not always give enough information on the solutions, nor on their energy level. Second, one may try to use Constrained Minimization, for example, minimize Rn |∇u 1 |2 + u 21 + |∇u 2 |2 + λu 22 on the set u ∈ H, u 1 ≡ 0, u 2 ≡ 0,
Rn
μ1 u 41
+ βu 21 u 22
= 1,
Rn
μ2 u 42
+ βu 21 u 22
=1 .
However, one easily sees that, contrary to the scalar case, this approach fails, since even if a minimizer exists, it gives rise to two (as opposed to one) Lagrange multipliers, which cannot be scaled out of the system. The third approach consists in determining, with the help of the equations we aim to solve, some subset of the energy space where all eventual solutions should belong, and
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then minimize the functional on this subset (note that E is easily seen not to be bounded below on the whole H ). The so-called Nehari manifold is defined by 2 2 2 (Mu , u ) . N0 := u ∈ H, (u 1 , u 2 ) ≡ (0, 0) : u H = Rn
This set has the same properties as the set M0 in Proposition 2.1, in particular, N0 is homeomorphic to the unit sphere in H . So, proving that the minimization problem A0 := inf E(u 1 , u 2 ) u∈N0
(9)
has a solution (which is a solution of (4)) is analogous to doing the same for Tλ,μ in Proposition 2.1, see Sect. 3.4. However, except in particular cases (these will be the cases from statements (iv) in our theorems), the minimizer for A0 can be standard, that is, N0 is too large, and minimization on it does not give anything interesting. This is where the idea appears to minimize on N - note that this set no longer has the properties that a Nehari manifold has in the scalar case. Finally, we make several remarks with respect to the general theory of elliptic systems, developed in recent years (see for example the survey paper [9], and the references there). System (4) is of the so-called gradient type, that is, it can be written in the vector form −u = ∇u f (u), here f (u) = μ1 u 41 + 2βu 21 u 22 + μ2 u 42 − 2u 21 − 2λu 22 . It is generally thought that gradient systems are not much different from scalar equations. We see here that we have an important example for which it would be wrong to think in this way, if we are interested in finding nonstandard solutions. The reason for this is the fact that system (4) is not fully coupled. A general notion of full coupling for nonlinear systems was given and analyzed in [6]; for a system −u = ∇ f (u) full coupling would be implied for example by f u 1 (0, s) > 0, f u 2 (s, 0) > 0 for s > 0. It is the semi-coupled nature of system (4) which causes the phenomena described in Theorems 1 and 2 - if the system were fully coupled (for instance, if there were a term u 1 u 2 in f (u)), then it would have nonstandard positive ground states for any positive values of its parameters. 1 4
3. Proofs of Theorems 1 and 2 The first point in the proofs is to use the functions wλ,μ from Proposition 2.1 in order to obtain an upper bound for A. Then we are going to use this bound in order to study the behaviour of the minimizing sequences for A and Ar . The proofs of Theorems 1 and 2 will be carried out jointly, to some extent. 3.1. An upper bound on A. √ √ Set wλ (x) = wλ,1 (x) = λw1 ( λx), respectively Tλ = Tλ,1 , Sλ = Sλ,1 (see Proposition 2.1 for the notations). We introduce the function h : R+ → R+ , defined by 2 2 n w (x)w (x) d x h(λ) := R 1 4 λ . Rn w1 (x) d x Note that h depends only on λ and n. The following proposition gives some bounds on h.
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B. Sirakov
Proposition 3.1. For any λ ≥ 1 we have n
h(λ) ≤ λ1− 4
(10)
and n
n
λ1− 2 ≤ h(λ) ≤ σ λ1− 2 ,
(11)
where σ = σ (n) is the universal constant w 2 (0) n w 2 (x) d x σ = 1 R4 1 . Rn w1 (x) d x Proof. We know that w1 is radial and strictly decreasing in |x|. This implies that for λ ≥ 1, x ∈ Rn , √ w1 (x) ≥ w1 ( λx). √ By using this, the change of variables x → λx and the Hölder inequality we obtain √ √ n n w12 (x)wλ2 (x) d x ≥ λ1− 2 w14 ( λx)d( λx) = λ1− 2 w14 (x) d x, Rn
Rn
Rn
and
Rn
w12 (x)wλ2 (x)
dx ≤ λ
1 Rn
n
= λ1− 4
w14 (x)
Rn
2
dx
Rn
w14 (
√
1 λx) d x
2
w14 (x) d x.
√ Finally, by the change of variables x → x/ λ we have h(λ) = λ
1− n2
Rn
√ w12 (x/ λ)w12 (x) d x n =: λ1− 2 h 1 (λ). 4 Rn w1 (x) d x
By the monotonicity properties of w1 the function h 1 (λ) is increasing, and, by Lebesgue monotone convergence, h 1 (λ) → σ as λ → ∞. Next, consider the following linear system in k, l ∈ R: ⎧ μ1 k + βh(λ)l = 1, ⎨ ⎩
n
n
(12)
βh(λ)k + μ2 λ2− 2 l = λ2− 2 .
Note that k and l are determined solely by the parameters in system (4). The use of system (12) is seen from the following simple lemma. Lemma 3.1. Suppose the parameters λ, μ1 , μ2 , β in √ (4) are √such that the linear system (12) has a solution k > 0, l > 0. Then the couple ( kw1 , lwλ ) belongs to N .
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Proof. Recall (Proposition 2.1) that n wλ4 = 4Tλ = Sλ2 = λ2− 2 S12 , wλ 2λ =
(13)
Rn
for all λ ≥ 1. Hence (12) and the definition of h(λ) imply ⎧ ⎨ Rn μ1 w14 k 2 + Rn βw12 wλ2 kl = k S12 = kw1 21 , ⎩
Rn
βw12 wλ2 kl + Rn μ2 wλ4 l 2 = l Sλ2 = lwλ 2λ ,
and the lemma follows.
By using this lemma we will obtain an upper bound for the infima we are working with. Recall A0 is defined in (9), A and Ar are defined in (5). We have the following estimate. Proposition 3.2. Suppose the parameters λ, μ1 , μ2 , β in system (4) are such that the linear system (12) has a solution k > 0, l > 0. Then n 1 0 < A 0 ≤ A ≤ Ar ≤ k + λ2− 2 l S12 . 4 Proof. We only have to prove the first and the last inequality in Proposition 3.2. We use the fact that 1 1 u 1 21 + u 2 2λ = (Mu 2 , u 2 ), for all u ∈ N0 ⊃ N . (14) E(u) = 4 4 Rn Then (13) and Lemma 3.1 imply √ √ n 1 k + λ2− 2 l S12 . Ar ≤ E( kw1 , lwλ ) = 4 Note that for each u ∈ N0 , by Hölder and Sobolev inequalities, 2 2 (Mu 2 , u 2 ) ≤ C0 (u 1 4L 4 + u 2 4L 4 ) ≤ C1 (u 1 41 + u 2 4λ ), u 1 1 + u 2 λ = Rn
so E is bounded uniformly away from zero on N0 , and A0 > 0.
Here, and in the sequel, c0 , C0 , C1 denote positive constants which depend only on the parameters in system (4) and on the dimension n. Finally, let us list for further reference the conditions under which the solutions of (12) are positive: k > 0 and l > 0 if either Dλ > 0 or
and
n
n
βh(λ) < min{μ2 , μ1 λ2− 2 } = λ1− 4 min{ν1 , ν2 }, n
n
βh(λ) > max{μ2 , μ1 λ2− 2 } = λ1− 4 max{ν1 , ν2 },
(15) (16)
where we have set n
n
ν1 = λ1− 4 μ1 , ν2 = λ 4 −1 μ2 ,
n
Dλ = μ1 μ2 λ2− 2 − β 2 h 2 (λ).
In view of the bounds on h we proved in Proposition 3.1, we see that the conditions (15) n √ and (16) are implied by either − μ1 μ2 < β < min{ν1 , ν2 } or β > λ 4 max{ν1 , ν2}.
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3.2. Behaviour of the minimizing sequences for A. Proof of Theorem 1 (i) and (iv). The main goal of this section is to find conditions under which each minimizing sequence for A is such that the L 4 -norms of both components of the members of the sequence are bounded uniformly away from zero. Careful study of the bounds on the minimizing sequences that we obtain will permit us to prove Theorem 1, parts (i) and (iv). For each λ ≥ 1, set g(λ) = λn/4−1 h(λ) (17) (g(λ) ≤ 1 by Proposition 3.1). We have the following result. Proposition 3.3. Let {u m } ⊂ N be a sequence such that E(u m ) → A as m → ∞. Then there exists a constant c0 > 0 such that u m,1 L 4 (Rn ) ≥ c0 and u m,2 L 4 (Rn ) ≥ c0 for all m, provided −∞ < β < ν 0 , (18) where ν 0 is the smaller root of the equation g(λ)(2 − g(λ))x 2 − (ν1 + ν2 )x + ν1 ν2 = 0. Remark 1. We will show that the hypothesis on β in Theorem 2 (i) can be replaced by β ∈ (−∞, ν 0 ). It is easy to see that the upper bound in the statement of Theorem 2 (i) implies β < ν 0 . Indeed, n
1 ≥ g(λ)(2 − g(λ)) ≥ λ− 4 , since we have, by Proposition 3.1, n
n
3n
2h(λ)λ1− 4 − h 2 (λ) ≥ λ1− 4 h(λ) ≥ λ2− 4 .
(19)
Note that in (19) one uses two inverse inequalities from Proposition 3.1, so β < ν 0 is a considerably better upper bound than the one in Theorem 2 (i). Remark 2. An elementary computation shows that for all λ ≥ 1, ν1 ν2 ν0 ∈ , min{ν1 , ν2 } . ν1 + ν2 Proof of Proposition 3.3. Let {u m } ⊂ N be a minimizing sequence for A, that is, by (14), 1 1 2 2 E(u m ) = u m,1 1 + u m,2 λ = (Mu 2m , u 2m ) −→ A, 4 4 Rn as m → ∞. It follows that {u m } is bounded in H . We recall that u m,i ≡ 0 for each m, i. Set 1/2 1/2 u 4m,1 , ym,2 = u 4m,2 . ym,1 = Rn
Rn
By the Sobolev and Holder inequalities, it follows from the definition of Sλ , Proposition 2.1 and u m ∈ N that 2 2 μ1 u 4m,1 + βu 2m,1 u 2m,2 ≤ (μ1 ym,1 + β + ym,1 ym,2 ), (20) S1 ym,1 ≤ u m,1 1 = Rn n 2 λ1− 4 S1 ym,2 ≤ u m,2 2λ = βu 2m,1 u 2m,2 + μ2 u 4m,2 ≤ (μ2 ym,2 + β + ym,1 ym,2 ), (21) Rn
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where β + = max{β, 0}. Proposition 3.3 immediately follows for β ≤ 0. So, from now on we shall suppose β > 0. Adding up (20) and (21) results in n S1 (ym,1 + λ1− 4 ym,2 ) ≤ (Mu 2m , u 2m ) = 4 A + o(1), Rn
(22)
where o(1) → 0 as m → ∞. 1 ym,i . Thanks to Proposition 3.2 from (20)–(22) we obtain the following Set z m,i = S1 inequalities (k and l denote the positive solutions of (12)) ⎧ n n ⎪ z m,1 + λ1− 4 z m,2 ≤ k + λ2− 2 l + o(1), ⎪ ⎪ ⎪ ⎨ μ1 z m,1 + βz m,2 ≥ 1, (23) ⎪ ⎪ ⎪ ⎪ n ⎩ βz m,1 + μ2 z m,2 ≥ λ1− 4 . We would like to infer from (23) that the two sequences {z m,1 }, {z m,2 } stay uniformly away from zero. For this it is enough to show that each two of the lines n
n
l1 = {z = (z 1 , z 2 ) ∈ R2 : z 1 + λ1− 4 z 2 = k + λ2− 2 l}, l2 = {z ∈ R2 : μ1 z 1 + βz 2 = 1},
n
l3 = {z ∈ R2 : βz 1 + μ2 z 2 = λ1− 4 },
meet, and their crossing points have strictly positive coordinates (these lines are determined by the parameters in system (4)). Indeed, for large m the point (z m,1 , z m,2 ) is arbitrarily close to the triangle (or segment, or point) between these crossing points. Since β < ν 0 ≤ min{ν1 , ν2 } ≤ ν1 ν2 = μ1 μ2 , (24) we see that we have to verify the following inequalities n
n
βλ1− 4 < μ2 ,
β < μ1 λ1− 4 ,
μ1 (k + λ μ2 (k + λ
2− n2
β(k + λ
2− n2
l) > 1,
(26)
2− n2
(27)
l) > λ
2− n2
(25)
l) < λ
,
1− n4
.
(28)
Inequalities (25) can be recast as β < min{ν1 , ν2 }, which is true by (24). Since
n n λ2− 2 μ2 + μ1 λ2− 2 − 2βh(λ) n , k + λ1− 4 l = n μ1 μ2 λ2− 2 − β 2 h 2 (λ) and the denominator of this fraction is positive (by (24) and Proposition 3.1), elementary computations show that (26) is equivalent to n
(μ1 λ2− 2 − βh(λ))2 > 0, while (27) is equivalent to (μ2 − βh(λ))2 > 0, so (26) and (27) hold, thanks to (25) and Proposition 3.1. Finally, by developing (28) we see that it is equivalent to n 2h(λ)λ1− 4 − h 2 (λ) (29) β 2 − (ν1 + ν2 )β + ν1 ν2 > 0, n λ2− 2 which is implied by (18). This finishes the proof of Proposition 3.3.
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Next, we are going to show how inequalities (23) lead to the statement of Theorem 1 (i) and (iv). Proof of Theorem 1 (i) and (iv). Set tm,1 = z m,1 − k, tm,2 = z m,2 − l. By using system (12) with λ = 1 we have from inequalities (23), which are valid for β ≥ 0, ⎧ tm,1 + tm,2 ≤ o(1), ⎨ μ1 tm,1 + βtm,2 ≥ 0, (30) ⎩ βt + μ t m,1 2 m,2 ≥ 0. Now, whenever β < min{μ1 , μ2 }
or
β > max{μ1 , μ2 },
the three half-spaces {t : t1 + t2 ≤ 0}, {t : μ1 t1 + βt2 ≥ 0}, {t : βt1 + μ2 t2 ≥ 0} meet at most in a triangle in the (t1 , t2 )-plane, and this triangle shrinks to t1 = t2 = 0 at the limit m → ∞, so we have z m,1 → k, z m,2 → l as m → ∞. Then, by passing to the limit in (22) with λ = 1, and by using A ≤ 41 (k + l)S12 (Proposition 3.2), we obtain √ √ 1 (k + l)S12 = E( kw1 , lw1 ). 4 Parts (i) and (iv) of Theorem 1 are proved. A=
3.3. Proof of Proposition 1.1 and Theorem 2 (i). Proof of Proposition 1.1. Our goal is to show that any minimizer of E restricted to N is such that d E(u) = E (u) = 0. We write N = N1 ∩ N2 , where Ni is the set of u ∈ H such that u 1 ≡ 0, u 2 ≡ 0, and G i (u) = 0, with 2 2 |∇u 1 | + u 1 − μ1 u 41 + βu 21 u 22 , G 1 (u) = n n R R 2 2 G 2 (u) = |∇u 2 | + λu 2 − βu 21 u 22 + μ2 u 42 . Rn
Rn
We have, for each ψ = (ψ1 , ψ2 ) ∈ H (setting λ1 = 1, λ2 = λ), < E (u), ψ > =
2
∇u i ∇ψi + λi u i ψi − μi u i3 ψi − βu i u 2j ψi ,
j = i,
i=1
< G i (u),
ψ >= 2
∇u i ∇ψi + λi u i ψi − 2μi u i3 ψi − βu i u j (u i ψ j + u j ψi ) ,
j = i.
By computing < G i (u), u > for u ∈ Ni we see that G i (u) = 0 for i = 1, 2 and u ∈ N (since u i ≡ 0 on Ni ). Hence, supposing that u = (u 1 , u 2 ) ∈ N is a minimizer for E restricted to N , standard minimization theory implies the existence of two Lagrange multipliers L 1 , L 2 ∈ R such that E (u) + L 1 G 1 (u) + L 2 G 2 (u) = 0. Setting G 1 (u) = 0 in the expression < E (u) + L 1 G 1 (u) + L 2 G 2 (u), (u 1 , 0) >= 0, we are led to μ1 u 41 + L 2 βu 21 u 22 = 0. (31) L1 Rn
Rn
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Similarly, setting G 2 (u) = 0 in < E (u) + L 1 G 1 (u) + L 2 G 2 (u), (0, u 2 ) >= 0, we obtain 2 2 βu 1 u 2 + L 2 μ2 u 42 = 0. (32) L1 Rn
Rn
The system (31)–(32) has a strictly positive determinant and hence the unique solution L 1 = L 2 = 0. The positivity of the determinant follows from the Hölder inequality when β 2 < μ1 μ2 and from the fact that this determinant is diagonally dominant (as was already noted in [17]) for β < 0, by G 1 (u) = G 2 (u) = 0. Proof of Theorem 2 (i). Suppose we have a minimizing sequence of radial couples {u m } ⊂ N for Ar . Then, by standard functional analysis and the compact embedding Hr1 (Rn ) → L 4 (Rn ) the sequences {u m,i } converge (up to a subsequence) weakly in H 1 (Rn ) and strongly in L 4 (Rn ) to a function u i ∈ H 1 (Rn ). We have, by (14) and standard results on weak convergence,
u 1 21 + u 2 2λ ≤ lim inf u m,1 21 + u m,2 2λ = 4 Ar . (33) m→∞
In the previous subsection we proved that the L 4 -norms of both {u m,1 }, {u m,2 } are bounded away from zero, so the strong limit u = (u 1 , u 2 ) is nonstandard. In addition, we have 2 2 (Mu , u ) = lim (Mu 2m , u 2m ) = 4 lim E(u m ) = 4 Ar . (34) m→∞ Rn
Rn
m→∞
Next, let s1 , s2 be the solutions of the linear system 2 4 2 2 Rn μ1 u 1 s1 + Rn βu 1 u 2 s2 = u 1 1 , 2 2 2 4 Rn βu 1 u 2 s1 + Rn μ2 u 2 s2 = u 2 λ .
(35)
This system has a unique solution, see the end of the proof of Proposition 1.1. If s1 = s2 = 1 we are done, since then u ∈ N and by (33) and (34) u is a minimizer, so Proposition 1.1 finishes the proof of Theorem 2 (i). Lemma 3.2. Under the hypotheses of Proposition 3.3 the solution of system (35) satisfies s1 > 0, s2 > 0. Before proving Lemma 3.2, let us show how s1 = s2 = 1 follows from it. Recall the range given in Theorem 2 (i) is included in (18). Suppose (s1 , s2 ) = (1, 1) and set ⎞ ⎛ 4 2 2 Rn μ1 u 1 Rn βu 1 u 2 ⎠. B = ⎝ 2 2 4 Rn βu 1 u 2 Rn μ2 u 2 Since G 1 (u m ) = G 2 (u m ) = 0, u m u in H and u m → u in L 4 × L 4 , we have 2 2 u 1 1 ≤ lim inf u m,1 1 = lim inf μ1 u 4m,1 + βu 2m,1 u 2m,2 m→∞ m→∞ Rn = μ1 u 41 + βu 21 u 2m,2 , Rn
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and, similarly,
u 2 2λ ≤
Hence
B
s1 s2
Rn
=
βu 21 u 22 + μ2 u 42 .
u 1 21 u 2 2λ
(36)
1 B . 1
√ √ Set v1 = s1 u 1 , v2 = s2 u 2 . Then by the definition of s1 , s2 the couple (v1 , v2 ) is on N but by (14) and (34), 1 1 s s1 s1 1 , 1 > , > = 4 Ar =< ,B s2 1 1 1 (the brackets denote scalar product), which is a contradiction with the minimality of Ar . Hence s1 = s2 = 1, so u attains Ar . Remark. Note that we could not use the fact that E(u) = 41 (u 1 21 + u 2 2λ ) on N to get a contradiction like in the scalar case, since we cannot1 infer from (35) that s1 u 1 21 + s2 u 2 2λ < u 1 21 + u 2 2λ . This is very much in contrast with the situation which one has when minimizing a scalar functional (see the proof of Proposition 3.5 in the next section). Proof of Lemma 3.2. The lemma is obvious if β ≤ 0. So we can suppose β > 0. For example, let us prove that s1 > 0. We need to show that μ2 u 42 > u 2 2λ βu 21 u 22 . u 1 21 Rn
Rn
By Sobolev and Hölder inequalities this is implied by 1/2 2 4 2 u2 > βu 2 λ μ2 u 1 1 Rn
⇐
μ2
Rn
1/2 u 42
>
1/2 Rn
u 41
β u 2 2λ . S1
By using (36) we see that the last inequality is implied by 1/2 β 4 2 2 4 u2 > βu 1 u 2 + μ2 u 2 μ2 n S1 Rn Rn R 1/2 1/2 β β 4 4 ⇐ 1> u1 + u2 . S1 μ2 Rn Rn 1 Indeed, there exist linear systems a x + a x = b , i = 1, 2, with positive coefficients and positive i1 1 i2 2 i solutions, such that ai1 + ai2 > bi , i = 1, 2, but b1 x1 + b2 x2 > b1 + b2 – for example 8x1 + 4x2 = 11, 2x1 + 2x2 = 3.
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Since u is the limit of a minimizing sequence for A, we can use what we proved in the previous section (inequalities (23)–(28)). With the notations used in (23)–(28), the last inequality above can be recast as β (37) lim β z m,1 + z m,2 < 1. m→∞ μ2 By using consecutively the first inequality in (25) and the first inequality in (23), we see that (37) is implied by (28), which we have already shown to hold under the hypothesis of Proposition 3.3. To show that s2 > 0 the argument is analogous, by using Sλ = λ1−n/4 S1 and the second inequality in (25). Finally, A = Ar was established in [16] for small positive values of β. The same proof can be shown to be valid in our case, thanks to what we already proved, in particular inequalities (23)–(28) and Lemma 3.2. The proof of Theorem 2 (i) is finished. Going through the proof of Theorem 2 (i) we see that we only needed the hypotheses of Proposition 3.3 and Proposition 1.1 (see also the remark following Proposition 3.3), so we can state the following result. Proposition 3.4. The value Ar is attained by a nonstandard solution of (4) (and A = Ar if β ≥ 0), provided β < ν0 , where ν0 is the smaller root of the equation (see (17)) g(λ)(2 − g(λ))x 2 − (ν1 + ν2 )x + ν1 ν2 = 0. 3.4. Proof of Theorem 2 (iv) and extensions. The idea of the proof of statements (iv) in Theorems 1 and 2 is rather simple: should it turn out that A0 < min{E(u 1 , 0), E(0, u 2 )} (38) (u 1 , u 2 are defined in (8), A0 is defined in (9)), then the minimizer for A0 cannot be standard and is a least energy solution (of course in this case A0 = A). Recall that (u 1 , 0), (0, u 2 ) have least energy among the standard nontrivial solutions. We have the following (basically known) fact. Proposition 3.5. The minimal value A0 > 0 is attained by a nontrivial (possibly standard) radial solution of (4), provided β ≥ 0. The fact that A0 is attained by a solution of (4) can be proven for example through the same argument as in Chapter 4 of [23], where the scalar case is considered. We will give here, for the reader’s convenience and to permit comparison with the proofs in the previous sections, a direct argument leading to Proposition 3.5. Before proceeding, we recall some facts about spherical rearrangement (Schwarz symmetrization), see for example [15]. Proposition 3.6. Suppose v1 , v2 ∈ H 1 (Rn ) and let v1∗ , v2∗ be the radial functions obtained by Schwarz symmetrization from v1 , v2 . Then for any p ∈ [2, 6] if n = 3, p ≥ 2 if n ≤ 2, vi∗ H 1 ≤ vi H 1 , vi∗ L p = vi L p , (v1∗ )2 (v2∗ )2 ≥ v12 v22 . Rn
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Proof of Proposition 3.5. Take a minimizing sequence {u m } ⊂ N0 for A0 . Then {u m 2H } tends to 4 A0 (by (14)) so {u m } is bounded in H . By Proposition 3.6 the sequence of rearrangements u ∗m = (u ∗m,1 , u ∗m,2 ) is bounded in H , and hence converges weakly in H and strongly in L 4 × L 4 to a couple u ∗ . Hence, by u m ∈ N0 and Proposition 3.6, u ∗ 2H ≤ lim inf u ∗m 2H ≤ lim inf u m 2H = lim inf (Mu 2m , u 2m ) m→∞ m→∞ m→∞ ≤ lim (M(u ∗m )2 , (u ∗m )2 ) = (M(u ∗ )2 , (u ∗ )2 ), m→∞
E(u ∗ ) ≤ lim inf E(u ∗m ) ≤ lim inf E(u m ) = A0 . m→∞
m→∞
By the Sobolev inequality, Proposition 3.6 and u m ∈ N0 we have 2 2 ∗ ∗ ∗ 2 u m,1 L 4 + u m,2 L 4 ≤ C0 u m H ≤ C0 M((u ∗m )2 , (u ∗m )2 ) ≤ C1 u ∗m 4L 4 ×L 4 , Rn
so u ∗ = (0, 0). If u ∗ 2H = Rn (M(u ∗ )2 , (u ∗ )2 ), A0 is attained by u ∗ . If not, that is u ∗ 2H < Rn (M(u ∗ )2 , (u ∗ )2 ), take s ∈ (0, 1) such that v = su ∗ ∈ N0 . Then by (14) and Proposition 3.6, E(v) =
1 1 1 1 v2H < u ∗ 2H ≤ lim inf u ∗m 2H ≤ lim inf u m 2H = A0 , 4 4 4 m→∞ 4 m→∞
a contradiction with the definition of A0 . ∗ Hence u is a minimizer and there exists a Lagrange multiplier L ∈ R such that d E(u) u=u ∗ + L d u2H − Rn (Mu 2 , u 2 ) u=u ∗ = 0. Evaluating this differential against u ∗ and u ∗ ∈ N0 give Lu ∗ 2H = 0, i.e. L = 0. Next, set 2 1 u 1 21 + u 2 2λ . J (u) = J (u 1 , u 2 ) = 4 Rn (Mu 2 , u 2 ) Lemma 3.3. Suppose β ≥ 0. We have A0 =
inf
u∈H \{(0,0)}
J (u) =
inf
u∈Hr \{(0,0)}
J (u).
(39)
Proof. It is easy to see, by the Sobolev inequality and Proposition 3.6, that the two infima in (39) are positive and equal. Let B0 be their value. If u ∗ is a minimizer for A0 then J (u ∗ ) = A0 by u ∗ ∈ N0 , hence B0 ≤ A0 . If B0 < A0 take v = (0, 0) such that J (v) < A0 . Let s > 0 be such that sv ∈ N0 . Then 1 2 s v2H = E(sv) ≥ A0 > J (v) 4 implies v2H < s 2 Rn (Mv 2 , v 2 ), a contradiction with sv ∈ N0 .
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Further, define the function f (k1 , k2 ) := J ( k1 w1 , k2 wλ ) =
(k1 S12 + k2 λ2−n/2 S12 )2 1 , 2 2 4 S1 (μ1 k1 + 2βh(λ)k1 k2 + μ2 λ2−n/2 k22 )
on the set K = {(k1 , k2 ) : k1 ≥ 0, k2 ≥ 0, (k1 , k2 ) = (0, 0)} (recall the definition of h(λ) and (13)). Since f (k1 , 0) =
1 2 S = E(u 1 , 0), 4μ1 1
f (0, k2 ) =
1 2− n 2 λ 2 S1 = E(0, u 2 ), 4μ2
we see that for (38) to hold it is sufficient that f does not attain its minimum over K on the lines k1 = 0 or k2 = 0. The function f is a fraction of two quadratic forms in (k1 , k2 ), and elementary analysis shows that the quantity (ak1 + bk2 )2 ck12 + 2γ k1 k2 + dk22
(a, b, c, d, γ > 0)
does not attain its minimum in K on the axes if and only if aγ − bc > 0,
ad − bγ < 0,
(40)
and then the minimum is attained for k1 = bγ − ad, k2 = aγ − bc. Applying this to f (k1 , k2 ) we see that (40) becomes n
βh(λ) − μ1 λ2− 2 > 0,
n
n
μ2 λ2− 2 − βh(λ)λ2− 2 < 0,
or, equivalently, βg(λ) = β
h(λ)
(41) n > max{ν1 , ν2 }. λ1− 4 Inequality (41) is implied by the hypothesis of Theorem 2 (iv) (by Proposition 3.1), so Theorem 2 (iv) is proved. √ √ Remark. Note that, in the case λ = 1, the fact that the couple ( kw1 , lw1 ) (defined in Theorem 1) is a minimizer for A was already proved in Sect. 3.2. Since (38) (which follows from √ √ (41)) implies that A0 = A for β > max{μ1 , μ2 }, λ = 1, the couple ( kw1 , lw1 ) is a minimizer for A0 as well. It is possible to give other conditions under which (38) holds. For instance, we can compute min
(k1 ,k2 )∈K
J ( k 1 w1 , k 2 w1 )
and
min
(k1 ,k2 )∈K
J ( k1 wλ , k2 wλ ).
We have, by (13), 2 1 k1 S12 + k2 (S12 + (λ − 1) Rn w12 ) . J ( k 1 w1 , k 2 w1 ) = 4 S12 (μ1 k12 + 2βk1 k2 + μ2 k22 )
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We introduce the following universal constant: σ0 =
w1 2L 2 (Rn ) w1 4L 4 (Rn )
=
1 w12 . S12 Rn
Since w1 2H 1 = Rn w14 , we have σ0 ∈ (0, 1). Then J ( k 1 w1 , k 2 w1 ) =
S12 [k1 + k2 (1 + σ0 (λ − 1))]2 , 4 μ1 k12 + 2βk1 k2 + μ2 k22
(42)
from which it follows that sufficient conditions for (38) are ⎧ −1 ⎪ ⎪ β > max{μ1 bλ , μ2 bλ }, with bλ := 1 + σ0 (λ − 1) ∈ [1, λ), ⎨ (43) λ−2+n/2 [2βbλ − (μ1 bλ2 + μ2 )]2 μ2 < 1. μ1 (βbλ − μ2 )2 + 2β(βbλ − μ2 )(β − μ1 bλ ) + μ2 (β − μ1 bλ )2 √ √ We have obtained (43) by using (40) applied to J ( k1 w1 , k2 w1 ), and by comparing the minimal value given by (40) with E(0, u 2 ). Note that in the fraction in (43) we are dividing a polynomial of degree 2 in β by a polynomial of degree 3 in β. In order to get simpler to state sufficient conditions for (38), one could minimize the fraction in (42), with σ0 replaced by 1 (since σ0 < 1). Then one obtains the following conditions for the corresponding minimum to be attained away from the axes and to be smaller than min{E(u 1 , 0), E(0, u 2 )}: setting ξ1 = μ1 λ, ξ2 = μ2 /λ, γ1 = β − ξ1 , γ2 = β − ξ 2 , ⎧ γ1 > 0, γ2 > 0, and ⎪ ⎪ ⎨ (44) (γ1 + γ2 )2 max{ξ1 , λn/2 ξ2 } ⎪ ⎪ < 1. ⎩ ξ1 γ22 + 2βγ1 γ2 + ξ2 γ12 ⎪ ⎪ ⎩
n
For instance, when ξ1 = ξ2 = ξ this condition reads β > (2λ 2 − 1)ξ . Similarly, carrying out the above argument for 2 1 k1 (Sλ2 − (λ − 1) Rn wλ2 ) + k2 Sλ2 J ( k 1 wλ , k 2 wλ ) = 4 Sλ2 (μ1 k12 + 2βk1 k2 + μ2 k22 ) =
Sλ2 (k1 (1 − (1 − 1/λ)σ0 ) + k2 )2 4 (μ1 k12 + 2βk1 k2 + μ2 k22 )
= σ0 /λ), we are led to (we have again used (13), together with wλ 2L 2 (Rn ) wλ −4 L 4 (R n ) the following sufficient conditions for (38): ⎧ β > max{μ1 cλ−1 , μ2 cλ }, with cλ := 1 − σ0 (1 − 1/λ) ∈ (1/λ, 1], ⎪ ⎪ ⎨ (45) λ2−n/2 [2βcλ − (μ1 + μ2 cλ2 )]2 μ1 ⎪ ⎪ ⎩ < 1. μ1 (β − μ2 cλ )2 + 2β(β − μ2 cλ )(βcλ − μ1 ) + μ2 (βcλ − μ1 )2
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Likewise, minimizing the fraction obtained by replacing cλ by 1 in the expression of √ √ J ( k1 wλ , k2 wλ ) and comparing to min{E(u 1 , 0), E(0, u 2 )} gives the following sufficient condition: setting δ1 = β − μ1 , δ2 = β − μ2 , ⎧ δ1 > 0, δ2 > 0, ⎪ ⎪ ⎨
and (46)
(δ1 + δ2 )2 max{λ2−n/2 μ1 , μ2 } ⎪ ⎪ < 1. ⎩ μ1 δ22 + 2βδ1 δ2 + μ2 δ12 n
In particular, if μ1 = μ2 = μ, this condition reduces to β > (2λ1− 2 − 1)μ. To summarize, we state the following proposition. Proposition 3.7. The infimum A0 is attained by a nonstandard radial solution of system (4) provided one of the conditions (41), (43), (44), (45), (46) holds (then A0 = A = Ar ).
3.5. Proofs of statements (ii) and (iii) in Theorems 1 and 2. Suppose we have a nonstandard solution u = (u 1 , u 2 ) of system (4), such that u 1 ≥ 0, u 2 ≥ 0 in Rn . Note that each of the functions u i satisfies a linear equation −u i + ci (x)u i = 0 in Rn , where c1 (x) = 1 − μ1 u 21 (x) − βu 22 (x), c2 (x) = λ − βu 21 (x) − μ2 u 22 (x). So by the Strong Maximum Principle (see for example [11]) each of the functions u 1 , u 2 is strictly positive in Rn . By the results in [5] u 1 and u 2 are radial with respect to some point in Rn . Note that solutions of (4) which are in H 1 (Rn ) are also in C 2 (Rn ) and tend to zero as x → ∞ – this can be proved with the help of a classical “bootstrap” argument. Next, we multiply the first equation in (4) by u 2 , the second equation by u 1 , and integrate the resulting equations over Rn . This yields
(∇u 1 .∇u 2 + u 1 u 2 ) = Rn u 1 u 2 (μ1 u 21 + βu 22 ), Rn (∇u 1 .∇u 2 + λu 1 u 2 ) = Rn u 1 u 2 (βu 21 + μ2 u 22 ), Rn
from which it follows that ! " u 1 u 2 (λ − 1) + (μ1 − β)u 21 + (β − μ2 )u 22 = 0. Rn
This equality is in a contradiction with the positivity of u 1 and u 2 , as long as the three constants (λ − 1), (μ1 − β), (β − μ2 ) are of the same sign or zero, and one of them is not zero. These are statements (ii) in Theorems 1 and 2. By Proposition 1.1 if a minimizer for A (or Ar ) exists and β 2 < μ1 μ2 then there is a positive solution of system (4) (see also Remark 3 after Theorem 1). So the existence of a minimizer for A (or Ar ) gives a contradiction whenever the hypotheses of (ii) are satisfied, and we obtain statements (iii).
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4. Systems of d Equations In this section we state results on existence and non-existence of least energy nonstandard solutions of systems of an arbitrary number of equations. This, in addition, will permit to us to review the key points in the proofs of Theorems 1 and 2, and to show the underlying symmetries in the conditions, which were somewhat obscured by the simplifications we made in the introduction. The results we shall state reduce to Theorems 1 and 2 when d = 2 and β ≥ 0. In all that follows matrices and vectors will have nonnegative entries. All (in)equalities between vectors will be understood to hold component-wise. Let M = (μi j )i,d j=1 , λ = (λ1 , . . . , λd ), μii > 0, λi > 0, i = 1, . . . , d, be the coefficients of (3). We want to give conditions for existence and nonexistence of least energy solutions of (3) in terms of M, λ, d, and n. Define the matrix d λj = h . λi i, j=1 n
Note the entries of depend only on λ and n. Note also that we have h(1/a) = a 2 −2 h(a), for a ∈ R+ . For any two matrices P = ( pi j )i,d j=1 , Q = (qi j )i,d j=1 ∈ Md (R), we denote Pi = ( pi j )dj=1 ∈ Rd , P · Q = ( pi j qi j )i,d j=1 ∈ Md (R). We set e = (1, . . . , 1) ∈ Rd , λa := (λa1 . . . , λad ), for a > 0. #d We define u := (u 1 , . . . , u d ), u 2 := (u 21 , . . . , u 2d ), u2H := i=1 u i 2λi , 1 1 < Mu 2 , u 2 >, E(u) = u2H − 2 4 Rn < Mu 2 , u 2 > , N0 : = u ∈ H, u ≡ (0, . . . , 0) : u2H = Rn ⎧ ⎫ d ⎨ ⎬ N = u ∈ H : u i ≡ 0 and u i 2λi = μi j u 2j u 2j , i = 1, . . . , d , ⎩ ⎭ Rn j=1
A0 : = inf E(u), u∈N0
A = inf E(u), u∈N
Ar =
inf
u∈N ∩Hr
E(u).
We next list our hypotheses on M and λ. Hypothesis 1. Suppose M and λ are such that the linear system in k ∈ Rd (M · )k = e
(47)
has a solution k > 0. This hypothesis guarantees an upper bound on A, as in Sect. 3.1. Hypothesis 2. For a solution k > 0 of system (47), all solutions z ∈ Rd of the system of d + 1 linear inequalities n M z ≥ λ1− 4 (48) n n < λ1− 4 , z > ≤ < λ2− 2 , k > are such that z > 0 (the brackets denote scalar product).
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This hypothesis guarantees that minimizing sequences for A are such that the L 4 norms of all their components are bounded uniformly away from zero. Note that when λ = e system (48) is equivalent to Mt ≥ 0, e.t ≤ 0, with t = z − k. Hypothesis 3. Given a symmetric matrix P ∈ Md (R) such that pi2j ≤ pii p j j for all √ d satisfies (48), we have det(M · P) = i, j ∈ {1, . . . , d}, and the vector z = (S1−1 pii )i=1 0. This hypothesis guarantees that a minimizer for A is a solution of system (3), as in the proof of Proposition 1.1. Hypothesis 4. Given P, z as in Hypothesis 3, and a vector v ∈ Rd such that 1− n4
S12 λi
z i ≤ vi ≤
d
μi j pi j ,
j=1 n
for all i = 1, . . . , d (note this condition implies λ1− 4 ≤ v ≤ M z), the solution of the linear system (M · P)s = v is such that s > 0. This hypothesis permits to complete the existence result, as in Sect. 3.3. Note that Hypotheses 1, 2 and 4 are only about systems of linear equalities and inequalities (that is, intersections of hyperplanes or half-spaces in Rd ), and as such are trivial to verify for any given set of parameters M, λ. Equivalent formulations of these hypotheses can be obtained through variants of Farkas’ Lemma, for example. Further, it is not difficult to give hypotheses on the off-diagonal entries of M with respect to the diagonal of M, under which Hypothesis 3 holds. For instance, it is easy to see that there exists a constant C(d) depending only on d, such that if μi j ≤ C(d)μii μ j j for all i, j, then pi2j ≤ pii p j j implies M · P is positive definite. We have C(2) = 1, C(3) = 13 , C(4) ≥ 17 , etc. Hypothesis 5. There exist indices i, j ∈ {1, . . . , d} such that either λi > λ j and Mi ≤ M j , or λi ≥ λ j and Mi ≤ M j , Mi = M j . Under this hypothesis it is possible to carry out the contradiction argument in Sect. 3.5. As in Sect. 3.4, setting J (u) = d
u4H 1 , 4 Rn < Mu 2 , u 2 >
for u ∈ H = [H 1 (Rn )] , we have A0 = inf u∈H \{0,...,0} J (u), and the infimum is attained by a nontrivial solution of (3). Hence if this infimum is strictly smaller than the energies of the vectors which have wλi ,μii as the i th component and whose other components are zero, for all i, then A0 is attained by a nonstandard solution of (3). A sufficient condition for this is given for example by the following hypothesis (compare to the argument after Lemma 3.3).
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Hypothesis 6. The infimum of the function f : [0, ∞)d → R, defined by n
< λ2− 2 , k >2 f (k) = < (M · )k, k > is strictly smaller than its values on the coordinate axes in Rd . So we can state the following extension of Theorem 2. Theorem 3. Let d ≥ 2. (i) If Hypotheses 1-4 hold, then A = Ar is attained by a solution u of (3), such that u > 0 in Rn . (ii) If Hypothesis 5 holds, then system (3) does not have a solution with nonnegative components, such that u i ≡ 0, u j ≡ 0. (iii) If Hypotheses 3 and 5 hold, then A and Ar are not attained. (iv) If Hypothesis 6 holds, then A0 = A = Ar is attained by a nonstandard radial solution of (3). Finally, it is clear that if (after eventual permutation of indices) there is a d1 ∈ {2, . . . , d} such that the hypotheses above hold for M replaced by the d1 -principal minor of M and λ replaced by (λ1 , . . . , λd1 , 0, . . . , 0), then Theorem 3 applies, giving nonstandard solutions with u d1 +1 = . . . = u d ≡ 0. 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. Comptes Rendus Mathematique 342(7), 453–458 (2006) 3. Bartsch, T., Wang, Z.-Q., Wei, J.: Bound states for a coupled Schrödinger system. Preprint 4. Buljan, H., Schwartz, T., Segev, M., Soljacic, M.: Christoudoulides, D.: Polychromatic partially spatially incoherent solitons in a noninstantaneous Kerr nonlinear medium. J. Opt. Soc. Am. B. 21(2), 397– 404 (2004) 5. Busca, J., Sirakov, B.: Symmetry results for semilinear elliptic systems in the whole space. J. Diff. Eq. 163(1), 41–56 (2000) 6. Busca, J., Sirakov, B.: Harnack type estimates for nonlinear elliptic systems and applications. Ann. Inst. H. Poincaré Anal. Non Lin. 21(5), 543–590 (2004) 7. Christodoulides, D., Eugenieva, E., Coskun, T., Mitchell, M., Segev, M.: Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media. Phys. Rev. E 63, 035601 (2001) 8. Coffman, C.V.: Uniqueness of the ground state solution for −u + u = u 3 and a variational characterization of other solutions. Arch. Rat. Mech. Anal. 46, 81–95 (1972) 9. de Figueiredo, D.G.: Nonlinear elliptic systems. Anais da Academia Brasileira de Ciências 72(4), 453– 469 (2000) 10. Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in Rn . Adv. Math. Studies 7A, 209–243 (1979) 11. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. 2nd edition. BerlinHeidelberg-Newyork: Springer-Verlag,1983 12. Hioe, F.T.: Solitary waves for N coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 82, 1152– 1155 (1999) 13. Kutuzov, V., Petnikova, V.M., Shuvalov, V.V., Vysloukh, V.A.: Cross-modulation coupling of incoherent soliton modes in photorefractive crystals. Phys. Rev. E 57, 6056–6065 (1998) 14. Kwong, M.K.: Uniqueness of positive solutions of −u+u = u p in Rn . Arch. Rat. Mech. Anal. 105, 243– 266 (1989) 15. Lieb, E., Loss, M.: Analysis. Providence, RI: Amer. Math. Soc. 1996 16. Lin, T.C., Wei, J.: Ground state of N coupled nonlinear Schrödinger equations in Rn , N ≤ 3. Commun. Math. Phys. 255, 629-653 (2005), see also Erratum, to appear in Commun. Math. Phys.
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17. Lin, T.C., Wei, J.: Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. H. Poincaré, Anal. Non-Lin. 22(4), 403–439 (2005) 18. Maia, L., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled Schrödinger system. J. Differ. Eqs. 229, 743–767 (2006) 19. Manakov, S.V.: On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Zh. Eksp. Teor. Fiz. 65, 505–516 (1973), English translation in J. Exp. Th. Phys. 38, 248–256 (1974) 20. Mitchell, M., Chen, Z., Shih, M., Segev, M.: Self-trapping of partially spatially incoherent light. Phys. Rev. Lett. 77, 490–493 (1996) 21. Mitchell, M., Segev, M.: Self-trapping of incoherent white light. Nature (London) 387, 880–882 (1997) 22. Rabinowitz, P.: Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series in Mathematics 65, Providence, RI:Amer. Math. Soc., 1986 23. Willem, M.: Minimax methods. Berlin-Heidelberg-Newyork, Springer, 1996 24. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media. Zh. Eksp. Teor. Fiz. 61, 118–134 (1971), English translation in J. Exp. Th. Phys. 34, 62–69 (1972) Communicated by P. Constantin
Commun. Math. Phys. 271, 223–246 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0182-2
Communications in
Mathematical Physics
Unitary Positive-Energy Representations of Scalar Bilocal Quantum Fields Bojko Bakalov1 , Nikolay M. Nikolov2,3 , Karl-Henning Rehren3 , Ivan Todorov2,3 1 Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA.
E-mail: [email protected]
2 Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72,
BG-1784 Sofia, Bulgaria. E-mail: [email protected], [email protected]
3 Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1,
D-37077 Göttingen, Germany. E-mail: [email protected] Received: 6 June 2006 / Accepted: 18 July 2006 Published online: 9 January 2007 – © Springer-Verlag 2007
Abstract: The superselection sectors of two classes of scalar bilocal quantum fields in D ≥ 4 dimensions are explicitly determined by working out the constraints imposed by unitarity. The resulting classification in terms of the dual of the respective gauge groups U(N ) and O(N ) confirms the expectations based on general results obtained in the framework of local nets in algebraic quantum field theory, but the approach using standard Lie algebra methods rather than abstract duality theory is complementary. The result indicates that one does not lose interesting models if one postulates the absence of scalar fields of dimension D − 2 in models with global conformal invariance. Another remarkable outcome is the observation that, with an appropriate choice of the Hamiltonian, a Lie algebra embedded into the associative algebra of observables completely fixes the representation theory. 1. Introduction An important tool in the study of globally conformal invariant (GCI) quantum field theory models in even space-time dimensions D ≥ 4 [21, 19, 20, 18] are bilocal fields, which arise in operator product expansions (OPE) as follows. Let φ(x) be a local scalar field of integer dimension d ≥ 2d0 := D − 2. We denote the contribution of twist D − 2 in the OPE of ((x1 − x2 )2 )d−d0 · φ ∗ (x1 )φ(x2 ) by W (x1 , x2 ) = W (x2 , x1 )∗
(1.1)
V (x1 , x2 ) = V (x2 , x1 ) = V (x1 , x2 )∗
(1.2)
if φ(x) is complex, and by
if φ(x) = φ ∗ (x) is real. This means that the expansion of W (x1 , x2 ) into local fields yields an infinite series of conserved symmetric traceless tensor currents, starting with the scalar field W (x, x) of dimension 2d0 (which may be zero), and including the
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stress-energy tensor. The similar expansion of V (x1 , x2 ) contains only tensors of even rank. It is a nontrivial consequence of GCI that the twist D − 2 fields W and V are bilocal in the sense of (strong = Huygens) local commutativity with respect to both arguments [17]. Depending on the scaling dimension d of the field φ(x) whose OPE produces the bilocal field, the latter may exhibit different singularities in its correlation functions. In the case d = D − 2 when the singularities have the lowest possible degree, one can derive the commutation relations of the bilocal field, and they involve another bilocal field with the same properties. If we assume uniqueness of the bilocal field, then it can be normalized in such a way that the commutation relations take the form [W (x1 , x2 ), W (x3 , x4 )] = Δ2,3 W (x1 , x4 ) + Δ1,4 W (x3 , x2 ) + N · Δ12,34
(1.3)
in the complex case, and [V (x1 , x2 ), V (x3 , x4 )] = Δ2,3 V (x1 , x4 ) + Δ2,4 V (x1 , x3 ) + Δ1,4 V (x2 , x3 ) + Δ1,3 V (x2 , x4 ) + N · (Δ12,34 + Δ12,43 ) (1.4) in the real case, where Δi,+ j = Δ+ (xi − x j ) and Δi, j = Δi,+ j − Δ+j,i are the two-point and commutator functions of a massless free scalar field (which has dimension d0 ), and Δ12,34 = Δ+1,4 Δ+2,3 − Δ+4,1 Δ+3,2 . The coefficients N are the normalizations of the four-point functions 0|W (x1 , x2 )W (x3 , x4 )|0 = N · Δ+1,4 Δ+2,3 , 0|V (x1 , x2 )V (x3 , x4 )|0 = N · (Δ+1,4 Δ+2,3 + Δ+1,3 Δ+2,4 ).
(1.5)
The above relations are such that if W (x1 , x2 ) satisfies (1.3), then V (x1 , x2 ) = W (x1 , x2 ) + W (x2 , x1 ) satisfies (1.4) with 2N instead of N . Our goal in this paper is to explore and rule out the possibility of realizations of the above commutator relations in GCI models, other than the free field realizations ∗
W (x1 , x2 ) = :ϕ (x1 ) · ϕ(x2 ): ≡
N
:ϕ p∗ (x1 )ϕ p (x2 ):
(complex),
(1.6)
p=1
where ϕ p (x) are N mutually commuting complex massless free fields of dimension d0 , and V (x1 , x2 ) = :ϕ(x1 ) · ϕ(x2 ): ≡
N
:ϕ p (x1 )ϕ p (x2 ):
(real),
(1.7)
p=1
where ϕ p (x) are N mutually commuting real massless free fields. (Free-field constructions of bilocal fields involving spinor or vector fields can also be given; their correlations exhibit higher singularities.) Clearly, the free field realizations exist only when N is a positive integer. Indeed, it was shown in [19] for the real case that due to Hilbert space positivity in the vacuum sector the coefficient N in (1.4) must be a positive integer and the bilocal field V (x1 , x2 ) is of the form (1.7) (or N = 0 corresponding to the trivial case V = 0). As a byproduct of our considerations, we establish this result of [19] both in the complex and the real case.
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In particular we deduce that, if φ(x) is a real scalar field of dimension D − 2, which is assumed to be the unique such field in the model, and V (x1 , x2 ) arises in the OPE of φ(x) with itself, then V (x, x) is a multiple of φ(x) and thus φ(x) is a sum of Wick squares. Dropping the uniqueness assumption, a similar statement is still true, but then a linear combination of bilocal fields of the form (1.7) with different positive coefficients may appear [17]. It is important to observe that expressions (1.6) and (1.7) are defined only on the Fock space of the free fields or on subspaces thereof. When discussing theories containing a scalar field of dimension D − 2 one has to envisage the possibility of different representations of the associated bilocal field occurring in the full Hilbert space H. A pertinent general structure theorem in the framework of algebraic QFT [7] states that all superselection sectors of a local QFT A are contained in the vacuum representation of a canonically associated (graded local) field extension F, and they are in a one-to-one correspondence with the irreducible unitary representations of a compact gauge group G (of the first kind) of internal symmetries of F, so that A ⊂ F consists of the fixed points under G. The gauge group is unitarily implemented in the vacuum Hilbert space H of F, and the central projections in U (G) decompose this Hilbert space into inequivalent representations of A ⊂ U (G) . 1 In the above free field representation, the bilocal fields are the fixed points under the gauge group U(N ) or O(N ) in the theory of N free complex or real massless scalar fields, respectively. Since the latter is known to have no nontrivial superselection sectors ([24, Sect. 3.4.5–6] and [5, App. A]), one can conclude that it coincides with the above canonical field net F, so that the sectors of the bilocal field are in correspondence with the representations of G = U(N ) or O(N ). However, the general theorem of [7] evokes a great amount of abstract group duality, and its application requires the passage from fields to nets of local algebras and back. Although this passage is well understood in the case of free fields, it would be desirable to see the comparatively simple assertion emerge by more elementary methods. We are interested in the unitary positive-energy representations of the infinite dimensional Lie algebras of local commutators (1.3) and (1.4). We shall pursue two alternative formulations of the problem. The first consists in defining energy positivity with respect to the conformal Hamiltonian canonically expressed in terms of the fields (Eq. (2.4) below). In this case, we do not assume in advance the free field realizations (1.6) and (1.7) in the vacuum sector, and not even that N ∈ N in the commutation relations; these properties will be deduced instead from unitarity (cf. [19]). In the second formulation, in order to deal with a more general class of “additively renormalized” Hamiltonians (see the next section for details), given the free field realizations in the vacuum sector, we assume that the representations are generated from the vacuum by (relatively) local fields. This allows us to use the Reeh–Schlieder theorem [23], according to which every local relation among Wightman fields which holds on the vacuum vector must hold in the full Hilbert space H generated by other (relatively) local fields. Apart from the commutation relations, there are polynomial relations of the form N +1 det W (xi , x j ) − contraction terms = 0, i, j=1
(1.8)
1 U (G) = {U (g) : g ∈ G} are the unitary representers of the gauge group, U (G) is the commutant of U (G), i.e., the algebra of bounded operators on H which commute with every element of U (G), and U (G) is the commutant of U (G) , which coincides with the weak closure of the linear span of U (G).
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which arise from (1.6) and (1.7) by expanding the left-hand-side of the identity N +1 : det ϕ ∗ (xi ) · ϕ(x j ) :=0 i, j=1
(1.9)
as a polynomial in the normal products :ϕ ∗ (xi ) · ϕ(x j ): in accord with Wick’s theorem (and similarly in the real case). The determinant relations (1.8) are valid in the Fock space representation, and hence, by the Reeh–Schlieder theorem, also in other superselection sectors. Therefore, the algebra of bilocal fields may be regarded as the quotient of the associative enveloping algebra of the Lie algebra (1.3) or (1.4) by the determinant relations (1.8) and possibly some additional relations.2 We shall demonstrate that in both formulations of the problem, one arrives at the same classification of unitary positive-energy representations (superselection sectors) of the Lie algebras (1.3), (1.4) of bilocal fields. We find precisely those representations that occur in the Fock space of N free scalar fields, and they are in a one-to-one correspondence with the irreducible unitary representations of U(N ) in the complex case or O(N ) in the real case. A possible application to the program of classification of GCI models is the following. If bilocal fields satisfying (1.3) or (1.4) appear in the OPE of some local fields of the model, then all other (relatively) local fields, which possibly intertwine different superselection sectors, may be regarded as Wick products of a free field multiplet transforming in some representation of U(N ) or O(N ), possibly tensored with some other GCI fields that decouple from the free scalar fields. The classification problem may then be focused on subtheories with the restrictive feature that they decouple from massless scalar free fields. This is the field theoretic formulation of the result in algebraic QFT [6] that every quantum field theory extension of A is contained in F ⊗ B, where F is the canonical field net associated with A as above, while B is an arbitrary (graded) local net. The Lie algebras (1.3) and (1.4) are isomorphic to u(∞, ∞) and sp(∞, R), respectively. For the classification of their unitary positive-energy representations, we use methods of highest-weight representations of finite-dimensional Lie algebras. In fact, we adapt the methods of [8, 9] developed for the proof of the Kashiwara–Vergne conjecture [14] (see below) in the finite-dimensional case, and ideas of [25] generalizing to the infinite-dimensional case. Thus, our study relates two independent earlier developments. The first is the general insight into the structure and origin of superselection sectors within Haag’s operator algebraic approach to QFT [10], culminating in the quoted results of Doplicher and Roberts [7] and subsequent work [6]. The second is the Kashiwara–Vergne conjecture [14], proved by Enright and Parthasarathy [8] and by Jakobsen [11], according to which all unitary highest-weight representations of certain reductive Lie algebras occur in tensor products of the Segal–Shale–Weil representation. This amply generalizes the seminal method of Jordan [12] and Schwinger [26] to embed the angular momentum algebra su(2) into the algebra of two real harmonic oscillators. In the present context, it is the analog (for finitely many degrees of freedom) of our free field Fock representation. It should be noted, however, that our assumptions do not precisely match those of [14, 8, 11]. In the latter, the authors enforce half-integrality of the Cartan spectrum by demanding integrability of the representations of u(n, n) and sp(2n, R) to representations of U(n, n) and of the metaplectic group Mp(2n, R) (the two-fold covering of 2 Our result shows that if additional relations exist, then they are satisfied automatically in every unitary positive-energy representation satisfying the determinant relations.
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Sp(2n, R)), respectively. While we have no direct quantum field theoretic motivation for such requirements, it turns out that in our approach the same constraints on the spectrum arise either by the choice of the canonical conformal Hamiltonian, or by the validity of the determinant relations (1.8) through the Reeh–Schlieder theorem. Let us point out that all of our results hold also for spacetime dimension D = 1. In this case the bilocal field W (x1 , x2 ) generates the vertex algebra W1+∞ with a central charge −N (corresponding to +N in the terminology of the present article, Eq. (1.3), and corresponding to the central charge c = 2N as defined in [18] in terms of the stressenergy tensor). The representation theory of W1+∞ was developed by Kac and Radul, and conclusions similar to ours were obtained in [13]. Despite the fact that we do not mention nor use conformal invariance in the main body of our paper, it should be stressed that the expansions of the bilocal fields W (x1 , x2 ) and V (x1 , x2 ) into local fields of twist D −2 include the conformal stress-energy tensor. This implies that the conformal Lie algebra so(D, 2) is embedded in the (suitably completed and centrally extended) Lie algebras u(∞, ∞) and sp(∞, R), thus generating the global conformal invariance of the bilocal fields. In order not to distract the reader’s attention from the main line of argument, we have relegated the construction of the stress-energy tensor and of the conformal generators to Appendix B. 2. Classification: the Complex Case We classify all irreducible unitary positive-energy representations (superselection sectors) of the Lie algebra (1.3) of the complex bilocal field W (x1 , x2 ). The completely analogous case of the real bilocal field V (x1 , x2 ) will be sketched in the next section. 2.1. Statement of the results. We first identify the commutation relations (1.3) with the Lie algebra u(∞, ∞) as follows. We choose an orthonormal basis of the one-particle space, i.e., a basis of functions f i (i = 1, 2, . . .) of positive energy such that Δ+ ( f¯i , f j ) = δi j , Δ+ ( f i , f¯j ) = Δ+ ( f i , f j ) = Δ+ ( f¯i , f¯j ) = 0.
(2.1)
Smearing W (x1 , x2 ) with these functions and their conjugates, we define the generators X i j = W ( f¯i , f¯j ),
X i∗j = W ( f j , f i ),
N δi j , E i+j = (E +ji )∗ = W ( f i , f¯j ) + 2 N ∗ ¯ δi j , E i−j = (E − ji ) = W ( f j , f i ) + 2
i, j = 1, 2, . . . .
(2.2)
Then the commutation relations (1.3) become equivalent to the ones of u(∞, ∞) (considered as a Lie algebra over the complex numbers equipped with the real structure given by the conjugation properties in (2.2)): − − + [E i+j , E kl ] = δ jk E il+ − δil E k+j , [E i−j , E kl ] = δ jk E il− − δil E k−j , [E i+j , E kl ] = 0, ∗ ∗ ] = δ jl X ki , [E i+j , X kl
[E i+j , X kl ] = −δil X k j ,
∗ [E i−j , X kl ] = δ jk X il∗ ,
[E i−j , X kl ] = −δik X jl ,
− ∗ ] = δik El+j + δ jl E ki . [X i j , X kl
(2.3)
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These relations depend neither on the space-time dimension nor on the parameter N occurring in (1.3), which has been absorbed by the shift of the generators E ii± . The Lie algebra alone “ignores” its field theoretic origin (1.6). This observation could create the impression that the classification problem does not depend on the parameter N at all. In fact, the parameter N will reappear either through the additional determinant relations (1.8), or through the canonical choice of the Hamiltonian defining the condition of positive energy, as discussed below. We say that a representation has positive energy if the Hamiltonian is well defined and diagonalizable, its spectrum is bounded from below and all of its eigenspaces are finite dimensional. Remark 1. In spite of the fact that the Lie algebra commutation relations (2.3) are equivalent to the commutation relations (1.3) of the bilocal field W (x1 , x2 ) via (2.2), it is not evident if they will fix, up to a unitary equivalence, the field action. It remains to fix in addition the action of the central modes. Such a situation arises, for instance, in the case of an abelian current in two-dimensional (chiral) conformal field theory. In the case at hand of a harmonic bilocal field, the d’Alembert equation entails that if a mode of the type of (2.2) is zero in the vacuum representation, then it will be zero also in any other representation where the d’Alembert equation is satisfied. But since we are interested only in representations which are locally intertwined with the vacuum sector, the d’Alembert equation does hold by virtue of the Reeh-Schlieder theorem. One can easily check the statement in the mode representation given in Appendix A. Hence, for harmonic bilocal Lie fields the Lie algebra commutation relations uniquely fix the whole theory. The canonical conformal Hamiltonian associated with the free field expression (1.6) is Hc =
∞
εi · (E ii+ + E ii− − N ),
(2.4)
i=1
provided we choose the above orthonormal basis { f i } to diagonalize the one-particle conformal Hamiltonian with eigenvalues εi . The latter are positive integers and occur with finite multiplicities depending on the space-time dimension. (An explicit diagonalization of the conformal Hamiltonian will be provided in Appendix A.) We also introduce the charge operator Q=
∞ (E ii+ − E ii− ),
(2.5)
i=1
which generates the center of the Lie algebra u(∞, ∞), and we demand that Q is well defined on the representation. We define a vacuum representation as an irreducible unitary positive-energy representation of the commutation relations (1.3) of the bilocal field W (x1 , x2 ), in which Q is well-defined and Hc has the eigenvalue 0 on the ground state |0 (the vacuum state). We shall show in Corollary 1 below that the vacuum representation exists and is unique, and that the condition Hc |0 = 0 is equivalent to the seemingly stronger requirement that the vacuum expectation value of W (x1 , x2 ) vanishes (and similarly for V (x1 , x2 )). Now we can state our first main result.
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Theorem 1. Consider irreducible unitary positive-energy representations of the commutation relations (1.3) of the bilocal field W (x1 , x2 ), or equivalently of the Lie algebra u(∞, ∞), with fixed N . We assume that the charge operator (2.5) is well defined. The condition of “positive energy” is with respect to the conformal Hamiltonian Hc (2.4). Then: (i) N is a nonnegative integer, and all irreducible unitary positive-energy representations of u(∞, ∞) are realized (with multiplicities) in the Fock space of N complex massless free scalar fields by (1.6). (ii) The ground states of equivalent representations of u(∞, ∞) in the Fock space form irreducible representations of the gauge group U(N ). This establishes a one-to-one correspondence between the irreducible representations of u(∞, ∞) occurring in the Fock space and the irreducible representations of U(N ). The case N = 0 is included, meaning that there is only the trivial representation. The important conclusion is that all superselection sectors of the bilocal field W (x1 , x2 ) are realized in the Fock space of N complex massless free scalar fields, as anticipated following the result of [7], obtained in Haag’s framework [10] using local nets of von Neumann algebras. Moreover, the multiplicity of a representation of W (x1 , x2 ) in the Fock space equals the dimension of the corresponding representation of the gauge group U(N ). We shall prove Theorem 1 in the remainder of this section. In distinction to [7], our proof proceeds in a very concrete way based on the Wightman framework rather than the framework of local von Neumann algebras. A remarkable consequence of Theorem 1 is that the determinant relations (1.9) are automatically satisfied in every unitary positive-energy representation of the Lie algebra u(∞, ∞). For completeness, we display the Fock space representation (1.6) of the generators (2.2) of u(∞, ∞): X i j = bi · a j , 2E i+j = ai ∗ · a j + a j · ai ∗ , 2E i−j = bi ∗ · b j + b j · bi ∗ ,
(2.6)
where ai = ϕ( f¯i ) and bi = ϕ ∗ ( f¯i ) are the annihilation operators for the fields ϕ and ϕ ∗ , respectively, and the (bold face) vector notation indicates that these fields are multiplets of size N . The creation operators are ai ∗ = ϕ ∗ ( f i ) and bi ∗ = ϕ( f i ), and together with the annihilation operators they satisfy the canonical commutation relations: p q p q p q ∗ (2.7) ai , a j = δ p,q δi, j = bi , b j ∗ , ai , b j (∗) = 0, etc. Next, we give an alternative formulation of the classification problem as follows. We start with the assumption that N is a positive integer and the bilocal field W (x1 , x2 ) has the free field realization (1.6) in the vacuum sector, while all other superselection sectors are generated from the vacuum one by (relatively) local fields. We again require that the charge operator (2.5) be well defined, but now we allow a general Hamiltonian of the form H=
∞
εi · (E ii+ + E ii− − gi ).
(2.8)
i=1
It is sufficient to assume that the energies εi form an increasing sequence of positive numbers, 0 < ε1 ≤ ε2 ≤ · · · , with finite degeneracies. The real parameters gi replacing the vacuum energy subtractions in (2.4) may be regarded as a (finite or infinite) additive
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renormalization of the Hamiltonian. They will be adjusted in such a way that the sum (2.8) converges in the representations under consideration. We shall see in Subsect. 2.6 that, using the Reeh–Schlieder theorem, this proviso eventually fixes the parameters gi up to a finite renormalization, which is of course irrelevant. We thus postulate that an operator (2.8) exists and is bounded from below with finite degeneracies. To apply the Reeh–Schlieder theorem, we consider the operators n Dn := det X i j i, j=1 , (2.9) which arise by smearing the multilocal fields det(W (xi , x j ))i,n j=1 with the conjugates f¯i (i = 1, . . . , n) of the first n basis functions in both arguments. In the vacuum representation, all multilocal determinant fields of the form (1.8) vanish, in particular D N +1 vanishes, while D N = 0. (The contraction terms in (1.8) are absent in this case because all f¯i carry negative energy. Infinitely many other determinant relations are generated from D N +1 by taking commutators with the generators, but will not be needed for our argument.) Appealing to the Reeh–Schlieder theorem, we shall require that D N +1 also vanish in the unitary representations of interest, while D N = 0. Theorem 2. Consider irreducible unitary positive-energy representations of the Lie algebra u(∞, ∞) on which the operator D N +1 vanishes, while D N = 0 for some N ∈ N. Assume that the charge operator (2.5) is well defined, and the condition of “positive energy” holds for the generalized Hamiltonian (2.8). Then the conformal Hamiltonian (2.4) is well defined and it differs from the generalized one by a finite additive constant. Therefore, the alternative assumptions of Theorem 2 lead to the same classification as in Theorem 1. The proof of Theorem 1 will be given in Subsect. 2.2–2.5, while Theorem 2 will be proven in Subsect. 2.6. 2.2. The ground state and the Cartan spectrum. We start with preliminary considerations which hold for a general Hamiltonian H of the form (2.8). Consider an irreducible unitary positive-energy representation of the Lie algebra u(∞, ∞) with commutation relations (2.3). The Cartan subalgebra of u(∞, ∞) is spanned by the generators E ii± , which commute with each other and with the Hamiltonian H . Since, by assumption, H is diagonalizable with finite-dimensional eigenspaces, it follows that the Cartan generators can be simultaneously diagonalized. By the commutation relations, the spectrum of the Cartan generators E ii± is integer-spaced and in particular discrete. A joint eigenvalue is a pair of sequences h + = − + − On a state (h +1 , h +2 , . . .), h − = (h − 1 , h 2 , . . .). We denote such a pair by h = (h , h ). ± ± | h with eigenvalues h i of E ii , the Hamiltonian H (2.8) has the eigenvalue i εi (h i+ + h i− − gi ). Since X i j lowers the eigenvalues of H by εi + ε j > 0 and H is bounded from below, there must be a ground state | h annihilated by all X i j . If εi < ε j , then E i±j lowers the eigenvalues of H by εi − ε j , hence these elements also annihilate the ground state. If εi = ε j , then E i±j | h has the same eigenvalue as | h , i.e., the ground state may be degenerate. Let an eigenvalue ε of the one-particle Hamiltonian be n-fold degenerate (n < ∞). Then the ground states form a representation of the Lie subalgebra u(n)⊕u(n) with generators E i±j (εi = ε j = ε). Choose for | h a highest-weight vector of this representation, so that E i±j | h = 0 whenever i < j (εi = ε j = ε). Because E i±j and
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231
± E kl commute whenever εi = ε j = εk = εl , the same can be done for all degenerate eigenvalues of the one-particle Hamiltonian simultaneously. Thus the ground state | h can be chosen to satisfy
X i j | h = 0 ∀ i, j,
E i±j | h = 0 for i < j, and E ii± | h = h i± | h .
(2.10)
Together with the commutation and conjugation relations, the pair of sequences h = (h + , h − ) of Cartan eigenvalues determines the inner product and hence the representation completely. Unitarity imposes conditions on h, some of which are elementary to obtain. Computing h |X i j X i∗j | h = h +j + h i− ,
(2.11)
we deduce from unitarity that h +j + h i− must be nonnegative for all i, j. Computing ± ± h |E i±j E ± ji | h = h i − h j ,
i < j,
(2.12)
± we conclude that both sequences h ± = (h ± 1 , h 2 , . . .) are weakly decreasing. Computing further recursively n h |(E i±j )n (E ± ji ) | h =
= n! (h i± − h ±j )(h i± − h ±j − 1) · · · (h i± − h ±j − n + 1),
(2.13)
we obtain from unitarity that all differences h i± − h ±j are nonnegative integers and ±
hi (E ± ji )
−h ±j +1
| h = 0,
i < j.
(2.14)
Thus, the eigenvalues h ± form a pair of integer-spaced weakly decreasing sequences such that h i+ + h i− ≥ 0. Therefore, both sequences h + and h − must stabilize at some + − values h +∞ , h − ∞ . The convergence of the eigenvalue i (h i − h i ) of Q on the ground state implies that h +∞ = h − ∞ =: h ∞ ≥ 0.
(2.15)
If we now specialize to the canonical Hamiltonian (2.4), then the convergence of the eigenvalue i εi · (h i+ + h i− − N ) of Hc implies 2h ∞ = N .
(2.16)
To prove that N is a nonnegative integer, let r be sufficiently large such that h i+ = − h i = h ∞ for i > r , and observe that by Eqs. (2.10), (2.14) one has E i±j | h = h ∞ δi j | h for all i, j > r . This allows one to compute recursively the norm square of r +n (r ) (r ) Dn ∗ | h , where Dn := det X i j i, j=r +1 . Instead of computing it explicitly, it suffices
to observe as in [19] that this norm square must be a polynomial pn (N ) of degree n in N = 2h ∞ , and that pn (N ) vanishes whenever N is a nonnegative integer smaller than n. Indeed, if N ∈ N0 , the representation under consideration, restricted to the subalgebra with generators X i j , X i∗j and E i±j with i, j > r , coincides with the vacuum representa-
tion of the free-field realization (1.6) with N complex scalar fields, where Dn(r ) vanishes manifestly for n > N . For the same reason, for n = N , pn (n) is positive. These facts taken together imply that pn (N ) ∼ N (N − 1) · · · (N − n + 1)
(2.17)
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B. Bakalov, N. M. Nikolov, K.-H. Rehren, I. Todorov
with an irrelevant positive coefficient. Then it is clear that nonnegativity of all pn (N ) for a given value of N = 2h ∞ implies that N must be a nonnegative integer. In particular, if we define the vacuum state requiring Hc |0 = 0, we must have h i± = h ∞ = N /2 for all i. Because the sequence of Cartan weights determines the representation, the vacuum representation is unique, and given by Eq. (2.6). Moreover, we have the following result. Corollary 1. The vacuum state |0 spans a one-dimensional representation of the Lie subalgebra u(∞) ⊕ u(∞) with generators E i±j , such that N δi j |0. (2.18) 2 In particular, Q = 0, and the vacuum expectation value of the bilocal field vanishes, E i±j |0 =
0|W (x1 , x2 )|0 = 0.
(2.19)
Note that (2.19) for a ground state |0 implies 0|Hc |0 = 0 and hence Hc |0 = 0 (see Proposition 2 in Appendix B). One could have expected (2.19) to be a part of the definition of the bilocal field as described in the introduction, but we see here that it follows from the seemingly weaker assumption that the vacuum state has zero energy. 2.3. Unitarity bounds from Casimir operators. To obtain further constraints on the admissible values of h, we shall consider certain finite-dimensional subalgebras of u(∞, ∞). Namely, for a positive integer n, we consider the Lie subalgebra u(n, n) spanned by the generators (2.2) with indices 1 ≤ i, j ≤ n. We choose n sufficiently large so that h +n = h − n = h∞. Clearly, unitarity of a representation of u(∞, ∞) implies unitarity of its restriction to g := u(n, n). We may then follow the strategy of [8], using results of [9]. Denote by k := u(n) ⊕ u(n) the Lie algebra of the maximal compact subgroup U(n) × U(n) of U(n, n). We adapt the conventions of [8] for the positive roots of these Lie algebras so that our ± lowest energy condition h ± condition. Introduce 1 ≥ · · · ≥ h n turns into a highest weight + + ± 2n an orthonormal basis ei (i = 1, . . . , n) of R such that h = i (h i ei + h i− ei− ). We define the positive roots to be the roots ei± − e±j (i < j) and −ei− − e+j associated with the annihilation operators E i±j (i < j) and X i j for the ground state; then the ground state | h is a highest-weight vector for g. Unitarity of an irreducible representation Ug(h) of g with a ground state | h is equivalent to the condition that the inner product on the Verma g-module Vg(h) is semi-definite. Then Ug(h) = Vg(h)/Ng(h) is the quotient of the Verma module by its (maximal) submodule of null vectors. We also introduce the Verma k-module Vk(h) and its quotient Uk(h) = Vk(h)/Nk(h) by the maximal submodule of null vectors Nk(h) = Ng(h) ∩ Vk(h). In fact, Nk(h) is generated by the null vectors from Eq. (2.14), and Uk(h) is the unitary representation of ± k specified as follows. Denote by h ± the finite sequence (h ± 1 , . . . , h n ). Then we have h i± − h ∞ = m i± ,
(2.20)
± ± ± m± 1 ≥ m 2 ≥ · · · ≥ m r ± > m r ± +1 = · · · = m n = 0
(2.21)
where
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233
are nonnegative integers. The restriction of the representation Uk(h) to su(n)⊕su(n) ⊂ k is given by the pair of Young diagrams Y ± with r ± rows of length m i+ and m i− (1 ≤ i ≤ r ± < n), respectively, while h ∞ determines the representation of the center u(1) ⊕ u(1) of k. To get a necessary condition for the unitarity of Ug(h), we exploit the eigenvalues of Casimir operators. The quadratic Casimir operator for k, Ck = (E i+j E +ji + E i−j E − (2.22) ji ), ij
has an eigenvalue (λ + , λ + ) − (, ) in any highest-weight representation of k with n highest weight λ, where = 21 i=1 (n + 1 − 2i)(ei+ + ei− ) is one-half the sum of all positive roots of k, and (·, ·) is the natural inner product in R2n . On the other hand, the Casimir operator for g, ∗ ∗ Cg = (E i+j E +ji + E i−j E − (2.23) ji − X i j X i j − X i j X i j ), ij
has an eigenvalue (h + δ, h + δ) − (λ, λ) in any highest-weight representations of g with n highest weight h, where δ = − n2 i=1 (ei+ + ei− ). Then writing the difference as Ck − Cg = 2 i j X i∗j X i j + i (E ii+ + E ii− ), it is easy to calculate 2 λ |X i∗j X i j | λ = (λ + δ, λ + δ) − (h + δ, h + δ) · λ | λ , (2.24) ij
whenever | λ is a highest-weight vector for k of weight λ within a highest-weight g-module with highest weight h. Assume now that the highest weight h gives rise to an irreducible unitary represen∗ ] ⊗ U (h) tation of g. Then Uk(h) induces the highest-weight g-module M(h) = C[X kl k ∗ ∗ ± ± spanned by vectors of the form X · · · X E · · · E | h , which is the quotient of the Verma module Vg(h) by the (nonmaximal in general) submodule C[X kl ] ⊗ Nk(h). ∗ } ⊗ U (h) of M(h) spanned by vectors of the form Consider the subspace C{X kl k ∗ ± ± X E · · · E | h . This is a k-module equivalent to Uk(e+1 + e− 1 ) ⊗ Uk(h), because the ∗ transform like a vector with respect to both u(n) factors of k. Let λ be generators X kl the highest weight of any k-subrepresentation of Uk(e+1 + e− 1 ) ⊗ Uk(h), and let | λ be ∗ } ⊗ U (h) ⊂ M(h). the corresponding vector in C{X kl k By the unitarity assumption, expression (2.24) must be nonnegative, and if it is positive, then λ | λ > 0 and (λ + δ, λ + δ) > (h + δ, h + δ). If instead (2.24) vanishes, then λ |X i∗j X i j | λ must vanish for all i, j. Since in M(h) we have X i j Uk(h) = {0}, the commutation relations (2.3) imply that the generators X i j map any element of ∗ } ⊗ U (h) into U (h). In particular, X | λ belongs to the Hilbert space U (h). C{X kl ij k k k Hence X i j | λ = 0, and therefore | λ is a highest-weight vector for g with weight λ within M(h). This implies that λ | λ = 0 and that λ + δ is a g-Weyl transform of h + δ [28, 3]. Then (λ + δ, λ + δ) = (h + δ, h + δ). We have obtained in both cases (λ + δ, λ + δ) − (h + δ, h + δ) =: γ ≥ 0.
(2.25)
± By the Littlewood–Richardson rule, Uk(e+1 + e− 1 ) ⊗ Uk(h) contains Uk(λ) with λ = ± ± h + er ± +1 , where r are the heights of the first columns of the Young diagrams Y ± defined by h ± according to (2.21). For this choice of λ we have γ = 2(2h ∞ − r + − r − )
(it can be shown that this is the minimal value of γ ). Then (2.25) gives the following necessary condition for unitarity: r + + r − ≤ 2h ∞ .
(2.26)
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2.4. Fock space representations. Kashiwara and Vergne [14] have shown that all highest-weight representations of su(n, n) satisfying the bound (2.26) with h +n = h − n = h ∞ half-integer are contained in the (r + + r − )-fold tensor power of the Segal–Shale– Weil representation, and Schmidt [25] has extended this result to n = ∞. We essentially reformulate these results in our setting. Recall that the bilocal field W (x1 , x2 ) is given on the Fock space by (1.6). Consequently, the generators (2.2) of u(∞, ∞) are given by (2.6). Clearly, the representation of u(∞, ∞) on the Fock space is unitary. We claim that every representation with a ground state | h satisfying the bound (2.26) with 2h ∞ = N ∈ N is contained in the Fock space of N complex free scalar fields. (The case N = 0 implies r + = r − = 0, and hence triviality of the representation by virtue of Eqs. (2.11) and (2.12).) It is sufficient to display a vector with the properties of | h within the Fock space. Let h i± = m i± + N /2 according to (2.20) (with n sufficiently large), and let Y ± be the associated Young diagrams with rows of length m i± . Denote the heights of the columns of these diagrams by r ± = r1± ≥ r2± ≥ · · · ≥ rm±± . 1
Consider the Fock space vector | h F =
m +1 k=1
a
∗∧rk+
m− 1
b
∗∧rl−
|0,
(2.27)
l=1
where a ∗∧r and b∗∧r stand for the components p p a ∗∧r = det ai ∗ p=1,...,r and b∗∧r = det bi ∗ p=N +1−r,...,N i=1,...,r
(2.28)
i=1,...,r
of the antisymmetric U(N ) tensors c1 ∗ ∧ · · · ∧ cr ∗ (c = a or b). Note that ai ∗ and bi ∗ transform like a U(N ) vector and a conjugate vector, respectively (see Subsect. 2.5). The vector (2.27) is annihilated by X i j for all i, j (because rk+ ≤ N −rl− , thanks to the bound (2.26) and 2h ∞ = N ) and by E i±j for i < j (by virtue of the antisymmetrizations). In addition, | h F has eigenvalues h i± for the operators E ii± . Therefore, | h F has all the properties of the ground state | h . This proves part (i) of Theorem 1. Because the Fock space representation is unitary, the presence of this ground state in the Fock space proves, in particular, that the necessary conditions for unitarity found above are in fact also sufficient [14]. We conclude that all superselection sectors of the complex bilocal field W (x1 , x2 ) are realized in the Fock space of N complex massless scalar fields. The sectors are classified by the Cartan eigenvalues of u(∞, ∞): h + = (m +1 + h ∞ , . . . , m r++ + h ∞ , h ∞ , . . .), − h − = (m − 1 + h ∞ , . . . , m r − + h ∞ , h ∞ , . . .),
(2.29)
± + − where h ∞ = N /2, m ± 1 ≥ · · · ≥ m r ± > 0 are integers, and r + r ≤ N .
2.5. Representations of the gauge group. It remains to relate the superselection sectors of the bilocal field W (x1 , x2 ) classified in the previous subsections to the unitary representations of the gauge group U(N ).
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The gauge group U(N ) is unitarily represented on the Fock space in such a way that the vacuum is invariant and the creation operators a∗ and b∗ transform like an N -vector and a conjugate N -vector, respectively, because ϕ transforms like a vector. In particular, the expressions (2.6) and the bilocal fields W (x1 , x2 ) given by (1.6) on the Fock space are gauge invariant. Because the gauge group and the fields commute with each other, it follows that a ground state | h F is a component of a U(N ) tensor representation whose dimension equals the multiplicity of the corresponding superselection sector within the Fock space. The ground state | h F displayed in (2.27) is in fact a common highest weight vector for the commuting actions of u(∞, ∞) and u(N ) on the Fock space. The latter is the Lie algebra of the gauge group with generators 2E pq =
∞ p q q p q p p q (ai ∗ ai + ai ai ∗ − bi ∗ bi − bi bi ∗ )
(2.30)
i=1
which annihilate | h F if p < q. (The infinite sum in (2.30) converges on all Fock vectors on which Q is finite.) − + The components k a ∗∧rk and l b∗∧rl in (2.27) belong to tensors transforming under the gauge subgroup SU(N ) in the representations given by the Young diagrams Y + and (Y − )∗ , respectively. (The latter is the conjugate diagram whose columns have heights N − rm−− ≥ · · · ≥ N − r1− .) The ground state | h F therefore belongs to a sub1
representation of the tensor product Y + ⊗ (Y − )∗ . The tracelessness of (2.27) implies that the only contribution comes from the irreducible representation whose Young diagram Y has column heights N − rm−− ≥ · · · ≥ N − r1− ≥ r1+ ≥ · · · ≥ rm+ + , obtained as the 1
1
juxtaposition of (Y − )∗ and Y + (recall that N − r1− ≥ r1+ by (2.26) and 2h ∞ = N ). Similarly, because a∗ carries charge 1 and b∗ carries charge −1, the U(1) transformation is specified by the eigenvalue of the charge operator Q on | h F given by q = |Y + | − |Y − | = |Y | − N m − 1 , where |Y | stands for the number of boxes of the − ∗ − diagram Y so that |(Y ) | = N m − 1 − |Y |. Therefore, the U(N ) transformation given by the pair (Y, q) is determined by the pair (Y + , Y − ). Conversely, every irreducible unitary representation of U(N ) is given by a pair (Y, q), where |q| ≤ |Y | and q = |Y | mod N . (Each U(N ) vector contributes 1 to |Y | and 1 to q, while each conjugate vector contributes N − 1 to |Y | and −1 to q.) The pair (Y, q) then determines a unique split of Y into Y + and (Y − )∗ such that q = |Y + | − |Y − |. This gives an explicit one-to-one correspondence between the data (2.29) defining the superselection sectors and the unitary irreducible representations of the gauge group U(N ). Moreover, since the Cartan eigenvalues determine the occupation numbers in the ground state, one can see that the above ground states (2.27) exhaust the multiplicity of the superselection sector representation in the Fock space, and hence the multiplicity space carries the corresponding representation of U(N ). This proves part (ii) of Theorem 1.
2.6. The determinant relations. To prove Theorem 2, we follow the line of argument of Subsect. 2.2 until (2.15) for which we did not need the special choice Hc of the Hamiltonian. We now assume that for some N ∈ N the operator D N +1 vanishes in the representation under consideration, while Dn (n ≤ N ) do not vanish. It is easy to compute, using
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expansion formulas for determinants and the commutation relations (2.3), that ∗ X nn Dn∗ | h = (h +n + h − n − n + 1)Dn−1 | h .
Hence
X 11 · · ·
X nn Dn∗ | h
=
n
(2.31)
(h +m
+ h− m
− m + 1) | h ,
(2.32)
m=1
and D N +1 can vanish only if h +m + h − m = m − 1 for some positive integer m ≤ N + 1.
(2.33)
In particular, since the Cartan eigenvalues are integer-spaced and h +∞ = h − ∞ , we conclude that all Cartan eigenvalues belong to 21 N0 . Setting N := 2h ∞ , we obtain N ≤ h +m + h − m = m − 1 ≤ N. Since the representation under consideration is determined by its Cartan eigenvalues h ± , and h ∞ = N /2, we know from Subsect. 2.3 and 2.4 that it is realized on the Fock space of N complex scalar fields. But in the Fock representation, Dn vanishes if n > N while Dn (n ≤ N ) are nontrivial. Therefore our assumption that D N does not vanish implies that N ≤ N . We conclude that N = N and 2h ∞ = N .
(2.34)
Turning to the Hamiltonian (2.8), we observe that its eigenvalue on the ground state can converge only if i εi (2h ∞ − gi ) is finite. Hence we must have (2.35) gi = 2h ∞ + δi = N + δi with arbitrary shifts δi such that ΔE = i εi δi is well-defined. Clearly, this constitutes just an irrelevant additive renormalization H = Hc − ΔE of the Hamiltonian. This proves Theorem 2. 3. Classification: the Real Case The classification of the superselection sectors of the real bilocal field V (x1 , x2 ) satisfying (1.4) proceeds in perfect analogy to the complex case discussed in the previous section. We shall just repeat the relevant steps and point out the differences. The generators of the Lie algebra are N δi j . X i∗j and E i j = E ∗ji = V ( f i , f¯j ) + 2 They satisfy the commutator relations of sp(∞, R): X i j = V ( f¯i , f¯j ) = X ji ,
(3.1)
[E i j , E kl ] = δ jk E il − δil E k j , ∗ ] [E i j , X kl
=
δ jk X il∗
∗ + δ jl X ki ,
[E i j , X kl ] = −δik X jl − δil X k j ,
∗ [X i j , X kl ]
= δ jk Eli + δ jl E ki + δik El j + δil E k j .
(3.2)
The Fock space representation is given by X i j = ai · a j
and
2E i j = ai ∗ · a j + a j · ai ∗ ,
where ai = ϕ( f¯i ) and ai ∗ = ϕ( f i ) (cf. (2.7)).
(3.3)
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237
The general Hamiltonian is H=
∞
εi · (E ii − gi ),
(3.4)
i=1
while the canonical conformal Hamiltonian is Hc =
∞ i=1
N . εi · E ii − 2
(3.5)
There is no charge operator in the real case. The determinant operators Dn are defined by the same formula (2.9) as in the complex case. Theorem 3. The statements of Theorems 1 and 2 hold if we replace everywhere the bilocal field W (x1 , x2 ) by V (x1 , x2 ), the Lie algebra u(∞, ∞) by sp(∞, R), complex free fields by real free fields, the gauge group U(N ) by O(N ), and omit the assumption about the charge operator. The important conclusion is that all superselection sectors are realized in the Fock space of N real massless free scalar fields by (1.7). In the remainder of this section we give a sketch of the proof of the theorem. The ground state | h is annihilated by all X i j and by E i j for i < j. Computing the same norms as in Subsect. 2.2, we conclude that the Cartan eigenvalues of the generators E ii are given by a single integer-spaced sequence h such that h 1 ≥ h 2 ≥ · · · ≥ 0. This sequence must stabilize at some value h ∞ . Finiteness of the canonical Hamiltonian Hc requires 2h ∞ = N . Exploiting the vanishing of E i j on the ground state whenever r +n h i = h j = h ∞ and i = j, we can determine the norms of the vectors det X i j i, j=r +1 | h and conclude that if they are nonnegative, then N must be a nonnegative integer. The unique vacuum representation is given by (3.3), and the obvious analog of Corollary 1 holds. For a positive integer n such that h n = h ∞ , consider the restriction of our representation of sp(∞, R) to a unitary representation of the maximal compact subalgebra k := u(n) of g := sp(2n, R) ⊂ sp(∞, R). Then the Cartan eigenvalues have the form h i = m i + h ∞ (i ≤ n), where m 1 ≥ · · · ≥ m r > 0 are integers and m r +1 = · · · = m n = 0. The Young diagram Y with rows of lengths m i determines the representation of su(n) ⊂ u(n) with highest weight m i ei , while h ∞ determines the action of the center of u(n). Considering the Casimir operators Ck = E i j E ji (3.6) ij
of k and C=
ij
1 E i j E ji − (X i∗j X i j + X i j X i∗j ) 2
of g, one arrives at λ |X i∗j X i j | λ = γ · λ | λ , γ = (λ + δ, λ + δ) − (h + δ, h + δ), ij
(3.7)
(3.8)
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whenever | λ is a highest-weight vector for k of weight λ within a highest-weight n g-module with highest weight h. Here, δ = − i=1 i · ei . By the same argument as in Subsect. 2.3, γ is nonnegative. ∗ } is given by U (2e ), hence The adjoint representation of k on the linear span of {X kl 1 we may choose for λ the highest weight of any irreducible subrepresentation of the tensor product U (2e1 ) ⊗ U (h). By the Littlewood–Richardson rule, we may choose λ = h + er +1 + es+1 , where r and s ≤ r are the heights of the first two columns of the Young diagram Y (i.e., r is the smallest number such that h r +1 = h ∞ and s is the smallest number such that h s+1 ≤ h ∞ + 1). This choice of λ gives the necessary condition for unitarity r + s ≤ 2h ∞ = N .
(3.9)
Next, we display a ground state | h F in the Fock space of N real scalar fields by | h F =
m1
a
∗∧rk
0 |0,
(3.10)
k=1
where rk are the heights of the columns of the Young diagram Y and a ∗∧r stands for the component p a ∗∧r = det ai ∗ p=1,...,r (3.11) i=1,...,r
of the antisymmetric O(N ) tensor a1 ∗ ∧ · · · ∧ ar ∗ (r ≤ N ), while [· · · ]0 stands for the corresponding component of the traceless part of the product tensor. The presence of this ground state implies that all representations with highest weights as specified above are realized in this Fock space, and are indeed unitary. The superselection sectors of the bilocal field V (x1 , x2 ) are thus classified by the Cartan eigenvalues of sp(∞, R): | h = (m 1 + h ∞ , . . . , m r + h ∞ , h ∞ , . . .)
with r + s ≤ N ,
(3.12)
where h ∞ = N /2, m 1 ≥ m 2 ≥ · · · ≥ m r > 0 are integers, and r and s are the heights of the first two columns of the Young diagram Y whose rows have lengths m i . The gauge group O(N ) acts unitarily on the Fock space by leaving the vacuum invariant and transforming the creation operators a ∗ like a vector. Therefore the ground state | h F belongs to the unitary representation of O(N ) given by the Young diagram Y . In fact, it is a common highest-weight vector for the commuting actions of sp(∞, R) and so(N ) (the Lie algebra of the gauge group) on the Fock space. By the unitarity bound (3.9), only those Young diagrams occur whose first two columns have total height r +s ≤ N . It remains to convince oneself that such Young diagrams give precisely all irreducible unitary representations of O(N ). The standard labeling [4] of the unitary representations of O(N ) is given by pairs (Y, ±), where Y is a Young diagram with at most N /2 rows determining the representation of the subgroup SO(N ), and ± stand for the two representations of the quotient group O(N )/SO(N ) ∼ = Z2 given by the determinant. Note that (Y, +) is equivalent to (Y, −) iff N even and Y has exactly N /2 rows. Since the completely antisymmetric rank r tensor representation of O(N ) whose diagram Yr consists of a single column of height r is equivalent to det ⊗Y N −r , the representation with diagram Y (such that r + s ≤ N ) is equivalent to (Y, +) if r ≥ N /2, and
Unitary Positive-Energy Representations of Scalar Bilocal Quantum Fields
239
to (Y , −) if r ≤ N /2, where Y arises from Y by replacing the first column of height r by a column of height N − r . One easily sees that this relabeling of the irreducible unitary representations is a bijection. This proves the analog of Theorem 1. Turning to the analog of Theorem 2, we proceed by exploiting the vanishing of the determinant operator (1.8) in every superselection sector. Since N +1
∗ X 11 · · · X N +1,N +1 D N +1 | h = 2(2h m − m + 1) | h , (3.13) m=1
the vanishing of D N +1 implies that 2h m = m − 1
for some positive integer m ≤ N + 1.
(3.14)
In particular, N := 2h ∞ is a nonnegative integer and N ≤ 2h m = m − 1 ≤ N . As a consequence, the representation is realized on the Fock space of N real scalar fields. Assuming that D N does not vanish, we conclude that N ≤ N , hence N = N and h ∞ = N /2. Convergence of the ground state energy requires gi = h ∞ up to an irrelevant finite renormalization, thus proving the analog of Theorem 2. 4. Concluding Remarks Finding all representations of an algebra is often a highly nontrivial problem. Great progress has been made in the mathematical theory of highest-weight representations of Lie algebras, and these methods have been successfully exploited for the classification of unitary positive-energy representations (superselection sectors) of conformal QFT models in two space-time dimensions. In four space-time dimensions these powerful methods were thought to be inapplicable, because scalar local quantum fields do not satisfy commutation relations of Lie type [2].3 However, bilocal quantum fields appearing in certain operator product expansions do have this property. By virtue of this observation, one can benefit from the theory of highest-weight modules of Lie algebras in order to study positive-energy representations in quantum field theory. This article illustrates the approach on a class of nontrivial examples, thus building the connection between two important developments in physics and in mathematics that have taken place unaware of each other during the last decades. One is the Doplicher–Haag–Roberts (DHR) theory of superselection sectors in the framework of algebraic QFT, which establishes the duality between sectors and gauge symmetry (of the first kind). The other is the classification of highest-weight unitary modules of certain simple Lie algebras including sp(2n, R) and su(n, n). These Lie algebras are found to be realized in the commutation relations of the simplest bilocal quantum fields occurring in globally conformal invariant QFT in D ≥ 4 (even) dimensions. Obtaining by Lie algebra methods the explicit classification of unitary positive-energy representations of the commutation relations satisfied by the bilocal fields, we prove that they are all realized in a Fock space representation, corresponding to the Segal–Shale– Weil representation in mathematical terminology. 3 There are examples of Poincaré covariant local Lie fields which violate, however, the spectrum condition; see [15].
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This outcome was expected from the corresponding abstract result obtained in the DHR theory. However, considerable technical difficulties are encountered in relating the field representations and their extensions with the representations of the corresponding nets. The merit of our study is that it gives an independent re-derivation of the DHR result directly in the field-theoretic framework for the special cases at hand. Moreover, we have shown that, with the canonical choice of the Hamiltonian, the embedded Lie algebras u(∞, ∞) and sp(∞, R) possess the same unitary positive-energy representations as the associative field algebras. On the other hand, our result facilitates the program of classifying globally conformal invariant quantum field theories in four dimensions, because it indicates that without loss of generality one can “decouple” scalar free fields from a model [18]. Acknowledgements. B.B., N.N. and I.T. thank for its hospitality the Institut für Theoretische Physik der Universität Göttingen, where this work was done. B.B. was supported in part by an FRPD grant from North Carolina State University. The work of N.N. and I.T. in Göttingen was made possible by an Alexander von Humboldt Research Fellowship and an AvH Research award, respectively, and was supported in part by the Research Training Network within FP 5 of the European Commission under contract HPRN-CT-2002-00325 and by the Bulgarian National Council for Scientific Research under contract PH-1406.
A. Mode Expansions of Local and Bilocal Fields The conformal Hamiltonian H is a central element of the Lie algebra so(D) ⊕ R of the maximal compact Lie subgroup of the conformal group SO(D, 2). The eigenfunctions of the conformal Hamiltonian form a basis of test functions over the compactified Minkowski space M ∼ = (S D−1 × S1 )/Z2 , and each eigenspace is a finite-dimensional representation of SO(D). In the complex parameterization [27, 16, 22] of M given by 4 (A.1) M ∼ = z ∈ C D : z = eiτ u, τ ∈ R, u ∈ S D−1 (⇒ z2 = (z 1 )2 + · · · + (z D )2 = e2iτ ), these eigenfunctions are the Fourier polynomials f n, ,μ (z) = (z2 )n h ,μ (z) = ei(2n+ )τ h ,μ (u) .
(A.2)
h
is a (real) basis of spherical harmonics Here n is an arbitrary integer and {h ,μ (u)}μ=1 D−1 , i.e., homogeneous harmonic polynomials of degree = 0, 1, . . .. The number on S h of spherical harmonics of degree in D-dimensional space-time equals D − 2 + 2 D − 2 + . (A.3) h = D−2+ D−2
The z-parameterization of M is conformally equivalent to the affine parameterization of the Minkowski space, and any GCI (poly)local field φ(x) can be transformed to a conformally covariant field in the z-coordinates [16, 22]. We introduce a system of modes φn, ,μ of φ(z) by 1 φn, ,μ := φ[ f n, ,μ ] = φ(z) f n, ,μ (z) d D z, (A.4) V M 2x j ( j = 1, . . . , D − 1) and 1+x 2 −2i x 0 2 1−x , where x = (x 0 , x 1 , . . . , x D−1 ) and x 2 = −(x 0 )2 + (x 1 )2 + · · · + (x D−1 )2 . This is a zD = 1+x 2 −2i x 0 4 The embedding of the Minkowski space M in M reads z j =
conformal map belonging to the connected complex conformal group.
Unitary Positive-Energy Representations of Scalar Bilocal Quantum Fields
241
where d D z is the (complex) volume form of C D restricted on the real submanifold M (this is well-defined only in the even space-time dimension D since otherwise M is nonorientable), and V is the (pure imaginary) volume of M. One can write the collection of all modes of the field φ(z) as a formal power series φ(z) =
∞
h ∞
n = −∞ = 0 μ = 1
f n, ,μ (z) φ−n− − D , ,μ . 2
(A.5)
The complex integral over M gives rise to a linear functional on the space of all formal power series of the above type, called the residue [1, Sect. 3] and given explicitly by Resz f n, ,μ (z) := δn,− D δ ,0 .
(A.6)
φn, ,μ = Resz φ(z) f n, ,μ (z),
(A.7)
2
We can write
provided we choose h ,μ (z) to be orthonormal with respect to the residue, i.e., D
Resz (z2 )− 2 − h ,μ (z) h ,μ (z) = δ , δμ,μ .
(A.8)
An important property of the residue is that it is translation invariant: Resz ∂z α f (z) = 0,
α = 1, . . . , D.
(A.9)
In addition, it satisfies the Cauchy formula [1]
− D Resz (z − w)2 + 2 f (z) = f (w)
for f (z) ∈ C[[z]],
(A.10)
where ((z − w)2 )n+ denotes the formal series resulting from the Taylor expansion of ((z − w)2 )n in w around 0. It is possible to characterize GCI fields φ(z) as formal power series of the above type, with properties equivalent to the Wightman axioms. The corresponding algebraic structure is a higher-dimensional vertex algebra [16, 22, 1]. For massless scalar fields of canonical scaling dimension d0 = (D − 2)/2 in even space-time dimension D, only the modes with n = 0 and n = − − d0 contribute in (A.5), which correspond to solutions of the wave equation. The mode expansion for a pair of conjugate fields then can be conveniently written as (cf. [22]): ϕ(z) = ϕ ∗ (z) =
h ∞ =0 μ=1 h ∞ =0 μ=1
(z2 )− −d0 ϕ +d0 , μ + ϕ− −d0 , μ h ,μ (z),
(A.11)
∗ ∗ (z2 )− −d0 ϕ +d h ,μ (z), + ϕ− −d 0, μ 0, μ
(∗) where the modes ϕ± ,μ are conjugate to each other: ∗ . (ϕ ,μ )∗ = ϕ− ,μ
(A.12)
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This corresponds to the conjugation law (ϕ(z))∗ = (z2 )−d0 ϕ ∗ (z/z2 ) reflecting the fact that we work in a complex parameterization of the real compactified Minkowski space. (∗) In terms of the modes ϕ ,μ , the canonical commutation relations
−d
−d ϕ(z), ϕ ∗ (w) = (z − w)2 + 0 − (w − z)2 + 0
(A.13)
become ∗ ϕ +d0 ,μ , ϕ− = −d ,μ 0
∗ d0 δ , δμ,μ = ϕ +d , ϕ− −d0 ,μ , 0 ,μ + d0
(A.14)
and all other commutators vanish. This follows from (A.11) and the orthogonal harmonic 0 decomposition of ((z − w)2 )−d [1, Sect. 3.3]: + 0 ((z − w)2 )−d +
=
h ∞ =0 μ=1
d0 (z2 )− −d0 h ,μ (z)h ,μ (w) . + d0
(A.15)
Choosing an enumeration, n = n( , μ) ∈ N, we define an infinite number of creation and annihilation operators an(∗) and bn(∗) for states of positive and negative charge by setting + d0 + d0 ∗ an = ϕ +d0 ,μ , bn = ϕ +d0 ,μ , d0 d0 (A.16) + d + d 0 ∗ 0 ϕ− −d0 ,μ , bn∗ = ϕ− −d0 ,μ . an∗ = d0 d0 These operators satisfy the canonical commutation relations am , an∗ = δm,n = bm , bn∗ , am , bn(∗) = 0, etc.
(A.17)
The conformal Hamiltonian H and the charge operator Q are then expressed as ∞
H =
εn an∗ an + bn∗ bn ,
Q =
n=1
∞ ∗
an an − bn∗ bn ,
(A.18)
n=1
the energy eigenvalues. where εn( ,μ) := + d0 ( = 0, 1, . . .) are Introducing the notation h n( ,μ) (z) :=
d0 +d0 h ,μ (z),
we rewrite (A.11) as follows: 2 )−εn a + b∗ , h (z) (z ϕ(z) = ∞ n n n=1 n ∗ 2 −εn b ϕ ∗ (z) = ∞ (A.19) n . n = 1 h n (z) an + (z )
Similarly, one can write the mode expansion of a complex bilocal field W (z1 , z2 ) satisfying the commutation relations (1.3) as ∞
W (z1 , z2 ) =
n 1 ,n 2 = 1
h n 1 (z1 ) h n 2 (z2 ) X n∗1 n 2 + (z12 )−εn1 (z22 )−εn2 X n 1 n 2 +
+ (z12 )−εn1 E n−2 n 1 −
N N δn 1 ,n 2 + (z22 )−εn2 E n+1 n 2 − δn 1 ,n 2 , 2 2
(A.20)
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(∗)
± = (E ± )∗ satisfy the commutation relations (2.3). where X mn and E mn nm The mode expansion of a real bilocal field V (z1 , z2 ) satisfying (1.4) looks exactly (∗) (∗) = X nm . the same, but without the superscripts ± on E nm and with the symmetry X mn
Remark A.1. Cutting the mode expansion to a given n provides an approximation u(n, n) of u(∞, ∞), which can be rendered invariant under the maximal compact subgroup Spin(D) × U(1) of the conformal group provided we restrict n to the values n=
L
h
(n = (L + 1)(2L + 3)(L + 2)/6 for D = 4).
(A.21)
=0
B. Stress-Energy Tensor and the Conformal Lie Algebra The stress-energy tensor in any conformal field theory is expected to be a local tensor field that gives rise to the space-time symmetry generators when integrated against certain functions suggested by the (classical) Lagrangian field theory. These integrals are usually ill defined in general axiomatic QFT. But in the presence of GCI the theory can be extended to the compactified Minkowski space as we stated in the previous appendix. Then one can introduce rigorously the notion of a stress-energy tensor without any further assumptions. We shall formulate the notion of a stress-energy tensor in a GCI QFT directly in the z-picture introduced in the previous appendix. It is a symmetric tensor field Tαβ (z) = Tβα (z), which is traceless: Tαα (z) = 0, and conserved: ∂z α Tαβ (z) = 0 (summation over repeated indices). It is assumed also to be a quasiprimary tensor field of a scaling dimension equal to the space-time dimension D. These assumptions can be conveniently reformulated using the following generating function of Tαβ : T (z; v) = Tαβ (z)v α v β .
(B.1)
Note that T (z; v) is a quadratic polynomial in v with coefficients that are operator-valued (formal) distributions. Then, the above postulates for Tαβ read as follows: ∂v2 T (z; v) = 0 (tracelessness), ∂z · ∂v T (z; v) = 0 (conservation law) .
(B.2) (B.3)
The statement that Tαβ is a quasiprimary tensor field reads: [Tα , T (z; v) ] = [H, T (z; v) ] = Ωαβ , T (z; v) =
∂z α T (z; v) , (z · ∂ + D) T (z; v) ,
α z z ∂z β − z β ∂z α + v α ∂v β − v β ∂v α T (z; v) , [Cα , T (z; v) ] = z2 ∂z α − 2 z α z · ∂z − 2D z α
+ 2 z · v ∂v α − 2 v α z · ∂v T (z; v) ,
(B.4) (B.5) (B.6) (B.7)
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where Tα , H , Ωαβ = −Ωβα and Cα are the generators of the conformal Lie algebra so(D, 2), which satisfy the relations: H, Ωαβ = 0 = Tα , Tβ = Cα , Cβ , Ωα1 β1 , Ωα2 β2 = δα1 α2 Ωβ1 β2 + δβ1 β2 Ωα1 α2 − δα1 β2 Ωβ1 α2 − δβ1 α2 Ωα1 β2 , [H, Tα ] Ωαβ , Tγ Ωαβ , Cγ Tα , Cβ
= Tα ,
[H, Cα ] = −Cα ,
= δαγ Tβ − δβγ Tα ,
(B.8)
= δαγ Cβ − δβγ Cα , = 2 δαβ H − 2 Ωαβ .
Since Tαβ is a tensor field it requires “tensor test functions”. For f (z; v) = f αβ (z) v α v β , we define T [ f ] :=
f αβ (z) ∈ C[z, 1/z2 ] ,
1 Resz f (z, ∂v ) T (z; v) . 2
(B.9)
(B.10)
Using the residue technique of [1] (see Appendix A), one can derive the following statement. Proposition 1. Let T (z; v) be a local field defined by (B.1) and satisfying relations (B.2)– (B.4) in a vertex algebra, which is not assumed to be conformal in advance. Introduce the operators X := T [ f X ] for X = Tα , H , Ωαβ , Cα (α, β = 1, . . . , D), where z·v g X (z, v) , gTα = v α , g H = v · z , z2 = z α v β − z β v α , gCα = z2 v α − 2 z α z · v .
fX = gΩαβ
(B.11)
Then these operators obey the conformal Lie algebra relations (B.8) if and only if Eqs. (B.5)–(B.7) hold. Given the bilocal field W (z, w) or V (z, w), we can define the stress-energy tensor T (z; v) by applying to the bilocal field a second order differential operator D = D(∂z , ∂w ; v) and equating the arguments [18]: 1 T (z; v) = D W (z, w)w = z , or T (z; v) = D V (z, w)w = z , 2 where
(D − 1)D(∂z , ∂w ; v) = d0 (v · ∂z )2 + (v · ∂w )2
−D v · ∂z v · ∂w + v2 ∂z · ∂w
(B.12)
(B.13)
and d0 := (D − 2)/2. Note that the second formula in (B.12) follows from the first for V (z, w) = W (z, w) + W (w, z). It is an easy exercise to verify that the harmonicity of W (z, w) (or of V (z, w)) implies both the tracelessness and the conservation of T (z; v); for instance,
(D − 1) ∂z + ∂w · ∂v D(∂z , ∂w ; v) W (z, w)
= −2 (v · ∂z ) ∂z2 + (v · ∂w ) ∂w2 W (z, w) = 0 . (B.14)
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Proposition 2. Let W (z1 , z2 ) be given by (A.20). Then T (z; v) defined by (B.12) generates a representation of the conformal Lie algebra by (B.11). Furthermore, W (z1 , z2 ) transforms under this representation as a scalar bilocal field of dimension (d0 , d0 ); in particular,
H, W (z1 , z2 ) = z1 · ∂z1 + z2 · ∂z2 + 2 d0 W (z1 , z2 ) . (B.15) A similar statement is valid for V (z1 , z2 ). The proposition shows that, if one wants to realize both u(∞, ∞) and so(D, 2) in the state space of the theory, one cannot absorb the central term in (1.3) involving the constant N by a redefinition of the field W (x1 , x2 ) → W (x1 , x2 ) − (N /2)(Δ+1,2 + Δ+2,1 ), without its reappearance in formula (B.12) for the generators of so(D, 2). References 1. Bakalov, B., Nikolov, N.M.: Jacobi identity for vertex algebras in higher dimensions. J. Math. Phys. 47, 053505 (2006) 2. Baumann, K.: There are no scalar Lie fields in three or more dimensional space-time. Commun. Math. Phys. 47, 69–74 (1976) 3. Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Structure of representations that are generated by vectors of highest weight (Russian). Funk. Anal. i Prilozen. 5, 1–9 (1971); English translation, Funct. Anal. Appl. 5, 1–8 (1971) 4. Boerner, H.: Representations of Groups, 2nd edition, Amsterdam: North-Holland Publishing Company 1970 5. Buchholz, D., Doplicher, S., Longo, R., Roberts, J.E.: A new look at Goldstone’s theorem. Rev. Math. Phys. SI1, 49–84 (1992) 6. Carpi, S., Conti, R.: Classification of subsystems for graded-local nets with trivial superselection structure. Commun. Math. Phys. 253, 423–449 (2005) 7. Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131, 51–107 (1990) 8. Enright, T.J., Parthasarathy, R.: A proof of a conjecture of Kashiwara and Vergne. In: Noncommutative harmonic analysis and Lie groups (Marseille, 1980), Lecture Notes in Math. 880, Berlin-New York: Springer, 1981, pp. 74–90 9. Enright, T.J., Howe, R., Wallach, N.: A classification of unitary highest weight modules. In: Representation theory of reductive groups (Park City, Utah, 1982), Progr. Math. 40, Boston, MA: Birkhäuser, 1983, pp. 97–143 10. Haag, R.: Local Quantum Physics. Berlin-New York: Springer 1992 11. Jakobsen, H.P.: The last possible place of unitarity for certain highest weight modules. Math. Ann. 256, 439–447 (1981) 12. Jordan, P.: Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrkörperproblem. Zeitschr. für Physik 94, 531 (1935) 13. Kac, V., Radul, A.: Representation theory of the vertex algebra W1+∞ . Transform. Groups 1, 41–70 (1996) 14. Kashiwara, M., Vergne, M.: On the Segal–Shale–Weil representations and harmonic polynomials. Invent. Math. 44, 1–47 (1978) 15. Lowenstein, J.H.: The existence of scalar Lie fields. Commun. Math. Phys. 6, 49–60 (1967) 16. Nikolov, N.M.: Vertex algebras in higher dimensions and globally conformal invariant quantum field theory. Commun. Math. Phys. 253, 283–322 (2005) 17. Nikolov, N.M., Rehren, K.-H., Todorov, I.T.: Harmonic bilocal fields generated by globally conformal invariant scalar fields. (In preparation) 18. Nikolov, N.M., Rehren, K.-H., Todorov, I.T.: Partial wave expansion and Wightman positivity in conformal field theory. Nucl. Phys. B 722, 266–296 (2005) 19. Nikolov, N.M., Stanev, Ya.S., Todorov, I.T.: Four dimensional CFT models with rational correlation functions. J. Phys. A: Math. Gen. 35, 2985–3007 (2002) 20. Nikolov, N.M., Stanev, Ya.S., Todorov, I.T.: Globally conformal invariant gauge field theory with rational correlation functions. Nucl. Phys. B670[FS], 373–400 (2003) 21. Nikolov, N.M., Todorov, I.T.: Rationality of conformally invariant local correlation functions on compactified Minkowski space. Commun. Math. Phys. 218, 417–436 (2001)
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22. Nikolov, N.M., Todorov, I.T.: Elliptic thermal correlation functions and modular forms in a globally conformal invariant QFT. Rev. Math. Phys. 17, 613–667 (2005) 23. Reeh, H., Schlieder, S.: Bemerkungen zur Unitäräquivalenz von Lorentz-invarianten Feldern. Nuovo Cim. 22, 1051–1068 (1961) 24. Roberts, J.E.: Lectures on algebraic quantum field theory. In: The Algebraic Theory of Superselection Sectors, D. Kastler (ed.), Singapore: World Scientific, 1990, pp. 1–112 25. Schmidt, M.U.: Lowest weight representations of some infinite dimensional groups on Fock spaces. Acta Appl. Math. 18, 59–84 (1990) 26. Schwinger, J.: On angular momentum. In: Quantum Theory of Angular Momentum, L.C. Biedenharn, H. Van Dam (eds.), New York: Academic Press, 1965, pp. 229–279 27. Todorov, I.T.: Infinite-dimensional Lie algebras in conformal QFT models. In: A.O. Barut, H.-D. Doebner (eds.), Conformal Groups and Related Symmetries. Physical Results and Mathematical Background, Lecture Notes in Physics 261, Berlin: Springer, 1986, pp. 387–443 28. Verma, D.-N.: Structure of certain induced representations of complex semisimple Lie algebras. Bull. Amer. Math. Soc. 74, 160–166 (1968); Errata, ibid. 628 Communicated by Y. Kawahigashi
Commun. Math. Phys. 271, 247–274 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0181-3
Communications in
Mathematical Physics
The Uncertainty of Fluxes Daniel S. Freed1 , Gregory W. Moore2 , Graeme Segal3 1 Department of Mathematics, University of Texas, 1 University Station, Austin, TX 78712-0257, USA.
E-mail: [email protected]
2 Department of Physics, Rutgers University, Piscataway, NJ 08855-0849, USA.
E-mail: [email protected]
3 All Souls College, Oxford OX1 4AL, United Kingdom.
E-mail: [email protected] Received: 6 June 2006 / Accepted: 15 August 2006 Published online: 31 January 2007 – © Springer-Verlag 2007
Abstract: In the ordinary quantum Maxwell theory of a free electromagnetic field, formulated on a curved 3-manifold, we observe that magnetic and electric fluxes cannot be simultaneously measured. This uncertainty principle reflects torsion: fluxes modulo torsion can be simultaneously measured. We also develop the Hamilton theory of self-dual fields, noting that they are quantized by Pontrjagin self-dual cohomology theories and that the quantum Hilbert space is Z/2Z-graded, so typically contains both bosonic and fermionic states. Significantly, these ideas apply to the Ramond-Ramond field in string theory, showing that its K -theory class cannot be measured. Fluxes in the classical theory of electromagnetism and its generalizations are real-valued and Poisson-commute. Our main result is a Heisenberg uncertainty principle in the quantum theory: magnetic and electric fluxes cannot be measured simultaneously. This observation applies to any abelian gauge field, including the standard Maxwell field theory in four spacetime dimensions as well as the B-field and Ramond-Ramond fields in string theories. It is the torsion part of the fluxes which experience uncertainty—the nontrivial commutator of torsion fluxes is computed by the link pairing on the cohomology of space, and there are always nontrivial commutators if torsion is present. We remark that torsion fluxes arise from Dirac charge/flux quantization. This Heisenberg uncertainty relation goes against the conventional wisdom that the quantum Hilbert space is simultaneously graded by the abelian group of magnetic and electric flux; in fact, it is only graded by the free abelian group of fluxes modulo torsion. The most interesting example is the Ramond-Ramond field in 10-dimensional superstring theory. Here the Dirac quantization law is expressed in terms of topological K -theory, and conventional wisdom holds that the quantum Hilbert space is graded by the integer K -theory of space. The main result proves this wrong: the grading is only by K -theory modulo torsion. Notice that there are still operators reflecting the quantization by the full K -theory group; the assertion is that these operators do not all commute among themselves if there is torsion.
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Our exposition in Section1 begins with the classical Maxwell equations. We work on a compact1 3-dimensional smooth manifold Y . We first define the classical fluxes and show that they Poisson commute. The Hamiltonian formulation of Maxwell’s equations has Poisson brackets which are not invertible, so do not derive from a symplectic structure, if the second cohomology of Y is nontrivial: the symplectic leaves of the Poisson structure are parametrized by the fluxes. The most natural quantization of this system is as a family of Hilbert spaces parametrized by the real vector space of fluxes. Dirac charge quantization is implemented in Maxwell theory by writing the electromagnetic field as the curvature of a T-connection, where T = U (1) is the circle group. There is now an action principle and the space of classical solutions is a symplectic manifold, the tangent bundle to the space C(Y ) of equivalence classes of T-connections on Y . Its quantization is a single Hilbert space, defined as the irreducible representation of the Heisenberg group built from the product of C(Y ) and its Pontrjagin dual. (The salient features of Heisenberg groups and their representations are reviewed in Appendix A.) Magnetic and electric fluxes are refined to take values in the abelian group H 2 (Y ; Z). The Heisenberg uncertainty relation, stated in Theorem 1.6, follows from the commutation relations in the Heisenberg group. Our second aim in this paper, carried out in Section2, is to establish an appropriate Hamiltonian quantization of generalized self-dual fields, such as the Ramond-Ramond field in superstring theory. We highlight the main issues with the simplest self-dual field: the left-moving string on a circle, which we simply call the self-dual scalar field. Its quantization, which does not quite follow from the usual general principles, serves as a model for the general case. The flux quantization condition for other gauge fields, both self-dual and non-self-dual, is expressed in terms of a generalized cohomology theory. The fields themselves live in a generalized differential cohomology theory. We briefly summarize the salient points of the differential theory. The data we give in Definition 2.2 is sufficient for the Hamiltonian theory developed here; the full Lagrangian theory requires a more refined starting point. As in the Maxwell theory one can write classical equations (2.10) and a Heisenberg group (Theorem 2.3), which now is Z/2Z-graded. The Hilbert space of the self-dual field is defined (up to noncanonical isomorphism) as a Z/2Z-graded representation of that graded Heisenberg group. The fact that the Hilbert space is Z/2Z-graded, so in general has fermionic states, is one of the novel points in this paper. The noncommutativity of quantum fluxes (2.12) in the presence of torsion is then a straightforward generalization of Theorem 1.6 in the Maxwell theory. Section 2 concludes by showing how some common examples, including the Ramond-Ramond fields, fit into our framework. We call attention to one feature which emerged while investigating self-dual fields in general. For non-self-dual generalized abelian gauge fields any generalized cohomology theory may be used to define the Dirac quantization law. However, for a self-dual field the cohomology theory must itself be Pontrjagin self-dual. See Appendix B for an introduction to generalized cohomology theories and duality. Pontrjagin self-duality is a strong restriction on a cohomology theory. Ordinary cohomology, periodic complex K -theory, and periodic real K -theory2 are all Pontrjagin self-dual and all occur in physics as quantization laws for self-dual fields. Pontrjagin self-duality is not satisfied by most cohomology theories. For example, if the cohomology of a point in a Pontrjagin
1 There is also an uncertainty principle for fluxes if space is noncompact; we hope to return to that topic in the future. 2 K O-theory is Pontrjagin self-dual with a shift; see Proposition B.4.
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self-dual theory contains nonzero elements in positive degrees, then there are nonzero elements in negative degrees as well if the duality is centered about degree zero. In this paper we confine ourselves to the Hamiltonian point of view. We only construct the quantum Hilbert space and the operators which measure magnetic and electric flux up to noncanonical isomorphism. In future work we plan to develop the entire Euclidean quantum field theory of gauge fields, both self-dual and non-self-dual. We emphasize that the Hamiltonian quantization we use for a self-dual field is a special definition; it does not follow from general principles of quantization—see the discussion at the beginning of Section2. Perhaps one should not be surprised that the Hamiltonian theory for self-dual fields requires a separate definition; after all, the same is true for the Lagrangian theory [W]. Our definition is motivated by some preliminary calculations for a full theory of self-dual fields as well as by the special case of the self-dual scalar field in two dimensions. The role of Heisenberg groups and the noncommutativity of fluxes we find in Maxwell theory was anticipated in [GRW]. Our point of view in this paper is unapologetically mathematical. The fields under discussion are free, their quantum theory is mathematically rigorous, whence our mathematical presentation. In particular, we use the representation theory of Heisenberg groups to define the quantum Hilbert space of a free field. Appendix A reviews these ideas in the generality we need. A companion paper [FMS] presents our ideas and fleshes out the examples in a more physical style. 1. Maxwell Theory Classical Maxwell equations. Let Y be a compact oriented3 Riemannian 3-manifold and M = R × Y the associated Lorentzian spacetime with signature (1, 3). The classical electromagnetic field F ∈ 2 (M) satisfies Maxwell’s equations d F = jB , d ∗ F = jE ,
(1.1)
where the magnetic and electric currents j B , j E ∈ 3 (M) are closed. The Hodge star operator ∗ is defined relative to the Lorentz metric. Let ⊂ Y be a closed oriented cl and classical electric flux E cl are defined by surface. The classical magnetic flux Bt, t, cl Bt, (F) =
{t}×
F,
cl Et, (F) =
{t}×
∗F.
cl (F) and E cl (F) are static—that is, independent of time t— It follows from (1.1) that Bt, t, if we impose j B = j E = 0. Let V be the vector space of smooth solutions to the vacuum Maxwell equations ( j B = j E = 0) of finite energy. Stokes’ theorem implies that cl (F) and E cl (F) depend only on the homology class of , so define functions both B
B cl , E cl : H2 (Y ) × V −→ R
(1.2)
3 The orientation assumption is only for convenience of exposition. It is easily removed by working with differential forms twisted by the orientation bundle. The Hodge star operator maps ordinary forms to twisted forms on a non-oriented manifold, so the electric current j E is a twisted form whereas the magnetic current j B is untwisted.
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which are homomorphisms on H2 (Y ) × {F} for all F ∈ V . Put differently, in this vacuum case B cl , E cl are H 2 (Y ; R)-valued functions on V , simply the de Rham cohomology classes of F, ∗F. To express Maxwell’s equations in Hamiltonian form, write F = B − dt ∧ E
(1.3)
for B = B(t) ∈ 2 (Y ) and E = E(t) ∈ 1 (Y ); we set the speed of light to one. Let be the Hodge star operator on Y (relative to its Riemannian metric). Then B and E are closed and their evolution equations are ∂B = −dY E, ∂t
∂(E) = dY B. ∂t
(1.4)
Let W = 2 (Y )closed × 2 (Y )closed . For each time t there is an isomorphism V → W obtained by evaluating the restriction of (F, ∗F) to Y at time t. The Hamiltonian of the electromagnetic field, its total energy, is 1 H= B ∧ B + E ∧ E. 2 Y To put (1.4) in Hamiltonian form we introduce a Poisson structure on W . Definition 1.1. A (translationally-invariant) Poisson structure on a real vector space W is a skew-symmetric pairing {·, ·} : W ∗ × W ∗ −→ R on the space of linear functions on W . The symplectic leaves are the affine translates in W of the annihilator of the kernel of {·, ·}. If {·, ·} is nondegenerate, then W is symplectic. In general there is a kernel, the subspace K = { ∈ W ∗ : {, } = 0 for all ∈ W ∗ },
(1.5)
and its annihilator is {w ∈ W : (w) = 0 for all ∈ K }. The Poisson structure induces a Poisson bracket on polynomial functions on W . The Poisson structure for Maxwell is most easily written in terms of the linear functionals η (B, E) =
η ∧ B, Y
η (B, E) =
η ∧ E,
η∈
Y
1 (Y ) . d0 (Y )
Namely, {η1 , η2 } = {η1 , η2 } = 0,
{η1 , η2 } =
dη1 ∧ η2 . Y
Then Eq. (1.4) takes the Hamiltonian form ∂B = {H, B}, ∂t
∂(E) = {H, E}. ∂t
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The kernel (1.5) of the Poisson bracket consists of η , η for closed η, and so its annihilator is the subspace W0 = d1 (Y ) × d1 (Y ) ⊂ W. Therefore, the symplectic leaves are the fibers of the map π in π
0 −→ W0 −→ W −→ H 2 (Y ; R) × H 2 (Y ; R) −→ 0
(1.6)
which assigns to (B, E) the pair ([B]dR , [E]dR ) of de Rham cohomology classes. The classical fluxes (1.2) may be viewed as H 2 (Y ; R)-valued linear functions on W , which together form the map π in (1.6). Classical Fact 1.2. The classical magnetic and electric fluxes B cl and E cl Poisson commute. To make this precise, for a closed 1-form η on Y define the linear functions B cl (η), E cl (η) : W → R by cl cl B (η)(B, E) = η ∧ B, E (η)(B, E) = η ∧ E. (1.7) Y
Y
Integration by parts shows {B cl (η1 ), E cl (η2 )} = 0. We pass to the quantum theory (without Dirac charge quantization) by quantizing the affine symplectic fibers in (1.6), thus obtaining a family of Hilbert spaces parametrized by H 2 (Y ; R) × H 2 (Y ; R). The parameter is the pair of fluxes, which varies continuously over this real vector space. For simplicity we discuss only the quantization of the fiber at (0, 0), the symplectic vector space W0 . Briefly, one writes W0 ⊗ C as a direct sum of lagrangian subspaces and the Hilbert space completes the polynomial functions (Fock space) on one of the summands. For finite dimensional symplectic vector spaces the resulting (projective) Hilbert space H is independent of the lagrangian splitting. For infinite dimensional symplectic vector spaces one needs to fix a polarization to specify the quantization. In Hamiltonian field theory the natural polarization is given by the energy operator i d/dt: the complexification of the space of classical solutions is the sum of positive energy solutions and negative energy solutions. The cleanest characterization of the quantum (projective) Hilbert space H is as the unique irreducible representation of associated Heisenberg group which is compatible with the polarization. We recall the definition of the Heisenberg group and leave further discussion to Appendix A. Let W0 be any symplectic vector space with symplectic form . The Heisenberg group is a central extension of the translation group W0 by the circle group T of unit complex numbers. It is defined as the set W0 × T with multiplication (w1 , λ1 ) · (w2 , λ2 ) = (w1 + w2 , λ1 λ2 exp(iπ (w1 , w2 ))) .
(1.8)
The quantum Hilbert space H0 is heuristically the space of L 2 functions on d1 (Y ), but rather than define a measure on this infinite dimensional vector space we appeal to the representation theory of the Heisenberg group.
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Semiclassical Maxwell theory: Dirac charge quantization. Our main concern is the modification of this discussion when Dirac charge quantization is taken into account. The quantization of charge leads to the quantization of flux. Thus on the spacetime M = R × Y the quantum magnetic and electric fluxes are constrained to live in a full lattice inside the vector space H 2 (Y ; R), namely the image of integer cohomology H 2 (Y ; Z) in H 2 (Y ; R). The map H 2 (Y ; Z) −→ H 2 (Y ; R)
(1.9)
has a kernel, the torsion subgroup Tors H 2 (Y ; Z) ⊂ H 2 (Y ; Z), so the lattice is naturally identified with H 2 (Y ; Z)/torsion. A geometric model which implements charge quantization takes the electromagnetic field F to be −1/2πi times the curvature of a connection A on a principal circle bundle4 P → M. This lifts the magnetic flux from H 2 (M; Z)/torsion to an element of the abelian group H 2 (M; Z), the Chern class of P. The curvature of a connection depends only its isomorphism class. The space C(M) of isomorphism classes of smooth connections forms an infinite dimensional abelian Lie group under the tensor product of circle bundles with connection. The geometry of C(M) is important to us, so we pause to elucidate it. q In this paragraph M is any smooth manifold. Let Z (M), q > 0, be the set of closed q-forms on M with integral periods. Also, let T denote the circle group. Then i
curvature
0 −→ H 1 (M; T) −→ C(M) −→ 2Z (M) −→ 0
(1.10)
is an exact sequence of abelian groups. Thus any closed 2-form with integral periods is realized as the curvature/(−2πi) of a connection and the kernel of the curvature map is the group of isomorphism classes of flat connections. The latter is an abelian Lie group whose identity component is the torus H 1 (M; Z) ⊗ T and whose group of components is the finite group Tors H 2 (M; Z). This is encoded in the exact sequence β
1 −→ H 1 (M; Z) ⊗ T −→ H 1 (M; T) −→ Tors H 2 (M; Z) −→ 1,
(1.11)
where β is the Bockstein homomorphism. Also, 2Z (M) is the union of affine spaces b¯ of closed forms whose de Rham cohomology class is b¯ ∈ H 2 (M; Z)/torsion. Let Cb¯ ⊂ C(M) be the preimage of b¯ ; then Cb¯ → b¯ is a principal bundle with structure group H 1 (M; T). Another view of C(M) is exhibited by the exact sequence 0 −→ 1 (M)
Chern
1Z (M) −→ C(M) −→ H 2 (M; Z) −→ 0;
(1.12)
any integral cohomology class of degree two is the Chern class of a principal circle bundle and the kernel of the Chern class map is the set of connections on the trivial bundle up to gauge equivalence. Let Cb denote the set of equivalence classes of connections on a bundle with Chern class b ∈ H 2 (M; Z) and b¯ the image of b in de Rham cohomology. Then Cb → b¯ is a principal bundle with structure group the torus H 1 (M; Z) ⊗ T. The group C(M) is naturally a Lie group. The quotient H 2 (M; Z) in (1.12) is its group of components and the kernel in that exact sequence is its identity component. 4 In this Hamiltonian setup one postulates that P be the pullback of a bundle on Y , i.e., time translation is lifted to P.
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There is an action principle for Maxwell theory on the space5 of fields C(M). Namely, 1 L(A) = − FA ∧ ∗FA , 2 where −2πi FA is the curvature of the connection A. Cauchy data at fixed time identifies the space M of solutions to the corresponding Euler-Lagrange equations as the tangent bundle to C(Y ). Now, in contrast to the classical theory considered above, the entire space M has a symplectic structure. The magnetic and electric fluxes are defined as in (1.7): for a closed 1-form η on Y we have the functions B scl (η), E scl (η) : M → R given by B scl (η)(A) = η ∧ FA , E scl (η)(A) = η ∧ ∗FA , Y
Y
where the integrals are computed at any fixed time. The semiclassical magnetic flux is quantized—if η represents an integral class then B scl is integer-valued—whereas E scl is not. Semiclassical Fact 1.3. The semiclassical magnetic and electric fluxes Poisson commute: {B scl (η1 ), E scl (η2 )} = 0. In fact, B scl (η1 ) is locally constant on M, since [FA ]dR takes discrete values, so B scl (η1 ) Poisson commutes with any function on M. To motivate our definition of the quantum electric flux below we observe that any function on a symplectic manifold generates an infinitesimal symplectic diffeomorphism. A linear function on a symplectic vector space generates an infinitesimal translation, and for E scl (η) it is infinitesimal translation by η, viewed as a static connection on the trivial bundle. Its equivalence class in C(Y ) lies in the torus H 1 (Y ; Z) ⊗ T. In the quantum version below this torus is augmented to the compact abelian Lie group H 1 (Y ; T). Quantum Maxwell theory. We quantize the semiclassical Maxwell theory. Recalling that the space M of classical solutions may be identified with the tangent bundle to C(Y ), we see that the Hilbert space HY is heuristically the space of L 2 functions on C(Y ). It is naturally graded by the group H 2 (Y ; Z) of components of C(Y ); the homogeneous subspaces consist of L 2 functions supported on a single component: Hb . (1.13) HY = b∈H 2 (Y ;Z)
This is the grading by magnetic flux. As for electric flux, observe that the group of flat connections H 1 (Y ; T) acts on C(Y ) by tensor product. This induces a representation of H 1 (Y ; T)on L 2 functions, and we decompose HY according to the group of characters6 Hom H 1 (Y ; T), T ∼ = H 2 (Y ; Z): He . (1.14) HY = e∈H 2 (Y ;Z) 5 In fact, the fields do not form a space but rather the groupoid of T connections on M. The lagrangian is gauge-invariant, so determines a function on the space C(M) of equivalence classes. 6 The identification of the characters with H 2 (Y ; Z) uses Poincaré duality, which is an important ingredient in the general picture.
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Main Observation 1.4. The gradings (1.13) and (1.14) do not necessarily induce a simultaneous grading of HY by magnetic and electric flux. In this heuristic picture, where HY is the space of L 2 functions on C(Y ), the main observation follows immediately from the fact that translation by an element of H 1 (Y ; T) does not preserve the components of C(Y ), which are labeled by H 2 (Y ; Z). In fact, the action on the group of components is by translation via the Bockstein homomorphism β in (1.11). In other words, the identity component of H 1 (Y ; T) does preserve the group of components, and the noncommutativity is measured strictly by the torsion subgroup Tors H 2 (Y ; Z) of the second cohomology. Amelioration 1.5. The Hilbert space HY is simultaneously graded by magnetic and electric fluxes modulo torsion, that is, by the abelian group H 2 (Y ; Z)/torsion × H 2 (Y ; Z)/torsion. The main observation is in force whenever Y has torsion in its second cohomology, or equivalently in its first homology. For example, take Y to be a three-dimensional lens space, such as real projective space RP3 . These observations can be formulated more sharply. For each ω ∈ H 1 (Y ; T) there are (scalar-valued) linear operators B qtm (ω), E qtm (ω). The operator B qtm (ω) is multiplication by b, ω on Hb ; the operators E qtm (ω) form the representation of H 1 (Y ; T) on HY . Theorem 1.6. For ω1 , ω2 ∈ H 1 (Y ; T) we have [ B qtm (ω1 ), E qtm (ω2 ) ] = (ω1 βω2 )[Y ] idHY ,
(1.15)
where β is the Bockstein in (1.11) and [Y ] the fundamental class of Y in homology. The left hand side of (1.15) is the group commutator B qtm (ω1 )E qtm (ω2 )B qtm (ω1 )−1 qtm E (ω2 )−1 . The pairing on the right-hand side of (1.15) is symmetric and depends only on βω1 , βω2 , so factors to a symmetric pairing τ : Tors H 2 (Y ; Z) × Tors H 2 (Y ; Z) −→ T,
(1.16)
the so-called link pairing or torsion pairing in cohomology. Poincaré duality implies that τ is a perfect pairing: Tors H 2 (Y ; Z) is its own Pontrjagin dual. These heuristics are made rigorous, and Theorem 1.6 is proved, by defining the quantum Hilbert space HY as a representation of a generalized Heisenberg group. (See Appendix A for more details.) Definition 1.7. Let A be an abelian group and ψ : A × A → T a 2-cocycle, that is, ψ(a1 , a2 )ψ(a1 + a2 , a3 ) = ψ(a1 , a2 + a3 )ψ(a2 , a3 ),
a1 , a2 , a3 ∈ A. (1.17)
The group G(A, ψ) attached to (A, ψ) is the set A × T with multiplication (a1 + a2 , λ1 λ2 ψ(a1 , a2 )) . Commutators in G(A, ψ) are measured by a map s : A × A → T: [(a1 , λ1 ), (a2 , λ2 )] = [0, s(a1 , a2 )] = [0, ψ(a1 , a2 ) ψ(a2 , a1 )−1 ],
a1 , a2 ∈ A. (1.18)
The map s is bimultiplicative—a homomorphism in each variable separately—and is alternating: s(a, a) = 1 for all a ∈ A. We say s is nondegenerate if for all a1 ∈ A there exists a2 ∈ A such that s(a1 , a2 ) = 1. If s is nondegenerate, then we call G(A, ψ) a Heisenberg group.
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If A is a Lie group and ψ is smooth, then G(A, ψ) is also a Lie group. The Heisenberg group associated to a symplectic vector space (V, ) is G(V, eiπ ); see (1.8). The Heisenberg group is a central extension of A by the circle group: 1 −→ T −→ G(A, ψ) −→ A −→ 0. For quantum Maxwell theory we take A = C(Y ) × C(Y ).
(1.19)
The pairing ψ is defined in terms of a symmetric pairing σ : C(Y ) × C(Y ) −→ T on connections, namely ψ (A1 , A2 ), (A1 , A2 ) = σ (A1 , A2 ).
(1.20)
Notice that ψ vanishes on H × H for the subgroups H = C(Y )×{0} and H = {0}×C(Y ), so the Heisenberg group is canonically split over these subgroups. The simplest way to define σ is to use the topological fact that every compact oriented 3-manifold Y with a pair of circle connections A1 , A2 bounds a compact oriented 4-manifold X over which the connections extend. Let −2πi F˜ j , j = 1, 2, denote the curvatures of the extended connections. Then ˜ ˜ (1.21) σ (A1 , A2 ) = exp 2πi F1 ∧ F2 . X
We remark that the diagonal value σ (A, A) is the Chern-Simons invariant of the connection A. For α1 ∈ 1 (Y )/ 1Z (Y ) (see (1.12)) this formula simplifies to σ (α1 , A2 ) = exp 2πi α1 ∧ F2 , Y
where −2πi F2 is the curvature of the connection A2 . If ω1 and ω2 are flat connections, i.e., live in the subgroup H 1 (Y ; T) (see (1.10)), then σ (ω1 , ω2 ) = (ω1 βω2 )[Y ] is the link pairing τ (βω1 , βω2 ) (see (1.16)). Proposition A.4 in Appendix A asserts that the Heisenberg group G(A, ψ) has a unique irreducible unitary representation up to isomorphism which is compatible with positive energy and on which the central circle acts by scalar multiplication. Definition 1.8. The quantum Hilbert space HY of Maxwell theory is this unique irreducible representation of G(A, ψ). Recall (1.20) that G(A, ψ) is canonically split over C(Y ) × {0} and {0} × C(Y ). The operators B qtm (ω) and E qtm (ω) for ω ∈ H 1 (Y ; T) are defined by restricting the representation in Definition 1.8 to the lifts of the subgroups H 1 (Y ; T) × {0} and {0} × H 1 (Y ; T) of C(Y ) × C(Y ) in G(A, ψ). Theorem 1.6, and so our main observation, now follows directly from the commutation relations of the Heisenberg group.
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2. Self-Dual Abelian Gauge Fields The simplest self-dual field is the self-dual scalar field in 2-dimensional field theory, which with Dirac charge quantization takes values in the circle T. It is also called a leftmoving string. Its motion is described by a T-valued function of time t and space x of the form f (t, x) = φ(x + t), which is a general solution to the first-order wave equation ∂ f /∂t = ∂ f /∂ x.
(2.1)
When space is a circle the classical motions are parametrized by the loop group LT = Map(S 1 , T). Now any φ ∈ LT may be uniquely decomposed as ⎛ ⎞ φ(x) = λeiwx exp ⎝i qn einx ⎠ , n =0
where w ∈ Z is the winding number, λ ∈ T, and q−n = qn are complex numbers. This defines a decomposition of the loop group LT ∼ =T×Z×V
(2.2)
as the product of the circle group, the integers, and a real vector space. In physics V is called the space of “oscillators”. There is a Hamiltonian description of the motion. The Poisson brackets define a nondegenerate pairing on V , but the overall structure is degenerate: the symplectic leaves are parametrized by T × Z. The quantization of this system does not follow standard rules, which would give a family of Hilbert spaces parametrized by T × Z. Rather, we observe that T and Z are Pontrjagin dual, and in fact the entire loop group (2.2) is Pontrjagin self-dual. So there is a Heisenberg central extension, the standard central extension of the loop group at level one. Furthermore, we introduce a Z/2Z-grading. Definition 2.1. A Z/2Z-grading on a topological group G is a continuous homomorphism : G → Z/2Z. A graded representation of a Z/2Z-graded group is a Z/2Zgraded Hilbert space H0 ⊕ H1 and a homomorphism G → G L(H0 ⊕ H1 ) such that even elements of G preserve the grading on H0 ⊕ H1 and odd elements reverse it. A Z/2Z-grading is constant on the identity component, so for a Lie group does not induce any structure on the Lie algebra. In particular, there are no sign rules for a grading on a group. For the loop group LT there is a unique nontrivial grading according to the parity of the winding number, and it lifts to a grading of the Heisenberg central extension. The unique irreducible unitary representation of the Heisenberg central extension is graded—see the discussion at the end of Appendix A—and is defined to be the quantum Hilbert space of the self-dual scalar. That the Hilbert space is Z/2Z-graded is expected from the Bose-Fermi correspondence in two dimensions. Another motivation for the grading is [S, (12.3)]. More complicated self-dual fields, such as the Ramond-Ramond fields of superstring theory, lead to a Pontrjagin self-dual abelian group which generalizes (2.2). Its definition requires new ideas. The Dirac quantization law is implemented by a cohomology theory; for Ramond-Ramond fields it is a flavor of K -theory. The abelian group which plays the role of (2.2) is then a differential cohomology group built by combining the cohomology theory with differential forms. For ordinary cohomology these groups were first introduced by Cheeger-Simons [CS] and are a smooth version of Deligne cohomology [D].
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The generalization we need is developed by Hopkins-Singer [HS].7 We summarize what we need and then go on to define self-dual fields and their quantum Hilbert space. Non self-dual fields quantized by ordinary cohomology, such as the B-field in Type II superstring theory, fit into our theory by doubling and considering them as self-dual (Example 2.6); there is an equivalent direct treatment as a non-self-dual field. Generalized differential cohomology. Let E be a multiplicative cohomology theory, and E R , E T the associated theories with real and circle coefficients. (See Appendix B for a brief introduction.) The real E-cohomology of a point VE• = E • (pt; R) is a Z-graded real vector space. For any space X there is a natural map E • (X ) −→ H (X ; VE )•
(2.3)
whose image is a full lattice and whose kernel is the torsion subgroup. The codomain— the space on the right-hand side of (2.3)—is a direct sum of ordinary real cohomology groups. The total degree is the sum of the cohomological degree and the degree in VE . In case E is ordinary cohomology, VH• equals R in degree zero and vanishes otherwise, and (2.3) is the map (1.9). In case E is complex K -theory, VK• ∼ = R[u, u −1 ] is a Laurent series ring with the inverse Bott element u of degree 2. Now suppose M is a smooth manifold. The generalized differential cohomology Eˇ • (M) is a Z-graded abelian Lie group. We content ourselves here with a description of its geometry, analogous to the discussion of the geometry of C(M) = Hˇ 2 (M) in Section1. First, the E-cohomology with circle coefficients is a compact abelian group whose identity component is a torus, as indicated in the sequence β
1 −→ E d−1 (M) ⊗ T −→ E Td−1 (M) −→ Tors E d (M) −→ 1, valid for any d ∈ Z. (The E-cohomology is graded by the integers and can be nonzero in negative degrees, as for example for K -theory.) Let E (M; VE )q denote the space of closed VE -valued differential forms of total degree q whose de Rham cohomology class lies in the image of (2.3). There are exact sequences 0 −→ E Td−1 (M) −→ Eˇ d (M) i
field strength
−→
E (M; VE )d −→ 0
(2.4)
and 0 −→ (M; VE )d−1
characteristic class d E (M; VE )d−1 −→ Eˇ d (M) −→ E (M) −→ 0. (2.5)
In the physics language, a gauge field of degree d up to gauge equivalence is an element of Eˇ d (M). Its field strength is computed using the labeled map in (2.4); the isomorphism classes of flat fields form the compact abelian group which is its kernel. A gauge field also has a magnetic flux, the characteristic class in (2.5), and the space of equivalence classes of gauge fields with trivial magnetic flux is a quotient space of differential forms. Because E is a multiplicative cohomology theory, the associated differential theory has a graded multiplication Eˇ d (M) ⊗ Eˇ d (M) −→ Eˇ d+d (M).
7 To define gauge fields and gauge transformations one needs to introduce groupoids which represent the differential cohomology groups. Several models are considered in [HS]. For the Hamiltonian theory as discussed here we need only the differential cohomology groups of gauge equivalence classes.
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It combines the wedge product on differential forms and the multiplication in E: the field strength of the product is the wedge product of the field strengths and the characteristic class of the product is the E-product of the characteristic classes. Also, if ω ∈ E Td−1 (M) and Fˇ ∈ Eˇ d (M), then the product i(ω) · Fˇ only depends on the characteristic class of Fˇ and is the usual cohomological product E Td−1 (M) ⊗ E d (M) −→ E Td+d −1 (M) −→ Eˇ d+d (M).
In particular, i(ω) · i(ω ) = ω · βω = τ (βω, βω ),
ω ∈ E Td−1 (M), ω ∈ E Td −1 (M),
where τ is the “link (torsion) pairing”
τ : Tors E d (M) ⊗ Tors E d (M) −→ Tors E Td+d −1 (M).
(2.6)
Similarly, if α lies in the kernel of (2.5) then the product α · Fˇ only depends on the field strength of Fˇ and is given by a wedge product. Integration (also called pushforward or direct image) is defined in generalized differential cohomology. Suppose M is a compact n-manifold which is oriented for ˇ E-cohomology. An orientation for differential E-cohomology includes an orientation for E-cohomology, but might involve more data. For ordinary cohomology an Hˇ -orientation is an H -orientation in the usual notion of orientation. For real K O-theory a K O-orientation is a spin structure on M. A differential K O ∨ -orientation is a spin structure together with a Riemannian metric. Similarly, for complex K -theory a K -orientation is a spinc structure, whereas a differential Kˇ -orientation is a spinc -structure together with a Riemannian metric and compatible covariant derivative on the spinc structure. For a ˇ compact E-oriented n-manifold M we have the integration map
: Eˇ • (M) −→ Eˇ •−n (pt). M
There is an extension to integration in fiber bundles and for arbitrary maps as well. These integrations satisfy the usual Stokes’ theorem and are compatible with the corresponding integration E-cohomology. However, in general it does not commute with the field strength: there is an invertible differential form Aˆ E (M) ∈ (M; VE )0 such that • ˇ ˇ ˇ for F ∈ E (M) the field strength of M F is
Aˆ E (M) ∧ F,
(2.7)
M
ˇ This form is derived from the Riemann-Roch theorem, where F is the field strength of F. which relates the pushforwards in E and ordinary cohomology. Thus when E is ordinary cohomology this form is 1; its value for real K -theory explains our choice of notation. There is an extension tointegration in fiber bundles. Finally, if Fˇ = i(ω) in (2.4), then ˇ = i(ω ) for ω = ˆ F M M A E (M) ∧ ω.
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Notice that Eˇ • (pt) is not concentrated in degree zero. For example, even in ordinary cohomology the exact sequences (2.4) and (2.5) imply ⎧ ⎪ ⎨Z, • = 0, • ˇ H (pt) = T, • = 1, ⎪ ⎩0, otherwise. A cohomology theory determines a Pontrjagin dual cohomology theory. The cohomology theories which express the quantization law for self-dual fields are quite special: they are Pontrjagin self-dual. See Appendix B, especially Definition B.1, for a discussion. Self-dual abelian gauge fields. We now describe a generalized self-dual field. Just as a σ -model depends on a target Riemannian manifold as well as a spacetime dimension, so too does a generalized self-dual field depend on external data. We begin by specifying enough of this data to define the Hamiltonian theory, then write the classical theory in a Lorentzian spacetime of the Hamiltonian form R × Y , and finally define the quantum Hilbert space of a self-dual field. We emphasize that our definition of the quantum Hilbert space and certain operators on it is only up to noncanonical isomorphism. This is enough to demonstrate the Heisenberg uncertainty principle for flux. Thus the following definition is only good enough for our noncanonical construction here; we need more precise data for the full quantum theory. Definition 2.2. The data which define a Hamiltonian self-dual generalized abelian gauge field are: (i) a Pontrjagin self-dual multiplicative cohomology theory (E • , i) with shift s ∈ Z; (ii) a dimension m and a multi-degree d, i.e., an ordered collection d = (d1 , d2 , . . . , dk ) of integer degrees; and (iii) a natural isomorphism φ : E d −→ E m−s+1−d
(2.8)
ˇ such that for any compact E-oriented manifold Y of dimension m the pairing sY : Eˇ d (Y )× Eˇ d (Y ) −→ T, ˇ Fˇ1 ) · Fˇ2 ( Fˇ1 , Fˇ2 ) −→ ıˇ φ(
(2.9)
Y
is skew-symmetric. In (i) s ∈ Z is a shift in degree—part of Pontrjagin self-duality—and i determines a homomorphism E −s (pt) → Z. (In all cases we know i is specified by that homomorphism.) In (ii) m is the dimension of space, so we are working in a theory with (m + 1)-dimensional spacetimes. The field strength of the self-dual field has degree d, which is a multi-degree in case there are several gauge fields. The notation for multidegrees is d + 1 = (d1 + 1, d2 + 1, . . . , dk + 1), etc. The physical meaning of (iii) is that φ induces a map of a non-self-dual gauge field to its dual; it appears in the classical equation of motion (2.10) satisfied by a self-dual gauge field. See Proposition B.2 for
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a description of the pairing in (2.9). The self-dual scalar is the case when E is ordinary cohomology, m = 1, d = 1, and φ is the identity map. Other examples are given in the next subsection. We remark that whereas the pairing (2.9) is skew-symmetric, it is not necessarily alternating—that is, its values on the diagonal may be nontrivial. It is always a perfect pairing by Proposition B.2. The Maxwell story of Section1 generalizes to self-dual gauge fields. (Example 2.6 below explains how to write Maxwell theory as a special case.) Let Y be an E-oriented Riemannian manifold of dimension m and M = R × Y the associated Lorentzian spacetime of signature (1, n − 1). The role of the electromagnetic field is played by a differential form of total degree d on spacetime: d −q F ∈ (M; VE )d = q (M; VE ). q∈Z
The analog of the classical Maxwell equations (1.1) with zero current are the self-duality equations d F = 0, φ(F) = ıˇ(∗F).
(2.10)
The second equation, the self-duality condition, takes values in (M; VE )m−s+1−d . For the notation observe first that (2.8) applied to Rq with compact supports gives, after d −q m−s+1− d −q tensoring with R, a map VE → VE . Thus on the component of F which is a differential form of degree q we have d −q
φ : q (M; VE
m−s+1− d −q
) −→ q (M; VE
).
On the right-hand side the Lorentzian Hodge star operator, d +q−m−1 d +q−m−1 ∗ ) −→ q M; (VE ) ∗ : m+1−q (M; VE appears, followed by the duality map (B.7) d +q−m−1 ∗ m−s+1− d −q . ıˇ : q M; (VE ) −→ q M; VE
(2.11)
We remark that if there is a twisting in the definition of the gauge field, then the de Rham differential in the first equation is also twisted; see Example 2.9. Write F = B − dt ∧ E as in (1.3); then (2.10) implies φ(B) = ıˇ(E), where is the Riemannian Hodge star operator on Y . In other words, up to some invertible algebraic maps the self-duality equates the electric field and the magnetic field. The classical flux, now both electric and magnetic combined, is an H (Y ; VE )d -valued function on the space of solutions to (2.10), defined simply as the de Rham cohomology class of F. There is a Poisson structure with symplectic leaves parametrized by H (Y ; VE )d and the classical fluxes Poisson commute. The self-duality equation (2.10) is a first-order linear hyperbolic equation, and so a solution is determined by the value at any fixed time, as for the special case of the self-dual scalar field in two dimensions—see (2.1). Specifically, d the space of solutions is isomorphic to the real vector space (Y ; VE )closed .
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In the semiclassical picture with Dirac charge quantization the field is a geometric representative of a class in Eˇ d (M). The space of classical solutions on M is the differential cohomology group Eˇ d (Y ). Our definition of its quantization is motivated by the discussion at the beginning of Section2 of the self-dual scalar in two dimensions, as well as computations in other examples. The pairing (2.9) is skew bimultiplicative, but not necessarily alternating. We apply Proposition A.2 of Appendix A to construct the corresponding graded Heisenberg group. Theorem 2.3. There exists a central extension 1 −→ T −→ GY −→ Eˇ d (Y ) −→ 0 of A = Eˇ d (Y ), unique up to noncanonical isomorphism, with graded commutator (2.9). Then Proposition A.4 and the remarks which follow imply Theorem 2.4. There exists an irreducible Z/2Z-graded unitary representation H = H0 ⊕ H1 of GY on which the central circle acts by scalar multiplication, and it is unique up to noncanonical isomorphism. The quantum Hilbert space of the self-dual field on Y is the irreducible representation HY in Theorem 2.4. The generalization of Theorem 1.6 to self-dual gauge fields is now immediate. The quantum fluxes are defined using the kernel torus in E d (Y ), lifted to the Heisenberg group GY . Namely, for ω ∈ E Td −1 (Y ) ⊂ Eˇ d (Y ) and ω˜ a lift to GY , define F qtm (ω) ˜ to be the corresponding unitary operator on HY . Then the commutation relation in GY implies [F qtm (ω˜ 1 ), F qtm (ω˜ 2 )] = sY (ω1 , ω2 ) idHY ,
ωi ∈ E Td −1 (Y ),
(2.12)
where sY is the pairing (2.9) and the left-hand side is the graded group commutator (A.1). The nondegeneracy of sY implies the following. Generalized Main Observation 2.5. If Tors E d (Y ) = 0, then not all fluxes commute. This is the generalization of Main Observation 1.4. Analogous to Amelioration 1.5 there is a grading of HY by E d (Y )/torsion. Examples. The lagrangian versions of the following examples are discussed in [F, Section3]. The first example demonstrates how the framework of self-dual fields encompasses ordinary gauge fields. Example 2.6 (Dual gauge fields). Fix a dimension m for space and a degree 1 ≤ d ≤ m for a gauge field. Set d = (d, m+1−d) and quantize using ordinary cohomology E = H . The natural isomorphism φ in (2.8) is ˇ Fˇ ) = (−1)d(m−d) Fˇ , (−1)d−1 Fˇ , φ( F, Fˇ ∈ Hˇ d (Y ), Fˇ ∈ Hˇ m+1−d (Y ), (2.13) for any compact oriented manifold Y . The homomorphism i : H 0 (pt) → Z is the natural isomorphism. (The signs in Eq. (2.13) are derived from [F, Example 3.17].)
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The Maxwell theory of Section1 is the case m = 3, d = 2. Then up to an overall sign (2.9) is equal to the commutator in that theory—see (1.18), (1.20), and (1.21). (Note that (1.21) is the product in Hˇ 2 .) In the discussion of Section1 the alternating form sY is written as the skew-symmetrization of a 2-cocycle ψ. For a general self-dual field the 2-cocycle ψ is not determined by the data, but rather its existence is guaranteed by Theorem 2.3 . The “lagrangian splitting” (1.19) gives rise to ψ as well as to a partitioning of fluxes into magnetic and electric. Example 2.7 (Standard self-dual gauge field [HS],[W]). Here m = 4k + 1 for an integer k ≥ 0, we use ordinary cohomology E = H , and there is a single degree d = 2k + 1. The automorphism φ is trivial in this case. The self-dual scalar in two dimensions is the case k = 0. Let L 3 be a three-dimensional lens space. For k ≥ 1 the manifold Y = L 3 × S 2k−1 × S 2k−1 exhibits noncommuting fluxes. Example 2.8 (Type II Ramond-Ramond field ( Bˇ = 0)). Here m = 9 and we quantize using complex K -theory E = K . We assume the B-field vanishes. The Ramond-Ramond field on a compact Riemannian spin manifold Y has an equivalence class in Kˇ d (Y ), where d = 0 in Type IIA and d = −1 in Type IIB. Recall that K • (pt) ∼ = Z[u, u −1 ] is a Laurent series ring with deg u = 2. The automorphism φ is essentially complex conjugation:8 5, Type IIA; ˇ ˇ φ( F) = u F, = (2.14) 6, Type IIB The homomorphism i : K 0 (pt) → Z is the augmentation, an isomorphism. To check that (2.9) is skew-symmetric in Type IIA we note that for any Fˇ1 , Fˇ2 ∈ Kˇ 0 (Y ), the element Fˇ1 Fˇ2 + Fˇ2 Fˇ1 is the complexification of the realification of Fˇ1 Fˇ2 . Using the periodicity of K -theory we can shift the integral in (2.9) to the right-hand vertical map in the commutative diagram Kˇ 0 (Y )
/ Kˇ O 0 (Y )
/ Kˇ 0 (Y )
Kˇ −9 (pt)
/ Kˇ O −9 (pt)
/ Kˇ −9 (pt)
Starting with Fˇ1 Fˇ2 in the upper left-hand corner we deduce the skew-symmetry, since the composition T → Z/2Z → T on the bottom line is zero. A similar argument works for Type IIB. We make the classical self-dual equations (2.10) explicit for Type IIA. The field 0 strength F ∈ R × Y [u, u −1 ] has an expansion F = F0 + F2 u −1 + F4 u −2 + F6 u −3 + F8 u −4 + F10 u −5 ,
(2.15)
where the subscript q indicates the degree of the scalar-valued differential form Fq , and the forms Fq are closed. Then φ(F) = F0 u 5 − F2 u 4 + F4 u 3 − F6 u 2 + F8 u − F10 8 Complex conjugation maps the inverse Bott element u to −u.
(2.16)
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and ıˇ(∗F) = ∗F0 + ∗F2 u + ∗F4 u 2 + ∗F6 u 3 + ∗F8 u 4 + ∗F10 u 5 .
(2.17)
The self-duality equation is then F6 = − ∗ F4 ,
F8 = ∗F2 ,
F10 = − ∗ F0 .
(2.18)
In the quantum theory the Generalized Main Observation 2.5 asserts that any manifold with torsion in its K -theory exhibits noncommuting fluxes. For K 0 , hence for Type IIA, there is torsion in K -theory on any manifold with finite abelian fundamental group. (Calabi-Yau manifolds with this property have been considered in the physics literature.) For K 1 , hence for Type IIB, one can take the product of a circle and a manifold with finite abelian fundamental group to obtain examples. Example 2.9 (Type II Ramond-Ramond field ( Bˇ = 0)). Now we allow the nonzero Bfield, which is a cocycle Bˇ representing an element of Hˇ 3 (Y ). Then the Ramond-Ramond ˇ denoted Kˇ d+ Bˇ . As field has an equivalence class in differential K -theory twisted by B, before d = 0 in Type IIA and d = −1 in Type IIB. The automorphism (2.14) maps to ˇ The de Rham differential in (2.10) is replaced by differential K -theory twisted by − B. −1 ˇ the “twisted” differential d + u H , where H ∈ 3 (Y )closed is the field strength of B. Equations (2.15)–(2.18) hold as in the untwisted case, but now the condition that F be closed is replaced by the equation (d + u −1 H )F = 0. In components this reads d F0 = 0,
d F2 + H ∧ F0 = 0,
d F4 + H ∧ F2 = 0,
etc.
Example 2.10 (Type I Ramond-Ramond field). The Type I Ramond-Ramond field, or ‘Bfield’, is quantized by periodic real K -theory, whose Pontrjagin self-duality is proved in Proposition B.4. The degree of the field strength is d = −1, as in Type IIB. Note the nonzero shift s = 4. The ring K O • (pt) has torsion; after tensoring over the reals we obtain VK O ∼ = R[u 2 , u −2 ], where deg u = 2 as in K -theory. The automorphism (2.8) is ˇ = λ−1 F, ˇ φ( F) where λ−1 ∈ K O 8 (pt) is the generator whose complexification is u 4 ∈ K 8 (pt). In fact, there is a background self-dual current—part of the Green-Schwarz mechanism—and the B-field is a ‘cochain’ of degree −1 which trivializes the current. See [F] for details on the lagrangian theory.9 Appendix A: Central Extensions of Abelian Groups We shall consider the class of abelian Lie groups A which fit into an exact sequence 0 −→ π1 (A) −→ V −→ A −→ π0 (A) −→ 0, where V — the Lie algebra of A — is a locally convex and complete topological vector space. We shall assume that the exponential map V → A is a local diffeomorphism which makes V a covering space of the connected component of A, and that the group 9 The Pontrjagin self-duality was expressed in that paper in terms of a hybrid of real and quaternionic K -theory, but should have been stated as it is here.
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π0 (A) of connected components, and the fundamental group π1 (A), are finitely generated discrete abelian groups. Requiring π0 and π1 to be finitely generated may seem unduly restrictive, but it includes the examples we are interested in, and going beyond it makes things decidedly more complicated. In particular, if F is a discrete closed subgroup of a topological vector space V it need not be true that a homomorphism F → R extends to a continuous homomorphism V → R if F is not finitely generated; for example, if F is the subgroup of a Hilbert space V generated by the elements of an orthonormal basis {en } then the map which takes each en to 1 does not extend continuously to V . We shall now classify the central extensions G of A by the circle group T — we shall call these groups generalized Heisenberg groups. In fact we shall describe the category EA of extensions, in which a morphism from G to G is an isomorphism G → G which makes the diagram T
/G
/A
T
/ G
/A
commute. The class of extensions we consider are those which as manifolds are smooth locally trivial circle bundles over A. It follows from Proposition A.1(ii) that these circle bundles are in fact globally trivial. A smooth bimultiplicative map ψ : A × A −→ T defines an extension G(A, ψ): as a manifold G(A, ψ) is the product A × T, and its multiplication is given by (a1 , λ1 ) (a2 , λ2 ) = (a1 + a2 , λ1 λ2 ψ(a1 , a2 )). (Notice that a bimultiplicative map automatically satisfies the cocycle condition (1.17).) Let us recall that a quadratic map f : A → T is a smooth map such that (a1 , a2 ) → ψ f (a1 , a2 ) = f (a1 + a2 ) f (a1 )−1 f (a2 )−1 is bimultiplicative. We introduce the category BA whose objects are the smooth bimultiplicative maps, and whose morphisms from ψ to ψ are the smooth quadratic maps f such that ψ f = ψ /ψ. There is a functor BA → EA , as a quadratic map f defines a homomorphism G(A, ψ) → G(A, ψψ f ) by (a, λ) → (a, λ f (a)). Proposition A.1. (i) An object of EA is determined up to isomorphism by its commutator map s : A × A −→ T, which is a bimultiplicative alternating map, and every bimultiplicative alternating map can arise. (ii) The functor BA → EA is an equivalence of categories.
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Proof. To prove that an extension is determined by its commutator map we observe that the category EA has a composition-law. For the fibre-product G1 ×A G2 over A of two extensions G1 and G2 is an extension of A by T × T, and we can define the desired composite G1 ∗ G2 as the quotient of the fibre-product by the antidiagonal subgroup of T × T — the image of T by λ → (λ, λ−1 ). The isomorphism-classes of EA form a group Ext(A) under this composition-law, for the inverse of G is the quotient of the product extension G × T of A by T × T by the diagonal subgroup of T × T. Using the composition-law, it is enough to show that an extension which is itself commutative is trivial. But any of the abelian groups of the type we are considering can be written noncanonically as a product K × V ×π , where K is a compact torus, V is a vector space, and π is discrete. (To see this, we first split off π , using the fact that the identity component of the group is divisible; then we split the Lie algebra of the identity-component as V times the Lie algebra of the finite-dimensional maximal compact subgroup K , using the version of the Hahn-Banach theorem which asserts that any continuous linear map V0 → R defined on a closed subspace of a locally convex topological vector space V can be extended to a continuous linear map V → R.) This reduces us to showing that any inclusion of a circle in a torus is split, which follows from Pontrjagin duality. To complete the proof of Proposition A.1, let Bim(A1 , A2 ) denote the group of bimultiplicative maps A1 × A2 → T, and let Alt(A) denote the group of alternating maps A × A → T. There is an obvious map Bim(A, A) → Alt(A) which takes ψ to the commutator (a1 , a2 ) → s(a1 , a2 ) = ψ(a1 , a2 )/ψ(a2 , a1 ) of the extension G(A, ψ). We need only show that Bim(A, A) → Alt(A) is surjective for all abelian groups A in our class. Using the fact that Alt(A1 ⊕ A2 ) ∼ = Alt(A1 ) ⊕ Alt(A2 ) ⊕ Bim(A1 , A2 ), together with the more obvious decomposition of Bim(A1 ⊕ A2 , A1 ⊕ A2 ) as a sum of four groups, we see that if the desired surjectivity holds for A1 and A2 then it holds for A1 ⊕ A2 . It is therefore enough to consider separately the cases when A is a torus, a topological vector space, and a discrete cyclic group, and each of these is trivial. In fact the abelian groups A which are important in the present work come equipped with skew bimultiplicative maps s : A × A → T rather than alternating ones — i.e. s(b, a) = s(a, b)−1 , and so s(a, a) has order 2, but we need not have s(a, a) = 1. Such a skew map s defines a mod 2 grading A = Aeven ∪ Aodd of A, as a → s(a, a) is a continuous homomorphism A → Z/2. In fact the group Skew(A) of these forms can be identified with the group Extgr (A) of isomorphism classes of graded central extensions, as we shall now explain. A central extension of a — not necessarily abelian — group A by T can be regarded as a rule that associates a hermitian line L a to each a ∈ A, and associative unitary isomorphisms m a,b : L a ⊗ L b → L a+b to each a, b ∈ A. (The elements of the extended group are then pairs (a, λ), with a ∈ A and λ ∈ L a of unit length.) To define a graded central extension we simply replace the lines L a by graded lines (with a mod 2 grading), and require the isomorphisms m a,b to preserve the grading. A graded line is either even or odd, and evidently the group A acquires a grading from the degree of L a . In fact a graded central extension is simply a central extension G of A by T which is at the same time a mod 2 graded group — i.e. is equipped with a homomorphism G → Z/2.
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The analogue of the commutator map s : A×A → T in the graded case is the graded commutator, which maps (a, b) ∈ A × A to the composite −1 s(a, b) = m b,a ◦ Ta,b ◦ m a,b : L a+b −→ L a+b ,
(A.1)
where Ta,b : L a ⊗ L b → L b ⊗ L a expresses the symmetry of the graded tensor product — i.e. it multiplies by −1 if both L a and L b are odd. We think of s(a, b) as an element of T: concretely, it is (−1)deg(L a ) deg(L b ) times the naive commutator in the extension. Graded central extensions of A possess a composition law (which takes {L a } and {L a } to {L a ⊗ L a }), and there is an obvious short exact sequence 0 −→ Ext(A) −→ Extgr (A) −→ Hom(A; Z/2) −→ 0. Comparing this with the other obvious exact sequence 0 −→ Alt(A) −→ Skew(A) −→ Hom(A; Z/2) −→ 0, we have gr
Proposition A.2. An object of EA is determined up to isomorphism by its graded commutator map s : A × A −→ T, which is a skew bimultiplicative map, and every skew bimultiplicative map can arise. We can also make an assertion analogous to Proposition A.1(ii) in the graded case: if we consider the category of graded extensions with a given grading on A the assertion is verbally the same as in the ungraded case. Representations. Our next task is to describe the irreducible unitary representations of a generalized Heisenberg group G(A, ψ). It will be enough to consider representations in which the central subgroup T acts by multiplication: the general case reduces easily to this one. The centre Z of the group G = G(A, ψ) is an extension of Z by T, where Z is the kernel {a ∈ A : s(a, b) = 1 for all b ∈ A} of the commutator map. Being abelian, the extension Z is split, but not canonically, for the cocycle ψ need not vanish on Z . By Schur’s lemma in any irreducible representation ρ of G the subgroup Z acts by scalar multiplication, i.e. by a homomorphism χ : Z → T which is a splitting of the extension. Proposition A.3. For a finite-dimensional generalized Heisenberg group G an irreducible unitary representation ρ is determined up to isomorphism by the splitting homomorphism χ : Z → T, and any such homomorphism can arise. Proof. If the Heisenberg group G is non-degenerate, i.e. its centre is exactly T, the assertion is that there is only one possible representation. This is essentially the classical theorem of Stone and von Neumann. Let us assume that it is known in the case when A is a finite-dimensional vector space. In general, let T be the kernel of the commutator map restricted to the identity component of A; this is necessarily a torus over which
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the extension G is canonically split (for the bimultiplicative cocycle ψ which defines G must vanish when restricted to a torus). Furthermore, T is Pontrjagin-dual to G/G1 , where G1 is the centralizer of T in G. Any Hilbert space H on which G acts unitarily can be decomposed according to the action of T as H = ⊕Hα , where α runs through the characters of T . Now H0 is a representation of G1 , and evidently H is the representation of G induced from the representation H0 of G1 , which must therefore be irreducible. In fact H0 is a representation of the non-degenerate Heisenberg group G1 /T , whose group of components is finite, and whose identity-component is a vector Heisenberg group. We have therefore reduced ourselves to treating the case of a Heisenberg group G arising from an abelian group A of the form V × π , where V is a vector space and π is a finite abelian group. In this case, let be a maximal abelian subgroup of the restriction of the extension to π . We can write = T × π , though non-canonically. In any representation of G the action of the subgroup π decomposes the Hilbert space into pieces according to the characters of π , which are permuted transitively by the conjugation-action of π . By the argument we have already used we see that the representation is induced from an irreducible representation of G0 × π in which π acts trivially. (Here G0 is the identity-component of G.) But G0 is a vector Heisenberg group, and so we have reduced ourselves to the case we have assumed to be known. When the group G is degenerate, the representation ρ is actually a representation of G/Z , where Z is the kernel of the homomorphism χ : Z → T. But G/Z is a non-degenerate Heisenberg group, and so we are back to the previous case. Turning now to infinite-dimensional groups, to have an analogue of the Stone-von Neumann theorem we must introduce the concept of a positive-energy representation, which is defined when the abelian group A is polarized. There are many versions of this concept. The following is a rather narrow one, but seems simplest for our purposes. We shall say A is polarized if there is a continuous action of the group R on the Lie algebra V of A by operators {u t }t∈R which preserve the skew form coming from the commutator and decompose the complexification VC into a countable sum of finitedimensional subspaces Vλ in which u t acts by multiplication by eiλt . Here each λ is real, and we assume that the algebraic sum of the Vλ is dense in V . In our applications {u t } will be the Hamiltonian flow on the phase space V induced by a positive quadratic energy-function — but we allow the symplectic structure on V to be degenerate. Then for a polarized group A we say a unitary representation of the Heisenberg group G on a Hilbert space H is of positive energy if there is a unitary action of R on H by operators Ut = ei H t whose generator H has discrete non-negative spectrum, and which intertwines with the action of G on H in the sense that the action of exp(u t (v)) for any v ∈ V is the conjugate by Ut of the action of exp(v). In our applications {Ut } will be the time-evolution of a quantum system, which we require to have positive energy in the usual quantum-mechanical sense. Now we have the following version of the Stone-von Neumann theorem. Proposition A.4. For a polarized generalized Heisenberg group G an irreducible unitary representation of positive energy is completely determined by the splitting homomorphism χ : Z → T, and any such homomorphism can occur. Proof. By exactly the same arguments as in the finite-dimensional case we reduce first to the case of a non-degenerate Heisenberg group, and then to a vector Heisenberg group. We then find, just as in [PS, Section9.5], that the unique positive energy irreducible unitary representation of the Heisenberg group formed from V is realized on the completion
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of the symmetric algebra S(W ), where VC = W ⊕ W¯ is the decomposition into positiveand negative-energy pieces. To conclude, we return to graded central extensions. For a mod 2 graded group G it is natural to consider unitary representations on mod 2 graded Hilbert spaces, and to require that the action of the even elements of G preserves the grading, while that of the odd elements reverses it. But if G is a graded generalized Heisenberg group every representation automatically has such a grading. To see this, we may as well assume that G is non-degenerate, for in any irreducible representation G acts through a non-degenerate quotient. If G is non-degenerate, the grading homomorphism A → Z/2 is necessarily of the form a → s(ε, a) for some ε ∈ A which has order 2. If εˆ is a lift of ε in G then εˆ commutes with the even elements of G and anticommutes with the odd elements. Furthermore, εˆ 2 acts as a scalar, so its eigenspaces define the desired mod 2 grading on the Hilbert space. (We do not have a preferred way of naming the eigenspaces odd and even; but that does not matter, as reversing the choice gives us an isomorphic graded representation.) Appendix B: Self-Dual Cohomology Theories Any cohomology theory E • arises from a loop-spectrum, i.e. a sequence E = {Eq }q∈Z of spaces with base-point such that E q (X ) is the set of homotopy classes of maps X → Eq . For a space X with base-point, the reduced cohomology E˜ q (X ) is the set of homotopy classes of base-point-preserving maps X → Eq . The term “loop"-spectrum reflects the existence of canonical homotopy equivalences Eq → Eq+1 — where denotes the based loop-space — which express the behaviour of cohomology groups under suspension. The spectrum is determined up to homotopy equivalence by the cohomology theory, and algebraic topologists usually mean the spectrum when they refer to a cohomology theory. The spectrum also defines a homology theory E • by E q (X ) = lim πq+i (X + ∧ Ei ), i
where X + denotes the space X — which is not assumed to have a given base-point — with a disjoint base-point adjoined.10 In particular, E q (pt) = E −q (pt). It may be helpful to notice that the spaces X + ∧ Eq form a spectrum, in the sense that there are natural maps X + ∧ Eq −→ (X + ∧ Eq+1 ). This is not a loop-spectrum, but one can make it into a loop-spectrum X + ⊗ E whose q th space is lim i (X + ∧ Eq+i ), i
and then E q (X ) is the homotopy group of X + ⊗ E. In general there is no way to calculate the groups E • (X ) algebraically from the groups E • (X ), although if X is compact and can be embedded as a neighbourhooddeformation-retract in Euclidean space R N we have q th
E q (X ) ∼ = E˜ N −q (D N X ), 10 For two spaces Y and Z with base-points y and z the wedge product Y ∧ Z denotes the spaces obtained 0 0 from the product Y × Z by collapsing the subspace (Y × z 0 ) ∪ (y0 × Z ) to a single point.
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where D N X is the N -dual of X , defined as the one-point compactification of an open neighbourhood of X in R N of which X is a deformation-retract, and E˜ • denotes reduced cohomology. This notion of duality — usually called S-duality — is explained by the fact that the category of spectra forms — when the morphisms are defined appropriately — a tensor category in which the tensor product is induced by the wedge-product of spaces and the sphere-spectrum {S q } is the neutral object. In this category the natural dual of a compact space X is the spectrum formed by the mapping-spaces {Map(X ; S q )}. The sequence of spaces {Dq X } forms a spectrum, and there is a natural map — the “scanning” map11 — Dq X → Map(X ; S q ) which induces an equivalence of the associated loop-spectra. If E • is a multiplicative theory, in the sense that X → E • (X ) is a contravariant functor to anticommutative graded rings, then the multiplication is induced by maps E p ∧ Eq → E p+q . These define maps E p (X ) × E q (X ) −→ E q− p (X ) taking f : X + → E p and g : S q+i → X + ∧ Ei to the composite S q+i −→ X + ∧ Ei −→ X + ∧ Ei ∧ E p −→ X + ∧ E p+i , which make E • (X ) a graded module over E • (X ). There is another way — essentially purely algebraic — to pass between cohomology theories and homology theories which from the point of view of spectra does not look at all natural. Let us recall that if I is a divisible abelian group such as R or T then the contravariant functor A → Hom(A; I ) from the category of abelian groups to itself takes exact sequences to exact sequences. So if E • is a homology theory we can define a cohomology theory e•I by q
e I (X ) = Hom(E q (X ); I ). Unfortunately this does not work when I = Z, as Z is not a divisible group. It is rea• as the theory that fits into a long exact sonable, however, to define the theory e• = eZ sequence q
q
. . . −→ eq (X ) −→ eR (X ) −→ eT (X ) −→ eq+1 (X ) −→ . . . , • → e• is induced by the obvious homomorwhere the transformation of theories eR T • • by spectra, and defining e• phism R → T. (This does entail representing eR and eT as the theory represented by the fibre of the map of spectra; but it is a fairly anodyne operation as we shall in the end have a short exact sequence
0 −→ Ext(E q−1 (X ); Z) −→ eq (X ) −→ Hom(E q (X ); Z) −→ 0.) Because E • (X ) is a module over E • (X ) it is easy to see that the cohomology theories e•I and e• are module-theories over E • . Let s be an integer termed a “shift”. If we choose once and for all an element of i ∈ es (pt) — which is simply Hom(E −s (pt); Z) 11 To define the scanning map, choose ε > 0 so that the ε-neighbourhood of X in Rq is contained in the open neighbourhood U X of X of which Dq (X ) is the compactification. Then the scanning map is the extension to the compactification of the map which takes x ∈ U X to the composite U X → Rq → Rq /(Rq − Ux ) ∼ = Sq , where Ux is the ε-neighbourhood of x in Rq .
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if E 1−s (pt) = 0, as is the case for all the theories we shall be concerned with — then we get a natural element in i X ∈ es (X ) for all spaces X , and hence a natural transformation E • (X ) −→ e•+s (X )
(B.1)
which takes ξ to ξ.i X . Definition B.1. The theory (E • , i) is called Pontrjagin self-dual if (B.1) defines an isomorphism of cohomology theories. This is a very strong constraint on a theory, as it implies, for instance, that the groups E q (pt) and E −s−q (pt) have the same rank for every q. Known examples include classical homology and periodic complex and real K -theory — the shift s is zero for the former two and we take s = −4 for the latter; see below. In treating the examples it is more convenient to use an equivalent version of the selfduality condition which is stated in terms of the theory E T• called “E • with coefficients in T". This is a module-theory over E • which fits into a long exact sequence q
q
. . . −→ E q (X ) −→ E R (X ) −→ E T (X ) −→ E q+1 (X ) −→ . . . ,
(B.2)
where — at least if certain finiteness conditions are satisfied which hold in the cases at • with real coefficients is defined simply by E q (X ) = E q (X ) ⊗ R. hand — the theory E R R The transformation E • → e•+s induced by a choice of i also leads to a transforma•+s • tion E T → eT , and the latter is an isomorphism if and only if the former is. To check •+s (pt), which is a self-duality, therefore, it is enough to prove that the map E T• (pt) → eT • map of modules over E (pt), is an isomorphism, i.e. that the module-action −q
E q−s (pt) × E T (pt) −→ E T−s (pt) −→ T i
(B.3)
is a perfect pairing. For a compact E-oriented m-manifold Y Poincaré duality there is a perfect pairing q
E m−s−q (Y ) ⊗ E T (Y ) −→ T
(B.4) q for each integer q. For Pontrjagin self-duality identifies E T (Y ) ∼ = Hom E q+s (Y ); T under which (B.4) becomes the composition E m−s−q (Y ) ⊗ Hom E q+s (Y ); T −→ Hom (E m (Y ); T) −→ T
• and evaluation on the fundamental class, and this is a of the module-action of E • on eT perfect pairing by Poincaré duality. Now let us turn to the differential theory Eˇ • associated to E • . If E • is Pontrjagin selfdual then we have the following very attractive differential version of Poincaré duality. We make the finiteness assumption alluded to after (B.2).
Proposition B.2. If Y is a compact m-dimensional manifold which is oriented for Eˇ • , and the theory (E • , i) is Pontrjagin self-dual, then we have an integration operation ıˇ : Eˇ m−s+1 (Y ) −→ T (B.5) Y
which, together with the natural multiplication, gives us a perfect pairing, Eˇ q (Y ) × Eˇ m−s−q+1 (Y ) −→ T.
(B.6)
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The proof relies on the following, which is also used in (2.11). Lemma B.3. For each q ∈ Z the natural pairing q−s
ıˇ : VE
−q
⊗ VE
→R
(B.7)
is nondegenerate. q
Proof. By the finiteness assumption VE = E q (pt) ⊗ R. The element i ∈ es (pt) induces a homomorphism E −s (pt) → Z, and so a linear map ıˇ VE−s → R and the pairing (B.7). q−s If there is a nonzero v ∈ VE in the kernel of the pairing, we can assume v is in the q−s q−s image of E (pt) → E (pt) ⊗ R, so defines an element e ∈ E q−s (pt)/torsion. −q −q Now E R (pt) → E T (pt) is onto the identity component, and the perfection of the −q pairing (B.3) implies that E q−s (pt)/torsion × E T (pt)id → T is also perfect, from which e = 0. Proof of Proposition B.2. First, (B.5) is defined as the composition
i 2.4 Y Eˇ m−s+1 (Y ) −→ Eˇ −s+1 (pt) = E T−s (pt) −→ T.
Suppose Fˇ ∈ Eˇ q (Y ) is in the kernel of (B.6) and let F ∈ E (M; VE )q be its field strength. Let Gˇ ∈ Eˇ m−s−q+1 (Y ) have trivial characteristic class, so it is the image of a differential form α in the exact sequence (2.5). Then by the remark following (2.6) and (2.7) et seq. we see that Y Fˇ · Gˇ is image of Y Aˆ E (Y ) ∧ F ∧ α under the natural map R → T. Decompose F and α as a sum of differential forms of fixed degree: q− j F= F j , F j ∈ j (VE ), j
α=
m−s−q−k
αk , αk ∈ k (VE
).
k
Then since Fˇ is assumed in the kernel of (B.6), and since Aˆ E (Y ) is invertible, it follows that ıˇ applied to q− j j−q−s F ∧α = F j ∧ αm− j ∈ VE ⊗ VE Y
j
Y
j
vanishes for all α. Then from Lemma B.3 we conclude F = 0. Thus Fˇ is the image q−1 in (2.4) of ω ∈ E T (Y ), and furthermore ω lies in the kernel of the pairing q−1
E m−s−q+1 (Y ) ⊗ E T (Y ) −→ T. But this is the perfect Poincaré-Pontrjagin pairing (B.4), whence ω = 0. The Pontrjagin self-duality of ordinary cohomology and complex K -theory are relatively easy, but for real K -theory it is more subtle. In fact, the Pontrjagin dual of K O-theory — the associated theory denoted e• above — is again K O-theory but with a shift of degree s = 4. Note K O −3 (pt) = 0 and K O −4 (pt) is generated by an element μ whose complexification is 2u −2 ∈ K −4 (pt). We choose the ele ment i ∈ Hom K O −4 (pt); Z to map μ → 1.
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Proposition B.4. (K O, i) is Pontrjagin self-dual. K Oq (pt) ∼ = K O −q (pt) Z (μ) 0 Z/2Z (η2 ) Z/2Z (η) Z 0 0 0 Z
q 4 3 2 1 0 −1 −2 −3 −4
q
K OT (pt) T 0 0 0 T 0 Z/2Z Z/2Z T
Proof. According to (B.3) it suffices to verify that the pairing −q
K O q−4 (pt) ⊗ K OT (pt) −→ K OT−4 (pt) −→ T i
is an isomorphism for all q. The chart above, together with Bott periodicity K O q+8 ∼ = K O q , reduces our task to the following four statements: K O 0 (pt) ⊗ K OT−4 (pt) −→ K OT−4 (pt) is an isomorphism; K O −4 (pt) ⊗ K OT0 (pt) −→ K OT−4 (pt) is an isomorphism; KO
−1
KO
−2
−3
(B.8) (B.9)
−4
(pt) ⊗ K OT (pt) −→ K OT (pt) is injective; (pt) ⊗ K OT−2 (pt) −→ K OT−4 (pt) is injective.
(B.10) (B.11)
To verify (B.8) and (B.9) we introduce quaternionic K -theory K Sp and use Bott period0 (pt) ∼ K Sp 0 (pt)⊗T and K O 0 (pt) ∼ K O 0 (pt)⊗T, icity K Spq ∼ = = = K O q+4 . Since K SpT T 0 0 it suffices to know that the natural pairing K O (pt) ⊗ K Sp (pt) → K Sp 0 (pt) is an isomorphism, which is clear: the generators of K O 0 (pt) and K Sp 0 (pt) are the trivial real and quaternionic lines, and the map is the tensor product over the reals. Now the generator of K O −1 (pt) ∼ = Z/2Z is a class η, and η2 generates K O −2 (pt), q whence (B.10) follows from (B.11). Let K OZ/2Z (X ) denote the K O-group with Z/2Zcoefficients. The diagram of short exact sequences /Z
0
2
/Z
/ Z/2Z
/0
/T
/0
1/2
/R
/Z
0
induces a diagram of long exact sequences 0
/ K O −2 (pt)
0
/ K O −2 (pt) R
0
/ K O −2 (pt) Z/2Z
/ K O −1 (pt)
0
/
/ K O −2 (pt) T
/ K O −1 (pt)
0
/
(B.12)
Each of K O −1 (pt) and K O −2 (pt) is cyclic of order two, from which K OZ−2 /2Z (pt) is either isomorphic to Z/4Z or Z/2Z×Z/2Z; we will see shortly that it is the former. Then
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the sequence shows that the generator of this group maps to the generator of K OT−2 (pt) ∼ = −3 Z/2Z. A similar argument with a different stretch of (B.12) shows that K OZ/2Z (pt) is cyclic of order two, and with yet another stretch of (B.12) we find K OZ−4 /2Z (pt) is cyclic
−4 of order two and K OZ−4 /2Z (pt) → K OT (pt) is injective. Hence we are reduced to the −4 statement that η2 K OZ−2 /2Z (pt) → K OZ/2Z (pt) is nonzero. Cohomology groups with
Z/2Z coefficients may be computed by smashing with RP2 and shifting degree by two,12 so we must show that −2
0
η2 : K O (RP2 ) −→ K O (RP2 )
(B.13)
is nonzero. Note that the former group is generated by H − 1, where H → RP2 is the 0 nontrivial real line bundle, and since w2 (H ⊕ H ) = 0 it follows that K O (RP2 ) is cyclic of order four, verifying the claim above. Finally, we deduce that (B.13) is nonzero by applying the long exact “Bott sequence”13 η −→ K˜ q−2 (RP2 ) −→ K O (RP2 ) −→ KO q
q−1
(RP2 ) −→ K˜ q−1 (RP2 ) −→ 0
−1
twice. First, set q = 0 and use K −1 (RP2 ) = 0 to deduce that η : K O (RP2 ) → KO −1 2 2 −3 (RP ) is surjective. Then set q = −1 and use K (RP ) = 0 to deduce that η: K O −2
(RP2 ) → K O (RP2 ) is an isomorphism. (Both groups are cyclic of order two.) Acknowledgements. The work of D.F. is supported in part by NSF grant DMS-0305505. The work of G.M. is supported in part by DOE grant DE-FG02-96ER40949. G.M. would like to thank the Institute for Advanced Study for hospitality and the Monell foundation for support during completion of this paper. This research was supported in part by the National Science Foundation under Grant No. PHY99-07949 to the Kavli Institute for Theoretical Physics. This is preprint NSF-KITP-05-119. We also thank the Aspen Center for Physics for providing a stimulating environment for discussions which led to this paper. We thank Dmitriy Belov, Jacques Distler, Sergey Gukov, Michael Hopkins, Stephan Stolz, and Edward Witten for informative conversations.
References [CS] [D] [F] [FMS] [GRW] [HS] [PS]
Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and topology (College Park, Md., 1983/84), Berlin: Springer, 1985, pp. 50–80 Deligne, P.: Théorie de hodge. II, Inst. Hautes Études Sci. Publ. Math. 40, 5–57 (1971) Freed, D.S.: Dirac charge quantization and generalized differential cohomology. Surv. Differ. Geom., VII (2000), Cambridge, MA: International Press, pp. 129–194 Freed, D.S., Moore, G.W., Segal, G.: Heisenberg groups and noncommutative fluxes, http://arxiv.org/list/hep-th/0605200,2006 Gukov, S., Rangamani, M., Witten, E.: Dibaryons, strings, and branes in AdS orbifold models. JHEP 9812, 025 (1998) Hopkins, M.J., Singer, I.M.: Quadratic functions in geometry, topology, and M-theory. J. Diff. Geom. 70, 329–452 (2005) Pressley, A., Segal, G.: Loop Groups. Oxford: Oxford University Press (1986)
12 For any cohomology theory E and any pointed space X , q E Z/2Z (X ) ∼ = E˜ q+2 (X + ∧ RP2 ). 13 One derivation begins with the fibration U/O → B O → BU and the Bott periodicity (U/O) ∼ Z × B O.
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[S]
Segal, G.: The definition of conformal field theory. In: Topology, geometry and quantum field theory U. Tillmann ed., London Math. Soc. Lecture Note Ser., Vol. 308, Cambridge: Cambridge Univ. Press, 2004, pp. 421–577 Witten, E.: Five-brane effective action in M-theory. J. Geom. Phys. 22, 103–133 (1997)
[W]
Communicated by N.A. Nekrasov
Commun. Math. Phys. 271, 275–287 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0185-z
Communications in
Mathematical Physics
Discrete and Embedded Eigenvalues for One-Dimensional Schrödinger Operators Christian Remling Mathematics Department, University of Oklahoma, Norman, OK 73019-0315, USA. E-mail: [email protected] Received: 12 June 2006 / Accepted: 1 August 2006 Published online: 12 January 2007 – © Springer-Verlag 2007
Abstract: I present an example of a discrete Schrödinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than exponentially. This settles a conjecture of Simon (in the negative). The potential is of von Neumann-Wigner type, with careful navigation around a previously identified borderline situation. 1. Introduction I am interested in one-dimensional discrete Schrödinger equations, u(n + 1) + u(n − 1) + V (n)u(n) = Eu(n), and the associated self-adjoint operators u(n + 1) + u(n − 1) + V (n)u(n) (H u)(n) = u(2) + V (1)u(1)
(1.1)
(n ≥ 2) (n = 1)
on 2 (N). We could also consider whole line operators (on 2 (Z)), and for the purposes of this paper, that would actually make very little difference. Recent work has shown that there are fascinating and unexpected relations between the discrete and essential spectrum of H . If V ≡ 0, then σac (H ) = [−2, 2], σsing (H ) = ∅. It turns out that perturbations that substantially change the character of the spectrum on [−2, 2] must also introduce new spectrum outside this interval. Indeed, Damanik and Killip [3] proved the spectacular result that if σ \ [−2, 2] is finite, then [−2, 2] ⊂ σ and the spectrum continues to be purely absolutely continuous on [−2, 2]. The situation when σ \ [−2, 2] is a possibly infinite set of discrete eigenvalues, with ±2 being the only possible accumulation points, was subsequently investigated by Damanik and myself [5] (honesty demands that I point out that we actually treated the analogous problems in the continuous setting).
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A natural question, which was not addressed in [3, 5], concerns the minimal assumptions that will still imply that the spectrum is purely absolutely continuous on [−2, 2] in these situations. Put differently: How fast can the discrete eigenvalues approach the edges of the essential spectrum σess = [−2, 2] if the essential spectrum is not purely absolutely continuous? Is there in fact any bound on this rate of convergence? So we assume that σ \ [−2, 2] = {E n }, and we introduce dn ≡ dist (E n , [−2, 2]). We also assume that {E n } is infinite and dn → 0. It would in fact be natural (but not really necessary) to arrange the eigenvalues so that d1 ≥ d2 ≥ . . .. The hunt for examples with some singular spectrum on [−2, 2], but rapidly decreasing dn ’s was opened by Damanik, Killip, and Simon in [4]. This paper has an example where dn e−cn and 0 ∈ σ pp . Based on this, Simon conjectured [8, Sect. 13.5] that the slightly stronger condition 1/ ln dn−1 < ∞ might suffice to conclude that the spectrum is purely absolutely continuous on [−2, 2]. The purpose of this paper is to improve on the counterexample from [4]; this will also show that the above conjecture needs to be adjusted. More precisely, we will prove: Theorem 1.1. There exists a potential V so that: (1) For E = 0, the Schrödinger equation (1.1) has an 2 solution u. 2 (2) dn ≤ e−cn for some c > 0. Such a potential V is in fact explicitly given by V (n) =
(−1)n n
2 1+ ln n
(n ≥ 3);
(1.2)
here 2 could be replaced by any other constant c > 1. If a complete definition of V is desired, we can put V (1) = V (2) = 0. However, the behavior of V on finite sets is quite irrelevant for what we do here. Note also in this context that by adjusting V (1), say, we can achieve that 0 ∈ σ pp . To motivate (1.2), let us for a moment consider the simpler potential V (n) = g(−1)n /n. This is basically a discrete variant of the classical von Neumann-Wigner potential [10]. The values g = ±1 for the coupling constant are critical in two senses: First of all, there exists a square summable solution to the Schrödinger equation (1.1) at energy E = 0 if and only if |g| > 1. Second, if |g| ≤ 1, then the operator H has no spectrum outside [−2, 2] [2, Prop. 5.9]. On the other hand, if |g| > 1, there must be infinitely many eigenvalues E n with |E n | > 2 by the result from [3] discussed above. A rather detailed analysis is possible, and V (n) = g(−1)n /n is in fact the example of Damanik-Killip-Simon mentioned above: The eigenvalues approach ±2 exponentially fast [4, Theorem 1]. So it seems to make sense to make g n-dependent and approach the threshold value g = 1 more cautiously. The aim of this paper is to show that this idea works. Of course, since there are no positive results beyond the Damanik-Killip theorem at this point, the question of what the fastest possible decay of the dn ’s is must remain open. The fact that the type of counterexample used here feels right together with some experimentation 2 with the V 2 /4 trick from [2] have in fact led me to believe that the rate dn e−cn might already be the correct answer, but this is probably too bold a claim.
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The plan of this paper is as follows: We will prove the estimate on the eigenvalues (part (2) of Theorem 1.1) in Sect. 2–4. We will use oscillation theory: Roughly speaking, it is possible to locate eigenvalues by counting zeros. Our basic strategy is in part inspired by the treatment of [7]. From a more technical point of view, the statement we formulate as Lemma 3.1 is very much at the heart of the matter. Part (1) of Theorem 1.1 will be proved in Sect. 5. We will use a discrete version of Levinson’s Theorem (compare [6, Theorem 1.3.1]) as our main tool. 2. Variation of Constants We will write V (n) = V0 (n) + V1 (n) with V0 (n) = (−1)n /n and E = 2 + and view V1 as well as as perturbations. This strategy seems especially appropriate here because one can in fact solve (1.1) with V = V0 and E = 2 (almost) explicitly. This observation, which is crucial for what follows, is from [2]. There is actually no need to discuss the equation for a general E ≥ 2 here. Rather, it suffices to have good control on the solutions for E = 2 because we can then refer to oscillation theory at a later stage of the proof. Let us begin with the unperturbed problem: So, consider (1.1) with V (n) = V0 (n) = (−1)n /n and E = 2. As observed in [2], if we define n 1 1+ , (2.1) ϕ2n = ϕ2n+1 = 2j − 1 j=1
then ϕn solves this equation. We find a second, linearly independent solution ψ to the same equation by using constancy of the Wronskian, ϕn ψn+1 − ϕn+1 ψn = 1,
(2.2)
and making the ansatz ψn = Cn ϕn . Plugging this into (2.2), we see that ψ will solve (1.1) if Cn+1 − Cn =
1 . ϕn ϕn+1
(2.3)
For later use, we record asymptotic formulae for these solutions. A warning may be in order here: What we call ϕ in Lemma 2.1 below differs from the ϕ defined in (2.1) by a constant factor. By the same token, the asymptotic formula for Cn of course implies a particular choice of the constant in the general solution of (2.3). Lemma 2.1. There exist solutions ϕ, ψn = Cn ϕn to (1.1) with V = V0 and E = 2 satisfying the following asymptotic formulae: ϕ2n = ϕ2n+1 = (2n)1/2 + O(n −1/2 ), Cn = ln n + O(1/n). Sketch of proof. Take logarithms in (2.1) and asymptotically evaluate the resulting sum by using Taylor expansions and approximating sums by integrals. Then use this information to analyze (2.3).
To analyze the full equation, with V given by (1.2), we use variation of constants. So write T0 (n) for the transfer matrix of the unperturbed problem, that is, ϕn ψn . T0 (n) = ϕn+1 ψn+1
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Then det T0 (n) = 1 because of (2.2), and ψn+1 −ψn . T0−1 (n) = −ϕn+1 ϕn It will also be convenient to write V (n) = V0 (n) + (−1)n Wn , so Wn =
2 . n ln n
(2.4)
Now let y be a solution of the Schrödinger equation (1.1) with E = 2 and potential (1.2), and introduce Dn ∈ C2 by writing
yn yn+1
= T0 (n)Dn .
A calculation then shows that Dn solves Dn − Dn−1 = An Dn−1 ,
Cn −1
An ≡ (−1)n Wn ϕn2
Cn2 . −Cn
(2.5)
We have used the fact that ψn = Cn ϕn . We notice that An+1 ≈ −An , so we expect at least partial cancellations. To exploit this, we perform two steps (rather than just one) in the iteration (2.5). Clearly, D2n+1 = (1 + A2n+1 )(1 + A2n )D2n−1 , so we define Mn = (1 + A2n+1 )(1 + A2n ). −2 , we find that Using the formulae ϕ2n+1 = ϕ2n and C2n+1 = C2n + ϕ2n 2 W2n W2n+1 )ϕ2n
C2n −1
Mn = 1 + (W2n − W2n+1 + −2 1 2C2n + ϕ2n + W2n W2n+1 −W2n+1 0 −1
2 C2n −C2n 1 C2n . 0 0
We see at this point already that the idea of doing two steps at once was a major success because this new matrix Mn differs from the unity matrix by a correction of order O(ln n/n) whereas An itself only satisfies An = O(ln n). For the following calculations, it will be convenient to introduce some abbreviations and write Mn in the form 1 + n − wn + ρn cn (n − 2wn + ρn − ρn ) , (2.6) Mn = −n /cn 1 − n + w n where 2 n = (W2n − W2n+1 + W2n W2n+1 )ϕ2n C2n ,
cn = C2n ,
ρn = W2n W2n+1 ,
wn = W2n+1 , W2n+1 ρn = 2 . ϕ2n C2n
(2.7) (2.8)
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3. Counting Zeros We now want to use the difference equations from the preceding section to derive upper bounds on the number of zeros (more precisely: sign changes) of the solution y on large intervals. Our goal in this section is to prove the following: Lemma 3.1. There exist n 0 ∈ N and A > 0 so that the following holds: If N1 , N2 ∈ N with N2 ≥ N1 ≥ n 0 and ln N2 ≤ ln N1 + A ln1/2 N1 , then there exists a solution y of (1.1) with E = 2 and potential (1.2) satisfying yn > 0 for N1 ≤ n ≤ N2 . We start out by finding the eigenvalues and eigenvectors of Mn from (2.6). The eigenvalues are given by ρn n ρ ρn ρ2 λ± (n) = 1 + ± wn 1 + 2n − + n2 . 2 wn wn 4wn We will need information on the asymptotic behavior. From Lemma 2.1 and the definitions (see (2.4), (2.7), and (2.8)), we obtain that 1 1 1 . (3.1) , ρn , ρn = O n = + O n n ln n n 2 ln2 n This shows, first of all, that 1 +O λ± (n) = 1 ± n ln 2n
1 n 2 ln n
.
Next, write the eigenvector corresponding to λ+ in the form v+ = from (2.6) that an satisfies −
(3.2)
1 −an
. It then follows
n + (1 − n + wn − λ+ (n))(−an ) = 0, cn
and by using the slightly more precise formula n ρn λ+ (n) = 1 + wn + +O 2wn
1 2 n ln2 n
instead of (3.2), we obtain from this that an =
1 1 − +O cn 4n ln2 n
1 n ln3 n
.
(3.3)
We are interested in the sign of yn , so obviously only the direction of Dn matters. Let θn be the angle that D2n−1 makes with the eigenvector v+ ; see also Fig. 1 below. Lemma 3.2. There exists n 0 ∈ N such that the following holds: If n ≥ n 0 and 0 ≤ θn ≤ π/2, then y2n−1 > 0 and y2n > 0.
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Fig. 1.
Proof. The condition on θn implies that we can write D2n−1 = kn −d1 n with kn > 0 and dn ≤ an . Now y2n−1 ϕ2n−2 ϕ2n−2 C2n−1 1 = T0 (2n − 1)D2n−1 = kn y2n ϕ2n ϕ2n C2n −dn ϕ (1 − C2n−1 dn ) . = kn 2n−2 ϕ2n (1 − C2n dn ) Since kn , ϕ2n−2 , ϕ2n > 0 and C2n > C2n−1 > 0 (we may have to take n sufficiently large here), we see that y2n−1 , y2n will certainly be positive if 1 −C2n dn = 1 − cn dn > 0 or, equivalently, 1/cn > dn . But (3.3) shows that 1/cn > an for large n, and, as noted above, dn ≤ an , so this condition holds.
To motivate the subsequent arguments, we now make some preliminary, informal remarks about the dynamics of the recursion D2n+1 = Mn D2n−1 : First of all, a calculation shows that the eigenvector v − = v− (n) associated with the small eigenvalue λ− < 1 is of the form v− = −b1 n with bn > an , so v− lies below v+ . However, bn − an = O(ln−2 n), so v+ and v− are almost parallel. Now an application of a 2 × 2 matrix moves the vector towards the eigenvector corresponding to the large eigenvalue. So in the case at hand, D will approach v+ , or, in other words, θn will decrease. At the same time, v+ (n) approaches the positive x-axis, but this is a comparatively small effect. Nevertheless, a crossing between D and v+ will eventually occur. Our task is to bound from below the number of iterations it takes (starting from θ = π/4, say) to reach this crossing. We will use the eigenvector v+ = v+ (n) = −a1 n and the orthogonal vector a1n as our basis of R2 . As θn was defined as the angle between D2n−1 and v+ (n), it follows that D2n−1 is a constant multiple of the vector a (3.4) cos θn v+ (n) + sin θn n . 1 Conversely, we can find θ using the fact that D has such a representation. More precisely, to compute θn+1 from θn , we apply the matrix Mn to the vector from (3.4) and
Discrete and Embedded Eigenvalues
281
then take scalar products with v+ (n + 1) and an+1 1 . These operations produce multiples of cos θn+1 and sin θn+1 , respectively. We omit the details of this routine calculation. The result is as follows: If we introduce tn = tan θn , then tn+1 =
sn tn + λ+ (n)(an+1 − an ) ,
sn tn + λ+ (n)(1 + an an+1 )
(3.5)
where an was defined above (see also (3.3)) and a sn = (an+1 , 1)Mn n , 1 a
sn = (1, −an+1 )Mn n . 1 From (2.6), (3.1), (3.3), and Lemma 2.1, we obtain the asymptotic formulae sn = 1 + an an+1 + O
1 n ln n
,
sn =
ln n +O n
1 . n
(3.6)
We will prove Lemma 3.1 by analyzing the recursion (3.5). As a preliminary, we observe the following: Lemma 3.3. There exists n 0 ∈ N so that the following holds: If n ≥ n 0 and 0 ≤ tn ≤ 1/ ln n, then also tn+1 ≤ 1/ ln(n + 1). Proof. Let f (x) =
sn x + λ+ (an+1 − an )
sn x + λ+ (1 + an an+1 )
be the function from (3.5). Then f (x) =
λ+ sn x + λ+ (1 + an an+1 ))2 (
sn (an+1 − an )) , (sn (1 + an an+1 ) −
and since an+1 − an = O(1/(n ln2 n)), the derivative is positive for large enough n. Therefore, tn+1 = f (tn ) ≤ f (1/ ln n), and by dividing through by 1 + an an+1 and using (3.2), (3.3), and (3.6), we see that 1 + O n ln1 n ln1n + O n ln12 n 1 1 1 . + O = 1 − tn+1 ≤ 1 1 n ln n n ln2 n n + 1 + O n ln n On the other hand, 1 1 ln(n + 1) = ln n + ln 1 + = ln n + O , n n so the claim follows.
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We are now ready for the proof of Lemma 3.1. We must show that when solving the basic recursion D2n+1 = Mn D2n−1 (or its variant (3.5)), at least as much time as specified in the statement is spent in the region where yn > 0. We will in fact show that tn spends such an amount of time already in the region where ln−3/2 n ≤ tn ≤ ln−1 n.
(3.7)
Lemma 3.2 shows that this condition indeed implies that y2n−1 , y2n > 0. In fact, (3.7) might look unnecessarily restrictive so that the whole analysis would appear to be rather crude. However, an argument similar to the one we are about to give shows that the time spent in the neglected regions is at most of the same order of magnitude. More precisely, if 0 ≤ tn ≤ ln−3/2 n for N1 ≤ n ≤ N2 , then these N1 , N2 also satisfy the estimate from Lemma 3.1. Moreover, if ln−1 n ≤ tn ≤ M for N1 ≤ n ≤ N2 , then N2 ≤ C N1 , with C independent of M > 0. These remarks, together with a more careful analysis of the crossing between D and v+ , show that the condition from Lemma 3.1 is sharp. Let us now proceed with the strategy outlined above. Assume that (3.7) holds. From (3.5), we obtain that tn+1 − tn =
sn tn2 [sn − λ+ (n)(1 + an an+1 )] tn + λ+ (n)(an+1 − an ) −
.
sn tn + λ+ (n)(1 + an an+1 )
Note that sn − λ+ (n)(1 + an an+1 ) = O(1/(n ln n)). Also, tn ≤ ln−1 n by assumption, so the first term in the numerator is of the order O(1/(n ln2 n)). As for the next term, recall that an+1 − an = O(1/(n ln2 n)). Finally, since we are assuming that tn ≥ ln−3/2 n, we have that
sn tn2 1/(n ln2 n), so up to a constant factor, this term is not smaller than the other two summands from the numerator. The denominator clearly is of the form 1 + o(1). So, putting things together, we see that tn+1 − tn ≥ −C
ln n 2 t , n n
with C > 0. It will in fact be convenient to write this in the form tn+1 − tn ≥ −C
ln n tn tn+1 , n
with an adjusted constant C > 0. We can then introduce rn = 1/tn , and this new variable obeys rn+1 − rn ≤ C
ln n . n
(3.8)
This was derived under the assumptions that n is sufficiently large and that we have the two bounds ln n ≤ rn ≤ ln3/2 n. Our final task is to use (3.8) to find an estimate on the first n for which the second inequality fails to hold. Recall also that Lemma 3.3 says that there can’t be any such problems with the first inequality. Suppose that N1 ∈ N is sufficiently large and r N1 = ln N1 .
Discrete and Embedded Eigenvalues
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We can proceed by induction: What we have just shown says that if r j ≤ ln3/2 j for j = N1 , N1 + 1, . . . , n − 1, then rn ≥ ln n and n−1
ln j . rn ≤ ln N1 + C j
(3.9)
j=N1
So we can keep going as long as the right-hand side of (3.9) is ≤ ln3/2 n. Now clearly n−1
ln j ln2 n − ln2 N1 , j
j=N1
so, recalling from the statement of Lemma 3.1 what we are actually trying to prove, we see that it suffices to show that given a constant C > 0, we can find A > 0, n 0 ∈ N so that if N1 ≥ n 0 , then the condition ln N2 ≤ ln N1 + A ln1/2 N1
(3.10)
ln N1 + C(ln2 N2 − ln2 N1 ) ≤ ln3/2 N2 .
(3.11)
implies that
Indeed, it will then follow that yn > 0 for n = 2N1 − 1, . . . , 2N2 , and a simple adjustment gives Lemma 3.1 as originally stated, without the factors of 2. But the above claim is actually quite obvious: By taking squares in (3.10), we obtain the estimate ln2 N2 − ln2 N1 ≤ 2 A ln3/2 N1 + A2 ln N1 ≤ 3A ln3/2 N1 (say), and if also 3AC < 1, then (3.11) follows at once.
4. Oscillation Theory In this section, we will use Lemma 3.1 to derive the desired estimate on the discrete eigenvalues. For the time being, we are concerned with eigenvalues E n > 2; in particular, we then have that dn = E n −2. We will need some standard facts from oscillation theory; for proofs, we refer the reader to [9, Chap. 4]. This reference gives a careful discussion of all the results we will need (and several others), but some caution is required when looking up results in [9] because the operators discussed there are the negatives of the operators generated by (1.1). We first state a couple of comparison theorems: If u 1 , u 2 ≡ 0 both solve the same Schrödinger equation (1.1), then the number of zeros (more precisely: sign changes) on any fixed interval differs by at most one [9, Lemma 4.4]. Also, if E ≤ E and u, u solve the Schrödinger equation with energies E and E , respectively, and these solutions have the same initial phase at N1 (i.e. u(N1 + 1)/u(N1 ) = u (N1 + 1)/u (N1 )), then for any N2 > N1 , u has at least as many zeros on {N1 , . . . , N2 } as u . This can be deduced from [9, Theorem 4.7]. Finally, the following is a more direct consequence of [9, Theorem 4.7]: If u solves (1.1), u(0) = 0, u(1) = 1, and u has N zeros on N, then E N ≤ E < E N −1 . Here, we of course implicitly assume that the eigenvalues E n are arranged in their natural order:
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E 0 > E 1 > E 2 > . . . (and E −1 := ∞). In other words, it is possible to locate discrete eigenvalues by counting zeros. Let us now use these facts to derive Theorem 1.1(2) from Lemma 3.1. Define Nn = exp(γ n 2 ) + xn , with 0 < γ < A2 /4, where A is the constant from Lemma 3.1; xn ∈ [0, 1) is chosen so that Nn ∈ N. Then ln Nn+1 ≤ ln Nn + A ln1/2 Nn for all sufficiently large n, so Lemma 3.1 and the facts reviewed in the preceding paragraphs imply that any nontrivial solution u of (1.1) with E = 2 has at most C +n zeros on {1, . . . , Nn }, where C is a fixed constant, independent of n. Moreover, the same holds for any nontrivial solution of (1.1) with E ≥ 2 because increasing the energy leads to fewer zeros by the comparison theorem quoted above. Now fix E > 2 and let d = E − 2; for convenience, also assume that d < 1/2, say. Then if m ≥ 3/d, then |V (m )| < d for all m ≥ m, and thus the operator on 2 ({m, m + 1, . . .}) doesn’t have any spectrum in [2 + d, ∞). As a consequence, any solution u of the Schrödinger equation (1.1) with E = 2 + d has at most one zero on {m, m + 1, . . .}. If this is combined with what has been observed above, it follows that an arbitrary non-trivial solution u to (1.1) with E = 2 + d has at most C + n(d) zeros on N; here n(d) must be chosen so that Nn(d) ≥ 3/d. In other words, it is possible to pick n(d) ln1/2 d −1 . To sum this up: Any non-trivial solution u to (1.1) with E = 2 + d has at most C1 + C2 ln1/2 d −1 ≤ C3 ln1/2 d −1 zeros on N. We can now use that part of oscillation theory that relates the location of the eigenvalues to the number of zeros. It follows that if N = [C ln1/2 d −1 ], then the N th eigenvalue, E N , satisfies E N ≤ 2 + d. By rearranging, we see that E N − 2 ≤ exp(−cN 2 ), as claimed in part (2) of Theorem 1.1. Of course, we haven’t talked about eigenvalues < −2 yet, but this part of the claim is established by a completely analogous analysis. We can use the fact that −E is an eigenvalue of the original problem if and only if E is an eigenvalue for the potential −V . This reduces matters to the discussion of the eigenvalues bigger than 2, but for the two potentials V and −V . We have just discussed V , and, not surprisingly, it turns out that the sign change is quite irrelevant and we can simply run the whole argument again, with only a few very minor modifications. So we will not give any details, and these general remarks conclude the proof of part (2) of Theorem 1.1. 5. Asymptotic Integration What we will do below is modelled on the treatment of similar problems in the continuous setting. See in particular [6, Sect. 4.3]. We want to analyze the solutions of the discrete Schrödinger equation (1.1) with potential (1.2) and E = 0. For the following computations, it will be convenient to write V (n) = (−1)n 2vn , so 2 1 1+ vn = 2n ln n for n ≥ 3. Given a solution y of yn+1 + yn−1 + (−1)n 2vn yn = 0, yn−1 introduce Yn = yn . Then Y solves 0 1 Y . Yn+1 = −1 (−1)n+1 2vn n
(5.1)
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We again use a variation of constants type transformation, treating V as the perturbation. So define a new variable Z by Yn = Tn Z n , where cos π(n − 1)/2 sin π(n − 1)/2 ; Tn = cos π n/2 sin π n/2 0 1 note that indeed Tn+1 = −1 0 Tn , that is, T solves the unperturbed equation. A calculation shows that Z obeys 0 1 + (−1)n+1 Zn . Z n+1 = Z n − vn 1 + (−1)n 0 Here, we have used the fact that the trigonometric functions only take the values 0, ±1 at integer multiples of π/2, and, for example, cos2 nπ/2 = (1 + (−1)n )/2. Next, we diagonalize the non-oscillating part of the perturbation: To this end, write 1 1 Wn . Zn = 1 −1 Then W solves
Wn+1 = Wn + vn
−1 (−1)n
(−1)n+1 Wn . 1
Finally, we can now approximately get rid of the oscillating part with the help of a transformation of the type Wn = (1 + vn An )Un . The matrix An will be chosen shortly; it will satisfy An = O(1). Since vn2 , vn+1 −vn ∈ 1 , we then have that (1 + vn+1 An+1 )−1 = 1 − vn An+1 + Bn with Bn ∈ 1 . It thus follows that −1 Un+1 = Un + vn An − An+1 + (−1)n with Rn ∈ 1 . This suggests that we take (−1)n An = 2 and this choice leads to the equation 1 − vn Un+1 = 0
0 −1
(−1)n+1 1
Un + R n Un ,
1 , 0
0 Un + R n Un 1 + vn
(5.2)
for U . We expect that the summable perturbation R does not change the asymptotics of the solutions. We are especially interested in the decaying solution, and we want to show 1 n−1 that, more specifically, there exists a solution U satisfying Un ≈ j= j0 (1 − v j ) 0 . To do this, we mimic the proof of Levinson’s Theorem. We will basically follow the
286
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presentation given in [1] (but see also [6, Sect. 1.4]). When appropriate, we will right away specialize to the case at hand although the underlying arguments are actually of a much more general character throughout. Consider the “integral equation” ∞ 1 0 1 1 Cn = − (5.3) j 1−vk R j C j ≡ e1 − (T C)n . 0 1 − vj 0 k=n 1+vk j=n
We will see later that this is basically a way of rewriting (5.2). Since R ∈ 1 , 0 < vn ≤ c < 1, and vn → 0, the sum from (5.3) defines a bounded operator on ∞ ({ j0 , j0 + 1, . . .}; C2 ) (bounded sequences on {n : n ≥ j0 } taking values in C2 ). More precisely,
T C ∞ ≤
∞
1 |R j | · C ∞ . 1 − v j0 j= j0
So if we take j0 sufficiently large, then in fact T < 1 and thus 1 + T is boundedly invertible on ∞ ({ j0 , j0 + 1, . . .}; C2 ). In particular, C ≡ (1 + T )−1 e1 is a bounded solution to (5.3). This boundedness of course also makes sure that the series from (5.3) converges. Moreover, it follows from (5.3) that Cn → e1 as n → ∞. Finally, as already announced, we obtain a solution to the original Eq. (5.2) from this C: define ⎡ ⎤ n−1 Un = ⎣ (1 − v j )⎦ Cn ≡ pn Cn . (5.4) j= j0
Then Un solves (5.2) for n ≥ j0 . To verify this claim, call the diagonal matrix from (5.2) n , so that (5.2) becomes Un+1 = ( n + Rn )Un . Next observe that pn+1 e1 = pn n e1 and 1 0 1 0 = pn n pn+1 j j 1−vk 1−vk . 0 0 k=n+1 1+vk k=n 1+vk So if Un is defined by (5.4) and Cn ∈ ∞ solves (5.3), then ∞
1 0 1 j 1−vk R j C j , Un = pn e1 − pn 1 − vj 0 k=n 1+vk j=n
thus
1 0 j 1−vk R j C j 0 k=n+1 1+vk j=n+1 ∞ 1 0 1 j 1−vk R j C j = n pn e1 − pn n 1−v j 0 k=n 1+vk j=n+1 ∞ 1 0 1 j 1−vk R j C j = n pn e1 − pn n 1−v j 0 k=n 1+vk j=n 1 0 1 Rn C n + pn n 1−v n n 0 1−v 1+vn
Un+1 = pn+1 e1 − pn+1
∞
1 1−v j
= n Un + pn Rn Cn = ( n + Rn )Un , as required.
Discrete and Embedded Eigenvalues
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We can now go back to the original variable Y : 1 1 Yn = Tn (1 + vn An )Un . 1 −1 See also (5.1). Since Un = pn (e1 + o(1)) as n → ∞, it follows that there is a solution y of the Schrödinger equation satisfying |yn | = (1 + o(1))
n
(1 − v j )
(n → ∞).
j= j0
Now ln
n
(1 − v j ) =
j= j0
n
ln(1 − v j ) = −
j= j0 n
=−
j= j0
n
v j + O(1)
j= j0
1 1 + 2j j ln j
1 + O(1) = − ln n − ln ln n + O(1), 2
so yn2
1 , n ln2 n
and y is indeed square summable. The proof of Theorem 1.1 is complete.
References 1. Behncke, H., Remling, C.: Uniform asymptotic integration of a family of linear differential systems. Math. Nachr. 225, 5–17 (2001) 2. Damanik, D., Hundertmark, D., Killip, R., Simon, B.: Variational estimates for discrete Schrödinger operators with potentials of indefinite sign. Commun. Math. Phys. 238, 545–562 (2003) 3. Damanik, D., Killip, R.: Half-line Schrödinger operators with no bound states. Acta Math. 193, 31–72 (2004) 4. Damanik, D., Killip, R., Simon, B.: Schrödinger operators with few bound states. Commun. Math. Phys. 258, 741–750 (2005) 5. Damanik, D., Remling, C.: Schrödinger operators with many bound states. To appear in Duke Math. J. 6. Eastham, M.S.P.: The Asymptotic Solution of Linear Differential Systems. Oxford: Oxford University Press (1989) 7. Kirsch, W., Simon, B.: Corrections to the classical behavior of the number of bound states of Schrödinger operators. Ann. Physics 183, 122–130 (1988) 8. Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory. Colloquium Publications 54, Providence, RI: Amer. Math. Soc., (2005) 9. Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs 72, Providence, RI: Amer. Math. Soc., (2000) 10. von Neumann, J., Wigner, E.: Über merkwürdige diskrete Eigenwerte. Z. Phys. 30, 465–467 (1929) Communicated by B. Simon
Commun. Math. Phys. 271, 289–373 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0165-3
Communications in
Mathematical Physics
Canonical Structure and Symmetries of the Schlesinger Equations Boris Dubrovin1 , Marta Mazzocco2 1 SISSA, International School of Advanced Studies, via Beirut 2-4, 34014 Trieste, Italy.
E-mail: [email protected]
2 School of Mathematics, The University of Manchester, Manchester M60 1QD, United Kingdom.
E-mail: [email protected] Received: 19 February 2004 / Accepted: 26 September 2006 Published online: 25 January 2007 – © Springer-Verlag 2007
Abstract: The Schlesinger equations S(n,m) describe monodromy preserving deformations of order m Fuchsian systems with n + 1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations S(n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Schlesinger Equations as Monodromy Preserving Deformations of Fuchsian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Levelt basis near a logarithmic singularity and local monodromy data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Monodromy data and isomonodromic deformations of a Fuchsian system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Hamiltonian Structure of the Schlesinger System . . . . . . . . . . . . . . . 3.1 Lie-Poisson brackets for Schlesinger system . . . . . . . . . . . . . . . 3.2 Symplectic structure of the isomonodromic deformations . . . . . . . . 3.3 Multi-time dependent Hamiltonian systems and their canonical transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Scalar Reductions of Fuchsian Systems . . . . . . . . . . . . . . . . . . . . 4.1 Special Fuchsian differential equations . . . . . . . . . . . . . . . . . . 4.2 Transformation of Fuchsian systems into special Fuchsian differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Inverse transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Darboux coordinates for Schlesinger system . . . . . . . . . . . . . . . 4.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
290 296 296 299 303 303 305 307 312 312 320 327 336 346
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5. Comparison of Spectral and Isomonodromic Coordinates . 5.1 Spectral coordinates . . . . . . . . . . . . . . . . . . . 5.2 Spectral coordinates versus isomonodromic coordinates 5.3 Canonical transformations . . . . . . . . . . . . . . . A. Algebro-Geometric Darboux Coordinates . . . . . . . . . .
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349 349 351 357 359
1. Introduction The Schlesinger equations S(n,m) [52] is the following system of nonlinear differential equations [Ai , A j ] ∂ Ai = , i = j, ∂u j ui − u j [Ai , A j ] ∂ Ai = − , ∂u i ui − u j
(1.1)
j=i
for m × m matrix valued functions A1 = A1 (u), . . . , An = An (u), u = (u 1 , . . . , u n ), the independent variables u 1 , . . . , u n must be pairwise distinct. The first non-trivial case S(3,2) of the Schlesinger equations corresponds to the famous sixth Painlevé equation [17, 52, 18], the most general of all Painlevé equations. In the case of any number n > 3 of 2 × 2 matrices A j , the Schlesinger equations reduce to the Garnier systems Gn (see [18, 19, 47]). The Schlesinger equations S(n,m) appeared in the theory of isomonodromic deformations of Fuchsian systems. Namely, the monodromy matrices of the Fuchsian system d Ak (u) , = dz z − uk n
z ∈ C\{u 1 , . . . , u n }
(1.2)
k=1
do not depend on u = (u 1 , . . . , u n ) if the matrices Ai (u) satisfy (1.1). Conversely, under certain assumptions for the matrices A1 , . . . , An and for the matrix A∞ := − ( A1 + · · · + An ) ,
(1.3)
all isomonodromic deformations of the Fuchsian system (1.2) are given by solutions to the Schlesinger equations (see, e.g., [54])1 . The solutions to the Schlesinger equations can be parameterized by the monodromy data of the Fuchsian system (1.2) (see the precise definition below in Sect. 2). To reconstruct the solution starting from given monodromy data one is to solve the classical Riemann - Hilbert problem of reconstruction of the Fuchsian system from its monodromy data. The main outcome of this approach says that the solutions Ai (u) can be continued analytically to meromorphic functions on the universal covering of (u 1 , . . . , u n ) ∈ Cn | u i = u j for i = j [38, 44]. This is a generalization of the celebrated Painlevé property of absence of movable critical singularities (see details in [26, 27]). In certain cases the technique based on the theory of Riemann–Hilbert problem gives a possibility to compute the 1 Bolibruch constructed non-Schlesinger isomonodromic deformations in [8]. These can occur when the matrices Ai are resonant, i.e. admit pairs of eigenvalues with positive integer differences.
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asymptotic behavior of the solutions to Schlesinger equations near the critical locus u i = u j for some i = j, although there are still interesting open problems [29, 14, 22, 10]. It is the Painlevé property that was used by Painlevé and Gambier as the basis for their classification scheme of nonlinear differential equations. Of the list of some 50 second order nonlinear differential equations possessing Painlevé property the six (nowadays known as Painlevé equations) are selected due to the following crucial property: the general solutions to these six equations cannot be expressed in terms of classical functions, i.e., elementary functions, elliptic and other classical transcendental functions (see [57] for a modern approach to this theory based on a nonlinear version of the differential Galois theory). In particular, according to these results the general solution to the Schlesinger system S(3,2) corresponding to the Painlevé-VI equation cannot be expressed in terms of classical functions. A closely related question is the problem of construction and classification of classical solutions to the Painlevé equations and their generalizations. This problem remains open even for the case of Painlevé-VI although there are interesting results based on the theory of symmetries of the Painlevé equations [50, 49, 2] and on the geometric approach to studying the space of monodromy data [14, 25, 42, 43]. The above methods do not give any clue to the solution of the following general problems: are solutions of S(n+1,m) or of S(n,m+1) more complicated than those of S(n,m) ? Which solutions to S(n+1,m) or S(n,m+1) can be expressed via solutions to S(n,m) ? Furthermore, which of them can ultimately be expressed via classical functions? Interest in these problems was one of the starting points for our work. We began to look at the theory of symmetries of Schlesinger equations, i.e., of birational transformations acting in the space of Fuchsian systems that map solutions to solutions. One class of symmetries is well known [30, 31]: they are the so-called Schlesinger transformations
A(z) =
n i=1
n Ai A˜ i ˜ = dG(z) G −1 (z) + G(z)A(z)G −1 (z) = → A(z) z − ui dz z − ui
(1.4)
i=1
with a rational invertible m × m matrix valued function G(z) preserving the class of Fuchsian systems. Clearly such transformations preserve the monodromy of the Fuchsian system. More general symmetries of the S(3,2) Schlesinger equations do not preserve the monodromy. They can be derived from the theory, due to K.Okamoto [48], of canonical transformations of the Painlevé-VI equation considered as a time-dependent Hamiltonian system (see also [39, 4] regarding an algebro-geometric approach to some of the Okamoto symmetries). Some of the Okamoto symmetries were later generalized to the Schlesinger systems S(n,2) with arbitrary n > 3 [50, 56] (see also [28]) using the Hamiltonian formulation of the related Garnier equations. The generalization of these symmetries to S(n,m) with arbitrary n, m was one of the motivations for our work. With this problem in mind, in this paper we present a canonical Hamiltonian formulation of Schlesinger equations S(n,m) for all n, m. Recall [32, 40] that Schlesinger equations can be written as Hamiltonian systems on the Lie algebra n g := ⊕i=1 gl(m) (A1 , . . . , An )
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with respect to the standard linear Lie - Poisson bracket on g ∼ g with quadratic time-dependent Hamiltonians of the form Hk :=
tr (Ak Al ) l=k
u k − ul
.
(1.5)
Because of isomonodromicity they can be restricted onto the symplectic leaves O1 × · · · × On ∈ g obtained by fixation of the conjugacy classes O1 ,…, On of the matrices A1 , …, An . The matrix A∞ given in (1.3) is a common integral of the Schlesinger equations. Applying the procedure of symplectic reduction [41] we obtain the reduced symplectic space {A1 ∈ O1 , . . . , An ∈ On , A∞ = given diagonal matrix} modulo simultaneous diagonal conjugations.
(1.6)
The dimension of this reduced symplectic leaf in the generic situation is equal to 2g where g=
m(m − 1)(n − 1) − (m − 1). 2
Our aim is to introduce “good” canonical Darboux coordinates on the generic reduced symplectic leaf (1.6). Actually, there is a natural system of canonical coordinates on (1.6): it is obtained in [53, 20] within the general framework of algebro-geometric Darboux coordinates introduced by S.Novikov and A.Veselov [58] (see also [1, 13]). They are given by the z- and w-projections of the points of the divisor of a suitably normalized eigenvector of the matrix A(z) considered as a section of a line bundle on the spectral curve det(A(z) − w ) = 0.
(1.7)
However, the symplectic reduction (1.6) is time–dependent. This produces a shift in the Hamiltonian functions that can only be computed by knowing the explicit parameterization of the matrices A1 , . . . , An by the spectral coordinates. The same difficulty makes the computation of the action of the symmetries on the spectral coordinates for m > 2 essentially impossible. Instead, we construct a new system of the so-called isomonodromic Darboux coordinates q1 , …, qg , p1 , …, pg on generic symplectic manifolds (1.6) and we give the new Hamiltonians in these coordinates. Let us explain our construction. The Fuchsian system (1.2) can be reduced to a scalar differential equation of the form y (m) =
m−1
dl (z)y (l) .
(1.8)
l=1
For example, one can eliminate the last m − 1 components of to obtain a m th order equation for the first component y := 1 . (Observe that the reduction procedure depends on the choice of the component of .) The resulting Fuchsian equation will have regular singularities at the same points z = u 1 , …, z = u n , z = ∞. It will also have other singularities produced by the reduction procedure. However, they will be apparent singularities, i.e., the solutions to (1.8) will be analytic in these points. Generically there
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will be exactly g apparent singularities (cf. [46]; a more precise result about the number of apparent singularities working also in the nongeneric situation was obtained in [7]); they will be the first part q1 , . . . , qg of the canonical coordinates. The conjugated momenta are defined by 1 pi = Resz=qi dm−2 (z) + dm−1 (z)2 , i = 1, . . . , g. 2 Our first result is Theorem 1.1. Let the eigenvalues of the matrices A1 , …, An , A∞ be pairwise distinct. Then the map ⎫ ⎧ Fuchsian systems with given poles ⎬ ⎨ and given eigenvalues of A1 , . . . , An , A∞ → (q1 , . . . , qg , p1 , . . . , pg ) (1.9) ⎭ ⎩ modulo diagonal conjugations gives a system of rational Darboux coordinates on a large Zariski open set2 in the generic reduced symplectic leaf (1.6). The Schlesinger equations S(n,m) in these coordinates are written in the canonical Hamiltonian form ∂qi ∂Hk = , ∂u k ∂ pi ∂Hk ∂ pi =− , ∂u k ∂qi with the Hamiltonians
1 Hk = Hk (q, p; u) = −Resz=u k dm−2 (z) + dm−1 (z)2 , k = 1, . . . , n. 2
Here rational Darboux coordinates means that the elementary symmetric functions σ1 (q), . . . , σg (q) and σ1 ( p), . . . , σg ( p) are rational functions of the coefficients of the system and of the poles u 1 , . . . , u n . Moreover, there exists a section of the map (1.9) given by rational functions Ai = Ai (q, p), i = 1, . . . , n,
(1.10)
symmetric in (q1 , p1 ), …, (qg , pg ) with coefficients depending on u 1 , . . . , u n and on the eigenvalues of the matrices Ai , i = 1, . . . , n, ∞. All other Fuchsian systems with the same poles u 1 , . . . , u n , the same exponents and the same ( p1 , . . . , pg , q1 , . . . , qg ) are obtained by simultaneous diagonal conjugation Ai (q, p) → C −1 Ai (q, p)C, i = 1, . . . , n, C = diag (c1 , . . . , cm ). In the course of the proof of Theorem 1.1, we establish that the same parameters ( p1 , . . . , pg , q1 , . . . , qg ) are rational coordinates in the space of what we call special Fuchsian equations, i.e., m th order Fuchsian equations with n + 1 regular singularities 2 A precise characterization of this large open set is given in Theorem 4.14 and Remark 4.21.
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with given exponents and g apparent singularities with the exponents3 0, 1, …, m − 2, m. We then prove that there is a birational map between special Fuchsian equations and Fuchsian systems. This allows us to conclude that ( p1 , . . . , pg , q1 , . . . , qg ) are rational coordinates in the space of Fuchsian systems. The natural action of the symmetric group Sn on the Schlesinger equations is described in the following Theorem 1.2. The Schlesinger equations S(n,m) written in the canonical form of Theorem 1.1 admit a group of birational canonical transformations S2 , . . . , Sm , S∞
⎧ q˜ = u + u − q , i = 1, . . . , g, ⎪ ⎪ ⎨ p˜i = −1 p , k i = i1, . . . , g, i i Sk : (1.11) ⎪ u˜ l = u 1 + u k − u l , l = 1, . . . , n, ⎪ ⎩ H˜ = −H , l = 1, . . . , n, l l ⎧ 1 q˜i = qi −u 1 , i = 1, . . . , g, ⎪ ⎪ ⎪ 2 ⎪ ⎪ p˜ i = − pi qi2 − 2mm−1 qi , i = 1, . . . , g, ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ u˜ l = ul −u 1 , l = 2, . . . , n, ⎨ u 1 → ∞, S∞ : (1.12) ⎪ ⎪ ∞ → u 1 , ⎪ ⎪ ⎪ H˜ 1 = H1 , ⎪ ⎪ ⎪ 0 ⎪ H˜ l = −Hl (u l − u 1 )2 + (u l − u 1 )(dm−1 (u l − u 1 ))2 − ⎪ ⎪ ⎩ 2 0 −(u l − u 1 ) (m−1)(mm −m−1) dm−1 (u l − u 1 ), l = 2, . . . , n, where 0 dm−1 (u k )
=
g s=1
1 1 m (m − 1) − . u k − qs 2 u k − ul l=k
The transformation Sk acts on the monodromy matrices as follows −1 Mk Mk−1 . . . M1 , M˜ 1 = M1−1 . . . Mk−1
M˜ j = M j , j = 1, k, −1 M˜ k = Mk−1 . . . M2 M1 M2−1 . . . Mk−1 , i = k + 1, . . . , n. The transformation S∞ acts on the monodromy matrices as follows M˜ ∞ = e−
2πi m
−1 C 1 M∞ M1 M∞ C1−1 ,
M˜ 1 = e m C1 M∞ C1−1 , M˜ j = C1−1 M j C1 , j = 1, ∞, 2πi
where C1 is the connection matrix defined in Section 2. 3 As it was discovered by H.Kimura and K.Okamoto [34] these are the exponents at the generic apparent singularities of the scalar reduction of a Fuchsian system.
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Our approach to the construction of Darboux coordinates seems not to work for nongeneric reduced symplectic leaves. The problem is that, for a nongeneric orbit the number of apparent singularities of the scalar reduction is bigger than half of the dimension of the symplectic leaf. The most striking is the example of rigid Fuchsian systems. They correspond to the extreme nongeneric case where the reduced symplectic leaf is zero dimensional. One could expect to have no apparent singularities of the scalar reduction; by no means is this the case (see [33]). The study of reductions of Schlesinger systems on nongeneric reduced symplectic leaves can reveal new interesting systems of nonlinear differential equations. Just one example of a nongeneric situation is basic for the analytic theory of semisimple Frobenius manifolds. In this case m = n, the monodromy group of the Fuchsian system is generated by n reflections [12]. The dimension of the symplectic leaves equals n(n − 1) n − . 2 2 We do not know yet how to construct isomonodromic Darboux coordinates for this case if the dimension of the Frobenius manifold is greater than 3. Our next result is the comparison of the isomonodromic Darboux coordinates with those obtained in the framework of the theory of algebro-geometrically integrable systems (dubbed here spectral Darboux coordinates). The spectral Darboux coordinates are constructed as follows. In the generic situation under consideration the genus of the spectral curve (1.7) equals g. The affine part of the divisor of the eigenvector A(z)ψ = wψ, ψ = (ψ1 , . . . , ψm )T of the matrix A(z) has degree g. Denote γ1 , . . . , γg the z-projections of the points of the divisor and µ1 , . . . , µg their w-projections. These are the spectral Darboux coordinates in the case under consideration. Theorem 1.3. Let us consider a family of Fuchsian systems Ai d = A(z), A(z) = dz z − ui n
i=1
depending on a small parameter . The matrices A1 , . . . , An , A∞ are assumed to be independent of . Then the isomonodromic Darboux coordinates of this Fuchsian system have the following expansion as → 0: qk = γk + O(),
pk = −1 µk + O(1), k = 1, . . . , g.
Here γk , µk are the spectral Darboux coordinates of the matrix A(z) defined above. We do not even attempt in this paper to discuss physical applications of our results. However one of them looks particularly attractive. According to an idea of N.Reshetikhin [51] (see also [23]) the well known Knizhnik–Zamolodchikov equations in conformal field theory can be considered as a quantization of Schlesinger equations. We believe that our isomonodromic canonical coordinates could play an important role in the analysis of the quantization procedure, somewhat similar to the role played by the spectral canonical coordinates in the Sklyanin scheme of quantization of integrable systems [55]. We plan to address this problem in subsequent publications.
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The paper is organized as follows. In Sect. 2 we recall the relationship between Schlesinger equations and isomonodromic deformations of Fuchsian systems. In Sect.3 we discuss the Hamiltonian formulation of Schlesinger equations. A formula for the symplectic structure of Schlesinger equations recently found by I.Krichever [35] proved to be useful for subsequent calculations with isomonodromic coordinates; we prove that this formula is equivalent to the standard one. In Sect. 4 we construct the isomonodromic Darboux coordinates and establish a birational isomorphism between the space of Fuchsian systems considered modulo conjugations and the space of special Fuchsian differential equations. In Sect. 5 we express the semiclassical asymptotics of the isomonodromic Darboux coordinates via spectral Darboux coordinates. The necessary facts from the theory of spectral Darboux coordinates are collected in the Appendix. Finally we apply the above results to constructing the nontrivial symmetries of Schlesinger equations. 2. Schlesinger Equations as Monodromy Preserving Deformations of Fuchsian Systems In this section we establish our notations, recall a few basic definitions and prove some technical lemmata that will be useful throughout this paper. The Schlesinger equations S(n,m) describe monodromy preserving deformations of Fuchsian systems (1.2) with n + 1 regular singularities at u 1 , . . . , u n , u n+1 = ∞: Ak d , = dz z − uk n
z ∈ C\{u 1 , . . . , u n },
(2.1)
k=1
Ak being m × m matrices independent on z, and u k = u l for k = l, k, l = 1, . . . , n + 1. Let us explain the precise meaning of this claim.
2.1. Levelt basis near a logarithmic singularity and local monodromy data. A system A(z) d = dz z − z0
(2.2)
is said to have a logarithmic, or Fuchsian singularity at z = z 0 if the m × m matrix valued function A(z) is analytic in some neighborhood of z = z 0 . By definition the local monodromy data of the system is the class of equivalence of such systems w.r.t. local gauge transformations A(z) → G −1 (z)A(z) G(z) + (z − z 0 )G −1 (z)∂z G(z)
(2.3)
analytic near z = z 0 satisfying det G(z 0 ) = 0. The parameters of the local monodromy can be obtained by choosing a suitable fundamental matrix solution of the system (2.2). The most general construction of such a fundamental matrix was given by Levelt [37]. We will briefly recall this construction in the form suggested in [12].
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Without loss of generality one can assume that z 0 = 0. Expanding the system near z = 0 one obtains d A0 (2.4) = + A1 + z A2 + . . . . dz z Let us now describe the structure of local monodromy data. Two linear operators , R acting in the complex m-dimensional space V , , R : V → V, are said to form an admissible pair if the following conditions are fulfilled: 1. The operator is semisimple and the operator R is nilpotent. 2. R commutes with e2πi , e2πi R = R e2πi .
(2.5)
Observe that, due to the last condition the operator R satisfies R(Vλ ) ⊂ ⊕k∈Z Vλ+k for any λ ∈ Spec ,
(2.6)
where Vλ ⊂ V is the subspace of all eigenvectors of with the eigenvalue λ. The last condition says that 3. The sum in the r.h.s. of (2.6) contains only non-negative values of k. A decomposition R = R0 + R1 + R2 + . . .
(2.7)
Rk (Vλ ) ⊂ Vλ+k for any λ ∈ Spec .
(2.8)
is defined where
Clearly this decomposition contains only a finite number of terms. Observe the useful identity z R z − = R0 + z R1 + z 2 R2 + . . . .
(2.9)
Theorem 2.1. For a system (2.4) with a logarithmic singularity at z = 0 there exists a fundamental matrix solution of the form (z) = (z)z z R , where (z) is a matrix valued function analytic near z = 0 satisfying det (0) = 0 and , R is an admissible pair.
(2.10)
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The formula (2.10) makes sense after fixing a branch of logarithm log z near z = 0. Note that z R is a polynomial in log z due to nilpotency of R. The proof can be found in [37] (cf. [12]). Clearly is the semisimple part of the matrix A0 ; R0 coincides with its nilpotent part. The remaining terms of the expansion appear only in the resonant case, i.e., if the difference between some eigenvalues of is a positive integer. In the important particular case of a diagonalizable matrix A0 , T −1 A0 T = = diag (λ1 , . . . , λm ) with some nondegenerate matrix T , the matrix function (z) in the fundamental matrix solution (2.10) can be obtained in the form + z 1 + z 2 2 + . . . . (z) = T The matrix coefficients 1 , 2 , …of the expansion as well as the matrix components R1 , R2 , …of the matrix R (see (2.7)) can be found recursively from the equations [ , k ] − k k = Bk − Rk +
k−1
k−i Bi − Ri k−i , k ≥ 1.
i=1
Here Bk := T −1 Ak T, k ≥ 1. If kmax is the maximal integer among the differences λi − λ j then Rk = 0 for k > kmax . Observe that vanishing of the logarithmic terms in the fundamental matrix solution (2.10) is a constraint imposed only on the first kmax coefficients A1 , …, Akmax of the expansion (2.4). It is not difficult to describe the ambiguity in the choice of the admissible pair of matrices , R describing the local monodromy data of the system (2.4). Namely, the diagonal matrix is defined up to permutations of diagonal entries. Assuming the order fixed, the ambiguity in the choice of R can be described as follows [12]. Denote C0 ( ) ⊂ G L(V ) the subgroup consisting of invertible linear operators G : V → V satisfying z G z − = G 0 + z G 1 + z 2 G 2 + . . . .
(2.11)
The definition of the subgroup can be reformulated [12] in terms of invariance of a certain flag in V naturally associated with the semisimple operator . The matrix R˜ obtained from R by the conjugation of the form R˜ = G −1 R G
(2.12)
will be called equivalent to R. Multiplying (2.10) on the right by G one obtains another fundamental matrix solution to the same system of the same structure R˜ ˜ ˜ (z) := (z)z z R G = (z)z z ,
˜ ˜ i.e., (z) is analytic at z = 0 with det (0) = 0. The columns of the fundamental matrix (2.10) form a distinguished basis in the space of solutions to (2.4).
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Definition 2.2. The basis given by the columns of the matrix (2.10) is called the Levelt basis in the space of solutions to (2.4). The fundamental matrix (2.10) is called the Levelt fundamental matrix solution. The monodromy transformation of the Levelt fundamental matrix solution reads (2.13) z e2πi = (z)M, M = e2πi e2πi R . To conclude this section let us denote C( , R) the subgroup of invertible transformations of the form G k z k and [G, R] = 0}. (2.14) C( , R) = { G ∈ G L(V ) | z G z − = k∈Z
˜ associated with equivalent matrices R and R˜ are The subgroups C( , R) and C( , R) conjugated. It is easy to see that this subgroup coincides with the centralizer of the monodromy matrix (2.13), G ∈ C( , R) iff G e2πi e2πi R = e2πi e2πi R G, det G = 0.
(2.15)
Denote C0 ( , R) ⊂ C( , R)
(2.16)
the subgroup consisting of matrices G such that the expansion (2.14) contains only non-negative powers of z. Multiplying the Levelt fundamental matrix (2.10) by a matrix G ∈ C0 ( , R) one obtains another Levelt solution to (2.4), R ˜ (z)z z R G = (z)z z .
(2.17)
In the next section we will see that the quotient C( , R)/C0 ( , R) plays an important role in the theory of monodromy preserving deformations. Example 2.3. For ⎛
−1 =⎝ 0 0
0 0 0
⎞ 0 0⎠ , 1
⎛
0 R = ⎝a c
0 0 b
⎞ 0 0⎠ 0
the quotient C( , R)/C0 ( , R) is trivial iff a b = 0. 2.2. Monodromy data and isomonodromic deformations of a Fuchsian system. (k) Denote λ j , j = 1, . . . , m, the eigenvalues of the matrix Ak , k = 1, . . . , n, ∞, where the matrix A∞ , is defined as A∞ := −
n
Ak .
k=1
For the sake of technical simplicity let us assume that (k)
λi
(k)
= λ j for i = j,
k = 1, . . . , n, ∞.
(2.18)
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Moreover, it will be assumed that A∞ is a constant diagonal m × m matrix with eigenvalues λ(∞) j , j = 1, . . . , m. Denote (k) , R (k) the local monodromy data of the Fuchsian system near the points z = u k , k = 1, . . . , n, ∞. The matrices (k) are all diagonal (k)
(k) = diag (λ1 , . . . , λ(k) m ), k = 1, . . . , n, ∞,
(2.19)
and under our assumptions (∞) = A∞ . Recall that the matrix G ∈ G L(m, C) belongs to the group C0 ( (∞) ) iff z −
(∞)
G z
(∞)
= G0 +
G1 G2 + 2 + .... z z
(2.20)
It is easy to see that our assumptions about eigenvalues of A∞ imply diagonality of the matrix G 0 . Let us also recall that the matrices (k) satisfy tr (1) + · · · + tr (∞) = 0. (k)
(2.21)
(k)
The numbers λ1 , . . . , λm are called the exponents of the system (1.2) at the singular point u k . Let us fix a fundamental matrix solution of the form (2.10) near all singular points u 1 , . . . , u n , ∞. To this end we are to fix branch cuts on the complex plane and choose the branches of logarithms log(z −u 1 ), . . . , log(z −u n ), log z −1 . We will do it in the following way: perform parallel branch cuts πk between ∞ and each of the u k , k = 1, . . . , n along a given (generic) direction. After this we can fix Levelt fundamental matrices analytic on z ∈ C \ ∪nk=1 πk , (k)
(2.22) (k)
k (z) = Tk ( + O(z − u k )) (z −u k ) (z − u k ) R , z → u k , k = 1, . . . , n, and
(z) ≡ ∞ (z) =
1 (∞) + O( ) z −A∞ z −R , as z → ∞. z
(2.23)
(2.24)
Define the connection matrices by ∞ (z) = k (z)Ck ,
(2.25)
where ∞ (z) is to be analytically continued in a vicinity of the pole u k along the positive side of the branch cut πk . The monodromy matrices Mk , k = 1, . . . , n, ∞ are defined with respect to a basis l1 , . . . , ln of loops in the fundamental group π1 (C\{u 1 , . . . u n }, ∞) . Choose the basis in the following way. The loop lk arrives from infinity in a vicinity of u k along one side of the branch cut πk that will be called positive, then it encircles
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u k going in the anti-clock-wise direction leaving all other poles outside and, finally it returns to infinity along the opposite side of the branch cut πk called negative. Denote l ∗j ∞ (z) the result of analytic continuation of the fundamental matrix ∞ (z) along the loop l j . The monodromy matrix M j is defined by l ∗j ∞ (z) = ∞ (z)M j , j = 1, . . . , n.
(2.26)
The monodromy matrices satisfy M∞ Mn · · · M1 = ,
M∞ = exp (2πi A∞ ) exp 2πi R (∞)
(2.27)
if the branch cuts π1 , …, πn enter the infinite point according to the order of their labels, i.e., the positive side of πk+1 looks at the negative side of πk , k = 1, . . . , n − 1. Clearly one has Mk = Ck−1 exp 2πi (k) exp 2πi R (k) Ck , k = 1, . . . , n. (2.28) The collection of the local monodromy data (k) , R (k) together with the central connection matrices Ck will be used in order to uniquely fix the Fuchsian system with given poles. They will be defined up to an equivalence that we now describe. The eigenvalues of the diagonal matrices (k) are defined up to permutations. Fixing the order of the eigenvalues, we define the class of equivalence of the nilpotent part R (k) and of the connection matrices Ck by factoring out the transformations of the form −1 Rk → G −1 k Rk G k , C k → G k C k G ∞ , k = 1, . . . , n,
G k ∈ C0 ( (k) ), G ∞ ∈ C0 ( (∞) ).
(2.29)
Observe that the monodromy matrices (2.28) will transform by a simultaneous conjugation Mk → G −1 ∞ Mk G ∞ , k = 1, 2, . . . , n, ∞. Definition 2.4. The class of equivalence (2.29) of the collection (1) , R (1) , . . . , (∞) , R (∞) , C1 , . . . , Cn
(2.30)
is called monodromy data of the Fuchsian system with respect to a fixed ordering of the eigenvalues of the matrices A1 , …, An and a given choice of the branch cuts. Lemma 2.5. Two Fuchsian systems of the form (1.2) with the same poles u 1 , . . . , u n , ∞ and the same matrix A∞ coincide, modulo diagonal conjugations if and only if they have the same monodromy data with respect to the same system of branch cuts π1 , . . . , πn . Proof. Let (1) ∞ (z) =
1 1 (∞) (∞) ˜ (∞) ˜ (∞) + O( ) z − z −R , (2) (z) = + O( ) z − z − R ∞ z z
be the fundamental matrices of the form (2.24) of the two Fuchsian systems. Using the ˜ (∞) = (∞) . Multiplying (2) assumption about A∞ we derive that ∞ (z) if necessary
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on the right by a matrix G ∈ C0 ( (∞) ), we can obtain another fundamental matrix of the second system with R˜ (∞) = R (∞) . Consider the following matrix: (1) −1 Y (z) := (2) ∞ (z)[∞ (z)] .
Y (z) is an analytic function around infinity: 1 Y (z) = G 0 + O , as z → ∞, z
(2.31)
(2.32)
where G 0 is a diagonal matrix. Since the monodromy matrices coincide, Y (z) is a single valued function on the punctured Riemann sphere C\{u 1 , . . . , u n }. Let us prove that Y (z) is analytic also at the points u k . Indeed, having fixed the monodromy data, we can (1) (2) choose the fundamental matrices k (z) and k (z) of the form (2.23) with the same connection matrices Ck and the same matrices (k) , R (k) . Then near the point u k , Y (z) is analytic:
−1 (2) (1) Y (z) = Tk ( + O(z − u k )) Tk ( + O(z − u k )) . (2.33) This proves that Y (z) is an analytic function on all C and then, by the Liouville theorem Y (z) = G 0 , which is constant. So the two Fuchsian systems coincide, after conjugation by the diagonal matrix G 0 . Remark 2.6. The connection matrices are determined, within their equivalence classes by the monodromy matrices if the quotients C( (k) , R (k) )/C0 ( (k) , R (k) ) are trivial for all k = 1, . . . , n. In particular this is the case when all the characteristic exponents at the poles u 1 , . . . , u n are non-resonant. From the above lemma the following result readily follows. Theorem 2.7. If the matrices Ak (u 1 , . . . , u n ) satisfy Schlesinger equations (1.1) and the matrix A∞ = −(A1 + · · · + An ) is diagonal then all the characteristic exponents do not depend on u 1 , . . . , u n . The fundamental matrix ∞ (z) can be chosen in such a way that the nilpotent matrix R (∞) and also all the monodromy matrices are constant in u 1 , . . . , u n . Moreover, the Levelt fundamental matrices k (z) can be chosen in such a way that all the nilpotent matrices R (k) and also all the connection matrices Ck are constant. Viceversa, if the deformation Ak = Ak (u 1 , . . . , u n ) is such that the monodromy data do not depend on u 1 , . . . , u n then the matrices Ak (u 1 , . . . , u n ), k = 1, . . . , n satisfy Schlesinger equations. At the end of this section we give a criterion that ensures that the “naive” definition of monodromy preserving deformations still gives rise to the Schlesinger equations. Theorem 2.8. Let Ak = Ak (u 1 , . . . , u n ), k = 1, . . . , n be a deformation of the Fuchsian system (1.2) such that the following conditions hold true.
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1. The matrix A∞ = −A1 − · · · − An is constant and diagonal. 2. The Fuchsian system admits a fundamental matrix solution of the form (2.24) with the u-independent matrix R (∞) . Denote ∞ (z; u) the fundamental matrix solution of the family of Fuchsian systems of the form (2.24). 3. The monodromy matrices M1 , . . . , Mn defined as in (2.26) with respect to the fundamental matrix ∞ (z; u) do not depend on u 1 , . . . , u n . Note that this implies constancy of the diagonal matrices (k) of exponents, k = 1, . . . , n. 4. The (class of equivalence of) local monodromy data ( (k) , R (k) ) does not depend on u 1 , . . . , u n . 5. The quotients C( (k) , R (k) )/C0 ( (k) , R (k) ) are zero dimensional for all k = 1, . . . , n. Then the deformation satisfies the Schlesinger equations. Moreover, under the assumption of validity of 1 - 4, if Condition 5 does not hold true, then there exist non-Schlesinger deformations preserving the monodromy matrices. Proof. The first statement of the theorem easily follows from Remark 2.6 and Theorem 2.7. To prove the second part, let us assume that the dimension of the quotient C( (k) , R (k) )/C0 ( (k) , R (k) ) is positive for some k. Here (k) , R (k) are local monodromy data of the Fuchsian system (1.2) with some poles u 1 , . . . , u n , ∞. Let us choose a nontrivial family of matrices G(s) ∈ C( (k) , R (k) )/C0 ( (k) , R (k) ) for sufficiently small s, G(0) = . We will now obtain a deformation of the Fuchsian system (1.2) in the following way. Let us deform the k th connection matrix Ck by putting Ck (s) := G(s)Ck . To reconstruct the deformation of the Fuchsian system, we are to solve the suitable Riemann - Hilbert problem. It will be solvable for sufficiently small s because of solvability for s = 0. At this point one can also deform the poles u i (s), u i (0) = u i . This deformation is obviously isomonodromic but not of the Schlesinger type. The theorem is proved. 3. Hamiltonian Structure of the Schlesinger System 3.1. Lie-Poisson brackets for Schlesinger system. The Hamiltonian description of the Schlesinger system can be derived [24] from the general construction of a Poisson bracket on the space of flat connections in a principal G-bundle over a surface with boundary using Atiyah - Bott symplectic structure (see [5]). Explicitly this approach yields the following well known formalism representing the Schlesinger system S(n,m) in Hamiltonian form with n time variables u 1 , . . . , u n and n commuting time–dependent Hamiltonian flows on the dual space to the direct sum of n copies of the Lie algebra sl(m), g := ⊕n sl(m) (A1 , A2 , . . . , An ) . The standard Lie-Poisson bracket on g∗ reads i i k k A p j , Aq l = δ pq δli A p j − δ kj Aq l .
(3.1)
(3.2)
(We identify sl(m) with its dual by using the Killing form (A, B) = Tr AB, A, B ∈ sl(m).) The following statement is well-known (see [30, 40]) and can be checked by a straightforward computation.
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Theorem 3.1. The dependence of the solutions Ak , k = 1, . . . , n, of the Schlesinger system S(n,m) upon the variables u 1 , . . . , u n is determined by Hamiltonian systems on (3.1) with time-dependent quadratic Hamiltonians Hk =
Tr (Ak Al ) l=k
u k − ul
,
∂ Al = {Al , Hk }. ∂u k
(3.3) (3.4)
To arrive from the Hamiltonian systems (3.3), (3.4) to the isomonodromic deformations one is to impose an additional constraint. Define A∞ := −(A1 + · · · + An ).
(3.5)
∂ A∞ = {A∞ , Hi } = 0, i = 1, . . . , n. ∂u i
(3.6)
It can be easily seen that
So, the matrix entries of A∞ are integrals of the Schlesinger equations. They generate the action of the group sl(m) on g by the diagonal conjugations Ak → G −1 Ak G, k = 1, . . . , n, G ∈ sl(m).
(3.7)
To identify the Hamiltonian equations (3.4) with isomonodromic deformations one is to apply the Marsden - Weinstein procedure of symplectic reduction [41]. In our setting this procedure works as follows. Let us choose a particular level surface of the moment map corresponding to the first integrals (3.6). We will mainly deal with the level surfaces of the form (∞) (∞) (∞) (∞) A∞ = diag λ1 , . . . , λm , λi = λ j , i = j (3.8) (∞)
(∞)
for some pairwise distinct numbers λ1 , . . . , λm . After restricting the Hamiltonian systems onto the level surface (3.8) there remains a residual symmetry with respect to conjugations by diagonal matrices Ak → D −1 Ak D, k = 1, . . . , n,
D = diag (d1 , . . . , dn ).
(3.9)
Denote m−1 ⊂ S L(m) Diag C∗ the subgroup of diagonal matrices acting on g by simultaneous conjugations (3.9). Definition 3.2. The quotient of the restriction of the Hamiltonian system (3.3), (3.4) onto the level surface of the form (3.8) w.r.t. the transformations (3.9) will be called the reduced Schlesinger system. From the above results it readily follows that the reduced Schlesinger system describes all nontrivial monodromy preserving deformations of the Fuchsian system (1.2).
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3.2. Symplectic structure of the isomonodromic deformations. The symplectic leaves of the Poisson bracket (3.2) are products of adjoint orbits (A1 , . . . , An ) ∈ O1 × · · · × On ⊂ g,
(3.10)
where Ok is the adjoint orbit of Ak . The symplectic structure ω H induced by (3.2) on the orbits can be represented in the following form [24]. Given two tangent vectors δ1 A =
δ1 A k δ2 A k , δ2 A = z − uk z − uk k
(3.11)
k
at the given point A(z) the value of the symplectic form can be computed by ω H (δ1 A, δ2 A) = − Tr(Uk(1) δ2 Ak ),
(3.12)
k (i)
(i)
where the matrices Uk are such that δi Ak = [Uk , Ak ]. Actually, the symplectic structure (3.12) was obtained in [24] from the general Atiyah - Bott Poisson structure [3] on the moduli space of flat S L(m) connections on the surface . It is obtained by projecting the gauge invariant symplectic form 1 ω AB (δ1 A, δ2 A) = Tr (δ1 A ∧ δ2 A) (3.13) 2πi onto the moduli space. In our case is the Riemann sphere without small discs around the poles of A(z). The eigenvalues of the matrices Ak are Casimirs of the Poisson bracket (3.2). We will mainly consider the generic case where these eigenvalues are distinct for all k = 1, . . . , n. Then the level sets of the Casimirs coincide with the symplectic leaves (3.10). Denote (Spec A1 , . . . , Spec An ) the collection of the eigenvalues of the matrices (A1 , . . . , An ) ∈ g. Generically these are the parameters of the symplectic leaves. Fixing the level surface of the moment map (3.8) and taking the quotient over the action (3.9) of the group Diag ⊂ S L(m) of diagonal matrices one obtains a manifold that we denote M (Spec A1 , . . . , Spec An ; A∞ ) .
(3.14)
The dimension of this manifold is equal to 2g where g=
m(m − 1)(n − 1) − (m − 1). 2
(3.15)
Indeed, the dimension of a generic adjoint orbit Oi is equal to m 2 − m. Choosing a level surface (3.8) of the momentum map (A1 , . . . , An ) → A∞ := −(A1 + · · · + An ) we impose only m 2 − 1 independent equations, since the trace of the matrix A∞ is equal to the sum of traces of A1 , …, An . Finally, subtracting the dimension m − 1 of the group Diag we arrive at (3.15). The manifold still carries a symplectic structure since
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the action of the Abelian group Diag preserves the Poisson bracket (3.2). One of the aims of this paper is to construct a system of canonical Darboux coordinates on generic manifolds of the form (3.14). For our aims the following approach to the Hamiltonian theory of monodromy preserving deformations developed recently by Krichever [36]will be useful. He has obtained a general formula for the symplectic structure on the space of isomonodromic deformations of a generic linear system of ODEs of the form d = A(z), dz where z is a variable on a punctured genus g algebraic curve and A(z) is any meromorphic matrix function of z with poles of any order at P1 , . . . , Pn . In the case of genus g = 0 the formula reads ωK = −
1 Resz=Pk Tr(δ A ∧ δ −1 ), 2
(3.16)
k
where the variations δ A and δ are independent. The corresponding Hamiltonian function describing the isomonodromic deformations in the parameter P j is 1 HK j = − Resz=P j Tr(A(z)2 ). 2
(3.17)
Due to gauge invariants [36] the symplectic form admits a restriction onto the manifold (3.10). It defines therefore a symplectic structure ω K on the product of adjoint orbits. Let us prove that the symplectic structures ω H and ω K coincide, up to a sign. Theorem 3.3. In the setting of isomonodromic deformations of the Fuchsian system (1.2) the two symplectic forms (3.12) and (3.16) coincide (up to a sign) ω K = −ω H .
(3.18)
Proof. The variations δ A tangent to the orbit are obtained by means of an infinitesimal gauge transform with G = + φ, i.e. δ A = G AG −1 +
dG −1 d G − A = − [A, φ] + φ + O(φ 2 ). dz dz
So, representing the tangent vectors (3.11) as δi A = −[A, φi ] + we obtain
d φi , i = 1, 2 dz
(i) δi Ak = − Ak , Uk ,
where Uk(i) is given by the first term of the expansion of φi at u k , (i)
φi (z) = Uk + O(z − u k ).
Canonical Structure and Symmetries of the Schlesinger Equations
Because of this ω H (δ1 A, δ2 A) = −
307
(1) (2) . Tr Ak Uk , Uk
k
In the formula (3.16) we can take δi −1 = φi , i = 1, 2. Indeed, the matrices φi and δi −1 satisfy the same equation. This follows from d δ = δ A + Aδ. dz Thus 1 Resz=u k Tr(δ1 A φ2 − φ1 δ2 A) 2 k (1) (2) − ω H (δ1 A, δ2 A). Tr Ak Uk , Uk =
ω K (δ1 A, δ2 A) −
k
The theorem is proved.
One can also prove that the Hamiltonians (3.17) correctly reproduce the formula (3.3), up to a minus sign (this is not a problem because also the signs of the two symplectic structures are opposite). We shall use the Krichever formula in the next section in order to compute Poisson brackets for a new set of Darboux coordinates that we call isomonodromic coordinates. 3.3. Multi-time dependent Hamiltonian systems and their canonical transformations. Let P be a manifold equipped with a Poisson bracket { , }. A function H = H (x; t) depending explicitly on time defines a time dependent Hamiltonian system of the form x˙ = {x, H }.
(3.19)
E(t) := H (x(t); t)
(3.20)
The total energy
computed on an arbitrary solution x = x(t) to (3.19) is not conserved. However, the following well known identity describes its dependence on time ∂ H (x; t) E˙ = |x=x(t) . ∂t
(3.21)
One can recast Eqs. (3.19), (3.21) into an (autonomous) Hamiltonian form using the following standard trick of introducing extended phase space: Pˆ := P × R2t,E with a Poisson bracket { , }ˆsuch that { , }ˆP = { , }, {x, t}ˆ = {x, E}ˆ = 0, {E, t}ˆ = 1.
(3.22)
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The Hamiltonian Hˆ := H (x; t) − E
(3.23)
yields the dynamics (3.19), (3.21) along with the trivial equation t˙ = 1. The new Hamiltonian Hˆ is a conserved quantity. One returns to the original setting considering dynamics on the zero level surface Hˆ (x; t, E) = 0. Let us now consider n functions on P depending on n times H1 = H1 (x; t), . . . , Hn = Hn (x; t), where t := (t1 , . . . , tn ). Assume that the time dependent Hamiltonian systems ∂x = {x, Hi }, i = 1, . . . , n ∂ti
(3.24)
commute pairwise, i.e. ∂ ∂x ∂ ∂x = ∂t j ∂ti ∂ti ∂t j
for all i = j.
(3.25)
Because of commutativity there exist common solutions x = x(t) of the family (3.24) of differential equations. We want to introduce an analogue of the extended phase space for these multi-time dependent systems. We begin with the following simple Lemma 3.4. The Hamiltonian systems (3.24) commute iff the functions ci j (x; t) :=
∂ Hj ∂ Hi − + {Hi , H j }, i = j ∂t j ∂ti
(3.26)
are Casimirs of the Poisson bracket, i.e. {x, ci j } = 0. The energy functions E i = E i (t) := Hi (x(t); t), i = 1, . . . , n
(3.27)
∂ Hj ∂ Ei = + ci j (x; t) . ∂t j ∂ti x=x(t)
(3.28)
∂ Hi ∂ Hi (x; t) := ∂t j ∂t j
(3.29)
satisfy
In these equations
are partial derivatives.
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309
Proof. Spelling out the left-hand side of (3.25) gives ∂ ∂x = Lie ∂ {x, Hi } ∂t j ∂t j ∂ti = Lie ∂ x, Hi + x, Lie ∂ Hi ∂t j ∂t j ∂ Hi = {x, H j }, Hi + x, + {Hi , H j } . ∂t j In this calculation we have used that the Hamiltonian vector fields are infinitesimal symmetries of the Poisson bracket. Substituting this expression into (3.25) and using the Jacobi identity we arrive at ∂ Hj ∂ Hi x, − + {Hi , H j } = 0. ∂t j ∂ti This proves the first part of the lemma. The second part easily follows from (3.26): ∂ Hj ∂ Ei ∂ Hi = {Hi , H j } + = + ci j . ∂t j ∂t j ∂ti The lemma is proved.
Definition 3.5. The functions H1 (x; t), . . . , Hn (x; t) on P × Rn define n multi-time dependent commuting Hamiltonian systems if they satisfy equations ∂ Hj ∂ Hi − + {Hi , H j } = 0, i, j = 1, . . . , n. ∂t j ∂ti
(3.30)
The energy functions E i = E i (t) of a multi-time dependent commuting family satisfy ∂ Hj ∂ Ei = |x=x(t) . ∂t j ∂ti
(3.31)
Remark 3.6. The evolution equations for the energy functions take more “natural” form ∂ Ei ∂ Hi = |x=x(t) , ∂t j ∂t j
(3.32)
similar to (3.21) under an additional assumption of commutativity of the Hamiltonians {Hi , H j } = 0, i = j as functions on the phase space P. Observe that the one-form = H1 (x(t))dt1 + · · · + Hn (x(t))dtn
(3.33)
is closed for any solution x(t) if the Hamiltonians commute. The commutativity holds true in the case of Schlesinger equations (see below). The closeness of the one-form (3.33) is crucial in the definition of the isomonodromic tau-function.
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Let us now define the extended phase space Pˆ := P × R2n ,
(3.34)
where the coordinates on the second factor will be denoted t1 , . . . , tn , E 1 , . . . , E n . The Poisson bracket { , }ˆon the extended phase space is defined in a way similar to (3.22), { , }ˆP = { , }, {x, ti }ˆ = {x, E i }ˆ = 0, {E i , t j }ˆ = δi j .
(3.35)
The Hamiltonians on the extended phase space are given by Hˆ i = Hi (x; t) − E i , i = 1, . . . , n.
(3.36)
On the extended phase space the multi-time dependent commuting Hamiltonian equations can be put into a form of autonomous commuting Hamiltonian systems. Namely, the following statement holds true. Lemma 3.7. For a multi-time dependent commuting family of Hamiltonian systems the Hamiltonians (3.36) commute pairwise. The corresponding Hamiltonian equations on the extended phase space (3.34) read ∂x = {x, Hˆ j }ˆ = {x, H j }, ∂t j ∂ Hj ∂ Ei = {E i , Hˆ j }ˆ = , ∂t j ∂ti ∂ti = {ti , Hˆ j }ˆ = δi j . ∂t j
(3.37)
Proof is straightforward. On the common level surface Hˆ 1 = 0, . . . , Hˆ n = 0 ˆ one recovers the original multi-time dependent dynamics. in P, Let us now consider the particular case of a symplectic phase space P of the dimension 2g. Introduce canonical Darboux coordinates q1 , . . . , qg , p1 , . . . , pg , such that the symplectic form ω on P becomes ω=
g
dpi ∧ dqi .
i=1
Then the extended phase space carries a natural symplectic structure ωˆ = ω −
n
d E i ∧ dti .
(3.38)
i=1
The canonical transformations of a multi-time dependent commuting Hamiltonian family are defined as symplectomorphisms of the extended phase space Pˆ equipped with the symplectic structure (3.38).
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311
˜ t) satisfying Example 3.8. A multi-time dependent generating function S = S(q, q, det
∂2S = 0 ∂qi ∂ q˜ j
defines a canonical transformation of the form ∂S ∂S ∂S , pi = − , E˜ k = E k − . p˜ i = ∂ q˜i ∂qi ∂tk Usually in textbooks the last equation is written as the transformation law of the Hamiltonians, i.e., the new Hamiltonians H˜ k are given by ∂S H˜ k = Hk − , k = 1, . . . , n. ∂tk We will stick to this tradition. Let us come back to Schlesinger equations. In this case the position of the poles u 1 , . . . , u n play the role of the (complexified) time variables. It is straightforward to prove that the Schlesinger equations on g∗ can be considered as a multi-time dependent commuting Hamiltonian family. Theorem 3.9. The Hamiltonians Hk of the form (3.3) on g∗ Poisson commute {Hk , Hl } = 0, ∀k, l = 1, . . . n. They also satisfy ∂ Hk ∂ Hl = . ∂u l ∂u k We end this section with the following simple observations about the Hamiltonians (3.3). First, these Hamiltonians are not independent. Indeed, n
Hi = 0.
(3.39)
i=1
Therefore the solutions to the Schlesinger equations depend only on the differences u i − u j . Moreover, ! n n 1 2 2 u i Hi = tr Ai A j = tr A∞ − Ai . (3.40) 2 i=1
i< j
i=1
So, for a solution to the Schlesinger equations n i=1
ui
∂ Ak = [Ak , A∞ ]. ∂u i
(3.41)
Thus the Hamiltonian (3.40) generates trivial dynamics on the reduced symplectic leaves. This implies that the solutions to the Schlesinger equations depend only on n − 2 combinations of the variables u 1 , …, u n invariant w.r.t. the action of one-dimensional affine group u i → a u i + b, i = 1, . . . , n, a = 0. Due to this invariance it is sometimes convenient to normalize the position of the poles of the Fuchsian systems by u 1 = 0, u 2 = 1.
(3.42)
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4. Scalar Reductions of Fuchsian Systems In this section we establish a birational transformation, that we call scalar reduction, between the space of all m × m Fuchsian systems of the form (1.2) considered modulo diagonal conjugations and the space of special Fuchsian differential equations, that we describe in the next sub-section.
4.1. Special Fuchsian differential equations. Recall [9] that a scalar linear differential equation of order m with rational coefficients is called Fuchsian if it has only regular singularities. Writing the differential equation in the form y (m) = a1 (z)y (m−1) + · · · + am (z)y,
(4.1)
one spells out the condition of regularity of a point z = z 0 in the form of existence of the limits bk (z 0 ) := − lim (z − z 0 )k ak (z), k = 1, . . . , m. z→z 0
The infinite point z = ∞ is regular if there exist the limits bk (∞) := − lim z k ak (z), k = 1, . . . , m. z→∞
All the solutions to Eq. (4.1) are analytic at the points of analyticity of the coefficients. Let z = z 0 be a pole of the coefficients of the Fuchsian equation. The indicial equation at the point z = z 0 reads λ(λ − 1) · · · (λ − m + 1) + b1 (z 0 )λ(λ − 1) · · · (λ − m + 2) + · · · + bm−1 (z 0 )λ + bm (z 0 ) = 0.
(4.2)
If the roots λ1 = λ1 (z 0 ), . . . , λm = λm (z 0 ) of this equation are non-resonant, i.e. none of the differences λi − λ j is a positive integer, then there exists a fundamental system of solutions of the form ! λj l y j (z) = (z − z 0 ) a jl (z − z 0 ) , j = 1, . . . , m, z → z 0 . (4.3) 1+ l>0
Therefore the roots of the indicial equation coincide with the exponents at the regular singularity z = z 0 . Similarly, the indicial equation at z = ∞ reads λ(−λ − 1) · · · (−λ − m + 1) + b1 (∞)λ(−λ − 1) · · · (−λ − m + 2) + · · · + bm−1 (∞)λ − bm (∞) = 0.
(4.4)
The roots of this equation give the exponents λi = λi (∞) at z = ∞. The corresponding fundamental system of solutions, in the non-resonant case is given by ! −λ j −l a jl z y j (z) = z 1+ , j = 1, . . . , m. , z → ∞. (4.5) l>0
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Remark 4.1. For a Fuchsian differential equation with non-resonant roots of (4.4) there exists a unique canonical, up to a permutation of the roots, basis of solutions (4.5). Therefore there exists a unique canonical normalization of the monodromy matrices, for a given choice of a basis in the fundamental group of the punctured plane. If the Fuchsian equation has N + 1 poles including infinity then the exponents satisfy the following Fuchs relation:
m
λi (z 0 )(N − 1)
all poles z 0 i=1
m(m − 1) . 2
(4.6)
In the resonant case logarithmic terms are to be added. They can be obtained by a method similar to the one described above for the case of Fuchsian systems. See [9] for the details. Remark 4.2. If λ is a root of the indicial equation (4.2) and λ + n is not a root for any positive integer n then there exists a solution ! λ l y(z) = (z − z 0 ) 1 + cl (z − z 0 ) , z → z 0 . l>0
Definition 4.3. A pole z = z 0 of the coefficients of the Fuchsian equation is called apparent singularity if all solutions y(z) are analytic at z = z 0 . A necessary condition for the pole z = z 0 to be an apparent singularity is that all the roots λ1 , . . . , λm of the indicial equation (4.2) must be non-negative integers. Absence of logarithmic terms impose additional constraints onto the coefficients of the Fuchsian equation. Let M be the maximum of the roots. Then the full set of constraints can be obtained by plugging into the equation the expansions (4.3) truncated at the term of order M and requiring compatibility of the resulting linear system for the coefficients a jl , j = 1, . . . , m, l = 1, …, M. Definition 4.4. A Fuchsian differential equation of order m is called special if it has n + 1 regular singularities at the points z = u 1 , . . . , z = u n , z = ∞ and also g apparent singularities z = q1 , . . . , z = qg , where g is given by the formula (3.15) with the indices 0, 1, . . . , m − 2, m. Observe that, due to Fuchs relation [3] for a special Fuchsian equation the sum of the indices at the points z = u 1 , …, z = u m , z = ∞ is equal to m − 1. For this reason we will denote, as above (i)
λ1 , . . . , λ(i) m the indices at z = u i and (∞)
λ1
(∞)
, λ2
+ 1, . . . , λ(∞) m +1
the indices at infinity. These numbers have zero sum (cf. (2.21) n m i=1 j=1
(i)
λj +
m
(∞)
λj
= 0.
(4.7)
j=1
We now describe in more details the behavior of the coefficients of the special Fuchsian equation near an apparent singularity.
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Lemma 4.5. Near an apparent singularity z = qi there exist m linear independent solutions y1 (z), . . . , ym (z) to the special Fuchsian equation (4.1) having expansions at z = qi of the following form: (i)
α1 (z − qi )m−1 + O(z − qi )m+1 , (m − 1)!
y1 = 1 +
(i)
y2 = (z − qi ) + ......... ym−1 = ym =
α2 (z − qi )m−1 + O(z − qi )m+1 , (m − 1)! (4.8) (i) αm−1
1 (z − qi )m−2 + (z − qi )m−1 + O(z − qi )m+1 , (m − 2)! (m − 1)! (i)
1 αm (z − qi )m + (z − qi )m+1 + O(z − qi )m+2 , m! (m + 1)!
(i) are some constant coefficients. where α1(i) , . . . , αm
The proof is obvious. Denote L := −
dm d m−1 + a (z) + · · · + am (z) 1 dz m dz m−1
the differential operator in the l.h.s. of (4.1). Lemma 4.6. The point z = qi is an apparent singularity of the special Fuchsian equation Ly = 0 if and only if the following conditions are satisfied. 1. The coefficients a1 (z), a2 (z), . . . , an (z) have at most simple poles at z = qi and Resz=qi a1 (z) = −1. (i)
(4.9)
(i)
2. There exist coefficients α1 , . . . , αm−1 such that, after the substitution of the expansions (4.8) into the differential equation one obtains Ly j = O(z − qi ),
j = 1, . . . , m − 1, z → qi .
(4.10)
Proof. Suppose that z = qi is an apparent singularity. The indicial equation (4.2) for z 0 = qi by assumption must have the roots 0, 1, …, m − 2, m. Because of it b1 (z 0 ) = −1 and bk (z 0 ) = 0 for k > 1. So ak (z) =
ck (1 + O(z − qi )) , z → qi , k = 2, . . . , m, (z − qi )k−1
for some constants c1 , . . . , cm . Let us prove that c3 = · · · = cm = 0 (only the non-trivial case m ≥ 3 is to be studied). Indeed substituting the solution y1 (z) from (4.8) into the equation one obtains that the l.h.s. behaves as Ly1 ∼
cm , z → qi . (z − qi )m−1
Hence cm = 0. Similarly, substituting y2 (z) one proves that, for m ≥ 4 cm−1 = 0. Continuing this procedure we prove that all the poles are at most simple.
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Validity of (4.10) means that the solutions y1 (z), . . . , ym−1 (z) corresponding to the roots 0, 1, . . . , m − 2 of the indicial equation at z = qi contain no logarithmic terms up to the order O(z − qi )m . As it was explained in the previous page, this implies absence of logarithmic terms also in the higher orders since the order of resonance by assumption is equal to m. It remains to observe that the holomorphic solution ym (z) corresponding to the maximal root m always exists (see Remark 4.2). The vice–versa is obvious. The main result of this section is a coordinate description of the space of all special Fuchsian equations with given indices. Denote
(s)
(s)
ps := −αm−1 + δ1 , s = 1, . . . g., n 1 1 m(m − 1) (s) (i) σ1 − , δ1 = + qs − qt qs − u i 2 σ1(i) =
t=s m
i=1
λ(i) j .
(4.11)
j=1
Theorem 4.7. Any special Fuchsian equation of order m with given indices λ(i) j , j = 1, . . . , m, i = 1, . . . , n, ∞ satisfying (4.7) must have the form
y (m) = a1 (z)y (m−1) + · · · + am (z)y,
(4.12)
where the coefficients are given by
a1 (z) =
g s=1
ak (z) =
−
⎡ ⎤ n m 1 1 ⎣ (i) m (m − 1) ⎦ + λj − , z − qs z − ui 2 i=1
j=1
g (s) cm−k+1 R(qs )k−1
z − qs
s=1
(∞) k n−n−k
+ βk
z
k = 2, . . . , m.
+ (−1)
k−1
+ Pk n−n−k−1 (z)
n (i) βk [R (u i )]k−1 z − ui i=1
1 , R(z)k−1 (4.13)
& Here R(z) = nk=1 (z − u k ), Pn−3 (z), P2n−4 (z), . . . , Pm n−m−n−1 (z) are some poly(s) (s) nomials labeled by their degrees, c1 , . . . , cm−1 are some numbers. The coefficients (k)
(k)
β1 , . . . , βm , k = 1, . . . , n, ∞ depend only on the indices. They are determined from
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the identities (i)
λ(λ − 1) . . . (λ − m + 1) − β1 λ(λ − 1) . . . (λ − m + 2) + m ' (i) (i) λ − λj , + β2 λ(λ − 1) . . . (λ − m + 3) − · · · − βm(i) =
(4.14)
j=1
⎡
⎤ m m(m + 1) (∞) −1+ λ(λ + 1) . . . (λ + m − 1) − ⎣ λ j ⎦ λ(λ + 1) . . . (λ + m − 2) − 2 j=1
(∞) − β2 λ(λ + 1) . . . (λ + m − 3) + · · · + (−1)m−1 βm(∞) m ' (∞) (∞) = (λ − λ1 ) (λ − λ j − 1). j=2
(4.15)
The coefficients of the polynomials Pn−3 (z), P2n−4 (z), . . . , Pm n−m−n−1 (z) and the (s) (s) parameters c1 , . . . , cm−1 are rational functions of q1 , . . . , qg , p1 , . . . , pg and u 1 , . . . , u n . Proof of the Theorem. The ansatz (4.13) follows from the definition of a Fuchsian equation and from the first of the claims of Lemma 4.6. The expressions (4.14), (4.15) via indices is nothing but the spelling of the indicial equations (4.2), (4.4). Let us now use the second statement of Lemma 4.6 in order to show that all the remaining coefficients are uniquely determined by qs and ps . Denote δk(s) the constant term in the Laurent expansion of ak (z) near the apparent singularity z = qs , s = 1, . . . , g, a1 (z) =
1 (s) + δ1 + O(z − qs ), z − qs
ak (z) = −
(s) cm−k+1
z − qs
(s)
+ δk + O(z − qs ), k = 2, . . . , m.
(4.16)
(s)
(s)
Lemma 4.8. 1. The coefficients c j in (4.13) coincide with α j in (4.8). 2. Equation (4.1) with coefficients given by (4.13) is special Fuchsian iff the following equations hold valid for all s = 1, . . . , g: (s)
(s) ps α1 + δm = 0, (s)
(s)
(s)
ps α2 + δm−1 − α1 = 0, ... ... ... (s) (s) ps αm−2 + δ3(s) − αm−3 = 0, (s)
(s)
(s)
(s)
(s)
ps αm−1 + δ2 − αm−2 = 0,
(4.17)
where ps := −αm−1 + δ1 , s = 1, . . . , g.
(4.18)
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Proof. Substituting the solution yl (z) for 1 ≤ l ≤ m − 2 into (4.1), by using (4.13), one obtains, modulo terms of order O(z − qs ) the nontrivial contributions only from the terms (m−1)
a1 (z)yl
(m−2)
+ a2 (z)yl
(l−1)
+ am−l+1 (z)yl
(l−2)
+ am−l+2 (z)yl
.
According to Lemma 4.6 this expression must be of the order O(z − qs ) for z → qs . Spelling this out gives ( ) (s) cm−1 1 (s) (s) (s) (s) + δ1 αl + − + δ2 αl (z − qs ) z − qs z − qs ( ( ) ) (s) (s) m−l cl−1 cl (s) (s) (z − qs ) (s) + − + δm−l+1 1 + αl + δm−l+2 (z − qs ) + − z − qs (m − l)! z − qs = O(z − qs ).
(4.19)
Expanding these equations for l = 1, . . . , m − 2 yields (s)
cl
(s)
= αl
and also the first m − 2 equations of (4.17). Analogously for l = m − 1 the nontrivial contributions in (4.13) arise only in the terms (m−1) (m−2) (m−3) + a2 (z)ym−1 + a3 (z)ym−1 . a1 (z)ym−1
Again, imposing that this expression must be of the order O(z − qs ) , one obtains (s)
(s)
cm−1 = αm−1 and also the last equation of (4.17). Due to Lemma 4.6 the equations (4.19) for l = 1, . . . , m − 1 are necessary and sufficient for the points z = qs to be apparent singularities of a special Fuchsian equation. The lemma is proved. End of the proof of Theorem 4.7. We have derived a system of linear equations (4.17) (s) (s) for the parameters α1 , . . . , αm−1 , s = 1, . . . , g and for the coefficients of the polynomials Pn−3 (z), P2n−4 (z), . . . , Pm n−m−n−1 (z). It is easy to see that the number of equations is equal to the number of unknowns. It remains to prove that the determinant of this linear system is not an identical zero. (s) Let us first eliminate the α’s. To this end we need to spell out the terms δ2(s) , . . . , δm of order zero in the expansions of a2 , . . . , am at z = qs : δk(s) = (k − 1)
(t)
αm−k+1 R(qt )k−1 R (qs ) (s) Pkn−n−k−1 (qs ) αm−k+1 − + f k (qs ) + , k−1 R(qs ) (qs − qt )R(qs ) R(qs )k−1 t=s
where f k (z) =
(−1)k−1
*n
(i)
βk i=1 z−u i
(∞) k n−n−k z
[R (u i )]k−1 + βk R(z)k−1
.
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The rational functions f 1 (z), . . . , f m (z) depend only on the positions of the poles u i and the indices. Using these notations we rewrite Equations (4.17) as follows: (s)
ps α1 +
(t)
(m − 1)R (qs ) (s) α1 R(qt )m−1 + f m (qs ) α1 − R(qs ) (qs − qt )R(qs )m−1 t=s
Pm n−m−n−1 (qs ) = 0, + R(qs )m−1 (s)
ps α2 +
(t)
(m − 2)R (qs ) (s) α2 R(qt )m−2 α2 − + f m−1 (qs ) R(qs ) (qs − qt )R(qs )m−2 t=s
Pmn−m−2n (qs ) (s) = α1 + R(qs )m−2 ... ... ... (t) αm−1 R(qt ) R (qs ) (s) Pn−3 (qs ) (s) (s) αm−1 − + f 2 (qs )+ α ps αm−1 + .(4.20) R(qs ) (qs − qt )R(qs ) R(qs ) m−2 t=s
Let us introduce the following vector notations. Denote q = (q1 , . . . , qg ), p = ( p1 , . . . , pg ). For any function f (z) introduce vector f (q) := ( f (q1 ), . . . , f (qg )). Similar notations will be used for functions of p. For example, p2 = ( p12 , . . . , pg2 ). (s)
We also introduce g-component vectors α j with the coordinates α j and δ1 with the (s)
(l)
coordinates δ1 . The last ingredient will be the g × g matrices M (l) = (Mi j ), l = 1, . . . , m − 1 with the matrix entries R(q j )m−l (1 − δi j ) R (qi ) (l) Mi j = pi + (m − l) δi j − . (4.21) R(qi ) R(qi )m−l (qi − q j ) Using these notations we can rewrite Eqs. (4.20) as follows: Pn−3 (q) , R(q) P2n−4 (q) αm−3 = M (m−2) αm−2 + f 3 (q) + , R(q)2 ... ... ... P(m−1)(n−1)−n−1 (q) α1 = M (2) α2 + f m−1 (q) + , R(q)m−2 Pm(n−1)−n−1 (q) 0 = M (1) α1 + f m (q) + . R(q)m−1 αm−2 = M (m−1) (δ1 − p) + f 2 (q) +
(4.22)
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319
Substituting the first equation into the second equation we obtain Pn−3 (q) P2n−4 (q) + f 3 (q) + . αm−3 = M (m−2) M (m−1) αm−1 + f 2 (q) + R(q) R(q)2 Continuing this process we express all α’s via the known functions and the coefficients of the polynomials Pn−3 (z), P2n−4 (z), . . . , Pm n−m−n−1 (z). On the last step we arrive at a linear equation for these coefficients: Pˆm(n−1)−n−1 (q) + M (1) Pˆ(m−1)(n−1)−n−1 (q) + · · · + M (1) M (2) · · · M (m−2) Pˆn−3 (q) = M (1) M (2) · · · M (m−1) [p − δ1 ] − M (1) M (2) · · · M (m−2) f 2 (q) − · · · − M (1) f m−1 (q) − f m (q),
(4.23)
where we denote Pk n−k−n−1 (z) , k = 2, . . . , m. Pˆk n−k−n−1 (z) := R(z)k−1 It remains to prove that the determinant of the linear operator in the left-hand side of this system does not identically vanish. Indeed, this determinant is a polynomial in p1 , . . . , pg . Let us compute the terms of highest degree in these variables. It is easy to see that those terms can be written down explicitly Wm,n (q1 , . . . , qg , pˆ 1 , . . . , pˆ g ) , [R(q1 ) . . . R(qg )]m−1 where the polynomial Wm,n in 2g variables is defined in (A.21), pˆ s := ps R(qs ). Clearly it is not an identical zero. This proves the theorem.
Corollary 4.9. The positions q1 , . . . , qg of the apparent singularities along with the auxiliary parameters p1 , . . . , pg are coordinates on a Zariski open subset in the space of all special Fuchsian equations with given indices and given Fuchsian singularities u 1 , . . . , u n , ∞. Observe that, in terms of the special Fuchsian equation the auxiliary parameters pi are defined by 1 2 (4.24) pi = Resz=qi a2 (z) + a1 (z) , i = 1, . . . , g. 2 (s) They are related to ρs := αm−1 by the shift δ1(s) (see (4.11)) of the form
∂ S(q, u) , ∂qs g n ' ' ' (i) m(m−1) 1 S(q, u) = log + log (qs − u i )σ1 − 2 . 2 ρs = − ps +
s=t
(4.25)
i=1 s=1
We will see below that the coordinates qs and ps are canonically conjugated variables for the Schlesinger equations. The shift (4.25) is a canonical transformation. So, the variables qs and −ρs are also canonically conjugated.
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Remark 4.10. The linear system (4.23) to be solved in order to reconstruct the special Fuchsian equation with given indices and poles and g given pairs (qi , pi ) is very similar to the linear equation (A.20) used in the Appendix below in order to reconstruct the spectral curve (A.4) starting from its behavior over z = u 1 , …, z = u n , z = ∞ and a given divisor (γ1 , µ1 ) + · · · + (γg , µg ) of the degree g. The essential difference is that, in matrix notations in (A.20)the powers of the diagonal matrix (µ1 , . . . , µg ) enter while in (4.23) this is to be replaced by the matrix M of the form (4.21). It is a surprise that the matrix M (l) for l = m coincides with the Lax matrix for the Calogero - Moser system [45]! At the moment we do not have an explanation of this coincidence. 4.2. Transformation of Fuchsian systems into special Fuchsian differential equations. In this section we will assign to a Fuchsian system (1.2) a special Fuchsian differential equation of the form y (m) =
m−1
dl (z)y (l) .
(4.26)
l=0
Note a change of notations with respect to (4.1): dl (z) = am−l (z), l = 1, . . . , m. The reduction of a system of differential equations to a scalar equation is given by the following well known classical construction. Denote by φ1 , φ2 , . . . , φm the components of the vector function . The m th order linear differential equation for the scalar function y := φ1 can be written in the determinant form ⎞ ⎛ y y1 . . . ym ⎜ y y1 . . . ym ⎟ ⎟ det ⎜ ⎝ . . . . . . . . . . . . ⎠ = 0. (m) (m) y (m) y1 . . . ym Here y1 , …, ym is the first row of a fundamental matrix solution for the system (1.2). Expanding the determinant one obtains the needed differential equation in the form W (z)y (m) = W (z)y (m−1) + Wm−2 (z)y (m−2) + · · · + W0 (z)y, where
⎛
y1 ⎜ y1 W (z) = det ⎜ ⎝ ... y1(m−1)
⎞ ... ym ... ym ⎟ ⎟ ... ... ⎠ . . . ym(m−1)
(4.27)
(4.28)
is the Wronskian of the functions y1 (z), . . . , ym (z), the functions Wl are certain determinants with the rows constructed from (y1 , . . . , ym ) and their derivatives. Therefore dm−1 (z) =
W (z) Wl (z) , dl (z) = , l ≤ m − 2. W (z) W (z)
(4.29)
It readily follows that the coefficients of the scalar differential equation can have poles only at zeroes of the Wronskian and at the points u 1 , . . . , u n . Let us call the Fuchsian
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321
equation (4.27) the 1-associated with the Fuchsian system (1.2). In a similar way one can obtain for any j = 1, . . . , m the j-associated Fuchsian equation for the j th component of . We will now give a more precise description of the poles of the scalar Fuchsian equation and the corresponding exponents. Let us begin with the poles that come from zeroes of the Wronskian. They are so-called apparent singularities of the Fuchsian differential equation (4.27). That is, the coefficients of the differential equation have poles at zeroes of W (z) but all solutions are analytic at these poles. Let us first compute exponents at the apparent singularities. Lemma 4.11. Let z = q be a zero of the Wronskian W (z) of the multiplicity k. Then, if k ≤ m − 1 the exponents of solutions to the Fuchsian equation are 0, 1, . . . , m −k −1, m − k + j1 , m −k + 1+ j1 + j2 , . . . , m −1 + j1 +· · · + jk , (4.30) where j1 , . . . , jk are nonnegative integers satisfying k j1 + (k − 1) j2 + · · · + jk = k;
(4.31)
if k ≥ m, then the exponents are 0, 1 + j1 , 2 + j1 + j2 , . . . , m − 1 + j1 + · · · + jm−1 ,
(4.32)
where j1 , . . . , jm−1 are nonnegative integers satisfying (m − 1) j1 + (m − 2) j2 + · · · + jm−1 = k.
(4.33)
Proof. All the exponents at an apparent singularity must be nonnegative integers 0 ≤ n 1 ≤ · · · ≤ n m . The corresponding basis of solutions must have the form y1 (z) = (z − q)n 1 (1 + O(z − q)), . . . , ym (z) = (z − q)n m (1 + O(z − q)). Therefore all the exponents are pairwise distinct: in the opposite case the difference of two basic solutions would have a higher exponent. Besides, necessarily n 1 = 0. Indeed, otherwise all elements of the first line of the fundamental matrix would vanish at z = q. This contradicts non-degeneracy of the determinant det (z) of the fundamental matrix solution to the Fuchsian system. Let us now spell out the indicial equation (4.2) at z = q, λ(λ − 1) . . . (λ − m + 1) − b1 λ(λ − 1) . . . (λ − m + 2) − · · · − bm = 0, where bm−i = lim (z − q)m−i di (z), i = 0, 1, . . . , m − 1. z→q
Because of (4.29) we always have b1 = k, where k is the multiplicity of z = q as a zero of the Wronskian. Besides, if k ≤ m − 1 then bk+1 = · · · = bm = 0.
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So the indicial equation factorizes λ(λ − 1) . . . (λ − m + k + 1) × [(λ − m + k) . . . (λ − m + 1) − k (λ − m + k) . . . (λ − m + 2) − · · · − bk ] = 0. The sum of the k roots of the second factor is equal to (m − k) + (m − k + 1) + · · · + (m − 1) + k. As these roots must be pairwise distinct positive integers different from zeroes of λ(λ − 1) . . . (λ − m + k + 1), they can be represented in the form (4.30), (4.31). Let us now consider the case of multiplicity k ≥ m. Since λ = 0 must be a root of the indicial equation, one has b0 = 0. Hence the indicial equation reads λ (λ − 1) . . . (λ − m + 1) − k (λ − 1) . . . (λ − m + 2) − · · · − bm−1 = 0. Again, the (m − 1) roots of the second factor are pairwise distinct positive integers with the sum equal to 1 + 2 + · · · + (m − 1) + k. So, they can be represented in the form (4.32), (4.33). The lemma is proved.
Corollary 4.12. If z = q is a simple root of the Wronskian W (z), then the exponents of the Fuchsian differential equation (4.27) are 0, 1, . . . , m − 2, m. Remark 4.13. In [34] Kimura and Okamoto claimed that, for an apparent singularity of multiplicity k the exponents are 0, 1, . . . , m − 2, m + k − 1. We were unable to reproduce the proof of this statement for k > 1. Theorem 4.14. Take a Fuchsian system of the form (1.2) with pairwise distinct (∞) (k) (k) nonresonant exponents at ∞ λi , i = 1, . . . , m. Denote λ1 , . . . , λm the exponents at u k , k = 1, . . . , n. Suppose that for some j = 1, . . . , m, n
Ak ji u k = 0, ∀ i = 1, . . . , m, i = j.
(4.34)
k=1
Then the scalar differential equation for the j th component y := φ j of the solution of the deformed system (1.2) possesses the following properties.
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1. The coefficients of the equation depend rationally on the matrix elements of Ak , k = 1, . . . , n. They can be represented in the form fl (z) , &g m−l (z − u ) k k=1 i=1 (z − qi )
dl (z) = &n where g=
(n − 1)m(m − 1) − (m − 1), 2
the functions fl (z) are polynomials of degree deg fl (z) = (n − 1)(m − l) + g, and q1 , . . . , qg are zeroes of the Wronskian (4.28). They are apparent singularities of the Fuchsian equation. 2. This differential equation has regular singularities at u 1 , . . . , u n with the exponents (k) (∞) (∞) (∞) λi , i = 1, . . . , m, at u k and also at ∞ with the exponents λ1 , λ2 + 1, . . . , λm +1. 3. If the numbers q1 , . . . , qg are pairwise distinct then the exponents at each apparent singularity are 0, 1, . . . , m − 2, m. Proof. Without loss of generality we may assume j = 1. The above construction gives a Fuchsian equation with regular singularities at u 1 , . . . , u n , ∞ and apparent singularities (i) (i) at the zeroes of the Wronskian. Denote λ˜ 1 , . . . , λ˜ m the exponents of the solutions to the scalar equation at the point z = u i for every i = 1, . . . , n. From the construction it immediately follows that for j = 1, . . . , m, (i) (i) (i) (i) λ˜ j = λ j + n j , for some n j ∈ N,
(4.35)
(∞) after a suitable labelling of the exponents. Let us now consider the exponents λ˜ 1 , . . . , (∞) λ˜ m . They satisfy modified equalities (∞) λ˜ 1(∞) = λ1(∞) , λ˜ (∞) = λ(∞) + 1 + n (∞) ∈ N, j = 2, . . . , m. j j j , for some n j
(4.36)
Indeed, because of diagonality of the matrix A∞ there exists a fundamental matrix solution of the Fuchsian system of the form 1 (∞) z −A∞ z −R . = +O z Looking at the first row of this matrix yields the above estimates. We will prove below that, doing if necessary a small monodromy preserving defor(i) mation the above equalities are satisfied with n j = 0 for all i = 1, . . . , n, ∞, j = 1, . . . , m. To this end we now proceed to considering the apparent singularities. We already know from Corollary 4.12 that, if z = q is a simple zero of the Wronskian then the exponents at the apparent singularity are 0, 1, . . . , m − 2, m. Let us now prove that, under the assumption (4.34) the Wronskian has exactly g zeroes (counted with their multiplicities).
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Denote n '
R(z) :=
(z − u k ).
k=1
Differentiating the linear system φi (z) =
m
Ai j (z)φ j (z),
(4.37)
j=1
where Ai j (z) =
n A ki j k=1
z − uk
and using Leibnitz rule we have for each l = 1, . . . , m, φi(l) (z) =
m l−1 l −1 Ai(l−1−k) (z)φ (k) j j (z). k
(4.38)
j=1 k=0
Denoting y = φ1 , we rewrite the system in the form y (l) (z) =
m
Pl+1, j (z)φ j (z) +
j=2
l−1
Ql+1,l−k (z)y (l−1−k) (z)
(4.39)
k=0
for l = 1, . . . , m. The coefficients Pl+1, j (z) and Ql+1,l−k (z) are rational functions of z such that R(z)l Pl+1, j (z) is a polynomial in z of degree nl − l − 1 and R(z)k+1 Ql+1,l−k (z) is a polynomial in z of degree (k + 1)(n − 1). They can be computed from the following recursion relations starting with Q2,1 (z) = A11 (z), P2, j (z) = A1 j (z). The recursion reads Pl+2, j (z) = Pl+1, j (z) +
m
Pl+1,s (z)As j (z)
j = 2, . . . , m,
s=2 Ql+2,l+1−k (z) = Ql+1,l−(k−1) (z) + Ql+1,l−k (z), k = 1, . . . , l − 1, Ql+2,l+1 (z) = Ql+1,l (z), m (z) + Pl+1,s (z)As1 (z). Ql+2,1 (z) = Ql+1,1 s=2
Observe that Ql+1,l (z) = A11 (z) for all l. The system (4.39) can be written in the form ⎞ ⎛ ⎞ ⎛ y φ1 ⎜ y ⎟ ⎜φ ⎟ P ⎝ 2 ⎠ = Q⎝ ... ⎠, ... φm y (m−1)
(4.40)
Canonical Structure and Symmetries of the Schlesinger Equations
where
325
⎛
⎞ 1 0 ... 0 ⎜ 0 P22 (z) . . . P2m (z) ⎟ P=⎝ ⎠, ... ... ... 0 Pm,2 (z) . . . Pm,m (z) ⎛ ⎞ 1 0 ... ... 0 1 0 ... 0 ⎟ ⎜ −Q21 (z) ⎜ ⎟ ... 0 ⎟. Q = ⎜ −Q31 (z) −Q32 (z) 1 ⎝ ... ... ... ... ...⎠ −Qm,1 (z) . . . . . . −Qm,m−1 (z) 1
(4.41)
Let us prove invertibility of the m × m matrix P. Lemma 4.15. det (P) where (z) is a polynomial of degree g = −
'
(z) R(z)
m(m−1) 2
,
(4.42)
(n−1)m(m−1) −(m −1) with leading coefficient 2
⎡ ⎤ m n ' (∞) (∞) ⎣ λi − λ j u k A k1 j ⎦ .
2≤i< j≤m
(4.43)
j=2 k=1
Proof. First of all we prove that, if pl+1, j z nl−l−1 is the leading term in R(z)l Pl+1, j , (∞) then the leading term in R(z)l+1 Pl+2, j is pl+2, j z n(l+1)−(l+1)−1 with pl+2, j = (−λ j − (l + 1)) pl+1, j . In fact, from the above recursion relations one obtains l+1 R(z)l+1 Pl+2, j (z) = R(z)l+1 Pl+1, j (z) + R(z)
m
Pl+1,s (z)As j (z)
s=2
∼ ((nl − l − 1) pl+1, j −nlpl+1, j )z nl−l+n + R(z)l+1 Pl+1, j (z)A j j (z) ! n Ak j j (z) − (l + 1) pl+1, j z n(l+1)−(l+1)−1 . ∼ k=1
Thus we have (∞)
(∞)
(∞)
pl+1, j = (−λ j − l) pl, j · · · = (−λ j − l) · · · (−λ j − 2) p2, j , * with p2, j = nk=1 Ak1 j u k because P2, j (z) = A1 j (z). Substituting these leading terms in the entries Pi j (z) we obtain that the leading term of P is RMP where P = diagonal 1, p2,2 , . . . , p2,m , ! z n−2 z (n−1)(i−1)−1 z (n−1)(m−1)−1 R = diagonal 1, ,..., ,..., , R(z) R(z)i−1 R(z)m−1 ⎞ ⎛ 1 0 ... 0 ⎟ ⎜ 0 1 ... 1 ⎟ ⎜ (∞) (∞) ⎟, 0 (−λ − 2) . . . (−λ − 2) M=⎜ m 2 ⎟ ⎜ ⎠ ⎝... ... ... ... & &m−1 (∞) (∞) m−1 (−λ − j) . . . (−λ − j) 0 m j=2 j=2 2
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and computing the determinant we obtain (4.42), (4.43). The lemma is proved.
Observe that the leading coefficient of the polynomial (z) cannot be zero, thanks to our hypothesis (4.34). Continuing the procedure used in the proof of the lemma it is easy to obtain explicit formulae for the coefficients of the scalar equation. Let I be the inverse matrix of P. Its leading term equals Ii j (z) ∼ M−1
ij
R(z) j−1 p2,i z (n−1)( j−1)−1
that is, Ii j (z) =
Di j (z)R(z) j−1 , (z)
where Di j (z) is a polynomial in z of degree (n−1)m(m−1) −(m −1)−(( j −1)(n −1)−1) 2 with coefficients depending on i, j. Solving the system (4.40) we obtain for i > 1, ⎞ ⎛ m ⎝− φi (z) = Iis (z)Qs j (z) + Ii j (z)⎠ y ( j−1) . (4.44) j=1
2≤s< j
Substituting (4.44) in (4.38) with l = m, we obtain ⎛ ⎞ ⎤ ⎡ j−1 m m ⎣ y (m) (z) = Pm+1,i (z) ⎝Ii j (z) − Iis (z)Qs j (z)⎠ + Qm+1, j (z)⎦ y ( j−1) (z) j=1
s=2
i=2
that is the requested differential equation y (m) (z) =
m−1
dl (z)y (l) (z),
l=0
with dl =
m
Pm+1,i (z) Ii,l+1 (z) −
l+1
! Iis (z)Qs,l+1 (z) + Qm+1,l+1 (z)
s=2
i=2
f 1,l (z) , (z)R(z)m−l
where f 1,l (z) are polynomials of degree (n − 1)(m − l) + g. It remains to prove the statement about the exponents of the scalar Fuchsian equation at the poles z = u i , z = ∞. Let us assume for simplicity that all the apparent singularities are pairwise distinct (the general case can be considered in a similar way). Taking the sum of the equalities (4.35), (4.36) we obtain the following estimate: ! m n (i) ˜λ(i) + λ˜ (∞) = m − 1 + nj . j j j=1
i=1
i, j
The sum of exponents at the apparent singularity z = qs is equal to 1 + 2 + · · · + (m − 2) + m =
m(m − 1) + m. 2
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327
So the total sum of exponents of the Fuchsian equation satisfies m(m − 1) exponents = m − 1 + g +m 2 (i) m(m − 1) (i) + n j (g + n − 1) nj . + 2 i, j
i, j
But, according to the Fuchs relation (4.6) the total sum of exponents over all g + n + 1 regular singularities must be equal just to (g + n − 1) m(m−1) . Therefore all non2 (i) negative integers n j appearing in the equalities (4.35), (4.36) must be zero. The theorem is proved. Remark 4.16. The above construction of the Fuchsian equation (4.26) for a given Fuchsian system is clearly invariant w.r.t. simultaneous diagonal conjugations of the coefficients of the latter. To make sure that (4.26) is a special Fuchsian equation for a generic Fuchsian system, we are to prove that, in the generic case all the roots of the polynomial (z) are pairwise distinct. This will follow from Theorem 4.7 claiming that, in the space of all special Fuchsian equations, the positions of the apparent singularities are independent variables and from the result of the next section that says that under a certain genericity assumption the Fuchsian system can be reconstructed from the special Fuchsian equation uniquely up to a conjugation by constant diagonal matrices. 4.3. Inverse transformation Theorem 4.17. Consider an m th order special Fuchsian equation of the form y (m) (z) =
m−1 l=0
fl (z) y (l) (z), (z)R(z)m−l
(4.45)
&g & − (m − 1) and where R(z) = nk=1 (z − u k ), (z) = i=1 (z − qi ), g = (n−1)m(m−1) 2 fl (z) are polynomials of degree (n − 1)(m − l) + g. Let the exponents of the pole u k , (k) (∞) (∞) (∞) k = 1, . . . , n, be λi , i = 1, . . . , m, and the ones of ∞ be λ1 , λ2 +1, . . . , λm +1 and let q1 , . . . , qg be pairwise distinct apparent singularities of exponents 0, 1, . . . , m−2, m. If the monodromy group of the Fuchsian equation (4.45) is irreducible, then there exists a m × m Fuchsian system of the form Ak d = dz z − uk n
k=1
(∞)
(∞)
(k)
(k)
with exponents λ1 , . . . , λm at ∞ and λ1 , . . . , λm at u k , and no apparent singularities, such that the first row of its fundamental matrix satisfies the given m th order Fuchsian equation. The matrix entries of the matrices Ak , k = 1, . . . , n, depend rationally on the coefficients of the polynomials fl and on q1 , . . . , qg , u 1 , . . . , u n . Moreover, if n d A˜ k ˜ ˜ = dz z − uk k=1
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is another Fuchsian system corresponding to the given special Fuchsian equation, then there exists a diagonal matrix D such that A˜ k = D −1 Ak D, k = 1, . . . , n. Proof. This proof follows essentially the proof due to Bolibruch of reconstruction of a Fuchsian system from a given Fuchsian equation [3, 7]. We need some extra machinery in our case to eliminate the apparent singularities q1 , . . . , qg . Lemma 4.18. The system dY = F(z)Y dz constructed from (4.45) by the transformation Y j = [(z)R(z)] j−1
d j−1 y , dz j−1
j = 1, . . . , m,
is Fuchsian at u 1 , . . . , u n , q1 , . . . , qg with the same exponents of (4.45) and has a regular singularity at ∞. The proof is straightforward and can be found in [3, 7]. First of all we want to eliminate the apparent singularities q1 , . . . , qg . By Lemma 4.5, near the point z = qi we can choose a basis of solutions y1 , . . . , ym such that dl−1 yk 1 (z − qi )k−l + O(z − qi )m−l , = dz l−1 (k − l)!
l ≤ k < m,
(i)
αk dl−1 yk (z − qi )m−l + O(z − qi )m+2−l , = l > k, k < m, l−1 dz (m − l)! 1 dl−1 ym (z − qi )m−l+1 + O(z − qi )m+2−l , = k = m, l ≤ m. l−1 dz (m − l + 1)! To eliminate all apparent singularities q1 , . . . , qg , we apply the following gauge transformation: Yˆ = (z)(z)−M Y, where M = diag (0, 1, . . . , m − 2, m) and (z) is a lower triangular matrix with all diagonal elements equal to 1 and all off-diagonal elements equal to zero apart from the last row which is given by (z)ml = −
R(z)m−1 gl (z), l = 1, . . . , m − 1. (z) (i)
where gl (z) is a degree g polynomial in 1z such that gl (qi ) = αl and gl (z) ∼ z −g as z → ∞. Let us show that the new matrix Yˆ is holomorphic and invertible at z = qi . In fact near z = qi , we have (i)
ml = −
αl(i) R(qi )m−1 + O(1), (z − qi ) (qi )
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1 diag 1, R(z), . . . , R(z)m−1 G(z), where and (z)−M Y = diag 1, . . . , 1, (z) G(z)lk = O(z − qi ), l = k, m, G(z)ll = 1 + O(z − qi ), l = m (i)
G(z)mk = αk + O(z − qi ), k = m, G(z)mm = (z − qi ) + O(z − qi )2 . This gives Yˆ (z) = Yˆ0 + O(z − qi ), m(m−1)
2 where det(Yˆ0 ) = R(qi ) (qi ) = 0, as we wanted to prove. We now need to study infinity. First, in the non-resonant case, we can choose a basis y1 , . . . , ym of solutions for the differential equation of the form
1 1+O , z 1 ∞ yk (z) = ak z −λk −1 1 + O , z ∞
y1 (z) = a1 z −λ1
k = 2, . . . , m,
for some arbitrary non-zero coefficients a1 , . . . , am . As a consequence we obtain that 1 1+O , Yl1 = (−1) + l − 2)z z z 1 ∞ (∞) (∞) (∞) , Ylk = (−1)l−1 ak (λk +1)(λk + 2) · · · (λk + l − 1)z (g+n)(l−1) z −λk −l 1 + O z k = 2, . . . , m, l−1
(∞) (∞) a1 λ1 (λ1
(∞) + 1) · · · (λ1
(g+n)(l−1) −λ∞ 1 −l+1
that gives Y (z) = z C G (∞) (z)z −
(∞)
,
∞ ∞ where (∞) = diagonal(λ∞ 1 , λ2 +1, . . . , λm +1), C = diagonal(0, (g+n−1), . . . , (g+ n − 1)(m − 1)) and G (∞) (z) is holomorphicly invertible at infinity such that G (∞) (∞) has all minors not equal to zero. In particular the arbitrary choice of the parameters a1 , . . . , am implies freedom of multiplication of Y by a diagonal matrix, diagonal(a1 , . . . , am ), from the right. In the resonant case, in a similar way one obtains
Y (z) = z C G (∞) (z)z −
(∞)
z −R
(∞)
.
(∞) (∞) To estimate the indices at infinity of our Yˆ (z) = (z)(z)−M z C G (∞) (z)z − z −R we want to use the following lemma proved in [3, 7] (see Lemma 4.1.2 in both references).
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Lemma 4.19. Let U (z) be a matrix holomorphicly invertible at ∞ and let all the principal minors of U (∞) be non-zero. Then for any integers k1 ≤ k2 ≤ · · · ≤ km there exists a lower triangular matrix (∞) (z) with elements on the principal diagonal equal to 1, (∞) (z) polynomial in z, and a matrix V (∞) (z) holomorphicly invertible in a neighborhood of ∞ such that (∞) (z)z K U (z) = V (∞) (z)z K , where K = diag (k1 , k2 , . . . , km ). (i) (z) depend only on the first S := km − k1 terms of the We add that V (∞) (z) and* −s near ∞. series expansion of U (z) = ∞ s=1 Us z To apply the above lemma, we first observe that ˜ (z)z −Mg+C = z −Mg+C (z), ˜ where (z) is a lower triangular matrix with all diagonal elements equal to 1 and all off-diagonal elements equal to zero apart from the last row which is given by ˜ ml ∼ z (n−1)(l−m)−g , (z)
l = 1, . . . , m − 1.
To apply Lemma 4.19 we need to introduce a permutation P such that P z −Mg+C P −1 = z K where, in the case m ≥ 3, K = diagonal ((m − 1)(n − 1) − g, 0, (n − 1), 2(n − 1), . . . , (m − 2)(n − 1)) , and in the case m = 2, K = diagonal(0, 1) and P = . Moreover, in the case m > 2, (∞) (∞)P −1 has all principal minors different from ˜ we need to show that P (∞)G ˜ zero. This is a straightforward consequence of the fact that (∞) = and G˜ = P G (∞) (∞)P −1 is given by + 1) . . . (λ1(∞) + m − 2), G˜ 11 = (−1)m−1 a1 λ1(∞) (λ(∞) 1 (∞) (∞) G˜ 1k = (−1)m−1 ak (λk + 1) . . . (λk + m − 1), k = 1, (∞) (∞) (∞) G˜ l1 = (−1)l−1 a1 λ1 (λ1 + 1) . . . (λ1 + l − 2), l = 1, m, (∞) (∞) G˜ lk = (−1)l−1 ak (λk + 1) . . . (λk + l − 1), l = 1, m, k = 1, G˜ mk = ak , k = 1, . . . , m.
We can then apply Lemma 4.19 to (∞) (∞) (∞) (∞) (∞) (∞) ˜ ˜ (z)z − z −R = z K P (z)G (z)P −1 P z − z −R . P Yˆ = Pz −Mg+C (z)G
We obtain a gauge transformation with the matrix (∞) (z) polynomial in z, such that the new fundamental matrix (∞ (∞) (∞) ˜ (z)P −1 P z − ) z −R , Y˜ = (∞) (z)P Yˆ = (∞) (z)z K P (z)G
factors as (∞) (∞) (∞) (∞) Y˜ (z) = V (∞) (z)z K P z − z −R = V (∞) (z)P −1 z −Mg+C z − z −R ,
with the matrix V (∞) (z) holomorphicly invertible in a neighborhood of ∞. The new (∞) (∞) (∞) exponents at ∞ are λ1 , λˆ m = λm + 1 + mg − (m − 1)(n + g − 1), and, for
Canonical Structure and Symmetries of the Schlesinger Equations (∞)
331
(∞)
j = 2, . . . , m − 1, λˆ j = λ j + 1 − ( j − 1)(n − 1). Their sum is zero, therefore ∞ is a Fuchsian singularity (see [AB]). So we have constructed a m × m Fuchsian system of the form n d ˜ A˜ k ˜ Y Y = dz z − uk k=1
(∞) (∞) (∞) (k) (k) with exponents λ1 , λˆ 2 , . . . , λˆ m at ∞ and λ1 , . . . , λm at u k , and no apparent singularities. We now want to map this system to a m × m Fuchsian system with exponents (k) (k) λ1(∞) , . . . , λ(∞) m at ∞ and λ1 , . . . , λm at u k . We need the following: (k)
(k)
Lemma 4.20. Given a Fuchsian system of the form (1.2), let λ1 , . . . λm be the eigenvalues of the matrix Ak for k = 1, . . . , n, ∞ and let Gk be its diagonalizing matrix, (k) . , . . . λ Gk−1 Ak Gk = diag λ(k) m 1
(k)
(k)
(k)
(k)
Assume that there are two eigenvalues, say λ1 and λm such that λm = λ1 + 1, (k) (k) (k) (k) (k) (k) λm = λ1 + 2, and for all l = 1, m, λ1 = λl − 1 and λm = λl + 1. If not all −1 entries in position m1 of the matrices Gk Al Gk , l = 1, . . . , n, are zero, then there exists a gauge transformation G k (z; A1 . . . , An , u 1 , . . . , u n ), rational in all arguments, such that the new matrices A˜ l , l = 1, . . . , n, l = k have the same eigenvalues as the old (k) (k) (k) (k) ones Al and the new matrix A˜ k has eigenvalues λ1 + 1, λ2 . . . , λm−1 , λm − 1. Moreover the gauge transformation G k (z; A1 . . . , An , u 1 , . . . , u n ) preserves the Schlesinger equations. (∞)
Proof. We give here the gauge transformation G ∞ (z) giving rise to the change λ1 → (∞) (∞) (∞) (∞) (∞) (∞) λ1 + 1, λm → λm − 1. So we assume that λ1 and λm are such that λm = (∞) (∞) (∞) (∞) (∞) (∞) (∞) λ1 + 1, λm = λ1 + 2, and for all l = 1, m, λ1 = λl − 1 and λm = λl + 1, and if not all entries in position m1 of the matrices Al , l = 1, . . . , n, are zero. Let us fix a fundamental matrix normalized at infinity ∞ =
1 2 + 2 +O + z z
1 z3
z −A∞ z −R
where = A∞ ,
R (∞) = R1 + R2 + . . . ,
(∞)
,
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(B1 )i j , λi = λ j + 1 , 0, otherwise B1 = − Ak u k ,
(R1 )i j =
k
( 1 )i j =
⎧ (B ) j ⎨ − λ −λ1 i−1 , λi = λ j + 1 i j ⎩
arbitrary,
,
otherwise
(B2 − 1 R1 + B1 1 )i j , λi = λ j + 2, 0, otherwise B2 = − Ak u 2k ,
(R2 )i j =
(4.46)
k
( 2 )i j =
⎧ (−B2 + 1 R1 −B1 1 )i j ⎨ , λi = λ j + 2 λi −λ j −2 ⎩
arbitrary,
.
otherwise
˜ Consider the following gauge transformation (z) = (I (z) + G)(z) where I (z) := Diagonal (z, 0, . . . , 0) , and 1 , G m1 if p = 1, m, G pp = 1, G 1 p = 1mp G 1m , if p, q = 1, p = q, G pq = 0, G 11 = G 1m 2m1 + 111 , and G mm = 0. G m1 = 1m1 ,
G 1m = −
G p1 = 1 p1 ,
(4.47)
In order to see that this gauge transformation is always well defined it is enough to observe that 1m1 (u) is never identically equal to zero if at least one of the (m, 1) matrix entries of the matrices A1 (u), . . . , An (u) is different from identical zero. Indeed, this follows from the linear equations ∂i 1 = −Ai , ∂i 2 = −Ai 1 − u i Ai ,
(4.48) (4.49)
which are a straightforward consequence of the equation ∂i ∞ (z; u) = −
Ai ∞ (z; u), i = 1, . . . , n, z − ui
describing the u-dependence of the fundamental matrix ∞ (z; u). Let us prove that this transformation maps the matrices A1 , . . . , An to new matrices A˜ 1 , . . . , A˜ n given by A˜ k := (I (u k ) + G)−1 Ak (I (u k ) + G), such that A˜ ∞ = −
n k=1
(∞) (∞) (∞) − 1 . A˜ k = diagonal λ1 + 1, λ2 , . . . , λm−1 , λ(∞) m
(4.50)
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In fact (I (z) + G)−1 = J (z) + G −1 , where J (z) := Diagonal (0, . . . , 0, z) , therefore A˜ k := G −1 Ak I (u k ) + G −1 Ak G + J (u k )Ak I (u k ) + J (u k )Ak G. Multiplying by G from the left and summing on all k we get that the condition (4.50) is satisfied if and only if ⎛ (∞) −g11 λ(∞) g12 . . . − λ 1 2 ⎜ ⎜ (∞) (∞) 0 ... ⎜ λ2 − λ1 − 1 g21 ⎜ ⎜ ... 0 ... ⎝ (∞) (∞) λm − λ1 − 1 gm1 0 ... ⎞ (∞) g1 m−1 λ(∞) − λm−1 − λ(∞) . . . λ(∞) m + 1 g1m 1 1 ⎟ ... ... 0 ⎟= ⎠ ... ... 0 ... ... 0 ⎞ ⎛* ⎞ ⎛ * g1m k Akm1 u 2k 0 . . . 0 k Ak11 u k 0 . . . 0 0 ... 0⎠ + ⎝ = ⎝* ... 0 0 ... 0⎠ + A u 0 . . . 0 . . . 0 ... 0 k km1 k * * * * ⎛ ⎞ g1m s k Akms u k gs1 . . . g1m s k Akms u k gsm 0 ... 0 ⎜ ⎟ +⎝ (4.51) ⎠. ... ... ... 0 ... 0 Observe that in our assumptions on the eigenvalues λ(∞) and λ(∞) m , these formulae are 1 clearly satisfied thanks to the fact that 1 , 2 and R (∞) are given by formulae (4.46). Let us prove that this gauge transformation preserves the Schlesinger equations. Differentiating A˜ k w.r.t. u j , with j = k and using the Schlesinger equations for A1 , . . . , An we get: (I (u k ) + G)−1 A j (I (u k ) + G) ∂ A˜ k −1 ∂G ˜ = = Ak , (I (u k ) + G) + ∂u j ∂u j uk − u j
A˜ k , A˜ j = uk − u j A j (I (u k ) − I (u j )) − Bk j A j (I (u j ) + G) −1 ∂G ˜ , + Ak , (I (u k ) + G) + ∂u j uk − u j where
⎛
Bk j
0 ⎜ 0 ⎜ =⎝ ... 0
... ... ... ...
0 0 ... 0
u k −u j gm1
⎞
0 ⎟ ⎟. ... ⎠ 0
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Given the formulae (4.47), it is straightforward to prove that the equation A j (I (u k ) − I (u j )) − Bk j A j (I (u j ) + G) ∂G + = 0, ∂u j uk − u j is equivalent to Eqs. (4.48), (4.49). This proves that also A˜ 1 , . . . , A˜ n satisfy the Schlesinger equations. (k) (k) (k) Analogous formulae can be derived for the transformation λ1 → λ1 + 1, λm → (k) λm − 1, for k = 1, . . . , n. In fact, suppose u 1 = 0 and k = 1. We can simply apply the conformal transformation z˜ = u1k − 1z . The new residue matrices are A˜ l = Al for * l = 1, ∞, A˜ 1 = − l Al , A˜ ∞ = Ak . We then need to diagonalize A˜ ∞ and apply the above gauge transformation to the new system. We show that it is possible to make a finite sequence of gauge transformations described in Lemma 4.20 in such a way that the final Fuchsian system has exponents (∞) (∞) (∞) (∞) λ1 , . . . , λm at infinity and λk , . . . , λk at u k , k = 1, . . . n. (∞) By means of a permutation, we choose the following ordering of the parameters λk : (∞) (∞) (∞) (∞) ≥ λ1 ≥ λ2 · · · ≥ λm−1 . λm (∞) (∞) (∞) (∞) We start with λˆ 2 → λˆ 2 + 1 and λˆ m → λˆ m − 1. We want to apply such gauge (∞) (∞) s = λ2 − λˆ 2 times. To do this we need to check that for all p = 0, 1, . . . , s − 1 and for all l = 1, . . . , m the following conditions are satisfied:
ˆ (∞) = 2 p + 1, 2 p + 2, λˆ (∞) m − λ2 (∞) λˆ l λˆ (∞) m
(∞) − λˆ 2 (∞) − λˆ l
(4.52)
= p + 1,
∀ l = 2, m,
(4.53)
= p + 1,
∀ l = 2, m.
(4.54)
(∞) (∞) (∞) (∞) > λ2 , λˆ m − λˆ 2 > To prove (4.52) we observe that since λm (∞) (∞) max(2 p + 2) = 2s. To prove (4.53) we observe that λˆ l − λˆ 2 is a negative (∞) (∞) > s. Therefore all condinumber. To prove (4.54) we observe that λˆ m − λˆ 1 tions (4.52), (4.53), (4.54) are satisfied and thanks to the hypothesis that the monodromy group of the Fuchsian equation (4.45) is irreducible (which implies that at each step at least one residue matrix has m1 entry non-identically 0) there exists a gauge transfor(∞) (∞) mation G 2 (z) such that the new Fuchsian system has exponents λˆ 2 + s, λˆ m − s, and (∞) λˆ j for all j = 1, 3, . . . , m − 1. (∞) (∞) (∞) (∞) , …, λˆ , At the j th step of this procedure the parameters are λ , . . . , λ , λˆ 1
j
j+1
m−1
(∞) (∞) m− j−1 λ˜ m = λm − 2 [(m + j −2)(n −1)−2]. We want to apply a gauge transformation (∞) (∞) (∞) (∞) G j+1 (z) that maps λˆ j+1 → λˆ j+1 + 1, λ˜ m → λ˜ m − 1, a number j (n − 1) − 1 = (∞) (∞) λˆ −λ of times. As above we need to verify that for all p = 0, 1, . . . , j (n − 1) − 2 j+1
j+1
Canonical Structure and Symmetries of the Schlesinger Equations
335
for all l = 1, . . . , j, j + 2, . . . , m − 1 the following conditions are satisfied: (∞) (∞) λ˜ m − λˆ j+1 = 2 p + 2, 2 p + 1,
(4.55)
λl(∞) − λˆ (∞) j+1 = p + 1,
(4.56)
(∞) (∞) − λl = p + 1, λ˜ m (∞) λ˜ m − λˆ s(∞) = p + 1.
(4.57) (4.58)
The proof that these conditions are fulfilled at each step is straightforward. Again we can use the hypothesis that the monodromy group of the Fuchsian equation (4.45) is irreducible to prove that at each step at least one residue matrix has m j entry nonidentically 0. Therefore we have obtained a gauge transformation G m (z)G m−1 (z) · · · G 2 (z) such (∞) (∞) that the new Fuchsian system has exponents λ1(∞) , . . . , λ(∞) m at infinity and λk , . . . , λk at u k , k = 1, . . . n. The new fundamental matrix at infinity is ˜ ∞ :=
m−1 '
G j+1 (z)V (∞) (z)z −Mg+C z
(∞)
zR.
j=1
In order to normalize it at infinity we need to perform one last gauge transform: ⎛ ⎞−1 m−1 ' ˜ =⎝ G j+1 (∞)V (∞) (∞)⎠ . j=1
The final new Fuchsian system Ak d = dz z − uk n
k=1
(∞)
(∞)
(k)
(k)
has exponents λ1 , . . . , λm at ∞ and λ1 , . . . , λm at u k , and no apparent singularities. The matrix entries of the matrices Ak , k = 1, . . . , n, depend rationally on the coefficients of the polynomials fl and on q1 , . . . , qg . This concludes the proof of & &m−1 existence. Due to the ambiguity m−1 j=1 G j+1 (z) → D j=1 G j+1 (z), where D is any constant diagonal matrix with non-zero entries, we have that from the differential equation (4.45) we have constructed not one Fuchsian system, but a family of them, all related by diagonal conjugation. We now prove the last statement of the theorem. Let us start from another Fuchsian system n d Aˇ k ˇ ˇ = dz z − uk
(4.59)
k=1
(∞)
(∞)
(k)
(k)
with exponents λ1 , . . . , λm at ∞ and λ1 , . . . , λm at u k , and no apparent singularˇ as usual, ities. Let us normalize its fundamental matrix 1 (∞) (∞) ˇ (∞) := z −A z −R . +O z
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Let us apply the reduction procedure described in Theorem 4.14. This means that we construct a gauge transformation G(z) = Q−1 P, ˇ = Y, Q−1 P where Y is the Wronskian matrix of the differential equations (4.45). Now from such an equation we constructed a Fuchsian system Ak d = dz z − uk n
k=1
with the same exponents as (4.59). By the above construction, is also related to Y by a ˇ ˇ ˇ This gauge transformation preserves gauge, = G(z)Y . Therefore = G(z)G(z) . the normalization at infinity by construction. All monodromy data are preserved and by ˇ the uniqueness Lemma 2.5 we conclude that G(z)G(z) must be diagonal and constant in z. In particular this proves that the first row of the fundamental matrix satisfies the given m th order Fuchsian equation. 4.4. Darboux coordinates for Schlesinger system. According to Corollary 4.9, the parameters qi , pi , i = 1, . . . , g are coordinates on a Zariski open subset in the space of all special Fuchsian equations with given indices and given u 1 , . . . , u n . Due to Theorems 4.14 and 4.45, the parameters qi , pi , i = 1, . . . , g can be used as coordinates on a Zariski open subset in the space of all Fuchsian systems with given u 1 , . . . , u n and given exponents, considered modulo diagonal conjugations. Indeed, for fixed u 1 , . . . , u n , the condition (4.34) defines a Zariski open set in the space of all Fuchsian systems with given exponents. In this section we will prove that these coordinates are canonically conjugated with respect to the isomonodromic symplectic structure ω K (see (3.16)) on (3.14). Remark 4.21. In order to apply our coordinates to the description of solutions to the Schlesinger equations, one has to make sure that the Zariski closed subset where the map ⎧ ⎫ Fuchsian systems with given poles ⎨ ⎬ and given eigenvalues of A1 , . . . , An , A∞ → (q1 , . . . , qg , p1 , . . . , pg ) ⎩ ⎭ modulo diagonal conjugations becomes singular, or equivalently condition (4.34) is violated, is never invariant under the monodromy preserving deformation.4 This can be proved under the following two assumptions: i) If A∞ has a resonance of order one then the corresponding logarithmic correction R1 is not zero (see (2.9)). ii) For at least one j, the entries of the j th row of the matrices A1 , …, An satisfy the following condition: for every i = j there exists k such that Ak ji = 0.
(4.60)
4 In the theory of iso-spectral deformations an analogous problem arises. In this case one needs to check that the dynamics on the Jacobian of the spectral curve is never tangent to the Theta-divisor.
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337
Under these assumptions, by performing a small monodromy preserving deformation, condition (4.34) is satisfied. *n In fact suppose by contradiction that l=1 u l Al ji (u) ≡ 0 for some i, j. A simple differentiation using the Schlesinger equations gives n ∂ (∞) (∞) Ak ji = 0. u l Al ji = − 1 + λ j − λi ∂u k
(4.61)
l=1
(∞)
(∞)
Now if 1 + λ j − λi = 0 then A∞ has a resonance of order one and since R1 ji = *n l=1 u l Al ji must be zero, assumption i) is contradicted. Therefore Ak ji = 0, but this contradicts assumption ii). Clearly the two assumptions are satisfied in a large Zariski open set in the space of solutions of the Schlesinger equations. Let us rewrite Eq. (4.45) in the matrix form d = B(z) , dz where ⎛
⎞ y ⎜ y ⎟ = ⎝ ... ⎠, y (m−1)
⎛
0 ⎜ 0 ⎜ B(z) = ⎜ 0 ⎝... d0
(4.62)
1 0 ... ... d1
0 1 0 ... ...
⎞ ... 0 0 ... ⎟ ⎟ 1 0 ⎟, ... ... ⎠ . . . dm−1
(4.63)
&n &g fl (z) with dl (z) := (z)R(z) m−l , R(z) = k=1 (z −u k ), (z) = i=1 (z −qi ) and deg fl (z) = (n − 1)(m − l) + g. Recall that the system (4.63) is obtained from the original Fuchsian system by a gauge transformation. Lemma 4.22. If the apparent singularities qi , i = 1, . . . , g in Eq. (4.45) are distinct, for each i = 1, . . . , g, the matrix B(z) has one and only one eigenvalue ρi (z) with a simple pole at qi . For each i = 1, . . . , g, we define i (z) , for qi = u k , ∀ k = 1, . . . , n, Resz=qi ρz−q i pi = Resz=qi ρi (z), for qi = u k . Then (qi ) f˜m−2 (qi ) f˜m−1 (qi ) − f˜m−1 (qi ) (qi ) f˜m−1 (qi ) (qi ) 1 = Resz=qi dm−2 (z) + dm−1 (z)2 , 2
pi =
where for l = 0, . . . , m − 1, - fl (z) m−l , for qi = u k , ∀ k = 1, . . . , n, ˜ fl (z) = R(z) fl (z) m−l , for qi = u k . R (z)
(4.64)
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Observe that as an immediate consequence of the second part of Eq. (4.64), one obtains that the momenta pi coincide with those defined in (4.11). Proof. The characteristic equation of B is (z)ρ(z)m =
m−1 l=0
fl (z) ρ(z)l . R(z)m−l
Let us define ρ(z) ˜ = R(z)ρ(z). Since the polynomials fl (z) are regular at z = qi , there is only one eigenvalue ρ˜i (z) that has a pole at qi . If qi = u k for all k = 1, . . . , n, this pole is simple. Let us expand ρ˜i at z = qi and compare the left and right-hand sides of the characteristic equation. We obtain ρ˜i (z) =
f m−1 (qi ) (qi )(z−qi )
+
f m−2 (qi ) f m−1 (qi )
+
f m−1 (qi ) (qi )
(qi ) − f m−1 (qi ) (qi ) + O(z − qi ).
˜ (qi ) + pi + O(z − qi ), where f˜l = R(z)flm−l . This proves (4.64) for Therefore ρi (z) f(qm−1 i )(z−qi ) qi = u k for all k = 1, . . . , n. Analogously if qi = u k for one value of k = 0, . . . , ∞, ˜
(qi ) pi then ρi (z) has a double pole at qi and ρi (z) (qfm−1 2 + (z−qi ) + O(1) and again we i )(z−qi ) obtain (4.64) as we wanted to prove. The second part of formula (4.64) is immediately obtained from the formula (4.13) for a1 (z) = dm−1 (z).
Definition 4.23. We call the set (q1 , . . . , qg , p1 , . . . , pg ) the isomonodromic coordinates of the Schlesinger equations. Theorem 4.24. On a generic reduced symplectic leaf O1 × · · · × On /Diag the quantities (q1 , . . . , qg , p1 , . . . , pg ) are canonical coordinates. The Schlesinger equations in these coordinates are written in the canonical form ∂qi ∂Hk = , ∂u k ∂ pi ∂Hk ∂ pi =− , ∂u k ∂qi where the Hamiltonians in canonical coordinates are given by the formula 1 2 Hk = −Resz=u k dm−2 (z) + dm−1 (z) , 2
(4.65)
where dm−2 (z) and dm−1 (z) are defined in Theorem 4.14. Corollary 4.25. The Hamiltonians (4.65) are given by ⎡ ⎤ g (s) (i) (k) δ1 − ps β2 R (u i ) β2 R (u k ) (∞) −β2 u n−2 + − − Pn−3 (u k )⎦ Hk = ⎣ k u k − qs uk − ui 2 s=1 i=k ⎧ ⎫ * m (m−1) ⎬ g ⎨ mj=1 λ(i) 1 1 j − 2 × − + ⎭ R (u k ) ⎩ u − qs uk − ui s=1 k i=k ⎡ ⎤ m m (m − 1) (i) ⎦, ×⎣ λj − 2 j=1
Canonical Structure and Symmetries of the Schlesinger Equations
339
where the coefficients of the polynomial Pn−3 (z) are rational functions of p1 , . . . , pg , q1 , . . . , qg uniquely determined by (4.23). Example 4.26. In* the 2 × 2 case the polynomial (z) coincides with the (1, 2)-matrix entry of A(z) = k Ak /(z − u k ), (z) = R(z)A12 (z). So the isomonodromic coordinates qi coincide with the spectral coordinates (see below). Our p1 , . . . , pg are slightly different from the usual momenta pˆ 1 , . . . , pˆ g defined for Garnier systems (see [28]). In fact in our case we imposed the trace of all matrices Ak to be zero, while in [28], the determinant is zero. The relation between our coordinates and [28] is given by * * n qi (2 nk=1 u k − nqi ) + 1≤k
Keeping track of this time-dependent canonical transformation, it is not difficult to verify that our Hamiltonian functions (4.65) coincide with the one given in [28]. Proof of the theorem. Since the system (4.63) is gauge equivalent to the original Fuchsian system (1.2), it suffices to perform all computations with (4.63). First of all formula (4.65) is obtained by straightforward computation applying formula (3.17) to the matrix (4.63). We want to show that qi , pi are canonical coordinates on the reduced symplectic leaf (3.14) and that in those coordinates the Hamiltonian is indeed (4.65). To this aim observe that we can always put equations of the form (4.45) in the matrix form (4.62), (4.63). The apparent singularities are poles of the eigenvectors of the matrix B(z). In this way the proof reduces to proving the following Lemma 4.27. If qi = u 1 , . . . , u n , ∞ for all i = 1, . . . , g, the symplectic structure (3.16) on the space of monodromy data of the linear system of ODEs of the form (4.62), (4.63) is ωK =
g
d pi ∧ dqi .
i=1
Proof. Due to gauge invariance of the form ω K we have to compute ωK = −
1 1 Resz=u k Tr(δB ∧ δ −1 ) − Resz=qi Tr(δB ∧ δ −1 ). 2 2 n
g
k=1
i=1
Observe that δBil is zero for all i = m and δBml = δdl−1 so that ω K depends only on the m th column of the matrix −1 , i.e. ωK
n m 1 =− Resz=u k δdl−1 ∧ δ l j −1 − jm 2 k=1
−
g 1
2
i=1
l, j=0
Resz=qi
m l, j=0
δdl−1 ∧ δ l j −1
jm
.
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Let us deal with the apparent singularities first. We have to compute the expansion of B(z) and (z) at qi . Choose m linear independent solutions y1 , . . . , ym having expansions at z = qi of the form described in Lemma 4.5 where the constants αl are determined by the differential equation (4.62) and are (i)
αl+1 = −Resz=qi dl (z), l = 0, . . . , m − 2, d (z) m−1 (i) = Resz=qi 2 + dm−2 (z) . (4.66) αm z − qi (i) (i) α −α (z) Comparing these with (4.64) we have pi = m 2 m−1 = Resz=qi dm−1 z−qi + dm−2 (z) . Observe that the fundamental matrix of the scalar equation (4.45) has matrix elements given by l j =
dl−1 y j , dz l−1
l, j = 1, . . . , m.
We can show that in the computation of the residue in (3.16) at qi , one can neglect O(z − qi )m in y1 , . . . , ym and in the coefficient dm−1 and O(1) in d0 , . . . , dm−2 . This follows by straightforward computations based on a list of observations: 1. −1 jm = O(z − qi )m− j for j = 1, . . . , m − 1 and −1 mm O(z − qi )−1 . 2. From (4.11) and (4.66) we have that (i) (i) n *m λ(k) − m(m−1) αm + αm−1 1 j=1 j 2 = + . 2 qi − q j qi − u k j=i
k=1
3. For l = 1, . . . , m − 1, (i)
δdl−1 = and δdm−1 =
(i)
−αl δqi δαl + O(1), − (z − qi )2 (z − qi )
δqi 1 (i) (i) δαm + δαm−1 + O(1)δqi + O(z − qi ). + 2 (z − qi ) 2
4. For 1 ≤ j ≤ m − 1 and l ≤ j − 1: (i)
δ l j = −
δα j δqi (z − qi ) j−l−1 + (z − qi )m−l − ( j − l − 1)! (m − l)! (i)
−
α j δqi (m − l − 1)!
(z − qi )m−l−1 + δqi O(z − qi )m−l+1 + O(z − qi )m−l+2 ,
for 1 ≤ j ≤ m − 1 and l ≥ j: (i)
δ l j =
δα j
(i)
α j δqi
(z − qi ) (z − qi )m−l−1 + − (m − l)! (m − l − 1)! + δqi O(z − qi )m−l+1 + O(z − qi )m−l+2 , m−l
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and for j = m: δ lm = − −
(i)
δqi δαm (z − qi )m−l + (z − qi )m−l+2 − (m − l)! (m − l + 2)! (i)
αm δqi (z − qi )m−l−1 + δqi O(z − qi )m−l+2 + O(z − qi )m−l+3 . (m − l − 1)!
From 1) and 3) we immediately see that only the terms with j = m −1, m can contribute to the residue. From 1), 2), 3) and 4) we have g
Resz=qi
i=1
=
δdl−1 ∧ δ l,m−1 −1
m−1,m
l g
Resz=qi δdm−1 ∧ δ m,m−1 −1
i=1
=
m−1,m
1 (i) δα ∧ δqi , 2 m−1
and g
Resz=qi
i=1
=
δdl−1 ∧ δ lm −1
l g
=
mm
Resz=qi δdm−1 ∧ δ mm −1
mm
i=1
+ δdm−2 ∧ δ m−1,m −1
mm
=
1 (i) (i) ∧ δqi + δαm−1 ∧ δqi , = − δαm 2 thus we obtain g g 1 1 (i) −1 (i) − ∧ δqi = Resz=qi δdl−1 ∧ δ l j =− δ αm−1 − αm jm 2 2
=
i=1 g
l, j
i=1
δpi ∧ δqi .
i=1
Let us now show that Resz=u k δdl−1 ∧ δ l j −1 l, j
jm
= 0,
∀ k = 1, . . . , n.
l−m , where c Let us expand dl−1 at u k . We have dl−1 = (z−ucl−1 l−1 are m−l+1 + O(z − u k ) k) uniquely determined by the indicial equation, thus by the exponents. As a consequence
δdl−1 = O(z − u k )l−m . Analogously to estimate l j = (k) λj
dl−1 y j dz l−1
, we can again normalize (k)
+ O(z − u k )λ j +1 so that δy j = (k) (k) (k) O(z−u k )λ j −1 . Thus δdl−1 ∧δ l j = O(z−u k )λ j −m . Now −1 jm = O(z−u k )m−λ j * so that near u k we have l, j δdl−1 ∧ δ l j −1 jm = O(1) and the pole u k does not contribute to the residue. the solutions y j at u k in such a way that y j = (z − u k )
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The above lemma proves that qi , pi are canonical coordinates on the reduced symplectic leaf (3.14). We now want to prove that in those coordinates the Hamiltonians are indeed given by formula (4.65). To this aim we need to extend the phase space. Let us consider the space of all matrices B of the form (4.63) with coefficients d0 , . . . , dm−1 , ⎡ ⎤ g n m 1 1 ⎣ (i) m (m − 1) ⎦ dm−1 (z) = + λj − z − qs z − ui 2 s=1 i=1 j=1 ( g (s) (i) n c R(qs )m−k−1 βm−k k+1 + (−1)m−k−1 [R (u i )]m−k−1 + dk (z) = − z − qs z − ui s=1 i=1 ) 1 (∞) (m−k)(n−1)−n + P(m−k)(n−1)−n−1 (z) , + βm−k z m−k−1 R(z) k = 0, . . . , m − 2. We recall that this means that Eq. (4.45) has n + 1 Fuchsian poles at u 1 , . . . , u n , ∞ with (k) indices λ(k) 1 , . . . , λm for k = 1, . . . , n, ∞ given by Eqs. (4.14) and (4.15) and it has simple poles at the points q1 , . . . , qg . Near each simple pole qs , the matrix B can be expanded as B= where
B(s) (s) 0 + B1 + O(z − qs ), z − qs
⎞ 0 0 ... 0 ⎜ ... ... ... ...⎟ (s) ⎟ B0 = ⎜ ⎝ 0 0 ... 0 ⎠, (s) 1 −c1(s) . . . −cm−1 ⎛ ⎞ 0 1 0 ... 0 ⎜ 0 0 1 0 ... ⎟ ⎜ ⎟ ⎜ 0 ... 0 1 0 ⎟, B(s) = 1 ⎜ ⎟ ⎝... ... ... ... ... ⎠ (s) (s) (s) δm δm−1 . . . . . . δ1 ⎛
(s)
(s)
where δ1 , . . . , δm are given in Eqs. (4.11). If we do not impose Eqs. (4.17), that is if we do not assume the singularities q1 , . . . , qg to be apparent, there exists a fundamental solution of the form (s)
= G (s) (z)(z − qs ) (z − qs ) R , (s)
(4.67)
(s)
where G (s) (z) = G 0 + G 1 (z − qs ) + O(z − qs )2 , = diagonal(0, . . . , 0, 1), R (s) is an off-diagonal matrix with all entries equal to zero, apart from the last row. The matrices (s) R (s) , G (s) 0 and G 1 are determined by the following equations: (s)
(s)
(s)
B0 G 0 = G 0 ,
(s)
(s)
(s)
(s)
(s)
(s)
(s)
B0 G 1 + B1 G 0 = G 1 + G 0 R (s) + G 1 .
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343
This fact is a simple consequence of the gauge formula applied to the gauge = ˜ that maps B to + R (s) . G (s) (z) z−qs Observe that the m − 1 equations (4.17) imposing that the simple pole qs is apparent, coincide with the m − 1 equations R (s) ≡ 0. We consider now the symplectic structure on the extended space of isomonodromic deformations of systems of the form (4.62), (4.63) where the entries of the matrices R (s) , s = 1, . . . , g are not necessarily null. The following lemma concludes the proof of Theorem 4.24, by proving that the Hamiltonians in the isomonodromic coordinates are indeed given by (4.65). Lemma 4.28. On the extended space S × X n , where S are the symplectic leaves and X n = Cn \ {diagonals} is the configuration space of n points, the symplectic structure (3.16) becomes ωK =
g
d pi ∧ dqi −
i=1
n
dHk ∧ du k .
k=1
Proof. There are two main differences with the previous proof. The first one is that now we have to take into account the variations δu k , k = 1, . . . , n, the second one is that now the entries of the matrices R (s) are not necessarily zero. Let’s first look at the term δ ∧ −1 near the point qs . Using formula (4.67) we obtain −1 (s) −1 δ −1 = δG (s) (z) G (s) (z) G (z) − G (s) (z) δqs − z − qs −1 R (s) − G (s) (z) G(z)(s) δqs z − qs because all resonances are of order one and (z − qs ) R (s) (z − qs )− = R (s) . Therefore, when computing the residue at qs of Tr(δB ∧ δ −1 ) we just need to R (s) (s) −1 δq to d p ∧ dq . G(z) add the contribution of the term −Tr δB ∧ G(z)(s) z−q s s s s We are now going to prove that this extra contribution is zero. In fact G(z)
(s) (s) −1 G R (s) (G 0 )−1 (s) (s) (s) −1 (s) (s) −1 R (s) G(z)(s) + = 0 − G 0 R (G 0 ) , G 1 (G 0 ) z − qs z − qs + O(z − qs ).
Only the last column contributes to the trace. It is not difficult to see that the last (s) (s) column of G 0 R (s) G 0 is zero and that the only non–zero element of the last col(s) (s) (s) (s) δqs + O(1), the umn of G 0 R (s) (G 0 )−1 , G 1 (G 0 )−1 is the last one. Since δdm = z−q s residue is zero. Let us now show that δdl−1 ∧ δ l j −1 = 2δHk ∧ δu k , ∀ k = 1, . . . , n. Resz=u k l, j
jm
The contribution of the matrices R (s) does not play any role here because we are expanding at u k . On the other side, this time we need to take into account the variations δu 1 , . . . , δu n .
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B. Dubrovin, M. Mazzocco
Let us expand dl−1 at u k . We have (k)
dm−1 =
β1 − m(m−1) (k) 2 + Dm−1 + O(z − u k ), z − uk (k)
dl−1 =
(−1)m−l βm−l+1 (z − u k )m−l+1
(k)
+
Dl−1 (z − u k )m−l
+ O(z − u k )l+1−m ,
(4.68)
where (cfr. (4.13) with dl−1 (z) = am−l+1 (z)) ⎡ ⎤ g m 1 1 m (m − 1) ⎦ (k) (i) ⎣ Dm−1 = + λj − , u k − qs uk − ui 2 s=1 i=k j=1 ⎡ (i) βm−l+1 (k) (∞) (n−1)(m−l+1)−n Dl−1 = ⎣(−1)m−l [R (u i )]m−l + βm−l+1 u k uk − ui i=k ⎤ 1 +P(m−l+1)(n−1)−n−1 (u k )⎦ , l = 0, . . . , m − 2. R (u k )m−l We need to introduce some notation: [λ]0 := 1,
[λ]1 := λ,
[λ]n := λ(λ − 1) . . . (λ − n + 1), ∀ n = 2, 3, . . . .
The indicial equations (4.14) read [λ
(k)
m−1 m(m − 1) (k) (k) [λ(k) ]m−1 + ]m = β1 − (−1)m−l βm−l+1 [λ(k) ]l−1 . (4.69) 2 l=1
(k)
(k)
To start with, we perform our computation in the case when the exponents λ1 . . . , λm of the pole u k are non-resonant. Thanks to (4.3), there exists a basis of solutions y1 , . . . , ym of the form (k)
(k)
yi (z) = (z − u k )λi (1 + ηi (z − u k ) + O(z − u k )2 ),
i = 1, . . . , m.
Therefore we have li (z) =
(k) (k) dl−1 yi (k) (k) (k) = [λi ]l−1 (z − u k )λi −l + [λi + 1]l−1 ηi (z − u k )λi −l+1 + l−1 dz (k)
+ O(z − u k )λi (k)
−l+2
,
(k)
where the constants η1 , . . . , ηm are determined by the following equations: m(m − 1) [λi(k) + 1]m−1 − [λi(k) + 1]m − β1(k) − 2 . m−1 m−1 (k) (k) (k) (k) m−l (k) (−1) βm−l+1 [λi + 1]l−1 ηi = Dl−1 [λi ]l−1 . − l=1
l=1
(4.70)
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345
As above, only the last column of the inverse matrix −1 enters in the computation of the symplectic structure ω, −1
im
(k)
(k) (z − u k )m−1−λi + O(z − u k )m−λi . =& (k) (k) j=i (λi − λ j )
Proceeding in a similar way as in the first part of this proof, we arrive at the formula -m−1 m (k) −1 −1 (k) λi Resz=u k δdl−1 ∧ δ li im = δ Dl−1 ∧ δu k + & (k) (k) l (λ − λ ) j=i i l,i i=1 l=1 j m(m − 1) (k) (k) (k) λi +1 + β1 − δηi ∧ δu k + m−1 2 ! . m−1 (k) (k) m−l (k) + (−1) (m − l + 1)βm−l+1 [λ + 1]l−1 δηi ∧ δu k . (4.71) l=1
Now using the indicial equation (4.69), we get m−1 m(m − 1) (k) (k) [λi(k) + 1]m−1 + β1 − (−1)m−l (m − l + 1)βm−l+1 [λ(k) + 1]l−1 = 2 l=1 m(m − 1) (k) (k) (k) [λ(k) + 1]m−1 − (λi − m + 1) [λ + 1]m − β1 − 2 . m−1 (k) − (−1)m−l βm−l+1 [λ(k) + 1]l−1 . (4.72) l=1 (k)
Observe that the right-hand-side of Eq. (4.72) is (λi (k) ηi in (4.70). Using this in Eq. (4.71), we get Resz=u k
δdl−1 ∧ δ li −1
im
l,i
=
m
&
i=1
+
m−1
− m + 1) times the coefficient of
−1 (k) (k) j=i (λi −λ j )
(k) (k) 2[λi ]m δ Dm−1 ∧ δu k +
(k) (k) (k) [λi ]l−1 (2λi +2−m −l)δ Dl−1
∧ δu k .
l=1
(4.73) To conclude we observe that m
2λi − m + 2 − l 2λ − m − l + 2 [λi ]l & resλ=λi [λ]l &n = (k) (k) j=1 (λ − λ j ) j=i (λi − λ j ) i=1 i=1 n
2λ − m − l + 2 = −resλ=∞ [λ]l &n j=1 (λ − λ j ) 0, for l = 0, 1, . . . , m − 3, = 2, for l = m − 2.
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Analogously m i=1
&
[λi(k) ]m
(k) j=i (λi
(k) − λj )
(k)
= β1 −
m(m − 1) . 2
Using these in (4.73), we get finally Resz=u k
δdl−1 ∧ δ li −1
im
l,i
m(m − 1) (k) (k) δ Dm−1 ∧ δu k − = −2 β1 − 2
(k)
− 2 δ Dm−2 ∧ δu k = 2 dHk ∧ du k , as we wanted to prove. (k) (k) To conclude, we observe that if some of the exponents λ1 . . . , λm of the pole u k are resonant, then by Theorem 2.1 there exists a fundamental solution (k)
(k)
= G (k) (z)(z − u k ) (z − u k ) R , * * (k) (k) (k) = where G (k) (z) = ∞ R j is a finite sum of off-diagonal j=0 G j (z − u k ), and R matrices such that (k)
(z − u k ) R (k) (z − u k )−
(k)
(k)
(k)
= R0 + R1 (z − u k ) + · · · .
By applying enough iterates of Lemma 4.20 we can increase the order of the resonances arbitrarily, i.e. we can always assume that (k)
(z − u k ) R (k) (z − u k )−
(k)
(k)
p p+1 = R (k) + · · · ., p (z − u k ) + R p+1 (z − u k )
with p large enough. Then the extra term in δ −1 due to R (k) is given by −1 p−1 −G (k) (z)(R (k) + O(z − u k ) p ) G(z)(k) δu k p (z − u k ) which does not contribute to the residue.
4.5. An example. As we already know, for m = 2 the isomonodromic coordinates coincide with the spectral ones. Starting from m = 3 they are different. In this subsection we give an explicit parametrization of special Fuchsian equations (4.26) in the first non-trivial case m = 3 and n = 3, in terms of our isomonodromic coordinates q1 , . . . , qg , p1 , . . . , pg and compute the Hamiltonians of the Schlesinger equations in the canonical coordinates. Note that g = 4 for m = 3 and n = 3. Starting from a Fuchsian system dY = A(z)Y, dz
A(z) =
A1 A2 A3 + + , z − u1 z − u2 z − u3
(i) (i) where Ai are 3 × 3 matrices with the eigenvalues λ(i) 1 , λ2 , λ3 , i = 1, 2, 3 satisfying (∞)
−(A1 + A2 + A3 ) = A∞ = diag(λ1
(∞)
, λ2
(∞)
, λ3
)
Canonical Structure and Symmetries of the Schlesinger Equations
347
we arrive at a third order Fuchsian equation with eight regular singularities with the following Riemann scheme: ⎧ ⎫ ∞ u 1 u 2 u 3 q1 q2 q3 q4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ λ(∞) λ(1) λ(2) λ(3) 0 0 0 0 ⎪ ⎬ 1 1 1 1 P (∞) (1) (2) (3) ⎪ λ2 + 1 λ 2 λ2 λ2 1 1 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (∞) ⎭ (1) (2) (3) λ3 + 1 λ 3 λ3 λ3 3 3 3 3 satisfying the additional constraint of absence of logarithmic terms at the points q1 , …, q4 . From the previous considerations it follows that the Fuchsian equation must have the form ( 4 ) 3 1 1 y = −3 y + z − qs z − ui s=1 i=1 ) ( 4 (s) 3 (i) c R(qs ) β2 R (u i ) y (∞) 2 + + + β2 z + h + z − qs z − ui R(z) s=1 i=1 ( 4 (s) ) 3 (i) c R 2 (qs ) β3 R 2 (u i ) y (∞) 3 2 1 . + − + + β3 z + a z + b z + c 2 z − qs z − ui R (z) s=1
i=1
(4.74) Let us spell out the notations. The polynomial R(z) is given by R(z) = (z − u 1 )(z − u 2 )(z − u 3 ). The coefficients βk(i) , βk(∞) are given by the following formulae: (i)
(i) (i)
(i) (i)
(i) (i)
(i)
(i) (i) (i)
β2 = λ1 λ2 + λ1 λ3 + λ2 λ3 − 5, β3 = λ1 λ2 λ3 , i = 1, 2, 3, (4.75) β2(∞)
=
(∞) 2 2 [λ(∞) 1 ] +[λ2 ]
+ [λ3(∞) ]2 −2λ(∞) 1 −5,
β3(∞)
= −λ1(∞) (λ2(∞)
+ 1)(λ3(∞)
+ 1).
In this example we assume all the matrices Ai to be traceless: tr Ai = 0, i = 1, 2, 3.
(4.76)
We also put c2(s) = − ps − 3
R (qs ) 1 (qs ) + R(qs ) 2 (qs )
as in Eq. (4.11). We introduce the following quantities: ps[k] = ps + k
R (qs ) , k = 0, 1, 2, 3, s = 1, . . . , 4 R(qs )
(4.77)
1 (qs ) 2 (qs )
(4.78)
and p˜ s[k] = ps[k] −
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where, as above the monic polynomial (z) is defined by (z) = (z − q1 ) . . . (z − q4 ) ≡ z 4 − σ1 z 3 + σ2 z 2 − σ3 z + σ4 .
(4.79)
Here σ1 , …, σ4 are just the elementary symmetric functions of q1 , . . . , q4 . In these (s) notations, c2 = − p˜ s[3] . The coefficients h, a, b, c and c1(s) = α1(s) , s = 1, . . . , 4 are to be expressed in terms of the canonical coordinates q1 , …, q4 , p1 , …, p4 and u 1 , u 2 , u 3 from the assumption of absence of logarithmic terms at the apparent singularities. This assumption yields a (s) linear 8 × 8 system for the above unknowns. Eliminating the unknowns c1 one arrives at the following system: a qs2 + b qs + c + h · Mst ( p [2] , q)R(qt ) = ws , s = 1, . . . , 4, (4.80) t
where the 4 × 4 matrix M( p, q) = (Mst ( p, q)) is defined by ⎧ t =s ⎨ ps , Mst ( p, q) = ⎩ 1 , t = s qt −qs and ws =
Mst ( p [2] , q)R(qt )Mtr ( p [1] , q)R(qr ) p˜r[0] −
t,r
−
Mst ( p [2] , q)R(qt ) f 2 (qt ) − f 3 (qs ), s = 1, . . . , 4
(4.81)
t
and ( f 2 (z) = −
3
(i) β2
i=1
f 3 (z) =
3
(i)
β3
i=1
) R (u i ) (∞) + β2 z , z − ui
R 2 (u i ) (∞) + β3 z 3 . z − ui
Denote D = D( p, q, u) =
4 s=1
p˜ s[2]
R(qs ) (qs )
the determinant of the linear system (4.80). Then ⎡ 4 R(qs ) ws ⎣ 1 qk − ql a=− + 2 sign(s, j, k, l) p [2] + j R(q j ) D 2 W (q) (qs ) s=1 j,k,l ⎤ 4 R(q ) 1 j (qs − q j )2 2 + (σ1 − qs − 3q j )⎦ , (qs ) (q j ) j=1
(4.82)
Canonical Structure and Symmetries of the Schlesinger Equations
b=
4 ws s=1
⎡
⎣1 D 2
sign(s, j, k, l) p [2] j R(q j )
j,k,l
349
qk2 − ql2 R(qs )(σ1 − 2qs ) + + W (q) 2 (qs )
⎤ 4 R(q ) 1 j (qs − q j )2 (σ 2 − σ2 − σ1 qs − 2σ1 q j + 2qs q j )⎦ , + (qs ) (q j )2 1 j=1
c=−
4 ws s=1
⎡
⎣1 D 2
sign(s, j, k, l) p [2] j R(q j )qk ql
j,k,l
qk − ql + W (q)
4 1 σ4 2 R(q j ) 2 2 + + q j σ1 − 4σ2 − 3qs + 2σ1 qs + 3 (qs − q j ) (qs ) qs q j 2 (q j ) j=1 R(qs ) + 2 (3qs2 − 2σ1 + 2σ2 ) , (qs )
h=
4 1 ws , D (qs )
(4.83)
s=1
where W (q) =
'
(qi − q j ),
(4.84)
i< j
1234 . s jk l Using (4.65) one obtains the following expression for the Hamiltonians of Schlesinger equations S(3,3) : ⎤ ⎡ 4 ( j) β R(q ) 1 ⎣ (∞) s 2 Hi = − p˜ s[3] + R (u j )⎦ β2 u i + h − R (u i ) u i − qs ui − u j
sign(s, j, k, l) is the sign of the permutation
R (u
s=1 (u i )
j=i
R (u
1 (i) 9 i) i) + β2 +3 − , i = 1, 2, 3, 2 R (u i ) (u i ) 2 R (u i )
(4.85)
where the rational function h = h( p, q, u) was defined in (4.83). Clearly of these three Hamiltonians only one is independent: the solutions depend only on the combination (u 3 − u 1 )/(u 2 − u 1 ). 5. Comparison of Spectral and Isomonodromic Coordinates 5.1. Spectral coordinates. We recall the construction of the algebro-geometric Darboux coordinates on the generic reduced symplectic leaves (3.14) following the scheme of [58, 1, 13, 20]. We call these algebro-geometric Darboux coordinates spectral coordinates.
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The spectral coordinates are defined as follows. Let us assume that all the matrices Ai (i) have pairwise distinct nonzero eigenvalues λ(i) 1 , . . . , λm , and that the diagonal matrix (∞)
A∞ := −(A1 + · · · + An ) = diag (λ1
, . . . , λ(∞) m )
has distinct nonzero diagonal entries. Consider the characteristic polynomial of the matrix A(z) of the form A(z) =
n i=1
Ai z − ui
(5.1)
with constant matrices A1 , . . . , An satisfying the following properties. Denote R(z, w) = det(w − A(z)) = wm + α1 (z)w m−1 + · · · + αm (z) the characteristic polynomial of the matrix A(z). Denote by )m(m−1) ( n ' ' (z − u i ) (wi (z) − w j (z)) D(z) :=
(5.2)
(5.3)
i= j
i=1
the discriminant of the polynomial R(z, w). In this formula w1 (z),…,wm (z) are roots of the equation R(z, w) = 0. The resulting expression is a polynomial in the coefficients α1 (z), …, αm (z). Under the assumption that the matrix A∞ has simple spectrum, the degree of the discriminant is equal to N = m(m − 1)(n − 1).
(5.4)
Assumption 1. The N roots of the discriminant are simple and pairwise distinct. Also we require that D(u i ) = 0, i = 1, . . . , n.
(5.5)
Due to this assumption the spectral curve R(z, w) = 0
(5.6)
of the matrix A(z) is smooth outside the lines z = u 1 , . . . , z = u n , z = ∞. These lines intersect the spectral curve in singular points of multiplicity m. Let us introduce the row vectors b0 , b1 (z), b2 (z), …by b0 = (1, 0, . . . , 0), bk (z) = b0 Ak−1 (z), k > 0. Denote B(z) the m × m matrix with the rows b0 , b1 (z), . . . , bm−1 (z), ⎛ ⎞ b0 ⎜ b1 (z) ⎟ ⎜ ⎟ · ⎜ ⎟ B(z) = ⎜ ⎟. · ⎜ ⎟ ⎝ ⎠ · bm−1 (z)
(5.7)
(5.8)
Put 0 (z) =
( n ' i=1
) m(m−1) 2
(z − u i )
det B(z).
(5.9)
Canonical Structure and Symmetries of the Schlesinger Equations
351
Assumption 2. All the roots γ1 , . . . , γg of the polynomial 0 (z) are pairwise distinct and the degree of 0 (z) is equal to the maximal value g = deg 0 (z) =
1 m (m n − m − n − 1) + 1. 2
(5.10)
We also assume that the roots γ1 , . . . , γg do not coincide with the poles z = u i of the Fuchsian system neither with the zeroes of the discriminant D(z). Under this assumption there exists, for any i = 1, . . . , g, a unique, up to normalization, eigenvector ψ i of the matrix B(γi ) with zero eigenvalue. The first component of the eigenvector vanishes. It is also an eigenvector of the matrix A(γi ) with some eigenvalue µi , B(γi )ψ i = 0, A(γi )ψ i = µi ψ i , ψ i (ψ1i , ψ2i , . . . , ψmi )T , ψ1i = 0, i = 1, . . . , g. Observe that the genus of the Riemann surface (5.6) is equal to g. The spectral curve (5.6) together with the divisor D=
g
(γi , µi )
(5.11)
i=1
determines the matrix A(z) uniquely up to a conjugation by a constant diagonal matrix. Moreover, the matrices A(z) satisfying the above assumptions form a Zariski open subset in the space of all matrices of the form (5.1). All these facts are rather standard for the theory of algebraically completely integrable systems. We give a sketch of proofs of these statements in the Appendix. Definition 5.1. We call the set (γ1 , . . . , γg , µ1 , . . . , µg ) the spectral coordinates on (3.14). Example 5.2. For the m = 3 case the polynomial (z) determining the isomonodromic coordinates reads
(z) = R(z) A12 A13 (A22 − A33 ) − A212 A23 + A213 A32 − A12 A13 + A12 A13 , & where, as*usual R(z) = i (z − u i ) and Ai j = Ai j (z) are the entries of the 3 × 3 matrix A(z) = i Ai /(z − u i ). The positions of the spectral coordinates are determined by the polynomial
0 (z) = R(z) A12 A13 (A22 − A33 ) − A212 A23 + A213 A32 . We see that, unlike the case m = 2 (see above Eg. 4.26) starting from m = 3 the spectral and isomonodromic coordinates do not coincide. 5.2. Spectral coordinates versus isomonodromic coordinates. In this subsection we prove that in a certain semi–classical limit the isomonodromic coordinates q1 , …, qg , p1 , . . . , pg converge to the algebro–geometric Darboux coordinates. Let us consider the following family of Fuchsian systems depending on a small parameter .
d = A(z) , = (φ1 (z), . . . , φm (z))T . dz
(5.12)
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B. Dubrovin, M. Mazzocco
Theorem 5.3. Under the assumptions of Theorem 4.14 the apparent singularities of the scalar reduction of the Fuchsian system admit the following expansion qk = γk + O(), → 0, k = 1, . . . , g.
(5.13)
Moreover, the scalar reduction can be written in the following form m y (m) + m−1 a1 (z, )y (m−1) + · · · + am (z, )y = 0,
(5.14)
where the functions a1 (z, ), . . . , am (z, ) are analytic in (z, ) for z = u i , z = γ j , || << 1 and ak (z, ) = αk (z) − βk (z) + O( 2 ), k = 1, . . . , m,
(5.15)
where βk (z) =
1 k−1 µ j + α1 (γ j )µk−2 + · · · + α (γ ) + O(1), z → γ j . k−1 j j z − γj
(5.16)
In particular, the parameters p j defined in (4.64) have the following expansion p j = −1 [µ j + α1 (γ j )] + O(1), → 0, j = 1, . . . , g.
(5.17)
Here γk , µk are the spectral coordinates of the matrix A(z). Lemma 5.4. The following formula holds true for any k ≥ 0, k y (k) = [bk (z) + b˜k (z) + O( 2 )] ,
(5.18)
where the row vectors bk (z) were defined in (5.7) and the row vectors b˜k (z) are defined by the following recursive procedure: b˜0 = 0, b˜k+1 (z) = b˜k (z)A(z) + bk (z), k ≥ 0.
(5.19)
Here and below it will be understood that the product of the row vector by the column vector is a scalar. Proof. For k = 0 (5.18) is obvious. Since the first row of A(z) is b1 (z), the first equation of the Fuchsian system can be recast into the form y = b1 (z) . This proves (5.18) for k = 1. Let us now assume (5.18) for k and prove it for k + 1. Differentiating both sides of (5.18) in z and multiplying by yields k+1 y (k+1) = [bk (z) + b˜k (z) + O( 2 )] (z) + [bk (z) + b˜k (z) + O( 2 )] A(z)(z) = [bk+1 (z) + (b˜k (z)A(z) + bk (z)) + O( 2 )] (z). The proof of the lemma is completed by induction.
Canonical Structure and Symmetries of the Schlesinger Equations
353
˜ Corollary 5.5. Define an m × m matrix valued function B(z) with the rows b˜0 (z), b˜1 (z), …, b˜m−1 (z). Then the scalar reduction of the Fuchsian system reads
−1 ˜ m y (m) = [bm (z) + b˜m (z) + O( 2 )] B(z) + B(z) + O( 2 ) yˆ , (5.20) where yˆ := (y, y , . . . , m−1 y (m−1) )T .
(5.21)
Proof. From (5.18) we obtain ⎛ ⎞ y y ⎜ ⎟ 2 ⎝ ⎠ = (B + B˜ + O( )) . ... m−1 y (m−1) This proves (5.20).
From the corollary the claim of the theorem about expansions (5.13) of the apparent singularities, readily follows. Analyticity of the coefficients (am (z, ), am−1 (z, ), . . . , a1 (z, )) =
−1 ˜ + O( 2 ) = = [bm (z) + b˜m (z) + O( 2 )] B(z) + B(z)
˜ = bm (z)B −1 (z) + b˜m (z)B −1 (z) − bm (z)B −1 (z) B(z) B −1 (z) + O( 2 ) (5.22) also follows for small and for z away from the poles z = u i and z = γ j . Let us now simplify the r.h.s. of the formula (5.22). Lemma 5.6. The leading term in the r.h.s. of (5.22) reads bm (z) B −1 (z) = −(αm (z), αm−1 (z), . . . , α1 (z)).
(5.23)
Proof. Using the Cayley - Hamilton theorem we obtain bm (z) B −1 (z) = b0 Am(z)B −1 (z) = −b0 αm (z) + αm−1 (z)A(z) + · · · + α1 (z)Am−1 (z) B −1 (z) = − αm (z)b0 + αm−1 (z)b1 (z) + · · · + α1 (z)bm−1 (z) B −1 (z) = −(αm (z), αm−1 (z), . . . , α1 (z)). In the last line we use the definition of the inverse matrix B −1 (z). The lemma is proved. We will now simplify the linear in term. We need the following simple Lemma 5.7. Let us introduce matrix T (z) by ⎛ 0 1 0 0 1 ⎜ 0 ⎜ ... T (z) = ⎜ ⎝ 0 0 −αm (z) −αm−1 (z)
⎞ ... 0 ... 0 ⎟ ⎟ ... ⎟. ... 1 ⎠ . . . −α1 (z)
(5.24)
Then T (z)B(z) = B(z)A(z).
(5.25)
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B. Dubrovin, M. Mazzocco
Proof. Use the definition of the matrix B(z) and the Cayley - Hamilton theorem.
Lemma 5.8. The following identity holds true −1 ˜ (z) b˜m (z)B −1 (z) − bm (z)B −1 (z) B(z)B m
ek B (z)B −1 (z) T m−k (z) + α1 (z)T m−k−1 (z) + · · · + αm−k (z) . (5.26) = k=1
Here e1 , . . . , em are row vectors of the standard basis in Cm ∗ , (ek )i = δi k . Proof. Using arguments of Lemma 5.6 we replace bm (z)B −1 (z) by −(αm (z), . . . , α1 (z)). Next, using induction we derive the following formula for the row vectors (5.19): b˜k (z) = e2 B (z)Ak−2 (z) + e3 B (z)Ak−3 (z) + · · · + ek B (z), k ≥ 2. Using identity (5.25) the last formula can be recast into the form
b˜k (z) = e2 B (z)B −1 (z)T k−2 (z) + e3 B (z)B −1 (z)T k−3 (z) + · · · + ek B (z)B −1 (z) B(z), k ≥ 2. This implies formula (5.26) with the summation in the r.h.s. starting from k = 2. Since the first row of the matrix B (z) identically vanishes, adding the term with k = 1 does not change the sum. We now want to compute the Laurent expansion of the coefficients of the scalar reduction (5.14) at the points z = γ j = q j + O(). Let γ be one of the zeroes of 0 (z), µ the eigenvalue of the matrix A(γ ) such that the corresponding eigenvector ψ = (ψ1 , ψ2 , . . . , ψm )T , A(γ )ψ = µ ψ satisfies ψ1 = 0. According to our assumptions the eigenvector is defined uniquely up to a scalar factor. As we know, ψ is also an eigenvector of the matrix B(γ ) with zero eigenvalue, B(γ )ψ = 0. Moreover, there exists an analytic function λ(z) defined for |z − γ | << 1 being an eigenvalue of B(z) s.t. λ(z) has a simple zero at z = γ , and λ(z) does not coincide with other eigenvalues of B(z). Denote ψ(z) = (ψ1 (z), . . . , ψm (z))T the analytic vector valued function s.t. B(z)ψ(z) = λ(z)ψ(z), |z − γ | << 1, λ(γ ) = 0, ψ(γ ) = ψ. We also introduce a left eigenvector ψ ∗ (z) = (ψ1∗ (z), . . . , ψm∗ (z)), ψ ∗ (z)B(z) = λ(z)ψ ∗ (z), |z − γ | << 1. Denote ψ ∗ := ψ ∗ (γ ).
(5.27)
Canonical Structure and Symmetries of the Schlesinger Equations
355
Lemma 5.9. ψ ∗ is a left eigenvector of T (γ ) with the eigenvalue µ, ψ ∗ T (γ ) = µψ ∗ .
(5.28)
In particular, it can be chosen in the form ψk∗ = µm−k + α1 (γ )µm−k−1 + · · · + αm−k (γ ), k = 1, . . . , m.
(5.29)
Proof. From (5.27) for z = γ it follows ψ ∗ B(γ ) = 0.
(5.30)
Let us prove that ψ ∗ T (γ ) is again a left eigenvector of B(γ ) with zero eigenvalue. Indeed, using (5.25) we obtain ψ ∗ T (γ )B(γ ) = ψ ∗ B(γ )A(γ ) = 0. The eigenvector of T (γ ) with the eigenvalue µ can be written in the form (5.29). Let us prove that this eigenvector satisfies (5.30). According to our assumptions all the eigenvalues of the matrix A(γ ) are pairwise distinct. Of course, they coincide with the eigenvalues of the matrix T (γ ). Therefore it suffices to prove that ψ ∗ B(γ )ψ = 0 for an arbitrary eigenvector ψ of the matrix A(γ ), A(γ )ψ = µ ψ . Indeed, from (5.29) we obtain ˆ ψ ∗ B(γ )ψ = b0 ψ R(µ, µ , γ ), where R(z, w) − R(z, w ) ˆ R(µ, µ , γ ) = |z=γ , w=µ, w =µ w − w for µ = µ and ∂R(z, w) ˆ R(µ, µ, γ ) = |z=γ , w=µ . ∂w ˆ It is clear that R(µ, µ , γ ) = 0 for µ = µ. So ψ ∗ B(γ )ψ = 0. For µ = µ we have ˆ R(µ, µ, γ ) = 0 (since γ is not a zero of the discriminant D(z)) but b0 ψ = 0 since ψ1 = 0. The lemma is proved. We will now compute the leading term of the Laurent expansion of the logarithmic derivative B (z)B −1 (z) at z → γ . Lemma 5.10. For z → γ , B (z)B −1 (z) =
1 B (γ )ψ ⊗ ψ ∗ + O(1). z − q ψ ∗ B (γ )ψ
(5.31)
356
B. Dubrovin, M. Mazzocco
Proof. Let ψ(z) ⊗ ψ ∗ (z) (5.32) ψ ∗ (z)ψ(z) be the projector of Cm onto the direction of the eigenvector ψ(z) parallel to the (m − 1)dimensional subspace spanned by other eigenvectors. Denote 2 (z) = id − 1 (z) the complementary projector and put 1 (z) =
B2 (z) := B(z)2 (z). We have B(z) = λ(z)1 (z) + B2 (z). All these matrix valued functions are analytic for z sufficiently close to γ and since B(z) has a unique zero eigenvalue, rank (B2 (γ )) = m − 1, and the image of B2 (z) is transverse to that of 1 . So B (z)B −1 (z) = (log λ(z)) 1 (z) + λ−1 (z)B2 (z)1 (z) + regular terms. Since B2 (z)1 (z) ≡ 0, we obtain B2 (z)ψ (z) ⊗ ψ ∗ (z) + ··· ψ ∗ (z)ψ(z) B(z)ψ (z) ⊗ ψ ∗ (z) + ··· . =− ψ ∗ (z)ψ(z) In the last equation dots denote terms analytic at z = γ . We obtain ψ ⊗ ψ∗ 1 B(γ )ψ (γ ) ⊗ ψ ∗ 1 B (z)B −1 (z) = − + O(1). z−γ ψ ∗ψ λ (γ ) ψ ∗ψ Using the well-known formula of the “perturbation theory” B2 (z)1 (z) = −B2 (z)1 (z) = −
λ (γ ) =
ψ ∗ B (γ )ψ ψ ∗ψ
(observe that the formula implies ψ ∗ B (γ )ψ = 0) and the identity B(γ )ψ (γ ) + B (γ )ψ = λ (γ )ψ, we complete the proof of the lemma. End of the proof of the Theorem 5.3. We are to compute the sum m
ek B (z)B −1 (z) T m−k (z) + α1 (z)T m−k−1 (z) + · · · + αm−k (z)
k=1
=
m 1 B (γ )ψ ⊗ ψ ∗ m−k T ek (γ ) + α1 (γ )T m−k−1 (γ ) + · · · + αm−k (γ ) + O(1) z−γ ψ ∗ B (γ )ψ k=1
m 1 B (γ )ψ ⊗ ψ ∗ m−k = µ ek + α1 (γ )µm−k−1 + · · · + αm−k (γ ) + O(1) ∗ z−γ ψ B (γ )ψ k=1 m ∗ 1 ψk B (γ )ψ k ∗ 1 ψ + O(1) = ψ ∗ + O(1), = z−γ ψ ∗ B (γ )ψ z−γ k=1
where the left eigenvector ψ ∗ is chosen in the form (5.29). The theorem is proved.
Canonical Structure and Symmetries of the Schlesinger Equations
357
5.3. Canonical transformations. Let us consider the isomonodromic coordinates (q1 , . . . , qg , p1 , . . . , pg ) obtained from the scalar reduction w.r.t. the first row of (1.2). Proposition 5.11. For every k = 2, . . . , n, the following transformation ⎧ q˜i = u 1 + u k − qi , i = 1, . . . , g, ⎪ ⎪ ⎪ p˜ i = − pi , i = 1, . . . , g, ⎪ ⎪ ⎪ ⎨ u˜ l = u 1 + u k − u l , l = 1, . . . , n, (k) (1) Sk : λ˜ j = λ j , j = 1, . . . , m, ⎪ ⎪ ⎪ ⎪ λ˜ (1) = λ(k) ⎪ j , j = 1, . . . , m, ⎪ ⎩ j H˜ l = −Hl , l = 1, . . . , n,
(5.33)
is a birational canonical transformation of the Schlesinger systems. This transformation acts on the monodromy matrices as follows: −1 Mk Mk−1 . . . M1 , M˜ 1 = M1−1 . . . Mk−1
M˜ j = M j , j = 1, k, −1 M˜ k = Mk−1 . . . M2 M1 M2−1 . . . Mk−1 , i = k + 1, . . . , n.
(5.34)
Proof. The transformation (5.33) is obviously birational. To show that (5.33) is a canonical transformation of the Schlesinger systems, we just observe that it is obtained by the conformal transformation ζ = u 1 + u k − z of the scalar reduction (4.45). In fact (4.45) is transformed to m−1 l dm y ˜l (ζ ) d y , = d dζ m dζ l l=0
& ˜ ˜ ) = nk=1 (ζ − u˜ k ), (ζ ) = where d˜l (ζ ) = (−1)l−m dl (u 1 +u k −ζ ) ˜ fl˜(ζ ) m−l with R(ζ (ζ ) R(ζ ) &g (m−l)(n−1)+g f (u +u −ζ ). To obtain p˜ we use formula ˜ l 1 k i i=1 (ζ − q˜i ) and fl (ζ ) = (−1) (4.64): ˜ (ζ ) p˜ i = Resζ =q˜i d˜m−2 (ζ ) + dm−1 = ζ −q˜i 1 +u k −ζ ) = Resζ =q˜i dm−2 (u 1 + u k − ζ ) − dm−1 (u = − pi . ζ −q˜i To obtain the formulae for the exponents, just observe that the conformal transformation ζ = u 1 + u k − z permutes u 1 with u k . Let us now prove the formula (5.34). The involution i k : u 1 ↔ u k changes the basis in the fundamental group π1 (C\{u 1 , . . . , u n , ∞}). In fact, as explained in Sect. 2.2, the cuts π1 , . . . , πn along which we take our basis l1 , . . . , ln , are ordered according to the order of the poles. Applying the transformation i k we then arrive at the new basis of loops −1 l1 = l1l2 · · · lk−1lk lk−1 · · · l2−1l1−1 ,
l j = l j ,
j = 1, k,
−1 lk = lk−1 · · · l2−1 l1l2 · · · lk−1 .
from these formulae we immediately obtain (5.34).
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Proposition 5.12. The following transformation: ⎧ 1 ⎪ ⎪ q˜i = qi −u 1 , i = 1, . . . , g, ⎪ 2 ⎪ ⎪ ⎪ p˜ i = − pi qi2 − 2mm−1 qi , i = 1, . . . , g, ⎪ ⎪ 1 ⎪ ⎪ u˜ l = u −u , l = 2, . . . , n, ⎪ 1 l ⎪ ⎪ ⎪ → ∞, u 1 ⎪ ⎪ ⎪ ⎪ ∞ → u 1 , ⎪ ⎪ ⎪ ⎨ λ˜ (∞) = λ(1) + m−1 , 1 1 m S∞ : (5.35) ˜ (∞) = λ(1) − 1 , j = 2, . . . , m, λ ⎪ j j m ⎪ ⎪ (1) (∞) ⎪ ⎪ λ˜ 1 = λ1 − m−1 ⎪ m , ⎪ ⎪ (1) (∞) 1 ⎪ ˜ ⎪ λ j = λ j + m , j = 2, . . . , m, ⎪ ⎪ ⎪ ⎪ H ⎪ ˜ 1 = H1 , ⎪ ⎪ 0 ⎪ ⎪ H˜ l = −Hl (u l − u 1 )2 + (u l − u 1 )(dm−1 (u l − u 1 ))2 − ⎪ ⎪ ⎩ (m−1)(m 2 −m−1) 0 −(u l − u 1 ) dm−1 (u l − u 1 ), l = 2, . . . , n m where 0 dm−1 (u k ) =
g s=1
1 1 m (m − 1) − u k − qs 2 u k − ul l=k
is a birational canonical transformation of the Schlesinger systems. This transformation acts on the monodromy matrices M1 and M∞ as follows: M˜ ∞ = e−
2πi m
−1 C 1 M∞ M1 M∞ C1−1 ,
M˜ 1 = e m C1 M∞ C1−1 , M˜ j = C1−1 M j C1 , j = 1, ∞, 2πi
(5.36)
Proof. The fact that the above transformation is birational is trivial. To show that it is a canonical transformation of the Schlesinger systems, we just observe that it is obtained 1 by a conformal transformation ζ = z−u and a gauge transformation y = g(ζ ) y˜ , 1 g(ζ ) = ζ
m−1 m
of the scalar reduction (4.45). In fact (4.45) is transformed to s m−1 m 1 d p g dm− p y˜ m−1 1 d p g ds− p y˜ dm y˜ s ˆs = − + , d p g(ζ ) dζ p dζ m− p p g(ζ ) dζ p dζ s− p dζ m p=1
s=0
p=0
where
dˆ0 = (−1)m ζ −2m d0 ( ζ1 + u 1 ), m+1 ζ s−m + (−1)m *m−1 ζ l+s−2m cl+1 d ( 1 + u ), dˆs = (−1)m+1 cs+1 1 l=s s+1 l ζ j −1 j −2 j . Using the above formula and (4.64) and ci := (−1) j−1 ( j − i)! i −1 i −2 it is a straightforward computation to obtain the formulae for q˜i , p˜ i and H˜ l in (5.35). The transformation law of the exponents is obtained in two stages: first the conformal transformation ζ = 1z + u 1 maps (∞)
λ1
(1)
(1)
(∞)
→ λ1 , λi (∞)
λ 1 → λ1
(1)
, λi
(1)
+ 1 → λi , i = 2, . . . , m, (∞)
→ λi
+ 1, i = 2, . . . , m,
Canonical Structure and Symmetries of the Schlesinger Equations
359 m−1
then the gauge transformation y = g(ζ ) y˜ , g(ζ ) = ζ m adds m−1 m to all exponents at infinity and subtracts the same quantity to all exponents at 0. To show (5.36) we proceed as in the previous proof: the involution i ∞ : u 1 ↔ ∞ changes the base point of the fundamental group (this is obtained by conjugating all monodromy matrices with the connection matrix C1 of M1 ), and it changes the basis of loops as in the previous proof with k replaced by ∞ and k − 1 by n. This implies immediately (5.36). Remark 5.13. Obviously we can obtain analogous birational canonical transformations acting on the isomonodromic coordinates obtained from the scalar reduction w.r.t. any row of (1.2). Remark 5.14. Apart from the above symmetries, there are other birational canonical ( j) ( j) ( j) ( j) transformations. In fact let us denote by (q1 , . . . , qg , p1 , . . . , pg ) the isomonodromic coordinates obtained from the scalar reduction w.r.t. the j th row. The trans( j) ( j) ( j) ( j) formation that maps (q1 , . . . , qg , p1 , . . . , pg ) to the isomonodromic coordinates obtained from the scalar reduction w.r.t. the i th row(q1(i) , . . . , qg(i) , p1(i) , . . . , pg(i) ) is by construction a birational canonical transformation. These transformations are the analogues of Okamoto’s w3 for the Painlevé sixth equation (see [48]). Acknowledgements. The authors are very grateful to A. Bolibruch for many helpful conversations. This work is partially supported by European Science Foundation Programme “Methods of Integrable Systems, Geometry, Applied Mathematics” (MISGAM), Marie Curie RTN “European Network in Geometry, Mathematical Physics and Applications” (ENIGMA), and by Italian Ministry of Universities and Researches (MIUR) research grant PRIN 2004 “Geometric methods in the theory of nonlinear waves and their applications”. The researches of M.M. are also supported by EPSRC, SISSA, ETH, IAS, and IRMA (Strasbourg).
Appendix: Algebro-Geometric Darboux Coordinates Here we outline the construction of the so-called algebro-geometric Darboux coordinates. Our presentation follows [13]. However, the idea of constructing canonical coordinates for integrable systems by using the projections of the points in the divisor of a suitable normalized line–bundle on the spectral curve appeared for the first time in a paper by H. Flaschka and D.W. McLaughlin [16] where the special cases of the Toda system and KdV equation were dealt with. Later S.P.Novikov and A.P.Veselov [58] generalized the construction to any hyperelliptic spectral curve and introduced a general class of finite-and infinite-dimensional Poisson brackets. The Flaschka–McLaughlin construction was then generalized to generic rational Lax pairs by M.R. Adams, J. Harnad and J. Hurtubise [1]. A quantum version of this method was initiated by E.Sklyanin [55]. A construction of the polynomial 0 (z) equivalent to ours was given in [53, 20]. Our Theorem A.2 that enables to construct rational Darboux coordinates on the reduced symplectic leaves seems to be new (cf. however the recent paper [6] where a similar approach to constructing the Darboux coordinates was developed). Let us rewrite the characteristic polynomial (5.2) of the matrix A(z) =
n i=1
ˆ Ai A(z) , = &n z − ui i=1 (z − u i )
ˆ A(z) = −A∞ z n−1 + O(z n−2 ), z → ∞
(A.1)
(A.2)
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in the form n ' ˆ w) ˆ R(z, ˆ = det wˆ − det A(z) = (z − u i )
!m R(z, w), wˆ = w
i=1
n ' (z − u i ). i=1
(A.3) Expanding the determinant one obtains a polynomial
ˆ w) R(z, ˆ = wˆ m + αˆ 1 (z)wˆ m−1 + · · · + αˆ m (z) =
ai j z i wˆ j ,
(A.4)
i+(n−1) j≤m(n−1)
where ai j = ai j (A1 , . . . , An ; u 1 , . . . , u n )
(A.5)
are some polynomials in the entries of the matrices Ak and in u l , a0m = 1, αˆ s (z) =
s (n−1)
ai ,m−s z i , s = 0, 1, . . . , m.
i=0
It is well known that for a Zariski open subset in the projective space P M−1 , M = (n − 1)
m(m + 1) +m+1 2
(A.6)
with the homogeneous coordinates ai j , i + (n − 1) j ≤ m(n − 1), the affine algebraic curve
ˆ w) R(z, ˆ =
ai j z i wˆ j = 0
(A.7)
i+(n−1) j≤m(n−1)
is smooth. Indeed, it suffices to check smoothness of one of the curves of the above family, e.g. of wˆ m = z m(n−1) − 1. Under the smoothness assumption the standard compactification of (A.7) gives a compact Riemann surface of the genus g = (n − 1)
m(m − 1) −m+1 2
(A.8)
(cfr. the formula (5.10)). The infinite part of is a divisor D∞ of the degree m. If the normal Jordan form of the matrix A∞ contains k Jordan blocks of the multiplicities m 1 , . . . , m k then the divisor D∞ has the form D∞ = m 1 ∞1 + · · · + m k ∞k .
(A.9)
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Here ∞1 , . . . , ∞k ∈ are the points added at infinity. In particular, if the spectrum of A∞ is simple then the divisor D∞ is a sum of m distinct points ∞1 , …, ∞m : ∞k := {z → ∞, wˆ → ∞,
wˆ (∞) → −λk }, k = 1, . . . , m. z n−1
(A.10)
(∞) are the eigenvalues of the matrix A∞ . Here λ1(∞) , …, λm Let us give an intrinsic characterization of algebraic curves of the form (A.7). The coordinate functions z and wˆ have poles at the divisors D∞ and (n − 1)D∞ respectively. Conversely, the following simple statement can be proved by using standard arguments based on the Riemann - Roch theorem.
Lemma A.1. Let be a Riemann surface of the genus (A.8). Let D∞ be a divisor of the degree m on such that dim H 0 (, O(D∞ )) = 2, dim H 0 (, O((n − 1)D∞ )) = n + 1. Then the Riemann surface can be represented in the form (A.7). Proof. The first of the assumptions implies existence of a non-constant meromorphic function z with poles at the points of the divisor D∞ . The second one yields existence of another function wˆ with poles at (n − 1)D∞ that cannot be represented as a polynomial in z. Let us now consider the space H 0 (, O (m(n − 1)D∞ )). The M monomials z i wˆ j , i + (n − 1) j ≤ m(n − 1)
(A.11)
belong to this space. Let us prove that these monomials are linearly dependent. To this end let us compute the dimension dim H 0 (, O (m(n − 1)D∞ )). First of all, the degree of the divisor D := m(n − 1)D∞ equals deg D = m 2 (n − 1) = 2g − 2 + m(n + 1) > 2g − 2. So the Riemann - Roch theorem gives dim H 0 (, O (m(n − 1)D∞ )) = deg D − g + 1 = M − 1. This proves linear dependence of M functions of the form (A.11). The lemma is proved. Observe that for n > 1 the eigenvalues λ1(∞) , …, λ(∞) m are determined by the Riemann surface uniquely up to permutations and common affine transformations (∞)
λi
(∞)
→ aλi
+ b, i = 1, . . . , m.
We will say that our Riemann surface is D∞ -generic if the eigenvalues are pairwise distinct, (∞)
λi
(∞)
= λ j
, i = j.
Instead of using the coefficients ai j as the homogeneous coordinates in the space of algebraic curves (A.7) we will construct another system of coordinates on a Zariski
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open subspace in P M−1 . Let us choose n + g pairwise distinct numbers u 1 , …, u n , (i) γ1 , …, γg and also n + 1 m-tuples of pairwise distinct numbers λ(i) 1 , …, λm for every i = 1, . . . , n, ∞, λr(i) = λ(i) s , s = r, i = 1, 2, . . . , m, ∞, satisfying the constraint n m
λr(i) +
i=1 r =1
m
λr(∞) = 0.
(A.12)
r =1
Finally, choose arbitrary g numbers µ1 , …, µg . Denote ' (i) (u i − u j )λr(i) , r = 1, . . . , m, i = 1, . . . , n, λˆ r :=
(A.13)
j=i
µˆ s =
n ' (γs − u i )µs , s = 1, . . . , g.
(A.14)
i=1
Theorem A.2. For generic values of the parameters (i)
u 1 , . . . , u n , λ1 , . . . , λ(i) m , i = 1, . . . , n, ∞, γ1 , . . . , γg , µ1 , . . . , µg
(A.15)
ˆ w) satisfying the constraint (A.12) there exists a unique curve R(z, ˆ = 0 of the form (A.7) with a0m = 1 passing through the points (i)
(u i , λˆ r ), r = 1, . . . , m, i = 1, . . . , n, wˆ z, wˆ → ∞, n−1 → −λr(∞) , r = 1, . . . , m, z (γs , µˆ s ), s = 1, . . . , g.
(A.16) (A.17) (A.18)
Proof. Let us denote (i)
(i)
σk := σk (λ1 , . . . , λ(i) m ), k = 1, . . . , m (i) the value of the k th elementary symmetric function of λ(i) 1 , . . . , λm , i = 1, . . . , n, ∞. The equation of the algebraic curve must have the form
wˆ m − R(z)
n (i) σ1 wˆ m−1 z − ui
(A.19)
i=1
) ( m n (i) σk (∞) k n−n−k k k−1 [R (u i )] + σk z + pk n−n−k−1 (z) wˆ m−k = 0. +R(z) (−1) z − ui k=2
i=1
Here, as above R(z) :=
n ' i=1
(z − u i ),
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the polynomials pn−3 (z), p2n−4 (z), …, pm n−m−n−1 (z) labeled by their degrees are to be determined later. Such a curve will pass through the points (A.16), (A.17). To have it passing also through (A.18) the following system of equations must be satisfied: m
pk n−n−k−1 (γs )µˆ sm−k + Q(γs , µˆ s ) = 0, s = 1, . . . , g,
(A.20)
k=2
where ( n m n (i) (i) σk σ1 wˆ m m−1 − Q(z, w) ˆ := wˆ + [R (u i )]k−1 (−1)k R(z) z − ui z − ui i=1 k=2 i=1 (∞) k n−n−k wˆ m−k . +σk z This is a linear system for the g coefficients of the polynomials pn−3 (z), p2n−4 (z), …, pm n−m−n−1 (z). Let us prove that the determinant of this linear system does not vanish identically. Indeed, this determinant is equal to the following polynomial in γ1 , …, γg , µˆ 1 , …, µˆ g , Wm,n (γ1 , . . . , γg , µˆ 1 , . . . , µˆ g ) (−1)|π | (µˆ i1 . . . µˆ in−3 )m−2 (µˆ j1 . . . µˆ j2n−4 )m−3 (µˆ k1 . . . µˆ k3n−5 )m−4 . . . := π
×V (γi1 . . . γin−3 )V (γ j1 . . . γ j2n−4 )V (γk1 . . . γk3n−5 )V (γl1 , . . . γlm n−m−n−1 ). (A.21) Here the summation is over the partitions π : {1, 2, . . . , g} = {i 1 , . . . , i n−3 } { j1 , . . . , j2n−4 } ×{k1 , . . . , k3n−5 } · · · {l1 , . . . , lm n−m−n−1 },
(A.22)
|π | stands for the parity of the permutation π ∈ Sg , V (x1 , . . . , xk ) :=
'
(xi − x j )
1≤i< j≤k
is the Vandermonde determinant. It is clear that this polynomial is not an identical zero. The theorem is proved. We want now to show that the same data used in Theorem A.2 determine the matrix valued polynomial A(z) in the determinant representation (A.3). We will now prove that any generic curve of the form (A.7) can be represented in the determinant form (A.3). Actually, this can be done in many ways; we will describe the parameters of all determinant representations of . ˆ Let be the spectral curve (A.4) of a matrix A(z). Assuming smoothness of the spectral curve, we will associate with the determinant representation a degree g divisor D on . Let us first consider the eigenvector line bundle L on , ˆ A(z)ψ = wψ, ˆ
(A.23)
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ψ = (ψ1 , . . . , ψm )T . Define the divisor Dˆ := {ψ1 = 0}.
(A.24)
In other words, Dˆ is the divisor of poles of the meromorphic functions ψ2 /ψ1 , ψ3 /ψ1 , …, ψm /ψ1 . Lemma A.3. The degree of the divisor Dˆ is equal to deg Dˆ = g + m − 1. See [21] for a simple proof. Denote D = Dˆ ∩ \ D∞ .
(A.25)
Lemma A.4. The point (z 0 , wˆ 0 ) ∈ D only if 0 (z 0 ) = 0. Here the polynomial 0 (z) was defined in (5.9). Conversely, for any root z 0 of the polynomial 0 (z) there exists a point (z 0 , wˆ 0 ) ∈ D. Proof. For the convenience of the reader we will give here the proof. Rewriting the equation of the divisor in the form < b0 , ψ >= 0 (here < , > stands for the natural pairing between row- and column-vectors, the row-vector b0 was defined in (5.7)) we also derive that, for k = 1, . . . , m − 1, < bk (z 0 ), ψ >=< b0 , Ak (z 0 )ψ >= w0k < b0 , ψ >= 0, w0 =
wˆ 0 . R(z 0 )
The determinant of this linear homogeneous system must be equal to 0. This gives 0 (z 0 ) = 0. Conversely, let z 0 be a zero of 0 . Using the identity (5.25) we derive that the subspace Ker B(z 0 ) of the m-dimensional space is invariant w.r.t. the linear operator A(z 0 ). Here the matrix B(z) was defined in the line after the formula (5.7). Therefore there exists an eigenvector ψ ∈ Ker B(z 0 ), A(z 0 )ψ = w0 ψ, < b0 , ψ >= ψ1 = 0. By definition the point (z 0 , w0 ) belongs to the divisor D ∈ . The lemma is proved. Let us now compute the degree of the polynomial 0 (z). Let ˆ A(z) = −A∞ z n−1 + C z n−2 + O(z n−3 ). Explicitly, the matrix C = (Ci j ) reads C = A∞ u¯ +
n i=1
u i Ai , u¯ =
n i=1
ui .
(A.26)
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Lemma A.5. The polynomial 0 (z) has the form ' (∞) (∞) (λi − λ j ) z g + O(z g−1 ), 0 (z) = (−1)m−1 C12 C13 . . . C1m 2≤i< j
where g is given by the formula (A.8). Proof. For the j th coordinate of the row-vector bˆk := b0 Aˆ k−1 (z) (cf. (5.7) one obtains ) ( (∞) (∞) k (λ1 )k − (λ j )k (∞) k k(n−1) k(n−1)−1 λ1 (bˆk ) j = (−1) δ1 j z − C1 j z + ··· , (∞) (∞) λ1 − λ j where dots stand for the terms of lower order in z. Computing the determinant of this matrix we obtain the proof of the needed formula. Corollary A.6. If the eigenvalues of the matrix A∞ are pairwise distinct and all the elements of the first row of the matrix (A.26) are not equal to zero then the degree of the divisor D is equal to g. The remaining points of the divisor Dˆ are at infinity, Dˆ − D = ∞2 + ∞3 + · · · + ∞m . The statement of the corollary is a formalization of the following asymptotic behavior ˆ of the eigenvectors ψ = (ψ1 , . . . , ψm )T of the matrix A(z) at z → ∞: wˆ 1 (∞) , z → ∞, n−1 → −λ j . ψk = δk j + O z z Such a normalized eigenvector will have g + m − 1 poles on \ D∞ . Under the above assumptions the first component ψ1 has simple zeroes at the points ∞2 , . . . , ∞m . Remark A.7. We observe that, as it was shown in the proof of Lemma 4.20, the element Ci j of the matrix (A.26) is identically equal to zero for an isomonodromic deformation Ak Ak (u 1 , . . . , u n ), if and only if (∞)
(1 − λi
(∞)
+ λj
)Aki j = −A∞i j ,
that is either 1 − λi(∞) + λ(∞) = 0 or Aki j = 0 for all k. j Denote γs = γs (A), µs = µs (A), s = 1, . . . , g,
(A.27)
the coordinates of the points of the divisor D, D = (γ1 , µˆ 1 ) + · · · + (γg , µˆ g ), µˆ s = µs
n ' (γs − u i ). i=1
(A.28)
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We want to show that, given the generic values of the functions γs (A), µs (A), s = 1, . . . , g together with the numbers u 1 , …, u n and the pairwise distinct eigenvalues λ(i) 1 , (i) …, λm , i = 1, . . . , n, ∞ satisfying the constraint (A.12), one can uniquely determine the conjugacy class of the n-tuple of matrices A = (A1 , . . . , An ) modulo diagonal conjugations and permutations. To this end we will prove, essentially using the technique of [11] the converse statement that shows that, for a Zariski open subset in the space P M−1 of algebraic curves of the form (A.7), the curve can be represented in the determinant form. We will also describe parameters of such determinant representations of a given curve. Let D be a divisor of the degree g on the Riemann surface . We will say that the divisor is D∞ -non-special if dim H 0 (, O(D + ∞i − ∞1 )) = 1, i = 1, . . . , m.
(A.29)
ˆ w) Theorem A.8. Any smooth affine curve R(z, ˆ = 0 of the form (A.7) can be repˆ resented in the determinant form (A.4) for a matrix A(z) of the form (A.2). For a D∞ -generic curve such representations, considered modulo diagonal conjugations ˆ ˆ K , K = diag(k1 , . . . , km ) A(z) → K −1 A(z) and permutations of coordinates ˆ ˆ A(z) → P −1 A(z) P, P ∈ Sm−1 preserving the vector (1, 0, . . . , 0) are in one-to-one correspondence with the degree g D∞ -non-special divisors D on . Proof. Let us order the infinite points of and choose nonzero sections ψk ∈ H 0 (, O(D + ∞k − ∞1 )), k = 2, . . . , m.
(A.30)
They are determined uniquely up to constant factors. Introduce a vector valued meromorphic function on putting ψ = (1, ψ2 , . . . , ψm )T . Introduce the m × m matrix (z) = ( k j (z)) of Laurent series in 1/z expanding the functions ψk near the infinite points ∞ j ∈ . Let W (z) = diag(W1 (z), . . . , Wm (z)) be the diagonal matrix obtained by taking the Laurent series of the function wˆ on near ∞1 , . . . , ∞m . Define a matrix of polynomials ˆ A(z) := (z)W (z) −1 (z) . (A.31) +
Here ( )+ means the polynomial part in z of the expansion. By construction m(n−1)−1 ˆ ). A(z) = −z m(n−1) diag(λ1(∞) , . . . , λ(∞) m ) + O(z
Let us prove that the vector function ψ on satisfies ˆ A(z)ψ = wˆ ψ.
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This can be done using the standard arguments of Krichever’s scheme [35]. Indeed, by construction all the components of the difference ˆ wψ ˆ − A(z)ψ are analytic at the infinite points ∞1 , . . . , ∞m . Therefore they have poles only at the points of the divisor D. Due to nonspeciality of D the difference is equal to zero. We have proved that coincides with the spectral curve of the polynomial matrix ˆ A(z). By construction the divisor D we started with coincides with the one defined above. It remains to observe that, choosing another basic section ψ˜ k ∈ H 0 (, O(D + ∞k − ∞1 )), ψ˜ k = ck ψk , k = 2, . . . , m, ˆ yields the diagonal conjugation of the polynomial matrix A(z), ˆ ˆ C −1 , C = diag(1, c2 , . . . , cm ). A(z) → C A(z) Moreover, changing the order of the infinite points preserving ∞1 implies a permutation. The theorem is proved. Corollary A.9. The map [A(z)] → S pec
(A.32)
is a birational isomorphism of the space of classes of equivalence of rational matrixvalued functions of the form A(z) =
n i=1
Ai , A∞ = −(A1 + · · · + An ) z − ui
with diagonal A∞ considered modulo diagonal conjugations and the space of spectral data with the coordinates u 1 , . . . , u n , Spec A1 , . . . , Spec An , Spec A∞ , γ1 , µ1 , . . . , γg , µg ∈ S pec. (A.33) In particular, (γ1 , µ1 , . . . , γg , µg ) are coordinates on the reduced symplectic leaves of the Poisson bracket (3.2). We will now prove that γi = γi (A), µi = µi (A), i = 1, . . . , g are canonical coordinates on the reduced symplectic leaves of the Poisson bracket. It will be convenient to represent the Poisson bracket (3.2) in the following well known r -matrix form (see [15] regarding the definitions and notations). Lemma A.10. The Poisson bracket (3.2) can be represented in the form A(z 1 ) ⊗ A(z 2 ) = [A(z 1 ) ⊗ + ⊗ A(z 2 ), r (z 1 − z 2 )] , ,
(A.34)
where r (z) is a classical r -matrix, i.e. a solution of the linearized Yang – Baxter equation, given by j
jl
rik (z) =
δil δk . z
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Equivalently, (A.34) reads
Aij (z 1 ), Alk (z 2 ) =
1 δli Akj (z 1 ) − Akj (z 2 ) − Ali (z 1 ) − Ali (z 2 ) δ kj . z1 − z2 (A.35)
Proof. By the definition
ik A(z 1 ) ⊗ A(z 2 ) : ,
jl
p
A p kj δli − A p li δ kj (z 1 − u p )(z 2 − u p )
.
So 1 1 1 A p kj δli − A p li δ kj = − (z 1 − u p )(z 2 − u p ) (z 2 − z 1 ) p z 1 − u p z2 − u p p 1 Akj (z 1 )δli − Ali (z 1 )δ kj + Ali (z 2 )δ kj − Akj (z 2 )δli = (z 2 − z 1 ) = [A(z 1 ) ⊗ + ⊗ A(z 2 ), r (z 1 − z 2 )]ikjl .
A p kj δli − A p li δ kj
This concludes the proof.
(k)
(k)
Theorem A.11. The functions λi = λi (A), i = 1, . . . , m, k = 1, . . . , n, ∞, γi = γi (A), µi = µi (A), g = 1, . . . , g on the space of m × m matrices (A1 , . . . , An ) =: A have the following canonical Poisson brackets w.r.t. the structure (A.35): {γi , µ j } = δi j , all other Poisson brackets vanish. (k)
Proof. We already know that the eigenvalues λi of the matrices Ak are Casimirs of the Poisson bracket. It remains to compute the Poisson brackets of the functions µi (A) and γ j (A). Let us introduce the following notations. Let z not be a ramification point for the Riemann surface . Let us fix some ordering of the sheets of the Riemann surface. Denote |a >, a = 1, . . . , m the basis of eigenvectors of the matrix A = A(z), A|a >= wa |a >
(A.36)
< b0 |a >= 1, b0 = (1, 0, . . . , 0).
(A.37)
normalized by the condition
Here wa = wa (z), a = 1, . . . , m
Canonical Structure and Symmetries of the Schlesinger Equations
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are the roots of the characteristic equation det(w − A(z)) = 0. Denote < a| the dual basis of row-vectors < a|b >= δab .
(A.38)
Due to (A.37) one has b0 =
m
< a|.
(A.39)
a=1
Lemma A.12. The following formulae for the Poisson brackets hold true: {log 0 (z), log 0 (z )} = 0, {wa (z), wb (z )} = 0, 1 {log 0 (z), wc (z )} = < a|c >< c |b > . (z − z )
(A.40) (A.41) (A.42)
a=b
In this formulae the primes mean that the corresponding function is computed at the point z , i.e. A(z )|c >= wc |c > where wc = wc (z ). Proof. The following well known variational formulae will be useful in the computations of the Poisson brackets δwa =< a|δ A|a >, < b|δ A|a > < b|δa >= , b = a, wa − wb < b|δ A|a > . < a|δa >= wb − wa
(A.43) (A.44) (A.45)
b=a
Here |δa > is the variation of the eigenvector |a >. In the derivation of the last formula we have used the normalization (A.37). Denote (z) the matrix with the columns |1 >, . . . , |m >. The rows of the inverse matrix −1 coincide with the bra-vectors < 1|, . . . , < m|. From (A.39) it easily follows that the matrix B(z) is equal to the product of the Vandermonde matrix of the pairwise distinct numbers w1 , . . . , wm by −1 (z). So ' det B(z) = (wi − w j )det −1 (z). i< j
Using the Liouville formula δ log(det ) = tr −1 δ =
m
< a|δa >
a=1
yields δ log 0 (z) =
1 < a|δ A|a > − < b|δ A|b > − < a|δ A|b > . 2 wa − wb a=b
(A.46)
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It is understood that the values of the variables u i are fixed during the variation. From the above formulae for δ0 (z), δwc (z ) we derive, by a straightforward calculation, the brackets (A.40) – (A.42). The lemma is proved. Proof of the theorem. From (A.40) and from the representation δ log 0 (z) = −
m δγi + regular terms z − γi i=1
it easily follows that {γi , γ j } = 0. The commutation rule {µi , µ j } = 0 follows from (A.41). Let us compute the brackets {γi , µ j }. Due to Theorem A.8 we may assume that the projections z = γi of the points of the divisor D onto the z-plane are all pairwise distinct, they are distinct from u j and from the ramification points of the Riemann surface (cf. Assumption 2 above). Consider first the case j = i. Assume that the numeration of the sheets of the Riemann surface at the neighborhoods of the points z = γi and z = γ j is done in such a way that the pole of the eigenvector ψ of the matrix A(z) belongs to the sheet labeled by c. That means that the ket-vector |c > has a simple pole at z → γi , |c >=
|c˜ > + O(1), z → γi . z − γi
(A.47)
All other ket- and bra-vectors are analytic in z near this point and the corresponding bra-vector < c | has a simple zero < c | = (z − γi ) < c| (A.48) ˜ + O (z − γi )2 . From the already proven commutation rule of the coordinates γi it follows that {γi , µ j } = − lim lim (z − γi ){log 0 (z), wc (z )} z→γi z →γ j
=−
1 lim lim (z − γi ) < a|c >< c |b > . z→γ 2(γi − γ j ) i z →γ j a=b
Due to (A.47), (A.48) the r.h.s. is analytic at the point z = γ j . The singularity at z = γi can come only from the terms with b = c. So, the singular part in the sum equals < a|c >< c |b >= < a|c >< c |c > + regular. a=c
a=b
Using the z-independent normalization (A.37) we rewrite the singular term in the form < a|c >< c |c >= − < c|c >< c |c > . a=c
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Using again (A.47), (A.48) we establish analyticity of the last expression also at z = γi . Therefore {γi , µ j } = 0 for i = j. To compute {γi , µi } we will first calculate the limit lim {log 0 (z), wc (z )}.
z →z
Observe that, for z = z the numerator of formula (A.42) vanishes ⎛ ⎞ ⎝ < a|c >< c |b >⎠ = < a|c >< c|b >= δac δcb = 0. a=b
a=b
z =z
a=b
So the needed limit is equal to the derivative ⎛ ⎞ d lim {log 0 (z), wc (z )} = − ⎝ < a|c >< c |b >⎠ dz z →z a=b
. z =z
Let us denote |c˙ >:=
d |c > |z =z , dz
< c| ˙ :=
d < c | |z =z . dz
We obtain, using again the z -independent normalization (A.37), ⎞ ⎛ d ⎝ ⎠ < a|c >< c |b > = < a| c ˙ > + < c|b ˙ > dz a=c a=b b=c z =z < c|b ˙ >. = − < c|c˙ > + b=c
Therefore
⎡ {γi , µi } = lim (z − γi ) ⎣− < c|c˙ > + z→γi
⎤ < c|b ˙ >⎦ .
b=c
The last term in the brackets is analytic at the point z = γi . For the first one we obtain, using (A.47), (A.48), < c|c˙ >= −
< c| ˜ c˜ > + regular terms. z − γi
The last step is to use the normalization < c|c >≡ 1 to derive that < c| ˜ c˜ >= 1. Hence {γi , µi } = 1. The theorem is proved.
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References 1. Adams, M.R., Harnad, J., Hurtubise, J.: Darboux coordinates and Liouville–Arnold integration in loop algebras. Commun. Math. Phys. 155, 385–413 (1933) 2. Andreev, F.V., Kitaev, A.V.: Transformations RS42 (3) of the ranks ≤ 4 and algebraic solutions of the sixth Painlevé equation. Commun. Math. Phys. 228(1), 151–176 (2002) 3. Anosov, D.V., Bolibruch, A.A.: The Riemann-Hilbert Problem. Volume E 22, Aspects of Mathematics, Braunsdiweig: Friedrich Vieweg & Sohn Verlag (1994) 4. Arinkin, D., Lysenko, S.: On the moduli of SL(2)-bundles with connections on P1 \ {x1 , . . . , x4 }. Int. Math. Res. Not. 1997(19), 983–999 (1977) 5. Audin, M.: Lectures on gauge theory and integrable systems. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 488, Dordrecht: Kluwer, 1997 6. Babelon, O., Talon, M.: Riemann surfaces, separation of variables and classical and quantum integrability. Phys.Lett. A 312, 71–77 (2003) 7. Bolibruch, A.A.: The 21-st Hilbert problem for linear Fuchsian systems. In: Developments in mathematics: the Moscow school. London: Chapman and Hall, 1993 8. Bolibruch, A.A.: On isomonodromic deformations of Fuchsian systems. J. Dynam. Control Systems 3, 589–604 (1997) 9. Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York-Toronto-London: McGraw-Hill Book Company, Inc., 1955 10. Costin, O., Costin, R.D.: Asymptotic properties of a family of solutions of the Painlevé equation VI. Int. Math. Res. Not. 22, 1167–1182 (2002) 11. Dubrovin, B.: Matrix finite-gap operators. J. Soviet Math. 28, 20–50 (1985) 12. Dubrovin, B.: Geometry of 2D topological field theories. Volume 1620 of Springer Lecture Notes in Math. Integrable Systems and Quantum Groups, M. Francaviglia, S. Greco, eds. Berlin-HeidelbergNew York:Springer 1996 13. Dubrovin, B., Diener, P.: Algebro-geometrical Darboux coordinates in R-matrix formalism. Preprint 88/94/FM, 1994 14. Dubrovin, B., Mazzocco, M.: Monodromy of certain Painlevé-VI transcendents and reflection groups. Invent. Math. 141, 55–147 (2000) 15. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian methods in the theory of solitons. Springer Series in Soviet Mathematics, Berlin: Springer-Verlag, 1987 16. Flaschka, H., McLaughlin, D.W.: Cononically conjugate variables for the Korteweg–de Vries equation and the Toda lattice with periodic boundary conditions. Progr. Theor. Phys. 55, 438–456 (1976) 17. Fuchs, R.: Lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63, 301–321 (1907) 18. Garnier, R.: Sur des équations différentielles du troisième ordre dont l’intégrale générale est uniforme et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale a ses points critiques fixes. Ann. Sci. École Norm. Sup. 29(3), 1–126 (1912) 19. Garnier, R.: Solution du probleme de Riemann pour les systemes différentielles linéaires du second ordre. Ann. Sci. École Norm. Sup. 43, 239–352 (1926) 20. Gekhtman, M.: Separation of variables in the classical SL(N) magnetic chains. Commun. Math. Phys. 167, 593–605 (1995) 21. Griffiths, P.A.: Linearizing flows and a cohomological interpretation of Lax equations. Math. Sci. Res. Inst. Publ. 2, 36–46 (1984) 22. Guzzetti, D.: The elliptic representation of the general Painlevé VI equation. Comm. Pure Appl. Math. 55(10), 1280–1363 (2002) 23. Harnad, J.: Quantum isomonodromic deformations and the Knizhnik–Zamolodchikov equations. In: Symmetries and integrability of difference equations (Esttrel, PQ, 1994), CRM Lecture Notes 9, Providence, RI: Amer. Math.Soc., 1996 24. Hitchin, N.: Frobenius manifolds (with notes by David Calderbank). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 488, 69–112 (1997) 25. Hitchin, N.: A lecture on the octahedron. Bull. London Math. Soc. 35(5), 577–600 (2003) 26. Ince, E.L.: Ordinary differential equations. New York: Dover Publications INC., 1956 27. Its, A.R., Novokshenov, V.Yu.:The isomonodromic deformation method in the theory of Painlevé equations, volume 1191 of Lecture notes in mathematics. Berlin-Heidelberg-New York: Springer, 1980 28. Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé, a Modern Theory of Special Functions. Volume E 16, Aspects of Mathematics, Branschweig: Friedrich Vieweg # sohn, 1991 29. Jimbo, M.: Monodromy problem and the boundary condition for some Painlevé equations. Publ. Res. Inst. Math. Sci. 18, 1137–1161 (1982) 30. Jimbo, M., Miwa, T.:Monodromy preserving deformations of linear ordinary differential equations with rational coefficients II. Physica 2D, 2(3), 407–448 (1981)
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31. Jimbo, M., Miwa, T.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients III. Physica 2D 4(1), 26–46 (1982) 32. Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients I. Physica 2D 2(2), 306–352 (1981) 33. Katz, N.: Rigid Local Systems. Volume 139 of Ann. Math. Studies. Princeton, NJ: Princeton University Press, 1996 34. Kimura, H., Okamoto, K.: On the isomonodromic deformation of linear ordinary differential equations of higher order. Funkcial. Ekvac. 26(1), 37–50 (1983) 35. Krichever, I.M.: Methods of algebraic geometry in the theory of non-linear equations. Russ. Math. Surv. 32, 185–213 (1977) 36. Krichever, I.M.: Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations. Mosc. Math. J. 2(4), 717–806 (2002) 37. Levelt, A.H.M.:Hypergeometric functions. Doctoral thesis, University of Amsterdam, 1961 38. Malgrange, B.: Sur les déformations isomonodromiques I. Singularités régulières. Volume 37 of Mathematics and Physics. Progr. Math. Boston: Birkhauser 1983 39. Manin, Yu.I.: Sixth Painlevé equation, universal elliptic curve, and mirror of P2 . Amer Math Soc Transl. Ser. 2, 186, 131–151 (1998) 40. Manin, Yu.I.: Frobenius manifolds, quantum cohomology and moduli spaces. Colloquium Publ. Volume 47, Providence, RI: Amer. Math. Soc. (1999) 41. Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Mathematical Phys. 5(1), 121–130 (1974) 42. Mazzocco, M.: Picard and Chazy solutions to the PVI equation. Math. Ann. 321(1), 131–169 (2001) 43. Mazzocco M.: Rational solutions of the Painlevé VI equation. J. Phys. A: Math. Gen. 34, 2281– 2294 (2001) 44. Miwa, T.: Painlevé property of monodromy preserving equations and the analyticity of τ -functions. Publ. RIMS 17, 703–721 (1981) 45. Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16, 197–220 (1975) 46. Ohtsuki, M.: On the number of apparent singularities of a linear differential equation. Tokyo J. Math. 5, 23–29 (1982) 47. Okamoto, K.: Isomonodromic deformations, Painlevé equations and the Garnier system. J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math. 33, 576–618 (1986) 48. Okamoto, K.: Studies on the Painlevé equations I, sixth Painlevé equation. Ann. Mat. Pura Appl. 146, 337– 381 (1987) 49. Okamoto, K.: Painlevé equations and Dynkin diagrams. In: Painlevé Transcendents, London: Plenum, 1992, pp. 299–313 50. Okamoto, K., Kimura, H.: On Particular solutions of the Garnier system and the hypergeometric functions of several variables. Quart. J. Math. Oxford 37, 61–80 (1986) 51. Reshetikhin, N.: The Knizhnik–Zamolodchikov system as a deformation of the isomonodromy problem. Lett. Math. Phys. 26, 167–177 (1992) 52. Schlesinger, L.: Ueber eine Klasse von Differentsial System Beliebliger Ordnung mit Festen Kritischer Punkten. J. Fur Math. 141, 96–145 (1912) 53. Scott, D.R.D.: Classical functional Bethe ansatz for SL(N): separation of variables for the magnetic chain. J. Math. Phys. 35, 5831–5843 (1994) 54. Sibuya, Y.: Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation, Volume 82 of Trans. of math. Monographs. Providence, RI: Amer Math. Soc., 1990 55. Sklyanin, E.K.: Separation of variables in Gaudin model. J.Soviet Math. 47, 2473–2488 (1989) 56. Tsuda, T.: Universal characters and integrable systems. PhD thesis, Tokyo Graduate School of Mathematics, 2003 57. Umemura, H.: Irreducibility of the first differential equation of Painlevé. Nagoya Math. J. 117, 231–252 (1990) 58. Veselov, A.P., Novikov, S.P.: Poisson brackets and complex tori. Trudy Mat. Inst. Steklov. 165, 49–61 (1984) Communicated by L. Takhtajan
Commun. Math. Phys. 271, 375–385 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0199-1
Communications in
Mathematical Physics
The Classification of Non-Local Chiral CFT with c < 1 Yasuyuki Kawahigashi1, , Roberto Longo2,, , Ulrich Pennig3, , Karl-Henning Rehren3, 1 Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan.
E-mail: [email protected]
2 Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1,
I-00133 Roma, Italy. E-mail: [email protected]
3 Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen,
Germany. E-mail: [email protected], [email protected]. uni-goe.de Received: 23 May 2005 / Accepted: 29 August 2006 Published online: 6 February 2007 – © Springer-Verlag 2007
Dedicated to Hans-Jürgen Borchers on the occasion of his 80th birthday Abstract: All non-local but relatively local irreducible extensions of Virasoro chiral CFTs with c < 1 are classified. The classification, which is a prerequisite for the classification of local c < 1 boundary CFTs on a two-dimensional half-space, turns out to be 1 to 1 with certain pairs of A-D-E graphs with distinguished vertices. 1. Introduction Non-local chiral conformal quantum field theories have gained renewed interest because they give rise to local CFT on the two-dimensional Minkowski half-space x > 0 (boundary CFT, BCFT), and vice versa [10]. More precisely, a BCFT contains chiral fields which generate a net A of local algebras on the circle, such that A+ (O) = A(I ) ∨ A(J ) are the chiral BCFT observables localized in the double cone O = I × J ≡ {(t, x) : t + x ∈ I, t − x ∈ J }, where I > J are two open intervals of the real axis (= the pointed circle). The two-dimensional local fields of the BCFT define a net of inclusions A+ (O) ⊂ B+ (O) subject to locality, conformal covariance, and certain irreducibility requirements. If A is assumed to be completely rational [7], then there is a 1 to 1 correspondence [10] between Haag dual BCFTs associated with a given chiral net A, and non-local chiral extensions B of A such that the net of inclusions A(I ) ⊂ B(I ) is covariant, irreducible and relatively local, i.e., A(I ) commutes with B(J ) if I and J are disjoint. The correspondence is given by the simple relative commutant formula B+ (O) = B(K ) ∩ B(L), Supported in part by JSPS.
Supported in part by EU network “Quantum Spaces - Noncommutative Geometry” HPRN-CT-2002-
00280.
Supported in part by GNAMPA and MIUR.
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where O = I × J as before, K is the open interval between I and J , and L is the interval spanned by I and J . Conversely, B+ (O). B(L) = I ⊂L , J ⊂L , I >J
BCFTs which are not Haag dual are always intermediate between A+ and a Haag dual BCFT. The classification of local BCFTs on the two-dimensional half-space is thus reduced to the classification of non-local chiral extensions, which in turn [9] amounts to the classification of Q-systems (Frobenius algebras) in the C ∗ tensor category of the superselection sectors [2] of A. The chiral nets A = Vir c defined by the stress-energy tensor (Virasoro algebra) with c < 1 are known to be completely rational, so the classification program just outlined can be performed. Local chiral extensions of Vir c with c < 1 have a direct interpretation as local QFT models of their own. Their classification has been achieved previously ([5], see Remark 2.1) by imposing an additional condition [9] on the Q-system involving the braided structure (statistics [2]) of the tensor category. Of course, the present non-local classification contains the local one. As in [5] we exploit the fact that the tensor subcategories of the “horizontal” and of the “vertical” superselection sectors of Vir c with c < 1 are isomorphic with the tensor categories of the superselection sectors of SU (2) current algebras. (The braiding is different, however.) We therefore first classify the Q-systems in the latter categories (Sect. 1), and then proceed from Q-systems in the subcategories to Q-systems in the tensor categories of all sectors of Vir c (Sect. 2). Thanks to a cohomological triviality result [6], the classification problem simplifies considerably, and essentially reduces to a combinatorial problem involving the Bratteli diagrams associated with the local subfactors A(I ) ⊂ B(I ), combined with a arithmetic argument concerning Perron-Frobenius eigenvalues. In the last section, we determine the vacuum Hilbert spaces of the non-local extensions and of the associated BCFT’s thus classified. 2. Classification of Irreducible Non-Local Extensions of the SU(2) k -Nets As an easy preliminary, we first classify all irreducible, possibly non-local, extensions of the SU (2)k -nets on the circle. Consider the representation category of the SU (2)k net and label the irreducible DHR sectors as λ0 , λ1 , λ2 , . . . , λk as usual. (The vacuum sector is labeled as λ0 .) Label this net as A and an irreducible extension as B. Since A is completely rational in the sense of [7], the index [B : A] is automatically finite by [5, Prop. 2.3]. We need to classify the irreducible B-A sectors B ι A , where ι is the inclusion map. Note that the A-A sector A ι¯ι A gives the dual canonical endomorphism of the inclusion and this decomposes into a direct sum of λ j by [9]. Suppose we have such a sector B ι A , and consider the following sequence of commuting squares: End( A id A ) ⊂ End( A λ1 A ) ⊂ End( A λ21 A ) ⊂ End( A λ31 A ) ⊂ · · · , ∩ ∩ ∩ ∩ End( B ι A ) ⊂ End( B ιλ1 A ) ⊂ End( B ιλ21 A ) ⊂ End( B ιλ31 A ) ⊂ · · · . The Bratteli diagram of the first row arises from reflections of the Dynkin diagram Ak+1 as in usual subfactor theory, that is, it looks like Fig. 1, where each vertex is
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Fig. 1. The Bratteli diagram of the first row
Fig. 2. The Bratteli diagram of the second row
labeled with irreducible sectors appearing in the irreducible decomposition of λk1 for k = 0, 1, 2, . . . . (See [3, Sect. 9.6] for appearance of such graphs in subfactor theory.) The Bratteli diagram of the second row also arises from reflections of some Dynkin diagrams having the same Coxeter number as Ak+1 and starts with a single vertex because of the irreducibility of B ι A . This gives a bipartite graph G, one of the A-D-E Dynkin diagrams, and its initial vertex v as an invariant of ι, but the vertex v is determined only up to the graph automorphism, so we denote the orbit of a vertex v under such automorphisms by [v]. (Note that the Dynkin diagrams An , Dn , and E 6 have non-trivial graph automorphisms of order 2.) Also note that the graph G is bipartite by definition. A B-A sector corresponding to an even vertex of G might be equivalent to another B-A sector corresponding to an odd vertex of G. Later it turns out that this case does not occur in the SU (2)k -case, but it does occur in the Virasoro case below. Figure 2 shows an example of G and v where G is the Dynkin diagram E 6 . Theorem 2.1. The pair (G, [v]) gives a complete invariant for irreducible extensions of nets SU (2)k , and an arbitrary pair (G, [v]) arises as an invariant of some extension. Proof. As in the proof of [1, Prop. A.3], we know that the paragroup generated by ι is uniquely determined by (G, [v]) and it is isomorphic to the paragroup of the Goodmande la Harpe-Jones subfactor given by (G, [v]), which was defined in [4, Sect. 4.5]. Then we obtain uniqueness of the Q-system for the extension, up to unitary equivalence, as in [6, Theorem 5.3]. (We considered only a local extension of SU (2)28 corresponding to E 8 and its vertex having the smallest Perron-Frobenius eigenvector entry there, but the same method works for any (G, [v]).)
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Any combination of (G, [v]) is possible as in the proof of [1, Lemma A.1]. (We considered only the case of E 7 and its vertex having the smallest Perron-Frobenius eigenvector entry there, but the same method works for any (G, [v]). Remark 2.1. Note that the pair (G, [v]) uniquely corresponds to an (isomorphism class of) irreducible Goodman-de la Harpe-Jones subfactor. So we may say that irreducible extensions of nets SU (2)k are labeled with irreducible Goodman-de la Harpe-Jones subfactors N ⊂ M such that the inclusions A(I ) ⊂ B(I ) of the localized algebras are isomorphic to N ⊂ M tensored with a type III factor. 3. Classification of Non-Local Extensions of the Virasoro Nets Vir c with c < 1 Let A be the Virasoro net with central charge c < 1. As studied in [5, Sect. 3], it is a completely rational net. We would like to classify all, possibly non-local, irreducible extensions of this net. For c = 1 − 6/m(m + 1), m = 3, 4, 5, . . . , we label the irreducible DHR sectors of the net A as follows. We have σ j,k , j = 0, 1, . . . , m − 2, k = 0, 1, . . . , m − 1, with identification of σ j,k = σm−2− j,m−1−k . (Our notation σ j,k corresponds to λ j+1,k+1 in [5, Sect. 3].) Our labeling gives that the identity sector is σ0,0 and the statistical dimensions of σ1,0 and σ0,1 are 2 cos(π/m) and 2 cos(π/(m + 1)), respectively. We have m(m − 1)/2 irreducible DHR sectors. We again need to classify the irreducible B-A sectors B ι A , where ι is the inclusion map. Take such B ι A for a fixed Vir c with c = 1 − 6/m(m + 1) and we obtain an invariant as follows. Consider the following sequence of commuting squares as in Sect. 2: 2 3 End( A id A ) ⊂ End( A σ1,0 A ) ⊂ End( A σ1,0 A ) ⊂ End( A σ1,0 A ) ⊂ · · · , ∩ ∩ ∩ ∩ 2 3 End( B ι A ) ⊂ End( B ισ1,0 A ) ⊂ End( B ισ1,0 A ) ⊂ End( B ισ1,0 A ) ⊂ · · · .
From the Bratteli diagram of the second row, we obtain a graph G 1 and its vertex v1 as in Sect. 2. The graph G 1 is one of the A-D-E Dynkin diagrams and has the Coxeter number m. We also use σ0,1 instead of σ1,0 in this procedure and obtain a graph G 2 and its vertex v2 . The graph G 2 is one of the A-D-E Dynkin diagrams and has the Coxeter number m + 1. The quadruple (G 1 , [v1 ], G 2 , [v2 ]) is an invariant for ι. (The notation [·] means the orbit under the graph automorphisms as in Sect. 2.) Note that one of the graphs G 1 , G 2 must be of type A because the D and E diagrams have even Coxeter numbers. Henceforth, it is more convenient to replace the pair m, m + 1 by the pair m, m where m is odd and m = m + 1 or m = m − 1. In the latter case, we only need to switch the symmetric roles of G 1 and G 2 . We then prove the following classification theorem. Theorem 3.1. The quadruple (G 1 , [v1 ], G 2 , [v2 ]) gives a complete invariant for irreducible extensions of nets Vir c , and every quadruple, subject to the conditions on the Coxeter numbers as above, arises as an invariant of some extension. We will distinguish certain sectors by their dimensions. For this purpose, we need the following technical lemma on the values of dimensions, which we prove before the proof of the above theorem. Lemma 3.1. Let m be a positive odd integer and G one of the A-D-E Dynkin diagrams having a Coxeter number m with |m − m| = 1. Take a Perron-Frobenius eigenvector
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(µa )a for the graph G, where a denotes a vertex of G. Set d j = sin( jπ/m)/ sin(π/m) for j = 1, 2, . . . , m − 1. Then the sets {µa /µb | a, b are vertices of G} and {d2 , d3 , . . . , dm−2 } are disjoint. Proof. If m = 1, 3, then the latter set is empty, so we may assume m ≥ 5. Note that the value 1 is not in the latter set. Suppose a number ω is in the intersection and we will derive a contradiction. Then ω is in the intersection of the cyclotomic fields Q(exp(πi/m)) and Q(exp(πi/m )). Since m is odd, we have Q(exp(πi/m)) = Q(exp(2πi/m)), then we have Q(exp(2πi/m)) ∩ Q(exp(πi/m )) = Q by [8, Chap. VI, Cor. 3.2] because (m, 2m ) = 1. Suppose d j is equal to this ω. We may and do assume 2 ≤ j ≤ (m − 1)/2. We have ω=
ζ j − ζ−j = ζ j−1 + ζ j−3 + ζ j−5 + · · · + ζ − j+1 , ζ − ζ −1
where ζ = exp(2πi/(2m)). First assume that j is even. We note (m − 2, 2m) = 1 since m is odd. Then the map σ : ζ k → ζ k(m−2) for k = 0, 1, . . . , 2m − 1 gives an element of the Galois group for the cyclotomic extension Q ⊂ Q(ζ ). We have σ (ω) = ζ ( j−1)(m−2) + ζ ( j−3)(m−2) + ζ ( j−5)(m−2) + · · · + ζ (− j+1)(m−2) . Here the set {ζ ( j−1)(m−2) , ζ ( j−3)(m−2) , ζ ( j−5)(m−2) , . . . , ζ (− j+1)(m−2) } has j distinct roots of unity containing ζ m−2 and it is a subset of Z = {ζ k | k = 1, 3, 5, . . . , 2m − 1}. The set {ζ j−1 , ζ j−3 , ζ j−5 , . . . , ζ − j+1 } is the unique subset having j distinct elements of Z that attains the maximum of j Re k=1 αk among all subsets {α1 , α2 , . . . , α j } having j distinct elements of Z . However, we have m > 3, which implies j ≤ (m − 1)/2 < m − 2, thus the complex number ζ m−2 is not in the above unique set, and thus the sum ζ ( j−1)(m−2) + ζ ( j−3)(m−2) + ζ ( j−5)(m−2) + · · · + ζ (− j+1)(m−2) cannot be equal to ζ j−1 + ζ j−3 + ζ j−5 + · · · + ζ − j+1 , which shows that ω is not fixed by σ , so ω is not an element of Q, which is a contradiction.
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Fig. 3. The graphs G 1 , G 2 , G
Next we assume that j is odd. We now have that ζ 2( j−1)/2 + ζ 2( j−3)/2 + · · · + ζ 2(1− j)/2 ∈ Q. Since (m, (m − 1)/2) = 1, the map σ : ζ 2k → ζ k(m−1) for k = 0, 1, . . . , m − 1 gives an element of the Galois group for the cyclotomic extension Q ⊂ Q(ζ 2 ). Since m − 1 > j − 1, σ (ω) contains a term ζ m−1 which does not appear in ω. Then by an argument similar to the above case of even j, we obtain a contradiction. We now start the proof of Theorem 3.1. Proof. As explained before, we may assume that m is odd and hence that the graph G 1 is Am−1 . Note that a sector corresponding to an even vertex of Am−1 can be equivalent to another sector corresponding to an odd vertex of Am−1 . The tensor category having the irreducible objects {σ0,0 , σ1,0 , . . . , σm−2,0 } is isomorphic to the representation category of SU (2)m−2 , thus each of the irreducible objects is labeled with a vertex of the Dynkin diagram Am−1 . Let σ j,0 be one of the two sectors corresponding to [v1 ]. We choose j to be even, and then j is uniquely determined. Let 2k be the set of the irreducible B-A sectors arising from the decomposition of B ισ1,0 A for all k. Note that is a subset of the vertices of G 1 . Let ι˜ be one of the B-A sectors in having the smallest dimension. By the Perron-Frobenius theory and the definition of the graph G 1 , which is now Am−1 , we know that such ι˜ is uniquely determined and that the set {d(λ)/d(˜ι) | λ ∈ } is equal to {sin(kπ/m)/ sin(π/m) | k = 1, 2, . . . , (m − 1)/2}. The situation is illustrated in Fig. 3 where we also have the graph G which will be defined below. The vertices corresponding to the elements in are represented as larger circles. n ) for n = 0, 1, 2, . . . , and obtain Now we consider the Bratteli diagram for End(˜ισ0,1 a graph G and an orbit [v] whose reflection gives this Bratteli diagram. The graph G is one of the A-D-E Dynkin diagrams and its Coxeter number m differs from m by 1. We want to show that the pairs (G 2 , [v2 ]) and (G, [v]) are equal as follows. We first claim that the irreducible decomposition of ι˜ι˜ contains only sectors among σ0,k , k = 0, 1, . . . , m − 2. Suppose that ι˜ι˜ contains σ2l,k with l > 0 on the contrary. By Frobenius reciprocity, we have 0 < ˜ιι˜, σ2l,k = ˜ισ0,k , ι˜σ2l,0 .
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By the above description of the graph G 1 , we know that ι˜σ2l,0 is irreducible and distinct from ι˜. The assumption that ι˜σ2l,0 appears in the irreducible decomposition of ι˜σ0,k means that the graphs G and G 1 have a common vertex other than ι˜ and this is impossible by Lemma 3.1. We have thus proved that the irreducible decomposition of ι˜ι˜ contains only sectors among σ0,k , k = 0, 1, . . . , m − 2. We know ι˜σ j,0 and ι are equivalent sectors since they both are irreducible. To show (G 2 , [v2 ]) = (G, [v]), we therefore need to compare the irreducible decompositions of k and ι˜σ k for k = 0, 1, 2, . . . . Suppose that λ is an irreducible sector appearing ι˜σ j,0 σ0,1 0,1 k for some k. We have in the decomposition of ι˜σ0,1 2 . λσ j,0 , λσ j,0 = λ¯ λ, σ j,0
¯ contains only sectors among σ0,l , l = 0, 1, . . . , m − 2 Now the decomposition of λλ ¯ as above. Thus the only irreducible sector appearing in decompositions of both λλ 2 and σ j,0 is the identity sector, which appears exactly once in both. We conclude that k and ι˜σ k for λσ j,0 is also irreducible. Thus the irreducible decompositions of ι˜σ j,0 σ0,1 0,1 k = 0, 1, 2, . . . are described by the same Bratteli diagram and (G, [v]) and (G 2 , [v2 ]) are equal. Then A ι˜ι˜ A decomposes into some irreducible sectors among σ0,0 , σ0,1 , . . . , σ0,m −2 . Since the tensor category having the irreducible objects {σ0,0 , σ0,1 , . . . , σ0,m −2 } is isomorphic to the representation category of SU (2)m−1 , we obtain uniqueness of ι˜ for a given (G, [v]) hence (G 2 , [v2 ]). Then ι = ι˜σ j,0 determines a Q-system uniquely, up to unitary equivalence. We next prove a realization of a given (G 1 , [v1 ], G 2 , [v2 ]). We continue to assume that G 1 is Am−1 . Let σ j,0 be one of the two sectors corresponding to [v1 ] as above. Using the tensor category having the irreducible objects {σ0,0 , σ0,1 , . . . , σ0,m −2 }, we have ι˜ corresponding to (G 2 , [v2 ]) as in the proof of Theorem 2.1. Set ι = ι˜σ j,0 . Then one can verify that this ι produces the quadruple (G 1 , [v1 ], G 2 , [v2 ]) by the same argument as the above one showing G = G 2 . Remark 3.1. By [5, Theorem 4.1], we already know that the local extensions among the above classification are labeled with (An−1 , An ), (A4n , D2n+2 ), (D2n+2 , A4n+2 ), (A10 , E 6 ), (E 6 , A12 ), (A28 , E 8 ) and (E 8 , A30 ) for (G 1 , G 2 ) and the vertices v1 , v2 are those having the smallest Perron-Frobenius eigenvector entries. Remark 3.2. As in Remark 2.1, we may say that irreducible extensions of the Virasoro nets with c < 1 are labeled with pairs of irreducible Goodman-de la Harpe-Jones subfactors having the Coxeter numbers differing by 1. Remark 3.3. The graphs G 1 and G 2 are by definition bipartite, thus excluding the tadpole graphs which also have Frobenius norm < 2. Tadpole diagrams arise by pairwise identification of the vertices of Am diagrams when m = 2n is even. Indeed, when ι : A → B equals σn−1, j : A → A, the even vertices of G 2 = A2n pairwise coincide as B-A sectors with the odd vertices, so that the fusion graph for multiplication by σ0,1 is Tn . The invariant G 2 in these cases is Am , nevertheless. As an example, consider the case m = 4, that is, c = 7/10. In this case, we have six irreducible DHR sectors for the net Vir 7/10 . The graphs G 1 and G 2 are automatically A3
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Fig. 4. Two cases of (G 1 , [v1 ], G 2 , [v2 ]) for m = 4, c = 7/10
Fig. 5. The tadpole graph T2
Fig. 6. The graph G 1 × G 2 with G 1 = A6 and G 2 = D5 . Different vertices may represent the same B-A-sector. ι may be any vertex of G
and A4 , respectively, so we have four possibilities for the invariant (G 1 , [v1 ], G 2 , [v2 ]). If (G 1 , [v1 ], G 2 , [v2 ]) is as in case (1) of Fig. 4, then the sector ι is given by σ1,1 , thus the four vertices of the graph A4 give only two mutually inequivalent B-A sectors. That is, the fusion graph of the B-A sectors for multiplication by σ0,1 is the tadpole graph as in Fig. 5. In case (2) of Fig. 4 we have four mutually inequivalent B-A sectors for the graph G 2 = A4 for the sector ι given by σ0,1 , and the fusion graph is also A4 .
4. The Canonical Endomorphism We want to determine, viewed as a representation of the subtheory A = Vir c , the vacuum Hilbert space of the local boundary conformal QFT associated with each of the non-local extensions B, classified in the previous section. This representation is given by a DHR endomorphism θ of A whose restriction to a local algebra A(I ) (where θ is localized in the interval I ) coincides with the canonical endomorphism ι¯ι of the subfactors A(I ) ⊂ B(I ) classified above [9, 10]. We are therefore interested in the computation of ι¯ι. By the equality of local and global intertwiners, and by reciprocity, the multiplicity of each irreducible DHR sector σ within θ equals the multiplicity of ι within ισ . We therefore need to control the decomposition of ισ into irreducibles (“fusion”) for all DHR sectors. Because every irreducible sector is a product σ j,k = σ j,0 σ0,k , and σ j,0 and σ0,k are obtained from the generators σ1,0 and σ0,1 by the recursion σ0,k+1 = σ0,k σ0,1 σ0,k−1 and likewise for σ j+1,0 , it suffices to control the fusion with the generators. We know from the preceding section that the fusion of ι with the generators σ1,0 and σ0,1 separately can be described in terms of the two bipartite graphs G 1 and G 2 such that the vertices of the graphs represent irreducible B-A-sectors and two vertices are linked if the corresponding sectors are connected by the generator. ι corresponds
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to a distinguished vertex in both graphs. Moreover, applying the argument in the proof of Theorem 3.1, leading to the conclusion that (G, [v]) and (G 2 , [v2 ]) are equal (cf. Fig. 3), to each of the vertices of G 1 and of G 2 , we conclude that the fusion of ι with both generators can be described by the “product graph” G = G 1 × G 2 with vertices λ = (v1 ∈ G 1 , v2 ∈ G 2 ) and “horizontal” edges linking (v1 , v2 ) with (v1 , v2 ) if v1 and v1 are linked in G 1 , and likewise for “vertical” edges according to the graph G 2 . Again, the vertices λ represent irreducible B-A-sectors, and ι is a distinguished vertex of the product graph. See Fig. 6 for an example. From the product graph G, the fusion of each of its vertices with any DHR sector can be computed in terms of vertices of G, i.e., λσ can be decomposed into irreducibles represented by the vertices of G. But different vertices may represent identical B-A-sectors; we only know that within each horizontal or vertical subgraph, the even vertices represent pairwise inequivalent sectors, and so do the odd vertices, cf. Remark 3.3. In order to compute the canonical endomorphism, we have to determine all identifications between vertices of G as B-A-sectors. We continue to assume that m is odd and hence G 1 = Am−1 . The even Coxeter number m of G 2 is either m + 1 or m − 1. We exploit the fact that σm−2,0 and σ0,m −2 represent the same DHR sector τ , and that τ is simple (it has dimension 1). Hence fusion with τ , as a horizontal sector σm−2,0 , yields an automorphism α1 of the graph G 1 such that α1 (v1 ) = v1 σm−2,0 , and, as a vertical sector, similarly yields an automorphism α2 of G 2 . It follows that the vertices λ = (v1 , v2 ) and α(λ) = (α1 (v1 ), α2 (v2 )) of the product graph represent the same sectors. Because τ connects even vertices of G 1 with odd ones, α1 must be the unique non-trivial automorphism of G 1 . To determine α2 , one may use the above-mentioned recursion σ0,k+1 = σ0,k σ0,1 σ0,k−1 to compute the fusion of the vertices of G 2 with τ = σ0,m −2 . Since the A graph coincides with the fusion graph of σ0,1 , τ acts like the nontrivial automorphism. In the D cases, if β and β are the extremal vertices on the “short legs”, one finds by induction that βσ0,k contains the trivalent vertex γ with multiplicity 1 but neither β nor β if k is odd. If k is even, then βσ0,k , γ = 0, and βσ0,k , β = 1, βσ0,k , β = 0 if k = 0 mod 4, and vice versa if k = 2 mod 4. The cases D3 with three short legs (m − 2 = 4) and E 6 (m − 2 = 10) are easily treated “by hand”. E 7 and E 8 do not possess nontrivial automorphisms. From this analysis, we conclude that α2 is the unique non-trivial automorphism of G 2 if G 2 is either an A graph or E 6 or D2n+1 , and it is trivial if G 2 is E 7 , E 8 , or D2n . We now claim that the identifications due to α give all pairs of vertices of G which represent the same B-A-sector. Proposition 4.1. The graph (G 1 × G 2 )/(α1 × α2 ) is the fusion graph of ι with respect to σ1,0 and σ0,1 , i.e., its vertices represent inequivalent irreducible B-A-sectors, and its horizontal and vertical edges correspond to fusion with the two generators. Proof. By the Perron-Frobenius theory, the dimensions of the B-A-sectors represented by the vertices λ = (v1 , v2 ) of G are common multiples of ν(v1 )µ(v2 ), where ν(v1 ) and µ(v2 ) are the components of the Perron-Frobenius eigenvectors ν of G 1 and µ of G 2 . Let now λ = (v1 , v2 ) and λ = (v1 , v2 ) be two vertices of G which represent the same B-A-sector. Then clearly ν(v1 )µ(v2 ) = ν(v1 )µ(v2 ).
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If v1 is at distance j from an extremal vertex v˜1 of G 1 , then λ˜ = (v˜1 , v2 ) is a subsector of λσ j,0 and consequently of λ σ j,0 . Hence λ˜ is equivalent to some subsector (v˜1 , v2 ) of λ σ j,0 , implying µ(v2 )/µ(v2 ) = ν(v˜1 )/ν(v˜1 ) = dk for some k. Lemma 3.1 tells us that this is only possible if dk = 1. It follows that µ(v2 ) = µ(v2 ) and ν(v1 ) = ν(v1 ). This means in particular that v1 = v1 or v1 = α1 (v1 ), and that v2 and v2 and α2 (v2 ) are all even or all odd. If v1 = v1 , then λ and λ are two even or two odd vertices within the same vertical subgraph representing the same sector. This is only possible if λ = λ . If on the other hand v1 = α1 (v1 ), then the same argument applies to α(λ) and λ , giving λ = α(λ). Having determined the fusion graph, it is now straightforward to compute (as described above) the canonical endomorphism for every possible position of ι as a distinguished vertex of the fusion graph, and hence to determine the vacuum Hilbert space for each local boundary conformal QFT with c < 1. We display below the canonical endomorphism θv˜1 ,v˜2 whenever v˜1 and v˜2 are extremal vertices of G 1 and G 2 . All other cases are then easily obtained by the following argument: If v1 is at distance j from an extremal vertex v˜1 of G 1 , then v1 = v˜1 σ j,0 . If v2 is at distance k from the extremal vertex v˜2 on the same “leg” of G 2 , then v2 = v˜2 σ0,k . (If v2 is the trivalent vertex of the D or E graphs, then this is true for each of the three legs.) It then follows that (v1 , v2 ) = (v˜1 , v˜2 )σ j,k , and hence 2 . θv1 ,v2 = θv˜1 ,v˜2 σ j,k
The canonical endomorphisms θv˜1 ,v˜2 for all pairs of extremal vertices of G 1 and G 2 are listed in the following table. Table 4.1. The canonical endomorphisms θv˜1 ,v˜2 for all pairs of extremal vertices of G 1 and G 2 . The entry in the third column indicates the distance of v˜2 from the trivalent vertex, i.e., the length of the “leg” of G 2 on which v˜2 is the extremal vertex G2
m
dist.
θv˜1 ,v˜2
An Dn Dn E6 E6 E7 E7 E7
n+1 2n − 2 2n − 2 12 12 18 18 18
− 1 n−3 1 2 1 2 3
E8
30
1
E8 E8
30 30
2 4
σ0,0 σ0,0 ⊕ σ0,4 ⊕ σ0,8 ⊕ . . . ⊕ σ0,4[n/2]−4 σ0,0 ⊕ σ0,2n−4 σ0,0 ⊕ σ0,4 ⊕ σ0,6 ⊕ σ0,10 σ0,0 ⊕ σ0,6 σ0,0 ⊕ σ0,4 ⊕ σ0,6 ⊕ σ0,8 ⊕ σ0,10 ⊕ σ0,12 ⊕ σ0,16 σ0,0 ⊕ σ0,6 ⊕ σ0,10 ⊕ σ0,16 σ0,0 ⊕ σ0,8 ⊕ σ0,16 σ0,0 ⊕ σ0,4 ⊕ σ0,6 ⊕ σ0,8 ⊕ 2σ0,10 ⊕ σ0,12 ⊕ 2σ0,14 ⊕ ⊕ σ0,16 ⊕ 2σ0,18 ⊕ σ0,20 ⊕ σ0,22 ⊕ σ0,24 ⊕ σ0,28 σ0,0 ⊕ σ0,6 ⊕ σ0,10 ⊕ σ0,12 ⊕ σ0,16 ⊕ σ0,18 ⊕ σ0,22 ⊕ σ0,28 σ0,0 ⊕ σ0,10 ⊕ σ0,18 ⊕ σ0,28
The local chiral extensions classified earlier [5] are precisely those cases where G 2 is A, D2n , E 6 , or E 8 , and both v1 and v2 are extremal vertices (on the respective longest leg in the D and E cases). In the non-local cases, the local algebras of the associated BCFT on the half-space are the relative commutants as described in the introduction. Note that, in order to determine the resulting factorizing chiral charge structure [10] of the local fields, more detailed information about the DHR category and the Q-system is needed, than the simple combinatorial data exploited in this work.
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Acknowledgements. A part of this work was done during visits of the first-named author to Università di Roma “Tor Vergata” and Universität Göttingen, and he thanks them for their hospitality. The authors gratefully acknowledge the financial support of GNAMPA-INDAM and MIUR (Italy), EU network “Quantum Spaces - Noncommutative Geometry”, and Grants-in-Aid for Scientific Research, JSPS (Japan).
References 1. Böckenhauer, J., Evans, D.E., Kawahigashi, Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210, 733–784 (2000) 2. Doplicher, S., Haag, R., Roberts, J.E., Local observables and particle statistics. I. Commun. Math. Phys. 23, 199–230 (1971); II. 35, 49–85 (1974) 3. Evans, D.E., Kawahigashi, Y.: Quantum symmetries on operator algebras. Oxford: Oxford University Press, 1998 4. Goodman, F., de la Harpe, P., Jones, V.F.R.: Coxeter graphs and towers of algebras. MSRI publications 14, Berlin: Springer, 1989 5. Kawahigashi, Y., Longo, R.: Classification of local conformal nets Case c < 1. Ann. of Math. 160, 493– 522 (2004) 6. Kawahigashi, Y., Longo, R.: Classification of two-dimensional local conformal nets with c < 1 and 2-cohomology vanishing for tensor categories. Commun. Math. Phys. 244, 63–97 (2004) 7. Kawahigashi, Y., Longo, R., Müger, M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001) 8. Lang, S.: “Algebra”, Revised third edition, Graduate Texts in Mathematics 211, Berlin-Heidelberg-New York: Springer-Verlag, 2002 9. Longo, R., Rehren, K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567–597 (1995) 10. Longo, R., Rehren, K.-H.: Local fields in boundary conformal QFT. Rev. Math. Phys. 16, 909–960 (2004) Communicated by A. Connes
Commun. Math. Phys. 271, 387–430 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0134-x
Communications in
Mathematical Physics
Rayleigh Scattering at Atoms with Dynamical Nuclei J. Fröhlich1, , M. Griesemer2, , B. Schlein3, 1 Theoretical Physics, ETH-Hönggerberg, CH-8093 Zürich, Switzerland.
E-mail: [email protected]
2 Fachbereich Mathematik, Universität Stuttgart, D-70569 Stuttgart, Germany.
E-mail: [email protected]
3 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA.
E-mail: [email protected] Received: 6 September 2005 / Accepted: 2 June 2006 Published online: 8 February 2007 – © Springer-Verlag 2007
Abstract: Scattering of photons at an atom with a dynamical nucleus is studied on the subspace of states of the system with a total energy below the threshold for ionization of the atom (Rayleigh scattering). The kinematics of the electron and the nucleus is chosen to be non-relativistic, and their spins are neglected. In a simplified model of a hydrogen atom or a one-electron ion interacting with the quantized radiation field in which the helicity of photons is neglected and the interactions between photons and the electron and nucleus are turned off at very high photon energies and at photon energies below an arbitrarily small, but fixed energy (infrared cutoff), asymptotic completeness of Rayleigh scattering is established rigorously. On the way towards proving this result, it is shown that, after coupling the electron and the nucleus to the photons, the atom still has a stable ground state, provided its center of mass velocity is smaller than the velocity of light; but its excited states are turned into resonances. The proof of asymptotic completeness then follows from extensions of a positive commutator method and of propagation estimates for the atom and the photons developed in previous papers. The methods developed in this paper can be extended to more realistic models. It is, however, not known, at present, how to remove the infrared cutoff. 1. Introduction During the past decade, there have been important advances in our understanding of the mathematical foundations of quantum electrodynamics with non-relativistic, quantummechanical matter (“non-relativistic QED”). Subtle spectral properties of the Hamiltonians generating the time evolution of atoms and molecules interacting with the quantized radiation field have been established. In particular, existence of atomic ground states Activities supported, in part, by a grant from the Swiss National Foundation.
Work partially supported by U.S. National Science Foundation grant DMS 01-00160. Supported by a NSF postdoctoral fellowship
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and absence of stable excited states have been proven, and the energies and life times of resonances have been calculated in a rigorously controlled way, for a variety of models; see [BFS98, BFSS99, HS95, Sk98, GLL01, LL03, Gr04, AGG05, BFP05], and references given there. Furthermore, some important steps towards developing the scattering theory for systems of non-relativistic matter interacting with massless bosons, in particular photons, have been taken. Asymptotic electromagnetic field operators have been constructed in [FGS00], and wave operators for Compton scattering have been shown to exist in [Pi03]. Rayleigh scattering, i.e., the scattering of photons at atoms below their ionization threshold, has been analyzed in [FGS02] for models with an infrared cutoff. The results in this paper are based on methods developed in [DG99]. Some earlier results on Rayleigh scattering have been derived in [Sp97]. Compton scattering in models with an infrared cutoff has been studied in [FGS04]. While it is understood how to calculate scattering amplitudes for various low-energy scattering processes in models without an infrared cutoff, all known general methods to prove unitarity of the scattering matrix on subspaces of states of sufficiently low energy, i.e., asymptotic completeness, require the presence of an (arbitrarily small, but positive) infrared cutoff. In [Sp97, DG99, FGS02], Rayleigh scattering has only been studied for models of atoms with static (i.e., infinitely heavy) nuclei. As suggested by this discussion, the main challenge in the scattering theory for nonrelativistic QED presently consists in solving the following problems: i) To remove the infrared cutoff in the analysis of Rayleigh scattering; ii) to remove the infrared cutoff in the treatment of Compton scattering of photons at one freely moving electron or ion; iii) to prove asymptotic completeness of Rayleigh scattering of photons at atoms or molecules with dynamical nuclei. In this paper, we solve problem iii) in the presence of an (arbitrarily small, but positive) infrared cutoff. In order to render our analysis, which is quite technical, as simple and transparent as possible, we consider the simplest model exhibiting all typical features and challenges encountered in an analysis of problem iii). We consider a hydrogen atom or a one-electron ion. The nucleus is described, like the electron, as a non-relativistic quantum-mechanical point particle of finite mass. We ignore the spin degrees of freedom of the electron and the nucleus. The electron and the nucleus interact with each other through an attractive two-body potential V , which can be chosen to be the electrostatic Coulomb potential. Electron and nucleus are coupled to a quantized radiation field. As in [FGS02], the field quanta of the radiation field are massless bosons, which we will call photons. Physically, the radiation field is the quantized electromagnetic field. However, the helicity of the photons does not play an interesting role in our analysis, and we therefore consider scalar bosons. As announced, we focus our attention on Rayleigh scattering; we only consider the asymptotic dynamics of states of the atom and the radiation field with energies below the threshold for break-up of the atom into a freely moving nucleus and electron, i.e., below the ionization threshold. Moreover, we introduce an infrared cutoff: Photons with an energy below a certain arbitrarily small, but positive threshold energy do not interact with the nucleus and the electron. While all our other simplifications are purely cosmetic, the presence of an infrared cutoff is crucial in our proof of asymptotic completeness (but not for most other results presented in this paper). In the model we study, the atom can be located anywhere in physical space and can move around freely, and the Hamiltonian of the system is translation-invariant. This feature suggests to combine and extend the techniques in two previous papers, [FGS02]
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(Rayleigh scattering with static nuclei) and [FGS04] (Compton scattering of photons at freely moving electrons), and this is, in fact, the strategy followed in the present paper. As in [FGS04], to prove asymptotic completeness we must impose an upper bound on the energy of the state of the system that guarantees that the center of mass velocity of the atom does not exceed one third of the velocity of light. This is a purely technical restriction which we believe can be replaced by one that guarantees that the velocity of the atom is less than the velocity of light. We note that, for realistic atoms, the condition that the total energy of a state be below the ionization threshold of an atom at rest automatically guarantees that the speed of the atom in an arbitrary internal state is much less than a third of the speed of light. Next, we describe the model studied in this paper more explicitly. The Hilbert space of pure states is given by H = L 2 (R3 , dxn ) ⊗ L 2 (R3 , dxe ) ⊗ F,
(1)
where the variables xn and xe are the positions of the nucleus and of the electron, respectively, and F is the symmetric Fock space over the one-photon Hilbert space L 2 (R3 , dk), where the variable k denotes the momentum of a photon. Vectors in F describe pure states of the radiation field. The Hamiltonian generating the time evolution of states of the system is given by Hg = Hatom + H f + g φ(G exe ) + φ(G nxn ) , (2) where Hatom =
pe2 p2 + n + V (xe − xn ) 2m e 2m n
(3)
is the Hamiltonian of the atom decoupled from the radiation field. Here pe = −i∇xe and pn = −i∇xn are the momentum operators of the electron and the nucleus, respectively, and m e and m n are their masses. The term V (xe − xn ) is the potential of an attractive twobody force, e.g., the electrostatic Coulomb force, between the electron and the nucleus; (V (x) is negative and tends to zero, as |x| → ∞, and it is assumed to be such that the spectrum of Hatom is bounded from below and has at least one negative eigenvalue). The operator H f on the r.h.s. of (2) is the Hamiltonian of the free radiation field. It is given by H f = dk |k|a ∗ (k)a(k), where |k| is the energy of a photon with momentum k, and a ∗ (k), a(k) are the usual boson creation- and annihilation operators: For every function f ∈ L 2 (R3 , dk), a ∗ ( f ) = dk f (k)a ∗ (k) and a( f ) = dk f (k)a(k) are densely defined, closed operators on the Fock space F, and, for f, h, ∈ L 2 (R3 , dk), they satisfy the canonical commutation relations [a( f ), a ∗ (h)] = ( f, h), [a ( f ), a (h)] = 0,
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where ( f, h) denotes the scalar product of f and h. For f ∈ L 2 (R3 , dk), a field operator φ( f ) is defined by φ( f ) = a( f ) + a ∗ ( f ). It is a densely defined, self-adjoint operator on F. The functions (form factors) G exe and G nxn on the right side of (2) are given by G exe (k) = e−ik·xe κe (k), G nxn (k) = e−ik·xn κn (k),
(4)
where κe and κn belong to the Schwartz space, and κe (k) = κn (k) = 0, for all k ∈ R3 with |k| ≤ σ, for some σ > 0 (infrared cutoff). In many of our results, we could pass to the limit σ = 0; but in our proof of asymptotic completeness of Rayleigh scattering, the condition that σ > 0 is essential. Finally, the parameter g on r.h.s. of (2) is a coupling constant; it is assumed to be non-negative and will be chosen sufficiently small in the proofs of our results. (It should be noted that we are using units such that Planck’s constant = 1 and the speed of light c = 1, and we work with dimensionless variables xe , xn , k chosen such that Hatom and H f are independent of g.) In the description of the atom, it is natural to use the following variables: X=
m e xe + m n xn , x = xe − xn . me + mn
Here X is the position of the center of mass of the atom, and x is the position of the electron relative to the one of the nucleus. Then p2 P2 + + V (x) + H f + g φ G eX + m n x + φ G nX − m e x , (5) Hg = M M 2M 2m where P = −i∇ X , p = −i∇x , M = m e + m n , and m = m e m n M −1 ; (center-of-mass momentum, relative momentum, total mass, reduced mass, respectively). Self-adjointness of Hg on H (under appropriate assumptions on V ) is a standard result. In this paper, we study the dynamics generated by Hg on the subspace of states in H whose maximal energy is below the ionization threshold ϕ, Hg ϕ , R→∞ ϕ∈D R ϕ, ϕ
ion = lim
inf
(6)
where D R = {ϕ ∈ H : χ (|x| ≥ R)ϕ = ϕ} is the subspace of vectors in H with the property that the distance between the electron and the nucleus is at least R. Vectors in H with a maximal total energy below ion exhibit exponential decay in |x|, the distance between the electron and the nucleus, see [Gr04]. Under our assumptions, −Cg 2 ≤ ion ≤ 0, for a finite constant C depending only on κe and κn . When g ↓ 0 then ion ↑ 0, which is the ionization threshold of a one-electron atom or -ion decoupled from the radiation field. Our choice of the Hamiltonian Hg , see (2) and (5), and of the form factors G exe and n G xn , see (4), makes it clear that the dynamics of the system is space-translation invariant: Let (7) P f = dk ka ∗ (k)a(k)
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denote the momentum operator of the radiation field, and let = P + P f be the total momentum operator. It is easy to check that [Hg , ] = 0
(translation invariance).
(8)
It is then useful to consider direct-integral decompositions of the space H and the operator Hg over the spectrum of the total momentum operator (which, as a set, is R3 ). Thus ⊕ H= d H , with H L 2 (R3 , d x) ⊗ F, (9) R3
and
Hg =
⊕
R3
d Hg ( ),
(10)
where the fiber Hamiltonian Hg ( ) is the operator on the fiber Hilbert space H given by Hg ( ) =
( − P f )2 + Hat + H f + g φ G em n x + φ G n− m e x , M M 2M
(11)
where − P f is the center-of-mass momentum of the atom, and Hat = p 2 /2m + V (x) is the Hamiltonian describing the relative motion of the electron around the nucleus. We are now in the position to summarize the main results proven in this paper for the model introduced above. In a first part, we analyze the energy spectra of the fiber Hamiltonians Hg ( ) below a certain threshold < min(ion , β ), where ion is given in (6), and β = E 0at + Mβ 2 /2; E 0at is the ground state energy of Hat , and β is a constant < 1 (=speed of light, in our units). The condition < ion guarantees that the electron is bound to the nucleus, with exponential decay in x, and < β<1 implies that, for a sufficiently small coupling constant g, the center-of-mass velocity of the atom is smaller than the speed of light. (For center-of-mass velocities > 1, the ground state energy of the atom decoupled from the radiation field is embedded in continuous spectrum, and the ground state becomes unstable when the coupling to the radiation field is turned on.) For realistic atoms, in particular for hydrogen, ion β=1/3 , so that < ion is the only relevant condition. We let E g ( ) = inf σ (Hg ( )) denote the ground state energy of Hg ( ), and we define 2 3 at ≤ . (12) B = ∈ R : E 0 + 2M We prove that, for every ∈ B , E g ( ) is a simple eigenvalue of Hg ( ), i.e., that the atom has a unique ground state, provided g is small enough. This is a result that is expected to survive the limit σ ↓ 0, provided the factors κe and κn are not too singular at k = 0. For the Pauli-Fierz model of non-relativistic QED, existence of a ground state can be proven under conditions similar to the ones described above, provided the total charge of electrons and nucleus vanishes; see [AGG05]. By appropriately modifying Mourre theory in a form developed in [BFSS99], we prove that the spectrum of Hg ( ) in the interval (E g ( ), ) is purely continuous. With relatively little further effort, our methods would also show that σ (Hg ( ))∩(E g ( ), ) is absolutely continuous. (These results, too, would survive the removal of the infrared cutoff, σ ↓ 0. This will not be shown in this paper; but see [FGSi05].)
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We denote the ground state of Hg ( ), ∈ B , by ψ ; (ψ is called the dressed atom (ground) state of momentum ). The space of wave packets of dressed atom states, Hdas , is the subspace of the total Hilbert space H given by Hdas = ψ( f ) : ψ( f ) =
⊕
d f ( )ψ , f ∈ L 2 (B , d ) .
(13)
This space is invariant under the time evolution. In fact, e−i Hg t ψ( f ) = ψ( f t ), where, for f ∈ L 2 (B , d ), f t ( ) = e−i E g ( )t f ( ) ∈ L 2 (B , d ), for all times t. In a second part of our paper, scattering theory is developed for the models introduced above. We first construct asymptotic photon creation- and annihilation operators
a± (h)ϕ = s − lim ei Hg t a (h t )e−i Hg t ϕ, t→±∞
(14)
where h t (k) = e−i|k|t h(k) is the free time evolution of a one-photon state h(k). To ensure the existence of the strong limit on the r.h.s. of (14), we assume that h ∈ L 2 (R3 , (1 + |k|−1 )dk), that ϕ belongs to the range of the spectral projection, E (Hg ), of Hg corresponding to the interval (−∞, ], with < (ion , β ), as above, and β < 1, and that the coupling constant g is so small (depending on ) that the velocity of the center of mass of the atom is smaller than one. The last condition ensures that the distance between the atom and a configuration of outgoing photons increases to infinity and hence the interaction between these photons and the atom tends to 0, as time t tends to +∞. The details of the proof of (14) are very similar to those in [FGS00]. From (13) and (14) we infer that, for ψ( f ) ∈ Hdas , and under the conditions of existence of the limit ∗ (h ) · · · a ∗ (h )ψ( f ) exist, and their time evolution is the in (14), vectors of the form a± 1 ± n one of freely moving photons and a freely moving atom: ∗ ∗ (h 1 ) . . . a± (h n )ψ( f ) = a ∗ (h 1,t ) · · · a ∗ (h n,t )ψ( f t ) + o(1), e−i Hg t a±
(15)
as t → ±∞. Furthermore, a± (h)ψ( f ) = 0, under the same assumptions. Equation ∗ (h ) . . . a ∗ (h )ψ( f ) scattering (15) provides the justification for calling the vectors a± 1 ± n states. We already know that the atom does not have any stable excited states. It is therefore natural to expect that the time evolution of an arbitrary vector in the range of the spectral projection E (Hg ), with < min(ion , β<1 ) as above, approaches a vector describing a configuration of freely moving photons and a freely moving atom in its ground state, as time t tends to ±∞. Thus, with (15) and (13), we expect that, for < min(ion , β<1 ), ∗ ∗ a± (h 1 ) . . . a± (h n )ψ( f ) : ψ( f ) ∈ Hdas , h j ∈ L 2 (R3 , (1 + 1/|k|)dk),
− j = 1, . . . , n, n = 1, 2, . . . } ⊃ RanE (Hg ) ,
(16)
where S denotes the linear subspace spanned by a set, S, of vectors in H, and S − denotes the closure of S in the norm of H. Property (16) is called asymptotic completeness of Rayleigh scattering. The main result of this paper is a proof of (16) under the supplementary condition that < β for some β < 1/3 (the proof of (16) is the only part of the paper where, for technical reasons, we need to assume β < 1/3; all other results only require β < 1). Next, we reformulate (16) in a more convenient language.
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We define a Hilbert space of scattering states as the space Hdas ⊗ F, and we introduce an asymptotic Hamilton operator, H˜ gdas , by setting H˜ gdas = Hgdas ⊗ 1 + 1 ⊗ H f ,
(17)
Hgdas ψ( f ) = ψ(E g (. ) f ),
(18)
where
for arbitrary f ∈ L 2 (B , d ), with < min(ion , β<1/3 ), as above. On the range of E ( H˜ gdas ), the operators ± , given by ∗ ∗ ± (ψ( f ) ⊗ a ∗ (h 1 ) . . . a ∗ (h n ) ) = a± (h 1 ) . . . a± (h n )ψ( f )
(19)
exist; see (14). The vector is the vacuum in the Fock-space characterized by the property that a(h) = 0, for h ∈ L 2 (R3 , dk). The operators + and − are called wave operators, and the scattering matrix is defined by S = ∗+ − .
(20)
From Eqs. (14) and (15) we find that ˜ das
e−i Hg t ± = ± e−i Hg t , and hence the ranges of + and − are contained in the range of E (Hg ). Using that a± (h)ψ( f ) = 0, for h ∈ L 2 (R3 , dk) and ψ( f ) ∈ Hdas , one sees that + and − are isometries from the range of E ( H˜ gdas ) into H. If we succeeded in proving that Ran ± |`RanE ( H˜ gdas ) = RanE (Hg ) (21) we would have established the unitarity of the S-matrix, defined in (20), on RanE ( H˜ das ), i.e., asymptotic completeness of Rayleigh scattering. In order to prove (21), we show that ± have right inverses defined on RanE (Hg ). Our proof is inspired by proofs of similar results in [DG99] and in [FGS02, FGS04]. It is based on constructing a so-called asymptotic observable W and then proving that W is positive on the orthogonal complement of Hdas in RanE (Hg ). The proof of this last result is, perhaps, the most original accomplishment in this paper and is based on some new ideas. Our paper is organized as follows. In Sect. 2, we define our model more precisely, and we state our assumptions on the potential V (x) and on the form factors G exe and G nxn . In Sect. 3, we study the spectrum of the fiber Hamiltonian Hg ( ): In Sect. 3.1, we prove the existence of dressed atom states, and, in Sect. 3.3, we prove two positive commutator estimates, from which we conclude that the spectrum of Hg ( ) above the ground state energy and below an appropriate threshold is continuous. In Sect. 4, we discuss the scattering theory of the system. First, in Sect. 4.1, we prove the existence of asymptotic field operators, we recall some of their properties, we prove the existence of the wave operators, and we state our main theorem. Then, in Sect. 4.2, we introduce a modified Hamiltonian, Hmod , describing “massive” photons, and we explain why it is enough to prove asymptotic completeness for Hmod instead of Hg . In Sect. 4.3, we construct asymptotic observables W and inverse wave operators W± . In Sect. 4.4, we
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prove positivity of our asymptotic observables when restricted to the orthogonal complement of Hdas (the space of wave packets of dressed atom states). Finally, in Sect. 4.5, we complete the proof of asymptotic completeness. In Appendix A, we introduce some notation, used throughout the paper, concerning operators on the bosonic Fock space. In Appendix B, we summarize bounds used to control the interaction between the electron (or the nucleus) and the radiation field. 2. The Model We consider a non-relativistic atom consisting of a nucleus and an electron interacting through a two-body potential V (x). The Hamiltonian describing the dynamics of the atom is the self-adjoint operator Hatom =
pn2 p2 + e + V (xn − xe ) 2m n 2m e
(22)
acting on the Hilbert space Hatom = L 2 (R3 , dxn ) ⊗ L 2 (R3 , dxe ), where xn and xe denote the position of the nucleus and of the electron, respectively, and pn = −i∇xn and pe = −i∇xe are the corresponding momenta. We assume that the interaction potential V (x) satisfies the following assumptions. Hypothesis (H0). The potential V is a real-valued, locally square integrable function on R3 with lim V (x) = 0.
|x|→∞
Remarks. 1) Hypothesis (H0) is satisfied by the Coulomb potential V (x) = −1/|x| and it is inspired by this potential. It guarantees that (a), V ∈ L 2 (R3 ) + L ∞ (R3 ), and that (b), 3/2 2 3 V ∈ R + L∞ ε (R ), where R denotes the Rollnik class. Note that L loc ⊂ L loc and that L 3/2 (R3 ) ⊂ R. By (a), V is infinitesimally small w.r.to p 2 = −, and hence p 2 /2m + V , for any m > 0, is self-adjoint on D( p 2 ) = H 2 (R3 ) and bounded from below. If inf σ ( p 2 /2m + V ) is an eigenvalue, then, by (b), it is non-degenerate [RS78, Theorem XIII.46]. Similarly the Hamiltonian Hatom with domain H 2 (R3xn × R3xe ) is a self-adjoint operator on the Hilbert space L 2 (R3 , dxn ) ⊗ L 2 (R3 , dxe ) and it is bounded from below. 2) With little more effort we could have covered a much larger class of locally square integrable potentials V where only the negative part V− (x) = max{−V (x), 0} is infinitesimally small with respect to and Hatom is self-adjointly realized in terms of a Friedrich’s extension. This would allow, e.g., for confining potentials that tend to ∞ as |x| → ∞. 3) Note that we are neglecting the degrees of freedom corresponding to the spin of the nucleus and of the electron, because they do not play an interesting role in the scattering process. Next we couple the atom to a quantized scalar radiation field. We call the particles described by the quantized field (scalar) photons. The pure states of the photon field are vectors in the bosonic Fock space over the one-particle space L 2 (R3 , dk), F= L 2s (R3n , dk1 . . . dkn ), (23) n≥0
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where L 2s (R3n ) denotes the subspace of L 2 (R3n ) consisting of all functions which are completely symmetric under permutations of the n arguments. The variables k1 , . . . kn denote the momenta of the photons. The dispersion relation of the photons is given by ω(k) = |k|, which characterizes relativistic particles with zero mass. The free Hamiltonian of the quantized radiation field is given by the second quantization of ω(k) = |k|, denoted by d(|k|). Formally, d(|k|) = dk |k| a ∗ (k)a(k), (24) where a ∗ (k) and a(k) are the usual creation- and annihilation operators on F, satisfying the canonical commutation relations [a (k), a (k )] = 0, [a(k), a ∗ (k )] = δ(k − k ). More notations for operators on Fock space that are used throughout the paper are collected in Appendix A. The total system, atom plus quantized radiation field, has the Hilbert space H = Hatom ⊗ F; its dynamics is generated by the Hamiltonian Hg = Hatom + d(|k|) + g φ(G exe ) + φ(G nxn ) , (25) where g is a real non-negative coupling constant (the assumption g ≥ 0 is not needed, it just makes the notation a little bit simpler), and where φ(G x ) = dk a ∗ (k)G x (k) + a(k)G x (k) . (26) The form factors G ex and G nx are square integrable functions of k with values in the multiplication operators on L 2 (R3 , dx). Clearly, G ex describes the interaction between the electron and the radiation field, and G nx couples the field to the nucleus. The next hypothesis specifies our assumptions on the form factors G ex and G nx . Hypothesis (H1). The form factors G ex and G nx have the form G ex (k) = e−ik·x κe (k) and G nx (k) = e−ik·x κn (k) ,
(27)
where κe , κn belong to Schwartz space S(R3 ), and κe (k) = κn (k) = 0 if |k| ≤ σ , for some σ > 0. The particular form of G ex and G nx given in (27) guarantees the translation invariance of the system (see the discussion after (37)). The presence of an infrared cutoff σ > 0 in κe and κn is used in the proofs of many of our results; but it is not necessary for the existence of the asymptotic field operators and for the existence of the wave operator in Sect. 4.1. Notice that, even though our main results require the coupling constant g to be sufficiently small, how small g has to be does not depend on the infrared cutoff σ , in the following sense. If we define κe (k) = κ˜ e (k)χ (|k|/σ ) and κn (k) = κ˜ n (k)χ (|k|/σ ), with κ˜ e,n ∈ C0∞ (R3 ), and with χ ∈ C ∞ (R) monotone increasing and such that χ (s) = 0 if s ≤ 1 and χ (s) = 1 if s ≥ 2, then Hypothesis (H1) is satisfied for every choice of σ > 0. Moreover how small g has to be is independent of the choice of the parameter σ . Assuming Hypotheses (H0) and (H1), the Hamiltonian Hg , defined on the domain H 2 (R3xe × R3xn ) ⊗ D(d(|k|)), is essentially self-adjoint and bounded from below. This follows from Lemma 20, which shows, using Hypothesis (H1), that the interaction φ(G exe ) + φ(G nxn ) is infinitesimal with respect to the free Hamiltonian H0 = Hatom + d(|k|).
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To study the system described by the Hamiltonian Hg , it is more convenient to use coordinates describing the center of mass of the atom and the relative position of the nucleus and the electron. We define X=
m n xn + m e xe , mn + me
x = xe − xn .
(28)
Then, the atomic Hamiltonian Hatom becomes Hatom =
p2 P2 + + V (x), 2M 2m
(29)
where P = −i∇ X is the center of mass momentum of the atom, and p = −i∇x is the momentum conjugate to the relative coordinate x. Moreover, M = m e + m n is the −1 −1 is the reduced mass. Expressed in the new total atomic mass, and m = (m −1 e + mn ) coordinates, the total Hamiltonian of the system is given by p2 P2 + + V (x) + d(|k|) + g φ(G eX +λe x ) + φ(G nX −λn x ) 2M 2m P2 p2 = + + V (x) + d(|k|) + gφ(G X,x ). 2M 2m
Hg =
(30)
Here λe = m n /M and λn = m e /M, and we use the notation G X,x (k) = G eX +λe x (k) + G nX −λn x (k) = e−ik·X Fx (k),
(31)
Fx (k) = e−iλe k·x κe (k) + eiλn k·x κn (k).
(32)
with
The fact that the form factors κe and κn contain an infrared cutoff (meaning that κn (k) = κe (k) = 0, if |k| ≤ σ ) implies that photons with very small momenta do not interact with the atom. In other words, they decouple from the rest of the system. We denote by χi (the subscript i stands for “interacting”) the characteristic function of the set {k ∈ R3 : |k| ≥ σ }. Then the operator (χi ), whose action on the n-particle sector of F is given by (χi ) = χi ⊗ χi ⊗ · · · ⊗ χi ,
(33)
defines the orthogonal projection onto states without soft bosons. The fact that soft bosons do not interact with the atom implies that Hg leaves the range of (χi ) invariant; Hg commutes with (χi ). Another way to isolate the soft, non-interacting, photons from the rest of the system is as follows. We have that L 2 (R3 ) = L 2 (Bσ (0))⊕ L 2 (Bσ (0)c ), where Bσ (0) is the open ball of radius σ around the origin and Bσ (0)c denotes its complement. Hence the Fock space can be decomposed as F Fi ⊗ Fs , where Fi is the bosonic Fock space over L 2 (Bσ (0)c ) (describing interacting photons), and Fs is the bosonic Fock space over L 2 (Bσ (0)) (describing soft, non-interacting, photons). Accordingly, the Hilbert space H = L 2 (R3 , dX ) ⊗ L 2 (R3 , dx) ⊗ F can be decomposed as H Hi ⊗ Fs
with
Hi = L (R , dX ) ⊗ L 2 (R3 , dx) ⊗ Fi . 2
3
(34)
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By U : H → Hi ⊗ Fs we denote the unitary map from H to Hi ⊗ Fs . The action of the Hamiltonian Hg on Hi ⊗ Fs is then given by U Hg U ∗ = Hi ⊗ 1 + 1 ⊗ d(|k|),
(35)
Hi = Hg |`Hi .
(36)
with
Note that in the representation of the system on the Hilbert space Hi ⊗ Fs , the projection (χi ) (projecting on states without soft bosons) is simply given by U (χi )U ∗ = 1⊗ P , where P denotes the orthogonal projection onto the vacuum in Fs . One of the most important properties of the Hamiltonian Hg is its invariance with respect to translations of the whole system, atom and field. More precisely, defining the total momentum of the system by = P + d(k),
(37)
we have that [Hg , ] = 0. Because of this property, it can be useful to rewrite the Hilbert space H = L 2 (R3 , dX ) ⊗ L 2 (R3 , dx) ⊗ F as a direct integral over fibers with fixed total momentum. Specifically, we define the isomorphism T : L 2 (R3 , dX ) ⊗ L 2 (R3 , dx) ⊗ F −→ L 2 (R3 , d ; L 2 (R3 , dx) ⊗ F)
(38)
as follows. For ψ = {ψ (n) (X, x, k1 , . . . , kn )}n≥0 ∈ L 2 (R3 , dX ) ⊗ L 2 (R3 , dx) ⊗ F, we define (n)
(T ψ)( ) = {(T ψ) (x, k1 , . . . kn )}n≥0 ∈ L 2 (R3 , dx) ⊗ F
(39)
(n) (T ψ)(n) (x, k1 , . . . kn ) = ψ ( − k1 − · · · − kn , x, k1 , . . . kn ),
(40)
with
where ψ (n) (P, x, k1 , . . . , kn ) =
1 (2π )3/2
dX e−i P·X ψ (n) (X, x, k1 , . . . , kn )
(41)
is the Fourier transform of ψ (n) with respect to its first variable. Because of its translation invariance, the Hamiltonian Hg leaves invariant each fiber with fixed total momentum ⊕ 2 3 (L (R , dx) ⊗ F)d . More of the Hilbert space L 2 (R3 , d ; L 2 (R3 , dx) ⊗ F) precisely, (T ∗ Hg T ψ)( ) = Hg ( )ψ( ) with Hg ( ) =
( − d(k))2 + d(|k|) + Hat + gφ(Fx ), 2M
(42)
where we put Hat = p 2 /2m + V . Recall that Fx (k) = e−iλe k·x κe (k) + eiλn k·x κn (k). Note that, for every fixed , the operator Hg ( ) is a self-adjoint operator on the fiber space L 2 (R3 , dx) ⊗ F. Our first results, stated in the next section, describe the structure of the spectrum of the fiber-Hamiltonian Hg ( ), for fixed values of .
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3. The Spectrum of Hg () 3.1. Dressed atom states. The first question arising in the analysis of the spectrum of Hg ( ) =
( − d(k))2 + d(|k|) + Hat + gφ(Fx ) 2M
(43)
concerns the existence of a ground state of Hg ( ): we wish to know whether or not E g ( ) = inf σ (Hg ( )) is an eigenvalue of Hg ( ). We will answer this question affirmatively, under the assumption that the energy E g ( ) lies below some threshold and that the coupling constant is sufficiently small. The restriction to small energies is necessary to guarantee that the center of mass of the atom does not move faster than with the speed of light (c = 1 in our units), and that the atom is not ionized. The following lemma (and its corollary) proves that an upper bound on the total energy is sufficient to bound the momentum of the center of mass of the atom (provided the coupling constant is sufficiently small) and to make sure that the electron is exponentially localized near the nucleus. Lemma 1. Assume that Hypotheses (H0) and (H1) are satisfied. i) Define E 0at = inf σ (Hat ), and fix β > 0. Suppose < β = E 0at + (M/2)β 2 . Then there is g,β > 0 such that
| − d(k)|
E (Hg ( )) ≤ β
M
(44)
for all g ≤ g,β (g ≥ 0), and for all ∈ R3 . In particular (|P|/M)E (Hg ) ≤ β. ii) Define the ionization threshold ion = lim
inf ϕ, Hg ϕ
R→∞ ϕ∈D R
with D R = {ϕ ∈ D(Hg ) : χ (|x| ≥ R)ϕ = ϕ}. Let , α ∈ R be such that + α 2 /(2m) < ion . Then sup eα|x| E (Hg ( )) < ∞.
(45)
∈R3
Proof. i) Fix ε > 0 such that + ε < β = E 0at + (M/2)β 2 . Choose χ ∈ C0∞ (R) such that χ (s) = 1 for s ≤ , and χ (s) = 0 if s > + ε. Then we have that (| − d(k)|/M)E (Hg ( )) ≤ (| − d(k)|/M)χ (Hg ( )) ≤ (| − d(k)|/M)χ (H0 ( )) + Cg, (46) where H0 ( ) = ( − d(k))2 /2M + Hat + d(|k|) is the non-interacting fiber Hamiltonian and where the constant C is independent of . To prove the last equation note that, if χ denotes an almost analytic extension of χ (see Appendix A in [FGS04] for a short introduction to the Helffer-Sjöstrand functional calculus), we have that 1 1 1 gφ(Fx ) , (47) χ (Hg ( )) − χ (H0 ( )) = (z) dxdy ∂z¯ χ π H0 ( ) − z Hg ( ) − z
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and therefore | − d(k)| (χ (Hg ( )) − χ (H0 ( ))) ≤ Cg| − d(k)|(H0 ( ) + i)−1 φ(Fx )(Hg ( ) + i)−1 ≤ Cg,
(48)
uniformly in . Next, since Hat ≥ E 0at = inf σ (Hat ), d(|k|) ≥ 0, and by the definition of χ , we have that ( − d(k))2 χ (H0 ( )) . χ (H0 ( )) = E +ε−E at 0 2M Since + ε − E 0at < (1/2)Mβ 2 , we conclude from (46) that
| − d(k)|
E (Hg ( )) ≤ β
M for g sufficiently small (independently of ). As for part ii), we use Theorem 1 of [Gr04] and an estimate from its proof. Given R ≥ 0 and ∈ R3 , let R = R ( ) =
inf
ϕ∈D R ,ϕ=1
inf
ϕ, Hg ϕ ,
ϕ∈D R, ,ϕ=1
ϕ, Hg ( )ϕ ,
where D R = {ϕ ∈ D(Hg ) : χ (|x| ≤ R)ϕ = 0} and D R, = {ϕ ∈ D(Hg ( )) : χ (|x| ≤ R)ϕ = 0}. Suppose for a moment that R ( ) ≥ R
(49)
for all ∈ R3 and all R ∈ R. Then lim R→∞ R ( ) ≥ ion and hence eα|x| E (Hg ( )) < ∞ by [Gr04, Theorem 1]. Moreover, the value of the parameter R in the proof of [Gr04, Theorem 1], in the case of the Hamiltonian Hg ( ), can be chosen independent of thanks to (49). It follows that the estimate for eα|x| E (Hg ( )) from that proof is also independent of . It thus remains to prove (49). To this end we proceed by contradiction, assuming that (49) is wrong. Then there exist 0 ∈ R3 and ε > 0 such that R ( 0 ) = R − ε. Hence, we find ϕ0 ∈ D R, 0 ⊂ L 2 (R3 , dx) ⊗ F with ϕ0 = 1 and with ϕ0 , Hg ( 0 )ϕ0 ≤ R −
ε . 2
(50)
Moreover, since the map → ϕ0 , Hg ( )ϕ0 , for fixed ϕ0 , is continuous in (it is just a quadratic function in ), there exists δ > 0 such that ϕ0 , Hg ( )ϕ0 ≤ R −
ε 4
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J. Fröhlich, M. Griesemer, B. Schlein
for all with | − 0 | ≤ δ. Next, we choose f ∈ L 2 (Bδ ( 0 )) (where Bδ ( 0 ) denotes the ball of radius δ around 0 ) with f = 1, and we define ϕ ∈ L 2 (R3 , d ; L 2 (R3 , dx) ⊗ F) by ϕ( ) = f ( )ϕ0 . From (50), we obtain that ϕ, Hg ϕ =
d ϕ( ), Hg ( )ϕ( ) ≤ ( R − ε/4),
(51)
because f = 1. Since ϕ ∈ D R (which is clear from the construction of ϕ), this contradicts the definition of R . In the next proposition we prove the existence of a simple ground state for Hg ( ), provided the energy is lower than a threshold energy and the coupling constant is small enough. Proposition 2. Assume Hypotheses (H0) and (H1) are satisfied. Fix β < 1 and choose < min(β , ion ) (see Lemma 1 for the definition of β and ion ). Then, for g sufficiently small (depending on β and ), E g ( ) = inf σ (Hg ( )) is a simple eigenvalue of Hg ( ), provided that E g ( ) ≤ . Remark. Since E g ( ) ≤ E 0at + 2 /2M it suffices that E 0at + 2 /2M ≤ and that g is small enough. Proof. The proof is very similar to the proof of Theorem 4 in [FGS04]. For completeness we repeat the main ideas, but we omit details. In order to prove the proposition, we consider the modified Hamiltonian Hmod (defined in Sect. 4.2) given, on the fiber with fixed total momentum , by Hmod ( ) =
p2 ( − d(k))2 + d(ω) + + V (x) + gφ(Fx ), 2M 2m
(52)
where the dispersion law ω(k) (with ω(k) = |k| if |k| ≥ σ and ω(k) ≥ σ/2 for all k) is assumed to satisfy Hypothesis (H2) of Sect. 4.2. Set E mod ( ) = inf σ (Hmod ( )). Note that Hmod ( ) and Hg ( ) act identically on the range of (χi ), the orthogonal projection onto the subspace of vectors without soft bosons. The proof of the proposition is divided into four steps. 1) Suppose E g ( ) ≤ . Then, for sufficiently small g (depending on β and ), we have that inf E g ( − k) + |k| − E g ( ) > 0 ,
|k|≥ε
(53)
for every ε > 0. This inequality follows by perturbation of the free Hamiltonian (see Lemma 35 in [FGS04] for details). 2) E g ( ) = E mod ( ). Moreover, if ψ is an eigenvector of Hg ( ) (or of Hmod ( )) corresponding to the eigenvalue E g ( ), then ψ ∈ Ran(χi ). In particular, ψ is an eigenvector of Hg ( ) corresponding to the eigenvalue E g ( ) if and only if ψ is an eigenvector of Hmod ( ) corresponding to the eigenvalue E mod ( ) = E g ( ).
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In order to prove these statements note that the Hamiltonians Hg ( ) and Hmod ( ) act on the fiber space H = L 2 (R3 ) ⊗ F = L 2 (R3 ) ⊗ Fi ⊗ Fs , where Fs is the Fock space of the soft bosons, Fs = ⊕n≥0 L 2s Bσ (0)×n ; dk1 . . . dkn . Thus H
(n) L 2s Bσ (0)×n , dk1 . . . dkn ; L 2 (R3 ) ⊗ Fi =: H .
n≥0
n≥0
(n) The restriction of Hg ( ) to the subspace H with exactly n soft bosons is given by
(Hg ( )ψ)(k1 , . . . , kn ) = H (k1 , . . . , kn )ψ(k1 , . . . , kn ) with H (k1 , . . . , kn ) =
( − d(k) − 2M
n
j=1 k j )
2
+ d(|k|) +
n
|k j | +
j=1
p2 2m
+ V (x) + gφ(Fx ) ⎛ ⎞ ⎛ ⎞ n n n n = Hg ⎝ − kj⎠ + |k j | ≥ E g ⎝ − k j ⎠ + k j j=1 j=1 j=1 j=1 > E g ( )
if (k1 , . . . kn ) = (0, . . . 0).
(54)
In the last inequality we used the result of part (1). This proves that E g ( ) = inf σ (Hg ( )) = inf σ (Hg ( )| L 2 (R3 )⊗Fi ) = inf σ (Hmod ( )| L 2 (R3 )⊗Fi ) ≥ E mod ( ) .
(55)
Since Hg ( ) ≤ Hmod ( ), we conclude that E g ( ) = E mod ( ). Equation (54) also proves that eigenvectors of Hg ( ) corresponding to the energy E g ( ), if they exist, belong to the range of (χi ). That the same is true for eigenvectors of Hmod ( ) corresponding to the energy E g ( ) follows from an inequality for Hmod ( ) analogous to (54). 3) If E g ( ) ≤ , and for g sufficiently small (depending on β and ) we have that ( ) = inf E g ( − k) + ω(k) − E g ( ) > 0 . k
(56)
For |k| > σ/4, this follows from part (1) (because ω(k) ≥ |k|), while for |k| ≤ σ/4, this inequality follows from E g ( − k) + ω(k) − E g ( ) = E g ( − k) + |k| − E g ( ) + (ω(k) − |k|) ≥ σ/4, by (53) and by construction of ω (see Sect. 4.2). 4) inf σess (Hmod ( )) ≥ min E g ( ) + ( ), mod ( ) ,
(57)
where mod ( ) is defined like ion with Hg replaced by Hmod ( ). (Recall from 2) that E mod ( ) = E g ( )).
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The proof of part 4) is very similar to the proof of Lemma 36 in [FGS04] with a small modification at the beginning. We first need to localize with respect to the relative coordinate x. That is, we choose J0 , J∞ ∈ C ∞ (R3 , [0, 1]) with J0 (x) = 1 for |x| ≤ 1, 2 = 1. Let J J0 (x) = 0 for |x| ≥ 2 and J02 + J∞ ,R (x) = J (x/R). Then Hmod ( ) = J0,R Hmod ( )J0,R + J∞,R Hmod ( )J∞,R + O(R −2 )
(58)
as R → ∞. As in [FGS04], one shows that 2 (E g ( ) + ( )) + K J0,R Hmod ( )J0,R ≥ J0,R
with K relatively compact w.r.t. Hmod ( ), while 2 ion ( ) + o(1) J∞,R Hmod ( )J∞,R ≥ J∞,R
as R → ∞ follows from the definition of mod ( ). Part 4) follows from these estimates applied to the r.h.s. of (58). Along with 1) and 2), and since mod ( ) ≥ ion for every (see (49) and its proof), this proves that E g ( ) is an eigenvalue of Hg ( ), provided that E g ( ) ≤ and g is sufficiently small (depending on ). The proof of the fact that E g ( ) is a simple eigenvalue is given in Corollary 6, below. From now on, for fixed β < 1 and such that E g ( ) ≤ < min(β , ion ), we denote by ψ the unique (up to a phase) normalized ground state vector of Hg ( ). The vector ψ is called a dressed atom state with fixed total momentum . The space of dressed atom wave packets, Hdas ⊂ H, is defined by T Hdas = ψ ∈ L 2 { : E g ( ) ≤ }; L 2 (R3 , dx) ⊗ F : ψ( ) ∈ ψ , where ψ is the one-dimensional space spanned by the vector ψ ; Hdas is a closed linear subspace left invariant by the Hamiltonian Hg . In fact, Hg commutes with the orthogonal projection Pdas , onto Hdas . This follows from (T Pdas T ∗ ϕ)( ) = Pψ ϕ( ). 3.2. The Fermi Golden Rule. From Hypothesis (H0), and from standard results in the theory of Schrödinger operators (see [RS78]), it follows that the spectrum of Hat =
p2 +V 2m
in the negative half-axis (−∞, 0) is discrete. We denote the negative eigenvalues of Hat by E 0at < E 1at < · · · < 0. The eigenvalues E atj can accumulate at zero only. If Hat has no eigenvalues (a possibility which is not excluded by our assumptions), then our results (which only concern states of the system for which the electron is bound to the nucleus) are trivial. We denote the (finite) multiplicity of the eigenvalue E atj by m j . For fixed j ≥ 0, we denote by ϕ j,α , α = 1, . . . m j , an orthonormal basis of the eigenspace of Hat corresponding to the eigenvalue E atj . By Hypothesis (H0), the lowest eigenvalue, E 0at , of Hat is simple (m 0 = 1, see Remark (1) after Hypothesis (H0)). The unique (up to a phase) ground state vector of Hat is denoted by ϕ0 . For every fixed , the free Hamiltonian, H0 ( ) =
( − d(k))2 + d(|k|) + Hat , 2M
(59)
Rayleigh Scattering at Atoms with Dynamical Nuclei
403
Π=M
Π
E ( Π) 0
Fig. 1. Eigenvalues of the free Hamiltonian H0 ( ) (the dashed curve represents inf σ (H0 ( )))
has eigenvalues E j ( ) = E atj + 2 /2M with multiplicity m j corresponding to the eigenvectors ψ j = ϕ j,α ⊗ (where ∈ F denotes the Fock vacuum), for every j ≥ 0. For | |/M ≤ 1, E 0 ( ) = inf σ (H0 ( )), while all other eigenvalues E j ( ), j ≥ 1, are embedded in the continuous spectrum (see Fig. 1). For | |/M > 1, all eigenvalues of H0 ( ) are embedded in the continuous spectrum, and the Hamiltonian H0 ( ) does not have a ground state. We restrict our attention to the physically more interesting case | |/M ≤ 1; (this will be ensured by the condition that the total energy of the system is less than β=1 = E 0at + (1/2)M and that the coupling constant g is sufficiently small). One then expects the embedded eigenvalues E j ( ), j ≥ 1, to dissolve and to turn into resonances when the perturbation φ(Fx ) is switched on. We prove that this is indeed the case, provided the lifetime of the resonances, as predicted by Fermi’s Golden Rule in second order perturbation theory, is finite. For given i, j ≥ 0 and k ∈ R3 , we define the m i × m j matrix (Ai j (k))α,α := ϕi,α , Fx (k)ϕ j,α .
(60)
For given j ≥ 0, we then define the resonance matrix by setting 2 ( − k)2 + |k| − + E iat − E atj , (61) dk A∗ji (k)A ji (k) δ j ( ) = 2M 2M i:i≤ j
where δ(.) denotes the Dirac delta-function. Note that j ( ) is an m j × m j matrix. According to second order perturbation theory (Fermi’s Golden Rule), the m j eigenvalues of j ( ) are inverses of the lifetimes of the resonances bifurcating from the unperturbed eigenvalue E j ( ) = E atj + 2 /2M. We put γ j ( ) = inf σ ( j ( )).
(62)
Instability of the eigenvalue E j ( ) is equivalent, in second order perturbation theory, to the statement that γ j ( ) > 0.
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J. Fröhlich, M. Griesemer, B. Schlein
Hypothesis (H2). For fixed β < 1 and < min(β , ion ), we assume that inf{γ j ( ) : ∈ R3 , j ≥ 1 and E j ( ) < } > 0 .
(63)
3.3. The positive commutator. In order to prove the absence of embedded eigenvalues we use the technique of positive commutators. We prove the positivity of the commutator between the Hamiltonian Hg ( ) and a suitable conjugate operator A. Then the absence of eigenvalues follows with the help of a virial theorem. We make use of ideas from [BFSS99], adapting them to our problem. For fixed j ≥ 1, we construct a suitable conjugate operator A, and, in Proposition 3, we prove the positivity of the commutator [Hg ( ), i A] when restricted to an energy interval containing the unperturbed eigenvalue E j ( ) but no other eigenvalues of H0 ( ). In Proposition 4, we then establish a commutator estimate on an energy interval around the ground state energy E 0 ( ) of H0 ( ). For fixed j ≥ 1 we define |ϕ j,α ϕ j,α | ⊗ P , P = | | . (64) Pj = α
By definition, P j is the orthogonal projection onto the eigenspace of H0 ( ) corresponding to the eigenvalue E j ( ) = 2 /2M + E atj . Our conjugate operator is a symmetric (but not self-adjoint!) operator given by A = d(a) + i D,
(65)
where a=
1 ˆ k k · y + y · kˆ , with kˆ = , y = i∇k , 2 |k|
(66)
and D = gθ P j a(Fx )Rε2 P j − gθ P j Rε2 a ∗ (Fx )P j .
(67)
In (67), we introduced the notation P j = 1 − P j and we used −1 Rε2 = (H0 ( ) − E j ( ))2 + ε2 .
(68)
Note that ε Rε2 → δ(H0 ( ) − E j ( )) strongly, as ε → 0. The real parameters θ and ε will be fixed later on. By a formal computation, the commutator Hg ( )i A − i AHg ( ) is given by [Hg ( ), i A] = N −
( − d(k)) ˆ − gφ(ia Fx ) − [Hg ( ), D] . · d(k) M
(69)
This expression is our definition of the operator [Hg ( ), i A]. In the proof of Proposition 3 we will see that ϕ, [Hg ( ), i A]ϕ defines a quadratic form that is bounded from below (but possibly +∞) on vectors from the spectral subspaces RanE (Hg ( )) of Proposition 3. On even smaller subspaces a rigorous connection between [Hg ( ), i A] and A will be established in the proof of the Virial Theorem, Proposition 5.
Rayleigh Scattering at Atoms with Dynamical Nuclei
405
Proposition 3. We assume that Hypotheses (H0)–(H2) are satisfied. We fix β < 1 and choose < min(β , ion ); (see Lemma 1 for the definition of β and ion ). Moreover, we suppose that the interval ⊂ (−∞, ) is such that E j ( ) ∈ and d := dist , σpp (H0 ( ))\{E j ( )} > 0 . (70) Then, for g > 0 sufficiently small (depending on β, and the distance d), one can choose ε and θ such that E (Hg ( ))[Hg ( ), i A]E (Hg ( )) ≥ C E (Hg ( )) ,
(71)
with a positive constant C. Remarks. 1) The choice of the parameter ε, θ and of the constant C depends on the value of g. We can choose, for example, θ O(g κ )
ε O(g α )
C O(g 2+κ−α ),
for 0 < κ < α < 1. 2) How small g has to be chosen depends on the values of β, on γ j ( ), and on the distance d in (70). We must have that g (1 − β), γ j ( ) g α−κ , and γ j ( )d 2 max(g α−κ , g 1−α ) (for an arbitrary choice of κ, α with 0 < κ < α < 1). Moreover, g has to be sufficiently small, in order for Eq. (44) to hold true (and thus g depends on the choice of the threshold ). 3) In this proposition, we do not need the infrared cutoff in the interaction (i.e., we can choose σ = 0). However, the infrared cutoff is needed in the proof of Proposition 5 (the Virial Theorem), and hence in the proof of the absence of embedded eigenvalues. Proof. Using Eq. (44), it is easy to check that, for as above, − d(k) ˆ E (Hg ( )) E (Hg ( )) N − · d(k) M ≥ (1 − β)E (Hg ( ))(1 − P )E (Hg ( )) .
(72)
Thus, defining B = (1 − β)(1 − P ) − gφ(ia Fx ) − [Hg ( ), D] ,
(73)
we conclude that E (Hg ( ))[Hg ( ), i A]E (Hg ( )) ≥ E (Hg ( ))B E (Hg ( )),
(74)
and it is enough to prove the positivity of the r.h.s. of the last equation to complete the proof. The advantage of working with B, instead of the commutator [Hg ( ), i A], is that B is bounded with respect to the Hamiltonian Hg ( ) while [Hg ( ), i A] is not (since the number operator N is not bounded with respect to Hg ( )). In order to prove that E (Hg ( ))B E (Hg ( )) ≥ C E (Hg ( )),
(75)
we first establish the inequality E (H0 ( ))B E (H0 ( )) ≥ C E (H0 ( )).
(76)
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To this end we may assume that 1−β λ0 := inf σ E (H0 ( ))B E (H0 ( ))|RanE (H0 ( )) ≤ , (77) 2 for otherwise (76) holds with C = (1 − β)/2. This assumption will allow us to apply the Feshbach map with projection P j to the operator E (H0 ( ))B E (H0 ( )) − λ0 . Indeed, this operator restricted to Ran P j is invertible for small g and g 2 θ ε−2 as is shown in Step 1. Step 1 through Step 5 prepare the proof of (76). Step 1. There exists a constant C > 0, independent of and g, such that g2 θ P j E (H0 ( ))B E (H0 ( ))P j ≥ 1 − β − C g + 2 P j E (H0 ( )). ε
(78)
In fact, P j E (H0 ( ))B E (H0 ( ))P j ≥ (1 − β)P j E (H0 ( ))(1 − P )E (H0 ( ))P j + g P j E (H0 ( ))φ(ia Fx )E (H0 ( ))P j − g 2 θ P j E (H0 ( ))a ∗ (Fx )P j a(Fx )Rε2 E (H0 ( ))P j
− g 2 θ P j E (H0 ( ))Rε2 a ∗ (Fx )P j a(Fx )E (H0 ( ))P j .
(79)
Applying Lemma 21 of Appendix B to bound φ(ia Fx ), using that P P j E (H0 ( )) = 0 (by the choice of the interval ), and that Rε2 ≤ ε−2 , inequality (78) follows easily. Step 2. We define
−1 E = P j B P j − P j B P j E (H0 ( )) (B − λ0 )|Ran P j E (H0 ( )) × E (H0 ( ))P j B P j ,
(80)
where λ0 is defined in (77) (note that, by (77) and by the result of Step 1, the inverse on the r.h.s. of (80) is well defined, if g and g 2 θ/ε2 are small enough). Then we have λ0 ≥ inf σ E|Ran P j . (81) The proof of (81) relies on the isospectrality of the Feshbach map and can be found, for example, in [BFSS99]. This inequality says that, instead of finding a bound on the operator B restricted to the range of E (H0 ( )), we can study the operator E restricted to the much smaller range of the projection P j (which is finite-dimensional). Using the assumption (77) and Eq. (78), we see that, if g and g 2 θ/ε2 are sufficiently small, (B − λ0 )|Ran P j E (H0 ( )) ≥
1−β . 4
(82)
(Later, when we will choose the parameter θ and ε, we will make sure that g 2 θ/ε2 is small enough, if g is small enough). This implies that the operator E is bounded from below by E ≥ Pj B Pj −
4 P j B P j E (H0 ( ))P j B P j . 1−β
Next, we study the second term on the r.h.s. of inequality (83).
(83)
Rayleigh Scattering at Atoms with Dynamical Nuclei
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Step 3. There exists a constant C > 0 independent of g, θ and ε such that g4θ 2 P j B P j E (H0 ( ))P j B P j ≤ Cg 2 P j + C g 2 θ 2 + 2 P j a(Fx )Rε2 a ∗ (Fx )P j . (84) ε In order to prove this bound, we note that, for any ψ ∈ H, ψ, P j B P j E (H0 ( ))P j B P j ψ = E (H0 ( ))P j B P j ψ2 .
(85)
Furthermore P j B P j = −g P j φ(ia Fx )P j − P j [Hg ( ) − E j ( ), D]P j = −g P j φ(ia Fx )P j + gθ P j (Hg ( ) − E j ( ))P j Rε2 a ∗ (Fx )P j , (86) because P j (Hg ( ) − E j ( ))P j = P j (H0 ( ) − E j ( ))P j + g P j φ(Fx )P j = 0. Hence we find that E (H0 ( ))P j B P j ψ = −g E (H0 ( ))P j φ(ia Fx )P j ψ + gθ P j E (H0 ( ))(H0 ( ) − E j ( ))Rε2 a ∗ (Fx )P j ψ + g 2 θ P j E (H0 ( ))φ(Fx )P j Rε2 a ∗ (Fx )P j ψ.
(87)
Using Lemma 21 and the bound (H0 ( ) − E j ( ))Rε ≤ 1, we conclude that E (H0 ( ))P j B P j ψ ≤ CgP j ψ + C gθ + g 2 θ/ε Rε a ∗ (Fx )P j ψ. (88) Taking the square of this inequality, we obtain (84). Step 4. We show that P j B P j = 2g 2 θ P j a(Fx )Rε2 a ∗ (Fx )P j .
(89)
Using that P j (1 − P )P j = 0 and P j φ(iaG x )P j = 0, we easily find that P j B P j = gθ P j Hg ( )P j Rε2 a ∗ (Fx )P j + gθ P j a(Fx )Rε2 P j Hg ( )P j .
(90)
Writing Hg ( ) = H0 ( ) + gφ(Fx ), using that P j commutes with H0 ( ), and that P j a ∗ (Fx ) = a(Fx )P j = 0 we find P j B P j = gθ P j a(Fx )P j Rε2 a ∗ (Fx )P j + gθ P j a(Fx )Rε2 P j a ∗ (Fx )P j .
(91)
Equation (89) now follows writing P j = 1 − P j and using that P j a(Fx )P j = P j a ∗ (Fx )P j = 0. From Step 3 and Step 4 and from (83) we get g4θ 2 P j a(Fx )Rε2 a ∗ (Fx )P j − Cg 2 P j E ≥ 2g 2 θ − C g 2 θ 2 + 2 (92) ε for a constant C independent of g, θ and ε.
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Step 5. Next, we claim that P j a(Fx )Rε2 a ∗ (Fx )P j ≥
γ j ( ) (1 + oε (1))P j , ε
(93)
where oε (1) → 0, as ε → 0. In order to prove (93), we use the pull-through formula for a(q)Rε2 and the fact that a(q)P j = 0. This yields P j a(Fx )Rε2 a ∗ (Fx )P j = dqdq P j Fx (q)a(q)Rε2 Fx (q )a ∗ (q )P j =
dqdq P j Fx (q)
−1 2 ( − q − d(k))2 2 + |q| + d(|k|) + Hat − E j ( ) + ε 2M
×Fx (q )a(q)a ∗ (q )P j 2 −1 ( − q − d(k))2 2 = dq P j Fx (q) + |q| + d(|k|) + Hat − E j ( ) + ε 2M ×Fx (q)P j . (94) m j Next, we write P j = P jat ⊗ P , where P jat = α=1 |φ j,α φ j,α | is the orthogonal projection onto the eigenspace of Hat corresponding to the eigenvalue E atj . Then we obtain that P j a(Fx )Rε2 a ∗ (Fx )P j 2 −1 2 ( − q)2 at at 2 + |q| − + Hat − E j = +ε dq P j Fx (q) 2M 2M ×Fx (q)P jat ⊗ P . (95) All operators involved in the expression on the r.h.s. of (95) act trivially on the Fock space, and we get the lower bound
−1 2 ( − q)2 2 at 2 + |q| − + Hat − E j +ε Fx (q)P jat dq 2M 2M 2 −1 ( − q)2 2 at at at 2 + |q| − + Hat − E j ≥ +ε dq P j Fx (q)Pm 2M 2M P jat Fx (q)
m≤ j ×Pmat Fx (q)P jat
( − q)2 2 = dq + |q| − + E mat − E atj 2M 2M m≤ j α,α × A∗m j (q)Am j (q) |ϕ j,α ϕ j,α |. α,α
−1
2 +ε
2
(96)
Rayleigh Scattering at Atoms with Dynamical Nuclei
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Using that ε(x 2 + ε2 )−1 = δ(x) + oε (1), as ε → 0, and recalling the definition of the matrix j ( ), we find that 1 (1 + oε (1))ψ, P j j ( )P j ψ ε γ j ( ) (1 + oε (1))P j ψ2 . ≥ ε
ψ, P j a(Fx )Rε2 a ∗ (Fx )P j ψ ≥
(97)
This proves Eq. (93). Proof of Eq. (76). From (92) and (93), we derive that γ j ( ) g4θ 2 2 2 2 2 E≥ (1 + oε (1)) − Cg P j . 2g θ − C g θ + 2 ε ε
(98)
Choosing ε = g α and θ = g κ , with 0 < κ < α < 1, we get E ≥ γ j ( )g 2+κ−α P j ,
(99)
for g sufficiently small. Note that, with this choice of ε and θ , g 2 θ/ε2 = g 2+κ−2α 1, and thus (82) is satisfied, if g is small enough. From (77) and (81) we then get that E (H0 ( ))B E (H0 ( )) ≥ γ j ( )g 2+κ−α E (H0 ( ))
(100)
which proves (76). Proof of Eq. (75). To prove (75), and hence complete the proof of the proposition, we must replace E (H0 ( )) by the spectral projection E (Hg ( )) of the full Hamiltonian Hg ( ). Given an interval ⊂ (−∞, ) with E j ∈ and dist , σpp (H0 ( ))\{E j } > 0, (101) we choose an interval ⊂ (−∞, ) such that ⊂ , and dist , σpp (H0 ( ))\{E j } > 0
(102)
and with δ = dist(, c ) > 0. Furthermore, we choose a function χ ∈ C ∞ (R) with the property that χ = 1 on and χ = 0 on c . We can assume that |χ (s)| ≤ Cδ −1 . Applying (100) with replaced by , and multiplying the resulting inequality with χ (H0 ( )) we get χ (H0 ( ))Bχ (H0 ( )) ≥ γ j ( )g 2+κ−α χ 2 (H0 ( )) .
(103)
Setting χ := χ (Hg ( )) and χ0 := χ (H0 ( )), we have that χ Bχ = χ0 Bχ0 + (χ − χ0 )Bχ0 + χ0 B(χ − χ0 ) + (χ − χ0 )B(χ − χ0 ) .
(104)
Using (χ − χ0 )Bχ0 + χ0 B(χ − χ0 ) ≥ −(1/2)χ0 Bχ0 − 2(χ − χ0 )|B|(χ − χ0 )
(105)
we find that χ Bχ ≥ 1/2χ0 Bχ0 − 3(χ − χ0 )|B|(χ − χ0 ).
(106)
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Next we use that
1 1 − dz ∂z¯ χ (z) Hg ( ) − z H0 ( ) − z 1 1 = g dz ∂z¯ χ (z) φ(Fx ) . Hg ( ) − z H0 ( ) − z
χ − χ0 =
(107)
From the definition of B it follows that (H0 ( ) − z 1 )−1 |B|(Hg ( ) − z)−1 ≤ C(1 + gθ ε−2 ) = C(1 + g 1+κ−2α ), with the choice ε = g α , θ = g κ . This implies that ψ, (χ − χ0 )|B|(χ − χ0 )ψ ≤ C g 2 + g 3+κ−2α ψ2 ,
(108)
where the constant C depends on δ (C is proportional to δ −2 ). Thus, with (106), χ Bχ ≥ (1/2)γ j ( )g 2+κ−α χ02 − C(g 2 + g 3+κ−2α ) ≥ (1/2)γ j ( )g 2+κ−α χ 2 − C(g 2 + g 3+κ−2α ).
(109)
Since 0 < κ < α < 1, we have g 2+κ−α g 2 and g 2+κ−α g 3+κ−2α . Therefore, multiplying from the left and the right with E (Hg ( )), and choosing g small enough, we find that E (Hg ( ))B E (Hg ( )) ≥ C E (Hg ( )),
(110)
for some positive constant C, which, with (74), completes the proof of the proposition. Proposition 3 and Proposition 5, below, prove absence of embedded eigenvalues of Hg ( ) on (−∞, ) with the exception of a small interval around the ground state energy, inf σ (H0 ( )). Absence of embedded eigenvalues near the ground state energy follows from our next proposition. Recall from (66) the notation 1 ˆ k · y + y · kˆ , a := 2 with kˆ = k/|k|, y = i∇k . A formal calculation shows that − d(k) ˆ − gφ(ia Fx ). · d(k) (111) M This expression is our definition of the operator [Hg ( ), id(a)]. The remarks after (69) apply equally to the connection between [Hg ( ), id(a)] and d(a). [Hg ( ), id(a)] = N −
Proposition 4. Assume Hypotheses (H0)–(H1). Fix β < 1, and choose < min(β , ion ), with β = E 0at + (M/2)β 2 . Suppose the interval ⊂ (−∞, ) is such that ⊂ (−∞, E 1 ( )) and d = dist (, E 1 ( )) > 0 .
(112)
(Recall that E 1 ( ) denotes the first excited eigenvalue of the free Hamiltonian H0 ( )). Then, if g ≥ 0 is sufficiently small (depending on β, and d), there exists C > 0 such that E (Hg ( ))[Hg ( ), id(a)]E (Hg ( )) ≥ (1 − β)E (Hg ( )) 1 − Pϕ0 ⊗ E (Hg ( )) − Cg E (Hg ( )).
(113)
Rayleigh Scattering at Atoms with Dynamical Nuclei
411
Proof. By definition of [Hg ( ), id(a)], E (Hg ( ))[Hg ( ), id(a)]E (Hg ( )) ≥ (1 − β)E (Hg ( ))N E (Hg ( )) − g E (Hg ( ))φ(ia Fx )E (Hg ( )) ≥ (1 − β − Cg)E (Hg ( )) − (1 − β)E (Hg ( ))P E (Hg ( )).
(114)
Here we use that, by Hypothesis (H1), E (Hg ( ))| − d(k)|/M ≤ β, and that, by Lemma 21, φ(ia Fx )E (Hg ( )) ≤ C. Next, we note that E (Hg ( ))P E (Hg ( )) = E (Hg ( )) χ (Hat = E 0at ) ⊗ P E (Hg ( )) + E (Hg ( )) χ (Hat ≥ E 1at ) ⊗ P E (Hg ( )) = E (Hg ( ))Pϕ0 ⊗ E (Hg ( )) + E (Hg ( ))χ (H0 ( ) ≥ E 1 ( ))E (Hg ( )), (115) where ϕ0 is the unique (up to a phase) ground state vector of the atomic Hamiltonian Hat (the fact that the ground state vector of Hat is unique follows from Hypothesis (H0), see Remark (1) after Hypothesis (H0)). Next we choose a function χ ∈ C ∞ (R) with χ (s) = 0 for s ≤ E 1 ( ) − d and χ (s) = 1 for s ≥ E 1 ( ). Equation (115) then implies that E (Hg ( ))P E (Hg ( )) ≤ E (Hg ( ))Pϕ0 ⊗ E (Hg ( )) + E (Hg ( ))χ (H0 ( ))E (Hg ( )). Note that
1 1 − dz∂z¯ χ (z) H0 ( ) − z Hg ( ) − z 1 1 φ(Fx ) . = Cg dz∂z¯ χ (z) H0 ( ) − z Hg ( ) − z
(116)
χ (H0 ( )) − χ (Hg ( )) =
(117)
Since, by definition of the interval , χ (Hg ( ))E (Hg ( )) = 0, we find that E (Hg ( ))χ (H0 ( ))E (Hg ( )) = E (Hg ( )) × χ (Hg ( )) − χ (H0 ( )) E (Hg ( )) ≤ Cg E (Hg ( )). (118) With (114) and (116), this shows that E (Hg ( ))[Hg ( ), id(a)]E (Hg ( )) ≥ (1 − β)E (Hg ( )) 1 − Pϕ0 ⊗ E (Hg ( )) − Cg E (Hg ( )).
(119)
Proposition 5 (Virial Theorem). Let Hypotheses (H0)–(H1) be satisfied, and assume that Hg ( )ϕ = Eϕ, for some ϕ ∈ L 2 (R3 ) ⊗ F with (χi )ϕ = ϕ, and for an energy E < ion . Then ϕ, [Hg ( ), i A]ϕ = 0, and ϕ, [Hg ( ), id(a)]ϕ = 0, where [Hg ( ), i A] and [Hg ( ), id(a)] are defined by (69) and (111), respectively.
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Proof. We choose ω ∈ C ∞ (R3 ) with ω(k) = |k|, if |k| > σ , ω(k) ≥ σ/2 for all k, and we define amod =
1 d(∇ω(k) · y + y · ∇ω(k)). 2
As in (111) and (69), the commutators [Hmod ( ), id(amod )] and [Hmod ( ), i Amod ] are defined in terms of symmetric operators − d(k) · d(∇ω) − gφ(iamod Fx ), M [Hmod ( ), i Amod ] := [Hmod ( ), id(amod )] − [Hmod ( ), D].
[Hmod ( ), id(amod )] := d(|∇ω|2 ) −
Since ω(k) = |k| for |k| ≥ σ , these operators coincide with [Hg ( ), id(a)] and [Hg ( ), i A] on states without soft bosons, that is, on the range of the projection (χi ). Therefore it is enough to prove that ϕ, [Hmod ( ), id(amod )]ϕ = 0
(120)
ϕ, [Hmod ( ), i Amod ]ϕ = 0.
(121)
and
This is done in the same way as in the proof of Lemma 40 in [FGS04]. Corollary 6. Assume Hypotheses (H0)–(H2). Fix β < 1 and choose < min(β , ion ), with β and ion as in Lemma 1. Then, for sufficiently small values of g > 0, σpp (Hg ( )) ∩ (−∞, ) = {E g ( )},
(122)
where E g ( ) is a simple eigenvalue, for all with E g ( ) ≤ . Remark. How small g has to be chosen depends on the choice of β (g 1 − β), on the choice of (we need (44) to hold true), on γ j ( ) (g inf{γ j ( ) : j ≥ 1, E g ( ) ≤ }), and it also depends on the distances between the eigenvalues of the atomic Hamiltonian (we must require that g 1/2 min{|E atj+1 − E atj | : 0 ≤ j ≤ n}, where n is such at ). that E nat ≤ < E n+1 Proof. We first prove that if g is sufficiently small, then σpp Hg ( )|`Ran(χi ) ∩ (−∞, ) = {E g ( )},
(123)
and that E g ( ) is a simple eigenvalue of Hg ( )|`Ran(χi ). To this end, we define 0 = (−∞, (E 1 ( ) + E 0 ( ))/2). We define intervals (E j ( ) + 2E j−1 ( )) (E j ( ) + E j+1 ( )) j = , , 3 2 for j = 1, . . . n, where n is such that E n−1 ( ) < ≤ E n ( ). Each interval j contains exactly one eigenvalue of the free Hamiltonian H0 ( ), and (−∞, ) ⊂ ∪nj=0 j . The absence of eigenvalues of Hg ( )|`Ran(χi ) inside j , for j ≥ 1 and for g sufficiently small, follows from Propositions 3 and 5. Next, suppose that ψ is a normalized eigenvector of Hg ( ) corresponding to an eigenvalue E ∈ 0 . Without loss of generality, we can assume that ψ, ϕ0 ⊗ is real; recall that ϕ0 is the unique (up to a phase)
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normalized ground state vector of Hat = p 2 /2m + V (x). Then, by Proposition 4 and Proposition 5, we have that 0 ≥ (1 − β)ψ, (1 − Pϕ0 ⊗ )ψ − Cg = (1 − β) 1 − ψ, ϕ0 ⊗ 2 − Cg ≥ (1 − β) (1 − ψ, ϕ0 ⊗ ) − Cg =
1−β ψ − ϕ0 ⊗ 2 − Cg. 2
(124)
Hence ψ − ϕ0 ⊗ 2 ≤
2Cg . 1−β
(125)
If there were two orthogonal eigenvectors of Hg ( )|`Ran(χi ), ψ1 and ψ2 , corresponding to eigenvalues in 0 then both would satisfy inequality (125), and, thus, we would conclude that 2Cg ψ1 − ψ2 ≤ 2 . (126) 1−β But this is impossible if g ≤ (1 − β)/4C. So, for g small enough, there can only be one eigenvector of Hg ( )|`Ran(χi ) corresponding to an eigenvalue in 0 . In Proposition 2, we have proven that Hg ( )|`Ran(χi ) has a ground state vector. This proves the fact that E g ( ) is a simple eigenvalue of Hg ( )|`Ran(χi ) as well as the fact that Hg ( )|`Ran(χi ) has no other eigenvalue in 0 . Hence (123) follows. To complete the proof of the corollary, we need to show that σpp Hg ( )|`(Ran(χi ))⊥ = ∅ . (127) To this end, we decompose F Fi ⊗ Fs ⊕n≥0 L 2s (Bσ (0)×n , dk1 . . . dkn ; Fi ), and we write L 2 (R3 , dx) ⊗ F ⊕n≥0 Hn , where Hn = L 2s (Bσ (0)×n , dk1 . . . kn ; L 2 (R3 , dx) ⊗ Fi ) is the space of vectors containing exactly n soft, non-interacting, bosons. The Hamiltonian leaves each Hn invariant, and the restriction of Hg ( ) on Hn is given by (Hg ( )|`Hn ψ)(k1 , . . . kn ) = H (k1 , . . . , kn )ψ(k1 , . . . kn ) H (k1 , . . . , kn ) = Hg ( − k1 − · · · − kn ) +
n
|k j | .
(128)
j=1
Here Hg ( − k1 − · · · − kn ) is an operator over L 2 (R3 , dx) ⊗ Fi , the space of states with no soft bosons. We know that the only eigenvalue of Hg ( − k1 − · · · − kn ) in (−∞, ) is its ground state energy E g ( − k1 − · · · − kn ) as long as E g ( − k1 − · · · − kn ) < . In particular the only eigenvalue of H (k1 , . . . , kn ) in (−∞, ) is given by E g ( −k1 −. . . kn )+|k1 |+· · ·+|kn | if this number is less than . Thus E ∈ (−∞, ) is an eigenvalue of Hg ( )|`Hn if and only if there exists a set M ⊂ Bσ (0)×n with positive measure, such that E = E g ( − k1 − · · · − kn ) + |k1 | + · · · + |kn | for all (k1 , . . . , kn ) ∈ M. But this is impossible because, by (44) and β < 1, |∇ E g ( )| = |ψ , ( − d(k))/Mψ | ≤ 1 for every with E g ( ) < and g small enough.
(129)
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4. Scattering Theory The proofs of most of the results in this section are similar to those of the corresponding results in [FGS04]. In order to give an idea of the structure of the proof of our main result (asymptotic completeness, Theorem 9), we repeat here the most important theorems, but we omit most of their proofs (we refer to the corresponding statements in [FGS04]). The main difference with respect to [FGS04] is encountered in the proof of the positivity of the asymptotic observable in Sect. 4.4: there, we propose some new ideas to control the internal degrees of freedom of the atom (which are not present in [FGS04], because there we considered free electrons coupled to the quantized radiation field).
4.1. The wave operator. The first step towards understanding scattering theory for the model studied in this paper consists in the construction of states with asymptotically free photons. This can be accomplished using asymptotic field operators, which are constructed in the next theorem. Note that in Theorems 7 and 8 we do not impose any infrared cutoff on the interaction; we can take σ = 0, provided the form factor κ(k) is smooth at k = 0. We use the notation L 2ω (R3 ) = L 2 (R3 , (1 + 1/|k|)dk). Theorem 7 (Existence of asymptotic field operators). Assume Hypotheses (H0)–(H1) are satisfied (but σ = 0 is allowed!). Fix β < 1 and choose < min(β , ion ) (with β as in Lemma 1). If g ≥ 0 is so small that (44) is true, then the following results hold. i) Let h ∈ L 2ω (R3 ) and let h t (k) = e−i|k|t h(k). Then the limit
a+ (h)ϕ = lim ei Hg t a (h t )e−i Hg t ϕ t→∞
exists for all ϕ ∈ RanE (Hg ). ii) Let h, g ∈ L 2ω (R3 ). Then
[a+ (g), a+∗ (h)] = (g, h) and [a+ (g), a+ (h)] = 0, in the sense of quadratic forms on RanE (Hg ) (a (h) means either a ∗ (h) or a(h)). iii) Let h ∈ L 2ω (R3 ), and let M := sup{|k| : h(k) = 0} and m := inf{|k| : h(k) = 0}. Then a+∗ (h)Ranχ (Hg ≤ E) ⊂ Ranχ (Hg ≤ E + M), a+ (h)Ranχ (Hg ≤ E) ⊂ Ranχ (Hg ≤ E − m), if E ≤ . iv) Let h i ∈ L 2ω (R3 ) for i = 1, . . . n. Put Mi = sup{|k| : h i (k) = 0} and assume n Mi ≤ we have that ϕ ∈ D(a+ (h 1 ) . . . a+ (h n )), ϕ ∈ RanE λ (Hg ). Then if λ+ i=1 the limits
a+ (h 1 ) . . . a+ (h n )ϕ = lim ei Hg t a (h 1,t ) . . . a (h n,t )e−i Hg t ϕ t→∞
exist, and
a+ (h 1 ) . . . a+ (h n )(Hg + i)−n/2 ≤ Ch 1 ω . . . h n ω .
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v) If ϕ ∈ E (Hg )Hdas and h ∈ L 2ω (R3 ), a+ (h)ϕ = 0. (Wave packets of dressed atom states are vacua of the asymptotic field operators.) The proof of this theorem is very similar to the one of Theorem 13 and Lemma 14 in [FGS04]. It relies on a propagation estimate for the center of mass of the atom (see Proposition 12 in [FGS04]), which guarantees that if the energy is smaller than β , then the asymptotic velocity of the atom is bounded above by β (here β < 1), and it exploits the fact that, because the energy is below the ionization threshold, the electron is exponentially bound to the nucleus. These two facts and the fact that the propagation speed of photons is the speed of light are sufficient to prove that the interaction between the atom and asymptotically freely propagating photons tends to zero, as t → ∞. The existence of asymptotic field operators allows us to introduce the wave operator + of the system. In order to define + , we add a new copy of the Fock space F describing states of free photons to the physical Hilbert space H = L 2 (R3 , dX ) ⊗ L 2 (R3 , dx) ⊗ F. We define the extended Hamiltonian H˜ g = Hg ⊗ 1 + 1 ⊗ d(|k|)
(130)
= H ⊗ F. In the next theorem, we establish the exison the extended Hilbert space H to a subspace of the tence of the wave operator + as an isometry from a subspace of H physical Hilbert space H. The “scattering identification map”, I , used in the definition of the wave operator + , is defined in Appendix A.6. Theorem 8 (Existence of the wave operator). Let Hypotheses (H0)–(H1) be satisfied (but σ = 0 is allowed). Fix β < 1, and choose < min(β , ion ) (with β , ion defined as in Lemma 1). Then if g ≥ 0 is small enough (depending on β and ) the limit ˜
+ ϕ := lim ei Hg t I e−i Hg t (Pdas ⊗ 1)ϕ t→∞
(131)
exists for an arbitrary vector ϕ in the dense subspace of RanE ( H˜ ) spanned by finite linear combinations of vectors of the form γ ⊗ a ∗ (h 1 ) . . . a ∗ (h n ) , where γ = E λ (Hg )γ , h i ∈ L 2ω (R3 ), and with λ+ i sup{|k| : h i (k) = 0} ≤ . If ϕ = γ ⊗a ∗ (h 1 ) . . . a ∗ (h n ) belongs to this space then + ϕ = a+∗ (h 1 ) . . . a+∗ (h n )Pdas γ .
(132)
Furthermore + = 1, and + has therefore a unique extension, also denoted by + , ˜ On (Pdas ⊗ 1)E ( H˜ g )H, ˜ the operator + is isometric. For all t ∈ R, to E ( H˜ g )H. ˜
e−i Hg t + = + e−i Hg t . For the proof of this theorem we refer to the proof of Theorem 15 in [FGS04], which is almost identical. From Eq. (132) we see that vectors in the range of + are limits of linear combination of vectors describing the wave packet of dressed atom states and configurations of finitely many asymptotically freely moving photons. Physically, it is expected that the asymptotic evolution of every state with an energy below the ionization threshold of the atom (that is < ion ) and so small that the atom does not propagate
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with a velocity larger than one (i.e., < β=1 ) can be approximated by linear combinations of vectors describing a dressed atom state and a configuration of finitely many freely propagating photons. More precisely, one expects that Ran + ⊃ RanE (Hg ), if < min(β=1 , ion ). This statement is called asymptotic completeness of Rayleigh scattering. Due to technical difficulties, we can only prove asymptotic completeness for states with energy less than a threshold energy < min(β=1/3 , exp ) and assuming that the coupling constant g is small enough. The following theorem is our main result. Theorem 9 (Asymptotic Completeness). Assume that Hypotheses (H0)–(H2) are satisfied (see Eqs. (27), and (63)). Fix β < 1/3, and choose < min(β , ion ) (with β and ion as in Lemma 1). Then, for g > 0 sufficiently small, Ran + ⊃ E (Hg )H. Remark. The allowed range of values of g depends on the value of (1/3 − β); (we need that g 1/3 − β), on the choice of (g must be small enough in order for Eq. (44) to hold true), on the value of α = inf{γ j ( ) : j ≥ 1, E g ( ) < } (g α), and on at (g 1/2 δ). As δ = min{|E atj+1 − E atj | : 0 ≤ j ≤ n}, with n such that E nat ≤ < E n+1 remarked in Sect. 2, the assumption that g is positive is not necessary, it only simplifies the notation (but g = 0 is not allowed, because in this case the fiber Hamiltonian Hg ( ) has embedded eigenvalues). Theorem 9 will be seen to follow from Lemma 11, where we show that it suffices to prove an analogous statement for a modified Hamiltonian Hmod (introduced in the next section) and from Theorem 19 in Sect. 4.4, where asymptotic completeness for Hmod is proven. 4.2. The modified Hamiltonian. The fact that the bosons are massless leads to some technical difficulties connected with the unboundedness of the operator N = d(1) with respect to the Hamiltonian. However, as long as the infrared cutoff is strictly positive, the number of bosons with energy below σ is conserved. This allows us to introduce a modified Hamiltonian, where the dispersion law of the soft, non-interacting, photons is changed. We define p2 P2 + + V (x) + d(ω) + gφ(G X,x ), 2M 2m and we assume that the dispersion law ω has the following properties: Hmod =
Hypothesis (H3). ω ∈ C ∞ (R3 ), with ω(k) ≥ |k|, ω(k) = |k|, for |k| ≥ σ , ω(k) ≥ σ/2, for all k ∈ R3 , supk |∇ω(k)| ≤ 1, and ∇ω(k) = 0 unless k = 0. Furthermore, ω(k1 + k2 ) ≤ ω(k1 ) + ω(k2 ) for all k1 , k2 ∈ R3 . (Here σ > 0 is the infrared cutoff defined in Hypothesis (H1).) The two Hamiltonians, Hg and Hmod , agree on states of the system without soft bosons. Recall that χi (k) is the characteristic function of the set {k : |k| ≥ σ } and that the operator (χi ) is the orthogonal projection onto the subspace of vectors describing states without soft bosons. It is straightforward to check that Hg and Hmod leave the range of the projection (χi ) invariant and that Hg |`Ran(χi ) = Hmod |`Ran(χi ).
(133)
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The same conclusion can be reached using the unitary operator U : H → Hi ⊗ Fs introduced in Sect. 2. On the factorized Hilbert space Hi ⊗ Fs , the Hamiltonians Hg and Hmod are given by U Hg U ∗ = Hi ⊗ 1 + 1 ⊗ d(|k|), U Hmod U ∗ = Hi ⊗ 1 + 1 ⊗ d(ω) with p2 P2 + + V (x) + d(|k|) + gφ(G X,x ), Hi = 2M 2m
(134)
and we see explicitly that the two Hamiltonians agree on states without soft bosons. The modified Hamiltonian Hmod , just like the physical Hamiltonian Hg , commutes with spatial translations, i.e., [Hmod , ] = 0, where = P + d(k) is the total momentum of the system. In the representation of the system on the Hilbert space L 2 (R3 ; L 2 (R3 , dx) ⊗ F), the modified Hamiltonian Hmod is given by (T Hmod T ∗ ψ)( ) = Hmod ( )ψ( ), Hmod ( ) =
p2 ( − d(k))2 + + V (x) + d(ω) + gφ(Fx ), 2M 2m
where T : H → L 2 (R3 , d ; L 2 (R3 , dx) ⊗ F) has been defined in Sect. 2. The fiber Hamiltonians Hg ( ) and Hmod ( ) commute with the projection (χi ) and agree on its range, Hg ( )|`Ran(χi ) = Hmod ( )|`Ran(χi ).
(135)
In the proof of Proposition 2 we have shown that if β < 1 and < min(β , ion ) then, for g small enough, inf σ (Hmod ( )) = inf σ (Hg ( )) = E g ( ), where E g ( ) is a simple eigenvalue of Hg ( ) and of Hmod ( ), as long as E g ( ) ≤ . Moreover, the corresponding dressed atom states coincide. Since the subspace Hdas is defined in terms of the dressed atom states ψ , it follows that vectors in Hdas also describe dressed atom wave packets for the dynamics generated by the modified Hamiltonian Hmod . We remark that σpp (Hmod ( )) ∩ (−∞, ) = {E g ( )}, for all ∈ R3 with E g ( ) ≤ , and for g sufficiently small; see Eq. (123) and Corollary 6. Next, we discuss the scattering theory for the modified Hamiltonian Hmod . As in Theorem 8 we fix β < 1 and we choose < min(β , ion ). Then, by the assumption that ω(k) = |k| for |k| ≥ σ , and since d(|k| − ω) commutes with Hg and Hmod we have that ei Hmod t a (e−iωt h)e−i Hmod t = ei Hg t e−id(|k|−ω)t a (e−iωt h)eid(|k|−ω)t e−i Hg t = ei Hg t a (e−i|k|t h)e−i Hg t , (136) for all t. It follows that the limit
amod,+ (h)ϕ = lim ei Hmod t a (e−iωt h)e−i Hmod t ϕ t→∞
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exists and that amod,+ (h)ϕ = a+ (h)ϕ, for all ϕ ∈ RanE (Hmod ) ⊂ RanE (Hg ) and for all h ∈ L 2ω (R3 ). This and the fact that vectors in Hdas describe dressed atom states for Hg and for Hmod show that the asymptotic states constructed with the help of the Hamiltonians Hg and Hmod coincide. On the extended Hilbert space H˜ = H ⊗ F, we define the extended modified Hamiltonian H˜ mod = Hmod ⊗ 1 + 1 ⊗ d(ω). ˜ mod In terms of Hmod and H˜ mod we also define a modified version, + , of the wave operator + introduced in Sect. 4.1. Lemma 10. Let Hypotheses (H0), (H1) and (H3) be satisfied (σ = 0 in Hypothesis (H1) is allowed, and then Hmod = Hg ). Fix β < 1 and < min(β , 0). Then if g ≥ 0 is sufficiently small, depending on β and , the limit i Hmod t −i H˜ mod t ˜ mod Ie ϕ + ϕ = lim e t→∞
(137)
˜ Moreover, the modified wave operator mod exists for all ϕ ∈ E ( H˜ mod )H. + , defined by mod mod ˜ + = + (Pdas ⊗ 1), agrees with + on RanE ( H˜ mod ). That is, mod + ϕ = + ϕ,
(138)
for all ϕ ∈ RanE ( H˜ mod ) ⊂ RanE ( H˜g ). We now extend the domain of + to include arbitrarily many soft, non-interacting bosons. As a byproduct we obtain a proof of (138). To start with, we recall the isomorphism U : F → Fi ⊗ Fs introduced in Sec. 2 and define a unitary isomorphism U ⊗ U : H˜ → Hi ⊗ Fi ⊗ Fs ⊗ Fs separating interacting from soft bosons in the ˜ With respect to this factorization the extended Hamiltonian extended Hilbert space H. H˜ becomes H˜ g = H˜ i ⊗ 1 ⊗ 1 + 1 ⊗ 1 ⊗ d(|k|) ⊗ 1 + 1 ⊗ 1 ⊗ 1 ⊗ d(|k|), where H˜ i = Hi ⊗ 1 + 1 ⊗ d(|k|). As an operator from Hi ⊗ Fi ⊗ Fs ⊗ Fs to Hi ⊗ Fs , the wave operator + acts as soft U + (U ∗ ⊗ U ∗ ) = int + ⊗ +
(139)
where int + : Hi ⊗ Fi → Hi is given by ˜
i Hi t −i Hi t int int Ie (Pdas ⊗ 1) + = s − lim e t→∞
(140)
while soft + : Fs ⊗ Fs → Fs is given by soft + = I (P ⊗ 1),
(141)
where P is the orthogonal projection onto the vacuum vector ∈ Fs . In view of (139) and (140), the domain of + can obviously be extended to RanE ( H˜ i ) ⊗ Fs ⊗ ˜ mod = Fs ⊃ RanE ( H˜ g ). For the modified wave operator mod + + (Pdas ⊗ 1), we have int mod soft + = +,mod ⊗ + , and from Hg |`Ran(χi ) = Hmod |`Ran(χi ) it follows that int mod is well defined on RanE ( H int ˜ i ) ⊗ Fs ⊗ Fs and +,mod = + . Consequently, also + mod = + . + We summarize the main conclusions in a lemma.
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Lemma 11. Let the assumptions of Lemma 10 be satisfied, and let + be defined on RanE ⊗ Fs ⊗ Fs , as explained above. Then Ran + ∼ = Ran int + ⊗ Fs
(142)
in the factorization H ∼ = Hi ⊗Fs . In particular, the following statements are equivalent: i) ii) iii) iv)
Ran + Ran + Ran + Ran +
⊃ E (Hg )H, ⊃ (χi )E (Hg )H, ⊃ E (Hmod )H, ⊃ (χi )E (Hmod )H.
4.3. Existence of the asymptotic observable and of the inverse wave operator. Fix β < 1 and choose < min(β , ion ) (recall from Lemma 1 that β = E 0at + Mβ 2 /2). We choose numbers β1 , β2 , β3 and γ such that β < β1 < β2 < β3 < γ . Definition. We pick a function χγ ∈ C∞ (R; [0, 1]) such that χγ ≡ 1 on [γ , ∞) and χγ ≡ 0 on (−∞, β3 ]. Our asymptotic observable W is defined by W = s − lim ei Hmod t f (Hmod )d(χγ (|y|/t)) f (Hmod )e−i Hmod t , t→∞
where the energy cutoff f is smooth and supported in (−∞, ). For the existence of this limit, see Proposition 13 below. The physical meaning of the asymptotic observable is easy to understand: W measures the number of photons that are propagating with an asymptotic velocity larger than γ . We will prove in Sect. 4.4 that W is positive when restricted to the subspace of vectors orthogonal to the space Hdas of wave packets of dressed atom states. Instead of inverting the wave operator + directly, we can then invert it with respect to the asymptotic observable W . More precisely, we define an operator W+ : H → H˜ = H ⊗ F, called + W+ . Then, using the positivity of W , we the inverse wave operator, such that W = can construct an inverse of + . In order to define W+ , we need to split each boson state into two parts, the second part being mapped to the second Fock-space of prospective asymptotically freely moving bosons. Definition. We define jt : h = L 2 (R3 , dk) → h⊕h as follows: let jt h = ( j0,t h, j∞,t h), where j,t (y) = j (|y|/t), j ∈ C ∞ (R; [0, 1]), j0 + j∞ ≡ 1, j0 ≡ 1 on (−∞, β2 ], supp( j0 ) ⊂ (−∞, β3 ] while j∞ ≡ 1 on [β3 , ∞) and supp( j∞ ) ⊂ [β2 , ∞). Then the inverse wave operator W+ is given by ˜
˘ jt )d(χγ (|y|/t)) f (Hmod )e−i Hmod t , W+ = s − lim ei Hmod t f ( H˜ mod )( t→∞
where f is a smooth energy cutoff supported in (−∞, ). See Appendix A.5 for the ˘ jt ). For the existence of this limit, see Proposition 14. definition of the operator ( Note that, since by definition β < γ , the photons which propagate with velocity larger than γ are asymptotically free. To prove this fact, notice first that Lemma 1 continues to hold with Hg replaced by Hmod . Hence, the assumption that supp f ⊂ (−∞, ) (where f is the energy cutoff appearing in the definition of W and W+ ) with < min(β , ion )
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guarantees, for sufficiently small g, that both the nucleus and the electron remain inside a ball of radius βt around the origin. In fact, using the assumption < β (and g small enough), we can prove, analogously to Proposition 12 in [FGS04], that s − lim h(|X |/t) f (Hmod )e−it Hmod = 0
t→∞ with h ∈
(143)
for any h ∈ C ∞ (R) C0∞ (R), supp h ⊂ (β, ∞) and for any f ∈ C0∞ (R) with supp f ⊂ (−∞, ). Recall that X is the coordinate of the center of mass of the atom. Moreover, the assumption that < ion implies that the electron and the nucleus remain exponentially bound for all times; therefore, both the electron and the nucleus are localized inside the ball of radius βt. As a consequence, the interaction strength between the nucleus (or the electron) and those bosons counted by d(χγ (|y|/t)) decays in t at an integrable rate. To establish this fact rigorously we need the following lemma, similar to Lemma 9 in [FGS04]. Lemma 12. Assume that Hypothesis (H1) is satisfied and that R > R > 0. Then, for every µ ≥ 0, there exists a constant Cµ such that sup e−α|x| χ (|X | ≤ R) χ (|y| ≥ R )G X,x ≤ Cµ (R − R)−µ .
(144)
X,x∈R3
Moreover, if < ion , we have φ(χ (|y| ≥ R )G X,x )χ (|X | ≤ R)E (Hmod ) ≤ Cµ (R − R)−µ .
(145)
Remark. In the proof of the existence of the operators W and W+ , where we use this lemma, typically R = βt and R = γ t. Hence the r.h.s. of (145) gives a decay in time which is integrable if we choose µ large enough. Proof. To prove (144), we first choose ε = (R − R)/2λ > 0, with λ = max(λn , λe ) (recall that λe = m n /M and λn = m e /M) and we observe that e−α|x| χ (|X | ≤ R)χ (|y| ≥ R )G X,x ≤ χ (|x| ≤ ε) χ (|X | ≤ R) ×χ (|y| ≥ R )G X,x + e−αε G X,x . (146) Using that G X,x (k) = e−i(X +λe x)·k κe (k) + e−i(X −λn x)·k κn (k), it follows that G X,x 2 ≤ 2 dk |κe (k)|2 + |κn (k)|2 .
(147)
Hence the second term on the r.h.s. of (146) can be bounded by Cε−µ = C(R − R)−µ (because εµ e−αε is bounded). Moreover the square of the first term on the r.h.s. of (146) can be estimated by χ (|x| ≤ ε) χ (|X | ≤ R) χ (|y| ≥ R )G X,x 2 ≤ 2 χ (|x| ≤ ε) χ (|X | ≤ R) dy χ (|y| ≥ R ) × |κˆ e (X + λe x − y)|2 + |κˆ n (X − λn x − y)|2 R − R |κˆ e (X + λe x − y)|2 ≤ 2 dy χ |X + λe x − y| ≥ 2 R − R |κˆ n (X − λn x − y)|2 + 2 dy χ |X − λn x − y| ≥ 2 2 2 ≤C dy | κ ˆ (y)| + | κ ˆ (y)| n e |y|≥ R 2−R
(148)
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for all X and x. Here we used that, from |y| ≥ R , |X | ≤ R, and since, by definition of ε, λe |x| ≤ λe ε ≤ (R − R)/2 and λn |x| ≤ λn ε ≤ (R − R)/2, we have that |X +λe x − y| ≥ |y| − |X | − λe |x| ≥ (R − R)/2 and analogously |X − λn − y| ≥ (R − R)/2. Since κe , κn ∈ C0∞ (R3 ), their Fourier transforms decay faster than any power, and hence (148) implies (144). To prove (145), we use that φ(χ (|y| ≥ R )G X,x )χ (|X | ≤ R)E (Hmod ) ≤ e−α|x| χ (|X | ≤ R)φ(χ (|y| ≥ R )G X,x )(N + 1)−1 (N + 1)eα|x| E (Hmod ) ≤ C sup e−α|x| χ (|X | ≤ R)χ (|y| ≥ R )G X,x
(149)
x,X
because (N + 1)eα|x| E (Hmod ) is finite (because < ion and by a simple commutation). Equation (145) then follows from (144). The decay of the interaction determined in the last lemma is one of the two key ingredients for proving the existence of W and W+ . The other one is a propagation estimate for the photons, analogous to Proposition 24 in [FGS04], but with the cutoff for x/t (in [FGS04], x is the position of the electron) replaced by a cutoff for the asymptotic velocity X/t of the center of mass of the nucleus-electron compound (the reason why we can introduce here a cutoff in X/t is that, because of (143), we know it can not exceed β). For the details of the proof of the next two proposition we refer to Theorems 26 and 28 in [FGS04]. Proposition 13 (Existence of the asymptotic observable). Assume that Hypotheses (H0), (H1) and (H3) are satisfied. Fix β < 1, and choose < min(β , ion ). Suppose that f ∈ C0∞ (R) with supp( f ) ⊂ (−∞, ). Let γ , and χγ be as defined above, and let χγ ,t be the operator of multiplication with χγ (|y|/t). Then, for g ≥ 0 small enough (in order for (44) to hold true), W = s − lim ei Hmod t f d(χγ ,t ) f e−i Hmod t t→∞
exists, W = W ∗ and W commutes with Hmod . Here f = f (Hmod ). Proposition 14 (Existence of W+ ). Assume Hypotheses (H0), (H1) and (H3) are satisfied. Fix β < 1 and choose < min(β , ion ). Suppose that f ∈ C0∞ (R) with supp( f ) ⊂ (−∞, ), and that χγ and jt are defined as described above. If g ≥ 0 is so small that (44) holds, then (i) the limit ˜ ˘ jt )d(χγ ,t ) f (Hmod )e−i Hmod t W+ = s − lim ei Hmod t f ( H˜ mod )( t→∞
˜
exists, and e−i Hmod s W+ = W+ e−i Hmod s , for all s ∈ R; (ii) (1 ⊗ χ (N = 0))W+ = 0; ˜ + W+ . (iii) W =
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4.4. Positivity of the asymptotic observable and asymptotic completeness. In this section we prove the positivity of the asymptotic observable W , restricted to the subspace of states orthogonal to wave packets of dressed atom states and not containing any soft bosons. We need the following lemma. Lemma 15. Assume Hypotheses (H0), (H1), (H3). Fix β < 1 and choose < min(β , ion ). Suppose, moreover, that f ∈ C0∞ (R) and supp( f ) ⊂ (−∞, ). Put a X = (1/2)(∇ω · (y − X ) + (y − X ) · ∇ω), where X is the position of the center of mass of the atom. Then, if g is so small that (44) holds true, we have that f (Hmod )[i Hmod , d(a X )] f (Hmod ) ≥ (1 − β) f (Hmod )N f (Hmod ) − Cg f (Hmod )2 (150) on the range of the projection (χi ). This lemma follows from a straightforward estimate of the commutator [Hmod , d(a X )], from (44), (45), and from Lemma 21. Theorem 16 (Positivity of the asymptotic observable). Assume that Hypotheses (H0)– (H3) are satisfied. Fix β < 1/3 and choose < min(β , ion ) (with β and ion as in Lemma 1). Let the operator W be defined as in Proposition 13 with supp f ⊂ (−∞, ). Then if g > 0 is sufficiently small we can choose γ > β in the definition of W such that ϕ, W ϕ ≥ C f (Hmod )ϕ2 ,
⊥ for all ϕ ∈ Ran Pdas (χi ) .
(151)
Here C is a positive constant depending on g, β, , but independent of ϕ. In particular, if ⊂ (−∞, ) and then f = 1 on , then W |RanE (Hmod )(χi )P ⊥ ≥ C > 0. das
(152)
⊥ . Since D is dense in Ran(χ )P ⊥ , it is Proof. Let D = D(d(a)) ∩ Ran(χi )Pdas i das enough to prove that
ϕ, W ϕ ≥ C f ϕ2
(153)
for every ϕ ∈ D. As before, we use the notation f = f (Hmod ). The first step consists in proving that there exists a constant C, depending only on , such that, for every ϕ ∈ D and for every ε > 0, 1−β t ϕt , f d(χγ ,t ) f ϕt ≥ C f ϕ−2 dsϕs , f N f ϕs − (γ + β + ε) t 0 2 ×ϕt , f N f ϕt − Cg f ϕ2 + o(1), as t → ∞. (154) This inequality follows from Lemma 15 by straightforward adaptations of arguments given in the proof of Theorem 27 in [FGS04]. Next, we observe that 1 t 1 t dsϕs , f N f ϕs ≥ f ϕ2 − ϕs , f P f ϕs , (155) t 0 t 0 where P denotes the orthogonal projection onto the Fock vacuum . The second term on the r.h.s. of the last equation can be written as an integral over fibers with fixed
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⊥ ϕ and of Fubini’s Theorem to total momentum. Making use of the fact that ϕ = Pdas interchange the integration over s and over , we obtain 1 t 1 t ds ϕs , f P f ϕs = d ds P f (Hmod ( ))e−i Hmod ( )s Pψ⊥ ϕ( )2 , t 0 t 0 (156)
where Pψ = |ψ ψ | is the orthogonal projection onto the dressed atom state ψ , and Pψ⊥ = 1 − Pψ is its orthogonal complement. For every fixed , the operator P f (Hmod ( )) is compact on L 2 (R3 , dx) ⊗ F, because eα|x| E (Hmod ( )) ≤ C; (since < ion , this follows from Lemma 1). By the continuity of the spectrum of Hmod ( ) on RanE (Hmod )Pψ⊥ (χi ) (see Corollary 6), and the RAGE Theorem (see, for example, [RS79]), it follows that 1 t ds P f (Hmod ( ))e−i Hmod ( )s Pψ⊥ ϕ( )2 → 0, (157) t 0 as t → ∞, pointwise in . Using Lebesgue’s Dominated Convergence Theorem, we conclude that 1 t ds ϕs , f P f ϕs → 0 , (158) t 0 as t → ∞. From (155) we obtain 1 t f ϕ2 ds ϕs , f N f ϕs ≥ t 0 2
(159)
for t large enough, where we can assume f ϕ = 0 without loss of generality. Equations (158) and (159) allow us to apply Lemma 17, with h 1 (s) = ϕs , f N f ϕs and h 2 (s) = ϕs , f P f ϕs (it is easy to check that h 1 and h 2 are bounded and continuous). We conclude that there exists a sequence {tn }n≥0 with tn → ∞, as n → ∞, such that 1 tn ds ϕs , f N f ϕs ≥ (1 − ε)ϕtn , f N f ϕtn , and tn 0 (160) ϕtn , f P f ϕtn → 0, as n → ∞. From (154) we infer that ϕtn , f d(χγ ,tn ) f ϕtn ≥ C f ϕ−2 (1 − 2β − γ − 2ε)2 ϕtn , f N f ϕtn 2 − Cg f ϕ2 + o(1) ,
(161)
as n → ∞. Choosing γ − β and ε > 0 sufficiently small, we conclude that (1 − 3β)2 f ϕ−2 ϕtn , f N f ϕtn 2 − Cg f ϕ2 + o(1) 2 (1 − 3β)2 f ϕ2 − 2ϕtn , f P f ϕtn ≥C 2 (162) − Cg f ϕ2 + o(1),
ϕtn , f d(χγ ,tn ) f ϕtn ≥ C
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as n → ∞. Hence, by (160), there are constants C1 > 0 and C2 < ∞, depending only on , such that ϕtn , f d(χγ ,tn ) f ϕtn ≥ C1 (1 − 3β − C2 g)2 f ϕ2 + o(1) ,
(163)
as n → ∞. If β < 1/3 and g is small enough, we arrive at (151) by taking the limit n → ∞. (Since we already know that the limit defining W exists, it is enough to prove its positivity on some arbitrary subsequence!) Lemma 17. Suppose h 1 and h 2 are positive, continuous, bounded functions on R, such that 1 t m 1 (t) := ds h 1 (s) ≥ C > 0 (164) t 0 for all t > 0 large enough, and m 2 (t) :=
1 t
t
dsh 2 (s) → 0 as t → ∞ .
(165)
0
Then, for every δ > 0, there exists a sequence {tn }n≥0 , with tn → ∞, as n → ∞, such that m 1 (tn ) ≥
1 h 1 (tn ) 1+δ
(166)
and h 2 (tn ) → 0 Proof. Define the sets ST := t ∈ [0, T ] : m 1 (t) <
as n → ∞.
1 h 1 (t) , 1+δ
(167)
for some δ > 0.
By the continuity of h 1 (t) and m 1 (t) the sets ST are measurable (with respect to Lebesgue measure on R), for all T . Denote by µ(A) the Lebesgue measure of a measurable set A ⊂ R. We show that lim inf T →∞
µ(ST ) < 1. T
In fact, if (168) were false, then (since µ(ST )/T ≤ 1 for all T ≥ 0) µ(ST ) = 1, T →∞ T lim
and hence, for arbitrary ε > 0, we could find a T0 such that µ(ST ) ≥ 1−ε, T
(168)
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for all T > T0 . This would imply that m 1 (T ) =
1 T
0
1+δ ≥ T
T
ds h 1 (s) ≥
ST
1 T
ds h 1 (s) ST
1+δ ds m 1 (s) ≥ T
T
ds m 1 (s) −
0
µ(STc ) m 1 ∞ , T
(169)
where m 1 ∞ denotes the supremum of the bounded function m 1 and STc = [0, T ]\ST is the complement of ST inside [0, T ]. Hence, we find m 1 (T ) ≥
1+δ T
T 0
for every T ≥ T0 . Put m 1 (T ) := (1/T ) m (T ) d 1 log m 1 (T ) = 1 = dT m 1 (T ) T
ds m 1 (s) − εm 1 ∞
T
0
(170)
ds m 1 (s). Then we have
m 1 (T ) 1 εm 1 ∞ −1 ≥ δ− . m 1 (T ) T m 1 (T )
(171)
By the assumption that m 1 (T ) ≥ C for all T large enough, we have m 1 (T ) ≥ C, and thus, choosing ε < Cδ/2m 1 ∞ , we find δ d log m 1 (T ) ≥ dT 2T
(172)
for all T large enough. This contradicts the boundedness of m 1 (T ) (which follows from the boundedness of m 1 (T )). This proves (168), and implies that there exist ε > 0 and a sequence {Tm }m≥0 converging to infinity such that µ(STm ) ≤ 1−ε, Tm
(173)
for all m ≥ 0. Hence µ(STcm ) ≥ εTm , for all m. Next, we show that there exists a sequence {tn }n≥0 , with tn → ∞ as n → ∞, such that tn ∈ ∪m≥0 STcm , for all n ≥ 0, and h 2 (tn ) → 0 ,
(174)
as n → ∞. Since, for all n ≥ 0, tn ∈ STcm , for some m ∈ N, the sequence tn automatically satisfies (166). Thus the lemma follows if we can prove (174). To this end we argue again by contradiction. If there were no sequence {tn }n≥0 ∈ ∪m≥0 STcm satisfying (174) then there would exist τ and α > 0 such that h 2 (t) ≥ α, for all t ∈ ∪m≥0 STcm ∩ [τ, ∞). But then, for an arbitrary m ∈ N with Tm ≥ τ , 1 Tm
0
Tm
ds h 2 (s) ≥
1 Tm
≥α
STcm ∩[τ,Tm ] µ(STcm ) ατ
Tm
−
ds h 2 (s) ≥ α
Tm
≥ αε −
ατ Tm
µ(STcm ∩ [τ, Tm ]) Tm (175)
for all m ∈ N with Tm ≥ τ . Taking m → ∞, this contradicts the assumption (165).
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4.5. Asymptotic completeness. Using the positivity of the asymptotic observable W , we can complete the proof of asymptotic completeness for the Hamiltonian Hmod . Our proof is based on induction in the energy. The following simple lemma is useful. Lemma 18. Assume that Hypotheses (H0)–(H3) are satisfied. Fix β < 1 and choose ˜ + and + are defined as in Lemma 10 and in < min(β , ion ). The wave operators Theorem 8, respectively. Suppose that Ran + ⊃ E η (Hmod )H, for some η < . Then, for every ϕ ∈ RanE ( H˜ mod ), there exists ψ ∈ RanE ( H˜ mod ) such that ˜ + (E η (Hmod ) ⊗ 1)ϕ = + ψ. ˜ If ⊂ (−∞, ) and ϕ ∈ E ( H˜ mod )H˜ then ψ ∈ E ( H˜ mod )H. The interpretation of this lemma is simple: If we know that asymptotic completeness holds for vectors with energy lower than η, then it continues to be true if we add asymptotically free photons to these vectors (no matter what the total energy of the new state is). For the proof of this lemma we refer to Lemma 20 of [FGS04]. Using this lemma we can prove asymptotic completeness for Hmod ; the proof is similar to the proof of Theorem 19 in [FGS04]. We repeat it here, because it is very short, and because it explains the ideas behind all the tools introduced in Sect. 4. Theorem 19. Assume that Hypotheses (H0)–(H3) are satisfied. Fix β < 1/3 and choose < min(β , ion ); (with β and ion defined as in Lemma 1). If g > 0 is sufficiently small, then Ran + ⊃ E (−∞,) (Hmod )H. Proof. The proof is by induction in energy steps of size m = σ/2. We show that Ran + ⊃ E (−∞,−km) (Hmod )H
(176)
holds for k = 0, by proving this claim for all k ∈ {0, 1, 2, . . .}. Since Hmod is bounded below, (176) is obviously correct for k large enough. Assuming that (176) holds for k = n + 1, we now prove it for k = n. Since Ran + is closed (by Theorem 8) and since Ran + ⊃ Hdas , it suffices to prove that ⊥ (χi )E (Hmod )H , Ran + ⊃ Pdas
with < for = (inf σ (Hg=0 ) − 1, − nm). Here we use Lemma 11. Fix ∞ < min(β , ion ) and choose f ∈ C0 (R) real-valued, with f ≡ 1 on and ). We define the asymptotic observable W in terms of f , as in supp( f ) ⊂ (−∞, ⊥ W P ⊥ (χ ) is strictly positive Proposition 13. By Theorem 16, the operator (χi )Pdas i das ⊥ on Pdas (χi )E (Hmod )H, and hence onto, if g is small enough. Given ψ in this space, ⊥ (χ )ϕ such that we can therefore find a vector ϕ = Pdas i ⊥ Pdas (χi )W ϕ = ψ.
˜ + W+ ϕ and W+ ϕ = E −nm ( H˜ mod )W+ ϕ. Furthermore, by By Proposition 14, W ϕ = part (ii) of Proposition 14, W+ ϕ has at least one boson in the outer Fock space, and thus an energy of at most − (n + 1)m in the inner one. That is, W+ ϕ = [E −(n+1)m (Hmod ) ⊗ 1]W+ ϕ.
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Hence we can use the induction hypothesis Ran + ⊃ E −(n+1)m (Hmod )H. By Lemma 18, ˜ + W+ ϕ = + γ for some γ ∈ E ( H˜ mod )H. We conclude that it follows that ⊥ + γ ψ = (χi )Pdas
= (χi ) + (1 ⊗ P ⊥ )γ
= + ((χi ) ⊗ (χi )P ⊥ )γ ,
where P ⊥ is the projection onto the orthogonal complement of the vacuum. This proves the theorem. A. Fock Space and Second Quantization Let h be a complex Hilbert space, and let ⊗ns h denote the n-fold symmetric tensor product of h. Then the bosonic Fock space over h, F = F(h) = h⊗s n , n≥0
is the space of sequences ϕ = (ϕn )n≥0 , with ϕ0 ∈ C, ϕn ∈ ⊗ns h, and with the scalar product given by ϕ, ψ := (ϕn , ψn ), n≥0
where (ϕn , ψn ) denotes the inner product in ⊗ns h. The vector = (1, 0, . . .) ∈ F is called the vacuum. By F0 ⊂ F we denote the dense subspace of vectors ϕ for which ϕn = 0, for all but finitely many n. The number operator N is defined by (N ϕ)n = nϕn . A.1. Creation- and annihilation operators. The creation operator a ∗ (h), h ∈ h, is defined on h⊗s n−1 by √ for ϕ ∈ h⊗s n−1 , a ∗ (h)ϕ = n S(h ⊗ ϕ), and extended by linearity to F0 . Here S denotes the orthogonal projection onto the symmetric subspace ⊗ns h ⊂ ⊗n h. The annihilation operator a(h) is the adjoint of a ∗ (h). Creation- and annihilation operators satisfy the canonical commutation relations (CCR) [a(g), a ∗ (h)] = (g, h),
[a # (g), a # (h)] = 0.
In particular, [a(h), a ∗ (h)] = h2 , which implies that the graph norms associated with the closable operators a(h) and a ∗ (h) are equivalent. It follows that the closures of a(h) and a ∗ (h) have the same domain. On this common domain we define the self-adjoint operator φ(h) = a(h) + a ∗ (h).
(177)
The creation- and annihilation operators, and thus φ(h), are bounded relative to the square root of the number operator: a # (h)(N + 1)−1/2 ≤ h . More generally, for any p ∈ R and any integer n, (N + 1) p a # (h 1 ) . . . a # (h n )(N + 1)− p−n/2 ≤ Cn, p h 1 · . . . · h n .
(178)
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A.2. The functor . Let h1 and h2 be two Hilbert spaces and let b ∈ B(h1 , h2 ). We define (b) : F(h1 ) → F(h2 ) by (b)|` ⊗ns h1 = b ⊗ · · · ⊗ b. In general (b) is unbounded; but if b ≤ 1 then (b) ≤ 1. From the definition of a ∗ (h) it easily follows that (b)a ∗ (h) = a ∗ (bh)(b), h ∈ h1 , (b)a(b∗ h) = a(h)(b), h ∈ h2 .
(179) (180)
If b∗ b = 1 on h1 then these equations imply that (b)a(h) = a(bh)(b), (b)φ(h) = φ(bh)(b),
h ∈ h1 , h ∈ h1 .
(181) (182)
A.3. The operator d(b). Let b be an operator on h. Then d(b) : F(h) → F(h) is defined by d(b)|` ⊗ns h =
n (1 ⊗ . . . b ⊗ . . . 1). i=1
For example N = d(1). From the definition of a ∗ (h) we infer that [d(b), a ∗ (h)] = a ∗ (bh) [d(b), a(h)] = −a(b∗ h), and, if b = b∗ , i[d(b), φ(h)] = φ(ibh). Note that d(b)(N
+ 1)−1
(183)
≤ b.
A.4. The tensor product of two Fock spaces. Let h1 and h2 be two Hilbert spaces. We define a linear operator U : F(h1 ⊕ h2 ) → F(h1 ) ⊗ F(h2 ) by U = ⊗ , U a ∗ (h) = [a ∗ (h (0) ) ⊗ 1 + 1 ⊗ a ∗ (h (∞) )]U
(184) for h = (h (0) , h (∞) ) ∈ h1 ⊕ h2 .
This defines U on finite linear combinations of vectors of the form a ∗ (h 1 ) . . . a ∗ (h n ) . From the CCRs it follows that U is isometric. Its closure is isometric and onto, hence unitary. A.5. Factorizing Fock space in a tensor product. Suppose j0 and j∞ are linear operators on h and j : h → h ⊕ h is defined by j h = ( j0 h, j∞ h), h ∈ h. Then j ∗ (h 1 , h 2 ) = ∗ h and consequently j ∗ j = j ∗ j + j ∗ j . We define j0∗ h 1 + j∞ 2 ∞ ∞ 0 0 ˘ j) = U ( j) : F → F ⊗ F, ( ˘ j) = ( j ∗ j) which is the ˘ j)∗ ( where ( j) is as defined in Sect. A.2. It follows that ( ∗ identity if j j = 1. In this case ˘ j)a # (h) = [a # ( j0 h) ⊗ 1 + 1 ⊗ a # ( j∞ h)]( ˘ j), ( ˘ ˘ ( j)φ(h) = [φ( j0 h) ⊗ 1 + 1 ⊗ φ( j∞ h)]( j).
(185) (186)
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A.6. The “Scattering Identification”. We define the scattering identification I : F ⊗ F → F by I (ϕ ⊗ ) = ϕ, I ϕ ⊗ a (h 1 ) · · · a ∗ (h n ) = a ∗ (h 1 ) · · · a ∗ (h n )ϕ, ∗
ϕ ∈ F0 ,
and extend it by linearity to F0 ⊗ F0 . (Note that this definition is symmetric with respect to the two factors in the tensor product.) There is a second characterization of I which can be useful. Let ι : h ⊕ h → h be defined √ by ι(h (0) , h (∞) ) = h (0) + h (∞) . Then ∗ I = (ι)U , with U as above. Since ι = 2, the operator I is unbounded, but it can be proved that I (N + 1)−k ⊗ χ (N ≤ k) is bounded, for any k ≥ 1. B. Bounds on the Interaction In this section we review standard estimates that are used throughout this paper to bound the interaction. Lemma 20. Let L 2ω (R3 ) := L 2 (R3 , (1 + 1/|k|)dk) and let h ∈ L 2ω (R3 ). Then 1/2 2 a(h)ϕ ≤ dk|h(k)| /|k| d(|k|)1/2 ϕ,
where h2ω =
a ∗ (h)ϕ ≤ hω (d(|k|) + 1)1/2 ϕ, √ φ(h)ϕ ≤ 2 hω (d(|k|) + 1)1/2 ϕ, 1 |h(k)|2 , α > 0, ±φ(h) ≤ αd(|k|) + dk α |k| dk (1 + 1/|k|)|h(k)|2 .
The next lemma is used to control the factor φ(ia Fx ) appearing in the commutators of Sect. 3.3. ˆ with kˆ = k/|k| Lemma 21. Assume Hypothesis (H0)–(H1). Let a = (1/2)(kˆ · y + y k) and choose < ion . Then there exists C < ∞ such that φ(ia Fx )E (Hg ( )) ≤ C ,
(187)
ˆ with Fx as in Eq. (32). For a X := (1/2)(kˆ · (y − X ) + (y − X ) · k), φ(ia X G X,x )E (Hg ) ≤ C ,
(188)
where G X,x (k) = e−ik·X Fx (k); (see Eqs. (31), (32)). Proof. Note that (a Fx )(k) = (i kˆ · ∇k + 2/|k|)(e−iλe k·x κe (k) + eiλn k·x κn (k)) = e−iλe k·x λe x · kˆ κe (k) + i kˆ · ∇κe (k) + 2/|k|κe (k) + eiλn k·x −λn x · kˆ κn (k) + i kˆ · ∇κn (k) + 2/|k|κn (k) .
(189)
Equation (187) follows from Lemma 20, because eα|x| E (Hg ( )) is bounded (see Lemma 1) and from Hypothesis (H1). Equation (188) follows from (187) because (a X G X,x )(k) = e−i X ·k (a Fx )(k) .
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Commun. Math. Phys. 271, 431–454 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0201-y
Communications in
Mathematical Physics
Spin-Flip Dynamics of the Curie-Weiss Model: Loss of Gibbsianness with Possibly Broken Symmetry Christof Külske1 , Arnaud Le Ny2 1 Department of Mathematics and Computing Sciences, University of Groningen, Blauwborgje 3,
9747 AC Groningen, The Netherlands. E-mail: [email protected]
2 Université de Paris-Sud, Laboratoire de Mathématiques, Bâtiment 425, 91405 Orsay Cedex, France.
E-mail: [email protected] Received: 14 October 2005 / Accepted: 14 September 2006 Published online: 8 February 2007 – © Springer-Verlag 2007
Abstract: We study the conditional probabilities of the Curie-Weiss Ising model in vanishing external field under a symmetric independent stochastic spin-flip dynamics and discuss their set of points of discontinuity (bad points). We exhibit a complete analysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending the results for the corresponding lattice model, where only partial answers can be obtained. For initial temperature β −1 ≥ 1, we prove that the time-evolved measure is always Gibbsian. For 23 ≤ β −1 <1, the time-evolved measure loses its Gibbsian character at a sharp transition time. For β −1 < 23 , we observe the new phenomenon of symmetry-breaking in the set of points of discontinuity: Bad points corresponding to non-zero spin-average appear at a sharp transition time and give rise to biased non-Gibbsianness of the time-evolved measure. These bad points become neutral at a later transition time, while the measure stays non-Gibbs. In our proof we give a detailed description of the phase-diagram of a Curie-Weiss random field Ising model with possibly non-symmetric random field distribution based on bifurcation analysis. 1. Introduction Recent years have seen a variety of situations where non-Gibbsian lattice-spin measures appear from proper Gibbs measures that are subjected to natural transformations [6, 11]. One particularly interesting dynamical phenomenon is the loss (and possible recovery) of the Gibbs property of an initial Gibbs measure that evolves according to a stochastic spin-flip dynamics. This was analyzed by van Enter et al. in [5] and related results for continuous spins that evolve diffusively starting from an initial Gibbs measure were recently obtained in [17]. Suppose a lattice spin system is in an equilibrium situation at initial temperature β −1 , described by a proper Gibbs measure in infinite volume. Suppose now the system undergoes a fast heating procedure, described by a high-temperature Glauber dynamics. A question arises: Is there a well-defined Hamiltonian for every time? Or, in the more catchy terms of [5]: Is there always a well-defined temperature? It turns out
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that there can be transition-phenomena of an ’in and out of Gibbsianness’ as a function of time. These phenomena are already present when we consider an infinite-temperature time evolution, i.e. a spin-flip that is independent from site to site. Let us highlight one particular phenomenon from [5]. Pick the plus-state of a low temperature Ising model in zero external magnetic field as initial measure µβ at time t = 0 and perform an unbiased spin-flip dynamics, independently over the sites, with rate 1. The following result is proved in [5] (Theorem 5.2) for the resulting probability measure µβ,t . Theorem 1.1 ([5]). Assume that the initial inverse temperature β is above the critical inverse temperature of the nearest neighbor Ising model. Then there exist t0 (β) ≤ t1 (β) such that 1. µβ,t is a Gibbs measure for all 0 ≤ t < t0 (β). 2. µβ,t is not a Gibbs measure for all t > t1 (β). There remain open questions in this picture: Is the transition between Gibbsianness and non-Gibbsianness sharp, i.e. t0 (β) = t1 (β), as conjectured in [5]? Can one understand the trajectory of the interactions of the measure µβ,t as time varies? These and related questions seem to be difficult to answer for lattice systems, even for the infinite-temperature dynamics. The purpose of this paper is to provide detailed answers for the corresponding mean-field model. In the course of our analysis we also find an interesting new mechanism of non-Gibbsianness that was not observed on the lattice. Recall that a measure µ on the lattice is Gibbs iff (a version of) its conditional probabilities µ(σx | · ) : σx c → µ(σx |σx c ) is continuous w.r.t. the product topology for any site x of the graph. This means that the influence of a perturbation of a conditioning σx c outside of a large volume tends to zero, when tends to Zd . To investigate this property for the time-evolved measure µβ,t , we relate its conditional probabilities to expectations w.r.t. a certain constrained measure, obtained by conditioning the measure at time t = 0 to be in a given spin configuration at time t. In this constrained measure the spin-configuration in the conditioning appears as an additional ‘frozen’ external magnetic field configuration. The investigation of such a constrained measure was performed in [5] as well as in [17] for the corresponding setup, and we perform it here for a mean-field model. The failure of the Gibbs-property for the time-evolved measure occurs if there is a sensitive dependence of the constrained model on the external magnetic field configuration, at certain ‘bad configurations’. The chessboard configuration, the simplest example of a ‘neutral configuration’, serves as such a bad configuration in the proof of the second part of Theorem 1.1 in the large β and large t region. In turn, the absence of such bad configurations implies the Gibbs property. To get a precise understanding of the constrained model is a difficult task for lattice models, where one usually cannot hope for exhaustive results. In the mean-field set-up, one can however relate Gibbsianness to continuity properties of conditional probabilities of an infinite-volume constrained model, that can be computed in terms of the rate-function for a standard quenched disordered model. To understand the structure of its minimizers is then equivalent to understand the phase-diagram of a Curie-Weiss random field Ising model with possibly non-symmetric random-field distribution of the quenched disorder, as we shall see. In this way, our analysis of the transitions between Gibbsian and non-Gibbsian behavior is reduced to the analytic problem of a bifurcation-analysis for this rate-function. We perform this analysis in the spirit of catastrophe theory and discover an interesting new structure where bad configurations (called points in our mean-field approach, see below) with non-zero spin-average appear in a spin-flip invariant model. We call this phenomenon biased non-Gibbsianness, as opposed to the more standard neutral
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non-Gibbsianness where the bad points correspond to configurations with zero spinaverage. It is a symmetry breaking in the set of bad configurations that happens in a regime of low enough temperatures, strictly smaller than the regime of temperatures for which there is a phase transition for the initial measure. It corresponds to the region below the so-called tricritical temperature of the Curie-Weiss random field Ising model where phase transition is possible for non-symmetric versions of this model. We can find a parametrization in terms of explicit functions of the boundary curve in the (β, t)-space of the region for which µβ,t is Gibbs. This curve consists of two different branches with different functional forms, depending whether they form the boundary to biased non-Gibbsianness, where spin-flip symmetry is broken during the loss of Gibbsianness, or to neutral non-Gibbsianness. Biased non-Gibbsianness has not been detected in other models yet. Can a similar property, i.e. non-Gibbsianness with symmetry breaking of bad configurations, also happen for a lattice model? We expect heuristically that the same phenomena should be present also in a Kac model at long enough (but finite) range, see also [7]. 2. Model, Results and Outline of the Proofs 2.1. Synopsis and strategy. The model and the time-evolved measure are defined in Sect. 2.2 and the new notions of Gibbsianness and non-Gibbsianness for mean-field models in Sect. 2.3. For lattice systems a Gibbs measure in infinite-volume is characterized by the fact that its conditional probabilities depend continuously on the conditioning, in the sense of the product topology. Clearly, this notion would yield meaningless answers for mean field models, since the finite-volume Gibbs measures of mean field models converge to convex combinations of product measures, by de Finetti’s theorem. Non-trivial convex combinations of product measures are known to have conditional probabilities that have every point as a point of discontinuity [8] and thus non-Gibbsianness in this naive approach would always be trivially true for mean-field models. Observing that mean-field models are first defined at finite volume without boundary conditions, we consider the conditioning at this level first, prescribing the magnetization outside (finite) subsets by points α of the interval [−1, 1], before performing an infinitevolume limit. By exchangeability, such a limit of constrained probabilities, if it exists, depends only on this empirical average; thus, good or bad points of the interval [−1, 1] arise naturally instead of (spins) configurations on the countable lattice. Similarly, the product topology has to be replaced by the standard Borel topology on [−1, 1]. Our approach is motivated by two main observations: Recent progress [9, 11, 16] on generalized Gibbs measures has highlighted the importance of continuity properties for Gibbs measures, and secondly prescribing the values of the magnetization in the conditioning has to do with the macroscopic character of the mean-field interaction. This strategy has moreover already been used with satisfactory results in [10, 15]. Correspondingly, we say that a measure is Gibbs if its limiting conditional probabilities are continuous at any point of prescribed outer magnetization, i.e. if all its points are good, neutral non-Gibbs if 0 is the only bad point, and biased non-Gibbs if there exists a non-zero bad point of prescribed outer magnetization (Definition 2.1). We are ready to state our main results on Gibbsianness during the time-evolution of our model in Sect. 2.4. First, we describe the three possible scenarii depending on the temperature in Theorem 2.2. The high temperature scenario is the simplest one: The time-evolved measure always stays Gibbs. For an intermediate range of temperatures, the phenomenon of neutral non-Gibbsianness appears at a sharp transition time depending on the
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temperature only, and stays neutral non-Gibbs forever. The low temperature scenario, below the tricritical temperature of the Curie-Weiss random field Ising model, is more peculiar: After a period of Gibbsianness, biased non-Gibbsianness appears first at a sharp transition time, and becomes neutral at a later sharp transition time. Transition times as functions of the temperature of the initial measure are given in Theorem 2.3. Biased non-Gibbsianness appears below the tricritical temperature of the Curie-Weiss random field Ising model because of the presence of metastable minima. This model corresponds to the constrained model in which a phase transition is required to get non-Gibbsianness as described in Sect. 3. In this section, we compute the limiting conditional probability in terms of the rate function of the constrained model and transform our Gibbsianness problem to an analysis of the structure of minima of this function. Using elements of catastrophe theory in Sect. 4, we give a detailed phase diagram of this model discovering three main regions of different qualitative behavior of the number and nature of these minima. Region 1 of the phase diagram (with temperature and dynamical strength of the disorder as parameters) where there is uniqueness of the minimizer for all α, i.e. for all random fields distributions, corresponds to the region of Gibbsianness for the time-evolved measure. Uniqueness of the minimizer indeed implies the absence of phase transition in the quenched model, in the sense that the empirical magnetization always weakly converges to the Dirac measure at this minimizer. The unique phase is thus always selected when one varies around any given value of α, so the conditional probabilities are always continuous and the time-evolved measure is Gibbs. This in particular proves short-times Gibbsianness, known in general for lattice systems [18]. Region 2 is the part of the phase diagram where the rate function of the symmetric case α = 0 admits two global minimizers. This is the region of phase transition for the symmetric Curie-Weiss random field Ising model where the empirical magnetization weakly converges to a mixture of Dirac measures at these symmetric minimizers: The two phases coexist and approaching α = 0 from above or below yields two different limits corresponding to the two phases. The point α = 0 appears thus to be a discontinuity point of the conditional probabilities and Region 2 is the region of neutral non-Gibbsianness. Region 3 is the region of the phase diagram where there exists a point α = 0 of the prescribed outer magnetization for which two global minimizers of the rate-function coexist. It strictly includes the region of uniqueness where non-zero metastable minima already appear at α = 0. Coexistence of two of the minima is then appearing at some other bigger αc , as we shall see. In the neighborhood of this αc it is possible to select different phases, similarly to Region 2 and one recovers a discontinuity of conditional probabilities, but at a non-neutral point of prescribed magnetization. This region is thus the region of biased non-Gibbsianness. We give in Sect. 4 the precise analysis of the phase diagram, in terms of the temperature and of the strength of the field, of non-symmetric Curie-Weiss random field Ising models. The extended phase diagram and the time evolution picture of Gibbsianness are related in Sect. 5. 2.2. The model. We start at time t = 0 with the Curie-Weiss Ising model in zero magnetic field at inverse temperature β > 0 whose finite-volume Gibbs measures on spin-configurations σ[1,N ] = (σi )i=1,...,N ∈ N := {−1, +1} N are given by 2 β N exp 2N i=1 σi µβ,N (σ[1,N ] ) = , (1) Z β,N
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where the normalization factor Z β,N is the standard partition function. This model shows a phase transition at the critical inverse temperature β = 1, distinguishing in the infinite-volume limit a region of uniqueness at high temperature (β ≤ 1) from a region of coexistence of a ’+’-like and a ’-’-like phase at low temperature (β > 1). The order parameter of the system is the empirical magnetization N 1 m = m N (σ ) = σi , N i=1
and we call point any possible value of its infinite-volume limit, i.e. a point of the interval [−1, 1]. Then we apply a stochastic spin-flip dynamics at rate 1, independently over the sites. Our object of interest is the resulting image measure µβ,t,N given by µβ,t,N (η[1,N ] ) :=
µβ,N (σ[1,N ] )
pt (σi , ηi ).
(2)
1 + e−2t 1 log , 2 1 − e−2t
(3)
σ[1,N ]∈ N
Here pt (σi , ηi ) =
eηi σi h t , with 2 cosh h t
N i=1
ht =
is the transition kernel from a spin-value σi at time t = 0 to a spin value ηi at t > 0. The product form in (2) is a special case of a Glauber dynamics for infinite temperature [5, 18, 19]. It is well known that, for fixed N , the time-evolved measure µβ,t,N tends to a spin-flip invariant product measure on {−1, +1} N , with t ↑ ∞. We shall however see that the large-N -behavior of the conditional probabilities of the time-evolved measure can be non-trivial, even when the corresponding measure is close to a product measure.
2.3. The notion of Gibbsianness for mean-field models. Definition 2.1. A point α0 ∈ (−1, 1) is said to be good if and only if 1. The limit γβ,t (η1 |α) := lim µβ,t,N (η1 |η[2,N ] ) N ↑∞
whenever lim
N ↑∞
N 1 ηi = α N
(4)
i=2
exists for all α in a neighborhood of α0 . 2. The function α → γβ,t (η1 |α) is continuous at α = α0 . A point is bad when it is not good. We call the mean-field model µβ,t,N Gibbs iff every point α is good, neutral non-Gibbs iff α = 0 is the only bad point and biased non-Gibbs iff there is a bad point α = 0. For more information and discussion we refer the reader to [10, 15]. Note that the eβαη1 initial mean-field model is Gibbs in this sense because its kernel γβ (η1 |α) = 2 cosh(βα) is a continuous function at every α.
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Fig. 1. Gibbs vs non-Gibbs time evolution picture
2.4. Main result. We are now ready to state our main result and describe the Gibbs vs. non-Gibbs character of the time-evolved measure depending on the temperature. Theorem 2.2. 1. If β −1 ≥ 1, the limiting conditional probabilities are continuous functions of the empirical mean α ∈ (−1, 1), for all t ≥ 0 and the time-evolved measure is Gibbs. 2. If 23 ≤ β −1 < 1, there exists a sharp value 0 < t0 (β) < ∞ such that: • For all 0 ≤ t < t0 (β), the limiting conditional probabilities are continuous, and the time-evolved measure is Gibbs. • For all t ≥ t0 (β), the limiting conditional probabilities are discontinuous at α = 0, and continuous at any α = 0. The time-evolved measure is neutral non-Gibbs. 3. If 0 < β −1 < 23 , then there exist sharp values 0 < t0 (β) < t1 (β) < ∞ such that: • For all 0 ≤ t < t0 (β), the limiting conditional probabilities are continuous and the time-evolved measure is Gibbs. • For all t0 (β) ≤ t < t1 (β), there exists αc = αc (β, t) ∈ ]0, 1[, such that the limiting conditional probabilities are discontinuous at the points αc and −αc , and continuous otherwise. The time-evolved measure is thus biased nonGibbs and for fixed β −1 , the function t → αc (β, t) is decreasing and we have limt↑t1 αc (β, t) = 0. • For t ≥ t1 (β), the limiting conditional probabilities are discontinuous at α = 0, continuous at any α = 0, and the time-evolved measure is neutral non-Gibbs. In this picture (see Fig. 1), we distinguish three regions with different qualitative behavior, one Gibbs region, one neutral non-Gibbs region and one region denoted by “NG*” of biased non-Gibbsianness, due to bad configurations with broken symmetry. In the next theorem we give the functional form of the two branches of the boundary curves of the Gibbs-region in time-temperature space. Theorem 2.3. 1. If
2 3
≤ β −1 < 1, the threshold time is given by t0 (β −1 ) = − 41 log(1 − β −1 )
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2. If 0 < β −1 < 23 , the boundary curve of the Gibbsian region has the parametrization, in terms of a parameter M > 0, 2y 3 + M(1 − y 2 )2 1 , t4 (M) = − log 4 2y + M(1 − y 2 ) y(2 + M y)(1 + y 2 ) , β4−1 (M) = 2M 2 y 3 + 2(y + y 3 ) + M(1 + 3y 2 + y 4 )
(5)
where y = y(M) = tanh M. We do not have an explicit curve for the line t1 (β) but an implicit characterization of the corresponding line of the phase diagram. In order to understand the symmetry breaking in the set of points of discontinuity in detail, we discuss the extended phase diagram of the quenched model, generalizing the known phase diagram from the symmetric case α = 0 [1, 21, 3] to non-symmetric distributions. It also has some interest in itself, beyond the study of non-Gibbsianness. Figure 1 then appears as the image of three of the lines of the extended phase diagram, by relating time in the original model to the coupling strength of the quenched fields in the quenched model via t = atanh(exp(−2h t )). An explicit expression for the trajectory t → γβ,t of the kernel in question in the "Gibbs" region is moreover given in Theorem 3.1. Theorem 2.3 is the result of an analysis of the underlying bifurcation problem. In Sect. 3 we derive explicit expressions for the limiting conditional probabilities of the time-evolved model in terms of the rate function of a mean-field model with possibly non-symmetric random field distribution. In Sect. 4 we discuss the bifurcation structure of this rate function and the phase diagram of the corresponding model. The proof of Theorems 2.2 and 2.3 follows immediately in Sect. 5 from the phase diagram.
3. Infinite-Volume Limit of Conditional Probabilities The existence of an infinite-volume limit for the conditional probabilities of the timeevolved model is not granted but must be deduced by an explicit analysis of the model. We show that the limit (4) exists and that a kernel γβ,t (η1 |α) is well-defined when a certain rate-function depending on the triple (β, t, α) has a unique minimizer. In such a case, we have an explicit expression in terms of the minimizer of this rate-function, implying continuity by a smooth dependence of the minimizer on small α-dependent perturbations of the rate-function. On the contrary, when this rate function has more than one minimizer, this dependence is not smooth anymore, leads to different left and right infinite-volume limits and thus gives rise to bad points.
3.1. Continuity result - Explicit expression in terms of a rate-function. Theorem 3.1. Denote by η˜ a random variable taking values ±1 with average α, and write Eα for its expectation. Put ε = εt := h t /β and suppose that the function β,ε,α : m → β,ε,α (m) =
1 m2 − Eα log cosh(β(m + εη)) ˜ 2 β
(6)
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has a unique minimizer m ∗ (β, ε, α), for the fixed choice of (β, ε, α). Then the conditional probabilities of the time evolved measure (2) have a well-defined infinite-volume limit γβ,t in the sense of (4) and we have the representation ∗ (β, ε , α) + ε η g (α) cosh β m β,t 1 t t e 1 , with gβ,t (α) = log γβ,t (η1 |α) = . (7) 2 cosh gβ,t (α) 2 cosh β m ∗ (β, ε , α) − ε t
t
The (t, α)-dependent effective field gβ,t (α) captures all information about the interaction at time t of the spin η1 with the empirical magnetization of the outside world α. The kernel (7) interpolates between the initial distribution and the final product measure. Assuming Theorem 3.1, the further study of discontinuous behavior of the limiting conditional probabilities then boils down to an analytical problem which will be treated in the next section. The result of the theorem itself relies on the following explicit expression of the conditional probabilities in finite-volume in terms of a quenched model. 3.2. Proof: Finite-volume representation in terms of quenched model. For any fixed configuration of ‘random fields’ η[1,N ] , let us introduce the following ‘quenched’ measure on the configurations σ[1,N ] : 2 N β N exp 2N + βε i=1 ηi σi i=1 σi . (8) µβ,ε,N [η[1,N ] ](σ[1,N ] ) := Z β,ε,N [η[1,N ] ] This is a Curie-Weiss Ising model with additional random fields η of strength that will be given by the spin-configuration at time t, constrained to have a fixed empirical distribution α at temperature β −1 . The single-site conditional probabilities of the timeevolved model can then be expressed in terms of an expectation w.r.t a quenched model of the form (8) on N-1 spins, depending on η[2,N ] . Proposition 3.2. For each finite-volume N we have the following representation of the conditional probabilities of the time-evolved model : µβ,t,N (η1 |η[2,N ] ) =
eη1 gβ,t,N (η[2,N ] ) , 2 cosh gβ,t,N (η[2,N ] )
where
1 N µ [η ] cosh β σ + ε β ,ε ,N −1 [2,N ] N i N N N i=2 N −1 1 gβ,t,N (η[2,N ] ) = log 1 N 2 µβ N ,ε N ,N −1 [η[2,N ] ] cosh β N N −1 i=2 σi − ε N
(9)
and
N −1 , βN εN = ht . (10) N Proof. The proof follows from the following rewriting: 2 N β N exp σ + βεσ + βε η σ 1 i=1 i i=2 i i σ[1,N ] µβ,t,N (η1 = +|η[2,N ] ) 2N . = 2 β N N µβ,t,N (η1 = −|η[2,N ] ) − βεσ1 + βε i=2 ηi σi i=1 σi σ[1,N ] exp 2N βN = β ·
Carrying out the σ1 -sums allows to recognize a quenched model on N-1 spins with frozen random fields η[2,N ] .
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Once we have recognized this quenched model, we can reduce its large-N analysis to the determination of the minimizers of a corresponding rate-function. For any N fixed configuration η[1,N ] obeying the constraint N1 i=1 ηi = α, we use a Gaussian transformation for the empirical σ -sum. The result is standard (see e.g. [2, 14]). N Proposition 3.3. Let the empirical spin sum N1 i=1 σi be distributed according to the quenched measure µβ,ε,N [η[1,N ] ] with random fields obeying the constraint 1 N i=1 ηi = α. Denote by G an auxiliary independent Gaussian N (0, 1) variable, N and put XN =
N 1 1 G. σi + √ N βN i=1
Then the distribution Pβ,ε,α;N of the variable X N has the Lebesgue-density given by e−β N β,ε,α (m) dm Pβ,ε,α;N [X N ∈ (m, m + dm)] = −β N (m) , β,ε,α e dm where the rate function β,ε,α is given by (6). The finite-volume distributions are uniquely defined due to permutation invariance. We got an identity for each finite N and the function β,ε,α (m) is the rate-function for X N . N So we have the convergence of N1 i=1 σi to the minimizer of when it is unique. Proof of Theorem 3.1. Put together the explicit representation of Proposition 3.2 of the conditional probabilities with the characterization of Proposition 3.3 of the limit of the quenched model. Then use continuity of the rate-function to get rid of the slight N -dependence (10) and to ignore the influence of the Gaussian variable G. 4. Extended Phase Diagram of Nonsymmetric Random Field Model - Bifurcation Analysis of the Rate-Function 4.1. Description of the extended phase diagram. We discuss now the structure of the minima of the rate function (6) when the parameters β −1 (temperature), ε (strength of the random fields), α (asymmetry of the distribution) vary. Recall that β −1 is the initial temperature of the system and = t = h t /β (3). The extended phase diagram is the decomposition of the parameter-space into regions where the structure of the rate-function does not change. By this we mean that there is no creation of minima (local bifurcations) and also no degeneracy with two global minima. While the potential function (6) depends on three parameters, the parameter α plays a special role to us here. We are interested in values of the parameters (β −1 , ε) for which there is a value of α = αc such that a degeneracy in the depth between two global minima occurs. For the actual computations it is convenient to make a change of variables E = βε,
M = βm and write
β,E,α (M) := β β,ε,α (m) =
M2 − Eα log cosh M + E η˜ . 2β
(11)
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Fig. 2. Extended phase diagram of quenched CW-RFIM
M → β,E,α=0 (M) is an even function that remains unchanged under the joint substitution of (α, M) by (−α, −M), thus we only consider the case α ∈ [0, 1]. We encounter two mechanisms of creation of new minima, a fold bifurcation and a pitchfork bifurcation [13, 20]. This analysis has partially been done for the case α = 0 [21, 1, 3] and our analysis incorporates these results. There are moreover interesting new phenomena arising when we look at general, possibly non-zero α’s. To state this ¯ β, ,α = β,β ,α and get the following Fig. 2 describing the different theorem, we write regions of the extended phase diagram : Region R40 : Uniqueness for all α; no phase transition and no degeneracy. Region R01 : Phase transition for α = 0; coexistence of two phases, the “+”/“-”-like phases. Uniqueness with possibly metastability for α = 0. Region R12 : Phase transition with metastability for α = 0; coexistence of “+”/“-”-like phases, and a “±”-metastable phase. Uniqueness for α = 0. Region R23 : Metastability without phase transition; unique neutral “±”-like phase with two “+”/“-”-like metastable phases when α = 0. Coexistence for αc = 0. Region R34 : Uniqueness with metastability for α = 0, degeneracy at some α = 0 and phase coexistence for αc = 0. ¯ β,ε,α has a unique minimizer M ∗ . Theorem 4.1. 1. If β −1 ≥ 1,
1 ≤ β −1 < 1, there exists a sharp value 0 ≤ ε0 (β −1 ) ≤ ε∗ = 23 · arctanh 3 : ∗ ¯ β,ε,0 has two global minimizers ±M , M = 0 is a local maximizer, • For ε < ε0 , ¯ β,ε,α has a unique minimizer and no maximizer for all α = 0. but ¯ β,ε,α has a unique minimizer M ∗ for all α ∈ [−1, +1]. • For ε ≥ ε0 , 2 −1 3. If 0 < β < 3 , there exist sharp values 2. If
2 3
0 < ε1 (β −1 ) < ε2 (β −1 ) < ε3 (β −1 ) < ε4 (β −1 ) s.t.
(12)
¯ β,ε,0 has two minimizers ±M ∗ and a local maximizer at M = 0; • For ε ≤ ε1 , ¯ β,ε,α has a unique global minimizer and no maximizer for α = 0. ¯ β,ε,0 has two minimizers ±M ∗ , M = 0 is a local minimizer • For ε1 < ε < ε2 , and there are two local maximizers ± Mˆ with 0 < Mˆ < M ∗ . All minimizers are ¯ β,ε,α has a unique global minimizer for all α = 0. global for ε = ε2 and
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¯ β,ε,0 has two local minimizers ±M ∗ , M = 0 is the unique • For ε2 < ε < ε3 , global minimizer and there are two local maximizers ± Mˆ with 0 < Mˆ < M ∗ . ¯ β,ε,0 has a unique global minimizer M = 0 and no local one. • For ε3 ≤ ε, ¯ β,ε,αc takes its global • For ε2 < ε < ε4 there exists αc ≡ αc (β −1 , ε) > 0 s.t minimum at precisely two global minimizers. ¯ β,ε,α has a unique global minimizer, for all α. • For ε ≥ ε4 , The functions εi describe the extended phase diagram (Fig. 2). The regions of the (β −1 , ε)-plane where the structure of the extrema of the rate-function changes are separated by the lines (L i )i=0,..,4 corresponding to (εi )i=0,..,4 . Theorem 4.2 (Characterization of the lines L i ). 1. There exists a tricritical point (β∗−1 , ε∗ ) with β∗−1 = 23 , where all the lines L i meet. 2. L 0 and L 1 are defined explicitly for β −1 ∈ [0, 1] by the same expression for i = 0, 1: εi (β −1 ) = β −1 · arctanh
1 − β −1 , ∀β −1 ∈ Di .
(13)
3. L 2 and L 3 can be parameterized by E > E ∗ = β∗ .ε1 (β∗ ) with, for i = 2, 3, ⎧ −1 ⎨ βi (E) = G i (Mi (E), E) ⎩
i (E)
= βi−1 (E) · E,
(14)
with G i explicitly known, Mi (E) = 0 implicitly via Fi (M, E) = 0, and Fi explicit. 4. The line L 4 is explicitly parameterized by M > 0 in the sense that
2 −1 β −1 1 −1 = β < (M) · , 0 < M < ∞ , 0 < β 4 E 4 (M) ε4 (β −1 ) 3
(15)
2y 3 + M(1 − y 2 )2 , E 4 (M) =arctanh 2y + M(1 − y 2 )
(16)
with
where y = y(M) = tanh(M), and β4−1 (M) was defined in (5). The explicit expressions of the Fi ’s and G i ’s entering in the parameterizations of the lines L i for i = 2, 3 are given in the proof (Eqs. (35), (36), (39), (40)). The well-known phase-diagram of the Curie-Weiss random field Ising model with symmetric distribution α = 0 was already described in [21]. They obtained L 0 analytically and L 2 numerically. A more complete analysis is required here; we need to know the nature of the extrema as a function of (β −1 , ε, α). For the proof of Theorem 2.2 on non-Gibbsianness, only the lines L 0 , L 2 and L 4 are relevant, but the study of the lines L 1 and L 3 is valuable to help the understanding of the bifurcation-structure of the rate-function.
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4.2. Bifurcation-structure of the rate function. The structure of the extrema of the ratefunction is best understood when we fix E = βε and vary the temperature β −1 , so that we are moving along lines of fixed slope E in the phase diagram. There are two fundamentally different cases: E ≤ E ∗ and E > E ∗ ,where √3 E ∗ = β∗ ε∗ = arctanh ≈ 0.6585 3 corresponds to the slope at the tricritical point, whose coordinates are √3 2 2 −1 β∗ = and ε∗ = arctanh ≈ 0.44. 3 3 3
(17)
This is due to the fact that the fourth derivative of the rate-function for α = 0 vanishes at M = 0 if and only if E = E ∗ . (A) Take E ≤ E ∗ , assume first α = 0 and start from the region above the line L 0 (high-temperature region). Then the rate function has a unique minimizer. When we lower β −1 we will eventually cross the line L 0 from the outside. At L 0 we encounter a pitchfork bifurcation: The unique minimizer M = 0 becomes a local maximizer and two symmetric minimizers are emerging from it. The symmetry is lifted when α = 0. (B) Let us now fix E > E ∗ . Put again α = 0 first and start from the region below L 0 -L 1 where there are two global minima and increase the temperature β −1 . Then we first see a pitchfork-bifurcation at the line L 1 that produces a new minimum at M = 0 and two local maxima. When we increase β −1 this minimum gets deeper to the depth of the non-zero minimum, with equal depth at the line L 2 . When we further increase the temperature β −1 the zero-minimum becomes the deepest. The non-zero minima will finally vanish at the line L 3 by a fold bifurcation, to be discussed below. When we vary α around 0, the degeneracy of the global minima for the non-zero minimizer is lifted in the open region below L 0 -L 2 . The global minimizer jumps from a positive value to a negative value when we pass α = 0 from positive values of α to negative ones. Suppose we are in the open region between L 2 and L 3 . By increasing α from zero, the minimum of the positive minimizer will become deeper w.r.t. the minimum around zero. At the critical value αc (β −1 , ε) the depths of the two minima become equal. This mechanism creates a bad point for the time-evolved measure at a non-zero value of α. This sort of degeneracy for the specifically chosen α = αc (β −1 , ε) is also possible in the whole open region between L 2 and L 4 , even though there is a unique minimizer between L 3 and L 4 for α = 0. If we are in particular on the line L 3 , increasing α yields a fold bifurcation driven by the parameter α at α = 0, for fixed (β −1 , ε), creating a new local minimizer for α positive. This also occurs in the region between L 3 and L 4 , but at strictly positive values of α. Further increasing the α will then create equal depth of these minima at α = αc (β −1 , ε). The line L 4 is the boundary curve in the parameter space for which this is possible. 2
4.3. General facts about the rate-function. Note first that β,E,α (M) ∼ M 2β as |M| ↑ ∞ so by continuity and boundedness its infimum is attained at one of its local minima. Lemma 4.3. For all (β, E, α), the function M → β,E,α (M) has at most three local minima. For β −1 ≥ 1, it has no maximum and its minimum is the only local extremum.
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Proof. To prove the desired bound of local minima, it suffices to show that the number of zeroes of the second derivative of (w.r.t. M) is bounded by 4. By an explicit computation, this follows from an elementary discussion of a polynomial in the variable z = e M+E . The other statement follows since the curvature is always positive. In order to discuss the phase diagram we will need to consider the first four derivatives k the k th derivative with respect to M, one has e.g. w.r.t. M. If we denote ∂ M ∂ M β,E,α (M) =
M − Eα tanh M + E η˜ , β
and any minimizer of the rate function must satisfy the so called mean-field equation M = Eα tanh M + E η˜ . β
(18)
For any real function f (M, E) we use the notations 1 1 f¯(M, E) = ( f (M + E) + f (M − E)) and f (M, E) = ( f (M + E) − f (M − E)) 2 2 (19) for the symmetric average and the discrete derivative around M. For example we write ∂ M β,E,α (M) =
M − tanh M, E − α tanh M, E . β
4.4 Elements of catastrophe theory 4.4.1 Catastrophe manifold. To understand the structure of the minima of the potential function M → β,E,α (M) when the three parameters (β, E, α) are varied is an elementary but not completely trivial analytical problem. We find it useful to introduce to this end some systematic basic notions from catastrophe theory going back to Thom and Zeeman (see [20]), to the level that we need them. Catastrophe theory allows for the classification of the possible singularities of local bifurcations of potential functions, as provided by Thom’s theorem. It makes rigorous that it suffices to look at polynomial approximations for the potential around critical points to understand locally the nature of bifurcations, i.e. the changes of the structure of the extrema of the potential. In our present case we recognize a butterfly-singularity which appears for a four-dimensional family, while we view it here from a three-dimensional sub-space. Indeed, our family has only one parameter α that distinguishes M from −M, while in the full butterflyunfolding one has two such “odd” parameters. As a consequence, not all symmetries are lifted in our present problem. However, more than just locally identifying the nature of the singularity around a critical point, we manage to provide a fairly explicit analysis over the whole manifold. This is possible by carefully exploiting the specific form of our potential. Taking an elementary but useful message from catastrophe theory, we use M as a coordinate and do not try to solve the appearing equations for M directly. This will be useful throughout the analysis, and allows us in particular to find the explicit parametric form of boundary curve between Gibbsian and non-Gibbsian parts of the phase diagram. Here the magnetization (or state-variable) M appears as a parameter of the curve.
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The catastrophe manifold X is the set of points in the product space of state space R and parameter (or control) space C = R+ × R+ × R satisfying the mean field equation X = (M, β −1 , E, α) ∈ R × C|∂ M β,E,α (M) = 0 . It is a smooth sub-manifold of R4 that can be characterized by the function Mβ −1 − tanh M, E −1 αMF (M, β , E) := ,
tanh M, E
(20)
well-defined since E = 0 implies that the denominator does not vanish. Hence X can be written as a graph of the smooth map αMF with the coordinates (M, β −1 , E). The points where this manifold folds “under”, when we view the direction of the state variable M as the vertical direction, are points of the vertical tangent plane, and 2 these are given by the vanishing of the curvature of the potential (∂ M β,E,α (M) = 0). The projection to the parameter space of this space, is the bifurcation set 2 β,E,α (M) = 0 . B := (β −1 , E, α)|∃M ∈ R, ∂ M β,E,α (M) = 0, ∂ M It is our aim to describe first this bifurcation set. For our analysis of non-Gibbsianness we are of course interested in the points of global bifurcations where two minima acquire equal depths. To understand this, we show how the analysis of the local bifurcations of the full three-parameter problem can be reduced to a bifurcation problem of a potential function of one variable and the remaining one-dimensional control variable E. We manage here to do this globally and not only near a critical point. To do so we are now going to exploit the linear nature of the potential as a function of the parameters (β −1 , α). Proposition 4.4. Define the functions 1 + tanh(M + E) tanh(M − E) 1 + M(tanh(M + E) + tanh(M − E))
(21)
M 1 − tanh2 M, E − tanh M, E . α12 (M, E) :=
tanh M, E + M tanh2 M, E
(22)
−1 β12 (M, E) :=
and
Then the bifurcation set has the parametric representation −1 B = α12 (M, E), β12 (M, E), E , E > 0, M ∈ R .
(23)
Remark. Equation (23) gives a parametric representation of the bifurcation set B. The geometry of this set, consisting of a two-dimensional surface with self-intersections, is best understood when we look at intersections of B with planes of fixed E. Performing parametric plots for various fixed values of E, with running variable M, we see a qualitative change of the curves in (β −1 , α)-space at E = E ∗ . In going from values E < E ∗ we see a cusp-like shape (Fig. 3) that unfolds into a pentagram-shaped curve for E > E ∗ (Fig. 4). This singularity is well known from the so-called butterfly unfolding (compare [20], −1 −1 (M, E) = β12 (−M, E), from p 178). We note that α12 (M, E) = −α12 (−M, E) and β12 so the curve is symmetric about the axis α = 0. The critical value E = E ∗ is explained 4 ∗ by the fact that ∂ M β −1 ,α=0,E (M = 0) changes sign at E = E .
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Fig. 3. Cusp: α vs. β −1 at fixed E = 0.6 < E ∗
Fig. 4. Pentagram: α vs. β −1 at fixed E = 1.3 > E ∗
Proof of the Proposition . Consider the equations of the bifurcation set B ⎧ M ⎪ ⎨ β = tanh M, E + α tanh M, E ⎪ ⎩
1 β
= 1 − tanh2 M, E − α tanh2 M, E .
For fixed M and E > 0, this is a linear system for β −1 and α and the proposition follows from standard arguments showing that it has always a unique solution of the −1 given form (β12 (M, E), α12 (M, E)). The point E ∗ , α = 0, β −1 = 23 is characterized by the fact that the first four derivatives of the potential at M = 0 vanish. Let us discuss the number of minima of the potential function β −1 ,α,E (M) for parameter values outside of B. For E < E ∗ , the potential has a single minimum outside the cusp-shaped region in Fig. 3. Within the cusp, the potential has two minima which are created by a fold bifurcation by crossing a branch of the cusp from the outside. This singularity is the well-known cusp-catastrophe, see [20] p. 174. For E > E ∗ , the situation is more complicated and well-described for instance in Fig. 9.12 in [20]. To the right of the pentagram-shaped curve the potential has a unique
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minimum. In the two triangle-shaped regions the potential has two minima. For the upper triangle, one of these minima is created by a fold crossing the upper boundary from above; the other one is created by crossing the right boundary from the left. In the four-cornered region the potential has three minima. Finally, in the region to the left, the potential has two minima. Since the number of minima is a constant for parameter values in the complement of the bifurcation set, we know the number of minima of the potential for any parameter values by extension from the known shape of the potential locally around the critical point, when we know that there are no other types of singularities appearing. That is, we want to prove now that over the whole phase diagram, and not only locally around (E ∗ , α = 0, β −1 = 23 ), this structure persists. 4.4.2. Exact form of L 0 , L 1 and L 4 from one-dimensional bifurcation problem. Additionally to fixing E, we fix now the parameter β −1 . We want to characterize the points α in the bifurcation set over the (β −1 , E)-plane. We will see that there are either 0, 1, 2, 3 or 4 points of such α, and no more, for any fixed (β −1 , E), as the plots of the fixed E-slices of B given in Fig. 3 and 4 suggest. We have from the proposition −1 α ∈ R|(α, β −1 , E) ∈ B = αMF (M, β −1 , E), ∃M ∈ R : β −1 = β12 (M, E) , (24) where α M F is given by (20). We thus see that by varying the independent parameter β −1 we change the number of solutions for M depending on the form of the function −1 (M, E) and the number of its local extrema. The corresponding M gives us a soluβ12 −1 (M, E) tion α = α12 (M, E) on the bifurcation set B. In particular β −1 ≥ sup M∈R β12 implies that there are no solutions for α, and hence the potential β −1 ,E,α has a unique minimizer, for any α. To understand the number of solutions for M in (24) we can obtain for different E, −1 (M, E), viewed as we need to proceed now with the bifurcation-analysis of M → β12 (the inverse of) an auxiliary potential function of the one-dimensional parameter E. Remark that one can rewrite −1 (M, E) = (1 − tanh2 (E)) · β12
1 + tanh2 (M) . tanh(M)(1 − tanh2 (E)) (25)
1 − tanh2 (M) tanh2 (E) + 2M
Let us denote for E ≤ E ∗ ,
−1 (M, E), β0−1 (E) := sup β12
(26)
−1 (M, E). β4−1 (E) := sup β12
(27)
M∈R
and for E ≥ E ∗ , M∈R
In Fig. 4 the temperature β4−1 (E) appears as the projection of the right corners of the pentagram to the β −1 -axis. We shall see that these functions indeed define the main critical line L 0 -L 4 of the phase diagram whose image to the time-temperature space is in fact the boundary curve of the Gibbsian region given in Fig. 1. One easily finds β0−1 (E) = 1 − tanh2 E for E < E ∗ , explaining the functional form (13) of the boundary curve L 0 . This functional form for the maximum over M
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−1 Fig. 5. Pitchfork-bifurcation driven by E at E = E ∗ : potential function β12 (X = E, Y = M, Z = β −1 )
is not true anymore for E > E ∗ , as we will see below. Instead we define for E ≥ E ∗ −1 the function β1−1 (E) := β12 (M = 0, E) = 1 − tanh2 E by this very same functional −1 form as β0 . This corresponds to the line L 1 of the phase diagram. This temperature appears as the leftmost corner in Fig. 4. To perform actual computations, we consider the potential β12 (M, E) with state variable M and control parameter E. Denote by Y the corresponding catastrophe set Y = {(M, E) ∈ R × R+ |∂ M β12 (M, E) = 0} . We can find an explicit representation of Y in terms of a smooth curve, writing β12 (M, E) = β1 (E) U0 (M) − (tanh2 E)U1 (M)
(28)
with the functions U0 (M) =
1 + 2M tanh M tanh2 M + 2M tanh M (M) = and U . 1 1 + tanh2 M 1 + tanh2 M
The following proposition describes the global structure of β12 (M, E). Proposition 4.5. The function M → β12 (M, E) undergoes a pitchfork bifurcation where two minima are created by increasing E at the point E ∗ . More precisely, the ¯ catastrophe set has the form Y = Y ∪ {0} × R+ , where Y¯ = M, E 4 (M) , M ∈ R is the graph of the function E 4 (M) =
1 + z 4 (M) 1 log , 2 1 − z 4 (M)
(29)
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where z 4 (M) ≡
z 42 (M) ≥ 0 is given by
z 42 (M) :=
U0 (M) 2 tanh3 M + M(1 − tanh2 M)2 = U1 (M) 2 tanh M + M(1 − tanh2 M)
(30)
for M = 0, and z 42 (M = 0) = 0. The bifurcation set of (28), the set of E’s for which 2 β (M, E) = 0, consists of the single point there exists an M s.t. ∂ M β12 (M, E) = ∂ M 12 E = E∗. Remark. Y¯ describes a line of global minimizers of M → β12 (M, E). Varying the parameter E, we thus have the following picture: If E ≤ E ∗ the function M → β12 (M, E) has its unique local minimum at the unique minimizer M = 0, and no other local extrema. If E > E ∗ the minimizer of the function M → β12 (M, E) are given by a pair ±M4 (E), with M4 (E) > 0. M = 0 is a local maximizer and there are no other local extrema. Proof. It is easy to show that a pitchfork bifurcation occurs at E = E ∗ . In fact, for 2 β this it suffices to show that, at E = E ∗ and M = 0 the derivatives satisfy ∂ M 12 = 2 4 0, ∂ E ∂ M β12 < 0 and ∂ M β12 > 0. We skip details here. More interesting is the global analysis. It implies in particular that there are no other bifurcations appearing and the local picture around E = E ∗ and M = 0 extends to the whole space. Note that ∂ M β12 (M, E) = 0 is equivalent to U0 (M) = (tanh2 E)U1 (M),
(31)
where the parameter E > 0 appears only in terms of multiplication by a function, so that z 2 = tanh2 E acts like a re-parameterized linear control variable. It is easy to check that this equation for M always has the solution M = 0. Next, we find all the other solutions (after dividing out M = 0) by putting tanh2 E = z 42 (M).
(32)
To show that there are no further bifurcation points other than E = E ∗ , realize that 2 β (M, E) = 0 imply that ∂ z 2 (M) = 0. But, given the equations ∂ M β12 (M, E) = ∂ M 12 M 4 the explicit form (30) of this function of one variable and no parameter, it is an elementary task to show that this implies that M = 0. However, from this follows by the second 2 β (M = 0, E) = 0 that in fact E = E is the only point in the bifurcation equation ∂ M 12 ∗ set. As an important consequence of this we get the desired exact form for L 4 . Theorem 4.6. The line L 4 has the parametric representation of the form 1 − z 42 (M) 1 + z 4 (M) 1 −1 (β4 (E), E), E ≥ E ∗ = ,M >0 , , log 1 − z 4 (M) U0 (M) − z 42 (M)U1 (M) 2 (33) where z 4 (M) is given by (30). Remark. As a corollary we obtain from here the simple form of L 4 that was given in Theorem 4.2 by a simple computation using the explicit expressions. Note that the parameter M is really β times the magnetization that the system acquires at the melting of the minima at αc (β, ε).
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−1 Proof. Recall that β4−1 (E) is the maximum over M of β12 (M, E) for E ≥ E ∗ . But from the previous proposition it follows that a local maximum is achieved along the curves provided by Y¯ in (M, E)-space, for E ≥ E ∗ . Since there are no other bifurcations appearing, and the functions go to zero with |M| ↑ ∞ these two symmetric local maxima must automatically be global maxima. So we may parameterize the curve in the −1 (β −1 , E)-space in terms of the corresponding M, with e.g. β4−1 (E) = β12 (M, E 4 (M)). From this the representation follows by substitution of the explicit expressions.
Having understood the form and bifurcation of β12 (M, E) it is now immediate to go back to the full bifurcation problem and understand the sets of solutions of M for the equation appearing on the r.h.s. of (24): −1 Corollary 4.7. Consider the equation β −1 = β12 (M, E). Fix E ≤ E ∗ . Depending −1 whether β0 (E) is strictly smaller (resp. equal, strictly bigger) than β −1 , the number of solutions M is 0 (resp. 1, 2). Fix now E > E ∗ . Then this number ranges from 0 to 5 depending on the value of β −1 relative to β1−1 (E) and β4−1 (E).
The set of α’s that lie in the bifurcation set above a point (β −1 , E) is given by applying the map α12 to the solutions of the above equation. Note that this map is not −1 (M, E)) has double-points always injective, indeed the curve M → (α12 (M, E), β12 for E > E ∗ (we see from the pentagram-shaped Fig. 4 that there are three double-points). Denote by M3 (E), for E > E ∗ , the positive solution of α12 (M, E) = 0. Then we have −1 (M3 (E), E). This is the point of fold-bifurcation at α = 0. β3−1 (E) = β12 4.4.3. Global bifurcations: Degeneracy of minima. Let us finally turn to the discussion of the set of α’s such that M → β,ε,α (M) has two different global minimizers. We recall that the temperature β2−1 (E) is the unique temperature of equal depth of all three minima of the potential at α = 0. Therefore it is clear that β0−1 (E) ≤ β2−1 (E) ≤ β3−1 (E). Exact characterizations of the later two temperatures will be given using different techniques in the next section. Proposition 4.8. For any β −1 ∈ (β2−1 (E), β4−1 (E)) there is a unique αc (β −1 , E) > 0 such that there are two different global minima of equal depth. For any other β −1 the global minimum is unique for any α = 0. Proof of the Proposition. Fix E > E ∗ . Now, for β = β2 (E), we have equal depth for the three minima of at α = 0. By increasing α, the corresponding minimizers move along smooth curves, until they vanish. The uniqueness of the value of αc now follows from the following lemma. Lemma 4.9. Suppose that, for fixed (β, E), there are two different smooth curves α → Mi (β, α), i = 1, 2, α running in the same interval α ∈ I , of local extreme points, where M1 (β, α) < M2 (β, α). Then the function α → β,E,α (M2 (β, α)) − β,E,α (M1 (β, α)) is decreasing in α. In particular, there is at most one value of α for which β,E,α (M1 (β, α)) = β,E,α (M2 (β, α)). Proof of the Lemma. Looking at the evolution of the values of the rate-function at both curves of extreme points with the parameter α we have cosh(Mi (β, α) + E) d 1 β,E,α (Mi (β, α)) = − log . dα 2 cosh(Mi (β, α) − E)
(34)
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cosh(M+E) It is elementary to see that the function M → − 21 log cosh(M−E) is decreasing with M (and odd in M), for any fixed E > 0. This implies the claim.
We start now at some point β −1 ∈ (β2−1 (E), β3 (E)−1 ) and α = 0: Precisely three local minimizers M˜ 0 = 0 and M˜ 1 = − M˜ −1 with M˜ 1 > 0 exist, with furthermore β −1 ,E,α=0 ( M˜ 0 ) < β −1 ,E,α=0 ( M˜ 1 ). Indeed, recall that β2 (E) is the boundary curve for ferromagnetic order in the phase-diagram of the well-known symmetric Curie-Weiss RFIM. Now, increasing α these minimizers move along curves M˜ i (α). The lemma implies that β −1 ,E,α ( M˜ 0 (α)) − β −1 ,E,α ( M˜ −1 (α)) and β −1 ,E,α ( M˜ 1 (α)) − β −1 ,E,α ( M˜ 0 (α)) decrease, along these curves. Hence M˜ −1 (α) plays no role for the discussion of the global minimum because the zero-type minimum is already deeper. It is well-known from the geometry of the butterfly bifurcation and the form of β −1 ,E,α in the pentagram-shaped region that the two left minima are vanishing by fold-bifurcations when increasing α. When the point M0 (α) vanishes by a fold bifurcation, at the corresponding point α ∗ we have β −1 ,E,α ∗ ( M˜ 1 (α ∗ )) − β −1 ,E,α ∗ ( M˜ 0 (α ∗ )) < 0. Hence there is a point αc (β −1 , E) when the two right-most minima have equal depth. It is unique because the lemma excludes double crossings. For the other regions of the phase diagram we use the same arguments. We see in that way that for any other β −1 the global minimum is unique for α = 0. Indeed, consider β −1 < β2−1 (E), E > E ∗ . Then the plus-type minimum is already deeper than the zero-type minimum and becomes even deeper with increasing α, and this excludes the degeneracy of the global minimum. 4.5. The line L 2 : degeneracy of non-zero minima and neutral minimum. When E > E∗, there can be at most two non-negative local minima and it is indeed the case when we cross L 1 . When β −1 increases, no other local extremum can merge but another type of degeneracy can occur, depending on the depth of the local minima. The region R12 between L 1 and L 2 is now characterized by the presence of two symmetric absolute minima at M and −M and a metastable minimum at 0, and all three acquire equal depth at L 2 . We first state that we indeed get a line for L 2 by stating the next lemma which says that degeneracies of the depths of the minima of the ‘same type’ can only occur for one specific value of β −1 , which also provides the sharpness of the transition times of Theorem 2.2. Lemma 4.10. Suppose that, for fixed (β, E, α), there are two different smooth curves β −1 → Mi (β, E, α), i = 1, 2, of local extreme points, where M12 (β, α) < M22 (β, α). Then the function β −1 → β,E,α (M2 (β, E, α)) − β,E,α (M1 (β, E, α)) is decreasing in β −1 . In particular, for fixed E > E ∗ , there is at most one value of β −1 for which β,E,α (M1 (β, E, α)) = β,E,α (M2 (β, E, α)). Proof. The proof is completely analogous to Lemma 4.9, the variation w.r.t. α has only to be replaced by the variation w.r.t β −1 . So we get the following two equations valid at L 2 for (M, E, β): ⎧ M ⎪ ⎨ β = tanh(M, E), ⎪ ⎩
M2 2β
− log(cosh)(M, E) = − log(cosh)(E).
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Dividing the two equations eliminates the linear variable β −1 and implies the following implicit relation of (M, E) characterizing the solutions F2 (M, E) = 0 with F2 (M, E) = M tanh(M, E) − 2 log cosh(M, E) + 2 log cosh(E).
(35)
Note that M = 0 is always a solution, for any E ∈ R+ . Looking however for E ∈ (E ∗ , ∞), we can numerically find a non-zero solution M2 = M2 (E). We also emphasize that there can only be one value of M = 0 for fixed E > E ∗ , since we have proved before that the rate-function can have at most 3 local minima, for any choice of the parameters. Having solved this (M, E)-relation, the curve L 2 is obtained immediately by solving the mean-field equation for β −1 . This gives us the parametric representation E → β2−1 (E) =
tanh(M2 (E), E) . M2 (E)
(36)
We obtain from this a parametric plot by putting ε2 (E) := β2−1 (E) · E. Let us show that the line has a representation as a function ε2 = ε2 (β −1 ), i.e. that d −1 d E β (E) < 0. To see this we take the derivative of the second equation w.r.t. E and note that the terms involving the M-derivative vanish, due to the mean-field equation. We get d −1 M(E)2 β (E) = tanh(M(E), E) − tanh(E). dE 2 The proof is now finished by showing that the r.h.s. is strictly less than zero, for all M > 0 and E > 0. This is seen by writing it in terms of a rational function of the variables y = e2M and b = e2E . 4.6. The line L 3 : Fold bifurcation. The region between the lines L 2 and L 3 is characterized by a global minimum at M = 0 and two symmetric local minima at non-zero values ±M ∗ , M ∗ > 0 for α = 0, non-symmetric for α = 0. Decreasing β −1 along the lines of fixed E > E ∗ a fold bifurcation occurs at the line L 3 : a minimum and a maximum are created from a saddle point by varying the parameter p into one direction. It occurs at a one-dimensional parameter p = p0 whenever ⎧ ∂ M p0 (M0 ) = 0, ⎪ ⎪ ⎪ ⎪ ⎨ ∂ 2 p (M0 ) = 0, 0 M (37) ⎪ ∂ ∂ ⎪ p M p0 (M0 ) = 0, ⎪ ⎪ ⎩ 3 ∂ M p0 (M0 ) = 0. The first equation just says that M0 is a local minimizer. The second arises because minimizer and maximizer of quadratic type are merging and creating a saddle point. The line L 3 is thus characterized by the mean field equation at M and furthermore 2 ∂M β,E,α=0 (M) = 0. These two equations are reexpressed as ⎧ ⎨ ⎩
M β
= tanh(M, E), (38)
β −1
=
1 − tanh2 (M,
E).
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Again, dividing the two equations eliminates the linear variable β −1 and implies an implicit (M, E)-relation of the form F3 (M, E) = 0 with (39) F3 (M, E) = M 1 − tanh2 (M, E) − tanh(M, E). We follow the same logic as for the line L 2 . Note that M = 0 is always a solution, for any E. For E ∈ (E ∗ , ∞) there is a unique solution M = M3 (E) and we can find it numerically. As before, this gives us the parametric representation E → β3−1 (E) = 1 − tanh2 (M3 (E), E)
(40)
and ε3 (E) := β3−1 (E).E. To show that the line has a representation as a function ε = ε3 (β −1 ), we proceed similarly to L 2 . Let us now vary the parameter α around α = 0 at the line L 3 . We have ∂α ∂ M β,E,α (M) = − tanh M, E < 0 if and only if M > 0. But this means that we have a fold bifurcation w.r.t. α, too. The creation of a minimum/maximum-pair is in the direction of increasing α, since the third derivative w.r.t. M of the rate-function has a positive sign. This will allow the extension of the non-Gibbs region to a wider range of temperature for fixed E, namely in the “NG*” region R34 between L 3 and L 4 . 5. Proof of the Main Results, Conclusions and Perspectives 5.1. Proof of Theorems 2.2 and 2.3. Let us summarize the picture. We are dealing with a quenched system depending on the parameters β −1 (initial temperature), ε (time, ≡ strength of the random fields) and α (conditioning of empirical average of spins at time t, ≡ tilting of the random field distribution). Equivalently we are looking at the parameter space spanned by (β −1 , E, α). To get the time evolution picture (Fig. 1) from the extended phase diagram (Fig. 2), and thus Theorems 2.2 and 2.3 from Theorems 4.1 and 4.2, we use the smooth relation (3) between time t and h t = E t . Since t −→ h t is a decreasing one-to-one map from [0, ∞) to [0, ∞), we get the expressions of the lines of Fig. 1 directly from those of Fig. 2 by inverting (3). The regions of the extended phase diagram with qualitatively different phase structure are mapped into the regions of Gibbsianness, of neutral non-Gibbsianness or of biased non-Gibbsianness. To prove monotonicity of the critical αc in NG* as a function of time, we look at the system ⎧ ⎨ β,E,α (M1 ) = β,E,α (M2 ), ∂ M β,E,α (M1 ) = 0, ⎩∂ M β,E,α (M2 ) = 0,
and show that
dα d E β −1 ≥
0. Taking the derivative of the first equation w.r.t. E and using the second and the third equation, the claimfollows because M → log cosh M, E is an increasing function and M → Eα tanh M + E η˜ η˜ is descreasing.
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5.2. Conclusions and open questions. In this study of the extension of the Gibbs property from lattice systems to mean-field models, we have recovered the results that correspond to what has been proved on the lattice. Completing the picture, we discovered a new phenomenon, biased non-Gibbsianness. This is another indicator of the relevance of the definition of Gibbsianness for mean-field models recently introduced in [10, 15] and used in this paper, but other results and examples would be welcome. It is indeed well known that mean-field results transfer only partially to lattice systems, with sometimes peculiarities (van der Waals theory, equivalence of ensembles, structure of the Gibbs measures, etc.). Not all analogies should be taken too literal: For lattice systems, in d = 2, the chessboard configuration is bad, while a neutral random configuration is good, but in the mean-field set-up the difference disappears and the conditioning leads to the same constrained measure. Care is thus needed to extend mean-field results to lattices, but they remain a good tool to detect new types of phenomena. These may also be recovered for limiting cases of lattice measures. In particular, it would be interesting to know whether the new phenomenon of biased non-Gibbsianness could occur on lattices. Kac models are good candidates to study this question and a positive answer, even partial, would support the relevance of the notion of Gibbsianness in the mean-field set-up. Acknowledgements. We thank the Laboratoire de Mathématiques d’Orsay (CK), Eurandom (ALN, CK) and University of Groningen (ALN) for invitations, crucial in the achievement of this work. We thank A. van Enter, R. Fernández, E. Orlandi, F. Redig, M. Vares and V. Zagrebnov for encouraging discussions, H. Broer for advice on catastrophe theory, and we are grateful to L.M. Le Ny for the help in scilab and matlab to draw the pictures.
References 1. Amaro de Matos, J.M.G., Patrick, A.E., Zagrebnov, V.: Random infinite-volume Gibbs states for the Curie-Weiss random field Ising model. J. Statist. Phys. 66, 139–164 (1992) 2. Bovier, A.: Statistical mechanics of disordered systems. A mathematical perspective, Cambridge Series in Statistical and Probabilistic Mathematics 18,Cambridge: Cambridge University Press, 2006 3. Cassandro, M., Orlandi, E., Picco, P., Vares, M.E.: One-dimensional random field Kac’s model: localization of the phases. Elec. J. Probab. 10, 786–864 (2005) 4. Ellis, R.S.: Entropy, Large Deviations and Statistical Mechanics. Newyork: Springer-Verlag 1985 5. van Enter, A.C.D., Fernández, R., den Hollander, F., Redig, F.: Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures. Commun. Math. Phys. 226, 101–130 (2002) 6. van Enter, A.C.D., Fernández, R., Sokal, A.D.: Regularity properties of position-space renormalization group transformations: Scope and limitations of Gibbsian theory. J. Statist. Phys. 72, 879–1167 (1993) 7. van Enter, A.C.D., Külske, C.: Two connections between random systems and non-Gibbsian measures. To appear in J. Statist. Phys., DOI 10.1007/s10955-006-9185-9 8. van Enter, A.C.D., Lörinczi, J.: Robustness of the non-Gibbsian property: some examples. J. Phys. A 29, 2465–2473 (1996) 9. van Enter, A.C.D., Verbitskiy, E.A.: On the variational principle for Generalized Gibbs measures. In: Proceedings of the workshop “Gibbs vs. non-Gibbs in statistical mechanics and related fields” (Eurandom, 2003), Markov proc. Relat. Fields 10(2), 2004 10. Häggström, O., Külske, C.: Gibbs property of the fuzzy Potts model on trees and in mean-field. In: Proceedings of the workshop “Gibbs vs. non-Gibbs in statistical mechanics and related fields” (Eurandom, 2003), Markov proc. Relat. Fields 10(2), 2004 11. Fernández, R.: Gibbsianness and non-Gibbsianness in Lattice random fields. In: Mathematical Statistical Physics. Proceedings of the 83rd Les Houches Summer School, July 2005, London: Elsevier, 2006 12. Georgii, H.-O.: Gibbs Measures and Phase Transitions, NY, de Gruyter 1988 13. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences, 42, Springer, NY 1990 14. Külske, C.: Metastates in disordered mean-field models: Random field and Hopfield models, J. Statist. Phys. 88(5/6), 1257–1293 (1997) 15. Külske, C.: Analogues of non-Gibbsianness in joint measures of disordered mean field models, J. Statist. Phys. 112, 1101–1130 (2003)
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16. Külske, C., Le Ny, A., Redig, F.: Relative entropy and variational properties of generalized Gibbs measures. Ann. Probab. 32(2), 1691–1726 (2004) 17. Külske, C., Redig, F.: Loss without recovery of Gibbsianness during diffusion of continuous spins. Probab. Theor. Relat. Fields. 135(3), 428–456 (2006) 18. Le Ny, A., Redig, F.: Short times conservation of Gibbsianness under local stochastic evolutions. J. Statist. Phys. 109(5/6), 1073–1090 (2002) 19. Liggett, T.M.: Interacting Particle Systems. NY Springer-Verlag, 1985 20. Poston, T., Stewart, I.: Catastrophe Theory and its Applications, Surveys and reference works in mathematics, London: Pitman (1978) 21. Salinas, S.R., Wreszinski, W.F.: On the mean-field Ising model in a random external field. J. Statist. Phys. 41(1/2), 299–313 (1985) Communicated by J.L. Lebowitz
Commun. Math. Phys. 271, 455–465 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0205-7
Communications in
Mathematical Physics
On the Motion of a Compact Elastic Body Robert Beig, Michael Wernig-Pichler Institut für Theoretische Physik der Universität Wien Boltzmanngasse 5, A-1090 Vienna, Austria. E-mail: [email protected] Received: 2 January 2006 / Accepted: 1 September 2006 Published online: 13 February 2007 – © Springer-Verlag 2007
Abstract: We study the problem of motion of a relativistic, ideal elastic solid with free surface boundary by casting the equations in material form (“Lagrangian coordinates”). By applying a basic theorem due to Koch, we prove short-time existence and uniqueness for solutions close to a trivial solution. This trivial, or natural, solution corresponds to a stress-free body in rigid motion. 1. Introduction In the problem of motion for classical continua with free surface boundary, despite its obvious physical relevance, there is a surprising scarcity of rigorous results. A recent review of known results with and without gravity can be found in [14]. In the case of a nonrelativistic, non-gravitating perfect fluid local wellposedness has only been proved very recently by Lindblad [12]. In the present paper we consider the analogous problem for relativistic elastic solids without self-gravity. The nonrelativistic case of our results (which is in principle known - see [15] - although even there a detailed treatment seems to be lacking) can be proved by similar methods. The nonrelativistic case for incompressible materials has, by different methods, been treated in [7]. The plan of this paper is as follows. In Sect. 2 we describe elasticity theory on an arbitrary curved background as a Lagrangian field theory. Section 3 is devoted to the notion of “natural state”, that-is-to-say a solution to the elastic equations corresponding to a configuration with zero stress. The existence of such a solution requires the elastic body to move along a geodesic, timelike Killing vector. We require the Killing vector to be also hypersurface-orthogonal (thus forcing spacetime to be “ultrastatic”)1 . In the case where the metric on the space orthogonal to the Killing vector is flat, we are in Special Relativity. In Sect. 4 we perform a 3+1 decomposition of the elastic equations corresponding to the natural space-time splitting afforded by the Killing vector. We then 1 In Special Relativity a geodesic, timelike Killing motion has to be a time translation, so the above assumption is superfluous in that case.
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write the elastic equations in “material” form (often called “Lagrangian representation”). This means that the maps making up the elastic configuration space - which go from spacetime to the 3 dimensional “material space” - are replaced by time dependent maps (“deformations”) from material space into physical space. In the material representation the boundary of the body is fixed, namely the boundary of material space. In Sect. 5 we state a corollary to the theorem of Koch [11], which is the version of existence theorem we are using. In Sect. 6 we state our basic constitutive assumption. This assumption, which is satisfied by elastic materials occurring in practice, implies the validity of the conditions in the Koch theorem. From this one concludes the main result, which is stated in Sect. 6. In the appendix we show that the corner conditions on the boundary, which initial data have to satisfy in order for the time evolved solution to be a classical solution, can be satisfied for a large class of initial data. We should emphasize that our requirement of the existence of a global relaxed state, and the associated restriction on the background spacetime would be unnecessary if we were dealing with well-posedness in the absence of a boundary, or inside a region of the body causally disconnected from the boundary. However in the presence of the boundary we have to ensure corner conditions using an implicit function argument, and for this we use the existence of a state - the global relaxed one - which satifies these corner conditions to all orders. In this sense our strong restriction on the existence of a global relaxed configuration is perhaps not completely essential, but very convenient. We can state the central result of this paper as follows: For initial data close to initial data for the natural deformation in the appropriate function space, there exists, for sufficiently short times, a unique solution to the elastic equations. This solution depends continuously on initial data, in particular tends to the natural deformation, when the initial data tend to that of the natural deformation. 2. The Theory We treat elasticity as a Lagrangian field theory in the manner of [2] or [4], see also [10]. In the language of standard elasticity this means that the material in question is “hyperelastic”. The dynamical objects of the theory are furnished by sufficiently regular maps f sending a closed region S¯ = S ∪ ∂S of spacetime M onto B¯ = B ∪ ∂B, with B a bounded domain in R3 with smooth boundary ∂B called the “body” or material space. The domain B is to be thought of as an abstract collection of points (“labels”) making up the elastic continuum, and f is to be thought of as the back-to-labels map sending spacetime events to the particles by which they are occupied. We endow B with ¯ Coordinates on M are written as x µ , with µ, ν a volume form which is smooth on B. etc. ranging from 0 to 3 and coordinates on B as X A with A, B etc. going from 1 to 3. The metric gµν is also taken to be smooth. There are additional requirements on the maps f . Namely the equation f A ,µ u µ = 0, uµ
gµν u µ u ν = −1
(1)
should have a solution which is unique up to sign. Thus particles move along trajectories in M with unit tangent u µ . Furthermore the function n on S¯ defined by the equation ABC ( f (x)) f A ,µ (x) f B ,ν (x) f C ,λ (x) = n(x)µνλρ (x)u ρ (x) (2) ¯ should be everywhere positive on S. It is well known that (2) implies that the continuity equation ∇µ (nu µ ) = 0 (3)
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is identically satisfied. The equations of motion are the Euler-Lagrange equations of the action 1 ρ( f, ∂ f, x) (−g) 2 d 4 x. (4) S[ f, ∂ f ] = S
The function ρ is required to be smooth in all its arguments. It plays the double role of being the Lagrangian as well as energy (i.e. rest mass plus internal elastic energy) density of the material. Diffeomorphism covariance demands that ρ be of the form ρ( f, ∂ f, x) = ρ( ˆ f, H AB )| H AB =h AB (∂ f,x) ,
(5)
where h AB (∂ f, x) = f A ,µ (x) f B ,ν (x)g µν (x). Thus ρ depends only on points X on B and positive definite contravariant tensors H AB (see [2]). (In nonrelativistic elasticity materials governed by such a Lagrangian are said to satisfy the condition of “material frame indifference”.) By abuse of notation we henceforth omit the hat from ρ. ˆ It is useful to know that the Euler-Lagrange equations are equivalent to divergence-lessness of the symmetric energy momentum tensor. More precisely, there holds the identity (see [2]) −∇ν Tµ ν = E A f A ,µ ,
(6)
∂ρ − ρgµν ∂g µν
(7)
where Tµν = 2 and −E A = (−g)
− 21
1 ∂µ (−g) 2
∂ρ ∂( f A ,µ )
−
∂ρ . ∂f A
(8)
The relation (6) shows that of the four conservation laws ∇ν Tµ ν = 0 only three are independent. This, of course, is due to the fact that within the present formalism the continuity equation is an identity. We now turn to boundary conditions. The boundary conditions usually considered in nonrelativistic elasticity are either the so-called “boundary conditions of place” - where, in our language, the set f −1 (∂B) is prescribed, or “traction boundary conditions” - where the normal traction, i.e. the components of the stress tensor normal to this surface are prescribed. When the normal traction is zero, one speaks of “natural” boundary conditions: these are the boundary conditions appropriate for a freely floating elastic body considered in the present work. They are employed e.g. in geophysics for the elastic motion corresponding to seismic waves, the free boundary in question corresponding to the surface of the earth (see [1] 2 ).While one could in the present framework in principle consider all the above boundary conditions, it is interesting to observe that these conditions - except for the natural condition - become inconsistent once one couples to the Einstein equations. Namely the standard junction conditions, together with the Einstein equations, imply that Tµ ν n ν | f −1 (∂ B) be zero, where n µ is the conormal of f −1 (∂B). However these conditions are precisely the natural boundary conditions. To see this, we have to first compute the right-hand side of Eq. (7). The result is Tµν = ρu µ u ν + tµν ,
(9)
2 Needless to say, these studies are confined to the linear approximation in which the problem of the free boundary disappears.
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where tµν is the (negative) Cauchy stress tensor given by tµν = nτ AB f A ,µ f B ,ν
(10)
ρ = n
(11)
and we have written ρ as ∂ . ∂ H AB
Note this makes sense, since n - whence - is a function purely of and τ AB = 2 AB f and H , as is apparent from the identity
6n 2 = H A A H B B H CC ABC A B C .
(12)
The function is the relativistic version of the “stored-energy-function” of standard elasticity. The quantity τ AB corresponds to the negative of the “second Piola-Kirchhoff stress” of nonrelativistic elasticity. Now to the boundary conditions, namely Tµ ν n ν | f −1 (∂ B) = 0.
(13)
It follows from the tangency of u µ to the inverse images of points of B under f , that (u, n) = 0. Consequently the contraction of Eq. (13) with u µ , using (9), is identically satisfied. The remaining components yield τ AB f B ,µ g µν n ν | f −1 (∂ B) = 0.
(14)
Equation (14) will turn out to be a Neumann-type boundary condition, but it has a free (i.e. determined-by- f ) boundary. 3. Natural Configuration We assume there exists a contravariant metric H0AB (X ), smooth and positive definite on ¯ such that B, ∂ =0 (15) |(X,H AB =H AB (X )) = 0 (X ) > 0, 0 ∂ HCD AB AB (X,H
=H0 (X ))
¯ The quantity H AB measures local distances in the relaxed body. The function on B. 0 0 (X ) is the density of rest mass in the stressfree configuration. If there exists a configuration f 0 such that H0AB ( f 0 (x)) = h AB (∂ f 0 (x), x) = f 0A ,µ (x) f 0B ,ν (x)g µν (x) =: h 0AB (x),
(16)
this map f 0 is called a natural or relaxed configuration. Note that spacetime has to be special in order for a natural configuration to exist. Namely it follows from (16) that µ h 0AB has to be constant along u 0 , the four velocity associated with f 0 , from which one µ infers that u 0 is a Born rigid motion (see [16]), i.e. Lu 0 (gµν + u 0 µ u 0 ν ) = 0.
(17)
One deduces from (6,7,9) and (15) that f 0 solves the equations of motion E A = 0 if and µ only if this rigid motion is geodesic. This in turn implies that u 0 is Killing. In this work
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µ
µ
we will assume that u 0 is in addition irrotational. Using coordinates in which u 0 ∂µ = ∂t and u 0 µ d x µ = −dt, the spacetime metric has thus to be of the form (i, j = 1, 2, 3) gµν d x µ d x ν = −dt 2 + gi j (x k )d x i d x j .
(18)
(In particular: when the positive definite three metric gi j is flat, we are in Minkowski spacetime.) With this choice of coordinates, the natural configuration f 0 is of the form f 0 (t, x) = f 0 (x) with f 0 (x) a smooth, invertible function S¯ ∩ {t = 0} → B¯ where, using coordinates on M and B¯ in which i jk and ABC are both positive, det ( f 0A ,i ) is positive on S¯ ∩ {t = 0}, the latter condition guaranteeing that n 0 is positive and h 0AB is positive definite on S¯ ∩ {t = 0}. The inverse of f 0 (x) will be denoted by 0 (X ). 4. 3 + 1 and the Material Picture In this section we switch to the material description of the elastic medium by changing to “Lagrangian” coordinates. In a first step we replace the four velocity u µ corresponding to the configuration f by the coordinate velocity v µ ∂µ = ∂t + V i ∂i , which is a multiple of u µ . Using (1) we find that V i is given in terms of f A by (note that f A ,i is non-degenerate by construction) f˙ A + f A ,i V i = 0,
(19)
where a dot denotes partial differentiation w.r. to t. It follows from (18) that h AB = f A ,µ f B ,ν g µν = f A ,i f B , j (g i j − V i V j ).
(20)
The timelike nature of u µ (equivalently: the positive definiteness of h AB ) requires that |V |2 = gi j V i V j < 1.
(21)
n = k (1 − |V |2 )1/2 ,
(22)
k i jk = f A ,i f B , j f C ,k ABC
(23)
We also see that where k is defined by
with i jk being the volume element of gi j . Consequently the action (4) takes the form S = (1 − |V |2 )1/2 k (det gi j )1/2 dt d 3 x. (24) S
We now pass over to the material representation. By this we mean that configurations f A are replaced by “deformations” x i = i (t, X ) defined by f A (t, (t, X ))) = X A
(25)
¯ It follows from (25) that with det (i ,A ) positive in B. f A ,i (t, (t, X )) i ,B (t, X ) = δ A B , and
i ,A (t, X ) f A , j (t, (t, X )) = δ i j
˙ i (t, X ) = 0 f˙ A (t, (t, X )) + f A ,i (t, (t, X ))
(26) (27)
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and, from (27) and (19), that ˙ i (t, X ). V i (t, (t, X )) =
(28)
In order to study the field equations in the material representation it will be extremely convenient to change representation directly in the action (24). Using (23) there holds S= γ −1 dt d 3 (29) {t}×B
with
1
γ −1 = (1 − |V |2 ) 2
(30)
1 2
and d 3 = ABC (X ) d X A ∧ d X B ∧ d X C = d 3 X . Now the field equations take the form ∂L ∂L ∂L + ∂A − ∂t = 0 in B, (31) ˙i ∂i ,A ∂i ∂ 1 ¯ Note that where L = γ −1 2 for functions such that det (i A ) is positive in B. ˙ j ). depends only on X A , gi j depends only on i and γ −1 depends only on (i , Finally h AB , i.e.
˙ j ], ˙ i h AB = F A i (k ,C )F B j (l ,D )[g i j () −
(32)
˙ j , k ,A ), so that depends where F A i ( j ,B ) is the inverse of i ,A , depends on (i , ˙ j , k ,B ). Using these facts, together with the identity on (X A , i ,
we find that
and
∂h AB = −2h C(A F B) i , ∂i ,C
(33)
1 ∂ ∂L 2 N γ −1 = −2 h AB F C i ∂i ,A ∂ H BC H D E =h D E
(34)
1 ∂L ∂ A B ˙ j. 2 γ gi j + 2 = − F F i j ˙i ∂ H AB H C D =h C D ∂
(35)
From (34) we infer that the boundary conditions (14) are simply equivalent to ∂L n A = 0, i ∂ ,A {t}×∂ B
(36)
where n A is a conormal of ∂B. We now turn to the natural deformation, which is the material version of the natural configuration described in Sect. 3. In the chosen foliation it is time-independent, namely of the form i (t, X ) = i0 (X ), (37) where 0 ( f 0 (x)) = x. Furthermore we have that F A i (0 k ,C )F B j (0 l ,D )g i j (0 (X )) = H0AB (X ),
(38)
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in short: F0 A i F0 B j g0 i j = H0AB . We know from Sect. 3 that the field equations and the boundary conditions are identically satisfied in the stress-free state. In the present context this fact takes the form ∂L ∂L ∂L = 0, = 0, = 0, (39) ˙i 0 ∂i ,A 0 ∂i 0 ∂ ˙ i ) = (0 i , 0). To where the subscript 0 means evaluation in the stress-free state (i , derive Eq. (39) we have used (15,34,35). The causality properties of this system are essentially governed by the nature of the .. j
coefficients M tt i j of and M AB i j of j ,AB . In order to be able to apply the theorem of Koch [11], we will need a “negativity” condition on M tt i j and a certain positivity (“coerciveness”) condition on M AB i j . We easily find that M0tt i j
=
∂2 L ˙ i ∂ ˙j ∂
1
= − 2 0 g0 i j .
(40)
0
For the coefficient of j ,AB , using (34), observe that M0AB i j =
∂2 L ∂i ,A ∂ j ,B
where
(41)
0
L 0 ABC D :=
1
= 4 2 L 0 C E D F H0AE H0B F F0 C i F0 D j ,
∂ 2 ∂ H AB ∂ H C D
H E F =H0E F
.
(42)
5. Koch Theorem For convenience we state here a corollary of the theorem in [11], which is the precise statement we need. Let i (t, X ) be maps from {t} × B to R3 , where B is an open set in R3 with smooth boundary ∂B. We are given a system of the form of ∂α Fα i = wi in {t} × B,
(43)
where α runs from 0 to 3 and with Fα i and w j all being smooth functions of (X A , i , j ,α ) on B¯ × R3 × R12 , together with the boundary conditions Fα i n α |{t}×∂ B = F A i n A |{t}×∂ B = 0.
(44)
We make the following further assumptions: ∂ Fα i (i) Symmetry: There hold the symmetries M αβ i j = M βα ji , where M αβ i j = ∂ j . ,β
(ii) Static solution: There is given a time independent function i0 (t, X ) = i0 (X ) ∈ ¯ satisfying C ∞ (B) F0α i = 0,
w0 i = 0
(45)
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and the following estimates (iii) Time components: M0tt i j ηi η j ≤ −κ|η|2
in B
for a positive constant κ. (iv) Space components: M0AB i j δi ,A δ j ,B d 3 X + ||δ||2L 2 (B) ≥ σ ||δ||2H 1 (B) B
(46)
(47)
¯ where H s denotes the Sobolev space H s,2 . We for σ > 0 and all δ ∈ C ∞ (B), remark that condition (47) implies the “strong ellipticity” or “Legendre-Hadamard” condition, namely that M0AB i j N A N B k i k j > 0 in B¯ for non-zero N A , k i . This latter condition is the relevant one for wellposedness in the pure initial value problem (see [9]). ˙ Theorem. Let ((0), (0)) lie in a small neighborhood of (0 , 0) in H s+1 (B) × s H (B), s ≥ 3 and satisfy the corner conditions of order s in the sense that ∂tr FiA n A |{t=0}×∂ B is in H s−r (B) ∩ H01 (B) for 0 ≤ r ≤ s 3 . Then there exists, for ¯ with sufficiently small t0 , a unique classical solution of (43,44) in C 2 ([0, t0 ) × B) r 2 ˙ ((0), (0)) as initial data. Furthermore ∂ (t) ∈ L (B) for 0 ≤ r ≤ s + 1. In the last expression ∂ r denotes all partial derivatives of order r . The evolved solution stays ˙ close to the static solution 0 in the sense that ((t), (t)) remains close to (0 , 0) in H s+1 (B) × H s (B) and ∂ r (t) remains close to ∂ r 0 ∈ L 2 (B) for 0 ≤ r ≤ s + 1. (This last statement is the “stability” part of the theorem.) 6. Main Result We now add our final constitutive assumption, sometimes called “pointwise stability” see [13], namely that L 0 ABC D N AB N C D ≥ σ (H0 C A H0 B D + H0 C B H0 AD ) N AB N C D in B,
(48)
where σ is a positive constant which only depends on the choice of coordinates and with H0 AB being the inverse of H0AB . Condition (48) is a convexity condition on ρ as a function of H AB . We note that this in general does not imply convexity with respect to the deformation gradient i ,A (see [5]). An important special case is where 40 L 0 ABC D = λH0 AB H0 C D + µ(H0 C A H0 B D + H0 C B H0 AD ) and µ(X ) > 0,
¯ 3λ(X ) + 2µ(X ) > 0 in B.
(49) (50)
The quantities µ and λ, when they are independent of X , are the Lamé constants of homogeneous, isotropic materials. We now invoke the Korn inequality (see e.g. [8] or [6]) of which we need a slight generalization due to [3], in the following form: Let A be vector field on (B, H0 AB ) in some chart. Let L() be defined by L() AB = 2H0 C(A C ,B) , 3 This condition says that (∂ r FA ) n | i A {t=0}×∂ B is zero for 0 ≤ r ≤ s In the appendix we show that it can t ˙ be met by a suitable choice of odd-order normal derivatives of ((0), (0)) on ∂B.
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i.e. the Lie derivative of H0 AB w.r. to to leading derivative-order in A . Then there is a positive constant σ such that L() 2L 2 (B) + 2L 2 (B) ≥ σ 2H 1 (B) .
(51)
We now assume coordinates X in B to be chosen so that 0 is the identity map. Combining (41) with (48) and using (51), there follows condition (iv), i.e. that M0AB i j δi ,A δ j ,B d 3 X + δ 2L 2 (B) ≥ σ δ 2H 1 (B) (52) B
for positive σ . The validity of the condition (iii) in the Koch theorem is immediate from Eq. (40). Furthermore the validity of (ii) has been checked in Sect. 4. The validity of (i) is obvious from the variational character of the equations. Thus all assumptions of the Koch theorem are met. ¯ But, when It remains to check the determinant condition det (i ,A ) > 0 in B. ˙ ((0), (0)) is close to (0 , 0) in H s+1 (B) × H s (B), s ≥ 3, this immediately fol¯ Now the precise form lows by Sobolev embedding and the positivity of det (i0 ,A ) in B. of our final statement can be read off from the Koch theorem in Sect. 5. Stated somewhat informally, our final result is as follows: Theorem. Let there be given a volume form (X A ), a spacetime metric gµν (x λ ) of the form (18) and an internal energy (X A , H BC ) satisfying (15), all smooth functions of their respective arguments. Suppose there exists a smooth natural (i.e. stress-less) conµ figuration such that the corresponding vector field u 0 is the static Killing field ∂t . Also suppose that the elasticity tensor for this natural configuration satisfies the pointwise ˙ stability condition (48). Let the initial data ((0), (0)) for the deformation be close to those for the natural deformation and satisfy the corner condition of the appropriate order. Then there exists, for these initial data, a solution (t) of the dynamical equations (31) with boundary conditions (36) for small times. This solution remains close to the natural deformation. We end with a remark on the possible generalization of the results presented here to the case of an elastic body (or bodies, if there are several) which are self-gravitating. The presence in G.R. of constraints and gauge freedom will then make things more complicated. Furthermore the fact that the elastic equations are written in material form, but the Einstein equations are equations on spacetime, means that one is now not dealing with a system of partial differential equations in the strict sense.
Acknowledgements. We are grateful to Bernd Schmidt for very useful remarks on the manuscript. One of us (R.B.) thanks Tom Sideris for pointing out the relevance of the work of Koch and Sergio Dain for helpful discussions on the Korn inequality. This work has been supported by Fonds zur Förderung der Wissenschaftlichen Forschung in Österreich, project no. P16745-NO2.
7. Appendix: Corner Conditions We study corner conditions for the system (43) with the boundary conditions (44). Sup˙ pose that ((0)|∂ B , (0)| ∂ B ) is sufficiently close to (0 |∂ B , 0). In order for obtaining a solution of the equations of motion, one has to be able to satisfy conditions on the ˙ behaviour of certain normal derivatives of ((0), (0)) on ∂B, which guarantee that,
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not only F A i n A |∂ B = 0 at t = 0, but also a sufficient number of time derivatives at t = 0 of that condition is satisfied. (It is understood that higher-than-second-order time derivatives are eliminated in terms of spatial and lower-order time derivatives, using the equations of motion: this by virtue of the negativity of M0tt i j is always possible for ˙ ((0)|∂ B , (0)| ∂ B ) close to (0 |∂ B , 0).) First recall that 0 (t, X ) = 0 (X ) identically satisfies the equations of motion and the boundary condition. Now we check that the corner condition of order 0, i.e. the undifferentiated-in-time Eq. (36) can be satisfied by a choice of ∂n (0)|∂ B (where ∂n denotes any derivative transversal to ∂B). To see this notice first that, by the above observation, (0 |∂ B , 0) solves the order-0 corner condition. Furthermore ∂2 L ∂2 L n = n n (53) A A B . j i j i ∂∂n ∂ ,A ∂ ,B ∂ ,A ∂B ∂B Then the result follows from the (finite-dimensional) implicit function theorem using that, by virtue of (48), the quadratic form M0AB i j n A n B |∂ B
(54)
is nonsingular, which in turn follows from the strong ellipticity condition (see Sect. 5). One now performs the process of taking (s ≥ 1 say) time derivatives of the boundary condition and eliminating ∂tm (0)|∂ B for s + 1 ≥ m > 1. Using that F0 i is zero for th ˙ ((0)|∂ B , (0)| ∂ B ) = (0 |∂ B , 0), the s order corner condition becomes an equation of the form ˙ j (0)|∂ B = ˆ lower order (55) M0AB i j n A n B ∂ns for odd s and ˆ lower order M0AB i j n A n B ∂ns+1 j (0)|∂ B =
(56)
for even s, where “= ˆ lower order” means expressions which depend on lower-order, ˙ even-numbered normal derivatives of ((0)|∂ B , (0)| ∂ B ), modulo terms which depend ˙ on normal derivatives of the same order, but which are zero when ((0)|∂ B , (0)| ∂B) = ˙ (0 |∂ B , 0). Equations (55,56) are identically satisfied if ((0)|∂ B , (0)| ∂ B ) and their normal derivatives appearing on both sides are replaced by those of (0 |∂ B , 0). Using the fact that M0AB i j n A n B depends only on first derivatives of , we are now able to ˙ recursively solve the corner conditions, with (0)|∂ B and (0)| ∂ B and their even normal ˙ derivatives given arbitrarily, provided that ((0)|∂ B , (0)| ∂ B ) is close to (0 |∂ B , 0). ˙ We have to - and by the above can - choose a large class of initial data ((0), (0)) close to (0 , 0) so that the corner conditions are fulfilled for arbitrary order and the negativity of M tt i j and the coerciveness of M AB i j are satisfied. References 1. Aki, K., Richards, P.G.: Quantitative Seismology. (Sausolito: University Science Books), (2002) 2. Beig, R., Schmidt, B.G.: Relativistic elasticity. Class. Quantum Grav. 20, 889–904 (2003) 3. Chen, W., Jost, J.: A Riemannian version of Korn’s inequality. Calc. Var. Partial Differ. Eq. 14, 517–530 (2002) 4. Christodoulou, D.: The Action Principle and Partial Differential Equations. (Princeton, NJ: Princeton University Press), 2000 5. Ciarlet, P.G.: Mathematical Elasticity, Volume 1: Three-Dimensional Elasticity. Austerdam: NorthHolland, 1988 6. Dain, S.: Generalized Korn’s inequality and conformal Killing vectors. http://arxiv.org/list/ gr-qc/0505022, 2005
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7. Ebin, D.: Deformations of Incompressible Bodies with Free Boundaries. Arch. Rat. Mech. Anal. 120, 61–97 (1992) 8. Horgan, C.O.: Korn’s inequalities and their applications in continuum mechanics. SIAM Rev. 37, 491–511 (1995) 9. Hughes, T., Kato, T., Marsden, J.: Well-posed quasilinear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rat. Mech. Anal. 63, 273–294 (1976) 10. Kijowski, J., Magli, G.: Relativistic elastomechanics as a Lagrangian field theory. J. Geom. Phys. 9, 207–223 (1992) 11. Koch, H.: Mixed problems for fully nonlinear hyperbolic equations. Math. Z. 214, 9–42 (1993) 12. Lindblad, H.: Wellposedness for the motion of a compressible liquid with free surface boundary. Commun. Math. Phys. 260, 319–392 (2005) 13. Marsden, J., Hughes, T.: Mathematical foundations of elasticity. (New York: Dover), 1994 14. Rendall, A.: Theorems on Existence and Global Dynamics for the Einstein Equations. Living Rev. Relativity 6 4.URL (cited on <9.December 2005>), available at http://relalivity.livingreviews/org/Articles/ Irr-2005-6/, 2005 15. Sideris, T.: Nonlinear hyperbolic systems and elastodynamics. In: Phase Space Analysis of PDES, Vol. II, Pisa: Scuole Normal Supèriore, 2004, pp. 451–485 16. Trautman, A.: Foundations and Current Problems of General Relativity. In: Lectures on General Relativity, edited by S. Deser, and K.W. Ford, Englewood Cliffs, NJ: Prentice-Hall, 1965 Communicated by G.W. Gibbons
Commun. Math. Phys. 271, 467–488 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0151-9
Communications in
Mathematical Physics
Determining a Magnetic Schrödinger Operator from Partial Cauchy Data David Dos Santos Ferreira1 , Carlos E. Kenig2 , Johannes Sjöstrand3 , Gunther Uhlmann4 1 2 3 4
Institut Galilée, Université Paris 13, 93430 Villetaneuse, France. E-mail: [email protected] Department of Mathematics, University of Chicago, Chicago, IL 60637, USA C.M.L.S., Ecole Polytechnique, 91128 Palaiseau Cedex, France Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA
Received: 19 January 2006 / Accepted: 5 April 2006 Published online: 31 January 2007 – © Springer-Verlag 2007
Abstract: In this paper we show, in dimension n ≥ 3, that knowledge of the Cauchy data for the Schrödinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic field and the electric potential. We follow the general strategy of [7] using a richer set of solutions to the Dirichlet problem that has been used in previous works on this problem. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2. Carleman Estimate . . . . . . . . . . . . . . . . . . . . . 3. Construction of Solutions by Complex Geometrical Optics 4. Towards Recovering the Magnetic Field . . . . . . . . . . 5. Moving to the Complex Plane . . . . . . . . . . . . . . . 6. Recovering the Potential . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Let ⊂ Rn be an open bounded set with C ∞ boundary; we are interested in the magnetic Schrödinger operator L A,q (x, D) =
n (D j + A j (x))2 + q(x) j=1
= D 2 + A · D + D · A + A2 + q
(1.1)
¯ Rn ) and bounded electric potenwith real magnetic potential A = (A j )1≤ j≤n ∈ C 2 (; tial q ∈ L ∞ (). As usual, D = −i∇. In this paper, we always assume the dimension to be ≥ 3. Let us introduce the
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Assumption 1. 0 is not an eigenvalue of the magnetic Schrödinger operator L A,q : H 2 () ∩ H01 () → L 2 (). Let ν be the unit exterior normal. Under Assumption 1, the Dirichlet problem L A,q u = 0 1 (1.2) u|∂ = f ∈ H 2 (∂) has a unique solution in H 1 (), and we can introduce the Dirichlet to Neumann map (DN) 1
1
N A,q : H 2 (∂) f → (∂ν + i A · ν)u|∂ ∈ H − 2 (∂) associated to the magnetic Schrödinger operator L A,q with magnetic potential defined by (1.1). The inverse problem we consider in this paper is to recover information about the magnetic and electric potential from the DN map measured on subsets of the boundary. As was noted in [12], the DN map is invariant under a gauge transformation of the magnetic potential: it ensues from the identities e−i L A,q ei = L A+∇,q , e−i N A,q ei = N A+∇,q ,
(1.3)
¯ is such that |∂ = 0. Thus N A,q carries that N A,q = N A+∇,q when ∈ C 1 () 1 information about the magnetic field B = d A. Sun showed in [12] that from this information one can determine the magnetic field and the electric potential if the magnetic potential is small in an appropriate class. In [8] the smallness assumption was eliminated for smooth magnetic and electric potentials and for C 2 and compactly supported magnetic potential and L ∞ electrical potential. The regularity assumption on the magnetic potential was improved in [13] to C 2/3+ , > 0, and to Dini continuous in [10]. Recently in [11] a method was given for reconstructing the magnetic field and the electric potential under some regularity assumptions on the magnetic potential. All of the above mentioned results rely on constructing complex geometrical optics solutions, with a linear phase, for the magnetic Schrödinger equation. We also mention that the inverse boundary value problem is closely related to the inverse scattering problem at a fixed energy for the magnetic Schrödinger operator. The latter was studied under various regularity assumptions on the magnetic and electrical potentials in [9], for small compactly supported magnetic potential and compactly supported electric potential. This result was extended in [3] for exponentially decaying magnetic and electric potentials with no smallness assumption. In this paper we extend one of the main results of [7] to the case of the magnetic Schrödinger equation. Let x0 ∈ Rn \ch() (where ch() denotes the convex hull of ), we define the front and back sides of ∂ with respect to x0 by F(x0 ) = {x ∈ ∂ : (x − x0 ) · ν(x) ≤ 0}, B(x0 ) = {x ∈ ∂ : (x − x0 ) · ν(x) > 0}. The following result was obtained in [7] when there are no magnetic potentials, i.e. A1 = A2 = 0 (we then write Nq j = N0,q j for the Dirichlet to Neumann maps): 1 Here A is viewed as the 1-form n j=1 A j d x j .
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Theorem (1.2 in [7]) (Kenig, Sjöstrand, Uhlmann). Let be an open bounded set with C ∞ boundary in Rn , n ≥ 3, let q1 , q2 be two bounded potentials on such that Assumption 1 is satisfied. Let x0 ∈ Rn \ch(), suppose that there exists a neighborhood F˜ of the front side F(x0 ) such that ˜ ∀ f ∈ H 2 (∂), Nq1 f (x) = Nq2 f (x) ∀x ∈ F, 1
then q1 = q2 . Let us now state the precise results of this article. Theorem 1.1. Let be a simply connected open bounded set with C ∞ boundary in Rn , ¯ and q1 , q2 be two bounded potentials n ≥ 3, let A1 , A2 be two real C 2 vector fields on on such that Assumption 1 is satisfied. Let x0 ∈ Rn \ch(), suppose that the Dirichlet to Neumann maps related to the operators L A1 ,q1 and L A2 ,q2 coincide on part of the boundary near x0 in the sense that there exists a neighborhood F˜ of the front side of ∂ with respect to x0 such that ˜ ∀ f ∈ H 2 (∂), N A1 ,q1 f (x) = N A2 ,q2 f (x) ∀x ∈ F, 1
(1.4)
then if A1 and A2 are viewed as 1-forms, d A1 = d A2 and q1 = q2 . Remark 1.2. We only use the simple connectedness of the set to obtain the equality d A1 = d A2 and to deduce that the two magnetic potentials differ from a gradient. If we already know that A1 − A2 = ∇, we don’t need the fact that is simply connected, in particular, Theorem 1.1 contains Theorem 1.2 of [7]. Nevertheless Theorem 1.1 in [7] improves on this result by restricting the Dirichlet-to-Neuman maps to a space of functions on the boundary with support in a small neighborhood of the back side B(x0 ). We have left the corresponding result in the magnetic case open. As in [7], we make the following definition of a strongly star shaped domain. Definition 1.3. An open set with smooth boundary is said to be strongly star shaped with respect to x1 ∈ ∂ if every line through x1 which is not contained in the tangent hyperplane cuts the boundary at precisely two distinct points x1 and x2 with transversal intersection at x2 . With this definition, Theorem 1.1 implies the following corollary Corollary 1.4. Under the assumptions on , the magnetic potentials A1 , A2 and the electric potentials q1 , q2 of Theorem 1.1, let x1 ∈ ∂ be a point of the boundary such that the tangent hyperplane of ∂ at x1 only intersects ∂ at x1 and such that is strongly star shaped with respect to x1 . Suppose that there exists a neighborhood F˜ of x1 in ∂ such that (1.4) holds then d A1 = d A2 and q1 = q2 .
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We proceed as in [7] by constructing some complex geometrical optics solutions using a Carleman estimate. The construction of these solutions is fairly similar to those presented in the latter paper, except for the changes due to the presence of the magnetic potential. However, the part concerned with the recovery of the potential and the magnetic field is new. The plan of this article is as follows. In the second section, we prove a Carleman estimate with boundary terms for the magnetic Schrödinger operator, which will be useful both for the construction of the complex geometrical optics solutions, and for estimating boundary terms in the limit when the semi-classical parameter tends to zero. In the third section, we construct complex geometrical optics solutions by solving an eikonal and a transport equation and using the Carleman estimate derived in the preceding section. The fourth and fifth sections are devoted to the analysis of the information obtained when passing to the limit when the semi-classical parameter tends to zero. This actually provides enough information on a certain Radon transform to determine the magnetic field. In the last section, once the magnetic field has been determined, we apply the same arguments to determine the electric potential. 2. Carleman Estimate Our first step is to construct solutions of the magnetic Schrödinger equation L A,q u = 0 of the form 1
u(x, h) = e h (ϕ+iψ) (a(x) + hr (x, h))
(2.1)
(where ϕ and ψ are real functions) by use of the complex geometrical optics method: of course, ψ and a will be sought as solutions of respectively an eikonal equation and a transport equation. In order to be able to go from an approximate solution to an exact solution, one wants the conjugated operator ϕ
ϕ
e h h 2 L A,q e− h
to be locally solvable in a semi-classical sense, which means its principal symbol2 pϕ (x, ξ ) = ξ 2 − (∇ϕ)2 + 2i∇ϕ · ξ
(2.2)
to satisfy Hörmander’s condition {Re pϕ , Im pϕ } ≤ 0 when pϕ = 0. Since we furthermore want to obtain solutions (2.1) for both the phases ϕ and −ϕ, we will consider phases satisfying the condition {Re pϕ , Im pϕ } = 0 when pϕ = 0.
(2.3)
˜ is said to be a limiting Definition 2.1. A real smooth function ϕ on an open set ˜ and if the symbol (2.2) satisfies Carleman weight if it has non-vanishing gradient on ˜ This is equivalent to say that condition (2.3) on T ∗ ().
ϕ ∇ϕ, ∇ϕ + ϕ ξ, ξ = 0 when ξ 2 = (∇ϕ)2 and ∇ϕ · ξ = 0. 2 Here and throughout this article, we are using the semi-classical convention.
(2.3 )
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The appropriate tool in deducing local solvability for the conjugated operator and in proving that the geometrical optics method is effective (meaning that indeed one gains one power of h in the former asymptotics) is a Carleman estimate. The goal of this section is to prove such an estimate. ˜ will denote an open set ˜ . We In this section, is as in the introduction and will use the following notations: (u|v) = u(x)v(x) ¯ d x, (u|v)∂ = u(x)v(x) ¯ dσ (x),
∂
√ and u = (u|u) denotes the L 2 norm on . We say that the estimate F(u, h) G(u, h) holds for all u ∈ X (where X is a function space, such as L 2 ()) and for h small if there exist constants C > 0 and h 0 > 0 (possibly depending on q and A) such that for all 0 ≤ h ≤ h 0 and for all u ∈ X , the inequality F(u, h) ≤ C G(u, h) is satisfied. We will make extensive use of the Green formula for the magnetic Schrödinger operator L A,q , which for the sake of convenience, we state as a lemma. ¯ and q ∈ L ∞ () then we have the Lemma 2.2. Let A be a real C 1 vector field on magnetic Green formula (L A,q u|v) − (u|L A,q¯ v) = u|(∂ν + iν · A)v ∂ − (∂ν + iν · A)u|v ∂ (2.4) for all u, v ∈ H 1 () such that u, v ∈ L 2 (). Proof. Integrating by parts, we have (L A,q u|v) = (∇u|∇v) + (Du|Av) + (Au|Dv) + (q + A2 )u|v − (∂ν + iν · A)u|v ∂
(2.5)
and by permuting u and v, replacing q by q, ¯ and taking the complex conjugate of the former, we get (L A,q¯ u|v) = (∇u|∇v) + (Du, Av) + (Au|Dv) + (q + A2 )u|v − u|(∂ν + iν · A)v ∂ . Subtracting the former to (2.5), we end up with (2.4). If ϕ is a limiting Carleman weight, we define ∂± = {x ∈ ∂ : ±∂ν ϕ ≥ 0}. ˜ let A be a C 1 vector Proposition 2.3. Let ϕ be a C ∞ limiting Carleman weight on , ∞ ¯ field on and q ∈ L (), the Carleman estimate ϕ
ϕ
ϕ
ϕ
−h(∂ν ϕ e h ∂ν u|e h ∂ν u)∂− + e h u2 + e h h∇u2 ϕ
ϕ
ϕ
h 2 e h L A,q u2 + h(∂ν ϕ e h ∂ν u|e h ∂ν u)∂+ ¯ ∩ H 1 () and h small. holds for all u ∈ C ∞ () 0
(2.6)
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In particular, when u ∈ C0∞ (), we have the Carleman estimate ϕ
ϕ
ϕ
e h u + he h ∇u he h L A,q u.
(2.6 )
The proof of the Carleman estimate follows standard arguments. We refer the reader to Chapters 17.2 and 28 in [6] and the references therein for a general study on L 2 Carleman estimates. ϕ
Proof. Taking v = e h u, it is equivalent to prove the following a priori estimate −h(∂ν ϕ ∂ν v|∂ν v)∂− + v2 + h∇v2 ϕ ϕ 1 2 (e h h 2 L A,q e− h )v2 + h(∂ν ϕ ∂ν v|∂ν v)∂+ h ϕ
(2.7)
ϕ
since he h ∇u v + h∇v and (e h ∂ν u)|∂ = ∂ν v|∂ . Conjugating the magnetic Schrödinger operator by the exponential weight gives rise to the following operator: ϕ
ϕ
e h h 2 L A,q e− h = P + i Q + R + h 2 (q + A2 ),
(2.8)
where P and Q are the self-adjoint operators P = h 2 D 2 − (∇ϕ)2 , Q = ∇ϕ · h D + h D · ∇ϕ, and R = h(A · h D + h D · A) + 2i h A · ∇ϕ. Our first remark concerns the fact that we may neglect the term h 2 (q + A2 ) since the right-hand side of (2.7) may be perturbed by a term bounded by h 2 v2 , which may be absorbed by the left-hand side if h is small enough. Omitting the term q + A2 gives rise to such an error. Hence we will prove the a priori estimate for the operator P + i Q + R. The same is not true of the term R because errors of order v2 + h∇v2 may not be absorbed into the left hand-side. Note that if p and q denote the principal symbol respectively of P and Q, the fact that ϕ is a limiting Carleman weight means that { p, q} = 0 when p + iq = 0. This condition is not enough to obtain an a priori estimate for P + i Q, one needs to have a positive Poisson bracket. Our first step is to remedy this by using a classical convexity argument. Consider the modified Carleman weight ϕ˜ = ϕ + h
ϕ2 , 2ε
where ε is a suitable small parameter to be chosen independent of h, and denote by p˜ ˜ Q, ˜ R˜ the corresponding operators, when ϕ and q˜ the corresponding symbols, and by P, has been replaced by ϕ. ˜ Then, we have h h h ∇ ϕ˜ = 1 + ϕ ∇ϕ, ϕ˜ = 1 + ϕ ϕ + ∇ϕ ⊗ ∇ϕ; ε ε ε
Determining a Magnetic Schrödinger Operator
473
therefore when ξ 2 = (∇ ϕ) ˜ 2 and ∇ ϕ˜ · ξ = 0, we have ˜ ∇ ϕ ˜ { p, ˜ q} ˜ = 4 ϕ˜ ξ, ξ + 4 ϕ˜ ∇ ϕ, 2 h 4h 1 + ϕ (∇ϕ)4 > 0, = ε ε
(2.9)
since ϕ is a limiting Carleman weight. Furthermore, if we restrict ourselves to the hyperplane Vx orthogonal to ∇ϕ, we get h 2 4h 1 + ϕ (∇ϕ)4 + a(x)(ξ 2 − (∇ ϕ) ˜ 2) { p, ˜ q}(x, ˜ ·)|Vx = ε ε with a(x) = 4h(∇ ϕ) ˜ 2 /ε − 4 ϕ˜ ∇ ϕ, ˜ ∇ ϕ/(∇ ˜ ϕ) ˜ 2 , and since this bracket is a quadratic polynomial with no linear part, this implies that there exists a linear form b(x, ξ ) in ξ such that h 2 4h 1 + ϕ (∇ϕ)4 + a(x) p˜ + b(x, ξ )q. { p, ˜ q} ˜ = ˜ ε ε This computation implies on the operator level that 2 2 ˜ ˜ Q] ˜ = 4h 1 + h ϕ (∇ϕ)4 + h (a P˜ + Pa) i[ P, ε ε 2 h ˜ w ) + h 3 c(x), + (bw Q˜ + Qb 2 where the first order differential operator bw is the semi-classical Weyl quantization3 of b. In fact, the positivity of the bracket (2.9) essentially induces the positivity of the ˜ Q] ˜ commutator i[ P, ˜ Q]v|v) ˜ i([ P, =
4h 2 ε
1+
h 2 ϕ (∇ϕ)2 v2 ε
>0
˜ ˜ w v) + h 3 (cv|v) + h Re(a Pv|v) + h Re( Qv|b
(2.10)
(recall that v|∂ = 0 which explains why there are no boundary terms). The former fact ˜ will be enough to obtain the a priori estimate on P˜ + i Q. Our last observation is that
˜ + (∇ϕ)2 v2 h∇v2 = ( Pv|v) leading to ˜ 2 + v2 . h∇v2 Pv Now, we turn to the proof of the estimate. We have 2 ˜ 2 + Qv ˜ 2 + i( Qv| ˜ Pv) ˜ − i( Pv| ˜ Qv) ˜ ˜ = Pv ( P˜ + i Q)v 3 The absence of the h 2 term is due to the use of the Weyl quantization.
(2.11)
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and the magnetic Green formula (2.4) (used in the straightforward case with no potential P˜ = h 2 L0,−(∇ ϕ) ˜ 2 / h 2 ), together with the fact that v|∂ = 0, gives ˜ Pv) ˜ = ( P˜ Qv|v) ˜ ˜ ν v)∂ ( Qv| − h 2 ( Qv|∂ ˜ = ( P˜ Qv|v) + 2i h 3 (∂ν ϕ˜ ∂ν v|∂ν v)∂ and similarly, since Q˜ is first order, we get ˜ Qv) ˜ = ( Q˜ Pv, ˜ v). ( Pv, Therefore we have 2 ˜ ˜ 2 + Qv ˜ 2 + i([ P, ˜ Q]v|v) ˜ ( P˜ + i Q)v = Pv
− 2h 3 (∂ν v|∂ν ϕ˜ ∂ν v)∂ , and using (2.10), we get 2 2 ˜ ˜ 2 + Qv ˜ 2 + C1 h v2 ( P˜ + i Q)v + 2h 3 (∂ν v|∂ν ϕ˜ ∂ν v)∂ ≥ Pv ε ˜ ˜ h∇v) − (Ch Pv v + Ch Qv
2 h2 C 1 1 ˜ 2 + Qv ˜ 2+ ≤ 2 Pv 2
2 2
v2 +h∇v2 ,
which combined with (2.11), gives when ε is small enough 2 ˜ ( P˜ + i Q)v + 2h 3 (∂ν v|∂ν ϕ˜ ∂ν v)∂
˜ 2 + Qv ˜ 2+ (1 − O(ε−1 h 2 ) Pv
h2 v2 + h∇v2 . ε
Thus, taking h and ε small enough 2 ˜ + 2h 3 (∂ν v|∂ν ϕ˜ ∂ν v)∂ ( P˜ + i Q)v
h2 v2 + h∇v2 . ε
(2.12)
˜ due to the The last part4 of the proof is concerned with the additional term Rv magnetic potential; from the former inequality we deduce ˜ 2 + h 3 (∂ν v|∂ν ϕ˜ ∂ν v)∂ ( P˜ + i Q˜ + R)v h2 ˜ 2 v2 + h∇v2 − O(ε)h −1 Rv ε ˜ 2 v2 + h∇v2 , we obtain and using the fact that h −1 Rv 2 ˜ 2 + h 3 (∂ν v|∂ν ϕ˜ ∂ν v)∂ h v2 + h∇v2 ( P˜ + i Q˜ + R)v ε 4 This is the main difference with respect to the proof of the Carleman estimate in [7].
Determining a Magnetic Schrödinger Operator
475 ϕ2
ϕ
if ε is chosen small enough. Finally, with v = e 2ε e h u, we get ϕ2
ϕ2
ϕ
ϕ
e 2ε e h u2 + h∇e 2ε e h u2
1 ϕ2 ϕ 2 e 2ε e h h L A,q u2 h2 ϕ2 ϕ ϕ + h e ε ∂ν (e h u)|∂ν ϕ˜ ∂ν (e h u) ∂ ,
this gives the Carleman estimate (2.6) since 1 ≤ eϕ
2 /2ε
≤ C,
∂ν ϕ˜ h 3 1 ≤ =1+ ϕ ≤ 2 ∂ν ϕ ε 2
¯ for all h small enough. on 1 () the semi-classical Sobolev space of order 1 on with assoWe denote by Hscl ciated norm
u2H 1
scl ()
= h∇u2 + u2
s (Rn ) the semi-classical Sobolev space on Rn with associated norm and by Hscl 2 s 2 ˆ )|2 dξ. u H s (Rn ) = h D u L 2 (Rn ) = (1 + h 2 ξ 2 )s |u(ξ scl
Changing ϕ into −ϕ, we may rewrite the Carleman estimate in the following convenient way: √ ϕ ϕ h ∂ν ϕ e− h ∂ν u L 2 (∂+ ) + e− h u H 1 () scl √ ϕ ϕ he− h L A,q u + h −∂ν ϕ e− h ∂ν u L 2 (∂− ) . (2.13) By regularization, this estimate is still valid for u ∈ H 2 ()∩ H01 (). A similar Carleman estimate gives the following solvability result: ˜ let A be a C 1 vector field on Proposition 2.4. Let ϕ be a limiting Carleman weight on , ¯ and q ∈ L ∞ (). There exists h 0 such that for all 0 ≤ h ≤ h 0 and for all w ∈ L 2 (), there exists u ∈ H 1 () such that ϕ
ϕ
h 2 L A,q (e h u) = e h w and hu H 1
scl ()
w.
Proof. We need the following Carleman estimate: ϕ
ϕ
v he− h L A,q e h v H −1 (Rn ) , ∀v ∈ C0∞ (). scl
(2.14)
ˆ ⊂ , ˜ assume that we have extended A to a C 1 vector field on ˆ and q to a Let ∞ ∞ ˆ ˆ L function on . Let χ ∈ C0 () equal 1 on . With the notations used in the proof of Proposition 2.3 we have ˜
h D−1 ( P˜ + i Q) h D = P˜ + i Q˜ + h R1 ,
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where R1 is a semi-classical pseudo-differential operator of order 1, therefore from estimate (2.12) we deduce h2 (v2H 1 (Rn ) − O(ε)R1 v2 ) scl ε h2 v2H 1 (Rn ) scl ε
˜ ( P˜ + i Q) h Dv2H −1 (Rn ) scl
ˆ if h and ε are small enough. Besides, we have for any v ∈ C0∞ (), ( R˜ + h 2 (q + A2 ))v H −1 (Rn ) hv, scl
therefore if ε is small enough, we have ϕ˜
ϕ˜
h Dv he− h L A,q e h h Dv H −1 (Rn ) scl
ˆ Hence if u ∈ C ∞ (), taking v = χ h D−1 u ∈ C ∞ () ˆ in the for any v ∈ C0∞ (). 0 former estimate, and using the fact that ∞ s = O(h )u (1 − χ ) h D−1 u Hscl
we obtain ϕ˜
ϕ˜
u he− h L A,q e h u H −1 (Rn ) , ∀u ∈ C0∞ (). scl
˜ h = eϕ /ε eϕ/ h . Classical arguments This gives the Carleman estimate (2.14) since eϕ/ involving the Hahn-Banach theorem give the solvability result. 2
3. Construction of Solutions by Complex Geometrical Optics The goal of this section is to construct solutions of the magnetic Schrödinger equation of the form (2.1). To do so we take ψ to be a solution of the eikonal equation p(x, ∇ψ(x)) + iq(x, ∇ψ(x)) = 0; such solutions exist since { p, q} = 0 when p = q = 0. More precisely, the eikonal equation reads (∇ψ)2 = (∇ϕ)2 , ∇ϕ · ∇ψ = 0.
(3.1)
In fact, in the remainder of this article, we fix the limiting Carleman weight to be 1 log(x − x0 )2 . (3.2) 2 For such a choice of ϕ, the second part of the eikonal equation is merely the fact that ψ is a function of the angular variable (x − x0 )/|x − x0 | and we can actually give an explicit solution of the eikonal equation ϕ(x) =
ω · (x − x0 ) π − arctan 2 (x − x0 )2 − (ω · (x − x0 ))2 x−x 0 ,ω , = d S n−1 |x − x0 |
ψ(x) =
(3.3)
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where ω ∈ S n−1 . Let us be more precise about the set where ω may vary, keeping in mind that we want this function to be smooth — in particular, we have to ensure that ¯ ω = (x − x0 )/|x − x0 | whenever x ∈ . ¯ ⊂ B(x0 , r0 ), let H denote For that purpose, let r0 > 0 be large enough so that ¯ (and a hyperplane separating x0 and ch(), and H + the open half space containing therefore x0 ∈ / H + ), we set = {θ ∈ S n−1 : x0 + r0 θ ∈ H + } ˇ and 0 be and ˇ the image of under the antipodal application. Let ω0 ∈ S n−1 \ ( ∪ ) n−1 ˇ a neighborhood of ω0 in S \ (∪ ), then the distance ×0 (θ, ω) → d S n−1 (θ, ω) ¯ ⊂ ˜ = x0 + R+ , hence we have (x − x0 )/|x − x0 | ∈ is a C ∞ function. Moreover, ˜ of , ¯ thus ψ depends smoothly on the variables for all x in the open neighborhood ˜ (x, ω) on × 0 . Remark 3.1. Suppose that x0 = 0 and ω = (1, 0, . . . , 0), which we can always assume by doing a translation and a rotation. Notice that by considering the complex variable z = x1 + i|x | ∈ C (with x = (x1 , x ) ∈ R × Rn−1 ), we have ϕ = log |z| = Re log z, ψ =
π Re z − arctan = Im log z 2 Im z
when Im z > 0 (note that ψ = arctan(Im z/ Re z) on the first quadrant Re z > 0, Im z > 0) hence ϕ + iψ = log z. With such ϕ and ψ, we have 1 1 h 2 L A,q e h (ϕ+iψ) = e h (ϕ+iψ) h(D + A) · (∇ψ − i∇ϕ)
+ h(∇ψ − i∇ϕ) · (D + A) + h 2 L A,q ,
thus we will have 1 1 h 2 L A,q e h (ϕ+iψ) a = O(h 2 )e h (ϕ+iψ) if a is a C 2 solution of the first transport equation, given by (D + A) · (∇ψ − i∇ϕ) + (∇ψ − i∇ϕ) · (D + A) a = 0. We write the latter as a vector field equation (∇ψ − i∇ϕ) · Da + (∇ψ − i∇ϕ) · Aa +
1 ( ψ − i ϕ)a = 0. 2i
(3.4)
We seek a under exponential form a = e , which means finding solution of 1 (∇ϕ + i∇ψ) · ∇ + i(∇ϕ + i∇ψ) · A + (ϕ + iψ) = 0 2 on . The function has C 2 regularity since the magnetic potential A is C 2 .
(3.5)
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Remark 3.2. Considering the complex variable z = x1 + i|x | as in Remark 3.1 with ϕ + iψ = log z, we may seek as a solution of the following Cauchy-Riemann equation in the z variable: ∂ (n − 2) 1 − + A · (e1 + ier ) = 0, ∂ z¯ 2(z − z¯ ) 2 where er = (0, θ ) is the unit vector pointing in the direction of the r -axis. Indeed, if we denote by (x1 , r, θ ) ∈ R × R+ × S n−2 a choice of cylindrical coordinates on Rn and z = x1 + ir , we have 2 ∂ ∂ log z ∂ ∂ log z ∂ = + ∂ x1 ∂ x1 ∂r ∂r z ∂ z¯ ∂2 1 ∂2 (n − 2) ∂ + (ϕ + iψ) and (ϕ + iψ) = + +
n−2 r ∂r r 2 S ∂ x12 ∂r 2 (n − 2)i n−2 ∂ log z = . = r ∂r rz Remark 3.3. Note that the set of solutions of (3.4) is invariant under the multiplication by a function g satisfying ∇(ϕ + iψ) · ∇ =
(∇ϕ + i∇ψ) · ∇g = 0. In the setting of Remark 3.1, this condition reads ∂g =0 ∂ z¯ on , i.e. g is a holomorphic function of z = x1 + i|x |. Having chosen the phase ϕ + iψ and the amplitude e , we obtain an approximate solution of the magnetic Schrödinger equation 1
ϕ
1
h 2 L A,q (e h (ϕ+iψ) e ) = e h (ϕ+iψ) h 2 L A,q e = O(h 2 )e h
(recall that is C 2 ) which we can transform into an exact solution thanks to Proposition 2.4; there exists r (x, h) ∈ H 1 () such that 1
1
h 2 e h (ϕ+iψ) L A,q r (x, h) = −e h (ϕ+iψ) hL A,q e and r H 1 () L A,q e . scl We sum up the result of this section in the following lemma. Lemma 3.4. Let x0 ∈ Rn \ch , there exists h 0 > 0 and r such that r H 1 () = O(1) scl and 1
u(x, h) = e h (ϕ+iψ) (e(x) + hr (x, h)) is a solution of the equation L A,q u = 0, when h ≤ h 0 , and ϕ is the limiting Carleman weight (3.2), ψ is given by (3.3) and is a solution of the Cauchy-Riemann equation (3.5). Note that with ϕ as in (3.2) the parts of the boundary ∂± delimited by the sign of ∂ν ϕ correspond to the front and back sides of the boundary ∂− = F(x0 ), ∂+ = B(x0 ).
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4. Towards Recovering the Magnetic Field Let x0 ∈ Rn \, suppose that the assumptions of Theorem 1.1 are fulfilled and consider Fε = {x ∈ ∂ : (x − x0 ) · ν(x) < ε|x − x0 |2 } ⊃ F(x0 ) ˜ therefore satisfying with ε > 0 small enough so that Fε ⊂ F, 1
N A1 ,q1 f (x) = N A2 ,q2 f (x), ∀x ∈ Fε , ∀ f ∈ H 2 (∂).
(4.1)
We may assume without loss of generality that the normal components of A1 and A2 are equal on the boundary A1 · ν = A2 · ν on ∂
(4.2)
since we can do a gauge transformation in the magnetic potential N A,q = N A+∇,q ¯ such that |∂ = 0 and ∂ν is a pre(see (1.3) in the introduction) with ∈ C 3 () 2 5 scribed C function on the boundary . We extend A1 and A2 as C 2 compactly supported6 functions in Rn . We consider two geometrical optics solutions 1
u j (x, h) = e h (ϕ j +iψ j ) (e j + hr j (x, h)) of the equations L A1 ,q¯1 u 1 = 0 and L A2 ,q2 u 2 = 0 constructed in the former section with phases ϕ2 (x) = −ϕ1 (x) = ϕ(x) = log |x − x0 |, x−x 0 ,ω , ψ2 (x) = ψ1 (x) = ψ(x) = d S n−1 |x − x0 |
(4.3)
˜ of (and ω varies in 0 ), and where 1 and 2 are defined on a neighborhood solutions of the equations 1 (∇ϕ − i∇ψ) · ∇1 + i(∇ϕ − i∇ψ) · A1 + (ϕ − iψ) = 0, 2 1 (∇ϕ + i∇ψ) · ∇2 + i(∇ϕ + i∇ψ) · A2 + (ϕ + iψ) = 0. 2
(4.4)
1 . Note that it implies the The remainders r j are bounded independently of h in Hscl following estimate on u j :
e−
ϕj h
u j H 1 = O(1). scl
By w we denote the solution to the equation L A1 ,q1 w = 0, w|∂ = u 2 |∂ 5 After the use of a partition of unity and a transfer to {x ≥ 0}, this is Theorem 1.3.3. in [6]. 1 6 Note that A and A do not necessarily agree on ∂. 1 2
(4.5)
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D. Dos Santos Ferreira, C. E. Kenig, J. Sjöstrand, G. Uhlmann
so that N A1 ,q1 (u 2 |∂ ) = (∂ν w)|∂ + i A1 · ν u 2 |∂ . The assumption (4.1) means that ∂ν (w − u 2 )(x) = 0, ∀x ∈ Fε (here we use the fact (4.2) that the normal components of the magnetic potentials coincide on the boundary). Besides, we have L A1 ,q1 (w − u 2 ) = −L A1 ,q1 u 2 = (L A2 ,q2 − L A1 ,q1 )u 2 = (A2 − A1 ) · Du 2 + D · (A2 − A1 )u 2 + (A22 − A21 + q2 − q1 )u 2 ,
(4.6)
hence we deduce L A1 ,q1 (w − u 2 )|u 1 = (A2 − A1 ) · Du 2 |u 1 + u 2 |(A2 − A1 ) · Du 1 1 (A2 − A1 ) · ν u 2 |u 1 ∂ + (A22 − A21 + q2 − q1 )u 2 |u 1 . + i
(4.7)
=0
The magnetic Green’s formula gives (L A1 ,q1 (w − u 2 )|u 1 )
= (w − u 2 |L A1 ,q¯1 u 1 ) − (∂ν + i A1 · ν)(w − u 2 )|u 1 )∂
=0
= −(∂ν (w − u 2 )|u 1 )∂\Fε ,
(4.8)
and combining (4.7) and (4.8), we finally obtain ∂ν (u 2 − w) u¯ 1 dσ (x) = (A22 − A21 + q2 − q1 )u 2 u¯ 1 d x ∂\Fε + (A2 − A1 ) · (Du 2 u¯ 1 + u 2 Du 1 ) d x. (4.9)
With our choice of ϕ2 = ϕ = log |x − x0 |, we have Fε ⊃ F(x0 ) = ∂−
thus
∂\Fε ⊂ ∂+ ,
and moreover ∂ν ϕ > ε on ∂\Fε , therefore the modulus of the left-hand side in (4.9) is bounded by ϕ 1 √ ∂ν ϕ e− h ∂ν (u 2 − w)∂+ × e1 + hr1 ∂+
ε
≤e1 ∂ +r1 H 1
scl
which, by virtue of the Carleman estimate (2.13), is bounded by a constant times
ϕ ϕ 1 √ he− h L A1 ,q1 (u 2 − w) + −∂ν ϕ e− h ∂ν (u 2 − w)∂− . √
ε =0 because of (4.1)
Determining a Magnetic Schrödinger Operator
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In view of (4.6) and of (4.5) the former expression is O(h − 2 ). Therefore we can conclude 1 that the right-hand side of (4.9) is O(h − 2 ). This constitutes an important difference with [7], where7 the corresponding term was O(h). More directly, the first and second right-hand side terms of (4.9) are respectively O(1) and O(h −1 ) as may be seen from (4.5). It turns out that the information obtained when disregarding the bounded term is enough to recover the magnetic field. We multiply (4.9) by h and let h tend to 0: lim
h→0
(A2 − A1 ) · h Du 2 u¯ 1 + u 2 (A2 − A1 ) · h Du 1 d x = 0.
Using the explicit form of the solutions u 1 and u 2 , this further means
¯
(A2 − A1 ) · (∇ϕ + i∇ψ)e1 +2 d x = 0.
(4.10)
Adding the complex conjugate of the first line of (4.4) to the second line, we see that ¯ 1 + ∇2 + i(A2 − A1 ) + (ϕ + iψ) = 0; (∇ϕ + i∇ψ) · ∇ this implies that ¯
(D + A2 − A1 ) · (∇ϕ + i∇ψ)(e1 +2 ) = 0.
(4.11)
As observed in Remark 3.3, in the expression for u 2 , we may replace e2 by e2 g if g is a solution of (∇ϕ + i∇ψ) · ∇g = 0. Then (4.10) can be replaced by
¯
(A2 − A1 ) · (∇ϕ + i∇ψ)e1 +2 g d x = 0.
From equation (4.11), we see that we can replace A2 − A1 by i∇ in the former equality
¯ g(x) ∇ · e1 +2 (∇ϕ + i∇ψ) d x = 0
for all functions g such that (∇ϕ + i∇ψ) · ∇g = 0 on . Remark 4.1. An integration by parts gives ∂
(∂ν ϕ + i∂ν ψ)e
hence we have
¯ 1 +2
g dσ −
¯
e1 +2 (∇ϕ + i∇ψ) · ∇g d x = 0
=0
∂ (∂ν ϕ
¯
+ i∂ν ψ)e1 +2 g dσ = 0.
7 See (5.16) and the subsequent lines in [7].
(4.12)
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5. Moving to the Complex Plane In this section, we follow Remark 3.1 and choose to work in the cylindrical coordinates. Let us be more precise: suppose that x0 = 0 ∈ / ch() and that we have picked n−1 ˇ ω ∈ S \( ∪ ) with the notations of Sect. 3. After a rotation, we assume that ω = (1, 0, . . . , 0), therefore we have ˜ ⊂ {x ∈ Rn : x = 0}. We choose the following cylindrical coordinates: t = x1 , r = |x | > 0, θ = (x) =
x ∈ S n−2 . |x |
By Sard’s theorem, the set of critical values of : → S n−2 is of measure 0, therefore the set θ0 = −1 (θ0 ) = {x ∈ : x = r θ0 , r > 0} is an open set with smooth boundary for almost every θ0 in (). The result obtained in the former section reads
θ
S n−2
¯ g(x) ∇x · e1 +2 (∇x ϕ + i∇x ψ) r n−2 dr dt dθ = 0,
and taking g = g1 (t, r ) ⊗ g2 (θ ) and varying g2 leads to θ
¯ g(t, r ) ∇x · e1 +2 (∇x ϕ + i∇x ψ) r n−2 dr dt = 0
(5.1)
for any function g such that (∇x ϕ + i∇x ψ) · ∇x g = 0 on θ , and this for almost every θ . Now we consider the complex variable z = t + ir ∈ C+ = {w ∈ C : Im w > 0} and write our results in this setting. Let us recall the results of the computations made in Remarks 3.1 and 3.2: 2 ∂ (n − 2)i and x (ϕ + iψ) = z ∂ z¯ rz (n − 2)i 2 ∂ ∇x · ∇x ◦ (ϕ + iψ) = + . z ∂ z¯ rz ∇x (ϕ + iψ) · ∇x =
and thus
(5.2)
Similarly, the functions j satisfy ∂ (n − 2) 1 ¯ 1 + 2 ) = ( + (A1 − A2 ) · (e1 + ier ). ∂ z¯ (z − z¯ ) 2
(5.3)
Finally, (5.1) reads θ
g(z)
(n − 2) ¯ 1 +2 1 ∂ − (e )(z − z¯ )n−2 d z¯ ∧ dz = 0, z ∂ z¯ (z − z¯ )
(5.4)
for any g ∈ H(θ ). Replacing the holomorphic function g/z on θ by g, we can drop the factor 1/z.
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If g is a holomorphic function, we have8 ¯ d e1 +2 g(z)(z − z¯ )n−2 dz ∂ ¯ (z − z¯ )n−2 e1 +2 g(z)d z¯ ∧ dz = ∂ z¯ n − 2 ¯ 1 +2 ∂ − e g(z)(z − z¯ )n−2 d z¯ ∧ dz, = ∂ z¯ z − z¯ therefore the Stokes’ formula implies ∂ (n − 2) ¯ 1 +2 (e g(z) )(z − z¯ )n−2 d z¯ ∧ dz − ∂ z¯ (z − z¯ ) θ ¯ = g(z)e1 +2 (z − z¯ )n−2 dz. ∂θ
Together with (5.4) this gives
¯
∂θ
g(z)e1 +2 (z − z¯ )n−2 dz = 0
(5.5)
for any g ∈ H(θ ). Lemma 5.1. There exists a non-vanishing holomorphic function F on θ , continuous ¯ θ , whose restriction to ∂θ is equal to (z − z¯ )n−2 e¯ 1 +2 . on ¯
Proof. We denote f (z) = (z − z¯ )n−2 e1 +2 and consider the Cauchy integral operator 1 f (ζ ) C( f )(z) = dζ, ∀z ∈ C\∂θ . 2πi ∂θ ζ − z The function C( f ) is holomorphic inside and outside θ and the Plemelj-SokhotskiPrivalov formula reads on the boundary lim C( f )(z) − lim C( f )(z) = f (z 0 ), ∀z 0 ∈ ∂θ .
z→z 0 z∈θ
z→z 0 z ∈ / θ
(5.6)
The function ζ → (ζ − z)−1 is holomorphic on θ when z ∈ / θ hence (5.5) implies that C( f )(z) = 0 when z ∈ / θ . The second limit in (5.6) is 0, thus F = C( f ) is a holomorphic function on θ whose restriction to the boundary agrees with f . It remains to prove that F does not vanish on θ . This is clear by the argument principle since vararg F = vararg f = 0 ∂θ
∂θ
and F is holomorphic. 8 The result of this computation is the transcription of the fact that the formal adjoint of (∇ϕ + i∇ψ) · ∇ is ∇ · (∇ϕ + i∇ψ) in the complex setting, where the measure is r n−2 d z¯ ∧ dz.
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In particular, with the former function, we have on the boundary ¯
(z − z¯ )n−2 e1 +2 = F(z, θ ), ∀z ∈ ∂θ . We want to prove that F admits a holomorphic logarithm9 ; to do so, we consider the one form on dF α= F (recall that F is non-vanishing on ) which is closed. Since is simply connected, α has a primitive α = da, which defines a logarithm of F. The equation F = ea implies that a is holomorphic with respect to z, hence one can define a holomorphic logarithm of F. This implies ¯ 1 + 2 + log(z − z¯ )n−2 = log F(z), ∀z ∈ ∂θ with log F a holomorphic function on θ , and therefore ¯ 1 + 2 + log(z − z¯ )n−2 dz = 0. g(z) ∂θ
An application of Stokes’ formula gives ∂ n − 2 ¯ 1 + 2 ) − ( d z¯ ∧ dz = 0, g(z) ∂ z¯ z − z¯ θ hence using equation (5.3), this implies g(z)(A1 − A2 ) · (e1 + ier ) d z¯ ∧ dz = 0. θ
(5.7)
With g = 1, the former equality reads (t + ir )(∇x ϕ + i∇x ψ) · (A1 − A2 ) dt dr = 0. θ
Denote by Pθ = span(ω, er ) the plane along the axis directed by ω = (1, 0, . . . , 0), and by Pθ+ the half plane where x · er > 0, then θ = ∩ {x = (x1 , x ) ∈ Rn : x = r θ, r > 0} = ∩ Pθ+ . Let πθ be the projection on Pθ and dλθ the measure on the plane, then (5.7) (with g = 1) implies πθ (A1 − A2 ) dλθ = 0, Pθ ∩
for almost every θ ∈ S n−2 , hence for all θ ∈ S n−2 by continuity. The former may be rephrased under the form ξ · 1 (A1 − A2 ) dλ P = 0, ∀ξ ∈ P x0 +P
for all linear planes P containing ω = (1, 0, . . . , 0). We can also let x0 vary in a small neighborhood of 0 ∈ / ch() and ω vary in a neighborhood 0 of (1, 0, . . . , 0) on the sphere S n−1 . 9 This is obvious when is simply connected, which is the case if is for instance convex. θ
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¯ If Lemma 5.2. Let A be a C 1 vector field on . ξ · A dλ P = 0, ∀ξ ∈ Tx (P)
(5.8)
P∩
for all planes P such that d((0, e1 ), T (P)) < δ then d A = 0 on . The proof of this lemma is based on the following microlocal version of Helgason’s support theorem. Theorem 5.3. Let f ∈ C 0 (Rn ), suppose that the Radon transform of f satisfies R f (H ) = f dλ H = 0 H
for all hyperplanes H in some neighbourhood of a hyperplane H0 , then N ∗ (H0 ) ∩ WFa ( f ) = ∅, where N ∗ (H0 ) denotes the conormal bundle of H0 . The proof of this result may be found in [2] (see Proposition 1) or in [5] (see Sect. 6). We will also need the microlocal version of Holmgren’s theorem (see [5], Sect. 1 or [6] Sect. 8.5). Theorem 5.4. Let f ∈ E (Rn ) then we have N (supp f ) ⊂ WFa ( f ), where N (supp f ) is the normal set of the support of f . These two results may be combined to provide a proof of Helgason’s support theorem (see [2] and [5]). We also refer to the book [4] for a review on Radon transforms. Proof of Lemma 5.2. Let χ ∈ C0∞ (|x| < 21 ) and χε = ε−n χ (·/ε) be a standard regularization, one has y ξ · (χε ∗ 1 A) dλ P = ε−n χ ξ · A dλ−y+P dy = 0 ε P (−y+P)∩ when d((0, e1 ), T (P)) < δ − ε. Therefore it suffices to prove the result when = Rn and A ∈ C0∞ (Rn ; Rn ) since d(χε ∗ 1 A) tends to d A as a distribution when ε tends to 0. Our first step is to prove that ι∗H d A = 0
(5.9)
for any subspace H ⊂ Rn of dimension 3 such that d((0, e1 ), T (H )) < δ. Here ι H denotes the injection of H in Rn . For any plane P ⊂ H such that d((0, e1 ), T (P)) < δ, we have d
d A, ξ ∧ η dλ P = η · A dλ P − ξ · A dλ P dt P tξ +P tη+P t=0
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when ξ, η ∈ Tx (H ). The space H is of dimension 3 so we can assume that either η or ξ belongs to Tx (P), thus the former expression is zero because of (5.8) and of the fact that whenever η ∈ Tx (P) ξ · A dλ P is constant. tη+P
Therefore, if R H denotes the Radon transform in H , we obtain R H ι∗H d A, ξ ∧ η (P) = 0 for any plane P ⊂ H such that d((0, e1 ), T (P)) < δ and for any ξ, η ∈ Tx (H ). Combining Theorems 5.3 and 5.4 we obtain N ∗ (P) ∩ N (supp ι∗H d A, ξ ∧ η) = ∅ for any plane P ⊂ H such that d((0, e1 ), T (P)) < δ. This gives (5.9) since such a family of planes sweeps other ∩ H and the support of A is on one side of at least one such plane. The result (5.9) implies in particular that
d A(x), ξ ∧ η = 0, ∀x ∈ Rn , ∀(ξ, η) ∈ S n × Rn , |ξ − e1 | < δ, and therefore d A = 0 by linearity. 6. Recovering the Potential End of the proof of Theorem 1.1. Applying this lemma, we finally obtain d A1 = d A2
on ,
therefore the difference of the two potentials is a gradient A1 − A2 = ∇ (recall that is simply connected). The identity (5.7) now reads g(z)∂z¯ (z, θ ) d z¯ ∧ dz = 0 θ
for any holomorphic function g on θ and by Stokes’ theorem we get g(z)(z, θ ) dz = 0. ∂θ
˜ ∈ Reasoning as in the beginning of Lemma 5.1, there exists a holomorphic function ˜ ˜ H(θ ) such that |∂θ = |∂θ . Now is real-valued, and since is real-valued on ∂θ and harmonic, it is real-valued everywhere. The only real-valued holomorphic ˜ and hence is constant on ∂θ . Varying x0 and functions are the constant ones, so ω, we get that is constant on the boundary ∂. Indeed, the fact that is constant on ∂θ means that X · = 0 when X is a tangent vector field to ∂θ , hence it suffices to show that when varying x0 and ω, one generates enough directions to span the tangent space to ∂ at a given point a ∈ ∂. Note that the plane Pθ belonging to the family of rotating planes passing through a ∈ ∂ is spanned by the vectors ω and er = πω (a − x0 )/|πω (a − x0 )|, where πω is the orthogonal projection onto the space ω⊥ . A variation of x0 in a small neighborhood of 0 induces a variation
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of er in a neighborhood of πω (a)/|πω (a)|. If one picks ω to be outside the tangent space Ta (∂), then the intersection of Ta (∂) with the union of planes generated by ω and er when er varies in a neighborhood of πω (a)/|πω (a)|, contains enough independent vectors to span Ta (∂). Therefore is constant on ∂, and we may as well assume that = 0 on ∂. By a gauge transformation, we may assume that = 0, thus A1 = A2 . We could almost directly apply the result in [7] to recover the identity of the two potentials q1 = q2 , if it were not for the presence of the two magnetic potentials in the equations. Instead we go back to the limit induced √ by (4.9). The second right-hand side term is now zero. The left-hand side is now O( h) since the O(h −1 ) term in (4.6) is zero and we can reproduce the arguments given after (4.9). Therefore we obtain lim (q2 − q1 )u 2 u¯ 1 d x = 0, (6.1) h→0
thus
¯
(q2 − q1 )e1 +2 d x = 0.
As observed in Remark 3.3, we may replace e2 by e2 g if g is a solution of (∇ϕ + i∇ψ) · ∇g = 0. Then the former can be replaced by ¯ (q2 − q1 )e1 +2 g(x) d x = 0.
Moving to the complex plane, as in Sect. 5, this reads ¯ (q2 − q1 )g(z)e1 +2 (z − z¯ )n−2 d z¯ ∧ dz ∧ dθ = 0
for any holomorphic function g on θ . But the transport equation (5.3) now reads ∂ ¯ (z − z¯ )n−2 e1 +2 = 0, ∂ z¯ ¯
therefore if we take g = (z − z¯ )−n+2 e−1 −2 , and write q = 1 (q1 − q2 ), we obtain q(t, r, θ )g(θ ) dt dr dθ = 0 (6.2) R2 ×S n−1
for any (say smooth) function g(θ ). Varying x0 slightly, this remains true for the translated functions q(· − y), when y is small, hence for regularisations of q, y q(x − y) dy, χε ∗ q = ε−n χ ε therefore it suffices to assume that q is smooth. When q is smooth, varying g, (6.2) implies q(t, r, θ ) dt dr = 0 (6.3) R2
for all θ ∈ S n−2 .
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As in Sect. 5, varying x0 and ω, (6.3) may be interpreted as q dλ P = 0 P
for any plane P such that d((0, e1 ), T (P)) < δ. This implies that for any subspace H of dimension 3 such that d((0, e1 ), T (H )) < δ we have R H q(P) = q dλ P = 0 P
for any plane such that d((0, e1 ), T (P)) < δ. Applying Theorems 5.3 and 5.4 as in the proof of Lemma 5.2, we get N ∗ (P) ∩ N (supp q| H ) = ∅, and therefore q = 0 on H leading to q = 0. This ends the proof of Theorem 1.1. Acknowledgement. The work of Carlos Kenig and Gunther Uhlmann was partially supported by NSF. Johannes Sjöstrand wishes to acknowledge the hospitality of the Department of Mathematics of the University of Washington, and David Dos Santos Ferreira the hospitality of the Department of Mathematics of the University of Chicago.
References 1. Boman, J., Quinto, T.: Support theorems for real-analytic Radon transforms. Duke Math. J. 55(4), 943–948 (1987) 2. Boman, J.: Helgason’s support theorem for Radon transforms – a new proof and a generalization. In: Mathematical methods in tomography (Oberwolfach, 1990), Lecture Notes in Math. 1497, BerlinHeidelberg-NewYork: Springer, 1991, pp. 1–5 3. Eskin, G., Ralston, J.: Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy. Commun. Math. Phys. 173, 199–224 (1995) 4. Helgason, S.: The Radon transform. 2nd ed., Progress in Math., Basel-Boston: Birkhäuser, 1999 5. Hörmander, L.: Remarks on Holmgren’s uniqueness theorem. Ann. Inst. Fourier 43(5), 1223–1251 (1993) 6. Hörmander, L.: The Analysis of Linear Partial Differential Operators. Classics in Mathematics, BerlinHeidelberg-New York: Springer, 1990 7. Kenig, C.E., Sjöstrand, J., Uhlmann, G.: The Calderón problem with partial data. To appear in Ann. of Math. 8. Nakamura, G., Sun, Z., Uhlmann, G.: Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field. Math. Ann. 303, 377–388 (1995) 9. Novikov, R.G., Khenkin, G.M.: The ∂-equation in the multidimensional inverse scattering problem. Russ. Math. Surv. 42, 109–180 (1987) 10. Salo, M.: Inverse problems for nonsmooth first order perturbations of the Laplacian. Ann. Acad. Scient. Fenn. Math. Dissertations, Vol. 139, 2004 11. Salo, M.: Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field. To appear in Comm. in PDE 12. Sun, Z.: An inverse boundary value problem for the Schrödinger operator with vector potentials. Trans. Amer. Math. Soc. 338(2), 953–969 (1992) 13. Tolmasky, C.F.: Exponentially growing solutions for nonsmooth first-order perturbations of the Laplacian. SIAM J. Math. Anal. 29(1), 116–133 (1998) Communicated by P. Constantin
Commun. Math. Phys. 271, 489–509 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0212-8
Communications in
Mathematical Physics
A Non-Variational Approach to Nonlinear Stability in Stellar Dynamics Applied to the King Model Yan Guo1 , Gerhard Rein2 1 Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University,
Providence, RI 02912, USA. E-mail: [email protected]
2 Mathematisches Institut der Universität Bayreuth, D-95440 Bayreuth, Germany
Received: 27 March 2006 / Accepted: 31 October 2006 Published online: 13 February 2007 – © Springer-Verlag 2007
Abstract: In previous work by Y. Guo and G. Rein, nonlinear stability of equilibria in stellar dynamics, i.e., of steady states of the Vlasov-Poisson system, was accessed by variational techniques. Here we propose a different, non-variational technique and use it to prove nonlinear stability of the King model against a class of spherically symmetric, dynamically accessible perturbations. This model is very important in astrophysics and was out of reach of the previous techniques.
1. Introduction In astrophysics a galaxy or a globular cluster is often modeled as a large ensemble of particles, i.e., stars, which interact only by the gravitational field which they create collectively, collisions among the stars being sufficiently rare to be neglected. The time evolution of the distribution function f = f (t, x, v) ≥ 0 of the stars in phase space is then given by the Vlasov-Poisson system: ∂t f + v · ∇x f − ∇x U · ∇v f = 0, U = 4πρ, lim U (t, x) = 0, |x|→∞ ρ(t, x) = f (t, x, v)dv.
(1.1) (1.2) (1.3)
Here t ∈ R denotes time, and x, v ∈ R3 denote position and velocity, U = U (t, x) is the gravitational potential of the ensemble, and ρ = ρ(t, x) is its spatial density. A well-known approach to obtaining steady states of the Vlasov-Poisson system is to make an ansatz of the form f 0 (x, v) = φ(E),
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where the particle energy E is defined as E :=
1 2 |v| + U0 (x). 2
Since for a time-independent potential U0 = U0 (x) the particle energy is conserved along the characteristics of the Vlasov equation (1.1) it remains to make sure that for the chosen φ the resulting semilinear Poisson equation 1 2 (1.4) |v| + U0 dv U0 = 4π φ 2 has a solution. For a very large class of ansatz functions φ this approach leads to steady states with finite mass and compact support, cf. [RR]. Steady states where as above the distribution of the particles in phase space depends only on the particle energy are usually called isotropic and are always spherically symmetric. The latter follows by applying the results in [GNN] to Eq. (1.4), cf. also [R1, Thm. 3]. A central question in stellar dynamics, which has attracted considerable attention in the astrophysics literature, cf. [BT, FP] and the references there, is the dynamical stability of such steady states. The Vlasov-Poisson system is conservative, i.e., the total energy 1 1 H( f ) := E kin ( f ) + E pot ( f ) = |v|2 f (x, v) dv d x − |∇U f (x)|2 d x 2 8π of a state f is conserved along solutions and hence is a natural candidate for a Lyapunov function in a stability analysis; U f denotes the potential induced by f . However, the energy does not have critical points, i.e., the linear part in an expansion about any state f 0 with potential U0 does not vanish: 1 2 1 |v| + U0 ( f − f 0 ) dv d x − H( f ) = H( f 0 ) + |∇U f − ∇U0 |2 d x. 2 8π To remedy this situation one observes that for any reasonable function the so-called Casimir functional C( f ) := ( f (x, v)) dv d x is conserved as well. If the energy-Casimir functional HC := H + C is expanded about an isotropic steady state, then (E + ( f 0 )) ( f − f 0 ) dv d x HC ( f ) = HC ( f 0 ) + 1 1 − |∇U f − ∇U0 |2 d x + ( f 0 )( f − f 0 )2 dv d x + . . . 8π 2 =: HC ( f 0 ) + DHC ( f 0 )[ f − f 0 ] + D 2 HC ( f 0 )[ f − f 0 ] + . . . . . Now one can try to choose in such a way that at least formally f 0 is a critical point of the energy-Casimir functional, i.e., ( f 0 ) = −E. If φ < 0 on the support of f 0 the
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former relation holds on this support if = −φ −1 . The formal second order variation in the above expansion then takes the form 1 1 1 2 2 g dv d x − (1.5) D HC ( f 0 )[g] = |∇x Ug |2 d x. 2 { f0 >0} −φ (E) 8π It is natural to expect that positive definiteness of this quadratic form should imply stability for f 0 . Ever since the seminal work of Antonov [An] there have been vigorous efforts in the astrophysics community to establish this positive definiteness and to derive stability results in this way. An important step in this direction was to show that the above quadratic form is positive definite on linearized, dynamically accessible perturbations. To make this precise we define the Lie-Poisson bracket of two functions f 1 , f 2 of x and v as { f 1 , f 2 } := ∇x f 1 · ∇v f 2 − ∇v f 1 · ∇x f 2 .
(1.6)
Then the following holds: Lemma 1.1. Let φ < 0 and let h ∈ Cc∞ (R6 ) be spherically symmetric with support in the set { f 0 > 0} and such that h(x, −v) = −h(x, v). Then h 2 1 2 1 2 2 D HC ( f 0 )[{ f 0 , h}] ≥ − φ (E) |x · v| E, + U0 h dv d x. 2 f0 >0 x ·v r Here U0 denotes the radial derivative of the steady state potential. Since U0 is radially increasing, the right-hand side in the estimate above is indeed positive for h = 0. We refer to [KS, SDLP] for astrophysical investigations where this result is used to analyze linearized stability. We do not go into the reasons why perturbations of the form { f 0 , h} are called dynamically accessible for the linearized system, but the interested reader can find background on this and related concepts in [M]. For the sake of completeness we provide a proof of this elegant result in the Appendix. Despite its significance the result is still quite a distance away from a true, nonlinear stability result. There are at least two serious mathematical difficulties. Firstly, it is very challenging to use the positivity of D 2 HC ( f 0 )[g] to control the higher order remainder in the expansion of the energy-Casimir functional [Wa]. This is due to the non-smooth nature of f 0 = φ(E) in all important examples. Secondly, even if one succeeds in controlling the higher order terms, the positivity of D 2 HC ( f 0 )[g] in the lemma is only valid for certain perturbations of the form g = { f 0 , h}. This class of perturbations is invariant under solutions of the linearized Vlasov-Poisson system, but it is not invariant under solutions of the nonlinear system. To overcome these difficulties a variational approach was initiated by Wolansky [Wo1] and then developed systematically by Guo and Rein [G1, G2, GR1, GR2, GR3, GR4, R2, R3, RG]. Their method entirely avoids the delicate analysis of the second order term D 2 HC ( f 0 ) in (1.5), and it has led to the first rigorous nonlinear stability proofs for a large class of steady states. More precisely, a large class of steady states is obtained as minimizers of energy-Casimir functionals under a mass constraint f = M, and their minimizing property then entails their stability. In particular, all polytropes f 0 (x, v) = (E 0 − E)k+ with 0 < k ≤ 7/2 are covered; here E 0 < 0 is a certain cut-off energy, and (·)+ denotes the positive part. For k > 7/2 the corresponding steady state has infinite mass and is therefore unphysical. In addition, many new stable galaxy models
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were established. The variational method has also been investigated in [DSS, H, LMR, SS, Wo2]. Despite its considerable success, the variational approach has drawbacks and limitations, the main one being that by its very nature it can not access the stability of steady states which are only local, but not global minimizers of the energy-Casimir functional. Since the existence of the steady state as a (global) minimizer is aimed for, certain growth conditions on the Casimir function are needed, which are not satisfied for all steady states with φ < 0. Most notably, the King model obtained by f 0 (x, v) = (e E 0 −E − 1)+ is the single most important model which is currently out of reach. It describes isothermal galaxies and is widely used in astrophysics. The corresponding Casimir function (2.2) has very slow growth for f → ∞, and as a result the variational method fails. The aim of the present paper is to develop a new approach to nonlinear stability results for steady states which need not be global minimizers of the corresponding energy-Casimir functional by exploiting Lemma 1.1. Although we are aiming for a general approach, we focus here on the King model and as a first step establish its nonlinear stability against spherically symmetric, dynamically accessible perturbations. An application of the present approach to the relativistic Vlasov-Poisson system for which the variational approach does not apply is given in [HR]. The paper proceeds as follows. In the next section we formulate our results. The nonlinear stability of the King model is an easy corollary of the following main theorem: In a certain neighborhood of the King model the potential energy distance of a perturbation can be controlled in terms of the energy-Casimir distance. In particular, within a certain class of perturbations, which is invariant under solutions of the nonlinear Vlasov-Poisson system, the King model is a local minimizer of the corresponding energyCasimir functional. The resulting stability estimate is more explicit than the ones obtained by the variational approach. The main part of the work is then done in Sect. 3 where the local minimizing property of the King model is established. In an appendix we give a proof of Lemma 1.1. 2. Main Results We start with a steady state f 0 with induced potential U0 and spatial density ρ0 , satisfying the relation
1 f 0 (x, v) = φ0 (E) := e E 0 −E − 1 , E := |v|2 + U0 (x). + 2
(2.1)
The cut-off energy E 0 < 0 is a given negative constant and (·)+ denotes the positive part. The corresponding Casimir function in the sense of the introduction is 0 ( f ) := (1 + f ) ln(1 + f ) − f.
(2.2)
The existence of such King models, i.e., of suitable solutions of the resulting semilinear Poisson equation (1.4), is established in [RR]. Such a model has compact support supp f 0 = {(v, v) ∈ R6 | E(x, v) ≤ E 0 } =: {E ≤ E 0 },
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and it is spherically symmetric. A state f is called spherically symmetric if for any rotation A ∈ SO(3), f (x, v) = f (Ax, Av), x, v ∈ R3 . It is well known that non-negative, smooth, and compactly supported initial data f (0) ∈ Cc1 (R6 ) launch unique global smooth solutions t → f (t) of the Vlasov-Poisson system [Pf, LP, Sch]. If the initial datum is spherically symmetric then this symmetry is preserved, and the modulus of the particle angular momentum squared, L := |x × v|2 = |x|2 |v|2 − (x · v)2 , is conserved along characteristics of the Vlasov equation. Hence for any smooth function such that (0, L) = 0, L ≥ 0, the functional ( f, L) dv d x is conserved along spherically symmetric solutions of the Vlasov-Poisson system; unless explicitly stated otherwise integrals always extend over R3 . We consider the following class of perturbations: S f0 := f ∈ L 1 (R6 ) | f ≥ 0 spherically symmetric, ( f, L) = ( f 0 , L) for all ∈ C 2 ([0, ∞[2 ) with
(0, L) = ∂ f (0, L) = 0, L ≥ 0, and ∂ 2f bounded . As noted above, the class S f0 ∩Cc1 (R6 ) is invariant under solutions of the Vlasov-Poisson system. Moreover, functions in S f0 are equi-measurable to f 0 , i.e., for every τ > 0 the sets { f > τ } and { f 0 > τ } have the same measure, in particular, || f || p = || f 0 || p for any L p -norm, p ∈ [1, ∞]. For the Casimir function 0 defined in (2.2) we define the energy-Casimir functional as in the introduction. Then for f ∈ S f0 we have HC ( f ) − HC ( f 0 ) = [0 ( f ) − 0 ( f 0 ) + (E − E 0 )( f − f 0 )] dv d x 1 − |∇U f − ∇U0 |2 d x; 8π notice that f = f 0 which allows us to bring in the term E 0 ( f − f 0 ). Now (E − E 0 )( f − f 0 ) ≥ −0 ( f 0 )( f − f 0 ) with equality on the support of f 0 , and hence for f ∈ S f0 , 0 ( f ) − 0 ( f 0 ) + (E − E 0 )( f − f 0 ) ≥
1 inf (τ ) ( f − f 0 )2 2 0≤τ ≤|| f0 ||∞ 0
≥ C 0 ( f − f 0 )2 ,
(2.3)
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where C0 := 1/(2 + 2|| f 0 ||∞ ); notice again that || f ||∞ = || f 0 ||∞ for any f ∈ S f0 . The deviation from the steady state is going to be measured by the quantity d( f, f 0 ) := [0 ( f ) − 0 ( f 0 ) + (E − E 0 )( f − f 0 )] dv d x 1 + (2.4) |∇U f − ∇U0 |2 d x, 8π which, as we have seen, controls both || f − f 0 ||2 and ||∇U f − ∇U0 ||2 , and satisfies the following relation to the energy-Casimir functional: 1 d( f, f 0 ) = HC ( f ) − HC ( f 0 ) + (2.5) |∇U f − ∇U0 |2 d x. 4π Our stability result is the following: Theorem 2.1. There exist constants δ > 0 and C > 0 such that for any solution t → f (t) of the Vlasov-Poisson system with f (0) ∈ S f0 ∩ Cc1 (R6 ) and d( f (0), f 0 ) ≤ δ the estimate d( f (t), f 0 ) ≤ C d( f (0), f 0 ) holds for all time t > 0. Remark. In order to better understand the perturbation class S f0 we show that it contains spherically symmetric, dynamically accessible perturbations by which we mean the following: Let H = H (x, v) ∈ C 2 (R6 ) be spherically symmetric, and let g = g(s, x, v) denote the solution of the linear problem ∂s g + ∇v H · ∇x g − ∇x H · ∇v g = 0, i.e., ∂s g(s) = {H, g(s)}, with initial datum g(0) = f 0 ; we assume that H is such that this solution exists on some interval I about s = 0. Then for any s ∈ I , g(s) is spherically symmetric and equi-measurable with f 0 . Moreover, for any function as considered in the definition of S f0 , ∂s (g(s), L) = ∂ f (g(s), L){H, g(s)}, and by a simple computation, {H, L} = 0. Hence d (g(s), L) dv d x = ∂ f {H, g(s)} + ∂ L {H, L} dv d x ds = {H, (g(s), L)} dv d x = 0 after an integration by parts. This means that g(s) ∈ S f0 for any such generating function H and any s. The only undesirable restriction in the class S f0 or in the generating functions H respectively is the spherical symmetry which hopefully can be removed in the future. The stability result Theorem 2.1 is easily deduced from the following theorem:
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Theorem 2.2. There exist constants δ0 > 0, and C0 > 0 such that for all f ∈ S f0 with d( f, f 0 ) ≤ δ0 the following estimate holds: HC ( f ) − HC ( f 0 ) ≥ C0 ||∇U f − ∇U0 ||22 . Before going into the proof of this theorem, which will occupy the rest of this paper, we conclude this section by deducing our stability result from it. Proof of Theorem 2.1. Let δ := δ0 (1 + 1/(4πC0 ))−1 with δ0 and C0 from Theorem 2.2. Consider a solution t → f (t) of the Vlasov-Poisson system with f (0) ∈ S f0 ∩ Cc1 (R6 ) and d( f (0), f 0 ) ≤ δ < δ0 . Then by continuity we can choose some maximal t ∗ ∈]0, ∞] such that d( f (t), f 0 ) < δ0 , t ∈ [0, t ∗ [. Now f (t) ∈ S f0 for all t, and hence Theorem 2.2, the relation (2.5) of d to the energyCasimir functional, and the fact that the latter is a conserved quantity yield the following chain of estimates for t ∈ [0, t ∗ [: 1 ||∇U f (t) − ∇U0 ||22 4π 1 ≤ HC ( f (t)) − HC ( f 0 ) + (HC ( f (t)) − HC ( f 0 )) 4πC0 1 = 1+ (HC ( f (0)) − HC ( f 0 )) 4πC0 1 ≤ 1+ d( f (0), f 0 ) < δ0 . 4πC0
d( f (t), f 0 ) = HC ( f (t)) − HC ( f 0 ) +
This implies that t ∗ = ∞, and Theorem 2.1 is established.
3. Proof of Theorem 2.2 Theorem 2.2 is proven by contradiction. There are two main ingredients: The first part (Subsect. 3.1) is a general argument to establish that if the estimate in the theorem fails, then there exists a non-zero function g such that D 2 HC ( f 0 )[g] ≤ 0. The second (Subsect. 3.2) is to use the measure-preserving property incorporated in our perturbation class S f0 to conclude that g = { f 0 , h} for some function h. This leads to a contradiction to Lemma 1.1 (Subsect. 3.3). 3.1. Existence of g = 0 with D 2 HC ( f 0 )[g] ≤ 0. The aim of this subsection is to prove the following result: Lemma 3.1. Assume that Theorem 2.2 were false. Then there is a function g ∈ L 2 (R6 ) which is spherically symmetric, supported in the set {E ≤ E 0 }, even in v, i.e., g(x, −v) = g(x, v), and such that 1 ||∇Ug ||22 = 1, (3.1) 8π
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D 2 HC ( f 0 )[g] =
1 2
{ f 0 >0}
0 ( f 0 ) g 2 dv d x − 1 ≤ 0,
∂ f ( f 0 , L) g dv d x = 0
(3.2) (3.3)
for all functions as specified in the definition of S f0 . Proof. If Theorem 2.2 were false, then for any n ∈ N there exists f n ∈ S f0 such that d( f n , f 0 ) < but HC ( f n ) − HC ( f 0 ) <
1 , n
1 ||∇U fn − ∇U0 ||22 , 8π n
(3.4)
in particular, f n = f 0 . We define 1 1 σn := √ ||∇U fn − ∇U0 ||2 , gn := ( fn − f0 ) σ 8π n
(3.5)
so that
1 ||∇Ugn ||22 = 1 8π and f n = f 0 + σn gn . By (2.5) and (3.5), σn2 ≤ d( f n , f 0 ) <
1 . n
(3.6)
(3.7)
Bounds on (gn ) and weak limit. As a first step in the proof of the lemma we need to establish bounds on the sequence (gn ) which allow us extract a subsequence that converges to some g which will be our candidate for the function asserted in the lemma. By (3.5), (2.4), (2.5), and (3.4) we find that 1 [0 ( f 0 + σn gn ) − 0 ( f 0 ) + (E − E 0 )σn gn ] dv d x − 1 σn2 1 1 2 = 2 d( f n , f 0 ) − |∇U fn − ∇U0 | d x σn 4π 1 1 1 1 ||∇U f n − ∇U0 ||22 = . (3.8) = 2 (HC ( f n ) − HC ( f 0 )) < 2 σn σn 8π n n Recalling the estimate (2.3) which is applicable since f n ∈ S f0 this implies that 1 1 1 1+ > 2 [. . .] dv d x ≥ C0 2 | f n − f 0 |2 dv d x = C0 |gn |2 dv d x, n σn σn which means that the sequence (gn ) is bounded in L 2 (R6 ). Moreover, since the integrand [. . .] in (3.8) is non-negative we find that 1 1 1 1+ > 2 [. . .] dv d x ≥ (E − E 0 ) gn dv d x, n σn {E≥E 0 } σn {E≥E 0 }
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where we used the fact that on the set {E ≥ E 0 } the steady state distribution f 0 and hence also 0 ( f 0 ) vanish while 0 ( f n ) ≥ 0. By (3.7), (E − E 0 ) gn dv d x ≤ 2σn → 0, n → ∞. (3.9) {E≥E 0 }
Now fix any E 0 < E 1 < 0. Since lim|x|→∞ U0 (x) = 0 it follows that E = E(x, v) > E 1 for x or v large so that the set {E ≤ E 1 } ⊂ R6 is compact. Hence (gn ) is bounded in L 1 ({E ≤ E 1 }). In addition E − E0 gn dv d x ≤ gn dv d x → 0; (3.10) {E≥E 1 } {E≥E 0 } E 1 − E 0 notice that gn ≥ 0 outside the support of f 0 , i.e., on the set {E > E 0 }. Thus we have shown that (gn ) is bounded in L 1 ∩ L 2 (R6 ). We extract a subsequence, denoted again by (gn ) such that gn g weakly in L 2 (R6 ). Since gn ≥ 0 on {E > E 0 } and since (3.10) holds for any E 0 < E 1 < 0 we conclude that g vanishes a. e. outside the set {E ≤ E 0 } as desired. Since the functions gn are spherically symmetric so is g. Proof of (3.1). In order to pass the weak convergence into (3.6) we need better bounds for the sequence (gn ). Indeed, we can bound its kinetic energy: 1 2 |v| + U0 (x) − E 0 gn dv d x = (E − E 0 ) gn dv d x → 0 {E≥E 0 } 2 {E≥E 0 } by (3.9) so that {E≥E 0 }
is bounded, while {E≤E 0 }
1 2 |v| gn dv d x 2
1 2 |v| |gn | dv d x ≤ (E 0 − U0 (0)) 2
{E≤E 0 }
|gn | dv d x
is bounded as well; recall that U0 is spherically symmetric and radially increasing. Now well known interpolation arguments imply that the sequence of induced spatial densities (ρgn ) is bounded in L 1 ∩ L 7/5 (R3 ), cf. [R4, Ch. 1, Lemma 5.1], so without loss of generality, ρgn ρg weakly in L 7/5 (R3 ). Let again E 0 < E 1 < 0 be arbitrary but fixed and choose R1 > 0 such that U0 (R1 ) = E 1 which implies that E(x, v) ≥ E 1 for |x| ≥ R1 . Then by (3.10), |ρgn | d x ≤ gn dv d x → 0. {|x|≥R1 }
{E≥E 1 }
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The fact that the sequence (ρgn ) remains concentrated in this way gives the desired compactness: ∇Ugn → ∇Ug strongly in L 2 (R3 ), cf. [R4, Ch. 2, Lemma 3.2]. Hence we can pass to the limit in (3.6) and find that g satisfies the condition (3.1) in the lemma. Proof of (3.2). Since ||gn ||2 is bounded it follows from (3.7) that ||σn gn ||2 ≤ Cσn → 0. Therefore, after extracting again a subsequence, σn gn → 0 almost everywhere. By Egorov’s Theorem there exists for every m ∈ N a measurable subset K m ⊂ {E ≤ E 0 } with the property that vol ({E ≤ E 0 } \ K m ) <
1 and lim σn gn = 0 uniformly on K m ; n→∞ m
note that the set {E ≤ E 0 } has finite measure. In addition we can assume that K m ⊂ K m+1 , m ∈ N. On the set {E ≤ E 0 }, [0 ( f n ) − 0 ( f 0 ) + (E − E 0 )σn gn ] =
1 1 ( f 0 )(σn gn )2 + ( f 0 + τ σn gn )(σn gn )3 2 0 6 0
for some τ ∈ [0, 1]. Since both f 0 and f 0 + σn gn = f n are non-negative the same is true for f 0 + τ σn gn , and we can use the estimate | 0 ( f )| =
1 ≤ 1, f ≥ 0, (1 + f )2
the estimate (3.8), and the fact that the integrand [. . .] in (3.8) is non-negative by (2.3) to conclude that 1 1 2 0 ( f 0 ) |gn | dv d x = 2 [. . .] dv d x σn Km 2 Km 1 1 0 ( f 0 + τ σn gn ) (σn gn )3 dv d x − 2 σn 6 Km 1 < 1 + + sup |σn gn | |gn |2 dv d x. n Km Taking the limit n → ∞ in this estimate we find that Km
1 ( f 0 ) g 2 dv d x ≤ 1. 2 0
If we observe the choice of the sets K m , let m → ∞, and recall the fact that g = 0 outside the set {E ≤ E 0 } the proof of (3.2) is complete.
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Proof of (3.3). To prove (3.3), the measure-preserving property of the set S f0 plays the crucial role. Let = ( f, L) be a function as specified in the definition of that set respectively in the lemma. By Taylor expansion with respect to the first argument, 1 ( f n , L) − ( f 0 , L) = ∂ f ( f 0 , L) σn gn + ∂ 2f ( f 0 + τ σn gn , L) (σn gn )2 2 for some τ ∈ [0, 1]. If we integrate this identity and observe that, since f n ∈ S f0 , ( f 0 , L) dv d x, ( f n , L) dv d x = it follows that 1 ∂ 2f ( f 0 + τ σn gn , L) gn2 dv d x → 0. ∂ f ( f 0 , L) gn dv d x = − σn 2 As to the latter limit we note that ∂ 2f is bounded, (gn ) is bounded in L 2 (R6 ), and σn → 0. On the other hand ∂ f ( f 0 , L) is supported on the compact set {E ≤ E 0 } and hence bounded. Since gn g weakly in L 2 (R6 ), the identity (3.3) follows as n → ∞. Conclusion of the proof of Lemma 3.1. The function g constructed above has all the properties required in the lemma, except that it need not be even in v. Hence we decompose it into its even and odd parts with respect to v: g = geven + godd . We claim that (3.1), (3.2), and (3.3) remain valid for the even part. Since ρg = g dv = geven dv = ρgeven we have ∇Ugeven = ∇Ug , and (3.1) remains valid. Since 0 ( f 0 ) is even in v,
1 1 2 1≥ 0 ( f 0 ) (geven + godd ) dv d x = 0 ( f 0 ) (geven )2 + (godd )2 dv d x 2 2 1 2 ( f 0 ) (geven ) dv d x, ≥ 2 0 i.e., (3.2) remains valid. Finally, let be as in the definition of the set S f0 . Then ∂ f ( f 0 , L) is even in v so that in (3.3) the odd part of g drops out, and the assertions of Lemma 3.1 hold for geven .
3.2. Characteristics and g = { f 0 , h}. In this subsection we construct a spherically symmetric function h such that g = { f 0 , h}. To this end we need to introduce variables which are adapted to the spherical symmetry: r := |x|, w :=
x ·v , L := |x × v|2 ; r
w is the radial velocity, and L has already been used above. Any spherically symmetric function of x and v such as the desired h can be written in terms of these variables, so h = h(r, w, L). Then L { f 0 , h} = φ0 (E){E, h} = −φ0 (E) w ∂r + 3 − U0 (r ) ∂w h, r
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and the equation we wish to solve for h reads 1 L w ∂r + 3 − U0 (r ) ∂w h = − g. r φ0 (E)
(3.11)
In order to analyze its characteristic system r˙ = w, w˙ =
L − U0 (r ) r3
we define for fixed L > 0 the effective potential L (r ) := U0 (r ) +
L 2r 2
and observe that in terms of the variables r, w, L the conserved particle energy takes the form E = E(x, v) = E(r, w, L) =
1 2 w + L (r ); 2
L only plays the role of a parameter here since L˙ = 0. We need to analyze the effective potential L . The boundary condition for U0 at infinity implies that limr →∞ L (r ) = 0, and clearly, limr →0 L (r ) = ∞. Moreover, 4π r m 0 (r ) U0 (r ) = 2 ρ0 (s) s 2 ds =: > 0, r > 0, r 0 r2 since U0 is increasing, thus U0 (0) < E 0 , and by (2.1), ρ0 (0) > 0. Now L (r ) = 0 ⇔
m 0 (r ) L L − 3 = 0 ⇔ m 0 (r ) − = 0, 2 r r r
and since the left-hand side of the latter equation is strictly increasing for L > 0 with limr →∞ (m 0 (r ) − L/r ) = M > 0 and limr →0 (m 0 (r ) − L/r ) = −∞ there exists a unique r L > 0 such that L (r L ) = 0, L (r ) < 0 for r < r L , L (r ) > 0 for r > r L . Moreover, since d dr
L L m 0 (r ) − = 4πr 2 ρ0 (r ) + 2 > 0 r r
the implicit function theorem implies that the mapping ]0, ∞[ L → r L is continuously differentiable. For r = r L , L (r L ) = −2
m 0 (r ) L L + 3 4 + 4πρ0 (r ) = 4πρ0 (r ) + 4 > 0. r3 r r
(3.12)
The behavior of L implies that for any L > 0 and L (r L ) < E < 0 there exist two unique radii 0 < r− (E, L) < r L < r+ (E, L) < ∞ such that L (r± (E, L)) = E, and L (r ) < E ⇔ r− (E, L) < r < r+ (E, L).
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Since L (r ) = 0 for r = r L it follows again by the implicit function theorem that the mapping (E, L) → r± (E, L) is C 1 on the set {(E, L) ∈ R×]0, ∞[| L (r L ) < E < 0}. Let τ → (r (τ ), w(τ ), L) be a characteristic curve with L > 0 and running in the set {E ≤ E 0 }, so that L (r L ) < E ≤ E 0 < 0. Then r (τ ) ∈ [r− (E, L), r+ (E, L)], w(τ ) = ± 2E − 2 L (r (τ )). By Eq. (3.11) which we want to solve, 1 d h(r (τ ), w(τ ), L) = − g(r (τ ), w(τ ), L), dτ φ0 (E) which we rewrite in terms of the parameter r as 1 d h(r, w(r ), L) = − g(r, w(r ), L), w(r ) = ± 2E − 2 L (r ). dr φ0 (E) w(r ) Hence for (r, w, L) ∈ {E ≤ E 0 } with L > 0 we define h as follows: We let E := 1 2 2 w + L (r ) so that r− (E, L) ≤ r ≤ r+ (E, L), and h(r, w, L) := −sign w
1 φ0 (E)
r
g(s,
r− (E,L)
√ 2E − 2 L (s), L) ds. √ 2E − 2 L (s)
(3.13)
Outside the set {E ≤ E 0 } we let h = 0. We need to make sure that we can consistently set h(r, 0, L) = 0, i.e., the integral in the above definition must vanish for r = r+ (E, L). To this end let be as specified in the definition of the class S f0 . We want to apply the change of variables (x, v) → (r, w, L) → (r, E, L) to the integral in (3.3). Now d x dv = 8π 2 dr dw d L = 8π 2 √
dr d E d L , 2E − 2 L (r )
(3.14)
where we note that g is even in v and hence in w so that we can restrict the integral to w > 0. We obtain the identity
0=
∞ ∞ ∞
∂ f ( f 0 , L) g dv d x = 8π ∂ f ( f 0 , L) g dr dw d L 0 0 0 √ r+ (E,L) g(r, 2E − 2 L (r ), L) dr ∂ f (φ0 (E), L) d E d L , = 8π 2 √ 2E − 2 L (r ) M r− (E,L) 2
where M := {(E, L)(x, v)| f 0 (x, v) > 0}. The class of test functions ∂ f (φ0 (E), L) is sufficiently large to conclude that for almost all E and L,
r+ (E,L)
r− (E,L)
as desired.
g(r,
√ 2E − 2 L (r ), L) dr = 0 √ 2E − 2 L (r )
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3.3. Contradiction to Lemma 1.1. As defined above, h need not be smooth or even integrable, so in order to derive a contradiction to Lemma 1.1 we need to regularize it. In order to do so it should first be noted that h as defined in (3.13) is measurable, which follows by Fubini’s Theorem and the fact that by the change of variables formula the function √ g(s, 2E − 2 L (s), L) (s, r, E, L) → 1[ L (r L ),E 0 ] (E)1[r− (E,L),r+ (E,L)] (s)1[0,r ] (s) √ 2E − 2 L (s) is integrable; r ≤ max{|x| | (x, v) ∈ supp f 0 }. The cut-off h. As a first step in regularizing h we define for m large the set 1 1 6 . m := (x, v) ∈ R | E ≤ E 0 − , L ≥ m m We want to approximate h by h1 m . In order to analyze this approximation the following lemma will be useful: Lemma 3.2. There exists a constant Cm > 0 such that for L ≥ E0 , r+ (E,L) dr < Cm . √ 2E − 2 L (r ) r− (E,L)
1 m
and L (r L ) < E ≤
Proof. We first establish the following auxiliary estimate: For all m ∈ N there exists a constant ηm > 0 such that for all L ≥ 1/m, L (r L ) < E ≤ E 0 , and r ∈ [r− (E, L), r+ (E, L)], | L (r )| ≥ ηm . (3.15) √ L (r ) − L (r L ) To see this let m ∈ N and L , E, r be as specified. Then E 0 ≥ E ≥ L (r ) = U0 (r ) +
L 1 ≥ U0 (0) + , 2 2r 2mr 2
and hence r ≥ (2m(E 0 − U0 (0))−1/2 . Let R := max{|x| | (x, v) ∈ supp f 0 }, i.e., U0 (R) = E 0 . Then L ≤ 2r 2 (E 0 − U0 (r )) ≤ 2R 2 (E 0 − U0 (0)) . ¯ in Hence if we assume that (3.15) were false, we can find a sequence (rn , L n ) → (¯r , L) the set [(2m(E 0 − U0 (0))−1/2 , R] × [1/m, 2R 2 (E 0 − U0 (0))] such that lim
n→∞
L n (rn ) L n (rn ) − L n (r L n )
= 0.
If r¯ = r L¯ it follows that L¯ (¯r ) > L¯ (r L¯ ) and L¯ (¯r ) = 0 which is a contradiction to the uniqueness of the minimizer r L¯ of L¯ . So assume that r¯ = r L¯ . Now recall that L n (r L n ) = 0. By Taylor expansion at r = r L n we find intermediate values θn and τn between rn and r L n such that
| L (θn ) (rn − r L n )| → 2| L¯ (r L¯ )| = 0, n → ∞, = n 1 L n (rn ) − L n (r L n ) 2 2 L n (τn ) (rn − r L n ) | L n (rn )|
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where we recall (3.12). This contradiction completes the proof of (3.15). To complete the proof of the lemma we split the integral as r+ (E,L) rL r+ (E,L) dr = ... + . . . =: I1 + I2 . √ 2E − 2 L (r ) r− (E,L) r− (E,L) rL √ In the first term, we make a change of variables u = L (r ) − L (r L ) so that du dr = 1 2u L (r ) < 0 on [r− (E, L), r L [. By (3.15), 0 1 dr du I1 = √ 2 du E− L (r L ) 2(E − L (r L ) − u ) √ √ E− (r ) √ 1 L L 2 du 2 ds ≤ = <∞ √ 2 ηm 0 ηm 0 1 − s2 E − L (r L ) − u √ by a further change of variables u = E − L (r L )s. The same type of estimate holds for the second part I2 of the integral under investigation, and the proof of the lemma is complete.
We now show that the cut-off function h1 m is square integrable and solves the equation { f 0 , h1 m } = g1 m in the sense of distributions, more precisely: Lemma 3.3. For any m ∈ N large, h1 m ∈ L 2 (R6 ), and for any spherically symmetric test function ψ = ψ(r, w, L) ∈ C 1 ([0, ∞[×R × [0, ∞[), { f 0 , ψ}h1 m = − g1 m ψ. E≤E 0
E≤E 0 2 6 L (R ). Since the integrand is even in v we can apply
Proof. We first prove that h1 m ∈ the change of variables (3.14): 1 m h 2 dv d x r+ (E,L) h 2 (r, 2E − 2 L (r ), L) √ = 8π 2 Sm
where
r− (E,L)
dr d E dL, 2E − 2 L (r )
1 1 Sm := (E, L) = (E, L)(x, v) | (x, v) ∈ supp f 0 , E ≤ E 0 − , L ≥ . (3.16) m m √ Let (E, L) ∈ Sm . In the estimates below we write w(r ) = 2E − 2 L (r ) and r± (E, L) = r± for brevity. Then by the definition (3.13) of h and the fact that 1/|φ0 (E)| ≤ 1 for E ≤ E 0 , r+ r+ r 1 dr ds 2 dr = h 2 (r, w(r ), L) g(s, w(s), L) w(r ) φ0 (E) r− w(s) w(r ) r− r− 2 r+ r+ dr ds ≤ g(s, w(s), L) w(s) w(r ) r− r− 2 r+ r+ dr dr ≤ g 2 (r, w(r ), L) w(r ) w(r ) r− r− r+ dr ; g 2 (r, w(r ), L) ≤ Cm2 w(r ) r−
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in the last two estimates we used the Cauchy-Schwarz inequality and Lemma 3.2. A further integration with respect to E and L and the change of variables (r, E, L) → (x, v) shows that ||h1 m ||2 is bounded in terms of Cm and ||g||2 . Now let ψ = ψ(r, w, L) be a test function as specified in the lemma. Along characteristic curves of (3.11), which as before we parameterize by r distinguishing between w > 0 and w < 0, a simple computation shows that d {E, ψ} = −sign w 2E − 2 L (r ) [ψ(r, sign w 2E − 2 L (r ), L)]. dr By √ the change of variables used repeatedly above, using the abbreviation w(r ) := 2E − 2 L (r ), and recalling the definition (3.16) we find that ... + ... { f 0 , ψ}h1 m dv d x = {w>0} {w<0} r+ dr d dE dL φ0 (E) w(r ) [ψ(r, w(r ), L)] h(r, w(r ), L) =− dr w(r ) Sm r− r+ d dr − φ0 (E) (−w(r )) [ψ(r, −w(r ), L)] h(r, −w(r ), L) dE dL dr w(r ) Sm r− r+ g(r, w(r ), L) dr d E d L φ0 (E) ψ(r, w(r ), L) − = φ0 (E) w(r ) Sm r− r+ g(r, −w(r ), L) − dr d E d L φ0 (E) ψ(r, −w(r ), L) φ0 (E) w(r ) S r− m =− ψg1 m dv d x; notice that h(r, ±w(r ), L) = 0 for r = r± (E, L), which together with the definition (3.13) of h along the characteristics was essential in the integration by parts above, and also that g is even in v respectively w. The proof of Lemma 3.3 is now complete.
Regularization of h1 m . The function h1 m is not smooth, hence Lemma 1.1 cannot be applied to it, and therefore we smooth it. For fixed m ∈ N the function h1 m is, as a function of r, w, L, supported in a set of the form Q m := [R0 , R1 ] × [−W0 , W0 ] × [L 0 , L 1 ]
(3.17)
with 0 < R0 < R1 , W0 > 0, and 0 < L 0 < L 1 ; it will be important that its support stays away both from r = 0 and L = 0. Let ζ ∈ Cc∞ (R3 ) be even in all three variables, i.e., ζ (z 1 , z 2 , z 3 ) = ζ (|z 1 |, |z 2 |, |z 3 |), ζ ≥ 0, ζ = 1, and define ζn := n 3 ζ (n·) for n ∈ N. For n sufficiently large we define ∞ ∞ ∞ ¯ ζn (r − r¯ , w − w, ¯ d L¯ d w¯ d r¯ . h n (r, w, L) = (h1 m )(¯r , w, ¯ L) ¯ L − L) 0
−∞ 0
Since m is strictly inside the set {E ≤ E 0 } with a positive distance from its boundary, h n ∈ Cc∞ (]0, ∞[×R×]0, ∞[), and without loss of generality we can assume that supp h n ⊂ Q m ∩ {E < E 0 }. Since h is odd in w respectively v so is h n , and clearly, h n → h1 m in L 2 ∩ L 1 . The crucial step is to show that lim { f 0 , h n } = g1 m
n→∞
(3.18)
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¯ and := ∞ ∞ ∞ for in L 2 . To this end we fix (r, w, L), write E¯ := E(¯r , w, ¯ L) 0 −∞ 0 brevity, and split the convolution integral into three parts: L { f 0 , h n } = −φ0 (E) w∂r + 3 − U0 (r ) ∂w h n r L ¯ 0 (E) − φ0 ( E)) ¯ = − (h1 m )(¯r , w, ¯ L)(φ w∂r + 3 − U0 (r ) ∂w r ¯ d L¯ d w¯ d r¯ ¯ L − L) ×ζn (r − r¯ , w − w, ¯ L ¯ 0 ( E) ¯ w∂ + (h1 m )(¯r , w, ¯ L)φ ¯ r¯ + 3 − U0 (¯r ) ∂w¯ r¯ ¯ d L¯ d w¯ d r¯ ¯ L − L) ×ζn (r − r¯ , w − w, ¯ L L ¯ 0 ( E) ¯ (w−w)∂ − (h1 m )(¯r , w, ¯ L)φ ¯ + − −U (¯ r )+U (r ) ∂w¯ r¯ 0 0 r¯ 3 r 3 ¯ d L¯ d w¯ d r¯ ¯ L − L) ×ζn (r − r¯ , w − w, =: −I1 + I2 − I3 . By Lemma 3.3, ¯ n (r − r¯ , w − w, ¯ d L¯ d w¯ d r¯ → g1 m in L 2 . ¯ L)ζ ¯ L − L) I2 = g1 m (¯r , w, We show that I1 and I3 tend to zero in L 2 . To do so we change the integration variables ¯ so that r¯ = r − r˜ /n, w¯ = r − w/n, into r˜ = n(r − r¯ ), w˜ = n(w − w), ¯ L˜ = n(L − L) ˜ L¯ = −3 4 ˜ ¯ ˜ ¯ ˜ r − L/n, d L d w¯ d r¯ = n d L d w˜ d r˜ , and ∂r ζn (r − r¯ , w − w, ¯ L − L) = n ∂r˜ ζ (˜r, w, ˜ L) with analogous formulas for the other derivatives. Thus ¯ φ (E) − φ0 ( E) ¯ 0 I1 = (h1 m )(¯r , w, ¯ L) 1/n L ˜ d L˜ d w˜ d r˜ , ˜ L) × w∂r˜ + 3 − U0 (r ) ∂w˜ ζ (˜r , w, r ¯ φ0 ( E) ¯ I3 = (h1 m )(¯r , w, ¯ L) L˜ r¯ −3 − r −3 U0 (¯r ) − U0 (r ) ˜ d L˜ d w˜ d r˜ . + ˜ L) × −w∂ ˜ r˜ + − 3 + L ∂w˜ ζ (˜r , w, r¯ 1/n 1/n Now ¯ φ0 (E) − φ0 ( E) → −φ0 (E) 1/n 3˜r r¯ −3 − r −3 → 4, 1/n r
L˜ r˜ L w w˜ + 2 − 3 + rU ˜ 0 (r ) , 2r r
U0 (¯r ) − U0 (r ) → −U0 (r ) r˜ 1/n
for n → ∞, and all these limits are uniform with respect to (r, w, L) ∈ Q m defined in ˜ ∈ supp ζ . Hence (3.17) and with respect to (˜r , w, ˜ L)
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I1 → −(h1 m )(r, w, L)φ0 (E) r˜ L L L˜ ˜ d L˜ d w˜ d r˜ = 0, ˜ L) ˜ 0 (r ) w∂r˜ + 3 − U0 (r ) ∂w˜ ζ (˜r , w, w w˜ + 2 − 3 + rU 2r r r I3 → (h1 m )(r, w, L)φ0 (E) L˜ 3L r˜ ˜ d L˜ d w˜ d r˜ = 0, −w∂ ˜ r˜ + − 3 + 4 − U0 (r )˜r ∂w˜ ζ (˜r , w, ˜ L) r˜ r where both limits are in L 2 and the zeroes result from integration by parts with respect to r˜ and w˜ and an exact cancellation. The assertion (3.18) is now established. Finally: The desired contradiction. The functions h n have all the properties required in Lemma 1.1, and we obtain the estimate
1 D HC ( f 0 )[{ f 0 , h n }] ≥ − 2 2
{E<E 0 }
φ0 (E)
1 U (r ) |h n |2 dv d x. r 0
By (3.18) and the fact that h n and g1 m are supported in a common compact set, ∇U{ f0 ,h n } → ∇Ug1 m in L 2 as n → ∞. Since r1 U0 (r ) is bounded on [0, ∞[ we obtain the estimate D 2 HC ( f 0 )[g1 m ] ≥ −
1 2
1 φ0 (E) U0 (r ) h 2 1 m dv d x. r {E<E 0 }
Since limm→∞ g1 m = g in L 1 ∩ L 2 (R6 ), there exists m 0 sufficiently large such that g1 m 0 = 0 and by Lemma 3.3, h1 m 0 = 0. For all m ≥ m 0 , m 0 ⊂ m and h 2 1 m 0 ≤ h 2 1 m . Since φ0 (E) < 0 and r1 U0 (r ) > 0 we have for all m ≥ m 0 , D 2 HC ( f 0 )[g1 m ] ≥ −
1 2
1 φ0 (E) U0 (r ) h 2 1 m 0 dv d x =: C > 0. r {E<E 0 }
Hence m → ∞ leads to a contradiction to Lemma 3.1, and the proof of Theorem 2.2 is complete.
4. Appendix: Proof of Lemma 1.1 Let Uh (x) :=
{ f 0 , h}dv
dy |x − y|
be the potential induced by −{ f 0 , h}, which is spherically symmetric. A short computation using the definition (1.6) of the Lie-Poisson bracket shows that
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{ f 0 , h}dv = ∇x ·
507
v φ0 (E) h(x, v) dv.
By the spherical symmetry of Uh and h, 1 Uh (r ) = 2 ∇x · v φ0 (E) h(x, v) dv r |x|≤r 1 x = 2 φ0 (E) h(x, v)v · dv dω(x) = 4π wφ0 (E) h(x, v) dv. r |x|=r r Therefore, by the Cauchy-Schwarz inequality, 1 − w 2 φ0 (E) dv − φ0 (E) h 2 dv d x. |∇x Uh |2 d x ≤ 2π 8π Since w 2 φ0 (E) = w
d φ0 dw
1 2 L w + 2 + U0 (r ) , 2 2r
an integration by parts with respect to w yields 1 2 π ∞ ∞ L 2 − w φ0 (E) dv = 2 w + 2 + U0 (r ) dw d L = ρ0 (r ). φ0 r 0 2 2r −∞ Hence 1 D HC ( f 0 )[{ f 0 , h}] ≥ − 2 2
φ0 (E) |{E, h}|2 − 4πρ0 h 2 dv d x.
Since h is odd in v respectively w the function µ(r, w, L) :=
1 h(r, w, L) rw
is smooth away from r = 0; in passing we notice that the functions h n to which we applied Lemma 1.1 in the proof of Theorem 2.2 have support bounded away from r = 0, but this is not necessary for Lemma 1.1. Since h = r wµ, {E, h} = r w{E, µ} + µ{E, r w}, and hence |{E, h}|2 = (r w)2 |{E, µ}|2 + r w{E, r w}{E, µ2 } + µ2 |{E, r w}|2 = (r w)2 |{E, µ}|2 + {E, µ2 r w{E, r w}} − µ2 r w{E, {E, r w}}. The first term is as claimed in Lemma 1.1. The second term leads to { f 0 , µ2 r w{E, r w}} whose integral with respect to x and v vanishes; if we cut a small ball of radius about x = 0 from the x-integral, then the surface integral appearing after the integration by parts with respect to x vanishes for → 0 since r µ2 ≤ C/r . By the Poisson equation, 1 {E, {E, r w}} = −2wU0 − w(rU0 ) = −r w 4πρ0 + U0 . r Hence the third term above becomes 4πρ0 h 2 + h 2 r1 U0 , and Lemma 1.1 is proven.
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Acknowledgement. The research of the first author is supported in part by an NSF grant. This article is dedicated to the memory of Xudong Liu.
References [An] [BT] [BG] [DSS] [FP] [GNN] [G1] [G2] [GR1] [GR2] [GR3] [GR4] [H] [HR] [KS] [LMR] [LP] [M] [Pf] [R1] [R2] [R3] [R4] [RG] [RR] [SS] [Sch] [SDLP] [Wa]
Antonov, V.A.: Remarks on the problem of stability in stellar dynamics. Sov. Astr. AJ. 4, 859– 867 (1961) Binney, J., Tremaine, S.: Galactic Dynamics. Princeton, NJ: Princeton University Press, 1987 Burchard, A., Guo, Y.: Compactness via symmetrization. J. Funct. Anal. 214, 40–73 (2004) Dolbeault, J., Sánchez, Ó., Soler, J.: Asymptotic behaviour for the Vlasov-Poisson system in the stellar-dynamics case. Arch. Ration. Mech. Anal. 171, 301–327 (2004) Fridman, A., Polyachenko, V.: Physics of Gravitating Systems I. New York: Springer-Verlag, 1984 Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979) Guo, Y.: Variational method in polytropic galaxies. Arch. Ration. Mech. Anal. 150, 209–224 (1999) Guo, Y.: On the generalized Antonov’s stability criterion. Contemp. Math. 263, 85–107 (2000) Guo, Y., Rein, G.: Stable steady states in stellar dynamics. Arch. Ration. Mech. Anal. 147, 225– 243 (1999) Guo, Y., Rein, G.: Existence and stability of camm type steady states in galactic dynamics. Indiana Univ. Math. J. 48, 1237–1255 (1999) Guo, Y., Rein, G.: Isotropic steady states in galactic dynamics. Commun. Math. Phys. 219, 607– 629 (2001) Guo, Y., Rein, G.: Isotropic steady states in stellar dynamics revisited. Commun. Math. Phys. 219, 607–629 (2001) Hadži´c, M.: Compactness and stability of some systems of nonlinear PDEs in galactic dynamics. Diploma thesis, University of Vienna, 2005 Hadži´c, M., Rein, G.: Global existence and nonlinear stability for the relativistic Vlasov-Poisson system in the gravitational case. http://arxiv.org/list/math-ph/0607012, 2006 Kandrup, H., Sygnet, J.F.: A simple proof of dynamical stability for a class of spherical clusters. Astrophys. J. 298, 27–33 (1985) Lemou, M., Mehats, F., Raphael, P.: Orbital stability and singularity formation for Vlasov-Poisson systems. Comptes Rendus Mathematique 341, 269–274 (2005) Lions, P.-L., Perthame, B.: Propagation of moments and regularity for the 3-dimensional VlasovPoisson system. Invent. Math. 105, 415–430 (1991) Morrison, P.J.: Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467–521 (1998) Pfaffelmoser, K.: Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. J. Differ. Eq. 95, 281–303 (1992) Rein, G.: Stability of spherically symmetric steady states in galactic dynamics against general perturbations. Arch. Ration. Mech. Anal. 161, 27–42 (2002) Rein, G.: Reduction and a concentration-compactness principle for energy-Casimir functionals. SIAM J. Math. Anal. 33, 896–912 (2002) Rein, G.: Nonlinear stability of Newtonian galaxies and stars from a mathematical perspective. In: Nonlinear Dynamics in Astronomy and Physics, Annals of the New York Academy of Sciences 1045, 103–119 (2005) Rein, G.: Collisionless Kinetic Equations from Astrophysics—The Vlasov-Poisson System. In: Handbook of Differential Equations, Evolutionary Equations. Vol. 3. Eds. C.M. Dafermos, E. Feireisl, Oxford: Elsevier, 2005 Rein, G., Guo, Y.: Stable models of elliptical galaxies. Monthly Not. Royal Astr. Soc. 344, 1396– 1406 (2003) Rein, G., Rendall, A.: Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Phil. Soc. 128, 363–380 (2000) Sánchez, Ó., Soler, J.: Orbital stability for polytropic galaxies. Ann. Inst. H. Poincaré Anal. Non Linéaire 23, 781–802 (2006) Schaeffer, J.: Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions. comm. Part. Differ. Eqs. 16, 1313–1335 (1991) Sygnet, J.F., Des Forets, G., Lachieze-Rey, M., Pellat, R.: Stability of gravitational systems and gravothermal catastrophe in astrophysics. Astrophys. J. 276, 737–745 (1984) Wan, Y-H.: On nonlinear stability of isotropic models in stellar dynamics. Arch. Ration. Mech. Anal. 147, 245–268 (1999)
Non-Variational Approach to Nonlinear Stability in Stellar Dynamics
[Wo1] [Wo2]
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Wolansky, G.: On nonlinear stability of polytropic galaxies. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 15–48 (1999) Wolansky, G.: Static solutions of the Vlasov-Einstein system. Arch. Ration. Mech. Anal. 156, 205– 230 (2001)
Communicated by H. Spohn
Commun. Math. Phys. 271, 511–522 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0172-4
Communications in
Mathematical Physics
Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm Equation A. Alexandrou Himonas1 , Gerard Misiołek1 , Gustavo Ponce2 , Yong Zhou3,4 1 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA.
E-mail: [email protected]; [email protected]
2 Department of Mathematics, University of California, Santa Barbara, CA 93106, USA.
E-mail: [email protected]
3 Department of Mathematics, East China Normal University, Shangai 200062, China.
E-mail: [email protected]
4 Institute des Hautes Éudes Scientifiques, 35, route de Chartres, F-91440 Bures-sur-Yvette, France
Received: 19 April 2006 / Accepted: 21 July 2006 Published online: 8 February 2007 – © Springer-Verlag 2007
Abstract: It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution. 1. Introduction This work is concerned with the nonperiodic Camassa-Holm equation ∂t u − ∂t ∂x2 u + 3u∂x u − 2∂x u∂x2 u − u∂x3 u = 0,
t, x ∈ R.
(1.1)
This equation appears in the context of hereditary symmetries studied by Fuchssteiner and Fokas [FF]. It was first written explicitly, and derived physically as a water wave equation by Camassa and Holm [CH], who also studied its solutions. Equation (1.1) is remarkable for its properties such as infinitely many conserved integrals, bi-hamiltonian structure or its non-smooth travelling wave solutions known as “peakons” (see formula (1.9)). It was also derived as an equation for geodesics of the H 1 -metric on the diffeomorphism group, see [Mi]. For a discussion of how it relates to the theory of hereditary symmetries see [F]. The inverse scattering approach to the Camassa-Holm equation has also been developed in several works, for example see [CH, CoMc, Mc1, BSS], and references therein. A considerable amount of work has been devoted to the study of the corresponding Cauchy problem in both nonperiodic and periodic cases. Among these results, of relevance to the present paper will be the fact that (1.1) is locally well-posed (in Hadamard’s sense) in H s (R) for any s > 3/2, see for example [LO, R, D]. The long time behaviour of solutions has been studied and conditions which guarantee their global existence and their finite blow up have been deduced. In particular, in [Mc1] a necessary
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and sufficient condition was established on the initial datum to guarantee finite time singularity formation for the corresponding strong solution. For further results in this direction we refer to [Mc1, CoE] and the survey article [Mo] and references therein. For well-posedness results in the periodic case we refer to [HM1, Mi] and [DKT], where the equation is studied in its integral-differential form (see (1.2) below) as an ODE on the space of diffeomorphisms of the circle. A recent result demonstrating that the solution map u 0 → u for the Camassa-Holm equation is not locally uniformly continuous in Sobolev spaces can be found in [HM2]. Also the Camassa-Holm equation has been studied as an integrable infinite-dimensional Hamiltonian system, and several works have been devoted to several aspects of its scattering setting, see [CH, CoMc, Mc1, BSS] and references therein. It is convenient to rewrite the equation in its formally equivalent integral-differential form 1 2 2 ∂t u + u∂x u + ∂x G ∗ u + (∂x u) = 0, (1.2) 2 where G(x) = 21 e−|x| . Our first objective here is to formulate decay conditions on a solution, at two distinct times, which guarantee that u ≡ 0 is the unique solution of Eq. (1.1). The idea of proving unique continuation results for nonlinear dispersive equations under decay assumptions of the solution at two different times was motivated by the recent works [EKPV1, EKPV2] on the nonlinear Schrödinger and the k-generalized Korteweg-de Vries equations respectively. In the recent works [Co, He] and [Z] it was shown that u cannot preserve compact support in a non-trivial time interval (i.e. for t ∈ [0, ], > 0) except if u ≡ 0. However, this result does not preclude the possibility of the solution having compact support at a later time. In fact, in [Z] the question concerning the possibility of a smooth solution of (1.1) having compact support at two different times was explicitly stated. In particular, our first result, Theorem 1.1, gives a negative answer to this question. Theorem 1.1. Assume that for some T > 0 and s > 3/2, u ∈ C([0, T ] : H s (R))
(1.3)
is a strong solution of the IVP associated to Eq. (1.2). If u 0 (x) = u(x, 0) satisfies that for some α ∈ (1/2, 1), |u 0 (x)| ∼ o(e−x ),
and
|∂x u 0 (x)| ∼ O(e−αx )
as x ↑ ∞,
(1.4)
and there exists t1 ∈ (0, T ] such that |u(x, t1 )| ∼ o(e−x )
as x ↑ ∞,
then u ≡ 0. Notation.We shall say that | f (x)| ∼ O(eax ) as x ↑ ∞
if
| f (x)| ∼ o(eax )
if
lim
| f (x)| = L, eax
lim
| f (x)| = 0. eax
x→∞
and as x ↑ ∞
x→∞
(1.5)
Propagation Speed for the C-H Equation
513
Remarks. (a) Theorem 1.1 holds with the corresponding decay hypothesis in (1.4)-(1.5) stated for x < 0. (b) The time interval [0, T ] is the maximal existence time interval of the strong solution. This guarantees that the solution is uniformly bounded in the H s -norm in this interval (see (2.12)), and that our solution is the strong limit of smooth ones such that the integration by parts in the proof (see (2.21), (2.29)) can be justified. The proof of Theorem 1.1 will be a consequence of the following result concerning some persistence properties of the solution of Eq. (1.2) in L ∞ -spaces with exponential weights. Theorem 1.2. Assume that for some T > 0 and s > 3/2, u ∈ C([0, T ] : H s (R))
(1.6)
is a strong solution of the IVP associated to Eq. (1.2) and that u 0 (x) = u(x, 0) satisfies that for some θ ∈ (0, 1), |u 0 (x)|, |∂x u 0 (x)| ∼ O(e−θ x ) as Then
x ↑ ∞.
|u(x, t)|, |∂x u(x, t)| ∼ O(e−θ x ) as
x ↑ ∞,
(1.7) (1.8)
uniformly in the time interval [0, T ]. The following result establishes the optimality of Theorem 1.1 and tells us that a strong non-trivial solution of (1.2) corresponding to data with fast decay at infinity will immediately behave asymptotically, in the x-variable at infinity, as the “peakon” solution vc (x, t) = c e−|x−ct| ,
t > 0.
(1.9)
Theorem 1.3. Assume that for some T > 0 and s > 3/2, u ∈ C([0, T ] : H s (R))
(1.10)
is a strong solution of the IVP associated to Eq. (1.2) and that u 0 (x) = u(x, 0) satisfies that for some α ∈ (1/2, 1), |u 0 (x)| ∼ O(e−x ),
|∂x u 0 (x)| ∼ O(e−αx ) as x ↑ ∞
(1.11)
for some α ∈ (1/2, 1). Then |u(x, t)| ∼ O(e−x ) as x ↑ ∞,
(1.12)
uniformly in the time interval [0, T ]. In the case when the solution u(x, t) possesses further regularity and its data u 0 has stronger decay properties we shall give a more precise description of its behavior at infinity in the space variable. As it was noted in both [Co] and [Z] in the case of compactly supported initial data u 0 the difference h(x, t) of the solution and its second derivative, i.e. h(x, t) = (1 − ∂x2 )u(x, t),
(1.13)
remains compactly supported. Thus, if u 0 is supported in the interval [a, b] in its lifespan, one has that h(x, t) has compact support in the time interval [η(a, t), η(b, t)], (for the definition of η(·, ·) see (2.38)).
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A. A. Himonas, G. Misiołek, G. Ponce, Y. Zhou
Theorem 1.4. Assume that for some T > 0 and s > 5/2, u ∈ C([0, T ] : H s (R))
(1.14)
is a strong solution of the IVP associated to Eq. (1.2). (a) If u 0 (x) = u(x, 0) has compact support, then for any t ∈ (0, T ], u(x, t) =
c+ (t) e−x , c− (t) e x ,
for for
x > η(b, t), x < η(a, t).
(1.15)
(b) If for some µ > 0, ∂x u 0 ∼ O(e−(1+µ)|x| ) j
as
|x| ↑ ∞
j = 0, 1, 2,
(1.16)
then for any t ∈ (0, T ], h(x, t) = (1 − ∂x2 ) u(x, t) ∼ O(e−(1+µ)|x| )
as
|x| ↑ ∞,
(1.17)
and lim e±x u(x, t) = c± (t),
x→±∞
(1.18)
where in (1.15), (1.18) c+ (·), c− (·) denote continuous non-vanishing functions, with c+ (t) > 0 and c− (t) < 0 for t ∈ (0, T ]. Furthemore, c+ (·) is a strictly increasing function, while c− (·) is strictly decreasing. Theorem 1.4 tells us that, as long as it exists, the solution u(x, t) is positive at infinity and negative at minus infinity regardless of the profile of a fast-decaying data u 0 = 0. Finally, as a consequence of some of the estimates obtained in the proofs of the previous results we shall show that any strong solution corresponding to a data with compact support blows up in finite time. For details on the structure of the blow-up we refer to [Mc1, Mc2] and references therein. Other blow up results are discussed in [Mo]. Corollary 1.1. Assume that for some T > 0 and s > 5/2, u ∈ C([0, T ] : H s (R))
(1.19)
is a strong solution associated to Eq. (1.2) with initial data u(x, 0) = u 0 (x) having compact support. Then the solution u(x, t) blows up in finite time, i.e. there exists a time interval [0, T ∗ ) such that u ∈ C([0, T ∗ ) : H s (R)) and
T∗ 0
∂x u(t) L ∞ dt = ∞.
(1.20)
Propagation Speed for the C-H Equation
515
2. Proof of the Results First, assuming the result in Theorem 1.2 we shall prove Theorem 1.1. Proof of Theorem 1.1. Integrating Eq. (1.2) over the time interval [0, t1 ] we get t1 t1 1 u∂x u(x, τ )dτ + ∂x G ∗ (u 2 + (∂x u)2 )(x, τ )dτ = 0. (2.1) u(x, t1 ) − u(x, 0) + 2 0 0 By hypothesis (1.4) and (1.5) we have u(x, t1 ) − u(x, 0) ∼ o(e−x ) as x ↑ ∞. From (1.4) and Theorem 1.2 it follows that t1 u∂x u(x, τ )dτ ∼ O(e−2αx ) as x ↑ ∞,
(2.2)
(2.3)
0
and so
t1
u∂x u(x, τ )dτ ∼ o(e−x ) as x ↑ ∞.
(2.4)
0
We shall show that if u = 0, then the last term in (2.1) is O(e−x ) but not o(e−x ). Thus, we have t1 t1 1 1 ∂x G ∗ (u 2 + (∂x u)2 )(x, τ ) dτ = ∂x G ∗ (u 2 + (∂x u)2 )(x, τ ) dτ 2 2 (2.5) 0 0 = ∂x G ∗ ρ(x), where by (1.4) and Theorem 1.2, 0 ≤ ρ(x) ∼ O(e−2αx ), Therefore
ρ(x) ∼ o(e−x ) as x ↑ ∞.
so that
1 ∞ sgn(x − y) e−|x−y| ρ(y) dy 2 −∞ x ∞ 1 1 = − e−x e y ρ(y) dy + e x e−y ρ(y) dy. 2 2 −∞ x
(2.6)
∂x G ∗ ρ(x) = −
From (2.6) it follows that ∞ ex e−y ρ(y) dy = o(1)e x x
∞
e−2y dy ∼ o(1)e−x ∼ o(e−x ),
(2.7)
(2.8)
x
and if ρ = 0 one has that
x
−∞
e y ρ(y) dy ≥ c0 ,
Hence, the last term in (2.5) and (2.7) satisfies c0 −x e , −∂x G ∗ ρ(x) ≥ 2
for x 1.
for x 1,
(2.9)
(2.10)
which combined with (2.1)-(2.3) yields a contradiction. Thus, ρ(x) ≡ 0 and consequently u ≡ 0, see (2.5).
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A. A. Himonas, G. Misiołek, G. Ponce, Y. Zhou
Proof of Theorem 1.3. This proof is similar to that given for Theorem 1.1 and therefore it will be omitted. We proceed to prove Theorem 1.2. Proof of Theorem 1.2. We introduce the following notations (∂x u)2 , 2
(2.11)
M = sup u(t) H s .
(2.12)
F(u) = u 2 + and t∈[0,T ]
Multiplying Eq. (1.2) by u 2 p−1 with p ∈ Z+ and integrating the result in the x-variable one gets ∞ ∞ ∞ u 2 p−1 ∂t u d x + u 2 p−1 u∂x u d x + u 2 p−1 ∂x G ∗ F(u) d x = 0. (2.13) −∞
−∞
−∞
The estimates ∞ 1 d 2p 2 p−1 d u(t) 2 p = u(t) 2 p u(t) 2 p u 2 p−1 ∂t u d x = 2 p dt dt −∞ and
∞ −∞
2p u 2 p−1 u∂x u d x ≤ ∂x u(t) ∞ u(t) 2 p
(2.14)
(2.15)
and Hölder’s inequality in (2.13) yield d u(t) 2 p ≤ ∂x u(t) ∞ u(t) 2 p + ∂x G ∗ F(u)(t) 2 p , dt and therefore, by Gronwall’s inequality t u(t) 2 p ≤ u(0) 2 p + ∂x G ∗ F(u)(τ ) 2 p dτ e Mt .
(2.16)
(2.17)
0
Since f ∈ L 1 (R) ∩ L ∞ (R) implies lim f q = f ∞ ,
q↑∞
(2.18)
taking the limits in (2.17) (notice that ∂x G ∈ L 1 and F(u) ∈ L 1 ∩ L ∞ ) from (2.18) we get t ∂x G ∗ F(u)(τ ) ∞ dτ e Mt . (2.19) u(t) ∞ ≤ u(0) ∞ + 0
Next, differentiating (1.2) in the x-variable produces the equation 1 ∂t ∂x u + u∂x2 u + (∂x u)2 + ∂x2 G ∗ u 2 + (∂x u)2 = 0. 2
(2.20)
Propagation Speed for the C-H Equation
517
Again, multiplying Eq. (2.20) by ∂x u 2 p−1 ( p ∈ Z+ ), integrating the result in the x-variable, and using integration by parts,
∞
−∞
u∂x2 u(∂x u)2 p−1 d x =
∞
−∞
∞ (∂ u)2 p 1 x dx = − ∂x u(∂x u)2 p d x, 2p 2 p −∞ (2.21)
u ∂x
one gets the inequality d ∂x u(t) 2 p ≤ 2 ∂x u(t) ∞ ∂x u(t) 2 p + ∂x2 G ∗ F(u)(t) 2 p , dt and therefore as before ∂x u(t) 2 p ≤ ∂x u(0) 2 p + 0
t
∂x2 G ∗ F(u)(τ ) 2 p dτ e2Mt .
(2.22)
(2.23)
Since ∂x2 G = G − δ, we can use (2.18) and pass to the limit in (2.23) to obtain t 2 ∂x G ∗ F(u)(τ ) ∞ dτ e2Mt . (2.24) ∂x u(t) ∞ ≤ ∂x u(0) ∞ + 0
We shall now repeat the above arguments using the weight ⎧ ⎪ x ≤ 0, ⎨1, ϕ N (x) = eθ x , x ∈ (0, N ), ⎪ ⎩eθ N , x ≥ N ,
(2.25)
where N ∈ Z+ . Observe that for all N we have 0 ≤ ϕ N (x) ≤ ϕ N (x)
a.e. x ∈ R.
(2.26)
Using notation in (2.11), from Eq. (1.2) we obtain ∂t (u ϕ N ) + (u ϕ N )∂x u + ϕ N ∂x G ∗ F(u) = 0,
(2.27)
while from (2.20) we get ∂t (∂x u ϕ N ) + u ∂x2 u ϕ N + (∂x u ϕ N )∂x u + ϕ N ∂x2 G ∗ F(u) = 0.
(2.28)
We need to eliminate the second derivatives in the second term in (2.28). Thus, combining integration by parts and (2.26) we find ∞ u ∂x2 u ϕ N (∂x u ϕ N )2 p−1 d x −∞ ∞ = u(∂x u ϕ N )2 p−1 (∂x (∂x u ϕ N ) − ∂x u ϕ N )d x −∞ ∞ ∞ (∂x u ϕ N )2 p = dx − u∂x u∂x u ϕ N (∂x u ϕ N )2 p−1 d x 2 p −∞ −∞
2p ≤ 2 u(t) ∞ + ∂x u(t) ∞ ∂x u ϕ N 2 p . (2.29)
518
A. A. Himonas, G. Misiołek, G. Ponce, Y. Zhou
Hence, as in the weightless case (2.19) and (2.24), we get u(t)ϕ N ∞ + ∂x u(t)ϕ N ∞ ≤ e2Mt ( u(0)ϕ N ∞ + ∂x u(0)ϕ N ∞ ) t ϕ N ∂x G ∗ F(u)(τ ) ∞ + ϕ N ∂x2 G ∗ F(u)(τ ) ∞ dτ. + e2Mt 0
(2.30) A simple calculation shows that there exists c0 > 0, depending only on θ ∈ (0, 1) (see (1.7) and (2.25)) such that for any N ∈ Z+ , ∞ 1 4 dy ≤ c0 = . (2.31) e−|x−y| ϕ N (x) ϕ (y) 1 − θ N −∞ Thus, for any appropriate function f one sees that ∞ −|x−y| 2 ϕ N ∂x G ∗ f 2 (x) = 1 ϕ N (x) sgn(x − y) e f (y) dy 2 −∞ ∞ 1 1 ≤ ϕ N (x) ϕ N (y) f (y) f (y) dy e−|x−y| 2 ϕ N (y) −∞ ∞ 1 1 −|x−y| ϕ N (x) dy ϕ N f ∞ f ∞ ≤ e 2 ϕ N (y) −∞ ≤ c0 ϕ N f ∞ f ∞ . (2.32) Since ∂x2 G = G − δ the argument in (2.32) also shows that ϕ N ∂x2 G ∗ f 2 (x) ≤ c0 ϕ N f ∞ f ∞ .
(2.33)
Thus, inserting (2.32)-(2.33) into (2.30) and using (2.11)-(2.12) it follows that there exists a constant c˜ = c(M; ˜ T ) > 0 such that
u(t)ϕ N ∞ + ∂x u(t)ϕ N ∞ ≤ c˜ u(0)ϕ N ∞ + ∂x u(0)ϕ N ∞ t
u(τ ) ∞ + ∂x u(τ ) ∞ u(τ )ϕ N ∞ + ∂x u(τ )ϕ N ∞ dτ + c˜ 0 t
≤ c˜ u(0)ϕ N ∞ + ∂x u(0)ϕ N ∞ + u(τ )ϕ N ∞ + ∂x u(τ )ϕ N ∞ dτ . 0
(2.34) Hence, for any N ∈ Z+ and any t ∈ [0, T ] we have
u(t)ϕ N ∞ + ∂x u(t)ϕ N ∞ ≤ c˜ u(0)ϕ N ∞ + ∂x u(0)ϕ N ∞
≤ c˜ u(0) max (1, eθ x ) ∞ + ∂x u(0) max (1, eθ x ) ∞ . (2.35) Finally, taking the limit as N goes to infinity in (2.35) we find that for any t ∈ [0, T ]
|u(x, t)eθ x | + |∂x u(x, t)eθ x | ≤ c˜ u(0) max (1, eθ x ) ∞ + ∂x u(0) max (1, eθ x ) ∞ (2.36) which completes the proof of Theorem 2.
Propagation Speed for the C-H Equation
519
It remains to prove Theorem 1.4. Proof of Theorem 1.4. A simple calculation shows that the solution u of Eq. (1.1) satisfies the identity (2.37) (1 − ∂x2 )u ◦ η (∂x η)2 = (1 − ∂x2 )u 0 , (it has a mechanical interpretation as conservation of spacial angular momentum). Here η = η(x, t) is the flow of u, that is ⎧ ⎨ dη(x, t) = u(η(x, t), t), (2.38) dt ⎩η(x, 0) = x, so that by the assumption and the standard ODE theory t → η(t) is a smooth curve of C 1 -diffeomorphisms of the line, close to the identity map and defined on the same time interval as u (see [Mi] for details in the periodic case). From (2.37) we then have x ∞ 1 ∞ −|x−y| 1 1 u(x, t) = e h(y, t)dy = e−x e y h(y, t)dy+ e x e−y h(y, t)dy, 2 −∞ 2 2 −∞ x (2.39) where (1 − ∂x2 )u 0 (η−1 (x, t)) h(x, t) = (1 − ∂x2 )u(x, t) = (2.40)
2 . ∂x η(η−1 (x, t), t) Let us first prove part (a). Thus, from (2.40) it follows that if u 0 has compact support in x in the interval [a, b], then so does h(·, t) in the interval [η(a, t), η(b, t)], for any t ∈ [0, T ]. Moreover, defining η(b,t) η(b,t) E + (t) = e y h(y, t) dy and E − (t) = e−y h(y, t) dy, (2.41) η(a,t)
η(a,t)
one has from (2.40) that 1 −|x| 1 e ∗ h(x, t) = e−x E + (t), 2 2
x > η(b, t),
(2.42)
1 1 −|x| e ∗ h(x, t) = e x E − (t), 2 2 Hence, it follows that for x > η(b, t),
x < η(a, t).
(2.43)
u(x, t) = and
u(x, t) =
u(x, t) = −∂x u(x, t) = ∂x2 u(x, t) =
1 −x e E + (t), 2
(2.44)
1 x e E − (t). 2
(2.45)
and for x < η(a, t), u(x, t) = ∂x u(x, t) = ∂x2 u(x, t) =
Next, integration by parts, (2.44), (2.45), and the equation in (1.1) yield the identities ∞ ∞ ∞ e y h(y, 0)dy = e y u 0 (y)dy − e y ∂x2 u 0 (y)dy E + (0) = −∞ −∞ −∞ ∞ ∞ = e y u 0 (y)dy + e y ∂x u 0 (y)dy = 0 (2.46) −∞
−∞
520
A. A. Himonas, G. Misiołek, G. Ponce, Y. Zhou
and d E + (t) dt =−
∞
−∞
e y u∂x u dy +
∞
−∞
e y ∂x2 (u∂x u) dy −
∞
−∞
e y ∂x F(u)dy
∞
∞ (∂x u)2 ∞ y y 2 |−∞ + = e ∂x (u∂x u) − u∂x u |−∞ − e u + e y F(u)dy 2 −∞ ∞ (∂x u)2 y 2 = dy > 0. (2.47) e u + 2 −∞ Therefore, in the life-span of the solution u(x, t), E + (t) is an increasing function. Thus, from (2.46) it follows that E + (t) > 0 for t ∈ (0, T ]. Similarly, it is easy to see that E − (t) is decreasing with E − (0) = 0, therefore E − (t) < 0 for t ∈ (0, T ]. Taking c± (t) = 21 E ± (t) we obtain (1.15). Next, let us consider part (b). Since h(x, t) = (1 − ∂x2 )u(x, t) satisfies the equation ∂t h(x, t) + u(x, t)∂x h(x, t) = −2∂x u(x, t)h(x, t),
(x, t) ∈ R × [0, T ], (2.48)
an argument similar to that given in the proof of Theorem 1.2 shows that (1+µ)|x| ˜ ∞ , sup h(t)e(1+µ)|x| ∞ ≤ c h(0)e
(2.49)
t∈[0,T ]
with c˜ depending only on M in (2.12) and T , and that for any θ ∈ (0, 1), ∂x u(t) ∼ O(e−θ|x| ) j
|x| ↑ ∞
as
for
j = 0, 1, 2.
(2.50)
Thus, the definitions in (2.41) make sense with the integrals extended to the whole real line and the computations in (2.46)-(2.47) can be carried out in the same fashion. Finally, using (2.49) in (2.39) we obtain (1.18). Proof of Corollary 1.1. Suppose that the support of u 0 is contained in some interval [a, b]. According to a result of McKean [Mc2] the corresponding solution u(x, t) persists for all time only if either h 0 (x) = (1 − ∂x2 )u 0 (x) is of one sign or 0 = h 0 (x) ≤ 0 when a ≤ x < xo
and
0 = h 0 (x) ≥ 0 when xo < x ≤ b
for some xo in (a, b). Otherwise there exists T ∗ < ∞ such that ∂x u(t) L ∞ → ∞ as t ↑ T ∗. Assuming that the solution is global in time and computing as in (2.46) we find that b b x e h 0 (x) d x = e−x h 0 (x) d x = 0. a
a
This implies that h 0 (x) must change sign. Furthermore, since e x is strictly increasing we have the inequalites xo b x0 b h 0 (x) d x > − e x h 0 (x) d x = e x h 0 (x) d x > e xo h 0 (x) d x −e xo a
a
xo
x0
Propagation Speed for the C-H Equation
and similarly xo −e−xo h 0 (x) d x < − a
xo
e−x h 0 (x) d x =
521
a
b
e−x h 0 (x) d x < e−xo
xo
b
h 0 (x) d x.
xo
These two sets of inequalities cannot both hold and so T ∗ < ∞ is the maximal existence time of the solution u. In order to prove (1.20), we can combine (2.16), (2.18) and (2.22) to get t
e−2 0 ∂x u(τ ) ∞ dτ u(t) ∞ + ∂x u(t) ∞ ≤ u(0) ∞ + ∂x u(0) ∞ t τ + e− 0 ∂x u(r ) ∞ dr ∂x G ∗ F(u)(τ ) ∞ 0
+ ∂x2 G ∗ F(u)(τ ) ∞ dτ for any t < T ∗ . Since
∂x G ∗ F(u) ∞ ≤ c ∂x u ∞ u ∞ + ∂x u ∞
and
∂x2 G ∗ F(u) ∞ ≤ c ∂x u ∞ u ∞ + ∂x u ∞ , we have e−2
t 0
∂x u(τ ) ∞ dτ
u(t) ∞ + ∂x u(t) ∞ ≤ u(0) ∞ + ∂x u(0) ∞ t τ +c ∂x u(τ ) ∞ e−2 0 ∂x u(r ) ∞ dr u(τ ) ∞ 0
+ ∂x u(τ ) ∞ dτ.
Applying Gronwall’s inequality again we obtain
u(t) ∞ + ∂x u(t) ∞ ≤ u(0) ∞ + ∂x u(0) ∞ exp c
0
t
∂x u(τ ) ∞ dτ ,
T∗
valid for any t < T ∗ . This estimate implies that if 0 ∂x u(t) ∞ dt were finite then the solution u could be extended beyond time T ∗ . The corollary follows. Acknowledgements. The authors would like to thank Prof. D. Holm for useful comments concerning this work. G. P. was supported by an NSF grant. Y. Z. was supported by NSFC under grant no. 10501012 and Shanghai Rising-Star Program 05QMX1417.
References [BSS] [CH] [Co] [CoE]
Beals, R., Sattinger, D., Szmigielski, J., Multipeakons and the classical moment problem. Adv. Math. 154(2), 229–257 (2000) Camassa, R., Holm, D.: An integrable shallow water equation with peaked solutions. Phys. Rev. Lett. 71, 1661–1664 (1993) Constantin, A.: Finite propagation speed for the Camassa-Holm equation. J. Math. Phys. 46(2), 4 (2005) Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26(2), 303–328 (1998)
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[CoMc] [CoS] [D] [DKT] [EKPV1] [EKPV2] [FF] [F] [He] [HM1] [HM2] [LO] [Mc1] [Mc2] [Mi] [Mo] [R] [Z]
A. A. Himonas, G. Misiołek, G. Ponce, Y. Zhou
Constantin, A., McKean, H.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 (1999) Constantin, A., Strauss, W.: Stability of peakons. Comm. Pure Appl. Math. 53, 603–610 (2000) Danchin, R.: A few remarks on the Camassa-Holm equation. Differ. Int. Eqs. 14, 953–988 (2001) De Lellis, C., Kappeler, T., Topalov, P.: Low-regularity solutions of the periodic Camassa-Holm equation. Comm. Partial Differential Equations (to appear) (2007) Escauriaza, L., Kenig, C. E., Ponce, G., Vega, L.: On unique continuation of solutions of Schrödinger equations. Comm. PDE 31, 1811–1823 (2006) Escauriaza, L., Kenig, C. E., Ponce, G., Vega, L.: On uniqueness properties of solutions of the k-generalized KdV equations. J. Funct. Anal. (to appear) (2006) Fuchssteiner, B., Fokas, A.: Symplectic structures, their backlund transformations and hereditary symmetries. Phys. D 4, 47–66 (1981/1982) Fuchssteiner, B.: Some tricks from the symmetry-toolbox for nonlinear equations: generalization of the Camassa-Holm equation. Physica D 95, 229–243 (1996) Henry, D.: Compactly supported solutions of the Camassa-Holm equation. J. Nonlinear Math. Phys. 12, 342–347 (2005) Himonas, A., Misiołek, G.: The Cauchy problem for an integrable shallow water equation. Differ. Int. Eqs. 14, 821–831 (2001) Himonas, A., Misiołek, G.: High-frequency smooth solutions and well-posedness of the Camassa-Holm equation. Int. Math. Res. Not. 51, 3135–3151 (2005) Li, Y., Olver, P.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Eqs. 162, 27–63 (2000) McKean, H.: Breakdown of the Camassa-Holm equation. Comm. Pure Appl. Math. 57, 416– 418 (2004) McKean, H.: Breakdown of a shallow water equation. Asian J. Math. 2, 767–774 (1998) Misiołek, G.: Classical solutions of the periodic Camassa-Holm equation. Geom. Funct. Anal. 12, 1080–1104 (2002) Molinet, L.: On well-posedness results for the Camassa-Holm equation on the line: A survey. J. Nonlin. Math. Phys. 11, 521–533 (2004) Rodriguez-Blanco, G.: On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal. 46, 309–327 (2001) Zhou, Y.: Infinite propagation speed for a shallow water equation. Preprint, available at http://www.fim.math.ethz.ch/preprints/2005/zhou.pdf, 2005
Communicated by P. Constantin
Commun. Math. Phys. 271, 523–559 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0180-4
Communications in
Mathematical Physics
Low-regularity Schrödinger maps, II: global well-posedness in dimensions d ≥ 3 Alexandru D. Ionescu1, , Carlos E. Kenig2, 1 Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, USA.
E-mail: [email protected]
2 Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514,
USA. E-mail: [email protected] Received: 16 May 2006 / Accepted: 4 August 2006 Published online: 13 February 2007 – © Springer-Verlag 2007
Abstract: In dimensions d ≥ 3, we prove that the Schrödinger map initial-value problem ∂t s = s × x s on Rd × R; s(0) = s0 d/2 is globally well-posed for small data s0 in the critical Besov spaces B˙ Q (Rd ; S2 ), Q ∈ S2 .
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . 2. Notation and Preliminary Lemmas . . . . . . . . 3. Proof of Theorem 1.2 . . . . . . . . . . . . . . . 4. Maximal Function and Local Smoothing Estimates 5. Dyadic Bilinear Estimates, I . . . . . . . . . . . . 6. Dyadic Bilinear Estimates, II . . . . . . . . . . . 7. Dyadic Bilinear Estimates, III . . . . . . . . . . . 8. A Dyadic Trilinear Estimate . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . .
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523 527 533 538 543 547 550 557 559
1. Introduction We consider the Schrödinger map initial-value problem ∂t s = s × x s on Rd × R, s(0) = s0 , The first author was supported in part by an NSF grant and a Packard fellowship.
The second author was supported in part by an NSF grant.
(1.1)
524
A. D. Ionescu, C. E. Kenig
where d ≥ 3 and s : Rd × R → S2 → R3 is a continuous function. The Schrödinger map equation has a rich geometric structure and arises naturally in a number of different ways; we refer the reader to [15] or [9] for details. In this paper, which is a continuation of our earlier work [5], we prove a global well-posedness result for the initial-value d/2 problem (1.1) for small data in the critical Besov spaces B˙ Q (Rd ; S2 ) defined below. Let C(Rd ) = { f : Rd → C : f is continuous and bounded}. Let Z+ = {0, 1, . . .}. For σ = d/2 + σ , σ ∈ Z+ , we define the Besov-type spaces1 N
B˙ σ = B˙ σ (Rd )={φ ∈ C(Rd ) : φ = lim
N →∞
φ B˙ σ =
max 2
(d) −1 F(d) φ · ηk (ξ ) in L ∞ (Rd ) and F(d)
k=−N
dk/2
(d) , 2σ k · F(d) (φ)(ξ ) · ηk (ξ ) L 2 < ∞},
k∈Z −1 where F(d) and F(d) denote the Fourier transform and the inverse Fourier transform (d)
(d)
on S (Rd ), and {ηk }k∈Z is a smooth partition of 1 with ηk supported in the set {ξ ∈ Rd : |ξ | ∈ 2k−1 , 2k+1 } (see Sect. 2 for precise definitions). Let B˙ ∞ = B˙ ∞ (Rd ) =
∞
B˙ d/2+σ (Rd ) with the induced metric.
σ =0
For σ as above or σ = ∞, and Q = (Q 1 , Q 2 , Q 3 ) ∈ S2 we define the complete metric spaces σ Rd ; S2 = { f : Rd → R3 : | f (x)| ≡ 1 and fl − Q l ∈ B˙ σ for l = 1, 2, 3}, (1.2) B˙ Q with the induced distance σ ( f, g) dQ
=
3
fl − gl B˙ σ .
(1.3)
l=1 ∞ (Rd ; S2 ). For any metric space X , x ∈ X , For Q ∈ S2 let f Q (x) ≡ Q, f Q ∈ B˙ Q and r > 0, let B X (x, r ) denote the open ball {y ∈ X : d(x, y) < r }. Our main theorem concerns global well-posedness of the initial-value problem (1.1) for small data d/2 s0 ∈ B˙ Q (Rd ; S2 ), Q ∈ S2 .
Theorem 1.1. (a) Assume d ≥ 3 and Q ∈ S2 . Then there are numbers 0 ≤ 0 ∈ (0, ∞) ∞ (Rd ; S2 ) ∩ B with the property that for any s0 ∈ B˙ Q ( f , ) there is a unique d/2 B˙ (Rd ;S2 ) Q 0 Q
solution
∞ ∞ s = SQ (s0 ) ∈ C(R : B˙ Q (Rd ; S2 ) ∩ B B˙ d/2 (Rd ;S2 ) ( f Q , 0 )) Q
of the initial-value problem (1.1). 1 For σ > d/2 one may replace the spaces B˙ σ with B˙ d/2 ∩ H˙ σ throughout the paper (only minor changes would be needed in Sect. 3), where H˙ σ are the usual homogeneous Sobolev spaces. One may also allow noninteger values of σ = σ − d/2 (in Theorem 1.1 (c) and Theorem 1.2 (c)) at the expense of some technical changes in Sect. 3. We use this definition of the spaces B˙ σ to measure higher smoothness of functions mostly for simplicity.
Global Well-Posedness of Schrödinger Maps
525
∞ (Rd ; S2 ) ∩ B (b) For any s0 , s0 ∈ B˙ Q ( f , ), d/2 B˙ (Rd ;S2 ) Q 0 Q
∞ ∞ sup d Q (S Q (s0 )(t), S Q (s0 )(t)) ≤ C · d Q (s0 , s0 ). d/2
d/2
(1.4)
t∈R
∞ (s ) extends uniquely to a Lipschitz mapping Thus the mapping s0 → S Q 0 d/2
S Q : B B˙ d/2 (Rd ;S2 ) ( f Q , 0 ) → C(R : B B˙ d/2 (Rd ;S2 ) ( f Q , 0 )). Q
Q
σ (Rd ; S2 ) ∩ B (c) In addition, for any σ = d/2 + σ , σ ∈ Z+ , and s0 ∈ B˙ Q d/2 B˙ (Rd ;S2 ) Q
( f Q , 0 ),
d/2 σ S Q (s0 ) ∈ C(R : B˙ Q (Rd ; S2 ) ∩ B B˙ d/2 (Rd ;S2 ) ( f Q , 0 )), Q
and, for any T ≥ 0, σ σ sup d Q (S Q (s0 )(t), f Q ) ≤ C(T, σ, d Q (s0 , f Q )). d/2
|t|≤T
(1.5)
Theorem 1.1 appears to be the first low-regularity global well-posedness result for the Schrödinger map initial-value problem. Its direct analogue in the setting of wave maps is the work of Tataru [23] (see also [10, 12, 24, 21, 22, 11, 18], and [25] for other local and global well-posedness theorems for wave maps). The initial-value problem (1.1) has been studied extensively (also in the case in which the sphere S2 is replaced by more general targets). It is known that sufficiently smooth solutions exist locally in time, even for large data (see, for example, [19, 2, 3, 13, 9] and the references therein). Such theorems for (local in time) smooth solutions are proved using variants of the energy method. For low-regularity data, the energy method cannot be applied, and the initial-value problem (1.1) has been studied indirectly using the “modified Schrödinger map equation” (see, for example, [15, 16, 8, and 7]). While existence and uniqueness theorems for this modified Schrödinger map equation in certain low-regularity spaces are known (at least in dimension d = 2), it is not clear whether such theorems can be transfered to the original Schrödinger map initial-value problem (see, however, [17]). In [5], the authors proved local well-posedness of the initial-value problem (1.1) for small data in the natural Sobolev spaces H Qσ (Rd ; S2 ), σ > (d + 1)/2. This was achieved by reducing the initial-value problem (1.1) to the nonlinear Schrödinger equation (1.7) below2 , and by analyzing the resulting equation using a direct perturbative argument. We follow the same approach in this paper. Slightly later and independently, Bejenaru [1] proved local well-posedness of the initial-value problem (1.7) for small data in the Sobolev spaces H σ , for σ in the full subcritical range σ > d/2. The resolution spaces used by Bejenaru [1] appear to be very different from the spaces used by us in [5] and in this paper. To reach the full subcritical range, Bejenaru observed, apparently for the first time in the setting of Schrödinger maps, that the gradient part of the nonlinearity in (1.7) has a certain null structure (similar to the null structure of the wave maps observed by Klainerman). We exploit this null structure through the identity (3.14). 2 Using the stereographic projection, such a reduction is possible due to the fact that the solutions take values only in a small part of the sphere; the models (1.1) and (1.7) are certainly not equivalent without such a smallness assumption.
526
A. D. Ionescu, C. E. Kenig
Theorem 1.1 can be restated using the stereographic projection. By rotation invariance, we may assume Q = (0, 0, 1). (1.6) Assume > 0 is small enough. For f = ( f 1 , f 2 , f 3 ) ∈ B B˙ d/2 (Rd ;S2 ) ( f Q , ) let Q
g = L( f ) =
f1 + i f2 . 1 + f3
Clearly, L( f ) : Rd → C is continuous and takes values in a small neighborhood of 0. For g ∈ B B˙ d/2 (0, ) we define L(g) = f = ( f1 , f2 , f3 ) =
g + g (−i)(g − g) 1 − gg , , . 1 + gg 1 + gg 1 + gg
Clearly,
L(g) : Rd → S2 is continuous and takes values in a small neighborhood of Q. A direct computation shows that u : Rd → {z ∈ C : |z| ≤ 1} is a smooth solution of the equation d 2u (i∂t + x )u = (∂x j u)2 on Rd × R, 1 + uu j=1
if and only if the function s : Rd → S2 ∩ {(x1 , x2 , x3 ) ∈ R3 : x3 ≥ 0}, s(t) =
L(u(t)), is a smooth solution of the Schrödinger map equation ∂t s = s × x s on Rd × R. Since B˙ σ , σ ∈ [d/2, ∞) are Banach algebras, in the sense that ||uv|| B˙ σ ≤ Cσ (||u|| B˙ σ ||v|| B˙ d/2 + ||u|| B˙ d/2 ||v|| B˙ σ ) for any σ ∈ [d/2, ∞) and u, v ∈ B˙ σ , for Theorem 1.1 it suffices to prove Theorem 1.2 below. Theorem 1.2. (a) Assume d ≥ 3. Then there are numbers 1 ≤ 1 ∈ (0, ∞) with the property that for any φ ∈ B˙ ∞ ∩ B B˙ d/2 (0, 1 ) there is a unique solution u=
S ∞ (φ) ∈ C(R : B˙ ∞ ∩ B B˙ d/2 (0, 1 )) of the initial-value problem (i∂t + x )u = 2u(1 + uu)−1 dj=1 (∂x j u)2 on Rd × R; u(0) = φ.
(1.7)
(b) For any φ, φ ∈ B˙ ∞ ∩ B B˙ d/2 (0, 1 ), sup
S ∞ (φ)(t) −
S ∞ (φ )(t) B˙ d/2 ≤ Cφ − φ B˙ d/2 . t∈R
Thus the mapping φ → S ∞ (φ) extends uniquely to a Lipschitz mapping
S d/2 : B B˙ d/2 (0, 1 ) → C(R : B B˙ d/2 (0, 1 )).
(1.8)
Global Well-Posedness of Schrödinger Maps
527
(c) In addition, for any σ = d/2 + σ , σ ∈ Z+ , and φ ∈ B˙ σ ∩ B B˙ d/2 (0, 1 ),
S d/2 (φ) ∈ C(R : B˙ σ ∩ B B˙ d/2 (0, 1 )), and, for any T ≥ 0, sup
S d/2 (φ)(t) B˙ σ ≤ C(T, σ, s0 B˙ σ ).
|t|≤T
(1.9)
By scale invariance, it suffices to construct the solution u = S ∞ (φ) on the time interval [−1, 1] and prove the bounds (1.8) and (1.9) for t ∈ [−1, 1]. The resolution spaces we construct in Sect. 2 are adapted to this restriction in time. This restriction creates a somewhat artificial distinction between frequencies that are ≤ 1 and frequencies that are ≥ 1. The benefit of this time restriction, however, is that the denominators in formulas such as (2.13) and (2.31) (and in many other places) do not vanish, and all of our integrals are absolutely convergent (in particular, changes of order of integration are justified). The direct use of scale-invariant spaces would lead to denominators such as τ + |ξ |2 , and the integrals containing such denominators would not converge absolutely. The rest of the paper is organized as follows: in Sect. 2 we define our main (dyadic) resolution spaces and establish some of their basic properties. These spaces are minor modifications of the resolution spaces already used by the authors in [5] (see also [4] for the one-dimensional analogues of these resolution spaces). In Sect. 3 we give the main argument that proves Theorem 1.2; the main ingredients in our perturbative argument are the four nonlinear estimates (3.8), (3.9), (3.10), and (3.12). In the remaining sections we prove these four nonlinear estimates. The key ingredients in these proofs are the scale-invariant L 2,∞ (maximal function) estimate in Lemma 4.1 and the scale-invariant e (local smoothing) estimate in Lemma 4.2. These two estimates have been used L e∞,2 before by the authors in [4] and [5]. The maximal function bound fails (logarithmically) in dimension d = 2, which is the main reason why we need to assume d ≥ 3. 2. Notation and Preliminary Lemmas In this section we summarize most of the notation, define our main normed spaces,3 and prove some of their basic properties. Let F and F −1 denote the Fourier transform and −1 the inverse Fourier transform operators on S (Rd+1 ). For l = 1, . . . , d let F(l) and F(l) denote the Fourier transform and the inverse Fourier transform operators on S (Rl ). We fix η0 : R → [0, 1] a smooth even function supported in the set {µ ∈ R : |µ| ≤ 8/5} and equal to 1 in the set {µ ∈ R : |µ| ≤ 5/4}. Then we define η j : R → [0, 1], j = 1, 2, . . . , (2.1) η j (µ) = η0 (µ/2 j ) − η0 (µ/2 j−1 ), (d)
and ηk
: Rd → [0, 1], k ∈ Z, (d)
ηk (ξ ) = η0 (|ξ |/2k ) − η0 (|ξ |/2k−1 ).
(2.2)
3 It is likely that only minor changes would be needed to guarantee that all of our normed spaces are in fact Banach spaces. We do not need this, however, since the limiting argument in the construction of solutions takes place in the Banach space C(R : B˙ d/2 ).
528
A. D. Ionescu, C. E. Kenig
For j1 , j2 ∈ Z, we also define η[ j1 , j2 ] = j1 ≤ j ≤ j2 η j (with the conventions η[ j1 , j2 ] ≡ 0 if j1 > j2 and η j ≡ 0 if j ≤ −1), η±j (µ) = η j (µ) · 1[0,∞) (±µ), η[±j1 , j2 ] (µ) = η[ j1 , j2 ] (µ) · 1[0,∞) (±µ), η≤ j = η[0, j] , η≥ j = 1 − η≤ j−1 . (d) For k ∈ Z, let Ik = {ξ ∈ Rd : |ξ | ∈ 2k−1 , 2k+1 }; for j ∈ Z+ , let I j = {µ ∈ R : |µ| ∈ 2 j−1 , 2 j+1 } if j ≥ 1 and I j = [−2, 2] if j = 0. For k ∈ Z and j ∈ Z+ , let (d)
Dk, j = {(ξ, τ ) ∈ Rd × R : ξ ∈ Ik
and |τ + |ξ |2 | ∈ I j } and Dk,≤ j =
Dk, j .
0≤ j ≤ j
For k ∈ Z we define first the normed spaces X k = { f ∈ L 2 (Rd × R) : f supported in Ik(d) × R and f Xk =
∞
2 j/2 βk, j η j (τ + |ξ |2 ) · f L 2 < ∞},
(2.3)
j=0
where, with k+ = max(k, 0), βk, j = 1 + 2( j−2k+ )/2 .
(2.4)
The spaces X k are not sufficient for our estimates, due to various logarithmic divergences. For any vector e ∈ Sd−1 , let Pe = {ξ ∈ Rd : ξ · e = 0} with the induced Euclidean measure. For p, q ∈ [1, ∞] we define the normed spaces p,q p,q L e = L e (Rd × R), p,q
Le
= { f ∈ L 2 (Rd × R) : p,q f L e = R
Pe ×R
| f (r e + v, t)|q dvdt
p/q dr
1/ p
< ∞}.
(2.5)
For k ≥ 100, j ∈ Z+ and k ∈ [1, k + 1] ∩ Z, let
e,k e,k Dk, j = {(ξ, τ ) ∈ Dk, j : ξ · e ∈ Ik ∩ [0, ∞)} and Dk,≤ j =
0≤ j ≤ j
e,k Dk, j .
For k ≥ 100, k ∈ [1, k + 1] ∩ Z, and e ∈ Sd−1 , we define the normed spaces
e,k and Yke,k = { f ∈ L 2 (Rd × R) : f supported in Dk,≤2k+10
f Y e,k = 2−k /2 γk,k · F −1 [(τ + |ξ |2 + i) · f ] L 1,2 < ∞}, e
k
where
γk,k = 22d(k−k ) .
(2.6)
Global Well-Posedness of Schrödinger Maps
529
For k ≥ 100 and e ∈ Sd−1 , we define the normed spaces
e,k Yke = { f ∈ L 2 (Rd × R) : f supported in ∪k+1 k =1 Dk,≤2k+10 and
f Yke =
k+1 k =1
f · ηk+ (ξ · e)Y e,k < ∞}.
(2.7)
k
For simplicity of notation, we also define Yke = {0} for k ≤ 99. We fix L = L(d) large and e1 , . . . , e L ∈ Sd−1 , el = el if l = l , such that for any e ∈ Sd−1 there is l ∈ {1, . . . , L} such that |e − el | ≤ 2−100 .
(2.8)
We assume in addition that if e ∈ {e1 , . . . , e L }, then −e ∈ {e1 , . . . , e L }. For k ∈ Z we define the normed spaces (2.9) Z k = X k + Yke1 + · · · + Yke L . The spaces Z k are our main normed spaces. −1 (m) ∈ We prove now several estimates. In view of the definitions, if m ∈ L ∞ (Rd ), F(d) L 1 (Rd ), and f ∈ Z k , then m(ξ ) · f ∈ Z k and −1 (m)|| L 1 (Rd ) · || f || Z k . ||m(ξ ) · f || Z k ≤ C||F(d)
(2.10)
We show first that the spaces Z k are logarithmic modifications of the spaces X k . Lemma 2.1. If k ∈ Z, j ∈ Z+ and f ∈ Z k , then f · η j (τ + |ξ |2 ) X k ≤ C f Z k .
(2.11)
Proof of Lemma 2.1. Clearly, we may assume k ≥ 100 and f = f e,k ∈ Yke , for some e ∈ {e1 , . . . , e L } and k ∈ [1, k + 1] ∩ Z. Let
h(x, t) = 2−k /2 F −1 [(τ + |ξ |2 + i) · f e,k ](x, t). Thus
(2.12)
f e,k (ξ, τ ) = 1 D e,k
k,≤2k+10
(ξ, τ ) ·
2k /2 F(h)(ξ, τ ). τ + |ξ |2 + i
(2.13)
In view of the definitions, for (2.11) it suffices to prove the stronger bound,
2k /2 2− j/2 1 D e,k (ξ, τ ) · F(h) L 2 ≤ Ch L 1,2 , ξ,τ
k, j
e
(2.14)
for any j ≤ 2k + 11. We write ξ = ξ1 e + ξ , x = x1 e + x , x1 , ξ1 ∈ R, x , ξ ∈ Pe . Let
h (x1 , ξ , τ ) = h(x1 e + x , t)e−i(x ·ξ +tτ ) d x dt. Pe ×R
By the Plancherel theorem, h L 1,2 = Ch L 1 e
2 x1 L ξ ,τ
.
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Thus, for (2.14), it suffices to prove that
(k − j)/2 2 h (x1 , ξ , τ )e−i x1 ξ1 d x1 1 D e,k (ξ, τ )· R
k, j
≤ C min(1, 2
k − j/2
L 2ξ
1 ,ξ ,τ
) · h L 1
2 x1 L ξ ,τ
.
(2.15)
This follows easily since for any (ξ , τ ) ∈ Pe × R the measure of the set {ξ1 : |ξ1 | ≈ 2k , |τ + ξ12 + |ξ |2 | ≤ 2 j+1 } is bounded by C min(2k , 2 j−k ).
The implicit gain of γk,k in the bound (2.15) shows that Yke,k → X k if k ≤ 9k/10. Let Tk = [9k/10, k + 1] ∩ Z. In view of the definitions, if f ∈ Z k then we can write ⎧ L ⎪ el ,k ⎪ f = g + f el ,k , g j supported in Dk, j , f el ,k supported in Dk,≤2k+10 ; ⎪ j ⎨ k ∈Tk l=1
j∈Z+
L j/2 ⎪ ⎪ ⎪ 2 βk, j ||g j || L 2 + || f el ,k || el ,k ≤ 2 f Z k . ⎩ Y k ∈Tk l=1
j∈Z+
k
(2.16) This is our main atomic decomposition of functions in Z k . In addition, the bound (2.15) shows that if k ∈ Z and f is supported in Ik(d) × R ∩ {(ξ, τ ) : ξ · e ≥ 2k−40 } for some e ∈ Sd−1 , then, for any j ≥ 0, || f · η j (τ + |ξ |2 )|| Z k ≤ C2−k/2 ||F −1 [(τ + |ξ |2 + i) · f ]|| L 1,2 . e
(2.17)
2 ∞ We prove now L ∞ t L x and L x,t estimates.
Lemma 2.2. If k ∈ Z, t ∈ R, and f ∈ Z k , then sup F −1 ( f )(., t) L 2x ≤ C f Z k .
(2.18)
≤ C2dk/2 f Z k . F −1 ( f ) L ∞ x,t
(2.19)
t∈R
Thus
Proof of Lemma 2.2. By the Plancherel theorem it suffices to prove that f (ξ, τ )eitτ dτ 2 ≤ C f Z k .
(2.20)
We use the representation (2.16). Assume first that f = g j . Then g j (ξ, τ )eitτ dτ 2 ≤ C||g j (ξ, τ )|| L 2 L 1 ≤ C2 j/2 ||g j || L 2 ,
(2.21)
R
R
Lξ
Lξ
ξ
τ
ξ,τ
which proves (2.20) in this case. This inequality also shows that F −1 (g j ) L ∞ ≤ C2dk/2 2 j/2 g j L 2 .
(2.22)
Global Well-Posedness of Schrödinger Maps
531
Assume now that k ≥ 100 and f = f e,k ∈ Yke,k , e ∈ {e1 , . . . , e L }, k ∈ Tk . We have to prove that f e,k (ξ, τ )eitτ dτ 2 ≤ C f e,k Y e,k . (2.23) R
Lξ
k
We define the function h as in (2.12). In view of (2.15),
η≥2k −49 (τ + |ξ |2 ) · f e,k X k ≤ C f e,k Y e,k .
(2.24)
k
Since the bound (2.23) was already proved for functions in X k (see (2.21)), for (2.23) it suffices to prove the stronger bound f e,k (ξ, τ ) · η≤2k −50 (τ + |ξ |2 ) · eitτ dτ 2 ≤ C f e,k Y e,k . (2.25) R
Lξ
k
We use the formula (2.13), and write ξ = ξ1 e + ξ , ξ1 ∈ R, ξ ∈ Pe . For (2.25) it suffices to prove that
η≤2k −50 (τ + |ξ |2 ) k /2 + itτ 2 η[k −1,k +1] (ξ1 ) · F(h)(ξ, τ ) · e dτ , 2 ≤ C||h|| L 1,2 2+i e Lξ τ + |ξ | R (2.26) for any t ∈ R. As in Lemma 2.1, for (2.26) it suffices to prove that
η≤2k −50 (τ + |ξ |2 ) k /2 + itτ 2 η[k −1,k +1] (ξ1 ) · h (ξ , τ ) · e dτ 2 ≤ C||h || L 2 , 2 ξ ,τ Lξ τ + |ξ | + i R (2.27) for any t ∈ R and h ∈ L 2 (Pe × R). We may assume t = 0 and let
η≤2k −50 (τ + µ) h (ξ , τ ) dτ. h (ξ , µ) = τ +µ+i R In view of the boundedness of the Hilbert transform on L 2 (R), ||h || L 2
ξ ,µ
Thus, for (2.27), it suffices to prove that
+ 2 2 2k /2 ||η[k −1,k +1] (ξ1 ) · h (ξ , ξ1 + |ξ | )|| L 2
ξ ,ξ1
which follows easily by changes of variables.
≤ C||h || L 2 . ξ ,τ
≤ C||h || L 2 , ξ ,µ
We consider now the action of multipliers of the form m ≤ j (τ + |ξ |2 ). Lemma 2.3. Assume m : R → C is a smooth function supported in the interval [−2, 2] and let m ≤ j (µ) = m(µ/2 j ), j ∈ Z+ . If k ≥ 100, k ∈ Tk , j ∈ [0, 2k − 80] ∩ Z, (d) e ∈ Sd−1 , and f is supported in Ik × R, then ||F −1 [m ≤ j (τ + |ξ |2 ) · f · ηk+ (ξ · e)]|| L 1,2 ≤ C||F −1 ( f )|| L 1,2 . e
e
(2.28)
Thus, if k ∈ Z, j ∈ Z+ , and f ∈ Z k , then ||η≤ j (τ + |ξ |2 ) · f || Z k ≤ C|| f || Z k .
(2.29)
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Proof. We write as before ξ = ξ1 e + ξ , x = x1 e + x , x1 , ξ1 ∈ R, x , ξ ∈ Pe . Using the Plancherel theorem, it suffices to prove that
2 (τ + |ξ | ) ei x1 ξ1 m ≤ j (τ + |ξ |2 + ξ12 ) · ηk+ (ξ1 ) dξ1 η≤2k+10 R
L 1x L ∞ ξ ,τ
≤ C. (2.30)
1
In view of the restriction j ≤ 2k − 80, we may assume that the supremum in (ξ , τ ) in (2.30) is taken over the set {(ξ , τ ) : −τ − |ξ |2 ∈ [22k −70 , 22k +10 ]}. Let M = M(ξ , τ ) = (−τ − |ξ |2 )1/2 , M ≈ 2k . By integration by parts ei x1 ξ1 m ≤ j ξ12 − M 2 · ηk+ (ξ1 ) dξ1 ≤ C R
2 j−k 2 , 1 + 2 j−k x1
if M ≈ 2k , which gives (2.30). The inequality (2.29) follows from (2.28) and (2.15).
We conclude this section with a representation formula for functions in Yke,k .
Lemma 2.4. If k ≥ 100, k ∈ Tk , e ∈ {e1 , . . . , e L }, and f e,k ∈ Yke,k , then we can write + f e,k ξ1 e + ξ , τ = η[k −1,k +1] (M)
2−k /2 · η≤k −80 (ξ1 − M) × e−i y1 ξ1 h(y1 , ξ , τ ) dy1 + g, ξ1 − M + i/2k R
(2.31)
e , where ξ1 , τ ∈ R, ξ ∈ Pe , h is supported in R × Sk,k
e 2 2k −10 2k +10 and |ξ | ≤ 2k+1 }, (2.32) ,2 Sk,k = {(ξ , τ ) ∈ Pe × R : −τ − |ξ | ∈ 2 M = M(ξ , τ ) = (−τ − |ξ |2 )1/2 , and ||g|| X k + ||h|| L 1
2 y1 L ξ ,τ
≤ (C/γk,k ) · || f e,k ||Y e,k .
(2.33)
k
Proof of Lemma 2.4. Let
h (x, t) = 2−k /2 F −1 [(τ + |ξ |2 + i) · f e,k ](x, t), so ⎧ ⎨ f e,k ξ e + ξ , τ = η+ 1 [k −1,k +1] (ξ1 ) · 1[0,2k+1 ] (|ξ |) · ⎩h L 1,2 ≤ (C/γk,k ) · || f e,k || e,k . e
Let
Yk
2k /2 F(h )(ξ1 e τ +|ξ |2 +i
+ ξ , τ ); (2.34)
Global Well-Posedness of Schrödinger Maps
h (y1 , ξ , τ ) =
533
Pe ×R
h (y1 e + y , t)e−i(y ·ξ +tτ ) dy dt.
As in Lemma 2.1 (see (2.15)),
||η≥2k −79 (τ + ξ 2 ) · f e,k || X k ≤ (C/γk,k ) · || f e,k ||Y e,k . k
Thus it remains to write η≤2k −80 (τ + ξ 2 ) · f η≤2k −80 τ + ξ 2 · f e,k ξ1 e + ξ , τ
+ = 2k /2 · η[k −1,k +1] (ξ1 ) · 1[0,2k+1 ] (|ξ |) ·
e,k
as in (2.31). Using (2.34),
η≤2k −80 (τ + ξ 2 ) τ + |ξ |2 + i
R
h (y1 , ξ , τ )e−i y1 ξ1 dy1 . (2.35)
Clearly, we may assume that approximate
h
e . Let is supported in R × Sk,k
+ η[k −1,k +1] (ξ1 ) · 1[0,2k+1 ] (|ξ |) ·
M = (−τ
− |ξ |2 )1/2
and
η≤2k −80 (ξ12 − M 2 )
ξ12 − M 2 + i η≤k −80 (ξ1 − M) 1 + + E(ξ1 , ξ , τ ), (2.36) = η[k −1,k +1] (M) · 1[0,2k+1 ] (|ξ |) · · 2M ξ1 − M + i/2k where, with µ = |ξ12 − M 2 | + 1 = τ + |ξ |2 + 1, η≤2k (µ) µ 1 + |E(ξ1 , ξ , τ )| ≤ C1[0,2k+1 ] (|ξ |)η[k · 2k + . −5,k +5] (ξ1 ) · µ µ 2 We substitute (2.36) into (2.35) and notice that the error term corresponding to E(ξ1 , ξ , τ ) can be bounded in X k (as in Lemma 2.1). The main term in the right-hand side of (2.36) leads to the representation (2.31), with
h = 1[0,2k+1 ] (|ξ |) · 2k · (2M)−1 · h .
3. Proof of Theorem 1.2 For σ = d/2 + σ , σ ∈ Z+ , we define the normed spaces F σ = {u ∈ C(R : B˙ ∞ ) : u F σ = max 2dk/2 , 2σ k · ηk(d) (ξ ) · F(u) Z k < ∞}, k∈Z
(3.1)
and N σ = {u ∈ C(R : B˙ ∞ ) : (d) u N σ = max 2dk/2 , 2σ k · ηk (ξ ) · (τ + |ξ |2 + i)−1 · F(u) Z k < ∞}.
(3.2)
k∈Z
For φ ∈ B˙ ∞ let W (t)φ ∈ C(R : B˙ ∞ ) denote the solution of the free Schrödinger evolution. Assume ψ : R → [0, 1] is an even smooth function supported in the interval [−8/5, 8/5] and equal to 1 in the interval [−5/4, 5/4]. We prove first two linear estimates.
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Lemma 3.1. If σ = d/2 + σ , σ ∈ Z+ , and φ ∈ B˙ ∞ , then ψ(t) · [W (t)φ] ∈ F σ and ψ(t) · [W (t)φ] F σ ≤ Cφ B˙ σ . Proof of Lemma 3.1. A straightforward computation shows that (τ + |ξ |2 ), F[ψ(t) · (W (t)φ)](ξ, τ ) = F(d) (φ)(ξ ) · ψ = F(1) (ψ). Then, directly from the definitions, where, for simplicity of notation, ψ (d) (τ + |ξ |2 ) Z k ψ(t) · [W (t)φ] F σ = max 2dk/2 , 2σ k ηk (ξ ) · F(d) (φ)(ξ ) · ψ k∈Z
≤
(τ + |ξ |2 ) X k max 2dk/2 , 2σ k ηk(d) (ξ ) · F(d) (φ)(ξ ) · ψ
k∈Z
≤C
max 2dk/2 , 2σ k ηk(d) (ξ ) · F(d) (φ)(ξ ) L 2
k∈Z
≤ Cφ B˙ σ , as desired.
Lemma 3.2. If σ = d/2 + σ , σ ∈ Z+ , and u ∈ N σ , then
t W (t − s)(u(s)) ds σ ≤ C||u|| N σ . ψ(t) · F
0
Proof of Lemma 3.2. A straightforward computation shows that
t F ψ(t)· W (t − s)(u(s))ds (ξ, τ ) = 0
c
R
F(u)(ξ, τ )
(τ − τ ) − ψ (τ + |ξ |2 ) ψ dτ . τ + |ξ |2
For k ∈ Z, let (d)
f k (ξ, τ ) = F(u)(ξ, τ ) · ηk (ξ ) · (τ + |ξ |2 + i)−1 . For f ∈ Z k , let
T ( f )(ξ, τ ) =
R
f (ξ, τ )
(τ − τ ) − ψ (τ + |ξ |2 ) ψ (τ + |ξ |2 + i) dτ . τ + |ξ |2
(3.3)
In view of the definitions, it suffices to prove that ||T || Z k →Z k ≤ C uniformly in k ∈ Z.
(3.4)
To prove (3.4) we use the representation (2.16). Assume first that f = g j is supported in Dk, j . Let g #j (ξ, µ ) = g j (ξ, µ − |ξ |2 ) and [T (g)]# (ξ, µ) = T (g)(ξ, µ − |ξ |2 ). Then,
[T (g)]# (ξ, µ) =
R
g #j (ξ, µ )
(µ − µ ) − ψ (µ) ψ (µ + i) dµ . µ
(3.5)
Global Well-Posedness of Schrödinger Maps
535
We use the elementary bound ψ (µ) (µ − µ ) − ψ (µ + i) ≤ C[(1 + |µ|)−4 + (1 + |µ − µ |)−4 ]. µ Then, using (3.5),
1/2 |g #j (ξ, µ )|2 dµ |T (g)# (ξ, µ)| ≤ C(1 + |µ|)−4 · 2 j/2
R + C1[−2 j+10 ,2 j+10 ] (µ) |g #j (ξ, µ )|(1 + |µ − µ |)−4 dµ . R
It follows from the definition of the spaces X k that ||T || X k →X k ≤ C uniformly in k ∈ Z+ ,
(3.6)
as desired. Assume now that f = f e ∈ Yke , k ≥ 100, e ∈ {e1 , . . . , e L }. We write f e (ξ, τ ) =
τ + |ξ |2 i f e (ξ, τ ) + f e (ξ, τ ). 2 τ + |ξ | + i τ + |ξ |2 + i
Using Lemma 2.1, ||i(τ + |ξ |2 + i)−1 f e (ξ, τ )|| X k ≤ C|| f e ||Yke . In view of (3.3) and (3.6), for (3.4) it suffices to prove that
e 2 ψ (τ + |ξ | f (ξ, τ ) ψ (τ − τ ) dτ + ) f e (ξ, τ ) dτ ≤ C|| f e ||Yke . R
Zk
R
Xk
(3.7) The bound for the first term in the left-hand side of (3.7) follows easily from the definition. The bound for the second term in the left-hand side of (3.7) follows from (2.23) with t = 0. We prove now several nonlinear estimates. The main ingredients are the dyadic estimates in Lemma 5.2, Lemma 6.1, Lemma 7.1, and Lemma 8.1. We reproduce these dyadic estimates below: • if k1 , k2 , k ∈ Z, k1 ≤ k2 + 10, f k1 ∈ Z k1 , and f k2 ∈ Z k2 , then (d) f k1 ∗ f k2 ) Z k ≤ C2−|k2 −k|/4 2dk1 /2 f k1 Z k1 · 2dk2 /2 f k2 Z k2 , 2dk/2 ηk (ξ ) · (
(3.8) −1 −1 −1
where F ( f k1 ) ∈ {F ( f k1 ), F ( f k1 )}. • if k1 , k2 , k ∈ Z, k1 ≤ k2 − 10, |k − k2 | ≤ 2, f k1 ∈ Z k1 , and f k2 ∈ Z k2 , then (d) f k1 ∗ [(τ2 + |ξ2 |2 + a · i) f k2 ] Z k 2dk/2 ηk (ξ ) · (τ + |ξ |2 + i)−1
(3.9) ≤ C 2dk1 /2 f k1 Z k1 · 2dk2 /2 f k2 Z k2 , where F −1 (
f k1 ) ∈ {F −1 ( f k1 ), F −1 ( f k1 )} and a ∈ [0, 1]. • if k1 , k2 , k ∈ Z, k1 ≤ k2 + 10, f k1 ∈ Z k1 , and f k2 ∈ Z k2 , then (d) 2dk/2 ηk (ξ ) · (τ + |ξ |2 + i)−1 [(τ1 + |ξ1 |2 + a · i) f k1 ] ∗ f k2 Z k ≤ C2−|k2 −k|/4 · 2dk1 /2 f k1 Z k1 · 2dk2 /2 f k2 Z k2 ,
(3.10)
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where a ∈ [0, 1]. • if k1 , k2 , k3 , k ∈ Z, f k1 ∈ Z k1 , f k2 ∈ Z k2 , f k3 ∈ Z k3 , and min(k, k2 , k3 ) ≤ k1 + 20,
(3.11)
then (d) 2k2 +k3 · 2dk/2 ηk (ξ ) · (τ + |ξ |2 + i)−1 · (
f k1 ∗
f k2 ∗
f k3 ) Z k ≤ C2−| max(k1 ,k2 ,k3 )−k|/4 · 2dk1 /2 f k1 Z k1 · 2dk2 /2 f k2 Z k2 · 2dk3 /2 f k3 Z k3 ,
(3.12) f kl ) ∈ {F −1 ( f kl ), F −1 ( f kl )}, l = 1, 2, 3. where F −1 (
The bounds (3.9) and (3.10) are stated in Lemmas 6.1 and 7.1 with a = 1. They follow, however, for any a ∈ [0, 1] using Lemma 2.1. For σ = d/2 + σ , σ ∈ Z+ , let F
σ
= {u ∈ C(R : B˙ ∞ ) : u ∈ F σ and ||u|| F σ = ||u|| F σ }.
Lemma 3.3. (a) If u, v ∈ F d/2 , then u · v ∈ F d/2 , and ||u · v|| F d/2 ≤ C||u|| F d/2 · ||v|| F d/2 . (b) If u, v ∈ F d/2 + F
d/2
, then u · v ∈ F d/2 + F
d/2
, and
||u · v|| F d/2 +F d/2 ≤ C||u|| F d/2 +F d/2 · ||v|| F d/2 +F d/2 . (c) If u ∈ F d/2 + F
d/2
and v, w ∈ F d/2 , then u · 2
d
l=1 ∂xl v
· ∂xl w ∈ N d/2 and
d u · 2 ∂xl v · ∂xl w N d/2 ≤ C||u|| F d/2 +F d/2 · ||v|| F d/2 · ||w|| F d/2 . l=1 (d)
(d)
Proof of Lemma 3.3. For part (a), let f k = ηk (ξ ) · F(u), gk = ηk (ξ ) · F(v), k ∈ Z. For part (a) it suffices to prove that for any k1 , k2 ∈ Z, (d) 2dk/2 ||ηk (ξ ) · ( f k1 ∗ gk2 )|| Z k ≤ C 2dk1 /2 || f k1 || Z k1 · 2dk2 /2 ||gk2 || Z k2 , k∈Z
which follows easily from (3.8). The proof of part (b) is similar, using only (3.8) and the definitions. For part (c), for k ∈ Z let f k = ηk(d) (ξ ) · F(u), u k = F −1 ( f k ), gk = ηk(d) (ξ ) · F(v), vk = F −1 (gk ), h k = ηk(d) (ξ ) · F(w), wk = F −1 (h k ). It suffices to prove that for any k1 , k2 , k3 ∈ Z, k∈Z
d (d) 2dk/2 ηk (ξ ) · (τ + |ξ |2 + i)−1 · F
u k1 · 2 ∂xl vk2 · ∂xl wk3 Z
k
l=1
≤ C 2dk1 /2 || f k1 || Z k1 · 2dk2 /2 ||gk2 || Z k2 · 2dk3 /2 ||h k3 || Z k3 ,
(3.13)
where
u k1 ∈ {u k1 , u k1 }. If min(k2 , k3 ) ≤ k1 + 20, then (3.13) follows directly from (3.12). Assume that min(k2 , k3 ) ≥ k1 + 20.
Global Well-Posedness of Schrödinger Maps
537
Using (3.12) again, we only need to bound the sum over k ≥ k1 + 20. In this case we use the identity 2
d
∂xl vk2 · ∂xl wk3 = H (vk2 · wk3 ) − wk3 · H vk2 − vk2 · H wk3 ,
(3.14)
l=1
where H = i∂t + x . We estimate the sum over k ≥ k1 + 20 corresponding to the term H (vk2 · wk3 ) using (3.8) and (3.9). We estimate the sums over k ≥ k1 + 20 corresponding to the terms wk3 · H vk2 and vk2 · H wk3 using (3.8), (3.9), and (3.10). The bound (3.13) follows easily. We complete now the proof of Theorem 1.2. For u ∈ C(R : B˙ ∞ ) let N (u) = N0 (u) · 2
d
(∂x j u)2 where N0 (u) = u(1 + uu)−1 .
j=1
It follows from Lemma 3.3 (b) that for u, v ∈ B F d/2 (0, ), 1, N0 (u) − N0 (v) F d/2 +F d/2 ≤ C||u − v|| F d/2 .
(3.15)
In addition, for any l ∈ {1, . . . , d}, σ ∈ 1, 2, . . . , and u ∈ B F d/2 (0, ) ∩ F d/2+σ ,
∂xσl N0 (u) F d/2 +F d/2 ≤ C∂xσl u F d/2 + C(σ , u F d/2+σ −1 ).
(3.16)
Using Lemma 3.3 (c), it follows that for u, v ∈ B F d/2 (0, ), ||N (u) − N (v)|| N d/2 ≤ C 2 ||u − v|| F d/2 . In addition, for any l ∈ {1, . . . , d}, σ ∈ 1, 2, . . . , and u ∈ B F d/2 (0, ) ∩ F
||∂xσl N (u)|| N d/2 ≤ C 2 ||∂xσl u|| F d/2 + C(σ , ||u|| F d/2+σ −1 ).
(3.17) d/2+σ
, (3.18)
The bounds (3.17) and (3.18), together with Lemmas 3.1 and 3.2, and the imbedding F d/2 → C(R : B˙ d/2 ) (which follows from Lemma 2.2) are sufficient to construct the solution u ∈ C([−1, 1] : B˙ ∞ ) in Theorem 1.2 and prove the bounds (1.8) and (1.9) on the time interval [−1, 1]. Indeed, we define recursively u 0 = ψ(t) · W (t)φ; t u n+1 = ψ(t) · W (t)φ + ψ(t) 0 W (t − s)(N (u n (s))) ds for n = 1, 2, . . . . As in [5, Sect. 5], one can combine the ingredients mentioned before to show that if φ ∈ B B˙ d/2 (0, ), 1, and n ∈ Z+ , then ⎧ ⎪ ⎨ u n F d/2 ≤ Cφ B˙ d/2 ; u n+1 − u n F d/2 ≤ 2−n · Cφ B˙ d/2 ; ⎪ ⎩ u n F d/2+σ ≤ C(σ , φ B˙ d/2+σ ) for σ = 1, 2, . . . . The solution u is then obtained by taking the limit of the sequence u n in C(R : B˙ d/2 ) (for the bound (1.8), one needs also u n − u n F d/2 ≤ Cφ − φ B˙ d/2 , if φ, φ ∈ B B˙ d/2 (0, ), 1, and n ∈ Z+ ).
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A. D. Ionescu, C. E. Kenig
The uniqueness of solutions in C(R : B˙ ∞ ∩ B B˙ d/2 (0, 1 )), for 1 sufficiently small, follows from the following simple observation: if u ∈ C([−T, T ] : B˙ ∞ ∩ B B˙ d/2 (0, 1 )) is a solution of Eq. (1.7) then there is δ = δ(||u|| L ∞ ˙ d/2+100 ) with the property that t B ||u|| F d/2 ([t0 −δ,t0 +δ]) ≤ C 1 for any t0 ∈ [−T + δ, T − δ], where, as usual, for any interval I and u ∈ C(I :
(3.19)
B˙ ∞ ),
inf u F d/2 . d in R ×I See, for example, [4, Sect. 10] for such an argument. The uniqueness of solutions then follows from (3.17) and Lemma 3.2. u F d/2 (I ) =
u ≡u
4. Maximal Function and Local Smoothing Estimates In this section we prove two lemmas that will be used extensively in the bilinear and the trilinear estimates in the following four sections. For l = 1, . . . , d and k ∈ Z+ , let (l) k = 2k · Zl . Let χ (1) : R → [0, 1] denote a fixed smooth function supported in the interval [−2/3, 2/3] with the property that χ (1) (ξ − n) ≡ 1 on R. n∈Z
χ
(1)
Let : R → [0, 1] denote a fixed smooth function supported in the interval [−4, 4] and equal to 1 in the interval [−3, 3]. Let χ (l) , χ
(l) : Rl → [0, 1],
(l) (ξ ) = χ
(1) (ξ1 ) · . . . · χ
(1) (ξl ). χ (l) (ξ ) = χ (1) (ξ1 ) · . . . · χ (1) (ξl ) and χ For k ∈ Z+ and n ∈
(l) k
(4.1)
we define
(l)
(l)
k,n (ξ ) = χ
(l) ((ξ − n)/2k ). χk,n (ξ ) = χ (l) ((ξ − n)/2k ) and χ Clearly,
(l)
χk,n ≡ 1
on Rl .
n∈k (d)
(d)
For simplicity of notation, we let χk,n = χk,n and k = k . We start with a maximal function estimate. Lemma 4.1. If d ≥ 3, k ∈ Z, f ∈ Z k , and e ∈ Sd−1 , then ||F −1 ( f )|| L 2,∞ ≤ C2(d−1)k/2 f Z k . e
In addition, if k ≥ 100, k1 ∈ [0, k + 10d] ∩ Z, and f ∈ X k , then ⎡ ⎤1/2 ⎣ ||F −1 (χk1 ,n (ξ ) · f )||2 2,∞ ⎦ ≤ C2(d−1)k1 /2 · 2(k−k1 )/2 f X k . n∈k1
L e
(4.2)
(4.3)
If k ≥ 100, k1 ∈ [0, k + 10d] ∩ Z, and f ∈ Z k , then ⎡ ⎤1/2 ⎣ ||F −1 [χk1 ,n (ξ ) · f · η≤k+k1 (τ + |ξ |2 )]||2 2,∞ ⎦ ≤ C2(d−1)k1 /2 ·2(k−k1 )/2 f Z k . n∈k1
L e
(4.4)
Global Well-Posedness of Schrödinger Maps
539
Proof of Lemma 4.1. We use the representation (2.16) and assume first that f = g j is supported in Dk, j . For (4.2), it suffices to prove that ||F −1 (g j )|| L 2,∞ ≤ C2(d−1)k/2 · 2 j/2 g j L 2 .
(4.5)
e
We define g #j (ξ, µ) = g j (ξ, µ − |ξ |2 ). The left-hand side of (4.5) is dominated by
2 g #j (ξ, µ)ei x·ξ e−it|ξ | dξ 2,∞ dµ. [−2 j+1 ,2 j+1 ]
Rd
L e
Thus, for (4.5) it suffices to prove that 2 h(ξ )ei x·ξ e−it|ξ | dξ Rd
L 2,∞ e
≤ C2(d−1)k/2 · ||h|| L 2 ,
(4.6)
ξ
for any function h supported in the set {ξ ∈ Rd : |ξ | ≤ 2k+1 }. To prove (4.6), using a standard T T ∗ argument, it suffices to show that 2 2 ei x1 ξ1 ei x ·ξ e−it ξ1 +|ξ | Rd−1 ×R × η0 (ξ1 /2k+1 ) · η0 (|ξ |/2k+1 ) dξ1 dξ 1 ∞ ≤ C2(d−1)k .
(4.7)
L x L x ,t 1
By stationary phase (one may also rescale to k = 0), for any x ∈ Rd−1 and x1 ∈ R, 2 ei x ·ξ e−it|ξ | η0 (|ξ |/2k+1 ) dξ ≤ C min 2(d−1)k , |t|−(d−1)/2 , Rd−1
2 ei x1 ·ξ1 e−itξ1 η0 ξ1 /2k+1 dξ1 ≤ C min(2k , |t|−1/2 ).
and
R
In addition, by integration by parts, if |x1 | ≥ 2k+10 |t| then 2 ei x1 ·ξ1 e−itξ1 η0 ξ1 /2k+1 dξ1 ≤ C2k (1 + 2k |x1 |)−2 . R
, x , t)
Let K (x1 denote the function in the left-hand side of (4.7). In view of the three bounds above, sup t∈R, x ∈Rd−1
|K (x1 , x , t)| ≤ C2dk (1 + 2k |x1 |)−2 + C2dk/2 |x1 |−d/2 · 1[2−k ,∞) (|x1 |).
The bound (4.7) follows since d ≥ 3. We prove now the bound (4.3), assuming k ≥ 100 and f = g j is supported in Dk, j . Clearly we may assume k1 ≤ k − 10d, and it suffices to prove that for any n ∈ k1 , ||F −1 (χk1 ,n (ξ ) · g j )|| L 2,∞ ≤ C2(d−1)k1 /2 2(k−k1 )/2 · 2 j/2 g j L 2 . e
By the same argument as before, for (4.8) it suffices to prove that 2 h(ξ )ei x·ξ e−it|ξ +n| dξ 2,∞ ≤ C2(d−1)k1 /2 2(k−k1 )/2 · ||h|| L 2 , Rd
L e
ξ
(4.8)
(4.9)
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A. D. Ionescu, C. E. Kenig
for any function h supported in the set {ξ ∈ Rd : |ξ | ≤ 2k1 } and any vector n ∈ Rd with |n| ≤ 2k+2 . As before, for (4.9), it suffices to show that 2 2 ei x1 ξ1 ei x ·ξ e−it ξ1 +|ξ | e−2itn 1 ξ1 e−2itn ·ξ Rd−1 ×R (4.10) × η0 (ξ1 /2k1 ) · η0 (|ξ |/2k1 ) dξ1 dξ 1 ∞ ≤ C2(d−1)k1 2k−k1 . L x L x ,t 1
x
Rd−1
and x1 ∈ R, By stationary phase, for any ∈ 2 ei x ·ξ e−2itn ·ξ e−it|ξ | η0 (|ξ |/2k1 ) dξ ≤ C min(2(d−1)k1 , |t|−(d−1)/2 ), Rd−1
2 ei x1 ξ1 e−2itn 1 ξ1 e−itξ1 η0 (ξ1 /2k1 ) dξ1 ≤ C min(2k1 , |t|−1/2 ).
and
R
In addition, by integration by parts, if |x1 | ≥ 2k+10 |t| then 2 ei x1 ·ξ1 e−2itn 1 ξ1 e−itξ1 η0 ξ1 /2k1 dξ1 ≤ C2k1 (1 + 2k1 |x1 |)−d/2 . R
Let K 1 (x1 , x , t) denote the function in the left-hand side of (4.10). In view of the three bounds above, sup t∈R, x ∈Rd−1
|K 1 (x1 , x , t)| ≤ C2dk1 ·1[0,2k−2k1 ] (|x1 |)+C2dk/2 |x1 |−d/2 ·1[2k−2k1 ,∞) (|x1 |).
The bound (4.10) follows since d ≥ 3. Assume now that f = f e,k ∈ Yke,k , k ≥ 100, k ∈ Tk , e ∈ {e1 , . . . , e L }. It suffices to prove the stronger bound (4.4), and we may assume k1 ≤ [30d, k − 30d] ∩ Z. We assume first k1 ≤ k . (4.11) We fix an arbitrary orthonormal basis in Pe and use it to define an isomorphism e : Rd−1 → Pe . (d−1)
For k ∈ Z+ let ek = e (k
) ⊆ Pe . For n ∈ ek let
(d−1) (d−1) e e e −1 ◦ (e )−1 and χ
k,n =χ
k,( , χk,n = χ e )−1 (n ) ◦ ( ) k,(e )−1 (n )
(4.12)
e ,χ e χk,n k,n : Pe → [0, 1] (compare with the notation at the beginning of the section). We write ξ = ξ1 e + ξ , x = x1 e + x , ξ1 , x1 ∈ R. ξ , x ∈ Pe . For (4.4) it suffices to prove that ⎡ ⎢ (1) ||F −1 [χke1 ,n (ξ ) · χk1 ,n 1 (ξ1 ) ⎣ (1) 1
n 1 ∈k , n 1 ∈[2k
−10
,2k
+10
e k+10 ] n ∈k1 , |n |≤2
⎤1/2
⎥ × f e,k · η≤k+k1 (τ + |ξ |2 )]||2 2,∞ ⎦
L e
≤ C2(d−1)k1 /2 2(k−k1 )/2 || f e,k ||Y e,k . k
(4.13)
Global Well-Posedness of Schrödinger Maps
541
We use Lemma 2.1 and Lemma 2.4 and notice that we may replace η≤k+k1 ξ12 − M 2 · η≤k −80 (ξ1 − M) by η≤k1 −50 (ξ1 − M), at the expense of C(k − k + 10) error terms in X k . The contributions of these error terms are controlled using (4.3) and the large factor γk,k in (2.6). For simplicity of notation, in the rest of this proof we let n 1 ,n denote the sum over n 1 and n as in (4.13). Using Lemma 2.4, for (4.13) it suffices to prove the stronger bound ⎡ ⎣
n 1 ,n
⎤1/2 2−k /2 · η≤k1 −50 (ξ1 − M) · h (ξ , τ )]||2 2,∞ ⎦ L e ξ1 − M + i/2k
(1) ||F −1 [χke1 ,n (ξ ) · χk1 ,n 1 (ξ1 ) ·
≤ C2(d−1)k1 /2 2(k−k1 )/2 ||h || L 2 ,
(4.14) (1)
(1)
for any h ∈ L 2 (Pe × R). We notice that χk1 ,n 1 (ξ1 ) · η≤k1 −50 (ξ1 − M) = χk1 ,n 1 (ξ1 ) · η≤k1 −50 (ξ1 − ⎡ C⎣
(1) M) · χ
k1 ,n 1 (M).
n 1 ,n
Thus the left-hand side of (4.14) is equal to
R×Pe ×R
(1)
(1)
ei x1 ξ1 ei x ·ξ eitτ χke1 ,n (ξ ) · χk1 ,n 1 (ξ1 ) · χ
k1 ,n 1 (M)
2 2−k /2 · η≤k1 −50 (ξ1 − M) × · h (ξ , τ ) dξ dξ dτ 2,∞ 1 L e ξ1 − M + i/2k
1/2 .
We may disregard the factor χk(1) (ξ ) in the integral above, and integrate the variable 1 ,n 1 1 ξ1 first. The left-hand side of (4.14) is dominated by ⎡ ⎣ C n1
,n
Pe ×R
ei x 1 M e
i x ·ξ
(1)
eitτ χke1 ,n (ξ ) · χ
k1 ,n 1 (M) · 2
−k /2
⎤1/2 2 h (ξ , τ ) dξ dτ 2,∞⎦ . L e
We make the substitutions τ = −µ2 − |ξ |2 (so M = µ) and h (ξ, µ) = 2−k /2 · µ · h (ξ , −µ2 − |ξ |2 ) (so ||h || L 2 ≈ ||h || L 2 ). For (4.14) it suffices to prove that ⎡ ⎣ n1
,n
Pe ×R
ei x 1 µ e
i x ·ξ
e
−it (µ2 +|ξ |2 )
⎤1/2 2 χke1 ,n (ξ ) · χ
k(1) (µ) · h (ξ , µ) dξ dµ 2,∞⎦ 1 ,n 1 L e
≤ C2(d−1)k1 /2 2(k−k1 )/2 ||h || L 2 , which follows from (4.9). If k ≤ k1 then, using the large factor γk,k in (2.6), it suffices to prove the bound (4.4) in the case k1 = k . This was already proved before.
We prove now a local smoothing estimate. Lemma 4.2. If k ∈ Z, e ∈ Sd−1 , l ∈ [−1, 40] ∩ Z, and f ∈ Z k , then F −1 [ f · η1 (ξ · e /2k−l )] L ∞,2 ≤ C2−k/2 f Z k . e
(4.15)
542
A. D. Ionescu, C. E. Kenig p,q
p,q
Proof of Lemma 4.2. Since L e ≡ L −e , for (4.15) it suffices to prove that F −1 [ f · η1+ (ξ · e /2k−l )] L ∞,2 ≤ C2−k/2 f Z k .
(4.16)
e
We use the representation (2.16). Assume first that f = g j . In view of the definitions, it suffices to prove that if j ≥ 0 and g j is supported in Dk, j , then
Rd
ei x·ξ eitτ g j (ξ, τ ) · η1+ (ξ · e /2k−l ) dξ dτ
L ∞,2 e
≤ C2−k/2 2 j/2 g j L 2 .
(4.17)
Let g #j (ξ, µ) = g j (ξ, µ − |ξ |2 ). By Hölder’s inequality, for (4.17) it suffices to prove that 2 ei x·ξ e−it|ξ | h(ξ ) · η1+ (ξ · e /2k−l ) dξ dτ ∞,2 ≤ C2−k/2 h L 2 , (4.18) Rd
L e
which follows easily using the Plancherel theorem and a change of variables. Assume now that k ≥ 100, f = f e,k ∈ Yke,k , k ∈ Tk , e ∈ {e1 , . . . , e L }. The estimates in Lemma 2.4 show that
+ + || f e,k · [η[k−50,k+1] (ξ · e ) − η[k−50,k+1] ((Me + ξ ) · e )]|| X k ≤ C|| f e,k ||Y e,k . k
Since (4.16) was already proved for f ∈ X k , it suffices to show that ei x1 ξ1 ei x ·ξ eitτ f e,k (ξ1 e + ξ , τ ) R×Pe ×R + ((Me + ξ ) · e ) dξ1 dξ dτ ∞,2 ≤ C2−k/2 f e,k Y e,k . ×η[k−50,k+1] L e
(4.19)
k
We use now the representation in Lemma 2.4, and integrate the variable ξ1 first in the left-hand side of (4.19). For (4.19) it suffices to prove the stronger bound + k+2 ei x1 M ei x ·ξ eitτ · η[k ) · 2−k /2 h (ξ , τ ) −1,k +1] (M) · η0 (|ξ |/2 Pe ×R + ((Me + ξ ) · e ) dξ dτ ∞,2 ≤ C2−k/2 h L 2 , (4.20) ×η[k−50,k+1] L e
ξ,τ
for any h ∈ L 2 (Pe × R). We may make the substitutions τ = −µ2 − |ξ |2 (so M = µ), and h (ξ , µ) = 2−k /2 · µ · h (ξ , −µ2 − |ξ |2 ) (so ||h || L 2 ≈ ||h || L 2 ). For (4.20) it suffices to prove that 2 2 + k+2 ei x1 µ ei x ·ξ e−it (µ +|ξ | ) · η[k ) · h (ξ , µ) −1,k +1] (µ) · η0 (|ξ |/2 Pe ×R + ((µe + ξ ) · e ) dξ dµ ∞,2 ≤ C2−k/2 h L 2 , ×η[k−50,k+1] L e
which follows from (4.18).
ξ,τ
Global Well-Posedness of Schrödinger Maps
543
5. Dyadic Bilinear Estimates, I In this section we prove several dyadic bilinear estimates. We assume in the rest of this section that d ≥ 3. We record first a simple L 2 estimate (see, for example, Lemma 6.1 (a) in [4] for the proof): if k1 , k2 , k ∈ Z, j1 , j2 , j ∈ Z+ , and gk1 , j1 , gk2 ,, j2 are L 2 functions supported in Dk1 , j1 and Dk2 , j2 , then gk1 , j1 ∗
gk2 , j2 )|| L 2 ≤ C2d·min(k1 ,k2 ,k)/2 2min( j1 , j2 , j)/2 ||gk1 , j1 || L 2 · ||gk2 , j2 || L 2 , ||1 Dk, j · (
(5.1) gkl , jl ) ∈ {F −1 (gkl , jl ), F −1 (gkl , jl )}, l = 1, 2. where F −1 (
For any k ∈ Z, j ∈ Z+ , and f k ∈ Z k we let f k,≤ j (ξ, τ ) = f k (ξ, τ ) · η≤ j (τ + |ξ |2 ) and f k,≥ j (ξ, τ ) = f k (ξ, τ ) · η≥ j (τ + |ξ |2 ). (5.2) We will often use the following simple estimate. Lemma 5.1. If k1 , k2 ∈ Z, k1 ≤ k2 + C, j1 , j2 ∈ Z+ , f k1 ∈ Z k1 , and f k2 ∈ Z k2 , then f k2 ,≥ j2 ) L 2
f k1 ,≥ j1 ∗
≤ C(2 j2 /2 + 2(k1 +k2 )/2 )−1 (βk1 , j1 · βk2 , j2 )−1 · (2dk1 /2 f k1 Z k1 ) · ( f k2 Z k2 ),
(5.3)
where F −1 (
f kl ,≥ jl ) ∈ {F −1 ( f kl ,≥ j1 ), F −1 ( f kl ,≥ jl )}, l = 1, 2. Proof of Lemma 5.1. If k2 ≥ 100 then, in view of (2.10), we may assume that × R ∩ {(ξ2 , τ2 ) : |ξ2 − v| ≤ 2k2 −50 } for some v ∈ Ik(d) . (5.4) f k2 is supported in Ik(d) 2 2 Let v = v/|v|. Then, for k2 ≥ 100, using Lemma 4.2 (and (4.17) when j2 ≥ 2k2 ), F −1 (
f k2 ,≥ j2 ) L ∞,2 ≤ C2−k2 /2 βk−1 f k2 Z k2 . 2 , j2 v
(5.5)
Using Lemma 4.1 (and (4.5) when j1 ≥ 2k1 ), F −1 (
f k1 ,≥ j1 ) L 2,∞ ≤ C2(d−1)k1 /2 βk−1 f k1 Z k1 . 1 , j1 v
(5.6)
Using the definition, F −1 (
f k2 ,≥ j2 ) L 2 ≤ C2− j2 /2 βk−1 f k2 Z k2 . 2 , j2
(5.7)
Finally, using Lemma 2.2 (and (2.22) when j1 ≥ 2k1 ), F −1 (
f k1 ,≥ j1 ) L ∞ ≤ C2dk1 /2 βk−1 f k1 Z k1 . 1 , j1
(5.8)
The bound (5.3) follows by using (5.5) and (5.6) when k1 + k2 ≥ j2 , and (5.7) and (5.8) when k1 + k2 ≤ j2 (if k2 ≤ 100 we always use (5.7) and (5.8)). Our next bilinear estimate is the main ingredient in the proof of the algebra properties in Lemma 3.3. Lemma 5.2. If k1 , k2 , k ∈ Z, k1 ≤ k2 + 10, f k1 ∈ Z k1 , and f k2 ∈ Z k2 , then (d)
f k1 ∗ f k2 ) Z k ≤ C2−|k2 −k|/4 (2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ), (5.9) 2dk/2 ηk (ξ ) · (
where F −1 (
f k1 ) ∈ {F −1 ( f k1 ), F −1 ( f k1 )}.
544
A. D. Ionescu, C. E. Kenig
Proof of Lemma 5.2. We may assume k ≤ k2 +20. The bound (5.12) follows easily from (5.1) if k2 ≤ 99 (compare with Case 1 below). Assume k2 ≥ 100. In view of (2.10), we may assume that (d)
(d)
f k2 is supported in Ik2 × R ∩ {(ξ2 , τ2 ) : |ξ2 − v| ≤ 2k2 −50 } for some v ∈ Ik2 . (5.10) With v as above, let v = v/|v| ∈ Sd−1 and
= max (k1 + k2 , 0) + 100. K For e ∈ {e1 , . . . , e L } let ηk,e (d)(ξ ) =
(d)
+ ηk (ξ ) · η[k−10,k+10] (ξ · e) (d) ηk (ξ )
if k ≥ 100; if k < 100.
(5.11)
In view of (2.10), for (5.12) it suffices to prove that for any e ∈ {e1 , . . . , e L }, (d) 2dk/2 ηk,e (ξ ) · (
f k1 ∗ f k2 ) Z k ≤ C2−|k2 −k|/4 2dk1 /2 f k1 Z k1 · 2dk2 /2 f k2 Z k2 . (5.12) Using (5.3) with j1 = j2 = 0, we estimate first (d) f k1 ∗ f k2 ) Z k 2dk/2 η≤ K −1 (τ + |ξ |2 ) · ηk,e (ξ ) · (
(d)
≤ C2dk/2 2 K /2 βk, K ηk,e (ξ ) · (
f k1 ∗ f k2 ) L 2
≤ C2dk/2 2 K /2 (1 + 2( K −2k+ )/2 ) · 2− K /2 · (2dk1 /2 f k1 Z k1 ) · ( f k2 Z k2 )
(5.13)
≤ C2d(k−k2 )/2 · 2k2 −k · (2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ). It remains to estimate (d)
f k1 ∗ f k2 ) Z k 2dk/2 η≥ K (τ + |ξ |2 ) · ηk,e (ξ ) · (
≤ C2−|k2 −k|/4 (2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ).
(5.14)
Using the atomic decomposition (2.16), we analyze several cases. Case 1. f k2 = gk2 , j2 ∈ X k2 , f k1 = gk1 , j1 ∈ X k1 . We have to prove that (d) gk1 , j1 ∗ gk2 , j2 Z k 2dk/2 η≥ K (τ + |ξ |2 ) · ηk,e (ξ ) ·
≤ C2−|k2 −k|/4 (2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ).
(5.15)
− 10. Using (5.1), the left-hand side of (5.15) is We may assume that max ( j1 , j2 ) ≥ K dominated by C2dk/2 2max( j1 , j2 )/2 βk,max( j1 , j2 )
sup
j≤max( j1 , j2 )+C
1 Dk, j · (
gk1 , j1 ∗ gk2 , j2 ) L 2
≤ C2dk/2 2max( j1 , j2 )/2 βk,max( j1 , j2 ) · 2dk1 /2 2min( j1 , j2 )/2 gk1 , j1 L 2 · gk2 , j2 L 2 , which gives (5.15) using the simple inequality (see the definition (2.4)) βk,max( j1 , j2 ) ≤ C2k2 −k βk2 ,max( j1 , j2 ) ≤ C2k2 −k βk1 , j1 βk2 , j2 .
(5.16)
Global Well-Posedness of Schrödinger Maps
545
Case 2. f k2 = gk2 , j2 ∈ X k2 , f k1 ∈ Yke1l , l ∈ {1, . . . , L}. We have to prove that (d) 2dk/2 η≥ K (τ + |ξ |2 ) · ηk,e (ξ ) · (
f k1 ∗ gk2 , j2 ) Z k
≤ C2−|k2 −k|/4 (2dk1 /2 f k1 Y el ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ).
(5.17)
k1
− 10 (otherwise the left-hand In view of the definitions, we may assume that j2 ≥ K side of (5.17) is equal to 0). We estimate the left-hand side of (5.17) by C2dk/2 2 j2 /2 βk, j2
f k1 ∗ gk2 , j2 L 2 ≤ C2dk/2 2 j2 /2 βk, j2 · F −1 (
f k1 ) L ∞ · gk2 , j2 L 2 , which gives (5.17), in view of (2.19).
Case 3. f k2 = f ke2l ,k ∈ Yke2l ,k , k ≤ k1 + 20, f k1 ∈ Yk1l , l, l ∈ {1, . . . , L}. In view of (2.31) and the analysis in Case 1 and Case 2 above, we may assume that
f k1 is supel ,k 2 2k +50 1 } and f k2 is supported in the set ported in the set {(ξ1 , τ1 ) : |τ1 + |ξ1 | | ≤ 2 e
{(ξ2 , τ2 ) : |τ2 + |ξ2 |2 | ≤ 2k+k ≤ 2k+k1 +20 }. In this case the left-hand side of (5.14) is equal to 0. The analysis in Case 1, Case 2, and Case 3 suffices to prove (5.14) if k1 ≥ k2 − 10. So we may assume from now on that k1 ≤ k2 − 10 and |k − k2 | ≤ 2.
(5.18)
Case 4. f k2 = f ke2l ,k ∈ Yke2l ,k , k ∈ Tk , k ≤ k1 + 20, f k1 = gk1 , j1 ∈ X k1 , l ∈ {1, . . . , L}. In view of (2.31) and the analysis in Case 1 above, we may assume that k ≥ 100 and f ke2l ,k is supported in the set {(ξ2 , τ2 ) : |τ2 + |ξ2 |2 | ≤ 2k+k −100 }. Then we may assume
− 10; for (5.14) it suffices to prove that j1 ≥ K
2dk/2 2 j1 /2 βk, j1 ||
gk1 , j1 ∗ f ke2l ,k || L 2 ≤ C 2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 L 2 · 2dk2 /2 f ke2l ,k
e ,k Yk l 2
.
(5.19)
We use Lemma 2.4, so we may assume ⎧ e ,k ⎨ f k l (ξ1 el + ξ , τ ) = 2−k /2 · 2 e ,k
⎩ ||h|| L 2 ≤ C|| f k2 || ξ ,τ
e ,k
Yk l
η≤k −100 (ξ1 −M) ξ1 −M+i/2k
· h(ξ , τ ) (5.20)
,
2
where ξ1 ∈ R, ξ ∈ Pel , M = M(ξ , τ ) = (−τ − |ξ |2 )1/2 , and h is supported in
Ske2l ,k = {(ξ , τ ) ∈ Pel × R : −τ − |ξ |2 ∈ [22k −80 , 22k +10 ], |ξ | ≤ 2k2 +1 }.
(5.21)
In view of (5.18), for (5.19), it suffices to prove that for h ∈ L 2 (Pel × R) and f ke2l ,k as in (5.20),
||
gk1 , j1 ∗ f ke2l ,k || L 2 ≤ C2dk1 /2 gk1 , j1 L 2 · ||h|| L 2 . ξ ,τ
(5.22)
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For later use, we prove (5.22) without using the restriction k ≤ k1 + 20. We estimate the L 2 norm in the left-hand side of (5.22) by duality: the left-hand side of (5.22) is bounded by 2−k /2 sup
gk1 , j1 (η1 el + η , β) · h(ξ , τ ) (R×Pel ×R)2
a L 2 =1
η≤k −100 (ξ1 − M) · a(ξ + η , ξ + η , τ + β) dξ dη dξ dη dτ dβ . 1 1 1 1 ξ1 − M + i/2k Using the boundedness of the Hilbert transform on L 2 and then Hölder’s inequality in the variables (ξ , τ, β), this is bounded by !1/2
C2−k /2 ||h|| L 2 · 2k /2 · |
gk1 , j1 (η1 el + η , β)|2 dβ dη1 dη , R×Pel
R
which easily gives (5.22).
Case 5. f k2 = f ke2l ,k ∈ Yke2l ,k , k ≥ k1 + 20, f k1 = f k1 ,≤ K ∈ Z k1 , l ∈ {1, . . . , L}. We may assume also k ≥ 100 and notice that
f ke2l ,k ∗
f k1 ,≤ K is supported in the set {(ξ, τ ) : ξ · el ∈ [2k −2 , 2k +2 ]}.
(5.23)
We have to prove that
(d) 2dk/2 η≥ K (τ + |ξ |2 ) · ηk,e (ξ ) · (
f k1 ,≤ K ∗ f ke2l ,k ) Z k
≤ C(2dk1 /2 f k1 ,≤ K Z k1 ) · (2dk2 /2 f ke2l ,k
e ,k
Yk l
).
(5.24)
2
We will use (2.29) implicitly in some of the estimates below, and write
−F −1 [(τ + |ξ |2 + i) · (
f k1 ,≤ K ∗ f ke2l ,k )]
= (i∂t + x − i)F −1 ( f ke2l ,k ) · F −1 (
f k1 ,≤ K ) +F −1 f ke2l ,k · (i∂t + x )F −1 (
f k1 ,≤ K ) +2∇x F −1 f ke2l ,k · ∇x F −1 (
f k1 ,≤ K ).
(5.25)
Thus, using (5.23),
(d)
f k1 ,≤ K ∗ f ke2l ,k ) Z k 2dk/2 η≥ K (τ + |ξ |2 ) · ηk,e (ξ ) · (
≤ C2dk/2 2−k /2 γk2 ,k · ||(i∂t + x − i)F −1 f ke2l ,k · F −1 (
f k1 ,≤ K )|| L 1,2 el
+C2 2 ||F ( f ke2l ,k ) · (i∂t + x )F −1 (
f k1 ,≤ K )|| L 2
/2 dk/2 − K −1 el ,k −1
+C2 2 ||∇x F ( f k2 ) · ∇x F ( f k1 ,≤ K )|| L 2 .
/2 dk/2 − K
−1
(5.26)
We estimate the first term in the right-hand side of (5.26) by
C2dk/2 2−k /2 γk2 ,k · ||(i∂t + x − i)F −1 ( f ke2l ,k ) L 1,2 · F −1 (
f k1 ,≤ K )|| L ∞ , el
Global Well-Posedness of Schrödinger Maps
547
which is bounded by the right-hand side of (5.24) in view of Lemma 2.2. We estimate the last two terms in the right-hand side of (5.26) by
f k1 ,≤ K || L 2 , C2dk/2 2− K /2 · 2 K f ke2l ,k ∗
which is bounded by the right-hand side of (5.24), using (5.3).
, l ∈ {1, . . . , L}. Case 6. f k2 = f ke2l ,k ∈ Yke2l ,k , k ≥ k1 + 20, f k1 = gk1 , j1 ∈ X k1 , j1 ≥ K Then, using Lemma 2.1, we decompose
,k el ,k f ke2l ,k = f ke2l ,≤ j1 −10 + f k2 ,≥ j1 +10 + X k2 .
In view of the analysis in Case 1, for (5.24) it suffices to prove that
(d)
,k 2dk/2 ηk,e (ξ ) · (
gk1 , j1 ∗ f ke2l ,≤ j1 −10 ) Z k
(d)
,k +2dk/2 η≥ j1 (τ + |ξ |2 ) · ηk,e (ξ ) · (
gk1 , j1 ∗ f ke2l ,≥ j1 +10 ) Z k el ,k dk1 /2 j1 /2 dk2 /2 ≤C 2 2 βk1 , j1 gk1 , j1 L 2 · 2 f k2 el ,k . Yk
(5.27)
2
The bound for the first term follows easily using the L 2 norm and (5.22). To control the second term in the right-hand side of (5.27) we use again the decomposition (5.25), as well as (5.23), and estimate it (as in (5.26)) by
,k −1 C2dk/2 2−k /2 γk,k ||(i∂t + x − i)F −1 ( f ke2l ,≥ gk1 , j1 )|| L 1,2 j1 +10 ) · F (
el
+C2
dk/2 − j1 /2
2
||F
−1
,k ( f ke2l ,≥ j1 +10 ) · (i∂t
+ x )F
−1
(
gk1 , j1 )|| L 2
,k
+C2dk/2 2− j1 /2 ||∇x F −1 ( f ke2l ,≥ j1 +10 ) · ∇x F −1 (
gk1 , j1 )|| L 2 .
(5.28)
We estimate the first term in the right-hand side of (5.28) by
,k −1 C2dk/2 2−k /2 γk,k ||(i∂t + x − i)F −1 ( f ke2l ,≥ gk1 , j1 )|| L ∞ , j1 +10 ) L 1,2 · F (
el
which is bounded by the right-hand side of (5.27) in view of Lemma 2.2. We estimate the last two terms in the right-hand side of (5.28) by
,k C2dk/2 2− j1 /2 · 2 j1 f ke2l ,≥ gk1 , j1 || L 2 , j1 +10 ∗
which is bounded by the right-hand side of (5.24), using (5.3).
6. Dyadic Bilinear Estimates, II In this section we prove our second main bilinear estimate: Lemma 6.1. If k1 , k2 , k ∈ Z, k1 ≤ k2 − 10, |k − k2 | ≤ 2, f k1 ∈ Z k1 , and f k2 ∈ Z k2 , then 2dk/2 ηk(d) (ξ ) · (τ + |ξ |2 + i)−1
f k1 ∗ [(τ2 + |ξ2 |2 + i) f k2 ] Z k ≤ C(2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ), f k1 ) ∈ {F −1 ( f k1 ), F −1 ( f k1 )}. where F −1 (
(6.1)
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In view of (2.10), we may assume that (d)
(d)
f k2 is supported in Ik2 × R ∩ {(ξ2 , τ2 ) : |ξ2 − v| ≤ 2k2 −50 } for some v ∈ Ik2 , (6.2) and let v = v/|v|. With ηk,e defined as in (5.11), e ∈ {e1 , . . . , e L }, for (6.1) it suffices to prove that (d) 2dk/2 ηk,e (ξ ) · (τ + |ξ |2 + i)−1
f k1 ∗ [(τ2 + |ξ2 |2 + i) f k2 ] Z k ≤ C(2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ),
(6.3)
for any e ∈ {e1 , . . . , e L }. We consider again several cases. Case 1. k1 ≤ 100 and f k2 = gk2 , j2 ∈ X k2 , j2 ≥ k1 + k2 + 100. In this case we may assume f k1 = gk1 , j1 ∈ X k1 . For (6.3) it suffices to prove that (d) 2dk/2 2 j2 ηk,e (ξ ) · (τ + |ξ |2 + i)−1
gk1 , j1 ∗ gk2 , j2 Z k ≤ C(2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ).
(6.4)
If | j1 − j2 | ≥ 500, then the left-hand side of (6.4) is dominated by gk1 , j1 ∗ gk2 , j2 L 2 C2dk/2 2 j2 2− max( j1 , j2 ) · 2max( j1 , j2 )/2 βk,max( j1 , j2 )
≤ C2dk/2 2 j2 2− max( j1 , j2 )/2 βk,max( j1 , j2 ) · 2dk1 /2 2min( j1 , j2 )/2 gk1 , j1 L 2 · gk2 , j2 L 2 , (6.5) using (5.1), which suffices for (6.4) in view of (5.16). If | j1 − j2 | ≤ 500 then, using (5.1), the left-hand side of (6.4) is dominated by 2− j/2 βk, j · 2 j/2 2dk1 /2 (gk1 , j1 L 2 · gk2 , j2 L 2 ) C2dk/2 2 j2 j≤ j2 +C
≤ C2
dk2 /2 dk1 /2 j2
2
2 · ( j2 + βk2 , j2 ) · (gk1 , j1 L 2 · gk2 , j2 L 2 ),
which suffices for (6.4) since βk1 , j1 ≈ 2 j1 /2 ≈ 2 j2 /2 . Case 2. k1 ≤ 100 and f k2 = gk2 , j2 ∈ X k2 , j2 ≤ k1 + k2 + 100. In this case we may assume f k1 = gk1 , j1 ∈ X k1 . For (6.3) it suffices to prove that (d) gk1 , j1 ∗ gk2 , j2 Z k 2dk/2 2 j2 ηk,e (ξ ) · (τ + |ξ |2 + i)−1
≤ C(2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 Z k1 ) · (2dk2 /2 2 j2 /2 gk2 , j2 L 2 ).
(6.6)
(d)
In view of (6.2), we may assume that gk2 , j2 is supported in Ik2 ∩ {(ξ2 , τ2 ) : |ξ2 − v| ≤ (d)
2k2 −40 } for some v ∈ Ik2 . Then we estimate the left-hand side of (6.6) (using the L 1,2 e norm and (2.17)) by 2dk/2 2 j2 · 2−k/2 ||F −1 (
gk1 , j1 )|| L 2,∞ · ||F −1 (gk2 , j2 )|| L 2,2 , e
which gives (6.6) in view of Lemma 4.1.
e
Global Well-Posedness of Schrödinger Maps
549
Case 3. k1 ≤ 100 and f k2 = f ke2l ,k ∈ Yke2l ,k , k ∈ Tk2 , l ∈ {1, . . . , L}. In view of the analysis in Cases 1 and 2, and (2.31), we may assume k ≥ 200. Then, using Lemma 2.2, (2.17), and the fact that
f k1 ∗ [(τ2 + |ξ2 |2 + i) f ke2l ,k ] is supported in the set
{(ξ, τ ) : ξ · el ∈ [2k −2 , 2k +2 ]}, we estimate (d) f k1 ∗ [(τ2 + |ξ2 |2 + i) f ke2l ,k ] Z k 2dk/2 ηk,e (ξ ) · (τ + |ξ |2 + i)−1
≤ C2dk/2 2−k /2 γk,k ||F −1 (
f k1 )|| L ∞ · F −1 [(τ2 + |ξ2 |2 + i) f ke2l ,k ] L 1,2 , el
which suffices for (6.3). Thus, from now on we may assume k1 ≥ 100 which implies that k2 ≥ 100.
(6.7)
As in the proof of Lemma 5.2, let
= k1 + k2 + 100, K and define f k2 ,≤ K −1 as in (5.2). Then, using Lemma 4.1 and (2.17), (d) 2dk/2 ηk,e (ξ ) · (τ + |ξ |2 + i)−1
f k1 ∗ [(τ2 + |ξ2 |2 + i) f k2 ,≤ K −1 ] Z k ≤ C2dk/2 · 2−k/2 ||F −1 (
f k1 ) · F −1 [(τ2 + |ξ2 |2 + i) f k2 ,≤ K −1 ]|| L 1,2 e
≤ C2dk/2 · 2−k/2 ||F −1 (
f k1 )|| L 2,∞ · ||F −1 [(τ2 + |ξ2 |2 + i) f k2 ,≤ K −1 ]|| L 2,2 e
≤ C2
dk/2 −k/2 (d−1)k1 /2
2
2
|| f k1 || Z k1 · 2
e
/2 K
|| f k2 ,≤ K −1 || Z k2 .
Thus, for (6.3) it suffices to prove that (d) 2dk/2 ηk,e f k1 ∗ [(τ2 + |ξ2 |2 + i) f k2 ,≥ K ] Z k (ξ ) · (τ + |ξ |2 + i)−1
≤ C(2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ).
(6.8)
To prove (6.8) we analyze several more cases:
and gk2 , j2 is Case 4. f k2 = gk2 , j2 ∈ X k2 , f k1 = gk1 , j1 ∈ X k1 . We may assume j2 ≥ K (d) (d) k −40 2 supported in Ik2 ∩{(ξ2 , τ2 ) : |ξ2 −v| ≤ 2 } for some v ∈ Ik2 . If | j1 − j2 | ≥ 10, then 2 the same L estimate as in Case 1 (see (6.5)) gives the desired estimate. If | j1 − j2 | ≤ 10, then we estimate the left-hand side of (6.8) (using the L 1,2 e norm and (2.17)) by C2dk/2 2 j2 · [( j2 − 2k2 )+ + 1]2−k/2 ||F −1 (
gk1 , j1 )|| L 2,∞ · ||F −1 (gk2 , j2 )|| L 2,2 e
e
2( j2 −k−k1 )/2 · [( j2 − 2k2 )+ + 1] ≤C · (2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ), βk1 , j1 · βk2 , j2 which is controlled by the right-hand side of (6.8).
. Using Case 5. f k2 = gk2 , j2 ∈ X k2 , f k1 ∈ Yke1l , l ∈ {1, . . . , L}. We may assume j2 ≥ K 2 the L norm, we estimate the left-hand side of (6.8) by C2dk/2 2 j2 · 2− j2 2 j2 /2 βk, j2 ||F −1 (
f k1 )|| L ∞ · ||F −1 (gk2 , j2 )|| L 2 , which suffices, in view of Lemma 2.2.
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Case 6. f k2 = f ke2l ,k ∈ Yke2l ,k , k ∈ Tk2 , f k1 ∈ Yk1l , l, l ∈ {1, . . . , L}. In view of (2.31)
≤ k2 + k , so k ≥ k1 + 10. Thus and the analysis in Cases 4 and 5, we may assume K el ,k f k1 is supported in the set {(ξ, τ ) : ξ · el ∈ [2k −2 , 2k +2 ]}, and we can estimate f k2 ∗
the left-hand side of (6.8) by ,k C2dk/2 2−k /2 γk,k ||F −1
f k1 ∗ [(τ2 + |ξ2 |2 + i) f kel ,≥
] || L 1,2 K e
el
2
≤ C2
dk/2 −k /2
2
γk,k ||F
−1
(
f k1 )|| L ∞ · ||F
−1
,k [(τ2 + |ξ2 | + i) f kel ,≥
]|| L 1,2 , K 2
2
el
which suffices, in view of Lemma 2.2. 7. Dyadic Bilinear Estimates, III In this section we prove our last dyadic bilinear estimate: Lemma 7.1. If k1 , k2 , k ∈ Z, k1 ≤ k2 + 10, f k1 ∈ Z k1 , and f k2 ∈ Z k2 , then (d) 2dk/2 ηk (ξ ) · (τ + |ξ |2 + i)−1 [ τ1 + |ξ1 |2 + i f k1 ] ∗ f k2 Z k
(7.1)
≤ C2−|k2 −k|/4 · (2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ). In view of (2.10), we may assume that (d)
(d)
f k2 is supported in Ik2 × R ∩ {(ξ2 , τ2 ) : |ξ2 − v| ≤ 2k2 −50 } for some v ∈ Ik2 , (7.2) and let v = v/|v|. With ηk,e as in (5.11), for (7.1) it suffices to prove that −1 (d) [(τ1 + |ξ1 |2 + i) f k1 ] ∗ f k2 Z k 2dk/2 ηk,e (ξ ) · τ + |ξ |2 + i ≤ C2−|k2 −k|/4 · (2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ),
(7.3)
for any e ∈ {e1 , . . . , el }. We consider again several cases. Case 1. k1 ≥ 100, k2 ≥ k1 + 10d, f k1 ∈ Z k1 , f k2 = gk2 , j2 ∈ X k2 . We may assume |k2 − k| ≤ 2, and let gk1 , j1 = f k1 · η j1 (τ + |ξ |2 ), j1 ∈ Z+ . Since 2 j1 /2 βk1 , j1 ||gk1 , j1 || L 2 ≤ C|| f k1 || Z k1 (see Lemma 2.1), for (7.3) it suffices to prove that (d) 2dk/2 2 j1 ηk,e (ξ ) · (τ + |ξ |2 + i)−1 gk1 , j1 ∗ gk2 , j2 Z k −1 ≤ C 1 + 2k1 − j1 /2 · (2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ). (7.4) We have several subcases depending on j1 and j2 . Assume first that j1 ≤ k1 + k2 + 10 and j2 ≤ k1 + k2 + 20.
(7.5)
(d) in the left-hand side of For (7.4) it suffices to prove that (with e as in the function ηk,e (7.4)) 2dk/2 2 j1 2−k/2 F −1 gk1 , j1 ∗ gk2 , j2 L 1,2 e −1 (7.6) ≤ C 1 + 2k1 − j1 /2 · (2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 gk2 , j2 L 2 ).
Global Well-Posedness of Schrödinger Maps
551
(1)
We use the cutoff functions χk1 ,n 1 and χke1 ,n defined in (4.1) and (4.12)) to decompose ⎧ ⎨ gk2 , j2 = n
(1) k −30 ,2k2 +10 ] 1 ∈k , n 1 ∈[2 2 1
n ∈ek , |n |≤2k2 +10 1
gkn21,,nj2 ;
(1) ⎩ g n 1 ,n (ξ e + ξ , τ ) = g e k2 , j2 (ξ1 e + ξ , τ ) · χk1 ,n 1 (ξ1 ) · χk1 ,n (ξ ). k2 , j2 1
(7.7)
In view of (7.5), we have the identity
h n 1 ,n (ξ1 e + ξ , τ ) := gk1 , j1 ∗ gkn21,,nj2 (ξ1 e + ξ , τ )
(1)
=χ
k1 ,n 1 (ξ1 ) · χ
ke1 ,n (ξ ) · η≤k1 +k2 +30d (τ + n 21 + |n |2 ) · h n 1 ,n (ξ1 e + ξ , τ ) (7.8)
=χ
ke1 ,n (ξ ) · η≤k1 +k2 +30d (τ + n 21 + |n |2 ) · h n 1 ,n (ξ1 e + ξ , τ ). For simplicity of notation, let n 1 ,n denote the sum over n 1 and n as in (7.7). Let
h n 1 ,n (x1 , ξ , τ ) = ei x1 ξ1 h n 1 ,n (ξ1 e + ξ , τ ) dξ1 , R
thus, using (7.8),
ei x1 ξ1 h n 1 ,n (ξ1 e+ξ , τ ) dξ1 = χ
ke1 ,n (ξ )·η≤k1 +k2 +30d (τ +n 21 +|n |2 )·
h n 1 ,n (x1 , ξ , τ ). R
We notice now that the supports in (ξ , τ ) of χ
ke1 ,n (ξ ) · η≤k1 +k2 +30d τ + n 21 + |n |2 and χ
ke1 ,m (ξ ) · η≤k1 +k2 +30d (τ + m 21 + |m |2 ) are disjoint unless |n 1 − m 1 | + |n − m | ≤ C2k1 (recall that n 1 , m 1 ≈ 2k2 ). Thus, for any x1 ∈ R, 2 h n 1 ,n 2 ≤ C
h n 1 ,n (x1 , ξ , τ )2L 2 . ei x1 ξ1 R
L ξ ,τ
n 1 ,n
ξ ,τ
n 1 ,n
Thus, using the Plancherel theorem, the left-hand side of (7.6) is dominated by
C2
dk/2 j1 −k/2
2 2
R
⎡ ⎣
⎤1/2 F
−1
(h
n 1 ,n
n 1 ,n
)(x1 e + x
, t)2L 2 ⎦ x ,t
d x1 .
(7.9)
We use now the definition of h n 1 ,n in (7.8) to estimate, for any x1 ∈ R,
F −1 (h n 1 ,n )(x1 e + x , t) L 2
x ,t
≤ CF
−1
(gk1 , j1 )(x1 e + x , t) L 2 · F −1 (gkn21,,nj2 )(x1 e + x , t) L ∞ .
x ,t
x ,t
Thus, the expression in (7.9) is dominated by C2dk/2 2 j1 2−k/2
R
⎡
F −1 (gk1 , j1 )(x1 e + x , t) L 2 · ⎣ x ,t
n1
,n
⎤1/2 F −1 (gkn21,,nj2 )(x1 e
+ x , t)2L ∞ ⎦ x ,t
d x1 .
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A. D. Ionescu, C. E. Kenig
By Hölder’s inequality in x1 , this is dominated by ⎡ C2dk/2 2 j1 2−k/2 gk1 , j1 L 2 · ⎣
n 1 ,n
⎤1/2 ,n
F −1 (gkn21, j2 )2 2,∞ ⎦ Le
.
(7.10)
Using the bound (4.3) in Lemma 4.1, this is dominated by C2dk2 /2 2 j1 2−k2 /2 gk1 , j1 L 2 · 2(d−1)k1 /2 · 2(k2 −k1 )/2 · (2 j2 /2 gk2 , j2 L 2 ),
(7.11)
which suffices for (7.6) since βk1 , j1 = 1 + 2 j1 /2−k1 . Assume now that j1 ≤ k1 + k2 + 10 and j2 ≥ k1 + k2 + 20.
(7.12)
For (7.4) it suffices to prove that 2dk/2 2 j1 2− j2 /2 βk, j2 gk1 , j1 ∗ gk2 , j2 L 2 −1 ≤ C 1 + 2k1 − j1 /2 · (2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ). (7.13) Using Lemma 5.1, we estimate the left-hand side of (7.13) by 2dk2 /2 2 j1 2− j2 /2 βk, j2 · 2− j2 /2 (βk1 , j1 · βk2 , j2 )−1 · (2dk1 /2 gk1 , j1 Z k1 ) · gk2 , j2 Z k2 , which suffices for (7.13). In this case we have proved the stronger bound −1 (d) 2dk/2 2 j1 ηk,e (ξ ) · (τ + |ξ |2 + i)−1 gk1 , j1 ∗ gk2 , j2 Z k ≤ C 1 + 2k1 − j1 /2 ×2(k1 −k2 )/2 · (2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ).
(7.14)
Assume now that j1 ≥ k1 + k2 + 10 and | j2 − j1 | ≥ 10.
(7.15)
Since the sequence 2− j/2 βk, j is decreasing in j, for (7.4) it suffices to prove the stronger bound 2dk/2 2 j1 2− j1 /2 βk, j1 · sup 1 Dk, j · (gk1 , j1 ∗ gk2 , j2 ) L 2 j∈Z+
≤ C2
(k1 −k2 )/2
· (2
dk1 /2 j1 /2
2
βk1 , j1 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ).
(7.16)
Using (5.1), we estimate the left-hand side of (7.16) by C2dk2 /2 2 j1 /2 βk, j1 · 2dk1 /2 2 j2 /2 · gk1 , j1 L 2 · gk2 , j2 L 2 , which suffices for (7.16). Finally, assume that j1 ≥ k1 + k2 + 10 and | j2 − j1 | ≤ 10.
(7.17)
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553
Using (2.17), for (7.4) it suffices to prove that 2dk/2 2 j1 · 2−k/2 [( j2 − 2k2 )+ + 1]F −1 (gk1 , j1 ∗ gk2 , j2 ) L 1,2 e
≤ C · (2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ).
(7.18)
Using (4.5), we estimate the left-hand side of (7.18) by 2dk2 /2 2 j1 2−k/2 [( j2 − 2k2 )+ + 1] · 2(d−1)k1 /2 2 j1 /2 gk1 , j1 L 2 · gk2 , j2 L 2 , which suffices for (7.18) since βk1 , j1 ≥ 2( j1 −k1 −k2 )/2 . Case 2. k1 ≥ 100, k2 ≥ k1 + 10d, f k1 ∈ Z k1 , f k2 = f ke2l ∈ Yke2l , l ∈ {1, . . . , L}. We may assume |k2 − k| ≤ 2, and let gk1 , j1 = f k1 · η j1 (τ + |ξ |2 ), j1 ∈ Z+ . Since 2 j1 /2 βk1 , j1 ||gk1 , j1 || L 2 ≤ C|| f k1 || Z k1 , for (7.3) it suffices to prove that (d)
2dk/2 2 j1 ηk,e (ξ ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ f ke2l ) Z k −1 ≤ C 1 + 2k1 − j1 /2 · (2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ) · (2dk2 /2 f ke2l Y el ). (7.19) k2
We consider two subcases. Assume first that j1 ≤ k1 + k2 + 10,
(7.20)
and define f ke2l ,≤k1 +k2 +20 and f ke2l ,≥k1 +k2 +21 as in (5.2). To estimate (d)
2dk/2 2 j1 ηk,e (ξ ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ f ke2l ,≤k1 +k2 +20 ) Z k we argue as in the proof of the bound (7.5) in Case 1. The only difference is that in passing from (7.10) to (7.11) we use the bound (4.4) in Lemma 4.1, instead of the bound (4.3). To estimate (d) 2dk/2 2 j1 ηk,e (ξ ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ f ke2l ,≥k1 +k2 +21 ) Z k
we define gk2 , j2 = f ke2l ,≥k1 +k2 +21 · η j2 (τ + |ξ |2 ), j2 ∈ [k1 + k2 + 20, 2k2 ] ∩ Z, and use the bound (7.14) and Lemma 2.1. Assume now that j1 ≥ k1 + k2 + 10, (7.21) and decompose f ke2l
=
f ke2l ,≤ j1 −10
+
2k2 j2 = j1 −9
f ke2l · η j2 (τ + |ξ |2 ).
The contribution of the sum over j2 in the expression above, which has at most k2 − k1 terms, can be estimated using (7.16) and (7.18). Then, we estimate the contribution of the function f ke2l ,≤ j1 −10 by C2dk/2 2 j1 · 2− j1 /2 βk, j1 ||gk1 , j1 ∗ f ke2l ,≤ j1 −10 || L 2 . The bound (7.19) follows from (5.27).
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Case 3. k2 ≤ C. In this case, k1 , k ≤ C and we may assume f k1 = gk1 , j1 ∈ X k1 and f k2 = gk2 , j2 ∈ X k2 . Since βk1 , j1 ≈ 2 j1 /2 , βk2 , j2 ≈ 2 j2 /2 , βk, j ≈ 2 j/2 , for (7.3) it suffices to prove that 1 Dk, j · gk1 , j1 ∗ gk2 , j2 L 2 2dk/2 2 j1 j≤max( j1 , j2 )+C
≤ C2
−|k2 −k|/4
· (2dk1 /2 2 j1 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 gk2 , j2 L 2 ).
(7.22)
This follows easily from (5.1). Case 4. k1 ≤ 100, k2 ≥ (k1 +10d)+ , f k2 = gk2 , j2 ∈ X k2 . We may assume f k1 = gk1 , j1 ∈ X k1 , βk1 , j1 ≈ 2 j1 /2 , and |k2 − k| ≤ 2. For (7.3) it suffices to prove that (d)
2dk/2 2 j1 ηk,e (ξ ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ gk2 , j2 ) Z k ≤ C(2dk1 /2 2 j1 /2 2 j1 /2 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ).
(7.23)
Assume first that
j1 ≤ k1 + k2 + 10. Then, using (4.5) and (2.17), we estimate the left-hand side of (7.23) by
(7.24)
C2dk/2 2 j1 · 2−k/2 [( j2 − 2k2 )+ + 1]F −1 (gk1 , j1 ∗ gk2 , j2 ) L 1,2 e
≤ C2dk2 /2 2 j1 2−k2 /2 [( j2 − 2k2 )+ + 1] · (2(d−1)k1 /2 2 j1 /2 gk1 , j1 L 2 ) · gk2 , j2 L 2 , (7.25) which suffices for (7.23). Assume now that (7.26) j1 ≥ k1 + k2 + 10 and | j2 − j1 | ≥ 10. Since the sequence 2− j/2 βk, j is decreasing in j, for (7.23) it suffices to prove that 2dk/2 2 j1 2− j1 /2 βk, j1 · sup 1 Dk, j · (gk1 , j1 ∗ gk2 , j2 ) L 2 j∈Z+
≤ C(2dk1 /2 2 j1 /2 2 j1 /2 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ).
(7.27)
Using (5.1), we estimate the left-hand side of (7.27) by C2dk2 /2 2 j1 /2 βk, j1 · 2dk1 /2 2 j2 /2 · gk1 , j1 L 2 · gk2 , j2 L 2 , which suffices for (7.27). Finally, assume that j1 ≥ k1 + k2 + 10 and | j2 − j1 | ≤ 10.
(7.28)
For (7.23) it suffices to prove that 2− j/2 βk, j · 1 Dk, j · (gk1 , j1 ∗ gk2 , j2 ) L 2 2dk/2 2 j1 j≤ j1 +20
≤ C · (2dk1 /2 2 j1 /2 2 j1 /2 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ). Using (5.1), we estimate the left-hand side of (7.18) by βk, j · 2dk1 /2 gk1 , j1 L 2 · gk2 , j2 L 2 , 2dk2 /2 2 j1 · j≤ j1 +20
which suffices for (7.29) since βk, j ≤ Cβk2 , j2 .
(7.29)
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Case 5. k1 ≤ 100, k2 ≥ (k1 + 10d)+ , f k2 = f ke2l ∈ Yke2l . We may assume f k1 = gk1 , j1 ∈ X k1 , βk1 , j1 ≈ 2 j1 /2 , k2 ≥ 100, and |k2 − k| ≤ 2. For (7.3) it suffices to prove that (d) (ξ ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ f ke2l ) Z k 2dk/2 2 j1 ηk,e
≤ C(2dk1 /2 2 j1 /2 2 j1 /2 gk1 , j1 L 2 ) · (2dk2 /2 f ke2l Y el ).
(7.30)
k2
If j1 ≤ k1 + k2 + 50 then (7.30) follows using an estimate similar to (7.25) in Case 4 (clearly, f ke2l L 2 ≤ f ke2l Y el , using Lemma 2.1). We assume k2
j1 ≥ k1 + k2 + 50, and decompose
(7.31)
f ke2l = f ke2l ,≤ j1 −10 + f ke2l ,≥ j1 +10 + X k2 .
In view of (5.27), (d)
2dk/2 2 j1 ηk,e (ξ ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ f ke2l ,≤ j1 −10 ) Z k ≤ C(2dk1 /2 2 j1 /2 2 j1 /2 gk1 , j1 L 2 ) · (2dk2 /2 f ke2l Y el ), k2
as desired. In addition, (d) 2dk/2 2 j1 ηk,e (ξ ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ f ke2l ,≥ j1 +10 ) Z k
≤ C2dk2 /2 2 j1
2k+10
2− j/2 1 Dk, j · (gk1 , j1 ∗ f ke2l ,≥ j1 +10 ) L 2
j= j1
⎛
2k+10
≤ C2dk2 /2 2 j1 ⎝
⎞ 2− j/2 ⎠ · 2− j1 /2 · 2− j1 /2 · (2dk1 /2 gk1 , j1 Z k1 ) · ( f ke2l Y el ), k2
j= j1
using Lemma 5.1, since j1 ≤ 2k2 . This completes the proof of (7.30). Case 6. k1 , k2 ≥ 100d, |k1 − k2 | ≤ 10d, f k1 ∈ Z k1 , f k2 = gk2 , j2 ∈ X k2 . Let gk1 , j1 = f k1 ·η j1 (τ +|ξ |2 ), j1 ∈ Z+ . Since 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ≤ C f k1 Z k1 , for (7.3) it suffices to prove that (d) 2dk/2 2 j1 ηk,e (ξ ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ gk2 , j2 ) Z k ≤ C(1 + 2k1 − j1 /2 )−1
×2−|k2 −k|/4 (2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ) · (2dk2 /2 2 j2 /2 βk2 , j2 gk2 , j2 L 2 ). (7.32) Using the definition, we estimate the left-hand side of (7.32) by (d)
C2dk/2 2 j1 ηk,e (ξ ) · η≤2k−201 (τ + |ξ |2 ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ gk2 , j2 )Yke (d)
+ C2dk/2 2 j1 ηk,e (ξ ) · η≥2k−200 (τ + |ξ |2 ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ gk2 , j2 ) Z k . (7.33) We estimate the first term in (7.33) (which is nontrivial only if k ≥ 100) by C2dk/2 2 j1 2−k/2 F −1 (gk1 , j1 ) L 2 · F −1 (gk2 , j2 ) L 2,∞ e
≤ C2dk/2 2 j1 2−k/2 gk1 , j1 L 2 · 2(d−1)k2 /2 gk2 , j2 Z k2 ,
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using Lemma 4.1. This suffices for (7.32) since βk1 , j1 = 1 + 2 j1 /2−k1 . For the second term in (7.33) we use L 2 estimates. Assume first that j1 ≤ 2k1 + 30d.
(7.34)
Then the second term in (7.33) is bounded by 2− j/2 βk, j 1 Dk, j · (gk1 , j1 ∗ gk2 , j2 ) L 2 , C2dk/2 2 j1 j
where the sum is over j ≥ max(0, 2k − 200) and j ≤ max(2k2 , j2 ) + C. Since βk, j ≈ 2 j/2−k+ , using Lemma 5.1 this expression is bounded by C2dk/2 2 j1 2−k+ [|k2 − k+ | + ( j2 − 2k2 )+ + 1] · 2−k2 · βk−1 2 , j2 ×(2dk1 /2 gk1 , j1 Z k1 ) · gk2 , j2 Z k2 , which suffices for (7.32) (recall that d ≥ 3). Assume now that j1 ≥ 2k1 + 30d and | j1 − j2 | ≤ 10.
(7.35)
Then the second term in (7.33) is bounded by 2− j/2 βk, j 1 Dk, j · (gk1 , j1 ∗ gk2 , j2 ) L 2 , C2dk/2 2 j1 j
where the sum is over j ≥ max(0, 2k − 200) and j ≤ j2 + C. Since βk, j ≈ 2 j/2−k+ , using Lemma 5.1 this expression is bounded by C2dk/2 2 j1 2−k+ (| j1 − 2k+ | + 1) · 2− j2 /2 · (βk1 , j1 βk2 , j2 )−1 ×(2dk1 /2 gk1 , j1 Z k1 ) · gk2 , j2 Z k2 , which suffices for (7.32). Finally, assume that j1 ≥ 2k1 + 30d and | j1 − j2 | ≥ 10.
(7.36)
Since 2− j/2 βk, j ≈ 2−k+ for j ≥ 2k+ , the second term in (7.33) is bounded by C2dk/2 2 j1 2−k+ gk1 , j1 ∗ gk2 , j2 L 2 ≤ C2dk/2 2 j1 2−k+ · 2− j1 /2 βk−1 · (2dk1 /2 gk1 , j1 Z k1 ) · gk2 , j2 Z k2 , 1 , j1 using Lemma 5.1, which suffices for (7.32). Case 7. k1 , k2 ≥ 100d, |k1 − k2 | ≤ 10d, f k1 ∈ Z k1 , f k2 = f ke2l ∈ Yke2l , l ∈ {1, . . . , L}. Let gk1 , j1 = f k1 · η j1 (τ + |ξ |2 ), j1 ∈ Z+ . Since 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ≤ C f k1 Z k1 , for (7.3) it suffices to prove that (d) 2dk/2 2 j1 ηk,e (ξ ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ f ke2l ) Z k ≤ C(1 + 2k1 − j1 /2 )−1
× 2−|k2 −k|/4 (2dk1 /2 2 j1 /2 βk1 , j1 gk1 , j1 L 2 ) · (2dk2 /2 f ke2l Y el ). k2
(7.37)
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Using the definition, we estimate the left-hand side of (7.37) by (d) C2dk/2 2 j1 ηk,e (ξ ) · η≤2k−201 (τ + |ξ |2 ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ f ke2l )Yke (d)
+ C2dk/2 2 j1 ηk,e (ξ ) · η≥2k−200 (τ + |ξ |2 ) · (τ + |ξ |2 + i)−1 (gk1 , j1 ∗ f ke2l ) Z k . (7.38) We estimate the first term in (7.38) (which is nontrivial only if k ≥ 100) by C2dk/2 2 j1 2−k/2 F −1 (gk1 , j1 ) L 2 · F −1 ( f ke2l ) L 2,∞ e
≤ C2dk/2 2 j1 2−k/2 gk1 , j1 L 2 · 2(d−1)k2 /2 f ke2l Y el , k2
using Lemma 4.1. This suffices for (7.37) since βk1 , j1 = 1 + term in (7.38) we use L 2 estimates. Assume first that
2 j1 /2−k1 .
For the second
j1 ≤ 2k1 + 30d.
(7.39)
Then the second term in (7.38) is bounded by C2dk/2 2 j1 2− j/2 βk, j 1 Dk, j · (gk1 , j1 ∗ f ke2l ) L 2 , j
where the sum is over j ≥ max(0, 2k − 200) and j ≤ 2k2 + C. Since βk, j ≈ 2 j/2−k+ , using Lemma 5.1 this expression is bounded by C2dk/2 2 j1 2−k+ [|k2 − k+ | + 1] · 2−k2 · (2dk1 /2 gk1 , j1 Z k1 ) · f ke2l Z k2 , which suffices for (7.37) (recall that d ≥ 3). Assume now that j1 ≥ 2k1 + 30d. Since
2− j/2 βk, j
≈
2−k+
(7.40)
for j ≥ 2k+ , the second term in (7.38) is bounded by
C2dk/2 2 j1 2−k+ gk1 , j1 ∗ f ke2l L 2 ≤ C2dk/2 2 j1 2−k+ · 2− j1 /2 βk−1 · gk1 , j1 Z k1 · (2dk2 /2 gk2 , j2 Z k2 ), 1 , j1 using Lemma 5.1, which suffices for (7.37). 8. A Dyadic Trilinear Estimate In this section we prove the following trilinear estimate: Lemma 8.1. If k1 , k2 , k3 , k ∈ Z, f k1 ∈ Z k1 , f k2 ∈ Z k2 , f k3 ∈ Z k3 , and min(k, k2 , k3 ) ≤ k1 + 20,
(8.1)
then 2k2 +k3 · 2dk/2 ηk(d) (ξ ) · (τ + |ξ |2 + i)−1 · (
f k1 ∗
f k2 ∗
f k3 ) Z k ≤ C2−| max(k1 ,k2 ,k3 )−k|/4 · (2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ) · (2dk3 /2 f k3 Z k3 ), (8.2) f kl ) ∈ {F −1 ( f kl ), F −1 ( f kl )}, l = 1, 2, 3. where F −1 (
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By symmetry, we may assume k2 ≤ k3 . We start with the following simple geometric observation: if w 1 , w 2 ∈ Sd−1 , then there is e ∈ {e1 , . . . , e L } such that 2 | ≥ 2−5 . e·w 1 ≥ 2−5 and |e · w
(8.3)
2 ≥ 0 (by possibly 2 with − w2 ) and To prove this, we may assume 1 · w w replacing w w1 + w take e ∈ {e1 , . . . , e L } with e − ( 2 )/| w1 + w 2 | ≤ 2−100 (compare with (2.8)). The bound (2.17) shows that if k ∈ Z and (d)
f is supported in Ik
× R ∩ {(ξ, τ ) : ξ · e ≥ 2k−40 } for some e ∈ {e1 , . . . , e L },
then, for any J ≥ 0,
|| f · η≤J (τ + |ξ |2 )|| Z k ≤ C (J − 2k+ )+ + 1 · 2−k/2 ||F −1 [(τ + |ξ |2 + i) · f ]|| L 1,2 . (8.4) e
In view of (2.10), we may assume that for i = 1, 2, 3, (d)
(d)
f ki is supported in Iki × R ∩ {(ξ, τ ) : |ξ − vi | ≤ 2ki −50 } for some vi ∈ Iki , (8.5) (d)
and it suffices to prove that for any v ∈ Ik , (d) 2k2 +k3 · 2dk/2 ηk (ξ ) · η0 (|ξ − v|/2k−50 ) · (τ + |ξ |2 + i)−1 · (
f k1 ∗
f k2 ∗
f k3 ) Z k
≤ C2−| max(k1 ,k2 ,k3 )−k|/4 · (2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ) · (2dk3 /2 f k3 Z k3 ). (8.6) Assume k1 ≤ k3 (in the case k1 ≥ k3 the bound (8.8) below still holds, by a similar argument). Let w 1 = v/|v|, w 2 = v3 /|v3 |, and fix e as in (8.3). Fix J = 2 max(k1 , k2 , k3 , 0) + 100.
(8.7)
Let f k1 ∗
f k2 ∗
f k3 ) F(ξ, τ ) = ηk(d) (ξ ) · η0 (|ξ − v|/2k−50 ) · (τ + |ξ |2 + i)−1 · (
and = (2dk1 /2 f k1 Z k1 ) · (2dk2 /2 f k2 Z k2 ) · (2dk3 /2 f k3 Z k3 ). Using (8.4), Lemma 4.1 and Lemma 4.2, 2k2 +k3 · 2dk/2 η≤J (τ + |ξ |2 ) · F Z k ≤ C2k2 +k3 2dk/2 × 2−k/2 (J − 2k+ )+ + 1 ||F −1 (
f k1 )|| L 2,∞ ||F −1 (
f k2 )|| L 2,∞ ||F −1 (
f k3 )|| L ∞,2 e e e ≤ C2k2 +k3 2dk/2 2−k/2 (J − 2k+ )+ + 1 · 2−(k1 +k2 +k3 )/2 2−d max(k1 ,k2 ,k3 )/2 · ,
(8.8)
which is dominated by the right-hand side of (8.6), provided that (8.1) holds and d ≥ 3. It remains to bound 2k2 +k3 · 2dk/2 η≥J +1 (τ + |ξ |2 ) · F Z k .
(8.9)
We use the atomic decomposition (2.16) for the functions f k1 , f k2 , and f k3 , and notice that the expression in (8.9) equal to 0 only if at least one of the functions f k1 , f k2 , is not or f k3 has modulation τ + |ξ |2 ≥ 2 J −10 . Let J ≥ J − 10 denote the highest of these
Global Well-Posedness of Schrödinger Maps
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modulations. Then we estimate 2k2 +k3 · 2dk/2 η≤J +10 (τ + |ξ |2 ) · F Z k as in (8.8). Using (4.5) or (4.17) for the function with the high modulation, the right-hand side of (8.8) is multiplied by at most −1 Cβmax(k · (|J − J | + 1) ≤ C, 1 ,k2 ,k3 ),J
which suffices to complete the proof of (8.6). Acknowledgement. We would like to thank Bejenaru for making his preprint [1] available to us.
References 1. Bejenaru, I.: On Schrödinger maps. Preprint 2006, available at arxiv.org 0604255 2. Chang, N.-H., Shatah, J., Uhlenbeck, K.: Schrödinger maps. Comm. Pure Appl. Math. 53, 590–602 (2000) 3. Ding, W.Y., Wang, Y.D.: Local Schrödinger flow into Kähler manifolds. Sci. China Ser. A 44, 1446– 1464 (2001) 4. Ionescu, A.D., Kenig, C.: Global well-posedness of the Benjamin–Ono equation in low-regularity spaces. J. Amer. Math. Soc. 50894-0347(06) 00551-0, published electronically 24 october 2006 5. Ionescu, A.D., Kenig, C.: Low-regularity Schrödinger maps. Preprint 2006, available at arxiv.org 0605210 6. Kato, J.: Existence and uniqueness of the solution to the modified Schrödinger map. Math. Res. Lett. 12, 171–186 (2005) 7. Kato, J., Koch, H.: Uniqueness of the modified Schrödinger map in H 3/4+ (R2 ). Preprint, 2005, available at arxiv.org 0508423 8. Kenig, C.E., Nahmod, A.: The Cauchy problem for the hyperbolic-elliptic Ishimori system and Schrödinger maps. Nonlinearity 18, 1987–2009 (2005) 9. Kenig, C.E., Pollack, D., Staffilani, G., Toro, T.: The Cauchy problem for Schrödinger flows into Kähler manifolds. Preprint, 2005, available at arxiv.org 0511701 10. Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46, 1221–1268 (1993) 11. Klainerman, S., Rodnianski, I.: On the global regularity of wave maps in the critical Sobolev norm. Internat. Math. Res. Notices 13, 655–677 (2001) 12. Klainerman, S., Selberg, S.: Remark on the optimal regularity for equations of wave maps type. Comm. Part. Differ. Eqs. 22, 901–918 (1997) 13. McGahagan, H.: An approximation scheme for Schrödinger maps. Preprint, 2005 14. Nahmod, A., Stefanov, A., Uhlenbeck, K.: On the well-posedness of the wave map problem in high dimensions. Comm. Anal. Geom. 11, 49–83 (2003) 15. Nahmod, A., Stefanov, A., Uhlenbeck, K.: On Schrödinger maps. Comm. Pure Appl. Math. 56, 114– 151 (2003) 16. Nahmod, A., Stefanov, A., Uhlenbeck, K.: Erratum: “On Schrödinger maps” [Comm. Pure Appl. Math. 56 114–151 (2003)], Comm. Pure Appl. Math. 57, 833–839 (2004) 17. Nahmod, A., Shatah, J., Vega, L., Zeng, C.: Schrödinger maps into Hermitian symmetric spaces and their associated frame systems, available at arxiv.org 0612481 18. Shatah, J., Struwe, M.: The Cauchy problem for wave maps. Int. Math. Res. Notices 11, 555–571 (2002) 19. Sulem, P.L., Sulem, C., Bardos, C.: On the continuous limit for a system of classical spins. Commun. Math. Phys. 107, 431–454 (1986) 20. Soyeur, A.: The Cauchy problem for the Ishimori equations. J. Funct. Anal. 105, 233–255 (1992) 21. Tao, T.: Global regularity of wave maps. I. Small critical Sobolev norm in high dimension. Internat. Math. Res. Notices 6, 299–328 (2001) 22. Tao, T.: Global regularity of wave maps. II. Small energy in two dimensions. Commun. Math. Phys. 224, 443–544 (2001) 23. Tataru, D.: Local and global results for wave maps. I. Comm. Part. Differ. Eq. 23, 1781–1793 (1998) 24. Tataru, D.: On global existence and scattering for the wave maps equation. Amer. J. Math. 123, 37– 77 (2001) 25. Tataru, D.: Rough solutions for the wave maps equation. Amer. J. Math. 127, 293–377 (2005) Communicated by P. Constantin
Commun. Math. Phys. 271, 561–575 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0196-4
Communications in
Mathematical Physics
Generalized Kähler Manifolds, Commuting Complex Structures, and Split Tangent Bundles Vestislav Apostolov1,3 , Marco Gualtieri2 1 Département de Mathématiques, UQAM, C.P. 8888, Succ. Centre-ville, Montréal (Québec), H3C 3P8,
Canada. E-mail: [email protected]
2 M.I.T. Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA.
E-mail: [email protected]
3 Institute of Mathematics and informatics, Bulgarian Academy of Sciences, Acad. G. Boncher St. bl.8,
1113 Sofia, Bulgaria Received: 23 May 2006 / Accepted: 12 October 2006 Published online: 13 February 2007 – © Springer-Verlag 2007
Abstract: We study generalized Kähler manifolds for which the corresponding complex structures commute and classify completely the compact four-dimensional ones. 1. Introduction The notion of a generalized Kähler structure was introduced and studied by the second author in [17], in the context of the theory of generalized geometric structures initiated by Hitchin in [19]. Recall that a generalized Kähler structure is a pair of commuting complex structures (J1 , J2 ) on the vector bundle T M ⊕ T ∗ M over the smooth manifold M 2m , which are: • integrable with respect to the (twisted) Courant bracket on T M ⊕ T ∗ M, • compatible with the natural inner-product ·, · of signature (2m, 2m) on T M ⊕T ∗ M, • and such that the quadratic form J1 ·, J2 · is definite on T M ⊕ T ∗ M. It turns out [17] that such a structure on T M ⊕T ∗ M is equivalent to a triple (g, J+ , J− ) consisting of a Riemannian metric g and two integrable almost complex structures J± compatible with g, satisfying the integrability relations c c d+c F+ + d− F− = 0, dd± F± = 0,
where F± = g J± are the fundamental 2-forms of the Hermitian structures (g, J± ), and c are the i(∂ − ∂) operators associated to the complex structures J . The closed 3-form d± ± c F is called the torsion of the generalized Kähler structure. H = d+c F+ = −d− − These conditions on a pair of Hermitian structures were first described by Gates, Hull and Roˇcek [13] as the general target space geometry for a (2, 2) supersymmetric sigma model. As a trivial example we can take a Kähler structure (g, J ) on M and put J+ = J , J− = ±J to obtain a solution of the above equations.
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The case of interest is when J+ = ±J− , i.e. when the generalized Kähler structure does not come from a genuine Kähler structure on (M, J ). In this paper, we refer to such generalized Kähler structures as non-trivial. One can then ask the following Question 1. When does a compact complex manifold (M, J ) admit a non-trivial generalized Kähler structure (g, J+ , J− ) with J = J+ ? Despite a growing number of explicit constructions [2, 8, 20, 21, 24, 27], the general existence problem for non-trivial generalized Kähler structures remains open. On the other hand, there are a number of known obstructions, or conditions that the existence of a generalized Kähler structure imposes on the underlying complex manifold, which we now describe. Firstly, it follows from the definition that for a complex manifold (M, J ) to admit a compatible generalized Kähler structure it must also admit a Hermitian metric whose ¯ fundamental 2-form is ∂ ∂-closed. This is trivially satisfied when (M, J ) is of Kähler type (i.e. (M, J ) admits a Kähler metric) and when M is compact and four-dimensional (m = 2) [14], but it imposes constraint on the complex structure in general [11]. Secondly, Hitchin [20] showed that if (M, J ) carries a generalized Kähler structure (g, J+ , J− , H ) such that J =J+ and J+ , J− do not commute, then the commutator [J+ , J− ] = J+ J− − J− J+ defines a holomorphic Poisson structure π = [J+ , J− ]g −1 on (M, J ). (When m = 2 this was proved in [2].) In the case when H 0 (M, ∧2 T M) = 0, for instance, this result implies that for any compatible generalized Kähler structure on (M, J ), the complex structures J+ and J− must commute, i.e. J+ J− = J− J+ . Thus motivated, we study in this paper non-trivial generalized Kähler structures (g, J+ , J− ) for which J+ and J− commute. In this case Q = J+ J− is an involution of the tangent bundle T M, and thus gives rise to a splitting T M = T− M ⊕ T+ M as a direct sum of the ±1-eigenspaces of Q. Our first main result proves an assertion first made in [13], which can be stated as follows: Theorem A. Let (g, J+ , J− ) define a generalized Kähler structure with [J+ , J− ] = 0. Then the ±1-eigenspaces of Q = J+ J− define g-orthogonal J± -holomorphic foliations on whose leaves g restricts to a Kähler metric. Combined with Hitchin’s result [20] mentioned above, Theorem A shows that there is a wealth of Kähler complex manifolds which do not admit non-trivial twisted generalized Kähler structures at all. As pointed out in [22], such examples include (locally) deRham irreducible compact Kähler–Einstein manifolds with c1 (M) < 0. In general, the splitting of T M as a direct sum of two integrable, holomorphic sub-bundles is a necessary but not sufficient condition for the existence of a non-trivial generalized Kähler structure of the considered type. Indeed, we have to also ensure that the leaves of the corresponding foliations are Kähler, and that there is a Hermitian ¯ metric on M with ∂ ∂-closed fundamental form with respect to either complex structure (see Remark 2). When M is four dimensional, however, these two extra conditions are automatically satisfied, and our second main result affirms the converse of Theorem A: Theorem B. Let (M, J ) be a compact complex surface whose holomorphic tangent bundle splits as the direct sum of holomorphic sub-bundles. Then (M, J ) admits a non-trivial generalized Kähler structure (g, J+ , J− ) with J+ = J and [J+ , J− ] = 0. Theorems A and B establish a one-to-one correspondence between compact 4-manifolds admitting non-trivial generalized Kähler structures with [J+ , J− ] = 0 and complex surfaces with split holomorphic tangent bundle. The latter class of complex surfaces has been studied by Beauville [5]. We use his classification and an argument from
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[35] to derive in Theorem 3 the complete list of compact complex surfaces admitting generalized Kähler structures of the considered type (which we call ambihermitian). We further refine our four dimensional classification by considering the untwisted case, i.e. when [H ] = 0 ∈ H 3 (M, R), and the twisted case, where [H ] is nonzero. We show, by using the fundamental results of Gauduchon [14, 15], that untwisted generalized Kähler structures on compact four-manifolds can only exist when the first Betti number is even; likewise in the twisted case, any generalized Kähler 4-manifold must have odd first Betti number (Corollary 1). 2. Hermitian Geometry In this section we present certain key properties of Hermitian manifolds which we will need in the later sections, giving special attention to the four-dimensional case. Let M be an oriented 2m-dimensional manifold. A Hermitian structure on M is defined by a pair (g, J ) consisting of a Riemannian metric g and an integrable almost complex structure J , which are compatible in the sense that g(J ·, J ·) = g(·, ·). The Hermitian structure (g, J ) is called positive if J induces the given orientation on M and negative otherwise. The complex structure J induces a decomposition T M ⊗ C = T 1,0 M ⊕ T 0,1 M of the complexified vectors into ±i eigenspaces, and hence defines the usual bi-grading of complex differential forms k (M) ⊗ C = p,q (M), p+q=k
T ∗M
by (J α)(X )= − α(J X ), so that it commutes with the where we let J act on Riemannian duality between vectors and 1-forms: (J α) = J α . The product structure ∧2 J induces a splitting of the real 2-forms into ±1 eigenspaces: 2 (M) = J,+ (M) ⊕ J,− (M), whose complexification is simply J,+ (M) ⊗ C = 1,1 (M) and J,− (M) ⊗ C = 2,0 (M) ⊕ 0,2 (M). Furthermore, the fundamental 2-form F = g J , a real (1, 1)-form of square-norm m, defines a g-orthogonal splitting J,+ (M) = C ∞ (M) · F ⊕ 0J,+ (M). In this way we obtain the U (m) irreducible decomposition of real 2-forms: 2 (M) = C ∞ (M) · F ⊕ 0J,+ (M) ⊕ J,− (M). On a positive Hermitian 4-manifold, the above U (2) splitting of 2 (M) is compatible with the S O(4) decomposition 2 (M) = + (M) ⊕ − (M) into self-dual and anti-self-dual forms, as follows: + (M) = C ∞ (M) · F ⊕ J,− (M); − (M) = 0J,+ (M).
(1)
For a negative Hermitian structure the rôles of + (M) and − (M) in the above identifications are interchanged. Thus, on an oriented Riemannian 4-manifold (M, g), we obtain the well-known correspondence between smooth sections in + (M) (resp. − (M)) of square-norm 2 and positive (resp. negative) almost Hermitian structures (g, J ). The Lee form θ ∈ 1 (M) of a Hermitian structure is defined by d F ∧ F m−2 =
1 θ ∧ F m−1 , (m − 1)
(2)
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or equivalently θ = J δ g F, where δ g is the co-differential with respect to the Levi–Civita connection D g of g. Since J is integrable, d F measures the deviation of (g, J ) from a Kähler structure (for which J and F are parallel with respect to D g ). We have the following expression for D g F (see e.g. [25, p.148]): g
2g((D X J )Y, Z ) = d c F(X, Y, J Z ) + d c F(X, J Y, Z ),
(3)
where d c = i(∂¯ − ∂), so that d c F = ∧3 J (d F) is a real 3-form of type (1, 2) + (2, 1). In four dimensions, (2) reads as d F = θ ∧ F,
(4)
D X F = 21 (X ∧ J θ + J X ∧ θ ),
(5)
and (3) becomes (see e.g. [14, 34]) g
where X = g(X ) denotes the g-dual 1-form to X . We see from this that a Hermitian 4-manifold is Kähler if and only if θ = 0. The existence of a Kähler metric on a compact complex manifold (M 2m , J ) implies the Hodge decomposition of the de Rham cohomology groups p,q Hdk R (M, C) ∼ H∂¯ (M), = p+q=k
p,q
where H∂¯
(M) denote the Dolbeault cohomology groups. This, together with the equalq, p ∼ = H (M), implies that the odd Betti numbers of a complex manifold
p,q H∂¯ (M)
ity ∂¯ admitting a Kähler metric must be even. When m = 2, it turns out that this condition is also sufficient. Theorem 1 [7, 26, 31, 33]. Let M be a compact four-manifold endowed with an integrable almost complex structure J . Then there exists a compatible Kähler metric on (M, J ) if and only if b1 (M) is even. This important result was first established by Todorov [33] and Siu [31], using the Kodaira classification of compact complex surfaces. Direct proofs were found recently by Buchdahl [7] and Lamari [26]. Since we also deal with complex manifolds of non-Kähler type (i.e. which do not ¯ admit any Kähler metric), we recall the definition of the ∂ ∂-cohomology groups: ¯ p − 1, q − 1)-forms}. H∂ ∂¯ (M) := {d-closed ( p, q)-forms}/∂ ∂{( p,q
Note that there is a natural map p,q
p,q
ι : H∂ ∂¯ (M) → H∂¯
(M).
¯ When (M, J ) is of Kähler type, the well-known ∂ ∂-lemma states that the above map is in fact an isomorphism. While the existence of Kähler metrics on a compact complex manifold (M, J ) is generally obstructed, a fundamental result of Gauduchon [14] states that on any compact conformal Hermitian manifold (M, c, J ), there exists a unique (up to scale) Hermitian metric g ∈ c, such that its Lee form θ is co-closed, i.e. satisfies δ g θ = 0. Such a metric
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is called a standard metric of c. By (2), a standard metric of (c, J ) can be equivalently defined by the equation 2i∂ ∂¯ F m−1 = dd c (F m−1 ) = 0. We now recall how, in four dimensions, the harmonic properties of the Lee form with respect to a standard metric are related to the parity of the first Betti number (compare with Theorem 1 above). Proposition 1 [14, 15]. Let M be a compact four-manifold endowed with a conformal class c of Hermitian metrics, with respect to an integrable almost complex structure J . Let g be a standard Hermitian metric in c. Then the following two conditions are equivalent: (i) The first Betti number b1 (M) is even. (ii) The Lee form θ of g is co-exact. Proof. For the sake of completeness we outline a proof of this result. Let M be a compact four-manifold endowed with a standard Hermitian structure (g, J ), and F and θ = J δ g F be the corresponding fundamental 2-form and Lee 1-form (with δ g θ = 0). We first prove that if b1 (M) is even, then θ is co-exact (this is [14, Théorème II.1]). Applying the Hodge ∗ operator to θ , this is equivalent to showing that d c F is exact. Recall that 2i∂ ∂¯ F = dd c F = 0 because g is standard. By Theorem 1, there exists a ¯ Kähler metric on (M, J ) and then, by the ∂ ∂-lemma, ¯ ∂¯ F = ∂ ∂α, for some (0, 1)-form α = ξ − i J ξ . We deduce d c F = dd c ξ , as required. In the other direction, we have to prove that if θ is co-exact then b1 (M) is even. We reproduce an argument from [15]. With respect to a standard metric g, the forms θ and J θ = −δ g F are both co-closed, and therefore the (0, 1)-form θ 0,1 := θ − i J θ is ¯ ∂-coclosed. In terms of Hodge decomposition, this reads as θ 0,1 = θh0,1 + ∂¯ ∗ , ¯ where ∈ 0,2 (M) and θh0,1 is the (∂¯ ∂¯ ∗ + ∂¯ ∗ ∂)-harmonic part of θ 0,1 . Note that = α + iβ, where α, β ∈ J,− (M) and α(·, ·) := −β(J ·, ·). We first claim that if θh0,1 = 0, then φ = F + β is a harmonic self-dual 2-form. Indeed, g g since J is integrable, it satisfies (D J X J )(J Y ) = (D X J )(Y ) (see (3)), and therefore g g J (δ β) = δ α, i.e. θ − i J θ = ∂¯ ∗ = δ g = δ g α + iδ g β. It follows that J θ = −δ g β, and thus δ g φ = J θ + δ g β = 0. By a well-known result of Kodaira (see e.g. [4]), a compact complex surface has even b1 (M) if and only if the dimension b+ (M) of the space of harmonic self-dual 2-forms on (M, g) is equal to 2h 2,0 (M) + 1, where h 2,0 (M) = dimC H∂¯2,0 (M); otherwise b+ (M) = 2h 2,0 (M). It follows that b1 (M) is even if and only if b+ (M) > 2dimC H∂¯2,0 (M).
Therefore, it suffices to show that θh0,1 = 0, provided that θ is co-exact (because φ will be then a harmonic self-dual 2-form which is not a real part of a holomorphic (2, 0)form). To this end, we consider the natural map κ : Hd1R (M) → H∂¯0,1 (M) ∼ = H 1 (M, O)
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from de Rham to Dolbeault cohomology given by ξ → ξ 0,1 on representatives. One easily checks that κ is well-defined and injective. Moreover, by the Noether formula (see e.g. [4]), κ is an isomorphism of (real) vector spaces if and only if b1 (M) is even; otherwise, the image of Hd1R (M) in H∂¯0,1 (M) is of real codimension one. For any element ξ 0,1 = ξ − i J ξ in the image of κ, we calculate its L 2 -hermitian product with θh0,1 : θh0,1 , ξ 0,1 L 2 = θ 0,1 , ξ 0,1 L 2 − ∂¯ ∗ , ξ 0,1 L 2 ¯ 0,1 L 2 = θ 0,1 , ξ 0,1 L 2 − , ∂ξ 1 i (θ, ξ ) L 2 + (J θ, α) L 2 2 2 i g 1 − (δ F, α) L 2 = (θ, ξ ) L 2 . 2 2
= θ 0,1 , ξ 0,1 L 2 = =
1 (θ, ξ ) L 2 2
It follows that θh0,1 , ξ 0,1 L 2 = 0, if θ is co-exact (because ξ is closed). Thus, in this case, the image of κ is contained in the complex subspace of H∂¯1 (M) which is orthogonal
to θh0,1 , and therefore would have real codimension at least 2, unless θh0,1 = 0.
Finally, we review some natural connections which are useful in the Hermitian context. An integrable almost complex structure J induces a canonical holomorphic structure on the tangent bundle T M, via the Cauchy–Riemann operator which acts on smooth sections X and Y of T M by ∂¯ X Y := 21 ([X, Y ] + J [J X, Y ]) = − 21 J (LY J )(X ). Identifying T M with the complex vector bundle T 1,0 M, this operator may be viewed as a partial connection and has the equivalent expression ∂¯ X Y = [X, Y ]1,0 ,
(6)
for any complex vector fields X and Y of type (0, 1) and (1, 0), respectively. In a similar way, any J -linear connection ∇ determines a partial connection ∂¯ ∇ on 1,0 T by projection, or acting on real vector fields by ∂¯ X∇ Y = 21 (∇ X Y + J ∇ J X Y ).
(7)
The operators ∂¯ and ∂¯ ∇ have the same symbol but do not coincide in general. However, it is well-known that for any Hermitian structure (g, J ), there exists a unique connection ∇, called the Chern connection of (g, J ), which preserves both J and g, and such that ¯ Note that the Chern connection ∇ has torsion, unless (g, J ) is Kähler. It is ∂¯ ∇ = ∂. related to the Levi–Civita connection D g by (see e.g. [16]): g
g(∇ X Y, Z ) = g(D X Y, Z ) + 21 d c F(X, J Y, J Z ).
(8)
In four dimensions, one uses (4) to rewrite (8) in the following form (cf. [14, 34]): g (9) ∇ X − D X = 21 X ⊗ θ − θ ⊗ X + J θ (X )J , where θ = g −1 (θ ) stands for the vector field g-dual to θ .
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3. Generalized Kähler Structures As described in the introduction, a generalized Kähler structure on a manifold M consists of a pair (J1 , J2 ) of commuting generalized complex structures such that J1 ·, J2 · determines a definite metric on T M ⊕ T ∗ M. The generalized complex structures J1 , J2 are integrable with respect to the Courant bracket on sections of T M ⊕ T ∗ M, given by [X + ξ, Y + η] H = [X, Y ] + L X η − L Y ξ − 21 d(i X η − i Y ξ ) + i Y i X H, which depends upon the choice of a closed 3-form H , called the torsion or twisting. The space of 2-forms b ∈ 2 (M) acts on T M ⊕ T ∗ M by orthogonal transformations via eb (X + ξ ) = X + ξ + i X b, and this action affects the Courant bracket in the following way [eb (W ), eb (Z )] H = eb [W, Z ] H +db . So, if (J1 , J2 ) is integrable with respect to the H -twisted Courant bracket, then (e−b J1 eb , e−b J2 eb ) is integrable for the (H + db)-twisted Courant bracket. A generalized complex structure J , because it is orthogonal and squares to −1, lies in the orthogonal Lie algebra, and therefore may be decomposed according to the splitting so(T M ⊕ T ∗ M) = ∧2 T M ⊕ End(T M) ⊕ ∧2 T ∗ M, or, in block matrix form,
J =
Aπ σ A
,
where π is a bivector field, A is an endomorphism of T M, and σ is a 2-form. Just as for an ordinary complex structure, the integrability of J may be expressed as the vanishing of a Nijenhuis tensor [J , J ] = 0 obtained by extending the Courant bracket. Restricted to ∧2 T M, this specializes to the usual Schouten bracket of bivector fields, requiring that [π, π ] = 0. This means that π is a Poisson structure. In [17], a complete characterization of the components of the generalized Kähler pair (J1 , J2 ) was given in terms of Hermitian geometry, which we now repeat here. Theorem 2 ([17], Theorem 6.37). For any generalized Kähler structure (J1 , J2 ), there exists a unique 2-form b and Riemannian metric g such that 1 J+ ± J− −(F+−1 ∓ F−−1 ) −b b e J1,2 e = , J+ ± J− 2 F+ ∓ F− where J± are integrable g-compatible complex structures and F± = g J± satisfy c F− = 0, dd+c F+ = 0. d+c F+ + d−
(10)
Conversely, any pair of g-compatible complex structures satisfying condition (10) define a generalized Kähler structure. Note that the pair (J1 , J2 ) is integrable with respect to the (H − db)-twisted Courant bracket where H = d+c F+ .
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An immediate corollary of this result and the preceding discussion is that the bivector fields π1 = −F+−1 + F−−1 , π2 = −F+−1 − F−−1 (11) are both Poisson structures, a fact first derived in [29] directly from (10). We also see from the theorem that by taking a triple (g, J+ , J− ) such that J+ = ±J− , c F and therefore (10) reduces to d c F = 0, which is nothing one obtains d+c F+ = d− − + + but the ordinary Kähler condition on (g, J+ ). We now proceed with an investigation of the class of generalized Kähler structures (g, J+ , J− ) for which the pair of complex structures commute but are unequal, i.e. which satisfy [J+ , J− ] = 0 and J+ = ±J− . Theorem A. Let (g, J+ , J− ) define a generalized Kähler structure with [J+ , J− ] = 0. Then the ±1-eigenspaces of Q = J+ J− define g-orthogonal J± -holomorphic foliations on whose leaves g restricts to a Kähler metric. Proof. Let T± M = ker(Q ∓ id) = ker(J+ ± J− ). Since ker(J+ ± J− ) = im(J+ ∓ J− ), we see that T± M coincide with the images of the Poisson structures π1 = (J+ − J− )g −1 ,
π2 = (J+ + J− )g −1
from (11). Therefore T± M are integrable distributions and determine transverse foliations of M. Since Q is an orthogonal operator, we see further that the foliations defined by its ±1 eigenvalues must be orthogonal with respect to the metric g. The complex structures induce decompositions T+ M ⊗ C = A ⊕ A and T− M ⊗ C = B ⊕ B, where A = T J1,0 M ∩ T J0,1 M, + −
B = T J1,0 M ∩ T J1,0 M + −
are themselves integrable since they are intersections of integrable distributions. We now show that A is preserved by the Cauchy-Riemann operator of J+ , proving that T+ M is a J+ -holomorphic sub-bundle. Let X be a (0, 1)-vector field for J+ and let Z ∈ C ∞ (A). Then ∂ X Z = [X, Z ]1,0 . Since T J1,0 M = A ⊕ B, we may project to these two components: + ∂ X Z = [X, Z ] A + [X, Z ] B . To show that A is J+ -holomorphic, we must show the vanishing of the second term, which upon expanding X = X A + X B , reads [X, Z ] B = [X A , Z ] B + [X B , Z ] B . The first term vanishes since A ⊕ A = T+ M ⊗ C is involutive, and the second term vanM is involutive. Therefore A is J+ -holomorphic. An identical ishes since A ⊕ B = T J0,1 − argument proves that B is J+ -holomorphic, and that both A, B are J− -holomorphic, as required. To show that g restricts to a Kähler metric on the leaves of T± M, observe that since c F . Similarly J+ = J− along the leaves of T− M, we have upon restriction d+c F+ = d− − c and along the leaves of T+ M we have J+ = −J− , so that upon restriction, d+c = −d− c c F+ = −F− , giving again d+ F+ = d− F− . But since the generalized Kähler condition c F , we conclude that both F are closed upon restriction to the forces d+c F+ = −d− − ± leaves of either foliation, therefore defining Kähler structures there.
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Remark 1. Along the above lines one can establish the following useful result: Let J+ and J− be a pair of commuting almost complex structures on a 2m-manifold M, such that J+ is integrable, and let T± M denote the sub-bundles of T M corresponding to (±1)-eigenspaces of Q = J+ J− . Then any two of the following three conditions imply the third. (a) T± M are integrable sub-bundles of T M; (b) T± M are holomorphic sub-bundles of T M with respect to J+ ; (c) J− is an integrable almost complex structure. cF =0 Under the above three conditions, one can show that the equation d+c F+ + d− − ± implies that Q is parallel with respect to the Chern connections ∇ of J± ; in other words, for a generalized Kähler structure with [J+ , J− ] = 0, the Chern connections ∇ ± have holonomy contained in U (m + ) × U (m − ), where dimR T± M = 2m ± .
Remark 2. Returning to the existence problem of generalized Kähler structures with [J+ , J− ] = 0, Theorem A implies that we must consider complex manifolds (M, J+ ) whose tangent bundle splits as a direct sum of two integrable, holomorphic sub-bundles T± M; the second complex structure J− is obtained from J+ by composing with Q, the product structure defining T± M. Thus, Question 1 from the Introduction reduces in this case to asking whether there is a Riemannian metric g on M which is compatible with the commuting pair (J+ , J− ), satisfying the generalized Kähler condition? Locally, the answer is always ‘yes’. Indeed, by using complex coordinates adapted to the transverse foliations, i.e. a neighborhood U = V × W ⊂ Cm 1 × Cm 2 such that T− U = T V, T+ U = T W , then for any Kähler metrics gV and gW on V and W , the product metric gU := gV × gW is Kähler with respect to both J± , and (gU , J± ) is a generalized Kähler structure. The global existence question is more subtle and just the splitting of T M as a direct sum of two integrable, holomorphic sub-bundles is a necessary but not sufficient condition: First of all, the fact that the leaves of the corresponding foliations must be Kähler leads to the obvious product example where one of the factors is of non-Kähler type: according to Theorem A, the resulting complex manifold cannot admit any compatible generalized Kähler metric g. Secondly, an example from [9] (combined with a standard averaging argument [6, 12]) shows there are compact complex manifolds whose holomorphic tangent bundle splits and defines two transversal foliations with Kähler leaves ¯ but which do not admit any Hermitian metric with ∂ ∂-closed fundamental form, and, therefore, has no compatible generalized Kähler metrics of the considered type. We end this section by showing that if there exists one generalized Kähler metric g on (M, J+ , J− ), then there is in fact a whole family parametrized by smooth functions. (This is similar to the variation of a Kähler metric by adding dd c f .) The construction is closely related to the potential theory developed in [13, 28]. We will use the integrable decomposition T M = T+ M ⊕ T− M, and the associated decomposition d = δ+ + δ− of the exterior derivative (induced by c = the ‘type’ decomposition ∧∗ T ∗ M = (∧∗ T+ M ∗ ) ⊗ (∧∗ T− M ∗ )), so that, defining δ± [J+ , δ± ], we have c c = ±δ+c + δ− . (12) d±
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Proposition 2. Let (M, g, J+ , J− ) be a generalized Kähler structure. Then, for any smooth function f ∈ C ∞ (M, R) and sufficiently small real parameter t, the 2-form c F˜+ = F+ + t (δ+ δ+c f − δ− δ− f)
(13)
defines a new Riemannian metric g˜ = − F˜+ J+ which is compatible with both J± , and such that (g, ˜ J± ) defines a generalized Kähler structure with unmodified torsion class [H ] ∈ H 3 (M, R). Proof. The J± -invariant 2-form in (13) defines the J− -fundamental form F˜− = g˜ J− , or c F˜− = F− + t (−δ+ δ+c f − δ− δ− f ). c F˜ = 0, since We now show that d+c F˜+ + d− − c c c c c ) + d− (−δ+ δ+c − δ− δ− ) = − δ+c δ− δ− + δ− δ+ δ+c d+c (δ+ δ+c − δ− δ− c c + δ+c δ− δ− − δ− δ+ δ+c = 0.
Finally, by the identity c c c ) = δ− δ+ δ+c − δ+c δ− δ− d+c (δ+ δ+c − δ− δ− c = (δ+ + δ− )δ+c δ− c c = dδ+ δ− ,
we see that d+c ( F˜+ − F+ ) is exact, showing that [d+c F+ ] = [d+c F˜+ ], completing the proof.
4. Generalized Kähler Four-Manifolds In dimensions divisible by four, generalized Kähler structures fall into two broad classes, defined by whether the complex structures ±J+ and ±J− induce the same or different orientations on the manifold. Definition 1. Let M be a manifold of dimension 4k. A triple (g, J+ , J− ), consisting of a Riemannian metric g and two g-compatible complex structures J± with J+ = ±J− , is called a bihermitian structure if J+ and J− induce the same orientation on M; otherwise, it is called ambihermitian. Similarly, an (am)bihermitian conformal structure is a triple (c, J+ , J− ), where c = [g] is a conformal class of (am)bihermitian metrics. In this section we will concentrate on the 4-dimensional case, where we have the following characterization of the generalized Kähler condition in terms of the Lee forms θ± . Proposition 3. Let (g, J± ) be an (am)bihermitian structure on a four-manifold M. Then c F = 0 is equivalent to θ + θ = 0 in the bihermitian case, the condition d+c F+ + d− − + − and to −θ+ + θ− = 0 in the ambihermitian case. The condition dd+c F+ = 0 means that g is a standard metric, i.e. δ g θ+ = 0. The twisting [H ] vanishes if and only if θ+ = δ g α for α ∈ 2 (M), i.e. the Lee form is co-exact.
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c F = (J θ ) ∧ F , so that Proof. By (4), we have d± ± ± ± ± c F− = (J+ θ+ ) ∧ F+ + (J− θ− ) ∧ F− . d+c F+ + d−
(14)
Note that in the bihermitian case F+ ∧ F+ = F− ∧ F− is twice the volume form vg , whereas in the ambihermitian case F+ ∧ F+ = −F− ∧ F− = 2vg . Therefore, applying the Hodge star operator ∗ to (14) and using the fact that δ g = − ∗ d∗ when acting on 2-forms, we obtain the result. As an immediate corollary of this result and Proposition 1 we obtain: Corollary 1. Let M be a generalized Kähler 4-manifold. If the torsion class [H ] ∈ H 3 (M, R) vanishes, then the first Betti number must be even (and hence M is of Kähler type); if [H ] = 0 then the first Betti number must be odd. Proof. We have to prove that for a compact generalized Kähler 4-manifold (M, g, J± ) the torsion class [H ] vanishes if and only if b1 (M) is even. By Proposition 3, [H ] = 0 if and only if the Lee forms θ± are coexact. Thus, our claim follows from Proposition 1. Note that in one direction, namely that [H ] = 0 implies b1 (M) is even, the result alternatively follows from the generalized Hodge decomposition for generalized Kähler structures proven in [18]. Bihermitian complex surfaces were studied in [1, 2, 8, 10, 20, 21, 24, 27, 30] and classified for even first Betti number in [2], where the classification of Poisson surfaces [3] is used, and existence is only partially proven. In fact, [2] provides enough to show that in this case, any bihermitian structure is conformal to a unique generalized Kähler structure, up to scale. Proposition 4. Let (c, J+ , J− ) be a bihermitian conformal structure on a compact four-manifold M with b1 (M) even. Then there is a unique (up to scale) metric g ∈ c such that (g, J+ , J− ) is generalized Kähler. Proof. By [2, Lemma 4], any standard metric g of (c, J+ ) (which is unique up to scale [14]) is standard for (c, J− ) as well, and furthermore θ+ + θ− = 0. By Proposition 3, this is equivalent to the generalized Kähler condition. In the case where the first Betti number is odd, bihermitian structures have been studied in [1, 2, 10, 30]. It follows from the results there that M must be a finite quotient 2 of (S 1 × S 3 )kCP , k ≥ 0. It is no longer true in this case that the standard metric provides a generalized Kähler metric in all cases. To the best of our knowledge, the only known examples of generalized Kähler structures on 4-manifolds with b1 (M) odd are given by standard metrics in the anti-self-dual bihermitian conformal classes described in [30]. We now turn to the ambihermitian case, where we establish a complete classification of generalized Kähler structures. We start with the following observation. Lemma 1. Let M be a four-manifold endowed with a pair (J+ , J− ) of almost complex structures inducing different orientations on M. Then, M admits a Riemannian metric compatible with both J± if and only if J+ and J− commute. In this case, the tangent bundle splits T M = T+ M ⊕ T− M (15) as an orthogonal direct sum of Hermitian complex line bundles defined as the ±1eigenbundles of Q = J+ J− .
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Proof. Let g be a Riemannian metric on M, compatible with J+ and J− . Fix the orientation on M induced by J+ . As discussed in § 2, the fundamental 2-forms F+ and F− are sections of + (M) and − (M), respectively. Since − (M) is in the +1eigenspace of ∧2 J+ , F− is J+ -invariant. Hence J+ and J− commute. The converse is elementary. Our next step is to identify the compact complex surfaces (M, J ) that admit a generalized Kähler metric (g, J+ , J− ) of ambihermitian type with J+ = J . Lemma 2. Let (g, J+ , J− ) be an ambihermitian structure on a 4-manifold M and let Q = J+ J− be the almost product structure it defines. Then the Lee forms satisfy θ+ = θ− if and only if T± M are holomorphic sub-bundles for J± . In this case, the standard metric in the conformal class defines a generalized Kähler metric. c F = 0. We claim that this Proof. If θ+ = θ− , then by Proposition 3, we have d+c F+ + d− − latter condition implies that Q = J+ J− is covariant constant with respect to the Chern connections ∇ ± (in fact this is true in any dimension). Then, T± M must be holomorphic. Since ∇ + J+ = 0 by definition, to prove ∇ ± Q = 0 it suffices to show that ∇ + J− = 0. From Eq. (3), we see that
∇ + − ∇ − = R, where R ∈ 1 (End(T M)) is given by c 2g(R X Y, Z ) = d+c F+ (X, J+ Y, J+ Z ) − d− F− (X, J− Y, J− Z ).
Consequently, ∇ + J− = ∇ − J− + [R, J− ]. By definition, ∇ − J− = 0, and expanding the commutator we obtain c 2g([R X , J− ]Y, Z ) = d+c F+ (X, J+ J− Y, J+ Z ) + d− F− (X, Y, J− Z ) c c +d+ F+ X, J+ Y, J+ J− Z ) + d− F− X, J− Y, Z ).
(16)
c F = 0. If Y is taken in T+ M and Z in T− M, then the terms cancel since d+c F+ + d− − + + If Y, Z ∈ T+ M, then trivially g((∇ X J− )Y, Z ) = g((∇ X J+ )Y, Z ) = 0 and similarly for Y, Z ∈ T− M. Hence ∇ + J− must vanish identically. Similarly, ∇ − J+ = 0, proving the claim. In the other direction, we use Eq. (9) and the fact that J− is skew-symmetric to express
∇ X+ J− = D X J− − 21 (X ∧ (J− θ+ ) + (J− X ) ∧ θ+ ), g
where α ∧ X = α ⊗ X − X ⊗ α for α ∈ T ∗ M and X ∈ T M. Finally, by (5), we obtain ∇ X+ J− = 21 (X ∧ J− (θ− − θ+ ) + J− X ∧ (θ− − θ+ ) ).
(17)
It is clear from Eq. (17) that ∇ + J− (and hence ∇ ± Q) vanishes if and only if θ+ = θ− . The final statement follows from Proposition 3. We are now ready to prove our second main result. Theorem B. Let (M, J ) be a compact complex surface whose holomorphic tangent bundle splits as the direct sum of holomorphic sub-bundles. Then (M, J ) admits a non-trivial generalized Kähler structure (g, J+ , J− ) with J+ = J and [J+ , J− ] = 0.
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Proof. Let T M = T+ M ⊕ T− M be the holomorphic splitting of T M. Since any holomorphic one-dimensional sub-bundle T± M ⊂ T M is automatically integrable, the almost complex structure J− = −J |T+ M + J |T− M is integrable (see Remark 1). By definition, J+ = J and J− commute, and J± induce different orientations. Clearly there are (many) Riemannian metrics g compatible with both J± , so that (g, J+ , J− ) defines an ambihermitian structure. Then the claim follows from Lemma 2, Proposition 3 and [14]. We now use the results from [5, 35] to give the complete list of ambihermitian generalized Kähler surfaces. Theorem 3. A compact complex surface (M, J ) admitting a generalized Kähler structure of ambihermitian type (g, J+ , J− ) with J+ = J is biholomorphic to one of the following: (a) a geometrically ruled complex surface which is a projectivization of a projectively flat holomorphic vector bundle over a compact Riemann surface; (b) a bi-elliptic complex surface, i.e. a complex surface finitely covered by a complex torus; (c) a compact complex surface of Kodaira dimension 1 and even first Betti number, which is an elliptic fibration over a compact Riemann surface, whose only singular fibres are multiple smooth elliptic curves; (d) a compact complex surface of general type, uniformized by the product of two hyperbolic planes H × H and with fundamental group acting diagonally on the factors; (e) a Hopf surface, with universal covering space C2 \ {(0, 0)} and fundamental group generated by a diagonal automorphism (z 1 , z 2 ) → (αz 1 , βz 2 ) with 0 < |α| ≤ |β| < 1, and a diagonal automorphism (z 1 , z 2 ) → (λz 1 , µz 2 ) with λ, µ primitive th roots of 1; (f) an Inoue surface in the family SM constructed in [23]. Proof. Let (M, g, J+ , J− ) be a compact generalized Kähler four-manifold of ambihermitian type. By Proposition 3 and Lemma 2, the holomorphic tangent bundle of (M, J+ ) must split as a direct sum of two holomorphic line bundles (T± M, J+ ). Complex surfaces with split tangent bundles were studied and essentially classified by Beauville [5]. We use his results to retrieve the list (a)–(f). When b1 (M) is even, the cases that occur according to [5] correspond to the surfaces listed in (a)–(d) of Theorem 3, modulo the fact that our description of the surfaces in (a) is slightly different from the one in [5, §5.5], and that the existence of a splitting of T M on any surface in (c) is not addressed in [5, §5.2]. To clarify these points, we notice that in the case of a ruled surface M = P(E) → , [5, Thm.C] implies that the universal cover is the product CP 1 × U, where U is the universal covering space of , and the diagonal action of π1 (M) = π1 () gives rise to a P G L(2, C) representation of π1 (), i.e. the holomorphic bundle E is projectively-flat as claimed in (a). Note that for any elliptic fibration f : M → as in (c), the base curve can be given the structure of an orbifold with a 2π/m i cone point at each point corresponding to a fibre of multiplicity m i (see, [5, § 5.2] and [35, § 7]). Since the Kodaira dimension of M is equal to 1, the orbifold Euler characteristic of must be negative, and therefore is a good orbifold uniformized by the hyperbolic space H. Since the first Betti number of M is even, it follows from [35, Thm.7.4] that the universal covering space of M is
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C × H, on which the fundamental group π1 (M) acts diagonally by isometries of the canonical product Kähler metric. When b1 (M) is odd, the possible cases are described in [5, §§ (5.2),(5.6),(5.7),(5.8)]. To prove that the only complex surfaces that really occur are those listed in (e) and (f) in Theorem 3 we have to exclude the possibility that (M, J+ ) is an elliptic fibration of Kodaira dimension 1, odd b1 (M), and with only multiple singular fibres with smooth reduction. It is shown in [5, § (5.2)] that for the holomorphic tangent bundle of such a surface to split, it must be covered by a product of simply connected Riemann surfaces on which the fundamental group acts diagonally. On the other hand, any elliptic surface M with Kodaira dimension 1 and b1 (M) odd is finitely covered by an elliptic fiber bundle M over a compact Riemann surface of genus > 1, which has trivial monodromy (cf. [35, p.139]). Since M (and hence M ) is not Kähler, b1 (M ) is odd too. Wall [35, p.141] showed that the universal cover of such an M is C × H on which π1 (M ) does not act diagonally. It then follows from Beauville’s result cited above that the holomorphic tangent bundle of M (and hence of M) does not split. Acknowledgements. We would like to thank P. Gauduchon, G. Grantcharov and N. J. Hitchin for their help and stimulating discussions. We are grateful to the referee for many valuable remarks and suggestions.
References 1. Apostolov, V: Bihermitian surfaces with odd first Betti number. Math. Z. 238, 555–568 (2001) 2. Apostolov, V., Gauduchon, P., Grantcharov, G.: Bihermitian structures on complex surfaces. Proc. London Math. Soc. 79(3), 414–428 (1999); Corrigendum, 92, 200–202 (2006) 3. Bartocci, C, Macrì, E: Classification of Poisson surfaces. Commun. Contemp. Math. 7, 89–95 (2005) 4. Barth, W., Hulek, K., Peters, C., Van de Ven, A.: Compact Complex Surfaces. 2nd ed., Heidelberg: Springer, 2004 5. Beauville, A.: Complex manifolds with split tangent bundle. In: ‘Complex analysis and algebraic geometry’. Berlin: de Gruyter, pp.61–70 (2000) 6. Belgun, F.: On the metric structure of non-Kähler complex surfaces. Math. Ann. 317, 1–40 (2000) 7. Buchdahl, N: On compact Kähler surfaces, Ann. Inst. Fourier 49, 287–302 (1999) 8. Bursztyn, H., Cavalcanti, G.R., Gualtieri, M.: Reduction of Courant algebroids and generalized complex structures. Adv. in Math., doi: 10.1016/j.aim.2006.09.008,2007 9. deBartolomeis, P., Tomassini, A.: On Solvable Generalized Calabi-Yau Manifolds. Ann. Inst. Fourier (Grenoble) 56(5), 1281–1296 (2006) 10. Dloussky, G.: On surfaces of class V I I0+ with numerically anticanonical devisor. Amer. J. Math. 128(3), 639–670 (2006) 11. Egidi, N.: Special metrics on compact complex manifolds. Diff. Geom. Appl. 14, 217–234 (2001) 12. Fino, A., Grantcharov, G.: Properties of manifolds with skew-symmetric torsion and special holonomy. Adv. Math. 189, 439–450 (2004) 13. Gates, S.J., Hull, C.M., Ro˘cek, M: Twisted multiplets and new supersymmetric nonlinear sigma models. Nuc. Phys. B 248, 157–186 (1984) 14. Gauduchon, P.: La 1-forme de torsion d’une variété hermitienne compacte. Math. Ann. 267, 495–518 (1984) 15. Gauduchon, P.: Le premier espace de cohomologie de deRham d’une surface complexe à premier nombre de Betti impair. Unpublished preprint 16. Gauduchon, P.: Hermitian connections and Dirac operators. Boll. Un. Mat. It. 11-B, Suppl. fasc. 257–288 (1997) 17. Gualtieri, M.: Generalized complex geometry. D. Phil. Thesis, University of Oxford, 2003, http://arxiv. org/list/math.DG/0401221 (2003) 18. Gualtieri, M.: Generalized geometry and the Hodge decomposition.http://arxiv.org/list/math.DG/ 0409093, 2004 19. Hitchin, N.J.: Generalized Calabi-Yau manifolds. Q. J. Math. 54, 281–308 (2003) 20. Hitchin, N.J.: Instantons and generalized Kähler geometry. Commun. Math. Phys. 265, 131–164 (2006) 21. Hitchin, N.J.: Bihermitian metrics on Del Pezzo Surfaces. http://arxiv.org/list/math.DG/0608213, 2006 22. Hitchin, N.J.: Private communication
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23. Inoue, M.: On surfaces of class V I I0 , Invent. Math. 24, 269–310 (1974) 24. Kobak, P.: Explicit doubly-Hermitian metrics. Differ. Geom. Appl. 10, 179–185 (1999) 25. Kobayashi, S., Nomizu, K.: Faundations of Differential Geometry. Vol.II, Newyork: Interscience Publishers, , 179 (1963) 26. Lamari, A.: Courants kählériens et surfaces compactes, Ann. Inst. Fourier 49, 263–285 (1999) 27. Lin, Y., Tolman, S.: Symmetries in generalized Kähler geometry. Commun. Math. Phys. 268, 199–222 (2006) 28. Lindström, U., Rócek, M., von Unge, R., Zabzine, M.: Generalized Kähler manifolds and off-shell supersymmetry. Commun. Math. Phys. 269, 833–849 (2007) 29. Lyakhovich, S., Zabzine, M.: Poisson geometry of sigma models with extended supersymmetry. Phys.Lett. B 548, 243–251 (2002) 30. Pontecorvo, M.: Complex structures on Riemannian four-manifolds. Math. Ann. 309, 159–177 (1997) 31. Siu, Y.-T.: Every K 3 surface is Kähler. Invent. Math. 73, 139–150 (1983) 32. Spindel, Ph., Sevrin, A., Troost, W., Van Proeyen, A.: Complex structures on parallelised group manifolds and supersymmetric σ -models. Phys. Lett. B 206, 71–74 (1988) 33. Todorov, A.: Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of K 3 surfaces. Invent. Math. 61, 251–265 (1980) 34. Vaisman, I.: Some curvature properties of complex surfaces, Ann. Mat. Pure Appl. 32, 1–18 (1982) 35. Wall, C.T.C.: Geometric structures on compact complex analytic surfaces. Topology 25, 119–153 (1986) Communicated by G.W. Gibbons
Commun. Math. Phys. 271, 577–589 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0191-9
Communications in
Mathematical Physics
The Classification of Static Electro–Vacuum Space–Times Containing an Asymptotically Flat Spacelike Hypersurface with Compact Interior Piotr T. Chru´sciel1, , Paul Tod2 1 LMPT, Fédération Denis Poisson, Parc de Grandmont, F-37200 Tours, France.
E-mail: [email protected]
2 Mathematical Institute and St John’s College, Oxford University, Oxford, United Kingdom.
E-mail: [email protected] Received: 8 November 2005 / Accepted: 5 October 2006 Published online: 22 February 2007 – © Springer-Verlag 2007
Abstract: We show that static electro–vacuum black hole space–times containing an asymptotically flat spacelike hypersurface with compact interior and with both degenerate and non–degenerate components of the event horizon do not exist. This is done by a careful study of the near-horizon geometry of degenerate horizons, which allows us to eliminate the last restriction of the static electro-vacuum no-hair theory.
1. Introduction A classical question in general relativity, first raised and partially answered by Israel [9], is that of classification of regular static black hole solutions of the Einstein–Maxwell equations. The most complete results existing in the literature so far are due to Simon [16], Masood–ul–Alam [12], Heusler [7, 8] and one of us (PTC) [2] (compare Ruback [15]) leading, roughly speaking, to the following: Suppose that ∀i, j
Qi Q j ≥ 0 ,
(1.1)
where Q i is the charge of the i-th connected degenerate component of the black hole. Then the black hole is either a Reissner-Norsdström black hole, or a Majumdar–Papapetrou black hole. The above results settled the classification question in the connected case. The general case, however, remained open. The aim of this work is to remove the sign conditions (1.1), finishing the problem. More precisely, we prove: Theorem 1.1. Let (M, g, F) be a static solution of the Einstein–Maxwell equations with defining Killing vector X . Suppose that M contains a connected and simply connected Partially supported by a Polish Research Committee grant 2 P03B 073 24.
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¯ of which is the union of an asymptotically flat space-like hypersurface , the closure end and of a compact set such that: 1. The Killing vector field X is timelike on . 2. The topological boundary ∂ ≡ \ of is a nonempty, two-dimensional, topological manifold, with gµν X µ X ν = 0 on ∂. Then, after performing a duality rotation of the electromagnetic field if necessary: (i) If ∂ is connected, then is diffeomorphic to R3 minus a ball. Moreover there exists a neighborhood of in M which is isometrically diffeomorphic to an open subset of the (extreme or non–extreme) Reissner–Nordström space–time. (ii) If ∂ is not connected, then is diffeomorphic to R3 minus a finite union of disjoint balls. Moreover the space–time contains only degenerate horizons, and there exists a neighborhood of in M which is isometrically diffeomorphic to an open subset of the standard Majumdar–Papapetrou space–time. The property that the set {gµν X µ X ν = 0} is a topological manifold, as well as simple connectedness of , will hold when appropriate further global hypotheses on M are made. In fact, Corollary 1.2 and Theorem 1.3 of [2], as well as the associated remarks, are valid now without the sign hypothesis (1.1), and will not be repeated here. The definitions and conventions used here coincide with those of the papers [1, 2], except for Sect. 2 where a different signature is used. The idea of the proof is to show that degenerate components of the horizon are only possible in standard Majumdar–Papapetrou space–times, as follows: we start by showing that the space-metric of a static degenerate horizon is spherical, with vanishing rotation one-form, and with constant “second-order surface gravity”. This leads to very precise information on the geometry of the orbit-space metric near the horizon. (This part of our work is inspired by the calculations in [14].) Let ϕ be the electric potential normalised so that ϕ tends to zero at infinity. One then uses two conformal transformations of Masood-ul-Alam to prove that this geometry is possible with ϕ = ±1 on a component of the horizon if and only if the metric is a Majumdar–Papapetrou metric. This, together with [3] (compare [6]), reduces the problem to one where |ϕ| is strictly bounded away from one, which has already been shown to lead to the Reissner-Nordström geometry in [2]. 2. The Near-Horizon Geometry of Static Electrovacuum Degenerate Killing Horizons In this section, we establish some results on the form of the metric of a static electrovac space-time near a degenerate Killing horizon. There is a range of formalisms available, and we shall exploit the Newman-Penrose spin-coefficient formalism1 as reviewed in [13] or [17]. To agree with the equations as they appear in these references, we shall in this section take the space-time signature to be (+ − −−). As in [4] we introduce Gaussian null coordinates near a component N of the event horizon, but with the signature changed so that the metric is g = r φdu 2 − 2dudr − 2r h a d x a du − h ab d x a d x b .
(2.1)
1 Alternatively, one could introduce the near-horizon geometry as in [14] (compare [4]), and use the discussion of Kundt’s class of metrics in [10]. We are grateful to H. Reall for this observation.
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The Killing vector X is ∂u with norm g(X, X ) = r φ and N is located at r = 0. The surface gravity is κ = −∂r (r φ) at r = 0, and degeneracy of N means that κ vanishes. It follows that φ = r A(r, x a ) for some A. We shall show the following: (i) The co-vector field h a defined on the spheres (r = r0 , u = u 0 ) vanishes to order r. (ii) The metric h ab on the spheres S = (r = 0, u = u 0 ) has constant Gauss curvature K. (iii) On N , A = K > 0, so that Ah ab |r =0 is the unit round metric on S 2 . √ (iv) In the purely electric case, the electrostatic potential ϕ satisfies ∂r ϕ = ± A at N . From [4] we know that staticity implies that, at N , h a is a gradient, say h a d x a |r =0 = dλ. Now in the metric (2.1), choose the coordinates x a so that they are isothermal on N and then introduce ζ = x 1 + i x 2 . Choosing m to be proportional to d ζ¯ at r = 0, the metric becomes g = r 2 Adu 2 − 2dudr − 2r (hdζ + hdζ )du − 2mm , (2.2) where m = − Z˚ d ζ¯ + O(r ) , ∂λ + O(r ) , h= ∂ζ in terms of functions λ (real) and Z˚ (complex) of ζ and ζ . Our goal for the next two pages is Eq. (2.11). A version of this equation has appeared in the literature before, as Eq. (50) of [11]. In the interest of making the current paper self-contained, and to introduce notation, we shall rederive it in the spin-coefficient formalism. We will calculate in the null tetrad (l µ , n µ , m µ , m µ ) by r2 A ∂r , 2 µ n ∂µ = = −∂r , l µ ∂µ = D = ∂u +
µ
m ∂µ = δ =
1 Z
∂ζ +
r Y
∂ζ −
rh Z
+
r 2h Y
∂r ,
where Z = Z˚ + O(r ). Here and elsewhere, a circle over a quantity indicates the value at r = 0. We follow the numbering of [13] in the following. The spin-coefficients are calculated following (A.2) with the result2 α=−
1 ∂ Z˚
−
1 h + O(r ) , 4 Z˚
2 Z˚ Z˚ ∂ζ 1 1 ∂ Z˚ − h + O(r ) , β= 2 Z˚ Z˚ ∂ζ 4 Z˚
2 Note that the term (α − β)π in (A.3g) is misprinted, and should read (α − β)π ¯ .
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1 ∂ + O(r ) , log Z Z −1 r =0 4 ∂r 1 = r A˚ + O(r 2 ) , 2 1 ∂ log(Z Z ) + O(r ) , µ=− r =0 2 ∂r 1 h + O(r ) , τ = 2 Z˚ γ =
together with π = −τ , ν = 0, λ = 1 + O(r ), ρ = O(r 2 ), κ = O(r 2 ), and σ = O(r 2 ). From these, we calculate the curvature components from (A.3), setting the scalar curvature to zero. For the Weyl spinor, we find 0 = O(r 2 ), 1 = O(r ), 3 = O(1), and 4 = O(1) together with two expressions for 2 : 1 1 2 = ∂ζ ∂ζ λ − ∂ζ λ∂ζ λ + O(r ) , (2.3) 2 2 Z˚Z˚ 1 ˚ 1 = A−K + ∂ζ λ∂ζ λ + O(r ), (2.4) 4 2 Z˚ Z˚ where K = − 1 ∂ζ ∂ζ log( Z˚ Z˚ ) , which is the Gauss curvature of S. Z˚ Z˚
For the Ricci spinor, we find 00 = O(r 2 ), 01 = O(r ) and the remaining components are O(1). In particular we have 1 1 ˚ A+K + ∂ζ λ∂ζ λ + O(r ) , (2.5) 11 = 4 2 Z˚ Z˚ ∂ζ ( Z˚ Z˚ ) 1 1 2 02 = ∂ζ λ − (∂ζ λ) + O(r ) . ∂ζ ∂ ζ λ − (2.6) 2 2 ( Z˚ Z˚ ) 2 Z˚ Since we are concerned with electrovac solutions, the Ricci spinor AB A B is obtained from the Maxwell spinor φ AB according to AB A B = kφ AB φ A B , where k =
2G . c4
(We shall often assume G = c = 1.) In particular this means that 00 = kφ0 φ 0 ; 02 = kφ0 φ 2 ; 11 = kφ1 φ 1 .
(2.7)
We saw above that 00 = O(r 2 ) so that, from the first equation in (2.7), we deduce that φ0 = O(r ) and since φ2 = O(1) we must have 02 = O(r ). By (2.6) this is ∂ ζ ∂ζ λ −
∂ζ ( Z˚ Z˚ )
1 ∂ζ λ − (∂ζ λ)2 = 0. 2 ( Z˚ Z˚ )
A second equation on λ follows from (2.3), (2.4) and (2.5) as 1 1 1 ∂ζ ∂ζ λ − ∂ζ λ∂ζ λ = kφ1 φ 1 − K . 2 2 2 Z˚ Z˚
(2.8)
(2.9)
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The component φ1 of the Maxwell field is constrained by the Maxwell equations, specifically by (A.5b) of [13] which here becomes δφ1 + 2π φ1 = O(r ) , or ∂ζ φ1 − φ1 ∂ζ λ = O(r ) .
(2.10)
This integrates at once to give φ1 = χ eλ + O(r ), where χ is holomorphic in ζ on S. It is also bounded (since it is the contraction of the self-dual part of the Maxwell field with the volume form of S), and so it must be constant (the value of this constant is proportional to the charge of the black hole). We use this in (2.9), and then we can write (2.8) and (2.9) jointly as a tensor equation on S as 1 ∇a ∇b λ − ∇a λ∇b λ = − R˚ ab + 2k|χ |2 e2λ h˚ ab , 2
(2.11)
where, as before, a circle over a quantity indicates the value at r = 0. As noted above, this is equivalent to Eq. (50) of [11] (also our (2.10) is equivalent to their (47)). We shall deduce from (2.11) that necessarily λ is constant and h˚ ab is the metric of a round sphere. First, introduce ψ = e−λ/2 so that (2.11) becomes ∇a ∇b ψ =
ψ ˚ Rab − k|χ |2 ψ −3 h˚ ab , 2
(2.12)
ψ ˚ R − 2k|χ |2 ψ −3 . 2
(2.13)
and from the trace of this find
ψ =
Take ∇ b on (2.12) and use (2.13) to find ∇a ψ 3 R˚ − 12k|χ |2 ψ −1 = 0 , so that c1 12k R˚ = 3 + 4 |χ |2 ψ ψ
(2.14)
for some constant c1 . Insert this into (2.13), and then into (2.12), to obtain c1 ψ + 8k|χ |2 , 2ψ 3 c1 ψ + 8k|χ |2 ˚ ∇a ∇b ψ = h ab . 4ψ 3
ψ =
(2.15) (2.16)
The possibility that c1 ≥ 0 or χ = 0 leads to ψ of constant sign, which is possible on a compact manifold only if c1 = χ = 0 and ψ is constant, leading to R˚ ab = 0. But there are no such metrics on S 2 , hence c1 < 0 and χ = 0 .
(2.17)
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Multiply (2.12) by ∇b ψ and integrate using (2.14) to find |∇ψ|2 = c2 −
2k c1 − 2 |χ |2 2ψ ψ
(2.18)
for some constant c2 . In order to analyse the critical set of ψ, consider any geodesic γ (s), s ∈ I , with unit tangent γ˙ . From (2.16) we find d 2ψ c1 ψ + 8k|χ |2 = . 2 ds 4ψ 3 Suppose that p is a critical point of ψ and that the right-hand-side vanishes at p. Then ψ(γ (s)) = ψ( p) is a solution satisfying the right initial data at p, and uniqueness of solutions of ODEs shows then that ψ is constant on all geodesics through p. It easily follows from (2.12) and (2.15) that both ψ and the metric are analytic in an appropriate chart, so this situation arises if and only if ψ is constant. Supposing it is not, we conclude that the Hessian of ψ is strictly definite at critical points, and the Laplacian does not vanish there. Consequently, the critical set of ψ is either a finite collection of points, or the whole sphere. In the former case, each critical point is a strict local extremum. In both cases the set := {∇ψ = 0} (2.19) is connected (possibly empty). We next show that c2 is negative. Let a := inf ψ , b := sup ψ , then 0 < a ≤ b < ∞ since ψ is positive and smooth. Suppose that a = b. At any p such that ψ( p) = b we have ∇ψ( p) = 0 and ψ( p) < 0.3 The latter together with (2.15) gives c1 b < −8k|χ |2 , while the former reads, in view of (2.18), c1 b = 2c2 b2 − 4k|χ |2 , leading to c2 b2 < −2k|χ |2 , implying c2 < 0. Our next target is to derive (2.21) below. Let p− be a minimum of ψ and let γ be any geodesic starting at p− , with tangent γ˙ of unit length. Then ψ ◦ γ is a solution of the Cauchy problem d 2ψ c1 ψ + 8k|χ |2 = , ψ(0) = a , 2 ds 4ψ 3
dψ (0) = 0 , ds
which shows that ψ depends only upon the geodesic distance from p− , and not on the direction of the geodesic. Thus, the level sets of ψ coincide with the geodesic spheres centred at p− , within the injectivity radius of p− . A similar conclusion holds at any maximum of ψ. 3 The inequality has to be strict, otherwise ψ is constant, either by the geodesic argument just given, or by the fact that ψ would again have constant sign.
Classification of Static Electro–Vacuum Space–Times
583
On , as defined in (2.19), we may use ψ locally as a coordinate, leading to the following form of the metric dψ 2 + H 2 (ψ, φ)dφ 2 , F 2 (ψ) where φ is a local coordinate on the level sets of ψ, and F 2 (ψ) = |∇ψ|2 = c2 −
c1 2k − 2 |χ |2 . 2ψ ψ
(2.20)
Equation (2.13) implies H 2 = F 2 (ψ)G(φ) and we may redefine φ to make G = 1, leading to the local form dψ 2 + F 2 (ψ)dφ 2 . (2.21) F 2 (ψ) Within the radius of injectivity of p− and away from p− we have dψ/F = dρ, where ρ is the distance function from p− . Note that F 2 (ψ) = c3 ρ 2 + o(ρ 2 )
(2.22)
for small ρ, for a constant c3 which can be read off from (2.20); this is compatible with elementary regularity provided that φ has appropriate periodicity. The integral curves of ∇ψ can be used to obtain a diffeomorphism between the level sets of ψ within , which shows that (2.21) provides a global representation of the metric on . Let p+ be any point in S such that ψ( p+ ) = b, then p+ ∈ , and likewise the level sets of ψ near p+ are geodesic spheres. This, and what has been said, implies that the set Sˆ := { p− } ∪ ∪ { p+ } is both open and closed in S, hence Sˆ = S. Furthermore, within the radius of injectivity of p+ and away from p+ , we have dψ/F = d ρ, ˆ where ρˆ is the distance function from p+ . Since the periodicity of φ has already been determined we must have F 2 (ψ) = c32 ρˆ 2 + o(ρˆ 2 )
(2.23)
for small ρ. ˆ Eliminating c3 between (2.22) and (2.23), a standard calculation leads to (F 2 ) (a) = −(F 2 ) (b) . Equivalently, c1 ab(a 2 + b2 ) = −8k|χ |2 (b3 + a 3 ) . Eliminating c2 from the equations F(a) = F(b) = 0 one finds c1 ab = −4k|χ |2 (a + b) . Substitute into the previous equation to obtain a = b, which is a contradiction. We conclude that regularity of the metric of S requires that |∇ψ| = 0, so ψ and therefore λ are constant, which establishes (i). From (2.11), if λ is constant then the
584
P. T. Chru´sciel, P. Tod
metric h˚ ab is that of a round sphere, establishing (ii). Next, from (2.3), with λ constant, 2 is zero at N and then from (2.4), A˚ = K , establishing (iii). Finally, recall that the electrostatic potential ϕ is defined by the equation dϕ = i X F , where F is the Maxwell two-form and i X denotes contraction with X . Since X = l + r 2 An/2, in the purely electric case we have ϕ1 =
1 1 ∂ϕ Fab l a n b = . 2 2 ∂r
But (see (2.7) and (2.5)) 11 = 2ϕ1 ϕ¯ 1 = √ so ∂r ϕ = ± A at N , establishing (iv).
1 ˚ A + O(r ) , 2
3. Proof of Theorem 1.1 As argued by Heusler [7] (compare [2, Lemma 3.2]), a duality rotation guarantees that the Maxwell field is purely electric. Following [1], we equip with the orbit space metric4 γ defined as γ (Y, Z ) = g(Y, Z ) −
g(X, Y )g(X, Z ) , g(X, X )
(3.1)
where X is the defining Killing vector, that is, the Killing vector which asymptotes ∂/∂t in the asymptotic regions, and satisfies the staticity condition. As in [12], we consider the functions ± =
(1 ± V )2 − ϕ 2 , 4
(3.2)
where −V 2 is the Lorentzian length of the Killing vector X , and the metrics g± := 2± γ .
(3.3)
(The interest in those metrics arises from the positivity of their scalar curvatures [12].) Proposition 3.4 of [2] shows that 0 ≤ |ϕ| ≤ 1 − V ,
(3.4)
hence the functions ± are non-negative, with the inequalities being strict in the interior unless the metric is locally a Majumdar–Papapetrou metric (compare [12]). From now on we suppose that this is not the case. The possibility that |ϕ| is strictly bounded away from one leads to the Reissner-Nordström solutions [2], so we assume, for contradiction, that there exists a component of the horizon N on which ϕ|r =0 =: ∈ {±1}, then N is degenerate [2, Prop. 3.4]. By the results in Sect. 2 and by (3.4) we have ˚ + O(r 2 )) , ϕ = (1 − Ar 4 In [1] the symbol h is used; to avoid a clash of notation with the previous section we use γ instead.
Classification of Static Electro–Vacuum Space–Times
585
√ ˚ + O(r 2 ) we obtain and since V = |guu | = Ar 1 ˚ + O(r 2 ))2 − (1 − Ar ˚ + O(r 2 ))2 = Ar ˚ + O(r 2 ) , (1 + Ar + = (3.5) 4 1 ˚ + O(r 2 ))2 − (1 − Ar ˚ + O(r 2 ))2 = O(r 2 ) . − = (1 − Ar (3.6) 4 We can use the space-times coordinates r and x a of Sect. 2 as coordinates on the orbit space near {r = 0}. From (3.1) and Sect. 2 we infer gr2u 1 = , 2 ˚ + O(r 3 ) guu Ar gr u gau O(r 2 ) − = , ˚ 2 + O(r 3 ) guu Ar
γrr = grr −
(3.7)
γra = gra
(3.8)
γab = gab −
gau gbu O(r 4 ) = h˚ ab + O(r ) + . ˚ 2 + O(r 3 ) guu Ar
(3.9)
This leads to the following form of the metric g+ : 2 2 g+ = + × guu γ = + dr 2 + O(r 2 )d x a dr + (guu gab + O(r 4 ))d x a d x b g guu uu = 1 + O(r ) dr 2 + O(r 2 )d x a dr + r 2 (1 + O(r )) A˚ h˚ ab + O(r 3 ) d x a d x b . (3.10) We want to think of the coordinate r above as a radial coordinate near the origin of R3 . First, since A˚ h˚ ab is the unit round metric on S 2 we have r 2 A˚ h˚ ab d x a d x b = (d x i )2 − dr 2 , 2 so this part of the metric combines with the leading
iparti of the dr term in (3.10) to give a smooth tensor field. Next, the form r dr = x d x is smooth with respect to the standard differentiable structure on R3 , and vanishes at the origin as O(| x |), so that the term O(r )dr 2 gives a contribution which, in the coordinates x i , vanishes at the origin as O(| x |), with bounded first derivatives, and second derivatives dominated by a multiple of r −1 . To understand the remaining terms, a seemingly straightforward approach is to use spherical coordinates on S 2 . However, those coordinates are singular at the z-axis, which leads to problems when one wishes to capture the regularity of the resulting metric. An alternative way of handling this proceeds as follows: Think of the sphere S 2 as a subset of R3 with global coordinates xˆ i , and let β be any smooth one-form on S 2 . Then β can be uniquely extended to a smooth one-form βˆi (xˆ j )d xˆ i defined on R3 \ {0} by requiring that
xˆ i ∂xˆ i βˆ j = 0 , βˆ j xˆ j = 0 , i S∗2 (βˆi d xˆ i ) = β , where i S∗2 is the pull-back map. Similarly any two-covariant tensor field α on S 2 can be uniquely extended to a smooth tensor field αˆ k (xˆ j )d xˆ k d xˆ defined on R3 \ {0} by requiring that xˆ i ∂xˆ i αˆ k = 0 , αˆ k xˆ k = αˆ k xˆ = 0 , i S∗2 (αˆ k d xˆ k d xˆ ) = α .
586
P. T. Chru´sciel, P. Tod
a) := B(0,
a) \ {0}
denote a punctured coordinate ball centred at 0,
of Let B ∗ (0, 3 radius a, in R and consider the map
a) → (0, ∞) × R3 , : B ∗ (0, i x i i x → r = (x i )2 , xˆ = . r A term βa d x a dr in the metric extends as above to a tensor field βˆi d xˆ i dr on (0, ∞)×R3 , and its pull-back by produces a term i dxi xi dxi x ∗ (βˆi d xˆ i ) = βˆi d = βˆi − βˆi 2 dr = βˆi . r r r r =0
This shows that the terms O(r 2 )d x a dr in (3.10) give contributions, in the coordinates x i , of the form xi j k r × smooth function of dx dx . r A similar analysis of the remaining d x a d x b terms shows that, in the coordinates x i , the metric g+ can be extended by continuity through the origin to a metric still denoted by the same symbol, of the form g+ = (δi j + O(| x |))d x i d x j ,
(3.11)
with derivatives satisfying, for some constant C, x |−1 . |∂ j (g+ )k | ≤ C , |∂i ∂ j (g+ )k | ≤ C|
(3.12)
The key fact in the remainder of the proof is the following: the positivity of the scalar curvatures of both metrics g± implies a differential inequality on the the quotient − /+ , which is incompatible with the vanishing of this quotient at the origin. Some technicalities are required because of the potential lack of smoothness of g± at the origin. We start by rescaling g+ so that the scalar curvature vanishes:
b)) ∩ Proposition 3.1. There exists b > 0 and a positive function ψ ∈ C(B(0, ∞ ∗ 4
C (B (0, b)), bounded away from zero, such that the scalar curvature of ψ g+ vanishes
b). on B ∗ (0,
b) of the equation Proof. We want to construct a solution ψ in B ∗ (0, R(ψ 4 g+ )ψ 5 = −8 g+ ψ + R(g+ )ψ = 0 . We look for ψ of the form ψ = 1 + u, where u vanishes on a coordinate sphere of radius b. The equation for u reads −8 g+ u + R(g+ )u = −R(g+ ) .
(3.13)
From (3.11)–(3.12) one finds that the scalar curvature R(g+ ) of the metric g+ satisfies R(g+ ) ≤
C | x|
(3.14)
Classification of Static Electro–Vacuum Space–Times
587
for some constant C . By scaling x → b−1 x we can assume b = 1, note that (3.14) becomes then C b R(g+ ) ≤ . (3.15) | x|
1)). To solve (3.13) one can It follows that the right-hand-side of (3.13) is in L 2 (B(0, proceed as follows: Let 0 < < 1; we wish, first, to show the existence of a solution
1) \ B(0,
) of (3.13) that vanishes both on the coordinate sphere of u ∈ C ∞ (B(0, radius one and on that of radius . This will follow from the standard theory if we can show that the solutions of the homogeneous equation, still denoted by u are unique. For this, extend u by zero to the interior ball of radius , and recall the Hardy inequality
B(0,1)
u2 ≤C r2
B(0,1)
|du|2
(note that the standard version thereof uses the flat metric, but by uniform ellipticity both the measure and the norm of du can be taken with respect to the current metric with an appropriately modified constant C). We then have
B(0,1)
|R(g+ )|u 2 ≤ C b
B(0,1)
u2 ≤ CC b r
B(0,1)
We can choose b small enough to obtain CC b ≤ 1, then 0= u (−8 g+ u + R(g+ )u )
B(0,1)\B( 0,) 2 2 = 8|du | + R(g+ )u ≥ 7
B(0,1)\B( 0,)
|du|2 .
B(0,1)\B( 0,)
|du |2 ,
giving uniqueness, as desired. In the case of the non-homogeneous equation the last calculation further gives 7 |du |2 ≤ u (−8 g+ u + R(g+ )u ) = − R(g+ )u
B(0,1)
B(0,1)
1 (R(g+ )r )2 + 2 B(0,1)
1 ≤ (R(g+ )r )2 + 2 B(0,1)
≤
B(0,1)
2 B(0,1)
C |du 2 | , 2 B(0,1)
u 2 r2
where we have used 2x y ≤ −1 x 2 + y 2 . Choosing = 2C −1 , we can carry the last term to the left-hand-side, which shows that the sequence u is bounded in H 1 . Stan 1). We can dard arguments imply the existence of a function u solving (3.13) on B ∗ (0, use Sobolev’s inequality and [18, Corollary 1.1, p. 29] to conclude that u is a weak
1). By the last calculation one has g+ u ∈ L 2 (in a weak solution of (3.13) on B(0, sense), and since the metric g+ is Lipschitz continuous we can invoke elliptic theory [5, Theorem 8.8] to conclude that u ∈ H 2 ⊂ C 0 . Since all the norms involved can be made arbitrarily small when b is small enough, we will have u L ∞ ≤ 1/2 for appropriate b, hence ψ ≥ 1/2.
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We turn our attention now to the metric g− ; as already pointed out, the sign of its scalar curvature will provide the desired contradiction. In the coordinates x introduced
b) we have above, on B ∗ (0, 2− 2− −4 2 g− = − γ = ψ ψ 4 g+ . g+ = 2+ 2+
=:gˆ =:ψˆ 4
with ψˆ nonnegative, and vanishing precisely at the origin. The transformation law of
b), the scalar curvature under conformal transformation gives, on B ∗ (0, ˆ ψ = R(g− )ψˆ 5 ≥ 0 . −8 gˆ ψˆ + R(g) =0
b) which equals ψˆ on the boundary of Let H be the gˆ –harmonic function in B(0, the ball. The function H can be constructed by minimising |d H |2gˆ dµgˆ
B(0,b)
b)). By the in the class of functions satisfying the boundary data, hence H ∈ H 1 (B(0, maximum principle [5, Theorem 8.1] (note that uniform ellipticity of the metric, guaranteed by (3.11), and the H 1 character of H , suffice for this) H is bounded away from zero.
b) − B(0,
e) for e < b tells us Applying the maximum principle on the set B(0, ˆ that ψ is greater than or equal to the harmonic function which equals ψˆ on the outer
and zero on the inner boundary Se (0).
5 In the limit as e goes to zero, boundary Sb (0) this harmonic function converges, uniformly on compact subsets, to a bounded function
b). By Serrin’s removable singularity Hˆ which solves the Laplace equation on B ∗ (0, theorem [18, Theorem 1.19, p. 30] (note again that (3.11), suffices for this) it holds that ∗
ˆ Hˆ = H | B ∗ (0,b)
. Thus, ψ ≥ H (x) in B (0, b). Since H (x) is strictly positive in B(0, b),
by the comparison principle [5, Theorem 8.1] we obtain ˆ and ψ(z) is continuous at 0,
ˆ ψ(0) > 0. This contradicts the fact that − / + tends to zero as x approaches the origin.
= ±1 is only possible for Majumdar–Papapetrou solutions, and the remarks Hence ϕ(0) at the end of the Introduction complete the proof. Acknowledgements. It is a pleasure to thank M. Anderson, R. Bartnik, H. Reall and especially H. Bray for helpful discussions. The authors acknowledge hospitality and support from the Newton Institute, Cambridge, during work on this paper.
References 1. Chru´sciel, P.T.: The classification of static vacuum space–times containing an asymptotically flat spacelike hypersurface with compact interior. Class. Quantum Grav. 16, 661–687 (1999), gr-qc/9809088 2. Chru´sciel, P.T.: Towards the classification of static electro–vacuum space–times containing an asymptotically flat spacelike hypersurface with compact interior. Class. Quantum Grav. 16, 689–704 (1999), gr-qc/9810022 5 The comparison argument in this paragraph has been pointed out to us by H. Bray.
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3. Chru´sciel, P.T., Nadirashvili, N.S.: All electrovacuum Majumdar–Papapetrou spacetimes with non–singular black holes. Class. Quantum Grav. 12, L17–L23 (1995), gr-qc/9412044 4. Chru´sciel, P.T., Reall, H.S., Tod, K.P.: On non-existence of static vacuum black holes with degenerate components of the event horizon. Class. Quantum Grav. 23, 549–554 (2005), gr-qc/0512041 5. Gilbarg D., Trudinger N.: Elliptic partial differential equations of second order. 2nd ed., Berlin-Heidelberg-New York, Springer Verlag, 1977 6. Hartle, J.B., Hawking, S.W.: Solutions of the Einstein–Maxwell equations with many black holes. Commun. Math. Phys. 26, 87–101 (1972) 7. Heusler, M.: On the uniqueness of the Reissner–Nordström solution with electric and magnetic charge. Class. Quantum Grav. 11, L49–L53 (1994) 8. Heusler, M.: On the uniqueness of the Papapetrou–Majumdar metric. Class. Quantum Grav. 14, L129– L134 (1997), gr-qc/9607001 9. Israel, W.: Event horizons in static electrovac space-times. Commun. Math. Phys. 8, 245–260 (1968) 10. Kramer D., Stephani H., MacCallum M., Herlt E.: Exact solutions of Einstein’s field equations. Cambridge: Cambridge University Press, 1980 11. Lewandowski, J., Pawłowski, T.: Extremal isolated horizons: A local uniqueness theorem. Class. Quantum Grav. 20, 587–606 (2003), gr-qc/0208032 12. Masood–ul–Alam, A.K.M.: Uniqueness proof of static charged black holes revisited. Class. Quantum Grav. 9, L53–L55 (1992) 13. Newman E.T., Tod K.P.: Asymptotically flat space–times. In: General Relativity and Gravitation, Held, A. ed., New York and London: Plenum 1980, pp. 1–36 14. Reall, H.S.: Higher dimensional black holes and supersymmetry. Phys. Rev. D 68, 024024 (2003), hep-th/0211290 15. Ruback, P.: A new uniqueness theorem for charged black holes. Class. Quantum Grav. 5, L155– L159 (1988) 16. Simon, W.: Radiative Einstein-Maxwell spacetimes and ‘no–hair’ theorems. Class. Quantum Grav. 9, 241–256 (1992) 17. Stewart J.: Advanced general relativity. Cambridge Monographs on Mathematical Physics, Cambridge: Cambridge University Press 1990 18. Véron L.: Singularities of solutions of second order quasilinear equations. Pitman Research Notes in Mathematics Series, Vol. 353, Harlow: Longman 1996 Communicated by G.W. Gibbons
Commun. Math. Phys. 271, 591–634 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0174-2
Communications in
Mathematical Physics
Noncommutative Instantons from Twisted Conformal Symmetries Giovanni Landi1 , Walter D. van Suijlekom2 1 Dipartimento di Matematica e Informatica, Università di Trieste, Via A. Valerio 12/1, I-34127 Trieste, Italy
and INFN, Sezione di Trieste, Trieste, Italy. E-mail: [email protected]
2 Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany.
E-mail: [email protected] Received: 7 February 2006 / Accepted: 31 March 2006 Published online: 22 February 2007 – © Springer-Verlag 2007
Abstract: We construct a five-parameter family of gauge-nonequivalent SU (2) instantons on a noncommutative four sphere Sθ4 and of topological charge equal to 1. These instantons are critical points of a gauge functional and satisfy self-duality equations with respect to a Hodge star operator on forms on Sθ4 . They are obtained by acting with a twisted conformal symmetry on a basic instanton canonically associated with a noncommutative instanton bundle on the sphere. A completeness argument for this family is obtained by means of index theorems. The dimension of the “tangent space” to the moduli space is computed as the index of a twisted Dirac operator and turns out to be equal to five, a number that survives deformation. Contents 1. Introduction . . . . . . . . . . . . . . . . . . 2. Connections and Gauge Transformations . . . 2.1 Connections on modules . . . . . . . . . 2.2 Gauge transformations . . . . . . . . . . 3. Toric Noncommutative Manifolds Mθ . . . . . 3.1 Deforming a torus action . . . . . . . . . 3.2 The manifold Mθ as a fixed point algebra 3.3 Vector bundles on Mθ . . . . . . . . . . . 3.4 Differential calculus on Mθ . . . . . . . . 4. Gauge Theory on the Sphere Sθ4 . . . . . . . . 4.1 The principal fibration Sθ7 → Sθ4 . . . . . 4.2 Associated bundles . . . . . . . . . . . . 4.3 Yang–Mills theory on Sθ4 . . . . . . . . . 5. Construction of SU(2)-Instantons on Sθ4 . . . . 5.1 The basic instanton . . . . . . . . . . . . 5.2 Twisted infinitesimal symmetries . . . . .
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G. Landi, W. D. van Suijlekom
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1. Introduction The importance of Yang–Mills instantons in physics and mathematics needs not be stressed. They have played a central role since their first appearance [7] and are most elegantly described via the so-called ADHM construction [5, 4]. The generalization in [30] of this method for instantons on a noncommutative space R4 has found several important applications notably in brane and superconformal theories. Toric noncommutative manifolds M were constructed and studied in [15]. One starts with any (Riemannian spin) manifold M carrying a torus action and then deforms the torus to a noncommutative one governed by a real antisymmetric matrix of deformation parameters. The starting example of [15] – the archetype of all these deformations – was a four dimensional sphere Sθ4 , which came with a natural noncommutative instanton bundle endowed with a natural connection. At the classical value of the deformation parameter, θ = 0, the bundle and the connection reduces to the one of [7]. The present sphere Sθ4 can be thought of [14] as a one point compactification of a noncommutative R4θ which is structurally different from the one considered in [30]. In [27] this basic noncommutative instanton was put in the context of an SU(2) noncommutative principal fibration Sθ7 → Sθ4 over Sθ4 . In the present paper, we continue the analysis and consider it in the setting of a noncommutative Yang–Mills theory. We then construct a five-parameter family of (infinitesimal) gauge-nonequivalent instantons, by acting with twisted conformal symmetries on the basic instanton. All these instantons will be gauge configurations satisfying self-duality equations – with a suitably defined Hodge ∗θ -operator on forms (Sθ4 ) – and will have a “topological charge” of value 1. A completeness argument on the family of instantons is provided by index theoretical arguments, similar to the one in [6] for undeformed instantons on S 4 . The dimension of the “tangent” of the moduli space can be computed as the index of a twisted Dirac operator which turns out to be equal to its classical value that is five. The twisting of the conformal symmetry is implemented with a twist of Drinfel’d type [18, 19] – in fact, explicitly constructed by Reshetikhin [31] – and gives rise to a deformed Hopf algebra Uθ (so(5, 1)). That these are conformal infinitesimal transformations is stressed by the fact that the Hopf algebra Uθ (so(5, 1)) leaves the Hodge ∗θ -structure of (Sθ4 ) invariant. The paper is organized as follows. In Sect. 2 we recall the setting of gauge theories (connections) and gauge transformations on finite projective modules (the substitute for vector bundles) over algebras (the substitute for spaces). The main objective is to implement a Bianchi identity that will be crucial later on for the self-duality equations. Section 3 deals with toric noncommutative manifolds. These were indeed named isospectral deformations in that they can be endowed with the structure of a noncommutative Riemannian spin manifold via a spectral triple (C ∞ (Mθ ), D, H) with the properties of [13]. For this class of examples, the Dirac operator D is the classical one and H = L 2 (M, S) is the usual Hilbert space of spinors on which the algebra C ∞ (Mθ ) acts in a twisted manner. Thus one twists the algebra and its representation while keeping
Noncommutative Instantons from Twisted Conformal Symmetries
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the geometry unchanged. The resulting noncommutative geometry is isospectral and all spectral properties are preserved including the dimension. Both the algebra and its action on spinors can be given via a “star-type” product. In Sect. 4 we specialize to gauge theories on the sphere Sθ4 and introduce a Yang–Mills action functional, from which we derive field equations (equations for critical points), as well as a topological action functional whose absolute value gives a lower bound for the Yang–Mills action. The heart of the paper is Sect. 5 where we explicitly construct instantons. As usual, these are gauge configurations which are solutions of (anti)self-duality equations and realize absolute minima of the Yang–Mills functional. We start from a basic instanton which is shown to be invariant under twisted orthogonal transformations in Uθ (so(5)). We then perturb it by the action of conformal operators in Uθ (so(5, 1)) − Uθ (so(5)) producing a five parameter family of new, not gauge equivalent instantons. A completeness argument is obtained by using an index theorem to compute the dimension of the tangent space of the moduli space of instantons on Sθ4 , which is shown to be just five. The relevant material from noncommutative index theory is recalled in the Appendix. Section 6 sketches a general scheme for gauge theories on four dimensional toric noncommutative manifolds.
2. Connections and Gauge Transformations We first review the notion of a (gauge) connection on a (finite projective) module E over an algebra A with respect to a given calculus; we take a right module structure. Also, we recall gauge transformations in this setting. We refer to [12] for more details (see also [24]). 2.1. Connections on modules. Let us suppose we have an algebra A with a differential calculus (A = ⊕ p p A, d). A connection on the right A-module E is a C-linear map ∇ : E ⊗A p A −→ E ⊗A p+1 A, defined for any p ≥ 0, and satisfying the Leibniz rule ∇(ωρ) = (∇ω)ρ + (−1) p ωdρ, ∀ ω ∈ E ⊗A p A, ρ ∈ A. A connection is completely determined by its restriction ∇ : E → E ⊗A 1 A,
(2.1)
∇(ηa) = (∇η)a + η ⊗A da, ∀ η ∈ E, a ∈ A,
(2.2)
which satisfies
and which is extended to all of E ⊗A p A using Leibniz rule. It is the latter rule that implies the A-linearity of the composition, ∇ 2 = ∇ ◦ ∇ : E ⊗A p A −→ E ⊗A p+2 A.
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Indeed, for any ω ∈ E ⊗A p A and ρ ∈ A it follows that ∇ 2 (ωρ) = ∇ (∇ω)ρ + (−1) p ωdρ = (∇ 2 ω)ρ + (−1) p+1 (∇ω)dρ + (−1) p (∇ω)dρ + ωd2 ρ = (∇ 2 ω)ρ. The restriction of ∇ 2 to E is the curvature F : E → E ⊗A 2 A,
(2.3)
of the connection. It is A-linear, F(ηa) = F(η)a for any η ∈ E, a ∈ A, and satisfies ∇ 2 (η ⊗A ρ) = F(η)ρ, ∀ η ∈ E, ρ ∈ A.
(2.4)
Thus, F ∈ HomA (E, E ⊗A 2 A), the latter being the collection of (right) A-linear homomorphisms from E to E ⊗A 2 A (an alternative notation for this collection that is used in the literature, is EndA (E, E ⊗A 2 A)). In order to have the notion of a Bianchi identity we need some generalization. Let EndA (E ⊗A A) be the collection of all A-linear endomorphisms of E ⊗A A. It is an algebra under composition. The curvature F can be thought of as an element of EndA (E ⊗A A). There is a map [∇, · ] : EndA (E ⊗A A) −→ EndA (E ⊗A A), [∇, T ] := ∇ ◦ T − (−1)|T | T ◦ ∇,
(2.5)
where |T | denotes the degree of T with respect to the Z2 -grading of A. Indeed, for any ω ∈ E ⊗A p A and ρ ∈ A, it follows that [∇, T ](ωρ) = ∇(T (ωρ)) − (−1)|T | T (∇(ωρ)) = ∇ T (ω)ρ − (−1)|T | T (∇ω)ρ + (−1) p ωdρ = ∇(T (ω)) ρ + (−1) p+|T | T (ω)dρ −(−1)|T | T (∇ω)ρ − (−1) p+|T | T (ω)dρ = ∇(T (ω)) − (−1)|T | T (∇ω) ρ = ([∇, T ](ω)) ρ, and the map in (2.5) is well-defined. It is straightforwardly checked that [∇, · ] is a graded derivation for the algebra EndA (E ⊗A A), [∇, S ◦ T ] = [∇, S] ◦ T + (−1)|S| S ◦ [∇, T ].
(2.6)
Proposition 1. The curvature F satisfies the Bianchi identity, [∇, F] = 0.
(2.7)
Proof. Since F is an even element in EndA (E ⊗A A), the map [∇, F] makes sense. Furthermore, [∇, F] = ∇ ◦ ∇ 2 − ∇ 2 ◦ ∇ = ∇ 3 − ∇ 3 = 0.
In Sect. II.2 of [11], such a Bianchi identity was implicitly used in the construction of a so-called canonical cycle from a connection on a finite projective A-module E. Connections always exist on a projective module. On the module E = C N ⊗C A N A , which is free, a connection is given by the operator ∇0 = I ⊗ d : C N ⊗C p A −→ C N ⊗C p+1 A.
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With the canonical identification C N ⊗C A = (C N ⊗C A) ⊗A A (A) N , one thinks of ∇0 as acting on (A) N as the operator ∇0 = (d, d, . . . , d) (N -times). Next, take a projective module E with inclusion map, λ : E → A N , which identifies E as a direct summand of the free module A N and idempotent p : A N → E which allows one to identify E = pA N . Using these maps and their natural extensions to E-valued forms, a connection ∇0 on E (called Levi-Civita or Grassmann) is the composition, λ
I⊗d
p
E ⊗A p A −→ C N ⊗C p A −→ C N ⊗C p+1 A −→ E ⊗A p+1 A, that is ∇0 = p ◦ (I ⊗ d) ◦ λ.
(2.8)
One indicates it simply by ∇0 = pd. The space C(E) of all connections on E is an affine space modeled on HomA (E, E ⊗A 1 A). Indeed, if ∇1 , ∇2 are two connections on E, their difference is A-linear, (∇1 − ∇2 )(ηa) = ((∇1 − ∇2 )(η))a, ∀ η ∈ E, a ∈ A, so that ∇1 − ∇2 ∈ HomA (E, E ⊗A 1 A). Thus, any connection can be written as ∇ = pd + α,
(2.9)
where α is any element in HomA (E, E ⊗A 1 A). The “matrix of 1-forms” α as in (2.9) is called the gauge potential of the connection ∇. The corresponding curvature F of ∇ is F = pd pd p + pdα + α 2 .
(2.10)
Next, let the algebra A have an involution ∗ ; it is extended to the whole of A by the requirement (da)∗ = da ∗ for any a ∈ A. A Hermitian structure on the module E is a map ·, · : E × E → A with the properties η, ξ a = ξ, η a, η, ξ ∗ = ξ, η , η, η ≥ 0, η, η = 0 ⇐⇒ η = 0,
(2.11)
for any η, ξ ∈ E and a ∈ A (an element a ∈ A is positive if it is of the form a = b∗ b for some b ∈ A). We shall also require the Hermitian structure to be self-dual, i.e. every right A-module homomorphism φ : E → A is represented by an element of η ∈ E, by the assignment φ(·) = η, ·, the latter having the correct properties by the first of (2.11). The Hermitian structure is naturally extended to an A-valued linear map on the product E ⊗A A × E ⊗A A by η ⊗A ω, ξ ⊗A ρ = (−1)|η||ω| ω∗ η, ξ ρ, ∀η, ξ ∈ E ⊗A A, ω, ρ ∈ A. (2.12) A connection ∇ on E and a Hermitian structure ·, · on E are said to be compatible if the following condition is satisfied [12]: ∇η, ξ + η, ∇ξ = d η, ξ , ∀ η, ξ ∈ E.
(2.13)
It follows directly from the Leibniz rule and (2.12) that this extends to ∇η, ξ + (−1)|η| η, ∇ξ = d η, ξ , ∀ η, ξ ∈ E ⊗A A.
(2.14)
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On the free module A N there is a canonical Hermitian structure given by η, ξ =
N
η∗j ξ j ,
(2.15)
j=1
with η = (η1 , . . . , η N ) and η = (η1 , . . . , η N ) any two elements of A N . Under suitable regularity conditions on the algebra A all Hermitian structures on a given finite projective module E over A are isomorphic to each other and are obtained from the canonical structure (2.15) on A N by restriction [12, II.1]. Moreover, if E = pA N , then p is self-adjoint: p = p ∗ , with p ∗ obtained by the composition of the involution ∗ in the algebra A with the usual matrix transposition. The Grassmann connection (2.8) is easily seen to be compatible with this Hermitian structure, d η, ξ = ∇0 η, ξ + η, ∇0 ξ .
(2.16)
For a general connection (2.9), the compatibility with the Hermitian structure reduces to αη, ξ + η, αξ = 0, ∀ η, ξ ∈ E,
(2.17)
which just says that the gauge potential is skew-hermitian, α ∗ = −α.
(2.18)
We still use the symbol C(E) to denote the space of compatible connections on E. Let EndsA (E ⊗A A) denote the space of elements T in EndA (E ⊗A A) which are skew-hermitian with respect to the Hermitian structure (2.12), i.e. satisfying T η, ξ + η, T ξ = 0,
∀ η, ξ ∈ E.
(2.19)
Proposition 2. The map [∇, · ] in (2.5) restricts to EndsA (E ⊗A A) as a derivation [∇, · ] : EndsA (E ⊗A A) −→ EndsA (E ⊗A A),
(2.20)
Proof. Let T ∈ EndsA (E ⊗A A) be of order |T |; it then satisfies T η, ξ + (−1)|η||T | η, T ξ = 0,
(2.21)
for η, ξ ∈ E ⊗A A. Since [∇, T ] is A-linear, it is enough to show that [∇, T ]η, ξ + η, [∇, T ]ξ = 0,
∀ η, ξ ∈ E.
This follows from Eqs. (2.21) and (2.14), [∇, T ]η, ξ + η, [∇, T ]ξ = ∇T η, ξ − (−1)|T | T ∇η, ξ + η, ∇T ξ −(−1)|T | η, T ∇ξ = ∇T η, ξ − ∇η, T ξ + η, ∇T ξ − (−1)|T | T η, ∇ξ = d T η, ξ + η, T ξ = 0.
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2.2. Gauge transformations. We now add the additional requirement that the algebra A is a Fréchet algebra and that E is a right Fréchet module. That is, both A and E are complete in the topology defined by a family of seminorms · i such that the following condition is satisfied: for all j there exists a constant c j and an index k such that ηa j ≤ c j ηk ak .
(2.22)
The collection EndA (E) of all A-linear maps is an algebra with involution; its elements are also called endomorphisms of E. It becomes a Fréchet algebra with the following family of seminorms: for T ∈ EndA (E), T i = sup {T ηi : ηi ≤ 1} . η
(2.23)
Since we are taking a self-dual Hermitian structure (see the discussion after (2.11)), any T ∈ EndA (E) is adjointable, that is it admits an adjoint, an A-linear map T ∗ : E → E such that ∗ T η, ξ = η, T ξ , ∀ η, ξ ∈ E. The group U(E) of unitary endomorphisms of E is given by U(E) := {u ∈ EndA (E) | uu ∗ = u ∗ u = idE }.
(2.24)
This group plays the role of the infinite dimensional group of gauge transformations. It naturally acts on compatible connections by (u, ∇) → ∇ u := u ∗ ∇u, ∀ u ∈ U(E), ∇ ∈ C(E), u∗
(2.25)
u∗
where is really ⊗ idA ; this will always be understood in the following. Then the curvature transforms in a covariant way (u, F) → F u = u ∗ Fu,
(2.26)
since, evidently, F u = (∇ u )2 = u ∗ ∇uu ∗ ∇u ∗ = u ∗ ∇ 2 u = u ∗ Fu. As for the gauge potential, one has the usual affine transformation, (u, α) → α u := u ∗ pdu + u ∗ αu.
(2.27)
Indeed, ∇ u (η) = u ∗ ( pd+α)uη = u ∗ pd(uη)+u ∗ αuη = u ∗ pudη+u ∗ p(du)η+u ∗ αuη = pdη +(u ∗ pdu +u ∗ αu)η for any η ∈ E, which yields (2.27) for the transformed potential. The “tangent vectors” to the gauge group U(E) constitute the vector space of infinitesimal gauge transformations. Suppose {u t }t∈R is a differentiable family of elements in EndA (E) (in the topology defined by the above sup-norms) and define X := (∂u t /∂t)t=0 . Unitarity of u t then induces that X = −X ∗ . In other words, for u t to be a gauge transformation, X should be a skew-hermitian endomorphism of E. In this way, we understand EndsA (E) as the collection of infinitesimal gauge transformations. It is a real vector space whose complexification EndsA (E) ⊗R C can be identified with EndA (E). Infinitesimal gauge transformations act on a connection in a natural way. Let the gauge transformation u t , with X = (∂u t /∂t)t=0 , act on ∇ as in (2.25). From the fact that (∂(u t ∇u ∗t )/∂t)t=0 = [∇, X ], we conclude that an element X ∈ EndsA (E) acts infinitesimally on a connection ∇ by the addition of [∇, X ], (X, ∇) → ∇ X = ∇ + t[∇, X ] + O(t 2 ), ∀ X ∈ EndsA (E), ∇ ∈ C(E).
(2.28)
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As a consequence, for the transformed curvature one finds (X, F) → F X = F + t[F, X ] + O(t 2 ),
(2.29)
since F X = (∇ + t[∇, X ]) ◦ (∇ + t[∇, X ]) = ∇ 2 + t[∇ 2 , X ] + O(t 2 ). 3. Toric Noncommutative Manifolds Mθ We start by recalling the general construction of toric noncommutative manifolds given in [15] where they were called isospectral deformations. These are deformations of a classical Riemannian manifold and satisfy all the properties of noncommutative spin geometry [13]. They are the content of the following result taken from [15]: Theorem 3. Let M be a compact spin Riemannian manifold whose isometry group has rank r ≥ 2. Then M admits a natural one parameter isospectral deformation to noncommutative geometries Mθ . The idea of the construction is to deform the standard spectral triple describing the Riemannian geometry of M along a torus embedded in the isometry group, thus obtaining a family of spectral triples describing noncommutative geometries. 3.1. Deforming a torus action. Let M be an m dimensional compact Riemannian manifold equipped with an isometric smooth action σ of an n-torus Tn , n ≥ 2. We denote by σ also the corresponding action of Tn by automorphisms – obtained by pull-backs – on the algebra C ∞ (M) of smooth functions on M. The algebra C ∞ (M) may be decomposed into spectral subspaces which are indexed by the dual group Zn = Tn . Now, with s = (s1 , . . . , sn ) ∈ Tn , each r ∈ Zn yields a n 2πis character of T , e → e2πir ·s , with the scalar product r ·s := r1 s1 +· · ·+rn sn . The r th spectral subspace for the action σ of Tn on C ∞ (M) consists of those smooth functions fr for which σs ( fr ) = e2πir ·s fr ,
(3.1)
and each f ∈ C ∞ (M) is the sum of a unique series f = r ∈Zn fr , which is rapidly ∞ convergent in the Fréchet topology of C (M) (see [33] for more details). Let now θ = (θ jk = −θk j ) be a real antisymmetric n × n matrix. The θ -deformation of C ∞ (M) may be defined by replacing the ordinary product by a deformed product, given on spectral subspaces by
fr ×θ gr := fr σ 1 r ·θ (gr ) = eπir ·θ·r fr gr , 2
(3.2)
where r · θ is the element in Rn with components (r · θ )k = r j θ jk for k = 1, . . . , n. The product in (3.2) is then extended linearly to all functions in C ∞ (M). We denote the space C ∞ (M) endowed with the product ×θ by C ∞ (Mθ ). The action σ of Tn on C ∞ (M) extends to an action on C ∞ (Mθ ) given again by (3.1) on the homogeneous elements. Next, let us take M to be a spin manifold with H := L 2 (M, S) the Hilbert space of spinors and D the usual Dirac operator of the metric of M. Smooth functions act on spinors by pointwise multiplication thus giving a representation π : C ∞ (M) → B(H), the latter being the algebra of bounded operators on H.
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There is a double cover c : Tn → Tn and a representation of Tn on H by unitary n operators U (s), s ∈ T , so that U (s)DU (s)−1 = D,
(3.3)
since the torus action is assumed to be isometric, and such that for all f ∈ C ∞ (M), U (s)π( f )U (s)−1 = π(σc(s) ( f )).
(3.4)
Recall that an element T ∈ B(H) is called smooth for the action of Tn if the map Tn s → αs (T ) := U (s)T U (s)−1 , is smooth for the norm topology. From its very definition, αs coincides on π(C ∞ (M)) ⊂ B(H) with the automorphism σc(s) . Much as was done before for the smooth functions, we shall use the torus action to give a spectral decomposition of smooth elements of B(H). Any such smooth element T is written as a (rapidly convergent) series T = Tr with r ∈ Zn and each Tr is homogeneous of degree r under the action of Tn , i.e. αs (Tr ) = e2πir ·s Tr , ∀ s ∈ Tn .
(3.5)
Tn so that we can Let (P1 , P2 , . . . , Pn ) be the infinitesimal generators of the action of write U (s) = exp 2πis · P. Now, with θ a real n × n anti-symmetric matrix as above, one defines a twisted representation of the smooth elements of B(H) on H by
L θ (T ) := (3.6) Tr U 21 r · θ = Tr exp πi r j θ jk Pk . r
r
Tn and preserves the spectral components The twist L θ commutes with the action αs of of smooth operators: αs (L θ (Tr )) = U (s) T U ( 21 r · θ ) U (s)−1 = U (s)T U (s)−1 U ( 21 r · θ ) = e2πir ·s L θ (Tr ). (3.7) Taking smooth functions on M as elements of B(H), via the representation π , the previous definition gives an algebra L θ (C ∞ (M)) which we may think of as a representation (as bounded operators on H) of the algebra C ∞ (Mθ ). Indeed, by the very definition of the product ×θ in (3.2) one establishes that L θ ( f ×θ g) = L θ ( f )L θ (g),
(3.8)
proving that the algebra C ∞ (M) equipped with the product ×θ is isomorphic to the algebra L θ (C ∞ (M)). It is shown in [33] that there is a natural completion of the algebra C ∞ (Mθ ) to a C ∗ -algebra C(Mθ ) whose smooth subalgebra – under the extended action of Tn – is precisely C ∞ (Mθ ). Thus, we can understand L θ as a quantization map from L θ : C ∞ (M) → C ∞ (Mθ ),
(3.9)
which provides a strict deformation quantization in the sense of Rieffel. More generally, in [33] one considers a (not necessarily commutative) C ∗ -algebra A carrying an action of Rn . For an anti-symmetric n × n matrix θ , one defines a star product ×θ between elements in A much as we did before. The algebra A equipped with the product ×θ gives rise to a C ∗ -algebra denoted by Aθ . Then the collection {Aθ }∈[0,1] is a continuous
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family of C ∗ -algebras providing a strict deformation quantization in the direction of the Poisson structure on A defined by the matrix θ . Our case of interest corresponds to the choice A = C(M) with an action of Rn that is periodic or, in other words, an action of Tn . The smooth elements in the deformed algebra make up the algebra C ∞ (Mθ ). The quantization map will play a key role in what follows, allowing us to extend differential geometric techniques from M to the noncommutative space Mθ . It was shown in [15] that the datum (L θ (C ∞ (M)), H, D) satisfies all properties of a noncommutative spin geometry [13] (see also [22]); there is also a grading γ (for the even case) and a real structure J . In particular, boundedness of the commutators [D, L θ ( f )] for f ∈ C ∞ (M) follows from [D, L θ ( f )] = L θ ([D, f ]), D being of degree 0 (since Tn acts by isometries, each Pk commutes with D). This noncommutative geometry is an isospectral deformation of the classical Riemannian geometry of M, in that the spectrum of the operator D coincides with that of the Dirac operator D on M. Thus all spectral properties are unchanged. In particular, the triple is m + -summable and there is a noncommutative integral as a Dixmier trace [17], (3.10) − L θ ( f ) := Tr ω L θ ( f )|D|−m , with f ∈ C ∞ (Mθ ) understood in its representation on H. A drastic simplification of this noncommutative integral is given by the lemma [20, Prop. 5.1]. Lemma 4. If f ∈ C ∞ (Mθ ) then f dν. − Lθ ( f ) = M
Proof. Any element f ∈ C ∞ (Mθ ) is given as an infinite sum of functions that are homogeneous under the action of Tn . Let us therefore assume that f is homogeneous of degree k so that σs (L θ ( f )) = L θ (σs ( f )) = e2πik·s L θ ( f ). From the tracial property of the noncommutative integral and the invariance of D under the action of Tn , we see that Tr ω σs (L θ ( f ))|D|−m = Tr ω U (s)L θ ( f )U (s)−1 |D|−m = Tr ω (L θ ( f )|D|−m ). In other words, e2πik·s Tr ω (L θ ( f )|D|−m ) = Tr ω (L θ ( f )|D|−m ) from which we infer that this trace vanishes if k = 0. If k = 0, then L θ ( f ) = f , leading to the desired result.
3.2. The manifold Mθ as a fixed point algebra. A different but equivalent approach to these noncommutative manifolds Mθ was introduced in [14]. There the algebra C ∞ (Mθ ) is identified as a fixed point subalgebra of C ∞ (M) ⊗ C ∞ (Tnθ ), where C ∞ (Tnθ ) is the algebra of smooth functions on the noncommutative torus. This identification was shown to be useful in extending techniques from commutative differential geometry on M to the noncommutative space Mθ . We recall the definition of the noncommutative n-torus Tnθ [32]. Let θ = (θ jk ) be a real n × n anti-symmetric matrix as before, and let λ jk = e2πiθ jk . The unital ∗algebra A(Tnθ ) of polynomial functions on Tnθ is generated by n unitary elements U k , k = 1, . . . , n, with relations U j U k = λ jk U k U j ,
j, k = 1, . . . , n.
(3.11)
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The polynomial algebra is extended to the universal C ∗ -algebra with the same generators. There is a natural action of Tn on A(Tnθ ) by ∗-automorphisms given by τs (U k ) = e2πisk U k with s = (sk ) ∈ Tn . The corresponding infinitesimal generators X k of the j action are algebra derivations given explicitly on the generators by X k (U j ) = 2πiδk . n n ∗ ∞ They are used [10] to construct the pre-C -algebra C (Tθ ) of smooth functions on Tθ , which is the completion of A(Tnθ ) with respect to the locally convex topology generated by the seminorms, |u|r :=
sup
r1 +···+rn ≤r
X 1r1 · · · X nrn (u),
(3.12)
and · is the C ∗ -norm. The algebra C ∞ (Tnθ ) turns out to be a nuclear Fréchet space and one can unambiguously take the completed tensor product C ∞ (M)⊗C ∞ (Tnθ ). Then, one σ ⊗τ −1 as the fixed point subalgebra of C ∞ (M)⊗C ∞ (Tnθ ) defines C ∞ (M)⊗C ∞ (Tnθ ) consisting of elements a in the tensor product that are invariant under the diagonal action of Tn , i.e. such that σs ⊗ τ−s (a) = a for all s ∈ Tn . The noncommutative manifold Mθ is defined by “duality” by setting σ ⊗τ −1 C ∞ (Mθ ) := C ∞ (M)⊗C ∞ (Tnθ ) . As the notation suggests, the algebra C ∞ (Mθ ) turns out to be isomorphic to the algebra L θ (C ∞ (M)) defined in the previous section. Next, let S be a spin bundle over M and D the Dirac operator on ∞ (M, S), the ∞ C (M)-module of smooth sections of S. The action of the group Tn on M does not lift directly to the spinor bundle. Rather, there is a double cover c : Tn → Tn and a group n homomorphism s → Vs of T into Aut(S) covering the action of Tn on M, Vs ( f ψ) = σc(s) ( f )Vs (ψ),
(3.13)
for f ∈ C ∞ (M) and ψ ∈ ∞ (M, S). According to [14], the proper notion of smooth C ∞ (Tnθ/2 ) sections ∞ (Mθ , S) of a spinor bundle on Mθ are elements of ∞ (M, S)⊗ Tn . Here s → τs is the canonwhich are invariant under the diagonal action V × τ −1 of ical action of Tn on A(Tnθ/2 ). Since the Dirac operator D commutes with Vs (remember that the torus action is isometric) one can restrict D ⊗ id to the fixed point elements ∞ (Mθ , S). Then, let L 2 (M, S) be the space of square integrable spinors on M and let L 2 (Tnθ/2 ) be the completion of C ∞ (Tnθ/2 ) in the norm a → a = tr(a ∗ a)1/2 , with tr the usual τ −1 of Tn extends to L 2 (M, S)⊗ L 2 (Tnθ/2 ) trace on C ∞ (Tnθ/2 ). The diagonal action V × (where it becomes U × τ ) and one defines L 2 (Mθ , S) to be the fixed point Hilbert subspace. If D also denotes the closure of the Dirac operator on L 2 (M, S), one still denotes by D the operator D ⊗ id on L 2 (M, S) ⊗ L 2 (Tnθ/2 ) when restricted to L 2 (Mθ , S). The triple (C ∞ (Mθ ), L 2 (Mθ , S), D) is an m + -summable noncommutative spin geometry. 3.3. Vector bundles on Mθ . Noncommutative vector bundles on Mθ , i.e. finite projective modules over C ∞ (Mθ ), were obtained in [14] as fixed point submodules of ∞ (M, E) ⊗ C ∞ (Tnθ ) under some diagonal action of the torus Tn . Here ∞ (M, E) denotes the C ∞ (M)-bimodule of smooth sections of a vector bundle E → M. We will
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presently give an equivalent description of these modules over C ∞ (Mθ ) in terms of a kind of ∗-product. Let E be a σ -equivariant vector bundle over M, that is a bundle which carries an action V of Tn by automorphisms, covering the action σ of Tn on M, Vs ( f ψ) = σs ( f )Vs (ψ), ∀ f ∈ C ∞ (M), ψ ∈ ∞ (M, E).
(3.14)
We also assume that the topology on the Fréchet C ∞ (M)-bimodule ∞ (M, E) is given in terms of a family of Tn -invariant seminorms · i . We define the C ∞ (Mθ )-bimodule ∞ (Mθ , E) as the vector space ∞ (M, E) but with deformed bimodule structure given by f θ ψ =
f k V 1 k·θ (ψ),
(3.15)
V− 1 k·θ (ψ) f k ;
(3.16)
2
k
ψ θ f =
k
2
∞ here f = k f k with f k ∈ C (M) homogeneous of degree k under the action of n T – as in (3.5) – and ψ is a smooth section of E. By using the explicit expression (3.2) for the star product and Eq. (3.14), one checks that these are indeed actions of C ∞ (Mθ ). Moreover, the invariance of · i under the action of Tn makes both actions continuous, turning ∞ (Mθ , E) into a Fréchet C ∞ (Mθ )-bimodule. The C ∞ (Mθ )-bimodule ∞ (Mθ , E) is finite projective [14] and still carries an action V of Tn with equivariance as in (3.14) for both the left and right action of C ∞ (Mθ ): the group Tn being abelian, one directly establishes that Vs ( f θ ψ) = σs ( f ) θ Vs (ψ), ∀ f ∈ C ∞ (Mθ ), ψ ∈ ∞ (Mθ , E),
(3.17)
and a similar property for the right structure θ . In fact, since the category of σ -equivariant finite projective module over C ∞ (Mθ ) is equivalent to the category of finite projectve modules over C ∞ (Mθ ) σ Tn (see [23]) for all θ (in particular θ = 0), the isomorphism C ∞ (Mθ )σ Tn C ∞ (M)σ Tn shows that all σ -equivariant finite projective modules over C ∞ (Mθ ) are of the above type [14, Proof of Prop. 5]. This also reflects the result in [34] that the K-groups of a C ∗ -algebra deformed by an action of Rn are isomorphic to the K-groups of the original C ∗ -algebra: as mentioned above, the noncommutative manifolds Mθ are a special case – in which the starting algebra is commutative and the action periodic – of the general formulation in [33] of deformations of C ∗ -algebras under an action of Rn . Although we defined the above left and right actions on sections with respect to an action of Tn on the vector bundle E, the same construction can be done for vector bundles carrying an action of the double cover Tn . We have already seen an example of this double cover action for the spinor bundle, for which we defined a left action of C ∞ (Mθ ) using the twisted representation (3.6). From the very definition of ∞ (Mθ , E) the following lemma is true: Lemma 5. If E F as σ -equivariant vector bundles, then ∞ (Mθ , E) ∞ (Mθ , F) as C ∞ (Mθ )-bimodules.
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3.4. Differential calculus on Mθ . It is straightforward to construct a differential calculus on Mθ . This can be done in two equivalent manners, either by extending to forms the quantization maps, or by using the general construction in [12] by means of the Dirac operator. Firstly, let ((M), d) be the usual differential calculus on M, with d the exterior derivative. This becomes a Fréchet algebra if we consider (M) as the space of smooth sections of a bundle over M. Moreover, it carries an action σ of Tn by automorphisms which commutes with the differential d. Again, we assume that the seminorms defining the Fréchet topology of (M) are Tn -invariant. The quantization map L θ : C ∞ (M) → C ∞ (Mθ ) is extended to (M) by imposing that it commutes with d. The image L θ ((M)) will be denoted (Mθ ) and becomes a Fréchet algebra with the induced seminorms from (M). Equivalently, (Mθ ) could be defined to be (M) as a vector space but equipped with an “exterior star product” which is the extension of the product (3.2) to (M) by the requirement that it commutes with d. Indeed, since the action of Tn commutes with d, an element in (M) can be decomposed into a sum of a rapidly convergent series of homogeneous elements for the action of Tn – as was done for C ∞ (M). Then one defines a star product ×θ on homogeneous elements in (M) as in (3.2) and denotes (Mθ ) = ((M), ×θ ). This construction is in concordance with the previous section, when (M) is considered as a C ∞ (M)-bimodule of sections. The extended action of Tn from C ∞ (M) to (M) is used to endow the space (Mθ ) with the structure of a C ∞ (Mθ )-bimodule with a left and right action given in (3.15)–(3.16). As mentioned, a differential calculus D (C ∞ (Mθ )) on C ∞ (Mθ ) can also be obtained from the general procedure [12] using the isospectral Dirac operator D on Mθ defined p above. The C ∞ (Mθ )-bimodule D (C ∞ (Mθ )) of p-forms is made of classes of operators j j j j ω= a0 [D, a1 ] · · · [D, a p ], ai ∈ C ∞ (Mθ ), (3.18) j
modulo the sub-bimodule of operators ⎫ ⎧ ⎨ ⎬ j j j j j j j D, b0 D, b1 · · · D,b p−1 : bi ∈ C ∞ (Mθ ), b0 D, b1 · · · D, b p−1 = 0 . ⎭ ⎩ j
(3.19) The exterior differential d D is given by ⎤ ⎡ ⎤ ⎡ j j j j j j D, a0 D, a1 · · · D, a p ⎦ , (3.20) a0 D, a1 · · · D, a p ⎦ = ⎣ dD ⎣ j
j
and satisfies d2D = 0. One also introduces an inner product on forms by declaring that forms of different degree are orthogonal, while for two p-forms ω1 , ω2 , the product is (ω1 , ω2 ) D = − ω1∗ ω2 . (3.21) Here the noncommutative integral is the natural extension of the one in (3.10), (3.22) − T := Tr ω T |D|−m ,
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with T an element in a suitable class of operators. Not surprisingly, these two constructions of forms agree [14], that is, the differential calculi (Mθ ) and D (C ∞ (Mθ )) are isomorphic. This allows us in particular to integrate forms of top dimension, by defining ω := − ω D , ω ∈ m (Mθ ), (3.23) Mθ
where ω D denotes the element in mD (C ∞ (Mθ )) corresponding to ω (replacing every d in ω by d D ). We have the following noncommutative Stokes theorem: Lemma 6. If ω ∈ m−1 (Mθ ) then
dω = 0. Mθ
Proof. From the definition of the noncommutative integral, (0) dω = − d D ω D = − d D L θ (ω D ), Mθ (0)
(0)
with ω D the classical counterpart of ω, i.e. ω = L θ (ω D ). At this point one remembers that D commutes with L θ (see Sect. 3.1), and realizes that there is an analogue of Lemma 4 for forms, i.e. L θ (T ) = M T . One concludes that the above integral vanishes since it vanishes in the classical case.
The next ingredient is a Hodge star operator on (Mθ ). Classically, the Hodge star operator is a map ∗ : p (M) → m− p (M) depending only on the conformal class of the metric on M. On the one end, since Tn acts by isometries, it leaves the conformal structure invariant and therefore, it commutes with ∗. On the other hand, with the isospectral deformation one does not change the metric. Thus it makes sense to define a map ∗θ : p (Mθ ) → m− p (Mθ ) by ∗θ L θ (ω) = L θ (∗ω), for ω ∈ (M).
(3.24)
With this Hodge operator, there is an alternative definition of an inner product on (Mθ ). Given that ∗θ maps p (Mθ ) to m− p (Mθ ), we can define for α, β ∈ p (Mθ ), (3.25) (α, β)2 = − ∗θ (α ∗ ∗θ β), since ∗θ (α ∗ ∗θ β) is an element in C ∞ (Mθ ). Lemma 7. Under the isomorphism D (C ∞ (Mθ )) (Mθ ), the inner product (·, ·)2 coincides with (·, ·) D . Proof. Let ω1 , ω2 be two forms in D (C ∞ (M)), so that L θ (ωi ) are two generic forms in D (C ∞ (Mθ )) L θ ((M)) = (Mθ ). Then, using Lemma 4 it follows that (3.26) − L θ (ω1 )∗ L θ (ω2 ) = − L θ (ω1∗ ×θ ω2 ) = − ω1∗ ×θ ω2 .
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Now, the inner product ( , ) D coincides with ( , )2 as defined by (3.25) in the classical case – under the above isomorphism D (C ∞ (M)) (M); see for example [12, VI.1]. It follows that the above expression equals ∗ (3.27) − ∗(ω1 ×θ (∗ω2 )) = − ∗θ (L θ (ω1 )∗ (∗θ L θ (ω2 ))), using Lemma 4 for forms once more, together with the defining property of ∗θ .
Lemma 8. The formal adjoint d∗ of d with respect to the inner product (·, ·)2 – i.e. so that (d∗ α, β)2 = (α, dβ)2 – is given on p (Mθ ) by d∗ = (−1)m( p+1)+1 ∗θ d ∗θ . Proof. Just as in the classical case, this follows from Stokes Lemma 6, together with the observation that ω = − ∗θ ω, ω ∈ m (Mθ ), Mθ
again established from the classical case by means of the mentioned analogue of Lemma 4 for forms.
Remark 9. The algebra (Mθ ) can also be defined as a fixed point algebra [14]. The σ ⊗τ −1 . Furaction σ of Tn on (M) allows one to define (Mθ ) by (M)⊗C ∞ (Tnθ ) thermore, since the exterior derivative d on (M) commutes with the action of Tn , the differential dθ , for the fixed point algebra is defined as dθ = d ⊗ id. Similarly, the Hodge star operator takes the form ∗θ = ∗ ⊗ id with ∗ the classical Hodge operator. 4. Gauge Theory on the Sphere Sθ4 We now apply the general scheme of noncommutative gauge field theories – as developed in Sect. 2 – to the case of the SU(2) noncommutative principal bundle Sθ7 → Sθ4 constructed in [27]. This will also make more explicit all the constructions above. It is worth stressing that what follows is valid for the more general θ -deformed G-principal bundle. We will come back to this point later in the paper. 4.1. The principal fibration Sθ7 → Sθ4 . The SU(2) noncommutative principal fibration Sθ7 → Sθ4 is given by an algebra inclusion A(Sθ4 ) → A(Sθ7 ). The algebra A(Sθ4 ) of polynomial functions on the sphere Sθ4 is generated by elements z 0 = z 0∗ , z j , z ∗j , j = 1, 2, subject to relations ∗ ∗ ∗ z µ z ν = λµν z ν z µ , z µ z ν∗ = λνµ z ν∗ z µ , z µ z ν = λµν z ν∗ z µ , µ, ν = 0, 1, 2, (4.1) ∗ together with the spherical relation µ z µ z µ = 1. Here θ is a real parameter and the deformation parameters are given by λµµ = 1 and
λ12 = λ21 =: λ = e2πiθ , λ j0 = λ0 j = 1,
j = 1, 2.
(4.2)
For θ = 0 one recovers the ∗-algebra of complex polynomial functions on the usual S 4 .
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The differential calculus (Sθ4 ) is generated as a graded differential ∗-algebra by the ∗ in degree 0 and elements dz , dz ∗ in degree 1 satisfying the relations, elements z µ , z µ µ µ dz µ dz ν + λµν dz ν dz µ = 0 z µ dz ν = λµν dz ν z µ ,
dz µ dz ν∗ + λνµ dz ν∗ dz µ = 0, z µ dz ν∗ = λνµ dz ν∗ z µ ,
(4.3)
with λµν as before. There is a unique differential d on (Sθ4 ) such that d : z µ → dz µ ∗ : (dω)∗ = dω∗ and and the involution on (Sθ4 ) is the graded extension of z µ → z µ ∗ ∗ ∗ d d (ω1 ω2 ) = (−1) 1 2 ω2 ω1 for ω j a form of degree d j . = e2πiθab ) a real antisymmetric matrix, the algebra A(S 7 ) of and (θab With λab θ polynomial functions on the sphere Sθ7 is generated by elements ψa , ψa∗ , a = 1, . . . , 4, subject to relations ψa ψb = λab ψb ψa , ψa ψb∗ = λba ψb∗ ψa , ψa∗ ψb∗ = λab ψb∗ ψa∗ ,
(4.4)
and with the spherical relation a ψa∗ ψa = 1. Clearly, A(Sθ7 ) is a deformation of the ∗-algebra of complex polynomial functions on the sphere S 7 . As before, a differential calculus (Sθ7 ) can be defined to be generated by the elements ψa , ψa∗ in degree 0 and elements dψa , dψa∗ in degree 1 satisfying relations similar to the ones in (4.3). In order to construct the noncommutative fibration over the given 4-sphere Sθ4 we need to select a particular noncommutative 7 dimensional sphere Sθ7 . We take the one corresponding to the following deformation parameters ⎛
λab
1 ⎜1 =⎝ µ µ
1 1 µ µ
µ µ 1 1
⎛ ⎞ 0 µ √ θ⎜0 µ⎟ , µ = λ, or θab = ⎝ 1⎠ 2 1 −1 1
0 0 −1 1
−1 1 0 0
⎞ 1 −1⎟ . (4.5) 0⎠ 0
The previous choice is essentially the only one1 that allows the algebra A(Sθ7 ) to carry an action of the group SU(2) by automorphisms and such that the invariant subalgebra coincides with A(Sθ4 ). The best way to see this is by means of the matrix-valued function on A(Sθ7 ) given by ⎞ −ψ2∗ ∗ ψ1 ⎟ . −ψ4∗ ⎠ ∗ ψ3
⎛
ψ1 ⎜ψ2 =⎝ ψ3 ψ4
(4.6)
Then the commutation relation of the algebra A(Sθ7 ) gives † = I2 and p = † is a projection, p 2 = p = p † , with entries in A(Sθ4 ). Indeed, the right action of SU(2) on A(Sθ7 ) is simply given by αw () = w,
w=
w1 w2
−w 2 w1
∈ SU(2), √
(4.7)
1 Compatibility requires that µ2 = λ; we drop the case µ = − λ since its “classical” limit would
correspond to “anti-commuting” coordinates.
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from which the invariance under the SU (2)-action of the entries of p follows at once. Explicitly, ⎛ ⎞ 1 + z0 0 z1 −µz 2∗ ∗ 1 + z0 z2 µz 1 ⎟ ⎜ 0 , (4.8) p = 21 ⎝ ∗ z1 z 2∗ 1 − z0 0 ⎠ −µz 2 µz 1 0 1 − z0 with the generators of A(Sθ4 ) given by z 0 = ψ1∗ ψ1 + ψ2∗ ψ2 − ψ3∗ ψ3 − ψ4∗ ψ4 = 2(ψ1∗ ψ1 + ψ2∗ ψ2 ) − 1 = 1 − 2(ψ3∗ ψ3 + ψ4∗ ψ4 ), z 1 = 2(µψ3∗ ψ1 + ψ2∗ ψ4 ) = 2(ψ1 ψ3∗ + ψ2∗ ψ4 ), z 2 = 2(−ψ1∗ ψ4 + µψ3∗ ψ2 ) = 2(−ψ1∗ ψ4 + ψ2 ψ3∗ ).
(4.9)
One straightforwardly computes that z 1∗ z 1 + z 1∗ z 1 + z 02 = 1 and the commutation rules z 1 z 2 = λz 2 z 1 , z 1 z 2∗ = λz 2∗ z 1 , and that z 0 is central. The relations (4.9) can also be expressed in the form, ∗ zµ = ψa∗ (γµ )ab ψb , zµ = ψa∗ (γµ∗ )ab ψb , (4.10) ab
ab
with γµ twisted 4 × 4 Dirac matrices given by 1 0 00 01 1 = 2 γ0 = , γ 1 µ0 −1 0 −1 00
,
γ2 = 2
0
0 −1 0 0
.
(4.11)
( j, k = 1, 2).
(4.12)
0µ 00
0
Note that as usual γ0 is the grading 1 γ0 = − [γ1 , γ1∗ ][γ2 , γ2∗ ]. 4 These matrices satisfy twisted Clifford algebra relations [14], γ j γk + λ jk γk γ j = 0,
γ j γk∗ + λk j γk∗ γ j = 4δ jk ;
There are compatible toric actions on Sθ4 and Sθ7 . The torus T2 acts on A(Sθ4 ) as, σs (z 0 , z 1 , z 2 ) = (z 0 , e2πis1 z 1 , e2πis2 z 2 ), s ∈ T2 .
(4.13)
This action is lifted to a double cover action on A(Sθ7 ). The double cover map p : T2 → 2 2 T is given explicitly by p : (s1 , s2 ) → (s1 + s2 , −s1 + s2 ). Then T acts on the ψa ’s as, " ! σ : (ψ1 , ψ2 , ψ3 , ψ4 ) → e2πis1 ψ1 , e−2πis1 ψ2 , e−2πis2 ψ3 , e2πis2 ψ4 . (4.14) Equation (4.9) shows that σ is indeed a lifting to Sθ7 of the action of T2 on Sθ4 . Clearly, this compatibility is built in the construction of the Hopf fibration Sθ7 → Sθ4 as a deformation of the classical Hopf fibration S 7 → S 4 with respect to an action of T2 , a fact that also dictated the form of the deformation parameter λ in (4.5). As we shall see, the previous double cover of tori comes from a spin cover Spinθ (5) of SOθ (5) deforming the usual action of Spin(5) on S 7 as a double cover of the action of SO(5) on S 4 . In which sense the algebra inclusion A(Sθ4 ) → A(Sθ7 ) is a noncommutative principal bundle was explained in [27] to which we refer for more details. Presently we shall recall the construction of associated bundles.
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4.2. Associated bundles. We shall work in the context of smooth functions on Sθ7 and Sθ4 as defined in general in Sect. 3. Let ρ be any finite-dimensional representation of SU(2) on the vector space W . The space that generalizes to the case θ = 0, the space of SU(2)-equivariant maps from S 7 to W , is given by, C ∞ (Sθ7 )ρ W " $ ! # := η ∈ C ∞ (Sθ7 ) ⊗ W : (αw ⊗ id)(η) = id⊗ρ(w)−1 (η),∀w ∈ SU(2) ,
(4.15)
where αw is the SU(2) action given in (4.7). This space is clearly a C ∞ (Sθ4 )-bimodule. We have proved in [27] that C ∞ (Sθ7 ) ρ W is a finite projective module. It is to be thought of as the module of sections of a “noncommutative vector bundle” associated to the principal bundle via the representation ρ. It is worth stressing that the choice of a projection for a finite projective module requires the choice of one of the two (left or right) module structures. Similarly, the definition of a Hermitian structure requires the choice of the left or right module structure. In the following, we will always work with the right structure for the associated modules. There is a natural (right) Hermitian structure on C ∞ (Sθ7 ) ρ W , defined in terms of a fixed inner product of W as, η, η := ηi ηi , (4.16) i
where we denoted η = i ηi ⊗ ei and = i ηi ⊗ ei , given an orthonormal basis {ei , i = 1, . . . , dim W } of W . One quickly checks that η, η is an element in C ∞ (Sθ4 ), and that , satisfies all conditions of a right Hermitian structure. The bimodules C ∞ (Sθ7 ) ρ W are of the type described in Sect. 3.3. The associated T2 induced from its action vector bundle E = S 7 ×ρ W on S 4 carries an action V of 7 on S , which is obviously σ -equivariant. By the very definition of C ∞ (Sθ7 ) and of ∞ (Sθ4 , E) in Sect. 3.3, it follows that C ∞ (Sθ7 ) ρ W ∞ (Sθ4 , E). Indeed, from the undeformed isomorphism, ∞ (S 4 , E) C ∞ (S 7 ) ρ W , the quantization map L θ of C ∞ (S 7 ), acting only on the first leg of the tensor product, establishes this isomorphism,
η
L θ : C ∞ (S 7 ) ρ W → C ∞ (Sθ7 ) ρ W.
(4.17)
The above is well defined since the action of T2 commutes with the action of SU (2). Also, it is such that L θ ( f θ η) = L θ ( f )L θ (η) = L θ ( f )L θ (η) for f ∈ C ∞ (S 4 ) and η ∈ C ∞ (S 7 ) ρ W , due to the identity L θ = L θ on C ∞ (S 4 ) ⊂ C ∞ (S 7 ). A similar result holds for the action θ . Proposition 10. The right C ∞ (Sθ4 )-module C ∞ (Sθ7 )ρ W admits a homogeneous module basis {eα , α = 1, . . . , N } – with a suitable N – that is, under the action V of the torus T2 , its elements transform as, Vs (eα ) = e2πis·rα eα , s ∈ T2 ,
(4.18)
with rα ∈ Z2 the degree of eα . Proof. The vector space W is a direct sum of irreducible representation spaces of SU(2) and the module C ∞ (Sθ7 ) ρ W decomposes accordingly. Thus, we can restrict to irreducible representations. The latter are labeled by an integer n with W Cn+1 .
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Consider first the case W = C2 . Using [27, Prop. 2], a basis {e1 , . . . , e4 } of the right module C ∞ (Sθ7 ) ρ C2 is given by the columns of † where is the matrix in (4.6): e1 :=
!
ψ1∗ −ψ2
"
, e2 :=
!
ψ2∗ ψ1
"
, e3 :=
!
ψ3∗ −ψ4
"
, e4 :=
!
ψ4∗ ψ3
"
.
(4.19)
Using the explicit action (4.14) it is immediate to compute the corresponding degrees, r1 = (−1, 0), r2 = (1, 0), r3 = (0, 1), r4 = (0, −1).
(4.20)
More generally, with W = Cn+1 a homogeneous basis {eα , α = 1, . . . , 4n } for the right module C ∞ (Sθ7 ) ρ Cn+1 can be constructed from the columns of a similar (n + 1) × 4n † matrix (n) given in [27].
The above property allows us to prove a useful result for the associated modules. Proposition 11. Let ρ1 and ρ2 be two finite dimensional representations of SU(2) on the vector spaces W1 and W2 , respectively. There is the following isomorphism of right C ∞ (Sθ4 )-modules, " ! " ! C ∞ (Sθ7 ) ρ1 W1 ⊗C ∞ (S 4 ) C ∞ (Sθ7 ) ρ2 W2 C ∞ (Sθ7 ) ρ1 ⊗ρ2 (W1 ⊗ W2 ). θ
Proof. Let {eα1 , α = 1, . . . , N1 } and {eβ2 , β = 1, . . . , N2 } be homogeneous bases for the right modules C ∞ (Sθ7 ) ρ1 W1 and C ∞ (Sθ7 ) ρ2 W2 respectively. Then, the right module C ∞ (Sθ7 ) ρ1 ⊗ρ2 (W1 ⊗ W2 ) has a homogeneous basis given by {eα1 ⊗ eβ2 }. We define a map ! " " ! ! " " ! " ! φ : C ∞ Sθ7 ρ1 W1 ⊗C ∞ S 4 C ∞ Sθ7 ρ2 W2 → C ∞ Sθ7 ρ1 ⊗ρ2 (W1 ⊗ W2 ) θ
by " ! " ! " ! φ eα1 ×θ f α1 ⊗ eβ2 ×θ f β2 = eα1 ⊗ eβ2 ×θ σrβ θ f α1 ×θ f β2 , with summation over α and β understood. Here rβ ∈ Z2 is the degree of eβ2 under the action of T2 , so that eβ2 ×θ σrβ θ ( f ) = f ×θ eβ2 for any f ∈ C ∞ (Sθ4 ). Note that this map is well-defined since " ! φ eα1 ×θ f α1 ×θ f ⊗ eβ2 ×θ f β2 − eα1 ×θ f α1 ⊗ f ×θ eβ2 ×θ f β2 = 0. Moreover, it is clearly a map of right C ∞ (Sθ4 )-modules. In fact, it is an isomorphism with its inverse given explicitly by φ −1
!! " " " " ! ! eα1 ⊗ eβ2 ×θ f αβ = e11 ⊗ eβ2 ×θ f 1β + · · · + e1N1 ⊗ eβ2 ×θ f N1 β
with f αβ ∈ C ∞ (Sθ4 ).
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Given a right C ∞ (Sθ4 )-module E, its dual module is defined by ! " ! "$ # E := φ : E → C ∞ Sθ4 : φ(η f ) = φ(η) f, ∀ f ∈ C ∞ Sθ4 ,
(4.21)
and is naturally a left C ∞ (Sθ4 )-module. In the case that E is also a left C ∞ (Sθ4 )-module, then E is also a right C ∞ (Sθ4 )-module. If E := C ∞ (Sθ7 ) ρ W comes from the SU(2)representation (W, ρ), by using the induced dual representation ρ on the dual vector space W given by ! " ρ (w)v (v) := v ρ(w)−1 v ; ∀ v ∈ W , v ∈ W, (4.22) we have that E C ∞ (Sθ7 ) ρ W # $ := φ ∈ C ∞ (Sθ7 ) ⊗ W : (αw ⊗ id)(φ) = (id ⊗ρ (w)−1 )(φ), ∀w ∈ SU(2) . (4.23) Next, let L(W ) denote the space of linear maps on W , so that L(W ) = W ⊗ W . The adjoint action of SU(2) on L(W ) is the tensor product representation ad := ρ ⊗ ρ on W ⊗ W . We define % C ∞ (Sθ7 ) ad L(W ) := T ∈ C ∞ (Sθ7 ) ⊗ L(W ) : : (αw ⊗id)(T ) = (id ⊗ad(w)
−1
& )(T ), ∀w ∈ SU(2) ,
(4.24)
and write T = Ti j ⊗ ei j with respect to the basis {ei j } of L(W ) induced from the basis dim W of W and the dual one {e }dim W of W . {ei }i=1 i i=1 On the other hand, there is the endomorphism algebra ! "$ # EndC ∞ (S 4 ) (E) := T : E → E : T (η f ) = T (η) f, ∀ f ∈ C ∞ Sθ4 . (4.25) θ
We will suppress the subscript C ∞ (Sθ4 ) from End in the following. As a corollary to the previous proposition, we have the following. Proposition 12. Let E := C ∞ (Sθ7 ) ρ W for a finite-dimensional representation ρ. Then there is an isomorphism of algebras End(E) C ∞ (Sθ7 ) ad L(W ). Proof. By Proposition 11, we have that C ∞ (Sθ7 )ad L(W ) ≡ C ∞ (Sθ7 )ρ⊗ρ (W ⊗W ) is isomorphic to E⊗C ∞ (S 4 ) E as a right C ∞ (Sθ4 )-module. Since E is a finite projective θ
C ∞ (Sθ4 )-module, there is an isomophism End(E) E⊗C ∞ (S 4 ) E , whence the result. θ
We see that the algebra of endomorphisms of the module E can be understood as the space of sections of the noncommutative vector bundle associated to the adjoint representation on L(W ) – exactly as it happens in the classical case. This also allows an identification of skew-hermitian endomorphisms Ends (E) – which were defined in general in (2.19) – for the toric deformations at hand.
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Corollary 13. There is an identification ∞ 7 (Sθ ) ad u(n), Ends (E) CR ∞ (S 7 ) denoting the subspace of self-adjoint elements in C ∞ (S 7 ) and u(n) conwith CR θ θ sists of skew-adjoint matrices in Mn (C) L(W ), with n = dim W .
Proof. Note that the involution T → T ∗ in End(E) reads in components Ti j → T ji so that, with the identification of Proposition 12, the space Ends (E) is made of elements X ∈ C ∞ (Sθ7 ) ad L(W ) satisfying X ji = −X i j . Since any element in C ∞ (Sθ7 ) can be written as the sum of two self-adjoint elements, X i j = X i j + i X i j , we can write X= X ii ⊗ ieii + X i j ⊗ (ei j − e ji ) + X i j ⊗ (iei j + ie ji ) = Xa ⊗ σ a, i
i= j
a
∞ (S 7 ) and {σ a , a = 1, . . . , n 2 } the generators of with X a generic elements in CR θ u(n).
Example 14. Of central interest in the following is the special case of the noncommutative instanton bundle first constructed in [15]. Now W = C2 and ρ is the defining representation of SU(2). The projection p giving the C ∞ (Sθ4 )-module C ∞ (Sθ7 ) ρ C2 N as a direct summand of C ∞ (Sθ4 ) for some N , is precisely given by the one in (4.8) and N = 4. Indeed, a generic element in C ∞ (Sθ7 ) ρ C2 is of the form † f for some f ∈ C ∞ (Sθ4 ) ⊗ C4 with defined in (4.6), and the correspondence is given by ! "4 C ∞ (Sθ7 ) ρ C2 p C ∞ (Sθ4 ) , † f ↔ p f.
(4.26)
Furthermore, End(E) C ∞ (Sθ7 )ad M2 (C). It is a known fact that M2 (C) decomposes into the adjoint representation su(2) and the trivial representation C while it is easy to see that C ∞ (Sθ7 ) id C C ∞ (Sθ4 ). Thus, we conclude that ! ! " " (4.27) End(E) ∞ ad(Sθ7 ) ⊕ C ∞ Sθ4 , where we have set ∞ (ad(Sθ7 )) := C ∞ (Sθ7 ) ad su(2). The latter C ∞ (Sθ4 )-bimodule will be understood as the space of (complex) sections of the adjoint bundle. It is the ∞ (S 7 ) su(2). complexification of the traceless skew-hermitian endomorphisms CR ad θ 4.3. Yang–Mills theory on Sθ4 . Let us now move to the main goal of this paper and discuss the Yang–Mills action functional on Sθ4 together with its equations of motion. We will see that instantons naturally arise as the local minima of this action. Before we proceed we recall the noncommutative spin structure (C ∞ (Sθ4 ), H, D, γ5 ) of Sθ4 with H = L 2 (S 4 , S) the Hilbert space of spinors, D the undeformed Dirac operator, and γ5 – the even structure – the fifth Dirac matrix. Let E = ∞ (Sθ4 , E) for some σ -equivariant vector bundle E on S 4 , so that there exists a projection p ∈ M N (C ∞ (Sθ4 )) such that E p(C ∞ (Sθ4 ) N . Recall from Sect. 2 that a connection ∇ on E = ∞ (Sθ4 , E) can be given as a map from E to E ⊗ 1 (Sθ4 ) satisfying a Leibniz rule with respect to the right multiplication of C ∞ (Sθ4 ) on E. It is also required
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to be compatible with a Hermitian structure on E. The Yang–Mills action functional is defined in terms of the curvature F of ∇, an element in HomC ∞ (S 4 ) (E, E ⊗ 2 (Sθ4 )). θ
Equivalently, it can be thought of as belonging to End(S 4 ) (E ⊗ (Sθ4 )) of degree 2. We θ define an inner product on the latter algebra following [12, III.3]. Any T ∈ End(S 4 ) (E ⊗ θ
(Sθ4 )) of degree k can be also understood as an element in pM N (k (Sθ4 )) p, since E ⊗(Sθ4 ) is a finite projective module over (Sθ4 ). A trace over internal indices together with the inner product in (3.25), defines the inner product (·, ·)2 on End(S 4 ) (E ⊗(Sθ4 )). θ In particular, we can give the following definition: Definition 15. The Yang–Mills action functional on the collection C(E) of compatible connections on E is defined by YM(∇) = F, F 2 = − ∗θ tr(F ∗θ F), for any connection ∇ with curvature F. Recall from Sect. 2 that gauge transformations are unitary endomorphisms U(E) of E. Lemma 16. The Yang–Mills action functional is gauge invariant, positive and quartic. Proof. From Eq. (2.26), under a gauge transformation u ∈ U(E) the curvature F transforms as F → u ∗ Fu . Since U(E) can be identified with the unitary elements in pM N (A) p, it follows that u YM(∇ ) = − ∗θ (u ji F jk ∗θ Fkl u li ) = YM(∇), i, j,k,l
using the tracial property of the Dixmier trace and the fact that u li u ji = δl j . Positiveness of the Yang–Mills action functional follows from Lemma 7 giving (F, F)2 = (FD , FD ) D = − FD∗ FD , which is clearly positive.
The Yang–Mills equations of motion (equations for critical points) are obtained from the Yang–Mills action functional by a variational principle. Let us describe how this principle works in our case. We consider a linear perturbation ∇t = ∇ + tα of a connection ∇ on E by an element α ∈ Hom(E, E ⊗C ∞ (S 4 ) 1 (Sθ4 )). The curvature Ft of ∇t is θ
readily computed as Ft = F + t[∇, α] + O(t 2 ). If we suppose that ∇ is a critical point of the Yang–Mills action functional, this linear perturbation should not affect the action. In other words, we need ' ∂ '' YM(∇t ) = 0. (4.28) ∂t 't=0 If we substitute the explicit formula for Ft , we obtain [∇, α], F 2 + [∇, α], F 2 = 0,
(4.29)
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613
using the fact that (·, ·)2 defines a complex scalar product on Hom(E, E ⊗ (Sθ4 )). Positive definiteness of this scalar product implies that (Ft , Ft ) = (Ft , Ft ), which when differentiated with respect to t, at t = 0, yields [∇, α], F 2 = [∇, α], F 2 ; hence, ([∇, α], F)2 = 0. Using the fact that α was arbitrary, we derive the following equations of motion: ( (4.30) [∇ ∗ , F = 0, where the adjoint [∇ ∗ , ·] is defined with respect to the inner product (·, ·)2 , i.e. ∗ (4.31) [∇ , α], β 2 = α, [∇, β] 2 for any α ∈ Hom(E, E ⊗ 3 (Sθ4 )) and any β ∈ Hom(E, E ⊗ 1 (Sθ4 )). From Lemma 8, it follows that [∇ ∗ , F] = ∗θ [∇, ∗θ F], so that the equations of motion can be written as the more familiar Yang–Mills equations, [∇, ∗θ F] = 0.
(4.32)
Clearly, connections with a self-dual or antiself-dual curvature, ∗θ F = ±F, are special solutions of the Yang–Mills equations. Indeed, in this case the latter equations follow directly from the Bianchi identity, [∇, F] = 0, given in Proposition 1. We will now establish a connection between the Yang–Mills action functional and the so-called topological action [12, VI.3] on Sθ4 . Suppose E is a finite projective module over C ∞ (Sθ4 ) defined by a projection p ∈ M N (C ∞ (Sθ4 )). The topological action for E is the pairing between the class of p in K-theory and the cyclic cohomology of C ∞ (Sθ4 ). For computational purposes, we define it in terms of the curvature of a connection on E, Definition 17. Let ∇ be a connection on E with curvature F. The topological action is given by Top(E) = (F, ∗θ F)2 = − ∗θ tr(F 2 ), where the trace is taken over internal indices and in the second equality we have used the identity ∗θ ◦ ∗θ = id on Sθ4 . Let us show that this does not depend on the choice of a connection on E. Since two connections differ by an element α in HomC ∞ (S 4 ) (E, E ⊗ 1 (Sθ4 )), we have to establish θ
that (F , ∗θ F )2 = (F, ∗θ F)2 , where F = F + t[∇, α] + O(t 2 ) is the curvature of ∇ := ∇ + tα, t ∈ R . By definition of the inner product (·, ·)2 we have that (F , ∗θ F )2 − (F, ∗θ F)2 = t (F, ∗θ [∇, α])2 + t ([∇, α], ∗θ F)2 + O(t 2 ) = t (F, [∇ ∗ , ∗θ α])2 + t ([∇ ∗ , ∗θ α], F)2 + O(t 2 ). With Eq. (4.31), this reduces to (F , ∗θ F )2 − (F, ∗θ F)2 = t ([∇, F], ∗θ α)2 + t (∗θ α, [∇, F])2 + O(t 2 ), which vanishes modulo t 2 due to the Bianchi identity [∇, F] = 0. The Hodge star operator ∗θ splits 2 (Sθ4 ) into a self-dual and antiself-dual space, 2 (Sθ4 ) = 2+ (Sθ4 ) ⊕ 2− (Sθ4 ).
(4.33)
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In fact, 2± (Sθ4 ) = L θ 2± (S 4 ) . This decomposition is orthogonal with respect to the inner product (·, ·)2 , a fact that follows from the property (α, β)2 = (β, α)2 ; thus we can write the Yang–Mills action functional as (4.34) YM(∇) = F+ , F+ 2 + F− , F− 2 . Comparing this with the analogue decomposition of the topological action, Top(E) = F+ , F+ 2 − F− , F− 2 ,
(4.35)
we see that YM(∇) ≥ |Top(E)|, with equality holding iff ∗θ F = ±F.
(4.36)
Solutions of these equations are called instantons. We conclude that instantons are absolute minima of the Yang–Mills action functional. 5. Construction of SU(2)-Instantons on Sθ4 In this section, we construct a set of SU(2) instantons on Sθ4 , with topological charge 1, by acting with a twisted infinitesimal conformal symmetry on the basic instanton on Sθ4 constructed in [15]. We will find a five-parameter family of such instantons. Then we prove that the “tangent space” to the moduli space of irreducible instantons at the basic instanton is five-dimensional, proving that this set is complete. Here, one has to be careful with the notion of tangent space to the moduli space. As will be discussed elsewhere [26], one can construct a noncommutative family of instantons, that is instantons parametrized by the quantum quotient space of the deformed conformal group SLθ (2, H) by the deformed spin group Spθ (2). It turns out that the basic instanton of [15] is a “classical point” in this moduli space of instantons. We perturb the connection ∇0 linearly by sending ∇0 → ∇0 + tα, where t ∈ R and α ∈ Hom(E, E ⊗ 1 (Sθ4 )). In order for this new connection still to be an instanton, we have to impose the self-duality equations on its curvature. After deriving this equation with respect to t, setting t = 0 afterwards, we obtain the linearized self-duality equations to be fulfilled by α. It is in this sense that we are considering the tangent space to the moduli space of instantons at the point ∇0 . 5.1. The basic instanton. We start with a technical lemma that simplifies the discussion. Let E = C ∞ (Sθ7 ) ρ W be a module of sections associated to a finite dimensional representation of SU(2), as defined in Eq. (4.15). Lemma 18. There is the following isomorphism of right C ∞ (Sθ4 )-modules, E ⊗C ∞ (S 4 ) (Sθ4 ) (Sθ4 ) ⊗C ∞ (S 4 ) E. θ
θ
Consequently, Hom(E, E ⊗C ∞ (S 4 ) (Sθ4 )) (Sθ4 ) ⊗C ∞ (S 4 ) Hom(E). θ
θ
Proof. Recall from Proposition 10 that the right C ∞ (Sθ4 )-module E has a homogeneous module-basis {eα , α = 1, . . . , N } for some N and each eα of degree rα . A generic element in E ⊗C ∞ (S 4 ) (Sθ4 ) can be written as a sum α eα ⊗C ∞ (S 4 ) ωα , with ωα an θ
θ
ω ∈ (Sθ4 ) – given element in (Sθ4 ). Now, for every ω ∈ (Sθ4 ) there is an element
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615
explicitly by ω = σrα ·θ (ω) – such that eα ω = ωeα , where the latter equality holds inside the algebra (Sθ7 ) (recall that (Sθ4 ) ⊂ (Sθ7 )). We can thus define a map T : E ⊗C ∞ (S 4 ) (Sθ4 ) (Sθ4 ) ⊗C ∞ (S 4 ) E, θ
θ
by T (eα ⊗C ∞ (S 4 ) ωα ) = ωα ⊗ eα ; it is a right C ∞ (S 4 )-module map: θ
" ω α ×θ T eα ⊗C ∞ (S 4 ) (ωα ×θ f ) = ( f ) ⊗C ∞ (S 4 ) eα = T (eα ⊗C ∞ (S 4 ) ωα ) ×θ f. !
θ
θ
θ
Since an inverse map T −1 is easily constructed, T gives the desired isomorphism.
Thus, we can unambiguously use the notation (Sθ4 , E) for the above right C ∞ (Sθ4 )module E ⊗C ∞ (S 4 ) (Sθ4 ). θ We let ∇0 = p ◦d be the canonical (Grassmann) connection on the projective module E = C ∞ (Sθ7 ) ρ C2 p(C ∞ (Sθ4 ))4 , with the projection p = † of (4.8) and is the matrix (4.6) (refer to Example 14). When acting on equivariant maps, we write ∇0 as ∇0 : E → E ⊗C ∞ (S 4 ) 1 (Sθ4 ), θ
(∇0 f )i = d f i + ωi j ×θ f j ,
(5.1)
where ω – referred to as the gauge potential – is given in terms of the matrix by ω = † d.
(5.2)
The above, is a 2×2-matrix with entries in 1 (Sθ7 ) satisfying ωi j = ω ji and i ωii = 0. Note here that the entries ωi j commute with all elements in C ∞ (Sθ7 ). Indeed, from (4.6) we see that the elements in ωi j are T2 -invariant and hence central (as one forms) in (Sθ7 ). In other words L θ (ω) = ω, which shows that for an element f ∈ E as above, we have ∇0 ( f )i = d f i + ωi j ×θ f j = d f i + ωi j f j which coincides with the action of the classical connection d + ω on f . The curvature F0 = ∇02 = dω + ω2 of ∇0 is an element of End(E) ⊗C ∞ (S 4 ) 2 (Sθ4 ) θ that satisfies [1, 14] the self-duality equation ∗θ F0 = F0 ;
(5.3)
hence this connection is an instanton. At the classical value of the deformation parameter, θ = 0, the connection (5.2) is nothing but the SU (2) instanton of [7]. Its “topological charge”, i.e. the values of Top(E) in Definition 17, was already computed in [15]. Clearly it depends only on the class [ p] of the bundle and can be evaluated as the index Top([ p]) = index(D p ) = − γ5 π D (ch2 ( p)) (5.4) of the twisted Dirac operator D p = p(D ⊗ I4 ) p.
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The last equality in (5.4) follows from the vanishing of the class ch1 ( p) of the bundle2 . On the other hand, one finds π D ch2 ( p)) = 3γ5 , which, together with the fact that 1 − 1 = Tr ω |D|−4 = , 3 on S 4 (see for instance [22, 24]), gives the value Top([ p]) = 1. We aim at constructing all connections ∇ on E whose curvature satisfies the selfduality equations and have topological charge equal to 1. We can write any such connection in terms of the canonical connection as in Eq. (2.9), i.e. ∇ = ∇0 +α with α a one-form valued endomorphism of E. Clearly, this will not change the value of the topological charge. Being interested in SU(2)-instantons, we impose that α is traceless and skewhermitian. Here the trace is taken in the second leg of End(E) C ∞ (Sθ7 ) ad M2 (C). When complexified, we get an element α ∈ 1 (Sθ4 )⊗C ∞ (S 4 ) ∞ (ad(Sθ7 )) =: 1 (ad(Sθ4 )) θ (cf. Example 14). As usual, we impose an irreducibility condition on the instanton connections, a connection on E being irreducible if it cannot be written as the sum of two other connections on E. We are interested only in the irreducible instanton connections on the module E. Remark 19. In [27], we constructed projections p(n) for all modules C ∞ (Sθ7 ) ρ Cn over C ∞ (Sθ4 ) associated to the irreducible representations Cn of SU(2). The induced Grassmann connections ∇0(n) := p(n) d, when acting on C ∞ (Sθ7 ) ρ Cn , were written as d + ω(n) , with ω(n) an n × n matrix with entries in 1 (Sθ7 ). A similar argument as above then shows that all ω(n) have entries that are central (as one forms) in (Sθ7 ); again, this means that L θ (ω(n) ) = ω(n) . In particular, this holds for the adjoint bundle associated to the adjoint representation on su(2)C C3 (as complex representation spaces), from (2) which we conclude that ∇0 coincides with [∇0 , ·] (since this is the case if θ = 0). 5.2. Twisted infinitesimal symmetries. The noncommutative sphere Sθ4 can be realized as a quantum homogeneous space of the quantum orthogonal group SOθ (5) [36, 14, 1]. In other words, A(Sθ4 ) can be obtained as the subalgebra of A(SOθ (5)) made of elements that are coinvariant under the natural coaction of SOθ (4) on SOθ (5). For our purposes, it turns out to be more convenient to take a dual point of view and consider an action instead of a coaction. We obtain a twisted symmetry action of the Lie algebra so(5) on Sθ4 . Elements of so(5) act as twisted derivations on the algebra A(Sθ4 ). This action is lifted to Sθ7 and the basic instanton ∇0 described above is invariant under this infinitesimal twisted symmetry. Different instantons are obtained by a twisted symmetry action of so(5, 1). Classically, so(5, 1) is the conformal Lie algebra consisting of the infinitesimal diffeomorphisms leaving the conformal structure invariant. The Lie algebra so(5, 1) is given by adding 5 generators to so(5). We explicitly describe its action on Sθ4 together with its lift 2 The Chern character classes and their realization as operators are in the Appendix.
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to Sθ7 as an algebra of twisted derivations. The induced action on forms leaves the conformal structure invariant, and when acting on ∇0 eventually results in a five-parameter family of instantons. In fact, what we are really describing are Hopf algebras Uθ (so(5)) and Uθ (so(5, 1)) which are obtained from the undeformed Hopf algebras U(so(5)) and U(so(5, 1)) via a twist of a Drinfel’d type. Twisting of algebras and coalgebras has been known for some time [18, 19, 21]. The twists relevant for toric noncommutative manifolds are associated to the Cartan subalgebra of a Lie algebra and were already introduced in [31]. Their use to implement symmetries of toric noncommutative manifolds like the ones of the present paper was made explicit in [35]. The geometry of multi-parametric quantum groups and quantum enveloping algebras coming from twists has been studied in [2, 3]. Since we do not explicitly need Hopf algebras, we postpone a full fledged analysis of them and of their actions which will appear elsewhere. Here we shall rather give the explicit actions of both so(5) and so(5, 1) as Lie algebras of twisted derivations (having however always in the back of our mind their origin as deformed Hopf algebras). The eight roots of the Lie algebra so(5) are two-component vectors r = (r1 , r2 ) of the form r = (±1, ±1) and r = (0, ±1), r = (±1, 0). There are corresponding generators Er of so(5) together with two mutually commuting generators H1 , H2 of the Cartan subalgebra. The Lie brackets are [H1 , H2 ] = 0, [H j , Er ] = r j Er , [E −r , Er ] = r1 H1 + r2 H2 , [Er , Er ] = Nr,r Er +r ,
(5.5)
+ r
is not a root. with Nr,r = 0 if r In order to give the action of so(5) on Sθ4 , for convenience we introduce “partial derivatives”, ∂µ and ∂µ∗ with the usual action on the generators of the algebra A(Sθ4 ) i.e, ∂µ (z ν ) = δµν , ∂µ (z ν∗ ) = 0, and ∂µ∗ (z ν∗ ) = δµν , ∂µ∗ (z ν ) = 0. With these we construct operators on A(Sθ4 ), H1 = z 1 ∂1 − z 1∗ ∂1∗ , E +1,+1 = z 2 ∂1∗ − z 1 ∂2∗ , E +1,0 =
√1 (2z 0 ∂ ∗ 1 2
H2 = z 2 ∂2 − z 2∗ ∂2∗ , E +1,−1 = z 2∗ ∂1∗ − z 1 ∂2 ,
− z 1 ∂0 ),
E 0,+1 =
√1 (2z 0 ∂ ∗ 2 2
(5.6)
− z 2 ∂0 ) ,
and E −r = (Er )∗ , with the obvious meaning of the adjoint. Proposition 20. The operators in (5.6) give a well defined action of so(5) on the algebra A(Sθ4 ) provided one extends them to the whole of A(Sθ4 ) as “twisted derivations” via the rules, 1
1
Er (ab) = Er (a)λ 2 (−r1 H2 +r2 H1 ) (b) + λ 2 (r1 H2 −r2 H1 ) (a)Er (b), H j (ab) = H j (a)b + a H j (b), for any two elements a, b ∈
A(Sθ4 );
here λ =
e2π iθ
(5.7)
is the deformation parameter.
Proof. With these twisted rules, one explicitly checks compatibility of the action (5.6) with the commutation relations (4.1) of A(Sθ4 ).
1
Remark 21. The operators λ± 2 ri H j in (5.7) are understood as exponentials of diagonal ∗ of A(S 4 ), the operators H and H can be written as matrices: on the generators z µ , z µ 1 2 θ finite dimensional matrices. A comparison with Eq. (4.13) shows that H1 and H2 in (5.6) are the infinitesimal generators of the action of T2 on Sθ4 .
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We can write the twisted action of so(5) on A(Sθ4 ) by using the quantization map L θ introduced in Sect. 3. For L θ (a) ∈ A(Sθ4 ) and t ∈ so(5) a twisted action is defined by T · L θ (a) = L θ (t · a),
(5.8)
where T is the “quantization” of t and t · a is the classical action of so(5) on A(S 4 ) (a better but heavier notation for the action T · would be t ·θ ). One checks that both of these definitions of the twisted action coincide. The latter definition allows one to define an action of so(5) on C ∞ (Sθ4 ) by allowing a to be in C ∞ (S 4 ) in Eq. (5.8). Furthermore, as 1
operators on the Hilbert space H of spinors, one could identify λ 2 (r1 H2 −r2 H1 ) = U ( 21 r ·θ ), with r = (r1 , r2 ), θ the antisymmetric two by two matrix with θ12 = −θ21 = θ and U (s) is the representation of T2 on H as in Sect. 3.1. The twisted action of the Lie algebra so(5) on A(Sθ4 ) is extended to the differential calculus ((Sθ4 ), d) by requiring it to commute with the exterior derivative, T · dω := d(T · ω) (Sθ4 ).
Then, we need to use the rule (5.7) on a generic form. For for T ∈ so(5), ω ∈ instance, on 1-forms we have, ) * ! 1 1 " Er Er (ak )d λ 2 (−r1 H2 +r2 H1 ) (bk ) +λ 2 (r1 H2 −r2 H1 ) (ak )d Er (bk ) , ak dbk = k
Hj
) k
k
* ak dbk
=
H j (ak )dbk + ak d H j (bk ) .
(5.9)
k
The representation of so(5) on Sθ4 given in (5.6) is the fundamental vector representation. When lifted to Sθ7 one gets the fundamental spinor representation: as we see from the quadratic relations among corresponding generators, as given in (4.9), the lifting amounts to take the “square root” representation. The action of so(5) on A(Sθ7 ) is constructed by requiring twisted derivation properties via the rule (5.7) – when acting now on any two elements a, b ∈ A(Sθ7 ) – so as to reduce to the action (5.6) on A(Sθ4 ) when using the defining quadratic relations (4.9). The resulting action on A(Sθ7 ) can be given as the action of matrices ’s on the ψ’s, ab ψb∗ , ab ψb , ψa∗ → (5.10) ψa → b
b
with the matrices = {H j , Er } given explicitly by, 1 −1 1 1 −1 1 H1 = 2 , , H2 = 2 −1 −1 1 0 0 1 0 0 0 E +1,+1 = , , E +1,−1 = −µ 0 0 00 −1 0 0 0 0 0 0 0 0µ E +1,0 = √1 µ 0 0 −1 , E 0,+1 = √1 0 1 0 0 , 2 2 0 00 0 00 and := σ σ −1 with σ :=
0 −1 1 0
0
0
0 −1 1 0
.
(5.11)
(5.12)
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619
Furthermore, E −r = (Er )∗ . With the twisted rules (5.7) for the action on products of any two elements a, b ∈ A(Sθ7 ), one checks compatibility of the above action with the 1
commutation relations (4.4) of A(Sθ7 ). Again, the operators λ± 2 ri H j in (5.7) are exponentials of diagonal matrices H1 and H2 given in the representation (5.11) and as above, 1 one could think of λ 2 (r1 H2 −r2 H1 ) as the operator U ( 21 r · θ ).
Remark 22. Compare the form of the matrices H1 and H2 in the representation (5.11) above with the lifted action σ of T2 on Sθ7 as defined in (4.14). One checks that σs = eπi((s1 +s2 )H1 +(−s1 +s2 )H2 ) , when acting on the spinor (ψ1 , . . . , ψ4 ). Notice that = − t at θ = 0. There is a beautiful correspondence between the matrices in the representation (5.11) and the twisted Dirac matrices introduced in (4.11), 1 ∗ 4 [γ1 , γ1 ] = 2H1 1 4 [γ1 , γ2 ] = (µ + µ)E +1,+1 √ 1 4 [γ1 , γ0 ] = 2E +1,0
1 ∗ 4 [γ2 , γ2 ] = 2H2 ∗ 1 4 [γ1 , γ2 ] = (µ + µ)E +1,−1 √ 1 4 [γ2 , γ0 ] = 2µE 0,+1 .
(5.13)
Remark 23. It is straightforward to check that the twisted Dirac matrices satisfy the following relations under conjugation by σ : (σ γ0 σ −1 )t = γ0 ;
(σ γ1 σ −1 )t = γ1 λ H2 ;
(σ γ2 σ −1 )t = γ2 λ H1 .
(5.14)
As for Sθ4 , the twisted action of so(5) on A(Sθ7 ) is straightforwardly extended to the and the property differential calculus ((Sθ7 ), d). Furthermore, due to the form of a2 = σab ψb∗ for the second column of the matrix in (4.6), we have also that so(5) acts on by left matrix multiplication by , and by right matrix multiplication on ∗ by the matrix transpose t as follows: ai →
b
ab bi ,
∗ ia →
∗ ab . ib
(5.15)
a
All this is used in the following Proposition 24. The instanton gauge potential ω is invariant under the action of so(5). Proof. From the above observations, the gauge potential transforms as, t −r H ω = ∗ d → ∗ λ 1 2 + λr2 H1 d, where λ−ri H j is understood in its representation (5.11) on A(Sθ7 ). Direct computation t λ−r1 H2 + λr2 H1 = 0, which finishes the proof.
for = {H j , Er } shows that
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5.3. Twisted conformal transformations. In order to have new instantonic configurations we need to use conformal transformations. The conformal Lie algebra so(5, 1) consists of the generators of so(5) together with the dilation and the so-called special conformal transformations. On R4 with coordinates {xµ , µ = 1, . . . , 4} they are given by the operators H0 = µ xµ ∂/∂ xµ and G µ = 2xµ ν xν ∂/∂ xν − ν xν2 (∂/∂ xν ), respectively [28]. Since we are deforming with respect to the two-torus coming from the Cartan subalgebra H1 , H2 of so(5), we write the Lie algebra so(5, 1) as so(5) together with five extra generators, H0 , G r , the latter labeled by the corresponding roots with respect to H1 and H2 , that is r = (±1, 0), (0, ±1). Besides the Lie brackets of so(5) given in (5.5), we have [H0 , Hi ] = 0, √ [H0 , G r ] = 2Er ,
[H j , G r ] = r j G r , √ [H0 , Er ] = ( 2)−1 G r ,
(5.16)
whenever r = (±1, 0), (0, ±1), and [G −r , G r ] = 2r1 H1 + 2r2 H2 , r,r G r +r , [Er , G r ] = N
[G r , G r ] = Nr,r Er +r , √ [E −r , G r ] = 2H0 ,
(5.17)
r,r = 0 if r + r does not with, as before, Nr,r = 0 if r + r is not a root of so(5) and N belong to {(±1, 0), (0, ±1)}. The action of so(5, 1) on A(Sθ4 ) is given by the operators (5.6) together with H0 = ∂0 − z 0 (z 0 ∂0 + z 1 ∂1 + z 1∗ ∂1∗ + z 2 ∂2 + z 2∗ ∂2∗ ),
G 1,0 = 2∂1∗ − z 1 (z 0 ∂0 + z 1 ∂1 + z 1∗ ∂1∗ + λz 2 ∂2 + λz 2∗ ∂2∗ ), G 0,1 = 2∂2∗ − z 2 (z 0 ∂0 + z 1 ∂1 + z 1∗ ∂1∗ + z 2 ∂2 + z 2∗ ∂2∗ ),
(5.18)
and G −r = (G r )∗ . Note that the introduction of the extra λ’s in G 1,0 (and G −1,0 ) are necessary for the Lie algebra structure of so(5, 1) – as dictated by the Lie brackets in (5.16) and (5.17) – to be preserved, that is in order to have a Lie algebra representation. Since the operators H0 and G r are quadratic in the z’s, one has to be careful when deriving the above Lie brackets and use the twisted rules (5.7). For instance, on the generator z 2 , we have [E −1,−1 , G 1,0 ](z 2 ) = E −1,−1 (−λz 1 z 2 ) + G 1,0 (z 1∗ ) = −λ(E −1,−1 (z 1 )λ H2 (z 2 ) + λ H1 (z 1 )E −1,−1 (z 2 )) + G 1,0 (z 1∗ ) = −z 2∗ z 2 + z 1 z 1∗ + 2 − z 1 z 1∗ = G 0,−1 (z 2 ). As for Proposition 20, a direct computation establishes the following: Proposition 25. The operators in (5.6) and (5.18) give a well defined action of so(5, 1) on the algebra A(Sθ4 ) provided one extends them to the whole of A(Sθ4 ) as twisted derivations via the rules (5.7) together with 1
1
G r (ab) = G r (a)λ 2 (−r1 H2 +r2 H1 ) (b) + λ 2 (r1 H2 −r2 H1 ) (a)G r (b), H0 (ab) = H0 (a)b + a H0 (b), for any two elements a, b ∈ A(Sθ4 ).
(5.19)
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Equivalently, so(5, 1) could be defined to act on A(Sθ4 ) by T · L θ (a) = L θ (t · a),
(5.20)
for T the operator deforming t ∈ so(5, 1) and L θ (a) ∈ A(Sθ4 ) deforming a ∈ A(S 4 ). Again, Eq. (5.20) makes sense for a ∈ C ∞ (S 4 ), which defines an action of so(5, 1) on C ∞ (Sθ4 ). As before, the action on the differential calculus ((Sθ4 ), d) is obtained by requiring commutation with the exterior derivative: T · dω = d(T · ω), for T ∈ so(5, 1) and ω ∈ (Sθ4 ). On products we shall have formulæ like the one in (5.9), ) * ! 1 1 " G r (ak )d λ 2 (−r1 H2 +r2 H1 ) (bk ) +λ 2 (r1 H2−r2 H1 ) (ak )d G r (bk ) , ak dbk = Gr k
H0
) k
k
* ak dbk
=
H0 (ak )dbk + ak d H0 (bk ) .
(5.21)
k
What we are dealing with are “infinitesimal” twisted conformal transformations: Lemma 26. The Hodge ∗θ -structure of (Sθ4 ) is invariant for the twisted action of so(5, 1), T · (∗θ ω) = ∗θ (T · ω). Proof. Recall that T (L θ (a)) = L θ (t · a) for a ∈ A(S 4 ) and T the “quantization” of t ∈ so(5, 1). Then, since so(5, 1) leaves the Hodge ∗-structure of (S 4 ) invariant and the differential d commutes with the action of so(5, 1), it follows that the latter algebra leaves the Hodge ∗θ -structure of (Sθ4 ) invariant as well.
Again, the action of so(5, 1) on Sθ4 can be lifted to an action on Sθ7 . And the latter action can be written as in (5.10) in terms of matrices ’s acting on the ψ’s, ψa → ab ψb∗ , ab ψb , ψa∗ → (5.22) b
b
where in addition to (5.11) we have also the matrices = {H0 , G r }, given explicitly by H0 = 21 (−z 0 I4 + γ0 ), G 1,0 = 21 (−z 1 λ−H2 + γ1 ),
G 0,1 = 21 (−z 2 + λ−H1 γ2 ),
(5.23)
= σ σ −1 . Notice the reappearance of the twisted Dirac with G −r = (G r )∗ and ∗ matrices γµ , γµ of (4.11) in the above expressions. In the above expressions, the operators λ−H j are 4 × 4 matrices obtained from the spin representation (5.11) of H1 and H2 and given explicitly by ) )µ * * µ √ µ µ −H1 −H2 = = (5.24) , λ , µ = λ. λ µ µ µ
µ
As for so(5), the action of so(5, 1) on the matrix is found to be by left matrix multiplication by and on ∗ by , ∗ ∗ ai → ab ib ab bi , ia → . (5.25) b
a
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Here we have to be careful with the ordering between and ∗ in the second term since the ’s involve the (not-central) z’s. There are the following useful commutation relations between the z µ ’s and : z 1 ai = (λ−H2 )ab bi z 1 , ∗ ∗ z 1 ia = ib (λ−H2 )ba z 1 ,
z 2 ai = (λ−H1 )ab bi z 2 , ∗ ∗ z 2 ia = ib (λ−H1 )ba z 2 ,
(5.26)
with λ−H j understood as the explicit matrices (5.24). Proposition 27. The instanton gauge potential ω = ∗ d transforms under the action of the extra elements of so(5, 1) as ω → ω + δi ω, where δ0 ω := H0 (ω) = −z 0 ω − 21 dz 0 I2 + ∗ γ0 d, δ1 ω := G +1,0 (ω) = −z 1 ω − 21 dz 1 I2 + ∗ γ1 d, δ2 ω := G 0,+1 (ω) = −z 2 ω − 21 dz 2 I2 + ∗ γ2 d, δ3 ω :=
−1,0 (ω)
= −ωz 1 − 21 dz 1 I2 + ∗ γ1∗ d,
δ4 ω := G 0,−1 (ω) = −ωz 2 − 21 dz 2 I2 + ∗ γ2∗ d, with γµ , γµ∗ the twisted 4 × 4 Dirac matrices defined in (4.11). Proof. The action of H0 on the instanton gauge potential ω = ∗ d takes the form H0 (ω) = H0 ( ∗ )d + ∗ d(H0 ()) = ∗ (−z 0 I4 + γ0 )d − 21 dz 0 ∗ , since z 0 is central. Direct computation results in the above expression for δ0 ω. Instead, the twisted action of G r on ω takes the form, ∗ ∗ ab ib G r : ωi j → (λ−r1 H2 )ac dcj + (λr2 H1 )ab ib ac dcj a,b,c r2 H1
+ (λ
∗ )ab ib (dac )cj ,
j = σ H j σ −1 = −H j . Let us consider the case r = (+1, 0). where we used the fact that H ∗ so that from the defiFirstly, note that the complex numbers (λ−H2 )ac commute with ib nition of and , and using (5.26), we obtain for the first two terms, ∗ ∗ −z 1 ( ∗ d)i j + 21 ib (σ γ1 σ −1 )cb (λ−H2 )cd dd j + 21 ib (γ1 )bc dcj .
The first term forms the matrix −z 1 ω whereas the second two terms combine to give ∗ (d ) reduces ∗ γ1 d, due to relation (5.14). Finally, using Eq. (5.26) the term ib ac cj 1 1 ∗ to − 2 dz 1 ib bj = − 2 dz 1 I2 . The formulae for r = (−1, 0) and r = (0, ±1) are established in likewise manner.
Remark 28. At first sight, the infinitesimal gauge potentials δ j ω given above do not seem to be su(2)-gauge potentials, in that they do not satisfy (δ j ω)kl = (δ j ω)lk and k (δ j ω)kk = 0. This is only due to the fact that the generators G r and H0 are the deformed analogues of the generators of the complexified Lie algebra so(5, 1) ⊗R C. One recovers su(2)-gauge potentials by acting with the real generators 21 (G r + G r∗ ), 1 1 1 ∗ 2i (G r − Fr ) and H0 . The resulting gauge potentials, δ0 ω, 2 (δ1 ω + δ3 ω), 2i (δ1 ω − δ3 ω), 1 1 2 (δ2 ω + δ4 ω) and 2i (δ2 ω − δ4 ω), are traceless skew-hermitian matrices with entries in 7 1 (Sθ ).
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623
The transformations of the gauge potential ω under the twisted symmetry so(5, 1), given in Proposition 27, induce natural transformations of the canonical connection ∇0 in (5.1) to ∇t,i := ∇0 + tδi ω + O(t 2 ). We shall presently see that these new connections are (infinitesimal) instantons, i.e. their curvatures are self-dual. In fact, this also follows from Lemma 26 which states that so(5, 1) acts by conformal transformation therefore leaving the self-duality equation ∗θ F0 = F0 for the basic instanton ∇0 invariant. 4 We start by writing ∇t,i in terms of the canonical connection on E p A(Sθ4 ) . Using the isomorphism, described in Example 14, between this module and the module of equivariant maps A(Sθ7 ) ρ C2 , we find that ∇t,i = pd + tδi α + O(t 2 ) with δ0 α = pγ0 (d p) p − 21 dz 0 ∗ , δ1 α = pγ1 (d p) p − 21 dz 1 ∗ , δ2 α = pγ2 (d p) p −
∗ 1 2 dz 2 ,
δ3 α = pγ1∗ (d p) p − 21 dz 1∗ ∗ , δ4 α =
pγ2∗ (d p) p
−
(5.27)
∗ ∗ 1 2 dz 2 ,
The δi α’s are 4 × 4 matrices with entries in the one-forms 1 (Sθ4 ) and satisfying conditions pδi α = δi αp = pδi αp = δi α, as expected from the general theory on connections on modules in Sect. 2. Indeed, using relations (5.26) one can move the dz µ ’s to the left of at the cost of some µ’s, so getting an expression like dz i p ∈ M4 (1 (Sθ4 )). From Eq. (2.10), the curvature Ft,i of the connection ∇t,i is given by Ft,i = F0 + t pd(δi α) + O(t 2 ).
(5.28)
In order to check self-duality (modulo t 2 ) of this curvature, we will express it in terms of the projection p and consider Ft,i as a two-form valued endomorphism on the module 4 E p A(Sθ4 ) . Proposition 29. The curvatures Ft,i of the connections ∇t,i , i = 0, . . . , 4, are given by Ft,i = F0 + tδi F + O(t 2 ), where F0 = pd pd p and δi F are the following 4 × 4-matrices of 2-forms: δ0 F = −2z 0 F0 , δ1 F = −2z 1 λ H2 F0 , δ2 F = −2z 2 λ
H1
F0 ;
δ3 F = −2z 1∗ λ−H2 F0 , δ4 F =
(5.29)
−2z 2∗ λ−H1 F0 .
Proof. A small computation yields for δi F = pd(δi α), thought of as an 2 (Sθ4 )valued endomorphism on E the expression, δi F = p(d p)γi (d p) p − pγi (d p)(d p) p, with the notation γ3 = γ1∗ and γ4 = γ2∗ , and using p(d p) p = 0. Then, the crucial property p(d pγi + γi d p)(d p) p = 0 all i = 0, . . . , 4 yields δi F = −2 pγi d pd pp. This is expressed as δi F = −2 pγi pdpdp by using d p = (d p) p + pd p. Finally, pγi p = ( ∗ γi ) ∗ , so that the result follows from the definition of the z’s in terms of the Dirac matrices given in Eq. (4.10), together with the commutation relations between them and the matrix in Eq. (5.26)
Proposition 30. The connections ∇t,i are (infinitesimal) instantons, i.e. ∗θ Ft,i = Ft,i
mod t 2 .
Moreover, the connections ∇t,i are not gauge equivalent to ∇0 .
(5.30)
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Proof. The first point follows directly from the above expressions for δi F and the selfduality of F0 . To establish the gauge inequivalence of the connections ∇t,i with ∇0 , we recall that an infinitesimal gauge transformation is given by ∇0 → ∇0 + t[∇0 , X ] for X ∈ ∞ (ad(Sθ7 )). We need to show that δi ω is orthogonal to [∇0 , X ] for any such X , i.e. ([∇0 , X ], δi ω)2 = 0, with the natural inner product on 1 (ad(Sθ7 )) := 1 (Sθ4 ) ⊗C ∞ (S 4 ) ∞ (ad(Sθ7 )). From θ Remark 19, it follows that "∗ ! (∇0(2) (X ), δi ω)2 = (X, ∇0(2) (δi ω))2 , which then should vanish for all X . From Eq. (5.20), we see that δi ω = Ti (ω) coincides with L θ (ti · ω(0) ) with ti and ω(0) the classical counterparts of Ti and ω, respectively. In the undeformed case, the infinitesimal gauge potentials generated by acting with elements in so(5, 1) − so(5) on the basic instanton gauge potential ω(0) satisfy (∇0(2) )∗ (δi ω(0) ) = 0 as shown in [6]. The result then follows from the observation that (2) ∇0 commutes with the quantization map L θ (cf. Remark 19).
5.4. Local expressions. In this section, we obtain “local expressions” for the instantons on Sθ4 constructed in the previous section; that is we map them to a noncommutative R4θ obtained by “removing a point” from Sθ4 . The algebra A(R4θ ) of polynomial functions on the 4-plane R4θ is defined to be the ∗-algebra generated by elements ζ1 , ζ2 satisfying ζ1 ζ2 = λζ2 ζ1 ;
ζ1 ζ2∗ = λζ2∗ ζ1
(5.31)
with λ = e2πiθ as above. At θ = 0 one recovers the ∗-algebra A(R4 ) of polynomial functions on the usual 4-plane R4 . The algebra A(R4θ ) can also be defined as the vector space A(R4 ) equipped with a deformed product ×θ as in Eq. (3.2). Indeed, the torus T2 acts naturally on the two complex coordinates of R4 C2 . This also allows us to define the smooth algebra Cb∞ (R4θ ) as the vector space Cb∞ (R4 ) of bounded smooth functions on R4 equipped with a deformed product ×θ . However, for our purposes it suffices to consider the polynomial algebra A(R4θ ) with one self-adjoint central generator ρ added together with relations ρ 2 (1 + |ζ |2 ) = (1 + |ζ |2 )ρ 2 = 1, where |ζ |2 := ζ1∗ ζ1 + ζ2∗ ζ2 (this enlargement was already done in [14]). In the following, we will denote the enlarged algebra by 4 ) and will also use the notation A(R θ ρ 2 = (1 + |ζ |2 )−1 =
1 . 1 + |ζ |2
(5.32)
Note that ρ 2 is an element in Cb∞ (R4θ ). 4 ) by One defines elements z µ , µ = 0, 1, 2 in A(R z 0 = (1 − |ζ |2 )(1 + |ζ |2 )−1 ,
z j = 2ζ j (1 + |ζ |2 )−1 j = 1, 2,
(5.33)
and checks that they satisfy the same relations as in (4.1) of the generators z µ of A(Sθ4 ). The difference is that the classical point z 0 = −1, z j = z ∗j = 0 of Sθ4 is not in the
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625
spectrum of z µ . We interpret the noncommutative plane R4θ as a “chart” of the noncommutative 4-sphere Sθ4 and Eq. (5.33) as the (inverse) stereographic projection from Sθ4 to R4θ . In fact, one can cover Sθ4 by two such charts with domain R4θ , and transition functions on R4θ \{0}, where {0} is the classical point ζ j = ζ j∗ = 0 of R4θ (cf. [14] for more details). A differential calculus ((R4θ ), d) on R4θ is obtained from the general procedure described in Sect. 3. Explicitly, (R4θ ) is the graded ∗-algebra generated by the elements ζµ of degree 0 and dζµ of degree 1 with relations, dζµ dζν∗ + λνµ dζν∗ dζµ = 0, dζµ dζν + λµν dζν dζµ = 0, ζµ dζν = λµν dζν ζµ , ζµ dζν∗ = λνµ dζν∗ ζµ ,
(5.34)
and λ12 = λ21 =: λ = e2πiθ . There is a unique differential d on (R4θ ) for which one has d : ζµ → dζµ and a Hodge star operator ∗θ : p (R4θ ) → 4− p (R4θ ), obtained from the classical Hodge star operator as before. In terms of the standard Riemannian metric on R4 , on two-forms we have, ∗θ dζ1 dζ2 = −dζ1 dζ2 ,
∗θ dζ1 dζ1∗ = −dζ2 dζ2∗ ,
∗θ dζ1 dζ2∗ = dζ1 dζ2∗ ,
(5.35)
and ∗2θ = id. These are the same formulae as the ones for the undeformed Hodge ∗ on R4 – since the metric is not changed in an isospectral deformation. Again, we slightly enlarge the differential calculus (R4θ ) by adding the self-adjoint central generator ρ. The differential d on ρ is derived from the Leibniz rule for d applied to its defining relation, ! " (dρ 2 )(1 + |ζ |2 ) + ρ 2 d(1 + |ζ |2 ) = d ρ 2 (1 + |ζ |2 ) = 0, so that ρdρ = 21 dρ 2 = − 21 ρ 4 d(1 + |ζ |2 ) = − 21 ρ 4 µ (ζµ dζµ∗ + ζµ∗ dζµ ). The enlarged (R4 ). differential calculus will be denoted by θ The stereographic projection from S 4 onto R4 is a conformal map commuting with the action of T2 ; thus it makes sense to investigate the form of the instanton connections on Sθ4 obtained in Proposition 27 on the local chart R4θ . As in [25], we first introduce a “local section” of the principal bundle Sθ7 → Sθ4 on the local chart of Sθ4 defined in (5.33). Let u = (u 1 , u 2 ) be a complex spinor of modulus one, u ∗1 u 1 + u ∗2 u 2 = 1, and define ∗ ψ1 u ψ3 ζ1 ζ2∗ u1 =ρ 1 , =ρ . (5.36) ψ2 u2 ψ4 −µζ2 µζ1 u2 Remark 31. Strictly speaking, the symbols ψa here denote elements in the algebra A(Sθ7 ) enlarged by an extra generator which is the inverse of 1 + z 0 = 2(1 + ψ1∗ ψ1 + ψ2∗ ψ2 ). Intuitively, this corresponds to “remove” the fiber S 3 in Sθ7 above the classical point z 0 = −1, z j = z ∗j = 0 of the base space Sθ4 . The commutation rules of the u j ’s with the ζk ’s are dictated by those of the ψ j : u 1 ζ j = µζ j u 1 , u 2 ζ j = µζ j u 2 ,
j = 1, 2.
(5.37)
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The right action of SU(2) rotates the vector u while mapping to the “same point” of Sθ4 , which, using the definition (4.9), from the choice in (5.36) is found to be z1, 2(ψ1 ψ3∗ + ψ2∗ ψ4 ) =
2(−ψ1∗ ψ4 + ψ2 ψ3∗ ) = z2 ,
2(ψ1∗ ψ1 + ψ2∗ ψ2 ) − 1 = z0 , (5.38)
and is in the local chart (5.33), as expected. " ! u −u ∗ By writing the unit vector u as a matrix, u = u 1 u ∗2 ∈ SU(2), we have 2
=ρ
I2 0 0 Z
u , u
with Z =
1
!
ζ1∗ ζ2∗ −µζ2 µζ1
"
.
(5.39)
Then, direct computation of the gauge potential ω = ∗ d yields uωu ∗ = ρ −1 dρ + ρ 2 Z ∗ dZ + (du)u ∗ ∗ ∗ 2(ζ dζ ∗ − dζ ζ ∗ ) 1 1 2 1 2 i ζi dζi − dζi ζi + (du)u ∗ , = ∗ ∗ ∗ ∗ 2 (1 + |ζ | ) 2(ζ2 dζ1 − dζ2 ζ1 ) i ζi dζi − dζi ζi
(5.40)
while its curvature F0 = dω + ω2 is 1 2dζ1 dζ2∗ dζ1 dζ1∗ − dζ2 dζ2∗ u F0 u ∗ = ρ 4 dZ ∗ dZ = ∗ ∗ ∗ . (5.41) 2 2 2dζ dζ −dζ dζ 2 1 1 1 − dζ2 dζ2 (1 + |ζ | ) From the expressions (5.35) for the Hodge operator on two forms, one checks that this curvature is self-dual, ∗θ (u F0 u ∗ ) = u F0 u ∗ , as expected. The explicit local expressions for the transformed – under infinitesimal conformal transformations – gauge potentials and their curvature can be obtained in a similar manner. As an example, let us work out the local expression for δ0 ω which is the most transparent one. Given the expression for δ0 ω in Proposition 27, a direct computation shows that its local counterpart is uδ0 ωu ∗ = −2ρdρ − 2ρ 4 Z ∗ dZ,
(5.42)
giving for the transformed curvature, u Ft,0 u ∗ = F0 + 2t (1 − 2ρ 2 )F0 + O(t 2 ).
(5.43)
It is clear that this rescaled curvature still satisfies the self-duality equation; this is also in concordance with Proposition 29, being z 0 = 2ρ 2 − 1. 5.5. Moduli space of instantons. We will closely follow the infinitesimal construction of instantons for the undeformed case given in [6] . This will eventually result in the computation of the dimension of the “tangent space” to the moduli space of instantons on Sθ4 by index methods. It will turn out that the five-parameter family of instantons constructed in the previous section is indeed the complete set of infinitesimal instantons on Sθ4 . Let us start by considering a family of connections on Sθ4 , ∇t = ∇0 + tα,
(5.44)
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627
where α ∈ 1 (ad(Sθ7 )) ≡ 1 (Sθ4 ) ⊗C ∞ (S 4 ) ∞ (ad(Sθ7 )) and after Example 14 we have θ
denoted ∞ (ad(Sθ7 )) = C ∞ (Sθ7 ) ⊗ad su(2). For ∇t to be an instanton, we have to impose the self-duality equation ∗θ Ft = Ft on the curvature Ft = F0 + t[∇, α] + O(t 2 ) of ∇t . This leads, when differentiated with respect to t, at t = 0, to the linearized self-duality equation P− [∇0 , α] = 0,
with P− :=
1 2 (1 − ∗θ )
(5.45)
the projection onto the antiself-dual 2-forms. Here [∇0 , α], [∇0 , α]i j = dαi j + ωik αk j − αik ωk j ,
(5.46)
is an element in 2 (ad(Sθ7 )) and has vanishing trace, due to the fact that ωik αk j = αk j ωik (cf. Eqs. (5.1), (5.2) and the related discussion). If the family ∇t were obtained from an infinitesimal gauge transformation we would have had α = [∇0 , X ], for some X ∈ ∞ (ad(Sθ7 )). Indeed, [∇0 , X ] is an element in 1 (ad(Sθ7 )) and P− [∇0 , [∇0 , X ]] = [P− F0 , X ] = 0, since F0 is self-dual. Hence, we have defined an element in the first cohomology group H 1 of the so-called self-dual complex, ! ! "" d ! ! "" d ! ! "" 0 1 (5.47) 0 → 0 ad Sθ7 −→ 1 ad Sθ7 −→ 2− ad Sθ7 → 0, where 0 (ad(Sθ7 )) = ∞ (ad(Sθ7 )) and d0 = [∇0 , · ], d1 := P− [∇0 , · ]. Note that these operators are Fredholm operators, so that the cohomology groups of the complex are finite dimensional. As usual, the complex can be replaced by a single Fredholm operator ! ! "" ! ! "" ! ! "" d0∗ + d1 : 1 ad Sθ7 −→ 0 ad Sθ7 ⊕ 2− ad Sθ7 , (5.48) with d0∗ the adjoint of d0 with respect to the inner product (3.25). Our goal is to compute h 1 = dim H 1 , the number of “true” instantons. This is achieved by calculating the alternating sum h 0 − h 1 + h 2 of Betti numbers from the index of this Fredholm operator, index(d0∗ + d1 ) = −h 0 + h 1 − h 2 ,
(5.49)
while showing that h 0 = h 2 = 0. By definition, H 0 consists of the covariant constant elements in ∞ (ad(Sθ7 )). Since (2) the operator [∇0 , · ] commutes with the action of T2 and coincides with ∇0 on 7 ∞ (ad(Sθ )) (cf. Remark 19), being covariantly constant means that [∇0 , X ] = ∇0(2) (X ) = 0.
(5.50)
If we write once more X = L θ (X (0) ) in terms of its classical counterpart and use the (2) fact that ∇0 commutes with L θ (cf. Remark 19) we find that this condition entails ! ! "" ! ! "" (2) (2) (5.51) ∇0 L θ X (0) = L θ ∇0 X (0) = 0. For the undeformed case, there are no covariant constant elements in ∞ (ad(S 7 )) for an irreducible self-dual connection on E, thus we conclude that h 0 = 0. A completely analogous argument for the kernel of the operator d1∗ shows that also h 2 = 0.
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5.6. Dirac operator associated to the complex. The next step consists in computing the index of the Fredholm operator d0∗ + d1 defined in (5.48). Firstly, this operator can be replaced by a Dirac operator on the spinor bundle S with coefficients in the adjoint bundle. For this, we need the following lemma, which is a straightforward modification of its classical analogue [6]. Recall that the Z2 -grading γ5 induces a decomposition of the spinor bundle S = S + ⊕ S − . Note also that S − coincides classically with the charge −1 anti-instanton bundle. Indeed, the Levi-Civita connection – when lifted to the spinor bundle and restricted to negative chirality spinors – has antiself-dual curvature. Similarly, S + coincides with the charge +1 instanton bundle. Then Remark 10 implies that the C ∞ (S 4 )-modules ∞ (S 4 , S ± ) have a module-basis that is homogeneous under the action of T2 . We conclude from T2 -equivariance that ∞ (Sθ4 , S − ) is isomorphic to the charge −1 anti-instanton bundle ∞ (Sθ7 ×SU(2) C2 ) on Sθ4 . Similarly ∞ (Sθ4 , S + ) is isomorphic to the charge +1 instanton bundle. Lemma 32. There are the following isomorphisms of right C ∞ (Sθ4 )-modules: 1 (Sθ4 ) ∞ (Sθ4 , S + ⊗ S − ) ∞ (Sθ4 , S + ) ⊗C ∞ (S 4 ) ∞ (Sθ4 , S − ), θ
0 (Sθ4 ) ⊕ 2− (Sθ4 ) ∞ (Sθ4 , S − ⊗ S − ) ∞ (Sθ4 , S − ) ⊗C ∞ (S 4 ) ∞ (Sθ4 , S − ). θ
Proof. Since classically 1 (S 4 ) ∞ (S 4 , S + ⊗ S − ) as σ -equivariant C ∞ (S 4 )bimodules, lemma 5 shows that 1 (Sθ4 ) ∞ (Sθ4 , S + ⊗ S − ) as C ∞ (Sθ4 )-bimodules. The observations above the Lemma indicate that S ± S 7 ×ρ ± C2 for the spinor representation ρ + ⊕ ρ − of Spin(4) SU(2) × SU(2) on C4 , so that ∞ (Sθ4 , S + ⊗ S − ) C ∞ (Sθ7 ) ρ + ⊗ρ − (C2 ⊗ C2 ) " ! " ! C ∞ (Sθ7 ) ρ + C2 ⊗C ∞ (S 4 ) C ∞ (Sθ7 ) ρ − C2 , θ
using Proposition 11 in the last line. This proves our claim. An analogous statement holds for the second isomorphism.
Let us forget for the moment the adjoint bundle ad(Sθ7 ). Since (Sθ4 ) (S 4 ) as vector spaces and both d and the Hodge ∗ commute with the action of T2 , the operator d∗ + P− d can be understood as a map from 1 (S 4 ) → 0 (S 4 ) ⊕ 2− (S 4 ) (see Sect. 3.4). Under the isomorphisms of the above lemma, this operator is replaced [6] by a Dirac operator with coefficients in S − , D : ∞ (Sθ4 , S + ⊗ S − ) → ∞ (Sθ4 , S − ⊗ S − ).
(5.52)
Twisting by the adjoint bundle ad(Sθ7 ) merely results in a composition with the projection p(2) defining the bundle ad(Sθ7 ). Hence, eventually the operator d0∗ + d1 is replaced by the Dirac operator ! ! " " D : ∞ Sθ4 , S + ⊗ S − ⊗ ad(Sθ7 ) → ∞ Sθ4 , S − ⊗ S − ⊗ ad(Sθ7 ) , (5.53) with coefficients in the vector bundle S − ⊗ ad(Sθ7 ) on Sθ4 . We have finally arrived at the computation of the index of this Dirac operator which we do by means of the Connes-Moscovici local index formula. It is given by the pairing, index(D) = φ, ch(S − ⊗ ad(Sθ7 )) = φ, ch(S − ) · ch(ad(Sθ7 )).
(5.54)
Noncommutative Instantons from Twisted Conformal Symmetries
629
In the Appendix we recall the expression for both the cyclic cocycle φ and the Chern characters, as well as their realization as operators π D (ch(·)) on the Hilbert space of spinors H. In [27] we computed these operators for all modules associated to the principal bundle Sθ7 → Sθ4 . In particular, for the adjoint bundle we found that π D ch0 (ad(Sθ7 )) = 3, π D ch1 (ad(Sθ7 )) = 0, π D ch2 (ad(Sθ7 )) = 4(3γ5 ). To compute the Chern character of the spinor bundle S − we use its mentioned identification with the charge −1 instanton bundle ∞ (Sθ7 ×SU(2) C2 ) on Sθ4 . It then follows from [15] (cf. also [27]) that π D ch0 (S − ) = 2, π D ch1 (S − ) = 0, π D ch2 (S − ) = −3γ5 . Combining both Chern characters and using the local index formula on Sθ4 , we have index(D) = 6 Res z −1 tr(γ5 |D|−2z ) + 0 + 21 (2 · 4 − 3 · 1)Res tr(3γ52 |D|−4−2z ), (5.55) z=0
z=0
with D identified with the classical Dirac operator on S 4 (recall that we do not change it in the isospectral deformation). Now, the first term vanishes due to index(D) = 0 for this classical operator. On the other hand γ52 = I4 , and 3Res tr(|D|−4−2z ) = 6 Tr ω (|D|−4 ) = 2, z=0
since the Dixmier trace of |D|−m on the m-sphere equals 8/m! (cf. for instance [22, 24]). We conclude that index(D) = 5 and for the moduli space of instantons on Sθ4 , we have the following Theorem 33. The tangent space at the base point ∇0 to the moduli space of (irreducible) SU(2)-instantons on Sθ4 is five-dimensional. 6. Towards Yang–Mills Theory on Mθ In this final section, we shall briefly describe how the Yang–Mills theory on Sθ4 can be generalized to any compact four-dimensional toric noncommutative manifold Mθ . With G a compact semisimple Lie group, let P → M be a principal G bundle on M. We take M to be a compact four-dimensional Riemannian manifold equipped with an isometrical action σ of the torus T2 . For the construction to work, we assume that this action can be lifted to an action σ of a cover T2 on P that commutes with the action of G. As in Sect. 3, we define the noncommutative algebras C ∞ (Pθ ) and C ∞ (Mθ ) as the vector spaces C ∞ (P) and C ∞ (M) with star products defined like in (3.2) with respect to the action of T2 and T2 respectively or, equivalently as the images of C ∞ (P) and ∞ C (M) under the corresponding quantization map L θ . Since the action of T2 is taken to commute with the action of G on P, the action α of G on the algebra C ∞ (P) given by αg ( f )( p) = f (g −1 · p)
(6.1)
induces an action of G by automorphisms on the algebra C ∞ (Pθ ). This also means that the inclusion C ∞ (M) ⊂ C ∞ (P) as G-invariant elements in C ∞ (P) extends to an inclusion C ∞ (Mθ ) ⊂ C ∞ (Pθ ) of the G-invariant element in C ∞ (Pθ ). Clearly, the action of G translates into a coaction of the Hopf algebra C ∞ (G) on C ∞ (Pθ ).
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Proposition 34. The inclusion C ∞ (Mθ ) → C ∞ (Pθ ) is a (principal) Hopf-Galois C ∞ (G) extension. Proof. One needs to establish bijectivity of the canonical map χ : C ∞ (Pθ ) ⊗C ∞ (Mθ ) C ∞ (Pθ ) → C ∞ (Pθ ) ⊗ C ∞ (G), f ⊗C ∞ (Mθ ) f → f R ( f ) = f f (0) ⊗ f (1) . Now, for the undeformed case the bijectivity of the corresponding canonical map χ (0) : C ∞ (P) ⊗C ∞ (M) C ∞ (P) → C ∞ (P) ⊗ C ∞ (G) follows by the very definition of a principal bundle. Furthermore, there is an isomorphism of vector spaces, T : C ∞ (Pθ ) ⊗C ∞ (Mθ ) C ∞ (Pθ ) → C ∞ (P) ⊗C ∞ (M) C ∞ (P) f ⊗C ∞ (Mθ ) f → fr ⊗C ∞ (M) σr θ ( f ), r
where f = r fr is the homogeneous decomposition of f under the action of T2 . (0) We claim that the canonical map is given as the composition χ = (L θ ⊗ id) ◦ χ ◦ T ; hence it is bijective. Indeed, (L θ ⊗ id) ◦ χ (0) ◦ T f ⊗C ∞ (Mθ ) f = L θ ( fr σr θ ( f (0) )) ⊗ f (1) r
= L θ ( f ×θ f (0) ) ⊗ f (1) = χ ( f ⊗C ∞ (Mθ ) f ), since the action of T2 on C ∞ (Pθ ) commutes with the coaction of C ∞ (G).
Noncommutative associated bundles are defined as in (4.15) by setting # $ E = C ∞ (Pθ ) ρ W := f ∈ C ∞ (Pθ ) ⊗ W |(αg ⊗ id)( f ) = (id ⊗ρ(g)−1 )( f ), ∀g ∈ G for a representation ρ of G on W . These C ∞ (Mθ ) bimodules are finite projective since they are of the kind defined in Sect. 3.3 (cf. Remark 10). Moreover, Proposition 12 generalizes and reads End(E) C ∞ (Pθ ) ad L(W ), where ad is the adjoint representation of G on L(W ). Also, one identifies the adjoint bundle as the module coming from the adjoint representation of G on g ⊂ L(W ), namely ∞ (ad(Pθ )) := C ∞ (Pθ ) ad g. For a (right) finite projective C ∞ (Mθ )-module E we define an inner product (·, ·)2 on HomC ∞ (Mθ ) (E, E ⊗C ∞ (Mθ ) (Mθ )) as in Sect. 4.3. The Yang–Mills action functional for a connection ∇ on E in terms of its curvature F is then defined by YM(∇) = (F, F)2 .
(6.2)
This is a gauge invariant, positive and quartic functional. The derivation of the Yang– Mills equations (4.32) on Sθ4 does not rely on the specific properties of Sθ4 and continues to hold on Mθ . The same is true for the topological action, and YM(∇) ≥ |Top(E)| with equality iff ∗θ F = ±F. In other words, instanton connections are minima of the Yang–Mills action. The explicit construction of instanton connections on Sθ4 carried over in Sect. 5 can of course not be generalized to any manifold Mθ . Local expressions could in principle be obtained on a “local chart” R4θ of Mθ if T2 acts on the corresponding local chart R4
Noncommutative Instantons from Twisted Conformal Symmetries
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of M. On the other hand, the infinitesimal construction of instantons on Mθ giving the dimension of the “tangent of the moduli space” can be generalized to any toric noncommutative manifold Mθ , again closely following [6]. It is essential however the existence of a “base point”, i.e. an instanton connection that can be linearly perturbed to obtain a family of infinitesimal instantons. Acknowledgements. We are grateful to an anonymous referee for an excellent review which led to a much improved version of the paper. We thank Paolo Aschieri and Marc Rieffel for useful remarks and suggestions. Part of the work was carried out while GL was visiting ESI in Vienna and INI in Cambridge.
A. Local Index Formula Suppose in general that (A, H, D, γ ) is an even p-summable spectral triple with discrete simple dimension spectrum. For a projection e ∈ M N (A), the operator De = e(D ⊗ I N )e is a Fredholm operator, thought of as the Dirac operator with coefficient in the module determined by e. The local index formula of Connes and Moscovici [16] provides a method to compute its index via the pairing of suitable cyclic cycles and cocycles. We shall recall the “even” case since it is the one that is relevant for the present paper. Let C∗ (A) be the chain complex over the algebra A; in degree n, Cn (A) := A⊗(n+1) . On this complex there are defined the Hochschild operator b : Cn (A) → Cn−1 (A) and the boundary operator B : Cn (A) → Cn+1 (A), satisfying b2 = 0, B 2 = 0, bB+Bb = 0; thus (b + B)2 = 0. From general homological theory, one defines a bicomplex CC∗ (A) by CC(n,m) (A) := CCn−m (A) in bi-degree (n, m). Dually, one defines CC ∗ (A) as functionals on CC∗ (A), equipped with the dual Hochschild operator b and coboundary operator B (we refer to [12] and [29] for more details on this). Theorem 35 (Connes-Moscovici [16]). (a) An even cocycle φ ∗ = k≥0 φ 2k in CC ∗ (A), (b + B)φ ∗ = 0, is defined by the following formulæ. For k = 0, φ 0 (a) := Res z −1 tr(γ a|D|−2z ); z=0
(A.1)
whereas for k = 0, φ 2k (a 0 , . . . , a 2k ) " ! := ck,α Res tr γ a 0 [D, a 1 ](α1 ) · · · [D, a 2k ](α2k ) |D|−2(|α|+k+z) , α
z=0
(A.2)
with ck,α = (−1)|α| (k + |α|) (α!(α1 + 1)(α1 + α2 + 2) · · · (α1 + · · · + α2k + 2k))−1 and T ( j) denotes the j th iteration of the derivation T → [D 2 , T ].
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(b) For e ∈ K 0 (A), the Chern character ch∗ (e) = k≥0 chk (e) is the even cycle in CC∗ (A), (b + B)ch∗ (e) = 0, defined by the following formulæ. For k = 0, ch0 (e) := tr(e);
(A.3)
whereas for k = 0, chk (e) := (−1)k
(2k)! 1 ei0 i1 − δi0 i1 ⊗ ei1 i2 ⊗ ei2 i3 ⊗ · · · ⊗ ei2k i0 . k! 2
(A.4)
(c) The index of the operator De is given by the natural pairing between cycles and cocycles, indexDe = φ ∗ , ch∗ (e).
(A.5)
For toric noncommutative manifolds, the above local index formula simplifies drastically [27]. Theorem 36. For a projection p ∈ M N (C ∞ (Mθ )), we have (−1)k 1 −1 −2z + Res tr γ p − [D, p]2k |D|−2(k+z) indexD p = Resz tr γ p|D| z=0 k z=0 2 k≥1
and now the trace tr comprises a matrix trace as well. Proof. Recall that the quantization map L θ preserves the spectral decomposition, for the toric action of Tn , of smooth operators (see equation (3.7)). Then, once extended the + deformed ×θ -product to C ∞ (Mθ ) [D, C ∞ (Mθ )] – which can be done unambiguously since D is of degree 0 – we write the local cocycles φ 2k in Theorem 35 in terms of the quantization map L θ : φ 2k L θ ( f 0 ), L θ ( f l ), . . . , L θ ( f 2k ) " ! = Res tr γ L θ f 0 ×θ [D, f 1 ](α1 ) ×θ · · · ×θ [D, f 2k ](α2k ) |D|−2(|α|+k+z) . (A.6) z=0
Suppose now that f 0 , . . . , f 2k ∈ C ∞ (M) are homogeneous of degree r 0 , . . . , r 2k respectively, so that the operator f 0 ×θ [D, f 1 ](α1 ) ×θ · · · ×θ [D, f 2k ] is a homogeneous j 0 1 2k element of degree r = 2k j=0 r . It is in fact a multiple of f [D, f ] · · · [D, f ] as can be established by working out the ×θ -products. Forgetting about this factor – which is a power of the deformation parameter λ – we obtain from (3.6) that L θ ( f 0 ×θ [D, f 1 ](α1 ) ×θ · · · ×θ [D, f 2k ]) = f 0 [D, f 1 ] · · · [D, f 2k ]U ( 21 r · θ ). (A.7) Each term in the local index formula for (C ∞ (Mθ ), H, D) then takes the form Res tr γ f 0 [D, f 1 ](α1 ) · · · [D, f 2k ](α2k ) |D|−2(|α|+k+z) U (s) z=0
for s = 21 r · θ ∈ Tn . The appearance of the operator U (s) here is a consequence of the close relation with the index formula for Tn -equivariant Dirac spectral triples. In [9] an even dimensional compact spin manifold M on which a (connected compact) Lie group G acts by isometries was studied. The equivariant Chern character was defined as an equivariant version of the JLO-cocycle, the latter being an element in equivariant entire
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cyclic cohomology. The essential point is that an explicit formula for the above residues was obtained. In the case of a Tn -action on M this is Res tr γ f 0 [D, f 1 ](α1 ) · · · [D, f 2k ](α2k ) |D|−2(|α|+k+z) U (s) z=0
2 = (|α| + k) lim t |α|+k tr γ f 0 [D, f 1 ](α1 ) · · · [D, f 2k ](α2k ) e−t D U (s) , (A.8) t→0
for every s ∈ Tn . Moreover, from Theorem 2 in [9] the limit vanishes when |α| = 0. This finishes the proof of our theorem.
By inserting the symbol π for the algebra representation, the components of the Chern character are represented as operators on the Hilbert space H by explicit formulæ, π D (chk (e))
(2k)! 1 (π(ei0 i1 ) − δi0 i1 )[D, π(ei1 i2 )][D, π(ei1 i2 )] · · · [D, π(ei2k i0 )], k! 2 (A.9) for k > 0, while π D (ch0 (e)) = π(ei0 i0 ). := (−1)k
References 1. Aschieri, P., Bonechi, F.: On the noncommutative geometry of twisted spheres. Lett. Math. Phys. 59, 133– 156 (2002) 2. Aschieri, P., Castellani, L.: Bicovariant Calculus on Twisted I S O(N ), Quantum Poincarè Group and Quantum Minkowski Space. Int. J. Mod. Phys. A11, 4513–4549 (1996) 3. Aschieri, P., Castellani, L.: Universal Enveloping Algebra and Differential Calculi on Inhomogeneous Orthogonal q-groups. J. Geom. Phys. 26, 247–271 (1998) 4. Atiyah, M.F.: The Geometry of Yang–Mills Fields. Fermi Lectures, Scuola Normale Pisa (1979) 5. Atiyah, M.F., Hitchin, N.J., Drinfel’d, V.G., Manin, Yu.I.: Construction of instantons. Phys. Lett. A65, 185–187 (1978) 6. Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. Lond. A362, 425–461 (1978) 7. Belavin, A., Polyakov, A., Schwarz, A., Tyupkin, Y.: Pseudoparticle solutions of the Yang–Mills equations. Phys. Lett. 59B, 85–87 (1975) 8. Chari, V., Pressley, A.: A guide to quantum groups. Cambridge: Cambridge University Press, 1994 9. Chern, S., Hu, X.: Equivariant Chern character for the invariant Dirac operator. Michigan Math. J. 44, 451– 473 (1997) 10. Connes, A.: C ∗ -algèbres et géométrie differentielle. C.R. Acad. Sci. Paris Ser. A-B, 290, A599– A604 (1980) 11. Connes, A.: Noncommutative differential geometry. IHES Sci. Publ. Math. 62, 257–360 (1985) 12. Connes, A.: Noncommutative Geometry. San Diego: Academic Press, 1994 13. Connes, A.: Gravity coupled with matter and the foundation of noncommutative geometry. Commun. Math. Phys. 182, 155–176 (1996) 14. Connes, A., Dubois-Violette, M.: Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples. Commun. Math. Phys. 230, 539–579 (2002) 15. Connes, A., Landi, G.: Noncommutative manifolds: The instanton algebra and isospectral deformations. Commun. Math. Phys. 221, 141–159 (2001) 16. Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 5, 174–243 (1995) 17. Dixmier, J.: Existence de traces non normales. C.R. Acad. Sci. Paris Sér A-B 262, A1107–A1108 (1966) 18. Drinfel’d, V.G.: Constant quasiclassical solutions of the Yang-Baxter quantum equation. Soviet Math. Dokl. 28, 667–671 (1983) 19. Drinfel’d, V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1, 1419–1457 (1990) 20. Gayral, V., Iochum B., Várilly, J.C. : Dixmier traces on noncompact isospectral deformations. J. Funct. Anal. 237, 507–539 (2006)
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21. Giaquinto, A., Zhang, J.J.: Bialgebra actions, twists and universal deformation formulas. J. Pure Appl. Algebra. 128, 133–151 (1998) 22. Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Boston, Birkhäuser 2001 23. Julg, P.: K-théorie équivariante et produits croisés. C.R. Acad. Sci. Paris 292, 629–632 (1981) 24. Landi, G.: An Introduction to Noncommutative Spaces and their Geometry. Berlin: Springer-Verlag, 1997 25. Landi, G.: Spin-Hall effect with quantum group symmetry. Lett. Math. Phys. 75, 187–200 (2006) 26. Landi, G., Pagani, C., Reina, C., van Suijlekom, W.: Work in progress 27. Landi, G., van Suijlekom, W.: Principal fibrations from noncommutative spheres. Commun. Math. Phys. 260, 203–225 (2005) 28. Lie, S.: Theorie der transformationsgruppen. New York: Chelsea, 1970 29. Loday, J.-L.: Cyclic Homology. Berlin: Springer-Verlag, 1992 30. Nekrasov, N., Schwarz, A.: Instantons on noncommutative R 4 , and (2,0) superconformal six dimensional theory. Commun. Math. Phys. 198, 689–703 (1998) 31. Reshetikhin, N.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett. Math. Phys. 20, 331–335 (1990) 32. Rieffel, M.A.: Non-commutative tori - A case study of non-commutative differentiable manifolds. Contemp. Math. 105, 191–212 (1990) 33. Rieffel, M.A.: Deformation Quantization for Actions of Rd . Memoirs of the Amer. Math. Soc. 506, Providence, RI: Amer. Math. Soc., 1993 34. Rieffel, M.A.: K -groups of C ∗ -algebras deformed by actions of R d . J. Funct. Anal. 116, 199–214 (1993) 35. Sitarz, A.: Twists and spectral triples for isospectral deformations. Lett. Math. Phys. 58, 69–79 (2001) 36. Várilly, J.C.: Quantum symmetry groups of noncommutative spheres. Commun. Math. Phys. 221, 511– 523 (2001) Communicated by A. Connes
Commun. Math. Phys. 271, 635–647 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0211-9
Communications in
Mathematical Physics
On Two-Dimensional Sonic-Subsonic Flow Gui-Qiang Chen1 , Constantine M. Dafermos2 , Marshall Slemrod3 , Dehua Wang4 1 Department of Mathematics, Northwestern University, Evanston, IL 60208, USA.
E-mail: [email protected]
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
E-mail: [email protected]
3 Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA.
E-mail: [email protected]
4 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA.
E-mail: [email protected] Received: 16 March 2006 / Accepted: 22 August 2006 Published online: 1 March 2007 – © Springer-Verlag 2007
Abstract: A compensated compactness framework is established for sonic-subsonic approximate solutions to the two-dimensional Euler equations for steady irrotational flows that may contain stagnation points. Only crude estimates are required for establishing compactness. It follows that the set of subsonic irrotational solutions to the Euler equations is compact; thus flows with sonic points over an obstacle, such as an airfoil, may be realized as limits of sequences of strictly subsonic flows. Furthermore, sonic-subsonic flows may be constructed from approximate solutions. The compactness framework is then extended to self-similar solutions of the Euler equations for unsteady irrotational flows.
1. Introduction Consider the two-dimensional Euler equations for steady irrotational flows: vx − u y = 0, (ρu)x + (ρv) y = 0,
(1.1) (1.2)
(ρu 2 + p)x + (ρuv) y = 0,
(1.3)
(ρuv)x + (ρv + p) y = 0,
(1.4)
2
where u and v are the two components of flow velocity and ρ is the density. For a polytropic gas with adiabatic exponent γ > 1, p = p(ρ) = ρ γ /γ is the normalized pressure. Equations (1.1) and (1.3)–(1.4) classically yield the normalized Bernoulli’s law (Courant-Friedrichs [10]): 1 γ − 1 2 γ −1 ρ = ρ(q ˆ 2 ) := 1 − q , 2
(1.5)
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where q =
G.-Q. Chen, C. M. Dafermos, M. Slemrod, D. Wang
√
u 2 + v 2 is the flow speed. The sound speed c is defined as c2 = p (ρ) = 1 −
At the sonic point q = c, (1.6) implies q 2 =
γ −1 2 q . 2
2 γ +1 .
(1.6)
We define the critical speed qcr as
qcr ≡
2 γ +1
and rewrite Bernoulli’s law (1.5) in the form 2 q 2 − qcr =
2 2 q − c2 . γ +1
Thus the flow is subsonic when q < qcr , sonic when q = qcr , and supersonic when q > qcr . For isothermal flow, p = c¯2 ρ, where c¯ > 0 is the constant sound speed. Then, in the place of (1.5), Bernoulli’s law yields 2 u + v2 2 (1.7) ρ = ρ(q ˆ ) := ρ0 exp − 2c¯2 for some constant ρ0 > 0. In this case, qcr = c. ¯ Bernoulli’s law plays the role of mechanical energy conservation in irrotational flow. One may opt to determine the flow either through system (1.2)–(1.4) or through system (1.1)–(1.2) and (1.5) or (1.7). The former option secures conservation of mass and momentum, while the latter option imposes irrotationality and requires conservation of mass and mechanical energy. Both approaches are mathematically equivalent for smooth flows, because, once inserted through the initial data, irrotationality persists and yields Bernoulli’s law. This is no longer the case in the presence of shocks, as vorticity may be created and mechanical energy may be converted into heat. The theory of isentropic thermodynamics favors retaining system (1.2)–(1.4) for rough flows. Nevertheless, system (1.1)–(1.2) and (1.5) has been more popular among aerodynamicists, as it is mathematically simpler and also brings out the analogy with the treatment of incompressible fluids (cf. Bers [2]). In any case, it is generally believed that, when shocks are weak, the solutions of the two systems are close. Accordingly, here we will deal with the system of potential flow, namely (1.1)–(1.2) and (1.5) or (1.7). In weak solutions of system (1.1)–(1.2) and (1.5), the left-hand sides of (1.3) and (1.4) do not necessarily vanish, but they are equal to distributions representing the (artificial and a priori unknown) body force that would have to be imposed in order to balance momentum. These equations have been ignored in traditional treatments of the system of potential flow. The purpose of this paper is to reveal and emphasize that (1.3) and (1.4), modified by the addition of the artificial body force and interpreted as entropy (in the mathematical–not the physical sense) balance equations, play a very useful and important role in the treatment of weak solutions of the system of potential flow. Indeed, with the help of these equations, we establish, in Sect. 2, a compensated compactness framework for sonic-subsonic approximate solutions to the system of two-dimensional potential flow that may contain stagnation points. In particular, in Sect. 3, we show that sets of sonic-subsonic irrotational flows that satisfy crude bounds are precompact. Thus
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flows with sonic points over an obstacle, such as an airfoil, may be realized as limits of sequences of strictly subsonic flows. Furthermore, sonic-subsonic flows may be constructed from approximate solutions to the potential flow system, under very crude estimates. Finally, in Sect. 4, we extend the compensated compactness framework to the realm of self-similar solutions to the Euler equations for unsteady irrotational flow. For the fundamental ideas and early applications of compensated compactness, see the classical papers by Tartar [36] and Murat [33]. For applications to the theory of hyperbolic conservation laws, see for example [6, 7, 11, 14, 15, 17, 34]. In particular, the compensated compactness approach has been applied in [5, 9, 12, 13, 27, 28] to the one-dimensional Euler equations for unsteady isentropic flow, allowing for cavitation. Particularly, relevant to the present work are the papers by Morawetz [30, 31] for two-dimensional steady transonic flow away from stagnation and cavitation points. In particular, the basic estimate used in our proof of Theorem 2.1 is contained in Morawetz’s papers (the remark following Identity I, page 505 of [30] or, equivalently, Eq. (5.5) of [31]) when applied to her case n = 1. However, in this work, we identify the relation of this estimate with balance of momentum, recognize its validity at stagnation points, realize that the direct use of conservation of momentum is a useful simplification, and point out how it can be applied in a simple and direct fashion. 2. Compensated Compactness Framework for Steady Flow As noted in the Introduction, the standard presentation of steady irrotational gas dynamics is to analyze (1.1)–(1.2) subject to Bernoulli’s law (1.5) or (1.7), and Eqs. (1.3)–(1.4) seem to have been lost from the story. However, (1.1)–(1.2) coupled with Bernoulli’s law in fact imply the conservation of linear momentum (1.3)–(1.4). Hence, in the language of conservation laws (cf. Dafermos [11] and Lax [26]), Eqs. (1.3)–(1.4) provide “entropy-entropy flux pairs”. Let a sequence of functions w ε (x, y) = (u ε , v ε )(x, y), defined on open subset ⊂ R2 , satisfy the following Set of Conditions (A): (A.1) q ε (x, y) = |w ε (x, y)| ≤ qcr a.e. in ; −1 (A.2) ∂x ηk (w ε ) + ∂ y qk (w ε ), k = 1, 2, 3, 4, are confined in a compact set in Hloc () for the momentum entropy-entropy flux pairs: (η1 , q1 ) = (ρu 2 + p, ρuv),
(η2 , q2 ) = (ρuv, ρv 2 + p),
(2.1)
and the two natural entropy-entropy flux pairs: (η3 , q3 ) = (v, −u),
(η4 , q4 ) = (ρu, ρv).
(2.2)
Then, by the div-curl lemma of Tartar [36] and Murat [33] and the Young measure representation theorem for a uniformly bounded sequence of functions (cf. Tartar [36]; also Ball [1]), we have the following commutation identity: ν(w), ηi (w)q j (w) − qi (w)η j (w) = ν(w), ηi (w)ν(w), q j (w) − ν(w), qi (w)ν(w), η j (w),
(2.3)
where ν = νx,y (w), w = (u, v), is the associated Young measure (a probability measure) for the sequence wε (x, y) = (u ε , v ε )(x, y). The main point in this section for the compensated compactness framework is to prove that ν is in fact a Dirac measure by using only the above momentum entropy pairs in (2.1), besides the two natural entropy pairs in (2.2). This in turn implies the compactness of the sequence wε (x, y) = (u ε , v ε )(x, y) in L 1 ().
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Theorem 2.1 (Compensated compactness framework). Let a sequence of functions w ε (x, y) = (u ε , v ε )(x, y) satisfy Framework (A). Then the associated Young measure ν is a Dirac mass and the sequence w ε (x, y) is compact in L 1 (); that is, there is a subsequence (still labeled) w ε that converges a.e. as ε → 0 to w = (u, v) satisfying q(x, y) = |w(x, y)| ≤ qcr
a.e. (x, y) ∈ .
Proof. For simplicity of notation, we drop the subscript (x, y). Since the Young measure ν is a probability measure, Eq. (2.3) with i = 1, j = 2, for ν yields ν(w1 ) ⊗ ν(w2 ), I (w1 , w2 ) = 0,
(2.4)
where I (w1 , w2 ) = (η1 (w1 ) − η1 (w2 ))(q2 (w1 ) − q2 (w2 )) − (q1 (w1 ) − q1 (w2 ))(η2 (w1 ) − η2 (w2 )) = (ρ1 u 21 + p1 − ρ2 u 22 − p2 )(ρ1 v12 + p1 − ρ2 v22 − p2 ) − (ρ1 u 1 v1 − ρ2 u 2 v2 )2 = −ρ1 ρ2 (u 1 v2 − u 2 v1 )2 + ( p1 − p2 )2 + ( p1 − p2 )(ρ1 q12 − ρ2 q22 ), w1 = (u 1 , v1 ) and w2 = (u 2 , v2 ) are two independent vector variables, and ν(w1 ) ⊗ ν(w2 ) should be understood as a product measure for (w1 , w2 ), i.e., ν(w1 ) ⊗ ν(w2 ), ϕ(w1 , w2 ) := ν(w1 ), ν(w2 ), ϕ(w1 , w2 ) for any test function ϕ(w1 , w2 ). For γ > 1, 1 2 1 − ρ γ −1 , ρ = (γ p) γ , q 2 = γ −1 and then ρq 2 =
1 2 (γ p) γ − γ p . γ −1
Thus, I (w1 , w2 ) = −ρ1 ρ2 (u 1 v2 − u 2 v1 )2 + ( p1 − p2 )2 1 1 2( p1 − p2 ) (γ p1 ) γ − γ p1 − (γ p2 ) γ + γ p2 + γ −1 1
1 1 γ +1 2γ γ ( p 1 − p 2 )2 + ( p1 − p2 ) p1γ − p2γ = −ρ1 ρ2 (u 1 v2 − u 2 v1 ) − γ −1 γ −1
2
1
−1 1 γ +1 2γ γ −1 − p˜ γ = −ρ1 ρ2 (u 1 v2 − u 2 v1 ) − ( p1 − p2 ) γ −1 γ −1 2 − q˜ 2 q γ + 1 ( p1 − p2 )2 2cr , = −ρ1 ρ2 (u 1 v2 − u 2 v1 )2 − 2 γ −1 γ −1 − q˜ 2
2
where p˜ = p(ρ) ˜ lies between p1 and p2 as determined by the mean-value theorem 1
on p γ , and q˜ is determined by ρ˜ through Bernoulli’s law (1.5). We notice that setting u = q cos θ and v = q sin θ implies u 1 v2 − u 2 v1 = q1 q2 sin(θ2 − θ1 );
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I (w1 , w2 ) ≤ 0 when qi ≤ qcr for i = 1, 2 (so that q˜ ≤ qcr and
2 γ −1
− q˜ 2 > 0);
and I (w1 , w2 ) = 0 if and only if either p1 = p2 > 0 and u 1 v2 − u 2 v1 = 0, or ρ1 = ρ2 = p1 = p2 = 0. This implies that supp(ν(w1 ) ⊗ ν(w2 )) ⊂ {w2 = ±w1 }, that is, supp ν ⊂ {±P0 }
for some point P0 = (u 0 , v0 ) with q0 ≤ qcr .
When q0 = 0, then P0 = −P0 and the Young measure is a Dirac mass concentrated at the stagnation point. When q0 = 0, the support of the Young measure ν consists of at most two points ±P0 in the (u, v)–plane, that is, there exists α ∈ [0, 1] such that ν = α δ P0 + (1 − α) δ{−P0 } , in the (u, v)–phase plane. Then, taking i = 3, j = 4 in (2.3) and using ρ > 0 as q ≤ qcr , we have q02 = (2α − 1)2 q02 , that is, α(1 − α)q02 = 0. Since q0 = 0, then either α = 0 or α = 1, which implies that the Young measure is a Dirac measure. Similarly, for isothermal flows, p = c¯2 ρ,
q 2 = −2c¯2 ln
ρ . ρ0
Then we have I (w1 , w2 ) = −ρ1 ρ2 (u 1 v2 − u 2 v1 )2 + c¯4 (ρ1 − ρ2 )2 + c¯2 (ρ1 − ρ2 )(ρ1 q12 − ρ2 q22 ) ρ1 ρ2 2 4 2 4 − ρ2 ln = −ρ1 ρ2 (u 1 v2 − u 2 v1 ) + c¯ (ρ1 − ρ2 ) − 2c¯ (ρ1 − ρ2 ) ρ1 ln ρ0 ρ0 ρ˜ 2 4 2 4 2 = −ρ1 ρ2 (u 1 v2 − u 2 v1 ) + c¯ (ρ1 − ρ2 ) − 2c¯ (ρ1 − ρ2 ) 1 + ln ρ0 = −ρ1 ρ2 (u 1 v2 − u 2 v1 )2 − c¯2 (ρ1 − ρ2 )2 c¯2 − q˜ 2 ≤ 0,
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when qi ≤ c¯ for i = 1, 2 (and thus q˜ ≤ c), ¯ and ρ˜ lies between ρ1 and ρ2 as determined by the mean-value theorem on 1 + ln ρρ0 . This implies that supp(ν(w1 ) ⊗ ν(w2 )) ⊂ {w2 = ±w1 }, that is, supp ν ⊂ {±P0 }
for some point P0 = (u 0 , v0 ) with q0 ≤ c. ¯
Then the same argument as for the case γ > 1 by using the other two entropy pairs (η j , q j ), j = 3, 4, yields that the Young measure is a Dirac measure and the strong convergence follows immediately.
We now consider a sequence of approximate solutions wε (x, y) to the Euler equations (1.1)–(1.2) with Bernoulli’s law (1.5) or (1.7). That is, besides Set of Conditions (A), the sequence wε (x, y) = (u ε , v ε )(x, y) further satisfies ε 2
ε
vxε − u εy = o1ε (1),
(2.5)
ε 2
(2.6)
ε
(ρ(|w ˆ | )u )x + (ρ(|w ˆ | )v ) y =
o2ε (1),
oεj (1)
where → 0, j = 1, 2, in the sense of distributions as ε → 0. Then, as a corollary of the compensated compactness framework (Theorem 2.1), we have Theorem 2.2 (Convergence of approximate solutions) . Let wε (x, y) = (u ε , v ε )(x, y) be a sequence of approximate solutions to the Euler equations (1.1)–(1.2) with Bernoulli’s law (1.5) or (1.7) in . Then there exists a subsequence (still labeled) wε (x, y) that converges a.e. as ε → 0 to a weak solution w = (u, v) of the Euler equations (1.1)–(1.2) with Bernoulli’s law (1.5) or (1.7) satisfying q(x, y) = |w(x, y)| ≤ qcr
a.e. (x, y) ∈ .
There are various ways to construct approximate solutions by either numerical methods such as finite difference schemes and finite element methods, or analytical methods such as vanishing viscosity and relaxation methods. Even though the flow may eventually turn out to be smooth, the point of considering here weak solutions is to demonstrate that such solutions may be constructed by merely using very crude estimates, which are available in a variety of approximating methods through basic energy-type estimates besides the L ∞ estimate. 3. Sonic Limit of Subsonic Flows As our principal application, we consider the sonic limit of subsonic flows past an obstacle P, such as an airfoil. We follow the presentation of Bers [3, 4] (also cf. [16 – 25]). Write z = x + i y, w = u − iv = qe−iθ , q = u 2 + v 2 ; u = q cos θ, v = −q sin θ. We consider a fixed simple closed rectifiable curve C (the boundary of the obstacle P) in the z-plane and a fixed point z T on it (the trailing edge). This curve may possess at z T a protruding corner or cusp, but should otherwise be a Lyapunov curve (a Lyapunov curve is a curve which possesses a tangent inclination which satisfies a Hölder condition with respect to the arc length). Let S be the length of C, and επ the opening of the corner
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at z T . If ε = 0, C has a cusp at z T ; if ε = 1, C possesses a tangent at z T ; otherwise, 0 < ε < 1; see Figs. 1–3. The profile C admits the parametric representation: s z = zT + ei (σ ) dσ, 0 ≤ s ≤ S. 0
The function (s) must satisfy the condition
(S) − (0) = (1 + ε)π,
0 ≤ ε ≤ 1,
(3.1)
and the Hölder condition | (s2 ) − (s1 )| ≤ k(s2 − s1 )α , 0 ≤ s1 < s2 ≤ S,
(3.2)
for some constants k > 0 and 0 < α < 1. Denote by D(C) the domain exterior to C. A pair of functions (u, v) ∈ C 1 (D(C)) is called a solution of Problem P, if (u, v) satisfy (1.1)–(1.2) with Bernoulli’s law (1.5) or (1.7), and the slip boundary condition (u, v) · n = 0
on C,
(3.3)
where n denotes the normal on C, and the limit w∞ = lim (u − iv) z→∞
exists and is finite. A pair of functions (u, v) defined on D(C) is said to satisfy the Kutta-Joukowski condition if u 2 + v2 → 0
as z → z T , if ε = 1
(a stagnation point at the trailing edge), or u 2 + v 2 = O(1)
as z → z T , if 0 ≤ ε < 1.
Fig. 1. Profile C of the obstacle P: ε = 0
Fig. 2. Profile C of the obstacle P: ε = 1
Fig. 3. Profile C of the obstacle P: 0 < ε < 1
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In fact, a solution of Problem P automatically satisfies the Kutta-Joukowski condition if 0 ≤ ε < 1; in particular, for such a function, u 2 + v2 = 0
at z T , if 0 < ε < 1.
Also, with every solution of Problem P, we associate the circulation as
= (u, v) · t ds, C
where t is the unit tangent to C. Bers [3, 4] then defined the following two boundary-value problems: Problem P1 (w∞ ). Find a solution of Problem P satisfying a Kutta-Joukowski condition and a prescribed limit w∞ as z → ∞. Problem P2 (w∞ , ). Find a solution of Problem P for which w∞ and are prescribed. Problem P2 is only considered in the case of a smooth profile ε = 1. We use Bers’s fundamental existence-uniqueness theorems whose proof simplified an earlier result of Shiffman [35] on the existence of solutions. Theorem 3.1 (Bers [4]). For a given w∞ , there exists a number qˆ < qcr , depending on the profile and the equation of state, such that Problem P1 (w∞ ) has a unique solution (u, v) for q∞ := |w∞ | < q. ˆ The velocity (u, v) is Hölder continuous on the profile and depends continuously on w∞ . The maximum speed qm of |w| takes on all values between 0 and some critical value qcr , and qm → 0 as q∞ → 0, qm → qcr as q∞ → q. ˆ A similar result holds for Problem P2 (w∞ , ). We note that Bers’s Theorem does not apply to the critical flows, that is, those flows for which q∞ = qˆ and which hence must be sonic at some point in D(C) ∪ C. In fact, the Gilbarg-Shiffman maximum principle [25] asserts that the sonic point should occur on C, which presupposed the existence of critical flows. In this regard, Gilbarg and Shiffman [25] remarked in footnote 8: “The actual existence of critical flows past finite profiles of bounded curvature has been proved by M. Shiffman (unpublished)”. Bers in [4] made a similar though less precise statement: “Shiffman also proved (unpublished) that for a smooth profile the solution of P2 (w∞ , ) converges to a critical flow for q∞ → q”. ˆ Here, we establish a more general result. ε < qˆ be a sequence of speeds at ∞, and let (u ε , v ε ) be the corTheorem 3.2 Let q∞ ε q, responding solutions to either Problem P1 (w∞ ) or P2 (w∞ , ). Then, as q∞ ˆ the solution sequence (u ε , v ε ) possesses a subsequence (still denoted by) (u ε , v ε ) that converges strongly a.e. in D(C) to a pair of functions (u, v) which is a weak solution of Eqs. (1.1)–(1.2) with Bernoulli’s law (1.5) or (1.7). Furthermore, the limit velocity (u, v) satisfies the boundary conditions (3.3) as the normal trace of the divergence-measure field (ρu, ρv) on the boundary (see [8] ).
Proof. The strong solutions (u ε , v ε ) satisfy (1.1)–(1.4) and are subsonic. Hence Theorem 2.1 immediately implies that the Young measure is a Dirac mass and the convergence is strong a.e. in D(C). The fact the boundary conditions (3.3) are satisfied for (u, v) in the sense of distributions is standard by multiplying (1.2) by a test function and applying the divergence theorem and the fact that the sequence of subsonic solutions does satisfy (3.3), which implies (u, v) satisfies the boundary conditions (3.3) actually as the normal trace of the divergence-measure field (ρu, ρv) on the boundary in the sense of Chen-Frid [8].
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4. Extension to Self-Similar Solutions for Unsteady Flow Here we are concerned with self-similar solutions depending on the variables (ξ, η) = (x/t, y/t) for the Euler equations for unsteady flows. Then the equations of mass and momentum conservation imply ⎧ ⎪ ⎨(ρ(u − ξ ))ξ + (ρ(v − η))η + 2ρ = 0, (4.1) (ρu(u − ξ ) + p)ξ + (ρu(v − η))η + 2ρu = 0, ⎪ ⎩(ρv(u − ξ )) + (ρv(v − η) + p) + 2ρv = 0. ξ η The last two equations are of course easily rewritten as (ρ(u − ξ )2 + p)ξ + (ρ(u − ξ )(v − η))η + 3ρ(u − ξ ) = 0, (ρ(u − ξ )(v − η))ξ + (ρ(v − η)2 + p)η + 3ρ(v − η) = 0. If we introduce the velocity potential : (U, V ) ≡ (u − ξ, v − η) = ∇, we again find the normalized Bernoulli relation for polytropic gas with γ > 1: Q2 1 ρ γ −1 ++ = , γ −1 2 γ −1
(4.2)
where Q 2 = U 2 + V 2 = |∇|2 . We write the Bernoulli equation (4.2) in the same form as (1.5): 1 γ −1 2 Q − (γ − 1) γ −1 , ρ = ρ(Q ˆ 2 ; ) := 1 − 2 write the two equations of momentum conservation as (ρU 2 + p)ξ + (ρU V )η + 3ρU = 0, (ρU V )ξ + (ρV 2 + p)η + 3ρV = 0,
(4.3)
(4.4)
and write the continuity equation as (ρU )ξ + (ρV )η + 2ρ = 0.
(4.5)
The Bernoulli relation (4.3) yields the sound speed c as c2 := p (ρ) = 1 −
γ −1 2 Q − (γ − 1). 2
Thus, (4.4) is elliptic when Q 2 < c2 , i.e., γ +1 2 Q + (γ − 1) < 1; 2
(4.6)
and (4.4) is hyperbolic when the above inequality (4.6) is reversed. Let a sequence of functions W ε (ξ, η) := (U ε , V ε )(ξ, η), defined on an open subset ⊂ R2 , satisfy the following Set of Conditions (B):
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(B.1) There exist ε (ξ, η) ≥ −M for some M ∈ (0, ∞) independent of ε such that ε → a.e. in as ε → 0, γ +1 ε (Q (x, y))2 + (γ − 1)ε (x, y) ≤ 1 2
(4.7) a.e. (x, y) ∈ ;
(4.8)
(B.2) The sequences ∂ξ ηk (W ε ; ε ) + ∂η qk (W ε ; ε ), k = 1, 2, 3, 4, are confined in −1 () with (ηk , qk ), k = 1, 2, 3, 4, defined as a compact set in Hloc (η1 , q1 )(W ; ) = (ρ(Q ˆ 2 ; )U 2 + p(ρ(Q ˆ 2 ; )), ρ(Q ˆ 2 ; )U V ), 2 2 2 (η2 , q2 )(W ; ) = (ρ(Q ˆ ; )U V, ρ(Q ˆ ; )V + p(ρ(Q ˆ 2 ; ))), and (η3 , q3 )(W ) = (V, −U ), (η4 , q4 )(W ; ) = (ρ(Q ˆ 2 ; )U, ρ(Q ˆ 2 ; )V ). Then we have the following compensated compactness framework. Theorem 4.1 (Compensated compactness framework). Let a sequence of functions W ε (ξ, η) := (U ε , V ε )(ξ, η) satisfy the Set of Conditions (B) with ε . Then the sequence W ε is compact in L 1 (); that is, there is a subsequence (still labeled) W ε that converges, as ε → 0, to W = (U, V ) a.e. satisfying (4.6) with . Proof. We proceed as in the proof of Theorem 2.1. First, Condition (B.1) implies −M ≤ ε (ξ, η) ≤
1 , γ −1
(Q ε (ξ, η))2 ≤
2 (1 + (γ − 1)M) , γ +1
which indicates that the sequence (W ε ; ε ) is uniformly bounded in L ∞ (R2 )3 . Then, for any continuous function g on R3 , we write g(W ε ; ε ) = g(W ε ; ε ) − g(W ε ; ) + g(W ε ; ) ∂g ε ¯ (W ε ; )( − ) + g(W ε ; ), = ∂
(4.9)
¯ lies between ε and as determined by the mean-value theorem for each ε. where ∂g is uniformly bounded on the range of (W ε ; ε ), then the first term on the Here, if ∂ right-hand side of the last equality in (4.9) goes to zero strongly as ε → 0, and the weak∗ limit of g(W ε ; ε ) in L ∞ () is w ∗ - lim g(W ε ; ε ) = ν(W ), g(W ; (ξ, η)), where ν = νξ,η (W ), W = (U, V ), is the associated Young measure with W ε (ξ, η), and we have used the strong convergence of ε in (4.7). On the other hand, Condition (B.2) indicates that ε 2 ε 2 ε 2 ) ; ε )(U ε )2 + p(ρ((Q ˆ ) ; ε )) ξ + ρ((Q ˆ ) ; ε )U ε V ε ρ((Q ˆ η (4.10) −1 is confined in a compact set in Hloc (),
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ε 2 ε 2 ε 2 ρ((Q ˆ ) ; ε )U ε V ε ξ + ρ((Q ˆ ) ; ε )(V ε )2 + p(ρ((Q ˆ ) ; ε )) η −1 is confined in a compact set in Hloc ().
(4.11)
Then we see that the nonlinear term g in the expressions above represents the terms of the form ρ(Q ˆ 2 ; )U 2 , ∂g etc., and ∂ is uniformly bounded as long as (W ; ) stay in the elliptic region. We then apply the commutation identity as in the proof of Theorem 2.1 and write it as
ν(W1 ) ⊗ ν(W2 ), I (W1 , W2 ; (ξ, η)) = 0, where Wi = (Ui , Vi ), i = 1, 2, and I = −ρ1 ρ2 (U1 V2 − U2 V1 )2 + ( p1 − p2 )2 + ( p1 − p2 )(ρ1 Q 21 − ρ2 Q 22 ). By virtue of the mean-value theorem and similar calculations to those in the proof of Theorem 2.1, we obtain 1 2(1 − (γ − 1)) −1 2 2 γ +1 γ I = −ρ1 ρ2 (U1 V2 − U2 V1 ) − ( p1 − p2 ) − (γ p) ˜ γ −1 γ −1 1 − (γ − 1) − γ 2+1 Q˜ 2 = −ρ1 ρ2 (U1 V2 − U2 V1 )2 − ( p1 − p2 )2 , ˜2 1 − (γ − 1) − γ −1 2 Q where p˜ = p(ρ) ˜ lies between p1 = p(ρ1 ) and p2 = p(ρ2 ) as determined by the 1 mean-value theorem on p γ , and Q˜ is determined by ρ˜ through the relation (4.3). The ellipticity condition (4.6) ensures I ≤ 0. Then, by using the other two natural entropy pairs (ηk , qk ), k = 3, 4, the proof is completed as in the proof of Theorem 2.1.
We now consider a sequence of approximate solutions (W ε ; ε )(ξ, η) to Vξ − Uη = 0, (ρU )ξ + (ρV )η + 2ρ = 0,
(4.12)
defined on an open subset ⊂ R2 , satisfying the Bernoulli relation (4.3) and the ellipticity constraint (4.6). That is, besides the Set of Conditions (B) with ε , the sequence (W ε ; ε )(ξ, η) = (U ε , V ε ; ε )(ξ, η) further satisfies (U ε , V ε ) = ∇ε + o1ε (1), ε 2
ε
ε
(4.13) ε 2
ε
ε
ε 2
ε
| ; )U )ξ + (ρ(|W ˆ | ; )V )η + 2ρ(|W ˆ | ; ) = (ρ(|W ˆ
o2ε (1),
(4.14)
where oεj (1) → 0, j = 1, 2, in the sense of distributions as ε → 0. We note that (4.13) implies Vξε − Uηε = o3ε (1), with o3ε (1) → 0 in the sense of distributions as ε → 0. Then, as a corollary of the compensated compactness framework (Theorem 4.1), we have
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Theorem 4.2 (Convergence of approximate solutions). Let (W ε ; ε )(ξ, η) = (U ε , V ε ; ε )(ξ, η) be a sequence of approximate solutions to the Euler equations (4.12) with the Bernoulli relation (4.3) in . Then there exists a subsequence (still labeled) W ε (ξ, η) that converges a.e. as ε → 0 to a weak solution W = (U, V ) = ∇ of the Euler equations (4.12) together with the Bernoulli relation (4.3) which satisfies the elliptic constraint (4.6) a.e. in . In particular, when (W ε ; ε )(ξ, η) = (∇ε ; ε )(ξ, η) are exact subsonic solutions to the Euler equations (4.12) with the Bernoulli relation (4.3) in and ε (ξ, η) ≥ −M for some M ∈ (0, ∞) independent of ε, ε and ∇ε are uniformly bounded and thus ε (ξ, η) → (ξ, η)
a.e. as ε → 0.
Therefore, the sequence of the exact, subsonic solutions (W ε , ε ) satisfies the Set of Conditions (B) with ε . Then Theorems 4.1–4.2 imply that the sequence W ε possesses a subsequence (still denoted by) W ε , which converges a.e. in to a function W such that W (ξ, η) = ∇(ξ, η) is a weak solution of Eqs. (4.12) with the Bernoulli relation (4.3). Finally, we remark that, in the isothermal case γ = 1, we have the same results just as we did in Theorems 4.1–4.2. Acknowledgements. Gui-Qiang Chen’s research was supported in part by the National Science Foundation under Grants DMS-0505473, DMS-0244473, and an Alexander von Humboldt Foundation Fellowship. Constantine Dafermos’ research was supported in part by the National Science Foundation grant DMS-0202888 and DMS-0244295. Marshall Slemrod’s research was supported in part by the National Science Foundation grant DMS-0243722. Dehua Wang’s research was supported in part by the National Science Foundation grant DMS-0244487 and the Office of Naval Research grant N00014-01-1-0446. The authors would like to thank C. Xie for pointing out to us a missing point in an earlier version of this paper and the other members of the NSF-FRG on multidimensional conservation laws: S. Canic, J. Hunter, T.-P. Liu, C.-W. Shu, and Y. Zheng for stimulating discussions.
References 1. Ball, J.M.: A version of the fundamental theorem for Young measures. Lecture Notes in Phys. 344, Berlin: Springer, 1989, pp. 207–215 2. Bers, L.: Results and conjectures in the mathematical theory of subsonic and transonic gas flows. Comm. Pure Appl. Math. 7, 79–104 (1954) 3. Bers, L.: Existence and uniqueness of a subsonic flow past a given profile. Comm. Pure Appl. Math. 7, 441–504 (1954) 4. Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. New York, John Wiley & Sons, Inc., London: Chapman & Hall, Ltd., 1958 5. Chen, G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III). Acta Math. Sci. 6, 75–120 (1986) (in English); 8, 243–276 (1988) (in Chinese) 6. Chen, G.-Q.: Compactness Methods and Nonlinear Hyperbolic Conservation Laws. AMS/IP Stud. Adv. Math. 15, Providence, RI: AMS, 2000, pp. 33–75 7. Chen, G.-Q.: Euler Equations and Related Hyperbolic Conservation Laws. In: Handbook of Differential Equations, Vol. 2, edited by C. M. Dafermos, E. Feireisl, Amsterdam: Elsevier Science B.V, pp. 1–104 8. Chen, G.-Q., Frid, H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Rational Mech. Anal. 147, 89–118 (1999)
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9. Chen, G.-Q., LeFloch, Ph.: Compressible Euler equations with general pressure law. Arch. Rational Mech. Anal. 153, 221–259 (2000); Existence theory for the isentropic Euler equations. Arch. Rational Mech. Anal. 166, 81–98 (2003) 10. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. New York: Springer-Verlag, 1962 11. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. 2nd edition, Berlin: SpringerVerlag, 2005 12. Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (I)–(II). Acta Math. Sci. 5, 483–500, 501–540 (1985) (in English); 7, 467–480 (1987), 8, 61–94 (1988) (in Chinese) 13. DiPerna, R.J.: Convergence of viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1– 30 (1983) 14. DiPerna, R.J.: Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82, 27–70 (1983) 15. DiPerna, R.J.: Compensated compactness and general systems of conservation laws. Trans. Amer. Math. Soc. 292, 383–420 (1985) 16. Dong, G.-C.: Nonlinear Partial Differential Equations of Second Order. Translations of Mathematical Monographs, 95, Providence, RI: American Mathematical Society, 1991 17. Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS-RCSM, 74, Providence, RI: AMS, 1990 18. Finn, R.: On the flow of a perfect fluid through a polygonal nozzle, I, II. Proc. Nat. Acad. Sci. USA. 40, 983–985, 985–987 (1954) 19. Finn, R.: On a problem of type, with application to elliptic partial differential equations. J. Rational Mech. Anal. 3, 789–799 (1954) 20. Finn, R., Gilbarg, D.: Asymptotic behavior and uniquenes of plane subsonic flows. Comm. Pure Appl. Math. 10, 23–63 (1957) 21. Finn, R., Gilbarg, D.: Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations. Acta Math. 98, 265–296 (1957) 22. Finn, R., Gilbarg, D.: Uniqueness and the force formulas for plane subsonic flows. Trans. Amer. Math. Soc. 88, 375–379 (1958) 23. Gilbarg, D.: Comparison methods in the theory of subsonic flows. J. Rational Mech. Anal. 2, 233– 251 (1953) 24. Gilbarg, D., Serrin, J.: Uniqueness of axially symmetric subsonic flow past a finite body. J. Rational Mech. Anal. 4, 169–175 (1955) 25. Gilbarg, D., Shiffman, M.: On bodies achieving extreme values of the critical Mach number, I. J. Rational Mech. Anal. 3, 209–230 (1954) 26. Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM: Philadelphia, 1973 27. Lions, P.-L., Perthame, B., Souganidis, P.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math. 49, 599–638 (1996) 28. Lions, P.-L., Perthame, B., Tadmor, E.: Kinetic formulation of the isentropic gas dynamics and p-systems. Commun. Math. Phys. 163, 169–172 (1994) 29. Morawetz, C.S.: The mathematical approach to the sonic barrier. Bull. Amer. Math. Soc. (N.S.) 6, 127– 145 (1982) 30. Morawetz, C.S.: On a weak solution for a transonic flow problem. Comm. Pure Appl. Math. 38, 797–818 (1985) 31. Morawetz, C.S.: On steady transonic flow by compensated compactness. Methods Appl. Anal. 2, 257–268 (1995) 32. Morawetz, C.S.: Mixed equations and transonic flow. J. Hyper. Diff. Eqs. 1, 1–26 (2004) 33. Murat, F.: Compacite par compensation. Ann. Suola Norm. Pisa (4), 5, 489–507 (1978) 34. Serre, D.: Systems of Conservation Laws, Vols. 1–2, Cambridge: Cambridge University Press, 1999, 2000 35. Shiffman, M.: On the existence of subsonic flows of a compressible fluid. J. Rational Mech. Anal. 1, 605– 652 (1952) 36. Tartar L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics, Heriot-Watt Symposium IV, Res. Notes in Math. 39, Boston-London: Pitman, 1979, pp. 136–212 Communicated by P. Constantin
Commun. Math. Phys. 271, 649–679 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0217-3
Communications in
Mathematical Physics
From Zwiebach Invariants to Getzler Relation A. Losev1, , S. Shadrin2,3, 1 Institute for Theoretical and Experimental Physics, Bolshaya Cheremushkinskaya 25, Moscow,
117218, Russia. E-mail: [email protected]
2 Department of Mathematics, Stockholm University, Stockholm, SE-10691, Sweden.
E-mail: [email protected]
3 Moscow Center for Continuous Mathematical Education, Boljshoi Vlasjevskii Pereulok 11, Moscow,
119002, Russia. E-mail: [email protected] Received: 3 April 2006 / Accepted: 15 August 2006 Published online: 2 March 2007 – © Springer-Verlag 2007
Abstract: We introduce the notion of Zwiebach invariants that generalize GromovWitten invariants and homotopical algebra structures. We outline the induction procedure that induces the structure of Zwiebach invariants on the sub-bicomplex, that gives the structure of Gromov-Witten invariants on sub-bicomplex with zero differentials. We propose to treat Hodge dGBV with 1/12 axiom as the simplest set of Zwiebach invariants, and explicitly prove that it induces WDVV and Getzler equations in genera 0 and 1 respectively. 1. Pre-Introduction In [1], Barannikov and Kontsevich have found a solution to the WDVV equation starting from the algebra of polyvector fields on Calabi-Yau manifolds. The algebraic properties of polyvector fields used in their construction are captured by an abstract algebraic structure called dGBV-algebra with Hodge property. One of the main results of this paper is a new interpretation of Barannikov-Kontsevich construction. We represent their solution as a sum over trivalent trees. Using this representation we give a new independent proof that this sum over trivalent trees satisfies the WDVV equation. Since we have a sum over trivalent trees, it is very natural to study the sum over graphs of higher genera (with the same tensor expressions associated to elements of graphs). We prove that in genus 1 our construction satisfied the Getzler elliptic equation [8]. But in order to prove this we have introduced a new surprising algebraic axiom (we call it 1/12 axiom, see Sect. 5.3). A. L. is partially supported by the Russian Federal Agency of Atomic Energy, by the federal program 40.052.1.1.1112 and by the grants INTAS-03-51-6346, NSh-8065.2006.2, RFBR-04-02-17227, NWO-RFBR047.011.2004.026 (RFBR-05-02-89000-NWO-a). S. S. is partially supported by the grants RFBR-04-02-17227, NSh-4719.2006.1, NWO-RFBR047.011.2004.026 (RFBR-05-02-89000-NWO-a), MK-5396.2006.1, by the Göran Gustafsson foundation and by the Pierre Deligne fund based on his 2004 Balzan prize in mathematics.
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Probably, the main problem for us was to find a proper explanation of this additional axiom. In fact, in order to obtain naturally the genus 0 part of our construction (i.e., Barannikov-Kontsevich solution in terms of trivalent trees) it is enough to study the BCOV-action written down in [3, 1, Appendix], [4, 6]. But then we have to introduce 1/12-axiom just for computational reasons, and Getzler’s relation in genus 1 comes over as a miracle. So, we have found another approach to the explanation of our results. It is a kind of an “operadic” homotopy extension of Gromov-Witten theory in the spirit of Getzler [9] and Zwiebach [16]. In this framework all our axioms including 1/12-axiom come very natural. Moreover, in this approach the relations coming from geometry of the moduli space of curves seem to be very expected. This is an amazing fact that there are two completely different natural approaches to the same construction: one is from the B-model side (Barannikov-Kontsevich) and the other one is from the A-model side (we call it the theory of Zwiebach invariants). For the introduction we have chosen the second approach, since it better explains our results. However, the origin of the idea to use trivalent trees is also hidden in the first approach and we explain this in the appendix.
2. Introduction String theory appeared in the beginning of the seventies as an attempt to find fundamental degrees of freedom that would form theory free of ultraviolet divergences and give gravity as a low energy effective theory. In its standard formulation the string theory computes g-loop scattering amplitudes of 2 particles into (n − 2)-particles as an integral over the moduli space of complex structures of the genus g surface with n marked points. The measure of integration is a correlator in a very specific conformal field theory that has an odd symmetry Q (such that Q 2 = 0) and so-called ghosts (due to gauge fixing of the diffeomorphism invariance). The energy-momentum tensor in such a theory is Q-exact. In the process of the study of string theory it was generalized to the so-called topological string theory. In topological string theory conformal theory with ghosts is replaced by a more general conformal theory with Q-symmetry and (co)exact energy-momentum tensor. The most impressive application of these ideas is the theory of geometric GromovWitten invariants (known in physics as type A topological strings). This theory attracted a lot of attention in the last decade since its amplitudes give answers to famous problems in enumerative algebraic geometry. Further generalization of these ideas involves the construction of the set of factorizable closed forms on the moduli spaces of complex structures on Riemann surfaces (so that the integral of the top form produces amplitudes). In this way we get generalized amplitudes that take values in the cohomology of the moduli space. Evaluation of these generalized amplitudes on the contractible cycles (together with the factorization property) leads to relations among amplitudes (WDVV and Getzler relations) that are very important in applications. This leads to the new definition of amplitudes as a system of factorizable maps from the tensor products of the vector space with bilinear pairing to cohomology of the moduli spaces of Riemann surfaces that we call simply Gromov-Witten invariants. Note that here we do not insist that amplitudes come from the integral over the moduli space of
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differential forms coming from conformal field theory; we study amplitudes on their own 1 . Note that formalization of general (irrational) conformal field theory produces objects that are rather difficult to deal with. One has to study infinite sums of tensor products of two irreducible representations of chiral algebras that have to obey additional conditions coming from the duality (see [2, 13]). Otherwise, one has to study various limits of rational conformal theories when the number of irreducible representation goes to infinity. It is really a challenge to develop such a theory in full generality and to find a reasonable amount of understandable examples (as far as we know only the theory on a torus and on its orbifolds are constructively known among irrational theories). Such an understanding would be a curious extension of differential geometry but it is out of reach for the moment. Therefore we have to wait a bit before we can say something constructive about general irrational conformal theories with Q-symmetry and the exact energy-momentum tensor. However, we can say something about the degeneration of this magnificent picture in the limit where the conformal theory degenerates so that the conformal dimensions of some fields tend to zero (note that there are fields with exactly zero dimension among them). It seems that we can write down tractible axioms on correlators of fields with nearly zero dimension (viewed as differential forms on the proper moduli spaces) at the point of degeneration – we will call these Zwiebach invariants 2 . One can show at the heuristic level that the amplitudes in the nearly degenerate theory can be obtained as a sum over graphs with Zwiebach invariants associated to vertices. Our next step would be to forget about the conformal field theory origin of the procedure and to study the theory of Zwiebach invariants (as a set of maps taking values in forms on the moduli spaces that obey some axioms) on their own. It is similar to forgetting the conformal field theory origin of the Gromov-Witten invariants. However, we will show that now we may also formalize the passage to the nearly degenerate theory, when the dimension of some fields is lifted. We will see that this leads to the procedure of induction of the structure of the Zwiebach invariant on the sub-bicomplexes. And Zwiebach invariants of bicomplex with zero differential turn out to be Gromov-Witten invariants. After presenting the outline of such a general construction we have to study the confirming example – and we really do it. Namely, we study the case when Zwiebach invariants take the simplest possible form – they are constructed from the Hodge dGBV algebra that satisfies the 1/12 axiom, and (possibly) some other conditions. Instead of looking for the formal proof that the set of these other conditions is empty (we admit that it would be nice to have such a proof) we just compute directly induced structures on cohomology of the bicomplex in genera zero and one. We show by explicit computation that these structures do solve WDVV and Getzler equations. All this should be compared with the theory of induction of the homotopical structure on the subcomplex. The simplest homotopical structure is the structure of differential graded (Lie) algebra. Therefore, we propose the generalization of this story to the bicomplexes with dgA being replaced by Hodge dGBV with 1/12 axiom, and homotopical algebra structure being replaced by Zwiebach invariants. 1 We call them just Gromov-Witten invariants in order to distinguish them from the geometrical GromovWitten invariants that follow from the theory of holomorphic maps. 2 We call these correlators Zwiebach invariants because of the inspiring work of Zwiebach [16] on related issues.
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The natural question to ask is whether all Zwiebach invariant can be obtained by induction from the simplest ones (like all homotopical algebras can be obtained by induction from the differential graded ones). We do not know the answer at the moment. We hope that the notion of Zwiebach invariants will help to understand why constructions of [15, 3, 1] lead to Gromov-Witten invariants. 2.1. Definition of Gromov-Witten invariants. By Gromov-Witten invariants we mean the set of maps m g,n : H0⊗n ⊗ C → R,
(1)
where H0 is a vector space and C is the space of cycles in the Deligne-Mumford compactification of the moduli space of genus g curves with n marked points Mg,n . This set of maps satisfies the following conditions [10]: (1) It is symmetric with respect to diagonal action of the symmetric group on factors of H0⊗n and cycles in Mg,n . (2) It vanishes when restricted to cycles that are zero in rational homologies of Mg,n (3) It satisfies the factorization property described below. The factorization property corresponds to degenerations of a surface of genus g with n marked points. First we consider the case when a surface degenerates into surfaces of genera g1 and g2 with n 1 and n 2 marked points respectively and that have a common point: m g,n (h 1 , . . . , h n )(c1 × c2 ) = ηi j m g1 ,n 1 +1 (h 1 , . . . , h n 1 , ei )(c1 )m g2 ,n 2 +1 (h n 1 +1 , . . . , h n , e j )(c2 ). (2) i, j
Here {e j } is a basis in H0 , ηi j is the inverse metric on H0 written in this basis, c1 and c2 are some cycles in Mg1 ,n 1 and Mg2 ,n 2 respectively, c1 × c2 is viewed as a cycle in Mg,n via the embedding Mg1 ,n 1 +1 × Mg2 ,n 2 +1 → Mg,n . Then we consider the degeneration of a curve of genus g into a curve of genus g − 1 with a double point. In this case the factorization property means m g,n (h 1 , . . . , h n )(c) = ηi j m g−1,n+2 (h 1 , . . . , h n , ei , e j )(c). (3) i, j
Here c is a cycle in Mg−1,n+2 considered also as a cycle in Mg,n via the natural mapping Mg−1,n+2 → Mg,n . 2.2. Set of factorizable maps from topological conformal field theory. In this and in the next subsections we assume some knowledge of conformal field theory (CFT). The reader that does not know CFT may skip this subsection and proceed to Subsect. 2.4 where we formalize insights coming from CFT. Consider CFT with odd symmetry. This means that the space of local observables Hc is a complex with the differential Q, correlators satisfy k v1 , . . . , Q(vi ), . . . , vk = 0, i=1
(4)
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both holomorphic and antiholomorphic energy-momentum tensors are Q-exact, Q(G) = T,
Q(G) = T ,
(5)
and the fields G and G do not have singularities in their mutual operator product. Consider the correlators v1 (z 1 ), . . . , vn (z n ), G(x1 ), . . . , G(x p ), G(y1 ), . . . , G(yq )
(6)
g,n of Riemann surfaces with germs as differential ( p, q)-forms on the moduli space M of local coordinates at marked points z 1 , . . . , z n . This means that we can contract such a form with a holomorphic (and antiholomorphic) vectors, tangent to the moduli space. A holomorphic tangent vector is determined by a Beltrami differential; so we can multiply G by the Beltrami differential and integrate over the surface. If n is not zero one can also multiply G by a holomorphic vector field in the neighbourhood of a marked point and integrate around it. Similarly, one can define contraction with an antiholomorphic tangent vector. This differential form descends down to the moduli space Mg,n if the correlator contains only the first order poles when x and y approach the set of the marked points. The second order pole in operator product expansion between G and v is called the action of the operator G 0 on v. Similarly, we define G 0 . Only the phase of the local coordinate corresponds to the noncontractable piece of the structure group of the bundle of germs of local coordinates over the moduli space of complex structures with the marked points. Therefore we only have to impose the condition G − (v) := (G 0 − G 0 )v = 0.
(7)
In order to get closed forms on the moduli space we impose Q(v) = 0.
(8)
Finally, we need (and this part of the construction is missing in [16]) our differential form to be extendable to the Deligne-Mumford compactification of the moduli space. One can show that this is satisfied if the fields v that are in the image of G − are not in the kernel of T0 + T 0 ; here T0 = Q(G 0 ) and T 0 = Q(G 0 ). 2.3. Topological string amplitudes in degenerating conformal theory. In the previous subsection we outlined the construction of amplitudes in an arbitrary topological conformal theory. However, in the so-called degenerating theories (like type B theory on Calabi-Yau at the infinite volume limit) life simplifies a bit, and the construction of amplitudes can be encoded in a tractable linear algebra data. By a degenerating theory we mean a family of theories parametrized by a parameter such that at = 0 the subset H ⊂ Hc of fields has zero conformal dimension: T0 H = T 0 H = 0.
(9)
In some cases (in particular, in the type B example) one can check that in this limit most of the correlators (6) vanish over the bounded domain of moduli of complex structures. However, this does not mean that the integrals over the moduli space vanish. What really happens is the following: the support of the correlation function moves towards
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the region where the surface degenerates. The good model of this phenomena is the ordinary integral: +∞ I () = exp(−t)dt. (10) 0
The value of this integral is independent of while the integrand tends to zero as goes to zero. The support of the integral is at t of order 1 . Thus we have a contribution from the boundary of the moduli space (see [15]). This contribution comes from the infinitely long tubes connecting components of the degenerating surface and equals to K =
G0G0 T0 + T 0
= G− G+,
(11)
where G+ =
G0 + G0 T0 + T 0
.
(12)
Note that G + has a regular limit as tends to zero, since {Q, G + } = 1 − 0 ,
(13)
where 0 is a projector to the space H0 of zero modes of T0 + T¯0 that presumably has a smooth limit as goes to zero. Note that H0 is the limit of the kernel rather than the kernel of the limiting operator (that coincides with H ). Note that the space H is equipped with the bilinear pairing: given by the two-point function: (v1 , v2 ) = v1 (z 1 ), v2 (z 2 ).
(14)
Since the conformal dimension of fields vi is zero, this correlation function is independent of the coordinates z i . Therefore, we obtain the rules for computation of the amplitude in the limiting theory. The contribution from the bulk of the moduli space is obtained by substitution of elements from H0 . The contribution from the degenerated surfaces is given by the sum over graphs, such that k-vertices of the graphs are labeled by k-point correlation functions (of different genera). A weight of a graph is given by the pairing between vertices, so-called propagators K (given by (11)) that correspond to edges, and elements of H0 that correspond to tails. Pairing is performed with the help of the bilinear form defined in (14). Note that vertices are paired with G − closed vectors, therefore vertices correspond to horizontal invariant forms on components of the moduli space and can be integrated over it. We make an attempt to formalize this in the next subsection. 2.4. Zwiebach invariants. In this section, we sketch the principal construction of Zwiebach invariants that motivates our purely algebraic constructions in the rest of the paper. Note that part of this was already presented in the work of Zwiebach [16], but he missed the Hodge condition. In different settings, but in a closed way, a piece of the algebraic structure that we finally get was also obtained in [9].
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2.4.1. Kimura-Stasheff-Voronov space. We consider the Kimura-Stasheff-Voronov compactification K g,n of the moduli space of curves of genus g with n marked point. It is a real blow-up of Mg,n ; we just remember the relative angles at double points. We can also choose an angle of the tangent vector at each marked point; this way we get the principal U (1)n -bundle over K g,n . We denote the total space of this bundle by S g,n . Let H be a bicomplex with two differentials denoted by Q and G − and with a scalar product (·, ·) invariant under the differentials: (Qv, w) = ±(v, Qw), (G − v, w) = ±(v, G − w). (k) Below we consider the action of Q and G − on H ⊗n . We denote by Q (k) and G − th the action of Q and G − respectively on the k component of the tensor product. 2.4.2. Definition. The Zwiebach invariants is the set {C g,n |g ≥ 0, n ≥ 0, 3g−3+n ≥ 0} of H ⊗n -valued differential forms on S g,n , satisfying the axioms: (1) (2) (3)
(4)
C g,n is (graded) symmetric under the interchange of factors in H ⊗n with the simultaneous renumeration of marked points; C g,n is totally closed, (Q + d)C g,n = 0 (Q = nk=1 Q (i) ); (k) C g,n is totally horizontal, (G − + ı k )C g,n = 0 for all 1 ≤ k ≤ n (we denote by ı k the substitution of the vector field generating the action on S g,n of the k th copy of U (1)) and C g,n is invariant under the action of U (1)n ; {C g,n } is the factorizable set of maps (cf. Eqs. (2), (3)), that is, C g,n |γ2 = [C g1 ,n 1 ∧ C g2 ,n 2 ], C g,n |γ1 = [C g−1,n+2 ].
(15) (16)
Here γ2 corresponds to the degeneration of the surface into two components, γ1 corresponds to the degeneration of a handle, and [·] denotes the contraction with the scalar product of the last factors in H ⊗n 1 +1 and H ⊗n 2 +1 in the first case and of the last two factors in H ⊗n+2 in the second case. It is useful to rewrite the last two axioms in local charts. Locally, S g,n is a product of K g,n and n circles. Then the horizontality axiom means that C g,n is represented as (1)
(n)
C g,n = (1 + dφ1 G − ) ∧ · · · ∧ (1 + dφn G − )C˜ g,n ,
(17)
where C˜ g,n is (the pull-back of) a form on K g,n and φi is the angle at the i th marked point. The factorization property in terms of C˜ g,n looks as follows: (n +1) ∧ C˜ g2 ,n 2 +1 , (18) C˜ g,n |γ2 = C˜ g1 ,n 1 +1 ∧ 1 + dψ G − 2 (n+2) C˜ g,n |γ1 = 1 + dψ G − ∧ C˜ g−1,n+2 . (19) Here γ2 corresponds to the degeneration of the surface into two components, γ1 corresponds to the degeneration of a handle, ψ denotes the relative angle at the double point, and [·] denotes the contraction with the scalar product of the last factors in H ⊗n 1 +1 and H ⊗n 2 +1 in the first case and of the last two factors in H ⊗n+2 in the second case. Indeed, we just use that ψ = φn 1 +1 + φn 2 +1 in the first case and ψ = φn+1 + φn+2 in the second case. Note that below we usually use C˜ g,n instead of C g,n just to make our calculations more transparent.
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2.4.3. Gromov-Witten invaiants. Zwiebach invariants on the bicomplex with zero differentials determine Gromov-Witten invariants. Indeed, in this case C g,n = C˜ g,n . Together with the factorization property this means that {C g,n } is lifted from the the blowdown of Kimura-Stasheff-Voronov spaces, i.e. it is determined by a set of continuous forms on Deligne-Mumford spaces. Therefore, integrating these forms along cycles in Mg,n we get Gromov-Witten invariants defined on the space dual to H . 2.4.4. Induced Zwiebach invariants. Induced Zwiebach invariants are obtained by contraction of an acyclic sub-bicomplex of (H, Q, G − ). Let H = H ⊕ H
such that (H , H
) = 0 and H
is an acyclic sub-bicomplex. We denote by G + the contraction operator. This means that G + H = 0, = {Q, G + } is the projection to H
along H , and {G + , G − } = 0. ind (or rather C ind ) on a modification of ˜ g,n We construct an induced Zwiebach form C g,n K g,n . Each degeneration of a curve gives us a boundary stratum γ that is a pricipal U (1) bundle over K g1 ,n 1 +1 × K g1 ,n 2 +1 or K g−1,n+2 /Z2 (Z2 exchanges the labels of the last two points). At each such component of the boundary we glue the cylinder γ × [0, +∞] such that γ in K g,n is identified with γ × {0} in the cylinder. So we take a form C˜ g,n , restrict it to H ⊗n , and extend it to the cylinder glued at γ as the restriction to H ⊗n of (n +1) C˜ g1 ,n 1 +1 ∧ e−t−dt·G + 1 + dφG − 2 ∧ C˜ g2 ,n 2 +1 (20) in the first case of curve degeneration or ˜ e−t−dt·G + 1 + dφG (n+2) ∧ C g−1,n+2 −
(21)
in the second case of curve degeneration. Here t is a coordinate along the cylinder and operators and G + in the formulas act at the same copy of H as G − . In terms of C g,n , this is just the same contraction as in Eqs. (15) and (16), but the scalar product is defined as (V, W )t = (V, e−t−dt·G + W ). ind (or C ind ) are Now it is a straightforward calculation to check that the forms C˜ g,n g,n (d + Q)-closed and satisfy the factorization property when restricted to the strata γ × {+∞}. This construction is not smooth and is defined not on K g,n , but on its extension. Nevertheless, one can easily turn this into a clear mathematical theory. We sketch the required construction in the next subsection. 2.4.5. Moduli spaces with cuffs. Instead of Zwiebach invariants on the spaces S g,n we can consider Zwiebach invariants on the moduli spaces with cuffs. That is, at each boundary stratum of S g,n of codimension 1 we glue the cylinder equal to this stratum multiplied by [0, +∞]. Then we consider the set of forms satisfying the same axioms as above, but we require the properties of horizontality and factorization on the “+∞” ends of the glued cylinders. Thus we obtain a slight generalization of the notion of Zwiebach invariants. If we have a system of Zwiebach invariants on S g,n , then we can lift it to cuffs. We just take the pull-backs of these forms under the mapping that keeps the moduli spaces and projects all cylinders to their “0” ends.
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Then when we consider the induced Zwiebach invariants, we glue new cylinders to the “∞” ends of the cuffs. Thus, at each boundary stratum we have two consequently glued cylinders. So, we choose a certain mapping, which identifies two glued cylinders with one cylinder. Then the theory of induced Zwiebach invariants is again the theory of Zwiebach invariants on the moduli spaces with cuffs. 2.4.6. Hodge case In the Hodge case, we assume that Q H = G − H = 0. Then the induced Zwiebach invariants determine Gromov-Witten invariants obtained by integrals over the fundamental cycles. What we get is a sum over graphs with vertices marked by the initial Zwiebach invariants (or rather their integrals over the fundamental cycles), internal edges correspond to the contraction of outputs with the scalar product (·, G − G + ·), and tails are marked by the elements of H . In this paper, we study the case where the unique nonvanishing integral of the initial Zwiebach invariants over the fundamental cycles exists for g = 0, n = 3. There are some obstructions for the existence of such initial Zwiebach invariants. We study them in the next subsection. 2.4.7. Obstructions. First, we choose C˜ 0,3 . It is a H ⊗3 -valued constant, so it determines a commutative multiplication on H . Since C˜ 0,3 is Q-closed, we have the Leibnitz rule: Q(ab) = Q(a)b + a Q(b). Now we try to choose C˜ 0,4 . From the factorization property, it follows that C˜ 0,4 is a sum of 0-form and 1-form. It is (d + Q)-exact. So, comparing values of the 0-form at two different boundary cycles of K0,4 , we obtain that our multiplication is homotopy associative, that is, the 3-form a ⊗ b ⊗ c → (ab)c − a(bc) is Q-exact. Another relation comes from an attempt to glue the 1-forms arising on the boundary of K0,4 due to the factorization property. Consider the total space S 0,4 . There are 7 distinguished 1-cycles, determined by the action of U (1) at a marked point and at double points. If we take an a ⊗ b ⊗ c ⊗ d-valued component of C g,n , then from the factorization property, if follows that the integrals over these cycles are equal to (G − (a)b, cd), (G − (b)a, cd), (G − (c)d, ab), (G − (d)c, ab) and (G − (ab), cd), (G − (ac), bd), (G − (ad), bc) (the cycles corresponding to marked points are taken in the fiber over one of the boundary points). A path along each cycle can be obtained as a Dehn twist along the corresponding cycle on a surface with 4 marked points. The relation among these Dehn twists [7] imply that there is a 2-dimensional surface in S 0,4 , whose boundary is the sum of these seven 1-cycles, and this gives us the 7-term relation up to homotopy: a ⊗ b ⊗ c → G − (abc) + G − (a)bc + G − (b)ac + G − (c)ab − G − (ab)c − G − (ac)b − G − (bc)a
(22)
is a Q-exact 3-form. Now we try to choose C˜ 1,1 . The Dehn twists along the cycles on a genus 1 surface with marked point also give us a new relation. There are three cycles, x and y are the basis in the first homology group of a torus, and z is the cycle around the marked point. If Dx , D y and Dz are the corresponding Dehn twists, then [Dx ] = [D −1 y ] in the homology −1 4 of K1,1 , and (D y Dx D y ) = Dz in the mapping class group [7]. Therefore, we obtain that the kernel of the linear function a → (12str (G − ◦ a·) − str ((G − a)·))
(23)
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A. Losev, S. Shadrin
contains the kernel of Q. Here str denotes the supertrace, and a· (resp., (G − (a))·) is the operator of multiplication by a (resp., G − (a)). From [7] it follows that no other relations can come from the relations among Dehn twists. But of course there can be other obstructions of different geometric origin. We are grateful to E. Getzler for the explanation of the geometric origin of the 7-term relation and 1/12-axiom.
2.5. dGBV algebras. The simplest solutions to the relations presented in the previous subsection are known as differential Gerstenhaber-Batalin-Vilkovisky (dGBV) algebras, see [1, 12]. They have naturally appeared in the paper of Barannikov and Kontsevich [1] as an axiomatic description of the properties of polyvector fields on Calabi-Yau. We refer to the Pre-Introduction and to the Appendix of this paper for the discussion of additional benefits from the ideas hidden in [1]. We have seen above that it is very natural to obtain relations coming from the geometry of the moduli space of curves in calculations with graph constructions in dGBV algebras. In the rest of the paper we give a formal algebraic proof of WDVV and Getzler relations for the potential corresponding to the simplest version of Zwiebach invariants. 3. Construction 3.1. Hodge dGBV algebra. A Hodge differential Gerstenhaber-Batalin-Vilkovisky algebra is a supercommutative associative C-algebra H with two odd linear operators Q, G − : H → H.
(24)
These operators must satisfy the system of axioms: (1) (2)
Q 2 = G 2− = QG − + G − Q = 0. H = H0 ⊕ H4 , where Q H0 = G − H0 = 0 and H4 is represented as a direct sum of subspaces of dimension 4 generated by eα , Qeα , G − eα , QG − eα for some vectors eα ∈ H4 , i. e. eα , Qeα , G − eα , QG − eα . (25) H4 = α
(3)
This axiom is called the axiom of Hodge decomposition. The ordinary dGBValgebra is the structure that we have without axiom (2). Q is a derivation: Q(ab) = Q(a)b + (−1)a˜ a Q(b).
(4)
(26)
Here and below, we denote by a˜ the parity of a ∈ H . G − is an operator of the second order: ˜
˜ bG − (ac) + (−1)a˜ aG − (bc) G − (abc) = G − (ab)c + (−1)b(a+1) ˜
˜ b abG − (c). − G − (a)bc − (−1)a˜ aG − (b)c − (−1)a+
Equation (27) is called the 7-term relation.
(27)
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659
3.2. Some notations. We define an operator G + : H → H . We set G + H0 = 0. On each subspace eα , Qeα , G − eα , QG − eα , we define G + as G + eα = G + G − eα = 0, G + Qeα = eα , and G + QG − eα = G − eα . Clearly, G + is an odd operator, G − G + + G + G − = 0, and 4 = QG + + G + Q is the projection to H4 along H0 . Denote by 0 the projection to H0 along H4 . Thus G + is the homotopy operator corresponding to the contraction of H to H0 . Note that we assume that this homotopy commutes with G − . 3.3. Integral. Let H be a Hodge dGBV algebra. An integral on H is an even linear
function : H → C such that ˜ Q(a)b = (−1)a+1 a Q(b), (28) (29) G − (a)b = (−1)a˜ aG − (b), and
a˜
G + (a)b = (−1)
aG + (b).
(30)
These properties
imply that G − G + (a)b = aG − G + (b), 4 (a)b = a4 (b), and 0 (a)b = a0 (b). We define a scalar product on H : (a, b) = ab. (31) We assume that this scalar product is non-degenerate. We call the full structure that we have here (a Hodge dGBV algebra and an integral determining a non-degenerate scalar product on H ) a cyclic Hodge dGBV algebra, or cH algebra for short. Further properties of this structure can be found in [12]. We would like to make two remarks on the scalar product (31): (1) (2)
Obviously, H0 is orthogonal to H4 . Using the non-degenerate scalar product (31), we may turn an operator A : H → H into the bivector (by bivector we call, for short, any element in H ⊗2 ). Below we denote this bivector by [A].
3.4. Variables. Let H0 be a finite dimensional space. Let e1 , . . . , en be its basis. Denote by T1 , . . . , Tn some independent variables. We take the parity of Ti equal to the parity of ei . 3.5. Construction of potential. We construct a formal power series F = F0 +F1 +F2 +. . . in variables T1 , . . . , Tn . We consider all trivalent graphs. This means that we consider graphs with vertices of index 3 only and with possible half-edges (leaves). We mark all leaves by elements from the set L = {e1 T1 , . . . , ek Tk }. We associate to each internal edge of a graph the bivector [G − G + ]. In our pictures, we denote this by thick black points on the edges. Each internal vertex (of index 3) corresponds to the 3-form m(a, b, c) = abc.
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Now each graph gives us a monomial in T1 , . . . , Tn as follows. At each vertex we have three incoming edges. They give three inputs for the corresponding 3-form m. Such input is either a “half” of [G − G + ]3 or an element of L. We take the product of values of 3-forms m on their inputs at all vertices of a graph. This is the monomial that we associate to the graph. We take each graph with the combinatorial coefficient that is equal to the inverse order of its group of automorphisms. ˜ Denote by J : H → H the operator J : h → (−1)h h.4 If we consider a graph with g loops, then at g edges we put the bivector [J G − G + ] instead of [G − G + ]. These g edges can be arbitrary ones, but with the only restriction: if we cut the graph at these edges, then we get a tree. Thus we obtain a Feynman diagram expansion of the integral discussed in the Appendix.
3.6. Examples. We give some examples. Let a, b, c be different elements of L. Consider the graph a
c
b .
(32)
a
b
The order of its group of automorphisms is equal to 2. So, it gives the monomial 1 [G − G + ] ⊗ [G − G + ], (ab∗) (∗c∗) (∗ab) 2 1 = ab · G − G + (c · G − G + (ab)) . (33) 2 Another example: a .
(34)
The order of its group of automorphisms of this graph is also equal to 2. We have the monomial 1 1 [J G − G + ], (∗ ∗ a) = str (G − G + ◦ a·) . 2 2
(35)
Here str is the supertrace. We recall that the supertrace of an operator A is defined as str (A) = tr (J A). Equation (35) means that this monomial is equal to the supertrace of the operator G − G + ◦ a· : H → H , h → G − G + (ah). 3 We note that from Sect. 3.3, it follows that this bivector is symmetric. 4 In physics, this operator is known as the fermionic parity operator and is usually denoted by (−1) F .
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661
3.7. Potential. We denote by F the formal sum of such monomials over all possible trivalent graphs with leaves marked by elements of L = {e1 T1 , . . . , ek Tk }. Of course, we identify isomorphic graphs. F is naturally represented as F0 + F1 + F2 + . . . , where Fi is the sum over graphs with i loops. We shall now draw the first few terms of F0 and F1 . For brevity, we denote by E the sum e1 T1 + · · · + ek Tk : 1 F0 = 6
F1 =
E E
E
1 2
1 + 8
E +
E
E
E
E E
1 4
1 + 8
+ E
E
E
E + . . . , (36)
E
E
1 E 4
E + ....
(37)
4. WDVV Equation We consider the moduli space M0,4 . The cohomology classes of any two points of M0,4 coincide. This gives a differential equation for the Gromov-Witten potential in genus zero. We check this differential equation in our construction. 4.1. Boundary points. We denote the classes of boundary points of M0,4 by 12|34 , 13|24 , and 14|23 :
1
2
3 12|34
4
1
3
2
4
1
4
13|24
2
3
(38)
14|23
We explain these pictures by the following example. The first picture denotes the moduli point of M0,4 represented by a two-component curve such that the marked points 1 and 2 lie on one component and the marked points 3 and 4 lie on the other component. We have 12|34 = 13|24 = 14|23 in homology of M0,4 . 4.2. Differential equations. This relation gives us some differential equations. We suppose that F0 is a formal power series in variables T1 , . . . , Tn , and ηi j is a metric on the space generated by T1 , . . . , Tn . If all variables are even, we have: ∂ 3 F0 ∂ 3 F0 ∂ 3 F0 ∂ 3 F0 ηkl = ηkl ∂ T1 ∂ T2 ∂ Tk ∂ Tl ∂ T3 ∂ T4 ∂ T1 ∂ T3 ∂ Tk ∂ Tl ∂ T2 ∂ T4 3 ∂ 3 F0 ∂ F0 = ηkl . ∂ T1 ∂ T4 ∂ Tk ∂ Tl ∂ T2 ∂ T3 (39) We have here three equations; each of them is called the Witten-Dijkgraaf-VerlindeVerlinde (WDVV) equation.
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4.3. Theorem. In our case, F0 is the sum over trees. The metric ηi j is given by the scalar product on H0 , ηi j = (ei , e j ). Theorem 1. F0 , ηi j satisfy the WDVV equation. We explain the simplest case of this theorem. Denote by [0 ] the 2-form corresponding to the operator 0 . We can put this bivector on an internal edge of a graph. We denote this by a thick white point on the edge. In the simplest case, the theorem states that the 4-form u v (t, u, v, w) →
(40) w
t
restricted to H0 is symmetric. In other words, for any t, u, v, w ∈ H0 , tu · 0 (vw) = tv · 0 (uw) = tw · 0 (uv).
(41)
We prove Theorem 1 in Sect. 9.4. The simplest case of Theorem. 1 (given by Eq. (41)) is discussed in detail in Sect. 7. 5. Getzler Relation Getzler elliptic relation [8] is a linear relation among some natural complex codimension 2 strata in the cohomology ring of the moduli space M1,4 . It gives a differential equation for Gromov-Witten potentials in genera zero and one. We prove that our construction satisfies this differential equation. 5.1. Cycles in M1,4 . We list the codimension two cycles entering the Getzler relation.
1
1
2,2
2,3
1
1 2,4
3,4 (42)
0,3
0,4
b
We use here the notations from [14]. A line marked by 1 corresponds to a genus one curve. An unmarked line corresponds to a genus zero curve. Notches correspond to the marked points.5 5 Note that here we use pictures with absolutely different meaning then in the rest of the paper. For instance, in all other pictures we put notches just to set operators on graphs.
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663
For example, the generic point of the stratum 2,2 is represented by a curve of genus one. It has no marked points, but it has two attached genus zero curves with two marked points on each of them. Each picture means that we label marked points by the numbers {1, 2, 3, 4} in all possible ways. For example, there are 3 variants for 2,2 and 12 variants for 2,3 . 5.2. Relation. Getzler elliptic relation: 12 2,2 − 4 2,3 − 2 2,4 + 6 3,4 + 0,3 + 0,4 − 2 b = 0.
(43)
We rewrite this relation as a differential equation for the formal power series F0 and F1 . If all variables are even, we have: 2,2
2,3
b
∂ 2 F1 ∂ 3 F0 ∂ 3 F0 ηi j ηkl ∂ T1 ∂ T2 ∂ Ti ∂ T j ∂ Tk ∂ Tl ∂ T3 ∂ T4
(44)
+
∂ 3 F0 ∂ 2 F1 ∂ 3 F0 ηi j ηkl ∂ T1 ∂ T3 ∂ Ti ∂ T j ∂ Tk ∂ Tl ∂ T2 ∂ T4
+
∂ 2 F1 ∂ 3 F0 ∂ 3 F0 ηi j ηkl , ∂ T1 ∂ T4 ∂ Ti ∂ T j ∂ Tk ∂ Tl ∂ T2 ∂ T3
∂ 3 F0 ∂ 3 F0 ∂ 2 F1 ηi j ηkl ∂ T1 ∂ Ti ∂ T j ∂ T2 ∂ Tk ∂ Tl ∂ T3 ∂ T4 +11 ter ms obtained by per mutations o f {1, 2, 3, 4}, .. . ∂ 4 F0 ∂ 4 F0 ηi j ηkl ∂ T1 ∂ T2 ∂ Ti ∂ Tk ∂ T3 ∂ T4 ∂ T j ∂ Tl +2 ter ms obtained by per mutations o f {1, 2, 3, 4}.
(45)
(46)
5.3. The 1/12-axiom. In our construction, F0 is the
sum over trees, F1 is the sum over graphs with one loop, and the metric is just ηi j = ei e j . Theorem 2. F0 , F1 , ηi j satisfy the Getzler relation, if G−
=
1 12
G−
.
(47)
We explain these pictures. On the left-hand side, we mark the loop by G − . This means that we put on the loop the bivector [G − ]. On the right-hand side, we put G − on the leaf and we have an empty loop. This means that we apply G − to the input on the leaf and that we put the bivector [Id] on the loop. In order to simplify the understanding and to explain our notations, we rewrite the 1/12-axiom (47) in terms of tensors and in terms of supertraces. In terms of tensors, the 1/12-axiom looks like 1 [J G − ], ∗ ∗ h = [J ], ∗ ∗ G − (h) . (48) 12
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A. Losev, S. Shadrin
In terms of supertraces, the 1/12-axiom means str (G − ◦ h·) =
1 str (G − (h)·) . 12
(49)
So, this is just a rigid version of the axiom (23) obtained from the relation among Dehn twists in the fundamental group of K1,1 . In fact, one can include this additional axiom in the definition of cH-algebra, since it has the same status as, say, the 7-term relation.
5.4. The simplest case. We describe the simplest case of Theorem 2. Let a, b, c, d be elements of {e1 T1 , . . . , en Tn }. At each picture below, we distribute a, b, c, d among leaves in all possible ways (in other words, we put the sum a + b + c + d at each leaf). Then we calculate 2,2 , . . . , b according to our rules and check the relation (43),6
2,2 =
1 16
2,3 =
1 4
2,4 =
1 8
3,4 =
1 4
0,3 =
1 4
0,4 =
1 16
b =
1 4
+
1 16
+
+
1 4
1 4
(50) +
+
+
1 4
1 2
1 4
+
1 16
6 We would like to note that the computations hidden behind these words are rather hard.
From Zwiebach Invariants to Getzler Relation
665
As usual, an internal vertex corresponds to the integral of all inputs, an edge with the thick black point corresponds to the bivector [G − G + ], and an edge with the thick white point corresponds to the bivector [0 ]. 5.5. Proof. We explain the proof of Theorem 2 in Sect. 9. The simplest case of Theorem 2 is discussed in Sect. 8 6. Strategy of Proofs We prove our theorems in two steps. For each theorem, the first step is the simplest case of a theorem. For both our theorems, Theorem 1 and Theorem 2, it is the case of degree 4 (4 marked points on a surface and 4 leaves in a graph). Studying Gromov-Witten invariants, it is enough to have a relation in M0,4 (or M1,4 ) to prove a differential equation in any degree. Indeed, a relation in M0,4 (M1,4 ) can be lifted to any M0,n (M1,n ), n ≥ 4 via the projection forgetting all but four marked points. It is not the case in our construction. Nevertheless, we have a general technique that allows us to extend an argument proving the simplest case of any relation to the argument that proves the corresponding differential equation in any degree. So, our proofs are organized in three sections. First, we prove the simplest case of Theorem 1; second, we prove the simplest case of Theorem 2; third, we explain how one can extend our arguments to have the full proofs. 7. The Simplest Case of Theorem 1 For the convenience of the reader, we explain the proof of the simplest case of Theorem 1 in terms of tensors and in terms of graphs simultaneously. This gives also a number of illustrations to the correspondence between the language of graphs and the language of tensors.
7.1. The simplest case. We formulate the simplest case of Theorem 1. Consider a, b, c, d ∈ L = {e1 T1 , . . . , ek Tk }. Theorem 1 states that b
c (51)
a
d
is symmetric under premutations of a, b, c, d. We prove this. We have the operator 0 on the internal edge. Since 0 = Id − QG + − G + Q, we have: b
c
a
d
b
c
a
d
=
b − a
QG+
c
b −
d
a
G+ Q
c . d
(52)
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A. Losev, S. Shadrin
Here we use a new object in our graphs, an internal vertex of index 4. A vertex of index k corresponds in our formulas to the k-form (53) m k (a1 , . . . , ak ) = a1 · · · · · ak . As usual, the inputs of this form correspond to the incoming edges and leaves. So, Eq. (52) can be rewritten just as ab · 0 (cd) = abcd − ab · QG + (cd) − ab · G + Q(cd). Since Q(x y) = Q(x)y + (−1)x˜ x Q(y) and write in terms of graphs that
˜ Q(x)y = (−1)x+1
(54)
x Q(y), we can
Q Q
+
+
=0
Q
(55)
(it is the case of even inputs on leaves). Thus we have: b
G+ Q
a
c
G+
+
d
Qa
c
b =
d
Qb
c
G+
=
a b
b
c
QG+
a
d Qc
G+
b
d
(56)
.
(57)
d G+
c
+ a
;
a
Qd
One can also rewrite these equations as 7
ab · QG + (cd) = G + (cd) · Q(a) · b + G + (cd) · Q(b) · a,
ab · G + Q(cd) = G + (ab) · Q(c) · d + G + (ab) · Q(d) · c.
(58) (59)
Since Qa = Qb = Qc = Qd = 0, we have b a
QG+
c
b =
d
a
G+ Q
c =0
(60)
d
7 Starting from here and up to the end of the paper we put the signs in formulas with graphs without any additional explanation. All signs in our formulas agree with each other. The choice of the sign at each picture is determined by the choice of the underlying tensor formula. So, we always put signs in the most convenient way, and one can check that the corresponding underlying tensor formulas agree with each other.
From Zwiebach Invariants to Getzler Relation
667
and therefore b
c
b
c
= a
. a
d
(61)
d
The last expression is obviously symmetric under permutations of a, b, c, d. The simplest case of Theorem 1 is proved.
7.2. The next to the simplest case. We proceed to the next to the simplest case of Theorem 1. We ought to do it since it is not clear from the previous calculations how the full system of axioms of dGBV algebra is used. Take a, b, c, d, e ∈ L = {e1 T1 , . . . , ek Tk }. Theorem 1 states that c
b
d
d
b +
a
e
+ a
c
b
b
d
d e
d
e
+ c
c
+ a
a
d
e
b
e
a
+
e
c
(62) c
a
b
is symmetric under permutations of a, b, c, d. We study the first summand of this expression. We have: c
b
d =
a
e c
d
a
e
b
c
d
a
QG+
e
−
b
b
c
d
a
G+ Q
e
−
.
(63)
Since Q(ab) = Q(a)b + a Q(b) = 0, the middle term of this expression in equal to 0. For the last term, we have: Q (c · G − G + (de)) = Q(c) · G − G + (de) + c · QG − G + (de) = −c · G − QG + (de) − c · G − G + Q(de) = −c · G − (de).
(64)
In particular, we use here 4 = QG + + G + Q, G − 4 = G − , Q(de) = 0. This allows us to rewrite Eq. (63) as b
c
d
c
d
a
e
= b a
e
b
c
d
a
G + G−
e
+
.
(65)
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A. Losev, S. Shadrin
In the same way we can write down the similar formulas for the next two summands of the expression (62): d
b
e
d
e
a e
c
a
d
= b a b
c
c
= b a
d
a
G+ G−
c
b
e
c
a
G+ G−
+
c e
b
+
d
e ,
(66)
.
(67)
d
For G − we can use the 7-term relation (27). Note that G − (c) = G − (d) = G − (e) = 0. This yields: G − (cde) = G − (cd)e + G − (ce)d + G − (de)c,
(68)
G + (G − (cd)e) + G + (G − (ce)d) + G + (G − (de)c) = −G − G + (cde).
(69)
and therefore
Using this, we see that the sum of the last summands of Eqs. (65), (66), and (67) is equal to e
a
− d
. c
(70)
b
Thus we have that the first line of Expression (62) is equal to c
d
d
a
e
c
a
a
c
e
a
c
b
− d
+ b
+ b
b
e
e
d
. (71)
The same argument proves that the second line of Expression (62) is equal to d
a
b
e
c
d
e
b e
c
d
e
a
c
a
c
− b
+ d
+ c
e
a
. (72)
b
a
d
d
a
d
b
b
e
a
e
Hence, Expression (62) is equal to c
d
a
e
+ c
+ b
b
. (73)
+ c
Obviously, it is symmetric under permutations of a, b, c, d. The next to the simplest case of Theorem 1 is proved.
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669
8. The Simplest Case of Theorem 2 We prove the simplest case of Theorem 2 in two steps. First, we represent each cycle as a linear combination of graphs P1 , . . . , P9 : P1 =
P2 =
P3 =
P4 =
P5 =
P6 =
P7 =
P8 =
P9 =
Then we substitute these expressions into the Getzler relation (43) and get zero.
8.1. The cycle 2,4 .. We recall that
2,4 =
1 8
+
1 4
.
(74)
Here we put on leaves the sum e = a + b + c + d of arbitrary four elements a, b, c, d ∈ L = {e1 T1 , . . . , ek Tk }. Since 0 = Id − QG + − G + Q, we have
=
−
− QG+
.
(75)
G+ Q
Using Eq. (55), we move Q to the neighbouring edges. The third summand of the right-hand side of Eq. (75) is equal to zero. Indeed, we move Q to leaves, and use that Q(e) = 0. We consider the second summand of the right-hand side of Eq. (75). There we move Q to the edge marked by 0 and to the edge marked by G − G + . In the first case we get zero, since Q0 = 0. In the second case, Q transforms G − G + into −G −
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A. Losev, S. Shadrin
and goes to leaves (we do the same with the third summand of the right-hand side of Eq. (63)). Finally, we have G− =
.
+
(76)
G+ The same argument shows that =
+
G− G+
.
(77)
Thus, we have
2,4 =
1 8
+
1 4 G−
1 8
+
1 4
G− G+
. (78)
G+ We consider the last two terms of this expression. We can apply here the 7-term relation (27). Since G − 0 = 0 and G − e = 0, it takes the form G− 1 8
+
1 4
G− G+
=−
1 8
(79)
G+ (G − jumps to the edge with G + and we get there −G − G + ; exactly the same argument is used to obtain Eq. (70)). Thus, we have 2,4 =
1 4
.
(80)
Now we start the same procedure with the next thick white point. We have = + G−
G+
G+ +
G−
(81)
From Zwiebach Invariants to Getzler Relation
671
Applying the 1/12-axiom (47), we have G−
G+
=−
1 12
.
(82)
From the 7-term relation (27), it follows that G − (e4 ) = 2e · G − (e3 ). Applying this, we have G+ G−
=−
1 2
.
(83)
So, the final formula for the cycle 2,4 is 2,4 =
1 4
−
1 8
−
1 48
. (84)
8.2. The other cycles. The same calculations with the other cycles express these cycles in terms of the graphs P1 , . . . , P9 : 2,2 = 2,3 = 2,4 = 3,4 = 0,3 = 0,4 = b =
1 1 1 1 P1 + P4 − P3 + P9 , 16 16 8 192 1 1 1 P1 + P5 − P2 , 4 4 4 1 1 1 P2 − P1 − P7 , 4 8 48 1 1 1 1 P3 − P2 − P6 + P7 , 4 12 48 144 1 1 1 P6 − P8 − P7 , 4 4 12 1 1 1 P9 + P8 − P6 , 16 4 8 3 1 1 P4 − P5 + P9 . 8 2 16
Substituting these expressions into the Getzler relation (43), we see that the coefficient at each Pi is equal to zero. This proves the simplest case of Theorem 2. 9. General Case of Both Theorems In this section, we will do the following. In order to prove our theorems in the general case, we must consider graphs with an arbitrary number of leaves in addition to the basic four leaves that we consider in the simplest case. The idea is to use the “self-repeating” structure of our graphs. It means that we replace each edge marked by a thick black point by the sum over all trivalent trees with two special leaves playing the role of the ends of the edge. In the similar way, we replace each edge marked by a thick white point by the sum over all trivalent trees with two special leaves playing the role of the ends of the edge and a special edge marked by a thick white point on the path connecting these two
672
A. Losev, S. Shadrin
leaves (all other edges are marked by a thick black points, of course). Also we replace each leaf by the sum over rooted trivalent trees with a special leaf that corresponds to the initial one. At the level of tensors this means that we replace in the formulas (50)–(51) for the simplest cases the operators 0 , G − G + and vectors a, b, c, d by certain operators O0 , Oc , and vectors Ol a, Ol b, Ol c, Ol d. We define all these operators (O0 , Oc , and Ol ) in Sect. 9.2. In order to give compact definitions of these operators, we introduce in Sect. 9.1 an auxiliary vector γ that is responsible, in a sense, for the self-repeating structure of our graphs. All our new operators, O0 , Oc , and Ol , are formal power series in the variables T1 , . . . , Tk . The degree zero part of these operators gives the simplest cases of our theorems. The degree one part of these operators gives the next to the simplest cases of our theorems. We give an example for this in Sect. 9.3. In Sect. 9.4 we complete the proof of Theorem 1, and in Sect. 9.5 we complete the proof of Theorem 2. 9.1. Vector γ . In this section, we define a vector γ ∈ H ⊗ C[[T1 , . . . , Tn ]]
(85)
and study its properties. We denote by E the sum E = e1 T1 + · · · + en Tn . We denote by γ the outcome at the root of the sum of all rooted trivalent trees with E on leaves and G − G + on edges:
γ =
E
E +
E
E
1 2
+
E +
E
1 E 2
E
E
E
E
1 8
E
E +
1 E 2
+ ...
(86)
Lemma 1. Vector γ satisfied two equations: G − (γ ) = 0, Q(γ ) + 21 G − (γ 2 ) = 0.
(87) (88)
In particular, our γ is a specific solution to the Maurer-Cartan equation defined in [1, Lemma 6.1] We prove Lemma 1. The first statement is obvious, since G − E = 0 and G 2− = 0. We prove the second statement. Since [Q, G − G + ] = −G − and Q E = 0, and using the self-repeating structure of our graphs, we have: Q(γ ) = −
∞ 1 . G−G+ γ · G−G+ γ · . . . G−G+ γ · G− γ 2
2 i=0
i
(89)
From Zwiebach Invariants to Getzler Relation
673
From the 7-term relation (27), it follows that 3γ · G − (γ 2 ) = G − (γ 3 ) + 3γ 2 · G − (γ ) = G − (γ 3 ), since G − (γ ) = 0. Substituting this in (89), we get 1 Q(γ ) = − G − (γ 2 ) 2 ∞ 1 . (90) − G−G+ γ · G−G+ γ · . . . G−G+ γ · G−G+G− γ 3
6 i=1
i−1
Since G − G + G − = 0, we have Q(γ ) = −(1/2)G − (γ 2 ). Lemma 1 is proved. 9.2. Some additional operators and vectors. In this section, we define some additional operators and vectors in terms of γ and study their properties. Define the operator , (h) = G − G + (γ · h), which obeys:
(91)
[Q, ](h) = −G − (γ · h) − G − G +
γ2 ·h . 2
(92)
Define the operator Ol as: Ol =
∞ i=0
◦ ◦· · · ◦ .
(93)
i
Consider a vector a ∈ H0 ⊗ C[[T1 , . . . , Tk ]]. Using Eq. (92), Lemma 1, and the 7-term relation (27), we have Q Ol (a) = −G − (γ · Ol (a)) .
(94)
We will use the vector Ol (a) instead of a on leaves, and relation (94) instead of Qa = 0. In terms of graphs the vector Ol (a) can be represented as: Ol (a) =
∞
γ
γ
γ
...
a
(95)
i=0
γ
(the sum is taken over the number of fragments Define the operator Oc ,
in graphs).
Oc = Ol G − G + .
(96)
Using Eq. (92), Lemma 1, and the 7-term relation (27), we have [Q, Oc ](h) = −G − (γ · Oc (h)) − Oc − (γ · G − (h)) − G − (h).
(97)
We will use the operator Oc instead of G − G + on edges, and relation (97) instead of [Q, G − G + ] = −G − . We draw the operator Oc in terms of graphs as: Oc =
∞ i=0
γ
γ
...
γ
(98)
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A. Losev, S. Shadrin
γ
(the sum is taken over the number of fragments Define the operator Or as:
in graphs).
Or (h) = h + γ · Ol G − G + (h).
(99)
Now consider the operator O0 defined by the formula O0 = Ol 0 Or .
(100)
By applying several times Eq. (92), Lemma 1, and the 7-term relation (27), we arrive at: O0 = Ol + Or − Id − [Q, Ol G + Or ] + Ol G + Or γ · G − − G − γ · Ol G + Or (101) (here we denote by γ · the operator of multiplication by γ ). We will use the operator O0 instead of 0 on edges, and relation (101) instead of 0 = Id − QG + − G + Q. We draw the operator O0 in terms of graphs: O0 =
∞
γ
γ
...
γ
γ
...
(102)
i, j=0
γ
(the sum is taken over the number of fragments
and
γ
in graphs).
9.3. Degree one case. We study the case of degree one for Theorem 1. If we replace the operator 0 by O0 and the vectors a, b, c, d by Ol a, Ol b, Ol c, Ol d, then we have the following picture: Ol b
Ol c
O0
.
Ol a
(103)
Ol d
The operators Ol and O0 are the formal power series in T1 , . . . , Tk . We write down the first two terms of the power series expansions of these operators: Ol (x) = Id(x) +
k
G − G + (ei Ti · x) + . . . ,
(104)
i=1
Oc (x) = 0 (x) +
k
(G − G + (ei Ti · 0 (x)) + 0 (ei Ti · G − G + (x))) . . . . (105)
i=1
Then we have the power series expansion of picture (103) Ol b Ol a
O0
Ol c
b
c
=
(106) a
Ol d
d b
c
d
+
b
d
E
+ a
E
a
c
From Zwiebach Invariants to Getzler Relation
675
E
b
c
+
d
a
b
+ a b
d
d
c
E
d
E E
a
+
+ c +...
c
a
b
Thus we see that the degree zero part of the power series expansion of (103) is the simplest case of Theorem 1 (see Sect. 7), and the degree one part of it is the next to the simplest case of Theorem 1 (see Sect. 7.2).
9.4. Proof of Theorem 1. First we reformulate Theorem 1 in terms of O0 and Ol . We claim that for any a, b, c, d ∈ L = {e1 T1 , . . . , ek Tk }, Ol b
Ol c
O0
(107)
Ol a
Ol d
is symmetric under permutations of a, b, c, d. Using Eqs. (101) and (94) we can prove this exactly by the same argument as we prove the simplest case of this theorem. Indeed, first we can use Eq. (101) (instead of the formula 0 = Id − QG + − G + Q). Using Eq. (55), we have Ol b Ol a
O0
Ol c
Ol b
Ol c
=−
(108) Ol a
Ol d
Ol b + Ol a Ol b
Ol d Ol
Ol c
Ol c +
Ol d
Ol
Ol d
Ol G+ Or γ · G−
Ol b (109) Ol a
Ol c
+
(110) Ol a Ol b
+
Ol d Ol G+ Or
Ol a Ol b + Ol a
G− (γ · Ol c) (111) Ol d
Ol G+ Or
G− (γ · Ol d) (112) Ol c
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A. Losev, S. Shadrin
Ol c
Ol b
Ol G+ Or γ · G−
(113)
+ Ol a
Ol d Ol c
Ol G+ Or
+
G− (γ · Ol b) (114) Ol a
Ol d Ol c
Ol G+ Or
+
G− (γ · Ol a) (115) Ol b
Ol d
(abusing notations, we denote by γ · the operator of multiplication by γ ). Applying the 7-term relation (27) to the summands (110), (111), and (112) and using G − (γ ) = G − (Ol c) = G − (Ol d) = 0, we get that the sum of these three summands is equal to Ol b
Ol G+ Or G− γ·
Ol a
Ol c .
(116)
Ol d
Note that Or G − = G − . Hence, Ol G + Or G − γ · = Ol G + G − γ · = −Ol G − G + γ ·. Note also that Ol G − G + γ · = Ol − Id. Hence, the sum of (110), (111), and (112) is equal to Ol b −
Ol
Ol a
Ol c
Ol b
Ol c .
+ Ol a
Ol d
(117)
Ol d
The same argument proves that the sum of (113), (114), and (115) is equal to Ol c −
Ol
Ol b
Ol b +
Ol a
Ol d
Ol c
Ol a
.
(118)
.
(119)
Ol d
Substituting these expressions in Eq. (108), we have Ol b Ol a
O0
Ol c
Ol b
Ol c
= Ol d
Ol a
Ol d
The right-hand side here is obviously symmetric under permutations of a, b, c, d. This proves Theorem 1.
From Zwiebach Invariants to Getzler Relation
677
9.5. On Theorem 2. We do not give here the detailed calculation proving the general case of Theorem 2. We just explain how to do this. It is obvious that our argument works, and calculations with Theorem 1 completely explain what to do. In order to have the full statement of Theorem 2, we change the markings on edges and leaves in pictures of the cycles 2,2 , . . . , b . So, we change 0 to O0 , we change G − G + to Oc , and we change e on leaves to Q l e. In order to prove Theorem 2, we express these new cycles 2,2 , . . . , b in terms of graphs P1 , . . . , P9 , where we also change G − G + to Oc and e to Q l e. Our calculations are just the same (like in the case of Theorem 1). But instead of the relation 0 = Id − QG + − G + Q we use Eq. (101), instead of [Q, G − G + ] = −G − we use Eq. (97), and instead of Qe = 0 we use Eq. (94). The expressions of cycles 2,2 , . . . , b in terms of graphs P1 , . . . , P9 are just the same as in the simplest case. Moreover, the intermediate step (Eq. (80) for 2,4 ) in calculations with each cycle is just the same as in the simplest case, but we must also change 0 , G − G + , and e to O0 , Oc , and Ol e in the intermediate pictures. Finally, this proves Theorem 2. We note that this argument works not only for Theorem 1 and Theorem 2. That is, if we have any PDE for our potential F, which is proved in its simplest case by the same argument as we have used for the simplest cases of Theorem 1 and Theorem 2 (to get out step by step of thick white points increasing the indices of vertices), then the argument described here immediately gives the full proof of this PDE. This corresponds in the theory of Gromov-Witten invariants to the lift of relations among strata in the moduli spaces of curves (for example, the Getzler relation in M1,4 gives us relations in M1,5 , M1,6 , and so on). Appendix A. BCOV-Action In this appendix, we explain how one can reformulate the results of Barannikov and Kontsevich in terms of graphs just by studying the BCOV-action proposed in the Appendix of their paper [3, 1]. A.1. Sums over trees. Let V be an arbitrary vector space. Our goal is to find a critical point and the critical value at this point of the following expression: A(v) = K 1 (v) +
1 1 1 K 2 (v, v) + K 3 (v, v, v) − B2 (v, v). 2 6 2
(120)
Here K 1 , K 2 , and K 3 are certain symmetric 1-, 2-, and 3-forms respectively, and B2 is a nondegenerate scalar product. We denote by b2 the inverse bivector of B2 . Our goal is to obtain a critical point of A(v) and the critical value at this point as a formal power series in K i . We consider the sum of rooted trees without leaves. We suppose that there are vertices of degree 1, 2, and 3, and the root is the vertex of degree 1. At each vertex (except the root) of degree i we put the i-form K i . At each edge we put the bivector b2 . Then, substituting the bivectors into the form according to the graph, we get a vector at the root. We also weight each graph with the inversed order of its automorphism group. We denote the vector represented in this way by vcr (we suppose that the sum over rooted trees converges). Lemma 2. vcr is a critical point of A(v).
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Now we consider the sum over trees without leaves and without a root. We suppose that there are vertices of degree 1, 2, and 3, and the root is the vertex of degree 1. At each vertex of degree i we put the i-form K i . At each edge we put the bivector b2 . Substituting the bivectors into the form according to the graph, we get a number. As usual, we weight each graph with the inversed order of its automorphism group. We denote the number obtained in this way by Acr (here we also suppose that the sum over trees converges). Lemma 3. Acr = A(vcr ). Both lemmas can be proved directly, by a simple linear algebra argument. A.2. BCOV-action. We consider a cH-algebra H . Barannikov and Kontsevich propose to study the action: 1 1 3 A(v) = (121) (E + G − v) − Qv · G − v. 6 2 We recall that E = e1 T1 + · · · + en Tn , n = dim H0 . This is an immediate generalization of the Kodaira-Spencer theory of Bershadsky, Cecotti, Ooguri, and Vafa. However, the 1/12-axiom is missing in [3] and in all subsequent papers [4, 6, 5]. Proposition 1. If vcr is the critical point of A(v), then γ = E +G − (vcr ) is the G − -closed solution of the Maurer-Cartan equation (see Eqs. (87)–(88), Sect. 9.1). Proposition 2. The critical value F0 = A(vcr ) is the solution of the WDVV equation (see Eq. (36), Sect. 3.7). Barannikov and Kontsevich formulate and prove both propositions without using the representations of γ and F0 in terms of graphs. However, these representations exist and are naturally provided by the linear algebra formalism explained in Sect. A.1. Let us demonstrate this. The graph representation of F0 is a direct corollary of the graph representation of γ . In order to obtain the graph representation of γ , we rewrite A(v) as 1 1 A(v) = (122) E3 + E 2 · G − (v) 6 2 1 1 1 + (123) E · G − (v)2 + G − (v)3 − Qv · G − v. 2 6 2 We recall that H = H0 ⊕ α eα , Qeα , G − eα , QG − eα . In fact, the scalar product B2 (v, v) = Qv · G − v is nondegenerate only on α eα . So there exists the bivector b2 inversed to B2 . We note that if we apply G − to both components of b2 , then we obtain the bivector [G − G + ]. Now we consider the sum over the rooted trees discussed in Sect. A.1. We have:
K 1 (v) = (1/2) E 2 · G − (v), K 2 (v, v) = E · G − (v)2 , and K 3 (v, v, v) = G − (v)3 . We see that we can move G − from vertices to edges. Then, if we consider the sum over rooted trees with one additional G − at the root, we obtain the following: (1)
At edges we put the bivector [G − G + ].
From Zwiebach Invariants to Getzler Relation
(2) (3) (4)
679
At vertices of degree 3 we put the 3-form (v1 , v2 , v3 ) → v1 v2 v3 .
At vertices of degree 2 we put the 2-form (v1 , v2 ) → v1 v2 E, i.e. we view it as the vertex of degree 3 with one leaf marked by E.
At vertices of degree 1 we put the 1-form v1 → v1 E 2 /2, i.e. we view consider it as the vertex of degree 3 with two leaves marked by E.
Thus we represented G − (vcr ) as a sum over the trivalent rooted trees with leaves. Moreover, E + G − (vcr ) is exactly the vector γ studied in Sect. 9.1. Acknowledgements. We are grateful to E. Getzler and M. Kontsevich for fruitful discussions and to the referee, who has encouraged us to add the Pre-introduction and the Appendix. Also, A. L. is grateful to A. Gerasimov for the explanation of the role of Hodge theory in string theory.
References 1. Barannikov, S., Kontsevich, M.: Frobenius manifolds and formality of Lie algebras of polyvector fields. Internat. Math. Res. Notices 1998 (4), 201–215 (1998) 2. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B241 (2), 333–380 (1984) 3. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165 (2), 311–427 (1994) 4. Dijkgraaf, R.: Chiral deformations of conformal field theories. Nucl. Phys. B 493, 588–612 (1997) 5. Gerasimov, A.: Towards integrability of topological strings. Preprint HMI-05-12, available at http://www.maths.tcd.ie/report_series/ as TCDMATH 05-07 6. Gerasimov, A., Shatashvili, S.: Towards integrability of topological strings I: three-forms on Calabi-Yau manifolds. J. High Energy Phys. 2004 (11), 074 (2004) 7. Gervais, S.: A finite presentation of the mapping class group of a punctured surface. Topology 40 (4), 703–725 (2001) 8. Getzler, E.: Intersection theory on M1,4 and elliptic Gromov-Witten invariants. J. Amer. Math. Soc. 10 (4), 973–998 (1997) 9. Getzler, E.: Batalin-Vilkovisky algebras and two-dimensional topological field theories. Commun. Math. Phys. 159, 265–285 (1994) 10. Kontsevich, M., Manin, Yu.: Gromov-Witten theory, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164 (3), 525–562 (1994) 11. Losev, A.: Hodge strings and elements of K. Saito’s theory of primitive form. In: Topological field theory, primitive forms and related topics (Kyoto, 1996), Progr. Math. 160, Boston, MA: Birkhäuser, 1998, pp. 305–335 12. Manin, Yu.I.: Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces. Providence, RI: Amer. Math. Soc., 2000 13. Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Comm. Math. Phys. 123 (2), 177–254 (1989) 14. Pandharipande, R.: A geometric construction of Getzler’s elliptic relation. Math. Ann. 313 (4), 715–729 (1999) 15. Witten, E.: Chern-Simons gauge theory as a string theory. In: The Floer memorial volume, Progr. Math., 133, Basel: Birkhäuser 1995, pp. 637–678 16. Zwiebach, B.: Closed string field theory: quantum action and the Batalin-Vilkovisky master equation. Nuclear Phys. B390 (1), 33–152 (1993) Communicated by N.A. Nekrasov
Commun. Math. Phys. 271, 681–697 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0207-5
Communications in
Mathematical Physics
Mean-Field- and Classical Limit of Many-Body Schrödinger Dynamics for Bosons Jürg Fröhlich1 , Sandro Graffi2 , Simon Schwarz1 1 Theoretische Physik, ETH Zürich, Zürich, Switzerland.
E-mail: [email protected]; [email protected]
2 Dipartimento di Matematica, Università di Bologna, Bologna, Italy. E-mail: [email protected]
Received: 8 May 2006 / Accepted: 10 October 2006 Published online: 22 February 2007 – © Springer-Verlag 2007
Abstract: We present a new proof of the convergence of the N −particle Schrödinger dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in , up to an exponentially small remainder. For = 0, the classical dynamics in the mean-field limit is given by the Vlasov equation. 1. Introduction and Statement of Results Consider the Schrödinger operator H N = H N0 + VN , H N0 = −
N i=1
2 1 i , VN = 2 N
(1.1) N
v(xi − x j ),
(1.2)
i< j
where v is a two-body potential independent of N . The operator H N acts on H(N ) := L 2S (R3N ), the totally symmetric part of L 2 (R3N ), which is the Hilbert space of pure state vectors for a system of N nonrelativistic bosons. We propose to study the dynamics described by the N −body Schrödinger equation i∂t N (t) = H N N (t),
(1.3)
for an inital condition N (t = 0) = N ,0 ∈ L 2S (R3N ). Under assumptions specified below, H N , defined on the symmetrized Sobolev space HS2 (R3N ), is a self-adjoint operator. Hence the unitary group U N (t) = e−i HN t/, t ∈ R, exists. Let p ≤ N , and let a ( p) ( p) be a bounded operator on L 2S (R3 p ). It defines an operator A N acting on H(N ) in the following way:
682
J. Fröhlich, S. Graffi, S. Schwarz
N (N − 1) · · · (N − p + 1) (PS a ( p) ⊗ I (N − p) PS )(x1 , . . . , x N), Np (1.4) (x1 , . . . , x N ) ∈ L 2S (R3N ), ( p)
(A N )(x1 , . . . , x N ) =
where PS is the projection onto the symmetric subspace L 2S (R3N ) of L 2 (R3N ). The ( p) operator A N may be viewed as an operator acting on p particles; the numerator on the right side of (1.4) is a combinatorial factor motivated by “second quantization”; the denominator is the correct scaling factor to take the N → ∞ limit. We are interested in the asymptotics of certain expectation values of the Heisenberg( p) picture operators ei HN t/ A N e−i HN t/, as N → ∞. If H N is chosen as in (1.1), (1.2), ( p) and A N is chosen as in (1.4), the limit N → ∞ is the usual mean-field limit; see [He, Sp1]. Our first main result is the following Theorem 1.1. Let > 0 and t ≥ 0 be fixed, and let v ∈ L ∞ (R3 ). If N ,0 (x1 , . . . , x N ) = ψ(x1 ) · · · ψ(x N ) is a normalized “coherent” (i.e., product) initial state, then ( p)
lim N ,0 , ei HN t/ A N e−i HN t/ N ,0 =
(1.5)
N →∞
( p)
lim N ,t , A N N ,t = p,t , a ( p) p,t =: a ( p) (ψt ).
N →∞
Here N ,t is again a coherent state, i.e., N ,t (x1 , . . . , x N ) = ψt (x1 ) · · · ψt (x N ), and p,t = N = p,t , where ψt is a solution of the Hartree equation i∂t ψt = −
2 ψt + (v ∗ |ψt |2 )ψt 2
(1.6)
with initial condition ψt=0 = ψ. Remark. 1. For large N , the quantum evolution e−i HN t/ N ,0 can be replaced by the nonlinear single-particle evolution N ,t (x1 , . . . , x N ). Particle interaction effects are translated into the nonlinearity of this evolution. This justifies interpreting the limit N → ∞ as a mean-field limit. 2. For sufficiently short times the corrections to the limit in (1.5) are O(1/N ) . Theorem 1.1 can be extended to yield a systematic expansion in powers of 1/N that converges for small enough |t|. 3. Since lim N (N − 1) · · · (N − p + 1)/N p = 1, the second equality in (1.5) follows N →∞
easily from (1.4), because ( p)
2(N − p)
lim N ,t , A N N ,t = p,t , a ( p) p,t L 2 (R3 p ) ψ L 2 (R3 ) .
N →∞
4. Theorem 1.1 was first proven in [He], see also [GiVe]. A new proof was given in [Sp1] and extended to more general classes of two-body potentials, including the Coulomb potential, in [EY, BGM, BEGMY]. The proof in our paper is quite different. It enables us to tackle the problem of obtaining convergence estimates uniform in Planck’s constant , as we now proceed to discuss.
Mean-Field- and Classical Limit of Many-Body Schrödinger Dynamics for Bosons
683
It is well known that, for VN ∈ C 2 , the classical dynamics of N particles tends to the dynamics defined by the Vlasov equation, in the limit N → ∞. More precisely, if ρ N denotes the empirical distribution, namely ρ N (d x, dξ ; t) =
N 1 δ(x − xi (t))δ(ξ − ξi (t)) d xdξ, N i=1
where (x1 (t), . . . , x N t); ξ1 (t), . . . , ξ N (t)) is a solution of the classical equations of motion, then, in the limit N → ∞, ρ N tends weakly to f t (x, ξ )d xdξ , where f t (x, ξ ) is a solution of the Vlasov equation: ∂t f t = −ξ · ∇x f t + ∇x Ve f f · ∇ξ f t , Ve f f (x, t) = v(x − y) f t (y, ξ )dydξ ,
(1.7) (1.8)
see [BH]. It is natural to ask whether this convergence result is related to that of Theorem 1.1. Our next result provides, under very restrictive assumptions on the two-body interactions, a partial answer to this question. First, we define a restricted class of interactions. For σ > 0, p = 1, 2, . . . we define the spaces L 1σ, p := { f ∈ L 1 (R6 p ) | eσ |z| f ∈ L 1 (R6 p )},
(1.9)
f ∈ L 1 (R6 p )}.
(1.10)
Aσ, p := { f ∈ L (R ) | e 1
6p
σ |s|
We denote
f σ, p := eσ |s| f L 1 =
R6 p
|eσ |s| f (s)| ds .
Here x j ∈ R3 , ξ j ∈ R3 , j = 1, . . . , p, and z := (X p , p ) ∈ R3 p × R3 p ; X p := (x1 , . . . , x p ), p := (ξ1 , . . . , ξ p ), |z| :=
p (|x j | + |ξ j |); j=1
f (s), s := (S, ) ∈ R3 p × R3 p is the Fourier transform of f . We further denote by tN : (X N ; N ) → (X N (t); N (t)) the flow generated by H Nc , where H Nc is the classical Hamilton function corresponding to the operator H N . Definition 1.1. We define by: 1. (1.11) W N N (X N , N ; , t) = (2π )−3N eiY N , N N (X N + Y N /2, t) N (X N − Y N /2, t) dY N , R3N
the Wigner distribution of the N -particle normalized wave function N (X N , t);
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2. N W j (X j , j ; , t) =
R3(N − j)
W N N (X N , N ; t) d X N − j d N − j ,
(1.12)
the j−particle Wigner function ((N − j)-marginal distribution of the N -particle Wigner distribution); 3. W(ψ)(x, ξ ; t) = (2π )−3
R3
eiy,ξ ψt (x + y/2)ψt (x − y/2) dy, (1.13)
the Wigner distribution of the solution ψt (x) of the Hartree equation. Consider now a velocity-dependent two-body potential v(x, ξ ) : R6 → R. Our second main result is Theorem 1.2. Let v ∈ Aσ,1 , for some σ > 0. Let N ,0 be a product state. Then, for fixed p and t ≥ 0, there is a constant C p > 0 independent of such that, as an equality between tempered distributions, W p N (X p , p ; , t) =
p
W(ψ)(x j , ξ j ; t) +
j=1
√ Cp + O e−1/ v σ,1 t , (1.14) N
as N → ∞. Remark. 1. Equation. (1.14) shows that, up to an exponentially small error independent of , the mean-field convergence towards a single-particle nonlinear dynamics holds uniformly in . The dependence of the potential on the velocity is here a critical assumption. The extension of this result to the N -body Schrödinger operators considered in Theorem 1.1 appears to be out of reach of the semiclassical microlocal analysis techniques employed in this paper. 2. It is known since some time that W(ψ)(x, ξ ; t) converges in S (R6 ) to a solution f t (x, ξ ) of the Vlasov equation, as → 0 (see e.g. [NS, Sp2, LP]). Under the additional scaling assumption = N −1/3 one also has [NS]: lim W p N (X p , p ; , t) =
N →∞
p
f t (x j , ξ j )
(1.15)
j=1
if v is a local two-body potential with Fourier transform vˆ of compact support. Moreover, in the case of the Kac potentials, (1.15) holds whenever N → ∞ entails → 0 [GMP]. Under the above scaling assumption = N −1/3 , the same result has been proved also for fermions [EESY]. More precisely, the validity of (1.14) without the error term has been proved for the corresponding Husimi functions. 3. The classical limit is equivalent to the limit of heavy particles. We set = 1 in (1.2), but let the particle mass m become large. We impose the condition that the kinetic √ energy per particle be independent of m, namely mvi2 = O(1), i.e., |vi | = O(1/ m), √ for all i. This suggests to rescale time as t = mτ . Then the Schrödinger equation becomes N N j i 1 − v(xi − x j ) N , N + √ ∂τ N = 2m N m j=1
i, j=1
√ which is equivalent to (1.1)–(1.3), for = 1/ m.
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2. The N → ∞ Limit: Convergence Estimates 2.1. Kinematical algebra of “observables”. The above systems can be described by a kinematical algebra of operators, the quantum mechanical analogue of the algebra of functions on phase space of a classical system. Let H( p) := L 2S (R3 p ), 0 < p < N , N ∈ N. Let a ( p) be a bounded operator on H( p) , and α ( p) (x1 , . . . , x p ; y1 , . . . , y p ) := α ( p) (X p ; Y p ) the tempered distribution kernel in S (R3 p × R3 p ) associated to a ( p) by the nuclear theorem: (a ( p) ϕ ( p) )(X p ) = α ( p) (X p ; Y p )ϕ ( p) (Y p ) dY p , (2.1) Rp
where ϕ ( p) (Y p ) ∈ L 2S (R3 p ). Then (a ( p) )∗ has the distribution kernel α ( p) (Y p ; X p ). To ( p) ( p) a ( p) we associate the operator A N on L 2S (R3N ) specified in (1.4). The operators A N span the algebra of all bounded operators on H(N ) . ( p) ( p) ( p) ( p) (N ) Ifa is bounded on H then A N (a ) is bounded on H . Since PS = 1 and N p! ≤ 1, we have that p Np ( p)
A N ϕ (N ) 2H(N ) ≤ a ( p) 2H( p) |ϕ (N ) (Y p ; X N − p )|2 dY p d X N − p = a
( p) 2
H( p) ϕ
R3(N − p)
(N ) 2
R3 p
H(N ) .
The following statement is easily verified. ( p)
Proposition 2.1. The map a ( p) → A N (a ( p) ) is linear, and ( p)
( p)
(A N (a ( p) ))∗ = A N ((a ( p) )∗ ), ( p)
A N (a ( p) ) B(H(N ) )
≤ a
( p)
(2.2)
B(H( p) ) .
(2.3) ( p)
( p)
2.2. The Schwinger-Dyson expansion. Given an operator A N = A N (a ( p) ), with ( p) a ( p) ∈ B(H( p) ), p ≤ N , we denote by At,N the corresponding Heisenberg-picture operator with respect to the free time evolution ei HN t/, i.e., 0
( p)
( p)
At,N = ei HN t/ A N e−i HN t/. 0
0
(2.4)
( p)
We further denote by A N (t) the corresponding operator with H N0 replaced by H N , namely ( p)
( p)
A N (t) := ei HN t/ A N e−i HN t/.
(2.5)
( p)
and by A I,N (t, s) the two-parameter operator family ( p)
( p)
A I,N (t, s) := ei HN t/e−i HN t/ As,N ei HN t/e−i HN t/. 0
0
(2.6)
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J. Fröhlich, S. Graffi, S. Schwarz
Then we obviously have ( p) ( p) A N (t) = A I,N (t, s)
s=t
.
(2.7)
We denote VN by V . Iterating the identity: i t i HN t1 / −i H 0 t1 / 0 ( p) ( p) ( p) N e e [Vt1 , As,N ]ei HN t1 /e−i HN t1 / dt1 (2.8) A I,N (t, s) = As,N + 0 we get that ( p)
( p)
A I,N (t, s) = As,N +
∞ n i n=1
0
t
dt1 . . . 0
tn−1
( p)
dtn [Vtn , . . . , [Vt1 , As,N ] . . .], (2.9)
and finally, setting s = t, we obtain the Schwinger-Dyson expansion ( p) A N (t)
=
( p) At,N
tn−1 ∞ i n t t1 ( p) ) + ··· [Vtn , . . . , [Vt1 , At,N ] . . .] dtn . . . dt1 . 0 0 0 n=1
(2.10) From now on, we drop the index N in the Heisenberg-picture operators with respect to ( p) ( p) the free evolution, i.e., we use the abbreviation: At,N := At . ( p)
The boundedness of A N and of the interactions Vti implies the boundedness of all multiple commutators, with 1 ( p) ( p)
[Vtn , . . . , [Vt1 , At ] . . .] H(N ) ≤ (2 V H(N ) /)n A N H(N ) n ≤ (2 V H(N ) /)n a ( p) H( p) , ( p)
( p)
for A N = A N (a ( p) ). By (1.2), V H(N ) ∝ N . Hence, for fixed N and , the series is tn norm-convergent, for all t ≥ 0. The time integrations yield a factor , so that the norm n! of the series in (2.10) is bounded by exp [2 V H(N ) |t|/] · a ( p) H( p) . These estimates are obviously not adequate to investigate the N → ∞ or the → 0 limit, let alone to prove uniformity in . 2.3. The N → ∞ limit. We exploit the structure of the commutators on the right-hand side of (2.10), the symmetry of wave functions in L 2S (R3N ), and the fact that each term ( p) in A N only acts on p arguments of a wave function, so that many commutators will vanish. Note that Vt =
N N 1 −i H 0 t/ i j i H 0 t/ 1 ij N e v e N = vt , N N i< j
(2.11)
i< j
where vt = ei(i + j )t /2 v i j e−i(i + j )t /2 , v i j = v(xi − x j ). ij
(2.12)
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Therefore N 1 i j ( p) [vs , A N ] = N
( p)
[Vs , A N ] =
i< j
p p N 1 ( p+1) i j ( p) 1 ( p) i j ( p) A N ([vs , a ]) + A N ([vs , a ]) = N N
=
i=1 j= p+1 p
N−p N
=
i< j
( p+1)
AN
i( p+1)
([vs
, a ( p) ]) +
i=1
p 1 ( p) i j ( p) A N ([vs , a ]). N
(2.13)
i< j
In more precise terms, the expression p p N − p ( p+1) i( p+1) ( p) 1 ( p) i j ( p) A N ([vs , a ]) + A N ([vs , a ]), N N
( p)
[Vs , A N (a ( p) )] =
i=1
i< j
(2.14) holds as an operator identity on H(N ) . In second-quantization language, the first sum in (2.14) corresponds to tree graphs, the second one to loop graphs. Next, we insert (2.14) in (2.8) and perform a second step, but only for the first sum in (2.14), leaving the second one unchanged. To keep our notation compact, it is useful to introduce the notion of tree amplitudes of n th order, recursively defined in the following way: g
(0, p)
=
( p) at ;
(n; p) gt1 ,...,tn
p+n−1 i i( p+n) (n−1; p) = [vtn , gt1 ,...,tn−1 ], n ≥ 1.
(2.15)
i=1
Then expression (2.14) becomes i N − p ( p+1) (1; p) i ( p) i j (0; p) ( p) [Vs , A N ] = A N (gs )+ A N ([vs , g ]). N N p
i< j
The first term is O(1), while the second one is of order p( p − 1)/N (for fixed ) and is therefore suppressed by a factor 1/N . Performing (k − 1) iterations only for the tree amplitudes, we conclude that ( p)
( p)
( p),k
ei HN t/ A N e−i HN t/ = At,N + Bt,N +
k 1 ( p),n ( p),k Q t,N + Rt,N , N
(2.16)
n=1
where ( p),k Bt,N
=
( p),n
Q t,N = ( p),k
Rt,N =
k−1 n=1 t
t
tn−1
(n; p)
(gt1 ,...,tn ) dtn . . . dt1 ,
(2.17)
ei HN tn /e−i HN tn / H (n − 1; p; N )ei HN tn /e−i HN tn / dtn . . . dt1 , 0
0
0
0 t 0
( p+n)
AN
0
0
...
tn−1
...
...
(2.18) tk−1
ei HN tk /e−i HN tk / H (k, p; N )ei HN tk /e−i HN tk / dtk . . . dt1 , 0
0
0
(2.19)
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J. Fröhlich, S. Graffi, S. Schwarz
with H (s, p; N ) :=
p+s−1
( p+s−1)
AN
(s; p)
ij
([vts , gt1 ,...,ts−1 ]).
(2.20)
i< j ( p+n−1)
Equation (2.16) is most easily verified as follows (think of A N ( p)
n−1, p
(gt1 ,...,tn−1 ) as a
p + n − 1-particle operator replacing At,N in (2.14) ): ( p+n−1)
i
[Vtn , A N
n−1, p
(gt1 ,...,tn−1 )]
1 N 1 N
p+n−1
p+n−1 (N − ( p + n − 1)) ( p+n) i p+n n−1, p = i A N ([vtn , gt1 ,...,tn−1 ])+ N i=1
( p+n)
([vtn , gt1 ,...,tn−1 ]) = A N
( p+n)
([vtn , gt1 ,...,tn−1 ]).
i AN
( p+n)
ij
n−1, p
ij
(n−1; p)
n, p
(gt1 ,...,tn )+
j>i=1 p+n−1
i AN
j>i=1
2.4. Control of the expansion, small time, fixed. First, we prove a bound on the norm ( p+n) (n; p) of A N (gt1 ,...,tn ) ( p+n)
A N
(n; p)
(gt1 ,...,tn ) H(N ) ≤
2 ( p+n−1) (n−1; p) ( p + n − 1) v ∞ A N (gt1 ,...,tn−1 ) H(N ) . (2.21)
This follows from the unitarity of the free time evolution and the boundedness of the interactions, v i j = v ∞ . The bound (2.21) then yields recursively n 2 ( p+n) (n; p) ( p)
v ∞ A N H(N )
A N (gt1 ,...,tn ) H(N ) ≤ ( p + n − 1)( p + n − 2) · · · ( p + 1) p n ( p + n)! 2
v ∞ a ( p) H( p) , ≤ (2.22) p! independently of all time indices. Considering the expansion (2.16), we have that
∞
tn−1
t
( p+n) (n; p) ... A N (gt1 ,...,tn ) dtn · · · dt1
0 0
≤
H(N )
n=1
n ∞ n ∞ 4 t ( p + n)! 2 ( p) ( v ∞ t)n A N H(N ) ≤ 2 p a ( p) H( p)
v ∞ t n! p! n=1
(2.23)
n=1
−1 4 ( p + n)! n+ p ≤2 . The series on the R.S. of (2.23) converges for |t|<
v ∞ . because p!n! The third term in (2.16) is bounded similarly. Let ( p+n)
( p+n)
A N ,I (i, j) := ei HN tn /e−i HN tn / A N 0
(n−1; p)
([gt1 ,...,tn−1 , vtn ])ei HN tn /e−i HN tn /. (2.24) ij
0
Mean-Field- and Classical Limit of Many-Body Schrödinger Dynamics for Bosons
689
Then
tn−1 ∞ p+n−1 t
1
( p+n)
··· A N ,I (i, j) dtn · · · dt1
(N ) ≤ N 0 0 H n=1
i< j=1
tn−1 t ∞ 1 ( p + n − 1)2 2 ( p+n−1) (n−1; p)
v ∞ ...
A N (gt1 ,...,tn−1 ) H(N ) dtn . . . dt1 ≤ N 2 0 0 n=1 n−1 ∞ 2 1 2 |t|n ≤ ( p + n − 1)2 ( p + n − 2) · · · p
v ∞
v ∞ A( p) H(N ) ≤ N n! n=1 n ∞ ∞
a ( p) H( p) p 4
a ( p) H( p) ( p + n)! 2
v ∞ t 2 ( ≤ ( v ∞ |t|)n . ≤ N n! p! N n=1
n=1
Therefore
k ∞
1
a ( p) H( p) p 4 ( p),n
2 Q ≤ ( v ∞ |t|)n . t,N
N N (N ) H n=1
n=1
The remainder term in (2.16) clearly vanishes, as k → ∞. To summarize, we have proven the following result. −1 4
v ∞ . Then Proposition 2.2. Let |t| < ( p)
( p)
ei HN t/ A N e−i HN t/ = At
+
∞ n=1 0
t
tn−1
... 0
( p+n)
AN
(n; p)
(gt1 ,...,tn ) dtn . . . dt1 + O(1/N ). (2.25)
2.5. Convergence for all times, fixed. We assume that the statement of Theorem 1.1 holds up to some time T independent of p, i.e., ( p)
lim N ,0 , ei HN T / A N e−i HN T / N ,0 H(N ) = a ( p) (ψ p,T ).
N →∞
(2.26)
Let us proceed one step further in time, with t < (4 v ∞ /)−1 . On account of (2.16), we have that ( p)
ei HN (T +t)/ A N e−i HN (T +t)/ = ( p)
( p)
(2.27) ei HN T /e−i HN t/ A N e−i HN t/e−i HN T / = ei HN T / A N (t)e−i HN T / N t tn−1 ( p+n) (n; p) ... ei HN T / A N (gt1 ,...,tn )e−i HN T / dtn . . . dt1 + O(1/N ). + n=1
0
0
690
J. Fröhlich, S. Graffi, S. Schwarz
This expansion is norm convergent, by (2.16) and the unitarity of ei HN T /. Taking expectation values in N ,0 we get, as above, ( p)
N ,0 , ei HN (T +t)/ A N e−i HN (T +t)/ N ,0 H(N ) = ( p)
= N ,0 , ei HN T / At e−i HN T / N ,0 H(N ) + tn−1 N t ( p+n) (n; p) ... N ,0 , ei HN T / A N (gt1 ,...,tn )e−i HN T / N ,0 H(N ) dtn . . . dt1 n=1
0
0
+ O(1/N ). Hence, by the inductive assumption and the norm convergence of the series ( p)
lim N ,0 , ei HN (T +t)/ A N e−i HN (T +t)/ N ,0 H(N ) = a ( p) (eit/ψT )+
N →∞ ∞
t
lim
n=1
N →∞
...
tn−1
0
0
( p+n)
N ,0 , ei HN T / A N
(n; p)
(gt1 ,...,tn )e−i HN T / N ,0 H(N ) dtn . . . dt1 .
We postpone to Sect. 3, below, the proof that actually a ( p) (eit/ψT )+ ∞ t lim ... N →∞
n=1
0
0
tn−1
( p+n)
N ,T , A N
(n; p)
(gt1 ,...,tn ) N ,T H(N ) dtn . . . dt1
(2.28)
= a ( p) (ψt+T ), and this ensures that the convergence is global in time. 2.6. Control of the expansion, uniformity with respect to . In this subsection, we set (X p , p ) =: (x, ξ ). Given a symbol τ (x, ξ ) ∈ Aσ, p , we denote by T the corresponding Weyl operator. Its action on vectors ψ ∈ S(R3 p ) is given by (T ψ)(x) =
1 τ ((x + y)/2, ξ )ei(x−y),ξ /ψ(y) dydξ. 3 p R 3 p R 3 p
(2.29)
In general, T is a semiclassical pseudodifferential operator. Let us recall some relevant results (see e.g. [Ro]). 1. If τ ∈ L 1 (R3 p × R3 p ) then T extends to a continuous operator on L 2 (R3 p ) with
T ≤ τ L 1 ; hence T L 2 →L 2 ≤ T σ := τ σ , where τ is the Fourier transform of τ , and |τˆ (s)|eσ |s| ds = τ σ, p .
τ σ := R6 p
Obviously, τ L 1 ≤ τ σ . We have dropped here the index p to simplify the notation.
Mean-Field- and Classical Limit of Many-Body Schrödinger Dynamics for Bosons
691
2. If v ∈ Aσ, p , g ∈ Aσ, p , then the symbol of the commutator [V, G]/i is the Moyal bracket {v, g} M . Hence the multiple commutator [V, [V, . . . [V, G] . . .]/(i)n has as its symbol the multiple Moyal bracket {v, {v, . . . , {v, g} M . . .} M . We recall that, given g, g ∈ Aσ, p , their Moyal bracket {g, g } M is defined as {g, g } M = g#g − g #g, where # is the composition of Weyl symbols. In the Fourier transform representation the explicit expression of the Moyal bracket is (see e.g. [Fo],§3.4): 2 ∧ g (s 1 )g (s − s 1 ) sin (s − s 1 ) ∧ s 1 /2 ds 1 , (2.30) ({g, g } M ) (s) = R2n where, given two vectors s = (v, w) and s 1 = (v 1 , w 1 ), s ∧ s 1 := w, v1 − v, w1 . 3. If the observable T has symbol τ (x, ξ ), then the Heisenberg observable Tt has symbol (τ ◦ 0t )(x, ξ ). Here 0t (x, ξ ) = (x + ξ t, ξ ) is the free flow with initial conditions (x, ξ ). In particular, (τ ◦ 0t )(x, ξ ) ∈ Aσ, p whenever τ ∈ Aσ, p . Under the present assumptions, it can be proven, starting from the expression (2.30) (see [BGP], Lemma 3.2 ), that the following estimate on the Moyal bracket holds:
{g, g } M σ −δ ≤
1 e2 δ 2
g σ g σ , 0 < δ < σ.
(2.31)
For the convenience of the reader we reproduce here the proof of (2.31). Since (s − s 1 ) ∧ s 1 = s ∧ s 1 , and |s ∧ s 1 | ≤ |s| · |s 1 |, by definition of the Aσ -norm and (2.30) we get:
({g, g } M ) σ −δ = 2 e(σ −δ)|s| ds |g(s ˆ 1 )gˆ (s − s 1 ) sin ((s − s 1 ) ∧ s 1 )/2| ds 1 R6 p R6 p 2 1 ≤ ds e(σ −δ)(|s|+|s |) |g(s) ˆ gˆ (s 1 ) sin ((s ∧ s 1 )/2| ds 1 R6 p R6 p 1 ≤ e(σ −δ)|s| |g(s)| ˆ ds e(σ −δ)|s | |gˆ (s 1 )s ∧ s 1 | ds 1 6 p 6 p R R 1 (σ −δ)|s| ≤ e |g(s)||s| ˆ ds e(σ −δ)|s | |gˆ (s 1 )||s 1 | ds 1 , R6 p
R6 p
1 , ∀ x > 0, ∀δ > 0. eδ Let now gr := {gr −1 , v} M , r > 1; g1 = {g, v} M . Then, applying (2.31) r times, we can write: 1 r
v rσ g σ . (2.32)
gr σ −r δ ≤ e2 δ 2 whence the assertion because xe−δx ≤
These results immediately yield the following bound. Lemma 2.1. Let the operator a ( p) be the Weyl quantization of a symbol τa (x, ξ ) ∈ Aσ, p for some σ > 0. Then there is L( p) > 0 independent of such that ( p+n)
A N
(n; p)
(gt1 ,...,tn ) H(N ) ≤ L n n!3 (2 v σ )n a ( p) σ .
(2.33)
692
J. Fröhlich, S. Graffi, S. Schwarz (n, p)
(n; p)
Proof. Denote by Gt1 ,...,tn the symbol of gt1 ,...,tn . Using definition (2.15) and the estimate (2.31) we get the uniform estimate corresponding to (2.22): (n, p)
Gt1 ,...,tn ;N σ −nδn ≤
2( p + n − 1) (n−1, p)
v σ Gt1 ,...,tn−1 σ , 0 < δn < σ. e2 δn2
(2.34)
The recursive definition (2.15) allows us to use the recursive estimate (2.32). We get (n, p)
Gt1 ,...,tn σ −nδn ≤ 2n (e2 δn2 )−n Setting δn :=
( p + n)!
v nσ a ( p) σ . p!
(2.35)
1 we get the bound (2.33) on account of the majorizations 2n ( p+n)
A N
(n; p)
(n, p)
(gt1 ,...,tn ) H(N ) ≤ Gt1 ,...,tn σ/2 , v L 2 →L 2 ≤ v σ .
This proves the lemma.
Remark. The uniform control in introduces an extra n!2 divergence with respect to the fixed- estimate (2.22). We now obtain uniform estimates of the three terms in expansion (2.16). Lemma 2.2. There exist constants M1 > 0, M2 > 0, M3 > 0, L 1 > 0, L 2 > 0, L 3 > 0, independent of (, t) and N , such that ( p),k
Bt,N H(N ) ≤ M1 a ( p) σ
k
L n1 n!2 ( v σ t)n ,
(2.36)
n=1 ( p),n
Q t,N H(N ) ≤ M3 a ( p) σ L n2 n!2 ( v σ t)n ,
(2.37)
( p),k
Rt,N H(N )
(2.38)
≤ M3 a
( p)
σ L k3 k!2 ( v σ
t) . k
Proof. Inserting the estimate (2.33) in the expressions (2.17, 2.18, 2.19) we get, on account of unitarity of U0 (t): ( p),k
Bt,N H(N ) ≤ a ( p) σ
k
(2L v σ )n n!3
n=1
≤ a ( p) σ
k
t
0
tn−1
...
dtn . . . dt1
0
(2L v σ |t|)n n!2 .
n=1
The last inequality comes from performing the time integrations, which are majorized by a factor |t|n /n! in (2.36, 2.37) and by a factor |t|k /k! in (2.38) (proven with the help of the same argument). This proves the lemma. Using this result, we can easily prove the uniform version of the expansion (2.16). Proposition 2.3. Let := v σ t. Then, under the same assumption on the operator a ( p) made in Lemma 2.1, there exists k = k(), = () such that ( p)
( p)
( p),k
( p),k
ei HN t/ A N e−i HN t = At,N + Bt,N + Rt,N +
, N
(2.39)
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693
where ( p),k Bt,N
=
k
t
...
tn−1
( p+n)
AN
0
n=1 0
(n; p)
(gt1 ,...,tn ) dtn . . . dt1 .
(2.40)
( p),k
Here Bt,N fulfills the majorization (2.36), and √
( p),k
Rt,N H(N ) ≤ M3 e−L 3 /
.
(2.41)
Proof. The estimate (2.37) and a standard Nekhoroshev-type argument show that the choice 1 1 (2.42) k() := √ = √
v σ t ( p),k
minimizes the divergence of Rt,N . A straightforward computation then yields (2.41). ( p),N
By definition of Q t,N
we get the uniform version of the estimate (2.21), whence
() := p2 p a ( p) σ
k()
√
n!2 (2)n ≤ p2 p −1/2 (e/2)−1/
.
n=1
3. Connection with the Hartree Equation and Proof of the Theorems We wish to prove that the representation of the evolution obtained in Proposition 2.2 coincides with the evolution generated by the Hartree equation in the limit N → ∞. For this purpose, we recall that the Hartree equation is Hamiltonian. We define the functional 2 H(ψ, ψ) = − |∇ψ(x)|2 d x + V(ψ, ψ). (3.1) 2 R3 for ψ ∈ H 1 (R3 ), where V(ψ, ψ) =
1 ψ(x)ψ(y)v(x − y)ψ(x)ψ(y) d xd y. 2 R 3 ×R 3
(3.2)
If ψ(x), ψ(y) are considered as canonical variables with Poisson brackets {ψ(x), ψ(y))} = iδ(x − y), {ψ(x), ψ(y)} = {ψ(x), ψ(y)} = 0, then (3.1) is the Hamiltonian functional generating a time evolution of functionals on phase space equivalent to the Hartree equation. Namely, if A(ψ) is a functional and A(t) denotes its time evolution, one has that ∂t A(t, ψ) =
1 {H, A(t)}(ψ).
Choosing A = φ, ψ, φ ∈ C0∞ (R3 ), then A(t, ψ) = A(ψt ), where ψt is a solution of the Hartree equation i∂t ψt = −
2 ψt + (v ∗ |ψt |2 )ψt . 2
(3.3)
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Define the free flow 0t (A) := At of A by At = A(eit/ψ), and denote by t (A) the interacting flow. Formally, the interacting flow is given by the Lie expansion in the interaction representation (analogous to the Schwinger-Dyson expansion of Sect. 2.2). Indeed we have the following result: Lemma 3.3. t (A) admits the formal expansion t (A) = At +
∞ n 1 n=1
t
tn
...
0
0
{Vtn . . . {Vt1 , At } . . .} dtn . . . dt1 .
(3.4)
Proof. To see this, we consider the dynamics in the interaction picture. We set A˜ t := t ◦ 0−t (A). Then ∂t A˜ t =
{Vt , A˜ t } ,
where Vt := 0t (V) is the free evolution of V. After integrating in time we get A˜ t = A +
t 0
{Vs , A˜ s } ds.
Noticing that A(t) = t (A) is given by t ◦ 0−t (At ), and iterating the above identity, we obtain the series (3.4), and this concludes the proof of the lemma. The desired identification is based on the following proposition: Proposition 3.4. . Let ψ ∈ H 1 (R3 ), and let be a product state, i.e. (x1 , . . . , xl ) =
l
ψ(xs ).
s=1
Then, for all N ≥ p, ( p + n) p+n (n; p) ( p+n) (n; p) n+ p , A p+n (gt1 ,...,tn )n+ p H(n+ p) = gt1 ,...,tn (ψ) := ( p + n)! n 1 {Vtn . . . {Vt1 , A} . . .}(ψ), where A = a ( p) (ψ) :=
ψ(x1 ) · · · ψ(x p )α ( p) (x1 , . . . , x p ; y1 , . . . , y p )ψ(y1 ) · · · ψ(y p )
p k=1
d xk dyk .
(3.5)
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Proof. We have that
t ◦ 0−t ({Vt , A}/)
t=0
˜ t=0 . = {V, A}/ = ∂t A|
Define the projection ρ := |ψψ|. Then, denoting ρ˜t := t ◦ 0−t (ρ), we have that, for all n ≥ 1, ˜ t=0 = ∂t Tr(Aρ˜t⊗n )|t=0 . A˜ t = Tr(Aρ˜t⊗n ); ∂t A| Therefore, in the interaction picture ∂t ρ˜t = Tr(ρ˜t⊗2 Vt12 − Vt12 ρ˜t⊗2 )/, and in the same way we get 1 {Vt , A} = Tr((Vtin+1 A − AVtin+1 )ρ ⊗ n+1 ). n
i=1
It is then easy to check that 1 (n; p) {Vt . . . {Vt1 , A} . . .}(ψ) = Tr(gt1 ,...,tn · ρ ⊗ n+ p ) n n ( p + n) p+n ( p+n) (n; p) n+ p , A p+n (gt1 ,...,tn )n+ p H(n+ p) = ( p + n)! (n; p)
= gt1 ,...,tn (ψ), and this concludes the proof of the proposition. We are now in a position to prove our main results. Proof of Theorem 1.1. Consider the expectation value of the expansion (2.25) in a coherent (i.e., product) state: ( p)
( p)
N , ei HN t/ A N e−i HN t/ N H(N ) = N , At,N N H(N ) + tn−1 N t ( p+n) (n; p) ... N , A N (gt1 ,...,tn ) N H(N ) dtn · · · dt1 + O(1/N ). 0
n=1 0
By definition of N , Np ( p) N , ei HN t/ A N e−i HN t/ N H(N ) = p , at p H( p) + N (N − 1) . . . (N − p + 1) tn−1 ∞ t (n; p) ... n+ p , gt1 ,...,tn n+ p H(n+ p) dtn · · · dt1 + O(1/N ). + n=1 0
0
Since the series is norm-convergent, the limits N → ∞ and n → ∞ can be interchanged. Then ( p)
lim N , ei HN t/ A N e−i HN t/ N H(N ) = p , at p H( p) + tn−1 ∞ t (n; p) ... n+ p , gt1 ,...,tn n+ p H(n+ p) dtn · · · dt1 = a ( p) (ψt ),
N →∞
n=1 0
0
where the last equality follows from formula (3.5).
(3.6)
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Proof of Theorem 1.2. If, instead of (2.25), the representation (2.39) is considered, the above argument yields Np ( p) ( p) k, p N , ei H N t/ A N e−i H N t/ N = p , at p + N , Rt,N ψ N N (N − 1) . . . (N − p + 1) t k() √ t n−1 (n; p) + ... n+ p , gt1 ,...,tn n+ p dtn · · · dt1 = a ( p) (ψt ) + O(e−1/ ).
(3.7)
0
n=1 0
Given any bounded operator A on L 2 (R3l ) with (Weyl) symbol σ A (x, ξ ) : S(R6l ) → R, where S is the Schwartz space of rapidly decreasing functions, its matrix elements can be expressed in terms of the symbol and of the Wigner function by the following well known formula (see e.g.[Fo]): , A = σ A (x, ξ )W (x, ξ ) d xdξ, (3.8) R3l ×R3l
where W (x, ξ ) is the Wigner function of the state . Therefore, in our case Np N , A N (t) N = σ A (X p , p )W N N (X p , p , t) d X p d p , N (N − 1) . . . (N − p + 1) R3 p ×R3 p
where W N N (X p , p , t) is the Wigner function corresponding to the time evolution, ei HN t/ N , of the product state N ,0 = ψ(x1 ) . . . ψ(x N ). The N − p variables (X N − p , N − p ) are integrated out. By (3.7) and (1.13), we can take the N → ∞ limit and write σ A (X p , p )W N N (X p , p , t) d X p d p = R 3 p ×R 3 p
R 3 p ×R 3 p
σ A (X p , p )
p
√
Wψ (xl , ξl ; t) d X p d p + O(e−1/
).
l=1
Since this formula holds for any σ A (X p , p ) ∈ S(R3 p × R3 p ) ∩ Aσ, p , the assertion is proved. Proof of formula (2.28). By (3.5), we have that ( p+n)
(n; p)
N , ei HN T / A N (gt1 ,...,tn )e−i HN T / N H(N ) = n 1 {Vtn . . . {Vt1 , A} . . .}(ψT ), which yields formula (2.28), by Lemma 3.3, on account of the uniform convergence of the series. References [BGM]
Bardos, C., Golse, F., Mauser, N.: Weak coupling limit of the n-particle Schrödinger equation. Methods Appl. Anal. 2, 275–293 (2000) [BEGMY] Bardos, C., Erdös, L., Golse, F., Mauser, N., Yau, H-T.: Derivation of the Schrödinger-Poisson equation from the quantum N-body problem. C.R. Acad. Sci. Paris 334, 515–520 (2002) [BGP] Bambusi, D., Graffi, S., Paul, T.: Normal forms estimates and quantization formulae. Commun. Math. Phys. 207, 173–195 (1999)
Mean-Field- and Classical Limit of Many-Body Schrödinger Dynamics for Bosons
[BH] [Do] [EESY] [EY] [Fo] [GMP] [GiVe] [He] [LP] [NS] [Ro] [Sp1] [Sp2]
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Braun, W., Hepp, K.: The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Commun. Math. Phys. 56, 101–113 (1977) Dobrushin R., L.: Vlasov equations. Sov. J. Funct. An. 13, 115–119 (1979) Elgart, A., Erdös, L., Schlein, B., Yau, H.-T.: Nonlinear Hartree equation as the mean field limit of weakly coupled fermions. J. Math. Pures Appl. 83, 1241–1273 (2004) Erdos, L., Yau, H.-T.: Derivation of the nonlinear Schrödinger equation from a many body coulomb system. Adv. Theor. Math. Phys. 5, 1169–2005 (2001) Folland G.: Harmonic Analysis in Phase Space. Princeton, NJ: Princeton University Press, 1989 Graffi, S., Martinez, A., Pulvirenti, M.: Mean field approximation of quantum systems and classical limit. Math. Meth. Models in Appl. Sci. 13, 55–63 (2003) Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations with nonlocal interaction. Math. Z. 170, 109–136 (1980) Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys 35, 265–267 (1974) Lions P., L., Paul, T.: Sur les mesures de Wigner. Revista Matematica Ibero Americana 9, 553– 618 (1993) Narnhofer, H., Sewell, G.: Vlasov hydrodynamics of a quantum mechanical model. Commun. Math. Phys. 79, 9–24 (1981) Robert D.: Autour de l’approximation semiclassique. Basel: Birkhäuser Verlag, 1987 Spohn, H.: Kinetic equations from hamiltonian dynamics: markovian limits. Rev. Mod. Phys. 53, 569–615 (1980) Spohn, H.: On the Vlasov hierarchy. Math. Meth. Models in Appl. Sci. 3, 445–455 (1981)
Communicated by H.-T. Yau
Commun. Math. Phys. 271, 699–722 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0216-4
Communications in
Mathematical Physics
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric Stefan Hollands1 , Akihiro Ishibashi2 , Robert M. Wald2 1 Institut für Theoretische Physik, Universität Göttingen, D-37077 Göttingen, Germany.
E-mail: [email protected]
2 Enrico Fermi Institute and Department of Physics, The University of Chicago, Chicago, IL 60637, USA.
E-mail: [email protected]; [email protected]; [email protected] Received: 2 June 2006 / Accepted: 18 October 2006 Published online: 1 March 2007 – © Springer-Verlag 2007
Abstract: A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is “rotating”—i.e., is such that the stationary Killing field is not everywhere normal to the horizon—must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, P. This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic.
1. Introduction Consider an n-dimensional stationary spacetime containing a black hole. Since the event horizon of the black hole must be mapped into itself by the action of any isometry, the asymptotically timelike Killing field t a must be tangent to the horizon. Therefore, we have two cases to consider: (i) t a is normal to the horizon, i.e., tangent to the null geodesic generators of the horizon; (ii) t a is not normal to the horizon. In 4-dimensions it is known that in case (i), for suitably regular non-extremal vacuum or Einstein-Maxwell black holes, the black hole must be static [42, 5]. Furthermore, in 4-dimensions it is known that in case (ii), under fairly general assumptions about the nature of the matter content but assuming analyticity of the spacetime and non-extremality of the black hole, there must exist an additional Killing field that is normal to the horizon. It can then be
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shown that the black hole must be axisymmetric1 as well as stationary [18, 19]. This latter result is often referred to as a “rigidity theorem,” since it implies that the horizon generators of a “rotating” black hole (i.e., a black hole for which t a is not normal to the horizon) must rotate rigidly with respect to infinity. A proof of the rigidity theorem in 4-dimensions which partially eliminates the analyticity assumption was given by Friedrich, Racz, and Wald [9, 32], based upon an argument of Isenberg and Moncrief [27, 20] concerning the properties of spacetimes with a compact null surface with closed generators. The above results for both cases (i) and (ii) are critical steps in the proofs of black hole uniqueness in 4-dimensions, since they allow one to apply Israel’s theorems [23, 24] in case (i) and the Carter-Robinson-Mazur-Bunting theorems [2, 36, 25, 1] in case (ii). Many attempts to unify the forces and equations of nature involve the consideration of spacetimes with n > 4 dimensions. Therefore, it is of considerable interest to consider a generalization of the rigidity theorem to higher dimensions, especially in view of the fact that there seems to be a larger variety of black hole solutions (see e.g., [7, 12, 15]), the classification of which has not been achieved yet.2 The purpose of this paper is to present a proof of the rigidity theorem in higher dimensions for non-extremal black holes. The dimensionality of the spacetime enters the proof of the rigidity theorem in 4-dimensions in the following key way: The expansion and shear of the null geodesic generators of the horizon of a stationary black hole can be shown to vanish (see below). The induced (degenerate) metric on the (n − 1)-dimensional horizon gives rise to a Riemannian metric, γab , on an arbitrary (n − 2)-dimensional cross-section, , of the horizon. On account of the vanishing shear and expansion, all cross-sections of the horizon are isometric, and the projection of the stationary Killing field t a onto gives rise to a Killing field, s a , of γab on . In case (ii), s a does not vanish identically. Now, when n = 4, it is known that must have the topology of a 2-sphere, S 2 . Since the Euler characteristic of S 2 is nonzero, it follows that s a must vanish at some point p ∈ . However, since is 2-dimensional, it then follows that the isometries generated by s a simply rotate the tangent space at p. It then follows that all of the orbits of s a are periodic with a fixed period P, from which it follows that, after period P, the orbits of t a on the horizon must return to the same generator. Consequently, if we identify points in spacetime that differ by the action of the stationary isometry of parameter P, the horizon becomes a compact null surface with closed null geodesic generators. The theorem of Isenberg and Moncrief [27, 20] then provides the desired additional Killing field normal to this null surface. In n > 4 dimensions, the Euler characteristic of may vanish, and, even if it is non-vanishing, if n > 5 there is no reason that the isometries generated by s a need have closed orbits even when s a vanishes at some point p ∈ . Thus, for example, even in the 5-dimensional Myers-Perry black hole solution [30] with cross section topology = S 3 , one can choose the rotational parameters of the solution so that the orbits of the stationary Killing field t a do not map horizon generators into themselves. One possible approach to generalizing the rigidity theorem to higher dimensions would be to choose an arbitrary P > 0 and identify points in the spacetime that differ 1 In this paper, by “axisymmetric” we mean that spacetime possesses one-parameter group of isometries isomorphic to U (1) whose orbits are spacelike. We do not require that the Killing field vanishes on an “axis.” 2 There have recently appeared several works on general properties of a class of stationary, axisymmetric vacuum solutions, including an n-dimensional generalization of the Weyl solutions for the static case (see e.g., [3, 6, 16, 17], and see also [26, 43] and references therein for some techniques of generating such solutions in 5-dimensions).
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by the action of the stationary isometry of parameter P. Under this identification, the horizon would again become a compact null surface, but now its null geodesic generators would no longer be closed. The rigidity theorem would follow if the results of [27, 20] could be generalized to the case of compact null surfaces that are ruled by non-closed generators. We have learned that Isenberg and Moncrief are presently working on such a generalization [22], so it is possible that the rigidity theorem can be proven in this way. However, we shall not proceed in this manner, but rather will parallel the steps of [27, 20], replacing arguments that rely on the presence of closed null generators with arguments that rely on the presence of stationary isometries. Since on the horizon we may write t a = na + s a ,
(1)
where n a is tangent to the null geodesic generators and s a is tangent to cross-sections of the horizon, the stationarity in essence allows us to replace Lie derivatives with respect to n a by Lie derivatives with respect to s a . Thus, equations in [27, 20] that can be solved by integrating quantities along the orbits of the closed null geodesics correspond here to equations that can be solved if one can suitably integrate these equations along the orbits of s a in . Although the orbits of s a are not closed in general, we can appeal to basic results of ergodic theory together with the fact that s a generates isometries of to solve these equations. For simplicity, we will focus attention on the vacuum Einstein’s equation, but we will indicate in Sect. 4 how our proofs can be extended to models with a cosmological constant and a Maxwell field. As in [18, 19] and in [27, 20], we will assume analyticity, but we shall indicate how this assumption can be partially removed (to prove existence of a Killing field inside the black hole) by arguments similar to those given in [9, 32]. The non-extremality condition is used for certain constructions in the proof (as well as in the arguments partially removing the analyticity condition), and it would not appear to be straightforward to generalize our arguments to remove this restriction when the orbits of s a are not closed. Our signature convention for gab is (−, +, +, · · · ). We define the Riemann tensor by Rabc d kd = 2∇[a ∇b] kc and the Ricci tensor by Rab = Racb c . We also set 8π G = 1. 2. Proof of Existence of a Horizon Killing Field Let (M, gab ) be an n-dimensional, smooth, asymptotically flat, stationary solution to the vacuum Einstein equation containing a black hole. Thus, we assume the existence in the spacetime of a Killing field t a with complete orbits which are timelike near infinity. Let H denote a connected component of the portion of the event horizon of the black hole that lies to the future of I − . We assume that H has topology R × , where is compact. Following Isenberg and Moncrief [27, 20], our aim in this section is to prove that there exists a vector field K a defined in a neighborhood of H which is normal to H and on H satisfies L L · · · L (L K gab ) = 0, m = 0, 1, 2, . . . ,
(2)
m times
where is a suitably chosen vector field transverse to H . As we shall show at the end of this section, if we assume analyticity of gab and of H it follows that K a is a Killing field.
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We also will explain at the end of this section how to partially remove the assumption of analyticity of gab and H . We shall proceed by constructing a candidate Killing field, K a , and then proving that Eq. (2) holds for K a . This candidate Killing field is expected to satisfy the following properties: (i) K a should be normal to H . (ii) If we define S a by Sa = t a − K a
(3)
then, on H , S a should be tangent to cross-sections3 of H . (iii) K a should commute with t a . (iv) K a should have constant surface gravity on H , i.e., on H we should have K a ∇a K b = κ K b with κ constant on H , since, by the zeroth law of black hole mechanics, this property is known to hold on any Killing horizon in any vacuum solution of Einstein’s equation. We begin by choosing a cross-section , of H . By arguments similar to those given in the proof of Proposition 4.1 of [5], we may assume without loss of generality that has been chosen so that each orbit of t a on H intersects at precisely one point, so that t a is everywhere transverse to . We extend to a foliation, (u), of H by the action of the time translation isometries, i.e., we define (u) = φu (), where φu denotes the one-parameter group of isometries generated by t a . Note that the function u on H that labels the cross-sections in this foliation automatically satisfies Lt u = 1 .
(4)
t a = na + s a ,
(5)
Next, we define n a and s a on H by
where n a is normal to H and s a is tangent to (u). It follows from the transversality of t a that n a is everywhere nonvanishing and future-directed. Note also that Ln u = 1 on H . Our strategy is to extend this definition of n a to a neighborhood of H via Gaussian null coordinates. This construction of n a obviously satisfies conditions (i) and (ii) above, and it also will be shown below that it satisfies condition (iii). However, it will, in general, fail to satisfy (iv). We shall then modify our foliation so as to produce a new foliation ˜ u) ( ˜ so that (iv) holds as well. We will then show that the corresponding K a = n˜ a satisfies Eq. (2). Given our choice of (u) and the corresponding choice of n a on H , we can uniquely define a past-directed null vector field a on H by the requirements that n a a = 1, and that a is orthogonal to each (u). Let r denote the affine parameter on the null geodesics determined by a , with r = 0 on H . Let x A = (x 1 , x 2 , . . . , x n−2 ) be local coordinates on an open subset of . Of course, it will take more than one coordinate patch to cover , but there is no problem in patching together local results, so no harm is done in pretending that x A covers . We extend the coordinates x A from to H by demanding that they be constant along the orbits of n a . We then extend u and x A to a neighborhood of H by requiring these quantities to be constant along the orbits of a . It is easily seen that the quantities (u, r, x A ) define coordinates covering a neighborhood of H . Coordinates that are constructed in this manner are known as Gaussian null coordinates and are unique up to the choice of and the choice of coordinates x A on . It 3 Note that as already mentioned above, since H is mapped into itself by the time translation isometries, t a must be tangent to H , so S a is automatically tangent to H . Condition (ii) requires that there exist a foliation of H by cross-sections (u) such that each orbit of S a is contained in a single cross-section.
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
follows immediately that on H we have a a ∂ ∂ a a , = , n = ∂u ∂r
703
(6)
and we extend n a and a to a neighborhood of H by these formulas. Clearly, n a and a commute, since they are coordinate vector fields. Note that we have a ∇a (n b b ) = b a ∇a n b = b n a ∇a b =
1 a n ∇a (b b ) = 0 , 2
(7)
so n a a = 1 everywhere, not just on H . Similarly, we have a (∂/∂ x A )a = 0 everywhere. It follows that in Gaussian null coordinates, the metric in a neighborhood of H takes the form gµν dx µ dx ν = 2 dr − r αdu − rβ A dx A du + γ AB dx A dx B , (8) where, again, A is a labeling index that runs from 1 to n − 2. We write βa = β A (dx A )a , γab = γ AB (dx A )a (dx B )b .
(9)
Note that α, βa , and γab are independent of the choice of coordinates, x A , and thus are globally defined in an open neighborhood of H . From the form of the metric, we clearly have βa n a = βa a = 0 and γab n a = γab a = 0. It then follows that γ a b is the orthogonal projector onto the subspace of the tangent space perpendicular to n a and a , where here and elsewhere, all indices are raised and lowered with the spacetime metric gab . Note that when r = 0, i.e., off of the horizon, γab differs from the metric qab , on the (n − 2)-dimensional submanifolds, (u, r ), of constant (u, r ), since n a fails to be perpendicular to these surfaces. Here, qab is defined by the condition that q a b is the orthogonal projector onto the subspace of the tangent space that is tangent to (u, r ); the relationship between γab and qab is given by qab = r 2 β c βc a b − 2rβ(a b) + γab .
(10)
However, since on H (where r = 0), we have γab = qab , we will refer to γab as the metric on the cross-sections (u) of H . Thus, we see that in Gaussian null coordinates the spacetime metric, gab , is characterized by the quantities α, βa , and γab . In terms of these quantities, if we choose K a = n a , then condition (2) will hold if and only if the conditions L L · · · L (Ln γab ) = 0 , m times
L L · · · L (Ln α) = 0 , m times
L L · · · L (Ln βa ) = 0 ,
(11)
m times
hold on H . Since the vector fields n a and a are uniquely determined by the foliation (u) and since φu [(u )] = (u + u ) (i.e., the time translations leave the foliation invariant), it follows immediately that n a and a are invariant under φu . Hence, we have
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Lt n a = Lt a = 0, so, in particular, condition (iii) holds, as claimed above. Similarly, we have Lt r = 0 and Lt u = 1 throughout the region where the Gaussian null coordinates are defined. Since Lt gab = 0, we obtain from Eq. (8), 0 = −2r Lt α ∇a u ∇b u − 2r Lt β(a ∇b) u + Lt γab .
(12)
Contraction of this equation with n a n b yields Lt α = 0 .
(13)
Lt βa = 0 ,
(14)
Lt γab = 0 .
(15)
Contraction with n a then yields
and we then also immediately obtain
The next step in the analysis is to use the Einstein equation Rab n a n b = 0 on H , in a manner completely in parallel with the 4-dimensional case [19]. This equation is precisely the Raychaudhuri equation for the congruence of null curves defined by n a on H . Since that congruence is twist-free on H , we obtain on H , d 1 θ =− θ2 − σab σ ab , dλ n−2
(16)
where θ denotes the expansion of the null geodesic generators of H , σab denotes their shear, and λ is the affine parameter along null geodesic generators of H with tangent n a . Now, by the same arguments as used to prove the area theorem [19], we cannot have θ < 0 on H . On the other hand, the rate of change of the area, A(u), of (u) (defined with respect to the metric qab = γab ) is given by
d ∂λ 1 √ A(u) = θ γ dn−2 x . (17) du 2 ∂u However, since (u) is related to by the isometry φu , the left side of this equation must vanish. Since ∂λ/∂u > 0 on H , this shows that θ = 0 on H . It then follows immediately that σab = 0 on H . Now on H , the shear is equal to the trace free part of Ln γab while the expansion is equal to the trace of this quantity. So we have shown that Ln γab = 0 on H . Thus, the first equation in Eq. (11) holds with m = 0. However, n a in general fails to satisfy condition (iv) above. Indeed, from the form, Eq. (8), of the metric, we see that the surface gravity, κ, associated with n a is simply α, and there is no reason why α need be constant on H . Since Ln γab = 0 on H , the Einstein equation Rab n a (∂/∂ x A )b = 0 on H yields Da α =
1 Ln βa , 2
(18)
(see Eq. (79) of Appendix A) where Da denotes the derivative operator on (u), i.e., Da α = qa b ∇b α = γa b ∇b α. Thus, if α is not constant on H , then the last equation in Eq. (11) fails to hold even when m = 0. As previously indicated, our strategy is to repair ˜ so that the corresponding n˜ a arising this problem by choosing a new cross-section from the Gaussian null coordinate construction will have constant surface gravity on H .
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
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˜ requires some intermediate constructions, to which we now The determination of this turn. First, since we already know that Lt γab = 0 everywhere and that Ln γab = 0 on H , it follows immediately from the fact that t a = s a + n a on H that Ls γab = 0 ,
(19)
on H (for any choice ). Thus, s a is a Killing vector field for the Riemannian metric γab = qab on . Therefore the flow, φˆ τ : → of s a yields a one-parameter group of isometries of γab , which coincides with the projection of the flow φu of the original Killing field t a to . We define κ to be the mean value of α on ,
1 √ α γ dn−2 x , (20) κ= A() where A() is the area of with respect to the metric γab . In the following we will assume that κ = 0, i.e., that we are in the “non-degenerate case." Given that κ = 0, we may assume without loss of generality, that κ > 0. We seek a new Gaussian null coordinate system (u, ˜ r˜ , x˜ A ) satisfying all of the above A properties of (u, r, x ) together with the additional requirement that α˜ = κ, i.e., constancy of the surface gravity. We now determine the conditions that these new coordinates would have to satisfy. Since clearly n˜ a must be proportional to n a , we have n˜ a = f n a ,
(21)
for some positive function f . Since Lt n˜ a = Lt n a = 0, we must have Lt f = 0. Since on H we have n a ∇a n b = αn b and α˜ is given by n˜ a ∇a n˜ b = α˜ n˜ b ,
(22)
α˜ = Ln f + α f = −Ls f + α f = κ .
(23)
we find that f must satisfy
The last equality provides an equation that must be satisfied by f on . In order to establish that a solution to this equation exists, we first prove the following lemma: Lemma 1. For any x ∈ , we have κ = lim
S→∞
1 S
S 0
α(φˆ τ (x)) dτ .
(24)
Furthermore, the convergence of the limit is uniform in x. Similarly, x-derivatives of S S −1 0 α(φˆ τ (x)) dτ converge to 0 uniformly in x as S → ∞. Proof. The von Neumann ergodic theorem (see e.g., [44]) states that if F is an L p function for 1 ≤ p < ∞ on a measure space (X, dm) with finite measure, and if Tτ is a continuous one-parameter group of measure preserving transformations on X , then
1 S F(Tτ (x)) dτ (25) F ∗ (x) = lim S→∞ S 0
∞ e−τ F(Tτ (x)) dτ (26) = lim →0+
0
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converges in the sense of L p (and in particular almost everywhere). We apply this the√ orem to X = , dm = γ dn−2 x, F = α, and Tτ = φˆ τ , to conclude that there is an p ∗ L function α (x) on to which the limit in the lemma converges. We would like to prove that α ∗ (x) is constant. To prove this, we note that Eq. (18) together with the facts that Lt βa = 0 and t a = n a + s a yields 1 Db α = − Ls βb . 2
(27)
Now let
S
α(φˆ τ (x)) dτ .
(28)
1 ∗ φˆ S βb (x) − βb (x) , 2
(29)
a(x, S) = 0
Then Db a(x, S) = − and thus |a(x, S) − a(y, S)| ≤ C sup{[D b a Db a(z, S)]1/2 ; z ∈ } ≤ C sup{[β b βb (z)]1/2 ; z ∈ } = C < ∞ ,
(30)
where C, C are constants independent of S and x, and where C is finite because is compact. Consequently, |a(x, S) − a(y, S)| is uniformly bounded in S ≥ 0 and in x, y ∈ . Thus, for all x, y ∈ , we have lim
S→∞
1 C |a(x, S) − a(y, S)| ≤ lim = 0. S→∞ S S
(31)
Let y ∈ be such that a(y, S)/S converges as S → ∞. (As already noted above, existence of such a y is guaranteed by the von Neumann ergodic theorem.) The above equation then shows that, in fact, a(x, S)/S must converge for all x ∈ as S → ∞ and that, furthermore, the limit is independent of x, as we desired to show. Thus, α ∗ (x) is constant, and hence equal to its spatial average, κ. The estimate (30) also shows that the limit (24) is uniform in x. Similar estimates can easily be obtained for the norm with respect to γab of [Dc1 · · · Dck a(x, S) − Dc1 · · · Dck a(y, S)], for any k. These estimates show that x-derivatives of a(x, S)/S converge to 0 uniformly in x. We now are in a position to prove the existence of a positive function f on satisfying the last equality in Eq. (23) on . Let
∞ f (x) = κ p(x, σ ) dσ, (32) 0
where p(x, σ ) > 0 is the function on × R defined by σ p(x, σ ) = exp − α(φˆ τ (x)) dτ .
(33)
0
The function f is well defined for almost all x because p(x, σ ) < e−σ (κ−) for any and sufficiently large σ , by Lemma 1. It also follows from the uniformity statement in
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
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Lemma 1 that f is smooth on . By a direct calculation, using Lemma 1, we find that f satisfies −Ls f (x) + α(x) f (x) = κ ,
(34)
as we desired to show. We now can deduce how to choose the desired new Gaussian null coordinates. The new coordinate u˜ must satisfy Lt u˜ = 1 ,
(35)
as before. However, in view of Eq. (21), it also must satisfy Ln u˜ = n a ∇a u˜ =
1 a 1 n˜ ∇a u˜ = . f f
(36)
Since n a = t a − s a , we find that on , u˜ must satisfy 1 . f
(37)
1 (Ls ln f − α) . κ
(38)
1 − Ls u˜ = Substituting from Eq. (34), we obtain Ls u˜ = 1 +
Thus, if our new Gaussian null coordinates exist, there must exist a smooth solution to this equation. That this is the case is proven in the following lemma. Lemma 2. There exists a smooth solution h to the following differential equation on : Ls h(x) = α(x) − κ .
(39)
Proof. First note that the orbit average of any function of the form Ls h(x), where h is smooth must vanish, so there could not possibly exist a smooth solution to the above equation unless the average of α over any orbit is equal to κ. However, this was proven to hold in Lemma 1. In order to get a solution to the above equation, choose > 0, and consider the regulated expression defined by
∞ h (x) = − e−τ α(φˆ τ (x)) − κ dτ . (40) 0
Due to the exponential damping, this quantity is smooth, and satisfies the differential equation Ls h (x) = α(x) − κ − h (x) .
(41)
We would now like to take the limit as → 0 to get a solution to the desired equation. However, it is not possible to straightforwardly take the limit as → 0 of h (x), for there is no reason why this should converge without using additional properties of α. In fact, we will not be able to show that the limit as → 0 of h (x) exists, but we will nevertheless construct a smooth solution to Eq. (39). To proceed, we rewrite Eq. (40) as
∞ κ h (x) = − (42) e−τ φˆ τ∗ α(x) dτ + , 0
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where φˆ τ∗ denotes the pull-back map on tensor fields associated with φˆ τ . Taking the gradient of this equation and using Eq. (27), we obtain
∞
1 ∞ −τ ∗ dh (x) = − e−τ φˆ τ∗ (dα)(x) dτ = e φˆ τ (Ls β)(x) dτ , (43) 2 0 0 where here and in the following we use differential forms notation and omit tensor indices. Since Ls clearly commutes with φˆ τ∗ and since Ls is just the derivative along the orbit over which we are integrating, we can integrate by parts to obtain
∞ −τ ∗ 1 φˆ τ β(x) dτ . dh (x) = − β(x) + e (44) 2 2 0 It follows from the von Neumann ergodic theorem4 (see Eq. (25)) that the limit
∞ lim e−τ φˆ τ∗ β(x) dτ = β ∗ (x) , (45) →0
0
L p (). Furthermore, the limit in the sense of
exists in the sense of L p () also exists of ˆ all x-derivatives of the left side. Indeed, because φτ is an isometry commuting with the derivative operator Da of the metric γab , we have ∞
∞ −τ ˆ ∗ φτ βa (x) dτ = D c1 · · · D ck e e−τ φˆ τ∗ Dc1 · · · Dck βa (x) dτ . (46) 0
0
The expression on the right side converges in theorem, meaning that 1 dh → − (β − β ∗ ) 2
L p , as
→ 0 by the von Neumann ergodic
in W k, p () as → 0 ,
(47)
for all k ≥ 0, p ≥ 1, where W k, p () denotes the Sobolev space of order (k, p). By the Sobolev embedding theorem, C m () ← W k, p () for k > m + (n − 2)/ p ,
(48)
where the embedding is continuous with respect to the sup norm on all the derivatives in the space C m , i.e., sup |D m ψ(x)| ≤ const.ψW k, p for all ψ ∈ C m . Thus, convergence of the limit (45) actually occurs in the sup norms on C m . Thus, in particular, β ∗ ∈ C ∞ = ∩m≥0 C m . Now pick an arbitrary x0 ∈ , and define F by
dh = F (x) , (49) h (x) − h (x0 ) = C(x)
4 Here, the theorem is applied to the case of a tensor field T of type (k, l) on a compact Riemannian manifold , rather than a scalar function, and where the measure preserving map is a smooth one-parameter family of isometries acting on T via the pull back. To prove this generalization, we note that a tensor field T of type (k, l) on a manifold may be viewed as a function on the fiber bundle, B, of all tensors of type (l, k) over that satisfies the additional property that this function is linear on each fiber. Equivalently, we may view T as a function, F, on the bundle, B , of unit norm tensors of type (l, k) that satisfies a corresponding linearity property. A Riemannian metric on naturally gives rise to a Riemannian metric (and, in particular, a volume element) on B , and B is compact provided that is compact. Since the isometry flow on naturally induces a volume preserving flow on B , we may apply the von Neumann ergodic theorem to F to obtain the orbit averaged function F ∗ . Since F ∗ will satisfy the appropriate linearity property on each fiber, we thereby obtain the desired orbit averaged tensor field T ∗ .
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
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where the integral is over any smooth path C(x) connecting x0 and x. This integral manifestly does not depend upon the choice of C(x), independently of the topology of . By what we have said above, the function F is smooth, with a smooth limit
1 F(x) = lim F (x) = − (β − β ∗ ) , (50) →0 2 C(x) which is independent of the choice of C(x). Furthermore, the convergence of F and its derivatives to F and its derivatives is uniform. Now, by inspection, F is a solution to the differential equation Ls F (x) = α(x) − κ − F (x) − h (x0 ) .
(51)
Furthermore, the limit
lim h (x0 ) = − lim
→0
→0
∞ 0
1 = κ − lim S→∞ S
e−τ α(φˆ τ (x0 )) − κ dτ
S 0
α(φˆ τ (x0 )) dτ = 0
(52)
exists by the ergodic theorem, and vanishes by Lemma 1. Thus, the smooth, limiting quantity F = lim→0 F satisfies the desired differential equation (39). We now define a new set of Gaussian null coordinates (u, ˜ r˜ , x˜ A ) as follows. Define u˜ on to be a smooth solution to Eq. (38), whose existence is guaranteed by Lemma 2. Extend u˜ to H by Eq. (35). It is not difficult to verify that u˜ is given explicitly by
u(x) ˜ = 0
u(x)
f (π ◦ φ−τ (x))
−1
dτ +
1 1 ln f (π(x)) − h(π(x)) , κ κ
(53)
where f and h are smooth solutions to Eqs. (23) and (39), respectively, on and π : H → , x → π(x)
(54)
is the map projecting any point x in H to the point π(x) on the cross section on ˜ denote the surface u˜ = 0 on H . Then our desired the null generator through x. Let A Gaussian null coordinates (u, ˜ r˜ , x˜ ) are the Gaussian null coordinates associated with ˜ The corresponding fields α, . ˜ β˜a , γ˜ab satisfy all of the properties derived above for α, βa , γab and, in addition, satisfy the condition that α˜ = κ is constant on H . Now let K a = n˜ a . We have previously shown that Ln˜ γ˜ab = 0 on H , since this relation holds for any choice of Gaussian null coordinates. However, since our new coordinates have the property that α˜ = κ is constant on H , we clearly have that Ln˜ α˜ = 0 on H . Furthermore, for our new coordinates, Eq. (18) immediately yields Ln˜ β˜a = 0 on H . Thus, we have proven that all of the relations in Eq. (11) hold for m = 0. We next prove that the equation L˜ Ln˜ γ˜ab = 0 holds on H . Using what we already know about β˜a , γ˜ab and taking the Lie-derivative Ln˜ of the Einstein equation Rab (∂/∂ x˜ A )a (∂/∂ x˜ B )b = 0 (see Eq. (82) of Appendix A), we get 0 = Ln˜ Ln˜ L˜γ˜ab + κ L˜γ˜ab , (55)
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˜ u), on H . Since t a = n˜ a + s˜ a , with s˜ a tangent to ( ˜ and since all quantities appearing in Eq. (55) are Lie derived by t a , we may replace in this equation all Lie derivatives Ln˜ by −Ls˜ . Hence, we obtain 0 = Ls˜ Ls˜ L˜γ˜ab − κL˜γ˜ab , (56) ˜ Now, write L ab = L ˜γ˜ab . We fix x0 ∈ ˜ and view Eq. (56) as an equation holding on . ∗ ˜ → ˜ now denotes the flow of at x0 for the pullback, φˆ τ L ab , of L ab to x0 , where φˆ τ : s˜ a . Then Eq. (56) can be rewritten as d κτ d −κτ ˆ ∗ (e (57) φτ L ab ) = 0 . e dτ dτ Integration of this equation yields eκτ
d −κτ ∗ (e φˆ τ L ab ) = −κCab , dτ
(58)
where Cab is a tensor at x0 that is independent of τ . Integrating this equation (and absorbing constant factors into Cab ), we obtain φˆ τ∗ L ab − eκτ L ab = (1 − eκτ )Cab .
(59)
However, since φˆ τ is a Riemannian isometry, each orthonormal frame component of φˆ τ∗ L ab at x0 is uniformly bounded in τ by the Riemannian norm of L ab , ˜ Consequently, the limit of Eq. (59) as τ → ∞ i.e., sup{(L ab L ab (x))1/2 ; x ∈ }. immediately yields L ab = Cab
(60)
from which it then immediately follows that φˆ τ∗ L ab = L ab .
(61)
Thus, we have Ls˜ L˜γ˜ab = 0, and therefore Ln˜ L˜γ˜ab = L˜ Ln˜ γ˜ab = 0 on H , as we desired to show. Thus, we now have shown that the first equation in (11) holds for m = 0, 1, and that the other equations hold for m = 0, for the tensor fields associated with the “tilde" Gaussian null coordinate system, and K = n. ˜ In order to prove that Eq. (11) holds for all m, we proceed inductively. Let M ≥ 1, and assume inductively that the first of Eqs. (11) holds for all m ≤ M, and that the remaining equations hold for all m ≤ M −1. Our task is to prove that these statements then also hold when M is replaced by M + 1. To show this, we apply the operator (L˜) M−1 Ln˜ to the Einstein equation Rab n˜ a ˜b = 0 (see Eq. (78)) and restrict to H . Using the inductive hypothesis, one sees that (L˜) M (Ln˜ α) ˜ = 0 on H , thus establishing the second equation in (11) for m ≤ M. Next, we apply the operator (L˜) M−1 Ln˜ to the Einstein equation Rab (∂/∂ x˜ A )a ˜b = 0 (see Eq. (81)), and restrict to H . Using the inductive hypothesis, one sees that (L˜) M (Ln˜ β˜a ) = 0 on H , thus establishing the third equation in (11) for m ≤ M. Next, we apply the operator (L˜) M Ln˜ to the Einstein equation Rab (∂/∂ x˜ A )a (∂/∂ x˜ B )b = 0 (see Eq. (82)), and restrict to H . Using the inductive hypothesis and the above results (L˜) M (Ln˜ α) ˜ =0
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(M+1) and (L˜) M (Ln˜ β˜a ) = 0, one sees that the tensor field L ab ≡ (L˜) M+1 γ˜ab satisfies a differential equation of the form (M+1)
Ln˜ [Ln˜ L ab
(M+1)
+ (M + 1)κ L ab
]=0
(62) (M+1)
on H . By the same argument as given above for L ab , it follows that Ln˜ L ab = 0. This establishes the first equation in (11) for m ≤ M + 1, and closes the induction loop. Thus far, we have assumed only that the spacetime metric is smooth (C ∞ ). However, if we now assume that the spacetime is real analytic, and that H is an analytic submanifold, then it can be shown that the vector field K a that we have defined above is, in fact, analytic. To see this, first note that if the cross section of H is chosen to be analytic, then our Gaussian null coordinates are analytic, and, consequently, so is any quantity defined in terms of them, such as n a and a . Above, K a was defined in terms of a certain special Gaussian null coordinate system that was obtained from a geometrically special cross section. That cross section was obtained by a change (53) of the coordinate u. Thus, to show that K a is analytic, we must show that this change of coordinates is analytic. By Eq. (53), this will be the case provided that f and h are analytic. We prove this in Appendix C. Since gab and K a are analytic, so is L K gab . It follows immediately from the fact that this quantity and all of its derivatives vanish at any point of H that L K gab = 0 where defined, i.e., within the region where the Gaussian null coordinates (u, ˜ r˜ , x˜ A ) a are defined. This proves existence of a Killing field K in a neighborhood of the horizon. We may then extend K a by analytic continuation. Now, analytic continuation need not, in general, give rise to a single-valued extension, so we cannot conclude that there exists a Killing field on the entire spacetime. However, by a theorem of Nomizu [31] (see also [4]), if the underlying domain is simply connected, then analytic continuation does give rise to a single-valued extension. By the topological censorship theorem [10, 11], the domain of outer communication has this property. Consequently, there exists a unique, single valued extension of K a to the domain of outer communication, i.e., the exterior of the black hole (with respect to a given end of infinity). Thus, in the analytic case, we have proven the following theorem: Theorem 1. Let (M, gab ) be an analytic, asymptotically flat n-dimensional solution of the vacuum Einstein equations containing a black hole and possessing a Killing field t a with complete orbits which are timelike near infinity. Assume that a connected component, H , of the event horizon of the black hole lies to the future of I − , is analytic and is topologically R × , with compact and that κ = 0 (where κ is defined by Eq. (20) above). Then there exists a Killing field K a , defined in a region that covers H and the entire domain of outer communication, such that K a is normal to the horizon and K a commutes with t a . The assumption of analyticity in this theorem can be partially removed in the following manner, using an argument similar to that given in [9]. Since κ > 0, the arguments of [34] show that the spacetime can be extended, if necessary, so that H is a proper subset of a regular bifurcate null surface H ∗ in some enlarged spacetime (M ∗ , g ∗ ). We may then consider the characteristic initial value formulation for Einstein’s equations [35, 29, 8] on this bifurcate null surface. Since the extended spacetime is smooth, the initial data induced on this bifurcate null surface should be regular. Since this initial data is invariant under the orbits of K a , it follows that the solution to which this data gives rise will be invariant under a corresponding one-parameter group of diffeomorphisms in the domain of dependence, D(H ∗ ), of H ∗ . Thus, if one merely assumes that the spacetime
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S. Hollands, A. Ishibashi, R. M. Wald
is smooth, existence of a Killing field in D(H ∗ ) holds. However, since D(H ∗ ) lies inside the black hole, this argument does not show existence of a Killing field in the domain of outer communications. Interestingly, if one assumes that the spacetime is analytic—so that existence of a Killing field in the domain of outer communications follows from the above analytic continuation arguments—then this argument shows that the Killing field known to exist in the domain of outer communications also can be extended to all of D(H ∗ ). 3. Proof of Existence of Rotational Killing Fields We proved in the previous section that if the quantity κ defined by Eq. (20) is nonvanishing, then there exists a vector field K a in a neighborhood of H which is normal to H and is such that Eqs. (2) hold. As explained at the end of the previous section, in the analytic case, this implies the existence of a Killing field normal to the horizon in a region containing the horizon and the domain of outer communication. Since we are considering the case where t a is not pointing along the null generators of H , the Killing field K a is distinct from t a . Hence, their difference S a ≡ s˜ a = t a − K a is also a nontrivial Killing field. There are two cases to consider: 1. The Killing field S a has closed orbits, or 2. The Killing field S a does not have closed orbits. Only the first case can occur in 4-dimensions. In the first case, it follows immediately that the Killing field S a corresponds to a rotation at infinity. The purpose of this section is to show that, in the second case, even though the orbits of S a are not closed, there must a , . . . , ϕ a , which possess closed exist N ≥ 2 mutually commuting Killing fields, ϕ(1) (N ) orbits with period 2π and are such that a a S a = 1 ϕ(1) + · · · + N ϕ(N ),
(63)
for some constants i , all of whose ratios are irrational. To simplify notation, throughout this section, we omit the “tildes” on all quantities, i.e., in this section a , n a , , u, r, α, βa , γab denote the quantities associated with our preferred Gaussian null coordinates. The Killing field S a satisfies a number of properties that follow immediately from the construction of K a = n a given in the previous section. First, since L K r = 0 = Lt r and since L K a = 0 = Lt a , it follows that S a also satisfies these properties, i.e., L S r = 0 and L S a = 0. Similarly, since L K u = 1 = Lt u, we also have L S u = 0. In addition, since K a commutes with t a , so does S a . Thus, S a is tangent to the surfaces of constant (u, r ), and commutes with a and t a . Finally, it follows immediately from Eq. (11) and Eqs. (13)–(15) that S a also satisfies the analog of Eq. (11). To proceed, we focus attention now on the Riemannian manifold (, γab ) and make arguments similar to those given in [21]. Let G denote the isometry group of (, γab ). Then G is a compact Lie group. Let H ⊂ G denote the one-parameter subgroup of G generated by the Killing field S a on (, γab ). Then the closure, H, of H is a closed subgroup of G, and hence is a Lie subgroup. Since H is abelian, so is H, and since G is compact, it follows that H is a torus. Let N = dim(H). Since the N -dimensional torus can be written as the direct product of N factors of U (1), it follows that the isometries a , . . . , ϕ a , which possess in H are generated by N commuting Killing vector fields, ϕ(1) (N ) closed orbits on with period 2π . Since the isometry subgroup H generated by S a is
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
713
dense in H, it follows that, on , S a must be a linear combination of these Killing vector fields of the form (63). Since, as we have noted above, S a satisfies the analog of Eq. (11), the diffeomorphisms on corresponding to every element of H leave invariant each tensor field on of the form
T(k)
⎧ ⎪ L ··· L γ ⎪ ⎪ ab ⎪ ⎪ ⎪ ⎪ m+1 times ⎨ L ··· L β = a ⎪ m times ⎪ ⎪ ⎪ ⎪ L · · · L α , ⎪ ⎪ ⎩
(64)
m times
for all m ≥ 0. Since H is dense in H, each T(k) also must be invariant under the diffeomorphisms corresponding to the elements of H. Consequently, the Killing fields a , . . . , ϕ a Lie derive all T a ϕ(1) (k) on . We now extend each ϕ( j) to a vector field defined (N ) in an entire neighborhood of H as follows. First, we Lie-drag ϕ(aj) from to H via the vector field K a = n a . Then we Lie-drag the resulting vector field defined on H off the horizon via the vector field a . The vector field (denoted again by ϕ(aj) ), which has now been defined in an entire neighborhood of H , satisfies the following properties throughout this neighborhood: (i) ϕ(aj) commutes with both n a and a . (ii) ϕ(aj) satisfies the analog of Eq. (11). Property (ii) implies that in the analytic case, ϕ(aj) is a Killing field of the spacetime metric. As was the case for K a , we may then uniquely extend ϕ(aj) as a Killing field to the entire domain of outer communication. That this extended Killing field (which we also denote by ϕ(aj) ) must have closed orbits can be seen as follows: The orbits of ϕ(aj) on are closed with period 2π . Thus, if we consider the flow of ϕ(aj) by parameter 2π , any point x ∈ will be mapped into itself, and vectors at x that are tangent to also will get mapped into themselves. Furthermore, since ϕ(aj) commutes with n a and a , tangent vectors at x that are orthogonal to also will get mapped into themselves. Consequently, the isometry on the spacetime corresponding to the action of ϕ(aj) by parameter 2π maps point x into itself and maps each vector at x into itself. Consequently, this isometry is the identity map in any connected region where it is defined. Thus, we have shown: Theorem 2. Let (M, gab ) be an analytic, asymptotically flat n-dimensional solution of the vacuum Einstein equations containing a black hole and possessing a Killing field t a with complete orbits which are timelike near infinity. Assume that a connected component, H , of the event horizon of the black hole lies to the future of I − , is analytic and is topologically R × , with compact and that κ = 0 (where κ is defined by Eq. (20) above). If t a is not tangent to the generators of H , then there exist mutually commuting a , . . . , ϕ a (where N ≥ 1) with closed orbits with period 2π which are Killing fields ϕ(1) (N ) defined in a region that covers H and the entire domain of outer communication. Each of these Killing fields commute with t a , and t a can be written as a a + · · · + N ϕ(N t a = K a + 1 ϕ(1) ),
(65)
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S. Hollands, A. Ishibashi, R. M. Wald
for some constants i , all of whose ratios are irrational, where K a is the horizon Killing field whose existence is guaranteed by Theorem 1. Theorem 2 shows that the null geodesic generators of the event horizon rotate rigidly with respect to infinity. a , . . . , ϕa Remarks. 1) As in the case of K a , in the non-analytic case the Killing fields ϕ(1) (N ) can be proven to exist in D(H ∗ ) (see the end of Sect. 2). 2) If the orbits of S a are closed on —i.e., equivalently, if the orbits of t a map each generator of H to itself after some period P—then the above argument shows that S a itself is a Killing field with closed orbits. As previously noted in the introduction, in the case of 4-dimensional spacetimes, the orbits of S a on are always closed. However, the orbits of S a on need not be closed when n > 4. For example, on the round 3-sphere S 3 , one can take an incommensurable linear combination of two commuting Killing fields with closed orbits to obtain a Killing field with non-closed orbits. This possibility is realized for S a in suitably chosen 5-dimensional Myers-Perry black hole solutions [30]. Our theorem shows that if the orbits of S a fail to be closed, then the spacetime must admit at least two linearly independent rotational Killing fields. 3) In 4-dimensions, if t a is normal to the horizon, then Thm. 3.4 of [42] (applied to the vacuum or Einstein-Maxwell cases) shows that the exterior region must be static. The proof of this result makes use of the fact that there exists a bifurcation surface (i.e., that we are in the non-degenerate case κ = 0), and that there exists a suitable foliation of the exterior region by maximal surfaces; the existence of such a foliation was proven in [5]. The arguments of [42] generalize straightforwardly to higher dimensions. Thus, the staticity of higher dimensional vacuum or Einstein-Maxwell stationary black holes with t a normal to the horizon must hold provided that the arguments of [5] also generalize suitably to higher dimensions.5 It should be noted that n-dimensional static vacuum (and Maxwell-dilaton) black hole spacetimes with a standard null infinity of topology I ∼ = S n−2 ×R were shown to be essentially unique by Gibbons et al. [13], and are, in particular, spherically symmetric. However, static spacetimes with a non-trivial topology at infinity do not have to have any extra Killing fields [14].6
4. Matter Fields The analysis of the foregoing sections can be generalized to include various matter sources, and we now illustrate this by discussing several examples. The Einstein equation with matter is Rab = Tab −
1 gab T c c . n−2
(66)
The simplest matter source is a cosmological constant Tab = −gab . The only significant change resulting from the presence of a cosmological constant is a change in the asymptotic properties of the spacetime. For the case of a negative cosmological constant, we can consider spacetimes that are asymptotically AdS rather than asymptotically flat. For asymptotically AdS spacetimes, I is no longer null, but is instead timelike. However, the only place where we used the character of I in our arguments was to conclude that the Killing field t a is nowhere vanishing on H , and therefore generates a suitable 5 It has recently been shown in [41] that this is indeed the case. 6 See also [37–39] for uniqueness results of higher dimensional static black holes and [28, 40] for uniqueness
results of some restricted class of 5-dimensional stationary black holes.
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
715
foliation (u) of H by cross sections. This argument goes through without change in the asymptotically AdS case, as do all subsequent arguments in our proof. Thus, the rigidity theorem holds without modification in the case of a negative cosmological constant. For a positive cosmological constant, I would have a spacelike character and it is not clear precisely what should be assumed about the behavior of t a near I . Nevertheless, our results apply straightforwardly to any “horizon” that is the boundary of the past of any complete, timelike orbit of t a and lies to the future of the orbit. In this sense, our rigidity theorem holds for both black hole event horizons and cosmological horizons. For Maxwell fields the field equations and stress tensor are 1 ∇ a Fab = 0 , ∇[a Fbc] = 0 , Tab = Fac Fb c − gab F cd Fcd . 4
(67)
We assume that both metric and Maxwell tensor are invariant under t a , i.e., Lt gab = 0 = Lt Fab . In parallel with the vacuum case, we wish to show that there exists a vector field K a tangent to the generators of the horizon satisfying L L · · · L (L K gab ) = 0 , L L · · · L (L K Fab ) = 0 , m = 0, 1, 2, . . . , m times
m times
(68) on H . To analyze these equations we introduce a Gaussian null coordinate system as above, and correspondingly decompose the field strength tensor as Fµν dx µ ∧ dx ν = S du ∧ dr + V A du ∧ dx A + W A dr ∧ dx A + U AB dx A ∧ dx B . (69) We write Va = V A (dx A )a , Wa = W A (dx A )a , Uab = U AB (dx A )a (dx B )b .
(70)
It follows from Lt Fab = 0 that Lt S = Lt Va = Lt Wa = Lt Uab = 0. As in the vacuum case, we take the candidate Killing field K a to be n a , where n a is the vector field associated with a suitable Gaussian null coordinate system to be determined. Equations (68) are then equivalent to Eqs. (11) together with L L · · · L (Ln S) = 0 , m times
L L · · · L (Ln Va ) = 0 , m times
L L · · · L (Ln Wa ) = 0 , m times
L L · · · L (Ln Uab ) = 0 .
(71)
m times
The Maxwell equations, the Bianchi identities, and the stress tensor are presented in Appendix B. The Raychaudhuri equation now gives Ln γab = 0 = Va on H . Then, the Bianchi identity (87) yields Ln Uab = 0. The Maxwell equation (84) yields Ln S = 0 on H . Furthermore, we have Tab n a (∂/∂ x A )b = 0 (see Eq. (92)), from which it follows in view of Einstein’s equation that also Rab n a (∂/∂ x A )b = 0 on H . We may now argue
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S. Hollands, A. Ishibashi, R. M. Wald
in precisely the same way as in the vacuum case that, by a suitable choice of Gaussian null coordinates, we can achieve that Ln α = 0 = Ln βa on H . It then follows from Eqs. (95) and (96) that Ln [Tab (∂/∂ x A )a (∂/∂ x B )b ] = 0 and Ln T a a = 0 on H , which in view of Einstein’s equation means that Ln [Rab (∂/∂ x A )a (∂/∂ x B )b ] = 0 on H . This may in turn be used to argue, precisely as in the vacuum case, that L Ln γab = 0 on H . Taking a Lie derivative Ln of the Maxwell equation (85) and the Bianchi identity (86) then leads to the equation Ln [Ln Wa + κ Wa ] = 0 ,
(72)
on H . Since Wa is Lie derived by t a and since t a = n a + s a as in the vacuum case, this equation may alternatively be written as Ls [Ls Wa − κ Wa ] = 0 .
(73)
φˆ τ∗ Wa − eκτ Wa = (1 − eκτ )Ca ,
(74)
Integration gives
where Ca is a 1-form field on independent of τ , and where φˆ τ is again the flow of s a . The same type of argument following Eq. (59) then implies that Ln Wa = 0 on H . We have thus shown that all Eqs. (71) and (11) for m = 0 and the first equation in (11) for m = 1 are satisfied on the horizon. The remainder of the argument closely parallels the vacuum case. Acknowledgements. We wish to thank Jim Isenberg and Vince Moncrief for valuable discussions and for making available to us the manuscript of their forthcoming paper [22]. We would also like to thank P. Chrusciel and I. Racz for useful discussions and important suggestions. We have benefited from workshops at Isaac Newton Institute (“Global Problems in Mathematical Relativity”), Oberwolfach (“Mathematical Aspects of General Relativity”), and at KITP UCSB (“Scanning New Horizons:GR Beyond 4 Dimensions”). This research was supported in part by NSF grant PHY 04-56619 to the University of Chicago.
A. Ricci Tensor in Gaussian Null Coordinates In this Appendix, we provide expressions for the Ricci tensor in a Gaussian null coordinate system. As derived in Sect. 2, in Gaussian null coordinates, the metric takes the form gab = 2 ∇(a r − r α∇(a u − rβ(a ∇b) u + γab , (75) where the tensor fields βa and γab are orthogonal to n a and a . The horizon, H , corresponds to the surface r = 0. We previously noted that γ a b is the orthogonal projector onto the subspace of the tangent space orthogonal to n a and a , and that when rβa = 0, it differs from the orthogonal projector, q a b , onto the surfaces (u, r ). It is worth noting that in terms of the Gaussian null coordinate components of γab , we have q ab = (γ −1 ) AB (∂/∂ x A )a (∂/∂ x B )b . It also is convenient to introduce the nonorthogonal projector pa b , uniquely defined by the conditions that pa b n b = pa b b = 0 and that pa b be the identity map on vectors that are tangent to (u, r ). The relationship between pa b and γ a b is given by pa b = −r a βb + γ a b .
(76)
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
717
In terms of Gaussian null coordinates, we have pa b = (∂/∂ x A )a (dx A )b , from which it is easily seen that Ln pa b = 0 = L pa b . It also is easily seen that q ac γcb = pa b and that pa b q b c = q a c . We define the derivative operator Dc acting on a tensor field T a1 ...ar b1 ...bs by the following prescription. First, we project the indices of the tensor field by q a b , then we apply the covariant derivative ∇c , and we then again project all indices using q a b . For tensor fields intrinsic to , this corresponds to the derivative operator associated with the metric qab . We denote the Riemann and Ricci tensors associated with qab as Rabc d and Rab . The Ricci tensor of gab can then be written in the following form: 1 1 1 n a n b Rab = − q ab Ln Ln γab + q ca q db (Ln γab )Ln γcd + α q ab Ln γab 2 4 2 r + · 4αL L α + 8αL α + (L α)q ab Ln γab 2 +q ab L γab · − Ln α − rq cd βc Ln βd +(rq cd βc βd + 2α)L (r α) + rq cd βc Dd α +2q ab Da {βb L (r α) + Db α − Ln βb } +q bc L (rβc ) · (rq e f βe β f + 2α)L (rβb ) −4Db α + 2Ln βb + 4rq ae βe D[a βb]
+2(L α)L (r 2 q ab βa βb ) + 4rq ab βa βb L α + 2rq ab βa βb L L α 1 cd ab +2q βa L (rβb ) · 2L (r α) − rq βc L (rβd ) 2
−1 2 ab −1 2 ab +2r L r q βa (Db α − Ln βb ) + 2r αL (r q βa βb ) , (77) 1 1 1 n a b Rab = −2L α + q ca q db (Ln γcd )L γab − q ab L Ln γab − α q ab L γab 4 2 2
1 ab r 1 ab − q βa βb + · − 2L L α − q L γab · 2L α + q cd βc L (rβd ) 2 2 2 ab ab ab (78) −q βa L βb − L {q βa L (rβb )} − q Da (L βb ) , 1 1 n b p c a Rbc = − p b a Db α + Ln βa + βa q bc Ln γbc − p d [a p e b] Dd (q bc Ln γce ) 2 4 1 bc r (q Ln γbc )L βa + Ln L βa + 2αL βa + · 2 2
+L (rβa ) · r −1 L (r 2 q bc βb βc ) + 2L α −2 p b a Db (L α)+L (q bc βb Ln γca )−2r −1 L r 2 q cd βc p b a D[b βd]
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S. Hollands, A. Ishibashi, R. M. Wald
1 − q bc L γbc · − (rq e f βe β f + 2α)L (rβa ) 2 +2 p d a Dd α − q bc βb Ln γca + 2rq e f βe p d a D[d β f ]
−2L (αβa ) − 2r (L α)L βa + p d a Db q bc βc L (rβd )
−2 p b a q cd Dd D[b βc] − q bc (L βb )Ln γca −q bc L (rβb ) · (rq e f βe β f + 2α)L γca + p d a Dc βd
+βc L (rβa ) − rq e f βc β f L γea
bc de +q (L γca ) · 2βb L (r α) + 2Db α − Ln βb + 2rq βe D[b βd] , (79)
1 1 a b Rab = − q ab L L γab + q ca q db (L γab )L γcd , 2 4
(80)
1 1 b p c a Rbc = − βa q bc L γbc − L βa + q bc βc L γab − p d [a p e b] Dd q bc L γce 4 2 r + · − L L βa + L q bc βc L γab 2 1 cd be (81) + (q L γcd ) −L βa + q βe L γab , 2
1 p c a p d b Rcd = −L Ln γab − αL γab + p c a p d b Rcd − p c (a p d b) Dc βd − βa βb 2
1 cd cd cd (q Ln γcd )L γab + (q L γcd )Ln γab + q L γd(a Ln γb)c − 4 r + · − 2αL L γab − p e a p f b Dc (q cd βd L γe f ) 2
1 − (q cd L γcd ) (rq e f βe β f + 2α)L γab + 2 p e (a p f b) De β f 2 −2(L α)L γab − r −1 {L (r 2 q e f βe β f )}L γab −rq e f βe β f L L γab − 2L { p c (a p d b) Dc βd } −2β(a L βb) − r (L βa )L βb − rq ce q d f βc βd (L γae )L γb f +2q cd βd L (rβ(a ) L γb)c + 2 p e (a p f b) q cd (Dd βe ) L γ f c cd ef (82) +q (rq βe β f + 2α)(L γca )L γdb .
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
719
B. Maxwell Equations in Gaussian Null Coordinates With the notation introduced in Appendix A and the definitions (69) and (70), the Maxwell equations, ∇a F ab = 0, are equivalent to the following equations: 1 0 = L S + Sq ab L γab − q ab βa Wb − q ab Da Wb 2 r − · 2L (q ab βa Wb ) + q ab q cd βa Wb L γcd , 2 1 0 = Ln S + Sq ab Ln γab + q ab Da Vb 2 r − · 2Ln (q ab βa Wb ) + q ab q cd βc Wd Ln γab 2
(83)
−2q ab Da SVb + 2αWb − q cd βd Ubc − 4rq ab q cd Da (βc β[d Wb] ) , (84)
0=
1 ab cd q q (Wb Ln γcd + Vb L γcd ) + Ln (q ab Wb ) + L (q ab Vb ) 2
+ q ab Sβb + 2αWb − q cd βd Ubc r − · − 2L q ab Sβb + 2αWb − q cd βd Ubc 2 −8q ab q cd βc β[d Wb] − 4L q ab q cd βc β[d Wb]
−q cd (L γcd ) · q ab Sβb + q e f βe U f b + 2αWb − 2rq e f βe β[b W f ] . (85) The Bianchi identities, ∇[a Fbc] = 0, are given by Ln Wa − L Va + p c a Dc S = 0 ,
(86)
Ln Uab − 2 p
= 0,
(87)
L Uab − 2 p c [a p d b] Dc Wd = 0 ,
(88)
= 0.
(89)
p
d
c
e
d
[a p b] Dc Vd
f
[a p b p c] Dd Ue f
The stress tensor, Tab = Fac Fb c − (1/4)gab F cd Fcd , is given by 1 n a n b Tab = q ab Va Vb + r · 2q ab βa Vb S + (rq ab βa βb + 2α)S 2 + α F cd Fcd , (90) 2 1 1 n a b Tab = − S 2 − q ac q bd Uab Ucd 2 4 + r · αq cd Wc Wd + q bc βc 2Wb S + q de We Ubd +
r ab cd q q − q ac q bd βa βb Wc Wd 2
n b p c a Tbc = −SVa + q bc Uab Vc
,
(91)
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S. Hollands, A. Ishibashi, R. M. Wald
a b Tab b p c a Tbc p c a p d b Tcd
T cc
1 + r · q bc βc (−Wa Vb + Uab S) − rq bc βb βc + 2α Wa S + βa F cd Fcd , 4 (92) = q ab Wa Wb , (93) bc bc = SWa + q Uab Wc − rq βc Wa Wb , (94) 1 1 2 cd cd ef = γab S + 2V(a Wb) − γab q Vc Wd + q Uac Ubd − γab q Uce Ud f 2 4 −r · γab · αq bc Wb Wc + q bc βc Wb S + q de We Ubd r cd e f ce d f βc βd We W f , + q q −q q 2 1 2 1 ab cd ab S − q Va Wb − q q Uac Ubd = (n − 4) 2 4 (n − 4) − · r · 2αq ab Wa Wb + 2q ab βa Wb S + q cd Wd Ubc 2 + r q ab q cd − q ac q bd βa βb Wc Wd ,
(95)
(96)
where 1 cd 1 1 F Fcd = q ab Va Wb − S 2 + q ac q bd Uab Ucd 4 2 4 cd + r · αq Wc Wd + q bc βc Wb S + q de We Ubd r + q ab q cd − q ac q bd βa βb Wc Wd 2
.
(97)
C. Analyticity of f and h In this Appendix, we prove the following lemma, which establishes that if the spacetime (M, gab ) and horizon H are analytic, then the candidate Killing field K a also is analytic. Lemma 3. If the spacetime as well as and H are analytic, then the functions f, h : → R given by Eqs. (32) and (39) are real analytic. Proof. Consider first the function p(x, σ ) on ×R defined above in Eq. (33). Let x0 ∈ be fixed, and choose Riemannian normal coordinates y 1 , . . . , y n−2 around x0 , so that the coordinate components γ AB by δ AB . If α = (α1 , . . . , αn−2 ) ∈ Nn−2 0 (x0 ) are given is a multi-index, we set |α| = αi , and α! = αi !, as well as ∂α =
∂ |α| . (∂ y 1 )α1 · · · (∂ y n−2 )αn−2
(98)
We will show that, for y in a sufficiently small ball around x0 , and for sufficiently large σ we have the following estimate: |∂ α p(y, σ )| ≤ α!C R −|α| e−σ κ/2 ,
(99)
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
721
with C and R being some constants independent of α. This implies that f (y) has a convergent power series representation near x0 . Using Einstein’s equation as in the proof of Lemma 1, we have ∂ p(y, σ ) = [β A (y) − (φˆ σ∗ β) A (y)] p(y, σ ) . ∂yA
(100)
Now we complexify and consider a complex multi-disk around x0 of radius R. Then, using the multi-dimensional version of the Cauchy inequalities, we furthermore have the estimate |∂ α β A (y)| ≤ α!C R −|α| ,
(101)
where C is now taken as the supremum of β A in the complex multi-disk around x0 . Furthermore, because is compact, and because φˆ σ is an isometry, we can find the same type of estimate also for ∂ α (φˆ σ∗ β) A (y) uniformly in σ and y in a ball around x0 . Finally, from Lemma 1, we have the estimate | p(y, σ )| ≤ e−σ (κ−) ,
(102)
for arbitrary small > 0 for any y in a ball around x0 , and for sufficiently large σ . Applying now further derivatives to Eq. (100) and using the above estimates, we obtain the estimate (99). In order to prove the analyticity of h, we must look at the explicit construction of that function given in the proof of Lemma 2. That construction shows that h will be analytic, if we can show that the vector field β ∗ (x) defined by the integral (45) is analytic. This follows from the fact that the Taylor coefficient of the integrand ∂ α (φˆ τ∗ β) A (y) satisfies an estimate of the form (101) uniformly in τ and y in a ball around x0 . References 1. Bunting, G.L.: Proof of the uniqueness conjecture for black holes. Ph.D. Thesis, Univ. of New England, Armidale, N.S.W., 1983 2. Carter, B.: Axisymmetric black hole has only two degrees of freedom. Phys. Rev. Lett. 26, 331–333 (1971) 3. Charmousis, C., Gregory, R.: Axisymmetric metrics in arbitrary dimensions. Class. Quant. Grav. 21, 527 (2004) 4. Chru´sciel, P.T.: On rigidity of analytic black holes. Commun. Math. Phys. 189, 1–7 (1997) 5. Chru´sciel, P.T., Wald, R.M.: Maximal hypersurfaces in asymptotically stationary space-times. Commun. Math. Phys. 163, 561 (1994) 6. Emparan, R., Reall, H.S.: Generalized Weyl solutions. Phys. Rev. D 65, 084025 (2002) 7. Emparan, R., Reall, H.S.: A rotating black ring in five dimensions. Phys. Rev. Lett. 88, 101101 (2002) 8. Friedrich, H.: On the global existence and the asymptotic behavior of solutions to the Einstein-MaxwellYang-Mills equations. J. Diff. Geom. 34, 275 (1991) 9. Friedrich, H., Racz, I., Wald, R.M.: On the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon. Commun. Math. Phys. 204, 691–707 (1999) 10. Galloway, G.J., Schleich, K., Witt, D.M., Woolgar, E.: Topological censorship and higher genus black holes. Phys. Rev. D 60, 104039 (1999) 11. Galloway, G.J., Schleich, K., Witt., D., Woolgar, E.: The AdS/CFT correspondence conjecture and topological censorship. Phys. Lett. B 505, 255 (2001) 12. Gibbons, G.W., Hartnoll, S.A., Pope, C.N.: Bohm and Einstein-Sasaki metrics, black holes and cosmological event horizons. Phys. Rev. D 67, 084024 (2003) 13. Gibbons, G.W., Ida, D., Shiromizu, T.: Uniqueness of (dilatonic) charged black holes and black p-branes in higher dimensions. Phys. Rev. D 66, 044010 (2002) 14. Gibbons, G.W., Ida, D., Shiromizu, T.: Uniqueness and non-uniqueness of static black holes in higher dimensions. Phys. Rev. Lett. 89, 041101 (2002)
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15. Gibbons, G.W., Lu, H., Page, D.N., Pope, C.N.: Rotating black holes in higher dimensions with a cosmological constant. Phys. Rev. Lett. 93, 171102 (2004) 16. Harmark, T.: Stationary and axisymmetric solutions of higher-dimensional general relativity. Phys. Rev. D 70, 124002 (2004) 17. Harmark, T., Olesen, P.: Structure of stationary and axisymmetric metrics. Phys. Rev. D 72, 124017 (2005) 18. Hawking, S.W.: Black holes in general relativity. Commun. Math. Phys. 25, 152–166 (1972) 19. Hawking S.W., Ellis G.F.R.: The large scale structure of space-time. Cambridge: Cambridge University Press, 1973 20. Isenberg, J., Moncrief, V.: Symmetries of cosmological Cauchy horizons with exceptonal orbits. J. Math. Phys. 26, 1024–1027 (1985) 21. Isenberg, J., Moncrief, V.: On spacetimes containing Killing vector fields with non-closed orbits. Class. Quantum Grav. 9, 1683–1691 (1992) 22. Isenberg, J., Moncrief, V.: Work in progress 23. Israel, W.: Event horizons in static vacuum space-times. Phys. Rev. 164, 1776–1779 (1967) 24. Israel, W.: Event horizons in electrovac vacuum space-times. Commun. Math. Phys. 8, 245–260 (1968) 25. Mazur, P.O.: Proof of uniqueness of the Kerr-Newman black hole solution. J. Phys. A, 15, 3173– 3180 (1982) 26. Mishima, T., Iguchi, H.: New axisymmetric stationary solutions of five-dimensional vacuum Einstein equations with asymptotic flatness. Phys. Rev. D 73, 044030 (2006) 27. Moncrief, V., Isenberg, J.: Symmetries of cosmological Cauchy horizons. Commun. Math. Phys. 89, 387– 413 (1983) 28. Morisawa, Y., Ida, D.: A boundary value problem for the five-dimensional stationary rotating black holes. Phys. Rev. D 69, 124005 (2004) 29. Müller zum Hagen, H.: Characteristic initial value problem for hyperbolic systems of second order differential systems. Ann. Inst. Henri Poincaré 53, 159–216 (1990) 30. Myers, R.C., Perry, M.J.: Black holes in higher dimensional space-times. Ann. Phys. 172, 304 (1986) 31. Nomizu, K.: On local and global existence of Killing vector fields. Ann. Math. 72, 105–120 (1960) 32. Racz, I.: On further generalization of the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon. Class. Quant. Grav. 17, 153 (2000) 33. Racz, I., Wald, R.M.: Extensions of spacetimes with Killing horizons. Class. Quantum Grav. 9, 2643– 2656 (1992) 34. Racz, I., Wald, R.M.: Global extensions of spacetimes describing asymptotic final states of black holes. Class. Quantum Grav. 13, 539–552 (1996) 35. Rendall, A.: Reduction of the characteristic initial value problem to the Cauchy problem and its application to the Einstein equations. Proc. Roy. Soc. Lond. A 427, 211–239 (1990) 36. Robinson, D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34, 905–906 (1975) 37. Rogatko, M.: Uniqueness theorem for static black hole solutions of sigma models in higher dimensions. Class. Quant. Grav. 19, L151 (2002) 38. Rogatko, M.: Uniqueness theorem of static degenerate and non-degenerate charged black holes in higher dimensions. Phys. Rev. D 67, 084025 (2003) 39. Rogatko, M.: Uniqueness theorem for generalized Maxwell electric and magnetic black holes in higher dimensions. Phys. Rev. D 70, 044023 (2004) 40. Rogatko, M.: Uniqueness theorem for stationary black hole solutions of sigma-models in five dimensions. Phys. Rev. D 70, 084025 (2004) 41. Rogatko, M.: Staticity theorem for higher dimensional generalized Einstein-Maxwell system. Phys. Rev. D 71, 024031 (2005) 42. Sudarsky, D., Wald, R.M.: Extrema of mass, stationarity, and staticity, and solutions to the Einstein Yang-Mills equations. Phys. Rev. D 46, 1453–1474 (1992) 43. Tomizawa, S., Morisawa, Y., Yasui, Y.: Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method. Phys. Rev. D 73, 064009 (2006) 44. Walters, P. An Introduction to Ergodic Theory. New York: Springer-Verlag 1982 Communicated by G.W. Gibbons
Commun. Math. Phys. 271, 723–773 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0187-x
Communications in
Mathematical Physics
Tensor Gauge Fields in Arbitrary Representations of G L( D, R): II. Quadratic Actions Xavier Bekaert1 , Nicolas Boulanger1,2, 1 Institut des Hautes Études Scientifiques, Le Bois-Marie, 35 route de Chartres, 91440 Bures-sur-Yvette,
France. E-mail: [email protected]
2 Université de Mons-Hainaut, Mécanique et Gravitation, 6 avenue du Champ de Mars, 7000 Mons, Belgium.
E-mail: [email protected] Received: 26 June 2006 / Accepted: 10 August 2006 Published online: 31 January 2007 – © Springer-Verlag 2007
Abstract: Quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary irreducible representations of the general linear group in Ddimensional Minkowski space are explicitly written in a compact form by making use of Levi–Civita tensors. The field equations derived from these actions ensure the propagation of the correct massless physical degrees of freedom and are shown to be equivalent to non-Lagrangian local field equations proposed previously. Moreover, these actions allow a frame-like reformulation à la MacDowell–Mansouri, without any trace constraint in the tangent indices. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Completely Symmetric Tensor Gauge Fields . . . . . . . . . . . . . 2.1 Bargmann–Wigner programme . . . . . . . . . . . . . . . . . . 2.1.1 Local, constrained approach of Fronsdal. . . . . . . . . . . 2.1.2 Curvature tensors of de Wit, Freedman and Weinberg. . . . 2.1.3 Non-local, unconstrained approach of Francia and Sagnotti. 2.1.4 Higher-derivative, unconstrained approach. . . . . . . . . . 2.2 Fierz–Pauli programme . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Non-local actions of Francia and Sagnotti. . . . . . . . . . . 2.2.2 Non-local actions in terms of differential forms. . . . . . . . 2.2.3 Non-local actions à la MacDowell and Mansouri. . . . . . . 3. Mixed-Symmetry Tensor Gauge Fields . . . . . . . . . . . . . . . . 3.1 Bargmann–Wigner programme . . . . . . . . . . . . . . . . . . 3.1.1 Local, constrained approach of Labastida. . . . . . . . . . . 3.1.2 Higher derivative, unconstrained approach. . . . . . . . . . Chargé de Recherches FNRS, Belgium
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724 726 727 727 728 730 731 733 733 734 736 740 740 740 741
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3.1.3 Non-local, unconstrained approach of de Medeiros and Hull. 3.1.4 Bargmann–Wigner equations. . . . . . . . . . . . . . . . . 3.2 Fierz–Pauli programme . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Local actions. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Non-local actions. . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Proof of Theorem 2. . . . . . . . . . . . . . . . . . . . . . A. Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . A.1 Young diagrams and tensorial representations . . . . . . . . . . A.1.1 Young diagrams and irreducible representations. . . . . . . A.1.2 Multiform and hyperform algebras. . . . . . . . . . . . . . A.2 Differential complexes . . . . . . . . . . . . . . . . . . . . . . A.2.1 Multicomplex of differential multiforms. . . . . . . . . . . A.2.2 Generalized complex of differential hyperforms. . . . . . . B. Technical Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . C. Light-cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Combining the principle of relativity with the rules of quantum mechanics implies that linear relativistic wave equations describing the free propagation of relativistic particles in Minkowski space are in one-to-one correspondence with unitary representations of the Poincaré group. Using the method of induced representations, Wigner showed in 1939 that the unitary irreducible representations (UIRs) of the Poincaré group I S O0 (3, 1) are completely characterized by two real numbers: the mass-squared m 2 and the spin1 s of the corresponding particle [1]. Physical considerations2 further impose m 2 0 (no tachyon) and 2s ∈ N (discrete spin). The Bargmann-Wigner programme amounts to associating, with any given UIR of the Poincaré group, a manifestly covariant differential equation whose positive energy solutions transform according to the corresponding UIR. In 1948, this programme was completed in four dimensions when, for each UIR of I S O0 (3, 1) , a relativistic wave equation was written whose positive energy solutions transform according to the corresponding UIR [2]. This programme is the first step towards the completion of the Fierz-Pauli programme which consists in writing a manifestly covariant quadratic action for each first-quantized elementary particle propagating in Minkowski spacetime. In four spacetime dimensions, the latter programme was initiated in 1939 [3] and completed in the seventies by Singh and Hagen for the massive case (m 2 > 0) [4] and by Fronsdal and Fang for the massless case (m 2 = 0) [5, 6]. The description of free massless (massive) gauge fields in D = 4 has thus been known for a long time and is tightly linked with the representation theory of S O(2) ∼ = U (1) (respectively Spin(3) ∼ = SU (2) ). This case is very particular because all non-trivial irreducible representations (irreps) of these compact groups are exhausted by the completely symmetric tensor-spinors, pictured by a one-row Young diagram with [s] columns for a spin-s particle (where [n] denotes the integer part of n). The Bargmann–Wigner programme generalizes to the Poincaré group I S O0 (D − 1, 1). When D > 4 , more complicated Young diagrams appear whose analysis requires 1 In the massless case, the discrete label s is more accurately called helicity, but we use the naming “spin” whenever the mass of the particle is positive or zero. 2 In this paper, we will not consider infinite-dimensional representations of the little group.
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appropriate mathematical tools, as introduced in [7–10]. For tensorial representations, the word “spin” will denote the number of columns possessed by the corresponding Young diagram. From now on, we restrict the analysis to massless UIRs induced by representations of the “little group” S O(D − 2) for D 5, because each massive representation in D − 1 dimensions may actually be obtained as the first Kaluza–Klein mode in a dimensional reduction from D down to D − 1 . There is no loss of generality because the massive little group S O(D − 2) in D − 1 dimension is identified with the D-dimensional massless little group. Such a Kaluza–Klein mechanism leads to a Stückelberg formulation of the massive field [11]. An analysis of the gauge structure for arbitrary mixed-symmetry tensor gauge fields φY was undertaken in [9, 10]. The results of Dubois-Violette and Henneaux [8] for rectangle-shaped Young-diagram tensor representations were extended to arbitrary tensor representations of G L(D, R) . Guided by the duality symmetry principle, through a systematic study, in [9] we proposed a general local field equation which applies to tensor gauge fields φY in arbitrary irreps of G L(D, R) and generalizes the Bargmann–Wigner equations [2] of D = 4 to any spacetime dimension D 3 . The fermionic case goes along the same lines [12]; for this reason, we will restrict ourselves to tensorial representations of the Poincaré group in this paper. In a work [13] on completely symmetric higher-spin (s > 2) tensor gauge fields φs , Francia and Sagnotti discovered that foregoing locality allows to relax the trace conditions of the Fronsdal formulation. They wrote a non-local field equation which involves the de Wit–Freedman curvature [14] and which was shown to be equivalent to Fronsdal’s field equation, after gauge-fixing.3 The authors of [15] followed another path: For completely symmetric tensor fields φs of rank s > 0 they constructed field equations derived from actions S ∼ d D x φs · G(φs ) , where the “Einstein tensor” G(φs ) is higher-derivative and divergence-free, ∂ · G(φs ) = 0 . It contains 2[ s+1 2 ] = s + ε(s) derivatives of the field (where ε(n) denotes the parity of the natural number n ∈ N: its value is zero if n is even, or one if n is odd). Subsequently, in [16] we proved that, restricted to completely symmetric tensor gauge fields φs , the field equation proposed in [9] was equivalent to Fronsdal’s field equation and we further conjectured the validity of the same field equation in the arbitrary mixed-symmetry tensor gauge field φY case. This conjecture was verified explicitly on a simple mixed-symmetry higher-spin tensor gauge field example.4 In the same work [16], we then showed that both works [13] and [15] were actually equivalent, provided one multiplied the higher-derivative Einstein-like tensor G(φs ) of [15] by an appropriate power of the non-local inverse d’Alembertian operator −1 , thereby recovering the nonlocal action of [13]. In light of this observation, the authors of [15] reconsidered their previous work in [19] and inserted the fermionic case along the lines of [13]. They also conjectured a schematic form of the Einstein-like tensor G(φY ), where φY , transforms in an arbitrary irrep. of G L(D, R) . In the present work we pursue this investigation and provide the explicit expression for the higher-derivative Einstein-like tensor G(φY ) corresponding to a field transforming in an arbitrary irrep. of G L(D, R) . The field equation derived from the action (s > 0) 3 Actually, Fronsdal’s action S [φ ] = d 4 x L F (φ ) trivially extends to D dimensions [14]: S [φ ] = s 4 s D s D F F d x L (φs ). The Lagrangian L (φs ) is independent of the dimension D . 4 That the aforementioned field equation is correct for an arbitrary mixed-symmetry tensor gauge field φY was finally proved in [17], thereby generalizing Bargmann–Wigner’s programme to arbitrary dimension D 4 . Actually, the latter programme had previously been completed in [18] with different equations.
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S[φY ] =
d D x φY ·
1 [
s−1 2 ]
G(φY )
(1)
is then shown to be equivalent to the field equation of [9, 16, 17] which propagates the correct massless physical degrees of freedom. The quadratic Lagrangian is always of second order but non-local for fields of higher-spin s > 2. The corresponding field equation sets to zero all traces of the generalized curvature tensor KY introduced in [9]: Tr KY ≈ 0 , where the weak equality X ≈ 0 means “X is equal to zero on the surface of the field equations” (or, “on-shell”). As a preliminary result of the present work, the non-local quadratic action [13] of Francia and Sagnotti is rewritten in a compact and suggestive form by using Levi-Civita tensors. Moreover, we express these actions in a frame-like fashion thereby providing a bridge between the local constrained approach of Vasiliev [20] and the non-local unconstrained approach. Indeed, we show that the latter action may be obtained as a flat spacetime limit of a MacDowell–Mansouri-like action in constant-curvature background, where the gauge fields and parameters are unconstrained, in contrast with Vasiliev’s formalism. The plan of the paper is as follows. In Sect. 2, we first review the various approaches to higher-spin symmetric tensor gauge fields in flat spacetime. Subsection 2.2.3 proposes an extension of the non-local action for the unconstrained frame-like approach to constant-curvature spacetimes. Mixed-symmetry tensor gauge fields φY are studied in Sect. 3, where we recall our results (Theorem 1) on the completion of the Bargmann– Wigner programme, writing in detail most of the intermediate steps in the proof.5 Our main result (Theorem 2) is presented in Subsect. 3.2.2 where a non-local second-order covariant quadratic action is given for each inequivalent UIR of the Poincaré group, thereby completing the Fierz–Pauli programme in arbitrary dimension D 4 . Three appendices follow. In Appendix A, we systematically introduce our notation by reviewing all the mathematical machinery on irreps necessary for our purpose. We also summarize some former results on the gauge structure of mixed-symmetry tensor fields. The proofs of some technical lemmas are relegated to Appendix B while Appendix C contains the proof of Theorem 1 which states that the Bargmann–Wigner equations presented in [9, 16, 17] restrict the physical components of a tensor gauge field φY to an UIR of the little group O(D − 2) . 2. Completely Symmetric Tensor Gauge Fields Completely symmetric tensors φµ1 ...µs = φ(µ1 ...µs ) of rank s correspond to a Young tableau6 made of one row with s cells. This is the simplest case of irreducible tensors under G L(D, R) associated with a Young diagram made of s columns, thus we fix the main ideas on this specific example since it already exhibits the prominent properties of the general case. Einstein’s gravity theory is a non-Abelian massless spin-2 field theory, the two main formulations of which are the “metric” and the “frame” approaches. In a very close analogy, there exist two main approaches to higher-spin (i.e. spin s > 2) field theories that are by-now referred to as “metric-like” [5, 14] and “frame-like” [21, 20]. In the 5 Because these lemmas and other intermediate results were either spread in the literature or not yet published in full detail. 6 The reader unfamiliar with Young tableaux may read the brief introduction to the tensorial irreps of G L(D, R) in Subsect. A.1.1.
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former approach, the components of the massless field φs transform in the irreducible representation of the general linear group which is labeled by a Young diagram Y made of s columns. Both metric-like and frame-like approaches may be divided into two subclasses called the “constrained” and “unconstrained” approaches according to whether trace constraints are imposed or not on the gauge fields and parameters. 2.1. Bargmann–Wigner programme. Not all covariant wave equations that would describe proper physical degrees of freedom are Euler-Lagrange equations for some Lagrangian. Therefore, we prefer to separate the discussion of the linear field equations from the discussion on quadratic Lagrangians for symmetric tensor gauge fields. 2.1.1. Local, constrained approach of Fronsdal. The local spin-s field equation of [5, 14] states that the Fronsdal tensor F vanishes on-shell Fµ1 ...µs ≡ φµ1 ...µs − s ∂ α ∂(µ1 φµ2 ...µs )α +
s(s − 1) ∂(µ1 ∂µ2 Tr φµ3 ...µs ) ≈ 0 , (2) 2
where Tr stands for the trace operator and curved (respectively square) brackets denote complete symmetrization (antisymmetrization) with strength one. The gauge transformations are δφµ1 ...µs = s ∂(µ1 µ2 ...µs ) .
(3)
s(s − 1)(s − 2) ∂(µ1 ∂µ2 ∂µ3 Tr µ4 ...µs ) , 2
(4)
Since (3) transforms F as δFµ1 ...µs =
the gauge parameter µ2 ...µs is constrained to be traceless, Tr = 0 , in order to leave the field equation (2) invariant. Eventually, the standard de Donder gauge-fixing condition Dµ2 ...µs ≡ ∂ α φαµ2 ...µs −
(s − 1) ∂(µ2 Tr φµ3 ...µs ) = 0 2
(5)
is used to reduce the Fronsdal equation (2) to its canonical form φµ1 ...µs ≈ 0 . In order that Dµ2 ...µs = 0 contains as many conditions as the number of independent components of the gauge parameter , the gauge potential φ must be double-traceless, Tr2 φ = 0 . As shown in [14], this gauge theory leads to the correct number of physical degrees of freedom, that is, the dimension of the irrep. of the little group O(D − 2) corresponding to the one-row Young diagram of length s. The main advantage of the Fronsdal approach to free massless fields is that it respects the following two requirements of orthodox quantum field theory: (i) Locality, (ii) Second-order field equations (for bosonic fields). Theories for which the second requirement is violated, i.e. the field equations contain the n th derivatives of the bosonic field with n > 2, are called “higher-derivative”. Roughly speaking, non-local theories are a particular case of higher-derivative theories where the order in the derivatives is infinite, n = ∞. Both requirements (i) and (ii) are related to the no-go theorem of Pais and Uhlenbeck on free quantum field theories with a higherderivative kinetic operator for the propagating degrees of freedom [22]. They proved that for such a kinetic operator, the quantum field theory cannot be simultaneously stable
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(bounded energy spectrum), unitary and causal. In modern language, one would say that the field theory contains “ghosts”. Notice that the Pais–Uhlenbeck no-go theorem does not imply that all higher-derivative theories are physically sick. For instance, at least three harmless violations of the requirements (i) or (ii) have been suggested in the physics literature: (a) “Gauge artifact”: The ghosts associated with the higher-derivatives correspond to spurious “gauge” degrees of freedom. More precisely, in a proper gauge, the physical degrees of freedom propagate according to local second-order field equations. For instance, the worldsheet non-local action of the non-critical bosonic string is obtained from the Polyakov action by integrating out the the massless scalar fields describing the coordinates of the string in the target space [23]. In the conformal gauge, it reduces to the local Liouville action for the scalar field associated with the conformal factor. (b) “Perturbative cure”: The theory admits a perturbative expansion with an orthodox free limit. One can prove that, if the higher-derivatives are present in the perturbative interaction terms only, then they may be replaced with lower-derivative terms order by order [24]. This perturbative cure is perfectly justified when the higher-derivative theory is the effective field theory of a more fundamental orthodox theory, the higher-derivative terms corresponding to perturbative corrections. A good example of perturbatively non-local effective field theory is Wheeler–Feynman’s electrodynamics in which the degrees of freedom of the electromagnetic field are frozen out. Another one is the α -expansion in string theory. (c) “Non-perturbative miracle”: The possibility remains that the higher-derivative quantum field theory is consistent in the non-perturbative regime but does not admit a reasonable free limit. Such a possibility has been raised for conformal gravity [25] which is of fourth order, but it has never been proved that such a scenario indeed works. 2.1.2. Curvature tensors of de Wit, Freedman and Weinberg. The main drawback of Fronsdal’s approach is the presence of algebraic constraints on the fields. They introduce several technical complications and are somewhat unnatural. To get rid of these trace constraints, it is necessary to relax one of the two requirements (i) or (ii) of orthodox quantum field theory in one of the harmless ways explained in the previous subsection. This is the path followed by higher-spin gauge fields in order to circumvent the conclusions of the Pais–Uhlenbeck no-go theorem. Indeed, all known formulations of free massless higher spin fields exhibit new features with respect to lower-spin (s 2) fields (e.g. trace conditions, non-locality or higher-derivative kinetic operators, auxiliary fields, etc). These unavoidable novelties of higher spins are deeply rooted in the fact that the curvature tensor, that is presumably the central object in higher-spin theory, contains s derivatives. A major progress of the recent approaches to higher-spin fields was to produce “geometric” field equations, i.e. equations written explicitly in terms of the curvature. The curvature tensor Rµ1 ...µs ; ν1 ...νs of de Wit and Freedman [14] and the curvature tensor Kµ1 ν1 |... | µs νs of Weinberg [26] are essentially the projection of ∂µ1 . . . ∂µs φν1 ...νs , the s th derivatives of the gauge field, on the tensor field irreducible under G L(D, R) with symmetries labeled by the Young tableau ... µ1 µ2 µs (6) ... ν1 ν2 νs .
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The Weinberg and de Wit–Freedman tensors are simply related by a choice of symmetry convention. In the case s = 2, the de Wit–Freedman curvature tensor precisely is the Jacobi tensor while the Weinberg tensor coincides with the Riemann tensor. In the case s = 3, they are related by Rµ1 ν1 ρ1 ; µ2 ν2 ρ2 = K
µ1 ν1 ρ1 (µ2 | ν2 | ρ2 ) ,
(7)
and Kµ1 ν1 | µ2 ν2 | µ3 ν3 = 2 R[µ1 [µ2 [µ3 ; ν1 ]ν2 ]ν3 ] ,
(8)
where the three antisymmetrizations are taken over every pair of indices (µi , νi ). (We refer to Appendix A.1.1 for the notations.) The Weinberg tensor is in the antisymmetric convention for which the projection is more easy to perform because, since ∂µ1 . . . ∂µs φν1 ...ν2 is already symmetric in all indices of the two rows of the Young tableau (6), it only remains to antisymmetrize over all pairs (µi , νi ). This corresponds to taking s curls of the symmetric tensor field φs . On the one hand, the Weinberg tensor is, by construction, antisymmetric in each of the s sets of two indices K[µ1 ν1 ] |... | µs νs = . . . = Kµ1 ν1 |... | [µs νs ] = Kµ1 ν1 |... | µs νs .
(9)
Moreover, the complete antisymmetrization over any set of three indices gives zero, so that the Weinberg tensor indeed belongs to the space irreducible under G L(D, R) characterized by a two-row rectangular Young diagram of length s . On the other hand, the de Wit–Freedman tensor is, by definition, symmetric in each of the two sets of s indices R(µ1 ...µs ) ; ν1 ...νs = Rµ1 ...µs ; (ν1 ...νs ) = Rµ1 ...µs ; ν1 ...νs .
(10)
Moreover, it obeys the algebraic identity R(µ1 ...µs ; ν1 )ν2 ...νs = 0 ,
(11)
so that it also belongs to the space irreducible under G L(D, R) characterized by a tworow rectangular Young diagram of length s. Both definitions of the curvature tensor are equivalent, in the sense that they define the same tensor space invariant under the action of G L(D, R). Due to these symmetries, the curvature tensors are strictly invariant under gauge transformations (3) with unconstrained gauge parameter µ1 ...µs−1 . Indeed, if the indices of two partial derivatives appear in the same column, the corresponding irreducible tensor vanishes. For the same reason, the irreducible components of the partial derivative of the de Wit–Freedman tensor ∂ρ Rµ1 ...µs ; ν1 ...νs which are labeled by the Young tableau µ1 µ2 ν1 ν2 ρ
... ...
µs νs
,
identically vanish. In terms of the Weinberg tensor, this translates into the “Bianchi identity” ∂[ρ Kµ1 ν1 ] |... | µs νs = 0 .
(12)
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A generalization of the Poincaré lemma states that the differential Bianchi-like identity (12) together with the previous algebraic irreducibility conditions on K imply that the Weinberg tensor is the s th derivative of a symmetric tensor field of rank s [7, 8]. The same theorem states that the most general pure-gauge tensor field for which the curvature vanishes identically is a symmetrized derivative of a symmetric tensor field of rank s − 1. The gauge structure of symmetric tensor gauge fields was elegantly summarized by Dubois-Violette and Henneaux in terms of generalized cohomologies [8] (see Sect. A.2 for a brief review of these concepts). 2.1.3. Non-local, unconstrained approach of Francia and Sagnotti. The field equations proposed by Francia and Sagnotti [13] for unconstrained completely-symmetric tensor gauge fields are non-local, but they are invariant under gauge transformations (3) where the trace of the completely-symmetric tensor gauge parameter is not constrained to vanish. They read s−2 ηµ1 µ2 . . . ηµs−1 µs − 2 Rµ1 ...µs ; ν1 ...νs ≈ 0 for s even , (13) s−1 ηµ1 µ2 . . . ηµs µs+1 − 2 ∂µs+1 Rµ1 ...µs ; ν1 ...νs ≈ 0 for s odd , where Rµ1 ...µs ; ν1 ...νs is the spin-s curvature tensor introduced by de Wit and Freedman. Putting it in words, the geometric equations (13) for completely symmetric tensor gauge fields φs are easily constructed: When s is odd one takes one divergence together with s−1 2 trace(s) of the tensor Rµ1 ...µs ; ν1 ...νs , and when s is even one just takes s/2 trace(s) [13]. So one constructs a gauge-invariant object with the symmetries of the field of rank s but containing s + ε(s) derivatives. Consequently, the authors of [13] further multiplied s+ε(s) by 1− 2 in order to get second-order field equations. Via algebraic manipulations, the field equations (13) for rank-s completely symmetric tensor fields have been shown [13] to be equivalent to Fµ1 µ2 µ3 µ4 ...µs − ∂(µ1 ∂µ2 ∂µ3 Hµ4 ...µs ) ≈ 0 ,
(14)
where the tensor Hµ1 ...µs−3 is a non-local function of the field φµ1 ...µs and its derivatives, whose gauge transformation is proportional to the trace of the gauge parameter. The gauge-fixing condition Hµ1 ...µs−3 = 0 leads to the Fronsdal equation (2). Therefore, this geometric formulation of higher-spin gauge fields falls into the class (a) of harmless non-locality. Basically, the main additional subtlety arising for spin s 4 is that the usual de Donder condition is reachable with a traceless gauge parameter if and only if the double trace of the field vanishes. Therefore, in the Fronsdal approach the field is constrained to have vanishing double trace (which is consistent with the invariance of the double trace of the field under gauge transformations with traceless parameter). As pointed out in [27], more work is therefore required in order to obtain the double-trace condition for spin s 4 in the unconstrained approach. A solution is to take a modified identically traceless de Donder gauge [27]. After this further gauge-fixing, the field equation implies the vanishing of the double trace of the field, thereby recovering the usual de Donder condition. Heuristically, one can also argue that the non-local field equations (13) are equivalent to local ones (2) by going in a traceless-transverse gauge (i.e. Tr φ = 0 and ∂ · φ = 0), because both equations reduce to the Klein–Gordon equation φ ≈ 0 since the powers of the d’Alembertian cancel in the non-local approach. Of course, rigorously speaking, we should prove that this rule applies for the formal object −1 . We take this opportunity to briefly discuss the meaning given to the inverse d’Alembertian in the non-local
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unconstrained approach, and in which sense local higher-derivative field equations may be equivalent to non-local second-order field equations. Regarding −1 , we note that an obvious way of defining a pseudodifferential operator (such as 1/) is through its Fourier transform, because the latter simply is a non-polynomial function of the momentum (such as −1/ p 2 ), a much less frightening object. The second comment is that any linear application A on a vector space V is invertible on the quotient V /Ker A ∼ = ImA. (More concretely, let w = Av be in ImA, then one may write v = A−1 w + u with u ∈ KerA.) The third comment is that the representatives in the quotient Kern /Ker for n > 1 are usually called “runaway solutions” because they are unbounded at infinity. These solutions are the classical counterparts of the ghosts in the quantum theory, so one rejects them on physical grounds. In mathematical terms, one requires the solutions to be in an appropriate functional class such that Kern = Ker (for all n > 1). In this restricted sense, the non-local equations (13) and the following higher-derivative equations: ηµ1 µ2 . . . ηµs−1 µs Rµ1 ...µs ; ν1 ...νs ≈ 0 for s even , (15) µ η 1 µ2 . . . ηµs µs+1 ∂µs+1 Rµ1 ...µs ; ν1 ...νs ≈ 0 for s odd , are thus equivalent at the level of sourceless free field equations. Nevertheless, this equivalence of the equations of motion does not imply the equivalence of the variational principle of course and, thus, does not contradict the Pais-Uhlenbeck no-go theorem on higher-derivative Lagrangians [22]. This being said, from now on we refer to (13) or (15) without any distinction. It is convenient to rewrite the Francia–Sagnotti equations (15) in terms of the Weinberg tensor in order to generalize them to mixed-symmetry tensor gauge fields more easily: η(ν1 ν2 . . . ηνs−1 νs ) Kµ1 ν1 |... | µs νs ≈ 0 for s even , (16) (ν η 1 ν2 . . . ηνs νs+1 ) ∂νs+1 Kµ1 ν1 |... | µs νs ≈ 0 for s odd , where the symmetrization over all indices ν of the Minkowski metrics is important in order to have the proper symmetries on the free indices µi , 1 i s . 2.1.4. Higher-derivative, unconstrained approach. The compensator field equation for symmetric tensor fields [13, 27] (generalized later to completely symmetric tensor-spinor fields [28]) Fµ1 µ2 µ3 µ4 ...µs −
s(s − 1)(s − 2) ∂(µ1 ∂µ2 ∂µ3 αµ4 ...µs ) ≈ 0 2
(17)
is the same as (14) except that the symmetric tensor αµ1 ...µs−3 of rank s − 3 is an independent field, called “compensator”. It is a pure-gauge field whose gauge transformation δαµ1 ...µs−3 = (Tr )µ1 ...µs−3
(18)
precisely cancels the contribution (4) coming from the Fronsdal tensor so that (17) is invariant under gauge transformations with unconstrained gauge parameter. The compensator field may be gauged away by using the freedom (18), which gives the Fronsdal equation (2). Fixing α = 0 is called the “Fronsdal gauge”, where the constraint Tr = 0 is imposed on the gauge parameter. Again, in order to recover the double trace constraint Tr2 φ = 0 on the gauge field more work is necessary [28]. The “Ricci curvature tensor” (Tr R)µ1 ...µs ; ν1 ...νs−2 is the trace of the de Wit–Freedman tensor. Its symmetries are encoded in the Young tableau
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µ1
µ2
...
µs−2 µs−1
ν1
ν2
...
νs−2
µs
.
(19)
The Damour–Deser identity [29] schematically written TrK = d s−2 F relates the Riccilike tensor TrR to the (s − 2)th curl of the Fronsdal tensor F . These curls are obtained by projecting the (s − 2)th partial derivative ∂ν1 . . . ∂νs−2 Fµ1 ...µs of the Fronsdal tensor on the irreducible component labeled by (19) via the antisymmetrization over the pairs (µi , νi ) for 1 i s − 2 . Consequently, the compensator equation (17) implies the higher-derivative “Ricci-flat” equation (TrR)µ1 ...µs ; ν1 ...νs−2 ≈ 0
⇐⇒
(TrK)µ1 ν1 |...... | µs−2 νs−2 | µs−1 | µs ≈ 0 .
(20)
Conversely, Eq. (20) and the Damour–Deser identity imply that the s − 2th curl of the Fronsdal tensor F vanishes on-shell. As was explained in [16], the generalized Poincaré lemma of [7, 8] shows7 the equivalence of this “closure” condition d s−2 F ≈ 0 of the Fronsdal tensor to its “exactness” expressed by the compensator equation (17). In other words, the field equations (17) and (20) are strictly equivalent in a flat spacetime with trivial topology. Notice that both of them are higher-derivative when s > 2, the compensator equation being of third order and the Ricci-flat-like equation being of s th order. Furthermore, the Ricci-flat-like equation (20) is equivalent to a set of first-order field equations. In D = 4, they correspond to the Bargmann–Wigner equations [2], originally expressed in terms of two-component tensor-spinors in the representation (s, 0) ⊕ (0, s) of S L(2, C). They were generalized to D > 4 in [9, 16] for arbitrary tensorial UIRs of the Poincaré group, and in [12] for spinoral UIRs. The main idea is to start with a tensor field that is (on-shell) irreducible under the Lorentz group O(D − 1, 1) with symmetries labeled by the Young tableau depicted by (6). The antisymmetric convention proves to be more convenient so one considers a (on-shell) traceless tensor field whose components Kµ1 ν1 |... | µs νs obey the G L(D, R) irreducibility conditions explained in Subsect. 2.1.2. One then requires that it also obeys the Bianchi-like identity (12), which is equivalent to the fact that the tensor K is the Weinberg curvature of a completelysymmetric tensor gauge field φs of rank s. The on-shell tracelessness Tr K ≈ 0 of the irreducible tensor is therefore equivalent to the Ricci-flat-like equation (20) if the tensor K obeys the differential Bianchi identity (12). Finally, one can also show that the (on-shell) O(D − 1, 1)-irreducibility conditions combined with the differential Bianchi identity imply that the tensor field is divergenceless on-shell ∂ · K ≈ 0 . In summary, the equations
∂[ρ Kµ1 ν1 ] | µ2 ν2 |... | µs νs = 0 , ∂ ρ Kρν1 | µ2 ν2 |... | µs νs ≈ 0
(21)
imposed on a tensor field K taking values in an irreducible representation of the group O(D − 1, 1), are equivalent to the Ricci-flat-like equations (20) and thereby to all other field equations of symmetric tensor gauge fields alike. 7 We insist on the fact that it was not necessary to make use of the de Wit–Freedman connections to derive this result since the Poincaré lemma allows a direct jump from the Ricci-flat-like equation to the compensator equation.
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2.2. Fierz–Pauli programme. Fronsdal was able to write down a local second-order action, quadratic in the double-traceless gauge field φ and invariant under the gauge transformations (3) with traceless parameter [5]. Moreover, Curtright pointed out that these requirements fix the Lagrangian uniquely, up to an overall factor [30]. The EulerLagrange equation derived from Fronsdal’s action is equivalent to (2). Notice that by introducing a pure gauge field (sometimes referred to as “compensator”), it is possible to write a local (but higher-derivative) action for spin-3 [13, 27] that is invariant under unconstrained gauge transformations. Very recently, this action was generalized to the completely symmetric spin-s case by further adding an auxiliary field associated with the double trace of the gauge field [31]. Retrospectively, the reference [32] may be interpreted as an older “non-minimal” version of it, as explained in more details in [28] (see also [33] for the fermionic counterpart of [32]). 2.2.1. Non-local actions of Francia and Sagnotti. In this subsection, we introduce a compact expression for the “Einstein tensors” of [13] by using Levi-Civita “epsilon” tensors. In this way, it is much simpler to write the Einstein-like tensor, and the Noether (sometimes referred to as “Bianchi”) identity is automatically satisfied without explicitly introducing the trace expansion as in [13]. Since the Levi-Civita tensors are involved it is natural to use the antisymmetric convention for Young tableaux, so the starting point is the Francia–Sagnotti equations (16) in terms of the Weinberg tensor K . It turns out to be convenient to introduce the symmetric tensor ηµ1 ...µ2n of rank 2n defined by η µ1 µ2 µ3 µ4 ... µ2n−1 µ2n := η(µ1 µ2 ηµ3 µ4 . . . ηµ2n−1 µ2n ) ,
(22)
for all integers n ∈ N, corresponding to the product of n metrics with all indices symmetrized. The Einstein-like tensor G µ1 µ2 ...µs−1 µs := s even , ε µ1 ν1 ...ρ1 σ1 τ1 . . . ε µs νs ...ρs σs τs ην1 ... νs . . . ηρ1 ...ρs Kσ1 τ1 |... | σs τs ε µ1 ν1 ...ρ1 σ1 τ1 . . . ε µs+1 νs+1 ...ρs+1 σs+1 τs+1 ην1 ... νs+1 . . . ηρ1 ...ρs+1 ηµs+1 τ1 ∂σ1 Kσ2 τ2 |... | σs+1 τs+1 s odd , (23)
is defined via traces of the Hodge dual on every set of antisymmetric indices of the Weinberg tensor. In the even spin case, the symmetry under the exchange of two µi indices is a consequence of the symmetry properties of the curvature tensor K under the exchange of pairs (σi , τi ) of antisymmetric indices together with the symmetry properties of the tensor η defined in (22). In the odd spin case, the symmetry is not automatic and, actually, one must understand that there is an implicit symmetrization over the µ indices in the second line of (23). By taking traces, etc., one may show that the Einstein-like equations G µ1 µ2 ...µs ≈ 0 are algebraically equivalent to Eqs. (16) of Francia and Sagnotti [13]. The Einstein-like tensor (23) is automatically gauge invariant under (3) because it is a linear combination of the curvature tensor. The Noether identity corresponding to the gauge transformations (3) with unconstrained parameters is the divergencelessness of the Einstein-like tensor, ∂µ1 G µ1 µ2 ...µs = 0 , which follows from the Bianchi-like identity (12) obeyed by the Weinberg tensor. The Einstein-like tensor contains a product of D − 3 symmetric tensors ην1 ...νs+ε(s) . One may rewrite the traces over the Levi-Civita tensors as products of Kronecker symbols ν ...ν
ν ]
ν
ν1 δµ11 ...µpp ≡ δµ[ν11 . . . δµpp = δ[µ . . . δµpp ] 1
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via the identity ν ...ν
εµ1 ...µ p ρ1 ...ρ D− p εν1 ...ν p ρ1 ...ρ D− p = − p ! (D − p)! δµ11 ...µpp .
(24)
This leads to an expansion of the Einstein-like tensor as a sum of the product of metrics times traces of the [ 2s ]th trace of the curvature tensor written in (16): for s even , ην1 ...νs Kµ1 ν1 |... | µs νs + . . . Gµ1 ...µs ∝ (25) ν 1 η ...νs+1 ∂νs+1 Kµ1 ν1 |... | µs νs + . . . for s odd . The coefficients in the expansion of the Einstein-like tensor were determined uniquely in [13] by imposing that the Noether identity be obeyed. Therefore, the Einstein-like tensor (23) must correspond to the one of Francia and Sagnotti, up to an overall coefficient. The conclusion of the discussion on the negative powers of the d’Alembertian in Subsect. 2.1.3 is that one cannot remove them in the Lagrangian of the non-local approach without introducing ghosts, but that one can remove them in the Euler-Lagrange equations provided that the ghosts are eliminated “by hand” by choosing an appropriate functional space of allowed solutions. The authors of [13] proposed an action of the 1 form d D x φ · [ s−1 G(φ) . In the form chosen here, this prescription leads to ]
2
d D x εµ1 ν1 ...ρ1 σ1 τ1 . . . εµs νs ...ρs σs τs ην1 ...νs . . . ηρ1 ...ρs φµ1 ...µs
S[φs ] =
1
×
∂σ1 . . . ∂σs φτ1 ...τs , (26) for even spin s, and to S[φs ] = d D x εµ1 ν1 ...ρ1 σ1 τ1 . . . εµs+1 ...τs+1 ητ1 µs+1 ην1 ...νs+1 . . . ηρ1 ...ρs+1 φµ1 ...µs ×
s 2 −1
1
s−1 2
∂σ1 . . . ∂σs+1 φτ2 ...τs+1 ,
(27)
for odd spin s. The kinetic operator is self-adjoint, thus the Einstein-like equations Gµ1 ...µs ≈ 0 are the Euler-Langrange equations of the quadratic action, and the action is manifestly gauge invariant. The fact that these properties are manifest allows a straightforward generalization to any mixed-symmetry tensor gauge field, as we explain in Sect. 3. 2.2.2. Non-local actions in terms of differential forms. Introducing letters from the beginning of the Latin alphabet in order to denote tangent space indices, one may rewrite the actions (26) and (27) in a frame-like fashion. In flat spacetime of course, the distinction between tangent and curved indices is somewhat irrelevant since the background coframe reads, in components, (e0 )aµ = δµa . However, making this distinction may suggest a natural generalization of the quadratic actions to curved spacetimes by using differential forms. To start with, we write the action for a symmetric spin-s field φs featuring only “tangent” indices except for D suitably chosen “exterior form” indices: S[φ] = d D x εµν...ρσ τ εa1 b1 ...c1 d1 f1 . . . εas−1 bs−1 ...cs−1 ds−1 fs−1 ηνb1 ...bs−1 . . . ηρc1 ...cs−1 × × φµ a1 ...as−1
1 s
2 −1
Kσ τ | d1 f1 |...|ds−1 fs−1 ,
(28)
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
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for s even, and S[φs ] = − d D x εa1 b1 ...c1 d1 f1 εµν...ρσ τ εa2 b2 ...c2 d2 f2 . . . εas bs ...cs ds fs × ×η f1 as ηνb1 b2 ...bs . . . ηρc1 c2 ...cs ∂d1 φµa1 a2 ...as−1
1
s−1 2
Kσ τ | d2 f2 |...|ds fs , (29)
for s odd. The action (29) has been obtained from (27) after one integration by part, all the other operations being a mere change of labels. Now, we introduce some tensor-valued differential forms. For instance the Weinberg tensor field K defines a tensor-valued two-form R1 via (R1 )a1 b1 |...|as−1 bs−1 =
1 Kµν|a1 b1 |...|as−1 bs−1 d x µ ∧ d x ν , 2
(30)
while the symmetric tensor gauge field φs defines a tensor-valued one-form e ∈ s−1 (R D∗ ) ⊗ 1 (R D ) by ea1 ...as−1 = φµ a1 ...as−1 d x µ . (31) Also, the background coframe defines a vector-valued one-form (e0 )a = δµa d x µ .
(32)
It is tempting to treat the spin-s field one-form ea1 ...as−1 as a sort of “vielbein” for higherspins perturbing the pure spin-two flat background e0a , as suggested by Vasiliev [21]. In this way, the curvature two-form (30) can be thought of as the generalization of the linearized Riemann two-form in the moving-frame formulation of gravity [20]. Actually, one may also introduce a “Lorentz connection” one-form (ω1 )a1 b1 | a2 ...as−1 = ∂[a1 φb1 ] µ a2 ...as−1 d x µ .
(33)
(The notation has been chosen in such a way as to easily make contact with the materials reviewed in Sect. 2 of [34].) In the even-spin case, the action can be written in the following “Einstein–Cartan– Weyl” form by making use of the former differential forms: S[φs ] = εa1 b1 ...c1 d1 f1 . . . εas−1 bs−1 ...cs−1 ds−1 fs−1 ηb2 ...bs−1 . . . ηc2 ...cs−1 × 1 d f |...|d f × e0b1 ∧ . . . ∧ e0c1 ∧ ea1 ...as−1 ∧ s −1 R11 1 s−1 s−1 , 2
(34)
while the odd-spin case goes as follows: S[φs ] = εa1 b1 ...c1 d1 f1 εa2 b2 ...c2 d2 f2 . . . εas bs ...cs ds fs η f1 as ηb1 b3 ...bs . . . ηc1 c3 ...cs × 1 a d | a a ...a d f |...|d f × e0b2 ∧ . . . ∧ e0c2 ∧ ω11 1 2 3 s−1 ∧ s−1 R12 2 s s . (35) 2 We implicitly understood everywhere that a symmetrization over all indices labeled by the same Latin letter should be performed. The writing of the actions (34) and (35) suggests that they might make sense in an arbitrary curved background at the condition that the linearized curvature be replaced with its full non-Abelian counterpart. As a preliminary step in this direction, we show in the next subsection that the above Einstein–Cartan–Weyl actions can be seen as a flat
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spacetime limit of a MacDowell–Mansouri-like [35] action quadratic in curvatures and torsions taking value in some (A)d S D higher-spin algebra when D 4. (For D = 3, the action looks more like a Chern–Simons action, in agreement with the fact that the theory is “topological” in the sense that there are no local physical degrees of freedom in three dimensions for s > 0.) 2.2.3. Non-local actions à la MacDowell and Mansouri. The isometry algebra of (A)d S D manifold is presented via its translation-like generators Pa and Lorentz generators Mab (a, b = 0, 1, . . . , D − 1) together with their commutation relations [ Mab , Mcd ] = i (ηac Mdb − ηbc Mda − ηad Mcb + ηbd Mca ) ,
(36)
[ Pa , Mbc ] = i (ηab Pc − ηac Pb ) ,
(37)
[Pa , Pb ] = i Mab .
(38)
By defining M Dˆ a := ()−1/2 Pa , it is possible to collect all generators into the generaˆ These generators M AB span a pseudo-orthogtors M AB where A = 0, 1, . . . , D − 1, D. onal algebra since they satisfy the commutation relations [M AB , MC D ] = i (η AC M D B − η BC M D A − η AD MC B + η B D MC A ) , where η AB is the mostly minus invariant metric of the corresponding pseudo-orthogonal algebra. This is easily understood from the geometrical construction of (A)d S D as the hyperboloid defined by X A X A = (d−1)(d−2) which is obviously invariant under the 2 pseudo-orthogonal group. It is possible to derive the Poincaré algebra io(D − 1, 1) from the (A)d S D isometry algebra by performing the Inönü-Wigner contraction → 0 , in which limit the generators Pa become commuting genuine translation generators. The constant-curvature spacetime algebras can be uniformly realized as follows: M AB = −i X [A
∂ , ∂ X B]
(39)
if one takes ∂ Dˆ ∼ 0 and X Dˆ ∼ 0 in the flat limit → 0. ∂X Since the gauge fields and parameters are unconstrained in the non-local formulation, it is natural to make use of the so-called off-shell constant-curvature spacetime higherspin algebras which were discussed recently in [36, 34] and which we will now review in many details according to the present perspective. These higher-spin algebras can be easily defined as the Lie algebras of polynomials in the operators (39) endowed with the commutator as Lie bracket. In more abstract terms, they are the Lie algebras coming from the realization of the universal enveloping algebra induced by the unitary representation (39) of the constant-curvature spacetime isometry algebra. In more concrete terms, we will consider the Weyl-ordered monomials in the isometry algebra generators defined by (39), Ta1 b1 |...|at bt |at+1 ... as−1 = Ma1 b1 . . . Mat bt Pat+1 . . . Pas−1 + perms,
(40)
as the most convenient basis of generators for our purpose (t ∈ N and s ∈ N0 ), where “perms” stands for the sum of all nontrivial permutations of the generators M and P.
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
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The symbol of the differential operators (40) is a tensor irreducible under G L(D, R) with symmetries labeled by the two-row Young tableau a1
...
at
b1
...
bt
at+1
...
as−1
.
(41)
In order to mimic MacDowell–Mansouri formulation, one defines a connection one-form ω taking values in the higher-spin algebra a b1 |...|at bt |at+1 ... as−1
ω(x µ , d x ν , M AB ) = −i d x ν ων 1
Ta1 b1 |...|at bt |at+1 ... as−1
and whose non-Abelian curvature is the two-form R = dω + ω2 . The component of ω linear in Pa is the moving frame ea of the spacetime manifold while the component linear in Mab is its Lorentz connection ωab . In the pure gravity case, the coefficient Rab of Mab in R is the sum Rab = R ab + ea ∧eb of the Riemann two-form plus cosmological terms while the coefficient T a of Pa in R is the torsion. The components ωa1 b1 |...|at bt |at+1 ... as−1 of the connection ω are assumed to be irreducible tensors under G L(D, R) described by the Young diagram (41), as can be done without loss of generality. In general, if a connection one-form is decomposed as a sum ω = ω0 + ω1 of a vacuum solution ω0 plus a small fluctuation ω1 , then its curvature can also be expanded in powers of the fluctuation: at order zero, one has R0 = dω0 + ω02 = 0 by assumption, and at order one, the linearized curvature reads R1 = D0 ω1 = dω1 + [ω0 , ω1 ]+ , where the background covariant derivative D0 = d + [ω0 , ]± is nilpotent, D02 = R0 = 0. The linearization of the gauge transformations δ ω = d + [ω, ]− reads δω1 = D0 and leaves the linearized curvature invariant. In the present case, the background higher-spin connection is assumed to be purely gravitational in the sense that ω0 = −i (e0a Pa + ω0ab Mab ) .
(42)
Moreover, if the gravitational background is assumed to be a vacuum solution of the constant-curvature spacetime algebra, then the background connection one-form describes the corresponding constant-curvature spacetime manifold, since R0 = 0 decomposes as R0ab = − e0a ∧e0b and T0a = 0 . In order to evaluate the action of the covariant derivative with respect to this background, it is sufficient to compute the commutator of P and M with any monomial T . A nice property of the Weyl ordering is that the commutator of a Lie algebra element with a Weyl-ordered element of the universal enveloping algebra preserves the Weyl ordering. Therefore the generators T transform as tensors under the adjoint action of the Lorentz algebra spanned by the Mab ’s and it is convenient to split the background covariant derivative into the sum D0 = D0L + [e0 , ]± , where D0L is the covariant derivative with respect to the background Lorentz connection. The commutator between a translation-like generator P and any generator T is easily computed
Pa , Tb1 c1 |......|bt ct |bt+1 ... bs−1 = 2i
t
Tb1 c1 |......|bi−1 ci−1 |bi+1 ci+1 |......|bt ct |bt+1 ... bs−1 [ci η bi ] a
i=1
+ i Tb1 c1 |......|bt ct |abt+1 |bt+2 ... bs−1 + . . . + i Tb1 c1 |......|bt ct |abs−1 |bt+1 ... bs−2 .
(43)
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X. Bekaert, N. Boulanger
While the background one-form (42) is assumed to contain the spin-two gauge fields (e0a , ω0ab ) only, the fluctuation one-form may contain the infinite tower of symmetric tensor gauge fields. In particular, the components along the pure translation-like generators in ω1 = ea1 ...as−1 Pa1 . . . Pas−1 + O(Mab ) (44) are frame-like one-forms ea1 ...as−1 given by (31) in some proper gauge. More precisely, the linearized gauge transformations δω1 = D0 read in components δ ea1 ...as−1 = D0L a1 ...as−1 + (e0 )c c(a1 |a2 ... as−1 )
(45)
and a b1 |...|at bt |at+1 ... as−1
δ ω11
= D0L a1 b1 |...|at bt |at+1 ... as−1 + (e0 )c a1 b1 |...|at bt |c(at+1 ... as−1 ) − Y A a2 b2 |...|at bt |at+1 ... as−1 [a1 e0b1 ] (46) − . . . + a1 b1 |...|at−1 bt−1 |at+1 ... as−1 [at e0bt ] a ...a
for t > 0, due to the commutation relations (43). The frame-like one forms eµ1 s−1 can be seen as rank-s tensors reducible under G L(D, R) which can be decomposed into the sum of two tensors irreducible under G L(D, R) respectively labeled by the Young tableau a1
...
as−1
a1
...
as−1
and
µ
.
µ
(47)
The gauge variation ηµc c(a1 |a2 ... as−1 ) can be chosen in such a way as to precisely cancel a ...a the “hook” part in eµ1 s−1 labeled by the Young tableau (47). In this metric-like gauge, the identification (31) holds. Pursuing the analogy with gravity, the other components of ω1 should be expressed in terms of these dynamical fields e via some torsion constraints on the curvature R . These constraints are only known at linearized order where they take the form a b |...|a b |a ... a R11 1 t t t+1 s−1 = 0 , for 0 t < s − 1 . (48) The commutation relations (43) lead to the following expression for the linearized curvatures, a ... as−1
R11
c(a1 |a2 ... as−1 )
= D0L ea1 ... as−1 + (e0 )c ∧ ω1
and a b1 |...|at bt |at+1 ... as−1
R11
a b1 |...|at bt |at+1 ... as−1
= D0L ω11
a b1 |...|at bt |c(at+1 ... as−1 )
+ (e0 )c ∧ ω11
a b2 |...|at bt |at+1 ... as−1 [a1
+ Y A ω12
∧ e0b1 ]
a b1 |...|at−1 bt−1 |at+1 ... as−1 [at
+ . . . + Y A ω11 for t = 0.
∧ e0bt ]
(49)
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
739
Therefore, the torsion constraints (48) are solved as a b1 |...|at bt | (at+1 ... as−1 ) ν]
(ω1 )[µ1
a b1 |...|at bt |at+1 ... as−1
= (D0L )[µ (ω1 )ν]1
+ O() .
(50)
In the metric-like gauge, these relations may be used recursively to express the auxiliary one forms with mixed symmetries in terms of the frame-like field. For instance, when t = 0 and = 0, Eq. (50) reproduces (33). Moreover, then the Riemann-like two-form a b |...|a b (R1 )µν1 1 s−1 s−1 may be identified with the Weinberg tensor according to (30). By using the expression (49) together with the former remarks, one can check that the MacDowell–Mansouri-like action S[φs ] =
1 εa b ...c d f . . . εas−1 bs−1 ...cs−1 ds−1 fs−1 ηb2 ...bs−1 . . . ηc2 ...cs−1 × 1 1 1 1 1 1 c a |a ...a d f |...|d f × e0b1 ∧ . . . ∧ R11 1 2 s−1 ∧ s −1 R11 1 s−1 s−1 , 2 (A)d S
(51)
reproduces the Einstein–Cartan–Weyl-like action (34) at order zero in , in the metriclike gauge. More precisely, one should first take the → 0 limit in the action (51) and then one uses the zero-torsion constraints to express the auxiliary one-forms in terms of the frame-like field. In the pure gravity case s = 2, one recovers the MacDowell–Mansouri action [35]. In the odd spin case, it is the action S[φs ] =
1 εa b ...c d f εa b ...c d f . . . εas bs ...cs ds fs η f1 as ηb1 b3 ...bs . . . ηc1 c3 ...cs × 2 1 1 1 1 1 2 2 2 2 2 1 a d |c a | a ...a d f |...|d f × e0b2 ∧ . . . ∧ R11 1 2 2 3 s−1 ∧ s−1 R12 2 s s , (52) 2 (A)d S
which can reproduce the action (35). We implicitly understood everywhere that a symmetrization over all indices labeled by the same Latin letter should be performed. The “d’Alembertian” in (anti) de Sitter is not determined uniquely from its flat spacetime limit. In general, (A)d S = ∇ 2 + O() , where the term O() is an operator acting on the spin degrees of freedom. A convenient requirement in order to remove this ambiguity could be that (A)d S should commute with the (A)d S covariant derivative, hence it is tempting to define (A)d S as the anticommutator [ D0 , D0† ]+ because it commutes with the differential D0 . The MacDowell–Mansouri-like actions (51)–(52) are automatically gauge invariant since the Lagrangian is quadratic in the linearized curvatures. Notice that these MacDowell–Mansouri-like actions may provide quadratic actions in constant-curvature spacetime within the unconstrained approach. This issue should be investigated further. We should also point out that these quadratic actions are of the same MacDowell-Mansouri form as the Lopatin–Vasiliev action [20] but the latter is local and has a different structure for the contraction of indices. This is possible because the tangent indices are not constrained to be traceless here and so more freedom is allowed in the contraction of indices. Let us conclude this subsection with some speculative observations. The appealing feature of the quadratic actions (51)–(52) is that the starting point of Vasiliev et al. in their construction [37] of cubic vertices, invariant under non-Abelian gauge
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X. Bekaert, N. Boulanger
transformations associated with the constrained (“on-shell”) higher-spin algebra, was the formulation of symmetric tensor gauge fields à la MacDowell–Mansouri via a local constrained frame-like formulation [20]. Therefore, by analogy, our result suggests that a non-linear Lagrangian for the non-Abelian higher-spin gauge theory with unconstrained (“off-shell”) higher-spin algebra – if any – could be of the non-local MacDowell–Mansouri-like form presented here. Although elusive, such a non-local expression quadratic in the curvatures has some precedents. Indeed, the expressions (51)–(52) are reminiscent of the two-dimensional non-local action S[g], quadratic in the worldsheet scalar curvature, which is obtained from the Polyakov action S P [g, X ] by integrating out the D massless Klein–Gordon scalars X µ (σ ) describing the position of the bosonic string in the target space [23]. The harmless non-locality of this action and of the free higher-spin actions fall into the same category. An analogous picture for the full MacDowell–Mansouri-like actions would be in agreement with the folklore stating that a non-Abelian gauge theory of higher-spin fields might be interpreted as the effective theory of some more fundamental theory describing extended objects. In any case, we believe that the frame-like actions presented here deserve to be explored further. 3. Mixed-Symmetry Tensor Gauge Fields In the present section we generalize the gauge theory of free rank-s symmetric tensor fields to the case of massless gauge fields with components transforming in an arbitrary irrep. of the general linear group, labeled by a Young diagram Y made of s columns. The reader is now assumed to have read Appendix A because the fundamental definitions are not repeated here. Following the terminology introduced in Sect. A.2.2, we say that Y (R D ). the gauge field φY is a (differential) hyperform of (s) 3.1. Bargmann–Wigner programme. 3.1.1. Local, constrained approach of Labastida. It is natural to try to generalize the work of Fronsdal (briefly reviewed in Subsect. 2.1.1) to arbitrary mixed-symmetry tensor gauge fields. In [38], Labastida conjectured some gauge invariances and determined a local gauge-invariant wave operator which was supposed to describe the proper degrees of freedom, but he was not able to prove that one may reach a gauge where the on-shell physical degrees of freedom provide the appropriate UIR of O(D − 2) . Labastida used a set of commuting oscillators [38] and thereby chose the symmetric convention for Young tableaux. Nevertheless, it turns out to be convenient for our later purposes to deal with fields in the antisymmetric convention. So, throughout the present Sect. 3.1, the gauge field φY is understood to be a (differential) multiform of [s]1 ,...,s (R D ) whose components are in the irrep. of G L(D, R) labeled by the Young diagram Y = (1 , . . . , s ). Each basis element di x µ of each exterior algebra ∧(R D∗ ) plays the role of a graded oscillator. We introduce the Labastida operator defined by F := − di di† +
1 di d j Tri j , 2
(53)
where there is always an implicit summation from 1 to s over all repeated Latin indices. Each term on the right-hand-side commutes with the operator Tri j ∗i , hence the Labastida operator F preserves the G L(D, R)-irreducibility conditions (105). In other words,
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
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Y (R D ) , the Young symmetrizer Y A commutes with the operator F, so that if φY ∈ (s) Y (R D ). then the mixed-symmetry Labastida tensor FY := FφY also belongs to (s) It is natural to postulate that the field equation is
FY ≈ 0 ,
(54)
and that the gauge transformations take the form δ φY = Y A di i ,
(55)
where i are differential multiforms belonging to [s]1 , ... , i −1 , ... , s (R D ). The gauge transformation of the Labastida tensor under (55) is given by δ FY =
1 Y A di d j dk (Tri j k ) , 2
(56)
due to the identity (115). Eq. (56) is the analogue of (4). The commutation relations (116) suggest to require that Tr(i j k) = 0 . Notice that this condition is weaker than the tracelessness of every parameter independently. The gauge invariance of the wave equation (54) was one of the requirements of Labastida in order to determine uniquely his relativistic wave operator in the symmetric convention [38]. One may easily check that the translation of Labastida’s requirements in the antisymmetric convention also fixes uniquely the wave operator. Hence the Labastida tensor in the symmetric convention of [38] must be equal to a linear combination of the Labastida tensor FY in the antisymmetric convention. The main technical problems in the local approach are of course the trace conditions to be imposed on the gauge field and the gauge parameters. In the general mixed-symmetry case φY , it is very difficult to determine them from first principle, contrary to the completely symmetric case, because there is now a wide variety of inequivalent ways to take traces. Moreover, a troublesome aspect in the construction of Labastida is that the double-trace constraints that he imposes on the gauge field φY are in general not invariant under his gauge transformations.8 For instance the analogue of the doubletrace constraint of Labastida reads Tr(i j Trkl) φY = 0 in the antisymmetric convention. But the identity Tr(i j Trkl) dm m = 4 d(i† Tr jk l) + dm Tr(i j Trkl) m shows that the former double-trace constraint is in general not preserved by gauge transformations (55), where the parameters are only subject to the trace constraint Tr(i j k) = 0. It is then fair to say that the problem of constructing a local action principle for arbitrary gauge fields φY is still open. 3.1.2. Higher derivative, unconstrained approach. The curvature tensor of Weinberg was appropriately generalized in [9] by extending the cohomological results of [8] to arbitrary mixed-symmetry tensor fields. The definitions and main properties of the curvature tensors in the general case under consideration are reviewed in Sect. A.2.2. Y (R D ) for the mixed-symmetry tensor gauge field The curvature tensor field KY ∈ (s) Y (R D ) is obtained by taking s curls, K = d . . . d φ and Y is the Young diaφY ∈ (s) 1 s Y Y gram obtained by adding a row of length s on top of the Young diagram Y . The curvature 8 We are grateful to A. Pashnev and M. Tsulaia for calling this fact to our attention.
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X. Bekaert, N. Boulanger
tensor is invariant under the gauge transformations (55) without any trace constraint on the gauge parameters i . The Bianchi-like identities are the set of equations di KY = 0 (i = 1, . . . , s). The commutation relation [ Tri j , di d j ]− = − di di† − d j d †j ,
(57)
where no sum on the indices i and j is understood, follows from (115) and implies in turn the operatorial identity Tr12 d1 . . . ds = d3 . . . ds F . Applied on the gauge field φY , this last identity leads to the generalization of the Damour–Deser identity for arbitrary mixed-symmetry fields Tr12 KY = d3 d4 . . . ds FY .
(58)
Therefore, the Labastida equation (54) implies the Ricci-flat-like equation Tr KY ≈ 0 ,
(59)
stating that the curvature tensor is traceless on-shell, in agreement with (106). In analogy with the situation reviewed in Subsect. 2.1.4, the Ricci-flat-like equation (59) implies the compensator equation FY ≈
1 Y A di d j dk αi jk , 2
(60)
where αi jk = α(i jk) are some (differential) hyperforms associated with the Young diagrams obtained by removing three boxes in distinct columns of Y . The compensator fields αi jk are pure-gauge fields expected to vary according to δ αi jk = Tr(i j k)
(61)
in order to compensate the variation (56) of the Fronsdal tensor in the third-order field equation (60). As one can see, the Labastida equation (54) arises as a partial gauge-fixing of the compensator equation. The results explained in the previous paragraph were announced in [17] but the complete proof was not presented there because of the lack of space. For the sake of completeness, we now sketch the subtle use of Poincaré lemmas that enables to relate the Ricci-flat-like equation (59) with the compensator equation (60) via the Damour–Deser identity (58). The argument is deeply rooted in the following lemma, the proof of which is given in Appendix B.1: Lemma 1. Let P be a differential hyperform of (s) (R D ) . Then, ds P = 0
=⇒
di P = 0 , ∀i ∈ {1, . . . , s} .
(62)
As a corollary of Lemma 1, we have the implication k
i=1
ds−k+i P = 0
Lemma 1
=⇒
di P = 0 , ∀I ⊂ {1, 2, . . . , s} | #I = k , i∈I
(63) for any integer k ∈ {1, . . . , s}, which can easily been proved by induction. The properties (63) and (121) combined together prove the following
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
743
Proposition 1. Let P be a differential hyperform of (s) (R D ) . Then, k
i=1
{i} P = 0, ds−k+i P = 0 =⇒ d
∀I ⊂ {1, 2, . . . , s} | #I = k ,
i∈I
(64) In other words, Proposition 1 provides a sufficient condition for the cocycle condition d k P = 0 of the generalized cohomology group (k) H (i1 ,...,is ) (d) associated with the operator d = d {1} + · · · + d {s} acting on the space of hyperforms (s) (R D ) . The generalized Poincaré lemma of [9] proves the triviality of the generalized cohomology groups (k) H (1 ,...,s ) (d) for 1 k s, 0 < and < D . The Ricci-flat-like equation (59) s 1 combined with the Damour–Deser identity (58) states that the Fronsdal tensor obeys the equation d3 d4 . . . ds FY ≈ 0 . Proposition 1 for k = s − 2 implies that d s−2 FY ≈ 0 . The triviality of (2) H Y (d) implies the exactness of the on-shell Fronsdal tensor, FY ≈ d 3 α , as expressed by the compensator equation (60). 3.1.3. Non-local, unconstrained approach of de Medeiros and Hull. As was pointed out in [16], Eqs. (16) of Francia and Sagnotti were generalized by Hull and de Medeiros in [15] as follows: Tr(12 Tr34 . . . Trs−1 s) KY ≈ 0
,
(65)
for s even. The sum of products of all possible traces over indices all belonging to distinct columns in (65) correspond in (16) to the contraction with the symmetrized powers η(µ1 µ2 . . . ηµs−1 µs ) of the metric tensor. For s odd, the equation may be written in two ways: † Tr(12 . . . Trs−2 s−1 Trs s+1) ds+1 KY = Tr(12 . . . Trs−2 s−1 ds) KY ≈ 0 ,
(66)
because of the fact that KY is of degree zero in the s + 1th set of antisymmetric indices. One can check explicitly that the operators Tri j ∗i commute with the operator Tr(12 . . . Tr2n−1 2n) when i and j belong to the set {1, . . . , 2n} [19]. Therefore, Eqs. (65)– (66) have the same symmetry properties as the corresponding tensor gauge field φY . As they are, it is not obvious that they describe the proper physical degrees of freedom because the light-cone gauge is hard to reach since the gauge transformations (55) involve many parameters and are highly reducible in general. As a preliminary, we show in the next paragraph that Eqs. (65)–(66) are equivalent to the following compensator-like equation: FY ≈
1 Y A di d j dk Hi jk , 2
(67)
generalizing Eq. (14). The essential difference between (67) and the compensator equation (60) is that the tensor fields Hi jk are non-local functions of the gauge field φY and its partial derivatives. Nevertheless, their gauge transformations are proportional to Tr(i j k) so that the gauge-fixing condition Hi jk = 0 leads to the Labastida equation (54). To prove the on-shell equivalence between the deMedeiros–Hull equations (65)–(66) and (67) we need a crucial identity.
744
X. Bekaert, N. Boulanger
Lemma 2. For any given natural number n ∈ N, Tr(12 . . . Tr2n−1 2n) d1 d2 . . . d2n−1 d2n=n−1 F −
n−1 n−2 d j dk Tr jk F + di d j dk Oi jk , 2n−1
where there is an implicit sum from 1 to 2n over every repeated index and Oi jk denotes a set of differential operators (1 i, j, k 2n). The proof is given in Appendix B.2. Applying the operator appearing in Lemma 2 for n = [ s+1 2 ] on the gauge field φY , one gets the on-shell equality n−1 FY −
n − 1 n−2 d j dk Tr jk FY + di d j dk i jk ≈ 0 , 2n − 1
(68)
for the multiforms i jk := Oi jk φY , by virtue of Eqs. (65)-(66). Taking a trace of both sides of Eq. (68), leads to n−1 Tri j FY ≈ dk σk ,
(69)
for some multiforms σk . Inserting (69) into (68) gives (67). 3.1.4. Bargmann–Wigner equations. Following the discussion in Subsect. 2.1.4, we stress that the s th -order Ricci-flat-like equation (59) is equivalent to a set of firstorder field equations for KY . Indeed, the vanishing of the Ricci-like tensor means that the on-shell Weinberg tensor field KY takes values in an irrep. of O(D − 1, 1). The Bargmann–Wigner equations are somehow the converse statement. Let KY be a differential hyperform with components in a tensorial irrep. of the Lorentz group O(D − 1, 1) whose symmetries are labeled by the Young diagram Y (in the antisymmetric convention). As explained in Appendix A.2.2, the Bianchi-like identities (122) imply that the hyperform KY is exact, which means that it is precisely the curvature tensor of a gauge field φY taking values in an irreducible representation of G L(D, R) labeled by the Young diagram Y . This proves the equivalence between the Ricci-flat-like equation (59) obeyed by the Weinberg tensor field, and the Bianchi-like equations (122) obeyed by an O(D − 1, 1)-irreducible tensor field with the same symmetries as the Weinberg tensor. Moreover, due to the commutation relation (115) the compatibility condition between the Bianchi-like identities (122) and the tracelessness property (59) are the transversality conditions di† KY ≈ 0 (i = 1, . . . , s) .
(70)
Equations (70) and (122) are called the Bargmann–Wigner equations since they generalize (21). They were proposed in [9, 16] as field equations for mixed-symmetry tensor gauge fields. By definition, the Bargmann–Wigner equations state that the differential hyperform KY is harmonic on-shell. Up to now, we have achieved to prove the equivalence of the Labastida equation (54), the Ricci-flat equations (59), compensator (60), the deMedeiros–Hull equations (65)– (66) and the Bargmann–Wigner equations (70) and (122). In order to prove that they describe the proper physical degrees of freedom, it is sufficient to do so for one of these equations: this is done in Appendix C for the Bargmann–Wigner equations. As a corollary, this completes the Bargmann–Wigner programme for arbitrary finite-component fields in any dimension, as summarized in the following theorem.
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
745
Theorem 1. (Bargmann–Wigner’s programme) [17]. Let Y be an allowed Young diagram (¯1 , . . . , ¯s ) with at least two rows of equal length s and Y := (¯1 − 1, . . . , ¯s − 1) be the Young diagram (1 , . . . , s ) obtained by removing the first row of Y . Any tensorial irreducible representation of the group O(D−1, 1) with finite-dimensional representation space V O(D−1,1) , where V = R D , provides a massless unitary irreducY ible representation of the group I O(D − 1, 1) associated with the Young diagram Y : Its infinite-dimensional representation space is the space of harmonic differential multiforms KY of spin s taking values in V O(D−1,1) . The latter space is isomorphic to the Y
Hilbert space HY of physical states ϕY ∈ L 2 (R D ) ⊗ VYO(D−2) that are solutions of ϕY ≈ 0 . Any single-valued massless unitary irreducible representation of I O(D − 1, 1) induced from a finite-dimensional irreducible representation of O(D − 2) is equivalent to a representation obtained in this way. 3.2. Fierz–Pauli programme. In the first subsection, we discuss the state of the art in order to clarify what is new in the present work with respect to the extensive literature on the subject. In the second subsection, a non-local Lagrangian for any mixed-symmetry tensor gauge field is written in compact form, two particular cases of which are exhibited in the third subsection. 3.2.1. Local actions. Local covariant Lagrangians have already been obtained for gauge fields labeled by the most general “hook” diagrams (1 , 1, . . . , 1) [39], “two-row” diagrams (2, . . . , 2, 1, . . . , 1) [40] and “two-column” diagrams (1 , 2 ) [41, 42] in approaches where trace constraints are imposed on the higher-spin fields. On the one hand, a decisive step towards the explicit completion of the Fierz–Pauli programme has been performed in the O Sp(1, 1|2) formalism [18]. The drawback of this formalism is that it requires some technically involved computations in order to write the quadratic action only in terms of the Sp(2) singlet variables (i.e. the constrained mixed-symmetry gauge field). This last step has never been performed explicitly for the mixed-symmetry case to our knowledge. On the other hand, in [43] Labastida introduced an explicit self-adjoint Einstein-like tensor corresponding to his field equation and conjectured that this Einstein-like tensor would provide the local constrained quadratic action for a tensor gauge field labeled by an arbitrary Young diagram. The problem of his approach is that he could not prove in full generality that his choice of trace constraints would lead to the proper physical degrees of freedom. More recently, an algorithm for the construction of quadratic actions for mixed-symmetry tensor gauge fields was given in the BRST approach [44]. Finally, de Medeiros and Hull conjectured in [19] the rough form of a non-local Einstein-like tensor but they did not give the precise coefficients of its expansion in powers of traces, neither did they prove that their Einstein-like equation describes the proper physical degrees of freedom. In this sense, the non-local second-order action that we write in Theorem 2 provides the first explicit realization of the Fierz–Pauli programme in full generality. More accurately, our analysis is restricted to Minkowski spacetime and to fields with a finite number of components. Incidentally, we should mention that for “massless” mixedsymmetry tensor gauge fields, the Bargmann–Wigner programme for the anti de Sitter group S O(D −1, 2) has already been examined in many details [45] and the Fierz–Pauli programme has recently experienced considerable progress [46]. Also, the completion of the Bargmann–Wigner programme has recently been extended to all massless irreps
746
X. Bekaert, N. Boulanger
(including infinite-component ones) of the Poincaré group I S O(D − 1, 1) [47]. The non-locality property of the action proposed here remains elusive and it would be pleasant to explicitly derive its local counterparts. Actually, the BRST algorithm of [44] indirectly ensures the existence of a local action invariant under unconstrained gauge transformations, but with many auxiliary fields. In the same way, the work of [18] may be interpreted as a proof of the existence of a local second-order action invariant under constrained gauge transformations. 3.2.2. Non-local actions. The main idea is that the use of the Levi-Civita tensors enables a straightforward generalization of the results of Subsect. 2.2.1 to the mixed-symmetry case. Still, one should make sure to take the appropriate traces and that the result is projected on the proper symmetry. Our main results are summarized in compact form in the following theorem. Subsequently, we provide two examples and then describe in more detail the construction of the non-local Lagrangian for arbitrary mixed-symmetry tensor gauge fields. Theorem 2. (Fierz–Pauli’s programme). Let s be a positive integer. The smallest even integer that is not smaller than s is denoted by s := 2[ s+1 2 ] = s + ε(s) . Let Y := (1 , . . . , s ) be a Young diagram with first row of length s (that is to say, s = 0 when s is odd) and such that 1 + 2 D − 2. Let φY be a gauge field with components in the tensorial irreducible representation of the group G L(D, R) with (finite-dimensional) G L(D,R) , where V = R D . representation space VY The second-order quadratic action S [ φY ] = φY | K | φY defined by the self-adjoint kinetic operator s
K = Tr (D−1) 2 −|Y| ◦ ∗s ◦
1 s
2
◦
s
di ,
(71)
i=1
with s
Tr
(D−1) 2s −|Y|
2
Tr j :=
s− j+1
D−1− j −s− j+1
,
j=1
is manifestly gauge-invariant under the transformations δ | φY =
s
di | i ,
(72)
i=1
where i are differential multiforms belonging to [s]1 , ... , i −1 , ... , s (R D ). Let Y be the Young diagram obtained by adding one row of length s to the Young diagram Y . The equation of motion derived from this action may be cast into the form δS [ φY ] ≈0 δ φY |
⇐⇒ | GY ≈ 0 ,
(73)
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
747
where the Einstein tensor GY is defined by GY :=
≈ 0 Y S Tr (D−1) 2 −|Y| K Y s+1
≈ 0 Y S Tr (D−1) 2 −|Y| d † K s
s+1
Y
for s even , for s odd ,
(74)
the dual of the curvature tensor K and with K Y the dual of the Young diagram Y . Y Y The (infinite-dimensional) space of field configurations extremizing the action S[ φY ] carries the massless unitary irreducible representation of the group I O(D − 1, 1) associated with the Young diagram Y : it is isomorphic to the Hilbert space HY of physical O(D−2) that are solutions of ϕY ≈ 0. states ϕY ∈ L 2 (R D ) ⊗ VY In order to help the reader to get used to the notations involved in Theorem 2 and to provide some flavor of the general proof, we present two particular examples with mixed symmetry gauge fields (one for each parity of the spin s). G L(D,R) An odd-spin example. We first consider the gauge field φY ∈ VY with the associated Young diagram Y = (1 , 2 , 3 , 0) = (2, 1, 1, 0). The spin is s = 3, hence 3 = 2[ 3+1 2 ] = 3 + ε(3) = 4. The tensor gauge field components read φµ11 µ21 µ31 ; µ12 . The ket | φY is assumed to be expressed in the symmetric convention, which means that it is totally symmetric in (µ11 , µ21 , µ31 ) and obeys φ(µ1 µ2 µ3 ; µ1 ) = 0 . The Young tableau 1 1 1 2 associated to φµ1 µ2 µ3 ; µ1 is depicted as follows: 1 1 1
2
Y
µ11 µ21 µ31
=
.
µ12
(75)
The curvature tensor KY will have components Kµ1 µ1 µ1 | µ2 µ2 | µ3 µ3 described by the 1 2 3 1 2 1 2 Young tableau µ11 µ21 µ31 µ12 µ22 µ32
=
Y
µ13
(76)
that is, as a Young diagram, Y = (3, 2, 2, 0) = (2, 1, 1, 0) + (1, 1, 1, 0). The curvature tensor is expressed in the antisymmetric convention because of the presence in the Lagrangian of the s = 4 Levi-Civita tensors εµ1 µ2 ... µ D εµ1 µ2 ... µ D εµ1 µ2 ... µ D εµ1 µ2 ... µ D 1 1
1
2 2
2
3 3
3
4 4
4
(77)
contracted with the s = 3 derivatives of the gauge field components ∂µ1 ∂µ2 ∂µ3 φµ1 µ2 µ3 ; µ1 . 3
2
2
1 1 1
2
In order for the Ricci-flat-like equation TrKY ≈ 0 to define a nontrivial theory, we must have D 1 + 2 + 2 = 5. We choose here D = 5. Continuing the construction of the Lagrangian, we have to act with an ε(3) = 1 extra derivative ∂µ4 on the gauge field φµ1 µ2 µ3 ; µ1 . The components of the bra φY | are 1
1 1 1
2
748
X. Bekaert, N. Boulanger
written, in the symmetric convention, as φµ4 µ3 µ2 ; µ4 and they correspond to the Young 4 5 5 5 tableau µ45 µ35 µ25 .
µ44
(78)
s
Finally, the trace operator Tr (D−1) 2 −|Y | = (Tr14 )2 (Tr23 )2 reads, in components, (ηµ1 µ4 ηµ1 µ4 ) (ηµ2 µ3 ηµ2 µ3 ) . 5 2
4 3
3 3
(79)
4 4
Summarizing, the action is explicitly written as 1 1 1 4 4 S[φY ] = d 5 x φµ4 µ3 µ2 ; µ4 (ηµ1 µ4 ηµ1 µ4 ηµ2 µ3 ηµ2 µ3 )(εµ1 ... µ5 . . . εµ1 ... µ5 ) 4 4 3 3 3 4 4 5 5 5 5 2 2 1 × ∂µ4 ∂µ1 ∂µ2 ∂µ3 φµ1 µ2 µ3 ; µ1 . 1 3 2 2 1 1 1 2 At this stage, it is instructive to draw the G L(5, R) Young diagram Z corresponding to the product (77), in which we mark by a “×” the cells corresponding to the components of the bra φY | (and ket | φY ) and by a “−” the cells corresponding to the partial derivatives. The components of the metric tensors are marked by a “◦”. It gives
Z =
µ11 µ21 µ31 µ41
×
×
×
−
µ12 µ22 µ32 µ42
×
−
−
◦
−
◦
◦
◦
µ14 µ24 µ34 µ44
◦
◦
◦
×
µ15 µ25 µ35 µ45
◦
×
×
×
=
µ13 µ23 µ33 µ43
.
(80) The differential multiform ds KY = d4 KY is labeled by the Young diagram Y + := Y + (1, 1, 1, 1) = (3, 2, 2, 1) . The ket ∗s ds | KY = (∗1 ∗2 ∗3 ∗4 ) d4 | KY enters in the Lagrangian with the following tensorial components, in the antisymmetric convention, ( ∗1 ∗2 ∗3 ∗4 d4 KY )µ4 µ5 | µ3 µ4 µ5 | µ3 µ4 µ5 | µ2 µ3 µ4 µ5 . 1 1
2 2 2
3 3 3
4 4 4 4
Only one G L(5, R)-irreducible component of the above tensor, also denoted by ( d4 K)Y , + survives inside the action. It is labeled by the Young diagram Y = (5, 5, 5, 5, 5) − +
(1, 2, 2, 3) = (4, 3, 3, 2) and the corresponding Young tableau reads ×
×
×
◦
×
◦
◦
◦
µ43 µ33 µ23
◦
◦
◦
µ42
◦
µ45 µ35 µ25 µ15 Y +
=
µ44 µ34 µ24 µ14
=
.
(81)
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
749
It may be obtained by rotating Z by 180 degrees and removing the cells of the Young tableau Y + corresponding to d4 KY . In terms of S L(5, R)-irreducible representations, this tensor is equivalent to d4 KY . The Euler-Lagrange derivatives are proportional to Y S (Tr1 4 )2 (Tr2 3 )2 ( (82) d4 K)Y ≈ 0 +
or, in components, δS δφµ4 µ3 µ2 ; µ4 5 5 5
∝ Y S (ηµ1 µ4 ηµ1 µ4 ηµ2 µ3 ηµ2 µ3 )( d4 K)µ4 µ4 µ4 µ4 | µ3 µ3 µ3 | µ2 µ2 µ2 | µ1 µ1 , 5 2
4 3
3 3
4 4
2 3 4 5
3 4 5
3 4 5
4 5
4
where Y S is the Young projector associated with the Young tableau (78). In the field equations (and also in the Lagrangian), only one G L(5, R)-irreducible component of the tensor product ηµ1 µ4 ηµ1 µ4 ηµ2 µ3 ηµ2 µ3 will contribute. It is the irreducible component 4 3 3 3 4 4 5 2 in its characterized by the Young tableau X such that the product X · Y contains Y +
decomposition. We find that X = (2, 2, 2, 2). Drawing the tableau,
=
X
◦
◦
◦
◦
◦
◦
◦
◦
(83)
.
Indeed, it is easy to check, using the Littlewood-Richardson rules, that
◦
◦
◦
◦
◦
◦
◦
◦
·
× ×
×
×
⊃
×
×
×
◦
×
◦
◦
◦
◦
◦
◦
.
◦ According to the definitions introduced in Appendix A.1.1, one may say that, on-shell, the field ( d4 K)Y takes values in a tensorial representation of S L(5, R) labeled by the + − Y of Y, where the subtraction of the Young diagram Y corresponds to difference Y +
the trace constraints (82) imposed by the equations of motion. Due to the isomorphism S L(5,R) ∼ S L(5,R) V , the former tensorial representation is equivalent to a tensorial = VY Y+
+
representation labeled by the difference Y + − Y corresponding to the field (d4 K)Y + on which are imposed the trace constraints Y S Tr1 4 Tr2 3 d4 KY ≈ 0 , (84) labeled by Y . This provides a group-theoretical proof of the fact that the Einstein-like equations (82) are equivalent to the deMedeiros–Hull equations (84). They respectively are particular instances of (73) and (66).
750
X. Bekaert, N. Boulanger G L(D,R)
An even-spin example. We next consider the gauge field φY ∈ VY with the associated Young diagram Y = (1 , 2 , 3 , 4 ) = (3, 2, 2, 2). We choose the dimension D = 7. The spin is s = 4, hence 4 = 2[ 4+1 2 ] = 4 + ε(4) = 4. The tensor gauge field components read φµ1 µ2 µ3 µ4 ; µ1 µ2 µ3 µ4 ; µ1 . The associated Young tableau is depicted as 1 1 1 1 2 2 2 2 3 follows: µ11 µ21 µ31 µ41 .
µ12 µ22 µ32 µ42 µ13
(85)
The curvature tensor KY has components Kµ1 µ2 µ3 µ4 ; µ1 µ2 µ3 µ4 ; µ1 µ2 µ3 µ4 ; µ1 described 1 1 1 1 2 2 2 2 3 3 3 3 4 by the Young tableau µ11 µ21 µ31 µ41 µ12 µ22 µ32 µ42 µ13
µ23
µ33
,
µ43
µ14
(86)
where Y = (4, 3, 3, 3) = (3, 2, 2, 2) + (1, 1, 1, 1). The curvature tensor is expressed in the antisymmetric convention because of the presence in the Lagrangian of the s = 4 Levi-Civita tensors εµ1 µ2 ... µ7 εµ1 µ2 ... µ7 εµ1 µ2 ... µ7 εµ1 µ2 ... µ7 1 1
1
2 2
2
3 3
3
4 4
4
(87)
contracted with the s = 4 derivatives of the gauge field components ∂µ2 ∂µ3 ∂µ4 ∂µ1 φµ1 µ2 µ3 µ4 ; µ1 µ2 µ3 µ4 ; µ1 . 3
3
3
4
1 1 1 1
2 2 2 2
3
With D = 7, the Ricci-flat-like equation Tr KY ≈ 0 defines a nontrivial theory, since 1 + 2 + 2 D. The components of the bra φY | are written, in the symmetric convention, as s φµ1 µ2 µ3 µ4 ; µ1 µ2 µ3 µ4 ; µ4 . Finally, the trace operator Tr (D−1) 2 −|Y | = Tr14 (Tr23 )2 reads, 7 7 7 7 6 6 6 6 5 in components, ηµ1 µ4 ηµ2 µ3 ηµ2 µ3 . 5 4
4 4
5 5
Summarizing, the action is explicitly written as 1 1 1 4 4 S[φY ]= d 7 x φµ1 µ2 µ3 µ4 ; µ1 µ2 µ3 µ4 ; µ4 (ηµ1 µ4 ηµ2 µ3 ηµ2 µ3 ) (εµ1 ... µ7 . . . εµ1 ... µ7 ) 7 7 7 7 6 6 6 6 4 4 5 5 4 5 5 2 1 Kµ1 µ2 µ3 µ4 ; µ1 µ2 µ3 µ4 ; µ1 µ2 µ3 µ4 ; µ1 . 1 1 1 1 2 2 2 2 3 3 3 3 4 This construction is more transparent when drawing the G L(7, R) Young diagram Z corresponding to the product (87), in which we mark by an “×” the cells corresponding to the components of the bra φY | (and ket | φY ) and by a “−” the cells corresponding
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
751
to the partial derivatives. The components of the metric tensors are marked by a “◦”. It gives
Z =
µ11 µ21 µ31 µ41
×
×
×
×
µ12 µ22 µ32 µ42
×
×
×
×
µ13 µ23 µ33 µ43
×
−
−
−
−
◦
◦
◦
µ15 µ25 µ35 µ45
◦
◦
◦
×
µ16 µ26 µ36 µ46
×
×
×
×
µ17 µ27 µ37 µ47
×
×
×
×
=
µ14 µ24 µ34 µ44
.
Only one G L(7, R)-irreducible component of the differential multiform ∗1 ∗2 ∗3 ∗4 KY
survives inside the action. The corresponding differential hyperform is denoted by K Y and is labeled by the Young diagram Y = (7, 7, 7, 7) − (3, 3, 3, 4) = (4, 4, 4, 3). The associated Young tableau reads ×
×
×
×
×
×
×
×
µ45 µ35 µ25 µ15
×
◦
◦
◦
µ44 µ34 µ24
◦
◦
◦
µ47 µ37 µ27 µ17 µ46 µ36 µ26 µ16
=
.
(88)
It has been obtained by rotating Z by 180 degrees and removing the cells corresponding to the Young diagram Y . In terms of S L(7, R)-irreducible representations, the tensor
is equivalent to K . K Y Y The Euler-Lagrange equations are 0≈
δS ∝ YS δφµ1 µ2 µ3 µ4 ; µ1 µ2 µ3 µ4 ; µ4 7 7 7 7 6 6 6 6 5
4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 , × (ηµ1 µ4 ηµ2 µ3 ηµ2 µ3 ) K µ µ µ µ |µ µ µ µ |µ µ µ µ |µ µ µ 5 4
4 4
7 6 5 4
5 5
7 6 5 4
7 6 5 4
7 6 5
(89)
where Y S is the projector on the symmetries of φµ1 µ2 µ3 µ4 ; µ1 µ2 µ3 µ4 ; µ4 . In the field 7 7 7 7 6 6 6 6 5 equations (and thus in the Lagrangian), only one G L(7, R)-irreducible component of the tensor product ηµ1 µ4 ηµ2 µ3 ηµ2 µ3 will contribute. It is the irreducible component 4 4 5 4 5 5 characterized by the Young tableau X such that the tensor product X · Y contains Y in its decomposition. We find X = (2, 2, 1, 1). Drawing the diagram,
X
=
◦
◦
◦
◦
◦
◦ .
752
X. Bekaert, N. Boulanger
It is easy to check, using the Littlewood-Richardson rule, that
◦
◦
◦
◦
◦
◦
·
×
×
×
×
×
×
×
×
⊃
×
×
×
×
×
×
×
×
× .
×
◦
◦
◦
◦
◦
◦
S L(7,R) ∼ S L(7,R) Due to the isomorphism V , the Einstein-like equations (89) are = VY Y equivalent to the deMedeiros–Hull equations (90) Y S Tr1 4 Tr2 3 KY ≈ 0 .
Equations (89) and (90) respectively provide a particular example of (73) and (65). 3.2.3. Proof of Theorem 2. The proof may be divided in three distinct parts. Firstly, we show that our definition of the action produces a result different from zero, which is a non-trivial statement due to the numerous contractions of various irreducible tensors. Secondly, the kinetic operator (71) is proven to be self-adjoint, which implies that the equations of motion indeed are (73). Thirdly, the Euler–Lagrange equations (73) are shown to be equivalent to the equations of Hull and de Medeiros. In light of the results of Sect. 3.1, this step ends the proof of Theorem 2. The simpler way to start the proof of Theorem 2 is to make explicit the construction of the Lagrangian step by step and exhibit the Young tableaux corresponding to the diverse objects involved, because the procedure is very simple even though the multiplicity of indices somehow casts a shadow on this quality. (1◦ ) The starting point is the product of the s Levi-Civita tensors corresponding to the operator ∗1 . . . ∗s in (71). In components, this product reads εµ1 µ2 ... µ D εµ1 µ2 ... µ D . . . εµ1 µ2 ... µ D . 1 1
1
2 2
s
2
s
s
(91)
Obviously, this product defines an irreducible representation of G L(D, R) labeled by the following Young tableau:
Z =
µ11
µ21
...
µs1
µ12
µ22
...
µs2
.. .
.. .
...
.. .
µ1D
µ2D
...
µsD
(92)
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All other indices present in the Lagrangian have to be contracted with the contravariant indices of the Levi-Civita tensors in (91), therefore we will have to “store” into the tableau (92) the indices of the components of the gauge fields, partial derivatives and metric tensors. Since the components of the tensor gauge field in the bra and in the ket are contracted with the Levi-Civita tensors in the action, the antisymmetrization is automatic so that one may assume without loss of generality that the only algebraic constraints on the gauge field is that it is totally symmetric in the indices appearing in the rows of Y . Only the G L(D, R)-irreducible components of φY will appear in the Lagrangian. For the ket | φY , the tensor gauge field components read φ
r
r
µ11 µ21 ... µs1 ; µ12 µ22 ... µ22 ; ...... ; µ1 µ2 ... µ 1 , 1
1
1
where ra is the length of the a th row in Y . The Young tableau (102) corresponding to the gauge field can be obtained by looking at the Young tableau Y included in the left upper corner of (92). The s partial derivatives in the operator d1 . . . ds read in components ∂µ1
1 +1
∂µ2
2 +1
. . . ∂µs
s +1
(∂µs+1 )ε(s) . 1
The contraction of the components ∂µ1
1 +1
∂µ2
2 +1
. . . ∂µs
s +1
(∂µs+1 )ε(s) φ 1
r
r
µ11 µ21 ... µs1 ; µ12 µ22 ... µ22 ; ...... ; µ1 µ2 ... µ 1
1
1 1
(93)
with the Levi-Civita tensors (91) in the Lagrangian projects the derivatives of the gauge field on the components of the curvature tensor whose symmetry properties are characterized by the Young diagram Y := (1 + 1, 2 + 1, . . . , s + 1) and Young tableau (124). This explains the appearance of the curvature tensor in the ket of the Euler-Lagrange equations (73). In the odd-spin case where ε(s) = 1 and s = s + 1, an extra partial derivative ∂µs+1 is applied on the curvature tensor. The index of this extra partial 1 derivative is not antisymmetrized with the index of any other partial derivative, as can be seen in (93) by the fact that no other partial derivative index µii +1 possesses the same column index: i = s. Therefore the contraction of (93) with (91) is nonzero. The first derivative of the odd-spin curvature tensor is characterized by the Young diagram Y + := (1 + 1, 2 + 1, . . . , s + 1, 1) . An important point to understand next is that the components corresponding to the bra φY | can be chosen as φ
1+ε(s)
µsD µs−1 D ...µ D
s−r2 +1
; µsD−1 µs−1 D−1 ...µ D
; ... ; µsD−
s−r +1
1 +1
...µ D−1 +1
.
(94)
1
It is easier to state the preceding point in terms of Young tableaux and diagrams. The previous ordering of the indices of the bra φY can be read off from (92): One rotates the Young diagram Y corresponding to φY by 180 degrees ( in the plan of the sheet of paper) and places it at the right-bottom corner of (92). The indices appearing in the cells of the rotated Y Young diagram coincide with the components of φY |. s The indices that remain uncontracted in (92) are traced by the operator Tr (D−1) 2 −|Y | , as indicated in (71). The resulting action is nonvanishing because no two indices µij and
µij with the same row index j are contracted by the same epsilon tensor. In the Lagrangian, all the indices with the same row label i could be totally symmetrized without giving
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a vanishing result. In fact, by construction of the Lagrangian, such an operation would be redundant. We have explained how the curvature tensor KY appeared in the Lagrangian, as well as the action of an extra derivative ∂µs+1 when ε(s) = 1. By contraction with the 1 epsilon-tensors (91), the curvature tensor KY is dualized on every column, giving ∗s KY for s even and ∗s+1 ds+1 KY for s odd. By construction, for s even, only the G L(D, R)-irreducible component of the differential multiform ∗s KY which is labeled by the Young diagram Y ∈ Ys will survive in the action. [See Appendix A.1.2 for the general definition of the dual Young diagram Y
∈ V G L D and and tensor.] The corresponding differential hyperform is denoted by K
Y Y
D V = R . The coordinates of Y are (D − s − 1, D − s−1 − 1, . . . , D − 1 − 1). One
and understand its appearance in the Lagrangian by can read the components of K Y inspecting the Young tableau (92): Mark with a “•” the cells of (92) which correspond to KY . Then rotate (92) by 180 degrees. The empty cells now sit at the top of the
. Y associated with the components of K rotated tableau and give the Young tableau Y
Now consider the Young tableau Y included in the left upper-corner of Y . From (94) and
the paragraph below (94), it corresponds to the Young tableau associated with the components of the bra φY |. The remaining indices of Y correspond to the components of s the operator Tr (D−1) 2 −|Y | . Note that the cells in which these remaining indices appear do not constitute a Young diagram. In the odd-spin case, the G L(D, R)-irreducible component of the differential multi . The corresponding form ∗s+1 ds+1 KY which survives in the Lagrangian is labeled by Y + GLD K) and transforms in V with V = R D . differential hyperform is denoted (d s+1 Y+
Y+
∈ Ys+1 are (D − 1, D − − 1, D − The coordinates of Y + s s−1 − 1, . . . , D − 1 − 1). Similarly as in the even-spin case, the Young tableau associated with the components of (d s+1 K)Y is obtained from (92) by marking with a “•” the cells of (92) which corre+ spond to (ds+1 K)Y + and rotating (92) by 180 degrees. The empty cells which sit now associated with the compoat the top of the rotated tableau give the Young tableau Y + nents of (d K) . Again, the Young tableau Y included in the left upper-corner of Y s+1
Y+
+
corresponds to the components of the bra φY | in (94). The remaining indices which correspond to the components sit below and at the right of the Young tableau Y ⊂ Y + (D−1) s+1 −|Y | 2 . The cells in which these remaining indices appear do of the operator Tr not constitute a Young diagram. (2◦ ) The detailed construction of the Lagrangian explained above enables us to provide a Young-diagrammatic proof of the self-adjoint property of the kinetic operator, K† = K. Take the rectangular Young diagram with D rows and s columns which underlies (92). Mark with an “×” the cells corresponding to | φY and φY | and fill with the symbol “−” the cells corresponding to the partial derivatives. Finally, mark with the s symbol “◦” the cells that remain, which correspond to the trace operators Tr (D−1) 2 −|Y | . Denote the resulting rectangular Young tableau by the symbol Z . Now rotate Z by 180 degrees . What appears is not yet Z , since each symbol “−” has to jump upward over the symbols “◦”. Because there is an even number s of “−” and because to each “◦” in the i th column there is a corresponding “◦” in the (s − i + 1)th column, there
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is an even number of jumps. The rotation of Z corresponds to taking the adjoint of φY | K | φY . A jump in a column corresponds to a transposition in the indices of a Levi-Civita tensor, therefore an even number of transposition brings a factor +1. Finally, there is an even number of integrations by part because s is even. We have thus showed that K† = K. Moreover, it is now obvious that the action is invariant under the gauge transformations (72) because it depends on the ket | φY only through the curvature. δS [ φ ] (3◦ ) The Euler-Lagrange equations δ φ Y | ≈ 0 are obtained by varying the acY tion with respect to the bra φY | , so by definition they have the symmetries of | φY . In the odd-spin case, it means that one sets (on-shell) to zero the component (D−1) s+1 −|Y | 2 (ds+1 K)Y which belongs to VYG L D . In the even-spin case, one sets of Tr + s
which belong to V G L D . The (on-shell) to zero the components of Tr (D−1) 2 −|Y | K Y Y operator between square brackets is in general reducible, it decomposes under G L(D, R) into a sum of irreducible powers of the metric tensor. However, only a certain G L(D, R)irreducible component labeled X even for s even (X odd for s odd) will survive in the field Y / X even contains Y and the component equation, the component for which the division / X odd contains Y (see Appendix A.1.1 for the division rule for which the division Y + with Young diagrams). As recalled in Appendix A.1.2, with respect to S L(D, R), the SL irreducible representations Y and Y are equivalent: V S L D ∼ = VY D (V = R D ). SimY and Y are equivalent irreps of S L(D, R). The dual S L(D, R)-irreducible ilarly, Y + + ) is called the contragredient S L(D, R)-irreducible representation Y (respectively Y +
representation of Y (of Y + ), see e.g. the third reference of [48]. Consequently, the field s equations for s even imply that the component of the trace Tr 2 KY which belongs to S L(D,R) is set to zero. In the odd-spin case, it means that the field equations set to zero VY s+1
S L(D,R)
. The latter two field the component of the trace Tr 2 ds+1 KY which belongs to VY equations are therefore equivalent to Eqs. (65) and (66), which in turn are equivalent to the Ricci-flat-like equations Tr KY ≈ 0. Acknowledgements. X.B. is grateful to D. Francia and J. Mourad for discussions on the issue of non-local Lagrangians. The authors thank the Institut des Hautes Études Scientifiques de Bures-sur-Yvette and the Université de Mons-Hainaut for hospitality. One of us (X.B.) is supported in part by the European Research Training Network contract 005104 “ForcesUniverse”.
A. Notation and Conventions In this section, we review former results, introduce the fundamental definitions and take the opportunity to fix the notation.
A.1. Young diagrams and tensorial representations. We essentially extracted the standard definitions and properties on irreps and Young diagrams from various “textbook” references such as [48] (see also [49] and the appendix of the second reference of [8]). A.1.1. Young diagrams and irreducible representations. A Young diagram Y is a diagram which consists of a finite number s > 0 of columns of identical squares (referred
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to as the cells) of finite non-increasing lengths 1 2 . . . s 0. The total number of cells of the Young diagram Y is denoted by |Y | = sj=1 j . The set of Young diagrams with at most s columns is denoted by Ys . We identify any Young diagram Y with its “coordinates” (1 , . . . , s ). For instance, ∈ Y3
Y ≡
is identified with the triple (4, 3, 1) ∈ N3 . A Young tableau is a Young diagram where each cell contains an index. Let Y be the Abelian group made of all formal finite sums of Young diagrams with integer coefficients. This group is N-graded by the number |Y | of boxes: Y = n∈N Yn . The famous “Littlewood–Richardson rule” defines a multiplication law which endows Y with a structure of graded commutative ring. The product of two Young diagrams X and Y is defined as X ·Y = mX Y | Z Z , Z
where the coefficients m X Y | Z = m Y X | Z are the number of distinct labeling of the Young diagram Z obtained from the Littlewood–Richardson rule. As one can see, |X · Y | = |X | + |Y |. A related operation in Y is the “division” of Z by Y defined as Z /Y = mX Y | Z X , X
where the sum is over Young diagrams X such that the product X · Y contains the term Z (with coefficient m X Y | Z ). Multilinear applications with a definite symmetry are associated with a definite Young tableau, while the symmetry in itself is specified by the Young diagram. Let V be a finitedimensional vector space of dimension D over a field K and V ∗ its dual. The dual of the n th tensor power V ⊗n of V is canonically identified with the space of multilinear forms of rank n: (V ⊗n )∗ ∼ = (V ∗ )⊗n . Let Y be a Young diagram whose first column has length 1 < D and let us consider that each of the |Y | copies of V ∗ in the tensor product (V ∗ )⊗|Y | is labeled by one cell of Y . The Schur module VYG L D is defined as the vector space of all multilinear forms T in (V ∗ )⊗|Y | such that : (i) T is completely antisymmetric in the entries of each column of Y , (ii) complete antisymmetrization of T in the entries of a column of Y and another entry of Y that is on the right-hand side of the column vanishes. The space VYG L D is an irreducible subspace invariant for the natural action of G L D on (V ∗ )⊗|Y | . Its elements were called hyperforms by P. J. Olver [7]. Let Y be a Young diagram and T an arbitrary multilinear form in (V ∗ )⊗|Y | ; one defines the multilinear form Y A (T ) ∈ (V ∗ )⊗|Y | by Y A (T ) = T ◦ AY ◦ SY with AY =
(−)ε(c) c , SY = r , c∈C
r ∈R
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
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where C is the group of permutations which permute the entries of each column, ε(c) is the parity of the permutation c, and R is the group of permutations which permute the entries of each row of Y. It can be proved that any Y A (T ) belongs to VYG L D and that the application Y A of End (V ∗ )⊗|Y | satisfies the condition Y A2 = λY A for some number λ = 0 . Thus Y A = λ−1 Y A is a projection of (V ∗ )⊗|Y | onto itself, i.e. Y2A = Y A , with image Im(Y A ) = VYG L D . The projector Y A is referred to as the Young symmetrizer in the antisymmetric convention for the Young diagram Y . Actually the construction of the Young symmetrizer introduced above by first symmetrizing the entries of the rows and then antisymmetrizing the entries of the columns of a given Young tableau could as well have been defined with antisymmetrization first followed by symmetrization. The corresponding irreducible G L D -modules are isomorphic and the corresponding projector is called the Young symmetrizer in the symmetric convention for the Young diagram Y and is denoted by Y S . The changes of convention Y S ◦ Y A and Y A ◦ Y S are mere changes of basis in the Schur module VYG L D . Notice that for Young diagrams Y made of one row (or one column), it is not necessary to specify the choice of convention because both symmetrizers produce the same result; and the corresponding hyperforms of the Schur module VYG L D are usually said to be completely (anti)symmetric tensors. In all other cases, the hyperforms are also called mixed-symmetry tensors in the literature. A
Example. The simplest instance of a mixed-symmetry tensor is the tensor Tµν|ρ of rank µ ρ
identified with the couple (2, 1) ∈ N2 . three associated with the “hook” tableau ν A A A We chose the antisymmetric convention so that Tµν|ρ = T[µν]|ρ and T[µν|ρ] = 0 , where square brackets always denote complete antisymmetrization over all indices with S strength one. In the symmetric convention, we would have a tensor Tµρ ; ν such that S
S
S
Tµρ ; ν = T(µρ) ; ν and T(µρ ; ν) = 0 , where curved brackets always denote complete symmetrization over all indices with strength one. We can switch from one convention S A A S to the other by the following changes of basis Tµρ ; ν = −Tν(µ ; ρ) and Tµν ; ρ = Tρ[µ ; ν] . If the vector space V is endowed with a non-degenerate symmetric bilinear form (i.e. O( p,q) a metric) with signature ( p, q), where p +q = D, then the subspace VY of traceless GLD hyperforms in the Schur module VY is irreducible under the group O( p, q). Whenever the sum of the lengths of the first two columns of Y is greater than D, 1 + 2 > D, then O( p,q) = {0}. So Young diagrams such that the irreducible space is identically zero: VY 1 + 2 D are said to be allowed. All non-zero finite-dimensional irreps of O( p, q) are uniquely characterized by the datum of an allowed Young diagram. Let Y> 0 be the Abelian monoid made of all formal finite sums of Young diagrams with non-negative integer coefficients. Finite direct sums of irreps of G L D may therefore be labeled by elements of Y> 0 via the rule VmG XL D+ n Y = m V XG L D ⊕ n VYG L D , where the positive integer coefficients m, n ∈ N must be interpreted as the multiplicity of the corresponding representations. The same is true for the groups O( p, q) . The evaluation of the Kronecker product of two irreps of G L D can be done by means of the
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Littlewood–Richardson rule which gives
V XG L D ⊗ VYG L D = V XG·YL D =
m X Y | Z VZG L D .
(95)
Z
A related operation is that of contraction of one set of contravariant indices of symmetry Z with a subset of a set of covariant tensor indices of symmetry Y to yield a sum of covariant tensors with indices of symmetry X given by the division rule
LD VZG L D / VYG L D = VZG/Y =
m X Y | Z V XG L D .
X
The irreps of G L D may be reduced to direct sums of irreps of O( p, q) by extracting all possible trace terms formed by contraction with products of the metric tensor and its inverse. The reduction is given by the branching rule
G L D ↓ O( p, q)
O( p,q)
VYG L D ↓ VY /
:
,
(96)
where is the formal infinite sum
= 1+
+
+
+
+
+ ...
+
corresponding to the sum of all possible powers of the metric tensor. The decomposition (96) actually has a useful converse
O( p, q) ↑ G L D
O( p,q)
:
VY
LD ↑ VYG/ −1 ,
(97)
because the series has an inverse
−1 = 1 −
+
−
−
+ ....
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The operation (97) leads to a formal finite sum of irreps, some of which with strictly negative integer coefficients that have to be interpreted as constraints on some trace of the corresponding tensor basis. (Remark: These constraints are not preserved by the full G L D group.) A.1.2. Multiform and hyperform algebras. The elements of the algebra ∧ (V ∗ ) of symmetric tensor products of antisymmetric forms ∈ ∧(V ∗ ) are called multiforms. The s ∗ subspace ∧(V ) of sums of symmetric products of s antisymetric forms is denoted by ∧[s] (V ) ≡ ∧(V ∗ ) · · · ∧(V ∗ ) . s factors
(98)
The D generators of the i th factor ∧(V ∗ ) are written di x µ ( i = 1, 2, . . . , s ). By definition, the multiform algebra ∧[s] (R D ) is presented by the commutation relations di x µ d j x ν = (−)δi j d j x ν di x µ ,
(99)
where the wedge product is not written explicitly. Let G be an Abelian group. The direct sum V∗ = ⊕g Vg is called the G-graded space associated with the family of vector spaces {Vg }g∈G . Moreover, if V is an algebra such that for any two elements α ∈ Vg and β ∈ Vh the product α β ∈ Vg·h , then V is said to be a G-graded algebra. As an example, the algebra ∧[s] (V ) is Ns -graded, ∧[s] (V ) =
(1 ,...,s )∈Ns
∧[s]1 ,2 ,...,s (V ) ,
(100)
where an element α of ∧[s]1 ,2 ,...,s (V ) reads
α =
1 1 α[µ1 ...µ1 ] | ...... | [µs ...µs ] d1 x µ1 ∧ · · · ∧ 1 1 s 1 1 ! . . . s ! ×d1 x
µ1
1
. . . . . . ds x µ1 ∧ · · · ∧ ds x µs . s
s
(101)
Each exterior algebra is Z2 -graded by the parity of the antisymmetric form. This induces a Z2 -grading of the algebra ∧[s] (V ) given by the parity ε(1 + · · · + s ) of the multiform α ∈ ∧[s]1 ,2 ,...,s (V ). The algebra of multiforms is therefore graded commutative [see Eq. (99)]. If (1 , . . . , s ) defines a Young diagram Y , then one can form a Young tableau by placing all the µij indices in (101) corresponding to the i th exterior algebra ∧(V ∗ ) in the
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i th column of Y : µ11
µ21
...
µ12
µ12
µ22
...
µ22
r
...
µs1
r
... .. .
.. .
µ1
2
.. .
µ2
2
.. . µ1
(102)
1
So the space ∧[s]1 ,2 ,...,s (V ) of multiforms is an eigenspace of the operator AY antisymmetrizing over the indices placed in the same column. Conversely, any hyperform in the antisymmetric convention can be seen as a multiform. This induces a natural product on the space of hyperforms. From now on, we will assume that V is equipped with a metric. Then the Hodge dual operations 1 ,...,i ,...,s 1 ,...,D−i ,...,s ∗i : ∧[s] (V ) → ∧[s] (V ) , 1 i s
(103)
in each subspace ∧i (V ∗ ) may be defined. In practice, the operator ∗i acts as the Hodge operator on the i th antisymmetric form in the tensor product. To remain in the space ⊗(V ∗ ) of covariant tensors requires the use of the metric in order to lower contravariant indices. Using the metric, another simple operation that can be defined is the trace. The convention is that we always take the trace over indices in two different columns, say the i th and j th . We denote this operation by ,...,i ,..., j ,...,s
1 Tri j : ∧ [s]
,...,i −1,..., j −1,...,s
1 (V ) → ∧ [s]
(V ), i = j.
(104)
Using the previous definitions of multiforms, Hodge dual and trace operators, we may reformulate the definition of the Schur module as follows: Let α be a multiform in ∧[s]1 ,...,s (V ). If j i < D , ∀ i, j ∈ {1, . . . , s} : i j , then one obtains the equivalence Tri j { ∗i α } = 0 ∀ i, j : 1 i < j s
⇐⇒
D α ∈ V(G1L,..., . s)
(105)
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Indeed, the condition (i) is satisfied since α is a multiform and the condition (ii) is simply rewritten in terms of tracelessness conditions. Let Y be an allowed Young diagram, 1 +2 D. The further irreducibility condition O(D−1,1) obeyed by a multiform α ∈ VY ⊂ VYG L D , is the vanishing of all possible traces. Using the irreducibility conditions (105) under G L D one may show that the vanishing of the trace over the indices placed in the first two columns implies the vanishing of all other possible traces: If α ∈ VYG L D , then: ⇐⇒
α∈
Tr α = 0
⇐⇒
Tri j α = 0 ∀ i, j ∈ {1, . . . , s}
O( p,q) VY ,
(106)
where we defined Tr ≡ Tr12 . Let Y = (1 , . . . , s ) be any Young diagram in Ys . We define the dual Young dia := (˜1 , . . . , ˜s ) by the following lengths of its columns: ˜i := D − s+1−i for gram Y ˜
˜
1 ,...,s 1 ,...,s (V ). One denotes by α ∈ ∧[s] (V ) i ∈ {1, . . . , s} . Let α be a multiform of ∧[s] the dual multiform defined by
α := ∗s α ,
where
∗s ≡
s
∗i .
i=1
The dual multiform α belongs to the same representation space of S L D as α . If αY ∈ VYG L D is a hyperform labeled by the Young diagram Y , then the dual multiform αY is in GLD
the irrep. of G L D associated with the dual Young diagram Y , i.e. αY ∈ VY , called the
contragradient representation of VYG L D . Actually, the representations are equivalent under S L D . If Y = (1 , 2 , . . . , s ) is an allowed Young diagram, 1 + 2 D, then the Young diagram Y ∗ = (D − 1 , 2 , . . . , s ) is also an allowed Young diagram called the assoO( p,q) is a hyperform in the irrep. of ciated Young diagram. In such case, if αY ∈ VY O( p, q) corresponding to the Young diagram Y , then the multiform ∗1 αY is in the irrep. O( p,q) . The two of O( p, q) labeled by the associated Young diagram Y ∗ , i.e. ∗1 αY ∈ VY ∗ irreps of O( p, q) become equivalent when they are restricted to S O( p, q). Notice that, for an allowed Young diagram, all columns but the first one have length i < D/2 (2 i s). Therefore each inequivalent finite-dimensional irreps of S O( p, q) is uniquely characterized by a Young diagram with columns of length smaller than D/2. The metric on V allows to endow the space ∧[s] (V ) of multiforms with a non-degenerate symmetric bilinear form ( , ) : ∧[s] (V ) ∧[s] (V ) −→ K
(107)
called the scalar product defined by taking the scalar product in each of the s exterior algebras ∧(V ∗ ). More explicitly, (α,β ) =
1 α 1 1 1 ! . . . s ! µ1 ...µ1
| ... | µs1 ...µss
β
µ11 ...µ1 | ... | µs1 ...µss 1
for two multiforms α and β which read in components as in (101). The scalar product is positive definite if and only if the metric on V is. Via the left multiplication in ∧[s] (V ) the generators di x µ can be interpreted as operators. Their adjoint (di x µ )† for the scalar
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product reproduces the interior product in each exterior algebra because the operators di x µ and (d j x ν )† satisfy the canonical graded commutation relations [di x µ , (d j x ν )† ]± = δi j ηµν ,
(108)
where [ , ]± stands for the Z2 -graded commutator, ηµν are the components of the µ (pseudo-Riemannian) metric on V and ηµλ ηλν = δν . The anticommutation relations (108) also imply that ∧[s] (V ) is isomorphic to a Fock space whose creation operators would be the di x µ ’s and the destruction operators the (di x µ )† ’s. In terms of the latter operators, the trace operators Tri j defined in (104) can be written as Tri j = ηµν (di x µ )† (d j x ν )† . If (1 , 2 , . . . , s ) defines a Young diagram Y , then the operators AY and SY are welldefined on ∧[s]1 ,2 ,...,s (V ). Moreover, they are self-adjoint, therefore the two Young symmetrizers are the adjoint of each other: Y†A = Y S . There is no ambiguity once a Young tableau is specified. This implies that one may define the non-degenerate product on the space of hyperforms ( , )Y : VYG L D VYG L D → K
(109)
( α , β )Y := ( α , Y A β ) ,
(110)
defined by
where α and β may be taken to be multiforms of ∧[s]1 ,2 ,...,s (V ) but the result depends
only on their irreducible component in VYG L D . One observes that β may naturally be assumed to be in the antisymmetric convention, and α in the symmetric convention. Indeed, ( α , β )Y = ( α , Y A β )Y = ( Y S α , β )Y because the symmetrizers are projectors and adjoint with respect to each other. In Dirac’s terminology, one may take the “bras” to be hyperforms in the symmetric convention and the “kets” in the antisymmetric convention.9 A.2. Differential complexes. The objective of the works presented in [7–10] was to construct complexes for irreducible tensor fields of mixed symmetries, thereby generalizing to some extent the calculus of differential forms. A.2.1. Multicomplex of differential multiforms. We start with basic definitions from homological algebra. A differential complex is defined to be an N-graded space V∗ = ⊕i∈N Vi with a nilpotent endomorphism d of degree one, i.e. there is a chain of linear transformations d
d
d
d
. . . −→ Vi−1 −→ Vi −→ Vi+1 −→ . . . such that d 2 = 0. A well-known example of such structure is the de Rham complex for which the vector space is the set ∗ (Rd ) of differential forms graded by the form degree. The role of the nilpotent operator is played by the exterior derivative d = d x µ ∂µ . One d can now define the quotient H ∗ (d) := Ker Imd called the cohomology of d. This space 9 We are grateful to Marc Henneaux for calling these observations to our attention.
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
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inherits the grading of V∗ . The elements of H (d) are called (co-)cycles. Elements of Imd are said to be trivial or exact (co)-cycles. A straightforward generalization of the previous definitions is to consider a more complicated grading. More specifically, one takes Ns as Abelian group (s 2). A multicomplex of order s ∈ N is defined to be an Ns -graded space V(∗,...,∗) = V(i1 ,...,is ) (i 1 ,...,i s )∈Ns
with s nilpotent endomorphisms d j (1 j s) such that d j V(i1 ,...,i j ,...,n s ) ⊂ V(i1 ,...,i j +1,...,is ) .
A multicomplex of order one is a usual differential complex. A concrete realization of this definition is the space of differential multiforms whose elements are sums of products of the generators d j x µ with smooth functions as coefficients. More precisely, the space of differential multiforms is the graded tensor product of C ∞ (R D ) with s symmetrized copies of the exterior algebra ∧(R D∗ ), where R D∗ is the dual space with basis di x µ (1 i s, thus there are s times D of them). We denote this multigraded space C ∞ (R D ) ⊗ ∧[s] (R D ) as
[s] (R D ) =
[s]1 ,2 ,...,s (R D ) , (111) (1 ,...,s )∈Ns
by analogy with the de Rham complex ∗ (R D ) = [1] (R D ). The tensor field αµ1 ...µ1 |...|µs ...µs (x) defines a multiform α ∈ [s]1 ,...,s (R D ) which explicitly reads 1
1
1
s
α =
1 α 1 1 1 ! . . . s ! µ1 ...µ1 ×d1 x
µ1
1
(x) d1 x µ1 ∧ · · · ∧ 1
| ... | µs1 ...µss
· · · ds x µ1 ∧ · · · ∧ ds x µs . s
s
(112)
In the sequel, when we refer to the differential multiform α we speak either of (112) or of its components. More generally, we call a (smooth covariant) tensor field any element of the space (R D∗ ) ⊗ C ∞ (R D ) . We endow [s] (R D ) with the structure of a multicomplex by defining s exterior derivatives 1 ,...,i ,...,s 1 ,...,i +1,...,s di : [s] (R D ) → [s] (R D ) , 1 i s ,
(113)
defined by taking the exterior derivative with respect to the i th set of antisymmetric indices. Naturally, for each label i (1 i s) one can define the cohomology group Kerd H ∗ (di ) ≡ Im i . The nilpotent operators d j ≡ d j x µ ∂µ generalize the exterior differdi ential of the de Rham complex. If the manifold R D is endowed with a metric then, by using the Hodge operators ∗i introduced previously, one may also define the coderivatives 1 ,...,i ,...,s 1 ,...,i −1,...,s (R D ) → [s] (R D ) , 1 i s. di† := (−)q+1+i (D−i +1) ∗i di ∗i : [s] (114)
As usual, the Laplacian or d’Alembertian may be defined by the anticommutator = [di , di† ]+ . A multiform α in [s] (R D ) is said to be harmonic if it is closed (di α = 0) and coclosed (di† α = 0) for all i ∈ {1, . . . , s} . Notice the very useful identities [ Tri j , dk ]± = 2 δk(i d †j) ,
(115)
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and [ di , d j ]± = 0 ,
[ di , d †j ]± = δi j .
(116)
A.2.2. Generalized complex of differential hyperforms. Let N be a natural number not smaller than 2. An N -complex is defined as a graded space V∗ = ⊕i Vi equipped with an endomorphism d of degree 1 that is nilpotent of order N 2: d N = 0. The generalized cohomology of the N -complex V∗ is the family of N − 1 graded spaces (k) H (d) with 1 k N − 1 defined by (k) H (d) = Ker(d k )/Im(d N −k ), i.e. (k) H ∗ (d) = ⊕i (k) H i (d), where (k) i H (d) = α ∈ Vi | d k α = 0, α ∼ α + d N −k β, β ∈ Vi+k−N . Proposition 2 [10] Any multicomplex structure of order N − 1 possesses a canonical N -complex structure. This fact plays a crucial role in the gauge structure of mixed-symmetry tensor gauge fields. The proof is rather simple. Proof In order to connect the two definitions one has to build an N-grading from the Ns -grading of the multicomplex V(∗,...,∗) = V(i1 ,...,is ) endowed with the s (i 1 ,...,i s )∈Ns
nilpotent endomorphisms d j . A simple choice is to consider the total grading defined by the sum i ≡ sj=1 i j . We introduce the operator dT ≡
s
dj
j=1
which possesses the nice property of being of total degree one. Two convenient cases arise: • [ di , d j ]+ = 0 : Usually the nilpotent operators d j are taken to be anticommuting and therefore d is nilpotent. This case is rather standard in homological perturbation theory. • [ di , d j ]− = 0 when i = j and di 2 = 0 : From our present perspective, commuting d j ’s are indeed quite interesting because, in that case, dT is in general nilpotent of order s + 1 and the space V is endowed with a (s + 1)-complex structure. Indeed, every term in the expansion of dTs+1 contains at least one of the d j twice. Due to the (anti)commutation relations (116), the second case in the proof is illustrated by the multicomplex [s] (R D ) of differential multiforms. The total cohomology group [10] is the generalized cohomology group (k) H (i1 ,...,is ) (dT ) associated with the operator dT and the Ns -grading, whose elements α ∈ V(i1 ,...,is ) satisfy the set of cocycle conditions
i∈I
di α = 0 ,
∀ I ⊂ {1, 2, . . . , s} | #I = k ,
(117)
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
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with the equivalence relation
J ⊂ {1, 2, . . . , s} #J = s − k + 1
j∈J
α ∼ α +
where β J belongs to V( j1 ,..., js ) with i jk ≡ k ik − 1
d j βJ ,
(118)
if k ∈ J , if k ∈ J .
This can be easily seen by decomposing the cocycle condition dTk α = 0 and the equivalence relation α ∼ α + dTs−k+1 β in Ns degree. A differential hyperform [7] is a G L(D, R)-irreducible tensor field, that is, an G L(D,R) Y (R D ) the space of differential . We denote by (s) element of C ∞ (R D ) ⊗ VY hyperforms associated with the Young diagram Y made of s columns. We also introduce the Ys -graded space Y
(s) (R D ) . (119)
(s) (R D ) = Y ∈Ys
In order to endow the space (s) (R D ) with a structure of multicomplex, one may introduce the maps [9, 10] ( ,...,i ,...,s )
1 d {i} : (s)
( ,...,i +1,...,s )
1 (R D ) → (s)
(R D ) ,
(120)
for 1 i s and i+1 > i . This operator is defined as follows: take the derivative of a {i} differential hyperform of Y(s) and consider the image in Y(s) , where Y {i} is the Young diagram obtained from Y by adding one more cell in the i th column. In other words, d {i} ≡ Y{i} ◦ ∂. Since hyperforms in the antisymmetric convention may also be seen as A multiforms, the action of an operator d {i} may be expressed as a linear combination of the action of the exterior derivatives d j . So we have the obvious property that, for any differential hyperform α of (s) (R D ) , di α = 0 , ∀I ⊂ {1, 2, . . . , s} | #I = k i∈I
=⇒
d {i} α = 0, ∀I ⊂ {1, 2, . . . , s}|#I = k.
(121)
i∈I
We proved in [9] the triviality of the generalized cohomology groups (k) H (1 ,...,s ) (d) for 1 k s, 0 < s and 1 < D, in the space of differential hyperforms (s) (R D ) with d = d {1} + · · · + d {s} , thereby extending the results of [7, 8]. In particular, for (1) H Y (d), where Y ∈ Ys is a Young diagram made of s columns, one may show10 that Y (R D ) are equivalent to the closure conditions of a hyperform KY ∈ (s) di KY = 0 , (i = 1, . . . , s) 10 See Corollary 1 of [9] for more details.
(122)
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and that they imply the following exactness of the differential hyperform: KY = d1 . . . ds φY ,
(123)
Y (R D ) with Y the Young diawhere φY is a differential hyperform belonging to (s) gram obtained by removing the first row of Y . Such an exact hyperform KY is called the curvature tensor of the gauge field φY . If the components φµ1 ...µ1 | ... | µs ...µs 1
1
1
s
of the gauge field are characterized by the Young tableau (102), then the components Kµ1 ...µ1 | ... | µs ...µs are described by the Young tableau 1
1 +1
1
s +1
µ11
µ21
...
µ12
r
...
µs1
µ12
µ22
...
µ22
r
...
µs2
µ13
µ23
...
µ32
r
... .. .
.. .
.. .
µ1 +1 µ2 +1 2 2
.. . µ1 +1 1
(124)
Analogously, for (s) H Y (d), where Y is a Young diagram made of s columns, one Y (R D ) is equivalent to may show that the closure condition of a hyperform φY ∈ (s) d1 . . . ds φY = 0 ,
(125)
and they imply the following exactness of the differential hyperform: φ Y = SY
s i=1
di i =
s
d {i} {i} ,
(126)
i=1
where the i are differential multiforms belonging to [s]1 ,...,i −1,...,s (R D ) while the {i} are differential hyperforms (or zero if they are not well-defined) belonging to ( ,...,i −1,...,s ) (R D ). Such an exact hyperform φY is called a pure gauge field.
(s)1 The norm of the functions in L 2 (R D ) together with the scalar product on ∧[s] (R D ) define a natural non-degenerate symmetric bilinear form on the space [s] (R D ) of differential multiforms, so that the codifferential di† in (114) becomes the adjoint of the exterior
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
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derivative di . This implies that one may define the following scalar product on the space of differential hyperforms Y Y , : (s) (R D ) (s) (R D ) → R
defined by
(127)
α | β :=
d D x ( α , β )Y ,
(128)
where ( α , β )Y is the scalar product (110) on VYG L(D,R) . We remind the reader that, without loss of generality, one may take the bras α | to be differential hyperforms in the symmetric convention and the kets | β in the antisymmetric convention. Given a non-degenerate symmetric bilinear form , on a functional space, a quadratic action for the field φ is a bilinear functional S[φ] = φ | K | φ entirely determined by the datum of a self-adjoint (pseudo)differential operator K called kinetic operator. Because of the non-degeneracy of the bilinear form, the action S[φ] is extremized for configurations obeying the field equation K | φ = 0. Translation invariance requires the kinetic operator K to be independent of the coordinates x, hence the field equation is a linear partial differential equation (PDE) with constant coefficients. Boundary conditions and regularity requirements should be specified when solving PDEs.11 For instance, in order to convert linear PDEs into algebraic equations by going to the momentum representation, we consider the gauge field φY either as a rapidly decreasing smooth function or as a tempered distribution, that is the ket | φY ∈ S(R D ) ⊗ VYG L(D,R) G L(D,R) and the bra φY | ∈ S (R D ) ⊗ VY . The action S [ φY ] is said to be gauge invariant under (72) if di i | K | φY = 0 for all i and φY . This gauge invariance property is equivalent to the Noether identity di† K = 0 since the bilinear form is non-degenerate. B. Technical Lemmas B.1. Proof of Lemma 1. We consider any two adjacent columns of the differential hyperform P...|µ1 ...µr |ν1 ...νq |... , and we want to show that the following implication holds (without making explicit the other columns this time; they play no role in the proof) Pµ1 ...µr |[ν1 ...νq ,ρ] = 0
=⇒
∂[ρ Pµ1 ...µr ]|ν1 ...νq = 0 ,
(129)
where a comma stands for a derivative. In the case where q = r , the above implication is trivial (P is then symmetric under the exchange of the two columns), so we assume q < r from now on. (A) Since P ∈ (s) (R D ) , one has P[µ1 ...µr |ν1 ]ν2 ...νq ≡ 0 which gives Pµ1 ...µr |ν1 ...νq ≡ r (−)r Pν1 [µ1 ...µr −1 |µr ]ν2 ...νq . Without bothering about coefficients, we write Pν1 [µ1 ...µr −1 |µr ]ν2 ...νq ∝ Pµ1 ...µr |ν1 ...νq .
(130)
(B) We antisymmetrize on the first (r + 2) indices of the differential hyperform P , yielding K[µ1 ...µr |ν1 ν2 ]ν3 ...νq ≡ 0 . Decomposing this identity, we see three classes of terms appearing, where ν1 and ν2 are 11 Throughout this article, we are sometimes sloppy concerning such technical issues of functional analysis because our main concern is algebraic. Practically, this means that we always implicitly assume that the functional space we work with is such that the objects we talk about and the operations we perform on them, are well defined. There is no lack of rigor in such assumption because they may be legitimated and we refer to textbooks such as [50] for details.
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1. both in the first column, 2. one in the first column, the second in the other, 3. both in the second column. Explicitly one finds 0 ≡ a Pν1 [µ1 ...µr −1 |µr ]ν2 ν3 ...νq − Pν2 [µ1 ...µr −1 |µr ]ν1 ν3 ...νq +b Pν1 ν2 [µ1 ...µr −2 |µr −1 µr ]ν3 ...νq + c Pµ1 ...µr |ν1 ...νq ] ,
(131)
for some non-vanishing coefficients a, b, c ∈ N0 . This allows us to write Pν1 ν2 [µ1 ...µr −2 |µr −1 µr ]ν3 ...νq as a linear combination of Pµ1 ...µr |ν1 ...νq and Pν1 [µ1 ...µr −1 |µr ]ν2 ν3 ...νq − Pν2 [µ1 ...µr −1 |µr ]ν1 ν3 ...νq . Using (130) one obtains Pν1 ν2 [µ1 ...µr −2 |µr −1 µr ]ν3 ...νq ∝ Pµ1 ...µr |ν1 ...νq .
(132)
(C) Starting this time from P[µ1 ...µr |ν1 ν2 ν3 ]ν4 ...νq ≡ 0 and using the relations (130) and (132), one obtains similarly Pν1 ν2 ν3 [µ1 ...µr −3 |µr −2 µr −1 µr ]ν4 ...νq ∝ Pµ1 ...µr |ν1 ...νq . At the end of the day one gets Pν1 ...νq [µq+1 ...µr |µ1 ...µq ] ∝ Pµ1 ...µr |ν1 ...νq .
(133)
As a consequence of our starting hypothesis, Eq. (129), we have Pν1 ...νq [µq+1 ...µr |µ1 ...µq , ρ] = 0 , and finally, using Relation (133), ∂[ρ Pµ1 ...µr ]|ν1 ...νq = 0 .
B.2. Proof of Lemma 2. The proof is somewhat tedious because it requires some care with the combinatorial gymnastic. By definition, Tr(12 . . . Tr2n−1 2n) =
1 (2n)! π ∈S2n
Trπ(2i−1)π(2i) .
i∈{1,...,n}
To start with, one makes use of (115) for i = j in order to rearrange the factors in the following sum over all permutations π of the set {1, . . . , 2n} Tr(12 . . . Tr2n−1 2n) d1 d2 . . . d2n−1 d2n = n
1 Trπ(2i−1)π(2i) dπ(2i−1) dπ(2i) . = (2n)! π ∈S2n
(134)
i=1
Then, one evaluates each factor † † Trπ(2i−1)π(2i) dπ(2i−1) dπ(2i) = − dπ(2i−1) dπ(2i−1) − dπ(2i) dπ(2i)
+ dπ(2i−1) dπ(2i) Trπ(2i−1)π(2i) ,
(135)
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
769
by using (57). Now, one inserts (135) into the products n
Trπ(2i−1)π(2i) dπ(2i−1) dπ(2i)
i=1
= n + n−1
2n
−
d j d †j +
j=1
+
n−2
2n−2
n
dπ(2i−1) dπ(2i) Trπ(2i−1)π(2i)
i=1
2n
† † dπ( j) dπ(k) dπ( j) dπ(k)
j=1 k= j+1+ε( j)
2n
+
di d j dk (. . .) . (136)
i, j,k=1
We evaluated and grouped the terms in (136) according to the number of d’Alembertians and curls, by making use of the commutation relation (116). More precisely, the decomposition in powers of the d’Alembertian goes as follows. n : The leading term comes from picking the d’Alembertian in each of the n factors in the product. n−1 : The terms come from choosing a factor and taking the d’Alembertian in the n − 1 remaining factors. Still, one may either choose in the right-hand-side of (135) one of the terms of the form d j d †j or the last term with the trace. n−2 : In degree n − 2, the terms are of two types: either they contain two curls and are of the form d j dk d †j dk† or they contain at least three curls. The first type of terms comes from choosing two factors in the product and one term of the form d d † in each of them. All other choices give rise to terms of the second type. n−3 : All terms of degree n − 3 or lower in the d’Alembertian include at least three curls. All such terms have been put together in the last term of (136). Eventually, one should perform the sum over all permutations of the 2n elements in the set {1, . . . , 2n}. The result is n
1 Trπ(2i−1)π(2i) dπ(2i−1) dπ(2i) (2n)! π ∈S2n
i=1
= n + n−1
−
2n j=1
+
d j d †j +
2n 1 d j dk Tr jk 2(2n − 1)
(137)
j,k=1
2n 2n n − 1 n−2 d j dk d †j dk† + di d j dk (. . .) , 2n − 1 j,k=1
i, j,k=1
because of the two identities n
dπ(2i−1) dπ(2i) Trπ(2i−1)π(2i) = n
π ∈S2n i=1
dπ(1) dπ(2) Trπ(1)π(2) ,
π ∈S2n
and 2n−2
2n
π ∈S2n j=1 k= j+1+ε( j)
† † dπ( j) dπ(k) dπ( j) dπ(k) = 2n(n − 1)
π ∈S2n
† † dπ(1) dπ(2) dπ(1) dπ(2) ,
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supplemented by the fact that for any object s jk symmetric in its indices j and k,
sπ(1)π(2) = (2n − 2)!
π ∈S2n
2n
s jk .
(138)
j,k=1
Finally, by making use of the definition (53) in (137) and going back to the departure equation (134), one obtains by straightforward algebra Tr(12 . . . Tr2n−1 2n) d1 . . . d2n = n−1 F +
2n n − 1 n−2 d j dk − Tr jk + d †j dk† 2n − 1 j,k=1
+
2n
di d j dk (. . .) ,
i, j,k=1
The commutation relation (115) ends the proof of Lemma 2. C. Light-Cone The proof of the theorem was already sketched in Appendix A of [42] but we present it here in full details in order to be self-contained. In physical terms, the proof amounts to show that on-shell fieldstrengths are essentially gauge fields in the light-cone gauge [18, 51]. Indeed, in order to prove Theorem 1 it is convenient to introduce a light-cone basis associated with any light-like vector p µ : we define it as a basis of R D−1,1 such that the light-like direction + is normalized along the vector while the light-like direction − is orthogonal and the remaining space-like directions define the transverse hyperplane R D−2 . Hence, p + = 1 is the only non-vanishing component of the vector p µ in this basis. Lemma 3 Let p µ be a given vector on the lightcone (defined by p 2 = 0) in Minkowski space R D−1,1 . This vector defines the operators pi = pµ di x µ and their adjoint pi† = pµ (di x µ )† . Any multiform | α of the Fock space ∧[s]1 ,2 ,...,s (R D ) with s > 0 such that pi | α = 0 ,
pi† | α = 0 ,
∀ i ∈ {1, . . . , s}
reads in the light-cone basis | α = p1 p2 . . . ps | β , 1 −1,2 −1,...,s −1 (R D−2 ) is a multiform on the transverse hyperplane where | β ∈ ∧[s] R D−2 .
Proof of Lemma 3 As explained in Appendix A.1.2, the space ∧[s] (R D ) is isomorphic to a Fock space whose creation operators are the di x µ and the destruction operators the (di x µ )† . In the light-cone basis, the condition pi† | α = (di x − )† | α = 0 states that the i th Fock space ∼ = ∧(R D ) is in the vacuum for the creation operator di x + .12 Thus the occupation number of di x + is zero for all integers i from 1 to s. 12 Because the metric is off-diagonal in the light-cone directions.
Tensor Gauge Fields in Arbitrary Representations of G L(D, R). II
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On the one hand, the condition pi | α = di x − | α = 0 states that the i th Fock space ∼ = ∧(R D ) has maximal occupation number for the creation operator di x − . For any fixed i, this operator is Grassmann-odd, thus its maximal occupation number is equal to one. This is true for all integers i, hence | α = d1 x − d2 x − . . . ds x − | β for some 1 −1,2 −1,...,s −1 multiform | β ∈ ∧[s] (R D ) . On the other hand, we have also shown that the occupation number is zero for all creation operators di x + , thus | β is transverse and belongs to ∧[s] (R D−2 ) . Proof of Theorem 1 The on-shell harmonicity of the differential hyperform KY implies that the massless Klein–Gordon equation KY ≈ 0 is obeyed. Let us Fourier transform the tensor field components KY (x) in such a way that the harmonicity conditions become algebraic.13 The d’Alembert equation implies that the support of the Fourier transform KY ( p) is on the mass-shell p 2 ≈ 0, so that the momentum vector p µ is light-like on-shell. For each Fourier mode of the tensor field KY ( p) associated with a momentum vector p µ , let us introduce a light-cone basis. As follows from Lemma 3, the harmonicity conditions impose that the components of each Fourier mode are on-shell equal to KY ( p) ≈ p 1 . . . p s φY ( p) for some transverse multiform φY ( p) labeled by the Young diagram Y . It is now easy to prove that the on-shell O(D − 1, 1)-irreducibility conditions of the components KY ( p) imply the O(D − 2)-irreducibility condition of the components of φY ( p) . Therefore the harmonicity conditions restrict the hyperform KY ( p) to carry an UIR of O(D − 2) labeled by the Young diagram Y . This conclusion is true for any Fourier mode, therefore it applies to the complete Fourier transform as well. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Wigner, E.P.: Annals Math. 40, 149 (1939) Nucl. Phys. Proc. Suppl. 6, 9 (1989) Bargmann, V., Wigner, E.P.: Proc. Nat. Acad. Sci. 34, 211 (1948) Fierz, M., Pauli, W.: Proc. Roy. Soc. Lond. A 173, 211 (1939) Singh, L.P.S., Hagen, C.R.: Phys. Rev. D 9, 898; 910 (1974) Fronsdal, C.: Phys. Rev. D 18, 3624 (1978) Fang, J., Fronsdal, C.: Phys. Rev. D 18, 3630 (1978) Olver, P.J.: “Differential hyperforms I.” Univ. of Minnesota report 82-101; “Invariant theory and differential equations.” In: Koh, S.: Invariant theory. Berlin-Heidelberg-New York: Springer-Verlag, 1987 p. 62 Dubois-Violette, M., Henneaux, M.: Lett. Math. Phys. 49, 245 (1999); Commun. Math. Phys. 226, 393 (2002) Bekaert, X., Boulanger, N.: Commun. Math. Phys. 245, 27 (2004) Bekaert, X., Boulanger, N.: Class. Quant. Grav. 20, S417 (2003) Govindarajan, T.R., Rindani, S.D., Sivakumar, M.: Phys. Rev. D 32, 454 (1985); Rindani, S.D., Sivakumar, M.: Phys. Rev. D 32, 3238 (1985); Aragone, C., Deser, S., Yang, Z.: Annals Phys. 179, 76 (1987); Pashnev, A.I.: Theor. Math. Phys. 78, 272 (1989); Rindani, S.D., Sahdev, D., Sivakumar, M.: Mod. Phys. Lett. A 4, 265, 275; (1989); Hallowell, K., Waldron, A.: Nucl. Phys. B 724, 453 (2005); Boulanger, N., Kirsch, I.: Phys. Rev. D 73, 124023 (2006)
13 Boundary conditions and regularity requirements should be specified when solving PDEs. In Theorem 1, O(D−2) we implicitly assumed that the “ket” | ϕY ∈ L 2 (R D ) ⊗ VY . This choice is convenient because (a) it |x|→∞
provides an obvious norm for HY , (b) it selects solutions such that | ϕY (x) | −→ 0, and (c) if we consider O(D−2) ϕY as a temperate distribution (since the “bra” ϕY | ∈ S (R D ) ⊗ VY ) then we are always allowed
to convert linear PDEs into algebraic equations by going to the momentum representation.
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12. 13. 14. 15. 16. 17.
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Commun. Math. Phys. 271, 775–789 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0186-y
Communications in
Mathematical Physics
Families Index Theorem in Supersymmetric WZW Model and Twisted K-Theory: The SU(2) Case Jouko Mickelsson1,2 , Juha-Pekka Pellonpää3 1 Department of Mathematics and Statistics, P.O. Box 68, University of Helsinki, FI-00014 Helsinki, Finland 2 Mathematical Physics, KTH, Stockholm SE-10691, Sweden. E-mail: [email protected] 3 Department of Physics, University of Turku, FI-20014 Turku, Finland. E-mail: [email protected]
Received: 28 June 2006 / Accepted: 11 August 2006 Published online: 31 January 2007 – © Springer-Verlag 2007
Abstract: The construction of twisted K-theory classes on a compact Lie group is reviewed using the supersymmetric Wess-Zumino-Witten model on a cylinder. The Quillen superconnection is introduced for a family of supercharges parametrized by a compact Lie group and the Chern character is explicitly computed in the case of SU (2). For large euclidean time, the character form is localized on a D-brane. 0. Introduction Gauge symmetry breaking in quantum field theory is described in terms of families index theory. The Atiyah-Singer index formula gives via the Chern character cohomology classes in the moduli space of gauge connections and of Riemann metrics. In particular, the 2-form part is interpreted as the curvature of the Dirac determinant line bundle, which gives an obstruction to gauge covariant quantization in the path integral formalism. The obstruction depends only on the K-theory class of the family of operators. In the Hamiltonian quantization odd forms on the moduli space become relevant, [CMM]. The obstruction to gauge covariant quantization comes from the 3-form part of the character. The 3-form is known as the Dixmier-Douady class and is also the (only) characteristic class of a gerbe; this is the higher analogue of the first Chern class (in path integral quantization) classifying complex line bundles. The next step is to study families of “operators” which are only projectively defined; that is, we have families of hamiltonians which are defined locally in the moduli space but which refuse to patch to a globally defined family of operators. The obstruction is given by the Dixmier-Douady class, an element of integral third cohomology of the moduli space. On the overlaps of open sets the operators are related by a conjugation by a projective unitary transformation. This leads to the definition of twisted K-theory, [DK, Ro]. In the present paper we shall first review the basic definitions of both ordinary K-theory and twisted K-theory in Sect. 1. In Sect. 2 the construction of twisted
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(equivariant) K-theory classes on compact Lie groups is outlined using a supersymmetric model in 1 + 1 dimensional quantum field theory. Finally, in Sect. 3 the Quillen superconnection formula is applied to the projective family of Fredholm operators giving a Chern character alternatively with values in Deligne cohomology on the base or in global twisted de Rham forms, [BCMMS]. The use of Quillen superconnection has been proposed in the general context of twisted K-theory in [Fr], but here we will give the details in simple terms using the supersymmetric Wess–Zumino-Witten model. The Quillen superconnection can be modified (Prop. 2 and Eq. (3.13)) to give a map from twisted K-theory to twisted cohomology on the base space. However, the (nonequivariant) twisted K-theory of compact Lie groups is all torsion [Br, Do] and the twisted cohomology vanishes. Nevertheless it turns out that, at least in the case of the group SU (2), the cocycle evaluated from the superconnection formula contains more information than its twisted cohomology class. We show that the de Rham form is integrally quantized as the dimension of the relevant SU (2) representation times the basic 3-form on the group manifold. We also construct a map (not restricted to the case of SU (2)) from twisted K-theory to gerbes over the base, as defined in terms of local line bundles, modulo the twisting gerbe. In the case of SU (2) this map agrees with the known identification K 1 (SU (2), k) = Z/kZ, [Ro]. In the limit → ∞ for the scaling parameter in the superconnection, the support of the form is localized on a ‘D-brane’, a quantized conjugacy class on SU (2), [GR]. 1. Twist in K-Theory by a Gerbe Class Let M be a compact manifold and P a principal bundle over M with structure group PU (H ), the projective unitary group of a complex Hilbert space H . We shall consider the case when H is infinite dimensional. The characteristic class of P is represented by an element ∈ H 3 (M, Z), the Dixmier-Douady class. Choose a open cover {Uα } of M with local transition functions gαβ : Uαβ → PU (H ) of the bundle P. In the case of a good cover we can even choose lifts gˆ αβ : Uαβ → U (H ), to the unitary group in the Hilbert space H, on the overlaps Uαβ = Uα ∩ Uβ , but then we only have gˆ αβ gˆ βγ gˆ γ α = σαβγ · 1 for some σαβγ : Uαβγ → S 1 , where Uαβγ = Uα ∩Uβ ∩Uγ and 1 is the identity operator. Complex K-theory classes on M may be viewed as homotopy classes of maps M → Fr ed, to the space of Fredholm operators in an infinite-dimensional complex Hilbert space H. This defines what is known as K 0 (M). The other complex K-theory group is K 1 (M) and this is defined by replacing Fr ed by Fr ed∗ , the space of self-adjoint Fredholm operators with both positive and negative essential spectrum. The twisted K-theory classes are here defined as homotopy classes of sections of a fiber bundle Q over M with model fiber equal to either Fr ed or Fr ed∗ . One sets Q = P × PU (H ) Fr ed, and similarly for Fr ed∗ , where the PU (H ) action on Fr ed is simply the conjugation by a unitary transformation gˆ corresponding to g ∈ PU (H ). We denote by K ∗ (M, []) the twisted K theory classes, the twist given by P.
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Using local trivializations a section is given by a family of maps ψα : Uα → Fr ed such that −1 (x)ψa (x)gˆ αβ (x) ψβ (x) = gˆ αβ
on the overlaps Uαβ . 2. Supersymmetric Construction of K (M,[]) We recall from [Mi1] the construction of the operator Q A as a sum of a ‘free’ supercharge Q and an interaction term Aˆ in (2.7) acting in H. The Hilbert space H is a tensor product of a ‘fermionic’ Fock space H f and a ‘bosonic’ Hilbert space Hb . Let G be a connected, simply connected simple compact Lie group of dimension N and g its Lie algebra. The space Hb carries an irreducible representation of the loop algebra Lg of level k where the highest weight representations of level k are classified by a finite set of G representations (the basis of Verlinde algebra) on the ‘vacuum sector’. In a Fourier basis the generators of the loop algebra are Tna , where n ∈ Z and a = 1, ..., dim G = N . The commutation relations are k c [Tna , Tmb ] = λabc Tn+m + nδab δn,−m , 4
(2.1)
where the λabc ’s are the structure constant of g; in the case when g is the Lie algebra of SU (2) the nonzero structure constants are completely antisymmetric and we use the normalization λ123 = √1 , corresponding to an orthonormal basis with respect to 2 −1 times the Killing form. This means that in this basis the Casimir invariant C2 = a,b,c λabc λacb takes the value −1. In a unitary representation of the loop group we have the hermiticity relations a (Tna )∗ = −T−n .
With this normalization of the basis, for G = SU (2), k is a nonnegative integer and 2 j = 0, 1, 2, ..., k labels the possible irreducible representations of SU (2) on the vacuum sector. The case k = 0 corresponds to a trivial representation and we shall assume in the following that k is strictly positive. In general the level k is quantized as an integer x times twice the square of the length of the longest root with respect to the dual Killing form (this unit is in our normalization equal to 1 in the case G = SU (2)); alternatively, we can write k = 2x/ h ∧ , where h ∧ is the dual Coxeter number of the Lie algebra g. The space H f carries an irreducible representation of the canonical anticommutation relations (CAR), ψna ψmb + ψmb ψna = 2δab δn,−m ,
(2.2)
a . The representation is fixed by the requirement that there is an irreducand (ψna )∗ = ψ−n ible representation of the Clifford algebra {ψ0a } in a subspace H f,vac such that ψna v = 0 for n < 0 and v ∈ H f,vac . The central extension of the loop algebra at level 2 is represented in H f through the operators
K na := −
1 4
b,c;m∈Z
b λabc ψn−m ψmc ,
(2.3)
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which satisfy 1 c + nδab δn,−m . [K na , K mb ] = λabc K n+m 2
(2.4)
We set Sna = 1 ⊗ Tna + K na ⊗ 1. This gives a representation of the loop algebra; in the case G = SU (2) the level is k + 2 in the tensor product H = H f ⊗ Hb . In the parametrization of the level by the integer x this means that we have a level shift x → x = x + h ∧ . Next we define the supercharge operator 1 a a a a ψn T−n + ψn K −n . (2.5) Q := i 3 a,n This operator satisfies Q 2 = h, where h is the hamiltonian of the supersymmetric Wess-Zumino-Witten model, k+2 1 N a a ˜ f + N , (2.6) · = h b + 2kh h := − : Tna T−n :+ : nψna ψ−n :+ 2 4 24 24 a,n a,n =:h b
=:h f
where k˜ := k+2 4 and the normal ordering :: means that the operators with negative Fourier a ψ b : = −ψ b ψ a index are placed to the right of the operators with positive index, : ψ−n n n −n if n > 0 and : AB := AB otherwise. In the case of the bosonic currents Tna the sign is + on the right-hand-side of the equation. Finally, the gauged supercharge operator Q A is defined as Q A := Q + i k˜ ψna Aa−n , (2.7) a,n
Aan ’s
where the are the Fourier components of the g-valued function A in the basis Tna . We denote the mapping A → Q A by Q • . The basic property of the family of self-adjoint Fredholm operators Q A is that it is equivariant with respect to the action of the central extension of the loop group LG. Any element w ∈ LG is represented by a unitary operator S(w) in H but the phase of S(w) is not uniquely determined. The equivariance property is S(w −1 )Q A S(w) = Q Aw with Aw = w −1 Aw + w −1 dw. For the Lie algebra we have the relations
λabc ψmb Acn−m . [Sna , Q A ] = i k˜ nψna +
(2.8)
(2.9)
b,c;m
The group LG can be viewed as a subgroup of the group PU (H ) through the projective representation S and occasionally we write w instead of S(w). [In order that the embedding is continuous in operator norm in the Hilbert space of a positive energy representation, one should replace LG by the Sobolev completion with respect to the weight 1/2, [PS].] The space A of smooth vector potentials on the circle is the total space for a principal bundle with fiber G ⊂ LG, the group of based loops at 1. Since now G ⊂ PU (H ), A may be viewed as a reduction of a PU (H ) principal bundle over
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G. The G action by conjugation on the Fredholm operators in H defines an associated fiber bundle Q over G and the family of operators Q A defines a section of this bundle. Thus {Q A } is a twisted K-theory class over G where the twist is determined by the level k + 2 projective representation of LG. The Dixmier-Douady class on G = S 3 is k + 2 times the basic class on S 3 . This is seen as follows. The transition function h on the equator S 2 in G = S 3 of the principal G bundle A → G represents the basic 2-homology class in G since the principal bundle is the classifying bundle for the based loop group (the total space is contractible). It follows that the pull back h ∗ of the basic 2-form on G is the basic 2-form on the sphere S 2 . This in turn transgresses to the basic 3-form on S 3 . The level k + 2 projective representation of G in a Hilbert space H defines the (k + 2)th power of the basic circle bundle on G and therefore the embedding G ⊂ PU (H ) defines a PU (H ) bundle over S 3 which has Dixmier-Douady class equal to k + 2 times the basic 3-cohomology class. There is additional gauge symmetry due to constant global gauge transformations. For this reason the construction above leads to elements in the G-equivariant twisted Ktheory K G∗ (G, []), where the G-action on G is the conjugation by group elements. The tr (g −1 dg)3 , where the trace is computed in the class [] is represented by the form (k+2) 24π 2 defining representation of SU (2). It happens that in the case of SU (2) the construction gives all generators for both equivariant and nonequivariant twisted K-theories, but not for other compact Lie groups. 3. Quillen Superconnection Let Q A be the supercharge associated to the vector potential A on the circle, with values in the Lie algebra g. Recall that this transforms as gˆ −1 Q A gˆ = Q Ag with respect to g ∈ LG ⊂ PU (H ). Consider the trivial Hilbert bundle over A with fiber H, the operators Q A acting in the fibers. Define a covariant differentiation ∇ acting on the sections of the bundle, ∇ := δ + ω, ˆ where δ is the exterior differentiation on A and ωˆ is a connection 1-form defined as follows. First, any vector potential on the circle can be uniquely written as A = f −1 d f for some smooth function f : [0, 1] → G such that f (0) = 1; here we parametrize S 1 with parameter y ∈ [0, 1] such that any element of S 1 is of the form e2πi y . A tangent vector δ f at f is then represented by a function v : [0, 1] → g such that v(0) = 0 with periodic derivatives at the end points and v = f −1 δ f. We set, [CM], ω f (δ f ) := f −1 δ f − α f −1 (δ f (1) f (1)−1 ) f,
(3.1)
where α is a fixed smooth real valued function on [0, 1] such that α(0) = 0, α(1) = 1 and all derivatives equal to zero at the end points. The point of the second term in (3.1) is that it makes the whole expression periodic so that ω takes values in Lg. Then ωˆ f (δ f ) is defined by the projective representation S of Lg in H. The gauge transformation A → A g corresponds to the right translation r g ( f ) = f g, which sends ω f (δ f ) to g −1 ω f (δ f )g. However, for the quantized operator ωˆ we get an additional term. This is because of the central extension LG which acts on ωˆ through the adjoint representation. One has gˆ −1 ωˆ gˆ = g −1 ωg + γ (ω, g)
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with γ (ω, g) :=
k+2 8π
S1
ω, dgg −1 K .
The bracket · , · K is the Killing form on g. But one checks that the modified 1-form k+2 ωˆ c | f := ωˆ f −
ω f , f −1 d f K 8π S 1 transforms in a linear manner, gˆ −1 ωˆ c gˆ = r g ωc .
(3.2)
Here r g denotes the right action of G on A and the induced right action on connection forms. We would like to construct characteristic classes on the quotient space A/G from classes on A using the equivariance property (3.2). First, we can construct a Quillen superconnecton [Qu] as the form of mixed degree Dt :=
√
1 t Q • + δ + ωˆ c − √ ψ, F, 4 t
(3.3)
where F is the algebra valued curvature form computed from the connection ω and Lie a , where F a ’s are the Fourier coefficients of F. Formally, this
ψ, F := ψna F−n −n expression is the same as the Bismut superconnection for families of Dirac operators, [Bi]. Here t is a free positive real scaling parameter. This is introduced since in the case of the Bismut superconnection one obtains the Atiyah-Singer families index forms in the limit t → 0 from the formula (3.4) or (3.5) below. When dim G is even, we define a family of closed differential forms on A from t := trs e
√ −( t Q • +δ+ωˆ c −
1 √
ψ,F)2 4 t
.
(3.4)
In this case, the supertrace is defined as trs (·) = tr (·). Here is the grading operator with eigenvalues ±1. It is defined uniquely up to a phase ±1 by the requirement that it anticommutes with each ψna and commutes with the algebra Tna . To get integral forms the n-form part of t should be multiplied by (1/2πi)n/2 . In the odd case the above formula has to be modified: t := trν e
√ −(ν t Q • +δ+ωˆ c −
ν √
ψ,F)2 4 t
,
(3.5)
where ν is an odd element, ν 2 = 1, anticommuting with odd differential forms and commuting with Q A , and the trace trν extracts the operator trace of the √coefficients of the linear term in ν. In this case the n-form part should be multiplied by 2i(1/2πi)n/2 . Proposition 1. The exponential e−Dt is an operator valued form with values in traceclass operators for any t > 0. 2
Proof. The form in the exponent, when evaluated in a direction of a given set of tangent vectors on A, is equal to −th + u t + vt , where u t is an element in the Lie algebra of a is a Clifford multiplication Lg acting in the representation space H and vt = ψna v−n by a vector of finite norm. Now tr e−th+u t is given by the character of the loop group in the tensor product representation in H = H f ⊗ Hb . This is a tensor product of a pair
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of highest weight representations and thus the trace is the product of two (converging) characters of the loop group. 2 Finally e−Dt = e−th+u t +vt , for the form component of any given degree, can be expanded in the perturbation vt by the Duhamel formula (see the proof of Theorem, end of Sect. 4) as a finite sum of trace-class operator forms. The problem with the expressions (3.4) and (3.5) is that they cannot be pushed down to the base G = A/ G. The obstruction comes from the transformation property √ 1 t Q Ag + δ + r g ωc − √ ψ, r g F 4 t √ 1 (3.6) = gˆ −1 ( t Q A + δ + ωˆ c − √ ψ, F)gˆ + gˆ ∗ θ, 4 t where θ is the connection 1-form on LG corresponding to the curvature form c on LG, defined by the central extension. Here gˆ is a local LG valued function on the base G, implementing a change of local section G → A. This additional term is the difference δg − gˆ −1 δ g, gˆ ∗ θ = g −1 ˆ where the first term on the right comes from the transformation of the connection form ωˆ c with respect to a local gauge transformation g. Taking the square of the transformation rule (3.6) we get (r g Dt )2 = gˆ −1 Dt2 gˆ + g ∗ c.
(3.7)
The last term on the right arises as δ gˆ ∗ θ + (gˆ ∗ θ )2 = gˆ ∗ δθ = gˆ ∗ c = g ∗ c, where in the last step we have used the fact that the curvature of a circle bundle is a globally defined 2-form c on the base, and thus does not depend on the choice of the lift gˆ to LG. Proposition 2. Let Uα and Uβ be two open sets in G with local sections ψα , ψβ to the total space A of the G principal bundle A → G. Let gαβ : Uα ∩ Uβ → G be the local gauge transformation tranforming ψα to ψβ . Then the pull-back forms tα and tβ are related on Uα ∩ Uβ as ∗
tβ = ψβ∗ t = e−gαβ c tα . Proof. Since the curvature is closed, δc = 0, the term g ∗ c on the right in (3.7) commutes with the rest and therefore can be taken out as a factor exp(−g ∗ c) in the exponential of the square of the transformed superconnection. Remark. It is an immediate consequence of Proposition 2 that the 1-form part t [1] of t is a globally defined form on the base G. We can view this as the generalization of the differential of the families η invariant, governing the spectral flow along closed loops in the parameter space; in fact, in the classical case of Bismut-Freed superconnection for families of Dirac operators this is exactly what one gets from the Quillen superconnection formula. We can write π −1/2 t [1] = h −1 dh/2πi with log h = 2πiη. Note that η is only continuous modulo integers. Thinking of η as the spectral asymmetry, we normalize it by setting η(A) = 0 for the vector potential A = 0, or on the base, for the trivial holonomy g = 1.
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In the odd case we can relate the calculation of t [3] to the computation of the Deligne class in twisted K-theory, [Mi2]. On the overlap Uαβ we have from the proposition: ∗ c ∧ tα [1]. tβ [3] = tα [3] − gαβ
(3.8)
∗ 2 tα [3] − tβ [3] = d(gˆ αβ θ ∧ t [1]) ≡ dωαβ .
(3.9)
This gives
Using gˆ αβ gˆ βγ gˆ γ α = σαβγ we get −1 2 2 2 ωαβ − ωαγ + ωβγ = (σαβγ dσαβγ ) ∧ t [1].
(3.10)
Choose a function h : G → S 1 as in the Remark, π −1/2 t [1] = h −1 dh/2πi. 2 } in (3.10) can be written as ˇ Next the Cech coboundary of the cochain {ωαβ 1 = d(log(σαβγ )h −1 dh). dωαβγ
(3.11)
Defining aαβγ δ = log(σβγ δ ) − log(σαγ δ ) + log(σαβδ ) − log(σαβγ ) we can write (∂ω1 )αβγ δ = h −aαβγ δ dh aαβγ δ ,
(3.12)
2 , ω1 , ˇ where ∂ denotes the Cech coboundary operator. Thus the collection { tα [3], ωαβ αβγ a αβγ δ } defines a Deligne cocycle on the manifold G with respect to the given open h covering {Uα }. The system of closed local forms obtained from the Chern character formula (3.5) can be modified to a system of global forms by multiplication
˜ t := e−θα ∧ t ,
(3.13)
where θα is the 2-form potential, dθα = on Uα . One checks easily that now ˜ t = 0. (d + )
(3.14)
Although the operator d + squares to zero and can thus be used to define a cohomology theory, [BCMMS], the Chern character should not be viewed to give a map to this twisted cohomology theory. In fact, the twisted cohomology over complex numbers vanishes for simple compact Lie groups. For this reason, in order to hope to get nontrivial information from the Chern character, one should look for a refinement of the twisted cohomology. In fact, there is another integral version of twisted cohomology proposed in [At]. In that version one studies the ordinary integral cohomology modulo the ideal generated by the Dixmier-Douady class . At least in the case of SU (2) it is an experimental fact that the twisted (nonequivariant) K-theory as an abelian group is isomorphic to the twisted cohomology in this latter sense. There is a similar result for other compact Lie groups, but the 3-cohomology class used to define the twisting in cohomology is in general not the original twisting gerbe class; both are integral multiples of a basic 3-form, but the coefficients differ, except for the case of SU (2), [Br, Do]. One can explicitly see why the integral cohomology mod is relevant for twisted K-theory by the following construction in the odd case. First we replace the space Fr ed∗ by the homotopy equivalent space U1 consisting of 1 + trace-class unitaries on H. An ordinary K-theory class on X is a homotopy class of maps m : X → U1 . In this
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representation the Chern character defines a sequence of cohomology classes on X by pulling back the generators tr (g −1 dg)2n+1 of the cohomology of U1 . In the twisted case −1 m α gαβ on we have only maps m α on open sets Uα which are related by m β = gαβ overlaps. In the case of G = SU (2) = S 3 we need only two open sets U± , the slightly extended upper and lower hemispheres, and a map g−+ on the overlap U−+ = U− ∩ U+ to the group PU (H ), of degree k. If now m − ≡ 1 identically and the set of points x with m + (x) = 1 (the support of m + ) is concentrated around the North pole, then the pair m ± is related by the conjugation by g = g−+ at the equator and at the same time it defines an ordinary K-theory class since the functions patch to a globally defined function on S 3 . Let us assume that the winding number of m : S 3 → U1 is k. Next form a continuous path m ± (t) of representatives of twisted K-theory classes starting from m(1) = m and ending at the trivial class represented by the constant function m(0) = 1. Let ρ be a smooth function on S 3 which is equal to 1 on the overlap U−+ and zero in small open neighborhoods V± of the poles. We can also extend the domain of definition of g(x) to a larger set U+ \ V+ . For 0 ≤ t ≤ 1 define [m − (t)](x) := e2πitρ(x)P0 and [m + (t)](x) := e2πitρ(x)P(x) , where P(x) = g −1 (x)P0 g(x), with P0 a fixed rank one projection, x ∈ U−+ . These are smooth functions on U± respectively and are related by the conjugation by g on the overlap. But for t = 0 both are equal to the identity 1 ∈ U1 . On the other hand, at t = 1 the integral 1 tr (m −1 dm)3 24π 2 S 3 is easily computed to give the value k. This paradox is explained by the fact that for the intermediate values 0 < t < 1 the functions m ± do not patch up to a global function on S 3 . Thus we have a homotopy joining a pair of (trivial) twisted K-theory classes corresponding to the pair of third cohomology classes 0, computed from the Chern character. This confirms the claim, at least in the case of X = S 3 , that the values of the Chern character should be projected to the quotient H ∗ (X, Z)/Z. 4. Quillen Superconnection in the Case of SU(2) In this section, we consider the case G = SU (2) and calculate a pull-back of t = 2 trσ e−Dt , with respect to a local section over U+ = SU (2) \ {−1}, in the limit t → ∞. The basic idea is that in this case one can naturally associate to the twisted K-theory class a K-theory class, defined modulo the twisting gerbe. This method was used in [Mi2] to calculate a characteristic class for twisted K theory on 3-manifolds. Actually, on the level of 3-cohomology that method can be generalized to arbitrary simply connected compact Lie groups as follows. Let {Uα } be an open cover of G which trivializes the bundle A → G. Let ψα : Uα → A be local sections. We may take {Uα } as the open set of holonomies such that the real number α is not in the spectrum of Q A (for any A in the fibers over Uα ). Let L αβ (A) be the top exterior power of the finite dimensional spectral subspace E αβ : α < Q A < β and denote L αβ = ψα∗ L αβ ⊗ (h ∗αβ c)−n β ,
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J. Mickelsson , J.-P. Pellonpää
where c is the level k˜ central extension cocycle of the loop group and h αβ is the transition function, ψβ = ψα h αβ . The integers n β will be determined below. We require that L αβ ⊗ L βγ ⊗ L γ α = C, canonically isomorphic to the trivial line bundle. First note that ψβ∗ L βγ = ψα∗ L βγ ⊗ (h ∗αβ c)n βγ by the projective action of the loop group on the spectral subspaces, where n βγ = dim E βγ for β < γ and n γβ = −n βγ . Therefore
L αβ ⊗ L βγ ⊗ L γ α = ψα∗ L αβ ⊗ ψα∗ L βγ ⊗ (h ∗αβ c)n βγ
⊗ ψα∗ L γ α ⊗ (h ∗αγ c)n γ α ⊗ (h ∗αβ c)−n β ⊗ (h ∗βγ c)−n γ ⊗ (h ∗γ α c)−n α = ψα∗ (L αβ ⊗ L βγ ⊗ L γ α ) ⊗ (h ∗αβ c)n βγ −n β ⊗ (h ∗βγ c)−n γ ⊗ (h ∗γ α c)−n γ α −n α . The line bundles L αβ form a cocycle as follows from E αγ = E αβ ⊕ E βγ and therefore the nontrivial part is a product of pull-backs of c. Using the group property of the central extension of the loop group we have h ∗αγ c = h ∗αβ c ⊗ h ∗βγ c, and collecting the factors the tensor product above becomes (h ∗αβ c)n βα −n β +n α ⊗ (h ∗βγ c)n γ α −n γ +n α , where we have taken into account n βγ + n γ α = n βα in the first factor, and this becomes trivial provided we can choose the locally constant functions n α such that n αβ = n α −n β on intersections. But since we assumed that the base G is simply connected, H 1 (G) = 0 and the solution exists. The solution is not uniquely defined: it is defined modulo adding to each n α a constant. This corresponds to modifying the gerbe defined by the line bundles L αβ by a power of the gerbe defined by the local line bundles h ∗αβ c. But these latter line bundles correspond to the Dixmier-Douady class . Thus we get a map from twisted K theory K 1 (G, ) to H 3 (G, Z)/Z. This map needs be neither injective nor surjective. However, in the case of G = SU (2) these groups are known to be equal. We now explain why the Quillen superconnection formalism has to give the same result. The calculation of the characteristic class in [Mi2] was based on the following observation. As a classifying space for self-adjoint Fredholm operators, with both positive and negative essential spectrum, one can also use the Lie group U1 consisting of unitary operators which differ from the unit by a trace class operator. The 3-cohomology of U1 is generated by tr (g −1 dg)3 . In general, the K 1 class of a map φ : X → U1 is mapped to cohomology as the odd Chern character given by the de Rham forms φ ∗ tr (g −1 dg)2i+1 . In the twisted case we do not have a map from X to U1 but a family of local maps which are related on intersections of open sets by conjugation by PU (H ) valued transition functions. But in the case X = SU (2) one can deform the twisted map such that it becomes a globally well-defined map SU (2) → U1 and one gets a characteristic class φ ∗ tr (g −1 dg)3 in twisted K theory. However, as explained in the end of the previous section, this class is defined only modulo the Dixmier-Douady class of the twisting bundle. Going back to the G = SU (2) case, in [Mi2] the map from twisted K-theory to untwisted K theory was explicitly performed by using the fact that the G bundle over G is trivialized over a pair of open sets, S+3 = S 3 \ {N } and D 3N , where S, N are the ‘South’ and ‘North’ poles on the 3-sphere and D 3N is a small disk around the North
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pole. The transition functions are defined on a thickened 2-sphere close to N . After a deformation, the map g+ : S+3 → U1 becomes constant equal to the unit element in U1 and thus defines a global K 1 class on S 3 by pull-back from U1 . Applying the Quillen superconnection to this class (using instead of U1 the space Fr ed∗ as a classifying space) must give the same cohomology class. However, we can do better than that: Using the global SU (2) gauge invariance of the construction of Fredholm operators, we know that the differential forms obtained from the Quillen superconnection must be invariant under conjugation by SU (2). The cohomology class does not depend on the scaling parameter t. On the other hand, in the limit t → ∞ the forms are supported only on the subset of parameters A such that the operator Q A has zero eigenvalue. The zeros of Q A are easily computed, [FHT2]. The elements of a simple compact Lie group G can be parametrized as g = exp(2πa), where a is in the Lie algebra of G. The constant vector potential A ≡ a on the circle S 1 has the holonomy g. Since the construction of the operators Q A is equivariant with respect to (constant) gauge transformations we may assume that a is in the Cartan subalgebra h of G. Actually, performing an additional gauge transformation one can require that ∨
(a + d, αi ) ≤ 0, where αi ’s are the simple roots of the affine Lie algebra based on G, ∨ means the duality transformation h → h∗ defined by the Killing form, and (·, ·) is the inner product in d h∗ coming from the Killing form. Here d = −i dθ is the derivation of the loop algebra. Then the only zeros of Q A are in the weight subspace λ+ρ, where λ is the highest weight of the G representation on the bosonic vacuum sector, ρ is half the sum of the positive roots (which is also a highest weight on the fermionic vacuum sector). The value of a corresponding to the nonzero kernel of Q A is given by ∨ k˜ a = −λ − ρ. ∨
˜ where 2 j = 0, 1, 2, . . . and in our In particular, for G = SU (2) we obtain a = −2 j/k, normalization k˜ = (k + 2)/4. Let ϕ ∈ [0, 2π [ and n ∈ S 2 . Define a local section ψ : SU (2) \ {N } → A by ψ(ϕ, n) = − 2i ϕn · σ , where σ = (σ1 , σ2 , σ3 ) and σa ’s are the Pauli matrices. At the boundary ϕ = 2π all the constant vector potentials ψ(2π, n) have holonomy −1 around the circle. At ϕ = 2π the operators Q ψ(ϕ,n) are invertible and for this reason at the limit t → ∞ the exponential of the Quillen superconnection vanishes: Thus in this limit the forms obtained from (3.5) become globally well-defined on SU (2) despite the fact that ψ : SU (2) → A is discontinuous at ϕ = 2π. To conclude, since the 3-form part of t invariant under conjugation by SU (2) and, in the limit t → ∞, ψ ∗ t [3] becomes concentrated on the 2-sphere, the form must be proportional to the area 2-form of the sphere. The proportionality constant is uniquely determined by the cohomology class. This justifies the results of the theorem. The forms t for different values of the parameter t represent the same cohomology class. The relevant transgression formula is exactly the same as in the case of families of Dirac operators:
2 d t Proposition 3. dt = −dA tr ddtDt e−Dt .
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Proof. The proof is exactly as in [BGV], Theorem 9.17. The only thing to note is that 2 e−Dt is a trace-class operator form, Prop. 1. Let K be a compact (Hausdorff) space, and let C(K ) be a Banach space of continuous functions u : K → C equipped with the sup norm. By the Riesz representation theorem, the topological dual M(K ) of C(K ) can be considered as a set of regular complex Borel measures on K . Equip M(K ) with the weak-star topology. In this topology, we say that a net {µ i }i∈I ⊆ M(K
), where I is a directed set, converges to a point µ ∈ M(K ) if limi∈I udµi = udµ for any u ∈ C(K ); we denote µ = w∗ -limi∈I µi . 2 2 ) denote the measure defined by the area 2-form area(S 2 ) of S 2 . Let area(S ) ∈ M(S Let δa ∈ M [0, 2π ] be the Dirac measure concentrated on a ∈ ]0, 2π [, β := k˜ 2 /(2π 2 ), and ϕ kj := 2π Theorem.
2j + 1 2j + 1 = 2π ∈ ]0, 2π [. k+2 4k˜
(1) For any t > 0, ϕ ∈ [0, 2π [, n ∈ S 2 , βt −βt (ϕ−ϕ kj ) i k˜ ∗ t e ϕ + 4(ϕ − ϕ kj ) sin2 (ϕ/2) ψ [3]ϕ,n = − √ · π π 1 × α(y)2 − α(y) dy × 0
× dϕ ∧ area(S 2 )|n + O(t −1/2 ). (2) Let V ⊂ [0, 2π [ be a neighborhood of ϕ kj , and let X , Y , Z be (continuous) vector fields of SU (2). Then lim sup ψ ∗ t [3]ϕ,n X ϕ,n , Yϕ,n , Z ϕ,n = 0. t→∞
ϕ ∈V / n∈S 2
(3) When ψ ∗ t [3] is interpreted as a measure on [0, 2π ] × S 2 , √ 1 ∗ t ∗ δ k ⊗ area(S 2 ). w -lim ψ [3] = − πi j + t→∞ 2 ϕj √ To get an integral form, we multiply t [3] by ( πi)−1 (2π )−1 . Recall that
the function α in item (1) is smooth and α(0) = 0 and α(1) = 1 so that the integral (α 2 − α) can be chosen arbitrary small by changing α. For completeness, next we represent the sketch of the direct but lengthy proof of the theorem taken from our previous (unpublished) version of the article, [MP]. Proof. (1) Since
1 tν − δ Q • + [Q • , ωˆ c ] + δ ωˆ c + ωˆ c2 − {Q • , ψ, F} 4 =:B1 (1-form) =:B2 (2-form)
ν − √ − δ ψ, F + [ ψ, F, ωˆ c ] = t Q 2• − E t , 4 t =:B3 (3-form)
Dt2 = t Q 2• +
√
Families Index Theorem in Supersymmetric WZW Model and Twisted K-Theory
where E t := −
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1 √ ν tν B1 + B2 − √ B3 , t 4 t
using the perturbation series expansion ν −t (Q 2• −E t )
tr e
ν −t Q 2•
= tr e
∞ + tn n=1
n
2 2 2 trν e−ts1 Q • E t e−ts2 Q • E t · · · e−tsn+1 Q • ds1 · · · dsn ,
where n is the standard n-simplex, one gets
2 2 2 trν e−ts1 Q • E t e−ts2 Q • ds1 t = trν e−t Q • + t 1
2 2 2 2 ν +t tr e−ts1 Q • E t e−ts2 Q • E t e−ts3 Q • ds1 ds2 2
2 2 2 2 3 +t trν e−ts1 Q • E t e−ts2 Q • E t e−ts3 Q • E t e−ts4 Q • ds1 ds2 ds3 . 3
The three form part of the above form is
√ 2 2 2 2 t [3] = t t tr e−ts1 Q • B1 e−ts2 Q • B1 e−ts3 Q • B1 e−ts4 Q • ds1 ds2 ds3 3
√ 2 2 2 2 2 2 + t tr e−ts1 Q • B1 e−ts2 Q • B2 e−ts3 Q • +e−ts1 Q • B2 e−ts2 Q • B1 e−ts3 Q • ds1 ds2 2
1 2 2 + √ tr e−ts1 Q • B3 e−ts2 Q • ds1 . 4 t 1 After long simplification (see, [MP]) one gets i k˜ 2 √ −t[2k˜ 2 ϕ˜ 2 −(2 j+1)k˜ ϕ+ ˜ j ( j+1)/2+1/8] × √ te 2 π 2 × ϕ + 4(ϕ − ϕ kj ) sin2 (ϕ/2) 1 × α(y)2 − α(y) dy dϕ ∧ area(S 2 )|n + O(t −1 ) 0 βt −βt (ϕ−ϕ kj ) i k˜ e ϕ + 4(ϕ − ϕ kj ) sin2 (ϕ/2) = −√ · π π 1 × α(y)2 − α(y) dy
ψ ∗ t [3]ϕ,n = −
0
× dϕ ∧ area(S 2 )|n + O(t −1/2 ), where ϕ˜ = ϕ/(2π ), β = k˜ 2 /(2π 2 ), and area(S 2 )|n := n 1 dn 2 ∧ dn 3 + n 2 dn 3 ∧ dn 1 + n 3 dn 1 ∧ dn 2 = sin θ dθ ∧ dφ is the area 2-form of S 2 at n (where (θ, φ) are the spherical coordinates).
(4.1)
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(2) To prove (2), it suffices to consider the case of coordinate vector fields X = ∂/∂ϕ, √ −r (ϕ−ϕ kj )2 Y = ∂/∂θ, and Z = ∂/∂φ. Since limr →∞ supϕ ∈V re = 0 for any neighbor/ hood V ⊂ [0, 2π [ of ϕ kj it follows from (4.1) that ∂ ∂ ∂ lim sup ψ ∗ t [3]ϕ,n , , = 0. t→∞ ϕ ∈V ∂ϕ ϕ ∂θ n ∂φ n / n∈S 2
(3) Using the formal notation δ(ϕ − a) for the Dirac measure δa concentrated on a ∈ ]0, 2π [ one gets r −r (ϕ−a)2 δ(ϕ − a) = w∗ -lim e r →∞ π and for any p ≥ 1, w∗ -lim r →∞
1 √ −r (ϕ−a)2 re = 0. rp
Hence, from (4.1) one sees that, when ψ ∗ t [3] is interpreted as a measure on [0, 2π ] × S 2 , √ 1 ∗ t ∗ δ k ⊗ area(S 2 ). w -lim ψ [3] = − πi j + t→∞ 2 ϕj Acknowledgement. We want to thank Varghese Mathai for discussions and critical remarks leading to improvements in the paper and the referees for careful reading and suggestions.
References [At]
Atiyah, M.: K theory past and present. Sitzungsberichte der Berliner Mathematischen Gesellschaft, Berlin: Berliner Math. Gesellschaft 2001, pp 411–417 [AS] Atiyah, M., Segal, G.: Twisted K-theory. Ukr. Mat. Visn. 1(3), 287–330 (2004). http://arxiv. org/list/math.KT/0407054 [Bi] Bismut, J.-M.: Localization formulas, superconnections, and the index theorem for families. Commun. Math. Phys. 103(1), 127–166 (1986) [BCMMS] Bouwknegt, P., Carey, A., Mathai, V., Murray, M.K., Stevenson, D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228, 17–49 (2002) [BGV] Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Berlin and Heidelberg: Springer-Verlag, (1992) [Br] Braun, V.: Twisted K theory of Lie groups. JHEP0403, 029 (2004) [CM] Carey, A.L., Mickelsson, J.: The universal gerbe, Dixmier-Douady class, and gauge theory. Lett. Math. Phys. 59, 47–60 (2002) [CMM] Carey, A.L., Mickelsson, J., Murray, M.K.: Index theory, gerbes, and hamiltonian quantization. Commun. Math. Phys. 183, 707–722 (1997) [DK] Donovan, P., Karoubi, M.: Graded brauer groups and K-theory with local coefficients. Inst. Hautes Études Sci. Publ. Math. No. 38, 5–25 (1970) [Do] Douglas, C.L.: On the twisted K-homology of simple lie groups. Topology 45(6), 955–988 (2006) [FdV] Freudenthal, H., de Vries, H.: Linear Lie Groups. Pure Appl. Math. 35, New York: Academic Press, 1969 [Fr] Freed, D.: Twisted K-theory and loop groups. In: the Proceedings of ICM2002, Beijing, Beijing: The Higher Education Press of China, 2002 [FHT] Freed, D., Hopkins, M., Teleman, C.: Twisted equivariant K-theory with complex coefficients. http://arxiv.org/list/math.AT/0206257, 2002; Twisted K-theory and loop group representations. http://arxiv.org/list/math.AT/0312155, 2003
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[GR] [Mi1] [Mi2] [MP] [PS] [Qu] [Ro]
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Gawedzki, K., Reis, N.: WZW branes and gerbes. Rev. Math. Phys. 14, 1281–1334 (2002) Mickelsson, J.: Gerbes, (twisted) K theory, and the supersymmetric WZW model. In: Infinite Dimensional Groups and Manifolds, ed. by T. Wurzbacher. IRMA Lectures in Mathematics and Theoretical Physics 5, Berlin: Walter de Gruyter, 2004 Mickelsson, J.: Twisted K theory invariants. Lett. Math. Phys. 71, 109–121 (2005) Mickelsson, J., Pellonpää, J.-P.: Families index theorem in supersymmetric WZW model and twisted K-theory: The SU (2) case. http://arxiv.org/list/hep-th/0509064, version 1, 2005 Pressley, A., Segal, G.: Loop Groups. Oxford: Clarendon Press, 1986 Quillen, D.: Superconnections and the Chern character. Topology 24(1), 89–95 (1985) Rosenberg, J.: Continuous-trace algebras from the bundle theoretic point of view. J. Austral. Math. Soc. Ser. A 47(3), 368–381 (1989)
Communicated by M.R. Douglas
Commun. Math. Phys. 271, 791–820 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0195-5
Communications in
Mathematical Physics
A Strong Szeg˝o Theorem for Jacobi Matrices E. Ryckman Department of Mathematics, University of California, Los Angeles, CA 90095, USA. E-mail: [email protected] Received: 30 June 2006 / Accepted: 18 September 2006 Published online: 6 February 2007 – © Springer-Verlag 2007
Abstract: We use a classical result of Golinskii and Ibragimov to prove an analog of the strong Szeg˝o theorem for Jacobi matrices on l 2 (N). In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary ∞ 2 and sufficient conditions on the spectral measure such that ∞ k=n bk and k=n (ak − 1) 2 2 lie in l1 , the linearly-weighted l space. 1. Introduction Let us begin with some notation. We study the spectral theory of Jacobi matrices, that is semi-infinite tridiagonal matrices ⎞ ⎛ b1 a1 0 0 ⎟ ⎜ ⎜a1 b2 a2 0⎟ ⎟ ⎜ J =⎜ ⎟, ⎜ 0 a2 b3 . . . ⎟ ⎠ ⎝ .. .. . . 0 0 where an > 0 and bn ∈ R. In this paper we make the overarching assumption that the sequences bn and an2 − 1 are conditionally summable. We may then define λn := − κn := −
∞ k=n+1 ∞ k=n+1
for n = 0, 1, . . . .
bk , (1.1) (ak2 − 1)
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Let dν be the spectral measure for the pair (J, δ1 ), where δ1 = (1, 0, 0, . . . )t , and assume that dν is not supported on a finite set of points (we will call such measures nontrivial). Let dν(x) (1.2) m(z) := δ1 , (J − z)−1 δ1 = x−z be the associated m-function, defined for z ∈ C\supp(ν). We will write {βn } ∈ ls2 if βl22 := s
|n|s |βn |2 < ∞,
n
and let H˙ 1/2 (∂D) denote the (homogeneous) Sobolev space of order 1/2 of functions defined on ∂D: f ∈ H˙ 1/2 if f 2H˙ 1/2 := fˆ(n)l22 = 1
|n|| fˆ(n)|2 < ∞.
n
If f is a function on [−2, 2], we say f ∈ H˙ 1/2 if f (2 cos θ ) ∈ H˙ 1/2 (∂D). Also, we will say v ∈ W if v(x) is supported in [−2, 2] and has one of the forms
4−
x2
±1
v0 (x) or
2−x 2+x
±1 v0 (x)
(1.3)
with log(v0 ) ∈ H˙ 1/2 . Our main result is: Theorem 1.1. Let J be a Jacobi matrix. The following are equivalent: (1) The sequences associated to J by (1.1) obey λ, κ ∈ l12 . (2) J has finitely-many eigenvalues that all lie in R\[−2, 2], and on [−2, 2] the spectral measure is purely absolutely continuous, dν(x) = v(x)d x, with v ∈ W. The main ingredient in the proof will be the following version of the strong Szeg˝o theorem1 . Theorem 1.2. (Golinskii-Ibragimov). Let dµ be a probability measure on ∂D that is not supported on a finite set of points, and let {αn } ⊆ D be the associated Verblunsky coefficients. The following are equivalent: (1) α ∈ l12 . dθ and log w ∈ H˙ 1/2 . (2) dµ = w 2π We now outline the proof of Theorem 1.1. To apply the strong Szeg˝o theorem we must move to the circle, so we must first remove all the eigenvalues in R \ [−2, 2]. To do so we use double commutation (see [5]): 1 The version we use is due to [7] and [11]. For relevant definitions see, for instance, [16].
A Strong Szeg˝o Theorem for Jacobi Matrices
793
Theorem 1.3. (Double Commutation). Let E ∈ R \ σ (J ), and let γ > 0. Define a new Jacobi matrix J by √ cn−1 cn+1 a˜ n = an , cn
a φ φ an φn φn+1 n−1 n−1 n b˜n = bn + γ , − cn−1 cn where J φ = Eφ, φ0 = 0, φ1 = 1 and n
cn = 1 + γ
|φ j |2 .
j=1
Then σ ( J) = σ (J ) ∪ {E}, E is a simple eigenvalue of J, and m(z) ˜ =
1
γ m(z) − . 1+γ z−E
Conversely, let |E| > 2 be a simple eigenvalue of J with eigenvector φ. Choose γ = −1/φ2 and define a new Jacobi matrix J as above. Then σ ( J) = σ (J ) \ {E} and m (z) = (1 + γ )m(z) +
γ . z−E
We prove an asymptotic integration result in Sect. 2, which we combine with the above theorem in Sect. 3 to prove Proposition 1.4. Let J be a Jacobi matrix, and let E be an isolated eigenvalue of J in R \ [−2, 2]. Let J be the Jacobi matrix obtained from J by removing the eigenvalue E using Theorem 1.3. Then (1) λ˜ , κ˜ ∈ l12 if and only if λ, κ ∈ l12 , (2) v˜ ∈ W if and only if v ∈ W. This proposition essentially allows us to consider Theorem 1.1 under the additional hypothesis σ (J ) ⊆ [−2, 2]. This allows us to move to the circle, as follows. Given a nontrivial probability measure dµ on ∂D that is invariant under complex conjugation, one can define a nontrivial probability measure dν on [−2, 2] by
2
−2
2π
g(x)dν(x) =
g(2 cos θ )dµ(θ ).
0
Similarly, given such a measure dν, one can define a measure dµ that is symmetric under complex conjugation by
2π 0
when h(−θ ) = h(θ ).
h(θ )dµ(θ ) =
2
−2
h(arccos(x/2))dν(x)
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E. Ryckman
The map dµ → dν is one of a family of four maps that we call the Szeg˝o mappings2 . We denote it by dν = Sz (e) (dµ). The other three maps are given by Sz (o) (dµ) = c2 (4 − x 2 )Sz (e) (dµ), 2 (2 ∓ x)Sz (e) (dµ), Sz (±) (dµ) = c±
c=
(1.4)
1
, 2(1 − |α0 |2 )(1 − α1 ) 1 . c± = √ 2(1 ∓ α0 )
(1.5)
If dµ is absolutely continuous with respect to Lebesgue measure we will write dµ(θ ) = dθ w(θ ) 2π and then Sz (∗) (dµ)(x) = v (∗) (x)d x. In this case the above relations become 1 w(arccos(x/2)), √ π 4 − x2 c v (o) (x) = 4 − x 2 w(arccos(x/2)), π 2∓x (±) w(arccos(x/2)). v (x) = c± 2±x v (e) (x) =
(1.6)
For ∗ ∈ {e, o, +, −}, we will write J (∗) for the Jacobi matrix determined by dν (∗) and a (∗) , b(∗) for its parameter sequences. The relationship between α and a (∗) , b(∗) is given by Proposition 1.5. (Direct Geronimus Relations3 ). Let dµ be a nontrivial probability measure on ∂D that is invariant under conjugation, and let dν (∗) = Sz (∗) (dµ). Then for all n ≥ 0, (e)
2 [an+1 ]2 = (1 − α2n−1 )(1 − α2n )(1 + α2n+1 ), (e)
bn+1 = α2n (1 − α2n−1 ) − α2n−2 (1 + α2n−1 ), (o)
2 )(1 − α2n+3 ), [an+1 ]2 = (1 + α2n+1 )(1 − α2n+2 (o)
bn+1 = −α2n+2 (1 + α2n+1 ) + α2n (1 − α2n+1 ), (±) 2 2 ] = (1 ± α2n )(1 − α2n+1 )(1 ∓ α2n+2 ), [an+1 (±)
bn+1 = ∓α2n+1 (1 ± α2n ) ± α2n−1 (1 ∓ α2n ). Since an > 0, there is no ambiguity in which sign to choose for the square root above. We always take α−1 = −1. The value of α−2 is irrelevant since it is multiplied by zero. From the Direct Geronimus Relations we see that decay of the α’s determines decay of the a’s and b’s. This allows us to prove one direction of Theorem 1.1 in Sect. 4. To prove the other direction, we will find certain relationships between the Verblunsky parameters and solutions of J u = Eu at E = ±2. We study asymptotics of these solutions in Sect. 5, then find the desired relationships in Sect. 6, which we term the Inverse Geronimus Relations. In Sect. 7 we review some Weyl theory, and in Sect. 8 we combine all these ideas to finish the proof. 2 The map Sz (e) is due to [17], while the other three are due to [2], then developed further in [13] and [16]. 3 The relationship between α and a (e), b(e) was first discovered by [4]. The other three were later found by
[2] using techniques similar to [4]. References [13] and [16] have a different proof using operator techniques.
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2. Asymptotic Integration Suppose J and J are related through double commutation (as in Theorem 1.3). In the next section we will relate λ˜ , κ˜ to λ, κ. By Theorem 1.3 we see4 ∞
2 c(k − 1)c(k + 1) |κ(n ˜ − 1) − κ(n − 1)| = a(k) − 1 , 2 c(k) k=n
∞ a(k − 1)φ(k − 1)φ(k) a(k)ψ(k)ψ(k + 1) . ˜ |λ(n − 1) − λ(n − 1)| = |γ | − c(k − 1) c(k) k=n
So to prove part (1) of Proposition 1.4, we must determine asymptotics for φ when E ∈ R \ [−2, 2]. To do so we use the theory of asymptotic integration as developed p in [8–10, 14] and particularly [1]. However, as we need ls control of the errors (rather than the usual o(1) control) we must modify their results. Throughout, we will use the notation x y if there is a constant c > 0 such that x ≤ cy. Also, if xn is a sequence, p p we write x = y + ls to indicate xn = yn + εn for some other sequence ε ∈ ls . Proposition 2.1. Let (k) = diag[λ1 (k), . . . , λn (k)] and suppose that there exists 0 < δ < 1 so that for a fixed i either λi (k) λi (k) ≤1−δ ≥ 1 + δ or (I I ) (I ) λ j (k) λ j (k)
(2.1)
for each j = i, where k ≥ k0 for some k0 . Suppose also that V (k) ∈ ls2 for some s ≥ 0. Then the system (k + 1) = [(k) + V (k)](k) (2.2) has a solution of the form i (k) =
k−1
λi (l) + Vii (l) (ei + ls2 ),
(2.3)
l=k0
where ei is the i th standard unit vector in Rn . As all norms on a finite dimensional space are equivalent, it does not matter which we p p mean when we write things like V (k) ∈ ls or (ei + ls ). We will prove Proposition 2.1 by using a Harris-Lutz transformation followed by a Levinson-type result. We will state and use these results, then prove them at the end of this section. Proposition 2.2. With the assumptions of Proposition 2.1, there exists a sequence of matrices Q(k) such that Q(k)ii = 0, Q(k) ∈ ls2 , and V (k) − diagV (k) + (k)Q(k) − Q(k + 1)(k) = 0. 4 In order to avoid excessive subscripting later in this section, we will write a(n) for a , etc. n
(2.4)
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Proposition 2.3. Say (k) satisfies the assumptions of Proposition 2.1, and suppose that R(k) ∈ ls1 for some s ≥ 0. Then the system x(k + 1) = [(k) + R(k)]x(k)
(2.5)
has a solution of the form xi (k) =
k−1
λi (l) (ei + ls2 ).
(2.6)
l=k0
Assuming Propositions 2.2 and 2.3 we have Proof of Proposition 2.1. Let Q(k) be as guaranteed by Proposition 2.2, and define x(k) by (k) = [I + Q(k)]x(k) (as Q(k) → 0, [I + Q(k)] is invertible for large k, so the above definition makes sense). Then is a solution of (2.2) if and only if x solves (k)]x(k), (k) + V x(k + 1) = [ where (k) = (k) + diagV (k), (k) = [I + Q(k)]−1 [V (k)Q(k) − Q(k + 1)diagV (k)]. V still satisfies the dichotomy condition (2.1). Moreover, as It is easy to see that (k) ∈ ls1 . So we may apply Proposition 2.3 V (k), Q(k) ∈ ls2 we have that V to the x-system to find a solution xi (k) =
k−1
λ˜ i (l) (ei + ε(k))
l=k0
for some ε(k) ∈ ls2 . But then i (k) = [I + Q(k)]xi (k) =
k−1
λi (l) + V (l)ii
ei + ε(k) + Q(k)ei + Q(k)ε(k) .
l=k0
By Proposition 2.2 we have that ε(k) + Q(k)ei + Q(k)ε(k) ∈ ls2 , as required. Next we prove Propositions 2.2 and 2.3. In doing so we will make frequent use of the following two lemmas. 2 Lemma 2.4. Let s ≥ 1, β, γ ∈ ls2 , and define a sequence ηn := ∞ k=n βk γk . Then η ∈ ls ∞ 1 2 and ηls2 ≤ βls2 γ ls2 . In particular, if τ ∈ ls then k=n τk ∈ ls .
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Proof. Throughout the proof, all norms refer to ls2 . By Cauchy-Schwarz we have η2 = ≤
∞
∞ ∞ ∞ 2
2 ns βk γk ≤ ns |βk γk |
n=1
k=n
∞
∞
ns
n=1
n=1
|βk |2
∞
k=n
≤ β2
∞ ∞
k=n
|γk |2 =
∞ ∞
n s |βk |2
∞
n=1 k=n
k=n
|γk |2 = β2
n=1 k=n
∞
|γk |2
k=n
k|γk |2 ≤ β2 γ 2 .
k=1
The last statement follows by applying the above argument to β = γ = |τ |1/2 . Lemma 2.5. Suppose γ (k) ∈ l12 and β > 1. Then
k−1 l=1
β 2(l−k) γ (l) ∈ l12 .
Proof. We must show that γ →
k−1
β 2(l−k) γ (l)
l=1
maps l12 → l12 . Equivalently we will show γ →
k−1
l=1
k 2(l−k) β γ (l) l
maps l 2 → l 2 . This is an integral operator with kernal h(l, k) = χ{1,...,k−1} (l)
k 2(l−k) β l
so by Schur’s Test this will be a bounded operator if we can show sup k
∞
h(l, k) ≤ C and sup l
l=1
∞
h(l, k) ≤ C
k=1
for some C ≥ 0. This is done by the following lemma.
Lemma 2.6. For any α ∈ R and ε > 0 we have sup l
∞
|k| + 1 α k=1
|l| + 1
e−ε|k−l| < ∞.
The proof is standard and proceeds by splitting the sum at k = l and bounding each piece separately. We omit the details.
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Proof of Proposition 2.2. Define Q(k) by Q(k)ii = 0 and Q(k)i j = −
∞ m V (m)i j λ j (l) if (i, j) ∈ (I ), λ j (m) λi (l)
m=k
Q(k)i j =
k−1 m=k0
l=k
V (m)i j λi (m)
m l=k
λi (l) if (i, j) ∈ (I I ). λ j (l)
As V (k) ∈ ls ⊆ l ∞ , Q(k)i j is dominated (in either case above) by a convergent geometric series, so the sums defining Q converge. By the above definition, (2.4) holds. p To show that Q(k) ∈ ls we argue as follows. For (i, j) ∈ (I ) we have that m 1 λi (l) ∈ l 2. λ (m) λ j (l) 1 j p
l=k
Similarly, for (i, j) ∈ (I I ) we have that λi (l) 1 k−1 |β|m−k λ (m) λ j (l) i l=m
p
for some |β| > 1. So by Lemmas 2.4 and 2.5 we see that Q ∈ ls .
Proof of Proposition 2.3. Define w(k) by x(k) =
k−1
λi (l) w(k).
l=k0
Then x solves (2.5) if and only if w solves the system w(k + 1) =
1 [(k) + R(k)]w(k). λi (k)
We’ll compare the w-system to the diagonal system y(k + 1) =
1 (k)y(k). λi (k)
The y-system has a fundamental matrix Y (k) = diag
k−1 l=k0
k−1 λn (l) λ1 (l) , . . . , 1, . . . , λi (l) λi (l) l=k0
with a 1 in the i th spot. Let P1 = diag[ p1 , . . . , pn ], where 1, (i, j) ∈ (I ) pj = 0, (i, j) ∈ (I I ) and let P2 = I − P1 . By the assumptions on (k) we see that for k0 ≤ l ≤ k − 1, Y (k)P1 Y (l + 1)−1 ≤ C and for k0 ≤ k ≤ l, Y (k)P2 Y (l + 1)−1 ≤ C for some C > 0.
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799
Now let k1 ≥ k0 to be chosen later, and consider the operator [T z](k) =
k−1
Y (k)P1 Y (l + 1)
−1
l=k1
∞
1 1 R(l)z(l) − R(l)z(l) Y (k)P2 Y (l + 1)−1 λi (l) λi (l) l=k
acting on l ∞ (N, Cn ). Choose k1 so that 2Cδ −1
∞
R(l) < ε < 1
l=k1
(which is possible because R(k) ∈ ls1 ). Then we see that
T zl ∞ ≤ 2Cδ
−1
R(l) zl ∞ ≤ εzl ∞
∞ l=k1
for all z ∈ l ∞ . Thus, T : l ∞ → l ∞ is a contraction. In particular, given y ∈ l ∞ , there exists a unique w ∈ l ∞ solving w = y + T w. Say y ∈ l ∞ and w = y + T w. By the definition of T , y is a solution of the y-system if and only if w is a solution to the w-system. In particular this holds for y = ei . It remains to show w = y + l12 , for which we consider each of the sums defining T w separately. As R(k) ∈ ls1 ⊆ ls2 and Y (k)P1 Y (l + 1)−1 λi1(l) w(l) 1, Lemma 2.5 shows the first sum is in l12 . Similarly, because R(k) ∈ l11 , Lemma 2.4 shows that the second sum is in l12 . Finally, we allow perturbed diagonalizable systems, rather than just the perturbed diagonal systems of Proposition 2.1. Proposition 2.7. Suppose A(k) has eigenvalues λi (k) satisfying (2.1) and supk |λ j (k)| ≤ C for all j. Let A(k) = S(k)−1 (k)S(k), where (k) = diag[λ1 (k), . . . , λn (k)], and suppose that S(k) → S(∞), where S(∞) is invertible and S(k + 1) − S(k) ∈ ls2 for some s ≥ 0. Finally, suppose V ∈ ls2 . Then the system (k + 1) = [A(k) + V (k)](k) has a solution of the form i (k) = S(k)
−1
k−1
λi (l) + V (l)ii (ei + ls2 ),
l=k0
where (k) = S(k)V (k)S(k)−1 + S(k + 1) − S(k) A(k) + V (k) S(k)−1 V (k) ∈ ls2 . so in particular V
(2.7)
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Proof. We’ll reduce to the case of Proposition 2.1. Define z(k) = S(k)(k), so is a solution of (2.7) if and only if z solves the system (k)]z(k), z(k + 1) = [(k) + V
(2.8)
is as in the statement of the proposition. Now where V (k) V (k) + S(k + 1) − S(k) A(k) + V (k) V V (k) + S(k + 1) − S(k) ∈ ls2 because S(∞) is invertible and sup j,k |λ j (k)| ≤ C. So by Proposition 2.1, there exists a solution to (2.8) of the form z i (k) =
k−1
(l)ii (ei + ls2 ). λi (l) + V
l=k0
Undoing the transformation we find a solution to (2.7) of the form i (k) = S(k)−1 z i (k) as desired.
3. The Double Commutation Result In this section we prove Proposition 1.4. Proof of Proposition 1.4(2). Suppose that J and J are related by double commutation at E ∈ R an isolated point of σ (J ). Write dν(x) = v(x)d x and recall that Lebesgue almost everywhere v(x) =
1 Im m(x + i0). π
By Theorem 1.3 m(z) ˜ =
γ 1
m(z) − . 1+γ z−E
But then v(x) ˜ =
1 1 1 Im m(x ˜ + i0) = Im m(x + i0) = v(x) π π(1 + γ ) 1+γ
almost everywhere. Clearly v˜ ∈ W if and only if v ∈ W.
Part (1) is more difficult, and will take the rest of this section to prove. We will use the asymptotic integration results obtained in Sect. 2. Lemma 3.1. Write E = β + β −1 with |β| > 1. The recurrence equation at E has solutions of the form ψ± (k) = c± β ±k (1 + l12 ) for some constants c± ∈ R\{0}.
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801
Proof. We will prove the result for E > 2, the proof for E < −2 being similar. We can write the recurrence equation a(k + 1)ψ(k + 1) + b(k) − E ψ(k) + a(k)ψ(k − 1) = 0 as the system where (k) =
(k + 1) = [A(k) + V (k)](k),
ψ(k) ψ(k − 1)
A(k) =
Let λ± (k) =
−1
E a(k+1)
1
−1 0
E 2 − 4a(k + 1)2 2a(k + 1) 1 1 S(k) = −1 λ+ − λ−
E±
(3.1)
V (k) =
−b(k) a(k+1)
λ (k) = + 0 −λ− . λ+
1−
0
a(k) a(k+1)
0
0 λ−
.
Then A(k) = S(k)−1 (k)S(k). If |E| > 2 and k is large enough, then (k) satisfies the dichotomy condition (2.1). It is easy to see that the rest of the hypotheses in Proposition 2.7 are satisfied for s = 1, so there are solutions of the form
k−1 (l)± λ± (k) + V e± + l12 , ± (k) = S(k)−1 l=k0
+ = V 11 , V − = V 22 , e+ = e1 , e− = e2 , and V (l) ∈ l 2 . We also have where V 1 (k)± = λ± (k) 1 ± r± (k) , λ± (k) + V where
a(k + 1) λ± (k) − λ± (k + 1) + λ∓ (k) a(k + 1) − a(k) − b(k) r± (k) = . a(k + 1) λ± (k + 1) − λ∓ (k + 1)
We now claim that ∞
r± (l) ∈ l12 ,
l=k
so in particular we can subsume the 1 + r terms into the error to write ± (k) = c±
k−1
λ± (l)
S(k)−1 e± + l12 .
l=k0
To see this is indeed the case, we make the following observations. First, a(k) → 1, λ(J ), κ(J ) ∈ l12 , and λ+ (k) and λ− (k) tend to different finite constants. In this way we see ∞ λ∓ (l) a(l + 1) − a(l) − b(l) ∈ l12 . a(l + 1) λ± (l + 1) − λ∓ (l + 1) l=k
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E. Ryckman
Second, we can write λ+ (k) − λ+ (k + 1) as 2 2 2 −a 2 2 ak+2 E 2 − 4ak+1 E 2 − 4ak+1 E 2 − 4ak+2 k+1 E − 4ak+2 − = 2ak+1 2ak+2 2ak+1 ak+2
a − a k+1 k+2 2 = E 2 − 4ak+1 2ak+1 ak+2 2 − E 2 − 4a E 2 − 4ak+2 k+1 + ak+1 . ak+1 ak+2 Because κ(J ) ∈ l12 , the first term is summable to be in l12 as well. To see the same is true √ of the second term, we do a Taylor expansion of E 2 − 4a 2 around the point E 2 − 4. After cancelling the constant terms we see that because κ(J ) ∈ l12 we have
E 2 − 4a 2 − E 2 − 4a ∞ k+1 k+2 ∈ l12 . ak+1 ak+1 ak+2 k=n
So the second term sums to be in l12 as well, proving the claim Now, E = β + β −1 and √ E ± E2 − 4 ±1 β = 2 so
E ± E 2 − 4a(k + 1)2 β ±1 λ± (k) = √ a(k + 1) E ± E2 − 4 β ±1 1 ± q± (k) . = a(k + 1) Arguing as we did for the r± terms we find ∞
q± (l) ∈ l12 ,
l=k
so we can subsume these products into the error term as well. Finally, using that κ(J ) ∈ l12 and taking the top row of ± we see ψ± (k) = c± β ±k 1 + l12 , as claimed. Proof of Proposition 1.4(1). Recall that
∞ c(k − 1)c(k + 1) , |κ(n ˜ − 1) − κ(n − 1)| = a(k)2 − 1 2 c(k) k=n
∞ a(k − 1)φ(k − 1)φ(k) a(k)φ(k)φ(k + 1) , ˜ |λ(n − 1) − λ(n − 1)| = |γ | − c(k − 1) c(k) k=n
A Strong Szeg˝o Theorem for Jacobi Matrices
803
where J φ = Eφ, φ(0) = 0, φ(1) = 1 and c(n) = 1 + γ
n
|φ( j)|2 .
j=1
Write φ as a linear combination of ψ+ and ψ− . Let us first suppose that φ is just a multiple of ψ− . As ψ− is geometrically decreasing, the same is true of
a(k)
2
c(k − 1)c(k + 1) −1 c(k)2
and
a(k − 1)φ(k − 1)φ(k) a(k)φ(k)φ(k + 1) − . c(k − 1) c(k)
So in this case it is easy to see that |κ(n ˜ − 1) − κ(n − 1)| and |λ˜ (n − 1) − λ(n − 1)| are 2 in l1 . Now suppose that φ is not just a multiple of ψ− . As ψ+ increases geometrically and ψ− decays geometrically, we see c(k) ∼ 1 + γ
k l=1
ψ(l)2 ∼ 1 + γ
k
˜ β 2l 1 + δ(l) ∼ 1 + β 2k 1 + δ(k) ,
(3.2)
l=1
˜ where δ(k), δ(k) represent some sequences in l12 , and “∼” indicates asymptotic equivalence (modulo multiplication by constants). Similarly ψ(k)ψ(k + 1) ∼ β 2k+1 1 + ε(k) for some ε ∈ l12 . Combining these shows a(k − 1)ψ(k − 1)ψ(k) a(k)ψ(k)ψ(k + 1) − c(k − 1) c(k) a(k − 1)β 2k−1 1 + ε(k − 1) − a(k)β 2k+1 1 + ε(k) c(k − 1)c(k)
β 4k−1 a(k − 1)1 + ε(k − 1) − a(k)1 + ε(k) + c(k − 1)c(k)
β 4k−1 a(k − 1)δ(k)1 + ε(k − 1) − a(k)δ(k − 1)1 + ε(k) . + c(k − 1)c(k) Because c(k − 1)c(k) ∼ β 4k−1 , the first term is geometrically decreasing, so okay by Lemma 2.4.
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Terms of the form a(k − 1)β 4k−1 ε(k − 1)δ(k) c(k − 1)c(k) are in l11 , being products of l12 sequences. Again, Lemma 2.4 shows this is fine. This leaves terms of the form β 4k−1 ε(k − 1) − ε(k) c(k − 1)c(k) for some sequence ε ∈ l12 . So it is sufficient to prove ∞ k=n
β 4k−1 ε(k − 1) − ε(k) ∈ l12 . c(k − 1)c(k)
Let C(k) =
β 4k−1 . c(k − 1)c(k)
Summing by parts shows ∞
∞ C(k) ε(k − 1) − ε(k) = C(n)ε(n − 1) + ε(k) C(k + 1) − C(k) .
k=n
(3.3)
k=n
The first term is clearly in l12 , so consider the second. Using (3.2) we can write |C(k + 1)−C(k)| β 4k+3 β 4k−1 − = c(k)c(k + 1) c(k)c(k + 1)
β 4k−1 β 4 − 1 + β 2k+2 δ(k − 1) − δ(k + 1) . c(k − 1)c(k)c(k + 1) As c(k − 1)c(k)c(k + 1) ∼ β 6k the first term is geometrically decaying and the second term is in l12 . Combining this with (3.3) and Lemma 2.4 shows that ∞ k=n
β 4k−1 ε(k − 1) − ε(k) ∈ l12 . c(k − 1)c(k)
This completes the proof for the λ’s. The proof for the κ’s is similar and simpler.
A Strong Szeg˝o Theorem for Jacobi Matrices
805
4. Proof of Theorem 1.1 ((2) ⇒ (1)) By assumption, J has finitely many eigenvalues, and they all lie in R \ [−2, 2]. By Theorem 1.3 and Proposition 1.4 we see it suffices to prove the theorem when σ (J ) ⊆ [−2, 2], which we now assume. Now, dν(x) = χ[−2,2] (x)v(x)d x and v(x) has one of the forms ±1
±1 2−x 2 4−x v0 (x) or v0 (x) 2+x with log v0 ∈ H˙ 1/2 . Define w(θ ) = cv0 (2 cos θ ) with c chosen to normalize w to be a probability measure on ∂D. Notice that dν = Sz (∗) (dµ), dθ and (∗) is one of (e), (o), (+), (−) according to which of the above where dµ = w 2π forms v(x) has. Therefore, α and a, b are related by one of the Direct Geronimus Relations of Proposition 1.5. By assumption log w ∈ H˙ 1/2 , so Theorem 1.2 shows that its Verblunsky coefficients satisfy α ∈ l12 . By Proposition 1.5 we see that
κn = α2n−1 + K (α)n , λn = α2n−2 + L(α)n , where K (α)n and L(α)n are sums from n to infinity of terms that are at least quadratic in α. So by Lemma 2.4 we see that λ, κ ∈ l12 too. 5. Asymptotic Integration Redux For this section we will make the standing assumption that the parameters defined by (1.1) obey κ(J ), λ(J ) ∈ l12 . In Sect. 7 we will need asymptotics on solutions at energies E = ±2. As before we will use asymptotic integration, but because the recurrence equation at E = ±2 yields a system with a Jordan anomaly, we cannot use the results of Sect. 2. Instead we construct a small solution ψs and big solution ψb : Proposition 5.1. There are solutions ψs and ψb to J ψ = Eψ at energy E = ±2 such that ψs (k) ψ (k) → 0 b
and ±
ψs (k + 1) = 1 + l12 . ψs (k)
Moreover, for either solution and for k sufficiently large, (±1)k ψ(k) > 0.
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E. Ryckman
The rest of this section is devoted to a proof of this statement for E = 2, the proof for E = −2 being analogous. Recall we can write the recurrence equation as 2−b(k+1) a(k) − a(k+1) a(k+1) (k + 1) = (k), 1 0 where
(k) =
ψ(k) . ψ(k − 1)
We begin with some preliminary transformations. Let 1 1/2 S= 1 −1/2 and let (k) = S(k). Then solves (k + 1) = [J + B(k)](k), where
⎡ B(k) =
1 J= 0
a(k+1)−1 + a(k)−1 −b(k) ⎢ 2a(k+1) ⎣ 2 a(k+1)−1 + a(k)−1 −b(k) a(k+1)
1 , 1
⎤
−3 a(k+1)−1 + a(k)−1 −b(k) ⎥ 4a(k+1) ⎦. −3 a(k+1)−1 + a(k)−1 −b(k) 2a(k+1)
In particular, notice that ∞ ∈ l 2. B(l) 1 l=k
Lemma 5.2. There exists a sequence of matrices 1 1 Q(k) = q(k) 1 such that q ∈ l12 ,
Q(k + 1)−1 [J + B(k)]Q(k) = L(k) + M(k),
and M(k) ∈ where
∞ l=k
l11 ,
1 + α(k) L(k) = 0
α(l) ∈ l12 ,
∞
1 + β(k) , 1 + γ (k)
β(l) ∈ l12 , γ (k) ∈ l12 .
l=k
In particular, if (k) = Q(k)x(k) then x(k + 1) = [L(k) + M(k)]x(k).
(5.1)
A Strong Szeg˝o Theorem for Jacobi Matrices
807
Proof. Define 1 , 1
1 Q(k) = q(k)
∞ where q(k) = − l=k B(l)21 ∈ l12 by Lemma 2.4. All the claimed properties are straightforward calculations. As we seek asymptotics as k → ∞, we need only consider systems for k larger than some k0 . In particular, we can choose k0 so that |α(k)|, |β(k)|, |γ (k)| < 1 for k ≥ k0 . In this case define x(k) = P(k)z(k), where
0 . k−1 j=1 1 + γ ( j)
1 P(k) = 0 This transforms the x-system into
z(k + 1) = [J (k) + R(k)]z(k),
(5.2)
where J (k) =
1 + α(k) 0
1 + β(k) 1
and R(k) ∈ l11 . We will compare this to the simpler system y(k + 1) = J (k)y(k). We begin by finding a basis of solutions to the y-system. Lemma 5.3. The y-system above has two solutions ys (k) =
u(k) v(k) and yb (k) = 0 1
such that u(k) =
k−1
1 + α( j) ,
j=1
0 < |u(k)| 1, |v(k)| ∼ k.
(5.3)
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E. Ryckman
Proof. Let u(1) = 1 and v(1) = 0. By the form of J (k) we see that u(k) =
k−1
1 + α( j) .
j=1
From the formula log(1 + x) = x + O(|x|2 ) one sees that if αk and |αk |2 are convergent, then (1 + αk ) converges to a nonzero finite ∞ number. 2 Because the α s are conditionally summable and l=k α(l) ∈ l1 , we see that αk and |αk |2 indeed converge, so the product defining u(k) converges to a nonzero finite number as k → ∞. As we have assumed that α(k) < 1, we also have u(k) = 0, which is (5.3). Now,
v(k + 1) = 1 + α(k) v(k) + 1 + β(k) =
k u(k + 1)
1 + β( j) . u( j + 1)
(5.4)
j=1
By (5.3) and β( j) → 0 we have |v(k)|
k
1 k.
l=1
Moreover, there is some j0 so that for k ≥ j ≥ j0 , u(k + 1)
1 + β( j) u( j + 1) is sign-definite. Without loss, assume that it is positive, so for j and k large enough we have u(k + 1)
1 + β( j) 1. u( j + 1) Thus, |v(k)| k too.
Now let
Y (k) =
u(k) 0
v(k) 1
be a fundamental matrix for the y-system. The next two lemmas construct the desired solutions to (5.2). Lemma 5.4. There is a bounded solution to the system (5.2) that has z(k + 1) − z(k) ∈ l12 . Moreover, z(k) is sign-definite for large enough k, and z(k) > 0.
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809
Proof. Consider the operator [T z](k) = −
∞
Y (k)Y (l + 1)−1 R(l)z(l)
(5.5)
l=k
acting on l ∞ (N; C2 ), with k ≥ k1 and k1 to be chosen momentarily. Notice that u(k) u(k) v(k) − u(l+1) v(l + 1) . Y (k)Y (l + 1)−1 = u(l+1) 0 1 By Lemma 5.3, u(k) u(l + 1) 1. By (5.4) we see that u(k)
u(k) v(k) − v(l + 1) = − 1 + β( j) u(l + 1) u( j + 1) l
j=k
so
v(k) − u(k) v(l + 1) |l − k|. u(l + 1)
Thus If z ∈
Y (k)Y (l + 1)−1 |l − k|. l∞
(5.6)
we see [T z](k)∞ z∞
∞
lR(l).
l=k1
Now, R(l) ∈ l11 , so by choosing k1 sufficiently large, we can ensure T z < εz for some ε < 1. Thus, T is a contraction on l ∞ , so in particular, given any y ∈ l ∞ there is a unique z ∈ l ∞ solving z = y + T z. If y = ys from Lemma 5.3, then by the form of T (and a lengthy but easy calculation) we see that this z solves the z-equation. Since T z < εz we see that ys ≤ z + T z < (1 + ε)z
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so z ys > 0 by Lemma 5.3. Moreover, because ys (k) is sign-definite for large enough k and T z < εz, we see the same is true of z. Next, notice that z(k + 1) − z(k) ≤ ys (k + 1) − ys (k) + [T z](k + 1) − [T z](k). For the first term we use Lemma 5.3 to see ys (k + 1) − ys (k) |α(k)| ∈ l12 . For the second term we use (5.5) to write [T z](k + 1) − [T z](k) = Y (k)Y (k + 1)−1 R(k)z(k) −
∞
[Y (k + 1) − Y (k)]Y (l + 1)−1 R(l)z(l). (5.7)
l=k+1
Now, [Y (k + 1) − Y (k)]Y (l + 1)−1 = !
u(k+1)−u(k) v(k + 1) − u(l+1) 0
u(k+1) u(l+1) v(l
+ 1) + v(k) − 0
u(k) u(l+1) v(l
" + 1)
By Lemma 5.3, u(k + 1) − u(k) 1. u(l + 1) For the other term in the matrix we use (5.4) to rewrite
u(k + 1) u(k) v(k + 1) − v(l + 1) + v(k) − v( j + 1) u( j + 1) u( j + 1)
=−
l l u(k + 1)
u(k)
1 + β( j) + 1 + β( j) u( j + 1) u( j + 1)
j=k+1
j=k
l
1 u(k)
1 + β(k) + u(k) − u(k + 1) 1 + β( j) = u(k + 1) u( j + 1) j=k+1
=
u(k)
1 + β(k) − α(k)u(k) u(k + 1)
l j=k+1
1 1 + β( j) . u( j + 1)
In particular we have [Y (k + 1) − Y (k)]Y (l + 1)−1 1 + |α(k)|l.
.
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811
Plugging this into (5.7) and using z ∈ l ∞ and (5.6) we find [T z](k + 1) − [T z](k) R(k) + R(k) +
∞ l=k+1 ∞
∞
R(l) + |α(k)|
lR(l)
l=k+1
R(l) + |α(k)|.
l=k+1
The first and third terms are clearly l12 , and by Lemma 2.4 so is the second. Thus z(k + 1) − z(k) ∈ l12 . Lemma 5.5. There is a solution
z (k) z b (k) = b1 z b2 (k)
to the z-system that is sign-definite for k large enough and has |z b1 (k)| ∼ k and |z b2 (k)| 1. Proof. Again, we compare the z-system to the y-system and use Lemma 5.3. This time, consider the operator k−1
[T z](k) =
Y (k)Y (l + 1)−1 R(l)z(l)
l=k1
with k1 ≥ 1 to be chosen momentarily. Let z 0 = yb and z j+1 = yb + T z j . Then z j+1 (k) − z j (k) = T [z j − z j−1 ](k) and z 1 (k) − z 0 (k) = T [yb ](k). By (5.6) and Lemma 5.3 we have [T y](k) ≤
k−1
Y (k)Y (l + 1)−1 · R(l) · y(l)
l=k1
k
k−1
R(l)l.
l=k1
We can choose k1 sufficiently large that [T y](k) < kε, where ε < 1. Then inductively we find that z j+1 (k) − z j (k) < kε j+1 . In particular, for each k, z j (k) → z b (k) as j → ∞ and z b = yb + T z b .
(5.8)
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By the form of T we see that because yb solves the y-equation, z b solves the z-equation. Moreover, as [T z b ](k) < kε and yb (k) ∼ k we have z b ∼ k.
(5.9)
Finally, because yb (k) is sign-definite for large enough k and [T z b ](k) < kε, we see the same is true for z b . To deduce the component bounds, we expand Y (k)Y (l + 1)−1 R(l)z b (l) and notice that the bottom component is bounded by |z b1 (l)R(l)21 | + |z b2 (l)R(l)22 |. Plugging this into (5.8) shows |z b2 (k)|
k−1
|z b1 (l)R(l)21 | + |z b2 (l)R(l)22 |
l=k1
k−1
lR(l) 1.
l=k1
Combining this with (5.9) yields the final bound. Proof of Proposition 5.1. Undoing the transformations we find that (k) = S Q(k)P(k)z(k), and therefore that 1 1 + q(k) z 1 (k) + 2 1 + γ ( j) z 2 (k) , 2 k−1
ψ(k) =
j=1
where z(k) =
z 1 (k) . z 2 (k)
Let ψs and ψb correspond to taking z to be z s and z b . All the claimed properties now follow from Lemmas 5.4 and 5.5.
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813
6. The Inverse Geronimus Relations Recall that the Direct Geronimus Relations provide formulas for a (∗) , b(∗) in terms of α. In this section we go the other way. We begin by determining whether a particular Jacobi matrix is in the range of the Szeg˝o maps based on the values of its m-function. Note that while Sz (e) maps onto all probability measures supported on [−2, 2], the ranges of the other three maps are given by $ dν(x) <∞ , Ran(Sz 2 −2 4 − x $ # 2 dν(x) <∞ . Ran(Sz (±) ) = dν : −2 2 ∓ x (o)
# ) = dν :
2
(6.1)
If x ∈ R, write m(x + i0) = lim m(x + iε). ε↓0
If either x ≤ inf σ (J ) or x ≥ sup σ (J ) we’ll write m(x) to indicate the value of the integral dν(t) t−x (which exists in R ∪ {±∞}). We begin by developing some elementary properties of the m-functions, which we then use to study the associated polynomials. Lemma 6.1. Let J be a Jacobi matrix with σ (J ) ⊆ [−2, 2]. Then J ∈ Ran(Sz (o) ) ⇔ m(−2) − m(2) < ∞, J ∈ Ran(Sz (±) ) ⇔ ∓m(±2) < ∞. Proof. The second line follows from the definition of the m-function and (6.1). For the first line note 2 2
1 dν(x) 1 + dν(x) = 4 m(−2) − m(2) = 2 2 + x 2 − x 4 −2 −2 − x and again use (6.1). As with the ranges, the normalization constants (1.5) have interpretations in terms of the m-function: Lemma 6.2. If m (∗) (x) is the m-function for dν (∗) then 1 , (1 ∓ α0 )(1 − α1 ) 1 ∓m (±) (±2) = . 2(1 ∓ α0 )
∓m (o) (±2) =
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E. Ryckman
Proof. By (1.6) we can write (2 − x)(2 + x) dν (e) (x), 2(1 − α02 )(1 − α1 ) 2∓x dν (e) (x). dν (±) (x) = 2(1 ∓ α0 )
dν (o) (x) =
The values of m (±) then follow from dν (e) being a probability measure. For the m (o) values we have 2 2 1 dν (o) (x) = m (o) (−2) = (2 − x)dν (e) (x) 2(1 − α02 )(1 − α1 ) −2 −2 2 + x 2π 1 2 − (z + z −1 )dµ(z) = 2(1 − α02 )(1 − α1 ) 0 2π 1 zdµ(z)) = (1 − (1 − α02 )(1 − α1 ) 0 1 − α0 = . 2(1 − α02 )(1 − α1 ) The value of −m (o) (2) follows similarly. We’ll need lower bounds on the m-function: Lemma 6.3. If σ (J ) ⊆ [−2, 2], then ∓m(±2) > 1/4. Proof. As J has no eigenvalues off [−2, 2], 2 dν(x) . m(E) = x −2 − E For t ∈ [−2, 2] and E > 2, t − E ≥ −4. Because dν is a probability measure that is not a point mass at t = 2, the Monotone Convergence Theorem implies −m(2) = lim −m(E) > 1/4. E↓2
Similar arguments show m(−2) > 1/4. (∗)
We now turn to the polynomials. Given dµ, write Pn (x) for the monic polyno(∗) mial of degree n with respect to the measure dν (∗) = Sz (∗) (dµ). Similarly, let Q n (x) (∗) be the second-kind polynomial for dν . That is, Q solves the same recurrence equation as P but with initial conditions Q −1 ≡ −1 and Q 0 ≡ 0. If |m(x)| < ∞, let Fn(∗) (x) = m(x)Pn(∗) (x) + Q (∗) n (x). Proposition 6.4. Let dµ be a nontrivial probability measure on ∂D that is invariant under conjugation, and let α be its Verblunsky parameters. Then (e)
Pn+1 (2) = (1 − α2n−1 )(1 − α2n )Pn(e) (2), (e)
Pn+1 (−2) = −(1 − α2n−1 )(1 + α2n )Pn(e) (−2),
A Strong Szeg˝o Theorem for Jacobi Matrices
815
(o)
Fn+1 (2) = (1 + α2n+1 )(1 + α2n+2 )Fn(o) (2), (o)
Fn+1 (−2) = −(1 + α2n+1 )(1 − α2n+2 )Fn(o) (−2), (+)
Fn+1 (2) = (1 + α2n )(1 + α2n+1 )Fn(+) (2), (+)
Pn+1 (−2) = −(1 + α2n )(1 − α2n+1 )Pn(+) (−2), (−)
Pn+1 (2) = (1 − α2n )(1 − α2n+1 )Pn(−) (2), (−) (−2) = −(1 − α2n )(1 + α2n+1 )Fn(−) (−2). Fn+1
Proof. The proof is by induction. As the arguments for any of the P’s are virtually identical, we only present the proof for the case P = P (e) . Similarly, we only present the argument for the F’s in the case F = F (+) . The desired relationship between P0 ≡ 1 and P1 (x) = x − b1 follows from Proposition 1.5: P1 (2) = 2 − b1 = 2 − α0 (1 − α−1 ) = 2(1 − α0 ) = (1 − α−1 )(1 − α0 )P0 (2). To deduce the desired relationship between F1 (2) and F0 (2), we argue as follows. By Lemma 6.3 and Lemma 6.1 we have that 41 < −m(2) < ∞, so F0 (2) = m(2). Next, recall that P−1 ≡ 0, P0 ≡ 1, Q −1 ≡ −1, and Q 0 ≡ 0. So F1 (2) = m(2)P1 (2) + Q 1 (2) (2 − b1 ) 2α0 − b1 =− +1=− 2(1 − α0 ) 2(1 − α0 ) −1 = (1 + α0 )(1 + α1 )F0 (2), = (1 + α0 )(1 + α1 ) 2(1 − α0 ) where we have used Proposition 1.5 and Lemma 6.2. Now, assume the formulas hold up to index n − 1. As Pn satisfies the three-term recurrence equation we have Pn+1 (2) = (2 − bn+1 )Pn (2) − an2 Pn−1 (2)
an2 Pn (2) = (2 − bn+1 ) − (1 − α2n−3 )(1 − α2n−2 ) = (1 − α2n−1 )(1 − α2n )Pn (2), where the second equality is by the inductive hypothesis, and the third equality is by Proposition 1.5. Similarly, Fn satisfies the three-term recurrence equation, so the same argument works: Fn+1 (2) = (2 − bn+1 )Fn (2) − an2 Fn−1 (2)
an2 Fn (2) = (2 − bn+1 ) − (1 + α2n−2 )(1 + α2n−1 ) = (1 + α2n )(1 + α2n+1 )Fn (2).
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E. Ryckman
Proposition 6.5. (Inverse Geronimus Relations5 ) Let dµ a nontrivial probability measure on ∂D that is invariant under conjugation, and let α be its Verblunsky parameters. Define A(∗) n =− Cn(∗) = −
(∗)
Pn+1 (−2) (∗)
Pn (−2) (∗)
Fn+1 (−2) (∗)
Fn (−2)
, Bn(∗) = , Dn(∗) =
(∗)
Pn+1 (2) (∗)
Pn (2)
,
(∗)
Fn+1 (2) (∗)
Fn (2)
.
If dν = Sz (e) (dµ), α2n =
(e)
(e)
(e) An
(e) Bn
An − Bn +
1 α2n−1 = 1 − (A(e) + Bn(e) ). 2 n
If dν = Sz (o) (dµ), −α2n+2 =
Cn(o) − Dn(o) (o) Cn
+
(o) Dn
1 − α2n+1 = 1 − (Cn(o) + Dn(o) ). 2
If dν = Sz (+) (dµ), −α2n+1 =
(+) A(+) n − Dn
A(+) n
+
Dn(+)
1 − α2n = 1 − (A(+) + Dn(+) ). 2 n
If dν = Sz (−) (dµ), α2n+1 =
(−)
Cn
Cn(−)
(−)
− Bn +
Bn(−)
1 α2n = 1 − (Cn(−) + Bn(−) ). 2
By Sturm oscillation theory and that Sz (∗) (dµ) is supported in [−2, 2], we see (∗) (∗) (±1)n+1 Pn (±2) and −(±1)n+1 Fn (±2) are strictly positive for all n > 0. In particular, the above ratios are all defined. Proof. This is a simple calculation based on Proposition 6.4.
7. Some Weyl Theory By Proposition 6.5 we see that decay of the Verblunsky parameters is controlled by decay of the sequences An , Bn , Cn , and Dn . By Proposition 5.1 we see that there is a solution at E = ±2 with the desired asymptotics. The following result connects these two ideas. Proposition 7.1. Let J be a Jacobi matrix with σ (J ) ⊆ [−2, 2], and let An , Bn , Cn , Dn be defined as above. Then m(−2) < ∞ implies Cn = 1 + l12 , and m(−2) = ∞ implies An = 1 + l12 . Similarly, −m(2) < ∞ implies Dn = 1 + l12 , and −m(2) = ∞ implies Bn = 1 + l12 . 5 The case dν = Sz (e) (dµ) is due to [4] (with an alternate proof given in [3]). The statement in the other three cases appears to be new (although anticipated in [16] and related to some formulas of [2]).
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817
Let us write pn and qn for the orthonormal versions of Pn and Q n , and then f n (z) = m(z) pn (z) + qn (z). Proposition 7.1 is a trivial consequence of Proposition 7.2. Let J be a Jacobi matrix with σ (J ) ⊆ [−2, 2]. Then m(−2) < ∞ implies (−1)n+1 f n (−2) = s +l12 , and m(−2) = ∞ implies (−1)n+1 pn (−2) = s +l12 , for some s ∈ R. Similarly, −m(2) < ∞ implies f n (2) = s + l12 , and −m(2) = ∞ implies pn (2) = s + l12 . To prove this, we will use some Weyl theory. Recall pn (z) and qn (z) are solutions to J u = zu with p−1 = q0 = 0 and p0 = −q−1 = 1. When z ∈ C \ R, the Weyl solution f n (z) = m(z) pn (z) + qn (z) is defined and satisfies f n (z)l22 =
Im m(z) . Im z
(7.1)
As the m-function and the solutions p and q will play prominent roles, we develop some of their key properties. To start, we relate the values of m at ±2 to its values at ±2 + iε. Lemma 7.3. Let J be a Jacobi matrix with σ (J ) ⊆ [−2, 2], m-function m, and spectral measure dν. Then 2 dν(t) < ∞ ⇒ m(−2) = m(−2 + i0), −2 2 + t 2 dν(t) = ∞ ⇒ |m(−2 + i0)| = ∞. −2 2 + t Similarly,
2
−2 2 −2
dν(t) <∞ ⇒ 2−t
m(2) = m(2 + i0),
dν(t) =∞ ⇒ 2−t
|m(2 + i0)| = ∞.
In particular, when m(±2) is finite, we may write m(±2) for m(±2+i0) and then f n (±2) for f n (±2 + i0). Notice that by Lemma 6.3, ∓m(±2) can only diverge to +∞. Proof. The first implication follows from the Dominated Convergence Theorem applied to 2 v(t)dt . −m(2 + iε) = (2 − t) + iε −2 The second implication follows from the Monotone Convergence Theorem applied to 2 2−t − Re m(2 + iε) = dν(t). 2 2 −2 (2 − t) + ε
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E. Ryckman
If L ∈ N and u(n; z) solves J u = zu, we define u(z)2L
=
L
|u(l; z)|2 .
l=0
For non-integer values of L we define u(z) L to be the linear interpolation between the values at L and L . Now suppose x ∈ R is fixed and m(x + i0) exists finitely. Let ε, y > 0 be related by sup |m(z) − m(x + i0)| + y =
0
ε2 , 4
where z = x + i y. Note that y is a monotone function of ε that goes to zero as ε goes to zero. Define L(ε) by |y |1/2 pn (z ) L(ε) = 1, where z = x + i y . For each y > 0, L(ε) exists because pn (z ) is not in l 2 . The following lemma is a discrete analog of Lemma 9 of [6]. The proof is a direct translation, so we omit it. Lemma 7.4. Let x ∈ R and suppose that m(x + i0) exists finitely. Then f n (x + i0) L(ε) <ε pn (x) L(ε) whenever ε is sufficiently small. Next we recall a result of [12]. ˜ Lemma 7.5. Let x ∈ R and define L(ε) by pn (x) L(ε) = ˜ qn (x) L(ε) ˜
1 . 2ε
˜ Then L(ε) is a well defined, monotonely decreasing continuous function that goes to infinity as ε goes to 0, and √ √ pn (x) L(ε) ˜ 5 − 24 5 + 24 ≤ . ≤ |m(x + iε)| qn (x) L(ε) |m(x + iε)| ˜ Proof of Proposition 7.2. Again, we will only prove the statements for E = 2. Suppose first that 1/4 ≤ −m(2) < ∞. Then by Lemma 7.3, m(2 + i0) is finite and nonzero too. So by Lemma 7.5 we have that pn (2) L qn (2) L remains finite and nonzero as L ↑ ∞. As solutions at E = 2 are of the form c1 ψb + c2 ψs for some ci ∈ R, we see that pn (2) and qn (2) must be simultaneously bounded or simultaneously unbounded. Because pn (2) and qn (2) form a basis for solutions at E = 2, we see they cannot both be bounded. So they are both unbounded.
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Now, by Lemma 7.4 we see that f n (2) L →0 pn (2) L as L ↑ ∞. So f n (2) cannot be unbounded, and so has the form cψs for some c ∈ R. Now Proposition 5.1 yields the desired result. Now suppose that −m(2) = ∞. Then by Lemma 7.3 we have that |m(2 + i0)| = ∞ too. Then by Lemma 7.5 we have pn (2) L →0 qn (2) L so we must have that pn (2) remains bounded. Thus, pn (2) = cψs (n) for some c ∈ R, so again we are done by Proposition 5.1. 8. Proof of Theorem 1.1 ((1) ⇒ (2)) By Proposition 5.1 we see that all solutions at E = ±2 eventually satisfy (±1)k ψ(k) > 0. So by the Sturm oscillation theorem for Jacobi matrices (see Chapter 4 of [19]), J has only finitely-many eigenvalues, all lying in R \ [−2, 2]. So by Proposition 1.4 it suffices to prove the theorem when σ (J ) ⊆ [−2, 2], which we now assume. Consider the values of the m function at E = ±2. We have four cases: Case 1. m(−2) = −m(2) = ∞. As Sz (e) is onto, dν ∈ Ran(Sz (e) ), so choose Rn (−2) = Pn (−2),
Rn (2) = Pn (2),
dµ = [Sz (e) ]−1 (dν).
Case 2. m(−2), −m(2) < ∞. By Lemma 6.1, dν ∈ Ran(Sz (o) ), so choose Rn (−2) = Fn (−2),
Rn (2) = Fn (2),
dµ = [Sz (o) ]−1 (dν).
Case 3. m(−2) = ∞, −m(2) < ∞. By Lemma 6.1, dν ∈ Ran(Sz (+) ), so choose Rn (−2) = Pn (−2),
Rn (2) = Fn (2),
dµ = [Sz (+) ]−1 (dν).
Case 4. m(−2) < ∞, −m(2) = ∞. By Lemma 6.1, dν ∈ Ran(Sz (−) ), so choose Rn (−2) = Fn (−2),
Rn (2) = Pn (2),
dµ = [Sz (−) ]−1 (dν).
In any case, let α be the Verblunsky parameters associated to dµ. By Proposition 7.1 we see that Rn+1 (−2) = 1 + l12 , Rn (−2)
Rn+1 (2) = 1 + l12 . Rn (2)
Then by Proposition 6.5 we see that α ∈ l12 . By Theorem 1.2 we see log w ∈ H˙ 1/2 , so by (1.6) we see v ∈ W. Acknowledgement. It is a pleasure to thank Rowan Killip for his helpful advice.
820
E. Ryckman
References 1. Benzaid, Z., Lutz, D.A.: Asymptotic representation of solutions of perturbed systems of linear difference equations. Stud. Appl. Math. 77, no. 3, 195–221 (1987) 2. Berriochoa, E., Cachafeiro, A., García-Amor, J.: Connection between orthogonal polynomials on the unit circle and bounded interval. J. Comput. Appl. Math. 177, no. 1, 205–223 (2005) 3. Damanik, D., Killip, R.: Half-line Schrödinger operators with no bound states. Acta Math. 193, no. 1, 31–72 (2004) 4. Geronimus, Ya.L.: Polynomials Orthogonal on a Circle and Their Applications. Amer. Math. Soc. Translation 104, Providence, RI: AMS, 1954 5. Gesztesy, F., Teschl, G.: Commutation methods for Jacobi matrices. J. Diff. Eq. 128, 252–299 (1996) 6. Gilbert, D.J., Pearson, D.B.: On subordinacy and analysis of the spectrum of one-dimensional Schr˝odinger operators. J. Math. Annal. Appl. 128, 30–56 (1987) 7. Golinskii, B.L., Ibragimov, I.A.: On Szeg˝o’s limit theorem. Math. USSR Izv. 5, 421–444 (1971) 8. Harris, W.A. Jr., Lutz, D.A.: On the asymptotic integration of linear differential systems. J. Math. Anal. Appl. 48, 1–16 (1974) 9. Harris, W.A. Jr., Lutz, D.A.: A unified theory of asymptotic integration. J. Math. Anal. Appl. 57, no. 3, 571–586 (1977) 10. Hartman, P., Wintner, A.: Asymptotic integrations of linear differential equations. Amer. J. Math. 77, 45– 86 (1955) 11. Ibragimov, I.A.: A theorem of Gabor Szeg˝o. Mat. Zametki 3, 693–702 (1968) 12. Jitomirskaya, S., Last, Y.: Power-law subordinacy and singular spectra I. Half-line operators. Acta Math. 183, 171–189 (1999) 13. Killip, R., Nenciu, I.: Matrix models for circular ensembles. Int. Math. Res. Not. no. 50, 2665–2701 (2004) 14. Levinson, N.: The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948) 15. Ryckman, E.: A spectral equivalence for Jacobi matrices. http://arxiv.org/list/math.sp/0604515, 2006 16. Simon, B.: Orthogonal Polynomials on the Unit Circle. American Mathematical Society Colloquium Publications 54, Parts 1 & 2, Providence, RI: AMS, 2005 17. Szeg˝o, G.: Orthogonal Polynomials. 4th edition, American Mathematical Society Colloquium Publications, Vol. XXIII., Providence, R.I.: AMS, 1975 18. Szeg˝o , G.: On certain Hermitian forms associated with the Fourier series of a positive function. Comm. Sém. Math. Univ. Lund. 1952, Tome Supplementaire, 228–238 (1952) 19. Teschl, G.: Jacobi Matrices and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs 72, Providence, RI: AMS, 2000 Communicated by B. Simon
Commun. Math. Phys. 271, 821–838 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0193-7
Communications in
Mathematical Physics
A New Bernstein’s Inequality and the 2D Dissipative Quasi-Geostrophic Equation Qionglei Chen1 , Changxing Miao1 , Zhifei Zhang2 1 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P. R. China.
E-mail: [email protected]; [email protected]
2 School of Mathematical Science, Peking University, Beijing 100871, P. R. China.
E-mail: [email protected] Received: 5 July 2006 / Accepted: 9 August 2006 Published online: 8 February 2007 – © Springer-Verlag 2007
Abstract: We show a new Bernstein’s inequality which generalizes the results of Cannone-Planchon, Danchin and Lemarié-Rieusset. As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and super-critical dissipation for the small initial data in the critical Besov space, and local well-posedness for the large initial data. 1. Introduction We are concerned with the 2D dissipative quasi-geostrophic equation ∂t θ + u · ∇θ + κ(−)α θ = 0, x ∈ R2 , t > 0, (QG)α θ (0, x) = θ0 (x).
(1.1)
Here α ∈ [0, 21 ], κ > 0 is the dissipative coefficient, θ (t, x) is a real-valued function of t and x. The function θ represents the potential temperature, the fluid velocity u is determined from θ by a stream function ψ, 1 ∂ψ ∂ψ , (−) 2 ψ = −θ. (u 1 , u 2 ) = − , (1.2) ∂ x2 ∂ x1 A fractional power of the Laplacian (−)β is defined by β f (ξ ) = |ξ |2β fˆ(ξ ), (−) where fˆ denotes the Fourier transform of f . We rewrite (1.2) as 1
1
u = (∂x2 (−)− 2 θ, −∂x1 (−)− 2 θ ) = R⊥ θ = (−R2 θ, R1 θ ),
822
Q. Chen, C. Miao, Z. Zhang
where Rk , k = 1, 2, is the Riesz transform defined by R k f (ξ ) = −iξk /|ξ | fˆ(ξ ). (QG)α is an important model in geophysical fluid dynamics, they are special cases of the general quasi-geostrophic approximations for atmospheric and oceanic fluid flow with small Rossby and Ekman numbers. There exists deep analogy between Eq. (1.1) with α = 21 and the 3D Navier-Stokes equations. For more details about its background in geophysics, see [8, 21]. The case α > 21 is called the subcritical case, the case α = 21 is critical, and the case 0 ≤ α < 21 is supercritical. In the subcritical case, Constantin and Wu [9] proved the existence of global in time smooth solutions. In the critical case, Constantin, Cordoba, and Wu [10] proved the existence and uniqueness of global smooth solution on the spatial periodic domain under the assumption of small L ∞ norm. Recently, Chae and Lee [5] studied the super-critical case and proved the 2−2α . Very recently, Corglobal well-posedness for small data in the Besov spaces B˙ 2,1 doba-Cordoba [13], Ning [17, 18] studied the existence and uniqueness in the Sobolev spaces H s , s ≥ 2 − 2α, α ∈ [0, 21 ]. Wu [24, 25] studied the well-posedness in general Besov space B sp,q , s > 2(1 − α), p = 2 N . Many other relevant results can also be found in [4, 11, 12]. One purpose of this paper is to study the well-posedness of the 2D dissipative quasi2
+1−2α
p geostrophic equation in the critical Besov space B p,q , p ≥ 2, q ∈ [1, ∞). If we use the standard energy method as in [5, 26], we need to establish the lower bound for the term generated from the dissipative part p 2α j θ | j θ | p−2 j θ d x ≥ 22α j j θ p , p ≥ 2, (1.3)
R2
where j is the frequency localization operator at |ξ | ≈ 2 j (see Sect. 2). For p = 2, this is a direct consequence of Plancherel formula. In the case α = 1, it is proved by Cannone and Planchon [3]. To generalize (1.3) to general index α, p, it is sufficient to show the following Bernstein’s inequality: cp2
2α j p
p
2
j f p ≤ α (| j f | 2 )2p ≤ C p 2
2α j p
j f p ,
p > 2,
(1.4)
which together with an improved positivity Lemma 3.3 in [18] (see also Sect. 3, Lemma 3.3) will imply (1.3). We should point out that (1.4) is proved by Lemarié-Rieusset [19] in the case α = 1, and by Danchin [14] when p is any even integer. On the other hand, Wu [26] gives a formal proof for general index. The first purpose of this paper is to present a rigorous proof of Theorem 3.4 in [26] which plays a key role in Wu’s paper. Theorem 1.1 (Bernstein’s inequality). Let p ∈ [2, ∞) and α ∈ [0, 1]. Then there exist two positive constants c p and C p such that for any f ∈ S and j ∈ Z, we have cp2
2α j p
p
2
j f p ≤ α (| j f | 2 )2p ≤ C p 2
2α j p
j f p .
(1.5)
The second purpose is to study the well-posedness of the 2D dissipative quasi2
+1−2α
p geostrophic equation in the critical Besov space B p,q Fourier localization technique.
by using Theorem 1.1 and
New Bernstein’s Inequality and 2D Dissipative Quasi-Geostrophic Equation
823
Theorem 1.2. Assume that (α, p, q) ∈ (0, 21 ] × [2, ∞) × [1, ∞). If θ0 belongs to B σp,q with σ = 2p + 1 − 2α, then there exists a positive real number T such that a unique solution to the 2D dissipative quasi-geostrophic equation θ (t, x) exists on [0, T ) × R2 satisfying 2
+1
p L 1 (0, T ; B˙ p,q ), θ (t, x) ∈ C([0, T ); B σp,q ) ∩
with the time T bounded from below by 2α j 1 sup T > 0 : (1 − e−κc p 2 T ) 2 2 jσ j θ0 p q (Z) ≤ cκ . Furthermore, if θ0 B˙ σ ≤ κ for some positive number , then we can choose T = +∞. p,q
Remark 1.1. It is pointed out that the homogeneous Besov space B˙ σp,q is important as it gives the important scaling invariant function space. In fact, if θ (t, x) and u(t, x) are solutions of (1.1), then θλ (t, x) = λ2α−1 θ (λ2α t, λx) and u λ (t, x) = λ2α−1 u(λ2α t, λx) are also solutions of (1.1). The B˙ σp,q norm of θ (t, x) is invariant under this scaling. Moreover, for the global existence result, the smallness assumption is imposed only on the homogenous norm of the initial data. Remark 1.2. The result of Theorem 1.2 for the case ( p, q) = (2, 1) corresponds to the 2−2α result of Chae and Lee [5] in the critical Besov space B2,1 . In the case ( p, q) = (2, 2), it corresponds to the result of Ning [17] in the Sobolev space H 2−2α . On the other hand, thanks to the embedding relationship: s s H s B2,q , for q > 2, B2,1
our result improves the results of [5] and [17]. Remark 1.3. Wu [24, 25] proved the well-posedness of (1.1) for the initial data in the sub-critical Besov space B sp,q with s > 2 − 2α, p = 2 N . We obtain the well-posedness 2
+1−2α
p in the critical Besov space B p,q
, and get rid of the restriction on p = 2 N .
Remark 1.4. Very recently, Miura [20] proved the local well-posedness of (1.1) for the large initial data in the critical Sobolev space H 2−2α . His result is a particular case of Theorem 1.2, and our proof is simpler (see Sect. 4.2). Notation. Throughout the paper, C denotes various “harmless” large finite constants, and c denotes various “harmless” small constants. We shall sometimes use X Y to denote the estimate X ≤ CY for some C. {c j } j∈Z denotes any positive series with q (Z) norm less than or equal to 1. We shall sometimes use the · p to denote L p (Rd ) norm of a function. 2. Littlewood-Paley Decomposition Let us recall the Littlewood-Paley decomposition. Let S(Rd ) be the Schwartz class of rapidly decreasing functions. Given f ∈ S(Rd ), its Fourier transform F f = fˆ is defined by d e−i x·ξ f (x)d x. fˆ(ξ ) = (2π )− 2 Rd
824
Q. Chen, C. Miao, Z. Zhang
Choose two nonnegative radial functions χ , ϕ ∈ S(Rd ), supported respectively in B = {ξ ∈ Rd , |ξ | ≤ 43 } and C = {ξ ∈ Rd , 43 ≤ |ξ | ≤ 83 } such that χ (ξ ) +
ϕ(2− j ξ ) = 1, ξ ∈ Rd ,
j≥0
ϕ(2− j ξ ) = 1, ξ ∈ Rd \{0}.
j∈Z
Setting ϕ j (ξ ) = ϕ(2− j ξ ). Let h = F −1 ϕ and h˜ = F −1 χ , we define the frequency localization operator as follows: j f = ϕ(2− j D) f = 2 jd
h(2 j y) f (x − y)dy, ˜ j y) f (x − y)dy. Sj f = k f = χ (2− j D) f = 2 jd h(2 Rd
Rd
k≤ j−1
Informally, j = S j − S j−1 is a frequency projection to the annulus {|ξ | ≈ 2 j }, while S j is a frequency projection to the ball {|ξ | 2 j }. One easily verifies that with our choice of ϕ, j k f ≡ 0 i f | j − k| ≥ 2 and j (Sk−1 f k f ) ≡ 0 i f | j − k| ≥ 5. (2.1) Now we give the definitions of the Besov spaces. Definition 2.1. Let s ∈ R, 1 ≤ p, q ≤ ∞, the homogenous Besov space B˙ sp,q is defined by B˙ sp,q = { f ∈ Z (Rd ); f B˙ s
p,q
< ∞}.
Here
f B˙ s
p,q
=
⎧ 1 ⎪ ⎪ q q jsq ⎪ 2 j f p , for q < ∞, ⎨ j∈Z
⎪ js ⎪ ⎪ ⎩ sup 2 j f p , j∈Z
for q = ∞,
and Z (Rd ) denotes the dual space of Z(Rd ) = { f ∈ S(Rd ); ∂ γ fˆ(0) = 0; ∀γ ∈ Nd multi-index} and can be identified by the quotient space of S /P with the polynomials space P. Definition 2.2. Let s ∈ R, 1 ≤ p, q ≤ ∞, the inhomogenous Besov space B sp,q is defined by B sp,q = { f ∈ S (Rd ); f B sp,q < ∞}.
New Bernstein’s Inequality and 2D Dissipative Quasi-Geostrophic Equation
Here f B sp,q =
825
⎧ 1 ⎪ ⎪ q q ⎪ ⎨ 2 jsq j f p + S0 ( f ) p , for q < ∞, j≥0
⎪ js ⎪ ⎪ ⎩ sup 2 j f p + S0 ( f ) p ,
for q = ∞.
j≥0
If s > 0, then B sp,q = L p ∩ B˙ sp,q and f B sp,q ≈ f p + f B˙ s . We refer to [1, 23] p,q for more details. Next let’s recall Chemin-Lerner’s space-time space which will play an important role in the proof of Theorem 1.2. Definition 2.3. Let s ∈ R, 1 ≤ p, q, r ≤ ∞, I ⊂ R is an interval. The homogeneous mixed time-space Besov space L r (I ; B˙ sp,q ) is the space of the distribution such that L r (I ; B˙ sp,q ) = { f ∈ D(I ; Z (Rd )); f L r (I ; B˙ s
p,r )
Here f (t) L r (I ; B˙ s
p,q )
< +∞}.
1 r sj r 2 = f (τ ) dτ j p
q (Z)
I
,
(usual modification if r, q = ∞). We also need the inhomogeneous mixed time-space Besov space L r (I ; B sp,q ), s > 0 whose norm is defined by p f (t) L r (I ;B sp,q ) = f (t) L r (I ;L x ) + f (t) L r (I ; B˙ s
p,q )
.
For convenience, we sometimes use L rT ( B˙ sp,q ) and L r ( B˙ sp,q ) to denote L r (0, T ; B˙ sp,q ) r s and L (0, ∞; B˙ p,q ), respectively. The direct consequence of Minkowski’s inequality is that L rt ( B˙ sp,q ) ⊆ L rt ( B˙ sp,q ) if r ≤ q and L rt ( B˙ sp,q ) ⊆ L rt ( B˙ sp,q ) if r ≥ q. We refer to [7] for more details. Let us state some basic properties about the Besov spaces. Proposition 2.1.
(i) We have the equivalence of norms D k f B˙ s
p,q
∼ f B˙ s+k , for k ∈ Z+ . p,q
(ii) Interpolation: for s1 , s2 ∈ R and θ ∈ [0, 1], one has , f ˙ θs1 +(1−θ)s2 ≤ f θ˙ s1 f (1−θ) ˙ s2 B p,q
B p,q
B p,q
and the similar interpolation inequality holds for inhomogeneous Besov space. (iii) Embedding: If s > dp , then B sp,q → L ∞ ; p1 ≤ p2 and s1 − B sp11 ,q1
d p1
→
> s2 − B sp22 ,q2 ,
d p2 ,
then
B sp,min( p,2) → H ps → B sp,max( p,2) .
Here H ps is the inhomogeneous Sobolev space. Proof. The proof of (i) − (iii) is rather standard and one can refer to [23].
826
Q. Chen, C. Miao, Z. Zhang
Finally we introduce the well-known Bernstein’s inequalities which will be used repeatedly in this paper. Lemma 2.2. Let C be a ring, and B a ball, 1 ≤ p ≤ q ≤ +∞. Assume that f ∈ S (Rd ), then for any |γ | ∈ Z+ ∪ {0} there exist constants C, independent of f , j such that ∂ γ f q ≤ Cλ
|γ |+d( 1p − q1 )
f p if supp fˆ ⊂ λB,
(2.2)
f p ≤ Cλ−|γ | sup ∂ β f p ≤ C f p if supp fˆ ⊂ λC. |β|=|γ |
(2.3)
Proof. The proof can be found in [6]. 3. A New Bernstein’s Inequality
Firstly, we will give certain kind of Bernstein’s inequality which can be found in [[19], Chapter 29]. Proposition 3.1. Let 2 < p < ∞. Then there exist two positive constants c p and C p such that for every f ∈ S and every j ∈ Z, we have 2j
2
p
2j
c p 2 p j f p ≤ ∇(| j f | 2 )2p ≤ C p 2 p j f p .
(3.1)
Naturally, we want to establish a generalization of (3.1) for the fractional differential operator α (0 < α < 1) which is defined by α f = F −1 (|ξ |α f ). However it seems p nontrivial, since for p > 2, the spectrum of | j f | 2 can’t be included in a ring although j supp j f is localized in |ξ | ≈ 2 . This section is devoted to prove Theorem 1.1. For this purpose, we first need the following priori lemma. Lemma 3.2. Let p ∈ [1, ∞), s ∈ [0, p) ∩ [0, 2). Suppose that , r, m satisfy 1 < ≤ r < ∞, 1 < m < ∞,
1 1 p−1 = + . r m
Then for f (u) = |u| p , the following estimate holds: p−1
f (z) B˙ s ≤ C p z B˙ 0 z B˙ s . ,2
(3.2)
r,2
m,2
Proof. Let us first recall the equivalence norm of Besov spaces: for 0 ≤ s < 2, 1 ≤ , q ≤ ∞, v B˙ s ,q
∞
−sq
t
sup τ+y v + τ−y v
|y|≤t
0
q dt − 2v
t
1 q
,
where τ±y v(x) = v(x ± y). In the special case when 0 ≤ s < 1, we also have v B˙ s ,q
∞
t 0
−sq
sup τ+y v
|y|≤t
q dt − v
t
1 q
.
New Bernstein’s Inequality and 2D Dissipative Quasi-Geostrophic Equation
827
It is not difficult to check that (|z 1 | p−[s]−1 + |z 2 | p−[s]−1 )|z 1 − z 2 |, | f [s] (z 1 ) − f [s] (z 2 )| ≤ C |z 1 − z 2 | p−[s] , p < [s] + 1,
p ≥ [s] + 1,
(3.3)
where f [s] (z) = Dz[s] f (z). For simplicity we set u ± τ±y u. We divide the proof of Lemma 3.2 into two cases. Case 1. p ≥ 2. We write τ y f (u) + τ−y f (u) − 2 f (u) = f (u + ) + f (u − ) − 2 f (u) 1 = f (u)(u + + u − − 2u) + (u ± − u) [ f (λu ± + (1 − λ)u) − f (u)]dλ, (3.4) 0
±
which together with (3.3) gives that | f (u + ) + f (u − ) − 2 f (u)| ≤ f (u)|u + + u − − 2u| + C
|u ± − u|2 {max(|u ± |, |u|)} p−2 .
±
Using the Hölder inequality, we have f (u + ) + f (u − ) − 2 f (u) p−1
≤ um u + + u − − 2ur + C
p−2
u ± − u22θ um ,
±
where θ =
mr m+r .
Then by the previous equivalence norm of Besov spaces, we have p−1
f (u) B˙ s ≤ Cu B˙ s um ,2
r,2
p−2
+ um u2
s
2 B˙ 2θ,4
.
Thanks to the interpolation inequality u2
s
2 B˙ 2θ,4
≤ u B˙ s u B˙ 0
m,∞
r,2
,
0 , we obtain and the inclusion map L m → B˙ m,∞ p−1
f (u) B˙ s ≤ Cu B˙ s um . ,2
(3.5)
r,2
Case 2. p < 2. (3.3) and (3.4) imply that | f (u + ) + f (u − ) − 2 f (u)| ≤ f (u)|u + + u − − 2u| + C
|u ± − u| p .
±
In the same way as leading to (3.5), we can deduce that p−1
f (u) B˙ s ≤ C(um u B˙ s + u ,2
r,2
p−1
p
p−1
≤ C(um u B˙ s + u B˙ s u B˙ 0 r,2
r,2
Collecting (3.5) and (3.6), the lemma is proved.
s
p B˙ p,2 p
m,∞
) p−1
) ≤ Cum u B˙ s . r,2
(3.6)
828
Q. Chen, C. Miao, Z. Zhang
Remark 3.1. In fact, the inequality holds for all p ∈ [1, ∞), s ∈ [0, p). But in order to make the presentation lighter, we only give the proof of the case s ∈ [0, p) ∩ [0, 2), and the other cases can be treated in the same way. Now let’s come back to the proof of Theorem 1.1. By homogeneity and scaling, it is enough to prove the inequality for j = 0. According to the definition of Besov spaces, we have p p α (|0 f | 2 )2 ∼ (3.7) = |0 f | 2 B˙ α . 2,2
Applying Lemma 3.2 to the right-hand side of (3.7) yields that for 2 ≤ p < ∞, α ∈ [0, 1], p
p
−1
|0 f | 2 B˙ α ≤ C p 0 f B2˙ 0 0 f B˙ α . 2,2
(3.8)
p,2
p,2
Since supp 0 f is localized in C, by Lemma 2.2, we infer that 0 f B˙ 0 , 0 f B˙ α ≤ C0 f p .
(3.9)
p,2
p,2
Collecting (3.7)–(3.9) implies that p
2
α (|0 f | 2 )2p ≤ C p 0 f p .
(3.10)
In order to prove the inverse inequality, we first use Proposition 3.1 to get p
p
c p f 0 p2 ≤ (| f 0 | 2 )2 ,
(3.11)
p
p
where f 0 0 f. To estimate (| f 0 | 2 )2 , we decompose (| f 0 | 2 ) into p p p p p (| f 0 | 2 ) = k (| f 0 | 2 ) + k (| f 0 | 2 ) P≥M (| f 0 | 2 ) + P<M (| f 0 | 2 ), k≥M
k<M
for a sufficiently large M which will be determined later. On the one hand, we write p
p
P≥M (| f 0 | 2 )2 = −ε 1+ε (P≥M | f 0 | 2 )2 , for a small enough ε > 0 such that 1 + ε <
p 2.
Thanks to Lemma 2.2, we get
p
p
p
−ε 1+ε (P≥M | f 0 | 2 )2 ≤ C p 2−Mε 1+ε (| f 0 | 2 )2 ≈ C p 2−Mε | f 0 | 2 B˙ 1+ε , 2,2
which together with Lemma 3.2 implies that p
p
P≥M (| f 0 | 2 )2 ≤ C p 2−Mε f 0 p2 .
(3.12)
On the other hand, using Lemma 2.2 again, we obtain p
p
P<M (| f 0 | 2 )2 = 1−α α (P<M | f 0 | 2 )2 p
≤ C p 2 M(1−α) α (| f 0 | 2 )2 ,
(3.13)
New Bernstein’s Inequality and 2D Dissipative Quasi-Geostrophic Equation
829
Combining (3.11)–(3.13) yields that p
p
p
p
c p f 0 p2 ≤ (| f 0 | 2 )2 ≤ P≥M (| f 0 | 2 )2 + P<M (| f 0 | 2 )2 p p −Mε M(1−α) α 2 2 ≤ Cp 2 f0 p + 2 (| f 0 | )2 . If we choose M such that C p 2−Mε ≤ 21 c p , we conclude that p
p
c p f 0 p2 ≤ α (| f 0 | 2 )2 .
(3.14)
This completes the proof of Theorem 1.1. Finally let us recall the following improved positivity lemma. Lemma 3.3. Suppose that s ∈ [0, 2], and f, s f ∈ L p (R2 ), p ≥ 2. Then p s 2 p−2 s |f| f f dx ≥ ( 2 | f | 2 )2 d x. 2 2 p R R Proof. The proof can be found in [18].
(3.15)
4. The proof of Theorem 1.2 In this section, we will prove Theorem 1.2. We divided it into two parts. 4.1. Global well-posedness for small initial data. Step 1. A priori estimates. Taking the operator j on both sides of (1.1), we have ∂t j θ + κ 2α j θ + u · ∇ j θ = [u, j ] · ∇θ. Multiplying by p| j θ | p−2 j θ and integrating with respect to x yield that d p j θ p + κ p 2α j θ | j θ | p−2 j θ d x + p u · ∇ j θ | j θ | p−2 j θ d x 2 2 dt R R p−2 =p [u, j ] · ∇θ | j θ | j θ d x. (4.1) R2
Since divu = 0, by integration by parts we infer that u · ∇ j θ | j θ | p−2 j θ d x = 0. R2
(4.2)
Thanks to Lemma 3.3 and Theorem 1.1, we deduce that α p 2 p 2α p−2 | j θ | 2 d x ≥ c p 22α j j θ p . (4.3) j θ | j θ | jθdx ≥ 2 p R2
R2
Summing up (4.1)–(4.3) and Hölder inequality yield that d j θ p + 2κc p 22α j j θ p ≤ C[u, j ] · ∇θ p , dt
830
Q. Chen, C. Miao, Z. Zhang
which together with Gronwall’s inequality implies that j θ p ≤ e−κc p t2
j θ0 p + Ce−κc p t2
2α j
2α j
∗ [u, j ] · ∇θ p ,
(4.4)
where the sign ∗ denotes the convolution of functions defined in R+ , in details t 2α j 2α j e−κc p (t−τ )2 f (τ )dτ. e−κc p t2 ∗ f 0
Taking the L r (0, T ) norm, 1 ≤ r ≤ ∞, T ∈ (0, ∞], and using Young’s inequality to obtain 2α j j θ L rT (L p ) ≤ e−κc p t2 L rT j θ0 p + C [u, j ] · ∇θ L 1 (L p ) . (4.5) T
Multiplying 2 jσ on both sides of (4.5), then taking q (Z) norm, we obtain θ κ −1/r θ0 B˙ σ + 2 jσ [u, j ] · ∇θ L 1 (R+ ,L p ) q (Z) , σ + 2α r L r ( B˙ p,q
)
(4.6)
p,q
where we used the fact that −κc t22α j e p and σ =
2 p
≤
L rT
1 − e−r κc p 2 r κc p 22α j
2α j T
1 r
,
for 1 ≤ r ≤ ∞,
(4.7)
+ 1 − 2α. On the other hand, it follows from Proposition 5.3 that
jσ 2 [u, j ] · ∇θ L 1 (R+ ,L p )
q (Z)
≤ Cu
2 +1−α
p L 2 ( B˙ p,q
)
θ
≤ Cθ L ∞ ( B˙ σ ) θ p,q
2 +1−α
p L 2 ( B˙ p,q
2 +1 p L 1 ( B˙ p,q )
)
,
(4.8)
where in the last inequality we have used the interpolation and the fact that u L r ( B˙ s
p,q )
= Rk θ L r ( B˙ s
p,q )
≤ Cθ L r ( B˙ s
p,q )
, for s ∈ R, (r, p, q) ∈ [1, ∞]3 , (4.9)
since j Rk θ p ≈ j Rk j θ p ≤ C j θ p for all 1 ≤ p ≤ ∞, here j = ( j−1 + j + j+1 ). Combining (4.6) and (4.8), we get θ κ −1/r θ0 B˙ σ + Cθ (4.10) 2 +1 . σ + 2α L ∞ ( B˙ σ ) θ r p L r ( B˙ p,q
)
p,q
p,q
L 1 ( B˙ p,q )
On the other hand, it follows from ([13], Corollary 2.6) that θ (t, x) p ≤ θ0 (x) p , t ≥ 0,
(4.11)
which together with (4.10) implies that θ (t) L ∞ (B σp,q ) + c1 κθ (t)
2 +1
p L 1 ( B˙ p,q )
≤ 2θ0 B σp,q + Cθ L ∞ (B σp,q ) θ
2 +1
p L 1 ( B˙ p,q )
.
(4.12)
New Bernstein’s Inequality and 2D Dissipative Quasi-Geostrophic Equation
831
Step 2. Approximation solutions and uniform estimates. Let us define the sequence {θ (n) , u (n) }n∈N0 by the following systems: ⎧ (n+1) ∂t θ + u (n) · ∇θ (n+1) + κ(−)α θ (n+1) = 0, x ∈ R2 , t > 0, ⎪ ⎪ ⎪ ⎨ (n) u = R⊥ θ (n) , ⎪ (n+1) (n+1) ⎪ (0, x) = θ (x) = j θ0 (x). θ ⎪ 0 ⎩
(4.13)
j≤n+1
Setting (θ (0) , u (0) ) = (0, 0) and solving the linear system, we can find {θ (n) , u (n) }n∈N0 for all n ∈ N0 . As in Step 1, we can deduce that θ (n+1) (t) L ∞ ( B˙ σ
p,q )
(n+1)
≤ 2θ0
+ c1 κθ (n+1) (t)
2 +1
p L 1 ( B˙ p,q )
B˙ σ + C2 (c1 κ)−1 θ (n) L ∞ ( B˙ σ
+ c1 κθ (n)
p,q )
p,q
× θ (n+1) L ∞ ( B˙ σ
p,q )
+ c1 κθ (n+1)
If we take > 0 such that θ0 B˙ σ ≤ κ, ≤ p,q
θ (n) (t) L ∞ ( B˙ σ
p,q )
In fact, assume that θ (k) L ∞ ( B˙ σ
2 +1 p L 1 ( B˙ p,q )
c1 8C2 ,
+ c1 κθ (n) (t)
p,q )
+ c1 κθ (k)
2 +1
p L 1 B˙ p,q )
.
(4.14)
then for all n, we will show 2 +1
p L 1 ( B˙ p,q )
2 +1
p L 1 ( B˙ p,q )
≤ 4θ0 B˙ σ .
(4.15)
p,q
≤ 4θ0 B˙ σ for k = 0, . . . , n. p,q
It follows from (4.14) that θ (n+1) L ∞ ( B˙ σ
p,q )
+ c1 κθ (n+1)
2 +1
p L 1 ( B˙ p,q )
≤ 2θ0 B˙ σ + C2 (c1 κ)−1 4θ0 B˙ σ p,q
p,q
≤ 2θ0 B˙ σ + p,q
(n+1) θ L ∞ ( B˙ σ
p,q )
+ c1 κθ (n+1)
1 (n+1) (n+1) θ 2 +1 , L ∞ ( B˙ σp,q ) + c1 κθ p 2 L 1 ( B˙ p,q )
2 +1 p L 1 ( B˙ p,q )
(4.16)
which implies (4.15). Summing up (4.11) and (4.15), we finally get for all n, (n) θ (n) (t) L ∞ (B σp,q ) + c1 κθ (t)
2 +1
p L 1 ( B˙ p,q )
≤ 4θ0 B σp,q .
(4.17)
Step 3. Compactness arguments and Existence. We will show that, up to a subsequence, the sequence {θ (n) } converges in D (R+ × R2 ) to a solution θ of (1.1). The proof is based on compactness arguments. First we show that ∂t θ (n) is uniformly bounded in (n+1) satisfies the equation the space L ∞ (B −2α p,q ). By (4.13), ∂t θ ∂t θ (n+1) = −∇ · (u (n) θ (n+1) ) − κ(−)α θ (n+1) .
832
Q. Chen, C. Miao, Z. Zhang
Then thanks to Proposition 5.1 with p = ∞, we get ∂t θ (n+1) L ∞ (B −2α θ (n+1) L ∞ (B 0p,q ) + u (n) L ∞ (L p ) θ (n+1) L ∞ (B σp,q ) p,q ) + θ (n+1) L ∞ (L p ) u (n) L ∞ (B σp,q ) θ (n+1) L ∞ (B σp,q ) + θ (n) L ∞ (B σp,q ) θ (n+1) L ∞ (B σp,q ) < ∞, where we have used the fact: for s > 0, B sp,q = L p ∩ B˙ sp,q , and the inclusion map B σp,q ⊂ B 0p,q . We remark that the above inequality can be obtained also by Proposition 5.2 with s = −2σ , let s1 be any number such that 0 < s1 < 2p . Now let us turn to the proof of the existence. Observe that for any χ ∈ Cc∞ (R2 ), the map: u → χ u is compact from B σp,q (R2 ) into L p (R2 ). This can be proved by noting that the map: u → χ u is compact from H ps into H ps for s > s, p < ∞, and the embedding relation − B σp,q → B σp,2 → H pσ − (by Proposition 2.1(iii)). Thus by the Lions-Aubin compactness theorem (see [22]), we can conclude that there exists a subsequence {θ (n k ) } and a function θ so that lim θ (n k ) = θ in L loc (R+ × R2 ). p
n k →+∞
Moreover, the uniform estimate (4.17) allows us to conclude that 2
+1
p L 1 (0, ∞; B˙ p,q ), θ (t, x) ∈ L ∞ (0, ∞; B σp,q ) ∩
and θ (t) L ∞ (B σp,q ) + θ (t)
2 +1
p L 1 ( B˙ p,q )
≤ 4θ0 B σp,q .
Then by a standard limit argument, we can prove that the limit function θ (t, x) satisfies Eq. (1.1) in the sense of distribution. We still have to prove θ (t, x) belongs to C(R+ ; B σp,q ). Our idea comes from [15]. We observe that ∂t j θ = −κ 2α j θ − j ∇ · (uθ ).
(4.18)
For fixed j, the right-hand side of (4.18) belongs to L ∞ (0, ∞; B σp,q ), which can be easily proved by using Lemma 2.2. Therefore, we infer that ∂t j θ ∈ L ∞ (0, ∞; B σp,q ) for fixed j, which implies that each j θ is continuous in time in B σp,q . On the other hand, note that θ L ∞ (B σp,q ) =
j∈Z
q sup 2 jσ j θ L p
1 q
< +∞,
t≥0
which implies that | j|≤n j θ converges uniformly in L ∞ (R+ ; B σp,q ) to θ (t, x). Hence, θ (t, x) ∈ C(R+ ; B σp,q ).
New Bernstein’s Inequality and 2D Dissipative Quasi-Geostrophic Equation 2
833 +1
p Step 4. Uniqueness. Assume that θ ∈ L ∞ (B σp,q ) ∩ L 1 ( B˙ p,q ) is another solution of (1.1) with the same initial data θ0 (x). Let δθ = θ − θ and δu = u − u . Then (δθ, δu) satisfy the following equations ⎧ α 2 ⎪ ⎨ ∂t δθ + u · ∇δθ + δu · ∇θ + κ(−) δθ = 0, x ∈ R , t > 0, (4.19) δu = R⊥ δθ, ⎪ ⎩ δθ (0, x) = 0.
Following the same way as a priori estimates, we can deduce that d j δθ p + 2κc p 22α j j δθ p ≤ C [u, j ] · ∇δθ p + j (δu · ∇θ ) p , dt which together with Gronwall’s inequality leads to 2α j (4.20) j δθ p ≤ Ce−κc p t2 ∗ [u, j ] · ∇δθ p + j (δu · ∇θ ) p . Choose a positive number η such that and interpolation, we get δu · ∇θ δθ δθ
2 −η
p L 1T ( B˙ p,q )
2 −η+ 2α p p p L T ( B˙ p,q 1 p 2 −η
˙p L∞ T ( B p,q )
)
2α p
<η<
δu θ
δθ
2 p.
Thanks to Proposition 5.2, (4.9),
2 −η+ 2α p
p p L T ( B˙ p,q
2 +1− 2α p p p L T ( B˙ p,q
1 p 2 −η+2α
p L 1T ( B˙ p,q
)
)
θ
2 +1− 2α p
p p L T ( B˙ p,q
)
)
θ
2 +1− 2α p
p p L T ( B˙ p,q
)
.
(4.21)
On the other hand, thanks to Proposition 5.3, (4.9) we have [u, j ] · ∇δθ L 1 (L p ) c j 2
− j ( 2p −η)
T
cj2
− j ( 2p −η)
u
2 +1−α
p L 2T ( B˙ p,q
)
δθ
θ
2 +1−α
p L 2T ( B˙ p,q
1 2 2 −η
˙p L∞ T ( B p,q )
δθ
)
δθ
2 −η+α
p L 2T ( B˙ p,q
1 2 2 −η+2α
p L 1T ( B˙ p,q
)
,
)
(4.22)
where c j q (Z) ≤ 1. Taking L ∞ (L 1 , respectively) norm on time, and using Young’s j ( 2 −η)
inequality, then multiplying 2 p then taking q (Z) norm, we have Z (T ) δθ
2 −η
˙p L∞ T ( B p,q )
(2
j ( 2p −η+2α)
+ δθ
, respectively) on both sides of (4.20), 2 −η+2α
p L 1T ( B˙ p,q
)
j ( 2 −η) 2 p [u, j ]∇δθ L 1 (L p ) q (Z) + δu · ∇θ θ
T
2 +1− 2α p p p L T ( B˙ p,q
)
+ θ
2 +1−α p L 2T ( B˙ p,q )
Z (T ),
2 −η
p L 1T ( B˙ p,q )
(4.23)
where we have used (4.21) and (4.22) in the last inequality. Now it is clear that two terms in the bracket of the right-hand side of (4.23) tend to 0 as T goes to 0. Therefore, if T has been chosen small enough, then it follows from (4.23) that Z ≡ 0 on [0, T ] which implies that δθ ≡ 0. Then by a standard continuous argument, we can show that δθ (t, x) = 0 in [0, +∞) × R2 , i.e. θ (t, x) = θ (t, x). This completes the proof of global well-posedness.
834
Q. Chen, C. Miao, Z. Zhang
4.2. Local well-posedness for large initial data. Now we prove the local well-posedness for the large initial data. As the existence result will be essentially followed from the a priori estimate. For simplicity, we only present the a priori estimate of the solution θ (t, x). j ( 2 +1−α) Let us return to (4.5). Taking r = 2 in (4.5), multiplying 2 p on both sides of (4.5), then taking q (Z) norm and applying Proposition 5.3 and (4.9), we get θ
2 +1−α
p L 2T ( B˙ p,q
)
≤ C3 κ
− 21
E j (T ) 21 2 jσ j θ0 p
≤ C3 κ
− 21
E j (T ) 21 2 jσ j θ0 p
+ u q (Z) q (Z)
+ θ
2 +1−α p L 2T ( B˙ p,q )
2
2 +1−α
p L 2T ( B˙ p,q
θ
)
2 +1−α p L 2T ( B˙ p,q )
,
(4.24)
where E j (T ) 1 − e−κc p 2
2α j T
.
Set 1 q κ q q . T0 sup T > 0; E j (T ) 2 2 jσ q j θ0 p ≤ 2C32 j∈Z Then the inequality (4.24) implies that there holds for T ∈ [0, T0 ], θ (t)
2 +1−α p L 2T ( B˙ p,q )
1 ≤ 2 E j (T ) 2 2 jσ j θ0 p q (Z) ,
which together with (4.6) and (4.8) leads to θ (t) L ∞ ( B˙ σ T
p,q )
+ c1 κθ (t)
2 +1
p L 1T ( B˙ p,q )
≤ 2θ0 B˙ σ + Cθ 2 p,q
2 +1−α
p L 2T ( B˙ p,q
)
≤ Cθ0 B˙ σ . p,q
Combining with (4.11), we obtain for T ∈ [0, T0 ], θ (t) L ∞ (B σp,q ) + c1 κθ (t) T
2 +1
p L 1T ( B˙ p,q )
This completes the proof of local well-posedness.
≤ Cθ0 B σp,q .
Acknowledgements. We would like to thank Professors H. Smith and T. Tao so much for their helpful discussion and suggestions. The authors are also deeply grateful to the referees for their valuable advice. Q. Chen and C. Miao were partly supported by the NSF of China (No.10571016), and Z. Zhang was partly supported by NSF of China (No.10601002).
New Bernstein’s Inequality and 2D Dissipative Quasi-Geostrophic Equation
835
5. Appendix Firstly, we recall the paradifferential calculus which enables us to define a generalized product between distributions, which is continuous in many functional spaces where the usual product does not make sense (see [2]). The paraproduct between u and v is defined by Tu v S j−1 u j v. j∈Z
We then have the following formal decomposition: uv = Tu v + Tv u + R(u, v), with R(u, v) =
(5.1)
j u j v and j = j−1 + j + j+1 .
j∈Z
The decomposition (5.1) is called the Bony’s paraproduct decomposition. Now we state some results about the product estimates in Besov spaces. Proposition 5.1. Let s > − dp , 2 ≤ p ≤ ∞, 1 ≤ q ≤ ∞. Then uv B sp,q u p v
d +s p
+ v p u
B p,q
d +s p
.
(5.2)
B p,q
Proof. Using Lemma 2.2, we have S0 (uv) p ≤ S0 (uv) p ≤ Cu p v p u p v 2
d +s p
.
(5.3)
B p,q
Then using the Bony’s paraproduct decomposition and the property of quasiorthogonality (2.1), for fixed j ≥ 0, we have j (Sk−1 uk v) + j (Sk−1 vk u) + j (k u k v) j (uv) = |k− j|≤4
|k− j|≤4
k≥ j−2
I + I I + I I I.
(5.4)
We shall estimate the above three terms separately. Using Young’s inequality and Lemma 2.2, we get d
j
j (Sk−1 uk v) p 2 p Sk−1 u p k v p . Thus we have 2s j I p u p
2
( j−k)( dp +s) k( dp +s)
2
k v p c j u p v
d +s p
,
(5.5)
B p,q
|k− j|≤4
where the q (Z) norm of c j is less than or equal to 1. Similarly to I I , we have 2s j I I p c j v p u
d +s p
B p,q
.
(5.6)
836
Q. Chen, C. Miao, Z. Zhang
Now we turn to estimate I I I . From Lemma 2.2, Young’s inequality, and Hölder inequality we have d
d
j j j (k u k v) p 2 p j (k u k v) p 2 p k u p k v p . 2
So, we get, 2s j I I I p u p
2
( j−k)( dp +s) k( dp +s)
2
k v p c j u p v
k≥ j−2
where we have used the fact s + desired inequality (5.2).
d p
+
1 r2
d p,2
≤ p ≤ ∞, 1 ≤ q ≤ ∞, r1 =
≤ 1, and u be a solenoidal vector field. Then u · ∇v L rt ( B˙ s
p,q )
If s1 =
d p
,
(5.7)
> 0. Summing up (5.3), (5.5)–(5.7), we obtain the
Proposition 5.2. Let s > − dp −1, s < s1 < 1 r1
d +s p
B p,q
r s1 ∇v u L 1 ( B˙ p,q ) t
s+ d −s1
r p L t 2 ( B˙ p,q
)
1 r1
.
+ r12 =
(5.8)
or s1 = s, q has to be equal to 1.
Proof. Throughout the proof, the summation convention over repeated indices i ∈ [1, d] is used. Similarly to the proof of Proposition 5.1, we will estimate separately each part of the Bony’s paraproduct decomposition of u i ∂i v. By Lemma 2.2, we have j (Sk−1 u i k ∂i v) L rt (L p ) Sk−1 u L r1 (L ∞ ) k ∇v L r2 (L p ) t t ( d −s )k s k 2 p 1 2 1 k u L r1 (L p ) k ∇v L r2 (L p ) t
k ≤k−2
2
( dp −s1 )k
r s1 k ∇v r2 u L 1 ( B˙ p,q ) L (L p ) , t
where the fact s1 <
d p
t
t
has been used in the last inequality. Hence, we get
2s j j (Sk−1 u i ∂i k v) L r (L p ) u L r1 ( B˙ sp,q 1 ) ×
2( j−k)s 2
(s+ dp −s1 )k
r s1 v c j u L 1 ( B˙ p,q )
k ∇v L r2 (L p ) t
| j−k|≤4 t
t
t
| j−k|≤4
s+ d +1−s1
r p L t 2 ( B˙ p,q
)
,
(5.9)
where c j q (Z) ≤ 1. Since divu = 0 and p ≥ 2, Lemma 2.2 applied yields that j (k u k ∂i v) L rt (L p ) 2
j ( dp +1)
j (k u i k v)
p
L rt (L 2 )
.
New Bernstein’s Inequality and 2D Dissipative Quasi-Geostrophic Equation
Thus by Hölder inequality and
837
+ 1 + s > 0, we have
d p
j ( d +1+s) 2s j j (k u i k ∂i v) L r (L p ) 2 p k u L r1 (L p ) k v L r2 (L p ) t
t
k≥ j−2
v
s+ dp +1−s1 r L t 2 ( B˙ p,q )
2
( j−k)( dp +1+s) ks1
2
k u L r1 (L p ) t
k≥ j−2
r s1 v c j u L 1 ( B˙ p,q ) t
t
k≥ j−2
s+ d +1−s1
r p L t 2 ( B˙ p,q
)
.
(5.10)
On the other hand, due to s1 > s, we get ( d +1+s−s )k 1 j (k u i Sk−1 ∂i k v) L rt (L p ) 2 p 2(s1 −s)k k v L r2 (L p ) k u L r1 (L p ) k ≤k−2 (s1 −s)k
2
Then we have
t
v
s+ d +1−s1
r p L t 2 ( B˙ p,q
)
t
k u L r1 (L p ) . t
2s j j (k u i Sk−1 ∂i k v) L r (L p ) t
| j−k|≤4
v
s+ dp +1−s1 r L t 2 ( B˙ p,q )
c j v
| j−k|≤4
s+ d +1−s1
r p L t 2 ( B˙ p,q
)
2s( j−k) 2s1 k k u L r1 (L p ) t
r s1 . u L 1 ( B˙ p,q )
(5.11)
t
Summing up (5.9)–(5.11), the desired inequality (5.8) is proved. Finally we give the commutator estimate. Proposition 5.3. Let 1 ≤ p, q ≤ ∞, r1 = r11 + solenoidal vector field. Assume in addition that
1 r2
≤ 1, ρ < 1, γ > −1 and u be a
2 d > 0 and ρ + > 0. ρ − γ + d min 1, p p Then the following inequality holds: [u, j ] · ∇v L rt (L p ) c j 2
− j ( dp +ρ−1−γ )
∇u
d +ρ−1
r p L t 1 ( B˙ p,q
)
∇v
d −γ −1
r p L t 2 ( B˙ p,q
)
,(5.12)
where c j denotes a positive series with c j q (Z) ≤ 1. In the above, we denote u i j ∂i v − j (u i ∂i v). [u, j ] · ∇v = 1≤i≤d
If ρ = 1, ∇u ∇v
d −γ −1
r p L t 2 ( B˙ p,q
d +ρ−1
r p L t 1 ( B˙ p,q
)
)
has to be replaced by ∇u
has to be replaced by ∇v
d −γ −1
r p L t 2 ( B˙ p,q
d +ρ−1
r p L t 1 ( B˙ p,q r
r
)∩L t 1 (L ∞ )
)∩L t 1 (L ∞ )
. If γ = −1,
.
Proof. The proof is a straightforward adaptation of Lemma A.1 in [16] which is a version of the commutator estimate in Besov space.
838
Q. Chen, C. Miao, Z. Zhang
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Commun. Math. Phys. 271, 839–851 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0188-4
Communications in
Mathematical Physics
Vafa-Witten Estimates for Compact Symmetric Spaces Sebastian Goette Mathematisches Institut, Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany. E-mail: [email protected] Received: 13 July 2006 / Accepted: 6 September 2006 Published online: 13 February 2007 – © Springer-Verlag 2007
Abstract: We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rk G − rk H ≤ 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement. Herzlich gave an optimal upper bound for the lowest eigenvalue of the Dirac operator on spheres with arbitrary Riemannian metrics in [9] using a method developed by Vafa and Witten in [14]. More precisely, he proved that for every metric g¯ on S n that is pointwise larger than the round metric g, the first eigenvalue λ1 ( D¯ 2 ) of the Dirac operator with respect to g¯ is not larger than the first Dirac eigenvalue λ1 (D 2 ) of the round sphere. Herzlich asked if there are other Riemannian manifolds with optimal Vafa-Witten bounds, in particular if the Fubini-Study metric on CP 2m−1 has this property. In the present note we give positive answers to both questions by generalising Herzlich’s results to symmetric spaces G/H of compact type, where rk G − rk H ≤ 1. In particular, we improve a recent estimate by Davaux and Min-Oo for complex projective spaces in [4], see Example 6.2 below. Theorem 0.1. Let M = G/H be a simply connected symmetric space of compact type with rk G − rk H ≤ 1 and assume that M is G-spin. Let g be a symmetric metric, and let D denote the corresponding Dirac operator on M. If g¯ is another metric with g¯ ≥ g on T M and D¯ is the corresponding Dirac operator, then λ1 ( D¯ 2 ) ≤ λ1 (D 2 ). In the case of equality, we have g¯ = g. For an arbitrary Riemannian metric g¯ such that c2 g¯ ≥ g for some suitable positive constant c2 , the theorem implies λ1 ( D¯ 2 ) ≤ c2 λ1 (D 2 ). Supported in part by DFG special programme “Global Differential Geometry”
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We combine the methods of [9] and [4] with related estimates in [7]. In particular, we compare D¯ to an operator D¯ 1 with nonvanishing kernel acting on the same sections 2 as D¯ 0 = D¯ ⊗ idRk . We use Parthasarathy’s formula to compute λ1 (D ), and we exhibit ¯ ¯ ¯ a similar formula to estimate D1 − D0 . Both formulas give the same value for g = g. To prove that D¯ 1 has a kernel, we use the invariance of the Fredholm index if rk H = rk G. If rk H = rk G − 1 we use the invariance of the mod-2-index as in [7]. Unfortunately, both approaches fail if rk G −rk H ≥ 2. Note that in [9], a spectral flow argument was used instead in the case rk H = rk G − 1. In [2], Baum applied the Vafa-Witten approach to Lipschitz maps f of high degree from a closed Riemannian spin manifold of dimension 2n to S 2n . We extend her result ˆ to Lipschitz maps of high A-degree from higher dimensional closed Riemannian spin manifolds to S 2n . Recall that if N n and M m are closed oriented manifolds, [N ] is the ˆ fundamental class of N and ω is a generator of H m (M; Z), then the A-degree is defined as ˆ N ) f ∗ ω [N ]. deg Aˆ f = A(T If n = m, then deg Aˆ is the usual degree. Let D N denote the untwisted Dirac operator on N , and let 0 ≤ λ1 (D 2N ) ≤ λ2 (D 2N ) ≤ . . . denote the eigenvalues of D 2N , where each eigenvalue is repeated according to its multiplicity. Theorem 0.2. Let N be a closed Riemannian spin manifold and let k ∈ N. Then there is no Lipschitz map N → S 2m of Lipschitz constant 1 with deg ˆ f > 2m−1 (k − 1) A unless λk (D 2N ) ≤ λ1 (D M ). If we replace the target manifold by a symmetric space G/H with rk G = rk H , then we can prove similar theorems where the degree condition on f is replaced by ˆ N ) f ∗ α > C(k − 1), [N ], A(T with α ∈ H ∗ (M; Z) and C > 0 depending only on M. Alternatively, there is no 1-Lipschitz map N → M with deg ˆ f > C(k − 1) A for some constant C depending on M, unless λk (D 2N ) ≤ c for some c depending only on M, with c > λ1 (D M ) in general. This will be discussed in Sect. 7. In the case rk G − rk H = 1 we need a K -theoretic condition on f instead of the cohomological ˆ A-degree condition. We may thus ask if even-dimensional spheres are the only manifolds that admit optimal Vafa-Witten bounds for Lipschitz spin maps of sufficiently large ˆ A-degree. Remark 0.1. The actual estimate and the index theoretic considerations involved in the proof of Theorem 0.1 are very similar to those used for the scalar curvature comparison result in [7]. Nevertheless, we need a stronger metric condition (g¯ ≥ g on T M, not just on 2 T M). This is due to the fact that the Vafa-Witten estimate is related to Gromov’s K -length, whereas scalar curvature comparison is related to K -area, see [8].
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Remark 0.2. Note that Theorem 0.1 still holds if we replace D¯ by D¯ = D¯ + A, where A is a symmetric endomorphism of the spinor bundle. This is because in the proof, we can ¯ ¯ ¯ ¯ then replace Di by Di = Di + A⊗ id for i = 0, 1. Then D1 still has a kernel, and of ¯ ¯ ¯ ¯ course D0 − D1 = D1 − D0 . Remark 0.3. Remark 0.2 in particular implies that on a symmetric space, λ1 (D + A) ≤ λ1 (D) for all symmetric endomorphisms A of the spinor bundle. In Sect. 8 we will exhibit an isotropy irreducible quotient G/H = SO(5)/SO(3) of compact Lie groups with a normal metric, for which the generalised estimate above does not hold. This shows in particular that the methods of the present paper do not readily generalise to normal homogeneous spaces. 1. The Smallest Dirac Eigenvalue of a Symmetric Metric Let M = G/H be a symmetric quotient of compact Lie groups of equal rank with Lie algebras h ⊂ g. We fix an Ad-invariant metric g on g and let p = h⊥ . The tangent bundle of M can be written as T M = G × H p, where H acts on p by the restriction of the adjoint action Ad G . The scalar product g induces a symmetric Riemannian metric on M that will also be denoted by g. Note that if M is symmetric, then the Levi-Civita connection on T M is precisely the reductive connection on G × H p. Let be a spinor module for the Clifford algebra of p. If we assume that M is G-spin, then the H -representation of H on p induces an action σ : H → End. The natural metric on is σ -invariant, so we obtain a G-equivariant metric g on S M. Equipped with this metric and the reductive connection, the G-equivariant vector bundle S M = G ×H → M can be identified with the spinor bundle of M, and the G-equivariant Dirac operator D acts as an essentially selfadjoint operator on L 2 (M; S M). Let Gˆ denote the set of equivalence classes of irreducible G-representations. We ˆ By Frobenius reciprocity and the Peter-Weyl write γ : G → EndV γ for all γ ∈ G. theorem, the L 2 -sections of S M can be decomposed G-equivariantly into a Hilbert sum V γ ⊗ Hom H (V γ , ). (1) L 2 (M; S M) = γ ∈Gˆ
The Dirac operator preserves this decomposition, and we have D|V γ ⊗Hom H (V γ ,) = id V γ ⊗γ D with γ
D=
n
γe∗i ⊗ ci
∈
End(Hom H (V γ , )).
(2)
i=1
Here e1 , . . . , en is a g-orthonormal base of p and ci denotes Clifford multiplication by ei .
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Let cG denote the Casimir operator of the G-action γ with respect to the metric g, which acts as a scalar on V γ . If M is symmetric, then the Casimir operator cσH of σ also acts as a scalar, even though σ is in general not irreducible. By Parthasarathy’s formula [12], γ
γ∗
D 2 = cG + cσH .
Proposition 1.1. The smallest eigenvalue of D 2 is given by γ∗ λ1 (D 2 ) = min cG + cσH γ ∈ Gˆ with Hom H (V γ , ) = 0 . A more explicit formula for the first eigenvalue in the case rk G = rk H has recently been given in [11]. 2. The Vafa-Witten Estimate Let M = G/H and g as before and consider an arbitrary Riemannian metric g¯ on M ¯ Let λ1 ( D¯ 2 ) denote the with g¯ > g. The corresponding Dirac operator will be denoted D. 2 smallest eigenvalue of D¯ . As mentioned in the introduction, we will estimate λ1 ( D¯ 2 ) by comparing the related operator D¯ 0 = D¯ ⊗ idC N to an operator D¯ 1 acting on the same space of sections. Assume that we are given vector bundles W ⊂ V = M × C N → M such that the twisted Dirac operator on S M ⊗ W has nonvanishing index; these bundles will be constructed in steps below and in Sect. 4 and 5. Let ∇ 0 be the trivial connection on V , and let ∇ 1 be another connection for which W ⊂ V is a parallel subbundle. Let D¯ 0 , D¯ 1 denote the corresponding twisted Dirac operators on S M ⊗ C N for the metric g. ¯ ¯ whereas D¯ 1 has nontrivial kernel. By Then D¯ 0 is just the direct sum of N copies of D, a Rayleigh quotient argument, thus ( D¯ 1 − D¯ 0 )s 2 2 ¯ 2 2 L ¯ ¯ ¯ 0 )2p D λ1 ( D ) = λ1 ( D 0 ) ≤ ≤ sup − D (3) ( 1 op s2L 2 p∈M for some 0 = s ∈ ker( D¯ 1 ). Here · L 2 denotes the L 2 -norm of sections, whereas · op ¯ ¯ denotes the pointwise operator norm of an endomorphism. The operator D1 − D0 is of ¯ 2 order zero, so ( D1 − D¯ 0 ) p is well-defined and can be estimated pointwise on M. op
In our current situation, let γ ∈ Gˆ be a G-representation such that Hom H (V γ , ) = 0
γ∗
λ1 (D 2 ) = cG + cσH ,
and
∗
cf. Proposition 1.1. Let γ ∗ denote the dual representation on V γ = (V γ )∗ and let ∗
V = G × H (V γ | H ) denote the corresponding vector bundle over G, then V is trivialized by the map V → M × Vγ
∗
with
[g, v] → (g H, γg∗ v).
To a map v : M → V γ corresponds the section g H → [g, (γ ∗ )−1 g v(g)] ∈ Vg H .
(4)
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Now let ∇ 0 denote the trivial connection and let ∇ 1 denote the reductive connection on V . Then D 0 , D 1 are the corresponding Dirac operators on S M ⊗ V . If we apply the reductive connection ∇ 1 on V to a section that is constant in the given trivialisation, then
1 ∗ ∗ −1 ∇[g,X ] v = g, γ X (γ )g v = (g H, γ∗ Ad g X v), where X ∈ p = h⊥ and [g, X ] =
∂ −t X ge H ∈ Tg H M. ∂t
Thus we can identify T p M with p and choose a g-orthonormal frame f 1 , . . . , f m of h. We recall that the action σ∗ of h on S M can be described in terms of Clifford multiplication and Lie brackets by n 1 [ei , e j ], f k ci c j . 4
σ∗ fk =
i, j=1
Let e¯1 , . . . , e¯n be a g-orthonormal ¯ base of T p M. We may assume that there exist 0 < µ1 , . . . , µn ≤ 1 such that e¯i = µi ei for all i ∈ {1, . . . , n}, where e1 , . . . , en is an orthonormal base with respect to g. We have to compute the operator norm of the operator C = D¯ 1 − D¯ 0 =
n
ci γe¯∗i =
i=1
n
µi ci γe∗i ,
i=1
which is the difference of two Dirac operators on S M ⊗ V with respect to the metric g. ¯ We note that this formula looks similar to (2) above. This is a special feature of symmetric spaces, which does not even generalise to normal homogeneous spaces, see Sect. 8 below. Because C is selfadjoint, we have Cv2 = C 2 v, v, and it suffices to estimate the eigenvalues of C 2 . We follow the proof of Parthasarathy’s formula [12]. Using [p, p] ⊂ h we compute C2 = −
n
µi2 (γe∗i )2 +
i=1
=−
i, j=1
2 m n 1 2 ∗ 2 ∗ γ fk + µi (γei ) + µi µ j [ei , e j ], f k ci c j 4
n i=1
−
m
(γ ∗fk )2
k=1 γ∗
n 1 µi µ j ci c j γ[e∗ i ,e j ] 2
≤ cG −
1 16
k=1 m
1 − 16
n i, j,k,l=1
k=1
i, j=1
2
µi µ j [ei , e j ], f k ci c j
i, j
µi µ j µk µl [ei , e j ], [ek , el ] ci c j ck cl .
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A term similar to the last one on the right-hand side has already been estimated in [7], Eq. (1.11). Because [p, p] ⊂ h, we have −
1 16
n
µi µ j µk µl [ei , e j ], [ek , el ] ci c j ck cl =
i, j,k,l=1
≤
1 8
n i, j=1
[ei , e j ]2 = −
m n
n 2 1 2 2 µi µ j [ei , e j ] 8 i, j=1
[ei , e j ], f k ci c j
2
= cσH . (5)
k=1 i, j=1
Combining the calculations above, we have the estimate ∗
γ C 2 = ( D¯ 1 − D¯ 0 )2 ≤ cG + cσH = λ1 (D 2 ).
(6)
3. Rigidity We will now give a short proof due to M. Listing that the equality λ1 ( D¯ 2 ) = λ1 (D 2 ) in Theorem 0.1 implies that g¯ = g. Let 0 = s ∈ ker( D¯ 1 ) as in (3). If λ1 ( D¯ 2 ) = λ1 (D 2 ), we conclude that ( D¯ 1 − D¯ 0 )s 2 2 = λ1 (D 2 ) s2 2 . L L by (3) and (6). This implies in particular that ( D¯ 1 − D¯ 0 )2p = λ1 (D 2 ) holds for op
all p ∈ supp(s). By [1], we know that s is nonzero on a dense open subset of M, so we have ¯ (7) ( D1 − D¯ 0 )2p = λ1 (D 2 ) op
for all p ∈ M, and we must have equality in (5). Since M is of compact type, g has trivial center. In particular, for each 1 ≤ i ≤ n there exists 1 ≤ j ≤ n such that [ei , e j ] = 0. Thus (7) implies by (5) that µ1 = · · · = µn = 1. The rigidity statement follows once we have shown that ker( D¯ 1 ) = 0. 4. The Equal Rank Case We prove Theorem 0.1 for rk G = rk H . It remains to show that the operator D¯ 1 has a kernel. Proposition 4.1 ([12, 5]). Let M = G/K be a symmetric space with rk G = rk K + k. Then the complex spinor bundle is locally induced by a representation σ of the Lie algebra k of K , which splits as k q σ =2 2 σi , i=1
where σ1 , . . . , σq are certain pairwise non-isomorphic irreducible complex representations of k.
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By our choice of γ in (4), we have Hom H (V γ , ) = 0. By Schur’s lemma, the H -representations γ | H and σ contain a common irreducible subrepresentation, say σ1 acting on 1 . Consider the vector bundle W = G × H 1∗ , then W is a parallel subbundle of V with respect to the reductive connection ∇ 1 . Let D1 be the Dirac operator acting on S M ⊗ W . As above, we now have L 2 (M; S M ⊗ W ) = V γ ⊗ Hom H (V γ , ⊗ 1∗ ). γ ∈Gˆ
Again by Parthasarathy’s formula ([12, 5]), the operator D12 acts on V γ , ⊗ 1 as γ
γ
γ
D12 = cG + cσH − cσH = cG .
γ
Because cG = 0 iff γ is the trivial representation, the kernel of D12 is precisely ker D1 = ker D12 = Hom H (C, ⊗ 1∗ ) ∼ = Hom H (1 , ) = Hom H (1 , 1 ) by Schur’s Lemma. In particular dim ker D1 = 1, whence ind D1 = ±1 = 0. To complete the proof of Theorem 0.1 in this case, we note that the Dirac operator on S M ⊗ W has nonzero index, and hence D¯ 1 has a kernel. The claim now follows from (3) and (6). 5. Mod-2-Indices and the Case rk H= rk G − 1 We now prove Theorem 0.1 for rk G = rk H + 1. We proceed similar as in [7], Sect.2.c. Assume first that n = dim M ≡ 1 mod 8. In this case, there exists a real vector bundle SR M such that S M = SR M ⊗R C, and the even part of the real Clifford algebra still acts on SR M. The bundle SR M is induced by a real representation σR of H . Let σR,i ∗ . denote its irreducible components, such that σi = σR,i ⊗R C, and let WR = G × H R ,1 Let ωR = c1 · · · cn denote the real Clifford volume element, then ωR is parallel, anti-selfadjoint, and commutes with D. The operator DR,1 = ωR D1 is anti-selfadjoint and has coefficients in the even part of the real Clifford algebra, so it acts on SR M. By the same reasoning as above, ∗ ker(ωR DR,1 ) = HomR,H (R, R ⊗R R ,1 ) = Hom R,H (R,1 , R,1 )
is one-dimensional. With respect to the metric g, ¯ we similarly construct the operator D¯ R,1 = ω¯ R D¯ 1 act¯ ing on SR M ⊗R VR . Its restriction to S R M ⊗R WR can be deformed into DR,1 through a family of elliptic, formally anti-selfadjoint operators. Since the parity of the dimension of the kernel is preserved by such a deformation (see [10]), the operator ω¯ R D¯ R,1 has
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nontrivial kernel. We regard V → M as a real vector bundle by forgetting the complex structure, then WR is again a parallel subbundle of V with respect to ∇ 1 , so D¯ 1 has a nontrivial kernel, too. Once again, our theorem follows from (3) and (6). Finally, if rk G − rk H = 1 and dim M ≡ 1 mod 8, we consider N = M × S 2m with dim N ≡ 1 mod 8. We fix a round metric g0 on S 2m and regard N with the metrics g ⊕ g0 and g¯ ⊕ g0 . Then our main theorem holds for N , and the result for M follows because λ1 ( D¯ 2M ) = λ1 ( D¯ 2N ) − λ1 (D S2 2m ) ≤ λ1 (D 2N ) − λ1 (D S2 2m ) = λ1 (D 2N ). 6. Examples We consider spheres and projective spaces. Example 6.1. For even-dimensional spheres S 2m = Spin(2m + 1)/Spin(2m), the restriction of the 2m -dimensional spinor representation σ of Spin(2m + 1) to H splits into two irreducible representations σ |Spin(2m) = σ + ⊕ σ − . In [9], the bundles V = S M = G ×H
⊃
W = S+ M = G ×H +
were used to prove the optimality of the Vafa-Witten estimate. Our method is a direct generalisation of this approach to other symmetric spaces. Example 6.2. The complex projective space CP n is spin iff n is odd, so we regard the odd-dimensional projective space M = CP 2m−1 = G/H with G = SU(2m) and H = U(2m − 1). The Dirac spectrum has been computed in [3] and [13]. To summarize, the spinor bundle splits as SM =
2m−1
0,q T ∗ M ⊗ τ m ,
q=0
where 0,q T ∗ M denotes the bundle of anti-holomorphic differential forms of degree q, and τ denotes the tautological bundle. Note that τ m is a square root of the canonical bundle. The smallest eigenvalue of D 2 on 0,q T ∗ M ⊗ τ m with respect to the Fubini-Study metric is given by
2 8m 2 − 4m(q + 1) for q < m, and λ1 D |0,q T ∗ M⊗τ m = (8) 8m 2 − 4m(2m − q) for q ≥ m. Thus the lowest eigenvalue is attained in the middle degrees q = m − 1, m, and is given by 4m 2 = (n + 1)2 . In contrast, Davaux and Min-Oo used the bundle τ m for q = 0 in [4], which explains their larger upper bound 8m 2 − 4m = 2n(n + 1). Let us give a geometric description of the bundles 0,q T ∗ M ⊗ τ m for q = m − 1, m. We have T CP 2m−1 = [x, v] ∈ S 2m−1 ⊗ C2m x = 0, x ⊥ v } with [z, v] = (zx, zv) z ∈ S 1 .
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For α ∈ 0,m T ∗ M, we note that α(zx, zv1 , . . . , zvm−1 ) = z¯ m α(x, v1 , . . . , vm−1 ) and α(zv1 , . . . , zvm ) = z¯ m α(v1 , . . . , vm ). Thus inserting representatives of tangent vectors into m-forms gives a natural trivialisation = m−1 ⊕ m : CP 2m−1 × 0,m C2m −→ (0,m−1 T ∗ M ⊕ 0,m T ∗ M) ⊗ τ m . 7. Lipschitz Maps of High Â-Degree We give a proof of Theorem 0.2. Let N be a closed Riemannian spin manifold and f : N → M a Lipschitz map of Lipschitz constant 1. Then f can be approximated by smooth (1 + ε)-Lipschitz maps for all ε > 0. We may therefore assume that f is smooth. As in [2, 9], consider the spinor bundles ± → S 2n equipped with the Dirac connection ∇ 1 with respect to the round metric g. Let ∇ 0 denote the trivial connection on + ⊕ − ∼ = S 2n ⊗ , where is the spinor module of R2n+1 . The Chern character of the spinor bundles is given by ch( ± ) = 2m−1 ± ω, where ω ∈ H 2m (S 2m ; Z) is a generator. Let D N ,1,± be the Dirac operator on N twisted by f ∗ ± , and let D N ,1 denote their direct sum. By the Atiyah-Singer index theorem, we have ˆ N ) f ∗ ch( ± ) [N ] = 2m−1 A(T ˆ N )[N ] ± deg ˆ f. ind(D N ,1,± ) = A(T A By combining both bundles, it is now easy to see that ˆ N )[N ], 2 deg ˆ f > 2m (k − 1) (9) dim ker(D N ,1 ) ≥ max 2m A(T A if deg Aˆ f > 2m−1 (k − 1). Let p ∈ N , then there exists an orthonormal frame e¯1 , . . . , e¯n of T p N and an orthonormal frame e1 , . . . , e2m of T f ( p) S 2m such that
µk ek if k ≤ 2m, and d p f (e¯k ) = 0 otherwise. If f is (1 + ε) Lipschitz, then µ1 , . . . , µ2m ≤ 1 + ε. Let D N ,0 denote the Dirac operator on N twisted by f ∗ , but with respect to the trivial connection ∇ 0 . Then D N ,1 − D N ,0 =
2m
c¯i γ∗µi ei .
i=1
Now a similar computation as in Sect. 2 gives D N ,1 − D N ,0 2 ≤ (1 + ε) (cγ + cσ ) = (1 + ε) λ1 (D 2 ). H G
(10)
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Combining (9) and (10) as before, we conclude at least 2m (k −1) eigenvalues of D N ,0 are not larger than (1 + ε) λ1 (D 2 ). Because D N ,0 consists of 2m copies of the Dirac operator D N , we conclude that at least k eigenvalues of D 2N (counted with multiplicities) are not larger than (1 + ε) λ1 (D 2 ). Since we can choose ε > 0 arbitrarily small, Theorem 2 is proved. We remarked in the Introduction that Theorem 0.2 does not hold unchanged for other symmetric spaces. Regard M = CP 2m−1 as in Example 6.2. By (8), the VafaWitten will be optimal only if we choose one of the bundles W = 0,m−1 T ∗ M ⊗ τ m or W ∗ = 0,m T ∗ M ⊗ τ m as twist bundle. To determine the Chern characters of these bundles, recall that T CP 2m−1 ⊕ C ∼ = 2m τ −1 , and thus in K -theory, [0,q T ∗ M] =
q q 2m [τ −i ]. (−1)q−i [i (2m τ −1 )] = (−1)q−i i i=0
i=0
We know that a = c1 (τ ) is a generator of H 2 (CP 2m−1 ), and by the above, we find ch(W ) =
m−1
(−1)
m−1−i
i=0
2m e(m−i)a . i
Already for CP 3 , the explicit classes are ch(W ) = 3 + 2a −
2 3 a 3
and
ch(W ∗ ) = 3 − 2a +
2 3 a . 3
On the other hand, we have seen that 2m W ⊕ W∗ ∼ =C m . Proceeding as above, we can prove the following result. Proposition 7.1. Let N be a closed Riemannian spin manifold and let k ∈ N. Then there is no Lipschitz map N → CP 2m−1 of Lipschitz constant 1 with m−1 (m−i)a 2m − 1 i 2m A(T ˆ N) f ∗ (k − 1), e (−1) −1 > m−1 i i=0
where a is a generator of H 2 (CP 2m−1 ), unless λk (D 2N ) ≤ λ1 (D 2M ).
Vafa-Witten Estimates for Compact Symmetric Spaces
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8. A Homogeneous Counterexample We now explain Remark 0.3. In general, the spectrum of a Dirac operator on a homogeneous space is hard to compute. On the Berger space M = SO(5)/SO(3) however, most of the relevant geometric structure can be described using octonions, see [6]. We will see that a straightforward adaptation of the arguments of Sects. 1–5 to the Berger space is not possible. We embed H = SO(3) in G = SO(5) via the irreducible SO(3)-representation of dimension 5. On g = so(5), we take the scalar product A, B = − 21 tr(AB). As above, let p denote the orthogonal complement of h = so(3). Then dim p = 7, and the isotropy action of SO(3) on p factors as SO(3) −→ G2 −→ Spin(7) −→ SO(7).
(11)
Let [·, ·]p denote the projection of the Lie bracket in g. Let I ⊂ O denote the imaginary octonions, let ∗ denote octonion multiplication, and let ∗I denote octonion multiplication followed by the projection onto I. Lemma 8.1 ([6]). With a suitable isometric, G2 -equivariant identification of p with the imaginary octonions, one has 1 [v, w]p = √ v ∗I w 5
for all
v, w ∈ p.
Let R again denote the real spinor module of p, then G2 also acts on R by (11). ∼ I as in Lemma 8.1. With respect to a suitable isoLemma 8.2 ([6]). We identify p = metric, G2 -equivariant identification R ∼ = O and a suitable orientation of p, Clifford multiplication p × R → R equals Cayley multiplication I × O → O from the right.
We fix an orthonormal base e1 , . . . , e7 of p ∼ = I such that ei ∗ ei+1 = ei+3 , where indices are taken mod 7. Clifford multiplication with ei will be abbreviated as ci . As in [5] and [6], let us introduce the notation
and
1 ci jk = [ei , e j ]p, ek = − √ (ei ∗ e j ) ∗ ek 5 7 ˜ p,X = 1 [X, ei ], e j ci c j . ad 4 i, j=1
Let us define 7 7 1 ˜ 1 A= ci adp,ei = ci jk ci c j ck . 3 12 i=1
i, j,k=1
It is easy to see that A is a symmetric, G2 -invariant endomorphism of R . Thus it induces a G-invariant symmetric endomorphism of SR M. One may compute that 1 7 0 : R ⊕ I → R ⊕ I. A= √ 2 5 0 −1
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Let D denote the real Dirac operator acting on (SR M). We want to consider the family of operators 3 λ A. (12) D = D + 3λ − 2 1 1 The operator D 2 is the Riemannian Dirac operator on M, whereas D˜ = D 3 is called the reductive or cubic Dirac operator.
Proposition 8.3. For λ sufficiently close to 21 , the smallest eigenvalue of (D λ )2 is
441 20
λ2 .
In particular there exists ε > 0 sufficiently small such that 1 λ1 (D 2 +ε )2 > λ1 (D 2 ). As stated in Remark 0.3, this implies that the Vafa-Witten technique cannot be used to prove sharp upper bounds for the first Dirac eigenvalue. For certain questions, the reductive Dirac operator D˜ on a normal homogeneous space plays the same role as the geometric Dirac operator on a symmetric space, but here clearly λ1 ( D˜ 2 ) < λ1 (D 2 ), so with respect to optimal Vafa-Witten estimates, the reductive Dirac operator is even worse than the geometric Dirac operator. Proof. The space of sections L 2 (M; SR M) can be decomposed as in (1). As in [5], we see that D λ acts on the isotypical components V γ ⊗ Hom H (V γ , R ) by id V γ ⊗γ D λ , where γ
Dλ =
7 7 7 λ ˜ p,ei = γe∗i ⊗ ci + λ id V γ ⊗ci ad γe∗i ⊗ ci + ci jk ci c j ck . 4 i=1
i=1
i, j,k=1
Let us write µ = 3λ − 1, then D λ = D˜ + µ id V γ ⊗A. The square of D˜ has been computed in [5] as ˜ 2 = γ + ρG 2 − ρ H 2 . (γ D) If we use integers p ≥ q ≥ 0 to index the irreducible representations of SO(5), then by [6], we have the explicit formula ˜ 2 = γ p,q + ρG 2 − ρ H 2 = p 2 + 3 p + q 2 + q + 49 on Hom H (V γ p,q , R ). (γ p,q D) 20 Let us now analyze all irreducible representations of SO(5). For the trivial representation, clearly γ0,0
21 D λ = 3λ γ0,0 D˜ = 3λ A|R = √ λ, 2 5
so
γ0,0
1 21 D = γ0,0 D 2 = √ . 4 5
The standard representation γ1,0 contains no SO(3)-irreducible subrepresentation isomorphic to R or I, so it does not contribute to the spectrum of D λ . The representation γ1,1 = 2 γ1,0 contains no trivial SO(3)-subrepresentation, but an SO(3)subrepresentation isomorphic to I, and we find that γ 1 21 λ1 ( 1,1 D) ≥ γ1,1 D˜ − 1 A|I = 13 √ − √ > √ = γ0,0 D . 2 2 5 4 5 4 5
Vafa-Witten Estimates for Compact Symmetric Spaces
851
For all other representations, we have p > 2, thus √ γ 249 7 21 λ1 ( p,q D) ≥ γ2,0 D˜ − 1 |A| = √ − √ > √ = γ0,0 D . 2 2 5 4 5 4 5 In particular, the smallest eigenvalue of D λ equals and is attained by a G-invariant spinor.
21 √ 2 5
λ for λ sufficiently close to 21 ,
Acknowledgements. The rigidity statement in Theorem 0.1 was pointed out to us by Listing. An anonymous referee helped to make this article more readable.
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