Communications in Mathematical Physics - Volume 240

Commun. Math. Phys. 240, 1–29 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0892-7

Communications in

Mathematical Physics

Energy Growth in Schr¨odinger’s Equation with Markovian Forcing M. Burak Erdo˜gan1 , Rowan Killip2 , Wilhelm Schlag3 1 2

Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA 3 Department of Mathematics 253-37, Caltech, Pasadena, CA 91125, USA

Received: 7 October 2002 / Accepted: 28 February 2003 Published online: 25 July 2003 – © Springer-Verlag 2003

Abstract: Schr¨odinger’s equation is considered on a one-dimensional torus with time dependent potential v(θ, t) = λV (θ )X(t), where V (θ ) is an even trigonometric polynomial and X(t) is a stationary Markov process. It is shown that when the coupling constant λ is sufficiently small, the average kinetic energy grows as the square-root of time. More generally, the H s norm of the wave function is shown to behave as t s/4 . 1. Introduction We study the long term behaviour of a quantum mechanical particle moving on the circle in the presence of a time-dependent potential. The evolution of the wave function ψ is described by the Schr¨odinger equation d 2ψ dψ (θ, t) = − 2 (θ, t) + v(θ, t)ψ(θ, t), (1) dt dθ where we regard the circle as R/2πZ. We are interested in how the kinetic energy,
π   ∂θ ψ(θ, t)2 dθ =
ψ
2 1 (t) −
ψ
2L2 (dθ) (2) H (dθ) i

−π

grows as a function of time. In fact, we will determine the behaviour of all Sobolev norms. Note that since v will be real,
ψ
L2 is conserved. In two recent papers ([3, 4] see also [5, Appendix 1]), Bourgain studied the case where v(θ, t) is analytic/smooth in θ and quasi-periodic/smooth in t. In particular, he showed [4] that if   sup ∂θα ∂tβ v(θ, t) < ∞ for all α, β = 0, 1, . . . , θ,t

then for any s,  > 0,
ψ
H s = O(t  ) as t → ∞. (This result holds in any number of space dimensions.) Conversely, it is shown [3] that energy may grow logarithmically

2

M.B. Erdo˜gan, R. Killip, W. Schlag

even for t-almost periodic and θ-smooth choices of v. Bourgain also gave an example of a random model exhibiting polynomial growth [4], which we will discuss in due course. In contradistinction to Bourgain, we are primarily interested in the case where v(θ, t) is not a smooth function of t. To show that energy growth is a generic phenomenon, there seems no real alternative than to consider a random model. We consider v(θ, t) = λX(t)V (θ ),

(3)

where X(t) is a stationary Markov process and V (θ ) is an even trigonometric polynomial and show that if the coupling constant λ is sufficiently small then, on average, the energy grows as the square root of time. One physical interpretation of this model is as a rigid rotator coupled to a classical heat bath. Let us describe our requirements for the process X(t). We will also take the opportunity to introduce some notation. Hypothesis 1. The process X(t) is a stationary Markov process with state space S ⊆ R and stationary distribution dµ. It is further assumed that: (i) the Markov process is generated by a self-adjoint operator B on L2 (S; dµ); (ii) the generator B is positive semi-definite with discrete spectrum; (iii) zero of B;  is a simple eigenvalue  (iv) x dµ(x) = 0 and x 2 dµ = 1; (v) the function u(x) = x belongs to the quadratic form domain of B; and (vi) the co-ordinate operator x : u(x) → xu(x) is relatively B-bounded with relative bound 0. Expectation with respect to this process will be denoted by E, the eigenvalues of B by 0 = κ0 < κ1 ≤ · · · ≤ κN ≤ · · · (repeated according to multiplicity) and the corresponding eigenvectors by uN . Of necessity, u0 (x) ≡ 1. Remarks. 1. By “generator of the Markov process” we mean the operator associated with the forward Kolmogorov equation. Many probability books use the generator associated with the backward equation. As the two operators are the adjoints of one another and we assume that B is self-adjoint, the distinction is moot here. The meaning of B and µ is perhaps best described with a sample calculation (see also (10) below). The probability that X(t) ∈ A is equal to µ(A) and  the probability density (with respect to µ) for X(t) given that X(t) ∈ A is χA (x) µ(A). For s > t,   the probability density for X(s) given that X(t) ∈ A is e−B(s−t) χA (x)/µ(A) and so the probability that the process passes through the sets A1 , A2 at times t1 < t2 is
  χA2 (x) e−B(t2 −t1 ) χA1 (x)dµ(x). 2. Regarding (ii) and (iii), the existence of a spectral gap (i.e., that zero is at an isolated point in the spectrum) is essential both for our analysis and, we believe, for the result. However, while ample for our interests, the assumption that the spectrum is discrete is presumably unnecessary. 3. Our arguments remain valid if one chooses X(t) to be the projection under a suitable map of a stationary Markov process on a more general state space. The simplest such modification is to retain the state space S but consider Y (t) = f (X(t)) for some f : R → R. Such a change would not affect the arguments  presented here  provided the analogues of (iv), (v), and (vi) held. That is, provided f dµ = 0, f 2 dµ = 1,

Energy Growth

3

f belongs to the quadratic form domain of B, and u(x) → f (x)u(x) is a relatively B-bounded operator with relative bound zero. Some remarks about the possibility of treating other functions f are offered at the end of this section. 4. Condition (iv) says that X(t) has mean zero and unit variance. 5. The requirement, (v), that u(x) = x is in the quadratic form domain of B ensures that sample paths of the process are not too rough. Specifically, it controls the highfrequency asymptotics of the power spectrum (see below). Exactly how this enters our analysis is described at the end of Subsect. 3.1. 6. The last condition, (vi), has been chosen to obviate functional analytic minutiae in the definition of Lλ in (14) below. Since λ will later be chosen small, merely assuming x is relatively B-bounded (with any bound) would permit the same elementary analysis. The most basic example is the following: take S = {−1, 1} and   1 −1 B= −1 1 in the natural basis for 2 (S). With these definitions, X(t) flips between +1, −1 with exponential waiting times. The Ornstein-Uhlenbeck process also obeys Hypothesis 1. Here S = R and B=−

∂2 ∂ +x . 2 ∂x ∂x

The stationary distribution is dµ = (2π)−1/2 exp{−x 2 /2} dx, the standardized Gaussian. The eigenvalues are κN = N with corresponding eigenvectors uN (x) = (N !)−1/2 HeN (x), the L2 -normalized Hermite polynomials (our notation is that of Abramowitz and Stegun [1]). The three term recurrence for these polynomials, √ √ x uN (x) = N uN−1 (x) + N + 1 uN+1 (x), (4) shows that x is indeed relatively bounded with bound zero. Theorem. Let X(t) be a process obeying Hypothesis 1 and let V (θ) be an even trigonometric polynomial. If the coupling constant λ is sufficiently small then for any initial wave-function ψ0 ∈ H s ,
∞

1 e−t/T E
ψ(·, t)
2H s dt λs T s/2
ψ0
2L2 +
ψ0
2H s (5) T 0

as T → ∞. The expectation is over possible trajectories of the Markov process X(t). In particular, the energy grows on average as the square-root of time. Remarks. 1. We write x y if and only if x
y and y
x. The notation x
y means that there exists C > 0 so that x ≤ Cy. 2. Since adding a constant (even if time-dependent) to the Hamiltonian has no physical  effect – it just changes the phase of the wave function – we may assume that V (θ) dθ = 0. 3. In the interest of clarity, we will not prove the theorem in this generality, but rather in the special case that V (θ ) = cos(θ ) and X(t) is the Ornstein–Uhlenbeck process. This simplifying assumption will be invoked at the beginning of Sect. 3, where the computational part of the proof begins. Some further remarks on the general case are given at the end of Subsect. 3.1.

4

M.B. Erdo˜gan, R. Killip, W. Schlag

4. Extending the method we present to arbitrary (i.e., not necessarily even) trigonometric polynomials, V , however, requires more than just computational stamina; in this setting, the operator H introduced in Sect. 3 is no longer a finite difference operator. For heuristic reasons that we will describe in a moment, it is natural to believe that the rate of energy growth is determined by the power spectrum of the forcing process. The power spectrum is the non-random function
2 1  T iωt  S(ω) = lim e X(t) dt   T →∞ 4T −T

ω ∈ R.

(6)

The limit is in the following sense: for any continuous function φ of compact support 1 4T



 

T

−T

e

iωt


2  X(t) dt  φ(ω) dω → φ(ω)S(ω) dω

almost surely [9, §X·7]. By the Wiener–Khinchin Theorem, S is the Fourier transform of the two-time corre lation function E X(t)X(t + τ ) so it is easy to show that S(ω) =

N

|xˆN |2

κN , ω2 + κN2

(7)

with respect to the B eigenfunctions: xˆN = where xˆN are the Fourier coefficients of x−2 xuN (x) dµ. This means that S(ω) |ω| as |ω| → ∞. (In fact, ω2 S(ω) → x|B|x which is finite by (v) of Hypothesis 1.) At small coupling, λ, significant energy growth can only occur as the result of a cumulative effect over a long time period. A particle of unit mass and momentum p experiences a cosine potential as being time-dependent with typical angular frequency ω = p. Now if the potential has a time dependent coupling constant given by a process X(t), only the part of that process with characteristic frequency ω will have a cumulative effect – the effect of the parts at other frequencies will average out to zero, at leading order. In this way, it is natural to intuit that the rate at which the particle gains energy is proportional to the amount of power in the process at that frequency. As energy is proportional to momentum squared we expect d 2 p ∝ S(p) p−2 dt which implies p˙ p −3 . This leads to p t 1/4 and so the prediction that energy should grow as the square-root of time. More generally, the argument of the preceding paragraph suggests that processes with S(ω) |ω|−γ give rise to p t 1/(2+γ ) . T. Spencer informed us that V. Zakharov predicted such behaviour during a private discussion of this problem. We are not privy to his reasoning. Bourgain’s random model [4, Part II] is far from being  stationary. In essence, his model may be described by V (θ ) = cos(θ ) and X(t) by gj γj (t), where gj are normalized Gaussian random variables and γj are disjointly supported bump functions. Although the functions γj are all of approximately unit norm in H s , they are not evenly spaced. Indeed, for s = 1, supp(γj ) is approximately [j 4/5 , (j + 1)4/5 ]. This means that

Energy Growth

5

X(t) is not uniformly locally H s . Consequently, it doesn’t offer a very clear view, to our eyes at least, of the role of temporal smoothness in the rate of energy growth. However, it is possible to fit Bourgain’s result into the heuristic described above by introducing a local power spectrum 
2   S(x, t) = E  X(t + τ )eiωτ φ(τ ) dτ  , where φ is positive, C ∞ , supported by [−2, 2], and equal to 1 on [−1, 1]. For the model under discussion, one finds that there are t 1/(2+2s) many γj ’s supported in a unit neighbourhood of t and that each |γˆ (ω)|2 is of size t −(2s+1)/(2s+2) over the interval [−t 1/(2s+2) , +t 1/(2s+2) ] and essentially zero outside this interval. By modifying the above argument to say p p˙ ∝ S(p, t), we find that p p˙ t −s/(s+1) and so
ψ 
t 1/(2s+2) as proved by Bourgain. Earlier we mentioned that the model under consideration may be interpreted as a rigid rotator coupled to a classical heat bath. We should say that for the case we treat, the bath is at an infinite temperature and so it is reasonable to expect that the energy increases indefinitely. The power spectrum of a finite temperature bath would have exponential tails. (For example, the well known Planck law: S(ω) ∝ ω3 /(eω/T − 1), where T denotes the temperature.) This leads us to ask what would happen in the presence of a finite temperature heat bath. A finite temperature bath may be modelled by a stationary mixing process all of paths are analytic in the strip R + i[−α, α]. (For example, X(t) = whose sample Y (t + s)(s 2 + 1)−1 ds, where Y (t) is the two-state process described above.) Naive physical intuition suggests that thermal equilibrium might be achieved and in particular, that while there may be rare excursions to high energy the energy should remain bounded on average. We believe that this is not the case and that in fact, the energy tends to infinity with probability one. This is what the heuristic given above predicts. The paper is arranged as follows: in the next section, we show that the combined evolution of the wave-function and the stochastic process is governed by a semigroup and derive an explicit expression for its generator, Lλ . It is further shown that the time averaged quantity discussed in the theorem admits a simple expression in terms of the resolvent of this generator. In Sect. 3 we study the properties of the resolvent of Lλ and, in particular, show that to leading order in λ it is given by the resolvent of a certain finite difference operator, H . It turns out that H is the generator of a continuous time random walk and so, at leading order in λ, the evolution may be described as a random walk in momentum space (cf. the first remark after Lemma 3.4). Note however that the hopping rate decays as the reciprocal of momentum squared. The main technical parts of the paper are the study of the long-term properties of the random walk with generator H and the derivation of estimates which permit us to control the discrepancy between this and the true evolution, which is generated by Lλ . The method we employ to achieve the former goal contains elements which appear to be new and which may be of interest to those studying random walks or diffusion. The division of this technical material between the final two sections, 4 and 5, is not by “random walk” vs. “discrepancy estimates” but rather by the nature of the estimates involved. This is more natural in terms of the proofs. Section 4 derives weighted-norm inequalities for the resolvent of H (Corollary 4.3) and for the difference between this and the resolvent of Lλ (Corollary 4.4). Section 5 uses

6

M.B. Erdo˜gan, R. Killip, W. Schlag

these norm estimates to obtain the pointwise estimates necessary to prove the theorem. The section closes by restating the theorem and finishing its proof. The material of Sect. 2 and of the beginning of Sect. 3 is essentially standard fare. However, it particularly parallels the work of Tcheremchantsev [12–14] since, in a crude sense, his model resembles the Fourier transform of ours. The question we address is by no means the only interesting problem of its type. It is natural to ask to what extent the heuristic described above gives the correct behaviour and, in particular, whether Zakharov’s prediction is true. While it is presumably impossible to give lower bounds for energy growth purely in terms of the smoothness properties of V (x, t), one might hope that such deterministic upper bounds are possible. It seems that the methods presented here could be extended to cover certain processes with rougher time dependence; see below. Treating processes with smoother time dependence would necessitate giving up the assumption of the existence of a spectral gap. This is a rather daunting proposition. For the analogous problem on Rn /Zn , n ≥ 2, the operator corresponding to H is rather complicated. (Even its domain, the kernel of L0 , is not simple; it depends on the number of ways an integer can be represented as a sum of n squares.) Studying the behaviour of its Green function would require a better understanding of its structure than we currently possess. Even more difficult is the corresponding problem in Rn . In the case that the potential is supported inside a compact time-independent interval, Bourgain [6] has shown that as a result of dispersion, the energy can be bounded by t  ,  > 0 no matter how rough the time dependence of the potential. We now wish to offer a few remarks about the possibility of treating processes of the form f (X(t)) when f is rather rough. Please note that these remarks are somewhat conjectural. Pursuing them would necessitate a considerable enterprise, one that we do not intend to undertake in the foreseeable future. Let X(t) be the Ornstein-Uhlenbeck process and consider Y (t) = f (X(t)) for some measurable function f : R → R that is not too large at infinity, say f ∈ L∞ (dµ). It is not difficult to show that the power spectrum of Y (t) is given by S(ω) =

∞

  fˆ(N )2 N=0

N2

N , + ω2

where fˆ(N) = uN |f (cf. (7)). If f is in the quadratic form domain of B then  N |fˆ(N)|2 < ∞ and S(ω) ω−2 . As promised in Remark 3 after Hypothesis 1, the analysis doesn’t change. If f is rough then it will not belong to the quadratic form domain of B (a second-order elliptic differential operator) and so the sum above will not converge. In the simplest case, V (θ) = cos(θ ), this affects the analysis in the following way: the off-diagonal coefficients of the operator H are given by an = 2λ2

N≥1

(N + β)|fˆ(N)|2 ∼ 2λ2 S(2n + 1) (N + β)2 + (2n + 1)2

(cf. (40)). If fˆ(N) N −α with α ∈ (1/2, 1), then S(ω) ω−2α and so an n−2α . This means that the large-n hopping rate is enhanced relative to the case where f is in the quadratic form domain of B (in that case, an n−2 ). Proceeding√formally, we infer that the weight ρ in Sects. 4 and 5 should be changed to c min{n, βλ−1 n1+α } and so ultimately that, on average, the kinetic energy grows as t σ with σ = 1/(1 + α).

Energy Growth

7

This is exactly the exponent one would predict from the power spectrum asymptotics by the heuristic given above. 2. Reformulation Just as classical probability distributions are described by normalized positive measures, i.e., normalized positive linear functionals on random variables, so quantum mechanical distributions are described by normalized positive linear functionals on the space of observables (bounded linear operators). These are called density matrices; they are the positive trace-class operators and are normalized to have trace equal to one. The expected value of an observable A is given by tr(Aρ). For example, a system in the quantum state |ψ is described by ρ = |ψ ψ|, the projection onto the linear space spanned  by |ψ . More generally, a system in states |ψi with probabilities pi is described by ρ =  i pi |ψi ψi |. Notice that the expected value of an observable A is given by tr(Aρ) = i pi ψi |A|ψi . The only natural choice! As is usual in this business, we consider the space of density matrices as a subset of the space of Hilbert–Schmidt operators. This affords us the pleasure of working in a Hilbert Space (the inner product is given by ρ|σ = tr(ρ † σ )). We will denote the space of Hilbert–Schmidt operators on L2 (R/2πZ) by I2 L2 (R/2π Z) or, more often, I2 for short. The state of the Markov process X isdescribed by its probability density u(x), a non-negative function in L2 (S; dµ) with u(x) dµ = 1. While the quantum system is described by its density matrix ρ, a non-negative operator in I2 with tr(ρ) = 1 (in particular, ρ ∈ I1 ). The state of the combined system of the Markov process and the quantum particle at any time t is described by an element of the Hilbert space L2 (S; dµ)⊗I2 . We will denote this element by P(t). For example, when the process and quantum system are independent of one another, P = u ⊗ ρ. The most general P is the limit of convex combinations of tensor products. A particular case is when the quantum system is in an initial state ψ0 and the process X is in its stationary distribution. This is represented by P(t = 0) = u0 ⊗ ρ0 ,

where ρ0 = |ψ0 ψ0 | and u0 (x) ≡ 1.

Note that at positive times, P need no longer be a tensor product. This is because the states of quantum particle and the forcing process become correlated. To help explain the meaning of P we now describe how to calculate expectations. A natural class of random variables/observables is f (X)A for some f ∈ L2 (dµ) and A ∈ I2 . In the independent case, P = u ⊗ ρ, it is clear that the average value of f (X)A is given by
    f (x)u(x) dµ(x) · tr Aρ = f¯ ⊗ A† P L2 (dµ)⊗I . 2

Linearity then forces the same choice for general P, namely  

 (8) E f (X(t))ψ(t)|A|ψ(t) = f¯ ⊗ A† P (t) L2 (µ)⊗I . 2  Similarly, the average value of more complicated observables such as j fj (X)Aj is determined by linearity. Of course, we are mainly interested in observables that are not Hilbert–Schmidt; for example, the kinetic energy. These can be dealt with using a simple approximation argument.

8

M.B. Erdo˜gan, R. Killip, W. Schlag

To determine the evolution equation for P, we perform the following calculation: let f ∈ L2 (dµ) be from the domain of B and let A ∈ I2 be such that [H0 , A] ∈ I2 , then       d E f X(t) ψ(t)Aψ(t) dt                 = E − [Bf ] X(t) ψ Aψ + f X(t) ψ˙ Aψ + f X(t) ψ Aψ˙            = E − [Bf ] X(t) ψ Aψ + if X(t) ψ (H0 + λX(t)V )Aψ      − if X(t) ψ A(H0 + λX(t)V )ψ            = E − [Bf ] X(t) ψ Aψ + if X(t) ψ [H0 , A]ψ      + iλX(t)f X(t) ψ [V , A]ψ ,

(9)

where we have used a dot to denote the time derivative, H0 to denote the free Hamiltod2 nian (the Laplacian, − dθ 2 ) and V represents the operator of multiplication by the spatial part of the potential, V (θ ) = cos(θ ). Also, [A, C] = AC − CA denotes the commutator of the operators A, C. In the first equation we used the fact that ∂t E{f (X(t))} = − E{[Bf ](X(t))}. Strictly speaking, this should be B † since

d ∂u f (x)u(x, t) dµ(x) = f (x) (x, t) dµ(x) dt ∂t
= − f (x)[Bu](x, t) dµ(x)
= − [B † f ](x)u(x, t) dµ(x). (10) However, we assumed B to be self-adjoint so there is no need to distinguish. Equation (9) shows that the time derivative of the average value of an observable f (X)A is given by the average value of another observable, −[Bf ](X)A + if (X)[H0 , A] + iλXf (X)[V , A]. In order that we obtain the same relation when calculating averages with P, as described in (8), we must have d        f¯ ⊗ A†  P = − (B f¯) ⊗ A† − i f¯ ⊗ [H0 , A]† − iλ(x f¯) ⊗ [V , A]† P dt     = − (B f¯) ⊗ A† + i f¯ ⊗ [H0 , A† ] + iλ(x f¯) ⊗ [V , A† ]P . (11) Taking adjoints in L2 (dµ) ⊗ I2 and then choosing f ⊗ A from a dense subset of this space, we find that

d P = − B ⊗ I + iI ⊗ [H0 , · ] + iλx ⊗ [V , · ] P, dt

(12)

where x denotes the operator u(x) → xu(x) in L2 (dµ). In passing from (11) to (12) we used the following observation: For any operator C on L2 (R/2π Z) the adjoint of map

Energy Growth

9

ρ → [C, ρ] on I2 is given by ρ → [C † , ρ]. The demonstration of this is simple: σ |[C, ρ] = tr{σ † (Cρ − ρC)} = tr{(σ † C − Cσ † )ρ} = [C † , σ ]|ρ .

(13)

In particular, if C is self-adjoint on L2 then [C, ·] is self-adjoint on I2 . A more general derivation of the master equation, (12), for a quantum system subjected to Markovian forcing is given in [11]. It is natural to call this a master equation from its similarity to the purely classical and purely quantum mechanical equations of the same name. For a highly readable account of both classical and quantum mechanical stochastic processes with a strong physical motivation, we recommend [10]. To simplify notation, we rewrite the evolution of P as d P = −Lλ P dt

where Lλ = B ⊗ I + iI ⊗ [H0 , · ] + iλx ⊗ [V , · ].

(14)

The linear operator Lλ so defined is m-accretive and has domain D(B) ⊗ H 2 . In fact, it is the sum of a non-negative operator, B ⊗ I , and a skew-adjoint (A† = −A) operator, iI ⊗ [H0 , · ] + iλx ⊗ [V , · ]. Maximality follows from (vi) of Hypothesis 1. This said, we see that Lλ generates a contraction semigroup so we can write P(t) = e−tLλ P(0). Combining the equation above with (8) we infer that for any Hilbert–Schmidt observable A,
∞

1 e−t/T E ψ(t)|A|ψ(t) dt T 0
∞    e−t/T u0 ⊗ A† P(t) dt = T1
0 ∞     e−t/T u0 ⊗ A†  e−tLλ P(0) dt = T1 0

  = u0 ⊗ A†  

  β  P(0) , Lλ + β

(15)

where β = T1 . (Accretiveness ensures that Lλ + β is invertible.) This trick, which is standard, is behind our choice of time averaging. As it diagonalizes the Laplacian, it is only natural that we choose to work in a Fourier basis. It will also simplify the study of the H s norms. Because V (θ) is assumed even, the Hamiltonian is symmetric under the reflection θ → −θ. Therefore, the natural choice of orthonormal basis for L2 (R/2πZ) is  −1/2  :n=0 (2π) −1/2 |n = π . (16) cos(nθ ) : n = 1, 2, . . .  π −1/2 sin(|n|θ ) : n = −1, −2, . . . The reflection symmetry of the Hamiltonian actually means that the odd and even subspaces are invariant under the evolution. This is at the origin of the invariant subspaces described in Lemma 3.1 below.

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M.B. Erdo˜gan, R. Killip, W. Schlag

The natural basis for B is in terms of its eigenfunctions uN , N = 0, 1, . . . . Therefore, we introduce the following orthonormal basis for the Hilbert space L2 (µ) ⊗ I2 : |N, n, m = uN (x) ⊗ |n m|

for n, m ∈ Z and N = 0, 1, . . .

(17)

with |n and m| defined as in (16). With this notation, we can now reformulate the behaviour of H s norms in terms of the operator Lλ : Proposition 2.1. For a quantum particle initially in the state ψ(t = 0) = ψ0 ∈ H s and for X(t = 0) chosen independently according to the stationary distribution,
∞  β   

 2   β e−βt E ψ(t)H s dt = (1 + |n|2s ) 0, n, n (18) P(0) , Lλ + β 0 n where P(0) = u0 ⊗ |ψ0 ψ0 |. Note that β plays the role of

1 T

in the theorem.

Proof. This follows immediately from (15): For N > 0, let A=

N

(1 + |n|2s ) |0, n, n 0, n, n|

n=−N

and then use the monotone convergence theorem to take N → ∞.

 

3. The Operator Lλ In the previous section, we reduced the study of the average behaviour of H s norms to the consideration of the resolvent of the operator Lλ . The purpose of this section is to isolate the dominant terms in this resolvent; the main technical estimates appear in the next two sections. For expository reasons, we make the following: Hypothesis 2. For the remainder of this paper we shall consider only the case where V (θ) = cos(θ ) and X(t) is the Ornstein–Uhlenbeck process. Some remarks about the general case are given after Lemma 3.4. To a first approximation, we will treat Lλ as a perturbation of L0 , (that is, Lλ with λ = 0). For this reason, we introduce the notation  1 (19) Lλ − L0 = x ⊗ [V , · ] which implies Lλ = L0 + iλJ. J = iλ Notice that J is a self-adjoint operator. For convenience, we will always take λ ≥ 0. Lemma 3.1. The basis |N, n, m of (17) diagonalizes the operator L0 . Specifically,

L0 |N, n, m = B ⊗ I + iI ⊗ [H0 , · ] |N, n, m

= N + i(n2 − m2 ) |N, n, m . (20) Also, each of the four subspaces span{|N, n, m : N ≥ 0, n, m ≥ 0}, span{|N, n, m : N ≥ 0, n, m < 0},

span{|N, n, m : N ≥ 0, n ≥ 0, m < 0}, span{|N, n, m : N ≥ 0, n < 0, m ≥ 0}

is invariant under J and hence also under Lλ = L0 + iλJ .

Energy Growth

11

Proof. The first part is self-evident (recall κN = N for Ornstein-Uhlenbeck). The second is a reflection of the fact that because V (θ ) is even, V |n and |n have the same parity: even if n ≥ 0 and odd if n < 0.   As we are interested in H s norms, Proposition 2.1 shows that we need only consider how Lλ acts in two of these invariant subspaces, namely span{|N, n, m : N ≥ 0, n, m ≥ 0}, which corresponds to the even part of the wave function, and span{|N, n, m : N ≥ 0, n, m < 0}, which corresponds to the odd part. In calculating (Lλ + β)−1 in each of these subspaces, a major role is played by the kernel of L0 . This is given by span{|0, n, n : n ∈ Z}. To isolate this part we define projections onto

span{|0, n, n : n ≥ 0},

P Q

onto onto

span{|N, n, m : n, m ≥ 0 and either N > 0 or n = m}, span{|0, n, n : n < 0}, and

Q⊥

onto

span{|N, n, m : n, m < 0 and either N > 0 or n = m}.

P ⊥

Notice that P P ⊥ = 0 and P + P ⊥ is the projection onto span{|N, n, m : N ≥ 0, n, m ≥ 0}, the invariant subspace associated with the even part of the wave function. Similarly, QQ⊥ = 0 and Q + Q⊥ projects onto span{|N, n, m : N ≥ 0, n, m < 0}. Further note that the ranges of both P and Q lie within the kernel of L0 . It is now easier to treat the two parts of Lλ separately. We begin with the “even” part that is the part invariant under P + P ⊥ . This is followed by a parallel but abbreviated discussion of the “odd” part. 3.1. Even. On the range of P + P ⊥ , we can write Lλ in block form   0 iλP J P ⊥ , (21) iλP ⊥ J P P ⊥ Lλ P ⊥  where we noted  that since V (θ ) dθ = 0, P J P = 0. (This may also be derived from the fact that x dµ = 0.) Of course, L0 P is also zero since P is a projection onto a subset of the kernel of L0 . By the well-known formulae for block inversion,  −1 P (Lλ + β)−1 P = β + λ2 P J P ⊥ Rλ⊥ (β)P ⊥ J P , (22) where Rλ⊥ (β) = (P ⊥ Lλ P ⊥ + β)−1 is the resolvent of the operator appearing in the bottom right corner. Here, as elsewhere in this section, we regard the operator inside the braces on the right-hand side of (22) as an operator acting on the range of P for the purposes of inversion. Similarly, Rλ is the inverse of an operator acting on the range of P ⊥. The formulae for block inversion also show that  −1 P (Lλ + β)−1 P ⊥ = −iλ β + λ2 P J P ⊥ Rλ⊥ (β)P ⊥ J P P J P ⊥ Rλ⊥ (β). (23) Note the occurrence of our earlier expression (22) as a factor here. Indeed, control of (23) will be a simple by-product (see Lemma 5.4) of the analysis of (22) to which we now direct our attention. (We do not need the other two matrix entries of (Lλ +β)−1 , that

12

M.B. Erdo˜gan, R. Killip, W. Schlag

is, those mapping onto the range of P ⊥ : These are irrelevant to (18) of Proposition 2.1 because |0, n, n is orthogonal to the range P ⊥ .) As we will eventually demonstrate, one can control (22) by replacing Rλ⊥ (β) by ⊥ R0 (β) and then treating the error as a small perturbation. With this as our inspiration, we define H : Range(P ) → Range(P ) by H = λ2 P J P ⊥ R0⊥ (β)P ⊥ J P .

(24)

(Notice that H depends on both λ and β.) Equation (22) can now be written as P (Lλ + β)−1 P  −1   = H + β + λ2 P J P ⊥ Rλ⊥ (β) − R0⊥ (β) P ⊥ J P =

∞



H +β

−1 

   −1 j − λ2 P J P ⊥ Rλ⊥ (β) − R0⊥ (β) P ⊥ J P H + β .

(25) (26)

j =0

The convergence of this infinite resolvent series will not be justified until we derive certain estimates in the next section; we merely offer it as a sign-post of where we are headed. In addition to H , the next two sections employ the operator D = λR0⊥ (β)P ⊥ J P

(27)

which is to be regarded as mapping Range(P ) → Range(P ⊥ ). The importance of this operator will become clearer as we proceed. We now give a trio of lemmas. The first describes the behaviour of J on the range of P + P ⊥ , which it leaves invariant. The others give explicit formulas for the way the operators D and H act on their domain, that is, on the range of P . Lemma 3.2. For any N, n, m ≥ 0,  √ J |N, n, m = N + 1 cn |N + 1, n + 1, m + cn−1 |N + 1, n − 1, m  − cm |N + 1, n, m + 1 − cm−1 |N + 1, n, m − 1 √  + N cn |N, n + 1, m + cn−1 |N, n + 1, m  − cm |N, n, m + 1 − cm−1 |N, n, m − 1 , where the co-efficients ck are given by   0 ck = √12  1 2

(28)

: k = −1 :k=0 . :k>0

Proof. Recall that J = x ⊗ [V , · ]. The proof is a mundane calculation based on the following two formulae: First, as was stated above, (4), √ √ x uN (x) = N uN−1 (x) + N + 1 uN+1 (x),

Energy Growth

13

and second,  1 √ :n=0   2 |1 cos(θ )|n = cn |n + 1 + cn−1 |n − 1 = √1 |0 + 21 |2 :n=1  1 2 1 2 |n + 1 + 2 |n − 1 : n ≥ 2 in the ket notation of (16).

 

Lemma 3.3. The operator D acts like a vector-valuedfirst order finite difference operator specifically, one with two components: if |ψ = ψn |0, n, n then   (29) 1, n + 1, n|D|ψ = αn ψn − ψn+1 ,   1, n, n + 1|D|ψ = α¯ n ψn+1 − ψn , (30) where

√ λ 1 2 :n=0 αn = :n>0 2 1 + β + i(2n + 1) 1

(31)

and all other co-efficients N, n, m|D|ψ are zero. Proof. This follows from the previous lemma plus the fact that R0⊥ (β)|N, n, m =

1 |N, n, m N + β + i(n2 − m2 )



Lemma 3.4. The operator H is a discrete second order difference operator  in diverψn |0, n, n gence form. Indeed, H = (1 + β)D † D or, more explicitly, if |ψ = then  a 0 ψ0 − a 0 ψ1 n=0 0, n, n|H |ψ = , (an + an−1 )ψn − an−1 ψn−1 − an ψn+1 n ≥ 1 where an = 2(1 + β)|αn |2 . Proof. Define an operator S by S|N, n, m = |N, m, n ; that is, S swaps n and m. From (28) one sees that SJ P = −J P and, by taking adjoints for example, P J S = −P J so H = λ2 P J P ⊥ R0⊥ (β)P ⊥ J P = =

λ P J SP R0⊥ (β)P ⊥ SJ P λ2 P J P ⊥ R0⊥ (β)† P ⊥ J P . ⊥

2

(32) (33) (34)

In the last line here we used SR0⊥ (β)S = R0⊥ (β)† which is easily checked since both are diagonal in the |N, n, m basis. Now we may write   H = 21 λ2 P J P ⊥ R0⊥ (β) + R0⊥ (β)† P ⊥ J P (35) = (1 + β)λ2 P J P ⊥ R0⊥ (β)† R0⊥ (β)P ⊥ J P

(36)

= (1 + β)D D.

(37)

†

14

M.B. Erdo˜gan, R. Killip, W. Schlag

Equation (36) requires some explanation: from (28) we know that the range of P ⊥ J P is spanned by vectors of the form |1, n, m and for such vectors we have 1 2



 R0⊥ (β) + R0⊥ (β)† |1, n, m =

1+β |1, n, m + (n2 − m2 )2

(38)

= (1 + β)R0⊥ (β)† R0⊥ (β)|1, n, m .

(39)

(1 + β)2

The explicit formula for H follows easily from H = (1 + β)D † D and the formula for D given in the previous lemma.   Remarks. 1. If we consider the matrix representation of H in the |0, n, n basis, the off-diagonal entries are non-positive and the sum of entries in each column (and row) is zero. This makes H the generator of a continuous time Markov chain. Indeed, from the state |0, n, n there are two possible transitions: to |0, n + 1, n + 1 , which occurs at rate an , and to |0, n − 1, n − 1 with rate an−1 . In this way one can interpret the Markov chain as a continuous time random walk in which the hopping rate decays as n → ∞. The physical interpretation of this is that H , which is a good approximation to P Lλ P , represents a spatially inhomogeneous random walk in momentum space. 2. For more general V and X, the expression (28) for operator J becomes far more complicated since now both N and either n or m can change by an arbitrary amount. (Notice that in (28), they change by one.) Apart from rendering the explicit formulae unreadable, this generality creates no significant mathematical problems. In particular, the operator D is still a vector-valued first order finite difference operator (cf. Lemma 3.3). Actually, it is better to regard it as a sequence of such operators, {DN : N > 0} corresponding to the final values of N such that D, as defined by (27), is equal to N DN . From (19),   xˆN cn,k |N, n + k, n − |N, n, n + k ,

P ⊥ J P |0, n, n =

N>0,k=0

 where cn,k = n + k|V |n and xˆN = uN |x|u0 = xuN (x) dµ. This should be compared with the formulae of Lemma 3.2 where cn,k and xˆN are only non-zero for two values of k and N respectively. Note also that xˆ0 = 0 so there is no need for an N = 0 term in the sum. Similarly, V (θ ) dθ = 0 implies cn,0 = 0. Because V is a trigonometric polynomial, only values of k smaller than the degree of V contribute to the sum given above. In this way, each DN can be regarded as a vector of 2 deg(V ) finite difference operators:   N  , n + k, n|DN |ψ = δN,N  αN,n,k ψn − ψn+k ,   N  , n, n + k|DN |ψ = δN,N  α¯ N,n,k ψn+k − ψn , where 1 ≤ k ≤ deg(V ) and αN,n,k =

λxˆN cn,k . κN + β + i(2nk + k 2 )

All other matrix entries of DN are zero. The derivation of these formulae uses the fact that for n ≥ k ≥ 1, cn+k,−k = cn,k .

Energy Growth

15

3. In the general setting, H = to hold, but with



N (κN



† + β)DN DN . In this way, Lemma 3.4 continues

deg(V )

an =

2(κN + β)|αN,n,k |2

k=1 N≥1

deg(V )

= 2λ2

|cn,k |2

k=1

N≥1

(κN + β)|xˆN |2 . (κN + β)2 + (2nk + k 2 )2

(40)

This of Hypothesis 1 enters our analysis: an n−2 as n → ∞ if and only  is how (v) 2 if κN |xˆN | < ∞. That is, if and only if u(x) = x belongs to the quadratic form domain of B. 3.2. Odd. For the odd portion of the wave-packet, we must study Q(Lλ +β)−1 Q. Equations (21)–(27) continue to hold if P and P ⊥ are replaced by Q and Q⊥ respectively. In particular, this defines new operators D and H associated to the odd subspace. The exact form of these operators differs very slightly from those in the even subspace. Indeed, they are slightly simpler as the following replacement for Lemmas 3.3 and 3.4 demonstrates: Lemma 3.5. The operator D associated to the odd subspace acts as follows: if |ψ =  ∞ n=1 ψn |0, −n, −n then   (41) 1, −n − 1, −n|D|ψ = αn ψn − ψn+1 ,   1, −n, −n − 1|D|ψ = α¯ n ψn+1 − ψn , (42) where αn =

λ 1 , 2 1 + β − i(2n + 1)

(43)

and all other co-efficients N, n, m|D|ψ are zero. Further, H = (1 + β)D † D so 0, −n, −n|H |ψ = (an + an−1 )ψn − an−1 ψn−1 − an ψn+1 ,

(44)

where an = 2(1 + β)|αn |2 for n ≥ 1 and a0 = 0. 4. Norm Estimates This section is devoted to deriving weighted norm estimates involving the operators D and H = (1+β)D † D associated to the even portion of the wave packet (see (24) and (27) in the last section). These are then used to obtain similar estimates for P (Lλ + β)−1 P . The operators D and H associated to the odd portion of the wave so closely resemble those for the even portion, that the same estimates can be derived with only cosmetic changes to the proofs. We will not discuss these operators again in this section. The derivation of the weighted norm inequalities follows the well-known scheme of Combes–Thomas [7] and Agmon [2]. In the section that follows this, we use these weighted norm results to obtain pointwise estimates on (H + β)−1 and P (Lλ + β)−1 P . However, the standard reduction from norm-estimates to pointwise estimates is unsuitable in our situation; the incorrect prefactor it gives renders it useless. Part of our remedy

16

M.B. Erdo˜gan, R. Killip, W. Schlag

to this problem involves obtaining derivative estimates for the Green function. Indeed, as D acts essentially as a (vector-valued) differential operator (cf. Lemma 3.3), this is how we encourage the reader to interpret (50)–(52). We will discuss this further in the next section. Given a weight function w : {0, 1, . . . } → (0, ∞) we consider two weighted Hilbert spaces: (i) Let 2w denote the range of P endowed with the norm  2

  ψn |0, n, n  2 = w(n) |ψn |2 ; (45)  w

n

n

and (ii) let 2w˜ denote the span of {|N, n, m : N, n, m ≥ 0} together with the norm  2

  ψN,n,m |N, n, m  2 = w(n, ˜ m) |ψN,n,m |2 , (46)  w˜

N,n,m

N,n,m

√ where w(n, ˜ m) = w(n)w(m). Notice that by Lemma 3.3, D : 2w → 2w˜ when w(n) ≡ 1. In fact, this is true for a much broader class of weights. The following lemma will be used in Corollary 4.4 to control the sum in (26). Lemma 4.1. Consider the weight function w(n) = e2ρ(n) with  ρ(n) = c min{n, λ−1 n2 β} and c > 0 a (small) constant. There is an operator E : 2w˜ → 2w˜ such that   −λ2 P J P ⊥ Rλ⊥ (β) − R0⊥ (β) P ⊥ J P = D † ED

(47)

and, for λ sufficiently small (not depending on β),  2 E  2 2
λ,  → w˜

w˜

where the implicit constant holds uniformly as c, λ, β → 0. Proof. From the resolvent formula, one finds ∞  j  Rλ⊥ (β) − R0⊥ (β) = (−iλ)j R0⊥ (β) P ⊥ J P ⊥ R0⊥ (β) .



j =1

This means that (47) holds with E=

∞

 j  −1 (−iλ)j +1 R0⊥ (β)† R0⊥ (β) P ⊥ J P ⊥ R0⊥ (β) P ⊥ J P ⊥

j =0



and, since R0⊥ (β)†

−1

R0⊥ (β) is a unitary operator in 2w˜ , that we need only show  ⊥  R (β)P ⊥ J P ⊥ 22 2
1. (48) 0  → w˜

w˜

We show this by employing the Schur test. Looking at the explicit formula, (28), for J shows that (in the |N, n, m basis) the matrix representing J has only finitely many non-zero elements in each row. Indeed the number of non-zero elements is at most eight

Energy Growth

17

(there could be fewer since c−1 = 0). The same holds for the columns as well since J is self-adjoint in the unweighted space. √ These matrix entries are of size N + 1 (indeed, J is not bounded); however, we have 1 |N, n, m R0⊥ (β)|N, n, m = N + β + i(n2 − m2 )  −1 and N +β +i(n2 −m2 ) ≤ 21 (N +1)−1 when N = 0 or n = m; that is, for all vectors in its domain. Therefore, each of the at most eight non-zero entries in every row/column of the matrix for R0⊥ (β)P ⊥ J P ⊥ is bounded by a uniform constant. The preceding discussion is sufficient to show that R0⊥ (β)P ⊥ J P ⊥ is bounded in the unweighted space. To obtain the weight estimate one further needs to know that         ˜ m) w(n ˜ ± 1, m) = |N, n ± 1, m  2
|N  , n, m  2 = w(n, w˜

w˜

and the corresponding result with m ± 1. This is easily checked to be the case because |ρ(n + 1) − ρ(n)|
1 for the weight in question.   Remark. By delving a little deeper into the specifics of the Ornstein–Uhlenbeck process, one can show that the main estimate holds with λ2 on the right-hand side rather than just λ. This need not be the case for more general processes. Proposition 4.2. There exists a (small) constant c > 0 so that for all λ < 1 the following hold:   † (D D + β)−1  2 2 ≤ 2β −1 , (49) w →w   D(D † D + β)−1 D †  2 2 ≤ 3, (50)  →  w˜ w˜  † −1 −1/2 D(D D + β)  2 2 ≤ 3β , (51)  →  w w˜  † −1 † −1/2 (D D + β) D  2 2 ≤ 3β , (52)  → w˜

w

√ where w(n) = e2ρ(n) and ρ(n) = c min{n, λ−1 n2 β}. Proof. The first estimate is standard Combes–Thomas/Agmon fare: Let e±ρ denote the multiplication operators associated with the corresponding function of n: e±ρ |0, n, n = e±ρ(n) |0, n, n .     Then eρ |0, n, n  = |0, n, n 2 and so w

 †  (D D + β)−1 

2w →2w

  = eρ (D † D + β)−1 e−ρ   −1  . =  eρ (D † D + β)e−ρ

Now by brute calculation, we have that eρ D † De−ρ = D † D − ξ + iη,

18

M.B. Erdo˜gan, R. Killip, W. Schlag

 where ξ and η are self-adjoint operators defined as follows: for |ψ = ψn |0, n, n ,     0, n, nξ ψ = ξn ψn+1 + ξn−1 ψn−1 ,  ρ(n) − ρ(n + 1)  ξn = 2|αn |2 sinh2 n ≥ 0, ξ−1 = 0,    2  0, n, nηψ = ηn ψn+1 + ηn−1 ψn−1 ,  2 ηn = |αn | sinh ρ(n) − ρ(n + 1) n ≥ 0, η−1 = 0. Recall that αn comes from the explicit formula for D (see Lemma 3.3). By Schur’s test,   λ2 2 ρ(n) − ρ(n + 1) sinh . 2 2 n n 1+n √ So, since |ρ(n + 1) − ρ(n)|
c min{1, λ−1 n β}, one can ensure that
ξ
≤ 21 β by choosing c small enough. As D † D is a positive operator, it follows that         1 1 −1      D † D + β  2 2 =  D † D − ξ + iη + β  ≤ 2β w →w  2 ξ  ≤ sup 2|ξn |2 ≤ sup

which proves (49). By identical arguments, one can show that for c small enough     1 −1    DD † + β  2 2 ≤ 2β .  → w˜

w˜

Equation (50) now follows from the following commutation formula: D

1 D†D

+β

D† =

1 DD †

+β

DD † = 1 −

β DD † + β

,

which we learned from Percy Deift (see [8] for √ other, unrelated, applications). Corresponding to the weight w(n, ˜ m) = w(n)w(m) we define ρ(n, ˜ m) = 21 ρ(n) + 21 ρ(m) ˜ so that w(n, ˜ m) = eρ(n,m) . As was the case for eρ we regard eρ˜ as a multiplication operator, ˜ |N, n, m . e±ρ˜ |N, n, m = e±ρ(n,m)

 To prove (51) we will use the following consequence of Lemma 3.3: if |ψ = ψn |0, n, n then   1, n + 1, n|e2ρ˜ De−2ρ |ψ = αn eρ(n+1)−ρ(n) ψn − eρ(n)−ρ(n+1) ψn+1 , (53)  ρ(n)−ρ(n+1)  2ρ˜ −2ρ ρ(n+1)−ρ(n) (54) |ψ = α¯ n e ψn+1 − e ψn , 1, n, n + 1|e De and all other co-efficients are zero. This shows, by Schur’s test, that e2ρ˜ De−2ρ = D + ζ , where ζ is an operator with   −ρ˜ ρ    e ζ e  ≤ sup 2|αn | sinh 1 ρ(n + 1) − 1 ρ(n)  ≤ β 1/2 . 2 2 n

Energy Growth

19

The second inequality here follows from our particular choice of ρ when c√is sufficiently small. In particular, notice that |ρ(n + 1) − ρ(n)|
c min{1, λ−1 n β} and |αn |
λ(1 + n)−1 , see (31). Now,  2  2    ρ˜  1 1 −ρ  D   (55)  D † D + β  2 2 = e D D † D + β e  w →w˜     −ρ 1 1 † 2ρ˜ −ρ  (56) e D D e = e   † † D D+β D D+β      −ρ  1 1 † † 2ρ −ρ  = e D e (57) D + D ζ e   † † D D+β D D+β      −ρ  1 † † ≤ 2β −1  e (58) D + D ζ eρ  D   † D D+β by employing e2ρ˜ D = (D + ζ )e2ρ and (49). We continue this chain by using, inter alia, β 1 D † D = 1 − D † D+β , the triangle inequality, and (49) again, D † D+β     !   −ρ   −ρ β 1 ρ † ρ˜   −ρ˜ ρ  e ≤ 2β −1 1 +  + (59) e e ζ e e D e    D†D + β   D†D + β    −ρ  1 † ρ˜  ≤ 6β −1 + β −1/2  e (60) D e   † D D+β    ρ˜  1 −ρ  ≤ 6β −1 + β −1/2  e (61) D e   † D D+β     1  ≤ 6β −1 + β −1/2  D (62)  D†D + β  2 2 .  → w

w˜

In passing from (60) to (61) we used the fact that the norm of an operator is equal to the norm of its adjoint. Now, (55)–(62) show that      1   x = β D † obeys x 2 ≤ 6 + x, D D +β 2 2 w →w˜

from which it follows that x ≤ 3. This completes the proof of (51). The proof of (52) is essentially the same as the above. Indeed, since the proof of (49)–(51) did not require that c be positive, only that it was small, it follows that (51) holds with w and w˜ replaced with their reciprocals. Taking the adjoint of the operator in this modified inequality (51) proves (52).   Since H = (1 + β)D † D, the following follows trivially from this proposition. Corollary 4.3. There exists a (small) constant c > 0 so that for all λ < 1 and all β < 1,   (H + β)−1  2 2 ≤ 4β −1 , (63) w →w   D(H + β)−1 D †  2 2 ≤ 6, (64) w˜ →w˜   D(H + β)−1  2 2 ≤ 6β −1/2 , (65) w →w˜   (H + β)−1 D †  2 2 ≤ 6β −1/2 , (66) w˜ →w √ where w(n) = e2ρ(n) and ρ(n) = c min{n, λ−1 n2 β}.

20

M.B. Erdo˜gan, R. Killip, W. Schlag

The operator H was introduced as the “main part” of P Lλ P . We now estimate the discrepancy between the two. Corollary 4.4. There exists a (small) constant c > 0 so that for λ sufficiently small and all β < 1,     1 1 −1  P  (67)  (L + β) P − H + β  2 2
λβ , λ w →w       1 1 D P (68) P − D†   2 2
λ, (Lλ + β) H +β w˜ →w˜      1 1  D P P −
λβ −1/2 , (69)  (Lλ + β) H + β 2 →2 w w˜      1 1 † −1/2  P , (70)  (L + β) P − H + β D  2 2
λβ λ w˜ →w √ where w(n) = e2ρ(n) and ρ(n) = c min{n, λ−1 n2 β}. Proof. By (26) and then Lemma 4.1, P

1 1 P− (Lλ + β) H +β ∞

 −1     −1 j = − λ2 P J P ⊥ Rλ⊥ (β) − R0⊥ (β) P ⊥ J P H + β H +β = =

j =1 ∞



−1 

j =1 ∞



−1

H +β H +β

 −1 j D † ED H + β

  j  −1 −1 D†E D H + β D†E D H + β ,

j =0

where the operator E obeys

 2 E  2

w˜ →2w˜


λ.

So, once λ is sufficiently small, (64) from Corollary 4.3 permits us to sum the series and prove (67)–(70).   5. Pointwise Estimates Given a weight function w(n) = e2ρ(n) and an estimate such as (63),   (H + β)−1  2 2 ≤ 4β −1  → w

w

there is a standard and simple way to obtain pointwise bounds for the Green function: 0, n, n|(H + β)−1 |0, m, m         ≤ e−ρ |0, n, n eρ (H + β)−1 e−ρ eρ |0, m, m 

≤ 4β −1 exp ρ(m) − ρ(n) .

(71) (72) (73)

Energy Growth

21

In terms of the exponential behaviour, estimates of this type cannot really be improved. However, as we will show, the factor β −1 in front can be significantly improved. As was remarked after Lemma 3.4, the operator H is the generator of a continuoustime Markov chain. Indeed, since H is the dominant term in the operator Lλ , this shows that to a good approximation, the quantum particle undergoes a random walk in momentum space. Note that the hopping amplitudes for this random walk are an n−2 and so the diffusion is slower near infinity than for a homogeneous random walk. As probability is conserved, for any m,

0, n, n|e−tH |0, m, m = 1 for all t ≥ 0, (74) n

where each summand is positive. (Those unfamiliar with continuous-time Markov chains may see Chapter VI of [9].) From (74),
∞

0, n, n|(H + β)−1 |0, 0, 0 = e−βt 0, n, n|e−tH |0, 0, 0 dt = β −1 . (75) 0

n

n

However, if√ we use the pointwise estimate that we derived above, (71)–(73), with ρ(n) = c min{n, n2 β}, then, ignoring the λ dependence, we have

0, n, n|(H + β)−1 |0, 0, 0 ≤ 4β −1 e−ρ(n) β −5/4 . n

n

The reason for the discrepancy is not in the choice of ρ, but rather that too much was given away in passing from (71) to (72). The weighted norm estimate on (H + β)−1 is of the correct size, however, the vectors that realize that norm are spread out, not localized as |0, m, m is. Indeed, H is a differential operator, so the vectors |ψ for which
H |ψ
is small must have slowly varying co-efficients. Another manifestation of this is the fact −1 that the norm as the derivative of the Green √ of D(H + β) , which should be regarded function, is β times smaller than the norm of (H + β)−1 . The following lemma shows how one may represent |0, n, n so as to utilize this extra information. √ Lemma 5.1. Let w(n) = e2ρ(n) with ρ(n) = c min{n, λ−1 n2 β} and c < 1 a constant. For each n ≥ 0, there are vectors |n ∈ 2w˜ and |n ∈ 2w so that |0, n, n = D † |n + |n and

 −1/4 β −3/8 : 0 ≤ n ≤ λ1/2 β −1/4    λ √  ∓ρ(n)  ±ρ e e |n 
λ−1/2 β −1/4 n : λ1/2 β −1/4 ≤ n ≤ λβ −1/2 ,  0 : λβ −1/2 < n  −1/4 β 1/8 : 0 ≤ n ≤ λ1/2 β −1/4    λ √  ∓ρ(n)  ±ρ −1/2 1/4 e β n : λ1/2 β −1/4 ≤ n ≤ λβ −1/2 , e |n 
λ  1 : λβ −1/2 < n

uniformly for β, λ ∈ (0, 1).

(76)

(77)

22

M.B. Erdo˜gan, R. Killip, W. Schlag

Proof. In the case λβ −1/2 < n we choose |n = |0, n, n and |n = 0. The inequalities are trivial in this case. For n ≤ λβ −1/2 , let |n =

n+N−1

 αn−1 1 −

k−n N



|1, k, k + 1 ,

k=n

where N ≈ min{λ1/2 β −1/4 , λn−1 β −1/2 } is an integer: "

# λ1/2 β −1/4 : 0 ≤ n ≤ λ1/2 β −1/4 # −1 −1/2 λn β : λ1/2 β −1/4 ≤ n ≤ λβ −1/2

N= "

(x is the least integer greater than or equal to x). For ease of reading, we use the symbol ≈ to indicate that two things are equal modulo the rounding of N to an integer. By Lemma 3.3, D † |n =

n+N



1−

k−n N



|0, k, k − |0, k + 1, k + 1

k=n

= |0, n, n −

n+N

1 N |0, k, k ,

k=n+1

 1 so we choose |n = n+N k=n+1 N |0, k, k . We now proceed to calculate the norms: n+N  2

 −ρ  −2 | = N e−2ρ(k) e n  k=n+1

≤ N −1 e−2ρ(n)
max{λ−1/2 β 1/4 , λ−1 nβ 1/2 } e−2ρ(n) ,

n+N  2

 ρ  e2ρ(k) e |n  = N −2

≤N

k=n+1 −1 2ρ(n+N)

e


max{λ−1/2 β 1/4 , λ−1 nβ 1/2 } e2ρ(n) . Here we used the fact that ρ(n+N ) ≤ ρ(n)+5. The justification of this is as follows: for −1/2 , ρ(n) = cλ−1 n2 √β and then either (a) 0 ≤ n ≤ λ1/2 β −1/4 in which case n ≤ λβ√ −1 2nN √β ≤ λ−1/2 2Nβ 1/4 ≈ 2; or (b) λ1/2 β −1/4 ≤ n ≤ λβ −1/2 λ−1 N 2 β ≈ 1 and λ√ √ √ in which case λ−1 N 2 β ≈ λ(n2 β)−1 ≤ 1 and λ−1 2nN β ≈ 2. This completes the proof of (77).

Energy Growth

23

For β < 1, we have |αn | ≥ λ(1 + n)−1 and so  2 n+N−1

 −2   −ρ  αn  1 − e |n  ≤

k−n 2 −2ρ(k) e N

≤ N λ−2 (n + N )2 e−2ρ(n) ,

k=n

 2 n+N−1

 −2   ρ  αn  1 − e |n  ≤

k−n 2 2ρ(k) e N

≤ N λ−2 (n + N )2 e2ρ(n+N) .

k=n

By using ρ(n + N) ≤ ρ(n) + 5 these simplify to  2   e∓2ρ(n) e±ρ |n 
N λ−2 (n + N )2 . Now, when n ≤ λ1/2 β −1/4 , we have n ≤ N ≈ λ1/2 β −1/4 and so λ−2 N (n + N )2
λ−1/2 β −3/4 . When λ1/2 β −1/4 ≤ n ≤ λβ −1/2 , we have n ≥ N ≈ λβ −1/2 n−1 from which it follows that λ−2 N (n + N )2
λ−1 nβ −1/2 . In both cases, this is just what is required to give (76).   √ Proposition 5.2. Let ρ(n) = c min{n, λ−1 n2 β} with c > 0 sufficiently small. For all λ sufficiently small and all β < 1,    0 < 0, n, n

 1  (β) −|ρ(n)−ρ(m)| e 0, m, m ≤ Cn(β) Cm H +β

(78)

and     0, n, nP  where (β)

Ck

 1 1  (β) −|ρ(n)−ρ(m)| e , P− 0, m, m  ≤ λCn(β) Cm Lλ + β H +β

(79)

 −1/4 β −3/8  : 0 ≤ k ≤ λ1/2 β −1/4 λ √ −1/2 −1/4 =C λ β k : λ1/2 β −1/4 ≤ k ≤ λβ −1/2  β −1/2 : λβ −1/2 < k

with some constant C which does not depend on β or on λ. Proof. The result depends only on the fact that the operators in question admit the estimates (63)–(66) and (67)–(70). We discuss only the operator H . By symmetry, it suffices to study the case n ≥ m. By Lemma 5.1 and Corollary 4.3,   −1  0, n, n H + β 0, m, m      −1 = n |D + n | e−ρ eρ H + β e−ρ eρ D † |m + |m        ≤ 6 e−ρ |n  + β −1/2 e−ρ |n  eρ |m  + β −1/2 eρ |m 



(β) ρ(m)−ρ(n)
Cn(β) Cm e .

Positivity in (78) follows from integrating 0, n, n|e−tH |0, m, m > 0.

 

24

M.B. Erdo˜gan, R. Killip, W. Schlag

Remark. This new pointwise estimate gives results in line with the conservation of probability (cf, (75)):

0, n, n|(H + β)−1 |0, 0, 0 n




λ−1/4 β −3/8 Cn(β) e−ρ(n)

n


β

−1


+


β −1 +




λβ −1/2 λ1/2 β −1/4 ∞

−1 n2 √β

λ−3/4 β −5/8 e−cλ

β −1 e−t

2

√

√ n dn

t dt

1


β −1 . In this computation we estimated the sum over 0 ≤ n ≤ λ1/2 β −1/4 by replacing Cn by its largest value, namely λ−1/4 β −3/8 . The sum over λβ −1/2 < n is of a geometric series and so can be computed exactly. Lastly, we changed variables in the integral according to n = λ1/2 β −1/4 t. (β)

Proposition 5.3. For all λ sufficiently small, β ≤ λ2 , m ≥ 0, and s ≥ 0,  1  

   n2s 0, n, n 0, m, m β −1 (m2s + λs β −s/2 ), H +β n

(80)

and

   n2s 0, n, nP

n

 1 1  P− 0, m, m
λβ −1 (m2s + λs β −s/2 ), Lλ + β H +β

(81)

where the constants do not depend on λ or β. Combining these two estimates shows that for λ sufficiently small   

  1   1 + n2s 0, n, nP P 0, m, m 1 + m2s + λs β −s/2 . (82) Lλ + β n Proof. The proof of (81) is the same as that of the upper bound in (80), differing only in that it uses (79) instead of (78). It is therefore omitted. We will use the following  simple observation repeatedly: for any function f with f (x) > 0 and f  (x)/f (x)
1, b

n=a




b

f (n)
f (x0 ) +

f (x) dx,

(83)

a

for any x0 ∈ [a, b]. The assumption that λ−1 β 1/2 ≤ 1 is sufficient to ensure the hypothesis on the logarithmic derivative in each of the instances below. We will also use the following simple inequalities:
∞
s nγ e−n dn
s γ e−s , nγ en dn
s γ es , (84) s

0

Energy Growth

25

and their corollaries
∞ 2 2 2 2 nγ +1 e−u n dn
u−2 s γ e−u s ,




s

nγ +1 eu

2 n2

dn
u−2 s γ eu

2s2

.

(85)

0

s

In (84) and (85), the implicit constants depend only on γ > −1. We now begin the proof of the
part of (80); it is much like the argument given in the remark above. The first step is to replace the matrix elements of the resolvent of H by the upper bounds given in (78) of Proposition 5.2. Now we need to calculate the sum over n. The reasoning necessary depends on the value of m and so we consider separately three cases: Case 1. 0 ≤ m ≤ λ1/2 β −1/4 . In this regime, ρ(m) ≤ c and so

(β) −|ρ(m)−ρ(n)| n2s Cn(β) Cm e
λ−1/4 β −3/8 n2s Cn(β) e−ρ(n) n

n


S1 + S2 + S3 , where S1 , S2 , and S3 are given by restricting the sum to those n in [0, λ1/2 β −1/4 ], [λ1/2 β −1/4 , λβ −1/2 ], and [λβ −1/2 , ∞) respectively. Using (83) with x0 = 0, λ1/2 β −1/4

−1/2 −3/4

S1
λ

β

n2s
λs β −1 β −s/2 .

n=0

Next, using (83) with x0 = λ1/2 β −1/4 and (85), we get −3/4 −5/8

S2
λ

β

−1/2 λβ

1

n2s+ 2 e−cβ

1/2 n2 /λ


λ−1/2 β −3/4 + β −1 λs β −s/2 .

n=λ1/2 β −1/4

But β ≤ λ2 and so the above implies S2
λs β −1 β −s/2 . Finally, in S3 , we can extend the sum from λβ −1/2 ≤ n < ∞ to all positive n: S3
λ−1/4 β −7/8

∞

n2s e−cn
λ−1/4 β −7/8
λs β −1 β −s/2 ,

n=0

where the last inequality uses β < λ2 . Case 2. λ1/2 β −1/4 ≤ m ≤ λβ −1/2 . In this regime, ρ(m) = cλ−1 m2 β 1/2 and we proceed as above, i.e., we divide the sum over all n into three pieces S1 , S2 , and S3 each corresponding to the same interval of ns as in Case 1. For the sum over 0 ≤ n ≤ λ1/2 β −1/4 we use (83) with x0 = 0, and the fact that 1/4 x e−cx
1 uniformly for x > 0 and so in particular when x = λ−1 m2 β 1/2 : −3/4 −5/8

S1
λ

β

m

1/2 −cλ−1 m2 β 1/2

e

λ1/2 β −1/4

n=0

s −s/2 −1/4 −7/8


λ β

λ

s −1 −s/2


λ β

β

β

.

m

1/2 −cλ−1 m2 β 1/2

e

n2s

26

M.B. Erdo˜gan, R. Killip, W. Schlag

Next, applying (83) twice with x0 = m and then both parts of (85),
m 1 1 −1 2 2 1/2 n2s+ 2 e−cλ (m −n )β dn S2
λ−1/2 β −1/2 m1/2 λ−1/2 m2s+ 2 + !0
∞ 1 −1 2 2 1/2 + n2s+ 2 e−cλ (n −m )β dn m −1


λ

m2s+1 β −1/2 + m2s β −1


β −1 m2s . In the last inequality, we used the fact that m ≤ λβ −1/2 . Finally, using (83) and (84), ∞

S3
λ−1/2 β −3/4 m1/2

−1 m2 β 1/2

n2s e−cn+cλ

n=λβ −1/2


λ−1/2 β −3/4 m1/2 λ2s β −s e−cλβ

−1/2 +cλ−1 m2 β 1/2


β −1 m2s . To obtain the last inequality, consider the cases s > 1/4 and s ≤ 1/4 separately. Case 3. λβ −1/2 < m. We proceed similarly: By (83) −1/4 −7/8 −cm

S1
λ

β

e

λ1/2 β −1/4

n2s

n=0 s −s/2 1/4 −9/8 −cm


λ β

λ

β

s −s/2 −1

m

−1 s −s/2

,


λ β
β

β

λβ

e

1/4 −cm

e

because x 1/4 e−cx
1. Extending the summation region down to n = 0, applying (83) with x0 = 0 and then (85),
λβ −1/2 1 −1 2 1/2 −1/2 −3/4 −cm S2
λ β e n2s+ 2 ecλ n β dn
β
β

0 −1 2s −s −c(m−λβ −1/2 )

λ β

−1

e

2s

m ,

where the last line follows from m ≥ λβ −1/2 . To complete the proof of Case 3 and so of the
part of (80) we estimate S3 by extending the sum to all n ≥ 0 and applying both parts of (84): S3


∞

n2s e−c|m−n|
β −1 m2s .

n=λβ −1/2

To prove the  part of (80) we use

0, n, n|(H + β)−1 |0, m, m = β −1 , n

which was derived earlier (see (75)).

(86)

Energy Growth

27

Let G(n, m) = 0, n, n|(H + β)−1 |0, m, m and, given  > 0, define  λ1/2 β −1/4 : m ≤ λ1/2 β −1/4 n0 = . m : m ≥ λ1/2 β −1/4

(87)

For  sufficiently small (independent of m, β and λ) one can check that

G(n, m) ≤ 21 β −1 ,

(88)

n
using computations similar to those performed earlier in this proof. By (86), this means that the sum over n ≥ n0 must be at least 21 β −1 and so ∞

n=0

n2s G(n, m) ≥

∞

−1 2s s −s/2 1 −1 2s n2s ). 0 G(n, m) ≥ 2 β n0  β (m + λ β

n=n0

The final inequality here should be checked separately for each case in (87).

 

2s , to be considered as a weight on the range of P , and Lemma 5.4. If w(n) = 1 + n√ W (n, m) = (1 + |n − m|2 )−1 w(n)w(m), to be considered as a weight on the range of P + P ⊥ , then   P J P ⊥ R ⊥ (β)P ⊥  1
1, λ  →1 W

w

when λ is sufficiently small. Proof. The resolvent formula implies that P J P ⊥ Rλ⊥ (β)P ⊥ = P J R0⊥ (β)P ⊥

∞ 

j − iλP ⊥ J R0⊥ (β)P ⊥ .

j =0

Thus, the claim will follow from  ⊥  J R (β)P ⊥  1 0 

1 W →W


1,

(89)

whose proof is very much like that of (48) in Lemma 4.1. For variety, we present the argument slightly differently. Notice that the weight W obeys W (n ± 1, m)
1 W (N, n, m)

and

W (n, m ± 1)
1, W (N, n, m)

and let N denote the multiplication operator N|N, n, m = (1 + N )|N, n, m . By Schur’s test, (28), and the above, J N−1 is a bounded operator on 1W .As R0⊥ (β)|N, n, m = [N + β + i(n2 − m2 )]−1 |N, n, m , the operator NR0⊥ (β)P ⊥ is also bounded on this space. Taking the product of these two bounded operators proves Eq. (89).   We are now ready to complete the proof of the theorem, which we restate for the reader’s convenience.

28

M.B. Erdo˜gan, R. Killip, W. Schlag

Theorem. Let X(t) be a process obeying Hypothesis 1 and let V (θ ) be an even trigonometric polynomial. If the coupling constant λ is sufficiently small then for any initial wave-function ψ0 ∈ H s ,
1 T

∞ 0



e−t/T E
ψ(·, t)
2H s dt λs T s/2
ψ0
2L2 +
ψ0
2H s

(90)

as T → ∞. The expectation is over possible trajectories of the Markov process X(t). In particular, the energy grows on average as the square-root of time. Proof. By Proposition 2.1, we need to estimate    1 + |n|2s 0, n, n

 n

 β  P(0) , Lλ + β

(91)

where P(0) = u0 ⊗ |ψ0 ψ0 | and β = T1 . Recall from Sect. 3 that |0, n, n is in the range of P , if n ≥ 0, or of Q, if n < 0. Recall also that by Lemma 3.1 the ranges of both P + P ⊥ and Q + Q⊥ are invariant subspaces for Lλ . Combining these two facts, we can re-write (91) as    1 + |n|2s 0, n, nP

  β  P + P ⊥ P(0) Lλ + β n≥0   

  β   1 + |n|2s 0, n, nQ Q + Q⊥ P(0) . + Lλ + β



(92)

n<0

As earlier, we will treat the even part of the wave function (the top line above); the odd part may be treated identically. By (22) and (23),   P (Lλ + β)−1 [P + P ⊥ ] = P (Lλ + β)−1 P P − iλP J P ⊥ Rλ⊥ (β)P ⊥ .

(93)

So let |φ1 = P |P(0) and |φ2 = −iλP J P ⊥ Rλ⊥ (β)P ⊥ |P(0) . We will write φj (m) = 0, m, m|φj , j = 1, 2, for their components. Of course, φ1 (m) = |ψˆ 0 (m)|2 , so by Proposition 5.3, Eq. (82),    1 + n2s 0, n, nP

 n≥0

  β  φ1 (m)(1 + m2s + λs β −s/2 ) P φ1

Lλ + β m  2  2

ψ0 H s + λs β −s/2 ψ0 L2 .

(94)

By similar reasoning,          β    |φ2 (m)|(1 + m2s + λs β −s/2 ) 1 + n2s 0, n, nP P φ2 
 Lλ + β  n≥0 m     
φ2 1 + λs β −s/2 φ2 1 , (95) w

Energy Growth

29

where w is as in Lemma 5.4 (we will also use the W notation found there). By this lemma and the fact that 0, n, m|P(0) = ψˆ 0 (m)ψˆ 0 (n),   φ2  1
λ
P(0)
1 w

W

 1/2  1/2 1 ˆ 0 (m)|2 ˆ 0 (n)|2 w(m)| ψ w(n)| ψ 1 + |n − m|2 n,m  2
λψ0 H s ,
λ

where the last inequality used the 2 -boundedness of the matrix with entries (1 + |n − m|2 )−1 . In the particular case that s = 0, the above argument says     φ2  1
λψ0 2 2 . L  Substituting these two estimates for φ2 into (95) gives         2    2 β    1 + n2s 0, n, nP P φ2 
λψ0 H s + λs+1 β −s/2 ψ0 L2 .  Lλ + β n≥0 

(96)

The combination of (94) and (96) shows that (90) does indeed hold once λ is chosen sufficiently small.   Acknowledgements. The authors are grateful to the Institute for Advanced Study (Princeton), where this work was commenced, and in particular, to Thomas Spencer for his interest in this problem. B.E. and R.K. were supported, in part, by NSF Grant DMS-9729992, W.S. was supported by NSF Grant DMS 0070538 and a Sloan fellowship.

References 1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Superintendent of Documents, Washington, DC: U.S. Government Printing Office, 1964 2. Agmon, S.: Lectures on Exponential Decay of Solutions of Second-order Elliptic Equations: Bounds on Eigenfunctions of N-body Schr¨odinger Operators. Princeton, NJ: Princeton Univ. Press, 1982 3. Bourgain, J.: Growth of Sobolev norms in linear Schr¨odinger equations with quasi-periodic potential. Commun. Math. Phys. 204(1), 207–247 (1999) 4. Bourgain, J.: On growth of Sobolev norms in linear Schr¨odinger equations with smooth time dependent potential. J. Anal. Math. 77, 315–348 (1999) 5. Bourgain, J.: Global Solutions of Nonlinear Schr¨odinger Equations. Providence, RI: Am. Math. Soc., 1999 6. Bourgain, J.: On long-time behaviour of solutions of linear Schr¨odinger euations with smooth timedependent potential. Preprint 2002 7. Combes, J.M., Thomas, L.: Asymptotic behaviour of eigenfunctions for multiparticle Schr¨odinger operators. Commun. Math. Phys. 34, 251–270 (1973) 8. Deift, P.A.: Applications of a commutation formula. Duke Math. J. 45(2), 267–310 (1978) 9. Doob, J.L.: Stochastic Processes. New York: Wiley, 1953 10. Gardiner, C.W.: Handbook of Stochastic Methods. Second edition, Berlin: Springer, 1985 11. Pillet, C.-A.: Some results on the quantum dynamics of a particle in a Markovian potential. Commun. Math. Phys. 102(2), 237–254 (1985) 12. Tcheremchantsev, S.: Markovian Anderson model. C. R. Acad. Sci. Paris S´er. I Math. 324(8), 907– 912 (1997) 13. Tcheremchantsev, S.: Markovian Anderson model: Bounds for the rate of propagation. Commun. Math. Phys. 187(2), 441–469 (1997) 14. Tcheremchantsev, S.: Transport properties of Markovian Anderson model. Commun. Math. Phys. 196(1), 105–131 (1998) Communicated by B. Simon

Commun. Math. Phys. 240, 31–51 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0879-4

Communications in

Mathematical Physics

Super Yangian Y (osp(1|2)) and the Universal R-matrix of Its Quantum Double D. Arnaudon, N. Cramp´e , L. Frappat
, E. Ragoucy Laboratoire d’Annecy-le-Vieux de Physique Th´eorique, LAPTH, CNRS, UMR 5108, Universit´e de Savoie, B.P. 110, 74941 Annecy-le-Vieux Cedex, France Received: 13 September 2002 / Accepted: 6 March 2003 Published online: 11 June 2003 – © Springer-Verlag 2003

Abstract: We present the Drinfel’d realisation of the superYangian Y (osp(1|2)), including the explicit expression for the coproduct. We show in particular that it is necessary to introduce supplementary Serre relations. The construction of its quantum double is carried out. This allows us to give the universal R-matrix of DY (osp(1|2)). 1. Introduction The Yangian Y (a) based on a simple Lie algebra a is defined as the homogeneous quantisation of the algebra a[u] = a ⊗ C[u] endowed with its standard bialgebra structure, where C[u] is the ring of polynomials in the indeterminate u. It was introduced by Drinfel’d [1–3]. The most elegant and concise presentation of the Yangian uses the FRT formalism [4], based on a certain evaluated R-matrix. For unitary algebras, the R-matrix is given by R(u) = I ⊗ I + P /u, where P is the permutation map. This can be extended to the superunitary series by considering the superpermutation instead [5, 6]. This formalism was extended to the orthogonal, symplectic and orthosymplectic cases by taking R(u) = I ⊗ I + P /u − K/(u + κ), where K is a partial (super)transposition of P and 4κ = (α0 + 2ρ, α0 ), see for example [7]. Although one can exhibit a suitable R-matrix for any simple Lie algebra or basic simple Lie superalgebra a for defining the Yangian Y (a), the obtained structure is a Hopf algebra, but not a quasi-triangular one. It is well known that the quantum double construction allows one to construct a quasi-triangular Hopf algebra from a Hopf algebra, and that this procedure leads to the universal R-matrix of the algebra under consideration. However, in order to find an explicit useful formula for this universal R-matrix, it is necessary to consider another realisation of the Yangian, given in terms of generators and relations similar to the description of a loop algebra as a space of maps. Unfortunately in this realisation no explicit formula for the comultiplication is known in general, except in the sl(2) case [8].


Member of Institut Universitaire de France

32

D. Arnaudon, N. Cramp´e, L. Frappat, E. Ragoucy

The aim of this paper is to extend this construction to the case of the superalgebra osp(1|2). Let us recall that the super Yangian Y (osp(1|2)) is defined by the relations R12 (u − v) L1 (u) L2 (v) = L2 (v) L1 (u) R12 (u − v)

(1.1)

C(u) = Lt (u − κ) L(u) = 13 ,

(1.2)

and

where the generators of Y (osp(1|2)) are encapsulated into the 3 × 3 matrix L(u) and R12 (u) is the R-matrix introduced in [7]. Writing a Gauss decomposition of L(u), we then introduce some specific combinations of the generators Lij (u), called e(u), f (u) and h(u), which define a Drinfel’d realisation of the super Yangian Y (osp(1|2)). More precisely, we show that the associative algebra A generated by e(u), f (u) and h(u) subjected to certain relations, and the super Yangian Y (osp(1|2)) are isomorphic as bialgebras. At this point, two remarks are in order. First, one is able to find an explicit formula for the comultiplication in terms of the Drinfel’d generators e(u), f (u) and h(u), thereby generalising Molev’s formula in the case of osp(1|2). Second, it is necessary to introduce supplementary Serre-type relations among the e(u), f (u), h(u) generators, as in the case of Uq (
osp(1|2)) [9]. These supplementary Serre-type relations are cubic in the Drinfel’d generators. Indeed, it appears that the quadratic exchange relations among the e(u), f (u), h(u) generators, derived from the RLL relations, lead to a superalgebra which is bigger than Y (osp(1|2)). Hence it is necessary to quotient this bigger structure by supplementary relations. The next step is the construction of the quantum double. The super Yangian Y (osp(1|2)) = Y + being given in terms of the Drinfel’d generators e(u), f (u), h(u) with positive modes, we introduce another set of generators e(u), f (u), h(u) with negative modes, generating a Hopf algebra Y − . We construct a Hopf pairing between Y + and Y − , such that Y − is isomorphic to the dual of Y + with opposite comultiplication. We prove that this Hopf pairing is not degenerate. This allows us to define the double super Yangian DY (osp(1|2)) in a proper way. In particular we are able to give a presentation of the double super Yangian DY (osp(1|2)) in terms of the Drinfel’d generators e(u), f (u), h(u), now Z moded, subjected to suitable quadratic exchange relations, and suitable supplementary Serre-type relations of cubic form. As we emphasised above, the quantum double procedure allows one to construct explicitly the universal R-matrix. {xi , i ∈ N} being a basis of Y + and {x i , i ∈ N} ∈ Y − its dual basis (i.e. < x i , xj >= δji ), the universal R-matrix of DY is given by R =  + and H + denote the unital subalgebras of Y + generated by the xi ⊗ x i . Let E+ , F − , E− and H − be the dual subalgebras. positive modes of e(u), f (u) and h(u), and let F Following the kind of arguments used in [10], we give Poincar´e–Birkhoff–Witt bases for ± in terms of the modes of the generators e(u) and f (u). We show that this leads to E± , F the usual nice factorised expression of the universal R-matrix of DY (osp(1|2)), namely − , RH ∈ H + ⊗ H − and RF ∈ F + ⊗ E− . R = RE RH RF , where RE ∈ E+ ⊗ F Finally, considering the action of the universal R-matrix R on evaluation representations of DY (osp(1|2)), we obtain the evaluated R-matrix of DY (osp(1|2)), which coincides with the one introduced in [7], up to a normalisation factor written as a ratio of 1 functions of period 2κ.

Drinfel’d Realisation of Super Yangian Y (osp(1|2))

33

2. The RTT Presentation of Super Yangian Y (osp(1|2)) We denote by V the 3-dimensional Z2 -graded vector space representation of osp(1|2). The first and the third basis vectors have the grade 0 (mod 2) whereas the second has the grade 1 (mod 2). The same gradation was used by Ding [9] in order to define Uq (
osp(1|2)). The multiplication for the tensor product is defined for a, b, α, β ∈ Y (osp(1|2)) by (a ⊗ α)(b ⊗ β) = (−1)[b][α] (ab ⊗ αβ),

(2.1)

where [a] ∈ Z2 denotes the grade of a. Let Eij be the elementary matrix with entry 1 in row i and column j and 0 elsewhere. The “usual” super transposition .T is defined   by AT = 3i,j =1 (−1)[j ]([i]+[j ]) Aj i Eij for any matrix A = 3i,j =1 Aij Eij . The super transposition .t we will use is a conjugation of the previous one: 3 

At =

(−1)[i][l]+[i] J ij Akj J lk Eil = J AT J −1 , where J = E31 + E22 − E13 .

i,j,k,l=1

(2.2) The super permutation P12 (i.e. X21 = P12 X12 P12 ) is defined by P12 = (−1)[j ] Eij ⊗ Ej i . The super R-matrix R12 (u) ∈ End(V ⊗ V ) is defined by:   P P t1 u R12 (u) = ρ(u) I⊗I+ − u+1 u u+κ  u+1 0 0 0 u 0  0 1 0 −1 0  u  0 0 u+κ−1 0 1  u+κ u+κ  1  0 u 0 1 0 u  2 +κu−κ 1 u  = ρ(u) 0 0 u+κ 0 u(u+κ) u+1  0 0 0 0 0   (2u+κ) 1  0 0 u(u+κ) 0 − u+κ   0 0 0 0 0 0 0 0 0 0

(2.3) 0 0 0 0 (2u+κ) 0 u(u+κ) 0 0 1 0 − u+κ 1 0 u+κ−1 0 u+κ 0 − u1 0 0

 0 0   0    0   0   (2.4)  1 0  u  0 0   1 0  0 u+1 u

0 0 0 0 0

where ρ(u) =

1 (u|2κ) 1 (u + κ − 1|2κ) 1 (u + κ + 1|2κ) 1 (u + 2κ|2κ) 1 (u + 1|2κ) 1 (u + κ|2κ)2 1 (u + 2κ − 1|2κ)

, κ=

3 , 2 (2.5)

.t1 is the super transposition in the first space and I is the 3 × 3 identity matrix. The x/ω

x  function 1 is defined by 1 (x|ω) = √ω  ω . 2πω

34

D. Arnaudon, N. Cramp´e, L. Frappat, E. Ragoucy

It is known (see for instance [7]) that: Proposition 2.1. The matrix R(u) satisfies R12 (u) R13 (u + v) R23 (v) = R23 (v) R13 (u + v) R12 (u), (super Yang–Baxter) (2.6) (2.7) R12 (u + κ)−1 = R12 (u)t1 , (crossing symmetry) R12 (u) R12 (−u) = ρ(u)ρ(−u) I ⊗ I, (unitarity). (2.8) We gave in our previous paper [7] the RTT presentation of superYangian Y (osp(1|2)). In the following, 1 will denote the unit of the algebra and 13 = 1 I. ij

Theorem 2.1. Let U(R) be the associative superalgebra generated by the elements L(n) ( 1 ≤ i, j ≤ 3, n ∈ Z>0 ) and 1, gathered in the formal series L(u) = 13 +

3  

L(n) u−n Eij = ij

3 

Lij (u) Eij

(2.9)

R12 (u − v) L1 (u) L2 (v) = L2 (v) L1 (u) R12 (u − v) .

(2.10)

i,j =1 n∈Z>0

i,j =1

subject to the defining relations

The superYangian Y (osp(1|2)) is then isomorphic to the associative superalgebra Y(R), a quotient of U(R) by the relation C(u) = Lt (u − κ) L(u) = 13 .

(2.11)

The Hopf algebra structure of Y(R) is given by

 ˙ L(u) i.e.,

L(u) = L(u) ⊗

3

 

Lij (u) = Lik (u), ⊗Lkj (u)

(2.12)

k=1

S(L(u)) = L(u)−1 ;

ε(L(u)) = I.

(2.13)

Note that the central element C(u) in U(R) plays the rˆole of the quantum determinant in Y (gl(N )). In particular, the relation C(u) = 13 will be essential in the proof of surjectivity in Theorems 3.1 and 4.1, which turns out to imply the existence of a Gauss decomposition of L(u). 3. The Drinfel’d Realisation of Y (osp(1|2)) Definition 3.1. Let A+ be the associative superalgebra generated by the odd elements ek , fk (k ∈ Z≥0 ), the even elements hk (k ∈ Z≥0 ), the unit1 and the defining rela∞ −k−1 , f (u) = tions, given in terms of the generating functions e(u) = k=0 ek u ∞ ∞ −k−1 −k−1 , h(u) = 1 + k=0 hk u : k=0 fk u

Drinfel’d Realisation of Super Yangian Y (osp(1|2))

[h(u), h(v)] = 0, h(v) − h(u) {e(u), f (v)} = , u−v (u − v + κ − 1)[h(u), e(u) − e(v)] [h(u), e(v)] = − (u − v)(u − v + κ) h(u)(e(v) + e(u) − e0 ) − (e(u) − e0 )h(u) + , u−v+κ (u − v − κ + 1)[h(u), f (u) − f (v)] [h(u), f (v)] = (u − v)(u − v − κ) h(u)(f (v) + f (u) + f0 ) − (f (u) + f0 )h(u) − , u−v−κ [e(u), e(v)] {e0 , e(u) − e(v)} (e(u) − e(v))2 {e(u), e(v)} = , − − 2 (u − v) (u − v) 2 (u − v)2 [f (u), f (v)] {f0 , f (u) − f (v)} (f (u) − f (v))2 {f (u), f (v)} = − , − − 2 (u − v) (u − v) 2 (u − v)2

35

(3.1) (3.2)

(3.3)

(3.4) (3.5) (3.6)

and the supplementary Serre relations e(u)3 = e(u){e(u), e0 } + [e02 , e(u)], f (u) = 3

−f (u){f (u), f0 } + [f02 , f (u)].

(3.7) (3.8)

The relations (3.1)–(3.6) are equivalent to the following commutation relations in terms of the modes ek , fk , hk (k ≥ 0) : • hk and hl :

[hk , hl ] = 0,

• ek and fl :

{ek , fl } − hk+l = 0,

(3.10)

• hk and el :

[h0 , el ] − el = 0,

(3.11)

2 [h1 , el ] − 2el+1 = {h0 , el },

(3.12)

2 [hk+2 , el ] + 2 [hk , el+2 ] − 4 [hk+1 , el+1 ] = [hk , el ] + {hk+1 , el } − {hk , el+1 },

(3.13)

[h0 , fl ] + fl = 0,

(3.14)

2 [h1 , fl ] + 2fl+1 = −{h0 , fl },

(3.15)

2 [hk+2 , fl ] + 2 [hk , fl+2 ] − 4 [hk+1 , fl+1 ] = [hk , fl ] − {hk+1 , fl } + {hk , fl+1 },

(3.16)

2{ek+2 , el } + 2{ek , el+2 } − 4{ek+1 , el+1 } = {ek , el } + [ek+1 , el ] − [ek , el+1 ],

(3.17)

2{fk+2 , fl } + 2{fk , fl+2 } − 4{fk+1 , fl+1 } = {fk , fl } − [fk+1 , fl ] + [fk , fl+1 ].

(3.18)

• hk and fl :

• ek and el :

• fk and fl :

(3.9)

36

D. Arnaudon, N. Cramp´e, L. Frappat, E. Ragoucy

The Serre relations (3.7) and (3.8) in terms of modes are conjectured to be (for k ≥ 0) [{ek , ek+1 }, ek ] = 2ek3 ,

(3.19)

1 1 [{ek , ek+1 }, ek+1 ] = −ek ek+1 ek − ek2 ek+1 − ek+1 ek2 , 2 2

(3.20)

2 2 [{ek , ek+1 }, ek+2 ] = −2ek+1 ek − 2ek+1 ek ek+1 − 2ek ek+1 ,

(3.21)

[{fk , fk+1 }, fk ] = −2fk3 ,

(3.22)

1 1 [{fk , fk+1 }, fk+1 ] = fk fk+1 fk + fk2 fk+1 + fk+1 fk2 , 2 2

(3.23)

2 2 [{fk , fk+1 }, fk+2 ] = 2fk+1 fk + 2fk+1 fk fk+1 + 2fk fk+1 .

(3.24)

This conjecture is supported by two results: on the one hand we have proved them in the graded algebra (to be defined below); on the other hand we checked explicitly the first nine relations. Proposition 3.1. The algebra A+ (resp. Y(R)), can be equipped with an ascending filtration with the degree of the generators defined by deg(ek ) = k, deg(fk ) = k, ij deg(hk ) = k (deg(L(k) ) = k − 1) and deg(xy) = deg(x) + deg(y), for x, y ∈ A+ (resp. x, y ∈ Y(R)). Let gr(A+ ) and gr(Y(R)) denote the corresponding graded algebras and osp(1|2)[u] denote the Lie super algebra of polynomials in an indeterminate u with coefficients in osp(1|2). The algebras gr(A+ ), gr(Y(R)) and U (osp(1|2)[u]) are isomorphic. Proof. We first recall the notion of graded algebras. We start with an algebra A equipped with a grading deg, i.e. a morphism from (A, .) to (N, +). One introduces Ak = {x ∈ A, deg(x) ≤ k}, k ≥ 0 and gr(Ak ) = Ak /Ak−1 for k ≥ 1, gr(A0 ) = A0 . Then the graded algebra of A is gr(A) = ⊕k≥0 gr(Ak ). The algebra gr(A+ ) is the algebra generated by ek , fk , hk (k ∈ Z) and the relations (3.9)–(3.24) where the right hand side of the equalities is substituted by zero. These equalities are equivalent to : [hk , hl ] = 0, {ek , fl } = hk+l , [hk , el ] = ek+l , [hk , fl ] = −fk+l , {en , em } = {e0 , en+m }, {fn , fm } = {f0 , fn+m } , [{el , em }, en ] = 0, [{fl , fm }, fn ] = 0,

(3.25) (3.26) (3.27) (3.28)

which are the relations of U (osp(1|2)[u]). Then, gr(A+ ) is isomorphic to U (osp(1|2)[u]). The isomorphism between gr(Y(R)) and U (osp(1|2)[u]) is proved in [7].  Theorem 3.1. The linear map φ:

A+ −→ Y(R) e(−u) −→ L33 (u)−1 L23 (u) (3.29) 32 33 −1 f (−u) −→ L (u)L (u) (3.30) h(−u) −→ L22 (u)L33 (u)−1 + L32 (u)L33 (u)−1 L23 (u)L33 (u)−1 (3.31)

is an isomorphism of algebra.

Drinfel’d Realisation of Super Yangian Y (osp(1|2))

37

Proof. The first step of the proof is to show that φ is a morphism of algebra: {φ(e(−u)), φ(f (−v))} = {L33 (u)−1 L23 (u), L32 (v)L33 (v)−1 }     = −L33 (u)−1 L32 (v) L23 (u), L33 (v)−1 + L33 (u)−1 L23 (u), L32 (v) L33 (v)−1   − L33 (u)−1 , L32 (v) L33 (v)−1 L23 (u) (3.32)   1 = (3.33) −L33 (u)−1 φ(h(−v))L33 (u) + L33 (u)−1 φ(h(−u))L33 (u) u−v 1 = (3.34) (φ(h(−u)) − φ(h(−v))) . u−v We used the relations (2.10) and [φ(h(−v)), L33 (u)] = 0. For the other relations, the proofs are similar once one remarks, in particular, that L21 (u)L33 (u)−1 = L22 (u)L33 (u)−1 φ(f (−u)) − φ(f (−u)) −[φ(f0 ), L22 (u)L33 (u)−1 ], L12 (u)L33 (u)−1 = φ(e(−u + 1)) − L22 (u)L33 (u)−1 φ(e(−u + 1)) −[φ(e0 ), L22 (u)L33 (u)−1 ], L31 (u)L33 (u)−1 = (φ(f (−u)))2 + {φ(f0 ), φ(f (−u))}, L13 (u)L33 (u)−1 = (φ(e(−u + 1)))2 − {φ(e0 ), φ(e(−u + 1))}.

(3.35) (3.36) (3.37) (3.38)

The second step consists in proving the surjectivity of φ. The relations (3.29)–(3.31), (3.35)–(3.38) and the following particular relations, coming from (2.11), C 22 (u) = L22 (u − κ)L22 (u) + L32 (u − κ)L12 (u) − L12 (u − κ)L32 (u) = 1, (3.39) C 33 (u) = L11 (u − κ)L33 (u) + L21 (u − κ)L23 (u) − L31 (u − κ)L13 (u) = 1, (3.40) constitute a system of nine equations. We can show that these equations are independent and allow us to express all the generators of Y(R) in terms of φ(en ), φ(hk ) and φ(fl ) (n, k, l ≥ Z≥0 ). This proves the surjectivity of φ. The final step is the proof of the injectivity of φ. The map φ preserves the filtration, therefore defines a surjective morphism between gr(A+ ) and gr(Y(R)). Since the injectivity of the latter morphism is given by Proposition 3.1, the injectivity of φ is proved.  Note that the RLL relations encode both the commutation relations and the Serre relations. + + Let E and F be the subalgebras of A+ , without the unit, generated by {ek , hl |k, l ≥ 0} and {fk , hl |k, l ≥ 0}, respectively. Let E + , H+ and F + be the subalgebras of A+ + and F + be the same generated by ek , hk and fk with k ≥ 0, respectively and E+ , H algebras with the unit. Proposition 3.2. φ provides A+ with a coalgebra structure given by : • counit: ε(e(u)) = 0 ,

ε(f (u)) = 0 ,

ε(h(u)) = 1 .

(3.41)

38

D. Arnaudon, N. Cramp´e, L. Frappat, E. Ragoucy

• coproduct :

 

(e(u)) = e(u) ⊗ 1 + h(u) ⊗ e(u) + [h(u), f0 ] ⊗ (e(u)2 − {e(u), e0 })   1 ⊗ 1 + X12 (u) , (3.42)   X12 (u) = (−1)k f (u − 1) ⊗ e(u) + (f (u − 1)2 k>0

k + {f (u − 1), f0 }) ⊗ (e(u)2 − {e(u), e0 }) ,  

(f (u)) = 1 ⊗ f (u) + f (u) ⊗ h(u) + (f (u)2 + {f (u), f0 }) ⊗ [h(u), e0 ]   1 ⊗ 1 + Y12 (u) , (3.43)   Y12 (u) = (−1)k f (u) ⊗ e(u + 1) k>0

k +(f (u)2 + {f (u), f0 }) ⊗ (e(u + 1)2 − {e(u + 1), e0 }) ,

(h(u)) = 1 ⊗ 1 + { (e(u)), f0 ⊗ 1 + 1 ⊗ f0 } .

(3.44)

) the coproduct of A+

Proof. To clarify this proof, we denote A (resp. Y (resp. Y(R)) and εA (resp. εY ) the counit of A+ (resp. Y(R)). We construct A thanks to the relation Y ◦ φ = (φ ⊗ φ) ◦ A . At first, we calculate

Y (f (u)). We begin by      −1 −1

Y (φ(f (−u))) = Y L32 (u)L33 (u) = Y L32 (u) Y L33 (u) (3.45) and  3   −1  −1   33 3k 33 −1 k3 33 −1 33 33 = L (u)L (u) ⊗ L (u)L (u) L (u) ⊗ L (u)

Y L (u) 

k=1

 = L33 (u)−1 ⊗ L33 (u)−1  2 n   3k 33 −1 k3 33 −1 L (u)L (u) ⊗ L (u)L (u) . n≥0

(3.46)

k=1

Using the results of the proof of the previous theorem, we get  

Y (φ(f (−u))) = (φ ⊗ φ) 1 ⊗ f (−u) + f (−u) ⊗ h(−u)



+(f (−u)2 + {f (−u), f0 } ⊗ [h(−u), e0 ])



1 ⊗ 1 + Y12 (−u)

= (φ ⊗ φ) ◦ A (f (−u)). By the injectivity of φ, we find (3.43). For A (e(u)), the proof is similar and for

A (h(u)), the equality (3.44) is obvious. For the counit, the proof is similar by using εA = εY ◦ φ. 

Drinfel’d Realisation of Super Yangian Y (osp(1|2))

39

Unfortunately, no explicit formula is known for the coproduct in terms of the modes. Note that (3.42)–(3.44) imply that:   + (3.47)

(e(u)) = e(u) ⊗ 1 + h(u) ⊗ e(u) + mod F ⊗ E + E + ,   +

(f (u)) = 1 ⊗ f (u) + f (u) ⊗ h(u) + mod F + F + ⊗ E , (3.48)  +  +

(h(u)) = h(u) ⊗ h(u) + mod E ⊗ F . (3.49) To prove (3.49), we need to calculate the anticommutator of the relation (3.44).

4. The Construction of the Double DY (osp(1|2)) In the following, we replace the notations L(u), Lij (u), e(u), f (u) and h(u) by L+ (u), ij L+ (u), e+ (u), f + (u) and h+ (u) respectively. 4.1. RTT presentation. Definition 4.1. Let DY(R) be the associative superalgebra generated by the elements ij L(n) (1 ≤ i, j ≤ 3, n ∈ Z), 1 and the defining relations, given in terms of formal series   3 ij + ij (u) E and L− (u) = L+ (u) = 13 + 3i,j =1 n∈Z>0 L(n) u−n Eij = ij i,j =1 L  3  ij ij 13 + i,j =1 n∈Z≤0 L(n) u−n Eij = 3i,j =1 L− (u) Eij : ± ± ± • R12 (u − v) L± 1 (u) L2 (v) = L2 (v) L1 (u) R12 (u − v) ,

(4.1)

•

(4.2)

− R12 (u − v) L+ 1 (u) L2 (v) = ± ±t ±

• C (u) = L

+ L− 2 (v) L1 (u) R12 (u − v) ,

(u − κ) L (u) = 13 .

(4.3)

The Hopf algebra structure of DY(R) is given by the relations (2.12) and (2.13) with the substitution L(u) → L± (u). Proposition 4.1. The bilinear form <, > between the two subalgebras of DY(R), ij ij Y − (R) = {L(n) |n ∈ Z≤0 } with opposite coproduct and Y + (R) = Y(R) = {L(n) |n ∈ Z>0 } given by: + −1 < L− 1 (u), L2 (v) > = R21 (v − u)  lj ij kl i.e. < L− (u), L+ (v) > = R −1 (v − u) , −

+

ki

< L (u), 13 > = < 13 , L (v) >=< 13 , 13 >= I

(4.4) (4.5)

is a Hopf pairing, i.e. satisfies the conditions for a, b ∈ Y − (R) and α, β ∈ Y + (R): < a, αβ > = (−1)[α][β] < (a), β ⊗ α >, < ab, α >=< a ⊗ b, (α) >, (4.6) ε(a) = < a, 1 >, ε(α) =< 1, α >, < S −1 (a), α >=< a, S(α) >, (4.7) < a ⊗ b, α ⊗ β > = (−1)[b][α] < a, α >< b, β > . (4.8)

40

D. Arnaudon, N. Cramp´e, L. Frappat, E. Ragoucy

Proof. The proof for the consistency of (4.4) and (4.5) with the defining relations (4.1)– (4.3) and the conditions (4.6) and (4.7) uses the same methods as in [11]. For example: + + + + < L− 0 (w), (R12 (u − v)L1 (u) L2 (v) − L2 (v) L1 (u)R12 (u − v)) > + − + = R12 (u − v) < L− 0 (w), L2 (v) >< L0 (w), L1 (u) >

+ − + − < L− 0 (w), L1 (u) >< L0 (w), L2 (v) > R12 (u − v)

=

−1 −1 R12 (u − v)R20 (v − w)R10 (u − w) −1 −1 −R10 (u − w)R20 (v − w)R12 (u − v)

(4.9) (4.10) (4.11)

= 0 (due to (2.6)) 

We will show below (see remark after Theorem 4.3) that this pairing is not degenerate.

4.2. Drinfel’d realisation. Definition 4.2. Let DA be the associative superalgebra generated by the unit 1, the even elements hk (k ∈ Z) and the odd elements ek , fk (k ∈ Z), gathered in the generating functions: e+ (u) =

∞ 

ek u−k−1 ,

k=0

e− (u) = − f + (u) = f − (u) =

−1 

ek u−k−1 ,

k=−∞ ∞  fk u−k−1 , k=0 −1  − fk u−k−1 , k=−∞

h+ (u) = 1 +

∞ 

hk u−k−1 ,

e(u) = e+ (u) − e− (u) ,

(4.12)

f (u) = f + (u) − f − (u),

(4.13)

h− (u) = 1 −

k=0

−1 

hk u−k−1 ,

(4.14)

k=−∞

satisfying the relations hα (u)hβ (v) = hβ (v)hα (u) where α, β = ±,

 {e(u), f (v)} = δ(u − v) h+ (u) − h− (u) with δ(u − v) =

(4.15) ∞ 

uk v −k−1 ,

(4.16)

k=−∞

(u − v − 1)(2u − 2v + 1)e(u)h± (v) = (u − v + 1)(2u − 2v − 1)h± (v)e(u), (u − v + 1)(2u − 2v − 1)f (u)h± (v) = (u − v − 1)(2u − 2v + 1)h± (v)f (u), (u − v − 1)(2u − 2v + 1)e(u)e(v) = −(u − v + 1)(2u − 2v − 1)e(v)e(u), (u − v + 1)(2u − 2v − 1)f (u)f (v) = −(u − v − 1)(2u − 2v + 1)f (v)f (u),

(4.17) (4.18) (4.19) (4.20)

Drinfel’d Realisation of Super Yangian Y (osp(1|2))

41

and the supplementary Serre relations  e0 , e± (u) e± (u)     5 2 ±  = e0 , e (u) − 2u e02 , e± (u) + e0 e± (u)e0 − {e0 , e1 }, e± (u) , (4.21) 2   ± ± f0 , f (u) f (u)      5 = − f02 , f ± (u) − 2u f02 , f ± (u) − f0 f ± (u)f0 − {f0 , f1 } , f ± (u) . (4.22) 2



The bialgebra structure of DA is given by (3.42), (3.43), (3.44) and (3.41), adding superscripts ± to e(u), f (u) and h(u). Remark 4.1. DA could be alternatively defined by the relations (3.1)–(3.6), adding a superscript to the generating functions with parameter u, and a superscript to the generating functions with parameter v, where , = ±. The Serre relations (3.7) and (3.8) with are also valid in DA, but not sufficient, because they do not couple enough positive modes with the negative ones. The commutation relations in DA between ek , fk and hk (k ∈ Z) are the same as the relations (3.9)–(3.18) with, in this case, k, l ∈ Z and with the following additional relations, for k ∈ Z : 2[h−1 , ek ] − 2ek−1 = [h−1 , ek−2 ] − {h−1 , ek−1 }, 2[h−1 , fk ] − 2fk−1 = [h−1 , fk−2 ] + {h−1 , fk−1 }.

(4.23) (4.24)

Similarly to Sect. 3, we conjecture that the Serre relations in terms of modes are of the form (3.19)–(3.24) with now k ∈ Z. As before, this conjecture is supported by explicit computations on the first orders. Moreover, we can define a graded algebra for DA, grad(DA), as in Proposition 3.1. Then the Serre relations in grad(DA) are the relations (3.19)–(3.24), for k ∈ Z, substituting the right hand side of the equalities by zero. Beside this, the expansion of (4.21), (4.22) in terms of modes shows that the remaining terms of the Serre relations in DA have strictly lower degree and are also cubic. These results are sufficient to prove the next theorems. Let A+ and A− be the subalgebras of DA generated respectively by {en , fn , hn |n ∈ Z≥0 } and {ek , fk , hk , E−1 , F−1 |k ∈ Z<0 }. Theorem 4.1. The linear maps  : DA −→ DY(R) and φ ± : A± −→ Y ± (R) given by e± (−u) −→ L± (u)−1 L± (u) 33

±

± 32

±

± 22

f (−u) −→ L h (−u) −→ L

23

(u)L

± 33

(u)L

± 33

(u)

−1

(u)

−1

(4.25) (4.26) +L

± 32

(u)L

± 33

(u)

−1 ± 23

L

(u)L

± 33

(u)

−1

, (4.27)

are isomorphisms of bialgebra. Proof. The proof is similar to the one of Theorem 3.1 and Proposition 3.2.



42

D. Arnaudon, N. Cramp´e, L. Frappat, E. Ragoucy

For later convenience, we introduce the following combinations in DA (for k ∈ Z): ek 2 , 4 fk 2 = {fk , fk+1 } − . 4

E2k+1 = {ek , ek+1 } −

(4.28)

F2k+1

(4.29)

In particular, E−1 and F−1 will be essential as well as their image in Y − (R) : 2    k   k 5  −  : E−1 −→ −  L23 L13 −L33 −L33 (0) (0) (0) (0) , 4 k≥0 k≥0 2  k  5  32   33 k  33 −L(0) −L L(0) + L31 . F−1 −→ − (0) (0) 4 k≥0

(4.30)

(4.31)

k≥0

Proposition 4.2. The Hopf pairing <, > between A− and A+ is given by : 1 −1 , < e− (u), f + (v) >= , u−v u−v (u − v − 1)(2u − 2v + 1) < h− (u), h+ (v) > = , (u − v + 1)(2u − 2v − 1) < F−1 , e0 2 > = 1, < E−1 , f0 2 >= 1,

< f − (u), e+ (v) > =

(4.32) (4.33) (4.34)

or in terms of generators (n, k ≥ 0): < 1, 1 > = 1, < f−k−1 , en >= −δn,k , < e−k−1 , fn >= δn,k ,      n 1 n−k 1 < h−k−1 , hn > = − . 2+ − k 3 2

(4.35) (4.36)

Proof. We use Theorem 4.1 to prove this proposition, for example < e− (−u), f + (−v) > = < L− (u)−1 L− (u), L+ (v)L+ (v)−1 > 33

− 33

23

−1

32

33

(u) ⊗ L (u),   −1  32 33 >

L+ (v) L+ (v)

=
(4.38)

= < L− (u), L+ (v)L+ (v)−1 > 23

32

33

− 22

+ 33

−1

= < L (u), L 1 = . u−v

(4.37)

− 23

(v) >

− 23


(u), L

(4.39) + 32

(v) > (4.40)

The difficult point lies in the step between equalities (4.38) and (4.39). It is done using the explicit form (3.46) and showing that only the first term of the sum contributes to the pairing. For < f − (u), e+ (v) >, the proof is similar. The identity < h− (u), h+ (v) >=< h− (u), {e+ (v), f0 } + 1 > and the two previous results allow us to obtain the relation (4.33).

Drinfel’d Realisation of Super Yangian Y (osp(1|2))

43

The pairings < F−1 , e0 2 > and < E−1 , f0 2 > are calculated thanks to the explicit expressions (4.30) and (4.31). To find the explicit form (4.36), we remark that (n ≥ 0): 1 < h− (u), h0 >= 1, < h− (u), h1 > = u + , (4.41) 2   1 < h− (u), hn+2 > − 2u + < h− (u), hn+1 > 2   1 +(u + 1) u − (4.42) < h− (u), hn > = 0. 2

n+2 2  + 3 u+ 21 (u+1)n+1 + A trivial induction shows that < h− (u), hn+2 >= 13 u − 21 1 n+1 , which gives the result.  3 (u + 1) −

−

4.3. Dual bases. Now, we look for bases of A+ and A− . Let E and F be the subalgebras of A− , generated by {ek , E−1 , hl |k, l < 0} and {fk , F−1 , hl |k, l < 0} respectively. Let E − , H− and F − be the subalgebras of A− , without the unit, generated − and F − by {ek , E−1 |k < 0}, {hk |k < 0} and {fk , F−1 |k < 0}, respectively and E− , H be the same algebras with the unit. + F + and a− ∈ F − E− . − H Proposition 4.3. (i) Let a± ∈ A± , then a+ ∈ E+ H ± ± ±    (ii) ∀e± ∈ E , h± ∈ H , f± ∈ F , < f− h− e− , e+ h+ f+ >= (−1)[e+ ][e− ] < f− , e+ >< h− , h+ >< e− , f+ > . (4.43) Proof. (i) We use a proof similar to the one given in [10]. We first consider A+ . As {ek , hl , fm |k, l, m ≥ 0} is a generating set of A+ , it is enough to prove that any monoj mial i=0 eki αi hli βi fmi γi , with j, ki , αi , li , βi , mi , γi ∈ Z≥0 , is a linear combination of  + F + . We make an induction on the degree p = j (αi + elements belonging to E+ H i=0 βi + γi ) of such a monomial. For p = 1, the assertion is obvious. Let’s assume the assertion is true for p ≥ 1 and consider an element j βi γi such that j (α + β + γ ) = p + 1. The last p generators αi i i i=0 eki hli fmi i=0 i can be ordered using the induction hypothesis on p. Then, three cases are possible depending on the first element: • It belongs to E+ . Then, the element is ordered. + . If the second generator belongs to H + or F + , the assertion for • It belongs to H p + 1 is proven. It remains the case where the second generator (say ek ) belongs to E + . We make an induction on the index, l, of the first generator hl . Let ap−1 be the ordered product of the last p − 1 generators. If l = 0 then h0 ek ap−1 = ek h0 ap−1 + ek ap−1 due to (3.11).

(4.44)

As the induction on p allows us to order h0 ap−1 , we can order this element. If l = 1 then h1 ek ap−1 = ek h1 ap−1 + ek+1 ap−1 + h0 ek ap−1 + ek h0 ap−1 due to (3.12). (4.45)

44

D. Arnaudon, N. Cramp´e, L. Frappat, E. Ragoucy

Using the case l = 0 for h0 ek ap−1 and the induction hypothesis on p for h1 ap−1 and h0 ap−1 , we can order this element. Let l ≥ 2 and assume that for l − 2 and l − 1 the elements can be ordered. Then, using the commutation relations (3.13), one gets hl ek ap−1 = (−hl−2 ek+2 + hl−1 ek /2 − hl−2 ek+1 /2 +hl−2 ek /2 + 2hl−1 ek+1 )ap−1 + bp+1 ,

(4.46)

where bp+1 can be ordered thanks to the induction on p. The other elements are ordered by the induction on l − 2 and l − 1. This ends the induction on l. + . We denote it fm . If the second element belongs to F + , the assertion • It belongs to F + , we prove the assertion for for p + 1 is proven. If the second element belongs to H p + 1 analogously to the previous case, using (3.14), (3.15) and (3.16). Finally, if the second element ek belongs to E+ , then fm ek ap−1 = −ek fm ap−1 + hm+k ap−1 due to (3.10)

(4.47)

and fm ap−1 , hm+k ap−1 are ordered thanks to the hypothesis on p. This ends the induction on p and (i) is proven for A+ . For A− , the proof is almost similar. However, an additional difficulty appears because of exchange relations between e−k and h−1 . The relation (4.23) allows us to order eˆ−l ≡ 2e−l − e−l−1 − e−l−2 (l ≥ 1) and h−1 . Fortunately, e−k can be expressed in terms of eˆ−l :     +∞ 1 l−k+1 1 1− − eˆ−l . (4.48) ∀k ≥ 1, e−k = 3 2 l=k

Therefore, we can order e−k and h−1 . Likewise, e−k and h−n can be ordered. − , e¯+ ∈ E+ H + , e− ∈ E − and f+ ∈ F + . One computes − H (ii) Let f¯− ∈ F < f¯− e− , e¯+ f+ > = < f¯− ⊗ e− , (e¯+ ) (f+ ) > + = < f¯− ⊗ e− , (e¯+ ⊗ 1 + mod(A+ ⊗ E )) (1 ⊗ f+ + mod(F + ⊗ A+ )) > = (−1)[e¯+ ][e− ] < f¯− , e¯+ >< e− , f+ >

(4.49) (4.50) (4.51)

using the following identities: +

< e− , E f+ > = (−1)[f+ ][E = (−1) = 0,

+

]

+

[f+ ][E ]

< (e− ), f+ ⊗ E

+

> −

< e− ⊗ 1 + mod(A ⊗ E − ), f+ ⊗ E

+

>

(4.52) + [e¯+ ][F + ] + ¯ ¯ < (f− ), F ⊗ e¯+ > < f− , e¯+ F > = (−1) − [e¯+ ][F + ] < 1 ⊗ f¯− + mod(F ⊗ A− ), F + ⊗ e¯+ > = (−1) = 0, (4.53) + + [F + ][A+ ] + + ¯ ¯ < (f− ), F ⊗ A > < f− , A F > = (−1) = (−1)[F = 0.

+ ][A+ ]

< 1 ⊗ f¯− + mod(F

−

⊗ A− ), F + ⊗ A+ > (4.54)

Drinfel’d Realisation of Super Yangian Y (osp(1|2))

45

Let f− ∈ F − , h− ∈ H− , e+ ∈ E + and h+ ∈ H+ such that f¯− = f− h− and e¯+ = e+ h+ . Then, we prove analogously that <  f¯− , e¯+ >=< f− h− , e+ h+ >=< f− , e+ >< h− , h+ > using (h+ ) = 1 ⊗ h+ + i hi ⊗ h i + mod(E + H+ ⊗ H+ F + ), where hi and h i ∈ H+ . Remarking that [e¯+ ] = [e+ ] and that for the unit the theorem is obvious, we prove the second assertion of the theorem.  − , Remark 4.2. The point (ii) of the previous theorem shows that the dual of E− (resp. H − + + + − + − +         F ) is F (resp. H , E ). In particular, one has < E , E >= 0 =< E , H >,  and the same relations changing E by F. − , F + and E− are respectively Theorem 4.2. Bases of E+ , F   BE + = e0 a0 E1 b0 e1 a1 E3 b1 . . . ek ak E2k+1 bk . . . |a0 , a1 . . . , b0 , b1 . . . ∈ Z≥0 ,  BF − = F−1 b0 f−1 a0 F−3 b1 f−2 a1 . . . F−2k−1 bk f−k−1 ak . . .  |a0 , a1 . . . , b0 , b1 . . . ∈ Z≥0 ,   BF + = F2k+1 bk fk ak . . . F3 b1 f1 a1 F1 b0 f0 a0 . . . |a0 , a1 . . . , b0 , b1 . . . ∈ Z≥0 ,  BE − = e−k−1 ak E−2k−1 bk . . . e−2 a1 E−3 b1 e−1 a0 E−1 b0 . . .  |a0 , a1 . . . , b0 , b1 . . . ∈ Z≥0 .

(4.55) (4.56) (4.57) (4.58)

− , F + and E− , respecProof. To prove that BE + , BF − , BF + and BE − generate E+ , F tively, the methods are the same as the ones used in the proof of the assertion (i) of Proposition 4.3, using the commutation relations between ek and el as well as the proved results on the Serre relations. The independence of the set of generators is given by their independence in the corresponding graded algebras which is obvious.  Theorem 4.3. For any element b+ of BE + (resp. BF + ), there is one and only one element b− in BF − (resp. BE − ) such that the pairing < b− , b+ > does not vanish. They are given by: k

<

−→ 

k

bn

F−2n−1 f−n−1

2an +cn

,

n=0

−→ 

em 2bm +cm E2m+1 am >

m=0 

= (−1)

0≤n<m≤k cn cm

k 

(−1)cl al ! bl !,

(4.59)

l=0 k

k

<

←− 

2an +cn

e−n−1

bn

E−2n−1 ,

n=0

←− 

F2m+1 am fm 2bm +cm >

m=0 

= (−1)

0≤n<m≤k cn cm

k 

al ! bl !,

(4.60)

l=0

where k ∈ Z≥0 , ak , bk ∈ Z≥0 , ck ∈ {0, 1}, k

−→  n=0

k

en = e0 e1 . . . ek−1 ek and

←−  n=0

en = ek ek−1 . . . e1 e0 .

46

D. Arnaudon, N. Cramp´e, L. Frappat, E. Ragoucy

Proof. We first consider the pairing < a− , ek > for a− ∈ BF − . If a− has degree at least

, a

∈ B − such that a = a a

. Using (4.35), we get 2, there are a− − F − − −

< a− , ek > = < a− ⊗ a− , (ek ) >

= < a− ⊗ a− , ek ⊗ 1+1 ⊗ ek +

k−1 

hl ⊗ ek−l−1 + mod(F

+

⊗ E +E +) >

l=0

= < a− ⊗ a− , ek ⊗ 1 + 1 ⊗ ek +

k−1 

hl ⊗ ek−l−1 >= 0,

l=0

This shows that the pairing of < a− , ek > is different from zero only for a− = f−k−1 and given by (4.35). Consider now < a− , ek2 >. If a− = f−m has degree 1, one computes: < f−m , ek2 > = < (f−m ), ek ⊗ ek > = < f−m ⊗ 1 + 1 ⊗ f−m +

k−1 

hl ⊗ f−m−l−1 , ek ⊗ ek >= 0. (4.61)

l=0

a

of degree at least 2, one has For a− = a− −



< a− , ek 2 > = < a− ⊗ a− , (ek )2 >=< a− ⊗ a− ,

k−1 

[ek , hl ] ⊗ ek−l−1 >

(4.62)

l=0

= < a− ⊗ a− , −ek ⊗ ek−1 +

k−1 

l (ek , ek+1 , . . . , ek+l ) ⊗ e−l−1 >,

l=1

(4.63) where l (ek , ek+1 , . . . , ek+l ) is a linear combination of ek , ek+1 , . . . , ek+l . Then, for

= f

k ≥ 1, < a− , ek 2 > is equal 1 for a− −k−1 and a− = f−k (i.e. for a− = F−2k−1 ) and 0 otherwise. For k = 0, by (4.34) and the previous calculation, we know that the pairing of e0 2 does not vanish only with F−1 . Similarly, < f−k−1 2 , a+ > is equal to 1 if a+ = E2k+1 , and to 0 in the other cases. Now, we show by induction that < F−2k−1 b , ek 2b >= b! and < F−2k−1 b f−k−1 , 2b+1 ek >= −b! and that the other pairings with ek 2b or ek 2b+1 are zero. We assume these assertions for b < b0 . < a− , ek 2b0 >

b0  k−1 

=< a− ⊗ a− , ek 2 ⊗ 1 + 1 ⊗ ek 2 + l (ek , ek+1 , . . . , ek+l ) ⊗ ek−l−1 >

⊗ a− , =< a−

p=0

=<

a−

⊗ a− ,

l=0

b0  

 b0 ek 2p ⊗ ek 2(b0 −p) > p  b0 ek 2p ⊗ ek 2(b0 −p) > . p

b 0 −1  p=1

(4.64)

Drinfel’d Realisation of Super Yangian Y (osp(1|2))

47

Due to the hypothesis, this pairing is non-zero only if a = F−2k−1 p and a

= F−2k−1 b0 −p and, in this case, is equal to     b0 b0 < F−2k−1 p , ek 2p >< F−2k−1 b0 −p , ek 2(b0 −p) >= p! (b0 − p)! = b0 !. p p (4.65) A similar result is proven for ek 2b0 +1 and then by induction on b, the assertions are proven. Similarly, < f−k−1 2a , E2k+1 a >= a! and < f−k−1 2a+1 , ek E2k+1 a >= −a!. We can sum up all these results by ∀a, b ∈ Z≥0 , c ∈ {0, 1}, < F−2k−1 b f−k−1 2a+c , ek 2b+c E2k+1 a >= (−1)c a! b! (4.66) and all other pairings with ek 2b+c E2k+1 a are zero. Similarly, we show that all other pairings with F−2k−1 b f−k−1 2a+c are also zero: l

<

−→ 

k

bn

F−2n−1 f−n−1

2an +cn

,

n=0

−→ 

em 2bm +cm E2m+1 am >

m=0 l



=< F−1 b0 f−1 2a0 +c0 ⊗

−→ 





F−2l−1 bl f−l−1 2al +cl ,

n=1 k

(e0

2b0 +c0

a0

E1 ) (

−→ 

em 2bm +cm E2m+1 am ) >

m=1 c0 (c1 +···+cl )

= (−1)

δa0 ,a0 δb0 ,b0 δc0 ,c0 (−1)c0 a0 ! b0 !

l

×<

−→  n=1

k

bl

F−2l−1 f−l−1

2al +cl

,

−→ 

em 2bm +cm E2m+1 am > .

m=1

Repeating this calculus k times, we prove (4.59) of the theorem. Equation (4.60) is proven along the same lines.  + is Abelian, one of its basis is {h0 a0 h1 a1 . . . hk ak . . . |a0 , a1 , . . . ∈ Remark 4.3. Since H − and H + is not degenerZ≥0 }. In addition, the pairing restricted of the subalgebras H ated. Remark 4.4. A corollary of the previous results is that the pairing between A− and A+ is not degenerated. Then, thanks to the isomorphisms φ ± and , neither is the pairing between Y − (R) and Y + (R). Theorem 4.4. Y + (R) ⊗ Y − (R) is the quantum double of Y + (R) with the multiplication between Y + (R) and Y − (R) defined by (4.2). Thus, it is isomorphic, as a Hopf algebra, to the quantum double of Y (osp(1|2)), denoted DY (osp(1|2)). Similarly, A+ (R)⊗A− (R) is the quantum double of A+ (R).

48

D. Arnaudon, N. Cramp´e, L. Frappat, E. Ragoucy

˙ ± (u)⊗L ˙ ± (u), the cross-multiplication in Proof. From ( ⊗ 1) (L± (u)) = L± (u)⊗L a quantum double is defined by + − + + − − + L− 2 (v)L1 (u) = < S(L2 (v)), L1 (u) > L1 (u)L2 (v) < L2 (v), L1 (u) > (4.67) − −1 = R12 (u − v)L+ 1 (u)L2 (v)R12 (u − v),

which is equivalent to (4.2). The other assertions are obvious.

(4.68)



5. Universal R-Matrix 5.1. Construction of the universal R-matrix. We express the universal R-matrix of double super Yangian DY (osp(1|2)) according to the generators of Drinfel’d basis. Since DA isthe quantum double of A+ , it admits a canonical universal R-matrix given by R = xi ⊗ x i , where {xi , i ∈ N} is the basis of A+ and {x i , i ∈ N} ∈ A− is the dual basis (i.e. < x i , xj >= δji ). Therefore, thanks to the explicit expression of the pairing, we have the following result. Theorem 5.1. The universal R-matrix can be factorised as R = RE RH RF ,

(5.1)

− , RH ∈ H + ⊗ H − and RF ∈ F + ⊗ E− . The explicit expressions where RE ∈ E+ ⊗ F of the universal factors RE and RF are RE = RF =

    exp ei 2 ⊗ F−2i−1 (1 ⊗ 1 − ei ⊗ f−i−1 ) exp E2i+1 ⊗ f−i−1 2 , (5.2)

−→  i≥0 ←− 

    exp F2i+1 ⊗ e−i−1 2 (1 ⊗ 1 + fi ⊗ e−i−1 ) exp fi 2 ⊗ E−2i−1 . (5.3)

i≥0

Proof. The factorisation of the the universal R-matrix is involved by the relation (ii) of Proposition (4.3). In addition, to prove the expression of RE , we expand the exponentials and the products RE =

=

 1  2ai ei ⊗ F−2i−1 ai (1 ⊗ 1 − ei ⊗ f−i−1 ) a !b ! i≥0 ai ,bi ≥0 i i   × E2i+1 bi ⊗ f−i−1 2bi

−→ 

−→  i≥0





  (−1)ci  ei 2ai +ci E2i+1 bi ⊗ F−2i−1 ai f−i−1 2bi +ci ai !bi ! a ,b ≥0, i i ci ∈{0,1}

  = 

(5.4) (5.5)

 

a0 ,b0 ≥0, a1 ,b1 ≥0, c0 ∈{0,1} c1 ∈{0,1}

 ... 

(−1)c0 +c1 +··· (−1)c0 c1 +··· a0 !b0 !a1 !b1 ! . . .

  × e0 2a0 +c0 E1 b0 e1 2a1 +c1 E3 b1 . . . ⊗ F−1 a0 f−1 2b0 +c0 F−3 a1 f−2 2b1 +c1 . . . . (5.6)

Drinfel’d Realisation of Super Yangian Y (osp(1|2))

49

 Therefore, RE can be written xi ∈B + xi ⊗ x i and < x i , xj >= δji due to (4.59). The E proof to find the explicit form of RF is similar.  Theorem 5.2. The factor RH of the universal R-matrix of DY (osp(1|2)) is given by      d      

 RH = exp , (5.7) K+ (u) ⊗ C T 1/2 K− v + 3n + 23  −i−1  du i n≥0

i≥0

where K± (u) = ln h± (u), C(q) = q+1+q −1 , T is the shift operator: Tf (u) = f (u+1) and (ψ(u))i = ψi for any function ψ(u) = i ψi u−i−1 . Proof. The proof is inspired by the results exposed in [12]. Starting from the pairing (4.33), a direct calculation shows that < K− (u), K+ (v) > = ln

(u − v − 1)(2u − 2v + 1) (u − v + 1)(2u − 2v − 1)

(5.8)

from which it follows < K− (u),

1 d 1 K+ (v) > = − − dv u − v − 1 u − v + 1/2 1 1 + + u − v + 1 u − v − 1/2 = (T −1 + T 1/2 − T − T −1/2 )

(5.9) −1 . u−v

Therefore one obtains  −1 d −1 < T −1 + T 1/2 − T − T −1/2 K− (u), K+ (v) > = . dv u−v

(5.10)

(5.11)

The formal inversion of the operator (T −1 + T 1/2 − T − T −1/2 )−1 is given by 

T −1 + T 1/2 − T − T −1/2

−1

=



T 3n+2 + T 3n+3/2 + T 3n+1 .

(5.12)

n≥0

Let B(q) be the q-analog of the symmetrised Cartan matrix of osp(1|2). We define C(q) by the relation1 B(q)−1 = [2κ]1 1/2 C(q), C(q) is a matrix with polynomial entries in q q

and q −1 and positive coefficients. One gets C(q) = q + 1 + q −1 . It follows that  n≥0

< C(T 1/2 ) K− (u + 3n + 3/2),

d −1 K+ (v) > = . dv u−v

(5.13)

Since the pairing (5.13) exhibits a duality relation in diagonal form, one gets immediately the expression (5.7) for the universal factor RH .  1 The presence of q 1/2 instead of q in the definition of C(q) is due to the normalisation of the fermionic simple root.

50

D. Arnaudon, N. Cramp´e, L. Frappat, E. Ragoucy

5.2. Evaluated R-matrix. Proposition 5.1. Let π be the fundamental 3-dimensional representation of osp(1|2) with representation space V and Vz a C-module. Then πz such that πz : DA en fn hn

−→ Vz ⊗ V n −→ zn E12 + z E23 n

n −→ z E21 − z E32 n n −→ zn E11 + (zn − z )E22 − z E33

(5.14) (5.15) (5.16)

is an evaluation representation of the double superYangian DY (osp(1|2)) for z = z+ 21 . Proof. The image by πz of the elements of DA have to satisfy the commutation relations (4.15)–(4.22). For example, we prove it for (4.19). We use that πz (e(u)) = δ(z − u)E12 + δ(z − u)E23 . Then, one gets : (u − v − 1)(2u − 2v + 1)πz (e(u))πz (e(v)) +(u − v + 1)(2u − 2v − 1)πz (e(v))πz (e(u)) = [(u − v − 1)(2u − 2v + 1)δ(z − u)δ(z − v) +(u − v + 1)(2u − 2v − 1)δ(z − v)δ(z − u)]E13 = [(z − z − 1)(2z − 2z + 1) + (z − z + 1)(2z − 2z − 1)E13 = 0. (5.17) The other commutation relations are proven analogously.



Theorem 5.3. Let πz and πw be two fundamental evaluation representations, then (πz ⊗ πw )R = R12 (z − w),

(5.18)

where the R-matrix R12 (z) is given by (2.4). Proof. (πz ⊗ πw )RE = 13 ⊗ 13 + +



πz (ei ej ) ⊗ πw (f−i−1 f−j −1 )



j >i≥0

πz (ei 2 ) ⊗ πw (F−2i−1 ) − πz (ei ) ⊗ πw (f−i−1 )

i≥0



+πz (E2i+1 ) ⊗ πw (f−i−1 2 ) =

E12 ⊗ E21 E23 ⊗ E21 E23 ⊗ E32 E12 ⊗ E32 + − − 1 z−w z−w z−w− 2 z − w + 21 4(z − w) + 3 + E13 ⊗ E31 . (z − w)(2(z − w) + 1)

The explicit form of RF is proven analogously. The calculation of RH is standard, but one has to use the following formula introduced in [12]:       γ −β+1   1   x − α + N n + 1   N , (5.19) =  ln exp  u−γ i x − β + N n + 1 −i−1   γ −α+1 n≥0

i≥0

where (ψ(u))i = ψi is defined as in Theorem 5.2.

N



Drinfel’d Realisation of Super Yangian Y (osp(1|2))

51

Acknowledgements. We warmly thank J. Avan and A. Molev for discussions and advice. Some preliminary computations were done using the symbolic manipulation program Form, by J. Vermaseren [13].

References 1. Drinfeld, V.G.: Hopf algebras and the quantumYang–Baxter equation. Sovt. Math. Dokl. 32, 254–258 (1985) 2. Drinfeld, V.G.: Quantum Groups. In: Proceedings Int. Cong. Math. Berkeley, California, USA, pp 798–820 (1986) 3. Drinfeld, V.G.: A new realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36, 212–216 (1988) 4. Faddeev, L.D., Reshetikhin, N.Yu., Takhtajan, L.A.: Quantization of Lie groups and Lie algebras. Leningrad Math. J. 1, 193–225 (1990) 5. Kulish, P.P., Sklyanin, E.K.: Solutions of the Yang–Baxter equation. Zap. Nauchn. Sem. LOMI 95, 129–160 (1980) and J. Sov. Math. 19, 1596–1620 (1982) 6. Nazarov, M.L.: Quantum Berezinian and the classical Capelli identity. Lett. Math. Phys. 21, 123–131 (1991) 7. Arnaudon, D., Avan, J.N., Cramp´e, L., Frappat, E., Ragoucy,: R-matrix presentation for super Yangians Y (osp(m|2n)). J. Math. Phys. 44, 302–308 (2003), math.QA/0111325 8. Molev, A.I.: Yangians and their applications. Handbook of Algebra 3, London-New York: Elsevier, to appear 9. Ding, J.: A remark on the FRTS realization and Drinfeld realization of quantum affine Superalgebra Uq (osp(1, ˆ 2)). math.QA/9905086 10. Rosso, M.: An analogue of P.B.W theorem and the universal R-matrix. Commun. Math. Phys. 124, 307–318 (1989) 11. Vladimirov, A.A.: A method for obtaining quantum doubles from the Yang-Baxter R-matrices. hep-th/9302042 12. Khoroshkin, S.M., Tolstoy, V.N.: Yangian double and rational R-matrix. hep-th/9406194 13. Vermaseren, J.: New features of FORM. math-ph/0010025 Communicated by L. Takhtajan

Commun. Math. Phys. 240, 53–73 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0907-4

Communications in

Mathematical Physics

The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus
Florin P. Boca1,2 , Radu N. Gologan2 , Alexandru Zaharescu1,2 1 2

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. E-mail: [email protected]; [email protected] Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 70700, Romania. E-mail: [email protected]

Received: 4 February 2003 / Accepted: 6 March 2003 Published online: 1 August 2003 – © Springer-Verlag 2003

Abstract: We consider a billiard in the punctured torus obtained by removing a small disk of radius ε > 0 from the flat torus T2 , with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity ω only. Let τ˜ε (ω), and respectively R˜ ε (ω), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when T2 is being identified with [0, 1)2 . We prove that the probability measures on [0, ∞) associated with the random variables ε τ˜ε and ε R˜ ε are weakly convergent as ε → 0+ and explicitly compute the densities of the limits. 1. Introduction and Main Results Various ergodic and statistical properties of the periodic Lorentz gas were studied during the last decades (see [1, 5–15, 17, 20, 22, 23] for a non-exhaustive list of references). In the case of periodically distributed circular scatterers of radius ε ∈ (0, 21 ) in R2 , one considers the region Zε = {x ∈ R2 ; dist(x, Z2 ) ≥ ε} and the first exit time (called free path length by some authors) defined as τε (x, ω) = inf{τ > 0 ; x + τ ω ∈ ∂Zε }, where x ∈ Yε = Zε /Z2 and ω belongs to the unit circle T, which will be steadily identified with [0, 2π ) throughout the paper. Equivalently, one can consider the free motion of a point-like particle in the billiard table Yε obtained by removing pockets of the form of quarters of a circle of radius ε. If we identify T2 = R2 /Z2 with [0, 1)2 , then Yε can be regarded as a punctured two-torus. The reflection in the side cushions of the table is 

Research partially supported by ANSTI grant C6189/2000.

54

F.P. Boca, R.N. Gologan, A. Zaharescu

ε

ε

ε

ε

ω

ε

ε

Oε

ε

Fig. 1. The trajectory of the billiard

specular and the trajectory between two such reflections is rectilinear. Assume that the particle has constant speed, say equal to 1, and leaves the table when it reaches one of the four pockets. In this setting τε (x, ω) denotes the exit time from the table (or equivalently the length of the trajectory). In this paper we will study the case where the trajectory starts at the origin O = (0, 0) with initial velocity ω. In this case the first exit time is τ˜ε (ω) := τ (O, ω) and one averages over ω only. We shall give very precise estimates on the average of τ˜ε and of the number R˜ ε (ω) of collisions of the particle with the side cushions as ε → 0+ . The related problem of estimating the moments 1 cr = 2π


2π τ˜εr (ω) dω,

r > 0,

0

was raised by Ya.G. Sinai in 1981. An answer was given in [3], where it was proved for any interval I ⊆ [0, π4 ] and any (small) δ > 0 that
εr

1

τ˜εr (ω) dω = cr |I | + Or,δ (ε 8 −δ )

as ε → 0+ ,

(1.1)

I

where 2 cr = ζ (2)


1/2 0

   1 − (1 − x)r 1 − (1 − x)r+1 x x r−1 + (1 − x)r−1 + − dx. rx(1 − x) (r + 1)x(1 − x)

Since all the probability measures µ˜ ε defined by 1 µ˜ ε (f ) = 2π


2π 0

  f ε τ˜ε (ω) dω,

  f ∈ Cc [0, ∞) ,

The Statistics of the Trajectory of a Billiard

55

have the support included into a common compact (see Lemma 3.1 in [3] or Lemma 2.1 in this paper), the equality (1.1) implies that µ˜ ε → µ˜ weakly as ε → 0+ , and that the moments of µ˜ are given by 1 2π


2π ωr d µ(ω) ˜ = cr . 0

However, this does not directly provide an explicit formula for the density of µ. ˜ The primary aim of this paper is to give a direct proof for this convergence, computing in the meantime the density of µ˜ in closed form. For obvious symmetry reasons we may only consider the interval [0, π4 ] instead of [0, 2π ]. To state the main result we consider the repartition function Hε (t) =

  π 4   ; ε τ˜ε (ω) > t   ω ∈ 0, π 4

of µ˜ ε and the function ψ(x) =

1−x x

 1 + ln

x 1−x

 x∈

,

1 2

 ,1 .

(1.2)

It is seen that ψ is non-decreasing and satisfies ψ( 21 ) = 1, ψ(1− ) = 0, and especially
1 ψ(x) dx =

ζ (2) − 1 π2 1 = − . 2 12 2

(1.3)

1/2

We can now state the main result. Theorem 1.1. For each t > 0, there exists H (t) = lim Hε (t). Moreover, one has ε→0+ 1

Hε (t) = H (t) + Oδ (ε 8 −δ ) for any (small) δ > 0, where  2t   if 0 < t < 21 ; 1 −  ζ (2)   
1 2 H (t) = ψ(x) dx if 21 < t < 1;    ζ (2) t     0 if t > 1. ˜ = h(t)dt and In particular µ˜ ε → µ˜ weakly as ε → 0+ , where d µ(t)   if 0 < t ≤ 21 ; 1 2 h(t) = −H  (t) = · ψ(t) if 21 ≤ t ≤ 1; ζ (2)  0 if t ≥ 1.

56

F.P. Boca, R.N. Gologan, A. Zaharescu 1

1.2 1

0.8

0.8 0.6 0.6 0.4 0.4 0.2

0.2 0.2 0.4 0.6 0.8

1

1.2 1.4

0.2 0.4 0.6 0.8

1

1.2 1.4

Fig. 2. The graphs of H (t), and respectively of h(t) = −H  (t)

This result deserves some comments. Firstly, since µ˜ is a probability measure on 1 [0, 1], one must have that 0 h(t) dt = 1. But this amounts to 1 + 2


1 ψ(t) dt =

ζ (2) , 2

1/2

that is to formula (1.3), and we got a first proof of that formula. 1 + However, one can prove (1.3) in other ways. For instance, writing ψ(x) = x1 ln 1−x 1+ln x ln(1 − x) + x − 1 − ln x, integrating each term from t to 1 and using the Taylor 1 expansion for x1 ln 1−x , we gather
1 t

  ln2 t 1−t ψ(x) dx = Li2 (1) − Li2 (t) − + (1 − t) ln −1 , 2 t

 tn where Li2 (t) = ∞ 1 n2 denotes the dilogarithm function. Plugging now t = and using the formula (cf. relation (1.6) in [21])   1 π2 ln2 2 ζ (2) − ln2 2 Li2 = − = , 2 12 2 2

(1.4) 1 2

in (1.4)

(1.5)

we arrive again at relation (1.3). Thus, the fact that µ˜ has mass one is equivalent to the (non-trivial) equality (1.5) concerning the dilogarithm. Secondly, Theorem 1.1 should be compared with the most recent results of the first two authors ([4]), who proved the existence of the limit repartition  π

 4   ; ετε (x, ω) > t   (x, ω) ∈ [0, 1)2 × 0, π 4

as ε → 0+ .

This repartition function is no longer compactly supported; it was proved in [4] to be of the form 4ζ (2)

∞  n=1

1 2n − 1 + Oδ (ε 8 −δ ), 2 2 n n (n + 1) (n + 2)t

∀ t ≥ 2.

The Statistics of the Trajectory of a Billiard

57

Interestingly, the density h coincides1 for t ∈ [0, 1] with the density of the limit measure of the geometric free path, proved to exist in the small-scatterer limit in [4], and computed independently in [13 and 4]. Another interesting feature of the free path lengths in the small-scatterer limit is that their limit repartition functions are closely related with dilogarithms in the homogeneous case treated in this paper, and with dilogarithms and possibly trilogarithms in the nonhomogeneous case treated in [4], where one averages over a phase space defined by both velocity and position, and where the limit measure has a tail at +∞. We finally estimate the repartition function Fε (t) =

  π 4   ; ε R˜ ε (ω) > t   ω ∈ 0, π 4

of the random variable εR˜ ε in the small-scatterer limit, and prove Theorem 1.2. For each t > 0, there exists F (t) = lim Fε (t). Moreover, one has ε→0+ 1

Fε (t) = F (t) + Oδ (ε 8 −δ ) for any (small) δ > 0, where 4 F (t) = π


π/4  H 0

 
π/2  4 t t dω = H √ dω. cos ω + sin ω π 2 sin ω π/4

In particular this implies that the probability measures ν˜ ε defined by
2π

1 ν˜ ε (f ) = 2π

  f ε R˜ ε (ω) dω,

  f ∈ Cc [0, ∞) ,

0

converge weakly to the probability measure ν˜ with √ repartition function F . It is clear that the support of ν˜ coincides with the interval [0, 2]. Moreover, the function F is linear on [0, 21 ] and on this interval one has 4 F (t) = π


π/2 π/4

 √ √ t 2 4t 2 π 1− dω = 1 + · ln tan ζ (2) sin ω π ζ (2) 8

√ √ 4 2 ln(1 + 2)t . =1− π ζ (2) One may replace the range [0, π4 ] of ω by an arbitrary interval I ⊆ [0, π4 ], proving the existence of the weak limit of the probability measures associated with the random variables ε τ˜ε and ε R˜ ε when ω ∈ I , and also computing their limits. This can be done only with minimal modifications in Sect. 4. 1

after scaling t by 2 since the factor ε is replaced by 2ε in [4]

58

F.P. Boca, R.N. Gologan, A. Zaharescu

2. Formulas for Sectors Ending at Farey Points For each integer Q ≥ 1, let FQ denote the set of Farey fractions of order Q, that is the set of irreducible rational numbers in [0, 1] with denominators less then or equal to Q.  It is well-known that if qa < qa  are consecutive elements in FQ , then one has a  q − aq  = 1

q + q  > Q.

and

Conversely, if q, q  ∈ {1, . . . , Q} and q + q  > Q, then there exist a ∈ {1, . . . , q − 1}  and a  ∈ {1, . . . , q  − 1} such that qa < qa  are consecutive in FQ . For each interval I ⊆ [0, 1], we consider the set FQ (I ) = I ∩ FQ of Farey fractions of order Q from I of cardinality NQ,I = #FQ (I ) =

Q2 |I | + O(Q ln Q). 2ζ (2)

For each h > 0 we consider the vertical scatterers of height 2h, V˜q,a,h = {q} × [a − h, a + h], and the set Z˜ h =



(q, a) ∈ Z2∗ ,

V˜q,a,h .

(q,a)∈Z2∗

The free path length in this model is then given by lh (ω) = inf{τ > 0 ; (τ cos ω, τ sin ω) ∈ Z˜ h }. We denote by Rh (ω) the number of reflections in the side cushions in the billiard model in the case of vertical scatterers considered above. We also define for each ε > 0, each interval I ⊆ [0, 1] and each integer Q ≥ 1 the quantities    t   ˜ Hε,I,Q (t) =  ω ; tan ω ∈ I, l 1 (ω) > Q ε  and

   t   ˜ Fε,I,Q (t) =  ω ; tan ω ∈ I, R 1 (ω) > . Q ε 

The repartition of Farey fractions in FQ (I ) will play a central role in the next section while proving an asymptotic formula for H˜ ε,I,Q (t) as ε → 0+ , |I | → 0 and Q → ∞ in a suitable way. In the remainder of this section Q ≥ 1 will be a fixed integer. A key remark is that every line of slope between 0 and 1 through the origin O will necessarily intersect the set  VQ = V˜ 1 a/q∈FQ

q,a, Q

The Statistics of the Trajectory of a Billiard

59

2 consisting in NQ = NQ,[0,1] vertical scatterers of height Q centered at the integer points a (q, a) with q irreducible fraction in FQ . This is contained in the next statement which is essentially Lemma 3.1 in [3].

Lemma 2.1. For any ω ∈ [0, π4 ] one has {(τ cos ω, τ sin ω) ; τ > 0} ∩ VQ = ∅. Proof. Let tP denote the slope of the line OP . We use the inequalities q + q  ≥ Q + 1 > max(q, q  ) to infer tA =

a − a ≤ tS  = q q

1 Q

< tN =

a+

1 Q

≤ tA =

q

a , q

1 1 where we set A = (q, a), A = (q  , a  ), N = (q, a + Q ), N  = (q  , a  + Q ), 1 1 1 1 S = (q, a − Q ), S  = (q  , a  − Q ), W = (q − Q , a), W  = (q  − Q , a  ). This clearly shows that for any ω ∈ [0, π4 ], the line of slope ω through the origin will necessarily intersect VQ .  

We need a few more elementary things concerning Farey fractions. Suppose next that  < qa < qa  are three consecutive fractions in FQ . Then the relation aq  − a  q = 1 yields that q  = a¯ (mod q), where a¯ denotes the multiplicative inverse of a (mod q). Since q  ∈ (Q − q, Q], then q  − a¯ is the unique multiple of q in the interval (Q − q − a¯ a, ¯ Q − a]. ¯ Hence q  − a¯ = q[ Q− q ], and so we get a q

q = q



 Q − a¯ + a. ¯ q

(2.1)

Making use of a  q − aq  = 1 and of q  ∈ (Q − q, Q], we arrive in a similar way at q  = q



 Q + a¯ − a. ¯ q

(2.2)

N (a’- 1 Q)q ) S’o(q, q’ A

N’ A’ q’ N’(q’, o (a+1 Q) q S’

)

N’

N

A’ S’

A S

O

O

Fig. 3. The cases q <

q

and respectively q > q 

S

60

F.P. Boca, R.N. Gologan, A. Zaharescu (1)

(2)

(3)

(4)

We consider the partition Iq ∪ Iq ∪ Iq ∪ Iq of [0, q], where Iq(1) = [max(2q − Q, 0), min(Q − q, q)];

Iq(2) = [max(2q − Q, Q − q), q];

Iq(3) = [0, min(2q − Q − 1, Q − q)];

Iq(4) = [Q − q, 2q − Q].

(1)

(2)

(3)

It is clear that Iq = ∅ unless q ≤ 2Q 3 , Iq = Iq = ∅ unless q ≥ 2Q unless q ≥ 3 . Taking into account (2.1) and (2.2) we see that

Q 2,

(4)

and Iq

=∅

q  > q ⇐⇒ a¯ ≤ Q − q and q  > q ⇐⇒ a¯ ≥ 2q − Q. Since a¯ cannot actually take the values 0 and q, one has 2Q Lemma 2.2. (i) min(q  , q  ) > q ⇐⇒ q ≤ and a¯ ∈ Iq(1) . 3 Q (ii) q  < q < q  ⇐⇒ q ≥ and a¯ ∈ Iq(2) . 2 Q (iii) q  < q < q  ⇐⇒ q ≥ and a¯ ∈ Iq(3) . 2 2Q (iv) q > max(q  , q  ) ⇐⇒ q ≥ and a¯ ∈ Iq(4) . 3 For each ω we put eω,Q = (q, a) if the half-line R+ ω first intersects the scatterer (q, a)+V1/Q among the NQ components of VQ . We denote by ωq,a the angle determined by the trajectories which end near the lattice point (q, a), that is    π   ωq,a =  ω ∈ 0, ; eω,Q = (q, a) . 4 Explicit formulas for ωq,a can be given using Lemma 2.2 and h + O(h2 ) 1 + x2 h = + O(h2 ), 1 + (x + h)2

arctan(x + h) − arctan x =

x ∈ [0, 1], h > 0 small, (2.3)

as follows

1 ( q, a+ Q- )

O

1111111 0000000 0000000 1111111 (q , a) 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 (q, a- 1- ) 0000000 1111111 0000000Q 1111111

(q", a"- -Q1 )

(q, a+ 1Q- )

(q’,a’- -Q1 )

(q," a"- -Q1 )

(q, a) 1

(q’,a’+ -Q1 ) O

Fig. 4. The cases q < min(q  , q  ) and q  < q < q 

(q , a- Q- )

The Statistics of the Trajectory of a Billiard

( q, a+ 1-Q ) (q, a)

(q", a"- Q1- )

61

1 (q, a+ - ) Q

(q’, a’+ 1-Q )

(q, a )

(q", a"- 1- ) Q

(q, a- Q1- )

1 (q, a- - ) Q

1 -) (q’, a’+ Q

O

O

Fig. 5. The cases q  < q < q  and q > max(q  , q  )

Lemma 2.3.

Q 2,

(i) If q ≤

(1)

ωq,a (ii) If

Q 2

ωq,a


(3)

= [0, q], Iq

(4)

= ∅, and

(4)

= ∅ and    1 2 (1)   +O if a¯ ∈ Iq = [2q − Q, Q − q];  2) 2q  Qq(1 + γ Q          Q − q + a¯ 1 (2) if a¯ ∈ Iq = [Q − q, q]; = + O  Qq a(1 ¯ + γ 2) Q2 q            Q − a¯ 1 (3)   + O if a¯ ∈ Iq = [0, 2q − Q]. Qq(q − a)(1 ¯ + γ 2) Q2 q  2Q 3 ,

then Iq

(1)

= ∅ and    1 Q − q + a¯ (2)   if a¯ ∈ Iq = [2q − Q, q]; +O  2) 2q   Qq a(1 ¯ + γ Q          Q − a¯ 1 (3) if a¯ ∈ Iq = [0, Q − q]; = + O 2) 2 q   Qq(q − a)(1 ¯ + γ Q           Q−q 1 1 (4)   if a¯ ∈ Iq = [Q − q, 2q − Q]. +O + Qa(q ¯ − a)(1 ¯ + γ 2) Q2 q  Q2 q 

(iii) If q >

ωq,a

2Q 3 ,

(2)

= Iq = Iq   2 1 = + O . Qq(1 + γ 2 ) Q2 q

then Iq

then Iq

  Proof. If q > Q 2 then one has, according to Lemma 2.2, that q < min(q , q ). In this case we infer from (2.3) that

ωq,a = arctan

a+ q

1 Q

− arctan

a− q

1 Q

=



2

 Qq 1 +

 +O (a− 1 )2 Q

q2

 1 . Q2 q 2

62

F.P. Boca, R.N. Gologan, A. Zaharescu

This implies, in connection with 1 1+

−

1 2 (a− Q ) 2 q

1 a2 q2

1+



1 2 ) (a − Q a2 2a 1 −   , 2 2 q q Qq Q

the desired formula for ωq,a . In (ii) and (iii) we split the discussion in the following three cases: 1) a¯ ∈ Iq , that is q  < q < q  . (2)

In this case

1 a− Q q

ωq,a = arctan

<

a+

Q−q

1 Q

q

1− =  qq  1 + =

1 a + Q q

<

a q

<

− arctan

a +

q−q  Q 1 2 (a  + Q ) 2 q + q

a 2 ) q 2  +q

Qqq  (1 +

 +O 

+O

1 a  − Q q 

<

1 a+ Q q

and we may write

1 Q

q  (1 −

q−q  2  Q ) qq 

(Q − q + q  )2 1 + 2  Qq q Q2 q 2 q 2



  Q−q (Q − q + q  )2 1 a  Q − q + q  a = . + +O −  + Qqq  (1 + γ 2 ) qq  q q Qq 2 q  Q2 q 2 q 2 The desired formula for ωq,a follows in this case from the formula above, the fact a¯ Q that q  = a¯ as a result of [ Q− q ] = 0, from q ≥ 2 , and from   Q − q + q a 2q  2 1 Q − q + q a = − < = 2   2 , q q q 2 q 2 q 2 q 2 q q Q q qq  1 1 1  3  ≤ 2 , Qq 2 q  Q q Q q (Q − q + q  )2 1 1 1  2 2  2 2 ≤ 2  . Q2 q 2 q 2 q q Q q Q q 2) a¯ ∈ Iq , that is q  < q < q  . (3)

¯ q ≥ In this case q  = q − a, Proceeding as in the case a¯ ∈ ωq,a

a  −

(2) Iq

Q 2,

and

1 a− Q q

<

1 a + Q  q

<

a q

<

1 a  − Q  q

as required.

1 a+ Q q

we find 

1 − q−q Q − arctan = = arctan +O  1 2 (a− Q ) q  q qq  1 + q 2   Q − a¯ 1 = + O , Qq(q − a)(1 ¯ + γ 2) Q2 q  1 Q

<

a−

1 Q



1 Q2 q 2



.

The Statistics of the Trajectory of a Billiard

3) a¯ ∈ Iq , that is q > max(q  , q  ).

63

(4)

In this case

1 a− Q q

<

1 a + Q  q

<

ωq,a = arctan

a q

<

a  − q 

1 a  − Q  q

1 Q

<

1 a+ Q q

− arctan

, and we write

a + q

1 Q

(1) (2) = ωq,a + ωq,a ,

with (1) ωq,a (2) ωq,a

  1 a Q−q , +O = arctan − arctan = q  q Qqq  (1 + γ 2 ) Q2 q 2   1 a + Q a Q−q 1 . = arctan − arctan = + O q q Qqq  (1 + γ 2 ) Q2 q 2 a  −

1 Q

Since in this case q  = q − a¯ and q  = a¯ (thus q  + q  = q) we arrive at   Q−q 1 1 , + + O ωq,a = Qa(q ¯ − a)(1 ¯ + γ 2) Q2 q 2 Q2 q 2  

as required.

3. The Case of Vertical Scatterers In this section we estimate H˜ ε,I,Q (t) as ε → 0+ in the hypothesis that |I | → 0 and Q → ∞ in a controlled way. More precisely we prove Proposition 3.1. Let θ, θ1 ∈ (0, 1) and suppose that I = [tan ω0 , tan ω1 ] is a subinterval of [0, 1] of size |I |  εθ , and Q is an integer such that Q = cosεω0 + O(ε −θ ). Then one has   as ε → 0+ , H˜ ε,I,Q (t) = cI H (t) + O Eθ,θ1 ,δ (ε) uniformly for t in a compact subset of (0, ∞) \ {1, 2}, where H has been explicitly defined in Theorem 1.1, and we denote
1 dt = ω1 − ω0 , Eθ,θ1 ,δ (ε) = εmin(2θ, 2 −2θ1 −δ,θ+θ1 −δ) . cI = 1 + t2 I

It is convenient to write

   t cos ω   ˜ Hε,I,Q (t) =  ω ; tan ω ∈ I, l 1 (ω) cos ω > . Q ε

(3.1)

We also define for each interval I ⊆ [0, 1] and each integer Q ≥ 1 the quantity     H˜ I,Q (t) =  ω ; tan ω ∈ I, l 1 (ω) cos ω > tQ . Q

Let Q− and Q+ be two integers such that Q− ≤

cos ω1 cos ω0 ≤ ≤ Q+ . ε ε

64

F.P. Boca, R.N. Gologan, A. Zaharescu

This also gives for any Q as in the statement of Proposition 3.1, Q± − 1 = O(ε θ ). Q

(3.2)

Since ε → H˜ ε,I,Q (t) is monotonically increasing, one has  +  − tQ tQ ˜ ˜ ˜ HI,Q ≤ Hε,I,Q (t) ≤ Hε,I,Q . Q Q

(3.3)

We now estimate H˜ I,Q (t) in Proposition 3.2. Let θ, θ1 ∈ (0, 1) and suppose that I = [tan ω0 , tan ω1 ] is a subinterval of [0, 1] of size |I |  Q−θ . Then one has   as Q → ∞, H˜ I,Q (t) = cI H (t) + O Eθ,θ1 ,δ (Q) where we denote 1

Eθ,θ1 ,δ (Q) = Qmax(2θ1 − 2 +δ,−θ−θ1 +δ) . The key technical tools we shall employ to estimate H˜ I,Q (t) are the following two lemmas from [3 and 2]. Lemma 3.3 ([3, Lemma 2.2]). Suppose that q ≥ 1 is an integer, I and J are intervals of length less than q, and that f is a C 1 function on I × J . Then one has

 ϕ(q) f (a, b) = 2 f (x, y) dx dy q a∈I , b∈J ab=1 (mod q)

I ×J



+ Oδ T q 2

1 2 +δ

f ∞ + T q

3 2 +δ

|I| |J | Df ∞ Df ∞ + T



for any integer T ≥ 1 and any δ > 0, where  ·  denotes the L∞ norm on I × J , ∂f Df = | ∂f ∂x | + | ∂y |, and ϕ is Euler’s totient function. Lemma 3.4 ([2, Lemma 2.3]). Suppose that 0 < a < b and that f is a C 1 function on [a, b]. Then one has   

 ϕ(k) 1  f (k) = f (x) dx + O ln b f ∞ + |f | . k ζ (2) b

a
a

b

a

We shall need the next two corollaries of these two lemmas. Corollary 3.5. 

(i) For each 0 < t1 < t2 ≤ 1 one has 

t1 Q
  1 cI (t2 − t1 ) = + O Eθ,θ1 ,δ (Q) . Qq(1 + γ 2 ) ζ (2)

The Statistics of the Trajectory of a Billiard

(ii) For each

1 2

65

< t1 < t2 ≤ 1 one has 



t1 Q
1 cI = 2 Qq(1 + γ ) ζ (2)





t1 Q

t2 t1

cI 1 = Qq(1 + γ 2 ) ζ (2)

  1−x dx + O Eθ,θ1 ,δ (Q) ; x


t2 t1

  1−x dx + O Eθ,θ1 ,δ (Q) . x

(iii) For each t ∈ ( 21 , 23 ) one has 



tQ
a∈qI a∈[2q−Q,Q−q] ¯ a a=1 ¯ (mod q)

Corollary 3.6. For each

1 = cI Qq(1 + γ 2 )

1 2


2/3 t

  2 − 3x dx + O Eθ,θ1 ,δ (Q) . x

< t1 < t2 ≤ 1, both sums





t1 Q


Q−q Qq a(1 ¯ + γ 2)



t1 Q
and

Q−q Qq(q − a)(1 ¯ + γ 2)

are of the form cI ζ (2)


t2 t1

  1−x x ln dx + O Eθ,θ1 ,δ (Q) . x 1−x

We only give the proof of Corollary 3.6. The proof of Corollary 3.5 is easier and we leave it as an exercise to the reader. Proof of Corollary 3.6. We only prove the first equality. The second one is proved in a similar way by changing a¯ to q − a. ¯ To estimate the inner sum we consider for each q ∈ (t1 Q, t2 Q] ⊂ ( Q , Q] the function 2 f (a, a) ¯ = fq (a, a) ¯ =

Q−q q(Q − q) = , 2 Qq a(1 ¯ +γ ) Q(q 2 + a 2 )a¯

Since Q − q + 1 ≤ a, ¯ one has    ∂f  1   1 f ∞ ≤ ,   Qq ∂a ∞ Qq 2

and

a ∈ qI, a¯ ∈ [Q − q + 1, q].

   ∂f  1    .  ∂ a¯  Qq(Q − q + 1) ∞

66

F.P. Boca, R.N. Gologan, A. Zaharescu

Applying Lemma 3.3 to f with T = [Qθ1 ], I = [Q − q + 1, q] and J = qI , we gather  a∈qI, a∈(Q−q,q] ¯ a a=1 ¯ (mod q)

  Q−q ϕ(q) = · WQ (q) + O FQ (q) , Qq a(1 ¯ + γ 2) q

(3.4)

where Q−q WQ (q) = Qq 2


qI


q

da 1+

a2 q 2 Q−q+1

d a¯ cI (Q − q) q = ln , a¯ Qq Q−q +1

and the error terms 1

FQ (q) = Q2θ1 −1 q − 2 +δ + Qθ1 −1

1 1 qQ−θ (2q − Q) Qq(Q−q+1) q 2 +δ + Q−q +1 Qθ1

1

3

≤ Q2θ1 − 2 +δ +

Qθ1 − 2 +δ Q−θ −θ1 + Q−q +1 Q−q +1

sum up over q to (we may take δ < θ1 ) 

1

1

FQ (q)  Q2θ1 − 2 +δ + Qθ1 − 2 +δ ln Q + Q−θ−θ1 ln Q  Fθ,θ1 ,δ (Q).

(3.5)

Q 2
Using the inequality ln(1 + x) ≤ x, we see immediately that  
t2 Q
t2 Q Q−q q 1 WQ (q) dq = cI ln dq + O Qq Q−q Q

t1 Q

t1 Q


t2 = cI t1

1−x x ln dx + O(Q−1 ). x 1−x

(3.6)

  δ−2 Finally, the inequalities WQ ∞,[t1 Q,t2 Q]  Qδ−1 and WQ ∞,[t1 Q,t2 Q]  q show that Lemma 3.4 applied to the function WQ yields

 t1 Q
ϕ(q) 1 WQ (q) = q ζ (2)


t2 Q WQ (q) dq + O(Q2δ−1 ).

(3.7)

t1 Q

Finally we put together (3.4), (3.5), (3.6) and (3.7) to get the desired estimate in the first formula of the statement.  

The Statistics of the Trajectory of a Billiard

67

Proof of Proposition 3.2. We use the explicit formulas for ωq,a found in Sect. 2. The crude estimate      1 1 1 max , , Q2 q a Q2 q  a Q2 q  a q ∈FQ (I )

≤

q ∈FQ (I )

q ∈FQ (I )

Q Q 1  #(qI ∩ Z) + 1 1  q|I | + 1 |I |   ln Q +  Q−θ−1 2 2 Q q Q q Q q=1

q=1

and Lemma 2.3 show that H˜ I,Q (t) = SI,Q (t) + O(Q−θ−1 ), where SI,Q (t) =



ω˜ q,a ,

a q ∈FQ (I )

q>tQ

with

ω˜ q,a =

 2     Qq(1 + γ 2)         Q − q + a¯      Qq a(1 ¯ + γ 2)

(1)

if a¯ ∈ Iq ; (2)

if a¯ ∈ Iq ; (3.8)

  Q − a¯      Qq(q − a)(1 ¯ + γ 2)        Q−q    Qa(q ¯ − a)(1 ¯ + γ 2)

if a¯ ∈

(3) Iq ;

(4)

if a¯ ∈ Iq .

It is clear that 


ω1



ω˜ q,a =

a q ∈FQ (I )

dω + O ω0

1 Q



= cI + O(Q−1 ).

(3.9)

Suppose now that 0 < t < 21 . In this case we may write in view of Lemma 2.3, (3.8), (3.9) and Corollary 3.5 (i)   SI,Q (t) = ω˜ q,a − ω˜ q,a a q ∈FQ (I )

= cI −



a q ∈FQ (I )

q≤tQ



0
= cI −

2 + O(Q−1 ) Qq(1 + γ 2 )

  2cI t + O Eθ,θ1 ,δ (Q) . ζ (2)

68

F.P. Boca, R.N. Gologan, A. Zaharescu

If t ≥ 23 , then Lemma 2.3 and (3.8) yield    SI,Q (t) = ω˜ q,a + tQ
(2)

tQ
+









tQ
a∈qI a∈(Q−q,2q−Q) ¯ a a=1 ¯ (mod q)

which can also be written as 

Q − a¯ Qq(q − a)(1 ¯ + γ 2)

Q−q , Qa(q ¯ − a)(1 ¯ + γ 2)



tQ


+

Q−q Qq a(1 ¯ + γ 2)



tQ
Q−q Qq(q − a)(1 ¯ + γ 2)





tQ
a∈qI a∈[0,Q−q]∪[2q−Q,q] ¯ a a=1 ¯ (mod q)

+

a∈qI, a∈I ¯ q a a=1 ¯ (mod q)

Q − q + a¯ Qq a(1 ¯ + γ 2)

tQ
+

 ω˜ q,a (4)

a∈qI, a∈I ¯ q a a=1 ¯ (mod q)





ω˜ q,a + (3)

a∈qI, a∈I ¯ q a a=1 ¯ (mod q)



=



1 . Qq(1 + γ 2 )

Applying Corollaries 3.6 and 3.5 (ii) we find that 2cI SI,Q (t) = ζ (2) =

2cI ζ (2)


1 t


1

1−x x

 1 + ln

x 1−x



  dx + O Eθ,θ1 ,δ (Q)

  ψ(x) dx + O Eθ,θ1 ,δ (Q) .

(3.10)

t

Finally, in the case where 21 < t < 23 , Lemma 2.3 yields     2 SI,Q (t) = ω˜ q,a = SI,Q + TI,Q (t), 3 tQ
where TI,Q (t) =

 tQ


(3.11)

a∈qI gcd(a,q)=1



ω˜ q,a + (1)

a∈qI, a∈I ¯ q a a=1 ¯ (mod q)



ω˜ q,a + (2)

a∈qI, a∈I ¯ q a a=1 ¯ (mod q)



 ω˜ q,a . (3)

a∈qI, a∈I ¯ q a a=1 ¯ (mod q)

The Statistics of the Trajectory of a Billiard

69

Using also (3.8) the sum TI,Q (t) can be successively written as 

 tQ


=

a∈qI a∈[2q−Q,Q−q] ¯ a a=1 ¯ (mod q)

tQ
+

 2 + 2 Qq(1 + γ )







a∈qI a∈(Q−q,q] ¯ a a=1 ¯ (mod q)

Q−q + Qq a(1 ¯ + γ 2)

a∈qI a∈(Q−q,q] ¯ a a=1 ¯ (mod q)



tQ
 Q − q + a¯ + 2 Qq a(1 ¯ +γ ) 

tQ
a∈qI a∈[0,2q−Q) ¯ a a=1 ¯ (mod q)

a∈qI a∈[0,q] ¯ a a=1 ¯ (mod q)





tQ


a∈qI a∈[0,2q−Q) ¯ a a=1 ¯ (mod q)



1 + Qq(1 + γ 2 )

Q − a¯ Qq(q − a)(1 ¯ + γ 2)

Q−q Qq(q − a)(1 ¯ + γ 2)

1 . Qq(1 + γ 2 )

a∈qI a∈[2q−Q,Q−q] ¯ a a=1 ¯ (mod q)

Applying Corollaries 3.5 and 3.6 we find that 2cI TI,Q (t) = ζ (2)


2/3 t

1−x x cI ln dx + x 1−x ζ (2)


2/3 t

2 − 3x 1+ x



  dx + O Eθ,θ1 ,δ (Q)


2/3   2cI ψ(x) dx + O Eθ,θ1 ,δ (Q) . = ζ (2) t

The desired estimate now follows from (3.10), (3.11) and the equality above.

 

Proof of Proposition 3.1. By Proposition 3.2, the mean value theorem and (3.2) we infer that  ±  ±   tQ tQ ˜ HI,Q = cI H + O Eθ,θ1 ,δ (Q) Q Q     = cI H (t) + O(ε θ ) + O Eθ,θ1 ,δ (θ )   1 = cI H (t) + O Q− min(2θ, 2 −2θ1 −δ,θ+θ1 −δ) . The statement in Proposition 3.1 now follows from this inequality and (3.3).

 

Proposition 3.7. Let θ, θ1 ∈ (0, 1) and suppose that I = [tan ω0 , tan ω1 ] is a subinterval of [0, 1] of size |I |  εθ , and Q is an integer such that Q = cosεω0 + O(ε −θ ). Then one has     t ˜ + O Eθ,θ1 ,δ (ε) Fε,I,Q (t) = cI H cos ω0 + sin ω0 uniformly for t in a compact subset of (0, 1) ∪ (1, 2) ∪ (2, ∞), with Eθ,θ1 ,δ (ε) as in Proposition 3.1. Proof. We write I = [tan ω0 , tan ω1 ]. Since |R1/Q (ω) − (q + a)| ≤ 1 whenever εω,Q = (q, a), it follows that |Rh (ω) − lh (ω)(cos ω + sin ω)| ≤ 1,

∀h, ∀ω.

70

F.P. Boca, R.N. Gologan, A. Zaharescu

This manifestly implies for all ω ∈ [ω0 , ω1 ],     t +ε t +ε H˜ ε,I,Q ≤ H˜ ε,I,Q cos ω1 + sin ω1 cos ω + sin ω     t =  ω ; tan ω ∈ I, l 1 (ω)(cos ω + sin ω) > + 1  Q ε ˜ ≤ Fε,I,Q (t)     t ≤  ω ; tan ω ∈ I, l 1 (ω)(cos ω + sin ω) > − 1  Q ε   t −ε = H˜ ε,I,Q cos ω + sin ω   t −ε . (3.12) ≤ H˜ ε,I,Q cos ω0 + sin ω0 By Proposition 3.1, the mean value theorem and t +ε t − = O(ε θ ), cos ω1 + sin ω1 cos ω0 + sin ω0 we get H˜ ε,I,Q



t +ε cos ω1 + sin ω1



    t +ε = cI H + O Eθ,θ1 ,δ (ε) cos ω1 + sin ω1       t θ = cI H + O(H ∞ ε ) +O Eθ,θ1 ,δ (ε) cos ω0 + sin ω0     t = cI H + O Eθ,θ1 ,δ (ε) . (3.13) cos ω0 + sin ω0

In a similar way one gets       t −ε t H˜ ε,I,Q = cI H + O Eθ,θ1 ,δ (ε) . cos ω0 + sin ω0 cos ω0 + sin ω0 The statement follows from (3.12), (3.13) and (3.14).

(3.14)

 

4. Proof of Theorems 1.1 and 1.2 We focus now on the case of circular scatterers of radius ε. For each integer lattice point (q, a) ∈ Z2∗ , let (q, a ± ε± ) be the intersection of the line x = q with the tangents from O to the circle of center (q, a) and radius ε (see Fig. 6). The quantities ε± are computed from   a − a±ε± · q  ε± q q , ε=  a±ε± 2 =  2 q + (a ± ε± )2 1+ q

The Statistics of the Trajectory of a Billiard

71

(q,a+ ε + )

(q,a)

(q,a- ε- )

Ο

Fig. 6. A circular scatterer

which eventually gives  ε q 2 (q 2 + a 2 ) − a 2 ε 2 aε 2 ε ε± (q, a) = ± = 2 2 2 2 q −ε q −ε cos arctan

 a q

+O

ε2 q

 .

(4.1)

Let I = [tan ω0 , tan ω1 ] ⊆ [0, 1] be an interval of size |I |  εθ with 0 < θ < 21 . It follows from (4.1) that there exists ε0 = ε0 (θ ) > 0 such that for all ε < ε0 and all a q ∈ FQ (I ) one has ε − ε 3/2 ε + ε 3/2 ≤ ε± (q, a) ≤ . cos ω0 cos ω1 Since θ <

1 2

(4.2)

it also follows that cos ω0 cos ω1 −  εθ−1 . ε + ε 3/2 ε − ε 3/2

Thus we find integers Q− and Q+ such that cos ω1 cos ω0 Q− ≤ ≤ ≤ Q+ 3/2 ε+ε ε − ε 3/2 By (4.2) and (4.3) it follows for ε < ε0 and

and a q

Q+ − Q−  εθ−1 .

(4.3)

∈ FQ (I ) that

1 1 ≤ ε± (q, a) ≤ − Q+ Q and cos ω0 + O(ε θ−1 ). ε Obvious monotonicity properties of    t   Hε,I (t) =  ω ; tan ω ∈ I, τε (ω) > ε  Q± =

(4.4)

72

F.P. Boca, R.N. Gologan, A. Zaharescu

and of H˜ ε,I,Q (t) imply that   Hε,I (t) ≤  ω ; tan ω ∈ I, l

1 Q+

(ω) >

  t − 2ε  = H˜ ε,I,Q+ (t − 2ε 2 ) ε

(4.5)

and Hε,I (t) = H˜ ε,I,Q− (t + 2ε 2 ).

(4.6)

Due to (4.4) we may apply Proposition 3.1 to get H˜ ε,I,Q± (t ∓ 2ε) = cI H (t ∓ 2ε) + O(ε min(2θ, 2 −2θ1 −δ,θ+θ1 −δ) ). 1

(4.7)

In view of (4.5), (4.6) and (4.7) we gather for any t = 1, 2 1

Hε,I (t) = cI H (t) + O(ε min(2θ, 2 −2θ1 −δ,θ+θ1 −δ) ). 1 N

(4.8)

Finally we choose a partition [0, 1] = ∪N j =1 Ij with intervals Ij of equal size |Ij | = θ  ε . Summing in (4.8) over I ∈ {I1 , . . . , IN } we arrive at 1 4H (t)  cj + O(ε min(θ, 2 −2θ1 −θ−δ,θ1 −δ) ). π

N

Hε (t) =

j =1

The proof of Theorem 1.1 is complete once we choose θ = θ1 = 18 . Proof of Theorem 1.2. Arguing as above we get from Proposition 3.7 taking θ = θ1 = 18 :   1 t Fε,J (t) = F˜ε,J,Q± (t ∓ 2ε 2 ) = cJ H + O(ε 4 −δ ) cos ω0 + sin ω0 for any interval J = [tan ω0 , tan ω1 ] ⊆ [0, 1]. Thus if the intervals Ij = [tan ωj , tan ωj +1 ], 1 1 1 ≤ j ≤ N = [ε− 8 ], are such that |Ij | = N1  ε 8 , then   N N 1 4 4 t Fε (t) = Fε,Ij (t) = (ωj +1 − ωj )H + O(ε 8 −δ ). π π cos ω + sin ω j =1

j =1

It only remains to compare the main terms in the relation above and in Theorem 1.2. This is achieved by merely appying the mean value theorem: 4 π


π/4  H 0

t cos ω + sin ω



ω  N
j +1  t 4 H = dω π cos ω + sin ω j =1 ωj

   N  1 4 t = (ωj +1 − ωj ) H + O(ε 8 ) π cos ωj + sin ωj j =1

=

  N 1 4 t + O(ε 8 ), (ωj +1 − ωj )H π cos ωj + sin ωj j =1

which concludes the proof of Theorem 1.2.

 

The Statistics of the Trajectory of a Billiard

73

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Commun. Math. Phys. 240, 75–96 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0891-8

Communications in

Mathematical Physics

Renormalization Horseshoe for Critical Circle Maps Michael Yampolsky
Mathematics Department, University of Toronto, 100 St. George Street, Toronto, Ontario, M5S 3G3, Canada Received: 3 October 2002 / Accepted: 7 March 2003 Published online: 10 July 2003 – © Springer-Verlag 2003

Abstract: We prove that the invariant horseshoe for the renormalization of critical circle maps is globally uniformly hyperbolic with one-dimensional unstable direction. This extends our earlier result on hyperbolicity of periodic orbits of the renormalization, and completes the proof of Lanford’s Program. With this paper we complete our description of the renormalization picture for critical circle maps, begun in [Ya1, Ya2, Ya3 and EY]. The renormalization theory of critical circle maps serves to explain the universality phenomena which are observed in the families of one-dimensional dynamical systems given by smooth homeomorphisms of the circle with a finite number of (non-flat) critical points. Historically, this is one of the two main examples of universality in one-dimensional dynamics, the other being the Feigenbaum universality. A renormalization operator for critical circle maps appeared in the papers of Ostlund et al [ORSS] and Feigenbaum et al [FKS]. The most general formulation of the main conjectures of the theory is due to Lanford [Lan1, Lan2], and became known as Lanford’s Program. The renormalization theory of critical circle maps has developed in parallel with the other one-dimensional renormalization theory, that of the unimodal maps of the interval. The recent spectacular progress in the latter began with the seminal work of Sullivan [Sul1, Sul2, MvS], who gave a conceptual explanation of the existence of the Feigenbaum fixed point using the methods of holomorphic dynamics and the Teichm¨uller theory. The results of Sullivan were adapted to the critical circle maps by de Faria [dF1, dF2], who, in particular, introduced an appropriate analogue of the Douady-Hubbard quadratic-like maps. McMullen’s approach to constructing the renormalization horseshoe for bounded type [McM2] has also been translated into the critical circle map setting by de Faria and de Melo [dFdM2]. In our works [Ya1, Ya2] we have extended the above mentioned results to all combinatorial types, and constructed the global renormalization horseshoe
The author gratefully acknowledges the financial support of NSERC and the Connaught Foundation of the University of Toronto.

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for critical circle maps. The two one-dimensional renormalization theories diverged at the next stage: proving the hyperbolicity of the horseshoe presented different challenges in the two contexts. In the unimodal setting it has been shown by Lyubich [Lyu4, Lyu5]. His proof roughly consists of two parts: endowing the space of quadratic-like maps with the structure of an analytic manifold, in which the stable sets of the renormalization operator become submanifolds; and showing the existence of a single unstable direction for the renormalization. The second step, which is the more difficult of the two, and involves Lyubich’s new Rigidity Theorem [Lyu3], for critical circle maps is elementary. The first step, on the other hand, can not be carried out for critical commuting pairs – which is the language in which the renormalization of critical circle maps has been defined. We have resolved this difficulty in [Ya3] by introducing a new, more natural, renormalization construction, the cylinder renormalization operator. The paper [Ya3] contains the proof of hyperbolicity of the periodic orbits of this transformation. In this paper we prove that the whole renormalization horseshoe is hyperbolic for the cylinder renormalization operator. The two novel ingredients are the construction of the stable lamination for the renormalization, made difficult in this case by the absense of the hybrid class – external map decomposition; and the study of the limiting case, the parabolic renormalization carried out in [EY].

1. Preliminaries Some notations. We will generally assume that the reader is familiar with the renormalization theory of critical circle maps, and will only recall its main points briefly. For a detailed discussion, we refer the reader to [Ya3]. We use dist and diam to denote the Euclidean distance and diameter in C. We shall say that two real numbers A and B are K-commensurable for K > 1 if K −1 |A| ≤ |B| ≤ K|A|. The notation Dr (z) will stand for the Euclidean disk with the center at z ∈ C and radius r. The unit disk D1 (0) will be denoted D. The plane (C \ R) ∪ J with the parts of the real axis not contained in the interval J ⊂ R removed will be denoted CJ . By the circle T we understand the affine manifold R/Z, it is naturally identified with the unit circle S 1 = ∂D. The real translation x → x + θ projects to the rigid rotation by angle θ , Rθ : T → T. For two points a and b in the circle T which are not diametrically opposite, [a, b] will denote the shorter of the two arcs connecting them. As usual, |[a, b]| will denote the length of the arc. For two points a, b ∈ R, [a, b] will denote the closed interval with endpoints a, b without specifying their order. The cylinder in this paper, unless otherwise specified will mean the affine manifold C/Z. Its equator is the circle {Im z = 0}/Z ⊂ C/Z. A topological annulus A ⊂ C/Z will be called an equatorial annulus, or an equatorial neighborhood, if it has a smooth boundary and contains the equator. By “smooth” in this paper we will mean “of class C ∞ ”, unless another degree of smoothness is specified. The notation“C ω ” will stand for “real-analytic”.

Critical circle maps. A critical circle map is an orientation preserving automorphism of T of class C 3 with a single critical point c. A further assumption is made that the critical point is of cubic type. This means that for a lift f¯ : R → R of a critical circle map f with critical points at integer translates of c, ¯ f¯(x) − f¯(c) ¯ = (x − c) ¯ 3 (const +O(x − c)). ¯

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We note that all the renormalization results will hold true if in the above definition “3” as the order of smoothness and the order of the critical point is replaced by any other odd number. To fix our ideas, we will always place the critical point of f at 0 ∈ T. Being a homeomorphism of the circle, a critical circle map f has a well-defined rotation number, denoted ρ(f ). It is useful to represent ρ(f ) as a continued fraction with positive terms 1

ρ(f ) =

1

r0 + r1 +

.

(1.1)

1 r2 + · · ·

Further on we will abbreviate this expression as [r0 , r1 , r2 , . . . ] for typographical convenience. Note that the numbers ri are determined uniquely if and only if ρ(f ) is irrational. In this case we shall say that ρ(f ) (or f itself) is of type bounded by B if sup ri ≤ B; it is of a periodic type if the sequence {ri } is periodic. For a map f such that the continued fraction (1.1) contains at least m + 1 terms, we denote {pm /qm } the best rational approximation of ρ(f ) given as the truncated continued fraction pm /qm = [r0 , r1 , . . . , rm−1 ], and set Im ≡ [0, f qm (0)]. Recall that an iterate f k (0) is called a closest return of the critical point if the arc [0, f k (0)] contains no iterates f i (0) with i < k. One verifies then that the iterates {f qm (0)} are closest returns of 0. By a classical result of Poincar´e every circle homeomorphism f with an irrational rotation number is semi-conjugate to the rigid rotation Rρ(f ) . Yoccoz [Yoc] has shown that in the case when f is a critical circle map, the semiconjugacy actually becomes a conjugacy.

1.1. Some function spaces. In this section we will introduce some spaces of functions which will play a role in what follows. Firstly, we recall the definition of the Epstein class, introduced by Eckmann and Epstein [EE]. An orientation preserving interval homeomorphism g : I = [0, a] → g(I ) = J belongs to the Epstein class E if it extends to an analytic three-fold branched covering map of a topological disk G ⊃ I onto the double-slit plane CJ˜ , where J˜ ⊃ cl J . The importance of E will lie in its invariance under the renormalization of commuting pairs (see below). We refer the reader to [Ya3] for a detailed discussion of E and further definitions. We will equip it with the Carath´eodory topology. For s ∈ (0, 1) we will denote Es a sequentially compact subset of E consisting of maps with a geometric bound s. Let us introduce some notation suppose B is a complex Banach space whose elements are functions of the complex variable. Let us say that the real slice of B is the real Banach space B R consisting of the real-symmetric elements of B. If X is a Banach manifold modelled on B with the atlas {γ } we shall say that X is real-symmetric if (B R ) ⊂ B R for any pair of indices γ1 , γ2 . The real slice of X is then defined as γ1 ◦γ−1 2 the real Banach manifold XR ⊂ X given by γ−1 (B R ) in a local chart γ . An operator A defined on a subset of X is real-symmetric if A(X R ) ⊂ XR . Denote π the natural projection C → C/Z. For an equatorial annulus U ⊂ C/Z let AU be the space of bounded analytic maps φ : U → C/Z, such that φ(T) is homotopic to T, equipped with the uniform metric. We shall turn AU into a real-symmetric complex Banach manifold as follows. Denote U˜ the lift π −1 (U ) ⊂ C. The space

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of functions φ˜ : U˜ → C which are analytic, continuous up to the boundary, and 1˜ + 1) = φ(z), ˜ periodic, φ(z becomes a Banach space when endowed with the sup norm. ˜ U . For a function φ : U → C/Z denote φˇ an arbitrarily chosen lift Denote that space A ˜ U . We use the ˇ −1 (z))) = φ. Observe that φ ∈ AU if and only if φ˜ = φˇ − Id ∈ A π(φ(π ˜ local homeomorphism between AU and AU given by φ˜ → π ◦ (φ˜ + Id) ◦ π −1 ˜ to define the atlas on AU . The coordinate change transformations are given by φ(z) → ˜ φ(z + n) + m for n, m ∈ Z, therefore with this atlas AU is a real-symmetric complex Banach manifold. Now let f : U → C/Z be an analytic critical circle map. By definition, there is a neighborhood of the equator in which 0 is the only critical point of f . Let f˜ : U˜ → C ˜ U is be a lift of f . The Argument Principle implies that for > 0 small enough, if g˜ ∈ A   real-analytic, with g˜ (0) = −1, g˜ (0) = 0, and ||f −g|| < , then g = π ◦(g˜ +Id)◦π −1 is a critical circle map as well. Let (f ) be the supremum of such , and set ˜ U , g˜  (0) = −1, g˜  (0) = 0, and ||g˜ − f˜|| < (f )}. CU = ∪f {π ◦(g˜ +Id)◦π −1 | g˜ ∈ A As shown in [Ya3], the space CU is a codimension 2 submanifold of AU . A tangent space ˜ U , given by to CU may be identified with the codimension 2 Banach subspace of A ˜ U with φ˜  (0) = 0, φ˜  (0) = 0}. BU = {φ˜ ∈ A We shall say that f is a critical cylinder map if f ∈ CU for some U ⊂ C/Z. Let us denote by CR U the real slice of CU : ¯ CR z)} = {f ∈ AU , such that f |T is a critical circle map}. U = {f ∈ CU with f (z) = f (¯ 2. Universality and Renormalization 2.1. Definition of renormalization of critical circle maps using commuting pairs. The strong analogy with universality phenomena in statistical physics and with the already discovered Feigenbaum-Collett-Tresser universality in unimodal maps, led the authors of [FKS] and [ORSS] to explain the existence of the universal constants by introducing a renormalization operator acting on critical circle maps. The definition is by no means straightforward, we refer the reader to [Ya3] for the motivating discussion. Definition 2.1. A commuting pair ζ = (η, ξ ) consists of two C 3 -smooth orientation preserving interval homeomorphisms η : Iη → η(Iη ), ξ : Iξ → ξ(Iξ ), where (I) Iη = [0, ξ(0)], Iξ = [η(0), 0]; (II) Both η and ξ have homeomorphic extensions to interval neighborhoods of their respective domains with the same degree of smoothness, which commute, η ◦ ξ = ξ ◦ η; (III) ξ ◦ η(0) ∈ Iη ; (IV) η (x) = 0 = ξ  (y), for all x ∈ Iη \ {0}, and all y ∈ Iξ \ {0}.

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The space of C r -smooth commuting pairs considered modulo an affine conjugacy can be further equipped with a natural C r distance (see [dFdM1]). The commutation condition allows one to iterate the extensions of the maps of a commuting pair. We can also use it to perform the following glueing construction. We can regard the interval I = [η(0), ξ ◦ η(0)] as a circle, identifying η(0) and ξ ◦ η(0) and define fζ : I → I by
fζ =

ξ ◦ η(x) for x ∈ [η(0), 0] . η(x) for x ∈ [0, ξ ◦ η(0)]

The mapping ξ extends to a C 3 -diffeomorphism of open neighborhoods of η(0) and ξ ◦ η(0). Using it as a local chart we turn the interval I into a closed one-dimensional manifold M. Condition (II) above implies that the mapping fζ projects to a well-defined C 3 -smooth homeomorphism Fζ : M → M. Identifying M with the circle by a diffeomorphism φ : M → T we recover a critical circle mapping f φ = φ ◦ Fζ ◦ φ −1 . The critical circle mappings corresponding to two different choices of φ are conjugated by a diffeomorphism, and thus we recovered a C 3 -smooth conjugacy class of circle mappings from a critical commuting pair. Let f be a critical circle mapping, whose rotation number ρ has a continued fraction expansion (1.1) with at least m + 1 terms, and let pm /qm = [r0 , . . . , rm−1 ]. The pair of iterates f qm+1 and f qm restricted to the circle arcs Im and Im+1 correspondingly can be viewed as a critical commuting pair in the following way. Let f¯ be the lift of f to the real line satisfying f¯ (0) = 0, and 0 < f¯(0) < 1. For each m > 0 let I¯m ⊂ R denote the closed interval adjacent to zero which projects down to the interval Im . Let τ : R → R denote the translation x → x + 1. Let η : I¯m → R, ξ : I¯m+1 → R be given by η ≡ τ −pm+1 ◦ f¯qm+1 , ξ ≡ τ −pm ◦ f¯qm . Then the pair of maps (η|I¯m , ξ |I¯m+1 ) forms a critical commuting pair corresponding to (f qm+1 |Im , f qm |Im+1 ). Henceforth we shall simply denote this commuting pair by (f qm+1 |Im , f qm |Im+1 ).

(2.1)

This allows us to readily identify the dynamics of the above commuting pair with that of the underlying circle map, at the cost of a minor abuse of notation. Following [dFdM1], we say that the height χ (ζ ) of a critical commuting pair ζ = (η, ξ ) is equal to r, if 0 ∈ [ηr (ξ(0)), ηr+1 (ξ(0))]. If no such r exists, we set χ (ζ ) = ∞, in this case the map η|Iη has a fixed point. For a pair ζ with χ (ζ ) = r < ∞ one verifies directly that the mappings η|[0, ηr (ξ(0))] and ηr ◦ ξ |Iξ again form a commuting pair. For a commuting pair ζ = (η, ξ ) we will denote ξ |Iξ ), where the tilde means rescaling by the linear factor λ = − |I1η | . by  ζ the pair ( η|Iη ,  Definition 2.2. The renormalization of a real commuting pair ζ = (η, ξ ) is the commuting pair r ◦ ξ |I ,  r (ξ(0))]). Rζ = (η
ξ η|[0, η


The non-rescaled pair (ηr ◦ ξ |Iξ , η|[0, ηr (ξ(0))]) will be referred to as the pre-renormalization pRζ of the commuting pair ζ = (η, ξ ).

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For a pair ζ we define its rotation number ρ(ζ ) ∈ [0, 1] to be equal to the continued fraction [r0 , r1 , . . . ], where ri = χ (Ri ζ ). In this definition 1/∞ is understood as 0, hence a rotation number is rational if and only if only finitely many renormalizations of ζ are defined; if χ (ζ ) = ∞, ρ(ζ ) = 0. Thus defined, the rotation number of a commuting pair can be viewed as a rotation number in the usual sense: Proposition 2.1. The rotation number of the mapping Fζ is equal to ρ(ζ ). For ρ = [r0 , r1 , . . . ] ∈ [0, 1] let us set


 1 G(ρ) = [r1 , r2 , . . . ] = , ρ

where {x} denotes the fractional part of a real number x (G is usually referred to as the Gauss map). As follows from the definition, ρ(Rζ ) = G(ρ(ζ )) for a real commuting pair ζ with ρ(ζ ) = 0. The renormalization of the real commuting pair (2.1), associated to some critical qm+2 |I
, f
qm+1 |I
). Thus for a given critical circle map f , is the rescaled pair (f
m+1 m+2 circle map f the renormalization operator recovers the (rescaled) sequence of the first return maps: qi+1 |I qi |I )}∞ . i , f {(f
i+1 i=1 A critical commuting pair is a commuting pair (η, ξ ) whose maps are real-analytic. We shall further require that η analytically extends to an open interval Pη such that η(Pη ) ⊃ [η(0), ξ(0)] in which it has two cubic critical points at 0 and at ξ(η−1 (0)) and no other critical points; and similarly that ξ analytically extends to an open interval Pξ 0 with ξ(Pξ ) ⊃ [η(0), ξ(0)], and has a single critical point 0 in this interval. We state the last two assumptions for technical convenience; they will automatically be satisfied for renormalizations of analytic critical circle maps as defined further. The space of critical commuting pairs modulo affine conjugacy will be denoted by C; its subset consisting of pairs ζ with χ (ζ ) = ∞ will be denoted by S∞ . Renormalization is a transformation R : C \ S∞ → C. It is not difficult to show that this transformation is injective: Proposition 2.2 ([Ya2]). The map R : C \ S∞ → C is one-to-one. Proof. Let ζ = (η, ξ ) be a pair in R(C \ S∞ ). The desired statement will follow if we show that there exists a unique value of r ∈ N such that there is an element ζ−1 ∈ C\S∞ with height χ (ζ−1 ) = r for which Rζ−1 = ζ . Indeed, in this case ζ−1 will be deter−r ◦ η). To prove the uniqueness of r consider mined uniquely as the rescaled pair (ξ, ξ
an arbitrary commuting pair ζ−1 as above. Note that for j < χ (ζ−1 ) the commuting pair −j ◦ η) does not have a critical point at σ ◦ γ −1 (0) and hence is not in (γj , σj ) = (ξ, ξ
j j C. On the other hand for j > χ (ζ−1 ) the pair of maps (γj , σj ) is not a commuting pair,  since σj has a singularity at γj (0).  A commuting pair ζ lies in the Epstein class if both of its maps are. It is clear that the Epstein class E is invariant under the action of R. It was shown by various people such as ´ ¸tek, Herman, andYoccoz, that the renormalizations Sullivan (in the unimodal case), Swia 3 of any C -smooth commuting pair with an irrational rotation number converge to the Epstein class, at a geometric rate in the C 2 -metric (see [dFdM1]). Moreover, there exists σ ∈ (0, 1) such that all limit points lie in Eσ .

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2.2. Renormalization of maps of the cylinder. The cylinder renormalization operator Rcyl was introduced in [Ya3] as a more natural way to renormalize analytic critical circle maps. It has two crucial advantages: the operator acts on maps themselves, rather than their conjugacy classes; and extends to an analytic operator in the Banach manifold CU of analytic critical circle maps defined in some equatorial neighborhood U ⊂ C/Z. We briefly recall the definition of Rcyl below; the reader is directed to [Ya3] for the detailed exposition. First note that in [Ya3] we constructed a closed equivalence relation denoted conf ∼ between commuting pairs in the Epstein class Eσ with the property that if ζ1conf ∼ζ2 , then the analytic extensions of the commuting pairs are conjugate in a specific neighborhood of the interval of definition by a conformal change of coordinates. In particular, ρ(ζ1 ) = ρ(ζ2 ). Moreover, there exists a value N ∈ N such that if ζ1 and ζ2 are N -times ∼RN ζ2 . This last property implies that periodic orbits of renormalizable, then RN ζ1conf N R constructed in Theorem 3.1 project to the quotient space. The cylinder renormalization operator defined in [Ya3] is a real-symmetric analytic operator Rcyl from an open subset R of CU to CU (thus Rcyl maps an open subset of M ≡ CR U into M), such that the following holds. For every f ∈ M such that the continued fraction of ρ(f ) has at least r terms, f ∈ R. For such maps the operator Rcyl changes the rotation number by shifting its continued fraction expansion r terms to the left. Moreover, for every r = ∞ there is a canonical immersion ιr

Sr ∩ Eσ /conf ∼ −→ M such that ρ(ιr (ζ )) = ρ(R(ζ )), and ιr ◦ RN ◦ ι−1 r ≡ Rcyl . Every map f in R posesses a domain Cf called a fundamental crescent, which is bounded by two simple curves l and f qN (l) which meet at two endpoints p1 , p2 , which are repelling fixed points of the iterate f qN . Moreover, the quotient of Cf ∪ f qN (Cf ) \ {p1 , p2 } by the iterate f qN is conformally isomorphic to C/Z. The first return map Rf of Cf under this isomorphism becomes Rcyl f . 2.3. Explanation of universality: Lanford’s Program. The original golden-mean universality hyperbolicity conjecture was described in [ORSS] and [FKS]. It was later generalized by Lanford [Lan2], and in its most general form became known as Lanford’s Program. Since stating a conjecture allows for a degree of vagueness, we simply use R for the renormalization operator without specifying which one we have in mind; the original Program was stated for critical commuting pairs, but we will prove it for Rcyl acting on critical circle maps. To outline Lanford’s Program, let us give an informal description of the action of renormalization on the space of critical circle maps. Let ρ = φ(θ ) : T → T be a monotone continuous function such that φ −1 (ρ) is a point for ρ irrational and an interval otherwise. Imagine the space of critical circle maps as an infinite-dimensional cylinder C = T × C , where the rotation number of a commuting pair ζ (θ, ·) with the equatorial coordinate θ ∈ T is ρ(θ ). The cylinder C is partitioned into strips (cf. Fig. 1) Sr = {ζ ∈ C|ρ(ζ ) = [r, r1 , . . . ]} for r = 1, . . . , ∞. A boundary component of the strip S∞ is the hypersurface P∞ ⊂ S∞ with the property that a pair ζ ∈ P∞ if and only if it has a single fixed point with unit eigenvalue ( ζ is a “parabolic pair”). The sets Sr accumulate on P∞ in the clockwise direction.

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S

S∞

Sr

R P

Rr

Fig. 1.

It is natural to think of the transformation R : C \ S∞ → C as being defined piecewise, given on each Sr , r = ∞ by the formula: Rr : (η, ξ ) → (η, ηr ◦ ξ ). The operator Rr expands the strip Sr in the equatorial direction, mapping it onto a thin equatorial cylinder, which intersects all the vertical strips. We may close the domain of R by adjoining the hypersurface P∞ to it and setting R on P∞ to an appropriately defined (multivalued) parabolic renormalization transformation P; with this convention the image of P∞ is also an equatorial cylinder. Lanford conjectured that this picture can be appropriately endowed by a smooth metric, so that the operator R is uniformly expanding in the θ direction and uniformly contracting in the complementary directions. Thus R posesses an infinite-dimensional “horseshoe” structure. The “boxes” Rr−1 (Sr−1 ∩ Rr−2 (Sr−2 ∩ · · · (Rr−n Sr−n ) · · · ) ∩ Sr0 −1 −1 ∩R−1 r0 (Sr1 ∩ Rr1 (Sr2 ∩ · · · Rrn−1 (Srn ) · · · )

are exponentially small in n, and their intersection is the R-invariant hyperbolic Cantor set A parametrized by bi-infinite sequences of ri ∈ N ∪ {∞}. The set A is a “horseshoe” attractor for the renormalization operator, dist(Rn ζ, A) → 0 at a geometric rate. The action of R on A is conjugate to the two-sided shift.

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Let us conclude this exposition by formalizing the Hyperbolicity Conjecture as follows: Renormalization Hyperbolicity Conjecture (Lanford’s Program). The renormalization operator R in the space of critical circle maps possesses a “horseshoe” attractor A on which its action is conjugated to the two-sided shift. Moreover, there exists a renormalization-invariant space of critical circle maps with the structure of an infinite dimensional smooth manifold, with respect to which A is a hyperbolic set with onedimensional expanding direction. If t denotes the expanded coordinate in a local chart, then the dependence t → ρ(t, ·) is continuous and (not strictly) monotone. 3. Outline of the Results 3.1. Known results. The early efforts to prove the Renormalization Hyperbolicity Conjecture culminated in an argument of Mestel [Mes], who established the existence and local hyperbolicity properties of a fixed point of R1 in C ∞ . The argument was based on rigorous computer estimates. This approach was developed by Lanford in the context of renormalization of unimodal maps. One of its major drawbacks is its local nature; neither the uniqueness of the fixed point, nor global hyperbolicity of R1 are established.

Construction of the renormalization horseshoe. In his address to ICM-86 in Berkeley Sullivan [Sul1] outlined a program of constructing the invariant set of renormalization of unimodal maps and its stable set by means of Teichm¨uller theory, which he subsequently carried out (see [Sul2, MvS]). Sullivan’s program has been adapted to the setting of critical circle mappings by de Faria [dF1, dF2]. De Faria has defined, in particular, extensions of Epstein commuting pairs to holomorphic dynamical systems in the plane, which are analogous to quadratic-like mappings in the unimodal renormalization theory. We discuss these extensions, called holomorphic commuting pairs, in §4.1. Using Sullivan’s Riemann surface laminations machinery, de Faria constructed a horseshoe attractor AB for the renormalization operator acting on the space of Epstein commuting pairs of type bounded by some constant B (this corresponds to restricting the operator R to the finite collection of vertical strips ∪Sr , r ≤ B in Fig. 1). In a recent paper [Ya2] we generalized the results of de Faria to Epstein commuting pairs of unbounded topological type. Before formulating the result, let us make some further notation. Let  be the space of a bi-infinite sequence of natural numbers, and denote by σ :  →  the shift on this space: ∞ σ : (ri )∞ −∞ → (ri+1 )−∞ . Let us also complement the natural numbers with the symbol ∞ with the conventions ¯ the space (N ∪ {∞})Z , and by  ¯ + the space ∞ + x = ∞, 1/∞ = 0; and denote by  N (N ∪ {∞}) . Our theorem completely settles the first part of Lanford’s Program: Theorem 3.1 (Renormalization horseshoe). There exists an R-invariant set I ⊂ E consisting of pairs with irrational rotation numbers with the following properties. The ¯ → : ¯ action of R on I is topologically conjugate to the two-sided shift σ :  i ◦ R ◦ i −1 = σ,

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and if ζ = i −1 (. . . , r−k , . . . , r−1 , r0 , r1 , . . . , rk , . . . ) then ρ(ζ ) = [r0 , r1 , . . . , rk , . . . ]. The set I is pre-compact, its closure A ⊂ E is the horseshoe attractor for the renormalization. That is, for any ζ ∈ E with irrational rotation number we have Rn ζ → A in Carath´eodory topology. Moreover, for any two pairs ζ , ζ  ∈ E with ρ(ζ ) = ρ(ζ  ) we have dist(Rn ζ, Rn ζ  ) → 0 for the uniform distance between analytic extensions of the renormalized pairs on compact sets. The proof of Theorem 3.1 combines the ideas of McMullen from [McM2] with a priori bounds obtained in [Ya1]. We consider the geometric limits of the rescaled sequences of renormalizations ζ, Rζ, . . . , Rn ζ. These limits which we call bi-infinite limiting renormalization towers (see §4.1 for the definition) should be thought of as bi-infinite sequences of nested holomorphic commuting pairs in the plane. They are naturally equipped with rotation numbers which are ¯ Each tower corresponds to a single point in the attractor A, bi-infinite sequences in . and the result follows from a Rigidity Theorem [Ya2]: ¯ there exists a unique Theorem 3.2 (Tower rigidity). For each bi-infinite sequence s¯ ∈  (up to a real homothety) bi-infinite tower with s¯ as its rotation number. An important difference with the situation considered by McMullen and its adaptation to the case of circle maps carried out in [dFdM2] is the presense of parabolic commuting pairs in a limiting tower. Parabolic elements of the attractor A are the pairs ζ in the closure of I on which renormalization is not defined. Necessarily, ρ(ζ ) = 0, and ζ posesses a unique fixed point, which has eigenvalue 1. We call any commuting pair with this property parabolic, and construct a family of natural extensions of R to parabolic commuting pairs, the parabolic renormalizations Pθ , θ ∈ T. We may extend Theorem A to parabolic pairs as follows. For every parabolic commuting pair ζ ∈ E and every ρ ∈ T there exists a unique θ (ζ, ρ) with the property that ρ(Pθ(ζ,ρ) ζ ) = ρ, and moreover, if ρ is rational then Pθ(ζ,ρ) ζ is also a parabolic pair. Let us define X = {ζ ∈ E, ρ(ζ ) ∈ R \ Q} ⊂ E, ¯ +. and blow-up the boundary of this space by replacing every boundary element ζ by ζ × −1 To topologize this space, let us replace every ζ ∈ X with ρ(ζ ) ∈ R \ Q with (ζ, i (ζ )), and in the case ζ ∈ ∂X , and ρ(ζ ) = [r1 , . . . , rk ], for s¯ = (s1 , s2 , . . . ) let us express the corresponding point in the blown-up space as (ζ, t¯), where t¯ = (r1 , . . . , rk , s1 , s2 , . . . ). ¯ + to topologize the blow-up of X , and Now we use the product topology on X ×  call the resulting space Xˆ . Let Aˆ ⊂ Xˆ be the blow-up of A. Define the generalized renormalization operator G : Xˆ → Xˆ setting G(ζ ) = R(ζ ) in the case ρ(ζ ) ∈ R \ Q, and G(ζ, (r1 , r2 , . . . )) = (Pθ (ζ ), (r2 , r3 , . . . )), where θ = θ(ζ, [r1 , r2 , . . . ]). With our definitions we have the following: Theorem 3.3. The set Aˆ is invariant under G. The map i :  → I extends homeomorˆ so that G ◦ i = i ◦ σ . The generalized renormalizations G n (x) ¯ → A, phically to  converge to Aˆ for any x ∈ Xˆ .

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In the case of bounded type rotation number, de Faria and de Melo [dFdM2] strengthened the original result of de Faria, utilizing a technique of McMullen [McM2] to show that the rate of converegence to AB is geometric: Theorem 3.4 ([dFdM2]). For every B ∈ N and every r ∈ N ∪ 0 there exists αr,B < 1 such that for every ζ ∈ E with ρ(ζ ) ∈ R \ Q of type bounded by B and for every ζˆ ∈ AB with ρ(ζˆ ) = ρ(ζ ) we have dist C r (Rn ζ, Rn ζˆ ) < const ·(αr,B )n . Despite the difficulty in imposing a manifold structure on the space of commuting pairs compatible with the analyticity of R, it has long been known how one may construct an unstable direction in such a manifold. A version of this construction was first shown to us by Folkert Tangerman, who participated in its discussion in Sullivan’s seminar at the Graduate Center of CUNY. Cylinder renormalization operator, and hyperbolicity of periodic orbits.. The main result of [Ya3] is the following: Theorem 3.5 (Hyperbolicity of Periodic Orbits of Rcyl ). There exists a real-symmetric analytic operator Rcyl from an open subset of CU to CU (thus Rcyl maps an open subset of M into M), such that the following holds. For every r = ∞ there is a canonical homeomorphism onto the image ιr

Sr ∩ Es /conf ∼ −→ M such that ρ(ιr (ζ )) = ρ(R(ζ )), and ιr ◦ RN ◦ ι−1 ≡ Rcyl . For every periodic point r p ∈ Sr of the operator RN there exists  ∈ N such that the point ιr (p) is a hyperbolic fixed point of Rcyl |M , with one-dimensional unstable direction. In addition, in the paper [EY] we have considered the cylinder version of the parabolic renormalization Pθ (which we will again denote Pθ to avoid a cumbersome notation). In this setting, we gave a much simplified argument for a part of Theorem 3.3. Moreover, denoting Gcyl the cylinder version of the generalized renormalization operator, we have the following: Theorem 3.6. The family of projections ιr can be complemented by

∼ → M, ι∞ : S∞ ∩ Xˆ /conf ¯ so that ι∞ ◦ G N ◦ ι−1 ∞ ≡ Gcyl . Let s¯ = (. . . , s−n , . . . , s0 , . . . , sn , . . . ) ∈  \  be a periodic sequence, and set κ = (ζs¯ , (s0 , . . . , sn , . . . )) = i(¯s ) ∈ Xˆ . Denote f = ιs0 (κ) ∈ M the corresponding critical cylinder map, which in this case posesses a parabolic periodic orbit. Let k be the period of f as a periodic point of Gcyl . Then there exists a codimension one complex-analytic submanifold of CU containing f k is an analytic contraction. in which Gcyl

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3.2. Statement of the main result. In the present paper we complete our investigation of Lanford‘s Program by showing the following: Theorem 3.7. The set Iˆ ≡



ir (I ∩ Sr ) ⊂ M

r

is uniformly hyperbolic for the analytic operator Rcyl , with one-dimensional unstable direction. A key ingredient of the proof is the construction of the stable lamination of the renormalization operator: Theorem 3.8. There exists a lamination L ⊂ CU of a neighborhood of the closure of Iˆ whose leaves are codimension one analytic submanifolds with the following properties. • For every point f ∈ Iˆ there is a unique leaf Lf passing through f . All the points in Lf are infinitely cylinder renormalizable, and lie in the stable set of f . • Moreover, every g ∈ Lf is quasiconformally conjugate to f on some neighborhood of the equator (which varies with g). • The collection of leaves {Lf } is dense in L, and leaves in L \ ∪Lf are submanifolds consisting of critical cylinder maps with parabolic periodic orbits. • Denoting Lˆ the blow-up of L constructed as in Theorem 3.3 we have that Lˆ is Gcyl -invariant and its leaves are again stable subsets of Gcyl . 4. Preliminary Considerations 4.1. Holomorphic commuting pairs. De Faria [dF1, dF2] introduced holomorphic commuting pairs to apply Sullivan’s Riemann surface lamination technique to renormalization of critical circle maps. They are suitably defined holomorphic extensions of critical commuting pairs which replace Douady-Hubbard polynomial-like maps [DH2]. A critical commuting pair ζ = (η|Iη , ξ |Iξ ) extends to a holomorphic commuting pair H if there exist four simply-connected R-symmetric domains , D, U , V such that ¯ U¯ , V¯ ⊂ , U¯ ∩ V¯ = {0}; U \ D, V \ D, D \ U , and D \ V are nonempty • D, connected and simply-connected regions, U ∩ R = Iη , V ∩ R = Iξ ; • mappings η : U → ( \ R) ∪ η(Iη ) and ξ : V → ( \ R) ∪ ξ(Iξ ) are onto and univalent; • ν ≡ η ◦ ξ : D → ( \ R) ∪ ν(ID ) is a three-fold branched covering with a unique critical point at zero, where ID = D ∩ R. We shall call ζ the commuting pair underlying H, and write ζ ≡ ζH . The domain D ∪ U ∪ V of a holomorphic commuting pair H will be denoted  or H , the range will be denoted  or H . The closure of the set of points whose orbit under H is contained in  will be referred to as the filled Julia set of H, denoted K(H). The Julia set of H is defined as J (H) = ∂K(H). It is easy to see directly from the definition (cf. [dF2]) that: Proposition 4.1. Let ζ be a commuting pair with χ (ζ ) < ∞. Suppose ζ is a restriction of a holomorphic commuting pair H, that is ζ = ζH . Then there exists a holomorphic commuting pair G with range H , such that ζG = Rζ .

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In this paper we will slightly generalize the definition of holomorphic commuting pairs. Namely, we will say that H is a generalized holomorphic commuting pair if H is conjugated to a holomorphic commuting pair G by a quasiconformal homeomorphism. All the relevant definitions naturally extend to this case. We say that a real commuting pair (η, ξ ) with an irrational rotation number has complex a priori bounds, if all its renormalizations extend to holomorphic commuting pairs with bounded moduli: mod( \ ) > µ > 0. For µ ∈ (0, 1) let H(µ) denote the space of holomorphic commuting pairs H : H → H , with mod(H \ H ) > µ, min(|Iη |, |Iξ |) > µ and diam(H ) < 1/µ. The existense of complex a priori bounds is a key analytic issue of renormalization theory. In the case of critical circle maps it is settled by the following theorem: Theorem 4.2. There exist universal constants µ > 0 and K > 1 such that the following holds. Let ζ ∈ C be a critical commuting pair with an irrational rotation number. Then there exists N = N (ζ ) such that for all n ≥ N the commuting pair Rn ζ extends to a holomorphic commuting pair Hn : n → n in H(µ). The range n is a Euclidean disk, and the regions n ∩ (±H) are K-quasidisks. We first proved this theorem in [Ya1] for commuting pairs ζ in an Epstein class Es , in which case N = N (s). Our proof was later adapted by de Faria and de Melo [dFdM2] to the case of a non-Epstein critical commuting pair. Let ζ be at least n times renormalizable critical commuting pair. For the lack of a better term, let us say that the pair of numbers τn (ζ ) = (rn−1 , rn−2 ) forms the history of the pair Rn ζ . Based on the above theorem and a detailed analysis of the shapes of the domains n we proved the following in [Ya1]: Theorem 4.3 ([Ya1]). There exists a universal constant K1 > 1 such that the following holds. Let ζ1 = (η1 , ξ1 ) and ζ2 = (η2 , ξ2 ) be two critical commuting pairs with irrational rotation numbers. Let n > max(N (ζ1 ), N (ζ2 )) + 1 as above. Assume that the nth renormalizations of ζ1 , ζ2 have the same rotation number and the same history. Then their holomorphic commuting pair extensions Hn1 , Hn2 are K1 −quasiconformally conjugate. The conjugating map is conformal on the filled Julia set. For commuting pairs of type bounded by B this theorem was first proved by de Faria [dF1, dF2], with “K1 " depending on the value of B. 4.2. Limiting renormalization towers. Sullivan’s original proof of the existence and uniqueness of the Feigenbaum fixed point used the Teichm¨uller theory for Riemann surface laminations associated to quadratic-like maps, which he himself developed. McMullen [McM2] has replaced this approach with an elegant argument based on a rigidity result for dynamical objects he called renormalization towers. The renormalization towers of critical commuting pairs were studied in [dFdM2] for pairs of bounded type, and later in [Ya2] without the restriction on the type. Their definition is as follows. Let {Hk }∞ 1 be a sequence of generalized holomorphic commuting pairs. Set ζk = ζHk . Suppose m(k) > n(k) → ∞ are such that ζk is m(k)-times renormalizable. Moreover, the pre-renormalizations pRn(k)+i ζk have holomorphic pair extensions Hki such that Hki → Hi ∈ H, for i ∈ Z. Then the bi-infinite sequence T = (Hi )∞ −∞ is referred to as a bi-infinite limiting renormalization tower. The rotation number ρ(T ) is naturally defined as the bi-infinite sequence of heights (χ (ζHi ))∞ −∞ . The precise formulation of Theorem 3.2 is:

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Tower Rigidity Theorem [Ya2]. For any bi-infinite sequence ρ¯ = (χn )∞ −∞ , χn ∈ N ∪ {∞} there exists a bi-infinite limiting renormalization tower T with ρ(T ) = ρ. ¯ 2 ∞ Moreover, for any two limiting renormalization towers T1 = (Hi1 )∞ −∞ , T2 = (Hi )−∞ with ρ(T1 ) = ρ(T2 ) we have ζH1 ≡ ζH2 for all i. i

i

4.3. Parabolic maps and their perturbations. Let us begin with a statement about limits of renormalizable maps in the Epstein class: Lemma 4.4 (Lemma 2.13, [Ya2]). Let ζ = (η, ξ ) ∈ E be a critical commuting pair with ρ(ζ ) = 0, which appears as a limit of a sequence {ζn } ⊂ E with ρ(ζn ) ∈ R \ Q. Then the map η has a unique fixed point in the interval Iη , which is necessarily parabolic, with multiplier one. We will call a commuting pair ζ = (η, ξ ) ∈ E parabolic if the map η has a unique fixed point in Iη , which has a unit multiplier; this point will usually be denoted pη . Note that by virtue of its uniqueness, pη has to be globally attracting on one side for the interval homeomorphism η|Iη , it is globally attracting on the other side under η−1 . A review of the theory of parabolic bifurcations, as applied, in particular, to maps in the Epstein class is given in [Ya2]; we summarize some basic facts here. Let us fix a parabolic pair η0 ∈ E and set p = pη0 . We will denote A : U A → C and R : U R → C the attracting and repelling Fatou coordinates of η0 , C A and C R the corresponding Fatou cylinders, and πA : U A → C A and πR : U R → C R the natural projections. Any conformal transit homeomorphism τ : C A → C R is a translation in suitable coordinates. Lifting it produces a map τ¯ : U A → C satisfying τ ◦ πA = πR ◦ τ¯ . We will sometimes write τ ≡ τθ , and τ¯ = τ¯θ , where R ◦ τ¯ ◦ (A )−1 (z) ≡ z + θ mod(z). Suppose for an Epstein map η in a sufficiently small neighborhood of η0 the parabolic point splits into a complex conjugate pair of repelling fixed points pη ∈ H and p¯ η with multipliers e2πi±α(η) . In this situation one may still speak of attracting and repelling petals UηA , UηR , and define the Douady coordinates A R R A η : U → C and η : Uf → C,

such that

A R R A η (η(z)) = η (z) + 1 and η (η(z)) = η (z) + 1.

The Fatou and Douady coordinates can be specified uniquely with an arbitrary choice of real basepoints a ∈ U A and r ∈ U R , by requiring that A (a) = A η (a) = 0, and R R R  (r) = η (r) = 0. With these normalization, the Douady coordinates A η , and η depend continuously on η with respect to the compact-open topology on the image (in fact, the dependence is analytic in the appropriate sense), and converge to the respective Fatou coordinates as η → η0 . Moreover, select the smallest n(η) ∈ N for which ηn(η) (a) ≥ r. Then −1 ηn(η) (z) = (R ◦ Tθ(η)+K ◦ A η) η

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wherever both sides are defined. In this formula Ta (z) denotes the translation z → z + a, θ(η) ∈ [0, 1) is given by θ (η) = 1/α(η) + o(1) mod 1, α(η)→∞

and the real constant K is determined by the choice of the basepoints a, r. Thus for n(η ) a sequence {ηk } converging to η0 , the iterates ηk k converge locally uniformly if and only if there is a convergence θ (η) → θ , and the limit in this case is a certain lift of the transit homeomorphism τθ for the parabolic map η0 . 4.4. Parabolic renormalization of maps of the cylinder. We again refer the reader to [Ya3] for the definition of the spaces AU , BU , the Banach manifold CU and the cylinder renormalization Rcyl . To define ι∞ as in Theorem 3.6, let ζ = (η, ξ ) ∈ Es have a parabolic fixed point p (as shown in [Ya2], such a point is necessarily unique). Let CζA , and CζR be the ´ attracting and repelling Ecalle-Voronin cylinders of this point, and denote Eζ the first return map from a neighborhood of T ⊂ CζR to CζA . Let us normalize it so that its critical point on T is at 0. Then Eζ is a critical circle map in CR U . As before, we see that it is independent of the choice of representative of [ζ ]conf . For θ ∈ C/Z denote ∼ τθ : C/Z → C/Z the transit isomorphism z → z + θ mod Z. Monotonicity consid¯ + there exists a unique choice erations (see [Ya2]) imply that for every element s¯ ∈  of θ = θ (¯s ) ∈ T for which the composition τθ ◦ Ef has rotation number with continued fraction s¯ , and when σ¯ ∈  + \  + , it has a parabolic periodic orbit in T. We set ι∞ (ζ, s¯ ) = fζ,¯s = τθ(¯s ) ◦ Eζ . Let πζA : UζA → CζA , πζR : UζR → CζR denote the natural projections. Let N ≥ 0 be such that ηN (ξ(0)) ∈ C A . Fix a branch πR−1 , and for every θ ∈ T take the smallest M for which ηM ◦ πR−1 ◦ τθ ◦ πA ◦ ηN ◦ ξ(0) ∈ [0, η(0)]. The composition γθ ≡ ηM ◦ πR−1 ◦ τθ ◦ πA ◦ ηN ◦ ξ has a well-defined extension to the whole interval [0, η(0)] which is independent of the choice of the branch of πR−1 . ¯ + \  + . The generalized renormalization of an element (ζ, s¯ ) ∈ Xˆ is Now let s¯ ∈  defined in [Ya2] as
G(ζ, s¯ ) = ((γθ(¯s ) |[η(0),0] , η|[0,γ (0)] ) . θ (¯s

Finally, let us define the generalized cylinder renormalization of a pair (fζ,¯s , σ (¯s )) as Gcyl (fζ,¯s , σ (¯s )) = fG N (ζ,¯s ) . Denote Z the subset of CU consisting of maps with a parabolic orbit having the same period as the parabolic orbit of fζ . As before, Gcyl extends to an analytic operator from a neighborhood of fζ in Z (which is an analytic codimension one submanifold of CU by the Implicit Function Theorem) to CU .

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5. Stable Lamination of Rcyl In this section we shall prove Theorem 3.8. For a positive reduced rational fraction p/q let us define Sp/q to be the set of elements g ∈ CU possessing an invariant equatorial quasi-circle Tg ⊂ U containing the critical point 0, and such that the rotation number ρ(g|Tg ) = p/q. Further, let Dp/q ⊂ Sp/q be the collection of maps g ∈ Sp/q with g q (0) = 0; and let Pp/q be the set of maps g ∈ Sp/q with a parabolic periodic point in Tg . Let us also define Vq ⊃ Dp/q as the subset of CU given by the condition g q (0) = 0. Our first lemma is the following: Lemma 5.1. There exists N ∈ N and an open set Y ⊂ CU , which contains CR U , such that for all p/q whose continued fraction expansion contains at least N + 2 terms the following holds: (I) the set Dp/q is open in Vq ∩ Y, and is a codimenion one submanifold of CU ; (II) for a given p/q the quasicircle Tg can be chosen to move holomorphically with g ∈ Dp/q ∩ Y. Proof. The argument we give makes use of the complex a priori bounds of [Ya1]. As follows from that work, there exists a Euclidean disk D ⊂ U around the origin, and N ∈ N such that for all p/q as above, and every g ∈ Dp/q ∩ CR U the following holds. Denote pi /qi the i th continued fraction approximation of p/q. Then there exist simplyg g connected domains 1 containing g −qN (0) ∩ Tg = T and 2 g −qN +1 (0) ∩ Tg = T g g g such that i
D, and moreover, mod(D \ (1 ∪ 2 )) > for some universal bound

> 0, and g

g

g qN (1 ) = D, g qN +1 (2 ) = D, where both maps are 3-to-1 on the interior. Now we may select an open neighborhood Y of CR U small enough so that for all maps g g ∈ Dp/q ∩ Y there are two domains i
D containing the appropriate preimages of 0 and having the same properties. Let Bg : Ug (0) → D denote the B¨ottcher coordinate of g, which conjugates g q to z → z3 on a neighborhood of the origin. We may choose a particular normalization of Bg so that the dependence g → Bg is analytic. Let lg denote the Bg -preimage of the straight segment [x, x 3 ) for some small x > 0. Inductively define lgm to be the g q -preimage of lgm−1 adjacent to it. Due to the choice of Y we have made, and since g q factors through g qN , g qN +1 , all lgm are contained in D, and thus exist for all m. For obvious reasons these curves do not intersect. Observe that they converge to a repelling periodic point pg of g. We set q−1

m ∞ mq n Lg = (∪∞ m=0 lg ) ∪ (∪m=0 g (lg )), and Tg = ∪n=0 g (Lg ).

Since Bg analytically depends on g, the dependence g → Tg is a holomorphic motion, and by construction, for g ∈ CR U , Tg = T. For every g ∈ Dp/q we may choose a neighborhood U (g) ⊂ Y small enough so that the orbit pg remains repelling throughout this neighborhood. Therefore, the above construction remains true for every f ∈ U (g) ∩ Vq , hence f ∈ Dp/q . By the Implicit Function Theorem, Dp/q ∩ Y is a codimension one submanifold of CU , and the proof is complete.  

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Let us give several further definitions. For a critical cylinder map g ∈ CU we denote MU (g) the space of Beltrami differentials µ on C/Z with ||µ||∞ < ∞ which satisfy the invariance condition g ∗ µ(z) = µ(g(z)) for almost every z, g(z) ∈ U. Equipped with L∞ -norm, MU (g) is a Banach space. As usual, σ0 ∈ MU (g) will stand for the trivial Beltrami differential in C/Z, specifying the standard complex structure. As before, we denote U˜ = π −1 (U ) ⊂ C, and identify the tangent space to CU at any point f with BU . Let C1 be an open cone in BU given by C1 = {v ∈ BU , such that inf Re v(x) > x∈R

1 ||v||}. 2

Let us take a rotation number p/q ∈ Q whose continued fraction has at least N +2 terms, and fix f ∈ Dp/q ∩ Y. To every tangent vector v ∈ Tf Dp/q we will associate an element of MU (f ) which produces the infinitesimal quasiconformal deformation of f in the direction of v. To do so, consider g ∈ Dp/q ∩ Y. Utilizing the complex bounds again, we see that pRN g has a generalized holomorphic commuting pair extension Hg : g → . Here the fundamental region Wg =  \ g is bounded by the arcs of the circle ∂, and the invariant quasicircle Tg , as well as a finite number of their preimages. In view of the previous lemma, the dependence g → Wg is a holomorphic motion, equivariant with respect to the dynamics of Hg . We may now select an analytic family of K-quasiconformal maps hg : Wf → Wg which conjugate the holomorphic commuting pairs Hg and Hf on the boundary. By the standard pull-back argument (see e.g. [Ya1]) we can extend hg to K-quasiconformal conjugacies of the holomorphic pairs Hf and Hg . Using the dynamics of f and g we can propagate this conjugacy to a K-quasiconformal conjugacy h˜ g mapping a fixed equatorial annulus V onto an annulus Vg around Tg . Note that by construction, the dependence g → h˜ g is analytic. Finally, we may extend h˜ g to a quasiconformal homeomorphism of C/Z, again so that the dependence on g is analytic, and denote µg ∈ MU (f ) its Beltrami differential. Now let v ∈ Tf Dp/q . Select an analytic family of maps gt ∈ Dp/q with the property gt = π(π ◦ f ◦ π −1 + Id +tv + o(t)), and construct h˜ gt and µgt as above. Differentiate the analytic map t → µgt to obtain µf,v ∈ MU (f ). A standard computation shows that v satisfies the following equivariance relation: ¯ = µf,v . π(Id +v) = −f  ϑ + ϑ ◦ f, where ∂ϑ

(5.1)

Lemma 5.2. A vector field v ∈ Tf Dp/q if and only if it satisfies the equivariance condition (5.1) with µf,v ∈ MU (f ). Proof. We have seen the “only if” part. To show the converse, starting with v satisfying (5.1) consider the family of Beltrami differentials µt = tµf,v . Denoting ψt the normalized solution of the Beltrami equation ψt∗ σ0 = µt , and gt = ψt ◦ f ◦ ψt−1 ∈ Dp/q , we have   gt = π(π ◦ f ◦ π −1 + Id +tv + o(t)).

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Lemma 5.3. There exists an open subset Y1 ⊂ Y, which again contains CR U , such that for all p/q with the continued fraction expansion of length at least N + 2, and every f ∈ Dp/q ∩ Y1 the tangent space Tf Dp/q ∩ C1 = ∅. The same is true for f ∈ Pp/q ∩ Y1 for an arbitrary p/q. Proof. Let h ∈ CR U ∩ Dp/q , and denote the sup-distance between f and h. Fix a subannulus V
U . Let 0 be such that if < 0 , and v ∈ Tf Dp/q , vˆ ∈ Th Dp/q , with unit norms, and µf,v ≡ µh,vˆ in  \ h , then ||v − v|| ˆ < 1/8. Let us now take Y1 to be the 0 neighborhood of CR U . Let us show that for f ∈ Y1 , and v ∈ C1 , v∈ / Tf Dp/q . Assume the contrary. Then we may choose µ = µh,vˆ ∈ MU (h) such that ¯ 1 = µ, π(Id +v) ˆ = −h ϑ1 + ϑ1 ◦ h, where ∂ϑ and inf Re v(x) ˆ >

x∈R

1 ||v| ˆ V ||. 4

(5.2)

Indeed, we may simply take µ ≡ µf,v on h \ h , and then extend it dynamically to a Beltrami differential on C/Z as above. The h-invariant Beltrami differential µ gives rise to an analytic curve {ht } ⊂ CR ˆ V ∈ BV . V beginning at h|V whose tangent vector is v| R Clearly, no vector field which satisfies (5.2) can be tangent to a map in CU , and the claim follows. For f ∈ Pp/q the result follows by continuity.   The last lemma implies that the analytic submanifolds Dp/q ∩Y1 uniformly converge to a codimension one lamination L ⊂ CU . It obviously satisfies the properties required in Theorem 3.8. 6. Proof of Global Hyperbolicity 6.1. Unstable direction. Let us select an annulus U   U such that for a renormalizable ˆ f ∈ I, Rcyl f ∈ CU  . ˆ Define a cone field C ⊂ BR U on I as follows:  Cf = {v ∈ Tf CR U ; v analytically extends to U and inf v(x) > 0}. x∈R

ˆ We first have Fix a renormalizable map f ∈ I. Proposition 6.1. The operator Df Rcyl : BU → BU is compact. Proof. See Proposition 9.1 of [Ya3].

 

Lemma 6.2. (I) The cone field C is renormalization-invariant: DRcyl : C → C. (II) Moreover, there exists α > 0 and k ∈ N such that for any f ∈ Iˆ and vector field v ∈ Cf ,

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inf DRkcyl (v(x)) > (1 + α) inf v(x).

x∈R

x∈R

(III) Finally, there exists  ∈ N such that if v ∈ v = 0, then DRcyl (v) ∈ C.

BR U

belongs to the closure of C, and

Proof. Let f˜ = π −1 ◦ fˆ ◦ π . As follows from an elementary computation, if f˜t (x) = f˜(x) + tv(x) + o(t), then f˜t2 (x) = f˜2 (x) + tv2 (x) + o(t) = f˜2 (x) + t (f˜ (f˜(x))v(x) + v(f˜(x))) + o(t),

(6.1)

and more generally, if we write f˜tn (x) = f˜n (x) + tvn (x) + o(t), then vn (x) = f  (f n−1 (x))vn−1 (x) + v(f n−1 (x))

(6.2)

Let us denote Ck
U the fundamental crescent of f corresponding to the k th cylinder renormalization Rkcyl f , and Jk = Ck ∩ T. Let φk (x) be the corresponding uniformizing

coordinate Ck → C/Z, and k : Cn → C its lift. In the local chart given by φk−1 , the k th cylinder renormalization Rkcyl f is represented by the iterate gk = f qmk for some m ∈ N. To prove the first claim, observe that by (6.2), inf vn (x) ≥ inf v(x) > 0.

(6.3)

DRcyl v = [(k ) ◦ gk · (vqmk |Jk )] ◦ (k )−1 .

(6.4)

x∈R

x∈R

The image

Since (k ) > 0, (I) follows. To prove (II), note that by the real a priori bounds, the sizes of the intervals Jk decrease geometrically with k. On the other hand, applying the Koebe Distortion Theorem to the conformal extension of k to Ck and its two neighboring iterates, we see that the distortion of k on the interval gk (Jk ) is bounded uniformly in k nd f . Hence, there exists α > 0 and l ∈ N independent of f such that for all k ≥ l, (k ) |gk (Jk ) > 1 + α, and (II) follows from (6.4). Finally, to see (III), observe that if v(x) ≡ 0, then there exist , n ≤ qm such that v(x) > 0 for every x ∈ f n (J ). The claim now follows from (6.2).   We conclude: Proposition 6.3. There exists k ∈ N, α > 0 such that for every f ∈ Iˆ the operator Df Rkcyl has an eigenvalue λf > 1 + α. Proof. Let k, α be as in the previous lemma. Since the spectrum of the compact operator Df Rkcyl is discrete, its every non-zero element is an eigenvalue. Hence Part (II) of the previous lemma implies that the spectral radius RSp (Df Rkcyl ) > 1 + α.  

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6.2. Geometric Contraction. Theorem 6.4. There exists ρ < 1, C > 0 such that for every f ∈ L, ˆ < Cρ k . distU (Rkcyl (f ), I) Moreover, there exists k ∈ N such that ∀f ∈ Iˆ for the differential ||DRkcyl |Lf || < ρ. Proof. The argument we use is due to Lyubich [Lyu4, Lyu5]. Firstly, note that the convergence to Iˆ is uniform in L. Indeed, otherwise there exists > 0 such that for an arbitrary large k ∈ N we may find two maps f and g in the same leaf of L whose first k renormalizations are -apart in the uniform metric. Using a normality argument, we may then construct two distinct bi-infinite limiting renormalization towers in the sense of [Ya2] with the same rotation number. This contradicts the Tower Rigidity Theorem of [Ya2]. By the Schwarz Lemma in a Banach manifold, the uniform convergence implies strict contraction in the leaves of L, and the claim follows.   ´ ¸tek [Sw] in 1988, with the use 6.3. Concluding remarks. It has been shown by Swia of the methods of real dynamics, that in smooth and monotone (∂/∂t > 0) families {ft } of smooth homeomorphisms of the circle with a finite number of critical points, all non-flat, the parameter values corresponding to irrational rotation numbers occupy a set of measure zero. In a straightforward way, it can be shown that the global hyperbolicity of the renormalization horseshoe proved in this paper implies this result for the case of critical circle maps (compare this with the case of unimodal maps, where the corresponding statement is proved by Lyubich in [Lyu5] with the use of the renormalization hyperbolicity). We remark that we fully expect the methods of our works on critical circle maps to generalize to the case of several critical points. A question indirectly involved in the analysis of the above problem is the size of the parameter intervals Hn for which ρ(ft ) = [n, . . . ]. We investigate it in [Ya4], and prove in effect that |Hn |  n2 . An important open question which remains in the theory of critical circle maps is: C 1+β -Rigidity Problem. When is the conjugacy between two critical circle maps with the same irrational rotation number which maps 0 to 0, C 1+β -smooth at the critical point? de Faria and de Melo [dFdM2] showed that if the two maps are analytic and of bounded type, then the conjugacy has the desired smoothness. Their argument stems from their result on geometric convergence of renormalizations for maps of bounded type. We expect that given the geometric convergence for all types obtained in this paper, the result can be extended to arbitrary analytic maps. It is worth noting that for non-analytic maps of unbounded type the result is known to be false (see [dFdM1, dFdM2]). Finally, it is worth noting that while the present paper finishes the description of the renormalization picture for the homeomorphisms of the circle with a critical point whose order is an odd integer, no comparable theory exists for other values of the critical exponent. The issues of non-analyticity in this case pose a formidable and exciting problem for further study.

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Acknowledgement. The author would like to express his gratitude to Mikhail Lyubich for many stimulating discussions.

References [Do] [DH1] [DH2] [dF1] [dF2] [dFdM1] [dFdM2] [Ep1] [EKT] [EY] [EE] [FKS] [He] [Keen] [Lan1] [Lan2] [Lyu2] [Lyu3] [Lyu4] [Lyu5] [LY] [McM1] [McM2] [Mes] [Mil] [MvS] [ORSS] [Sul1]

Douady, A.: Does a Julia set depend continuously on the polynomial? In: Complex dynamical systems: The mathematics behind the Mandelbrot set and Julia sets. Devaney, R.L. (ed), Proc. of Symposia in Applied Math., Vol 49, Providence RI: Amer. Math. Soc. 1994, pp. 91–138 Douady, A., Hubbard, J.H.: Etude dynamique des polynˆomes complexes, I-II. Pub. Math. d’Orsay, 1984 ´ Douady, A., Hubbard, J.H.: On the dynamics of polynomial-like mappings. Ann. Sci. Ec. Norm. Sup. 18, 287–343 (1985) de Faria, E.: Proof of universality for critical circle mappings. Thesis, CUNY, 1992 de Faria, E.: Asymptotic rigidity of scaling ratios for critical circle mappings. Ergodic Theory Dynam. Systems 19(4), 995–1035 (1999) de Faria, E., de Melo, W.: Rigidity of critical circle mappings I. J. Eur. Math. Soc. (JEMS) 1(4), 339–392 (1999) de Faria, E., de Melo, W.: Rigidity of critical circle mappings II. J. Am. Math. Soc. 13(2), 343–370 (2000) Epstein, A.: Towers of finite type complex analytic maps. PhD Thesis, CUNY, 1993 b sin(2π x) with Epstein, A., Keen, L., Tresser, C.: The set of maps Fa,b : x → x + a + 2π any given rotation interval is contractible. Commun. Math. Phys. 173, 313–333 (1995) Epstein, A., Yampolsky, M.: The universal parabolic map. To appear in Ergodic Theory Eckmann, J.-P., Epstein, H.: On the existence of fixed points of the composition operator for circle maps. Commun. Math. Phys. 107, 213–231 (1986) Feigenbaum, M., Kadanoff, L., Shenker, S.: Quasi-periodicity in dissipative systems. A renormalization group analysis. Physica 5D, 370–386 (1982) Herman, M.: Conjugaison quasi-symmetrique des homeomorphismes analytiques du cercle a des rotations. Manuscript Keen, L.: Dynamics of holomorphic self-maps of C∗ . In: Holomorphic functions and moduli I Drasin D. et al. (eds.), New York: Springer-Verlag, 1988 Lanford, O.E.: Renormalization group methods for critical circle mappings with general rotation number. In: VIIIth International Congress on Mathematical Physics (Marseille,1986), Singapore: World Sci. Publishing, 1987, pp. 532–536 Lanford, O.E.: Renormalization group methods for critical circle mappings. Nonlinear evolution and chaotic phenomena. In: NATO Adv. Sci. Inst. Ser. B:Phys. 176, New York: Plenum, 1988, pp. 25–36 Lyubich, M.: Renormalization ideas in conformal dynamics. In: Cambridge Seminar “Current Developments in Math.”, May 1995. Cambridge, MA: International Press, 1995, pp. 155– 184 Lyubich, M.: Dynamics of quadratic polynomials, I-II. Acta Math. 178, 185–297 (1997) Lyubich, M.: Feigenbaum-Coullet-Tresser Universality and Milnor’s Hairiness Conjecture. Ann. Math. (2) 149(2), 319–420 (1999) Lyubich, M.: Almost every real quadratic map is either regular or stochastic. Ann. Math.(2) 156(1), 1–78 (2002) Lyubich, M., Yampolsky, M.: Dynamics of quadratic polynomials: Complex bounds for real maps. Ann. l’Inst. Fourier 47(4), 1219–1255 (1997) McMullen, C.: Complex dynamics and renormalization. Ann. Math. Studies, V.135, Princeton, NJ: Princeton Univ. Press, 1994 McMullen, C.: Renormalization and 3-manifolds which fiber over the circle. Ann. Math. Studies, Princeton, NJ: Princeton University Press, 1996 Mestel, B.D.: A computer assisted proof of universality for cubic critical maps of the circle with golden mean rotation number. PhD. Thesis, University of Warwick, 1985 Milnor, J.: Dynamics in one complex variable. Introductory lectures. Braunschweig: Friedr. Vieweg & Sohn, 1999 de Melo, W., van Strien, S.: One dimensional dynamics. Berlin-Heidelberg-New York: Springer-Verlag, 1993 Ostlund, S., Rand, D., Sethna, J., Siggia, E.: Universal properties of the transition from quasi-periodicity to chaos in dissipative systems. Physica 8D, 303–342 (1983) Sullivan, D.: Quasiconformal homeomorphisms and dynamics, topology and geometry. Proc. ICM-86, Berkeley, V.II, pp. 1216–1228

96 [Sul2] [Sh] [Sw] [Ya1] [Ya2] [Ya3] [Ya4] [Yoc]

M. Yampolsky D.Sullivan. Bounds, quadratic differentials, and renormalization conjectures. AMS Centennial Publications. 2: Mathematics into Twenty-first Century (1992). Shishikura, M.: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. Math. (2) 147(2), 225–267 (1998) ´ ¸tek, G.: Rational rotation numbers for maps of the circle. Commun. Math. Phys. 119, Swia 109–128 (1988) Yampolsky, M.: Complex bounds for renormalization of critical circle maps. Erg. Th. & Dyn. Systems 19, 227–257 (1999) Yampolsky, M.: The attractor of renormalization and rigidity of towers of critical circle maps. Commun. Math. Phys. 218(3), 537–568 (2001) ´ Yampolsky, M.: Hyperbolicity of renormalization of critical circle maps. Publ. Math IHES, to appear Yampolsky, M.: On the eigenvalues of a renormalization operator. To appear in Nonlinearity Yoccoz, J.-C,: Il n’ya pas de contre-example de Denjoy analytique. C.R. Acad. Sci. Paris 298 s´erie I, 141–144 (1984)

Communicated by G. Gallavotti

Commun. Math. Phys. 240, 97–122 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0901-x

Communications in

Mathematical Physics

Holonomy and Skyrme’s Model Dave Auckly
, Lev Kapitanski

Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA Received: 6 November 2002 / Accepted: 10 March 2003 Published online: 25 July 2003 – © Springer-Verlag 2003

Abstract: In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat L2loc connection; the local developing maps for such connections need not be continuous.

1. Introduction In 1961 T. H. R. Skyrme introduced a model to describe self-interacting meson fields, [22–24]. For a review of the physical and mathematical literature on the Skyrme model see [10, 8, 11]. The static fields of the original Skyrme model may be described as maps u from R3 into the group of unit quaternions, Sp(1)
SU (2). It is required that these maps satisfy the boundary condition u(∞) = 1 and have finite energy
E(u) =




1 −1 1 |u du|2 + |u−1 du ∧ u−1 du|2 dx , 4 R3 2

The first author was partially supported by NSF grant DMS-0204651. The second author was partially supported by NSF grants DMS-9970638, and DMS-0200670

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where |u−1 du|2 =

3  j =1

−1

|u

du ∧ u

−1

|u−1

∂u 2 | , ∂x j

   −1 ∂u −1 ∂u |2 , du| = | u ,u ∂x j ∂x k 2

j
4

and |q|2 = q · q¯ = j =1 (qj )2 . In addition, the space of smooth maps satisfying the boundary condition splits into infinitely many different components classified by the degree of the map. In the very first paper [22], p.129, Skyrme noted that u−1 du is a flat connection. In terms of the connection a = u−1 du the energy takes the form
1 1 2 E[a] = |a| + |[a, a]|2 dx . 3 16 R 2 Some generalizations of this functional have been considered previously [14, 13]. It is natural to generalize the original setting in several directions. First, one may replace R3 with an arbitrary Riemannian 3-manifold. Second, one may consider maps into arbitrary Lie groups. Third, the model may be described in terms of flat connections on principal G-bundles. If the manifold M 3 is non-compact, the existence of ground states is a difficult open problem. On the other hand, if M 3 is compact (and G = SU (2)), the existence of ground states is much easier and has been established in [11]. The purpose of the present paper is to understand the underlying geometry of the space of maps and/or the space of connections together with the interaction between this geometry and the analytical behavior of the Skyrme fields. All the geometric features are present for closed 3-manifolds. For this reason we restrict to the case of closed 3-manifolds. The density 1 −1 du|2 + 41 |u−1 du ∧ u−1 du|2 requires a metric on the Lie group. Compact Lie 2 |u groups always admit bi-invariant metrics. This leads to a symmetry of the Skyrme functional. In addition, maps into a compact Lie group are automatically bounded. For these reasons we restrict our attention to compact Lie groups. In order to retain similarity with the original model, any generalization should be invariant under the group. For maps this means that E(u · g) = E(u) for any constant g ∈ G. For connections, this means that the constant gauge transformations should contain a copy of the group G and the energy should be invariant under the constant gauge transformations. The original functional is also invariant under left multiplication. However, this symmetry disappears at the level of connections and does not induce any new identifications on the components of the configuration space. A connection on a principle G-bundle, P → M is a Lie algebra valued 1-form, 1 A, on the total space of the bundle, [12]. The energy density 21 |A|2 + 16 |[A, A]|2 is thus a function on the total space, P . This function is not G-equivariant, so it does not descend to a function on M. One way to resolve this problem is to pick a reference connection, B, on P , and write A = B + a, where a may uniquely be identified as an 1 element of (T ∗ M ⊗ Ad∗ P ), [6]. Thus, L(a) = 21 |a|2 + 16 |[a, a]|2 is a well-defined density on M. The gauge group is the group of automorphisms of the bundle. It may be identified with (0 M ⊗ Ad P ). Then a gauge transformation, g, acts on a twisted 1-form, a, via a · g = g −1 ag + g −1 dB g. Note that the density L(a) is not gauge invariant unless the gauge transformation is B-covariantly constant ( dB g = 0). Indeed,

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1 L(0 · g) = L(0) = 0 implies 21 |dB g|2 + 16 |[dB g, dB g]|2 = 0. The energy density is invariant under B-covariantly constant gauge transformations. The B-covariantly constant gauge transformations can be identified with the centralizer of the holonomy of B, [6]. Thus, the energy density is invariant under the group exactly when B has central holonomy. By the holonomy reduction theorem, [12], there is a subbundle Q of P with structure group Z, the center of G, so that Ad∗ P ∼ = Q×Z g. Since Z acts trivially on the Lie algebra g, we have an isomorphism Q ×Z g ∼ = M × g, i.e., [(q, X)] → ([q], X). In other words, the gauge group becomes Maps(M, G) in our case. The flat connections on P correspond to the set {a ∈ (T ∗ M ⊗ g) | da + 21 [a, a] + FB = 0}, where FB is the curvature of the reference connection B regarded as an element of (2 M ⊗ g). Given a unitary representation µ : G → U (n), the associated complex vector bundle 1 P ×α Cn has first Chern class − 2πi Trace(µ(FB )). It follows that the curvature of B is a closed central 2-form with integral periods. Conversely, any closed central 2-form with integral periods arises as the curvature of the central connection on the bundle. To summarize, we have two distinct variational problems, one for equivalence classes of maps from a closed 3-manifold to a compact Lie group, and one for flat connections A = B + a modulo B-covariantly constant gauge transformations. These two problems lead to many new questions. In this paper we begin the investigation of these questions.

• We classify the path components of the space of smooth maps from a closed 3-manifold to a compact Lie group as is and modulo the group action. • We classify the path components of the space of smooth connections with fixed holonomy on bundles, P , as described above. • We also define analytical invariants that uniquely label these path components and show that these invariants are well defined for finite Skyrme energy fields. • We prove that in any dimension a distributionally flat L2loc -connection is locally trivial. • We prove that, given any set of invariants, there exists a map minimizing the Skyrme energy among all maps with these invariants. We prove the analogous statement for connections. The result about the L2loc -connections is the cornerstone of this analysis. It allows us to define the holonomy of such connections and to label sectors of not necessarily smooth finite Skyrme energy maps. It may be useful in other problems as well. We now describe the contents of the paper in more detail. In Sect. 2.1, Proposition 1, we prove that the homotopy classes of maps from M to G, [M, G], are in bijective cor˜ × H 1 (M, H1 (G0 )). Here G0 is the identity respondence with G/G0 × H 3 (M, H3 (G)) ˜ is the universal covering group of G0 . Since the energy, E(u), component of G, and G is G-invariant, the relevant set of classes of maps is [M, G]/G. This set is isomorphic ˜ × H 1 (M, H1 (G0 )) by Corollary 1. An analytic description of the to H 3 (M, H3 (G)) invariants is given in Proposition 2. The H 3 invariants are easily extended to finite Skyrme energy maps. The H 1 invariant for non-smooth maps is much more technical and is introduced in Sect. 4, Definition 2. The topological type of a smooth connection modulo B-covariantly constant gauge is specified by a Chern-Simons invariant and a holonomy representation. These correspond ˜ and H 1 (M, H1 (G0 )), respectively. The Chern-Simons invariant is to H 3 (M, H3 (G)) well defined for connections with finite Skyrme energy. In Sect. 3.3 we give a definition of holonomy that is valid for distributionally flat L2loc connections. In particular, our definition is valid for flat connections with bounded Skyrme energy.

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This generalization of holonomy requires a nonlinear version of Poincar´e’s lemma which may be useful in other contexts. More precisely, in Sect. 3.2 we prove that any distributionally flat L2loc connection is locally trivial. For more regular connections (namely, A ∈ W 1,3/2 ), this follows from a theorem of K. Uhlenbeck [27]. In Sect. 4 we reinterpret the H 1 invariant for maps as a holonomy, which, by the results of Sect. 3 is well defined for maps in W 1,2 . It follows readily that for maps with bounded Skyrme energy all invariants are well defined. We also describe the space of path components of flat connections with fixed holonomy, Remark 7. The results of Sect. 3 together with Lemma 7 show that the corresponding invariants are well defined for Skyrme potentials. In Sect. 5 we consider the minimization problems for the Skyrme functionals. In particular, we prove that for any fixed set of invariants there exists a map attaining these values that minimizes the Skyrme energy E(u) in this class. Moreover, we also solve the analogous problem in the space of flat connections. 2. Homotopy Classes of Maps Let M be a closed 3-manifold. Let G be a comact Lie group, G0 its identity component, ˜ the universal covering group of G0 , with covering projection p : G ˜ → G0 . and G Denote by [M, G] the free homotopy classes of continuous maps from M to G.

2.1. Algebraic description. Proposition 1. As sets, ˜ × H 1 (M; H1 (G0 )) . [M, G] ∼ = G/G0 × H 3 (M; H3 (G)) Proof. Pick a point x0 ∈ M. There is an isomorphism φ0 : [M, G] → G/G0 × [M, G0 ] given by φ0 ([u]) = ([u(x0 )], [u(x0 )−1 · u]). Now we will regard [M, G0 ] as based homotopy classes. The homotopy classes of maps into a Lie group form a group under pointwise mul˜ → G0 , induces a homomorphism p∗ : tiplication. The covering projection, p : G ˜ → [M, G0 ] by composition. The homomorphisms from π1 (M) to π1 (G0 ) [M, G] form a group under pointwise multiplication. The natural map, π1 : [M, G0 ] → Hom(π1 (M, x0 ), π1 (G0 , 1)), is a group homomorphism. Consider the sequence p∗ π1 ˜ → [M, G0 ] → Hom(π1 (M), π1 (G0 )) → 1 . 1 → [M, G]

(1)

If p∗ (u) ¯ = 1, there exists a homotopy H : M × I → G0 to the constant map making the following diagram commute: u¯ - ˜ G . .. . .. ¯ . H . p i0 .. .. . ? ?. - G0 M ×I H M

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By the homotopy lifting theorem [25], there is a homotopy H¯ that makes the extended diagram commute. By the unique lifting theorem, [25], H¯ 1 is constant; hence, p∗ is injective. ˜ is simply connected, π1 (p∗ ◦ u) ¯ is the trivial homomorphism for all u¯ ∈ Since G ˜ So, the image of p∗ lies in the kernel of π1 . If π1 (u) = 1, there is a map u¯ so [M, G]. that p∗ ◦ u¯ = u, by the lifting theorem [25]. Thus, the sequence (1) is exact at [M, G0 ]. Any closed, connected 3-manifold admits a Heegaard splitting [20, 15], and, therefore, a CW decomposition with exactly one 0-cell and one 3-cell. Given a homomorphism, α : π1 (M, x0 ) → π1 (G0 , 1), construct a map uα : M → G0 as follows. On (0) the 0-skeleton, define uα (x0 ) = 1. Any 1-cell of M specifies an element of π1 (M). (1) Pick a representative of the corresponding class in π1 (G0 ) and define uα on the 1cell via this representative. The attaching map of any 2-cell is trivial in π1 (M), so the (1) composition with uα is trivial in π1 (G0 ) and thus extends to a map of the disk into (2) G0 . Define uα on the 2-cell via this map. The composition of the attaching map of (2) the 3-cell with uα is trivial in π2 (G0 ) since π2 of any Lie group is trivial [4]. Thus, the composition extends to a map of the 3-disk that may be used to define the map (3) uα = uα . By construction, π1 ([uα ]) = α, i.e., π1 is surjective. Thus, the sequence (1) is exact. Given any short exact sequence of groups 1 → K → G → H → 1, there is a ˜ × Hom(π1 (M, x0 ), π1 (G0 , 1)). bijection G ∼ = [M, G] = K × H . Thus, [M, G0 ] ∼ (k) Let M denote the k-skeleton of M with base point x0 . Let SX denote the suspension of (X, x∗ ), i.e., the cylinder X×[0, 1] with both ends and the segment {x∗ }×[0, 1] collapsed to a single point. Consider the sequence M (2)

i2 -

M (3)

q∂ 2 M (3) /M (2) 2- SM (2)

- SM (3) .

(2)

(3) (2) is homeomorphic to D 3 /S 2 . Under this identification, ∂ (x) = The 2  space M  /M x (3) 3 (3) 2 (2) f ( |x| ), |x| for x ∈ D , where f : S → M is the attaching map for the 3-cell. The sequence (2) is co-exact [25]. Therefore, it induces the exact sequence

˜ → [M (3) /M (2) , G] ˜ → [M, G] ˜ → [M (2) , G] ˜ = 0. [SM (2) , G] Similarly, the sequence M (1)

- M (2)

- M (2) /M (1)

- SM (1)

- SM (2)

- S(M (2) /M (1) )

- S 2 M (1)

(3)

induces the exact sequence ˜ → [S(M (2) /M (1) ), G] ˜ → [SM (2) , G] ˜ → [SM (1) , G] ˜ = 0. [S 2 M (1) , G]

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These exact sequences may be spliced together to give ˜ [SM (2) , G] 6

∂2∗ q2∗ - [M (3) /M (2) , G] - [M, G] ˜ ˜

- 0



q1∗ ˜ [S(M (2) /M (1) ), G] Thus,

˜ ∼ [M, G] = [M (3) /M (2) , ∼ = [M (3) /M (2) ,

˜ ˜ G]/Ker q2∗ ∼ ∂2∗ = [M (3) /M (2) , G]/Im ˜ G]/Im (∂2∗ ◦ q1∗ ) .

The sequences (2), (3) induce ∂∗3

q2∗

H3 (M (3) ) → H3 (M (3) /M (2) ) → H2 (M (2) ) and

∂∗2

q1∗

H2 (M (2) ) → H2 (M (2) /M (1) ) → H1 (M (1) ) , which give q1∗ ◦∂∗3

H3 (M (3) /M (2) ) → H2 (M (2) /M (1) ) and the commutative diagram ˜ Hom(H2 (M (2) /M (1) ), π3 (G)) 6

(q1∗ ◦ ∂∗3)∗

F2 ˜ [S(M (2) /M (1) ), G]

˜ Hom(H3 (M (3) /M (2) ), π3 (G)) 6 F3

∂2∗ ◦ q1∗

(4)

- [M (3) /M (2) , G] ˜

Here F3 ([u])[γ ] = [u ◦ r3−1 (γ )] and F2 ([u])(γ ) = [u ◦ r2−1 ◦ (∂∗s )−1 (γ )], where ∂∗s : H3 (S(M (2) /M (1) )) → H2 (M (2) /M (1) ) is the suspension isomorphism, and r3 : π3 (M (3) /M (2) ) → H3 (M (3) /M (2) ) and r2 : π3 (S(M (2) /M (1) )) → H3 (S(M (2) /M (1) )) are the Hurewicz isomorphisms. Note that the vertical arrows in (4) are isomorphisms. Thus, ˜ ∼ ˜ ∼ ˜ . [M, G] = coKer(∂ ∗ ◦ q ∗ ) ∼ = H 3 (M; π3 (G)) = H 3 (M; H3 (G)) ∼ Since the fundamental group of a Lie group is abelian, π1 (G0 ) = H1 (G0 ; Z). It follows that Hom(π1 (M), π1 (G0 )) ∼ = Hom(H1 (M), H1 (G0 )). The universal coefficient theorem, [4], gives 0 → Ext1Z (H0 (M), H1 (G0 )) → H 1 (M; H1 (G0 )) → Hom(H1 (M), H1 (G0 )) → 0 . Since Ext∗Z (Z, —) = 0, we have Hom(H1 (M), H1 (G0 )) ∼ = H 1 (M; H1 (G0 )).  Recall that the Skyrme functional is invariant under right multiplication by elements of G. We are therefore interested in the classification of maps from M to G up to homotopy and right translation in G. The following is a corollary of Proposition 1.

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103

Corollary 1. ˜ × H 1 (M; H1 (G0 )) . [M, G]/G ∼ = H 3 (M; H3 (G)) 2.2. Analytical description. We will now develop analytical expressions for the ho˜ motopy invariants starting with H 3 (M; H3 (G)). Since the second homology of any ˜ ∼ 3-manifold is torsion free, the universal coefficient theorem gives H 3 (M; H3 (G)) = ˜ . The universal covering group is the direct product of Rn HomZ (H3 (M), H3 (G)) together with a finite collection of compact, simply - connected, simple Lie groups, ˜1 × ... × G ˜N . ˜ ∼ G = Rn × G

(5)

˜ is It is known, [2], that π3 of any compact simple Lie group is Z. Therefore, H3 (G) 3 (G; ˜ ∼ ˜ free, and, by the universal coefficient theorem, H3 (G) Hom (H Z), Z). = Z We are interested in maps from M to G, and not every map can be lifted to a map ˜ We would like to identify H 3 (G; ˜ Z) with a subgroup of H 3 (G; R) in from M to G. 3 order to express the homotopy H -invariant as the integral over M of the pull-back of a 3-form. The needed identification is the topic of the next lemma. Let Z be the center of G0 and Z0 be the identity component of Z. Let Gk be the ˜ k . Let d k be the connected Lie subgroup of G0 corresponding to the Lie algebra of G k k ˜ degree of the cover G → G . ˜ → G0 , induces a surjective homomorphism Lemma 1. The covering map, p : G ˜ R) , p ∗ : H 3 (G0 ; R) → H 3 (G; which restricts to an isomorphism

N  3 k 1 ˜ Z) . H G ; k Z → H 3 (G; d k=1

Proof. Consider the diagram: - G ˜

p−1 (Z0 ) p

p

? Z0

? - G0

iZ

jZ ˜ G/p−1 (Z0 ) q

(6)

? - G0 /Z0

The homomorphism iZ∗ : π1 (Z0 ) → π1 (G0 ) is injective, [3], so π1 (p −1 (Z0 )) = 1 and p−1 (Z0 ) ∼ = Rn . Therefore, the homotopy exact sequence applied to the top row of
0 )) ∼ ˜ ∼ ˜ −1 (Z0 )). Thus, H3 ((G/Z ˜ −1 (Z0 )) ∼ (6) implies that πm (G) = πm (G/p = π3 (G/p = ∼ ˜ ˜ π3 (G) = H3 (G). Now, G0 /Z0 being semisimple, has finite fundamental group. Define a transfer map τ : H3 (G0 /Z0 ) → H3 ((G
0 /Z0 )) by sending every simplex to the sum of its lifts. The compositions τ ◦ q∗ and q∗ ◦ τ are multiplications by |π1 (G0 /Z0 )|, ˜ k ; R) the order of the group. Thus, p∗ is surjective. The transfer map from H 3 (G 1 ˜ k ; Z) ∼ to H 3 (Gk ; R) induces an isomorphism H 3 (G = H 3 (Gk ; d k Z). The result now follows from the K¨unneth formula. 

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˜ k in the decomposition (5), we have a bi-invariant 3-form on G ˜ k defined For each G by ˜ k (Xg , Yg , Zg ) = −



  KG˜ k Tr ad Lg −1 ∗ Xg , Lg −1 ∗ Yg ad Lg −1 ∗ Zg . 2 32π

˜ k is closed because the Killing form is bi-invariant. Integrating it over any The form

˜ k , we see that it is non-trivial. The constants K ˜ k are chosen so that the SU (2) in G G ˜ k . The value of this constant for integral of this 3-form is 1 on any primitive S 3 in G every simple, simply connected, compact Lie group is listed in Table 1. The derivation of these constants is included in the Appendix. Table 1. Normalizing constants group, G

KG

An

Bn

Cn

Dn

SU (n + 1)

Spin(2n + 1)

Sp(n)

Spin(2n)

2 n+1

1 2n−1

2 n+1

1 2n−2

E6

E7

E8

F4

G2

1 24

1 36

1 240

1 9

1 2

˜ k. Recall that Gk is the connected subgroup of G0 with universal covering group G Let k be the 3-form on G given by      1   K Tr ad L ,

k (Xg , Yg , Zg ) = − −1 ∗ Xg , Lg −1 ∗ Yg ad Lg −1 ∗ Zg k ˜ g G 32π 2 where now Xg , Yg , and Zg are tangent vectors on G, and Lg −1 ∗ Wg is the orthogonal k ˜ k is a d k -fold projection of Lg −1 ∗ Wg onto the Lie algebra of G . Note that since G k k cover of G , the form is not in general an integral class. Summarizing, we have the following lemma. ˜ Z), is the free abelian group generated by Lemma 2. The cohomology group, H 3 (G; ˜ k → G0 is the restriction of the covering projection, and (ι ◦ p)∗ k , where p : G ι : G0 → G is the inclusion. Before we can get numerical expressions for the three dimensional part of the homotopy invariants, we need to consider the second invariant that corresponds to H 1 (M; H1 (G0 )), see Corollary 1. ˜ a map u : M → G Using the identification H1 (G0 ) ∼ = π1 (G0 ) ∼ = {p −1 (1)} ⊂ G, −1 ˜ induces a G-connection u du. Given a loop γ : ([0, 1], {0, 1}) → (M, x0 ), solve γ γ d γ the system of ordinary differential equations, dt gt = −u−1 du(γ˙ (t)) gt , g0 = 1 to ˜ The element of H 1 (M; H1 (G0 )) corresponding to u is obtain a path g γ : [0, 1] → G. γ αu , where αu ([γ ]) = g1 . This is just the holonomy of the connection. We take up the full description of holonomy in Sect. 3. Note that H 1 (M; H1 (G0 )) is a finitely generated abelian group. Let α1 , . . . , . . . , αb be its generators, and r α = 0 be the relations, where r1 | r2 | . . . | rb . There is no canonical choice for the classes α , but it is not difficult to construct a set given any closed connected 3-manifold. By the proof of Proposition 1, there are maps v : M → G0 with (v )∗ = α : H1 (M) → H1 (G0 ). It is usually possible to construct such maps explicitly.

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Given a map u : M → G, one obtains the map u0 = u(x0 )−1 u from M into G0 .  This map induces a map u0∗ : H1 (M) → H1 (G0 ). We write u0∗ = b =1 a α , where a = a (u) ∈ Zr . We are in position now to state the analytical description of [M, G]/G. Proposition 2. Given any element [u] of [M, G]/G, define the numbers a (u) and   −1 ∗ b  u  k . v a


ck (u) =



M

1

We have a (u) ∈ Zr and ck (u) ∈ Z. Any (b + N )-tuple (a1 , . . . , ab ; c1 , . . . , cN ) ∈ Zr1 × . . . × Zrb × ZN , is obtained from some map u : M → G. Two maps are equivalent if and only if they produce the same (b + N )-tuple. Remark 1. The three dimensional part of this invariant is a torsor, i.e., an affine space modeled on integers with no canonical choice of 0. Since


(uw)∗ k =

M




u ∗ k + M




w ∗ k ,

M

 we could encode the information contained in ck (u) as M u∗ k . However, these num  bers are no longer integers, but the differences M u∗ k − M w ∗ k always are. The  integrals M u∗ k are well defined for maps u : M → G with finite Skyrme energy, E(u), since u−1 du ∈ L2 and [u−1 du, u−1 du] ∈ L2 . Remark 2. The fact that the one-dimensional part of the invariant, a , is well defined for Sobolev maps in W 1,2 (M, G) follows from Sect. 2 of Burstall’s paper [5]. However, later we will need analogous one-dimensional invariants for L2loc connections. The result of [5] is not sufficient to define these invariants for connections. In Sect. 4 we will address this more general situation and obtain a proof independent of [5] that all of the numbers a and ck (u) are well defined for maps with finite Skyrme energy.

3. Connections and Holonomy In the previous section we described the one dimensional component of the homotopy invariant using the solutions of a system of ordinary differential equations. While this is legitimate for smooth maps, it does not directly apply to all maps with finite Skyrme energy. To circumvent this problem, we will give a new definition of holonomy in the ˇ spirit of Cech cohomology that is well defined for distributionally flat L2 -connections and reduces to the usual definition for smooth connections.

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3.1. Function spaces. Every compact Lie group has a faithful unitary representation [3]. 2 Pick such a representation and identify G with its image, a subgroup of U (N ) ⊂ CN . Use the corresponding representation to identify the Lie algebra of G, g, with a sub2 algebra of u(N ) ⊂ CN . These identifications allow us to consider G and g valued 2 maps as maps into CN . Given a Riemannian manifold , denote by W s,p ( , Cm ) the Sobolev space of m C -valued functions on . When s is a nonnegative integer, the norm can be chosen as
 p uW s,p ( ,Cm ) = |∇ s u|p d vol .

0≤k≤s

2

We define W s,p ( , G) to be the space of maps u ∈ W s,p ( , CN ) with u(x) ∈ G for almost all x ∈ . We define W s,p ( , g) similarly. Notice that W s,p ( , G) ⊂ 2 L∞ ( , CN ), since G is compact. Remark 3. The space W 1,p ( , G) forms a group under pointwise multiplication. We will sometimes shorten the notation to W s,p and will use the same notation for the corresponding spaces of differential forms on . Different faithful representations of G lead to equivalent norms. For compact manifolds, different Riemannian metrics also lead to equivalent norms provided the boundary is sufficiently regular. 3.2. Nonlinear Poincar´e Lemma. In this section we prove an important lemma that we use to define the holonomy for Sobolev connections. Let I m denote a unit cube in Rm . Lemma 3. Given any L2 g-valued 1-form A on I m such that dA +

1 [A, A] = 0 2

(7)

in the sense of distributions, there exists u ∈ W 1,2 (I m , G) such that u−1 ∈ W 1,2 (I m , G) and A = u−1 du. Furthermore, for any two such maps, u and v, there exists g ∈ G so that u(x) = g · v(x), for almost every x ∈ I m . Proof. Choose coordinates parallel to the edges of the cube. In coordinates, the g-valued 1-form A can be written as Ak (x) dx k , where each Ai (x) is a matrix-valued function. We then have [A, A] = (Ai Aj − Aj Ai ) dx i ∧ dx j . Here and in what follows we use the summation convention. The following observation is important for our construction. Let f be a scalar function in Lp (I m ), for some 1 ≤ p < ∞. By Fubini’s theorem, for almost all values of x 1 ∈ I 1 the function of m − 1 variables f (x 1 , ·) is in Lp (I m−1 ). We will extend f outside of I m by 0. Denote by T f the mollification of f defined as follows:
(T f )(x 1 , . . . , x m ) = ζ (x 1 − y 1 ) · . . . · ζ (x m − y m )f (y 1 , . . . , y m ) d m y , where ζ (t) =  −1 ζ ( −1 t) with ζ a smooth, even, compactly supported bump-function with integral 1. Choose a sequence k → 0 so that ∞
 |Tk f − f |p d m x < ∞ . m k=1 I

Holonomy and Skyrme’s Model

Then,


 ∞
I1

107



m−1 k=1 I

|Tk f (x 1 , ·) − f (x 1 , ·)|p dx 2 . . . dx m

dx 1 < ∞ .

This implies that there is a subset I˜1 of full measure in I 1 for each point x 1 of which Tk f (x 1 , ·) converges to f (x 1 , ·) in Lp (I m−1 ). If there is a finite number of functions f , the set I˜1 can be chosen to accommodate all of them. We now return to the connection A ∈ L2 (I m , G). Using the above observation we translate the coordinates in I m as follows. For almost every point x01 of the interval I 1 , the restrictions of all Ai to the hyperplane x 1 = x01 lie in L2 (I m−1 ) and (Ai )k (x01 , ·) = (Tk Ai )(x01 , ·) converge to Ai (x01 , ·) in L2 (I m−1 ). In addition, for almost every such x01 , the restrictions of all Lie brakets [Aj , A ] to the hyperplane x 1 = x01 lie in L1 (I m−1 ) and ([Aj , A ])k (x01 , ·) = (Tk [Aj , A ])(x01 , ·) converge to [Aj , A ](x01 , ·) in L1 (I m−1 ). Fix one such point x01 inside I 1 . Similarly (sparsing the sequence k , if necessary) we may fix values x02 ,. . . , x0m so that the restrictions of all Ai  to the slices {x01 }×. . .×{x0k }×I m−k are in L2 (I m−k ) and the mollifications ζ (x0k − y k )Ai (x01 , . . . , x0k−1 , y k , ·) dy k converge to Ai (x01 , . . . , x0k−1 , x0k , ·) in L2 (I m−k ). In addition, the restrictions of all [Aj , A ] to the slices {x01 } × . . . × {x0k } × I m−k are  in L1 (I m−k ) and the mollifications ζ (x0k − y k )[Aj , A ](x01 , . . . , x0k−1 , y k , ·) dy k converge to the braket [Aj , A ](x01 , . . . , x0k−1 , x0k , ·) in L1 (I m−k ). Almost every point in I m satisfies these “slice" conditions. By translating the coordinates we set x0 = 0. After these preliminary remarks we turn to the construction of u. We obtain u(x) as un (x), where uk : {0} × I k → G is defined inductively, setting u0 (0) = 1 and defining successive terms via the system of ordinary differential equations d v(t, x n−k , . . . , x n ) = v(t, x n−k , . . . , x n )An−k−1 (0, t, x n−k , . . . , x n ) dt v(0, x n−k , . . . , x n ) = uk (x n−k , . . . , x n ) ,

(8)

setting uk+1 = We will prove that 2 . It is not hard to see that, for system (8) has a unique solution in L2 with dv also in L dt any i ≥ n − k, the quantity wi = ∂i uk+1 − uk+1 Ai formally satisfies the differential equation (x n−k−1 , x n−k , . . . , x n )

v(x n−k−1 , x n−k , . . . , x n ).

∂j wi = ∂i (∂j uk+1 − uk+1 Aj ) + wi Aj + uk+1 (∂i Aj − ∂j Ai + Ai Aj − Aj Ai ) , on the slice {0} × I k+1 with j = n − k − 1. Using the defining differential equation for uk+1 and hypothesis (7), one obtains ∂j wi = wi Aj . The initial condition wi (xj = 0) = 0 follows from the induction step. So, formally, wi = 0, which implies that uk+1 ∈ W 1,2 . We now make this argument precise. The initial value problem d v(t) = v(t) a(t) , dt

v(0) = v0 ∈ G ,

(9)

with a ∈ L2 (I 1 , g), has a unique solution v ∈ W 1,2 (I 1 , G). To prove this, we mollify a to get a (t) = (ζ ∗ a)(t) and solve the smooth system d v (t) = v (t) a (t) , dt

v (0) = v0 ,

(10)

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obtaining smooth functions v with values in G. Since G is compact, the functions v are uniformly bounded, and they are equicontinuous on I 1 , since t t |v (t2 ) − v (t1 )| = | t12 v (s) a (s) ds| ≤ C1 t12 |a (s)| ds 1  1 t 1 2 ≤ C1 |t2 − t1 | 2 t12 |a (s)|2 ds ≤ C1 aL2 (I 1 ;g) |t2 − t1 | 2 . Hence, there is a sequence k → 0 for which vk converges uniformly on I 1 to a contind uous G-valued function v. Also, v a → v a in L2 . This implies that dt vk converges 2 in L ; the limit is the distributional derivative of v. This shows that v ∈ W 1,2 (I 1 ; G), and that v is a solution of (9). Note that for all t we have
t v(s)a(s) ds , (11) v(t) = v0 + 0

and this equation is equivalent to (9). If w ∈ W 1,2 (I 1 ; G) is another solution of (10), t then the difference, p(t) = v(t) − w(t), satisfies |p(t)| ≤ 0 |p(s)| |a(s)| ds , and is therefore identically zero. This argument establishes the base case of the induction, namely, u1 (0, ·) ∈  W 1,2 (I 1 , G). To complete the induction, we will prove that uk+1 A {0}×I k+1 = ∂ uk+1  for ≥ n − k − 1 and uk+1 ∈ W 1,2 ({0} × I k+1 , G) when ∂i uk = uk Ai  k for {0}×I

i ≥ n − k and uk ∈ W 1,2 ({0} × I k , G). We first need to show that the equation    1   A {0}×I k+1 , A {0}×I k+1 = 0 dA {0}×I k+1 + 2

(12)

holds in the sense of distributions. We prove this inductively. By the hypothesis of the lemma, we have

(Aj ∂i η − Ai ∂j η) d n x = [Ai , Aj ] η d n x (13) In

In

for any smooth scalar function η supported in the interior of the cube. When both i and j are larger than 1, Eq. (13) will also hold for scalar functions of the form
ζ (−x 1 )ζ (y 2 − x 2 ) . . . ζ (y n − x n ) φ(y 2 , . . . , y n ) d n y . η (x 1 , . . . , x n ) = Substitute this η in (13), integrate by parts to move all derivatives onto φ, and change the order of integration to obtain  2 n 2 n n I n (Aj ) (0, y , . . . , y ) ∂y i φ − (Ai ) (0, y , . . . , y ) ∂y j φ d y 2 n n = I n ([Ai , Aj ]) (0, y , . . . , y ) φ d y . Pass to the limit as  goes to 0 along the sequence chosen in the beginning of the proof. The result will be

n−1 (Aj (0, ·) ∂i φ − Ai (0, ·) ∂j φ) d x = [Ai (0, ·), Aj (0, ·)] φ d n−1 x . I n−1

I n−1

Repeating this argument, in a finite number of steps we obtain (12).

Holonomy and Skyrme’s Model

109

We next consider a mollified version of system (8),  d  v = v  (An−k−1 {0}×I k+1 ) (t, x n−k , . . . , x n ), dt v  (0) = (uk ) (x n−k , . . . , x n ) .

(14)

Let  (t, s; x n−k , . . . , x n ) be the solution of  d   (t, s; ·) =  (t, s; ·) (An−k−1 {0}×I k+1 ) (t, ·) , dt

 (s, s; ·) = 1 .

In terms of  , the solution of (14) can be written as v  (x n−k−1 , . . . , x n ) = (uk ) (x n−k , . . . , x n )  (x n−k−1 , 0; x n−k , . . . , x n ) , and it is all of the variables. For any i ≥ n − k, set wi = ∂i v  −  a smooth function ofk+1   . Differentiate Eq. (14) to obtain (with j = n − k − 1) v (Ai {0}×I k+1 ) on {0} × I   ∂j wi = ∂i (∂j v  − v  (Aj {0}×I k+1 ) ) + wi (Aj {0}×I k+1 )    + v  ∂i (Aj {0}×I k+1 ) − ∂j (Ai {0}×I k+1 )      + (Ai {0}×I k+1 ) (Aj {0}×I k+1 ) − (Aj {0}×I k+1 ) (Ai {0}×I k+1 ) ,  on the slice {0} × I k+1 . Notice that ∂j v  − v  (Aj {0}×I k+1 ) = 0 by (14) and   Fij := ∂i (Aj {0}×I k+1 ) − ∂j (Ai {0}×I k+1 )     +(Ai {0}×I k+1 ) (Aj {0}×I k+1 ) − (Aj {0}×I k+1 ) (Ai {0}×I k+1 )     = (Ai {0}×I k+1 ) (Aj {0}×I k+1 ) − (Aj {0}×I k+1 ) (Ai {0}×I k+1 )    − (Ai Aj − Aj Ai ){0}×I k+1 , 

(15)

by Eq. (12). Also,

 wi (0, x n−k , . . . , x n ) = ∂i (uk ) − (uk ) (Ai {0}×I k+1 )  = (∂i uk − uk Ai {0}×I k )   + (uk Ai {0}×I k ) − (uk ) (Ai {0}×I k )   + (uk ) ((Ai {0}×I k ) − (Ai {0}×I k+1 ) )   = (uk Ai {0}×I k ) − (uk ) (Ai {0}×I k )    , + (uk ) ((Ai {0}×I k ) − (Ai {0}×I k+1 ) ) =: wi0

by the induction hypothesis. Thus, wi (x n−k−1 , x n−k , . . . , x n ) is the solution of the following initial value problem: ∂j wi = wi (Aj ) + v  Fij ,

 wi |x j =0 = wi0 .

Therefore,
 wi (x j , ·) = wi0  (x j , 0; ·) +

0

xj

(v  Fij )|x=(0,s,·)  (x j , s; ·) ds .

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D. Auckly, L. Kapitanski

  → 0 in L1 ({0}×I k ), since u ∈ W 1,2 ({0}×I k , G), and (A  Notice that wi0 k i {0}×I k ) →  Ai {0}×I k in L2 ({0} × I k , g), and, due to the choice of the coordinates at the beginning    of the argument, (Ai {0}×I k+1 ) x j =0 → Ai {0}×I k in L2 ({0} × I k , g). In addition,  is smooth and uniformly bounded, so →0

  wi0  (x j , 0; ·)L1 ({0}×I k+1 ) → 0 .

Since (A ) → A in L2 ({0}×I k+1 ), we have Fij → 0 in L1 (see Eq. (15)). Both v  2

and  are smooth and uniformly bounded, thus wi tends to 0 in L1 ({0}×I k+1 , CN ), i.e.,  ∂i v  − v  (Ai {0}×I k+1 ) L1 ({0}×I k+1 ,CN 2 ) → 0 . (16)  Because the L1 -norm of v  (Ai {0}×I k+1 ) is uniformly bounded, (16) implies that the L1 -norm of ∂i v  is bounded, while Eq. (14) shows that ∂j v  is bounded in L1 as well. Thus, v  is bounded in W 1,1 (I k+1 , G). Since in our case the embedding W 1,1 → L1 is compact, there exists a sequence m → 0 so that v m converges strongly in L1 and almost everywhere in I k+1 to some v ∈ L1 (I k+1 , G). In addition, we have ∂i v = v Ai (0, ·) in the sense of distributions for all n − k − 1 ≤ i ≤ n. Hence, v ∈ W 1,2 (I k+1 , G). We still need to check the initial conditions in (8). Taking a further subsequence if necessary, we will have v m (t, ·) → v(t, ·) in L1 (I k ) for almost every t ∈ I 1 . Indeed, let m be such that v m − vL1 (I k+1 ) ≤ 2−m . Then, 
 ∞
∞
 m k |v − v| d x dt = |v m − v| d k+1 x ≤ 1 , I1

k m=1 I

k+1 m=1 I

which proves the convergence almost everywhere. To simplify the notation we will omit the subscript m on the subsequence in the future. Multiply Eq. (14) by an arbitrary η ∈ L2 (I k ) and an arbitrary smooth ϕ and integrate to obtain  η) ϕ(t)(v  (t), η) = ϕ(0)(v  t (0), + 0 ((v  (s), η)ϕ  (s) + (v  (s)(Aj ) (0, s), η)ϕ(s)) ds,  where (ξ, η) = I k ξ η d k x. Each of the terms on the right hand side converge, as  → 0, to the corresponding term with v . Hence, the left side must have a limit as well. For almost all t ∈ I 1 this limit must be ϕ(t)(v(t), η). As a result, v is a weakly continuous function from I 1 to L2 (I k ). It satisfies the differential equation

d (v(t), η) = (v(t) Aj (0, t), η) dt almost everywhere, and v(t) → v(0) = uk weakly in L2 (I k ). The fact that v is the unique solution to Eq. (8), follows from the argument given in the base case. Thus, we have du = u A. Recall, that u(x) ∈ G ⊂ U (N ) for almost every x. Hence, u−1 = u∗ is in W 1,2 (I n , G). This concludes the proof of the existence of u ∈ W 1,2 (I n , G) with u−1 du = A. Now assume that u˜ −1 d u˜ = u−1 du for some other u˜ ∈ W 1,2 (I n , G). Since −1 u˜ u belongs to W 1,2 (I n , G), we compute: d(u˜ u−1 ) = d u˜ u−1 − uu ˜ −1 duu−1 = −1 −1 −1 u˜ (u˜ d u˜ − u du) u = 0. Hence, u(x) ˜ = g u(x) for some constant g ∈ G. Now the lemma is proved. 

Holonomy and Skyrme’s Model

111

3.3. The holonomy representation for L2loc connections. Let be an n-dimensional Riemannian manifold. Let P be a principal G-bundle over . Let {ϕν : Uν × G → P }ν∈N be a bundle atlas. The transition functions, ψνµ : Uν ∩ Uµ → G , are specified by ϕν (x, g) = ϕµ (x, ψµν (x) · g). The associated local sections, σν : Uν → P , are given by σν (x) = ϕν (x, 1). Let A be a connection on P . Locally we may express A via Aν = σν∗ A. The −1 A ψ −1 local connection forms satisfy Aν = ψµν µ µν + ψµν dψµν . We say that A is 2 2 an Lloc -connection if all Aν are locally L 1-forms with values in g. The curvature of A is defined as FA = dA + 21 [A, A]. Locally, we have (FA )ν = σν∗ FA = dAν + 21 [Aν , Aν ], which can be interpreted as a distribution for an L2loc -connection. A connection A is flat if FA = 0. For smooth flat connections, the usual holonomy map is a map from π1 ( , x0 ) to G. This map depends on a base point above x0 in P . Changing the base point in P will conjugate the holonomy map by an element of G. The equivalence class of the holonomy map up to conjugation is called the holonomy representation. We are about to generalize the notion of holonomy representation to distributionally flat L2loc connections. Note that the value of the local gauge, u, from the nonlinear Poincar´e lemma in the previous subsection, is not defined at every point. This is why we do not generalize the holonomy map. To generalize the holonomy representation, we will triangulate the manifold and use the edge-path group in place of the fundamental group. We now review the definition of the edge-path group of a pointed simplicial complex [25]. Let K be a simplicial complex. Denote the vertex set of K by K (0) . Select a vertex p0 as the base point. In general, vertices will be denoted by letters p, q, . . ., the oriented edges will be denoted by e = [p, q] , where i(e) = p is the initial vertex and f (e) = q is the final vertex of e. The closed star of the vertex p will be denoted by st(p). An edge-path, ζ , is a non-empty finite sequence of oriented edges [e1 , . . . , e ] with f (ej ) = i(ej +1 ). A closed, pointed edge-path is one with i(e1 ) = f (e ) = p0 . Two edge-paths, ζ1 and ζ2 , are called simply equivalent if one can be obtained from the other by replacing a single edge, e = [p, r] by a two-edge path [p, q][q, r] (or vice versa), where p, q, and r are (not necessarily distinct) vertices of the same 2-simplex in K. Two edge-paths are equivalent if one can be obtained from the other by a finite sequence of such moves. The set of equivalence classes of all closed pointed edge-paths forms a group, called the edge-path group, E(K, p0 ). This group is canonically isomorphic to the fundamental group, π1 (|K|, p0 ), [25], Theorem 3.6.17. It is known that any smooth manifold admits a unique PL-structure compatible with the smooth structure, [17]. In particular, there exists a simplicial complex, K, whose topological realization, |K|, is homeomorphic to . Furthermore, we may assume that any nonempty intersection of closed stars of vertices is piecewise smoothly equivalent to the unit cube in Rn . Any two such complexes have a common subdivision. Given any open cover of , subdividing the complex, if necessary, we may assume that the closed star of any vertex is contained in some element of the cover. Fix a complex K, retaining the properties described above, subordinate to a bundle atlas and fix a vertex, p0 , as the base point. Let φ : E(K, p0 ) → π(|K|, p0 ) be the canonical isomorphism. For each vertex p choose an index ν(p) with st(p) ⊂ Uν(p) . Denote ψν(q)ν(p) by ψqp , ϕν(p) by ϕp , and Aν(p) by Ap . Let us define the holonomy representation of a distributionally flat L2loc connection, A. By the nonlinear Poincar´e lemma, there exists up ∈ W 1,2 (st(p), G) so that Ap = u−1 p dup . Pick such up for each vertex p. A collection of such functions is called

112

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a local developing map. We have, −1 u−1 du ψ −1 (uq ψqp )−1 d(uq ψqp ) = ψqp q qp + ψqp dψqp q −1 −1 = ψqp Aq ψqp + ψqp dψqp = u−1 p dup .

From the second part of the nonlinear Poincar´e lemma it follows that there exists a g[p,q] ∈ G so that up = g[p,q] uq ψqp . Definition 1. The holonomy representation of a flat connection, A, is the conjugacy class of the map ρA : π1 ( , p0 ) → G given by ρA (φ([e1 , . . . , e ])) = ge1 · . . . · ge . Lemma 4. This definition is independent of the representative of the edge-path group, the choice of up , ϕp , and the triangulation K. Proof. Consider a simple equivalence of two edge-paths generated by [p, r] ↔ [p, q] [q, r]. Observe that ψrq ψqp = ψrp . Since p, q, and r lie within a single simplex, the set Vp,q,r = st(p) ∩ st(q) ∩ st(r) is nonempty. By the definition of ge , g[p,r] · ur |Vp,q,r · ψrp = up |Vp,q,r = g[p,q] · uq |Vp,q,r · ψqp = g[p,q] · g[q,r] · ur |Vp,q,r · ψrq · ψqp = g[p,q] · g[q,r] · ur |Vp,q,r · ψrp . Thus, g[p,q] · g[q,r] = g[p,r] and ρA ◦ φ respects simple equivalence. By the nonlinear Poincare lemma, any other choice of local developing map, up , is related to up by up = hp · up , for some constants hp . The corresponding edge labels  are given by g[p,q] = hp · g[p,q] · h−1 q . This implies that ρA changes by conjugation  by hp0 , i.e., ρA (γ ) = hp0 · ρA (γ ) · h−1 p0 . Let K  be a subdivision of K, and let φ  : E(K  , p0 ) → π1 (|K  | = |K|, p0 ) be the corresponding isomorphism. Any loop in is represented by an edge-path ζ = [e1 , . . . , e ] in K. In K  the same loop is represented by a subdivision ζ  = [e1,1 , . . . , e1,k1 , . . . , e ,k ] . An edge, [p, q], of ζ is subdivided into a subpath [[p = r1 , r2 ], . . . , [rk−1 , rk = q]]. The complex K  is subordinate to the same bundle atlas as K. Thus, we may choose ν(r1 ) = . . . = ν(rm ) = ν(p) and ν(rm+1 ) = . . . = ν(rk ) = ν(q), and also, ur1 = . . . = urm = up , urm+1 = . . . = urk = uq , and  φr1 = . . . = φrm = φp , φrm+1 = . . . = φrk = φq . This implies g[p=r · ... · 1 ,r2 ]  g[rk−1 ,rk =q] = 1 · . . . · 1 · g[p,q] · 1 · . . . · 1. Thus, ρA does not depend on the triangulation. Let {ϕν : Uν × G → P } be a second bundle atlas. By taking intersections, if necessary, we assume that it has the same open cover {Uν }. Subdividing, if necessary, we may assume that the complex K is the same. We may therefore pick the same indexing functions, ν(p). There exist transition functions ϑν : Uν → G such that ϕν (x, g) = ϕν (x, ϑν (x) g). We will put primes on all objects computed using the  second bundle chart. The transition functions satisfy ψµν = ϑµ−1 ψµν ϑν . The local connection forms satisfy Ap = ϑp−1 Ap ϑp + ϑp−1 dϑp −1 −1 = ϑp−1 u−1 p dup ϑp + ϑp up up dϑp −1 = (up ϑp ) d(up ϑp ) ,

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 and we have up = up ϑp . Also, g[p,q] = g[p,q] . Indeed,    −1 = up ψpq (uq )−1 = up ϑp ψpq ϑq−1 u−1 g[p,q] q = up ψpq uq = g[p,q] .

This completes the proof of the lemma.



Remark 4. One can prove that for smooth connections our definition of the holonomy representation agrees with the usual one. 4. Configuration Spaces and the Skyrme Functional In this section we generalize the original Skyrme model to two separate settings. In the first setting the fields are maps from a 3-manifold to a compact Lie group. In the second setting, the fields are flat connections on a principle bundle over a 3-manifold. To define the Skyrme functional for maps, fix a Riemannian metric on the base manifold M, and fix a faithful unitary representation of the compact Lie group G. Pick a bi-invariant norm on the Lie algebra g. With the help of the Riemannian metric, this extends to a norm on Lie algebra valued forms in a standard way. The Skyrme functional for maps is defined to be
1 −1 1 E(u) = |u du|2 + |u−1 du ∧ u−1 du|2 d volM . 4 M 2 The configuration space of Skyrme maps is defined to be S G (M) = {u ∈ W 1,2 (M, G) | E(u) < ∞}/ ∼ , where the equivalence is given by u(·) ∼ u(·) g, for g ∈ G. To define the Skyrme functional for connections, we need a special type of reference connection. We start with the description of the admissible reference connections. Let P be a principal G-bundle over M, and let B be a (not necessarily flat) smooth connection on P . Given a base point x0 ∈ M and a lift xˆ0 ∈ P of x0 , we define a holonomy homomorphism B : x0 → G, where x0 is the based loop group of M, as follows. Given a based loop γ in M, construct the horizontal lift γ˜ : [0, 1] → P with γ˜ (0) = xˆ0 . By definition, B (γ ) is the unique element of G satisfying γ˜ (1) · B (γ ) = xˆ0 . Note that the horizontal lift γ˜ may be constructed from any lift γˆ starting at xˆ0 as γ˜ (t) = γˆ (t) g(t), where g : [0, 1] → G is the solution of the equation g(t) ˙ + iγ˙˜ (t) B g(t) = 0 with g(0) = 1. Remark 5. In this paper we have used three flavors of holonomy: the holonomy homomorphism, the holonomy map, and the holonomy representation. The holonomy homomorphism is defined for arbitrary smooth connections B. If the connection B is flat, then the holonomy homomorphism B depends only on the homotopy class of γ , and therefore reduces to the holonomy map, ρB : π1 (M, x0 ) → G. If the lift of the base point xˆ0 is changed, then the holonomy map changes by conjugation. The equivalence class of ρB up to conjugacy is the holonomy representation. A connection B is called central if B ( x0 ) is contained in the center, ZG , of G. A principal bundle is called central if it admits a central connection. Clearly, any bundle with abelian structure group is central. In general, a principal G-bundle P is central if and only if P ×G (G/ZG ) is a trivial bundle. For connected Lie groups,

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G )) which vanishes if and only if the bundle there is an obstruction in H 2 (M; π1 (G/Z P ×G (G/ZG ) is trivial. Let P be a central bundle with central connection B. Any other connection on P may be expressed as A = B + pr∗ a, where a is a g-valued 1-form on M. Our generalization of the Skyrme functional uses this description:
1 2 1 E[a] = |a| + |[a, a]|2 d volM . 16 M 2 The corresponding configuration space of Skyrme potentials is defined to be S B,G [M] = {a ∈ L2 (M, g) | da +

1 [a, a] + FB = 0, E[a] < ∞}/ ∼ , 2

where a ∼ g −1 a g for g ∈ G. Remark 6. The Skyrme functional is not gauge invariant. It is only invariant under constant gauge transformations. This is why the configuration space of Skyrme fields is infinite dimensional. The transformation u → u−1 du takes the configuration space of Skyrme maps into the configuration space of Skyrme potentials with the trivial reference connection. We may view this as a transformation D : S G (M) → S θ,G [M], or as a transformation ˜ ˜ ˜ is the trivial connection on M ×G D˜ : S G (M) → S θ,G [M], where θ (respectively, θ) ˜ We will reconsider the map ˜ We have E(u) = E[Du] = E[Du]. (respectively, M × G). D at the end of this section. In Sect. 2 we described the path components of the space of smooth maps from M to G up to multiplication by G on the right. It turns out that the numerical invariants specifying these components (Proposition 2) are well defined for Sobolev maps. To show this, we combine the description of these invariants given before Proposition 2 with the general definition of the holonomy representation from Sect. 3. We need to relate the holonomy of a connection on the trivial bundle M × G to the holonomy of the corre˜ The following lemma addresses a sponding connection on the trivial bundle M × G. ˜ → G is replaced by a Lie slightly more general situation in which the standard map G group homomorphism H → G. Lemma 5. Let f : H → G be a homomorphism of Lie groups such that f∗ : h → g is an isomorphism. Let fˆ : M × H → M × G be the obvious map. If A is a flat connection on M × G, then A˜ = f∗−1 fˆ∗ A is a flat connection on M × H , and the holonomy representations are related by the formula: ρA = f ◦ ρA˜ . Proof. Let Rk denote right multiplication by k in a given Lie group. Let Ls denote left multiplication by an element s of any space with a right group action. First note that A˜ is right invariant. Indeed, Rh∗ A˜ = f∗−1 Rh∗ fˆ∗ A = f∗−1 fˆ∗ Rf∗ (h) A = f∗−1 fˆ∗ (Ad f (h)−1 )∗ A = (Ad h−1 )∗ A˜ , where we have used the equalities fˆ ◦Rh = Rf (h) ◦ fˆ and (Ad f (h))◦f = f ◦(Ad h). Now notice that A˜ satisfies the second condition in the definition of a connection. Indeed, ˜ (x,h) ∗ X) = f∗−1 A(fˆ∗ L(x,h) ∗ X) = f∗−1 A(L(x,f (h)) ∗ f∗ X) = f∗−1 f∗ X = X , A(L

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for any X in h, since fˆ ◦ L(x,u) = L(x,f (h)) ◦ f . The connection A˜ is flat because A is. To compute the holonomy representations we use the trivial bundle atlases and a fixed triangulation. Let p be a vertex of the triangulation and σ˜ p denote the local section σ˜ p : st(p) → M × H which is simply σ˜ p (x) = (x, 1). Then, σp = fˆ ◦ σ˜ p . Let u˜ p ˜ The following computation shows that be the associated local developing map for A. up = f ◦ u˜ p is a local developing map for A: u−1 ˜ −1 ˜ p = f (u˜ −1 ˜ p u˜ −1 ˜ p = f (u˜ −1 ˜ p∗ f∗−1 fˆ∗ A p dup = f (u p ) f∗ d u p ) f∗ u p du p ) f∗ (Lu˜ p )∗ σ −1 −1 ∗ −1 −1 ∗ = f (u˜ p ) f∗ (Lu˜ p )∗ f∗ σp A = f (u˜ p ) (Lf (u˜ p ) )∗ f∗ f∗ σp A = σp∗ A . Here we used f ◦ Lu˜ p = Lf (u˜ p ) ◦ f . It follows that ge = f (g˜ e ), and so, ρA˜ = f ◦ ρA establishing the lemma.  Returning to the construction of invariants for Sobolev maps, we will apply the previ˜ → G0 → G. When u is an element of S G (M), ous lemma to the standard map τ : G the form A = θ + pr∗ Du is a connection on M × G. Here θ is the trivial connection on M × G. Notice that u is a local developing map for A associated to the trivial bundle atlas. The corresponding holonomy representation, ρA , is therefore trivial. Now, ˜ = τ∗−1 τˆ ∗ A, the previous lemma implies ρ ˜ = τ ◦ ρA = 1. Since for A˜ = θ˜ + pr∗ Du A τ −1 (1) ∼ = π1 (G0 ) ∼ = H1 (G0 ), we have ρA˜ (π1 (M)) ⊂ ZG˜ . In general, the holonomy is only well defined up to conjugation, but in this case, the holonomy is a well-defined map, ρA˜ : π1 (M) → H1 (G0 ). Since H1 (G0 ) is abelian, this must factor through a map from H1 (M) to H1 (G0 ). Definition 2. For any u ∈ W 1,2 (M, G) define αu to be the element of the group H 1 (M; H1 (G0 )) corresponding to ρA˜ : π1 (M) → H1 (G0 ). The constants a (u) from Proposition 2 are just the coordinates of αu in the basis {α } of H 1 (M; H1 (G0 )). Given any element α ∈ H 1 (M; H1 (G0 )), there exists a smooth map vα : M → G0 with (vα )∗ = α. Fix such a map for each cohomology class.  Combining the above discussion with Remark 1, we see that the integral ck (u) = M (u (vαu )−1 )∗ k is well defined for Sobolev maps. It is an integer for smooth maps, and we expect it to be an integer for finite Skyrme energy maps. For the special case G = SU (2), this is a fact proved in [7] using earlier results of [26] and [16], see also [11]. We partition the configuration space of maps into the following sectors:
G G Sα;c1 ,...,cN (M) = {u ∈ S (M) | αu = α, (u (vαu )−1 )∗ k = ck } . M

Two maps in the same path component of S G (M) have the same invariants. We do not yet know that two maps with the same invariants live in the same path component. This is true for smooth maps. We expect that this is true for maps with finite Skyrme energy. We now turn to Skyrme potentials. As we have shown, the holonomy is well defined for L2loc distributionally flat connections. We first partition S B,G [M] using the holonomy representation as follows: SρB,G [M] = {a ∈ S B,G [M] | ρB+a = ρ} . In the smooth case it is well known that two flat connections are gauge equivalent if and only if they have the same holonomy. To describe the path components of the

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space of smooth flat connections with fixed holonomy, it is convenient to fix a reference connection in this class. Let B + b be one such connection. Any other connection, B + a, is obtained from B + b by a gauge transformation, B + a = (B + b)u = B + u−1 b u + u−1 dB u, where u is a map from M to G. Conversely, given any such u, one obtains a connection (B + b)u with the same holonomy. The connections with fixed holonomy modulo constant gauge transformation are in one-to-one correspondence with the maps u : M → G modulo right multiplication by elements of G. The path components of the space of equivalence classes of smooth flat connections with fixed holonomy are determined by the one dimensional invariant and the three dimensional invariant of the gauge, u, as described above. We will now show that these invariants are well defined for Skyrme potentials with fixed holonomy. We start by showing that two L2loc distributionally flat connections have the same holonomy if and only if they are gauge equivalent. We only prove this for connections on central bundles. Let be a Riemannian manifold with sufficiently smooth boundary. For central 1,2 principal G-bundles over we define the gauge group to be G = Wloc ( , G), see Remark 3. Lemma 6. Two L2loc distributionally flat connections on a central bundle are gauge equivalent if and only if they have the same holonomy. Proof. Let B be a smooth central reference connection. Let A = B + a be an L2loc distributionally flat connection, and take g ∈ G. The element g acts on A by Ag = B + g −1 a g + g −1 dB g. To compute the holonomy representation of each connection, we fix a central bundle atlas and a triangulation. On the star of the vertex p we have g Ap = Bp + ap and Ap = Bp + g −1 ap g + g −1 dg. Pick a developing map up for g g −1 Ap , i.e., up dup = Ap . Then up = up g will be a developing map for Ap . Now, up = g[p,q] uq ψqp , so up · g = g[p,q] uq ψqp · g = g[p,q] uq g ψqp . Thus, the holonomy representation is the same. Now, assume that A1 = B + a 1 and A2 = B + a 2 have the same holonomy representation. We fix a central bundle atlas and a triangulation. Pick local developing 1 2 maps u1p and u2p , and let g[p,q] and g[p,q] denote the corresponding edge labels. The holonomy representation is defined up to conjugacy. Multiplying u2 at the base vertex p0 by a constant we may assure that the holonomy maps of A1 and A2 are the same. Choose a maximal tree in the 1-skeleton of the triangulation with root p0 . Any vertex is connected to the root by a unique path, (p0 , p1 , . . . , pm ). We inductively modify the local developing map u2 as follows. Set u¯ 2p0 = u2p0 , and recursively define 1 u¯ 2pi+1 = g[p g2 u2 . i+1 ,pi ] [pi ,pi+1 ] pi+1

The required gauge transformation is defined on each star by g = (u1p )−1 u¯ 2p . On the overlaps corresponding to the edges of the tree, these functions agree by construction. Any edge, e, not in the maximal tree belongs to a circuit, say, [[q1 , q2 ], . . . , [qm−1 , qm ], [qm , q1 ]], with q1 = p0 and each edge [qi , qi+1 ], except e, in the tree. Let g¯ f2 be the edge labels constructed from u¯ 2 . When f is in the tree, g¯ f2 = gf1 by construction. The 1 1 2 2 product g[q . . . g[q g1 is equal to g¯ [q . . . g¯ [q g¯ 2 , because 1 ,q2 ] m−1 ,qm ] [qm ,q1 ] 1 ,q2 ] m−1 ,qm ] [qm ,q1 ] 2 1 2 1 the holonomies agree. Hence, g¯ e = ge , and so g¯ f = gf for every edge f . Now, 1 2 (u1p )−1 u¯ 2p = (u1q )−1 g[q,p] g¯ [p,q] u¯ 2q = (u1q )−1 u¯ 2q , for any two neighboring vertices p

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and q. This implies that the function g = (u1 )−1 u¯ 2 is well defined globally on . 1,2 It belongs to Wloc ( , G), and it is easy to check that (A1 )g = A2 . The lemma is proved.  Returning to Skyrme potentials on a closed 3-manifold M, fix a reference connection B and holonomy ρ. Fix b ∈ SρB,G as a reference field. Given any other field a ∈ SρB,G [M], by the previous lemma, there exists u ∈ W 1,2 (M, G) such that B +a = (B +b)u . The one dimensional invariant of a is nothing more than αu described −1 . previously. In this setting we will denote it α[a]. Define u to be u = uvα[a] The three dimensional invariants are given by
K ˜k −1 du, u −1 du]) ad(u −1 du)) . ck [a] = − G 2 Tr(ad([u 192 π M ˜ k of the covering group G, ˜ Here k is the index corresponding to the simple factor G and the constants KG˜ k are those presented in Table 1, see Sect. 2.2. The over u−1 du means orthogonal projection onto the Lie algebra of Gk . We include vα−1 to ensure that the invariants are integral in the smooth case. Remark 7. The path components of the space of smooth flat connections with fixed holonomy ρ are uniquely specified by the invariants α and ck . Lemma 7. The invariant ck [a] is well defined for Skyrme potentials with fixed holonomy. Proof. Given a reference potential b ∈ SρB,G , any other potential, a, with the same holonomy satisfies a = u−1 bu + u−1 du, for some u ∈ W 1,2 (M, G). Since −1 du, u −1 du]) ad(u −1 du))| ≤ c |Tr(u−1 du ∧ u−1 du ∧ u−1 du)| , |Tr(ad([u

for some constant independent of u, we will show that Tr(u−1 du ∧ u−1 du ∧ u−1 du) is in L1 . Using u−1 du = a − u−1 bu, we compute Tr(u−1 du ∧ u−1 du ∧ u−1 du) = Tr(a ∧ a ∧ a − a ∧ a ∧ u−1 bu − a ∧ u−1 bu ∧ a − u−1 bu ∧ a ∧ a + u−1 bu ∧ u−1 bu ∧ a + u−1 bu ∧ a ∧ u−1 bu + a ∧ u−1 bu ∧ u−1 bu − u−1 bu ∧ u−1 bu ∧ u−1 bu) . (17) When ω2 is a matrix-valued 2-form and ω1 is a matrix-valued 1-form, Tr(ω1 ∧ ω2 ) = Tr(ω2 ∧ω1 ). Recall that a, b, a∧a, and b∧b are all in L2 , and u, u−1 ∈ L∞ . It follows −1 −1 that each term in (17) is in L1 . Observe that u−1 du = vα[a] (u−1 du − vα[a] dvα[a] )vα[a] . Hence, Tr(u−1 du ∧ u−1 du ∧ u−1 du) = Tr(u−1 du ∧ u−1 du ∧ u−1 du −1 dvα[a] −3 u−1 du ∧ u−1 du ∧ vα[a] −1 −1 −1 +3 u du ∧ vα[a] dvα[a] ∧ vα[a] dvα[a] −1 −1 −1 −vα[a] dvα[a] ∧ vα[a] dvα[a] ∧ vα[a] dvα[a] ) . By the previous computation, the first term is in L1 . Each of the remaining terms is in L1 because u−1 du ∈ L2 and vα[a] is smooth. This proves the lemma. 

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Partition the space of Skyrme potentials with fixed holonomy into the following sectors: B,b,G B,G Sρ,α;c [M] | α[a] = α, ck [a] = ck } . 1 ,...,cN [M] = {a ∈ Sρ As in the case of Skyrme maps, any two L2 Skyrme potentials from the same path component are in the same sector. Once again, we expect that the sectors are the path components. We conclude this section with a remark about D. Remark 8. The map θ,0,G G D : Sα;c 1 ,...,cN (M) → S1,α;c1 ,...,cN [M]

is well defined. Indeed, by the very definition, D preserves the invariants. To show that Du is distributionally flat for any u ∈ W 1,2 (M, G), consider in a chart the approximating sequence (u−1 ) d(u) , where (·) is the usual mollification. This sequence converges to u−1 du in the sense of distributions. The differential, d((u−1 ) d(u) ) = (−u−1 du u−1 ) ∧ (du) , converges to −u−1 du ∧ u−1 du in L1 , hence, in the sense of distributions. 5. Existence of Minimizers In this section we prove that each sector of the configuration space of Skyrme maps contains a minimizer and each sector of the configuration space of Skyrme potentials contains a minimizer as well. We use the direct method of the calculus of variations, i.e., we prove that for suitable minimizing sequences the topological invariants are preserved and the energy is lower semicontinuous. We begin by analyzing the holonomy. The following lemma establishes convergence of the holonomy and, therefore, one-dimensional invariants. Let be an n-dimensional Riemannian manifold. We define a topology on the space of representations π1 ( , x0 ) → G as follows. We say that ρn converges to ρ, if for every finite subset {γ1 , . . . , γk } ⊂ π1 ( , x0 ) there exist representatives ρ¯n , ρ¯ so that ρ¯n (γj ) → ρ(γ ¯ j ) in G for every j . It is not hard to see that the topology defined by this convergence is Hausdorff when π1 ( , x0 ) is finitely generated. We next prove a simple lemma about the holonomy representation. Lemma 8. If A(n) is a sequence of L2loc distributionally flat connections that converges weakly to a distributionally flat connection, A, then there exists a subsequence A(nk ) so that ρA(nk ) → ρA . Proof. We are in the setting from Sect. 3. Let ψqp be transition functions for the under(n) lying bundle. As in the definition of holonomy, there exist functions up and group (n) (n) (n) −1 (n) (n) (n) (n) (n) elements g[p,q] so that Ap = (up ) dup and up = g[p,q] uq ψqp . Since Ap are uniformly bounded in L2 (st(p)), after taking a subsequence we may assume that (n) up converges weakly in W 1,2 , strongly in L2 , and almost everywhere to some up . At (n) the same time, since G is compact, we may also assume that g[p,q] converges to some g[p,q] . It follows that Ap = (up )−1 dup and up = g[p,q] uq ψqp . Since we only need to check convergence on a finite set of group elements, this concludes the proof. 

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In order to address the three-dimensional invariants, we will need a special case of Tartar’s div − curl lemma, see [19]. This lemma will also be used later in the proof of the two main theorems. Lemma 9. Let M be a smooth 3-dimensional Riemannian manifold. Let ω1m ∈ L2 be a sequence of matrix-valued 1-forms and ω2m ∈ L2 be a sequence of matrix-valued 2-forms on M. If ω1m converges weakly in L2 to a form ω1 and ω2m converges weakly −1,2 in L2 to a form ω2 , and if each sequence dω1m and dω2m is precompact in Wloc (M), m m then ω1 ∧ ω2 converges to ω1 ∧ ω2 in the sense of distributions. Returning to the settings of Lemma 7, work on a closed 3-manifold M with a fixed reference field b ∈ SρB,G . B,b,G [M] and a ∈ S B,b,G [M] such that a  a Lemma 10. Given a sequence an ∈ Sρ,α n ρ,α 2 and an ∧ an  a ∧ a in L (M), there exists a subsequence such that ck [an ] → ck [a]. B,b,G [M], by Lemma 6, there exists a gauge transformaProof. Given any field an ∈ Sρ,α 1,2 −1 tion un ∈ W (M, G) such that an = u−1 n bun + un dun . Recall that un L∞ is uniformly bounded, therefore dun is uniformly bounded in L2 . Hence, upon taking a subsequence, we may assume that there exists u ∈ W 1,2 (M, G) such that un  u in W 1,2 (M, G), un → u in L2 (M, G). Note that this implies a = u−1 bu + u−1 du and un → u in L2 as well. From the definition of the invariant ck , it is clear that distribu−1 −1 −1 −1 −1 tional convergence of Tr(u−1 n dun ∧un dun ∧un dun ) to Tr(u du∧u du∧u du) implies the convergence of the invariants. As in the proof of Lemma 7 we have −1 −1 −1 Tr(u−1 n dun ∧ un dun ∧ un dun ) = Tr(an ∧ an ∧ an − 3an ∧ an ∧ un bun −1 +3un b ∧ bun ∧ an − b ∧ b ∧ b) .

Note that dan = −an ∧ an − FB is bounded in L2 . The Bianchi identity implies that d(an ∧ an ) = 0. By hypotheses, an  a and an ∧ an  a ∧ a in L2 (M). Hence, an ∧ an ∧ an converges to a ∧ a ∧ a in the sense of distributions by the div - curl −1 in L2 and (taking a subsequence) bu lemma. Since u−1 n n converges strongly to u 2 −1 converges weakly to bu in L , the product u−1 n bun converges to u bu in the sense −1 of distributions. However, u−1 n bun L2 = u buL2 = bL2 , so, upon taking a −1 −1 further subsequence, un bun converges to u bu weakly in L2 , and, hence, strongly in L2 . Since an ∧ an  a ∧ a in L2 (M), we see that Tr(an ∧ an ∧ u−1 n bun ) converges to Tr(a ∧a ∧u−1 bu) distributionally. Similarly (on a further subsequence), u−1 n b ∧bun converges to u−1 b ∧ bu strongly in L2 (recall: b ∧ b ∈ L2 ). Since an  a, we see −1 that Tr(u−1 n b ∧ bun ∧ an ) converges distributionally to Tr(u b ∧ bu ∧ a). Transition from u to u is as in Lemma 7. This proves the lemma.  We are now ready for the existence theorems. Recall that B is a central reference connection, ρ is a representation of π1 (M) into G, and b is a Lie algebra valued 1-form so that the holonomy of B + b is ρ, see Definition 1. The holonomy of the ˜ associated G-connection is encoded in α ∈ H 1 (M, H1 (G0 )), see Definition 2. The numbers c1 through cN encode the Chern-Simons invariants of the connection relative to B + b, see Lemma 7. B,b,G Theorem 1. Each sector Sρ,α;c 1 ,...,cN [M] contains a minimizer of the Skyrme energy.

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B,b,G Proof. Let an ∈ Sρ,α;c 1 ,...,cN [M] be a minimizing sequence. As in the proof of the

previous lemma, dan = −an ∧ an − FB is bounded in L2 and d(an ∧ an ) = 0. After taking a subsequence, we may assume that an  a in L2 (M) and an ∧ an converges weakly to some 2-form in L2 (M, g). By the div - curl lemma, an ∧ an converges to a ∧ a in the sense of distributions and, hence, weakly in L2 . By Lemma 8, ρB+a = ρB+an = ρ. As in the proof of Lemma 10, taking a subsequence, we produce a sequence un ∈ W 1,2 (M, G) weakly converging to u ∈ W 1,2 (M, G) such −1 −1 −1 2 that an = u−1 n bun + un dun , and un dun  u du in L (M, g). By definition, −1 −1 α[a] = ρθ˜ +pr∗ D˜ u , where u satisfies a = u bu + u du. Applying Lemma 8 to ˜ n we conclude that α = α[an ] → α[a], so that α[a] = α. By Lemma 10, θ˜ + pr∗ Du ck [a] = ck . This completes the proof.  G Corollary 2. Each sector Sα;c 1 ,...,cN (M) of the Skyrme maps contains a minimizer. θ,0,G G Proof. By Remark 8, the map D maps Sα;c 1 ,...,cN (M) into S1,α;c1 ,...,cN [M]. It is surjective by Lemma 6. Also, D preserves energy, i.e., E(u) = E[Du]. Any map u in the θ,0,G inverse image of a minimizer in S1,α;c  1 ,...,cN [M] minimizes E(·). End of proof.

6. Appendix: Normalizing Constants We now summarize how the constants in Table 1 are determined. The first observation is that π3 of every simple, simply connected, compact Lie group is generated by a homomorphic image of Sp(1) = SU (2) = Spin(3). To see this, one uses the homotopy exact sequence of the fibrations U (n) → U (n + 1) → S 2n+1 for An , for Bn and Dn , and

and SU (n) → U (n) → S 1

SO(m) → SO(m + 1) → S m+1 Sp(n) → Sp(n + 1) → S 4n+3

for Cn . The exceptional groups are treated separately beginning with the inclusions (see [1]) Spin(3) → Spin(9) → F4 → → Spin(10) ×Z4 S 1 → E6 → Spin(12) ×Z2 Sp(1) → E7 → SO(16) → E8 . Let γ denote the generator of π3 (Spin(3)). We have already seen that π3 (Spin(9)) and π3 (Spin(10) ×Z4 S 1 ) are generated by γ . It follows that π3 (F4 ) is generated by γ as well. Notice that Spin(3) is naturally included in the Spin(12) factor of Spin(12) ×Z2 Sp(1). It follows that γ is primitive in π3 (Spin(12) ×Z2 Sp(1)), thus π3 (E6 ) is generated by γ . Once again, γ generates π3 (SO(16)), and, hence, generates π3 (E7 ). Finally, the homotopy exact sequence of the fibration SO(16) → E8 → E8 /SO(16) shows that γ generates π3 (E8 ) since the perversely named octooctonionic projective plane E8 /SO(16) is 4-connected, [18], p.361. For the group G2 we use 6 the homotopy exact sequence of the fibration SU  (3) → G2 → S . ˜ A straightforward evaluation of the integral SU (2) shows that KSU (2) = 1. For the ˜ k be a homomorphism ˜ over γ . Let h : SU (2) → G other groups, one must integrate

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121

˜ k ). The constant is then the ratio of the value of the SU (2)-Killing form generating π3 (G ˜ k -Killing form of its image under h∗ . of a non-zero vector in su(2) to the value of the G For the classical groups these ratios may be inferred from [21] pp. 197, 199, 201, and 203, for example. For E6 , E7 , and E8 , these ratios may be found in [1] p. 87, 77, and 43. We could not find references listing the constants for F4 and G2 , so we compute them here. A nice description of G2 may be found in [9]. By definition, G2 is the group of endomorphisms of the purely imaginary octonions preserving the forms Re(x ∗ y) and Re([x, y]∗ z). Its Lie algebra may be realized as the space of 7 by 7 matrices of the form   0 −λ2 −λ3 −λ4 −λ5 −λ6 −λ7  λ2 0 −µ3 −µ4 −µ5 −µ6 −µ7    0 µ5 − λ6 −λ7 − µ4 λ4 − µ7 λ5 + µ6   λ3 µ3   0 −ν5 −ν6 −ν7   λ 4 µ4 λ6 − µ 5 λ µ λ + µ ν5 0 −λ2 − ν7 ν6 − λ3   5 5  7 4  λ6 µ6 −λ4 + µ7 ν6 λ2 + ν 7 0 −µ3 − ν5  λ7 µ7 −λ5 − µ6 ν7 λ3 − ν 6 µ 3 + ν 5 0 and the basis of the Lie algebra is obtained by setting each of the 14 parameters in turn to 1 and the others to 0. The Lie algebra of the SU (3) mentioned above is obtained by setting all of the λ’s to 0. The homomorphism h induces a homomorphism of Lie algebras. The image of   i 0 v= ∈ su(2) 0 −i in the Lie algebra of G2 is obtained by substituting ν5 = 1 and setting the rest of the parameters to 0. The 14 by 14 matrix for ad(h∗ (v)) in the given basis is obtained by direct computation, and the trace of its square is (−16), thus, giving KG2 = −8/−16 = 1/2. A matrix representation for F4 would require 52 by 52 matrices. Instead, we use generators and relations. Recall that the m-dimensional Clifford algebra CLm has m generators, e1 , . . . , em , with relations ei2 = −1, and ei ej = −ej ei for i = j . The Lie algebra spin(m) has ei ej , i < j , as a basis, together with the usual Lie bracket, [ei ej , ek e ] = ei ej ek e − ek e ei ej . Note that the only nonzero brackets are of the form [ei ej , ei e ] = 2ej e = −[ei ej , e ei ] when j = . One may use this to recover the constants KSpin(m) . We identify the 9-dimensional spinors with the positive 10-dimensional spinors, i.e., even ⊗ C | e + 2j −1 e2j ⊗ i · a = −a, j = 1, 2, 3, 4, 5}. 10 = 9 = {a ∈ CL10 even is the subspace of CL generated by products of even numbers of e ’s. Here CL10 10 j Define j = 1 − e2j −1 e2j ⊗ i and ωj = e2j −1 + e2j ⊗ i. Then { 1 2 3 4 5 , ω1 ω2 3 4 5 , ω1 2 ω3 4 5 , ω1 2 3 ω4 5 , ω1 2 3 4 ω5 , 1 ω2 ω3 4 5 , 1 ω2 3 ω4 5 , 1 ω2 3 4 ω5 , 1 2 ω3 ω4 5 , 1 2 ω3 4 ω5 , 1 2 3 ω4 ω5 , ω1 ω2 ω3 ω4 5 , ω1 ω2 ω3 4 ω5 , ω1 ω2 3 ω4 ω5 , ω1 2 ω3 ω4 ω5 , 1 ω2 ω3 ω4 ω5 } is a basis for 9 . As a vector space, the Lie algebra f4 is spin(9) ⊕ 9 . The Lie bracket of f4 restricted to spin(9) is the usual spin(9)-bracket. The Lie bracket between the spin(9) and 9 factors is given by [a, v] = a v, the usual representation induced from Clifford multiplication. For the remaining brackets see [1], Lemma 6.2. The Lie

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algebra of the primitive SU (2) in f4 is generated by e1 e2 , e1 e3 , and e2 e3 . A direct computation generates the 52 by 52 matrix for the f4 adjoint representation of e1 e2 . The trace of the square of this matrix is −72, giving KF4 = −8/ − 72 = 1/9. References 1. Adams, J.F.: Lectures on exceptional Lie groups. Chicago and London: The University of Chicago Press, 1996 2. Bott, R.: On torsion in Lie groups. Proc. Nat. Acad. Sci. U.S.A. 40, 586–588 (1954) 3. Br¨ocker, Th., tom Dieck, T.: Representation of compact Lie groups. New York: Springer-Verlag, 1985 4. Brown, K.S.: Cohomology of groups. New York: Springer-Verlag, 1982 5. Burstall, F.E.: Harmonic maps of finite energy for non-compact manifolds. J. Lond. Math. Soc. 30, 361–370 (1984) 6. Donaldson, S.K., Kronheimer, P.B.: The geometry of four-manifolds. New York: Oxford University Press, 1990 7. Esteban, M.J., M¨uller, S.: Sobolev maps with integer degree and applications to Skyrme’s problem. Proc. R. Soc. Lond. A 436, 197–201 (1992) 8. Gisiger, T., Paranjape, M.B.: Recent mathematical developments in the Skyrme model. Phys. Rep. 306, 109–211 (1998) 9. Gross, K.I.: The Plancherel transform on the nilpotent part of G2 and some applications to the representation theory of G2 . Trans. Am. Math. Soc. 132, 411–446 (1968) 10. Zahed, I., Brown, G.E.: The Skyrme model. Phys. Rep. 142, 1–102 (1982) 11. Kapitanski, L.: On Skyrme’s model. In: Nonlinear Problems in Mathematical Physics and Related Topics II: In Honor of Professor O. A. Ladyzhenskaya, Birman et al., (eds.), Dordrecht: Kluwer, 2002, pp. 229–242 12. Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vols. I, II. New York: John Wiley & Sons, Inc., 1996 13. Loss, M.: The Skyrme model on Riemannian manifolds. Lett. Math. Phys. 14, 149–156 (1987) 14. Manton, N.S., Ruback, P.J.: Skyrmions in flat space and curved space. Phys. Lett. B 181, 137–140 (1986) 15. Moise, E.: Affine structures in 3-manifolds, V: The triangulation theorem and Hauptvermutung. Ann. Math. 56 (2), 96–114 (1952) 16. M¨uller, S.: Higher integrability of determinants and weak convergence in L1 . J. reine angew. Math. 412, 20–34 (1990) 17. Munkres, J.R.: Elementary differential topology. Revised edition. Annals of Mathematics Studies, No. 54 Princeton, NJ: Princeton University Press, 1966 18. Mimura, M., Toda, H.: Topology of Lie groups, I and II. Transl. Am. Math. Soc., Providence, Rhode Island: Am. Math. Soc., 1991 19. Robbin, J.W., Rogers, R.C., Temple, B.: On weak continuity and the Hodge decomposition. Trans. AMS 303 (2), 609–618 (1987) 20. Rolfsen, D.: Knots and links. Mathematics Lecture Series, Vol. 7, Houston, TX: Publish or Perish, 1990 21. Simon, B.: Representations of finite and compact groups. Graduate Studies in Math., Vol 10, Providence, RI: AMS, 1996 22. Skyrme, T.H.R.: A non-linear field theory. Proc. R. Soc. Lond. A 260 (1300), 127–138 (1961) 23. Skyrme, T.H.R.: A unified theory of mesons and baryons. Nucl. Phy. 31, 556–569 (1962) 24. Skyrme, T.H.R.: The origins of Skyrmions. Int. J. Mod. Phys. A3, 2745–2751 (1988) 25. Spanier, E.H.: Algebraic topology. New York: Springer, 1966 ˇ ak, V.: Regularity properties of deformations with finite energy. Arch. Rat. Mech. Anal. 100, 26. Sver´ 105–127 (1988) 27. Uhlenbeck, K.K.: The Chern classes of Sobolev connections. Commun. Math. Phys. 101, 449–457 (1985) Communicated by L. Takhtajan

Commun. Math. Phys. 240, 123–148 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0896-3

Communications in

Mathematical Physics

On the Anisotropic Walk on the Supercritical Percolation Cluster Alain-Sol Sznitman Departement Mathematik, ETH-Zentrum, 8092 Z¨urich, Switzerland Received: 4 November 2002 / Accepted: 11 March 2003 Published online: 18 July 2003 – © Springer-Verlag 2003

Abstract: We investigate in this work the asymptotic behavior of an anisotropic random walk on the supercritical cluster for bond percolation on Zd , d ≥ 2. In particular we show that for small anisotropy the walk behaves in a ballistic fashion, whereas for strong anisotropy the walk is sub-diffusive. For arbitrary anisotropy, we also prove the directional transience of the walk and construct a renewal structure.

0. Introduction We investigate here the asymptotic behavior of an anisotropic random walk on the supercritical infinite cluster of Zd for bond percolation. Much work has been devoted to the study of a random walk on the supercritical cluster, when at each step the particle jumps with equal probability to one of the neighboring sites on the cluster, see for instance [3, 15, 19–21]. Much less is known in the anisotropic situation, where some preferred direction of jump is present. Conjectures on the behavior of such a walk can be found in the theoretical physics literature, see for instance Havlin-Bunde [16], pp. 136–139. In contrast to the naive intuition, these conjectures predict a phase transition from a ballistic to a sub-ballistic behavior, as the bias of the walk increases. One object of the present work is to investigate this effect. Before returning to this question, we first describe the model more precisely. We let Bd stand for the set of nearest neighbor bonds (or edges) on Zd , d ≥ 2, and
= {0, 1}Bd for the set of configurations. Thus a bond b ∈ Bd is open (resp. closed) in the configuration ω ∈
, when ω(b) = 1, (resp. ω(b) = 0), and ω naturally induces a partition of Zd into open clusters (or connected components). We denote by P the product measure on
endowed with its canonical σ -algebra, under which the canonical coordinates are Bernoulli variables with success probability p ∈ (0, 1). The anisotropic walk with direction
 ∈ S d−1 and strength λ > 0 in the configuration ω is then the Markov chain

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on Zd , with transition probability rω (x, y), x, y ∈ Zd , such that: rω (x, x) = 1, if all edges incident to x are closed, e·(y−x) , if y ∼ x, (i.e. y is a neighbor of x), and ω({x, y}) = 1, nω (x) rω (x, y) = 0, in all other cases, (0.1)

rω (x, y) =

where we have used the notations  = λ
, and nω (x) =

(0.2)



e

·(z−x)

, if some edge incident to x is open,

z∼x,ω({x,z})=1

= 1, otherwise .

(0.3)

We denote by Px,ω the canonical law on (Zd )N of the above Markov chain starting at x ∈ Zd , and by (Xn )n≥0 , the canonical process. It is plain that the open clusters are the irreducible components of this Markov chain, that admits the reversible measure mω (x) = e2·x nω (x), x ∈ Zd , i.e.

(0.4)

mω (x) rω (x, y) = mω (y) rω (y, x), x, y ∈ Zd , ω ∈
.

(0.5)

It is also convenient to introduce the semi-direct product measure Px = P × Px,ω , x ∈ Zd .

(0.6)

If pc (d) ∈ (0, 1), denotes the critical probability for bond percolation on Zd , cf. [14], we are chiefly interested in the supercritical regime when p > pc (d) ,

(0.7)

so that P-a.s., ω induces a unique infinite cluster C, see [14], p. 98. Our main purpose is the investigation of the asymptotic behavior of the walk on C, and we consider the event of positive P-probability: I = there is an infinite open cluster which is unique and contains 0,

(0.8)

as well as the conditioned measure: P = P0 [· | I] .

(0.9)

We now turn to the description of the main results of this article. We show in Theorem 1.2, that the walk is always transient in the direction
, that is: P-a.s., for all x ∈ C, Px,ω [lim Xn ·
 = ∞] = 1 . n

(0.10)

The proof of (0.10) relies on certain energy estimates. With the help of (0.10), we define regeneration times for the walk under the measure P . This is somewhat in the spirit of Shen [23], or Sznitman-Zerner [28]. There is however a special twist due to the conditioning present in (0.9), and in contrast to [23, 28], our regeneration times are configuration dependent. In this fashion we obtain a renewal structure for the walk under P , regardless of the strength and direction of the anisotropy, see Theorem 2.4. Such a renewal structure

Anisotropic Walk on the Supercritical Percolation Cluster

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is a powerful tool as the recent developments on random walks in random environment clearly demonstrate, cf. [26–29]. Let us then discuss the influence of the strength of the anisotropy. For weak anisotropy we show that the walk behaves in a ballistic fashion. Namely for 0 < λ < λw (d, p), we prove in Theorem 3.4 both a law of large numbers: P-a.s., for all x ∈ C, Px,ω -a.s., lim n

Xn = v, n

(0.11)

with v deterministic and v ·  > 0, and a functional central limit theorem governing the correction to the law of large numbers. Denoting by D(R+ , Rd ) the space of right continuous Rd -valued functions on R+ with left limits, endowed with the Skorohod topology, cf. [13], we prove that under P , 1 B.n = √ (X[·n] − [·n]v) converges in law on D(R+ , Rd ) to a n Brownian motion with non-degenerate covariance matrix.

(0.12)

The main ingredients in the proof of (0.11), (0.12), are the above mentioned renewal structure, as well as upper bounds on the probability of occurrence of low principal Dirichlet eigenvalues for the walk on C killed when exiting a large box, and controls on the exit distribution from a large box under P , see Lemma 3.1, 3.2. As alluded to above, an interesting feature of the model is that strengthening the anisotropy does not speed up the walk. We show in Theorem 4.1 that for λ > λs (d, p), P-a.s. for x ∈ C, Px,ω -a.s., lim n

Xn = 0. nλs /λ

(0.13)

Thus strong anisotropy leads to sub-ballistic and even sub-diffusive behavior. Intuitively, the effect is due to the presence of long dangling ends on the infinite cluster, which depending on their general direction can turn into powerful traps where the walk tends to spend a very long time, leading to a massive slowdown of the walk. This scenario can be found in the theoretical physics literature, see [16] or Dhar-Stauffer [11] and references therein. This has a similar flavor to what is known to happen for certain one-dimensional random walks in random environment, cf. Solomon [25], KestenKozlov-Spitzer [17], Sinai [24], or for the random craters models of Bramson-Durrett [7], Bramson [6], or for some random walks on inhomogeneous trees, cf. Lyons-Pemantle-Peres [18], or for some models related to the investigation of the aging phenomenon, see [5]. Let us then describe the structure of the present article. In Sect. 1, we introduce further notations and develop the necessary energy estimates that enable to prove transience in the direction
 for arbitrary non-vanishing anisotropy, see Theorem 1.2. In Sect. 2, we define certain regeneration times, see (2.13), (2.23), and prove the key renewal property in Theorem 2.4. We also dominate the displacement of the walk in the direction
 at the first regeneration time, in Proposition 2.5. This control is later used in the analysis of the walk in the weak anisotropy regime. Section 3 discusses the ballistic nature of the walk in the weak anisotropy regime. The law of large numbers and the central limit theorem appear in Theorem 3.4. The necessary bounds on the occurrence of low principal Dirichlet eigenvalues and on the exit distribution from a large box are provided in Lemmas 3.1, 3.2. Section 4 studies the strong anisotropy regime. The main result showing the subdiffusive nature of the walk for large λ can be found in Theorem 4.1.

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Finally let us explain the convention used concerning constants. Throughout the text we denote by c a positive constant depending only on d and p, with value changing from place to place. The dependence on additional parameters is otherwise mentioned in the notation, so that for instance c(λ) denotes a positive constant depending on d, p, λ. After finishing this article we learnt of an independent work [4], by N. Berger, N. Gantert and Y. Peres, which at the time was in the process of being completed. This article studies a random walk on the two-dimensional supercritical cluster, that is biased along the first coordinate axis. In this context the authors prove the directional transience of the walk corresponding to (0.10); they also show that for small bias the walk has a non-vanishing velocity, and that for large enough bias the walk has null limiting velocity. Their proofs use a strategy quite distinct from the one followed here.

1. Directional Transience In this section we introduce further notations and then prove with the help of energy estimates that the walk on the infinite cluster is always transient in the direction
, cf. (0.10) or Theorem 1.2. We tacitly assume (0.7) throughout the article. We begin with some additional notations and recall some classical facts. We denote by | · | the Euclidean distance on Rd and by (ei )1≤i≤d the canonical basis of Rd . For U a subset of Zd , |U | stands for the cardinality of U and ∂U for the boundary of U : ∂U = {x ∈ Zd \U, ∃y ∈ U, |x − y| = 1} .

(1.1)

For x ∈ Zd we will sometimes consider the discrete half-spaces: Hx+ = {x ∈ Zd , z ·  ≥ x · }, Hx− = {z ∈ Zd , z ·  ≤ x · } .

(1.2)

For a bond b = {x, y} ∈ Bd , and z ∈ Zd , we write b + z for {x + z, y + z} ∈ Bd , and denote by (tz )z∈Zd the spatial shift on
. From time to time we will consider the measurable set with full P-measure:
1 = {ω ∈
, there is an infinite open cluster and it is unique},

(1.3)

and for ω ∈
1 , we will denote by C the infinite open cluster. For C ⊆ Bd , OC stands for the event: OC = {ω ∈
, ω(b) = 1, for all b ∈ C} .

(1.4)

Given ω ∈
, an open path will stand for a nearest neighbor path on Zd , for which each step corresponds to an open bond of ω. Since p > pc (d), percolation occurs on Zd+ as well, see for instance [14], pp. 148, 304, and the following events have positive P-measure: I ± = {ω ∈
, 0 belongs to an infinite open cluster of H0± induced by the restriction of ω to edges between vertices in H0± } ,

Ix± = (tx )−1 (I ± ), Ix = (tx )−1 (I), x ∈ Zd , (cf. (0.8) for the notation) .

(1.5) (1.6)

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127

We let (θn )n≥0 , and (Fn )n≥0 , respectively stand for the canonical shift and filtration U , TU respectively denote the entrance time, the hitting on (Zd )N . For U ⊆ Zd , HU , H time, and the exit time of (Xn ) in or from U : U = inf{n ≥ 1, Xn ∈ U } , HU = inf{n ≥ 0, Xn ∈ U }, H TU = inf{n ≥ 0, Xn ∈ / U }.

(1.7)

x in place of H{x} , H {x} , for x ∈ Zd . We will write Hx , H The reversible character of the walk with transition probability (0.1) plays an important role. For ω ∈
, b = {x, y} ∈ Bd , we define the weight of the edge b in the configuration ω: wω (b) = mω (x) rω (x, y) = mω (y) rω (y, x) = exp{ · (x + y)} ω(b) .

(1.8)

When no confusion arises we drop the subscript ω and write L2 (m) and (·, ·)m for the Hilbert space of square mω -integrable functions on Zd , and its associated scalar product. We denote by R the transition kernel of the walk, so that for a function f on Zd :  R f (x) = rω (x, y) f (y), x ∈ Zd . (1.9) y

The Dirichlet form is then defined for f, g ∈ L2 (m) via:       1 E(f, g) = f, (I − R)g m = f (y) − f (x) g(y) − g(x) wω ({x, y}) . 2

|x−y|=1

(1.10) For U a non-empty subset of Zd , and ω ∈
, the walk reflected in U will be the Markov chain with state space U and transition probability:  U rω (x, x)=1, x ∈ U , if for all z ∼ x, z ∈ U, ω({x, z}) = 0 ,      wω ({x, y})  U , if x, y ∈ U , x ∼ y, and ω({x, y}) = 1, rω (x, y)=

(1.11) wω ({x, z})   z∈U,z∼x     U rω (·, ·) =0, otherwise . The measure on U : mU ω (x) =



wω ({x, y}), x ∈ U ,

(1.12)

y∈U

is then reversible for this Markov chain and the corresponding Dirichlet form (defined in analogy with (1.10)) is then:   1   E U (f, g) = f (y) − f (x) g(y) − g(x) wω ({x, y}), f, g ∈ L2 (mU ω). 2

x,y∈U x∼y

(1.13) Incidentally note that when U = Zd , rωU (·, ·) coincides with rω in (0.1) but mU ω may differ from mω in (0.4), due to the possible presence of isolated sites.

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We now turn to the proof of transience in the direction
. The crucial controls come from the next lemma. To state the lemma we still need to introduce the stopping times: Tu = inf{n ≥ 0, Xn ·
 ≥ u}, Tu = inf{n ≥ 0, Xn ·
 ≤ u}, u ∈ R ,

(1.14)

D+ = inf{n ≥ 1, Xn ·
 ≥ X0 ·
}, D− = inf{n ≥ 1, Xn ·
 ≤ X0 ·
} .

(1.15)

Lemma 1.1. P0 [D+ > T−L ] ≤ e−c(λ)L , L > 1 ,

 P0 {D− > TL } ≥ c(λ) ,

(1.16) (1.17)

L>1

(see the end of the introduction for the convention used to denote positive constants). Proof. Both estimates will follow from energy considerations. We begin with (1.16). We choose (fi )1≤i≤d , a basis of Rd , with f1 =
,

(1.18)

 > 1, consider the discrete box: and for L, L  , U0 = {z ∈ Zd , −L < z · f1 < 0, sup |z · fi | < L}

(1.19)

i≥2

as well as the positive and negative parts of ∂U0 : ∂+ U0 = {z ∈ ∂U0 , z · f1 ≥ 0}, ∂− U0 = {z ∈ ∂U0 , z · f1 ≤ −L} .

(1.20)

For ω ∈
, we will consider the walk reflected in U , see (1.11), where U = U0 ∪ ∂U0 .

(1.21)

It follows from Dirichlet’s principle, see (85) in Chapter 3 of Aldous-Fill [1], that  U  IU (ω) = mU (1.22) ω (x) Px,ω [H∂− U0 < H∂+ U0 ], with x∈∂+ U0

IU (ω) = inf{E U (f, f ), f|∂+ U0 = 1, f|∂− U0 = 0} ,

(1.23)

U the canonical law of the walk reflected in U starting at x. Clearly I (·) increases and Px,ω U when ω is replaced by ω with

ω(b) = 1, for all b ∈ Bd .

(1.24)

Thus integrating (1.22) over ω, we find:    U IU (ω) ≥ E mU ω (x) Px,ω [X1 ∈ U0 , H∂− U0 ◦ θ1 < H∂+ U0 ◦ θ1 ] ∂+ U0

≥



x∈∂+ U0 ,y∈U0 x∼y

  e·(x+y) E ω({x, y}) Py,ω [H∂− U0 = TU0 ] ,

(1.25)

Anisotropic Walk on the Supercritical Percolation Cluster

129

where we have used (1.8), (1.11), (1.12), the Markov property and the

fact that the reflected walk and the walk coincide up to time TU0 . Thus for M > 0, and  denoting  − M, we see that: the sum over x ∈ ∂+ U0 , y ∈ U0 , x ∼ y, with supi≥2 |y · fi | < L  

IU (ω) ≥ e−λ  E ω({x, y}) Py,ω [T−L < D+ , sup |(Xn − X0 ) · fi )| < M] ≥ c(λ)



i≥2,n≤T−L

Py [T−L < D+ ,

sup

i≥2,n≤T−L

|(Xn − X0 ) · fi | < M] .

(1.26)

Let us give some explanations on the last step. To this end for x, y, as above, we denote by E the event which appears under the Py,ω -probability in (1.26), and by ω∗ the configuration which agrees with ω for all bonds different from {x, y} and such that ω∗ ({x, y}) = 0. Note that when E occurs, the path up to time T−L only touches {x, y} at time 0. Thus for ω with ω({x, y}) Py,ω [E] > 0, we find:   Py,ω∗ [E] ≤ Py,ω [E] max rω∗ (y, z)/rω (y, z) , z

where the maximum runs over the (non-empty) collection of neighbors z of y with z ·
 < y ·
 and ω({y, z}) > 0. From (0.1), this maximum is at most 1 + e2λ . Hence for ω as above we obtain Py,ω∗ [E] ≤ Py,ω [E](1 + e2λ ) , and this inequality immediately extends to an arbitrary ω, since Py,ω [E] = 0 implies Py,ω∗ [E] = 0. As a result, singling out the role of the variable ω({x, y}) in the Pexpectation, we see that    Py [E] = E ω({x, y}) Py,ω [E] + E[1{ω({x, y}) = 0} Py,ω [E]   1−p  = E ω({x, y}) Py,ω [E] + E[ω({x, y}) Py,ω∗ [E] p     1−p 2λ ≤ 1+ (1 + e ) E ω({x, y}) Py,ω [E] , p which yields the last inequality of (1.26). From (1.22) with ω, we analogously see that



U IU (ω) ≤  e·(x+y) Py,ω [H∂− U0 < H∂+ U0 ] +  e·(x+y) 

 ≤ c(λ) (Py,ω [T−L < T0 ]

    Xn − X0 ) · fi | ≥ M +  1 , sup +Py,ω i≥2,n≤T0 ∧T−L

(1.27)



with  defined analogously as  in (1.26), imposing instead that supi≥2 |y · fi | ≥  − M. From (1.26), (1.27), letting L  and then M tend to infinity, and using translation L invariance, we obtain: P0 [T−L < D+ ] ≤ c(λ) P0,ω [T−(L−1) < T1 ] .

(1.28)

d

Note that P0,ω is the law of a random walk with drift E0,ω [X1 − X0 ] = 1 sinh( ·

ei )ei / d1 cosh( · ei ), that has a scalar product with
 bounded below by c(λ), the

130

A.-S. Sznitman

right-hand-side of (1.28) is smaller than c(λ)e−c(λ)L , see for instance (1.22) of [26]. The claim (1.16) easily follows.  > 1, We now turn to the proof of (1.17). We define for L, L  U  = U1 ∪ ∂U1 , U1 = {z ∈ Zd , 0 < z · f1 < L, sup |z · fi | < L}, i≥2

∂+ U1 = {z ∈ ∂U1 , z · f1 ≥ L}, ∂− U1 = {z ∈ ∂U1 , z · f1 ≤ 0} . It follows from Dirichlet’s principle that for ω ∈
:   U  mU IU  (ω) = ω (x) Px,ω [H∂+ U1 < H∂− U1 ], with

(1.29)

(1.30)

∂− U1 

IU  (ω) = inf{E U (g, g) g|∂− U1 = 1 g|∂+ U1 = 0} ,

(1.31)

U

and Px,ω stands for the canonical law of the reflected process in U  starting at x. We let πj (i), 1 ≤ j ≤ N (ω), 1 ≤ i ≤ mj , denote a maximal collection of edge-disjoint open self-avoiding paths in U  starting in ∂+ U1 , ending in ∂− U1 . Then for g as in (1.31), summing over 1 ≤ j ≤ N (ω), 1 ≤ i < mj ; and using Cauchy-Schwarz’s inequality, one finds:      2  g πj (i + 1) − g πj (i) ≤ E U (g, g) wω−1 ({πj (i), πj (i + 1)}) N(ω)2 = i,j U

≤ c(λ) E (g, g)



i,j

e

−2·x

U

≤ c(λ) E (g, g) |∂− U1 | .

(1.32)

x∈U 

From (1.30), minimizing over g, and integrating over ω, we find:     U ∂− U1 ] . E Px,ω [H∂+ U1 < H c(λ) |∂− U1 |−1 E[N (ω)2 ] ≤

(1.33)

∂− U1

 − M, the We pick M > 1, and observe that for x ∈ ∂− U1 with supi≥2 |x · f1 | < L expectation in the right-hand-side of (1.33) is smaller than: ∂− U1 , c(λ)(Px [TL < H + P0 [

sup

i≥2,n≤TL ∧T−1

sup

|(Xn − X0 ) · fi | < M]

i≥2,n≤TL |Xn · fi | ≥

M − 1]) .

(1.34)

 tend to infinity in (1.33), we obtain Letting L c(λ) lim

 L→∞

  E[N 2 (ω)] ≤ P0 [TL < D− ] + P0 |Xn · fi | ≥ M − 1 . sup 2 |∂− U1 | i≥2,n≤TL ∧T−1 (1.35)

From Theorems 7.68 and 11.22 in Grimmett [14], we know that the left-hand-side of (1.35) is bigger than c(λ). The claim (1.17) will follow once we show that the rightmost term of (1.35) tends to 0 as M tends to infinity. To this end note that for any i ≥ 2, under P0 , once |Xn · fi | reaches a new maximum, the walk has conditionally on its past a probability at least c(λ, L) of exiting the strip {z : −1 < z · f1 < L}, within the next c L steps, (with c in fact depending solely on d). The claim then easily follows.   We are now ready to prove directional transience.

Anisotropic Walk on the Supercritical Percolation Cluster

131

Theorem 1.2. P-a.s., for all x ∈ C, Px,ω [lim Xn ·  = ∞] = 1 . n

(1.36)

Proof. Recall the notation (1.5), and observe it suffices to prove that P0 [lim Xn ·  = ∞ | I − ] = 1 . n

(1.37)

Indeed from the ergodic theorem, (1.37) and the uniqueness of the infinite cluster, we see that P-a.s., for infinitely many y ∈ Zd , Py,ω [limn Xn ·  = ∞] = 1, and Py,ω [Hx < ∞] > 0, for all x ∈ C. The claim (1.36) then follows from the strong Markov property. We thus prove (1.37). We pick L > 1, and note that P0 [D+ = ∞ = T−L ] ≤ P0 [D+ = ∞, 0 belongs to a finite cluster] (1.38) +P [D = ∞ = T , sup |X · f | = ∞] = 0 , 0

+

−L

n

i

i≥2,n≥0

where we use for the last term a similar argument as for the control of the rightmost term of (1.35). Keeping in mind (1.16), we see that P0 [D+ < ∞] = 1 .

(1.39)

We can then consider the sequence Dk , k ≥ 0, of iterates of D+ : D0 = 0, Dk+1 = D+ ◦ θDk + Dk , k ≥ 0 .

(1.40)

Note that P0 -a.s., the (Dk )k≥0 , are finite, increasing and XDk ·  is non-decreasing. We first show that P0 [· | I − ]-a.s., {XDk , k ≥ 0} is an infinite subset of Zd .

(1.41)

Indeed otherwise for some finite A ⊆ Zd , P0 [{XDk , k ≥ 0} = A ⊆ C] > 0. Since P-a.s., on {A ⊆ C}, for some (in fact any) x0 ∈ C, with supx∈A x ·  < x0 · , inf x∈A Px,ω [Hx0 < ∞] > 0, it would follow that P0 [supn Xn · > supk≥0 XDk ·] > 0, which is impossible. We thus introduce Vi , i ≥ 0, the sequence of successive times of visit of new sites by XDk , k ≥ 0: V0 = 0, Vi+1 = inf{Dk > Vi ; XDk = XDk , for all k  < k}, for i ≥ 0 .

(1.42)

Let us then fix e0 ∈ Zd , with |e0 | = 1, such that: e0 ·
=

 1  e ·
 ≥√ , |e|=1,e∈Zd d max

(1.43)

(for some directions , there may be more than one single choice for e0 ). Then P0 [· | I − ]a.s., for i ≥ 0,  P0 [XVi +1 = XVi + e0 |FVi ∩ I − ] = P0 [XVi +1 = x + e0 , XVi = x|FVi ∩ I − ] x∈Zd

so that from (0.1) and (1.43), 1  P0 [ω({x, x + e0 }) = 1, XVi = x|FVi ∩ I − ], ≥ 2d d x∈Z

and since under P0 , ω({x, x + e0 }) is independent of FVi ∩ {XVi = x} ∩ I − ,  p p = P0 [XVi = x|FVi ∩ I − ] = . 2d 2d d x∈Z

(1.44)

132

A.-S. Sznitman

Using Borel-Cantelli’s lemma, cf. Durrett [12], p. 207, we find that P0 [· | I − ]-a.s., XVi +1 = XVi + e0 , for infinitely many i ≥ 0, (and thus lim XVi ·  = ∞) .

(1.45)

i

We then choose M  > M > 1, and define, cf. (1.14) for the notations: V  = TM  , and for k ≥ 0, Nk =

k 

1{Xn ·
 ≤ M} .

(1.46)

n=0

With a slight variation of the argument in (1.44), we see that P0 [· | I − ]-a.s.,   

{TL < D− } | FV  ∩ I − P0 {XV  +1 = XV  + e0 } ∩ θV−1 +1 ≥





L>1

P0 XV  = x, ω({x, x + e0 })

x∈Zd

 1   P0 {TL < D− } | ω({−e0 , 0}) = 1 2d L>1   p {TL < D− } | ω({−e0 , 0}) = 1 ≥ c(λ) , P0 = 2d

= 1 | FV  ∩ I −

(1.47)

L>1

using (1.17) and an argument similar as below (1.26) in the last step. As a result, for n ≥ 0,   (1.48) P0 [N∞ = ∞, NV  ≥ n | I − ] ≤ 1 − c(λ) P0 [NV  ≥ n | I − ] . Letting M  and then n tend to infinity, it follows that P0 [N∞ = ∞ | I − ] = 0 . Since M > 1 is arbitrary, (1.37) and thus (1.36) follow.

(1.49)  

2. The Renewal Structure In this section we take advantage of the transience in the direction
 of the walk on the infinite cluster, see Theorem 1.2, and introduce certain regeneration times. These times enable us to construct a renewal structure for the walk under the measure P , see (0.9). The approach has a similar spirit to Shen [23], or Sznitman-Zerner [28], however there is a special twist due to the conditioning present in the definition of P . In particular the regeneration times defined here are configuration dependent. We begin with some notations. For x ∈ Zd , ω ∈ Ix , cf. (1.6), j ≥ 0, we consider Pj,x (ω) = the set of open self-avoiding C-valued paths, (π(i))i≥0 , with π(0) = x and π(i) ∈ Hx− , for i ≥ j . Hx− ,

Since p > pc , see (0.7), percolation takes place on Ix , Pj,x (ω) is not empty for large j . We thus define:

(2.1)

see above (1.5), and P-a.s., on

Jx (ω) = inf{j ≥ 0; Pj,x (ω) = ∅}, if ω ∈ Ix , = ∞, if ω ∈ / Ix .

(2.2)

Anisotropic Walk on the Supercritical Percolation Cluster

133

We then introduce the sequence of configuration dependent stopping times and corresponding successive maxima of the walk in the direction
, cf. (1.14) for notations, W0 = 0, m0 = JX0 (ω) ≤ ∞, and by induction , , n ≤ Wk+1 } + 1 ≤ ∞, for all k ≥ 0 . Wk+1 = 2 + Tmk ≤ ∞, mk+1 = sup{Xn ·
(2.3) As a result of Theorem 1.2, we see that P-a.s., for all x ∈ C, Px,ω -a.s., Wk < ∞, for all k ≥ 0 .

(2.4)

With the choice of e0 as in (1.43) we introduce the collection of bonds: B = {b ∈ Bd ; b = {−e0 , e − e0 }, with e any unit vector of Zd such that e ·
 = e0 ·
},

(2.5)

as well as the configuration dependent stopping times S1 = inf{Wk ; k ≥ 1, XWk = XWk −1 + e0 = XWk −2 + 2e0 , and ω(b) = 1, for all b ∈ B + XWk } .

(2.6)

Lemma 2.1. P-a.s., for all x ∈ C, Px,ω -a.s., S1 < ∞ .

(2.7)

Proof. Observe that for k ≥ 0, P0 [I, S1 > Wk+1 ] =



E[I, P0,ω [S1 > Wk , XTmk = x](1 −

x∈Zd

Px,ω [X1 = x + e0 , X2 = x + 2e0 ] 1OB+x+2e0 )] .

(2.8)

Moreover P-a.s., 1I P0,ω [S1 > Wk , XTmk = x] = 1{J0 ≤x·
} P0,ω [S1 > Wk , XTmk = x] / B + x + 2e0 )-measurable. On the other hand the second is σ (ω(b), b = {x, x + e0 }, b ∈ 1 2 factor in the above expectation is smaller than 1 − ( 2d ) 1OB+x+2e0 ∪{{x,x+e0 }} , which is independent from the above σ -algebra. Therefore:    1 2 P0 [I, S1 > Wk+1 ] ≤ P0 [I, S1 > Wk ] 1 − p |B|+1 2d   1 2 k+1 ≤ 1− p |B|+1 , 2d

(2.9)

for k ≥ 0, using induction in the last step. As a result we see that P0 -a.s., in I, S1 < ∞, and (2.7) follows.   We then define D = inf{n ≥ 0, Xn ·
 < X0 ·
} ,

(2.10)

134

A.-S. Sznitman

as well as the configuration dependent stopping times Sk , k ≥ 0, Rk , k ≥ 1, and the levels Mk , k ≥ 0: S0 = 0, M0 = X0 ·
, and for k ≥ 0 , Sk+1 = S1 ◦ θTMk + TMk ≤ ∞, Rk+1 = D ◦ θSk+1 + Sk+1 ≤ ∞ ,

Mk+1 = sup Xn ·
 + 1 ≤ ∞.

(2.11)

n≤Rk+1

In view of Theorem 1.2, and Lemma 2.1, it follows that P-a.s., for all x ∈ C, Px,ω -a.s., for k ≥ 1, Sk+1 < ∞ on {Rk < ∞} .

(2.12)

We can now define the basic regeneration time: τ1 = SK , with K = inf{k ≥ 1, Sk < ∞ and Rk = ∞} .

(2.13)

Observe that in contrast to [28, 23], τ1 is configuration dependent. Lemma 2.2. P-a.s., for all x ∈ C, Px,ω -a.s., τ1 < ∞ .

(2.14)

Proof. It is straightforward to deduce from (1.17), that P0 [D = ∞|OB ] ≥ P0 [D− = ∞ | OB ] ≥ c(λ) . We now use an argument with a similar flavor as in (2.8). For k ≥ 1, we have:    P0 [I, Rk < ∞] = E I, P0,ω [Sk < ∞, XSk = x] Px,ω [D < ∞] .

(2.15)

(2.16)

x∈Zd

We introduce Ex+ = the collection of b = {y, z} in Bd with y or z in Hx+ and y, z = x − e0 , (2.17)

Ex− = Bd \Ex+ ,

(2.18)

and observe that up to a P-negligible set, 1I P0,ω [Sk < ∞, XSk = x] is σ (ω(b), b ∈ Ex− )-measurable. Thus conditioning on this σ -algebra in the expectation in (2.16), we obtain P0 [I, Rk < ∞] = P0 [I, Sk < ∞] P0 [D < ∞ | OB ] ≤ (1 − c(λ))k , with c(λ) ∈ (0, 1) ,

(2.19)

using induction and (2.15). Thus P0 -a.s., on I, τ1 < ∞, and (2.14) easily follows.

 

− Under P0 , I−e and OB ∩ {D = ∞} are independent and have positive probability, 0 see (1.5), (2.15). The following conditional measure, that plays an important role for the renewal property, is thus well defined: − Q = P0 [· | I−e , OB , D = ∞] . 0

(2.20)

We denote by E P , E Q the expectations under the respective measures P and Q. The next proposition is the key step for the renewal property.

Anisotropic Walk on the Supercritical Percolation Cluster

135

Proposition 2.3. Let f be a bounded σ (ω(b), b ∈ E0+ )-measurable function, f be a bounded σ (ω(b), b = {x, y}, x ·
 ≤ 0, y ·
 ≤ 0)-measurable function and g, h be bounded measurable functions on (Zd )N , then E P [fh(Xτ1 ∧· ) g(Xτ1 +· − Xτ1 ) f ◦ tXτ1 ] = E P [fh(Xτ1 ∧· )] E Q [g(X. )f ] ,

(2.21)

(cf. (0.9), and above (1.6) for the notations). Proof. The left-hand-side of (2.21) multiplied by P0 (I) equals:  E0 [I, τ1 = Sk , fh(XSk ∧· ) g(XSk +· − XSk ) f ◦ tXSk ] =



 E I, f

k≥1,x∈Zd  ×E0,ω [h(XSk ∧· ), Sk < ∞, XSk = x] f ◦ tx Ex,ω [D = ∞, g(X. − x)] . (2.22)

k≥1

The above expression will not change if we replace ω in the Ex,ω -expectation with ωx+ , which coincides with ω on Ex+ and satisfies ω+ (b) = 1, for b ∈ Ex− . Further note that 1I fE0,ω [h(XSk ∧· ), Sk < ∞, XSk = x] is σ (ω(b), b ∈ Ex− )-measurable. Thus the above equals    E I, fE0,ω [h(XSk ∧· ), XSk = x] P0 [D = ∞|OB ]E0 [f g(X. )|OB , D = ∞] . k≥1,x∈Zd

Using the above equality with f = 1, g = 1, we see that the above equals P0 (I) E P [fh(Xτ1 ∧· )] E0 [f g(X. ) | OB , D = ∞] . Our claim (2.21) follows straightforwardly since the last term coincides with E Q [f g(X)].   With the help of the above proposition we can define the sequence of P -a.s. finite times τ0 = 0 < τ1 < τ2 < · · · < τk < . . . , via the following procedure (with hopefully obvious notations): τk+1 = τ1 + τk (Xτ1 +· − Xτ1 , tXτ1 ω), k ≥ 0 .

(2.23)

Note that for any x, P -a.s., {Xτ1 = x} ⊆ Ix− ⊆ {Jx = 0}, and from (2.3), (2.6), (2.11), τk (Xτ1 +· − Xτ1 , tXτ1 ω), P -a.s. coincides with a measurable function of Xτ1 +· − Xτ1 and (tXτ1 ω)(b), b ∈ E0+ , thus in particular the P -a.s. finiteness of the τk follows from Proposition 2.3. We now come to the renewal property: Theorem 2.4. Under P , (Xτ1 ∧· ), (X(τ1 +·)∧τ2 − Xτ1 ), . . . , (X(τk +·)∧τk+1 − Xτk ), . . . , are independent and except for the first process distributed like (Xτ1 ∧· ) under Q. Proof. Consider k ≥ 2, and h1 , . . . , hk bounded measurable functions on (Zd )N . We recall that τ0 = 0, see above (2.23). It follows from Proposition 2.3 that EP

k 

hi (X(τi−1 +·)∧τi − Xτi−1 )

i=1

= E P [h1 (Xτ1 ∧· )]E Q

k  i=2

hi (X(τi−2 +·)∧τi−1 − Xτi−2 ) .

(2.24)

136

A.-S. Sznitman

− Observe that P -a.s., {D = ∞} = {D > τ1 } and Q = P [· | D > τ1 , OB , I−e ], so the 0 last factor in the right-hand-side of (2.24) equals − E P [D > τ1 , OB , I−e , h2 (X·∧τ1 ) 0  − × hi+1 (X(τi−1 +·)∧τi − Xτi−1 )]/P [D > τ1 , OB , I−e ]. 0 2≤i
Applying Proposition 2.3 again, we see that the above expression equals  hi (X(τi−3 +·)∧τi−2 − Xτi−3 )] , E Q [h2 (Xτ1 ∧· )] E Q [ 3≤i≤k

(where the last factor is of course absent when k < 3). Thus a repeated application of Proposition 2.3 yields that the left-hand-side of (2.24) equals: E P [h1 (Xτ1 ∧· )]

k 

E Q [hi (Xτ1 ∧· )] .

i=2

Our claim then follows.

 

We close this section with a stochastic domination result that will be useful in the next section, when controlling moments of supn≤τ1 |Xn | under Q. We introduce the probability −  = P0 [· | I−e Q , OB , D < ∞] . 0

(2.25)

− Proposition 2.5. Under P0 [· | I−e , OB ], Xτ1 ·
 is stochastically dominated by J , 0 where 0 = 0, and for k ≥ 1:

k = (2 + M 1 + 4H 1 ) + · · · + (2 + M k + 4H k ) ,

(2.26)

for independent variables M i , H i , i ≥ 1, J , with M i , H i respectively distributed like  and M = sup (Xn ·
 − X0 ·
) under Q,

(2.27)

n≤D def

H a geometric variable on N∗ ( = N\{0}) with parameter 1 − p |B|+1 /(2d)2 , (2.28) and J an independent geometric variable on N∗ with parameter P0 [D < ∞ | OB ]. Proof. We consider a non-decreasing, non-negative, bounded function f on R and for simplicity write − A = I−e ∩ OB . 0

With a similar calculation as in (2.22), we find:  E0 [A, f (Xτ1 ·
)] = P0 [D = ∞ | OB ] E0 [A, Sk < ∞, f (XSk ·
)] . k≥1

(2.29)

(2.30)

Anisotropic Walk on the Supercritical Percolation Cluster

137

We thus consider for k ≥ 2: E0 [A, Sk < ∞, f (XSk ·
)]      (2.11) ≤ E0 A, Rk−1 < ∞, f Mk−1 + 1 + (XS1 − X0 ) ·
 ◦ θTMk−1  ≤ E0 [A, Rk−1 < ∞, f (Mk−1 + 4j + 1), θT−1 (S1 = Wj )] M k−1

j ≥1

=



E0 [A, Rk−1 < ∞, f (Mk−1 + 4(j + 1) + 1)

j ≥1

−f (Mk−1 + 4j + 1), θT−1 M

k−1

(S1 > Wj )] + E0 [A, Rk−1 < ∞, f (Mk−1 + 5)],

so using similar considerations as in (2.8), (2.9)  ≤ E0 [A, Rk−1 < ∞, f (Mk−1 + 4(j + 1) + 1) j ≥1

 p |B|+1 j − f (Mk−1 + 4j + 1)] 1 − + E0 [A, Rk−1 < ∞, f (Mk−1 + 5)] (2d)2 = E × E0 [A, Rk−1 < ∞, f (Mk−1 + 4H k + 1)] , (2.31) where P stands for the probability governing the variables M i , H i , i ≥ 1, J , and E for the corresponding expectation. In view of (2.11), the above expression is smaller than: E × E0 [A, Sk−1 < ∞, D ◦ θSk−1 < ∞, f (XSk−1 ·
 + M ◦ θSk−1 + 4H k + 2)]   = E × E A, P0,ω [Sk−1 < ∞, XSk−1 = x]Ex,ω [f (x ·
+M x∈Zd

 +4H k + 2), D < ∞] , using a similar argument as in (2.22) = E × E0 [A, Sk−1 < ∞, f (XSk−1 ·
 + M k + 4H k + 2)] P0 [D < ∞ | OB ] and by induction ≤ E × E0 [A, f (XS1 ·
 + k − 1 )] P0 [D < ∞ | OB ]k−1 .

(2.32)

Repeating the argument in (2.31), we see that the left-hand-side of (2.31), for k ≥ 2, is smaller than: P0 (A) E[f ( k )] P0 [D < ∞ | OB ]k−1 , and this inequality remains true for k = 1. Inserting these inequalities in (2.30), we readily obtain our claim.   3. Weak Anisotropy and Ballistic Behavior Our main object in this section is to prove for small λ, a law of large numbers with non-degenerate limiting velocity and a functional central limit theorem governing the corrections to the law of large numbers, see Theorem 3.4. With the help of the renewal

138

A.-S. Sznitman

property in Theorem 2.4, the main task is to show that for small λ, τ1 has a finite second moment under the measure Q of (2.20). We will obtain such an estimate by controlling the P-probability of occurrence of certain low principal Dirichlet eigenvalues in suitable large boxes, cf. Lemma 4.2, and the exit measure of the walk from certain large boxes under P , cf. Lemma 4.1. We will be interested in large boxes of the form, (see (1.18) for the notation),  L, L  > 1, U = {x ∈ Zd , |x · f1 | < L, sup |x · fi | < L},

(3.1)

2≤i≤d

 ≈ L 2 , cf. (3.28) below. We denote by specifically we will later choose L large and L ∂+ U the part of the boundary of U : 7

∂+ U = {x ∈ ∂U, x · f1 ≥ L} .

(3.2)

An important role will be played by the principal Dirichlet eigenvalue of I −R, in U ∩C, for ω ∈
1 , cf. (1.3), (1.10): ω (U ) = inf{E(f, f ), f|(U ∩C )c = 0, f L2 (m) = 1}, when U ∩ C = ∅ , = ∞, by convention when U ∩ C = ∅ . (3.3) Note that when U ∩C = ∅, ω (U ) ≤ 1. We will derive for suitably large boxes U , upper bounds on P[ω (U ) ≤ γ ] for small γ (of order L−5 , see (3.28)) and on P [XTU ∈ / ∂+ U ]. These bounds will be instrumental for the control of moments of τ1 under Q. We are first going to derive a lower bound for ω (U ) in terms of a geometric quantity, and provide an upper bound on P0 [XTU ∈ / ∂+ U ] in terms of P[ω (U ) ≤ γ ], for a suitable γ . The geometric quantity mentioned above, stems from renormalization techniques in percolation, see for instance chapter 7 Sect. 4 in [14], as we now explain. If for all 1 ≤ i, j ≤ d, |
 · ei | ≤ 2|
 · ej |, we choose u, u orthogonal of the form ±ei ,  1 ≤ i ≤ d, with u ·
 > 0, u ·
 > 0, and define the two-dimensional discrete quadrant:
L = Nu + Nu .

(3.4)

On the other hand, if for some j , in the notations of (1.43),
 · e0 > 2| · ej |, we choose u = e0 , u = ej , and define instead
L = {au + bu , 0 ≤ |b| ≤ a} .

(3.5)

It then follows for instance from static renormalization, see Theorems 7.61, 7.65 in [14], that we can choose an integer K and a number c depending only on d, p, so that setting L = K
L + [−K, K]d ,

(3.6)

P-a.s., there is an infinite cluster in y + L, for each y ∈ Zd ,

(3.7)

(i.e. for each y the restriction of ω to bonds between sites in y + L, induces an infinite connected component), and E[ec ] < 2, with

(3.8)

(ω) = inf{k ≥ 0, [−k, +k]d meets an infinite cluster of L} .

(3.9)

Anisotropic Walk on the Supercritical Percolation Cluster

139

Thus for ω ∈
1 , see (1.3), and y ∈ C, we can define D(y, ω) = the minimal distance in C of y to an infinite cluster in y + L ,

(3.10)

(i.e. the minimal number of steps of a nearest neighbor self-avoiding path in C along  > 1: open edges, starting at y and ending in an infinite cluster of y + L), and for L, L D(U, ω) = max D(y, ω) (= 0, by convention when C ∩ U = ∅) . y∈C ∩U

(3.11)

We are now ready to prove Lemma 3.1. (0 < λ ≤ 1) e−2λD(U,ω)  > 1, ω ∈
1 . , for L, L D(U, ω)2d + L4  = 4L/√γ , with the notation (0.9), For L > 1, 0 < γ ≤ 1, L ω (U ) ≥ c

P [XTU ∈ / ∂+ U ] ≤ P[ω (U ) ≤ γ ] + c Ld γ −

(d+1) 2

e−λL

(3.12)

(3.13)

(we refer to the end of the introduction for the convention about constants). Proof. We begin with the proof of (3.12). Without loss of generality, we assume that C ∩ U = ∅. For each x ∈ C ∩ U , we pick a self-avoiding open path (πx (i))0≤i≤x , starting at x and remaining in U except for the terminal point which lies in ∂U . We denote by x the length of this path. We then define the maximal backtracking of these paths in the direction
: . H = max max(x − πx (i)) ·
x∈C ∩U

i

(3.14)

Following a now classical argument, see Saloff-Coste [22], p. 369, we write for f as in (3.3),      2 1= f 2 (x) mω (x) = f πx (i + 1) − f πx (i) mω (x) x

x

Cauchy−Schwarz

≤

 x

x

i

 

  2  mω (x) . f (πx (i + 1) − f πx (i)

(3.15)

i

Note that using λ ≤ 1, we find that mω (x)/wω ({πx (i), πx (i + 1)}) ≤ c e2λH , see (0.3), (1.8), and hence the above expression is smaller than    2 c e2λH f (z) − f (y) w(b, ω) × max x , b

b={y,z}

x∈C ∩U,b∈πx

with the notation b ∈ πx , meaning that b = {πx (i), πx (i + 1)}, for some i. As a result we obtain the lower bound: ω (U ) ≥ c

max b

e−2λH

x∈C ∩U,b∈πx

x

.

(3.16)

140

A.-S. Sznitman

We now specify the choice of paths πx . For each x ∈ C ∩ U we pick an open path in C of length D(x, ω) connecting x to an infinite cluster of x + L, see (3.10), and then continue this path with an infinite self-avoiding open path in x + L. At some point the concatenation of these two paths exits U , and we can extract a self-avoiding path between x and this exit point, and thus obtain the desired πx . Observe that i) H ≤ D(U, ω) + c , L is in essence two-dimensional) , ii) x ≤ D(U, ω) + c L2 , for x ∈ C ∩ U , (recall
iii) for b = {y, z}, y, z ∈ U : #{x ∈ C ∩ U, b ∈ πx } ≤ #{x ∈ C ∩ U, y or z ∈ x + L} + #{x ∈ C ∩ U,   ≤ c L2 + D(U, ω)d .

d 

|xi − yi | ≤ D(U, ω)}

1

(3.17) Note that 

x ≤ c(D(U, ω) + L2 )(D(U, ω)d + L2 ) ≤ c(D(U, ω)2d + L4 ) ,

x∈C ∩U,b∈πx

and λ ≤ 1, so that (3.12) follows from (3.16), (3.17). We then turn to the proof of (3.13). For n ≥ 1, we have / ∂+ U ] ≤ P[ω (U ) ≤ γ ] + P [TU > n, ω (U ) > γ ] P [XTU ∈ + P [XTU ∈ / ∂+ U, TU ≤ n].

(3.18)

For ω ∈ I, see (0.8), one sees with the help of Perron-Frobenius’ theorem that 1U ∩C R 1U ∩C has an operator norm on L2 (m) given by its maximum positive eigenvalue, namely 1 − ω (U ). So for n ≥ 1, and x ∈ C ∩ U , using the spectral theorem we find: mω (x) Px,ω [TU > n] = (1{x} , (1U ∩C R 1U ∩C )n 1U )L2 (m) ≤ and hence Px,ω [TU > n] ≤

   n mω (x) mω (U ) 1 − ω (U ) ,

 mω (U ) −nω (U ) , ω ∈ I, x ∈ C ∩ U, n ≥ 1 . e mω (x)

(3.19)

Thus the second term in the right-hand-side of (3.18), using λ ≤ 1, is smaller than  c L2 L 1

(d−1) 2

eλL−nγ .

(3.20)

Further for ω ∈
1 , x, y ∈ C, with dC (x, y) the distance on the infinite cluster between x and y, i.e. the minimum number of steps of an open path between x and y, Carne’s estimate [8] yields:   d (x, y)2  mω (y) C Px,ω [Xn = y] ≤ 2 exp − , n ≥ 1, (3.21) mω (x) 2n

Anisotropic Walk on the Supercritical Percolation Cluster

141

and hence the last term in the right-hand-side of (3.18) is smaller than:   2   (d−1) e−λL + LL (d−2) exp − L + λL c L 2k k≤n

  2  (d−1) e−λL + LL (d−2) exp − L + λL . ≤ cn L 2n 4L  √ If we now choose n = [ 2L γ ] + 1, L = γ , (note that to (3.18), (3.20), (3.22), we obtain (3.13).  

2L γ

≥ 1, so n ≤

4L γ ),

(3.22) coming back

We now derive an upper bound on the P-probability of occurrence of certain low eigenvalues, see (3.13).  = 4L/√γ , Lemma 3.2. (0 < λ ≤ 1). For L > 1, 0 < γ ≤ 1, L P[ω (U ) ≤ γ ] ≤ c Ld γ −

(d−1) 2

[(c γ L4 ) λ + e−cL c

2/d

].

(3.23)

Proof. From the results in Antal-Pisztora [2], see Lemma 2.14 in [10] for the precise version we use here, there is a suitable ρ(d, p) > 0, such that P[ for some z, z ∈ [−k, k]d ∩ C, dC (z, z ) > ρk] ≤ e−ck , for k ≥ 0,

(3.24)

with the same notation as in (3.21). Hence using (3.8), we see that for u > 0,   u u + P (ω) > ≤ c e−cu . P[I, D(0, ω) > u] ≤ P I, D(0, ω) > u, (ω) ≤ ρ ρ (3.25) Thus for U as above we find: (3.12)  P[ω (U ) ≤ γ ] ≤ P c

e−2λ D(U,ω) ≤ γ , U ∩ C =  ∅ D(U, ω)2d + L4 ≤ P[D(U, ω)2d ≥ L4 , U ∩ C = ∅] + P[e−2λ D(U,ω) ≤ c γ L4 , U ∩ C = ∅]   1    1 ,I ≤ |U | P[D(0, ω) ≥ L2/d , I] + P D(0, ω) ≥ log 2λ cγ L4

(3.25)

(d−1) 2

which proves (3.23).

 

≤ c Ld γ −

(e−c L

2/d

+ (c γ L4 )c/λ ) ,

(3.26)

We are now ready to derive moment estimates on τ1 . The definition of Q appears in (2.20). Proposition 3.3. There exist a non-increasing sequence of constants λm , m ≥ 1, in (0, 1] depending on d and p such that for 0 < λ ≤ λm : E Q [τ1m ] < ∞ .

(3.27)

142

A.-S. Sznitman

Proof. We begin with a similar estimate for supn≤τ1 |Xn | in place of τ1 . We write for L > 1,  = 4L 27 = 4L/√γ , γ = L−5 . UL = U in (3.1), with L

(3.28)

Then for u > 0, with the notation (1.18), (2.29), P0



  |Xn · fi | ≥ u, A ≤ P0

sup n≤τ1 ,1≤i≤d

sup n≤τ1 ,1≤i≤d

|Xn · fi | ≥ u, Xτ1 ·
<

u2

 u2

7 + P0 Xτ1 ·
≥ , A = α1 + α2 . 4

7

4 (3.29)

2

Setting L(u) = ( u4 ) 7 , we have for m ≥ 1, and large u, / ∂+ UL(u) ] ≤ u−m , α1 ≤ P [XTUL(u) ∈

(3.30)

provided 0 < λ ≤ λm (d, p), thanks to (3.13), (3.23). Moreover using Proposition 2.5 and Chebyshev’s inequality we find α2 ≤

 u − 2 m 7

4

 u − 2 m 7

E0 [(Xτ1 ·
)m , A]

E[( J )m ] ≤

 u − 2 m 7

E[J m ] E[(M 1 + 4H 1 + 2)m ] 4 4 (using the independence of J and M i , H i , i ≥ 1 under P ), ≤

m  ≤ c(λ,
, m) u− 7 m (1 + E[H 1 ] + E Q [M m ]) . 2

(3.31)

m

From (2.28), E[H 1 ] is obviously finite, and for λ < λm+d and large k, we obtain   k ≤ M < 2k+1 ] ≤ c(λ) P [XTU Q[2  + P XTU

2k

∈ ∂+ U2k , T0 ◦ θTU

2k

2k

∈ / ∂+ U 2 k ] < T2k+1 ◦ θTU



,

2k

so that summing over all positions of XTU k , and using translation invariance 2

≤ c(λ)(1 + |∂+ U2k |) P [XTU

7

2k

7

∈ / ∂+ U2k ] ≤ c(λ) 2 2 (d−1)k 2−(m+d) 2 k

≤ c(λ) 2−(m+1)k , thanks to (3.30) with u = 4 · 2 2 k . 7

(3.32)



Hence E Q [M m ] < ∞, and coming back to (3.29), (3.30), (3.31), we see that for λ ≤ λ7m+d , P0 [ sup |Xn | ≥ u, A] ≤ u−m , when u is large . n≤τ1

(3.33)

Anisotropic Walk on the Supercritical Percolation Cluster

143 1

Thus for m ≥ 1, 0 < λ ≤ λm (d, p), for t a large integer, and u = t 7 : P0 [τ1 > t, A] ≤ P0 [ sup |Xn | ≥ u, A] + P0 [TUu > t, I] n≤τ1

(3.33),(3.19)

≤

7d

−5

u−m + P[ω (Uu ) ≤ u−5 ] + c u 4 eλu−tu

≤ 3u−m = 3t − 7 , m

(3.34)

with the help of (3.23) in the last step. Our claim (3.27) now follows straightforwardly.   We are now ready to prove the main result of this section pertaining to the ballistic nature of the walk when the anisotropy is weak. Theorem 3.4. For 0 < λ ≤ λ2 , (cf. Proposition 3.3), P-a.s., for all x ∈ C, Px,ω -a.s., lim n

v=

Xn = v, with n

E Q [Xτ1 ] , so that v ·  > 0, (see (2.20) for the definition of Q) . E Q [τ1 ]

(3.35) (3.36)

Moreover under P , the D(R+ , Rd )-valued processes B.n = √1n (X[· n] − [· n]v) converge in law towards a Brownian motion with non-degenerate covariance matrix (3.37) A=

E Q [(Xτ1 − τ1 v)(Xτ1 − τ1 v)t ] . E Q [τ12 ]

(3.38)

Proof. Note that (3.35) is equivalent to Xn → v, as n → ∞, with v in (3.36) . (3.39) n With the renewal property stated in Proposition 2.3, the proofs of (3.39) and (3.37) are merely repetitions of the proofs of Proposition 2.1 of Sznitman-Zerner [28] and Theorem 4.1 of Sznitman [26], once we know that P -a.s.,

E Q [τ12 ] < ∞ .

(3.40)

However (3.40) is ensured by choosing λ ≤ λ2 in view of Proposition 3.3. There remains to prove the non-degeneracy of A. The proof uses a rather similar argument as in Theorem 4.1 of [26]. Namely it suffices to show that only w = 0 satisfies: Q[w · (Xτ1 − τ1 v) = 0] = 1 .

(3.41)

But if Q[2 < S1 , XS1 = x] > 0, it is straightforward to prove with similar arguments as in Lemma 2.2 and possibly modifying the path by inserting several back and forth crossings, just after time 2, of the edge the path crosses at the second step, that Q[Xτ1 = x, τ1 = n] > 0 for an unbounded set of integers n. Thus (3.41) implies that w · v = 0 and w · x = 0 for all x with Q[XS1 = x, S1 > 2] > 0. Taking limits over such sites, one obtains that w · y = 0, for any y ∈ Rd orthogonal to . Since w · v = 0 and v ·  > 0, this implies w = 0. This finishes the proof of Theorem 3.4.  

144

A.-S. Sznitman

4. Strong Anisotropy and Sub-Diffusive Behavior We will now study the asymptotic behavior of the walk when λ is large. We will see that unlike the small λ regime where the walk on the infinite cluster has non-degenerate velocity, the large λ regime leads to a drastic slowdown. This effect is related to the presence of certain long but finite arms in the infinite cluster roughly pointing in the direction
, that are powerful traps when λ is large. The flavor of the results presented here is similar to Bramson-Durrett [7], see also Bramson [6]. Theorem 4.1. There exists λs (d, p) ≥ 1, such that for λ > λs , P0 -a.s., lim n

|Xn | = 0, nλs /λ

(4.1)

and consequently P-a.s., for all x ∈ C, Px,ω -a.s., lim n

|Xn | = 0. nλs /λ

(4.2)

Proof. Note that (4.2) is an immediate consequence of (4.1), (it is in fact equivalent), and we only need to prove (4.1). Using the notation (1.18), we define for L > 1: L = {z ∈ Zd ,

d 

|z · fi | ≤ L} .

(4.3)

1

Without loss of generality we assume from now on that λ ≥ 1. Given L0 > 1, and a sequence (δk )k≥0 in [1, ∞), we are going to construct an increasing sequence (Lk )k≥0 , such that for k ≥ 0: Lk+1 = L0 + c

k 

(4.4)

δi ,

i=0

Lk ∪ ∂Lk ⊆ Lk+1 ,

(4.5)

for any x ∈ ∂Lk , there is a self-avoiding nearest neighbor path (4.6) (xi )0≤i≤K , with K ≥ 2, x0 = x, xi ∈ Lk+1 \(Lk ∪ ∂Lk ), i ≥ 1, and i) K ≤ c δk ,  ≥ δk , ii) (xK − x0 ) ·
iii)

K 

e2·(x0 −xi ) ≤ c(λ) ,

i=0

(we refer to the end of the introduction for the convention concerning constants). Indeed consider L > 1 and δ ≥ 1. For any x ∈ ∂L , we can find nx =

d  1

i fi , with i ∈ {−1, 1}d ,

(4.7)

Anisotropic Walk on the Supercritical Percolation Cluster

145

such that the half-space {z ∈ Zd , nx · z ≤ nx · x} contains L , and since nx can only take finitely many values and does not belong to R− f1 , we can choose vx = v(nx ) ∈ S d−1 , with vx · nx > 0, and vx · f1 ≥ c0 (d) > 0, (recall f1 =
) . (4.8) For some i0 ≥ 1, depending on d, we can then construct a nearest neighbor path √ x 0 = x, x 1 , . . . , x i0 , with x i , i ≥ 1, outside L ∪∂L and x i0 at least at distance 2 + d from the above mentioned half-space. We then pick a nearest neighbor path in the set of vertices √ of the cubes y + [0, 1]d , y ∈ Zd which intersect the segment [x i0 , x i0 + √ vx (δ + d + 2i0 )c0−1 ] ⊆ Rd , starting at x i0 and ending in a cube containing x i0 +vx (δ+ d +2i0 )c0−1 . Note that all vertices of this cube lie at distance at least δ + i0 from x. Extracting a self-avoiding path (xi )0≤i≤K , from the concatenation of the above two paths we have 2 ≤

K ≤ c δ, x0 = x, xi ∈ / L ∪ ∂L , for i ≥ 1. Moreover we see that (xK − x0 ) ·
 ≥ δ, K and i=0 exp{−2 · (xi − x0 )} ≤ c(λ). We now see that for a suitable c > 0, for any L0 , (δk )k≥0 as above, setting Lk+1 = Lk + c δk , k ≥ 0, we can realize (4.4), (4.5), (4.6). We thus consider L0 > 1, (δk )k≥0 a sequence in [1, ∞), which will later be specified in (4.23), and the corresponding sequence (Lk )k≥0 . We define for k ≥ 0,

k = Lk+1 \(Lk ∪ ∂Lk ) , Nk = TLk , the exit time from Lk ,

(4.9) (4.10)

and for each x ∈ ∂Lk , with (xi )0≤i≤K as in (4.6), we consider the event: Jx = {ω : ω({xi , xi+1 }) = 1, 0 ≤ i < K, and ω(b) = 0, for any b = {xi , z}, with i ≥ 1, and z = xi−1 , xi+1 } .

(4.11)

Note that from (4.6) follows that for k ≥ 0, and x ∈ ∂Lk , Jx is σ (ω(b), b ∈ / Ek )-measurable ,

(4.12)

where Ek stands for the set of nearest neighbor-edges with at least one end-point in Lk . We will use Lemma 4.2. (λ ≥ 1). For k ≥ 0, n ≥ 1, P0 -a.s., P0 [Nk+1 ≥ Nk + n | FNk ] ≥ c(λ)(1 − c(λ) e−2λδk )n+ (p(1 − p))cδk .

(4.13)

Proof. We first prove that P0 -a.s., on {Nk < ∞}, the left-hand-side of (4.13) is bigger than c(λ)

inf

x∈∂Lk

Px1 [T k ≥ n, Jx ], (with the notations of (4.6)) .

(4.14)

Indeed if h is a bounded non-negative FNk -measurable function vanishing on {Nk = ∞}, E0 [Nk+1 ≥ Nk + n, h]    E[E0,ω h, XNk = x] Px,ω [Nk+1 ≥ n] = x∈∂Lk

≥



x∈∂Lk

≥ c(λ)

  E E0,ω [h, XNk = x], Jx , Px,ω [X1 = x1 , T k ◦ θ1 ≥ n] 

x∈∂Lk

  E E0,ω [h, XNk = x], Jx , Px1 ,ω [T k ≥ n] .

(4.15)

146

A.-S. Sznitman

Observe that E0,ω [h, XNk = x] is σ (ω(b), b ∈ Ek )-measurable, whereas the remaining terms inside the P-expectation are σ (ω(b), b ∈ / Ek )-measurable. Hence the left-handside of (4.15) is bigger than c(λ) E0 [h]

inf

x∈∂Lk

Px1 [T k ≥ n, Jx ] ,

and the claim in (4.14) follows. Since (4.13) is obvious on {Nk = ∞}, we only need to provide a suitable lower bound for (4.14). Note that for x ∈ ∂Lk , on Jx , xi , 1 ≤ i ≤ K, is in essence a one-dimensional segment in k , which the walk can only exit through x0 = x. Hence with the notations of (1.7), it follows that xK < Hx0 )n ] . Px1 [T k ≥ n, Jx ] ≥ E[Jx , Px1 ,ω (HxK < Hx0 ) PxK ,ω (H

(4.16)

Then using for instance Chung [9], Chapter I §12, we have on Jx Px1 ,ω [HxK < Hx0 ] =

γ0 − γ 1 = (γ0 − γK )−1 , γ0 − γ K

xK < Hx0 ] = 1 − γK−1 − γK , where PxK ,ω [H γ0 − γ K γ0 = 0, γi = −



π0,j , for 0 < i ≤ K, with π0,0 = 1, and

(4.17) (4.18)

(4.19)

0≤j
π0,j =

rω (xj , xj −1 ) rω (x1 , x0 ) ... , 1≤j
(4.20) wω ({x0 , x1 }) = = exp{ · (x0 + x1 − xj − xj +1 )}, (recall ω ∈ Jx ) . wω ({xj , xj +1 })

(1.8)

Note that from (4.6) ii) and iii), (and recall x0 = x ∈ ∂Lk ), 1 = γ0 − γ1 ≤ γ0 − γK ≤ c(λ), and γK−1 − γK = π0,K−1 ≤ c(λ) e−2λδk .

(4.21)

Coming back to (4.16) we thus find Px1 [T k ≥ n, Jx ] ≥ c(λ)(1 − c(λ) e−2λδk )n+ P[Jx ], and using (4.6) i) and (4.11), our claim (4.13) follows.

(4.22)

 

We now specify L0 and the sequence (δk )k≥0 , via L0 = 2, δk = c1 log(k ∨ 1) + 1, k ≥ 0 ,

(4.23)

where c1 is a small enough constant which only depends on d and p, so that with the help of (4.13), we find that for large k, P0 [Nk+1 ≥ Nk + e2λδk | FNk ] Lk ≤ c k log k .

P0 -a.s. − 1 ≥ k 2 , and

(4.24) (4.25)

Anisotropic Walk on the Supercritical Percolation Cluster

147

It follows that for large k, P0 [Ni+1 < Ni + e

2λδ k 2

, i=

√ k  k 1 k , . . . , k − 1] ≤ 1 − k − 2 2 ≤ e− 2 , 2

(4.26)

and from Borel-Cantelli’s lemma, that P0 -a.s., for large k, Nk ≥ e

2λδ[ k ] 2

≥ c(λ) k 2c1 λ .

(4.27)

Taking (4.25) into account, it follows that P0 -a.s., lim n

Picking λs =

1 c1 ,

the claim (4.1) follows.

|Xn | 1

= 0.

(4.28)

n c1 λ  

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Commun. Math. Phys. 240, 149–159 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0897-2

Communications in

Mathematical Physics

Symplectic Structures on Moduli Spaces of Parabolic Higgs Bundles and Hilbert Scheme Indranil Biswas1 , Avijit Mukherjee2 1

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India. E-mail: [email protected] 2 Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany. E-mail: [email protected] Received: 11 November 2001 / Accepted: 12 March 2003 Published online: 1 August 2003 – © Springer-Verlag 2003

Abstract: Parabolic triples of the form (E∗ , θ, σ ) are considered, where (E∗ , θ) is a parabolic Higgs bundle on a given compact Riemann surface X with parabolic structure on a fixed divisor S, and σ is a nonzero section of the underlying vector bundle. Sending such a triple to the Higgs bundle (E∗ , θ) a map from the moduli space of stable parabolic triples to the moduli space of stable parabolic Higgs bundles is obtained. The pull back, by this map, of the symplectic form on the moduli space of stable parabolic Higgs bundles will be denoted by d . On the other hand, there is a map from the moduli space of stable parabolic triples to a Hilbert scheme Hilbδ (Z), where Z denotes the total space of the line bundle KX ⊗ OX (S), that sends a triple (E∗ , θ, σ ) to the divisor defined by the section σ on the spectral curve corresponding to the parabolic Higgs bundle (E∗ , θ). Using this map and a meromorphic one–form on Hilbδ (Z), a natural two–form on the moduli space of stable parabolic triples is constructed. It is shown here that this form coincides with the above mentioned form d .

1. Introduction In [2], we proved that the algebraic two–form on the moduli space of triples of the form (E, θ, σ ), where (E, θ ) is a stable Higgs bundle on Riemann surface X and σ a nonzero section of E, obtained by pulling back the natural symplectic form on the moduli space of stable Higgs bundles coincides with the pullback of the symplectic structure on the Hilbert scheme of zero dimensional subschemes of the total space of the holomorphic cotangent bundle KX of X. Our aim here is to establish an analogous result in the context of parabolic triples. Let X be a compact connected Riemann surface of genus g. Fix a finite subset S of X. A parabolic vector bundle of rank two with parabolic structure over S consists of a holomorphic vector bundle E, a line Fs ⊂ Es , and a real number 0 < λs < 1 for each s ∈ S. A Higgs field on this parabolic bundle is a holomorphic section θ of the

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vector bundle End(E) ⊗ KX ⊗ OX (S) such that θ (s) is nilpotent with respect to the flag 0 = F 0 ⊂ F 1 = Fs ⊂ F 2 = Es of the fiber Es for each s ∈ S. We fix the parabolic weights {λs }s∈S and consider parabolic Higgs bundles of rank two and of fixed positive degree d, with d > 6(g − 1) + #S. Let MsH denote the moduli space of stable parabolic Higgs bundles. For such stable parabolic Higgs bundles the dimension of the space of all holomorphic sections of the underlying vector bundle remains constant over the moduli space (Lemma 3.1). Therefore, there is a projective bundle φ : PH −→ MsH whose fiber over any point representing a parabolic Higgs bundle ((E, {Fs }), θ ) is the projective space PH 0 (X, E). In other words, PH is the moduli space of triples of the form ((E, {Fs }), θ, σ ), where ((E, {Fs }), θ ) is a stable parabolic Higgs bundle and σ is a nonzero section of E. See [1 and 5], where the notion of triples were introduced, for a detailed study and many interesting results on triples. The moduli space MsH has a natural holomorphic one–form, which we call , such that d is a symplectic form on MsH . The total space of the holomorphic cotangent bundle of the moduli space of stable parabolic bundles sits inside MsH as a Zariski open dense subset. The restriction of  to this open subset coincides with the canonical one– form on the total space of any cotangent bundle. The main result proved here relates the form φ ∗  on PH with a certain one–form on a Hilbert scheme. Let Z denote the surface defined by the total space of the line bundle KX ⊗ OX (S) over X. Given a stable parabolic Higgs bundle ((E, {Fs }), θ ), there is a spectral curve, which is a divisor on Z, and a rank one torsionfree sheaf L on it such that γ∗ L ∼ = E, where γ is the projection map of the spectral curve to X. Since γ is a finite map, we have H 0 (X, E) identified with the space of all holomorphic sections of L. Considering the divisor of the section of L corresponding to σ ∈ H 0 (X, E) \ {0} we get a map from PH to the Hilbert scheme Hilbδ (Z) of zero dimensional subschemes of Z of length δ = d + 2(g − 1) + #S. This map, which we will denote by f , sends a parabolic triple ((E, {Fs }), θ, σ ) to the divisor on the spectral curve for ((E, {Fs }), θ ) defined by σ . Note that using the inclusion map of a spectral curve in Z, a divisor on a spectral curve is a zero dimensional subscheme of Z. (See Sect. 3.1 for the details.) Using the fact that Z is the total space of KX ⊗OX (S), there is a natural meromorphic one–form δ on Hilbδ (Z). The pullback f ∗ δ is a holomorphic one–form on PH . We prove that f ∗ δ coincides with φ ∗  (Theorem 3.2). Although we have restricted ourselves to rank two parabolic bundles, the extension of Theorem 3.2 to higher rank cases is straightforward. The reason for restriction to the rank two case is the ensuing notational simplification. 2. Preliminaries 2.1. Parabolic Higgs bundles. Let X be a compact connected Riemann surface of genus g. Fix a finite subset S := {s1 , s2 , · · · , sn } ⊂ X . If g = 0, then take n ≥ 4, and n ≥ 1 if g = 1. A parabolic vector bundle of rank two over X with parabolic structure over S consists of the following [8]:

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1. a holomorphic vector bundle E of rank two over X; 2. for each point s ∈ S, a line Fs ⊂ Es of the fiber Es ; 3. for each point s ∈ S, a real number λs with 0 < λs < 1. The numbers {λs } are called parabolic weights. See [8] for the definition of parabolic (semi)stability. We fix real numbers {λs }s∈S and an integer d. Henceforth, by a parabolic vector bundle we will always mean a parabolic vector bundle over X of rank two and degree d with parabolic structure over S and with parabolic weight λs for each s ∈ S. Let KX denote the holomorphic cotangent bundle of X. A Higgs structure on a parabolic vector bundle E∗ := (E, {Fs }) is a holomorphic section θ ∈ H 0 (X, End(E) ⊗ KX ⊗ OX (S))

(2.1)

with the property that for each s ∈ S, the image of the homomorphism θ (s) : Es −→ (E ⊗ KX ⊗ OX (S))s is contained in the subspace Fs ⊗ (KX ⊗ OX (S))s ⊂ (E ⊗ KX ⊗ OX (S))s and θ(s)(Fs ) = 0 [6, 3, 4]. In other words, θ (s) is nilpotent with respect to the flag 0 ⊂ Fs ⊂ Es . A parabolic Higgs bundle (E∗ , θ) as above is called stable if for every line subbundle L of E with θ (L) ⊆ L ⊗ KX ⊗ OX (S) ⊂ E ⊗ KX ⊗ OX (S), the following inequality is satisfied degree(L) +


s∈S 

λs <

par-deg(E∗ ) , 2

where S  := {s ∈ S | Ls = Fs }; if the strict inequality is replaced by partial inequality, then (E∗ , θ ) is called semistable. The moduli space of semistable parabolic Higgs bundles will be denoted by MH . It is an irreducible normal quasiprojective variety, and MsH is the moduli space of stable parabolic Higgs bundles, Zariski open dense smooth. 2.2. Symplectic structure on moduli space. The moduli space MsH has a natural holomorphic symplectic structure [3, Sect. 6]; we will briefly recall it here. Take a stable parabolic Higgs bundle (E∗ , θ) := ((E, {Fs }), θ ) represented by a point in MsH . Define End1 (E) ⊂ End(E) by the condition that for any s ∈ S and v ∈ End1 (E)s we have v(Fs ) ⊆ Fs . Let End0 (E) ⊂ End1 (E) be defined by the condition v(Fs ) = 0. Consider the two term complex C. of sheaves defined by [−,θ]

C0 := End1 (E) −→ C1 := End0 (E) ⊗ KX ⊗ OX (S), where Ci is at the i th position.

(2.2)

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The tangent space T(E∗ ,θ) MsH of the variety MsH at the point represented by the parabolic Higgs bundle (E∗ , θ) is identified with the hypercohomology H1 (C. ) [3, Lemma 6.1]. Consider the homomorphism [−,θ]

C0 −→ C1     1 End (E) −→ 0 This induces a homomorphism f : H 1 (C. ) −→ H 1 (X, End1 (E)) .

(2.3)

Now observe that End1 (E)∗ ∼ = End0 (E) ⊗ OX (S) with the duality pairing defined by (ω , α) −→ trace(ωα) ∈ C , where α ∈ End1 (E)x and ω ∈ (End0 (E) ⊗ OX (S))x , and x is any point of X. Now the Serre duality says H 1 (X, End1 (E))∗ ∼ = H 0 (X, End0 (E) ⊗ KX ⊗ OX (S)) . Consequently, we have a functional θ  ∈ H 1 (C. )∗ defined by β −→ (θ , f (β)), where f is constructed in (2.3). Let  denote the algebraic one–form on the variety MsH that sends any tangent vector β ∈ T(E∗ ,θ) MsH , where (E∗ , θ) ∈ MsH , to θ  (β) constructed above. The two–form d is a symplectic form on MsH . The restriction of d to the Zariski open subset of MsH defined by the total space of the cotangent bundle of the moduli space of parabolic bundles coincides with the canonical symplectic structure on cotangent bundles. (See [3, 4].)

2.3. Spectral data for parabolic Higgs bundles. Let (E∗ , θ) be a parabolic Higgs bundle. So we have trace(θ ) ∈ H 0 (X, KX ) and trace(θ 2 ) ∈ H 0 (X, KX⊗2 ⊗ OX (S)) since θ (s) is nilpotent for each s ∈ S. Set H := H 0 (X, KX ) ⊕ H 0 (X, KX⊗2 ⊗ OX (S)) .

(2.4)

Hitchin defined a map ψ : MH −→ H

(2.5)

that sends any semistable parabolic Higgs bundle (E∗ , θ) to (trace(θ ) , trace(θ 2 )) ∈ H [7, 6, 4]. This map ψ is known as the Hitchin map, and H is known as the Hitchin space. Let Z denote the total space of the line bundle KX ⊗ OX (S), which is a quasiprojective complex surface. The natural projection of Z to X will be denoted by γ . Since S is an effective divisor, there is a natural homomorphism from KX⊗2 ⊗ OX (S) to KX⊗2 ⊗ OX (2S); this homomorphism will be denoted by q.

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Take any point (α , β) ∈ H. Consider the map from Z to the total space of the line bundle KX⊗2 ⊗ OX (2S) defined by z −→ z ⊗ z + q(z ⊗ α(γ (z))) + q(β(γ (z))) ∈ (KX⊗2 ⊗ OX (2S))γ (z) .

(2.6)

The inverse image (in Z) by this map of the zero section of KX⊗2 ⊗OX (2S) is the spectral curve associated to the point (α , β) of the Hitchin space (see [7, 4]). For any point h = (α , β) ∈ H, the corresponding spectral curve will be denoted by Yh . The restriction to Yh of the projection γ to X will also be denoted by γ . Note that the map γ : Yh −→ X

(2.7)

is of degree two. This map is evidently ramified over every point of S. Given a parabolic Higgs bundle, there is a corresponding spectral curve and a torsionfree sheaf on it [7, 4]. To describe the construction, take a semistable parabolic Higgs bundle (E∗ , θ ) ∈ MH on X. The spectral curve associated to it is the one defined by ψ((E∗ , θ )) ∈ H by the above construction, where ψ is the Hitchin map defined in (2.5). Denote the point ψ((E∗ , θ)) by h. There is a torsionfree sheaf L of rank one on the spectral curve Yh such that γ∗ L ∼ = E

(2.8)

(the underlying vector bundle of the parabolic bundle), where γ as in (2.7). The spectral curve can be thought of as the eigenvalues of the endomorphism θ . The sheaf L is defined by the corresponding eigenvectors (see [7] for the details). Since the map γ is ramified over any point of s ∈ S, the direct image γ∗ L has a filtration over s. This filtration is defined by the order of the vanishing (at γ −1 (s)) of a (locally defined) section of L. In the isomorphism of γ∗ L with E, the filtration of γ∗ L at any s ∈ S coincides with the filtration Fs ⊂ Es for the parabolic structure. 3. Parabolic Triples and Hilbert Scheme In Sect. 2.1 we fixed the degree of a parabolic vector bundle to be d. Henceforth, we will assume that the integer d, which is the degree of a parabolic vector bundle, satisfies the condition d > 6g − 6 + n, where n = #S. Lemma 3.1. For any semistable parabolic Higgs bundle (E∗ , θ ) ∈ MH over X, we have H 1 (X, E) = 0, where E is the underlying vector bundle. Consequently, dim H 0 (X, E) = d + 2(1 − g). Proof. Take any (E∗ , θ) ∈ MH . Let E be the underlying vector bundle for the parabolic vector bundle E∗ . Since H 1 (X, E) ∼ = H 0 (X, E ∗ ⊗ KX )∗ , it suffices to show H 0 (X, E ∗ ⊗ KX ) = 0. Assume that τ ∈ H 0 (X, E ∗ ⊗ KX ) \ {0} is a nonzero section.

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So τ defines a nonzero homomorphism from E to KX , which will be denoted by τ . The kernel of τ is a torsionfree coherent subsheaf of E. Therefore, it defines a line bundle over X, which will be denoted by L. Now we have a diagram ι

−→

L

E  θ

τ

−→

KX (3.1)

τ ⊗Id

L ⊗ KX ⊗ OX (S) −→ E ⊗ KX ⊗ OX (S) −→ KX⊗2 ⊗ OX (S) ι⊗Id

We will show that the composition (τ ⊗ Id) ◦ θ ◦ ι = 0. To prove this, first note that the top sequence in (3.1) shows that degree(L) ≥ degree(E) − degree(KX ) = d − 2g + 2 .

(3.2)

On the other hand, degree(KX⊗2 ⊗ OX (S)) = 2g − 4 + n, where n is the cardinality of the set S. Since d is assumed to be at least 6g − 5 + n, we have degree(L) > degree(KX⊗2 ⊗ OX (S)) . Consequently, there is no nonzero homomorphism from L to KX⊗2 ⊗OX (S). In particular, the composition (τ ⊗ Id) ◦ θ ◦ ι = 0. Let L denote the line subbundle of E generated by L (note that ι may not be fiberwise injective). Since (τ ⊗ Id) ◦ θ ◦ ι = 0, it follows immediately, that θ(L ) ⊆ L ⊗ KX ⊗ OX (S). Finally, we have degree(L ) ≥ degree(L) ≥ d − 2g + 2 =

1 d + (d − 4g + 4) 2 2

d n par-deg(E∗ ) + = , 2 2 2 where the second inequality was obtained in (3.2) and the third one follows from the assumption d > 6g − 6 + n; the first inequality follows from the fact that there is a nonzero homomorphism from L to L . The above inequality shows that the line subbundle L of E contradicts the semistability condition of the parabolic Higgs bundle (E∗ , θ). Therefore, we have H 1 (X, E) = 0. Now the Riemann–Roch says that dim H 0 (X, E) = d + 2(1 − g). This completes the proof of the lemma.  >

The above lemma says that dim H 0 (X, E) remains constant over MsH . Therefore, there is a natural projective bundle φ : PH −→ MsH

(3.3)

of relative dimension d − 2g + 1 such that the fiber over any point (E∗ , θ) ∈ MH is the projective space PH 0 (X, E) consisting of all lines in H 0 (X, E); as before, E denotes the underlying vector bundle for the parabolic bundle E∗ . Therefore, PH is the moduli space of triples of the form (E∗ , θ , σ ), where (E∗ , θ) is a stable parabolic Higgs bundle and σ ∈ H 0 (X, E) \ {0} a nonzero section. By a parabolic triple we will mean a triple (E∗ , θ , σ ) of the above type. Consequently, PH is the moduli space of all parabolic triples. In Sect. 2.2 we defined a symplectic structure d on MsH . Define the algebraic one–form  := φ ∗ 

(3.4)

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on PH , where φ is defined in (3.3). So, d = φ ∗ d is the pullback to PH of the symplectic form  on MsH . 3.1. Parabolic triples and spectral data. Take a parabolic triple (E∗ , θ , σ ) ∈ PH . Its image ψ((E∗ , θ )) ∈ H will be denoted by h, where ψ is the Hitchin map defined in (2.5). As we saw in Sect. 2.3, the parabolic Higgs bundle (E∗ , θ ) gives a spectral curve Yh and a torsionfree sheaf L of rank one on Yh . It was noted in (2.8) that γ∗ L ∼ = E, where γ is the projection of Yh to X. Now, since γ is a finite map, the natural homomorphism H i (Yh , L) −→ H i (X, γ∗ L) = H i (X, E) ∼ PH 0 (X, E). The point is an isomorphism for all i ≥ 0. Therefore, PH 0 (Yh , L) = 0 0 in PH (Yh , L) corresponding to the point σ ∈ PH (X, E) will be denoted by σ  . In particular, σ  is a divisor on Yh . We will calculate the degree of the divisor defined by σ  . The Hitchin map ψ in (2.5) is a complete integrable system for the symplectic structure d on MH and the fiber of ψ over any h ∈ H is the Jacobian of the corresponding spectral curve Yh . Consequently, the genus of Yh coincides with dim MH /2 = 4g − 3 + n. Since degree(L) − (4g − 3 + n) + 1 = χ (L) = χ (E) = d + 2(1 − g), we conclude that degree(L) = d +n+2(g −1). Hence the degree of the divisor defined by the section σ  of L, which coincides with the degree of L, is d + n + 2(g − 1), where n = #S. Set δ = d +n+2(g−1). Let Hilbδ (Z) denote the Hilbert scheme of zero dimensional subschemes of the surface Z (the total space of KX ⊗ OX (S)) of length δ. The divisor of σ  is a zero dimensional subscheme of Yh of length δ. Taking the image of σ  by the inclusion map of the spectral curve Yh in Z, we get a zero dimensional subscheme of Z of length δ. Therefore, we have an element of Hilbδ (Z). Let f : PH −→ Hilbδ (Z)

(3.5)

be the map that sends any parabolic triple to the zero dimensional subscheme of Z defined by the divisor of the corresponding section on the spectral curve for the underlying parabolic Higgs bundle. In other words, f sends any (E∗ , θ , σ ) ∈ PH to the element of Hilbδ (Z) defined by the divisor of σ  on Yh . 3.2. Forms on moduli of triples. Using the map f defined in (3.5) we will construct an algebraic one–form on PH , and for that we will first show the existence of a natural meromorphic one–form on Hilbδ (Z). We start by observing that the variety Z has a natural meromorphic one–form with pole, of order at most one, along the divisor γ −1 (S), where γ , as before is the projection of Z to X. The γ ∗ OX (S) valued one–form sends any tangent vector v ∈ Tz Z to z ⊗ dγ (z)(v), where dγ (z) : Tz Z −→ Tγ (z) X is the differential of γ at z. Note that since z is an element of the fiber (KX ⊗ OX (S))γ (z) , the tensor product z ⊗ dγ (z)(v) gives an element of the fiber (OX (S))γ (z) after contracting (KX )γ (z) with Tγ (z) X. Since

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S is an effective reduced divisor, a γ ∗ OX (S) valued one–form on Z is a meromorphic one–form on Z with a pole along γ −1 (S) of order at most one. The meromorphic one–form on Z defined above will be denoted by Z . Using Z , a meromorphic one–form will be constructed on the Hilbert scheme Hilbk (Z) of zero dimensional subschemes of Z of length k, where k ≥ 1. Consider the Zariski open dense subset Uk ⊂ Hilbk (Z) consisting of distinct k points of Z. Let z = {z1 , z2 , · · · , zk } ∈ Hilbk (Z) be a point of Uk , that is, all zi are distinct. Then we have Tz Hilbk (Z) =

k 

Tzi Z .

i=1

Therefore, the meromorphic one–form Z on Z defines a meromorphic one–form on the Zariski open subset Uk of Hilbk (Z). In other words, this form sends any tangent vector {v1 , v2 , · · · , vk } ∈ Tz Hilbk (Z), where vi ∈ Tzi Z, to k


Z (zi )(vi )

i=1

whenever the sum makes sense. Evidently, this one–form is regular on the complement of the divisor on Uk defined by all points {z1 , z2 , · · · , zk } such that {γ (z1 ), γ (z2 ), · · · , γ (zk )} ∩ S = ∅ as the above sum makes sense on the complement. More precisely, the pole of the one– form on Uk is over this divisor, and the order of the pole is one. Since Uk is a Zariski open dense subset of Hilbk (Z), the meromorphic one–form on Uk defines a meromorphic one–form on Hilbk (Z). The meromorphic one–form on Hilbk (Z) defined above will be denoted by k . Note that Hilb1 (Z) = Z and 1 = Z . Consider the meromorphic one–form f ∗ δ on PH , where the map f is defined in (3.5). It was noted in Sect. 2.3 that for any spectral curve Yh and any s ∈ S, we have γ −1 (s) ∩ Yh = {0} (that is, the spectral curve is totally ramified over s and passes through 0). From this it follows immediately that f ∗ δ is a holomorphic one–form on PH . Indeed, for the origin 0 ∈ (KX ⊗ OX (S))x , where x ∈ X, the form Z vanishes at 0. Therefore, f ∗ δ is a holomorphic one–form on PH . Recall the one–form  = φ ∗  on PH constructed in (3.4). Theorem 3.2. The one–form f ∗ δ on PH coincides with the one–form  . In particular, df ∗ δ coincides with the pullback φ ∗ d of the symplectic form d on MsH . This theorem will be proved in the next section.

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4. Identification of One–Forms We start with the following lemma. Lemma 4.1. There is a holomorphic one–form ω on MsH such that φ ∗ ω coincides with f ∗ δ , where φ is the projection defined in (3.3). Proof. Recall that PH is a projective bundle over MsH with φ being the projection map. Since there is no nonzero holomorphic one–form on a projective space, the pullback of f ∗ δ by the inclusion map of a fiber of φ must vanish identically. Let T φ ⊂ T PH be the relative tangent bundle. In other words, T φ is the kernel of the differential dφ : T PH −→ φ ∗ T MsH . Since the evaluation of f ∗ δ on T φ vanishes, there is a homomorphism ω : T PH /T φ −→ OPH such that f ∗ δ coincides with the composition of the natural projection of T PH to T PH /T φ with the homomorphism ω ; here OPH is the structure sheaf of PH , or equivalently, the sheaf defined by the trivial line bundle. For any point ζ ∈ MsH , the restriction of T PH /T φ to the fiber φ −1 (ζ ) is a trivial vector bundle. In fact, T PH /T φ is identified with the pullback φ ∗ T MsH . Since φ −1 (ζ ) is compact and connected, the homomorphism ω (p) is independent of p ∈ φ −1 (ζ ) (with ζ fixed). In other words, there is a holomorphic one–form ω on MsH such that ω is the pullback of ω. This completes the proof of the lemma.  Recall the one–form  on MsH constructed in Sect. 2.2. Since  = φ ∗  (see (3.4)), in view of Lemma 4.1, to prove Theorem 3.2 it suffices to establish that the two one–forms ω and  on MsH coincide. Consider the Hitchin map ψ defined in (2.5). We want to show that there is a holomorphic one–form H on the Hitchin space H such that  − ω = ψ ∗ H .

(4.1)

To prove the existence of such a form H , take a point h ∈ H such that the corresponding spectral curve Yh is smooth. We recall that there is a nonempty Zariski open subset U of H such that for any point h ∈ U the corresponding spectral curve Yh is smooth. The fiber ψ −1 (h) is identified with the Picard variety Jh := Picd+2(1−g) (Yh ) of degree d + 2(1 − g) line bundles on Yh . (The degree d + 2(1 − g) was computed in the proof of Lemma 3.1.) Let jh : Jh −→ MsH be the inclusion map of the fiber of ψ. From the constructions of  and ω it follows that jh∗  = jh∗ ω. Therefore, exactly as in the proof of Lemma 4.1, we conclude that for any point z ∈ ψ −1 (h), the homomorphism ( − ω)(z) : Tz MsH −→ C factors through the projection dψ(z) : Tz MsH −→ Tψ(z) H defined as the differential of ψ at the point h. Consequently, there is a holomorphic one–form H on U such that  − ω = ψ ∗ H on ψ −1 (U ), where U , as before, is the open subset of H defined by the points corresponding to smooth spectral curves. Since  − ω on ψ −1 (U ) extends to MsH , U is a Zariski open dense subset of H and the map ψ is a submersion everywhere, it follows immediately that H extends to H and the equality in (4.1) is valid on MsH .

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Lemma 4.2. The one–form H on H vanishes identically. Proof. Recall that H = H 0 (X, KX ) ⊕ H 0 (X, KX⊗2 ⊗ OX (S)). Set H := H \ {0}, the nonzero vectors. For any nonzero complex number c, consider the automorphism of H that sends any point (α , β) to (cα , c2 β), where α ∈ H 0 (X, KX ) and β ∈ H 0 (X, KX⊗2 ⊗ OX (S)) . So we have a free action of C∗ on H defined this way. The quotient space Q := H /C∗ is a weighted projective space. Let ρ : H −→ Q

(4.2)

be the quotient map. In Sect. 2.3, given a point of H we constructed a spectral curve, which is a divisor on Z, the total space of KX ⊗ OX (S). We want to describe the above action of C∗ on H in terms of spectral curves. On Z there is an action of C∗ defined by the condition that the action of any c ∈ C∗ sends a point z to cz, where the scalar multiplication is defined by the vector space structure of the fibers of the line bundle KX ⊗ OX (S). It is easy to see that for any h ∈ H and c ∈ C∗ , the spectral curve corresponding to the point ch coincides with the image of the spectral curve corresponding to the point h by the automorphism of Z defined action of c. On the other hand, the meromorphic one–form Z on Z (constructed in Sect. 3.2) evidently has the property that it vanishes along the orbits of C∗ on Z. In other words, for the projection γ of Z to X, the pullback of Z by the inclusion map of a fiber of γ vanishes identically. Furthermore, for any c ∈ C∗ , if Tc denotes the automorphism of Z defined by the multiplication by c, then Tc∗ Z = cZ . From these observations it follows immediately that there is a one–form Q on the weighted projective space Q such that ρ ∗ Q = H on H , where ρ is the projection in (4.2). A weighted projective space does not admit any nonzero holomorphic one–form. Hence we have Q = 0. Since ρ ∗ Q = H , it follows immediately that H = 0, and the proof of the lemma is complete.  The above results clearly combine together to imply Theorem 3.2. Proof of Theorem 3.2. Lemma 4.2 (4.1) together imply that  = ω on MsH . So,  := φ ∗  = φ ∗ ω . Now Lemma 4.1, which says that φ ∗ ω = f ∗ δ , completes the proof of the theorem. 

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4. Faltings, G.: Stable G-bundles and projective connections. J. Alg. Geom. 2, 507–568 (1993) 5. Garc`ıa-Prada, O.: Dimensional reduction of stable bundles, vortices and stable pairs. Int. J. Math. 5, 1–52 (1994) 6. Hitchin, N.: The self–duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59–126 (1987) 7. Hitchin, N.: Stable bundles and integrable systems. Duke Math. J. 54, 91–114 (1987) 8. Mehta, V., Seshadri, C.S.: Moduli of vector bundles on curves with parabolic structure. Math. Ann. 248, 205–239 (1980) Communicated by R.H. Dijkgraaf

Commun. Math. Phys. 240, 161–170 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0904-7

Communications in

Mathematical Physics

Gaps in the Spectrum of the Maxwell Operator with Periodic Coefficients N. Filonov Physical Department, St. Petersburg State University, 1 Ulyanovskaya, Petrodvorets, St. Petersburg 198904, Russia. E-mail: [email protected] Received: 15 December 2002 / Accepted: 13 March 2003 Published online: 1 August 2003 – © Springer-Verlag 2003

Abstract: The periodic Maxwell operator is considered. Piecewise constant coefficients are constructed in such a way that the spectrum of operator has the gaps.

1. Introduction It is well known that the spectrum of a self-adjoint differential operator with periodic coefficients has a band structure. Bands can overlap and fill all the real axis (or semiaxis), or can be separated by the gaps. In this paper we show that occurrence of gaps in the spectrum of periodic Maxwell operator is possible. This question is important for the theory of photonic crystals (see [7]). Many authors studied the possibility of the opening of gaps in the spectra of scalar problems, usually with piecewise constant coefficients ([4–6]). In particular a two-dimensional case of the Maxwell system has been investigated. This case can be reduced to the scalar equations (see [4]). However there were no results in dimension three. We show (see Theorem 2.1 below) that one can find piecewise constant periodic coefficients ε, µ (dielectric and magnetic permeabilities) in such a way that the spectrum of the Maxwell operator has a given finite number of gaps. We use an idea of [9, 10] about specific structure of the operator in the case when the product of coefficients is constant, εµ = const. The absolute continuity of the spectrum of this operator is a by-product of our analysis. Unfortunately our result has little or no physical sense: a hypothetic medium has to consist of four different materials, the ratio of characteristics of which is greater than ten. On the other hand the magnetic permeability µ is always near one. The question of occurrence of gaps with µ = 1 is still an open question. In Sect. 2 we describe the Maxwell operator in R3 and formulate the main result. Then we introduce a quadratic form r in the space of divergence-free functions, which corresponds to the square of the Maxwell operator. We extend it to another form b in the whole space L2 (R3 , C3 ). It is enough to treat the spectrum of the corresponding

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operator. In Sect. 3 the domain of b is investigated. In Sect. 4 we establish some auxiliary general facts. In Sect. 5 we reduce our question to a one-dimensional problem: first to an operator acting in the space L2 (R3 , C) and then to an operator in L2 (R, C). The spectrum of the last operator can be explicitly calculated. We use the following standard notations: H s is the Sobolev space W2s , Dom A and σ (A) are the domain and the spectrum of an operator A. 2. Maxwell Operator in the Whole Space Recall the theory of the Maxwell operator developed in the works [1, 2, 11] and others. Let functions ε, µ be defined in R3 , c0 ≤ ε, µ ≤ c1 , where c0 , c1 are positive constants. Generally speaking one can consider matrix-valued ε, µ, but we restrict ourselves to the scalar coefficients. Let s = ε or s = µ. The space of vector-functions L2 (R3 , C3 , s dx) is decomposed into the sum L2 (R3 , C3 , s dx) = G ⊕s J (s).

(1)

Here G is the closure of the set {∇ϕ : ϕ ∈ C0∞ (R3 , C)} in L2 , J (s) = {u ∈ L2 : div(su) = 0}. The sum (1) is orthogonal with respect to the scalar product with the weight s dx. The Maxwell operator M acts in the space J := J (ε) ⊕ J (µ) with the scalar product
  εu1 , u2 C3 + µv1 , v2 C3 dx. (u1 ; v1 ), (u2 ; v2 )J = R3

The operator M is given by the formula M=



0 −iµ−1 rot

iε−1 rot 0

 (2)

on the domain Dom M = (ε) ⊕ (µ), We can write (see [2]) M=

(s) = {u ∈ J (s) : rot u ∈ L2 }. 

0 R

 R∗ , 0

(3)

(4)

where the operator R = −iµ−1 rot acts from J (ε) to J (µ). So the Maxwell operator is self-adjoint, M = M ∗ . Now we formulate our main result. Theorem 2.1. Let ρ be a parameter, ρ > 1, χ (t) =



1, t ∈ [2kπ, (2k + 1)π ), ρ 2 , t ∈ [(2k − 1)π, 2kπ ),

ε(x) = χ (x1 )χ (x2 )χ (x3 ),

k ∈ N,

(5)

µ(x) = ε(x)−1 .

(6)

Gaps in the Spectrum of the Maxwell Operator with Periodic Coefficients

163

Let the Maxwell operator M be defined by formulas (2), (3), (5),(6). Then √ √ a) the intervals (−1+η, − 3 η) and ( 3 η, 1−η), where η = π2 arctg ρ1 do not contain any point of the spectrum σ (M) if ρ > ctg √π ≈ 1, 54; 2( 3+1) b) for any natural number n the spectrum σ (M) has no less than n gaps for large enough ρ; c) the spectrum of M is absolutely continuous. The remainder of this paper is devoted to the proof of Theorem 2.1. The representation  (4) yields  that the operators M and −M are unitary equivalent (by I 0 the transformation ). So the spectrum of the Maxwell operator is symmetric 0 −I with respect to zero, λ ∈ σ (M) ⇐⇒ −λ ∈ σ (M). Then it is enough to consider its square  ∗ R R M2 = 0

 0 . ∗ RR

(7)

It is easy to see that the kernels ker R, ker R ∗ are trivial. So the operators R ∗ R and RR ∗ are unitarily equivalent, therefore {λ2 : λ ∈ σ (M)} = σ (M 2 ) = σ (R ∗ R). by

(8)

Consider the quadratic form r in the space J (ε) defined on the domain Dom r = (ε)
µ−1 | rot u|2 dx. r[u] = Ru 2µ = R3

R ∗ R.

It corresponds to the positive operator The functions in J (ε) satisfy the relation div(εu) = 0. We develop now an idea from [3] to remove this relation. Consider in L2 (ε dx) the quadratic form a on the domain Dom a = {u ∈ L2 (R3 , C3 ) : rot u ∈ L2 , div(εu) ∈ L2 },

−1 2 a[u] = µ | rot u| dx + α| div(εu)|2 dx, R3

R3

where α is a positive bounded function. The restriction of a on the subspace J (ε) coincides with the form r. The following lemma is immediate. Lemma 2.1. Let u ∈ Dom a, v = PJ (ε) u, w = PG u, where PJ (ε) and PG are orthoprojectors in L2 on the corresponding subspaces. Then u = v + w,

v, w ∈ Dom a,

a[v, w] = 0.

Thus the decomposition (1) is orthogonal with respect to form a also. Hence the selfadjoint operator A corresponding to the positive form a is an extension of the operator R ∗ R, (9) A = R ∗ R ⊕ A0 , here A0 is an operator in the subspace G. Therefore it is enough to investigate the spectrum of A.

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√ √ Let us next consider the operator of multiplying by ε, (U u)(x) = ε(x)u(x). It is an isometric isomorphism of the space with weight L2 (ε dx) on the space without weight, U : L2 (ε dx) → L2 (dx). In L2 (dx) we define a form b,

−1 −1 −1/2 2 b[f ] := a[U f ] = µ | rot(ε f )| dx + α| div(ε1/2 f )|2 dx (10) R3

R3

on the domain Dom b = {f ∈ L2 (R3 , C3 ) : rot(ε−1/2 f ) ∈ L2 , div(ε1/2 f ) ∈ L2 }. The corresponding operator B acts in L2 B = U AU ∗ ,

(R3 , C3 )

(11)

and

σ (R ∗ R) ⊂ σ (A) = σ (B).

(12)

Thus in what follows we consider just the form b. 3. Set Dom b Recall that the functions ε, µ are defined by formulas (5), (6). The class of functions Dom b permits an explicit description. Divide R3 into the cubes Qn with faces parallel to the coordinate planes, Qn = (n1 π, (n1 + 1)π ) × (n2 π, (n2 + 1)π ) × (n3 π, (n3 + 1)π ), n = (n1 , n2 , n3 ) ∈ Z3 . Note that ε and µ are constant on each cube Qn . We will use the following conditions of correlation for the functions h such that h |Qn ∈ H 1 (Qn ). Note by (e1 , e2 , e3 ) the standard basis in R3 . Condition 3.1. If S⊥x1 is a common face of cubes Qn and Qn+e1 , then the traces of functions (ε1/2 h) |Qn and (ε 1/2 h) |Qn+e1 on S⊥x1 coincide. If S⊥xj , j = 2, 3, is a common face of cubes Qn and Qn+ej , then the traces of functions (ε −1/2 h) |Qn and (ε−1/2 h) |Qn+ej on S⊥xj coincide. Condition 3.2. If S⊥x2 is a common face of cubes Qn and Qn+e2 , then the traces of functions (ε1/2 h) |Qn and (ε1/2 h) |Qn+e2 on S⊥x2 coincide. If S⊥xj , j = 1, 3, is a common face of cubes Qn and Qn+ej , then the traces of functions (ε −1/2 h) |Qn and (ε−1/2 h) |Qn+ej on S⊥xj coincide. Condition 3.3. If S⊥x3 is a common face of cubes Qn and Qn+e3 , then the traces of functions (ε1/2 h) |Qn and (ε 1/2 h) |Qn+e3 on S⊥x3 coincide. If S⊥xj , j = 1, 2, is a common face of cubes Qn and Qn+ej , then the traces of functions (ε −1/2 h) |Qn and (ε−1/2 h) |Qn+ej on S⊥xj coincide. Introduce the following notations: Hˆ 1 = {f ∈ L2 (R3 , C3 ) : f |Qn ∈ H 1 (Qn ),


n∈Z3

|∇f |2 dx < ∞, Qn

fi satisf ies condition (3.i), i = 1, 2, 3}, Here |∇f |2 =

3

Cˆ ∞ = {f ∈ Hˆ 1 : f |Qn ∈ C ∞ (Qn )}.

i,j =1 |∂i fj |

2.

Gaps in the Spectrum of the Maxwell Operator with Periodic Coefficients

165

Lemma 3.1. The sets Dom b and Hˆ 1 coincide. Proof. The inclusion Hˆ 1 ⊂ Dom b is clear. Show that if f ∈ Dom b then f ∈ Hˆ 1 . A partition of unity reduces the problem to the model case when f is supported in a ball with center in a vertex of some cube Qn and with radius less than the length of edge. Let supp f ⊂ K3 := {x ∈ R3 : |x| < 3}, ε(x) = ρ 2(θ(−x1 )+θ(−x2 )+θ(−x3 )) , where θ (t) = 1 if t > 0, θ (t) = 0 if t < 0, and rot(ε−1/2 f ), div(ε 1/2 f ) ∈ L2 (K3 ). It is enough to prove that f |K+ ∈ H 1 (K+ ),

K+ := {x ∈ K3 : xi > 0, i = 1, 2, 3}.

Consider in K+ functions f

ω1 ω2 ω3

(x) =

ω1 f1 (ω1 x1 , ω2 x2 , ω3 x3 ) ω2 f2 (ω1 x1 , ω2 x2 , ω3 x3 ) ω3 f3 (ω1 x1 , ω2 x2 , ω3 x3 )

,

where ωi are 1 or −1. Put F 1 2 3 =



1−ω1 2

1

1−ω2 2

2

1−ω3 2

3

f ω1 ω2 ω3 ,

(13)

ωi =±1

where i takes the values 1/ρ or −ρ. The integrals over K+ of squares of rot and div of eight functions F 1 2 3 are finite. The coefficients in the decomposition (13) are chosen in such a way that a tangent (normal) component of F 1 2 3 is zero on the face xi = 0 in a generalized sense if i = −ρ (if i = 1/ρ). Reference to [1, 2] shows hence that all functions F 1 2 3 belong to H 1 (K+ ). Ensembles of coefficients in (13) are linear independent so f ω1 ω2 ω3 ∈ H 1 (K+ ).     Lemma 3.2. Cˆ ∞ is dense in Hˆ 1 with respect to metrics n Qn |∇f |2 dx. Proof. With reference to a partition of unity as in Lemma 3.1 one can consider only a model situation. Let f ∈ Hˆ 1 and supp f ⊂ K3 . We have to approximate each component fi by smooth functions satisfying Condition 3.i. For definiteness consider f1 . The function g(x) = ρ −θ(x1 )+θ(x2 )+θ(x3 ) f1 (x) ◦

belongs to H 1 (K3 ). If G is a smooth function which is near g in the H 1 -norm, then the function ρ θ(x1 )−θ(x2 )−θ(x3 ) G(x) approximates f1 and satisfies Condition 3.1.   Choose the function α in the definition (10) as α = µ = ε −1 . Then we can establish a more convenient representation of the form b. Lemma 3.3. For f ∈ Cˆ ∞ we have b[f ] =


n∈Z3

|∇f |2 dx. Qn

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Proof. The equality α = µ = ε−1 yields  
 | rot f |2 + | div f |2 dx b[f ] = n∈Z3

=



Qn








n∈Z3

|∇f |2 dx Qn


+ 2 Re









∂i fi ∂j fj − ∂i fj ∂j fi dx  .

(14)

Qn (i;j )=(1;2),(2;3),(3;1)

Furthermore
Qn



 ∂i fi ∂j fj − ∂i fj ∂j fi dx =





Sjn+

∂i fi fj dS −

n Si+

∂i fi fj dS







−

Sjn−

∂j fi fj dS +

n Si−

∂j fi fj dS,

n (S n ) is “upper” (“lower”) face of Q orthogonal to x . All the integrals over where Si+ n i i− n in the summation in (14) can be cancelled by virtue of Conditions 3. faces Si±  

We summarize the results of this section: Theorem 3.1. Let ε, µ be defined by (5), (6), α = µ = ε −1 and the form b by formulas (10), (11). Then 
and b[f ] = |∇f |2 dx ∀f ∈ Dom b. Dom b = Hˆ 1 n∈Z3

Qn

4. Some Facts of Abstract Operator Theory First of all recall the spectral theorem in the multiplication operator form. Theorem 4.1. Any self-adjoint operator T is unitary equivalent to the operator of multiplication by a real finite a.e. function f in the space L2 (), where  is a space with a finite measure dm. If we note
|ϕ(x)|2 dm(x) F (λ, ϕ, T ) = x∈:f (x)<λ

for ϕ ∈ L2 (), then F is a nondecreasing function of λ, F (−∞, ϕ, T ) = 0, F (+∞, ϕ, T ) = ϕ 2 < ∞. The spectrum of the operator T is absolutely continuous if and only if F (λ, ϕ, T ) is an absolutely continuous function of a real variable λ for all ϕ ∈ L2 (). Lemma 4.1. Let M = M ∗ be a self-adjoint operator in a Hilbert space H . The spectrum of M is absolutely continuous if and only if the spectrum of M 2 is absolutely continuous.

Gaps in the Spectrum of the Maxwell Operator with Periodic Coefficients

167

Proof. In both cases ker M = {0}. There exists a decomposition H = H+ ⊕ H− such that subspaces H± are invariant with respect to M and the operators ±M |H± are positive. Thus it is enough to prove the assertion only for positive operators, M > 0. By Theorem 4.1 M is unitarily equivalent to multiplication by a function f in L2 () with f > 0 a.e. in ; the operator M 2 is unitarily equivalent to multiplication by f 2 . The equality F (λ, ϕ, M) = F (λ2 , ϕ, M 2 ), ϕ ∈ L2 (), yields the result.   Lemma 4.2. Let k = 1, ..., n, Tk = Tk∗ be a self-adjoint operator in a Hilbert space Hk with the domain Dom Tk . Let Ik be an identical operator in Hk and  = T1 ⊗ I2 ⊗ ... ⊗ In + I1 ⊗ T2 ⊗ ... ⊗ In + I1 ⊗ I2 ⊗ ... ⊗ Tn be the operator in the space H1 ⊗ ... ⊗ Hn with domain Dom T1 ⊗ ... ⊗ Dom Tn . Then a) the operator  is essentially self-adjoint and σ () = {λ = λ1 + ... + λn : λk ∈ σ (Tk )}; b) if the spectrum of T1 is absolutely continuous then the spectrum of  is also absolutely continuous. Proof. For part a) see [8] (Corollary of Theorem VIII.33). The proof of part b) is based on the same idea. It is enough to consider the case of two terms, n = 2. By virtue of Theorem 4.1 we can assume that Hk = L2 (k , dmk ) and Tk is the operator of multiplication by fk . The function F (λ, ϕ, T1 ) is an absolutely continuous function of λ for all ϕ ∈ L2 (1 ), i.e. ∀ε > 0∃δ > 0: if

p1 < q1 < ... < pN < qN

and

N 

(qj − pj ) < δ,

j =1

then

N  

 F (qj , ϕ, T1 ) − F (pj , ϕ, T1 ) < ε.

j =1

The operator  is the operator of multiplication by function (f1 (x) + f2 (y)) in the space L2 (1 × 2 , dm1 × dm2 ). We have to consider the function
|ω(x, y)|2 dm1 (x)dm2 (y), (15) F (λ, ω, ) = (x;y):f1 (x)+f2 (y)<λ

ω ∈ L2 (1 × 2 ). If ω(x, y) = ϕ1 (x)ϕ2 (y), ϕk ∈ L2 (k ), then
F (λ − f2 (y), ϕ1 , T1 )|ϕ2 (y)|2 dm2 (y). F (λ, ϕ1 ϕ2 , ) = 2

Now for all ε > 0 there exists δ > 0 such that if N  

N

j =1 (qj

− pj ) < δ then

 F (qj − f2 (y), ϕ1 , T1 ) − F (pj − f2 (y), ϕ1 , T1 ) <

j =1

⇒

N  

ε

ϕ2 2

 F (qj , ϕ1 ϕ2 , ) − F (pj , ϕ1 ϕ2 , ) < ε.

j =1

∀y ∈ 2

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Furthermore the absolute continuity of F (λ, ω, ) for a linear combination of type ω=

m 

(16)

ϕk (x)ψk (y)

k=1

is a corollary of the inequality m



m   F q, ϕk ψk ,  − F p, ϕk ψ k ,  k=1 m 

≤m

k=1

(F (q, ϕk ψk , ) − F (p, ϕk ψk , )) .

k=1

Finally the absolute continuity of function (15) for arbitrary ω ∈ L2 (1 × 2 ) follows from the density of the set of linear combination (16) in this space and from the inequality N  

 F (qj , ω, ) − F (pj , ω, )

j =1

 N    ≤ 2 ˜ ) − F (pj , ω, ˜ ) + ω − ω ˜ 2L F (qj , ω,

2 (1 ×2 )

j =1

 .

 

5. Reduction to a One-Dimensional Problem Theorem 3.1 yields that the “vector” form b can be decomposed into three “scalar” forms bi , i = 1, 2, 3, Dom bi = {fi ∈ L2 (R3 , C) : fi |Qn ∈ H 1 (Qn ), fi satisf ies 3.i, 
bi [fi ] := |∇fi |2 dx < ∞}. n∈Z3

Qn

Then b = b1 ⊕ b2 ⊕ b3 and B = B1 ⊕ B2 ⊕ B3 , where Bi is the operator corresponding to bi , Dom Bi = {g ∈ L2 (R3 , C) : g |Qn ∈ H 2 (Qn ), g satisf ies 3.i, traces of (ε−1/2 ∂i g) |Qn and (ε −1/2 ∂i g) |Qn+ei on S⊥xi coincide, traces of (ε1/2 ∂i g) |Qn and (ε 1/2 ∂i g) |Qn+ej on S⊥xj coincide, j = i, 
|g|2 dx < ∞}, n∈Z3

Qn

  (Bi g) |Qn = − g |Qn .

The operators B1 , B2 , B3 are unitarily equivalent, so σ (B) = σ (B1 ) ∪ σ (B2 ) ∪ σ (B3 ) = σ (B1 ). The operators Bi act on different variables separately by virtue of (6). Consider in L2 (R) the self-adjoint operator T given on the domain

Gaps in the Spectrum of the Maxwell Operator with Periodic Coefficients

169

Dom T = {ψ ∈ L2 (R, C) : ψ |(kπ,(k+1)π) ∈ H 2 (kπ, (k + 1)π ), ψ(2kπ − 0) = ρψ(2kπ + 0),

ψ  (2kπ − 0) = ρ −1 ψ  (2kπ + 0),

ψ((2k + 1)π − 0) = ρ −1 ψ((2k + 1)π + 0), ψ  ((2k + 1)π − 0) = ρψ  ((2k + 1)π + 0), 
(k+1)π |ψ  |2 dx < ∞}

k ∈ Z,

(17)

k∈Z kπ

by the formula

  (T ψ) |(kπ,(k+1)π) = − ψ |(kπ,(k+1)π) .

(18)

Now Lemma 4.2 says that the operator T ⊗ I ⊗ I + I ⊗ T ⊗ I + I ⊗ I ⊗ T is essentially self-adjoint on the domain Dom T ⊗ Dom T ⊗ Dom T . On the other hand it is contained in the self-adjoint operator W ∗ B1 W , where W is the operator of shift along x1 , (Wg)(x) = g(x1 + π, x2 , x3 ). Hence W ∗ B1 W coincides with its closure. Therefore σ (B) = {λ = λ1 + λ2 + λ3 : λi ∈ σ (T )}.

(19)

It remains to calculate the spectrum of T . Recall that the matrix of monodromy on an interval (p, q) for a differential equation of second order is the matrix which transforms Cauchy conditions of the solution on the left end (ψ(p), ψ  (p)) into the conditions on the right end (ψ(q), ψ  (q)). For example, the monodromy matrix for the equation −ψ  = λψ on (0, π ) 

is V0 (λ) =

√ √  √ sin( λπ cos( λπ √ √) √ )/ λ . − λ sin( λπ ) cos( λπ )

The trace of the monodromy matrix for our equation T ψ = λψ, λ > 0, on a period is equal to     −1  0 ρ ρ 0 tr V (λ) = tr V0 (λ) V0 (λ) −1 0 ρ 0 ρ √ 2 −2 2 = 2 − (ρ + ρ + 2) sin ( λπ ). A number λ belongs to the spectrum σ (T ) if and only if | tr V (λ)| ≤ 2. In our case it means √ 4ρ 2 sin2 ( λπ ) ≤ 2 . (ρ + 1)2 Thus we proved the following Lemma 5.1. Let T be the operator defined by (17), (18). Then

∞     2 2 2 σ (T ) = [0, η ] (k − η) , (k + η) , k=1

where η=

1 2ρ 2 1 arcsin 2 = arctg . π ρ +1 π ρ

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Now we can complete the Proof of Theorem 2.1. a) is a corollary of Lemma 5.1 and formulas (8), (12), (19). b) Choose a parameter η little enough (for example η < (13N )−1 ). Then the difference of any number in the set σ (T ) ∩ [0, N ] and the nearest integer will be lesser than 1/6. Therefore the numbers of type m + 1/2 with integer m < N will be out of the spectrum σ (B). c) The spectrum of the operator B1 (and therefore the spectrum of B) is absolutely continuous by virtue of Lemma 4.2. The formulas (7), (9) and (12) imply the absolute continuity of the spectrum of M 2 . It remains to refer to Lemma 4.1.   Acknowledgements. The author is grateful to l’Universit´e Paris Nord where the basic result was obtained, and very thankful to Prof. F. Klopp for helpful discussions and to Prof. T. Suslina who has given me the text of [9] before its publication. The work was supported by FNS 2000 “Programme Jeunes Chercheurs” under the Project “M´ethodes math´ematiques en physique de la mati`ere condens´ee”.

References 1. Birman, M., Solomyak, M.: The Maxwell operator in domains with nonsmooth boundary. Siberian Math. J. 28(1), 12–24 (1987) 2. Birman, M., Solomyak, M.: L2 -theory of the Maxwell operator in arbitrary domains. Russ. Math. Surv. 42(6), 75–96 (1987) 3. Birman, M., Solomyak, M.: Weyl asymptotics of the spectrum of the Maxwell operator for domain with a Lipschitz boundary. Vestnik Leningrad. Univ. Math. 20(3), 15–21 (1987) 4. Figotin, A., Kuchment, P.: Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model. SIAM J. Appl. Math. 56, 68–88 (1996); II. Two-dimensional photonic crystals. SIAM J. Appl. Math. 56, 1561–1620 (1996) 5. Friedlander, L.: On the density of states of periodic media in the large coupling limit. Comm. Partial Diff. Eqs. 27, 355–380 (2002) 6. Hempel, R., Lienau, K.: Spectral properties of the periodic media in large coupling limit. Comm. Partial. Diff. Eqs. 25, 1445–1470 (2000) 7. Kuchment, P.: The mathematics of photonic crystals. In: Mathematical Modeling in Optical Science, Frontiers in Applied Mathematics 22, Philadelphia, PA: SIAM, 2001, Chap. 7, pp. 207–272 8. Reed, M., Simon, B.: Methods of modern mathematical physics. Vol. I, New York: Academic Press, 1978 9. Suslina, T.: Absolute continuity of the spectrum of the periodic Maxwell operator in a layer. Zap. Nauch. Sem. POMI (Russian) 288, 232–255 (2002), English translation to appear 10. Suslina, T.: Absolute continuity of the spectrum of periodic operators of mathematical physics. Journ´ees Equations aux d´eriv´ees partielles. Nantes, 5–9 juin 2000 11. Weck, N.: Maxwell’s boundary value problem on Riemannian manifolds with nonsmooth boundaries. J. Math. Anal. Appl. 46(2), 410–437 (1974) Communicated by B. Simon

Commun. Math. Phys. 240, 171–196 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0894-5

Communications in

Mathematical Physics

R-Operator, Co-Product and Haar-Measure for the Modular Double of Uq (sl(2, R)) A.G. Bytsko, J. Teschner Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany. E-mail: [email protected]; [email protected] Received: 25 October 2002 / Accepted: 16 March 2003 Published online: 25 July 2003 – © Springer-Verlag 2003

Abstract: A certain class of unitary representations of Uq (sl(2, R)) has the property of being simultanenously a representation of Uq˜ (sl(2, R)) for a particular choice of q(q). ˜ Faddeev has proposed to unify the quantum groups Uq (sl(2, R)) and Uq˜ (sl(2, R)) into some enlarged object for which he has coined the name “modular double”. We study the R-operator, the co-product and the Haar-measure for the modular double of Uq (sl(2, R)) and establish their main properties. In particular it is shown that the Clebsch-Gordan maps constructed in [PT2] diagonalize this R-operator. 1. Introduction Quantum groups have become an indispensable tool in many areas of mathematical physics and mathematics. In a broad class of quantum theoretical models it has turned out that finding a relation to a quantum group is the key for obtaining exact information about the spectrum or the correlation functions. So far most of the vast amount of work devoted to quantum group theory and their physical applications was concerned with quantum groups that can be studied in a purely algebraic manner. This is the case e.g. if the relevant representations are highest weight representations, as is often assumed. However, in physical applications to quantum theoretical models the choice of a scalar product on the space of states usually determines the hermiticity relations for the representatives of the quantum group generators. In many cases like those corresponding to the so-called non-compact quantum groups it turns out that the corresponding unitary representations are always infinite-dimensional and generically neither of highest nor lowest weight type. In order to exploit the information provided by the appearance of such a quantum group it is clearly important to have efficient mathematical tools for analyzing the corresponding representation theory. Unfortunately there are comparatively few results about the representation theory of non-compact quantum groups. This seems to be an important obstacle for making

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further progress in many quantum integrable models. Nevertheless there is an interesting example that was first studied independently in [PT1, F2 and PT2]. These references were considering a particular class of infinite dimensional unitary representations of Uq (sl(2, R)). This class of representations, henceforth denoted Ps , s ∈ R, was also the first to be used in a concrete physical application: Understanding the tensor products of the above-mentioned representations was crucial for obtaining exact results on the quantum Liouville theory [PT1, T]. A rather remarkable duality phenomenon was observed in [F1, PT1 and F2]. The representations in question are simultaneously representations of the two quantum groups 2

πi

Uq (sl2 ) and Uq˜ (sl2 ) with deformation parameters q = eπib and q˜ = e b2 respectively. This duality turns out to be deeply related to the quantum field theoretical self-duality of Liouville theory under the change of the coupling constant b into b−1 [T]. Moreover, it is this duality under b → b−1 that allows one to cover the so-called strong-coupling regime where |b| = 1 by analytic continuation of the results for real values of b [PT1, FKV, T, FK2]. The results of the present paper clarify the origin of this phenomenon to a certain extent. Given the operators X representing one of the two algebras, say Uq (sl2 ), one may obtain the representatives of the second algebra Uq˜ (sl2 ) as nonpolynomial operator functions of the operators X. This is found to be consistent with the respective co-products. In particular, restricting attention to only one of the two algebras does not lead to any degeneracy as is sometimes suggested in the literature. Faddeev has proposed to unite the quantum groups Uq (sl2 ) and Uq˜ (sl2 ) into some enlarged object for which he has coined the name “modular double”. The proposal of [F2] as refined in [KLS] amounts to defining it as the product of Uˆ q (sl2 ) and Uˆ q˜ (sl2 ), where, roughly speaking, Uˆ q (sl2 ) is obtained from Uq (sl2 ) by adjoining a sign to the center generated by the Casimir of Uq (sl2 ). We feel that this definition for the modular double has a disadvantage, though. Most representations of Uˆ q (sl2 ) ⊗ Uˆ q˜ (sl2 ) are simply tensor products of representations of the two factors. The representations Ps on the contrary are distinguished by the fact that they do not factorize as a tensor product of representations for Uq (sl2 ) and Uq˜ (sl2 ). This is what makes the duality under b → b−1 a nontrivial statement. Since the category formed by the representations Ps is closed under tensor products [PT2] it seems natural to look for the group-like object that contains the interesting representations Ps only. We therefore propose to look for a definition of the “modular double” that excludes the representations of Uq (sl2 )⊗Uq˜ (sl2 ) which factorize trivially. This definition should still capture the duality phenomenon mentioned above. As we have indicated, this naturally leads us to consider nonpolynomial functions of the generators. The basic objects underlying our approach to the modular double will be an algebra A of bounded operators, a coproduct  on A, an invariant integration (Haar-measure) on A and the R-operator proposed in [F2]. We are going to establish the main properties satisfied by these objects, which are all self-dual under b → b−1 . The algebra A can be thought of as being generated from operators that represent Uq (sl2 ) in a similar (in fact, closely related) way as the algebra of bounded operators on L2 (R) is related to the usual quantum mechanical position and momentum operators x and p. Our point of view is inspired by the one of Woronowicz [W1], which has stimulated the development of a theory for noncompact quantum groups in a C ∗ -algebraic framework, see e.g. [KV] and references therein. However, although we believe that our results represent substantial progress towards a proof that the modular double fits into such a C ∗ -algebraic framework, it was not our aim to actually carry out such a proof here.

R-Operator, Co-Product and Haar-Measure for the Modular Double

173

We also clarify the relation between the R-operator proposed in [F2] and the calculus of Clebsch-Gordan and Racah-Wigner-coefficients of [PT2]. Establishing this relation is important for the following reason. In [T] it was shown that certain families of representations of the Virasoro algebra and of the quantum group Uq (sl(2, R)) behave equivalently under the respective product operations (fusion and tensor product). Together with the results of the present paper it follows that the respective braiding operations are equivalent as well. To be specific we will mostly consider the case that the deformation parameter is of 2 the form q = eπ ib , where b ∈ (0, 1). However, our results will carry over to the “strong coupling regime” |b| = 1, see the remarks in Subsect. 2.11. Note added. The referee has brought to our attention the work [KK] where a quantum group related to SU (1, 1) was studied. These papers address similar issues as treated in our paper. It would be rather interesting to understand if there is a more direct relation between the example studied in [KK] (where the 2 deformation parameter q is real) and the example studied in our paper (where q = eπib , (1−|b|)b = 0).

2. Definitions and Main Results 2.1. Star algebra Uq (sl(2, R)). Uq (sl(2, R)) is a Hopf-algebra with generators: E,

F,

K,

K −1 ;

relations: KE = qEK,

KF = q −1 F K,

star-structure: K ∗ = K,

E ∗ = E,

[E, F ] =

F∗ = F .

K 2 − K −2 ; q − q −1 (2.1)

The center of Uq (sl(2, R)) is generated by the q-Casimir C = FE +

qK 2 + q −1 K −2 − 2 . (q − q −1 )2

(2.2)

Compared to the definition used in [PT2] we have redefined F → −F . This will allow us to realize F by positive operators.

2.2. The representations Ps of Uq (sl(2, R)). In the present paper we will study a oneparameter class Ps , s ∈ R, of representations of Uq (sl(2, R)). They are constructed as follows: The representation will be realized on the space Ps of entire analytic functions f (x) that have a Fourier-transform f˜(ω) which is meromorphic in C with the possible poles contained in Ss ≡




±ω = s+i

Q 2

  + nb + mb−1 , n, m ∈ Z≥0 ,

(2.3)

where Q = b + b−1 . The representation of Uq (sl(2, R)) on Ps is then defined by choosing the representatives πs (X) for X = E, F, K to be the following finite difference operators

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A.G. Bytsko, J. Teschner

πs (E) ≡ Es ≡ e+πbx πs (F ) ≡ Fs ≡ e

−πbx

cosh πb(p − s) +πbx e sin πb2 cosh πb(p + s) −πbx e sin πb2

πs (K) ≡ Ks ≡ e−πbp , (2.4)

where p and x are self-adjoint operators satisfying [p, x] = (2π i)−1 . By embedding Ps as a dense subspace into the Hilbert space L2 (R) one obtains a unitary representation of Uq (sl(2, R)) generated from the self-adjoint operators E, F and K [S]. 2.3. The representations H and K of Uq (sl(2, R)). We will find it convenient to formulate our results in a “universal” setting. Let us define K ≡ L2 (R × R). The algebra B(K) of bounded operators on K is generated by two pairs (xi , pi ), i = 1, 2 satisfying [pi , xi ] = (2π i)−1 . The action of Uq (sl(2, R)) on K is defined by πK (E) ≡ E = eπb(x1 −x2 ) πK (F ) ≡ F = eπb(x2 −x1 )

cosh πbp2 πb(x1 −x2 ) e , sin πb2 cosh πbp1 πb(x2 −x1 ) e , sin πb2

πK (K) ≡ K = e

πb 2 (p2 −p1 )

.

(2.5)

This representation of Uq (sl(2, R)) on K is reducible: The operator s = 21 (p1 + p2 ) commutes with E, F, K and determines the representation of the Casimir via C ≡ πK (C) =

cosh2 πbs . sin2 πb2

(2.6)

The action of (2.5) on an eigenspace of s reduces to the action (2.4) on Ps upon identification p = 21 (p1 − p2 ) and x = x1 − x2 . This means that K decomposes into the representations Ps as follows:  ⊕ K  ds Ps . (2.7) R

The representations Ps and P−s are unitarily equivalent [PT2]: There exists a unitary operator Js : Ps → P−s such that X−s Js = Js Xs for all X ∈ Uq (sl(2, R)). The operator Js defines an operator J : K → K if one considers the operator function J ≡ Js . It will sometimes be convenient to consider instead of K a space H in which Ps and P−s are identified: H = {v ∈ K; (id − J)v = 0}. ⊕ H is of course isomorphic to R+ ds Ps . 2.4. Operator functions of E, F, K. It is important to also consider nonpolynomial functions of the operators E, F, K. Let us first note that standard functional calculus for positive selfadjoint operators allows one to consider complex powers of the generators such as Eγ , γ ∈ C. The following result offers a partial explanation for the phenomenon of modular duality.

R-Operator, Co-Product and Haar-Measure for the Modular Double

Lemma 1.

175

˜ s , F˜ s , K˜ s obtained by replacing b → b−1 in (2.4), (i) The operators E −1 (p − s) −1 cosh πb −1 e+πb x E˜ s ≡ e+πb x sin πb−2 −1 (p + s) −1 cosh πb −1 e−πb x F˜ s ≡ e−πb x sin πb−2

−1 K˜ s ≡ e−πb p ,

(2.8)

−2 generate a representation of Uq˜ (sl(2, R)) with q˜ = eπib . The generators E˜ s , F˜ s , K˜ s commute with the operators Es , Fs , Ks on Ps . (ii) For γ = b−2 we have γ  = sin(π b−2 ) E˜ s , sin(π b2 )Es γ (2.9) Ks = K˜ s .   γ 2 −2 ˜ sin(π b )Fs = sin(π b ) Fs ,

Being operator functions of Es , Fs , Ks , the operators E˜ s , F˜ s , K˜ s do not commute with Es , Fs , Ks in the usual sense (commutativity of the spectral projections). We shall now define an algebra of bounded operators that can be considered as operator functions of E, F, K. To begin with, let Os , s ∈ R be a family of bounded operators on L2 (R) such that sup Os < ∞.

s∈R

A bounded operator O on K can be defined for each such family (Os )s∈R by means of (2.7). These operators O form a subalgebra B0 of the algebra of all bounded operators on H. Let B be the C ∗ subalgebra obtained as the completion of B0 w.r.t. the operator norm. This algebra can be thought of as being generated from the unbounded elements p, x, s. However, there is no canonical way to define sgn(s) as a function of E, F, K. The center of the algebra of bounded operators generated from E, F, K should be generated from operator functions of the Casimir, or equivalently |s|, cf. Eq. (2.6). This is closely related to the fact that the representations Ps and P−s are unitarily equivalent. Elements of the “true” algebra A ⊂ B should therefore commute with the operator J which establishes the equivalence between Ps and P−s , A ≡ {O ∈ B; J−1 OJ = O}.

(2.10)

This amounts to considering only those elements of B that leave H invariant. 2.5. The Hopf algebra structure. A co-product is defined on Uq (sl(2, R)) via (E) =E ⊗ K + K −1 ⊗ E ,

(K) = K ⊗ K .

(F ) =F ⊗ K + K −1 ⊗ F ,

(2.11)

In the following we shall adopt the convention to denote (X) ≡ (πH ⊗ πH ) ◦ (X)

for

X ∈ Uq (sl(2, R)) .

(2.12)

It follows from [PT2, Theorem 2] that (E), (F) and (K) are self-adjoint and positive and therefore generate a representation of Uq (sl(2, R)) on H⊗H. The following Lemma proven in Sect. 3 establishes consistency of the co-product with modular duality:

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A.G. Bytsko, J. Teschner

Lemma 2. The co-product (2.11),(2.12) acts on the dual part of the modular double as follows: ˜ = E ˜ ⊗ K˜ + K˜ −1 ⊗ E, ˜ (E) ˜ ˜ = F˜ ⊗ K˜ + K˜ −1 ⊗ F, (F)

˜ = K˜ ⊗ K. ˜ (K)

(2.13)

A representation of the co-product on the algebra A can be defined by means of the Clebsch-Gordan maps defined in [PT2]. These maps yield a three parameter family of maps C[s3 |s2 , s1 ] : Ps2 ⊗ Ps1 → Ps3 that satisfy the intertwining property C[s3 |s2 , s1 ] ◦ (πs2 ⊗ πs1 ) ◦ (X) = πs3 (X) ◦ C[s3 |s2 , s1 ]

(2.14)

and extends to a two-parameter family of unitary operators C[s2 , s1 ] : L2 (R2 ) → H. Let us introduce the operators s1 = id ⊗ s, s2 = s ⊗ id on H ⊗ H respectively. The identification (2.7) allows us to consider C[s2, s1] as a unitary operator C : H ⊗ H → H ⊗ Hspec , where the operators si , i = 1, 2 are realized on the space Hspec  L2 (R+ × R+ ) as multiplication operators. For each element X ∈ A we may now define (X) by   (X) ≡ C† ◦ X ⊗ id ◦ C.

(2.15)

Since C is unitary and X is bounded we clearly have boundedness of (X) : H ⊗ H → H ⊗ H. Theorem 1. The coproduct  is coassociative on A, i.e. (id ⊗ ) ◦ (X) = ( ⊗ id) ◦ (X) for any X ∈ A. The antipode consistent with (2.11) is defined as an anti-automorphism of Uq (sl(2, R)) such that σ (K) = K −1 ,

σ (E) = −qE ,

σ (F ) = −q −1 F .

(2.16)

The action of the antipode on nonpolynomial functions of E, F and K can be introduced by means of σ (p) = −p,

σ (s) = −s,

σ (x) = x + 2i Q.

(2.17)

The fact that x is shifted by an imaginary amount means that σ is not defined on all of A. This unboundedness of the antipode is not unexpected [KV].

R-Operator, Co-Product and Haar-Measure for the Modular Double

177

2.6. The Haar-measure. Let us first note that the decomposition (2.7) induces a family of projections πs : A → B(L2 (R)). We shall often use the shorthand notation Os ≡ πs (O). Definition 1. Define linear functionals hl and hr on dense subsets Alh and Arh of A respectively by ∞ hl (O) = 0 dm(s) Tr(e−2πQp Os ), (2.18) ∞ +2πQp hr (O) = 0 dm(s) Tr(e Os ), where the measure m is defined by dm(s) ≡ 4 sinh 2πbs sinh 2π b−1 s ds. Theorem 2.

(2.19)

(i) The Haar-measures hl and hr are left and right invariant respectively, (id ⊗ hl ) ◦ (O) = hl (O) id ,

(2.20)

(hr ⊗ id) ◦ (O) = hr (O) id , where we assume O to be taken from the respective domains of definition. (ii) For any X ∈ Uq (sl(2, R)), the Haar-measures satisfy, respectively hl (adXl O) = hl (O) (X) , hr (adXr O)

(2.21)

= hr (O) (X) ,

where (X)is the co-unit, and the left and actions are as right q-adjoint  defined l (Y ) =  Y σ (X  ) and adr (Y ) =  )Y X  if (X) =  ⊗ X  . adX X σ (X X i i i i i X i i i i Remark 1. We believe that the triple (A, , h) that we have defined above constitutes a somewhat more satisfactory definition of the modular double, although more work is needed to show that it fits into the axiomatics for noncompact quantum groups of [KV]. The self-duality under b → b−1 is manifest in this formulation. It also becomes clear that the modular double can not be considered as a deformation of a classical group: The Haar-measure has no classical limit b → 0 due to the factor Q = b + b−1 that appears in the definition of h. 2.7. The R-operator. To begin with, we introduce the special function gb (x) that will be used to define the R-operator. It may be defined via (recall that Q = b + b−1 )  log gb (x) = − R+i0

tQ

t

dt e 2 x 2π ib . t (1 − ebt )(1 − et/b )

(2.22)

Let us furthermore introduce an anti-self-adjoint element H such that K = q H . Define   R = q H⊗H gb 4(sin πb2 )2 E ⊗ F q H⊗H , (2.23) where H ≡ πH (H ). As we will explain below (see Corollary 2), R coincides with the R-operator proposed by L. Faddeev in [F2]. Notice that the property (3.5) implies that |gb (x)| = 1 for x ∈ R+ . This makes R manifestly unitary.

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A.G. Bytsko, J. Teschner

Theorem 3. The operator R has the following properties: (i) R (X) =  (X) R , (2.24) (ii) (id ⊗ )R = R13 R12 , ( ⊗ id)R = R13 R23 , (2.25) (iii) (σ ⊗ id)R = R−1 , (id ⊗ σ )R = R−1 , (σ ⊗ σ )R = R . (2.26) Remark 2. The R-operator allows us to introduce the braiding of tensor products of the representations Ps . Specifically, let the operator B : Ps2 ⊗Ps1 → Ps1 ⊗Ps2 be defined by Bs2 ,s1 ≡ PRs2 ,s1 , where P is the operator that permutes the two tensor factors. Property (i) from Theorem 3 implies as usual that Bs2 ,s1 ◦ (X) = (X) ◦ Bs2 ,s1 . 2.8. Integral operator representation. The operator R can clearly be projected to an operator Rs2 s1 ≡ (πs2 ⊗ πs1 ) R on Ps2 ⊗ Ps1 . The action of this operator admits a representation by means of a distributional kernel:  2πi(k1 x1 +k2 x2 ) ψ(x , x ) be a Fourier transform ˜ 2 , k1 ) = Theorem 4. Let ψ(k 2 1 R dx2 dx1 e of ψ(x2 , x1 ) ∈ Ps2 ⊗ Ps1 . The action of the R-operator on Ps2 ⊗ Ps1 admits the following representations as an integral operator in “coordinate” and “momentum” space respectively:    Rs2 s1 ψ (x2 , x1 ) = dx2 dx1 Rs2 s1 (x2 , x1 |x2 , x1 ) ψ(x2 , x1 ) , (2.27) R    ˜ s s (k2 , k1 |k  , k  ) ψ(k ˜ 2 , k1 ) , Rs2 s1 ψ˜ (k2 , k1 ) = dk2 dk1 R (2.28) 2 1 2 1 R

with the kernels given by Rs2 s1 (x2 , x1 |x2 , x1 ) =e

 1 2   2πi s1 (x1 −x1 )+s2 (x2 −x2 )+ iQ 2 (x2 +x2 −x1 −x1 )+s1 s2 + 4 Q

Q

   i   + 2i (s1 + s2 ) + i(x2 − x1 ) Gb Q 2 − 2 (s1 + s2 ) + i(x2 − x1 ) ×   ,  Gb Q + 2i (s1 − s2 ) + i(x2 − x1 ) Gb Q + 2i (s2 − s1 ) + i(x2 − x1 ) (2.29) Gb

2

˜ s s (k2 , k1 |k  , k  ) R 2 1 2 1 =

δ(k2

+ k1





e−πi(k1 k2 +k1 k2 ) wb (s1 + k1 ) wb (s2 − k2 ) − k2 − k1 ) . (2.30) Gb (Q + i(k1 − k1 )) wb (s1 + k1 ) wb (s2 − k2 )

The functions wb (x) and Gb (x) are close relatives of the function gb (x) that will be −1 defined in Subsect.  3.1 below, and  G (Q + ix) is taken as a short notation for the distribution G−1 Q + i(x + i0) . 2.9. Highest weight representations. In order to demonstrate that the R-operator we are considering here indeed deserves to be called “universal” we are now going to show that the usual R-matrix for highest weight representations of Uq (sl(2, R)) can be extracted from the analytic properties of the matrix elements given in Theorem 4.

R-Operator, Co-Product and Haar-Measure for the Modular Double

179

As a preparation let us consider the representation of Uq (sl(2, R)) on the dual space Ps of Ps . An interesting class of elements of Ps is furnished by the (complexified) delta-functionals δk , δk , f  = f (k). The δk are well-defined for all k ∈ C \ Ss , and the action of Uq (sl(2, R)) is realized by Ets δk = + Fst δk =

Q

2b

Q − 2b

 − bi (k − s) q δk+ib  + bi (k + s) q δk−ib

Kst δk = e−πbk δk ,

(2.31)

2

t) is the standard definition of a q-number, and the superscript “t” where [t]q ≡ sin(πb sin(πb2 ) on the generators indicates transposition. attention to the Let us restrict  

set Ds of functionals δk for which k is an element of ≥0 . It is easy to verify that (2.31) realizes a highest weight k = −s + i Q + nb , n ∈ Z 2 representation on Ds .

Theorem 5. The action of Rt on H ⊗ Ds is given by R+ s

=q

H⊗Hs

1 2 ∞  n q 2 (n −n)  n (q − q −1 )E ⊗ Fs q H⊗Hs . k=1 [k]q

(2.32)

n=0

2.10. Diagonalization of the R-operator. Theorem 6. The Clebsch-Gordan maps C[s3 |s2 , s1 ] diagonalize the R-operator in the following sense: C[s3 |s1 , s2 ] Bs2 s1 = (s3 |s2 , s1 ) C[s3 |s2 , s1 ] ,

(2.33)

with eigenvalue (s3 |s2 , s1 ) given as (s3 |s2 , s1 ) = e−πi(hs3 −hs2 −hs1 ) ,

hs ≡ s 2 −

Q2 4 .

2.11. The strong coupling regime |b| = 1. We would finally like to point out that our results carry over to the strong coupling regime b = eiθ , θ ∈ [0, π/2). This is almost obvious for those results whose proof relies mainly on the properties of the special functions gb (x), Gb (x) and wb (x). In this case the operators E, F and K are normal (as follows from Eq. (3.19) below), and the hermitian conjugation acts as ˜ E† = E,

˜ F† = F,

˜ K† = K.

Concerning the results that rely on [PT2] one may note that they all amount to certain identities between distributions that are defined by a standard analytic regularization in terms of the meromorphic functions gb (x), Gb (x) and wb (x). The relevant analytic properties underlying the validity of these identities all remain intact upon analytically continuing from the case of real b to |b| = 1.

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A.G. Bytsko, J. Teschner

3. Preliminaries and Auxiliary Results 3.1. Special functions. The Double Gamma function 2 (x|ω1 , ω2 ) was introduced and studied by Barnes [Ba]. In what follows we will be dealing with (recall that Q = b+b−1 ) Gb (x) ≡ e

πi 2 x(x−Q)

2 (x|b−1 , b) . 2 (Q − x|b−1 , b)

(3.1)

This function is closely related to the remarkable special functions introduced under the names of “quantum dilogarithm” in [FK1], “hyperbolic G-function” in [Ru] and “quantum exponential function” in [W2]. Gb (x) is a meromorphic function that has the following properties [Ba, Sh]: self-duality Gb (x) = Gb−1 (x) , functional equation Gb (x + b) = (1 − e

(3.2) 2πibx

reflection property Gb (x) Gb (Q − x) = e

(3.3)

) Gb (x) , πix(x−Q)

(3.4)

,

πi x(Q− ¯ x) ¯

complex conjugation Gb (x) = e Gb (x) ¯ ,  ζb for (x) → +∞ , asymptotics Gb (x) ∼ ζb eπix(x−Q) for (x) → −∞ Gb (x) has poles at x = −nb − mb−1 n, m ∈ Z≥0 , Gb (x) has zeros at x = Q + nb + mb−1

(3.5) (3.6) (3.7)

−2

where ζb = e 4 + 12 (b +b ) . By Proposition 5 in [Sh], the Gb -function admits for b2 > 0 the following infinite product representation πi

πi

2

∞ 2πib−1 (x−nb−1 ) ) n=1 (1 − e . Gb (x) = ζ b ∞ 2πib(x+nb) ) n=0 (1 − e

(3.8)

We are also going to use two other functions that are closely related to Gb (x), namely gb (x) ≡

Gb

Q 2

ζb +

1 2πib

log x

,

and wb (x) ≡ e

πi 2

 Q2 4

+x 2

 Gb

Q 2

− ix .

(3.9)

The representation (2.22) for gb introduced above follows easily from the integral representation for the Double Gamma function introduced in [Sh]. For the reader’s convenience we shall also list the relevant properties of wb (x) that follow from (3.2)–(3.7): self-duality wb (x) = wb−1 (x) , functional equation wb (x + ib) = 2 wb (x) sin π b reflection property wb (x) wb (−x) = 1 ,

Q 2



− ix ,

¯ , complex conjugation wb (x) = wb (−x) wb (x) has poles at x = wb (x) has zeros at x =

−1 −i ( Q 2 + nb + mb ) Q −1 i ( 2 + nb + mb )

(3.10) (3.11) (3.12) (3.13)

n, m ∈ Z≥0 .

(3.14)

Notice that |wb (x)| = 1 if x is real. Hence wb (X) is unitary if X is a self-adjoint operator.

R-Operator, Co-Product and Haar-Measure for the Modular Double

181

3.2. Operator algebraic preliminaries. Lemma 3. Let A and B be self-adjoint operators such that [A, B] = 2π i. Let ϕ(t) be a function on the positive real axis and let γ = b12 . Then we have    b    ϕ(u + v) = wb 2π(A − B) ϕ e 2 (A+B) wb 2π(B − A) , (u + v)γ = uγ + v γ ,

(3.15) (3.16)

where u = ebA , v = ebB . A+B 1 Proof. It is convenient to introduce p ≡ B−A 2π and x ≡ 4π . Observe that [p, x] = 2πi ; so that we have     (3.17) f p eπbx = eπbx f p − i b2

for any function f (t) that is bounded and analytic in the strip b2 ≤ (p) ≤ 0. Using the Baker-Campbell-Hausdorff formula for Weyl-type operators and the properties of the function wb , we may calculate as follows: (3.12)

wb (−p) e2πbx wb (p) =

wb (p + i b2 ) πbx 1 (3.17) e e2πbx wb (p) = eπbx wb (p) wb (p − i b2 )

(3.11)

= 2eπbx (cosh πbp) eπbx  b  b b b = e 4 (A+B) e 2 (B−A) + e 2 (A−B) e 4 (A+B) = u + v.

The last expression is therefore unitarily equivalent to the positive self-adjoint operator e2π bx . Our claim (3.15) follows by applying the standard functional calculus of self-adjoint operators. Relation (3.16) can be proven along the same lines taking into account that, thanks to self-duality (3.10), wb obeys also the equation    wb (x + bi ) = 2 wb (x) sin πb Q (3.18) 2 − ix . Therefore, for ϕ(t) = t γ we have (u + v)γ

= wb (−p) e (3.18)

= 2e

π bx

2π b x

(3.17)

π

wb (p) = e b x

wb (p + wb (p −

i 2b ) πb x e i 2b )

π bx

(cosh πb p) e  1  1 1 1 = e 4b (A+B) e 2b (B−A) + e 2b (A−B) e 4b (A+B) = uγ + v γ .

 

Remark 3. Another way to prove relation (3.16) in Lemma 3 is to use the b-binomial formula (B.4) that we derive in the Appendix. When t approaches the value −iγ , the b-binomial coefficient (B.5) vanishes unless τ takes special values determined by (3.7). Furthermore, for t = −iγ the b-binomial coefficient has nonvanishing residues only at τ = 0 and τ = −iγ . The contributions from these two poles yield the two terms on the r.h.s. of (3.16). Similar consideration for t approaching −inγ , n > 1 shows that the b-binomial coefficient has nonvanishing residues at τ = 0, −iγ , . . . , −inγ . Therefore (u + v)nγ can be represented as sum of (n + 1) terms which is analogous to the q-binomial formula in the compact case.

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The proven lemma leads to useful representations for the generators and the R-operator of Uq (sl(2, R)). For brevity, we denote eb ≡ (2 sin π b2 )E and fb ≡ (2 sin π b2 )F, whereas e 1 and f 1 will stand for their counterparts with b replaced by b1 . b

b

Lemma 4. πH (E) and πH (F ) admit the following representation: eb = wb (p2 ) e2πb(x1 −x2 ) wb (−p2 ) ,

fb = wb (p1 ) e2πb(x2 −x1 ) wb (−p1 ) . (3.19)

R may be represented as follows:     R = q H⊗H wb (p2 ) ⊗ wb (p1 ) gb e2πb(x1 −x2 ) ⊗ e2πb(x2 −x1 )   × wb (−p2 ) ⊗ wb (−p1 ) q H⊗H .

(3.20)

1 1 (A − B) = pn+1 and 4π (A + B) = xn − xn+1 , Proof. In Lemma 3, we can identify 2π where n = 1, 2 (with the convention that n + 2 ≡ n). Then, as seen from the definition (2.5), we have u + v = eb for n = 1 and u + v = fb for n = 2. Therefore, (3.19) is just a particular case of (3.15). Furthermore, for functions ϕ(t) defined on R+ we have   ϕ(eb ) = wb (p2 ) ϕ e2πb(x1 −x2 ) wb (−p2 ) , (3.21)  2πb(x −x )  2 1 ϕ(fb ) = wb (p1 ) ϕ e wb (−p1 ) .

In particular, we can take ϕ(t) = gb (x) (recall that |gb (x)| = 1 for x ∈ R+ ). For this choice of ϕ(t), the representation (3.20) for R follows immediately from the definition (2.23).   Corollary 1. For γ =

1 b2

we have (eb )γ = e 1 ,

(fb )γ = f 1 .

b

b

(3.22)

Proof. Notice that, if u and v are identified as in the proof of Lemma 4, then uγ + v γ = e 1 b for n = 1 and uγ + v γ = f 1 for n = 2. Thus, relations (3.22) are a particular case b of (3.16) .   This proves the relations (2.9) from Lemma 1. Corollary 2. The definition of the R-operator proposed in [F2], ∞ 2n+1 e ⊗ f ) −2 b b n=0 (1 + q R = q H⊗H ∞ q H⊗H , q˜ = e−iπb , 2n+1 (1 + q ˜ e ⊗ f ) 1 1 n=0 b

which is valid for b =

eiϑ ,

ϑ ∈ (0,

π 2 ),

(3.23)

b

coincides with our definition (2.23).

Proof.   (2.22)    −1 (2.23) 1 + log(e ⊗ f ) q −H⊗H R q −H⊗H = gb eb ⊗ fb = ζ b Gb Q b b 2 2πib ∞ Q 2πib( 2 +nb) eb ⊗ f b ) (3.8) n=0 (1 − e =  1 −1 ) ∞ 2πib−1 ( Q −nb 2 (eb ⊗ fb ) b2 ) n=1 (1 − e ∞ 2n+1 e ⊗ f ) (3.22) b b n=0 (1 + q .   = ∞ 2n+1 e 1 ⊗ f 1 ) n=0 (1 + q˜ b

b

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183

Corollary 3. πH (E) and πH (F ) admit the following representation: eb = e2πb(x1 −x2 )+bψb (p2 ) ,

fb = e2πb(x2 −x1 )+bψb (p1 ) ,

(3.24)

where ψb (t) ≡ i ∂t (log wb (t)). Proof. Equations (3.21) for ϕ(t) = log(t) yield log(eb ) = wb (p2 ) 2πb(x1 − x2) wb (−p2 ) = 2πb(x1 − x2) + 2πb [wb (p2 ), (x1 − x2)] wb (−p2 )  1  = 2πb(x1 − x2) + ib∂t wb (t)  t=p2 wb (p2 ) and, analogously, log(fb ) = 2πb(x2 − x1) + bψb (p1 ). Exponentiating these relations, we obtain (3.24).   Remark 4. Alternatively, Eqs. (3.24) can be derived from (2.5) with the help of the BakerCampbell-Hausdorff formula. Observe also that (log eb +log fb ) = b(ψb (p1 )+ψb (p2 )) commutes with C and H. We may now give the proof of Lemma 2. In Lemma 3, let us choose A = (2π(x1 − x2)+ψb (p2 ))⊗1+1⊗ π2 (p2 −p1 ) and B = 1⊗(2π(x1−x2)+ψb (p2 ))+ π2 (p1 −p2 )⊗1. In view of (3.24) this implies the identification u = eb ⊗Kb and v = Kb−1 ⊗eb . Therefore, we have 1   1  1 (2.11)  (3.22)  (e 1 ) =  (eb ) b2 = (eb ) b2 = eb ⊗ Kb + Kb−1 ⊗ eb b2 b 1  1 (3.22)  (3.16)  = eb ⊗ Kb b2 + Kb−1 ⊗ eb b2 = e 1 ⊗ K 1 + K−1 1 ⊗ e1 . b

The relation for F˜ in (2.13) is proven similarly.

b

b

b

 

Fα

with α = 0 be defined on H in the sense of

[E, Fα ] = [α]q [2H + α − 1]q Fα−1 ,

[Eα , F] = [α]q [2H − α + 1]q Eα−1 , (3.25)

Lemma 5. Let the powers (3.21). Then we have

Eα

and

where the q-numbers are defined as in Theorem 5. Proof. (3.19)

[E, Fα ] =

(3.17)

=

(3.11)

= =

(2.5)

=

  e2πb(x1 −x2 ) e2παb(x2 −x1 ) wb (p2 ) w (−p ) , w (p ) w (−p ) b 2 b 1 b 1 2 sin πb2 (2 sin π b2 )α  w (p + ib) w (p ) w (p + (1 − α)ib) w (p + αib) b 1 b 1 b 2 b 2 − wb (p1 ) wb (p2 − ib) wb (p1 − αib) wb (p2 + (α − 1)ib) e2π(α−1)b(x2 −x1 ) × wb (p1 ) wb (−p1 ) (2 sin πb2 )α+1 



Q ip1  Q ip2  ip1  Q ip2  Q α−1 − + − − α − − α + 2b b q 2b b q 2b b q 2b b q F

1  α−1  i [α]q Q = [α]q ib (p2 − p1 ) + α − 1 q Fα−1 b − α + b (p2 − p1 ) q F

 [α]q 2H + α − 1 q Fα−1 .

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In the fourth line we used the definition of q-number; in the fifth line we used the identities [x]q [y]q − [x − α]q [y − α]q = [α]q [x + y − α]q and [t + b−2 ]q = −[t]q . The second formula in (3.25) is derived analogously.   Lemma 6. Let A and B be self-adjoint operators such that [A, B] = 2π i. Then for the function gb (x) defined in (2.22) we have gb (u) gb (v) = gb (u + v) , gb (v) gb (u) = gb (u) gb (q −1 uv) gb (v) ,

(3.26) (3.27)

2

where u = ebA , v = ebB and q = eiπb . Furthermore, (3.26) ⇔ (3.27). In the literature, Eqs. (3.26) and (3.27) are often referred to as the quantum exponential and the quantum pentagon relations. They also hold for the function sq (x) = ∞ 2n+1 ) which is the compact counterpart of g (x). For s (x), the quantum b q n=0 (1 + xq exponential relation has been known for a long time [Sch] and the quantum pentagon relation was found in [FV]. Since (3.26) and (3.27) are equivalent, it suffices to prove one of them. Proofs of the quantum pentagon relation were given in [FKV] and [W2]. Nevertheless, we find it instructive to give another proof of the quantum exponential relation in Appendix B since it will allow us to introduce the notion of b-binomial coefficients. 3.3. Alternative representations of the R-operator. Lemma 7. R and R−1 may be decomposed into powers of E ⊗ F as follows:  2 2  it e−πib t R=b dt q H⊗H 4(sin πb2 )2 E ⊗ F q H⊗H , Gb (Q + ibt)

(3.28)

R

R−1 = b



dt R

 it e−πbQt q −H⊗H 4(sin π b2 )2 E ⊗ F q −H⊗H , Gb (Q + ibt)

(3.29)

where the integration contour goes above the pole at t = 0. Proof. By Lemma 15 in [PT2] (see also [FKV, Ka]) we have:  Gb (α + iτ ) Gb (α) Gb (β) dτ e−2πτβ = . G (Q + iτ ) Gb (α + β) b R

(3.30)

1 The function Gb (Q+iτ ) has a pole at τ = 0 and is analytic in the upper half-plane. The integration contour in (3.30) goes above this pole. Considering the asymptotics of (3.30) for α → −∞ and α → +∞, using the properties (3.4) and (3.6), and making a change of variables, we obtain the following Fourier transformation formulae  2 2 e−πib t ζb 2πibtr b = dt e (3.31) Q  = gb (e2πbr ) , Gb (Q + ibt) Gb 2 − ir

R+i0



b R+i0

dt e2πibtr

   −1 e−πbQt = ζ b Gb Q − ir = gb (e2πbr ) . 2 Gb (Q + ibt)

(3.32)

R-Operator, Co-Product and Haar-Measure for the Modular Double

Lemma 7 follows if we put here r = with the definition (2.23).  

1 2πb

185

log(4(sin π b2 )2 E ⊗ F) and compare the result

We now come to the proof of Theorem 4. Consider the product representation of the R-operator of Lemma 4 projected to Ps2 ⊗ Ps1 by means of the reduction described in Subsect. 2.3,     (3.20) Rs2 s1 = q H2 ⊗H1 wb (s2 − p) ⊗ wb (s1 + p) gb e2πbx ⊗ e−2πbx −1 H ⊗H  q 2 1 × wb (s2 − p) ⊗ wb (s1 + p)  2  e−πiτ e2πiτ x ⊗ e−2πiτ x (3.31) H2 ⊗H1  = q dτ wb (s2 − p) ⊗ wb (s1 + p) Gb (Q + iτ ) R −1 H ⊗H  q 2 1. (3.33) × wb (s2 − p) ⊗ wb (s1 + p) It is now easy to compute the “matrix elements” of Rs2 s1 on the states |k2 , k1  = |k2  ⊗ |k1 , where |k ≡ e2πixk . Taking into account that p|k = k|k and k|k   = δ(k  − k), we find ˜ s s (k2 , k1 |k  , k  ) = k2 , k1 |Rs s |k  , k   R 2 1 2 1 2 2 1 1  2 +k k +k  k  ) −πi(τ 1 2 1 2 wb (s1 + k1 ) wb (s2 − k2 ) e = dτ Gb (Q + iτ ) wb (s1 + k1 ) wb (s2 − k2 ) R

×δ(k2 − k2 + τ ) δ(k1 − k1 − τ )

(3.34)

which gives us the kernel (2.30) of the “momentum” representation in Theorem 4. The kernel of the “coordinate” representation (2.27) can be obtained as a Fourier transform of (3.34): Rs2 s1 (x2 , x1 |x2 , x1 )      ˜ s s (k2 , k1 |k  , k  ) dk2 dk1 dk2 dk1 e2πi(x2 k2 +x1 k1 −x2 k2 −x1 k1 ) R = 2 1 2 1 R



    eπi (τ (k2 −k1 )−2k1 k2 ) 2πi (τ (x2 −x1 )+k  (x1 −x  )+k  (x2 −x  )) 1 1 2 2 e = Gb (Q + iτ ) R wb (s1 + k1 − τ ) wb (s2 − k2 − τ ) × . wb (s1 + k1 ) wb (s2 − k2 )

dτ dk2 dk1

The remaining integrations are performed by using relation (3.30) three times. The result of this straightforward but tedious calculation is given by (2.29). 4. Proofs of the Main Results 4.1. Proof of Theorem 1: Co-associativity. First, it is straightforward to write out (id ⊗ ) ◦ (X) and ( ⊗ id) ◦ (X) in terms of the Clebsch-Gordan maps C[s3 |s2 , s1 ]: (πs3 ⊗ πs2 ⊗ πs1 ) ◦ (id ⊗ ) ◦ (X) dm(s4 )dm(s21 ) C†3(21) (s21 ) · X · C3(21) (s21 ), = R+

(πs3 ⊗ πs2 ⊗ πs1 ) ◦ ( ⊗ id) ◦ (X) dm(s4 )dm(s32 ) C†(32)1 (s32 ) · X · C(32)1 (s32 ), = R+

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A.G. Bytsko, J. Teschner

where we have introduced   C3(21) (s21 ) ≡ C[s4 |s3 , s21 ] · id ⊗ C[s21 |s2 , s1 ] ,   C(32)1 (s32 ) ≡ C[s4 |s32 , s1 ] · C[s32 |s3 , s2 ] ⊗ id . Proposition 7 in [PT2] is equivalent to  C3(21) (s21 ) = ds32 ss13 where that

s1

s2 s21 s3 s4 s32 b

R+

s2 s21 s4 s32 b

(4.1)

C(32)1 (s32 ),

are the b-Racah-Wigner coefficients introduced in [PT2]. It follows

(πs3 ⊗πs2 ⊗  πs1 ) ◦ (id ⊗ ) ◦(X) s1  dm(s4 )dm(s21 ) dm(s32 )dm(s32 ) s3 = R+

s2 s21 ∗ s1 s2 s21  s4 s32 b s3 s4 s32 b

·

R+

· C†(32)1 (s32 ) · X · C(32)1 (s32 ).  , and using formula (89) from [PT2], Exchanging the integrations over s21 and s32 , s32  s1 s2 s21 ∗ s1 s2 s21   dm(s21 ) s3 s4 s32 = m(s32 )δ(s32 − s32 ), b s3 s4 s32 b

R+

yields the claim.

4.2. Proof of Theorem 2: Invariance of the Haar-measure. We shall consider the left invariant Haar measure hl only, the proof for the case of hr being completely analogous. A few preparations are in order. The elements of Alh can be represented as integral operators: If a vector ψ ∈ H is realized by a function ψ(k, s), then  (Oψ)(k, s) = dk  KO (k, k  |s)ψ(k  , s). R

In terms of the kernel KO

(k, k  |s) 

hl (O) =

R+

one may write the defintion of hl as  dm(s) dk e−2πQk KO (k, k|s). R

(4.2)

The distributional matrix elements of an operator O ∈ A are always of the form s, k|O|s  , k   ≡ δ(s − s  )k|O|k   s .

(4.3)

By using an analogous notation for operators in A ⊗ A one may represent the distributional matrix elements of (O) as  k2 , k1 |(O)|k2 , k1  s2 s1   

s = dm(s3 ) dk3 dk3 KO (k3 , k3 |s3 ) k33 R+

R

s2 s1  ∗ s3 s2 s1  k3 k2 k1 . k2 k1

(4.4)

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187

In order to make the justification for the following manipulations more transparent, we are going to employ the following regularization for the distributions involved:

s3 s2 s1 

s3 s2 s1  k3 k2 k1 = lim k3 k2 k1  ,

s3

s2 s1  k3 k2 k1 

↓0

= e−

3

i=1 |ki |

δ (k3 − k2 − k1 )Cs3

s2

s1  k2 k1 ,

where δ (x) = δ (−x) is a symmetric regularization of the delta-distribution. Let us furthermore note that it suffices to check the invariance property on a dense subset T of the domain of hl . Consider the matrix element     ψ2 , k1 |(O)|ψ2 , k1  s2 s1 := dk2 dk2 ψ(k2 )ψ  (k2 )k2 , k1 |(O)|k2 , k1  s2 s1 , R

ψ2 , ψ2

are smooth functions with compact support. Assuming that KO (k, k  |s) where has exponential decay w.r.t. k and k  it is not difficult to show that the matrix element  ψ2 , k1 |(O)|ψ2 , k1  s2 s1 will also have exponential decay w.r.t. k1 , k1 that can be made as large as one likes by choosing the subset T ⊂ Ah appropriately. Combining (4.2) and (4.4) leads to the following representation for the distributional matrix elements of (id ⊗ hl )(O):     k2 |(id ⊗ hl )(O)|k2  s2 = dm(s3 ) dk3 dk3 KO (k3 , k3 |s3 ) + R R  

s s s  ∗ s3 s2 s1  × dm(s1 ) dk1 e−2πQk1 k33 k22 k11 k3 k2 k1 . R+

R

(4.5)

We are going to use the following result: Proposition 1. The following equation holds as an identity between tempered distributions:   

s s s  ∗ s3 s2 s1  2πk3 Q e dm(s1 ) dk1 e−2πQk1 k33 k22 k11 k3 k2 k1 =

R+

δ(k3

R

− k3 )δ(k2 − k2 ).

(4.6)

Proof. The proof of the proposition will be based on the following important symmetries of the Clebsch-Gordan kernel:

s s s  Lemma 8. The Clebsch-Gordan kernel k33 k22 k11  has the following symmetries: 

s1 −s2 s3  s3 s2 s1  ∗ = e+πQ(k1 −k3 ) e−πihs2 −k , k3 k2 k1  1 −k2 −k3  (4.7) 



 s3 s2 s1  ∗ −πQ(k2 −k3 ) −πihs1 s2 s3 −s1 = e e . k3 k2 k1  −k2 −k3 −k1  Proof. One may verify directly that for si ∈ R, i = 1, 2, 3,  s  ∗ 

−s2 s3 1 s3 s2 s1  = e−πihs2 x ∗ −i Q x ∗ x ∗ −i Q , x3 x2 x1 1 2 3 2 2 

 s  ∗ s3 −s1 2 s3 s2 s1  −πihs1 Q Q = e . ∗ ∗ ∗ x3 x2 x1 x +i x +i x 2

2

3

2

(4.8)

1

The lemma follows by taking the Fourier-transformation of (4.8), taking into account that our regularization is compatible with the symmetry (4.7).  

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With the help of Eq. (4.7) we may rewrite the left hand side of (4.6) as follows:   

∗

s1 −s2 s3  s1 −s2 s3  π(k3 −k3 )Q lim e dm(s1 ) dk1 −k −k1 −k  −k   . −k −k 1 2 3  R+

↓0

2

R

3

(4.9)

The proposition now follows by using the Fourier-transform of [PT2, Corollary 1].

 

Inserting (4.6) into (4.5) yields  k2 |(id ⊗ hl )(O)|k2  s2 = δ(k2 − k2 )

 R+

 dm(s3 )

R

dk3 e−2πQk3 KO (k3 , k3 |s3 ).

Recognizing the definition of the Haar-measure on the right-hand side completes the proof of the left invariance property of hl . To prove the property (ii) in Theorem 2 we observe that definition (2.18) can be rewritten as  ∞ dm(s) Tr(K2 K˜ 2 Os ) . hl (O) = 0

Now it is straightforward to verify the first formula in (2.21) for X = E, F, K using the definition of the adjoint action, the relations (2.1), and the cyclicity of trace. For instance, Tr(K2 K˜ 2 adKl (Os )) = Tr(K2 K˜ 2 KOs K−1 ) = Tr(K2 K˜ 2 Os ), Tr(K2 K˜ 2 adEl (Os )) = Tr(K2 K˜ 2 (EOs K−1 −qK−1 Os E)) = Tr((KE−qEK)K˜ 2 Os ) = 0. Further, we notice that l (O)) = h (adl (adl (O))) = h (O) (X) (Y) = h (O) (XY). Together with hl (adXY l l l X Y the linearity of trace this implies that (2.21) extends to any element of Uq (sl(2, R)). 4.3. Proof of Theorem 3. Let us adopt the following notations: X1 ≡ X ⊗ 1 ⊗ 1, X2 ≡ 1 ⊗ X ⊗ 1, and X3 ≡ 1 ⊗ 1 ⊗ X. Property (i). First, we compute with the help of Lemma 5:     q H⊗H (E1 F2 )it q H⊗H K1−1 E2 − K1 E2 q H⊗H (E1 F2 )it q H⊗H

 = q H⊗H (E1 F2 )it , E2 q H⊗H  (3.25) = −q H⊗H [it]q [2H2 + it − 1]q Eit1 F2it−1 q H⊗H .

(4.10)

Next, we find     q H⊗H (E1 F2 )it q H⊗H E1 K2 − E1 K2−1 q H⊗H (E1 F2 )it q H⊗H  = q H⊗H (E1 F2 )it E1 K22 − E1 K2−2 (E1 F2 )it q H⊗H  = q H⊗H (2i q it sin πb2 ) [2H2 + it]q Eit+1 F2it q H⊗H . 1

(4.11)

Let us write down the integral representation (3.28) of R in the following form:   it R = dt ρ(t) q H⊗H E ⊗ F q H⊗H , (4.12) R

R-Operator, Co-Product and Haar-Measure for the Modular Double

189

−π ib2 t 2

where ρ(t) ≡ b Geb (Q+ibt) (2 sin πb2 )2it . Observe that (3.3) implies that ρ(t) satisfies the following functional equation: [it + 1]q ρ(t − i) = (2i q it sin π b2 ) ρ(t) .

(4.13)

Adding (4.10) with (4.11), we derive R (E) −  (E) R = R (E1 K2 + K1−1 E2 ) − (E1 K2−1 + K1 E2 ) R   (4.12) = dt q H⊗H ρ(t) (2i q it sin πb2 ) [2H2 + it]q Eit+1 F2it 1 R −[it]q [2H2 + it − 1]q Eit1 F2it−1 q H⊗H   = dt q H⊗H (2i q it sin πb2 ) ρ(t) R (4.13) −[it + 1]q ρ(t − i) [2H2 + it]q Eit+1 F2it q H⊗H = 0 . 1 Thus, we have proven (2.24) for X = E. The proof for F goes along the same lines with the help of the second formula in (3.25). And for K the proof is trivial because (K) commutes with (E ⊗ F)it . Property (ii). Recall that the rescaled generators eb and fb were introduced before Lemma 4. To prove the first formula in (2.25), we use the quantum exponential relation (3.26) from Lemma 6 with identification u = e1 K2−1 f3 and v = e1 f2 K3 ,   (2.23) (id ⊗ )R = (id ⊗ ) q H1 H2 gb (e1 f2 ) q H1 H2   (2.11) H1 H2 +H1 H3 = q gb e1 f2 K3 + e1 K2−1 f3 q H1 H2 +H1 H3 = q H1 H2 +H1 H3 gb (e1 K2−1 f3 ) gb (e1 f2 K3 ) q H1 H2 +H1 H3

(3.26) (2.1)

= q H1 H3 gb (e1 f3 ) q H1 H3 q H1 H2 gb (e1 f2 ) q H1 H2 = R13 R12 .

The second formula in (2.25) is proved in the same way. Property (iii). First, we derive     (σ ⊗ id) q H1 H2 (E1 F2 )it q H1 H2 = (σ ⊗ id) q H1 H2 (E1 )it q H1 (H2 +it) (F2 )it = q −H1 (H2 +it) (−qE1 )it q −H1 H2 (F2 )it = q −H1 H2 q −itH1 (−qE1 )it q itH1 (F2 )it q −H1 H2 = eπ b(ibt

2 −Qt)

q −H1 H2 (E1 F2 )it q −H1 H2 .

(4.14)

This means that, acting with (σ ⊗ id) on the r.h.s. of (3.28), we obtain the r.h.s. of (3.29). Thus, we have proven the first formula in (2.25). The second formula is verified analogously. Finally, acting with (id ⊗ σ ) on the last line in (4.14) and performing similar manipulations, we find that   2 (id ⊗ σ ) eπb(ibt −Qt) q −H1 H2 (E1 F2 )it q −H1 H2 = q H1 H2 (E1 F2 )it q H1 H2 which together with (4.14) implies the last formula in (2.26).

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4.4. Proof of Theorem 5: R-operator the highest weight representations. We first need to discuss the analytic continuation of    ˜ 2 , k1 |k  , k  ) f (k  , k  ) δk2 ⊗ δk1 , Rs2 s1 f = dk2 dk1 R(k (4.15) 2 1 2 1 Q



R

to the values k1 = −s1 + i 2 + nb , n ∈ Z≥0 . To begin with, one may trivially perform e.g. the integral over k2 to get an expression of the form    ˜ k (k1 |k  ) f (k − k  , k  ), dk1 R (4.16) δk2 ⊗ δk1 , Rs2 s1 f = 1 1 1 R

  where k = k2 + k1 . The analytic continuation of (4.16) to k1 = −s1 + i Q 2 + nb can be defined by deforming the contour of integration over k1 , R, in (4.16) into R + i Q 2 + −1  nb) + i0 plus a sum of small circles around the poles from the factor wb (s1 + k1 ) in  ˜ k (k1 |k  ) that lie between R and R + i Q + nb) + i0. R 1 2   We are now in the position to take the limit k1 → −s1 + i Q 2 + nb . The factor ˜ k (k1 |k  ), cf. (2.30), makes most of the terms vanish except wb (s1 + k1 ) that appears in R 1   for the terms from the poles at k1 = −s1 + i( Q 2 + bn ), 0 ≤ n ≤ n. The resulting expression is of the following form: Rts2 s1

δk2 ⊗ δk1 =

n 

Gl e

−πi((k1 −ibl)k2 +k1 k2 ) wn−l

wn

l=0

 wb (s2 − k2 )  , wb (s2 − k2 ) k  =k2 +ibl

(4.17)

2

where Gn := Resx=nb G−1 wb−1 (x). b (Q + x) and wn = Resx=i Q 2 +inb It remains to calculate the relevant residues. It is easy to derive from (3.2)–(3.4) that 2

Gb (x) Gb (−x) = −

eπix . 4 sin πbx sin π b−1 x

(4.18)

Hence limx→0 (xGb (x))2 = (2π)−2 . In fact, using the modular property of the Dedekind η-function, it is straightforward to compute the limit directly for the product representation (3.8) (as was done in [Sh]); which yields lim x Gb (x) =

x→0

1 . 2π

(4.19)

Hence, taking into account the properties (3.2)–(3.4), we find that 1 Gb (Q + z) n m  1  =− (1 − q 2k )−1 (1 − q˜ −2l )−1 2π

Res

k=1

at z = nb + mb−1 ,

(4.20)

l=1

−2

where n, m ∈ Z≥0 and q˜ = e−iπb . To complete the proof of Theorem 5 is now a matter of straightforward calculation using the functional relation (3.11), formula (4.20), as well as (2.31).

R-Operator, Co-Product and Haar-Measure for the Modular Double

191

4.5. Proof of Theorem 6. Let us first note that the left-hand side of (2.33) satisfies the intertwining property C[s3 |s1 , s2 ] Bs2 s1 ◦ (πs2 ⊗ πs1 ) ◦ (X) = πs3 (X) ◦ C[s3 |s1 , s2 ] Bs2 s1 .

(4.21)

A unitary operator that maps Ps2 ⊗Ps1 → Ps3 and satisfies (2.14) must be proportional to C[s3 |s2 , s1 ]. This is a consequence of the analysis used to prove Theorem 2 in [PT2]. It follows that there exists a function (s3 |s2 , s1 ) such that the statement of Theorem 6 holds. We are left with the task to calculate (s3 |s2 , s1 ) explicitly. To this aim let us note that Theorem 6 is equivalent to an identity between meromorphic functions. To write this identity down, let us assume that Ps2 ⊗ Ps1 is realized by ˜ 2 , k1 ). C[s3 |s1 , s2 ] is then realized as an integral operator: functions ψ(k   

s s s  ˜ 2 , k1 ). C[s3 |s1 , s2 ]ψ˜ (k3 ) = dk2 dk1 k33 k22 k11 ψ(k R

s s s  The explicit expression for the distributional kernel k33 k22 k11 can be found in Appendix A. For the moment it will be enough to note that it can be factorized as

s3 s2 s1 

s s  = δ(k1 + k2 − k3 ) Cs3 k22 k11 , (4.22) k3 k2 k1

s s  where Cs3 k22 k11 is a meromorphic function by Lemma 20 of [PT2]. It is then easy to see that Theorem 6 is equivalent to the identity 

s s s   ˜ s s (k2 , k1 |k  , k  ) = (s3 |s2 , s1 ) ks3 ks2 ks1 . dk2 dk1 k33 k11 k22 R (4.23) 2 1 2 1 3 2 1 R

In order to see that Eq. (4.23) is indeed equivalent to an identity between meromorphic functions let us note that (4.22) and (2.30) allow one to split off the distributional factors. What remains on the left-hand side is a convolution of two meromorphic functions, so is itself meromorphic (cf. Lemma 3 in [PT2]). Let us note that both sides of (4.23) have a pole at k2 = −s2 + i Q 2 . In the case of the right-hand side this is a consequence of Lemma 20 of [PT2]. Concerning the left-hand side of (4.23) one may as in the proof of Lemma 3 of [PT2] identify the above-mentioned pole as the consequence of the pinching of the contour by a collision of

s s ofs integration  two poles of the integrand. First we have the pole of k33 k11 k22 at k2 = −s2 +i Q 2 . Second −1    ˜ let us note that the factor Gb (Q+i(k1 −k1 )) appearing in Rs2 s1 (k2 , k1 |k2 , k1 ) produces ˜ s s (k2 , k1 |k  , k  ) has support only for a pole at k2 = k2 if one takes into account that R 2 1 2 1   k1 − k1 = k2 − k2 . The residue of the resulting pole on the left-hand side of (4.23) is

s s s  ˜ s s (k2 , k1 |k  , k  ) simply given by the product of the relevant residues of k33 k11 k22 and R 2 1 2 1 respectively. The equality of the residues of the two sides of Eq. (4.23) implies the following identity: Res

k2 =−s2 +i Q 2

Cs3

s1

s2  πQk1 2πis2 k1 e k1 k2 e

= (s3 |s2 , s1 )

Res

k2 =−s2 +i Q 2

Cs3

s2

s1  k2 k1 .

By evaluating the relevant residues we may therefore calculate (s3 |s2 , s1 ).

(4.24)

192

A.G. Bytsko, J. Teschner

Lemma 9.





s1 s2 s3 k1 k2 Q k2 =−s2 +i 2 − π2i (hs3 −hs2 −hs1 )

2π i Res

C

=e

2π i Res

k2 =−s2 +i Q 2

Cs3

s2

e− 2 Qk1 e−πis2 k1 π

s1 

wb (k1 − s1 )wb (s3 + s1 − s2 ) wb (k1 + s3 − s2 + i Q 2)

,

k2 k1

= e+ 2 (hs3 −hs2 −hs1 ) e+ 2 Qk1 e+πis2 k1 π

πi

wb (k1 − s1 )wb (s3 + s1 − s2 )

. wb (k1 + s3 − s2 + i Q 2)

s s  Proof. In order to exhibit the singular behavior of Cs3 k11 k22 near k2 = −s2 + i Q 2 one may deform the contour of integration in (A.6) into the union of a small circle around the pole of the integrand at s = 0 and a contour that separates the pole at s = 0 from all the other poles in the upper half-plane, approaching asymptotically ±i∞. The contribution from the residue of the pole at s = 0 exhibits the pole at k2 = −s2 + i Q 2 explicitly, whereas the rest is nonsingular.

s2 s1  Similarly, to analyze the singular behavior of Cs3 k2 k1 near k2 = −s2 + i Q 2 one needs to deform the contour in (A.6) into a small circle around the pole at s = −R3 together with a contour separating that pole from all the other poles in the lower half plane. It is then straightforward to calculate the values of the corresponding residues from (A.6).   Appendix A. The Clebsch-Gordan Coefficients for the Modular Double

 Definition 2. Define a distributional kernel xs33 xs22 xs11 (the “Clebsch-Gordan coefficients”) by an expression of the form

s3 s2 s1 

 ≡ lim xs33 xs22 xs11  , (A.1) x3 x2 x1 ↓0

where the meromorphic function

−s



3 s2 s1 x3 x2 x1 

s3

s1 

s2 x3 x2 x1 

is defined as

= e− 2 (hs3 −hs2 −hs1 ) πi

× Db (σ32 ; y32 + i)Db (σ31 ; y31 + i)Db (σ21 ; y21 + i), (A.2) hs = s 2 + 41 Q2 , the distribution Db (σ ; y) is defined in terms of the function wb (y) as Db (σ ; y) =

wb (y − 2i Q) , wb (y + σ )

(A.3)

and the coefficients yj i , σj i , j > i ∈ {1, 2, 3} are given by y32 =x2 − x3 + 21 (s3 + s2 ), y31 =x3 − x1 + 21 (s3 + s1 ), y21 =x2 − x1 +

1 2 (s2

+ s1 − 2s3 ),

σ32 =s1 − s2 − s3 , σ31 =s2 − s3 − s1 , σ21 =s3 − s2 − s1 .

(A.4)

R-Operator, Co-Product and Haar-Measure for the Modular Double

193

It is often useful to consider the Fourier-transform of the b-Clebsch-Gordan symbols defined by  3

s3 s2 s1 

 dx3 dx2 dx1 e2πi l=1 kl xl xs33 xs22 xs11 . = (A.5) −k3 k2 k1 The distribution

s2 s1  k3 k2 k1

s3

s3 k3

R

can be factorized as

s s2 s1  = δ(k1 + k2 − k3 )Cs3 k22 k2 k1

s1  k1 ,

s s  where Cs3 k22 k11 is a meromorphic function. A straightforward calculation using [PT2, Lemma 15] yields the following expression: C−s3

s2

s1  k2 k1



Q

e− 2 β21 β e 2 π(k1 −k2 ) eπi(k1 s2 −k2 s1 ) wb (σ32 )wb (σ31 )wb (σ21 ) πi

=

ds e−πsβ

3  wb (s + Rl ) l=1

R+i0

wb (s + Sl )

,

(A.6) where we used the abbreviations β = and

Q 2

R1 = − s2 + k2 , R2 = − s 1 − k1 , R3 =s3 − s2 − s1 , The analytic properties of

s3

s2 s1  k3 k2 k1

+ i(s1 + s2 + s3 ), β21 =

Q 2

+ i(s1 + s2 − s3 )

S1 =i Q 2 + R1 − σ32 , S2 =i Q 2 + R2 − σ31 ,

(A.7)

S3 =i Q 2.

can be summarized as follows:

s3 s2 s1  Lemma 10 (Lemma 20 in [PT2]). k3 k2 k1 depends meromorphically on all of its argu−1 ≥0 ments, with poles at ±iki = Q 2 + isi + nb + mb , n, m ∈ Z , i = 1, 2, 3 only.

Appexdix B. Quantum Exponential Function and b-Binomial Coefficient The definition (2.22) and the property (3.3) imply that the function gb (x) obeys the following functional equation: gb (qx) = (1 + x)−1 gb (q −1 x) .

(B.1)

For a pair of Weyl-type variables, uv = q 2 vu, a consequence of (B.1) is  −1  −1 u + v = gb (qu−1 v) u gb (qu−1 v) = gb (quv −1 ) v gb (quv −1 ) .

(B.2)

It is now obvious that  −1 gb (u + v) = gb (qu−1 v) gb (u) gb (qu−1 v)  −1 = gb (quv −1 ) gb (v) gb (quv −1 ) .

(B.3)

These relations allow us to prove the equivalence of (3.26) and (3.27) stated in Lemma 6. For instance, let us show that (3.27) together with the first relation in (B.3) implies (3.26).

194

A.G. Bytsko, J. Teschner ˜

˜ = 2π i. Introduce V ≡ qu−1 v. Notice that V = ebB , where B˜ = B − A so that [A, B] Then we have  −1 (3.27) (B.3) = gb (u) gb (q −1 uV ) = gb (u) gb (v) . gb (u + v) = gb (V ) gb (u) gb (V ) The inverse implication, (3.26) ⇒ (3.27), is proven similarly. Now we want to prove (3.26). First, we represent (u + v)it in an integral form: −1  (B.2) (u + v)it = gb (qu−1 v) uit gb (qu−1 v)  Gb (−ibτ1 ) (3.31) 2 = b dτ1 dτ2 eπbQ(τ1 −τ2 ) (qu−1 v)iτ1 uit (qu−1 v)iτ2 G (Q + ibτ ) b 2 R 2 2 2 2 = b dτ1 dτ2 eπbQ(τ1 −τ2 )−iπb (τ1 +τ2 ) +2iπb tτ1 R

Gb (−ibτ1 ) i(t−τ1 −τ2 ) i(τ1 +τ2 ) u v Gb (Q + ibτ2 )  2 2 = b2 dτ dτ2 eπbτ (Q+2ibt)−iπb τ −2πbτ2 (Q+ibt) ×

R

Gb (ibτ2 − ibτ ) i(t−τ ) iτ u × v , Gb (Q + ibτ2 ) where we introduced τ ≡ τ1 +τ2 . Computing the integral over τ2 with the help of (3.30), we derive an analogue of the binomial formula:   t it (u + v) = b dτ ui(t−τ ) v iτ , (B.4) τ b R+i0 where the b-binomial coefficient is given by 2  e2πib τ (t−τ ) Gb (Q + ibt) t . (B.5) = τ b Gb (Q + ibt − ibτ ) Gb (Q + ibτ ) We see that the function Gb is a b-analogue of the factorial. The b-binomial coefficients satisfy the q-Pascal identity:      t −i t t t t = q −2iτ + = + q 2i(τ −t+i) (B.6) τ τ b τ +i b τ +i b τ b b which can be easily verified with the help of (3.3). Using the b-binomial coefficients and the integral representation (3.31) of gb (x), we derive the quantum exponential relation:  2 2 e−πib t (3.31) gb (u + v) = b dt (u + v)it Gb (Q + ibt) R  2 2  e−πib t (B.4) 2 t = b ui(t−τ ) v iτ dt dτ τ b Gb (Q + ibt) R  2 2 2 2 e−πib (t−τ ) −πib τ (B.5) 2 ui(t−τ ) v iτ = b dt dτ Gb (Q + ib(t − τ ))Gb (Q + ibτ ) R   2 2 2 2 e−πib T e−πib τ (3.31) iT = b dT u b dτ v iτ = gb (u) gb (v) . Gb (Q + ibT ) Gb (Q + ibτ ) R R This completes the proof of Lemma 6.

R-Operator, Co-Product and Haar-Measure for the Modular Double

195

Remark 5. After this manuscript was written we were informed that a different proof of the quantum exponential relation and of relation (3.16) is given in [V]. Acknowledgement. We are grateful to L. Faddeev for useful comments. We thank A.Volkov for providing us with a draft version of [V] prior to its publication. A.B. was supported by Alexander von Humboldt Foundation. J.T. was supported by DFG SFB 288. A part of this work was carried out during A.B.’s visit to the Department of Mathematics, University of York.

References [Ba]

Barnes, E.W.: Theory of the double gamma function. Phil. Trans. Roy. Soc. A196, 265–388 (1901) [BR] Buffenoir, E., Roche, Ph.: Harmonic analysis on the quantum Lorentz group. Commun. Math. Phys. 207, 499–555 (1999); and: Tensor product of principal unitary representations of quantum Lorentz group and Askey-Wilson polynomials. J. Math. Phys. 41, 7715–7751 (2000) [F1] Faddeev, L.D.: Discrete Heisenberg-Weyl group and modular group. Lett. Math. Phys. 34, 249– 254 (1995) [hep-th/9504111] [F2] Faddeev, L.D.: Modular double of a quantum group. Math. Phys. Stud. 21, 149–156 (2000) [math.QA/9912078] [FK1] Faddeev, L.D., Kashaev, R.M.: Quantum dilogarithm. Mod. Phys. Lett. A9, 427–434 (1994) [hep-th/9310070] [FKV] Faddeev, L.D., Kashaev, R.M., Volkov, A.Y.: Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality. Commun. Math. Phys. 219, 199–219 (2001) [hepth/0006156] [FK2] Faddeev, L., Kashaev, R.: Strongly coupled quantum discrete Liouville Theory. II: Geometric interpretation of the evolution operator. J. Phys. A35, 4043–4048 (2002) [hep-th/0201049] [FV] Faddeev, L.D., Volkov, A.Y.: Abelian current algebra and the Virasoro algebra on the lattice. Phys. Lett. B315, 311–318 (1993) [hep-th/9307048] [Ka] Kashaev, R.M.: The non-compact quantum dilogarithm and the Baxter equations. J. Stat. Phys. 102, 923–936 (2001) [KK] Koelink, E., Kustermans, J.: A locally compact quantum group analogue of the normalizer of SU (1, 1) in SL(2, C). Commun. Math. Phys. 233, 231–296 (2003) ˜ (1, 1) and its Pontryagin dual. In: Locally compact Koelink, E., Kustermans, J.: Quantum SU quantum groups and groupoids, Vainerman, L. (ed.) IRMA Lectures on Mathematics and Mathematical Physics, Berlin-New York: Walter de Gruyter, 2003, pp. 49–78 [KLS] Kharchev, S., Lebedev, D., Semenov-Tian-Shansky, M.: Unitary representations of Uq (sl(2, R)), the modular double, and the multiparticle q-deformed Toda chains. Commun. Math. Phys. 225, 573–609 (2002) [hep-th/0102180] [KV] Kustermans, J., Vaes, S.: The operator algebra approach to quantum groups. Proc. Natl. Acad. Sci. USA. 97(2), 547–552 (2000) [PT1] Ponsot, B., Teschner, J.: Liouville bootstrap via harmonic analysis on a non-compact quantum group. [hep-th/9911110] [PT2] Ponsot, B., Teschner, J.: Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of Uq (sl(2, R)). Commun. Math. Phys. 224, 613–655 (2001) [math.QA/0007097] [PW] Pusz, W., Woronowicz, S.L.: Representations of quantum Lorentz group on Gelfand spaces. Rev. Math. Phys. 12, 1551–1625 (2000) [Ru] Ruijsenaars, S.N.M.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069–1146 (1997) [S] Schm¨udgen, K.: Operator representations of Uq (sl(2, R)). Lett. Math. Phys. 37, 211–222 (1996) [Sch] Sch¨utzenberger, M.P.: Une interpr`etation de certaines solutions de l’`equation fonctionnelle: F (x + y) = F (x)F (y). C. R. Acad. Sci. Paris 236, 352–353 (1953); Cigler, J.: Operatormethoden f¨ur q-Identit¨aten. Monatsh. Math. 88, 87–105 (1979) [Sh] Shintani, T.: On a Kronecker limit formula for real quadratic fields. J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 24, 167–199 (1977) [T] Teschner, J.: Liouville theory revisited. Class. Quant. Grav. 18, R153–R222 (2001) [hep-th/ 0104158] [W1] Woronowicz, S.: Unbounded elements affiliated with C ∗ -algebras and non-compact quantum groups. Commun. Math. Phys. 136, 399–432 (1991)

196 [W2] [V]

A.G. Bytsko, J. Teschner Woronowicz, S.L.: Quantum exponential function. Rev. Math. Phys. 12, 873–920 (2000) Volkov, A.Y.: Noncommutative hypergeometry. To appear

Communicated by L. Takhtajan

Commun. Math. Phys. 240, 197–241 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0893-6

Communications in

Mathematical Physics

Residues of q-Hypergeometric Integrals and Characters of Affine Lie Algebras Atsushi Nakayashiki Faculty of Mathematics, Kyushu University, Ropponmatsu 4-2-1, Fukuoka 810-8560, Japan. E-mail: [email protected] Received: 18 October 2002 / Accepted: 24 March 2003 Published online: 18 July 2003 – © Springer-Verlag 2003

Abstract: We study certain subspaces of solutions to the sl2 rational qKZ equation at level zero. Each subspace is specified by the vanishing of the residue at a certain divisor which stems from models in two dimensional integrable field theories. We determine the character of the subspace which is parametrized by the number of variables and the sl2 weight of the solutions. The sum of all characters with a fixed weight gives rise to the branching functions of the irreducible representations of sl2 in the level one integrable
2 . It is written in the fermionic form. highest weight representations of sl 1. Introduction In this paper we study a problem arising from the study of two dimensional integrable quantum field theories. It is related to the calculation of the character of the space of local operators. Let us first recall the known simplest example to give a preliminary insight on what kind of problem we are going to study. For the two dimensional massive Ising field theory in zero magnetic field, the problem of classifying local operators reduces to classifying the solutions to the equation [2, 3, 6] f (x1 , · · · , xn−2 , x, −x) = 0,

(1)

where f (x1 , · · · , xn ) is a symmetric polynomial. If we denote Rn the ring of symmetric polynomials of xj ’s, the space of solutions of (1) is given by Mn :=

Rn
+ n,


+ n

=

n 

(xi + xj ).

i<j

Specifying deg xi = 1, ∀i, Mn decomposes into the direct sum of the degree d subspaces Mn (d). The generating function of the dimension of Mn (d) is calculated as

198

A. Nakayashiki

ch Mn :=



1

q d dim Mn (d) =

d

q 2 n(n−1) , (q)n

(q)n =

n 

(1 − q j ).

j =1

Let M (i) , i = 0, 1 be the direct sum of Mn with n being even for i = 0 and odd for i = 1. Then ch M

(i)

=

 n≡i mod. 2

1

q 2 n(n−1) . (q)n

These infinite sums can be rewritten into the infinite product form [3, 6] ch M (0) = ch M (1) =

∞ 

(1 + q j ).

j =1

The last expression is the character of the Fock space of free fermions. The result is expected because the two dimensional Ising model in a zero magnetic field is described, at the critical point, by the free fermion conformal field theory. The two particle S-matrix of the Ising model is −1. To go beyond the Ising model there are two possibilities. One is to consider models with more complicated but diagonal S-matrices. In this case similar calculations were carried for several models in [6]. The other is to consider models with non-diagonal S-matrices. Up to now no similar calculations have been successful in such cases. In this paper we carry out similar calculations for the simplest model with a nondiagonal S-matrix, the SU (2) invariant Thirring model [16]. In this case the symmetric polynomial f in (1) is replaced by a tensor valued function satisfying a certain system of equations. The solutions to this system of equations are given by multi-dimensional q-hypergeometric integrals. For these integrals we have to solve the equation corresponding to (1). Thus the problem we consider looks far more complicated than the Ising case. Nevertheless we can solve this problem completely as shown in this paper. Let us explain our results supplying more precise statements. Let V be a two dimensional vector space and S(β) the linear operator acting on V ⊗2 defined by ˆ S(β) = S0 (β)S(β),

β − πiP ˆ , S(β) = β − πi

S0 (β) =

−β ( πi+β 2πi )( 2πi ) β ( πi−β 2πi )( 2πi )

,

where P is the operator which permutes the components of V ⊗2 . In general for a linear operator A acting on V ⊗2 , we denote Aij the operator on V ⊗n which acts for i th and j th components as A and other components as an identity. Consider the system of equations for a V ⊗n -valued function f : Pi,i+1 Si,i+1 (βi − βi+1 )f (β1 , · · · , βn ) = f (· · · , βi+1 , βi , · · · ),

(2)

n 2

Pn−1,n Pn−1,n−2 · · · P1,2 f (β1 − 2πi, · · · , βn ) = (−1) f (β2 , · · · , βn , β1 ). (3) This set of equations implies the sl2 rational qKZ equation at level zero [4, 15] and it can be solved in terms of multi-dimensional q-hypergeometric integrals [8, 11, 14, 16]. In this description of solutions, meromorphic solutions of (2) and (3) are parametrized by anti-symmetric polynomials P (X1 , · · · , X ) of Xa ’s of degree less than n in each variable with the coefficients in the space Cn of symmetric and 2π i-periodic

Residues of q-Hypergeometric Integrals

199

meromorphic functions of βj ’s; here  specifies the sl2 weight of a solution. We restrict ourselves to the case where the coefficients are in the space Rn of symmetric polynomials of xj = eβj , 1 ≤ j ≤ n and write P as P (X1 , · · · , X |x1 , · · · , xn ) indicating xj dependence explicitly. We denote by P the solution corresponding to P . It is known [12] that P , as a function of βn , has at most a simple pole at βn = βn−1 + π i. Then the condition corresponding to (1) is Resβn =βn−1 +πi f (β1 , · · · , βn ) = 0.

(4)

This condition should be rewritten as the condition for P . In fact we have derived equations for P which imply (4). They are P (X1 , · · · , X−1 , ±x −1 |x1 , · · · , xn−2 , x, −x) = 0.

(5)

We conjecture that these equations are equivalent to (4). Let Un, be the space of solutions of (5). Since P is an anti-symmetric polynomial of Xa ’s, it can be considered as an element of the th exterior product of a free Rn -module with the monomials Xj as a basis: P ∈ ∧ H (n) ,

j H (n) = ⊕n−1 j =0 Rn X .

There exist two special solutions 1 ∈ Un,1 ,

2 ∈ Un,2 ,

for which 1 and 2 vanish identically [11, 17, 18]. Moreover a polynomial P with the coefficients in Cn such that P vanishes identically belongs to the space [18] ∧−1 Hˆ (n) ∧ 1 + ∧−2 Hˆ (n) ∧ 2 ,

j Hˆ (n) = ⊕n−1 j =0 Cn X .

In this respect we consider the space Un, , Un,−1 ∧ 1 + Un,−2 ∧ 2

Mn, := and, for a non-negative integer λ, (i)

Mλ = ⊕n≡i mod. 2, n−2=λ Mn, . We define a degree, denoted by deg1 , assigning deg1 Xa = −1, deg1 xj = 1. If P is homogeneous by deg1 , P becomes homogeneous of degree deg2 P =

n2 + deg1 P , 4

P ∈ Un, .

(i)

We introduce a grading on Mλ by deg2 . Then Theorem 1. (i) ch Mλ

=

 n−2=λ, n≡i mod. 2

where

n2

q4 (q)n

     n n , −  q −1 q

  (q)n n = .  q (q) (q)n−

(6)

200

A. Nakayashiki

For a non-negative half integer S let χS (z) =

z2S+1 − z−(2S+1) z − z−1

be the character of the 2S + 1-dimensional irreducible representation of sl2 . An easy calculation shows 

(i) χλ/2 (z) ch Mλ

λ≡i mod. 2

=



2

n n  zn−2 q 4 . (q) (q)n−

n≡i mod. 2 =0

The last expression is the fermionic form of the character of the level one integrable
2 presented in [5, 10]. highest weight representation V ( i ) of the affine Lie algebra sl The formula (6) gives the branching function of the λ + 1-dimensional irreducible representation of sl2 in V ( i ). Finally we remark that, for the Ising model, it is possible to construct a local operator, in the form of a set of form factors, to each element of M (i) [2, 3]. The problem of (0) constructing form factors corresponding to the elements of Mλ will be studied in the subsequent paper. The present paper is organized in the following manner. In Sect. 2 the null residue equation (5) for a polynomial P is derived. Sections 3 to 9 are devoted to the case n even. We study the solution of (5) with  = 1 in Sect. 3. The module U2n,1 is proved to be a free R2n -module by giving its basis explicitly. In Sect. 4 the space U2n,2 is studied. Again we prove that U2n,2 is a free R2n -module by constructing a basis. In Sect. 5 we present a candidate of a basis of Un, in the space of exterior product of solutions for  = 1 and 2. Some combinatorial identity related to q-binomials and q-tetranomials are proved as a preparation of the study of Un, for general . In Sect. 6 the structure theorem of U2n, is given and a proof of it is presented assuming the linear independence of the basis. Sections 7 and 8 are devoted to the proof of the linear independence of the basis. The character formula (6) with n even is derived in Sect. 9. In Sect. 10 the results on the case n odd are presented and proved. In Appendix A the details of calculations of the residue in Sect. 2 is given.

2. Null Residue Equation Let Rn be the ring of symmetric polynomials of x1 , ..., xn . We consider the space of polynomials in X of degree at most n − 1 with the coefficients in Rn : k H (n) := ⊕n−1 k=0 Rn X .

The th exterior product ∧ H (n) is identified with the space of anti-symmetric polynomials P in X1 , ..., X of degree less than n in each variable with the coefficients in Rn . The identification is given by the map Xi1 ∧ · · · ∧ X i → Asym(X1i1 · · · Xi ) =

 σ ∈S

sgn σ Xσi1(1) · · · Xσi() .

Residues of q-Hypergeometric Integrals

201

In the following we suppose that the variables Xa , xj and αa , βj are related by Xa = e−αa , xj = eβj . To each element P ∈ ∧ H (n) and the index set M = (m1 , · · · , m ), 1 ≤ m1 < · · · < m ≤ n, we associate the integral:  IM (P ) :=

C

 

 

P (X1 , · · · , X |x1 , · · · , xn ) , φn (αa )gM  n a=1 j =1 (1 − Xa xj ) a=1

dαa

a=1

(7)

where φn (α) =

α−βj +πi n  ( −2πi ) j =1

gM =

  a=1

α−β



( −2πij )

,

 m a −1 αa − βj + πi 1   αa − βma αa − β j j =1



(αa − αb + π i).

1≤a
The integration contour C is defined as follows. It goes from −∞ to +∞, separating two sets {βj , βj − 2π i, βj − 4πi, · · · |1 ≤ j ≤ n} and {βj − π i, βj + π i, βj + 3π i, · · · |1 ≤ j ≤ n}. The set of integrals {IM (P )} satisfies a certain system of equations. Let us describe it. To this end we define a tensor valued function whose component is IM (P ). Let V be the vector space with the basis v± : V = Cv+ ⊕ Cv− . We consider V as the vector representation of sl2 by the identification v+ = t (1, 0) and v− = t (0, 1). To each M = (m1 , · · · , m ), 1 ≤ m1 < · · · < m ≤ n, we define the element vM in V ⊗n by vM = v1 ⊗ · · · ⊗ vn ,

M = {j | j = − }.

Using them define the V ⊗n -valued function ψP by  ψP = IM (P )vM . M

To write down the equations we need the operator acting on V ⊗2 defined by β − πiP ˆ S(β) = , β − πi

P (v1 ⊗ v2 ) = v2 ⊗ v1 .

In general for a linear operator A acting on V ⊗2 , writing  A= Ba ⊗ Ca , Ba , Ca ∈ End(V ), a

we define the linear operator Aij acting on V ⊗n by  1 ⊗ · · · ⊗ Ba ⊗ · · · ⊗ Ca ⊗ · · · ⊗ 1, Aij = a

where Ba and Ca are in i th and j th components respectively.

202

A. Nakayashiki

Then ψP satisfies the following system of equations: Pi,i+1 Sˆi,i+1 (βi − βi+1 )ψP (β1 , · · · , βn ) = ψP (· · · , βi+1 , βi , · · · ), Pn−1,n Pn−1,n−2 · · · P1,2 ψP (β1 − 2πi, · · · , βn ) = ψP (β2 , · · · , βn , β1 ).

(8) (9)

Here we remark on the solutions to Eqs. (2) and (3) in the introduction. By multiplying a scalar function we define P by  n n P = e 4 j =1 βj ζ (βj − βk )ψP , (10) j
where ζ (β) is a certain meromorphic function. See [12] and references therein for the precise description of ζ (β). We simply remark here that ζ (β) is holomorphic and nonzero at β = −π i. The function P solves Eqs. (2) and (3). Conversely every solution of (2) and (3) can be written using P . Let us give more detailed and precise statements (n) on this fact. Let ± (X) = nj=1 (1 ± Xxj ) and set [1, 17, 18] (n)

(n)

(n)

21 (X) = + (X) + (−1)n−1 − (X),  X − X 1 2 (n) (n) (n) (n) (n) 22 (X1 , X2 ) = + (X1 )+ (X2 ) − − (X1 )− (X2 ) X1 + X 2   (n) (n) (n) (n) +(−1)n + (X1 )− (X2 ) − + (X2 )− (X1 ) .

(11)

(12)

Notice that the space of meromorphic solutions of (8) and (9) is a vector space over the field Cn of symmetric and 2πi-periodic meromorphic functions of β1 , . . . , βn . The ring Rn becomes a subring Cn . Consider the scalar extension of H (n) from Rn to Cn : j Hˆ (n) = Cn ⊗Rn H (n) = ⊕n−1 j =0 Cn X .

Then Theorem 2 ([18]). (i) Any meromorphic solution of (2) and (3) which takes the value in the space of sl2 highest weight vectors in V ⊗n with the weight n − 2 can be written as P for some P ∈ ∧ Hˆ (n) . (ii) For P ∈ ∧ Hˆ (n) , P = 0 if and only if (n) (n) P ∈ ∧−1 Hˆ (n) ∧ 1 + ∧−2 Hˆ (n) ∧ 2 .

The function P has a nice homogeneity property. Let us define degrees, denoted by deg1 , of elements in ∧ H (n) by assigning deg1 X = −1,

deg1 xj = 1.

Then Proposition 1. If P is homogeneous by deg1 , P (β1 + θ, · · · , βn + θ ) = edeg2 (P )θ P (β1 , · · · , βn ), where n2 + deg1 P . 4 Proof. Notice that αa , βj appear in the integrand of IM (P ) in the form αa − βj or αa − αb except in P . The proposition immediately follows from this.  deg2 (P ) =

Residues of q-Hypergeometric Integrals

203

In the following the degree of an element of ∧ H (n) is measured by deg1 unless otherwise stated. As a function of βn , ψP has at most a simple pole at βn = βn−1 + π i [12]. We shall study the space of P such that Resβn =βn−1 +πi ψP = 0.

(13)

By the remark below (10) this is equivalent to (4). Let us find the condition for P such that (13) holds. For a polynomial P (X1 , · · · , X |x1 , · · · , x2n ) of xi s and Xj s, define ρ± (P ) = P (X1 , · · · , X−1 , ±x −1 |x1 , · · · , x2n−2 , x, −x),

x = xn−1 ,

and for a set M = (m1 , · · · , m ), 1 ≤ m1 < · · · < m ≤ n, define M (a) = (m 1 , · · · , m −1 ) = (m1 , · · · , ma−1 , ma+1 , · · · , m ). Let us set g˜ M = gM |βn =βn−1 +πi ,

Dn (X) =

n 

(1 − Xxj ),

j =1

J˜M (P ) =

−1 

P

φn−2 (αb ) · g˜ M · −1

2 2 b=1 Dn−2 (Xb )(1 − Xb xn−1 )

b=1

Using these notations we define (1) IM (P )

=

 

n−2  βn−1 − βj (−1)−1 βn−1 − βma − πi βn−1 − βj − π i

a=1

 × 

(2)

IM (P ) =

(4)

−1 

−1 

C −1 b=1



IM (P ) =

C −1 b=1

C −1 b=1



(3)

IM (P ) =

j =1

−1 

−1 

C −1 b=1

dαb J˜M (a) (P )

a−1

b=1 (αb − βn−1 + 2π i) , −1 b=a (βn−1 − αb )

dαb J˜M () (P ), dαb J˜M () (P )

−1  b=1

dαb J˜M (−1) (P )

αb − βn−1 , αb − βn−1 + π i

βn−1 − α−1 + π i , α−1 − βn−1 + π i

and (1)

fn−1 = (2)

−(−2πi) φn−1 (βn−1 − πi) , −1 2 Dn−2 (−xn−1 ) (4)

fn−1 = fn−1 = (−2πi)−1 (3)

fn−1 = (−2πi)−1

φn−1 (βn−1 ) −1 Dn−2 (xn−1 )

,

n−2 φn−1 (βn−1 + πi)  βn−1 − βj + 2π i . −1 βn−1 − βj Dn−2 (−xn−1 ) j =1

.

204

A. Nakayashiki

Then Proposition 2. For P ∈ ∧ H (2n) , the residue of IM is given by 1 Resβn =βn−1 +πi IM (P ) 2π i (1) (1) / M, = fn−1 IM (ρ− (P )) if n − 1, n ∈ (1)

(1)

(2)

(2)

/ M, = fn−1 IM (ρ− (P )) − fn−1 IM (ρ+ (P )) if n − 1 ∈ M, n ∈ =

3 

(j )

(j )

(j )

(j )

fn−1 IM (ρ(−1)j (P )) if n ∈ M, n − 1 ∈ / M,

j =1

=

4 

fn−1 IM (ρ(−1)j (P )) if n − 1, n ∈ M,

j =1

where ±1 are identified with ±. Corollary 1. If ρ+ (P ) = ρ− (P ) = 0,

(14)

then (13) is satisfied. Remark. The residue of IM (P ) can vanish for some M although ρ+ (P ) or ρ− (P ) is not zero. The P ’s corresponding to form factors of local operators give such examples [16, 12]. We conjecture that if the residue of IM (P ) vanishes for all M (14) holds. In other words we conjecture that (13) and (14) are equivalent. It is true for  = 1. Let us write down Eq. (14) in component forms. To this end we introduce a notation. For a polynomial f (x1 , · · · , xn ) we define f¯ by f¯ = f (x1 , · · · , xn−2 , x, −x), Expand P ∈ ∧ H (2n) as P =



x = xn−1 .

Pi1 ···i X i1 ∧ · · · ∧ X i ,

such that the coefficients Pi1 ···i ’s are anti-symmetric with respect to indices: Piσ1 ···iσ = sgn σ · Pi1 ···i , Then (14) is equivalent to  i : even

Pi1 ···i x −i =



σ ∈ S .

Pi1 ···i x −i = 0.

(15)

i : odd

Let Un, be the space of solutions of (14): Un, = { P ∈ ∧ H (n) | ρ+ (P ) = ρ− (P ) = 0 }. It is obviously an Rn -module. In the subsequent sections we shall study the structure of Un, .

Residues of q-Hypergeometric Integrals

205

3. Solutions for
= 1 In this section we study U2n,1 . Eq. (15) for P = n−1 

P2i (x)x −2i =

i=0

n 

2n−1

Pi (x)X i are

i=0

P2i−1 (x)x −(2i−1) = 0.

(16)

i=1

Thus the sets of polynomials {P2i } and {P2i−1 } satisfy the same equation independently. The solutions of Eq. (16) can be found easily as we shall see below. (n) Let ek be the elementary symmetric polynomial defined by n 

(1 + xj t) =

j =1

n 

(n)

ek t k .

k=0

(n)

(0)

We set ek = 0 for k > n, k < 0 or n < 0 and e0 = 1. Example 3.1. The polynomial (2n)

P = e1

(2n)

X + e3

(2n)

X 3 + · · · + e2n−1 X 2n−1

is a solution of (16). To verify this we use (n)

(n−2)

ek = ek

(n−2)

− x 2 ek−2 ,

(17)

which is valid for all integers k. Then n 

e2s−1 x −(2s−1) = (2n)

s=1

=

n  

(2n−2)

(2n−2)

e2s−1 − x 2 e2s−3

s=1 n 

e2s−1 x −(2s−1) − (2n−2)



n 

s=1

x −(2s−1) e2s−1 x −(2s−1) = 0. (2n−2)

s=1

Beginning from the solution in Example 3.1 we can construct a set of solutions of (2n) (16). Let Pr,s , 1 ≤ r ≤ n, s ∈ Z be defined by the following recursion relations: (2n)

(2n)

P1,s = e2s−1 , (2n)

(2n)

(2n)

(2n) = Pr−1,s+1 − e2s Pr−1,1 Pr,s

for r ≥ 2.

(18)

We easily have (2n) Pr,s = 0,

s ≤ 0.

(19)

Proposition 3. We have n 

Pr,s x −2(s−1) = 0. (2n)

s=1

Proof. Let us prove this equation by the induction on r. The case r = 1 is verified in Example 3.1.

206

A. Nakayashiki (2n)

Lemma 1. Pr,s

= 0, s ≥ n + 1.

This lemma is easily proved by the induction on r. (2n)

Lemma 2. Pr,s

(2n−2)

(2n−2)

= Pr,s

− x 2 Pr,s−1 .

Proof. Let us prove the lemma by the induction on r. If r = 1, the assertion is (17) and there is nothing to prove. For r ≥ 2, using (18) and (17) we have (2n)

Pr,s

(2n)

(2n)

(2n)

= Pr−1,s+1 − e2s Pr−1,1

  (2n−2) (2n−2) (2n−2) (2n−2) (2n−2) = Pr−1,s+1 − x 2 Pr−1,s − e2s − x 2 e2s−2 Pr−1,1   (2n−2) (2n−2) (2n−2) (2n−2) (2n−2) (2n−2) Pr−1,1 − x 2 Pr−1,s − e2s−2 Pr−1,1 = Pr−1,s+1 − e2s (2n)

(2n−2) − x 2 Pr,s−1 . = Pr,s

 Using Lemma 1, 2 and (19) we have, for r ≥ 2, n 

n  

Pr,s x −2(s−1) = (2n)

(2n−2)

(2n−2) Pr,s − x 2 Pr,s−1

s=1



x −2(s−1) = 0.

s=1



Thus Proposition 3 is proved. (2n)

Define the elements vr vr(2n)

=

(2n)

, wr

n 

of H (2n) by wr(2n) = Xvr(2n) .

(2n) 2(s−1) Pr,s X ,

(20)

s=1 (2n)

Corollary 2. The elements vr

(2n)

, wr

are solutions of (16). (2n)

Theorem 3. The space U2n,1 is a free R2n module of rank 2n with {vr basis: U2n,1 = ⊕nr=1 R2n vr(2n) ⊕nr=1 R2n wr(2n) . (2n)

For the proof let us define the n by n matrix X (2n) = (Xij ) by (2n)

Xij Proposition 4. Let
+ n =

n

(2n)

= Pi,j ,

i<j (xi

1 ≤ i, j ≤ n.

+ xj ). Then

det X (2n) =
+ 2n . We prepare several lemmas in order to prove Proposition 4.

(2n)

, wr

} as a

Residues of q-Hypergeometric Integrals

207

Lemma 3. As a polynomial of x1 , . . . , x2n , det X(2n) is homogeneous of degree (2n)

2n 2

.

(2n)

Proof. It can be easily checked that Xij = Pi,j is homogeneous of degree 2i +2j −3. Thus for any permutation (j1 , ..., jn ) of (1, ..., n) we have (2n)

(2n)

deg(X1j1 · · · Xnjn ) = 2

n 

i+2

i=1

n 

j − 3n = n(2n − 1).

j =1

 Lemma 4. det X (2n) is divisible by
+ 2n . Proof. Since det X(2n) is a symmetric polynomial, it is sufficient to prove det X(2n) = 0.

(21) −2(j −1)

In det X(2n) we successively add the j th column multiplied by x2n−1 column. Then the i th row of the first column becomes n 

to the first

Pi,j x −2(j −1) . (2n)

j =1



Thus (21) follows from Proposition 3.

Lemma 5. The expression of det X(2n) as a polynomial of e1 ,...,e2n is 1

n−1 det X(2n) = (−1) 2 n(n−1) e1n e2n + ··· , (2n)

where ek = ek

n−1 and · · · part does not contain e1n e2n . (2n)

Proof. It can be easily proved that Pr,s is at most of degree 1 in e2n , is of degree 0 for s < n − r + 2 and (2n)

Pr,n−r+2 = −e1 e2n + (terms of degree 0 in e2n ). Thus

lim

1

e2n →∞ en−1 2n

det X(2n)

which proves the lemma.

   e1 e3 · · · e2n−3 e2n−1    0 −e1   0 0 ···  0 0 · · · −e ∗  = (−1) 21 n(n−1) en , 1 =  1 .. ..   .. . .  .  0 −e · · · · · · ∗ 1



Lemma 6. The expression of
+ 2n as a polynomial of e1 ,...,e2n is 2 n(n−1) e n e n−1 + · · · .
+ 1 2n 2n = (−1) 1

208

A. Nakayashiki

Proof. The statement follows easily from (see [9] for example)
+ 2n = det(e2n−2i+j )1≤i,j ≤2n−1 . (2n)

 

Now Proposition 4 is obvious from Lemma 4, 5 and 6. (2n) (2n) By Proposition 4, vi , wj , i, j

Proof of Theorem 3. = 1, ..., n are linearly independent over R2n . Thus it is sufficient to prove that any solution P =

n 

Ps X 2(s−1)

s=1 (2n)

of (16) is written as a linear combination of vi , 1 ≤ i ≤ n with the coefficients in R2n . Let us consider the following linear equation for Qi , i = 1, ..., n: P =

n 

(2n)

Qi vi

,

i=1

which, in the matrix form, is (P1 , · · · , Pn ) = (Q1 , · · · , Qn )X (2n) . By Cramer’s formula, using Proposition 4, this equation is solved as Qi =

1 (2n) det Xi ,
+ 2n

(2n)

where the matrix Xi is defined by replacing the i th row in X (2n) by (P1 , · · · , Pn ). In (2n) a similar way to Lemma 4 one can prove that Xi is divisible by
+ 2n . Thus Qi ∈ R2n and the theorem is proved.  (2n)

Notice that if ψP = 0 the residue of ψP is zero. Thus 1 fact we have the relation (2n)

1

(2n)

= w1

should be in U2n, . In (22)

,

which is obvious from their definitions. 4. Solutions for
= 2 In this section we solve Eq. (14) in the case  = 2. In general, for solutions uj , 1 ≤ j ≤ m, of (14) with  = j , it is obvious that their exterior product  u1 ∧ · · · ∧ um = sgn σ u1 (Xσ1 , · · · , Xσ1 ) · · · um (Xσ1 +···+m−1 +1 , · · · , Xσ ), σ

is a solution of (14) with  = 1 + · · · + m . Thus, from the results of the previous section, (2n)

vi1

(2n)

∧ vi2 ,

(2n)

vi

(2n)

∧ wj

,

(2n)

wj1

(2n)

∧ wj2 ,

are solutions of (14) with  = 2 for any i1 , i2 , i, j , j1 , j2 . In fact there exist solutions which can not be expressed as an R2n -linear combination of those solutions. Let us give a construction of them.

Residues of q-Hypergeometric Integrals

209

Set (2n)

v0

=

n 

(2n)

e2j X 2j .

j =0 (2n)

Although v0

are not in H (2n) it satisfies (2n)

ρ± (v0

) = 0.

(2n)

we define, for 1 ≤ j ≤ n,  X1 − X2  (2n) (2n) (2n) (2n) (2n) 2ξj (X1 , X2 ) = v0 (X1 )wj (X2 ) + v0 (X2 )wj (X1 ) X1 + X 2

Using v0

(2n)

−v0 Since

(2n)

v0

(2n)

(X1 )wj (2n)

(−X) = v0

(2n)

(X2 ) + v0 (2n)

(X),

wj

(2n)

(X1 ).

(2n)

(X),

(X2 )wj

(−X) = −wj

(23)

ξj is an anti-symmetric polynomial of X1 and X2 with the coefficients in R2n . The definition of ξj is motivated by the relation (2n)

(2n)

= ξ1

2

(24)

,

which can be easily checked. (2n)

Proposition 5. ξj

∈ U2n,2 ,

1 ≤ j ≤ n.

Proof. Applying ρ± to (23) we see that  X1 ∓ x −1  (2n) (2n) (2n) (2n) (2n) ρ± (2ξj ) = v (X )ρ (w ) + ρ (v )w (X ) 1 ± ± 0 1 j j X1 ± x −1 0 (2n)

−v0

(2n)

(X1 )ρ± (wj

(2n)

) + ρ± (v0

(2n)

)wj

(X1 )

= 0, (2n)

where v0 (X1 ) means, for example, that we take bar of each coefficient of the power of X1 . (2n) (2n) ≤ 2n − 1, i = 1, 2. Obviously degXi ξj ≤ 2n, It remains to prove degXi ξj 2n i = 1, 2. Let us calculate the coefficient of X1 as a polynomial of X1 as (2n)

2ξj

lim

X12n  1− = lim X1 →∞ 1 +

X1 →∞

(2n)

−

v0

X12n

(2n)

(2n)

= e2n wj (2n)

Thus degX1 ξj 

(X1 )

X2 X1 X2 X1



(2n)

v0

(X1 )

X12n

(2n)

(2n) (2n) wj (X2 ) + v0 (X2 ) (2n)

(2n) (2n) wj (X2 ) + v0 (X2 ) (2n)

(2n)

(X2 ) − e2n wj

wj

(X1 )

wj

(X1 )



X12n



X12n

(X2 ) = 0. (2n)

≤ 2n − 1. By the anti-symmetry of ξj

(2n)

we get degX2 ξj

≤ 2n − 1.

210

A. Nakayashiki

Theorem 4. The module U2n,2 is a free R2n module of rank elements as a basis: (2n)

(2n)

vi1

∧ vi2

(2n) vi

(2n) ∧ wj

(2n)

(1 ≤ i1 < i2 ≤ n),

wj1

(2n)

∧ wj2

(1 ≤ i ≤ n, 1 ≤ j ≤ n − 1),

2n 2

with the following

(1 ≤ j1 < j2 ≤ n),

(2n) ξk

(1 ≤ k ≤ n).

(25)

The linear independence of (25) is a special case of Theorem 6. Let us prove the theorem assuming the linear independence.   First of all one can easily check that the number of elements (25) is 2n 2 . We number 2n the elements (25) from 1 to 2 and name the r th element by Qr . Let  Qr;ij X i ∧ X j , Qr;ij ∈ R2n . Qr = i<j

We set Qr;ij = −Qr;j i for i ≥ j . We introduce lexicographic order from the left on the set J2 = { (ij ) | 0 ≤ i < j ≤ 2n − 1 }. 2n 2n   (2n) Define the 2 by 2 matrix X (2n) = (Xr;ij ), 1 ≤ r ≤ 2n 2 , (ij ) ∈ J2 by (2n)

Xr;ij = Qr;ij . Lemma 7. As a polynomial of x1 , ..., x2n , det X(2n) is homogeneous of degree n(2n − 1)(4n − 3). This is a special case of Proposition 7. 4n−3 . Lemma 8. det X (2n) is divisible by (
+ 2n )

This lemma is also a special case of Proposition 8 below. Since the proof of the lemma generalizes the case  = 1 and illustrates the proof in the general case, we shall give a proof of it here. Proof. In det X (2n) , to each (0, j )th column with j = 1, ..., 2n − 1, we successively −2i for 2 ≤ 2i < j and the (j, 2i)th column times add the (2i, j )th column times x2n−1 −2i for j < 2i. Then, to each (1, j )th column with j = 2, ..., 2n − 1, we add the −x2n−1 −(2i−1)

(2i − 1, j )th column times x2n−1

−(2i−1) times −x2n−1 for j < 2i (r; 0, j )th component Q r;0,j

Q r;0,j =

n−1 

for 3 ≤ 2i − 1 < j and the (j, 2i − 1)th column

− 1 successively. Then, in the resulting determinant, the and the (r; 1, j )th component Q r;1,j becomes

−2i Qr;2i,j x2n−1 ,

Q r;1,j =

i=0

n 

−(2i−1)

Qr;2i−1,j x2n−1

,

i=1

respectively. Since Qr is a solution of (15), Q r;0,j = Q r;1,j = 0. This means that Q r;0,j and Q r;1,j are divisible by x2n−1 + x2n . Thus det X (2n) is divisible by (x2n−1 + x2n )4n−3 . Because det X (2n) is a symmetric polynomial of x1 , ..., x2n , 4n−3 . it is divisible by (
+  2n )

Residues of q-Hypergeometric Integrals

211

The following lemma is obvious from the linear independence of (25) and Lemma 7, 8. Lemma 9. There is a non-zero constant c such that 4n−3 det X (2n) = c(
+ . 2n )

The proof of Theorem 4 is similar to that of Theorem 3 using Lemma 3 and 4. 5. q-Tetranomial Identity From now on we use the multi-index notations like vI = vi1 ∧ · · · ∧ vil ,

for I = (i1 , · · · , il ).

Consider the following elements of U2n, : (2n)

(2n)

(2n)

vI ∧ w J ∧ ξ K , I = (i1 , · · · , i1 ), 1 ≤ i1 < · · · < i1 ≤ n, J = (j1 , · · · , j2 ), 1 ≤ j1 < · · · < j2 ≤ n − 1 − 3 , K = (k1 , · · · , k3 ), 1 ≤ k1 ≤ · · · ≤ k3 ≤ n − 1 − 3 + 1, 1 + 2 + 23 = .

(26) (27) (28) (29) (30)

If  = 1, 2, these are the basis of U2n,1 and U2n,2 given in Theorem 3 and 4 respectively. Let us count the number N2n, of such elements. Obviously it is given by      n n −  1 n −  1 − 3 N2n, = . 1 3 2 1 +2 +23 =

In the right-hand side     n n −  1 n −  1 − 3 n! = , 1 3 2 1 !2 !3 !(n − 1 − 2 − 3 )! which is the tetranomial coefficient denoted by   n . 1 2 3

(31)

Therefore we have (1 + x + x + x 2 )n =

2n 

N2n, x  .

=0

The left-hand side is (1 + x)2n =

2n    2n  x .  =0

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A. Nakayashiki

Thus we have

 N2n, =

 2n = rank ∧ H (2n) , 

(32)

and the identity   2n = 

 1 +2 +23 =

    n n −  1 n −  1 − 2 , 1 2 3

(33)

here we used the symmetry of (31) with respect to the permutations of 1 , 2 , 3 . By the definition of deg1 we easily have (2n)

deg1 vi

(2n)

= 2i − 1,

deg1 wj

= 2j − 2,

(2n)

deg1 ξk

= 2k − 2.

Thus deg1



(2n) vI

(2n) ∧ wJ

(2n)  ∧ ξK

=

1 

(2ir − 1) +

r=1

2 

(2jr − 2) +

r=1

3 

(2kr − 2),

r=1

(34) if the element is non-zero. We denote the right-hand side of (34) by dI,J,K . Let us consider the generating function of dI,J,K :  ch2n, := q dI,J,K , (35) I,J,K

where I, J, K run the index sets satisfying (27) – (30). It is easy to evaluate it explicitly. To write it we use the q-integer notations:   [n]p ! n [n]p = 1 − p n , [n]p ! = [1]p [2]p · · · [n]p , = . r p [r]p ![n − r]p ! Then we easily obtain ch2n, =



q 1 +2 (2 −1) 2

1 +2 +23 =



n 1





q2

n − 1 3

 q2



n −  1 − 3 2

 . (36) q2

In the right-hand side       [n]q 2 ! n − 1 n −  1 − 2 n = 2  1 q 2 3 [1 ]q 2 ![2 ]q 2 ![3 ]q 2 ![n − 1 − 2 − 3 ]q 2 ! q2 q2 is the q-tetranomial coefficient written as 

n  1 2 3

 .

(37)

q2

If some i ≤ −1, we set (37) = 0. Notice that (37) is symmetric with respect to the permutations of 1 , 2 , 3 . Now we prove the q-analogue of (33).

Residues of q-Hypergeometric Integrals

Proposition 6.   2n =  q



q

213

21 +2 (2 −1)



1 +2 +23 =

n 1



 q2

n − 1 2



 q2

n −  1 − 2 3

 . (38) q2

Proof. Let us denote the left and the right-hand sides of (38) by an, and bn, respectively. By direct calculations one can easily prove (38) for  = 0, 1, 2. It is obvious that a1, = b1, = 0 for  ≥ 3. Thus (38) holds for n = 1. Let us prove the identity by the induction on n. To this end we shall show that an, and bn, satisfy the same recursion relation. The following recursion relations hold for q-binomial coefficients:       m−1 m−1 m = + p m−r (39) r −1 p r r p p     m−1 r m−1 =p + . (40) r r −1 p p Using (39) and (40) we have an, = an−1, + (q 2n−−1 + q 2n−1 )an−1,−1 + q 2n− an−1,−2 .

(41)

Let us prove the same recursion relation for bn, . Using (39) for the first q-binomial coefficient in the product in bn, we have        2 n−1 n − 1 n −  1 − 2 bn, = q 1 +2 (2 −1) (42) 1 q 2 2 3 2 2 q q 1 +2 +23 =  2 + q 2n−1 q (1 −1) +2 (2 −1)  ×

1 +2 +23 =

n−1 1 − 1





q2

n − 1 2

 q2



n −  1 − 2 3

 .

(43)

q2

It is obvious that (43) = q 2n−1 bn−1,−1 . Applying (39) to the second q-binomial coefficient in (42) we have        n − 1 − 1 n −  1 − 2 21 +2 (2 −1) n − 1 q (44) (42) = 1 q 2 2 3 q2 q2 1 +2 +23 =    n−1 21 +2 (2 +1)+2(n−1−1 −2 ) + q . (45)  1 2  3 q 2 1 +2 +23 =−1

We again use (39) to the third product of q-binomial coefficients in (44) we get    n−1 21 +2 (2 −1)+2n−−(1 +2 ) (44) = bn−1, + q . (46) 1 2 3 q 2 1 +2 +23 =−2

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A. Nakayashiki

Therefore we have bn, = bn−1, + q 2n−1 bn−1,−1 + (45) + (46). Using the symmetry of 1 and 2 of the q-tetranomial coefficient we see that    2 n−1 q 1 +2 (2 −1)+23 +2n−1− . (45) = 1 2 3 q 2

(47)

(48)

1 +2 +23 =−1

There is an obvious identity of the form     n n [3 ]q 2 = [1 + 1]q 2 ,  1 2  3 q 2 1 + 1  2 3 − 1 q 2 which is also valid for 3 = 0. We rewrite (49) as       n n n 23 = − q 1 2 3 q 2 1 2 3 q 2  1 + 1  2 3 − 1 q 2   n + q 2(1 +1) .  1 + 1  2 3 − 1 q 2

(49)

(50)

We substitute (50) into (48) and get (45) = q 2n−1− bn−1,−1   2 − q 1 +2 (2 −1)+2n−1−

 n−1 (51)  1 + 1  2 3 − 1 q 2 1 +2 +23 =−1    n−1 21 +2 (2 −1)+2n−1−+2(1 +1) + q . (52)  1 + 1  2 3 − 1 q 2 1 +2 +23 =−1

If 1 = −1, the sum of the summand of (51) and (52) becomes zero. Therefore the sum in (51) and (52) can be taken for 1 ≥ −1, 2 ≥ 0, 3 ≥ 1. Then shifting the summation variables 1 and 3 we have (51) + (52) = −(46) + q 2n− bn−1,−2 . We substitute this into (47) and get the recursion relation (41) for bn, . Thus by the induction on n Proposition 6 is proved.  6. Solutions for General
We shall show that the solutions of (14) for  ≥ 3 are generated by the solutions for  = 1, 2. More precisely we prove   Theorem 5. The module U2n, is a free R2n -module of rank 2n  with the set of elements (2n) (2n) (2n) {vI ∧ wJ ∧ ξK } as a basis: (2n)

U2n, = ⊕I,J,K R2n vI

(2n)

∧ wJ

(2n)

∧ ξK ,

where I, J, K runs over all index sets satisfying (27)–(30).

Residues of q-Hypergeometric Integrals

215

The strategy of the proof is similar to the case  = 1, 2. The most complex part of the proof is in the following theorem. (2n)

(2n)

(2n)

Theorem 6. The elements {vI ∧ wJ ∧ ξK }, where I, J, K runs over all index sets satisfying (27)–(30), are linearly independent over R2n . The proof of this theorem is given in Sect. 8. In this section we prove Theorem 5 assuming Theorem 6.   (2n) (2n) (2n) th We number the elements {vI ∧ wJ ∧ ξK } from 1 to 2n  and denote the r element by Qr . Expand Qr as  Qr = Qr;i1 ···i X i1 ∧ · · · ∧ X i . (53) 0≤i1 <···
We extend the index (i1 , · · · , i ) of Qr;i1 ···i to 0 ≤ i1 , · · · , i ≤ 2n − 1 such that Qr;i1 ···i is anti-symmetric with respect to the permutation of i1 , · · · , i . We introduce lexicographic order from the left on the set J2n, = { (i1 , · · · , i ) | 0 ≤ i1 < · · · < i ≤ 2n − 1 }.   2n (2n,) by Let us define the 2n  by  matrix X   2n (2n,) , (i1 , · · · , i ) ∈ J2n, , X(2n,) = (Xr;i1 ···i ), 1 ≤ r ≤ 

(54)

(2n,)

Xr;i1 ···i = Qr;i1 ···i . Proposition 7. As an element of R2n , det X (2n,) is homogeneous of degree 2n−2 −1 .

2n2n−1 2

−1

+

Proof. By the definition of deg1 , if Qr = vI ∧ wJ ∧ ξK , deg1 Qr;s1 ···s = dI,J,K + s1 + · · · + s . Thus, as in the proof of Lemma 3, det X (2n,) is homogeneous of degree   d (2n,) := dI,J,K + (s1 + · · · + s ). 0≤s1 <···<s ≤2n−1

I,J,K

The proposition follows from the following lemma. Lemma 10.



d

(2n,)

    2n − 1 2n − 2 + . −1 −1

2n = 2

Proof. Let us set



an,

2n := 

 . q

By Proposition 6, (35), (36) an, = ch2n, .

(55)

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A. Nakayashiki

It follows that d (2n,) =

dan, |q=1 + dq



(s1 + · · · + s ).

(56)

0≤s1 <···<s ≤2n−1

We denote the first term of the right-hand side of (56) by An, and the second term by bn, . The sum bn, is expressed in terms of An, as follows. Notice that  1 q s1 +···+s = q 2 (−1) an, . 0≤s1 <···<s ≤2n−1

Then bn,

   1 d  2n (−1) 2 = an, )|q=1 = + An, , (q 2  dq

and d (2n,) = 2An, + Let us define cn, by cn, =

    2n . 2 

        2n 2n − 1 2n − 2  2n + − , 2 −1 −1 2 

(57)

Then the lemma is equivalent to cn, = 2An, .

(58)

Let us prove (58) by the induction on n. For n = 1, (58) is easily verified. Let us prove that cn, and 2An, satisfy the same recursion relation. Notice that an, here is the same as an, in the proof of Proposition 6. By differentiating the recursion relation (41) of an, one obtains the recursion relation for 2An, : 2An, = 2An−1, + 2(2An−1,−1 ) + 2An−1,−2     2n − 2 2n − 2 +2(4n −  − 2) + 2(2n − ) . −1 −2 We prove the same recursion relation for cn, . Set      2n 2n − 1 2n − 2 (1) cn, = + , 2 −1 −1 Using

(1)

(2)

cn,

    2n . 2 

        n n−2 n−2 n−2 = +2 + , m m m−1 m−2 (2)

in each binomial coefficient in cn, and cn, , we get (1) cn,

(2)

cn, =

(59)

   2n − 1 2n − 2 = + (4n − 3) + , −1 −2     2n − 2 2n − 2 (2) (2) (2) = cn−1, + 2cn−1,−1 + cn−1,−2 + 2( − 1) + (2 − 3) . −1 −2 (1) cn−1,

(1) + 2cn−1,−1



(1) + cn−1,−2

Taking the difference of these equations and using the induction hypothesis we get the desired recursion relation. 

Residues of q-Hypergeometric Integrals

217 2n−1

2n−2

( −1 )+( −1 ) . Proposition 8. The determinant det X (2n,) is divisible by (
+ 2n ) Proof. In det X(2n,) , for each 1 ≤ j2 < · · · < j ≤ 2n−1, we add the (2j, j2 , · · · , j )th −2j −2j column times x2n−1 or −x2n−1 to the (0, j2 , · · · , j )th column successively so that, in the resulting determinant, the (r; 0, j2 , · · · , j )th component becomes Q r;0,j2 ···j

=

n−1 

−2j

Qr;2j,j2 ···j x2n−1 .

j =0

Further, for each 2 ≤ j2 < · · · < j ≤ 2n − 1, we add the (2j − 1, j2 , · · · , j )th column −(2j −1) −(2j −1) times x2n−1 or −x2n−1 to the (1, j2 , · · · , j )th column successively so that the

th (r; 1, j2 , · · · , j ) component becomes Q r;1,j ···j = 2



n  j =1

−(2j −1)

Qr;2j −1,j2 ···j x2n−1

.

Since {Qr;i1 ···i } satisfy (15), we have Q r;0,j2 ···j = Q r;1,j ···j = 0. 2

It follows that is divisible by

Q r;0,j2 ···j

and

Q r;1,j ···j 2 



are divisible by x2n−1 + x2n . Thus det X (2n,) 2n−1

2n−2

(x2n−1 + x2n )( −1 )+( −1 ) . Since det X(2n,) is a symmetric polynomial of x1 , ..., x2n , it is divisible by 2n−1

2n−2

( −1 )+( −1 ) . (
+ 2n )  Corollary 3. There is a constant c such that 2n−1

2n−2

( −1 )+( −1 ) . det X(2n,) = c (
+ 2n ) Proof of Theorem 5. It is sufficient to prove that any P ∈ U2n, can be written as a linear combination of Qr ’s with the coefficients in R2n . Let us write  P = Pj1 ···j X j1 ∧ · · · ∧ X j . 0≤j1 <···<j ≤2n−1

We solve the following linear equations for Sr , r = 1, · · · ,  Sr Qr , P = r

which, in the matrix form, (Pj1 ,··· ,j ) = (Sr )X (2n,) ,

2n 

:

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A. Nakayashiki

where (Pj1 ,··· ,j ) and (Sr ) are both row vectors. Since {Qr } are linearly independent over R2n by Theorem 6, c = 0 in Corollary 3. Then by Cramer’s formula 2n−1

2n−2

−( −1 )−( −1 ) Sr = c−1 (
+ det Xr(2n,) , 2n ) (2n,)

is defined by replacing the r th row in X (2n,) by (Pj1 ,··· ,j ). where the matrix det Xr (2n,) is proved to be divisible by In a similar way to the proof of Proposition 8, det Xr 2n−1

2n−2

( −1 )+( −1 ) . (
+ 2n ) Thus Sr ∈ R2n as desired.



7. Equivalent Conditions for Linear Independence Let  (2n) be the 2n-dimensional vector space with the basis αi , βj , i, j = 1, · · · , n:  (2n) = ⊕ni=1 Cαi ⊕nj=1 Cβj . Define ωk ∈ ∧2  (2n) by ωk =

n 

αr−k+1 ∧ βr .

1 ≤ r ≤ n,

(60)

r=1

Here and in the following the indices of αi , βj , ωk are extended to integers and they are read by modulo n in the representative {1, · · · , n}. In this section we set (2n)

ek = ek

(2n) Pr,s = Pr,s ,

,

(2n)

vi = vi

(2n)

wj = wj

,

,

(2n)

ξk = ξk

,

since n always remains the same. Theorem 7. The following conditions are equivalent: (i) {αI ∧ βJ ∧ ωK } is linearly independent over C, (ii) {αI ∧ βJ ∧ ωK } linearly generate ∧  (2n) , (iii) {αI ∧ βJ ∧ ωK } is a basis of ∧  (2n) , (iv) {vI ∧ wJ ∧ ξK } is linearly independent over R2n , where I, J, K satisfy (27)–(30). Proof. By (32), it is obvious that (i), (ii) and (iii) are equivalent. Let us prove the equivalence of (i) and (iv). To this end we have to prepare several lemmas and propositions. Proposition 9. The following expressions for ξk hold: ξk =



−

k−1 

n−k+i 

+

i=1 r=n−k+1

ξk =



−

k−1 

n−k+i 

i=1 r=n−k+1

n−k  n  

e2(k−i+r) X 2r+1 ∧ vi ,

(61)

i=k r=i−k

+

n−k  n  

e2(k−i+r) wi ∧ X 2r .

i=k r=i−k

The proof of this proposition is given at the end of this section.

(62)

Residues of q-Hypergeometric Integrals

219

We consider the specialization e1 = −e2n = 1, Lemma 11. At (63) we have  1, Pr,s = 0,

ej = 0,

j = 1, 2n.

(63)

1 ≤ s ≤ n and s = n − r + 2 mod. n otherwise.

Proof. Let us prove the statement by the induction on r. For r = 1, P1,s = e2s−1 , and the statement is obvious. For r ≥ 2, Substituting (63) into (18) and using the hypothesis of the induction we have  P1,n+1 − e2n P1,1 = 1, r = 2 Pr,n−r+2 = Pr−1,n−(r−1)+2 − e2(n−r+2) Pr−1,1 = 1, r ≥ 3. For s = n − r + 2, Pr,s = Pr−1,s+1 − e2s Pr−1,1 = 0, 

by the induction hypothesis. For any integer i we set

0 ≤ j ≤ 2n − 1,

< i >= Xj ,

i ≡ j mod. 2n.

Corollary 4. At (63) we have vi =< 2(n − i + 1) >,

wi =< 2(n − i + 1) + 1 >,

1 ≤ i ≤ n.

The following lemma follows from Proposition 9 and Corollary 4. Lemma 12. At (63), ξk =

n 

< 2(n + r − k) + 1 > ∧vr =

r=1

n 

wk−r+1 ∧ vr .

r=1

At (63) we set αi = v2−i ,

βj = wj .

Then we have ωk =

n 

αr−k+1 ∧ βr = −ξk .

r=1

Thus at (63), {vI ∧ wJ ∧ ξK } = {±αI ∧ βJ ∧ ωK }, where ± means that + or − is determined to each (I, J, K). Consequently, if {αI ∧ βJ ∧ ωK } are linearly independent, for the matrix X (2n,) defined in (54), det X(2n,) = 0, at (63) and {vI ∧ wJ ∧ ξK } are linearly independent over R2n . Therefore (iv)⇒(i) in Theorem 7 is proved. Let us prove the converse. We have to use the following lemma which can be easily verified.

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A. Nakayashiki

Lemma 13. At (63), we have 2n−1

1

2n−2

2 n(n+1)(( −1 )+( −1 )) .
+ 2n = (−1)

Now, if {vI ∧ wJ ∧ ξK } are linearly independent over R2n , c = 0 in Corollary 3. Then, at (63), we get 2n−1

1

2n−2

det X(2n,) = c(−1) 2 n(n+1)(( −1 )+( −1 )) = 0. Thus {αI ∧ βJ ∧ ωK } are linearly independent.



Proof of Proposition 9. Let us first prove (61). In this proof we extend the index r of Pr,s to r ∈ Z by defining Pr,s = e2(s+r)−3 ,

r ≤ 0.

For r ≥ n + 1, Pr,s is defined by the recursion relation (18). Then the recursion relation (18) holds for all r, s ∈ Z. The following lemma is obtained by a simple calculation using the definitions (20) and (23). Lemma 14. (X1 + X2 )ξk (X1 , X2 ) =

n n  

2j

(e2j Pk,i − e2i Pk,j )X12i X2 .

i=0 j =0

Let us denote the right-hand side of (61) by ηk and expand it as ηk =

n−1 

(k)

aij



2j

2j

X12i+1 X2 − X22i+1 X1

 .

i,j =0 (k)

We set aij = 0 unless 0 ≤ i, j ≤ n − 1. Then (X1 + X2 )ηk =



    (k) 2j 2j +1 (k) (k) (k) ai−1,j − aj −1,i X12i X2 + aij − aj i X12i+1 X2 .

i,j

i,j

Comparing this with Lemma 14 we see that (61) is equivalent to the following equations: (k)

(k)

aij = aj i , (k) ai−1,j

(k) − aj −1,i

(64) = e2j Pk,i − e2i Pk,j . (k)

In order to prove these equations we calculate aij more explicitly. Lemma 15. We have  (k) aij =

i e P , r=0 2(i−r) k+r,j +1 − n−i e P r=1 2(i+r) k−r,j +1 ,

0≤i ≤n−k n − k + 1 ≤ i ≤ n − 1.

(65)

Residues of q-Hypergeometric Integrals

221

Proof. If 0 ≤ i ≤ n−k, X2i+1 ∧X2j appears only from the second sum of the right-hand side of (61). Its coefficient is calculated as (k)

aij =

k+i 

e2(k−r+i) Pr,j +1 =

i 

e2(i−r) Pk+r,j +1 .

r=0

r=k

Similarly, if n − k + 1 ≤ i ≤ n − 1, X2i+1 ∧ X 2j appears only from the first sum term and its coefficient is given by (k)

aij = −

k−1 

e2(k−r+i) Pr,j +1 = −

r=k−n+i

n−i 

e2(i+r) Pk−r,j +1 .

r=1

 (k)

Let us rewrite aij more. Lemma 16.     − e2r Pk−1,s , 0 ≤ i ≤ n − k (k) aij =  r+s=i+j +2, r≤i r+s=i+j +2, r≥j +1  − n − k + 1 ≤ i ≤ n − 1. e r+s=i+j +1, r≤j r+s=i+j +1, r≥i+1 2r Pk,s , Proof. First we suppose that 0 ≤ i ≤ n − k. Using Lemma 15 we have (k)

aij =

i 

e2(i−r) Pk+r,j +1

r=0

=

i 

e2(i−r) (Pk+r−1,j +2 − e2(j +1) Pk+r−1,1 )

r=0

=

i 

e2(i−r) Pk+r−1,j +2 + e2i Pk−1,j +2 − e2(j +1)

r=1

i 

e2(i−r) Pk+r−1,1 .

r=0

By Lemma 15 we have, for the first term, i 

(k)

e2(i−r) Pk+r−1,j +2 = ai−1,j +1 .

r=1

As to the third term, we see i 

e2(i−r) Pk+r−1,1 = Pk+i−1,1 + e2 Pk+i−2,1 + · · · + e2i Pk−1,1

r=0

= (Pk+i−2,2 − e2 Pk+i−2,1 ) + e2 Pk+i−2,1 + · · · + e2i Pk−1,1 = Pk+i−2,2 + e4 Pk+i−3,1 + · · · + e2i Pk−1,1 .. . = Pk−1,i+1 .

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A. Nakayashiki

Thus we have (k)

(k)

aij = ai−1,j +1 + e2i Pk−1,j +2 − e2(j +1) Pk−1,i+1 =

(k) a0,i+j

+

i−1 

(e2(i−r) Pk−1,j +2+r − e2(j +1+r) Pk−1,i+1−r ).

(66)

r=0

Notice that (k)

a0,i+j = Pk,i+j +1 = Pk−1,i+j +2 − e2(i+j +1) Pk−1,1 , which is precisely the r = i term in the second term of (66). Therefore (k)

aij =

i 

(e2(i−r) Pk−1,j +2+r − e2(j +1+r) Pk−1,i+1−r ),

r=0

which proves the first equation of the lemma. Next we consider the case n − k + 1 ≤ i ≤ n − 1. Here we use the recursion relation of Pr,s in the form Pr,s = Pr+1,s−1 + e2(s−1) Pr,1 .

(67)

Using (67) we have n−i 

(k)

aij = −

r=1 n−i 

=−

r=1 n−i 

=−

e2(i+r) Pk−r,j +1 e2(i+r) (Pk−r+1,j + e2j Pk−r,1 ) e2(i+r) Pk−r+1,j − e2(i+1) Pk,j − e2j

r=2

n−i 

e2(i+r) Pk−r,1 ,

r=1

where we have −

n−i 

(k)

e2(i+r) Pk−r+1,j = ai+1,j −1 ,

r=2

and n−i 

e2(i+r) Pk−r,1 = Pk,i+1 +

r=1

n−i 

e2(i+r) Pk−r,1 − Pk,i+1

r=1

= Pk−n+i,n+1 − Pk,i+1 = −Pk,i+1 , where we use Lemma 1. Therefore (k)

(k)

aij = ai+1,j −1 + e2j Pk,i+1 − e2(i+1) Pk,j (k)

= an−1,i+j +1−n +

n−i−2  r=0

(e2(j −r) Pk,i+1+r − e2(i+1+r) Pk,j −r ).

(68)

Residues of q-Hypergeometric Integrals

223

Here (k)

an−1,i+j +1−n = −e2n Pk−1,i+j +2−n , which is the r = n − i − 1 term in the second term of (68). Thus we have (k)

aij =

n−i−1 

(e2(j −r) Pk,i+1+r − e2(i+1+r) Pk,j −r ),

r=0



which completes the proof of Lemma 16.

Proof of (64). We can assume i < j . The proof is divided into three cases. (i) The case 0 ≤ i < j ≤ n − k: We use the first equation in Lemma 16 and notations like  (r ≤ i)1 = e2r Pk−1,s . (69) r+s=i+j +2, r≤i

Then we have (k)

aij = (r ≤ i)1 − (r ≥ j + 1)1 , and (k)

(k)

aj i = (r ≤ j )1 − (r ≥ i + 1)1 = (r ≤ i)1 − (r ≥ j + 1)1 = aij . (ii) The case n − k + 1 ≤ i < j ≤ n − 1: In this case we use notations like  (r ≤ j )2 = e2r Pk,s .

(70)

r+s=i+j +1, r≤j

Then (k)

(k)

aij = (r ≤ j )2 − (r ≥ i + 1)2 = (r ≤ i)2 − (r ≥ j + 1)2 = aj i . (iii) The case 0 ≤ i ≤ n − k < j ≤ n − 1: In this case we have     (k) aj i = e2r Pk,s − 

r+s=i+j +1, r≤i



= 

r+s=i+j +1, r≥j +1

−

r+s=i+j +1, r≤i

 e2r (Pk−1,s+1 − e2s Pk−1,1 )

r+s=i+j +1, r≥j +1



=



r+s=i+j +2, r≤i, s≥1





−

e2r (Pk−1,s − e2(s−1) Pk−1,1 ).

r+s=i+j +2, r≥j +1, s≥1

Notice that, if s = 0, Pk−1,s − e2(s−1) Pk−1,1 = 0. Then

 (k) aj i

=

(k) aij

−

 r+s=i+j +1, r≤i

−

 r+s=i+j +1, r≥j +1

 e2r e2s Pk−1,1 .

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A. Nakayashiki

The second term in the right-hand side vanishes because r ≤ i is equivalent to s ≥ j + 1 under the condition r + s = i + j + 1. Thus (64) is proved. Proof of (65). Since both hand sides of (65) are anti-symmetric with respect to i and j , we can assume i < j . (i) The case 0 ≤ i ≤ n − k: We use the notation (69) in which r + s = i + j + 2 is changed to r + s = i + j + 1. Then we have (k)

ai−1,j = (r ≤ i − 1)1 − (r ≥ j + 1)1 , (k)

(k)

aj −1,i = ai,j −1 = (r ≤ i)1 − (r ≥ j )1 . Thus (k)

(k)

ai−1,j − aj −1,i = −(r = i)1 + (r = j )1 = −e2i Pk−1,j +1 + e2j Pk−1,i+1 = −e2i Pk,j + e2j Pk,i , where to derive the last equation we have used the recursion relation of Pr,s . (ii) The case n − k + 1 ≤ i < j ≤ n − 1: We use the notation (70) in which r + s = i + j + 1 is changed to r + s = i + j . Then (k)

(k)

ai−1,j = aj,i−1 = (r ≤ i − 1)2 − (r ≥ j + 1)2 , (k)

aj −1,i = (r ≤ i)2 − (r ≥ j )2 . Therefore (k)

(k)

ai−1,j − aj −1,i = −(r = i)2 + (r = j )2 = −e2i Pk,j + e2j Pk,i , which completes the proof of (65). Thus (61) is proved. The expression (62) follows from (61) in the following manner. Let ηk denote the right-hand side of (62). Notice that wi (X) is obtained from vi (X) by the replacement X2r → X 2r+1 ,

0 ≤ r ≤ n − 1.

Then it is obvious that ηk is obtained from −ηk by the replacement X2r → X 2r+1 , Since ηk = we have

X2r+1 → X 2r , 

0 ≤ r ≤ n − 1.

(k)

aij X 2i+1 ∧ X 2j ,

 (k) aij X 2i ∧ X 2j +1 ηk = −  (k) = aj i X 2i+1 ∧ X 2j  (k) = aij X 2i+1 ∧ X 2j = ηk ,

which proves (62). Thus the proof of Proposition 9 is completed.



(71)

Residues of q-Hypergeometric Integrals

225

8. Linear Independence In this section we prove Theorem 8. The elements {αI ∧ βJ ∧ ωK } linearly generate ∧  (2n) , where I, J, K satisfy (27)–(30). As a consequence of Theorem 8 and 7, Theorem 6 is proved. Let B be the linear span of {αI ∧ βJ ∧ ωK } in ∧  (2n) . We use the symbol ≡ for the equality modulo B . For the sake of simplicity, in the following, we omit the symbol ∧ of the exterior product and simply write the product for the exterior product. The conditions (27)–(30) for the product αI βJ , I = (i1 , · · · , i1 ), J = (j1 , · · · , j2 ) are 1 ≤ i1 ≤ · · · ≤ i1 ≤ n,

1 ≤ j1 ≤ · · · ≤ j2 ≤ n − 1 ,

1 + 2 = .

The elements αI βJ which does not satisfy (27)–(30) are classified by the number of k of βj such that j > n − 1 . We write such j in the form j = n − r with 0 ≤ r ≤ 1 − 1 for later convenience. With these notations, in order to prove Theorem 8, it is sufficient to prove αi1 −1 · · · αi0 βj1 · · · βj2 −k−1 βn−rk · · · βn−r0 ≡ 0, 1 ≤ i1 −1 < · · · < i0 ≤ n, 1 ≤ j1 < · · · < j2 −k−1 ≤ n − 1 , 0 ≤ r0 < · · · < rk ≤ 1 − 1, 0 ≤ k ≤ 1 − 1;

(72)

here we changed the way of numbering the indices of α for the sake of later convenience. We set J = (j1 , · · · , j2 −k−1 ). Let us consider the element αi1 −1 · · · α   irk · · · α ir0 · · · αi0 βJ ωn−rk −irk +1 · · · ωn−r0 −ir0 +1 ,

(73)

where α  irk means that αirk is removed. Since is ≥ 1 − s,

0 ≤ s ≤ 1 − 1,

we have n + 1 − rs − irs ≤ n + 1 − 1 ,

0 ≤ s ≤ k.

Notice that, in (73), the number of α is 1 − k − 1 and the number of ω is k + 1. Then the indices of (73) satisfy (27)–(30) if and only if the indices of βj , ωk appearing in (73) satisfy j ≤ n − 1 and k ≤ n + 1 − 1 . Thus (73) ≡ 0.

(74)

Let us prove (72) by the induction on k. The case k = 0 is proved in Proposition 10 below.

226

A. Nakayashiki

Suppose that k ≥ 1 and (72) is valid until k − 1. Then it is possible to expand (73) using (60) as (73) ≡ αi1 −1 · · · α   irk · · · α ir0 · · · αi0 βJ  1 −1

≡±

p0 ,...,pk =0

p0 ,...,pk =n−1 +1

αpk +rk +irk βpk · · · αp0 +r0 +ir0 βp0

αi1 −1 · · · αirk +rk −pk · · · αir0 +r0 −p0 · · · αi0 βJ βn−pk · · · βn−p0 



≡±

n 

0≤p0 <···
sgn σ · αi1 −1 · · · αirk +rk −pσk · · · αir0 +r0 −pσ0 · · · αi0

×βJ βn−pk · · · βn−p0 ,

(75)

where the equation is valid for k = 0 without any assumption and we use the induction hypothesis for k ≥ 1. The sign ± is determined from 2 , k, {rs }. Since it is irrelevant to the following argument, we omit its explicit description. By (74) and (75) we have 0 ≡ αi1 −1 · · · αi0 βJ βn−rk · · · βn−r0 (76)  + sgn σ · αi1 −1 · · · αirk +rk −rσk · · · αir0 +r0 −rσ0 · · · αi0 βJ βn−rk · · · βn−r0 (77) σ =1



+ where



0≤p0 <···


σ (the

summand of (75)),

means that (p0 , · · · , pk ) = (r0 , · · · , rk ) is excluded.

Lemma 17. If a summand of (77) is proportional to αi1 −1 · · · αi0 βJ βn−rk · · · βn−r0 ,

(78)

it coincides with (78). Proof. It is sufficient to prove that, if (irk + rk − rσk , · · · , ir0 + r0 − rσ0 ) is a permutation of (irk , · · · , ir0 ), then (irk + rk − rσk , · · · , ir0 + r0 − rσ0 ) = (irσk , · · · , irσ0 ).

(79)

Notice that irk + Rk ≤ · · · ≤ ir1 + R1 ≤ ir0 ,

Rs = rs − r0 .

(80)

We set j0 = ir0 for the sake of simplicity and write (irk + Rk , · · · , ir0 + R0 ) = ((j0 − N )MN · · · (j0 − 1)M1 j0M0 ), where (j0 − s)Ms = j0 − s, · · · , j0 − s

(Ms terms).

MN = 0,

Residues of q-Hypergeometric Integrals

227

Then (irk , · · · , ir0 ) = (j0 − N − Rk , · · · , j0 ),

(81) (irk + rk − rσk , · · · , ir0 + r0 − rσ0 ) = ((j0 − N )MN · · · j0M0 ) − (Rσk , · · · , Rσ0 ).

Suppose that (81) is a permutation of (irk , · · · , ir0 ). Notice the term which coincides with irk . If j0 − a − Rs = irk = j0 − N − Rk , for some 0 ≤ a ≤ N , then obviously we have a = N and s = k. Next consider the term which coincides with irk−1 and so on. In this way we see that, for each 0 ≤ s ≤ N , (σk−MN −···−Ms , · · · , σk−MN −···−Ms−1 +1 ) is a permutation of (k − MN − · · · − Ms , · · · , k − MN − · · · − Ms−1 + 1). This means ((j0 − N )MN · · · j0M0 ) − (Rσk , · · · , Rσ0 ) = (j0 − N − Rσk , · · · , j0 − N − Rσk−MN +1 , · · · ) = (irσk , · · · , irσ0 ).  Let M − 1 be the number of (78) appearing in (77). Then we can solve Eq. (76) as Mαi1 −1 · · · αi0 βJ βn−rk · · · βn−r0   ≡ − 0≤p0 <···
(83)

Notice that in the left-hand side J does not appear. We omit J because J remains the same in all the arguments below. Let us consider the map among elements of the form (83) and 0: (i1 −1 , · · · , i0 |n − rk , · · · , n − r0 ) → (i1 −1 , · · · , irk + rk − pσk , · · · , ir0 + r0 − pσ0 , · · · , i0 |n − pk , · · · , n − p0 ), (84) where {rs }, {ps }, σ , {is } should satisfy 0 ≤ r0 < · · · < rk ≤ 1 − 1, LHS = ±RHS, 1 ≤ i1 −1 < · · · < i0 ≤ n.

0 ≤ p0 < · · · < pk ≤ 1 − 1,

(85) (86)

228

A. Nakayashiki

For a set of integers (i1 −1 , · · · , i0 ) we define h(i1 −1 , · · · , i0 ) = max{i1 −1 , · · · , i0 } − min{i1 −1 , · · · , i0 }. For (83) we set h(i1 −1 , · · · , i0 |n − rk , · · · , n − r0 ) := h(i1 −1 , · · · , i0 ). Lemma 18. Suppose that the right-hand side of (84) is written as ±(i 1 −1 , · · · , i0 |n − pk , · · · , n − p0 ),

i 1 −1 ≤ · · · ≤ i0 .

Then i1 −1 ≤ i 1 −1 ,

i0 ≤ i0 .

In particular h(i 1 −1 , · · · , i0 |n − pk , · · · , n − p0 ) ≤ h(i1 −1 , · · · , i0 |n − pk , · · · , n − p0 ), (87) and the equality holds if and only if i1 −1 = i 1 −1 ,

i0 = i0 .

Proof. Notice that is + s ≤ is + s ,

0 ≤ s ≤ s ≤ 1 − 1.

It follows that, for any 0 ≤ s ≤ 1 − 1, i1 −1 ≤ i1 −1 + 1 − 1 − pσs ≤ irs + rs − pσs ≤ i0 − pσs ≤ i0 , which proves the lemma.



We denote the left-hand and the right-hand sides of (87) by h and h respectively. By Lemma 18, the case h = h occurs only in the following four cases. (i) (ii) (iii) (iv)

rk rk rk rk

= 1 − 1, r0 = 1 − 1, r0 = 1 − 1, r0 = 1 − 1, r0

= 0. = 0 and there exists k1 such that irk1 + rk1 = i0 . = 0 and there exists k1 such that irk1 + rk1 − (1 − 1) = i1 −1 . = 0 and there exist k1 , k2 such that irk1 + rk1 − (1 − 1) = i1 −1 ,

irk2 + rk2 = i0 .

Proposition 10. After finitely many applications of the maps (84), each element is mapped to zero. Proof. Let us prove the proposition by the induction on 1 . For 1 = 1, there is nothing to prove. Suppose that 1 > 1 and the statement holds until 1 − 1. Let us prove the 1 case by the induction on h = h(i1 −1 , · · · , i0 ). If h ≤ 1 − 2, there is no (i1 −1 , · · · , i0 ) satisfying i1 −1 < · · · < i0 . Suppose that h ≥ 1 − 1. We denote the value of h of the right-hand side of (84) by h . By Lemma 18 h ≤ h. If h < h, the statement holds by the induction hypothesis. Let us consider the case h = h which is divided into (i) to (iv) above. We first consider the case (iv).

Residues of q-Hypergeometric Integrals

229

In this case pσk1 = 1 − 1, pσk2 = 0. Therefore σk1 = k,

σk2 = 0,

pk = 1 − 1,

p0 = 0.

We have irk + rk − pσk = i1 −1 + 1 − 1 − pσk = irk1 + rk1 − pσk , ir0 + r0 − pσ0 = i0 − pσ0 = irk2 + rk2 − pσ0 . For the value of σ0 , σk we have four cases. (a) (b) (c) (d)

k1 k1 k1 k1

= k, k2 = k, k2 = k, k2 = k, k2

= 0: We have σs = 0, k for any 1 ≤ s ≤ k − 1. = 0: We have σ0 = 0, k. = 0: We have σk = 0, k. = 0: We have σ0 , σk = 0, k.

It follows that the map (84) induces the map (i1 −2 , · · · , i1 |n − rk−1 , · · · , n − r1 )

→ ±(· · · , irk−1 + rk−1 − pσk−1 , · · · , ir1 + r1 − pσ1 , · · · |n − pk−1 , · · · , n − p1 ), (88) where σ is the permutation of (1, · · · , k − 1) specified according as the cases (a)–(d) above as σs = σs for 1 ≤ s ≤ k − 1, σk 2 = σ0 , σs = σs (otherwise), σk 1 = σk , σs = σs (otherwise), σk 1 = σk , σk 2 = σ0 , σs = σs (otherwise).

(a) (b) (c) (d)

Notice that 1 ≤ r1 < · · · < rk−1 ≤ 1 − 2,

1 ≤ p1 < · · · < pk−1 ≤ 1 − 2,

and LHS of (88) = ±(RHS of (88)) is equivalent to LHS of (84) = ±(RHS of (84)). Let us set Is−1 = is ,

rs − 1 = rs−1 ,

ps − 1 = ps−1 ,

σs − 1 = τs−1 .

Then

irs + rs − pσs = Irs−1 + rs−1 − pτ s−1 ,

≤ 1 − 3, 0 ≤ r1 < · · · < rk−2

0 ≤ p1 < · · · < pk−1 ≤ 1 − 3,

and τ is the permutation of (0, · · · , k − 2). We have

LHS of (88) = (I1 −3 , · · · , I0 |n − 1 − rk−2 , · · · , n − 1 − r0 ),

RHS of (88) = ±(· · · , Irk−2 + rk−2 − pτ k−2 , · · · , Ir0 + r0 − pτ 0 , · · · |

, · · · , n − 1 − p0 ), n − 1 − pk−2

230

A. Nakayashiki

Thus the map (88) can be considered as the map (84) with the replacements 1 → 1 − 2,

n → n − 1,

k → k − 2.

Notice that, in case (iv), the map beginning from the image of the map (84) is again of the form (iv) if the image is not zero and the value of h is invariant. Therefore, by the induction hypothesis on 1 , if we apply the maps (84) of the type (iv) sufficiently many times to an element we have zero. Thus the statement is proved in this case. The cases (ii) and (iii) can be proved in a similar manner. Things we should notice here are the following. In case (ii) if the image is non-zero and the next map preserves the value of h, the next map is of type (ii) or (iv). Similarly for case (iii) the next map is of type (iii) or (iv) under the same condition. Let us consider case (i). We divide the case into three. (i-i) pk = 1 − 1, p0 = 0: The next map is in case (ii) for which the statement is already proved as above. (i-ii) pk = 1 − 1, p0 = 0: The next map is in case (iii) for which the statement is already proved as above too. (i-iii) pk = 1 − 1, p0 = 0: In this case k ≤ 1 − 3 and the map (84) induces the map (i1 −2 , · · · , i1 |n − rk , · · · , n − r0 ) → (i1 −2 , · · · , irk + rk − pσk , · · · , ir0 + r0 − pσ0 , · · · , i1 |n − pk , · · · , n − p0 ). (89) We set Is−1 = is ,

rs − 1 = rs ,

ps − 1 = ps .

Then LHS of (89) = (I1 −3 , · · · , I0 |n − 1 − rk , · · · , n − 1 − r0 ), RHS of (89) = (I1 −3 , · · · , Irk + rk − pσ k , · · · , Ir0 + r0 − pσ 0 , · · · | n − 1 − pk , · · · , n − 1 − p0 ).

Thus (89) can be considered as the map (84) with the modification 1 → 1 − 2,

n → n − 1.

Therefore sufficiently many compositions of the maps of type (i–iii) lead each element to zero by the induction hypothesis. Let us summarize what is proved in case (i). To this end we introduce case (v) which is (ii) or (iii) or (iv). Then we have the diagram like (i) → (v) → (v) → · · · · · · · · · · · · · · · → 0  (i) → (v) → · · · · · · · · · · · · · · · → 0  .. . 

(i) → (v) → · · · → 0  0 ,

Residues of q-Hypergeometric Integrals

231

where the maps directed from NW to SE represent the composition of maps of type (i-iii) and those from W to E represent the composition of maps of type (v). This shows that in case (i) the statement is proved. Thus for 1 the statement is proved by the induction on h. Therefore the proposition is proved by the induction on 1 .  Finally let us complete the proof of Theorem 8. For k = 0, (82) holds. Then by Proposition 10, LHS of (82) ≡ 0,

(90)

which proves (72) for k = 0. If k ≥ 1, (75) holds by the induction hypothesis. Consequently (82) is valid. Again by Proposition 10 we have (90) which proves (72) for k. Thus the proof of Theorem 8 is completed.  9. Fermionic Character Formula For a graded vector space W = ⊕k W (k) such that W (k) is finite dimensional, we define its character by  ch W = q k dim W (k). k

We introduce a grading on U2n, by deg2 : U2n, = ⊕∞ k=0 U2n, (k),

(91)

where U2n, (k) is the subspace of elements with degree k. By Proposition 6, Theorem 5, (35) and (36) we have   2 qn 2 2n ch U2n, = q n ch R2n · ch2n, = . (92) [2n]q !  q We consider the R2n -module M2n, =

U2n, , U2n,−1 ∧ w1 + U2n,−2 ∧ ξ1 (2n)

(93)

here and in what follow of this section we omit (2n) of vi , etc. The denominator of the right-hand side of (93) is introduced because for any element P in the denominator, ψP vanishes identically due to Theorem 2 (ii), (22) and (24). Since w1 and ξ1 are homogeneous, M2n, is also graded. In this section we shall calculate the character of M2n, . To this end we construct an R2n free resolution of M2n, . Define the R2n -linear map ϕ by ϕ : U2n, ⊕ U2n,−1 −→ U2n,+1 ⊕ U2n, , ϕ (a, b) = (a ∧ w1 + (−1) b ∧ ξ1 , b ∧ w1 ), and the map ψ by ψ : U2n, ⊕ U2n,−1 −→ U2n,+1 , ψ (a, b) = a ∧ w1 + (−1) b ∧ ξ1 ,

232

A. Nakayashiki

where we set Un,0 = Rn and Un, = 0 for  < 0. We denote p the natural projection: p : U2n, −→ M2n, . The relation ϕ ϕ−1 = 0

(94)

can be easily verified. Theorem 9. The following sequence is exact for 0 ≤  ≤ n: ϕ0

ϕ1

ϕ−2

0 −→ U2n,0 −→ U2n,1 ⊕ U2n,0 −→ · · · −→ U2n,−1 ⊕ U2n,−2 ψ−1

p

−→ U2n, −→ M2n, −→ 0. Proof. We first prove Lemma 19. The sequence ϕm−1

ϕm

U2n,m−1 ⊕ U2n,m−2 −→ U2n,m ⊕ U2n,m−1 −→ U2n,m+1 ⊕ U2n,m

(95)

is exact at the middle term for any 0 ≤ m ≤ n − 1. Proof. We have to prove Ker ϕm = Im ϕm−1 . Lemma 20. For any 0 ≤ m ≤ n − 1, we have ch (Imϕm ) = ch U2n,m . Proof. We set

U2n,m+1 = ⊕1∈J R2n vI ∧ wJ ∧ ξK ⊂ U2n,m+1 ,

= ⊕1∈J U2n,m−1 / R2n vI ∧ wJ ∧ ξK ⊂ U2n,m−1 ,

where, in both cases, the sum is taken for I, J, K satisfying (27)–(29). Define the map

ϕ˜m : U2n,m+1 ⊕ U2n,m−1 −→ Im ϕm ,

ϕ˜m (a, b) = (a + (−1)m b ∧ ξ1 , b ∧ w1 ). Notice the following three properties of the basis Bas = {vI ∧ wJ ∧ ξK } of U2n, given in Theorem 5. The first is that, if vI ∧ wJ ∧ ξK is in Bas−1 and 1 ∈ / J , then vI ∧ w1 ∧ wJ ∧ ξK is in Bas . We denote the subset of Bas consisting of such elements by Bas+ . The second is that, if vI ∧ wJ ∧ ξK is in Bas+1 and J = (1, J ), then vI ∧ wJ ∧ ξK is in Bas . We denote the subset of Bas consisting of such elements by Bas− . The third property is Bas = Bas+ Bas− .

(96)

It follows that ϕ˜m is bijective. Moreover ϕ˜m preserves the degree since deg1 w1 = deg1 ξ1 = 0. Thus

ch (Im ϕm ) = ch U2n,m+1 + ch U2n,m−1 = ch U2n,m ,

where we use (96) to derive the last equation. Thus the lemma is proved.



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233

We have the exact sequence: 0 −→ Ker ϕm −→ U2n,m ⊕ U2n,m−1 −→ Im ϕm −→ 0.

(97)

ch (Im ϕm ) = ch U2n,m + ch U2n,m−1 − ch (Kerϕm ).

(98)

Then For two polynomials f (q), g(q) of q we denote f (q) ≥ g(q), if all the coefficients of f (q) − g(q) are non-negative. By (94) we have Im ϕm−1 ⊂ Ker ϕm . Then, by (98) and Lemma 20, we have ch (Im ϕm ) ≤ ch U2n,m + ch U2n,m−1 − ch (Im ϕm−1 ) = ch U2n,m = ch (Im ϕm ). Thus which proves Lemma 19.

Im ϕm−1 = Ker ϕm ,

(99)

Ker ψ−1 = Im ϕ−2 .

(100)



Let us prove It is sufficient to prove Ker ψ−1 ⊂ Im ϕ−2 . Suppose that ψ−1 (a, b) = a ∧ w1 + (−1)−1 b ∧ ξ1 = 0,

a ∈ U2n,−1 ,

b ∈ U2n,−2 .

By taking the exterior product with w1 we have (w1 ∧ b) ∧ ξ1 = 0. Since the map ξ1 ∧ : ∧m H (2n) −→ ∧m+2 H (2n) ,

0 ≤ m ≤ n − 1,

is injective as proved in [18] using the representation theory of sl2 , w1 ∧ b = 0. Thus we have ϕ−1 (a, b) = (0, 0). Therefore, by Lemma 19, we have (a, b) ∈ Im ϕ−2 . Consequently (100) is proved. Finally the equation Ker p = Im ψ−1 is obvious by the definition of M2n, . Thus Theorem 9 is proved.



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A. Nakayashiki

Corollary 5. We have 2

chM2n,

qn = [2n]q !



2n 





q

2n − −1

  . q

Proof. By Theorem 9 ch M2n, = ch U2n, −

−1 

  (−1)r ch U2n,−1−r + ch U2n,−2−r

r=0

= ch U2n, − ch U2n,−1 . Then the corollary follows from (92).



For a non-negative integer λ let us consider the direct sum of M2n, with n −  being fixed: (0)

M2λ := ⊕n−=λ M2n, . Then, by Corollary 5, we get (0) ch M2λ

=



2

n−=λ

qn [2n]q !



2n 



 q

2n − −1

  .

(101)

q


2 , V ( i ) be the level one irreducible Let i , i = 0, 1 be the fundamental weights of sl highest weight representation with the highest weight i , ei ,

fi ,

hi ,

i = 0, 1


2 and d the scaling element [7]. Consider the subspace of the Chevalley generators of sl V ( i ) consisting of sl2 highest weight vectors with the sl2 weight µ: V ( i |µ) = {v ∈ V ( i ) | e1 v = 0, h1 v = µv }. Then ch M2λ = trV ( 0 |2λ) (q −d ), (0)

which is the branching function of 2λ + 1-dimensional irreducible representation of sl2
2 . In fact if we multiply (101) by the character χλ (z) in the representation V ( 0 ) of sl of the 2λ + 1-dimensional irreducible representation of sl2 we get ∞ 

(0)

χλ (z) ch M2λ =

λ=0

2n ∞   n=0 =0

2

z2n−2 q n , []q ! [2n − ]q !

which is the fermionic form of the character of V ( 0 ) presented in [5, 10]. The space V ( 0 |2λ) becomes the irreducible representation of the Virasoro algebra with c = 1 and h = λ2 [7]. It follows that [13] 2

trV ( 0 |2λ) (q

−d

q λ (1 − q 2λ+1 ) )= , (q : q)∞

(z : q)∞ =

∞  j =0

(1 − zq j ).

Residues of q-Hypergeometric Integrals

235

10. The Case of Odd Number of Variables In this section we study the case where the number of xj ’s is odd. The structure is similar to the even case. But there are some differences in the construction of a basis. Thus we consider this case separately. We omit the proofs of the statements if they are quite similar to the even case. (2n+1) Let us define Pr,s , 1 ≤ r ≤ n + 1, s ∈ Z by the same recursion relation as (18): (2n+1)

(2n+1)

= e2s−1 ,

P1,s

(2n+1)

(2n+1)

(2n+1) Pr,s = Pr−1,s+1 − e2s (2n+1)

for r ≥ 2.

(102)

= 0 for s ≤ 0.

Then Pr,s

(2n+1)

Proposition 11. (ii)

(2n+1)

Pr−1,1

(2n+1) Pr,s

=

(2n−1) Pr,s (2n+1)

Let us define vi (2n+1)

v0

= 0 for r ≥ 2 and s ≥ n + 1.

(i) Pr,s

=

n 

(2n−1) − x 2 Pr,s−1 . (2n+1)

, wi

(2n+1)

e2j

, 0 ≤ i ≤ n by (2n+1)

X 2j ,

vi

=

j =0 (2n+1)

wi

(2n+1)

Pi,j

X 2(j −1) (i = 0),

j =1

n+1 

=

n+1 

Pi+1,j X 2j −1 . (2n+1)

j =1

By Proposition 11,  (2n+1)   (2n+1)  ρ ± vi = ρ± wi = 0,

0 ≤ i ≤ n.

Notice that (2n+1)

w0

(2n+1)

∈ / H (2n+1) ,

vi

(2n+1)

, wj

∈ H (2n+1) ,

j = 0.

We set, for 1 ≤ k ≤ n, (2n+1)

2ξk

(X1 , X2 ) =

 X1 − X2  (2n+1) (2n+1) (2n+1) (2n+1) (X1 )wk−1 (X2 ) + v0 (X2 )wk−1 (X1 ) v0 X1 + X 2 (2n+1)

+v0

(2n+1)

(X1 )wk−1

Then (2n+1)

ξk which means

∈ H (2n+1) , (2n+1)

ξk

(2n+1)

(X2 ) − v0

(2n+1)

(X2 )wk−1

(X1 ).

 (2n+1)  = 0, ρ± ξk

∈ U2n+1,2 .

We have the relation (2n+1)

1

(2n+1)

= v0

,

(2n+1)

2

(2n+1)

= ξ1

.

We set

 (2n+1) (2n+1) (2n+1)  ∧ wJ ∧ ξK , Baso− = vI

(2n+1)

Baso+ = v0

o− ∧ Bas−1 ,

(103)

236

A. Nakayashiki

where I, J, K satisfy (27)–(30) and define Baso = Baso+ Baso− . It follows from the even case that the cardinality of Baso is       2n 2n 2n + 1 o− o− o Bas = Bas−1 + Bas = + = . −1   The degrees of elements are given by (2n+1)

deg1 v0

(2n+1) deg1 wj

(2n+1)

= 0,

deg1 vi

= 2i − 1 (i = 0),

(2n+1) deg1 ξk

= 2j,

= 2k − 2.

The structure of U2n+1, is given by the following theorem.   Theorem 10. (i) The module U2n+1, is a free R2n+1 -module of rank 2n+1 with Baso  as a basis. (ii) The character ch2n+1, of U2n+1, with respect to deg1 is given by   2n + 1 ch2n+1, = .  q Let us first prove (ii) assuming (i). Let 

± bn, =

q 1 +2 (2 ±1) 2



1 +2 +23 =

n 1 2 3

 . q2

By Proposition 6 we have − bn,



2n = 

 . q

We easily have + + ch2n+1, = bn,−1 + bn, ,

(104)

where in the right-hand side the first and the second terms are the characters of Baso+ and Baso− respectively. Notice a similar identity to (49):     n n [2 ]q 2 = [3 + 1]q 2 ,  1  2 3 q 2  1 2 − 1  3 + 1 q 2 which implies  q 22

n  1  2 3

 q2

 =

n  1 2 3





n   − 1 3 + 1 1 2 2 q   n 2(3 +1) +q .  1 2 − 1  3 + 1 q 2 −

 q2

Residues of q-Hypergeometric Integrals

237

+ Using this relation we rewrite bn, as



+ bn, =



q 1 +2 (2 −1)+22 2

1 +2 +23 =

 =

2n 

n 1 2 3





q

q2

 n − q 1 2 3 q 2 1 +2 +23 =+1,3 ≥1    2 n + q 1 +2 (2 +1)+23 .  1  2 3 q 2 

21 +2 (2 +1)



(105) (106)

1 +2 +23 =+1,3 ≥1

We easily have  (105) + (106) = q +1

2n +1



+ − bn,+1 .

q



The desired result follows from (40).

The proof of (i) consists of two parts, the linear independence and the generation of the space U2n+1, . Let us first prove the linear independence. It is proved by the specialization argument as in the even case using the result of the even case. Consider the specialization of variables: (2n+1)

(2n+1)

e2n+1 = −e2n Proposition 12.

(2n+1)

= 1,

ej

= 0,

j = 2n, 2n + 1.

(107)

(i) At (107) we have  (2n+1) Pr,s =

s =n+2−r otherwise,

1, 0,

(ii) (2n+1)

v0

(2n+1)

= 1 − X 2n ,

vi

= X 2(n+1−i) ,

(iii) (2n+1)

ξk

=−

n 

(2n+1)

vk−r

i = 0,

(2n+1)

wj

= X 2n+1−2j ,

∧ wr(2n+1) ,

r=1 (2n+1) where the index k−r of vk−r is read by modulo n in the representative {1, 2, . . .

, n}.

At (107) we define γ , αi , βi , ωi , 1 ≤ i ≤ n by (2n+1)

γ = v0

(2n+1)

βi = wi

α1 = vn(2n+1) ,

, ,

ωi =

n 

(2n+1)

αi = vn+1−i ,

i ≥ 2,

αr−i+1 ∧ βr ,

r=1

where the indices of αi , βj are considered by modulo n in the representative {1, 2, . . . , n}. By Proposition 12, {αi , βj , γ } are linearly independent.

238

A. Nakayashiki

Corresponding to Baso− we define  (2n+1) (2n+1) (2n+1)  SBaso− = αI , ∧ βJ ∧ ωK where I, J, K satisfy (27)–(30). By Theorem 7 and 8, elements of SBaso− are linearly independent. Thus elements of o− SBaso− γ ∧ SBas−1

are linearly independent. Consequently elements of Baso are linearly independent over R2n+1 . Next let us prove that elements of Baso generate U2n+1, over R2n+1 . Number the   elements in Baso from 1 to 2n+1 and name the r th element Qr . Expand Qr as in (53) 2n+1 2n+1 and define the  by  matrix X(2n+1,) by the similar formula to (54). The assertion follows from Proposition 13. We have 2n )+(2n−1 (−1 −1 ) , det X(2n+1,) = c · (
+ 2n+1 )

for some non-zero constant c. Let (2n+1)

o = deg1 vI dI,J,K

(2n+1)

∧ wJ

(2n+1)

∧ ξK

(108)

,

and d (2n+1,) =





o dI,J,K +

(s1 + · · · + s ),

0≤s1 ≤···≤s ≤2n

I,J,K

(2n+1)

where (I, J, K) runs index sets such that vI Proposition 13 follows from

(2n+1)

∧ wJ

(2n+1)

∧ ξK

∈ Baso . Then

Lemma 21.  d (2n+1,) =



   2n 2n − 1 + . −1 −1

2n + 1 2

The proof of this lemma is similar to that of Lemma 10. In this way (i) of Theorem 10 is proved.  (2n+1)

In the following we omit (2n + 1) of vi Let M2n+1, =

, etc.

U2n+1, . U2n+1,−1 ∧ v0 + U2n+1,−2 ∧ ξ1

We define the maps ϕo , ψo , po replacing U2n, by U2n+1, , w1 by v0 and M2n, by M2n+1, in the definition of ϕ , ψ and p . We introduce a grading on M2n+1, by deg2 .

Residues of q-Hypergeometric Integrals

239

Then (i) The following sequence is exact for 0 ≤  ≤ n:

Theorem 11.

ϕ0o

o ϕ−2

ϕ1o

0 −→ U2n+1,0 −→ U2n+1,1 ⊕ U2n+1,0 −→ · · · −→ U2n+1,−1 ⊕ U2n+1,−2 o ψ−1

po

−→ U2n+1, −→ M2n+1, −→ 0. (ii) We have 1

ch M2n+1,

2

q 4 (2n+1) = [2n + 1]q !



2n + 1 

 q



2n + 1 − −1

  . q

Similarly to the even case if we set (1)

M2λ+1 = ⊕n−=λ M2n+1, , we have (1)

ch M2λ+1 =

    q 41 (2n+1)2   2n + 1  2n + 1 −  −1 q [2n + 1]q ! q

n−=λ

which is equal to 1

1

trV ( 1 |2λ+1) (q −d+ 4 ) =

2

q 4 (2λ+1) (1 − q 2λ+2 ) . (q : q)∞

A. Proof of Proposition 2 Let us write the integrand of IM (P ) by JM (P ):  IM (P ) =

 

C  a=1

dαa JM (P ).

We simply write Res instead of writing Resβn =βn−1 +πi . We divide the case into four: (I) n − 1, n ∈ / M, (II) n − 1 ∈ M, n ∈ / M, (III) n − 1 ∈ / M, n ∈ M, (IV) n − 1 ∈ M, n ∈ M. As in the proof of Proposition 3 in [12] one can calculate the residue in the following way according as the four cases: (I)  1 Res IM (P ) = Res 2π i 



−1 a=1 C

(Resαa =βn−1 −πi JM (P ))

 b=a

dαb .

240

A. Nakayashiki

(II)  1 Res IM (P ) = Res 2π i 



−1 a=1 C

(Resαa =βn−1 −πi JM (P ))

C −1

dαb

b=a

 −Res



(Resα =βn−1 JM (P ))

−1 

dαb .

b=1

(III)  1 Res IM (P ) = Res 2π i 



−1 a=1 C

(Resαa =βn−1 −πi JM (P ))

+Res

C −1

dαb

b=a

 −Res



(Resα =βn−1 JM (P ))

−1 

dαb

b=1

   −1 a=1 C



(Resαa =βn−1 +πi JM (P ))

dαb .

b=a

(IV)  1 Res IM (P ) = Res 2π i 



−1 a=1 C

−Res +Res

(Resαa =βn−1 −πi JM (P ))

−1 a=−1, C

−1 a=1 C

dαb

b=a

 

  





(Resαa =βn−1 JM (P ))

dαb

b=a



(Resαa =βn−1 +πi JM (P ))

dαb .

b=a

The theorem can be proved by the calculations using these formulae. References 1. Babelon, O., Bernard, D., Smirnov, F.: Null-vectors in integrable field theory. Commun. Math. Phys. 186, 601–648 (1997) 2. Cardy, J., Mussardo, G.: Form factors of descendent operators in perturbed conformal field theories. Nucl. Phys. B340, 387–402 (1990) 3. Christe, P.: Factorized characters and form factors of descendant operators in perturbed conformal systems. Int. J. Mod. Phys. A6, 5271–5286 (1991) 4. Frenkel, I., Reshetikhin, N.: Quantum affine algebras and holonomic difference equations. Commun. Math. Phys. 146, 1–60 (1992) 5. Kedem, R., McCoy, B., Melzer, E.: The sums of Rogers, Schur and Ramanujan and the BoseFermi correspondence in 1 + 1-dimensional quantum field theory. In: Recent Progress in Statistical Mechanics and Quantum Field Theory, P. Bouwknegt et al. (eds), Singapore: World Scientific, 1995, pp. 195–219 6. Koubek, A.: The space of local operators in perturbed conformal field theories. Nucl. Phys. B. 435, 703–734 (1995) 7. Kac, V.: Infinite Dimensional Lie Algebras. Third edition, Cambridge: Cambridge University Press, 1992

Residues of q-Hypergeometric Integrals

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8. Kirillov, A.N., Smirnov, F.: A representation of the current algebra connected with the su(2)-invariant Thirring model. Phys. Lett. B 198, 506–510 (1987) 9. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Second edition, Oxford: Oxford University Press, 1995 10. Melzer, E.: The many faces of a character. Lett. Math. Phys. 31, 233–246 (1994) 11. Nakayashiki, A., Pakuliak, S., Tarasov, V.: On solutions of the KZ and qKZ equations at level zero. Ann. Inst. Henri Poincar´e 71, 459–496 (1999) 12. Nakayashiki, A., Takeyama, Y.: On form factors of SU(2) invariant Thirring model. In: MathPhys Odyssey 2001, Integrable Models and Beyond- in honor of Barry, M. McCoy, Kashiwara, M. Miwa, T., (eds), Progr. Math. Phys., Basel-Bostan: Birkhauser, ¨ 2002, pp. 357–390 13. Rocha-Caridi, A.: Vacuum vector representations of the Virasoro algebra. In: Vertex operators in Mathematics and Physics, Leowsky, J. et al. (eds), Berlin: Springer, 1985, pp. 451–473 14. Smirnov, F.: Lectures on integrable massive models of quantum field theory. In: Nankai Lectures on Mathem. Physics, Ge, M-L., Zhao, B-H. (eds), Singapore: World Scientific, 1990, pp. 1–68 15. Smirnov, F.: Dynamical symmetries of massive integrable models. Int. J. Mod. Phys. A 7, Suppl. 1B, 813–837 (1992) 16. Smirnov, F.: Form factors in Completely Integrable Models of Quantum Field Theories. Singapore: World Scientific, 1992 17. Smirnov, F.: Counting the local fields in SG theory. Nucl. Phys. B 453, 807–824 (1995) 18. Tarasov, V.: Completeness of the hypergeometric solutions of the qKZ equations at level zero. Am. Math. Soc. Translations, Ser. 2 201, 309–321 (2000) Communicated by L. Takhtajan

Commun. Math. Phys. 240, 243–280 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0890-9

Communications in

Mathematical Physics

Spectral Boundary Conditions for Generalizations of Laplace and Dirac Operators Gerd Grubb Copenhagen Univ. Math. Dept., Universitetsparken 5, 2100 Copenhagen, Denmark. E-mail: grubbmath.ku.dk Received: 2 October 2002 / Accepted: 25 March 2003 Published online: 18 July 2003 – © Springer-Verlag 2003

Abstract: Spectral boundary conditions for Laplace-type operators on a compact manifold X with boundary are partly Dirichlet, partly (oblique) Neumann conditions, where the partitioning is provided by a pseudodifferential projection; they have an interest in string and brane theory. Relying on pseudodifferential methods, we give sufficient conditions for the existence of the associated resolvent and heat operator, and show asymptotic expansions of their traces in powers and power-log terms, allowing a smearing function ϕ. The leading log-coefficient is identified as a non-commutative residue, which vanishes when ϕ = 1. The study has new consequences for well-posed (spectral) boundary problems for first-order, Dirac-like elliptic operators (generalizing theAtiyah-Patodi-Singer problem). It is found e.g. that the zeta function is always regular at zero. In the selfadjoint case, there is a stability of the zeta function value and the eta function regularity at zero, under perturbations of the boundary projection of order − dim X. Introduction Spectral boundary conditions (involving a pseudodifferential projection  on the boundary) were first employed by Atiyah, Patodi and Singer [APS75] in their seminal work on Dirac operators D on manifolds X with boundary, introducing the eta-invariant of the tangential part of D as an extremely interesting new geometric object. For such a ∗ D (the square D 2 in the selfadjoint case) is a Dirac realization D , the operator D   Laplace operator with a boundary condition of the form γ0 u = 0,

(I − )(γ1 u + Bγ0 u) = 0;

(0.1)

∂ j ) u|∂X , and B denotes a first-order operator on ∂X (in the case derived here γj u = ( ∂n from D it is the tangential part of D). In this work we present a new analysis of Laplacetype operators with boundary conditions (0.1), under quite general choices of  and B,

244

G. Grubb

showing existence of heat trace expansions and meromorphic zeta functions, and analyzing the leading logarithmic and nonlocal terms. The methods developed here moreover lead to new results on the corresponding questions for Dirac operator problems. In a physics context, spectral boundary conditions are used in studies of axial anomalies (see e.g. Hortacsu, Rothe and Schroer [HRS80], Ninomiya and Tan [NT85], Niemi and Semenoff [NS86], Forgacs, O’Raifertaigh and Wipf [FOW87], see also Eguchi, Gilkey and Hanson [EGH80]), in quantum cosmology (see e.g. D’Eath and Esposito [EE91]), and in the theory of the Aharonov-Bohm effect (see e.g. Beneventano, De Francia and Santangelo [BFS99]). For a long time, applications of spectral boundary conditions were limited to fields of half-integer spin whose dynamics are governed by first-order operators of Dirac type. It is natural (and even required by supersymmetry arguments) to extend this scheme to integer spin fields, and, therefore, to operators of Laplace type. Indeed, a first step in this direction has been taken in Vassilevich [V01, V02], where spectral boundary conditions are formulated for bosonic strings to describe some collective states of open strings and Dirichlet branes. Local cases of (0.1), where  is a projection morphism and B is a differential operator, have been treated earlier (see e.g. Avramidi and Esposito [AE99] and its references); such cases have a mathematical foundation in Greiner [Gre71], Grubb [G74], Gilkey and Smith [GiS83], whereas the general global case is a new subject. Establishing heat trace asymptotics is very important in quantum field theory since they are related to ultraviolet divergences and quantum anomalies (see e.g. the surveys of Avramidi [A02] and Fursaev [F02]). A vanishing of the leading logarithmic term means that the standard definition of a functional determinant through the zeta function derivative at 0 is applicable, and the road is open to renormalization of the effective action. Overview of the contents. The problems for Laplace-type operators P are considered in Sects. 1–4. In Sect. 2 we give sufficient conditions for the existence of the resolvent (PT − λ)−1 in an angular region, with an explicit formula (Th. 2.10), and we use this to show asymptotic expansions of traces Tr(F (PT − λ)−m ) (for sufficiently large m) in powers and log-powers of λ (Th. 2.13); here F is an arbitrary differential operator. In Sect. 3 we show corresponding expansions of the heat trace Tr(F e−tPT ) in powers and log-powers of t (Cor. 3.1), and establish meromorphic extensions of zeta functions ζ (F, PT , s) = Tr(F PT−s ) with simple and double real poles in s. (These constructions do not need a differential operator square root of P as discussed in [V01].) The leading logarithmic term and nonlocal term are analyzed in Sect. 4 when F is a morphism or a first-order operator. This is done by determining which part of the resolvent actually contributes to these values (Th. 4.1). It follows that the log-coefficient identifies with a certain non-commutative residue, vanishing e.g. when F = I (Th. 4.2, 4.5). Then the zeta function is regular at 0, and the value at zero can be set in relation to an eta-invariant associated with  (Th. 4.9, Def. 4.10). Special results are also obtained when F is a first-order operator and certain symmetry properties hold. ∗ D defined from a Dirac-type In Sect. 5 we draw some conclusions for operators D  operator D (of the form σ (∂xn + A + perturb.) near ∂X with A selfadjoint elliptic on ∂X) together with a pseudodifferential orthogonal projection  on ∂X defining the boundary condition γ0 u = 0. Atiyah, Patodi and Singer [APS75] considered such problems with  equal to the nonnegative eigenprojection ≥ (A), and increasingly general choices have been treated through the years: Douglas and Wojciechowski [DW91], M¨uller [M94], Dai and Freed [DF94], Grubb and Seeley [GS95, GS96] gave results on perturbations

Spectral Boundary Conditions

245

≥ (A) + S by finite rank operators S; [W99] allowed smoothing operators S. Br¨uning and Lesch [BL99] introduced a special class of other projections (θ ), and [G99, G01 ] included all projections satisfying the well-posedness condition of Seeley [S69 ]. The methods of the present study lead to new results both for zeta and eta functions. ∗ D , s) = Tr((D ∗ D )−s ) is shown to be regular at s = 0 for The zeta function ζ (D    any well-posed  such that the principal parts of  and A2 commute (Cor. 5.3). This was known previously for perturbations of ≥ (A) and (θ ) of order − dim X (cf. [G01 ]); the new result includes in particular the perturbations of ≥ (A) and (θ ) of order −1. ∗ D , 0) is determined from  and ker D modulo local terms (Cor. 5.4). The value ζ (D   Restricting the attention to cases with selfadjointness at ∂X, assuming σ A = −Aσ,

σ 2 = −I,

 = −σ ⊥ σ,

(0.2)

∗ D )− 2 ): It has at most a we get new results for the eta function η(D , s) = Tr(D(D  simple pole at s = 0 for general well-posed  (Cor. 5.8); this includes perturbations of ≥ (A) and (θ) of order −1, where it was previously known for order − dim X in the selfadjoint product case. Moreover, the residue at s = 0 is locally determined, and stable under perturbations of  of order ≤ − dim X (Th. 5.9). In particular, in the selfadjoint product case, the vanishing of the simple pole, shown for special cases in [DW91, M94, W99, BL99], is stable under perturbations of order − dim X. (For the last result, see also Lei [L02].) There are similar stability results for the value of zeta at 0 (Th. 5.7). The author is grateful to D. Vassilevich, R. Mazzeo, P. Gilkey, R. Melrose, E. Schrohe and B. Booss-Bavnbek for useful conversations. s+1

1. Boundary Conditions with Projections Consider a second-order strongly elliptic differential operator P acting on the sections of an N-dimensional C ∞ vector bundle E over a compact C ∞ n-dimensional manifold X with boundary X = ∂X. X is provided with a volume element and E with a hermitian metric defining a Hilbert space structure on the sections, L2 (E). We denote E|X = E  . A neighborhood of X  in X has the form Xc = X × [0, c[ (the points denoted x = (x  , xn )), and there E is isomorphic to the pull-back of E  . On Xc , there is a smooth volume element v(x)dx  dxn , and v(x  , 0)dx  is the volume element on X  (defining L2 (E  )). We assume that P is principally selfadjoint, i.e., P − P ∗ is of order ≤ 1. Moreover, to have simple ingredients to work with, we assume that P is of the following form near X : Assumption 1.1. On Xc , P has the form P = −∂x2n + P  + xn P2 + P1 ,

(1.1)

where P  is an elliptic selfadjoint nonnegative second-order differential operator in E  (independent of xn ) and the Pj are differential operators of order j in E|Xc . Let 1 be a classical pseudodifferential operator (ψdo) in E  of order 0 with 21 = 1 , i.e., a projection operator, and denote the complementing projection by 2 : 2 = I −  1 .

(1.2)

246

G. Grubb

Let B be a first-order differential or pseudodifferential operator in E  . Then we consider the boundary condition for P : 1 γ0 u = 0,

2 (γ1 u + Bγ0 u) = 0,

(1.3)

j

where the notation γj u = (∂xn u)|X is used. In short, T u = 0, where T = {1 γ0 , 2 (γ1 + Bγ0 )}.

(1.4)

We shall study the resolvent and the heat operator defined from P under this boundary condition, when suitable parameter-ellipticity conditions are satisfied. More precisely, with H s (E) denoting the Sobolev space of order s (with norm us ), we define the realization PT in L2 (E) determined by the boundary condition (1.3) as the operator acting like P and with domain D(PT ) = {u ∈ H 2 (E) | T u = 0}; then we want to construct the resolvent (PT − λ)−1 and the heat operator e−tPT and analyze their trace properties. In particular, we want to show a heat trace expansion

  k k Tre−tPT ∼ ak t 2 + (1.5) −ak log t + ak t 2 . −n≤k<0

k≥0

Such expansions are known to hold for normal differential boundary conditions (Seeley [S69, S69 ], Greiner [Gre71]) without the logarithmic terms, and for pseudo-normal ψdo boundary conditions [G99], but the condition (1.3) is in general not of these types. Example 1.2. Assumption 1.1 holds if P = D ∗ D for an operator type including Dirac operators, namely a first-order elliptic differential operator D from E to another N-dimensional bundle E1 over X such that D = σ (∂xn + A1 ) with A1 = A + xn A11 + A10 on Xc ,

(1.6)

as considered in numerous works, originating in Atiyah, Patodi and Singer [APS75]; here σ is a unitary morphism from E  to E1 , A is a selfadjoint first-order elliptic operator in E  independent of xn , and the A1j are xn -dependent differential operators in E  of order j . Then we can take P  = A2 . However, for the study in the following one does not need to have a differential operator “square root” of P  . (And, if P is derived from (1.6) near X , the factorization need not extend to all of X.) See Vassilevich [V01] for a discussion of how one can find such D when P is a Laplacian. When P = D ∗ D, one can for example take as 1 the orthogonal projection ≥ (A) onto the nonnegative eigenspace of A and let B = A1 |xn =0 , also denoted A1 (0). Then ∗ D , where D is the realization of D under the boundary condition PT equals D≥ ≥ ≥ ≥ (A)γ0 u = 0, and (1.5) is known from [GS95] (also with certain finite rank perturbations of ≥ (A)). Boundary conditions (1.3) with such choices of 1 are often called spectral boundary conditions. The paper [G99] allows much more general realizations D1 of D, where ≥ (A) is replaced by a ψdo projection 1 that is “well-posed” with respect to D. Well-posedness (introduced by Seeley [S69 ]) means that the principal symbol π10 at each (x  , ξ  ) ∈ S ∗ (X  ) maps N+ (x  , ξ  ) bijectively onto the range of π10 (x  , ξ  ) in CN , where N+ (x  , ξ  ) is the space of boundary values of null-solutions: N+ (x  , ξ  ) = { z(0) ∈ CN | d 0 (x  , 0, ξ  , Dxn )z(xn ) = 0, z ∈ L2 (R+ )N },

Spectral Boundary Conditions

247

d 0 denoting the principal symbol of D. As shown in [S69 ], the closed-range operator defining the boundary condition may always be taken to be a projection; it may even be taken orthogonal. When 1 is orthogonal, then (D1 )∗ D1 = PT with B = A1 (0) in (1.3)–(1.4), and (1.5) holds. (Further details in Sect. 5.) In [V01], Vassilevich points to the need for considering other choices of B – and not just B = 0 but moreover cases unrelated to A1 and having complex coefficients, where PT is not selfadjoint. He inquires about heat kernel results for such cases, and that is precisely what we want to develop here. The conditions in (1.3) can be defined without reference to a factorization as in Example 1.2; 1 can be a general pseudodifferential projection in E  unrelated to P  . For the case where 1 is a local projection (a projection morphism in E  ranging in a subbundle F0 ), the question of heat trace expansions is covered by Greiner [Gre71], see also [G74, G96]. Since P is strongly elliptic, it satisfies the G˚arding inequality, so we can assume (possibly after addition of a constant) that Re(P u, u) ≥ c0 u21 , for u ∈ C0∞ (X ◦ ),

(1.7)

with c0 > 0. Then the Dirichlet realization PD (the realization of P in L2 (E) with domain D(PD ) = {u ∈ H 2 (E) | γ0 u = 0}) is invertible. We shall use the following notation for regions in C:

θ = { µ ∈ C \ {0} | | arg µ| < θ },

= π2 ,

θ,r = { µ ∈ θ | |µ| > r }. (1.8)

Since the principal symbol of P is positive selfadjoint, the spectrum of PD is for any δ > 0 contained in a set δ,R = δ ∪ {|λ| ≤ R(δ)}

(1.9)

for some R = R(δ), in addition to being contained in {Re λ > 0}. We denote (PD − λ)−1 = RD (λ), the resolvent of the Dirichlet problem, i.e. the solution operator for the semi-homogeneous problem (P − λ)u = f in X,

γ0 u = 0 on X ;

(1.10)

/ δ,R . The other RD maps H s (E) continuously into H s+2 (E) for s > − 23 , when λ ∈ semi-homogeneous Dirichlet problem (P − λ)u = 0 in X,

γ0 u = ϕ on X ,

(1.11)

is likewise uniquely solvable for λ ∈ / δ,R ; the solution operator will be denoted KD (λ). This is an elementary Poisson operator (in the notation of Boutet de Monvel [BM71]), 1 mapping H s (E  ) continuously into H s+ 2 (E) for all s ∈ R. We have Green’s formula (P u, v)X − (u, P ∗ v)X = (γ1 u + σ0 γ0 u, γ0 v)X − (γ0 u, γ1 v)X , with a certain morphism σ0 in E  .

(1.12)

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G. Grubb

2. Resolvent Constructions Our way to construct the heat operator for PT goes via the resolvent, which is particularly well suited to calculations in the pseudodifferential framework, since the spectral √ parameter λ (or, rather, the square root µ = −λ ) enters to some extent like a cotangent variable. In the following, we shall freely use the notation and results of Grubb and Seeley [GS95], Grubb [G01]. To save space, we do not repeat many details here but refer to these papers or to the perhaps simpler resum´e of the needed parts of the calculus in [G02, Sect. 2]. Let us just recall the definition of the symbol space S m,d,s (Rn−1 ×Rn−1 , ) (denoted S m,d,s () for short), where  is a sector of C \ {0} and m, d, s ∈ Z: A C ∞ function p(x  , ξ  , µ) lies in S m,0,0 (Rn−1 ×Rn−1 , ) when, for every closed subsector  , ∂|z| p(x  , ξ  , 1/z) ∈ S m+j (Rn−1 ×Rn−1 ) uniformly for |z| ≤ 1, 1/z ∈  ; j

here S k (Rn−1 ×Rn−1 ) is the usual symbol space of functions q(x  , ξ  ) with |∂x  ∂ξα q(x  , ξ  )| ≤ Cα,β ξ  k−|α| β

1

for all α, β ∈ Nn−1 ; we write x = (1 + |x|2 ) 2 and N = {0, 1, 2, . . . }. Moreover, S m,d,s (Rn−1 ×Rn−1 , ) = µd (|µ|2 + |ξ  |2 )s/2 S m,0,0 (Rn−1 ×Rn−1 , ). In the applications, we often need p and its derivatives to be holomorphic in µ ∈ ◦ for |µ| + |ξ  | ≥ ε > 0; such symbols will just be said to be holomorphic (in µ). The space denoted S m,d () in [GS95] is the space of holomorphic symbols in m,d,0 S (). The third upper index s was added in [G01] for convenience; one has in view of [GS95, Lemma 1.13] that S m,d,s () ⊂ S m+s,d,0 () ∩ S m,d+s,0 () for s ≤ 0, S m,d,s () ⊂ S m+s,d,0 () + S m,d+s,0 () for s ≥ 0,

(2.1)

and the s-index saves us from keeping track of a lot of sums and intersections. The ψdo with symbol p(x  , ξ  , µ) is defined by the usual formula: OP (p) v(x  ) →

 R2(n−1)







ei(x −y )·ξ p(x  , ξ  , µ)v(y  ) dy  d/ξ  ,

where d/ξ  stands for (2π)1−n dξ  ; the analogous definition on Rn is indicated by OP. do’s in bundles over manifolds are defined by use of local trivializations. moreover have expansions in homogeneous terms, p ∼  The symbols, we consider, m−j,d,s (), homogeneous in (ξ  , µ) of degree m−j +d +s. p with p ∈ S m−j j ∈N m−j In the general, so-called weakly polyhomogeneous case, the homogeneity takes place for |ξ  | ≥ 1, but if it extends to |ξ  | + |µ| ≥ 1 (in such a way that the symbol behaves as a standard classical symbol in the non-parametrized calculus with an extra cotangent variable |µ| entering on a par with ξ  in the estimates), the symbol is called strongly polyhomogeneous.

Spectral Boundary Conditions

249

The composition rules for these spaces are straightforward: When pi ∈ S mi ,di ,si () for i = 1, 2, then p1 p2 and p1 ◦ p2 ∈ S m1 +m2 ,d1 +d2 ,s1 +s2 (); the latter is the symbol of the composed operator OP (p1 ) OP (p2 ), satisfying
(−i)|α| α   α   (p1 ◦ p2 )(x  , ξ  , µ) ∼ (2.2) α! ∂ξ  p1 (x , ξ , µ)∂x  p2 (x , ξ , µ). α∈Nn−1

Before discussing the construction of the resolvent, we shall introduce some auxiliary 1 pseudodifferential operators on X . When λ ∈ C \ R+ , we write µ = (−λ) 2 , defined such that µ ∈ (cf. (1.8)). Definition 2.1. The operator A(λ) is defined for λ ∈ C \ R+ by 1

A(λ) = (P  − λ) 2 ; it is a ψdo in E  of order 1. 1

As a function of µ = (−λ) 2 ∈ , A is a strongly polyhomogeneous ψdo with symbol in S 0,0,1 ( )N×N in local trivializations, its principal symbol being equal to 1

a0 (x  , ξ  , µ) = (p (x  , ξ  ) + µ2 ) 2 . 0

(2.3)

This follows essentially from Seeley [S69], since p  (x  , ξ  ) + e2iθ t 2 is a classical elliptic symbol of order 2 with respect to the cotangent variables (ξ  , t). Moreover, A(−µ2 ) is invertible for µ ∈ , and parameter-elliptic (as defined in [G96]), and A−1 has symbol in 1 S 0,0,−1 ( )N×N in local trivializations, with principal part (a0 )−1 = (p 0 (x  , ξ  )+µ2 )− 2 . r 1−2r The symbols are holomorphic in µ. We observe furthermore that since ∂λ A = cr A , ∂λr A has symbol in S 0,0,1−2r ( )N×N for r ∈ N. The indication by an upper index N ×N means that the symbols are N ×N -matrix valued. The statement “has symbol in S m,d,s ( )N×N in local trivializations” will be written briefly as: “∈ OP S m,d,s ( )”. In the case considered in Example 1.2, A(λ) is the operator called Aλ in [GS96] and [G02], Aµ in [GS95], and A(0) = |A|. Definition 2.2. The Dirichlet-to-Neumann operator ADN is defined for λ ∈ / δ,R by: ADN (λ) = γ1 KD (λ);

(2.4)

cf. (1.11) ff. The operator ADN is a ψdo of order 1 for each λ; this is a well-known fact in the calculus of pseudodifferential boundary problems (cf. Boutet de Monvel [BM71], Grubb [G96]). We observe moreover: Lemma 2.3. The Dirichlet-to-Neumann operator ADN (λ) is a strongly polyhomogeneous ψdo in E  of order 1, which is principally equal to −A(λ), i.e., ADN (λ) = −A(λ) + ADN (λ),

(2.5)

where ADN (−µ2 ) ∈ OP S 0,0,1 ( ) and ADN (−µ2 ) ∈ OP S 0,0,0 ( ), with holomorphic symbols.

250

G. Grubb

Proof. Clearly, ADN (−µ2 ) is strongly polyhomogeneous of order 1, in the terminology of [GS95, G01], since its symbol can be found from the corresponding calculation in the case where µ = eiθ t is replaced by eiθ ∂xn+1 . Then its symbol is in S 0,0,1 ( )N×N . Formula (2.5) is derived from the fact that the principal part of P + µ2 at xn = 0 equals the principal part of −∂x2n + P  + µ2 . For the latter operator considered on X × R+ , the Poisson operator solving the semi-homogeneous Dirichlet problem as in (1.11) is the mapping ϕ(x  ) → z(x  , xn ) = e−xn A(λ)ϕ (as in [G02, Prop. 2.11] with A2 replaced by P  ); application of ∂xn followed by restriction to xn = 0 gives the mapping ϕ → −A(λ)ϕ. Then ADN (λ) and −A(λ) are principally equal, so their difference ADN is strongly polyhomogeneous of order 0, hence has symbol in S 0,0,0 ( )N×N .   In particular, ADN (−µ2 ) is parameter-elliptic, hence invertible for large enough µ, the inverse having symbol in S 0,0,−1 ( )N×N (see Prop. 2.8 below). Note that ∂λr ADN (λ) is strongly polyhomogeneous of degree 1 − 2r, hence lies in OP S 0,0,1−2r ( ). In our construction of the resolvent below, we need to be able to commute ADN and 1 with an error having symbol in S 0,0,0 . For this we introduce Assumption 2.4. The principal symbols of 1 and P  commute. This holds of course if 1 commutes with P  (as in Example 1.2 with P  = A2 , 1 = ≥ or < ); it holds for general choices of 1 if P  has scalar principal symbol. Proposition 2.5. Under Assumption 2.4, [ADN , 1 ] = ADN 1 −1 ADN and [A, 1 ] = A1 − 1 A are in OP S 0,0,0 ( ), with holomorphic symbols. Moreover, for any r ≥ 0, the r th λ-derivatives are in OP S 0,0,−2r ( ). Proof. The main effort lies in the treatment of the case r = 0. Note first that since ADN and 1 are in OP S 0,0,0 , so is their commutator [ADN , 1 ], so in view of (2.5), what we have to show is that [A, 1 ] is in OP S 0,0,0 . Denote P  + µ2 = P , with symbol p (x  , ξ  ) + µ2 in local coordinates. The powers of P are defined for low values of s by  s i P = 2π s (P − )−1 d, C

where C is a curve in C \ R− encircling the spectrum of P ; we let it begin with a ray with 1 angle δ and end with a ray with angle −δ, for some δ ∈ ]0, π2 [ . (The location of (r +µ2 ) 2 when r ∈ R+ is discussed in Remark 2.11 below.) Then since [(P − )−1 , 1 ] = (P − )−1 [1 , P  ](P − )−1 ,  s i [P , 1 ] = 2π s (P − )−1 [1 , P  ](P − )−1 d. C

Here, by the commutativity of the principal symbols, [1 , P  ] = M is a first-order ψdo (independent of  and µ). The integral makes good sense for s < 1 (converges in the 1

norm of operators from H 1 (E  ) to L2 (E  )), so we can write, since P 2 = A,  1 i [A, 1 ] = 2π  2 (P − )−1 M(P − )−1 d. C

(2.6)

The symbol of [A, 1 ] in a local coordinate system can be found (modulo smoothing terms) from this formula. It is represented by a series of terms obtained from (2.6) by

Spectral Boundary Conditions

251

insertion of the symbol expansions of the involved operators (P − )−1 and M and applications of the composition rule (2.2). We have that the symbol of (P − )−1 is an (N × N)-matrix r(x  , ξ  , µ, ) whose entries rij are series of strongly homogeneous 1 functions of (ξ  , µ,  2 ), whereas the symbol m(x  , ξ  ) of M is a matrix whose elements mij are series of functions homogeneous in ξ  outside a neighborhhod of 0. The composition will give rise to integrals of products γ

γ

γ 

∂x  ,ξ  rij ∂x  ,ξ  mj k ∂x  ,ξ  rkl ; there are finitely many contributions to each degree of homogeneity. The important thing is that in each homogeneous contribution, we can take the factor coming from mj k outside the integral sign, since it does not depend on  (nor on µ). What is left is a completely homogeneous integrand, which after integration gives a strongly homogeneous function of (ξ  , µ). The factors coming from m are of degree ≤ 1, hence lie in S 1,0,0 , and the contributions from the integration are of degree ≤ −1, hence lie in S 0,0,−1 (moreover, ξ  -differentiation of order α lowers the former to S 1−|α|,0,0 and the latter to S 0,0,−1−|α| ). The full contributions are then in S 1,0,0 · S 0,0,−1 ⊂ S 0,0,0 , and can be collected in a series of terms of falling degrees. Thus [A, 1 ] has a symbol series in S 0,0,0 . There is still the question of whether the remainder R, the difference between the operator defined from (P − )−1 M(P − )−1 and an operator S defined from a superposition of the homogeneous terms, also has the right kind of symbol. It takes a certain effort to prove this. We know a` priori, since A has symbol in S 0,0,1 and 1 has symbol in S 0,0,0 , that [A, 1 ] has symbol in S 0,0,1 ⊂ S 1,0,0 + S 0,1,0 . The remainder R will be of order −∞ in this class, hence have symbol in S −∞,1,0 , and we have to show that this can be reduced to S −∞,0,0 . Recall that an operator R with symbol in S −∞,1,0 has a kernel expansion
Kj (x  , y  )µ1−j , K(R)(x  , y  , µ) ∼ j ≥0



with K(R) − j <J Kj µ1−j = O( µ1−J ) for all J , where the Kj (x  , y  ) are C ∞ kernels, by [GS95, Prop. 1.21]. With R0 denoting the operator with kernel K0 , we can  write R = µR0 + R , where R has the kernel expansion j ≥1 Kj µ1−j , hence symbol in S −∞,0,0 , so we are through if we show that R0 = 0. Let µ run on the ray R+ , and let ε ∈ ]0, 21 [ . Then  1 −ε −ε i P [A, 1 ] = 2π  2 (P − )−1 P M(P − )−1 d. C

s P  +µ2 , P f 

t,µ is the Since P = L2  f H 2s,µ (uniformly in µ for µ ≥ 1), where H t,µ n−1 Sobolev space defined by localization from the case H (R ) with norm uH t,µ =  (ξ  , µ)t u(ξ ˆ  )L2 , t ∈ R. From the resolvent estimates

(P − )−1 f L2 ≤ C1 f L2 , (P − )−1 f H 2,µ  P (P − )−1 f L2 ≤ C2 f L2 ,

(2.7)

that are valid for  ∈ C, it then follows by interpolation that 1

|| 2 +ε (P − )−1 f H 1−2ε,µ ≤ C3 f L2 .

(2.8)

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G. Grubb

We also have that P

−ε

MvL2 ≤ C4 MvH −2ε,µ ≤ C5 MvH −2ε ≤ C6 vH 1−2ε ≤ C7 vH 1−2ε,µ ,

using that 1 − 2ε ≥ 0. Thus P

−ε



[A, 1 ]L(L2 ) ≤ C

1

C

|| 2 (P − )−1 L(L2 ) P

−ε

(2.9)

ML(H 1−2ε,µ ,L2 )

× (P − )−1 L(L2 ,H 1−2ε,µ ) |d|  1 1 ≤ C  || 2 −1− 2 −ε |d| ≤ C  , C

(2.10)

where we applied (2.7) to the first factor, (2.9) to the middle factor and (2.8) to the last factor. We have that [A, 1 ] = S +µR0 +R , where S +R is bounded in L2 uniformly −ε in µ for µ ≥ 1, hence P (S + R ) is a` fortiori so. Thus, by (2.10), µR0 f H −2ε,µ is bounded in µ for µ ≥ 1, for any f . But if R0 f is a nonzero function, µR0 f H −2ε,µ cannot be bounded for µ → ∞ (recall that ε < 21 ). Thus R0 = 0, and the proof for the case r = 0 is complete. For r ≥ 1, we observe that ∂λr A = cr A1−2r and that [1 , A−1 ] = A−1 A1 A−1 − A−1 1 AA−1 = A−1 [A, 1 ]A−1 ∈ OP S 0,0,−2 ; [1 , A−k ] = 1 A−1 A−k+1 − A−1 1 A−k+1 + A−1 1 A−k+1 − · · · − A−k 1
= A−j [1 , A−1 ]A−k+1+j ∈ OP S 0,0,−1−k (2.11) 0≤j ≤k−1

by the result for r = 0 and the composition rules. The first calculation shows the assertion on ∂λr [1 , A] for r = 1; the second calculation shows it for general r ≥ 2, when we take k = 1 − 2r. The result now likewise follows for ∂λr [1 , ADN ], when we use that ∂λr ADN = −∂λr A + ∂λr ADN , where ∂λr ADN ∈ OP S 0,0,−2r since it is strongly polyhomogeneous of degree −2r.   The resolvent (PT − λ)−1 is the solution operator for the problem (P − λ)u = f on X, 1 γ0 u = 0 on X  , 2 (γ1 u + Bγ0 u) = 0 on X  ,

(2.12)

where λ runs in a suitable subset of C. The problem of constructing the resolvent will be transformed by some auxiliary constructions: First, we can extend P to a strongly elliptic principally selfadjoint differential oper on an n-dimensional compact manifold X  in which X is smoothly imbedded, ator P  is contained modifying the definition of R(δ) in (1.9) such that also the spectrum of P  − λ and in δ,R . Assume in the following that λ ∈ / δ,R . Let Q(λ) be the inverse of P  to X, e+ is the let Q+ = r + Qe+ be its truncation to X (r + is the restriction from X  \ X). It is well-known that RD (λ) and KD (λ) (cf. (1.10), (1.11)) extension by zero on X are connected by the formula RD = Q+ − KD γ0 Q+ ,

(2.13)

Spectral Boundary Conditions

253

for λ ∈ / δ,R . Let v = RD f,

z = u − v,

ψ = −2 γ1 v,

(2.14)

then (2.12) may be replaced by the problem for z: (P − λ)z = 0 on X, 1 γ0 z = 0 on X  , 2 (γ1 z + Bγ0 z) = ψ on X  .

(2.15)

Here one can let f ∈ L2 (E) so that the solution u is sought in H 2 (E); then v ∈ H 2 (E), 3 z ∈ H 2 (E), and γj v and γj z are in H 2 −j (E  ), and so are i γj v and i γj z, since the projections j are bounded in H s (E  ) for all s ∈ R. One could also carry out the whole calculation for C ∞ sections while keeping track of the orders of the involved operators. We now make a reduction of (2.15) using ADN . When z ∈ H 2 (E) with (P − λ)z = 0, 3 z is in 1-1 correspondence with γ0 z running through the space H 2 (E  ): z = KD γ0 z,

(2.16)

in view of the unique solvability of the Dirichlet problem. Denote γ0 z = ϕ; then we can replace γ0 z by ϕ in (2.15). Moreover, γ1 z = ADN ϕ. So, with z = KD ϕ (assuring the validity of the first line in (2.15)), we arrive at the problem for ϕ: 1 ϕ = 0, 2 (ADN + B)ϕ = ψ.

(2.17)

1

Since the mapping γ1 : D(PD ) → H 2 (E  ) is surjective, γ1 v runs through all of H (E  ) when f runs through L2 (E). So the reduced problem (2.17) must be solved for 1 all ψ ∈ 2 (H 2 (E  )). When f is given in L2 (E) and v and ψ are defined by (2.14), then, 3 if ϕ ∈ H 2 (E  ) solves (2.17) and we set z = KD ϕ, u = v + z, we find that u ∈ H 2 (E) solves (2.12). 1 2

Lemma 2.6. For λ ∈ / δ,R , define S(λ) = ADN + [ADN , 1 ] + 2 B2 , S  (λ) = ADN − [ADN , 1 ].

(2.18)

1

When ψ runs through 2 H 2 (E  ), the problem (2.17) is uniquely solvable with ϕ ∈ 3 3 1 H 2 (E  ) for those λ for which S(λ) and S  (λ) are invertible from H 2 (E  ) to H 2 (E  ). When this holds, the solution is ϕ = S(λ)−1 ψ.

(2.19)

Uniqueness of solution holds when merely S(λ) is injective. Proof. Assume first that ϕ is a solution of (2.17). By the first equation, ϕ = 2 ϕ, so we can rewrite the second equation as: ADN 2 ϕ + [2 , ADN ]ϕ + 2 B2 ϕ = ψ.

(2.20)

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G. Grubb

Now we compose the first equation of (2.17) with ADN and add it to (2.20); this gives that ϕ satisfies (ADN + [2 , ADN ] + 2 B2 )ϕ = ψ. Since [2 , ADN ] = [ADN , 1 ], the operator in the left-hand side equals S(λ) defined in (2.18), so ϕ satisfies S(λ)ϕ = ψ.

(2.21)

When S(λ) is injective, there is at most one solution ϕ of (2.21), hence of (2.17); when S(λ) is bijective, the only possibility is that it equals S(λ)−1 ψ. Now assume that also S  (λ) is invertible, and, for a given ψ with 1 ψ = 0, define ϕ by (2.19). Then of course (2.21) holds, and we shall show that ϕ solves (2.17). Since 1 2 = 0 and 21 = 1 , an application of 1 to (2.21) gives: 0 = 1 Sϕ = 1 (ADN ϕ + [ADN , 1 ]ϕ + 2 B2 ϕ) = 1 ADN ϕ + 1 ADN 1 ϕ − 21 ADN ϕ = 1 ADN 1 ϕ = (ADN − [ADN , 1 ])1 ϕ = S  (λ)1 ϕ.

(2.22)

It follows that 1 ϕ = 0, so ϕ satisfies the first equation in (2.17). The second equation is then retrieved from this and (2.21): Since 2 ϕ = ϕ, ψ = (ADN + [ADN , 1 ] + 2 B2 )ϕ = ADN 2 ϕ + [2 , ADN ]ϕ + 2 Bϕ = 2 (ADN + B)ϕ.

(2.23)

  When Assumption 2.4 holds, the commutator terms in (2.18) have symbol in S 0,0,0 . Then we will show that S  (λ) can be inverted for large |λ| within the weakly polyhomogeneous calculus, and that the same holds for S(λ) when the following assumption is satisfied: Assumption 2.7. There is a θ ∈ ]0, π2 ] such that, with bh (x  , ξ  ) and πih (x  , ξ  ) denoting the strictly homogeneous principal symbols of B and the i , a0 − π2h bh π2h is invertible for ξ  ∈ Rn , µ ∈ θ ∪ {0} with (ξ  , µ) = (0, 0).

(2.24)

This will allow construction of a resolvent family for large λ with | arg(−λ)| < 2θ . If, moreover, θ > π4 , we can also construct a heat operator family. Note that when B is a differential operator, its symbol is a polynomial, so bh equals the usual principal symbol b0 . For 2 , the strictly homogeneous principal symbol π2h is in general not continuous at ξ  = 0, but when it is multiplied by bh , which is O(|ξ  |), we get a continuous function at ξ  = 0 (taking the value 0 there). Proposition 2.8. Let Assumption 2.4 hold. (i) For each θ  ∈ ]0, π2 [ there is an r(θ  ) ≥ 0 such that ADN (−µ2 ) and S  (−µ2 ) are invertible for µ ∈ θ  ,r . (ii) Let moreover Assumption 2.7 hold. Then for each θ  ∈ ]0, θ [ there is an r(θ  ) ≥ 0 such that S(−µ2 ) is invertible for µ ∈ θ  ,r . Here the operator families ADN (−µ2 )−1 and S  (−µ2 )−1 belong to OP S 0,0,−1 ( ), and S(−µ2 )−1 belongs to OP S 0,0,−1 ( θ ), with holomorphic symbols. For each r ∈ N, ∂λr map them into operators in OP S 0,0,−1−2r ( θ ).

Spectral Boundary Conditions

255

Proof. Let us go directly to the proof of (ii), the statements in (i) are proved by easier variants. We have on one hand that S(−µ2 ) is composed of operators of the type considered in [G96], with the µ-dependent factors of regularity ∞ and the i of regularity 0, B of regularity ∞, resp. 1, if it is a differential, resp. pseudodifferential, operator. Removing the commutator term [ADN , 1 ] from (2.20) for a moment, we have an operator S  (−µ2 ) = ADN + 2 B2 = −A + ADN + 2 B2 ,

(2.25)

which is of regularity 1 (since 2 B2 is so), so that the invertibility of the strictly homogeneous principal symbol −a0 + π2h bh π2h assures parameter-ellipticity in the sense of [G96], cf. Prop. 2.1.12 there. Then S  (−µ2 ) is invertible for µ ∈ θ  ,r with a sufficiently large r, the inverse being continuous from H s,µ (E) to H s+1,µ (E) for s ∈ R, by [G96, Th. 3.2.11]. Since the commutator term has L2 -norm bounded in µ by Proposition 2.5, we find by a Neumann series argument that S(−µ2 ) itself is invertible for large r with an inverse that is bounded from L2 (E) to H 1,µ (E). Now A−1 S is likewise invertible for the considered µ, and lies in OP S 0,0,0 ( θ ) since A−1 lies in OP S 0,0,−1 ( ). Then the “spectral invariance theorem” [G99, Th. 6.5], applied to A−1 S, shows that its inverse S −1 A belongs to our weakly polyhomogeneous calculus and lies in OP S 0,0,0 ( θ ). It follows that S −1 = (S −1 A)A−1 ∈ OP S 0,0,−1 ( θ ). Since holomorphy is preserved under composition, the resulting symbols are holomorphic. The statements on λ-derivatives follows by successive applications of the formula ∂λ S −1 = −S −1 ∂λ S S −1 , using that ∂λ S = ∂λ ADN + ∂λ [ADN , 1 ], with properties described in Lemma 2.3 ff. and Proposition 2.4. Similar proofs work for the other operators without the complication due to the presence of B (so any θ  ∈ ]0, π2 [ is allowed there).   Remark 2.9. It should be noted that the proof of Lemma 2.6 only shows the necessity of unique solvability of (2.21) for ψ in the range of 2 , so that Assumption 2.7 (assuring solvability for general ψ) may seem too strong. However, when Assumption 2.4 holds, 1 solvability of (2.21) for ψ ∈ 2 H 2 (E  ), µ = reiθ0 , r ≥ r0 (some r0 ≥ 0, |θ0 | < θ), 1 implies solvability for all ψ ∈ H 2 (E  ), r ≥ r1 with some r1 ≥ r0 : 1 Uniqueness of course holds. Existence is assured as follows: Let ψ ∈ H 2 (E  ) and write ψ = 1 ψ + 2 ψ. Define −1  S(λ)ψ = A−1 DN 1 ψ + S 2 ψ,

for r so large that also ADN is invertible. Then −1 −1 S Sψ = SA−1 DN 1 ψ + 2 ψ = ψ + [ADN , 1 ]ADN 1 ψ + 2 B2 ADN 1 ψ = ψ + [ADN , 1 ]A−1 1 ψ + 2 B2 [A−1 , 1 ]ψ = (I +  S1 )ψ, DN

DN

−1 −1 0,0,−1 ( ). where  S1 = [ADN , 1 ]A−1 DN 1 +2 B2 ADN [1 , ADN ]ADN has symbol in S So, like the operators treated in the proof of Proposition 2.8, I +  S1 is invertible for large enough µ on the ray, and ϕ =  S(I +  S1 )−1 ψ solves the equation Sϕ = ψ.

We can now describe the resolvent. Here we shall use the terminology of weakly polyhomogeneous pseudodifferential boundary operators worked out in [G01] (the relevant parts summed up in [G02, Sect. 2]), extending the calculus of Boutet de Monvel [BM71]. One can get quite far with linear combinations of compositions of elementary operators as in [GS95, G99], but when the expressions get increasingly complicated, it

256

G. Grubb

seems advantageous to use the systematic calculus. We shall here just recall the basic n definitions in the case where X, X  are replaced by R+ , Rn−1 . n to functions on Rn−1 ), One considers Poisson operators K (mapping functions on R+ n ) and sinn−1 trace operators T of class 0 (mapping functions on R to functions on R+ n n ), of gular Green operators G of class 0 (mapping functions on R+ to functions on R+ the form     ˜  , xn , ξ  , µ)v(y  ) dy  d/ξ  , ˜ : v(x  ) → ei(x −y )·ξ k(x K = OPK(k) 2(n−1)  ∞  R    ei(x −y )·ξ t˜(x  , xn , ξ  , µ)u(y  , xn ) dxn dy  d/ξ  , T = OPT(t˜) : u(x) → 2(n−1) 0  ∞ R    ei(x −y )·ξ g(x ˜  , xn , yn , ξ  , µ)u(y) dy d/ξ  . G = OPG(g) ˜ : u(x) → R2(n−1) 0

(2.26) Note that the usual ψdo definition is used with respect to the x  -variable. In fact, we can view OPK, OPT and OPG as OPK n OP , OPTn OP , resp. OPGn OP , where OPK n , etc. stand for the application of (2.26) with respect to xn -variables alone. ˜ t˜ and g˜ are called the symbol-kernels of K, T , resp. G. There is also The functions k, ˜  , xn , ξ  , µ) is replaced by a “complex formulation”, where e.g. the symbol-kernel k(x −1   ˜ the symbol k(x , ξ , ξn , µ) = Fxn →ξn k, and Fξn →xn is included in the definition of the operator (F denotes Fourier transformation). We say that k˜ ∈ S m,d,s (, S+ ), resp. g˜ ∈ S m,d,s (, S++ ), when   j ˜  , |z|un , ξ  , 1/z))| ≤ C ξ  m+j , resp. sup |∂|z| (zd κ −s−1 zξ  l−l uln ∂ul n k(x

un ∈R+

sup

un ,vn ∈R+









|∂|z| (zd κ −s−2 zξ  l−l +k−k uln ∂ul n vnk ∂vkn g(x ˜  , |z|un , |z|vn , ξ  , 1/z))| j

≤ C ξ  m+j ,

(2.27)

for all indices, uniformly for |z| ≤ 1,

1 z

in closed subsectors of , with similar estimates 1

for the derivatives ∂ξα ∂x  with m replaced by m − |α|. Here κ = (|µ|2 + |ξ  |2 ) 2 = 1 ˜ (| 1z |2 + |ξ  |2 ) 2 . t˜ is estimated like k. Again the symbol-kernels are said to be holomorphic (in µ), when they and their derivatives are holomorphic for µ ∈ ◦ , |µ| + |ξ  | ≥ ε > 0. The motivation for the scaling xn = |z|un is explained in [G01], where complete and satisfactory composition rules are worked out. The operators are defined on X, X  by standard localization methods. We recall one further operation, that of taking the normal trace tr n : When G is a singular Green operator as above, the normal trace tr n G is the ψdo on Rn−1 with symbol  ∞ (tr n g)(x ˜  , ξ  , µ) = g(x ˜  , xn , xn , ξ  , µ) dxn (2.28) β

0

(called g˜˜ in [G96]). Here the symbol map tr n acts as follows: tr n S m,d,s−1 (, S++ ) → S m,d,s ().

(2.29)

Spectral Boundary Conditions

257

For operators of trace class, Tr Rn+ G = Tr Rn−1 (tr n G)

(2.30)

Tr X G = Tr X (tr n G).

(2.31)

∞ (if G has the kernel K(x, y, µ) then tr n G has the kernel 0 K(x  , xn , y  , xn , µ) dxn ), and there is a similar rule for the operators carried over to the manifold situation, when the symbol-kernel of G is supported in Xc and the volume element on Xc is taken of the form v(x  )dx  dxn :

One has that G − χ Gχ is smoothing and O(|µ|−M ) for µ → ∞ in closed subsectors of , all M, when χ ∈ C0∞ ( ] − c, c[ ) and is 1 near xn = 0 (cf. e.g. [G01, Lemma 7.1]). Thus in the trace expansion calculations for singular Green operators on X, we can replace G by χ Gχ and use (2.31) to reduce to a calculation for a ψdo on X  ; here [GS95, Th. 2.1] can be applied. We shall express the fact that “the symbol-kernel is in S m,d,s ( θ , S+ )N×N , resp. m,d,s S ( θ , S++ )N×N in local trivializations” more briefly by saying that the operator lies in OPK S m,d,s ( θ , S+ ), OPT S m,d,s ( θ , S+ ), resp. OPG S m,d,s ( θ , S++ ). The symbol-kernels we consider,  moreover have expansions in appropriately quasihomogeneous terms, e.g., g˜ ∼ j ∈N g˜ m−j with g˜ m−j ∈ S m−j,d,s ( θ , S++ ). There is the same distinction between weakly polyhomogeneous symbol-kernels (with homogeneity for |ξ  | ≥ 1) and strongly polyhomogeneous symbol-kernels (with homogeneity for |ξ  | + |µ| ≥ 1, etc.) as for ψdo symbols. The following elementary examples are basic in our calculations: (1) KD (−µ2 ) is a strongly polyhomogeneous Poisson operator in OPK S 0,0,−1 ( , S+ ). (2) γ0 Q(−µ2 )+ and γ1 Q(−µ2 )+ are strongly polyhomogeneous trace operators of class 0 in OPT S 0,0,−2 ( , S+ ) resp. OPT S 0,0,−1 ( , S+ ). As mentioned in the proof of Lemma 2.3, KD is principally the same as the operator KA v → e−xn Av, for xn ∈ Xc (cf. [G02, Prop. 2.11]). We also have (by [G02, (1.17), (4.14)] with A2 replaced by P  ) that γ0 Q+ and γ1 Q+ act principally, for functions supported in Xc , like the operators 21 A−1 TA resp. 21 TA, where  ∞ TAu = e−xn Au(x  , xn ) dxn . (2.32) 0

Simple examples of singular Green operators are KD γ0 Q+ (whose negative is the singular Green part of RD , cf. (2.13)) and KD γ1 Q+ ; they are strongly polyhomogeneous and belong to OPG S 0,0,−3 ( , S++ ), resp. OPG S 0,0,−2 ( , S++ ). 2KD γ1 Q+ is (on Xc ) principally equal to GA, which acts as follows:  ∞ GAu = e−(xn +yn )Au(x  , yn ) dyn . (2.33) 0

For the latter, tr n is easy to determine by functional calculus:  ∞ e−2xn A dxn = (2A)−1 ; tr n GA =

(2.34)

0

this kind of calculation plays a role in our analysis of trace coefficients in Sect. 4.

258

G. Grubb

Theorem 2.10. Let Assumptions 1.1, 2.4 and 2.7 hold. Then for each θ  ∈ ]0, θ [ there is an r = r(θ  ) ≥ 0 such that for µ ∈ θ  ,r , PT + µ2 = PT − λ is a bijection from D(PT ) to L2 (E) with inverse (PT − λ)−1 = RT (λ) of the form RT (λ) = Q(λ)+ + G(λ), G(λ) = −KD (λ)γ0 Q(λ)+ + KD (λ)[S0 (λ)γ0 + S1 (λ)γ1 ]Q(λ)+ ;

(2.35)

here S0 and S1 (given in (2.36) below) are weakly polyhomogeneous ψdo’s in E  lying in OP S 0,0,0 ( θ ), resp. OP S 0,0,−1 ( θ ), and hence G is a singular Green operator of class 0 in OPG S 0,0,−3 ( θ , S++ ). Proof. For a θ  ∈ ]0, θ [ , take r = r(θ  ) such that ADN is well-defined and the operators S(λ) and S  (λ) are invertible for λ = −µ2 , µ ∈ θ  ,r . Then (2.17) is solved uniquely by (2.19), and it remains to draw the conclusions for the original problem (2.12). In view of (2.13), (2.14) and (2.19), the solution is: u = v + z = RD f + K D ϕ = RD f − KD S −1 2 γ1 RD f = Q+ f − KD γ0 Q+ f − KD S −1 2 γ1 Q+ f + KD S −1 2 ADN γ0 Q+ f. This shows (2.35) with S0 = S −1 2 ADN ,

S1 = −S −1 2 ;

(2.36)

they lie in OP S 0,0,0 ( θ ), resp. OP S 0,0,−1 ( θ ), by the rules of calculus. The statement on G now follows from the information given before the theorem on KD , γ0 Q+ and γ1 Q+ , and the composition rules.   Remark 2.11. Let us give some sufficient conditions for the validity of Assumption 2.7. Consider, in a local trivialization, a point (x  , ξ  ) with |ξ  | = 1 (the result is carried over to general ξ  = 0 by homogeneity, and for ξ  = 0 the assumption is trivially satisfied). Let 0 < λ1 (x  , ξ  ) ≤ λ2 (x  , ξ  ) ≤ · · · ≤ λN (x  , ξ  ) be the eigenvalues of the matrix p  0 (x  , ξ  ), associated with the orthonormal system of eigenvectors e1 (x  , ξ  ), . . . , eN (x  , ξ  ) in CN , and denote  ai (x  , ξ  ) = λi (x  , ξ  ) √ (all equal to c, when p  0 = cI ). Denote θ,± = {µ ∈ θ | Im µ ∈ R± }. When µ runs 1 through θ,± ∪ {0}, then (a12 + µ2 ) 2 runs through a convex subset Va1 ,± of θ,± lying to the right of a curve Ca1 ,± passing through a1 on the real axis. 1 Since aj ≥ a1 for j ≥ 1, we also have that (aj2 + µ2 ) 2 lies in Va1 ,± when µ ∈

θ,± ∪ {0}. We denote Va1 ,+ ∪ Va1 ,− = Va1 . It is important that although Va1 is not in general convex, the Va1 ,± are so. Let µ ∈ θ,+ ∪ {0}. Then for general v ∈ CN with norm 1, decomposed as v = c1 e1 + · · · + cN eN with |c1 |2 + · · · + |cN |2 = 1, 1

a0 v · v¯ = (p  + µ2 ) 2 v · v¯ 0

=

N


1

(ai2 + µ2 ) 2 ci ei · cj ej =

i,j =1

N
i=1

1

(ai2 + µ2 ) 2 |ci |2 ∈ Va1 ,+ ,

Spectral Boundary Conditions

259

since Va1 ,+ is convex. There is a similar argument for µ ∈ θ,− , showing altogether that 1

(p + µ2 ) 2 v · v¯ ∈ Va1 , when µ ∈ θ ∪ {0}, |v| = 1. 0

(2.37)

Now (2.24) is obtained, if at each (x  , ξ  ) with |ξ  | = 1, 1

¯ ≥ δ|v|2 , |((p + µ2 ) 2 − π2h bh π2h )v · v| 0

v ∈ CN ,

(2.38)

for some δ > 0. In view of (2.37), this holds if π2h bh π2h v · v¯ lies in a set in C with distance δ from Va1 when |v| = 1. We list some special cases where this holds; here we assume that π2h is an orthogonal projection. (1) Let bh be the principal symbol of a scalar first-order differential operator with real coefficients. Then bh is purely imaginary and π2h bh π2h v · v¯ = bh |π2h v|2 ∈ iR,

(2.39)

which certainly has positive distance from Va1 . More generally, we can take bh such that bh v · v¯ ranges in the sectors around iR consisting of complex numbers with π argument in ]θ1 , π2 − θ1 [ or ]π + θ1 , 3π 2 − θ1 [ for some θ1 ∈ ]θ, 2 ] (allowing also pseudodifferential choices). (2) Let bh = ib1 , where b1 is the principal symbol of a scalar first-order differential operator with real coefficients. Then bh is real, and the real number π2h bh π2h v · v¯ = bh |π2h v|2 has positive distance from Va1 for |v| = 1 if |bh | < a1 .

(2.40)

More generally, we can take bh real selfadjoint with numerical range in [−a1 + ε, +∞[ for some ε ∈ ]0, a1 [ , i.e., bh v · v¯ ≥ −a1 + ε

for |v| = 1.

(2.41)

When B is a differential operator, this in fact requires that |bh v · v| ¯ ≤ a1 −ε, since the symbol bh is odd in ξ  . This case seems to have an interest in brane theory according to [V01]. (3) Let r1 = dist(Va1 , 0). Then it suffices that the norm of bh is < r1 . Since RT is of order −2 and X is compact, the powers RTm are trace-class when m > n2 . They can be studied by composition or by differentiation, in view of the fact that ∂λm−1 (PT − λ)−1 = (m − 1)!(PT − λ)−m . Corollary 2.12. Under the hypotheses of Theorem 2.10, the resolvent powers RTm have the structure, for any m ≥ 1: RTm = (Qm )+ + G(m) =

m−1 1 RT (m−1)! ∂λ

=

m−1 1 Q)+ (m−1)! (∂λ

+

m−1 1 G, (m−1)! ∂λ

(2.42) (m−1)

1 ∂λ where G(m) = (m−1)! 0,0,−2m−1 ( θ , S++ ). OPG S

G is a singular Green operator of class 0 lying in

260

G. Grubb

Proof. One proof consists of applying the rules of calculus ([G01, G02]) to the compositions (Q+ + G) . . . (Q+ + G). Another proof is to use the exact formula we found in Theorem 2.10, combined with the fact that all the factors have the property that a differentiation with respect to λ lowers the s-index by 2.   Theorem 2.13. Assumptions as in Theorem 2.10. (i) Let ϕ be a morphism in E and let m > n2 . Then ϕRTm (λ) is trace-class and the trace has an expansion for |λ| → ∞ with arg λ ∈ ]π − 2θ, π + 2θ [ (uniformly in closed subsectors):
  k a˜ k (ϕ)(−λ)− 2 −m Tr ϕRTm (λ) ∼ −n≤k<0

+




 k a˜ k (ϕ) log(−λ) + a˜ k (ϕ) (−λ)− 2 −m .

(2.43)

k≥0

The coefficients a˜ k and a˜ k are locally determined.  m (ii) Let F be a differential operator in E of order m and let m > n+m 2 . Then F RT (λ) is trace-class and the trace has an expansion for |λ| → ∞ with arg λ ∈ ]π − 2θ, π + 2θ[ (uniformly in closed subsectors):
  m −k Tr F RTm (λ) ∼ a˜ k (F )(−λ) 2 −m −n≤k<0

+




 m −k a˜ k (F ) log(−λ) + a˜ k (F ) (−λ) 2 −m ,

(2.44)

k≥0

with locally determined coefficients a˜ k and a˜ k . If m is odd, a˜ −n = 0. Here, if F is tangential (differentiates only with respect to x  ) on Xc , the log-coefficients a˜ k with 0 ≤ k < m vanish, and the a˜ k with 0 ≤ k < m are locally determined. Proof. (i) is the special case of (ii) where m = 0, so we can treat them at the same time. It is well-known that the “interior” ψdo term F Qm + produces a series of powers ∞ m −k −m 2 for λ → ∞ in closed subsectors of C \ R+ , with coefficients k=−n c˜k (−λ) determined from the successive homogeneous terms in the symbol; here the terms with k − n − m odd vanish since they are produced by terms that are odd in ξ . Now consider the singular Green term F G(m) . By Corollary 2.12 and the composition   rules it lies in OPG S 0,0,m −2m−1 ( θ ) in general, and in OPG S m ,0,−2m−1 ( θ ) if F is tangential. With a cut-off function χ as after (2.31), we have that Tr(F G(m) −χ F G(m) χ ) is O(|µ|−M ) for µ → ∞ in closed subsectors of θ , all M; this difference does not contribute to the trace expansion, so it suffices to treat χ F G(m) χ . This operator can be 0 = X  ×R , with the same symbol-kernel considered as a singular Green operator on X+ + estimates as indicated above. Now we take the normal trace, obtaining an operator 

S = tr n (χ F G(m) χ ) ∈ OP S 0,0,m −2m ( θ ) 



⊂ OP (S m −2m,0,0 ( θ ) ∩ S 0,m −2m,0 ( θ ))

(2.45)

by (2.29) and (2.1). If F is tangential, the last indication is replaced by   OP (S m −2m,0,0 ( θ ) ∩ S m ,−2m,0 ( θ )).

Spectral Boundary Conditions

261

Here we apply [GS95, Th. 2.1]. It is an important point in that theorem that one gets a sum of two expansions

cj µM0 −j + (ck log µ + ck )µM1 −k , (2.46) j ≥0

k≥0

where M0 equals the dimension (here n−1) plus the order of the operator (here m −2m), and M1 equals the lowest d-index associated with the operator. The coefficients cj and ck are local (each stems from a specific homogeneous term in the symbol), whereas the ck are global in the sense that they depend on the full structure. So in the application to  S, the first series begins with a term cµn−1+m −2m , and the second series begins with a  −2m   m  for general F , (c log µ + c )µ−2m for tangential F . Addterm (c log µ + c )µ 1 ing the expansions and inserting µ = (−λ) 2 , we find the expansion (2.44) ff. after an adjustment of the indexation.   Remark 2.14. One has in particular that when all the occurring operators are strongly polyhomogeneous, there is only the first expansion in (2.46), no logarithmic or global terms. This applies to the special case where 1 is a morphism in E  and B is a differential operator; then
  m −k a˜ k (F )(−λ) 2 −m . (2.47) Tr F RTm (λ) ∼ k≥−n

3. Heat Operators and Power Operators The heat operator associated with PT is the solution operator u0 (x) → u(x, t) = e−tPT u0 for the problem ∂t u + P u = 0 on X × R+ , 1 γ0 u = 0 on X  × R+ , 2 (γ1 u + Bγ0 u) = 0 on X  × R+ , u|t=0 = u0 on X.

(3.1)

When θ > π4 in the above constructions, the heat operator can be defined from the resolvent powers or derivatives (recall (2.42)) by the formula  −tPT −m i e = t 2π e−tλ ∂λm (PT − λ)−1 dλ; (3.2) C

here C 

is a positively oriented curve in C going around the spectrum (like the boundary of δ,R in (1.9) with δ ∈ ]0, π2 [ ; one can take δ = π − 2θ  for a θ  ∈ ] π4 , θ [ ). One could construct the heat operator directly instead of passing via the resolvent as we did above; one advantage of our approach is that we can compose our λ-dependent operators pointwise in λ, whereas calculations with respect to the time-variable t need convolutions. (The passage from the λ-framework to the t-framework is essentially an inverse Laplace transformation; here products are turned into convolutions, as is usual for such integral transforms.) As shown e.g. in [GS96] (or see Sect. 2 of [G97]), the transition formula likewise applies to the trace expansions, carrying the expansions in Theorem 2.13 over to heat trace expansions with logarithms. In the resulting statement, we repeat our hypotheses for the convenience of the reader:

262

G. Grubb

Corollary 3.1. Let PT be the realization of P defined by the boundary condition (1.3); let Assumptions 1.1 and 2.4 hold, and let Assumption 2.7 hold with θ > π4 . Then the heat operator e−tPT is well-defined and its trace has the asymptotic expansion for t → 0+, for any morphism ϕ in E:

 k   k ak (ϕ)t 2 + (3.3) −ak (ϕ) log t + ak (ϕ) t 2 . Tr ϕe−tPT ∼ −n≤k<0

k≥0

The coefficients ak and ak are locally determined. Moreover, when F is a differential operator in E of order m there is a trace expansion for t → 0+:


  Tr F e−tPT ∼

ak (F )t

k−m 2

+

−n≤k<0




 k−m −ak (F ) log t + ak (F ) t 2 ,

(3.4)

k≥0

the coefficients ak and ak being locally determined. If m is odd, a−n = 0. Here, if F is tangential on Xc , the log-coefficients ak with 0 ≤ k < m vanish, and the ak with 0 ≤ k < m are locally determined. The coefficients ak , ak , ak are related to the coefficients a˜ k , a˜ k , a˜ k in Theorem 2.13 by universal nonzero proportionality factors; in particular, a0 (F ) = a˜ 0 (F ) and a0 (F ) = a˜ 0 (F ).

(3.5)

Observe that as in Remark 2.14, the expansion simplifies to
  k−m ak (F )t 2 , Tr F e−tPT ∼

(3.6)

k≥−n

when 1 is a morphism and B is a differential operator. The power function F PT−s (defined as 0 on the nullspace of PT ) is derived from the resolvent by the formula  i PT−s = 2π(s−1)...(s−m) λm−s ∂λm (PT − λ)−1 dλ, (3.7) C

where C is a curve in C \ R− around the nonzero spectrum; here we do not need the extra hypothesis θ > π4 . (Further details on transition formulas are found e.g. in [GS96] or [G97].) Then one can also deduce from Theorem 2.13 that there is the following pole structure of the zeta function ζ (F, PT , s) = Tr(F PT−s ) for s ∈ C:


ak (ϕ) Trϕ0 (PT )
 −s  ak (ϕ) ak (ϕ)

(s)Tr ϕPT ∼ + , (3.8) − + s s + 2k (s + 2k )2 s + 2k  

(s)Tr F PT−s ∼

−n≤k<0




−n≤k<0

+

k≥0

ak (F )

k≥0

k−m 2 ak (F )  2 + k−m 2 )

s+




(s

TrF 0 (PT ) − s +

ak (F ) s+

k−m 2

;

(3.9)

Spectral Boundary Conditions

263

here ak , ak and ak are derived from a˜ k , a˜ k and a˜ k by the same universal formulas as in Corollary 3.1. In particular, (3.5) holds. Consider the case where F is a first-order operator D1 with the structure D1 =  ψ(∂  xn +−sB1 ) on X for some morphism ψ. Here we get the generalized eta-function Tr D1 PT , which has the pole structure  

(s)Tr D1 PT−s ∼

With s =

s  +1 2

ak (D1 ) TrD1 0 (PT ) − k−1 s s + 2 −n≤k<0


ak (D1 ) ak (D1 ) + . + 2 (s + k−1 s + k−1 2 ) 2 k≥0

(3.10)

this takes the more customary form

 Tr




−s D1 PT

 +1 2



∼




2ak (D1 ) 2TrD1 0 (PT ) − s + k s + 1 −n≤k<0   (D )
 4a  (D1 ) 2a 1 k . + + k (s  + k)2 s +k 1



( s +1 2 )



(3.11)

k≥0



−1 is regular at s  = 0, this function in general has a double pole Note that since ( s +1 2 ) at s  = 0. Moreover, a−n (D1 ) = 0 since the principal interior symbol is odd in ξ .

4. The First log-Term and Nonlocal Term The hypotheses of Theorem 2.10 are assumed throughout this section. It is important to investigate whether the first log-term a0 in (3.3), resp. a˜ 0 in (2.43) vanishes, equivalently (cf. (3.8)) whether the zeta function ζ (ϕ, PT , s) is regular at s = 0. The values of a˜ 0 and a0 are of great interest too, and similar questions can be asked with ϕ replaced by F , in particular for (3.11). In view of (3.5), we can interchange a˜ 0 (F ) and a˜ 0 (F ) freely with a0 (F ) resp. a0 (F ). The coefficient a0 is known to vanish in the Atiyah-Patodi-Singer case described in Example 1.2, where 1 = ≥ , B = A1 (0) and ϕ = I , by a slightly tricky argument ([G92] proved it using [APS75]; see also [GS96, pf. of Cor. 2.3]). It also vanishes in cases of local boundary conditions, as mentioned in Remark 2.14. As indicated in the proof of Theorem 2.13, the first power where log-terms and global coefficients appear is determined from the possible values of d, the second upper index in the symbol spaces S m,d,0 that the normal trace of the singular Green operator belongs to. Consider first the case where F = ϕ, a morphism in E (taken to be constant in xn on Xc ). Then Theorem 2.13 tells us that the index d can be taken equal to −2m. We shall in the following strive to isolate the part of G(m) that contributes nontrivially to the first log-term and global term, in a way that gives information on the value. Theorem 4.1. Let ϕ be a morphism in E, independent of xn on Xc (its restriction to X  0 = likewise denoted ϕ). Consider χ ϕG(m) χ (cf. Theorem 2.13) as an operator on X+

264

G. Grubb

X  × R+ . With the notation introduced around (2.32), we have that χ ϕG(m) χ = tr n (χ ϕG(m) χ ) =

∂λm−1 −1 (m−1)! ϕ2 KAA TA + G1 ∂λm−1 1 −2 (m−1)! 2 ϕ2 A

+ G2 , (4.1)

+ S1 + S2 ,

where G1 is a strongly polyhomogeneous singular Green operator of degree −2m − 1, G2 ∈ OPG S 1,0,−2−2m ( θ , S++ ), S1 is a strongly polyhomogeneous ψdo in E  of degree −2m, and S2 ∈ OP S 1,0,−1−2m ( θ ). Proof. By Theorem 2.10 and (2.42), we have that ∂ m−1

λ χ ϕKD γ0 Q+ χ + χ ϕG(m) χ = − (m−1)!

∂λm−1 (m−1)! χ ϕKD [S0 γ0

+ S1 γ1 ]Q+ χ .

The first term is a G1 . For the second term, we first note that, since (1 − χ )∂λr KA and ∂λr TA(1 − χ) are smoothing and O(|µ|−M ) in closed subsectors of for all M (cf. e.g. [G01, Lemma 7.1]), ∂λr χ KD − ∂λr KA ∈ OPK S 0,0,−2−2r ( , S+ ),

∂λr γ0 Q+ χ − ∂λr 21 A−1 TA ∈ OPT S 0,0,−3−2r ( , S+ ), ∂λr γ1 Q+ χ

− ∂λr 21 TA

∈ OPT S

0,0,−2−2r

(4.2)

( , S+ ),

in view of the information around (2.32)–(2.34). Thus, using the formulas for S0 , S1 and S and replacing ADN by −A, χ ϕG(m) χ =

∂λm−1 −1 1 −1 (m−1)! ϕKAS 2 [(−A) 2 A

− 21 ]TA + G1 + G2 ,

∂ m−1

λ ϕKA(−A + 2 B2 )−1 2 TA + G1 + G2 , = − (m−1)!

(4.3)

where G2 and G2 are in OPG S 0,0,−2−2m ( θ , S++ ). We have furthermore: (−A + 2 B2 )−1 2 = −A−1 2 + (−A + 2 B2 )−1 2 B2 A−1 2 .

(4.4)

Here ∂λm−1 of A−1 2 is in OP S 0,0,1−2m ( ). Since ∂λr (−A + 2 B2 )−1 is in OP S 0,0,−1−2r ( θ ), 2 B2 ∈ OP S 1,0,0 ( ) and ∂λr A−1 2 ∈ OP S 0,0,−1−2r ( ), ∂λm−1 of the last term is in OP S 1,0,−2m ( θ ). Thus, if we use Proposition 2.5 to replace ∂λr A−1 2 by ∂λr 2 A−1 , we can write χϕG(m) χ =

∂λm−1 −1 (m−1)! ϕKA2 A TA + G1

+ G2 ,

(4.5)

with G2 like G2 in the theorem. Note that B has disappeared from the main term! It will be convenient to do one more commutation, placing 2 in front of the Poisson operator KA, which likewise gives an error like G2 . For this we use that e−xn A = OPKn ((A + iξn )−1 ) in the complex formulation. Here [(A + iξn )−1 , 2 ] = (A+iξn )−1 [2 , A](A+iξn )−1 , which by application of OPK n gives a Poisson operator in OPK S 0,0,−2 ( θ , S+ ) (with (m − 1)st derivative in OPK S 0,0,−2m ( θ , S+ )), so that we can write ∂λm−1 (4) χ ϕG(m) χ = (m−1)! ϕ2 KAA−1 TA + G1 + G2 , (4)

with G2 like G2 . This shows the first formula in (4.1).

Spectral Boundary Conditions

265

By functional calculus as in (2.34), we have that tr n (KA(A)−1 TA) = 21 A−2 . Then also tr n (ϕ2 KA(A)−1 TA) = 21 ϕ2 A−2 , showing the second formula in (4.1); the symbol properties follow from (2.29). (Placing the ψdo ϕ2 in front of the singular Green operator KA(A)−1 TA, and taking it outside tr n , is justified by the point of view where OPG = OPGn OP .)   The last commutation in the proof is not needed when ϕ = I , for then we can instead ∂ m−1

λ use circular permutation in the treatment of (m−1)! KA2 A−1 TA:  m−1  m−1  m−1 ∂λ ∂λ ∂λ KA2 A−1 TA = Tr X (m−1)! 2 A−1 TAKA = 21 Tr X (m−1)! 2 A−2 , Tr X0 (m−1)! +

(4.6) since TAKA = (2A)−1 . Observe that ∂ m−1

∂ m−1

λ λ A−2 = ϕ2 (m−1)! (P  − λ)−1 ; ϕ2 (m−1)!

(4.7)

it lies in OP S 0,0,−2m ( ). Let us list the three trace expansions connected with this operator family (which follow from [GS95, Th. 2.7]): 
k ∂λm−1  −1 1  Tr X 2 ϕ2 (m−1)! (P − λ) ∼ c˜k (ϕ)(−λ)− 2 −m k≥n−1

+  Tr X

−tP  1 2 ϕ2 e






   c˜2l (ϕ) log(−λ) + c˜2l (ϕ) (−λ)−l−m ,

l≥0

∼




k

ck (ϕ)t 2 +

k≥1−n

(s)Tr X

1

2 ϕ2 (P

 −s

)



∼


ck (ϕ)




   −c2l (ϕ) log t + c2l (ϕ) t l ,

l≥0 Tr[ 21 ϕ2 0 (P  )]

− s s + 2k   (ϕ)
c (ϕ) c2l 2l + ; (s + l)2 s+l

k≥1−n

(4.8)

l≥0

again with c0 (ϕ) = c˜0 (ϕ), c0 (ϕ) = c˜0 (ϕ). Theorem 4.2. Let ϕ be as in Theorem 4.1. In a comparison of (2.43) with the trace ∂ m−1

λ expansion of 21 ϕ2 (m−1)! (P  − λ)−1 in (4.8), we have that

a˜ 0 (ϕ) = c˜0 (ϕ), a˜ 0 (ϕ) = c˜0 (ϕ) + local contributions,

(4.9)

(at X  ), ϕ

where the local contributions are defined from P and the strictly homogeneous terms in the symbol of 2 of orders {0, −1, . . . , 1 − n}. In particular, a˜ 0 (ϕ) equals the residue c0 (ϕ) at 0 of the zeta function ζ ( 21 ϕ2 , P  , s) = Tr X ( 21 ϕ2 (P  )−s ), also identifiable with the non-commutative residue of 21 ϕ2 times 1 2: a˜ 0 (ϕ) = a0 (ϕ) = c0 (ϕ) = Ress=0 Tr( 21 ϕ2 (P  )−s ) =

1 4

res(ϕ2 ).

(4.10)

266

G. Grubb

Moreover, when θ > π4 , a0 (ϕ) in (3.3) equals the coefficient c0 (ϕ) of log t in the trace  expansion of 21 ϕ2 e−tP in (4.8). Proof. As noted in Theorem 2.13, the part ϕ(Qm )+ of ϕRTm contributes only a local power expansion, and ϕG(m) contributes the same expansion as χ ϕG(m) χ , so it suffices to study Tr X tr n (χ ϕG(m) χ ), where we can use (4.1). Here the strongly polyhomogeneous part S1 gives an expansion with purely local power terms (as in (2.47) with k ≥ 1 − n and m = 0), and the term S2 with symbol in S 1,0,−1−2m ( θ ) ⊂ S −2m,0,0 ( θ ) ∩ S 1,−1−2m,0 ( θ ) gives an expansion as in (2.46) with M0 = n − 2m and M1 = −1 − 2m. So the non-local and log-contributions from S1 + S2 begin with a term 1 (c log(−λ) + c )(−λ)− 2 −m , implying (4.9). The identification of c0 (ϕ) with 21 times the noncommutative residue res( 21 ϕ2 ) refers to the work of Wodzicski [W84, W84 ], where it was shown that for a classical integer-order ψdo C in E  , the following formula:   (tr E  c(x  , ξ  ))1−n dσ (ξ  ) (4.11) res(C) = ord Q Ress=0 Tr(CQ−s ) = X

|ξ  |=1

can be given a sense (the trace Tr is defined as a meromorphic extension from large Re s to s ∈ C, Q is an auxiliary invertible elliptic ψdo of positive order, tr E  is the fiber trace, and subscript 1 − n indicates the term of degree 1 − n = − dim X  in the symbol); the functional res vanishes on commutators. If P  is invertible, the last equality sign in (4.10) follows directly by taking Q = P  . If P  is not invertible but has a (necessarily finite dimensional) nullspace V0 (P  ), we use that (P  )−s is defined to be 0 on that nullspace, and that a replacement of P  by the invertible operator Q = P  + 0 (P  ) in (4.8) leaves the coefficient

c0 (ϕ)

(4.12)

invariant (whereas

c0 (ϕ)

is changed).

 

These formulas have consequences for the original Atiyah-Patodi-Singer problem, that we take up in Sect. 5. Let us also observe: Theorem 4.3. Let D1 be a first-order differential operator on X, of the form D1 = ψ(∂xn + B1 ) on Xc , where B1 is tangential and ψ is a morphism independent of xn . Then ∂ m−1

λ χ D1 G(m) χ = −ψ2 (m−1)! KATA + G1 + G2 ,

(4.13)

∂ m−1

λ tr n (χ D1 G(m) χ ) = − 21 ψ2 (m−1)! A−1 + S1 + S2 ,

where G1 is a strongly polyhomogeneous singular Green operator of degree −2m, G2 ∈ OPG S 1,0,−1−2m ( θ , S++ ), S1 is a strongly polyhomogeneous ψdo in E  of degree ∂ m−1

∂ m−1

λ λ −2m+1 and S2 ∈ OP S 1,0,−2m ( θ ). Here − 21 ψ2 (m−1)! A−1 = − 21 ψ2 (m−1)! (P  − 1

λ)− 2 has a trace expansion: 
1−k ∂λm−1  − 21 1  ∼ d˜k (ψ)(−λ) 2 −m −Tr X 2 ψ2 (m−1)! (P − λ) k≥n−1

+




 1−k d˜k (ψ) log(−λ) + d˜k (ψ) (−λ) 2 −m ,

k≥0

(4.14)

Spectral Boundary Conditions

267

where the d˜k and d˜k are zero for k odd. Hence in a comparison of (2.44) for F = D1 with (4.14), we have that a˜ 0 (D1 ) = d˜0 (ψ),

(4.15)

a˜ 0 (D1 ) = d˜0 (ψ) + local contributions,

where the local contributions are defined from P and D1 (at X ), ψ and the strictly homogeneous terms in the symbol of 2 of orders {0, −1, . . . , 1 − n}. (The same formulas hold for a0 (D1 ), resp. a0 (D1 ).) The coefficient d˜0 (ψ) is proportional to the noncommutative residue of ψ2 : d˜0 (ψ) = α res(ψ2 ),

(4.16)

for some nonzero constant α depending only on m and the dimension. Proof. Equation (4.13) follows from (4.1), when we use that the tangential operator B1 lifts the first upper index in our symbol classes by 1, ∂xn e−xn A = −Ae−xn A, and [D1 , χ ] is supported away from xn = 0. Now (4.14) holds by application of [GS95, Th. 2.1] (with µ2 = −λ as usual); the more precise information that the log-terms and nonlocal terms only occur for even k – which is not needed for our main purposes here – follows from [GH02] (or from an analysis as in [G02, Th. 5.2]). Since S1 produces only local power terms, and the logarithmic and global terms for S2 begin with the power −m, the result (4.15) on the zeroth coefficients follows. For (4.16) one applies an analysis as in [G02, proof of Th. 5.2] to (4.14). We have that ∂λm−1  (m−1)! (P

1

1

1

1

− λ)− 2 = cm (P  − λ)− 2 −m = (−λ)− 2 −m (P  + 1)− 2 −m ,

 = −λ−1 . (4.17)

Now 1

ψ2 (P  + 1)− 2 −m ∼


− 1 −m 2

j ≥0

j

j ψ2 (P  )j

(4.18)

similarly to [G02, (5.6)], and the subsequent analysis there gives that the first log-term comes from the term with j = 0 and has coefficient   c (tr[ψ(x  )2 (x  , ξ  )])1−n dσ (ξ  ) = c res(ψ2 ), (4.19) X

|ξ  |=1

with nonzero universal constants c and c .

 

The result has an interest for the analysis of the eta function associated with the APS-problem, which we take up in Sect. 5. Remark 4.4. Let us comment on what is meant by “local contributions” in (4.9) and (4.15). It is taken in a rather strict sense, based on the derivation of asymptotic formulas in [GS95, Th. 2.1] (also recalled in [G02, Th. 2.10]). The local contributions at a certain index k come from the homogeneous symbol terms with the degree that matches the index.

268

G. Grubb

(m) is of order −2m, local contributions at k = 0 (in (4.9)) Since ϕRTm = ϕQm + + ϕG come from terms of homogeneity degree −2m − n in (ξ, µ) in the symbol of ϕQm and terms of homogeneity degree −2m − n + 1 in (ξ  , µ) in the symbol of tr n ϕG(m) , in local trivializations. (m) is of order −2m + 1, local contributions at k = 0 Since D1 RTm = D1 Qm + + D1 G (in (4.15)) come from terms of homogeneity degree −2m + 1 − n in (ξ, µ) in the symbol of D1 Qm and terms of homogeneity degree −2m − n + 2 in (ξ  , µ) in the symbol of tr n D1 G(m) , in local trivializations. These contributions can be traced back to the symbols of ϕ and P , resp. ψ, P and D1 at xn = 0, and the xn -derivatives up to order n at xn = 0 of the symbol of P (resp. P and D1 ), together with the first n homogeneous terms (down to order 1 − n) in the symbol of 2 (or 1 ).

In many cases one can show that the first log-coefficient vanishes and get some information on the nonlocal content of the coefficient behind it. We first observe: Theorem 4.5. We always have that a0 (I ) = 0. Moreover, a0 (ϕ) = 0 if ϕ2 is a projection, and a0 (D1 ) = 0 if ψ2 is a projection. Proof. This follows from the fact, observed already by Wodzicki in [W84, W84 ], that when  is a ψdo projection, then res() = 0. The three statements in the theorem follow by applying this to the formulas shown above: a0 (I ) = c0 (I ) = a0 (ϕ) = c0 (ϕ) =

a0 (D1 )

=

d0 (ψ)

1 4 res(2 ), 1 4 res(ϕ2 ),

(4.20)

= α res(ψ2 );

they give 0 when the operator to which res is applied is a projection.

 

Other systematic results can be obtained when 1 is a spectral projection or a suitable perturbation of such an operator. Definition 4.6. A pseudodifferential orthogonal projection  in L2 (E  ) will be said to be a spectral projection associated with C, when C is a selfadjoint first-order elliptic classical ψdo in E  and  satisfies  = > (C) + V0 ,

(4.21)

where > (C) is the orthogonal projection onto the positive eigenspace of C and V0 is the orthogonal projection onto a subspace V0 of V0 (C) (the nullspace of C). We denote V0 (C)  V0 = V0 . It is shown below in Proposition 4.8 that for any orthogonal ψdo projection  there exists an invertible C such that  = > (C). A particular result is the following: Theorem 4.7. Let 1 = > (C) + S,

(4.22)

where C is a selfadjoint elliptic differential operator of order 1 and S is a ψdo of order ≤ −n. If n is odd, then res(ϕ2 ) and res(ψ2 ) are zero and hence a˜ 0 (ϕ) and a˜ 0 (D1 ) in (4.9), (4.15) are zero.

Spectral Boundary Conditions

269

Proof. We have that ϕ2 = ϕ≤ (C) − ϕS, where   1 C ≤ (C) = |C|−C +  (C) = +  (C) , with C  = C + 0 (C). I −   0 0 2 2|C | |C | (4.23) Here 21 ϕ, ϕ0 and ϕS have residue 0, since they have no 1 − n-degree term in the symbol. The symbol of |CC | has even-odd parity (the terms of even degree of homogeneity order are odd in ξ  and vice versa; more on such symbols e.g. in [G02, Sect. 5]), and so does the symbol composed with ϕ. Thus, when the interior dimension n is odd, the term of order 1 − n in the symbol is odd in ξ  , so the integration with respect to ξ  in (4.11) gives zero.   Before considering the more delicate results on the term a0 , we include some words about ψdo projections. Proposition 4.8. (i) When  is a ψdo projection in L2 (E  ), then ort = ∗ [∗ + (I − ∗ )(I − )]−1

(4.24)

is an orthogonal ψdo projection with the same range. Moreover, R =  + (I − ort )(I − )

(4.25)

is an invertible elliptic zero-order ψdo (with inverse ort + (I − )(I − ort )) such that ort = RR −1 .

(4.26)

(E  ). There exists a selfadjoint invertible

(ii) Let  be an orthogonal ψdo projection in L2 elliptic ψdo C of order 1 in E  such that  = > (C).

Proof. (i) The formula (4.24) is known from Birman and Solomyak [BS82], details of verification can also be found in Booss-Bavnbek and Wojciechowski [BW93, Lemma 12.8]. The statements on R are easily checked. (ii) If the principal symbol π 0 equals the identity or 0, we are in a trivial case, so let us assume that π 0 = I and 0; then  and ⊥ both have infinite dimensional range. Let C1 be a selfadjoint positive first-order elliptic ψdo with scalar principal symbol c10 (x  , ξ  ) (e.g. = |ξ  |). Let C  = C1  − ⊥ C1 ⊥ . C  is a ψdo of order 1 with principal symbol c = π 0 c10 π 0 − (I − π 0 )c10 (I − π 0 ) = (2π 0 − I )c10 , 0

which is invertible since 2π 0 −I and c10 are so. Clearly, C  is selfadjoint, and C1  ≥ 0, ⊥ C1 ⊥ ≥ 0 in view of the positivity of C1 . Moreover,  commutes with C  . Since C  is selfadjoint elliptic, it has a spectral decomposition in smooth orthogonal finite dimensional eigenspaces Vk with mutually distinct eigenvalues λk , k ∈ Z, such that λk < 0 for k < 0, λk > 0 for k > 0, λ0 = 0 (V0 may be 0). The positive resp. negative eigenspace of C  is V> = ⊕k>0 Vk resp. V< = ⊕k<0 Vk . The eigenspaces are invariant under : For uk ∈ Vk , C  uk = C  uk = λk uk = λk uk ;

270

G. Grubb

hence uk ∈ Vk . Now if uk ∈ Vk \ {0} with k < 0, then uk must be zero, for otherwise (C  uk , uk ) = λk uk 2 < 0, in contrast with (C  uk , uk ) = (C1 uk , uk ) = (C1 uk , uk ) ≥ cuk 2 > 0; here c > 0 is the lower bound of C1 . Thus V< ⊂ R(⊥ ). Similarly, V> ⊂ R(). Finally, let 0+ be the orthogonal projection onto V0 , let 0− be the orthogonal projection onto ⊥ V0 (note that V0 = V0 ⊕ ⊥ V0 since V0 ⊂ V0 ), and set C = C  + 0+ − 0− ; it is injective. Then V> (C) = V> (C  )⊕V0 ⊂ R() and V< (C) = V< (C  )⊕⊥ V0 ⊂ R(⊥ ), so since they are complementing subspaces, they equal R() resp. R(⊥ ), and C is as asserted.   Parts of the above proof details are given in Br¨uning and Lesch [BL99, Lemma 2.6]. They can be used to show the fact that res() = 0 for any ψdo projection, that we used above in Theorem 4.5. In fact, with the notation of the proposition, we have since res vanishes on commutators, res() = res(R −1 ort R) = res(ort ) = res(> (C)) = =

1 2

res(C|C|−1 ) =

1 2

Ress=0 Tr(C|C|−s−1 ) =

1 2

1 2

res(I + C|C|−1 )

Ress=0 η(C, s) = 0, (4.27)

where the last equality follows from the vanishing of the eta residue of C shown by Atiyah, Patodi and Singer [APS76] (odd dimensions) and Gilkey [Gi81]. (The relation between the vanishing of the noncommutative residue on projections, and the vanishing of eta residues, enters also in [W84].) Theorem 4.9. Let 1 be an orthogonal pseudodifferential projection, and let C be a first-order selfadjoint elliptic ψdo such that 1 = > (C) + V0 as in Definition 4.6. Then 2 (P  − λ)−m − 2 (C 2 − λ)−m ∈ OP S 2,0,−2m−2 ( ).

(4.28)

The power function 21 2 (C 2 )−s corresponding to 21 2 (C 2 − λ)−m (cf. (3.7)) satisfies 2 −s 1 2 2 (C ) Tr[ 21 2 (C 2 )−s ]

= 41 ((C 2 )−s − C|C|−2s−1 ), = 41 ζ (C 2 , s) − 41 η(C, 2s).

(4.29)

In particular, in (4.8), c0 (I ) = − 41 (η(C, 0) + dim V0 − dim V0 ) + local contributions.

(4.30)

It follows that when 1 is the projection entering in the construction in Theorems 2.10 and 2.13, then a˜ 0 (I ) = − 41 (η(C, 0) + dim V0 − dim V0 ) + local contributions.

(4.31)

Spectral Boundary Conditions

271

Proof. It follows from [GS95] (adapted to the present notation) that (C 2 − λ)−m ∈ OP S 0,0,−2m ( ), and that 2 (P  − λ)−m − 2 (C 2 − λ)−m ∂ m−1

λ = 2 (m−1)! [(P  − λ)−1 (C 2 − P  )(C 2 − λ)−1 ] ∈ OP S 2,0,−2m−2 ( ), (4.32)

so this difference contributes no log-terms or nonlocal terms at the powers (−λ)−m and 1 (−λ)−m− 2 . (This reflects the fact that in the residue construction, one can replace the auxiliary operator P  by C 2 .) Now in (4.29), the first line follows from (4.23) when we recall that the powers are defined to be zero on V0 (C); the second line follows by taking the trace. Then since Tr[ 21 2 0 (C)] = 21 dim V0 and ζ (C 2 , 0) = − dim V0 (C)+ a local coefficient, we have at s = 0: c0 (I ) −

1 2

dim V0 = Tr[ 21 2 (C 2 )−s ]s=0 = 41 ζ (C 2 , 0) − 41 η(C, 0) = − 41 dim V0 (C) − 41 η(C, 0) + a local coefficient,

which implies (4.30) since − 41 dim V0 (C)+ 21 dim V0 = − 41 (dim V0 −dim V0 ). Finally, (4.31) follows in view of (4.9).   We can give a name to the “nonlocal part of a0 (I )” appearing in this way: Definition 4.10. In the situation of Definition 4.6, we define the associated eta-invariant ηC,V0 by: ηC,V0 = η(C, 0) + dim V0 − dim V0 .

(4.33)

Note in particular that ηC,V0 = η(C, 0) + dim V0 (C), ηC,V0 = η(C, 0),

if

dim V0

if 1 = ≥ (C), =

1 2

(4.34)

dim V0 (C).

(4.35)

Note also that since ζ (PT , 0) = a0 (I ) − dim V0 (PT ), we have in the situation of Theorem 4.9 that ζ (PT , 0) = − 41 ηC,V0 − dim V0 (PT ) + local contributions.

(4.36)

Under special circumstances, we can show that c˜0 and d˜0 are purely local: Theorem 4.11. Let 1 be an orthogonal pseudodifferential projection (so that 2 = ⊥ 1 ), and assume that there exists a unitary morphism σ such that σ 2 = −I, Then

 Tr

X

 −Tr X

∂λm−1  1 2 2 (m−1)! (P

∂λm−1  1 2 σ 2 (m−1)! (P

σ P  = P  σ,

− λ) − λ)

−1

− 21

⊥ 1 = −σ 1 σ.



 =

1  4 Tr X

=

− 41 Tr X



∂λm−1  (m−1)! (P

 σ

(4.37)

− λ)

∂λm−1  (m−1)! (P

−1



− λ)

, − 21



(4.38) .

272

G. Grubb

Thus in (4.8), c˜0 (I ) (= c0 (I )) is locally determined (from the symbol of P  ), and in (4.14) with ψ = σ , d˜0 (σ ) = 0,

d˜0 (σ ) is locally determined

(4.39)

(from the symbol of P  and σ ). It follows that in the situation of Theorems 4.2 and 4.3 with D1 = σ (∂xn + B1 ), a˜ 0 (D1 ) = a0 (D1 ) = 0;

(4.40)

a˜ 0 (I ), a0 (I ) and a˜ 0 (D1 ) are locally determined

(depending only on finitely many homogeneous terms in the symbols of P and T , resp. P , T and D1 ). Proof. Introduce the shorter notation Rm,1 =

∂λm−1  (m−1)! (P

− λ)−1 ,

Rm,2 =

∂λm−1  (m−1)! (P

1

− λ)− 2 ;

note that σ Rm,i = Rm,i σ by (4.37). Then we have for the traces on X , using moreover that the trace is invariant under circular permutation: Tr(2 Rm,1 ) = −Tr(σ 1 σ Rm,1 ) = −Tr(1 Rm,1 σ 2 ) = Tr((1 − 2 )Rm,1 ) = Tr(Rm,1 ) − Tr(2 Rm,1 ).

(4.41)

It follows that Tr(2 Rm,1 ) = 21 Tr(Rm,1 ).

(4.42)

Tr(σ 2 Rm,2 ) = −Tr(σ 2 1 σ Rm,2 ) = Tr(1 σ Rm,2 ) = Tr((1 − 2 )σ Rm,2 ) = Tr(σ Rm,2 ) − Tr(σ 2 Rm,2 ),

(4.43)

Similarly,

implying Tr(σ 2 Rm,2 ) = 21 Tr(σ Rm,2 ).

(4.44)

This shows (4.38). It is classically known that Tr(Rm,1 ) has an expansion in powers with local coefficients; this shows the assertion on c˜0 (I ) = c0 (I ). For i = 2, Tr(σ Rm,2 ) has an expansion in powers of (−λ) with only local coefficients, since σ Rm,2 is strongly polyhomogeneous, cf. [GS95] or [G02, Th. 2.10]. This implies the assertions on d˜0 (σ ) and d˜0 (σ ). The last assertion now follows from (4.9) and (4.15).   The proof also shows that c˜0 (I ) = a˜ 0 (I ) = 0, but we know that already from Theorem 4.5.

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273

5. Consequences for the APS Problem The preceding results have interesting new consequences for the realizations of firstorder operators in Example 1.2, which we now consider in detail. Let D satisfy (1.6) and let  be a well-posed orthogonal ψdo projection for D. Then in view of Green’s formula: (Du, v)X − (u, D ∗ v)X = −(σ γ0 u, γ0 v)X , for u ∈ C ∞ (E), v ∈ C ∞ (E1 ),

(5.1)

the adjoint (D )∗ is the realization of D ∗ (of the form (−∂xn + A + xn A21 + A20 )σ ∗ on Xc ) defined by the boundary condition ⊥ σ ∗ γ0 v = 0 (associated with the well-posed projection  = σ ⊥ σ ∗ for D ∗ ). It follows that D ∗ D is of the form (1.1) with P  = A2 , ∗ D is the realization of D ∗ D defined by the boundary condition and that D  γ0 u = 0,

⊥ (γ1 u + A1 (0)γ0 u) = 0.

(5.2)

∗ D is of the type P considered in Sects. 1–4, with P = D ∗ D, P  = A2 , Thus D  T 1 =  and B = A1 (0). Note that the symbol considered in Assumption 2.7 is here 1

((a 0 )2 + µ2 ) 2 − (I − π h )a 0 (I − π h ).

(5.3)

When the principal symbols of  and A2 commute, Assumption 2.7 is essentially equivalent with well-posedness. More precisely, we have: Lemma 5.1. Let  be an orthogonal ψdo projection in L2 (E  ) and let PT be the realization of D ∗ D under the boundary condition (5.2), and assume that the principal symbols of  and A2 commute. 1◦ When  is well-posed for D, then Assumption 2.7 holds for {P , T } with θ = π2 . 2◦ If Assumption 2.7 holds for {P , T } with some θ > 0 and π 0 (x  , ξ  ) has rank N/2, then  is well-posed for D. Proof. 1◦ Fix x  , |ξ  | ≥ 1, and consider the model realization dπ0 0 (defined for the ordinary differential operator d 0 = σ (x  )(∂xn +a 0 (x  , ξ  )) in L2 (R+ , CN ) by the boundary condition π 0 (x  , ξ  )u(0) = 0), and the model realization pt00 (defined similarly from principal symbols). The well-posedness assures that dπ0 0 is injective, hence pt00 = (dπ0 0 )∗ dπ0 0 is selfadjoint positive, as an unbounded operator in L2 (R+ , CN ). It follows that pt00 − λ is bijective from its domain to L2 (R+ , CN ), for all λ ∈ C \ R+ . Using that π 0 commutes with (a 0 )2 , we can carry out the calculations in the proof of Lemma 2.6 for the model problem (without commutation error terms), which allows us to conclude that the equation in CN : 1

[((a 0 )2 − λ) 2 − (I − π 0 )a 0 (I − π 0 )]ϕ = ψ,

(5.4)

is uniquely solvable for ψ ∈ R(I − π 0 ). Moreover, the calculations in Remark 2.9 on the model level extend the solvability of (5.4) to all ψ ∈ CN . The invertibility property extends readily to the strictly homogeneous symbols for ξ  = 0, it is obvious for ξ  = 0 with λ = 0. 2◦ Assumption 2.7 gives for µ = 0, |ξ  | ≥ 1, that pt00 = (dπ0 0 )∗ dπ0 0 is bijective. This implies injectiveness of dπ0 0 , i.e., injective ellipticity of {d 0 , π 0 γ0 }. Then well-posedness holds exactly when π 0 has rank N/2. (One may consult [G99, p. 55].)  

274

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Example 5.2. When  is taken as ≥ + S with S of order −1 (cf. Example 1.2), π 0 commutes with a 0 itself, and we see directly that Assumption 2.7 holds simply because −(I − π h )a 0 (I − π h ) ≥ 0. – For the projections (θ ) = P (θ) introduced by Br¨uning and Lesch in [BL99], Assumption 2.7 is also directly verifiable, since the conditions of [BL99] assure that −(I − π h )a 0 (I − π h ) − c|a 0 | ≥ 0 for some c > −1. Here (θ ) commutes with A2 . Again, perturbations of order −1 are allowed. ∗ D with = . So there are expansions Thus the results of Sects. 2–4 apply to D  θ (2.43), (3.3) and (3.8) for ∗ Tr(ϕ(D D − λ)−m ),

∗

Tr(ϕe−tD D ),

∗ ∗ and (s)Tr(ϕ(D D )−s ) = (s)ζ (ϕ, D D , s),

and, with the choice F = D ( a morphism from E to E1 ), there are expansions as in (2.44), (3.4) and (3.10)–(3.11) for ∗ Tr(D(D D − λ)−m ), ∗ and ( s+1 2 )Tr(D(D D )

∗

Tr(De−tD D ),

− s+1 2

(s)Tr(D(D ∗ D )−s ),

) = ( s+1 2 )η(, D , s).

Such expansions were shown in [G99] by a different procedure where D was regarded as part of a first-order system of the double size. We get new results by drawing some consequences for the coefficients at k = 0 from Sect. 4. ∗ . It is easily checked that σ ∗ DD ∗ σ Before doing this, let us also briefly look at D D ∗ ∗ is of the form (1.1), and that σ D D σ is the realization of it with boundary condition ⊥ γ0 u = 0,

(γ1 u − (A + A20 (0))γ0 u) = 0.

(5.5)

∗ , a composition to the left with σ and to In the consideration of trace formulas for D D ∗ the right with σ leaves the formulas corresponding to (2.43), (3.3) and (3.8) unchanged

if ϕ = I . Theorems 4.5 and 4.7 imply immediately:

∗ D , where D is as in Example 1.2,  is well-posed for D, Corollary 5.3. Let PT = D  and the principal symbols of  and A2 commute.

(i) For the expansions (2.43), (3.3), (3.8), related to the zeta function, a˜ 0 (I ) = a0 (I ) = 0.

(5.6)

Moreover, a˜ 0 (ϕ) = a0 (ϕ) =

1 4

res(ϕ⊥ );

(5.7)

it is zero in the following cases (a) and (b): (a) ϕ⊥ is a projection, (b) n is odd and ⊥ = > (C) + S for some first-order selfadjoint elliptic differential operator C of order 1, S of order −n. (ii) For the expansions (2.44), (3.4), (3.10)–(3.11) with F = D, related to the eta function, a˜ 0 (D) = a0 (D) = α res(σ ⊥ ); it is zero if σ ⊥ is a projection, or if (b) holds.

(5.8)

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275

∗ D is regular at zero. Since the Note that (5.6) means that the zeta function of D  hypotheses assuring this (once D is taken of the form (1.6)), are entirely concerned with principal symbols, we have in particular: The regularity of the zeta function at s = 0 is preserved under perturbations of  of order −1. When [π 0 , (a 0 )2 ] = 0, this is a far better result than that of [G01 ], where it was shown for perturbations of order −n. We also have from Theorem 4.9 and the following considerations:

Corollary 5.4. Assumptions as in Corollary 5.3. Let  be a spectral projection as in Definition 4.6, with the notation introduced there and in Definition 4.10. Then in the expansions (2.43), (3.3), (3.8), a0 (I ) = − 41 ηC,V0 + local contributions, ∗ D , 0) = − 41 ηC,V0 − dim V0 (D ) + local contributions. ζ (D

(5.9) (5.10)

∗ ; here  is replaced by ⊥ =  (−C) +   There is a similar result for D D > V0 ∗ . So in this case, Theorem 4.9 gives: in view of the remarks above on D D ∗ a0 (I )(D D ) = − 41 (−η(C, 0) + dim V0 − dim V0 ) + local contributions

= 41 ηC,V0 + local contributions,

(5.11)

Observe moreover that since ∗

∗

∗ ∗ index D = Tre−tD D − Tre−tD D = a0 (I )(D D ) − a0 (I )(D D ), (5.12)

we find: Corollary 5.5. In the situation of Corollary 5.4, index D = − 21 ηC,V0 + local contributions.

(5.13)

For the case where  = ≥ (A) (cf. (4.34)), (5.13) is known from [APS75], and (5.9) is known from [G92]; for  = > (A) + V0 in the product case, cf. [GS96, Cor. 3.7]. We believe that it is an interesting new result that for rather general projections, the non-locality depends only on the projection, not the interior operator, in this sense. Now we turn to cases with selfadjointness properties. We are here both interested in truly selfadjoint product cases and in nonproduct cases where D is principally selfadjoint at X  . Along with D we consider the operator of product type D0 defined by D0 = σ (∂xn + A) on Xc , so that D = D0 + σ (xn A11 + A10 ).

(5.14)

In addition to the requirements that σ be unitary and A be selfadjoint, we now assume that E1 = E and that D0 is formally selfadjoint on Xc when this is provided with the “product” volume element v(x  , 0)dx  dxn ; this means that σ 2 = −I,

σ A = −Aσ.

(5.15)

D0 can always be extended to an elliptic operator on X (e.g. by use of D); let us denote the extension D0 also. If the extension is selfadjoint, we call this a selfadjoint product case.

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When  is an orthogonal projection in L2 (E1 ) = L2 (E  ), it is well-posed for D if and only if it is so for D0 . For D0 in selfadjoint product cases, some choices of  will lead to selfadjoint realizations D0, , namely (in view of (5.1)) those for which  = −σ ⊥ σ.

(5.16)

The properties (5.15) and (5.16) imply (4.37) with P  = A2 , 1 = , so we can ∗ D (and D 2 ). apply Theorem 4.11 to D  0, As pointed out in Appendix A.1 of Douglas and Wojciechowski [DW91], it follows from Ch. 17 (by Palais and Seeley) of Palais [P65] that when (5.15) holds and n is odd, there exists a subspace L of V0 (A) such that σ L ⊥ L and V0 (A) = L ⊕ σ L. M¨uller showed in [M94] (cf. (1.5)ff. and Prop. 4.26 there) that such L can be found in any dimension. Denoting the orthogonal projection onto L by L , we have that + = > (A) + L

(5.17)

satisfies (5.16). The projections (θ ) introduced by Br¨uning and Lesch [BL99] likewise satisfy (5.16). We here conclude from Theorem 4.11: Corollary 5.6. In addition to the assumptions of Corollary 5.3, assume that E1 = E and ∗D ,a   that (5.15), (5.16) hold. Then in (2.43), (3.3), (3.8) for PT = D  ˜ 0 (I ) (= a0 (I )) ∗ is locally determined. Equivalently, the value of ζ (D D , s) at zero satisfies: ∗ D , 0) = − dim V0 (D ) + local contributions. ζ (D

(5.18)

We use this to show for the zeta function: Theorem 5.7. In addition to the hypotheses of Corollary 5.6, assume that  =  + S,

(5.19)

where  is a fixed well-posed projection satisfying (5.16) and S is of order ≤ −n. ( can in particular be taken as + in (5.17) or (θ ) from [BL99].) Then the a˜ 0 -terms (and a0 -terms) in (2.43), (3.3), (3.8) for D and D are the same, ∗ ∗ a˜ 0 (I )(D D ) = a˜ 0 (I )(D D ),

(5.20)

∗ ∗ D , 0) + dim V0 (D ) = ζ (D D , 0) + dim V0 (D ), ζ (D

(5.21)

so

and in particular, ∗ ∗ ζ (D D , 0) = ζ (D D , 0) (mod Z).

(5.22)

Proof. We shall combine the fact that a˜ 0 (I ) is locally determined with order considerations. Let ∗ D − λ)−m , RTm = (D

∗ RTm = (D D − λ)−m .

(5.23)

Note that they have the same pseudodifferential part (D ∗ D − λ)−m + , so their difference R m − RTm is a singular Green operator. It is shown in [G01 , proof of Th. 1] that when S T is of order −n, the ψdo tr n (R m − RTm ) on X  has symbol in S −m−n,−m,0 ∩ S −n,−2m,0 . T

Spectral Boundary Conditions

277

The total order is −n − 2m, so the highest degree of the homogeneous terms in the symbol is −n − 2m. As noted in Remark 4.4, the local contribution to the terms with index k = 0 in the trace expansion of this difference comes from homogeneous terms of degree 1 − n − 2m (recall that dim X  = n − 1), so since the terms consist purely of local contributions, they must vanish. This shows (5.20), and the other consequences are immediate.   In a language frequently used in this connection, the statement means that ∗ D , 0) + dim V (D ) is constant on the Grassmannian of ψdo projections satζ (D  0  isfying (5.16) and differing from  by a term of order ≤ − dim X. When D equals D0 in a selfadjoint product case, Wojciechowski shows a result like this in [W99, Sect. 3] for perturbations of + of order −∞, assuming that D0,+ and D0, are invertible. The non-invertible case is treated by Y. Lee in the appendix of 2 [PW02]; he shows moreover that ζ (D0, , 0) + dim V0 (D0,+ ) = 0, so we conclude + 2 that ζ (D0, , 0) + dim V0 (D0, ) = 0 when  = + + S. ∗ D )− We can also discuss the eta function η(D , s) = Tr(D(D  meromorphically as in (3.11). First we conclude from Theorem 4.11:

s+1 2

), extended

Corollary 5.8. Assumptions of Corollary 5.6. In (2.44), (3.4), (3.10)–(3.11) for PT = ∗ D , D = D, one has that a D ˜ 0 (D) = a0 (D) = 0, and a˜ 0 (D) (= a0 (D)) is locally  1 determined. In other words, the double pole of η(D , s) at 0 vanishes and the residue at 0 is locally determined. (It may be observed that when (5.15) holds, the entries in the second line of (4.38) 1 vanish identically, since Tr X (σ ∂λm−1 (P  − λ)− 2 ) = 0.) It is remarkable here that the hypotheses, besides (5.15)–(5.16), only contain requirements on principal symbols (the well-posedness of  for D and the commutativity of the principal symbols of  and A2 ). So the result implies in particular that the vanishing of the double pole of the eta function is invariant under perturbations of  of order −1 (respecting (5.16)). Earlier results have dealt with perturbations of order −∞ [W99] or order −n [G01 ]. Now consider the simple pole of η(D , s) at 0. Here we can generalize the result of Wojciechowski [W99] on the regularity of the eta function after a perturbation of order −∞, to perturbations of order −n of general : Theorem 5.9. Assumptions of Theorem 5.7. ∗D In (2.44), (3.4), (3.10)–(3.11) with D1 = D, the a˜ 0 -terms (and a0 -terms) for D  ∗ D are the same: and D  ∗ ∗ D ) = a˜ 0 (D)(D D ); a˜ 0 (D)(D

in other words, Ress=0 η(D , s) = Ress=0 η(D , s).

(5.24)

278

G. Grubb

∗ D ) = 0 (this holds for  and for certain (θ ) if D In particular, if a˜ 0 (D)(D +  ∗ D ) = 0, i.e., the eta function equals D0 in a selfadjoint product case), then a˜ 0 (D)(D  η(D , s) is regular at 0.

Proof. As in the proof of Theorem 5.7, we combine the fact that a˜ 0 (D) is locally determined with order considerations. Consider DR m and DRTm , cf. (5.23). Since they have T m m the same pseudodifferential part D(D ∗ D − λ)−m + , their difference DRT − DRT is a  singular Green operator. It is shown in [G01 , proof of Th. 1] that when S is of order −n, the ψdo tr n (DRTm − DRTm ) on X has symbol in S −m−n,1−m,0 ∩ S −n,1−2m,0 . The total order is 1 − n − 2m, so the highest degree of the homogeneous terms in the symbol is 1 − n − 2m. Now the local contribution to the terms with index k = 0 in the trace expansion of this difference comes from homogeneous terms of degree 2 − n − 2m (cf. Remark 4.4), so since the terms contain only local contributions, they must vanish. This shows (5.24). ∗ D ) vanishes if a ∗ D ) does so; then the eta funcIn particular, a˜ 0 (D)(D ˜ 0 (D)(D   tion for D is regular at 0. The eta regularity for the case  = + , D equal to D0 and selfadjoint on X with product volume element on Xc , was shown in [DW91] under the assumptions n odd and D compatible; this was extended to general n and not necessarily compatible D in M¨uller [M94]. It was shown for certain (θ ) in [BL99, Th. 3.12].   In a frequently used terminology, the theorem shows that the residue of the eta function is constant on the Grassmannian of ψdo projections satisfying (5.16) and differing from  by a term of order ≤ − dim X. We do not expect that the order can be lifted further in general. The result on the regularity of the eta function at s = 0 for (−n)-order perturbations of the product case with  = + has been obtained independently by Yue Lei [L02] at the same time as our result, by another analysis based on heat operator formulas. The above results on the vanishing of the eta residue are concerned with situations where D equals D0 in a selfadjoint product case. However, the fact from Corollary 5.8 that Ress=0 η(D , s) is locally determined also in suitable non-product cases, should facilitate the calculation of the residue then. For example, if D = D0 + xnn+1 A31 on Xc with a first-order tangential differential operator A31 , and the volume element satisfies j ∂xn v(x  , 0) = 0 for 1 ≤ j ≤ n, then by [G02, proof of Th. 3.11], the local terms with index k ≤ 0 in the difference between the resolvent powers are determined entirely from the interior operators D and D0 . Then when n is even, the contributions to k = 0 vanish simply because of odd parity in ξ ; this gives examples where the eta function is regular at 0 in a non-product situation. References [APS75] [APS76] [A02] [AE99] [BFS99]

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Communicated by M. Aizenman

Commun. Math. Phys. 240, 281–327 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0895-4

Communications in

Mathematical Physics

Critical (Φ4 )3,
D.C. Brydges1,2,
, P.K. Mitter3,

, B. Scoppola4,


1

Department of Mathematics, University of British Columbia, 121-1984 Mathematics Rd, Vancouver, B.C., Canada V6T 1Z2. E-mail: [email protected] 2 Department of Mathematics, University of Virginia, Kerchof Hall, P. O. Box 400137, Charlottesville, VA 22904-4137, USA 3 Laboratoire de Physique Math´ematique, Universit´e Montpellier 2, Place E. Bataillon, Case 070, 34095 Montpellier Cedex 05, France. E-mail: [email protected] 4 Dipartimento di Matematica, Universit´a “La Sapienza” di Roma, Piazzale A. Moro 2, 00185 Roma, Italy. E-mail: [email protected] Received: 29 October 2002 / Accepted: 25 March 2003 Published online: 18 July 2003 – © Springer-Verlag 2003

Abstract: The Euclidean (φ 4 )3,  model in R3 corresponds to a perturbation by a φ 4 interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter ε in the range 0 ≤ ε ≤ 1. For ε = 1 one recovers the covariance of a massless scalar field in R3 . For ε = 0, φ 4 is a marginal interaction. For 0 ≤ ε < 1 the covariance continues to be Osterwalder-Schrader and pointwise positive. We consider the infinite volume critical theory with a fixed ultraviolet cutoff at the unit length scale and we prove that for ε > 0, sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. We construct the stable critical manifold near this fixed point and prove that under Renormalization Group iterations the critical theories converge to the fixed point. 1. Introduction, Model, RG Transformation 1.1. Introduction. Let φ be a mean zero Gaussian scalar random field on R3 with covariance
d 3 p ip·(x−y) 2 −(3+)/4 const = e (p ) . (1.1) E(φ(x)φ(y)) = (2π )3 |x − y|(3−)/2 Here p2 = |p|2 is the standard Euclidean norm in R3 and p · x is the standard scalar product. ε is a real parameter which we take in the region 0 ≤  ≤ 1. Note that for  = 1 we have the standard massless free scalar field in R3 . This covariance (−)−(3+)/4 has interesting properties. In particular, scaling limits of suitable perturbations (see below) of these theories should be Euclidean quantum field theories, because (−)−(3+)/4 is


Partially supported by NSERC of Canada Laboratoire Associ´e au CNRS, UMR 5825


Partially supported by CNR, G.N.F.M. and MURST



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D.C. Brydges, P.K. Mitter, B. Scoppola

Osterwalder- Schrader positive not only for  = 1 (see [GJ]) but also for 0 ≤  < 1. The latter follows from the convergent integral representation
∞ −(3+)/4 (−) (x − y) = 1/c ds s −(3+)/4 (− + s)−1 (x − y), (1.2) 0

∞

where c = 0 ds s −(3+)/4 (1 + s)−1 and 0 ≤  < 1. The scaling limit should be constructible and it will not be a generalized free field, because the result in this paper shows that there is a non-Gaussian Infrared fixed point. From the same integral representation one also finds that the kernel of (−)−(3+)/4 is pointwise positive. Furthermore (−)−(3+)/4 is the Green’s function (or potential) for a stable L´evy process in R3 with parameters (α, β) in the L´evy-Khintchine notation [KG], with the characteristic exponent α = (3+)/2, and β = 0. The sample paths have discontinuities and the process diffuses very fast. We hope, in the future, to exploit this to study self-avoiding Levy processes by representing them as supersymmetric versions of the functional integral of this paper, following the program set out in [BI]. These properties make the study of the above gaussian random field and its non-linear perturbations worthwhile. In particular we will study here the critical properties of a model corresponding to the partition function
(1.3) Z = dµC (φ) e−V0 (φ) , where dµC is the Gaussian measure with covariance C and C is (−)−(3+)/4 with a fixed unit scale ultraviolet cutoff described below and

V0 (φ) = V (φ, C, g0 , µ0 ) = g0 d 3 x : φ 4 :C (x) + µ0 d 3 x : φ 2 :C (x) (1.4) and the coupling constant g is held strictly positive. Moreover in order to define the model completely we must also introduce a volume cutoff. However as soon as we have described how to parametrise the theories that comprise the domain of the Renormalization Group, there is a clear definition and candidate for the infinite volume limit and that is the object under consideration in this paper. These cutoffs will be introduced presently when we give a precise definition of the covariance C in (1.4) as well as that of the model. From (1.1) we can read off the canonical scaling dimension [φ] of φ: [φ] = (3−)/4. This, together with (1.3), implies that we can assign to g, µ the dimensions [g] = , [µ] = (3 + )/2. Note that we have not put in a term 2z d 3 x|∇φ(x)|2 in (1.4). This is because the dimension [z] = −(1 − ε)/2. Hence for ε < 1 this is a candidate for an irrelevant (stable) direction. The point of this paper is to prove that Wilson’s theory of critical phenomena [KW] is correct for this model: namely, that for  > 0 the critical (infra-red) properties of the model are dominated by a non-Gaussian fixed point of Renormalization Group iterations with g = g∗ = O() provided the unstable parameter µ is fine tuned to a critical function µc (g) which determines the stable (critical) manifold of the fixed point. In the present work we will prove the existence of the non-Gaussian fixed point for  > 0 held sufficiently small and, on the way, construct the stable manifold. There are open problems, which may be accessible to these methods. The first is to verify that the fixed point describes a scaling limit which satisfies the Osterwalder-Schrader axioms (see

Critical ( 4 )3, 

283

discussion of lattice cutoffs below) and the second is to prove that the critical manifold connects the ultraviolet and infrared fixed points. The construction of standard (non-critical) φ34 theory was a tour de force of the Constructive Field Theory program. The Renormalization Group was (either implicitly or explicitly) behind this construction. The key steps and history are summarized in [GJ]. The mathematical analysis of Renormalization Group (henceforth denoted RG) transformations has by now a long history [F, BG]. In particular, critical (φ)44 was constructed in [GK] by RG methods, and in [FMRS] by related phase cell methods. Our particular line of attack is influenced by a series of works which started with [BY], developed further in [DH1, DH2, BDH-est, BDH-eps], with more recent developments in [MS]. We shall be concerned here with these latter developments. In [MS] fluctuation covariances of finite range were exploited, and this simplifies considerably the RG analysis. In particular the analysis of the fluctuation integration becomes a matter of geometry and one no longer needs the cluster expansion and analyticity norms. In the continuum approach of [MS] that is also adopted here the existence of fluctuation covariances with finite range follows easily from a judicious choice of a class of ultraviolet cutoffs. This raises the general question of which gaussian random fields can be decomposed into sums of fluctuation fields with covariances with finite range. A partial answer which includes the standard massless Euclidean field with lattice cutoff and lattice analogues of the covariances of this paper will be given in [BGM]. Lattice cutoffs are useful because they preserve Osterwalder-Schrader positivity. Only the existence of multiscale decompositions with the above finite range property (together with some regularity and positivity properties) is required in what follows. The present work borrows many technical considerations introduced in [BDH-est]. Although these are independent of the manner of treating the fluctuation step, we repeat some of them because the simpler norms in this paper allow easier proofs. We also borrow some ideas from [BDH-eps] where a related model (which however does not possess the physical properties mentioned earlier) was considered and the existence of a non-Gaussian fixed point was proved. We use this paper as an opportunity to improve previous arguments. As in all other work on these problems we get around the failure of perturbation theory to converge by using the first few terms of perturbation theory to generate an approximate renormalized action together with a remainder represented by polymers. This is done in Sect. 4. We have improved on the analogous steps in [BDH-eps], but it is still not “clean”. Our aim is to formulate the weak estimates on the remainder that are nevertheless strong enough to prove that the approximate renormalized action remains dominant under the action of the renormalization group. We want weak estimates because they should be easy to prove and allow more general perturbations away from Gaussian. Other authors [MP, AR] have the opposite philosophy, namely to search for explicit formulas for remainders in terms of convergent polymer expansions. They have in mind numerical applications and a preference for explicit versus inductively defined expansions.

1.2. Multiscale decomposition. We introduce a special type of ultraviolet cutoff as follows: Let g be a non-negative, C ∞ , O(3) invariant function of compact support in R3 such that g(x) vanishes for |x| ≥ 1/2. Define u = g ∗ g. Thus u is positive, C ∞ , and of compact support: u(x) = 0 for |x| ≥ 1. First we note that const = |x − y|(3−)/2




∞ 0

dl −(3−)/2 x−y u( l ) l l

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D.C. Brydges, P.K. Mitter, B. Scoppola

by scaling the variable of integration. Define


∞

C(x − y) = 1

  dl −(3−)/2 x−y l . u l l

(1.5)

Because the lower limit is 1 and not 0 this C is pointwise positive and C ∞ . Let L ≥ 2 be an integer. Let


(x − y) =

L

1

  dl −(3−)/2 x−y u l . l l

(1.6)

Clearly is C ∞ , pointwise positive and of finite range: (x − y) = 0 : |x − y| ≥ L.

is our fluctuation covariance and it generates a key scaling decomposition: C(x − y) = (x − y) + L

−(3−)/2

 C

x−y L

 .

(1.7)

Moreover because of our choice of the form of u, namely u = g ∗ g, and C are generalized positive definite. Now it can be shown (see e.g. Sect. 1.1 of [MS]) that under these conditions for any compact set N in R3 there exists a Gaussian measure dµC of mean 0 and covariance C supported on the Sobolev space Hs (N ) for any s > 0. Choosing s > 3/2+2 is sufficient for our purpose. Then, by Sobolev embedding, sample paths are at least twice differentiable. Define the compact set N = [− 21 LN , 21 LN ]3 ⊂ R3 . Our model with an ultraviolet cutoff and volume cutoff is now defined by the partition function
Z=

dµC (φ) Z0 (N , φ),

where

Z0 (N , φ) = e−V (N ,φ)

(1.8)

and the potential has now been restricted to the volume N . The Wick ordering in the potential is with respect to the infinite volume covariance C and this is well defined because the Wick constants are finite. Remark. We can exhibit C in the traditional form with an ultraviolet cutoff function. Let uˆ be the Fourier transform of u. Because of O(3) invariance we can write u(p) ˆ = v(p2 ). Then it is easy to see from the above that
C(x − y) =

d 3 p ip.(x−y) e F (p 2 )(p 2 )−(3+)/4 , (2π)3

(1.9)

where
F (p ) = 1/2 2

∞

p2

ds (3+)/4 v(s). s s

(1.10)

2 From this we see that F is positive, continuous and of fast decrease (since v(p 2 ) = |g(p)| ˆ and g is of compact support) and can be thus identified as the ultraviolet cutoff function.

Critical ( 4 )3, 

285

1.3. Renormalization Group transformation. From (1.7), if we now define the rescaled field Rφ(x) = φL−1 (x) = L−(3−)/4 φ(x/L) we get




(1.11)


dµC (φ)Z0 (N , φ) =

where

dµC (φ)Z1 (N−1 , φ),

(1.12)


Z1 (N−1 , φ) =

dµ (ζ )Z0 (N , ζ + φL−1 ).

(1.13)

The iteration of (1.13) will constitute our RG transformations. After n steps we have

dµC (φ)Z0 (N , φ) = dµC (φ)Zn (N−n , φ), (1.14) where


Zn (N−n , φ) =

dµ (ζ )Zn−1 (N−n+1 , ζ + φL−1 ).

(1.15)

After N − 1 steps we arrive at ZN−1 (1 , φ), where 1 is the L- block [− 21 L, 21 L]3 . In order to analyse the RG transformations, it is essential to write the partition function density in a form that is preserved by the RG operations. This is achieved by the following polymer gas representation. Pave R3 with closed unit blocks denoted henceforth . Now let  ⊂ R3 be the volume after a certain number of RG steps. We take  with the induced paving. A connected polymer X is a connected union of a subset of these closed unit blocks and is thus closed. A polymer activity K(X, φ) is a map X, φ → R, where the fields φ depend only on the points of X. We shall only consider polymer activities supported on connected polymers. We then write, suppressing the dependence on φ, ∞  1 −V (\X) Z() = e N! N=0



N 

K(Xj ),

(1.16)

X1 ,..,XN j =1

where the connected polymers Xj are disjoint, X = ∪N 1 Xj and V (Y ) = V (Y, φ, C, g, µ) is given by (1.4) with parameters g, µ and integration over Y . The initial purely exponential density is the special case where K is identically zero. It is possible to rewrite the above in a more compact form if we extend the polymer algebra by cells as done in [BY]. A cell may be the interior of a block, an open face, an open edge or a vertex. A polymer, which is a union of closed blocks, can be uniquely written as a disjoint union of cells, but the point is that all other sets generated by our manipulations, such as complements of polymers, can also be uniquely written as disjoint unions of cells. We define a commutative product,denoted ◦, on functions of sets (unions of cells), in the following way (F1 ◦ F2 )(X) = Y,Z:Y ◦ Z=X F1 (Y )F2 (Z), ∪ where X = Y ∪◦ Z iff X = Y ∪Z and Y ∩Z = ∅. The ◦ identity I is defined by I(X) = 1 if X = ∅ and otherwise vanishes. Finally we can define an Exponential operation on functions of unions of cells as the usual power series based on the ◦ product and the

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D.C. Brydges, P.K. Mitter, B. Scoppola

definition of the ◦ identity I. This has the properties of the usual exponential. We also define a space filling function
by
(X) = 1 if X is a cell and otherwise vanishes. We then have Exp
= 1 and it easy to see that (1.16) can be rewritten as Z() = [Exp(
e−V + K)]().

(1.17)

1.4. The Formal Infinite Volume Limit. By writing the integrands in the form (1.17), the i th RG transformation induces a map fN−i : (Ki−1 , Vi−1 ) → (Ki , Vi ).

(1.18)

The subscript N − i is there because the initial partition function was defined in a finite volume region N and consequently the i th RG transformation is a map from polymer activities supported on subsets of N−i+1 to polymer activities supported on subsets of N −i . The represention by polymer activities permits us to define (in Sect. 3) a candidate action of the RG in the infinite volume limit. This is an autonomous action on a Banach space of polymer activities supported on subsets of R3 . We will prove this action has a fixed point and a stable manifold. This ability to work directly in the infinite volume is an attractive feature, but one must work harder to prove results on infinite volume correlation functions defined as limits of finite volume correlations. This would include analysis of the RG near the boundary of , possibly by including in V a surface integral of relevant local monomials over the boundary of N−i . 2. Regulators, Derivatives and Norms 2.1. Regulators. To describe the Banach space of polymer activities we introduce a large field regulator which measures the growth of polymer activities in the fields φ, actually in ∂φ: 2

Gκ (X, φ) = eκ φ X,1,σ ,

(2.1)

where

φ 2X,1,σ =



∂ α φ 2X .

(2.2)

1≤|α|≤σ

Here φ X is the L2 norm and α is a multi-index. We take σ > 3/2 + 2 so that this norm can be used in Sobolev inequalities to control φ and its first two derivatives pointwise. After the function u is fixed, the parameter κ > 0 is fixed, for the whole paper, by a choice depending only on u, so that for all L ≥ 1 the large field regulator satisfies the stability property
dµ (ζ ) Gκ (X, ζ + φ) ≤ 2|X| G2κ (X, φ), (2.3) where X is a polymer and |X| is the number of unit blocks in X. This is proved in the same way as in the proof of the stability property of the large field regulator in Sect. 2.4 of [BDH-est].

Critical ( 4 )3, 

287

Now hold L sufficiently large and recall that  < 1. Then we get after rescaling
(2.4) dµ (ζ ) Gκ (X, ζ + φL−1 ) ≤ 2|X| Gκ (L−1 X, φ) because from the scaling property (1.11) of the fields φ we have φL−1 2X,1,σ ≤ L−(1−)/2 φ 2L−1 X,1,σ . Next we introduce a large set regulator. Let X be a connected polymer. We define Ap (X) = 2p|X| L(D+2)|X| ,

(2.5)

where for us the dimension of space D = 3, and p ≥ 0 is an integer. Call a connected polymer small if |X| ≤ 2D . A connected polymer which is not small is called large. Let X¯ L be the L-closure of X. This is the smallest union of L-blocks containing X. We have from Lemma 1 of [BDH-est] the following two facts: Fix any integer p ≥ 0, let L be sufficiently large depending on p, then for any connected polymer X, A(L−1 X¯ L ) ≤ cp A−p (X).

(2.6)

For X a large connected polymer, A(L−1 X¯ L ) ≤ cp L−D−1 A−p (X).

(2.7)

Here cp = O(1) is a constant independent of L. 2.2. Derivatives in fields and polymer activity norms. The following definitions are motivated by those in [BDH-est], the main difference being that we will need only a finite number of field derivatives and for them only natural C n0 norms. For a polymer activity K(X, φ) define the nth derivative in the direction (f1 , ..., fn ) as: (D n K)(X, φ; f ×n ) =

n 

(∂/∂sj ) K(X, φ +



sj fj ) |sj =0 ∀j ,

(2.8)

j =1

where the shorthand notation f ×n stands for f1 , f2 , ..., fn . The functions fj belong to C 2 (X) and we assume that such derivatives exist for n ≤ n0 for some n0 . We will measure the size of such derivatives, which are multilinear functionals on C 2 (X), by the norm:

(D n K)(X, φ) =

sup

fj C 2 (X) ≤1 ∀j

|(D n K)(X, φ; f ×n )|.

(2.9)

Let h > 0 be a real parameter. We define the following set of norms:

K(X, φ) h =

n0  hj j =0

j!

(D j K)(X, φ)

(2.10)

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D.C. Brydges, P.K. Mitter, B. Scoppola

and then

(K(X) h,Gκ =

sup K(X, φ) h G−1 κ (X, φ).

(2.11)

φ∈C 2 (X)

These norms differ from norms used in [BDH-est] in having the supremum over φ outside the sum over derivatives, as well as involving only finitely many derivatives. They are easier to use and retain the product property K1 (X1 , φ)K2 (X2 , φ) h ≤

K1 (X1 , φ) h K2 (X2 , φ) h which was the basis for proofs in [BDH-est]. We assume as earlier that the activity K is supported on connected polymers. We then define 

K h,Gκ ,A = sup

(K(X) h,Gκ A(X). (2.12)  X⊃

In addition we define kernel norms: |K(X)|h =

n0  h

(D j K)(X, 0) , j!

(2.13)

j =0

where h is a real parameter and h ≥ 0. We define  |K(X)|h A(X). |K|h ,A = sup

(2.14)

 X⊃

These definitions are also different from the kernel norms in [BDH-est, BDH-eps]), but retain the product property. When using these norms for our model we will choose n0 = 9, h = ε−1/4 and either h = h or h = h∗ , where h∗ = L(3−2[φ])/2 = L(3+ε)/4 . The kernel norms with h = h∗ will be useful for controlling flow coefficients. 3. RG Step In this section we describe the RG step. This consists of two parts: Fluctuation integration and rescaling, followed by the extraction of relevant parts. 3.1. Fluctuation integration and rescaling. The integration over the fluctuation field exploits the independence of ζ (x) and ζ (y) when |x − y| ≥ L. To do this we pave  by blocks of side L called L−blocks so that each L−block is a union of the original 1−blocks. Let X¯ L denote the L−closure of a set X, namely the smallest union of closed L−blocks containing X. The polymers will be combined into larger L−polymers which by definition are closed, connected and unions of L−blocks. The combination is performed in such a way that the new polymers are associated to independent functionals of ζ . We start with the representation (1.17) of Sect. 1.3, Z() = [Exp(
e−V + K)](), with K(X) = 0 unless X is closed and connected. By definition we have: N  1  e−V (\∪Xj ) K(Xj ) , Exp(
e−V + K)() = (Xj ) j =1 N! N

where the sum is over sequences of disjoint polymers (Xj ) = (X1 .....XN ).

(3.1)

(3.2)

Critical ( 4 )3, 

289

Let us define: ∪Xj = X and  \ X = Xc . Xc is an open set. We denote by X¯ c its closure. Obviously, V (Xc ) = V (X¯ c ), since V (Xc ) is given by a Lebesgue integral over Xc . Hence we can write:  e−V (Xc ,ζ +φ) = e−V (,ζ +φ) . ¯ ⊂Xc

Define the polymer activity P, supported on closed unit blocks, as: ˜

P (, ζ, φ) = e−V (,ζ +φ) − e−V (,φ)

(3.3)

with V˜ to be chosen. In the following V , K has field argument ζ + φ whereas V˜ depends only on φ. The dependence of P on ζ, φ is as defined above. Now write:  ˜ e−V (Xc ) = [e−V () + P ()], ¯ ⊂Xc

then expand the product and insert the expansion into (3.2): N  1  M ˜ Exp(
e−V + K)() = e−V (X0 ) K(Xj ) P (i ), (Xj ),(i ) j =1 i=1 N!M! (3.4) where X0 =  \ (∪Xj ) ∪ (∪i ). Let Y be the L−closure of (∪Xj ) ∪ (∪i ) and let Y1 , . . . , YP be the connected components of Y . These are L−polymers. Let f be the function that maps π := (Xj ), (i ) into {Y1 , . . . , YP }. Now we perform the sum over (Xj ), (i ) in (3.4) by summing over π ∈ f −1 ({Y1 , . . . , YP }) and then {Y1 , . . . , YP }. The result is: ˜

Exp(
e−V + K)() = ExpL (
e−V + BK)(),

(3.5)

where the subscript on ExpL indicates that that the domain is functionals of L−polymers and  N M 1  ˜ (BK)(Y ) = e−V (X0 ) K(Xj ) P (i ), (Xj ),(i )→{Y } j =1 i=1 N !M! N+M≥1

(3.6) where X0 = Y \ (∪Xj ) ∪ (∪i ) and the → is the map f . This representation (3.5) is a ˆ sum over products of polymer activities (B K)(Z j ), where the closed disjoint polymers Zj are necessarily separated by a distance ≥ L and the spaces between the polymers ˜ are filled with e−V which are independent of the fluctuation field ζ . The covariance

(x − y) = 0 if |x − y| ≥ L. So the fluctuation integral factorises and we obtain:
˜ dµ (ζ )[Exp(
e−V + K)](, ζ + φ) = ExpL (
L e−V + (BK  )(, φ), (3.7) where the superscript  (“sharp”) denotes integration with the measure dµ (ζ ). Now we perform the rescaling. This is accomplished by replacing φ by the rescaled field Rφ where, as in (1.11), (Rφ)(x) = φL−1 (x) = L−[φ] φ(x/L) and (RK) (L−1 X, φ) = K(X, φL−1 ). Let S = RB, then:
˜ dµ (ζ )[Exp(
e−V + K)](, ζ + φL−1 ) = Exp(
e−VL + (SK) )(L−1 , φ), (3.8)

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D.C. Brydges, P.K. Mitter, B. Scoppola

where V˜L (, φ) = V˜ (L, φL−1 ) and the superscript  (“natural”) denotes integration with the measure dµ L (ζ ) and L (x − y) = L2[φ] (L(x − y)). We have returned to a functional of the form Exp(
e−V + K)() with (V , K) → (V˜ , (SK) ). Thus the operation is an evolution of the interaction described in coordinates V , K. We will call this the fluctuation map. It has a locality property: (SK) (X)) is a functional of K(Y ), Y ⊂ X and V (),  ⊂ X. 3.2. Extraction. The RG map is the composition of the fluctuation map with a change of (V , K) coordinates described in this section. The object is to improve on the fluctuation map of the last section by exploiting the fact that, for  fixed, a functional Z(, φ) has more than one (V , K) representation. The situation is that of the implicit function theorem: how must K change in response to a change in V so that Exp(
e−V + K)() remains fixed as a functional on C n0 (). Having answered this we can choose a perturbation of V so that the corresponding change in K cancels parts of K that expand under the action of RG. In this way, expanding functionals are all in V where we can study their evolution explicitly using perturbation theory, whereas anything left in K is a contracting error term. Let P (φ(x)) be a local polynomial, which means that it is a polynomial in φ(x) and derivatives of φ(x) at x. We consider a change in V of the form 
VF (Y ) = dx αP (x)P (φ(x)), (3.9) Y

P

where the sum ranges over finitely many local polynomials and, for each such P , αP (x) has the form  αP (X, x) (3.10) αP (x) = X⊃x

such that αP (X, x) = 0 if x is not in the interior of X and αP (X, x) = 0 if X ⊂ . The corresponding change in K is given in terms of 
dx αP (X, x)P (φ(x)), F (X) = P

F (X, ) =


P

(3.11) dx αP (X, x)P (φ(x)).



Extraction Formula. Given VF0 and VF1 as above, with VF0 field independent, there exists E(K, F0 , F1 ) such that Exp(
e−V + K)() = eVF0 () Exp(
e

−VF

1

+ E(K, F0 , F1 ))(),

(3.12)

where the linearization of E is E1 (K, F0 , F1 ) = K − (F0 + F1 )e−V ,

VF 1 = V − VF .

(3.13)

This is a restatement of Theorem 6 in Sect. 4.2 of [BDH-est]. The estimates in Theorem 6 on the norms of E(K, F ) will play an essential role and are stated in Sect. 5. In the proof of Theorem 6 in [BDH-est], some changes are necessary since we here use

Critical ( 4 )3, 

291

polymer activities supported on closed sets instead of unions of open blocks. Thus, in [BDH-est], p. 780, Definition 2 and Lemmas 10 through 13 are unchanged apart from the nature of the sets: in Lemma 11 the set X in (X) becomes an arbitrary union of cells such as the complement of a set of closed polymers and the set Y in F (Y) is a closed polymer. We here write α(X, x) in place of the α(X, , x) of [BDH-est] because the  is redundant, being determined by x. Since polymers in this paper are connected, condition 3 in Lemma 13 is equivalent to ordinary connectedness of union of {Xi , Zk }. αP (X, x) in (3.10) is chosen later in Sect. 4. This choice will be local, in the sense that it is determined by V (),  ⊂ X and by K(X). Lemma 13 and Eq. (112) of [BDH-est] imply that E(K, F0 , F1 ))(X) also is local: it is determined by K(Y ), Y ⊂ X ˜ where X˜ is a neighbourhood of X, namely the union of X with all and V (),  ⊂ X, small sets that intersect X. Therefore, in (1.18), fN−i (K, V )(X, φ) is independent of ˜ Thus limN→∞ fN−i (K, V )(X, φ) N for all N large enough so that N−i contains X. exists pointwise in X. In this paper we are studying the action of this pointwise infinite volume limit.

3.3. Appendix. For convenient later reference we collect here the notations associated with rescaling that will be used in this paper. Some of them refer to objects not yet introduced. Rφ(x) = φL−1 (x) = L−[φ] φ(x/L), (RK)(L−1 X, φ) = KL (L−1 X, φ) = K(X, φL−1 ), VL (, φ) = V (L, φL−1 ),

L (x − y) = L2[φ] (L(x − y)). In general, for an integral kernel u(x − y), e.g. a covariance, we define a rescaled version uL (x − y) = L2[φ] u(L(x − y)) = L

3−ε 2

u(L(x − y)).

(3.14)

The scaling definitions imply that V˜L (, g, µ) = V (, φ, C, gL , µL ),

gL = Lε g, µL = L

3+ε 2

µ.

(3.15)

A set X is said to be small if X is connected and |X| ≤ 2D . In the present case the dimension D = 3. X¯ L is the L-closure of X, which by definition is the smallest union of closed L-blocks containing X. The superscript  (“natural”) denotes integration with the measure dµ L (ζ ) and the superscript  (“sharp”) denotes integration with the measure dµ (ζ ). 4. Application of Renormalization Group Step In this section we specify a particular RG map by making choices for V˜ , F in (3.3) and (3.13). We determine V˜ by first order perturbation theory, calculate the second order perturbative approximation to the RG map and find a formula for the error after second order. F is chosen so that expanding (relevant) parts of the second order approximation and error are removed from K.

292

D.C. Brydges, P.K. Mitter, B. Scoppola

Recall:





d 3 x : φ 4 :C (x) + µ

V (, φ) = V (, φ, C, g, µ) = g 

d 3 x : φ 2 :C (x). (4.1) 

Guided by first order perturbation theory we define V˜ by changing the normal ordering

3 4 ˜ V (, φ) = V (, φ, CL−1 , g, µ) = g d x : φ :CL−1 (x) + µ d 3 x : φ 2 :CL−1 (x). 



(4.2) Recall that the RG acts on functionals written in the form Exp(
e−V + K)(). In this section we refine this description by writing K = Qe−V + R,

(4.3)

where Q is an explicit polymer activity which we will call the “second order polymer activity”. It is motivated by second order perturbation theory in powers of g and is defined as follows: Q is supported on connected polymers X, |X| ≤ 2. We write Q(X, φ) = Q(X, φ; C, w, g) = g 2

3 

˜ φ; C, w(4−j ) ), nj Q(j,j ) (X,

(4.4)

j =1

where (n1 , n2 , n3 ) = (48, 36, 8) and w = (w (1) , w(2) , w(3) ) is a triple of integral kernels to be obtained inductively and   if X =   ×  ˜ X = (1 × 2 ) ∪ (2 × 1 ) if X = 1 ∪ 2 , (4.5)  0 otherwise ˜ φ; C, w(4−m) ) = Q(m,m) (X,

1 2


d 3 xd 3 y : (φ m (x)

X˜ m

− φ (y))2 :C w (4−m) (x − y) for m = 1, 2
1 (3,3) ˜ (1) Q (X, φ; C, w ) = d 3 xd 3 y : φ 3 (x)φ 3 (y) :C w (1) (x − y). (4.6) 2 X˜ Unlike the similar formulas in [BDH-eps], these objects are completely normal ordered. Next we define the second order approximation to the RG map. We say that an activity p(X) is supported on (closed/open) unit blocks if p(X) = 0 for X not a (closed/open) unit block. A block is closed by default. Let p be the activity supported on unit blocks defined by p(, ζ, φ) = V (, ζ + φ) − V˜ (, φ) = pg + pµ , where




(4.7)

d 3 x : ζ 4 : (x) + 4φ(x) : ζ 3 : (x)

pg = g 

+ 6 : φ 2 :CL−1 (x) : ζ 2 : (x) + 4 : φ 3 :CL−1 (x)ζ (x) ,


pµ = µ d 3 x 2φ(x)ζ (x)+ : ζ 2 : (x) . 

(4.8)

Critical ( 4 )3, 

293

We insert a parameter λ into our previous definitions in such a way that (i) at λ = 1 our λ dependent objects correspond with the previous definitions. (ii) The expansion through order λ2 is second order perturbation theory in g counting µ = O(g 2 ). (iii) Powers of λ are determined so as to correspond with leading powers of g buried inside polymer activities. (iv) All functions will turn out to be norm analytic in λ. Thus  1 ˜ P (λ) = e−V − λpg − λ2 pµ + λ2 pg2 + λ3 r1 , 2

(4.9)

˜

where r1 is defined by the condition P (λ = 1) = P = e−V − e−V . Similarly, we define   ˜ ˜ K(λ) = λ2 e−V Q + λ3 [e−V − e−V ]Q + R

(4.10)

which, for λ = 1 coincides with K = e−V Q + R. Corresponding to (3.6) we define B(λ, K)(Y ) =

 N+M≥1

N M 1  ˜ e−V (X0 ) K(λ, Xj ) P (λ, i ), (Xj ),(i )→{Y } j =1 i=1 N!M! (4.11)

where X0 = Y \ (∪Xj ) ∪ (∪i ). Let S(λ, K) = R ◦ B(λ, K), where R is the rescaling defined in the last section. The RG evolution for K with parameter λ is fK : K → E(S(λ, K) , F (λ)), where F (λ) = λ2 FQ + λ3 FR

(4.12)

will be specified later, and  is the rescaled integration over ζ , and, as usual, F (λ) = F , when λ = 1. Given a function f (λ) let 1 Tλ f = f (0) + f  (0) + f  (0) 2

(4.13)

be the Taylor expansion to second order evaluated at λ = 1. Then the second order (≤2) (≤2) approximation to the RG map is f (≤2) = (fK , fV ) with (≤2)

fK

(K, V ) = Tλ E(S(λ, K) , F (λ)) = E1 (Tλ S(λ, K) , FQ )

(≤2)

fV

(K, V ) = VF . (4.14)

Only the linearized E1 intervenes, because the nonlinear part of extraction generates terms only at order λ3 or higher. Proposition 4.1. There is a choice of FQ such that the form of Q remains invariant under ˜  the RG evolution at second order. In more detail, f (≤2) (V , Qe−V ) = (V(≤2) , Q(≤2) e−VL ),   = V (, C, g(≤2) , µ(≤2) ) evolved according to where the parameters in V(≤2)  = Lε g(1 − Lε ag) g(≤2)

µ(≤2) = L

3+ε 2

µ − L2ε bg 2 .

(4.15)

The parameters in Q(≤2) = Q(C, w , gL ) evolved according to w  = v + wL ,

v (1) = L ,

v (p) = (CL )p − (C)p ,

2 ≤ p ≤ 4.

(4.16)

294

D.C. Brydges, P.K. Mitter, B. Scoppola

The constants a, b are







a = 36

d 3 y v (2) (y),

b = 48

d 3 y v (3) (y).

(4.17)

ˆ supported on connected polymers X with |X| ≤ 2, Proof. We define a polymer activity Q 1 2 |X| =   2 (pλ (, ζ, φ))    ˆ Q(X, ζ, φ) = 1  . (4.18) pλ (1 , ζ, φ)pλ (2 , ζ, φ) |X| = 2, connected  2    ,  1 2 1 ∪2 =X

It is easy to check that Tλ S(K, λ) = −pL e

−V˜L

 + e

−V˜L

 −V˜L ˆ QL . QL + e

(4.19)

Let ˜ Q(C, v, gL ) = gL2

3 

˜ (j,j ) (X, ˜ φ; C, v (4−j ) ), nj Q

(4.20)

j =0

where (n0 , n1 , n2 , n3 ) = (12, 48, 36, 8) and
˜ φ; C, u) = 1 ˜ (m,n) (X, d 3 xd 3 y : φ m (x)φ n (y) :C u(x − y). Q 2 X˜ We then have

(4.21)

˜ ˜ v, gL ) + Q(C, wL , gL ) . Tλ S(K, λ) = e−VL Q(C,

(4.22)

˜ v, gL ) − Q(C, v, gL ). FQ = Q(C,

(4.23)

Define

Then we have from (4.22) and (4.23)   ˜ ˜ E1 Tλ S(λ, K) , F = Tλ S(λ, K) − FQ e−VL = e−VL Q(C, w , gL ).

(4.24)

This establishes the invariant form of Q and (4.16). Next we show that this choice of FQ has the right form (3.11) for extraction. We write FQ (X, φ) = F1,Q (X, φ) + F0,Q (X),

(4.25)

where F0,Q is the field independent part of FQ . From (4.23) ˜ (0,0) (C, v (4) ), F0,Q = 12gL2 Q   ˜ (4,0) (C, v (2) ) + 48 Q ˜ (2,0) (C, v (3) ) , F1,Q = gL2 36 Q

(4.26)

Critical ( 4 )3, 

295

and from (4.20) and (4.21),


(4)

F1,Q (X) = 36gL2

d 3 x : φ 4 :C (x)f1,Q (X, x)
(2) 2 +48gL d 3 x : φ 2 :C (x)f1,Q (X, x)

(4.27)

with  0 x ∈ X       (m) 3 (m ) (x − y) f1,Q (X, x) = X = , d yv       3 (m ) (x − y) X =  ∪  ,connected,   x  d y v

(4.28)

where m = 4 − m/2. This shows that this choice of FQ has the right form (3.11) for extraction. Next we find the contribution to the flow of V . Referring to (3.10) we introduce (m)

f1,Q (x) =



(m)

f1,Q (X, x)

X

which equals

3 (m ) d y v (x − y) + x

3



d yv

(m )


(x − y) =



d 3 y v (m ) (x − y),

 =x (, ) connected

where x  x. The equality holds because we can extend the sum on  to all the   =  since v (m ) (x − y) = 0 for |x − y| ≥ 1. From (4.17) and (3.9) we have

V (F1,Q , ) = agL2 d 3 x : φ 4 :C (x) + bgL2 d 3 x : φ 2 :C (x). (4.29) 



By (3.13) and scaling relations (3.15), this establishes the claims for the second order flow in V .   The exact RG evolution for K = Qe−V + R. The exact map K → K  = fK (λ, K, V )|λ=1 = E(S(λ, K) , F (λ))|λ=1

(4.30)

induces an evolution of the remainder R which is studied by Taylor series around λ = 0 with remainder written using the Cauchy formula: fK (λ = 1) =

3 (j )  f (0) K

j =0

j!

1 + 2πi

 γ

dλ

fK (λ). λ4 (λ − 1)

296

D.C. Brydges, P.K. Mitter, B. Scoppola (≤2)

The terms j = 0, 1, 2 are the second order part fK

. In the j = 3 term there are no terms f

(3)

(0)

mixing R with Q, P because of the λ3 in front of R. Therefore it splits K3! = R1 +R2 into the third order derivative at R = 0, which we write using the Cauchy formula as    dλ 1 −V  E S(λ, Qe ) , FQ (λ) (4.31) R1 ≡ Rmain = 2πi γ λ4 and terms linear in R:

  ˜ R2 ≡ Rlinear = S1 R − FR e−VL ,  −1 ˜ e−VL (Z\L X) RL (L−1 X). S1 R(Z) =

(4.32)

X:L−1 X¯ L =Z

The remainder term in the Taylor expansion is  1 dλ E(S(λ, K) , F (λ)). R3 = 4 2πi γ λ (λ − 1)

(4.33)

In Proposition 4.1 the coupling constant in Q is not the same as the coupling constant in V (≤2) . Furthermore, the coupling constant in V (≤2) will further change because of the contribution from FR . To take this into account we introduce V  () = V (, C, g  , µ ), g  = Lε g(1 − Lε ag) + ξR , 

µ =L

3+ε 2

(4.34)

µ − L bg + ρR , 2ε

2

where the remainders ξR , ρR anticipate the effects of a yet-to-be-specified FR . Then we set ˜



R4 = e−V Q(C, w , g  ) − e−VL Q(C, w , gL ).

(4.35)

With these definitions we have written the RG as a map 



Exp(
e−V + Qe−V + R)() → Exp(
e−V + Q e−V + R  )(L−1 ) with Q = Q(C, w , g  ) and the RG map fK has induced a map fR R  ≡ fR (V , R) = Rmain + Rlinear + R3 + R4 .

(4.36)

Construction of FR . To complete the RG step we must specify the relevant part from the remainder FR . The goal is to choose FR so that the map R → Rlinear will be contractive. As we will later prove, this will be the case provided the small set part of Rlinear is normalized so that certain derivatives with respect to φ vanish at φ = 0. We say that a functional J is normalized if, for all small sets X, J (X, 0) = 0, D J (X, 0; 1, 1) = 0, D 2 J (X, 0; 1, xµ ) = 0, 2

D 4 J (X, 0; 1, 1, 1, 1) = 0.

(4.37)

Critical ( 4 )3, 

297

For given coefficients α˜ P (X), we define 
F˜R (X, φ) = d 3 x α˜ P (X)P (φ(x), ∂φ(x)),

(4.38)

X

P

where P runs over the relevant monomials which in this model are P = 1, φ 2 , φ 4 , φ∂µ φ with µ = 1, 2, 3. Choose the coefficients α˜ P so that ˜ J = R  − F˜R e−V

(4.39)

is normalized (details are given below). We define the relevant part, supported on small sets, by  FR (Z) = (4.40) F˜R,L (L−1 X). X:small sets L−1 X¯ L =Z

Then, from the definition of Rlinear in (4.32),  −1 ˜ Rlinear (Z) = e−VL (Z\L X) JL (L−1 X) + X:small sets L−1 X¯ L =Z



˜

−1 X)

e−VL (Z\L

JL (L−1 X).

X:large sets L−1 X¯ L =Z

(4.41) Therefore the first sum in Rlinear is also normalized because normalization is preserved under multiplication by smooth functionals of φ and rescaling. Having defined FR , we must show that it has the form (3.11) required for the extraction operation. For each monomial αP (X)P in (4.38) we define  αP (Z, x) = α˜ P (X)L−[P ]+3 1L−1 X (x), (4.42) X small set L−1 X¯ L =Z

where [P ] is the dimension of the monomial P , (n[φ] for φ n and 2[φ] + 1 for φ∂φ). Then 
FR (Z, φ) = d 3 x αP (Z, x)P (φ(x), ∂φ(x)) (4.43) P

as required in (3.11). Next we compute VFR following (3.9). By translation invariance,  αP (Z, x) αP :=

(4.44)

Z⊃x

is a.e. independent of x, (a.e. because equality fails for x in boundaries of blocks). This can be written more simply as  αP = L−[P ]+3 α˜ P (X). (4.45) X small set:L−1 X⊃x

Therefore V (FR , ) =:








d x α0 + α2,0 φ + α4,0 φ 3



2

4

,

(4.46)

298

D.C. Brydges, P.K. Mitter, B. Scoppola

where the term α2,1 φ(x)∂µ φ(x) is absent because α2,1 = 0 by reflection invariance. We have to rewrite this in a C Wick ordered basis in order to compute V   
V (FR , ) = d 3 x β0 + ρR : φ 2 :C (x) + ξR : φ 4 :C (x) , (4.47) 

where β0 = α0 + C(0)α2,0 + 3C 2 (0)α4,0 ,

ρR = α2,0 + 6C(0)α4,0 ,

ξR = α4,0 , (4.48)

which are formulas for the error terms in (4.35). Determining coefficients from (4.38). The following computations are only used in the proof of Lemma 5.17 where we will learn that shifts in the coupling constants generated by R can be bounded by the kernel norm. Note that the odd derivatives D j J (X, 0; f ×j ), j =odd integer, vanish identically by φ  −φ symmetry. Taking derivatives of (4.39) we get ˜

J (X, 0) = R  (X, 0) − F˜R (X, 0)e−V (X,0) , ˜ D 2 J (X, 0; f ×2 ) = D 2 R  (X, 0; f ×2 ) − D 2 F˜R (X, 0; f ×2 )e−V (X,0) ˜ +F˜R (X, 0)D 2 V˜ (X, 0; f ×2 )e−V (X,0) ˜ D 4 J (X, 0; f ×4 ) = D 4 R  (X, 0; f ×4 ) + D 4 F˜R (X, 0; f ×4 )e−V (X,0) ˜ +4D 2 F˜R (X, 0; f ×2 )D 2 V˜ (X, 0; f ×2 )e−V (X,0) ˜ +F˜R (X, 0)D 4 V˜ (X, 0; f ×4 )e−V (X,0) ˜

− 3F˜R (X, 0)(D 2 V˜ (X, 0; f ×2 ))2 e−V (X,0) . Note that from (4.38), F˜R (X, 0) = α˜ 0 (X)|X|, ˜ D FR (X, 0; 1, 1) = 4|X|α˜ 2,0 (X), D 2 F˜R (X, 0; 1, xµ ) = |X|α˜ 2,1 (X, µ), D 4 F˜R (X, 0; 1, 1, 1, 1) = 24|X|α˜ 4 (X). 2

Now imposing successively the conditions (4.38) we get 1  ˜ R (X, 0)e−V (X,0) , |X|  1 1 −V˜ (X,0)  2  α˜ 2,0 (X) = D R (X, 0; 1, 1) + R  (X, 0)D 2 V˜ (X, 0; 1, 1) , e 4 |X|  1 −V˜ (X,0)  2  α˜ 2,1 (X, µ) = D R (X, 0; 1, xµ ) + R  (X, 0)D 2 V˜ (X, 0; 1, xµ ) , e |X| 1 1 −V˜ (X,0)  4  α˜ 4 (X) = e D R (X, 0; 1, 1, 1, 1) 24 |X|

+D 2 V˜ (X, 0; 1, 1) [D 2 R  (X, 0; 1, 1) + R  (X, 0)D 2 V˜ (X, 0; 1, 1)

 + R  (X, 0) D 4 V˜ (X, 0; 1, 1, 1, 1) − 3(D 2 V˜ (X, 0; 1, 1))2 . (4.49) α˜ 0 (X) =

Critical ( 4 )3, 

299

Note that the leading contributions to the α˜ P (X) are obtained by setting V˜ = 0 in the above formulae. Resume. We have thus produced at the end of the RG step the promised map : (V , Q, R) → (V  , Q , R  ). V  is the same as V with evolved coupling g → g  µ → µ given by the flow (4.35), with a, b given in (4.17) and ξR , ρR in (4.48). Q is the same as Q with the change w → w , (4.16), and R  is given by (4.36) with intervening quantities defined earlier. 5. Estimates We will assume L large but fixed and then ε sufficiently small depending on L . O(1) denotes a constant independent of L and ε. Constants C are independent of ε but may depend on L. These constants may change from line to line. It will not be necessary to keep track of these changes. Throughout we will assume that w at a generic step has been obtained by successive iterations (4.16) with initial w0 = 0. We make in this section the following hypothesis in terms of the norms introduced in Sect. 2.2: Hypothesis. |g − g| ¯ ≤ ε 3/2 , |µ| ≤ ε2−δ ,

(5.1)

R h,Gκ ,A ≤ ε3/4−η ,

(5.2)

|R|h∗ ,A ≤ ε11/4−η ,

(5.3)

where δ, η = O(1) > 0 are very small fixed numbers, say 1/64, and h = cε −1/4 with c = O(1) a very small number. Furthermore we take h∗ = L(3+ε)/4 and choose n0 = 9. g¯ is the approximate fixed point in the flow of the coupling constant g obtained from the first equation in (4.35) by ignoring the remainder ξRˆ . Namely, Lε − 1 = O(ε) > 0 (5.4) L2ε a for ε sufficiently small depending on L. We have used the estimate a = O(log L) > 0 which is proved below in Lemma 5.12 (independent of the Hypothesis). Note that the Hypothesis now implies that g = O(ε). Recall the definitions of ρR and ξR from (4.48). We will prove in this section the following g¯ =

Theorem 1. Given the Hypothesis above we have for L sufficiently large and then ε sufficiently small |ξR | ≤ Lε ε 11/4−η , |ρR | ≤ L(3+ε)/2 ε 11/4−η , ¯ ≤ ε 3/2 , |µ | ≤ Cε2−δ , |g  − g| |g  − g| ≤ ε 2 ,

R  h,Gκ ,A ≤ L−1/4 ε 3/4−η ,

(5.5) (5.6) (5.7) (5.8) (5.9)

|R  |h∗ ,A ≤ L−1/4 ε 11/4−η .

(5.10)

300

D.C. Brydges, P.K. Mitter, B. Scoppola

The following long series of lemmas, except Lemmas 5.1–5.4 and 5.14, are proved under the Hypothesis above, and will serve to prove Theorem 1. Lemmas 5.21, 5.22, 5.23 and 5.27 are the major parts of the program. Rmain is bounded in Lemma 5.21 and this result determines the qualitative form of the bound on the remainder. R3 and R4 are seen, in Lemmas 5.22, 5.23 to be negligible in comparison. Rlinear is the crux of the program and it is bounded in Lemma 5.27. The remaining lemmas are auxiliary results on the way to these lemmas. These auxiliary lemmas implement some the following principles: in bounds by G, h, A norms, a fluctuation field ζ (x) contributes CL and a field φ contributes O(1)g −1/4 . In bounds by the 1, A norms, fluctuation fields and φ fields contribute O(1). Lemma 5.1. Let Z be a 1-polymer, Y be a L−1 -polymer or ∅, Y ⊂ Z and vol(Z\Y ) ≥ 21 . Choose γ = O(1) > 0, κ = O(1) > 0. Let σ be sufficiently large. Then for any x ∈ Z, there exists a constant O(1) depending on κ, γ , j such that j

φ C 2 (Z) ≤ O(1)g − 4 eγ g j

 Z\Y

d 3 y|φ(y)|4

Gκ (Z, φ).

(5.11)

Proof. This is a simple variant of an analogous lemma in [BDH-est]. Write
1 φ(x) = d 3 y(φ(y) + φ(x) − φ(y)) vol(Z\Y ) Z\Y and bound



1 1 d 3 y|φ(y)| + d 3 y|φ(x) − φ(y)| vol(Z\Y ) Z\Y vol(Z\Y ) Z\Y   ≤ O(1) φ L4 (Z\Y ) + φ Z,1,σ ,

|φ(x)| ≤

where the first term was bounded using the H¨older inequality. The second term was bounded by writing the difference as the integral of ∇φ and using |∇φ(x)| ≤ O(1) φ ,1,σ for   x, which is the Sobolev embedding theorem, valid for σ > 3/2+2. We also have, under the same condition on σ , that ∇ 2 φ C(Z) ≤ O(1) φ Z,1,σ . Hence 

3 4 j j j

φ C 2 (Z) ≤ O(1) φ L4 (Z\Y ) + φ Z,1,σ ≤ O(1)g −j/4 eγ g Z\Y d y|φ(y)| Gκ (Z, φ), where O(1) depends on κ, γ , j .

 

For fluctuation fields ζ , we will have occasion to use the following lemmas. Define ˜ κ,α (X, ζ ) = eα ζ L2 (X) Gκ (X, ζ ), G 2

α, κ > 0,

(5.12)

κ is O(1) and will be held sufficiently small. The choice of α is dictated by Lemma 5.3 below. Lemma 5.2. For any x ∈ X ˜ κ,α (X, ζ ), |ζ (x)|j ≤ Cα,j G

(5.13)

where Cα,j = (α)−(j/2) O(1) and O(1) depends on j and κ. We have isolated out the α dependence in the bound.

Critical ( 4 )3, 

301

Lemma 5.3.


˜ κ,α (X, ζ ) ≤ 2|X| dµ (ζ )G

(5.14)

for κ = O(1) > 0 sufficiently small and α(L) = L−(3−2[φ]) κ  = L−(3+ε)/2 κ  and κ  = O(1) > 0 is held sufficiently small. The proof of Lemma 5.2 follows the lines of Lemma 5.1 except that we replace the L4 norm there by the L2 norm in the appropriate places and Y = ∅ and Z =   x. The proof of Lemma 5.3 is the same as the one referenced for (2.3). It is convenient, for the control of norms of our polymer activities in intermediate steps, to introduce some new regulators and some intermediate norms. Define ˜ κ,α (X, ζ ), ˆ κ,α (X, ζ, φ) = Gκ (X, ζ + φ)Gκ (X, φ)G G

(5.15)

ˆ κ,α is a regulator. G Lemma 5.4.


ˆ κ,α (X, ζ, φ) ≤ 2|X| G3κ (X, φ) dµ (ζ )G

(5.16)

for α = α(L) > 0 sufficiently small and κ = O(1) > 0 sufficiently small. Proof. Use Cauchy-Schwartz, stability of Gκ , (2.3) and Lemma 5.3.

 

For polymer activity K(X, ζ, φ) define the norms ˆ −1

K(X) h,Gˆ κ,α = sup K(X, ζ, φ) h G κ,α (X, ζ, φ),

(5.17)

˜ −1

K(X) h∗ ,G˜ κ,α = sup K(X, ζ, 0) h∗ G κ,α (X, ζ ),

(5.18)

φ,ζ

ζ

where in (5.17) and (5.18) the functional derivatives in K(X, ζ, φ) h and in

K(X, ζ, 0) h∗ are computed with respect to the field φ. These norms are needed because before fluctuation integration we will encounter activities K(X, ζ, φ) which are not just functions of ζ + φ. The following lemma is a variant of Theorem 1 [BDH-est] adapted to our purposes. Lemma 5.5. For V (Y, φ, ζ ) = V (Y, φ + ζ, C, g, µ) or V (Y, φ + ζ, CL−1 , g, µ),

e−V (Y,φ+ζ ) h ≤ 2|Y | e−g/4

 Y

d 3 x(φ+ζ )4 (x)

,

e−V (Y,φ+ζ ) h∗ ≤ 2|Y | for ε > 0 sufficiently small, and, for the second bound, depending on L.

(5.19)

(5.20)

302

D.C. Brydges, P.K. Mitter, B. Scoppola

Proof. V˜ (, φ) = V (, φ, CL−1 , g, µ) = g









d 3 x : φ 4 :CL−1 (x)+µ



d 3 x : φ 2 :CL−1 (x).

Undo the Wick ordering. Wick constants are finite and O(1). Recall from the initial Hypothesis that g = O(ε) and |µ| ≤ O(ε2 ), and h = cε−1/4 , with c small enough. Hence

V˜ (, φ)) = g d 3 x φ 4 (x) − O(1)g d 3 x φ 2 (x) − O(ε),  


g g 3 4 3 4 ˜ V (, φ) − d x φ (x) = d x φ (x) − O(1)g d 3 x φ 2 (x) − O(ε) ≥ −O(ε). 2  2   Hence

˜

e−V (,φ) ≤ (1 + O(ε))e−g/2

Compute now the derivatives

Dk

˜ e−V . We



d

3 xφ 4 (x)

.

get for 1 ≤ k ≤ n0 ,

j k    hk  1   (ε −1/4 )li   ˜  k −V˜  li ˜  k e )(, φ) ≤ c (D D V (, φ) e−V (,φ)  k! j ! 1≤l ≤4 li ! j =1

 i li =k

i=1

j  k    (ε −1/4 )l  1 ˜    ≤ ck D l V˜ (, φ) e−V (,φ) j! l! j =1

≤ O(1)ck e

1≤l≤4

− g2



d

3 xφ 4 (x)



e

1≤l≤4

(ε1/4 )l l!

    l˜ D V (,φ)

.

Take the expression for V˜ where the Wick ordering has been undone. Then it is easy to see that

3     (ε −1/4 )l   g  1/4 j   l˜ 3 d x φ ≤ d 3 xφ 4 (x) + O(1). V (, φ) ≤ O(1) ε   D l! 4   1≤l≤4

j =0

Hence

 g  hk  3 4   k −V˜ (D e )(, φ) ≤ O(1)ck e− 4  d xφ (x) . k! The sum over k is O(c) if c is small enough. The proof of (5.19) follows easily. The proof of (5.20) follows the same lines but we must take ε sufficiently small depending on L.   Lemma 5.6. Let pg (, ζ, φ), pµ (, ζ, φ) be as given in (4.8), h = cε −1/4 , g, µ as in the inductive hypothesis, and h∗ as defined earlier. Then for any α > 0, κ = O(1) > 0, ξ = O(1) > 0, 0 ≤ s < 1, ˜ κ,α (, ζ )Gκ (, φ)eg(1−s)ξ

pg (, ζ, φ) h ≤ Cα ε 1/4 (1 − s)−3/4 G



˜ κ,α (, ζ )Gκ (, φ)eg(1−s)ξ

pµ (, ζ, φ) h ≤ Cα  7/4−δ (1 − s)−1/2 G

d



3x

d

φ 4 (x)

3x

, (5.21)

φ 4 (x)

, (5.22)

Critical ( 4 )3, 

303

˜ κ,α (, ζ ),

pg (, ζ, 0) h∗ ≤ Cα,L ε G

(5.23)

˜ κ,α (, ζ ).

pµ (, ζ, 0) h∗ ≤ Cα,L ε 2−δ G (5.24)   Proof. Undoing the Wick ordering we have pg (, ζ, φ) = g  d 3 x 3j =1 aj ζ 4−j (x) φ j (x), where the constants aj are O(1),
3      hk D k pg (, ζ, φ; f ×k ) ≤ ε−k/4 g d 3x aj |ζ 4−j (x)||φ j −k (x)| f ×k . C 2 () 

j =1

Note that since pg is a third degree polynomial in φ, derivatives with k > 3 vanish. Moreover on the right hand side j ≥ k. Now use Lemma 5.1, 5.2 to get     hk D k pg (, ζ, φ) ≤ Cα gε −k/4 g −

3−k 4

˜ κ,α (, ζ )Gκ (, φ)eg(1−s)ξ (1 − s)−3/4 G



d

3x

φ 4 (x)

.

1 Use g = O(ε), multiply by k! and take the sum over k to obtain (5.21). The remaining parts are obtained along the same lines.  

Define p(s) = p(s, , φ, ζ ) by p(s) = spg + s 2 pµ . Then r1 = r1 (, φ, ζ ) defined by (4.9) is given by
 1 1 ˜ r1 = ds(1 − s)2 e−p(s)−V − p  (s)3 + 6p  (s)pµ (5.25) 2 0 with p (s) =

d ds p(s)

= pg + 2spµ and p  (s) = 2pµ .

Lemma 5.7.

r1 () h,Gˆ κ,α ≤ Cα ε 3/4 ,

(5.26)

r1 () h∗ ,G˜ κ,α ≤ Cα,L ε 3−δ .

(5.27)

Proof. The hypotheses for g, µ also hold for sg, s 2 µ and for [1 − s]g, [1 − s 2 ]µ. Since V + p(s) = V1 (s) + V2 (s) with V1 (s) = V (, φ + ζ, C, sg, s 2 µ), V2 (s) = V (, φ, CL−1 , [1 − s]g, [1 − s 2 ]µ),  
1 1 2 −V1 (s) −V2 (s)  3  ds(1−s) e

h e

h p (s) h +6 p (s) h pµ h .

r1 (, ζ, φ) h ≤ 2 0  By Lemma 5.5, the exponential terms are bounded by 4 exp(−(g[1 − s]/4) φ 4 ). Use κ α Lemma 5.6 choosing for the regulators the constants 4 and 3 to obtain

r1 (, ζ, φ) h ≤ Cα ε

3/4

ˆ κ,α G




1

9

g

ds(1 − s)2 (1 − s)− 4 e− 4 (1−s)



d

3x

φ 4 (x) g(1−s)3ξ

e

0

Choose 0 < ξ <

1 12

to get (5.26). Equation (5.27) is proved similarly.

 



d

3x

φ 4 (x)

.

304

D.C. Brydges, P.K. Mitter, B. Scoppola

Lemma 5.8. Consider P (λ) given in (4.9). Then for C independent of ε, but dependent on L,

P h,Gˆ κ,α ,A ≤ CL |λε 1/4 | for |λε 1/4 | ≤ 1,

(5.28)

P h∗ ,G˜ κ,α ,A ≤ CL |λε 1−δ/2 | for |λε 1−δ/2 | ≤ 1.

(5.29)

Proof. Equations (5.28) and (5.29) are immediate from Lemma 5.7, noting that λε1−δ/2 in (5.29) is the largest of the several combinations of λ, ε that arise. It comes from the term involving µλ2 .   Estimates for Qe−V . We now turn to the estimate of Qe−V . From (4.4), Q(C, w, g) = g

2

3 

am Q(m,m) (C, w (4−m) ),

(5.30)

m=1

where the am are numerical coefficients and the Q(m,m) are given in (4.6). Under an  (p) iteration, see Proposition 4.1, we have w(p) → w (p) = v (p) + wL , where p = 1, 2, 3 (p) and the v (p) are given in Proposition 4.1. Starting with w0 = 0 we get after n iterations (p)

wn

=

n−1  j =0

(p)

(5.31)

vLj .

(p)

(p)

We need to first estimate wn and the limit limn→∞ wn under appropriate norms. We consider Banach spaces Wp of measurable functions with norms · p , p = 1, 2, 3, defined as follows:

6p+1

f p = ess. sup |x| 4 |f (x)| . (5.32) x

We define the Banach space W1 × W2 × W3 consisting of vectors w with the norm

w = sup wp p . p

Then we have Lemma 5.9. For L sufficiently large and ε > 0 sufficiently small there exists a constant c = O(1) such that wn ≤ c/4 ∀n and wn+1 − wn ≤ c/8 L−n/4 so that wn → w∗ in the norm · , and w∗ ≤ c/4. Proof. Let us note first some weak uniform (in L) bounds. Recall that [φ] = | L (x)| ≤ O(1)|x|−2[φ] ,

|C(x)| ≤ O(1).

To see this observe that from the definition of (see Sect. 1)
∞ dl −2[φ]  x  | L (x)| ≤ l . u l l 0

3−ε 4 ,

(5.33)

Critical ( 4 )3, 

305

Let x = 0. Then, using support properties of u,
∞ 1 dl −2[φ] 2 | L (x)| ≤ l

u ∞ =

u ∞ 3 − ε |x|2[φ] |x| l which proves the first bound in (5.33). To prove the second recall
∞ dl −2[φ]  x  |C(x)| ≤ l u  ≤ O(1) u ∞ l l 1 which proves the second bound in (5.33). For p = 1, 2, 3, p

|v (p) (x)| = |CL (x) − C p (x)| ≤ p sup | L (x)|q |C(x)|p−q  ≤ O(1)

1≤q≤p

|x|−p2[φ] if |x| ≥ 1

0

,

where we exploited CL = L + C and L (x) = 0 if |x| ≥ 1 and (5.33). Hence

v (p) p ≤ O(1) and by scaling x = Lj x  ,

vLj p ≤ Ljp2[φ] L−j (p)

(6p+1) 4

v (p) p ≤ cp /8 L−j/4 .

(5.34)

Define the constant c = supp cp . Then the above estimates lead immediately to the proof of Lemma 5.9, because of (5.31).   Lemma 5.10. Q(X, φ)e−V (X,φ) satisfies the bounds    −V  ≤ Cp ε 1/2 , Qe 

(5.35)

h,Gκ ,Ap

   −V  Qe 

h∗ ,Ap

≤ Cp ε 2 .

(5.36)

Proof. Qe−V = g 2

3 

am Q(m,m) (C, w(4−m) )e−V ,

m=1 3              |am | Q(m,m) (C, w (4−m) , X, φ) e−V (X,φ)  . Q(X, φ)e−V (X,φ)  ≤ g 2 h

m=1

h

h

(5.37) Here X is a small set, because of the support property of Q. The last factor in the sum will be estimated by Lemma 5.5. From (4.6)
˜ φ; C, w(1) ) = 1 Q(3,3) (X, d 3 xd 3 y : φ 3 (x)φ 3 (y) :C w (1) (x − y), 2 X˜

306

D.C. Brydges, P.K. Mitter, B. Scoppola

Undo the Wick ordering, which produces lower order terms with finite coefficients. Apply hk D k with h = cε −1/4 and use Lemmas 5.1, 5.9 and g = O(ε). We get  hk   k (3,3) ˜  (X, φ; C, w(1) ) D Q k!
 3 4 1 ≤ O(1)g −3/2 w (1) 1 d 3 xd 3 y Gκ/4 (X, φ)eg/4 X d xφ (x) . (5.38) 7/4 |x − y| X˜ Next turn to Q(m,m) , m = 1, 2, again in (4.6) (m,m)

Q

3


2

˜ φ; C, w(4−m) ) = − m (X, 2




1

ds1 ds2

µ,ν=1 0

X˜

d 3 xd 3 y,

(x−y)µ (x−y)ν w (4−m) (x−y) : (φ m−1 ∇µ φ)(y+s1 (x−y))(φ m−1 ∇ν φ)(y+s2 (x−y)) :C . We consider in turn the cases m = 2, 1. We apply hk D k with h = aε −1/4 and use Lemmas 5.1–5.9 and the Sobolev inequality to dominate the ∇φ pointwise by the large field regulators.  hk    k (2,2) ˜ (X, φ; C, w(2) ) D Q k! ≤ O(1)g −1/2 w (2) 2 3
  |(x − y)µ ||(x − y)ν | g/4 X d 3 xφ 4 (x) × d 3 xd 3 y G (X, φ)e κ/4 |x − y|13/4 X˜ µ,ν=1

≤ cg −1/2 Gκ/4 (X, φ)eg/4

 X

d 3 xφ 4 (x)

.

(5.39)

We can estimate in the same way the case m = 1. Note that
|(x − y)µ ||(x − y)ν | d 3 xd 3 y ≤ O(1). |x − y|19/4 X˜ We have

  3 4 hk   k (1,1) ˜  (X, φ; C, w(3) ) ≤ cGκ/4 (X, φ)eg/4 X d xφ (x) . D Q k! From (5.38), (5.39) and (5.40) we get    3 4   g 2 Q(m,m) (C, w (4−m) ) ≤ cε1/2 Gκ/4 (X, φ)eg/4 X d xφ (x) . h

(5.40)

(5.41)

We estimate the r.h.s. of (5.37) using (5.41) and Lemma 5.5,     ≤ O(1)ε 1/2 . Q(X)e−V (X) 

(5.42)

Since the Q are supported on small sets, we get from (5.42)    −V  ≤ Cp ε 1/2 . Qe 

(5.43)

h,Gκ

h,Gκ ,Ap

This proves (5.35). To prove (5.36) we estimate the r.h.s. of (5.37) at φ = 0 after undoing  the Wick ordering, set h = h∗ , using Lemma 5.1.  The following lemma applies to Qe−V treated as a function of ζ, φ.

Critical ( 4 )3, 

307

Lemma 5.11.

   −V  Qe 

ˆ κ,α ,Ap h,G

≤ Cp ε 1/2 ,

(5.44)

≤ Cp ε 2 .

(5.45)

   −V  Qe 

˜ κ,α ,Ap h∗ ,G

ˆ κ,α ≥ Gκ . For (5.45) we bound Proof. Equations (5.44) follows from (5.35), since G −V derivatives with respect to φ of (Qe )(ζ + φ, X) at φ = 0     Q(X, ζ )e−V (X,ζ ) 

h∗

≤ g 2 O(1)

3     (m,m)  (C, w (4−m) , X, ζ ) Q

h∗

m=1

˜ α/2 (X, ζ ), G

  since e−V (X,ζ ) h ≤ 2|X| and X is a small set. Now we proceed as in Lemma 5.10, but ∗ dominate the ζ using Lemma 5.2.   We now control the perturbative flow coefficients a, b given in (4.17). Lemma 5.12. a > 0, b > 0 and moreover a = O(log L),


d 3 x v (4) (x) = O(L3 )

b = O(L3/2 ),

(5.46)

for L → ∞ and  = o(L). Proof. From (4.16), for p = 2, 3, using CL = L + C, p

p−1

v (p) = CL − C p = L ( L

p−2

+ p L

C + δp,3 3C 2 )

(5.47)

with pointwise multiplication. The positivity in Fourier space of the integral kernels on the right hand side implies that a > 0, b > 0 as claimed. The common factor of

L implies that v (p) (x) has support in the unit ball. This, together with (5.33), implies p that the leading divergence is in the contribution from L . Therefore it is sufficient to calculate
I (p) = d 3 x ( L (x))p . Recall that:


L (x) = Let p = 2 so that I (p) = 2!


L−1 ≤l1 ≤l2 ≤1

1 L−1

dl −2[φ] x u l . l l

dl1 dl2 (l1 l2 )−2[φ] l1 l2


d 3x u

(5.48)

    x x u . l1 l2

Suppose that L = 2N−1 for some integer N and break up the range of integration into disjoint regions Rn = {2−n−1 < l1 ≤ l2 ≤ 2−n }

n = 0, ...., N − 1

308

D.C. Brydges, P.K. Mitter, B. Scoppola

so that I

(p)

=

N−1 

I (p) (Rn ),

n=0

where


I (p) (R) = 2! R

dl1 dl2 (l1 l2 )−2[φ] l1 l2


d 3x u

    x x u . l1 l2

By scaling l1 , l2 , x by 2n , I (p) (Rn ) = 22p[φ]−3 I (R0 ). Therefore I (p) = I (p) (R0 )

N−1 

2(2p[φ]−3)n = O(1)

n=0

N−1 

2(2p[φ]−3)n = O(1)

n=0

1 − L2p[φ]−3 . 1 − 22p[φ]−3

This is also true for p = 3, 4, by the same argument with the appropriate multiple integral expression for I (p) (R0 ). For p = 2, 2p[φ] − 3 = −4ε → 0 and therefore I (p) = O(log L) by L’Hospital’s rule. For p = 3, 2p[φ] − 3 →= 3/2 and therefore I (p) = O(L3/2 ), etc.   Lemma 5.13.   ˜   Q(e−V − e−V )

ˆ κ,α ,Ap h,G

  ˜   Q(e−V − e−V )

≤ Cp ε 3/4 ,

(5.49)

≤ Cp ε 3 .

(5.50)

˜ κ,α ,Ap h∗ ,G

Proof. Q(X) is supported on connected polymers with size |X| ≤ 2. Without loss of generality we do the estimates for |X| = 1. We can write: ˜

Q(, ζ + φ)(e−V (,ζ +φ) − e−V (,φ) )
1/2 1 1 ˜ ds p(, ζ, φ)e−( 2 −s)V (,ζ +φ)−s V (,φ) = Q(, ζ + φ)e− 2 V (,ζ +φ)
+

0 1

˜

ds Q(, ζ + φ)p(, ζ, φ)e−(1−s)V (,ζ +φ)−s V (,φ)

1/2

whence

  ˜   Q(, ζ + φ)(e−V (,ζ +φ) − e−V (,φ) ) ≤ O(1)(A + B), h

where  
1   A = Q(, ζ + φ)e− 2 V (,ζ +φ) 

h 0

and


B=

1 1/2

1/2

˜

ds p(, ζ, φ) h e−s V (,φ) h

˜

ds Q(, ζ + φ) h p(, ζ, φ) h e−s V (,φ) h .

(5.51)

Critical ( 4 )3, 

309

To estimate A, use Lemma 5.11, still true for V replaced by 21 V , Lemma 5.6, (5.21) with 1 − s replaced by s, δ = 18 , and Lemma 5.5 with g replaced by sg. Observe that s −3/4 is integrable. We get ˆ κ,α (, ζ, φ). A ≤ Cα ε 3/4 G

(5.52)

To estimate B we use again Lemma 5.5 with g replaced by sg. We estimate p h using 1 Lemma 5.6, (5.21) with 1 − s replaced by s, δ = 16 . We estimate Q(, ζ + φ) h as in the proof of Lemma 5.10 with the following change. Expand out polynomials in ζ + φ, and dominate the ζ using Lemma 5.3. We then get a modified estimate (5.41) replacing g ˆ κ,α (X, ζ, φ) together with an overall multiplicative factor with 4s g, and Gκ (X, φ) with G Cα s −3/2 which is integrable in the range under consideration. Then we use Lemma 5.5 with g replaced by sg. Putting all this together we get ˆ κ,α (, ζ, φ). B ≤ Cα ε 3/4 G Using (5.52) and (5.53) we get for (5.51),   ˜   ˆ κ,α (, ζ, φ). Q(, ζ + φ)(e−V (,ζ +φ) − e−V (,φ) ) ≤ Cα ε 3/4 G h

(5.53)

(5.54)

It is easy to show that the same estimate holds if |X| = 2, connected. Hence   ˜   ≤ Cα,L ε 3/4 . Q(e−V − e−V ) ˆ κ,α ,A h,G

This proves (5.49). The proof of (5.50) is similar except that we have only fluctuation fields ζ to dominate using Lemma 5.3.   Lemma 5.14. K(λ) given by (4.10), satisfies the bounds

K(λ) h,Gˆ κ,α ,A ≤ Cα |λε 1/4−η/3 |2 for |λε 1/4−η/3 | ≤ 1,

(5.55)

K(λ) h∗ ,G˜ κ,α ,A ≤ Cα,L |λε 11/12−η/3 |2 for |λε 11/12−η/3 | ≤ 1.

(5.56)

Proof. This follows from Lemmas 5.11 and 5.13 and the hypotheses (5.2) and (5.3) on R. The λε1/4−η/3 and λε 11/12−η/3 originate from λ3 R contributions.   Lemma 5.15. For any polymer activity K:   ˜ α,κ (X, ζ ) |K(X)|h∗ + h−n0 hn∗0 K(X) h,Gκ ,

K(X, ζ ) h∗ ≤ O(1)G

K(Y, φ) h ≤ O(1)eγ g

 Z\Y

d 3 y|φ(y)|4

  Gκ (Z, φ) |K(Y )|h + L−n0 [φ] K(Y ) L[φ] h,Gκ ,

  |K  (X)|h∗ ≤ O(1)2|X| |K(X)|h∗ + h−n0 hn∗0 K(X) h,Gκ ,

(5.57)

(5.58) (5.59)

˜ α,κ is as defined in (5.12), and n0 is the maximum number of derivatives appearwhere G ing in the definition of Kernel and h norms. In (5.58), Y, Z, γ are as described in Lemma 5.1.

310

D.C. Brydges, P.K. Mitter, B. Scoppola

Recall that n0 = 9. The superscript  stands for dµ (ζ ) integration. α is chosen as in Lemma 5.3. Note that we have then α = α(L) =

κ . h2∗

(5.60)

Proof. First observe: 1 ˜ α,κ (X, ζ ) K(X) h,Gκ ,

(D n0 K)(X, ζ ) ≤ h−n0 G n0 !

(5.61)

˜ α,κ (X, ζ ). Now let n < n0 . We expand in Taylor series with remainsince Gκ (X, ζ ) ≤ G der (D n K)(X, ζ ; f ×n ) =

n0 −n−1

1 (D n+m K)(X, 0; f ×n , ζ ×m ) m! m=0
1 1 ds(1 − s)n0 −n−1 (D n0 K)(X, sζ ; f ×n , ζ ×n0 −n ) + (n0 − n − 1)! 0

whence

(D n K)(X, ζ ) n0 −n−1

1

ζ m

(D n+m K)(X, 0) C 2 (X) m! m=0
1 1 ds(1 − s)n0 −n−1 ζ nC02−n G (X, sζ ) (D n0 K)(X) Gκ . + (X) κ (n0 − n − 1)! 0 √ By Lemma 5.2, with ζ replaced by 1 − s 2 ζ , and (5.60),

1 −p p ˜ α,κ X, 1 − s 2 ζ , h∗ ζ C 2 (X) ≤ O(1)G (1 − s 2 )p/2 ≤

where O(1) depends on κ  and p and 0 ≤ s < 1. With s = 0 this bound is applied to the terms in the sum over m. For the Taylor remainder term we note that (1 − s)(n0 −n)/2−1 is integrable since n0 > n. Hence: !"n −n−1 # 0 (n + m)! hn∗ ˜ α,κ (X, ζ )

(D n K)(X, ζ ) ≤ O(1)G |K(X)|h∗ n! m!n! m=0 $ n0 !h−n0 hn∗0 +

K(X) h,Gκ n!(n0 − n − 1)!   ˜ α,κ (X, ζ ) |K(X)|h∗ + h−n0 hn∗0 K(X) h,Gκ . (5.62) ≤ O(1)G Summing (5.62) over 0 ≤ n ≤ n0 − 1 and adding (5.61) after multiplication by hn∗0 proves (5.57). Equation (5.59) follows from (5.57) using Lemma 5.3. Equation (5.58) is proved in the same way as (5.57) with Lemma 5.1 in place of Lemma 5.2, h∗ , h replaced by h, Lh.  

Critical ( 4 )3, 

311

The next lemma gives bounds on S(λ, K) = R ◦ B(λ, K) given in (4.11). Lemma 5.16. For any q > 0, there exists cL such that, for L large,

S(λ, K) h,Gκ ,Ap ≤ q |S(λ, K) |h∗ ,Ap ≤ q

when |λε 1/4−η/3 | ≤ cL ,

(5.63)

when |λε 11/12−η/3 | ≤ cL .

(5.64)

When R = 0 we may set η = 0 in (5.63) and replace λε11/12−η/3 by λε 1−δ/2 in (5.64). Proof. From the definition of reblocking (see (4.11)) and subsequent rescaling 

S(λ, K)(Z, φ, ζ ) h ≤

N+M≥1 N

× ×

1  ˜

e−V (X0 ,φL−1 ) h (X ),( )→LZ N !M! j i

j =1

M

i=1

K(λ, Xj , φL−1 , ζL−1 ) h

P (λ, i , φL−1 , ζL−1 ) h .

(5.65)

We rewrite V˜ (X0 , φL−1 ) = V˜L (L−1 X0 , φ) and apply Lemma 5.5 (the rewriting gives a better bound by saving factors of 2), ˆ κ,α (LZ, ζL−1 , φL−1 )

S(λ, K)(Z, φ, ζ ) h ≤ 2|Z| G



1  (Xj ),(i )→LZ N !M!

N+M≥1 M

K(λ, Xj ) hL ,Gˆ κ,α

j =1 i=1

N

×

P (λ, i ) hL ,Gˆ κ,α , (5.66)

where hL = L−(3−ε)/4 h. Using Lemma 5.4 and G3κ (LZ, φL−1 ) ≤ Gκ (Z, φ) for L large, 

1  (Xj ),(i )→LZ N !M! N+M≥1 M N ×

K(λ, Xj ) hL ,Gˆ κ,α

P (λ, i ) hL ,Gˆ κ,α .

S(λ, K) (Z) h,Gκ ≤ 22|Z|

j =1

i=1

Multiply by A2+p (Z) on the left and, using A2+p (L−1 X¯ L ) ≤ O(1)A−3 (X) (Lemma 1 %M % in [BDH-est]), by O(1) N j =1 A−3 (Xj ) i=1 A−3 (i ) on the right. Fix any  and sum both sides over Z  . The spanning tree argument of Lemma 7.1 of [BY] controls the sums over N, M, Z, (Xj ), (i ) → LZ with the result

S(λ, K) h,Gκ ,Ap ≤ O(1)



 N O(1)N L3N K(λ) h,Gˆ κ,α ,A + P (λ) h,Gˆ κ,α ,A .

N≥1

The proof of (5.63) is completed by Lemmas 5.8 and 5.14. When R = 0 we can replace Lemma 5.14 by Lemmas 5.13 and 5.8 which gives the result with η = 0.

312

D.C. Brydges, P.K. Mitter, B. Scoppola

For (5.64) we use (5.65) with φ = 0 and replace h by h∗ . We estimate the ζ depen˜ κ,α introduced in (5.12), to obtain, in the place of (5.66), dence by the regulator G 

˜ κ,α (LZ, ζL−1 , ζL−1 )

S(λ, K)(Z, 0, ζ ) h∗ ≤ 2|Z| G ×

N j =1

N+M≥1

K(λ, Xj ) h∗ ,G˜ κ,α

1  (Xj ),(i )→LZ N !M!

M i=1

P (λ, i ) h∗ ,G˜ κ,α . (5.67)

Then Lemma 5.15 is used to estimate the  in |S(λ, K) (Z)|h∗ , and the rest is as before.   Estimates on relevant parts and flow coefficients from the remainder. Let (α˜ P ) be the coefficients (α˜ 0 , α˜ 2,0 , α˜ 2,1 , α˜ 4 ) defined in (4.38) and (4.49). The flow coefficients ξR , ρR and β0 are given in (4.48). Lemma 5.17.

R  h,G3κ ,A−2 ≤ ε3/4−η ,

(5.68)

|R  |h∗ ,A−3 ≤ O(1)ε 11/4−η ,

(5.69)

|α˜ P |A ≤ O(1)ε 11/4−η ,

(5.70)

|β0 | ≤ Cε11/4−η ,

(5.71)

|ξR | ≤ Cε11/4−η ,

(5.72)

|ρR | ≤ Cε11/4−η .

(5.73)

Proof. Recall that α˜ P (X), are supported on small sets. Equation (5.68) follows from the hypothesis (5.2) and the stability of the large field regulator Gκ . Equation (5.69) follow from the hypotheses (5.3) and Lemma 5.16 with n0 = 9 and ε sufficiently small depending on L. Equation (5.70) follows from (5.69), and (4.49). In fact the dominant contribution comes by setting V˜ = 0 because the difference gives additional powers of ε. Then we have )

α˜ P A ≤ O(1)n(P )!h−n(P |1S R  |h∗ ,A , ∗ where n(P ) is the number of fields in the monomial P and 1S is the indicator function on small sets. Now use (5.69) to get (5.70). Equations (5.71), (5.72), (5.73) follow from (5.70), and the definitions (4.48), (4.45) and Wick coefficients are O(1).   Corollary 5.18. For ε sufficiently small ¯ ≤ ε3/2 , |µ | ≤ Cε2−δ , |g  − g|

(5.74)

|g  − g| ≤ ε 2 .

(5.75)

Critical ( 4 )3, 

313

Proof. It is easy to check from the first of the flow equations (4.15) and the definition 2ε ag) + ξ . of g¯ that g  − g¯ = (g − g)(1 ¯ − L2ε ag) + ξR and g  − g = (g − g)(−L ¯ R a = O(log L) > 0 and the initial Hypothesis implies g = O(ε) so that for ε sufficiently small 0 < 1 − L2ε ag < 1 − ag. The domain of g in the Hypothesis and the bound (5.72) of Lemma 5.17 imply, for ε sufficiently small, that ξR is smaller than the other terms, which gives the two bounds concerning g  . The bound on µ follows from the second of the flow equations (4.15), the hypothesis on µ, and the bound (5.73) on ρR .   The following lemma proves the stability of V with respect to perturbations by relevant parts F in our model. We state it in the form enunciated as the stability hypothesis in Sect. 4.2, (103), [BDH-est]. This lemma will be very useful for the control of the extraction formula, as explained in the reference above. 2 3 Recall from (4.12) that  F (λ) = λ FQ + λ FR and from (3.11) that (each part of) F decomposes: F (X) = ⊂X F (X, ). Lemma 5.19. For any R > 0 and ξ := R max(|λ2 |ε, |λ3 |ε 7/4−η ) sufficiently small, 

˜

e−VL ()−

X⊃ z(X)F (λ,X,)

h,Gκ ≤ 22 ,

(5.76)

where z(X) are complex parameters with |z(X)| ≤ R. Proof. First note that Lemma 5.5 still holds if we replace V˜ by V˜L provided ε is sufficiently small. We then have ˜



e−VL ()−

h ≤ 2 e−gL /4

X⊃ z(X)F (λ,X,)



d

3 xφ 4 (x)+ X⊃ R F (λ,X,) h

.

(5.77)

Recall that the F (X, ) are supported on small sets X. The proof now follows easily from the following Claim. For ε sufficiently small, 


F (λ, X, ) h ≤ CL ξ 

 d 3 xεφ 4 (x) + ε 1/2 φ 2,1,σ + 1 ,

(5.78)

where φ 2,1,σ is the norm defined in (2.2). Proof of the Claim. We have F (λ, X, ) h ≤ |λ|2 FQ (X, ) h + |λ|3 FR (X, ) h . From (4.25)–(4.28),     
 (m) 2 3 m  

FQ (X, ) h ≤ O(1)ε  d x : φ :C (x) sup |f ˜ (X, x, )| 

m=2,4

˜ (0,0)

+ |Q

(X, C, v

(4)

h x∈



1,Q

)| .

Undoing the Wick ordering produces lower order terms with O(1) coefficients. Now for m = 2, 4, 

  3 m  ε  d x : φ :C (x) ≤ O(1)ε d 3 xφ 4 (x) + O(1).  

h



314

D.C. Brydges, P.K. Mitter, B. Scoppola

By Lemma 5.12, supx∈ |f

(m) ˜ (X, x, )| 1,Q

˜ (0,0) (X, C, v (4) )| ≤ CL . Therefore ≤ CL , |Q


 3 4 |λ |R FQ (X, ) h ≤ CL R|λ| ε ε d xφ (x) + 1 . 2

2

(5.79)



Next consider FR , supported on small sets, defined in (4.38), (4.40). Recall (4.43), 
FR (X, φ) = d 3 x αP (X, x)P (φ(x), ∂φ(x)). 

P

By Lemma 5.17 and (4.42) |αP (X, x)| ≤ CL ε 11/4−η , so that 
|λ|3 |z(X)| FR (X, ) h ≤ CL R|λ|3 ε 11/4−η d 3 x P (φ(x), ∂φ(x)) h 

P


 ≤ CL R|λ|3 ε 7/4−η ε d 3 x φ 4 (x) + ε 1/2 φ 2,1,σ + 1 . 

The claim follows by combining this with (5.79). Note that in the above inequality the Sobolev norm arises only when estimating the j -term corresponding to φ∂µ φ. For this we can bound |φ∇φ| ≤ 1/2(|φ|2 + |∇φ|2 ) and then use the Sobolev embedding inequality.   Lemma 5.20. For any R > 0 and ξ := R max(|λ2 |ε 2 , |λ3 |ε 11/4−η ) sufficiently small, 

˜

|e−VL ()−

X⊃ z(X)F (λ,X,)

|h∗ ≤ 22 ,

(5.80)

where z(X) are complex parameters with |z(X)| ≤ R. Proof. This is the same as the last proof except that we can use the estimate |F (λ, X, )|h∗ ≤ CL ξ in place of (5.78). (No need to ensure R|F (λ, X, )|h∗ is smaller than εφ 4 because stability away from φ = 0 is not an issue with the kernel norm.)   Recall (4.31) Rmain =

1 2πi

 γ

  dλ −V  E S(λ, Qe ) , F (λ) . Q λ4

(5.81)

Lemma 5.21.

Rmain h,Gκ ,A ≤ CL ε 3/4 ,

(5.82)

|Rmain |h∗ ,A ≤ CL ε 3−3δ/2 .

(5.83)

Proof. Let J (λ) = S(λ, Qe−V ) . Suppose that F (λ) splits, F (λ) = F0 (λ)+F1 (λ), into a field independent part F0 and F1 satisfies stability as in (5.76). According to Theorem 6 on p. 780 of [BDH-est],  

E(J (λ), F0 , F1 ) h,Gκ ,A ≤ O(1) J (λ) h,Gκ ,A2 + f A4 , (5.84)

Critical ( 4 )3, 

315

provided the norms on the right hand side are less than a small constant R −1 = O(1). In the above |f (X)| ≤ 2|z(X)|−1 , where the z(X) are the complex parameters introduced in Lemma 5.19. The f (X) are supported on small sets. We choose |λ| = cL ε −1/4 . By Lemma 5.19 we have stability (5.76) if ε is small. Therefore (5.84) holds and by combining it with Lemma 5.16,

E(J (λ), F0 , F1 ) h,Gκ ,A ≤ q + O(R −1 ) ≤ O(1). Equation (5.82) follows by choosing the contour γ in (5.81) to be a circle of radius cL ε −1/4 . The proof of (5.83) follows the same steps but with contour γ being a circle of radius cL ε −1+δ/2 chosen so that ξ in Lemma 5.20 is small and the hypothesis of Lemma 5.16 is satisfied.   Recall from (4.33) that R3 =

1 2πi



dλ

γ

λ4 (λ − 1)

E(S(λ, K) , F (λ)).

(5.85)

Lemma 5.22.

R3 h,Gκ ,A ≤ CL ε 1−4η/3 ,

|R3 |h∗ ,A ≤ CL ε 11/3−4η/3 .

(5.86)

Proof. This proof follows the same steps as the proof of Lemma 5.21 with contours |λ| = cL ε 1/4−η/3 and |λ| = cL ε 11/12−η/3 .   Lemma 5.23. R4 as defined in (4.35) satisfies

R4 h,Gκ ,A ≤ Cε3/2 ,

|R4 |h∗ ,A ≤ Cε 3 .

Proof. From (4.35)     −V  −V˜L   −V˜L    Q(C, w , g ) + e Q(C, w , g ) − Q(C, w , gL ) . R4 = e −e First we observe from (4.35), Lemma 5.17 and Lemma 5.9 the following bounds: |g  − gL | ≤ Cε2 ,

(5.87)

|µ − µL | ≤ Cε2 ,

(5.88)

w ≤ c/4.

(5.89)

We estimate in turn the two terms in the expression for R4 above. Because of Q each term is supported on small sets. We write the first term as 


1   ˜ e−V − e−VL Q(C, w , g  ) = Q(C, w , g  )e− 2 V

1 0

ds(V˜L − V  )e−( 2 −s)V 1

 −sV L

.

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D.C. Brydges, P.K. Mitter, B. Scoppola

Then we bound     −V   −V˜L     e Q(C, w − e , g )   ≤ Q(C, w , g  )e

− 21 V 

h,Gκ 1




h,Gκ



ds V˜L − V  h e−( 2 −s)V h e−sVL h . (5.90) 1

0

Using (5.87) and (5.88) we can bound  3 4 ε

V˜L (X) − V  (X) h ≤ C eO(1)γ ε X d xφ (x) γ

(5.91)

for any γ = O(1) > 0. By Lemma 5.5 we can bound 1

e−( 2 −s)V

 (X)

g

1

h ≤ 2|X| e−( 2 −s) 4

e−s)VL (X) h ≤ 2|X| e

 −s g4



X

 X

d 3 xφ 4 (x)

d 3 xφ 4 (x)

,

.

(5.92) (5.93)

We now plug into (5.90) the bounds (5.92) and (5.93). We then write the s-integration in (5.90) as the union of the intervals [0, 41 and 41 , 1]. In the first interval we insert the bound (5.91) with γ replaced by ( 21 − s)γ . In the second interval we insert the same bound with γ replaced by sγ and in both cases take γ = O(1) sufficiently small. Then the s-integral factor in (5.90) is bounded by Cε. On the other hand the first factor in 1 (5.90) is bounded by O(1)ε 2 by virtue of Lemma 5.10. (The factor of 21 in the exponent does not make a difference.) Putting these bounds together and recalling that the Q are supported on small sets we obtain     −V   3 −V˜L     e Q(C, w − e , g ) ≤ Cε 2 . (5.94)   h,Gκ ,A

We can now easily bound the second term in the expression for R4 by noting that  |g 2 − gL2 | ≤ Cε3 and then using Lemma 5.10. We again get the bound     −V˜  3 e L Q(C, w , g  ) − Q(C, w , gL )  ≤ Cε 2 . (5.95)   h,Gκ ,A

Adding together (5.94) and (5.95) finishes the proof of the first bound in the lemma. The second bound is easy to prove since all derivatives in the h∗ norm are at φ = 0.   Lemma 5.24. Let X be a small set and let J be normalized as in (4.38). Then we have −(7−ε)/2 |D 2 J (X, 0; fL×2

D 2 J (X, 0) −1 )| ≤ O(1)L

−(4−ε) |D 4 J (X, 0; fL×4

D 4 J (X, 0) −1 )| ≤ O(1)L

Proof. See Lemma 15 [BDH-est].

2 

fj C 2 (L−1 X) ,

(5.96)

fj C 2 (L−1 X) .

(5.97)

j =1

4  j =1

Critical ( 4 )3, 

317

Corollary 5.25. Let Y = L−1 X where X is a small set, Z = L−1 X¯ L and let J be normalized as in (4.38). Then |JL (Y )|h ≤ O(1)L−(7−ε)/2 |J (X)|h ,

(5.98)

& $ ˜

JL (Y )e−VL (Z\Y ) h,Gκ ≤ O(1)L−(7−ε)/2 |J (X)|h + J (X) h,G3κ .

(5.99)

Proof. Equation (5.98) follows immediately from Lemma 5.24. For (5.99) we write ˜

˜

JL (Y, φ)e−VL (Z\Y,φ) h ≤ JL (Y, φ) h e−VL (Z\Y,φ) h $ & ≤ O(1)Gκ (Z, φ) |JL (Y )|h + L−n0 [φ] JL (Y ) L[φ] h,Gκ , where we used Lemmas 5.5 and Lemma 5.15. By (5.98), and rewriting the second term by moving the scaling from J to the norm, $ & ˜

JL (Y, φ)e−VL (Z\Y,φ) h ≤ O(1)Gκ (Z, φ) L−(7−ε)/2 |J (X)|h +L−n0 [φ] J (X) h,G3κ . Equation (5.99) follows by multiplying both sides by G−1 κ (Z, φ) and taking the supremum over φ.   Lemma 5.26. ˜

F˜R e−V h,Gκ ,A ≤ O(1)ε 7/4−η ,

(5.100)

˜ |F˜R e−V |h∗ ,A ≤ O(1)ε 11/4−η ,

(5.101)

˜ and J = R  − F˜R e−V satisfies on small sets the bounds 3

11

J h,G3κ ,A ≤ O(1)ε 4 −η |J |h∗ ,A ≤ O(1)ε 4 −η . Proof. First we prove (5.100). Take the definition of F˜R given in (4.38) and (4.39). F˜R is supported on small sets. We have
 ˜

FR (X, φ) h ≤ |α˜ P (X)| d 3 x P (φ(x), ∂φ(x)) h P

≤ O(1)

X



|α˜ P (X)|ε

−1

X

P

≤ O(1)




 3 4 1/2 2 ε d x φ (x) + ε φ X,1,σ + 1

|α˜ P (X)|ε −1 Gκ (X, φ)eγ g

 X

d 3 yφ 4 (y)

P

for any γ = O(1) > 0. Hence, using Lemma 5.5, ˜

˜



F˜R (Xφ)e−V (Xφ) h ≤ F˜R (Xφ) h e−V (Xφ) h ≤ O(1)

P

2|X| |α˜ P (X)|ε −1 Gκ (X, φ).

318

D.C. Brydges, P.K. Mitter, B. Scoppola

We thus obtain (remembering that α˜ P are supported on small sets) on using (5.70) ,  ˜

α˜ P A ≤ O(1)ε 7/4−η .

F˜R (Xφ)e−V (Xφ) h,Gκ ,A ≤ O(1)ε −1 P

This proves (5.100). Now we turn to the proof of (5.101). As observed in the proof −n(P ) of Lemma 5.17, for ε sufficiently small (depending on L), |α˜ P |A ≤ n(P )!h∗ −n(P ) 11/4−η |1S R  |h∗ ,A ≤ O(1)h∗ ε . We have from the definition of F˜R given in (4.38)  n |F˜R (X)|h∗ ≤ O(1) P |α˜ P (X)|h∗P , whence  |F˜R |h∗ ,A ≤ |α˜ P |A hn∗P ≤ O(1)ε 11/4−η P

which proves (5.101). ˜ ˜ To get these bounds for J = R  − F˜R e−V we apply (5.100) and (5.101) to the F˜R e−V  part. We bound R by Lemma 5.17.   Lemma 5.27.

Rlinear h,Gκ ,A ≤ O(1)L−(1−ε)/2 ε 3/4−η ,

(5.102)

|Rlinear |h∗ ,A ≤ O(1)L−(1−ε)/2 ε 11/4−η .

(5.103)

˜ Proof. We recall from (4.39) that J = R  − F˜R e−V is normalized. Let 1s.s (X) be the indicator function of the event that X is small. Referring to (4.39), the first term in Rlinear (Z) is  ˜ e−VL (Z\Y ) 1s.s (X)JL (Y ), (5.104) Rlinear,s.s (Z) := X:L−1 X¯ L =Z

where Y = L−1 X. By Corollary 5.25 this is bounded in h, Gκ norm by $ &  −(7−ε)/2 O(1)L 1s.s (X) |J (X)|h + J (X) h,G3κ .

(5.105)

X:L−1 X¯ L =Z

Multiply both sides by A(Z), using A(Z) ≤ A(X) on the right hand side. Then sum over Z to get the A norm and use the bounds on J in Lemma 5.17 and Lemma 5.26 to obtain

Rlinear,s.s h,Gκ ,A ≤ O(1)L−(1−ε)/2 ε 3/4−η ,  where we used an argument on p. 790 of [BDH-est] to control the X:L−1 X¯ L =Z by LD=3 times the sum over X in the definition of the A norm. Similarly the bound on the kernel norm by O(1)L−(1−ε)/2 ε 11/4−η comes from the kernel norm bound in Lemma 5.17, Lemma 5.26 and Lemma 5.15. Let 1l.s (X) be the large set indicator function. The second term in Rlinear (Z) is  −1 ˜  e−VL (Z\L X) 1l.s (X)RL (L−1 X), (5.106) Rlinear,s.s (Z) := X:L−1 X¯ L =Z

where we have used J (X) = R  (X) because the subtraction is supported on small sets. This is bounded in the same way as above using Lemma 5.17 except that the necessary L−(1−ε)/2 is obtained for a different reason: For large sets, by (2.7), A(Z) ≤ cp L−4 A−p (X), where cp = O(1).  

Critical ( 4 )3, 

319

Proof of Theorem 1 Concluded. From (4.36), R is the sum of Ri , where i = 1, 2, 3, 4. By Lemmas 5.21, 5.22, 5.23 and 5.27 with L large and ε small depending on L, the sum satisfies bounds (5.9) and (5.10).   6. Stable Manifold and Convergence to Non-Gaussian Fixed Point 6.1. Domains. Let g¯ be the approximate fixed point of the g flow given by (5.4). Let us define g˜ = g − g¯

(6.1)

u = (g, ˜ µ, R, w).

(6.2)

and

Then the RG iteration given by (4.35), (4.36), (4.16) can be written as u = f (u)

(6.3)

g˜  = fg (u) = α()g˜ + ξ˜ (u),

(6.4)

with components

µ = fµ (u) = L

3+ε 2

µ + ρ(u), ˜

(6.5)

R  = fR (u) =: U (u),

(6.6)

w = fw (u) = v + wL ,

(6.7)

u0 = (g˜ 0 , µ0 , 0, 0).

(6.8)

with initial

Here

α() = 2 − Lε = 1 − O(log L)ε, ξ˜ (u) = −L2ε a g˜ 2 + ξR (u),

(6.9)

ρ(u) ˜ = −L2ε b(g + g) ˜ 2 + ρR (u).

(6.10)

Note that the w flow is autonomous and solved by (5.31). In Lemma 5.9 it was proved that the w tends to the fixed point w∗ in an appropriate norm. Let E be the Banach space consisting of elements u with the (box) norm

u = sup(ε−3/2 |g|, ˜ ε−(2−δ) |µ|, ε−(11/4−η) |||R|||, c−1 w ).

(6.11)

|||R||| = sup(ε2 R h,G,A , |R|h∗ ,A )

(6.12)

Here

320

D.C. Brydges, P.K. Mitter, B. Scoppola

and

w = sup w (p) p . p

The · p and w norms were defined in (5.32) and the constant c is that of Lemma 5.9. Let B(r) ⊂ E be the closed ball of radius r, centered at the origin: B(r) = {u ∈ E : u ≤ r}.

(6.13)

Let D be the domain of (g, µ, R) specified in (5.1)–(5.3) of the hypothesis stated at the beginning of Sect. 5. Then we have u ∈ B(1) ⇒ (g, µ, R) ∈ D

(6.14)

and then Theorem 1 of Sect. 5 holds. Theorem 1, together with the autonomous Lemma 5.9, shows that the ball B(1) would be stable under the RG flow f , except for the unstable direction µ (as evident from (6.5)). The initial unstable parameter µ0 will have to be fine tuned to a critical function µ0 = µc (g0 ) which would determine the critical or stable submanifold in which we expect to get a contraction to a fixed point. This is part of the stable manifold theorem in the theory of hyperbolic dynamical systems [S]. In the following we put ourselves in this framework (see Appendix 2, Chapter 5 of [S] and Sect. 5 of [BDH-eps]). Suppose u ∈ B(1). Then from Theorem 1 of Sect. 5 we have for ε > 0 sufficiently small (depending on L), which implies in particular Lε = O(1), |ξR (u)| ≤ O(1)ε 11/4−η , |ρR (u)| ≤ O(1)L3/2 ε 11/4−η , |||U (u)||| ≤ L−1/4 ε 11/4−η .

(6.15)

From (6.9), (6.10) we have by virtue of (6.16) and the estimates a = O(log L), b = O(L3/2 ), see Lemma 5.12, for u ∈ B(1): |ξ˜ (u)| ≤ O(1)ε 11/4−η , |ρ(u)| ˜ ≤ ε 2−δ .

(6.16)

ξ˜ , ρ, ˜ U, w satisfy the following Lipshitz bounds in B(1/4) ⊂ B(1). Lemma 6.1. Let u, u ∈ B(1/4). Then we have the Lipshitz bounds: (i) (ii) (iii) (iv)

|ξ˜ (u) − ξ˜ (u )| ≤ O(1)ε 11/4−η u − u , |ρ(u) ˜ − ρ(u ˜  )| ≤ ε5/2−δ u − u , |||U (u) − U (u )||| ≤ O(1)L−1/4 ε 11/4−η u − u ,

fw (u) − fw (u ) ≤ cL−1/4 u − u .

(6.17)

Proof. We shall use the fact that ξR , ρR , U are analytic in B(1). This follows from the algebraic operations in Sect. 4 together with the analyticity of the extraction map (Theorem 5 [BDH-est]). (iii) Consider first the case u−u ≤ 1/100. Let u = u−u . By the Cauchy integral formula,

Critical ( 4 )3, 

321

U (u) − U (u ) =

1 2πi

  γ

  U (u + zu) 1 1 1 − U (u + zu) dz = dz, z−1 z 2π i γ (z − 1)z

where γ is the closed contour γ : z = reiϑ , r = 1/4 u −1 . The contour encloses the poles because u − u ≤ 1/100 implies r ≥ 25. Also

u + zu ≤ u + r u ≤

1 1 1 + = . 4 4 2

Therefore, for z ∈ γ , u + zu ∈ B(1), and hence from (6.16), |||U (u + zu)||| ≤ L−1/4 ε 11/4−η , so that, by estimating the Cauchy integral, |||U (u) − U (u )||| ≤ O(1)L−1/4 ε 11/4−η u − u . Case u − u > 1/100. By (6.16) |||U (u) − U (u )||| ≤ |||U (u)||| + |||U (u )||| ≤ O(1)L−1/4 ε 11/4−η which is less than O(1)L−1/4 ε 11/4−η 100 u − u . (i) On using (6.16) we have in the same way |ξ˜ (u) − ξ˜ (u )| ≤ O(1)ε 11/4−η u − u . (ii) To get the Lipshitz bound for ρ˜ we first obtain |ρR (u) − ρR (u )| ≤ O(1)L3/2 ε 11/4−η u − u . by the same methods. Then from (6.10) and b = O(L3/2 ) (Lemma 5.12), |ρ(u) ˜ − ρ(u ˜  )| ≤ O(L3/2 )|g˜  + g˜ + 2g|| ¯ g˜  − g| ˜ + |ρR (u) − ρR (u )|

≤ O(L3/2 )O(ε)e3/2 + O(1)L3/2 ε 11/4−η u − u ≤ ε5/2−δ u − u . (iv) From (6.7) and the definition of the norms in (5.32) we have

6p+1 (p) (p)

fw (u) − fw (u ) ≤ sup ess supx |x| 4 |wL (x) − wL  (x)| . 1≤p≤3

(p)

Now wL (x) = L2[φ] w (p) (Lx). We then get easily

fw (u) − fw (u ) ≤ L−1/4 w − w ≤ cL−1/4 u − u , and we are done.

 

Consider now the RG flow (6.3): uk = f (uk−1 ) with initial condition u0 = (g˜ 0 , µ0 , 0, 0) g˜ 0 = g0 − g. ¯

322

D.C. Brydges, P.K. Mitter, B. Scoppola

Theorem 6.2. There exists µ0 such that for u0 ∈ B(1/32), uk = f (uk−1 ) ∈ B(1/4) for all k ≥ 1. Remark. The following proof of existence of global solutions is a textbook argument in the theory of dynamical systems adapted to the present context. Proof. The strategy is to derive an equation that a global RG trajectory contained in B(r) must solve, if it exists. Then we prove that this equation has a solution by the contraction mapping principle. This solution can then be shown to be an RG trajectory. From the flows (6.4), (6.5) we easily derive after n steps of the RG, k−1 

g˜ k = α()k g˜ 0 +

α()k−1−j ξ˜ (uj ),

1 ≤ k ≤ n,

j =0

µk = L−

3+ε 2 (n−k)

µn −

n−1 

L−

3+ε 2 (j +1−k)

ρ(u ˜ j ),

0 ≤ k ≤ n − 1.

j =k

Let us fix µn = µf and take n → ∞. We have k−1 

g˜ k = α()k g˜ 0 +

α()k−1−j ξ˜ (uj ),

k ≥ 1,

(6.18)

j =0

µk = −

∞ 

L−

3+ε 2 (j +1−k)

ρ(u ˜ j ),

k ≥ 0,

(6.19)

j =k

together with Rk = U (uk−1 ),

k ≥ 1.

(6.20)

We can take the autonomous w flow, given by (6.7), as solved by (5.31) and need no longer consider it as a flow variable. Note that for ε sufficiently small (depending on L) 0 < α() < 1.

(6.21)

Then for uj ∈ B(1) the infinite sum of (6.19) converges by (6.21) and (6.16). So µ0 has now been determined provided (6.18)–(6.20) has a solution. It is easy to verify that any solution of (6.18)–(6.19), together with the autonomous w flow, is a solution of the RG flow uk = f (uk−1 ). Now write (6.18)–(6.20) in the form uk = Fk (u),

(6.22) (g)

(µ)

(R)

where u = (u0 , u1 , u2 , ...) and Fk has components (Fk , Fk , Fk ) given by the r.h.s. of (6.18), (6.19) and (6.20) respectively.

Critical ( 4 )3, 

323

If we write F(u) = (F0 (u), F1 (u), ...), then (6.22) can be written as a fixed point equation u = F(u).

(6.23)

Consider the Banach space E of sequences u = (u0 , u1 , u2 , ...) with norm

u = sup uk ,

(6.24)

k≥0

and the closed ball B(r) ⊂ E B(r) = {u : u ≤ r}.

(6.25)

We shall seek a solution of (6.23) in the closed ball B(1/4) with initial data u0 = (g˜ 0 , µ0 , 0, 0) in B(1/32) and g˜ 0 held fixed. The existence of a solution follows by the standard contraction mapping principle and the next lemma.   Lemma 6.3. u ∈ B(1/32) ⇒ F(u) ∈ B(1/16).

(6.26)

Moreover, for u, u ∈ B(1/4),

F(u) − F(u ) ≤

1

u − u . 2

(6.27)

Proof. First we prove (6.26), and thus take u ∈ B(1/32). From (6.18) and the estimates in (6.16) we have ε−3/2 |Fk (u)| ≤ α() (g)

 1 α()k−1−j + O(1)ε 5/4−η 32 k−1

j =0

≤

ε 5/4−η

1 1 ε 1/4−η 1 + O(1) ≤ + O(1) ≤ , 32 α() − 1 32 log L 16

since η < 41 and ε is sufficiently small. Similarly from (6.19) and (6.16) we have ε −(2−δ) |Fk (u)| ≤ (µ)

∞ 

L−

3+ε 2 (j +1−k)

≤ L−

3+ε 2

(1 − L−

j =k

for L sufficiently large. Finally from (6.20) and (6.16) ε −(11/4−η) |Fk (u)| ≤ L−1/4 ≤ (R)

This proves (6.26).

1 . 16

3+ε 2

)≤

1 16

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D.C. Brydges, P.K. Mitter, B. Scoppola

To prove (6.27), take u, u ∈ B(1/4). We can then use the Lipshitz estimates of Lemma 6.1. Note that the initial coupling g0 is held fixed. Then we have ε

−3/2

(g) (g) |Fk (u) − Fk (u )|

≤

k−1 

α()k−1−j ε −3/2 |ξ˜ (uj ) − ξ˜ (uj )|

j =0

≤ O(1)ε5/4−η u − u

k−1 

α()k−1−j

j =0

≤ O(1)ε 1/4−η u − u ≤

1

u − u . 2

Similarly, ε−(2−δ) |Fk (u) − Fk (u )| ≤ L− (µ)

(µ)

3+ε 2

∞ 

L−

3+ε 2 (j −k)

ε −(2−δ) |ρ(u ˜ j ) − ρ(u ˜ j )|

j =k

≤ L−

3+ε 2

O(1)ε 1/2 u − u ≤

1

u − u . 2

Finally ε−(11/4−η) |||Fk (u) − Fk (u )||| = ε −(11/4−η) |||U (uk−1 ) − U (uk−1 )||| 1 ≤ O(1)L−1/4 u − u ≤ u − u . 2 (R)

(R)

Thus (6.27) has also been proved.

 

6.2. Stable manifold and convergence to fixed point. Write E = E1 × E2 with u ∈ E represented as u = (u1 , u2 ). Here u1 = (g, ˜ R, w) and u2 = µ. E1 and E2 thus represent the contracting and expanding directions for the RG map f . Let pi , i = 1, 2, denote the projector onto Ei and fi = pi ◦ f . Note that the norm · on E being a box norm we also have

u = sup( u1 , u2 ). The following Lemma 6.4, the definition 6.5 of the stable manifold W s and our final Theorem 6.6 are Irwin’s proof of the stable manifold theorem as presented in Appendix 2, chapter 5 of [S]. Part of Irwin’s proof is replaced by Theorem 6.2. Lemma 6.4. Let u, u ∈ B(1/4). Then

f1 (u) − f1 (u ) ≤ (1 − ε) u − u

(6.28)

and, if u2 − u2 ≥ u1 − u1 then

f2 (u) − f2 (u ) ≥ (1 + ε) u − u .

(6.29)

Critical ( 4 )3, 

325

Proof. Because u, u ∈ B(1/4) we can use Lemma 6.1 throughout. As always L is sufficiently large and then ε sufficiently small. First we prove (6.28). f1 has components (fg , fR , fw ). From (6.4) fg (u) = α()g˜ + ξ˜ (u). Thus using Lemma 6.1 ε −3/2 |fg (u) − fg (u )| ≤ α() u − u + ε−3/2 |ξ˜ (u) − ξ˜ (u )| ≤ (α() + O(1)ε 5/4−η ) u − u ≤ (1 − ε) u − u for ε sufficiently small. Since fR (u) = U (u), we have from Lemma 6.1 ε −(11/4−η) |||fR (u) − fR (u )||| ≤ (1 − ε) u − u for L sufficiently large. Finally from the same lemma c−1 fw (u) − fw (u ) ≤ (1 − ε) u − u . These three inequalities prove (6.28). Next we turn to (6.29). In this case by assumption u2 − u2 ≥ u1 − u1 and hence, since our norms are box norms, we have

u − u = u2 − u2 = ε−(2−δ) |µ − µ |. From (6.5) fµ (u) = L

3+ε 2

µ + ρ(u). ˜

Then, using Lemma 6.1, ε −(2−δ) |fµ (u) − fµ (u )| ≥ L

3+ε 2

u − u − ε−(2−δ) |ρ(u) ˜ − ρ(u ˜  )|

3+ε

≥ (L 2 − ε 5/2−δ ) u − u ≥ (1 + ε) u − u , which proves (6.29).

 

Let f k be the k-fold composition of the RG map f . Definition 6.5. The stable manifold of f is defined by W s (f ) = {u ∈ B(1/32) : f k (u) ∈ B(1/4) ∀k ≥ 0}.

(6.30)

Write the initial points u as u = (u1 , u2 ) with u1 = (g˜ 0 , 0, 0) and u2 = µ0 . Observe that Theorem 6.2 says that there exists for u ∈ B(1/32) a u2 such that f k (u) ∈ B(1/4) ∀k ≥ 0. We now have Theorem 6.6. W s (f ) is the graph {(u1 , h(u1 )} of a function u2 = h(u1 ) with h Lipshitz continuous with Liph ≤ 1. Moreover f |W s (f ) contracts distances and hence has a unique fixed point which attracts all points of W s (f ). The fixed point is non-trivial.

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D.C. Brydges, P.K. Mitter, B. Scoppola

Proof. To prove the first statement it is enough to prove that if in W s (f ) we take two points u = (u1 , u2 ) and u = (u1 , u2 ) then

u2 − u2 ≤ u1 − u1

(6.31)

because then for a given u1 we would have at most one u2 , and by Theorem 6.2 there exists such a u2 . This means that W s (f ) is the graph of a function h, u2 = h(u1 ), and moreover

h(u1 ) − h(u1 ) ≤ u1 − u1 . Suppose (6.31) is not true. Then

u2 − u2 > u1 − u1 .

(6.32)

Then by (6.29) followed by (6.28) gives

f2 (u) − f2 (u ) ≥ (1 + ε) u − u > (1 − ε) u − u ≥ f1 (u) − f1 (u ) and hence Now

f (u) − f (u ) ≥ (1 + ε) u − u .

f 2 (u) − f 2 (u ) = f (f (u)) − f (f (u )) ,

and by the above and the second part of Lemma 6.4,

f 2 (u) − f 2 (u ) ≥ (1 + ε) f (u) − f (u ) ≥ (1 + ε)2 u − u . By induction we can prove for all k ≥ 0,

f k (u) − f k (u ) ≥ (1 + ε)k u − u . Since u, u ∈ W s (f ) the l.h.s. is bounded above by 21 and hence for k → ∞ we have a contradiction because u = u under (6.32). Hence (6.31) is true and the first statement of Theorem 6.6 has been proved. Now we prove that f |W s (f ) is a contraction. Note that if u, u ∈ W s (f ), then

f2 (u) − f2 (u ) ≤ f1 (u) − f1 (u ) .

(6.33)

We can prove this just as we proved (6.31). Namely assume the contrary and then show in the same way

f k (u) − f k (u ) ≥ (1 + ε)k−1 f (u) − f (u ) . The l.h.s. is bounded by 21 and so as k → ∞ we get a contradiction because f (u) = f (u ) under the negation of (6.33). This proves (6.33) which now implies

f (u) − f (u ) = f1 (u) − f1 (u ) ≤ (1 − ε) u − u by Lemma 6.4, and f |W s (f ) is a contraction. Note that Theorem 6.6 tells us that µ0 = h(g˜ 0 ) = µc (g0 ) which defines µc . If g˜ ∗ = g∗ − g¯ is one of the coordinates of the fixed point u∗ then g∗ = 0 since u∗ ∈ B(1/4). In fact the latter implies 1 |g∗ − g| ¯ ≤ ε 3/2 4 and we know g¯ = O(ε) and this excludes g∗ = 0. So our fixed point is non trivial (non-Gaussian).  

Critical ( 4 )3, 

327

References [AR]

Abdesselam, A., Rivasseau, V.: An Explicit Large Versus Small Field Multiscale Cluster Expansion. Rev. Math. Phys. 9, 123–199 (1997) [BDH-est] Brydges, D., Dimock, J., Hurd, T.R.: Estimates on Renormalization Group Transformation. Canad. J. Math. 50(4), 756–793 (1998) [BDH-eps] Brydges, D., Dimock, J., Hurd, T.R.: A Non-Gaussian Fixed Point for φ 4 in 4- Dimensions. Commun. Math. Phys. 198, 111–156 (1998) [BG] Giuseppe Benfatto, Giovanni Gallavotti: Renormalization Group. Princeton, NJ: Princeton University Press, 1995 [BI] Brydges, D., Imbrie, J.: End-to-end Distance from the Green’s Function for a Hierarchical Self-Avoiding Walk in Four Dimensions. http://arxiv.org/abs/math-ph/0205027, and Green’s Function for a Hierarchical Self-Avoiding Walk in Four Dimensions. http://arxiv.org/abs/math-ph/0205028 [BY] Brydges, D., Yau, H.T.: Grad φ Perturbations of Massless Gaussian Fields. Commun. Math. Phys. 129, 351–392 (1990) [DH1] Dimock, J., Hurd, T.R.: A Renormalisation Group Analysis of Correlation Functions for the Dipole Gas. J. Stat. Phys. 66, 1277–1318 (1992) [DH2] Dimock, J., Hurd, T.R.: Sine-Gordon Revisited. Ann. Henri Poincar´e 1(3), 499–541 (2000) [F] J¨urg Fr¨ohlich: Scaling and Self-Similarity in Physics. Boston, MA: Birkh¨auser Boston Inc, 1983 [FMRS] Feldman, J., Magnen, J., Rivasseau, V., S´en´eor, R.: Construction and Borel Summability of Infrared (φ)44 by a Phase Space Expansion. Commun. Math. Phys. 109, 437–480 (1987) [GJ] James Glimm, Arthur Jaffe: Quantum Physics. A Functional Integral Point of View. Second Edition. New York: Springer-Verlag, 1987 [GK] Gawedzki, K., Kupiainen, A.: Massless (φ)44 theory: Rigorous control of a renormalizable asymptotically free model. Commun. Math. Phys. 99, 197–252 (1985) [BGM] Brydges, D., Guadagni, G., Mitter, P.K.: Finite range Decomposition of Gaussian Processes. http://arxiv.org/abs/math-ph/0303013 [KG] Kolmogoroff, A.N. Gnedenko, B.V.: Limit Distributions of sums of independant random variables. Cambridge, MA: Addison Wesley, 1954 [KW] Wilson, K.G., Kogut, J.: The Renormalization Group and the  Expansion. Phys. Rep. (Sect C of Phys Lett.) 12, 75–200 (1974) [MP] Mack, G., Pordt, A.: Convergent Perturbation Expansions in Euclidean Quantum Field Theory. Commun. Math. Phys. 97, 267–298 (1985) [MS] Mitter, P.K., Scoppola, B.: Renormalization Group Approach to Interacting Polymerised Manifolds. Commun. Math. Phys. 209, 207–261 (2000) [S] Shub, M.: Global Stability of Dynamical Systems. New York: Springer-Verlag, 1987 Communicated by J.Z. Imbrie

Commun. Math. Phys. 240, 329–375 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0884-7

Communications in

Mathematical Physics

Stability of Quantum Systems at Three Scales: Passivity, Quantum Weak Energy Inequalities and the Microlocal Spectrum Condition Christopher J. Fewster1 , Rainer Verch2,
1

Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom. E-mail: [email protected] 2 Institut f¨ ur Theoretische Physik, Universit¨at G¨ottingen, Bunsenstr. 9, 37073 G¨ottingen, Germany Received: 16 April 2002 / Accepted: 1 April 2003 Published online: 23 June 2003 – © Springer-Verlag 2003

Abstract: Quantum weak energy inequalities have recently been extensively discussed as a condition on the dynamical stability of quantum field states, particularly on curved spacetimes. We formulate the notion of a quantum weak energy inequality for general dynamical systems on static background spacetimes and establish a connection between quantum weak energy inequalities and thermodynamics. Namely, for such a dynamical system, we show that the existence of a class of states satisfying a quantum weak inequality implies that passive states (e.g., mixtures of ground- and thermal equilibrium states) exist for the time-evolution of the system and, therefore, that the second law of thermodynamics holds. As a model system, we consider the free scalar quantum field on a static spacetime. Although the Weyl algebra does not satisfy our general assumptions, our abstract results do apply to a related algebra which we construct, following a general method which we carefully describe, in Hilbert-space representations induced by quasifree Hadamard states. We discuss the problem of reconstructing states on the Weyl algebra from states on the new algebra and give conditions under which this may be accomplished. Previous results for linear quantum fields show that, on one hand, quantum weak energy inequalities follow from the Hadamard condition (or microlocal spectrum condition) imposed on the states, and on the other hand, that the existence of passive states implies that there is a class of states fulfilling the microlocal spectrum condition. Thus, the results of this paper indicate that these three conditions of dynamical stability are essentially equivalent. This observation is significant because the three conditions become effective at different length scales: The microlocal spectrum condition constrains the short-distance behaviour of quantum states (microscopic stability), quantum weak energy inequalities impose conditions at finite distance (mesoscopic stability), and the existence of passive states is a statement on the global thermodynamic stability of the system (macroscopic stability).
Current address: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany. E-mail: [email protected]

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C.J. Fewster, R. Verch

1. Introduction At small scales and high energies, the behaviour of matter is governed by quantum field theory, and it is widely known that the energy-momentum tensor in quantum field theory violates the energy-positivity conditions that are generically fulfilled for classical matter in general relativity. For instance, in flat spacetime, there is in any quantum field theory a large class of physical states for which the expectation value of the energy density is a function on spacetime assuming negative as well as positive values [10]. However, as pointed out by Ford [17], one may argue on heuristic grounds that there must be constraints on the intensity and spatio-temporal extension of negative values of the energy density, as otherwise one could produce situations in which macroscopic violations of the second law of thermodynamics occur. This idea was further developed by Ford and several other authors (see, e.g., [18, 19, 32, 12, 11, 13, 14, 16, 47, 15, 30]) and led to a form of such constraints which are now called quantum weak energy inequalities (abbreviated, QWEIs) in the terminology of [15], or often simply quantum inequalities (QIs). To explain their nature, consider some quantum field propagating on a Lorentzian spacetime (M, g), and let
Tµν (x)ω be the expectation value of the energy-momentum tensor in a state (expectation functional) ω at some spacetime point x. For any smooth, timelike curve γ in spacetime, denote by ρω (τ ) = γ˙ µ (τ )γ˙ ν (τ )
Tµν (γ (τ ))ω the mean energy density in the state ω along the curve, parametrized by proper time τ . Then one says that the states of the quantum field fulfill a QWEI if for any timelike curve γ and any real-valued smooth, compactly supported test-function g there holds a bound of the form
inf dτ g 2 (τ )ρω (τ ) ≥ cg,γ > −∞, (1.1) ω

where the infimum is taken over the set of states ω of the quantum field theory in which ρω can be reasonably defined as a locally integrable function (typically, this is a dense set of all physical states for the quantum field theory). The important point is that the constant cg,γ bounding the weighted integral of the energy density along γ from below may depend on the weight-function g and the curve γ , but is state-independent. To comment on this constraint, we note first that an inequality of the form (1.1) has been shown to hold for the class of Hadamard states ω of the free Klein-Gordon field and the Dirac field on arbitrary, globally hyperbolic spacetimes [12, 15]. Moreover, in more special situations, the inequality (1.1) was obtained in a more specific form. For example, for the massless free scalar field in d-dimensional Minkowski spacetime and the specific choice of a (not compactly supported) Lorentzian weight function g 2 (τ ) = t0 /[π(τ 2 +t02 )] (t0 > 0), and taking for γ any straight timelike line, one obtains a bound
t0 d ρω (τ ) ≥− d (1.2) dτ 2 π τ + t02 t0 for all states ω of finite particle number and energy, where d is a state-independent universal constant depending only on the spacetime dimension [19, 16, 11]. Thus the intensity of the weighted negative energy is at most proportional to an inverse power of its mean duration, i.e. the width t0 of the Lorentzian weight function. This considerably limits the possibility of negative energies to build up to macroscopic violations

Stability of Quantum Systems at Three Scales

331

of the second law of thermodynamics. Also, other macroscopic dynamical instabilities are hampered in this way when taking expectation values of a quantum field’s energymomentum tensor as the right-hand side of Einstein’s equations of gravity: It can be shown that appropriate forms of averaged energy conditions such as (1.1) or (1.2) imply similar statements about the dynamical behaviour of solutions to Einstein’s equations as do the pointwise energy positivity conditions for classical matter. Examples include singularity theorems, or the impossiblity of exotic spacetimes with closed timelike curves or “warp-drive” scenarios [46, 20, 31]. The argument sketched above – that a bound on negative energy densities in quantum field theory of the type (1.2) really prevents a macroscopic violation of the second law of thermodynamics – is, of course, somewhat heuristic. The purpose of the present article is to show that such a conclusion may be drawn rigorously under fairly general circumstances. To this end, we shall specialize our setting a bit and consider a quantum system (which may, but need not, be a quantum field) situated in a static, spatially compact spacetime (the assumption of spatial compactness is mainly made for convenience). Thus, the underlying spacetime (M, g) is of the form M =R× , where is a smooth, s-dimensional compact manifold endowed with a Riemannian metric h, and the static Lorentzian metric g on M has the line element ds 2 = gab dx a dx b = g00 dt 2 − hij dx i dx j

(1.3)

with a smooth, strictly positive function g00 on . Consequently, when a point x ∈ M is represented as x = (t, x) with t ∈ R and x ∈ , then g is independent of t. The variable t is to be viewed as “time”-variable, whereas contains all the spatial locations of the system at instances of “equal time”. We will then assume that the quantum system on this spacetime can be modelled such that the observables of the system are elements of a C ∗ -algebra1 A, and that the time-evolution of the system can be described by a one-parameter group {αt }t∈R of automorphisms of A. A technical assumption is also made, namely that the automorphism group is strongly continuous in time: This means that ||αt (A) − A|| → 0 as t → 0 for each A ∈ A; the norm appearing here is the C ∗ -norm on A. In technical terms (cf. [6]), the pair (A, {αt }t∈R ) is a C ∗ -dynamical system. Here the time-parameter t is thought of as having the significance of the time-parameter of the spacetime, so that {αt }t∈R describes the time-evolution of the system on the underlying spacetime. That is, if the system is formed by a family A(O) ⊂ A, O ⊂ R × , of sub-C ∗ -algebras of A fulfilling the isotony condition O1 ⊂ O ⇒ A(O1 ) ⊂ A(O), then we assume that αt (A(O)) = A(τt (O)), where τt denotes the time-shift τt (t , x) = (t + t , x) on the underlying spacetime. These assumptions are very general (cf. [23]); apart from some points of technical detail (such as replacing the C ∗ -algebra by a more general type of ∗algebra, or relaxing the assumption of strong continuity of the dynamics), there is a vast range of quantum systems on static spacetimes that can be modelled as C ∗ -dynamical systems. Some comments on the extent to which the assumption of strong continuity is realized in the quantum field theoretic system may nevertheless be helpful. The typical description of such systems in the present setting of a static spacetime M starts with a C ∗ -algebra 1 We shall understand by a C ∗ -algebra always a C ∗ -algebra containing a unit element, generically denoted by 1.

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A generated by elements G(f ), where f ranges over C0∞ (M). These algebraic generators are subject to certain relations characterizing dynamical properties of the system, and a time evolution can usually be defined on A as a one-parametric group {α˜ t }t∈R of automorphisms of A via α˜ t (G(f )) = G(f ◦ τ−t ). However, thinking in particular of the case of a free bosonic field, where G(f ) are the Weyl-generators of a CCR-algebra2 , the corresponding {α˜ t }t∈R will, in general, fail to be strongly continuous. This difficulty can be overcome by passing to sufficiently “regular” Hilbert-space representations π of A in which {α˜ t }t∈R is weakly continuous. We will give a very detailed account of this construction for the example of the scalar Klein-Gordon field in Sect. 4; to prepare the ground for this, we now outline the basic idea. To simplify the discussion a little bit, and to help the reader who is not too familiar with C ∗ -dynamical systems to see the connection with the more common Hilbert space approach, we assume that there is a weakly continuous unitary group {Vt }t∈R on H, the representation Hilbert space of π , so that Vt π(A)Vt∗ = π(α˜ t (A)) for all A ∈ A. Then αt (A) = Vt AVt∗ , A ∈ π(A), defines a group of automorphisms of π(A) ⊂ B(H) which need not be strongly continuous but only weakly continuous, i.e.
ψ, (αt (B) − B)φ → 0 as t → 0 for all choices of vectors φ, ψ ∈ H. But this form of continuity implies that for each h ∈ C0∞ (R) and each A ∈ π(A) the weak (or Bochner-) integral
αh A = dt h(t)αt (A) (1.4) exists as an element in B(H); more precisely, αh A is contained in π(A)

, the von Neumann algebra generated by π(A).3 Moreover, it is easy to see (cf. Sect. 4) that ||αt (αh A) − αh A|| → 0 for t → 0. Hence, if we denote by A the C ∗ -algebra which is generated by all the αh A, where A ∈ π(A) and h ∈ C0∞ (R), then (A, {αt }t∈R ) forms a C ∗ -dynamical system. (The local algebras A(O) are then defined as A ∩ {π(G(f )) : f ∈ C0∞ (O)}

.) Therefore one sees that C ∗ -dynamical systems arise naturally as a means of describing quantum field systems as soon as one considers representations in which the time-evolution of the system is modelled by a weakly continuous unitary group (and also under more general circumstances, see Sect. 4), and this is surely a very general situation, irrespective of whether a quantum field is interacting or free. We should also emphasize that, once such a representation π is chosen, it doesn’t matter if one describes the system in terms of the algebra π(A) or the algebra A since, for each unit vector ψ ∈ H, the expectation value
ψ, Aψ of A ∈ π(A) can be approximated as closely as desired by the expectation value
ψ, αh Aψ of αh A ∈ A through sharply peaking h around 0. Conversely, αh A is weakly approximated by elements of π(A). For further discussion of the relation between states on A and states on A in the case of the CCR-algebra, see towards the end of Sect. 4.1 and Appendix A.5. To make contact with thermodynamics, we now turn to the notion of passivity introduced by Pusz and Woronowicz [33]. Let (A, {αt }t∈R ) be a C ∗ -dynamical system. The idea of Pusz and Woronowicz was to consider the behaviour of the system when the 2 3

See Sect. 4.1 for a fuller discussion of the CCR-algebra in the context of the free Klein-Gordon field. For a subset B of B(H), B denotes the commutant of B, i.e. B = {C ∈ B(H) : CB = BC ∀ B ∈ B}.

Stability of Quantum Systems at Three Scales

333

external conditions change in time. To this end, we note first (cf. Sect. 2) that there exists a norm-dense ∗-subalgebra D(δ) of A so that   d , A ∈ D(δ) , (1.5) αt (A) δ(A) = dt t=0 exists in A. This derivation is the generator of the dynamics when the external conditions remain unchanged. A change in the external conditions can be modelled via replacing δ by a time-dependent dynamics-generator δt (A) = δ(A) + i[Ht , A] ,

A ∈ D(δ) ,

where Ht = Ht∗ is a smooth function of t ∈ R having values in A. Now assume that Ht = 0 for t < 0 and for t > T , where T is some positive number. Then the dynamics of the system remains unchanged before t = 0 and after t = T . In other words, the system undergoes a cyclic change of external conditions during which it is thermally isolated. (For t < 0 and t > T , the system is closed.) It can be shown that there is a unique smooth family UtH , t ∈ R, of unitary elements in A solving the initial value problem 1 d H Ut = αt (Ht )UtH , U0H = 1 , dt i

(1.6)

and that, consequently, the family αtH , t ∈ R, of automorphisms of A given by αtH (A) = UtH ∗ αt (A)UtH solves the initial value problem d H α (A) = αtH (δt (A)) , α0H (A) = A , dt t for all A ∈ D(δ). We recall that a state on a C ∗ -algebra A is a continuous linear functional ω : A → C which is positive, i.e. ω(A∗ A) ≥ 0 for all A ∈ A, and fulfills ω(1) = 1. If the system is initially in the state ω, then the work done on the system under the cyclic change of external conditions is   
T H H dHt L (ω) = , dt ω αt dt 0 and, as shown in [33], it holds that LH (ω) =

1 ω(UTH ∗ δ(UTH )) . i

(1.7)

Pusz and Woronowicz call a state ω on A passive if LH (ω) ≥ 0 for all smooth H : R  t → Ht = Ht∗ ∈ A that have Ht = 0 for t outside [0, T ] with some T > 0. Then it follows from (1.7) that ω is passive if and only if inf

U ∈U0 (δ)

1 ω(U ∗ δ(U )) ≥ 0, i

where U0 (δ) denotes the set of all unitary U in D(δ) which are in A continuously connected to 1. Passive states may thus be viewed as states of the system which are “in equilibrium” in the sense that, if the system is in such a state, then it is impossible to extract energy (gain

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work) from the system under cyclic changes of the external conditions of the system (while keeping it thermally isolated). It is in this sense that the second law of thermodynamics is valid for passive states. However, the set of passive states is really larger than the class of thermal equilibrium states, as those have a definite temperature: Each mixture of thermal equilibrium states (KMS-states) at arbitrary temperatures (including zero, corresponding to ground states) is a passive state. On the other hand, passive states are invariant under {αt }t∈R , and some (in most circumstances, mild) additional conditions such as clustering, or complete passivity, imply that passive states are, in fact, thermal equilibrium states at a definite temperature, i.e. KMS-states or ground states. We refer to [33] for a detailed discussion of these matters. We should like to mention that, in the context of quantum field theory in Minkowski-spacetime, Kuckert [28] has recently introduced a weaker notion of passivity, called semipassivity, and has shown that for stationary and homogeneous states semipassivity is equivalent to the KMS-condition for a particular inertial observer. There are also more recently investigated interconnections between energy-compactness, semi-passivity, and thermal equilibrium properties of states in quantum field theory [29, 22, 1] to which we would like to direct the reader’s attention. In the present work, we shall show that the existence of passive states for C ∗ -dynamical systems modelled on a static, spatially compact spacetime can be deduced from a suitable form of a QWEI. For this purpose, we suppose that there exists an energy density whose spatial integral yields the generator of the dynamics, and we assume QWEIs for this energy-density to hold for a suitable dense set of states on A. The precise assumptions will be discussed in Sect. 2. Given these assumptions, along with some other physically well-motivated conditions, we obtain in Sect. 2 several assertions about the existence of passive states for the C ∗ -dynamical system. While the investigations in Sect. 2 are based on general, model-independent assumptions, Sect. 4 concerns the example of the free scalar Klein-Gordon field on a static, spatially compact spacetime. In particular, we will show that in the GNS-representation of each quasifree Hadamard state one obtains a C ∗ -dynamical system and an energy density on a suitable domain such that all the general assumptions of Sect. 2 are satisfied. It is of course well-known that quasifree Hadamard ground- and KMS-states exist for the free scalar field on static, spatially compact spacetimes [49], but we emphasise that our arguments do not make use of this fact. Thus there is the prospect that the results of Sect. 2 have a wider applicability beyond the realm of free quantum fields. Instead, our arguments rely heavily on the characterization of the Hadamard property in terms of a condition on the wavefront set of the two-point function of a state for the free field, called microlocal spectrum condition, which was established in [34] (cf. also [3, 38]). However, we shall work with a new version of that characterization taken in part from [40] and related to the concept of “domain of microlocal smoothness” in [2]. Thus, Sect. 4 contains several apparently new microlocal techniques in quantum field theory on curved spacetime. Some background material on microlocal analysis, including a convenient calculus for distributions taking values in Banach and Hilbert spaces is presented in Sect. 3. To conclude this introduction, we would like to point out that the results obtained in this work, together with results previously obtained, indicate an intimate connection – and apart from points of technical detail, equivalence – between the following classes of states of a free quantum field (on static spacetimes): (i) states fulfilling the microlocal spectrum condition (Hadamard states), (ii) states fulfilling QWEIs, (iii) states induced by operations on passive states (corresponding to local, finite energy excitations of passive states) .

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Namely, it was shown in [12] (see also [15]) that free quantum field states fulfilling the microlocal spectrum condition satisfy QWEIs. In the present work, we show that QWEIs imply the existence of passive states. On the other hand, [37] indicates that passive states fulfill the microlocal spectrum condition (and this result carries over to a suitable dense set of states in the GNS-representation of passive states, cf. [45]). This observation is of considerable significance since the regularity properties on the classes of states (i), (ii) and (iii) are imposed on very different length scales. The states (i) fulfill a constraint on their microscopic short-distance behaviour, the condition on the states (ii) is imposed at finite length scales while the specification of the class of states (iii) is a global condition on their thermodynamic stability. We conjecture that this connection between dynamical stability conditions at different length scales is a generic feature and extends to physical systems described by more general types of quantum fields. 2. Passivity from Quantum Weak Energy Inequalities We will consider a quantum system situated in a static spacetime of the form R × , where is an s-dimensional compact manifold; the variable t ∈ R is interpreted as a “time”-variable, whereas contains all spatial locations of the system. Furthermore, we assume that the spacetime is endowed with a static metric of the form (1.3). Then this metric induces a preferred measure on the Borel-sets of which we will denote by i dµ; it is the volume measure of the Riemannian  metric h on . In local coordinates (x ) for , dµ(x) has the coordinate expression det(hij ) d s x. (We note that everywhere in the discussion to follow one could replace dµ by any other Borel-measure on , but the choice just made is convenient and natural.) We shall make some general, model-independent assumptions concerning the mathematical description of the dynamical properties of the system, suited to discuss the connection between QWEIs and passivity. In Sect. 4 we will show that all these assumptions are fulfilled in the case of the free scalar Klein-Gordon field on a static, globally hyperbolic spacetime. Our quantum system will be modelled by a C ∗ -dynamical system (A, {αt }t∈R ) as discussed in the Introduction. Since {αt }t∈R is strongly continuous, there exists a norm-dense (maximal) domain D(δ) in A so that the derivation   d δ(A) = αt (A) dt t=0 exists in A for all A ∈ D(δ). In fact, D(δ) is a norm-dense ∗-subalgebra of A, as is the set D ∞ (δ) of all elements in A such that t → αt (A) is C ∞ at t = 0 (i.e., the common domain for δ n , for all n ∈ N), since defining for each A ∈ A and h ∈ C0∞ (R) their convolution αh A with respect to {αt }t∈R by (1.4) (with the integral now converging in norm because of strong continuity) one finds that αh A ∈ D ∞ (δ). Next, we need to introduce an energy-density in order to connect our dynamical system with QWEIs. Since energy-densities are in many examples unbounded operators or quadratic forms, we need an appropriate domain for such a quantity with convenient algebraic and density properties. Therefore, we shall now make the assumption that there is a norm-dense subspace W∞ ⊂ D ∞ (δ) which has the property of being stable under taking adjoints and under convolution with test-functions with respect to the action of

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{αt }t∈R . In other words, for all A ∈ W∞ and f ∈ C0∞ (R), αf A is also contained in W∞ . We shall denote by A∞ the ∗-algebra generated by W∞ . It will turn out to be convenient to introduce the set U∞ ⊂ D ∞ (δ) of all unitaries U which are of the form U = eiA1 · · · eiAN , where N ∈ N and A1 , . . . , AN are hermitian elements in A∞ . Lemma 2.2 below will show that it suffices to consider U of this form when discussing general cyclic changes to the system such as that described by UtH in Eqs. (1.6) and (1.7). We denote by U∞ alg the ∗-algebra which U∞ generates; this is a sub-∗-algebra of D ∞ (δ). (In fact, as a con∞ sequence of Lemma 2.2 below, the closures of U∞ alg and D (δ) in the graph-norm of δ coincide.) Now we can formulate our assumptions on the existence of an energy density generating the dynamics: We assume that there exists a set S which is a subset of the set of states on A and has the property of being closed under finite convex combinations and ∞ A operations induced by elements of U∞ alg , i.e. if ϕ ∈ S and A ∈ Ualg , then the state ϕ on A defined by ϕ A (B) := ϕ(A∗ BA)/ϕ(A∗ A) ,

B ∈ A,

is also contained in S. We denote by V the vector-space generated by S and U∞ alg , that ∗ is, the subset of all elements  in the continuous dual space A of A that arise as finite linear combinations of elements of S operated upon by elements of U∞ alg from right and left: (B) =

n 

ϕi (Ai BCi )

n ∈ N, Ai , Bi ∈ U∞ alg and ϕi ∈ S .

(2.1)

i=1

The energy density of the system is then defined to be a linear map
taking elements in V to C 1 -functions on R× . That is, given  ∈ V, then
[] is a C 1 -function on R× , and the assignment  →
[] is linear. We will find it convenient to write (
(t, x)) for
[](t, x). Moreover, we will write ([
(t, x), A]) to denote
[A − A ](t, x), where the functionals A and A  are given by A (B) = (BA) ,

A (B)

= (AB) .

The quantity ϕ(
(t, x)) ought to be viewed as the expectation value of the energy-density with respect to the preferred time-coordinate in the state ϕ ∈ S at (t, x). The following assumptions are in line with this point of view: (i) It will be assumed that
d dµ(x)([
(t, x), αt (A)]) = −i (αt (A)) , dt

A ∈ U∞ alg ,  ∈ V .

(ii) It will also be assumed that the integrated generating term of the energy density is conserved:
d dµ(x) ([
(t, x), A]) = 0 , A ∈ U∞ alg ,  ∈ V . dt

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We note that, in the presence of (i), condition (ii) may be equivalently expressed by saying that the generators of the time-evolutions are independent of t, d (αt (A)) = (δ(αt (A))) , dt

A ∈ U∞ alg ,  ∈ V .

With the assumptions formulated so far, the notion of a quantum weak energy inequality in the present general setting can now be precisely defined. Definition 2.1. Assume that a physical system on the static spacetime R × modelled by A, {αt }t∈R , W∞ , A∞ , S and
with the properties stated above is given, and let ω ∈ S. (a) We say that ω fulfills a static quantum weak energy inequality with respect to S if there is for each real-valued g ∈ C0∞ (R) a locally integrable4 non-negative function  x → q(g; x) such that

dt g 2 (t) ϕ(
(t, x)) − dt g 2 (t)ω(
(t, x)) ≥ −q(g; x) (2.2) R

R

holds for all states ϕ ∈ S and all x ∈ . (b) We say that ω fulfills a limiting static QWEI (with respect to S) if ω fulfills the static QWEI (2.2), and in addition each x ∈ has an open neighbourhood U such that
1 dµ(x ) q(gλ ; x ) < ∞, (2.3) U := sup lim sup 2 g λ→0+ gλ L1 U where the supremum is taken over real-valued g ∈ C0∞ (R) with g 2 L1 = 0, and gλ (t) = g(λt). (c) A state ω ∈ S will be called quiescent (with respect to S) if ω fulfills a limiting static QWEI in which each U = 0. (Since q(g, x) is non-negative this is in fact equivalent to the assertion that each x ∈ has an open neighbourhood U such that
dµ(x ) q(gλ ; x ) = 0 lim λ λ→0+

for each real-valued g ∈

U

C0∞ (R).)

Remarks. (i) It is easy to see that if there exists a state ω ∈ S fulfilling a static QWEI with respect to S, then all states ω ∈ S satisfy a static QWEI with respect to S as well. Therefore, it is actually more appropriate to say that the set of states S fulfills a static QWEI. Within the class of states S, the condition of static QWEI is thus independent of the individually chosen state. In contrast, conditions (b) and (c) are state-dependent. (ii) The element x ∈ appearing in q(g; x) ought to viewed as a label for the timelike curve γx (t) = τt ((0, x)), i.e. the orbit of the point (0, x) ∈ M (identified with x ∈ ) under the one-parametric group of time-shifts. Thus q(g; x) (or rather,  q(g; ˜ x) = q(g; x) + dt g 2 (t)ω(
(t, x))) in (2.2) plays exactly the role of cg,γx in (1.1). 4

That is, each x ∈ should have an open neighbourhood on which q(g; ·) is integrable.

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(iii) The condition (a) given above is a purely local statement about the dynamical system, and (b) and (c) are limits thereof which are still “spatially local”. However, since is compact and q(g; ·) is non-negative, one may easily draw global consequences: if a static QWEI holds, then q(g; ·) ∈ L1 ( , dµ), while if a limiting static QWEI holds then (2.3) also holds for U = , with  = 0 in the quiescent case. (Conversely, these global properties of course imply the local statements given above.) It is these global statements which will appear in the arguments presented below. (iv) It will be shown in Sect. 4 that any state for the scalar Klein-Gordon field on a static, globally hyperbolic spacetime possessing a two-point function that is stationary and of Hadamard form fulfills a limiting static QWEI with
q(g; x) = du | g (u)|2 Q(u, x), R

where  g denotes the Fourier-transform of g, and Q is a non-negative measurable function on R × so that Q(u, x) is polynomially bounded in u for each fixed x ∈ . Moreover, Q(u, x) is dµ-integrable over with respect to x for each  u ∈ R, and dµ(x) Q(u, x) is polynomially bounded in u. This then implies that all states of the scalar Klein-Gordon field on a stationary, globally hyperbolic spacetime whose two-point functions are of Hadamard form (or, synonymously, satisfy the microlocal spectrum condition) fulfill a static QWEI. We have now collected all the assumptions relevant for the present section. Before presenting our results on passivity properties of states satisfying static weak energy inequalities for systems obeying the assumptions given above, we put on record an auxiliary lemma. Recall (cf. Introduction) that U0 (δ) denotes the set of all unitaries in D(δ) which are continuously connected to the unit 1, i.e. those unitaries U in D(δ) so that there is a norm-continuous curve [0, 1]  t → U (t) of unitaries in A so that U (0) = 1, U (1) = U . Lemma 2.2. U∞ is dense in U0 (δ) with respect to the graph-norm of δ; i.e. for each U ∈ U0 (δ) there is a sequence Un ∈ U∞ , n ∈ N, with || Un − U || + || δ(Un ) − δ(U ) || → 0 as n → ∞ . Proof. Let A be a hermitian element in D(δ). We show that there is a sequence of hermitian elements in A∞ approximating A in the graph-norm of δ. To this end, pick  > 0 arbitrarily. Then choose some real-valued, non-zero f ∈ C0∞ (R) so that ||αf A − A|| + ||αf δ(A) − δ(A)|| < /2 . Because W∞ ⊂ A∞ is norm-dense in D(δ) and stable under taking adjoints, one may also choose some hermitian A ∈ W∞ so that  ||A − A|| < (||f ||L1 + ||f˙||L1 )−1 , 2 where f˙ is the derivative of f . Then one estimates ||αf A − A|| + ||δ(af A ) − δ(A)|| ≤ ||αf A − αf A|| + ||αf A − A|| + ||δ(αf A ) − δ(αf A)|| + ||αf δ(A) − δ(A)|| ≤ /2 + ||f ||L1 ||A − A|| + ||f˙||L1 ||A − A|| ≤  ,

Stability of Quantum Systems at Three Scales

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by making use of αf δ(B) = δ(αf B) = −αf˙ (B) for all B ∈ D(δ) and ||αg B|| ≤ ||g||L1 ||B|| for g ∈ C0∞ (R). As W∞ is by assumption stable under convolution with test-functions with respect to the dynamical automorphism group, this shows that each hermitian A ∈ D(δ) can be approximated by a sequence of hermitian elements in W∞ ⊂ A∞ in the graph-norm of δ. As argued in the bottom part of p. 279 in [33] (cf. also Thm. 5.4.28 in [7]), for each U ∈ U0 (δ) there exist finitely many hermitian elements A1 , . . . , AN ∈ D(δ) with ||Aj || < π (j = 1, . . . , N) so that U = eiA1 · · · eiAN . (n)

Since each Aj may be approximated by a sequence of hermitian elements Aj , n ∈ N, (n) iA1

(n) AN

···e approximates U in the in A∞ , it is quite easy to see that also Un = e graph-norm of δ, and each Un is contained in U∞ . This proves the lemma.   Now, with the notation and assumptions introduced prior to Lemma 2.2, our first result reads as follows. Theorem 2.3. Suppose that a state ω ∈ S satisfies a static QWEI. Then there exists a state ωp on A which is passive for the automorphism group {αt }t∈R , i.e. it fulfills 1 p ∗ ω (U δ(U )) ≥ 0 i for all the unitaries U in U0 (δ). Moreover, if ω is a quiescent state, then it is a passive state for the automorphism group {αt }t∈R . The proof of the first part of that result is based on a simple Lemma 2.4. Let ω be a state on A so that cω := inf

U ∈U∞

1 ω(U ∗ δ(U )) > −∞ , i

then there exists a state ωp on A which is passive for {αt }t∈R . Proof of Lemma 2.4. If cω ≥ 0, then ω is already passive. Therefore, we assume that cω ∈ n ∈ U∞ , n ∈ N, so that 1 limn→∞ ω(U n∗ δ(U n )) = (−∞, 0). There exists a sequence U i ∗ n AU n ) possesses, cω . The sequence of states ωn , n ∈ N, on A given by ωn (A) = ω(U by the Banach-Alaoglu-theorem [35], weak-* limit points, i.e. there is a state ωp on A and a subnet {ωn(σ ) }σ ∈S of {ωn }n∈N so that limσ ωn(σ ) (A) = ωp (A) for all A ∈ A. σ , we obtain for all U ∈ U∞ , n(σ ) by U Abbreviating U 1 1 p ∗ σ∗ U ∗ δ(U )U σ ) ω (U δ(U )) = lim ω(U i i σ

 1 σ )∗ δ(U U σ )) − ω(U σ∗ δ(U σ )) = lim ω((U U i σ 1 σ )) − 1 lim ω(U σ∗ δ(U σ )) σ )∗ δ(U U ≥ lim inf ω((U U σ i i σ ≥ cω − cω = 0 .

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In view of Lemma 2.2, this relation entails that inf

U ∈U0 (δ)

showing that ωp is passive.

1 p ∗ ω (U δ(U )) ≥ 0 , i

 

Proof of Thm. 2.3. In view of conditions (i) and (ii) one obtains for all U ∈ U∞ and any real g ∈ C0∞ (R) with ||g 2 ||L1 > 0,
1 ∗ dµ(x) ω(U ∗ [
(t, x), U ]) ω(U δ(U )) = i



 1 2 = dt g (t) dµ(x) ω(U ∗
(t, x)U ) − ω(
(t, x)) 2 ||g ||L1 R



 1 = dµ(x) dt g 2 (t) ω(U ∗
(t, x)U ) − ω(
(t, x)) 2 ||g ||L1 R
−1 ≥ dµ(x)q(g; x) . ||g 2 ||L1 If ω is quiescent, we replace g by gλ in the last term and take the limit as λ → 0+ in order to conclude that ω is passive by Lemma 2.2. Otherwise, we nonetheless have inf U ∈U∞ i −1 ω(U ∗ δ(U )) > −∞; the existence of the passive state ωp follows from Lemma 2.4.   It is shown in [33] that if ω is a passive state which is weakly clustering in time (and non-central) then ω is a ground state or a KMS-state at positive inverse temperature for the time-evolution {αt }t∈R . Here we add another observation, assuming that a state fulfilling a limiting static QWEI is weakly clustering in time for the integrated energy density. We denote by
dµ(x) (
(t, x)) ,  ∈ V , (
(t)) =

the integrated energy density at time t in the functional , and we will say that a state ω ∈ S is weakly clustering in time for the integrated energy density if for every B ∈ U∞ alg there exists a sequence{λn } of positive real numbers converging to 0 and some real-valued g ∈ C0∞ (R) with dt gn2 (t) = 1 so that
dt λn g 2 (λn t){ω(B
(t)) − ω(B)ω(
(t))} → 0 as n → ∞ . (2.4) R

We need another piece of notation. Recall that any state ω on a C ∗ -algebra A induces the so-called GNS-representation (H, π, ) consisting of a Hilbert-space H, a linear ∗-representation π of A by bounded linear operators on H, and a unit vector  which is cyclic for π(A) in the sense that π(A) is dense in H, and which fulfills ω(A) =
, π(A) for all A ∈ A. If in addition ω is invariant with respect to the automorphism group {αt }t∈R , ω ◦ αt = ω, then there is a strongly continuous unitary group Vt , t ∈ R, on H which implements the action of αt , t ∈ R, in the GNS-representation: Vt π(A)Vt∗ = π(αt (A)), A ∈ A; moreover, Vt  =  for all t (see, e.g. [6] for a thorough discussion of these matters). The selfadjoint operator H in H with Vt = eiH t is called the generator of the implementing unitary group.

Stability of Quantum Systems at Three Scales

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Theorem 2.5. Let ω ∈ S be a state which is weakly clustering in time for the integrated energy density. (a) If ω is a quiescent state, then it follows that ω is a ground state for {αt }t∈R . (b) If ω is invariant with respect to the automorphism group {αt }t∈R and fulfills a limiting static QWEI, then the generator H of the implementing unitary group is bounded below by − , defined by Eq. (2.3) [see also remark (iii) following Def. 2.1]. ∗ Proof. Choose any A ∈ U∞ alg with ω(A A) = 1 and let {λn } be a sequence of positive real numbers converging to 0 as well as g ∈ C0∞ (R) with ||g 2 ||L1 = 1 such that (2.4) √ holds with B = A∗ A. Then, setting gn (t) = λn g(λn t) it follows that
n := dt gn2 (t){ω(
(t)) − ω(A∗ A
(t))} → 0 as n → ∞ .

R

Recalling that ω(A∗ [
(t), A]) is independent of t, there results the following chain of relations: 1 ω(A∗ δ(A)) = ω(A∗
(t)A) − ω(A∗ A
(t)) i
= dt gn2 (t){ω(A∗
(t)A) − ω(A∗ A
(t))}
R = dt gn2 (t){ω(A∗
(t)A) − ω(
(t))} + n R
q(gλn ; x) ≥− dµ(x) + n . ||gλ2n ||L1 In the limit n → ∞, this gives 1 ω(A∗ δ(A)) ≥ − lim sup i λ→0+


dµ(x)

q(gλ ; x) ≥ − , gλ2 L1

(2.5)

since in either case ω obeys a limiting static QWEI. Case (a). As ω is assumed to be quiescent,  = 0, so 1 ω(A∗ δ(A)) ≥ 0 i ∞ ∗ holds for all A ∈ U∞ alg with ω(A A) = 1, hence also for all A ∈ Ualg . The proof of ∞ Lemma 2.2 shows that the closure of Ualg in the graph-norm of δ contains D(δ), and thus ω is a ground state.

Case (b). In view of the definition of  , Eq. (2.5) entails
A, H A =

1 ω(A∗ δ(A)) ≥ − i

2 2 for all A ∈ U∞ alg with ||A|| = 1, and hence also for all A ∈ D(δ) with ||A|| = 1. Since D(δ) contains a core for H , it follows that H is bounded below by − .  

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Theorem 2.3 shows that the existence of passive states for the given system may be deduced from the existence of states satisfying a static quantum weak energy inequality; however, this result leaves room for further sharpening. In particular, one would like to know if the passive state ωp is normal to the state ω assumed to fulfill a static quantum weak energy inequality. It turns out that such an assertion can be made under the assumption that ω satisfies a certain condition of “energy compactness”. Let us fix the required assumptions in detail. We assume that we are given a Hilbert-space H and a strongly continuous unitary group {Vt }t∈R on H with selfadjoint generator H , i.e. Vt = eitH . Furthermore, we assume that there is a C ∗ -subalgebra A of L(H) which contains the unit operator 1 and is left invariant under the automorphism group αt := AdVt , t ∈ R. In addition, it will be assumed that {αt }t∈R acts strongly continuously on the elements of A, and that, for each E ≥ 0, the spectral projector PE of H corresponding to the spectral interval [−E, E], is also contained in A. With these conventions, the definitions of δ, D(δ), U∞ and U∞ alg , etc., are as before. A vector  ∈ H will be said to be energy-compact if for each finite E > 0 the set PE A(1)  is a pre-compact subset of H. Here A(1) is the set of all elements in A whose norm is bounded by 1, and we recall that a subset J of H is called pre-compact if each sequence {χn }n∈N in J possesses a sub-sequence {χn(k) }k∈N which converges strongly to some element χ ∈ H, i.e. ||χn(k) − χ || → 0 as k → ∞. Energy-compactness conditions were introduced by Haag and Swieca in quantum field theory [24]; they impose restrictions on the energy-level density of quantum states. The original approach of Haag and Swieca has been considerably extended and refined to so-called “nuclearity conditions” in quantum field theory in a series of works by Buchholz and collaborators. It is interesting to note that there is a close connection between such nuclearity conditions and decent thermodynamical properties of quantum field systems. In fact, this was one of the central motivations for the introduction of nuclearity conditions in [4]. We recommend [23] and [4, 5, 41, 42] for further discussions and references on that subject, and we would like to refer also to [1, 22, 28, 29] for some recent developments. Here, we just mention that related energy-compactness conditions have been shown to hold for energy-ground states (vacuum states) in several quantum field theoretical models (see the quoted works, and references quoted therein); notably, they have also been established for linear quantum field theories on ultrastatic curved spacetimes [8, 44]. The assumptions listed above now lead to Theorem 2.6. Let  be a unit vector in H which is energy-compact, and let ω(A) :=
, A be the vector-state on A induced by . If cω :=

inf

U ∈U0 (δ)

1 ω(U ∗ δ(U )) > −∞ , i

then there exists a unit vector p ∈ A so that the vector-state ωp induced by p is passive. n∗ δ(U n )) = cω . n , n ∈ N, in U∞ so that limn→∞ 1 ω(U Proof. There is a sequence U i Since Un , n ∈ N, is a bounded sequence in H and possesses a weakly converging sub n  = p for some unit sequence, it is no loss of generality to assume that w-limn→∞ U p vector  ∈ H. Now let A1 , . . . , AN be finitely many hermitian elements in D ∞ (δ), define U as in the proof of Lemma 2.2, and write UE := eiPE A1 PE · · · eiPE AN PE ,

E > 0.

Stability of Quantum Systems at Three Scales

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Then it holds that UE = 1 + GE , where ||GE || ≤ 2 and GE = PE GE PE . Introducing the bounded operator FE := (1 + G∗E )[H, GE ] = −iUE∗ δ(UE ), we note that FE = PE FE PE . Moreover, by assumption PE A(1)  is a pre-compact set, and there n  ∈ PE A(1)  possesses a subsequence PE U n(k)  fore the sequence of vectors PE U p converging strongly to PE  (since Un  is already known to converge weakly to p ). This allows us to write (with ωp ( · ) =
p , · p ) 1 p ∗ ω (UE δ(UE )) = ωp (FE ) = ωp (PE FE PE ) i n(k) , FE PE U n(k)  = lim
PE U k→∞

n(k)  = lim 1 ω(U ∗ UE∗ δ(UE )U n(k) , FE U n(k) ) = lim
U n(k) k→∞ k→∞ i 1 ∗ δ(U n(k) ))} n(k) )∗ δ(UE U n(k) )) − ω(U lim {ω((UE U = n(k) i k→∞ 1 n(k) )) n(k) )∗ δ(UE U n(k) )) − lim 1 ω(U ∗ δ(U ≥ lim inf ω((UE U n(k) k→∞ i k→∞ i ≥ cω − cω = 0 . This shows that 1i ωp (UE∗ δ(UE )) ≥ 0 for all E > 0, and it is easy to check that also limE→∞ ωp (UE∗ δ(UE )) = ωp (U ∗ δ(U )). Therefore we conclude that inf

U ∈U0 (δ)

proving passivity of ωp .

1 p ∗ ω (U δ(U )) ≥ 0 , i

 

Under the assumptions of Thm. 2.6 one finds, by combining it with Thm. 2.3, the following Corollary 2.7. Let  be a unit vector in H which is energy compact, and suppose that the corresponding vector state ω fulfills a static quantum weak energy inequality. Then there exists a unit vector in A inducing a passive state for {αt }t∈R . 3. Some Techniques from Microlocal Analysis In the next section, we will show that the real scalar field on a static spacetime is a system satisfying the assumptions made in the previous section and to which our results therefore apply. Our principal tools will be drawn from microlocal analysis [25, 26], which provides powerful and geometrically natural techniques for dealing with the singular structure of distributions. We now proceed to describe the microlocal analysis used in Sect. 4 and the Appendix, adopting a slightly more intrinsic approach than usual, which avoids the explicit introduction of coordinates. Let X be a smooth manifold and denote by T˙ ∗ X = T ∗ X\Z the cotangent bundle of X with its zero section Z removed. Given a distribution u ∈ D (X), an element (x, k) ∈ T˙ ∗ X is called a regular directed point for u if there exists a set O and a map φ obeying5 5 Condition (A) below permits the introduction of coordinates on O by y µ =
ζ µ , φ(y), where ζ µ is any fixed basis for Tx∗ X, but such coordinates will rarely be necessary in our discussion.

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(A) O is an open neighbourhood of x, and φ : O → Tx X is a smooth map with nondegenerate tangent mapping T φ obeying T φ(x) = idTx X , and such that (B) there exists χ ∈ C0∞ (O) with χ (x) = 0 and a neighbourhood E of k in Tx∗ X such that     as λ → +∞ for each N ∈ N0 . (3.1) λN sup u(χ eiλ
,φ ) → 0 ∈E

The quantity inside the modulus signs can be interpreted as a local Fourier transform6 of χu evaluated at λ: To see this, it may help to write this quantity as
a u(χ eiλ
,φ ) = dvol(x)χ (x)u(x)eiλa φ (x) , where u(x) is the distributional kernel of u with respect to some volume measure dvol on X. As is well known, the Fourier transform of any smooth compactly supported function decays rapidly at infinity, so every (x, k) ∈ T˙ ∗ X is regular directed for any distribution which may be identified with a smooth function. It may also be shown that if (x, k) is a regular directed point for u then condition (B) will be satisfied for any pair (O, φ) obeying condition (A). Moreover, if condition (B) holds, it continues to hold if χ is replaced by ψχ for any ψ ∈ C0∞ (X). We are now in a position to define the central object of the theory. Definition 3.1. The wave-front set WF(u) of a distribution u ∈ D(X) is the complement in T˙ ∗ X of the set of regular directed points for u. Two important facts will be used extensively in this work: first, that the wave-front set of a distribution u is empty if and only if u can be identified with a smooth function; second, if P is a partial differential operator with smooth coefficients then WF(P u) ⊂ WF(u). Microlocal techniques have recently found many applications in the theory of quantum fields on curved spacetimes following Radzikowski’s discovery [34] that the Hadamard condition (see Sect. 4.2) can be reformulated as a condition on the wave-front set of the two-point function of the field. Recently, this criterion has been simplified by Strohmaier, Wollenberg and Verch [40], who consider a generalisation of the wave-front set to distributions taking values in a Hilbert space. Generalising this slightly further, if (B,  · ) is a Banach space, let D (X, B) be the space of distributions on X taking values in B, i.e., linear functionals T : D(X) → B such that f → 0 in D(X) implies T (f ) → 0. Then the wave-front set of T may be defined as for scalar valued distributions, but with the Banach space norm replacing the modulus signs in (3.1). With this definition, a convenient calculus may be constructed as follows. Proposition 3.2. Let X be a C ∞ -manifold, (H,
·, ·) a Hilbert space and (Bi ,  · i ) be Banach spaces (i = 1, 2). (i) If T ∈ D (X, B1 ) and S : D(X) → B2 is a linear map obeying S(f )2 ≤ cT (f )1 , for some c ≥ 0, then S ∈

D (X, B2 )

f ∈ D(X)

and WF(S) ⊂ WF(T ).

Throughout this paper, we adopt the nonstandard convention f(k) = transform on Rn . 6

 n d x f (x)eik·x for the Fourier

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(ii) If T ∈ D (X, H) and ψ ∈ H then f →
ψ, T (f ) and f →
T (f ), ψ define scalar distributions in D (X) with WF(
ψ, T ( · )) ⊂ WF(T ( · )) and WF(
T ( · ), ψ) ⊂ WF(T ( · ))† , where, for any  ⊂ T ∗ X,  † = {(x, k) ∈ T ∗ X | (x, −k) ∈ }. (iii) If S, T ∈ D (X, H) then U : (f, g) →
S(f ), T (g) defines U ∈ D (X × X) with

WF(U ) ⊂ WF(S)† ∪ Z × (Z ∪ WF(T )) . Proof. (i) The bound S(f )2 ≤ cT (f )1 implies that S ∈ D (X, B2 ) and moreover that any regular directed point for T is a regular directed point for S. The result follows on taking complements. (ii) The statements regarding f →
ψ, T (f ) follow immediately from (i) and the Cauchy-Schwarz inequality. To study f →
T (f ), ψ, it is convenient to prove an auxiliary result first. Lemma 3.3. Suppose B is a Banach space equipped with a conjugation  and that S ∈ D (X, B). Defining S † (f ) = S(f ) ,

f ∈ D(X) ,

we have S † ∈ D (X, B) and WF(S † ) = WF(S)† . Proof. Since S † (f ) = S(f ) and f → 0 if and only if f → 0, we have S † ∈ D (X, B). Furthermore, it is easy to see that (x, k) is a regular direction for S, (x, −k) is a regular direction for S † , so WF(S † ) ⊂ WF(S)† ; since (S † )† = S, we must in fact have equality.   Now any Hilbert space admits a conjugation, and the Cauchy-Schwarz inequality gives |
T (f ), ψ| ≤ ψ T (f ) = ψ T † (f ) , so we apply (i) and Lemma 3.3 to complete the proof of (ii). (iii) Applying Cauchy-Schwarz, |U (f, g)| ≤ S(f ) T (g) = S(f )†  T (g) = (S † ⊗ T )(f, g) , where S † ⊗ T ∈ D (X × X, H ⊗ H) is defined by (S † ⊗ T )(f, g) = S † (f ) ⊗ T (g). The same arguments which bound the wave-front set of a tensor product of scalar distributions may be used to show that

WF(S † ⊗ T ) ⊂ WF(S † ) ∪ Z × (Z ∪ WF(T )) [actually, there is a tighter bound than this]; using (i) and Lemma 3.3 the required result is obtained.  

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A further important property required below is the behaviour of distributions under the pull-back operation. Let X1 and X2 be smooth manifolds and let χ : X1 → X2 be a C ∞ -map. To this map one can associate its conormal bundle Nχ ⊂ T ∗ X2 where, by definition, (y, η) is in Nχ if and only if there is x ∈ X1 with y = χ (x) and t T χ (x)η = 0, t T χ(x) denoting the transpose of the tangent map of χ at x. If F : X2 → C is a smooth map, one obtains via the pull-back by χ a smooth map χ ∗ F = F ◦ χ : X1 → C. It can be shown (cf. Thm. 8.2.4 in [25]) that one can (uniquely) extend the pull-back operation to distributions u ∈ D (X2 ) – through approximating distributions by test functions – provided that Nχ ∩ WF(u) = ∅ . In this case, the pull-back χ ∗ u of u by χ has the property WF(χ ∗ u) = χ ∗WF(u) , where for any subset V ⊂ T ∗ X2 , χ ∗ V is the subset of T ∗ X1 defined as follows: χ ∗ V = {(x, t T χ (x)η) : (f (x), η) ∈ V } . Moreover (see again Thm. 8.2.4 in [25]), χ ∗ induces a continuous linear map from DV (X2 ) into Dχ ∗ V (X1 ) if V ∩ Nχ = ∅. Here, for each closed conic subset V ⊂ T ∗ X2 , the set DV (X2 ) is a linear subspace of the distribution space D (X2 ) which is defined as DV (X2 ) = {u ∈ D (X2 ) : WF(u) ⊂ V }. As we will discuss below, there is a notion of convergence in DV (X2 ) with respect to which DV (X2 ) is a closed subset of D (X2 ) (cf. also Def. 8.2.2 in [25]). This notion of convergence is sometimes referred to as the “H¨ormander pseudo-topology” of DV (X2 ) and it is this sense in which χ ∗ is continuous, Dχ ∗ V (X1 ) being analogously defined. Convergence in the H¨ormander pseudo-topology may be defined as follows. Suppose ur is a sequence in DV (X) and u ∈ DV (X). Then ur → u in DV (X) if the two following conditions hold: (i) ur → u weakly in D (X) (i.e., ur (f ) → u(f ) for each test function f ) (ii) for all (x, k) ∈ T˙ ∗ X\V , there exists (O, φ) obeying condition (A) above, χ ∈ C0∞ (O) with χ (x) = 0 and a neighbourhood E of k in Tx∗ X obeying  ∗  φ (φ(suppχ ) × E) ∩ V = ∅ and such that the quantities sup sup λN |ur (χ eiλ
,φ )|

λ∈R+ ∈E

are uniformly bounded in r for each N = 1, 2, . . . . Any distribution u ∈ D (X) may be arbitrarily well approximated in DV (X), for any V ⊃ WF(u), by a sequence of test functions ur ∈ D(X); it is this which permits the definition of pull-backs to be extended to distributions by continuity from the definition for functions. The notion of H¨ormander pseudo-topology is easily extended to distributions taking values in a Banach space B simply by replacing modulus signs with Banach norms where appropriate, and denoting by DV (X, B) the set of distributions in D (X, B) whose wave-front sets are contained in V .

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It is useful to have some simpler sufficient conditions for convergence in D V (X, B). It is not hard to show that, for example, if ur is a sequence converging weakly to u in D (X, B) and ur (f ) ≤ v(f ) for all r and some v ∈ D (X, B) then ur → u in

DWF(v) (X). The following is a slight elaboration of this observation. Proposition 3.4. Let X and Y be C ∞ -manifolds and (Bi ,  · i ) be Banach spaces (i = 1, 2). Suppose that ur is a sequence in D (X × Y, B1 ) converging weakly to u. Suppose further that there exists v ∈ D (X × Y, B2 ) such that ur (f, g)1 ≤ v(f, g)2

(3.2)

for all f ∈ D(X) and g ∈ D(Y ). Then ur → u in the H¨ormander pseudo-topology on

DWF(v) (X × Y ). Remark. The result continues to hold if Eq. (3.2) is generalised to ur (f, g)1 ≤

n 

vi (f, g)i

i=1

for Banach spaces (Bi ,  · i ) and distributions vi ∈ D (X × Y, Bi ), (i = 1, . . . , n) by applying the proposition to v = v1 ⊕ v2 ⊕ · · · ⊕ vn ∈ D (X × Y, ni=1 Bi ). Proof. Suppose (x0 , y0 ; k0 , l0 ) is a regular directed point for v. Let (OX , φX ) and (OY , φY ) obey condition (A) above for points x0 ∈ X and y0 ∈ Y , and define φ(x, y) = φX (x) ⊕ φY (y) ∈ Tx0 X ⊕ Ty0 Y ∼ = T(x0 ,y0 ) (X × Y ) . Then (OX × OY , φ) obeys condition (A) for (x0 , y0 ); furthermore, by continuity of the pull-back φ ∗ and the fact that WF(v) is closed, there exists open O ⊂ OX × OY and a neighbourhood G of (k0 , l0 ) in T(x∗ 0 ,y0 ) X × Y such that 

 φ ∗ (φ(O) × G) ∩ WF(v) = ∅ .

Since (x0 , y0 ; k0 , l0 ) is regular directed for v, there exists χ ∈ C0∞ (O) with χ (x0 , y0 ) = 0 and a neighbourhood E of (k0 , l0 ), contained (without loss of generality) in G such that 

   as λ → +∞ (3.3) λN sup v χ eiλ
(k,l),φ  → 0 (k,l)∈E

2

for each N . Choose smooth functions ηX and ηY such that η(x, y) = ηX (x)ηY (y) is compactly supported in the interior of suppχ , with η(x0 , y0 ) = 0. Then  η(x, y)/χ (x, y) (x, y) ∈ suppη ψ(x, y) = 0 otherwise is smooth and compactly supported and Eq. (3.3) continues to hold if χ is replaced by ψχ = η, thereby yielding 

   as λ → +∞ λN sup v ηX eiλ
k,φX  , ηY eiλ
l,φY   → 0 (k,l)∈E

2

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for each N. Together with Eq. (3.2) this immediately implies that (x0 , y0 ; k0 , l0 ) is a reg

ular directed point for each ur and u. It follows that ur and u belong to DWF (X, B1 ). (v) Furthermore,   

     sup sup λN ur (χ eiλ
,φ ) ≤ sup sup λN v ηX eiλ
k,φX  , ηY eiλ
l,φY   , λ∈R+ (k,l)∈E

1

λ∈R+ (k,l)∈E

2

the right-hand side of which is easily seen to be finite. This provides the required uniform

bound to ensure that ur → u in DWF(v) (X, B1 ).  

4. Quantum Fields on Static Backgrounds We will now describe how the structural assumptions made in Sect. 2 may be justified for the case of real scalar field theory on a globally hyperbolic static spacetime (M, g) with compact spatial sections. The assumptions to be checked are: • the existence of a C ∗ -dynamical system along with a suitable sub-∗-algebra A∞ and a generating linear space W∞ which is stable under convolutions; • the identification of a set of states S closed under finite convex combinations and ∞ operations induced by elements of the algebra U∞ alg constructed from A ; • the existence of an energy density [defined for every state in S] whose spatial integral generates the dynamics; • the existence of states satisfying a suitable static QWEI. Each assumption will be treated in turn in the following subsections. Most details are postponed to the Appendix. It is worth mentioning that we will also prove a converse to Thm. 2.5(a) for the free scalar field: namely, we will show in Thm. 4.8 that a non-degenerate ground state with mass gap and vanishing one-point functions is necessarily quiescent. 4.1. The Dynamical System. We begin by reviewing the quantisation of the real scalar field on a globally hyperbolic static spacetime (M, g). Such a spacetime is diffeomorphic to R × with line element ds 2 = gab dx a dx b = g00 dt 2 − hij dx i dx j , where h is a (positive definite) Riemannian metric and g00 is a smooth strictly positive function on . As before, we will assume that is s-dimensional and compact; the √ preferred measure on is dµ(x) = hd s x, where h = det h. The Killing vector ∂/∂t a (µ = 0, . . . , s) with will be denoted ξ . We will also introduce an orthonormal frame eµ −1/2

e0a = g00 ξ a . The Klein–Gordon equation on (M, g) is

g ab ∇a ∇b + m2 ϕ = 0 , for which the corresponding classical stress-energy tensor is 1 1 Tab = ∇a ϕ∇b ϕ − gab g cd ∇c ϕ∇d ϕ + m2 gab ϕ 2 . 2 2

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Integrating over a surface of constant “time” {t} × (t ∈ R) we obtain the classical energy
H = Tab (t, x)na ξ b dµ(x) , −1/2 g00 ξ a

where = = is the future-pointing unit normal to ; this is conserved by virtue of the Klein–Gordon equation and Gauss’ theorem. In addition, the classical energy density seen by an observer with velocity e0a is   s  1 a e0a e0b Tab = m2 ϕ 2 + (4.1) (eµ ∇a ϕ)2  . 2 na

e0a

µ=0

The quantisation of this system proceeds as follows (cf. [9]): first, let (S, σ ) be the symplectic space of smooth real-valued Klein–Gordon solutions where the symplectic form is given by


 σ (u, v) = dµ(x) ue0a ∇a v − ve0a ∇a u .

The CCR-algebra (or Weyl algebra) A[S, σ ] over (S, σ ) is the (unique up to C ∗ -isomorphism) unital C ∗ -algebra generated over C by unitary elements W(u) (u ∈ S) with W(0) = 1 subject to the Weyl relations W(u)W(v) = e−iσ (u,v)/2 W(u + v) ,

u, v ∈ S .

For each open relatively compact O ⊂ M, let A(O) be the sub-C ∗ -algebra of A[S, σ ] generated by elements of the form W(Ef ), where f ∈ C0∞ (O; R) and E : C0∞ (M) → C ∞ (M) is the advanced-minus-retarded fundamental solution for the Klein–Gordon equation. Then O → A(O) is an isotonous net of C ∗ -algebras which is also local in the sense that A1 A2 = A2 A1 holds for all A1 ∈ A(O1 ) and A2 ∈ A(O2 ) whenever the regions O1 and O2 in M cannot be connected by any causal curve. Hence, the net O → A(O) is the essential building block of a local quantum field theory on the curved spacetime (M, g) (cf. [23, 48]). Since the time translations τt (t , x) = (t + t , x) induce a symplectomorphism of (S, σ ) there is a 1-parameter group of ∗-automorphisms on A[S, σ ] given by α˜ t (W(u)) = W(τt ∗ u) ,

u ∈ S, t ∈ R,

where τt ∗ u = u ◦ τt−1 is the push-forward.7 The C ∗ -algebraic net O → A(O) then has the covariance property [9] α˜ t (A(O)) = A(τt (O)) . However, {α˜ t }t∈R is not strongly continuous (because W(u) − W(v) = 2 for all u = v). This obstacle can be circumvented as we shall explain, but let us first give a brief description of how we proceed. We will, in the following subsection, work in GNS-representations of quasifree Hadamard states which may be regarded as states in quantum field theory on curved spacetimes whose short-distance behaviour is close to that of vacuum states or thermal equilibrium states. We shall denote by SqH the set of all quasifree Hadamard states on A[S, σ ], and starting from this class we shall define the 7

∗ u = u ◦ (τ ) = u ◦ τ −1 = τ u. The push-forward τt ∗ and pull-back τt∗ are related by τ−t −t t∗ t

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underlying C ∗ -dynamical system, the sub-∗-algebra A∞ and the set of states S. In the remainder of this subsection we will discuss the C ∗ -dynamical system, and the algebra A∞ . Consider a state ω on A[S, σ ] and let (Hω , πω , ω ) be its GNS-representation. Then we call such a state weakly covariant if there exists on the von Neumann algebra (ω) Mω = πω (A[S, σ ])

a one-parameter group {αt }t∈R of automorphisms (leaving Mω invariant) so that (ω)

αt

◦ πω (A) = πω ◦ α˜ t (A) ,

t ∈ R, A ∈ A[S, σ ] .

(4.2)

A special case is a covariant state ω, where αt (A) = Vt AVt ∗ with a strongly (ω) (ω) continuous unitary group {Vt }t∈R on Hω . Then (Mω , {αt }t∈R ) is a W ∗ -dynamical system (cf. [6]), but we need to pass to a C ∗ -dynamical system whose definition we now describe, following the strategy outlined in the Introduction. Consider the operators αf A (ω) defined by (1.4) with αt ≡ αt for f ∈ C0∞ (R) and A ∈ πω (A[S, σ ]), where the integral is understood in the weak topology, so αf A ∈ Mω . It is straightforward to check that (ω) (ω) ||αf A|| ≤ ||f ||L1 ||A|| and αt (αf A) = αf ( . −t) A, so that ||αt (αf A) − αf A|| → 0 ∗ for t → 0. Now we define Aω ⊂ Mω as the C -closure of the ∗-algebra generated by (ω) (ω) all these αf A; then (Aω , {αt }t∈R ) is a C ∗ -dynamical system (where αt should here (ω) strictly be read as αt
Aω ). Now this dynamical system depends on the chosen state ω, but for our discussion, this dependence is spurious, as we shall explain. As mentioned above, we denote by SqH the set of quasifree Hadamard states on A[S, σ ] (see next subsection for a definition). At the present stage of discussion it is important to know that, whenever ω1 and ω2 are contained in SqH , then ω1 and ω2 are quasi-equivalent [43] (because of spatial compactness of the underlying spacetime). This means in the notation just introduced that there are von Neumann-algebra isomorphisms β21 : Mω1 → Mω2 and β12 : Mω2 → Mω1 such that β21 ◦ πω1 = πω2 and β12 ◦ πω2 = πω1 . We recall also that all quasifree states on A[S, σ ] are faithful, and so −1 are their GNS-representations. Thus β21 = β12 . Moreover, if any state ω1 ∈ SqH turns out to be covariant, then all ω2 ∈ SqH are weakly covariant, too, with (ω)

(ω2 )

αt

(ω1 )

= β21 ◦ αt

(ω)

(ω)

◦ β12 .

It is also straightforward to check that Aω2 = β21 (Aω1 ). In this sense, the C ∗ -dynamical (ω) system (Aω , {αt }t∈R ) is independent of the chosen state ω ∈ SqH once it is known that there exist quasifree Hadamard states for A[S, σ ] which are weakly covariant. This, however, can also be concluded from the fact that each pair of states in SqH is quasiequivalent. To see this one simply notes that for each ω ∈ SqH the time-shifted state ω ◦ αt is again in SqH – this is a consequence of the fact that the wave-front set of the two-point function (cf. Sect. 4.2) of a quasifree Hadamard state is left invariant under the isometries τt . Thus, since πω ◦ α˜ t and πω◦α˜ t are canonically unitarily equivalent by the uniqueness of the GNS-representation, there is for each t a von Neumann algebraic (ω) isomorphism αt : Mω → Mω with the covariance property (4.2). Hence, starting from the class of states SqH , we see that there is (up to isomorphism) a unique C ∗ -dynamical system associated with it. Now, for ω ∈ SqH we define the sub-vector space W∞ ω of Aω as the vector space generated by all αf πω (W(u)), where f ∈ C0∞ (R) and u ∈ S, and denote by A∞ ω the

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∗-algebra generated by W∞ ω . As a consequence of the Weyl-relations, it is straightforward to check that W∞ is norm-dense in Aω and stable under taking adjoints; it is ω stable under convolution with test-functions with respect to αt by its very definition. ∞ Moreover, one may easily check that A∞ ω = β21 (Aω ) for all ω1 , ω2 ∈ SqH . The vector ∞ are uniquely2 associated1with S space W∞ and algebra A qH up to isomorphism, just ω ω as the C ∗ -dynamical system was. In this light, we shall henceforth adopt the following conventions: − we choose an arbitrary, quasifree Hadamard state ω0 and keep it fixed, − we denote by (A, {αt }t∈R ), W∞ and A∞ the C ∗ -dynamical system, dense subspace and ∗-algebra associated with ω0 as just described. In applying the abstract results of Sect. 2, we will take (A, {αt }t∈R ) to be the C ∗ -dynamical system of interest. The results of Sect. 2 then assert the existence of passive states ωp on A as a consequence of suitable forms of static quantum weak energy inequalities (which will be established in the following sections) and the question might arise under which conditions the states ωp can be interpreted as passive states on the original Weyl algebra A[S, σ ]. The following lemma shows that these states always induce states on A[S, σ ] (recalling that a passive state is always invariant under the time-evolution). Lemma 4.1. Let ω be an {αt }t∈R -invariant state on A. Then ω induces a {α˜ t }t∈R -invariant state ωA on A[S, σ ]. Proof. For all f ∈ C0∞ (R) and all A = πω0 (A) ∈ πω0 (A[S, σ ]) the estimate |ω(αf A)| ≤ ||f ||L1 ||A|| entails that, for fixed A, f → ω(αf A) extends to a continuous linear functional on L1 (R). Consequently, there exists a function LA ∈ L∞ (R) so that
ω(αf A) =

dt LA (t)f (t) ,

f ∈ L1 (R) .

Now as ω is {αt }t∈R -invariant, it follows easily that LA must be constant (almost everywhere). Let us denote this constant by ω(A), then it holds that
ω(αf A) = ω(A)

dt f (t) ,

f ∈ L1 (R) ,

 showing that ω(A) = ω(αf A) whenever dt f (t) = 1 . The assignment A → ω(A) is obviously linear and we need to show that it fulfills state-positivity. Let B = A∗ A be a positive element in  πω0 (A[S, σ ]). All we need to demonstrate is the existence of some f ∈ L1 (R) with dt f (t) = 1 so that αf B is a positive  element in A (whereupon ω(B) = ω(αf B) ≥ 0). Choosing any f ≥ 0 in L1 (R) with dt f (t) = 1, it is clear that αf B is a positive element in B(Hω0 ). But since A inherits the ∗-operation and C ∗ -norm of B(Hω0 ), it follows that αf B is also a positive element in A (cf. Lemma 2.2.9 in [6]). Moreover, ω(αt (A)) = ω(αf (αt A)) = ω(αt (αf A)) = ω(αf A) = ω(A) so ω is an {αt }t∈R -invariant state on πω0 (A[S, σ ]). Thus ωA = ω ◦ πω0 is an {α˜ t }t∈R -invariant state on A[S, σ ].  

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However, it should be noted that one has no information regarding the continuity of ωA with respect to the time-evolution, in other words, there is no reason why the functions t → ωA(Aα˜ t (B)), A, B ∈ A[S, σ ], should be continuous, and therefore it is unclear if ωA is passive (in a W ∗ -sense) on A[S, σ ] if ω is a passive state on A. This can be concluded if ω fulfills further regularity conditions. A sufficient condition to that effect is that ω = ωp be a normal state on Mω0 , and we have seen in Cor. 2.7 that a certain energy-compactness condition ensures this normality. We should also like to point out that one can generalize the notion of n-point correlation functions so that it is applicable to states on A in the sense that sufficently regular states (“C ∞ -regular states”) on A possess n-point correlation functions for all n ∈ N, inducing states on the algebra of abstract Klein-Gordon field operators. These matters will be discussed in Appendix A.5. The result of this discussion again shows that there is hardly any difference in working with A[S, σ ] or A as long as “sufficiently regular” states are considered, and thus our passing from A[S, σ ] to A can rightfully be regarded as made purely for technical convenience. 4.2. The state space S. The states to be considered are drawn from the class of Hadamard states on A[S, σ ], for which the renormalised energy density may be defined by point-splitting. They are defined as follows. Suppose a state ω is sufficiently regular that the function (s, t) → ω(W(sEf )W(tEg)) is twice continuously differentiable for each (ω) pair of real-valued test functions f, g and that, moreover, the two-point function w2 defined by   ∂2 (ω) , f, g ∈ D(M; R) ω(W(sEf )W(tEg)) w2 (f, g) = − ∂s∂t s,t=0 extends (by complex linearity in its arguments) to a distribution in D (M × M). Then (ω) ω is said to be Hadamard if the corresponding 2-point correlation function w2 takes (ω) the so-called Hadamard form [27], which completely fixes w2 modulo smooth terms; in particular, the difference between any two Hadamard two-point functions is smooth. In [34], Radzikowski showed that this condition could be replaced by the requirement that the wave-front set of the two-point function should satisfy (ω) WF(w2 ) = {(x, k; x , −k ) ∈ T˙ ∗ (M × M) : (x, k) ∼ (x , k ), k ∈ Nx+ } ,

(4.3)

where Nx+ is the cone of (non-zero) future-pointing null covectors at x and (x, k) ∼ (x , k ) if there is a null geodesic connecting x and x to which k and k are cotangent at x and x respectively, with k being the parallel transport of k. [In the case x = x , we require k = k .] For future reference we will also use Nx− to denote the cone of non-zero ± past-pointing null covectors at x and N = x∈M Nx± for the future and past null cones in T ∗ M. Radzikowski’s criterion has been simplified recently by Strohmaier, Wollenberg and Verch [40], who consider Hilbert-space valued distributions induced by the field operators. Their characterisation is essentially the following. Theorem 4.2. A state ω on A[S, σ ] is Hadamard if and only if the following conditions hold in some GNS representation [not necessarily that induced by ω] (H, π, ) of A[S, σ ]:

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a) ω is represented by a vector ψ ∈ H, i.e., ω(A) =
ψ, π(A)ψ for all A ∈ A[S, σ ]; b) the function t → π(W(tEf ))ψ is differentiable for all f ∈ D(M; R); c) the H-valued functional f → (f )ψ := −id/dt π(W(tEf ))ψ|t=0 extends by complex-linearity to a Hilbert-space valued distribution (·)ψ ∈ D (M, H) obeying WF((·)ψ) ⊂ N− .

(4.4)

Remark. Condition a) may always be satisfied by using the GNS representation induced by ω, but it is convenient to allow for other representations. Note that the assignment f → (f )ψ defines the field operator (f ) in the GNS-Hilbert-space representation (H, π, ) on the domain D((f )) of all ψ ∈ H for which −id/dtπ(W(tEf ))ψ|t=0 exists as strong limit in H. Now let ω be any quasifree Hadamard state state on A[S, σ], and denote by (Hω ,πω ,ω) the corresponding GNS-representation, and by ω (f ) = −id/dtπω (W(tEf ))|t=0 the field operators, defined on a densedomain D(ω (f )) ⊂ Hω for f ∈ D(M). We define Had(ω) as the set of vectors ψ ∈ f ∈D(M) D(ω (f )) having the property that ω (·)ψ belongs to D (M, Hω ) and obeys the Hadamard condition (4.4). For every quasifree Hadamard state ω, the vectors ψ ∈ Had(ω) induce states and, more generally, continuous linear functionals on the C ∗ -algebra A = Aω0 of our dynamical system. Namely, let βωω0 : Mω0 → Mω be the von Neumann-algebra isomorphism with βωω0 ◦ πω0 = πω , then ω[ψ] (A) =
ψ, βωω0 (A)ψ , A ∈ Aω0 , is a state on A = Aω0 . We now define the state space S as the set of finite convex combinations of states induced by vectors in Had(ω), ω ∈ SqH . In other words, a state ω˜ on A is contained in S iff there are finitely many quasifree Hadamard states ωi ∈ SqH  (i = 1, . . . , N) together with unit vectors ψi ∈ Had(ωi ) and λi > 0, N i=1 λi = 1, such that N  [ψ ] ω(A) ˜ = λi ωi i (A) , A ∈ A . i=1

Theorem 4.2 guarantees that all the states in S are Hadamard states. The state space S has the following properties, as will be proved in Appendix A.1: Proposition 4.3. SqH ⊂ S; and S is closed under finite convex combinations and operations in U∞ alg . 4.3. The energy density. Let ω ∈ SqH and define Fω to consist of all linear functionals (not, in general, states)  on Mω0 ⊃ A given by (B) =
ψ, βωω0 (B)ϕ ,

B ∈ Mω0 ,

(4.5)

for some ψ, ϕ ∈ Had(ω). We also denote by F the set of all linear combinations of finitely many functionals i ∈ Fωi , ωi ∈ SqH and – as in Sect. 2 – use V to denote the vector space of functionals on A generated (as in Eq. (2.1)) by S and U∞ alg . In view of Prop. 4.3 (see also Thm. A.1), V is necessarily a subset of F. Thus, when investigating properties of the energy density, it is actually enough to consider elements in Fω for arbitrary ω ∈ SqH .

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Accordingly, let ω ∈ SqH be arbitrarily chosen. Then we define the one-point function as a linear map from Fω to D (M) by [](f ) =
ψ, ω (f )ϕ , for  ∈ Fω as in (4.5), and this is necessarily a weak solution to the Klein–Gordon equation. Similarly, the two-point function is a weak bisolution defined by ⊗2 [](f, g) =
ω (f )ψ, ω (g)ϕ ,

(4.6)

which satisfies the commutator property ⊗2 [](f, g) − ⊗2 [](g, f ) = iE(f, g)(1) as may be seen by a short argument using the Weyl relations and Leibniz’ rule. The microlocal properties of the one- and two-point functions are easily determined using the calculus of Prop. 3.2. Starting with the observation that
(f )ψ, ϕ = [](f ) =
ψ, (f )ϕ , for ψ, ϕ ∈ Had(ω), the Hadamard condition (4.4) and Prop. 3.2(ii) imply (N− )† ⊃ WF(( · )ψ) ⊃ WF([]) ⊂ WF(( · )ϕ) ⊂ N− so the one-point function obeys WF([]) ⊂ N+ ∩ N− = ∅ and is therefore smooth. Turning to the two-point function, Eq. (4.6), Prop. 3.2(iii) and the Hadamard condition (4.4) give

WF(⊗2 []) ⊂ WF(( · )ψ)† ∪ Z × (WF(( · )ϕ) ∪ Z) ⊂ (N+ ∪ Z) × (N− ∪ Z) . In the special case in which  is a state, the above inclusion and the commutator property combine to yield the stronger result that WF(⊗2 []) is contained in the right-hand side of Eq. (4.3) and that the two-point function therefore takes the Hadamard form. By polarisation, it follows that the normal ordered two-point function : ⊗2 : [] = ⊗2 [] − (1)⊗2 [ω0 ] (relative to the reference state ω0 fixed in Sect. 4.1) can be identified with a smooth function on M × M for each  ∈ F. The point-split normal ordered energy density is defined in terms of this quantity by   s  1 a a

: T : [](x, x ) = m2 + eµ ∇a ⊗ e µ ∇a  : ⊗2 : [](x, x ) (4.7) 2 µ=0

and is also smooth on M × M; finally, the normal ordered energy density itself is given (cf. (4.1)) by
[](x) = g00 (x)1/2 : T : [](x, x) . All the quantities defined so far clearly extend to finite linear combinations of functionals  in Fω as ω ranges over SqH , and hence to  ∈ F. In particular,
is defined on S. As will be proved in Sect. A.2. the spatial integral of this quantity generates the dynamics.

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Proposition 4.4. For all t ∈ R we have 
 1 d (αs A) dµ(x)([
(t, x), A]) = , i ds s=0

A ∈ U∞ alg ,  ∈ V ,

(4.8)

where V is, as in Sect. 2, the vector space generated by S and U∞ alg . Equation (4.8) clearly implies that both the assumptions (i) and (ii) made on the dynamics in Sect. 2 are satisfied: to derive (ii) one simply observes that the right-hand side is t-independent, while (i) follows on replacing A by αt A. 4.4. The quantum weak energy inequality. The last step in justifying the structural assumptions of Sect. 2 for our model is the identification of a state ω obeying a suitable QWEI. For this purpose, ω may be chosen to be any state in S whose 2-point function ⊗2 [ω](x, y) is invariant under x → τt x, y → τt x for any t ∈ R.8 Proposition 4.5. Let the (unrenormalised) point-split energy density T0 ∈ D (M × M) be defined by   s 1 2  a

a m + e µ ∇a ⊗ e µ ∇a  ⊗2 [ω] T0 = 2 µ=0

and define x : R → M × M by x (t) = (t, x; 0, x). Then: i) the pull-back x∗ T0 exists as an element of D (R) with WF(x∗ T0 ) ⊂ {(t, ζ ) | ζ > 0} ; ii) x∗ T0 is positive-type in the sense that x∗ T0 (f
f ) ≥ 0

for all f ∈ D(R) ,

(4.9)

where f (t) = f (−t). Furthermore, x∗ T0 is a tempered distribution whose Fourier transform is a positive measure with respect to which (−∞, u] has finite measure, polynomially bounded in u. Proof. i) is a direct calculation, using the fact that WF(T0 ) is contained in WF(⊗2 [ω]) which (since all covectors contained therein are null) has trivial intersection with the conormal bundle Nx = {((t, x; 0, x), (0, ξ ; ζ , ξ )) : ζ ∈ R, ξ , ξ ∈ Tx∗ } of x . Part ii) follows because x∗ T0 (f
g ) = γx

(2)∗

is defined by Nγx , where

(2) γx (t, t )

(2)

T0 (f ⊗g), where γx

: R2 → M ×M

= (t, x; t , x). (This map has conormal bundle Nγ (2) = Nγx × x

Nγx = {(t, x; 0, ξ ) : t ∈ R, ξ ∈ Tx∗ }

8 Such states certainly exist: for example, one could use a ground- or KMS-state, but only the invariance and Hadamard properties are needed below.

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is the conormal bundle for γx : t → (t, x). But Nγx contains no null covectors, so the (2)∗

pull-back γx T0 is well-defined.) Since T0 is positive type in the sense that T0 (F ⊗F ) ≥ (2)∗ 0 for F ∈ D(M), it follows by Theorem 2.2 in [12] that γx T0 is also positive type in this sense and that (4.9) holds. The remaining statements follow from Theorem A.11 in Sect. A.3., a variant of the Bochner-Schwartz theorem.   With the above definitions, the arguments of Sect. 5 of [12] may be adapted straightforwardly9 to show that ω obeys a static QWEI in the sense described in Sect. 2 with respect to the set of states S, where
q(g, x) = du | g (u)|2 Q(u, x) R

and Q(u, x) =

1 2π 2


(−∞,u)

∧  dζ x∗ T0 (ζ ) .

(4.10)

In fact, because  g is smooth, the static QWEI would be unchanged if we had instead used the integration range (−∞, u] to define Q; however, the above definition is technically more convenient, as it entails that Q(u, x) is left-continuous in u for each fixed x. We also note that Q is a well-defined nonnegative measurable function on R × as a consequence of Prop. 4.5(ii); a further consequence of which is that Q(u, x) is polynomially bounded in u for each fixed x ∈ . The final property required of Q is proved in Sect. A.4. 1 Proposition  4.6. For each u ∈ R, Q(u, ·) ∈ L ( , dµ); furthermore, the function Q(u) := dµ(x)Q(u, x) is monotonically increasing, left-continuous and polynomially bounded in u.

This then implies that q(g; . ) ∈ L1 ( , dµ), thus ω fulfills a static QWEI, but we even have Theorem 4.7. The state ω fulfills a limiting static QWEI (with respect to S), and all states in S fulfill a static QWEI. Proof. Note that


1 1 dµ(x)q(gλ ; x) = 2 dµ(x) du | g (u)|2 Q(λu, x) g L1 gλ2 L1 R
1 = 2 du | g (u)|2 Q(λu) , (4.11) g L1 R where Fubini’s theorem has been used. Since Q is polynomially bounded and  g is of rapid decrease, for any  > 0 there exists U > 0 such that
du | g (u)|2 Q(λu) < g 2 L1 , λ ∈ (0, 1) . |u|>U

9

There are two main differences: first, a change of parametrisation in the worldline; second, in [12], the state ω [there denoted ω0 ] was additionally assumed to be a ground state of the time evolution, which has the effect of limiting the ζ integration in (4.10) to [0, u), but is not otherwise needed in the derivation.

Stability of Quantum Systems at Three Scales

Thus

357



1 1 2 du | g (u)| Q(λu) ≤ du | g (u)|2 Q(λu) +  g 2 L1 R g 2 L1 |u|
= 2πQ(λU ) +  for all λ ∈ (0, 1), using monotonicity and left-continuity of Q. Putting this together with Eq. (4.11) and taking the limit λ → 0+ , we have
1 lim sup 2 dµ(x)q(gλ ; x) ≤ lim sup Q(u) +  = lim Q(u) +  , u→0+ λ→0+ gλ L1 u→0+ using monotonicity of Q(u) again. Since  was arbitrary, the left-hand side is bounded uniformly in g by Q(0+) := limu→0+ Q(u). Thus ω fulfills a limiting static QWEI with 0 ≤  ≤ Q(0+); the remaining statement follows as indicated in Remark (i) following Def. 2.1.   Finally, we strengthen the link between quiescence and ground states as follows: Theorem 4.8. If ω ∈ S is a non-degenerate ground state with a mass gap and vanishing one-point function then ω is quiescent. Proof. In the GNS representation (Hω , πω , ω ) induced by ω, the dynamics is generated by a self-adjoint operator H with spectrum σ (H ) ⊂ {0} ∪ [m0 , ∞) for some m0 > 0 and such that 0 is a simple eigenvalue with eigenvector . Thus for any F, G ∈ D(M), ⊗2 [ω](F, τt ∗ G) =
ω (F )ω , eiH t ω (G)ω 
= eiζ t d
ω (F )ω , Eζ ω (G)ω  , where dEζ is the spectral measure for H . Due to the spectral properties of H , the nondegeneracy of ω and the vanishing one-point functions
ω , ω (G)ω , the (finite) measure d
ω (F )ω , Eζ ω (G)ω  is in fact supported in [m0 , ∞). Similarly, since time translation commutes with ∇a ,
T0 (F, τt ∗ G) = eiζ t dρF,G (ζ ) , where dρF,G is finite and supported in [m0 , ∞). Now take any f, g ∈ D(R) and a sequence χn → δx in DT ∗ ( ). Put Fn = f ⊗ χn , Gn = g ⊗ χn . Then for any h ∈ D(R), we have

 h(ζ )dρFn ,Gn (ζ ) = dt h(t)T0 (Fn , τt ∗ Gn ) = T0 (Fn , Hn ) → γx(2)∗ T0 (f, h
g) = x∗ T0 (f
h

g) ∗ g  = (2π)−1  h) , x T0 (f 

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C.J. Fewster, R. Verch

where Hn = [h
g] ⊗ χn . Furthermore, a similar argument shows that
∗ g )  dρFn ,Gn (ζ ) → (2π)−1  x T0 (f  which, considering the case f = g first and then polarising, implies 
      ≤ Cf,g sup | h(ζ )dρ (ζ ) h| . Fn ,Gn   This provides the necessary uniformity to conclude that
∗   g h(ζ )dρFn ,Gn (ζ ) →  h) x T0 (f  for all h ∈ S (R). Choosing h so that supp h ∩ [m0 , ∞) = ∅, and using the fact that linear combinations of functions of the form f g are dense in S (R), we deduce that ∗ T is supported in [m , ∞).   0 x 0 It follows from this that Q(u, x) (for each x) and Q(u) are supported in [m0 , ∞). Since we already know that ω fulfills a limiting static QWEI with 0 ≤  ≤ Q(0+) we may now conclude that  = 0 and that ω is therefore quiescent.   5. Conclusion We have now seen that a C ∗ -dynamical system (A, {αt }t∈R ), together with a suitable class of states and an energy-density can be constructed for the free scalar field on a static, spatially compact globally hyperbolic spacetime such that the assumptions relevant to Sect. 2 are fulfilled. Consequently, we obtain from Thm. 2.3 that passive states ωp exist for the dynamical system (A, {αt }t∈R ). It may be worth mentioning again that A does not coincide with the Weyl-algebra A[S, σ ] for the free scalar field on the given static spacetime, so it is not a priori clear if ωp induces a passive state (in a W ∗ -sense) on A[S, σ ]. This would follow if ωp were found to be normal to any quasifree Hadamard state, and this is in fact expected to hold in view of the results of [37]. Furthermore, as shown by Thm. 2.6 and Cor. 2.7, ωp is normal when energy-compactness holds which is believed to be generically fulfilled in quantum field theoretical models relevant to particle physics. It should be noted that this technical complication does not arise in the case of the free Dirac field, and we should like to emphasize that our methods apply also in this case, so that also for the case of the Dirac field on a static, spatially compact and globally hyperbolic spacetime one would be led to the conclusion that there is a C ∗ -dynamical system together with a class of states and an energy density fulfilling the assumptions made in Sect. 2. In the course of this work, we have also proved various new results concerning the free scalar field. In particular, we have demonstrated that the static QWEI bounds obtained in [12] exhibit sufficient spatial regularity to be integrable. We hope also to have demonstrated the utility of the reformulation (introduced in [40] and developed further here) of the Hadamard condition as a wave-front set condition on Hilbert space-valued distributions. We expect both this and the microlocal calculus of Banach space-valued distributions given in Prop. 3.2 to have further applications. The results of Sect. 4 combine with those of Sect. 2 to give substance to the connection between the three conditions of dynamical stability mentioned towards the end of

Stability of Quantum Systems at Three Scales

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the Introduction. This connection corroborates the point of view originally advocated by Ford [17] that there should be local constraints on the amount and duration of negative energy densities in physical quantum states in order that no violations of the second law of thermodynamics can build up at macroscopic scales. However, we re-emphasise that our results indicate a still deeper equivalence between natural conditions of dynamical stability at microscopic, mesoscopic and macroscopic scales. Of course the question arises how far such an equivalence may be extended from the situation studied in the present article so as to apply to interacting quantum fields in generic curved spacetimes without time-symmetries, and hence be regarded as universal. The main difficulty seems to lie in a missing counterpart of the concept of passivity when the dynamics of a system is no longer described by a group of automorphisms with time-independent generators. A further study of possible generalizations of the passivity concept applicable to quantum field theory on generic curved spacetimes and their interconnections to microlocal spectrum condition and QWEIs appears, in the light of the present results, to offer an interesting line of investigation. A. Technical Appendix A.1. Stability of S Theorem A.1. Let ω be a quasifree Hadamard state. Then Had(ω) is invariant under the action of any element in βωω0 (B), where B is the ∗-algebra finitely generated by Weyl operators, elements of A∞ and elements of U∞ . Furthermore, if A ∈ βωω0 (B) then [ω (·), A] ∈ D (M, L(Hω )) with empty wave-front set. In particular, Aω ∈ Had(ω) for any A ∈ βωω0 (B). Proof. We will use the following conventions. We will write W (f ) for the Weyl operators πω (W(Ef )) in the GNS-representation of ω, and (f ) for the corresponding generators ω (f ) (ei(f ) = W (f )). Moreover, we identify elements A ∈ Mω0 (via βωω0 ) with elements in Mω , i.e., we write A in place of βωω0 (A). Observing these conventions, it is sufficient to prove that Had(ω) is invariant under Weyl operators, elements of A∞ or operators of the form eiA for A = A∗ ∈ A∞ and that the required wave-front set condition holds for the commutators of  with such operators. This will be accomplished in a series of lemmas, starting with the case of Weyl operators. Lemma A.2. For each g ∈ D(M) and h ∈ D(M; R), we have W (h)D((g)) ⊂ D((g)) and [(g), W (h)]ψ = −E(g, h)W (h)ψ ,

ψ ∈ D((g))

(A.1)

and [(·), W (h)] ∈ D (M, L(Hω )) with empty wave-front set. Proof. The domain property and (A.1) are standard. Since Eq. (A.1) implies [(g),W (h)] = |E(g, h)|, the remaining statements follow immediately from Prop. 3.2(i) and the fact that WF(E(·, h)) = ∅.   Lemma A.3. For each g ∈ D(M) and A ∈ A∞ , we have AD((g)) ⊂ D((g)) and [(g), A]ψ = B(g)ψ ,

(A.2)

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where B(g) ∈ A∞ is given by
B(g) =

(A.3)

dt f (t)E(τt ∗ h, g)W (τt ∗ h) .

Furthermore, B(·) ∈ D (M, L(Hω )) and WF(B(·)) = ∅. Proof. Suppose that g is real-valued and that A is of the form αf W (h) [the extension to finite linear combinations and products of such operators is immediate]. To establish the domain property, it suffices to show that s → s −1 (W (sg) − 1)Aψ is bounded for ψ ∈ D((g)). Now if X is any bounded operator one may check straightforwardly that
XAψ ≤ dt |f (t)| Xαt (W (h))ψ . (A.4)   Since (W (sg) − 1)W (h) = W (h) e−isE(g,h) W (sg) − 1 , we therefore have
s −1 (W (sg) − 1)Aψ ≤ s −1 dt |f (t)| (e−isE(g,τt ∗ h) W (sg) − 1)ψ
  −1 ≤s dt |f (t)| (W (sg)−1)ψ+|e−isE(g,τt ∗ h) −1| ψ   ≤ f 1 (g)ψ + sup |E(g, τt ∗ h)| ψ

.

(A.5)

t∈suppf

(Note that the supremum exists as t → E(g, τt ∗ h) is continuous.) Thus we have shown that Aψ ∈ D((g)) for any real-valued g: since in general D((g)) = D((Re g)) ∩ D((Im g)), the result extends immediately to general g ∈ D(M). Equations (A.2) and (A.3) are now easily checked using the identity W (f )∗ (g)W (f ) = (g) + E(f, g)1,

(A.6)

which holds on D((g)) (g ∈ D(M)). Next, define K(g) ∈ C0∞ (R) by K(g)(t) = f (t)E(τt ∗ h, g) .

(A.7)

Lemma A.4. K(·) ∈ D (M, L1 (R)) with WF(K) = ∅. Proof. Note first that K(g)L1 ≤ f L1 sup |E(τt ∗ h, g)| ≤ C sup |g(x)| t∈suppf

(A.8)

x

so K(·) ∈ D (M, L1 (R)). For each (x, k) ∈ T˙ ∗ M, choose O and φ obeying condition (A) of Sect. 3 and define coordinates on O via y µ =
ζ µ , φ(y), where ζ µ is an arbitrary fixed basis of Tx∗ M. Then for each χ ∈ C0∞ (O), E(τt ∗ h, χ eiλ
η,φ ) is the usual Fourier transform10 at λµ (where Tx∗ M   = µ ζ µ ) of Ft = −| det g|1/2 (χ Eτt ∗ h) ◦ κ −1 , where κ : y → y µ is the coordinate map. Since Ft ∈ C0∞ (κ(O)) with derivatives varying 10

See footnote 6.

Stability of Quantum Systems at Three Scales

361

continuously in t, and since suppf is compact, for any relatively compact neigbourhood V of k in Tp∗ M there exist constants CN such that sup |E(τt ∗ h, χ eiλ
,φ )| ≤

∈V

CN , 1 + λN

λ ∈ R+ , t ∈ suppf ,

(A.9)

for each N = 1, 2, . . . . Accordingly, each (x, k) ∈ T˙ ∗ M is a regular directed point for K, so WF(K) = ∅.   It now follows by Prop. 3.2(i) and the bound B(g) ≤ K(g)L1 obtained from Eq. (A.3) that B(·) ∈ D (M, L(Hω )) and WF(B(·)) = ∅.   Lemma A.5. For each g ∈ D(M) and A ∈ A∞ , we have eiA D((g)) ⊂ D((g)) and [(g), eiA ]ψ = C(g)ψ ,

ψ ∈ D((g)) ,

(A.10)

where C(g) is defined by the norm convergent series C(g) =

∞ k  k−1  i Aj B(g)Ak−j −1 ; k! k=1

(A.11)

j =0

we have C(·) ∈ D (M; L(Hω )) and WF(C(·)) = ∅. Proof. By the previous lemma, we may deduce (g)Ak ψ = Ak (g)ψ +

k−1 

Aj B(g)Ak−j −1

(A.12)

j =0

for A ∈ A∞ , k ≥ 1 and ψ ∈ D((g)). Norm convergence in (A.11) follows from the observation ∞ k−1  1  Ak−1 B = eA B(g) < ∞ . k! k=1

(A.13)

j =0

Moreover, the same argument shows that C(g)ψ ≤ eA B(g)ψ. By Prop. 3.2(i), we now conclude that C ∈ D (M; L(Hω )) with WF(C(·)) = ∅ since B also enjoys these properties.   The proof of Theorem A.1 is complete.

 

Proof of Proposition 4.3. The property SqH ⊂ S is obvious since for each ω ∈ SqH the GNS-vector ω is contained in Had(ω). Also, S is closed under finite convex combinations by its very definition. Now U∞ alg is a sub-∗-algebra of B, and therefore its action on Had(ω) preserves Had(ω). As ω ∈ SqH was arbitrary, this implies immediately that S is preserved under operations in U∞   alg .

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A.2. The integrated energy density as the dynamical generator Let γt : → M × M be defined by γt (x) = (t, x; t, x). Proposition A.6. Let ω ∈ SqH and let  ∈ Fω . Then for all t0 ∈ R we have    1/2

∗ 1 d (αs A) γt0 ([: T :, A]) (g00 ) = i ds s=0

(A.14)

for any A ∈ βωω0 (B). Given this, Prop. 4.4 follows easily. Proof of Proposition 4.4. Noting that U∞ alg ⊂ B and that any  ∈ V is a finite convex combination of functionals i ∈ Fωi , ωi ∈ SqH , Eq. (A.14) certainly holds for all A ∈ U∞ alg and  ∈ V. To complete the proof, we observe that since ([: T :, A]) is smooth we have


∗  1/2 γt0 ([: T :, A]) (g00 ) = dµ(x) ([
(t0 , x), A]) and the result follows.

 

Proof of Prop. A.6. In the following subsections, we first establish Eq. (A.14) for Weyl operators and then extend to A∞ and U∞ alg . As in A.1, we work in the GNS representation of ω, and write W (f ) for πω (W(Ef )), (f ) for ω (f ), and we identify A ∈ Mω0 with βωω0 (A) ∈ Mω without writing the βωω0 . A.2.1. Weyl operators. Let us first observe that, for any A ∈ B, we have ([: ⊗2 :, A])(f, g) =
(f )ψ, [(g), A]ϕ +
[(f ), A∗ ]ψ, (g)ϕ , as may be seen by a straightforward calculation. In the particular case A = W (h) for h ∈ D(M; R), this gives ([: ⊗2 :, W (h)])(x, y) = −[W (h) ](x)Eh(y) − Eh(x)[W (h) ](y) ,

(A.15)

and, noting that each term in (A.15) is a product of smooth Klein–Gordon solutions, we use the following lemma. Lemma A.7. Suppose u and v are C ∞ solutions to the Klein–Gordon equation on (M, g) and define   s  1 a a ∇a u)|x (eµ ∇a v)|x + m2 u(x)v(x) . ρ(x) = g00 (x)1/2  (eµ 2 µ=0

Then for any t0 ∈ R,


dµ(x)ρ(t0 , x) =

1 σ (ξ a ∇a u, v) , 2

and in the particular case u = Eh,

1 dµ(x)ρ(t0 , x) = dvolg (x)v(x)(ξ a ∇a h)(x) . 2 M

Stability of Quantum Systems at Three Scales

363

Proof. We have
σ (ξ a ∇a u, v) =

1/2




=

1/2










(e0a ∇a u)(e0b ∇b v) − v(e0a ∇a )2 u

dµ(x)g00



(e0a ∇a u)(e0b ∇b v) − δ ij v∇a eia ejb ∇b u + m2 uv

dµ(x)g00

=2



dµ(x)ρ(t, x) ,

where in the second step we have used the Klein–Gordon equation and in the third step, Gauss’ theorem. Applying this in the particular case where u = Eh for h ∈ D(M) and using the fact that ξ a ∇a commutes with E, we obtain

dµ(x)ρ(t, x) = σ (ξ a ∇a Eh, v) = σ (Eξ a ∇a h, v) = dvolg (x)v(x)(ξ a ∇a h)(x) 2

M

as required.

 

Using this result and (A.15) we have
1 dµ(x)([
(t, x), W (h)]) = − [W (h) + W (h) ](ξ a ∇a h) 2 ! 1 = − ψ, {(ξ a ∇a h), W (h)}ϕ 2   1 d = . (αs W (h)) i ds s=0 Here we have used Proposition A.8. If ψ, ϕ ∈ Had(ω) then s →
ψ, αs W (f )ϕ is continuously differentiable with derivative ! 1 d
ψ, αs W (f )ϕ = −i ψ, {(ξ a ∇a τs ∗ f ), αs W (f )}ϕ . (A.16) ds 2 Proof. It is enough to establish differentiability and the form of the derivative for s = 0. We begin with two observations. First, if ϕ ∈ Had(ω) and f → 0 in D(M; R) then W (f )ϕ → ϕ since W (f )ϕ − ϕ ≤ sup t −1 (W (tf ) − 1)ϕ = (f )ϕ → 0 t∈R\{0}

using the fact that (·)ϕ ∈ D (M, Hω ). Second, τs ∗ f = f − sξ a ∇a f + R(s) , where s −1 R(s) → 0 in D(M) as s → 0. Using the Weyl relations,

s −1 (W (τs ∗ f ) − W (f )) ϕ = s −1 eiE(τs ∗ f,f )/2 − 1 W (τs ∗ f − f )W (f )ϕ +s −1 [W (τs ∗ f − f ) − 1]W (f )ϕ

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C.J. Fewster, R. Verch

and if ϕ ∈ Had(ω) we may use the above observations and the invariance of Had(ω) under Weyl operators to conclude that the first term on the right-hand side converges to −(i/2)E(ξ a ∇a f, f )ϕ as s → 0. A further application of the Weyl relations allows us to rewrite the second term in the form s −1 [W (τs ∗ f − f ) − 1]W (f )ϕ = s −1 e−isν W (R(s))[W (−sξ a ∇a f ) − 1]W (f )ϕ +s −1 (W (R(s)e−isν − 1)W (f )ϕ , (A.17) where ν = E(R(s), ξ a ∇a f )/2. Estimating the second term of Eq. (A.17), we see that s −1 (W (R(s))e−isν − 1)W (f )ϕ ≤ s −1 (W (R(s)) − 1)W (f )ϕ + s −1 |e−isν − 1| ϕ 1 ≤ (s −1 R(s))W (f )ϕ + |E(R(s), ξ a ∇a f )| ϕ 2 which tends to zero as s → 0 by our two observations. The first term of Eq. (A.17) approaches a finite limit as s → 0:
ψ, W (R(s))s −1 [W (−sξ a ∇a f ) − 1]W (f )ϕ =
W (−R(s))ψ, s −1 [W (−sξ a ∇a f ) − 1]W (f )ϕ → −i
ψ, (ξ a ∇a f )W (f )ϕ , so we have established that   d i = −i
ψ, (ξ a ∇a f )W (f )ϕ − E(ξ a ∇a f, f )
ψ, ϕ ,
ψ, αs W (f )ϕ ds 2 s=0 which can be put into the required form using (A.1), which is valid on Had(ω). The proof is completed by noting that s → αs W (f )η = W (τs ∗ f )η and s → (ξ a ∇a τs ∗ f )η are continuous for η ∈ Had(ω) (in the first case owing to our first observation and in the second because (·)η ∈ D (M, Hω ) and s → ξ a ∇a τs ∗ f is continuous from R → D(M)). Thus the right-hand side of (A.16) is continuous in s.   A.2.2. The case A ∈ A∞ . To extend Eq. (A.14) to the case A ∈ A∞ it suffices (by Theorem A.1) to consider A = αf W (h) for f ∈ C0∞ (0, ∞), h ∈ D(M; R). Now   1 d = −i(δ(αf W (h))) = −i(α−f˙ W (h)) (αs A) i ds s=0
= i dt f˙(t)(αt W (h)) . Owing to Prop. A.8, we may integrate by parts to find 
  1/2

1 d 1 = (αs A) dt f (t) γt∗0 ([: T :, αt W (h)]) (g00 ), i ds i s=0 using Eq. (A.14) for αt W (h). Our goal is now to show that


 1/2

 1/2 dt f (t) γt∗0 ([: T :, αt W (h)]) (g00 ) = γt∗0 ([: T :, αf W (h)]) (g00 ) . (A.18)

Stability of Quantum Systems at Three Scales

To this end, we observe that

365




([: T :, αf W (h)])(F, G) =

dt f (t)([: T :, αt W (h)])(F, G) ,

F, G ∈ D(M)

in which the integrand is compactly supported and – by Prop. A.8 – continuous. Accordingly, we may approximate the integral by Riemann sums. Defining 1  TN (F, G) = f (n/N )([: T :, αn/N W (h)])(F, G) , N n∈Z

only finitely many terms contribute for any given N , so TN ∈ D (M ×M). Since the Riemann sums approximate the integral, we have TN → T = ([: T :, αf W (h)]) weakly, but we require the following stronger convergence property, whose proof is deferred to the end of this subsection. Lemma A.9. TN → T in DV (M × M), where V = (N+ × Z) ∪ (Z × N− ). Now the conormal bundle of the map γt0 is Nγt0 = {(t, x, α, ξ ; t, x, β, −ξ ) | α, β ∈ R, (x, ξ ) ∈ T ∗ }

(A.19)

and has trivial intersection with V . It therefore follows that γt∗0 TN → γt∗0 T in Dγ ∗ V ( ) t0

and in particular that (γt∗0 TN )(g00 ) → (γt∗0 T )(g00 ) = (γt∗0 ([: T :, αf W (h)]))(g00 ) . 1/2

1/2

1/2

But we also have

 1/2 1  f (n/N ) γt∗0 ([: T :, αn/N W (h)] (g00 ) N n∈Z


 1/2 −→ dt f (t) γt∗0 ([: T :, αt W (h)] (g00 )

(γt∗0 TN )(g00 ) = 1/2

since the final integrand is continuous. Thus (A.18) is established and the proof is complete.   Proof of Lemma A.9. It suffices to show that UN → U in DV (M × M), where UN and U are defined by analogy with TN and T , but with : ⊗2 : replacing : T :. It is clear that UN → U weakly, so we need only show that UN converges in DV (M × M). Now ([: ⊗2 :, αt W (h)])(F, G) = E(τt ∗ h, F )
ψ, W (τt ∗ h)(G)ϕ + E(τt ∗ h, G)
ψ, (F )W (τt ∗ h)ϕ and since the two terms each define a smooth distribution on M × M, their contributions to UN may be treated separately. Accordingly, we let 1  (1) UN (F, G) = f (n/N )E(τn/N ∗ h, F )
ψ, W (τn/N∗ h)(G)ϕ , N n∈Z

and observe that (1)

|UN (F, G)| ≤

1  |K(F )(n/N )| (G)ϕ , N n∈Z

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C.J. Fewster, R. Verch

where K(·) ∈ D (M, L1 (R)) was defined in Eq. (A.7). Since t → K(F )(t) is smooth and compactly supported, we therefore have (1)

|UN (F, G)| ≤ 2K(F )L1 (G)ϕ for all sufficiently large N . Putting this estimate together with the Hadamard condition, (1) Lemma A.4 and Prop. 3.4, we conclude that UN converges in DZ ×N − (M × M). An analogous argument applied to (2)

UN (F, G) =

1  f (n/N )E(τn/N ∗ h, G)
ψ, (F )W (τn/N∗ h)ϕ , N n∈Z

shows that UN converges in DN + ×Z (M × M). Thus UN = UN + UN converges in

 DV (M × M) as required.  (2)

(1)

(2)

A.2.3. The case A ∈ U∞ alg . The last step in the proof of Prop. A.6 is to extend the result to show that the energy density generates the dynamics for operators in U∞ alg . Here, it is enough to establish Eq. (A.14) with A replaced by an operator of the form eiA for A = A∗ ∈ A∞ , which may be arbitrarily well approximated in the graph norm of δ by the sequence N n  i n A SN = n! n=0

of partial sums. Since SN ∈

A∞ ,

and     1 d 1 d iA   (αs SN ) (αs e ) −→ i ds i ds s=0 s=0

it remains only to show that

 1/2

∗ 1/2 γt0 ([: T :, SN ]) (g00 ) −→ γt∗0 ([: T :, eiA ]) (g00 ) . Lemma A.10. Suppose A ∈ A∞ . Then   |(Aj [: ⊗2 :, A]Ak )(f, g) ≤ Aj +k B(g) (f )ψ + B(f ) (g)ϕ +(j + k)Aj +k−1 B(f ) B(g) for any j, k ∈ N0 . Proof. We have (Aj [: ⊗2 :, A]Ak )(f, g) =
(f )A∗j ψ, B(g)Ak ϕ −
B(f )∗ A∗j ψ, (g)Ak ϕ . Now (g)Ak ϕ = Ak (g)ϕ +

k−1  r=0

Ar B(g)Ak−1−r ϕ

Stability of Quantum Systems at Three Scales

367

so (g)Ak ϕ ≤ Ak (g)ϕ + kAk−1 B(g) . Putting this together with the analogous estimate on (f )A∗j ψ, the result is proved by Cauchy-Schwarz.   In consequence, and using ([: ⊗2 :, An ]) =

n 

(Aj [: ⊗2 :, A]An−1−k )

j =0

we obtain   |([: ⊗2 :, SN ])(f, g)| ≤ C B(g) (f )ψ + B(f ) (g)ϕ +C B(f ) B(g) for constants C, C independent of N , f and g. Applying the remark following Prop. 3.4, the Hadamard condition and the fact that B(·) has empty wave-front set, we see that ([: ⊗2 :, SN ]) → ([: ⊗2 :, eiA ]) in DV (M × M) as required. Proposition A.6 is thus proved.   A.3. A variation on the Bochner-Schwartz Theorem Theorem A.11. Suppose S ∈ D (R) is of positive type with WF(S) ⊂ {(τ, ζ ) | ζ > 0} .

(A.20)

S is a polynomially bounded positive measure; ii) (−∞, u] has Then i) S ∈ S (R) and  finite (necessarily polynomially bounded) measure with respect to  S; iii) if fn is any sequence of Schwartz test functions with fn (ζ ) monotonically increasing to χ(−∞,u) (ζ ) for each ζ ∈ R then  S(fn ) is monotonically increasing and
S(fn ) = dζ  S(ζ ) . (A.21) lim  n→∞

(−∞,u)

Proof. Part (i) is the usual Bochner-Schwartz theorem (Theorem IX.10 in [36]), while (iii) follows by the monotone convergence theorem if (ii) holds. It is enough to prove (ii) for u = 0. To this end, let f ∈ C ∞ (R) be 1) and nonnegative with suppf ⊂ (−∞, ∞ f = 1 on R− . Decomposing f as f (ζ ) = ∞ n=0 g(ζ + n), where g ∈ C0 (−1, 1) is nonnegative, we claim that
dζ  S(ζ )g(ζ − η) → 0 (A.22) rapidly as η → −∞. Thus ∞
 n=0

dζ  S(ζ )g(ζ + n) < ∞

(A.23)

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C.J. Fewster, R. Verch

and since each term in this series is positive,   the monotone convergence theorem entails that dζ  S(ζ )f (ζ ) < ∞. Accordingly, R− dζ  S(ζ ) < ∞ and the result is proved. It remains to prove our claim (A.22). Let G(ζ ) = ( S
g )(ζ ) =  S(g(· − ζ )), where we have written g (ζ ) = g(−ζ ). Then G is smooth and polynomially bounded, with polynomially bounded derivatives. Moreover, by (i), since g is nonnegative, G is also nonnegative and we may write
G(ζ ) = dη  S(η)g(η − ζ ) . (A.24) Using the convolution theorem, ∨

S(χ g eη ) =


dζ G(ζ ) χ (η − ζ )

(A.25)

for any χ ∈ D. Choose χ so that χ  is nonnegative and χ (ζ ) > 1 for |ζ | < 1. Then for any  ∈ (0, 1) we have 0≤

inf

ζ ∈(η−,η+)

G(ζ ) ≤ (2)−1 S(χ g ∨ eη ) ,

η ∈ R,

(A.26)

and since G has polynomially bounded first derivative, there exists C > 0 and r > 0 such that 0 ≤ G(η) ≤ (2)−1 S(χ g ∨ eη ) + C(1 + |η|)r ,

η ∈ R, 0 <  < 1 .

(A.27)

In particular, taking  = (1 + |η|)−(N+r+1) and using the hypothesis (A.20) on WF(S), it follows that (1 + |η|)N G(η) → 0 as η → −∞ for each N ≥ 0, thereby establishing our claim.   A.4. Integrability of Q(u, ·) Proof of Proposition 4.6. Let fn be a sequence of Schwartz test functions such that fn is a sequence of nonnegative (Schwartz) functions monotonically increasing to χ(−∞,u) . Then by Theorem A.11, ∗  2  x∗ T0 (fn
f n ) = (2π)−1  x T0 (|fn | ) → π Q(u, x)

(A.28)

for each x ∈ . Choosing gn ∈ D(R) such that gn − fn → 0 in S (R) and using the positive type property of x∗ T0 , we have 0 ≤ x∗ T0 (gn
g n ) → π Q(u, x) ,

x ∈ .

(A.29)

Lemma A.12. For each h ∈ D(R), x∗ T0 (h) ∈ L1 ( , dµ(x)) and
dµ(x)x∗ T0 (h) = T(h) ,

(A.30)

where −1/2

T(h) =  ∗ T0 (h ⊗ g00

)

(A.31)

and  : M → M × M is given by (t, x) = (t, x; 0, x). The distribution T is of positive type, with wave-front set contained in the right-hand side of (A.20).

Stability of Quantum Systems at Three Scales

369

Applying the lemma to h = gn
g n , each x∗ T0 (gn
g n ) is an L1 ( , dµ(x)) function with norm (= integral) converging to
lim

n→∞

dµ(x) x∗ T0 (gn
g n ) = lim T(gn
g n ) = lim T(fn
f n ) n→∞ n→∞
= dζ  T(ζ ) , (A.32) (−∞,u)

where we have again used Theorem A.11, now applied to T. Putting this together with (A.29) and applying Fatou’s lemma [35], we conclude that Q(u, ·) ∈ L1 ( , dµ) with
dµ(x)Q(u, x) ≤

0≤

1 π




dζ  T(ζ ) ,

(A.33)

(−∞,u)

the  right-hand side of which is polynomially bounded by Theorem A.11. Thus Q(u) = dµ(x)Q(u, x) is polynomially bounded; monotonicity and left-continuity follow from the same properties of Q(·, x) and the monotone convergence theorem.   Proof of Lemma A.12. Note first that x∗ T0 = γx∗  ∗ T0 , where γx : R → M is given by γx (t) = (t, x). Now, considering the definition of the distributional pull-back (cf. [25], Thms. 8.2.10, 8.2.12) we have   x∗ T0 (h) = γx∗  ∗ T0 (h) = (h ⊗ δx ) ∗ T0 (1M ) .

(A.34)

On the other hand, define ( ∗ T0 )h ∈ D ( ) by ( ∗ T0 )h (H ) =  ∗ T0 (h ⊗ H ). Then WF(( ∗ T0 )h ) = ∅ and so −1/2

 ∗ T0 (h ⊗ g00


)=

 dµ(x) ( ∗ T0 )h δx (1 )

(A.35)



−1/2

[the g00 arises because the preferred densities on R, and M = R× which identify distributions and distributional densities are related by ρM (t, x) = g00 (x)1/2ρR (t)ρ (x)]. Finally, if δn is a sequence in D( ), converging to δx in the H¨ormander pseudo-topology

on DWF ( ) we may calculate (δ ) x

 ( ∗ T0 )h δx (1 ) = lim ( ∗ T0 )h (δn ) n→∞

 = lim (h ⊗ fn ) ∗ T0 (1M ) n→∞   = (h ⊗ δx ) ∗ T0 (1M ) .



(A.36)

Putting this together with (A.34) and (A.35) we obtain (A.30). It is clear from (A.30) that T is of positive type, and a direct calculation of its wavefront set shows that WF(T) is contained in the right-hand side of (A.20).  

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C.J. Fewster, R. Verch

A.5. Correlation functions for states on A We begin our discussion of n-point correlation function for states on A with the following definitions: Let n ∈ N and F ∈ S (Rn ), and define
(α) WF (f1 , . . . , fn ) = dt1 · · · dtn F (t1 , . . . , tn )αt1 (W (f1 )) · · · αtn (W (fn )) for fj ∈ C0∞ (M, R), where we work in the defining representation πω0 of A with W (fj ) = πω0 (W(Efj )); hence the expression on the right-hand side exists as a weak integral in B(Hω0 ). Next we claim that the just defined objects are contained in A. This (k) can be seen by noting that there are sequences hj ∈ S (R), k ∈ N, so that N 

(k)

n h1 ⊗ · · · ⊗ h(k) n −→ F in S (R ) and N→∞

k=1

∞ 

(k)

||h1 ⊗ · · · ⊗ h(k) n ||∞ < ∞ .

k=1 (α)

This implies that αh(k) W (f1 ) · · · αh(k) W (fn ) converges in norm to WF (f1 , . . . , fn ) as n

1

(α)

k → ∞, and since clearly each αh(k) W (fj ) is in A, WF (f1 , . . . , fn ) is also contained j

in A. Now recall the definition of the n-point correlation functions of a state ω on the Weyl-algebra A[S, σ ]11 : One says that ω is C ∞ -regular if, for each n ∈ N, the map Rn × C0∞ (M, R)n  (t1 , . . . , tn ; f1 , . . . , fn ) → ω(W (t1 f1 ) · · · W (tn fn )) is C ∞ with respect to the tj and if the derivatives  ∂  (ω) n ∂ wn (f1 , . . . , fn ) = (−i) ··· ω(W (t1 f1 ) · · · W (tn fn )) ∂t1 ∂tn tj =0 induce, by requiring complex-linearity, distributions wn ∈ D (M n ). These distributions are then called the n-point correlation functions of ω. As is well-known (essentially by Wightman’s reconstruction theorem, cf. [39]), a C ∞ -regular state ω induces a state ω on the algebra F of abstract Klein-Gordon field operators. This algebra F is a ∗-algebra generated by a unit element 1 and a family of elements φ(f ), f ∈ C0∞ (M), subject to the following relations: (ω)

(1) (2) (3) (4)

f → φ(f ) is C-linear, φ(f )∗ = φ(f ), φ((g ab ∇a ∇b + m2 )f ) = 0, [φ(f1 ), φ(f2 )] = iσ (Ef1 , Ef2 )1 .

The state ω is then defined on F by setting ω(a1) = a (a ∈ C), ω(φ(f1 ) · · · φ(fn )) = wn(ω) (f1 , . . . , fn ) , and by requiring complex linearity. By the properties of the GNS-representation, the alge d  bra F is ∗-isomorphic to the algebra of field operators ω (f ) = −i dt π (W(tEf )) ω t=0 by identifying φ(f ) and ω (f ) (and by identifying unit operators). 11

More precisely, ω is to be viewed here as a state on the represented Weyl-algebra πω0 (A[S, σ ]).

Stability of Quantum Systems at Three Scales

371

Proceeding along these lines, one can define an analogue of n-point correlation functions also for states ω on A. We will say that a state ω on A is C ∞ -regular if for each n ∈ N and all F ∈ S (Rn ) the map Rn × C0∞ (M, R)n  (t1 , . . . , tn ; f1 , . . . , fn ) → ω(WF (t1 f1 , . . . , tn fn )) (α)

is C ∞ with respect to the tj and if the derivatives  ∂  (α) (ω) n ∂ w n (F ; f1 , . . . , fn ) = (−i) ··· ω(WF (t1 f1 , . . . , tn fn )) ∂t1 ∂tn tj =0 induce distributions w n ∈ (S (Rn ) ⊗ D(M n )) . Notice that there are many vector states with respect to the defining representation of A which are C ∞ -regular states on A: For example, all states on A induced by vectors W (f )ω0 , f ∈ C0∞ (M, R), have this property. We now proceed to establish the following result. (ω)

Theorem A.13. Let ω be a state on A which is C ∞ -regular. Let F ∈ C0∞ (Rn ) with  dt1 · · · dtn F (t1 , . . . , tn ) = 1 and define F (λ) (t1 , . . . , tn ) = λ−n F (t1 /λ, . . . , tn /λ) (λ > 0) so that F (λ) approximates the n-dimensional Dirac-distribution as λ → 0. Then for all n ∈ N the limits w (ω) n(ω) (F (λ) ; f1 , . . . , fn ) , n (f1 , . . . , fn ) = lim w λ→0

fj ∈ D(M) ,

exist, and induce distributions w n ∈ D (M n ) which will be called n-point correlation functions of ω. (ω) Moreover, the w n , n ∈ N, induce a state ω on the algebra F of field operators upon setting ω(a1) = a (a ∈ C) and (ω)

ω(φ(f1 ) · · · φ(fn )) = w (ω) n (f1 , . . . , fn ) ,

fj ∈ D(M) ,

and by requiring linearity. Proof. Let s = (s1 , . . . , sn ) ∈ Rn and define, for F ∈ S (Rn ), Fs (t1 , . . . , tn ) = F (t1 − s1 , . . . , tn − sn ) . (α)

Owing to the definition of WF (f1 , . . . , fn ), it holds that (α)

(α)

WFs (f1 , . . . , fn ) = WF (τs1 ∗ f1 , . . . , τsn ∗ fn ) , and this implies w n(ω) (Fs ; f1 , . . . , fn ) = w n(ω) (F ; τs1 ∗ f1 , . . . , τsn ∗ fn ) . Let us now denote, for simplicity of notation, the distribution in (S (Rn ) ⊗ D(M n ))

(ω) induced by w n simply by w. Then the last equation implies for this distribution the relation w((F ∗ G) ⊗ ϕ) = w(F ⊗ (G
ϕ)) ,

F ∈ S (Rn ) , G ∈ D(Rn ) , ϕ ∈ D(M n ) , (A.37)

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C.J. Fewster, R. Verch

where F ∗ G is the usual convolution of functions on Rn and
(G
ϕ)(t1 , x 1 , . . . , tn , x n ) = ds1 · · · dsn G(s1 , . . . , sn )ϕ(t1 −s1 , x 1 , . . . , tn −sn , x n ) . Now we define ((1) ϕ)(t1 , x 1 , . . . , tn , x n ) = −(∂t21 + · · · + ∂t2n )ϕ(t1 , x 1 , . . . , tn , x n ) and (G)(t1 , . . . , tn ) = −(∂t21 + · · · + ∂t2n )G(t1 , . . . , tn ) . Then relation (A.37) implies for all F ∈ S (Rn ), G ∈ D(Rn ), ϕ ∈ D(M n ) the following chain of equations: w((F ∗ G) ⊗ ϕ) = w((1 − )−m F ∗ (1 − )m G ⊗ ϕ) = w((1 − )−m F ⊗ (1 − )m G
ϕ) = w((1 − )−m F ⊗ G
(1 − (1) )m ϕ) = w((1 − )−m F ∗ G ⊗ (1 − (1) )m ϕ) which is valid for all m ∈ N. Letting G tend to the n-dimensional Dirac-distribution, we find w(F ⊗ ϕ) = w((1 − )−m F ⊗ (1 − (1) )m ϕ) for all F ∈ S (Rn ), ϕ ∈ D(M n ) and m ∈ N. Hence, exploiting the regularizing property of (1 − )−m , one may choose m so large that lim w(F (λ) ⊗ ϕ) = lim w((1 − )−m F (λ) ⊗ (1 − (1) )m ϕ)

λ→0

λ→0

exists (uniformly in ϕ) owing to the continuity of the functional w; this then implies that the resulting limit is a distribution in D (M n ) with respect to ϕ. In order to show that the above definition of ω really defines a state on F , one needs to check first that ω induces a linear functional on F , i.e. that it respects the (ω) relations expressed in (1)–(4) above. Relation (1) is fulfilled since the w n induce distributions (and are thus multilinear). The next relation (2) is essentially a consequence of [(it)−1 (W (tf ) − 1)]∗ = (i/t)(W (−tf ) − 1) and so its proof is completely analogous to showing that the n-point correlation functions of a C ∞ -regular state on the Weyl-algebra (ω) induce a state on F . Moreover, w n (f1 , . . . , fn ) = 0 if any of the fj is in the range of the Klein-Gordon operator since, in this case, W (tj fj ) = 1, so that ω respects relation (3). Finally, to check the CCR, note that (α)

(α)

WF (f1 , . . . , fj , . . . , fk , . . . , fn ) = WF ·sj k (f1 , . . . , fk , . . . , fj , . . . , fn ) with sj k (t1 , . . . , tn ) = e yields

iσ (Eτtj ∗ fj ,Eτtk ∗ fk )

(ω)

. Inserting this into the definition of the w n

n(ω) (F ; f1 , . . . , fk , . . . , fj , . . . , fn ) w n(ω) (F ; f1 , . . . , fj , . . . , fk , . . . , fn ) − w (ω)

x

x

=w n−2 (G; f1 , . . . , f j , . . . , f k , . . . , fn ),

Stability of Quantum Systems at Three Scales

373

where an x over a symbol means that the corresponding entry doesn’t appear, and the function G is given by
x x G(t1 , . . . , t j , . . . , t k , . . . , tn ) = dtj dtk F (t1 , . . . , tn )iσ (Eτtj ∗ fj , Eτtk ∗ fk ) . These equations establish that ω respects the CCR. What remains to be checked is the positivity of ω. To this end, let " #∗ " # N N   (k) (k) a0 1 + P = a0 1 + ak φ(f1 ) · · · φ(fm(k) ) ak φ(f1 ) · · · φ(fm(k) ) k k k=1

k=1 (k)

with a0 , . . . , aN ∈ C and fj it holds that

∈ C0∞ (M, R) be a generic positive element of F . Then

ω(P ) = |a0 | + 2Re a0 2

N 

(k)

(k) w (ω) mk (f1 , . . . , fmk )

k=1

+

N 

(ω)

(k)

()

a k a w mk +m (fm(k) , . . . , f1 , f1 . . . . , fm() ) k

k,=1

= lim lim ω(Q(h, t)∗ Q(h, t)), h→δ t→0

where we define for h ∈ C0∞ (R) with Q(h, t) = a0 1 +

N  k=1

" ak



dt h(t) = 1,

# " # (k) (k) αh W (tf1 ) − 1 αh W (tfmk ) − 1 ··· . it it

Hence the expressions over which the limits are taken are non-negative, and thus ω(P ) ≥ 0.   Acknowledgements. We would like to thank Bernd Kuckert for discussions related to passivity, and for comments on the manuscript. We also thank him for having made his recent results available to us prior to publication. We also thank Muharrem K¨usk¨u for drawing our attention to some typographical errors in the preprint of this paper. The work of CJF was assisted by grant NUF-NAL/00075/G from the Nuffield Foundation and, in its later stages, by EPSRC grant GR/25019/01.

References 1. Bros, J., Buchholz, D.: Towards a relativistic KMS-condition. Nucl. Phys. B 429, 291 (1994) 2. Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000) 3. Brunetti, R., Fredenhagen, K., K¨ohler, M.: The microlocal spectrum condition and Wick polynomials in curved spacetime. Commun. Math. Phys. 180, 633 (1996) 4. Buchholz, D., Wichmann, E.H.: Causal independence and the energy-level density of states in local quantum field theory. Commun. Math. Phys. 106, 321 (1986) 5. Buchholz, D., Porrmann, M.: How small is the phase space in quantum field theory? Ann. Inst. H. Poincar´e 52, 237 (1990) 6. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics, Vol. 1, 2nd edn. Berlin-Heidelberg-New York: Springer-Verlag, 1987

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7. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics, Vol. 2, 2nd edn. Berlin-Heidelberg-New York: Springer-Verlag, 1997 8. D’Antoni, C., Hollands, S.: Nuclearity, local quasiequivalence and split property for Dirac quantum fields in curved spacetime. arXiv:math-ph/0106028 9. Dimock, J.: Algebras of local observables on a manifold. Commun. Math. Phys. 77, 219 (1980) 10. Epstein, H., Glaser, V., Jaffe, A.: Nonpositivity of the energy density in quantized field theories. Nuovo Cimento 36, 1016 (1965) 11. Fewster, C.J., Eveson, S.P.: Bounds on negative energy densities in flat spacetime. Phys. Rev. D 58, 084010 (1998) 12. Fewster, C.J.: A general worldline quantum inequality. Class. Quantum Grav. 17, 1897 (2000) 13. Fewster, C.J., Teo, E.: Bounds on negative energy densities in static spacetimes. Phys. Rev. D 59, 104016 (1999) 14. Fewster, C.J., Teo, E.: Quantum inequalities and quantum interest as eigenvalue problems. Phys. Rev. D 61, 084012 (2000) 15. Fewster, C.J., Verch, R.: A quantum weak energy inequality for Dirac fields in curved spacetime. Commun. Math. Phys. 225, 331 (2002) ´ E.: ´ Quantum inequalities in two-dimensional Minkowski spacetime. Phys. Rev. D 56, 16. Flanagan, E. 4922 (1997) 17. Ford, L.H.: Quantum coherence effects and the second law of thermodyamics. Proc. Roy. Soc. Lond. A 364, 227 (1978) 18. Ford, L.H., Roman, T.A.: Averaged energy conditions and quantum inequalities. Phys. Rev. D 51, 4277 (1995) 19. Ford, L.H., Roman, T.A.: Restrictions on negative energy density in flat spacetime. Phys. Rev. D 55, 2082 (1997) 20. Ford, L.H., Roman, T.A.: Quantum field theory constrains traversable wormhole geometries. Phys. Rev. D 53, 5496 (1996) 21. Fulling, S.A., Narcowich, F.J., Wald, R.M.: Singularity structure of the two-point function in quantum field theory in curved spacetime, II. Ann. Phys. (N.Y.) 136, 243 (1981) 22. Guido, D., Longo, R.: Natural energy bounds in quantum thermodynamics. Commun. Math. Phys. 218, 513 (2001) 23. Haag, R.: Local quantum physics. 2nd Edn., Berlin: Springer Verlag, 1996 24. Haag, R., Swieca, J.A.: When does a quantum field theory describe particles? Commun. Math. Phys. 1, 308 (1965) 25. H¨ormander, L.: The analysis of linear partial differential operators I. Berlin: Springer Verlag, 1983 26. H¨ormander, L.: Fourier integral operators. I. Acta Math. 127, 79 (1971) 27. Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing Horizon. Phys. Rep. 207, 49 (1991) 28. Kuckert, B.: Covariant thermodynamics of quantum systems: Passivity, semipassivity, and the Unruh effect. Ann. Phys. (N.Y.) 295, 216 (2002) 29. Kuckert, B.: β-boundedness, semipassivity, and the KMS-condition. Commun. Math. Phys. 229, 369 (2002) 30. Pfenning, M.J.: Quantum inequalities for the electromagnetic field. Phys. Rev. D 65, 024009 (2002) 31. Pfenning, M.J., Ford, L.H.: The unphysical nature of ‘warp drive’. Class. Quantum Grav. 14, 1743 (1997) 32. Pfenning, M.J., Ford, L.H.: Scalar field quantum inequalities in static spacetimes. Phys. Rev. D 57, 3489 (1998) 33. Pusz, W., Woronowicz, S.L.: Passive states and KMS states for general quantum systems. Commun. Math. Phys. 58, 273 (1978) 34. Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory in curved spacetime. Commun. Math. Phys. 179, 529 (1996) 35. Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. 1. San Diego: Academic Press, 1975 36. Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. 2. San Diego: Academic Press, 1975 37. Sahlmann, H., Verch, R.: Passivity and microlocal spectrum condition. Commun. Math. Phys. 214, 705 (2000) 38. Sahlmann, H., Verch, R.: Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys. 13, 1203 (2001) 39. Steater, R.F., Wightman, A.S.: PCT, spin and statistics, and all that. New York: Benjamin, 1964 40. Strohmaier, A., Verch, R., Wollenberg, M.: Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems. J. Math. Phys. 43, 5514 (2002)

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41. Summers, S.J.: Normal product states for fermions and twisted duality for CCR- and CAR-type algebras with applications to the Yukawa2 quantum field model. Commun. Math. Phys. 86, 111 (1982) 42. Summers, S.J.: On the independence of local algebras in quantum field theory. Rev. Math. Phys. 2, 201 (1990) 43. Verch, R.: Local definiteness, primarity and quasiequivalence of quasifree Hadamard quantum states in curved spacetime. Commun. Math. Phys. 160, 507 (1994) 44. Verch, R.: Nuclearity, split property, and duality for the Klein-Gordon field in curved spacetime. Lett. Math. Phys. 29, 297 (1993) 45. Verch, R.: Wavefront sets in algebraic quantum field theory. Commum. Math. Phys. 205, 337 (1999) 46. Visser, M., Barcelo, C.: Energy conditions and their cosmological implications. arXiv:gr-qc/0001099 47. Vollick, D.N.: Quantum inequalities in curved two dimensional spacetime. Phys. Rev. D 61, 084022 (2000) 48. Wald, R.M.: Quantum field theory in curved spacetime and black hole thermodynamics. Chicago: University of Chicago Press, 1994 49. Weinless, M.: Existence and uniqueness of the vacuum for linear quantized fields. J. Funct. Anal. 4, 350 (1969) Communicated by J.L. Lebowitz

Commun. Math. Phys. 240, 377–395 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0889-2

Communications in

Mathematical Physics

Well Posed Constraint-Preserving Boundary Conditions for the Linearized Einstein Equations Gioel Calabrese1 , Jorge Pullin1 , Oscar Reula2 , Olivier Sarbach1 , Manuel Tiglio1 1 2

Department of Physics and Astronomy, Louisiana State University, 202 Nicholson Hall, Baton Rouge, LA 70803-4001, USA Facultad de Matem´atica, Astronom´ıa y F´ısica, Universidad Nacional de C´ordoba, Ciudad Universitaria, 5000 C´ordoba, Argentina

Received: 9 September 2002 / Accepted: 1 April 2003 Published online: 2 July 2003 – © Springer-Verlag 2003

Abstract: In this paper we address the problem of specifying boundary conditions for Einstein’s equations when linearized around Minkowski space using the generalized Einstein-Christoffel symmetric hyperbolic system of evolution equations. The boundary conditions we work out guarantee that the constraints are satisfied provided they are satisfied on the initial slice and ensures a well posed initial-boundary value formulation. We consider the case of a manifold with a non-smooth boundary, as is the usual case of the cubic boxes commonly used in numerical relativity. The techniques discussed should be applicable to more general cases, as linearizations around more complicated backgrounds, and may be used to establish well posedness in the full non-linear case. I. Introduction In the Cauchy problem of general relativity, Einstein’s equations split into a set of evolution equations and a set of constraint equations. Given initial data that satisfies the constraints one can show that the evolution equations imply that the constraints continue to hold in the domain of dependence of the initial slice. However, in numerical relativity, one usually solves the field equations in a domain of the form (t, x i ) ∈ [0, T ] ×
, with
a bounded space-like surface with boundary ∂
. In this case the domain of dependence of the initial slice is “too small” and one wants to evolve beyond it. Then the question that arises is what boundary conditions to give at the artificial boundary [0, T ] × ∂
. This problem has two aspects to it. First, one wishes to prescribe boundary conditions that make the problem a well posed one which preserves the constraints. In addition to that, one might desire to embed in the boundary conditions some physically appealing property (for instance that no gravitational radiation enters the domain). These two aspects are in principle separate. In this paper we will concentrate on the first one, namely, how to prescribe consistent boundary conditions. The construction we propose ends up giving “Dirichlet or Neumann-like”

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G. Calabrese, J. Pullin, O. Reula, O. Sarbach, M. Tiglio

boundary conditions on some components of the metric, with some free sources. The latter allow freedom to consider boundary conditions for any spacetime in any slicing. If the field equations are cast into a first order symmetric hyperbolic form there is a known prescription for achieving well posedness for an initial-boundary value problem (this is discussed in detail in Sect. II), namely maximal dissipative boundary conditions [1]. Well posedness is a necessary condition for implementing a stable numerical code in the sense of Lax’s theorem (this is well known in the numerical analysis literature [2] and was recently emphasized in the context of the Einstein equations in [3]). However, additional work is needed if one wishes to have a code that is not only stable but that preserves the constraints throughout the domain of evolution. As we stated above, this implies giving initial and boundary data that guarantees that the constraints are satisfied. Traditionally, most numerical relativity treatments have been careful to impose initial data that satisfies the constraints. However, very rarely boundary conditions that lead to well posedness are used, and much less frequently are they consistent with the constraints. Only recently, following Friedrich and Nagy [4], has work with numerical relativity in mind started along these lines [5–9]. Of particular interest is the recent paper of Szilagyi and Winicour [10], which outlines a construction with several points in common with the one we describe here. In this article we derive well posed, constraint-preserving boundary conditions for the generalization presented in [11] of the Einstein-Christoffel (EC) [12] symmetric hyperbolic formulation of Einstein’s equations, when linearized around flat spacetime. The procedure consists in studying the evolution system for the constraint variables and making sure that the corresponding initial-boundary value problem is well posed through appropriate boundary conditions. There is some freedom in the specification of the latter. Next, we translate them into boundary conditions for the variables of the main evolution system. The main difficulty consists in making use of the freedom that one has in the boundary conditions for the evolution system of the constraint variables in such a way that the resulting boundary conditions for the main evolution system ensure well posedness. We consider the case of a non-smooth cubic boundary as is of interest in numerical relativity. This involves the additional complication of being careful about ensuring compatibility at the edges joining the faces. To our knowledge, this is the first detailed analysis of the initial-boundary value problem with a non-smooth boundary in the gravitational context. The organization of this paper is as follows: in Sect. II we review the basic technique of energy estimates used to prove well posedness. In Sect. III we review the generalized EC symmetric hyperbolic formulation, write down the evolution system for the constraint variables and analyze under what conditions the latter is symmetric hyperbolic. In Sect. IV we compute the characteristic variables for the main evolution system and for the system that evolves the constraint variables. These characteristic variables are a key element for prescribing the boundary conditions that lead to well posedness. In Sect. V we write down the constraint preserving boundary conditions in terms of the variables of the main evolution system. In Sect. VI we derive the necessary energy estimates to show that all the systems involved in our constraint-preserving treatment are well posed. In Sect. VII we summarize the constraint-preserving construction of this paper. We end with a discussion of possible future improvements to and applications of the boundary conditions introduced in this paper. Some technical details are summarized in an appendix.

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II. Basic Energy Estimates In this section we review some basic notions of energy estimates for systems of partial differential equations, as discussed, for instance, in [13]. Consider a first order in time and space linear evolution system of the form ∂t u = Aj ∂j u,

(1)

where u = u(t, x i ) is a vector valued function, t ≥ 0, x i ∈
, and where the matrices A1 , A2 and A3 are constant. The symbol P ( a ) is defined as P := Ai ai , where a is an
2 arbitrary unit vector ( i=1...3 ai = 1). The system is symmetric (or symmetrizable) hyperbolic if there is a positive definite, Hermitian matrix H independent of a i such that H P = P † H for all ai . H is called a symmetrizer for P . The initial-boundary value problem consists in solving Eq. (1) given initial data for u at t = 0 and boundary data for u for x ∈ ∂
. In order to show well posedness of the initial-boundary value problem, one usually derives a bound for the energy  E(t) = (u, H u) d 3 x , (2)


where (., .) denotes the standard scalar product. Taking a time derivative of (2), using Eq. (1), the fact that H Aj are symmetric matrices, and Gauss’ theorem, one obtains    d j 3 j 3 2(u, H A ∂j u) d x = ∂j (u, H A u) d x = (u, H P ( n)u) dσ, E(t) = dt

∂
where n is the unit outward normal to the boundary ∂
of the domain. For simplicity let us assume that P ( n) has only the eigenvalues ±1 and 0. Let u(+) , u(−) , and u(0) denote the projections of u onto the eigenspaces corresponding to the eigenvalues 1, −1 and 0, respectively. That is, u = u(+) + u(−) + u(0) and P ( n)u = u(+) − u(−) . Then (u, H P ( n)u) = (u(+) , H u(+) ) − (u(−) , H u(−) ). If we impose homogeneous maximal dissipative boundary conditions, which have the form u(+) = Ru(−) with R “small enough” so that R T H R ≤ H 1 , it follows that E(t) ≤ E(0) for all t ≥ 0. These kind of energy estimates are a key ingredient in well posedness proofs. One can generalize this result to inhomogeneous maximal dissipative boundary conditions, (3) u(+) = Ru(−) + g, where g = g(t, x A ) is a prescribed function at the boundary and R T H R ≤ H , as before. In this case, an energy estimate can be obtained as follows: We first choose a function ψ that satisfies ψ (+) = Rψ (−) + g at the boundary. Then, we consider the variable u˜ ≡ u − ψ instead of u which now satisfies the homogeneous boundary condition u˜ (+) = R u˜ (−) and the modified evolution equation ∂t u˜ = Aj ∂j u˜ + F, 1

In the sense that (u, R T H Ru) ≤ (u, H u) ∀u.

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G. Calabrese, J. Pullin, O. Reula, O. Sarbach, M. Tiglio

where F = Aj ∂j ψ − ∂t ψ is a forcing term. One now repeats the estimate for the energy defined in (2), with u replaced by u, ˜ and obtains  d ˜ H F )d 3 x ≤ 2u ˜ · F , E(t) ≤ 2 (u, dt
 where u ˜ ≡ E 1/2 , F  ≡ ( (F, H F )d 3 x)1/2 and where we have used Schwarz’s inequality. Therefore, we obtain the estimate  t u(t, ˜ .) ≤ u(0, ˜ .) + F (s, .)ds. 0

Similar estimates can be obtained for systems with non-constant coefficients, as is the case, for instance, of linearizations around a given non-Minkowski metric, as in black hole perturbations. In the non-linear case the proofs of bounds are only for finite amounts of time, since solutions can blow up in a finite time starting from initial data with finite energy. The non-linear case also requires different norms from the simple L2 one considered here, and most results are limited to quasi-linear equations [14]. The existence of an energy estimate implies that the initial-boundary value problem is well posed. By this we mean that there exists a unique smooth solution to the problem for which the energy estimate holds [14, 15]. It should be noted that these results are in principle only valid for smooth boundaries. Later in this paper we will discuss boundaries that are non-smooth. It should be understood that in those cases the energy estimates we derive do not necessarily imply the existence and uniqueness of smooth solutions; the results presented in this article only guarantee that the solution is in L2 (for instance, using the Riesz’ lemma, see [16].) III. The Field Equations and Their Constraint Propagation In this section we present the field equations that we will use through this paper, and analyze the constraints propagation. In the first subsection we present the evolution equations for the main variables within the generalized Einstein-Christoffel symmetric hyperbolic formulation of Einstein’s equations. In the second subsection we analyze, in the fully non-linear case, the evolution equations for the constraint variables within this formulation and derive necessary conditions for the latter to be symmetrizable. We will need this system to be symmetric hyperbolic in order to derive an energy estimate in the way we sketched in Sect. II. Rather surprisingly, it turns out that in the original EC system the constraints’ propagation does not seem to be symmetrizable. Imposing the symmetric hyperbolicity condition naturally restricts the free parameter of the system to an open interval. As a side note, there seems to be some correlation between the stability properties of the system found in numerical experiments and this natural choice. A. The formulation. In [11] the following symmetric hyperbolic system 2 of evolution equations for the three-metric (gij ), the extrinsic curvature (Kij ), and some extra variables fkij that are introduced in order to make the system first order in space is derived: ∂0 gij = −2Kij , ∂0 Kij = −∂ fkij + l.o. , ∂0 fkij = −∂k Kij + l.o. , k

2

This system corresponds to System 3 of [11] with their parameter zˆ set to zero.

(4) (5) (6)

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381

where ∂ k ≡ g kl ∂l and where ∂0 = (∂t − £β )/N is the derivative operator along the normal to the spatial t = const. slices. The shift β i is an a priori prescribed function on spacetime while the lapse N is determined by N = g 1/2 eQ , with Q a priori prescribed. Here and in the following l.o. stands for “lower order terms”. These terms depend on gij , Kij , fkij , Q and β i but not on the derivatives of gij , Kij or fkij . Provided that the constraints (see Eq. (8) below) are satisfied, the spatial derivatives, dkij ≡ ∂k gij , of the three-metric are obtained from   η−4   dkij = 2 fkij + η gk(i fj )s s − f sj )s + (7) gij fks s − f sks , 4 where η is a free parameter with the only restriction η = 0. The evolution system (4,5,6), besides being symmetric hyperbolic, has the additional feature that all characteristic speeds (with respect to the normal derivative operator ∂0 ) are either 0 or ±1, so that all characteristic modes lie either along the light cone or along the orthogonal to the hypersurfaces t = const. direction. The particular case with η = 4 corresponds to the system derived in [12]. As we will show shortly, the latter does, however, not seem to admit a symmetric hyperbolic formulation for the evolution of the constraint variables. B. The evolution of the constraints. In order to solve Einstein’s vacuum equations, one has to supplement the evolution equations (4,5,6) with some constraints. These are the Hamiltonian and momentum constraints and additional constraints that make sure that the variables dkij correspond to the first order spatial derivatives of the three-metric. We write the constraints as C = 0,

Cj = 0,

Ckij = 0,

Clkij = 0,

(8)

where the constraint variables C, Cj , Ckij , Clkij are defined by 1 rs k g ∂ (drsk − dkrs ) + l.o. , 2 s Cj ≡ ∂ Ksj − g rs ∂j Krs + l.o. , Ckij ≡ dkij − ∂k gij , Clkij ≡ ∂[l dk]ij . C≡

(9)

(For the purpose of this article, the explicit definitions for the lower order terms appearing in the expressions for C and Cj are not needed. They can be found in [11].) Using the evolution system (4,5,6) for the main variables, one obtains the following principal part for the evolution system for the constraint variables [11]: η ∂0 C = ∂ k Ck + l.o. , (10) 4 4 − 2η ∂ 0 Ci = ∂i C − ∂ k C skis − ∂ k Ckiss + l.o. , (11) η (12) ∂0 Ckij = l.o. , η η η−4 ∂0 Clkij = gi[k ∂l] Cj + gj [k ∂l] Ci + gij ∂[l Ck] + l.o. (13) 2 2 4 A system is symmetrizable if we can find a transformation that brings the principal part into symmetric form. In order to investigate under which conditions this system is symmetrizable hyperbolic, we split Clkij in its trace and trace-less parts:

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 1 1 gl(i Bj )k − gk(i Bj )l + gij Wlk , 2 3 where Elkij is trace-less with respect to all pair of indices and where in terms of the traces Ski ≡ C s(ki)s , Aki ≡ C s[ki]s and Vlk ≡ Clkss , Clkij = Elkij +

4 12 4 9 12 Ski − Aki − Vki , Alk . Wlk = Vlk + 3 5 5 5 5 Rewriting the principal part in terms of these variables, we find η ∂0 C = ∂ k Ck + l.o. , 4 4 − 2η ∂0 Ci = ∂i C − ∂ k Ski − ∂ k Aki − ∂ k Vki + l.o. , η  3η 1 s ∂0 Ski = − ∂(k Ci) − gki ∂ Cs + l.o. , 4 3 ∂0 Aki = (1 − η) ∂[k Ci] + l.o. ,  7η ∂0 Vki = − 3 ∂[k Ci] + l.o. , 4 ∂0 Ckij = l.o. , ∂0 Elkij = l.o. Bki =

(14) (15) (16) (17) (18) (19) (20)

Having split all the variables into their trace and trace-less parts the only natural transformations that remain are rescaling of the variables or linear combinations of the antisymmetric symbols Aki and Vki . First, looking at the terms depending on Ci in the evolution equation for C and vice-versa, we see that η < 2 is needed in order to make the corresponding block in the principal part symmetric by a rescaling of C. Next, looking at the terms involving Ski in the evolution equation for Ci and vice-versa, through a similar reasoning, we obtain the condition η > 0. Finally, if 0 < η < 2, it is easy to see that the transformation V˜ki = (7η/4 − 3)Aki + (η − 1)Vki , A˜ ki = Aki + Vki , brings the corresponding block into manifestly symmetric form. We can summarize the result as follows: If 0 < η < 2, the principal part of the system (14–20) is symmetric with respect to the inner product associated with 4 ki 4 16 − 8η S Ski + A˜ ki A˜ ki CC + C i Ci + (U, U ) ≡ 2 η 3η 8 − 3η +V˜ ki V˜ki + C kij Ckij + E lkij Elkij , (21) where U = (C, Ci , Ski , A˜ ki , V˜ki , Elkij )T . It is interesting to notice that in a numerical empirical search for a value of η that improves the stability of a single black hole evolution, Kidder, Scheel and Teukolsky found the value η = 4/33, which lies inside the interval 0 < η < 2 [11]. On the other hand, the evolution of the original EC (η = 4), for which we were not able to find a symmetrizer, according to [11] does not perform very well in 3D black hole evolutions. Also of interest is that in the recent work of Lindblom and Scheel [17] they note that the previously mentioned range of η is also preferred. (See Fig. 7 of their paper; the range 0 < η < 2 translates to −∞ < γ < −0.5). In that work they also study the dependence on another parameter zˆ , which corresponds to a rescaling of a variable and therefore its effects do not influence the principal part of the equations.

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IV. Characteristic Variables Here we discuss the characteristic variables of the main evolution system (4,5,6) and of the evolution system (14–20) for the constraint variables. The characteristic variables are needed in order to give maximal dissipative boundary conditions, which yield a well posed initial-boundary value problem. From now on we will concentrate on linearized (around Minkowski spacetime in Cartesian coordinates) gravity for simplicity. That is, the background metric is ds 2 = −dt 2 + δij dx i dx j , where δij is the flat space metric. We also assume that our perturbations have vanishing shift and vanishing linearized densitized lapse. In these cases, all lower order terms vanish (with the obvious exception of the right-hand side (RHS) of (4)) and the evolution equations simplify considerably. We also choose our spatial domain to be a box
= [xmin , xmax ] × [ymin , ymax ] × [zmin , zmax ], though the analysis below could be generalized to other domains. A. Characteristic variables for the main system. When linearized around flat spacetime, the main evolution system reduces to ∂t gij = −2Kij , ∂t Kij = −∂ k fkij , ∂t fkij = −∂k Kij .

(22) (23)

Since gij does not appear in the evolution equations for Kij and fkij , we do not consider its evolution equation in the following. Therefore, the system we consider has the simple form ∂t u = Aj ∂j u, with u = (Kij , fkij )T . The characteristic variables with respect to an arbitrary unit vector a i are such that the symbol P ( a ) = Aj aj is diagonal. They can be obtained by first finding a complete set, e1 , ..., e24 , of eigenvectors of the symbol (the ej ’s are called characteristic modes) and then expanding u with respect to these vectors. The coefficients in this expansion are called the characteristic variables. For the above system there are six characteristic variables with speed 1, six with speed −1 and twelve with zero speed. They are given by vij

(+)

= Kij − faij ,

(speed +1),

(−) vij (0) vkij

= Kij + faij ,

(speed −1),

(24)

= fkij − ak faij , (speed 0),

where faij ≡ fkij a k . In terms of these variables, the evolution system becomes (±)

∂t vij

(0)

∂t vkij

(±)

(0)

= ±∂a vij − δ kl ∂kT vlij ,  1  (+) (−) = − ∂kT vij + vij , 2

(25) (26)

where ∂a denotes the derivative in the direction of a and ∂kT ≡ ∂k − ak ∂a are the derivatives with respect to the directions that are orthogonal to a.

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Gauge, physical, and constraint-violating modes. Before we proceed, it is interesting to give the following interpretation to the characteristic variables of the main system with respect to a fixed unit direction a k . Consider a Fourier mode of u with wave-vector j along a k , i.e. assume that the spatial dependence of u has the form eiaj x . In this case, the i i linearized constraints assume the form L(a )u = 0, where L(a ) is a constant matrix. Also, since in the case we are considering the non principal terms vanish, characteristic modes of this form solve the evolution equations. Now we can check which characteristic modes (or combination thereof) satisfy the constraint equations and which do not. It turns out that all modes violate the constraints, (±) (±) (±) except for the ones corresponding to the characteristic variables vaa and vˆAB ≡ vAB − C (±) δAB v C /2, where A, B denote directions which are orthogonal to a. Next, consider an infinitesimal coordinate transformation x µ → x µ + Xµ . With respect to it, Kij →

Kij − ∂i ∂j X t , gij →

gij + 2∂(i Xj ) . j

Assuming that Xt and Xj are proportional to eiaj x and using (7) we find that (±) (±) vaa

→ vaa + X t ± Xa ,

  η−4 (0) (0) vAij → vAij − δA(i aj ) Xa − Xj ) + δij XA , 4η

while the remaining variables are invariant. In particular, we can gauge away the vari(+) (−) (±) ables vaa and vaa . The characteristic variables vˆAB are gauge-invariant and satisfy the constraints. This suggests the following classification: We call (±)

• the variables vaa gauge variables. (±) • the variables vˆAB physical variables. B (±) (±) • and the remaining variables v B and vaB constraint violating variables. Notice that the constraint violating variables are not uniquely determined since one could add some physical variables to them. We make the classification unique by requiring that the splitting is orthonormal with respect to the metric induced by δ ij . We stress that this classification is exact only if all the fields depend on the variable ak x k and the time coordinate t only. (In particular, it is exact in 1 + 1 dimensions since then there is only one space dimension.) Nevertheless, this classification sheds light on the boundary treatment below: For example, we will see that giving boundary data that is consistent with the constraints will fix a combination of the in- and outgoing constraint violating variables while we will be free to choose any data for some combination of the in- and outgoing gauge and physical variables (see Eq. (39) below). B. Characteristic variables for the evolution system of the constraint variables. For our purpose, it is sufficient to find the characteristic constraint variables that have non-zero speed, since these are the ones that enter the boundary condition (3). In the system considered here there are three characteristic variables with speed 1 and three with speed −1, 2η − 4 (±) Vi = −Ci ± (27) Cai ± (Sai + A˜ ai ), η while the remaining variables have zero speed.

Boundary Conditions for Linearized Einstein Equations

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V. Constraint-Preserving Boundary Conditions Here we start with boundary conditions for the constraint variables that ensure that the constraints are preserved through evolution and then translate them into the boundary conditions for the main evolution system. In order to do so we write the in- and outgoing characteristic constraint variables in terms of (derivatives of) the characteristic variables of the main system. Well posedness of the resulting initial-boundary value problem is shown in the Sect. VI. In the following, we consider the boundary of the domain which in our case consists of the faces of a cube. Each face is characterized by its unit outward normal n. As discussed in Sect. II, the relevant characteristic variables at the n-face are the ones defined with respect to n. In order to preserve the constraints we impose homogeneous maximal dissipative boundary conditions for their evolution (inhomogeneous boundary data would not preserve the constraints). That is, (+)

Vi

j

(−)

= Li Vj

(28)

,

where the matrix L is constant. As discussed in Sect. II, the coupling matrix L must be “small enough” if we want to obtain a useful energy estimate. We will analyze this question in the Sect. VI. Next, our task is to translate the conditions (28) into conditions on the main variables. Using the definition of the constraint variables, Eqs. (9), we find η C = ∂ k vk , (29) 4 Ci = ∂ j Kij − ∂i K , (30) 

η l s l Sni + A˜ ni = ∂ fnil − ∂i f sn + (31) ni ∂ vl − 3∂i vn , 4 where vk = fks s − f ssk . Using Eqs. (29,30,31) and the definition of the characteristic variables, Eqs. (24,27), one obtains    1 3η (±) (0) (0) (+) (−) (±) B (±) 1− ∂ B 2vkkB − 2vBkk − vnB + vnB , (32) Vn = ∂n vBB − ∂ vBn ± 2 4    1 3η (±) (±) (±) (±) (0) (+) (−) VA = −∂n vnA − ∂ B vAB + ∂A vkk ∓ 1− ∂A 2vkkn + vBB − vBB , (33) 2 4 where the capital indices A, B = 1, 2 refer to transverse directions (i.e. directions orthogonal to n), and we sum over repeated indices irrespective of if they are up or down. Indices are raised and lowered with the identity metric. A first problem that arises in the above expressions is that with maximally dissipative boundary conditions one does not control normal derivatives ∂n of the incoming characteristic variables at the boundary and, therefore, cannot impose the above conditions, Eqs. (32,33). However, since in Eqs. (32,33) the normal derivatives are present only on variables with speed ±1, we can use Eqs. (25) to trade normal derivatives by time and transverse derivatives. Doing so one obtains (±)

(±)

(0)

Vn(±) = ±∂t vBB − ∂ B vBn ± ∂ C vCBB   1 3η (0) (0) (+) (−) ± 1− ∂ B 2vkkB − 2vBkk − vnB + vnB , 2 4 (±)

VA

(±)

(±)

(±)

(0)

= ∓∂t vnA − ∂ B vAB + ∂A vkk ∓ ∂ C vCnA

(34)

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∓

   1 3η (0) (+) (−) 1− ∂A 2vkkn + vBB − vBB . 2 4

(35)

The second problem concerns the main evolution system. Conditions (28) together with Eqs. (34,35) do not directly translate into maximal dissipative boundary conditions for the main variables since transverse derivatives appear in the expressions (34,35) 3 . In order to get a well posed initial-boundary problem we look for appropriate linear combinations of in- and outgoing main variables and an appropriate coupling matrix L such that the conditions (28) with (34,35) can be incorporated in a closed set of evolution system that is intrinsic to the boundary (in the sense that it does not involve normal derivatives of variables at the boundary). This is discussed next. A. Neumann boundary conditions: Evolution system on each face. Consider the com(+) (−) (+) (−) binations dij = vij − vij and sij = vij + vij (dij stands for difference 4 , and sij for sum). Noticing that Vn(+) + Vn(−) = ∂t dBB − ∂ A sAn , (+)

−VA

(−)

+ VA

(0)

= ∂t sAn + ∂ B d AB − ∂A dBB − ∂A dnn + 2∂ C vCnA   3η (0) + 1− ∂A 2vBBn + dBB , 4

we see that one way of imposing boundary conditions such that the constraints are satisfied is through 0 = Vn(+) + Vn(−) , 0=

(+) −VA

(−) + VA

(36) .

(37)

Equations (36,37) amount to giving as boundary conditions for the constraints’ propagation a coupling between the incoming and outgoing characteristic constraint variables as j in Eq. (28) with (Li ) = diag(−1, 1, 1). These equations can also be seen as equations for dBB and snA at the n-face, where dˆAB and dnn are a priori prescribed functions (i.e. they are not fixed by the constraints’ treatment). Here dˆAB is defined as the traceless part of dAB . That is, 1 dˆAB = dAB − δAB dCC , 2 where δAB = 1 if A = B and zero otherwise. In fact, Eqs. (36,37) do not contain as dynamical variables only dBB and snA , but also some zero speed variables. Therefore, in order to get a closed system, we need evolution (0) equations for the zero speed variables vABn . These equations can be obtained from the evolution system of the main variables, Eq. (26): (0)

0 = ∂t vABn +

1 ∂A snB . 2

(38)

3 Note that this difficulty does not arise in 1 + 1 dimensions since then there are no transverse deriv(+) (+) atives. In that case one can simply set the ingoing constraint violating variables vBB and vnA to zero. 4 Notice that d is used for this quantity as well as for d kij = ∂k gij , equation (7). There does not seem to be risk of confusion since both objects have different number of indexes.

Boundary Conditions for Linearized Einstein Equations

387

Below we will show that Eqs. (36,37,38) constitute a symmetrizable hyperbolic sys(0) tem of evolution equations at the n-face for the variables (dBB , sAn , vABn ), where dˆAB and dnn can be freely prescribed. Since the domain we are interested in is a box, Eqs. (36,37,38) need to be evolved at each face. In order to do so one needs boundary conditions at each edge (intersection of two faces). We will show below how compatibility conditions naturally fix most of the boundary conditions that are needed at the edge. It will turn out that one still has freedom in specifying a quantity at each edge. This is not entirely surprising since non-smooth points carry with them ambiguities that have to be fixed by hand (for instance, the normal to the boundary is not well defined at a point of non-smoothness). Assuming one already has the solution to Eqs. (36,37,38) at each face, the boundary conditions for the main variables, which are of the required form (3), are the following: (+)

(−)

(+)

(−)

vBB = vBB + dBB , vnA = −vnA + snA , (+) (−) (+) (−) = vnn + dnn , vˆAB = vˆAB + dˆAB , vnn

(39)

where dBB and snA are obtained from the evolution system (36–38) at the n-face and where we can specify the gauge variable dnn and the physical variable dˆAB freely. Note that dnn = −2fnnn , dˆAB = −2fnAB + δAB fnCC . (40) In view of Eq. (7) and the fact that if the constraints are satisfied we have dkij = ∂k gij , we see that these are Neumann conditions on some components of the three-metric. Now we go back to the evolution system (36–38) at the face and look at it in detail. (0) In order to make the notation more compact we write d = dBB , sA = sAn , hAB = vABn , (0) h = vBBn for the variables associated with the n-face. Then we have ∂ t d = ∂ A sA ,

(41)

4 − 3η 2 − 3η ∂t sA = −2∂ hBA − ∂A h − ∂A d − ∂ B dˆAB + ∂A dnn , 2 4 1 ∂t hAB = − ∂A sB . 2 B

(42) (43)

One can check that as long as η = 2 this system is symmetric hyperbolic with respect to the inner product associated with

4 − 3η 2 d + 8hˆ AB hˆ AB , (B, B) ≡ (d)2 + 2sA s A + (6 − 3η) h + 2

(44)

where B = (d, sA , hAB ) and hˆ AB denotes the trace-less part of hAB , hˆ AB = hAB − (δAB hCC )/2. At this point, a question that might arise is if, possibly, the evolution system (41,42,43) itself is subject to some constraints. In fact, the constraints CABij = ∂[A dB]ij = 0 are intrinsic to the n-face since they only involve derivatives of the main variables that are tangential to the boundary. The only components of CABij that can be expressed purely in terms of the variables that appear in (41,42,43) are CABnD = 2∂[A hB]D −

η ∂[A δB]D (d + 2h). 4

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G. Calabrese, J. Pullin, O. Reula, O. Sarbach, M. Tiglio

It is easy to check that by virtue of Eqs. (41,43) we have ∂t CABnD = 0. Therefore, the constraint variables CABnD are constant in time and no special constraint preserving boundary treatment needs to be made at the edges. The only issues that arise at the edges are compatibility issues that ensures that the solution is continuous. This is discussed next. B. Neumann boundary conditions: Compatibility conditions at the edges of the faces. The system we just introduced is defined on each of the six n-faces. The faces themselves have boundaries: the edges that join them. We need to ensure that the system on each face is well posed taking into account the boundary conditions at the edges. Since each edge is shared by two faces, this will translate into compatibility conditions among the various systems. √ The system of interest has, at each edge, √ two ingoing modes with speeds 3/2 and 1 and two outgoing modes with speeds − 3/2 and −1. At the n-face, the corresponding characteristic variables with respect to a direction mA are (we introduce a superscript (n) to make clear to which face the variables refer to), 

1 2 (n) 1 4 − 3η (n) 2 − 3η (n) (n) (n) √ w± 3/2 = ± , s − 2hmm + h + d 2 3 m 3 2 4 (n)

w±1 = ±sp(n) − 2h(n) mp , where p is a transverse direction that is orthogonal to m. (n) (n) (n) (n) In terms of the original variables, the variables d (n) , sA , hAB , dnn , dˆAB on the n-face are (n)

(n)

d (n) = −2fnBB , sA = 2KnA , hAB = fABn , (n) (n) = −2fnnn , dˆAB = −2fnAB + δAB fnCC . dnn Now let us consider the edge defined by the intersection of the n-face and the m-face. We then have the following compatibility conditions: (n) sm = sn(m) , ˆ (m) −2h(n) mp = dpn ,

h(n) pm

=

h(m) pn

.

On the other hand, we have, by definition of the characteristic variables,  2 (n) (n) (n) (n) (n) w+√3/2 − w−√3/2 = w+1 + w−1 = −4h(n) s , mp . 3 m Therefore, the correct boundary conditions at the edges are  2 (m) (n) (n) √ √ w+ 3/2 = w− 3/2 + s , 3 n (n) (n) (m) . w+1 = −w−1 + 2dˆnp

(45) (46) (47)

(48) (49)

Notice that up to now the quantities that were freely specified were dnn and dˆAB on each of the n-faces. Therefore, in order to fix the boundary data for the systems that are (m) (n) defined on these faces we also have to a priori specify the quantities sn = 2Knm = sm at the edges defined by the intersection of the n-face and the m-face.

Boundary Conditions for Linearized Einstein Equations

389

Imposing these boundary conditions automatically implies that the compatibility conditions (45) and (46) are satisfied. On the other hand, we have at the edge joining the n-face and the m-face,    1  (n) (m) (m) ∂t h(n) = 0, pm − hpn = − ∂p sm − sn 2 where the last equality follows from imposing the boundary conditions at the edges. Therefore, these boundary conditions also imply that the compatibility condition (47) is satisfied through evolution provided it does so initially. C. Dirichlet boundary conditions. In a similar way to the Neumann case, one can obtain (0) (0) (0) a closed system at the boundary for the variables (sBB , dAn , vAnn , vABB , vBBA ) by requiring 0 = Vn(+) − Vn(−) , 0=

(+) VA

(−) + VA

(50) (51)

,

where (0)

Vn(+) − Vn(−) = ∂t sBB − ∂ A dAn + 2∂ A vABB   3η (0) (0) + 1− ∂A 2vBBA − 2vAkk − dnA , 4 (+)

−VA

(−)

− VA

= ∂t dAn + ∂ B sAB − ∂A sBB − ∂A snn .

One also has to take into account the evolution equations for the zero speed variables: (0)

0 = ∂t vAij +

1 ∂A sij . 2

(52)

In this case, one can freely specify snn and sˆAB , which corresponds to Dirichlet conditions on some components of the extrinsic curvature: snn = 2Knn ,

sˆAB = 2KAB − δAB KCC .

(53)

Assuming one already has the solution to Eqs. (50,51) at each face, the boundary conditions for the main variables, which are of the required form (3), are the following: (+)

(−)

(+)

(−)

vBB = −vBB + sBB , vAn = vAn + dAn , (+) (−) (+) (−) = −vnn + snn , vˆAB = −vˆAB + sˆAB . vnn (0)

(54) (0)

(0)

In terms of the variables s = sBB , dA = dAn , hA = 3η vABB +(4−3η)(vBBA −vAnn ), we can rewrite the boundary system (50–52) as 8 − 3η A 1 ∂ dA − ∂ A h A , 4 2 1 ∂t dA = ∂A s + ∂A snn − ∂ B sˆBA , 2  4 + 3η 4 − 3η  ∂t hA = − ∂A s + ∂A snn − ∂ B sˆBA . 4 2 ∂t s =

(55) (56) (57)

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G. Calabrese, J. Pullin, O. Reula, O. Sarbach, M. Tiglio

This system is symmetrizable hyperbolic with respect to the inner product associated with   4 − 3η A 4 − 3η 2 A A hA − (58) d dA , (B, B) = 4s + 8d dA + h − 2 2 where √ B = (s, dA , hA ). There is one ingoing and one outgoing mode, with speeds ± 3/2, respectively. The corresponding variables in a direction mA are   1 8 − 3η √ w± 3/2 = 3/2 s ± dm − hm . 2 2 Again, one can check that the only constraints that are intrinsic to the boundary and can be expressed purely in terms of the quantities that appear in Eqs. (55,56,57) are CABCC =

3η − 4 1 ∂[A dB] + ∂[A hB] . 4 2

As before, Eqs. (56,57) imply ∂t CABCC = 0 and no constraint-preserving treatment is needed at the edges. We now turn to the compatibility at the edges of the faces. In terms of the original variables one has (n)

s (n) = 2KBB , (n) = 2Knn , snn

dA = −2fnnA ,

(n)

hA = 3η fABB + (4 − 3η)(fBBA − fAnn ),

(n)

sˆAB = 2KAB − δAB KCC .

From these expressions one can see that the only compatibility conditions at the intersection between the n-face and the m-face are   (m) (n) − sˆmm , (59) s (n) = 2 smm (m) (m) (n) (n) smm + 2ˆsnn = snn + 2ˆsmm .

Equation (59) fixes the boundary data for the ingoing variable:  (n) (n) (m) (n) w+√3/2 = −w−√3/2 + 4 3/2(snn − sˆmm ).

(60)

(61)

This condition will be used in order to derive the energy estimate (64) for the system at each face in the Dirichlet case. On the other hand, Eq. (60) is a compatibility condition at the intersection of the n-face and the m-face for the free boundary data. VI. Well Posedness We start by showing that the conditions (36,37) and (50,51), for the Neumann and Dirichlet case, respectively, imply that the initial-boundary value problem for the constraint variables is well posed. Then we derive energy estimates for the closed system of evolution equations at each face, and using these estimates we show that the initial-boundary value problem for the main evolution system is well posed as well. Since we have already shown that all the evolution equations involved in our constraint-preserving treatment are symmetric hyperbolic, and since we have already cast all boundary conditions in maximally dissipative form, the main purpose of this section is to explicitly show that the different couplings are “small enough” with respect to the corresponding symmetrizers, in the sense discussed in Sect. II.

Boundary Conditions for Linearized Einstein Equations

391

A. Constraint propagation. Here we derive an estimate for the growth of the energy  Econstraints = (U, U ) d 3 x,


where (U, U ) is defined in (21). Taking a time derivative and using Eqs. (14–20) we obtain, after integrations by parts,

 d 4 − 2η i i ˜ Econstraints = 2 CCn − C Sni − C Ani dσ, dt η ∂
where nk is the unit outward normal to the boundary ∂
of the domain. Expressing the integrand in terms of characteristic variables defined in (27), we obtain    d 1 (+) (+) (−) (−) dσ = 0, Econstraints = δ ij Vi Vj − Vi Vj dt 2 ∂
where the last equation follows from the conditions (36,37) and (50,51) in the Neumann and Dirichlet cases, respectively. Therefore, the initial-boundary value problem for the constraints is well posed. In particular, this implies that zero initial data for the constraints implies that the constraints are zero at later times as well. B. Face systems. 1. Neumann conditions. In order to show well posedness for each system defined on the n-face n we consider the corresponding energy norm for the Neumann case,  E( n ,N) = (B, B) dσ, n

where (B, B) is defined in (44). Taking a time derivative and using the above evolution equations we obtain         d 3 2√ 2√ 2 2 6 − w− − w−1 + 4 w+1 ds E( n ,N) = w+ 3/2 3/2 dt 2 ∂ n    +4 (62) s A ∂A dnn − ∂ B dˆAB dσ. n

We use the boundary conditions (48,49), with sn = 0 and dˆnp = 0 for the moment, to get rid of the first term on the RHS of Eq. (62). Using Schwarz’s inequality, the second term is estimated as follows:    1/2 4 s A ∂A dnn − ∂ B dˆAB dσ ≤ 4 f E( n ,N) , (m)

(m)

n



where f2 =



∂A dnn − ∂ B dˆAB



 ∂ A dnn − ∂C dˆ AC dσ.

n

Therefore, we end up with the estimate E( n ,N) (t)1/2 ≤ E( n ,N) (0)1/2 + 2



t

f (s) ds

(63)

0

and see that the energy is bounded and the bound is determined by the norm f of the free data on the boundary.

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(m) (m) The assumptions sn = 0 = dˆnp can be easily relaxed with an argument similar to the one we presented in the paragraph following Eq (3). That is, one defines a new variable that satisfies homogeneous boundary conditions and obtains an energy estimate for the new variable. Well posedness follows immediately.

2. Dirichlet conditions. The Dirichlet case is similar to the Neumann one. The energy is now given by  E( n ,D) =

(B, B) dσ, n

with (B, B) given by Eq. (58). We obtain the estimate     d 1/2 2√ 2√ w+ ds + 2 f˜E( n ,D) , E( n ,D) ≤ 8/3 − w − 3/2 3/2 dt ∂ n where f˜2 =





∂A snn − ∂ B sˆAB

(64)



 ∂ A snn − ∂C sˆ AC dσ,

n

and we can proceed as in the Neumann case, using in this case the boundary condition (61). C. Main system. Having obtained a bound for each closed system defined on a face we can obtain a bound for the main evolution variables by the standard techniques described in Sect. II, where we now have the boundary conditions (39) and (54) for the Neumann and Dirichlet cases, respectively, which are of the required form (3). VII. Summary The well posed, constraint preserving boundary conditions presented in this paper can be summarized as follows. A. Neumann case. • Free data: At each face (say, the n-one), the three quantities dnn = −2fnnn and dˆAB = −2fnAB + δAB δ CD fnCD must be a priori defined (subject to standard compatibility conditions with the initial data). The variables fnij can be computed from spatial derivatives of the three-metric using the inverse of the transformation (7):   η−4   1 fkij = dkij + δk(i dj )s s − d sj )s + δij dkss − d sks , 2 4η where dkij = ∂k gij . In the sense discussed in Sect. IV, the variable dnn is gauge and (n) the other two are physical. In addition, the quantities sm = 2Knm should be prescribed at each edge defined by the intersection of the n-face and the m-face. These (m) (n) quantities must satisfy the compatibility conditions sn = sm . • Evolution systems on faces: The 2D symmetric hyperbolic 7 × 7 system (41,42,43) is evolved on each face. This system needs boundary conditions at the edges of the corresponding face. They are given by Eqs. (48, 49).

Boundary Conditions for Linearized Einstein Equations

393 (n)

(n)

The solution to each of these systems provides the three quantities dBB and snA at each of the six n-faces. • Main evolution system: The main system (22,23) is evolved in the 3D domain. This system needs boundary conditions, at each face, for the six incoming characteristic modes. These boundary conditions at, say the n-face, are given by Eq. (39), where the needed information for three of these boundary conditions is provided by the a priori specified dnn and (n) (n) dˆAB , while the other three are given by dBB and snA . B. Dirichlet case. • Free data: At each face (say, the n one), now the three quantities snn = 2Knn and sˆAB = 2KAB − δAB δ CD KCD must be a priori given. They correspond to the time derivative of the normal and transversal, trace-less part of the three metric. These three quantities have to satisfy the standard compatibility conditions with the initial data, but also some compatibility conditions at edges, Eqs. (59,60). They are also gauge and physical variables, in the sense discussed in Sect. IV. • Evolution systems on faces: The 2D symmetric hyperbolic 5 × 5 system (55,56,57) is evolved on each face. This system needs boundary conditions at each edge. They are given by Eq. (61). (n) (n) The solution to each of these systems provides the three quantities sBB and dAn at each of the six n-faces. • Main evolution system: The main system (22,23) is evolved in the 3D domain. This system needs boundary conditions, at each face, for the six incoming characteristic modes. These boundary conditions are given by (54), where the needed information in three of these boundary conditions is provided by the a priori specified snn and sˆAB , while the other three are (n) (n) given by sBB and dAn . VIII. Conclusions We have studied the system of evolution equations for the constraints for a subfamily of the generalized Einstein-Christoffel symmetric hyperbolic system. We have shown how to give boundary data for the constraints in such a way that it translates into boundary data for the main system that yields a well posed problem, both for the main system and the system of evolution equations for the constraints. We have studied the case of a boundary that is not smooth, as is the case of the usual cubic boxes used in numerical relativity. This required additional care at the boundaries of each face, with the ensuing compatibility conditions. It should be noted that the energy estimates derived do not necessarily guarantee the existence of a smooth solution in the presence of corners even with the compatibility conditions we presented. Further work is needed to establish smoothness of the solution. Our analysis was carried out for the case of linearized gravity around Minkowski space-time. It is expected that similar techniques will be useful in the case of other background space-times and also in the non-linear case. We will discuss these generalizations in future papers. Also, since we have followed a systematic approach and have not taken any advantage of the gauge choice, in principle it should be possible to apply the same procedure to symmetric hyperbolic formulations with live gauges [18].

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We have also found that, at least with the formulation of Einstein’s equations here used (the generalized EC), the Neumann and Dirichlet cases are in fact the only ones for which well posedness can be established through the techniques used in this paper (see the appendix). More specifically, we have found that these two cases are the only ones in which closed systems at the faces can be obtained. However, this does not mean that these are the only well posed cases, since in the initial-boundary value problem an energy estimate is a sufficient but not necessary condition for well posedness. Also, giving the ˆ appropriate (dnn , d (n) AB , snm ) data in the Neumann case, or the appropriate (snn , sˆAB ) in the Dirichlet case, one should be able to recover any solution to the linearized Einstein equations in any coordinates. This is because our constraint-preserving treatment makes sure that one is solving not only Einstein’s evolution equations but also the constraints. But it does not make any restriction on the space of solutions to the Einstein equations. However, it is not clear how to choose these “appropriate” boundary conditions, without any a priori knowledge of the solutions, in order to model an isolated source, given that the boundaries are at a finite distance. This same problem appears in similar approaches [7, 10]. One possible solution is to provide these functions through Cauchy-characteristic [19, 10] or Cauchy-perturbative matching [20], or to resort arguments using the peeling property. Another possibility would be to impose a “no incoming radiation” condition. However, it is not clear how to do this within formulations that have as extra variables first but not second spatial derivatives of the three metric. This might be remedied by repeating this construction for other systems of evolution equations where the variables that represent gravitational radiation at the boundary play a more central role, as in [4]. For instance, a system of evolution equations of higher order where the Weyl tensor is the fundamental variable, would be suitable for this purpose. With the results of this paper one can now assure that both the evolution equation for the main variables of the problem and the evolution equations for the constraint variables are well posed on a manifold with (non smooth) boundary. This allows to evolve initial data that satisfy the constraints beyond their domain of dependence, as is of interest in numerical simulations of the binary black hole problem. Moreover, well posedness opens the possibility of constructing numerical schemes for which numerical stability can be rigorously proved. Acknowledgement. We wish to thank Gabriel Nagy for useful discussions and Alan Rendall for reading the manuscript. This work was supported in part by grants NSF-PHY-9800973, NSF-INT-0204937, the Swiss National Science Foundation, by Fundaci´on Antorchas and the Horace C. Hearne Jr. Institute of Theoretical Physics.

Appendix A: Closing the Boundary System In Sect. V we showed how to construct a closed system at the boundary by taking appropriate linear combinations of characteristic variables. We also chose a particular combination of ingoing and outgoing constraint variables and showed that it gives rise to a system of partial differential equations that lives on the boundary. To close this system we had to include the evolution of some zero speed modes. A question that arises is whether there are other ways of closing the boundary system, apart from the Neumann (41–43) and the Dirichlet (55–57) case. To answer this question we will make no assumptions on the coupling matrices R and L. The boundary condition for the system of the constraints is assumed to be (+)

Vi

(−)

= Li j Vj

,

(A1)

Boundary Conditions for Linearized Einstein Equations j

395

(±)

where Li is a 3 × 3 coupling matrix and Vi is given in (34) and (35). At the boundary data must be given to the ingoing variables. We will assume that they satisfy (+)

(−)

vij = Rij kl vkl + bij ,

(A2)

where Rij kl is a 6 × 6 coupling matrix and bij is the boundary data. If we insert (A2) into (A1) we obtain a system which contains derivatives of the boundary data bij , of (−) (0) the outgoing variables vij , and of the zero speed variables vT ij . This system can be solved for the time derivatives of three of the bij , namely bBB and bnA . To the remaining bij one can give arbitrary data and consider them as source terms. In order to close (−) the system the coefficients that multiply terms containing outgoing variables vij must vanish. After imposing this condition the coupling matrices R and L, which in general depend on 45 parameters, depend on one parameter only (apart from η). The zero speed variables that appear in the RHS of this system cannot be eliminated by any choice of this parameter. Therefore in order to close the system one has to enlarge it by including the evolution of at least the zero speed variables that appear in the RHS. The requirement that the evolution of these zero speed variables do not contain any spatial derivatives of outgoing variables, forces the couplings R and L to be the ones we used in Subsect. (V A) and (V C). Summarizing, the Neumann and the Dirichlet cases are the only ways that one can obtain a closed system at the boundary. Furthermore, as we have shown in this paper, the boundary system is symmetric hyperbolic and the coupling matrices R and L are “not too large”. Any other choice of coupling matrices would lead to a system for which the techniques used in this paper to prove well posedness cannot be applied. References 1. Lax, P.D., Phillips, R.S.: Commun. Pure Appl. Math. 13, 427 (1960) 2. Gustafsson, B., Kreiss, H.O., Oliger, J.: Time dependent problems and difference methods. New York: Wiley, 1995 3. Calabrese, G., Pullin, J., Sarbach, O., Tiglio, M.: Phys. Rev. D 66, 041501 (2002) 4. Friedrich, H., Nagy, G.: Comm. Math. Phys. 201, 619 (1999) 5. Stewart, J.M.: Class. Quantum Grav 15, 2865 (1998) 6. Iriondo, M.S., Reula, O.A.: Phys. Rev. D 65, 044024 (2002) 7. Szilagyi, B., Schmidt, B., Winicour, J.: Phys. Rev. D 65, 064015 (2002) 8. Bardeen, J.M., Buchman, L.T.: Phys. Rev. D 65, 064037 (2002) 9. Calabrese, G., Lehner, L., Tiglio, M.: Phys. Rev. D 65, 104031 (2002) 10. Szilagyi, B., Winicour, J.: Well Posed Initial-Boundary Evolution in General Relativity. arXiv: gr-qc/0205044 11. Kidder, L.E., Scheel, M.A., Teukolsky, S.A.: Phys. Rev. D 64, 064017 (2001) 12. Anderson, A., York, Jr, J.W.: Phys. Rev. Lett. 82, 4384 (1999) 13. Kreiss, H.O., Lorenz, J.: Initial-Boundary Value Problems and the Navier-Stokes Equations. London-New York: Academic Press, 1989 14. Secchi, P.: Diff. Int. Eq. 9, 671 (1996); Arch. Rat. Mech. Anal. 134, 595 (1996) 15. Rauch, J.: Trans. Am. Math. Soc. 291, 167 (1985) 16. Fritz, J.: Partial differential equations, fourth edition. Applied Mathematical Sciences 1, Berlin: Springer Verlag, 1982 17. Lindblom, L., Scheel, M.: Phys. Rev. D 66, 084014 (2002) 18. Sarbach, O., Tiglio, M.: Phys. Rev. D 66, 064023 (2002) 19. Winicour, J.: Living Reviews in Relativity. 4, 3 (2001) 20. Rezzolla, L., et al.: Phys. Rev. D 59, 064001 (1999); Rupright, M.E., et al.: Phys. Rev. D 58, 044005 (1998); Abrahams, A., et al.: Phys. Rev. Lett. 80, 1812 (1998) Communicated by H. Nicolai

Commun. Math. Phys. 240, 397–421 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0917-2

Communications in

Mathematical Physics

Topological Field Theory Interpretation of String Topology Alberto S. Cattaneo2 , Jurg ¨ Fr¨ohlich1 , Bill Pedrini1 1

Institut f¨ur Theoretische Physik, ETH H¨onggerberg, 8093 Z¨urich, Switzerland. E-mail: [email protected]; [email protected] 2 Institut f¨ ur Mathematik, Universit¨at Z¨urich, Winterthurerstrasse 190, 8057 Z¨urich, Switzerland. E-mail: [email protected] Received: 26 March 2002 / Accepted: 22 August 2002 Published online: 19 August 2003 – © Springer-Verlag 2003

Abstract: The string bracket introduced by Chas and Sullivan is reinterpreted from the point of view of topological field theories in the Batalin–Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern–Simons and BF theories) are considered together with generalized Wilson loops. The latter generate a (Poisson or Gerstenhaber) algebra of functionals with values in the S 1 -equivariant cohomology of the loop space of the manifold on which the theory is defined. It is proved that, in the case of GL(n, C) with standard representation, the (Poisson or BV) bracket of two generalized Wilson loops applied to two cycles is the same as the generalized Wilson loop applied to the string bracket of the cycles. Generalizations to other groups are briefly described.

1. Introduction In this paper we study the “string homology” defined by Chas and Sullivan [1] (see also [2]) and its algebraic structure from the cohomological point of view of topological field theory (TFT) [3, 4]. String homology provides new topological invariants for general, oriented d-dimensional manifolds without boundary. The topological field theory underlying our analysis is a genera-lization of three-dimensional Chern-Simons theory, [5]. It can be defined over an arbitrary differentiable, oriented, d-dimensional manifold, M, without boundary. Its formulation requires the data of a Lie group G and a connection, A, on a principal G-bundle, P , over M. In the main body of this paper we focus our attention on the example where G = GL(n, C), P is the trivial bundle, P = M × G, and where A is a flat connection on P . But, in the last section of this paper, we sketch the necessary extensions of our arguments to cover more general situations. We shall study the classical version of our “topological field theory”; but a few remarks on its quantization are contained in the last section.

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Our topological field theory is constructed by making use of the Batalin-Vilkovisky forma-lism or the BRST formalism, depending on whether d is odd or even; see e.g. [8]. For the convenience of the reader we recall some key features of these formalisms. The BV formalism has been invented as a tool to quantize field theories in the Lagrangian formalism with a large (infinite) number of (infinitesimal) symmetries, for example gauge theories. The space, C0 , of classical field configurations of such a theory is first augmented by introducing ghosts, and second by introducing antifields for fields and ghosts in equal number as the fields and the ghosts. The extended configuration space, C, thus obtained can be viewed as an (odd-symplectic) supermanifold, the fields, ghosts and antifields for fields and ghosts being local even or odd (Darboux) coordinates on it. The superfunctions on C form the supercommutative algebra of “preobservables”, denoted by O. This algebra is equipped with a natural Z2 -grading, | · |, and is furnished by construction with a non-degenerate, odd bracket, {·; ·}, {·; ·} : O × O −→ O ( O1 , O2 ) → {O1 ; O2 }

(1)

satisfying graded versions of antisymmetry, of the Leibniz rule, and of the Jacobi identity. This is equivalent to saying that (O, {·; ·}) is a Gerstenhaber algebra. Choosing † local “Darboux coordinates”, φ a , φa , on C, for example interpreting the φ a ’s as “fields” † (fields and ghosts) and the φa ’s as “antifields” (antifields for fields and ghosts),1 the bracket can be expressed as ←

→

←

→

∂ ∂ ∂ ∂ )2 − O 1 O . {O1 ; O2 } = O1 a † † a 2 ∂φ ∂φ ∂φa ∂φ a

(2)

In classical theory, one attempts to construct an action functional S of degree zero satisfying the classical master equation {S; S} = 0.

(3)

Such an action functional equips O with the structure of a differential algebra. The differential, δ, is given by δO = {S; O} (O ∈ O).

(4)

Because the bracket is odd and |S| = 0, |δO| = |O| + 1.

(5)

The classical master equation for S and the graded Jacobi identity imply that δ is nilpotent, i.e., (6) δ 2 = 0. The cohomology of δ, Hδ∗ , is called the algebra of “observables” of the theory. Thanks to the graded Leibniz rule it is indeed an algebra. The master equation and the graded Jacobi identity can be used to show that the bracket descends to cohomology, and Hδ∗ thus has the structure of a Gerstenhaber algebra. 1

† φ a and φa are assigned opposite Grassmann parity.

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The structure described above is well suited to formulate a topological field theory yielding the cohomological version of the results of Chas and Sullivan, provided the dimension d of the underlying manifold M is odd. When d is even we must actually follow the (Hamiltonian) BRST formalism. The latter was developed to quantize theories with (first-class) constraints. The classical phase space, C0 , is augmented by introducing ghosts and antighosts in equal number. The extended space, C, thus obtained can be considered as a supermanifold, the fields, ghosts and antighosts being (even or odd) coordinates on it. The algebra, O, of preobservables is defined to be the algebra of superfunctions on C. By construction, O is furnished with a non-degenerate, even bracket. Thus the algebra O has the structure of a super-Poisson algebra. The action S, now more appropriately called BRST generator, is odd (|S| = 1). The differential δ on the algebra of preobservables is still defined by (4), it has degree 1 and is nilpotent. The cohomology Hδ∗ of δ now has the structure of a super-Poisson algebra. (Observe that Hδ0 describes the algebra of functions on the reduced phase space, but in general other cohomology groups may be nontrivial, too.) The Lagrangian BV formalism and Hamiltonian BRST (or BFV) formalism are related to each other: after gauge fixing of the BV master action, which requires the elimination of the antifields by expressing them as appropriate functions of the fields, one finds an action for which the Legendre transformation to pass to the Hamiltonian formalism can be pursued; the Hamiltonian so obtained has BRST symmetry, and the BRST generator can be constructed. For more details we refer the reader to Appendix D, where the connection between the two formalisms is illustrated for our topological field theory. In this paper we start directly from an extended field space C and a master action (BRST generator) S satisfying the classical master equation, see Sect. 2, without asking whether the theory comes from a classical Lagrangian (or Hamiltonian) theory. Field configurations of our theory are differential forms, C, on M with values in the tensor product of a supercommutative algebra, E, with the metric2 Lie algebra ᒄ of the Lie group G. For simplicity, we suppose that the metric on ᒄ is given by the trace in a representation ρ0 . The forms C have total degree |C| = 1, where the mod 2 grading | · | takes account of both the form degree and the E-degree. The space of field configurations, C, can be considered as a supermanifold with a natural odd (even) bracket; this gives the space of (E-valued) superfunctions, O, the structure of a Gerstenhaber (super-Poisson) algebra. The action functional, S, is chosen to be the “Chern-Simons” action  
1 1 S[C] = (7) tr ρ0 CdA C + C 3 , 2 3 M where dA is the covariant exterior derivative (w.r.t. the flat connection A) over M. Of course, in the integrand of (7) only the part of total form degree d contributes. It is not hard to show that the action is even (odd), |S| = 0, (|S| = 1), and that it satisfies the master equation, {S; S} = 0. Observables of these theories can be constructed as follows. Let LM denote the space of marked, parametrized loops in M. It carries an obvious circle action. String space, SM, is defined as the quotient of LM by this circle action; see Sect. 3. From the connection A and the forms C one can construct, using Chen’s iterated integrals (“Dyson series”), generalized holonomies, holA (C), in a fairly obvious way explained in Sect. 4. 2 A Lie algebra endowed with a non-degenerate, Ad-invariant inner product is called metric. In particular, semi-simple Lie algebras with the Killing form are metric. But so are abelian Lie algebras with any non-degenerate inner product.

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The trace, hρ;A (C) = tr ρ holA (C), also called generalized Wilson loop, then defines a (generalized) preobservable with values in E ⊗ ∗ (SM), i.e., a differential form on SM whose components take values in a supercommutative algebra E. If a represents a cycle in string homology, H∗ M, as described in [1], then one can pair a with hρ;A (C) by integration,
hρ;A (C). (8) a   We shall see in Sect. 4 that a hρ;A (C) is an observable of the theory, i.e., δ a hρ;A (C) = 0, for arbitrary [a] ∈ H∗ M. The main result of this paper, proven in Sect. 7, is the following theorem. Theorem. Let G = GL(n, C), n = 1, 2, 3, . . ., and let ρ denote its standard representation (as matrices on Cn ). Let A be a flat connection on M × G. Then 


h; h = h, (9) a

a¯

{a;a} ¯

where {a; a} ¯ is the Chas-Sullivan bracket, see [1], defined on string homology, and h is a shorthand notation for hρ;A (C).   The definition of the Chas-Sullivan bracket on string homology and some of its properties are explained in Sect. 5. The special role played by the groups GL(n, C) is explained in Sect. 6. As sketched in Sect. 8, more general Lie groups can be accommodated by replacing the string space by a “space of chord diagrams” on the manifold M. Section 8 also contains a sketch of various other generalizations (e.g. to nontrivial principal G-bundles).

2. A TFT with Generalized Gauge Fields In this section, we introduce the topological field theories described in the Introduction in a mathematically precise fashion. We first describe the space of field configurations, then we introduce algebras of preobservables and define the bracket between two preobservables, and, finally, we define an “action functional“ satisfying the classical master equation. 2.1. Field configurations. The field theory is defined over a differentiable, oriented, d-dimensional manifold M. Let P = M × G be a (for simplicity, trivial) principal bundle over M with structure group G. Denote by ᒄ the Lie algebra of G, by Uᒄ the corresponding universal enveloping algebra, and by κ(·, ·) an invariant bilinear form on ᒄ, which, for notational simplicity, we suppose to be given by the trace in some representation ρ0 : κ(·, ·) = trρ0 [· ·]. Let A be a flat connection on P , i.e., A ∈ 1 (M, ᒄ) with dA + 21 [A, A] = 0. We require the following mathematical objects and concepts. A superalgebra X (over R) is an algebra furnished with a mod 2 grading | · |, such that, as a vector space, it has the structure X = X0 ⊕ X1 , with |xi | = i for xi ∈ Xi , and such that |x1 x2 | = |x1 | + |x2 |. A superalgebra is supercommutative if x1 x2 = x2 x1 (−1)|x1 ||x2 | . Next, let E be a supercommutative algebra (e.g. the algebra of supernumbers [11]). A superalgebra X is an E-bimodule if E acts on X from the left and the right, with

Topological Field Theory Interpretation of String Topology

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εx = xε(−1)|x||ε| and |εx| = |ε| + |x|, for arbitrary ε ∈ E and x ∈ X. E is clearly an E-bimodule. Any superalgebra X can be turned into an E-bimodule by considering XE = E ⊗R X and defining the grading |ε ⊗ x| = |ε| + |x|, the left action ε1 (ε2 ⊗ x) = (ε1 ε2 ) ⊗ x, the right action (ε2 ⊗ x)ε1 = (ε1 ε2 ) ⊗ x(−1)|x||ε2 | , and the product (ε1 ⊗ x1 )(ε2 ⊗ x2 ) = ε1 ε2 ⊗ x1 x2 (−1)|x1 ||ε2 | . For notational simplicity, one writes ε ≡ ε ⊗ 1, x ≡ 1 ⊗ x and εx ≡ ε ⊗ x. Given two superalgebras X1 and X2 which are E-bimodules, one may define a tensor product bimodule X1 · X2 = X1 ⊗E X2 , which becomes a superalgebra by defining the grading as |x1 ⊗ x2 | = |x1 | + |x2 | and the product as (x1 ⊗ x2 )(y1 ⊗ y2 ) = x1 y1 ⊗ x2 y2 (−1)|x2 ||y1 | . For notational simplicity one writes x1 ≡ x1 ⊗ 1, x2 ≡ 1 ⊗ x2 and x1 x2 ≡ x1 ⊗ x2 . Clearly one has that E · X = X. Let C G = ∗ (M)E · ᒄE . The space of field configurations is defined as  C G = {C ∈ C G |C| = 1}. (10) 1

We note that the components, Cµa 1 ...µk (x) ∈ E, of a field configuration C ∈ C1G , are bosonic for odd k and fermionic for even k; (a labels a basis in ᒄ). 2.2. Preobservables. A generalized preobservable is a functional on the space of field configurations with values in a superalgebra X which is also an E-bimodule; i.e., it is an element of (11) OG (X) ≡ 0 (C1G , X). G O (X) is clearly an E-bimodule, the grading being given by the grading on X. We shall not indicate the group G if not necessary. The space of (ordinary) preobservables is O ≡ O(E). Though not strictly necessary, the concept of generalized preobservables turns out to be very convenient in the following. The (tensor) product of two preobservables is defined as a map from O(X1 ) × O(X2 ) to O(X1 · X2 ) in the obvious way. 2.3. Bracket between preobservables. We begin by defining the two operators ←

→

δ δ : O(X) −→ O(X · ∗ (M)E · ᒄE ) , δC δC as follows:  →  ← 

d  δ δ d(d+|O|) O(C + tη) = tr ρ0 η O = (−1) tr ρ0 O η , dt t=0 δC δC M M

(12)

(13)

for O ∈ O(X) and arbitrary η ∈ C1 . The signs are chosen in such a way that these two operators act from the left/right as operators of degree d + 1, i.e., such that the Leibniz rules 

→ 

→ → δ δ δ |O1 |(d+1) (14) O1 (O1 O2 ) = O1 O2 + (−1) O2 , δC δC δC



← ← ← δ δ δ |O2 |(d+1) O1 = (−1) O2 + O1 O2 (15) (O1 O2 ) δC δC δC hold. Moreover, one has

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A.S. Cattaneo, J. Fr¨ohlich, B. Pedrini ←

→

δ δ O = (−1)(d+1)|O|+1 O . δC δC

(16)

Next, we define the bracket, {·; ·}, by {·; ·} :

O(X1 ) × O(X2 ) −→ O(X1 · X2 ) |O1 |d






( O1 , O2 ) → {O1 ; O2 } = (−1)

M

tr ρ0

δ δ O2 . (17) O1 δC δC ← →

The signs are chosen in such a way that, for d even, {·; ·} is an even bracket, while for d odd it is an odd bracket. In fact, {·; ·} has the following properties: (1) Antisymmetry, {O1 ; O2 } = −(−1)(|O1 |+d)(|O2 |+d) {O2 ; O1 },

(18)

a consequence of (16); (2) Leibniz rule {O1 ; O2 O3 } = {O1 ; O2 }O3 + (−1)|O2 |(|O1 |+d) O2 {O1 ; O3 },

(19)

a consequence of (14); (3) Jacobi identity {O1 ; {O2 ; O3 }} = {{O1 ; O2 }; O3 } + (−1)(|O1 |+d)(|O2 |+d) {O2 ; {O1 ; O3 }},

(20)

which can be checked by using (16), (14) and the definition (17). We observe that, for a manifold A, for multivector fields vi ∈ ∗ (A) and for generalized preobservables Oi ∈ O(∗ (A)E ) the contraction (≡ infinitesimal integration of chains with given orientation) can be understood as an operator, ιv acting from the left and of degree |v|, namely ιv1 {O1 ; O2 } = {ιv1 O1 ; O2 },

ιv2 {O1 ; O2 } = (−1)|v2 |(d+|O1 |) {O1 ; ιv2 O2 }.

(21)

An explicit calculation on O reveals that {Cµa 1 ...µk (x); Cµb k+1 ...µd (y)} = (−1)k δ (d) (x − y)κ ab εµ1 ...µk µk+1 ...µd .

(22)

2.4. BRST/BV generator and observables. We define an “action” functional, S, by  
1 1 3 S[C] = (23) tr ρ0 CdA C + C ∈ O. 2 3 M This functional has total degree d + 1 and is constructed so as to satisfy the BV/BRST master equation, {S; S} = 0. (24) It is thus to be thought of as a classical master action in the Lagrangian formalism, for d odd, or as a classical BRST generator in the Hamiltonian formalism, for d even. Being independent of the choice of a metric on M, the field theoretical model is

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called topological3 . One can check that, in a situation where M [d+1] = M [d] ×R, d even, S [d] is the BRST generator corresponding to S [d+1] after gauge fixing; (see Appendix D). S defines an odd differential, δ, on the algebra of preobservables by δ : O(X) −→ O(X) . O → {S; O}

(25)

We wish to mention another important property of S: The bracket between S and a field component C is given by {S; C} = (−1)d (dA C + C 2 ),

(26)

{S; Cµa 1 ...µk (x)} = (−1)d+k (dA C + C 2 )aµ1 ...µk (x).

(27)

or, more explicitly,

This is a key equation for proving the fundamental identity (39), below. The cohomology of δ, Hδ∗ , defines the algebra of generalized observables of the topological field theory. Because of (19) and (20), respectively, product and bracket descend to cohomo-logy; the generalized observables thus have the structure of  a super-Poisson algebra (even bracket), for d even,  a Gerstenhaber algebra (odd bracket), for d odd. 3. The String Space of a Manifold In this section we define the loop space of a manifold, and, subsequently, the string space as the quotient of the former by a circle action. Moreover, we describe how to define local coordinates on loop- and string space. One may define the loop space of a manifold M as LM = {γ (·) : S 1 −→ M, γ piecewise differentiable}.

(28)

Observe that S 1 has a marked point, 0, if we interpret S 1 as R/Z. Therefore a loop can be thought of as a parametrized closed curve in M with a marked point and a tangent vector in almost every point, the parameter t ranging from 0 to 1. Let (x µ )µ=1...d be local coordinates on a coordinate patch U ⊂ M. Then µ (γ (t))µ=1...d,t∈S 1 are corresponding local coordinates on the patch LU ⊂ LM. (For loops which extend over different patches, there is a similar construction of local coordinates; but it is not needed for the purposes of this paper). Loop space carries an obvious circle action S 1 × LM −→ LM . (s, γ (·)) −→ γ (· + s)

(29)

3 There is a sigma-model construction of S and {·; ·}, obtained by considering the fields C as maps T M −→ ᒄ (see [9]), where  reverses the parity of the fiber in a vector bundle.

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Fig. 1. Constructing local coordinates on S M

The string space, SM is defined as the quotient of LM by this action4 S 1 → LM ↓ πS 1 SM.

(30)

A string can thus be thought of as a closed curve in M with a tangent vector in almost every point. Local coordinates on SM can be constructed by choosing a local section SM −→ LM and then using local coordinates on LM; see Fig. 1. More precisely, let σ˜ ∈ SU ⊂ SM be a nonconstant string and p a point on it such that σ˜˙ (p) = 0. Let ψ be a function on M defined in a neighborhood of p such that ψ(p) = 0 and σ˜˙ (p); dψ(p) = 0. A local section sψ,p : SM −→ LM in a neighborhood of σ˜ is uniquely defined by the requirement that ψ(sψ,p (σ¯ )(t = 0)) = 0, for any string σ¯ that is a sufficiently small deformation of σ˜ . The functions (σ µ (t))µ=1...d,t∈S 1 , defined as σ µ (t) = γ µ (t) ◦ sψ,p , are then local coordinates on SM in a neighborhood of σ˜ . We denote by H∗ M the string homology, properly defined as the S 1 -equivariant loop space homology. We denote by d− the differential on both loop- and string space. 4 The string space is a singular manifold, with singularities arising at the constant and at the n-fold strings, which correspond to loops with nontrivial stabilizers w.r.t. the circle action.

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405

4. Generalized Holonomies and Wilson Loops In this section we define generalized Wilson loops as generalized observables with values in string cohomology. As such, they can be paired with cycles in string homology, yielding observables of the topological field theory. t We introduce standard simplices n |tfi = {(t1 , . . . , tn ) ∈ Rn |ti ≤ t1 ≤ . . . ≤ tn ≤ tf }, n = n |10 , and define the evaluation maps evn,k : n × LM −→ M (t1 , . . . , tn ; γ ) −→ γ (tk )

1 ≤ k ≤ n.

(31)

The nth order generalized parallel transporter is given by


tf t n  holA (C) t = holA |tt1i ev∗n,1 C holA |tt21 . . . holA |ttnn−1 ev∗n,n C holA |tfn . (32) tf i

n |ti

t t = P exp tkk+1 ιγ˙ (t) A, of the flat In this definition the parallel transporter, holA |tk+1 k connection A is a function n × LM −→ UᒄE ; (P denotes path ordering). For an expression in local coordinates, see Appendix C. t Thus, holnA tf is an element of O(∗ (LM)E · UᒄE ). We define generalized parallel i

t

transporters, holA |tfi , by t

holA (C)|tfi =

∞ 

t holnA (C)tf , i

(33)

n=0

and generalized holonomies by holA (C) = holA (C)|10 .

(34)

Furthermore, generalized “Wilson loops” in a representation ρ are defined by hρ;A (C) = tr ρ holA (C).

(35)

It is worth remarking that the degree of generalized parallel transporters and generalized Wilson loops is zero, i.e.,   |holA | = hρ;A  = 0. (36) Under a gauge transformation, g : M −→ G, one finds that holA (C) = g −1 holg(A+d)g −1 (gCg −1 )g,

hρ;A (C) = hρ;g(A+d)g −1 (gCg −1 ).

(37)

The tangent vectors, γ˙ , that generate the circle action on LM define a section of T LM. The contraction ιγ˙ hρ;A clearly vanishes. Moreover, one finds [6] that d− hρ;A =




1 0

  dτ tr ρ holA (C)|τ0 ιγ˙ ev∗τ (dA C + C 2 ) holA (C)|1τ ,

(38)

where evτ : LM −→ M, γ −→ γ (τ ). This implies that the Lie derivative Lγ˙ hρ;A = ιγ˙ d− hρ;A vanishes, too. The form hρ;A is thus horizontal and invariant with respect to the circle action, and thus defines a form on string space.

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Comparing (38) and (26), we find the fundamental identity [6, 7] ((−1)d δ + d− )hρ;A = 0,

(39)

which implies that the trace of the generalized holonomy is an observable with values in string cohomology, hρ;A ∈ Hδ∗ O(H∗ M), (40) and, for a cycle a ∈ H∗ M in string homology, the pairing
 a, hρ;A  := hρ;A ∈ Hδ∗ O (41) a

defines an observable. 5. The String Bracket In this section we recall how to define a bracket {·; ·} : H∗ M × H∗ M −→ H∗ M

(42)

on string homology. This definition is taken from the article of Chas and Sullivan [1], but we give a slightly simplified exposition. Define SM × ⊂ SM × SM as the space of pairs of strings which intersect transversally at at least one point. This space is a cycle of codimension d − 2, with n − 1-fold self intersections when the two strings intersect n times. We propose to construct the current corresponding to SM × . The d-form ω× = δ(x 1 − x¯ 1 ) . . . δ(x d − x¯ d )(dx 1 − d x¯ 1 ) . . . (dx d − d x¯ d ) ∈ d (M × M) (43) is the current for the diagonal in M × M. We define
C× = (ev∗1,1 × ev ¯ ∗1,1 )ω× ,

(44)

S 1 ×S¯ 1

which is a (d − 2)-current on LM × LM. It is closed, since ω× is closed, and the integration domain, S 1 ×S¯ 1 , in the above formula has no boundaries. In local coordinates, it reads
s=1
s¯=1 d−1  (−1)d+1 ds d s¯ δ (d) (γ (s) − γ¯ (¯s )) εν1 ν2 ...νk ν¯ k+1 ν¯ k+2 ...¯νd C× = (k − 1)!(d − k − 1)! s=0 s¯ =0 k=1

× γ˙ ν1 (s)d− γ ν2 (s) . . . d− γ νk (s)γ˙¯

ν¯ k+1

(¯s )d− γ¯ ν¯ k+2 (¯s ) . . . d− γ¯ ν¯ d (¯s ).

(45)

From this expression it is easy to see that it is horizontal, and thus also invariant with respect to the two circle actions on the two factors of LM × LM. Hence, C × defines a closed (d − 2)-current on SM × SM. Let (σ, σ¯ ) be a point in SM × , with p the (single) intersection point. In suitable coordinates on M σ˙ (p) = ∂1 (p) and σ¯˙ (p) = ∂d (p). We ¯ = x d (·) − x d (p), define local coordinates on SM using ψ(·) = x 1 (·) − x 1 (p) and ψ(·) as explained in Sect. 3. At (σ, σ¯ ), we then find the local expression × C(σ, σ¯ ) =

d−1 

(−1)k ε1ν ...ν ν¯ ...¯ν d (k − 1)!(d − k − 1)! 2 k k+1 d−1

k=1 − ν2

× d σ (0) . . . d− σ νk (0)d− σ¯ ν¯ k+1 (0) . . . d− σ¯ ν¯ d−1 (0) × δ(σ 2 (0) − σ¯ 2 (0)) . . . δ(σ d−1 (0) − σ¯ d−1 (0)). We must check that this is the current corresponding to

SM × ;

(see Appendix A).

(46)

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1. C × is localized on SM × , since, as one can see from (45), it vanishes when the two strings do not intersect. 2. A tangent vector, v + v, ¯ at (σ, σ¯ ) is parallel to SM × iff there exist real numbers α and α¯ such that v(0) + α σ˙ (0) = v(0) ¯ + α¯ σ¯˙ (0). (47) A simple calculation shows that C × is transverse to SM × , i.e., for all vectors π = v + v¯ fulfilling (47), one has × ιπ C(σ, σ¯ ) = 0.

(48)

3. Comparing (46) to Eq. (99) in Appendix A, we see that the regular part of C × at (σ, σ¯ ) is given by × C (σ,σ¯ ) =

d−1  k=1

(−1)k ε1ν ...ν ν¯ ...¯ν d (k − 1)!(d − k − 1)! 2 k k+1 d−1

¯ . . . d− σ¯ ν¯ d−1 (0), × d− σ ν2 (0) . . . d− σ νk (0)d− σ¯ ν¯ k+1 (0)

(49)

and the localization functions are given by f1 = σ 2 (0) − σ¯ 2 (0)

...

fd−2 = σ d−1 (0) − σ¯ d−1 (0).

(50)

It is easy to see that at (σ, σ¯ ), × − 2 | · ; C ¯ 2 (0)) . . . d− (σ d−1 (0) − σ¯ d−1 (0))|. (σ,σ¯ ) | = | · ; d (σ (0) − σ

(51)

 : SM × −→ SM

(52)

Let

be the map that associates to two intersecting strings their concatenation, with an appropriate scaling of the velocity vectors, as shown in Fig. 2. This map is nearly everywhere well-defined, namely on pairs of strings with one self-intersection, but n-valued when the two strings intersect n times.

Fig. 2. The map 

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Fig. 3. The definition of the string bracket

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The string bracket is defined on string homology by5 (see also Fig. 3) M {·; ·} : Hi M × Hi¯ M −→ Hi+i+2−d ¯

  ¯ (a, a) ¯ −→ {a; a} ¯ = (−1)i(d+i)  (a × a) ¯ ∩C × SM × .

(53)

The rˆole of C × is to orient the cycle obtained by intersecting an appropriately transversal representative a × a¯ with SM × ; see Appendix A. The sign factor appearing in (53) is chosen in such a way that the bracket is even, for even d, and odd, for odd d; in fact, it then satisfies: 1. Antisymmetry ¯ {a; ¯ a}, {a; a} ¯ = −(−1)(|a|+d)(|a|+d)

(54)

as can be checked by exchanging the factors in (53), and using Ex∗ C × = (−1)1+d C × , with Ex the map that permutes the factors in SM × SM. 2. Jacobi identity {a; {b; c}} = {{a; b}; c} + (−1)(|a|+d)(|b|+d) {b; {a; c}},

(55)

(see Appendix B for a proof). Here the degree | · | of a cycle is its dimension. Consider the symmetric algebra S(H∗ M) over H∗ M, with the grading given by | · |. Extending the bracket as a superderivation, namely in such a way that the 3. Leibniz rule

{a, bc} = {a, b}c + (−1)|b|(|a|+d) b{a, c}

(56)

is fulfilled, one finds that S(H∗ M) is  a super-Poisson algebra (even bracket), for d even,  a Gerstenhaber algebra (odd bracket), for d odd. 6. A Peculiarity of GL(n, C) In this section we highlight a property of GL(n, C) which will be needed in Sect. 7. Let G = GL(n, C), and let ρ denote its standard representation. We define an invariant bilinear form κ as the trace in this representation:

It then follows that

κab = κ(Ta , Tb ) = tr [ρ(Ta )ρ(Tb )] .

(57)

κ ab ρ(Ta ) ⊗ ρ(Tb ) v ⊗ w = w ⊗ v,

(58)



where v and w are vectors in the representation space of ρ. In components with respect to a basis in this space the above identity reads

p (59) κ ab ρ(Ta )rp ⊗ ρ(Tb )sq = δqr δs . 5 Our definition differs from that described by Chas and Sullivan by a sign given by {a; a} ¯ = {a; ¯ a}Chas−Sullivan .

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Fig. 4. Pictorial representation of (61)

To prove this identity, we define a basis {Eij |i, j = 1..n} of ᒄᒉ(n, C) by setting ρ(Eij )rs = δir δj s . For this basis, one finds that κ(Eij , Ekl ) = δil δj k . Equation (59) then follows immediately. In the following, expressions of the form tr ρ [A1 Ta A2 ] κ ab tr ρ [B1 Tb B2 ]

(60)

will appear, where ρ is a representation of G, {Ta } is a basis of ᒄ, and A. , B. are elements of Uᒄ. For G = GL(n, C) and ρ the standard representation, such expressions can be simplified using (59), as pictorially represented in Fig. 4: tr ρ [A1 Ta A2 ] κ ab tr ρ [B1 Tb B2 ] = tr ρ [A1 B2 B1 A2 ] .

(61)

7. An Algebra Homomorphism from S(H∗ M) to Hδ∗ O In this section we show that the map Sh : S(H∗ M) −→ Hδ∗ O a1 . . . ak −→ a1 , hρ;A  . . . ak , hρ;A ,

(62)

which associates to a cycle in string homology the corresponding observable of the topological field theory, based on the group GL(n, C) in the standard representation, is a super-Poisson/Gerstenhaber algebra homomorphism. This is accomplished by establishing the following properties: i) |a| = |a, hρ;A |, ii) {a; a}, ¯ hρ;A  = {a, hρ;A ; a, ¯ hρ;A }.

(63) (64)

Property i) follows from (36). Property ii), is proven in several steps: Step 1. Applying (21), one finds that ¯ ¯ = (−1)|a|(d+|a|) ¯ {a, h; a, ¯ h} a × a, ¯ {h; h}.

(65)

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¯ on LU × LU . First one verifies that Step 2. We derive a local expression for {h, h}  d  h(C + tη) dt t=0 d
s=1   tr hol(C)|s0 = k=0 s=0

 1 γ˙ µ1 (s)dsd− γ µ2 (s) . . . d− γ µk (s)ηµ1 µ2 ...µk (γ (s)) hol(C)|1s . (k − 1)! (66) →

←

δ h, h δ ∈ O(∗ (LM) · M · Uᒄ ), Using (13), one finds the local expressions for δC E E E δC namely →
s=1 d  (−1)(k+1)(d+1) δ h= ds δ (d) (γ (s) − x)εν1 ν2 ...νk µk+1 ...µd δC (k − 1)!(d − k)! s=0 k=1

×dx µk+1 . . . dx µd γ˙ ν1 (s)d− γ ν2 (s) . . . d− γ νk (s)   ×tr hol(C)|s0 Ta hol(C)|1s ⊗ κ ab Tb ,

(67)

and
s=1 d−1  (−1) δ h ds δ (d) (γ (s) − x)εµ1 ...µk νk+1 νk+2 ...νd = δC (k)!(d − k − 1)! s=0 ←

k=0

×dx µ1 . . . dx µk γ˙ νk+1 (s)d− γ νk+2 (s) . . . d− γ νd (s)   ×tr hol(C)|s0 Ta hol(C)|1s ⊗ κ ab Tb .

(68)

¯ ∈ O(∗ (LM × LM)E ) Equation (17) yields the local expression for {h; h}
s=1 d−1  (−1)d+1 ¯ = ds {h; h} (k − 1)!(d − k − 1)! s=0 k=1
s¯=1 d s¯ δ (d) (γ (s) − γ¯ (¯s )) εν1 ν2 ...νk ν¯ k+1 ν¯ k+2 ...¯νd × s¯ =0 ν1

ν¯ ×γ˙ (s)d− γ ν2 (s) . . . d− γ νk (s)γ˙¯ k+1 (¯s )d− γ¯ ν¯ k+2 (¯s ) . . . d− γ¯ ν¯ d (¯s )  s¯ 1      s 1 ab ¯ ¯ ×tr hol|0 Ta hol|s κ tr hol Tb hol . 0

s¯

(69)

¯ defines a form on SM × SM. It can be written Step 3. From (69) one sees that {h; h} using the current C × , i.e. ¯ = C × · H, {h; h} (70) × where H is a form on SM × SM, such that its restriction on SM reads  ×   ×  ¯ Tb ; H = tr hol Ta κ ab hol (71) ¯ × the generalized here hol× denotes the generalized holonomy of the first string and hol holonomy of the second string, both starting from their common intersection point. For G = GL(n, C) in the standard representation one finds, using (61),   ¯ × . (72) H = tr hol× hol

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Step 4. From (70) and (103) one finds that ¯ = a × a, a × a, ¯ {h; h} ¯ C × · H = (a × a) ¯ ∩C × SM × , H.

(73)

Moreover, one has that

  ¯ {a; a}, ¯ h = (−1)|a|(d+|a|)  (a × a) ¯ ∩C × SM × , h.

(74)

Thus, to prove (64), we simply have to show that   (a × a) ¯ ∩C × SM × , H =  (a × a) ¯ ∩C × SM × , h,

(75)

which holds, as described in (104), if   ¯ × (σ,σ¯ ) = ∗ , h(σ,σ¯ ) , tr hol× hol

(76)

for any (σ, σ¯ ) ∈ SM × and any parallel multivector  ∈ ∗ T(σ,σ¯ ) SM × . The validity of the latter follows immediately from the reparametrization invariance of hol. The theorem is thus proven. 8. Outlook In this section we outline various extensions and generalizations of the results proven in this paper. 8.1. Generalizations to other groups. We start by describing some ideas about how to generalize the results of this article by replacing GL(n, C) with an arbitrary Lie group. Inspiration is taken from [10]. A chord diagram (see Fig. 5) is a union of disjoint oriented S 1 -circles and disjoint arcs, with the endpoints of the arcs on the circles. A chord diagram on a manifold M (see Fig. 5) is a (continuous) map from a chord diagram to M such that each arc is mapped to a single point in M (that is, each arc is mapped to an intersection of strings in M), modulo the obvious action of S 1 on any circle. Let ch(M) be the space of chord diagrams on M. It can be viewed as a “manifold” with singularities (just like for SM), and boundaries when two different crossings between circles approach one another along one of the circles (see Fig. 6). One then defines a boundary operator, ∂ ch(M) on cells in ch(M) in such a way that the so called 4T -relation, represented in Fig. 7, is respected.

Fig. 5. A chord diagram on M

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413

Fig. 6. Approaching a boundary on ch(M)

Fig. 7. 4T -relations

The chord homology H∗ch M is the homology of ch(M) with respect to ∂ ch(M) . In analogy to SM × ⊂ SM × SM one defines ch(M)× ⊂ ch(M) × ch(M) as the space of pairs of chord diagrams on M whose strings intersect at least once. Similarly to  : SM × −→ SM one defines the (generally multivalued) map ch : ch(M)× −→ ch(M),

(77)

which associates to a pair of chord diagrams on M with one intersection point the union of the two chord diagrams with a new arc corresponding to the intersection (and in an analogous way for multiple intersection points). As in Eq. (53), one defines a bracket ch {·; ·} : Hich M × Hich M ¯ M −→ Hi+i+2−d ¯ ¯

(a, a) ¯ −→ (−1)i(i+d) ch ((a × a) ¯ ∩C × ch(M)× ),

(78)

which is a bracket/antibracket for d even/odd; the current C × on ch(M)× can be constructed in a similar way as in Sect. 5. Similarly to S(H∗ M), it is possible to define a super-Poisson/Gerstenhaber algebra S(H∗ch M). In analogy to (62), we define a map

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Fig. 8. The form h associated to (a part of) a chord diagram; t1 and t2 refer to a S 1 -parametrization of the circles

S(hch,G ) : S(H∗ch M G ) −→ Hδ∗ OG a1 . . . ak −→ a1 , hch  . . . ak , hch ,

(79)

where H∗ch M G denotes the homology of chord diagrams with circles labeled by representations of G. The form hch is defined as explained in Fig. 8. The map (79) is a super-Poisson-/Gerstenhaber algebra homomorphism. This can be proved by the same reasoning as that in Sect. 7 and in [10]. The content of [10] concerns the special case of the above construction for manifolds M of dimension d = 2 and for H0ch M ⊂ H∗ch M. The symmetric algebra on string homology, S(H∗ M), is obtained by taking the quotient of S(H∗ch M) by the ideal I generated by the diagrams of Fig. 9. One then sees that the following diagram is commutative:

S(H∗ch M) πI ? S(H∗ M)

) S(hch,GL(n,C)Hδ∗ OGL(n,C) *     

Fig. 9. The “GL(n, C)”-ideal I

Topological Field Theory Interpretation of String Topology

415

8.2. Generalization to nontrivial principal bundles. In this subsection we explain how to extend methods and results of this paper to the situation where P is a non-trivial bundle with base space M and thus not necessarily admits a flat connection. A principal bundle is determined by its “transition functions” tij : Ui ∩ Uj −→ G

(80)

defined on intersections of two coordinate patches of M, and with the property that tij tj k = tik

on Ui ∩ Uj ∩ Uk .

(81)

Two sets of transition functions t, t˜ describe the same bundle iff there exist “gauge transformations” gi : Ui −→ G (82) such that

tij = gi t˜ij gj−1 ,

on Ui ∩ Uj .

(83)

A connection on P associates to every patch a ᒄ-valued one-form Ai ∈ 1 (Ui ) ⊗ ᒄ, such that

Ai = tij Aj tij−1 + tij dtij−1

on Ui ∩ Uj .

(84) (85)

The curvature, F , of the connection A is given, on every patch, by a ᒄ-valued two-form 1 Fi = dAi + [Ai , Ai ] ∈ 2 (Ui ) ⊗ ᒄ, 2 such that

Fi = tij Fj tij−1 ,

on Ui ∩ Uj .

(86)

(87)

The forms C are ᒄE -valued forms. On every coordinate patch, C is given by Ci ∈ ∗ (Ui ) ⊗ ᒄE , with the property that

Ci = tij Cj tij−1 ,

on Ui ∩ Uj .

(88) (89)

A principal bundle is trivial iff one can choose trivial transition functions: tij = 1, for all Ui , Uj , with Ui ∩ Uj = ∅. The connection, the curvature and the forms C are then globally defined on M. We now turn our attention to the master action and the bracket of the topological field theory. The forms on the patches   1 1 si = tr ρ Ci (Fi + dAi Ci + Ci2 ) ∈ ∗ (Ui )E (90) 2 3 satisfy si = sj on Ui ∩ Uj , and thus yield a globally defined form s on M. We may therefore define a master action, S, by  

1 1 S= s= tr ρ C(F + dA C + C 2 ) ∈ ∗ (M)E . (91) 2 3 M M

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The bracket is well defined, since one has {Ci ; Ci } = {Cj ; Cj },

(92)

a consequence of the invariance of the bilinear form κ under the adjoint action of G on

ᒄ. The master action still satisfies the master equation {S; S} = 0. Furthermore,

{S; Ci } = δCi = (−1)d (Fi + dAi Ci + Ci2 ).

(93)

We now address the task of defining generalized parallel transporters and generalized Wilson loops. They can be defined as elements of ∗ Tγ LM, for each loop γ ∈ LM. Let 0 = t0 < t1 < . . . < tk−1 < tk = 1, and let U1 , . . . , Uk = U1 be patches on M such that γ (t) ∈ Ui , for t ∈ [ti−1 , ti ]. One then defines the trace of the generalized Wilson loop as t2  t  hρ;A (C) = tr ρ holA1 (C1 )01 t12 holA2 (C2 ) . . . holAk−1 t1 1  tk−1 . (94) (Ck−1 )|tk−2 tk−1,k holAk (Ck )t k−1

t The factors holAi (Ci )ti are defined as in (33). It is easy to see that this definition does i−1 not depend on the choice of the charts and is invariant under gauge transformations. One then shows that
1   d− hρ;A = (95) dτ tr ρ holA (C)|τ0 ιγ˙ ev∗τ (F + dA C + C 2 ) holA (C)|1τ , 0

where the τ -integral has to be split, as in (94), if the loop crosses different patches. Comparing (94) and (93), one finds that the fundamental identity (39) is fulfilled: ((−1)d δ + d− )hρ;A = 0.

(96)

8.3. Remarks on quantization. The construction we have described in this paper yields, in the case of an even-dimensional manifold M, a Poisson algebra of observables (related to the string topology of M if we choose GL(n) as our Lie group). It is then natural to ask if and how this Poisson algebra may be quantized. We sketch in this section a few approaches that might help understanding this problem. 8.3.1. Path-integral quantization. If d = dim M is even, our approach describes the BRST formalism for a field theory in the Hamiltonian formalism with the functional S [d] as the BRST generator. If we want to quantize this theory using path-integrals, we must first move to the Lagrangian formalism. As explained in Appendix D, the corresponding action functional on N = M × I is S [d+1] . In the case d = 2, this is the BV action for Chern–Simons theory, and this is in accordance with the fact that Chern–Simons theory provides a quantization of the Goldman [12] bracket (the 2-dimensional version of the string bracket), see [13]. In higher dimensions, S [d+1] defines new topological quantum field theories (TQFT), among which we have the so-called BF theories [3, 4] which can be obtained by particular choices of the metric Lie algebra. Our observables for strings on M have then to be lifted to the corresponding observables on N = M × I (or, more generally, on a (d + 1)-dimensional manifold N ).

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The formulae we have given in odd dimensions describe this algebra of observables. Notice however that, in order to avoid singularities in the computation of expectation values, one has to restrict oneself to imbedded strings in N (and possibly also to introduce a framing). In the particular case of BF theories, the expectation values of these observables correspond to the cohomology classes of imbedded strings considered in [14], as shown in [6, 7]. As a consequence, the quantization of the string topology of M must be related to the homology of the space of imbedded strings in M × I . This space must then be endowed with the structure of the associative algebra in such a way that its commutator yields, in the classical limit, the Poisson bracket of the projections of the strings to M. 8.3.2. Deformation quantization. For d = 2 and M non-compact, the ideas described above have an explicit realization in terms of deformation quantization (i.e., working with formal power series in h ¯ ), as described in [13]. The construction is based on the Kontsevich integral for link invariants [15] which is the perturbative formulation of Chern–Simons theory in the holomorphic gauge studied in [16]. The higher-dimensional generalization of this approach should be obtained by considering perturbative expansions, in a suitable gauge, of the corresponding TQFTs. This should be related to (a generalization of) the diagram technique (“graph homology”) developed by Kontsevich [17]. 8.3.3. Geometric quantization. In some cases (e.g., BF theories), the Poisson subalgebra of functionals commuting with S [d] is the algebra of a reduced phase space of generalized gauge fields on M. This space inherits a symplectic structure and one may try to quantize it using deformation quantization and produce a TQFT in Atiyah’s sense. In the 2-dimensional case, when the reduced phase space turns out to be the space of flat connections on M modulo gauge transformations, this program works (at least for compact groups). One may regard quantum groups as one of its outcomes. It would be very interesting to understand if the higher-dimensional case produces interesting generalizations thereof.

A. Intersection of Cycles and Currents In this section we explain some concepts and manipulations used in the proof of Eq. (64) in Sect. 7. Let A be a manifold and A× an oriented immersion of codimension n, which defines an element of the homology, H∗ A, of A. Let C × be the current that localizes on this immersion, i.e., a singular n-form on A with the following properties: 1. The form localizes on A× , i.e. for any point p not in A× one has Cp× = 0.

(97)

2. The form is transverse, i.e., for every point p on A× and an arbitrary parallel tangent vector P (p) ∈ Tp A× one has ιP (p) Cp× = 0.

(98)

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(c) Let A× be defined, locally, as the zero-set of functions f1 , . . . , fn , with df1 . . . dfn = 0. Then the current C × is given by × δ(f1 ) . . . δ(fn ), C× = C

(99)

× is a regular form, and for every point p in A× and every multivector where C V ∈ Tp A,    ×  = |V , df1 . . . dfn | . (100) V , C In particular, C × defines an orientation on the normal bundle, N (A× ), of A× in A. Given an i-cycle a ∈ Hi (A), one can define a new cycle by considering the intersection a ∩C × A× ∈ Hi−n (A).

(101)

As a set, it is obtained by intersecting an appropriate representative of a with A× . The orientation is defined as follows: Let p be a point in this intersection, P ∈ i−n Tp (a ∩ A× ) the multivector that is the infinitesimal version at p of a ∩ A× , T ∈ n Tp a the multivector in the normal bundle to A× such that T ∧ P is the infinitesimal version of a at p. Then one defines or a∩C × A× (P ) = or a (T ∧ P ) · or N(A× ) (T ),

(102)

where or N(A× ) is given by the current C × . For any closed form H on A, one has that a, C × H  = a ∩C × A× , H .

(103)

Next, let  be a map from A× into some other manifold B, and h a closed form on B. If for an arbitrary point p in A× and any parallel multivector P ∈ ∗ Tp A× , one has that

then

P , H p = ∗ P , h(p) ,

(104)

a ∩C × A× , H  = (a ∩C × A× ), h.

(105)

B. The Jacobi Identity for the String Bracket In this appendix we show how to prove the Jacobi identity for the string bracket of Sect. 5. We first rewrite the Jacobi identity as (−1)η(abc) {{a; b}; c} + cycl.(abc) = 0,

(106)

where the sign factor is η(abc) = (|a| + d)(|c| + d). We can define the first term as (−1)η(abc) {{a; b}; c} = (−1)η(abc)+σ (abc) 



× (1,23) a (1) × b(2) × c(3) ∩C ×(12) ∧C ×(13) SM ×(12) ∩ SM ×(13)



 + (2,13) a 1 × b2 × c3 ∩C ×(12) ∧C ×(23) SM ×(12) ∩ SM ×(23) . (107)

Topological Field Theory Interpretation of String Topology

419

Let us first explain the objects that appear in the above definition. The sign factor is σ (abc) = (|b|(d + |a|) + |c|(|a| + |b|), which follows from the definition of the string bracket, (53). a (1) ×b(2) ×c(3) is a cycle in SM (1) ×SM (2) ×SM (3) . A point in SM ×(ij ) is a triple of strings, (σ1 , σ2 , σ3 ) ∈ SM (1) × SM (2) × SM (3) , such that the i th and the j th intersect at least once. C ×(ij ) is the corresponding current. (i,j k) is the map (i,j k) : SM ×(ij ) ∩ SM ×(ik) −→ SM ×

(108)

which opens the intersections between the i th and the j th and between the i th and the k th string, in the same way as the map  in (52) does. Now consider the two terms appearing in (106) corresponding to the cycle a intersecting both the cycles b and c. The first term corresponds to the first term in (107). The second one appears in (−1)η(cab) {{c; a}; b} and reads (−1)η(cab)+σ (cab)



×(2,13) c(1) × a (2) × b(3) ∩C ×(12) ∧C ×(23) SM ×(12) ∩ SM ×(23) . (109) To prove that the Jacobi identity holds, we only have to prove that two such terms add up to zero. We first write the second term, rearranging the indices and bringing the cycles into a convenient order, i.e., (−1)η(cab)+σ (cab) (−1)|c|(|a|+|b|)



×(1,32) a (1) × b(2) × c(3) ∩C ×(31) ∧C ×(12) SM ×(12) ∩ SM ×(23) ;

(110)

then we bring the currents into a convenient form (−1)η(cab)+σ (cab) + (−1)|c|(|a|+|b|)+1



× (1,23) a (1) × b(2) × c(3) ∩C ×(12) ∧C ×(13) SM ×(12) ∩ SM ×(23) , (111) using that |C ×(ij ) | = d and C ×(ij ) = (−1)d+1 C ×(j i) . What remains to be shown is thus that ! η(abc) + σ (abc) + η(cab) + σ (cab) + |c|(|a| + |b|) + 1 = 1, (112) which is easily seen to hold. C. Local Expression for the Generalized Parallel Transporters In local coordinates (γ µ (t))t∈S 1 the generalized holonomy reads tf holnA (C)ti =




∞ 

tf

n1 ,...,nn =1 (t1 ,...,tn )∈ n |ti 1

1

× holA |t1i γ˙ µ1 (t1 )dt1 d− γ µ2 (t1 ) . . . d− γ ... t

n

n

µ1n

1 (t1 )C 1 1 µ1 µ2 ...µ1n1 (γ (t1 )) n

t

holA |t21 tf

× holA |tnn−1 γ˙ µ1 (tn )dtn d− γ µ2 (tn ) . . . d− γ µnn (tn )Cµn µn ...µnn (γ (tn )) holA |tn , n 1 2 (113) t

where d− is the differential on LM.

420

A.S. Cattaneo, J. Fr¨ohlich, B. Pedrini

D. BV/BRST In this appendix we explain the relationship between S [d+1] and S [d] , where d is an even number. We follow [8]. For notational simplicity, we omit the Lie algebra part of the forms. Let N  x be an oriented manifold with dim N = d even, and M = I × N  (t, x) with the product orientation. Let us write the fields on M as C = dtCt + D =

d  k=0

 1 1 dx i1 . . . dx ik Cti1 ...ik + dx i1 . . . dx ik Di1 ...ik . (114) k! k! d

dt

k=0

From (22) it follows that, in the BV-formalism, one can choose as fields and corresponding antifields, respectively, Di1 ...ik ←→

1 ε i1 ...ik ik+1 ...id Ctik+1 ...id . (d − k)!

(115)

After choosing a gauge in which the connection A has vanishing time component, At = 0, the master action in the Lagrangian formalism reads

1 ˙ [d+1] t (116) S [C , D] = dt (dA D + D 2 )Ct + DD. 2 I N A gauge-fixing functional [D] (|| = 1) defines a gauge-fixed action  [d+1] [D] S

=S

[d+1]

Ctik+1 ...id

→ 1 i1 ...ik ik+1 ...id δ = , D . ε k! δDi1 ...ik

(117)

For a gauge-fixing functional adapted to the “space-time” split M = I × N of the form
(118) [D] = − dt K[D], I

where K is some functional of D, with D interpreted as a form on N , one finds that  

1 [d+1] [d] ˙ DD , (119) S [D] = dt {S , K}N + 2 N I with S

[d]

1 [D] = 2




2 DdD + D 3 . 3 N

(120)

We remark that the gauge fixed action (119) is already in Hamiltonian form, since it is of first order in time derivatives. Since   → = (−1)k (dD + D 2 )i1 ...ik (t, x) (121) {S [d+1] ; Di1 ...ik (t, x)} δ Ct = δD 

and {S [d] ; Di1 ...ik (x)} = (−1)k (dD + D 2 )i1 ...ik (x),

(122)

Topological Field Theory Interpretation of String Topology

421

S d can be interpreted as the BRST-generator in the Hamiltonian formalism, and (119) is the gauge fixed action for a theory with vanishing Hamiltonian: the first term is the gauge-fixing term, while the second term can be written as

1 1 (−1)k i1 ...ik ik+1 ...id ˙ (123) Dik+1 ...id (t, x) Di1 ...ik (t, x), ε dt    2 I N k! (d − k)!   (t,x)  ˙ (t,x)

which is exactly the desired expression (considering  and  as conjugate variables), as can be inferred from (22): (−1)k i1 ...ik ik+1 ...id j j {Dj1 ...jk (x); Dik+1 ...id (y)} = δ (d) (x − y)δi11 . . . δikk . ε    (d − k)!    (x)

(124)

(y)

Acknowledgements. B. P. thanks Carletto Rossi for useful discussions about generalized holonomies. A. S. C. acknowledges a three-month invitation at Harvard University during the Fall Term 2001, and thanks Raoul Bott and David Kazhdan for stimulating discussions. A. S. C. acknowledges partial support by SNF Grant No. 20-63821.00.

References 1. Chas, M., Sullivan, D.: String Topology. Preprint math.GV/9911309 2. Cohen, R.L., Jones, J.D.S.: A Homotopy Theoretic Realization of String Topology. Preprint math.GT/0107187 3. Schwarz, A.S.: The Partition Function of Degenerate Quadratic Functionals and Ray–Singer Invariants. Lett. Math. Phys. 2, 247 (1978) 4. Blau, M., Thompson, G.: Topological Gauge Theories of Antisymmetric Tensor Fields. Ann. Phys. 205, 130 (1991) 5. Witten, E.: Quantum Field Theory and the Jones Polynomial. Commun. Math. Phys. 121, 351 (1989) 6. Cattaneo, A.S., Rossi, C.A.: Higher Dimensional BF Theories in the Batalin-Vilkovisky formalism: The BV action and Generalized Wilson Loops. Commun. Math. Phys. 221, 591 (2001) 7. Cattaneo, A.S., Cotta-Ramusino, P., Rossi, C.A.: Loop Observables for BF Theories in any Dimension and the Cohomology of Knots. Lett. Math. Phys. 51, 301 (2002) 8. Henneaux, M., Teitelboim, C.: Quantisation of Gauge Theories. Princeton, NJ: Princeton University Press, 1992 9. Aleksandrov, M., Schwarz, A., Zaboronsky, O., Kontsevich, M.: The Geometry of the Master Equation and Topological Quantum Field Theory. Int. J. Mod. Phys. A. 12, 1405 (1997) 10. Andersen, J.E., Mattes, J., Reshetikhin, N.: The Poisson Structure on the Moduli Space of Flat Connections and Chord Diagrams. Topology 35, 1069 (1996) 11. DeWitt, B.: Supermanifolds. Cambridge: Cambridge University Press, 1984 12. Goldman, W.: Invariant Functions on Lie Groups and Hamiltonian Flows of Surface Group Representations. Invent. Math. 85, 263 (1986) 13. Andersen, J.E., Mattes, J., Reshetikhin, N.: Quantization of the Algebra of Chord Diagrams. Math. Proc. Camb. Phil. Soc. 124, 451 (1998) 14. Cattaneo, A.S., Cotta-Ramusino, P., Longoni, R.: Configuration Spaces and Vassiliev Classes in Any Dimension. Preprint math.GT/9910139 15. Kontsevich, M.: Vassiliev’s Knot Invariants. Adv. Sov. Math. 16, 137 (1993) 16. Fr¨ohlich, J., King, C.: The Chern–Simons Theory and Knot Polynomials. Commun. Math. Phys. 126, 167 (1989) 17. Kontsevich, M.: Feynman Diagrams and Low-Dimensional Topology. In: First European Congress of Mathematics, Paris 1992, Volume II, Progress in Mathematics 120, Boston: Birkh¨auser, 1994, p. 97 Communicated by R.H. Dijkgraaf

Commun. Math. Phys. 240, 423–446 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0913-6

Communications in

Mathematical Physics

Berezin Quantization and K-Homology Erik Guentner
,

Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202-3216, USA Received: 18 October 2001 / Accepted: 13 January 2003 Published online: 13 August 2003 – © Springer-Verlag 2003

Abstract: The E-theory defined by Connes and Higson provides a realization of K-homology, the generalized homology theory dual to K-theory, based on the notion of asymptotic homomorphisms. With this realization it becomes possible to associate a K-homology element to a quantization scheme. In this article we associate an asymptotic homomorphism and K-homology element to the Berezin quantization of a bounded symmetric domain. Further, we identify this element with the element of K-homology defined by the Dolbeault operator of the domain. 1. Introduction The E-theory defined by Connes and Higson [CH90] provides a realization of K-homology, the generalized homology theory dual to K-theory, based on the notion of asymptotic homomorphisms between C ∗ -algebras. The theory has found a number of applications to index theory through the work of Higson [Hig93] and Higson-Kasparov-Trout [HKT98]. Further, it plays an important role in the recent proof given by Higson-Kasparov of the Baum-Connes Conjecture for amenable groups [HK97, HK01]. The E-theory groups are defined to be certain groups of homotopy classes of asymptotic homomorphisms. In this paper we study the possibility of associating asymptotic homomorphisms, and hence elements of E-theory groups to quantization schemes. In this way E-theory groups become the receptacle of topological invariants of quantization schemes. In an earlier paper we initiated the study of the relationship between E-theory and quantization by analyzing the Berezin-Wick quantization of the complex plane from this perspective [Gue00]. The results of that work are rather satisfying. To the quantization scheme we associate an asymptotic homomorphism and thereby an element of an


The author was supported by the NSF through an MSRI Postdoctoral Fellowship and other grants. Current address: Department of Mathematics, University of Hawai‘i, M¯anoa, 2565 The Mall, Keller 401A, Honolulu, HI 96822, USA. E-mail: [email protected]



424

E. Guentner

E-theory group. We further identify this element with the E-theory element defined by ¯ the ∂-operator on the complex plane. In this paper we continue this investigation by studying the Berezin quantization of bounded symmetric domains. We obtain results analogous to those described above for the Berezin-Wick quantization; the quantization defines an element of an appropriate E-theory group, and this element is identified with the element defined by the Dolbeault operator of the domain. Our main theorems are: Theorem A. Let  be a bounded symmetric domain. The Berezin–Toeplitz quantization defines an element of the E-homology of : [ Berezin ] ∈ E(C0 (), C).   Theorem B. Let  be a bounded symmetric domain. The E-homology class of the Berezin quantization equals the E-homology class of the Dolbeault operator: [ Berezin ] = [ Dolbeault ] ∈ E(C0 (), C).   We remark that these classes are nonzero; indeed the E-theory group E(C0 (), C) is isomorphic to the integers, and is generated by the class of the Dolbeault operator. The main property of the Berezin quantization that is used in defining the E-homology element is that the Toeplitz operators used in its definition commute asymptotically as the value of Planck’s constant tends to zero. This property was proven for a slightly restricted class of symbols in a paper of Borthwick-Lesniewski-Upmeier [BLU93, Thm. 2.2] through purely analytic means, and relying on the description of bounded symmetric domains in terms of Jordan algebras. The case of the Poincar´e disk had been considered earlier by Klimek-Lesniewski [KL92, Theorem VI.2]. In the course of defining the E-homology element of the Berezin quantization we prove the following variant of these results using differential geometric techniques (we refer to Sect. 5 for definitions): Theorem C. Let  be a bounded symmetric domain. Let φ and ψ be continuous bounded functions on  and assume that φ admits a continuous extension to . Then T
(φψ) − T
(φ)T
(ψ) → 0,

as
→ 0.  

I would like to thank Nigel Higson and Mohan Ramachandran for interesting discussions on the subject of this paper. 2. Bounded Symmetric Domains We do not assume much familiarity with bounded symmetric domains, and include a brief summary of the relevant aspects of the theory. A more thorough introduction can be found in the books of Helgason and Krantz [Hel78, Kra92]; for more detailed information we refer to the books of Pijatetski-Shapiro, Hua, Loos and Mok [PS69, Hua63, Loo77, Mok89]. A domain  is an open connected subset of Cn . Let  be a bounded domain in n C . Let L2 () be the space of measurable functions, square integrable with respect to Lebesgue measure. The Bergman space is denoted H 2 () and is the subspace of L2 () consisting of the holomorphic functions. Since the domain is bounded H 2 () contains

Berezin Quantization and K-Homology

425

the polynomials in the variables z1 , . . . , zn (holomorphic polynomials). Thus, H 2 () is an infinite dimensional subspace of L2 (). Further, it is closed. The Bergman kernel function of  is defined by the formula K(z, w) ¯ =

∞


φn (z)φn (w),

{φn } an orthonormal basis of H 2 ().

n=0

The Bergman kernel function is independent of the choice of orthonormal basis used to define it, and is holomorphic in z and w. ¯ Note that K(z, z¯ ) =




|φn (z)|2 ,

{φn } an orthonormal basis of H 2 ()

(1)

i

√ is a real-valued function. Since we may take φ1 (z) = 1/ || we conclude that K(z, z¯ ) is bounded below by ||−1 . The Bergman kernel function is used to define the infinitesimal Bergman metric, the Hermitian structure on  defined by the bilinear form h(z) =




hij dzi ⊗ d z¯ j ,

hij (z) =

ij

∂ 2 log K(z, z¯ ) . ∂zi ∂ z¯ j

(2)

(It follows from (1) that the form is Hermitian and positive semi-definite; the proof that it is positive definite is more complicated [Kra92], [Hel78, Ch. VIII, Prop. 3.4].) The associated (1, 1)-form of the infinitesimal Bergman metric is √ √ −1 ¯ −1
ω= hij dzi ∧ d z¯ j . ∂ ∂ log K(z, z¯ ) = 2 2

(3)

ij

A Hermitian structure is K¨ahler if its associated (1, 1)-form is closed. Thus, with its Hermitian structure  is a K¨ahler manifold. When applied to a bounded domain  differential geometric terms (isometry, completeness, etc) will always be interpreted with respect to the Riemannian structure underlying this K¨ahler structure. Proposition 2.1 ([Hel78, Ch. VIII, Prop. 3.5]). Let  be a bounded domain in Cn . Holomorphic automorphisms of  are isometries. A bounded domain  in Cn is symmetric if each point of the domain is the isolated fixed point of an involutive holomorphic automorphism of the domain; it is homogeneous if the group of its holomorphic automorphisms acts transitively. Every bounded symmetric domain is homogeneous [Loo77, Mok89]. Finally, observe that when equipped with the infinitesimal Bergman metric a bounded symmetric domain is a Hermitian symmetric space and, in particular, a complete manifold. Indeed, by the previous proposition, holomorphic automorphisms are isometries. We require two further facts concerning the geometry of bounded symmetric domains. The first is that the volume form of a bounded symmetric domain is, up to a constant, the product of the Bergman kernel function and Lebesgue meausure. We can safely ignore the constant when considering only one domain at a time. Note that the following lemma remains valid in the broader setting of homogeneous domains.

426

E. Guentner

Lemma 2.2 ([Ber74, Ch. VIII, Prop. 2.5 & 3.6, Hel78, Thm. 5.1].). Let  be a bounded symmetric domain and let dvol denote its volume form computed with respect to the induced Riemannian structure. Then dvol = k K(z, z¯ ) dx1 ∧ dy1 ∧ . . . dxn ∧ dyn ,  

where k is a constant depending only on the domain .

The second and final fact we require is the following result of Donnelly (compare [Don97, Prop. 3.2] which is actually more general than the result stated here). By combining it with an elegant method of Gromov [Gro91], Donnelly proved a vanishing theorem for the Dolbeault operator. We will use the result to provide estimates on spectral functions of twisted Dolbeault operators. We thank M. Ramachandran for bringing this result to our attention; since it is not completely explicit in Donnelly’s paper we include a sketch of the proof. Theorem 2.3. Let  be a bounded symmetric domain. The 1-form ∂¯ log K(z, z¯ ) is bounded (pointwise uniformly with respect to the metric on the cotangent space induced by the K¨ahler metric on ). Sketch of Proof. Let α(z) = ∂¯ log K(z, z¯ ). The idea of the proof is to use the transitive group of holomorphic automorphisms of  and the transformation law for the Bergman kernel with respect to such automorphisms [Hel78, Ch. VIII.3] to equate the norm of α(z) to the norm of α(0) plus a correction term. Precisely, if φz is a holomorphic automorphism of  mapping the origin to z then K(u, w) ¯ = K(φz (u), φz (w))Jφz (u)Jφz (w),

for all u, w ∈ ,

where Jφz is the complex Jacobian of φz . In particular,

α(z) = (φz∗ α)(0) = ∂¯ log K(φz (w), φz (w))(0)

≤ 2 ∂¯ log Jφz (0) + α(0) . Up to this point we have used only that the domain is homogeneous; it remains to show that for a bounded symmetric domain ∂¯ log Jφz (0) is bounded independently of z ∈ . This follows from more general facts as described by Donnelly [Don97]. Alternatively, when dealing with a specific domain, direct and elementary verification of this fact is often possible using precise knowledge of φz .   3. Berezin Quantization In this section we review the construction of the Berezin quantization of bounded symmetric domains. The primary sources for this material are the original papers of Berezin [Ber74, Ber75a, Ber75b]. Let  be a bounded symmetric domain and let K(z, z¯ ) be the Bergman kernel of  defined in the previous section. Define a family of measures on  by dµ
= c(
) K(z, z¯ )1−1/
dλ, where dλ is the ordinary Lebesgue measure and c(
) is a normalization constant that insures that the µ
-measure of  is one. It is worth noting that for
= 1 we obtain the normalized Lebesgue measure on ; as
→ 0 the measure concentrates at the origin.

Berezin Quantization and K-Homology

427

For each of the measures dµ
we consider the space of measurable square integrable functions, as well as the subspace of holomorphic (square-integrable) functions: L2
() = { functions square-integrable with respect to dµ
}, H
2 () = { holomorphic functions in L2
() }. Let C0 () denote the C ∗ -algebra of continuous functions on  vanishing at infinity. For φ ∈ C0 () we define the Toeplitz operator on H
2 () with symbol φ to be the composition multiply by φ project H
2 () −−−−−−−−−→ L2
() −−−−→ H
2 ().

(4)

Denoting the Hilbert space projection from L2
() to H
2 () by Q
we have T
(φ)(u) = Q
(φu), for all u ∈ H
2 (). The Berezin quantization is defined by associating to the function φ ∈ C0 () the family of Toeplitz operators on H
2 (): B(φ) = { T
(φ) }.

(5)

(This definition is not quite the same as the one given by Berezin [Ber74, Ber75a]; instead of considering Toeplitz operators themselves he associates to them their contravariant symbols, functions on . In this way he constructs a family of products on Cc∞ () which form his quantization.) Our first goal is to define the E-homology class of the Berezin quantization by associating to it a generalized asymptotic morphism. The content of this statement is: (i) T
: C0 () → K(H
2 ()) is a ∗-linear contraction, (ii) T
(φ)T
(ψ) − T
(φψ) → 0 as h → 0 (in norm in the respective B(H
2 ())), and further that we can endow the collection of Hilbert spaces H
2 () with the structure of a continuous field of Hilbert spaces { H
2 () } in such a way that (iii) the family of Toeplitz operators { T
(φ) } is a continuous section of the field of elementary C ∗ -algebras associated to the field { H
2 () }.

Note that Property (i) is a standard consequence of the Toeplitz construction. Each of properties (ii) and (iii) will be established in Sect. 5 (see Thms. 5.7 and 5.8); the discussion will be based on spectral properties of a family of twisted Dolbeault operators on . We pause to remark that, although we will not do so, it is possible to give analytic proofs of properties (ii) and (iii) as well. Indeed, property (ii) is a weak form of the correspondence principle proved by Borthwick-Lesniewski-Upmeier [BLU93, Thm. 2.2]. 4. Vanishing Theorems Let  be a bounded symmetric domain in Cn . As usual, consider  equipped with the infinitesimal Bergman metric (2) with respect to which it is a complete K¨ahler manifold. Define a family of Hermitian holomorphic line bundles E
on . As holomorphic line bundles these are all the trivial bundle. A Hermitian structure on such a line bundle is defined by a real-valued function giving the square of the length of 1 ∈ C at the point z ∈ . The bundles E
differ in their Hermitian structures, which are defined by −1 c(
) · K(z, z¯ )−1/ h . |1|2
(z) = k

428

E. Guentner

Equip the E
with the unique connexions compatible with their complex and Hermitian structures. Denote the curvature form of E
with this connexion by
. Lemma 4.1. The curvature forms
of the bundles E
satisfy √ −1 1
= ω. 2


(6)

Proof. This follows straightforwardly from the definitions. If s is the section of E
whose value at z ∈  is 1 ∈ C then both (i) |s(z)|2 = constant · K(z, z¯ )−1/
, and ¯ log |s(z)|2 . (ii)
= ∂∂ Comparing with (3) we obtain the result.

 

We are interested in the Dolbeault operator of  twisted by the line bundles E
and recall the definition of these operators. Denote by pq

Ac = { compactly supported (p, q)-forms }, pq Ac (E
) = { compactly supported E
-valued (p, q)-forms } ¯ (these spaces differ only in their pre-Hilbert inner products). The ∂-operator has a canonical extension to forms with values in any holomorphic vector bundle. In particular, we have for all p = 0, . . . , n,  pq  pq Ac (E
) → Ac (E
), ∂¯ : q

q

and ∂¯ maps Ac (E
) to Ac (E
). Each of the spaces Ac (E
) has an Hermitian inner product induced by the Hermitian structures of  and E
. The operator ∂¯ has a formal adjoint which we shall denote by ∂¯
∗ . Notice that although ∂¯ does not depend on the Hermitian structure ∂¯
∗ does. Finally, the Hilbert space completions of the spaces of compactly supported forms will be denoted pq

p,q+1

pq

Apq = Hilbert space of (p, q)-forms, A (E
) = Hilbert space of E
-valued (p, q)-forms. pq

The following two special instances of these notations are worth mentioning explicitly; it follows from Lemma 2.2 that L2
() = A00 (E
),





(7)

H
2 () = { holomorphic sections of E
} = ker ∂¯ : A00 (E
) → A01 (E
) . (8) Since  is a complete manifold the formally self-adjoint operator ∂¯ + ∂¯
∗ is essentially self-adjoint. The twisted Dolbealt operators D
are the closures of the operators ∂¯ + ∂¯
∗ . They are self-adjoint unbounded operators (one for each p = 0, . . . , n):   Apq (E
) → Apq (E
). D
: q

q

Berezin Quantization and K-Homology

429

The Dolbeault Laplacian is

= D
2 . It preserves the bidegree of forms and when pq restricted to (p, q)-forms is denoted

. We are interested in vanishing theorems for the twisted Dolbeault operators D
. They have the consequence that the kernel of this operator is concentrated in degree zero, so that by (8) the quantization space H
2 () is in fact the kernel of D
(p = 0). The vanishing theorem is obtained using the standard Bochner method; following Roe [Roe88] we employ an adaptation of the Bochner method to the current setting of complete manifolds. We give a short review for the purposes of which we suppress the subscript in all notations, writing simply E for a Hermitian holomorphic line bundle, for the curvature of its canonical connexion, etc. As is standard, denote by L the operator of exterior product with ω, the (1, 1)-form associated to the K¨ahler structure of . Further, denote the unique connexion on E compatible with its Hermitian and complex structure ¯ Using square brackets to denote commutators the K¨ahler identities [GH78] by ∇ + ∂. (compare [GH78, Ch. 0.7, p. 111]) are √ √ −1 ∗ −1 ¯ ∗ ∗ ¯ ∗ [L , ∂] = − and [L , ∇] = ∇ ∂ . 2 2  2 ¯ it is a calculation to obtain the From these and the fact that = ∇ + ∂¯ = ∇ ∂¯ + ∂∇ basic Bochner identity √ (9)
pq = ∇ ∗ ∇ + ∇∇ ∗ − 2 −1[L∗ , ]. ¯ As a final bit of notation denote the space of ∂-harmonic (p, q)-forms with values in E by Hpq (E); pq

Hpq (E) = kernel of
E . Theorem 4.2 (First Vanishing Theorem). Let E be a Hermitian holomorphic line bundle on . If the curvature of the canonical connection of E satisfies √ −1 = λω 2 for some λ > 0 then the spectrum of the Dolbeault Laplacian on (p, q)-forms
pq is bounded below by 4λ(p + q − n). In particular, Hpq (E) = 0 if p + q > n. Proof. Combining our assumption with the identity [GH78, Ch. 0.7, p. 121] [L∗ , L] = (n − p − q),

on (p, q)-forms

and the basic Bochner identity (9) we obtain
pq = ∇ ∗ ∇ + ∇∇ ∗ − 4λ[L∗ , L] = ∇ ∗ ∇ + ∇∇ ∗ + 4λ(p + q − n),

on (p, q)-forms.

 

Theorem 4.3 (Second Vanishing Theorem). Let D
denote the Dolbeault operator (p = 0) of  with values in the Hermitian holomorphic line bundle E
,
∈ (0, 1). Then H0q (E
) = 0,

for q > 0

 and on the orthogonal complement of its kernel D
is bounded below by 2 1/
− 1.

430

E. Guentner

Proof. Use the canonical line bundle and its dual; K =

n0

 = C{ dz1 ∧ · · · ∧ dzn },

and

∗ K

=C

∂ ∂ ∧ ··· ∧ ∂z1 ∂zn

.

∗ induced from the K¨ The metric on K ahler metric on  is given by    ∂ ∂ 2  ∧ · · · ∧ (z) = (constant) · K(z, z¯ ),  ∂z ∂zn 
1 √

and its curvature form therefore satisfies 2−1 K ∗ = −ω. Hence the curvature form of E
⊗ K ∗ satisfies √ √

−1 −1 1 E
⊗K ∗ = − 1 ω. (
+ K ∗ ) = 2 2
Apply the previous theorem to conclude (the second equality is just a definition) H0q (E
) = H0q (E
⊗ K ∗ ⊗ K) = Hnq (E
⊗ K ∗ ) = 0, if q > 0. The remainder of the proposition follows from Convergence Transfer  ([Roe88] or Sect. 3 of [Gue98]); if

= D
2 is bounded below by 4(1/
− 1) on q=1,3,... Apq (E
) then   D
is bounded below by 2 1/
− 1 on the orthogonal complement of its kernel.  Corollary 4.4. The quantization spaces H
2 () for the Berezin quantization are the kernels of the twisted Dolbeault operators D
; H
2 () = ker D
= H00 (E
). Proof. This follows immediately from the previous theorem and (8).

 

5. E-Theory Elements This section is devoted to the construction of an E-theory element corresponding to the Berezin quantization. We will realize this E-theory element as the homotopy class of a generalized asymptotic morphism as defined in the appendix. To obtain a generalized asymptotic morphism we provide the collection of Hilbert spaces Hh2 (), for
∈ (0, 1], with the structure of a continuous field of Hilbert spaces. We do this in a very explicit and concrete manner. Although the results are actually stronger than required for our immediate goal of defining and analyzing the E-theory element associated to the quantization they do allow an elementary analysis based on estimates of functions of the twisted Dolbeault operators; we believe them to be of independent interest. We introduce several convenient pieces of notation. Denote by E the trivial Hermitian holomorphic line bundle for which |1|(z) = 1 and by   A= A0q , and A
= A0q (E
), q

q

Berezin Quantization and K-Homology

431

the Hilbert spaces of E and E
-valued differential forms. The maps E
→ E of multiplication by the functions −1/2

u
= k

c(
)1/2 · K(z, z¯ )−1/2


are unitary bundle isomorphisms and induce unitary isomorphisms U
: A
→ A of the Hilbert spaces of forms which preserve the spaces of compactly supported forms. The U
will be used to define and trivialize the field { H
2 () }. We must, however, address the fact that the functions u
are not holomorphic and the U
do not preserve the subspaces of holomorphic sections, and in particular do not preserve the quantization spaces H
2 (). Recall that the interior product with a smooth differential form τ is the negative of the adjoint of the exterior product with τ . We introduce the notation τ  (·) for the interior product with τ . Proposition 5.1. Define V : A → A on the domain of smooth compactly supported forms by     pq V σ = ∂¯ log K(z, z¯ ) ∧ σ − ∂¯ log K(z, z¯ )  σ, σ ∈ Ac . The following diagram, in which all unbounded operators are defined on the domain of smooth compactly supported forms, U


A
−−−−→  ¯ ∂¯ ∗  D
=∂+


A  D+V /2


A
−−−−→ A. U


commutes (the domains are preserved by U
). Proof. Calculate for a smooth compactly supported form σ ; ¯ ∗ (σ ) = U
∂(u ¯ −1 σ ) U
∂U

−1 ¯ ¯ = U
(∂u−1
∧ σ + u
∂σ ) ¯ ¯ −1 ∧ σ + ∂σ = u
∂u
¯
∧ σ + ∂σ ¯ = −u−1 ∂u


¯ = −(∂¯ log u
) ∧ σ + ∂σ, from which follows ¯ ¯ ∗ = −(∂¯ log u
) ∧ (·) + ∂. U
∂U


(10)

Taking the adjoint in (10) we obtain U
∂¯
∗ U
∗ = (∂¯ log u
)  (·) + ∂¯ ∗ , where, of course, ∂¯ ∗ is the adjoint of ∂¯ on the space A. Notice that −1 ¯ −1/2 ∂ log K(z, z¯ ), ∂¯ log u
= ∂¯ log k c(
)1/2 K(z, z¯ )−1/2h = 2


(11)

432

E. Guentner

so that for smooth compactly supported forms σ we have 1 V σ = −∂¯ log u
∧ σ + ∂¯ log u
 σ, 2


pq

σ ∈ Ac (E).

(12)

Adding (10) and (11) and employing (12) we obtain U
D
U
∗ = U
(∂¯ + ∂¯
∗ )U
∗ = −(∂¯ log u
) ∧ (·) + (∂¯ log u
)(·) + D 1 =D+ V, 2
where, of course, D is the ordinary Dolbeault operator of  computed with respect to its K¨ahler metric.    0q The domain q Ac is a common core for the operators 1 V. 2
They extend to self-adjoint unbounded operators on (a common domain in) A. U
D
U
∗ = D +

Proposition 5.2. For
∈ (0, 1) the projections P
onto the kernels of D + 21
V form a norm continuous family of projections on the Hilbert space A. Proof. The operators D + 21
V have the same spectral properties as the D
(see Theorem 4.3). Consequently, the projections P
can be realized as continuous spectral functions of these operators, and in fact, they can be realized simultaneously (i.e., using a single continuous function); given
0 < 1, since the operators D + 21
V are bounded below on the orthogonal complements of their kernels independently of
∈ (0,
0 ], there exists f ∈ C0 (R) such that  1  P
= f D + (13) V , for all
∈ (0,
0 ]. 2
The proposition follows from this equality together with the following two lemmas (Lemma 5.4 allows us to apply Lemma 5.3).   Lemma 5.3. Let T be a self-adjoint, unbounded operator and A be a self-adjoint bounded operator on a Hilbert space. For all f ∈ C0 (R) the operator-valued function t → f (T + tA) : (0, ∞) → B(H ) is continuous in norm. Proof. The set { f ∈ C0 (R) : f (T + tA) is continuous in t } C ∗ -subalgebra

is a of C0 (R). The proof is concluded by showing that it is all of C0 (R) which follows from showing that it contains the resolvent functions r± (x) =  √ −1 x ± −1 . This is a simple calculation; the norm of   r± (T + tA) − r± (T + t  A) = r± (T + tA) t  − t Ar± (T + t  A) is bounded by |t − t  | A .

 

Berezin Quantization and K-Homology

433

Lemma 5.4. The potential V is pointwise uniformly bounded on . We will reduce the lemma to Theorem 2.3 but must prepare for the proof by introducing some notation. For ξ ∈ TR∗  ⊗ C denote c(ξ ) and  c(ξ ) the sum and difference of exterior and interior multiplication with ξ , respectively; c(ξ ) = ξ ∧ (·) + ξ (·),  c(ξ ) = ξ ∧ (·) − ξ (·).

 Each of c(ξ ) and  c(ξ ) are complex linear endomorphisms of ∗ TR∗  ⊗ C. Note that ξ ∧ (·) is complex-linear in ξ whereas ξ (·) is complex-antilinear in ξ . From this simple observation it follows that √ √ √  c( −1ξ )σ = ( −1ξ ) ∧ σ − ( −1ξ )σ √ √ = −1ξ ∧ σ + −1ξ σ √ = −1c(ξ )σ, √ √ √ √ so that  c( −1ξ ) = −1c(ξ ) and c( −1ξ ) = −1 c(ξ ). For ξ and η ∈ TR∗  we have c(ξ )2 = − ξ 2 ,  c(ξ )2 = ξ 2 , c(ξ ) c(η) +  c(η)c(ξ ) = 0.

(14) (15) (16)

√ Now TR∗  ⊗ C = TR∗  ⊕ −1TR∗  is an orthogonal sum (in the complexified metric coming from the Riemannian metric on TR∗  underlying its K¨ahler metric) and the inclusions of TR∗  into the first and second factors are isometric and real-linear. Combining all the above facts we conclude that (14) holds also for complex cotangent vectors;  2 √ √ c(α + −1β)2 = c(α) + c( −1β)  2 √ = c(α) + −1 c(β) √ = c(α)2 −  c(β)2 + −1 (c(α) c(β) +  c(β)c(α)) 2 2 = − α − β

√ = − α + −1β 2 . √ √ Similarly one shows that  c(α + −1β)2 = α + −1β 2 . The consequence of this discussion that we require is that if f is a real-valued smooth ¯ ) of ⊕n Anc satisfies function on  then the endomorphism  c(∂f ¯ )2 = ∂f ¯ 2 =  c(∂f

1

df 2 . 2

(17)

Proof of Lemma 5.4. We reduce the statement to Theorem 2.3. In the notation of the previous discussion V =  c(∂¯ log K(z, z¯ )). It follows from (17)that V 2 = ∂¯ log K(z, z¯ ) 2 ,

434

E. Guentner

meaning that the square of the bundle endomorphism V is multiplication by the function. Thus, the norm of V is bounded by the supremum norm of ∂¯ log K(z, z¯ ) which by Theorem 2.3 is finite.   We begin the procedure of associating a generalized asymptotic morphism to the Berezin quantization by endowing the collection of Hilbert spaces H
2 () with the structure of a trivial continuous field. We have defined a family of unitary isomorphisms U
: A
→ A which we restrict to isometries H
2 () → A. We have observed in Corollary 4.4 that ker(D
) = H
2 () ⊂ A
are a family of closed linear subspaces, and further in Proposition 5.2 that there exists a norm continuous family of projections { P
} on A such that U
(H
2 ()) = P
A. In other words, the range projections of the U
are the P
and these form a norm continuous family of projections on A. Thus the hypotheses of Lemma 7.1 are satisfied and we have proven the following Proposition 5.5. Let  be the collection of functions x of
∈ (0, 1) such that x(
) ∈ H
2 () and U
x(
) is a continuous function of
, where we view U
as the isometry H
2 () → P
A → A. Then  defines the structure of a trivial field of Hilbert spaces { H
2 () }.   Remark. The continuous field defined in the proposition is generated (in the sense of [Dix70, 10.2.3]) by the collection of constant functions of
valued in the holomorphic polynomials. The triviality of the field follows from the general theory of continuous fields once we note that each of the spaces H
2 () is infinite dimensional [Dix70, 10.8.7]. For the remainder of this section denote by K
and K the C ∗ -algebras of compact operators on A
and A, respectively. As an immediate consequence of Lemma 7.2 we obtain the following characterization of the continuous sections of the field of elementary C ∗ -algebras associated to the field { H
2 () }. Proposition 5.6. Employ the notation of Proposition 5.5. A function K(
) such that K(
) ∈ K
is a continuous section of the field of elementary C ∗ -algebras { K
} associated to the field { H
2 () } if and only if U
K(
)U
∗ is a continuous function of
with values in K.   We come to the main theorem of this section; we associate to the Berezin quantization a generalized asymptotic morphism. Recall that the Berezin quantization is defined by associating to φ ∈ C0 () the family of Toeplitz operators { T
(φ) } on the family of Hilbert spaces { H
2 () }. We denote this family by B(φ) = { T
(φ) }. Theorem 5.7. The Berezin quantization defines a generalized asymptotic morphism φ → B(φ) = { T
(φ) } : C0 () → Cb { K(H
2 ()) }. As described in the appendix, this generalized asymptotic morphism determines an element of the E-homology of ; t ∈ E0 (). [B] = homotopy class of φ

Berezin Quantization and K-Homology

435

Proof. Via the inclusion H
2 () = ker(D
) → A
the Toeplitz operator T
(φ) is viewed as the compression of the multiplication operator Mφ to the subspace H
2 (): T
(φ) = Q
Mφ Q
, where Q
is the projection onto the kernel of D
. As the compression of the ∗-homomorphism C0 () → B(A
) associating to φ ∈ C0 () the operator Mφ , each T
is contractive and ∗-linear. Since, by Theorem 4.3, the operator Q
for
∈ (0, 1) may be realized as f (D
) for some f ∈ C0 (R), standard arguments ([Gue98, Lem. 3.5], for example) show that T
(φ) is a compact operator. We have shown that for φ ∈ C0 () the family B(φ) defines a bounded section of the continuous field { K
} depending ∗-linearly on φ ∈ C0 (). It remains to prove that (i) for φ ∈ C0 (), B(φ) is a continuous section, and (ii) B satisfies the asymptotic multiplicativity axiom. For (i) we use the characterization of continuous sections of the field { K
} given in Proposition 5.6. Since the unitary operators U
: A
→ A are themselves multiplication operators they conjugate a multiplication operator on A to one on A
. Therefore, viewing the operators U
as isometries, we have a commutative diagram U


H
2 () −−−−→   T
(φ)

A  P M P . 
φ


U


H
2 () −−−−→ A. Since the family of projections P
on A is norm continuous, the family of conjugates U
T
(φ)U
∗ = P
Mφ P
is a norm continuous family of compact operators on A. We turn finally to (ii), the asymptotic multiplicativity. We use not only the fact that the operators D
have gaps in their spectra (to write projections as continuous spectral functions of these operators as in (13)) but also the fact that these gaps become wider as
→ 0. We must show that

T
(φψ) − T
(φ)T
(ψ) → 0,

as
→ 0,

(18)

the norms being taken in B(H
2 ()). We work with the conjugated operators P
Mφ P
on A. From the simple calculation

P
Mφ Mψ P
− P
Mφ P
Mψ P
≤ P
Mφ − Mφ P


Mψ

it follows that it suffices to show that

[P
, Mφ ] → 0,

as
→ 0,

(19)

where P
is the projection onto the kernel of D
. Let f ∈ C0 (R) be supported in [−1, 1] and satisfy f (0) = 1. Let  s(
) be a continuous function increasing to infinity as √
→ 0 and such that s(
) ≤ 2 1/
− 1 for all 0 <
≤ 1/2 (for example, s(
) = 1/
). The functions f
(x) = f (s −1 x),

436

E. Guentner

are supported in [−s, s] and satisfy f
(0) = 1. Hence, by the properties of the spectra of D
outlined in Theorem 4.3, we have P
= f
(D
). The proof concludes with the observation that for all g ∈ C0 (R) we have

[g
(D
), Mφ ] → 0,

as
→ 0.

This is proved by observing that the set of such g ∈ C0 (R) is a ∗-subalgebra which, by virtue of the fact that s → ∞ as
→ 0 and the calculation

[r± (s −1 D
), Mφ ] ≤ s −1 [Mφ , D
] ≤ s −1 gradφ ,  √ −1 contains the resolvent functions r± (x) = x ± −1 .

 

We close with a few remarks regarding a result of Borthwick-Lesniewski-Upmeier [BLU93, Thm. 2.2] which states that (18) holds for continuous and bounded functions φ and ψ, one of which is compactly supported. In the course of the proof of Theorem 5.7 we have proven the following generalization of their result, which for clarity we restate as Theorem 5.8. Let φ and ψ be continuous bounded functions on . Assume that φ has a continuous extension to . Then T
(φ)T
(ψ) − T
(φψ) → 0,

as
→ 0,

the norm being of bounded operators on the respective quantization spaces H
2 (). Proof. The crux of the argument given above is that (19) holds provided φ is continuously differentiable on  with bounded gradient. This clearly holds for φ continuously differentiable on a neighborhood of . Finally, if φ is continuous on  it can be approximated in the uniform norm by continuously differentiable φ.   6. The Equality The purpose of this section is to prove our second main theorem; the E-theory class of the Berezin quantization defined in the previous section is equal to the E-theory class of the Dolbeault operator of . We freely employ the notations of Sects. 4 and 5. In particular,   A0q , and A
= A0q (E
) A= q

q

are the Hilbert spaces of E and E
-valued differential forms on  and D : A → A,

and D
: A
→ A


are ordinary Dolbeault operator and the Dolbeault operator twisted by E
. Recall that via the unitary isomorphisms U
: A
→ A the twisted Dolbeault operator D
is unitary equivalent to the operator D + 21
V on A, where V is the potential introduced in Sect. 4. By abuse of notation we denote this operator by D
as well. In order to define the E-homology class associated to the Dolbeault operator we require one additional piece of structure; the Hilbert spaces A and A
are graded by the

Berezin Quantization and K-Homology

437

decomposition into the spaces of even and odd forms. Denote by γ the grading operator. With these notations established we recall that the E-homology class [D] ∈ E0 () of the Dolbeault operator of  is defined to be the homotopy class of the asymptotic morphism C0 (R) ⊗ C0 () → C0 (R) ⊗ K(A) defined on basic tensors by f ⊗ φ −→ f (t −1 D + xγ )φ,

for all f ∈ C0 (R) and φ ∈ C0 ().

This construction appeared in the original unpublished manuscript of Connes and Higson [CH89]; for details of the construction we refer to [Gue98]. Theorem 6.1. The E-homology classes of the Berezin quantization and Dolbeault operator are equal: [B] = [D] ∈ E0 (). Remark. The classes [B] and [D], and in particular the E-homology group E0 (), are nonzero. We are unable to find a reference for this elementary fact in the literature (but compare [BD82]), so provide the following simple argument. Extension by zero defines a ∗-homomorphism C0 () → C0 (Cn ) which induces a homomorphism E0 (Cn ) → E0 (). This is an isomorphism. Indeed, if B ⊂  is a small ball we similarly have E0 () → E0 (B) and the composite map E0 (Cn ) → E0 (B) is an isomorphism; since all groups in question are isomorphic to Z the result follows. To complete the argument recall that the class of the Dolbeault operator on Cn is nonzero in E0 (Cn ); indeed it generates E0 (Cn ) ∼ = Z (compare [Ati68]). Further, it follows from [Gue99a] that its image under the map E0 (Cn ) → E0 () is [D] ∈ E0 (). We proceed to the proof of Theorem 6.1 which will occupy the remainder of the section. Along the way we encounter the E-homology classes of a number of other asymptotic morphisms. We mention two explicitly. Proposition 6.2. The family of functions αt defined by (we write
= t −1 ), αt (f ⊗ φ) = f (D
+ xγ )φ,

for all f ∈ C0 (R) and φ ∈ C0 ()

extends to an asymptotic morphism αt from C0 (R) ⊗ C0 () to C0 (R) ⊗ K(A). Furthermore, αt represents the E-homology class of the Berezin quantization: [αt ] = [B] ∈ E0 (). Proof. We sketch the proof that αt defines an asymptotic morphism, following closely the proof of Theorem 3.4 of [Gue98]. We must show (i) f −→ f (D
+xγ ) defines a continuous family of ∗-homomorphisms from C0 (R) to C0 (R, B(A)), and (ii) [φ, f (D
+ xγ )] → 0 as
→ 0, for f ∈ C0 (R) and φ ∈ C0 ().

438

E. Guentner

The proof of (i) is slightly easier than in [Gue98] by virtue of the fact that the identity

V 1 1 r± (D
+ xγ ) − r± (D
 + xγ ) = r± (D
+ xγ ) − r± (D
 + xγ ), 2

is simpler than its counterpart in [Gue98]. The proof of (ii) is somewhat more difficult and is accomplished by decomposing the operator f (D
+ xγ ) with respect to the decomposition of A into ker(D
) and its orthogonal complement. For f ∈ C0 (R) we have f (D
+ xγ ) − f (x)P
→ 0,

as
→ 0

(20)

by virtue of the spectral properties of D
outlined in Theorem 4.3. The result follows easily since we observed in the proof of Theorem 5.7 that

[P
, Mφ ] → 0,

as
→ 0.

(21)

We now turn to the equality in the statement of the proposition. The zero element of E0 () is represented by the zero homomorphism C0 () → K(H ). Recall that the family of projections P
is a norm continuous family, and hence the same is true of the projections 1 − P
. Thus, by Lemma 7.1 the zero element of E0 () is represented by the zero generalized asymptotic morphism C0 () → Cb ({ K((1 − P
)A) }). Adding this class to the Berezin class we see that the latter is represented by the asymptotic morphism φ −→ P
Mφ P
: C0 () → K(A). Further, by (21) this asymptotic morphism is asymptotically equivalent to the asymptotic morphism φ −→ P
Mφ : C0 () → K(A). But, the suspension of this asymptotic morphism is in turn asymptotically equivalent to αt by (20).   Proposition 6.3. The family of functions βt defined on basic tensors by βt (f ⊗ φ) = f (t −1/4 (D + V /2) + xγ ),

for all f ∈ C0 (R) and φ ∈ C0 ()

extends to an asymptotic morphism from C0 (R)⊗C0 () to C0 (R)⊗K(A). Furthermore, βt represents the E-homology class of the Dolbeault operator of : [βt ] = [D] ∈ E0 (). Proof. The proof that βt defines an asymptotic morphism is identical to the proof of Theorem 3.4 in [Gue98]. In fact, βt is a simple rescaling of the asymptotic morphism defining the E-homology class of the operator D + V /2 and hence represents that class. But there is an equality [D] = [D + V /2] ∈ E0 () ([Gue98], Prop. 3.7) and the desired result follows.

 

Berezin Quantization and K-Homology

439

With these propositions in hand we can complete the proof of Theorem 6.1: Proof of Theorem 6.1. In the notation of the previous two propositions it suffices to show that the asymptotic morphisms αt and βt from C0 (R) ⊗ C0 () to C0 (R) ⊗ K(A) represent the same E-homology class: [αt ] = [βt ] ∈ E0 (). We construct an explicit homotopy from αt to βt . Define a family of functions ηt by   ηt (f ⊗ φ) = f σ −1 Dτ −1 + xγ φ, for all f ∈ C0 (R) and φ ∈ C0 (), where we have defined for s ∈ [0, 1] and t ≥ 1, σ 4 = ((1 − s) + st) ,

and

τ = ((1 − s)t + s) .

It is immediate from these definitions that αt and βt are obtained by composing ηt with evaluation at s = 0 and s = 1, respectively. It remains only to check that ηt defines an asymptotic morphism from C0 (R) ⊗ C0 () to C0 (R × [0, 1], K(A)). This follows from Lemma 7.1 of [Gue98] once we prove: from C0 () into Cb (R × [0, 1], B(A)), (i) φ −→ 1 ⊗1 ⊗ Mφ is a ∗-homomorphism  −1 (ii) f −

→ f σ Dτ −1 + xγ is a continuous family of ∗-homomorphisms from C0 (R) to C0 (R × [0, 1], B(A)),   (iii) for fixed s ∈ [0, 1] and t ≥ 1 the product f σ −1 Dτ −1 + xγ Mφ is a compact operator on A, and   (iv) for all f ∈ C0 (R) and φ ∈ C0 () the commutator [Mφ , f σ −1 Dτ −1 + xγ ] tends to zero as t → ∞ (considered as an element of C0 (R × [0, 1], K(A))). Of these, (i) is obvious, (iii) follows from standard arguments (see the proof of Theorem 5.7), and (ii) ane (iv) are treated in the following lemmas.   Lemma 6.4. The assignment   f −→ f σ −1 Dτ −1 + xγ ,

for all f ∈ C0 (R)

(22)

defines a continuous family of ∗-homomorphisms from C0 (R) to C0 (R × [0, 1], B(A)). Proof. We must verify that for fixed t ≥ 1 the expression in (22) is continuous in (x, s) ∈ R × [0, 1] and vanishes at infinity. Further, we must verify that the resulting element of C0 (R × [0, 1], B(A)) varies continuously with t ≥ 1. It suffices to consider the case f = r± is one of the resolvent functions. By virtue of the identity (aDb + xγ )2 = a 2 Db2 + x 2 we see that the spectrum of aDb + xγ lies in the complement of (−x, x), independently of a and b ∈ R. Thus, simple calculations show that

r± (aDb + xγ ) ≤ √

1 x2

+1

,

(23)

independently of a and b ∈ R. From this we conclude that for each t ≥ 1 the resolvent r± (σ −1 Dτ −1 + xγ ) vanishes at infinity in (x, s) ∈ R × [0, 1]. We record for later use the similar fact that, independently of a, b, and x ∈ R,

(aDb + xγ ) r± (aDb + xγ ) ≤ 1.

(24)

440

E. Guentner

Define for s  ∈ [0, 1] and t  ≥ 1,   σ14 = (1 − s  ) + s  t  ,

and

  τ1 = (1 − s  )t  + s  ,

(25)

and introduce the convenient shorthand D = σ −1 Dτ −1 + xγ ,

and D1 = σ1−1 Dτ −1 + x1 γ . 1

From the resolvent identity r± (D) − r± (D1 ) = r± (D1 ) (D − D1 ) r± (D),

(26)

we are lead to calculate the difference D − D1 . Since

 τ   τ1  τ − τ1 Dτ −1 − Dτ −1 = D + V − D + V = V, 1 2 2 2 we have σ −1 Dτ −1 − σ1−1 Dτ −1

τ − τ1 V = σ −1 Dτ −1 − σ1−1 Dτ −1 2 1

σ1−1 (τ − τ1 ) V 2 σ −1 (τ − τ1 ) = (1 − σ σ1−1 )σ −1 Dτ −1 + 1 V 2   σ −1 (τ − τ1 ) = (1 − σ σ1−1 ) σ −1 Dτ −1 + xγ − (1 − σ σ1−1 )xγ + 1 V 2 σ −1 (τ − τ1 ) = (1 − σ σ1−1 )D − (1 − σ σ1−1 )xγ + 1 V. 2 We therefore conclude that = (σ −1 − σ1−1 )Dτ −1 +

D − D1 = (1 − σ σ1−1 )D − (1 − σ σ1−1 )xγ +

σ1−1 (τ − τ1 ) V + (x − x1 )γ . 2

(27)

Collecting (26) and (27) we conclude that for all x ∈ R, s ∈ [0, 1] and t ≥ 1 the norm of r± (D) − r± (D1 ) is bounded by the sum of four terms, |1 − σ σ1−1 | r± (D1 ) Dr± (D) , |1 − σ σ1−1 | |x| r± (D1 ) r± (D) ,    σ −1 (τ − τ )  1   1   V r± (D1 ) r± (D) ,   2 |x − x1 | r± (D1 ) r± (D) . Employing now (23) and (24) we bound the sum of these four terms, and hence the norm of r± (D) − r± (D1 ) by

r± (D) − r± (D1 ) ≤ C1 |1 − σ σ1−1 | + C2 |σ1−1 (τ − τ1 )| + K|x − x1 |, where C1 , C2 and K are constants independent of (x, s) ∈ R × [0, 1] and t ≥ 1.

(28)

Berezin Quantization and K-Homology

441

To conclude the proof we require some further elementary estimates. We have |τ − τ1 | ≤ (1 + t)|s  − s| + |t  − t|,

(29)

and a similar estimate for |σ 4 − σ14 |. Further, σ 4 = (1 − s) + st ≥ 1 so that σ ≥ 1 and σ −1 ≤ 1. We conclude that |σ 4 − σ14 | = |σ 2 − σ12 | |σ 2 + σ12 | ≥ 2|σ − σ1 | |σ + σ1 | ≥ 4|σ − σ1 |, and further that 4|1 − σ σ1−1 | = 4|σ1 − σ | |σ1−1 | ≤ (1 + t)|s  − s| + |t  − t|.

(30)

For our final estimate we combine (28), (29) and (30) to obtain a bound on the norm of r± (D) − r± (D1 ) in terms of (x, s) ∈ R × [0, 1] and t ≥ 1:  

r± (D) − r± (D1 ) ≤ C (1 + t)|s  − s| + |t  − t| + K|x − x1 |, where C and K are constants independent of (x, s) ∈ R × [0, 1] and t ≥ 1. The content of this last estimate is that for a fixed T ≥ 1 the resolvent r± (D) is uniformly continuous in the variables (x, s) ∈ R × [0, 1] and t ∈ [1, T ]. The desired result follows.   Lemma 6.5. For all f ∈ C0 (R) and φ ∈ C0 () the commutator    Mφ , f σ −1 Dτ −1 + xγ ,

(31)

tends to zero as an element of C0 (R × [0, 1], K(A)) as t → ∞. Proof. The proof of this proposition has much in common with the proof of Proposition 6.2. In particular, we rely on the fact that for all φ ∈ C0 (), [Mφ , Pt −1 ] → 0,

as t → ∞

(32)

(compare to (21)). Our task is to show that the norm of the commutator (31) tends to zero as t → ∞ uniformly in (x, s) ∈ R × [0, 1]. We consider the cases s ≤ 1/2 and s ≥ 1/2 separately, beginning with the former. Retaining the notation of the previous proof, simple calculations show that for all f ∈ C0 (R) and φ ∈ C0 (),

[Mφ , f (D) ≤ 2 φ

f ⊗ Pτ −1 − f (D) + [Mφ ⊗ 1, Pτ −1 ⊗ f ] .

(33)

We proceed as in the proof of Proposition 6.2 to estimate the norm of the difference   f (x)Pτ −1 − f (D) = f (x)Pτ −1 − f σ −1 Dτ −1 + xγ (34) by breaking the space A into the direct sum of Pτ −1 A = ker Dτ −1 and its orthogonal complement. On ker Dτ −1 this difference is zero. On its orthogonal complement Pτ −1 is zero and the spectral properties of the operators Dτ −1 outlined in Theorem 4.3 imply that  2 σ −1 Dτ −1 + xγ = σ −2 Dτ2−1 + x 2 ≥ σ −2 Dτ2−1 ≥ 4σ −2 (τ − 1),

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independently of x ∈ R. We conclude that the norm of (34) is bounded by √ sup{ |f (y)| : |y| ≥ 2σ −1 τ − 1 }, independently of x ∈ R. To show that this expression √ tends to zero uniformly in s ∈ [0, 1/2] as t → ∞ we show that the expression 2σ −1 τ − 1 tends to infinity uniformly in s ∈ [0, 1/2] as t → ∞. This follows from simple estimates regarding τ and σ , each independent of s ∈ [0, 1/2]: (i) σ −1 ≥ t −1/4 , (ii) 4(τ − 1) ≥ 2(t − 1) ≥ t, for t ≥ 2. Estimate (i) follows from σ 4 = (1 − s) + st ≤ t and (ii) from τ − 1 = (t − 1)(1 − s). Combining (i) and (ii) we conclude that for t ≥ 2, √ 2σ −1 τ − 1 ≥ t 1/4 , independently of s ∈ [0, 1/2]. For such s we also have τ = (1 − s)t + s ≥ t/2, so that it follows from (32) that [Mφ , Pτ −1 ] → 0,

as t → ∞

uniformly in s ∈ [0, 1/2]. Combining what we have thus far, we have shown that the norm of (33) tends to zero as t → ∞ uniformly in (x, s) ∈ R × [0, 1/2]. We now turn our attention to the complementary interval s ∈ [1/2, 1]. On this interval we reduce to consideration of the resolvent functions f = r± and smooth and compactly supported φ. We use the identity [Mφ , r± (D)] = r± (D)[D, Mφ ]r± (D) = r± (D)σ −1 [Dτ −1 , Mφ ]r± (D),

(35)

from which follows that

[Mφ , r± (D)] ≤ σ −1 gradient of φ . It thus remains only to verify that σ → ∞ uniformly in s ∈ [1/2, 1] as t → ∞. It is, however, immediate from the definition of σ that σ 4 ≥ t/2 independently of s ∈ [1/2, 1]. The lemma is thereby proved.   7. Appendix: Continuous Fields In establishing notation and conventions for continuous fields of Hilbert spaces and C ∗ -algebras we follow Dixmier [Dix70, Ch. 10]. A continuous field of Hilbert spaces over a topological space T consists of a family of Hilbert spaces Ht , t ∈ T together with a vector space  of functions x(t) satisfying certain axioms [Dix70, 10.1.2]. We denote a continuous field by ({ Ht }, ), although whenever convenient we shall omit  from the notation, denoting the continuous field by { Ht }. If each Ht is equal to a fixed Hilbert space H and  is the set of continuous functions on T with values in H the field is called constant. We shall consistently denote the constant field by { H }. A field isomorphic to a constant field is called trivial.

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Lemma 7.1. Let H be a Hilbert space. For all t in a locally compact topological space T let Ht be a Hilbert space and Ut : Ht → H be an isometry. Assume that the family of range projections Pt of Ut is strongly continuous. The collection  = { x(t) ∈ Ht : Ut x(t) is continuous } defines the structure of a continuous field { Ht }. Further, if T is an interval and the Pt are norm continuous the field is trivial. Proof. To see that  defines the structure of a continuous field we verify the axioms directly. It is clear that  is a linear space and that for x ∈  the function x(t) is continuous. To show that { x(s) : x ∈  } = Hs let v ∈ Hs be given. Define w = Us (v) ∈ H and x(t) = Ut∗ w. It is easily verified that x(s) = v and further that Ut x(t) = Pt w is continuous in t. Finally let y(t) ∈ Ht be a function that is the local uniform limit of elements of . To see that y ∈  let s be given and show that Ut y(t) is continuous at s. Let ε > 0 be given and obtain an open neighborhood O of s in T and an x ∈  such that

y(t) − x(t) < ε,

for all t ∈ O.

By reducing to a smaller neighborhood of s if necessary we further arrange that

Ut x(t) − Us x(s) < ε,

for all t ∈ O.

It is then straightforward to verify that

Ut y(t) − Us y(s) ≤ 3ε,

for all t ∈ O.

We turn to the triviality of the field in the case that T is an interval and the Pt are norm continuous. In this case there exists a norm continuous family of unitaries Vt such that for all t, Vt Pt Vt∗ = P1 = P . It follows that P H = Vt Pt Vt∗ H = Vt Pt H and we therefore may view the product Vt Ut as a unitary operator Ut

Vt

Ht −−−−→ Pt H −−−−→ P H. This collection of unitary isomorphisms provides the desired trivialization; it is readily verified that x ∈  if and only if Vt Ut x(t) is a continuous H -valued function.   Lemma 7.2. Employ the notation of Lemma 7.1. Let Kt be the C ∗ -algebra of compact operators on Ht and ({ Kt },   ) the continuous field of elementary C ∗ -algebras associated to { Ht }. The collection of continuous sections of { Kt } is   = { K(t) ∈ Kt : Ut K(t)Ut∗ is continuous }. Proof. It suffices to show that   ⊂   (compare [Dix70]). Recall that   is the closure with respect to local uniform convergence of the linear span of the rank one families associated to continuous sections of { Ht }; x,y (t) = x(t),y(t) = ·, x(t)y(t). It therefore suffices to show that

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(i) x,y ∈   for all x and y ∈ , and (ii) the collection   is closed under local uniform convergence. The first assertion follows from the simple estimate

x,y − x  ,y  ≤ x

y − y  + x − x 

y 

and calculation Ut x,y (t)Ut∗ = Ut x(t),Ut y(t) . The second assertion follows as in the proof of the previous lemma.

 

8. Appendix: E-Theory Let A and B be C ∗ -algebras. An asymptotic morphism from A to B is a family of functions {φt } : A → B indexed by t ∈ T = [1, ∞) satisfying the continuity condition t → φt (a) is a continuous B-valued function for all a ∈ A as well as the asymptotic conditions    φt (a) + λφt (a  ) − φt (a + λa  )  φt (a)φt (a  ) − φt (aa  ) = 0, lim t→∞   φt (a)∗ − φt (a ∗ )

for all a, a  ∈ A and λ ∈ C.

Asymptotic morphisms {φt } and {ψt } are asymptotically equivalent if lim (φt (a) − ψt (a)) = 0,

t→∞

for all a ∈ A.

Denote by B[0, 1] the C ∗ -algebra of continuous B-valued functions on the closed interval [0, 1]. Asymptotic morphisms {φt } and {ψt } are homotopic if there is an asymptotic morphism {αt } : A → B[0, 1] from which they may be recovered upon composition with evaluation at zero and one. Both asymptotic equivalence and homotopy are equivalence relations on the set of asymptotic morphisms from A to B. The set of homotopy classes is denoted [[A, B]]. Let K be the C ∗ -algebra of compact operators on a separable, infinite dimensional Hilbert space. Let S be the C ∗ -algebra of continuous functions on R vanishing at infinity and for any C ∗ -algebra A denote the suspension of A by SA = S ⊗ A. The bivariant E-theory groups are defined by E(A, B) = [[SA, SB ⊗ K]]. Our primary concern is with the E-homology groups defined by E 0 (A) = E(A, C) = [[SA, S ⊗ K]], although when speaking about commutative C ∗ -algebras, A = C0 (X), where X is a locally compact metrizable space, it is customary to denote these groups by E0 (X) = E 0 (C0 (X)). Remark. There are many seemingly different, but nonetheless equivalent, versions of E-theory [Dad94, Gue99b, GHT00]. The equivalence of our definition with the original definition of Connes-Higson is proven by a slight adaptation of the arguments in [Hig87]. The equivalence of our definition with the one employed in [Gue98, Gue99a] follows immediately from Bott Periodicity (compare [CH89]); we will use several results from these references.

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In defining the E-theory class associated to the Berezin quantization it is convenient to use a slightly generalized notion of asymptotic morphism. The benefit of this slightly generalized notion is primarily one of notational convenience. Let { H
} be a continuous field of Hilbert spaces on the interval
∈ (0, 1), together with a trivialization { U
}. In particular, the U
are unitary isomorphisms from the H
to a fixed Hilbert space H and the continuous sections of { H
} are precisely the translates of continuous functions of
with values in H . The associated field of elementary C ∗ algebras, denoted K({ H
}) is trivialized by { adU
}, where adU
: K(H
) → K(H ) is conjugation with U
. Denote by Cb ({ K
}) and C0 ({ K
}) the C ∗ -algebras of continuous bounded sections and continuous sections vanishing at zero of K({ H
}), respectively. A generalized asymptotic morphism is a function φ from A into the set of sections of K ({ H
}) satisfying the continuity condition φ(a) ∈ Cb ({ K
}) for all a ∈ A as well as the obvious asymptotic conditions    φ(a) + λφ(a  ) − φ(a + λa  )  φ(a)φ(a  ) − φ(aa  ) ∈ C0 ({ K
}), for all a, a  ∈ A and λ ∈ C.   ∗ ∗ φ(a) − φ(a ) t } via the A generalized asymptotic morphism φ gives an asymptotic morphism {φ prescription t (a) = adU1/t (φ(a)(1/t)), φ

for all a ∈ A.

Lemma 8.1. The homotopy class of the asymptotic morphism associated to the generalized asymptotic morphism φ is independent of the chosen trivialization of the field { H
}. Proof. Let { U
} be an isomorphism of constant fields { H } ∼ = { H } and φ : A → Cb { K(H ) } be a generalized asymptotic morphism. We prove that the asymptotic morphisms associated to φ and U φ are homotopic, where U φ(a)(
) = U
(φ(a)(
)),

for all a ∈ A and
∈ (0, 1].

 t to (U A homotopy from φ φ)t is defined by ∗ αt (a)(s) = Ust+(1−s) φt (a)Ust+(1−s) ,

for all a ∈ A and s ∈ [0, 1].

The required continuity properties of αt follow from elementary facts about the unitary group U of H equipped with the strong operator topology; U is a metrizable topological space and acts on K as a topological transformation group. In particular, the map (U, K) −→ U KU ∗ : U × K → K is continuous so that the function Ur φt (s)Ur∗ of s ∈ [0, 1] and r ≥ 1 is uniformly continuous on the product of [0, 1] with any compact initial segment of the ray r ≥ 1. The asymptotic properties of αt are also straightforwardly verified.   We summarize the result from this appendix used in the defining the E-theory class of the Berezin quantization. Proposition 8.2. A generalized asymptotic morphism φ : A → Cb ({ K
}) defines an element of the E-homology group E(A, C). This element is independent of the choice of trivialization used to define it.  

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References [Ati68] [BD82] [Ber74] [Ber75a] [Ber75b] [BLU93] [CH89] [CH90] [Dad94] [Dix70] [Don97] [GH78] [GHT00] [Gro91] [Gue98] [Gue99a] [Gue99b] [Gue00] [Hel78] [Hig87] [Hig93] [HK97] [HK01] [HKT98] [Hua63] [KL92] [Kra92] [Loo77] [Mok89] [PS69] [Roe88]

Atiyah, M.F.: Bott periodicity and the index of elliptic operators. Quart. J. Math. Oxford Ser. (2) 19, 113–140 (1968) MR 37 #3584 Baum, P., Douglas, R.: K-homology and index theory. In: Operator Algebras and Applications, R. Kadison (ed.), Proceedings of Symposia in Pure Mathematics, Vol. 38, Providence, RI: American Mathematical Society, 1982, pp. 117–173 Berezin, F.A.: Quantization. Math. USSR Izvestija 8(5), 1109–1165 (1974) Berezin, F.A.: Quantization in complex symmetric spaces. Math. USSR Izvestija 9(2), 341– 379 (1975) Berezin, F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975) Borthwick, D., Lesniewski, A., Upmeier, H.: Non-perturbative deformation quantization of Cartan domains. J. Funct. Anal. 113, 153–176 (1993) Connes, A., Higson, N.: Almost homomorphisms and KK-theory. Unpublished manuscript, http://math.psu.edu/higson/Papers/CH.dvi, 1989 Connes, A., Higson, N.: D´eformations, morphismes asymptotiques et K-th´eorie bivariante. C. R. Acad. Sci. Paris, S´erie I 311, 101–106 (1990) Dadarlat, M.: A note on asymptotic homomorphisms. K-Theory 8, 465–482 (1994) Dixmier, J.: C ∗ -algebras. Amsterdam: North Holland, 1970 Donnelly, H.: L2 -cohomology of the Bergman metric for weakly pseudoconvex domains. Illinois J. Math. 41, 151–160 (1997) Griffiths, P., Harris, J.: Principles of algebraic geometry. Pure and Applied Mathematics, New York: John Wiley & Sons, 1978 Guentner, E., Higson, N., Trout, J.: Equivariant E-theory for C ∗ -algebras. Memoirs of the AMS, Vol. 703, Providence, RI: American Mathematical Society, 2000 Gromov, M.: K¨ahler hyperbolicity and L2 -Hodge theory. J. Differ. Geom. 33, 263–292 (1991) Guentner, E.: Boundary calculations in relative E-theory. Mich. Math. J. 45, 159–188 (1998) Guentner, E.: Boundary calculations in E-theory for operators of Dirac type. Preprint, 1999 Guentner, E.: Relative E-theory. K-Theory 17, 55–93 (1999) Guentner, E.: Wick quantization and asymptotic morphisms. Houston J. Math. 26, 361–375 (2000) Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, Vol. 80, New York: Academic Press, 1978 Higson, N.: A characterization of KK-theory. Pacific J. Math. 126(2), 253–276 (1987) Higson, N.: On the K-theory proof of the index theorem. Comtemp. Math 148, 67–86 (1993) Higson, N., Kasparov, G.: Operator K-theory for groups which act properly and isometrically on Hilbert space. Electronic Research Announcements of the AMS 3, 131–142 (1997) Higson, N., Kasparov, G.G.: E-theory and KK-theory for groups which act properly and isometrically on Hilbert space. Invent. Math. 144(1), 23–74 (2001) Higson, N., Kasparov, G., Trout, J.: A Bott periodicity theorem for infinite dimensional Euclidean space. Adv. Math. 135, 1–40 (1998) Hua, L.K.: Harmonic analysis of functions of several complex variables in the classical domains. Translations of Mathematical Monographs, Vol. 6, Providence, RI.: American Mathematical Society, 1963 Klimek, S., Lesniewski, A.: Quantum Riemann surfaces I: The unit disc. Commun. Math. Phys. 146, 103–122 (1992) Krantz, S.: Function theory of several complex variables. 2nd ed., Pacific Grove, CA: Wadsworth & Brooks/Cole, 1992 Loos, O.: Bounded symmetric domains and Jordan pairs. Irvine: Univ. of California, 1977 Mok, N.: Metric rigidity theorems on hermitian locally symmetric manifolds. Series in Pure Mathematics, Vol. 6, Singapore: World Scientific, 1989 Pyatetskii-Shapiro, I.I.: Automorphic functions and the geometry of classical domains. New York: Gordon and Breach, 1969 Roe, J.: An index theorem on open manifolds, II. J. Diff. Geom. 27, 115–136 (1988)

Communicated by A. Connes

Commun. Math. Phys. 240, 447–456 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0921-6

Communications in

Mathematical Physics

Spectral Triples and Associated Connes-de Rham Complex for the Quantum SU (2) and the Quantum Sphere Partha Sarathi Chakraborty1,
, Arupkumar Pal2 1 2

Indian Statistical Institute, 203, B. T. Road, Calcutta 700 108, India. E-mail: parthasc [email protected] Indian Statistical Institute, 7, SJSS Marg, New Delhi 110 016, India. E-mail: [email protected]

Received: 20 September 2002 / Accepted: 22 March 2003 Published online: 19 August 2003 – © Springer-Verlag 2003

Abstract: In this article, we construct spectral triples for the C ∗ -algebra of continuous functions on the quantum SU (2) group and the quantum sphere. There have been various approaches towards building a calculus on quantum spaces, but there seem to be very few instances of computations outlined in Chapter 6, [5]. We give detailed computations of the associated Connes-de Rham complex and the space of L2 -forms. 1. Introduction Given a noncommutative space, there is no general method for constructing a spectral triple on it. Even though there are general results asserting the existence of enough unbounded Kasparov modules ([1]), in concrete examples, it is often difficult to carry out this prescription. In [2], the authors characterized all spectral triples for the C ∗ -algebra A of continuous functions on SUq (2) represented on its L2 -space, assuming equivariance under the (co-)action of the group itself. In the present article, we take the more standard representation of A on H = L2 (N) ⊗ L2 (Z) (see (1.3) below), and impose an equivariance condition under the action of the group S 1 × S 1 . Employing similar techniques as in [2], we arrive at a spectral triple of dimension 2. One advantage of this triple is that it is relatively easy to compute the associated Connes-de Rham complex, which we give in Sect. 3. This complex is supported on {0, 1}, and thus captures the topological dimension, which can be seen to be 1 from the following well-known exact sequence: 0 −→ K ⊗ C(S 1 ) −→ A −→ C(S 1 ) −→ 0.

(1.1)

The complex of square integrable forms were introduced by Fr¨ohlich et al. in [8]. We also present calculations of these L2 -forms for this spectral triple.
The first author would like to acknowledge support from the National Board for Higher Mathematics, India.

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In the last section, we briefly indicate how to carry out a similar construction of a 2 . spectral triple and the associated calculus for the quantum sphere Sqc ∗ Let us start with a brief description of the C -algebra of continuous functions on the quantum SU (2), to be denoted by A. This is the canonical C ∗ -algebra generated by two elements α and β satisfying the following relations: α ∗ α + β ∗ β = I, αα ∗ + q 2 ββ ∗ = I, αβ − qβα = 0, αβ ∗ − qβ ∗ α = 0, β ∗ β = ββ ∗ .

(1.2)

The C ∗ -algebra A can be described more concretely as follows. Let {ei }i≥0 and {ei }i∈Z be the canonical orthonormal bases for L2 (N) and L2 (Z) respectively. We denote by the same symbol N the operator ek → kek , k ≥ 0, on L2 (N) and ek → kek , k ∈ Z, on L2 (Z). Similarly, denote by the same symbol  the operator ek → ek−1 , k ≥ 1, e0 → 0 on L2 (N) and the operator ek → ek−1 , k ∈ Z on L2 (Z). Now take H to be the Hilbert space L2 (N) ⊗ L2 (Z), and define π to be the following representation of A on H:
π(α) =  I − q 2N ⊗ I, π(β) = q N ⊗ . (1.3) Then π is a faithful representation of A, so that one can identify A with the C ∗ subalgebra of L(H) generated by π(α) and π(β). The image of π contains K ⊗C(S 1 ) as an ideal with C(S 1 ) as the quotient algebra, that is we have a useful short exact sequence i

σ

0 −→ K ⊗ C(S 1 ) −→ A −→ C(S 1 ) −→ 0.

(1.4)

We will denote by Af the *-subalgebra of A generated by α and β. Let  α i β j β ∗k if i ≥ 0, j ∗k αi β β := ∗ −i j ∗k (α ) β β if i < 0. Then {αi β j β ∗k : i ∈ Z, j, k ∈ N} is a basis for Af . The Haar state h on A is given by, h : a → (1 − q 2 )

∞ 

q 2i ei0 , aei0 .

i=0

Remark 1.1. The representation π admits a nice interpretation. Let M be a compact topological manifold and E, a Hermitian vector bundle on M. Let (M, E) be the space of continuous sections. Then (M, E) is a finitely generated projective C(M) module. Define an inner product on (M, E) as  s1 , s2 := (s1 (m), s2 (m))m dν(m), where ν is a smooth measure on M and (·, ·)m is the inner product on the fibre on m. Let HE be the Hilbert space completion of (M, E). Then we have a natural representation of C(M) in L(HE ). The same program can be carried out in the noncommutative context also. Let B be a C ∗ -algebra and E a Hilbert B-module with its B valued inner product ·, · B . Let τ be a state on B. Consider the inner product on E given by e1 , e2 = τ (e1 , e2 B ). If we denote by HE the Hilbert space completion of E, then we get a natural representation of B in L(HE ). Now in the context of SUq (2), let p = |e0 e0 | ⊗ I ∈ A. Then it is easy to verify that HE = l 2 (N) ⊗ l 2 (Z) for E = Ap

Spectral Triples and Connes-de Rham Complex for Quantum SU (2)

449

with its natural left Hilbert A-module structure. Moreover, the associated representation is nothing but the representation of A described above. Also, viewed this way, one can think of the representation of A on L2 (h) given in [2] as being a countable direct sum of representations each of which looks like π (just think of A as ⊕Api , where pi = |ei ei | ⊗ I ). 2. S 1 ×S 1 -Equivariant Spectral Triples The group G = S 1 × S 1 has the following action on A:  α → zα τz,w : β → wβ.

(2.1)

Let U be the following representation of G on H: Uz,w = zN ⊗ wN . Then for any ∗ π(a)U a ∈ A, one has π(τz,w (a)) = Uz,w z,w , i.e. the action τ is implemented through this representation U of G. A self-adjoint operator with discrete spectrum equivariant under this G-action must be of the form D : eij → dij eij .

(2.2)

It is easy to see that if D is such an operator, then [D, α] and [D, β] are given by
[D, α]eij = (di−1,j − dij ) 1 − q 2i ei−1,j , (2.3) [D, β]eij = (di,j −1 − dij )q i ei,j −1 .

(2.4)

Employing arguments very similar to those used in the proofs of Propositions 3.1 and 3.2 in [2], we now get the following results. Proposition 2.1. Let D be an operator of the form eij → dij eij . Then [D, a] is bounded for all a ∈ Af if and only if dij ’s satisfy the following two conditions: |di−1,j − dij | = O(1), |di,j −1 − dij | = O(i + 1).

(2.5) (2.6)

Corollary 2.2. Let (Af , H, D) be a spectral triple equivariant under the action of S 1 × S 1 . Then D can not be p-summable if p < 2. Proof. This is a consequence of the following growth restriction on the dij ’s: dij = O(i + |j | + 1), which follows from the last proposition.

(2.7)



That there indeed exists a spectral triple that is 2-summable is easy to see, by just taking D to be the operator

where S =

 i≥0 j ≥0

D = N ⊗ S + I ⊗ N,  |eij eij | − i≥0 |eij eij |. j <0

(2.8)

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Remark 2.3. The obstruction element given by Voiculescu ([12]) turns out to be zero for the ideals L(p,∞) where p < 2, (note that by Proposition 1.7, [12], it is enough to look at positive finite-rank contractions from the commutant U (G) in order to calculate this obstruction). Though one can not conclude anything definite from this, it is possible that by dropping the condition of S 1 × S 1 -equivariance, Dirac operators of lower summability might be achievable. Proposition 2.4. Let D be as in the previous proposition. Assume that D has compact resolvent. Then up to a compact perturbation, we have  1. for each j ∈ Z, all the dij ’s are of the same sign,   2. there is a big enough integer M such that (2.9) (a) all the dij ’s for j ≥ M are of the same sign,   (b) all the dij ’s for j ≤ −M are of the same sign. Proof. Again, the proof is very similar to the proof of Proposition 3.2 in [2], and hence is omitted.  This proposition says in particular that if (A, H, D) is a G-equivariant Fredholm I +sign D must be one of the following, module, then upto a compact perturbation, P = 2 where E is some finite subset of {−M + 1, −M + 2, . . . , M − 1}:     |eij eij | + |eij eij |, P2 = |eij eij | + |eij eij |, P1 = i≥0 j ≤−M

i≥0 j ∈E

P3 =

 i≥0 j ∈E

i≥0 j ≥M

|eij eij |, P4 =



i≥0 j ∈E

|eij eij |.

i≥0 j ∈E c

We will prove below that the D given by (2.8) is in some sense the unique nontrivial Dirac operator for the representation π of A. Theorem 2.5. Let D be a G-equivariant Dirac operator. Then the Kasparov module associated with D is either trivial or is the same as the one associated with D or −D. Proof. Let u = χ{0} (β ∗ β)(β−I )+I . First, observe that [u], (A, H, D) = index SuS = 1. Since the K-groups for SUq (2) are free abelain, by the results of Rosenberg & Schochet ([10]), it is now enough to show that [u], (A, H, D ) is either 0 or ±1 if I +sign D P := is one of the Pi ’s above. Since [u], (A, H, D ) = index P uP , di2 rect calculation now tells us that if P is P3 or P4 , the above pairing would be zero; it would be −1 if P = P1 , and it is 1 if P = P2 .  

α −qβ ∗ that comes in the definition of SUq (2) has nonThe canonical unitary β α∗ trivial K-theory class (see the remark following Theorem 5, [6]). One can verify that by computing its pairing with D ⊗ I (acting on H ⊗ C2 ). The following proposition can be derived as a corollary to Proposition 4.3, [2]. But the proof presented there was just by computing pairings between appropriate elements and does not give an insight as to why it is true. We give a different proof here that sheds light on this. Proposition 2.6. Given any m ∈ K 1 (SUq (2)) = Z, there exists a Kasparov module (L2 (h), F ) which induces this element.

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451

Proof. Using Remark 1.1, one could look at L2 (h) as ⊕Api . Representation of A by left multiplications in each piece looks like π . Now given m in Z, one has to pick m copies of π, and define F to be (sign m)S on each of these pieces and I on others. Then (L2 (h), F ) would be the required module.  3. Connes-de Rham Complex Let • (Af ) = ⊕n n (Af ) be the universal graded differential algebra over Af , i.e. n (Af ) = span{a0 (δa1 ) . . . (δan ) : ai ∈ Af , δ(ab) = a(δb) + (δa)b}. The universal differential algebra is not very interesting from the cohomological point of view. Interesting cohomologies are obtained from the representations of the algebra. For the spectral triple (Af , H, D), one has the standard Connes-de Rham complex of noncommutative exterior forms •D (Af ), given by • (A) := • (A)/(K + δK) ∼ = π(• (A))/π(δK), D

where K = ⊕p≥0 Kp is the two sided ideal of • (A) given by Kp = {ω ∈ p (A) : π(ω) = 0}. But often, the explicit computation of this complex is rather difficult. What we will do is the following. We will compute the complex obtained from the representation θ ◦ π : • (A) → Q(H), where θ : L(H) → Q(H) = L(H)/K(H) is the projection onto the Calkin algebra. More specifically, let d : Af → L(H) be = [D, π(a)]. Define πn : n (Af ) → L(H) by πn (a0 (δa1 ) . . . (δan )) = given by da n ). Define d = θ ◦ d, ψn = θ ◦ πn , and ψ := ⊕ψn : ⊕n → Q(H). π(a0 )(da1 ) . . . (da Let Jn = ker ψn . Define nd (Af ) = n (Af )/(Jn + δJn−1 ). Then nd (Af ) = ψ(n (Af ))/ψ(δJn ). We will compute these cohomologies n d (Af ). Before entering the computations, it should be stressed here that by computing these rather than the standard complex, we do not lose much. Because, first, since for a compact operator K one has Trω (K|D|−2 ) = 0, Proposition 5, p. 550, [5] concerning the Yang-Mills functional holds in our present case. Second, in the context of the canonical spectral triple associated with a compact Riemannian spin manifold this prescription also gives back the exterior complex. First, we need the following lemma which will be very useful for the computations. Lemma 3.1. Assume a, b ∈ Af and c ∈ K(H). If a(I ⊗ S) + b = c, then a = b = 0. Proof. For a functional ρ on L(L2 (N)), and T ∈ L(H), denote by aρ the operator (ρ ⊗ id)T . Now observe that for any a ∈ Af and any functional ρ, aρ  = aρ .

(3.1)

Write P = 21 (I + S). It is easy to see that the given condition implies that (bρ − aρ ) + 2aρ P = cρ , which in turn implies that (bρ − aρ )ei = cρ ei (bρ + aρ )ei = cρ ei

∀i < 0, ∀i ≥ 0.

Now from (3.1) and (3.2), it follows that for any i, j ∈ Z and j < 0, (bρ − aρ )ei  = (bρ − aρ )j −i ej  = j −i (bρ − aρ )ej  = (bρ − aρ )ej  = cρ ej .

(3.2) (3.3)

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Since c is compact, limj →−∞ cρ ej  = 0. Hence (bρ − aρ )ei = 0 for all i. In other words, (bρ − aρ ) = 0. Since this is true for any ρ, we get a = b. Using this equality, together with Eqs. (3.1) and (3.3), a similar reasoning yields a = 0.  Lemma 3.2. Let Iβ denote the ideal in Af generated by β and β ∗ . Then for n ≥ 1, we have ψ(n (Af )) = (I ⊗ S)n Af + (I ⊗ S)n+1 Iβ .

(3.4)

Proof. Let us first prove the equality for n = 1. Let Zk = q N+k (N + k), Bj k = j i=j −k+1 |ei−1 ei |, and  j −1 i=0 |ei ei−1 | ifj ≥ 1, Cj = 0 ifj = 0. It is easy to check that [D, α] = α(−I ⊗ S), [D, β] = q N N ⊗ [S, ∗ ] + β.

(3.5)

It follows from these that [D, αi β j β ∗ k ] = −i(I ⊗ S)αi β j β ∗ k + (j − k)αi β j β ∗ k + 2(Zi ⊗ Cj )αi β j −1 β ∗ k −2(Zi ⊗ Bj k )αi β j β ∗ k−1 .

(3.6)

Hence d(αi β j β ∗ k ) = −i(I ⊗ S)αi β j β ∗ k + (j − k)αi β j β ∗ k . Thus for any a ∈ Af , da = (I ⊗ S)b + c,

where b ∈ Af ,

c ∈ Iβ .

(3.7)

Note that for any a ∈ Af , ψ(a )(I ⊗ S) = (I ⊗ S)ψ(a ) in Q(H). Hence ψ(a (δa)) is again of the form (I ⊗S)b+c, where b ∈ Af , c ∈ Iβ , i.e. is a member of (I ⊗S)Af +Iβ . Thus ψ(1 (Af )) ⊆ (I ⊗ S)Af + Iβ . For the reverse inclusion, observe that (I ⊗ S) = (1 − q 2 )−1 ((dα)α ∗ + q 2 (dα ∗ )α), β = dβ and β ∗ = −dβ ∗ . The inductive step follows easily from (3.7).  Lemma 3.3. J0 = {0}, and for n ≥ 1, we have ψ(δJn ) = (I ⊗ S)n+1 Af + (I ⊗ S)n+2 Iβ .

(3.8)

Proof. By Lemma 3.1, ψ : Af → Q(H) is faithful. Hence it follows that J0 = {0}. We will prove here (3.8) by induction. From Lemma 3.2, we have ψ(δJ1 ) ⊆ ψ(2 (Af )) = Af + (I ⊗ S)Iβ . Let us show that I , (I ⊗ S)β and (I ⊗ S)β ∗ are all members of ψ(δJ1 ). Choose ω ∈ 1 (Af ) such that ψ(ω) = (I ⊗ S). Let ωk = kαk ω − δ(αk ), k = ±1. Then it follows from (2.3) that ψ(ωk ) = kαk (I ⊗ S) − kαk (I ⊗ S) = 0, so that ωk ∈ J1 . ψ(δωk ) = ψ(k(δαk )ω) = k 2 αk = αk ∈ ψ(δJ1 ), i.e. both α and α ∗ are in ψ(δJ1 ). It follows from this that I ∈ ψ(δJ1 ). Next we show that (I ⊗ S)β ∈ ψ(δJ1 ). Take ω = 21 (α(δβ) − δ(αβ) + qβ(δα)). Then ψ(ω) = 0 and ψ(δω) = (I ⊗ S)αβ. So (I ⊗ S)αβ ∈ ψ(δJ1 ). Similarly taking ω = 21 (α ∗ (δβ) − δ(α ∗ β) + q −1 β(δα ∗ )), it follows that (I ⊗ S)α ∗ β ∈ ψ(δJ1 ). These two together imply (I ⊗ S)β ∈ ψ(δJ1 ).

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A similar argument shows that (I ⊗ S)β ∗ is also in ψ(δJ1 ). Thus Af + (I ⊗ S)Iβ = ψ(δJ1 ). For the inductive step, notice that ψ(δJn ) ⊆ ψ(n+1 (Af )) = (I ⊗ S)n+1 Af + (I ⊗ n+2 S) Iβ . We will show that the following are all elements of ψ(δJn ): (I ⊗ S)n+1 α, (I ⊗ S)n+2 αβ, (I ⊗ S)n+2 αβ ∗ , (I ⊗ S)n+1 α ∗ , (I ⊗ S)n+2 α ∗ β, (I ⊗ S)n+2 α ∗ β ∗ . From the right Af -module structure of ψ(δJn ), it will then follow that (I ⊗ S)n+1 , (I ⊗ S)n+2 β and (I ⊗ S)n+2 β ∗ are in ψ(δJn ), giving us the other inclusion. Choose ω ∈ Jn−1 such that ψ(δω) = (I ⊗ S)n . Take ωk = kω(δαk ), k = ±1. Then ωk ∈ Jn and ψ(δωk ) = (I ⊗ S)n+1 αk . Similarly choosing ω such that ψ(δω) = (I ⊗S)n+1 β and ωk as before, we get ωk ∈ Jn and ψ(δωk ) = q −k (I ⊗S)n+2 αβ. Finally, take ω such that ψ(δω) = (I ⊗S)n+1 β ∗ and ωk as before to show that (I ⊗S)n+2 αk β ∗ ∈ ψ(δJn ).  Theorem 3.4.

 nd (Af )

=

Af ⊕ Iβ if n = 1, {0} if n ≥ 2.

Proof. Proof follows from Lemmas 3.2 and 3.3.



Remark 3.5. The differential d : Af → 1d (Af ) = Af ⊕ Iβ is given by d(αi β j β ∗k ) = −iαi β j β ∗k ⊕ (j − k)αi β j β ∗k . 4. L2 -Complex of Frohlich et al. In this section we will compute the complex of square integrable forms for the spectral triple (Af , H, D). For that we begin with similar computations for the spectral triple (C[z, z−1 ], H0 = L2 (Z), D0 = N ) associated with the algebra C[z, z−1 ]. Here we consider the embedding π0 : C[z, z−1 ] → L(H) that maps z to . n (C[z, z−1 ]) = 0, for n ≥ 2, Lemma 4.1. (i)  D0 1 (C[z, z−1 ]) = C[z, z−1 ]. (ii)  D0  Proof. (i) Let ω = αn0 ,··· ,nk zn0 δzn1 · · · δznk ∈ k (C[z, z−1 ]), where the sum is a finite one and δ is the universal differential. Then it is easily verified that   k ∗  k  n1 · · · nk αn0 ,··· ,nk z 0 nj n1 · · · nk αn0 ,··· ,nk z 0 nj dz, (ω, ω)D0 = where dz is the Lebesgue measure on the circle. Therefore, ∈ k (C[z, z−1 ]) : (ω, ω)D0 = 0} Kk (C[z, z−1 ]) := {ω   = αn0 ,··· ,nk zn0 δzn1 · · · δznk :  = 0, ∀r .

n0 +···+nk =r

n1 · · · nk αn0 ,··· ,nk

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Consequently we have, k

zn0 δzn1 · · · δznk − n1 · · · nk z 0 ni −k δz · · · δz ∈ Kk (C[z, z−1 ]), (4.9) r r −1 δz δz · · · δz − rz δz · · · δz ∈ Kk (C[z, z ]), (4.10) 1 zr δz · · · δz − δzr+1 δz · · · δz ∈ Kk−1 (C[z, z−1 ]). (4.11) r +1 From (4.11) we get δzr δz · · · δz ∈ δKk−1 (C[z, z−1 ]). Combining this with (4.9) and (4.10) we get, zn0 δzn1 · · · δznk ∈ Kk (C[z, z−1 ]) + δKk−1 (C[z, z−1 ]) for large n0 . Since Kk (C[z, z−1 ]) + δKk−1 (C[z, z−1 ]) is a bimodule we have zn0 δzn1 · · · δznk ∈ Kk (C[z, z−1 ]) + δKk−1 (C[z, z−1 ])

∀ n 0 , · · · , nk .

This proves (i). (ii) It suffices to note that zn0 δzn1 − n1 zn0 +n1 −1 δz ∈ K1 (C[z, z−1 ]). 0 (C[z, z−1 ]) → C[z, z−1 ] is given by d(zn ) = nzn . The induced d :  D0



Now we are in a position to compute the complex of square integrable forms. n (Af ) = 0 for n ≥ 2. Theorem 4.2. (i)  D n (Af ) = C[z, z−1 ] for n = 0, 1 (equality as an Af bimodule), and the differ(ii)  D 1 (Af ) is given by d(αi β j β ∗k ) = −izi . ential d : Af →  D Proof. Note that the homomorphism σ in (1.4) induces a surjective homomorphism denoted by the same symbol from Af to C[z, z−1 ]. We have the following short exact sequence σ

0 −→ Iβ −→ Af −→ C[z, z−1 ] −→ 0. Let σk : k (Af ) → (C[z, z−1 ]) be the induced surjective map. One easily verifies that (ω, ω)D = (σk (ω), σk (ω))D0 . Therefore, Kk (Af ) = {ω ∈ k (Af ) : (ω, ω)D = 0} = σk−1 (Kk (C[z, z−1 ])). We have the following commutative diagram: K0 = Iβ

−→

K1 (Af )

−→

K2 (Af )

−→ ... −→

Kn (Af )

Af ↓ 1 (Af ) ↓ 2  (Af ) n (Af )

σ

−→ σ1

−→ σ2

−→ ... σn −→

C[z, z−1 ] ↓ 1 (C[z, z−1 ]) ↓ 2  (C[z, z−1 ]) n (C[z, z−1 ])

−→ −→ −→ ... −→

π0 (C[z, z−1 ]) ↓ ˜ 1 (C[z, z−1 ])  D0 ↓ 2 ˜ D0 (C[z, z−1 ]) ... ˜ n (C[z, z−1 ]).  D0

This along with the previous lemma proves the theorem. We will only illustrate (i). Let ωn ∈ n (Af ), then by the previous lemma σn (ωn ) = ω1,n + δω2,n−1 , where

ω1,n ∈ Kn (C[z, z−1 ]), ω2,n−1 ∈ Kn−1 (C[z, z−1 ]). Let ω1,n = σn−1 (ω1,n ), ω2,n−1 = −1

σn−1 (ω2,n−1 ), then σn (ωn − ω1,n − δω2,n−1 ) = 0 implying ωn ∈ Kn + δKn−1 . 

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5. Computations for the Quantum Sphere In this section we will briefly indicate how to carry out all the earlier constructions for the quantum spheres. We will be sketchy because most of the arguments are very similar to the case of SUq (2). The quantum sphere was introduced by Podle´s in [9]. This is the 2 ), generated by two elements A and B subject universal C*-algebra, denoted by C(Sqc to the following relations: A∗ = A, B ∗ B = A − A2 + cI, BA = q 2 AB, BB ∗ = q 2 A − q 4 + cI. Here the deformation parameters q and c satisfy |q| < 1, c > 0. For a later purpose we also note down two irreducible representations whose direct sum is faithful. Let H+ = l 2 (N), H− = H+ . Define π± (A), π± (B) : H± → H± by π± (A)(en ) = λ± q 2n en

1 1 1/2 , ± c+ 2 4 2 where c± (n) = λ± q 2n − (λ± q 2n ) + c. where

π± (B)(en ) = c± (n)1/2 en−1

λ± =

Since π = π+ ⊕ π− is a faithful representation, an immediate corollary follows. Theorem 5.1 (Sheu [11]). 2 ) ∼ C ∗ (T ) ⊕ C ∗ (T ) := {(x, y) : x, y ∈ C ∗ (T ), σ (x) = σ (y)} where (i) C(Sqc = σ ∗ C (T ) is the Toeplitz algebra and σ : C ∗ (T ) → C(S 1 ) is the symbol homomorphism. (ii) We have a short exact sequence i

α

2 ) −→ C ∗ (T ) −→ 0. 0 −→ K −→ C(Sqc

(5.12)

Proof. (i) An explicit isomorphism is given by x → (π+ (x), π− (x)). (ii) Define α((x, y)) = x then ker α = K.  2 )) = K 0 (C(S 2 )) = Z ⊕ Z. Corollary 5.2. (i) K0 (C(Sqc qc 2 )) = K 1 (C(S 2 )) = 0. (ii) K1 (C(Sqc qc

Proof. The six term exact sequence associated with (5.12) along with the KK-equivalence of K and C ∗ (T ) with C proves the result.  2 ) generated by A and B. Then Proposition 5.3. Let Af in be the *-subalgebra of C(Sqc





0 N 1 0 , γ = Af in , H = H+ ⊕ H− , D = N 0 0 −1

is an even spectral triple. Proof. We only have to show that [D, a] is bounded for a ∈ Af in . For that it is enough to note that are bounded, (i) Nπ± (A), π± (A)N √ (ii) n(c± (n)1/2 − c) is bounded as n becomes large, (iii) [N, ] = . 

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Remark 5.4. This spectral triple has nontrivial Chern character. This can be seen as fol2 ), then applying Proposition 4, p. 296, [5], we get lows: let P0 = i(|e0 e0 |) ∈ C(Sqc the index pairing [P0 ], [(Af in , H, D, γ )] = −1, implying nontriviality of the spectral triple. Now we will briefly indicate the computations of the complex (•d (Af in ), d) introduced at the beginning of Sect. 3. Theorem 5.5. (i) nd (Af in ) = 0 for n ≥ 2. (ii) 1d (Af in ) = C[z, z−1 ], here also equality is as an Af in bimodule. Proof. Let π be the associated representation of • (Af in ) in L(H). Then straightforward verification gives (i) [D, A] is compact, (ii)  [D, B] =  ⊗ κ + compact, and (iii) 01 ∗ ∗ [D, B ] = − ⊗ κ + compact, where κ = . Therefore, modulo compacts 10 π(2k+1 (Af in )) = Cf∗ in (T ) ⊗ κ π(2k (Af in )) = Cf∗ in (T ) ⊗ I2 , where Cf∗ in (T ) is the *-subalgebra of C ∗ (T ) generated by the shift operator. Now for (i), note that ωn = BδB ∗ δB · · δB +B ∗ δB δB · · δB  ·  · n−2 times

n−2 times

satisfies (a) π(ωn ) is compact and (b)π(δωn ) = 2I is invertible, hence (i) follows. For (ii), observe that if a ∈ Af in and π(a) is compact then N a and aN are both compact. Hence, 1d (Af in ) = π(1 (Af in )) = C[z, z−1 ] because modulo compact C[z, z−1 ] is C ∗ (T ).  References 1. Baaj, S., Julg, P.: Th´eorie bivariante de Kasparov et op´erateurs non born´es dans les C ∗ -modules Hilbertiens. C. R. Acad. Sci., Paris, s´er. I Math 296(21), 875–878 (1983) 2. Chakraborty, P.S., Pal, A.: Equivariant spectral triples on the quantum SU (2) group. K-Theory 28(2), 107–126 (2003) 3. Connes, A.: Noncommutative geometry and Physics. In Gravitation et quantifications (Les Houches, 1992), Amsterdam: North-Holland, 1995, pp. 805–950 4. Connes, A.: On the Chern character of θ summable Fredholm modules. Commun. Math. Phys. 139(1), 171–181 (1991) 5. Connes, A.: Noncommutative Geometry. London-New York: Academic Press, 1994 6. Connes, A.: Cyclic cohomology, quantum group symmetries and the local index formula for SUq (2). math.QA/0209142 7. Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 5(2), 174–243 (1995) 8. Fr¨ohlich, J., Grandjean, O., Recknagel, A.: Supersymmetric Quantum Theory and Non-Commutative Geometry. Commun. Math. Phys 203, 119–184 (1999) 9. Podle´s, P.: Quantum spheres. Lett. Math. Phys. 14(3), 193–202 (1987) 10. Rosenberg, J., Schochet, C.L.: The K¨unneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor. Duke Math. J 55(2), 431–474 (1987) 11. Sheu, A.J-L.: Quantization of the Poisson SU (2) and its Poisson homogeneous space – the 2-sphere. With an appendix by Jiang-Hua Lu and Alan Weinstein. Commun. Math. Phys. 135(2), 217–232 (1991) 12. Voiculescu, D.: Some results on norm-ideal perturbation of Hilbert space operators I. J. Operator Theory 2, 3–37 (1979) 13. Woronowicz, S.L.: Twisted SU (2) group. An example of a noncommutative differential calculus. Publ. RIMS, Kyoto University 23(1), 117–181 (1987) Communicated by A. Connes

Commun. Math. Phys. 240, 457–482 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0905-6

Communications in

Mathematical Physics

Free Energy as a Dynamical Invariant (or Can You Hear the Shape of a Potential?)
Mark Pollicott1 , Howard Weiss2 1 2

Department of Mathematics, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK. E-mail: [email protected] Department of Mathematics, 402 McAllister, Pennsylvania State University, University Park, State College, PA 16802, USA. E-mail: [email protected]

Received: 24 December 2002 / Accepted: 4 April 2003 Published online: 1 August 2003 – © Springer-Verlag 2003

Abstract: Classical lattice spin systems provide an important and illuminating family of models in statistical physics. An interaction
on a lattice L ⊂ Zd determines a lattice spin system with potential A
. The pressure P (A
) and free energy FA
(β) = −(1/β)P (βA
) are fundamental characteristics of the system. However, even for the simplest lattice spin systems, the information about the potential that the free energy captures is subtle and poorly understood. We study whether, or to what extent, (microscopic) potentials are determined by their (macroscopic) free energy. In particular, we show that for a one-dimensional lattice spin system, the free energy of finite range interactions typically determines the potential, up to natural equivalence, and there is always at most a finite ambiguity; we exhibit exceptional potentials where uniqueness fails; and we establish deformation rigidity for the free energy. The proofs use a combination of thermodynamic formalism, algebraic geometry, and matrix algebra. In the language of dynamical systems, we study whether a H¨older continuous potential for a subshift of finite type is naturally determined by its periodic orbit invariants: orbit spectra (Birkhoff sums over periodic orbits with various types of labeling), beta function (essentially the free energy), or zeta function. These rigidity problems have striking analogies to fascinating questions in spectral geometry that Kac adroitly summarized with the question “Can you hear the shape of a drum?”. Contents I. Motivations and Introduction . . . . . . . . . . . . . . . . . . . . . . . . . II. The Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. The Free Energy and Beta Functions . . . . . . . . . . . . . . . . . . . . .

458 464 469

 The work of the second author was partially supported by a National Science Foundation grant DMS0100252. This work began during the second author’s sabbatical visit at IPST, University of Maryland, and he wishes to thank IPST for their gracious hospitality. This work was completed during a visit by the first author to Penn State as a Shapiro Fellow.

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III. 1. Locally constant functions . . . . . . . . . . . . . . . . . . . . . . . III. 2. H¨older continuous functions . . . . . . . . . . . . . . . . . . . . . . IV. The Unmarked Orbit Spectrum . . . . . . . . . . . . . . . . . . . . . . . . App. I. Uncountably Many H¨older Continuous Functions Sharing the Same Unmarked Orbit Spectrum (and Free Energy) . . . . . . . . . . . . . App. II. Locally Constant Functions with the Same Unmarked Orbit Spectrum (and Free Energy) . . . . . . . . . . . . . . . . . . . . . . App. III. Uncountably Many H¨older Continuous Functions Sharing the Same Weak Orbit Spectrum but Having Different Unmarked Orbit Spectra .

469 474 475 477 478 480

I. Motivations and Introduction I.1. Physical motivation: Does the (macroscopic) free energy of a lattice spin system determine the (microscopic) potential? A fundamental problem in statistical physics is to recover the microscopic interactions between particles from macroscopic thermodynamic functions. We study this problem for the one-dimensional lattice spin system. Classical lattice spin systems provide an important and illuminating family of models in statistical physics. An interaction
on Zr determines a lattice spin system, and the pressure P (
) and free energy F
(β) are fundamental characteristics of the system. However, even for the simplest lattice spin systems, the information about the interaction or the potential that the free energy captures is subtle and poorly understood. We study whether, or to what extent, potentials for certain model systems are determined by their free energy. Following Ruelle [Rue, pp. 36–38], we consider the lattice Zd and the full shift d space  m
{1, 2, · · · , m}Z of configurations equipped with the product topology. m d For S ⊂ Z define S
S
{1, 2, · · · , m}S . A lattice spin system is specified by an interaction
– a function defined on configurations restricted to all finite subsets  ⊂ Zd and which satisfies some regularity condition. For example, the interaction
for the general Ising model is defined by 1   if  = {x} −h(x)ξx
 (ξ ) = −J (x, y)ξx ξy if  = {x, y}  0 otherwise, where ξx denotes the restriction of the configuration ξ to the site x. One then constructs a partition function for each finite subset  ⊂ Zd . Ruelle found that it is often useful to study the associated potential function A
on  m defined by  1
(ξ |Y ), A
(ξ )
− #Y 0∈Y

where #Y denotes the cardinality of the set Y . This potential is essentially the contribution of the lattice site 0 to the energy in the configuration ξ , and the regularity requirement on 1

The (formal) Hamiltonian in general for the Ising model is   J (x, y)ξx ξy − h(x)ξx . H (ξ ) = − (x,y)∈Zd

x∈Zd

To prove convergence of the key quantities of interest, one is essentially forced to first define these quantities on finite sets and then take a (thermodynamic) limit.

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459


ensures that A
is well defined. Using this potential function and the d−dimensional family of shift maps {σ x , x ∈ Zd }, one can construct an analogous partition function for each finite subset  ⊂ Zd . More generally, for any continuous function A on  m and finite subset  ⊂ Zd , one can define a partition function     x ∗ Z (A)
exp A(σ ξ ) , ∗ ξ ∈

x∈

∗ = {ξ ∈  | there exists ξ ∗ ∈  m such that ξ = ξ ∗ |} and where, for each where   ∗ ξ ∈  one makes an arbitrary choice of ξ ∗ ∈  such that ξ ∗ | = ξ . We stress that this definition of partition function is well defined for all d, and in particular, for the case d = 1. To define the pressure and free energy of A one needs to compute the thermodynamic limit as  → ∞: 1 1 P (A)
lim log Z (A) and FA (β)
− P (βA). “∞” # β

The pressure and free energy are the two fundamental objects of study in statistical physics of lattice spin systems. For instance, phase transitions correspond to non-differentiability for some derivative of free energy. Establishing the existence of these limits is a non-trivial task and regularity restrictions on the function A are required for the thermodynamic limit to exist. A physically important class of interactions are the finite range interactions. We study to what extent potentials (especially those related to finite range interactions) are determined by their free energy. In particular, we show that for a one-dimensional lattice spin system the free energy of finite range interactions typically determines the potential, up to natural equivalence and there is always at most a finite ambiguity (Theorem 3 and Theorem 4); we exhibit exceptional potentials where uniqueness fails (Proposition 3.1); and we establish deformation rigidity for the free energy (Theorem 5). I.2. Mathematical motivation. Since free energy plays such an essential role in statistical physics, it is natural to study this quantity for dynamical systems and Riemannian manifolds. In particular, one would like to have a geometric/topological interpretation of the free energy. In the special case of locally constant functions for subshifts of finite type, Tuncel [Tun1] introduced a quantity closely related to free energy. This quantity was introduced as an invariant in coding theory for Markov chains and appears to have been studied only in this context. The term beta function is a bit confusing, since in thermodynamics, beta usually denotes inverse temperature and there is another beta function in dynamics which is the mapping of the unit interval defined by x → βx mod 1. We will follow the established nomenclature in the dynamics literature. For a general smooth dynamical system T : X → X one can define the pressure of a continuous function f : X → R using the variational principle: 

P (f ) = sup hµ (T ) + f dµ µ is a T − invariant probability measure , X

where hµ (T ) denotes the measure theoretic entropy with respect to the measure µ. By analogy with statistical physics, we define the free energy for f by Ff (β) = −(1/β) exp P (βf ).

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Since the first factor −(1/β) in the definition of free energy plays no further role in our analysis, it is notationally convenient to replace the free energy by the beta function for f defined by βf (t) = exp P (tf ). For certain classes of dynamical systems, e.g., subshifts of finite type and hyperbolic maps, the beta function can be defined using Birkhoff sums (or Birkhoff averages) of the function f over periodic orbits. This is one of a hierarchy of several natural periodic orbit invariants, including the zeta function, the marked orbit spectrum (the set of Birkhoff sums around periodic orbits labeled by the periodic orbit), the unmarked orbit spectrum (the set of Birkhoff sums around periodic orbits labeled by the period), and the orbit spectrum (the unlabeled set of Birkhoff sums around periodic orbits). The weak orbit spectrum seems to be a less natural and less useful invariant than the unmarked orbit spectrum in the context of subshifts of finite type. The main objectives of this paper are to study the relations between these invariants and to show that in many cases the function f can be recovered, up to some unavoidable natural ambiguities, from these various spectrum. Results on subshifts of finite type and H¨older functions can be easily reformulated in terms of one-dimensional Axiom A flows (as we will elaborate at the end of this subsection). We observe that such rigidity problems in the study of dynamical systems have striking similarities to fascinating questions in length geometry (spectral geometry) which Kac adroitly summarized with the question “Can you hear the shape of a drum?”. Given a compact hyperbolic surface, the unmarked length spectrum consists of the set of lengths of all closed geodesics, and the marked length spectrum (the analogue of the marked orbit spectrum) consists of the lengths of closed geodesics labeled by the free homotopy class of the geodesic. The marked length spectrum determines the hyperbolic surface up to isometry [Ota], but the unmarked length spectrum does not [Vig, Sun, Bus]. By analogy, for subshifts of finite type, the marked orbit spectrum determines the function up to a natural equivalence (Lemma 1.2), while the unmarked length spectrum does not (Proposition A.II.1). The unmarked length spectrum for a hyperbolic surface typically does determine the surface [Wol] and there is a uniform bound, depending only on the genus of the surface, on the number of non-isometric hyperbolic surfaces having the same unmarked length spectrum [McK]. By analogy, for subshifts of finite type, the unmarked orbit spectrum for a locally constant function typically determines the function (Theorem 6) and that there is a uniform bound, depending only on the number of coordinates of the locally constant function, on the number of locally constant functions having the same unmarked orbit spectrum (Theorem 6). Finally, Guillemin and Kazhdan [GK] showed that the unmarked length spectrum for hyperbolic surfaces is deformation rigid, i.e., there are no smooth curves of non-isometric surfaces. We show the analogue of this result for the unmarked orbit spectrum for H¨older functions in Theorem 7. W. Parry has posed and studied related questions over a number of years [Par]. We summarize these analogies in Table 1, where the hyperbolic surfaces (with fixed genus) are determined up to isometry, and the locally constant functions (with fixed number of coordinates) are defined up to coboundary and automorphism of the shift. In a slightly different direction, we show that there exist uncountably many inequivalent H¨older continuous functions sharing the unmarked orbit spectrum (Proposition A.I.1). Our results for the unmarked length spectrum are intimately related to corresponding results for the beta and zeta functions. For example, knowledge of the unmarked length spectrum is equivalent to knowledge of the zeta function (Proposition 2.2). In contrast, the beta function is a more subtle invariant.

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Table 1. Summary of results on orbit spectrum and comparisons with corresponding results for hyperbolic surfaces Hyperbolic surfaces

Locally constant functions

Marked length spectrum determines surface [Otal] (true for negatively curved surfaces)

Marked orbit spectrum determines function [Livsic] (true for H¨older functions)

Unmarked length spectrum does not determine surface [Vigneras/Sunada/Buser]

Unmarked orbit spectrum does not determine function [Proposition A.II.1]

Unmarked length spectrum typically determines surface [Wolpert]

Unmarked orbit spectrum typically determines function [Theorem 6]

Uniform bound on number of surfaces with same unmarked length spectrum [McKean]

Uniform bound on number of functions with same unmarked orbit spectrum [Theorem 6]

No smooth arc with same unmarked length spectrum [Kazdan-Guillemin] (true for negatively curved surfaces)

No smooth arc with same unmarked orbit spectrum [Theorem 7] (true for H¨older functions)

Unmarked length spectrum never simple [Randol]

Unmarked orbit spectrum never simple [Proposition 4.2]

This paper is organized into separate sections on rigidity for the zeta function (§II), rigidity for the beta function (§III), and rigidity for the unmarked orbit spectrum (§IV). In the three appendices we construct functions which exhibit various types of degeneracy or (strong) nonrigidity. For instance, in Appendix I we construct an uncountable family of H¨older continuous functions which share the same unmarked orbit spectrum and thus have the same free energy. We also observe that Theorem 5, deformation rigidity of free energy, naturally extends to smooth hyperbolic maps. Whereas the analogy between length spectrum rigidity for geodesic flows and the problems we consider is a useful guide, it is not possible to translate results directly from one setting to the other. For example, the height functions over subshifts of finite type corresponding to geodesic flows on hyperbolic surfaces form a very small subclass of all H¨older functions. Furthermore, the automorphism group of a Riemann surface is typically trivial (and always finite), while the automorphism group for a subshift of finite type is usually quite large. This crucial disparity arises from the fundamental difference in the topology of the spaces involved. More generally, we consider the situation we are studying as the Axiom A analogue of the results for geodesic flows, at least in the case where the functions are positive. More precisely, we recall that for any subshift of finite type σ : X → X and any H¨older continuous function f : X → R we can construct a space Xf = {(x, t) ∈ X × R : 0 ≤ t ≤ f (x)}, where we identify (x, r(x)) ∼ (σ x, 0), and a flow ψ defined locally by ψ(x, t) = (x, t + u), subject to the identifications. The following result is a corollary to a theorem

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of Bowen and shows the relationship with Axiom A flows. Any Axiom A flow (restricted to a basic set) is called one dimensional if the basic set has a cross section which is a Cantor set. Proposition 1.1 [Bow]. For any one-dimensional Axiom A flow we can associate a subshift and H¨older continuous function for which the length spectrum coincides with the orbit spectrum. Conversely, given any H¨older continuous function and any r ≥ 1 there is a C r Axiom A flow with a basic set whose length spectrum coincides with the orbit spectrum. In particular, we see that the questions we consider about H¨older continuous functions could be equally well formulated in terms of the properties of one-dimensional Axiom A flows. This helps to reinforce the analogies with the problems for surfaces and geodesic flows. I.3. Subshifts of finite type and their periodic orbit invariants. Let A be a n×n aperiodic (transition) matrix with entries 0 or 1. We define

 ∞ + A {1, 2, . . . , n} : A(xk , xk+1 ) = 1 = x∈ k=0

which is compact, totally disconnected, and zero dimensional in the Tychonoff product + + → A be the subshift of finite type defined by (σ x)n = xn+1 . topology. Let σ : A Since A is aperiodic, the shift map is topologically mixing and has a dense set of periodic points. We let Aut(σ ) be the group of shift commuting homeomorphisms τ (i.e., σ ◦ τ = τ ◦σ ). This group is always countable and except in cases of small n contains free groups. This is in stark contrast to the situation for hyperbolic surfaces where the group of automorphisms is always finite and typically trivial. This crucial disparity arises from the fundamental difference in the topology of the spaces involved. In particular, it is natural + that the zero dimensional space A allows a much larger space of automorphisms. ∞ + n We define a metric on A by d(x, y)
n=0 (1 − δxn yn )/2 that enables us to + define the Banach space of H¨older continuous functions on A . An important class of H¨older continuous functions are the locally constant functions, i.e., functions that only depend on finitely many coordinates. We let LC(n) be the n-dimensional vector space of locally constant functions which depend on only the first n coordinates. These spaces are nested, i.e., LC(n) ⊂ LC(n + 1) for all n ∈ N. In the physical nomenclature, locally constant functions correspond to finite range interactions which form an important class of potentials for lattice spin systems. If f is a locally constant function, then after recoding if necessary, we can always assume that f (x) = f (x0 x1 ), i.e., f is in LC(2) for some subshift of finite type. For such functions the thermodynamic formalism reduces to matrix algebra. + Let f : A → R be a H¨older continuous function and let Sn f denote the sequence of Birkhoff sums n−1  Sn f (x) = f (σ k x). k=0

We can associate to f the unmarked orbit spectrum Lf = {(Sn f (x), n) : σ n x = x}

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and the weak orbit spectrum Wf = {Sn f (x) : σ n x = x}. Since the weak orbit spectrum does not contain the periods of the orbits, it is a weaker invariant than the unmarked orbit spectrum. In Appendix III we construct an uncountable family of pairwise inequivalent H¨older continuous functions with different unmarked orbit spectrum, but all sharing the same weak orbit spectrum. The following observation shows two natural ways for functions to have the same orbit spectrum. The proof is obvious. Lemma 1.1. + (i) If f1 and f2 are cohomologous (f1 ∼ f2 ), i.e., there exists a function u ∈ C(A , R) with f1 − f2 = u ◦ σ − u, then Lf1 = Lf2 ; (ii) If f2 = f1 ◦ τ , where τ ∈ Aut(σ ) is a shift commuting homeomorphism (i.e. σ ◦ τ = τ ◦ σ ), then Lf1 = Lf2 .

Thus when studying the orbit spectrum, it is natural to ignore, or to factor out, these two type of trivial relations. + Definition. We define two functions f1 and f2 on A to be equivalent, f1 ≈ f2 , if f2 ∼ f1 ◦ τ , where τ ∈ Aut(σ ). We say two functions are non-equivalent if they are not equivalent .

The simplest type of such a shift commuting homeomorphism τ0 ∈ Aut(σ ), at least in the case of a full shift, is given by some perturbation of the alphabet. For one sided subshifts of finite type these questions were studied by Hedlund [Hed], who showed that the automorphism group of the (one-sided) full shift on two symbols is simply generated by the shift map and permutations of blocks of symbols. In contrast, he showed that for the (one-sided) full shift on three or more symbols the automorphism group is more complex. We can also associate to f the marked periodic spectrum, Mf = {(Sn f (x), x) : σ n x = x}. The following observation shows that the marked orbit spectrum essentially determines the function f up to cohomology. Lemma 1.2. Two functions f1 and f2 are cohomologous if and only if Lf1 = Lf2 Proof. This is a version of Livsic’s Theorem [Liv]; see [PP] for this precise statement and proof.  + Two other important invariants of a function f : A → R are the zeta-function, ζf , defined by the power series

ζf (z, t)
exp

∞ n   z n=1

n

exp(tSn f (x)),

σ n x=x

and the beta-function, βf , defined by βf (t) being the reciprocal of the radius of convergence of the zeta function ζf (z, t), i.e.,    1 log βf (t) = P (tf ) = lim log exp (tSn f (x)) , (1) n→∞ n n σ x=x

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where P (g) denotes thermodynamic pressure of a general H¨older continuous function g defined by    1 P (g) = lim log exp (Sn g(x)) . n→∞ n n σ x=x

For subshifts of finite type this definition of pressure is equivalent to the variational definition given in I.B. If f is a locally constant function, after recoding if necessary we can always assume that f (x) = f (x0 x1 ). A routine calculation [PP] shows that ζf (z, t) =

1 , det(I − zAt·f )

(2)

where Af denotes the n × n matrix with entries Af (i, j ) = A(i, j ) exp(f (i, j )). The hierarchy of these four invariants is illustrated by the following diagram: f1 ≈ f2 ⇒ Lf1 = Lf2 ⇒ ζf1 = ζf2 ⇒ βf1 = βf2 . In this manuscript we investigate under what conditions these arrows can be reversed. II. The Zeta Function The main result in this section is that the zeta function for locally constant functions typically determines the equivalence class of the function. We begin with the following simple lemma for matrix algebra. We recall that P is a permutation matrix if exactly one entry in each row and column is equal to 1 and all the others are 0. Lemma 2.1. Let B = (bij ) be a n × n matrix with non-negative entries and let B (t) = t ) denote the Hadamard t th power of B with characteristic polynomial q(z, t) = (bij det(tI − B (t) ). (1) The polynomial q is invariant under conjugation of B by any permutation matrix P , i.e., q(z, t) = det(tI −(P −1 BP )(t) ) for all t ∈ R. There are precisely n! permutation matrices of size n × n. (2) For integer values of t, the polynomial q is invariant under conjugation of B by any diagonal matrix D, i.e., q(z, n) = det(nI − (D −1 BD)(n) ) for n ∈ Z. Proof. The proof is a straightforward calculation.



Since the characteristic polynomial q(z, n) of B (t) is invariant under conjugating B by a permutation matrix, Lemma 2.1 implies there is an inherent finite ambiguity in trying to recover B from q(z, n). At best, one can recover B only up to conjugation by a permutation matrix, and there are n! permutation matrices. The following is a generalization of a lemma on Newton’s identities [Wae] allowing negative terms, and is a special case of a result on Dirichlet series due to Mandelbrojt which will appear in Proposition 2.2. Lemma 2.2. Consider an expression of the form s(t) = λt1 + λt2 + · · · + λtm − λtm+1 − λtm+2 − · · · − λtm+n , where λk > 0 and t ∈ R. Then one can obtain the numbers λk (up to permutation) from s(t).

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Proof. In the special case that s(t) = λt1 + λt2 + · · · + λtm is a sum of powers, the lemma follows immediately from Newton’s identities, in which case one only needs to know s(1), . . . , s(m). Now assume that in s(t) no two of the numbers λk coincide; the most general case requires a trivial modification to this argument. Then

 0, if 0 ≤ r < min λ   m m+n  k   r t   r t k , = − lim r t s(−t) = lim t→∞ t→∞   ±∞, if r ≥ min λk λk λk k=1

k=m+1

k

where the minimizing λk belongs to {λ1 , . . . , λm } if the limit is +∞ and the minimizing λk belongs to {λm+1 , . . . , λm+n } if the limit is −∞. Hence one can first detect the smallest λk as the jump point of the limit, remove that λk from the sum, then detect the next smallest λk , remove it from the sum, and so on.  When studying the zeta function of a function f ∈ LC(2) it is notationally con(t) (t) venient to work with the characteristic polynomial qf (z, t) = det(zI − Af ) of Af instead of ζf (z, t). Using the expression for the zeta function in (2) one can easily see the relationship qf (z, t) = zn ζf (1/z, t)−1 . Proposition 2.1. There exists an explicit uniform bound C = C(n) > 0 on the number of n×n aperiodic matrices with non-negative entries (up to conjugation by diagonal and permutation matrices) with the same characteristic polynomial q(z, t). Furthermore, the typical such n × n matrix is actually determined by its characteristic polynomial q(z, t) (up to conjugation by diagonal and permutation matrices). Proof. Let us first consider the case where the entries of B are positive. We only need to work with E1 (B (t) ), E2 (B (t) ), and E3 (B (t) ), the first three principal minors for the matrix B (t) . It is well known [HJ] that these three minors are the coefficients of the first three terms (in z) in qf , i.e., q(z, t) = zn −E1 (B (t) )zn−1 +E2 (B (t) )zn−2 −· · ·±En (B (t) ), where Ei (B (t) ) denotes the i th principal minor of B (t) and that the principal minors of B are themselves invariant under conjugation by diagonal and permutation matrices. By conjugating B by a suitable diagonal matrix, we can assume that each entry in the first column of B is 1 except (possibly) the entry b11 . The first principal minor,  E1 (B (t) ) = trace(B (t) ), is the sum ni=1 biit , which by Lemma 2.2 determines the unordered list of diagonal entries {b11 , b22 , . . . , bnn }. Since we are only interested in recovering the matrix B up to conjugation by permutation matrices, we can assume we know the ordered list of diagonal entries. The general term in the second principal minor corresponds to the determinant of the t − bt bt . In particular, when special principal matrix B (t) {i, j } and is of the form biit bjj ij j i t t t . Let us assume the generici = 1 and j ≥ 2, the terms are of the form b11 bjj − b1j t = bt bt for all i, j, k, l. Then Lemma 2.2 allows us to obtain the ity condition biit bjj kl lk unordered list of all double products {bij bj i }. Included in this unordered list are all the entries from the first row {b1j bj 1 } = {b1j }. The general term in the third principal minor corresponds to the determinant of the t bt −bt bt bt −bt bt bt − special principal matrix B (t) {i, j, k} and is of the form biit bjj kk ii j k kj j i ij kk t t t t t t t t t bki bik bjj + bj i bik bkj + bki bij bj k . Let us also assume the degeneracy condition that for all {i, j, k} all six terms are distinct and do not cancel with any terms in the determinant of other special principal matrices B (t) {q, r, s}. The matrices which do not satisfy all our

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nondegeneracy assumptions are easily seen to form an algebraic variety of codimension one. t bt bt , where we already know the Included in E3 (B (t) ) are terms of the form bkk ij j k t diagonal elements bkk . This observation, along with our genericity assumptions and Lemma 2.2, allows us to obtain, without any ambiguity, all double products {bij bj i }, including all the entries of the first row {b1j }. We now show how to recover the general entry bj k . The components of E3 (B) include all terms bj 1 b1k bkj for j ≥ 2, and thus terms b1k bkj . Let τ be one of the terms in E3 (B), and suppose that bj k bkj τ = b1k bkj = b1k . bj k Inverting this expression, we obtain bj k bkj , τ where we know the terms b1k and bj k bkj . Since E3 (B) contains terms of the form bki bij bj k , it contains the special terms of the form bk1 b1j bj k , and thus terms b1j bj k for k ≥ 2. Thus if we multiply the expression obtained just above for bj k by the known term b1j , we must obtain a term in E3 (B). If we do not, then our genericity assumption implies that τ = b1k bkj . This lets us determine the product b1j bj k , and since we know the element b1j , it lets us determine the entry bj k . If B is non-generic, it may a priori happen that the product b1j bj k (in the previous paragraph) is a term in E3 (B), even though τ = b1k bkj . Dividing by b1j would give the wrong value of bj k . However, very crudely, one can not obtain more than #E3 (B) ≤ 6n3 2 wrong values for bj k . Thus, again very crudely, there are at most C(n) = (6n3 )n such matrices with the same characteristic polynomial q(z, t). For a general matrix B with non-negative entries, by assumption, there exists an integer M ≥ 1 such that B M has all positive entries. Since the characteristic polynomial for B determines the characteristic polynomial for B M , we can apply the preceding argument to the characteristic polynomial for B M , to obtain the matrix B M up to conjugation by diagonal and permutation matrices. By extracting the (unique) M th root of B M we can recover the matrix B up to conjugation by diagonal and permutation matrices.  bj k = b1k

We remark that Proposition 2.1 need not hold for a general non-aperiodic matrix with non-negative entries. Consider the matrix B defined by   10 e . f B = 0 1 0 0 1−e−f Since the matrix B and hence B (t) are upper triangular, the characteristic polynomial q(z, t) = (z − 1)2 (z − (1 − e − f )t ). Hence one can obtain e + f from q(z, t) but there is no way to obtain e and f separately. We also remark that C(2) = 1. After conjugating by a diagonal matrix we can assume that   ab B= . 1d The entry b is positive since B is aperiodic. Knowledge of the trace of B (t) together with Lemma 2 gives the diagonal entries a and d, up to permutation. Since we can conjugate

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B by a permutation matrix, we can assume that we know a and d precisely. If det B = 0, then the knowledge of the determinant of B (t) together with Lemma 2 gives b. Since the only way for det B = 0 is for b = ad, we can obtain b precisely when det B = 0. We now show that conjugating the matrix representing a locally constant function by a permutation matrix results in a new function which differs from the original function by an automorphism of the shift. Lemma 2.3. Consider a locally constant function f represented by the n × n matrix Af . Let P be a n × n permutation matrix and let g denote the locally constant function represented by the matrix Bg = P −1 Af P = P T Af P , where P T denotes the transpose of P (we are using the fact that P −1 = P T for any permutation matrix). Then the functions f and g are related by an automorphism of the shift determined by permuting the n letters of the alphabet by the permutation defined by P . Proof. Let us write the permutation matrix    r1 r1,1 r2  r2,1   P =  ...  =  ... rn

rn,1

 r1,2 · · · r1,n r2,2 · · · r2,n  . .. . . ..  . .  . rn,2 · · · rn,n

The matrix P induces a permutation τ : {1, . . . , n} → {1, . . . , n} by defining rk = eτ (k) , where ek denotes the row vector (0, 0, . . . , 0, 1, 0, . . . 0) which contains a single 1 in the kth place and 0 in all other places. This permutation defines an automorphism + σ of the subshift A simply by replacing every occurrence of the letter k in a word by the letter τ (k).   The (i, j ) entry of the matrix Bg = P −1 Af P is nk=1 nl=1 r k,i rl,j exp(f  (k, l)), and thus the (i, j ) entry of the matrix for the function g ◦ σ is nk=1 nl=1 rk,i rl,j exp(f (τ (k), τ (l))). From definitions, we see that all of the products rk,i rl,j vanish unless i = τ (k) and j = τ (l), in which case the (i, j ) entry of the matrix for g ◦ σ is exp(f (i, j )).  It is implicit in the hypothesis of Lemma 2.3 that conjugation by the permutation matrix P preserves the transition matrix of the subshift of finite type. If this is not the case then the two functions f and g are defined on different subshifts of finite type. The next lemma says that in all but the trivial case the matrix obtained by conjugating a column stochastic matrix by a diagonal matrix is not column stochastic. Lemma 2.4. Let A denote an aperiodic n × n non-negative column stochastic matrix and D a diagonal matrix such that D −1 AD is a column stochastic matrix. Then D is the identity matrix. Proof. If A = {Aij } and D = diag{d1 , . . . dn }, then D −1 AD = {(dj /di )aij }. Since the matrix D −1 AD is invariant under multiplication of D by a scaler, wecan assume that d1 = 1. Since A is column stochastic, the column sum must satisfy ni=1 aij = 1 for −1 j = 1, . . . n, and thus (1, . . . , 1) is a left eigenvector of A with eigenvalue 1. For D AD to be column stochastic, the columns must satisfy ni=1 di aij = dj for j = 1, . . . n, and thus (d1 , . . . , dn ) is a left eigenvector of A with eigenvalue 1. By simplicity of the maximal eigenvalue (we are assuming that the transition matrix is aperiodic) we deduce that (d1 , . . . , dn ) = (1, . . . , 1).  From the above results we conclude the main result of this section.

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+ Theorem 1. Let A be a n × n transition matrix, (A , σ ) be a mixing one-sided subshift of finite type, and f ∈ LC(m) be a locally constant function. Then there are at most C(nm ) non-equivalent locally constant functions in LC(m) with the same zeta function. Furthermore, for generic f ∈ LC(m) (i.e., on the complement of a codimension one algebraic set) the zeta function determines the function (up to conjugation by diagonal and permutation matrices).

Proof. By recoding, we can assume that f ∈ LC(2) for a new subshift on nm symbols. We apply Proposition 2.1 to the matrix Af and use Lemmas 2.3 and 2.4 to show that the ambiguity in the conjugacy corresponds to equivalence of functions.  Corollary 1.1. Given any locally constant function f there are at most countably many non-equivalent locally constant functions g with the same zeta function. Proof. It suffices to observe that if f ∈ LC(m) and g ∈ LC(m + l) for l ≥ 0, and f and g share the same zeta function, then Theorem 1 shows that there are finitely many non-equivalent classes of such functions g. By considering the union over l the result follows.  In Appendix II we construct examples of non-equivalent locally constant functions with the same zeta function. In particular, this shows that C(m) ≥ 2 for some m. In the more general context of H¨older continuous functions, it is easy to see that knowledge of the zeta function is equivalent to knowing the unmarked orbit spectrum. Proposition 2.2. The zeta function for a H¨older continuous function determines the unmarked orbit spectrum, i.e., ζf1 = ζf2 implies that Lf1 = Lf2 . Thus ζf1 = ζf2 if and only if Lf1 = Lf2 .   n Proof. Let us write ζ (z, t) = ∞ σ n x=x exp(tSn f (x)). n=1 (z /n) an (t), where an (t) = The power series ζ (z, t) defines a holomorphic function in z in the disk of radius exp β(t). This implies that the functions an (t) are all uniquely determined and can be obtained by differentiating the power series for fixed t. For each n we can apply Lemma 2.2 to the sum of exponentials to obtain the set of numbers {Sn f (x): σ n (x) = x}, and thus we can recover the entire unmarked orbit spectrum.  The zeta function, being a function of two variables, seems to contain a great deal of information about the function. Below we show that if we fix the variable z = 1, the zeta function still captures the weak unmarked orbit spectrum. Proposition 2.3. The restricted zeta function ζf (1, t) determines the weak orbit spectrum, i.e., ζf1 (1, t) = ζf2 (1, t) implies that Wf1 = Wf2 . Proof. We can expand a restricted zeta function as a Dirichlet series ζf (1, t) =

∞ 

an exp (−µn t),

n=1

where µ1 ≤ µ2 ≤ . . . → ∞ are real numbers corresponding to the unmarked orbit spectrum and an ∈ C. This series converges on a half-plane containing some point c ∈ R. Then by a result of Mandelbrojt [Man, p. 388] we have for each ν ∈ R,  ∞  1  ζf (1, c + it) exp ν(c + it) dt if ν > µ  1 2π (c + it)2 (ν − µn ) =  −∞ µn <ν 0 if ν ≤ µ1 .

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In particular, this allows us to recover the coefficients an and the exponents µn uniquely (up to permutation).  Finally, we show that the zeta function is deformation rigid. We call this property isozetal rigidity. + Theorem 2. Let (A , σ ) be a mixing one-sided subshift of finite type. Assume that + , R) is a C 2 family of H¨older continuous functions with (−, )  λ → fλ ∈ C α (A identical zeta functions (i.e., ζfλ = ζf0 for all − < λ < ). Then the deformation is trivial, i.e., fλ ∼ f0 for all − < λ < .

Since we will prove a stronger result on beta functions in Sect. III.B, we omit the proof of this theorem. III. The Free Energy and Beta Functions III.1. Locally constant functions. We remind the reader that we shall use the terms free energy and beta function interchangeably, since they are trivially related. Our formulation will be in terms of beta functions. We begin by showing that it is not always true that the beta function determines the zeta function. Proposition 3.1. There exist locally constant functions f1 , f2 with βf1 = βf2 but ζf1 = ζf1 .  Proof. Let 2+ = ∞ n=0 {1,2}. We can choose two rationally independent numbers 0 < a < b and define locally constant functions   a if x0 = 1 a if (x0 , x1 ) = (1, 1) or (2, 2) f1 (x) = and f2 (x) = . b if x0 = 2 b if (x0 , x1 ) = (1, 2) or (2, 1) We easily see that βf1 (t) = βf2 (t) = exp (ta) + exp (tb). Using (2), the associated zeta functions are given by   1 − z exp (ta) −z exp (ta) −1 = 1 − z(exp (ta) + exp (tb)) ζf1 (z, t) = det −z exp (tb) 1 − z exp (tb) and

 1 − z exp (ta) −z exp (tb) −z exp (tb) 1 − z exp (ta) = 1 − 2z exp (ta) + z2 (exp (2ta) − exp (2tb)),

ζf2 (z, t)−1 = det



which are clearly different provided a = b.



We now discuss the ambiguity with which the beta function determines the function. There do exist functions f for which the beta function βf does determine the equivalence class of the function f . A trivial example is a coboundary f = u ◦ σ − u. In this case, the beta function βf = 0. Ruelle observed (see Lemma 3.3) that the lack of strict convexity of the pressure function implies that the function must be a coboundary. We now show that typically the beta function for a locally constant function f determines the equivalence class of the locally constant function.

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+ Theorem 3. Let (A , σ ) be a mixing one-sided subshift of finite type with incidence matrix A. For each m ∈ N, for generic locally constant functions f ∈ LC(m) the β-function determines the equivalence class of the function in LC(m).

The strategy for the proof of Theorem 3 is as follows. Proposition 3.2 below says that typically the beta function determines the characteristic polynomial. Proposition 2.1 says that typically the characteristic polynomial determines the function. Combining these results then gives Theorem 3. More precisely, we show that typically the characteristic polynomial qf (z, t) is minimal and thus the beta function determines qf (z, t). We begin with a few preliminaries. Suppose that βf1 = βf2 , for f1 , f2 ∈ LC(m). By recoding if necessary, we can assume that f1 , f2 ∈ LC(2). Both functions have the same pressure P = log β(1). Let A be the associated n × n transition matrix. If µi is the unique equilibrium state for fi , let gi denote the Jacobian function defined by gi (x0 , x1 ) = dσ ∗ µi /dµi (x0 , x1 ), whenever A(x0 , x1 ) = 1, and zero otherwise. There exists u(x) = u(x0 ) such that fi = gi + u ◦ σ − u + P . By construction βg1 = βg2 and P (g1 ) = P (g2 ) = 0. Thus the function gi can be viewed as a canonical representative in the cohomology class of fi . It suffices to show that generically g1 is cohomologous to g2 . We can associate to g1 and g2 non-negative column stochastic matrices P1 = (P1 (r, s))ij and P2 = (P2 (r, s))ij , where Pi (r, s) = A(r, s) exp gi (r, s) for 1 ≤ r, s ≤ n. (t) The function βgi (t) is the maximal eigenvalue for Pi
(Pi (r, s)t ), the matrix given (t) by raising all of the entries to the power t. The matrices Pi ∈ Mn (R), where Mn (R) denotes the ring of n × n matrices with entries in the ring

 k  t ni ai : ni ∈ Z, ai > 0 , R= i=1

and the beta function βgi (t) is a zero of the characteristic polynomial qi (z, t) = det(zI − (t) Pi ) ∈ Z[R]. Proposition 3.2. The beta function βf (t) for a locally constant function defined by a n × n non-negative column stochastic matrix A typically determines the characteristic polynomial qf (z, t). The minimal polynomial p(z) for βf (t) is the (unique) monic polynomial in Z[R] of smallest degree for which p(βf (t)) = 0. Clearly the beta function always determines its minimal polynomial. We begin by recalling the following result. Lemma 3.1. The minimal polynomial p(z) for βf (t) divides the characteristic polynomial qf (z, t). In particular, if qf (z, t) is minimal, then it is determined by the beta function. Proof. We include a sketch of the proof (due to Tuncel cf. [Tun2]) for completeness. Let us consider the field F of rational fractions on R. We define an ideal I ⊂ F[z] by I = {f (z, t) ∈ F[z] : f (β(t), t) = 0}. Since F[z] is a principal ideal domain [Fra, p. 282] there exists an element p(z, t) ∈ F[z] such that I = p(z, t)F[z]. The element p(z, t) must have minimal non-zero degree in I since it is generating. In particular, we can write qf (z, t) = p(z, t) · s(z, t), for some s(z, t) ∈ F[z]. Suppose that qf (z, t) = zl − q1 zl−1 − . . . − ql−1 z − ql , p(z, t) =

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zd − p1 zd−1 − . . . − pd−1 z − pd and s(z, t) = zm − s1 zm−1 − . . . − sm−1 z − sm , where the coefficients qi ∈ R and pi , si ∈ F. Let S be the group ring over Z generated using all the exponentials from the pi , and exponentials from the numerators and denominators of pi and si . We can assume that pi = p˜ i /p0 and si = s˜i /s0 , say, where p˜ i , p0 , s˜i , s0 ∈ S. We can then rewrite qf (z, t) = p(z, t) · s(z, t) as an identity in S:   s0 p0 zl − q1 zl−1 − . . . − ql−1 z − ql    = p˜ 0 zd − p˜ 1 zd−1 − . . . − p˜ d−1 z − p˜ d s0 zm − s˜1 zm−1 − . . . − s˜m−1 z − s˜m . However, since S is a unique factorization domain each irreducible factor of p0 s0 must divide one of the two terms multiplied on the right-hand side. For example, if it divides the first term, then it divides each term p˜ 0 , . . . , p˜ d , and is thus invertible (since they can be assumed to be coprime). Thus p0 is a monomial and p(z, t) ∈ R[z]. We deduce that p is a minimal polynomial of β(t).  By a typical matrix A we mean that no non-trivial product of integer powers of entries aij is equal to 1 (or equivalently, the numbers log aij are rationally independent). Lemma 3.2. A characteristic polynomial qf (z, t) for a typical n × n non-negative column stochastic matrix A is minimal, i.e., qf (z, t) = a(z, t)·b(z, t), where a(z, t), b(z, t) ∈ Z[R] are non-constant functions. Proof. If we assume for a contraction that the characteristic polynomial is not minimal and we write it as a product qf (z, t) = a(z, t) · b(z, t), then we can multiply out the coefficients and obtain a contradiction by comparing equations from the coefficients of the two polynomials multiplied together. First, consider for the purposes of illustration the case of degree 2. The general case is similar. Assume for a contradiction that we can write qf (z, t) = z2 − tr(A(t) )z −det(A(t) )      = z− li λti z − mj µtj  j  i    = z2 −  li λti + mj µtj  z + li mj λti µtj .





i

j

i,j

p 1−p Let A = 1−q . Then tr(A(t) ) = pt + q t and det(A(t) ) = pt q t − (1 − p)t (1 − 1)t . q Comparing the z coefficients we see that {λi , µj } = {p, q}. However, comparing the constant terms we see that we have {λi µj } = {pq, (1 − p)(1 − q)}. However, this means that pq = (1 − p)(1 − q), p2 = (1 − p)(1 − q) or q 2 = (1 − p)(1 − q), which imposes relations on p and q. For typical functions this condition fails. This contradiction shows that qf (z, t) is minimal in the case n = 2. For the general case, let us consider a typical matrix   a11 a12 · · · a1n a21 a22 · · · a2n   A=  ... ... . . . ...  .

an1 an2 · · · ann

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Then we can write  qf (z, t) = zn −

n 









aiit zn−1 + 

aijt ajt i  zn−2 −. . .±

i=j

i=1



t t t a1τ (1) a2τ (2) · · · anτ (n) ,

τ ∈Sn

where Sn denotes the set of permutations on n symbols. Let us assume for a contradiction that qf (z, t) = a(z, t) · b(z, t), where a(z, t), b(z, t) ∈ Z[R]. If we write 





a(z, t) = z + d

li λsi

b(z, t) = z

n−d

+

+ ... +

 



i 



z

d−1

 mi µsr

r

kj αis  zn−d−1

+ ... +

, 





j

pl ηls

(3) ,

l

then we see that  qf (z, t) = zn −

 i

li λti +











ki αit zn−1 + . . . + 

i

mi pl µtr ηlt  .

(4)

r,l

Comparing the coefficients of (3) and (4) we see the following: (1) By comparing the constant terms we have that ±

 τ ∈Sn

t t t a1τ (1) a2τ (2) · · · anτ (n) =



mr pl µtr ηlt ,

r,l

and thus there is a correspondence between the terms µr ηl and the terms a1τ (1) a2τ (2) · · · anτ (n) for some τ ∈ Sn . (2) By comparing the zd term we have that each ηr must be of the form a1τ (1) a2τ (2) · · · anτ (n) , ai1 i1 . . . ain−d in−d

(5)

for some permutation τ ∈ Sn , which also precisely fixes the d-terms i1 , . . . , id (i.e., (5) represents the terms for all the fixed points of the permutation). This is easily seen since in the expansion of det(zI −P s ) the zd contribution comes from d entries on the diagonal corresponding to rows (and columns) i1 , i2 , , . . . id , say. (3) Similarly, by comparing the zn−d term we have that each µr must be of the form a1τ (1) a2τ (2) · · · anτ (n) , ai1 i1 . . . ain−d in−d

(6)

for some permutation τ ∈ Sn , which additionally precisely fixes the (n − d)-terms i1 , . . . , in−d . Consider any term a1τ (1) a2τ (2) · · · anτ (n) where τ ∈ Sn has no fixed point. This must occur in the constant term described in (1). However, this cannot be written as a product

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of a term ηl of the form (5) and a term µr of the form (6). 2 This contradiction shows that qf (z, t) is irreducible.  Remark. For n ≤ 4 one can also show these results using the quadratic, cubic, and quartic formulae. In the case n = 2 let   a 1−b A= . 1−a b The characteristic polynomial of A(t) is q(z, t) = z2 − trA(t) z + det A(t) . For this expression to be factorized into two linear non-trivial polynomials over R we require that the square root of the discriminant lie in R, i.e., tr 2 A(t) − 4 det A(t) =

(a t + bt )2 − 4(a t bt − (1 − a)t (1 − b)t ) ∈ R.

This can only hold when a + b = 1. Since β(t) is the maximal eigenvalue of A(t) it follows that 2βf (t) = trA(t) +

tr 2 A(t) − 4 det A(t) .

√ If tr 2 A(t) = 4 det A(t) , the beta function determines both trA(t) and tr 2 A(t) − 4 det A(t) . Substituting trA(t) into the term in the square root, one sees that the beta function also determines det A(t) , and thus q(z, t). The nondegeneracy condition is equivalent to the condition a + b = 1. The following theorem shows that for any f ∈ LC(m) there are only finitely many functions in LC(n) with the same beta function. The proof uses an interesting interplay between ring theory and thermodynamic formalism. + , σ ) be a mixing one-sided subshift of finite type. For each m ∈ N Theorem 4. Let (A and every locally constant function f ∈ LC(m), there are at most finitely many nonequivalent locally constant functions in LC(m) with the same beta function.

Proof. Given the beta function associated to f we can consider the family of stochastic matrices given by 

n  S= P: P (i, j ) = 1, for all j and det(β(t)I − P (t) ) = 1, for all t ∈ R . i=1

By analyticity considerations this is equivalent to 

n  (k) S= P: P (i, j ) = 1, det(β(t)I − P ) = 1, for all j and for all k ∈ N . i=1

It is convenient to use this latter formulation to think of S as being an algebraic set in 2 Rn given by an infinite set of polynomials. However, any such algebraic set can always be defined by only finitely many polynomials by the Hilbert Basis Theorem [Ful, p. 13]. In particular, there are two possibilities: 2 To illustrate this, consider the case of a full shift on 4 symbols and n = 4. The “factors” of degrees d = n − d = 2. The constant term of det(zI − At ) is a sum of 24 terms of the form ±a1τ (1) a2τ (2) a3τ (3) a4τ (4) , where τ ∈ S4 is a permutation on 4 symbols. This includes a12 a23 a34 a41 corresponding to the cyclic permutation (1234). The coefficient of z2 must be a sum of the terms aij aj i and aii ajj , since each z2 contribution eliminates 2 rows (and columns) and the corresponding coefficient is the determinant of the remaining 2 × 2 matrix. In particular, we need that a12 a23 a34 a41 = aij aj i ars asr , say, which is impossible for a typical matrix.

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(i) S is a finite set; or (ii) S contains non-trivial connected components. However, case (ii) cannot occur, since it contradicts Theorem 5 (Deformation Rigidity). We therefore conclude that S is finite, as claimed.  The proof of the following corollary is very similar to the proof of Corollary 1.1. Corollary 4.1. Given any locally constant function f there are at most countably many non-equivalent locally constant functions g with the same beta function. III.2. H¨older continuous functions. In this section we prove the beta function is deformation rigid. In particular, this implies that there are no connected sets of isobetal functions. We call this phenomena isobetal rigidity. We observe that our proof easily extends to the beta function for smooth hyperbolic maps and H¨older continuous functions. Let us briefly recall Ruelle’s derivative formulas for pressure [PP, Rue]. + Lemma 3.3. Let f and g be H¨older continuous functions on A . (a) The first derivative of pressure can be expressed as ! ∂ !! P (f + sg) = gdµf , + ∂s !s=0 A

where µf is Gibbs measure for potential f . (b) The second derivative of pressure can be expressed as ! ∂ 2 !! P (f + sg) = var(g), ∂s 2 !s=0

 where µf is the Gibbs measure for potential f (i.e., P (f ) = hµf (σ ) +  + f dµf ), A and 2  n−1 1 i g(σ x) − gdµf dµf (x) ≥ 0. var µf (g) = lim + n→∞ n  A A k=0

(c) The expression var µf (g) = 0 if and only if g ∼ c, where c is a constant. This brings us to the statement of isobetal rigidity. + , σ ) be a mixing one-sided subshift of finite type. Assume that Theorem 5. Let (A + , R) is a C 2 family of H¨older continuous functions with (−, )  λ → fλ ∈ C α (A identical beta functions (i.e., βfλ = βf0 for all − < λ < ). Then fλ ∼ f0 for all − < λ < .

Proof. For any s0 ∈ (−, ) we can use the C 2 assumption to make the expansion + (s − s0 )2 fs(2) + O((s − s0 )3 ), fs = fs0 + (s − s0 )fs(1) 0 0 + where fs0 , fs0 ∈ C α (A ). The hypothesis implies that the beta functions for this one parameter family all coincide. Thus P (−tfs ) = 0 for all t ∈ R. From Lemma 3.3 we obtain that ! ∂ !! P (−tf ) = −t fs(1) dµ (7) 0= s 0 + ∂s !s=s0 A (1)

(2)

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and 0=

475

! ∂ 2 !! ), P (−tf ) = −t fs(2) dµ + t 2 var µ (fs(1) s 0 0 + ∂ 2 s !s=s0 A

(8)

where µ = µ−tfs0 denotes the Gibbs measure for the potential −tfs0 .  (2) Choose arbitrarily small t having the opposite sign of  + fs0 dµ. The expression A

(1)

on the right-hand side of (8) is thus non-negative, and it follows that var(fs0 ) = 0. (1) Lemma 3.3(c) allows us to conclude that fs0 ∼ c, and (7) implies that c = 0. Thus for + all s0 ∈ (−, ) there exists hs0 ∈ C α (A , R) such that ! ∂fs !! = hs0 ◦ σ − hs0 . ∂s !s=s0 We integrate both sides with respect to s, sum over each periodic orbit, and apply the Livsic theorem to conclude that for each s the function fs ∼ f0 .  For a smooth hyperbolic map we can apply the proof of Theorem 5 directly on the manifold to obtain the following extension of isobetal deformation rigidity to Axiom-A diffeomorphisms. Corollary IV.1. Let M be a smooth manifold and suppose that  ⊂ M is a basic set for an Axiom-A diffeomorphism T : M → M. Assume that (−, )  λ → fλ ∈ C α (, R) is a C 2 family of H¨older continuous functions on  with identical beta functions (i.e., βfλ = βf0 for all − < λ < ). Then fλ ∼ f0 for all − < λ < . IV. The Unmarked Orbit Spectrum We now turn to the final type of periodic orbit invariant we shall consider. This is the analogue of the unmarked length spectrum for hyperbolic surfaces where one labels the closed geodesics by word length. Recall that the length spectrum for a hyperbolic surface is never simple, and in fact has unbounded multiplicity [Ran]. By contrast, a transversality argument easily shows that the length spectrum of a non-constant negatively curved manifold is generically simple [Abr]. We say that the orbit spectrum for the function f is simple if the elements of the set Wf are all distinct. The next result shows that the orbit spectrum is generically simple, as analogous with the case of negatively curved surfaces. Proposition 4.1. Fix α > 0. There exists a dense Gδ subset of α−H¨older continuous functions for which the unmarked orbit spectrum is simple. + Proof. The space C α (A , R) of α−H¨older continuous functions is a complete metric space, and hence a Baire space. Let Bn denote the set of α−H¨older continuous functions which give distinct weights to periodic orbits up to period n. This is clearly an open dense set. It follows from the Baire category theorem that the intersection of all Bn will be a dense Gδ set. 

Even though we know by Proposition 4.1 that generically the spectrum is simple, every locally constant functions has non-simple spectrum, by analogy with the case of hyperbolic surfaces.

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Proposition 4.2. Every locally constant function has non-simple unmarked orbit spectrum with unbounded multiplicity, i.e., for each m sufficiently large, there exists M periodic points of the same period with the same Birkhoff sum. Proof. Let f ∈ LC(n), say, have range {α1 , . . . , αk }, then the values {Sm f (x) : for σ m x = x} are contained in the set

 k k   li αi : li = m , i=1

i=1

which has cardinality at most mk . However, since the number of periodic orbit of period m grows exponentially fast the result easily follows.  We now consider the extent to which the unmarked orbit spectrum determines the equivalence class of the function and prove the analogs of the theorems of Wolpert and McKean for hyperbolic surfaces (i.e., that typically the unmarked length spectrum determines the surface and there are uniform bounds depending on the genus of non-isometric surfaces having the same unmarked length spectrum). + , σ ) be a mixing one-sided subshift of finite type. If f ∈ LC(n) is Theorem 6. Let (A a locally constant function then there are at most C(n) non-equivalent locally constant functions in LC(n) with the same unmarked orbit spectrum. Furthermore, for generic f ∈ LC(n) (i.e., on the complement of a codimension one algebraic set) the unmarked orbit spectrum determines the function in LC(n) (up to conjugation by diagonal and permutation matrices).

Proof. This follows immediately from Theorem 1 and Proposition 2.2.



We now observe that the unmarked orbit spectrum is deformation rigid. + , σ ) be a mixing one-sided subshift of finite type. Assume that Theorem 7. Let (A + , R) is a C 2 family of H¨older continuous functions with (−, )  λ → fλ ∈ C α (A identical orbit spectra. Then the deformation is trivial, i.e., fλ ∼ f0 for all − < λ < .

We have already proved a stronger result on beta functions in Sect. III.B. Finally, although for generic f ∈ LC(n) we know that the beta function does determine the unmarked orbit spectrum, the following result shows that exceptional examples exist. Proposition 4.6. There exist non-equivalent f1 , f2 ∈ LC(2) with βf1 = βf2 , but Lf1 = Lf2 .  Proof. Let 2+ = ∞ n=0 {1,2}. Choose a < b such that exp[a] + exp[b] = 1 and define locally constant functions  a if (x0 , x1 ) = (1, 2) or (2, 1) f1 (x) = , b if (x0 , x1 ) = (1, 1) or (2, 2)  f2 (x) =

a if (x0 , x1 ) = (1, 1) or (2, 2) b if (x0 , x1 ) = (1, 2) or (2, 1)

.

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We can easily compute 

 exp t (Sn f1 (x)) = trace

σ n x=x

exp tb exp ta exp ta exp tb

n

= (exp ta + exp tb)n + (exp tb − exp ta)n  (exp ta + exp tb)n as n → ∞ and  σ n x=x



exp ta exp tb exp tSn f2 x)) = tr exp tb exp ta

n

= (exp ta + exp ta)n + (exp ta − exp tb)n  (exp ta + exp tb)n as n → ∞. It immediately follows that βf1 = βf2 . By considering periodic points of period 3, we see that 3β ∈ Lf1 and 3α ∈ / Lf1 , while 3α ∈ Lf2 and 3β ∈ / Lf2 . Thus f2 ∼ f1 ◦ τ , for any shift automorphism τ . 

Appendix I: Uncountably Many H¨older Continuous Functions Sharing the Same Unmarked Orbit Spectrum (and Free Energy) In contrast to the case of locally constant functions, the following result shows that for some H¨older continuous functions there are uncountably many mutually non-equivalent H¨older continuous functions with the same unmarked orbit spectrum. Proposition A.I.7. There exists an uncountable family of mutually inequivalent H¨older continuous functions which share the same unmarked orbit spectrum. Proof. Let 3+ denote the full one-sided shift on the three symbols 0, 1 and 2. To construct uncountably many functions with the same unmarked length spectrum we choose as our N index sequences i = (in )∞ older continuous functions n=1 ∈ {0, 2} , and define the H¨  fi (x) =

θn 0

if x ∈ [in in−1 . . . i1 1] otherwise,

where [in in−1 . . . i1 1] denotes the obvious cylinder set and 0 < θ < 1. It is easy to see that for a fixed i the function fi (y) = 0 for every periodic point y having period n that is not of the form y = yi = y0 . . . yn1 ik1 . . . i1 1ym2 . . . yn2 ik2 . . . i1 1 . . . ymr . . . ynr ikr . . . i1 1, where (i) ynj = ikj +1 , for j = 1, . . . , kr ; (ii) yl ∈ {0, 2} for mj ≤ l ≤ nj and j = 1, . . . , r (where m1 = 0); (iii) 0 ≤ k1 , . . . , kr and k1 + . . . + kr + r ≤ n.

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If one considers the Birkhoff sum over the periodic orbit for such a point y, an easy calculation shows that Sn fi (y) =

1 − θ kr 1 − θ k1 + ... + . 1−θ 1−θ

Let us observe that for the two periodic points (for different maps) yi = y0 . . . yn1 ik1 . . . i1 1ym2 . . . yn2 ik2 . . . i1 1 . . . ymr . . . ynr ikr . . . i1 1, yj = y0 . . . yn1 jk1 . . . j1 1ym2 . . . yn2 jk2 . . . j1 1 . . . ymr . . . ynr jkr . . . j1 1, the functions fi (yi ) = fj (yj ) and Sn fi (yi ) = Sn fj (yj ). This observation allows us to conclude that for each n ≥ 1 the sets {Sn fi (y) : σ n y = y} coincide for all i ∈ {0, 2}N . It immediately follows that for all i ∈ {0, 2}N the unmarked length spectra for fi coincide. Since different functions must necessarily have different marked length spectra, none of these functions differ by a coboundary. To see that there are uncountably many nonequivalent functions we need only recall that the space of automorphisms is countable. A simple cardinality argument completes the proof.  Corollary A.I.II. There exists an uncountable family of mutually inequivalent H¨older continuous functions which share the same beta function.

Appendix II: Locally Constant Functions with the Same Unmarked Orbit Spectrum (and Free Energy) We continue our study of the ambiguity in recovering functions from their unmarked orbit spectrum. Here we work by close analogy with a basic construction of isospectral non-isometric hyperbolic surfaces using a construction of Buser [Bus] involving cospectral graphs. Buser’s construction is really a reinterpretation of Sunada’s common covering surface construction [Sun]. The goal of this appendix is to prove the following result. Proposition A.II.1. There exist non-equivalent locally constant functions f1 and f2 on the one-sided full shift on six symbols with Lf1 = Lf2 . To make the proof of this result as self-contained as possible, we present some preparatory material. If we construct distinct cospectral graphs G1 and G2 , i.e., two directed graphs with the same number of closed loops (or cycles) of any given period, then it would immediately follow that the corresponding subshifts of finite type have the same number of periodic points of any given period. + + Consider the shift on n symbols σ : A → A , and let G be a finite group with a finite set of n generators G0 = {A1 , . . . , An }. We can associate to G0 ⊂ G a directed graph G = (V, E), called the Cayley graph [Bus, Bol] having the following properties: (a) There is a 1-1 correspondence between the vertices V of G and the elements of G; (b) The edge set E contains the edge from vertex g1 to vertex g2 provided there is a g ∈ G0 such that g1 = gg2 . There is a natural (right) action on the graph, by which g ∈ G carries a vertex g1 to the vertex g1 g, and the edge g1 to g2 is carried to an edge from g1 g to g2 g. Given a subgroup H ⊂ G we can also consider the quotient graph G/H . A trivial example is G0 = G/G,

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which consists of both a single vertex and a directed edge for each element (and inverse) in G0 . In particular, the corresponding subshift of finite type is the full shift. Given any directed graph G, let Nn (G) be the number of closed loops of length k (k ≥ 1). Two directed graphs G1 and G2 are called cospectral if Nk (G1 ) = Nk (G2 ), for all k ≥ 1. Two subgroups H1 , H2 ⊂ G are called almost conjugate if #{[g] ∩ H1 } = #{[g] ∩ H2 } for every conjugacy class [g], for every g ∈ G. Lemma A.II.1. The quotient graphs G/H1 and G/H2 of two almost conjugate subgroups are cospectral. Proof. The argument can easily be extracted from the proof of Sunada’s theorem [Bus, Sun]. Let Gk0 denotes words of length k in elements of G0 . Observe that Nk (Gi ) = #{(g, g0 ) ∈ G × Gk0 : g0 gHi = gHi }  = #{g ∈ G: g −1 g0 gHi = Hi } g0 ∈Gk0

=

1  #{g ∈ G: g −1 g0 g ∈ Hi }. #Hi k g0 ∈G0

For g, h ∈ G, the expression g −1 g0 g = h−1 g0 h holds if and only g0 = gh−1 g0 hg −1 = (gh−1 )g0 (gh−1 )−1 , which holds if and only if gh−1 ∈ Cg0 (or equivalently g ∈ Cg0 h), where Cg0 = {g ∈ G: gg0 g −1 = g0 } denotes the centralizer of g0 in G. Hence for fixed g1 = g −1 g0 g, one has that #{h ∈ G: h−1 g0 h = g1 } = #{gCg0 } = #Cg0 . Thus #{[g0 ] ∩ Hi } · #Cg0 = #{g ∈ G: g −1 g0 g ∈ Hi }, and we obtain Nk (Gi ) =

1  #{[g0 ] ∩ Hi } · #Cg0 . #Hi n g0 ∈G0

It immediately follows that Nk (G1 ) = Nk (G2 ) if and only if the subgroups H1 and H2 are almost conjugate.  Example ([Bus, Lub]). Consider the group G = SL(3, Z2 ) and G0 = {A, B} where 1 0 0 0 1 1 A = 0 1 0 and B = 0 0 1 . 100

011

There are two almost conjugate subgroups " ∗ ∗ ∗  # " #  ∗00 0 ∗ ∗ ∈ SL(2, Z2 ) and H2 = H1 = ∗ ∗ ∗ ∈ SL(2, Z2 ) , 0∗∗

∗∗∗

which are not conjugate. The quotient graphs G/H1 and G/H2 are non-isomorphic3 cospectral graphs each with seven vertices. See Fig. 1. 3 Two graphs A and B are isomorphic if there exists a one-to-one mapping φ from the vertex set of A onto the vertex set of B such that whenever vertices P and Q of A are connected by exactly k edges, then φ(P ) and φ(Q) are also connected by exactly k edges.

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Fig. 1. Cayley graphs for H1 and H2

Remark. Suppose G is a finite group that acts freely and isometrically on a compact Riemannian manifold M. Sunada showed that if H has two almost conjugate subgroups H1 and H2 , then the quotient manifolds M/H1 and M/H2 are isospectral (for both lengths of closed geodesics and eigenvalues of the Laplacian). We now prove Proposition A.II.1. Proof of Proposition A.II.1. We can colour the edges of the two non-isomorphic quotient Cayley graphs G/H1 and G/H2 in the above example according to the generator A (solid edges in Fig. 1) or B (dashed edges in Fig. 1) which corresponds to that edge. In Lemma A.II.1 we have seen that for each n, the number of closed loops of length n, coincide for these two graphs, and it is easily seen that the number of closed loops of length n which have the same colouring, also coincides for each n. We can extend each of these two graphs G/H1 by adding additional edges so that every vertex is connected to every other vertex. The corresponding subshift of finite type is then the full shift (on 6 symbols). For the larger graphs, we declare that the edges labeled by A have length α, the edges labeled by B have length β, and the additional edges have length 1, where 1, α and β are rationally independent. We can define f1 , f2 ∈ LC(2) by declaring that fi (x0 x1 ) equals the length of the edge connecting vertices x0 to x1 for the extension of G/Hi . Since the graphs G/H1 and G/H2 are non-isomorphic, the functions f1 , f2 are non-equivalent, however, by design, Lf1 = Lf2 . 

Appendix III: Uncountably Many H¨older Continuous Functions Sharing the Same Weak Orbit Spectrum but Having Different Unmarked Orbit Spectra In the following proposition we construct an uncountable family of pairwise inequivalent H¨older continuous functions with different unmarked orbit spectrum, but all sharing the same weak orbit spectrum. The construction is in the same spirit as the construction on Appendix I. Proposition A.III. There exists an uncountable family of pairwise inequivalent H¨older continuous functions with different unmarked orbit spectra, but all sharing the same weak orbit spectrum.

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Proof. Let 2+ denote a full shift on the two symbols 0 and 1. For any n ≥ 0 we denote [0n 1] := {x ∈ 2+ : xi = 0, 0 ≤ i ≤ n − 1 and xn = 1}. Let 0 < θ < 1 and then we can define a function  θ n if x ∈ [0n 1] f (x) = 0 if x = (0, 0, 0, . . . ). To write down the orbit spectrum it is convenient to relate this to the shift on infinitely many symbols [0n 1] (with allowed transitions [0n 1] → [0n−1 1] and [1] → [0n 1]). The representation of the locally constant f on this shift with infinitely many symbols is given by f ([0n 1]) = θ n . By a simple calculation we see that the length spectrum is the semi-group generated by the values 1 + θ + θ 2 + . . . + θ n = (1 − θ n+1 )/(1 − θ). As a first step consider for any k ≥ 1 the function defined by  if x ∈ [0n 1], n ∈ {k, k + 1, k + 2} θn     k k+1  if x ∈ [0k 1] θ + θ k+1 fk (x) = −θ if x ∈ [0k+1 1]    θ k+1 + θ k+2 if x ∈ [0k+2 1]    0 if x = (0, 0, 0, . . . ). It is easy to see that for k ≥ 2 the weak orbit spectra agree (i.e., Wfk = Wf1 ) although the unmarked orbit spectrums differ (i.e., Lfk = Lf1 ). It immediately follows from the latter observation that these functions are not mutually cohomologous. Moreover, it is a simple matter to modify this construction so that the function is either changed in a similar way, or left unchanged, for k = 3, 6, 9, . . . . In this way we can construct an uncountable family of functions having the same weak orbit spectrum as the original function. This is easily checked to be H¨older. First observe that none of these functions differ by a coboundary. This is easily seen by observing that different functions must necessarily have different marked length spectra. For example, the weighting for f and fk of the closed orbit of period k + 1 in the cylinder [0k 1] are 1 + θ + . . . + θ k + θ k and 1 + θ + . . . + θ k + θ k+1 , respectively. A similar observation applies in other cases. Secondly, observe that there are uncountably many non-equivalent functions; we need only recall that the space of automorphisms is countable. A simple cardinality argument completes the proof.  References [Abr] [BS]

Abraham, R.: Bumpy Metrics. Proceedings Sym. Pure Math. 14, 1–3 (1966) Bowen, R., Series, C.: Markov Maps Associated with Fuchsian Groups. Inst. Hautes Etudes Sci. Publ. Math. 50, 153–170 (1979) [Bol] Bollob´as, B.: Modern Graph Theory. Graduate Texts in Mathematics 184, Berlin-HeildelbergNew York: Springer Verlag, 1998 [Bow] Bowen, R.: One-dimensional hyperbolic sets for flows. J. Diff. Eq. 12, 173–179 (1972) [Bus] Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Basal-Boston: Birkh¨auser, 1992 [Fra] Fraleigh, J.: A First Course in Abstract Algebra. 8th ed, Reading, MA: Addison Wesley Longman, 1998 [Ful] Fulton, W.: Algebraic Curves. New York: Benjamin/Cummings, 1969 [GK] Guillemin, V., Kazhdan, D.: Some Inverse Spectral Results for Negatively Curved 2-Manifolds. Topology 19, 301–312 (1980)

482 [Hed] [HJ] [Liv] [Lub] [Man] [McK] [Mil] [Ota] [Par] [PP] [Ran] [Rue] [Sun] [Tun1] [Tun2] [Vig] [Wae] [Wol]

M. Pollicott, H. Weiss Hedlund, G.: Endomorphisms and Automorphisms of the Shift Dynamical System. Math. Syst. Th. 3, 320–375 (1969) Horn, R., Johnson, C.: Matrix Analysis. Cambridge: CUP, 1985 Livsic, A.: Cohomology Properties of Dynamical Systems. Math. USSR-Izv 6, 1278–1301 (1972) Lubotzky, A.: Discrete groups, Expanding Graphs and Invariant Measures. Progress in Mathematics 125, Basal-Boston: Birkhauser, 1994 Mandelbrojt, S.: Selecta. Paris: Gauthier-Villars, 1981 McKean, H.: Selberg’s Trace Formula as Applied to a Compact Riemann Surface. Comm. Pure Appl. Math. 25, 225–246 (1972) Milnor, J.: A note on curvature and fundamental group. J. Diff. Geom. 2, 1–7 (1968) Otal, J.P.: Le Spectre Marqu´e des Longueurs des Surfaces a` Courbure Negative. Ann. Math. 131, 151–162 (1990) Parry, W.: Private Communication Parry, W., Pollicott, M.: Zeta Functions and the Periodic Orbit Structures of Hyperbolic Dynamics. Ast´erisque 187–188 (1990) Randol, B.: The Length Spectrum of a Riemann Surface is Always of Unbounded Multiplicity. Proc. Am. Math. Soc. 78, 455–456 (1980) Ruelle, D.: Thermodynamic Formalism. Reading, MA: Addison-Wesley, 1978 Sunada, T.: Riemannian Coverings and Isospectral Manifolds. Ann. Math. 121, 69–186 (1985) Tuncel, S.: Conditional Pressure and Coding. Israel J. Math. 39, 101–112 (1981) Tuncel, S.: Coefficient Rings for Beta Function Classes of Markov Chains. Erg. Th. & Dyn. Sys. 20, 1477–1493 (2000) Vigneras, M.F.: Vari´et´es Riemanniennes Isospectrales et non Isometriques. Ann. Math. 112, 21–32 (1980) van der Waerden, B.L.: Algebra. vol 1, Ungar, 1970 Wolpert, S.: The Length Spectra as Moduli for Compact Riemann Surfaces. Ann. Math. 109, 323–351 (1979)

Communicated by J.L. Lebowitz

Commun. Math. Phys. 240, 483–500 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0909-2

Communications in

Mathematical Physics

Asymptotic Stability of the Stationary Solution to the Compressible Navier–Stokes Equations in the Half Space
Shuichi Kawashima1 , Shinya Nishibata2 , Peicheng Zhu1 1 2

Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan

Received: 23 January 2003 / Accepted: 7 April 2003 Published online: 13 August 2003 – © Springer-Verlag 2003

Abstract: We investigate the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for the compressible Navier–Stokes equation in a half space. The main concern is to analyze the phenomena that happens when the fluid blows out through the boundary. Thus, it is natural to consider the problem in the Eulerian coordinate. We have obtained the two results for this problem. The first result is concerning the existence of the stationary solution. We present the necessary and sufficient condition which ensures the existence of the stationary solution. Then it is shown that the stationary solution is time asymptotically stable if an initial perturbation is small in the suitable Sobolev space. The second result is proved by using an L2 -energy method with the aid of the Poincar´e type inequality. 1. Introduction The main purpose of the present paper is to investigate the phenomena which happens when the fluid blows out from the boundary. It is called the outflow problem in [9]. Especially, we are interested in the asymptotic stability of a stationary solution of the compressible Navier–Stokes equation with this boundary condition. The one-dimensional and isentropic motion of compressible viscous gas is formulated in the Eulerian coordinate as ρt + (ρu)x = 0,
 2 (ρu)t + ρu + p(ρ) = µuxx , x

(1.1a) (1.1b)

where the unknown functions ρ and u stand for the mass density and the velocity of gas, respectively. The positive constant µ is called the viscosity coefficient. The pressure p  The second author’s work was supported in part by Grant-in-Aid for Scientific Research (C)(2) 14540200 of the Ministry of Education, Culture, Sports, Science and Technology and the third author’s work was supported by JSPS postdoctoral fellowship under P99217.

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is assumed to be a function of the density ρ given by p = p(ρ) = Kρ γ ,

(1.2)

where the constants are supposed to satisfy K > 0 and γ ≥ 1. We study the initial boundary value problem of the system (1.1) in the first half space R+ := {x > 0}. Thus, we prescribe the initial and boundary conditions as (ρ, u)(0, x) = (ρ0 , u0 )(x), inf ρ0 (x) > 0,

(1.3)

ρ+ > 0,

x∈R+

u(t, 0) = ub ,

lim (ρ0 , u0 )(x) = (ρ+ , u+ ),

x→∞

ub < 0,

(1.4)

where u+ , ρ+ and ub are constants. In addition, we assume that the compatibility conditions on u(t, x) of order 0 and 1 hold at the origin (t, x) = (0, 0). The inequality in (1.4) implies that the fluid blows out through the boundary {x = 0}. Hence, this initial boundary value problem is called the outflow problem. It is worth noticing that only one boundary condition (1.4) on {x = 0} is necessary and sufficient for the wellposedness of this problem since the characteristic speed of the first hyperbolic equation (1.1a) is negative around the boundary {x = 0}, due to the inequality in (1.4). The opposite case ub > 0 needs two boundary conditions for the wellposedness, that is, ρ(t, 0) = ρb > 0 and u(t, 0) = ub > 0. This case is called the inflow problem and studied by Matsumura and Nishihara in [11]. They considered this problem in the Lagrangian coordinate. Let us notice that the method in the present paper is also applicable directly to this inflow problem formulated in the Eulerian coordinate and we can obtain a similar theorem as in [11]. At last, for the no flow case ub = 0, the stationary solution becomes the constant state (ρ+ , u+ ). (See (1.7) and (1.9).) The stationary solution (ρ, ˜ u)(x) ˜ is a solution to (1.1), independent of the time variable t. Thus, (ρ, ˜ u) ˜ should satisfy the equations (ρ˜ u) ˜ x = 0,  ρ˜ u˜ + p(ρ) ˜ = µu˜ xx .




(1.5a)

2

(1.5b)

x

In addition, (ρ, ˜ u) ˜ is supposed to satisfy the same conditions as (ρ, u), u(0) ˜ = ub ,

lim (ρ, ˜ u)(x) ˜ = (ρ+ , u+ ),

x→∞

inf ρ(x) ˜ > 0.

x∈R+

(1.6)

Integrate (1.5) over [x, ∞) for x > 0, respectively and use (1.6) to obtain the Rankine–Hugoniot condition ρ˜ u˜ = ρ+ u+ , 2

ρ˜ u˜ + p(ρ) ˜ − µu˜ x =

ρ+ u2+

+ p(ρ+ ).

(1.7a) (1.7b)

Here, we have also used the fact that u˜ x (ξ ) → 0 as ξ → ∞ due to (1.6). Owing to (1.4), we see from (1.7) that the condition, u+ < 0,

(1.8)

is necessary for the existence of a solution to (1.5) and (1.6). Letting x ↓ 0 in (1.7a) and using the boundary condition (1.6), we have that ρ(0)u ˜ b = ρ+ u+ .

(1.9)

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Due to (1.9), the magnitude of the stationary wave (ρ, ˜ u) ˜ is measured by δS := |ub − u+ |.

(1.10)

The property of the stationary solution is determined by the relation between a sound speed c+ and the Mach number M+ at spatial asymptotic states. They are defined by   |u+ | γ −1  c+ := p (ρ+ ) = γ Kρ+ , M+ := . (1.11) c+ The existence and the property of the stationary wave (ρ, ˜ u), ˜ satisfying (1.5) and (1.6), are summarized in the next lemma. Lemma 1.1. Let the condition (1.8) hold. Then the stationary problem (1.5) and (1.6) has a smooth solution (ρ, ˜ u), ˜ if and only if M+ ≥ 1 and wc u+ > ub , where wc is a positive constant defined in (2.4). The solution (ρ, ˜ u) ˜ is monotonic, that is, u˜ x ≶ 0 and ρ˜ x ≷ 0 if ub ≷ u+ . If M+ > 1, u(x) ˜ converges to u+ exponentially as x tends to infinity. Precisely, there exist constants σ > 0 and C > 0 such that ˜ − u+ )| ≤ CδS e−σ x for k = 0, 1, 2, · · · . |∂xk (u(x)

(1.12a)

If M+ = 1, u(x) ˜ is monotonically increasing and converges to u+ algebraically as x tends to infinity. Precisely, there exists a constant C > 0 such that |∂xk (u(x) ˜ − u+ )| ≤ C

δS k+1 (1 + δS x)k+1

Proof. This lemma is proved in Sect. 2.

for k = 0, 1, 2, · · · .

(1.12b)



The main purpose of this research is to show the time asymptotic stability of the stationary solution constructed in Lemma 1.1. This main result is stated in the next theorem. Theorem 1.2. Let σ ∈ (0, 1). Suppose that (1.8), M+ ≥ 1 and wc u+ > ub hold. In addition, let the initial data (ρ0 , u0 ) ∈ B 1+σ × B 2+σ and the boundary data ub satisfy (1.3), (1.4), and the compatibility conditions of order 0 and 1, as well as (ρ0 − ρ+ , u0 − u+ ) ∈ H 1 (R+ ). Then there exists a constant ε0 such that if δS + (ρ0 − ρ+ , u0 − u+ ) 1 ≤ ε0 , then the initial boundary value problem (1.1), (1.3) and (1.4) has a unique solution 1+σ/2,1+σ 1+σ/2,2+σ (ρ, u)(t, x) satisfying (ρ, u) ∈ BT × BT for an arbitrary T > 0 and 1 (ρ − ρ+ , u − u+ ) ∈ C([0, ∞); H (R+ )). Moreover, the solution (ρ, u)(t, x) converges to the stationary wave (ρ, ˜ u)(x) ˜ as time tends to infinity. Precisely, it holds that lim

sup |(ρ, u)(t, x) − (ρ, ˜ u)(x)| ˜ = 0.

t→+∞ x∈R

(1.13)

+

Proof. This theorem is proved at the end of Sect. 4.



In order to take the boundary effect into account for studying the compressible Navier– Stokes equation, it is natural and necessary to formulate the problem in the Eulerian coordinate, since the point of view moves subject to the fluid flow in the Lagrangian mass coordinate. In the latter coordinate, the boundary also moves except for the special case, zero flow on the boundary, i.e. ub = 0. Since our main concern is the out flow

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problem, ub < 0, we must formulate the problem in the Eulerian coordinate. This formulation although gives some difficulties to the study of the time asymptotic analysis. One of them is easily seen from the fact that the equation in the Eulerian coordinate is more complicated than that in the Lagrangian coordinate. To avoid this difficulty, the preceding research in [11] transformed the half space inflow problem in the Eulerian to that in the Lagrangian since it considered the case that the boundary {x = 0} is mapped to the half line, which is determined a priori by the boundary data ρb and ub > 0. However, for the outflow problem ub < 0, this transformation results in a free boundary problem, whose analysis is more complicated. Hence, we apply the energy method directly to the equations in the Eulerian coordinate in order to obtain the a priori estimate in both the H¨older space and the H 1 -Sobolev space. This computation is established as the following procedure. First we derive the local existence of the solution in H¨older space. Then we obtain the H 1 -a priori estimate for the solution with H¨older regularity by using the energy method with the aid of the difference quotient. Then it is shown that the a priori estimate in the H¨older space follows from the H 1 -estimate. Related results. After the pioneering work in 1960 by Il’in and Oleinik in [2], the study of the stability of nonlinear waves to the Cauchy problems of scalar conservation laws has a long history of many mathematicians’ researches. The study of the half space problem to the conservation laws started in the 1990s in [7, 8]. These results should be generalized to a more physically meaningful system of equations such as the compressible Navier–Stokes and the Boltzmann equations. The research in [9] formulated the classification of the possible asymptotic states for the compressible Navier–Stokes equation. According to this classification, they constitute twenty three cases subject to the relationship between the boundary condition and the spatial asymptotic condition. Some cases are solved in the preceding research [10, 11]. This kind of half space problem is also studied for the discrete Boltzmann equation in [3, 12, 13]. The research in [3, 13] studied the stability of the stationary solution to the general system of the discrete Boltzmann equation by the energy method with the aid of the Poincar´e type inequality. The present paper is in debt for some ideas to these papers. The research in [12] studied the stability of the traveling wave moving away from the boundary for the Broadwell model. Recent results [5, 6] have shown the stability of the rarefaction wave, and the superposition of the stationary and the rarefaction waves to the compressible Navier–Stokes equation under the boundary conditions other than the present paper. Outline of the paper. The remaining part of the present paper is organized as follows. In Sect. 2, we start detailed analysis with the proof of the existence of the stationary solution. The proof is based on an elementary observation with the phase plane analysis. Sections 3 and 4 are devoted to proving the stability theorem of the stationary wave. In Sect. 3, we first derive the system of equations for the perturbation from the stationary wave. Then applying the energy method on this system, we obtain the a priori estimate in the H 1 -Sobolev space. Here, to derive the estimate for the first order derivatives, we need to use the difference quotient since the existence of higher derivatives in the classical sense is not assumed. In addition, we need the a priori assumption that the solution is contained in the suitable H¨older space. This H¨older regularity is ensured in Sect. 4, where we derive the a priori estimate in the H¨older norm. We show that once the H 1 -a priori estimate is obtained, it gives the

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487

desired H¨older estimate by applying the Schauder theory for parabolic equations. The combination of the a priori estimate in the H¨older space and the local existence theorem completes the proof of the existence of the time global solution. Finally, we show the uniform estimate in the H 1 Sobolev space is sufficient for proving that the solution converges to the stationary wave in supremum norm as time tends to infinity. Notation. For a nonnegative integer l ≥ 0, H l (R+ ) denotes the l th order Sobolev space in the L2 sense, equipped with the norm · l . We note H 0 = L2 and · := · 0 . We denote by C k (I ; H l (R+ )) the space of k-times continuously differentiable functions on the interval I with values in H l (R+ ). L2 (I ; H l (R)) is the space of L2 -functions on I with values in H l (R). For 0 < σ < 1, B l+σ denotes the H¨older space of continuous functions which have the l th order derivatives of H¨older continuity with exponent σ . | · |l+σ is its norm. For a domain QT ⊆ [0, T ] × R+ , B α, β (QT ) denotes the H¨older space of continuous functions with the H¨older exponents α and β with respect to t and x, respectively. For integers k and l, B k+α, l+β (QT ) denotes the space of the functions j satisfying ∂ti u, ∂x u ∈ B α, β (QT ) for integers 0 ≤ i ≤ k, 0 ≤ j ≤ l. | · |k+α, l+β, QT is its norm. If QT = [0, T ] × R+ , we abbreviate B k+α, l+β (QT ) and | · |k+α, l+β, QT by k+α, l+β and | · |Tk+α, l+β , respectively. BT 2. Existence of the Stationary Solution This section is devoted to proving the existence of the stationary solution to the problem (1.5) and (1.6), which is stated in Lemma 1.1. Substituting u˜ = (ρ+ u+ )/ρ, ˜ which follows from (1.7a), in (1.7b) yields that

F (w) ˜ :=

γ  Kρ+ w˜ −γ

˜ µu+ w˜ x = F (w),  u(x) ˜ ρ+ − 1 + ρ+ u2+ (w˜ − 1), w(x) ˜ := = > 0, u+ ρ(x) ˜

(2.1a) (2.1b)

where we have also used (1.2). The boundary condition for w˜ is derived from (1.6), w(0) ˜ =

ub , u+

lim w(x) ˜ = 1.

x→∞

(2.2)

It is apparent that the existence of the solution to (1.5) and (1.6) follows from that to (2.1) and (2.2). In order to show the latter, we study the property of the function F . Differentiating (2.1b) in w, ˜ we have 2 2 F  (w) ˜ = ρ+ c+ (M+ − w˜ −(γ +1) ),

2 −(γ +2) F  (w) ˜ = (γ + 1)ρ+ c+ w˜ > 0,

(2.3)

where c+ and M+ are the sound speed and the Mach number, which are defined in (1.11). Owing to the convexity of F with F (1) = 0 and limw˜ →0+ F (w) ˜ = limw˜ →+∞ F (w) ˜ = +∞, an equation F (w) ˜ = 0 has another solution, which is denoted by wc . Namely,
 γ −γ (2.4) F (wc ) = K ρ˜ + wc − 1 + ρ+ u2+ (wc − 1) = 0. Then it holds that M+  1 if and only if wc  w∗  1, where w∗ is a unique zero point −2/(γ +1)

of F  , i.e., w∗ := M+

.

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First, we consider the case M+ > 1, i.e., wc < w∗ < 1. Let w(x) ˜ ≤ wc . Then F (w(x)) ˜ ≥ 0, i.e., w˜ x (x) ≤ 0 due to (2.1a). Thus, w(x) ˜ is decreasing. So, if w(0) ˜ ≤ wc , w(x) ˜ can not approach 1 as x → ∞. Consequently, there does not exist a solution to (2.1) and (2.2). Let wc < w(x) ˜ < 1, then F (w(x)) ˜ < 0, i.e., w˜ x > 0. Thus, it is easy to see that if wc < w(0) ˜ < 1, (2.1) and (2.2) has a solution w, ˜ which is monotonically increasing. At last, if 1 < w(x), ˜ then F (w(x)) ˜ > 0, i.e. w˜ x (x) < 0. We see that there exists a monotonically decreasing solution w˜ to (2.1) and (2.2). Consequently, we see that the boundary value problem (2.1) and (2.2) has a solution w(x) ˜ provided that 1 < w(0). ˜ Thus, we have proved that there exists a solution if and only if wc < w(0) ˜ for the case M+ > 1. Since F (w) ˜ is strictly convex, we obtain the following estimate from (2.1): |∂xk (w(x) ˜ − 1)| ≤ C|w(0) ˜ − 1|e−σ x

for k = 1, 2, · · · ,

σ := −

F  (1) > 0, µu+ (2.5)

where C is a positive constant. The case M+ = 1, i.e., wc = w∗ = 1, is handled by a similar observation as above. Namely, there exists a solution w, ˜ if and only if wc = 1 < w(0). ˜ The solution w˜ is monotonically decreasing. Moreover, although F  (1) = 0, the right inequality in (2.3) and F  < 0 gives the estimate |∂xk (w(x) ˜ − 1)| ≤ C

˜ − 1)k+1 (w(0) ˜ − 1)x)k+1 (1 + σ1 (w(0)

for

k = 1, 2, · · · ,

˜ F  (w(0)) σ1 := − > 0, 2µu+

(2.6)

where C is a positive constant depending on k. At last, we study the case M+ < 1, i.e., wc > w∗ > 1. From the convexity of F (w), ˜ F  (w) ˜ < 0 holds for w˜ in a suitable neighborhood of 1. There it holds that w˜ x ≶ 0 if and only if w˜ ≶ 1, owing to (2.1a). Thus, w(x) ˜ can not approach 1 as x → ∞. Consequently, there does not exist a solution satisfying (2.2). Summarizing the above observation, we have the next lemma. Lemma 2.1. The boundary value problem (2.1) and (2.2) has a unique smooth solution w˜ if and only if M+ ≥ 1 and wc < w(0). ˜ Moreover, if w(0) ˜ ≶ 1, then w˜ x ≷ 0. If M+ > 1, then the solution w˜ satisfies the estimate (2.5). If M+ = 1, then w˜ is monotonically decreasing and satisfies (2.6). The above lemma immediately gives the proof of Lemma 1.1.

3. The Energy Estimates In order to prove the stability result in Theorem 1.2, it is convenient to regard the solution (ρ, u) as a perturbation from the stationary solution (ρ, ˜ u). ˜ Thus, we introduce new unknown functions as φ(t, x) := ρ(t, x) − ρ(x), ˜

ψ(t, x) := u(t, x) − u(x). ˜

(3.1)

Asymptotic Stability of the Stationary Solution to NS Equations

489

Subtracting (1.5) from (1.1) yields that φt + uφx + ρψx = f,

(3.2a)



ρ(ψt + uψx ) + p (ρ)φx − µψxx = g,

(3.2b)

where f := −(ψ ρ˜x + φ u˜ x ),

g := −(φψ + φ u˜ + ψ ρ) ˜ u˜ x − (p  (ρ) − p  (ρ)) ˜ ρ˜x , |f |, |g| ≤ C|(φ, ψ)||u˜ x |.

(3.3a) (3.3b)

In deriving (3.3b), we have used the last equality in (2.1b). The initial and boundary conditions to the system (3.2) are derived from (1.3) and (1.4), φ(x, 0) = φ0 (x) := ρ0 (x) − ρ(x), ˜

ψ(x, 0) = ψ0 (x) := u0 (x) − u(x), ˜

ψ(t, 0) = 0.

(3.4) (3.5)

The local existence of the solution (φ, ψ) to the initial boundary value problem (3.2), (3.4) and (3.5) is proved by the standard method, using an iteration. Lemma 3.1. Assume that the same conditions in Theorem 1.2 hold. Then there exists a positive constant T0 , depending only on |φ0 |1+σ and |ψ0 |2+σ , such that the initial boundary value problem (3.2), (3.4) and (3.5) has a unique solution (φ, ψ) in the space: 1+σ/2, 1+σ

φ ∈ BT0

1+σ/2, 2+σ

ψ ∈ BT0

,

(φ, ψ) ∈ C([0, T0 ]; H 1 (R+ )),

,

inf

(t,x)∈[0,T0 ]×R+

ρ(t, x) > 0,

φx ∈ L2 (0, T0 ; L2 (R+ )),

ψx ∈ L2 (0, T0 ; H 1 (R+ )).

(3.6a) (3.6b)

Next, we prove the a priori estimate in the Sobolev space, which is stated in Proposition 3.2. To show this estimate, it is convenient to use notations  N(t) := sup (φ, ψ)(τ ) 1 ,

t

M(t)2 :=

0≤τ ≤t

0

φx (τ ) 2 + ψx (τ ) 21 + |φ(τ, 0)|2 dτ.

Proposition 3.2. Let (φ, ψ) be a solution to (3.2), (3.4) and (3.5) in a time interval [0, T ], which has the same regularity as in Lemma 3.1. Then there exist positive constants ε0 and C, such that if N (T ) + δS ≤ ε0 , then the following estimate holds for an arbitrary t ∈ [0, T ]: 

(φ, ψ)(t) 21 +

t 0

≤ C (φ0 , ψ0 ) 21 .

φx (τ ) 2 + ψx (τ ) 21 + |(φ, φx )(τ, 0)|2 dτ (3.7)

Proof. The proof is divided into three steps, which are stated in Lemmas 3.4, 3.5 and 3.6, respectively. Then combining the uniform estimates proved in these three lemmas gives the desired estimate (3.7). 

490

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The smallness assumption on N (T ) in Proposition 3.2 ensures that if ε0 is sufficiently small, then there exist certain positive constants cρ and Cρ such that 0 < cρ ≤ ρ(t, x) ≤ Cρ

for t ∈ [0, T ],

(3.8)

since ρ˜ ≥ c > 0 for a certain constant c. First, we establish the basic energy estimate. To this end, we define an energy ρE as   ρ p(η) 1 2 u + (ρ) , (ρ) := dη. (3.9) ρE := ρ 2 η2 Here, (ρ) is a strictly convex function with respect to the specific volume v := 1/ρ, since v (ρ) = −p(ρ),

vv (ρ) = ρ 2 p  (ρ) > 0.

(3.10)

The energy ρE satisfies the equation (ρE)t + (ρuE + pu)x = (µuux )x − µu2x owing to (1.1) and (3.10). Next, we introduce an energy form ρE, which is defined by  1 2 ˜ , (u − u) ˜ + (ρ, ρ) ρE := ρ 2   ρ

1 p(η) − p(ρ) ˜ 1 (ρ, ρ) ˜ := (ρ) − (ρ) ˜ − v (ρ) − = ˜ dη. 2 ρ ρ˜ η ρ˜

(3.11)

(3.12a) (3.12b)

In deriving the right equality in (3.12b), we have used (3.10). Since p  (ρ) > 0, we see from (3.12b) that  > 0. It is easy to see from (3.12b) that

 p(ρ) − p(ρ) ˜ 1 1  ˜ = ,  (ρ, ρ) ˜ = p ( ρ) ˜ − . (3.13) ρ (ρ, ρ) ρ˜ ρ2 ρ ρ˜ ˜ ≤ c. Hence, the energy form Notice that ρ(ρ, ρ) ˜ is equivalent to |ρ − ρ| ˜ 2 for |ρ − ρ| 2 ρE is equivalent to |(ρ − ρ, ˜ u − u)| ˜ . Namely, there exist positive constants c1 and C1 such that ˜ u − u)| ˜ 2 ≤ ρE ≤ C1 |(ρ − ρ, ˜ u − u)| ˜ 2 c1 |(ρ − ρ,

(3.14)

for |(ρ − ρ, ˜ u − u)| ˜ ≤ c. By the straightforward computation with (3.11), we see that ρE satisfies the equation ˜ − u)} ˜ x (ρE)t + {ρuE + (p(ρ) − p(ρ))(u = {µ(u − u)(u ˜ − u) ˜ x }x − µ{(u − u) ˜ x }2 + R, (3.15a)

R := − (ρu − ρ˜ u)(u ˜ − u) ˜ + (p(ρ) − p(ρ) ˜ − p  (ρ)(ρ ˜ − ρ)) ˜ u˜ x 1 − (ρ − ρ)(u (3.15b) ˜ − u)p( ˜ ρ) ˜ x ρ˜   ˜ − p  (ρ)(ρ ˜ − ρ)) ˜ u˜ x = − ρ(u − u) ˜ 2 + (p(ρ) − p(ρ) µ ˜ − u) ˜ u˜ xx . − (ρ − ρ)(u (3.15c) ρ˜

Asymptotic Stability of the Stationary Solution to NS Equations

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Because of (1.7a), it holds from (3.15b) that |R| ≤ C|(ρ − ρ, ˜ u − u)| ˜ 2 |u˜ x | ≤ C|(φ, ψ)|2 |u˜ x |

(3.16)

for |(φ, ψ)| ≤ c. Before proving the uniform estimate (3.7), we show the Poincar´e type inequality, which is proved by a similar method as [13] and [3]. Lemma 3.3. Assume the same conditions as Proposition 3.2 hold. (i) Let M+ > 1. There is a positive constant ε0 such that if N (T ) + δS ≤ ε0 , then the following estimates hold:  t ∞  t j |∂xk u| ˜ j |φ|2 dx dτ ≤ CδS |φ(τ, 0)|2 + φx (τ ) 2 dτ, (3.17a) 0 0 0  t ∞  t j |∂xk u| ˜ j |ψ|2 dx dτ ≤ CδS

ψx (τ ) 2 dτ (3.17b) 0

0

0

for t ∈ [0, T ] and k, j = 1, 2, . . .. (ii) Let M+ = 1. There is a positive constant ε0 such that if N (T ) + δS ≤ ε0 , then the following estimates hold:  t ∞  t (k+1)j −2 k j 2 |∂x u| ˜ |φ| dx dτ ≤ CδS |φ(τ, 0)|2 + φx (τ ) 2 dτ, (3.18a) 0 0 0  t ∞  t (k+1)j −2 |∂xk u| ˜ j |ψ|2 dx dτ ≤ CδS

ψx (τ ) 2 dτ (3.18b) 0

0

0

for t ∈ [0, T ] and k, j = 1, 2, . . . except k = j = 1. Proof. We prove (3.17a). Notice the identity



φ(τ, x) = φ(τ, 0) +

x

φx (τ, y) dy.

(3.19)

0

  x √ Take the absolute value of (3.19), use the inequality  0 φx (τ, y)dy  ≤ x φx (τ ) , ˜ j , and successively integrate the resultant inequality in x ∈ R+ . Thus, multiply by |∂xk u| we have  ∞  ∞ |∂xk u| ˜ j |φ|2 dx ≤ C |∂xk u| ˜ j (|φ(τ, 0)|2 + x φx (τ ) 2 ) dx. (3.20) 0

0

Then substitute the estimate (1.12) in (3.20) and compute the integration in x ∈ R+ . Consequently, we arrive at the desired estimate (3.17a). The other assertions in this lemma are proved similarly as above, by applying (1.12) and (3.5).  We are in a position to prove the basic energy estimate. Lemma 3.4. There exist positive constants ε1 and C such that if N (T ) + δS ≤ ε1 , then it holds that  t

(φ, ψ)(t) 2 +

ψx (τ ) 2 + |φ(τ, 0)|2 dτ 0

≤ C{ (φ0 , ψ0 ) 2 + (N (t) + δS )M(t)2 } for an arbitrary t ∈ [0, T ].

(3.21)

492

S. Kawashima, S. Nishibata, P. Zhu

Proof. Integrate Eq. (3.15a) over the region [0, t] × R+ and use the boundary condition (3.5) to obtain that  t  t  ∞ (ρuE)(τ, 0) dτ (ρE)(t, x) dx + µ

ψx (τ ) 2 dτ − 0 0 0  ∞  t ∞ = R(τ, x) dx dτ. (3.22) (ρE)(0, x) dx + 0

0

0

Due to (3.12a), the first terms on the left and the right-hand sides of (3.22) are equivalent to (φ, ψ)(t) 2 and (φ0 , ψ0 ) 2 , respectively. The last term on the left-hand side of (3.22) is handled by using (1.4) as ˜ ≥ c|φ(τ, 0)|2 , −(ρuE)(τ, 0) = ρ(τ, 0)|ub |(ρ(τ, 0), ρ(0))

(3.23)

where c is a certain positive constant. Hence, it suffices to estimate the last term on the right-hand side of (3.22) for completion of the proof of Lemma 3.4. This part is divided into two cases, M+ > 1 and M+ = 1. First, let M+ > 1. Then it is estimated by using Lemma 3.3 and (3.16) as   t ∞  t    ≤ CδS  R(τ, x) dx dτ

ψx (τ ) 2 + φx (τ ) 2 + |φ(τ, 0)|2 dτ. (3.24)   0

0

0

Next we treat the case M+ = 1. The first term on the right-hand side of (3.15c) is negative since p > 0 and u˜ x > 0, due to Lemma 1.1. Thus, we can neglect this term. Taking the absolute value of the second term in (3.15c), we see that it is bounded by C|(φ, ψ)|2 |u˜ xx |. So, integration of this term is estimated by using Lemma 3.3 with k = 2 and j = 1. Hence, we see that the estimate (3.24) also holds for this case. At last, substitution of the inequalities (3.23) and (3.24) in (3.22) completes the proof.  Next, we derive L2 -estimates for the first derivatives, φx and ψx . To this end, we define a difference quotient of a function ϕ(x) with respect to x as ϕh = ϕh (x) :=

ϕ(x + h) − ϕ(x) h

for

h > 0.

By straightforward computation, it holds that   ∞ 1 α+h ϕh (x) dx = − ϕ(x) dx for α ≥ 0, ϕ ∈ L1 (R+ ). h α α

(3.25)

(3.26)

In addition, the following identity holds (ϕω)h = ϕh ωh + ϕωh ,

(3.27)

where we have abbreviated ϕ h := ϕ(x + h) and ϕ := ϕ(x). Lemma 3.5. There exist positive constants ε2 (≤ ε1 ) and C such that if N (T )+δS ≤ ε2 , then the following estimate holds for any t ∈ [0, T ]:  t

φx (τ ) 2 + |φx (τ, 0)|2 dτ

φx (t) 2 + 0

≤ C{ (φ0 , ψ0 ) 21 + (N (t) + δS )M(t)2 }.

(3.28)

Asymptotic Stability of the Stationary Solution to NS Equations

493

Proof. We divide the proof of this lemma into two steps. 1st step. Take the difference quotient of (3.2a) and then apply (3.27), to obtain that φht + uφhx + ρψhx = fh − uh φxh − ρh ψxh . Dividing (3.29) by ρ, we have



 φh φh +u + ψhx = H, ρ t ρ x H :=

fh − uh φxh − ρh ψxh + ux φh , ρ

(3.29)

(3.30a) (3.30b)

where we have used (1.1a). Multiplying (3.30a) by φh /ρ, integrating resultant equality over [0, t] × R+ and applying integration by parts, we obtain that

    t ∞   1 ∞ φh 2 ux φh 2 |ub | t φh 2 (t, x) dx − dx dτ + (τ, 0) dτ 2 ρ 2 0 ρ ρ 2 0 0 0  t ∞    t ∞ H φh 1 ∞ φh 2 ψhx φh dx dτ. (3.31) (0, x) dx + dx dτ = + ρ ρ ρ 2 0 0 0 0 0 Make h ↓ 0 in (3.31), with (3.6) taken into account, to obtain that  2  t ∞

    1 ux φx 2 |ub | t φx 2  φx (t) − dx dτ + (τ, 0) dτ  2ρ 2 ρ 2 0 ρ 0 0  2  t ∞  t ∞ 1 φ0 x  ψxx φx H 0 φx   + dx dτ =  dx dτ, (3.32) +  ρ 2 ρ ρ 0

0

0

0

0

where (u˜ x φ + ρ˜ x ψ)x + ρx ψx , h↓0 ρ |H0 | ≤ C (|(φ, ψ)||u˜ xx | + |(φx , ψx )||u˜ x | + |φx ||ψx |) .

H0 := lim H = −

(3.33a) (3.33b)

In deriving (3.33b), we have used (2.1b). 2nd step. Next, we eliminate the last term on the left-hand side of (3.32) by using (3.2b). Multiply (3.2b) by φh /ρ to get that (ψφh )t − (ψφt )h + ψh φth + uψx φh +

gφh p  (ρ) ψxx φh φx φh − µ = , ρ ρ ρ

(3.34)

where we have used the identity ψt φh = (ψφh )t − ψφht = (ψφh )t − (ψφt )h + ψh φth . Integrate (3.34) over [0, t] × R+ , apply (3.26) and successively make h ↓ 0 to obtain that  ∞  t ∞  t ∞  p (ρ) 2 φx dx dτ ψφx dx + (φt + uφx )ψx dx dτ + ρ 0 0 0 0 0  t ∞  ∞  t ∞ ψxx φx gφx −µ dx dτ = dx dτ. (3.35) ψ0 φ0 x dx + ρ ρ 0 0 0 0 0

494

S. Kawashima, S. Nishibata, P. Zhu

Let us note that to obtain (3.35), we have used the fact that  t  h  t 1 ψφt dx dτ → (ψφt )(τ, 0)dτ = 0 0 h 0 0

as

h ↓ 0,

where the equality holds owing to (3.5). Substitute φt + uφx = f − ρψx , which follows from (3.2a), in (3.35). Multiply (3.32) by µ. Then, sum up the resultant two equations to obtain that 2  ∞ 

2    µ φx µ t ∞  φx (t) + ψφx (t, x) dx − ux dx dτ  2 ρ 2ρ 0 0 0  t ∞  t ∞    p (ρ) 2 µ|ub | t φx 2 (τ, 0) dτ − ρψx2 dx dτ + φx dx dτ + 2 ρ ρ 0 0 0 0 0 2  ∞   t ∞ φ0 x  (µH0 + g)φx µ   + ψ0 φ0 x dx + − f ψx dx dτ. (3.36) =  2 ρ0  ρ 0 0 0 We estimate each term in (3.36). The second term on the left-hand side is estimated by the Schwarz inequality as   ∞    ψφx dx  ≤ Cε ψ 2 + ε φx 2 , (3.37)  0

where ε is an arbitrary positive constant and Cε is a constant depending only on ε. Note that the first term on the right-hand side of (3.37) has been estimated in (3.21). The third term on left-hand side of (3.36) is handled as follows. Substitute ux = ψx + u˜ x and then use Lemma 3.3 and the Sobolev inequality, |ψx (τ )| ≤ C ψx (τ ) 1 , as well as the Schwarz inequality. The result is      t ∞ φ 2 x   ux dx dτ     0 0 ρ  t  t ≤ CδS

φx (τ ) 2 dτ + C

ψx (τ ) 1 φx (τ ) 2 dτ 0 0  t  t ≤ CδS

φx (τ ) 2 dτ + CN (t)

ψx (τ ) 21 + φx (τ ) 2 dτ. (3.38) 0

0

Owing to (3.8), the fifth term is estimated as   t ∞  t   2 ≤C  ρψ dx dτ

ψx (τ ) 2 dτ. x   0

0

(3.39)

0

Due to (1.2) and (3.8), it holds  t  t ∞  p (ρ) 2 φx dx dτ ≥ c

φx (τ ) 2 dτ. ρ 0 0 0

(3.40)

Finally, the last term in (3.36) is handled by using (3.3b), (3.33b), Lemma 3.3 and the second inequality in (3.38) as   t ∞   (µH0 + g)φx  − f ψx dx dτ   ρ 0 0  t ≤ C(δS + N (t))

ψx (τ ) 21 + φx (τ ) 2 + |φ(τ, 0)|2 dτ. (3.41) 0

Asymptotic Stability of the Stationary Solution to NS Equations

495

Therefore, substituting the estimates (3.37)–(3.41) in (3.36) and taking ε suitably small, we arrive at the desired estimate (3.28).  To complete the proof of Proposition 3.2, it suffices to derive the estimate for ψxx , which is stated in Lemma 3.6. In order to prove this lemma, we utilize the backward difference quotient, that is, ϕ−h = ϕ−h (x) :=

ϕ(x) − ϕ(x − h) , h

for 0 < h < x.

(3.42)

If ϕxx (x) exists, it holds that ϕxx (x) = lim(ϕ−h )h . h↓0

(3.43)

Lemma 3.6. There exist positive constants ε3 (≤ ε2 ) and C such that if N (T ) + δS ≤ ε3 , then it holds that for t ∈ [0, T ], 

t

ψx (t) 2 + 0

ψxx (τ ) 2 dτ ≤ C{ (φ0 , ψ0 ) 21 + (δS + N (t))M(t)2 }. (3.44)

Proof. Multiply Eq. (3.2b) by −(ψ−h )h /ρ and then use the identity 1 h = −(ψt ψ−h )h + (ψh2 )t , −ψt (ψ−h )h = −(ψt ψ−h )h + ψth ψ−h 2

(3.45)

h = ψ . Consequently we have which follows from ψ−h h

p  (ρ) µψxx 1 2 (ψh )t − (ψt ψ−h )h − uψx (ψ−h )h − φx (ψ−h )h + (ψ−h )h 2 ρ ρ g(ψ−h )h . (3.46) =− ρ Integrating (3.46) over [0, t] × (h, ∞) yields that 1 2

 t  2h  t ∞ 1 µψxx (ψ−h )h dx dτ ψh2 (t, x) dx + ψt ψ−h dx dτ + ρ h 0 h h 0 h    t ∞

1 ∞ 2 p  (ρ) g = uψx + φx − (ψ−h )h dx dτ. (3.47) ψh (0, x) dx + 2 h ρ ρ 0 h



∞

Notice that the second term converges to zero as h ↓ 0 by virtue of (3.5). Using this fact, we make h ↓ 0 in (3.47) to obtain that  t ∞ 2 µψxx 1

ψx (t) 2 + dx dτ 2 ρ 0 0   t ∞

1 p  (ρ) gψxx = ψ0 x 2 + φx ψxx − dx dτ. uψx + 2 ρ ρ 0 0

(3.48)

496

S. Kawashima, S. Nishibata, P. Zhu

The second term on the left-hand side of (3.48) is strictly positive owing to (3.8). On the other hand, the first integration on the right-hand side is estimated by applying the Schwarz inequality as   t ∞

   p  (ρ)   uψ ψ + dx dτ φ x x xx   ρ 0 0  t  t ≤ε

ψxx (τ ) 2 dτ + Cε

ψx (τ ) 2 + φx (τ ) 2 dτ, (3.49) 0

0

where ε is an arbitrary positive constant and Cε is a constant depending on ε. Finally, the last integral in (3.48) is estimated similarly as the previous lemma, using (3.3b) as   t ∞  t   gψxx   dx dτ  ≤ CδS

ψx (τ ) 21 + φx (τ ) 2 + |φ(τ, 0)|2 dτ. (3.50)  ρ 0 0 0 At last, substitute the estimates (3.49) and (3.50) in (3.48). Then make ε suitably small. Consequently, we arrive at the desired estimate (3.44).  Combination of Lemmas 3.4, 3.5 and 3.6 with N (t)+δS ≤ ε0 := ε3 = min{ε1 , ε2 , ε3 } yields the desired estimate (3.7) in Proposition 3.2. In these proofs, we have used the H¨older regularity (3.6a). The existence time of the solution (φ, ψ) depends on the H¨older norm of the initial data (φ0 , ψ0 ). Thus, to complete the proof of the global existence in Theorem 1.2, it needs to show the a priori estimate in the H¨older space. This part is established in the next section. 4. The H¨older Estimates Once the Sobolev estimates are obtained in the previous section, we can proceed to derive the H¨older estimates. For this purpose, it is easier to consider the problem in the Lagrangian coordinate than in the Eulerian coordinate. Thus, we rewrite the system (1.1) in the half space into the system in the Lagrangian coordinate with a free boundary. The Eulerian coordinate (x, t) and the Lagrangian (ξ, t) are related by the equation x = X(t, ξ ).

(4.1)

Here X(t, ξ ) is defined by an ordinary differential equation ∂t X(t, ξ ) = u(t, X(t, ξ )),

X(0, ξ ) = X0 (ξ ),

where X0 (ξ ) is the inverse of a monotone function X0 → ξ defined by  X0 ξ= ρ0 (y) dy.

(4.2)

(4.3)

0

Due to the boundary condition (1.4), the total amount of the fluid blown out through the boundary {x = 0} in the time interval [0, t] is given by  t ρ(s, 0) ds. B(t) := −ub 0

Since this amount is not determined a priori, the out flow problem in the Euler coordinate becomes a free boundary problem in the Lagrangian coordinate. By the coordinate

Asymptotic Stability of the Stationary Solution to NS Equations

497

change (t, x) → (t, ξ ), the domain [0, T ] × R+ is mapped to T = {(t, ξ ) : 0 ≤ t ≤ T , B(t) < ξ }. In addition,  := {(t, ξ ) : ξ = B(t)} is a free boundary in the Lagrangian coordinate. By the standard argument with (1.1a), (1.4), (4.2) and (4.3), we have  ξ 1 X(t, ξ ) = dη for ξ > B(t). (4.4) B(t) ρ(t, X(t, η)) Substituting ρ(t, ˆ ξ ) := ρ(t, X(t, ξ )),

u(t, ˆ ξ ) := u(t, X(t, ξ )),

(4.5)

in (1.1), we have the compressible Navier–Stokes equation in the Lagrangian coordinate ρˆt + ρˆ 2 uˆ ξ = 0, uˆ t + p(ρ) ˆ ξ − µ(ρˆ uˆ ξ )ξ = 0.

(4.6a) (4.6b)

The initial and boundary data for (ρ, ˆ u) ˆ are derived from (1.3) and (1.4) (ρ, ˆ u)(0, ˆ ξ ) = (ρˆ 0 , uˆ 0 )(ξ ) := (ρ0 , u0 ) (X0 (ξ )) for u(t, ˆ ξ ) = ub

ξ > 0,

for (t, ξ ) ∈ .

(4.8)

Let ξ = (t, x) be the inverse of x = X(t, ξ ). Then it is given by  t  x ρ(t, y) dy − ub ρ(s, 0) ds, (t, x) = 0

(4.7)

(4.9)

0

where we have used (4.4), (4.6a) and (4.8). The solution (ρ, u) in the Eulerian coordinate is expressed, with (4.9), by ρ(t, x) = ρ(t, ˆ (t, x)),

u(t, x) = u(t, ˆ (t, x)).

(4.10)

Lemma 4.1. Let σ ∈ (0, 1). Under the same assumptions as Proposition 3.2, it holds |ρ|T1+σ/2, 1+σ ,

|u|T1+σ/2, 2+σ ≤ C(T ),

(4.11)

where C(T ) is a constant depending only on T , |ρ0 |1+σ , |u0 |2+σ and E0 := (φ, ψ) 1 . Proof. The proof is divided into four steps. 1st step. We note that (3.7) implies |(ρ, u)|T1/4, 1/2 ≤ C(E0 ).

(4.12)

In fact the estimate for the density ρ is derived from (3.7) by a similar method as in [4] by applying the Sobolev inequality and the Fubini theorem. Namely, it holds that for x, x  > 0 and t, t  ∈ [0, T ], there exists a constant C(E0 ) such that 1

1

|ρ(t, x) − ρ(t  , x  )| ≤ C(E0 )(|t − t  | 4 + |x − x  | 2 ).

(4.13)

This immediately means the estimates for ρ in (4.12). The velocity u is handled in the same way.

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2nd step. Then we show that (4.12) means |(ρ, ˆ u)| ˆ 1/4, 1/2, T ≤ C(E0 , T ).

(4.14)

Using (4.13) we have |ρ(t, ˆ ξ  ) − ρ(t, ˆ ξ )| = |ρ(t, X(t, ξ  )) − ρ(t, X(t, ξ ))| 1

≤ |ρ|T1/4, 1/2 |X(t, ξ  ) − X(t, ξ )| 2 .

(4.15)

Since Xξ = ρ −1 due to (4.4), |X(t, ξ  ) − X(t, ξ )| ≤ C(E0 )|ξ − ξ  | holds. Substituting this inequality in (4.15), we have |ρ(t, ˆ ξ  ) − ρ(t, ˆ ξ )| ≤ C(E0 )|ξ  − ξ |1/2 .

(4.16)

From (4.13), it holds |ρ(t ˆ  , ξ ) − ρ(t, ˆ ξ )| ≤ |ρ|T1/4, 1/2 (|t  − t|1/4 + |X(t  , ξ ) − X(t, ξ )|1/2 ).

(4.17)

By using (4.2) and (4.12), we have |X(t  , ξ ) − X(t, ξ )| ≤ C(E0 )|t  − t|. Thus, (4.17) gives 1

|ρ(t ˆ  , ξ ) − ρ(t, ˆ ξ )| ≤ C(E0 , T )|t  − t| 4 .

(4.18)

Combining the estimates (4.16) and (4.18), we have the the estimates for ρˆ in (4.14). Since uˆ is handled similarly, (4.14) has been proved. 3rd step. Next, we derive the H¨older estimates for the derivatives of the quantities in the Lagrangian coordinate. Namely, we show that for 0 < σ < 1, |ρ| ˆ 1+σ/2, 1+σ, T , |u| ˆ 1+σ/2, 2+σ, T ≤ C(T ),

(4.19)

where C(T ) is a constant depending only on T , |ρˆ0 |1+σ , |uˆ 0 |2+σ and E0 . Suppose that 0 < σ ≤ 1/2 for the moment. Solve (4.6a) with respect to ρˆ t and substitute it in (4.6b). The result is



 ρˆξ γ ρˆξ µ + γ K ρˆ (4.20) + uˆ t = 0. ρˆ t ρˆ Solve (4.20) for (ρˆ ξ /ρ)(t, ˆ ξ ), multiply the result by ρˆ and then apply (4.14) to obtain that for 0 < σ ≤ 1/2, |ρˆ ξ |σ/2, σ, T ≤ C(E0 , T )(|ρˆ 0 ξ |σ + 1) ≤ C(T ).

(4.21)

Equation (4.6b) is rewritten as uˆ t = (µρ) ˆ uˆ ξ ξ + (µρˆξ )uˆ ξ − γ K ρˆ γ −1 ρˆξ .

(4.22)

We may regard (4.22) as a linear parabolic equation for uˆ whose coefficients and nonhomogeneous term are in B σ/2, σ (T ) owing to (4.14) and (4.21). Hence, we can apply the Schauder theory for parabolic equations (e.g., see [1]) to get that for 0 < σ ≤ 1/2, |u| ˆ 1+σ/2, 2+σ, T ≤ C(T ).

(4.23)

Asymptotic Stability of the Stationary Solution to NS Equations

499

The H¨older estimates for ρˆt are derived from (4.6a), (4.14) and (4.23). Thus, we have proved that for 0 < σ ≤ 1/2, |ρ| ˆ 1+σ/2, 1+σ, T ≤ C(T ).

(4.24)

This proves (4.19) for 0 < σ ≤ 1/2. For 1/2 < σ < 1, repeat the above argument in this step by using (4.19) with σ = 1/2 in place of (4.14). This observation concludes that (4.19) is also valid for 1/2 < σ < 1 and thus for 0 < σ < 1. 4th step. We translate the H¨older regularities in the Lagrangian coordinate (t, ξ ), which is obtained in the 3rd step, into those in the Eulerian coordinate (t, x). From (4.9) and (4.10), we have ρx (t, x) = ρˆξ (t, (t, x))x (t, x) = ρˆξ (t, (t, x))ρ(t, x), ρt (t, x) = ρˆξ (t, (t, x))t (t, x) + ρˆt (t, (t, x)) = ρˆξ (t, (t, x))(ρu)(t, x) + ρˆt (t, (t, x)). Estimating the above equalities with (4.12) and (4.19), we see that |ρt |Tσ/2, σ , |ρx |Tσ/2, σ ≤ C(T ). Similarly, we can prove that |ut |Tσ/2, σ , |ux |Tσ/2, σ ≤ C(T ). Finally, |uxx |Tσ/2, σ ≤ C(T ) is shown by estimating the equality uxx (t, x) = uˆ ξ ξ (t, (t, x))ρ 2 (t, x) + uˆ ξ (t, (t, x))ρx (t, x) utilizing (4.19). Therefore, the proof of Lemma 4.1 is completed.



B∞ ,

the desired H¨older estimate of the perturbaSince the stationary wave (ρ, ˜ u) ˜ is tion (φ, ψ) follows from (4.11). As the existence time of the local solution constructed in Lemma 3.1 depends on the H¨older norm, the global existence of the solution (φ, ψ) follows from the combination of Lemma 3.1 and Lemma 4.1 by the standard continuation argument. Hence, in order to complete the proof of Theorem 1.2, it remains to show the convergence (1.13). It is done by applying the uniform estimate in Proposition 3.2. Proof of Theorem 1.2. We prove (1.13). Since (φ, ψ)(t) is bounded due to (3.7), it suffices to show that

(φx , ψx )(t) → 0

as

t →∞

(4.25)

by virtue of the Sobolev inequality. Let P (t) := (φx /ρ)2 (t) . We see from (3.32) that the function P (t) is differentiable at almost every t ∈ (0, ∞). Thus, it holds that for almost every t ∈ (0, ∞),

2  ∞ 2 φx φx d P (t) = ux dx + ub (t, 0) dt ρ ρ 0  ∞  ∞ ψxx φx H0 φ x dx + 2 dx. (4.26) −2 ρ ρ 0 0 Take the absolute value of (4.26) and then integrate over (0, ∞) estimating the right∞ d hand side by (3.7), to obtain that 0 | ds P (t)|dt ≤ C. Combining this estimate with ∞ 0 |P (t)|dt ≤ C, which follows from (3.7), we conclude that P (t) converges to zero as t tends to infinity. This means φx (t) converges to zero as t tends to infinity because of (3.8). The convergence of ψx (t) in (4.25) is proved similarly as above by using (3.48). 

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References 1. Friedman, A.: Partial Differential Equations of Parabolic Type. Englewood Cliffs, NJ: Prentice Hall, Inc., 1964 2. Il’in, A., Oleinik, O.: Asymptotic behavior of the solutions of Cauchy problems for certain quasilinear equations for large time (Russian). Mat. Sb. 51, 191–216 (1960) 3. Kawashima, S., Nishibata, S.: Stationary waves for the discrete Boltzmann equation in the half space with the reflective boundaries. Commun. Math. Phys. 211, 183–206 (2000) 4. Kawashima, S., Nishida, T.: Global solutions to the initial value problem for the equations of onedimensional motion of viscous polytropic gases. J. Math. Kyoto Univ. 21(4), 825–837 (1981) 5. Kawashima, S., Zhu, P.: Asymptotic stability of the rarefaction waves of compressible Navier–Stokes equations in the half space. To appear 6. Kawashima, S., Zhu, P.: Asymptotic stability of the superposition of nonlinear waves of compressible Navier–Stokes equations in the half space. To appear 7. Liu, T., Matsumura, A., Nishihara, K.: Behaviors of solutions for the Burgers equations with boundary corresponding to rarefaction waves. SIAM J. Math. Anal. 29, 293–308 (1998) 8. Liu, T., Nishihara, K.: Asymptotic behavior for scalar viscous conservation laws with boundary effect. J. Diff. Eq. 133, 296–320 (1997) 9. Matsumura, A.: Inflow and outflow problems in the half space for a one-dimensional isentropic model system for compressible viscous gas. In: Proceedings of IMS Conference on Differential Equations from Mechanics. Hong Kong, 1999 10. Matsumura, A., Mei, M.: Asymptotics toward viscous shock profile for solution of the viscous p–System with boundary effect. Arch. Rat. Mech. Anal. 146, 1–22 (1999) 11. Matsumura, A., Nishihara, K.: Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional isentropic model system for compressible viscous gas. Commun. Math. Phys. 222, 449–474 (2001) 12. Nishibata, S.: Asymptotic stability of traveling waves to a certain discrete velocity model of the Boltzmann equation in the half space. SIAM J. Math. Anal. 34, 555–572 (2002) 13. Nikkuni,Y., Kawashima, S.: Stability of stationary solutions to the half-space problem for the discrete Boltzmann equation with multiple collisions. Kyushu J. Math. 54, 233–255 (2000) Communicated by P. Constantin

Commun. Math. Phys. 240, 501–507 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0910-9

Communications in

Mathematical Physics

Remarks on the Helicity of the 3-D Incompressible Euler Equations Dongho Chae Department of Mathematics, Seoul National University, Seoul 151-742, Korea. E-mail: [email protected] Received: 24 January 2003 / Accepted: 9 April 2003 Published online: 1 August 2003 – © Springer-Verlag 2003

Abstract: In this note we obtain a sufficient condition on the regularity of the weak solutions to guarantee the conservation of helicity for the 3-D incompressible Euler equations. As a corollary we obtain a lower bound of the vorticities for a weak solution of the Euler equations. 1. Introduction The Euler equations for the homogeneous incompressible fluid flows in D ⊂ R3 are   ∂v (x, t) ∈ D × (0, T ) ∂t + (v · ∇)v = −∇p, (E) div v = 0,  v(x, 0) = v0 (x), where v = (v1 , v2 , v3 ), vj = vj (x, t), j = 1, 2, 3 is the velocity of the fluid flows, p = p(x, t) is the scalar pressure, and v0 is the given initial velocity satisfying div v0 = 0. Our domain D is the whole of R3 , or T3 , which is [0, 1]3 with the periodic boundary condition, and T > 0 is given. The (total) helicity of the inviscid incompressible fluid flow in D is defined by  v(x, t) · ω(x, t)dx, H (t) = D

where v is the vorticity. As well as the more familiar kinetic energy, E(t) =  ω = curl 1 2 dx, the helicity is also an important conserved quantity in the study of |v(x, t)| 2 D the incompressible Euler equations. (See e.g. [1, 4, 5], and references therein.) In particular, the helicity is closely related to the topological invariants, e.g. the knottedness of vortex lines or vortex tubes (See [1] for the detailed discussion of these aspects). Thus, helicity conservation for the classical solutions of (E) implies that the topological configurations of the vortex lines/tubes for smooth inviscid incompressible fluid flows do

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not change in time. For the weak solutions (representing singular flows), however, that might not hold true. Heuristically, non-conservation of helicity for singular fluid flows could happen by the change of knottedness of the vortex lines/tubes, which should be preceded by crossings of them. To the author’s knowledge we do not yet have any rigorous example of such singular flows showing non-conservation of the helicity. We remark that, on the contrary, there is a sophisticated example of energy non-conservation for the weak solutions of the 3-D Euler system, discovered by Shnirelman [6]. Before this work Constantin, E and Titi obtained a sufficient condition on the regularity of weak solutions of (E) guaranteeing the conservation energy [2] (see [3] also for its refinements). In this paper, we are concerned with the conservation of helicity for the weak solutions of the 3-D Euler system. Following closely the arguments due to Constantin-E-Titi [2], we obtain a sufficient condition on the regularity of weak solutions of (E) guaranteeing the conservation helicity. By a weak solution of (E) in D × (0, T ) we mean a vector field v = (v1 , v2 , v3 ) ∈ C([0, T ); L2loc (D)) with its vorticity ω = curl v (in the sense of distribution) belonging 3

to L1 ((0, T ); L 2 (D)), satisfying the integral identity: 



 ∂φ(x, t) − v(x, t) · v(x, 0) · φ(x, 0)dx dxdt − ∂t D D 0  T − v(x, t) ⊗ v(x, t) : ∇φ(x, t)dxdt T

0

 − 

T

0



0

T



D

div φ(x, t)p(x, t)dxdt = 0,

(1.1)

D

v(x, t) · ∇ψ(x, t)dxdt = 0

(1.2)

D

for every vector test function φ = (φ1 , φ2 , φ3 ) ∈ C0∞ (D × [0, T )), and for every scalar test function ψ ∈ C0∞ (D × [0, T )). Here we used the notation (u ⊗ v)ij = ui vj ,  and A : B = 3i,j =1 Aij Bij for 3 × 3 matrices A and B. In the above definition we 3

imposed further regularity than the standard one, namely ω(·, t) ∈ L 2 (D) for almost every t ∈ [0, T ] in order to define the helicity for the weak solution. The following is our main theorem. Theorem 1.1. Let α >

1 3

be given. Suppose v is a weak solution of the 3-D Euler equa3

tions with curl v ∈ C([0, T ]; L 2 (D)) ∩ L3 (0, T ; B α,∞ (D)), where the derivatives are 9 5

in the sense of distribution. Then, the helicity is preserved in time, namely   v(x, t) · ω(x, t)dx = v(x, 0) · ω(x, 0)dx D

(1.3)

D

for all t ∈ [0, T ). Remark. We note that our sufficient condition of regularity of the weak solution is stronger than those obtained by Constantin-E-Titi [2]. Thus, there is a possibility that for some weak solutions of (E) the energy is preserved, but the helicity is not. It would be very interesting to find any weak solution of (E), regularity of which should be lower than described in Theorem 1.1, which shows non-conservation of the helicity.

Helicity of 3-D Incompressible Euler Equations

503

As an application of the above theorem we have the following estimate from below of the vorticity by a constant depending on the initial data for the weak solutions of the 3-D Euler equations. Corollary 1.1. Let α >

1 3

be given. Suppose v is a weak solution of the 3-D Euler 3

equations with curl v ∈ C([0, T ]; L 2 (D)) ∩ L3 (0, T ; B α,∞ (D)). Then, we have the 9 5

following estimate: ω(·, t)2 3

L 2 (D)

≥ CH0 ,

∀t ∈ [0, T ),

(1.4)

 where H0 = D v(x, 0) · ω(x, 0)dx is the initial helicity of the fluid flows, and C is an absolute constant. 2. Proof of the Main Results Proof of Theorem 1.1. We follow closely [2]. Let ϕ(x) ∈ C0∞ (R3 ) be the standard mollifier with ϕ ≥ 0, supp ϕ ⊂ {x ∈ R3 | |x| ≤ 1}. Let ϕ ε (x) = ε13 ϕ( xε ). Given f ∈ L1loc (R3 ), we denote by f ε (x) = (f ∗ ϕ ε )(x). Let us also denote δy v(x) = v(x − y) − v(x). We have the following identity: (u ⊗ v)ε = uε ⊗ v ε + rε (u, v) − (u − uε ) ⊗ (v − v ε ), where we set

 rε (u, v) =

D

∀u, v ∈ L2loc (R3 ), (2.1)

  ϕ ε (y) δy u(x) ⊗ δy v(x) dy.

We recall the following useful inequalities for the Besov spaces. ∇uε Lp ≤ Cεα−1 uBpα,∞ ,

(2.2)

u(· + y) − u(·)Lp ≤ C|y|α uBpα,∞ ,

(2.3)

u − uε Lp ≤ Cε α uBpα,∞ .

(2.4)

Suppose v(x, t) is a weak solution of (E), and ω = curl v. Let ξ(t) ∈ C0∞ ([0, T )). Given y ∈ D, choosing the test function φ(x, t) = ξ(t)(ϕ ε (x −y), ϕ ε (x −y), ϕ ε (x −y)) and ψ(x, t) = ξ(t)ϕ ε (x − y) in (1.1)–(1.2), we obtain div v ε = 0 and ∂v ε + div (v ⊗ v)ε = −∇p ε . ∂t

(2.5)

Also, choosing the text function, φ(x, t) = ξ(t)curlx (ϕ ε (x − y), ϕ ε (x − y), ϕ ε (x − y)) in (1.1), we obtain by integration by parts ∂ωε + div (v ⊗ ω)ε − div (ω ⊗ v)ε = 0. ∂t

(2.6)

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D. Chae

We compute d dt





∂v ε · ωε dx + ∂t



∂ωε dx ∂t D D   =− div (v ⊗ v)ε · ωε dx − v ε · div (v ⊗ ω)ε dx D D  + v ε · div (ω ⊗ v)ε dx

v ε · ωε dx = D

vε ·

D

= I + I I + I I I.

(2.7)

From (2.5) and (2.6), integrating by parts, and using the formula (2.1), we derive  I = =



(v ⊗ v)ε : ∇ωε dx D



v ⊗ v : ∇ω dx + rε (v, v) : ∇ωε dx D  − (v − v ε ) ⊗ (v − v ε ) : ∇ωε dx, ε

ε

ε

D

(2.8)

D

 II = =



(v ⊗ ω)ε : ∇v ε dx D



v ⊗ ω : ∇v dx + rε (v, ω) : ∇v ε dx D  − (v − v ε ) ⊗ (ω − ωε ) : ∇v ε dx, ε

ε

ε

D

(2.9)

D

 III = −

(ω ⊗ v)ε : ∇v ε dx





D

=−

D

ωε ⊗ v ε : ∇v ε dx − D

rε (ω, v) : ∇v ε dx

(ω − ω ) ⊗ (v − v ) : ∇v ε dx

+

ε

ε

(2.10)

D

respectively. Since div v ε = 0, we have by integration by part,  v ε ⊗ v ε : ∇ωε dx = D

3  i,j =1 D

=−

viε vjε

3 

i,j =1 D

viε

∂ωjε ∂xi ∂vjε ∂xi

dx  ωjε dx

=−

v ε ⊗ ωε : ∇v ε dx. D

(2.11)

Helicity of 3-D Incompressible Euler Equations

505

Also, using the fact div ωε = 0, we have by integration by parts 3 

 ω ⊗ v : ∇v dx = ε

ε

ε

D

i,j =1 D

=−

ωiε vjε

3 

i,j =1 D

ωiε

∂vjε ∂xi ∂vjε ∂xi

dx  vjε dx = −

ωε ⊗ v ε : ∇v ε dx = 0. D

(2.12) In view of (2.11)–(2.12), we find that the sum of the first terms of I, I I and I I I cancels out, and after rearrangement of the remaining terms we obtain,  I + II + III =



rε (v, v) : ∇ω dx + rε (v, ω) : ∇v ε dx D   − rε (ω, v) : ∇v ε dx − (v − v ε ) ⊗ (v − v ε ) : ∇ωε dx D D − (v − v ε ) ⊗ (ω − ωε ) : ∇v ε dx D + (ω − ωε ) ⊗ (v − v ε ) : ∇v ε dx ε

D

D

= {1} + {2} + {3} + {4} + {5} + {6}.

(2.13)

We estimate (2.13) term by term beginning with {1}:

 





ε ε

|{1}| =

ϕ (y)(v(x − y) − v(x)) ⊗ (v(x − y) − v(x))dy : ∇ω (x)dx

D D   ε 2 ε ≤ ϕ (y) |v(x − y) − v(x)| |∇ω (x)|dx dy D D ≤ ∇ωε  9 ϕ ε (y)v(· + y) − v(·)2 9 dy L5 D L2  ≤ C∇ωε  9 ∇v(· + y) − ∇v(·)2 9 ϕ ε (y)dy L5 D L5  ≤ Cεα−1 ωB α,∞ ω(· + y) − ω(·)2 9 ϕ ε (y)dy 9 L5 D 5  ≤ Cεα−1 ω3B α,∞ |y|2α ϕ ε (y)dy ≤ Cε 3α−1 ω3B α,∞ , (2.14) 9 5

9 5

D

9

9

where we used Sobolev’s lemma W 1, 5 (D) → L 2 (D) for D ⊂ R3 in the fourth line, and the Calderon-Zygmund inequality, ∇v 9 ≤ Cω 9 in the fifth line respectively. L5

L5

506

D. Chae

We estimate {2} as follows:

 





|{2}| =

ϕ ε (y)(v(x − y) − v(x)) · (ω(x − y) − ω(x))dy · ∇v ε (x)dx

D D   ε ≤ |v(x − y) − v(x)||ω(x − y) − ω(x)||∇v ε (x)|dx dy ϕ (y) D  D ε ≤ ∇v  9 ϕ ε (y)v(· + y) − v(·) 9 ω(· + y) − ω(·) 9 dy L2 D L2 L5  ≤ Cωε  9 ∇v(· + y) − ∇v(·) 9 ω(· + y) − ω(·) 9 ϕ ε (y)dy L2 D L5 L5  ≤ C∇ωε  9 ω(· + y) − ω(·)2 9 ϕ ε (y)dy L5 D L5  ≤ Cεα−1 ω3B α,∞ |y|2α ϕ ε (y)dy ≤ Cε 3α−1 ω3B α,∞ . (2.15) 9 5

9 5

D

The estimate of {3} is similar to that of {2}, and we have |{3}| ≤ Cε3α−1 ω3B α,∞ .

(2.16)

9 5

We estimate {4} as follows:





ε ε ε

|{4}| = [(v − v ) ⊗ (v − v )] : ∇ω dx

D ≤ |v − v ε |2 |∇ωε |dx ≤ v − v ε 2 9 ∇ωε  L2 α−1

D

≤ C∇v − ∇v ε 2 9 ∇ωε  L5

≤ Cε

3α−1

9

L5

≤ Cε

9

L5

ω − ωε 2 9 ωB α,∞ 9 5

L5

ω3B α,∞ . 9

(2.17)

5

Next, we estimate {5}:





|{5}| =

[(v − v ε ) ⊗ (ω − ωε )] : ∇v ε dx

D ≤ |v − v ε ||ω − ωε ||∇v ε |dx ≤ v − v ε  D

≤ C∇v − ∇v ε 

9

L5 ε 2

≤ Cε α−1 ω − ω 

ω − ωε  9

L5

9

L5

ωε 

ωB α,∞ ≤ Cε 9 5

9

L2 3α−1

9

L2

ω − ωε 

9

L5

∇v ε 

≤ Cω − ωε 2 9 ∇ωε  L5

ω3B α,∞ .

9

L2

9

L5

(2.18)

9 5

The estimate of {6} is similar to that of {5}, and we have |{6}| ≤ Cε3α−1 ω3B α,∞ . 9 5

(2.19)

Helicity of 3-D Incompressible Euler Equations

507

Combining (2.7)–(2.19), and integrating (2.7) over [0, T ], we have

 



ε ε

v ε (x, t) · ωε (x, t)dx − v (x, 0) · ω (x, 0)dx



D

D



T

≤ Cε3α−1 0

as ε → 0 for α > 13 .

ω(·, t)3B α,∞ dt → 0 9 5



Proof of Corollary 1.1. We estimate the helicity,  v(x, t) · ω(x, t)dx ≤ v(·, t)L3 ω(·, t) D

≤ C∇v(·, t)

3

L2

ω(·, t)

3

L2

3

L2

≤ Cω(·, t)2 3 ,

(2.20)

L2

where we used the Sobolev inequality and the Calderon-Zygmund inequality. Combining (2.20) with Theorem 1.1, we obtain the conclusion of Corollary 1.1.  Acknowledgements. This research is supported partially by the grant no.2000-2-10200-002-5 from the basic research program of the KOSEF.

References 1. Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Berlin-Heidelberg-New York: Springer-Verlag Inc., 1998 2. Constantin, W.E., Titi, E.: Onsager’s Conjecture on the Energy Conservation for Solutions of Euler’s Equations. Commun. Math. Phys. 165(1), 207–209 (1994) 3. Duchon, J., Robert, R.: Inertial Energy Dissipation for Weak Solutions of Incompressible Euler and Navier-Stokes Equations. Nonlinearity 13, 249–255 (2000) 4. Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge: Cambridge Univ. Press, 2002 5. Moffat, H.K., Tsinober, A.: Helicity in Laminar and Turbulent Flow. Ann. Rev. Fluid Mech. 24, 281–312 (1992) 6. Shnirelman, A.: Weak Solutions with Decreasing Energy of Incompressible Euler Equations. Commun. Math. Phys. 210, 541–603 (2000) Communicated by P. Constantin

Commun. Math. Phys. 240, 509–529 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0902-9

Communications in

Mathematical Physics

Strange Attractors in Periodically-Kicked Limit Cycles and Hopf Bifurcations Qiudong Wang1,
, Lai-Sang Young2,
1 2

Dept. of Math., University of Arizona, Tucson, AZ 85721, USA. E-mail: [email protected] Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012, USA. E-mail: [email protected]

Received: 26 November 2002 / Accepted: 10 April 2003 Published online: 19 August 2003 – © Springer-Verlag 2003

Abstract: We prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with SRB measures when certain simple dynamical systems are kicked at periodic time intervals. The settings considered include limit cycles and stationary points undergoing Hopf bifurcations. This paper is about a mechanism for producing chaos. The scheme consists of periodic kicks interspersed with long periods of relaxation. We apply it to some very tame dynamical settings, such as limit cycles and stable equilibria undergoing Hopf bifurcations, and prove the appearance of chaotic behavior under reasonable conditions. The results in this paper, beginning with the statements in Sect. 1, are rigorous. The rest of this introduction is devoted to a nontechnical discussion of some of the ideas and issues surrounding this work. Main results. In Theorem 1, we prove that when suitably kicked, all limit cycles can be turned into strange attractors with strong stochastic properties. Theorems 2 and 3 have to do with Hopf bifurcations. In the absence of forcing, the picture for a supercritical Hopf bifurcation is classical and well known: a stable fixed point loses its stability when a pair of complex conjugate eigenvalues crosses the imaginary axis, resulting in the appearance of a limit cycle which increases in diameter as it moves away from the newly unstable fixed point. Subjecting this system to periodic kicks, we prove that if there is a sufficiently strong “twist” at the fixed point and the forcing is of a suitable type, then in lieu of the limit cycle, a strange attractor sometimes emerges from the bifurcation. The appearance of horseshoes is also proved in both situations. The results above are related as follows. For arbitrary limit cycles, even though the geometric principles are clear, it is difficult to formulate a quantitative statement


This research is partially supported by a grant from the NSF.

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without detailed knowledge of the local geometry. In Hopf bifurcations, this information is contained in the first three derivatives at the bifurcating point. It can be condensed, in fact, into a single number called the “twist factor”. Horseshoes and strange attractors. We clarify what is meant by “horseshoes” and “strange attractors”. “Horseshoes” as introduced by Smale [S] are invariant Cantor sets on which the map is hyperbolic and has positive topological entropy. Since they are not attracting, these sets represent, from the observational point of view, transient chaos. The term “strange attractor” is used in this paper to summarize a number of precisely defined dynamical properties that together imply sustained, observable chaos. These properties include, for example, positive Lyapunov exponents starting from almost all initial conditions in the basin, statistical coherence in the sense of orbits in large sets organizing themselves according to certain special invariant measures called SRB measures, and strong stochastic properties such as exponential correlation decay for sequences of observations of the type ϕ, ϕ ◦ F, · · · , ϕ ◦ F n , · · · . (See Sect. 1.1 for more detail.) Horseshoes are created by “stretch-and-fold” type actions. They are robust; once they develop, they persist. The creation of strange attractors with SRB measures requires a balance that is considerably more subtle. For the logistic family, it has been proved that two kinds of maps with opposite behaviors – those with invariant densities and those with periodic sinks – partition parameter space in a complicated way; see e.g. [L]. There is evidence of a similar picture for invertible maps in 2D; see e.g. [N, WY2]. Though yet unsubstantiated, current thinking is that outside of the Axiom A category, SRB measures in general is a “positive probability phenomenon” rather than one that occurs on open sets of parameter space, even when the map has stretch-and-fold geometry. Our results in Theorems 1–3 reinforce this emerging picture: Our a priori conditions for the existence of strange attractors are more stringent than those for horseshoes. When these conditions are met, we prove the presence of horseshoes for open sets of parameters, and strange attractors for maps corresponding to parameters in a positive measure set. A mechanism for the production of chaos. Our scheme relies on the natural forces of shear to exaggerate the irregularities brought on (deliberately) by the kick. We explain – on an intuitive level – how this works in Hopf bifurcations in 2D. √ For argument’s sake, let 0 be the fixed point and  = {r = µ} be the emerging limit cycle, µ being the bifurcation parameter. Suppose we give the system a kick in the radial direction, distorting the shape of  as shown in Fig. 1a. Generically, the unforced system has a “twist”, meaning points at different distances from 0 rotate at different speeds. During the relaxation period, some points on this distorted circle rotate ahead of others. A stretch-and-fold action results if the twist is sufficiently strong; see Fig. 1b. Supporting analysis. We focus on strange attractors as the corresponding proofs for horseshoes are quite simple. Our analysis is based on [WY1] and [WY3], which together contain one of the very few rigorous theories of strange attractors besides Axiom A theory. We discuss briefly below the approach in these two papers and how to apply the results. [WY1] and [WY3] are about maps with attracting sets on which there is strong dissipation and (in most places) a single direction of instability. Two-parameter families {Ta,b } are considered. Roughly speaking, a is a parameter that allows us to effect changes

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(a)

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(b) Fig. 1a, b. A Hopf attractor

along the unstable direction and b is the inverse of “dissipation”. The idea is to try to pass to the singular limit at b = 0 to obtain a one-parameter family of 1D maps. Now 1-dimensional objects are considerably simpler than n-dimensional objects, and the theory of 1D maps is fairly well developed. We proved in [WY1] and [WY3] that if the singular limit above makes sense, and if the resulting family of 1D maps has certain good properties, then some of these properties can be passed back to b > 0, and they in turn allow us to prove the desired results on strange attractors for a positive measure set of a. Parts of the analysis in [WY1] and [WY3] have their origins in [BC], which contains the pioneering work on H´enon attractors. Some of the ideas in [BC] in turn come from the theory of 1D maps; see [J] in particular. We emphasize that in contrast to earlier results, the theory in [WY1] and [WY3] is “generic”, in the sense that the conditions under which it holds pertain only to certain general characteristics of the maps and not to specific formulas or contexts. To prove Theorems 1–3, we will show that the present setting fits into the framework of [WY1] and [WY3], and then leverage the results in these two papers. Relation to previous works. An application of [BC] to show the presence of strange attractors (in a weaker sense than explained above) in 2D homoclinic bifurcations is given in [MV]. The first evidence of chaotic behavior in periodically forced systems goes back to the work of Cartwright and Littlewood on the van der Pol oscillator [CL]. Levinson [Ln] made considerable progress on a simplified (linearized) model of this system, for which thirty years later Levi [Li] proved the existence of horseshoes. Numerical work by Zaslavsky [Z] suggested the presence of strange attractors for an even simpler second order equation. Not knowing about [Z], we studied again the same equation recently and proved (rigorously) the existence of horseshoes and strange attractors [WY2]. To our knowledge, the forcing of limit cycles had not been studied previously in the generality of Theorem 1 of this paper. Also, to our knowledge, chaotic behavior in connection with Hopf bifurcations had not been observed or predicted prior to this work. We mention also a connection of a different kind. Not all of our perturbations are small; large kicks are sometimes needed to “break” limit cycles. But under suitable circumstances, such as when the twist factor in a Hopf bifurcation is large, we prove that

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strange attractors may result from mild disturbances applied at infrequent time intervals. In this regard our results are in the direction of Ruelle and Takens [RT]; see also [NRT]. Our mechanism, which relies on shear, is different and perhaps more natural. 1. Precise Statements of Results 1.1. Horseshoes and strange attractors. In this subsection we isolate and define precisely a number of dynamical properties commonly associated with chaos. These are the properties which will appear in our results stated in Sects. 1.2–1.4. The following is a slight generalization of Smale’s horseshoe as introduced in [S]: (i) Let f : M → M be a C 1 embedding of a Riemannian manifold M into itself with a compact invariant set . We say f | is uniformly hyperbolic if there is a continuous splitting of the tangent bundle over  into DT -invariant subbundles E u ⊕ E s such that Df |E u is expanding and Df |E s is contracting (i.e. ∃C > 0 and λ > 1 s.t. for all v ∈ E u , Df n (v) ≥ Cλn v ∀n ≥ 0 etc.). (ii) Let n = {1, 2, · · · , n}Z , and let σ : n → n be the shift operator. Then (n , σ ) is called the full shift on n symbols. An embedding f : M → M is said to have a horseshoe if (H) for some N, n ∈ Z+ , f N has a uniformly hyperbolic invariant set  ⊂ M such that f N | is topologically conjugate to (n , σ ). Horseshoes are robust in the sense that they persist under perturbations of f . Having a horseshoe is a notion of topological chaos. It implies in particular that f has positive topological entropy. But since horseshoes usually have Lebesgue measure zero, it is entirely possible for a map to have a horseshoe and at the same time to have the orbit of Lebesgue-almost every point tend to a stable equilibrium. Next we describe a dynamical picture which is chaotic not only from the topological but also from the probabilistic point of view; it represents a stronger form of chaos. Let f be an embedding such that f (U¯ ) ⊂ U for some open set U . In this paper, we refer to  := ∩n≥0 f n (U¯ ) as an attractor and U as its basin. Our dynamical results include a number of properties frequently associated with “strange attractors”. (We regard the term “strange attractor”, which embodies a wide range of ideas, as descriptive rather than technically defined.) We give precise definitions of the relevant properties below, and label them as (SA1)–(SA4). Later on, to remind the reader what (SA1)–(SA4) stands for, we will refer to an attractor f | having these properties as “a strange attractor with (SA1)–(SA4)”. First, we recall the definition of SRB measures. An invariant Borel probability measure ν for f is called an SRB measure if f has a positive Lyapunov exponent ν-almost everywhere and the conditional measures of ν on unstable manifolds are equivalent to the Riemannian volume on these leaves. See [Y] for more information. The following are properties we associate with the idea of strange attractors: (SA1) Positive Lyapunov exponents. For Lebesgue-a.e. x ∈ U , the orbit of x has a positive Lyapunov exponent, i.e. lim

n→∞

1 log Df n (x) > 0. n

(This property is important enough that we state it separately; it, in fact, follows from (SA2).)

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(SA2) Existence of SRB measures and basin property. (a) f admits at least one and at most finitely many ergodic SRB measures all of which have no zero Lyapunov exponents; we denote them by ν1 , · · · , νr ; (b) for Lebesgue-a.e. x ∈ U , ∃j = j (x) such that for every continuous function ϕ : U → R,  n−1 1
i ϕ(f x) → ϕdνj . n i=0

(SA3) Statistical properties of dynamical observations. (a) For every ergodic SRB measure ν and every H¨older continuous function ϕ :  → R, the sequence {ϕ ◦ f i }i=0,1,··· obeys a Central Limit Theorem,   ϕ ◦ f i converges in distribution to the nori.e. if ϕdν = 0, then √1n n−1 0 mal distribution, and the variance is strictly positive unless ϕ◦f = ψ ◦f −ψ for some ψ. (b) Suppose that for some N ≥ 1, f N has an SRB measure ν that is mixing. Then given a H¨older exponent η, ∃τ = τ (η) < 1 such that for all ϕ, ψ :  → R H¨older with exponent η, ∃K = K(ϕ, ψ) such that ∀n ≥ 1,        (ϕ ◦ f nN )ψdν − ϕdν ψdν  ≤ K(ϕ, ψ)τ n .  We remark that all ergodic SRB measures with no zero Lyapunov exponents are mixing up to a finite factor. The first part of (SA2) can sometimes be strengthened to (SA4) Uniqueness of SRB measure and ergodic properties. (a) f admits a unique (and hence ergodic) SRB measure ν; (b) (f, ν) is mixing or, equivalently, isomorphic to a Bernoulli shift. 1.2. Chaotic behavior in periodically kicked limit cycles. A periodic orbit γ of a flow is called a limit cycle if it is attractive, a hyperbolic limit cycle if the eigenvalues of its section maps have moduli < 1. It is well known that hyperbolic limit cycles are robust, so that given a flow ϕt with such a cycle and a time T , if κ is a small perturbation, then under the iteration of ϕT ◦ κ, all points continue to be attracted to a simple closed curve. Theorem 1 describes some possible scenarios if larger perturbations are permitted. Theorem 1 (Creation of strange attractors from limit cycles). Let ϕt be a C 4 flow on an n-dimensional Riemannian manifold M. Assume that ϕt has a hyperbolic limit cycle γ . For n = 2, assume also that the normal bundle to γ is orientable. Let U be a neighborhood of γ , and let Emb3 (U, M) be the space of C 3 embeddings of U into M. Then there is an open set E ⊂ Emb3 (U, M) such that the following hold for every κ ∈ E: (i) ϕT ◦ κ has a horseshoe (i.e. Property (H)) for all large T ; (ii) ϕT ◦ κ has a strange attractor with (SA1)–(SA4) for a positive measure set of T . Remark. The ideas embodied in Theorem 1 can be expressed in many different ways. The presence of a parameter is important, for as we will see in Sect. 2, our conclusion will invariably be that maps corresponding to a positive measure set of parameters have strange attractors. In Theorem 1, we have chosen – for convenience – this parameter

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to be T , the time interval between consecutive kicks. One can, for example, fix T and use instead one or more parameter families of kick maps with suitable properties. We leave these variants of our results to the reader, our intent here being only to illustrate the general idea. 1.3. Periodically forced Hopf bifurcations in 2D. The following are the general equations for a Hopf bifurcation in 2D, written in normal form in complex coordinates: z˙ = λµ z + aµ z2 z¯ + bµ z3 z¯ 2 + · · · ,

(1)

where µ ∈ R is a parameter, aµ , bµ ∈ C are constants, and λµ = µ + (ω + µγµ )i, γµ , ω ∈ R, ω = 0. That is to say, at the bifurcation parameter µ = 0, the linearized equation at the bifurcation point z = 0 is z˙ = iωz, and that λµ crossses the imaginary axis from left to right as µ increases past zero. For our purposes, it is convenient to write these equations in polar coordinates: r˙ = (µ − αµ r 2 )r + r 5 gµ (r, θ ) , θ˙ = ω + γµ µ + βµ r 2 + r 4 hµ (r, θ ) .

(2)

Here ω, αµ , βµ , γµ ∈ R are constants, and gµ and hµ are functions which we will assume to be of class C 4 ; all depend smoothly on µ. (ω and γ are as in Eq. (1), while a = −α + iβ.) In addition to assuming ω = 0, we consider in this paper the case α > 0, i.e. the case of a generic supercritical Hopf bifurcation. From the main terms of  Eq. (3), one sees that as µ increases from 0, an attracting invariant circle of radius ≈

µ α

appears. For references on Hopf bifurcations, see e.g. [H, GH, MM]. From here on we omit the subscript µ in the constants in our equations except where the dependence on µ is at issue. To the system described by Eq. (3), we give a kick which leaves 0 fixed and which is radial in space and periodic in time, resulting in r˙ = (µ − αr 2 )r + r 5 g(r, θ ) + r(θ )

∞
n=0

θ˙ = ω + γ µ + βr 2 + r 4 h(r, θ ) .

δ(t − nT ) , (3)

Here δ(·) is the usual δ-function,  : S 1 → R is a C 3 function, and T is the period of the kick. Equation (3) is to be interpreted as follows: Since the kicks are radial, they do not affect the θ -coordinate. At times t = nT , n = 0, 1, 2, · · · , the r-coordinate changes abruptly from r− (nT ) to r+ (nT ), where r+ (nT ) := limε→0 rε (ε) and rε (t) is the solution of r˙ε = 1ε rε (θ ) with rε (0) = r− (nT ), that is to say, r+ (nT ) = r− (nT ) e(θ(nT )) . During the time interval (nT , (n + 1)T ), the system evolves according to (2) with initial position (r+ (nT ), θ (nT )) and final position (r− ((n + 1)T ), θ ((n + 1)T )). For each µ, let Fµ,T : R2 → R2 denote the time-T -map of the system defined by (3). Theorem 2. Assume αβ00 = 0, and let 0 : S 1 → R be a C 3 function with nondegenerate critical points. We consider forcing functions  of the form  = A0 , A ∈ (0, ∞). In (1) and (2) below, we assert that if | αβ00 |A is sufficiently large, then there is a set of parameters (µ, T ) for which Fµ,T has chaotic behavior.

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(1) (Existence of horseshoes). There exists K0 = K0 (0 ) such that if | αβ00 |A > K0 , then for all µ in some interval (0, µ0 ), there exist T0 (µ) = O( µ1 ) and an open set 0 (µ) ⊂ (T0 , ∞), 0 (µ) roughly 2π ω -periodic, such that for all T ∈ 0 (µ), Fµ,T has a horseshoe, i.e. Property (H). If | αβ00 |A is sufficiently large, then 0 (µ) = (T0 , ∞). (2) (Existence of strange attractors). There exists K1 = K1 (0 ) such that if | αβ00 |A > K1 , then the following hold for all µ in some interval (0, µ1 ): There exist T1 (µ) = O( µ1 ) and a positive Lebesgue measure set 1 (µ) ⊂ (T1 , ∞), 1 (µ) roughly 2π ω periodic, such that for all T ∈ 1 (µ), Fµ,T has a strange attractor with (SA1)– (SA3). If | αβ00 |A is sufficiently large, then Fµ,T has property (SA4) as well. In general, the larger the twist factor | αβ00 |, the weaker the forcing required. If | αβ00 | >> 1, then very mild disturbances (with e.g. (θ ) = ε sin θ ) at periodic time intervals can give rise to strange attractors. Also, T1 is usually >> T0 . Remark. Our purpose here is to bring to light the phenomenon of chaos appearing in periodically-kicked Hopf bifurcations, and to call the reader’s attention to a relevant set of mathematical tools. No attempt has been made to formulate the most general results. In Theorems 1 and 2, for example, kicks that are not radial can be considered. To gain more insight into what kind of kicks are suitable, see the intuitive explanation in the introduction. For alternate formulations regarding parameters, see the Remark following Theorem 1. 1.4. Hopf bifurcations in higher dimensions. The following is a direct generalization of our results in 2D: Consider as before a 1-parameter family of equations on Rn having a stationary solution at 0. Suppose that for µ < 0, the eigenvalues at 0 all have strictly negative real parts, and that at µ = 0, a pair of complex conjugate eigenvalues crosses the imaginary axis. We decompose the tangent space at 0 into Rn = V c ⊕ V s , where V c is the 2-dimensional subspace corresponding to the leading pair of eigenvalues and V s is the invariant subspace corresponding to the rest. The existence of a center manifold W c tangent to V c is well known. We assume that the bifurcation is supercritical, and that the unforced equation restricted to W c is in normal form. (See Sect. 4.5 for the precise meaning of this last condition.) To this bifurcation, we add a forcing that is radial. Let S n−1 = {u ∈ Rn  : u = 1}, ∞ u ¯ ¯ 0 (u)u 0 : S n−1 → R, and u¯ = u . We consider a forcing of the form A ¯ n=0 δ(t − nT ). Theorem 3. Let α and β be as in Eq. (3) for the unforced equation restricted to W c , and ¯ 0 |(S n−1 ∩V c ) is C 3 with nondegenerate critical points. Then results analogous assume  to those in Theorems 2 hold. 2. Sufficient Conditions for Strange Attractors (and Horseshoes) In this section we recall some results on the existence of strange attractors of certain types. The framework here is considerably more general than that in Sect. 1; it is unrelated to Hopf bifurcations or limit cycles. We will focus primarily on conditions that guarantee (SA1)–(SA4). Weaker conditions that guarantee the presence of horseshoes are discussed in Sect. 2.3.

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2.1. Conditions for (SA1)–(SA4). We recall in this subsection some earlier results which will be used in this paper. [WY1], which contains a 2D version of the results below, is the backbone of the strange attractor results in all of our theorems. A couple of useful modifications of the conditions in [WY1] are pointed out in [WY2]. The n-dimensional version of [WY1] is proved in [WY3]. This version is used in the proofs of Theorems 1 and 3. We consider a family of maps Ta,b : M = S 1 × D → M, where D is the closed unit disk in Rn−1 , a ∈ [a0 , a1 ] ⊂ R and b ∈ B0 , where B0 is any subset of R \ {0} with an accumulation point at 0.1 Points in M are denoted by (x, y) with x ∈ S 1 and y ∈ D. Very roughly, the parameters a and b can be thought of as having the following meanings: a moves points along S 1 × {0}, which, in most places, is a direction of instability, 1 while |b| can be interpreted as a measure of dissipation. In particular, Ta,0 has infinite dissipation; it sends all of M to a one-dimensional object (see (C1) below). (C0) Regularity conditions (i) For each b ∈ B0 , the function (x, y, a) → Ta,b (x, y) is C 3 ; (ii) each Ta,b is an embedding of M into itself; (iii) there exists K > 0 independent of (a, b) such that for all (a, b), | det DTa,b (z)| ≤ K | det DTa,b (z )|

∀z, z ∈ S 1 × D.

(C1) Existence of singular limit. There exist Ta,0 : M → S 1 × {0}, a ∈ [a0 , a1 ], such that as b → 0, the maps (x, y, a) → Ta,b (x, y) converge in the C 3 norm to (x, y, a) → Ta,0 (x, y). Identifying S 1 × {0} with S 1 , we refer to Ta,0 as well as its restriction to S 1 × {0}, i.e. the family of 1D maps fa : S 1 → S 1 defined by fa (x) = Ta,0 (x, 0), as the singular limit of Ta,b . The rest of our conditions are imposed on the singular limit alone. The next condition in [WY1] or [WY3] is the existence of a ∗ ∈ [a0 , a1 ] such that f = fa ∗ satisfies the so-called Misiurewicz condition. In practice, we have found that (C2) below is more directly checkable, albeit a little more cumbersome to state. That (C2) implies the condition in [WY1] and [WY3] is proved in [WY2], Appendix A. (C2) Existence of a sufficiently expanding map from which to perturb. There exists a ∗ ∈ [a0 , a1 ] such that f = fa ∗ has the following properties: There are numbers c1 > 0, N1 ∈ Z+ , and a neighborhood I of the set C := {f  = 0} such that (i) f is expanding on S 1 \ I in the following sense: (a) if x, f x, · · · , f n−1 x ∈ I, n ≥ N1 , then |(f n ) x| ≥ ec1 n ; (b) if x, f x, · · · , f n−1 x ∈ I and f n x ∈ I , any n, then |(f n ) x| ≥ ec1 n ; (ii) f n x ∈ I ∀x ∈ C and n > 0; (iii) in I , the derivative is controlled as follows: (a) |f  | is bounded away from 0; (b) by following the critical orbit, every x ∈ I \ C is guaranteed a recovery time n(x) ≥ 1 with the property that f j x ∈ I for 0 < j < n(x) and |(f n(x) ) x| ≥ ec1 n(x) . 1 The formulation here (with b = 0) together with (C1) is equivalent to that in [WY1] and [WY3]. We have elected to state (C1) as a separate condition because in applications, the definition of Ta,b for b = 0 usually comes for free while the existence of the singular limit Ta,0 has to be proved.

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Next we introduce the notion of smooth continuations. Let Ca denote the critical set of fa . For x = x(a ∗ ) ∈ Ca ∗ , the continuation x(a) of x to a near a ∗ is the unique critical point of fa near x. If p is a hyperbolic periodic point of fa ∗ , then p(a) is the unique periodic point of fa near p having the same period. It is a fact that in general, if p is a point whose fa ∗ -orbit is bounded away from Ca ∗ , then for a sufficiently near a ∗ , there is a unique point p(a) with the same symbolic itinerary under fa . (C3) Parameter transversality. For each x ∈ Ca ∗ , let p = f (x), and let x(a) and p(a) denote the continuations of x and p respectively. Then d d fa (x(a)) = p(a) da da

at a = a ∗ .

(C4) Nondegeneracy at “turns”. ∂ Ta ∗ ,0 (x, 0) = 0 ∂y

∀x ∈ Ca ∗ .

(C5) Conditions for mixing. (i) ec1 > 2 where c1 is in (C2). (ii) Let J1 , . . . , Jr be the intervals of monotonicity of fa ∗ , and let P = (pi,j ) be the matrix defined by  1 if f (Ji ) ⊃ Jj , pi,j = 0 otherwise. Then there exists N2 > 0 such that P N2 > 0. Theorem A ([WY1], [WY3]). Suppose {Ta,b } satisfies conditions (C0)–(C4). Then for all sufficiently small b ∈ B0 , there is a positive measure set of a for which Ta,b has properties (SA1), (SA2) and (SA3). Theorem B ([WY1, WY3] and appendix of [WY2]). In the sense of Theorem A, (C0) − (C5) ⇒ (SA1) − (SA4). 2.2. Model singular limit maps. In this subsection, we consider an (abstractly defined) class of 1D maps which satisfy Conditions (C2), (C3) and (C5) in Sect. 2.1. The maps in this class will be shown later on to arise as singular limits in the situations of interest. Proposition 2.1. Let  : S 1 → R be a C 3 function with nondegenerate critical points. Then there exist L1 and δ depending on  such that if L ≥ L1 and  : S 1 → R is a C 3 function with C 2 ≤ δ and C 3 ≤ 1, then the family fa (θ ) = θ + a + L((θ ) + (θ )),

a ∈ [0, 1],

satisfies (C2) and (C3) in Sect. 2.1. (C5) holds if L1 is sufficiently large. This is a slightly more general setting than that treated in Sects. 5.2 and 5.3 of [WY2], but the proofs are identical.

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2.3. Conditions for horseshoes. It is a general fact that the existence of an SRB measure with nonzero Lyapunov exponents implies the presence of horseshoes. This follows from Theorem 2 ([K]). Let f : M → M be a C 1+α diffeomorphism of a compact Riemannian manifold, and let µ be an invariant probability measure with (i) nozero Lyapunov exponents and (ii) hµ (f ) > 0. Then f has a horseshoe. In general, horseshoes appear considerably before strange attractors. We give a sufficient condition in the spirit of Sect. 2.1. The following applies to one value of a at a time, so let a = a0 = a1 in the notation of Sect. 2.1. 1 3 Lemma 2.1. Assume (C0)(i),(ii) and (C1) with  C replaced by C , and also (C2)(i)(a). Let I be as in (C2). If there is a point in n≥0 f −n (S 1 \ I ) which is not eventually periodic, then Ta,b has a horseshoe for all small b.

Proof. Relaxing c1 to c21 , (C2)(i)(a) continues to hold (for the same N1 ) if I in (C2) is replaced by a slightly smaller neighborhood I˜ of the critical set. This implies there exists ε > 0 such that for all x with x, f x, . . . , f n x ∈ S 1 \ I , there is a small interval J containing x such that J, f (J ), . . . .f n (J ) ⊂ S 1 \ I˜ and f n (J ) = [f n x − ε, f n x + ε]. Let N ≥ N1 be such that all intervals having the same N -itinerary have lengths < ε. Consider {x : f i x ∈ S 1 \ I˜ for 0 ≤ i ≤ N}, and let J1 , . . . , Jr be the components of this set corresponding to distinct itineraries. Define a transition matrix A on {1, 2, . . . , r} 1 by letting aij = 1 if and only if f N Ji ⊃ Jj , and let + = + N,A = {x ∈ S : ∀j ≥ 0, ∃i(j ) s.t. f Nj x ∈ Ji(j ) and ai(j )i(j +1) = 1}. We claim that

f −n (S 1 \ I ) ⊂ + ⊂

n≥0



f −n (S 1 \ I˜).

n≥0

The first containment above is a consequence of our choice of ε and N : if x is such that x, f x, . . . , f N x ∈ (S 1 \I ), x ∈ Ji , f N x ∈ Jj , then f N Ji ⊃ [f N x−ε, f N x+ε] ⊃ Jj . Our assumption on ∩f −n (S 1 \ I ) ensures that f N |+ is a nontrivial shift of finite type. From this it follows from a standard argument that for some k, n, f Nk |+ has an invariant set topologically conjugate to (n+ , σ ), the one-sided full shift on n symbols. The second containment in the displayed expression is obvious; it ensures that f N |+ is expanding. The existence of a horseshoe now follows immediately from invariant cones arguments.   3. Periodically-Kicked Limit Cycles This section contains geometric ideas leading to the proof of Theorem 1.

3.1. Existence of singular limits. The setting is as follows. Let ϕt be a flow on a Riemannian manifold M with a periodic orbit γ . We assume γ is a hyperbolic limit cycle, i.e. if  is a codimension one disk transversal to γ at x, and g :  →  is the return map, then all the eigenvalues of Dg(x) have modulus < 1. We consider a periodically kicked system represented by the iteration of ϕT ◦ κ, where κ : M → M is a map and T is a long period of relaxation. We will prove in this subsection that under very mild conditions, singular limits in the sense of Sect. 2.1 exist for ϕT ◦ κ as T → ∞.

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First we introduce the relevant geometric objects. Let V = {x ∈ M : ϕt (x) → γ as t → ∞}. For each x ∈ γ , let W ss (x) denote the strong stable manifold through x, i.e. W ss (x) = {y ∈ M : d(ϕt (x), ϕt (y)) → 0 as t → ∞}. Then V is foliated by W ss -leaves each one of which is an immersed codimension one submanifold meeting γ in exactly one point. Let π : V → γ be the projection map obtained by sliding along W ss -leaves, i.e. for y ∈ V , π(y) is the unique point x ∈ γ such that y ∈ W ss (x). Next we introduce a family of maps in the spirit of Sect. 2.1. Let p be the period of γ , and let bn , n = 1, 2, . . . , be a (any) monotonically decreasing sequence of numbers accumulating at 0. For n = 1, 2, . . . and a ∈ [0, p), we define Ta,bn = ϕnp+a ◦ κ, where κ is the “kick”. Proposition 3.1. Let ϕ : M × R → M be any C 4 flow with a hyperbolic limit cycle γ , and let U be a tubular neighborhood of γ . We assume κ : U¯ → M is a C 3 embedding with κ(U¯ ) ⊂ V . Then for all large n, Ta,bn (U ) ⊂ U and the following are true: (i) {Ta,bn } is of the form specified in the paragraph before (C0) in Sect. 2.1; (ii) conditions (C0) and (C1) in Sect. 2.1 are satisfied; and (iii) the singular limit Ta,0 is of the form Ta,0 = ϕa ◦ π ◦ κ . Proof. First, we argue that pointwise for each y ∈ U , Ta,bn (y) → Ta,0 (y) as n → ∞: For y ∈ U , we have z := κ(y) ∈ V , and by definition of W ss , d(ϕt (z), ϕt (π(z))) → 0

as t → ∞.

That ϕnp+a (π(z))) = ϕa (π(z)) follows immediately from the fact the period of the cycle is p. Since κ(U¯ ) is a compact subset of V , the convergence above is, in fact, exponentially fast and uniform for all y ∈ U¯ . To prove (C0)(iii), let J be such that | det(Dϕp (z))| = e−J for all z ∈ γ . The exponential convergence above implies that | det(DTa,bn )(z)| differs from e−J n by at most a constant for all z ∈ U¯ . The verification of (C0) is complete. (C1) requires that we prove a stronger form of convergence than that in the last paragraph. Observe that (a, z) → Ta,bn (z) can be written in the composite form (a, z) → (a, κ(z)) → (a, (ϕp )n (κ(z))) → ϕa ((ϕp )n (κ(z))) . Since the first map is C 3 and the last is C 4 , to prove the asserted C 3 -convergence, it suffices to show that (ϕp )n converges in C 3 to π as n → ∞. Recall from standard stable manifolds theory that for a C 4 flow, each W ss (x)-leaf is locally a C 4 embedded disk [HPS]. This together with the fact that all the leaves of W ss are ϕt -images of W ss (x) for some x implies that as a foliation, W ss is also C 4 . The required convergence is thus tantamount to the following calculus lemma: Lemma 3.1. Let I = (0, 1), D = {y ∈ Rn−1 : |y| < 1}, and consider H : I × D → I × D of the form H (x, y) = (x, h(x, y)), where

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(i) h(x, 0) = 0 for all x ∈ I , (ii) for all i, |∂yi h| ≤ λ for some λ < 1, and (iii) H C 4 < K for some K > 0. Then as n → ∞, H n converges in the C 3 -norm to H0 , where H0 (x, y) = (x, 0). Proof of Lemma 3.1. For notational simplicity, we give a proof only in two dimensions. Let us write H n (x, y) = (x, hn (x, y)). Since |hn (x, y)| ≤ λn , convergence in C 0 is assured as explained earlier. Computing recursively, we obtain ∂1 hn (·) =

n−1


∂2 h(H n−1 ·) · · · ∂2 h(H k+1 ·)∂1 h(H k ·),

k=0

∂2 hn (·) = ∂2 h(H n−1 ·)∂2 h(H n−2 ·) · · · ∂2 h(·) . It follows immediately that |∂2 hn | ≤ λn . Using also |∂1 h(x, y)| ≤ K|y|, we see that each term in ∂1 hn (·) is bounded above by λn−k−1 · Kλk , so that |∂1 hn | ≤ Knλn−1 . Moving on to second derivative estimates, since ∂ij hn is computed by differentiating each factor in each term of ∂j hn , we observe that ∂ij hn has ≤ 2n2 terms, and each term has ≤ (n + 1) factors. Moreover, if ∂2 h(H k ·) is differentiated, then our previous estimate of |∂2 h(H k ·)| ≤ λ is replaced by one of the following: |∂12 h(H k ·)| ≤ K, |∂22 h(H k ·)∂1 hk (·)| ≤ K 2 kλk−1 (K in the case k = 0), or |∂22 h(H k ·)∂2 hk (·)| ≤ Kλk . If ∂1 h(H k ·) is differentiated, then our previous estimate of |∂1 h(H k ·)| ≤ Kλk is replaced by |∂11 h(H k ·)| ≤ Kλk or estimates identical to those above for |∂12 h(H k ·)|. We conclude that |∂ij hn | ≤ 2n2 · const · n2 λn−2 . A similar argument gives |∂ij k hn | ≤ const nα λn−3 for some α. (The boundedness of the fourth derivative of H is used to control ∂111 h.)   3.2. Creation of strange attractors from limit cycles. We now use the ideas developed in Sect. 3.1 to prove Theorem 1, which says that when suitably kicked, any hyperbolic limit cycle will exhibit chaotic behavior. See Sect. 1.2 for the precise statement. Proof of Theorem 1. First we produce an open set E ⊂ Emb3 (U, M) consisting of suitable kicks. This step is not necessary, but some readers may find it helpful to first “straighten out” the W ss -foliation. More precisely, we may assume, via a C 4 change of coordinates, that γ = {y = 0} and the W ss -manifolds are codimension one planes perpendicular to γ . That this can be done is explained in Sect. 3.1. One way to choose E is to begin by selecting a C 3 map  : S 1 → R with nondegenerate critical points. Proposition 2.1 gives an open set of L and  such that fa (θ ) := θ + a + L((θ ) + (θ )) satisfies Conditions (C2), (C3) and (C5). These choices of L and  give rise to a collection of f0 , which constitutes an open set Eˆ0 of C 3 maps from S 1 to itself. From Eˆ0 , we construct an open set E0 ⊂ Emb3 (γ , M) consisting of κ0 such that π ◦ κ0 = f0 . Given f0 , the existence of κ0 is trivial in dimensions > 2: simply lift the image of κ(γ ) in the “vertical” direction to avoid self-intersections. An argument (which we leave as an exercise for the reader) is needed for 2D: use (i) f0 has degree one and (ii) our limit cycle γ has an orientable normal bundle. From E0 we construct E ⊂ Emb3 (U, M), where κ ∈ E is obtained by extending κ0 ∈ E0 to U in such a way that ∂yi (π ◦ κ) = 0 for some i at points whose θ -coordinates are near the critical points of π ◦ κ0 . This completes our construction of E.

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For each κ ∈ E, we now introduce, as in Sect. 3.1, a 2-parameter family of maps from a tubular neighborhood of γ to itself, namely Ta,bn = ϕnp+a ◦ κ, where p is the period of the cycle and a ∈ [0, p). Proposition 3.1 says that this family is of the type considered in Sect. 2.1, and that (C0) and (C1) are valid provided κ(U ) remains in the basin of attraction of γ . By design, the singular limit is what we started with in the last paragraph, so (C2), (C3), (C4) and, if we so desire, (C5), are met. Thus [WY3] (the relevant portion of which is summarized in Theorems A and B in Sect. 2.1) applies.2 [WY3] tells us that for each sufficiently large n, there is a positive Lebesgue measure set n ⊂ [0, p) such that for all a ∈ n , Ta,bn has a strange attractor with the properties in (SA1)–(SA4). Thus the dynamical description in Theorem 1(2) holds for all T ∈ {np + a : a ∈ n , n ≥ n0 for some n0 ∈ Z+ }. (There is no relation between“T ”, the period of the kicks, and Ta,bn ; we regret the unfortunate notation.) As for Theorem 1(1), if L in the singular limit is sufficiently large, then Lemma 2.1 says that horseshoes are present for all a provided that bn is sufficiently small, i.e. they are present for all T ≥ some T0 .   3.3. More on production of chaos: Example and Discussion. Section 3.2 contains an abstract existence result. We now turn to a more practical question: given an arbitrary limit cycle (the way it is embedded in the ambient manifold), what kinds of kicks will result in chaotic behavior? Example 3.1. A linear model. Consider θ˙ = 1 + σ · y , y˙ = −y + A(θ )v(θ )

(4) ∞


δ(t − nT ),

n=0

where θ ∈ ∈ all of whose eigenvalues have strictly positive real parts. For simplicity, we assume the kicks are perpendicular to the limit cycle {y = 0}, with the amplitude of the kicks at (θ, y) given by A0 (θ ) and the direction by v ∈ S n−2 . The W ss -manifolds here are a family of parallel codimension one planes. A simple computation gives S1, y

Rn−1 , σ

is a fixed vector in Rn−1 , and  is an (n−1)×(n−1) matrix

fa (θ ) = θ + a + A(θ )σ t −1 v(θ ) . Observe that the effect of the kick is magnified by σ t −1 v, an amount determined by the competition between shear and rate of contraction in the direction of v. In particular, for n = 2, where σ, λ =  ∈ R and v = +1 or −1, we have the following: Proposition 3.2. (cf. [WY2], Theorems 2 and 3). In dimension 2, given a C 3 function 0 with nondegenerate critical points, there exists K1 = K1 (0 ) such that if shear σ A := · amplitude of kick > K1 , λ contraction then the time-T -map of (4) has a horseshoe, i.e. Property (H), for all large T and a strange attractor with (SA1)–(SA4) for a positive measure set of T . This follows from a direct application of [WY1] (see Sect. 2.1). For (C2), (C3) and (C5), see Proposition 2.1. (C4) is satisfied since ∂y Ta,0 = σλ = 0. 2 Orientability of the normal bundle to γ is assumed in the setup in [WY1] and [WY3], although it is not essential in the proofs.

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(θ, ΑΦ0 (θ))

(θ, 0)

ss

W − leaves

σ

(θ + λ−ΑΦ0 (θ), 0)

Fig. 2. Geometric view of singular limit for Eq. (4) in dimension 2

Remark on general situation. Consider now a completely arbitrary limit cycle in any dimension. For the linearized equation, the W ss -plane through a point can be computed from the cumulative action as the point moves once around the cycle. When a kick-force is added, its effect is, as in the example above, determined by the angles the W ss -planes make with the periodic orbit γ and the directions of the kick in relation to the W ss planes. An explicit solution may not be available, but the geometric principles behind it are clear. The picture rendered by the linearized equation, however, may not be an accurate reflection of that for the nonlinear flow: The smaller the angles between the limit cycle and the W ss -leaves, the more prominent is the role played by curvature (or second derivatives). If these angles are small, and if to obtain the singular limit we have to slide a nontrivial distance along (curved) W ss -leaves (see Sect. 3.1), then the information given by the linearized equation is even less meaningful. Finally, for our scheme to work, care must be taken to ensure that the kick does not take us outside of the basin of attraction. From the discussion above, we see that in general, the answer to when chaotic behavior arises depends on fairly detailed information along the limit cycle. In the case of Hopf bifurcations, this information is contained in the first few derivatives at a single point. This together with the frequent occurrence of Hopf bifucations makes it a natural setting for the type of results formulated here. 4. Proof of Results on Hopf Bifurcations For the 2D result, the situation can be summarized as follows: 1. The dynamical properties in our theorems are derived from the “abstract theory” in [WY1] and [WY3]. In these two papers, we proved that these and other properties are enjoyed by maps satisfying a certain set of conditions. The aim of this section is to prove that these conditions are met by the system defined by Eq. (3). 2. The conditions in [WY1] and [WY3], which for the convenience of the reader we have reproduced in Sect. 2.1, are primarily of two types: the first concerns the existence of a singular limit; the second concerns the properties of the maps in the singular limit. These two aspects are discussed separately in the next two paragraphs. 3. The existence of a singular limit has been proved in a much more general setting than is needed here; see Proposition 3.1. 4. Instead of verifying the remaining conditions directly for Eq. (3), we have identified a class of model 1D maps and proved that if a singular limit belongs in this class, then it has many of the desired properties. See Proposition 2.1.

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5. In this section we will prove that the singular limits of Eq. (3) belong in the class in Propositin 2.1. The delicateness of the situation stems from the fact that we are dealing with a degenerate problem. The degeneracy here is twofold: as µ → 0, the limit cycle degenerates to a point, at the same time that it loses its hyperbolicity. The remainder of this section is organized as follows: The 2D case is treated in Sects. 4.1–4.4. We will focus on proving the presence of strange attractors. The proof for horseshoes follows from a (considerably simpler) version of our arguments here together with Lemma 2.1. The reduction of the n-dimensional result to 2D is carried out in Sect. 4.5. 4.1. Standardizing coordinates. We begin by blowing up the neighborhoods of 0. The purpose of this coordinate  change is to standardize the size and position of the limit cycle for all µ. Let y = r µα − 1. Then Eq. (3) in Sect. 1.3 becomes θ˙ = ωˆ +

β ˆ y) , µ(y + 2)y + µ2 h(θ, α

y˙ = −µ(y 2 + 3y + 2)y + µ2 g(θ, ˆ y) + (y + 1)(θ )

(5) ∞


δ(t − nT ),

n=0

where y ∈ (−1, ∞), θ ∈ R/(2πZ), and

 β ωˆ = ω + γ + µ, α 

1 µ 5 g(θ, ˆ y) = 2 (y + 1) g θ, (y + 1) , α α

 ˆh(θ, y) = 1 (y + 1)4 h θ, µ (y + 1) . α2 α

(6)

Observe that the presence of the ω-term ˆ in Eq. (5) prevents us from getting rid of the degeneracy at µ = 0 by a simple rescaling of time. 4.2. Singular limit in the absence of higher order terms. In this subsection, we set gˆ = hˆ = 0, and consider the flow ϕt generated by the unforced equation θ˙ = ωˆ + σ (y)y , y˙ = −λ(y)y, where

(7)

β µ(y + 2) , λ(y) = µ(y 2 + 3y + 2) . α Let µ be fixed for the rest of this subsection. As explained in Sect. 3.1, the singular limit . maps are related to limn→∞ ϕtn , where tn = 2nπ ωˆ σ (y) =

Proposition 4.1. For all small µ > 0 and −1 < y < ∞,

 β lim ϕt (θ, y) = θ + ln(y + 1), 0 . n→∞ n α

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Let (θ (t), y(t)) denote the solution of (7) with initial conditions (θ0 , y0 ). Proposition 4.1 follows immediately from Lemma 4.1 by letting n → ∞. Lemma 4.1. ˆ + θ (t) = θ0 + ωt



β y0 + 1 . ln y(t) + 1 α

Proof. The reader can verify that this is the solution by direct differentiation. We arrived 1 at the formula above by formally substituting ds = − λ(y) dy into the integral in 

t0

ˆ + θ (t) = θ0 + ωt

σ (y(s))ds, 0

obtaining  0

t

 σ (y(s))yds = −

y(t) y0

β σ (y) dy = − λ(y) α



y(t)

y0

1 dy . 1+y

 

Let π be the projection map in Sect. 3.1. We observe: (i) π(θ, y), which is the limit in Proposition 4.1, is defined for all θ ∈ S 1 and y > −1. (ii) From the formula for π, we deduce the following geometric information about the W ss -foliation: its leaves are invariant under translations in the θ -direction; their slopes have the same sign (either positive or negative) everywhere; near y = 0, the slopes are ≈ − βα ; they tend to +∞ or −∞ as y → ∞ and to 0 as y → −1. (iii) As µ → 0+ , π(θ, y) → (θ + αβ00 ln(y + 1), 0), where α0 and β0 are the values of α and β at µ = 0. This is a strong indication that in spite of the weakening hyperbolicity, the W ss -structure remains robust up to µ = 0. 4.3. Effects of higher order terms. We continue to consider the unforced equation. Let ϕt and ϕ˜t denote respectively the flows with and without higher order terms. When it is useful to identify the parameter µ, we will write ϕµ,t and ϕ˜µ,t . In this subsection and the next, if X is a quantity or object pertaining to ϕt , then the corresponding quantity or ˜ object for ϕ˜t is denoted by X. For each small µ > 0, let γµ denote the limit cycle for ϕt , and call its period pµ . To compare ϕt and ϕ˜t , we make a time change for ϕ˜t to synchronize their periods, i.e. to set p˜ µ = pµ . From the magnitudes of the higher order terms, we see that this time change, which will be assumed for the rest of the proof, is of order 1 ± O(µ2 ). We also introduce ιµ : γ˜µ → γµ by letting ιµ (ϕ˜t (0, 0)) = ϕt (0, y0 ), where (0, y0 ) is the point in γµ whose θ-coordinate is 0. Next we fix a compact domain of the infinite cylinder S 1 × R in which all the action will take place: let −1 < A0 < emin  − 1 and A1 > emax  − 1, and let A = S 1 × [A0 , A1 ]. The main result of this subsection is Proposition 4.2. For µ > 0 sufficiently small, πµ is defined on A, and (ι−1 ˜ µ C 3 → 0 as µ → 0 . µ ◦ πµ ) − π

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Even though the higher order terms in Eq. (5) tend to zero as µ → 0, this alone is insufficient justification for Proposition 4.2 because as µ → 0, the unforced part of Eq. (5) tends to the totally degenerate, completely integrable system θ˙ = ω, Let µ = ϕµ,[

1 µpµ ]pµ

˜ µ = ϕ˜ and  µ,[

y˙ = 0 .

1 µpµ ]pµ

, where [x] is the greatest integer ≤ x.

Lemma 4.2. There exists M such that for all small µ > 0, the following hold on A: ˜ µ C 4 ≤ M; (a) µ C 4 ,  ˜ (b) µ − µ C 3 = O(µ). Proof of Proposition 4.2 assuming Lemma 4.2. The task here is to deduce singular limit information, which depends on an infinite number of iterates, from the finite-time information provided by Lemma 4.2. ˜ µ leave points on their limit cycles fixed, and have uniform Observe that both µ and  “hyperbolic” estimates, i.e. the smaller eigenvalues are uniformly bounded away from 1 and the angles between the stable and neutral directions are uniformly bounded away from 0. Let qµ = (0, y0 ) ∈ γµ . From Lemma 4.2(a) and the uniformness of hyperbolicity, we see that there exists εˆ > 0 such that for all small µ > 0, the local stable manifolds of µ ˜ µ through (0, 0) are well defined as graphs of τµ , τ˜µ : [−ˆε , εˆ ] → S 1 . through qµ and  We claim that τµ − τ˜µ C 3 → 0 as µ → 0. ˜ µ ; µ ∈ (0, µ0 )} is bounded This is true because from Lemma 4.2(a), the set N := {µ ,  4 in the C -norm, and so it is relatively compact with respect to the C 3 -topology. The mapping G → τG is continuous with respect to the C 3 topologies for both G and τG (see [HPS]). Hence it is uniformly continuous on N . The convergence to 0 of τµ − τ˜µ C 3 now follows from Lemma 4.2(b). ˜ µ run from top to As noted in the last paragraph of Sect. 4.2, the W ss -leaves of  ss bottom of the annular region A. Since each of these (long) W -leaves is contained in the (ϕ˜t )−1 -image of graph(τ˜µ ) for some t ≤ T1 , it follows that the leaves of the W ss foliation for ϕt behave similarly and that the two foliations are asymptotically close in C 3 as µ → 0 (meaning there exist diffeomorphisms which converge to the identity in C 3 carrying the leaves of one to those of the other). By an argument similar to that in the last paragraph, we see also that as µ → 0, γµ converges in C 3 to γ˜µ = {y = 0}, and  ιµ converges in C 3 to the identity map. The assertion in Proposition 4.2 follows.  The proof of Lemma 4.2 uses the following elementary fact: Lemma 4.3. Let  ∈ RN be a convex open domain, and let W and Z be C 1 vector fields on . Suppose that for t ∈ [0, t0 ], ξ(t), η(t) ∈  are solutions of dξ = W (ξ ) and dt with ξ(0) = η(0). Then for all t ∈ [0, t0 ],

dη = Z(η) dt

C1 C2 t (e − 1), C2  where C1 := supx∈ W (x) − Z(x) and C2 := N i=1 supx∈ DZi (x), Zi being the component functions of Z. ξ(t) − η(t) ≤

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Proof. Writing dξ dη − = (W (ξ ) − Z(ξ )) + (Z(ξ ) − Z(η)), dt dt we see that ξ(t) − η(t) ≤ x(t) where x(t) satisfies the growth condition C1 C2 t C1 + C2 x, x(0) = 0. The solution of this equation is x(t) = C (e − 1).   2

dx dt

=

Proof of Lemma 4.2. We rescale time (for both equations) by letting t  = µt but continue to write t instead of t  , i.e. ϕt is now the flow generated by θ˙ =

ωˆ β ˆ y), + (2 + y)y + µ h(θ, µ α

(8)

ˆ y). y˙ = −(2 + 3y + y )y + µ g(θ, 2

The analogous time change is made for the equation with no higher order terms. First we verify that for t ∈ [0, 1) and µ ∈ (0, µ0 ), the first four derivatives of ϕt |A are uniformly bounded. For ϕt itself, we have |y(t) − y(0)| = O(µ). Let X denote the vector field in Eq. (8). Since 

t

Dϕt (·) = I +

DX(ϕs (·))Dϕs (·)ds 0

ωˆ , does not appear), and DX and ϕt are uniformly bounded (the only unbounded term, µ it follows that Dϕt is uniformly bounded. Bootstrapping our way up, we see that the same result holds for D i (ϕt ), i = 2, 3, 4. A similar argument works for ϕ˜t , proving (a). Next we wish to apply Lemma 4.3 with dξ dt = W (ξ ) representing the zeroth through dη third variational equations of ϕt , dt = Z(η) the corresponding equations for ϕ˜t , and t0 = 1. We claim that C1 = O(µ). This is because all the terms involving gˆ or hˆ have a copy of µ in front, and the first time change (made at the beginning of Sect. 4.3) creates a difference of O(µ) in the lower degree terms: before the second time change, this difference is O(µ2 ); it gets multiplied by µ1 in the second time change. The uniform boundedness of DZi  is justified above. The conclusion of Lemma 4.3 is precisely the claim in Lemma 4.2(b).  

4.4. Completing the proof. We now include back the forcing term in the equation. In the coordinates of Sect. 4.1, we see from Sect. 1.3 that the effect of the kick at time 0 is given by κ(θ, y) := (θ, y + ) = (θ, (y + 1)e(θ) − 1) , so that starting from |y| small, (ϕt ◦ κ)(θ, y) ∈ A for all t > 0. We continue to use ϕµ,t and ϕ˜µ,t to denote the time-t-maps of the flows with and without higher order terms, synchronizing for each µ the periods of the limit cycles as before. Proof of Theorem 2. Let Ta,0,µ = limn→∞ (ϕµ,npµ +a ◦ κ), and define T˜a,0,µ , fa,µ and f˜a,µ accordingly. As explained in the beginning of this section, it suffices to verify

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for given  that Ta,0,µ and fa,µ satisfy Conditions (C2)–(C5) on a parameter interval (0, µ0 ). Combining the formula for κ with Proposition 4.1, we have that for a ∈ [0, pµ ),

 β ˜ Ta,0,µ (θ, y) = θ + a + (ln(1 + y) + (θ )), 0 , (9) α so that β β f˜a,µ (θ ) = θ + a + (θ ) = θ + a + A0 (θ ) . α α

(10)

Identifying γ˜µ = S 1 × {0} with S 1 and using ιµ : γ˜µ → γµ as conjugating map, we obtain fa,µ : S 1 → S 1 as fa,µ (θ ) = (ι−1 µ ◦ πµ ) ◦ (κ ◦ ιµ )(θ ) + a . By Proposition 4.2 and the fact that κ ◦ ιµ → κ, we have that fa,µ − f˜a,µ C 3 → 0 as µ → 0. Thus if | βα |A is sufficiently large, then fa satisfies the conditions in Proposition 2.1. Also, ∂y Ta,0 ≈ ∂y T˜a,0 = βα . Part (1) of Theorem 2 now follows from Lemma 2.1; Part (2) follows from Theorems A and B (see Sect. 2.1).   4.5. Hopf bifurcations in n-dimensions. The hypotheses and notation are as in Sect. 1.4. First we make precise what is meant by “the unforced equation restricted to W c is in normal form”. Let h : V c → V s be such that W c = graph(h), and let  : Rn = V c ⊕ V s → V c be the projection map. Let ϕt be the given flow on Rn , and let ϕt∗ be the flow on V c defined by ϕt∗ (z) =  ◦ ϕt (z, h(z)). Our assumption is that the equation for ϕt∗ has the form of Eq. (3). The twist condition in Theorem 4 is computed from this equation. Proof of Theorem 3. We carry out in detail the strange attractor part of the proof, leaving the horseshoe part (which is considerably simpler) to the reader as before. I. Structure of unperturbed flow near 0. There exist µ0 > 0 and neighborhoods R of 0 in Rn such that the following hold for all µ ∈ [0, µ0 ): (i) ϕt (R) ⊂ R for all t > 0. (ii) Defined everywhere on R is a codimension two Dϕt -invariant strong stable subbundle roughly parallel to V s . We denote this subbundle by E s,2 and the invariant manifolds tangent to it by W s,2 . By the Invariant Section Theorem, W s,2 as a foliation is C 3 assuming the flow is C 4 (see [HPS]). Let π 2 : R → W c be projection by sliding along W s,2 -leaves. (iii) ϕt has a limit cycle γ contained in W c . Through each point in γ passes a codimension one strong stable manifold which we denote by W s,1 . (These are the W ss -manifolds in previous sections.) Let π 1 be the projection onto γ by sliding along W s,1 -leaves. Note that wherever W s,1 is defined, its leaves contain those of W s,2 . II. Reduction of problem. For each µ, let Ta,bn be defined as in Sect. 3.1. We are guaranteed for general reasons the existence of a well defined singular limit Ta,0 . The problem is reduced to proving (C2)–(C5) for this singular limit. Since Ta,0 alone matters, and (C4) requires only that we guarantee a nonzero derivative for Ta,0 in some direction normal to γ , while (C2), (C3) and (C5) pertain to the restriction of Ta,0 to γ , it may be sufficient to restrict the domain of Ta,0 to W c ,

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Q. Wang, L.-S. Young

that is to say, the problem is reduced to one in 2D, involving the flow ϕt |W c and kick κˆ := π 2 ◦ κ|W c . Now the hypotheses of Theorem 3 are on the kick-system (ϕt∗ , κ|V c ) (note that κ leaves V c fixed). To make use of this information, we project the kick-system (ϕt |W c , κ) ˆ to V c , resulting in (ϕt∗ , κ ∗ ), where the kick map κ ∗ : V c → V c is given by κ ∗ (z) =  ◦ π 2 ◦ κ(z, h(z)). The problem is thus reduced to comparing the two systems (ϕt∗ , κ|V c ) and (ϕt∗ , κ ∗ ), the objective being to deduce singular limit information about the second from that of the first. III. Magnified coordinates. The proof of √ Theorem 2 is carried out in blown-up coordinates. Accordingly, we consider an O( µ)-neighborhood of 0 in Rn and magnify coordinates (in all directions) by a factor ∼ √1µ , obtaining for ϕt∗ a limit cycle of radius ≈ O(1). Since magnification decreases higher derivatives, W c and the W s,2 -leaves are increasingly “straight” as µ → 0. More precisely, in coordinates magnified by ∼ √1µ , hC 3 and  ◦ π 2 − C 3 tend to 0 as µ → 0. As for κ, since this map is scale invariant, meaning κ(rz) = rκ(z), r > 0, we have, in magnified coordinates, κ|A C 3 = O(1), where A = { 43 < |z| < 45 }. (It is necessary to bound the domain away from 0 because κ is not differentiable at 0.) IV. Comparison of (ϕt∗ , κ|V c ) and (ϕt∗ , κ ∗ ) in magnified coordinates. We write κ ∗ − κ|V c as  ◦ π 2 ◦ κ(z, h(z)) − κ(z, 0) = [( ◦ π 2 − ) ◦ κ(z, h(z))] + [ ◦ (κ(z, h(z)) − κ(z, 0))] .

(11)

From the fact that hC 3 = o(1), κC 3 = O(1) in the relevant region, and  ◦ π 2 − C 3 = o(1), we see that κ ∗ − κ|V c C 3 = O(1) and κ ∗ − κ|V c C 2 = o(1) as µ → 0. It follows that in these coordinates, the singular limit maps Ta,0 corresponding to the two kick-systems are also C 3 -bounded and C 2 -near each other. By our assumptions on (ϕt∗ , κ|V c ) and from our proof in the 2D case, we know that for this system fa is in the model class considered in Sect. 2.2. By Proposition 2.1, this model class is robust under the type of perturbations above, and maps in this class satisfy (C2), (C3) and (C5). Similarly, information for (C4) is passed from one system to the other. The desired result for the n-dimensional system is proved.

 

Acknowledgement. The authors thank K. Lu and M. Shub for helpful conversations.

References [BC] [CL] [GH] [HPS] [H]

Benedicks, M., Carleson, L.: The dynamics of the H´enon map. Ann. Math. 133, 73–169 (1991) Cartwright, M.L., Littlewood, J.E.: On nonlinear differential equations of the second order. J. Lond. Math. Soc. 20, 180–189 (1945) Guckenheimer, J., Holmes, P.: Nonlinear oscillators, dynamical systems and bifurcations of vector fields. Appl. Math. Sciences 42, Berlin-Heidelberg-New York: Springer-Verlag, 1983 Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds. Lecture Notes in Math., 583 Berlin-Heidelberg-New York: Springer Verlag, 1977 Hopf, E.: Abzweigung einer periodischen L¨osung von einer station¨aren L¨osung eines Differentialsystems. Ber. Verh. S¨achs, Acad. Wiss. Leipzig Math. Phys. 94, 1–22 (1942)

Strange Attractors and Hopf Bifurcations [J] [K] [Li] [Ln] [L] [MM] [MV] [N] [NRT] [RT] [S] [WY1] [WY2] [WY3] [Y] [Z]

529

Jakobson, M.: Absolutely continue invariant measures for one-parameter families of onedimensional maps. Commun. Math. Phys. 81, 39–88 (1981) Katok, A.: Lyapunov exponents, entropy and periodic points for diffeomorphisms. Publ. IHES 51, 137–173 (1980) Levi, M.: Qualitative analysis of periodically forced relaxation oscillations. Mem. AMS 214, 1–147 (1981) Levinson, N.: A second order differential equation with singular solutions. Ann. Math. 50(1), 127–153 (1949) Lyubich, M.: Almost every real quadratic map is either regular or stochastic. Ann. Math., (2001) Marsden, J., McCracken, M.: The Hopf bifurcation and its applications. Appl. Math. Sci. 19 Berlin-Heildelberg-New York: Springer-Verlag, 1976 Mora, L., Viana, M.: Abundance of strange attractors. Acta. Math. 171, 1–71 (1993) Newhouse, S.: “Lectures in Dynamical Systems”. In: CIME Lectures, Bressanone, Italy, June 1978, Bascal-Britian: Birkh¨auser, 1980, pp. 1–114 Newhouse, S., Ruelle, D., Takens, F.: Occurrence of strange Axiom A attractors near quasiperiodic flows on T m , n ≥ 3. Commun. Math. Phys. 64, 35–40 (1978) Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971) Smale, S.: Differentiable dynamical systems. Bull. AMS 73, 717–817 (1967) Wang, Q., Young, L.-S.: Strange attractors with one direction of instability. Commun. Math. Phys. 218, 1–97 (2001) Wang, Q., Young, L.-S.: From invariant curves to strange attractors. Commun. Math. Phys. 225, 275–304 (2002) Wang Q., Young, L.-S.: Strange attractors with one direction of instability in n-dimensional spaces. 2002 preprint Young, L.-S.: What are SRB measures, and which dynamical systems have them? To appear In a volume in honor of D. Ruelle and Ya. Sinai on their 65th birthdays, J Stat. Phys. (2002) Zaslavsky, G.: The simplest case of a strange attractor. Phys. Lett. 69A(3), 145–147 (1978)

Communicated by G. Gallavotti

Commun. Math. Phys. 240, 531–538 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0908-3

Communications in

Mathematical Physics

Ruelle Inequality Relating Entropy, Folding Entropy and Negative Lyapunov Exponents
Pei-Dong Liu LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China. E-mail: [email protected] Received: 15 March 2003 / Accepted: 10 April 2003 Published online: 13 August 2003 – © Springer-Verlag 2003

Abstract: In this paper we prove an inequality conjectured by Ruelle relating the entropy, folding entropy and negative Lyapunov exponents of a differentiable map on a compact manifold, under a set of conditions on degenerate points of the map.

1. Statement of the Result Let M be a connected compact Riemannian manifold without boundary, f : M → M a C 1 map and µ an f -invariant measure. Let hµ (f ) be the (measure-theoretic) entropy of (f, µ) and for µ-a.e. x let −∞ ≤ λ(1) (x) ≤ λ(2) (x) ≤ · · · ≤ λ(dim M) (x) < +∞ be the Lyapunov exponents of f at x. A well-known inequality (Ruelle [4]) is the following hµ (f ) ≤


 M

λ(i) (x)+ dµ(x)

(1.1)

i

(a + := max{a, 0}) which is called Ruelle (or Margulis-Ruelle) inequality. For the purpose of studying positivity of entropy production in nonequilibrium statistical mechanics, Ruelle [5, P. 18] conjectured another inequality hµ (f ) ≤ Fµ (f ) −


 M

λ(i) (x)− dµ(x)

(1.2)

i

(a − := min{a, 0}) where Fµ (f ) := Hµ (|f −1 ),  being the partition of M into single points, and it is called the folding entropy of (f, µ) (see Ruelle [5] for more about this notion and see Rokhlin [3] for the conditional entropy theory). In this paper we confirm


This work is supported by SFMSBRP and NSFDYS

532

P.-D. Liu

this conjectured inequality in the above set-up, under some conditions on differentiability and degenerate points of the map f . We will denote f = {x : det Tx f = 0} and define M(r) = max{|(Tx f )−1 | : x ∈ M\B(f , r)}, where B(f , r) is the r-neighborhood of f . Our main result is the following theorem (see Remark 1.4 for some different assumptions) whose proof will be given in Sect. 2. Theorem 1.1. Let f : M → M be of class C 1+α (α > 0) and let µ be an f -invariant measure. Assume that there exist a ≥ 1, b ≥ 1 such that M(r) < r −a ,

f B(x, r) ⊃ B(f x, r b )

(1.3)

for all x ∈ M\B(f , r) when r is sufficiently small. Assume moreover that µ(B(f , r)) log r → 0 as r → 0, µ(f n f ) = 0 and log+ #{f −n x} ∈ L1 (M, µ) for all n ≥ 1. Then the inequality (1.2) holds true. Remark 1.2. Under the condition µ(f n f ) = 0, log #{f −n x} is lower semi-continuous on M\f n f and hence is Borel measurable. The author does not know if the µ-integrability of log+ #{f −n x} could be implied by that of log+ #{f −1 x}. Remark 1.3. For C k (k ≥ 2) f one can possibly analyze the validity of Condition (1.3) by using the Taylor expansion formula. Remark 1.4. (i) For a C 1 map f : M → M and an f -invariant measure µ the inequality (1.2) holds true if f −1 f ⊂ f , f n f f = ∅, µ(f n f ) = 0 and log+ #{f −n x} ∈ L1 (M, µ) for all n ≥ 1. The proof is a simplification of that of Theorem 1.1. The author hopes that the various assumptions in Theorem 1.1 could be weakened. (ii) For a continuous map f : M → M with M possibly having boundary , the statement of Remark 1.4 (i) is true if f is defined to be the closure of the set {x : f is  not C 1 at x or otherwise det Tx f = 0} . Remark 1.5. Inspired by Ledrappier and Young [1], one may guess the following: if, in  addition to the conditions of Theorem 1.1, f is of class C 2 and log | det Tx f |dµ(f ) > −∞, then there holds the generalized formula
 hµ (f ) = Fµ (f ) − ρi (x)γi (x)dµ(x), M

i

where −∞ < ρ1 (x) < ρ2 (x) < · · · < ρs(x) (x) < 0 are all the different negative Lyapunov exponents of f at x and γi (x) := di (x) − di−1 (x) with di (x) (d0 (x) := 0) being the supposed dimension of µ on the stable manifold W s,i (x) corresponding to ρi (x); further, the equality
 hµ (f ) = Fµ (f ) − λ(i) (x)− dµ(x) M

i

holds if and only if µ has absolutely continuous conditional measures on the stable manifolds of (f, µ). As we alluded to above, we will prove Theorem 1.1 in the next section. The author believes that Ruelle actually knows such a proof, possibly under some different assumptions (see the remark on Ruelle [5, P.18]), but the author thinks that the proof deserves an explicit exposition.

Ruelle Inequality

533

2. Proof of Theorem 1.1 Let (f, µ) be as given in Theorem 1.1. We will assume that f = ∅. When f = ∅ the   proof is much simpler (see also Remark 1.4 (i)). We will also assume that M i λ(i) (x)− dµ(x) > −∞ since otherwise (1.2) is obviously true. By the Oseledec multiplicative ergodic theorem,

λ(i) (x)dµ(x), log | det Tx f |dµ(x) = M

i

and hence log | det Tx f | is µ-integrable. Let s0 > 0 be given sufficiently small and let V0 be the open s0 -neighborhood of f f . Put G0 = M\f −1 V0 . Since G0 is closed and Tx f is nondegenerate for every x ∈ G0 , there are δ0 > 0 and r0 > 0 such that for every x ∈ G0 the map f |B(x,δ0 ) : B(x, δ0 ) → f B(x, δ0 ) is a diffeomorphism and f B(x, δ0 ) ⊃B(f x, r0 ). Then one p can choose disjoint open sets D1 , D2 , · · · , Dp such that G0 ⊂ i=1 Di ⊂ B(G0 , a20 ), where a0 = d(G0 , f ), for every i there is xi ∈ G0 satisfying Di ⊂ B(xi , δ20 ), and p i=1 ∂Di has µ-measure 0. We also assume that δ0 > 0, b0 > 1 are given such that for every x ∈ M and any y, z ∈ B(x, δ0 ) one has −1 b0−1 d(y, z) ≤ | exp−1 x y − expx z| ≤ b0 d(y, z).

Choose tn 0 and let Un be the open tn -neighborhood of f . Put Gn = M\Un . Recall that M(tn ) := max{|(Tx f )−1 | : x ∈ Gn } and we may assume that M(tn ) ≥ 1 for all n ≥ 1 since M(tn ) → +∞ as n → +∞. Since f is assumed to be C 1+α , by elementary computation one can choose a constant L depending only on M and f so that the numbers   1 δn δ0 r0 , rn = , , δn = min 2M(tn ) 4 4 LM(tn ) α2 (n)

satisfy the following requirements: for every x ∈ Gn the map fx := f |B(x,δn ) : (n) B(x, δn ) → f B(x, δn ) is a diffeomorphism, f B(x, δn ) ⊃ B(f x, rn ) and gx := (n) −1 (fx ) |B(f x,rn ) has the following expression: −1 (n) −1 G(n) + Rx(n) (·) : x := expx ◦gx ◦ expf x = (Tx f ) {ξ ∈ Tf x M : |ξ | < rn } → {η ∈ Tx M : |η| < δn } (n)

with Lip(Rx (·)) ≤ 1; moreover, d(y, z) > 2δn for any y, z ∈ Gn with fy = f z. Let εn = min{rn /10, tn } and we may assume that εn < δ0 /[2(1+b0 +M(tn ))]. Take a maximal εn -separated set Eεn of M, i.e., a subset Eεn of M such that d(x, y) > εn for any x, y ∈ Eεn with x = y and for any z ∈ M there is an element x ∈ Eεn satisfying d(x, z) ≤ εn . We then define a finite measurable partition αn = {αn (x) : x ∈ Eεn } of M such that αn (x) ⊂ int(αn (x)) and int(αn (x)) = {z ∈ M : d(z, x) < d(z, y) for all y ∈ Eεn with y = x}. For every αn define a new measurable partition βn of M such that, element of a partition β containing  denoting by β(x) the  x, βn (x) = αn (x) if αn (x) Un = ∅ and βn (x) = y:αn (y)∩Un =∅ αn (y) if αn (x) Un = ∅. Since tn → 0, εn → 0 and µ(f ) = 0 one has µ(βn (x)) → 0 for x ∈ Un . Then, modifying the proof of Walters [6, Theorem 8.3], one has hµ (f ) = lim hµ (f, βn ). n→+∞

534

P.-D. Liu

In what follows we estimate hµ (f, βn ). First, m−1



hµ (f, βn ) = lim Hµ βn

f −k βn ≤ Hµ (βn |f −1 βn ). m→+∞

k=1

Let γn = (f −1 βn )|Di for 1 ≤ i ≤ p. Then 
µ[βn (x) (f −1 βn )(x)] −1 dµ(x) − log Hµ (βn |f βn ) = µ[(f −1 βn )(x)] M p
(i)  µ[γn (x)] − log = dµ(x) µ[(f −1 βn )(x)] i=1 Di  p
 µ[βn (x) (f −1 βn )(x)] − log + dµ(x) (i) µ[γn (x)] i=1 Di 
µ[βn (x) (f −1 βn )(x)] dµ(x) − log + µ[(f −1 βn )(x)] M\∪i Di (i)

(2) (3) =: (1) n + n + n .  (1) Claim 2.1. limn→+∞ n = ∪i Di I (x)dµ(x) ≤ Hµ (|f −1 ), where I (x) = − log µ(f −1 )(x) ({x}) and µ(f −1 )(x) is the conditional measure of µ on (f −1 )(x).

To prove this claim we use the following lemma which is a generalization of Man´e [2, Theorem 0.5.4]. Lemma 2.2. Assume that {Pn }n≥1 is a sequence of countable measurable partitions of a Lebesgue space (X, B, µ) and P is another measurable partition of (X, B, µ) such that Pn ≤ P for all n ≥ 1. Let An , A be respectively the σ -algebra generated by Pn , P. Assume that for every A ∈ A and ε > 0 there exists an integer N = N (A, ε) > 0 such that for all n ≥ N there is a set An ∈ An satisfying µ(AAn ) < ε. Then for every bounded B-measurable function g one has as n → +∞, E(g|An ) −→ E(g|A) in measure. Proof. Since

(2.1)

E(E(g|A)|An ) = E(g|An ),

it is sufficient to assume that g is A-measurable. Man´e [2, Theorem 0.5.4] asserts that for every A ∈ A,  µ[A Pn (x)] −→ χA in measure, µ[Pn (x)] and hence (2.1) holds true for the characterization function χA . Put L = {g : g is A-measurable, bounded and such that (2.1) holds}. Clearly 1 ∈ L and c1 g1 + c2 g2 ∈ L for any g1 , g2 ∈ L and c1 , c2 ∈ R. Moreover, if gk ∈ L, 0 ≤ gk  g and g is bounded, L1

then E(g|An ) → g and hence (2.1) holds for g. The canonical methods in measure theory show that L = {g : g is A-measurable and bounded}. This proves the claim.  

Ruelle Inequality

535

Proof of Claim 2.1. Defining φ(x) = −x log x for x > 0 and φ(0) = 0, one has p (i)   µ[γn (x)] (1) −1 µ[(f βn )(x)]φ n = µ[(f −1 βn )(x)] −1 =

i=1 (f p
 i=1

βn )(x)

M

φ(E(χDi |f −1 βn ))dµ(x).

Put Pn = f −1 βn and P = f −1 . Then Pn and P satisfy the requirements of Lemma 2.2, and hence one can choose a subsequence {nk }k≥1 such that for all 1 ≤ i ≤ p, E(χDi |f −1 βnk ) → E(χDi |f −1 ) µ−a.e. as k → +∞. Since φ(x) is bounded on [0, 1], one has (1) nk

→

p
 M

i=1


=

∪i Di


≤

φ(E(χDi |f −1 ))dµ(x)

I (x)dµ(x)

I (x)dµ(x) = Hµ (|f −1 ).

M

This proves the claim since one can start with every subsequence {mk }k≥1 of {n}n≥1 in place of {n}n≥1 .   (2)

We next deal with n and we use the ideas and techniques of Ruelle [4]. First,  (i) p
(i)  µ[γn (x)] µ[βn (x) γn (x)] (2) dµ(x) n ≤ φ  (i) (i) γn (x)] µ[γn (x)] i=1 Di µ[βn (x) p
 ≤ log Nn(i) (x)dµ(x), i=1

Di

 (i) (i) where Nn (x) is the number of elements of βn which intersect γn (x) = (f −1 βn )(x) Di. We fix x0 ∈ f . Let now x ∈ Di \f −1 [βn (x0 )]. Due to the choice of Di ’s, one has  (i) γn (x) ⊂ Cn (x) := (f −1 βn )(x) B(x, δn ) and hence
(2) log N¯ n (x)dµ(x), n ≤ ∪i Di

where N¯ n (x) is the number of elements of βn which intersect Cn (x). Note that Cn (x) ⊂ gx(n) expf x B¯ f x (0, 2εn ) ⊂ expx B¯ x ((Tx f )−1 B¯ f x (0, 2εn ), 2εn ),  where B¯ z (Q, δ) is the closed δ-neighborhood of Q ⊂ Tz M in Tz M. Thus, if βn (y) Cn (x) = ∅ with y ∈ Eεn , one has

ε  n ⊂ expx B¯ x ((Tx f )−1 B¯ f x (0, 2εn ), 2(1 + b0 )εn ) B¯ y, 2

536

P.-D. Liu

and then

 −1 εn B¯ x exp−1 ⊂ B¯ x ((Tx f )−1 B¯ f x (0, 2εn ), 2(1 + b0 )εn ). x y, b0 2 εn ¯ Since B(y, 2 ), y ∈ Eεn are mutually disjoint, one has

N¯ n (x) ≤ C

dim M

max{δ¯i ((Tx f )−1 ), 1} =: C(x),

i=1

where C is a constant depending only on dim M and b0 , and δ¯i (A), 1 ≤ i ≤ d for a d × d matrix A are defined by a decomposition A = Q1 Q2 with Q1 , Q2 being unitary operators and  being a diagonal matrix whose diagonal elements are δ¯i (A).  (i) Consider now Nn (x) for x ∈ Di f −1 [βn (x0 )]. Noting that βn (x0 ) contains at most [Kεn− dim M ] elements of αn with K being a constant depending only on the manifold M and, by the first condition in (1.3), M(tn ) < tn−a when n is sufficiently large, one then has for large n, Nn(i) (x) ≤ Kεn− dim M · C max (y) y∈Di

2

= K(max{20LM(tn )1+ α , −(1+ α2 )a dim M

≤ K  tn

1 dim M }) · C max (y) y∈Di tn

· max (y) y∈Di

with K  being another constant. Thus,
lim sup (2) n ≤ n→+∞

log (x)dµ(x) + log C M −(1+ α2 )a dim M

+ lim sup µ(B(f , 2tn )) log[K  tn n→+∞
= log (x)dµ(x) + log C.

· max (y)] y∈∪i Di

M

¯ µ) Let f¯ : (M, ¯ ← be the natural extension of f : (M, µ) ←. Define A(x) ¯ = (Tx−1 f )−1 dim M ∧ for x¯ = (· · · , x−1 , x0 , x1 , · · · ) ∈ M¯ and |A(x) ¯ | = 1 + i=1 |A(x) ¯ ∧i | (see Ruelle [4] for the related definitions). Then one further has
lim sup (2) n ≤ n→+∞


=

log |((Tx f )−1 )∧ |dµ(x) + log C M M¯

log |A(x) ¯ ∧ |d µ( ¯ x) ¯ + log C.

We that log |A(x) ¯ ∧ | is µ-integrable ¯ since  remark + −1 log |(Tx f ) |dµ(x) < +∞.



log | det Tx f |dµ(x) > −∞ implies

Ruelle Inequality

537 (3)

We now proceed to estimating n . Since ∪i ∂Di has µ-measure 0 and G0 ⊂ ∪i Di , one has M\ ∪i Di ⊂ f −1 V0 µ-mod 0. Thus for large n,   

µ[βn (x) (f −1 βn )(x)] (3) − log dµ(x) ≤ + n µ[(f −1 βn )(x)] βn (x0 ) f −1 [βn (x0 )]∩Gn  
µ[βn (x) (f −1 βn )(x) B(x, δn )]  + dµ(x) − log µ[(f −1 βn )(x) B(x, δn )] Gn \[f −1 [βn (x0 )]∪∪i Di ] 
µ[(f −1 βn )(x) B(x, δn )] + dµ(x) − log µ[(f −1 βn )(x)] Gn \[f −1 [βn (x0 )]∪∪i Di ]   
µ[βn (x0 ) (f −1 βn )(x)] ≤ dµ(x) φ µ[(f −1 βn )(x)] M +µ(f −1 [βn (x0 )]) log[Kεn− dim M · C max (y)] y∈Gn
+  log(N¯ n (x) + 1)dµ(x) 


x∈Gn \∪i Di (f

−1 β )(x) n

B(x,δn )

log(#{f −1 x} + 1)dµ(x)

+ B(V0 ,2εn )

≤ C  + µ(B(f , 2tn )) log[Kεn− dim M · CM(tn )dim M ]

−1 ∧ + log |((Tx f ) ) |dµ(x) + log(#{f −1 x} + 1)dµ(x) 

B(f −1 V0 ,δn )

B(V0 ,2εn )

(C = max φ(x) + log 2 + log C) 0≤x≤1

log |((Tx f )−1 )∧ |dµ(x) + → C + f −1 V0

log(#{f −1 x} + 1)dµ(x). V0

Noting that µ(f f ) = 0 by assumption, V0 can be chosen arbitrarily small and f −1 V0 ⊂ f −1 V0 , we then have (2)  hµ (f ) ≤ lim sup Hµ (βn |f −1 βn ) ≤ lim sup((1) n + n ) + C n→+∞ n→+∞
−1 ≤ Hµ (|f ) + log |A(x) ¯ ∧ |d µ( ¯ x) ¯ + log C + C  . M¯

 f −k  . Define T = Finally we consider f m for m ≥ 1. Clearly f m = m−1 k=0 m−1 f −k maxx∈M {|Tx f |, 1}. For each r > 0 one has B(f m , r) = k=0 B(f f , r) ⊂ m−1 −k B(f , T m r) and hence k=0 f µ(B(f m , r)) log r → 0 as r → 0. When r is small, by the second assumption of (1.3) one can verify that for x ∈ M\B(f m ,r) i k and 0 < k ≤ m − 1, if f i x ∈ B(f , r b ) for i = 0, · · · , k − 1, then f k x ∈ B(f , r b ). Hence max{|(Tx f m )−1 | : x ∈ M\B(f m , r)} ≤ M(r)M(r b ) · · · M(r b ≤r

−a

=r

−a(1+b+···+bm−1 )

·r

−ab

· ··· · r

m−1

−abm−1

)

538

P.-D. Liu m−1

and f m B(x, r) ⊃ B(f m x, r b ) for every x ∈ M\B(f m , r). Clearly µ((f m )n f m ) =  m k 0 for all n ≥ 1 since f m f m ⊂ m k=1 f f . Thus we can replace f by f for any m ≥ 1 in the arguments above, and this yields
−m m log |Am (x) ¯ ∧ |d µ( ¯ x) ¯ + log C + C  , hµ (f ) ≤ Hµ (|f ) + M¯

where Am (x) ¯ := A(f¯−(m−1) x) ¯ ◦ · · · ◦ A(x). ¯ Therefore
1 1 −m Hµ (|f ) + lim log |Am (x) ¯ ∧ |d µ( ¯ x) ¯ hµ (f ) ≤ lim m→+∞ m m→+∞ m M¯
 = Hµ (|f −1 ) − λ(i) (x)− dµ(x), M

i

which completes the proof of Theorem 1.1.

 

Acknowledgement. The author is very grateful to Professor David Ruelle for his comments which lead to a substantial weakening of assumptions for (1.2) imposed in a previous version of the paper.

References 1. Ledrappier, F., Young, L.-S.: The metric entropy of diffeomorphisms. Part I: Characterization of measures satisfying Pesin’s formula. Part II: Relations between entropy, exponents and dimension. Ann. Math. 122, 509–574 (1985) 2. Man´e, R.: Ergodic Theory and Differentiable Dynamics. Berlin-Heidelberg-New York: SpringerVerlag 1987 3. Rokhlin, V.A.: Lectures on the theory of entropy of transformations with invariant measures. Russ. Math. Surveys 22(5), 1–54 (1967) 4. Ruelle, D.: An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Math. 9, 83–87 (1978) 5. Ruelle, D.: Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys. 85(1/2), 1–23 (1996) 6. Walters, P.: An Introduction to Ergodic Theory. New York: Springer, 1982 Communicated by G. Gallavotti

Commun. Math. Phys. 240, 539–551 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0898-1

Communications in

Mathematical Physics

Classification of Harish-Chandra Modules over the Higher Rank Virasoro Algebras
Yucai Su Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030, P.R. China. E-mail: [email protected] Received: 9 April 2001 / Accepted: 27 May 2003 Published online: 25 July 2003 – © Springer-Verlag 2003

Abstract: We classify the Harish-Chandra modules over the higher rank Virasoro and super-Virasoro algebras: It is proved that a Harish-Chandra module, i.e., an irreducible weight module with finite weight multiplicities, over a higher rank Virasoro or superVirasoro algebra is either a module of the intermediate series, or a finitely-dense module. As an application, it is also proved that an indecomposable weight module with finite weight multiplicities over a generalized Witt algebra is either a uniformly bounded module (i.e., a module with weight multiplicities uniformly bounded) with all nonzero weights having the same multiplicity, or a finitely-dense module, as long as the generalized Witt algebra satisfies one minor condition. 1. Introduction The notions of the higher rank Virasoro algebras [13] and the higher rank super-Virasoro algebras [17] appear as natural generalizations of the well-known Virasoro algebra [4] and super-Virasoro algebras (the Neveu-Schwarz superalgebra [12], the Ramond superalgebra [14]). As the universal central extension of the complex Lie algebra of the linear differential operators over the circle (the Witt algebra), the Virasoro algebra Vir, closely related to Kac-Moody algebras [5–7], is of much interest to both mathematicians and physicists, partly due to its relevance to string theory [15] and 2-dimensional conformal field theory [3]. The Virasoro algebra Vir can be defined as a Lie algebra with basis 3 {Li , c | i ∈ Z} such that [Li , Lj ] = (j − i)Li+j + i 12−i δi,−j c, [Li , c] = 0 for all i, j ∈ Z. Let n be a positive integer. Let M be an n-dimensional Z-submodule of C, and let s ∈ C, be such that 2s ∈ M. A rank n Virasoro algebra (or a higher rank Virasoro algebra if n ≥ 2) [13] is a complex Lie algebra Vir [M] with basis {Lµ , c | µ ∈ M}, such that  This work is supported by a NSF grant 10171064 of China and two grants “Excellent Young Teacher Program” and “Trans-Century Training Programme Foundation for the Talents” from Ministry of Education of China.

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[Lµ , Lν ] = (ν − µ)Lµ+ν −

µ3 − µ δµ,−ν c, [Lµ , c] = 0, ∀ µ, ν ∈ M. 12

(1.1)

A rank n super-Virasoro algebra (or a higher rank super-Virasoro algebra if n ≥ 2) [17] is the Lie superalgebra SVir [M, s] = SVir 0 ⊕ SVir 1 , where SVir 0 = Vir [M] has a basis {Lµ , c | µ ∈ M} and SVir 1 has a basis {Gη | η ∈ s + M}, with the commutation relations (1.1) and [Lµ , Gη ] = (η − µ2 )Gµ+η , [Gη , Gλ ] = 2Lη+λ − δη,−λ 31 (η2 − 41 )c, [Gη , c] = 0,

∀ µ ∈ M, η, λ ∈ s + M.

The irreducible representations of the Virasoro and super-Virasoro algebras, which play very important roles in the theory of vertex operator (super)algebras and mathematical physics, are well developed (see for example [2, 6–9, 16, 18]). However for the higher rank case, not much has been known except modules of the intermediate series [17, 19, 20], or Verma-like modules over the centerless higher rank Virasoro algebras [10]. A weight module is a module V with weight space decomposition: V = ⊕ Vλ , Vλ = {v ∈ V | L0 v = λ0 v, cv = hv}, where λ = (λ0 , h) ∈ C2 . λ∈C2

A weight module is quasi-finite if all weight spaces Vλ are finite dimensional. A module of the intermediate series is an indecomposable quasi-finite weight module with all the dimensions of weight spaces (and in the super case all the dimensions of weight spaces of “even” or “odd” part) are ≤ 1. It is proved [17] that a module of the intermediate series over a higher rank Virasoro algebra Vir [M] is Aa,b , Aa  , Ba  or one of their quotients for suitable a, b, a  ∈ C, where Aa,b , Aa  , Ba  all have a basis {xµ | µ ∈ M} such that c acts trivially and Aa,b : Lµ xν = (a + ν + µb)xµ+ν , Aa  : Lµ xν = (ν + µ)xµ+ν , ν = 0, Lµ x0 = µ(µ + a  )xµ , Ba  : Lµ xν = νxµ+ν , ν = −µ, Lµ x−µ = −µ(µ + a  )x0 ,

(1.2a) (1.2b) (1.2c)

for µ, ν ∈ M; and that a module of the intermediate series over the higher rank superVirasoro algebras SVir [M, s] is one of the three series of the modules SAa,b , SAa  , SB a  or their quotient modules for suitable a, b, a  ∈ C, where SAa,b , SAa  have basis {xµ | µ ∈ M} ∪ {yη | η ∈ s + M} and SB a  has basis {xη | η ∈ s + M} ∪ {yµ | µ ∈ M} such that c acts trivially and SAa,b : Lµ xν = (a + ν + µb)xµ+ν , Lµ yη = (a + η + µ(b − 21 ))yµ+η , Gλ xν = yλ+ν ,

Gλ yη = (a + η + 2λ(b − 21 ))xλ+η ,

SAa  : Lµ xν = (ν + µ)xµ+ν , ν = 0, Lµ x0 = µ(µ + a  )xµ , Lµ yη = (η + µ2 )yµ+η , Gλ xν = yλ+ν , ν = 0, Gλ x0 = (2λ + a  )yλ , Gλ yη = (η + λ)xλ+η , SB a  : Lµ xη = (η + µ2 )xµ+η , Lµ yν = νyµ+ν , ν = −µ, Lµ y−µ = −µ(µ + a  )y0 , Gλ xη = yλ+η , η = −λ, Gλ x−λ = (2λ + a  )y0 , Gλ yν = νxλ+ν , for µ, ν ∈ M, λ, η ∈ s + M. A uniformly bounded module is a quasi-finite weight module with all weight multiplicities being uniformly bounded. It is proved [19, 20] that a uniformly bounded irreducible

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module is simply a module of the intermediate series. An irreducible quasi-finite weight module is called a Harish-Chandra module. It is a deep fact (first conjectured in [6], then proved in [9] and partially proved in [2, 8, 16]) that a Harish-Chandra module over the Virasoro algebra is either a module of the intermediate series or else a highest or lowest weight module. The same is true for the super-Virasoro algebras [18]. Unlike the Virasoro or super-Virasoro case, a nontrivial highest or lowest weight module, or more generally, a nontrivial Verma-like module [10] for a higher rank Virasoro or super-Virasoro algebra is not a quasi-finite weight module. The classification of Harish-Chandra modules is definitely an important problem in the representation theory of Lie algebras. Since no Harish-Chandra modules other than modules of the intermediate series have been found for the higher rank Virasoro or super-Virasoro algebras, a natural question is: Does there exist any Harish-Chandra module other than modules of the intermediate series over the higher rank Virasoro or super-Virasoro algebras? The aim of this paper is to answer this question. Mainly we have the following theorem (for the Q-Virasoro algebra, this theorem was obtained in [11]). Theorem 1.1. A Harish-Chandra module over a higher rank Virasoro or super-Virasoro algebra is either a module of the intermediate series, or a finitely-dense module (cf. (2.10)). Let m be a positive integer. Let  be a nondegenerate additive subgroup of Cm , that is,  contains a C-basis of Cm . Let πi : Cm → C be the natural projection from an element of Cm to its i th coordinate, i = 1, 2, ..., m. Let A = C[] be the group algebra of  with basis {x α | α ∈ }. Define the derivations ∂i of A such that ∂i (x α ) = πi (α)x α . Set D = span{∂i | i = 1, 2, ..., m}. Then the tensor space A ⊗ D can be defined as a Lie algebra under the bracket [x α ∂i , x β ∂j ] = x α+β (πi (β)∂j − πj (α)∂i ), called a generalized Witt algebra, denoted by W (m, ) (see for example [23]). Note that a generalized Witt algebra defined in [10] is simply a Lie algebra W (1, ) with  ⊂ C being a free Z-module of finite rank, i.e., it is a centerless higher rank Virasoro algebra using our notion (1.1)). Observe that a generalized Witt algebra W (m, ) has a (unique) nontrivial central extension if and only if m = 1, in this case the universal central extension of
(1, ). W (1, ) is referred to as a generalized Virasoro algebra in [22], denoted by W As an application of Theorem 1.1, we obtain Theorem 1.2. Suppose there is a group injection Z × Z →  (in particular, if m ≥ 2 there always exists a group injection Z × Z →  by nondegenerateness of ). Then
(1, ) is either a an indecomposable quasi-finite weight module over W (m, ) or W uniformly bounded module with all nonzero weights having the same multiplicity, or a finitely-dense module. We believe that Theorem 1.2 will certainly be important to the classification problem of the Harish-Chandra modules over the generalized Witt algebras (especially the ± ], the derivation algebras of the Laurent classical Witt algebras Wn = Der C[t1± , ..., tm polynomial algebras of m variables, which is closely related to toroidal Lie algebras, cf. [1]). This is also one of our motivations to present results here. The paper is organized as follows. In Sect. 2, after collecting some necessary information on representations of the higher rank Virasoro algebras, we give the proof of Theorem 1.1 for the case n = 2. Then we complete the proofs of Theorems 1.1 and 1.2 in Sect. 3.

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2. Preliminaries and the Case n = 2 First observe that by regarding a module over a higher rank super-Virasoro algebra as a module over a higher rank Virasoro algebra, using Theorem 1.2, one obtains that a not-finitely-dense Harish-Chandra module over a higher rank super-Virasoro algebra is uniformly bounded, and thus using results in [20], it is a module of the intermediate series. Thus in the following, we shall only consider the “unsuper” case. So, let Vir [M] be a higher rank Virasoro algebra, where M is an n-dimensional Z-submodule of C with n ≥ 2. Following [19], a Vir [M]-module V is called a GHW module (here GHW stands for “generalized highest weight”) if V is generated by a nonzero weight vector v and there exists a Z-basis B = {b1 , ..., bn } of M such that Lµ v = 0 for all 0 = µ =

n 

mi bi ∈ M with all coefficients mi ∈ Z+ .

(2.1)

i=1

The vector v is called a GHW vector with respect to Z-basis B or simply a GHW vector. Highest or lowest weight modules, no matter how the ordering on M is defined [11, 13], are GHW modules. The following three lemmas are taken from [19, 21, 9] respectively. Lemma 2.1. (1) A uniformly bounded Harish-Chandra Vir [M]-module is a module of the intermediate series. (2) A Harish-Chandra Vir [M]-module is either a module of the intermediate series or a GHW module. Lemma 2.2. Let W be a quasi-finite weight Vir-module such that λ = (λ0 , h) is a weight. For any i ∈ Z, there exist only a finite number of primitive vectors with weights λ + j = (λ0 + j, h) such that j ≥ i (a primitive vector is a nonzero weight vector v such that Li v = 0 for i ≥ 1). Lemma 2.3. Let W be a quasi-finite weight Vir-module. Suppose 0 = v ∈ W has weight λ such that Li v = 0 for i >> 0. Then there exists a primitive vector with weight λ + j for some j ≥ 0. By Lemma 2.1, the proof of Theorem 1.1 is equivalent to proving that a nontrivial not-finitely-dense quasi-finite irreducible GHW module does not exist. In this section, we shall prove Theorem 1.1 for the special case n = 2. Thus suppose V is a nontrivial not-finitely-dense quasi-finite irreducible GHW Vir [M]module generated by a GHW vector v with weight  = (0 , h) ∈ C2 , where M = Zb1 + Zb2 such that b1 , b2 ∈ C are Z-linear independent and Li,j v = 0 for all (i, j ) ∈ Z2 with 0 = (i, j ) ≥ 0, where Li,j = Lib1 +j b2 .

(2.2)

Here in general, we define (i, j ) ≥ (k, l) if i ≥ k, j ≥ l and define (i, j ) > (k, l) if i > k, j > l. For m ∈ M, S ⊂ M, we denote by Vir [M] or Vir [S] the subalgebra of Vir [M] generated by {L±m , L±2m } or {L±m , L±2m | m ∈ S} respectively. In particular, if m ∈ M\{0}, then Vir [M] is a rank one Virasoro subalgebra isomorphic to Vir. If ∅ = S ⊂ M\{0}, then Vir [S] is a rank one or rank two Virasoro subalgebra of Vir [M]. For any subalgebra L of Vir [M], we denote by U (L) the universal enveloping algebra of L. Since for an irreducible weight module, the central element c must act as a scalar h for some fixed h ∈ C, we shall always omit h and simply denote a weight λ = (λ0 , h)

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by λ = λ0 ∈ C. Let P be the set of weights of V . Then P ⊂  + Zb1 + Zb2 . Thus when b1 , b2 are fixed, we can define an injection φb1 ,b2 : P → Z2 such that φb1 ,b2 ( + ib1 + j b2 ) = (i, j ).

(2.3)

We extend φb1 ,b2 to φb1 ,b2 :  + Zb1 + Zb2 → Z2 . To better understand the following discussion, one can regard Z2 as the set of integral points in the Oxy-plane. For m ∈ Z, we denote (−∞, m] = {i ∈ Z | i ≤ m} and [m, ∞) = {i ∈ Z | i ≥ m}. Lemma 2.4. For any v ∈ V , there exists p >> 0 such that Li,j v = 0 for all (i, j ) ≥ (p, p). Proof. Since v = uv for some u ∈ U (Vir [M]), then u is a linear combination of elements of the form u = Li1 ,j 1 · · · Lin ,jn . Thus  without loss of generality, we can suppose u = u . Take p = max{− iq <0 iq , − jq <0 jq } + 1. Using relation [u, u1 u2 ] = [u, u1 ]u2 +u1 [u, u2 ] for u ∈ Vir [M] and u1 , u2 ∈ U (Vir [M]), and using relations (1.1) and by (2.2), we obtain Li,j v = [Li,j , u]v = 0 for all (i, j ) ≥ (p, p) by induction on n.   For any p ∈ Z, it is easy to see that B  = {b1∗ = (p + 1)b1 + (p + 2)b2 , b2∗ = pb1 + (p + 1)b2 },

(2.4)

is also a Z-basis of M. Thus by Lemma 2.4 and definition (2.1), by choosing p >> 0, one sees that every nonzero weight vector is a GHW vector with respect to the Z-basis B . Lemma 2.5. Let Vir [S] be any rank two Virasoro subalgebra of Vir [M] and let b1∗ , b2∗ be any Z-basis of M. Then Vir [M] is generated by Vir [S] ∪ {Lb1∗ , Lb2∗ }. Proof. Since Vir [S] is a rank two Virasoro subalgebra of Vir [M], there exists (i, j ) > 0 such that L−ib1∗ −j b2∗ ∈ Vir [S]. Clearly L−b1∗ , L−b2∗ can be generated by L−ib1∗ −j b2∗ , Lb1∗ , Lb2∗ . Since Vir [M] is generated by L±b1∗ , L±b2∗ , it is also generated by Vir [S]∪{Lb1∗ , Lb2∗ }.   Lemma 2.6. Let Vir [S] be a rank two Virasoro subalgebra of Vir [M]. Then any nonzero Vir [S]-submodule of V is nontrivial. Proof. Suppose Cv = {0} is a trivial Vir [S]-submodule. Choose p >> 0 satisfying Lemma 2.4 and let b1∗ , b2∗ be as in (2.4). Then Vir [S] ∪ {Lb1∗ , Lb2∗ } acts trivially on Cv. Since V is irreducible, by Lemma 2.5, V = Cv is a trivial Vir [M]-module, a contradiction with the assumption that V is nontrivial.   Lemma 2.7. For any v ∈ V \{0} and any (i, j ) ∈ Z2 with (i, j ) > 0, we have L−i,−j v = 0. Proof. Suppose L−i,−j v = 0 for some (i, j ) > 0. Let p be as in Lemma 2.4, then L−i,−j , Lpi+1,pj , Lpi,pj +1 act trivially on v, but Vir [M] is generated by these three elements, a contradiction with that V is a nontrivial irreducible module.   Lemma 2.8. Let (i, j ) ∈ φb1 ,b2 (P ) and let (k, l) > 0. Then {x ∈ Z | (i, j ) + x(k, l) ∈ φb1 ,b2 (P )} = (−∞, m] for some m ≥ 0.

(2.5)

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Proof. Denote by I the set in the left-hand side of (2.5). By Lemma 2.7, if x1 ∈ I , then (−∞, x1 ] ⊂ I . Suppose I = Z. For any x2 ∈ Z, fix a weight vector vx2 of weight φb−1 ((i, j ) + x2 (k, l)). By Lemma 2.4, when x >> 0, Lx vx2 = 0, where 1 ,b2 Lm = Lmk,ml , m ∈ Z span a rank one Virasoro subalgebra Vir [kb1 + lb2 ]. Thus by ((i, j ) + x3 (k, l)) is a Vir [kb1 + lb2 ]Lemma 2.3, there exists x3 ≥ x2 such that φb−1 1 ,b2 primitive weight. This contradicts Lemma 2.2.   By Lemma 2.8, we can suppose {x ∈ Z | x(1, 1) ∈ φb1 ,b2 (P )} = (−∞, p − 2] for some p ≥ 2. Then (p − 1, p − 1) ∈ / φb1 ,b2 (P ) and Lemma 2.8 again shows that (i, j ) ∈ / φb1 ,b2 (P ) for all (i, j ) ≥ (p, p) which is > (p − 1, p − 1). Taking b1∗ , b2∗ to be as in (2.4), by (2.3), we have ∗ ∗ φb−1 ∗ ,b∗ (i, j ) =  + ib1 + j b2 1

2

=  + (i(p + 1) + jp)b1 + (i(p + 2) + j (p + 1))b2 = φb−1 (i(p + 1) + jp, i(p + 2) + j (p + 1)) for (i, j ) ∈ Z2 . 1 ,b2 Thus by Lemmas 2.7 and 2.8, for (i, j ) ∈ Z2 , we have (i, j ) ∈ / φb1∗ ,b2∗ (P ) if 0 = (i, j ) ≥ 0, and (i, j ) ∈ φb1∗ ,b2∗ (P ) if (i, j ) ≤ 0, and (2.6) / φb1∗ ,b2∗ (P ), and (k, l) ∈ / φb1∗ ,b2∗ (P ) for all (k, l) ≥ (i, j ) if (i, j ) ∈ (k, l) ∈ φb1∗ ,b2∗ (P ) for all (k, l) ≤ (i, j ) if (i, j ) ∈ φb1∗ ,b2∗ (P ),

(2.7)

and for any (i, j ) ∈ φb1∗ ,b2∗ (P ) and any (k, l) ≥ 0, {x ∈ Z | (i, j ) + x(k, l) ∈ φb1∗ ,b2∗ (P )} = (−∞, m] for some m ≥ 0.

(2.8)

In the following, for convenience, we redenote b1∗ , b2∗ by b1 , b2 respectively and from now on we simply denote φb1 ,b2 by φ. The following rather technical lemma is crucial in obtaining our classification. Lemma 2.9. For any (i, j ), (k, l) ∈ Z2 , there exists p ∈ Z such that {x ∈ Z | (i, j ) + x(k, l) ∈ φ(P )} = (−∞, p] or [p, ∞). Proof. Let I = {x ∈ Z | (i, j ) + x(k, l) ∈ φ(P )}. Suppose (k, l) = d(k1 , l1 ) for some d, k1 , l1 ∈ Z. Clearly if the result holds for (k1 , l1 ), then it holds for (k, l) too. Thus by replacing (k, l) by (k1 , l1 ) we can suppose k, l are coprime and k ≤ 0. Suppose I = (−∞, p] and I = [p, ∞) for any p. Then by (2.8) we must have k < 0 < l. Case 1. I = ∅. Since (0, 0) ∈ φ(P ), we obtain (i, j ) ∈ / Z(k, l). Furthermore, since k, l are coprime, there do not exist p, q ∈ Z\{0} such that p(k, l) + q(i, j ) = 0, i.e., (i, j ), (k, l) are Z-linear independent. So Vir [b1 , b2 ] is a rank two Virasoro subalgebra of Vir [M], where b1 = kb1 + lb2 , b2 = ib1 + j b2 . Then I = ∅ means that  + nb1 + b2 is not a weight for any n ∈ Z. Let W = ⊕λ∈+Zb1 Vλ , which is a Vir [b1 ]-submodule of V .

(2.9)

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Claim 1. W is a uniformly bounded Vir [b1 ]-module. Let 0 = w ∈ V+mb1 for some m ∈ Z. If L−mb1 −b2 w = 0, then Cw is a trivial Vir [b1 , b2 ]-submodule since Vir [b1 , b2 ] can be generated by the set {L−mb1 −b2 , Lnb1 +b2 | n ∈ Z} which acts trivially on w by (2.9). This is contrary to Lemma 2.6. Thus we obtain a linear injection L−mb1 −b2 |V+mb : V+mb1 → V−b2 . Thus dim V+mb1 ≤ 1 dim V−b2 for all m ∈ Z, i.e., W is uniformly bounded and the claim is proved. Since a uniformly bounded module has only a finite number of composition factors, we can take an irreducible Vir [b1 ]-submodule W  of W . Let W  be the Vir [b1 , b2 ]submodule of V generated by W  . Then W  is a quotient module of a finitely-dense generalized Verma module [10], thus V is a finitely-dense module in the following sense. So this case does not occur. Definition 2.10. A module V over a Lie algebra L is a finitely-dense module if there exist a rank 2 Virasoro subalgebra L of L and a L -submodule V  of V such that V  is a quotient of a finitely-dense generalized Verma module. A module V over a rank 2 Virasoro algebra Vir [M] is a finitely-dense generalized Verma module if there exist a Z-basis {b1 , b2 } of M and an intermediate series Vir [b1 ]-submodule V  of V such that V = IndVir [M]

Vir [b1 ]⊕Vir+ [M]

V

= U (Vir [M]) ⊗U (Vir [b1 ]⊕Vir+ [M]) V  ∼ = U (Vir− [M]) ⊗C V  ,

(2.10)

where Vir ± [M] = span{Lmb1 +nb2 | m ∈ Z, ±n ∈ Z+ \{0}}. Case 2. There exist x1 , x2 , x3 ∈ Z, −x1 < x2 < x3 such that −x1 , x3 ∈ I but x2 ∈ / I, i.e., / φ(P ), (i, j ) + x2 (k, l) ∈ φ(P ), (i, j ) + x3 (k, l) ∈ / φ(P ). (2.11) (i, j ) − x1 (k, l) ∈ Replacing x2 by the largest x < x3 such that (i, j ) + x(k, l) ∈ φ(P ), and then replacing x3 by x2 + 1 and (i, j ) by (i, j ) + x2 (k, l), we can suppose −x1 < x2 = 0, x3 = 1. Choose a nonzero weight vector vλ with weight λ = φ −1 (i, j ). Then (2.11) means that Lk,l vλ = 0 = L−x1 k,−x1 l vλ , thus also L−k,−l vλ = 0. Choose p, q >> 0 such that Lp,q vλ = 0 (cf. Lemma 2.4). Then S = {b1 = kb1 + lb2 , b2 = pb1 + qb2 } is a Z-linear independent subset of M since kq − lp < 0 (note that k < 0 < l). Note that for any mb1 + nb2 ∈ M, if n > 0, then Lmb1 +nb2 can be generated by L±b1 , Lb2 . Thus Lmb1 +nb2 vλ = 0 if n > 0. For any a ∈ U (Vir [S]) with weight mb1 + nb2 such that n > 0, we can write a as a linear combination of the form Li1 b1 +j1 b2 · · · Lis b1 +js b2 with js > 0. Thus avλ = 0. Let V  = U (Vir [S])vλ , this shows that λ + mb1 + nb2 is not a weight of V  for any n > 0. Now as discussed in Case 1, we obtain that V  is not quasi-finite. Thus this case does not occur either. Case 3. I = (−∞, p]∪[q, ∞) for some p, q ∈ Z. Since k, l are coprime and k < 0 < l, we can choose k  , l  such that kl  − lk  = −1

and

k < 0 < l.

(2.12)

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Then

{b1 = kb1 + lb2 , b2 = k  b1 + l  b2 },

forms a Z-basis of M. Then b1 , b2 are shown as in Fig. 1(i).

(k, l) A K (i)

(k  , l  )

A b YH H b1 H Ao 2

   b1  : (i, j ) + r(k, l) + (t + 1)(k , l )  (i, j ) + x(k, l)  9
AK Y H H K tb
    H A  b2 A (t + 1)b2

9 

HA 2 AK  sH H
tb2 A A 
H (ii) s −H 1 A HA
/   + x(k1 b1 + l1 b2 ) rH  
H   + xb1

 + xb2

q

pH

H

Figure 1

Assume that there exists t > 0 such that (i, j ) + r(k, l) + (t + 1)(k  , l  ) ∈ / φ(P ) for all r ∈ I . This means that {x ∈ Z | (i  , j  )+x(k, l) ∈ φ(P )} ⊂ Z\I

for

(i  , j  ) = (i +(t +1)k  , j +(t +1)l  ).

But Z\I is either empty or a finite set {p + 1, ..., q − 1}. This falls to Case 1 or Case 2, which is impossible. Thus, we obtain that for any t > 0, there always exists r ∈ I such that (i, j ) + r(k, l) + (t + 1)(k  , l  ) ∈ φ(P ).

(2.13)

Fix x ∈ Z such that x >> 0, and let s = (k  − l  )x, t = (l − k)x. Since k < 0 < l, k  < 0 < l  , we have t >> 0, s << 0. In particular, we can assume s < p, so s ∈ I . Since (m, n) ∈ / φ(P ) for all (m, n) > 0, and by (2.12) we have s(k, l) + t (k  , l  ) = x(1, 1) >> 0, thus we can assume (i, j ) + s(k, l) + t (k  , l  ), (i, j ) + (s − 1)(k, l) + t (k  , l  ) ∈ / φ(P ).

(2.14)

This proves that we can choose r, s ∈ I, t ≥ 0 such that (2.13), (2.14) hold. Say r < s −1 (otherwise the proof is similar). Now take b1 = (s − r)b1 − b2 , b2 = (s − r − 1)b1 − b2 . Then {b1 , b2 } is still a Z-basis of M (cf. Fig. 1(ii)). By (2.13), we can choose a nonzero weight vector v with the weight  = φ −1 ((i, j ) + r(k, l) + (t + 1)(k  , l  )). Then by (2.14), Lb1 v = Lb2 v = 0, and thus Lmb1 +nb2 v = 0 for all (m, n) > 0. Using this, one sees that v is a GHW vector with respect to the Z-basis {b1 + b2 , b1 + 2b2 } of M; thus as in the proofs in Lemmas 2.4, 2.8, for any v ∈ V , we have Lmb1 +nb2 v = 0 for all (m, n) >> 0, and for any (k1 , l1 ) > 0, we have  + x(k1 b1 + l1 b2 ) is not a weight for x >> 0.

(2.15)

 From Fig. 1(ii), one can easily see that when x >> 1 b1 +  l1 b2 ) corresponds to a point located below and to the left of some point (i, j ) + y(k, l) for some y ∈ I , which by (2.7) is in φ(P ), contrary to (2.15). Thus this case does not occur either. This completes the proof of Lemma 2.9.  

0, under mapping (2.3),  +x(k

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Proof of Theorem 1.1 for the case n = 2. For any i ∈ Z+ \{0}, since (−i, −j ) ∈ φ(P ) for all j ∈ Z+ (cf. Lemma 2.7), by Lemma 2.9, we can let yi = max{y ∈ Z | (−i, y) ∈ φ(P )}, xi = max{x ∈ Z | (x, −i) ∈ φ(P )}.

(2.16)

By (2.7), we have yi+1 ≥ yi ≥ 0, xi+1 ≥ xi ≥ 0. For y ∈ Z, if y ≤ yi then since (0, 0), (−i, y) ∈ φ(P ), by Lemma 2.9, all integral points lying between (0, 0) and (−i, y) and lying on the line linking them must be in φ(P ), i.e., Z2 ∩ {x(−i, y) | x ∈ Q, 0 ≤ x ≤ 1} ⊂ φ(P ) for i ∈ Z+ \{0}, y ∈ Z+ , y ≤ yi . (2.17) Let j, t ∈ Z+ \{0}. If ytj ≥ t (yj + 1), then t (−j, yj + 1) = (−tj, t (yj + 1)) ∈ φ(P ), and by (2.17) it gives (−j, yj + 1) ∈ φ(P ), contrary to definition (2.16). This means ytj < t (yj + 1) for all t, j ∈ Z+ \{0}.

(2.18)

Since (0, 1) ∈ / φ(P ) (cf. (2.6)), but (−j, yj ) ∈ φ(P ), by Lemma 2.9, we also have Z2 ∩ {(0, 1) + x(−j, yj − 1) | x ∈ Q, x ≥ 1} ⊂ φ(P ) for j ∈ Z+ \{0}.

(2.19)

In particular, taking x = t ∈ Z+ \{0}, it implies (−tj, t (yj − 1) + 1) ∈ φ(P ), i.e., ytj ≥ t (yj − 1) + 1 for t, j ∈ Z+ \{0}.

(2.20)

Using (2.18), (2.20), we obtain j (yi −1)+1 ≤ yij < i(yj +1) for i, j ∈ Z+ \{0}. From this we deduce yj yj i+j −1 yi i+j −1 − < < + for all i, j ∈ Z+ \{0}. j ij i j ij This shows that the following limits exist: α = lim

i→∞

yi xi , β = lim , i→∞ i i

(2.21)

where the second equation is obtained by symmetry. Since (0, 1) ∈ / φ(P ) and (−j, yj ) ∈ φ(P ), we have (−xj, x(yj −1)+1) = (0, 1)+x(−j, yj −1) ∈ / φ(P ) for all x ∈ (−∞, 0] (cf. (2.19)). In particular, taking x = −t and by definition (2.16), it means xt (yj −1)−1 < tj for all t, j ∈ Z+ \{0} such that t (yj − 1) − 1 ≥ 1.

(2.22)

Note that Lemma 2.9 means that for any i ∈ Z+ \{0} there exists j ∈ Z+ \{0} such that (−j, i) ∈ φ(P ), i.e., limj →∞ yj = ∞. Dividing (2.20) by tj and taking lim t→∞ , we obtain α > 0. Dividing (2.22) by t (yj − 1) − 1 and taking limj →∞ , we obtain β ≤ α −1 . For j ∈ Z+ \{0}, we denote Ij = {x ∈ Q | x ≥ 0, −(1, 1) + x((−j, yj ) + (1, 1)) ∈ φ(P )}. Taking x = j + 1, we have −(1, 1) + (j + 1)((−j, yj ) + (1, 1)) = (−j 2 , −1 + (j + 1)(yj + 1)) ∈ / φ(P ),

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because −1 + (j + 1)(yj + 1) ≥ j (yj + 1) > yj 2 by (2.18). Thus j + 1 ∈ / Ij . So Ij is a finite set, and then Lemma 2.9 implies Z2 ∩ {−(1, 1) + x((−j, yj ) + (1, 1)) | x ∈ Q, x < 0} ⊂ φ(P ). In particular, taking x = −t ∈ Z, it implies (t (j − 1) − 1, −(t (yj + 1) + 1) ∈ φ(P ), i.e., xt (yj +1)+1 ≥ t (j − 1) − 1 for t, j ∈ Z+ \{0}.

(2.23)

Dividing (2.23) by t (yj + 1) + 1 and taking limj →∞ , it gives β ≥ α −1 . Thus β = α −1 . Assume that α = pq is a rational number, where p, q ∈ Z+ \{0} are coprime. By Lemma 2.9, there exists some m ∈ Z such that (mp, −mq −1) = (0, −1)+m(p, −q) ∈ / φ(P ). Say, m > 0. Then again by Lemma 2.9, (0, 0) + i(−mp, mq + 1) ∈ φ(P ) for all i ∈ [0, ∞) since (0, 0) ∈ φ(P ). But taking i >> 0, we have yimp < i(mq + 1) because yimp limi→∞ imp = α = pq . This contradicts the definition of yimp in (2.16). Thus α is not a rational number. We define a well order >α on Z2 as follows: (i, j ) >α (k, l) ⇔ (i − k, j − l) >α (0, 0) and (i, j ) >α (0, 0) ⇔ j > −iα, i.e., (i, j ) >α (0, 0) if it is located above the line {x(−1, α) | x ∈ R} on the Oxy-plane. First we claim Claim 2. For any ∈ R+ \{0}, there exist p, q ∈ Z+ or p, q ∈ Z− such that 0 < q−pα < . It suffices to prove by induction on n that there exist pn , qn ∈ Z such that (such pn , qn must have the same sign)  n 1 |pn α − qn | < for n = 1, 2, . . . . (2.24) 2 Clearly, we can take p1 = 1 and choose q1 ∈ Z to satisfy |α −q1 | < 21 . Suppose (2.24) holds for n. Let αn = |pn α − qn |. Choose rn ∈ Z such that |rn αn − 1| < 21 αn . Let  n+1 . This pn+1 = rn pn , qn+1 = rn qn + 1, then we have |pn+1 α − qn+1 | < 21 αn < 21 proves the claim. Let + = {(i, j ) ∈ Z2 | (i, j ) >α (0, 0)}. Then + is a standard Borel subset of Z2 in the sense [10] that if (i, j ), (k, l) ∈ + such that (i, j ) >α (k, l), then there exists n ∈ Z+ such that n(k, l) >α (i, j ). First assume that + ∩ φ(P ) = ∅, that is, there is no weight located above the line {x(−1, α) | x ∈ R}. Then it can be verified that V is a Verma module defined in [10], thus is not quasi-finite. To be self-contained, we simply prove its non-quasi-finiteness as follows: For any n > 0, using Claim 2, we can choose (pi , qi ) ∈ Z2 such that (0, −1) < α (pi , qi ) < α (0, 0) for i = 1, 2, ..., n. Denote µi = −(pi , qi ), νi = −(0, −1) − µi >α (0, 0). Recall notation (2.2) that Lµ means Lib1 +j b2 if µ = (i, j ), we claim that vi = L−νi L−µi v , i = 1, 2, ..., n, all having the weight φ −1 (0, −1), are linear independent: suppose n 

ci L−νi L−µi v = 0 for some ci ∈ C.

(2.25)

i=1

Say, with respect to the ordering >α , ν1 is the largest element among all µi , νi with ci = 0. Claim 2 shows that one can choose γ ∈ Z2 such that all µi , νi < α γ except

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that γ < α ν1 . Applying Lγ to (2.25), using that + ∩ φ(P ) = ∅, we obtain an equation which has the form c1 Lγ −ν1 L−µ1 v + c Lγ −ν1 −µ1 v = 0 for some c ∈ C.

(2.26)

By applying Lν1 +µ1 −γ and Lµ1 Lν1 −γ to (2.26), we obtain v = 0, a contradiction. Next assume that there exists (i, j ) ∈ + ∩ φ(P ). Then for any (k, l) < α (i, j ), we must have (k, l) ∈ φ(P ), otherwise by Lemma 2.9 we would have (t (i − k) + i, t (j − l) + j ) = (i, j ) + t (i − k, j − l) ∈ φ(P ) for all t ≥ 0, which gives the following contradiction: when t >> 0, either (t (i − k) + i, t (j − l) + j ) ≥ 0, or t (i − k) + i < 0 but yt (k−i)−i < t (j − l) + j by definition of α in (2.21), or t (j − l) + j < 0 but xt (l−j )−j < t (i − k) + i by definition of β. We have i < 0 < j or i > 0 > j by (2.6). Say i < 0, and rewrite (i, j ) as (−i, j ). For a given n > 0, let = n1 (j − iα) > 0. Choose p, q to be as in Claim 2. Then we obtain (0, 0) < α (−p, q) and (−np, nq) < α (−i, j ). Note that since q > αp, when m >> 0 we have ymp < mq if p, q > 0 or x−mq < −mp if p, q < 0, i.e., (−mp, mq) ∈ / φ(P ). So we can let m ≥ n be the largest integer such that (−mp, mq) ∈ φ(P ). Let vλ be a nonzero weight vector with weight λ = φ −1 (−mp, mq), then vλ generates a nontrivial highest weight Vir [−pb1 + qb2 ]-submodule V  of V , where Vir [−pb1 + qb2 ] = span{Li = L−ip,iq | i ∈ Z} (if V  is trivial, then it is a trivial module over a rank two Virasoro algebra Vir [−pb1 + qb2 , kb1 + kb2 ] which is generated by {L±(−pb1 +qb2 ) , Lkb1 +kb2 } for some k >> 0, contradicting Lemma 2.6), such that  = φ −1 (0, 0) = λ − m(−pb1 + qb2 ) is a weight with weight multiplicities at least m by the well-known property of a nontrivial highest weight module over the Virasoro algebra. Since m ≥ n and n is arbitrary, we obtain that the weight multiplicity of  is infinite and so V is not quasi-finite. This proves that a nontrivial irreducible GHW Vir [M]-module does not exist and thus we obtain Theorem 1.1 for the special case n = 2.   3. Proofs of the Theorems in General Case Now we can complete the proofs of Theorems 1.1, 1.2 as follows. Proof of Theorem 1.1. Let V be a not-finitely-dense Harish-Chandra module over the higher rank Virasoro algebra Vir [M], where M has rank n ≥ 2. Suppose V is not a module of the intermediate series. Then by Lemma 2.1(2), there exists a GHW vector v and some Z-basis B = {b1 , ..., bn } of M such that (2.1) holds. Assume that V is nontrivial, then there exists at least some i, 1 ≤ i ≤ n such that L−bi v = 0. Choose any j = i, then v generates a nontrivial GHW Vir [bi , bj ]-submodule V  of V such that v is a nontrivial GHW vector. Therefore at least one of composition factors of the Vir [bi , bj ]module V  is a nontrivial irreducible GHW Vir [bi , bj ]-module, contradicting the result we obtained for the case n = 2.   Proof of Theorem 1.2. Suppose V is a not-finitely-dense indecomposable quasi-finite
(1, ), where m ≥ 2 or there exists a group weight module over W (m, ) or over W injection Z × Z → , i.e., there exist Z-linear independent elements α1 , α2 of . Below
(1, ) is the central extension version we shall only consider the case W (m, ) as W of W (1, ). Using notation as in the Introduction, W (m, ) = span{x α ∂i | α ∈ , i = 1, 2, ..., m} such that [x α ∂i , x β ∂j ] = x α+β (πi (β)∂j − πj (α)∂i ) for α, β ∈ , i, j = 1, 2, ..., m,

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where πi : Cm → C is the natural projection and  ⊂ Cm is a nondegenerate subgroup. Let D = span{∂i | i = 1, 2, ..., m} be the Cartan subalgebra of W (m, ). Then V has the weight space decomposition: V = ⊕ Vλ , Vλ = {v ∈ V | ∂ v = λ(∂)v for ∂ ∈ D}, λ∈D ∗

such that all Vλ are finite dimensional, where D∗ is the dual space of D. We regard  as an additive subgroup of D∗ by defining α(∂i ) = πi (α) for α ∈ , i = 1, 2, ..., m. For any µ ∈ D∗ , we let V [µ] = ⊕λ∈µ+ Vλ . Then V [µ] is a submodule of V such that V is a direct sum of different V [µ]. Thus V = V [µ] for some µ ∈ D∗ . If 0 is a weight of V , we take µ = 0. Suppose λ is a weight. Define a linear transformation Tλ : Vλ → V  = Vµ + Vµ+α1 + Vµ+α2 as follows: If λ = µ, we take Tµ = 1Vµ to be the embedding. Suppose λ = µ. Then also λ = 0. Let α = µ − λ ∈ . We choose β = α + α1 or α + α2 such that α, β are Z-linear independent. From linear algebra, there exists ∂ ∈ D with b1 = α(∂), b2 = β(∂) ∈ C being Z-linear independent and λ(∂) ∈ C\{0}. Define Tλ : Vλ → V  such that Tλ (v) = (x α ∂) v + (x β ∂) v ∈ V  for v ∈ Vλ .

(3.1)

We claim that Tλ is an injection. Suppose v = 0 such that Tλ (v) = 0. Let   = {mα + nβ | m, n ∈ Z} be a rank two subgroup of . Then Vir[  ] = span{Lγ = x γ ∂ | γ ∈   } is a rank two (centerless) Virasoro algebra isomorphic to the centerless version of Vir [b1 , b2 ]. Take a Z-basis {γ1 = α + β, γ2 = α + 2β} of   . Then from Lα v = Lβ v = 0, we obtain Lmγ1 +nγ2 v = 0 for all n, m ≥ 0 such that (m, n) = (0, 0). Thus v is a GHW vector. Since L0 v = ∂ v = λ(∂)v = 0, v generates a nontrivial GHW Vir[  ]-submodule of V , a contradiction with the result we obtained for the case n = 2. Thus Tλ is an injection. Therefore we have dim Vλ ≤ dim V  for all λ ∈ D∗ , i.e., V is a uniformly bounded module. Now assume that dim Vµ+α = dim Vµ+β for some α, β ∈  such that α = β and µ + α = 0 = µ + β. Choose ∂ ∈ D such that (µ + α)(∂), (β − α)(∂), (µ + β)(∂) = 0. Form a rank one centerless Virasoro subalgebra Vir[β − α] = span{x i(β−α) ∂ | i ∈ Z} and let V  = ⊕i∈Z Vµ+α+i(β−α) be a Vir[β − α]-submodule of V . Then regarding V  as a Vir[β − α]-module, each subspace Vµ+α+i(β−α) is exactly the weight space of weight (µ + α + i(β − α))(∂) with respect to ∂ ∈ Vir[β − α] (since for different i, (µ + α + i(β − α))(∂) have different values, and note that C∂ is the Cartan subalgebra of Vir[β − α]). Thus V  is also a uniformly bounded module over Vir[β − α]. Thus V  must only have a finite composition factors (and each nontrivial composition factor is a module of the intermediate series), and we see that all the nonzero weights (with respect to ∂) of V  have the same multiplicity. This is a contradiction with dim Vµ+α = dim Vµ+β .   References 1. Allison, B.N., Azam, S., Berman, S., Gao, Y., Pianzola, A.: Extended affine Lie algebras and their root systems. Mem. Am. Math. Soc. 126(603), (1997) 2. Chari, V., Pressley, A.: Unitary representations of the Virasoro algebra and a conjecture of Kac. Compositio Math. 67, 315–342 (1988) 3. Friedan, D., Qiu, Z., Shenker, S.: Conformal invariance, unitarity and two-dimensional critical exponents. In: Vertex Operators in Mathematics and Physics, J. Lepowsky, S. Mandelstam, I. M. Singer, (eds.), New York-Berlin: Springer, 1985

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4. Gelfand, I.M., Fuchs, D.B.: Cohomologies of the Lie algebra of vector fields on the circle. Funct. Anal. Appl. 2, 92–39 (1968) (English translation 114–126) 5. Goddard, P., Olive, D.: Kac-Moody and Virasoro algebras in relation to quantum physics. Internat. J. Mod. Phys. A1, 303–414 (1986) 6. Kac, V.G.: Some problems of infinite-dimensional Lie algebras and their representations. Lecture Notes in Mathematics 933, Berlin-Heidelberg-New York: Springer, 1982, pp. 117–126 7. Kac, V.G.: Infinite Dimensional Lie Algebras. 3rd ed., Cambridge: Cambridge Univ. Press, 1990 8. Martin, C., Piard, A.: Indecomposable modules over the Virasoro Lie algebra and a conjecture of V Kac. Commun. Math. Phys. 137, 109–132 (1991) 9. Mathieu, O.: Classification of Harish-Chandra modules over the Virasoro Lie algebra. Invent. Math. 107, 225–234 (1992) 10. Mazorchuk, V.: Verma modules over generalized Witt algebras. Compositio Math. 115, 21–35 (1999) 11. Mazorchuk, V.: Classification of simple Harish-Chandra modules over Q-Virasoro algebra. Math. Nachr. 209, 171–177 (2000) 12. Neveu, A., Schwarz, J.H.: Factorizable dual model of pions. Nucl. Phys. B 31, 86–112 (1971) 13. Patera, J., Zassenhaus, H.: The higher rank Virasoro algebras. Commun. Math. Phys. 136, 1–14 (1991) 14. Ramond, P.: Dual theory of free fermions. Phys. Rev. D 3, 2451–2418 (1971) 15. Rebbi, C.: Dual models and relativistic quantum strings. Phys. Rep. 12, 1–73 (1974) 16. Su, Y.: A classification of indecomposable sl2 (C)-modules and a conjecture of Kac on irreducible modules over the Virasoro algebra. J. Alg. 161, 33–46 (1993) 17. Su, Y.: Harish-Chandra modules of the intermediate series over the high rank Virasoro algebras and high rank super-Virasoro algebras. J. Math. Phys. 35, 2013–2023 (1994) 18. Su, Y.: Classification of Harish-Chandra modules over the super-Virasoro algebras. Commun. Alg. 23, 3653–3675 (1995) 19. Su, Y.: Simple modules over the high rank Virasoro algebras. Commun. Alg. 29, 2067–2080 (2001) 20. Su, Y.: Simple modules over the higher rank super-Virasoro algebras. Lett. Math. Phys. 53, 263–272 (2000) 21. Su, Y.: On indecomposable modules over the Virasoro algebra. Science in China A 44, 980–983 (2001) (also available at Math.QA/0012013) 22. Su, Y., Zhao, K.: Generalized Virasoro and super-Virasoro algebras and modules of the intermediate series. J. Alg. 252, 1–19 (2002) 23. Xu, X.: New generalized simple Lie algebras of Cartan type over a field with characteristic 0. J. Alg. 224, 23–58 (2000) Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 240, 553–586 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0918-1

Communications in

Mathematical Physics

Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs Motohico Mulase
, Andrew Waldron

Department of Mathematics, One Shields Avenue, University of California, Davis, CA 95616–8633, USA. E-mail: [email protected]; [email protected] Received: 23 July 2002 / Accepted: 27 May 2003 Published online: 19 August 2003 – © Springer-Verlag 2003

Abstract: We present an asymptotic expansion for quaternionic self-adjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. We show that the 2N × 2N Gaussian Orthogonal Ensemble (GOE) and N × N Gaussian Symplectic Ensemble (GSE) have exactly the same expansion term by term, except that the contributions from graphs on a non-orientable surface with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials, demonstrating that this duality is equivalent to Poincar´e duality of graphs drawn on a compact surface. Another consequence of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble. Contents 1. Introduction . . . . . . . . . . . . 2. Matrix Integrals . . . . . . . . . . 3. Graphical Expansion . . . . . . . 4. Examples . . . . . . . . . . . . . 5. Duality for Matrix Integrals . . . . 6. Characteristic Polynomial Duality 7. The Penner Model . . . . . . . . . 8. The Central Limit Theorem . . . . 9. Conclusions . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . .




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Research supported by NSF Grant DMS-9971371 and the University of California, Davis. Research supported by the University of California, Davis.

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A. Quaternionic Feynman Calculus . . . . . . . . . . . . . . . . . . . . . . . B. Generalized Penner Model . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction The purpose of this paper is to establish an asymptotic expansion for quaternionic selfadjoint matrix integrals in terms of Feynman diagrams and to give a new topological proof of the various characteristic polynomial dualities discovered by [2, 6, 7, 17]. Recent developments in the theory of random matrices exhibit particularly rich structures. Although originally introduced by Wigner as a model for heavy nuclei, random matrices appear almost ubiquitously in modern mathematics. Mathematical applications pertain, for example, to number theory, combinatorics, probability theory, and geometry of moduli spaces of Riemann surfaces (see for example, [1, 3, 10, 11, 15, 16, 21, 25, 27] and articles in [4] and references cited therein). In physics, ’t Hooft’s discovery [24] that quantum chromodynamics (QCD) simplifies in the limit where the number of colors N (i.e. gauge group SU (N )) is large relied on a graphical expansion in terms of “fat” or “ribbon” graphs. Hermitian matrix integrals appear in this context as generating functions for oriented ribbon graphs [3, 5]. Graphical expansions of gauge theories with other gauge groups were studied in the early 1980s (for example, see [18, 8, 9]). In particular, it was recognized that SO(2N )gauge theory and Sp(N )-gauge theory are identical in their graphical expansions, except that the parameter N in the SO(2N )-theory has to be replaced with −N [18]. This duality is also noted in more recent works (see for example, [20, 28]). A characteristic feature that distinguishes these gauge theories from the SU (N )-gauge theory is the appearance of graphs drawn on non-orientable surfaces. Real symmetric matrix integrals have been used as generating functions for these non-orientable ribbon graphs [10, 14, 23, 26]. Although the integration in the GOE and GSE matrix integrals is over real and quaternionic self-adjoint matrices, rather than so(2N ) and sp(N ) Lie algebra valued fields of the gauge theory case, on the basis of the Sp(N )-gauge theory and “SO(−2N )-gauge theory” equivalence, an N ×N quaternionic self-adjoint matrix integral should also give a generating function for non-orientable ribbon graphs identical with the 2N × 2N real symmetric matrix integral, with the parameter N replaced by −N . We show that this is indeed the case and our method also implies a simple graphical proof of the gauge theory result of [18]. As discussed in the Conclusion, since our proof is based on the construction of a new topological invariant of punctured surfaces, it generalizes to a large class of models. In this article, we develop a graphical expansion technique for an N × N self-adjoint quaternionic matrix integral, and directly verify its duality with a real symmetric matrix integral of size 2N . As an immediate consequence of the graphical expansion formulas, we give a new topological proof of the known duality for k-fold correlations of characteristic polynomials of N × N matrices for Gaussian Orthogonal, Gaussian Unitary, and Gaussian Symplectic Ensembles [2, 6, 7, 17]. For the GUE model expressed in terms of ribbon graphs, this N-k duality [6, 7] is precisely the Poincar´e duality of graphs drawn on a compact oriented surface. Similarly, the relation between GOE and GSE correlations stems from the combination of Poincar´e duality, this time including non-orientable surfaces, and our graphical expansion formula. It is interesting to note that the machinery of fermionic integrations employed in [6, 7] is equivalent to a very simple switch from a graph on a surface to its dual graph.

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An N × N matrix X with entries in the ring of quaternions H = R ⊕ iR ⊕ j R ⊕ kR is self-adjoint if X  = X, where X denotes the quaternionic conjugation of X defined by x + iy + j z + kw = x − iy − j z − kw ∈ H. Our result for the self-adjoint quaternionic matrix integral is     2Nt [dX] exp − N tr X 2 + j j j tr X j   log  [dX] exp(−N tr X 2 ) =

(−2N )χ(S ) v (j ) tj  . |Aut()|

∈G

(1.1)

j

Exact conventions are given later, at present it suffices to indicate that the sum is over all graphs  drawn on compact orientable and non-orientable surfaces S , and χ (S ) is the Euler characteristic of the surface S uniquely defined by the graph . Our proof of this result is based on viewing the GOE integral over real symmetric matrices as fundamental. The crucial observation is that the contribution of any given graph  in the GOE, GUE and GSE is a topological invariant of the surface S with f marked points on it, where f denotes the number of faces of the cell-decomposition of S defined by the graph . The connectivity of the space of triangulations of two dimensional surfaces then allows any graphical contribution to be calculated from a simple representative graph for any given topology. Writing the results for all three ensembles in a uniform notation (see (5.2)) makes the expected duality  GOE ←→ GSE GUE ←→ GUE
GSE ←→ GOE

(1.2)

manifest. The middle line for the GUE is a (trivial) self-duality. The tilde on the right hand side indicates that equality holds upon doubling/halving the matrix size and an overall sign change for contributions of graphs where the Euler characteristic χ (S ) is odd. Although not discussed in the main text, it is indeed possible to generalize the usual Schwinger trick to quaternionic source terms, and represent a Gaussian symplectic integral as non-commutative quaternionic differentiations. The result is a sum over both orientable and non-orientable ribbon graphs, and is easily verified to agree with ours for simple graphs. This method is described in Appendix A. We also note that a partial result for quaternionic expansions has been obtained in [13]. If we reduce our integral (1.1) to a symplectic Penner model by setting t1 = t2 = 0 and j

tj = −z 2 −1 ,

j ≥ 3,

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M. Mulase, A. Waldron

then we can explicitly compute the asymptotic expansion in z of      j

N 2m ki  z j/2−1 2α dki |(k)| exp − N i=1 j =2 j αN   R lim log     2 

k m→∞ N i 2α dk exp − |(k)| N i i=1 R 2

(1.3)

utilizing the Selberg integration formula and the asymptotic analysis technique of [19], where α is either a positive integer or its reciprocal. We demonstrate that the duality for GOE and GSE of (1.1) extends to the same type of duality between an arbitrary positive integer α and 1/α for (1.3) with the sign change for all terms with odd powers of N in the asymptotic expansion. The orthogonal Penner model gives the orbifold Euler characteristic of the moduli spaces of smooth real algebraic curves with an arbitrary number of marked points [10]. Also, the original Penner model [21] provides the orbifold Euler characteristic of the moduli spaces of pointed algebraic curves over C [11]. The GOE-GSE duality shows that the symplectic Penner model is identical to the orthogonal Penner model, except for doubling the matrix size and an overall sign change for contributions from surfaces of odd Euler characteristic. From the graphical expansion formulas for matrix integrals, one can uniformly derive the Central Limit Theorem for Gaussian random matrix ensembles. This result follows as a direct consequence of ’t Hooft’s original large N limit in which planar ribbon graphs dominate: we derive a precise limiting formula for GOE, GUE and GSE matrix ensembles in terms of planar two-vertex ribbon graphs. The formula is the same for all three ensembles except for an overall constant, which is parallel to the equivalence of SO(N ), SU (N ) and Sp(N )-gauge theories it large N . The material is organized as follows: In Sect. 2 we introduce the matrix integrals studied in this paper. Our conventions for the topological data of surfaces are given in Sect. 3 as well as our theorem and its proof for the graphical expansion of matrix integrals. Examples, including a comparison with the first few terms of the Penner model, are given in Sect. 4. The GOE-GSE duality appears in Sect. 5. The central formula of this paper is Eq. (5.2) which gives the graphical expansion for the GOE, GUE and GSE simultaneously in a manifestly duality invariant form. Its application to characteristic polynomial duality is in Sect. 6. The extended version of the duality for Penner type models is found in Sect. 7 while detailed derivations of the formulæ there are presented in Appendix B. Section 8 concerns the Central Limit Theorem for Gaussian random matrix ensembles. In the Conclusions (Sect. 9) we discuss possible further generalizations. In particular, the construction of a graphical topological invariant of surfaces with marked points necessary for the proof of our main result is rather general and may be applied to higher algebraic structures. 2. Matrix Integrals The object of our study is the integral over self-adjoint matrices1     tj j tr X [dX](β) exp − 41 tr X 2 + ∞ j =1 2j   Z (β) (t, N ) =  1 [dX](β) exp − 4 tr X 2

(2.1)

1 We employ various normalizations throughout the paper, so it is convenient to divide through by the free matrix integral.

Duality for Matrix Integrals

557

as a function of the “coupling constants” t = (t1 , t2 , t3 , . . . ) and the size N of the matrix variable X. Here X=S+

β−1

(2.2)

ei A i

i=1

is built from real, N × N, symmetric and antisymmetric matrices S and Ai , respectively. The parameter β takes values 1, 2 or 4 depending on whether we study real, complex or quaternionic self-adjoint matrices and in turn Gaussian orthogonal, unitary or symplectic ensembles (GOE, GUE, GSE). The imaginary units ei are then drawn from one of three sets,  

∅, 2 = −1} , {i : i ei ∈  {i, j, k : i 2 = j 2 = k 2 = ij k = −1} ,

β = 1, β = 2, β = 4.

(2.3)

The self-adjoint condition 

X† ≡ X = X ,

ei = −ei ,

(2.4)

is implied by antisymmetry of the matrices Ai . Finally, the measure [dX](β) is the translation invariant Lebesgue measure of the vector space of real dimension 21 N (β(N −1)+2) spanned by independent matrix elements of S and Ai . This measure is invariant, respectively, under orthogonal, unitary and symplectic transformations X −→ U † XU,

(2.5)

β−1 where U † U = 1 and U = U0 + i=1 ei Ui for real N × N matrices U0 and Ui . The matrix integral (2.1) is a holomorphic function in (t1 , t2 , . . . , t2m ) if we fix m > 0 and restrict the coupling constants to satisfy Re(t2m ) < 0

and

t2m+1 = t2m+2 = t2m+3 = · · · = 0.

Under this restriction, Z β (t, N ) has a unique Taylor expansion in (t1 , t2 , . . ., t2m−1 ) and an asymptotic expansion in t2m as t2m → 0 while keeping Re(t2m ) < 0. Let us introduce a weighted degree of the coupling constants by deg(tn ) = n. Then the asymptotic expansion of the truncated integral Z β (t, N ) has a well-defined limit as m → ∞ in the ring (Q[N ]) [[t1 , t2 , t3 , . . . ]] of formal power series in infinitely many variables with coefficients in the polynomial ring of N with rational coefficients [19]. The subject of our study in what follows is this asymptotic expansion of Z β (t, N ) as a function in infinitely many variables.

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M. Mulase, A. Waldron

3. Graphical Expansion The graphs appearing in our asymptotic expansion of the matrix integrals (2.1) are those drawn on orientable as well as non-orientable surfaces. To avoid confusion with an already well-established convention that ribbon graphs are drawn on orientable surfaces, we propose the terminology M¨obius graphs. Let us recall that a ribbon graph  is a graph with a cyclic order chosen at each vertex for half-edges adjacent to it. Equivalently, it is a graph drawn on a compact oriented surface S giving a cell-decomposition of it. The complement S \  of the graph  on S is the disjoint union of f open disks (or faces) of the surface. Since a ribbon graph  defines a unique oriented surface on which it is drawn as the 1-skeleton of a cell-decomposition, we denote the surface by S . Similarly, a M¨obius graph is drawn on a compact surface, orientable or non-orientable, giving a cell-decomposition of the surface. It can be viewed as a ribbon graph with twisted edges. A M¨obius graph  also uniquely defines the surface S in which it is embedded. Let G be the set of connected M¨obius graphs. A graph  ∈ G consists of a finite (j ) number of vertices and edges. Let v denote the number of j -valent vertices of . Then the number of vertices and edges are given by (j ) 1 (j ) v = v and e = j v . (3.1) 2 j

j

The unique compact surface S has f faces and its Euler characteristic is χ (S ) = v − e + f .

(3.2)

We will also need the number of faces with a given number of edges, so denote the (j ) number of j -gons in the cell-decomposition of S by f whereby the Poincar´e dual formulæ to Eqs. (3.1) are (j ) 1 (j ) f = f and e = jf . (3.3) 2 j

j

A M¨obius graph  also determines the orientability of S and we define  1 S orientable ,  = −1 S non-orientable .

(3.4)

By genus of S we mean g(S ) = 1 − 2−

1+ 2

(3.5)

χ (S ) .

Thus χ (S ) = 2 − 2g(S ) for an orientable surface and χ (S ) = 1 − g(S ) if it is non-orientable. We also define the parity of χ (S ) by  = (−1)χ(S ) ,

(3.6)

while many results can be written compactly in terms of   0  orientable ,  1  = 1 +  −  = 1  non-orientable, χ (S ) odd ,  2  non-orientable, χ (S ) even . 2 

Our main result is:

(3.7)

Duality for Matrix Integrals

559

Theorem 3.1. The logarithm of the asymptotic expansion of the matrix integral Z (β) (t, N ) is expressed as a sum over connected M¨obius graphs: 1

1

 (−4 + 6β − β 2 )1− 2  − 2 χ(S ) (2 − β)  β f −1 N f v (j ) (β) log Z (t, N ) = tj  |Aut()| 

∈G

∈ (Q[N]) [[t1 , t2 , t3 , . . . ]].

j

(3.8)

Remark. (1) We define (2 − β)  = 1 when β = 2 and  = 0.

v (j ) (2) For every connected M¨obius graph , the monomial j tj  is a finite product of total degree 2e . (3) The only reason to consider log Z β (t, N ) is because it yields a compact formula (3.8): Z β (t, N ) itself has an expansion in terms of graphs although a given summand may have a mixture of orientable and non-orientable connected components. The automorphism group Aut() of a M¨obius graph  is a group of automorphisms of the cellular complex S consisting of v vertices, e edges and f faces. When S is orientable, the group Aut() may contain orientation-reversing automorphisms as well. We note that a cyclic rotation of half-edges around a vertex corresponds to the invariance of the trace under a cyclic permutation tr (M1 M2 M3 · · · Mn ) = tr (Mn M1 M2 · · · Mn−1 ),

(3.9)

and an orientation-reversing flip of vertex with adjacent half-edges corresponds to the invariance of the trace of symmetric matrices tr (S1 S2 S3 · · · Sn ) = tr (Sn Sn−1 Sn−2 · · · S1 )

(3.10)

reversing the order of multiplication. The proof of the theorem involves two main ingredients. The first is to view GUE and GSE matrix integrals as the coupling of a singlet or triplet of skew-symmetric matrix integrals to the fundamental GOE integral. When β = 1, Eq. (3.8) is the M¨obius graphical expansion of a symmetric matrix integral [10, 26, 23, 14]:

Fig. 3.1. Two equivalent M¨obius graphs consisting of two vertices, three edges, and one face. The graphs are interchanged by a vertex flip

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M. Mulase, A. Waldron

  log Z (β=1) (t, N ) = ∈G

v (j ) N f tj  . |Aut()|

(3.11)

j

This formula follows immediately from the fact that the Wick contraction of any pair of symmetric matrices S = (Sab ) obeys Sab Scd = δac δbd + δad δbc ,

(3.12)

which is denoted graphically as an edge of a M¨obius graph (see Fig. 3.2). These edges connect vertices of the type tj tj tr S j = 2j 2j

N

Sa1 a2 Sa2 a3 · · · Saj a1 ,

(3.13)

a1 ,... ,aj =1

as depicted in Fig. 3.3. The factor (2j )−1 in (3.13) is precisely the one required to cancel the over-counting implied by the identities (3.9) and (3.10). Therefore the overall weight of any given graph is as quoted in (3.11). In contrast, for a pair of antisymmetric matrices A = (Aab ) the Wick contraction yields Aab Acd = δac δbd − δad δbc .

(3.14)

This is depicted in Fig. 3.4.

c d

b a

Sab

Scd

+ c d

b a

Fig. 3.2. Propagator for symmetric matrices

a a j-1 j

aj

b2

a1

b3

V

a4

a3

a3

a2

a1

b2

a2

b1

V’

b3 b1

Fig. 3.3. Two vertices connected at the half edge labeled by a1 a2 to the half edge labeled by b2 b1 . For the GOE a second graph with a twisted edge connecting a1 a2 to b1 b2 is also present since the ribbon edges are not directed

Duality for Matrix Integrals

561

c d

b a

-

A ab

A cd c d

b a

Fig. 3.4. Propagator for antisymmetric matrices

The minus sign for untwisted edges will play a crucial rˆole in what follows. Notice that since the exponent of the Gaussian part of (2.1) is 1 2 1 − Saa − 4 2



N

a=1



2 Sab

+

1≤a
β−1

 A2iab ,

(3.15)

i=1

there are no non-vanishing Wick contractions between symmetric and antisymmetric matrices. Therefore the model (2.1) is a sum over graphs with edges of either of the two types (3.12) or (3.14). Furthermore Ai Aj is only non-vanishing when i = j so edges “emitted” at vertices are only connected when they carry the same imaginary units ei . If we call e0 = 1 and A0 = S, a j -valent vertex now looks like β−1 1 1 j tr X = eα Aα = tr 2j 2j α=0

{(α1 ,... ,αj )}/∼

j ρ(α1 , . . . , αj )

tr Aαk eαk . 2j

(3.16)

k=1

The last sum runs over all j -tuples of integers 0, 1, . . . , β − 1 modulo the equivalence relation (α1 , . . . , αj ) ∼ (β1 , . . . , βj ) iff

tr (Aα1 · · · Aαj ) = tr (Aβ1 · · · Aβj ) .

(3.17)

The multiplicities ρ(α1 , . . . , αj ) are non-zero only when the product eα1 · · · eαj = ±1 .

(3.18)

In other words, the vertex (3.16) is real. Observe that the numbers 2j ρ(α1 , . . . , αj )

(3.19)

count the number of automorphisms of a vertex with a given configuration of units eα sprinkled at every edge, with respect to rotations and vertex flips. For example, for β = 4 and j = 3, 1 1 tr X3 = tr (1.S)3 6 6  1  + tr 1.S i.A1 i.A1 + 1.S j.A2 j.A2 + 1.S k.A3 k.A3 2  1  + tr i.A1 j.A2 k.A3 . 1

(3.20)

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M. Mulase, A. Waldron

On the first line, the trace of three identical symmetric matrices can be cycled and reversed, so 2j = 6 is the correct automorphism factor. On the second line the terms with a single symmetric and a pair of antisymmetric matrices can only be flipped yielding two automorphisms. The term on the third line has no automorphisms. We will keep track of the matrix type by “sprinkling” units {1, ei } over the set of M¨obius graphs (see Fig. 3.5). Orchestrating the above observations yields the following:   Lemma 3.2. The matrix integral log Z (β) (t, N ) may be computed as a sum over M¨obius graphs  with weight µ N f |Aut()|

(3.21)

multiplied by a single power of tj for each j -valent vertex in . The factor µ is calculated by (1) Writing down all possible configurations of units eα ∈ {1, e1 , . . . eβ−1 } at each vertex such that their product is ±1. (2) Counting these signed configurations with an additional minus sign for every untwisted edge with imaginary units ei at each end. The proof of this lemma follows from our previous remarks and by examining the Wick contraction for antisymmetric matrices (3.14). A sample computation of µ is given in Fig. 3.5. Our proof is completed by computing the numbers µ , which requires a second major ingredient: Lemma 3.3. The quantity µ is a topological invariant of the n-punctured surface S \ {p1 , p2 , . . . , pn }, where the removed points are all distinct and n = f is the number of faces of the cell decomposition defined by . The proof of this crucial lemma is almost a triviality. Graphs are identified if they are equivalent under rotations and flips of their vertices (Fig. 3.1), so it suffices to show that µ is invariant under contraction of an untwisted edge. This operation is depicted in Fig. 3.6. This relation is obvious since the products of units at either vertex must be real and the Wick contraction corresponding to an untwisted edge with units eα at either end comes with an overall factor +1. We remark that an analogous relation involving a vertex flip for the contraction of a twisted edge follows. We now compute the invariant µ for a graph of arbitrary topology. Since µ is topological, we may employ a standard graph for each distinct topology labeled by the orientability  , number of faces f and genus g(S ). The key theorem from topology we need here is the connectivity of the space of all triangulations of a compact surface with a fixed number of vertices under the action of a diagonal flip [12] (see Fig. 3.7). A diagonal flip of a triangulation is exactly the fusion move of the dual M¨obius graph. For orientable surfaces, the connectivity of concern implies the path connectivity of the moduli space Mg,n of smooth complete algebraic curves defined over C with n marked points. The connectivity of triangulations of non-orientable surfaces has been also established. Now note that because of the invariance of µ under edge contraction, we can expand all vertices of valence greater than 3 and contract all vertices of valence 1 and 2 to create

Duality for Matrix Integrals

563

1 1

=

1 1

+1

1

+

i

1 i

+

1

j

1 j

+1

+

+

1

k 1

+

i

j

k k

–1

+

k

–1

j k

+

i

+1

i

+1

k k

+

j

+1

i j

+1

j

+

k

i

i

k

+

j

i

+

–1

k j

–1

j

1 k

j i

j

1

–1

i

+

+

j

–1

i

+1

j 1

i

k 1

+1

i 1

+

1

k

i

k i

+

k

+1

j

j k

+1

Fig. 3.5. Computation of µ for a non-orientable M¨obius graph . Here χ (S ) = 0. The surface S is a Klein bottle. The contributions ±1 add to yield µ = 4 = (2 − β)2 (for β = 4)

564

M. Mulase, A. Waldron

e1 e1 e2

ei ei

e3

e6 e5

e2

e6

e3

e5

=

e4

e4 Fig. 3.6. Edge contraction

Fig. 3.7. The diagonal flip operation for a triangulation

Fig. 3.8. A standard M¨obius graph that represents an orientable surface of genus g with n marked points. It consists of n − 1 tadpoles on the left and g bi-petal flowers on the right

a trivalent graph. Since the dual of a trivalent graph is a triangulation of the surface and the number of vertices of the triangulation is the number n = f of faces of the M¨obius graph , the connectivity of the space of triangulations implies that our invariant µ is a constant for each topology of a given n-punctured surface. In the actual computation, it is convenient to use the following representatives for each of the three topological classes: (1) Orientable ( = 1); a standard graph is given in Fig. 3.8. (2) Non-orientable ( = −1), odd genus; with standard graph given in Fig. 3.9. (3) Non-orientable ( = −1), even genus; a standard graph is in Fig. 3.10. Finally, it is easy to calculate µ for each one particle irreducible component (subgraphs which remain connected after cutting a single edge): =β,

(3.22)

Duality for Matrix Integrals

565

Fig. 3.9. A standard M¨obius graph drawn on a non-orientable surface of genus g = 2k +1 with n marked points. In addition to n − 1 tadpoles on the left it has k orientable bi-petal flowers on the right and a non-orientable one added at the top

Fig. 3.10. A standard M¨obius graph representing a non-orientable surface of genus g = 2k with n marked points. It has n − 1 orientable tadpoles on the left, k bi-petal orientable flowers on the right and a non-orientable tadpole added at the top

= 1 + 3(β − 1) − (β − 1)(β − 2) ,

(3.23)

=2−β,

(3.24)

= (2 − β)2 .

(3.25)

Each of the above calculations is rather similar: A single line emitted from a tadpole subgraph can only ever carry the unit e0 = 1. As an example, (3.23) arises from (i) placing all 1’s on the remaining four lines, (ii) 1’s on one loop and imaginary units ei on the other which can be done in 2(β − 1) ways, (iii) the same imaginary unit on both loops giving β − 1 possibilities, (iv) different imaginary units which can only be achieved for

566

M. Mulase, A. Waldron

the quaternions and in 6 = (β − 1)(β − 2) different ways incurring a minus sign since −1 = ij ij (say). Therefore we find µ = (−4 + 6β − β 2 )

1− 21  − 21 χ(S )

(2 − β)



β

f −1

(3.26)

.

This result combined with the first lemma above completes the proof of our main theorem. Finally, we prove the gauge theory result of [18]. In the SO(2N ) case the integration is over antisymmetric matrix valued gauge fields. The expansion is again in terms of M¨obius graphs with a relative minus sign for each twisted edge. As shown in [18], in the Sp(N ) theory, a twisted edge gets a minus sign as well as a directed factor J = −J  (the 2N × 2N symplectic metric) on each line. Therefore the symplectic and orthogonal graphs differ only by a possible relative minus sign which can again be computed for the standard graphs in Figs. 3.8–3.10 since this relative sign is also invariant under edge contraction. The result is, of course, a factor (−)χ(S ) . 4. Examples A simple consistency check is the case β = 2. Clearly, non-orientable graphs give a vanishing contribution as required for hermitian matrix integrals. To obtain a more conventional normalization, we rescale X −→ β 1/2 X in (2.1) and absorb all but one power of β in the couplings t, so that     βtj j [dX](β) exp − β4 tr X 2 + ∞ tr X j =1 2j    log   β [dX](β) exp − 4 tr X 2 1

1

(−4 + 6β − β 2 )1− 2  − 2 χ(S ) (2 − β)  β χ(S )−1 N f v (j ) = tj  . |Aut()| ∈G

(4.1)

j

Hence when β = 2,

   tj j [dX](2) exp − 21 tr X 2 + ∞ tr X j =1 j    log   1 [dX](2) exp − 2 tr X 2 

=

∈G

v (j ) 2N tj  . |Aut()| f

(4.2)

j

We note that ∈G

f f

v (j )

v (j ) 2N N tj  = tj  , |Aut()| |Aut R()| j

∈R

(4.3)

j

where R denotes the set of all connected ribbon graphs and Aut R() the automorphism group of a ribbon graph disallowing orientation-reversing automorphisms. To see (4.3), let  be an oriented ribbon graph. Then either (a)  and its flip ˇ are isomorphic as a ribbon graph; or (b) they are different ribbon graphs. In case (a), we have

Duality for Matrix Integrals

567

|Aut()| = 2|Aut R()|. If (b) is the case, then Aut() ∼ = Aut R(), but  and ˇ appear as different graphs on the right hand side while they are the same as a M¨obius graph in G. Together, Eqs. (4.3) and (4.2) constitute the well-known result for the hermitian matrix integral [3]. A less trivial test for the β = 4, GSE model is a comparison with the Penner model. To begin with we re-express (3.8) in yet another normalization, together with a specialization t1 = t2 = 0:    tj j [dX](4) exp − 21 tr X 2 + ∞ tr X j =3 j    log   1 [dX](4) exp − 2 tr X 2 

=

(−1)χ(S ) f v(j ) (2N ) tj . |Aut()|

(4.4)

j ≥3

∈G

Here we have employed the useful identity (−1)  = (−1)χ(S ) .

(4.5)

The Penner substitution tj −→ −zj/2−1 ,

j ≥3

(4.6)

yields the graphical expansion    zj/2−1 j [dX](4) exp − 2m tr X j =2 j    lim log   m→∞ 1 [dX](4) exp − 2 tr X 2 

=

(−1)e f (2N ) (−1)χ(S ) (−z)e −v . |Aut()|

(4.7)

∈G

Equation (4.7) is valid as an asymptotic expansion of the integral for z → 0 while keeping z > 0, as an element of a formal power series ring (Q[N ])[[z]]. The integral on the left hand side can be evaluated explicitly (see (7.2) and (7.3) of Sect. 7). The leading terms as an expansion in z are    zj/2−1 [dX](4) exp − 2m tr X j j =2 j    lim log   m→∞ 1 2 [dX](4) exp − 2 tr X  1 1 2  = − N − N 2 + N 3 z + O(z2 ) . 12 2 3 

(4.8)

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M. Mulase, A. Waldron

On the other hand, our graphical expansion yields (−)e (−2N )b ze −v |Aut()|

(4.9)

∈G



(−1)3 + 4

(−1)2 = (−2N ) z 4  (−1)2 + (−2N)2 z 2    2 (−1) +  8   (−1)2 + (−2N) z 4 3

(−1)2 4

+ + =



(−1)2

  

8

 

−



(−1)3 + 12

+

(−1)3 2

+

(−1)3 4

+

(−1)3 4

+

(−1)3 4

+

(−1)3 12

+ O(z2 )

1 2  1 N − N 2 + N 3 z + O(z2 ) . 12 2 3

Needless to say agreement is perfect. In fact, as we shall show in Sect. 7, agreement to all orders amounts to known results for the orbifold Euler characteristic of the moduli space of real algebraic curves. 5. Duality for Matrix Integrals An additional change of variables X −→ N 1/2 X in (4.1), absorption of all but a single N in the couplings t, as well as the substitution β = 2α ,

(5.1)

yields   ∞ Nαtj 2 j [dX](2α) exp − Nα j =1 j tr X 2 tr X +    log   2 [dX](2α) exp − Nα tr X 2 1/2 χ(S ) = 2 (α N ) 

∈G

×

(3 − α −1 − α)

1− 21  − 21 χ(S )

|Aut()|

(α −1/2 − α 1/2 )



j

v

(j )

tj  .

(5.2)

Duality for Matrix Integrals

569

This formula is invariant under α −→ α −1 and N −→ −αN .

(5.3)

Remark. (1) The duality holds graph by graph2 . (2) The α = 1 GUE model is self-dual since χ (S ) is even for orientable graphs. (3) The graphical expansion of the α = 2, N × N GSE model is identical to that of the α = 1/2 GOE model if the size of the matrices are doubled and the contribution of every M¨obius graph embedded in a non-orientable surface of odd Euler characteristic is multiplied by −1. One might wonder whether matrix integrals exist whose graphical expansion coincides exactly with the image of (5.2) under the duality (5.3). Although we have no definite answer to this question at this point, in the next section we show that the combination of Poincar´e duality and the one discovered here underly the dualities for correlations of characteristic polynomials [2, 6, 7, 17]. 6. Characteristic Polynomial Duality The average of products of characteristic polynomials obey dualities between GOE and GSE models [7]:    k [dS]N×N exp − N2 tr S 2 =1 det N×N (λ − S) (1)    [dS]N×N exp − N2 tr S 2 (1)  √ [dX]k×k exp(−N tr X 2 ) HdetN k×k( − −1 X) (4)  = , (6.1) [dX]k×k exp(−N tr X 2 ) (4)

as well as a self duality for the GUE case3 [6]:    k [dX]N×N exp − N2 tr X 2 =1 det N×N (λ − X) (2)    [dX]N×N exp − N2 tr X 2 (2)    √ k×k [dY ] exp − N2 tr Y 2 det N k×k ( − −1 Y ) (2)   = .  [dY ]k×k exp − N2 tr Y 2

(6.2)

(2)

The duality relates expectations of k-fold products of distinct characteristic polynomials of N × N matrices to averages over the N th power of determinants of certain k × k matrices. Here,  = diag(λ1 , . . . , λk ) is a diagonal k × k matrix of real entries. The quaternionic determinant H det in (6.1) is defined by 1/2

Hdet k×k M = det2k×2k C(M) , 2

(6.3)

Physicists would call this a T -duality – valid order by order in perturbation theory. One√might question the reality of the integrals on the right-hand sides of (6.2) and (6.1) since an explicit −1 appears in the determinants. It is clear, however, from both the graphical expansions below and the original derivation in [6, 7] that both integrals are real. 3

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where the 2k × 2k matrix C(M) is obtained from the k × k quaternion valued matrix M by replacing the quaternionic units by their representation in terms of Pauli matrices4 1 → I2×2 , ei → iσi (i = 1, 2, 3). To begin with, we demonstrate that the N -k duality for GUE models follows from the usual ribbon graph expansion along with Poincar´e duality of graphs on a compact oriented surface. The first step is to represent the determinants as vertices of the graphical expansion. Let us assume that the parameter λ satisfies λ > λ > 0 for every and some positive λ, and let λ denote the set of all N × N hermitian matrices whose eigenvalues are contained in the bounded interval [−λ, λ]. Then for every X ∈ λ , we have a convergent power series expansion in λ−1 :  det I −

X λ



     ∞ 1 −j X = exp − λ tr X j  . = exp tr log I − λ j j =1

Therefore, 

N 2 [dX]N×N e− 2 tr X (2)

k

 det N×N

=1

 =

[dX]N×Ne λ

− N2 tr

(2)

X2



 ∞ 1 trk×k −j trN×N X j  exp − j 

j =1

 +

X I− λ

k

N 2 [dX]N×Ne− 2 tr X det N×N (2) c λ =1



X I− λ

 ,

(6.4)

where cλ is the complement of λ in the space of all N × N hermitian matrices. Since λ is a compact space, the first integral on the right-hand side of (6.4) is a conver−1 −1 −j gent power series in λ−1 1 , λ2 , . . ., λk . Set tj = −tr  . Then Re(tj ) < 0, and as λ → +∞, tj → 0. Thus the ribbon graph expansion provides each coefficient of the power series expansion of this integral in tj as λ → +∞. The second integral on the right-hand side of (6.4) is a polynomial in λ−1 whose coefficients converge to 0 as λ goes to infinity since cλ approaches the empty set and the integrand is bounded. Therefore, we obtain an asymptotic expansion formula      k X [dX]N×N exp − N2 tr X 2 I − det N×N =1 λ (2)    log   N 2 N×N exp − 2 tr X [dX] (2)

(j ) 1 = (−1)v N f −e (tr −j )v |Aut R()| ∈R

∈ 4

j

(Q[N ])[[λ−1 1 ,...

, λ−1 k ]] .

(6.5)

The Pauli matrices are  σ1 =

01 10



 ,

σ2 =

0 −i i 0



 ,

σ3 =

10 01

 .

Duality for Matrix Integrals

571

The computation of    √ [dY ]k×k exp − N2 tr Y 2 det N(I − −1 Y −1 ) (2)    log   [dY ]k×k exp − N2 tr Y 2 

(6.6)

(2)

can be performed similarly: First we decompose the space of all k ×k hermitian matrices into two pieces, one consisting of matrices with eigenvalues in [−λ, λ], and the other √ its complement. If λ > λ for every , then det N(I − −1 Y −1 ) can be expanded as −1 before. Asymptotically as an element of (Q[N ])[[λ−1 1 , . . . , λk ]], we have 

 N  √ [dY ]k×k exp − tr Y 2 det(I − −1 Y −1 )N (2) 2   √    N ( −1)j tr (Y −1 )j  . (6.7) tr Y 2 exp −N = [dY ]k×k exp − (2) j 2 j

The appearance of the term tr (Y −1 )m instead of tr Y m occurring in (6.7) replaces products of traces over identity matrices N f =

(j ) (tr I j )f

(6.8)

j

incurred in (3.8) as one travels around each face of the graph , by

(j ) (tr −j )f .

(6.9)

j (j )

(Recall that f denotes the number of j -gons in the cell-decomposition of S defined by the graph .) Therefore,     √ [dY ]k×k exp − N2 tr Y 2 det N(I − −1 Y −1 ) (2)    log   N k×k 2 exp − 2 tr Y [dY ] (2)       √ j −1 )j [dY ]k×k exp − N2 tr Y 2 exp −N j ( −1) tr (Y  j (2)    = log   N [dY ]k×k exp − 2 tr Y 2 (2)

√ (j ) 1 v v −e = ( −1)2e (tr −j )f (−1) N |Aut R()| ∈R

=



∈R

j

1 |Aut R()|

(j ) (−1)f N v −e (tr −j )f , j

where we used (−1)v −e = (−1)χ(S )−f = (−1)f .

(6.10)

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M. Mulase, A. Waldron

Let us denote by  ∗ the dual graph of a ribbon graph  drawn on a compact oriented surface S . We note that Aut R() ∼ = AutR( ∗ ) and  (j ) (j )  v = f ∗ , (6.11) e = e ∗ ,  (j )  (j ) f = v ∗ . Since one and two valent vertices are included in the set of ribbon graphs R, the map ∗ : R −→ R is a bijection. [Contrast this situation to the Penner model in Sect. 4, where the couplings t1 = t2 = 0 and Poincar´e duality does not apply.] Therefore, ∈R

=

1 |Aut R()| ∈R

=



∈R

(−1)v N f −e



(j )

(tr −j )v

j

(j ) 1 v ∗ f ∗ −e ∗ −j v ∗ N (tr  ) (−1) |Aut R( ∗ )| j

1 |Aut R()|

(j ) (tr −j )f . (−1)f N v −e

(6.12)

j

This implies that the matrix integrals (6.5) and (6.6) have the same asymptotic expansion. The N-k duality in Eq. (6.2) is a polynomial identity of degree N k in (Q[N ])[λ1 , λ2 , . . . , λk ], where we define deg(λ ) = 1. We must now consider also disconnected graphs, since there is no logarithm. The coefficient of the degree N k − d term of (6.2) is therefore determined by a partition d = 2(e1 + e2 + · · · + em ) corresponding to the product of m connected graphs consisting of ei edges. The contributions of connected graphs are computed in (6.5) and (6.6). We note that the duality (6.12) holds for every surface even when the number of edges is fixed. Therefore, the asymptotic equality we have derived implies the polynomial identity (6.2). In other words, the N -k duality of [6] is a simple consequence of the Poincar´e duality of graphs on a compact oriented surface. Our derivation of the characteristic polynomial duality between the GOE and GSE models goes quite similarly. Here again we see that the duality is a consequence of our graphical expansion formula (3.8) and Poincar´e duality: Using the same trick for characteristic polynomials as in the GUE case, from the expansion formula (3.8) we obtain an asymptotic expansion formula for the GOE side of the duality      k S 2 [dS]N×N exp − 2N tr S det I − =1 4 λ (1)    log   N N×N 2 [dS] exp − 4 tr S (1)       2 tr −j [dS]N×N exp − 2N tr S 2 exp − ∞ tr S j j =1 4 2j (1)    = log   2 [dS]N×N exp − 2N tr S 4 (1)

(j ) 1 = (−1)v 2v −e N f −e (tr −j )v . (6.13) |Aut()| ∈G

j

Duality for Matrix Integrals

573

(Note the non-standard normalization of the Gaussian exponent yields the factor 2−e .) Its GSE counterpart requires some care: The “characteristic polynomial” of a k × k quaternionic matrix X is defined by √ √ H det( − −1 X) = det1/2 (I2k×2k − −1 C(X)) and tr X j =

1 tr C(X)j . 2

Thus if all eigenvalues of X are in [−λ, λ] and λ > λ > 0, then   √ ∞ j √ 2 ( −1) 1 Hdet(I − −1 X−1 ) = exp − tr (X−1 )j  . 2 j j =1

Therefore, we have an asymptotic expansion   √ −1 [dX]k×k exp(−N tr X 2 ) HdetN k×k(I − −1 X ) (4)   log  [dX]k×k exp(−N tr X 2 ) (4)   √     2( −1)j [dX]k×k exp − Nk tr X 2 exp −N ∞ tr (X−1 )−j j =1 2j k (4)  = log    2 [dX]k×k exp − Nk k tr X =

∈G

=



∈G

(4)

(j ) 1 (−1)  +e +v 2f −e N v −e (tr −j )f |Aut()| 1 (−1)f 2f −e N v −e |Aut()|



j

(j )

(tr −j )f ,

(6.14)

j

where we have used the fact that (−1)  +e +v = (−1)f that follows from (4.5). (In addition the non-standard Gaussian exponent normalization now accounts for the absence of explicit factors k in the graphical expansion.) We now see that (6.13) and (6.14) are equal again through the dual construction of a M¨obius graph. The polynomial identity (6.1) follows from the equality of the asymptotic expansions. This time each term of (6.1) may have contributions from both orientable and non-orientable graphs, but since the dual graph construction works for each surface, the equality holds. 7. The Penner Model The Penner model for the hermitian matrix integral provides an effective tool to compute the orbifold Euler characteristic of the moduli space of smooth algebraic curves defined over C with an arbitrary number of marked points [21, 11]. It was discovered in [10] that the Penner model of the real symmetric (or GOE) matrix integral yields the

574

M. Mulase, A. Waldron

orbifold Euler characteristic of the moduli spaces of real algebraic curves. Due to the GOE-GSE duality (5.3), we see that the symplectic Penner model is identical to the GOE case, except for the matrix size and an overall sign for contributions of non-orientable surfaces. As we shall see, of the three main classes of M¨obius graphs – oriented, nonorientable odd χ (S ) and non-orientable even χ (S ) – only the first two survive the Penner substitution for the couplings, or in other words, the orbifold Euler characteristic vanishes when χ (S ) is even. Therefore the third symplectic Penner type model is not an independent topological quantity. We also show that the generalized Penner model expressed in terms of Vandermonde determinants to powers in 2(N ∪ 1/N) exhibits an extended duality that agrees with (5.3) when the power of the Vandermonde is restricted to 1, 2 or 4. Many explicit formulæ and derivations are reserved for Appendix B. The symplectic Penner model introduced in Sect. 4 reads    zj/2−1 tr X j [dX](4) exp − 2m j =2 j    lim log   m→∞ [dX](4) exp − 21 tr X 2 

=

(−1)e f (2N ) (−1)χ(S ) (−z)e −v , |Aut()|

(7.1)

∈G

to be viewed as an element of the formal power series ring (Q[N ])[[z]]. This integral is indeed explicitly computable. Symplectic invariance of the measure and integrand allows us to diagonalize the matrix variable X −→ diag(k1 , k2 , . . . , kN ) so that:    zj/2−1 tr X j [dX](4) exp − 2m j =2 j    lim log   m→∞ [dX](4) exp − 21 tr X 2     

N 2m zj/2−1 j 4  (k) i=1 exp − j =2 j ki dki   RN = lim log    , 

N ki2 m→∞ 4  (k) exp − dk N i i=1 R 2 

(7.2)

where (k) =



(ki − kj )

i<j

is the Vandermonde determinant. Using the asymptotic expansion technique established in [19], the Selberg integral formula and the Stirling formula, an explicit asymptotic expansion, even valid for every α ∈ N, can be computed for the integral    

2m zj/2−1 j 2α (k) N exp −  k i=1 j =2 i dki  j  RN K(z, N, α) = lim log     . (7.3) 2 

m→∞ 2α (k) N exp − ki dk  N i i=1 R 2 

(In what follows K(z, N, α) should be regarded as the asymptotic expansion, not the underlying integral; an explicit expression is given in Appendix B.) Specializing to the

Duality for Matrix Integrals

575

α = 1 hermitian case yields a very compact result corresponding to the original formula of Penner [21]:

K(z, N, 1) =

g≥0,n>0 2−2g−n<0

=

∈R

(2g + n − 3)!(2g − 1) b2g N n (−z)2g+n−2 (2g)!n!

(−1)e N f (−z)e −v . |Aut R()|

(7.4)

Identifying n with f and g as the genus yields the well known generating function of the Euler characteristic χ (Mg,n ) of the moduli of complex algebraic curves of genus g and n marked points. For the α = 2 symplectic case a similar simplification occurs and (7.3) can be written in terms of the α = 1 hermitian result plus additional terms corresponding to non-orientable surfaces of even genus g = 2q with m + 1 − 2q marked points K(z, N, 2) =

1 K(z, 2N, 1) 2 (2q + n − 2)!(22q−1 − 1) 1 − b2q (2N )n (−z)2q+n−1 . 2 (2q)! n! q≥0,n>0 1−2q−n<0

(7.5) Comparing (7.5) with (7.1), we obtain : f =n, g(S )=2q

1 (2q + n − 2)!(22q−1 − 1) (−1)e = b2q , |Aut()| 2 (2q)! n!

(7.6)

where the summation is over all connected non-orientable M¨obius graphs with n > 0 faces that are drawn on a non-orientable surface of genus 2q satisfying a hyperbolicity condition 1 − 2q − n < 0. The formula (7.6) is in exact agreement with the formula for the orbifold Euler characteristic of the moduli space of smooth real algebraic curves of genus 2q with n marked points that can be found in [10 and 20]. To study the GOE-GSE duality for the Penner model, we need the expansion of the analog of formula (7.3) valid for the single power of the Vandermonde determinant relevant to the GOE model. Indeed an integral formula for   J (z, N, γ ) = lim log m→∞

RN

   

2m zj/2−1 j |(k)|2/γ N exp − k i=1 j =2 i dki  j    , (7.7) 2 

N ki 2/γ i=1 exp − 2 dki RN |(k)|

valid for every positive integer γ ∈ N (i.e. for all powers 2/N of the Vandermonde) was derived in [10] in order to compute (7.6). (See Appendix B.) The matrix integrals K(z, N, α) and J (z, N, γ ) are closely related: Obviously J (z, N, 1) = K(z, N, 1) .

(7.8)

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M. Mulase, A. Waldron

The identity   1 1 J (z, 2N, 1) − K(z, N, 2) − K(z, 2N, 1) 2 2 1 = J (z, 2N, 1) 2 1 (2q + n − 2)!(22q−1 − 1) + b2q (2N )n (−z)2q+n−1 , (7.9) (2q)! n! 2

J (2z, 2N, 2) =

q≥0,n>0 1−2q−n<0

expresses the γ = 2 GOE case in terms of the γ = 1 oriented hermitian result and a sum over non-orientable contributions with odd Euler characteristic. Notice, as claimed above, in the graphical expansions of the orthogonal Penner model J (2z, 2N, 2) and the symplectic Penner model K(z, N, 2), non-orientable surfaces of odd genera do not contribute. This corresponds to the fact that the orbifold Euler characteristic of the moduli space of smooth real algebraic curves of odd genus 2q + 1 with n marked points is 0 for any value of q ≥ 0 and n > 0. More importantly, observe that the GSE and GOE formulæ (7.5) and (7.9) almost coincide except that the GOE formula is for matrix size 2N and the non-orientable odd Euler characteristic terms differ by an overall sign. (The appearance of 2z rather than z will be cured by the appropriate normalization given below and in the master formula (5.2).) This is precisely the duality (5.3). Finally, we show that the duality between GOE and GSE extends to arbitrary positive integers for the two types of Penner integrals introduced in this section. Let r ∈ N ∪ 1/N and set I (z, N, r)



 = lim log  m→∞

RN

    j 2m ki  z j/2−1 dki exp − i=1 j =2 j rN     2

k N i 2r i=1 exp − 2 dki RN |(k)|

|(k)|2r 

N

∈ (Q[N, N −1 , r, r −1 ])[[z]]. Then we have

 I (z, N, r) =

 z  K  αN , N, α , r = α ∈ N , J γNz , N, γ , r = γ −1 ∈ 1/N .

(7.10)

(7.11)

From inspection of the explicit asymptotic expansion formulæ of (B.1) and (B.9) presented in Appendix B, we obtain an extended duality I (z, N, r) = I (z, −rN, r −1 )

(7.12)

for an arbitrary positive integer r. This is in agreement with the duality (5.3) for r = 1, 2. 8. The Central Limit Theorem To prove a central limit theorem for large matrix size N , we need to show that the leading dependence is Gaussian in the coupling constants tj . More precisely, define the Gaussian

Duality for Matrix Integrals

577

expectation value of f (X) for GOE (α = 1/2), GUE (α = 1), and GSE (α = 2) as    2 f (X) [dX](2α) exp − Nα tr X 2   f (X) (N,α) = , (8.1)  2 [dX](2α) exp − Nα tr X 2 and consider



  αtj j exp tr X j j (N,α) .  V (t, N, α) =   αtj  j exp j j tr X

(8.2)

(N,α)

The expansion formula   (5.2) shows  that the contribution of a connected M¨obius graph αtj j  ∈ G to log exp is j j tr X (N,α)

2α

1 2 χ(S )

N

χ(S )−v

(3 − α −1 − α)

1− 21  − 21 χ(S )



(α −1/2 − α 1/2 )

|Aut()|

Since

αtj  j

j

tr X j



v

(j )

tj  .

(8.3)

j

 (N,α)

is the sum of all contributions from 1-vertex M¨obius graphs, we see that log V (t, N, α) has no terms coming from 1-vertex graphs. In particular, it has no terms with a positive power of N . Indeed, the power of N in (8.3) is strictly positive only when χ (S ) = 2 and v = 1, i.e.  is an orientable planar graph with one vertex. Therefore, limN→∞ log V (t, N, α) consists of contributions from graphs that have two or more vertices and χ(S ) − v = 0. But this is possible only when χ (S ) = v = 2. In other words,  is an orientable planar graph with exactly two vertices contributing (see (8.3)) 2α t j tj , |Aut()| 1 2 where j1 and j2 are the valences of the two vertices of . Altogether, we have established the following Theorem 8.1. The Central Limit Theorem for GOE (α = 1/2), GUE (α = 1) and GSE (α = 2) ensembles is     αtj j j j tr X  exp (N,α)    lim log       N→∞ αtj j exp j j tr X =

 connected, oriented, planar 2-vertex ribbon graph

(N,α)

α tj tj , |Aut R()| 1 2

(8.4)

where j1 and j2 are the valences of the two vertices of the ribbon graph . We notice that the formula is the same for all three ensembles except for the overall factor of α. In particular, only oriented planar ribbon graphs contribute in the large N limit. This mechanism was observed long ago by ’t Hooft in the hope that large N quantum chromodynamics could be solved exactly [24].

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M. Mulase, A. Waldron

9. Conclusions Let us tabulate the patina of results gathered here: • The asymptotic expansion of the three Gaussian random matrix ensembles is expressed as a sum over M¨obius graphs. • These expansions are related by a duality α −→ α −1 and N −→ −αN.

(9.1)

The α = 1 GUE model is self-dual and sums over only ribbon graphs. The duality between α = 1/2 GOE and α = 2 GSE models amounts to an equality of graphical expansions up to a factor (−1)χ(S ) for any graph . • When specialized to Penner model couplings, the Selberg integral representation yields an asymptotic expansion for all α ∈ N ∪ 1/N and the duality (9.1) holds for this extended set of α’s. Therefore, the first and probably most interesting question one might pose is whether our graphical expansion formulæ can also generalized to the extended set of α ∈ N ∪ 1/N, i.e. '

$

Matrix Integrals α = 1/2, 1, 2 &

'



$

Penner Model α ∈ N ∪ 1/N &

% @ @ @ R @ '

$

Graphical Expansion α = 1/2, 1, 2, ?? %

&

%

Let us briefly postpone a discussion of this issue while enumerating several other questions for which we have no immediate answers: (1) Do there exist matrix models where the duality holds exactly, without a factor (−1)χ(S ) ? (2) What is the significance of the minus sign in the transformation N → −αN ? Is there an interpretation where traces over N × N matrices are replaced by a supertrace and in turn bosonic matrix integrals by fermionic ones (cf. [9])?

Duality for Matrix Integrals

579

(3) Why the factor α in the transformation N → −αN ? Is there a generalization of the GSE dual to the GOE at odd values of N ? After this disquisitive interlude, we return to the postponed question. Let us examine the generality of Lemma 3.3 which claimed that a topological invariant of a punctured surface S was obtained by counting signed configurations of units {1, e1 . . . , e2α−1 } on the associated graph . Its proof relied on the following: (i) The units all square to ±1 and {±1, ±ei } is a group. (ii) At any vertex, their product was ±1 and therefore real. (iii) Units whose square was (−)1 were joined by Wick contractions of (anti)symmetric matrices. Therefore generalized matrix models whose graphical expansion is completely determined via our methods can be written down based on a larger group of “imaginary” units {±1, ±fa ; ±ei : fa2 = 1, ei2 = −1}. Simple examples are generated by considering elements drawn from Clifford algebras. i.e., the Pauli matrix representation of the quaternions can be generalized to larger sets of higher dimensional Dirac matrices. Although these Clifford type models seem not to generate theories with the new values of α exhibited in the Penner model, it would be interesting to investigate whether new matrix models of this type can indeed be constructed. Acknowledgement. The authors thank G. Kuperberg, M. Penkava, A. Schilling, A. Schwarz, A. Soshnikov, W. Thurston and J. Yu for both stimulating and useful discussions.

Appendix A. Quaternionic Feynman Calculus Since the quaternions are the last real associative division algebra, it is natural to develop a manifestly quaternionic Feynman calculus. Again, let us consider GOE, GUE and GSE models all at once via the unified notation X=S+

β−1

ei Ai ,

(A.1)

i=1

where the N × N matrices X are real, complex or quaternionic self-adjoint 

X† ≡ X = X ,

ei ≡ −ei ,

depending on the value of β = 1, 2 or 4, respectively. To begin, we need a shift identity   1    exp f (B) = f (X + B) , tr B X + XB 2

(A.2)

(A.3)

where the “background variable” B≡S+

i

ei A i = B †

(A.4)

580

M. Mulase, A. Waldron

and the N × N matrix of derivatives ∂ is given by     ∂ 1 ∂   2 ∂ S ab − i ei ∂ Aab , a = b , i ∂ab =   ∂ , a = b. ∂ S aa

(A.5)

The identity (A.3) holds thanks to the commutation relations !1   " tr ∂  X + X∂  , B = X . 2

(A.6)

Note also that ! 1  " ∂, tr B  X + XB  = X , 2

(A.7)

although [∂, B] = I only for the real and complex cases β = 1, 2. We may now rewrite matrix integration as differentiation5 

∞  1  tj [dX](4) exp − tr X 2 + tr X j 2 j j =1

 =



[dX](4)exp −

1 tr X 2 + 2

# ## tr (X + B)j ## j #

∞ tj j =1

# # ∞  1  2 1    # tj  2 j # = [dX](4)exp − tr X − ∂ + tr ∂ exp tr B # 2 2 j # j =1 # B=O  ∞  1   ##  1 tj 2 2 j # exp tr ∂ exp tr B # = [dX](4)exp − tr X . (A.8) 2 2 j # j =1 

B=O

B=O

The first factor on the last line is just an overall normalization while the two exponentials can be expanded in terms of Feynman diagrams: the nth order term in the expansion of each exponential is interpreted as either n edges or n vertices, respectively. Let us give some details: The operator 21 tr ∂ 2 acting on a quantum variable, yields 1 [ tr ∂ 2 , B] = ∂  , 2

(A.9)

which is represented graphically as attaching a ribbon edge to a vertex since the operator 1 2 2 tr ∂ is to be viewed as an edge. Note that by this rule, a vertex emitting a Bab is replaced with one emitting ∂ba which amounts to a twist. Attaching the other end to an adjacent vertex yields   1 1 ∂ab Bcd = β δac δbd + (2 − β) δad δbc . (A.10) 2 2 5 The result is equivalent to the one obtained using a Schwinger source term. The reformulation presented here is often called the background field formalism; a simple account may be found in the on-line textbook [22].

Duality for Matrix Integrals

581

Note that the brackets on the left hand sides above indicate that we are computing the derivative rather than allowing it to continue acting to the right as an operator. In particular, observe that for the GUE case, β = 2, so no twisted ribbon graphs can appear. For the GOE and GUE models we are done, one simply attaches all possible edges to vertices using the above rules and finds the usual known results. The symplectic case is more subtle however, thanks to the quaternionic non-commutativity of ∂ and B. In particular 

     ∂ab f (B) Bcd =  ∂ab f (B) Bcd + f (B) ∂ab Bcd

(A.11)

for some function f of the quaternionic matrix B. i.e., the quaternion valued operator ∂ does not satisfy the Leibniz rule. However, a generalized Leibniz rule does apply: First note that for any

Q=S+



eα A α ,

(A.12)

α

where the real matrices S and Aα need not have any definite symmetry properties (so that Q is not necessarily self adjoint) we have6

eα Q eα = − Q + (2 − β) Q .

(A.13)

α

Therefore we have the generalized Leibniz rule     1 ∂ab Q Bcd − ∂ab Q Bcd = δac δbd Q − (2 − β)(δac δbd − δad δbc )Q . 2

(A.14)

  Note that for β = 1, 2 the right hand side is equal to Q ∂ab Bcd , expressing the commutativity of real and complex numbers. This relation is the central graphical rule for our quaternionic Feynman calculus and is depicted in Fig. A.1. Note that it reverts to the rule (A.10) when Q = 1. It is now possible to compute any term in the expansion of (A.8) in terms of graphs. However, for quaternionic matrices, when connecting vertices with ribbon edges, intermediate vertices and unconnected edges may be twisted and/or flipped according to (A.1). It is easy, but tedious to verify that the results for simple graphs coincide with our general formula (5.2). 6 When β = 1, the sum on the left hand side is empty and equal to zero, while the right-hand side vanishes for real Q.

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M. Mulase, A. Waldron

∂



Q B $ %& '



 B Q $ %& '



∂

−

2−β 2 Q

Q $ %& '

2−β 2 Q

$ %& '

=

−

$ %& '

+

Fig. A.1. Quaternionic Feynman rule. Observe how connecting vertices with edges change vertices yet to be connected

B. Generalized Penner Model In this Appendix, we derive the asymptotic expansion formula K(z, N, α)



 = lim log 

RN

m→∞

=

∞ m=1

   

2m zj/2−1 j 2α (k) N exp − k i=1 j =2 i dki  j    2 

N ki 2α RN  (k) i=1 exp − 2 dki ∞

1 b2m N z2m−1 + (−1)m α m N m zm 2m (2m − 1) 4m m=1

m

∞ [2] (−1)m (m − 1)!b2q m 1−2q 1 α (α + − 1)N m+1−2q zm (2q)!(m + 1 − 2q)! 2 m=1 q=0 m

−

m+1

[ 2 ] [ 2 ]−q ∞ m=1 q=0

s=0

(−1)m (m − 1)!b2q b2s α m+1−2q N m+2−2q−2s zm , (2q)!(2s)!(m + 2 − 2q − 2s)! (B.1)

which is valid for every positive integer α. Here the bn ’s are the Bernoulli numbers defined by ∞ n=0

bn

tn t = t . n! e −1

The key techniques are the Selberg integration formula, Stirling’s formula for (1/z) and the asymptotic analysis of [19]. First we note that as an asymptotic series in z when

Duality for Matrix Integrals

583

z → 0 while keeping z > 0, we have 



lim log 

m→∞

2α (k)

RN

N

exp −



1 2

1 z

z e z

1 z

2m j/2−1 z j =2

i=1

( = log



n z

j

j ki





dki 



αN(N−1) 2

2α

 (k) [0,∞)N

N

ki

1 z

) (B.2)

e ki dki .

i=1

(For the mechanism changing the integration from RN to [0, ∞)N , we refer to [19].) This integral can be calculated by the Selberg integration formula:  2α (k)

N

[0,∞)N

1

eki kiz dki =

N−1

(1 + α + j α)(1 +

+ j α)

(1 + α)

j =0

i=1

1 z

.

Therefore, we have 



lim log 

m→∞

=c+

2α

RN

 (k)

N



exp −

2m j/2−1 z

j

j =2

i=1

j ki





dki 

 N−1

 N 1 N N αN(N − 1)  1 + + jα , log z + + log z + log z + log 2 z z 2 z j =0

(B.3) where c is the constant term independent of z. Since (B.1) does not have any constant term relative to z, here and below we ignore all constant terms independent of z (but possibly N dependent). The product of -functions can be calculated by the recursion formula, noticing that α is an integer: N −1

j =0



1  1 + + jα z

 = (1/z)

N

iα  N−1



i=0 j =0

1 + iα − j z



 N N−1 N−i−1 α 

1 N 1 + iα + j = (1/z) z z i=0 j =1

 N N−1

α−1

 1 + z(1 + (i − 1)α + j ) N−i N 1 = (1/z) . z z i=1 j =0

(B.4)

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M. Mulase, A. Waldron

We now apply Stirling’s formula for log (1/z) to obtain, up to a constant term: log

N −1

j =0



1  1 + + jα z



∞

=−

b2m N N N + log z + N z2m−1 log z − z 2 2m(2m − 1) z m=1

−N log z − +

αN(N − 1) log z 2

∞ N−1 α−1 1 (−1)m−1 (N − i)(1 + (i − 1)α + j )m zm. m

m=1 i=1 j =0

(B.5) We note that all negative powers of z and log z related terms in (B.3) cancel out using (B.5). Finally, we obtain K(z, N, α) =

∞ m=1

+

b2m N z2m−1 2m(2m − 1)

α ∞ N−1 1 (−1)m−1 (N − 1 − i)(iα + j )m zm . m

(B.6)

m=1 i=0 j =1

This last sum of powers can be calculated using Bernoulli polynomials, from which (B.1) follows. Using a formula for Bernoulli numbers, n   n   2n 2n (1 − 2n)b2n = b2q b2n−2q = b2q b2n−2q 22q , n = 1, 2q 2q q=0

q=0

and noting that b2 = 1/6, we recover the original formula of Penner for α = 1 [21]: m

[2] ∞ ∞ b2m (m − 1)!(2q − 1) 2m−1 K(z, N, 1) = − Nz b2q N m+2−2q (−z)m + 2m (2q)!(m + 2 − 2q)! m=1

=

m=1 q=0



g≥0,n>0 2−2g−n<0

=

∈R

(2g + n − 3)!(2g − 1) b2g N n (−z)2g+n−2 (2g)!n!

(−1)e N f (−z)e −v . |Aut R()|

(B.7)

For α = 2, (B.1) simplifies again: K(z, N, 2) = −

∞ b2m N z2m−1 2m

m=1

m

∞ [2] (m − 1)!(2q − 1) 1 b2q 2m+2−2q N m+2−2q (−z)m + 2 (2q)!(m + 2 − 2q)! m=1 q=0

Duality for Matrix Integrals

585

m

∞ [2] (m − 1)!(22q−1 − 1) 1 − b2q 2m+1−2q N m+1−2q (−z)m . 2 (2q)!(m + 1 − 2q)!

(B.8)

m=1 q=0

Note that the first two lines of (B.8) are identical to the Penner model 21 K(z, 2N, 1). The following integral formula, again valid for every positive integer γ ∈ N, has been established in [10]: J (z, N, γ )

   

N 2m zj/2−1 j 2/γ |(k)| exp − k i=1 j =2 i dki  j  RN = lim log     

N ki2 m→∞ 2/γ dk exp − |(k)| N i i=1 R 2  m  2m−1 ∞ ∞ b2m 1 N z m m z = (−1) N + 2m (2m − 1) γ γ 4m γ 

m=1

−

1 2

m=1

[m ∞ 2] m=1 q=0 m

−

(−1)m (m − 1)!b2q 1 1 − 1−2q (2q)!(m + 1 − 2q)! γ

m+1

[ 2 ] [ 2 ]−q ∞ m=1 q=0



s=0

 N m+1−2q

 m z γ

(−1)m (m − 1)!b2q b2s 1 · 1−2s N m+2−2q−2s (2q)!(2s)!(m + 2 − 2q − 2s)! γ

 m z . γ (B.9)

Our duality (7.12) for the generalized Penner model follows from comparing (B.1) with (B.9). References 1. Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12(4), 1119–1178 (1999) 2. Baker, T.H., Forrester, P.J.: The Calogero-Sutherland model and generalized classical polynomials. Comm. Math. Phys. 188, 175–216 (1997) 3. Bessis, D., Itzykson, C., Zuber, J.-B.: Quantum field theory techniques in graphical enumeration. Adv. Appl. Math. 1, 109–157 (1980) 4. Bleher, P.M., Its, A.R., (eds.): Random matrix models and their applications. Mathematical Sciences Research Institute Publications, Cambridge: Cambridge University Press, 2001 5. Br´ezin, C., Itzykson, C., Parisi, G., Zuber, J.-B.: Planar diagrams. Comm. Math. Phys. 59, 35–51 (1978) 6. Br´ezin, E., Hikami, S.: Characteristic polynomials of random matrices. Comm. Math. Phys. 214(1), 111–135 (2000) 7. Br´ezin, E., Hikami, S.: Characteristic polynomials of real symmetric random matrices. Comm. Math. Phys. 223(2), 363–382 (2001) 8. Cicuta, G.M.: Topological expansion for SO(N ) and Sp(2N ) gauge theories. Lett. Nuovo Cimento 35, 87 (1982) 9. Cvitanovic, P., Kennedy, A.D.: Spinors in negative dimensions. Physica Scripta 26, 5 (1982) 10. Goulden, I.P., Harer, J.L., Jackson, D.M.: A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves. Trans. Am. Math. Soc. 353(11), 4405–4427 (2001) 11. Harer, J.L., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85, 457–485 (1986) 12. Hatcher, P.: On triangulations of surfaces. Topology and it Appl. 40, 189–194 (1991) 13. Itoi, C., Sakamoto, Y.: Diagrammatic method for wide correlators in gaussian orthogonal and symplectic random matrix ensembles. cond-mat/9702156

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