Commun. Math. Phys. 277, 1–44 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0347-7
Communications in
Mathematical Physics
From the N-body Schrödinger Equation to the Quantum Boltzmann Equation: a Term-by-Term Convergence Result in the Weak Coupling Regime D. Benedetto1 , F. Castella2 , R. Esposito3 , M. Pulvirenti1 1 Dipartimento di Matematica, Università di Roma ‘La Sapienza’ P.ale A. Moro 5, 0085 Roma, Italia.
E-mail:
[email protected];
[email protected]
2 IRISA & IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France.
E-mail:
[email protected]
3 Dipartimento di Matematica pura ed applicata, Università di L’Aquila, Coppito - 67100, L’Aquila, Italia.
E-mail:
[email protected] Received: 25 September 2006 / Accepted: 14 May 2007 Published online: 30 October 2007 – © Springer-Verlag 2007
Abstract: In this paper we analyze the asymptotic dynamics of a system of N quantum particles, in a weak coupling regime. Particles are assumed statistically independent at the initial time. Our approach follows the strategy introduced by the authors in a previous work [BCEP1]: we compute the time evolution of the Wigner transform of the one-particle reduced density matrix; it is represented by means of a perturbation series, whose expansion is obtained upon iterating the Duhamel formula; this approach allows us to follow the arguments developed by Lanford [L] for classical interacting particles evolving in a low density regime. We prove, under suitable assumptions on the interaction potential, that the complete perturbation series converges term-by-term, for all times, towards the solution of a Boltzmann equation. The present paper completes the previous work [BCEP1]: it is proved there that a subseries of the complete perturbation expansion converges uniformly, for short times, towards the solution to the nonlinear quantum Boltzmann equation. This previous result holds for (smooth) potentials having possibly non-zero mean value. The present text establishes that the terms neglected at once in [BCEP1], on a purely heuristic basis, indeed go term-by-term to zero along the weak coupling limit, at least for potentials having zero mean. Our analysis combines stationary phase arguments with considerations on the nature of the various Feynman graphs entering the expansion. 1. Introduction As it is well known, a large particle system in a rarefaction regime should be described by a Boltzmann equation, be it in the context of quantum or classical mechanics. However, while the rigorous validity of the Boltzmann equation has been proved for classical systems for short times [L], or globally in time for special situations [IP] (see Ref. [CIP]
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D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
for further comments), there is no fully rigorous analysis for the equivalent quantum systems. The problem is physically relevant yet, because quantum effects, although usually negligible at ordinary temperatures (except for few light molecules), happen to play a role in the applications at a mesoscopic level. We refer, for example, to the treatment of electron gases in semiconductors (for physical references, see the textbooks [RV, AM, Ch], as well as [Bo, CTDL]—see [MRS] for a mathematically oriented presentation). Therefore, establishing a well founded quantum kinetic theory is certainly interesting not only from a conceptual viewpoint but also from a practical one. In fact, kinetic descriptions for quantum systems, beside dilute gases, include dense weakly interacting systems, as e.g. electron gas in semiconductors, whose classical analogues rather yield diffusion processes in velocity, described by the Landau equation. One pragmatic way to introduce the quantum Boltzmann equation (see e.g. [CC]) is to solve the scattering problem in quantum mechanics and then to replace, in the classical Boltzmann equation, the classical cross section with the quantum one. A better logically founded approach is to consider the kinetic-like equation for the Wigner transform of a quantum state associated to a particle system. Such a transform, defined on the classical phase space, should converge towards the solution of a nonlinear quantum Boltzmann equation along the natural asymptotic process. The mathematical challenge is to prove such a convergence result. This is the strategy we adopted in the previous papers [BCEP1, BCEP2, BCEP3]. In [BCEP1], we consider a quantum N particles system in the weak coupling regime (see below for the definition). The key point is, we prove the Wigner transforms of the reduced density matrices satisfy a hierarchy of equations that is similar to the so-called BBGKY hierarchy encountered in the context of classical interacting particles. Besides we prove that the arguments developed by Lanford [L] in the classical context may be partly reproduced: the hierarchy may be solved iteratively, and we may write an explicit expression for the j-particles reduced Wigner transforms at any time t, as a function of the initial state of the system; this expression involves a huge series expansion which is naturally indexed by graphs; the latter are of the Feynman graph type; they take into account all possible particles’ interactions between time 0 and time t. This basic observation is the common ingredient in [BCEP1, BCEP2, BCEP3], and the present text. Now, the situation is as follows. In the classical context, Lanford [L] has shown it is possible to pass to the limit directly in the so obtained series expansion, and the convergence is uniform for short time. We refer to the book [CIP] for additional remarks. One important feature naturally is that the trajectories of classical particles are quite explicitly known in the classical context. In the quantum context particles become delocalized and their various interactions are much more delicate to enumerate. In particular, the delocalization effect gives rise to various highly oscillatory phase factors which have to be analyzed. Due to these new analytical difficulties, the result we proved in [BCEP1] is not a complete convergence result: we prove a subseries (of the complete series expansion expressing the state of the system at time t) converges, along the weak coupling limit, towards the solution of the desired quantum Boltzmann equation. The convergence is uniform for short times. We also present in [BCEP1] arguments of heuristic nature which tend to establish the terms we neglect at once, when passing from the complete series expansion to the retained subseries, indeed go to zero along the weak coupling limit. Starting from the same observation, the analysis is extended in [BCEP3] to tackle the more delicate low density regime (see below for the definition). There, the new difficulty lies in the identification of the so-called cross-section entering the limiting Boltzmann
Term-by-Term Convergence
3
equation, and we refer to this text for the details. The result in [BCEP3] again is a partial result: a subseries of the complete series expansion (expressing the solution at time t) is proved to converge, uniformly for short times, towards the natural limit. Last, while the works [BCEP1] and [BCEP3] tackle the case of statistically independent particles, the analysis is also extended in [BCEP2] to handle the physically realistic case of bosons or fermions: such particles are not statistically independent (they obey the Bose-Einstein, resp. Fermi-Dirac statistics), and the limiting Boltzmann equation needs to be modified accordingly [UU] (see also [HL, ESY, H])). Again, a partial convergence result towards the conveniently corrected quantum Boltzmann equation is established in [BCEP2] along similar lines. Note that the situation is much better understood in the linear context, namely that of a single particle in a given random field: here the limiting equation is a linear Boltzmann equation, which has been rigorously derived for short times ([Sp]) and, more recently, globally in time (see [EY1, EY2], see also [EE] for the low density regime). With this overall picture in mind, the main contribution of the present text is the following: as in [BCEP1], we consider the weak coupling limit for statistically independent particles; we first write down the complete series expansion, or more precisely the Feynman graph expansion, which relates the state of the system at time t as a function of its initial value; we analyze the term-by-term limit of the full expansion; we eventually establish the limiting value satisfies the correct quantum Boltzmann equation. In other words, we are able to prove here that the terms we neglected at once in [BCEP1], when passing from the full expansion to a subseries, indeed go to zero term-by-term along the weak coupling limit. Two important new points allow us the present extension of the analysis performed in [BCEP1]. On the one hand, we are able to put forward a general stationary phase argument that allows us to treat in full generality the highly oscillatory phase factors entering the complete series expansion. On the other hand, we are able to characterize the various Feynman graphs entering in the expansion. Unfortunately, our analysis still has two weaknesses. First, the present result states a term-by-term convergence only, and uniform bounds are still missing. This is mainly due to the following fact: the graphs that naturally index the complete series expansion are “too numerous” at each order of the expansion; this gives rise to “too large” combinatorial factors in the analysis; in turn, this phenomenon prevents us from being able to provide a clear uniform bound. In any circumstance, we remark that the term-by-term convergence is a conceptually delicate point: the irreversible nature of the Boltzmann equation compared with the time-reversible character of the Schrödinger evolution emerges exactly there. This aspect of the analysis is clear in the Lanford proof, performed for classical particles in the low density regime. In that case, the term-by-term convergence follows easily by “direct inspection”, due to the fact that the classical evolution is somehow explicit. At variance, and as pointed out by Uchiyama (see [U], and [CIP] for further comments), the natural Hamiltonian dynamics formally leading to the two dimensional Broadwell model, a very similar model with similar convergence issues, fails to converge at the fourth order term of the perturbation expansion, showing the validation of the Broadwell equation to be false. The second point is, our result requires the elementary interaction potential has zero mean value. This condition appears naturally in the analysis. It prevents a singularity associated with “collisions” corresponding to an exchange of zero momentum. This singularity is probably the analog of the grazing collision singularity which occurs in the classical case. However, while it is a standard fact that one can remove this singularity
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D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
in the classical framework, upon assuming the potential is short range, in the quantum framework the short range assumption is not enough because of the particles’ delocalization: the latter makes the effective interaction range actually infinite, even for compactly supported potentials. Now, the observed singularity that comes up in our analysis, which is associated with particles exchanging zero momentum, should in principle be removed by compensation between the gain and loss terms. Yet making this statement rigorous requires regularity conditions which are hard to propagate along a general graph. For this reason, we give up with such a strategy and make the restrictive hypothesis of a potential having zero average. This is certainly not optimal, but it has the advantage of providing a sufficiently simple proof. Finally, we point out that the results we present here are limited to the unphysical case of particles obeying the Maxwell-Boltzmann statistics. In other words our analysis, based on computations that are fully expressed in classical phase space, requires particles to be described by fully factorized initial states of the form (see (2.17) below) f jN (t = 0, x1 , v1 , . . . , x j , v j ) ≡
j
f 0 (x j , v j ),
(1.1)
k=1
where we refer to Sect. 2 for the precise definitions and notation, and simply mention f jN (0) roughly is the j-particles distribution function of the underlying j-particles system at the initial time, while f 0 is a given one-particle density function. We wish to discuss here how our approach may be extended to the physical situation of particles obeying either the Bose-Einstein or the Fermi-Dirac statistics. As shown in [BCEP2], taking into account the statistical dependence between particles requires to modify in a deep way the structure of the initial state. Indeed, product states of the form (1.1) are not relevant for fermionic nor bosonic particles, because the associated statistics creates correlations. More precisely, free Bosons or Fermions are usually described by states which are usually called “quasi-free” and involve, in some sense, the minimal correlations that are compatible with the quantum statistics. The remarkable fact is, one may fully characterize these states in classical phase space: the j-particles distribution function of a quasi-free state is of the form π f jN (0) ≡ (±1)s(π ) f jN (0) , (1.2) π
where π is the generic permutation between indices {1, 2, . . . , j}, s(π ) is its signature, ±1 ≡ 1 for bosons, and ±1 ≡ −1 for fermions, while we have set π 1 N f j (0) (x1 , v1 , . . . , x j , v j ) ≡ dy1 . . . dy j dw1 . . . dw j (2π )3 j j x + x x −x y +y ε k π(k) i yk ·vk +iwk · k επ(k) −iwk · k 2π(k) − (yk − yπ(k) ), wk . e × f0 2 4 k=1
(1.3) Here, ε is the small parameter entering the weak-coupling limit we discuss throughout this text (see Sect. 2). Note that expression (1.3) is formula (2.29) in [BCEP2]. A simple π
stationary phase analysis shows each f jN (0) goes to zero weakly as ε → 0 whenever identity obviously reduces to the tensor product in (1.1). In π = identity, while f jN (0)
Term-by-Term Convergence
5
π other words, the correlation terms f jN (0) with π = identity are definitely small in a weak topology due to the fact they involve fast oscillating phase factors. Yet at later times t > 0, correlations do induce finite macroscopic effects caused by the interaction between particles: the expected limiting kinetic equation in the case of Fermions/Bosons is the Uehling-Uhlembeck equation, which corrects the structure of the usual collision operator, a quadratic term in the distribution function, by cubic contributions. We have actually proved in [BCEP2] that the quantum dynamics indeed agrees, in the weak-coupling limit, with the Uehling-Uhlembeck equation up to the second order terms (in the potential) of the perturbative expansion. In this picture, the generalization of the present paper’s result to Bosons and Fermions is highly nontrivial. Indeed, our basic approach relies on a complete Feynman graph expansion of the solution at time t, where each term of the expansion turns out to be associated with a specific highly oscillatory phase factor, the latter being analyzed using algebraic and stationary phase analytic arguments. The set of graphs (and associated phase factors) to be considered is obviously much larger in the case of Fermions/ Bosons, due to the contribution of all permutations π in (1.2)-(1.3): the present paper only estimates those terms stemming from the permutation π = identity. However, the analysis in [BCEP1, BCEP2], and that in the present paper, allow us to conjecture what is the family of graphs which gives the correct equation, and how one may show the other terms are vanishing. In this direction yet, there is still another difficulty that is specific to the case of fermionic/bosonic particles: as observed in [BCEP2], some of the new graphs associated with correlation terms are actually diverging as ε → 0, and only subtle cancellation effects between such terms allow us to prove the latter still do not contribute to the limit. In conclusion, although the case of bosonic/fermionic particles can certainly be approached with the present techniques, it is still open at this stage, and its solution requires a technical effort supported by new algebraic and analytical ideas. The paper is organized as follows. In Sect. 2, we first present the basic model, the associated scaled Schrödinger equation, and its Wigner transform. The latter solves an N −particles kinetic-like equation. We next derive the quantum BBGKY hierarchy that is associated with this kinetic model. We solve it iteratively as discussed before, and express its solution as a complete series expansion. This part of the discussion gathers arguments previously developed in [BCEP1]. On the other hand, we also present the limiting quantum Boltzmann equation that is to be derived, and express its solution as a complete series expansion as well, using an iterative argument. At this level, the solution of the interacting particles system, and the solution of the Boltzmann equation, are given through two different series expansions. With these expressions at hand, we state our main term-by-term convergence result. The remainder part of the paper is dedicated to the proof of our main result. Section 3 is mainly devoted to presenting the Feynman graph interpretation of the various series expansions, and Sect. 4 is devoted to sorting out the graphs. Each Feynman graph represents a different interaction history, between time t and the initial time. Section 5 presents three lemmas which allow for a very precise description of the various impulse exchanges along the particles’ interactions, depending on the type of Feynman graph (or: collision history) we are dealing with. Last, Sects. 6 through 9 are devoted to the analysis of four different types of graphs, corresponding to a natural partition of all possible graphs. The first two types (Sects. 6 and 7) are treated using a general stationary phase argument, in conjunction with the combinatorial information discussed in the previous Sect. 5. The last two types (Sects. 8 and 9) are treated using the fact the potential has zero mean value: it allows to balance
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a natural singularity in the problem, and to conclude at once. Again, this part of the analysis uses the combinatorial information discussed in Sect. 5. The present study holds in any dimension d ≥ 3. 2. The Model and its Scaling Limit – Statement of our Main Result 2.1. The N -body Schrödinger equation in the weak coupling regime. We consider a quantum system of N identical particles in Rd , located at the positions xi ∈ Rd (i = 1, . . . , N ). We assume the mass of the particles, as well as , are normalized to unity. We also assume all pairs of particles interact through the same two body potential φ. The potential energy of the full N -particles system is U (x1 , . . . , x N ) =
φ(xi − x j ).
(2.1)
i< j
The N -body Schrödinger equation associated with the potential U reads 1 i∂t Ψ (t, X N ) = − N Ψ (t, X N ) + U (X N ) Ψ (t, X N ), (2.2) 2 N where N = i=1 xi and xi is the Laplacian with respect to variable x i , while X N ∈ Rd N is a shorthand notation for the collection of positions X N = (x1 , . . . , x N ). Equation (2.2), when supplemented with the appropriate initial datum Ψ (0, X N ), completely determines the dynamics of the quantum N -particles system under consideration. We are interested in the asymptotic behavior of the system (2.2) in the weak coupling regime, which we now describe. On the one hand, we rescale the Schrödinger equation (2.2) according to the hyperbolic space-time scaling x → εx,
t → εt,
(2.3)
where ε is a small dimensionless parameter. Simultaneously, we also rescale the potential as φ →
√
εφ.
(2.4)
Last, we impose the following relation between the large number N of particles, and the small parameter ε, namely N = ε−d .
(2.5)
The whole rescaling (2.3), (2.4), (2.5) means the density of particles, i.e. the typical number of particles per unit volume, is kept fixed of the order of unity. It also means we are interested in the behavior of the N -particles system over long times, √ of the order of 1/ε, in the case when the interaction potential φ is weak, of the order of ε. Incidentally, our time rescaling forces the associated space rescaling: particles travel at a speed of the order of unity, and we need to look at the behavior of the system over long distances of the order 1/ε as well.
Term-by-Term Convergence
7
The resulting scaled form of Eq. (2.2) is ε2 iε∂t Ψ ε (t, X N ) = − N Ψ ε (t, X N ) + Uε (X N )Ψ ε (t, X N ), (2.6) 2 x √ , N = ε−d . φε (xi − x j ), φε = ε φ where Uε (x1 , . . . , x N ) = ε i< j
(2.7) Again, Ψ ε (t, X N ) is fully determined by Eq. (2.6) and the initial datum Ψ ε (0, X N ), which is specified later on (see (2.17) below). The aim of this text is to perform the limit ε → 0 in (2.6)–(2.7), and to identify the asymptotic dynamics. We wish to prove the above system tends to be well described by a nonlinear Boltzmann equation (2.29) in the limit ε → 0, as physically expected in view of the comments below. The present limit is usually called a weak-coupling limit. It is characterized by the fact √ the potential interaction is weak, of order ε, and the density of particles is 1. Therefore the number of collision per unit time is ε−1 . Since the quantum mechanical cross–section in the Born approximation (justified because the potential is small) is quadratic in the potential interaction, the cumulated effect is of the order number of collisions × [potential interaction]2 = 1/ε × ε = 1. We also comment on the weak coupling regime for classical systems. Here a test particle suffers ε−1 small collision per unit time. Now the limiting equation is expected to be the so-called Fokker-Plank-Landau equation which describes a diffusion in velocity. No rigorous result is known up to now in this direction, although the linear case (a single particle moving under the action of an external random field) is well understood ([KP, DGL]), and exhibits the expected diffusive behavior. The different structure of the one particle kinetic equation for classical (diffusion in velocity) and quantum (jumps in velocity, as given by a Boltzmann equation) is a consequence of the different nature of the scattering mechanism for a single classical and quantum particle. Indeed a quantum particle has a small probability to be deflected by any angle for each collision, and the cumulative effect is a jump. On the contrary, a classical particle is always deflected by a small angle, and the cumulative effect yields a diffusion. Another possible scaling to be considered is the low-density limit. In this case φ = O(1) is unscaled but N = O(ε−d+1 ). This again results in a cumulated effect of the order of unity, yet the picture is different: particles “collide” only once per unit time in this scaling, but each “collision” now has a dominant effect, of the order unity at once. In the classical context this is nothing but the Boltzmann-Grad limit (see e.g. [CIP]). We also refer to [C] for the analysis of a low-density situation in a linear context. The present paper is only concerned with the weak coupling limit. As mentioned in the introduction, we prove here a term-by-term convergence result. It completes the partial convergence result previously obtained in [BCEP1]. On the other hand, we prove in [BCEP3] a partial convergence result concerning the low density regime. Due to the fact the potential interactions somehow are weaker in the weak coupling regime than in the low density regime, the former case only involves φ at lower order (the cross-section obtained in the eventual Boltzmann equation is proportional with φ 2 , see below), while the latter requires to consider a full series expansion in φ (the so-called Born series expansion). It in turn needs to be identified and summed up in the appropriate way, and the difficulties linked with the necessary control and identification of the Born series
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D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
expansion are treated in [BCEP3]. We do not know whether the techniques we develop in the present text may be adapted to transform the partial convergence result of [BCEP3] into a complete term-by-term convergence result, valid in the low density regime as well. 2.2. Transforming the N -body Schrödinger equation into a hierarchy of kinetic equations. In order to tackle the asymptotic analysis of the scaled Schrödinger equation (2.6)–(2.7), we follow the approach we introduced in [BCEP1]. We consider the Wigner function f N (t, X N , VN ) associated with the wave function ε Ψ (t, X N ) solution to the N -body Schrödinger equation (2.6)–(2.7). It is defined, see [W], as d N 1 ε ε ε N f (t, X N , VN ) = dY N eiY N ·VN Ψ t, X N + Y N Ψ ε t, X N − Y N , 2π 2 2 Rd N
(2.8) and VN ∈ Rd N is a shorthand notation for the velocity variable VN = (v1 , . . . , v N ). The quantity f N roughly is the distribution function, in phase-space, of the N -particles system under study, although a delicate point is f N does not have a definite sign yet. We refer e.g. to [Ba] for the general properties of the Wigner function and [LP] for more sophisticated considerations on the mathematical side. A standard computation establishes that f N = f N (t, X N , VN ) satisfies a kinetic transport equation, namely
N 1 vk · ∇xk f N = √ TNε f N , (2.9) ∂t + ε k=1
N
where the operator ∂t + k=1 vk · ∇xk is the usual free stream operator, and we have introduced the crucial operator ε TNε = Tr, , with (2.10)
1≤r <≤N ε Tr,
f
N
×f
N
(t, X N , VN ) = −i
σ =±1
σ Rd
dh i h·(xr −x )/ε (h) φ e (2π )d
h h t, X N , v1 , . . . , vr −1 , vr − σ , vr +1 , . . . , v−1 , v + σ , v , . . . , v N . 2 2 (2.11)
The reader may note the scaled Schrödinger equation (2.6)–(2.7) is not a semi-classical equation: the interaction potential is φ(./ε) instead of φ. This explains the fact that Eq. (2.9) is not similar to the transport equation along the Hamiltonian flow associated with φ: the interaction is here varying on the quantum scale, not on the semi-classical one. Equation (2.9) actually asserts the dynamics of f N is governed by two effects: free transport on the one hand (this is the left-hand side of (2.9)), and internal “collisions” inside the particles’ system on the other hand (this is the right-hand side of (2.9)). The
Term-by-Term Convergence
9
ε describes the “collision” of particle r with particle , and the total operoperator Tr, ε ator TN takes all possible “collisions” into account. “Collisions” involve a momentum transfer h. They may occur at distant places (xr = x ), which typically is a quantum feature. This fact is balanced by the highly oscillatory factor exp(i h · (xr − x )/ε), a natural counterpart. Here and below, f denotes the Fourier transform of f , normalized as follows: dh −i h·x f (h) = (Fx f )(h) = d x e f (x), f (x) = ei h·x f (h). (2.12) (2π )d
Rd
Rd
Still following the approach introduced in [BCEP1], we next introduce the partial traces of the Wigner transform, according to the formula f jN (t, X j , V j ) = d x j+1 . . . d x N dv j+1 . . . dv N Rd(N − j) N
Rd(N − j)
× f (t, X j , x j+1 , . . . , x N ; V j , v j+1 , . . . , v N ),
(2.13)
whenever 1 ≤ j ≤ N − 1. We also set the convention f NN ≡ f N , f NN+1 ≡ 0. In relation with f N , the function f jN roughly is the distribution function in phase-space of the j-particles subsystem, inside the larger N -particles system under study. From now on we shall suppose that, due to the fact that particles are identical, the objects we have introduced (Ψ ε , f jN ) are all symmetric in the exchange of particles. This assumption is satisfied by fully uncorrelated quantum particles, since they obey the Maxwell-Boltzmann statistics. The symmetry actually makes all j-particles subsystems equivalent (they do not depend on the very j particles that have been selected). This justifies in passing the notation “ f jN ” without further reference to the particles’ names. Proceeding then as in the derivation of the BBKGY hierarchy for classical systems (see [CIP]), it is readily deduced from (2.9) that f jN = f jN (t, X j , V j ) satisfies the following hierarchy of equations: ⎛ ⎞ j 1 N−j N ⎝∂t + vk · ∇k ⎠ f jN = √ T jε f jN + √ C εj+1 f j+1 , (1 ≤ j ≤ N ), (2.14) ε ε k=1
where T jε has been defined before (2.10)–(2.11), and the new operator C εj+1 , is C εj+1 =
j
ε Ck, j+1 ,
with
(2.15)
k=1
ε N Ck, j+1 f j+1 (t, X j , V j ) = −i σ σ =±1
×e
i hε (xk −x j+1 )
N f j+1
Rd
dh (h) φ (2π )d
Rd
d x j+1
dv j+1
Rd
h h . t, X j , x j+1 , v1 , . . . , vk−1 , vk −σ , vk+1 , . . . , v j , v j+1 +σ 2 2 (2.16)
ε Note that (2.9) is recovered from (2.14) upon setting j = N . The operator Ck, j+1 describes the “collision” of particle k, belonging to the j-particle subsystem, with a
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D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
generic particle outside the subsystem, conventionally denoted by the number j + 1. This conventional numbering uses the fact all particles are identical. The total operator C εj+1 takes all such collisions into account. As usual [CIP], Eq. (2.14) shows the dynamics of the j-particle subsystem is governed by three effects: the free-stream operator, the collisions “inside” the subsystem (the T term), and the collisions with particles “outside” the subsystem (the C term). Last, there remains to determine the initial value { f jN (0)} Nj=1 of the solution { f jN (t)} Nj=1 . We assume { f jN (0)} Nj=1 is factorized, that is, for all j = 1, . . . , N , ⊗j
f jN (0) = f 0 ,
(2.17)
where f 0 is a one-particle Wigner function which we also assume to be a probability distribution. These assumptions require comments. The factorization property simply means we are assuming, as usual, the particles be fully uncorrelated at the initial time. Moreover we recall a quantum state whose Wigner transform is a probability distribution is not in general a wave function but rather a density matrix. As a consequence the evolution equation we have to start with is not the Schrödinger equation (2.2) but the associated Heisenberg equation for the density matrix. In both cases the corresponding Wigner equation is anyhow Eq. (2.9), and the analysis remains unchanged. We are now considering f 0 as fixed, but we could also assume an initial condition depending on ε, as happens, for instance, if the one particle state is a suitable smooth superposition of coherent states. Our analysis would change in a minor way. Finally we remark that Eq. (2.17) is only compatible with the Maxwell-Boltzmann statistics, but not with the Bose-Einstein nor Fermi-Dirac statistics, for which the derivation of kinetic equations involves extra difficulties related to quantum correlations. The analysis we present here cannot be adapted in a direct way to the case of fermionic nor bosonic particles. New difficulties arise in these two situations and the limiting Boltzmann equation (2.29) below actually needs to be modified then. Roughly speaking for Bosons and Fermions we need to consider many more relevant Feynman diagrams. We may quote [BCEP2] for a rigorous analysis up to the second order term. We also refer to the introduction on that point. 2.3. Solving the hierarchy. Up to now we have simply transformed the original Schrödinger Eq. (2.6) into a hierarchy of equations (2.14). Our approach lies in performing the asymptotic process ε → 0 on the hierarchy itself, rather than on the original Eq. (2.6). Let us first develop some preliminary considerations in order to have an idea of the size of the operators entering the hierarchy (2.6). Expanding f jN (t) as a perturbation of the free flow S(t) defined as (S(t) f j )(X j , V j ) = f j (X j − V j t, V j ),
(2.18)
we find N−j f jN (t) = S(t) f j0 + √ ε
t 0
1 N S(t −t1 )C εj+1 f j+1 (t1 )dt1 + √ ε
t
S(t −t1 )T jε f jN (t1 )dt1 .
0
(2.19)
Term-by-Term Convergence
11
We now try to keep information on the limit behavior of f jN (t). To do so, we assume for the moment that the time evolved j-particles distributions f jN (t) are smooth, in the sense that the derivatives are uniformly bounded in ε. x −x First, setting r = k ε j+1 in Eq. (2.16), we recover N C εj+1 f j+1 (X j ; V j ; t1 ) = − iεd
j
σ
k=1 σ =±1
dh (h) φ (2π )d
dr
dv j+1 ei h·r
h h N X j , xk − εr ; v1 , . . . vk − σ , . . . v j+1 + σ × f j+1 2 2 =O(εd+1 ),
(2.20)
N is uniformly bounded. Indeed, setting ε = 0 in the integrand, the inteprovided Dv2 f j+1 gration over r produces a Dirac mass δ(h), hence the integrand is independent of σ and the sum over σ vanishes. Observing next 1 N−j = O(ε−d+ 2 ), √ ε
(2.21)
the second term in the right-hand side of (2.19) is hence seen to give a vanishing contribution in the limit ε → 0. Second it also is possible to prove 1 √ ε
t
S(t − t1 )T jε f jN (t1 )dt1
(2.22)
0
is weakly vanishing (see [BCEP1] for a proof), upon using a stationary phase argument, at least provided f jN is smooth. We are now facing the following alternative: either the limit is trivial, or the time evolved distributions f jN (t) are not smooth. As a consequence, and since we believe the limit is not trivial (actually we expect to get the Boltzmann equation, according to the previous discussion), a rigorous convergence proof seems problematic. The difficulty in obtaining a-priori estimates on the regularity of the f jN ’s induces us to exploit the full series expansion of the solution, in order to rather make use of the regularity of the initial datum itself: it allows us to keep the advantage of the oscillating phases, and to control the possibly diverging powers of ε. Let us come to the technical details. As in the case of the Boltzmann-Grad limit for classical systems, we first express the solution f jN (t) to the hierarchy (2.14) as a complete series expansion obtained upon iterating the Duhamel formula, namely f jN (t)
=
N −j n=0
(N − j) . . . (N − j − n + 1) εn/2
ε ×Sint (t
ε − t1 )C εj+1 Sint (t1
t 0
tn−1 dt1 . . . dtn 0
ε − t2 )C εj+2 . . . Sint (tn−1
⊗( j+n)
ε − tn )C εj+n Sint (tn ) f 0
. (2.23)
12
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
This expresses the state of the j-particles subsystem at time t as an explicit function of ε (t)g is the j-particle interacting flow, namely the solution the initial value f 0 . Here Sint j to the initial value problem ⎛ ⎞ j 1 ε ε ε ⎝∂t + vk · ∇xk ⎠ Sint (t)g j = √ T jε Sint (t)g j , Sint (0)g j = g j . (2.24) ε k=1
Note Eq. (2.24) is nothing else than Eq. (2.14), with the C term removed. ε (t)g as a perturbation of the free flow S(t)g . In doing We may in turn expand Sint j j so we find ε Sint (t) f j
=
1
m≥0
εm/2
t
t1
dtm S(t − t1 )T jε
dt2 . . .
dt1 0
tm−1
0
0
×S(t1 − t2 )T jε . . . S(tm−1 − tm )T jε S(tm ) f j .
(2.25)
We insert (2.25) into (2.23). In order to keep reasonably short formulae, we first rewrite (2.23) and (2.25) as, respectively, N −j
(N − j) · · · (N − j − n + 1) εn/2 n=0 ⊗( j+n) ε ε ε ε f0 × Sint C εj+1 Sint C εj+2 · · · Sint C εj+n Sint ,
f jN (t) =
ε Sint =
m≥0
1 (S T j )m S. εm/2
(2.26)
Here, we make an obvious abuse of notation, namely, the integral
in (2.26) stands t
for the integration over the set 0 ≤ tn ≤ tn−1 ≤ · · · ≤ t1 ≤ t, i.e. for
t1 dt2 · · · .
dt1 0
0
Now, insertion of (2.23) into (2.25) eventually results in the following formula: f jN (t)
=
N −j
n=0 m 0 ≥0 m 1 ≥0
···
(N − j) · · · (N − j − n + 1) ε(n+m 0 +···+m n )/2
m n ≥0
ε m1 ε × (S T jε )m 0 S C εj+1 (S T j+1 ) S C εj+2 · · · (S T j+n−1 )m n−1 ⊗( j+n) ε × S C εj+n (S T j+n )m n S f 0 .
(2.27)
Remark 1. For any given value of N and j, the normalizing prefactor in (2.27) satisfies (N − j) · · · (N − j − n + 1) ∼ ε−dn−(n+m 0 +···+m n )/2 , ε→0 ε(n+m 0 +···+m n )/2 due to the weak-coupling scaling N ∼ ε−d .
(2.28)
Term-by-Term Convergence
13
2.4. The limiting Boltzmann equation. In order to state our main result precisely, we next need to introduce some additional notations, in relation with the limiting Boltzmann equation which is to be derived. It is physically expected that the distribution function f jN (t) converges towards a tensor product f (t)⊗ j as ε → 0. This is the so-called “propagation of chaos”: initially ⊗j uncorrelated particles f jN (0) = f 0 tend to remain uncorrelated for all times as ε → 0, although the original dynamics tends to actually create correlations between particles through the interaction potential φ. There remains to describe the behavior of the one particle distribution function f (t) ≡ f (t, x, v). In turn, it is physically expected that f (t) satisfies the following nonlinear Boltzmann equation:1 ∂t f (t, x, v) + v · ∇x f = Q( f, f )(t, x, v), f (0, x, v) = f 0 (x, v), where (2.29) Q( f, f )(t, x, v) = dv1 dω B(ω, v − v1 ) f (t, x, v ) f (t, x, v1 ) − f (t, x, v) f (t, x, v1 ) , R3 ×S2
(2.30) and the cross-section is
B(ω, v) =
1 (ω (ω · v))|2 . |ω · v| |φ 8π 2
(2.31)
Here we have used the standard notations, namely the so-called impact parameter is ω ∈ S2 , the incoming or pre-collisional velocities are v ∈ R3 and v1 ∈ R3 , and the outgoing or post-collisional velocities are v = v − [v − v1 ] · ω ω,
v1 = v1 + [v − v1 ] · ω ω.
(2.32)
We stress the cross-section B is the only quantum factor in the otherwise purely classical Eqs. (2.29)–(2.30). It retains the quantum features of the elementary “collision events”, in that it depends on the microscopic interaction potential φ through formula (2.31). This relation is known as the “Fermi Golden Rule”. At this level, we have completely determined the physically expected limit f (t)⊗ j of f jN (t) as ε → 0, for any given value of j. Since we aim at passing to the limit in f jN (t) in the form given in (2.27), we last need to express f (t)⊗ j in a form that is close to the above expansion (2.27). Computations similar to those performed below for f jN show that the function f j (t, X j , V j ) ≡ f (t)⊗ j
(2.33)
satisfies a hierarchy of equations (as does f jN (t)), known under the name of the “Boltzmann hierarchy” see [CIP]. We do not write down the Boltzmann hierarchy for sake of simplicity. Needless to say, the fact that f j and f jN do satisfy parallel hierarchies actually is the main motivation for the kinetic point of view we adopted at once. Now, the Boltzmann hierarchy is easily solved iteratively, as we did for (2.27). Without giving 1 The equation is written here in dimension d = 3 only, for simplicity. Going to the general dimension d only affects the prefactor 1/(8π 2 ) × |ω · v| in (2.31), while the integral that defines Q( f, f ) carries over Rd × Sd−1 .
14
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
further details (see [BCEP1]), we simply mention f j (t) is given by the following series expansion:
f j (t, X j , V j ) =
t
tn−1 dt1 · · · dtn S(t − t1 ) C j+1
n≥0 0
0 ⊗( j+n)
× S(t1 − t2 ) C j+2 · · · S(tn−1 − tn ) C j+n S(tn ) f 0
, (2.34)
where the operator S(t) is the free flow defined earlier, and the classical collision operator C j+1 that is attached with the Boltzmann equation (2.29)-(2.30) is2 C j+1 =
j
Ck, j+1 ,
k=1
where
Ck, j+1 f j+1 t, X j , V j :=
× f j+1
(2.35)
dω B(ω, vk − v j+1 )
dv j+1 R3
S2
X j , xk , v1 , . . . , vk−1 , vk , vk+1 , . . . , v j , v j+1
− f j+1 X j , xk , v1 , . . . , vk−1 , vk , vk+1 , . . . , v j , v j+1 .
(2.36)
The cross-section B has been defined in (2.31), and the pre-collisional velocities are vk = vk − [vk − v j+1 ] · ω ω, v j+1 = v j+1 + [vk − v j+1 ] · ω ω, as in (2.32). Eventually, and using the same abuse of notations as in (2.27) as far as time intet grals dt1 etc. are concerned, we arrive at the following formula for the j-particle 0
distribution functions f j (t) ≡ f (t)⊗ j at time t, associated with the solution f (t) of the Boltzmann Eq. (2.29)–(2.30): f j (t) =
n≥0
⊗( j+n)
S C j+1 S C j+2 . . . S C j+n S f 0
.
(2.37)
Note that the convergence of the series expansion (2.37), at least for small values of time t, is proved in [BCEP1], following [L]. Moreover, proving that f jN (t) “converges” towards f j (t), now amounts to comparing the associated series expansions (2.27) (for f jN ) resp. (2.37) (for f j ).
2.5. Statement of the result. We are in position to state our main result. Main Theorem. Assume the initial state f 0 is smooth, in the sense that the following norm: α α (2.38) Nα ( f 0 ) = sup (1 + ξ 2 + η2 ) 2 (1 − ξ − η ) 2 f 0 (ξ, η) ξ, η
2 Again, formulae are given in dimension d = 3 only for simplicity.
Term-by-Term Convergence
15
is finite for some α > 2d. Assume the potential φ is smooth, i.e. the following norm3 α α (h) (2.39) Nα (φ) = sup (1 + h 2 ) 2 (1 − h ) 2 φ h
if finite for some α > d. Last, assume the interaction potential φ has zero mean value, namely (0) = 0. φ
(2.40)
Then, for any given j ≥ 1, and for any time t ≥ 0, the series expansion (2.27) that relates the value of f jN (t) = f jN (t, X j , V j ) converges term-by-term towards the series expansion (2.37) that relates the value of f j (t) = f j (t, X j , V j ). The convergence that is mentioned here refers to the fact the Fourier transform f jN (t, j , H j ) goes to f j (t, j , H j ) term-by-term, uniformly in j and H j . 3. A Feynman Graph Formulation of the Problem Before coming to the proof of our main theorem, we first need to reformulate the expansion (2.27) in more appropriate terms. On the one hand, the expansion (2.27) is more naturally indexed by graphs. They represent all possible interaction histories amongst the N particles, between the initial time t = 0 and the final time t; this is a Feynman diagram expansion. On the other hand, the asymptotic procedure is more easily performed on the Fourier transform of f jN (see the statement of the main theorem). This point necessitates to reformulate the collision operators T ε and C ε in Fourier variables. The present section is devoted to setting up these two aspects. For obvious reasons, we shall always restrict our analysis to the one particle distribution function f 1N (t) = f 1N (t, x1 , v1 ). Needless to say, the computations can be easily rephrased to prove the more general convergence of f jN (t) towards f (t)⊗ j . 3.1. Graphical representation. To begin with, since our√goal is to pass to the limit termby-term in (2.27), we observe all prefactors (N − j)/ ε, etc. involved in (2.27) may √ safely be replaced by N / ε at once (all other parameters are considered fixed). Hence, recalling that the weak coupling regime imposes N ∼ ε−d , all these prefactors may be replaced by the simpler and equivalent value ε−d−1/2 . This being settled, the generic term of the expansion (2.27), called T (t) in the sequel, always has the form T (t) = ε
−d(m−1)−n/2
t 0
tn−1 dt1 . . . dtn S(t − t1 )O1ε S(t1 − t2 )O2ε · · · Onε S(tn ) f m0 , 0
(3.1) where f m0 = f 0⊗m is the initial datum and each Okε is either a C ε or a T ε operator. We refer to (2.10)–(2.11), resp. (2.15)–(2.16) for the very definition of T ε , resp. C ε . 3 We use the same notation N for the two norms involved in Eqs. (2.38) and (2.39) though, strictly α speaking, they do not act on the same function spaces.
16
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
We note that, written in that way, the term T (t) involves m − 1 operators of type C ε . This freezes the value m. It also involves n − m + 1 operators of type T ε . This freezes the value n which is also the total number of “collisions”. The “collision times” are t1 , t2 , . . ., tn . Now, recall the two operators T ε and C ε admit the expansions Tkε =
k
ε Tr, ,
ε Ck+1 =
k
ε Cr,k+1 ,
(3.2)
r =1
1≤r <≤N
corresponding to all possible particle names r and that are involved in the interactions of each type. As a consequence, keeping the letter O ε for the generic C ε or T ε operator, each Okε in (3.1) may be split into Okε =
rk <k
Orεk ,k ,
(3.3)
where Orεk ,k denotes either Trεk ,k or Crεk ,k . For that reason, the whole string {S O1ε · · · S Onε S} in (3.1) may in turn be split into a similar sum. Since the operators ε with k > i “creates” the particle k, it will be called “creation operator” in the Ci,k ε will be called “recollision operator” since the particles i sequel. On the other hand, Ti,k and k have already delivered an interaction. With the above notations, the study of the term-by-term asymptotic behavior of the series expansion (2.27) reduces to that of the generic term T (t, x1 , v1 ) = ε
−d(m−1)−n/2
t dt1 . . . 0
tn−1 × dtn S(t − t1 )Orε1 ,1 S(t1 − t2 )Orε2 ,2 · · · Orεn ,n S(tn ) f m0 , (3.4) 0
for any value of the parameters m, n, and any sequence of indices {(rk , k )}nk=1 (rk < k ). Here, each operator Orεk ,k denotes either Trεk ,k or Crεk ,k . The generic term T (t) is completely determined by the sequence of indices {(rk , k )}nk=1 that is involved in formula (3.4). In this notation the pair (rk , k ) denotes the indices of the particles that actually “collide” at the collision time tk , be it a “creation” (operator of type C ε ) or a “recollision” (operator of type T ε ). In the sequel, we shall systematically identify the generic term T (t), the associated collision sequence {(rk , k )}nk=1 , and the associated graph (see below). It is useful to introduce the following graphical representation of the generic term T (t) in (3.4), in the spirit of Feynman diagram expansions. We refer to Fig. 1 for an illustrative example in the particular case of the collision sequence {(rk , k )}10 k=1 = {(1, 2), (1, 2), (1, 2), (2, 3), (1, 2), (3, 4), (3, 4), (3, 2), (4, 5), (4, 5)} , for which m = 5 (recall we restrict our attention to the study of f 1N , i.e. the number of particles under consideration at the final time t is one), and the total number of collision events of type T ε or C ε is n = 10.
Term-by-Term Convergence
17
t
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10
t=0 1
3
5
4
2
Fig. 1. The graph associated with the collision sequence {(1, 2), (1, 2), (1, 2), (2, 3), (1, 2), (3, 4), (3, 4), (3, 2), (4, 5), (4, 5)}
In Fig. 1 and as a general convention, the legs of the graph denote the particles themselves. Each leg represents one particle that is actually involved in the generic term (3.4). It carries the index of the corresponding particle. The nodes correspond to the creation of a particle, i.e. to a C ε operator. The horizontal segments correspond to a recollision, i.e. to a T ε operator. In both cases the node or the horizontal segment connect the two legs rk and k , corresponding to the particles actually involved in the associated collision event at time tk . The straight lines between two successive collision times tk , tk+1 represent the free flight S(tk − tk+1 ) entering the Duhamel expansion (3.4). Note that time runs backwards, from time t on the top, to time 0 at the bottom. To be complete, let us last mention a point of terminology: whenever an operator Crεk ,k is involved in (3.4), we shall say particle k is a son of particle rk , while rk is its father. For instance, in Fig. 1, particle 3 is the son of particle 2, etc. In the same vein, we shall typically call particle 2 an ancestor of particles 4 or 5. Note this terminology again uses the fact that time is thought of as running backwards in these diagrams, from time t at the top to time 0 at the bottom. In this spirit, we shall conventionally say a collision occurring at time ti occurs before collision occurring at time t j whenever ti > t j , i.e. i < j: the words “before” and “after” will be systematically used in reference to the indices of the collisional times, rather than in reference to the actual values of the latter. Amongst all the indices 1, . . . , n of the collision times {tk }nk=1 we shall distinguish those times that correspond to the creation of a new particle. We shall denote these indices by {z p }mp=2 . In other words, z p = j whenever particle p has been created at time t j = tz p .
(3.5)
In the example of Fig. 1, we have z 2 = 1, z 3 = 4, z 4 = 6, z 5 = 9. We shall set Z = {z 2 , . . . , z m }. If p is the index of a particle, we define the “cluster of p” denoted by C p , as being C p = {q | particle p is an ancestor of particle q} ∪ { p}.
(3.6)
18
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
In other words, the set C p is the list of all descendants of particle p (including p). In the example of Fig. 1, we have C1 = {1, 2, 3, 4, 5} (note the cluster of particle one always includes all the created particles), C2 = {2, 3, 4, 5}, C3 = {4, 5}, C4 = {4, 5}, C5 = {5}. Given C p we also denote by E p the set of indices of recollision times that involve one descendent of p with one particle that is not a descendent of p, i.e. recollisions that are somehow “external” to the cluster C p . In other words, we set / Z | collision number j involves particles q1 and q2 with q1 ∈ C p but q2 ∈ / C p }. Ep ={ j ∈ (3.7) In the example of Fig. 1, we have E 1 = ∅ (note E 1 is always void), E 2 = {2, 3, 5}, E 3 = {8}, E 4 = {7}, E 5 = {10}. 3.2. Fourier transform of the distribution function. In what follows, it will be more convenient to pass to the limit on the Fourier transform of the generic term T (t, x1 , v1 ) given by (3.4). For that reason, we introduce several notations that appear on the Fourier side. We define the Fourier transform of the f k0 ’s as f k0 (ξ ; η) = d xdv e−iξ ·x−iη·v f k0 (x; v), R2dk
f k0 (x; v) =
1 (2π )2dk
f k0 (ξ ; η), dξ dη e+iξ ·x+iη·v
(3.8)
R2dk
with ξ = (ξ1 , . . . , ξk ) ∈ Rdk and similarly for η. A simple computation shows the operators T ε and C ε , when written in Fourier variables, take the form σ −i ε r, (h) ei 2 h·(η −ηr ) f k0 (ξ ; η) = 1/2 σ dh φ T ε σ =±1
Rd
h h × f k0 ξ1 , . . . , ξr − , . . . , ξ + , . . . ξk ; η1 , . . . , ηk , (3.9) ε ε σ −i ε 0 (ξ ; η) = r,k+1 (h) e−i 2 h·ηr C f k+1 σ dh φ εd+1/2 σ =±1
Rd
h h 0 × f k+1 ξ1 , . . . , ξr − , . . . , ξk , ; η1 , . . . , ηk , 0 . ε ε
(3.10)
Similarly, the free streaming flow S(t) f k0 (x, v) = f k0 (x − vt, v) is given in terms of Fourier transform by S(t) f k0 (ξ ; η) = f k0 (ξ ; η + tξ ).
(3.11)
Note the vector (0, . . . , 0, −h, 0, . . . , 0, +h, 0, . . . , 0), where the “−h” is in position r , and the “+h” is in position , plays a particular role in formula (3.9) (it then should be seen as a vector in Rdk ) and in formula (3.10) (it then is a vector in Rd(k+1) ). As a consequence, it is natural to introduce the vectors θr, (h) ∈ Rdm , defined as θr, (h) = (0, . . . , −h, . . . , h, . . . , 0) ∈ Rdm ,
(3.12)
Term-by-Term Convergence
19
where −h and h are in the r th and th position respectively. In other words, if for p = 1, . . . , m, the notation e p designates the natural projector defined by e p : X = (x1 , . . . , xm ) ∈ Rdm −→ e p (X ) = e p · X = x p ∈ Rd , then θr, is defined through er θr, (h) = −h, e θr, (h) = +h, and
(3.13)
e p θr, (h) = 0 whenever p = r, . (3.14)
In (3.13) and later, the two notations e p (X ) and e p · X are used indifferently. In a similar way, we also introduce the vectors θk ∈ Rm defined as θk = (0, . . . , −1, . . . , 1, . . . , 0) ∈ Rm , where −1 and 1 are placed in the rk and k positions respectively, and (rk , k ) is the pair of colliding particles at the collision time tk .
(3.15)
Note also that, setting θk (h) = θrk ,k (h),
(3.16)
θk (h) · θk (h ) = (h · h )(θk · θk ).
(3.17)
we have the identity
In a sense, θk (h) is a multiplication of θk by h, so that we shall sometimes make the abuse of notation θk (h) = θk h.
(3.18)
The conclusion is that, using the above notations, the generic term T (t, x1 , v1 ) satisfies T(t, ξ1 , η1 ) = ε
−d(m−1)−n/2
(−i)
n
t
n
σj
σ1 ,...σn =±1 j=1
⎡
⎤
t1 dt1
0
tn−1 dt2 · · · dtn
0
0
n H T Sε 0 ⎣ ⎦ φ (h j ) f m ξ + ; η+tξ + exp i , × dh 1 dh 2 · · · dh n ε ε 2ε
j=1
Rnd
(3.19) where ξ = (ξ1 , 0, . . . , 0) ∈ Rdm , η = (η1 , 0, . . . , 0) ∈ Rdm , and H=
n
θjh j ∈ R
j=1
dm
,
T =
n
t j θ j h j ∈ Rdm .
(3.20)
j=1
The phase Sε in (3.19) is given by Sε = σ1 θ1 (h 1 ) · ηε(1) + σ2 θ2 (h 2 ) · ηε(2) + · · · + σn θn (h n ) · ηε(n) ,
(3.21)
20
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
where ηε(1) = ε [η + (t − t1 )ξ ] ,
ηε(2) = ε [η + (t − t2 )ξ ] + (t1 − t2 )θ1 (h 1 ), ... n−1 ηε(n) = ε [η + (t − tn )ξ ] + (tr − tn )θr (h r ).
(3.22)
r =1
Note that the structure of the graph identifying the term T under consideration, enters through the definition of H , T and Sε , namely through the form of the vectors θ j . Remark 2. In the sequel, the variables {h k }nk=1 are called the exchanged momenta (at the collision times {tk }nk=1 ). Remark 3. The arguments of f m0 in (3.19) may as well be written H T 0 f m ξ + ; η + tξ + ε ε Hm Tm H1 H2 T1 T2 0 , ,..., ; η1 + tξ1 + , , . . . , , = f m ξ1 + ε ε ε ε ε ε
(3.23)
up to denoting H p = e p · H and T p = e p · T . With these notations, an obvious yet very important property is the following: m p=1
H p = 0,
m
T p = 0.
(3.24)
p=1
These two identities come from the fact that for each j, the vector h j only appears in the two factors Hr j = ∗ − h j and H j = ∗ + h j , where the “∗”’s denotes some functions that depend on the h k ’s for k = j only. The two opposite signs give the result. Remark 4. The arguments of f m0 in (3.19) can be recovered upon looking at the exchanged momenta in the graph. For instance, in the case of Fig. 2, we have ⎛ ⎞ ξ1 + ε−1 (−h 1 − h 2 − h 3 − h 6 − h 9 ) ⎜ ⎟ ε−1 (h 1 + h 3 − h 4 − h 5 ) ⎜ ⎟ −1 ⎜ ⎟ H ε (h + h + h − h ) 2 4 6 8 ⎜ ⎟ (3.25) =⎜ ξ+ −1 ⎟ ε (h 5 − h 7 − h 10 ) ε ⎜ ⎟ ⎝ ⎠ ε−1 (h 7 + h 8 ) ε−1 (h 9 + h 10 )
and
⎛ ⎜ ⎜ ⎜ T η + tξ + = ⎜ ⎜ ε ⎜ ⎝
⎞ η1 + tξ1 + ε−1 (−h 1 t1 − h 2 t2 − h 3 t3 − h 6 t6 − h 9 t9 ) ⎟ ε−1 (h 1 t1 + h 3 t3 − h 4 t4 − h 5 t5 ) ⎟ ⎟ ε−1 (h 2 t2 + h 4 t4 + h 6 t6 − h 8 t8 ) ⎟ . (3.26) ⎟ ε−1 (h 5 t5 − h 7 t7 − h 10 t10 ) ⎟ −1 ⎠ ε (h 7 t7 + h 8 t8 ) ε−1 (h 9 t9 + h 10 t10 )
Term-by-Term Convergence
21
h1
t1 h2
t2
h3 h4
t3 t4
h5
t5 h6
t6
h7
t7 t8
h8 h9
t9
h10
t10 1
6
4
5 3
2
Fig. 2. The exchanged momenta
In that case, we also have z 2 = 1, z 3 = 2, z 4 = 5, z 5 = 7, z 6 = 9, as well as C1 = {1, 2, 3, 4, 5, 6}, C2 = {2, 4, 5}, C3 = {3}, C4 = {4, 5}, C5 = {5}, C6 = {6}, and last E 1 = ∅, E 2 = {3, 4, 8, 10}, E 3 = {4, 6, 8}, E 4 = {8, 10}, E 5 = {8}, E 6 = {10}.
4. Organizing the Graphs At the heuristic level, we know from [BCEP1] that there is only one class of T (t)’s, i.e. only one class of graphs or collision sequences {(rk , k )}nk=1 , that give rise to a non-vanishing contribution in the limit ε → 0. This is the class of collision-recollision sequences, i.e. graphs of the form (r , ) = (1, 2), (r2 , 2 ) = (1, 2), (r3 , 3 ) = (r3 , 3), (r4 , 4 ) = (r3 , 3), "# $ ! "# $ !1 1
%
. . . . . . , (r2 j−1 , 2 j−1 ) = (r2 j−1 , j + 1), (r2 j , 2 j ) = (r2 j−1 , j + 1), . . . . . . . ! "# $ (4.1) Equivalently, collision-recollision sequences correspond to contributions of the form ε ε SCrε ,3 STrε ,3 . . . SCrε2 j−1 , j+1 STrε2 j−1 , j+1 . . . S f m0 , ST1,2 SC1,2 ! "# $ ! 3 "# 3 $ ! "# $
where particle 2 is created from particle 1 (at time t1 ), then immediately recollides with particle 1 (at time t2 ), next particle 3 is created from particle r3 = 1 or 2 (at time t3 ), then immediately recollides with particle r3 (at time t4 ), etc. Such a graph is illustrated in Fig. 3, in the case of the specific sequence {(1, 2), (1, 2), (2, 3), (2, 3), (2, 4), (2, 4), (1, 5), (1, 5)}.
22
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
t1 t2 t3 t4 t5 t6 t7 t8 1
5 3
4
2
Fig. 3. A collision-recollision sequence
We actually proved in [BCEP1] that the subseries of f 1N (t) that is constituted of all collision-recollision sequences, namely g N (t) :=
N −1
k=0 1≤r1 ≤1 1≤r2 ≤2
···
1≤rk ≤k+1 ⊗(k+1)
· · · SCrεk ,k+1 STrεk ,k+1 S f 0
,
SCrε1 ,2 STrε1 ,2
SCrε2 ,3 STrε2 ,3
(4.2)
indeed converges, uniformly for short times, towards f (t), solution to the Boltzmann equation. In this perspective, our main theorem is proved once we are able to prove that all other graphs are vanishing as ε → 0. In order to do so, we need to classify the graphs in an appropriate fashion. ε is called proper whenever it involves two particles r and Definition 1. A recollision Tr, , r < , such that is a son of r .
Definition 2. A graph is called a right graph (see Fig. 3) whenever the following conditions are fulfilled: n = m − 1, or, in other words n − m − 1 = m − 1 (the number of “creations” C ε (1) 2 equals that of “recollisions” T ε ). ε , the graph also involves exactly one recollision T ε between (2) for any creation Cr, r, the same particles, ε ε (3) all creation-recollision events Cr, and Tr, involving the same two particles are ε occurs at the collision time t consecutive (i.e. the recollision Tr, j+1 whenever the ε creation Cr, occurs at the collision time t j ). Definition 3. A graph that satisfies the above properties (1) and (2) without satisfying property (3) is called right non-ordered graph (see Fig. 4). Definition 4. A graph that is neither right, nor right non-ordered, is called a wrong graph.
Term-by-Term Convergence
23
1
5 3
4
2
Fig. 4. A right non-ordered graph
Remark 5. In that language, the subseries g N (t) in (4.2) is made up of all right graphs. With these conventions, and on the basis we proved in [BCEP1], we arrive at the Proposition 1. In order to prove our main theorem, it suffices to prove that the generic term T(t, ξ1 ; η1 ) given by (3.19) goes to zero uniformly with ξ1 and η1 as ε → 0, whenever the associated graph is either right non-ordered, or wrong. The remainder part of this text is devoted to the proof of Proposition 1. 5. Three Basic Lemmas We give here three lemmas which are the basic ingredients to establish the desired Proposition 1, when conveniently combined with a stationary phase analysis we provide later in this text. The basic observation is the following. The arguments of f m in (3.19) (see Remarks 3 and 4), namely H T Hm Tm H1 H2 T1 T2 = f m0 ξ1 + , ,..., ; η1 + tξ1 + , , . . . , , f m0 ξ + ; η + tξ + ε ε ε ε ε ε ε ε involve the momenta {H p }mp=2 . The key point is, one can express the creation momenta {h z p }mp=2 , i.e. the exchanged momenta at the creation times {tz p }mp=2 , as an explicit function of the arguments {H p }mp=2 , and {h j } j ∈Z / . Quantitatively, the statement is the following Lemma 1. For any particle p = 2, . . . , m, the following identity holds true: Hq + π p, j h j , hz p = q∈C p
(5.1)
j∈E p
where π p, j = +1 if j is the collision index between a particle q ∈ C p and r ∈ / Cq with r > q, and π p, j = −1 when r < q (see Fig. 5)
24
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
tz q
tzq tzr tj
tz r
πp , j =+1
q r
p
πp ,j =−1
tj
p
q
r
Fig. 5. Value of π p, j
Proof of Lemma 1. Since
⎧ ⎪ ⎨−h i if p = ri , θi h i · e p = +h i if p = i , ⎪ ⎩0 otherwise,
(5.2)
we have the first stage identity Hp = hz p −
h zq −
q∈S p
* π p, h ,
∈R p
where S p is the set of particles directly generated by p (the sons of p), and R p is the set of recollisions of particle p, namely R p = { j | particle p recollides at time t j }. Last, π˜ p, is the sign of the collision with respect to particle p, i.e. π˜ p, = +1 if the particle which recollides with p at time t , say particle r , has been created after p, namely z p < zr , and π˜ p, = −1 otherwise. As a consequence, we directly obtain hz p = Hp + π˜ p, h + h z p1 . (5.3) ∈R p
p1 ∈S p
The idea now is to iterate (5.3) up to the next “generation”. Namely, we may write ⎛ ⎞ ⎛ ⎞ ⎝ H p1 + hz p = ⎝Hp + π˜ p, h ⎠ + π˜ p1 , h + hz p ⎠ ∈R p
⎛ = ⎝Hp +
⎞ π˜ p, h ⎠ +
∈R p
+
p1 ∈S p
p1 ∈S p p2 ∈S p1
p1 ∈S p
⎛
⎝ H p2 +
⎛
∈R p1
⎝ H p1 +
⎞
π˜ p1 , h +
p3 ∈S p2
= ···
2
π˜ p1 , h ⎠
∈R p1
∈R p2
p2 ∈S p1
⎞ h z p3 ⎠ (5.4)
and continue up to the elimination of the last h j , j ∈ Z , i.e. up to the last generation. The result is hz p = Hq + π˜ q, h . (5.5) q∈C p
q∈C p ∈Rq
Term-by-Term Convergence
25
t
tz
p
tz
q
i (p)
p q Fig. 6. Illustration of Lemma 2
Finally (5.1) is a consequence of the following fact. If j is the index of an internal recollision for p, i.e. of a collision between p and q with q ∈ C p , then the exchanged momentum h j appears exactly two times in the last sum of (5.5), yet with opposite signs (see also Remark 3): they sum up to zero. Hence the only terms surviving in the last sum of (5.5) are those involving an external collision of the cluster of p. They are indexed by the elements of E p . Recalling the definition of π p, j , Lemma 1 is now proved. We next give two lemmas which characterize, in the particular case when m−1 = n/2, i.e. in the case when the number m − 1 of creations equals the number n − (m − 1) of recollisions, the graphs that are either wrong or right non-ordered. Lemma 2. (Right non-ordered graphs). Let m > 2. Assume the graph T (t) is right non-ordered, and hence m − 1 = n/2. For each r = 2, . . . , m, denote by i(r ) the (unique) recollision index of particle r with the particle which created it (i.e. his father). Then the graph T (t) involves at least two particles p and q, p < q, such that z p < z q < i( p)
(5.6)
Remark 6. In other words, the fact that T (t) is non-ordered implies there are two particles p and q such that particle q is created after p, yet before particle p recollides with its father. Such a situation is excluded in the case of right graphs: there, each recollision has to immediately follow the associated creation. n Lemma 3. (Wrong graphs with m − 1 = n/2). Let m > 2 and m − 1 = . Assume 2 T (t) is a wrong graph. Then, the graph T (t) involves at least one particle p for which (here |E p | denotes the cardinality of E p ) i) |E p | ≥ 2, ii) ∃i ∈ E p for which π p,i = −1. Remark 7. In other words, the fact T (t) is wrong implies there is a particle p such that the set formed by all its descendants is involved at least twice in collisions with particles that do not descend from p. Besides, at least one amongst these collisions involves a descendent of p and another particle q (which is not a descendent of p), such that particle q is created before p. (Recall in passing that in the chosen terminology, p is considered a descendent of himself).
26
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
tz
r
tz
p
tz
q
p q
r
Fig. 7. Illustration of Lemma 3
Proof of Lemma 2. Since the graph is right non-ordered, there is a particle p, created at time tz p and recolliding with its father at time ti( p) , such that at least one other collision event occurs between these two times tz p and ti( p) (the time ti( p) is well-defined for any value of p by the very definition of right non-ordered graphs, and the wrong ordering gives the second part of the statement). If this distinguished collision event corresponds to the creation of a particle q, which necessarily occurs at time tzq , then the lemma is proved since z p < z q < i( p). If the event is a recollision between a particle q, with associated creation index z q , and its father, then the recollision necessarily happens at time ti(q) . There are now two possibilities: either particle q has been created after particle p, and then the lemma is proved, or it has been created before, so that z q < z p < i(q) and the lemma is proved with q and p interchanged. Proof of Lemma 3. When m = 3 the lemma is easily proved by direct inspection. It is indeed an easy exercise to write down the explicit list of the 10 wrong graphs with m = 3 (and n = 4). When m > 3 we prove the lemma by induction, assuming it is true for the value m −1. We distinguish three cases, depending on whether |E m | ≥ 2, |E m | = 1 or |E m | = 0. Case (i) - |E m | ≥ 2. In that case we necessarily have πm,i = −1 for any collision time ti for which particle m is involved. This simply comes from the fact m is the last created particle. The induction is proved in the case |E m | ≥ 2. Case (ii) - |E m | = 0. In that case we first build up, starting from the graph T (t), a lower order graph, say T*(t), that involves m − 1 particles, m − 2 creations, and m − 2 recollisions. To do so, we pick up a collision time ti that corresponds to a recollision. To fix the notations, let us say the collision time ti involves the two particles p and q with q < p (hence p > 1). The new graph T*(t) is then simply obtained upon erasing the horizontal segment corresponding to the recollision occurring at time ti , together with the whole leg corresponding to particle m. This is illustrated in Fig. 8. If the graph T*(t) obtained in this way is wrong, we may use the induction hypothesis. If T*(t) is right or right non-ordered, then particle p is involved in at least two external recollisions inside the larger graph T (t): (a) the proper one, namely the one that involves p and its father, which happens at time ti( p) (this collision necessarily happens because p > 1). We have π p,i( p) = −1. (b) The collision with particle q, which occurs at time ti , say. It satisfies π p,i = −1 as well (simply because q < p). In any circumstance, the induction is proved in the case |E m | = 0. Case (iii) - |E m | = 1. In order to fix the ideas, let us say E m = {i}. In that situation, we may build up again a lower order graph T*(t) (we use the same symbol not to overweight notations) which involves m −1 particles, m −2 creations, and
Term-by-Term Convergence
27
m
q
p
*(t) starting from T (t) - the case |E m | = 0 Fig. 8. Building up the new graph T
ti m *(t) starting from T (t): the case when |E m | = 1 and i is a proper recollision Fig. 9. Building up T
m − 2 recollisions. It is obtained upon erasing the horizontal segment corresponding to the recollision occurring at time ti , together with the whole leg corresponding to particle m. This construction is relevant since particle m is only involved in the collision with index i, by our very assumption. There are now two subcases. Let us assume first i is the index of a proper recollision, see Fig. 9. If the lower order graph T*(t) is right, resp. right non-ordered, then the full graph T (t) is right, resp. right non-ordered as well. This is not possible (we assumed T (t) is wrong). Hence T*(t) is necessarily wrong. We are thus in position to use the induction hypothesis. Second, in the case when i is the index of a non proper recollision, see Fig. 10 and Fig. 11, we may assume, to fix notations, this recollision involves particles m and p, where p is not m’s father. If T*(t) is wrong, we may use the induction hypothesis. If T*(t) is right, or right non-ordered, the situation is as follows. On the one hand, if p > 1, see Fig. 10, then particle p is involved in at least two external recollisions inside the larger graph T (t), namely the proper one, which occurs at the collision time ti( p) , and the collision occurring at time ti . Naturally, we have π p,i( p) = −1. If p = 1, see Fig. 11, we have to modify the argument a bit since p = 1 does not have any father. However, since particle 1 is not m’s father (i is the index of a non-proper recollision), there exists a particle q = 1 which is an ancestor of m. As a consequence, the set E q obtained from the larger graph T (t) contains at least the index i, plus the proper recollision that occurs at time ti(q) . Naturally, we have πq,i(q) = −1. This finishes the induction in the case |E m | = 1. Lemma 3 is proved.
28
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
ti
m
p
*(t) starting from T (t): the case when |E m | = 1 and i is not a proper recollision ( p > 1) Fig. 10. Building up T
ti
1
m
q
*(t) starting from T (t): the case when |E m | = 1 and i is not a proper recollision ( p = 1) Fig. 11. Building up T
6. Graphs for which m − 1 < n/2 In this section, we analyze in formula (3.19) the contribution of graphs for which m −1 < n/2 (i.e. for which the number of recollisions is larger than the number of creation). We prove these graphs vanish as ε → 0. This necessitates to prove the integral over t1 , . . ., tn , h 1 , . . ., h n in (3.19) has size smaller than ε+d(m−1)+n/2 as ε → 0. This piece of information is obtained upon analyzing the fast oscillations induced by the phase factor exp(i Sε /2ε) in (3.19), in combination with an appropriate stationary phase argument. The main result of this section is the following Proposition 2. Let T (t) be a graph such that m − 1 < n/2. Then, under the smoothness assumptions of our main theorem, there is a constant C = C(d, m, n, φ, f 0 , t), which depends on all the mentioned arguments but not on ε, such that + + + + +T (t, ξ1 ; η1 )+ ∞ d d ≤ C εn/2−(m−1) −→ 0. L (R × R )
ε→0
Remark 8. The dependence of C upon the various parameters in Proposition 2 is of the form L 1 + φ L 1 n Nα ( f 0 )m C ≤ c(d)n φ where c(d) only depends on the dimension.
t m−1 , (m − 1)!
Term-by-Term Convergence
29
(0) = 0. Remark 9. In this part of the proof we do not require the assumption φ The remainder of this section is devoted to the proof of Proposition 2, which is organized into four steps. First step: isolating the quadratic part of the phase factor Sε in T(t, ξ1 , η1 ). We start from Eq. (3.19), which yields the value of T(t, ξ1 , η1 ). It provides the estimate t
T (t, ξ1 , η1 ) ≤ ε−d(m−1)−n/2
t1 dt1
σ1 ,...,σn =±1 0 n
× dh 1 dh 2 · · · dh n
j=1
Rnd
tn−1 dt2 · · · dtn
0
0
H T (h j ) φ f m0 ξ + ; η + tξ + ε ε
Sε , exp i 2ε
where H and T are defined in (3.20), and the phase factor is Sε = ε
n
σ j θ j h j · (η + (t − t j )ξ )
j=1
+ σ2 θ2 h 2 · θ1 h 1 (t1 − t2 ) + σ3 θ3 h 3 · (θ1 h 1 (t1 − t3 ) + θ2 h 2 (t2 − t3 )) + ... σn θn h n · (θ1 h 1 (t1 − tn ) + θ2 h 2 (t2 − tn ) + · · · + θn−1 h n−1 (tn−1 − tn )). (6.1) For later convenience, it is natural to introduce the time differences s j = t j−1 − t j
( j = 2, . . . , n),
(6.2)
and express the phase in these new variables. With these notations, we recover T (t, ξ1 , η1 ) ≤ ε−d(m−1)−n/2
t
σ1 ,...,σn =±1 0
t1 dt1
t2 dt2
0
0
tn−1 dt3 · · · dtn 0
n H T L Q 0 . φ (h j ) f m ξ + ; η + tξ + exp i exp i × dh 1 dh 2 · · · dh n ε ε 2 2ε Rnd
j=1
(6.3) Here, Q denotes the part of Sε which is quadratic in the {h k }nk=1 ’s. This is the important term. Its explicit value is Q = s2 θ1 h 1 · (σ2 θ2 h 2 + σ3 θ3 h 3 + · · · + σn θn h n ) + s3 (θ1 h 1 + θ2 h 2 ) · (σ3 θ3 h 3 + · · · + σn θn h n ) +... ×sn (θ1 h 1 + θ2 h 2 + · · · + θn−1 h n−1 ) · σn θn h n .
(6.4)
30
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
On the other hand, L is the part of Sε which is linear in the {h k }nk=1 ’s. Its value is essentially irrelevant. However, for sake of completeness, we give L = σ1 θ1 h 1 · (η + (t − t1 )ξ ) +
n
σ j θ j h j · (η + (t − s1 − · · · − s j )ξ ).
(6.5)
j=2
Second step: separating the fast and slow variables. Now we want to perform the integrals with respect to dh j , exploiting in (6.3) the oscillatory factor exp(i Q/(2ε)). However, since f m0 depends on the “fast” variables H j /ε, we need first to rescale H by ε, while carefully separating the H dependence of the phase factors. This is where Lemma 1 plays a crucial role as we now show. To be more specific, we split the {h k }nk=1 ’s into the two sets of variables {h k }nk=1 = {h k }k∈Z ∪ {h k }k ∈Z / , i.e. we separate the exchanged momenta into those associated with a recollision event, and those associated with a creation event. Since the creation momenta {h k }k∈Z are essentially related with the {H p }mp=2 ’s, we also change variables {h k }k∈Z → {H p }mp=2
(6.6) m in (6.3). Our new integration variables in (6.3) are thus the {h k }k ∈Z / , {H p } p=2 ’s. The reader should be cautious about the fact that the variable H1 is put apart here, since particle 1 does not stem fromany creation event. Yet H1 is anyhow recovered from the m {H p }mp=2 ’s through formula i=1 Hi = 0 (see Remark 3 and Eq. (3.24)). On the other hand, thanks to Lemma 1, we have hz p = Hq + π p, j h j , (6.7) q∈C p
j∈E p
and the second sum only involves the {h j } j ∈Z / ’s, while the first sum involves a triangular structure Hq = H p + Hq . q∈C p ,q> p
q∈C p
For this reason, the mapping (6.6) is one-to-one, and it has Jacobian +1. We are now in position to give expressions for T(t), and for the fast phase Q, where m the role of the {h k }k ∈Z / ’s and that of the {H p } p=2 ’s are clearly sorted out. n First, the definition of H = j=1 θ j h j provides the equality θ1 h 1 + · · · + θ j−1 h j−1 = H − (θ j h j + · · · + θn h n ). From this we deduce a more symmetric expression of the phase factor Q, namely Q=−
n
s j σ j θ j h j + · · · + σn θn h n · θ j h j + · · · + θn h n
j=2
+H ·
n j=2
!
s j (σ j θ j h j + · · · + σn θn h n ) . "#
=:L 1
$
(6.8)
Term-by-Term Convergence
31
This serves as a definition of the function L 1 , whose precise value is irrelevant. The important point lies in the fact L 1 is linear in the {h k }nk=1 ’s. Note that in (6.8), the m {h k }k∈Z ’s are thought of as (linear) functions of the ({h k }k ∈Z / , {H p } p=2 )’s, according to formula (6.7). We keep on using this convention from now on. We have obtained Q=−
n σ + σ j θ · θ j S∧ j h j · h + H · L 1 , 2
(6.9)
j,=2
where S j = s2 + · · · + s j (= t1 − t j ),
i ∧ j = min(i, j).
(6.10)
Second, we may sort out again in (6.9) the dependence of Q on the {h k }k ∈Z / ’s and on the {H p }mp=2 ’s, upon writing n
··· =
j,∈Z /
j,=2
and h z p =
··· +
j ∈Z / , ∈Z
Hq +
q∈C p
··· +
··· +
j∈Z , ∈Z /
···
j,∈Z
π p, j h j ,
j∈E p
see Lemma 1. The second equality holds for any creation momentum h k with k ∈ Z . This procedure provides the identity (note that for any ( j, p) such that j ∈ E p , we necessarily have j ∈ / Z) Q=− A j, h j · h + H · L 1 + H · L 2 + q(H, H ), (6.11) j,∈Z /
where L 2 is linear in the {h k }nk=1 , q(H, H ) is quadratic in H , the exact value of L 2 and q is irrelevant, and the important factor is A, j =
σz p + σ j σ + σ j π p, (θ · θ j ) S∧ j + (θz p · θ j ) Sz p ∧ j 2 2 +
p | ∈E p
πq, j
σzq + σ 2
q | j∈E q
+
π p, πq, j
p,q | ∈E p , j∈E q
(θzq · θ ) Szq ∧
σz p + σz q 2
(θz p · θzq ) Sz p ∧zq .
(6.12)
It may be useful to write down the diagonal term’s explicit value: A, = 2σ S + π p, (σz p + σ ) (θz p · θ ) Sz p p | ∈E p
+
π p, πq,
p,q | ∈E p ∩E q
σz p + σz q 2
(θz p · θzq ) Sz p ∧zq .
(6.13)
Third, plugging (6.11) into (6.3), and rescaling *p , H p → ε H
( p = 2, . . . , m),
(6.14)
32
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
* = (ε H1 , ε H2 , . . . , ε Hm )) gives the formula (here H
T (t, ξ1 , η1 ) ≤ ε−n/2
t
t1 dt1
σ1 ,...,σn =±1 0
t2 dt2
0
tn−1 dt3 · · · dtn
0
0
m *; η + tξ + T ,p (h j ) (h j ) (6.15) φ φ × dh j dH f m0 ξ + H ε p=2 j ∈Z / j ∈Z / j∈Z ⎛ ⎞ i i * * * * ⎝ ⎠ L + H · L 1 + H · L 2 + εq( H , H ) exp − × exp A j, h j · h . 2 2ε j,∈Z /
Finally we remove the T /ε dependence of f m0 , upon introducing the Fourier inverse *j }m as well as transform , f m0 with respect to the η variables. Since T is linear in { H j=2 in the {h j } j ∈Z / ’s, this operation simply changes the linear part of the phase. The final structure of the term under consideration is
T (t, ξ1 , η1 ) ≤ ε−n/2
t
t1 dt1
σ1 ,...,σn =±1 0
t2 dt2
0
tn−1 dt3 · · · dtn
0
0
m m , (h j ) (h j ) φ φ × dh j d Hp dv j j ∈Z /
p=2
⎛
j=1
j ∈Z /
⎞
j∈Z
⎞ i , 0 * ⎝ ⎠ ⎝ ⎠ β j h j exp − A j, h j · h , × f m ξ + H ; Vm exp iα + i 2ε j ∈Z /
⎛
j,∈Z /
(6.16) where Vm = (v1 , . . . vm ), while α and the {β j } j ∈Z / ’s are suitable (real-valued) functions H2 , . . . Hm ), and the times s1 , . . . , sn . Their explicit value is irrelevant in of ε, Vm , ( , the sequel. The only important point is that α and the {β j } j ∈Z / ’s do not depend on the {h j } j ∈Z / ’s. Unfortunately we still cannot apply directly the stationary phase theorem because the integration over the {h j } j ∈Z / ’s would produce a factor d
ε 2 (n−m+1) d
| det A| 2
,
(6.17)
d n . The factor ε 2 (n−m+1) is more than we need to kill ε− 2 . Indeed where A = Ai j i, j ∈Z / we know from [BCEP1] that the right graphs are O(1) and estimate (6.17) would give d us the contradictory information that such contribution should behave as O(ε 4 n ). The d point is, | det A|− 2 is not integrable in general with respect to the variables s j , as follows by analyzing simple examples. Thus we have to proceed more carefully. The main and somehow surprising point is that we can avoid to detect explicitly the singular manifold of | det A|−1 , by using a suitable interpolation technique (see the next step). For the moment we outline the basic features of the matrix A.
Term-by-Term Convergence
33
Lemma 4. Setting q = n − m + 1. Then A is a q × q symmetric matrix such that i) if i < j, Ai j depends, at most, on the time differences s2 , . . . , si . ii) Aii = 2σi si + G i (s2 , . . . si−1 ), where G i is a suitable function of the i − 2 time differences s2 , . . . , si−1 . Proof of Lemma 4. By direct inspection using that i ∈ E p implies i > z p .
Third step: applying the stationary phase theorem. For later convenience we renumber the entries of the matrix A whose indices are j1 , . . . jq , i.e. the indices of the recollision times, by setting Di =
1 Aj j . ε i
(6.18)
We also denote ki = h ji . We finally rescale the recollision time differences by setting sj µi = i . ε With these new notations we may estimate
T (t, ξ1 , η1 ) ≤ εq−n/2 ×
d Vm
m
t
dtz 3 · · ·
dtz 2
σ1 ,...σn =±1 0 q
0
dtz m 0
q q (k j ) (h j ) φ φ dµ j dk j
,p dH
p=2
tz m−1
tz2
Rq j=1
⎛
j=1
⎛
j=1
j∈Z
⎞ q q i , 0 * ⎝ ⎠ ⎝ ⎠ × f m ξ + H ; Vm exp i B j · k j exp − Di j ki · k j , 2 j=1
⎞
(6.19)
i, j=1
*j }m and where B = β j and the {h j } j∈Z ’s have to be thought of as functions of { H j=2 q {k j } j=1 , according to (6.7). Note that q − n/2 = n/2 − (m − 1) > 0. Therefore it is enough to show that the integrals in the right-hand side of (6.19) are uniformly bounded. In that direction, we establish the following Proposition 3. Let F : Rdq → C be a smooth function (in the sense that the norm defined in (6.21) below is finite), B ∈ Rdq , D be the q × q matrix defined in (6.18). Then the following estimate holds true: ⎞ ⎛ q q q dµ j dk j F(k1 , . . . kq ) exp ⎝i Bj · kj⎠ j=1 Rdq j=1 Rq j=1 ⎛ ⎞ q i × exp ⎝− Di j ki · k j ⎠ 2 i, j=1 *(F), ≤ CN
(6.20)
34
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
where *(F) = max N
I ⊂Iq
d K Iq \I d X I F I F(K Iq \I , X I ).
(6.21)
Here Iq = {1, . . . q} and F I F(K Iq \I , X I ) denotes the Fourier transform of F with respect to the variables indexed by I . Moreover C is a positive constant independent of * and the creation times. Vm , H Proof of Proposition 3. We begin the proof by establishing the following lemma which will be useful in the sequel. k k Lemma 5. Let {ai }i=1 and {xi }i=1 be two collections of real numbers. Let p > 1. Then, we have the estimate dz dz 1 . (6.22) p ≤ p k k k (1 + |z|) p |a | + |a z + x | |a | i i i i i=1 i=1 i=1 R R
Remark 10. While it is trivial to estimate the left-hand-side of (6.22) by const/|a i | for any given i, the above lemma allows us to keep the improved estimate by const/( i |ai |) p . Proof of Lemma 5. It is enough to consider the case when i |ai | > 0. We first estimate dz dz p , ∗ p ≤ ∗ k k |ai | + |ai z + xi | |ai | + |ai z + xi | R
i=1
where the symbol
R
i=1
∗
designates
. Next, by convexity, we may write
i:ai =0 ∗
|ai z + xi | =
∗
xi ≥ |ai | z + |ai |
∗
|ai |
⎞ ⎛ ∗ ⎟ ⎜ xi ⎟ ⎜ ⎟ . ∗ ⎜ z + ⎝ |ai | ⎠ ! "# $ =:x¯
The change of variables z + x¯ → z in the above integral finishes the proof.
By the Parseval formula and the explicit form of the Fourier transform of a complex Gaussian, we have q q i q I := dk j F(k1 , . . . kq )ei j=1 B j ·k j e− 2 i, j=1 Di j ki ·k j Rdq
j=1 dq
=
d
(2π ) 2 e−iπ 4 sign(D) | det D|
d 2
i
dξ e 2
ij
Di−1 j ξi ·ξ j
(ξ − B) , F
(6.23)
Rdq
where sign(D) = n + − n − , n ± being the number of positive and negative eigenvalues of D respectively. Therefore Cq L1 . F (6.24) I ≤ d | det D| 2
Term-by-Term Convergence
35
More generally, for any subset I ⊂ Iq of indices (including the case I = ∅, for which det D I = 1 and F I = identity), we also have Cq F I F L 1 , (6.25) I ≤ d | det D I | 2 where D I = Di j i, j∈I . As a consequence
2 *(F) d2 C q , | det D I | |I| d ≤ N
I ⊂Iq
from which we finally deduce * (F) CqN I ≤ d . 2 I ⊂Iq | det D I |
(6.26)
Let now I ⊂ Iq be such that q ∈ I . We evaluate the determinant of D I by using Lemma 4. Developing along the last row, we compute: det D I = 2σ jq µq det D I \{q} + g I \{q},q ,
(6.27)
where g I \{q},q is a function of the µ j ’s with j < q only (the value of this function depends on the choice of the index q, as well as on the choice of the subset I \ {q}). Therefore | det D I | = | det D I | + | det D I | I ⊂Iq
I : q ∈I /
=
I : q∈I
| det D I | +
I : q ∈I /
=
|2σ jq µq det D I \{q} + g I \{q},q |
I : q∈I
| det D I | +
I : q ∈I /
|2σ jq µq det D I + g I,q |,
(6.28)
I : q ∈I /
where the last equality comes from changing the summation index in the second sum. Now, from (6.26) and (6.28), we recover 1 *(F) dµ j I ≤ C q N dµ j dµq d 2 j
×
dµq
1 I : q ∈I /
| det D I | +
I : q ∈I /
Applying Lemma 5 therefore yields * (F) dµ j I ≤ C q N dµ j j
|2σ jq µq det D I + g I,q |
1 I : q ∈I /
| det D I |
d . 2
d . 2
(6.29)
36
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
There remains to iterate the procedure and to integrate successively over µq−1 , µq−2 , . . . up to µ1 . The iteration ends by using det D∅ = 1. This finishes the proof of Proposition 3. Fourth step: concluding the proof of Proposition 2. Inserting estimate (6.20) in (6.19), with ⎛ ⎞ q m *r + (ki ) ⎝ε F(k1 , . . . kq ) = φ φ π p, j k ⎠ , (6.30) H p=2
i=1
r ∈C p
: j ∈E p
* (F) is conveniently estimated. we readily see that the proposition is proved once N We first observe, setting G A (K ) = G(K )ei A·K , where A ∈ Rdq and K = (k1 , . . . , kq ), that *(G). *(G A ) = N N
(6.31)
This follows by an elementary computation. Now, thanks to (6.30), we have q m −i(a+bK )·Y (ki ), φ dY φ(yi ) e F(K ) = Rd(m−1)
i=2
(6.32)
i=1
m ’s. where a ∈ Rd(m−1) and b ∈ Rdq×d(m−1) are suitable functions of ε and the {, Hi }i=2 Using (6.31), we thus obtain
q m −ibK ·Y *(F) ≤ * e (ki ) φ N dY |φ(yi )| N i=2 Rd(m−1) m−1 q * q ≤ φ L 1 c N (φ )
i=1
* (φ ) . ≤c N n
n
In conclusion, we have established n t m−1 L 1 n * T (t, ξ1 , η1 ) ≤ ε 2 −m+1 f 0 m−1 cn φ L 1 + φ L 1 f0 L 1 . (m − 1)!
(6.33)
This concludes the proof of Proposition 2, and ends this section. 7. Right Non-ordered Graphs The previous analysis readily establishes the estimate T(t, ξ1 , η1 ) ≤ O(εn/2−(m−1) ) in full generality. In the particular case when m − 1 = n/2, this reduces to T(t, ξ1 , η1 ) ≤ O(1). An additional argument is needed to prove that right non-ordered graphs, or wrong graphs with m − 1 > n/2, are actually vanishing as ε → 0. We start by considering right non-ordered graphs, recalling that they involve as many n = m − 1). Besides, for any creation event, they creation as recollision events (i.e. 2 involve exactly one associated recollision between the same particles (i.e. between the father and the son). However the time ordering of the recollision is not the right one: at least one recollision does not happen immediately after the associated creation. It turns out that a slight improvement of the arguments developed in the previous section allows us to exploit this wrong time ordering so as to prove T(t, ξ1 , η1 ) ≤ O(εγ ), for some γ > 0. Lemma 2 is crucial in that respect. Before coming to the proof, we first state the main result of this section.
Term-by-Term Convergence
37
Proposition 4. Let T (t) be a right non-ordered graph such that m − 1 = n/2. Then, under the smoothness assumptions of our main theorem, there is a constant C = C(d, m, n, φ, f 0 , t), which depends on all the mentioned arguments but not on ε, such that + + d −2 + + . +T (t, ξ1 ; η1 )+ ∞ d d ≤ C εγ , with 0 < γ < L (R × R ) 2 Remark 11. The dependence of C upon the various parameters in Proposition 4 is of the form L 1 + φ L 1 n Nα ( f 0 )m C ≤ c(d)n φ
t m−2 , (m − 2)!
where c(d) only depends on the dimension. (0) = 0. Remark 12. In this part of the proof we do not require the assumption φ Proof of Proposition 4. The idea is to come back to the computation of the diagonal elements of the quadratic phase Q, see (6.13). For any given particle p, denote by i( p) the index of the (unique) recollision between particle p and p’s father. This notation is well-defined since the graph is assumed right non-ordered. With these notations, the index i( p) belongs to the set E p (and to no other set E q with q = p). Besides, we have θz p = θi( p) , and π p,i( p) = −1. As a consequence, the diagonal coefficient Ai( p),i( p) has the simple value (see (6.13)) Ai( p),i( p) = 2σi( p) Si( p) − 2(σz p + σi( p) )Sz p + 2σz p Sz p = 2σi( p) (Si( p) − Sz p )2σi( p) sz p +1 + sz p +2 + · · · + si( p) .
(7.1)
The important point is, not only the recollision time si( p) appears explicitly in (7.1), but also the intermediate times sz p +1 , sz p +2 , etc. Now, since the graph is assumed right non-ordered, we may apply Lemma 2: there is a particle q and a particle p, with q < p, such that z p < z q < i( p).
(7.2)
For this particular choice of p (and q), we recover Ai( p),i( p) = 2σi( p) sz p +1 + sz p +2 + · · · + szq + · · · + si( p) = 2σi( p) szq + α ,(7.3) where α is a sum of positive terms. We may now improve Proposition 3. By using (6.26), summing only over I = {i( p)} and I = ∅ we obtain, for I given in (6.23), the following bound: |I| ≤
cN0 (F) cN0 (F) s d . d ≤ sz q 2 2 zq 1+2 ε +α 1+2 ε
(7.4)
Note that szq is a creation time, and hence it is not rescaled. Obviously F in (7.4) is given by (6.30). Interpolating Eq. (7.4) and Eq. (6.26) we arrive at |I| ≤
cN0 (F) , d d (1−β) szq 2 β 2 1+2 ε I | det D I |
(7.5)
38
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
where 0 < β < d−2 d . Inserting (7.5) in (6.19) we proceed as in Sect. 6 replacing d 2 (1 − β) > 1. The final result follows by t 0
d 2
by
d d ds ≤ c ε 2 β t 1− 2 β . d β 1 + 2 εs 2
Remark 13. A remark is in order. In [BCEP1] we learnt that a quantum collision in a weak coupling regime, as expressed by an operator Ci,ε j , is not “completed”: it needs an operator Ti,ε j , expressing a proper recollision, and occurring approximatively at the same time, in order to give a contribution O(1). Therefore one may conceive that a right non-ordered graph forces the creation times occurring between pairs of creation–recollision times, to give a relevant contribution on a set of small measure only. Unfortunately we cannot fully exploit this feature since we are not able to explicitly characterize the singular manifold of (det D)−1 . 8. Graphs for which m − 1 > n/2 The previous two sections use a stationary phase approach to balance the factor ε−d(m−1)−n/2 in the definition of T, and to eventually prove T → 0 as ε → 0. The main point is (up to the slight refinement of Sect. 7) the quadratic phase Q allows to gain one factor ε per recollision time. This approach seems difficult to adapt in the case m − 1 > n/2: it would require a very fine control of various coefficients in the quadratic phase Q, combined with suitable cancellation effects due to the signs σ j , in order to recover additional factors ε from creation times as well. For that reason, in this section and the next one we renounce to control the phase and (0) = 0 to balance some we follow a quite different route: we exploit the assumption φ divergence arising from small exchanged momenta. The main result of this section is Proposition 5. Let T (t) be a graph such that m − 1 > n/2. Then, under the smoothness (0) = 0, there assumptions of our main theorem, and provided the potential φ satisfies φ is a constant C = C(d, m, n, φ, f 0 , t), which depends on all the mentioned arguments but not on ε, such that + + + + +T (t, ξ1 ; η1 )+
L ∞ (R d × R d )
≤ C εm−1−n/2 −→ 0. ε→0
Remark 14. The dependence of C upon the various parameters in Proposition 5 is of the form C ≤ c(d)n Nα (φ)n Nα ( f 0 )m where c(d) only depends on the dimension.
t n−(m−1) , (n − (m − 1))!
Term-by-Term Convergence
39
Proof of Proposition 5. To begin with, we first bound T, as given through formula (3.19), by n |T(ξ1 , η1 , t)| ≤ 2n f 0 ∞ ε−d(m−1)− 2
×
dh 1 dh 2 · · · dh n
n
(h j )| |φ
j=1
Rnd
t
t1 dt1
0 m
f0
0
k=2
tn−1 dt2 · · · dtn 0
ek · H ek · T , . ε ε
(8.1)
Note the phase factor exp(i Sε /ε) has been estimated by one here: stationary phase considerations are completely put apart from now on. Now, the point is, the variables ek ·T involve a particular triangular structure. Namely, for all k = 2, . . . , m, we have ek · T = tz k h z k + Ak ,
(8.2)
where Ak is a linear combination of t j and h j with j > z k . In particular, variable tz 2 is only involved in e2 · T , variable tz 3 is only involved in e2 · T and e3 · T , etc. For that reason, we may successively perform the integrations in tz 2 , . . . , tz m in (8.1), as follows. The integration in tz 2 produces the factor dtz 2
2 − α2 H2 H t h + A ε 2 z z 2 2 2 ≤ cNα ( f 0 ) . 1 + f0 ε , ε |h z 2 | ε
f 0 (e3 ·H/ε, e3 ·T /ε), . . . , f 0 (em · Note also that the integration in tz 2 leaves all other factors H/ε, em · T /ε) unchanged in (8.1). Next, the integration in tz 3 again produces a factor dtz 3
2 − α2 H3 H t h + A ε 3 z z 3 3 3 ≤ cNα ( f 0 ) , 1 + f0 ε , ε |h z 3 | ε
f 0 (em · H/ε, em · T /ε) unchanged while leaving all other factors f 0 (e4 · H/ε, e4 · T /ε), ... in (8.1). Performing this procedure until tz m , and last integrating over all other time variables {t j } j ∈Z / as well, we eventually obtain t n−(m−1) (n − (m − 1))! 2 − α2 m |φ (h j )| Hk (h j )| |φ . 1 + |h j | ε
|T(ξ1 , η1 , t)| ≤ cn Nα ( f 0 )m ε−d(m−1)− 2 +(m−1) n
× Rnd
dh 1 dh 2 · · · dh n
j∈Z
j ∈Z /
(8.3)
k=2
Recall variables Hk = H · ek are thought of as functions of (h 1 , . . . , h n ) as before. (0) = 0, in conjunction with the smoothThere remains to observe the assumption φ , readily gives ness of φ (h)| |φ L ∞ . ≤ D φ |h|
(8.4)
40
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
From this it follows t n−(m−1) (n − (m − 1))! 2 − α2 m Hk (h j )| |φ . 1 + ε
m −d(m−1)− 2 +(m−1) m−1 |T(ξ1 , η1 , t)| ≤ cn D φ L ∞ Nα ( f 0 ) ε n
×
dh 1 dh 2 · · · dh n
j ∈Z /
Rnd
(8.5)
k=2
Last, we rescale the creation momenta as in (6.6) and (6.14). We arrive at m − 2 +(m−1) m−1 |T(ξ1 , η1 , t)| ≤ cn D φ L ∞ Nα ( f 0 ) ε n
×
m k=2
t n−(m−1) (n − (m − 1))!
− α 2 2 , , (h j )|. d Hk 1 + | Hk | dh j |φ j ∈Z /
Rd
Rd
Hence, recalling that α > d, |T(ξ1 , η1 , t)| ≤ c(d)n Nα ( f 0 )m Nα (φ)n This finishes the proof of Proposition 5.
n t n−(m−1) ε− 2 +(m−1) . (n − (m − 1))!
(8.6)
9. Wrong Graphs with m − 1 D n/2 The previous section establishes the estimate T(t, ξ1 , η1 ) ≤ O(εm−1−n/2 ) in full gener(0) = 0. In the particular case when m − 1 = n/2, ality, provided the potential satisfies φ this estimate is useless. An additional argument is needed to prove wrong graphs with m − 1 = n/2 actually have vanishing contribution as ε → 0. We remind that wrong graphs with m − 1 = n/2 are characterized by Lemma 3 established in Sect. 5. Now, a slight modification of the argument used in the previous section allows us to exploit this specific feature, and prove T(t, ξ1 , η1 ) = O(ε). The main result of this section is Proposition 6. Let T (t) be a wrong graph such that m − 1 = n/2. Then, under the smoothness assumptions of our main theorem, and provided the potential φ satisfies (0) = 0, there is a constant C = C(d, m, n, φ, f 0 , t), which depends on all the menφ tioned arguments but not on ε, such that + + + + +T (t, ξ1 ; η1 )+ ∞ d d ≤ C ε −→ 0. L (R × R )
ε→0
Remark 15. The dependence of C upon the various parameters in Proposition 5 is of the form C ≤ c(d)n Nα (φ)n Nα ( f 0 )m where c(d) only depends on the dimension.
t n−(m−1) , (n − (m − 1))!
Term-by-Term Convergence
41
Proof of Proposition 6. We simply modify the argument of the previous section by using Lemma 3. According to Lemma 3, there is a particle p such that |E p | ≥ 2, and there exists one index i ∈ E p such that π p,i = −1. On the other hand, and as before, for any k = 2, . . . , m, we have ek · T = tz k h z k + Ak ,
(9.1)
where Ak is a linear combination of t j and h j with j > z k . Yet the situation is a bit more precise in the special case k = p. For this particular value, we may write indeed e p · T = tz p h z p + ti h i + , A p,
(9.2)
where , A p is a linear combination of t j ’s and h j ’s, j > z p , j = i. Notice that both variables tz p and ti only appear in quantities of the form e j · T with j < p: due to the information π p,i = −1, we know indeed the partner particle in the i th recollision is created before particle p. This information allow us to perform successively the integrations in the variables tz 2 , tz 3 , . . . , tz p , ti , tz p+1 , . . . , tz m in the estimate
n f 0 ∞ ε−d(m−1)− 2 |T(ξ1 , η1 , t)| ≤ cn
t1 dt1
0
×
t
dh 1 dh 2 · · · dh n
n j=1
Rnd
0
tn−1 dt2 · · · dtn 0
m ek · H ek · T (h j )| f , |φ . 0 ε ε k=2
We already discussed the effect of the integrations over the tz k ’s (k = p) in the previous section. On the other hand, integration over tz p and ti produces the factor
dtz p R
R
f0 dti
Ap e p · H tz p h z p + ti h i + , , ε ε
ε2 ≤ Nα ( f 0 ) area(h z p , h i )
− α e p · H 2 2 , 1 + ε
(9.3)
where area (h z p , h i ) denotes the area of the two-dimensional parallelogram defined by the two vectors h z p and h i . Its value is, . / hz p hz p area(h z p , h i ) = |h z p | h i − h i · |h z p | |h z p | . / hi hz p hz p h i . = |h z p | |h i | − · |h i | |h i | |h z p | |h z p |
(9.4)
42
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
Hence the whole integration procedure over variables tz 2 , tz 3 , . . . , tz p , ti , tz p+1 , . . . , tz m eventually gives the estimate n t n−(m−1)−1 |T(ξ1 , η1 , t)| ≤ cn Nα ( f 0 )m ε−d(m−1)− 2 +(m−1)+1 (n − (m − 1) − 1)! |φ (h j )| (h j )| × dh 1 dh 2 · · · dh n |φ |h j |
j∈Z , j=z p
Rnd
j ∈Z / , j=i
2 − α2 m (h i )| |φ (h z p )| Hk |φ × . 1 + |area(h z p , h i )| ε
(9.5)
k=2
The role of indices i and p is clearly put apart here. Here and below, the reader may keep in mind the relation m − 1 = n/2. Let us now estimate the right-hand-side of (9.5). To do so, let be the second external recollision in E p (see Lemma 3). Separating the role of as well, and rescaling the creation momenta Hk (k = 2, . . . , m) as in (6.6) and (6.14), we recover |T(ξ1 , η1 , t)| ≤ cn Nα ( f 0 )m Nα (φ)m−2 ε
t m−2 (m − 2)!
dh j
j ∈Z /
d, Hk
k∈Z
m α (h i )||φ (h z p )||φ (h )| |φ 2 (h j )| 1 + |, Hk |2 . |φ |area(h z p , h i )|
j ∈Z / , j=i,
(9.6)
k=2
Recall here that the variables {h k }k∈Z ’s are thought of as linear functions of the {h j } j ∈Z / ’s m , and the { Hk }k=2 ’s. In that respect, we actually know from Lemma 1 that hz p = ε
,q − h i + π p, h + H
π p, j h j .
(9.7)
j∈E p , j=i,
q∈C p
This suggests the change of variable h −→ k = ε
q∈C p
,q − h i + π p, h + H
π p, j h j
(9.8)
j∈E p , j=i,
in (9.6). Indeed, changing variables in this way we arrive at the estimate t m−2 |T(ξ1 , η1 , t)| ≤ cn Nα ( f 0 )m Nα (φ)m−2+n−(m−1)−2 ε (m − 2)! (k)| |φ (h )| (h i )| |φ |φ dh i dk × . |area(k, h i )| R2d
(9.9)
Term-by-Term Convergence
43
*j ’s with j ∈ C p , k, and some h j ’s with j ∈ E p . There Here h is a function of the H remains to observe the estimate (here we denote k := k/|k|, hi := h i /|h i |), (k)| |φ (h )| (h i )| |φ |φ dh i dk |area(k, h i )| R2d
=
dh i dk R2d
≤ Nd (φ)3
(k)| |φ (h )| (h i )| |φ |φ |k| |h i | hi − (hi · k) k 1 dσ dσ |σ − (σ · σ ) σ |
Sd−1 ×Sd−1 3
≤ c(d) Nd (φ) ,
(9.10)
where c(d) is some universal constant, depending on the dimension d only. Here we used the fact that d ≥ 3. Eventually we have proved |T(ξ1 , η1 , t)| ≤ c(d)n Nα ( f 0 )m Nα (φ)n ε This finishes the proof of Proposition 6.
t m−2 . (m − 2)!
(9.11)
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Ashcroft, N.W., Mermin, N.D.: Solid state physics. Philadelphia: Saunders, 1976 Balescu, R.: Equilibrium and Nonequilibrium Statistical Mechanics. New-York: John Wiley & Sons, 1975 Benedetto, D., Castella, F., Esposito, R., Pulvirenti, M.: Some considerations on the derivation of the nonlinear quantum boltzmann equation. J. Stat. Phys. 116(1–4), 381–410 (2004) Benedetto, D., Castella, F., Esposito, R., Pulvirenti, M.: On the weak-coupling limit for bosons and fermions. Math. Mod. Meth. Appl. Sci. 15(12), 1811–1843 (2005) Benedetto, D., Castella, F., Esposito, R., Pulvirenti, M.: Some considerations on the derivation of the nonlinear quantum boltzmann equation ii: the low-density regime. J. Stat. Phys. 124(2– 4), 951–996 (2006) Bohm, A.: Quantum Mechanics. Texts and monographs in Physics, Berlin-Heidelberg-New York: Springer-Verlag, 1979 Castella, F.: From the von neumann equation to the quantum boltzmann equation ii: identifying the born series. J. Stat. Phys. 106(5/6), 1197–1220 (2002) Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases. Cambridge: Cambridge Univ. Press, 1970 Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. Applied Mathematical Sciences, Vol. 106, New York: Springer-Verlag, 1994 Chuang, S.L.: Physics of optoelectronic devices. Wiley series in pure and applied optics, New-York: Wiley, 1995 Cohen-Tannoudji, C., Diu, B., Laloë, F.: Mécanique Quantique, I et II. Enseignement des Sciences, Vol. 16, Paris: Hermann, 1973 Dürr, D., Goldstain, S., Lebowitz, J.L.: Asymptotic motion of a classical particle in a random potential in two dimensions: landau model. Commun. Math. Phys. 113(2), 209–230 (1987) Eng, D., Erdös, L.: The linear boltzmann equation as the low density limit of a random schrödinger equation. Rev. Math. Phys. 17(6), 669–743 (2005) Erdös, L., Yau, H.T.: Linear Boltzmann Equation as Scaling Limit of Quantum Lorentz Gas. In: Advances in differential equations and mathematical physics (Atlanta, GA, 1997), Contemp. Math. 217, Providence, RI: Amer. Math. Soc., 1998, pp. 137–155 Erdös, L., Yau, H.-T.: Linear boltzmann equation as the weak coupling limit of a random schrödinger equation. Commun. Pure Appl. Math. 53(6), 667–735 (2000)
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[ESY] [H] [HL] [IP] [L] [KP] [LP] [MRS] [RS] [RV] [Sp] [U] [UU] [W]
D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti
Erdös, L., Salmhofer, M., Yau, H.-T.: On the quantum boltzmann equation. J. Stat. Phys. 116(1– 4), 367–380 (2004) Hugenholtz, M.N.: Derivation of the boltzmann equation for a fermi gas. J. Stat. Phys. 32, 231–254 (1983) Ho, N.T., Landau, L.J.: Fermi gas in a lattice in the van hove limit. J. Stat. Phys. 87, 821–845 (1997) Illner, R., Pulvirenti, M.: Global Validity of the Boltzmann equation for a two-dimensional rare gas in the vacuum. Commun. Math. Phys. 105, 189–203 (1986), Erratum and improved result. Commun. Math. Phys. 121, 143–146 (1989) Lanford, O. III: Time evolution of large classical systems. Lecture Notes in Physics, Vol. 38, E.J. Moser ed., Berlin-Heidelberg-New York: Springer-Verlag, 1975, pp. 1–111 Kesten, H., Papanicolaou, G.C.: A limit theorem for stochastic acceleration. Commun. Math. Phys. 78(1), 19–63 (1980) Lions, P.L., Paul, T.: Sur les mesures de wigner. Revista Mat. Ibero Amer. 9(3), 553–618 (1993) Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor equations., Vienna: SpringerVerlag, 1990 Reed, M., Simon, B.: Methods of modern mathematical physics III. Scattering theory. New York-London: Academic Press, 1979 Rosencher, E., Vinter, B.: Optoelectronique. Paris: Dunod, 2002 Spohn, H.: Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17(6), 385–412 (1977) Uchiyama, K.: On the boltzmann-grad limit for the broadwell model of the boltzmann equation. J. Stat. Phys. 52(1/2), 331–355 (1988) Uehling, E.A., Uhlembeck, G.E.: Transport phenomena in einstein-bose and fermi-dirac gases. i. phys. Rev. 43, 552–561 (1933) Wigner, E.P.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)
Communicated by H. Spohn
Commun. Math. Phys. 277, 45–67 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0356-6
Communications in
Mathematical Physics
Existence and Regularity of Weak Solutions to the Quasi-Geostrophic Equations in the Spaces − 1/ 2 L p or H˙ Fabien Marchand Département de Mathématiques, Université d’Evry Boulevard F. Mitterrand, 91025 Evry Cedex, France. E-mail:
[email protected] Received: 2 October 2006 / Accepted: 27 March 2007 Published online: 23 October 2007 – © Springer-Verlag 2007
Abstract: In this paper we study the 2D quasi-geostrophic equation with and without dissipation. We give global existence results of weak solutions for an initial data in the space L p or H˙ −1/2 . In the dissipative case, when the initial data is in L p , p > 2, we give a regularity result of these solutions. 1. Introduction The 2D quasi-geostrophic equation (QG)α , 0 < α ≤ 1, for a function θ (t, x) defined on [0, +∞) × R2 is ⎧ ⎨ θt + u · ∇θ + k2α θ = 0 (1) u = −R⊥ θ = −(−R2 θ, R1 θ ), ⎩ θ (0, ·) = θ0 1
where k ≥ 0 and is the operator (−) 2 , defined at the Fourier level by f (ξ ) = |ξ | f (ξ ). Thus, 2α f (ξ ) = |ξ |2α f (ξ ), where the Fourier transform f = F( f ) is defined by f (x)e−i x·ξ d x. f (ξ ) = R2
The Riesz transforms R1 and R2 are defined by iξk Rk f (ξ ) = − f (ξ ). |ξ |
46
F. Marchand
These equations come from more general quasi-geostrophic models of atmospheric and ocean fluid flow; the scalar θ represents the temperature and u the divergence free velocity field. For the non-dissipative case, i.e. k = 0, we will use the notation (QG) for the corresponding equation. The mathematical study of the non-dissipative case has first been proposed by Constantin, Majda and Tabak in [3] where it is shown to be an analogue for the 3D Euler equations. The dissipative cases have then been studied by Constantin and Wu in [4] when α > 1/2 and global existence in Sobolev space is studied by Constantin, Cordoba and Wu in [2] when α = 1/2. The case 1/2 < α ≤ 1, is called subcritical since smooth solutions are known to exist globally in time whereas the uniqueness of weak L 2 solutions is an unsolved problem. According to [4], (QG)1/2 describes the evolution of the temperature on the 2D boundary of a rapidly rotating half-space with small Rossby and Ekman numbers. This case is called critical since there is a balance between the dissipation and the non-linear term, therefore is a good model for the 3D Navier-Stokes equations. Remark 1.1. In a recent paper [11], Kiselev, Nazarov and Volberg show that any C ∞ periodic initial data give rise to a unique C ∞ solution to (QG)1/2 . The case 0 < α < 1/2 is called supercritical and is harder to deal with compared to the other cases. The aim in studying the fractional power 0 < α ≤ 1 is to provide a better understanding of the analysis of the 3D Navier-Stokes equations. We now present our main results (Theorems 1.1, 1.2 and 1.4); the problem we are concerned with is the existence and regularity of global weak solutions to the quasi-geostrophic equations when the initial data belongs to L p or to H˙ −1/2 . When k = 0, the quasi-geostrophic equation (QG) is an analogue to the 3D Euler equations, this analogy is described in [3]. The problem of the existence of weak L 2 solutions for the 3D Euler equations is still an open problem, and for the 2D Euler equations we have some results when we add other assumptions on the vorticity of the initial data (see the book of Majda and Bertozzi [16] or Lions [15]); one of the most famous results in that direction is Delort’s result [7] which states the existence of vortex sheets with fixed sign. For the quasi-geostrophic equation, we have a better structure in the non-linear term than for 3D Euler equations, and we can construct global weak solutions. It is proved in [21] that on the 2D torus, the non-dissipative quasi-geostrophic equation has global weak L 2 solutions when the initial data is in L 2 . In Sect. 2 we give a crucial remark on the non-linear term which allows us to define the non-linearity for a function θ in L 2 ((0, T ), L p (R2 )), p > 4/3 or in L 2 ((0, T ), H˙ −1/2 (R2 )). With the help of this remark we are also able to extend Resnick’s result, with a different proof, when the spatial domain is R2 and the initial data is in L p (R2 ), p > 4/3: Theorem 1.1. Let T ∈ (0, ∞], p > 4/3 and θ0 ∈ L p (R2 ); there exists a weak solution θ on (0, T ) × R2 to (QG) which belongs to L ∞ ((0, T ), L p ) and satisfies θ (t) p ≤ θ0 p for all t ∈ (0, T ). In order to prove this kind of result, the classical method is to consider a sequence of smooth functions which are solutions of a regularization for (QG) and show that this sequence converges towards a solution of (QG). In this approach, the difficulty relies on proving the convergence of the non-linear term; here, we decompose each term of the non-linearity in low and high frequencies. Terms with low frequencies are easy to
Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2
47
deal with and we use the key remark (Lemma 2.1) to handle the term with the product of high frequencies. When we have some dissipation, the same kind of ideas allows us to construct weak solutions with an initial data in the space L 4/3 or H˙ −1/2 : Theorem 1.2. Let T ∈ (0, ∞], 0 < α ≤ 1. (i) If θ0 ∈ L p , 4/3 ≤ p < ∞ there exists a weak solution θ on (0, T ) × R2 to (QG)α such that θ ∈ L ∞ ((0, T ), L p ) and satisfies the global inequality: t p p θ (t) p + pk (2) sgn(θ )|θ | p−1 2α θ d xds ≤ θ0 p 0
for all t ∈ (0, T ). (ii) If θ0 ∈ H˙ −1/2 (R2 ), there exists a weak solution θ on (0, T ) × R2 to (QG)α such that θ ∈ L ∞ ((0, T ), H˙ −1/2 ) ∩ L 2 ((0, T ), H˙ α−1/2 ) and satisfies the global inequality: t 2 θ (t) H˙ −1/2 + 2k |α−1/2 θ |2 d xds ≤ θ0 2H˙ −1/2 (3) R2
0
for all t ∈ (0, T ). Remark 1.2. In the case where p ≥ 2 the solutions given in (i) belong in particular to p the space L 2 ((0, T ), L 1−α ) (see Sect. 2.3). Let us mention that for the sub-critical case 1/2 < α ≤ 1, Wu proves in [26], by means of a fixed point method, the global existence and uniqueness of regular solutions 2 to (QG)α in L q (L p ) spaces with an initial data in L p , p > 2α−1 and 1p + αq = α − 21 . p In the dissipative case, we can expect that the weak L solutions constructed by a limiting process (as in the proof of Theorem 1.1) have a better regularity than in the non-dissipative case. When p = 2 using the classical Leray’s proof for the weak L 2 solutions of the Navier-Stokes equations, we can easily show: Proposition 1.3. Let T ∈ (0, ∞], 0 < α ≤ 1 and θ0 ∈ L 2 (R2 ); there exists a weak solution θ on (0, T ) × R2 to (QG)α such that θ ∈ L ∞ ((0, T ), L 2 ) ∩ L 2 ((0, T ), H˙ α ) and satisfies the global inequality: t θ (t)22 + 2k |α θ |2 d xds ≤ θ0 22 (4) 0
R2
for all t ∈ (0, T ). For p = 2, the issue of regularity (in space) of the weak L p solutions constructed in Theorem 1.1 is much more difficult. Lemma 3.1 gives information on |θ | p/2 and thanks to the Sobolev embedding we can easily deduce a Lebesgue regularity but we do not have any other immediate information on the smoothness of θ . The issue of global Sobolev regularity seems out of reach and we are able (at least when p is large enough) to give a local smoothing property which is a uniformly locally Sobolev regularity of the solutions: Theorem 1.4. Let T ∈ (0, ∞], 0 < α ≤ 1, p ∈ (max(2, 1/α), ∞) and θ0 ∈ L p (R2 ). The solutions constructed in Theorem 1.2 satisfy the following uniform local estimate: T sup |α (θ )|2 d xdt < ∞ (5) x0 ∈R2 0
|x−x0 |<1
α ) for all T ∈ (0, T ], for all T ∈ (0, T ], T < ∞. In particular, θ ∈ L 2 ((0, T ), Hloc
T < ∞.
48
F. Marchand
α ) is The fact that estimate (5) implies in particular that θ belongs to L 2 ((0, T ), Hloc shown in the Appendix. In Sect. 8 we discuss why estimate (5) is not always valid in the super-critical case. The proof of (5) relies on uniformly local energy estimates, more precisely we show that whenever (θk )k≥1 is a collection of solutions to an approximation of the quasi-geostrophic equations, it satisfies T sup |α (φ j θk )|2 d xdt < ∞ j∈Z2 0
with φ j = φ0 (· − j), φ0 ∈ D, φ0 ≥ 0 and j∈Z2 φ j = 1. This kind of estimate is in the spirit of Lemarié-Rieusset’s work, [13] or [14], on the weak infinite-energy solutions for the Navier-Stokes equations in R3 . We first give in Sect. 2 a key remark on the structure of the non-linear term. In the short Sect. 3, we give a new proof of the maximum principle which admits an interesting generalization. In §4 we define the notion of weak solution that we will use in our work. In Sect. 5 we prove the existence of weak L p solutions, 4/3 < p < ∞, for the quasi-geostrophic equations (dissipative or not). In Sect. 6 we use the proof of Theorem 1.1 to get global weak H˙ −1/2 solutions and extend the range of p to min (1, 4/3(1 − α)) < p < ∞. The proof of Theorem 1.4 appears in §7, and finally, we collect additional comments in §8. We conclude this introduction by establishing a few notations and definitions used in this paper. First of all, except in Sect. 3, the functional spaces for the space variable will be always considered on R2 . The norm in L p (R2 ), 1 ≤ p ≤ ∞, will be denoted by · p , D(R2 ) will denote the usual space of C ∞ functions which are compactly supported and H ∞ = ∩k≥0 H k . We now define the very convenient tool of Littlewood-Paley analysis. Let ϕ ∈ D(R2 ) be a non-negative function so that ϕ(ξ ) = 1 for |ξ | ≤ 1/2 and ϕ(ξ ) = 0 for |ξ | ≥ 1. For j ∈ Z, we define the j th dyadic block of the Littlewood-Paley decomposition of a tempered distribution f by j f = F −1 ψ(ξ/2 j )F( f ) with ψ(ξ ) = φ(ξ/2) − φ(ξ ); we also define the low-frequencies and high-frequencies cutting operators by
S j f = F −1 φ(ξ/2 j )F( f ) = k f, H j f = (I d − S j ) f =
k≤ j−1
k f.
k≥ j
We will make some use of Besov spaces: Definition 1.1 (Inhomogeneous Besov spaces). For s ∈ R, ( p, q) ∈ [1, ∞] and N ∈ Z,
s,q the distribution f ∈ S belongs to the Besov spaces B p if and only if
q f B s,q = S N f p + ( 2 jqs j f p )1/q < ∞; p j≥N s,q
with this norm B p is a Banach space.
Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2
49
Definition 1.2 (Homogeneous Besov space). Let (s, p, q) ∈ R × [1, ∞] × [1, ∞] such that s < 2p or s = 2p and q = 1. Then, a tempered distribution f is said to belong to
s,q the homogeneous Besov space B˙ p if j∈Z j f converges in S and ⎛ f B˙ s,q =⎝ p
⎞1/q 2 jqs j f p ⎠ q
< ∞.
j∈Z
An interesting aspect of Besov spaces is that they contain Sobolev spaces: B2s1 ,2 = H s1 and B˙ 2s2 ,2 = H˙ s2 for all s1 ∈ R and s2 < 1. For further details on Besov spaces the reader can consult [25]. p Finally, we recall that L uloc , p ∈ [1, ∞), is the space of uniformly locally L p integrable functions: p
L uloc =
1/ p
f ∈ S , f L p
uloc
p
= sup
x∈R2
|x−y|<1 − 2 ,∞
and we have the embedding L uloc (R2 ) → B∞ p
| f (y)| p dy
<∞ ,
(R2 ).
2. A Remark on the Structure of the Non-Linearity We first start with the following key remark (which appears in [1]) on the structure of the non-linearity: Lemma 2.1. Let θ ∈ L 2 such that θˆ is equal to zero in a neighborhood of 0 and ϕ ∈ D(R2 ), we have R2 θ [, ∂1 ϕ] (−1 θ ) d x − R1 θ [, ∂2 ϕ] (−1 θ ) d x, 2 θ R⊥ (θ ) · ∇ϕ d x = where [, ∂i ϕ] is the commutator between and ∂i ϕ.
(6)
Remark 2.1. As we will see in the next lemma, the operators [, ∂i ϕ] are operators of order zer o if ∇ϕ is smooth enough; the interest of 2.1 can thus be seen as a gain in derivative on θ with a loss on ϕ. Proof. On one hand, we write θ R⊥ (θ ) · ∇ϕ d x = R⊥ (θ ) · ∇ϕ(−1 θ ) d x. On the other hand, we have θ R⊥ (θ ) · ∇ϕ d x = − θ R(θ ) · ∇ ⊥ ϕ d x;
(7)
50
F. Marchand
x −2 we then use an approximation of the identity, ρ (x) = ρ( ) with ρ ∈ D and ρ d x = 1, and write
⊥
ρ ∗ θ R(ρ ∗ θ ) · ∇ ϕ d x =
=
∇(ρ ∗ θ ) −1 (ρ ∗ θ ) · ∇ ⊥ ϕ d x R(ρ ∗ θ ) · (−1 (ρ ∗ θ )∇ ⊥ ϕ) d x.
Then, letting go to 0, we get θ R⊥ (θ ) · ∇ϕ d x = Rθ · (−1 θ ∇ ⊥ ϕ) d x.
(8)
We then sum the two formulations (7) and (8). In order to deal with identity (6) , we need to study the continuity properties of the commutator [, ∂i ϕ] which can be seen as a Calderón commutator: Lemma 2.2 (Calderón commutator). The singular integral operator Ti = [, ∂i ϕ] is bounded on every L p with p in (1, ∞) and on every H˙ η with −1 < η < 1: Ti ( f ) p ≤ C p ∇ 2 ϕ∞ f p
for all f ∈ L p with p ∈ (1, ∞)
and Ti ( f ) H˙ η ≤ Cη max(∇ 2 ϕ∞ , ∇ϕ B˙ 1,∞ ) f H˙ η for all f ∈ H˙ η with η ∈ (−1, 1). 2
Proof. Ti is actually a singular integral operator since its kernel on R2 × R2 − D (where D is the diagonal x = y) is the function K i (x, y) = C
∂i ϕ(x) − ∂i ϕ(y) |x − y|3
and ∇ 2 ϕ is bounded. It is easy to see that Ti has the weak boundedness property; then, since Ti (1) = −Ti (1) = ∂i ϕ = −R1 ∂1 ∂i ϕ − R2 ∂2 ∂i ϕ ∈ B M O, the T (1) theorem of David and Journé [6] gives that Ti is a Calderón-Zygmund operator with a norm less than ∇ 2 ϕ∞ . In particular, it is bounded on every L p with p in (1, ∞). In order to prove the continuity on Sobolev spaces we first observe that, by duality, we only need to prove it when the regularity is positive. Then, according to a result of M. Meyer[20], Ti is bounded on H˙ η , 0 < η < 1, if and only if Ti (1) satisfies πTi (1) ( f ) = j∈Z j Ti (1)S j−2 f ∈ H˙ η for all f ∈ H˙ η . η−1,2 We thus take a function f ∈ H˙ η ; the Bernstein inequalities give that f ∈ B˙ ∞ . If we write
2 j (η−1) S j f ∞ ≤ 2( j−k)(η−1) 2k(η−1) k f ∞ , k≤ j
since η − 1 is negative, we get that
( 22 j (η−1) S j f 2∞ )1/2 ≤ C f B˙ η−1,2 . j∈Z
∞
Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2
51
We then write j ∂i ϕ S j−2 f 2 ≤ S j−2 f ∞ j ∂i ϕ2 , and get that πTi (1) ( f ) H˙ η ≤ C∂i ϕ B˙ 1,∞ f H˙ η ≤ Cϕ B˙ 3,∞ f H˙ η . 2
2
Since ϕ ∈ D, ϕ B˙ 3,∞ is finite. 2
Remark 2.2. A general statement on the Lebesgue continuity of Calderón commutators can be found in [13], pp. 99. The previous identity (6) allows us to define the non-linear term ∇ · (θ R⊥ (θ )) as a distribution when θ ∈ H˙ −1/2 or in L p , 4/3 ≤ p < 2 : Proposition 2.3. For every function f in the space H˙ −1/2 (R2 ) or L p , 4/3 ≤ p < 2 , we define the tempered distribution ∇ · ( f R⊥ ( f )) by: ∇ · ( f R⊥ ( f )) = ∇ · ( f R⊥ (S j f )) + ∇ · (S j f R⊥ (H j f )) + ∇ · (H j f R⊥ (H j f )), where j ∈ Z and ∇ · (H j f R⊥ (H j f )) is the distribution defined by: R2 (H j f ) [, ∂1 ϕ] (−1 (H j f )) d x ∇ · (H j f R⊥ (H j f ))|ϕ = − R1 (H j f ) [, ∂2 ϕ] (−1 (H j f )) d x for all ϕ ∈ D(R2 ). When f ∈ H˙ −1/2 (R2 ) we have: |∇ · (H j f R⊥ (H j f ))|ϕ| ≤ C max(∇ 2 ϕ∞ , ∇ϕ B˙ 1,∞ )H j f 2H˙ −1/2 , 2
and when f ∈ L p , 4/3 < p < 2, we have: |∇ · (H j f R⊥ (H j f ))|ϕ| ≤ C 2− j (3−4/ p) ∇ 2 ϕ∞ f 2p . Proof. Since S j f ∈ L ∞ , the first two terms are well defined tempered distributions and we just have to check it for the last term. Let us first consider the case where f ∈ H˙ −1/2 (R2 ); the continuity of Riesz transforms on H˙ −1/2 gives: |∇ · (H j f R⊥ (H j f ))|ϕ| ≤ C H j f H˙ −1/2 [, ∂i ϕ] −1 (H j f ) H˙ 1/2 . Then, the continuity of [, ∂i ϕ] on H˙ 1/2 (see the proof of Lemma 2.2) gives, |∇ · (H j f R⊥ (H j f ))|ϕ| ≤ C max(∇ 2 ϕ∞ , ∇ϕ B˙ 1,∞ )H j f 2H˙ −1/2 . 2
Now, in the case where f ∈ L p , 4/3 ≤ p < 2, Lemma 2.2 allows us to write: |∇ · (H j f R⊥ (H j f ))|ϕ| ≤ C∇ 2 ϕ∞ H j f p −1 H j f q with
1 p
+
1 q
= 1.
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F. Marchand
Thanks to the Sobolev embedding H˙ 1−2/q → L q , we obtain, 2
−2
−1 H j f q ≤ C p
H j f 2 .
We then write, 2
p
−2
4
H j f 2 = p
−3
1− 2p
≤ C2
− j (3− 4p )
≤ C2
− j (3− 4p )
≤ C2
− j (3− 4p )
H j f 2
f
−2( 1p − 21 ),2
B˙ 2
f B˙ 0,2 p
f p,
which gives the desired estimate: |∇ · (H j f R⊥ (H j f ))|ϕ| ≤ C∇ 2 ϕ∞ 2− j (3−4/ p) f 2p . Finally, we observe that the definition is clearly independent of j ∈ Z. Remark 2.3. The proof above shows that ∇ · (H j f R⊥ (H j f )) can be seen as a distribution in the Besov space B˙ 2−3,∞ or in the Sobolev space H −3− for all > 0 ( B˙ 21,1 (R2 ) → L ∞ (R2 )). It also shows that we can still give sense to ∇ · ( f R⊥ ( f )) when the distribution f
−j belongs to the more general space E N = { f ∈ S / S N f ∈ L ∞ and j≥N 2 j 2 f 2 < ∞} for a N ∈ Z. We have the immediate corollary, Corollary 2.4. Let f be a function as in Proposition 2.3. We have ∇ · ( f R⊥ ( f )) = lim
j→+∞
∇ · ( f R⊥ (S j f )) + ∇ · (S j f R⊥ (H j f ))
in S (R2 ). 3. Maximum Principle In this section we give a new proof of the maximum principle property [2, 10, 5]; we have the following inequality: Lemma 3.1. Let 0 < α ≤ 2, 2 ≤ p < ∞ and θ be a smooth function on Rn . We have: α p/2 2 2 | (|θ | )| d x ≤ p sgn(θ )|θ | p−1 2α θ d x, where sgn(θ ) denotes the sign of θ .
Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2
53
Proof. The integrals in the desired inequality can be rewritten in the following form: 2
|α (|θ | p/2 )|2 d x = 2
2α (|θ | p/2 )|θ | p/2 d x = −
d −t2α p/2 2 e |θ | 2 |t=0+ dt
and sgn(θ )|θ | p−1 2α θ d x = −
p
d −t2α p θ p |t=0+ . e dt
Let F(t) = e−t θ p and G(t) = e−t |θ | p/2 22 . We need to prove that F (0) ≤
G (0). p First of all, we have F(0) = G(0) = θ p . Then, if we denote by K t the kernel of 2α the operator e−t (K t ≥ 0 and K t (x)d x = 1), we can write 2α
|(e
−t2α
2α
p
θ )(x)| ≤
K t (x − y)|θ |(y)dy ≤
K t (x − y)|θ |
p/2
2/ p (y)dy .
Taking the L p norm we obtain,
|e−t θ | p d x ≤ 2α
|e−t |θ | p/2 |2 d x, 2α
i.e. F(t) ≤ G(t) for all t ≥ 0. Remark 3.1. This proof can be readily generalized in the following way: let m : R+ → R such that m(0) = 0 and the kernel K t of e−tm() ( Kˆ t (ξ ) = e−tm(|ξ |) ) is non-negative then, for every smooth function θ on Rn we have, p/2 2 2 | m()(|θ | )| d x ≤ p sgn(θ )|θ | p−1 m()(θ )d x. α
The fact that the Fourier transform of x → e−|x| α is non-negative is a consequence of the Bochner theorem (see [22]) and that x → e−|x| is positive definite when 0 < α ≤ 2 (see [23]). For the case where 1 ≤ p < 2 we recall a result that can be found in [5]: Lemma 3.2. Let 0 < α ≤ 2, 2 ≤ p < ∞ and θ be a smooth function on Rn . We have: sgn(θ )|θ | p−1 2α θ d x ≥ 0. We can now state the maximum principle for the quasi-geostrophic equations:
54
F. Marchand
Corollary 3.3 (Maximum principle). Let 1 ≤ p < ∞ and θ be a smooth solution of (QG)α ; we have: t p p θ (t) p + 2k |α (|θ | p/2 )|2 d xds ≤ θ0 p if 2 ≤ p, 0
θ (t) p ≤ θ0 p if
1 ≤ p < 2,
and θ (t)∞ ≤ θ0 ∞ for all t ≥ 0. Proof. d p p−2 ⊥ 2α θ (R θ · ∇θ − k θ )d x = −kp |θ | p−2 θ 2α θ d x |θ | d x = p |θ | dt and Lemma 3.1, 3.2 lead to the conclusion for the first and the second inequalities. The last inequality is a consequence of the first one since θ (t) p → θ (t)∞ as p goes to +∞. 4. Weak Solutions for the Quasi-Geostrophic Equations In this section, we give the definitions of solutions to the quasi-geostrophic equations that we will use in our work. In all of this section T is a fixed positive time possibly equal to +∞. We first start with the definition of a weak solution to the quasi-geostrophic equation with no dissipation. Because of the Riesz transforms in the non-linear term, we cannot consider, as in the 2 ((0, T ) × R2 ) to define Navier-Stokes equations, the case of a distribution θ (t, x) ∈ L loc weak solutions to (QG). In order to avoid the definition of the Riesz transform on bad spaces, we define the space F2 by the space of all functions f ∈ L 2uloc (R2 ) such that S j f
goes to 0 in S when j → −∞ and such that R1 (I d − S j ) f and R2 (I d − S j ) f admit lim
its in S which belong to L 2uloc (R2 ); for such a function we can write R1 f ∈ L 2uloc (R2 ) and R2 f ∈ L 2uloc (R2 ). Definition 4.1. A weak solution to the non-dissipative quasi-geostrophic equation (QG) 2 ((0, T ), F ) or in L 2 ((0, T ), H −1/2 ) on (0, T ) × R2 is a distribution θ (t, x) ∈ L loc 2 loc
which satisfies in D ((0, T ) × R2 ): ∂t θ − ∇ · (θ R⊥ θ ) = 0.
(9)
Now, in the dissipative case, we assume without loss of generality that k = 1 and we give two definitions of solutions. There exist several ways to define a solution to the equation (QG)α , for example we can find in [8] six different definitions to the Navier-Stokes equations (which are shown to be equivalent). Here we will use only two of them, the first one (weak solution) being the most physical one and the second one (mild solution) which is quite practical for some computations. We can now state the definitions:
Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2
55
Definition 4.2. Let 0 < α ≤ 1, a. A weak solution to the quasi-geostrophic equation (QG)α on (0, T )×R2 is a distri2 ((0, T ), F ) or L 2 ((0, T ), H −1/2 ) which satisfies in D ((0, T ) × bution θ (t, x) in L loc 2 loc 2 R ): ∂t θ − ∇ · (θ R⊥ θ ) + 2α θ = 0. (10) b. A mild solution to the quasi-geostrophic equation (QG)α on (0, T ) × R2 with
initial data θ0 ∈ S (R2 ) is a distribution θ (t, x) in ∩0
∩0
−t2α
t
θ0 +
e−(t−s) ∇ · (θ R⊥ θ )ds. 2α
(11)
0
Remark 4.1. The assumptions on θ are not restrictive; Leray’s solutions (θ (t) ∈ L ∞ ((0, T ), L 2 ) ∩ L 2 ((0, T ), H˙ α )), the weak L p solutions described in this paper (θ (t) ∈ p L ∞ ((0, T ), L p ) ∩ L 2 ((0, T ), L 1−α )) and the mild solutions that we obtain in [17] and
[18] are all in L 2 ((0, T ), F2 ) for every T < T . The H˙ −1/2 − solutions (see Definition 8.1) given in this paper which belong to 2 ((0, T ), H −1/2 ) but are L ∞ ((0, T ), H˙ −1/2 ) ∩ L 2 ((0, T ), H˙ α−1/2 ) are obviously in L loc
also in L 2 ((0, T ), F2 ) (for all T < T ) when 1/2 ≤ α ≤ 1.
Proposition 4.1. Let θ be in ∩0
4/3 Let θ0 ∈ L p with p > 4/3, for each parameter 0 < ≤ 1 we consider the approximated flow: ⎧ ⎨ θt + ∇ · (θ u) − θ = 0, (x, t) ∈ R2 × [0, ∞) (12) u = −R⊥ θ = −(−R2 θ, R1 θ ) ⎩ θ (x, 0) = (ω ∗ (χ1/ θ0 ))(x), x ∈ R2 with ω (x) = −2 ω( x ), ω ∈ D(R2 ), ωd x = 1 and χ R (x) = 1 if |x| ≤ R and χ R (x) = 0 otherwise. Using standard techniques, it is easy to prove (see [18] or [2]) that there exists a global solution θ ∈ C ∞ ((0, ∞) × R2 ) of (12) such that θ (t) ∈ H ∞ for all t > 0. Then, according to Remark 3.1 sgn(θ )|θ | p−1 (−)θ d x ≥ 0 and we get a family (θ ) of solutions to (12) with the uniform bound θ (t) p ≤ θ0 p .
(13)
Since L ∞ (R∗+ , L p (R2 )) = L 1 (R∗+ , L q (R2 )) , with 1p + q1 = 1, we can extract from those solutions θ a sequence (θk )k≥1 which is ∗-weakly convergent to some function
θ in the space L ∞ (R∗+ , L p (R2 )) (which implies convergence in D (R∗+ × R2 )). Weak
56
F. Marchand
convergence is not enough to control the non-linear term ∇ · (θk R⊥ (θk )) and we only
have ∂t θ + limk→∞ ∇ · (θk R⊥ (θk )) = 0 in D (R∗+ × R2 ).
We now show that ∇ ·(θk R⊥ (θk )) converges toward ∇ ·(θ R⊥ (θ )) in D (]0, ∞[×R2 ). Let ϕ be a function in D((0, ∞) × R2 ) and T , R two non-negative real numbers such that R). We are going to show that the ⊥support of ϕ is included in (0, T )× B(0, θk R θk · ∇ϕ dtd x converges towards θ R⊥ θ · ∇ϕ dtd x. We first start with the case p ≥ 2; we will need the following technical lemma which is proved in the appendix: Lemma 5.1 (Regularizing operators). Let T be a bounded operator from L p to H ps with −η −η s > 0 and from Hq to Hq , for η > 0 and q ≥ 2, then (T (θk ))k≥1 converges strongly 2 ((0, ∞) × R2 ). toward T (θ ) in L loc We choose ρ ∈ D(R2 ) such that ρ is equal to one on B(0, R) and equal to zero outside of B(0, 2R). We note γ (x) = ρ(x/10). We write θk R⊥ θk · ∇ϕ(t, x) dtd x = (1) + (2) with
(1) =
and
θk ρR⊥ ((1 − γ )θk ) · ∇ϕ(t, x) dtd x
(2) =
(γ θk ) R⊥ (γ θk ) · ∇ϕ(t, x) dtd x.
Applying Lemma 5.1 to T : f → ρ R j ((1 − γ ) f ) ( j = 1 or 2), we get θρR⊥ ((1 − γ )θ ) · ∇ϕ(t, x) dtd x. lim (1) = k→∞
In order to deal with (2) we define θ˜k = γ θk and θ˜ = γ θ . Let j ∈ Z, we can decompose (2) in θ˜k R⊥ (S j θ˜k ) · ∇ϕ dtd x + S j θ˜k R⊥ (H j θ˜k ) · ∇ϕ dtd x + H j θ˜k R⊥ (H j θ˜k ) · ∇ϕ dtd x. Using Lemma 5.1 with R⊥ S j and S j , we readily get ⊥ ˜ ˜ θ˜ R⊥ (S j θ˜ ) · ∇ϕ(t, x) dtd x θk R (S j θk ) · ∇ϕ(t, x) dtd x = lim k→∞
and lim
k→∞
S j θ˜k R⊥ (H j θ˜k ) · ∇ϕ(t, x) dtd x =
S j θ˜ R⊥ (H j θ˜ ) · ∇ϕ(t, x) dtd x.
Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2
57
With the help of Lemma 2.1 the last term can be rewritten in the following form: R2 H j θ˜k [, ∂1 ϕ] (−1 H j θ˜k ) d xdt − R1 H j θ˜k [, ∂2 ϕ] (−1 H j θ˜k ) d xdt; and we use Lemma 2.2 to get |
H j θ˜k R⊥ (H j θ˜k ) · ∇ϕ(t, x) dtd x|
≤ C T ∇x2 ϕ∞ H j θ˜k L ∞ (L 2 ) −1 H j θ˜k L ∞ (L 2 ) ≤ C(ϕ)2− j H j (γ θk )22 ≤ C(ϕ)2− j θ0 2p . In the same way, we have | H j θ˜ R⊥ (H j θ˜ ) · ∇ϕ(t, x) dtd x| ≤ C(ϕ)2− j θ0 2p . Finally, we obtain lim sup | θk R⊥ θk · ∇ϕ dtd x − θ R⊥ θ · ∇ϕ dtd x| ≤ C(ϕ)2− j θ0 2p , k→∞
and when j goes to +∞ we get the result. We now turn to the case 4/3 < p < 2; in that case we don’t need to localize in space but, as before, for a j ∈ Z we make a decomposition in low and high frequencies. The convergence of the terms with low frequencies are obtained by the regularizing effect of S j : Lemma 5.2. Let j ∈ Z. The sequence (S j θk )k≥1 (resp. (R⊥ (S j θk ))k≥1 ) converges p
strongly towards S j θ (resp. R⊥ (S j θ )) in L p−1 ((0, ∞) × R2 ). We postpone the proof of this lemma to the Appendix. We then obtain, lim θ R⊥ (S j θ ) · ∇ϕ(t, x) dtd x θk R⊥ (S j θk ) · ∇ϕ dtd x = k→∞
and
lim
k→∞
S j θk R⊥ (H j θk ) · ∇ϕ dtd x =
S j θ R⊥ (H j θ ) · ∇ϕ dtd x.
We now deal with the integral with the product of high frequencies: Ik = H j θk R⊥ H j θk · ∇ϕ dtd x. The estimate of Ik has already been done in Proposition 2.3: |Ik | ≤ C(ϕ)2
− j (3− 4p )
θ0 2p ,
which is enough to conclude as in the case p ≥ 2.
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F. Marchand . −1/2
6. Weak Solutions of ( QG)α with Initial Data in H
We now prove Theorem 1.2; we assume 0 < α ≤ 1, and first consider the case of an initial data in H˙ −1/2 . We follow the same lines as in Sect. 2.5 with the approximated flow: ⎧ ⎨ θt + ∇ · (θ u) + 2α θ − θ = 0, (x, t) ∈ R2 × [0, ∞) (14) u = −R⊥ θ = −(−R2 θ, R1 θ ) ⎩ θ (x, 0) = (ω ∗ θ0 )(x), x ∈ R2 with ω (x) = −2 ω( x ), ω ∈ D(R2 ) and ωd x = 1. We get a family (θ ) of smooth solutions to (14) with the uniform bound θ
(t)2H˙ −1/2
t +2 0
|α−1/2 θ |d xds ≤ θ0 2H˙ −1/2 .
(15)
We can extract from those solutions θ a sequence (θk )k≥1 which is ∗-weakly convergent to some function θ in the space L ∞ ((0, T ), H˙ −1/2 (R2 )). Then, as in the proof of Theorem 1.1, for a j ∈ Z we decompose θk R⊥ θk · ∇ϕ(t, x) dtd x in the three terms: ⊥ θk R (S j θk ) · ∇ϕ dtd x + S j θk R⊥ (H j θk ) · ∇ϕ dtd x + H j θk R⊥ (H j θk ) · ∇ϕ dtd x. The convergence of the first two terms is obtained by the regularizing effect of S j : Lemma 6.1. Let j ∈ Z; the sequence (S j θk )k≥1 (resp. (R⊥ (S j θk ))k≥1 ) converges 1/2 strongly towards S j θ (resp. R⊥ (S j θ )) in Hloc ((0, ∞) × R2 ). The proof of this lemma is postponed to the Appendix. −1/2 Using the inclusion L ∞ ((0, ∞), H˙ −1/2 ) → Hloc ((0, ∞) × R2 ), we get : lim
k→∞
⊥
θk R (S j θk ) · ∇ϕ(t, x) dtd x =
θ R⊥ (S j θ ) · ∇ϕ(t, x) dtd x
and lim
k→∞
S j θk R⊥ (H j θk ) · ∇ϕ(t, x) dtd x =
S j θ R⊥ (H j θ ) · ∇ϕ(t, x) dtd x.
We now turn to the convergence of the product of the “high frequencies”. More precisely, we need the uniform (in k) convergence of the integrals Ik = towards zero when j goes to +∞.
H j θk R⊥ H j θk · ∇ϕ dtd x
Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2
59
With the help of the uniform L 2 ( H˙ α−1/2 ) bound and Lemma 2.2, we can write: ∞ H j θk H˙ α−1/2 [, ∂i ϕ] −1 (H j θk ) H˙ 1/2−α dt |Ik | ≤ C 0 ∞ H j θk H˙ α−1/2 H j θk H˙ −1/2−α dt ≤ C(ϕ) 0 ∞ H j θk 2H˙ α−1/2 dt ≤ C(ϕ)2−2α j 0
≤ C(ϕ)2
−2α j
θk 2L 2 ( H˙ α−1/2 )
≤ C(ϕ)2−2α j θ0 2H˙ −1/2 . The case where θ0 ∈ L 4/3 is a direct consequence of the case θ0 ∈ H˙ 1/2 since → H˙ −1/2 .
L 4/3
7. Regularity Estimates for Weak Solutions In this section we prove Theorem 1.4 and more precisely estimate (5). Let T ∈ (0, ∞], 0 < α ≤ 1, p ∈ (max(2, 1/α), ∞) and θ0 ∈ L p (R2 ). Without loss of generality, we assume k = 1. We proceed as in the proof of Theorem 1.1 and we consider a sequence (θk )k≥1 which
converges in D towards a weak solution θ ∈ L ∞ ((0, T ), L p ) on (0, T ) × R2 to (QG)α which satisfies inequality (4). We will need the following useful tool to localize L 2 norms: Let φ0 ∈ D(R2 ) such that φ0 ≥ 0 and j∈Z2 φ0 (x − j) = 1. We define φ j (x) = φ0 (x − j) for all j ∈ Z2 .
Proposition 7.1. For all T ∈ (0, T ], T < ∞, we have
Ak (T ) = sup
j∈Z2 0
T
|α (φ j θk )|2 d xdt ≤ C(α, p, φ0 , θ0 , T ),
(16)
where C(α, p, φ0 , θ0 , T ) is finite for all T > 0.
Proof. Let j ∈ Z2 and T ∈ (0, T ] with T < ∞. If we multiply the evolution equation of θk by φ 2j θk , after an integration over R2 we get the evolution equation of the local energy: 1 d 2 2 2 2α |φ j θk | d x = − φ j θk u k · ∇θk d x − φ j θk θk d x + k φ 2j θk θk d x, 2 dt which can be rewritten as, 1 d |φ j θk |2 d x + |α (φ j θk )|2 d x − k φ 2j θk θk d x = β + γ 2 dt with
β=
φ 2j θk u k · ∇θk d x = −
1 2
θk2 φ j u k · ∇φ j d x
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F. Marchand
(because div u = 0), and
γ =
φ j θk 2α (φ j θk ) − φ j 2α θk d x.
Then, after an integration over time, we can write:
T
α
T
| (φ j θk )| d xdt + 2 k
2
2
0
φ 2j |∇θk |2 d xdt
0
= (φ j θk )(0, ·)2 − (φ j θk )(T , ·)2 T
T 2 2 + k β dt + 2 θk φ j d xdt + 2 0
0
Let j ∈ Z2 , if we note Ak, j (T ) =
T 0
T
2 Ak, j (T ) ≤ C( p, φ0 , θ0 ) + k
T
γ dt.
0
|α (φ j θk )|2 d xdt, we have:
|θk2 φ 2j |d xdt
0
We start with the easiest bound: T |θk |2 |φ 2j | d xdt ≤ C(φ0 ) 0
T
T
+2
(|β| + |γ |) dt.
0
supp φ j
0
|θk |2 d xdt
≤ C( p, φ0 )T θ0 p
≤ C( p, φ0 , θ0 )T . Estimates of β and γ are much more complicated: Lemma 7.2. There exists 0 < η < 1 such that |
T
0
1−η
β dt| ≤ C( p, φ0 , θ0 , T )Ak
(T ).
Proof. We first assume α = 1/2; since p is greater than 2, we have a uniform control 2+ ) for all small enough, we fix for example = p−2 . Let I be of (θk )k≥1 in L ∞ (L loc j 2 the smaller finite subset of Z2 such that i∈I j φi ≡ 1 on the support of φ j (the cardinal number of I j does not depend on j); the Hölder inequality gives
|β| ≤
|∇φ j · u k |
2+
1 2+
|θk |
dx
≤ C(φ0 )
supp(φ j )
|u k |
2+
1 2+
dx
2 4− 1+
|φ j |
2+ 1+
1+ 2+
dx
|φi θk |
2 4− 1+
1+ 2+
dx
.
i∈I j
The boundedness of Riesz transforms on L p and the Sobolev embedding H˙ 1/2−σ → 2 imply L 4− 1+ with σ = 4+2 1
|β| ≤ C( p, φ0 )θ0 p2
i∈I j
1/2−σ (φi θk )22 .
Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2
61
Then, the interpolation inequality 1/2 (φi θk )1−2σ 1/2−σ (φi θk )2 ≤ φi θk 2σ 2 2
and the Hölder inequality give
T
|β| dt ≤ C( p, φ0 , θ0 )
0
i∈I j
T
φi θk 22 dt
0
≤ C( p, φ0 , θ0 )(T θ0 p )
2σ
2σ
sup i∈Z
T
1−2σ
1/2
0 T
(φi θk )22 1−2σ
1/2
0
(φi θk )22
≤ C( p, φ0 , θ0 , T ) Ak (T )1−2σ . The case α > 1/2 is a direct consequence of the case α = 1/2 since we write ≤ C(φ0 )α (φi θk )1−2σ . 1/2 (φi θk )1−2σ 2 2 For the case 0 < α < 1/2, since p > 1/α, we have a uniform control of (θk )k in 1
+
α ), then we follow the proof of the case α = 1/2. L ∞ (L loc
Lemma 7.3. |
T
0
γ dt| ≤ C(α, p, φ0 , θ0 , T )(1 + A1/2 (T )).
Proof. We have the following formula (see [24] or more recently [5]) to compute 2α , when f ∈ H ∞ and 0 < α < 1: f (x − y) − f (x) 2α ( f ) = Cα p.v. dy. |y|2+2α We point out that in the case 0 < α < 1/2 we do not need the principal value ( p.v.) to define the integral; in the following to clarify the exposition we will omit the principal value. Let us assume 0 < α < 1; the formula allows us to compute 2α (θk φ j ): 2α (θk φ j ) = θk 2α φ j + φ j 2α θk + C2α (θk , φ j ) with
C2α (θk , φ j ) =
(17)
(δ y θk )(δ y φ j ) dy |y|2+2α
and δ y f (x) = f (x − y) − f (x). This identity shows that in order to control γ , we need to control the following two integrals : (1) = φ j θk2 2α φ j d x, δ y θk (x)δ y φ j (x) d yd x. (2) = φ j (x)θk (x) |y|2+2α
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F. Marchand
The control of (1) is obvious: |(1)| ≤ φ j 2α φ j ∞
supp φ j
|θk |2 d x ≤ C(α, p, φ0 , θ0 ).
To deal with (2), we split the integral in two parts, |y| > 1 and |y| ≤ 1. We first have |y|>1
φ j (x)θk (x)
Thus,
δ y θk (x)δ y φ j (x) d yd x = − |y|2+2α
δ y φ j (x) d yd x |y|2+2α |y|>1 δ y φ j (x) φ j (x)θk (x)θk (x−y) 2+2α d yd x. + |y| |y|>1
|y|>1
δ y φ j (x) φ j (x)θk2 (x) 2+2α d yd x |y|
φ j (x)θk2 (x)
≤
2φ j 2∞ θk 2L 2 (supp φ ) j
|y|>1
dy |y|2+2α
≤ C(α, p, φ0 , θ0 ). We then note that x →
1 p |y|>1 |y|2+2α θk (x − y)dy belongs to L , since it is the convolution (θk ) and a L 1 function. We thus get:
between a L p function δ y φ j (x) φ j (x)θk (x)θk (x − y) 2+2α d yd x ≤ C(α, p, φ0 , θ0 ). |y| |y|>1
2 We then deal with the |y| ≤ 1 part. Let I j be the smaller finite subset of Z such that χ j (x) = i∈I j φ0 (x − i) = 1 if dist (x, supp(φ j )) ≤ 1 (the cardinal number of I j does not depend on j), we thus get
δ y θk (x)δ y φ j (x) d yd x |y|2+2α |y|≤1 δ y (χ j θk )(x)δ y φ j (x) φ j (x)θk (x) d yd x. = |y|2+2α |y|≤1
I =
φ j (x)θk (x)
The Cauchy-Schwartz inequality gives
|δ y φ j |2 (x) |δ y (χ j θk )|2 (x) |I | ≤ |θk | (x) φ j (x)d yd x φ j (x)d yd x |y|2+2α |y|2+2α |y|≤1 |y|≤1 φ j (x)|θk |2 (x) |δ y (χ j θk )|2 (x) d yd x d yd x ≤ φ0 ∞ ∇φ0 2∞ |y|2α |y|2+2α |y|≤1 2
2
≤ C(α, p, φ0 , θ0 )α (χ j θk )22
≤ C(α, p, φ0 , θ0 ) α (φi θk )22 , i∈I j
which gives the estimate on γ .
Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2
63
Now, collecting all the previous estimates, we get the inequality
1/2
1−η
2 Ak (T ) ≤ C1 + C2 Ak (T ) + C3 Ak
(T ),
where C1 , C2 and C3 depends on α, p, φ0 , θ0 and T , and are finite for all T > 0. Finally, using the Young inequality we get
2 Ak (T ) ≤ C(α, p, φ0 , θ0 , T ) +
1
Ak (T ) + (1 − η)Ak (T ), 2
which gives the desired estimate for Ak (T ). The case α = 1 follows the same ideas but is easier since we can write: (θk φ j ) = θk φ j + φ j θk + 2∇φ j · ∇θk , and the rest of the proof is essentially the same as before. Finally, Proposition 7.1 implies estimate (5) in an obvious way.
8. Comments We gather in this concluding section some additional comments and observations. A. The critical power α = 1/2. We first explain why we need p > 1/α in the estimate (5); the reason appears in the non-linear estimate of Lemma 7.2. In the proof of this lemma we have seen that in the critical case α = 1/2 there is a perfect balance between the nonlinear term and the dissipation when we have a L 2+ control, whereas is the subcritical case α > 1/2 we have much more dissipation than we need. In the super-critical case 0 < α < 1/2, in order to get the same balance we need a better control than L 2+ 1 which is L α + . Now, one may wonder that in the super-critical case, when the initial data is in L 2 , we are able to construct weak L 2 solutions which belong to L ∞ ((0, T ), L 2 ) ∩ L 2 ((0, T ), H˙ α ) and thus the uniformly local estimate holds,
A(T ) = sup
j∈Z2
T
|1/2 (φ j θ )|2 d xdt ≤ C(φ0 , θ0 , T ),
0
whereas 2 < 1/α. The deep reason of this fact is that when we construct a L 2 solution we use the following cancellation property: θ R⊥ θ · ∇θ d x = 0. When we localize this cancellation does not hold and we have to estimate φ 2j θ R⊥ θ · ∇θ d x. B. Uniqueness in the sub-critical case. We now briefly discuss the uniqueness of the weak solutions given in Theorem 1.4 for the sub-critical case. We recall the uniqueness result given by Constantin and Wu in [4] which can be seen as an analog to Serrin’s uniqueness result for the Navier-Stokes equations:
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F. Marchand
Definition 8.1 ( H˙ −1/2 -solution). A H˙ −1/2 -solution on (0, T ) to the quasi-geostrophic equation (QG)α with initial value θ0 ∈ H˙ −1/2 is a weak solution to (QG)α on (0, T ) × R2 such that: θ ∈ L ∞ ((0, T ), H˙ −1/2 ) ∩ L 2 ((0, T ), H˙ α−1/2 ) with the inequality, θ (t, ·)2H˙ −1/2
t + 2k 0
R2
|α−1/2 θ |2 d xds ≤ θ0 2H˙ −1/2
(18)
for all t ∈ (0, T ). Theorem 8.1. Let θ0 ∈ H˙ −1/2 , assume that there exists a weak solution to (QG)α on (0, T ) × R2 with initial value θ0 such that: θ ∈ L ∞ ((0, T ), H˙ −1/2 ) ∩ L 2 ((0, T ), H˙ α−1/2 ) and θ ∈ L q ((0, T ), L p ) with ( p, q) ∈ [1, ∞) and 1p + αq = α − 1/2. Then θ satisfies the inequality (18) and is the unique H˙ −1/2 -solution (in the sense of Definition 8.1) on (0, T ) × R2 with initial data θ0 . Let 1/2 < α ≤ 1 and consider the case of an initial data in L p ∩ H˙ −1/2 . We have seen in Sect. 6 that we are able to construct weak solutions which are in particular H˙ −1/2 2 solution and belong to L ∞ ((0, T ), L p ). According to Theorem 8.1, when p > 2α−1 these solutions are unique in the class of the H˙ −1/2 -solutions. 2 When q = ∞ and p = 2α−1 , it is not hard to prove that we still have the same conclusion. This is the analogue to the result of Kozono, Sohr [12] for the Navier-Stokes equations. Uniqueness in the critical case is harder to deal with; however we show in [19] that we can still have some results in that case. 9. Appendix Lemma 9.1 (Regularizing operators). Let (θk )k≥1 be a sequence of solutions of (12) which is bounded in L ∞ ((0, ∞), L p ) for a 2 ≤ p < ∞ which is -weakly convergent towards a function θ ∈ L ∞ ((0, ∞), L p ). If T is a bounded operator from L p in H ps −η −η with s > 0 and from Hq in Hq , for η > 0 and q ≥ 2; then (T (θk ))k≥1 converge 2 ((0, ∞) × R2 ). strongly towards T (θ ) in L loc Proof. Let ϕ ∈ D((0, T ) × R2 ) and let us call Fk (τ, ξ ) the Fourier transform in time and space of ϕT (θk ); using the first assumption we immediately get that |Fk (τ, ξ )|(1 + |ξ |2 )s dτ dξ < A(ϕ). On the other hand, we can write ∂t (ϕT (θk )) = ∂t (ϕ)T (θk ) + ϕT (∂t θk ) = ∂t (ϕ)T (θk ) + k ϕT (θk ) − ϕT (∇ · (θk R⊥ θk ));
Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2
65
then, the second assumption leads to the following estimate: |Fk (τ, ξ )|(1 + τ 2 )(1 + |ξ |2 )−3 dτ dξ < B(ϕ). Interpolating the weights we get |Fk (τ, ξ )|(1 + τ 2 )λ (1 + |ξ |2 )−3λ+s(1−λ) dτ dξ < C(ϕ) for all λ ∈ [0, 1]. We can deduce in particular that (ϕT (θk ))k is uniformly bounded in s s H s+4 (R × R2 ); then, the compact embedding of H s+4 (supp(ϕ)) in L 2 (supp(ϕ)) implies that (ϕT (θk ))k is relatively compact in L 2 (supp(ϕ)). Since ϕT (θk ) converges weakly toward ϕT (θ ) we have proved the lemma. Lemma 9.2. Let (θk )k≥1 be a sequence of solutions of (12) which is bounded in L ∞ ((0, ∞), L p ) for a 1 < p < 2 and -weakly convergent towards a function θ ∈ L ∞ ((0, ∞), L p ). For every j ∈ Z, the sequence (S j θk )k≥1 (resp. (R⊥ (S j θk ))k≥1 ) p
converges strongly towards S j θ (resp. R⊥ (S j θ )) in L p−1 ((0, ∞) × R2 ). Proof. The proof is essentially the same as for Lemma 9.1; for a ϕ ∈ D((0, T ) × R2 ), with the regularizing effect of S j it is easy to obtain that (ϕ S j θk )k is uniformly bounded 2− p
2− p
p
in H p (R × R2 ); then, the compact embedding of H p (supp(ϕ)) in L p−1 (supp(ϕ)) p implies that (ϕ S j θk )k is relatively compact in L p−1 (supp(ϕ)). Lemma 9.3. Let (θk )k≥1 be sequence of solutions of (14) which is bounded in L ∞ ((0, ∞), H˙ −1/2 ) and -weakly convergent towards a function θ ∈ L ∞ ((0, ∞), H˙ −1/2 ). For every j ∈ Z, the sequence (S j θk )k≥1 (resp. (R⊥ (S j θk ))k≥1 ) converge strongly towards S j θ 1/2 (resp. R⊥ (S j θ )) in Hloc ((0, ∞) × R2 ). Proof. For a ϕ ∈ D((0, T ) × R2 ), with the regularizing effect of S j we obtain a uniform bound of (ϕ S j θk )k in H s (supp(ϕ)) for a s > 1/2. Then, the compact embedding of H s (supp(ϕ)) in H 1/2 (supp(ϕ)) implies that (ϕ S j θk )k is relatively compact in H 1/2 (supp(ϕ)).
Lemma 9.4. Let ϕ ∈ D(R2 ) and 0 < T < ∞; estimate (16) implies that (α (ϕθk ))k
is uniformly bounded in L 2 ((0, T ) × R2 ). Proof. There exists a finite subset I of Z2 such that α (ϕθk ) =
α (ϕφi θk ).
i∈I
Then, using formula (17) we get α (ϕθk ) = ϕ
i∈I
α (φi θk ) + α ϕ
i∈I
φi θk +
(δ y ϕ)(δ y φi θk ) dy. |y|2+α i∈I
66
F. Marchand
We shall prove that each of these terms is uniformly bounded in L 2 ((0, T ) × R2 ); for the first two terms, the bounds are easy, T T
α (φi θk )|2 d xdt ≤ ϕ2∞ #I sup |ϕ α (φi θk )|2 d xdt 0
i∈Z 0
i∈I
T 0
|α ϕ
≤ C(ϕ, φ0 )A(T ), φi θk |2 d xdt ≤ α ϕ2∞ #I sup
i∈Z 0
i∈I
T
|φi θk |2 d xdt
≤ C(ϕ, φ0 )T θ0 p , where #I is the cardinal number of I . For the last term, we split the integral in two parts, |y| > 1 and |y| ≤ 1, T (δ y ϕ)(δ y φi θk ) 2 dy d xdt |y|2+α |y|>1 0 T |δ y ϕ(x)|2 |δ y (φi θk )(x)|2 ≤C d yd xdt |y|2+α |y|>1 0 T 2 ≤ 4Cϕ∞ α/2 (φi θk )22 ≤ C(ϕ) 0
0
T
1/2
φi θk 22
T 0
α (φi θk )22
1/2
1/2
≤ C(ϕ, φ0 )T 1/2 θ0 p A1/2 (T ). For the other part we write T (δ y ϕ)(δ y φi θk ) 2 dy d xdt |y|2+α 0 |y|≤1 T |(φi θk )(x − y) − (φi θk )(x)|2 ≤ C(ϕ) d xd ydt |y|1+α |y|≤1 0 T dy ≤ C(ϕ)( ) φi θk 22 dt 1+α |y|≤1 |y| 0 ≤ C(ϕ, φ0 )T θ0 p .
References 1. Berselli, L.: Vanishing viscosity limit and long-time behavior for 2D quasi-geostrophic equations. Indiana Univ. Math. J. 51, 905–930 (2002) 2. Constantin, P., Córdoba, D., Wu, J.: On the critical dissipative quasigeostrophic equation. Indiana Univ. Math. J. 50, 97–107 (2001) 3. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) 4. Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. Siam J. Math. Anal. 30, 937–948 (1999)
Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2
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5. Córdoba, A., Córdoba, D.: A maximum principle applied to Quasi-Geostrophic equations. Commun. Math. Phys. 249, 511–528 (2004) 6. David, G., Journé, J.-L.: A boundedness criterion for generelized Calderón-Zygmund operators. Ann. of Math. 120, 371–397 (1984) 7. Delort, J.-M.: Existence de nappes de tourbillon en dimension deux. J Amer. Math. Soc. 4, 553–586 (1991) 8. Dubois, S.: What is a solution to the Navier-Stokes equations? C. R. Acad. Sci. Paris 335, 27–32 (2002) 9. Furioli, G., Lemarié-Rieusset, P.G., Terraneo, E.: Unicité dans L 3 (R3 ) et d’autres espaces limites pour Navier-Stokes. Revista Mat. Iberoamer. 16, 605–667 (2000) 10. Ju, N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophique equations. Commun. Math. Phys. 255, 161–181 (2005) 11. Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasigeostrophic equation, arXiv, 2006 12. Kozono, H., Sohr, H.: Remark on the uniqueness of weak solutions to the Navier-Stokes equations. Analysis 16, 255–271 (1996) 13. Lemarié-Rieusset, P.G.: Recent developments in the Navier-Stokes problem. Research Notes in Maths. 431, London: Chapman and Hall ICRC Press, 2002 14. Lemarié-Rieusset, P.G.: Solutions faibles d’énergie infinie pour les équations de Navier-Stokes dans R3. C. R. Acad. Sci. Paris 328, 1133–1138 (1999) 15. Lions, P.L.: Mathematical topics in fluid mechanics. Vol. 1, Oxford: Oxford Science Publication, 1996 16. Majda, A., Bertozzi, A.: Vorticity and incompressible fluid flow, Cambridge texts in applied mathematics, Cambridge: Camb. Univ. Press, 2002 17. Marchand, F., Lemarié-Rieusset, P.G.: Solutions auto-similaires non-triviales pour l’équation quasigéostrophique dissipative critique. C. R. Acad. Sci. Paris 341, 535–538 (2005) 18. Marchand, F.: Propagation of Sobolev regularity for the critical dissipative quasi-geostrophic equation. Asymp. Analysis 49(3/4), 275–293 (2006) 19. Marchand, F.: Weak-strong uniqueness criterions for the critical dissipative quasi-geostrophic equation. Preprint 20. Meyer, M.: Une classe d’espaces fonctionnels de type BMO. Application aux intégrales singulières. Ark. Math. 27, 305–318 (1989) 21. Resnick, S.: Dynamical Problem in Nonlinear Advective Partial Differential Equations. Ph. D. thesis, University of Chicago, 1995 22. Rudin, W.: Fourier analysis on groups. New York: Interscience Publ., 1963 23. Schoenberg, I.J.: Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44, 522–536 (1938) 24. Stein, E.M.: The characterization of functions arising as potentials I. Bull. Amer. Math. Soc. 67, 102–104 (1961) 25. Triebel, H.: Theory of function spaces. Monographs in Mathematics 78, Basel: Birkhäuser Verlag, 1983 26. Wu, J.: Dissipative quasi-geostrophic equations with L p data. Electronic J. Differ. Eq. 56, 1–13 (2001) Communicated by P. Constantin
Commun. Math. Phys. 277, 69–81 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0350-z
Communications in
Mathematical Physics
The Product Over All Primes is 4π 2 E. Muñoz García1 , R. Pérez Marco2 1 Institute for the International Education of Students, IES, Avda Seneca 7, 28040-Madrid, Spain.
E-mail: [email protected]
2 LAGA, CNRS UMR 7539, Université Paris 13, 93430-Villetaneuse, France.
E-mail: [email protected] Received: 24 October 2006 / Accepted: 27 May 2007 Published online: 19 October 2007 – © Springer-Verlag 2007
Abstract: We generalize the classical definition of zeta-regularization of an infinite product. The extension enjoys the same properties as the classical definition, and yields new infinite products. With this generalization we compute the product over all prime numbers answering a question of Ch. Soulé. The result is 4π 2 . This gives a new analytic proof, companion to Euler’s classical proof, that the set of prime numbers is infinite. 1. Introduction Christophe Soulé did ask several years ago (and the question is proposed in [SABK, p.101]) to give a meaning and find a value for the zeta-regularized product over all prime numbers, similar to the classical zeta-regularized product √ ∞! = 1.2.3. . . . = 2π . Regularization of infinite products arises in geometry [RS], arithmetic geometry [SABK], theoretical physics [EORBZ], and, more recently, in analytic number theory [De]. One of the main applications is the computation of regularized determinants of infinite dimensional operators, as pseudo-differential operators on manifolds. In a previous article [MG-PM1] we carry a computation “à la Euler” in order to show that p = 4π 2 . p
The computation in [MG-PM1] that we reproduce below is not rigorous. Its aesthetics “à la Euler” leaves no doubt about the correction of the end result. The aim of the present article is to make rigorous this computation. For this purpose, we generalize the classical definition of zeta-regularization of infinite products. The new definition extends the classical one and preserves its main properties.
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Given an increasing sequence 0 < λ1 ≤ λ2 ≤ λ3 ≤ . . . one defines the regularized (or zeta-regularized) infinite product as +∞
λn = exp(−ζλ (0)),
n=1
where ζλ is the zeta function associated to the sequence (λn ), +∞
ζλ (s) =
λ−s n
n=1
(see [SABK], Chap. V, Definition 5, p. 97). Implicitly this assumes that the zeta function converges in a half plane and has an analytic extension up to 0. This definition can be extended to arbitrary sequences of complex numbers for which the procedure makes sense. Observe that the definition applies to a finite product, and gives the expected result. In [MG-PM1] we took a “liberal” view on this definition, just assuming that we have some way of computing or making sense of ζλ (0). This was necessary since, as observed in [SABK], the zeta function associated to the sequence of primes does not extend meromorphically to a neighborhood of 0 since the imaginary axes is a line of singularities. This fact was first proved by E. Landau and A. Walfisz [LW]. For this reason the regularization of this infinite product was believed to be impossible (see for example the introduction of [Il]). The computation in [MG-PM1] runs as follows: Recall that (this is used for example in the definition of the Artin-Hasse exponential, see for example [Ko], Chap. IV) exp(X ) =
+∞
(1 − X n )−
µ(n) n
,
n=1
where µ is Moebius function. This can be proved taking logarithmic derivatives and using The Moëbius inversion formula. From this we get ep
−s
=
+∞
(1 − p −ns )−
µ(n) n
.
n=1
We consider now the zeta function associated to the sequence of primes P(s) = So it follows that eP (s) =
ep
1 . ps p
−s
p
=
+∞
(1 − p −ns )−
p n=1
µ(n) n
The Product Over All Primes is 4π 2
71
= =
+∞
(1 − p −ns )−
n=1 p +∞
ζ (ns)
µ(n) n
µ(n) n
,
n=1
where we have used in the last equality Euler’s product for the Riemann zeta function (and following Euler’s tradition we don’t need to justify the product exchange, but it can be done here). This formula eP (s) =
+∞
ζ (ns)
µ(n) n
n=1
can be found in [LW] and in [Da] (in this last reference there is a typo in the formula: n in the denominator of the exponent is missing). Now, taking the logarithmic derivative we get P (s) =
+∞ µ(n) nζ (ns) n=1
n
ζ (ns)
=
+∞
µ(n)
n=1
ζ (ns) , ζ (ns)
thus
P (0) =
+∞
µ(n)
n=1
ζ (0) . ζ (0)
Of course the infinite sum does not converge, but we recall that +∞ µ(n) n=1
ns
=
1 , ζ (s)
thus P (0) =
1 ζ (0) = −2 log(2π ). ζ (0) ζ (0)
We conclude that
p = e−P (0) = (2π )2 = 4π 2 .
p
After this “explanation” à la Euler, we developed in [MG-PM2] a generalization of classical regularization of infinite products that makes rigorous the previous computation. We present this theory.
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2. Super-Regularization of Infinite Products 2.1. Definition. Consider a sequence λ = (λn )n≥1 of complex numbers and define, as before, its associated zeta function ζλ (s) =
+∞
λ−s n .
n=1
We assume that ζλ is absolutely convergent in the half plane Re s > s0 . Note that we have to make a choice of the branch of each λ−s n . The end result will not depend on the choice made for a finite number of these terms, but the analytic extension properties and the result may depend on infinitely many choices. Obviously when (λn ) is a positive real sequence, which is the case on most of the applications (in particular when we consider the spectrum of an hermitian operator) we make the unique choice for which λ−s n is real for s > 0. For a directed sequence (λn ) as considered in [Il], it is also natural to take a compatible branch for all the λn . Nevertheless we do not make any particular assumption on the choice of branches. We consider now an extension of ζλ to two complex variables into a double Dirichlet series: ζλ (s, t) =
+∞
−t cn,m λ−s n µm .
n,m=1
We assume that this series is absolutely convergent in U0 = {Re s > s0 } × {Re t > t0 }, and defines in this domain U0 ⊂ C2 a meromorphic function. We also assume that t0 < 0 and that ζλ (s, 0) = ζλ (s). Now the function (s, t) →
∂ζλ (s, t) ∂s
is meromorphic in U0 . We assume that there exists t1 ≥ t0 such that for each t ∈ C with Re t > t1 the meromorphic function of one complex variable s →
∂ζλ (s, t) ∂s
that is meromorphic in Re s > s0 , does extend meromorphically to a half plane {Re s > s1 }, s1 < 0, that is a neighborhood of s = 0. We denote by s → exts
∂ζλ (s, t) ∂s
this extension. Remarks. 1. Rothstein’s theorem (see [Siu]) shows that the function of two complex variables (s, t) → exts
∂ζλ (s, t) ∂s
The Product Over All Primes is 4π 2
73
t
t
1 s 0 t
s0
0 singularities Fig. 1.
is meromorphic, but we won’t need this fact. 2. In the definition of [SABK] of the classical regularized product, it is assumed that the zeta function extends meromorphically to the whole complex plane. Not so much is necessary since only the derivative at 0 matters, thus it is natural to request a meromorphic extension to 0. But this simple assumption would not prove to be a good definition. We may have isolated singularities of the extension with non-trivial monodromy. We don’t know if actually there are Dirichlet series where this can happen. It is unlikely that this happens for Dirichlet series of arithmetic origin. These in general are meromorphic on half planes. Clearly, just assuming meromorphic extension to a half plane containing 0 avoids the monodromy problem and gives a good definition. It is this approach that we generalize. Now we assume that the function ∂ζλ (0, t) ∂s defined in the half plane Re t > t1 has a meromorphic extension to a neighborhood of {t = 0} not identically infinite. Denote this extension by ∂ζλ t → extt lim exts (s, t) . s→0 ∂s t → exts
We assume, for the scope of this article, that this extension has no pole at t = 0, but one can figure out a reasonable extension also in that situation. This procedure that circumvents the eventual singularities in {s = 0} is illustrated by Fig. 1. If it can be carried over, we define: Definition 1 (Super-regularized product). The super-regularized product of the sequence λ = (λn )n≥1 is by definition, provided that the limits and meromorphic extensions exist, ∂ζλ (s, t) . λn = exp − lim extt lim exts t→0 s→0 ∂s n We write in short n
∂ζλ (0, t) . λn = exp − lim t→0 ∂s
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E. Muñoz García, R. Pérez Marco
Zeta-regularization being a procedure cherished by physicists, the terminology “superregularization” seems appropriate. The idea of avoiding singularities by “going through the complex” is not new in Theoretical Physics, nor in Pure Mathematics, but the procedure of increasing the complex dimension appears to be new in the theory of regularization of infinite products and resummation of divergent series. Nevertheless some classical procedures as Abel’s summation can be viewed as going from complex dimension 0 to complex dimension 1 (one can consult [Bo] and [Ha] for classical procedures, and [Ra] for modern extensions). The introduction of supplementary variables has proved to be useful in other classical problems in Number Theory (see [Re] for several examples described by a specialist of several complex variables, in particular the discussion of Eisenstein’s trick in Sect. I.3.4). Several justifications of this definition are in order. First, we prove that the end result is independent of the two complex variable extension and the whole procedure. Proposition 2. Let (s, t) → ζˆλ (s, t) be another two complex variable extension of s → ζλ (s) satisfying the same properties as (s, t) → ζλ (s, t), so that the procedure described above to compute the super-regularized product can be carried over. Then ∂ ζˆλ ∂ζλ (0, t) = lim (0, t), t→0 ∂s t→0 ∂s lim
thus the super-regularization is independent of the choice of the complex extension. In particular, the super-regularization coincides with the classical zeta-regularization when this last one is well defined. Proof. We assume that ζˆ (s, t) is absolutely convergent in the domain Uˆ 0 = {Re s > sˆ0 } × {Re t > tˆ0 }, with tˆ0 < 0, and that in the half plane {Re t > tˆ1 } the meromorphic function s →
∂ ζˆλ (s, t) ∂s
extends meromorphically to a half plane {Re s > s1 }, s1 < 0, neighborhood of {s = 0}. Now, since t0 < 0 and tˆ0 < 0, the difference ζˆλ −ζλ is meromorphic in V0 = {Re s > max(s0 , sˆ0 )} × {Re t > max(t0 , tˆ0 )} and vanishes in {Re s > max(s0 , sˆ0 )} × {t = 0}, thus there exists a meromorphic function g defined in V0 such that ζˆλ (s, t) − ζλ (s, t) = t g(s, t). Taking partial derivatives, ∂ ζˆλ ∂ζλ ∂g (s, t) − (s, t) = t (s, t). ∂s ∂s ∂s Using this equation, we have that for Re t > max(tˆ1 , t1 ), s →
∂g (s, t) ∂s
extends meromorphically to a neighborhood of s = 0, and exts
∂ ζˆλ ∂ζλ ∂g (s, t) − exts (s, t) = t exts (s, t). ∂s ∂s ∂s
The Product Over All Primes is 4π 2
75
Again using this equation and the properties of ζˆλ and ζλ we get that t → exts
∂g (0, t), ∂s
which is well defined for Re t > max(tˆ1 , t1 ), has a meromorphic extension to t = 0, and ∂ ζˆλ ∂ζλ ∂g extt lim exts (s, t) − extt lim exts (s, t) = t. extt lim exts (s, t) , s→0 s→0 s→0 ∂s ∂s ∂s and making t → 0 in this equation we get the result. The super-regularization does extend the classical regularization because we can always consider the trivial extension ζλ (s, t) = ζλ (s).
2.2. Properties. The main properties of the classical regularization are preserved. Proposition 3. If λ = (λn ) is a finite sequence, then the super-regularization of the product coincides with the classical finite product. More generally, if λ = (λn ) is an infinite sequence for which the infinite product can be regularized then the super-regularization of the product coincides with the classical regularized product. Proposition 4. The super-regularized product is finitely associative. That is, if we partition the sequence λ into N parts λ = λ(1) ∪ λ(2) ∪ . . . ∪ λ(N ) ( j)
with λ( j) = (λn )n≥1 , and we assume that the super-regularized product exists for each sequence λ( j) , then the super-regularized product exists for the sequence λ, and we have the formula (where we assume all infinite products to be super-regularized products) +∞ +∞ +∞ +∞ (1) (2) (N ) λn = λn . λn . . . λn . n=1
n=1
n=1
n=1
Proof. Just observe that ζλ =
N
ζλ( j) .
j=1
Proposition 5. Let a ∈ C∗ . We assume that the super-regularized product of the sequence λ is well defined. Then the super-regularized product of the sequence λa = (λan )n≥0 exists and we have +∞ a +∞ a λn = λn . n=1
n=1
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E. Muñoz García, R. Pérez Marco
Proof. We have ζλa (s) = ζλ (as).
The next property is more involved. It relies on the analytic property that we can give a meaning to ζλ (0). We do strengthen the assumptions requesting not only that ∂ζλ (s, t), ∂s
s →
extends meromorphically to a half plane {Re s > s1 } with s1 < 0, but also that s → ζλ (s, t), has this property. Then we assume that for s = 0, we also have a meromorphic extension of t → ζ (0, t), to t = 0 and that t = 0 is not a pole. In practice the analytic extension of ζ and not just ∂ζ /∂s always happens and it almost follows from the first assumption. One has to be careful when only using the first assumption because of the possible different branches of the extension of ζ obtained by integration due to the fact that s →
∂ζλ (s, t) ∂s
may have poles which introduce a monodromy around them for the primitive. One can overcome these difficulties by assuming that the branch of the extension chosen depends continuously on the parameter t. More precisely, at t ∈ C, Re t > t1 , we consider the meromorphic extension ζλ (s, t) =
γt
∂ζλ (s, t) ds, ∂s
where we integrate over a path γt ⊂ C × {t} from a base point (t, s˜0 ), where Re s˜0 > s0 , and we assume that the path γt depends continuously on t and avoids the poles of the meromorphic extension of s →
∂ζλ (s, t) ∂s
(note that the integral does not really depend on γt but only on the homotopy class of paths out of the poles). The existence of such a continuous choice of paths is problematic when we have poles that escape to infinity in finite time. We assume in what follows that we have such a well behaved extension.
The Product Over All Primes is 4π 2
77
Theorem 6. Let a ∈ C∗ . If the super-regularized product of the sequence λ = (λn )n≥1 exists with the stronger assumption mentioned before, then the super-regularized product of the sequence aλ = (aλn )n≥0 exists and we have +∞ +∞ ζλ (0) (aλn ) = a λn , n=1
n=1
where ζλ (0) has to be interpreted as lim lim ζλ (s, t),
t→0
s→0 Re t>t1
which is well defined by successive analytic extensions and does not depend on the two complex variable extension. Note that indeed the right-hand side depends on the choice of the branch of the power of a. But also the left-hand side depends on the choice of the branches of the (aλn )−s in order to compute the zeta function. The claim of the theorem is that there is such a choice of a ζλ (0) , compatible with the implicit choices in order to compute the left-hand side, so that the formula holds. Lemma 7. Denoting by (s, t) → ζλ (s, t) the extension defined, we have that the limit lim ζλ (0, t)
t→0
is independent of the two complex variable extension (s, t) → ζλ (s, t) chosen. Proof. For another two complex variable extension (s, t) → ζˆ (s, t), we have as before, once the meromorphic extensions are performed, ζˆλ (0, t) − ζλ (0, t) = tg(0, t), thus lim ζˆλ (0, t) = lim ζλ (0, t)
t→0
as claimed.
t→0
Proof of Theorem 6. We prove first that the super-regularization of the infinite product of the sequence aλ exists. We can define the two variable complex extension of the zeta function associated to the sequence aλ as ζaλ (s, t) = a −s ζλ (s, t). Then in the region {Re s > s0 } × {Re t > t1 } we have ∂ζλ ∂ζaλ (s, t) = (s, t) − (log a)a −s ζλ (s, t). ∂s ∂s We have that for Re t > t1 fixed, s →
∂ζaλ (s, t) ∂s
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E. Muñoz García, R. Pérez Marco
extends meromorphically to s = 0, and ∂ζλ ∂ζaλ (0, t) = (0, t) − (log a)ζλ (0, t). ∂s ∂s Now we have a meromorphic extension of this last equation to a neighborhood of t = 0. This proves the existence of the super-regularized product associated to the sequence aλ. Moreover, ∂ζaλ ∂ζλ (0, t) = lim (0, t) − (log a) lim ζλ (0, t), t→0 ∂s t→0 ∂s t→0 lim
which gives the formula stated.
3. Application Theorem 8. The super-regularized product over all prime numbers is 4π 2 , 1 × 2 × 3 × 5 × 7 × 11 × 13 × · · · = p = 4π 2 . p
Remark. Many reasons indicate that 1 should be considered also as a prime number. But this, obviously, does not matter here. Proof. Consider the zeta function associated to the sequence of prime numbers p −s . P(s) = p
We have seen in the introduction that P(s) =
+∞ µ(n) n=1
n
log ζ (ns),
where ζ is Riemann zeta function. We define the extension to two complex variables P(s, t) =
+∞ µ(n) n=1
n 1+t
log ζ (ns).
We have P(s, 0) = P(s), and P(s, t) is a Dirichlet series of s and t (expand the logarithm). For {Re s > 0}, (log ζ (ns))n≥1 decays geometrically to 0. Thus P(s, t) is a meromorphic in {Re s > 0} × C. Now we can compute µ(n) ζ (ns) ∂P (s, t) = , ∂s n t ζ (ns) +∞
n=1
The Product Over All Primes is 4π 2
79
Fig. 2.
which for Re t > t1 = 1 fixed has a meromorphic extension to a neighborhood of s = 0. We compute the value at s = 0, +∞ µ(n) ζ (0) ∂P (0, t) = ∂s nt ζ (0) n=1
1 = log(2π ). ζ (t) Now we have a meromorphic extension to t = 0, which gives the result 1 ∂P (0, t) = log(2π ) = −2 log(2π ). t→0 ∂s ζ (0) lim
So
p = 4π 2 .
p
Corollary 9. The set of prime numbers is infinite. Proof. If the set was finite then the super-regularized product would be the usual product by Proposition 3. But 2 × 3 × 5 × 7 > 4π 2 because π < 4, as we show inscribing a circle inside a square. We can also proceed using the slightly less elementary argument that π 2 is not an integer.
Therefore the following picture gives a geometric proof that the set of prime numbers is infinite. Remark. 1. It is interesting to note the relation between the non-integer value of π 2 and the infinitude of prime numbers. An unexpected relation between two main topics
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of “The Elements” of Euclid. The first known proof of the infinitude of prime numbers is given by Euclid in Book IX, Prop. 20 of his “Elements” ([Euc], Vol. 2, p. 412). In book XII the “method of exhaustion” discovered by Eudoxus is presented ([Euc], Vol.3, p. 365). Historically this is the first algorithm that provides arbitrarily accurate approximations of the number π . 2. The first analytic proof of the existence of infinitely many prime numbers was given by L. Euler by finding Euler’s prime factorization of the Riemann zeta function ([Eul], Chap. XV, p. 271), and showing that the series 1 1 1 1 + + + ··· = = ∞. 1 2 3 p p is infinite ([Eul], Chap. XV, Example 1, p. 277) where this formula appears. Note that Euler’s formula also contains p = 1 as a prime number. We don’t know of any other analytic proof other than Euler’s and the one given above. It is also interesting to observe that in the “Introductio in Analysin Infinitorum” ([Eul]) we can find this basic result about prime numbers together with many formulas involving π , and the value of π to many decimal places (that apparently was due to De Lagny). The notation π was consolidated after the publication of the “Introductio” (but apparently was first used by W. Jones in 1706). Euler’s approach is equivalent to the fact that the Riemann zeta function has a pole at s = 1. Observe the noteworthy difference with our approach where only the value at s = 0 matters. Euler’s idea was exploited by P.G.L. Dirichlet in order to prove his theorem on the infinitude of primes in arithmetic progressions. For other non-analytic proofs of the infinitude of primes see the first “proof from the book” in [AZ]. 3. One may ask if the formula √ ∞! = 2π does prove the infinitude of positive integers (!?). We leave to the reader this metaphysical question. 4. We observe that the value 4π 2 obtained for the super-regularized product over all prime numbers coincides with the regularized determinant of the Laplacian on the circle. Acknowledgements. We are grateful to Christophe Soulé for his remarkable insight, for telling us about the problem, for the stimulating conversations on the subject, and his encouragement to publish our result. We thank Jesús Muñoz Díaz and the referee for their comments and observations that improved this article. We thank the IHES for its support and hospitality where this research was conducted during a visit of the authors in 2003.
References [AZ] [Bo] [Da] [De] [EORBZ]
Aigner, M., Ziegler, G.M.: Proofs from the book. Berlin-Heidelberg, New York: Springer Verlag, 2nd Edition, 2000 Borel, É.: Leçons sur les series divergentes. Paris: Gauthier-Villars, 1928 Dahlquist, G.: On the analytic continuation of eulerian products. Arkiv för Matematik 1, 533–554 (1951) Deninger, C.: Some analogies between Number Theory and Dynamical Systems on foliated spaces. Doc. Mat. J. DMV Extra Volume ICM I, 163–186 (1998) Elizalde, E., Odintsov, S.D., Romeo, A., Bytsenko, A.A., Zerbini, S.: Zeta regularization techniques with applications. Singapore, World Scientific, 1994
The Product Over All Primes is 4π 2
[Euc]
81
Euclid, L.: The Thirteen Books of the Elements, Translated with introduction and commentary by Thomas L. Heath, Vol. 1, 2, 3, New York: Dover Publications, 1956 [Eul] Euler, L.: Introductio in Analysin Infinitorum. Facsimil edition and commented translation of the edition of 1748 stored in the library of the Real Instituto y Observatorio de la Armada en San Fernando, eds. A.J. Duran Guardeño, F.J. Pérez Fernández Bacelona: Real Socieded Mathematic Espazola, 2000 [Ha] Hardy, G.H.: Divergent series. Oxford: Clarendon Press, 1963 [Il] Illies, G.: Regularized products and determinants. Commun. Math. Phys. 220, 69–94 (2001) [Ko] Koblitz, N.: p-adic numbers, p-adic analysis and zeta functions. Graduate Texts in Mathematics 58, 2nd edition, Berlin Heidelberg-New York: Springer, 1998 [LW] Landau, E., Walfisz, A.: Über die nichtfortsetzbarkeit einiger durch dirichletsche reihen definierter funktionen. Rendiconti del Circolo Matematico di Palermo 44, 82–86 (1920) [MG-PM1] Muñoz Garcia, E., Pérez Marco, R.: The product over all prime numbers is 4π 2 . Preprint IHES M/03/34, www.ihes.fr, May 2003 [MG-PM2] Muñoz Garcia, E., Pérez Marco, R.: Super-regularization of infinite products. Preprint IHES M/03/52, www.ihes.fr, August 2003 [Ra] Ramis, J.-P.: Séries divergentes et théories asymptotiques. In: Panoramas et synthèses, Paris: Société Mathématique de France, 1993 [Re] Remmert, R.: Classical topics in Complex Function Theory. Graduate Texts in Mathematics 172, Berlin, Heidelberg-New York: Springer Verlag [RS] Ray, D., Singer, I.: Analytic torsion for analytic manifolds. Ann. Math. 98, 154–177 (1973) [SABK] Soulé, C., Abramovich, D., Burnol, J.-F., Kramer, J.: Lectures on Arakelov Geometry. Cambridge Studies in Advanced Mathematics, Cambridge: Cambridge Univ. Press, 1992 [Siu] Siu, Y.-T.: Techniques of extension of analytic objects. Lecture Notes in Pure and Applied Mathematics, New York: Marcel Dekker, 1974 Communicated by A. Connes
Commun. Math. Phys. 277, 83–100 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0358-4
Communications in
Mathematical Physics
AdS/CFT Correspondence in the Euclidean Context Hanno Gottschalk1 , Horst Thaler2 1 Institut für angewandte Mathematik, Universität Bonn, Bonn, Germany.
E-mail: [email protected]
2 Dipartimento di Matematica e Informatica, Università di Camerino, Camerino, Italy.
E-mail: [email protected] Received: 1 November 2006 / Accepted: 21 May 2007 Published online: 25 October 2007 – © Springer-Verlag 2007
Abstract: We study two possible prescriptions for the AdS/CFT correspondence by means of functional integrals. The considerations are non-perturbative and reveal certain divergencies which turn out to be harmless, in the sense that reflection positivity and conformal invariance are not destroyed.
1. Introduction In this article we investigate the AdS/CFT correspondence for scalar fields within the Euclidean approach. Originally, this conjecture was formulated within the string theoretic context [20]. Soon afterwards it was discovered that it makes perfect sense in a purely quantum field theoretic setting [29]. This conjecture states that a quantum field theory (QFT) on AdS space gives rise to a conformal QFT (CFT) on its boundary and vice versa. Within the algebraic approach to QFT this correspondence can be made precise. The idea is to identify algebras of observables in wedge-like regions on AdS space with corresponding algebras in double cones on the boundary, see [24]. We hope that this work can contribute to the recent discussion on the mathematical status of non-algebraic AdS/CFT. We are interested in the passage from AdS-QFT to CFT by means of functional integrals. Without taking recourse to perturbative arguments we succeed in constructing functional integrals within the infinite dimensional setting. The Euclidean field theory of an interacting QFT is described through a probability measure dµ = e−V dµC/ e−V dµC , defined on an appropriate distribution space on the Riemannian counterparts of AdS spaces, which are hyperbolic spaces. The Gaussian measure dµC with covariance C specifies the underlying free theory and the density e−V accounts for the interaction. The measure dµ should satisfy the Osterwalder-Schrader axioms in order to make a passage from hyperbolic to AdS-spaces possible [5,17].
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On hyperbolic spaces there are two choices of invariant covariance operators, denoted G ± , since there are two linearly independent fundamental solutions to the equation (−∆ + m 2 )G(x, x ) = δ(x, x ). This follows from the fact that, due to invariance, G has to be a function of the geodesic distance d(x, x ), therefore the resulting equation for G(d) involves only the radial part of the Laplacian which can be transformed to a hypergeometric equation possessing two linearly independent solutions. This work is inspired by the ideas in [9] (see also [25]) where two natural prescriptions for the AdS/CFT correspondence are compared and shown to essentially agree. One way is to define a Laplace transform where the source term is restricted to the boundary, i.e. Z˜ ( f )/ Z˜ (0) =
e−V (φ) e∂φ( f ) dµC (φ)/ Z˜ (0).
(1)
At this place ∂φ means the restriction of the bulk field to the boundary. Below we shall see how to make this definition rigorous using a proper scaling. It turns out that in general nontrivial results for (1) can be obtained only through the multiplication with a regularizing factor which nonetheless doesn’t destroy reflection positivity and conformal invariance. Another possibility is to fix the values of the bulk field on the boundary by insertion of a delta function, so that heuristically we set Z ( f )/Z (0) =
e−V (φ) δ(∂φ − f )dµC (φ)/Z (0).
(2)
It will turn out that the correct choice for C is to take G + in case (1) and G − for (2). Essentially, the definition of (2) rests on the splitting of G − into a “bulk-part” and a “boundary-part”. For a related discussion about quantum fields on manifolds with a boundary, look at [15]. Another viewpoint on the relation between bulk and boundary fields, using representation theoretic arguments, can be found in [8]. The construction makes it also explicit that the two functionals agree up to the multiplication of test functions with a constant factor when both are defined. In Sect. 2 we introduce various propagators which serve as building blocks for the functional integrals, in particular the splitting of G − is proven. In Sect. 3 we show how to give a rigorous meaning to expressions (1) and (2). Then in Sect. 4 we treat P(φ)2 models for concreteness. In Sect. 5 we go over to discuss the two basic axiomatic properties of reflection positivity and conformal invariance.
2. Propagators on the Hyperbolic Space There are various propagators needed for the definition of AdS/CFT functional integrals, which we introduce in this section. Let us consider the upper half-space model of the (d + 1)-dimensional hyperbolic space Hd+1 := {(z, x) ∈ Rd+1 : z > 0}, equipped with the Riemannian metric 1/z 2 (dz 2 + d x12 + · · · + d xd2 ). The Green’s functions G ± are explicitly given by G ± (z, x; z , x ) = γ± (2u)−∆± F(∆± , ∆± +
1−d 2 ; 2∆±
+ 1 − d; −2u −1 ),
(3)
AdS/CFT Correspondence in the Euclidean Context
where u =
(z−z )2 +(x−x )2 , 2zz
Γ (∆± ) . 2π d/2 Γ (∆± +1− d2 )
∆± =
±
d 2
85 1 2
√ d 2 + 4m 2 =:
d 2
± ν, ν > 0 and γ± =
F is the hypergeometric function which for ζ ∈ C with |ζ | < 1 is
given by the absolutely convergent series F(a, b; c; ζ ) = 1 +
ab a(a + 1)b(b + 1) 2 ζ+ ζ + ··· . c c(c + 1)
(4)
Its analytic continuation to C\[1, ∞) is given by the integral representation 1 Γ (c) F(a, b; c; ζ ) = t b−1 (1 − t)c−b−1 (1 − ζ t)−a dt, if Rec > Reb > 0. Γ (b)Γ (c − b) 0 It should be noted that G + is the integral kernel of the inverse (−∆ + m 2 )−1 in L 2 (Hd+1 ). We would like to obtain a conformal theory on the boundary at infinity (z → 0). On the level of propagators this is achieved by taking appropriate scaled limits. From (3) we get as pointwise limits the bulk-to-boundary propagators ∆± z −∆± H± (z, x; x ) = lim z G ± (z, x; z , x ) = γ± 2 , z + (x − x )2 z →0 and the boundary propagators α± (x, x ) = lim z −∆± H± (z, x; x ) = γ± (x − x )−2∆± . z→0
(5)
Since 2∆+ ≥ d, the kernels α+ have a non-integrable singularity. They will be understood to be regularized by analytic continuation to values ν = 0, 1, 2, . . ., see [11]. Hence, whenever α+ is involved in some argument, statements will hold with the exception of singular points. Notation. The Fourier transform is defined as fˆ(k) = 1/(2π )d/2 Rd f (x)−ikx d x. We use the notation |ζ | for the absolute value of a complex number ζ , as well as |k| for the Euclidean norm of a vector k ∈ Rd . Tuples (z, x) will also be denoted by x. Then the Fourier transforms of H± (z, x; x ) and α± (0, x ) with respect to x ∈ Rd read ±ν d 1 ikx |k| ˆ e z 2 K ν (|k|z), (6) H± (x, k) = d 2 (2π ) 2 Γ (1 ± ν) and αˆ ± (k) =
Γ (∓ν) d 2
|k| 2
±2ν
=: C−ν |k|±2ν ,
2(2π ) Γ (1 ± ν) where K ν is the modified Bessel function of the second kind which is given by 2 1 ζ ν ∞ e−t−ζ /4t π K ν (ζ ) = dt, |argζ | < , Reζ 2 > 0. ν+1 2 2 t 2 −∞ −ν For small arguments it behaves like K ν (ζ ) ∼ 21 Γ (ν) ζ2 , ν > 0. Lemma 1. With c := 2ν we have G − (x, x ) = G + (x, x ) +
Rd
Rd
H+ (x, y)c2 α− (y, y )H+ (x , y )dydy .
(7)
(8)
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Proof. In [21] it was shown that G − (x, x ) = G + (x, x ) + c
Rd
H+ (x, y)H− (x , y)dy.
Let α− ( · ) be the function y → α− (0, y ), then for ν > 0, α− (y, y )H+ (x , y )dy = (α− ( · ) ∗ H+ (z , x ; · ))(y), Rd
(9)
(10)
where ∗ means convolution. Therefore, for 2ν < d, eiky αˆ − (k) Hˆ + (z , x ; k)dk (α− ( · ) ∗ H+ (z , x ; · ))(y) = Rd
−ν d 1 ik(x +y) |k| = e z 2 K ν (|k|z )dk. c(2π )d Γ (1 − ν) Rd 2 On the other hand, for 2ν < d, 1 1 eiky Hˆ − (x , k)dk. H− (x , y) = d d c c(2π ) 2 R
(11)
(12)
Using Morera’s theorem [1], it is not difficult to see that for fixed x , y the left-hand sides of (11) and (12) are holomorphic as functions of the parameter ν > 0 and, because they agree for 2ν < d, the result follows. Remark 1. Equation (8) presents the splitting of G − into a “bulk-part” and a “boundary-part”. Although it is a covariance on Hd+1 , the “boundary-part” is named like this because it contains the boundary covariance α− . We note that the “bulk-part” vanishes with respect to the scaling z −∆− . Moreover, a splitting for G + like that in Lemma 1 into a sum of two covariances is not possible. In order to get the right boundary covariance α+ (x, x ) = lim z→0 z −2∆+ G + (z, x; z, x ) the “bulk-part” should scale like z a , in any argument, with a > ∆+ in order to vanish with respect to the scaling z −∆+ , but such a covariance is not available among the solutions of (−∆ + m 2 ) f = δ (= 0). 3. Construction and Definition of Functional Integrals First we try to give a meaning to the functional integral (2). For 2ν < d, α− is a positive covariance and in this parameter range the splitting given in (8) entails the corresponding splitting for the random fields,
φ− (x) = φ+ (x) + cH+ φα (x),
where H+ φα (x) := Rd H+ (x, y)φα (y)dy, and φ− , φ+ , φα are the Gaussian random fields with covariances G − , G + and α− respectively. More precisely, φ+ , φα have to be understood as the first and second component of the following product measure space: (D(Hd+1 ) × S(Rd ) , B(D(Hd+1 ) ) ⊗ B(S(Rd ) ), µG + ⊗ µα− ). D(Hd+1 ) stands for the space of infinitely differentiable real-valued functions with compact support on Hd+1 and S(Rd ) denotes the Schwartz space of rapidly decreasing
AdS/CFT Correspondence in the Euclidean Context
87
real-valued functions on Rd . The primes indicate the topological duals, or distribution spaces. Finally, B stands for the Borel σ -algebras obtained from the respective weak-∗ topologies. Then we have EµG + ⊗µα− [(φ+ ( f ) + cH+ φα ( f ))(φ+ (g) + cH+ φα (g))] = EµG + [φ+ ( f )φ+ (g)] + c2 Eµα− [H+ φα ( f )H+ φα (g)], because the other terms vanish due to the product measure and the fact that the expectations of the fields vanish. But the last line is just the splitting (8). For this reason we may write for ν < d2 ,
D
F(φ− )dµG − (φ− ) =
D ×S
F(φ+ + cH+ φα )d(µG + ⊗ µα )(φ+ , φα ).
(13)
So far, F can be a general integrable function. Usually one considers the form F = e−V with V a local potential (with or without cut-offs). Remark 2. The bound −ν > − d2 for the field φ− is dictated by the positivity of α− . For d = 1 this is larger than the unitary bound −ν > −1 (in our notation). The same bound is needed when α− is asked to be reflection positive, see [12, Theorem 6.2.4]. For d ≥ 2 reflection positivity imposes the usual unitary bound −ν > −1. In order to cope with the delta function we shall boil down things to a finite dimensional approximation for the boundary field, insert the delta function in this case, perform integration over the (finite-dimensional) boundary field and then remove the approximation again. This is done in two steps. Step 1. We approximate the boundary covariance operator α− by covariance operators which possess bounded inverses in L 2 (Rd ). First we note that Rd
Rd
f (x)α− (x, y) f (y)d xd y = C−ν
Rd
|k|−2ν | fˆ|2 dk,
f ∈ S(Rd ).
From (14) we see that the bounded approximations can be defined as follows: n f ) := C−ν ( f, α−
Rd
χn (|k|)| fˆ|2 dk, (n ∈ N),
where ⎧ 2ν ⎨n , χn (|k|) := |k|−2ν , ⎩ −2ν n ,
for |k| ≤ n1 , for n1 < |k| ≤ n, for |k| > n.
Obviously, for their inverses we obtain n −1 ) f ) = (C−ν )−1 ( f, (α−
Rd
(χn (|k|))−1 | fˆ|2 dk.
(14)
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Step 2. Next we consider finite dimensional approximations for the boundary field φα n n . This approximation is performed by p φ n , where (n arbitrary) with covariance α− m α pm is the projection on the subspace spanned by the first m basis elements of a Hilbert n e ) be defined, space basis (ei )i≥1 of L 2 (Rd ). In order that the matrix elements (ei , α− j we choose the basis elements to be Schwartz functions, which is possible, since Schwartz spaces are separable. Making in addition the identification η : pm φα → ψα = ((φα )(e1 ), . . . , (φα )(em ))t ∈ Rm , we see that the integral (13) takes the form 1 C A− F(φ+ + cH+ (η−1 ψα ))dµG + (φ+ )e− 2 (ψα ,A− ψα ) dψα , Rm
D
n p η−1 )−1 = η( p α n p )−1 η−1 and C where A− := (ηpm α− m m − m A− =
1
|det A− | 2 d
(2π ) 2
. Now it is
possible to insert the delta function and we get 1 δ(ψα − ηpm f )F(φ+ + cH+ (η−1 ψα ))dµG + (φ+ )e− 2 (ψα ,A− ψα ) dψα C A− Rm
D 1
= C A− e− 2 ( f,( pm α− pm ) n
−1
f)
D
F(φ+ + cH+ ( pm f ))dµG + (φ+ ) =: Z m,n ( f ).
(15)
We notice that in the quotient Z m,n ( f )/Z m,n (0) the constant C A− drops out. The uniform n p )−1 f → (α n )−1 f , due to the boundedness of convergence pm → 1 leads to ( pm α− m − operators, see [28, Theorem 5.11]. Let H+ (z; · ) denote the function x → H (z, x; 0), then we have (H+ pm f )(z, · )2 = H+ (z; · ) ∗ pm f 2 ≤ H+ (z; · )1 pm f 2 by Young’s inequality. Moreover, z → H+ (z; · )1 remains bounded if z varies in a bounded subset of (0, ∞). Therefore, under the assumption that F( · + cH+ ( pm f ) − F( · + cH+ ( pn f )) L 1 (µG + ) ≤ constc(H+ ( pm f − pn f ))|Λ 2 , (16) with a bounded Λ ⊂ Hd we get convergence of the integral (15) as pm → 1. Finally n f ) → ( f, α f ) we take the limit n → ∞. From the definitions it is clear that ( f, α− − −1 n )−1 f ) → ( f, α f ). These considerations justify the following rigorous and ( f, (α− − definition of the generating functional (2), −1 − 12 ( f,α− f) Z ( f )/Z (0) := e F(φ+ + cH+ f )dµG + (φ+ )/Z (0). (17) D
We now come to a second possible prescription for the AdS/CFT-correspondence. Let us define −∆+ eφ(z (δz ⊗ f )) F(φ)dµG + (φ)/Y˜ (0), Z˜ ( f )/ Z˜ (0) = lim (Y˜ ( f )/Y˜ (0))z := lim z→0 D
z→0
where Z˜ (0) = Y˜ (0) and δz ⊗ f ∈ H −1 is the distribution defined by f (x)g(z, x)d x, f ∈ S(Rd ), g ∈ C0∞ (R>0 × Rd ). (δz ⊗ f )(g) =
(18)
Rd
We would like to compare the functional (18) with the one found in (17). To this end we rewrite (18) a little bit using the quasiinvariance of Gaussian measures with respect
AdS/CFT Correspondence in the Euclidean Context
89
to shifts by elements from H 1 . Applying the general result on quasiinvariance, proven e.g. in [2,4], we thus get with f z := z −∆+ (δz ⊗ f ) , dµG + ( · − G + f z ) = eφ((−∆+m = eφ( f z ) e
2 )G
+ fz )
1
e− 2 (G + f z ,(−∆+m
− 12 (G + f z , f z )
2 )G
+ fz )
dµG + ( · )
dµG + ( · ).
It should be noted that the random field φ( f ) can be extended to all f ∈ H −1 . Using this in (18) we arrive at the following expression: 1 (Y˜ ( f )/Y˜ (0))z = e 2 (G + f z , f z ) F(φ + G + f z )dµG + (φ)/Y˜ (0). D
(19)
Before being able to perform the limit z → 0 we have to take a closer look at the behavior of the term (G + f z , f z ). In Appendix A it is shown that in this limit we have to subtract certain divergent terms, more precisely, α+ (x, y) f (x) f (y)d xd y Rd Rd = lim z −d−2ν G + (z, x; z, y) f (x) f (y)d xd y z→0
Rd
Rd
2
[ν] −2(ν− j) j z (−1) a | fˆ(k)|2 |k|2 j dk √ j d 1 d π Γ (ν + ) 2 R (2π ) 2 j=0 =: lim z −d−2ν G + (z, x; z, y) f (x) f (y)d xd y − (Corr(z) f, f ). −
1
z→0
21−ν
Rd
Rd
In order to get nontrivial results in the limit we have to regularize the exponential prefactor in (19) by multiplying it with exp −(Corr(z) f, f ). From (3), (4) it is readily seen that, as z → 0, G + f z converges to H+ f uniformly on every bounded subset. Hence assuming that for some bounded Λ, F( · + G + f z ) − F( · + H+ f ) L 1 (µG + ) ≤ const(G + f z − H+ f )|Λ L p ,
(20)
for some p, we see that the integral in (19) converges, which shows that the correct definition for (18) reads Z˜ ( f )/ Z˜ (0) = lim e−(Corr(z) f, f ) (Y˜ ( f )/Y˜ (0))z z→0 1 = e 2 (α+ f, f ) F(φ + H+ f )dµG + (φ)/ Z˜ (0). D
(21)
In conclusion, we now obtain a proof of the duality conjecture: Theorem 1. Suppose that V is such that F = e−V fulfills (16) and (20). We then get Z ( f )/Z (0) = Z˜ (c f )/ Z˜ (0) when ν < d2 . −1 Proof. From (7) we see that α− = −c2 α+ . Compare now (17) and (21) to conclude.
Clearly, an ultra-violet and infra-red regularized local interaction VΛ fulfills the assumptions of the above theorem in any dimension. In the following section we show that this also holds for the case of models with polynomial interaction without ultra-violet cut-off on AdS with d + 1 = 2.
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H. Gottschalk, H. Thaler
4. P(φ)2 Fields on H2 We shall now address the existence of P(φ)2 models with interaction restricted to some bounded region Λ. We look at FΛ (φ + G + f z ) = e−VΛ (φ+G + f z ) with potentials ⎛ ⎞ n n n : φ j (x) :G + f j (x)d x ≡ : φ j : ( f j 1Λ ) =: ⎝ : φ j : ( f j )⎠(1Λ ), VΛ (φ) = Λ j=0
j=0
j=0
where : · :G + denotes Wick-ordering with respect to G + . n n n Since : (φ + f ) :G + (g) = j=0 j : φ j :G + (g f n− j ), a polynomial interaction VΛ is transformed into such under shifts and we may study the polynomial interaction itself. Proposition 1. Let VΛ be a polynomial interaction as above with n even and let f i be radial L 2 -functions. Then (a)
VΛ L p (µG + ) ≤ const( p, n)
n
f i 2 , 1 ≤ p < ∞,
i=0
and
(b)
where N ( f ) = (c)
n 2
e−VΛ (φ) dµG + (φ) ≤ econst( f n ∞ [N ( f )+(ln(M( f )+1)) ] ,
n−1 i=0
n/(n−i)
f i / f n n/(n−i) , M( f ) =
n
i=1 f i n/(n−i) .
lim e−VΛ ( · +G + f z ) − e−VΛ ( · +H+ f z ) L p (µG + ) = 0, 1 ≤ p < ∞.
z→0
Proof. The proof is just an adaption of the arguments given in [7] and [12, Chap. 8] in that Fourier transformation on H2 is used, see Appendix B. Here we repeat the main steps. Let us consider the expression N : φε (yµ )n µ : w(y1 , . . . , y N )dy, (n = (n 1 , . . . , n N ) ∈ N0N ), Rε (w, n) = µ=1
(22) where φε (y) := (φ χε )(y) and χε (y) := a(ε)χ (εy), ε > 0, is an approximate unity with χ ∈ C0∞ (H2 ) being a radial function with support in the unit ball and the factors a(ε) are chosen such that 2π H2 χε (r ) sinh r dr = 1 for all ε. We assume that the support of w is contained in B1 × · · · × B N , where the Bi are balls in H2 . The integral of (22) with respect to dµG + can be calculated as a sum of vacuum graphs. The graphs in the present case are built as follows: Consider N vertices each having n µ (1 ≤ µ ≤ N ) legs and combine arbitrary pairs of legs from different vertices to lines to obtain a graph. Vacuum graphs comprise the subset of graphs where all legs are paired. Denoting [I ] the set of all legs and Γ0 (I ) the set of all vacuum graphs, the integral of (22) can be estimated as (0 ≤ ρ ≤ 1) R(w, n)ε dµG ≤ M(ρ, n, G + )w2 + 1−ρ ρ × χµ,k 1 χˆ µ,k (m 2 + 41 + λ2 )−δ/4 ∞ , (23) (µ,k)∈[I ]
AdS/CFT Correspondence in the Euclidean Context
91
where χµ,k = χε and the constant is given as (n ∗ = supµ n µ , p = M(ρ, n, G + ) = |B|
G∈Γ0 (I ) l∈G
1−ρ
p p−1 ,
δ ≤ 2)
ρ
(ζl− ⊗ ζl+ )G + 2n ∗ (ζl− ⊗ ζl+ )G + B
(2n ∗ ) ,δ
.
(24)
The tuples (l− , l+ ) refer to some ordering of vertices (smaller, larger) and ζµ,k = ζµ is any radial C0∞ (H2 ) function which is identically one on {x : dist(x, Bµ ) ≤ 1}. In (24) we have used the norm δ
ζ ψBr,δ := (1 + |λ|2 ) 2 (ζ ψ) L r . Using the coordinate characterization of Sobolev spaces we see that G + ∈ H −1 × H −1 implies ζ G + ∈ H −1 × H −1 . The Fourier space characterization of Sobolev spaces then ρ shows that the norms (ζl− ⊗ ζl+ )G + B are finite. Moreover, using (23) and noting (2n ∗ ) ,δ
that χˆ ε (m 2 + 41 + λ2 )−δ/4 ∞ ≤ constχˆ ε ∞ ≤ constχε 1 and (χˆ ε − χˆ ε )(m 2 + λ2 )−δ/4 ∞ ≤ O(1)(ε ∧ ε )−δ/2 , see Appendix B, one derives
and
1 4
+
Rε (w, n) L p (µG + ) ≤ const( p, n)w2
(25)
Rε (w, n) − Rε (w, n) L p (µG + ) ≤ const( p, n)(ε ∧ ε )−δ/2 w2 .
(26)
is a Cauchy-sequence in L p (µ
The latter inequalities show that Rε G + ) with limit R(w, n) obeying the bound (25). Applying this to the special case R(w, n) = VΛ we get statement (a). In 2 dimensions there is just a logarithmic singularity G + (x, x ) ∼ const| ln d(x, x )| for small distances. With the aid of (25) and (26), by employing the arguments given in [12, Theorem 8.6.2], we see that also (b) holds true. In order to prove (c), we write e−V ( · +G + f z )(1Λ ) − e−V ( · +H+ f )(1Λ )) 1 = e−V ( · +G + f z )(s1Λ ) 0
×(V ( · + H+ f )(1Λ ) − V ( · + G + f z )(1Λ ))e−V ( · +H+ f )((1−s)(1Λ )) ds. The L p (µG + )-norm of the latter integral can be estimated as sup e−V ( · +G + f z )(s1Λ ) L 3 p (µG + ) e−V ( · +H+ f )((1−s)(1Λ )) L 3 p (µG + ) 0≤s≤1
×V ( · + G + f z )(1Λ ) − V ( · + H+ f )(1Λ ) L 3 p (µG + ) , which by (a) and (b) proves the assertion.
5. Reflection Positivity and Invariance In this section we probe the functional integrals for reflection positivity and conformal invariance. These two properties are essential to qualify them as providing us a conformal field theory on the boundary. The following considerations are valid for d ≥ 1, if we assume that a local (hence reflection positivity preserving) interaction exists for bounded Λ and limits of the generating functionals exist for Λ Hd+1 . The existence and related questions of uniqueness are left to future work.
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For simplicity let us consider the reflection θ with respect to coordinate x1 of Rd , i.e. θ (x1 , x2 , . . . , xd ) = (−x1 , x2 , . . . , xd ). Let Λ ⊂ Rd+1 be reflection-symmetric, where the action of θ is extended to Rd+1 via θ (x) = (z, θ x). We want to verify that the integrals (Y˜ ( f )/Y˜ (0))z are reflection positive, i.e. the finite matrix m i j = (Y˜ ( f i + θ f j )/Y˜ (0))z is positive-semidefinite for arbitrary f i ∈ S(Rd ) with support at x1 > 0. Note that our formulation of reflection positivity refers to the Laplace transform of measures and not to their Fourier transform. For local interactions the latter are restrictions of a reflection positive generating functional, see [12,6], in the sense that −∆+ ˜ ˜ (Y ( f )/Y (0))z = lim eφ(z gn ) FΛ (φ)dµG + (φ)/Y˜ (0), n→∞ D
for a sequence gn converging to δz ⊗ f in H −1 . In order that Z˜ ( f )/ Z˜ (0) be reflection positive, we want to verify that the correcting factor exp(−Corr(z) f, f ) is reflection positive. This means that the matrix [exp(−Corr(z)( f i + θ f j ), ( f i + θ f j ))]i j is positive-semidefinite. The quadratic forms in Corr(z) are given, up to constants, by ˆ = ( f, (−∆) j g). Naturally, at this place −∆ is the Laplacian on Rd . Now, ( fˆ, |k|2 j g) (θ f, (−∆) j g) = 0, for f, g ∈ S(Rd ) with support at x1 > 0. Using the fact, that for positive-semidefinite matrices Ai j , Bi j , the matrix Di j = Ai j Bi j is again positive-semidefinite, the claim follows. Therefore, reflection positivity holds also for z → 0 and then for Λ Hd+1 . The basic implication of the AdS/CFT correspondence is that covariance of the bulk functional integral translates into a conformal invariance on the boundary. On geometrical grounds the isometry group Iso(Hd+1 ) acts by conformal transformations on the boundary, see [18]. Here we allow also non-orientation preserving isometries and conformal transformations. This means in particular that ∂g(x) g ∗ d x = det d x, (27) ∂x where d x is the standard volume form on Rd and ∂g(x)/∂ x denotes the Jacobian matrix. In order to take into consideration the transformations (27), we regard our functionals as functions of d-forms ω with compact support, i.e., ω = f d x with f ∈ C0∞ (Rd ) such that the support of f doesn’t contain a point, which potentially is mapped to infinity by g. Suppose that Z˜ lim (ω) := limΛHd+1 Z˜ ( f )/ Z˜ (0) exists uniquely, then conformal invariance means the property that Z˜ lim (gω) = Z˜ lim (λg · ω), g ∈ Iso(Hd+1 ),
(28)
− ∆+ d with action gω := g −1∗ ω and conformal density λg (x) = det ∂g(x) . For (28) to ∂x hold the bulk-to-boundary propagator has to fulfill the following intertwining property. Lemma 2. For g ∈ Iso(Hd+1 ) let g(z, x) = (z g (z, x), x g (z, x)) ≡ (z g , x g ) denote the action of g. Then with g(x) = lim z→0 x g (z, x) we have −1 ∆d+ ∂g (x ) −1 H+ (g(z, x); x ) = det H+ (z, x; g (x )). ∂x
AdS/CFT Correspondence in the Euclidean Context
93
Proof. We note that H+ (z, x; x ) = lim z →0 z −∆+ γ+ zz (z−z )2 +(x−x )2
zz (z−z )2 +(x−x )2
∆+
Now,
is invariant with respect to isometries and therefore
z H+ (g(z, x); x ) = lim
−∆+
zz g−1
γ+
∆+
(z − z g−1 )2 + (x − x g −1 )2
−∆+
∆+ zz g−1 z = lim (z −1 )−∆+ γ+ . z g−1 (z − z g−1 )2 + (x − x g −1 )2 z →0 g z →0
In order to see the effect of the transformation g −1 we use its action on the isometric model of Hd+1 given by 2 2 Ld+1 := {ζ ∈ Md+1,1 | ζ12 + · · · + ζd+1 − ζd+2 = −1, ζd+2 > 0},
equipped with the metric induced from Minkowski space Md+1,1 with metric dζ12 + 2 − dζ 2 . For Ld+1 the isometry group is by definition O + (d + 1, 1) and the · · · + dζd+1 d+2 isometry map η : Hd+1 → Ld+1 is given by xi , 1 ≤ i ≤ d, z 1 1 = − (z 2 + x 2 − 1), ζd+2 = (z 2 + x 2 + 1), 2z 2z
ζi = ζd+1 with inverse
z=
1 ζi , xi = . ζd+1 + ζd+2 ζd+1 + ζd+2
Thus, an arbitrary isometry on Hd+1 can be cast into the form η−1 ◦ g ◦ η, g ∈ O + (d + 1, 1). Using this fact, one easily shows that z g−1 , ∂z g−1 /∂ xi and ∂ xi g−1 /∂z tend to zero as z → 0, whereas ∂ x g −1 /∂ x → ∂g −1 (x )/∂ x and ∂z g−1 /∂z ∼ z g−1 /z . Moreover,
invariance of the volume measure z −d−1 dzd x, up to a possible sign, implies −1 ∂g (z , x ) −d−1 −d−1 z = (z g−1 ) . det ∂(z , x ) Combining all this gives
lim
z →0
z g−1 z
−1 d1 ∂g (x ) = det , ∂x
which shows the statement of the lemma. We may summarize the above findings in
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Theorem 2. Let θ be the reflection with respect to a hyperplane of Rd containing 0. If the limit Z˜ lim (ω) := limΛHd+1 Z˜ ( f )/ Z˜ (0) exists for a sequence of reflection-invariant Λ s, then it is reflection positive, in the sense that the finite matrix Mi j = Z˜ lim (ωi + θ ω j ), θ ωi = (θ f i )d x, is positive-semidefinite for arbitrary ωi with f i ∈ S(Rd ), having support in the positive half-space. Moreover, if the limit exists uniquely, then conformal invariance holds in the sense of Eq. (28). Remark 3. When the functional Z˜ lim is analytic at 0, reflection positivity of Z˜ lim entails reflection positivity of the corresponding Schwinger functions (Sn )n∈N0 . This also holds in the case when Z˜ lim is not stochastically positive, see [13, Prop. 6.1]. Note that the correction term (Corr(z) f, f ) potentially destroys stochastic positivity in the limit z → 0. Remark 4. The case of conformal symmetry is not treated in the standard version of the Osterwalder-Schrader reconstruction theorem, cf. e.g. [22,23]. The required extension of the reconstruction theorem can easily be accomplished in the same way as the relation of rotationand Lorentz invariance, writing for example, dilatation invariance infinitesn imally as j=1 x j · ∇x j − n∆ Sn (x 1 , . . . , x n ) = 0, ∆ being the conformal weight, and then representing Sn as a Laplace transform of the Fourier transformed Wightman function. Via taking the differential operator under the integral transform and integration by parts, dilatation invariance of the Fourier transformed Wightman functions with weight ∆−d follows which is equivalent to dilatation invariance of Wightman functions with weight ∆. The same argument applies for special conformal transformations. We have thus completed the proof of AdS/CFT for Euclidean quantum fields up to the infra-red problem Λ Hd+1 (for d + 1 = 2). Due to the different nature of source terms, which include bulk-to-boundary propagators that increase if one approaches the conformal boundary in the direction of the source, this infra-red problem is different from, and probably much harder than the related one in [12] where sources are rapidly decaying. We will come back to this point elsewhere. A. Divergencies in lim z→0 z −2∆+ (G + f z , f z ) In investigating this limit we shall use the following integral representation, see [3,19]:
G + (z, x; z , y) = (zz )
(2π )
d 2
(2π )
∞
Rd
0
1
d/2
= (zz )
1
d/2
d 2
∞ 0
1 eik(x−y) dk Jν (zω)Jν (z ω)ωdω ω2 + |k|2
Cω (x − y)Jν (zω)Jν (z ω)ωdω,
(29)
where Cω is the integral kernel of (−∆ + ω2 )−1 in Rd . In addition, for Reν > − 21 , Jν can be represented as Jν (u) = √
21−ν π Γ (ν +
1 2)
u
ν
1 0
1
(1 − t 2 )ν− 2 cos(ut)dt.
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95
Then with f ∈ S(Rd ) we get
Rd
G + (z, x; z, y) f (x) f (y)d xd y =
Rd
×
∞
0
z d+2ν
√
2
π Γ (ν + 21 ) 2 cos(zωt)dt ω2ν+1 dω. d
(2π ) 2
1 1 | fˆ(k)|2 dk (1 − t 2 )ν− 2 2 2 Rd ω + |k| 0
21−ν
Employing the geometric series expansion
1 1 = 2 2 2 ω + |k| ω
1 1 + |k|2 /ω2
=
[ν] |k|2 j (−|k|2 /ω2 )[ν]+1 (−1) j 2 j+2 + ω ω2 + |k|2 j=0
we obtain z −d−2ν
×
⎩
√
21−ν
2
π Γ (ν + 21 ) 1 2 ∞ j 2(ν− j)−1 2 ν− 21 (−1) ω cos(zωt)(1−t ) dt dω | fˆ(k)|2 |k|2 j dk
Rd
⎧ [ν] ⎨
G + (z, x; z, y) f (x) f (y)d xd y =
1
Rd
0
j=0
+(−1)[ν]+1
d
(2π ) 2
Rd
0
∞ 0
| fˆ(k)|2 |k|2[ν]+2 dk 2 2 Rd ω + |k| 1
×ω2(ν−[ν])−1
cos(zωt)(1 − t 2 )
ν− 12
⎫ ⎬
2
dω . ⎭
dt
0
(30)
On the one hand, the terms
∞
ω
0
=z
1
2(ν− j)−1
−2(ν− j)
2 ν− 12
cos(zωt)(1 − t )
2 dt
0
∞ 1
0
2 ν− 21
cos(ωt)(1 − t )
dω 2
dt
ω2(ν− j)−1 dω =: z −2(ν− j) a j
0
diverge as z → 0. On the other hand, using dominated convergence, one can show that the last term in (30), for z → 0 converges to constant times
1
2 ν− 21
(1 − t )
2
∞
dt
0
0
| fˆ(k)|2 |k|2[ν]+2 dk ω2(ν−[ν])−1 dω. 2 2 Rd |k| + ω
Using the formula
1
t 0
2a+1
1 (1 − t ) dt = 2 2 b
Γ (a + 1)Γ (b + 1) Γ (a + b + 2)
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H. Gottschalk, H. Thaler
with a = − 21 , b = ν − 21 we thus get ⎧ ⎨ G + (z, x; z, y) f (x) f (y)d xd y lim z −d−2ν z→0 ⎩ Rd Rd
1
21−ν
2
[ν]
⎫ ⎬
z −2(ν− j) (−1) j a j | fˆ(k)|2 |k|2 j dk √ 1 ⎭ d π Γ (ν + ) R (2π ) 2 j=0 2
∞ 2−ν Γ ( 21 ) 1 | fˆ(k)|2 [ν]+1 = (−1) |k|2[ν]+2 dk ω2(ν−[ν])−1 dω. √ d d ω 2 + |k|2 π Γ (ν + 1) 2 0 R (2π ) −
d 2
(31) Let us perform the ω-integration in (31) first. With the aid of ∞ a−1 x π , 0 < a < b, dx = b 1 + x b sin(aπ/b) 0 where a = 2(ν − [ν]), b = 2, we get for the integral π |k|2(ν−[ν])−2 ˆ π | f (k)|2 |k|2[ν]+2 dk = (−1)[ν] | fˆ(k)||k|2ν dk, 2(ν−[ν])π d d 2 sin νπ R 2 sin R 2 and therefore (31) simplifies to −
π d 2 sin(νπ ) (2π ) 2 1
1 2ν Γ (ν + 1)
2 Rd
| fˆ(k)|2 |k|2ν dk.
(32)
Comparing (32) with (7) and exploiting relations Γ (ν)Γ (1 − ν) = π/ sin(νπ ) and Γ (1 − ν) = −νΓ (−ν) we see that the latter expression equals ( f, α+ f ). B. Fourier and Spherical Fourier Transform on Hd Hyperbolic spaces belong to the class of Riemannian symmetric spaces which can be represented in the form X = G/K with G a noncompact semisimple Lie group and K a maximal compact subgroup, i.e. Hd+1 S O0 (d + 1, 1)/S O0 (d + 1). For these types of spaces there is an analogue of the Fourier transform in Rd . Let g = k ⊕ p be the Cartan decomposition of the Lie algebra g of G. Then we have the following Iwasawa decomposition g = k ⊕ a ⊕ n, where a is a maximal abelian subspace of p, n := α∈Σ+ gα with Σ+ being a choice of positive roots with respect to (g, a). The norm induced from the Killing-form on p will be denoted by · . There is a corresponding Iwasawa decomposition for the Lie group G = K AN = N AK and every g ∈ G can be written as g = k(g) exp H (g)n(g) with unique elements k(g) ∈ K , H (g) ∈ a, n(g) ∈ N . Let M denote the centralizer of A in K , B := K /M, and let A(x, b) ∈ a be the vector A(x, b) := A(k −1 g) := −H (g −1 k) for x = g K ∈ X and b = k M ∈ B. The Fourier transform of a function f ∈ C0∞ (X ) is now defined as [16] f (x)e(−iλ+ρ)A(x,b) d x, λ ∈ a∗C , b ∈ B, (33) fˆ(λ, b) := X
AdS/CFT Correspondence in the Euclidean Context
97
where ρ = 21 α∈Σ+ m α α, m α = dimgα . Let us have a closer look at the space H2 which can be represented as the open disk D := {w ∈ C : |w| < 1}, equipped with the Riemannian metric g D = 4(1 − |w|2 )−2 (dw12 + dw22 ), which in turn is diffeomorphic to the homogenous space G/K , where the Lie group a b 2 2 G = SU (1, 1) = g = ¯ : |a| − |b| = 1 b a¯ acts on D by g·w =
aw + b , ¯ + a¯ bw
and the isotropy group of 0 is K = S O(2). In this picture the Fourier transform of a function on D is given by 1 f (w)e(−iλ+ 2 )w,b dσ (w), λ ∈ C, b ∈ ∂ D = B, fˆ(λ, b) = D
where dσ is the volume form related to g D and w, b denotes the geodesic distance from 0 to the circle which passes through w, and at b, is tangential to the boundary ∂ D of D. The spherical Fourier transform is defined by fˆ(λ) = f (w)φ−λ (w)dσ (w), (34) D
where φλ is the spherical function
φλ (w) =
1
∂D
e(iλ+ 2 )w,b db.
−1 In the general case spherical functions are given by φλ (g) = K e(iλ+ρ)A(k g) dk and obey φλ (e) = 1 and −∆φλ = (λ2 + ρ2 )φλ . We notice that for radial functions f , i.e. f (w) = f (|w|), the transform (34) may be written as ∞ fˆ(λ) = 2π f (tanh r2 )φλ (tanh r2 ) sinh r dr, (r = d(0, w)). 0
Moreover, since ew,b = write φλ (tanh
r 2)
1−|w|2 , |w−b|2
1 = 2π
with the substitutions w = tanh r2 , b = eiθ , we may
π
−π
1
(cosh r − sinh r cos θ )−(iλ+ 2 ) dθ,
and setting further u = tanh 21 θ, 21 dθ = (1 + u 2 )−1 du, we get φλ (tanh
r 2)
1 = π
(iλ− 21 ) 1 − u2 du cosh r + sinh r . 2 1 + u 1 + u2 −∞ ∞
Because of the group structure we may consider the convolution f 1 (h · o) f 2 (h −1 g · o)dh, o = eK . ( f 1 f 2 )(g · o) := G
(35)
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For radial functions f 1 , f 2 one gets ( f 1 f 2 )(λ) = fˆ1 (λ) fˆ2 (λ), whenever both sides exist. We also need the following estimate: (χˆ ε − χˆ ε )(λ2 +
1 4
δ
δ
+ m 2 )− 4 ∞ ≤ O(1)(ε ∧ ε )− 2 .
To see this let us regard χˆ ε as a function of λε by setting gε (λ/ε) := χˆ ε (λ). Then d d d(λ/ε) (gε ) = ε dλ (χˆ ε ) and with the aid of (35) and the substitution y = εr we get that |χˆ ε (λ) − χˆ ε (λ)| ≤ O(1)
|λ| ε ∧ ε
δ
2
for |λ| ≤ ε ∧ ε and |χˆ ε (λ) − χˆ ε (λ)| ≤ O(1) for |λ| > ε
∧ ε .
C. Sobolev Spaces In this section we introduce Sobolev spaces on hyperbolic spaces. For β ≥ 0 let us define the Sobolev space of order β as in [26] β H β := u ∈ L 2 (Hd+1 ) : u = (−∆ + m 2 )− 2 v, v ∈ L 2 (Hd+1 ) with u H β := v L 2 (Hd+1 ) . For β < 0 we define H β := u ∈ D : u = (−∆ + m 2 )k v, v ∈ H 2k+β with k such that 2k + β > 0 , and norm u H β := v H 2k+β . By definition the maps −∆ + m 2 : H β → H β−2 and (−∆ + m 2 )−1 : H β → H β+2 are isomorphisms of Hilbert spaces. The spaces H β can be identified with the completion of C0∞ (Hd+1 ) in the norm f H β = (−∆ + β
m 2 ) 2 f L 2 (Hd+1 ) . In Sect. 3 we have used the distribution f z = δz ⊗ f with f ∈ S(Rd ). Using the explicit expression (29) and a proper smoothing with an approximate unit one sees that f z ∈ H −1 . A second equivalent definition of Sobolev spaces uses local coordinates, see [27]. β For this one first considers the space Ho of distributions which are supported in a ball of fixed radius r, B(o, r ), around some fixed point o equipped with geodesic coordinates and defines the norm u H β as the pull-back of the H β (Rd+1 ) norm in the chosen o
coordinates. For another a ∈ Hd+1 and distribution f supported in B(a, r ) one defines f H β := f ◦ g H β , where g is an isometry with g(o) = a. Then points (ak )k∈N are a o chosen in order to obtain a locally finite covering by the balls B(ak , r ). Finally, employing a partition of unity (ϕk )k∈N w.r.t. the balls B(ak , r ) one says that u ∈ H β (Hd+1 ) if ϕk u H β < ∞. k
ak
AdS/CFT Correspondence in the Euclidean Context
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A third definition can be given using Fourier transforms, where we follow [10]. For this we define the Schwartz space S(Hd+1 ) = S(X ), consisting of complex-valued C ∞ -functions f on X satisfying τ D,m ( f ) = sup (1 + |g|)m φ0 (g)−1 |D f (g)| < ∞, g∈G
for all m ∈ N0 and differential operators D invariant under the left action of G. The norm of g is defined as |g| = | exp X k| = X , X ∈ p, k ∈ K . The space S(X ) becomes a Fréchet space when topologized by means of the seminorms τ D,m . Let S(a∗ × K /M) be the complex-valued C ∞ -functions on a∗ × K /M such that ν E,J,r ( f ) = sup (1 + λ)r |(E J f )(λ, k M)| < ∞, λ,k M
for all differential operators E on a∗ and J invariant on K /M and r ∈ N0 . With these seminorms S(a∗ × K /M) becomes a Fréchet space. The Fourier transform (33) establishes a topological isomorphism between S(X ) and S(a∗ × B)W = S(a∗+ × B), where the subscript W denotes the quotient space under the action of the Weyl group on a∗ . Moreover, the Fourier transform extends to an isometry of L 2 (X ) onto L 2 (a∗+ × B, |c(λ)|−2 dλdb), where c(λ) is the Harish-Chandra c-function. From the )(λ, b) = (λ2 + ρ2 ) fˆ(λ, b) we see that H β (X ) equals the space of property (−∆f u ∈ S(X ) such that |u| ˆ 2 (λ2 + ρ2 + m 2 )β |c(λ)|−2 dλdb < ∞. a∗+
B
It should be noted that in the case H2 we have a∗+ = R+ , ρ2 = |c(λ)|−2 = (2π )−1 λ tanh π λ.
1 4
and
References 1. Berenstein, C.A., Gay, R.: Complex Variables. New York: Springer, 1991 2. Berezanskii, Iu. M., Kondratiev, Iu. M.: Spectral Methods in Infinite-Dimensional Analysis. Vol. 1, Dordrecht: Kluwer Academic Publishers, 1995 3. Bertola, M., Bros, J., Moschella, U., Schaeffer, R.: Decomposing quantum fields on branes. Nucl. Phys. B 581, 575–603 (2000) 4. Bogachev, V.I.: Gaussian Measures. RI: Amer. Math. Soc., Providence, 1998 (translated from Russian) 5. Bros, J., Epstein, H., Moschella, U.: Towards a general theory of quantized fields on the anti-de Sitter spacetime. Commun. Math. Phys. 231, 481–528 (2002) 6. Dimock, J.: Markov quantum fields on a manifold. Rev. Math. Phys. 16, 243–256 (2004) 7. Dimock, J., Glimm, J.: Measures on Schwartz distribution space and applications to P(φ)2 field theories. Adv. Math. 12, 58–83 (1974) 8. Dobrev, V.K.: Intertwining operator realization of the AdS/CFT correspondence. Nucl. Phys. B 553, 559–582 (2004) 9. Dütsch, M., Rehren, K.H.: A comment on the dual field in the AdS-CFT correspondence. Lett. Math. Phys. 62, 171–184 (2002) 10. Eguchi, M., Okamoto, K.: The Fourier transform of the Schwartz space on a symmetric space. Proc. Japan Acad. 53, Ser. A, 237-241 (1977) 11. Gelfand, I.M., Shilov, G.E.: Generalized Functions, Vol.1. Properties and Operations. New York-London: Academic Press, 1964 (1977) 12. Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View, 2nd edition. New-York: Springer, 1987 13. Gottschalk, H.: Die Momente gefalteten Gauß-Poissonschen weißen Rauschens als Schwingerfunktionen. Diploma thesis, Bochum, 1995
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14. Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from noncritical string theory. Phys. Lett. B 428, 105–114 (1998) 15. Haba, Z.: Quantum field theory on manifolds with a boundary. J. Phys. A 38, 10393–10401 (2000) 16. Helgason, S.: Groups and Geometric Analysis. Mathematical Surveys and Monographs, Vol. 83, Providence, RI: Amer. Math. Soc., 2000 17. Jaffee, A., Ritter, G.: Quantum field theory on curved backgrounds II: Spacetime symmetries. http://arxiv. org/list/hep-th/0704.0052, 2007 18. Kniemeyer, O.: Untersuchungen am erzeugenden Funktional der AdS-CFT-Korrespondenz. Diploma thesis, Univ. Göttingen, 2002 19. Hong, L., Tseytlin, A.A.: On four point functions in the CFT/AdS correspondence. Phys. Rev. D 59, 086002 (1999) 20. Maldacena, J.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998) 21. Mück, W., Wiswanathan, K.S.: Regular and irregular boundary conditions in the AdS/CFT Correspondence. Phys. Rev. D 60, 081901 (1999) 22. Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. Commun. Math. Phys. 31, 83–112 (1973) 23. Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. II. With an appendix by S. Summers. Commun. Math. Phys. 42, 281–305 (1975) 24. Rehren, K.-H: Algebraic holography. Ann. Henri Poincarè 1, 607–623 (2000) 25. Rehren, K.-H.: QFT lectures on AdS-CFT. In: Proc. of 3rd Summer School on Modern Math. Phys., Belgrade:Inst. of Phys., pp. 95–118, 2005 26. Strichartz, R.S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52, 48–79 (1983) 27. Tartaru, D.: Strichartz estimates in the hyperbolic space and global existence for the nonlinear wave equation. Trans. Am. Math. Soc. 353, 795–807 (2000) 28. Weidmann, J.: Lineare Operatoren. B.G. Teubner, Stuttgart, 1976 29. Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998) Communicated by J.Z. Imbrie
Commun. Math. Phys. 277, 101–125 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0359-3
Communications in
Mathematical Physics
Measures on Banach Manifolds and Supersymmetric Quantum Field Theory Jonathan Weitsman Department of Mathematics, University of California, Santa Cruz, CA 95064, USA. E-mail: [email protected] Received: 3 November 2006 / Accepted: 11 April 2007 Published online: 31 October 2007 – © Springer-Verlag 2007
Dedicated to the memory of Raoul Bott Abstract: We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. These measures are mathematically well-defined objects inspired by the formal path integrals appearing in the physics literature on quantum field theory. We give three concrete examples of our construction. The first example is a family µs,t P of measures on a space of functions on the two-torus, parametrized by a polynomial P (the Wess-Zumino-Landau-Ginzburg model). The second is a family µs,t G of measures on a space G of maps from P1 to a Lie group (the Wess-Zumino-NovikovWitten model). Finally we study a family µs,t M,G of measures on the product of a space of connections on the trivial principal bundle with structure group G on a three-dimensional manifold M with a space of g-valued three-forms on M. We show that these measures are positive, and that the measures µs,t G are Borel probability measures. As an application we show that formulas arising from expectations in the measures µs,1 G reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar computation for the measures µs,t M,SU (2) , where M is a homology three-sphere, will yield the Casson invariant of M. 1. Introduction During the past two decades techniques and ideas arising from quantum field theory have played an increasing role in geometry and topology. The work of the school of mathematical physicists led by E. Witten on applications of supersymmetric quantum field theory and later of string theory to problems in mathematics has in many cases led to results that can be stated as mathematically precise conjectures, and often rigorously proved as theorems. However, the methods used by the physicists, among which formal pathintegral methods of integration over infinite-dimensional spaces are prominent, have not in the main been accessible to mathematical interpretation; for actual computation one Supported in part by NSF grant DMS 04/05670.
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proceeds by inspired analogy to finite-dimensional cases and by physical intuition. The purpose of this paper is to produce measures on infinite-dimensional spaces associated to supersymmetric quantum field theories which maintain the formal analogies to finite dimensions used by the physicists, but which are mathematically well-defined.
1.1. Finite dimensional analogs of supersymmetric quantum field theory. In this paper we give a construction of measures on Banach manifolds associated to supersymmetric quantum field theories. In order to approach this problem, we begin with finite-dimensional analogs of these theories and the associated measures. In the simplest case, given a map of finite-dimensional vector spaces, we give three constructions that coincide: That of the degree of the map, of the pull-back by this map of a Gaussian differential form of top degree, and of a measure obtained by pushforwards by local inverses. The third construction is the one which we generalize to Banach manifolds. The physics literature makes use of formal analogy to finite dimensions to work with the first and second constructions only. 1.1.1. Measures arising from maps of vector spaces. Let V, W be real, n-dimensional Hermitian vector spaces. Let ηW denote the invariant volume form on W , and let ξ ∈ H n (W ) denote the differential form ξ = 1 n/2 exp(−|w|2 /2)ηW . Given a proper (2π ) smooth map f : V −→ W, the pullback f ∗ ξ gives a signed measure on V. In terms of the natural volume form ηV this measure is given by f ∗ξ =
1 (2π )n/2
exp(−| f |2 /2)det(∇ f )ηV .
(1.1)
If f grows rapidly at infinity the integral of f ∗ ξ converges and gives the LeraySchauder degree of the map f, so that if 0 is a regular value of f , we have f ∗ξ = x , (1.2) V
x∈ f −1 (0)
where x = det(∇ f |x )/|det(∇ f |x )|. Equation (1.2) can be seen to follow formally from the expression (1.1) by replacing f by t f and then letting t −→ ∞. Formulas similar to (1.1) and (1.2) hold where V, W are complex Hermitian vector spaces and f : V −→ W is a proper smooth map. If W is a complex vector space of complex dimension n, the Euclidean volume form on W can be written as ηW = ωn /n!, where ω is a two-form on W given in terms of a set (w1 , . . . , wn ) of holomorphic coordinates on W by ω = i dwi ∧ d w¯ i . It follows that if f : V −→ W is holomorphic and x ∈ V is a regular point of f then det(∇ f |x )/|det(∇ f |x )| = 1. An expression for the measure given by f ∗ ξ can also be given as follows. Let x ∈ V be a regular point of f, and let U be a neighborhood of x where f is invertible; denote this local inverse by g. Let µW be the Gaussian measure on W corresponding to ξ, and consider its restriction to f (U ). We define a measure on U by µU := g∗ (µW | f (U ) ).
(1.3)
This gives a measure f ∗ µW on V by taking the union over all neighborhoods U and by assigning measure zero to the singular points of f. The measure f ∗ µW can be
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obtained from the signed measure associated to f ∗ ξ by taking absolute values of the determinant of ∇ f ; in other words, for any φ ∈ Cc (V ), 1 (1.4) φ exp(−| f |2 /2)det(∇ f )ηV = f ∗ µW (φ Phase det(∇ f )), (2π )n/2 where Phase det(∇ f )(x) is given by Phase det(∇ f )(x) := det(∇ f |x )/|det(∇ f x )| if x is a regular point of f and is extended arbitrarily to the singular set of f where the measure f ∗ µW is zero. Two features of the finite-dimensional case will recur in the generalization of this construction to Banach manifolds. These are the subtlety involved in the definition of the phase of the determinant of ∇ f, and the issue of the finiteness of the measure f ∗ µW . In the case of Banach manifolds, the measure analogous to f ∗ µW will have a straightforward construction, but the phase of the determinant may not be well-defined. This phenomenon is known in the physics literature as the appearance of an anomaly; the geometry underlying anomalies was studied by Atiyah-Singer [5] and Quillen [32]. Similarly, in finite dimensions, the finiteness of the measure f ∗ µW depends on the growth properties of f at infinity, and must be studied by hard analysis. The same will be true in the case of Banach manifolds. Remark 1.1. The reason supersymmetry arises in the context of integrals of the type given in Eq. (1.2) is the following. Using Berezin integrals over odd variables ψ, ψ (see for example [7]) the integral appearing in (1.2) can be written 1 ¯ f )ψ), (1.5) d ψ¯ dψ ηV exp(−| f |2 /2 + ψ(∇ (2π )n/2 where the Berezin integral over ψ and ψ gives rise to the determinant det ∇ f. Thus the degree can be seen as the integral of the exponential exp(S), where the action S given by ¯ ¯ := −| f (x)|2 /2 + ψ(∇ f (x))ψ S(x, ψ, ψ) involves both even and odd variables. These variables may be viewed as coordinates on a supermanifold, and a continuous group of symmetries of the function f may give rise to an invariance of S under the action of a graded Lie algebra. The physics literature studies infinite-dimensional analogs of this setting. In that context the invariance of S under the action of a graded Lie algebra is called supersymmetry, and the analog of the map f is called the Nicolai map [29]. The fact that supersymmetric field theories are analogs of integrals of the type (1.1) and are related to the degree of the map corresponding to f is due to Witten in [45, 46]. 1.1.2. Measures arising from sections of vector bundles. Finite-dimensional analogs of supersymmetric quantum field theories also arise in a slightly different context. Let M n be a compact Riemannian manifold, let E −→ M be an n-dimensional Hermitian vector bundle over M, and let ∇ be a connection on E. Let s : M −→ E be a section of E. In this case Mathai and Quillen [27] give an expression for the Euler number χ (E) of E as χ (E) =
1 (2π )n/2
exp(−|s|2 /2)det(∇s) M
n k=1
Ck ;
(1.6)
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here Ck is a sequence of terms involving ∇s and the curvature of E. In terms of local coordinates the Mathai-Quillen formula is given by the integral of a sum of such terms multiplied by a phase of a determinant with respect to a measure of the type defined in the previous section. 1.2. Outline of the paper. The physics literature on topological quantum field theory is concerned with formal infinite-dimensional integrals corresponding morally to the differential form (1.1), where the map f : V −→ W is replaced by a map F of infinite dimensional spaces. It then seeks to apply ideas from differential geometry (such as localization), as well as novel techniques, to gain insight about the solutions of the equation F = 0, and more generally about the formal properties of the underlying measure. While some of the results of this approach can be stated mathematically and sometimes verified by various mathematical methods, there have been so far no satisfactory constructions of measures on infinite dimensional spaces which would approach the physicists’ formal constructions. In this paper we will describe a method of constructing measures on Banach manifolds analogous to the pullback measures f ∗ µW in a number of examples motivated by supersymmetric quantum field theories. Starting with a supersymmetric model based on a map F of infinite-dimensional spaces, we will find a family of Banach manifolds X s , s > s0 , a Banach space Y, and a family of maps Fs : X s −→ Y, where the set Fs = 0 is independent of s and matches formally the set F0 = F = 0. The Banach space Y in our examples will be equipped with a white noise measure—the infinite-dimensional analog of the Gaussian measure µW above. We will construct an analog of the pullback measure f ∗ µW and of the phase of the determinant. We do not know of a general theorem that would give convergence of the integral of the phase of the determinant; instead we study some examples paralleling supersymmetric quantum field theories in the literature. In these examples we can give specific conjectures about the finiteness of the measure and the value of the integral of the phase of the determinant. In one example we will be able to prove the analogs of these conjectures. This paper is structured as follows. In Sect. 2, we first recall some facts about white noise measures. These are the infinite dimensional analogs of the Gaussian measures µW above. We then study maps F : X −→ Y , where X is a Banach manifold and Y is a Banach space equipped with a white noise measure. If F : X −→ Y is a differentiable (that is, C 1 ) map, we obtain a measure on X analogous to the measure f ∗ µW arising in (1.3). This is done in Proposition 2.2. In Sect. 3, we give examples of this construction of measures where the maps are motivated by supersymmetric quantum field theories appearing in the physics literature. In these examples we also give a construction of the phase of the determinant using a generalization of Fredholm determinants. We study three families of measures. The first family is denoted µs,t P , where P is a polynomial in one variable and s, t are parameters. These are measures on a Banach space of functions on the torus, and are related to objects known as the Wess-Zumino-Landau-Ginzburg models (Theorem 1). We next consider 1 a family µs,t G of measures on a group G of maps from P to a complex Lie group; these measures are related to the Wess-Zumino-Novikov-Witten models (Theorem 2). Finally, given a compact Lie group G and a three-manifold M, we consider measures µs,t M,G on the product of a space of connections on a principal G bundle on M with a space of g-valued three-forms on M. These measures are related to three-dimensional quantum s,t gauge theory (Theorem 3). We conjecture that the measures µs,t P and µ M,G extend to Borel probability measures. We also give precise conjectures about the values of the
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integrals of the phases of appropriate determinants (Conjecture 3.3 and Conjecture 3.7). In the case of three-dimensional gauge theory the conjecture is that µs,t M,SU (2) should yield the Casson invariant of homology three-spheres. At the end of this section we briefly describe how our methods can be used in a few other examples of maps of Banach manifolds inspired by supersymmetric quantum field theories. In Sect. 4 we turn to the measures µs,t G , where we are able to prove analogs of the s,t conjectures we have made about µ P and µs,t M,G . Using results of Atiyah and Bott [3], is a Borel probability measure on the space G (Theorem 4). We then we show that µs,t G compute (Theorem 5) some expectations of functions in this measure to recover formulas discovered by Frenkel and Zhu [13] in the context of vertex operator algebras. 2. Maps and Measures on Banach Manifolds In this section we consider some general constructions of measures on Banach manifolds. We begin with Gaussian measures and then present a method of producing non-Gaussian measures from Gaussian measures and maps of Banach manifolds.1 2.1. White Noise and Gaussian Measures. Let M d be a smooth compact Riemannian manifold and let V −→ M be a smooth Hermitian vector bundle over M. Let (V ) denote the space of smooth sections of V. The Hermitian metric on V and the Riemannian structure on M give rise to an L 2 inner product <, > on (V ). For λ ∈ R, let Hλ (M) denote the distributions on M of Sobolev class λ and let λ (V ) = (V )⊗ Hλ (M) denote the space of sections of V of Sobolev class λ. Let α be a connection on V, and let α be the corresponding Laplacian on sections of V. This Laplacian gives rise to a hermitian inner product on (V ) given by s, s λ =< s, (− α + 1)λ s > for s, s ∈ (V ). This inner product extends to λ (V ) and makes λ (V ) into a separable Hilbert space. If λ > 0 the space −λ (V ) may be identified with the dual of λ (V ); we continue to denote the pairing of −λ (V ) with λ (V ) by <, >. If λ = 0 we write 0 (V ) = L 2 (M, V ), and the pairing <, > gives rise to the L 2 metric on 0 (V ). If λ < −d/2, the space λ (V ) comes equipped with a finite Borel measure µV called white noise measure, which is characterized by the following property: Let φ ∈ −λ (V ) and let E φ : λ (V ) −→ C denote the function given by E φ (σ ) = exp(iRe < φ, σ >). Then 1 µV (E φ ) = exp(− ||φ||2L 2 ). 2
(2.1)
In particular, µV (1) = 1. Given t > 0, let Rt : λ (V ) −→ λ (V ) be the scaling function Rt (v) = tv. We write µtV = (Rt −1 )∗ µV for the scaled white noise measure. Then for every > 0, we have µtV (B (0)) > 0 for t sufficiently large. 1 In this paper we consider both the case of complex Banach spaces and that of real Banach spaces. The results presented in this section hold in either the real or complex category, and all formulas should be interpreted in the appropriate setting.
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We now define more general Gaussian measures. Choose s ∈ R. Then (− α + 1)−s : λ (V ) −→ λ+2s (V ) is an isomorphism. Let λ < −d/2 as before; then we may push the white noise measure µtV forward to obtain a Gaussian measure µtV,s := ((− α + 1)−s )∗ µtV on ν (V ), where ν < 2s − d/2.2 Proposition 2.1 (Cameron-Martin). Let λ < 2s − d/2 and let v ∈ (V ). Let Tv :
λ (V ) −→ λ (V ) be the translation given by Tv (a) = v + a. Then (Tv )∗ µtV,s is absolutely continuous with respect to µtV,s , and the Radon Nikodym derivative may be written as d(Tv )∗ µtV,s 1 2 2 2s 2 (2.2) t = exp t < (− + 1) v, · > − ||v|| α 2s . 2 dµtV,s The translate of the measure µtV,s by an arbitrary element v ∈ λ (V ) may not be absolutely continuous with respect to µtV,s . 2.2. Pullbacks of measures by differentiable maps and the main construction. Let X be a separable metric space. Let {(Uα , µα )}α∈A be a collection of pairs (Uα , µa ). where {Uα }α∈A is an open cover of X and µα is a finite Borel measure on Uα for each α. Suppose that for each α, β ∈ A, µα |Uα ∩Uβ = µβ |Uα ∩Uβ . Since X is Lindelöf, we obtain a σ -finite measure µ on X.3 We will call such a measure a Borel probability measure if ∞ > µ(X ) > 0. Let X, Y be separable Banach manifolds and let µ be a finite Borel measure on Y. Suppose F : X −→ Y is a differentiable map, and let U ⊂ X be the (open) set of points p ∈ X such that δF p is an isomorphism. For each p ∈ U , we can find a neighborhood U p of p, where F|U p is invertible. Denote this local inverse by G p , and define a finite Borel measure νU p on U p by νU p := G p ∗ (µ|F (U p ) ). Then the collection {(U p , νU p )} gives a measure ν on U. Let i : U −→ X be the inclusion. We define the measure F ∗ µ on X by F ∗ µ = i ∗ ν. We therefore obtain the following result, which is our main tool for constructing measures on Banach manifolds. Recall that a differentiable map F : X −→ Y of separable Banach spaces is a Fredholm map if δF|x is Fredholm for all x ∈ X. 2 In the literature these measures are usually regarded as living on the spaces Z (V ) of sections of V of Zygλ mund class λ < 2s − d/2. However, an extension of the Sobolev embedding theorem (see [21], Prop. 8.6.10) shows that there exist continuous inclusions
Z λ+ (V ) ⊂ λ (V ) ⊂ Z λ−d/2 (V ) for any > 0, so that our measures can just as well be regarded as living on λ (V ). This will be convenient for various purposes in this paper. 3 To do this, extend the measure to any countable subcover of the cover {U } α α∈A . This extension is independent of the choice of countable subcover.
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Proposition 2.2. Let X, Y be separable Banach manifolds. Let µ be a finite Borel measure on Y. Let F : X −→ Y be a differentiable map. Then the map F induces a σ -finite Borel measure F ∗ µ on X, supported on the open set U ⊂ X where δF is an isomorphism. In particular, let M d be a compact Riemannian manifold of dimension d, and let V be a vector bundle over M. Let λ < −d/2, and let F : X −→ λ (V ) be a differentiable map. Suppose that F is a Fredholm map and that ind δF|x = 0 for all x ∈ X. Then for any t > 0 the map F induces a σ -finite Borel measure F ∗ µtV on X, supported on the set of regular points of F. Suppose that there exists p ∈ X which is a regular point of F and satisfies F( p) = 0. Then for t sufficiently large, there exists a neighborhood W ⊂ X of p such that ∞ > F ∗ µtV (W ) > 0. The methods we have used in this section cannot decide whether F ∗ µtV gives a Borel probability measure on X. This has to be checked by analysis in each example. We will do this in one example in Sect. 4.
3. Examples of Measures on Banach Manifolds In this section we give examples of maps which can be used to construct measures on Banach manifolds by the construction of Proposition 2.2. These maps are inspired by formal path integrals appearing in the physics literature.
3.1. Some analytical preliminaries. In this section we collect some analytical facts we will use in our constructions. 3.1.1. Sobolev bounds. In order to study nonlinear maps on Banach manifolds, we will use the following result on the regularity of a product of distributions. Proposition 3.1. [see e.g. [21], Prop. 8.3.1]. Suppose s1 + s2 ≥ 0. Then f 1 f 2 ∈ Hs (M d ) if
4
Let f 1 ∈ Hs1 (M d ), f 2 ∈ Hs2 (M d ).
s ≤ si , i = 1, 2 and s < s1 + s2 − d/2. The product f 1 f 2 is bounded by || f 1 f 2 ||s < K || f 1 ||s1 || f 2 ||s2 ,
(3.1)
where K is a constant depending on s, s1 , s2 . This result immediately implies a similar result for sections of vector bundles of the appropriate Sobolev classes. 4 We will use the bound (3.1) only in the situation where s , s > d/2; in other words, when the Sobolev 1 2 spaces Hsi (M d ) are spaces of continuous functions.
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3.1.2. Schatten classes and regularized determinants. To define phases of determinants of some of the operators we will encounter, we will use regularized determinants [12, 39]. Let H be a separable Hilbert space, and let L(H) denote the ring of bounded operators on H. For k ≥ 1, write Sk (H) for the k th Schatten ideal of H; that is, the set of compact operators K : H −→ H such that (K ∗ K )k/2 is trace class. The space Sk (H) is a two-sided ideal of L(H). The function K −→ ||K ||k := tr(K ∗ K )k/2 defines a norm on Sk (H) which makes Sk (H) into a Banach space, and the inclusion Sk (H) −→ L(H) is continuous. Similarly, if k < k , Sk (H) ⊂ Sk (H), and the inclusion Sk (H) −→ Sk (H) is continuous. An example of an element of a Schatten ideal is given as follows. Let M be a compact Riemannian manifold of dimension d and let V be a Hermitian vector bundle over M. Let α be a connection on V, let (− α + 1) : λ+2 (V ) −→ λ (V ) be the Laplacian, and let j : λ+2 (V ) −→ λ (V ) denote the inclusion. Then j ◦ (− α + 1)−1 ∈ Sr ( λ (V ))
(3.2)
for all r > d/2. Recall that if K ∈ S1 (H), the Fredholm determinant det (1 + K ) is defined [18] by the convergent series det (1 + K ) :=
∞
tr ∧n K .
(3.3)
n=0
This construction was generalized to operators in Sk (H) by Poincaré [31]; we follow Simon [39]. Given K ∈ L(H), define Rk (K ) ∈ L(H) by Rk (K ) := [(1 + K ) exp(
k−1 (−K )n n=1
n
)] − 1.
(3.4)
Simon shows that if K ∈ Sk (H), then Rk (K ) ∈ S1 (H). Define the regularized determinant detk (1 + K ) by5 detk (1 + K ) := det (1 + Rk (K )).
(3.5)
The function detk (1+·) : Sk (H) −→ C is continuous. If K is trace class, the regularized determinant is related to the Fredholm determinant det (1 + K ) by k−1 (−1)n tr K n ). detk (1 + K ) = det (1 + K ) exp( n
(3.6)
n=1
The regularized determinant has multiplicativity properties which can be deduced from (3.6) and the multiplicativity of the Fredholm determinant by approximating elements of the Schatten classes by trace-class operators (which are dense in Sk (H)). From these multiplicativity properties it follows that if 1 + K is invertible, the regularized determinant detk (1 + K ) is nonzero. 5 The motivation for this definition is Plemelj’s formula (valid when K ∈ S (H) and ||K || < 1) 1 1
det (1 + K ) = exp(tr log(1 + K )). The appearance of Rk (K ) in (3.5) amounts to removing the first k − 1 terms from the power series for log(1 + K ) in Plemelj’s formula. The remaining terms are multiples of K n where n ≥ k and so are trace class if K ∈ Sk (H).
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3.2. Wess-Zumino-Landau-Ginzburg models on the torus. Let = S 1 × S 1 be the torus, equipped with a Kähler structure, and let V = × C −→ be the trivial bundle over . If λ ∈ R, the space λ (V ) is given by λ (V ) = Hλ (). Choose λ < −1 and s > −λ/2. Let µV be the white noise measure on λ (V ). By the Sobolev embedding theorem, the space λ+2s+1 (V ) consists of continuous functions. Let P be a polynomial in one variable. Define Fs : λ+2s+1 (V ) −→ λ (V ) by Fs (φ) = (− + 1)s (∂φ + P (φ)).
(3.7)
Theorem 1. The map Fs is well-defined and differentiable. Its derivative is a Fredholm operator of index 0. Therefore, for each t > 0, the pullback measure ∗ t µs,t P := Fs µV
exists and is a σ -finite measure on λ+2s+1 (V ), supported on the set of regular points of Fs . If the polynomial P has a transversal zero at y ∈ C, then for t sufficiently large, there exists a neighborhood W of the constant function φ0 = y such that ∞ > µs,t P (W ) > 0. Proof. Proposition 3.1 shows that the map Fs is well-defined and differentiable. The derivative δFs is a compact perturbation of the isomorphism (− + 1)s (∂ + 1) : λ+2s+1 (V ) −→ λ (V ), and so is Fredholm of index 0. The existence of the measure follows by Proposition 2.2. If y ∈ C is a point where P (y) = 0 and P (y) = 0, the constant function φ0 := y satisfies Fs (φ0 ) = 0, and δFs |φ0 = (− + 1)s (∂ + P (y)) is an isomorphism. The positivity of the measure Fs∗ µtV in a neighborhood of φ0 for t sufficiently large follows, again, by Proposition 2.2. We make the following conjecture. Conjecture 3.2. The measure µs,t P satisfies µs,t P (1) < ∞ for t sufficiently large. Suppose the polynomial P has a transversal zero. Then for t sufficiently large, Conjecture 3.2 implies that the measure µs,t P is a Borel probability measure on λ+2s+1 (V ). 3.2.1. The phase of the determinant of δFs . The derivative δFs |φ is given by the Fredholm operator Dφ := (− + 1)s (∂ + P (φ)) : λ+2s+1 (V ) −→ λ (V ). If P is nonconstant and y is a point where P (y) = 0, then if φ0 := y is the constant function, Dφ0 is an isomorphism. Then (Dφ0 )−1 Dφ : λ+2s+1 (V ) −→ λ+2s+1 (V ) can be written as a compact perturbation of the identity (Dφ0 )−1 Dφ = 1 + K (φ0 , φ),
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where K (φ0 , φ) = (∂ + P (φ0 ))−1 ◦ j ◦ M(P (φ) − P (φ0 )). Here for ψ ∈ λ+2s+1 (V ), we have denoted by M(ψ) : λ+2s+1 (V ) −→ λ+2s+1 (V ) the operator given by multiplication by ψ, j : λ+2s+1 (V ) −→ λ+2s (V ) is the inclusion, and (∂ + P (φ0 ))−1 is the inverse of the isomorphism ∂ + P (φ0 ) : λ+2s+1 (V ) −→ λ+2s (V ). Since ||M(ψ)|| ≤ ||ψ||2s+λ+1 by Proposition 3.1, the estimate (3.2) shows that K (φ0 , φ) ∈ S3 ( λ+2s+1 (V )) for all φ. The regularized determinant det3 (1 + K (φ0 , φ)) is therefore well-defined for all φ, and is nonzero at all regular points of Fs . Let U be the set of regular points of Fs . Define (φ0 , ·) : U −→ S 1 by (φ0 , φ) =
det3 (1 + K (φ0 , φ)) . |det3 (1 + K (φ0 , φ))|
(3.8)
Morally, the function (φ0 , φ) gives the ratio of the “phase of the determinant of δFs |φ ” to the “phase of the determinant of δFs |φ0 .” This method of defining determinants is used in the physics literature; by analogy with the case of maps of finite-dimensional vector spaces given in Eqs. (1.2) and (1.4) in the introduction, we make the following conjecture. Conjecture 3.3. Assume that Conjecture 3.2 holds. Then for t sufficiently large, |µs,t P ((φ0 , ·))| = (deg P − 1). Morally this conjecture says that for t large, the measure µs,t P is concentrated at the solutions of the nonlinear equation Fs = 0. Remark 3.4. The formal path integral for the Wess-Zumino-Landau-Ginzburg model appearing in the physics literature corresponds to formula (1.1) applied (formally) to F0 , and should morally count the solutions of the partial differential equation F0 = 0. This path integral corresponds to the action whose Bosonic part is given by WZ ¯ 2 + |P (φ)|2 . S B (φ) = |∂φ|
Since the solutions of the equation Fs = 0 are exactly those of the equation F0 = 0, the computations associated formally with the path integral corresponding to F0 should morally be replicated for the measure µs,t P . The supersymmetric field theory corresponding to these path integrals was constructed using classical methods of constructive quantum field theory in [22], and the resulting partition function does count (for generic P) the number of solutions of F0 = 0, which correspond to constant functions with values given by the zeros of P. The idea of using an infinite-dimensional degree to find solutions of nonlinear partial differential equations goes back to Leray-Schauder, who considered proper maps. The
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case of measures induced by nonlinear maps on λ (V ), where V is a vector bundle over a one-dimensional manifold was considered in [16] using the methods of [24, 34, 25, 15]. In this case the pullback measure on λ (V ) is a perturbation of a Gaussian measure, and the analog of our operator K is Hilbert-Schmidt. This is not the case if the underlying manifold is of dimension greater than one. 6 3.3. Wess-Zumino-Novikov-Witten models on P1 . Let G be a compact semisimple Lie group. We consider the corresponding complex Lie group as a subgroup G C ⊂ S L(n, C) of S L(n, C). Then G C and gC ⊂ sl(n, C) are subsets of the space Mn (C) of n × n matrices, and gC acquires a Hermitian inner product from this inclusion.7 Consider P1 , equipped with the standard Kähler metric, and choose a base point in P1 . Let λ < −2 and s > (−λ + 1)/2, and let G := Map∗,λ+2s+1 (P1 , G C ) be the space of maps from P1 to G C of Sobolev class λ + 2s + 1 which carry the base point in P1 to the identity in G C . We have chosen s sufficiently large so that the space G is a Lie group, with the group structure given by pointwise multiplication. As a Hilbert manifold G is modelled on the Lie algebra Lie(G) given by the space Lie(G) := 0∗,λ+2s+1 (P1 , gC ) := ∗,λ+2s+1 (V ) of sections of the trivial bundle V := P1 × gC −→ P1 of Sobolev class λ + 2s + 1 which 1 C 0,1 1 C 1 vanish at the base point of P1 . For ν ∈ R, let 0,1 ν (P , g ) := (P , g ) ⊗ Hν (P ) be the space of gC -valued (0, 1)-forms on P1 of Sobolev class ν; if W = T ∗ P1 ⊗ gC , 0,1 1 C 1 C then 0,1 ν (P , g ) = ν (W ). The space λ (P , g ) = λ (W ) is equipped with a white noise measure µW . 1 C The group G acts freely on the space Aλ+2s := 0,1 λ+2s (P , g ) by the formula 0,1 −1 −1 1 1 C g · A = g Ag + ∂¯ gg for g ∈ G, A ∈ Aλ+2s . Let : ν+2 (P , gC ) −→ 0,1 ν (P , g ) denote the Laplacian. 1 C Define the map Fs : G −→ 0,1 λ (P , g ) by Fs (g) = (− + 1)s ∂¯ gg −1 .
(3.9)
Theorem 2. The map Fs is well-defined and differentiable. Its derivative is a Fredholm operator of index 0. The map δFs |g is an isomorphism for all g ∈ G. Therefore, for each t > 0, the pullback measure ∗ t µs,t G := Fs µW
exists and is a σ -finite measure on G. For t sufficiently large, this measure is positive. In the next section we will prove that the measure µs,t G is a Borel probability measure for all t; see Theorem 4. Proof. An application of Proposition 3.1 shows that Fs is well-defined and differentiable. For g ∈ G, the tangent space T Gg can be identified with 0∗,λ+2s+1 (P1 , gC ). The map Fs is the composite of the map F : G −→ Aλ+2s given by F(g) = ∂¯ gg −1 with the 6 The appearance of regularized determinants is one reason we have chosen to work with Sobolev spaces and Hilbert manifolds rather than with Zygmund spaces and Banach manifolds. While there exist theories of determinants on Banach spaces (see e.g. [18, 37]) I am not aware of a theory of regularized determinants in this context. 7 The only reason for the restriction G C ⊂ S L(n, C) is to allow products and the inverse map on the space G to be estimated directly using Proposition 3.1. It is not very difficult to remove this restriction; see [30].
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1 C isomorphism (− + 1)s : Aλ+2s −→ 0,1 λ (P , g ). The map F gives a smooth action of G on F(G), which is the restriction to F(G) of the action of G on Aλ+2s . In terms of this group action, F(G) is the orbit of the point 0 ∈ Aλ+2s . The derivative of δF|e 1 C at the identity e ∈ G is δF|e = ∂¯ : 0∗,λ+2s+1 (P1 , gC ) −→ 0,1 λ+2s (P , g ), which is an isomorphism. Therefore the derivative of δF|g is an isomorphism at any point g ∈ G. Compare [3], Lemma 14.6 and (14.7).
3.3.1. The phase of the determinant of δFs . The derivative δFs |g is given by the Fredholm operator Dg := (− + 1)s (∂¯ − ∂¯ gg −1 ) : 0∗,λ+2s+1 (P1 , gC ) −→ λ0,1 (P1 , gC ). Choose a smooth base point g0 ∈ G. Then (Dg0 )−1 Dg : 0∗,λ+2s+1 (P1 , gC ) −→ 0∗,λ+2s+1 (P1 , gC ) can be written as a perturbation of the identity (Dg0 )−1 Dg = 1 + K (g0 , g), where K (g0 , g) = (∂¯ − ∂¯ g0 g0−1 )−1 ◦ M(∂¯ g0 g0−1 − ∂¯ gg −1 ). 1 C Here for ψ ∈ 0,1 λ+2s (P , g ), we have denoted by 1 C M(ψ) : 0∗,λ+2s+1 (P1 , gC ) −→ 0,1 λ+2s (P , g )
the wedge product by ψ, and (∂¯ − ∂¯ g0 g0−1 )−1 is the inverse of the isomorphism 1 C (∂¯ − ∂¯ g0 g0−1 ) : 0∗,λ+2s+1 (P1 , gC ) −→ 0,1 λ+2s (P , g ).
We will show that K (g0 , ·) ∈ S3 (0∗,λ+2s+1 (P1 , gC )) almost everywhere with respect to the measure µs,t G . Since λ < −2, the white noise measure on λ (W ) is supported ∗ t on λ+1 (W ) ⊂ λ (W ). Thus the pullback measure µs,t G = Fs µW is supported on −1 Map∗,λ+2s+2 (P1 , G C ) ⊂ G. It follows that (∂¯ g0 g − ∂¯ gg −1 ) ∈ 0,1 (P1 , gC ) for g 0
λ+2s+1
lying in the complement of a set of measure zero in the measure µs,t G . Then, for almost all g,
ˆ ∂¯ g0 g −1 − ∂¯ gg −1 ), K (g0 , g) = (∂¯ − ∂¯ g0 g0−1 )−1 ◦ j ◦ M( 0 1 C where for ψ ∈ 0,1 λ+2s+1 (P , g ), we have denoted by 1 C ˆ M(ψ) : 0∗,λ+2s+1 (P1 , gC ) −→ 0,1 λ+2s+1 (P , g ) 0,1 1 C (P1 , gC ) −→ 0,1 the wedge product by ψ, and j : λ+2s+1 λ+2s (P , g ) is the inclusion. ˆ Then by Proposition 3.1, || M(ψ)|| ≤ ||ψ||λ+2s+1 , and by the estimate (3.2),
K (g0 , g) ∈ S3 (0∗,λ+2s+1 (P1 , gC ))
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for almost all g. The regularized determinant det3 (1+ K (g0 , g)) is therefore well-defined almost everywhere, and since 1 + K (g0 , g) is invertible, det3 (1 + K (g0 , g)) is nonzero for all g where it is defined. Define (g0 , ·) : G −→ S 1 almost everywhere by (g0 , g) =
det3 (1 + K (g0 , g)) . |det3 (1 + K (g0 , g))|
(3.10)
Remark 3.5. The expression (3.6) for the determinant of a trace-class operator gives a conjecture for the value of the function . Approximate K by a sequence K n of traceclass operators. Since Fs is a holomorphic map, we expect that morally the phase of the determinant det (1 + K n ) will tend to +1 as n −→ ∞. In view of Eq. (3.6) the phase of the regularized determinant det3 (1 + K ) should be given morally by = exp(i limn−→∞ Im tr(K n 2 )), so that we might expect to be the exponential of a (possibly nonlocal) quadratic polynomial in ∂¯ gg −1 . This quantity is known as the anomaly in the physics literature, and often turns out to be well-defined when K is taken as a limit of trace-class operators, even though K is not Hilbert-Schmidt.8 See also Remark 4.3. 3.4. Three-dimensional Gauge Theory. Let M be a compact, smooth Riemannian 3-manifold. Let G be a compact Lie group, and choose an invariant Hermitian inner product on g. Choose λ < −3/2 and s > (−λ + 1/2)/2 and let A = Aλ+2s+1 be the space of connections on the trivialized principal G-bundle on M of Sobolev class λ + 2s + 1; in the notation of Sect. 2, this is given by A = λ+2s+1 (T ∗ M ⊗ g). For A ∈ A denote the curvature of A by FA . After a choice of a base point in M, connections A on M which satisfy the equation FA = 0 correspond by the monodromy representation to representations of π1 (M) in G. Choose such a flat connection A0 . Let iν (M, g) = ν (i (T ∗ M) ⊗ g) be the space of de Rham forms on M with values in g of Sobolev class ν. The space 0λ (M, g) ⊕ 2λ (M, g) is given, in the notation of Sect. 2, by λ (V ) where V = g ⊕ (2 T ∗ M) ⊗ g, and where λ < −3/2. It is therefore equipped with a white noise measure µV . Denote by ∗ the Hodge ∗ operator on differential forms with values in g obtained from the Riemannian pairing on differential forms on M and the hermitian inner product on g. Consider the map Fs,A0 : Aλ+2s+1 ⊕ 3λ+2s+1 (M, g) −→ 0λ (M, g) ⊕ 2λ (M, g) be given by Fs,A0 (A, ξ ) = (− A0 + 1)s (∗d A0 ∗ (A − A0 ) + FA + ∗d A ∗ ξ ).
(3.11)
Theorem 3. The map Fs,A0 is well-defined and differentiable. Its derivative is a Fredholm operator of index 0. Therefore, for each t > 0, the pullback measure ∗ t 0 µs,t,A M,G := Fs,A0 µV
exists and is a σ -finite measure on Aλ+2s+1 ⊕ 3λ+2s+1 (M, g), supported on the set of regular points of Fs,A0 . Suppose that A0 is chosen to be a flat connection that corresponds to an isolated conjugacy class of irreducible representations of π1 (M) in G. Then for t sufficiently large, there exists a neighborhood W of (A0 , 0) such that 0 ∞ > µs,t,A M,G (W ) > 0.
8 The physics literature uses a type of approximation of K by a trace-class operator called dimensional regularization.
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Proof. Again, we can use Proposition 3.1 to show that Fs,A0 is well-defined and differentiable. The derivative δFs,A0 is a compact perturbation of the operator (− A0 +1)s (∗d A0 ∗+d A0 ) : 1λ+2s+1 (M, g)⊕3λ+2s+1 (M, g) −→ 0λ (M, g)⊕2λ (M, g) and hence is Fredholm. Since the dimension of M is odd, this operator has index zero. If A0 is a flat connection corresponding to an isolated conjugacy class of irreducible representations of π1 (M) in G, the kernel of δFs,A0 |(A0 ,0) is zero; it follows that the cokernel is also zero, so that δFs,A0 is an isomorphism at (A0 , 0). As for the Wess-Zumino-Landau-Ginzburg model we have the following conjecture, which may be viewed as giving a stochastic version of the Uhlenbeck compactness theorem for solutions of the Yang-Mills equations. 0 Conjecture 3.6. The measure µs,t,A M,G satisfies 0 µs,t,A M,G (1) < ∞
for t sufficiently large. Suppose A0 is a flat connection that corresponds to an isolated conjugacy class of irreducible representations of π1 (M) in G. Conjecture 3.6 implies that for t suffi0 ciently large, the measure µs,t,A M,G is (up to normalization) a Borel probability measure on Aλ+2s+1 ⊕ 3λ+2s+1 (M, g). 3.4.1. The phase of the determinant of δFs,A0 . The derivative δFs,A0 |(A,ξ ) is given by the Fredholm operator D(A,ξ ) (α, η) = (− A0 + 1)s (∗d A0 ∗ α + d A α + ∗d A ∗ η + ∗[α, ∗ξ ]). Suppose that (A , 0) is a regular point of Fs,A0 . Then D(A ,0) is invertible, and (D(A ,0) )−1 D(A,ξ ) = 1 + K ((A , 0); (A, ξ )), where K ((A , 0); (A, ξ )) : 1λ+2s+1 (M, g) ⊕ 3λ+2s+1 (M, g) −→ 1λ+2s+1 (M, g) ⊕ 3λ+2s+1 (M, g) is given by K ((A , 0); (A, ξ ))(α, η) = (D(A ,0) )−1 (− A0 + 1)s ([A − A , α] + ∗[A − A , ∗η] + ∗[α, ∗ξ ]). A similar computation to that done for the Wess-Zumino-Landau-Ginzburg and WessZumino-Novikov-Witten models shows that K ((A , 0); (A, ξ )) ∈ S4 (1λ+2s+1 (M, g) ⊕ 3λ+2s+1 (M, g)) for all (A, ξ ). Thus the regularized determinant det4 (1 + K ((A , 0); (A, ξ ))) is well defined, and is nonzero at all regular points (A, ξ ) of Fs,A0 . Let U be the set of regular points of Fs,A0 . Define ((A , 0); ·) : U −→ S 1 by ((A , 0); (A, ξ )) =
det4 (1 + K ((A , 0); (A, ξ ))) . |det4 (1 + K ((A , 0); (A, ξ )))|
Then the analog of Conjecture 3.3 is the following.
(3.12)
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Conjecture 3.7. Assume that Conjecture 3.6 holds. Define the partition function 0 t Z st (M, G) by Z st (M, G) := µs,t,A M,G (((A , 0), ·)). Then for t sufficiently large, Z s (M, G) is independent of t, and |Z st (M, G)| is a topological invariant of M. If M is a homology three-sphere and A = A0 is chosen to be a flat connection corresponding to a conjugacy class of isolated irreducible representations of π1 (M) in SU (2), then |Z st (M, SU (2))| = |χ (M)|, where χ (M) is the Casson invariant of M. 0 Morally this conjecture says that for t large, the measure µs,t,A M,G should be concentrated at the transversal solutions of the nonlinear equation Fs,A0 = 0, (or, equivalently, F0,A0 = 0), which correspond to conjugacy classes of isolated irreducible representations of π1 (M) in G. These are the representations which enter into the definition of the Casson invariant (see [1]).
Remark 3.8. As in the case of the Wess-Zumino-Novikov-Witten model, we expect that a nontrivial phase of the determinant arises from the correction terms in the regularized determinant. Since the regularized determinant appearing in (3.12) is det4 , Eq. (3.6) leads us to expect this “anomaly” in (A) to be the exponential of a cubic polynomial in A. It seems reasonable to conjecture that this cubic polynomial is related to the Chern-Simons invariant of A. Remark 3.9. Note that if one attempts to write down the formal analog of the integral of the density (1.1), one obtains a formal expression of the type exp(−S(A))det(δFs,A0 ), (3.13) A
where S(A) = |(− A0 + 1)s FA |2 . If one attempts to interpret the integral (3.13) as a perturbation of a Gaussian measure, writing S(A) = |(− A0 + 1)d A|2 + I (A), a direct computation of Feynman diagrams shows that the expectation of |I (A)|2 in the appropriate Gaussian measure diverges. In the language of the physics literature, the integral (3.13) is not regularized.9 Our method of constructing measures on these spaces therefore differs in an essential way from standard renormalization theory. In those cases where standard renormalization theory would predict the existence of a quantum field theory, it is an important challenge to show that our regularization method will produce the same theory. For the purposes of producing topological invariants from these quantum field theories, one may expect that a very broad range of choices of regularization methods will produce the same invariant.10 9 Naive power counting indicates that the expectation of |I (A)|2 converges. This does not imply convergence since naive power counting is ineffective for theories with derivative interactions. One example is the expectation in the Gaussian measure with covariance (− A0 + 1)−(2s+1) of the square of the cubic vertex | < ((− A0 +1)s d A), ((− +1)s (A2 )) > |2 , which is divergent but for which naive power counting predicts convergence. 10 Readers familiar with the physics literature might be surprised by the appearance of the non-gauge-invariant Lagrangian (3.13) and by the absence of Faddeev-Popov ghost terms. The first problem may be remedied by replacing the map Fs,A0 by the map F˜ s,A0 given by
F˜ s,A0 (A, ξ ) = (− A + 1)s (∗d A0 ∗ (A − A0 ) + F A + ∗d A ∗ ξ ),
(3.14)
which has the same zeros. However, since this map also corresponds to a gauge-fixed theory, there is little to choose between the two maps, and we have preferred the simpler map Fs,A0 . The absence of Faddeev-Popov ghosts is due to the cancellation of the ghost determinants between bosons and fermions in a supersymmetric theory.
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3.5. Gauge theory in two and four dimensions. 3.5.1. Two dimensional gauge theory. Let be a Riemannian two-manifold, let G be a compact Lie group. Choose an invariant Hermitian inner product on g. Let λ < −1, let s > −λ/2, and let Aλ+2s+1 be the space of connections on the trivialized principal G-bundle over of Sobolev class λ + 2s + 1. Let A0 denote the product connection. Let iν (, g) denote the space of de Rham forms on of Sobolev class ν with values in g, and let denote the Hodge star operator on differential forms with values in g obtained from the Riemannian pairing on differential forms and the hermitian inner product on g. Let E be a finite-dimensional vector space of dimension 6g − 6, and let f : Aλ+2s+1 −→ E be a map. Let Fs : Aλ+2s+1 −→ 0λ (, g) ⊕ 2λ (, g) ⊕ E be given by Fs (A) = (− A0 + 1)s (FA ⊕ d A (A − A0 )) ⊕ f.
(3.15)
Again, using Proposition 3.1, Fs is a differentiable map of Banach manifolds, and its derivative is a Fredholm map. The index of this map depends on f ; for f = 0 this index is 6g − 6, so we may expect the index of Fs for generic f to be zero, and so to yield a measure on the space of connections Aλ+2s+1 . The zeros of Fs correspond to flat connections on satisfying conditions given by the map f and so, for appropriate choices of E and f, the resulting measure may be expected to be related to intersection numbers on a moduli space of vector bundles. 3.5.2. Four dimensional gauge theory Let M be a Riemannian four-manifold, let G be a compact Lie group. Let P be a principal G-bundle over M. Choose an invariant Hermitian metric on ad(P). Let λ < −2, let s > (−λ + 1)/2, and let Aλ+2s+1 be the space of connections on P of Sobolev class λ + 2s + 1. Let A0 be a smooth point of Aλ+2s+1 . Let iν (M, ad(P)) denote the space of de Rham forms on M of Sobolev class ν and values in ad(P). Let denote the Hodge star operator on ad(P)valued differential forms and let 2ν,+ (M, ad(P)) denote the space of self-dual de Rham forms of Sobolev class ν. Let E be a finite-dimensional vector space of dimension 8 p1 (P) − 3(b2+ (M) − b1 (M) + 1), and let f : Aλ+2s+1 −→ E be a map. Let Fs : Aλ+2s+1 −→ 0λ (M, ad(P)) ⊕ 2λ,+ (M, ad(P)) ⊕ E be given by Fs (A) = (− A0 + 1)s (FA+ ⊕ d A (A − A0 )) ⊕ f,
(3.16)
where FA+ is the self-dual part of the curvature of A. Again, using Proposition 3.1 one can show that Fs is a differentiable map of Banach manifolds. Its derivative is a Fredholm map, the composition of (− A0 + 1)s with a compact perturbation of the standard elliptic operator d +A + d A : 1λ+2s+1 (M, ad(P)) −→ 0λ+2s (M, ad(P)) ⊕ 2λ+2s,+ (M, ad(P)). Again, for generic f, we expect the index of δFs to be zero, and hence to yield a measure on the space of connections Aλ+2s+1 . The zeros of Fs correspond to self-dual connections on M satisfying conditions given by the map f and so, for appropriate choices of E and f, the resulting measure may be expected to be related to Donaldson invariants of M. Remark 3.10. The formal analogy between the path integrals appearing in the physics literature on gauge theory and the Mathai-Quillen formula was studied in [4].
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Remark 3.11. There are other possible choices of the map Fs in this example. One alternative to the definition (3.16) is F˜s (A) = (− A + 1)s (FA+ ⊕ d A (A − A0 )) ⊕ f.
(3.17)
This choice makes contact with the work of [4] as follows. Write A for the space of connections on P and G = Aut (P). In the spirit of [4], the formal path integrals of [47] morally compute the Euler number of the vector bundle11 2 V := + (ad(P)) ×G A −→ A/G by a formula involving the section A −→ FA+ , corresponding to the map F˜0 . The same computation applied to the section A −→ (− A + 1)s (FA+ ), corresponding to the map F˜s , should morally give the same result, as it is a section of the same vector bundle. In algebraic geometry there is the familiar idea of computing the number of zeros of a section of a vector bundle by finding a different section of the same vector bundle where the computation is simpler. In our setting a different choice of section takes us from the realm of formal path integrals to that of well-defined measures! It is possible to give a construction similar to the one we have given in this paper for measures corresponding to sections of the vector bundle V −→ A/G rather than for maps of Banach spaces. For this purpose the map F˜s (rather than Fs ) is crucial, since it gives rise to a section of the appropriate vector bundle. Similar constructions exist for gauge theory in dimensions 2 and 3, and indeed in higher dimensions. Details will appear elsewhere.
3.6. Quantum cohomology. Let be a Kähler two-manifold, and let M be a Kähler manifold; denote by T and T M the holomorphic tangent bundles of and M, respectively. Let λ < −1 and s > (−λ+1)/2. Let Mapλ+2s+1 (, M) be the space of maps from to M of Sobolev class λ + 2s + 1. Let V be the vector bundle over Mapλ+2s+1 (, M) whose fibre at φ ∈ Mapλ+2s+1 (, M) is the space λ (φ ∗ T ∗ M ⊗ T ) of sections of φ ∗ T ∗ M ⊗ T of Sobolev class λ. Let Fs be the section of V be given by ¯ Fs (φ) = (− + 1)s ∂φ. In a sufficiently small neighborhood of any map φ, the vector bundle V can be trivialized as a space of maps into a vector space. The section Fs can then be used to push forward the white noise measure on these vector spaces to a measure on Mapλ+2s+1 (, M). The analog of the Mathai-Quillen formula (1.6) then differs from this measure by the usual phase of the determinant as well as by an infinite series analogous to the series Ck appearing in (1.6); this infinite series can be shown to converge. Details will appear elsewhere. Remark 3.12. The analogy between path integrals associated to quantum cohomology in the physics literature and the Mathai-Quillen formula associated formally to F0 was studied in [50]. 4. The Wess-Zumino-Novikov-Witten Models on P1 In this section we make a more detailed study of the measure µs,t G constructed in s,t Theorem 2. First we prove that µG is a Borel probability measure on G. Since µs,t G (G) = 1, 11 We follow [4] and ignore the locus where the action of G is not free.
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there is no analog for µs,t G of Conjecture 3.3 or Conjecture 3.7. Instead we compute more subtle quantities given by expectation values of functions in the measure µs,t G . We show that these expectation values give formulas which agree with formulas discovered by Frenkel and Zhu [13] in the context of the theory of vertex operator algebras. 4.1. The measure µs,t G is a probability measure. Recall that we are working with a subgroup G C ⊂ S L(n, C), and this identification induces a hermitian inner product on gC = Lie(G C ). Choose λ < −2 and s > (−λ+1)/2. Recall that G := Map∗,λ+2s+1 (P1,G C ) is the space of based maps from P1 to G C of Sobolev class λ + 2s + 1. Let G˜ := Mapλ+2s+1 (P1 , G C ) denote the space of all maps from P1 to G C of Sobolev class λ + 2s + 1; like G, this is a Lie group with the group structure given by pointwise 1 C multiplication. Let B := 0,1 λ (P , g ) be the space of anti-holomorphic one forms on 1 C ˜ P1 of Sobolev class λ, and let Aλ+2s := 0,1 λ+2s (P , g ). The group G acts on Aλ+2s by the gauge action g · A = g Ag −1 + ∂¯ gg −1 ˜ A ∈ Aλ+2s . Recall that B = λ (W ), where W = T ∗ P1 ⊗ gC , and that B is for g ∈ G, therefore equipped with a white noise measure µW . Recall that the map Fs : G −→ B is given by Fs (g) = (− + 1)s ∂¯ gg −1 . The map Fs is the composite of the map F : G −→ Aλ+2s given by F(g) = ∂¯ gg −1 with the isomorphism (− + 1)s : Aλ+2s −→ B. Since F : G −→ Aλ+2s gives the orbit of 0 ∈ Aλ+2s ∈ A, and since G acts freely on Aλ+2s , Fs is injective. To determine the complement of the image of Fs , we make use of the stratification of the space Aλ+2s given in Atiyah-Bott [3], referring to the work of Shatz [38], Narasimhan-Seshadri [28], and Harder-Narasimhan [20]. See also [19, 2, 9, 8, 33]. Proposition 4.1. The image of Fs consists of the complement of a countable union of smooth, locally closed subvarieties Vi of B. Each of these subvarieties is of the form Vi = (− + 1)s G˜ · Ai , where Ai is a smooth element of 0,1 (P1 , gC ). The subvariety Vi has positive codimension in B, and the normals to Vi at (− + 1)s Ai are smooth. Proof. Since the map (− + 1)s : Aλ+2s −→ B is an isomorphism, it is enough to prove an analogous result for the image of the map F : G −→ Aλ+2s . This the orbit of ˜ the point 0 ∈ Aλ+2s under the action of G. The results of [3] show that the space Aλ+2s can be stratified as Aλ+2s = µ A(µ) , where the A(µ) are a countable family of G˜ orbits which are smooth locally closed subvarieties of Aλ+2s , each of finite codimension in Aλ+2s . On P1 these strata are given ˜ by G-orbits of a countable collection of elements Aµ ∈ Aλ+2s . (See [3], Sect. 14, pp. 608–609, [8], or [33], Theorem 2 and its Corollary). ˜ The G-orbit of the trivial connection A0 corresponding under our identification to 0,1 1 0 ∈ λ+2s (P , gC ) is the image of the map F. We denote the corresponding stratum by A(0) . Each stratum A(µ) contains a smooth point by Lemma 14.8 of [3]. The normals to A(µ) at a smooth point A ∈ A(µ) are given by the kernel of the adjoint D ∗A of the elliptic operator D A := ∂¯ + A : 0 (P1 , gC ) −→ 0,1 (P1 , gC ), and are therefore smooth (see [3], Sect. 14, p. 608). The kernel of D ∗A at smooth points was computed in [3], where
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it is shown to be zero for the stratum A(0) and nonzero for every other stratum; see Proposition 5.4, (5.10), (7.15), and (10.7) of [3]. In the Yang-Mills context each of these strata is the orbit under the complexified gauge group of a solution of the Yang-Mills equation. The minimum of the Yang-Mills functional, given by the trivial connection, has the open orbit F(G). We therefore obtain the following result. Proposition 4.2. The map Fs is a diffeomorphism of G with the complement of a set of measure zero with respect to the white noise measure µtW on B for any t. Proof. Since the map (− + 1)s : Aλ+2s −→ B is an isomorphism, it suffices to show that the subvariety Z i ⊂ Aλ+2s given by Z i = G˜ · Ai has measure zero with respect to 1 C the Gaussian measure µtW,s on the space Aλ+2s = 0,1 λ+2s (P , g ). Let a ∈ Z i . By the inverse function theorem, we can find > 0 such that the orthogonal projection B (a) ∩ Z i −→ T Z i |q is a diffeomorphism for all q ∈ B (a).12 ˜ Ai , the point a is given by a = g· Ai for some g ∈ G. ˜ Let U = B/2 (a). Since Z i = G· ˜ ˜ Let g ∈ G be a smooth element of G sufficiently close to g so that g · Ai ∈ U ∩ Z i . Then p := g · Ai is smooth. The choice of is such that the orthogonal projection B (a) ∩ Z i −→ T Z i | p is a diffeomorphism. Let ν ∈ T Z i |⊥ p . Then ν is smooth, and there exists α > 0 so that if δ, δ ∈ (0, α) and δ = δ , then (U ∩ Z i + δν) ∩ (U ∩ Z i + δ ν) = ∅. Suppose that µtW,s (U ∩ Z i ) = η > 0. By the Cameron-Martin formula (Proposition 2.1), the Radon-Nikodym derivative d(Tcν )∗ µtW,s /dµtW,s is bounded from below for q ∈ U and c > 0 by d(Tcν )∗ µtW,s dµtW,s
1 (q) = exp(ct 2 < (− + 1)2s ν, q > − c2 t 2 ||ν||22s ) 2 1 ≥ exp(−ct 2 ||ν||−λ+2s ||q||λ+2s − c2 t 2 ||ν||22s ). 2
Since for q ∈ U, ||q||λ+2s < ||a||λ+2s + , there exists β > 0 such that for all q ∈ U and all c ∈ (0, β), d(Tcν )∗ µtW,s dµtW,s
(q) >
1 . 2
Then if c ∈ (0, β), µtW,s (U ∩ Z i + cν) >
1 η. 2
12 Given a subspace T ⊂ A λ+2s , denote by πT : Aλ+2s −→ T the orthogonal projection onto T. Define f : Z i × Z i −→ T Z i |a × T Z i |a by
f ( p, q) = (πT Z i |a ◦ πT Z i |q p, πT Z i |a q). Then δ f (a,a) is an isomorphism. By the inverse function theorem, f is a diffeomorphism in a neighborhood of (a, a). It follows that the orthogonal projection πT Z i |q | Z i is a diffeomorphism near a for q near a.
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Let γ = min {α, β}. Then µtW,s (
∞
(Z i ∩ U +
n=1
γ ν)) = ∞, 2n
which is impossible. So it must be that µtW,s (U ∩ Z i ) = 0. Thus every point p ∈ Z i has a neighborhood in Z i of measure zero. Since the space Aλ+2s has a countable basis for its topology, so does the subspace Z i . Thus Z i is Lindelöf, and can be covered by countably many open sets of measure zero. It follows that µtW,s (Z i ) = 0. As a corollary we get the following theorem. Theorem 4. The measure µs,t G is a Borel probability measure on G for all t. 4.2. The Frenkel-Zhu formula. We now focus on the case G = SU (n), so that G C = S L(n, C). To compare expectations in the measure µs,t G to results arising from the theory of vertex operator algebras, we introduce some functions on G. Recall that 1 C ∗ 1 C 1 0,1 λ+2s (P , g ) = λ+2s (W ), where W = T P ⊗g . We have equipped P with a Kahler metric, and the Lie algebra gC comes with an inner product given by (a, b) −→ tr a ∗ b. Let us denote by : 0,1 (P1 ) −→ 1,0 (P1 ) the Hodge star operator. Also, let Cz : T ∗ P1 |z ⊗ (T ∗ P1 |z )∗ −→ C denote the contraction. The inner product on gC along with the Riemannian structure on P1 give rise to a pairing 0,1 1 C 1 C 1 , : 0,1 λ+2s (P , g ) × λ+2s (P , g ) −→ Hλ+2s (P );
here we have used Proposition 3.1 to see that the inner product of elements in 1 C 1 0,1 λ+2s (P , g ) is in Hλ+2s (P ). Similarly if we are given any hermitian matrix x we obtain from the pairing on gC given by (a, b) −→ tr a ∗ xb a pairing 0,1 1 C 1 C 1 , x : 0,1 λ+2s (P , g ) × λ+2s (P , g ) −→ Hλ+2s (P ).
Let x ∈ g be a hermitian matrix and let z ∈ P1 . We define a function Fˆx,z : G −→ R by Fˆx,z (g) = ∂¯ gg −1 , ∂¯ gg −1 x (z).
(4.1)
We also define Fx,z : G −→ R by ˆ Fx,z := Fˆx,z − µs,1 G ( Fx,z ). Similarly, let v ∈ 0,1 (P1 , gC ), let z ∈ P1 , and define f v,z : G −→ C by f v,z (g) = v, ∂¯ gg −1 (z). We recall some definitions from [13]. Given a finite set A, let P1 (A) denote the set of partitions of A into cycles and chains containing precisely one chain. Now let
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A = {1, . . . , n}. Given α ∈ P1 (A), let us write α explicitly as a collection of k cycles and one chain of length m: α = {(a(1, 1), . . . , a(1, j1 )), . . . , (a(k, 1), . . . , a(k, jk )), [b(1), . . . , b(m)]}. Let Cs (z, z ) denote the Green’s kernel of the self-adjoint operator (− + 1)2s on 1 ∗ 1 ∗ ∗ 1 0,1 0 (P ). Then C s (z, z ) ∈ (T P |z ) ⊗ T P |z . Let v, v ∈ 0,1 (P1 , gC ). We may write v = φξ, v = φ ξ , where φ, φ ∈ ∞ C (P1 , gC ) and ξ, ξ ∈ 0,1 (P1 ). Let x1 , . . . , xn ∈ g, and let z 1 , . . . , z n , z, w ∈ P1 . We define f v,v ,α;s (z, w; z 1 , . . . , z n ) =
k
tr(xa( p,1) . . . xa( p, j p ) )×tr(φ(z)∗ xb(1) . . . xb(m) φ (w))×
p=1 k
Cza( p,1) . . . Cza( p, j p ) (Cs (z a( p,1) , z a( p,2) ) ⊗ · · · ⊗ Cs (z a( p, j p ) , z a( p,1) )) ×
p=1
Cz Cz b(1) . . . Cz b(m) Cw ((ξ )(z) ⊗ Cs (z, z b(1) ) ⊗ Cs (z b(1) , z b(2) ) ⊗ . . . Cs (z b(m−1) , z b(m) ) ⊗ Cs (z b(m) , w) ⊗ ξ (w)). Then we have the following result. Theorem 5. Let x1 , . . . , xn ∈ g and let z 1 , . . . , z n , z, w ∈ P1 . Let v, v ∈ 0,1 (P1 , gC ). Then ¯ µs,1 f v,v ,α;s (z, w; z 1 , . . . , z n ). (4.2) G ( f v,z f v ,w Fx1 ,z 1 . . . Fxn ,z n ) = α∈P1 (A)
Proof. By Theorem 4.2, we may transfer the computation to the space B, equipped with its white noise measure µW , using the map Fs . For any x ∈ g, z ∈ P1 , we have Fx,z = Fs∗ Hx,z , where Hx,z : G −→ C is given by Hx,z := Hˆ x,z − µW ( Hˆ x,z ), and where Hˆ x,z : B −→ C is given by Hˆ x,z (A) = (− + 1)−s A, (− + 1)−s A x (z)
(4.3)
for A ∈ B. Similarly, if v ∈ 0,1 (P1 , gC ) and z ∈ P1 , f v,z = Fs∗ h v,z , where h v,z : B −→ C is given by h v,z (A) = v, (− + 1)−s A (z). Thus ¯ ¯ µs,1 G ( f v,z f v ,w Fx1 ,z 1 . . . Fxn ,z n ) = µW (h v,z h v ,w Hx1 ,z 1 . . . Hxn ,z n ). The expectation µW (h¯ v,z h v ,w Hx1 ,z 1 . . . Hxn ,z n ), like that of any polynomial, can 0,1 1 C be computed by using formula (2.1). Let φi ∈ −λ (P , g ), i = 1, . . . , 2k. Let t1 , . . . , t2k ∈ C and let ζ := i ti φi . For i = 1, . . . , 2k, let L φi : B −→ C be given by L φi (A) =< φi , A > . Applying formula (2.1) to E ζ gives a generating function µW (E ζ ) for the expectations of all polynomials in the L φi . Explicitly, we have Wick’s
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theorem: Let Q2k be the set of partitions of the finite set {1, . . . , 2k} into a collection of pairs (a, b), where a ∈ {1, . . . , k} and b ∈ {k + 1, . . . , 2k}. Then µW (
k i=1
L φi
2k i=k+1
L¯ φi ) =
< φa , φb > .
(4.4)
q∈Q2k (a,b)∈q
Applying Eq. (4.4) to the polynomial h¯ v,z h v ,w Hx1 ,z 1 . . . Hxn ,z n gives (4.2).
The formula on the right-hand side of Eq. (4.2), when evaluated at s = 0, is (up to a constant) the formula given in Theorem 2.3.1 of Frenkel and Zhu [13], who work with the theory of vertex operator algebras. In our context we have obtained a type of analytic continuation of this formula from a commutative algebra of functions on G. Remark 4.3. The formal path integral corresponding to the model we have described is given by an action whose Bosonic part is S BW Z N W (g) = |∂¯ g g −1 |2 , P1
where g is a map from P1 to S L(n, C). The Wess-Zumino-Novikov-Witten model as it appears in the physics literature differs from the one given by this formal path integral (arising by formally applying the formula (1.1) to the map F0 ) in two ways. First, the maps appearing in the Wess-Zumino-Novikov-Witten path integral in the physics literature are maps with values in the compact Lie group G (rather than G C ); second, the partition function is given by the integral with respect to this measure of a power of a certain function . Regarding the first issue, the work of Wendt [44] shows that the quantization of the loop group LG formally corresponds to a path integral over a space of maps into G C , so that a path integral over maps into G C may be the correct model for the study of the quantization of LG. Another way of seeing this is to repeat our construction for manifolds with boundary, using the results of Donaldson [10] to substitute for Proposition 4.1. Details will appear elsewhere. We now discuss the role of the function . Recall that the analogy with finite dimensions leads us to believe that one should compute expectations in the measure µs,1 G (·), where is the phase of the regularized determinant calculated in (3.10). In fact the physics literature refers to a calculation of an expectation of the type µ(k ·), where k ∈ Z, µ is morally a measure on the space Gu := Map∗ (P1 , G), and the function i 1 : Gu −→ S is given by = exp( 12π B tr (γ −1 dγ )3 ), where B is a three-manifold with ∂ B = P1 , and γ is an arbitrary extension of g to B. The function can be interpreted as a flat section of the restriction to the orbit Gu · 0 ⊂ Aλ+2s of a hermitian line bundle with connection L −→ Aλ+2s [35] on the space Aλ+2s . This section cannot be extended to G · 0 as a flat section since L is not flat on G · 0. However, since the space Aλ+2s is affine, one can define a function ˆ : Aλ+2s −→ S 1 by choosing a base point A0 ∈ Aλ+2s and letting (A) ˆ be the holonomy of the connection on L along the straight line path from A0 to A. Now the determinant line bundle [32] on Aλ+2s is given by Lh , where h is the dual Coxeter number of G, and the curvature of this line bundle is given [32] by a quadratic Kähler ˆ is the exponential of potential. This means it is reasonable to expect that the function the pull back to G of a quadratic function on Aλ+2s . In this picture the phase of the ˆ h of the determinant determinant appearing in (3.10) should be replaced by the section
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line bundle Lh , which arises from a method of regularizing determinants different from that of [39]. A result similar to that of Theorem 5 holds also if the measure µs,1 G is s,1 ˆ k+h ˆ = exp(i Q(∂¯ gg −1 )) replaced by the complex valued measure µG ( ·), where (g) and Q is any quadratic function on Aλ+2s . It is reasonable to expect that the function : Aλ+2s −→ S 1 defined in Eq. (3.10) is also a section of Lh . Note that if this is the case, the ideas of Remark 3.5 would again lead us to believe that should be of the form (g) = exp(i Q(∂¯ gg −1 )), where Q is a (possibly nonlocal) quadratic function on Aλ+2s . 5. Conclusions 5.1. Analytic continuation. The measures constructed on Banach spaces in the Wess-Zumino-Novikov-Witten model give analytic continuations of correlation functions arising in the associated theory of vertex operator algebras. This is reminiscent of the case of correlation functions in Minkowskian quantum field theory, which arise as analytic continuations of Euclidean correlation functions which can be computed by Euclidean path integrals.
5.2. Higher derivatives. The path integrals in the examples studied above correspond to partial differential equations of higher order than those arising in the problems studied by the physicists. One may wonder if Lagrangians with higher derivatives may be useful as models in physics.
5.3. Supersymmetry. Our construction uses supersymmetry in an essential way. From this point of view the supersymmetric models are more fundamental than the bosonic models; indeed one could try to construct the bosonic model by showing that the inverse of an appropriate determinant is integrable with respect to the supersymmetric measure.
5.4. Quantum field measures are not perturbations of Gaussian measures. The measures appearing in our work are not in general perturbations of Gaussian measures; they are locally pushforwards of such measures. In particular, factors of the type (− + 1)s , though they resemble dimensional regularization, may not suffice to regularize the theory. In effect the singularities of quantum field theory fall into two categories: Singularities arising from attempting to define nonlinear functions on spaces of distributions, which disappear when higher-order partial differential equations are considered, and singularities arising from the fact that a pullback measure may not be absolutely continuous with respect to Gaussian measure (as the Cameron-Martin theorem shows is the case for translations), which must be treated non-perturbatively (see Remark 3.9). The physics literature makes no distinction between these two types of singularities. Therefore the predictions of standard renormalization theory, that gauge theory in dimensions higher than four should not exist, and that sigma-models should exist only for manifolds of positive curvature, may not apply to supersymmetric models constructed by our methods. In fact gauge theory in six dimensions [36, 11, 43] and eight dimensions [6] has been studied in the physics literature, and corresponds to mathematical results, as is the case for quantum cohomology for manifolds which do not have positive curvature.
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5.5. Anomalies. The physics literature assigns a special role to the phase of the determinant of the tangential operator δFs . The invariants of maps in the finite dimensional case involve the phase of the determinant of the tangential operator. But in infinite dimensions, it is only the regularized determinant that can be defined. The construction of this determinant involves arbitrary choices (in our case, a choice of a base point) and its phase may therefore not lead to a topological invariant, or else fail to have the same symmetries as the map Fs . In the physics literature the appearance of such an “anomaly” is held to preclude the existence of a well-defined theory. The fact that we obtain a canonical construction of a measure but that the construction of the phase of the determinant involved arbitrary choices is in line with physicists’ intuition, which singled out the importance of these anomaly terms.
5.6. Noncommutative and commutative algebras. Comparison with the Frenkel-Zhu formula shows that a result arising from a noncommutative vertex operator algebra can be viewed in terms of the path integral as coming from a commutative algebra. The original Feynman-Kac formula is another reflection of this principle. It may be hoped that a better understanding of the role of path integrals may be helpful in interpreting algebraic structures arising in mathematics motivated by quantum field theory. For example, a construction of the Wess-Zumino-Novikov-Witten model in genus one should give a proof of the Kac character formula [23] as adumbrated in Wendt [44].
5.7. Equivariance. In the presence of a group action it should be possible to define equivariant versions of the measures Fs ∗ µ; this will involve a convergent infinite series of the type appearing in (1.6). The equivariant integral should satisfy a localization formula as in finite dimensions. In the case of the Wess-Zumino-Novikov-Witten model the equivariant extension was written down by Wendt [44]; it is closely related to the coset models studied by Gawedzki and Kupiainen [14].
5.8. Interpretation of the partition function. It would be interesting to have a geometric interpretation of finite dimensional analogs of the partition functions Fs∗ µ( k ) analogous to the interpretation of Fs∗ µ() as a degree in formula (1.2). References 1. Akbulut, S., McCarthy, J.: Casson’s Invariant for Oriented Homology 3-Spheres: An Exposition. Princeton, NJ: Princeton University Press, 1990 2. Atiyah, M.: Proc. Lond. Math. Soc. 7, 414–452 (1957) 3. Atiyah, M., Bott, R.: Philos. Trans. Roy. Soc. London Ser. A 308(1505), 523–615 (1983) 4. Atiyah, M., Jeffrey, L.: J. Geom. Phys. 7, 119–136 (1990) 5. Atiyah, M., Singer, I.: Proc. Nat. Acad. Sci. USA 81, 2597–2600 (1984) 6. Baulieu, L., Kanno, H., Singer, I.: Commun. Math. Phys. 194, 149–175 (1998) 7. Berezin, F.A.: Introduction to superanalysis. Dordrecht: D. Reidel Publishing Co., 1987 8. Daskalopoulos, G.: J. Differ. Geom. 36(3), 699–746 (1992) 9. Donaldson, S.: J. Differ. Geom. 18(2), 269–277 (1983) 10. Donaldson, S.: J. Geom. Phys. 8, 89–122 (1992) 11. Donaldson, S., Thomas, R.: Gauge theory in higher dimensions. In: The geometric universe (Oxford, 1996), Oxford: Oxford Univ. Press 1998, pp. 31–47 12. Dunford, N., Schwartz, J.: Linear Operators, Part II: Spectral Theory. New York: Interscience, 1963 13. Frenkel, I., Zhu, Y.: Duke Math. J. 66, 123–168 (1992)
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Communicated by J.Z. Imbrie
Commun. Math. Phys. 277, 127–160 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0357-5
Communications in
Mathematical Physics
Optimal Estimation of Qubit States with Continuous Time Measurements M˘ad˘alin Gu¸ta˘ 1 , Bas Janssens1 , Jonas Kahn2 1 University of Nottingham, School of Mathematical Sciences, University Park,
Nottingham NG7 2RD, UK. E-mail: [email protected]
2 Université Paris-Sud 11, Département de Mathématiques, Bât 425, 91405 Orsay Cedex, France
Received: 7 November 2006 / Accepted: 16 May 2007 Published online: 27 October 2007 – © Springer-Verlag 2007
Abstract: We propose an adaptive, two step strategy, for the estimation of mixed qubit states. We show that the strategy is optimal in a local minimax sense for the trace norm distance as well as other locally quadratic figures of merit. Local minimax optimality means that given n identical qubits, there exists no estimator which can perform better than the proposed estimator on a neighborhood of size n −1/2 of an arbitrary state. In particular, it is asymptotically Bayesian optimal for a large class of prior distributions. We present a physical implementation of the optimal estimation strategy based on continuous time measurements in a field that couples with the qubits. The crucial ingredient of the result is the concept of local asymptotic normality (or LAN) for qubits. This means that, for large n, the statistical model described by n identically prepared qubits is locally equivalent to a model with only a classical Gaussian distribution and a Gaussian state of a quantum harmonic oscillator. The term ‘local’ refers to a shrinking neighborhood around a fixed state ρ0 . An essential result is that the neighborhood radius can be chosen arbitrarily close to n −1/4 . This allows us to use a two step procedure by which we first localize the state within a smaller neighborhood of radius n −1/2+ , and then use LAN to perform optimal estimation. 1. Introduction State estimation is a central topic in quantum statistical inference [6,28,31,32]. In broad terms the problem can be formulated as follows: given a quantum system prepared in an unknown state ρ, one would like to reconstruct the state by performing a measurement M whose random result X will be used to build an estimator ρ(X ˆ ) of ρ. The quality of the measurement-estimator pair is given by the risk Rρ (M, ρ) (1.1) ˆ = E d(ρ(X ˆ ), ρ)2 , where d is a distance on the space of states, for instance the fidelity distance or the trace norm, and the expectation is taken with respect to the probability distribution PρM
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of X , when the measured system is in state ρ. Since the risk depends on the unknown state ρ, one considers a global figure of merit by either averaging with respect to a prior distribution π (Bayesian setup) ˆ = π(dρ)Rρ (M, ρ), ˆ (1.2) Rπ (M, ρ) or by considering a maximum risk (pointwise or minimax setup) Rmax (M, ρ) ˆ = sup Rρ (M, ρ). ˆ ρ
(1.3)
An optimal procedure in either setup is one which achieves the minimum risk. Typically, one measurement result does not provide enough information in order to significantly narrow down on the true state ρ. Moreover, if the measurement is “informative” then the state of the system after the measurement will contain little or no information about the initial state [35] and one needs to repeat the preparation and measurement procedure in order to estimate the state with the desired accuracy. It is then natural to consider a framework in which we are given a number n of identically prepared systems and look for estimators ρˆn which are optimal, or become optimal in the limit of large n. This problem is the quantum analogue of the classical statistical problem [49] of estimating a parameter θ from independent identically distributed random variables X 1 , . . . , X n with distribution Pθ , and some of the methods developed in this paper are inspired by the classical theory. Various state estimation problems have been investigated in the literature and the techniques may be quite different depending on a number of factors: the dimension of the density matrix, the number of unknown parameters, the purity of the states, and the complexity of measurements over which one optimizes. A short discussion on these issues can be found in Sect. 2. In this paper we give an asymptotically optimal measurement strategy for qubit states that is based on the technique of local asymptotic normality introduced in [22,23]. The technique is a quantum generalisation of Le Cam’s classical statistical result [40], and builds on previous work of Hayashi and Matsumoto [25,29]. We use an adaptive two stage procedure involving continuous time measurements, which could in principle be implemented in practice. The idea of adaptive estimation methods, which has a long history in classical statistics, was introduced in the quantum set-up by [7], and was subsequently used in [21,26,30]. The aim there is similar: one wants to first localize the state and then to perform a suitably tailored measurement which performs optimally around a given state. A different adaptive technique was proposed independently by Nagaoka [46] and further developed in [16]. In the first stage, the spin components σx , σ y and σz are measured separately on a small portion n˜ n of the systems, and a rough estimator ρ˜n is constructed. By standard statistical arguments (see Lemma 2.1) we deduce that with high probability, the true state ρ lies within a ball of radius slightly larger than n −1/2 , say n −1/2+ with > 0, centered at ρ˜n . The purpose of the first stage is thus to localize the state within a small neighborhood as illustrated in Fig. 1 (up to a unitary rotation) using the Bloch sphere representation of qubit states. This information is then used in the second stage, which is a joint measurement on the remaining n − n˜ systems. This second measurement is implemented physically by two consecutive couplings, each to a bosonic field. The qubits are first coupled to the field via a spontaneous emission interaction and a continuous time heterodyne detection
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Fig. 1. After the first measurement stage the state ρ lies in a small ball centered at ρ˜n
measurement is performed in the field. This yields information on the eigenvectors of ρ. Then the interaction is changed, and a continuous time homodyne detection is performed in the field. This yields information on the eigenvalues of ρ. We prove that the second stage of the measurement is asymptotically optimal for all states in a ball of radius n −1/2+η around ρ˜n . Here η can be chosen to be bigger that n > 0 implying that the two stage procedure as a whole is asymptotically optimal for any state as depicted in Fig. 2. The optimality of the second stage relies heavily on the principle of local asymptotic normality or LAN, see [49], which we will briefly explain below, and in particular on the fact that it holds in a ball of radius n −1/2+η around ρ˜n rather than just n −1/2 as it was the case in [22]. Let ρ0 be a fixed state. We parametrize the neighboring states as ρu/√n , where u = (u x , u y , u z ) ∈ R3 is a certain set of local parameters around ρ0 . Then LAN entails that ⊗n√ of n identical qubits converges for n → ∞ to a Gaussian the joint state ρnu := ρu/ n u u state of the form N ⊗ φ , in a sense explained in Theorem 3.1. By N u we denote a classical one-dimensional normal distribution centered at u z . The second term φ u is a Gaussian state of a harmonic oscillator, i.e. a displaced thermal equilibrium state with
Fig. 2. The smaller domain is the localization region of the first step. The second stage estimator is optimal for all states in the bigger domain
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displacement proportional to (u x , u y ). We thus have the convergence ρnu ; N u ⊗ φ u , to a much simpler family of classical – quantum states for which we know how to optimally estimate the parameter u [32,55]. The idea of approximating a sequence of statistical experiments by a Gaussian one goes back to Wald [53], and was subsequently developed by Le Cam [40] who coined the term local asymptotic normality. In quantum statistics the first ideas in the direction of local asymptotic normality for d-dimensional states appeared in the Japanese paper [27], as well as [25] and were subsequently developed in [29]. In Theorem 3.1 we strengthen these results for the case of qubits, by proving a strong version of LAN in the spirit of Le Cam’s pioneering work. We then exploit this result to prove optimality of the second stage. A different approach to local asymptotic normality has been developed in [23] to which we refer for a more general exposition on the theory of quantum statistical models. A short discussion on the relation between the two approaches is given in the remark following Theorem 3.1. From the physics perspective, our results put on a more rigorous basis the treatment of collective states of many identical spins, the keyword here being coherent spin states [33]. Indeed, it has been known since Dyson [13] that n spin- 21 particles prepared in the spin up state |↑⊗n behave asymptotically as the ground state of a quantum oscillator, when considering the fluctuations of properly normalized total spin components in the directions orthogonal to z. We extend this to spin directions making an “angle” of order n −1/2+η with the z axis, as illustrated in Fig. 3, as well as to mixed states. We believe that a similar approach can be followed in the case of spin squeezed states and continuous time measurements with feedback control [19]. In Theorem 4.1 we prove a dynamical version of LAN. The trajectory in time of the joint state of the qubits together with the field converges for large n to the corresponding trajectory of the joint state of the oscillator and field. In other words, time evolution preserves local asymptotic normality. This insures that for large n the state of the qubits
√
n
Fig. 3. Total spin representation of the state of n 1 spins: the quantum fluctuations of the x and y spin directions coincide with those of a coherent state of a harmonic oscillator
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“leaks” into a Gaussian state of the field, providing a concrete implementation of the convergence to the limit Gaussian experiment. The punch line of the paper is Theorem 6.1 which says that the estimator ρˆn is optimal in local minimax sense, which is the modern statistical formulation of optimality in the frequentist setup [49]. Also, its asymptotic risk is calculated explicitly. The paper is structured as follows: in Sect. 2, we show that the first stage of the measurement sufficiently localizes the state. In Sect. 3, we prove that LAN holds with radius of validity n −1/2+η , and we bound its rate of convergence. Sections 4 and 5 are concerned with the second stage of the measurement, i.e. with the coupling to the bosonic field and the continuous time field-measurements. Finally, in Sect. 6, asymptotic optimality of the estimation scheme is proven. The technical details of the proofs are relegated to the appendices in order to give the reader a more direct access to the ideas and results. 2. State Estimation In this section we introduce the reader to a few general aspects of quantum state estimation after which we concentrate on the qubit case. State estimation is a generic name for a variety of results which may be classified according to the dimension of the parameter space, the kind or family of states to be estimated and the preferred estimation method. For an introduction to quantum statistical inference we refer to the books by Helstrom [31] and Holevo [32] and the more recent review paper [6]. The collection [28] is a good reference on quantum statistical problems, with many important contributions by the Japanese school. For the purpose of this paper, any quantum state representing a particular preparation of a quantum system, is described by a density matrix (positive selfadjoint operator of trace one) on the Hilbert space H associated to the system. The algebra of observables is B(H), and the expectation of an observable a ∈ B(H) with respect to the state ρ is Tr(ρa). A measurement M with outcomes in a measure space (X , ) is completely determined by a σ -additive collection of positive selfadjoint operators M(A) on H, where A is an event in . This collection is called a positive operator valued measure. The distribution of the results X when the system is in state ρ is given by Pρ (A) = Tr(ρ M(A)). We are given n systems identically prepared in state ρ and we are allowed to perform a measurement Mn whose outcome is the estimator ρˆn as discussed in the Introduction. The dimension of the density matrix may be finite, such as in the case of qubits or d-levels atoms, or infinite as in the case of the state of a monochromatic beam of light. In the finite or parametric case one expects that the risk converges to zero as n −1 and the optimal measurement-estimator sequence (Mn , ρˆn ) achieves the best constant in front of the n −1 factor. In the non-parametric case the rates of convergence are in general slower than n −1 because one has to simultaneously estimate an infinite number of matrix elements, each with rate n −1 . An important example of such an estimation technique is that of quantum homodyne tomography in quantum optics [52]. This allows the estimation with arbitrary precision [12,41,42] of the whole density matrix of a monochromatic beam of light by repeatedly measuring a sufficiently large number of identically prepared beams [47,48,56]. In [1,9] it is shown how to formulate the problem of estimating infinite dimensional states without the need for choosing a cut-off in the dimension of the density matrix, and how to construct optimal minimax estimators of the Wigner function for a class of “smooth” states.
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If we have some prior knowledge about the preparation procedure, we may encode this by parametrizing the possible states as ρ = ρθ with θ ∈ some unknown parameter. The problem is then to estimate θ optimally with respect to a distance function on . Indeed, one of the main problems in the finite dimensional case is to find optimal estimation procedures for a given family of states. It is known that if the state ρ is pure or belongs to a one parameter family, then separate measurements achieve the optimal rate of the class of joint measurements [45]. However for multi-dimensional families of mixed states this is no longer the case and joint measurements perform strictly better than separate ones [21]. In the Bayesian setup, one optimizes Rπ (Mn , ρˆn ) for some prior distribution π . We refer to [2,5,14,15,24,36,39,44] for the pure state case, and to [3,4,11,38,43,51,57] for the mixed state case. The methods used here are based on group theory and can be applied only to invariant prior distributions and certain distance functions. In particular, the optimal covariant measurement in the case of completely unknown qubit states was found in [4,29] but it has the drawback that it does not give any clue as to how it can be implemented in a real experiment. In the pointwise approach [6,7,17,21,26,29,30,45] one tries to minimize the risk for each unknown state ρ. As the optimal measurement-estimator pair cannot depend on the state itself, one optimizes the maximum risk Rmax (Mn , ρˆn ), (see (1.3)), or a local version of this which will be defined shortly. The advantage of the pointwise approach is that it can be applied to arbitrary families of states and a large class of loss functions provided that they are locally quadratic in the chosen parameters. The underlying philosophy is that as the number n of states is sufficiently large, the problem ceases to be global and becomes a local one as the error in estimating the state parameters is of the order n −1/2 . The Bayesian and pointwise approaches can be compared [20], and in fact for large n the prior distribution π of the Bayesian approach becomes increasingly irrelevant and the optimal Bayesian estimator becomes asymptotically optimal in the minimax sense and vice versa. 2.1. Qubit state estimation: the localization principle. Let us now pass to the quantum statistical model which will be the object of our investigations. Let ρ ∈ M2 (C) be an arbitrary density matrix describing the state of a qubit. Given n identically prepared qubits with joint state ρ ⊗n , we would like to optimally estimate ρ based on the result of a properly chosen joint measurement Mn . For simplicity of the exposition we assume that the outcome of the measurement is an estimator ρˆn ∈ M2 (C). In practice however, the result X may belong to a complicated measure space (in our case the space of continuous time paths) and the estimator is a function of the “raw” data ρˆn := ρˆn (X ). The quality of the estimator at the state ρ is quantified by the risk Rρ (Mn , ρˆn ) := Eρ (d(ρ, ρˆn )2 ), where d is a distance between states. The above expectation is taken with respect to the distribution Pρ (d x) := Tr(ρ M(d x)) of the measurement results, where M(d x) represents the associated positive operator valued measure of the measurement M. In our exposition d will be the trace norm
ρ1 − ρ2 1 := Tr(|ρ1 − ρ2 |), but similar results can be obtained using the fidelity distance. The aim is to find a sequence of measurements and estimators (Mn , ρˆn ) which is asymptotically optimal in the local minimax sense: for any given ρ0 ,
Optimal Estimation of Qubit States with Continuous Time Measurements
lim sup
sup
n→∞ ρ−ρ0 1 ≤n −1/2+
n Rρ (Mn , ρˆn ) ≤ lim sup
sup
n→∞ ρ−ρ0 1 ≤n −1/2+
133
n Rρ (Nn , ρˇn ),
for any other sequence of measurement-estimator pairs (Nn , ρˇn ). The factor n is inserted because typically Rρ (Mn , ρˆn ) is of the order 1/n and the optimization is about obtaining the smallest constant factor possible. The inequality says that one cannot find an estimator which performs better than ρˆn over a ball of size n −1/2+ centered at ρ0 , even if one has the knowledge that the state ρ belongs to that ball! Here, and elsewhere in the paper will appear in different contexts, as a generic strictly positive number and will be chosen to be sufficiently small for each specific use. At places where such notation may be confusing we will use additional symbols to denote small constants. As set forth in the Introduction, our measurement procedure consists of two steps. The first one is to perform separate measurements of σx , σ y and σz on a fraction n˜ = n(n) ˜ of the systems. In this way we obtain a rough estimate ρ˜n of the true state ρ which lies in a local neighborhood around ρ with high probability. The second step uses the information obtained in the first step to perform a measurement which is optimal precisely for the states in this local neighborhood. The second step ensures optimality and requires more sophisticated techniques inspired by the theory of local asymptotic normality for qubit states [22]. We begin by showing that the first step amounts to the fact that, without loss of generality, we may assume that the unknown state is in a local neighborhood of a known state. This may serve also as an a posteriori justification of the definition of local minimax optimality. Lemma 2.1. Let Mi denote the measurement of the σi spin component of a qubit with i = x, y, z. We perform each of the measurements Mi separately on n/3 ˜ identically prepared qubits and define ρ˜n =
1 (1 + r˜ σ ), 2
if |˜r | ≤ 1,
where r˜ = (˜r x , r˜y , r˜z ) is the vector average of the measured components. If |˜r | > 1 then we define ρ˜n as the state which has the smallest trace distance to the right-hand side expression. Then for all ∈ [0, 2], we have P ρ˜n − ρ 21 > 3n 2−1 ≤ 6 exp(− 21 nn ˜ 2−1 ), ∀ρ. Furthermore, for any 0 < κ < /2, if n˜ = n 1−κ , the contribution to the risk E( ρ˜n −ρ 21 ) √ brought by the event E = [ ρ˜n − ρ 1 > 3n −1/2+ ] satisfies E ρ˜n − ρ 21 χ E ≤ 24 exp(− 21 n 2−κ ) = o(1). Proof. For each spin component σi we obtain i.i.d coin tosses X i with distribution P(X i = ±1) = (1 ± ri )/2 and average ri . Hoeffding’s inequality [50] then states that for all c > 0, we have P(|X i − X˜ |2 > c) ≤ 2 exp(− 21 nc). ˜ By using this inequality three times with c = n 2−1 , once for each component, we get 3 2 2−1 P |˜ri − ri | > 3n ˜ 2−1 ) ∀ρ, ≤ 6 exp(− 21 nn 1
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which implies the statement for the norm distance since ρ˜n − ρ 21 = i |˜ri − ri |2 . The bound on conditional risk follows from the previous bound and the fact that
ρ − ρ˜n 21 ≤ 4. In the second step of the measurement procedure we rotate the remaining n − n˜ qubits such that after rotation the vector r˜ is parallel to the z-axis. Afterwards, we couple the systems to the field and perform certain measurements in the field which will determine the final estimator ρˆn . The details of this second step are given in Sects. 4 and 5, but at this moment we can already prove that the effect of errors in the first stage of the measurement is asymptotically negligible compared to the risk of the second estimator. Indeed by Lemma 2.1 we get that if n˜ = n 1−κ , then the probability that the first stage gives a “wrong” estimator (one which lies outside the local neighborhood of the true state) is of the order exp(− 21 n 2−κ ) and so is the risk contribution. As the typical risk of estimation is of the order 1/n, we see that the first step is practically “always” placing the estimator in a neighborhood of order n −1/2+ of the true state ρ, as shown in Fig. 2. In the next section we will show that for such neighborhoods, the state of the remaining n − n˜ systems behaves asymptotically as a Gaussian state. This will allow us to devise an optimal measurement scheme for qubits based on the optimal measurement for Gaussian states. 3. Local Asymptotic Normality The optimality of the second stage of the measurement relies on the concept of local asymptotic normality [22,49]. After a short introduction, we will prove that LAN holds for the qubit case, with radius of validity n −1/2+η for all η ∈ [0, 1/4). We will also show that its rate of convergence is O(n −1/4+η+ ) for arbitrarily small . 3.1. Introduction to LAN and some definitions. Let ρ0 be a fixed state, which by rotational symmetry can be chosen of the form µ 0 , (3.1) ρ0 = 0 1−µ for a given 21 < µ < 1. We parametrize the neighboring states as ρu/√n , where u = (u x , u y , u z ) ∈ R3 such that the first two components account for unitary rotations around ρ0 , while the third one describes the change in eigenvalues µ + vz 0 U (v)∗ , ρv := U (v) (3.2) 0 1 − µ − vz with unitary U (v) := exp(i(vx σx + v y σ y )). The “local parameter” u should be thought of as having a bounded range in R3 or may even “grow slowly” as u ≤ n η . ⊗n√ Then, for large n, the joint state ρnu := ρu/ of n identical qubits approaches a n u u Gaussian state of the form N ⊗ φ with the parameter u appearing solely in the average of the two Gaussians. By N u we denote a classical one-dimensional normal distribution centered at u z which relays information about the eigenvalues of ρu/√n . The second term φ u is a Gaussian state of a harmonic oscillator which is a displaced thermal
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equilibrium state with displacement proportional to (u x , u y ). It contains information on the eigenvectors of ρu/√n . We thus have the convergence ρnu ; N u ⊗ φ u , to a much simpler family of classical - quantum states for which we know how to optimally estimate the parameter u. The asymptotic splitting into a classical estimation problem for eigenvalues and a quantum one for the eigenbasis has been also noticed in [4] and in [29], the latter coming pretty close to our formulation of local asymptotic normality. The precise meaning of the convergence is given in Theorem 3.1 below. In short, ⊗n√ there exist quantum channels Tn which map the states ρu/ into N u ⊗ φ u with vann ishing error in trace norm distance, and uniformly over the local parameters u. From the statistical point of view the convergence implies that a statistical decision problem concerning the model ρnu can be mapped into a similar problem for the model N u ⊗ φ u such that the optimal solution for the latter can be translated into an asymptotically optimal solution for the former. In our case the problem of estimating the state ρ turns into that of estimating the local parameter u around the first stage estimator ρ˜n playing the role of ρ0 . For the family of displaced Gaussian states it is well known that the optimal estimation of the displacement is achieved by the heterodyne detection [32,55], while for the classical part it is sufficient to take the observation as best estimator. Hence the second step will give an optimal estimator uˆ of u and an optimal estimator of the initial √ state ρˆn := ρu/ ˆ n . The precise result is formulated in Theorem 6.1 3.2. Convergence to the Gaussian model. We describe the state N u ⊗ φ u in more detail. N u is simply the classical Gaussian distribution N u := N (u z , µ(1 − µ)),
(3.3)
with mean u z and variance µ(1 − µ). The state φ u is a density matrix on H = F(C), the representation space of the harmonic oscillator. In general, for any Hilbert space h, the Fock space over h is defined as F(h) :=
∞
h ⊗s · · · ⊗s h,
(3.4)
n=0
with ⊗s denoting the symmetric tensor product. Thus F(C) is the simplest example of a Fock space. Let φ := (1 − p)
p k |kk|,
(3.5)
k=0
be a thermal equilibrium state with |k denoting the k th energy level of the oscillator and p = 1−µ µ < 1. For every α ∈ C define the displaced thermal state φ(α) := D(α) φ D(−α),
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where D(α) := exp(αa ∗ − αa) ¯ is the displacement operator, mapping the vacuum vector |0 to the coherent vector |α = exp(−α 2/2)
∞ αk √ |k. k! k=0
Here a ∗ and a are the creation and annihilation operators on F(C), satisfying [a, a ∗ ] = 1. The family φ u of states in which we are interested is given by φ u := φ( 2µ − 1αu ), u ∈ R3 , (3.6) with αu := −u y + iu x . Note that φ u does not depend on u z . We claim that the “statistical information” contained in the joint state of n qubits ⊗n√ ρnu := ρu/ , n
(3.7)
is asymptotically identical to that contained in the couple (N u , φ u ). More precisely: ⊗n Theorem 3.1. Let ρnu be the family of states (3.2) on the Hilbert space C2 , let N u u be the family (3.3) of Gaussian distributions, and let φ be the family (3.6) of displaced thermal equilibrium states of a quantum oscillator. Then for each n there exist quantum channels (trace preserving CP maps) Tn : T ((C2 )⊗n ) → L 1 (R) ⊗ T (F(C)), Sn : L 1 (R) ⊗ T (F(C)) → T ((C2 )⊗n ) with T (H) the trace-class operators on H, such that, for any 0 ≤ η < 1/4 and any > 0, sup N u ⊗ φ u − Tn ρnu 1 = O(n −1/4+η+ ), (3.8)
u ≤n η
sup ρnu − Sn N u ⊗ φ u 1 = O(n −1/4+η+ ).
u ≤n η
(3.9)
Moreover, for each 2 > 0 there exists a function f (n) of order O(n −1/4+η+ ) such that the above convergence rates are bounded by f (n), with f independent of ρ 0 as long as | 21 − µ| > 2 . Remark. Note that Eqs. (3.8) and (3.9) imply that the expressions on the left side converge to zero as n → ∞. Following the classical terminology of Le Cam [40], we will call this type of result strong convergence of quantum statistical models (experiments). Another local asymptotic normality result has been derived in [23] based on a different concept of convergence, which is an extension of the weak convergence of classical (commutative) statistical experiments. In the classical set-up it is known that strong convergence implies weak convergence for arbitrary statistical models, and the two are equivalent for statistical models consisting of a finite number of distributions. A similar relation is conjectured to hold in the quantum set-up, but for the moment this has been shown only under additional assumptions [23]. These two approaches to local asymptotic normality in quantum statistics are based on completely different methods and the results are complementary in the sense that the weak convergence of [23] holds for the larger class of finite dimensional states while
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the strong convergence has more direct consequences as it is shown in this paper for the case of qubits. Both results are part of a larger effort to develop a general theory of local asymptotic normality in quantum statistics. Several extensions are in order: from qubits to arbitrary finite dimensional systems (strong convergence), from finite dimensional to continuous variables systems, from identical system to correlated ones, and asymptotic normality in continuous time dynamical set-up. Finally, let us note that the development of a general theory of convergence of quantum statistical models will set a framework for dealing with other important statistical decision problems such as quantum cloning [54] and quantum amplification [10], which do not necessarily involve measurements. Remark. The construction of the channels Tn , Sn in the case of fixed eigenvalues (u z = 0) is given in Theorem 1.1 of [22]. It is also shown that a similar result holds uniformly over
u < C for any fixed finite constant C. In [23], it is shown that such maps also exist in the general case, with unknown eigenvalues. A classical component then appears in the limit statistical experiment. In the above result we extend the domain of validity of these theorems from “local” parameters u < C to “slowly growing” local neighborhoods
u ≤ n η with η < 1/4. Although this may be seen as merely a technical improvement, it is in fact essential in order to insure that the result of the first step of the estimation will, with high probability, fall inside a neighborhood u ≤ n η for which local asymptotic normality still holds (see Fig. 2). Proof. Following [22] we will first indicate how the channels Tn are constructed. The technical details of the proof can be found in Appendix A. ⊗n The space C2 carries two unitary representations. The representation πn of SU (2) is given by πn (u) = u ⊗n for any u ∈ SU (2), and the representation π˜ n of the symmetric group S(n) is given by the permutation of factors π˜ n (τ ) : v1 ⊗ · · · ⊗ vn → vτ −1 (1) ⊗ · · · ⊗ vτ −1 (n) ,
τ ∈ S(n).
As [πn (u), π˜ n (τ )] = 0 for all u ∈ SU (2), τ ∈ S(n), we have the decomposition
C2
⊗n
=
n/2
j
H j ⊗ Hn .
(3.10)
j=0,1/2
The direct sum runs over all positive (half)-integers j up to n/2. For each fixed j, Hj ∼ = C2 j+1 is an irreducible representation U j of SU (2) with total angular momenj representation of the symmetric tum J 2 = j ( j + 1), and Hn ∼ = Cn j is the irreducible
n group S(n) with n j = n/2− j − n/2−n j−1 . The density matrix ρnu is invariant under permutations and can be decomposed as a mixture of “block” density matrices ρnu =
n/2
pn,u ( j) ρ uj,n ⊗
j=0,1/2
1 . nj
(3.11)
The probability distribution pn,u ( j) is given by [4]: pn,u ( j) :=
n n nj + j+1 2 j+1 1 − pu , (1 − µu ) 2 − j µu2 2µu − 1
(3.12)
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√ with µu := µ + u z / n, pu :=
1−µu µu .
We can rewrite pn,u ( j) as
pn,u ( j) := Bn,µu (n/2 + j) × K ( j, n, µ, u), where
n k Bn,ν (k) := ν (1 − ν)n−k , k
(3.13)
k = 0, . . . , n
is a binomial distribution, and the factor K ( j, n, µ, u) is given by √ 2 j+1 n + (2( j − jn − nu z ) + 1)/(2µu − 1) K ( j, n, µ, u) := 1 − pu , √ n + ( j − jn − nu z + 1)/µu jn := n(µ − 1/2). Now K ( j, n, µ, u) = 1 + O(n −1/2+ ) on the relevant values of j, i.e. the ones in an interval of order n 1/2+ around jn , as long as µu is bounded away from 1/2, which is automatically so for big n. As Bn,µu (k) is the distribution of a sum of i.i.d. Bernoulli random variables, we can now use standard local asymptotic normality results [49] to conclude that if j is distributed according to pn,u , then the centered and rescaled variable √ j gn := √ − n(µ − 1/2), n converges in distribution to a normal N u , after an additional randomization has been performed. The latter is necessary in order to “smooth” the discrete distribution into a distribution which is continuous with respect to the Lebesgue measure, and with convergence to the Gaussian distribution in total variation norm. The measurement “which block”, corresponding to the decomposition (3.11), provides us with a result j and a posterior state ρ uj,n . The function gn = gn ( j) (with an conadditional randomization) is the classical part of the channel Tn . The randomization√ sists of “smoothening”√ with a Gaussian kernel of mean g ( j) and variance 1/(2 n), n
√ i.e. with τn, j := (n 1/4 / π ) exp − n(x − gn ( j))2 . Note that this measurement is not disturbing the state ρnu in the sense that the average state after the measurement is the same as before. The quantum part of Tn is the same as in [22] and consists of embedding each block state ρ uj,n into the state space of the oscillator by means of an isometry V j : H j → F(C), V j : | j, m → | j − m, where {| j, m : m = − j, . . . , j} is the eigenbasis of the total spin component L z := (i) i σz , cf. Eq. (5.1) of [22]. Then the action of the channel Tn is Tn :
j
pn,u ( j)ρ uj,n ⊗
1 → pn,u ( j) τn, j ⊗ V j ρ uj,n V j∗ . nj j
The inverse channel Sn performs the inverse operation with respect to Tn . First the oscillator state is “cut-off” to the dimension of an irreducible representation and then a block obtained in this way is placed into the decomposition (3.10) (with an additional normalization from the remaining infinite dimensional block which is negligible for the states in which we are interested). The rest of the proof is given in Appendix A.
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4. Time Evolution of the Interacting System In the previous section, we have investigated the asymptotic equivalence between the states ρnu and N u ⊗ φ u by means of the channel Tn . We now seek to implement this in a physical situation. The N u -part will follow in Sect. 5.2, the φ u -part will be treated in this section. We couple the n qubits to a Bosonic field; this is the physical implementation of LAN. Subsequently, we perform a measurement in the field which will provide the information about the state of the qubits; this is the utilization of LAN in order to solve the asymptotic state estimation problem. In this section we will limit ourselves to analyzing the joint evolution of the qubits and field. The measurement on the field is described in Sect. 5.
4.1. Quantum stochastic differential equations. In the weak coupling limit [18] the joint evolution of the qubits and field can be described mathematically by quantum stochastic differential equations (QSDE) [34]. The basic notions here are the Fock space, the creation and annihilation operators and the quantum stochastic differential equation of the unitary evolution. The Hilbert space of the field is the Fock space F(L 2 (R)) as defined in (3.4). An important linearly complete set in F(L 2 (R)) is that of the exponential vectors e( f ) :=
∞ ∞
1 1 √ f ⊗n := √ | f n , n! n! n=0 n=0
f ∈ L 2 (R),
(4.1)
with inner product e( f ), e(g) = exp( f, g). The normalized exponential states | f := e− f, f /2 e( f ) are called coherent states. The vacuum vector is | := e(0) and we will denote the corresponding density matrix || by . The quantum noises are described by the creation and annihilation martingale operators A∗t := a ∗ (χ[0,t] ) and At := a(χ[0,t] ) respectively, where χ[0,t] is the indicator function for [0, t] and a( f ) : e(g) → f, ge(g). The increments d At := a(χ[0,t+dt] ) − a(χ[0,t] ) and d A∗t play the role of non-commuting integrators in quantum stochastic differential equations, in the same way as one can integrate against the Brownian motion in classical stochastic calculus. We now consider the joint unitary evolution for qubits and field defined by the quantum stochastic differential equation [8,34]: 1 dUn (t) = (an d A∗t − an∗ d At − an∗ an dt)Un (t), 2 where Un (t) is a unitary operator on (C2 )⊗n ⊗ F(L 2 (R)), and an := √
n 1 (k) (k) σ+ , σ+ := 1 ⊗ · · · ⊗ (σx + iσ y )/2 ⊗ · · · ⊗ 1, 2 jn k=1
jn := (µ−1/2)n.
√ As we will see later, the “coupling factor” 1/ jn of the order n −1/2 , is necessary in order to obtain convergence to the unitary evolution of the quantum harmonic oscillator and the field.
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We remind the reader that the n-qubit space can be decomposed into irreducible representations as in (3.10), and the interaction between the qubits and field respects this decomposition Un (t) =
n/2
U j,n (t) ⊗ 1,
j=0,1/2 j
where 1 is the identity operator on the multiplicity space Hn , and U j,n (t) : H j ⊗ F(L 2 (R)) → H j ⊗ F(L 2 (R)), is the restricted cocycle 1 dU j,n (t) = (a j d A∗t − a ∗j d At − a ∗j a j dt)U j,n (t), 2
(4.2)
with a j acting on the basis | j, m of H j as a j | j, m = j − m ( j + m + 1)/2 jn | j, m + 1, a ∗j | j, m = j − m + 1 j + m/2 jn | j, m − 1. Remark. We point out that the lowering operator for L z acts as creator for our cutoff oscillator since the highest vector | j, j corresponds by V j to the vacuum of the oscillator. This choice does not have any physical meaning but is only related with our convention µ > 1/2. Had we chosen µ < 1/2, then the raising operator on the qubits would correspond to the creation operator on the oscillator. By (3.11) the initial state ρ ⊗n decomposes in the same way as the unitary cocycle, and thus the whole evolution decouples into separate “blocks” for each value of j. We do not have explicit solutions to these equations but based on the conclusions drawn from LAN we expect that as n → ∞, the solutions will be well approximated by similar ones for a coupling between an oscillator and the field, at least for the states in which we are interested. As a warm up exercise we will start with this simpler limit case where the states can be calculated explicitly. 4.2. Solving the QSDE for the oscillator. Let a ∗ and a be the creation and annihilation operators of a quantum oscillator acting on F(C). We couple the oscillator with the Bosonic field and the joint unitary evolution is described by the family of unitary operators U (t) satisfying the quantum stochastic differential equation 1 ∗ ∗ ∗ dU (t) = ad At − a d At − a adt U (t). 2 We choose the initial (un-normalized) state ψ(0) := e(z) ⊗ |, where z is any complex number, and we shall find the explicit form of the vector state of the system and field at time t: ψ(t) := U (t)ψ(0). We make the following ansatz: ψ(t) = e(αt ) ⊗ e( f t ), where f t (s) := f (s)χ[0,t] (s) for some f ∈ L 2 (R). For each β ∈ C, g ∈ L 2 (R), define I (t) := e(β) ⊗ e(g), ψ(t). ¯ We then have I (t) = exp(βα(t) + g, f t ), so that it satisfies d
d I (t) = β¯ dt α(t) + g(t) ¯ f (t) I (t)dt . (4.3)
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141
d We now calculate dt I (t) with the help of the QSDE. Since At e( f ) = χ[0,t] , f e( f ), we have, for continuous g, d At e(g) = g(t)e(g)dt. However, since As e( f t ) is constant for s ≥ t, we have d At e( f t ) = 0. Thus
d I (t) = e(β) ⊗ e(g), (ad A∗t − a ∗ d At − 21 a ∗ adt)ψ(t) ¯ (t)dt . = (g(t)α(t) ¯ − 1 βα(t))I 2
(4.4)
d Equating (4.3) with (4.4) for all t, β and continuous g, we find f (s) = α(s), dt α(t) =
− 21 α(t).
1
1
Thus α(t) = α(0)e− 2 t , f t (s) = α(0)χ[0,t] (s)e− 2 s with α(0) = z. 1
1
In conclusion ψ(t) = e(ze− 2 t ) ⊗ e(ze− 2 s χ[0,t] (s)). For later use we denote the 2 normalized solution by ψz (t) := U (t)|z ⊗ | = e−|z| /2 U (t)e(z) ⊗ |.
4.3. QSDE for large spin. We consider now the unitary evolution for qubits and field: 1 dUn (t) = (an d A∗t − an∗ d At − an∗ an dt)Un (t). 2 It is no longer possible to obtain an explicit expression for the joint vector state ψn (t) at time t. However we will show that for the states in which we are interested, a satisfactory explicit approximate solution exists. The trick works for an arbitrary family of unitary solutions of a quantum stochastic differential equation dU (t) = G dt U (t), and the general idea is the following: if ψ(t) is the true state ψ(t) = U (t)ψ and ξ(t) is a vector describing an approximate evolution t (ψ(0) = ξ(0)) then with Ut+dt := U (t + dt)U (t)−1 we get t t ψ(t + dt) − ξ(t + dt) = ψ(t + dt) − Ut+dt ξ(t) + Ut+dt ξ(t) − ξ(t) + ξ(t) − ξ(t + dt) t = Ut+dt [ψ(t) − ξ(t)] + [U (t + dt) − U (t)]U (t)−1 ξ(t) +[ξ(t) − ξ(t + dt)] t = Ut+dt [ψ(t) − ξ(t)] + G dt ξ(t) − dξ(t).
By taking norms we get d ψ(t) − ξ(t) ≤ G dt ξ(t) − dξ(t) .
(4.5)
The idea is now to devise a family ξ(t) such that the right side is as small as possible. We apply this technique block-wise, that is to each unitary U j,n (t) acting on H j ⊗ F(L 2 (R)) (see Eq. (4.2)) for a “typical” j ∈ Jn (see Eq. (A.1)). By means of the isometry V j we can embed the space H j into the first 2 j + 1 levels of the oscillator and for simplicity we will keep the same notions as before for the operators acting on F(C). As initial states for the qubits we choose the block states ρ uj,n . Theorem 4.1. Let ρ uj,n (t) = U j,n (t) ρ uj,n ⊗ U ∗j,n (t) be the j th block of the state of qubits and field at time t. Let φ u (t) := U (t) φ u ⊗ U (t)∗ be the joint state of the oscillator and field at time t. For any η < 1/6, for any > 0, sup
sup sup ρ uj,n (t) − φ u (t) 1 = O(n −1/4+η+ , n −1/2+3η+ ).
j∈Jn u ≤n η
t
(4.6)
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Proof. From the proof of the local asymptotic normality Theorem 3.1 we know that the initial states of the two unitary evolutions are asymptotically close to each other, sup ρ uj,n − φ u 1 = O(n −1/4+η+ ).
sup
j∈Jn u ≤n η
(4.7)
The proof consists of two estimation steps. In the first one, we will devise another initial state ρ˜ uj,n which is an approximation of φ u and thus also of ρ uj,n : sup
sup ρ˜ uj,n − φ u 1 = O(e−n ).
j∈Jn u ≤n η
(4.8)
In the second estimate we show that the evolved states ρ˜ uj,n (t) and φ u (t) are asymptotically close to each other sup
sup sup ρ˜ uj,n (t) − φ u (t) 1 = O(n −1/4+η+ , n −1/2+3η+ ).
j∈Jn u ≤n η
(4.9)
t
This estimate is important because the two trajectories are driven by different Hamiltonians, and in principle there is no reason why they should stay close to each other. From (4.7), (4.8) and (4.9), and using the triangle inequality we get sup
sup sup ρ uj,n (t) − φ u (t) 1 = O(n −1/4+η+ , n −1/2+3η+ ).
j∈Jn u ≤n η
t
The following diagram illustrates the above estimates. The upper line concerns the time evolution of the block state ρ uj,n and the field. The lower line describes the time evolution of the oscillator and the field. The estimates show that the diagram is “asymptotically commutative” for large n. Id j ⊗
U j,n (t)
Id⊗
U (t)
S(H j ) −−−−→ S(H j ⊗ F) −−−−→ S(H j ⊗ F) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ V j ·V j∗ S(F(C)) −−−−→ S(F(C) ⊗ F) −−−−→ S(F(C) ⊗ F) For the rest of the proof, we refer to Appendix B. We have shown how the mathematical statement of LAN (the joint state of qubits converges to a Gaussian state of a quantum oscillator plus a classical Gaussian random variable) can in fact be physically implemented by coupling the spins to the environment and letting them “leak” into the field. In the next section, we will use this for the specific purpose of estimating u by performing a measurement in the field. 5. The Second Stage Measurement We now describe the second stage of our measurement procedure. Recall that in the first stage a relatively small part n˜ = n 1−κ , 1 > κ > 0, of the qubits is measured and a rough estimator ρ˜n is obtained. The purpose of this estimator is to localize the state within a small neighborhood such that the machinery of local asymptotic normality of Theorem 3.1 can be applied.
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In Theorem 4.1 the local asymptotic normality was extended to the level of time evolution of the qubits interacting with a bosonic field. We have proven that at time t the joint state of the qubits and field is n/2
ρnu (t) :=
j=0,1/2
pn,u ( j)
√ 1 2 2 dz e−|z− 2µ−1αu | /2s exp(−|z|2 ) × 2 2π s C
|e(ze−t/2 ) j e(ze−t/2 ) j | ⊗ |e(ze−u/2 χ[0,t] (u))e(ze−u/2 χ[0,t] (u))| +O(n η−1/4+ , n 3η−1/2+ ), for u ≤ n η . The index j serves to remind the reader that the first exponential states live in different copies F(C) j of the oscillator space, corresponding to H j via the isometry V j . We will continue to identify H j with its image in F(C) j . We can now approximate the above state by its limit for large t, since exp(−|z|2 )e(ze−t/2 ) j | j, je(ze−u/2 χ[0,t] (u)) | e(ze−u/2 ) = exp(−|z|2 e−t ).
(5.1)
As we are always working with u ≤ n η , the only relevant z are bounded by n η+δ for small δ. (The remainder of the Gaussian integral has an exponentially decreasing norm, as discussed before.) Thus, for large enough time (i.e. for t ≥ ln(n)), we can write ρnu (t) = ρnu (∞) + O(n η−1/4+ , n 3η−1/2+ ) with ρnu (∞)
n/2
:=
pn,u ( j)| j, j j, j| ⊗
j=0,1/2
√ 1 −|z− 2µ−1αu |2 /2s 2 −u/2 −u/2 2 dz e |e(ze )e(ze )| exp(−|z| ) . 2π s 2 C (5.2)
Thus, the field is approximately in the state φ u depending on (u x , u y ), which is carried by the mode (u → e−u/2 χ[0,∞) (u)) ∈ L 2 (R) denoted for simplicity by e−u/2 . The atoms end up in a mixture of | j, j states with coefficients pn,u ( j), which depend only on u z , and are well approximated by the Gaussian random variable N u as shown in Theorem 3.1. Moreover since there is no correlation between atoms and field, the statistical problem decouples into one concerning the estimation of the displacement in a family of Gaussian states φ u , and one for estimating the center of N u . For the former problem, the optimal estimation procedure is known to be the heterodyne measurement [32,55]; for the latter, we perform a “which block” measurement. These measurements are described in the next two subsections. 5.1. The heterodyne measurement.√A heterodyne measurement √ is a “joint measurement” of the quadratures Q := (a + a ∗ )/ 2 and P := −i(a − a ∗ )/ 2 of a quantum harmonic oscillator which in our case represents a mode of light. Since the two operators do not commute, the price to pay is the addition of some “noise” which will allow for an approximate measurement of both operators. The light beam passes through a beamsplitter having a vacuum mode as the second input, and then one performs a homodyne (quadrature) measurement on each of the two emerging beams. If Qv and Pv are the vac√ uum quadratures then we measure the following output quadratures Q1 := (Q+Qv )/ 2
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√ and P2 := (P − Pv )/ √ 2, with [Q1 , P2 ] = 0. Since the two input beams are independent, between the distribution of Q and the disthe distribution of 2Q1 is the convolution √ tribution of Qv , and similarly for 2P2 . In our case we are interested in the mode e−u/2 which is in the state φ u , up to a factor√of order O(n η−1/4+ , n 3η−1/2+ ). From (3.6) we√obtain that the distribution of Q is N ( 2(2µ − 1)u x , 1/(2(2µ − 1))), that of P is N ( 2(2µ − 1)u y , 1/(2(2µ − 1))), and the joint distribution of the rescaled output (Q + Qv )/ 2(2µ − 1) , (P − Pv )/ 2(2µ − 1) , is N (u x , µ/(2(2µ − 1)2 )) × N (u y , µ/(2(2µ − 1)2 )).
(5.3)
We will √ denote by (u˜ x , u˜ y ) the result of the heterodyne measurement rescaled by the factor 2µ − 1 such that with good approximation (u˜ x , u˜ y ) has the above distribution and is an unbiased estimator of the parameters (u x , u y ). Since we know in advance that the parameters (u x , u y ) must be within the radius of validity of LAN we modify the estimators (u˜ x , u˜ y ) to account for this information and obtain the final estimator (uˆ x , uˆ y ): u˜ i if |u˜ i | ≤ 3n η uˆ i = . (5.4) 0 if |u˜ i | > 3n η Notice that if the true state ρ is in the radius of validity of LAN around ρ, ˜ then u ≤ n η , so that |uˆ i − u i | ≤ |u˜ i − u i |. We shall use this when proving optimality of the estimator. 5.2. Energy measurement. Having seen the φ u -part, we now move to the N u -part of the equivalence between ρnu and N u ⊗ φ u . This too is a coupling to a bosonic field, albeit a different coupling. We also describe the measurement in the field which will provide the information on the qubit states. The final state of the previous measurement, restricted to the atoms alone (without the field), is obtained by a partial trace of Eq. (5.2) (for large time) over the field τnu =
n/2
pn,u ( j)| j, j j, j| + O(n η−1/4+ , n 3η−1/2+ ).
j=0,1/2
We will take this as the initial state of the second measurement, which will determine j. A direct coupling to the J 2 does not appear to be physically available, but a coupling to the energyJz is realizable. This suffices, because the above state satisfies j = m (up to order O(n η−1/4+ , n 3η−1/2+ )). We couple the atoms to a new field (in the vacuum state |) by means of the interaction dUt = {Jz (d A∗t − d At ) − 21 Jz2 dt}Ut ,
with Jz := √1n nk=1 σz . Since this QSDE is ‘essentially commutative’, i.e. driven by a single classical noise Bt = (A∗t − At )/i, the solution is easily seen to be Ut = exp(Jz ⊗ (A∗t − At )).
Optimal Estimation of Qubit States with Continuous Time Measurements
Indeed, we have d f (Bt ) = f (Bt )d Bt +
1 2
145
f (Bt )dt by the classical Itô rule, so that
d exp(i Jz ⊗ Bt ) = {i Jz d Bt − 21 Jz2 dt} exp(i Jz ⊗ Bt ). For an initial state | j, m ⊗ |, this evolution gives rise to the final state √ Ut | j, m ⊗ = | j, m ⊗ exp((m/ n)(A∗t − At )) √ = | j, m ⊗ |(m/ n)χ[0,t] , where | f ∈ F(L 2 (R)) denotes the normalized vector exp(− f, f /2)e( f ). Applying this to the states | j, j j, j| in τnu yields Ut τnu ⊗ Ut∗ =
n/2
√ √ pn,u ( j)| j, j j, j| ⊗ | j/ nχ[0,t] j/ nχ[0,t] |
j=0,1/2
+O(n η−1/4+ , n 3η−1/2+ ). The final state of the field results from a partial trace over the atoms; it is given by n/2
√ √ pn,u ( j) |( j/ n)χ[0,t] ( j/ n)χ[0,t] | + O(n η−1/4+ , n 3η−1/2+ ) .
(5.5)
j=0,1/2
We now perform a homodyne measurement√on the field, which amounts to a direct measurement of (At + A∗t )/2t. In√ the state |( j/ nχ[0,t] , this yields the value of j with √ certainty for large time (i.e. t n). Indeed, for this state, E((At + A∗t )/2t) = j/ n, whereas Var(At + A∗t )/2t) = 1/(4t). Thus the probability distribution pn,u is reproduced up to order O(n η−1/4+ , n 3η−1/2+ ) in L 1 -distance. The following is a remider from the proof of Theorem 3.1 If we start with j distributed √ according to pn ( j) and we smoothen √jn − n(µ − 1/2) with a Gaussian kernel, then we obtain a random variable gn which is continuously distributed on R and converges in distribution to N (u z , µ(1 − µ)), the error term being of order O(n η−1/2 ) + O(n −1/2 ). For j distributed according to the actual distribution, as measured by the homodyne detection experiment, we can therefore state that gn is distributed according to N (u z , µ(1 − µ)) + O(n η−1/4+ , n 3η−1/2+ ) + O(n η−1/2 ) + O(n −1/2 ).
(5.6)
As in the case of (uˆ x , uˆ y ), we take into account the range of validity of LAN by defining the final estimator uˆ z =
gn 0
if |gn | ≤ 3n η if |gn | > 3n η .
(5.7)
Similarly, we note that if the true state ρ is in the radius of validity of LAN around ρ, ˜ then u ≤ n η , so that |uˆ z − u z | ≤ |u˜ z − u z |.
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6. Asymptotic Optimality of the Estimator In order to estimate the qubit state, we have proposed a strategy consisting of the following steps. First, we use n˜ := n 1−κ copies of the state ρ to get a rough estimate ρ˜n . Then we couple the remaining qubits with a field, and perform a heterodyne measurement. Finally, we couple to a different field, followed by homodyne measurement. From the measurement outcomes, we construct an estimator ρˆn := ρuˆ n /√n . This strategy is asymptotically optimal in a global sense: for any true state ρ even if we knew beforehand that the true state ρ is in a small ball around a known state ρ0 , it would be impossible to devise an estimator that could do better asymptotically than our estimator ρˆn on a small ball around ρ. More precisely: Theorem 6.1. Let ρˆn be the estimator defined above. For any qubit state ρ0 different from the totally mixed state, for any sequence of estimators ˆ n , the following local asymptotic minimax result holds for any 0 < < 1/12: lim sup
sup
n→∞ ρ−ρ0 1 ≤n −1/2+
n R(ρ, ρˆn ) ≤ lim sup
sup
n→∞ ρ−ρ0 1 ≤n −1/2+
n R(ρ, ˆ n ).
(6.1)
Let (µ0 , 1 − µ0 ) be the eigenvalues of ρ0 with µ0 > 1/2. Then the local asymptotic minimax risk is lim sup
sup
n→∞ ρ−ρ0 1 ≤n −1/2+
n R(ρ, ρˆn ) = Rminimax (µ0 ) = 8µ0 − 4µ20 .
(6.2)
Proof. We write the risk as the sum of two terms corresponding to the events E and E c that ρ˜n is inside or outside the ball of radius n −1/2+ around ρ. Recall that LAN is valid inside the ball. Thus R(ρ, ρˆn ) = E( ρ − ρˆn 21 χ E c ) + E( ρ − ρˆn 21 χ E ), ˆn where the expectation comes from ρˆn being random. The distribution of the result r ho of our measurement procedure applied to the true unknown state ρ depends on ρ. We bound the first part by R1 and the second part by R2 as shown below. R1 equals P(E c ) times the maximum error, which is 4 since for any pair of density matrices ρ and σ , we have ρ − σ 21 ≤ 4. Thus R1 = 4P( ρ − ρ˜n 1 ≥ n −1/2+ ). According to Lemma 2.1 this probability goes to zero exponentially fast, therefore the contribution brought by this term can be neglected. We can now assume that ρ˜n is in the range of validity of local asymptotic normality and we can write ρ ⊗n = ρnu with u the local parameter around ρ˜n . We get the following inequalities for the second term in the risk:
Optimal Estimation of Qubit States with Continuous Time Measurements
E( ρ − ρˆn 21 χ E ) ≤ E ρˆn − ρ 21 ≤ ≤
sup
ρ−ρ0
sup
ρ−ρ0
+ ≤
ρ˜n − ρ 1 ≤ n −1/2+ E ρˆn − ρ 21 ρ˜n = ρ0 Eρnu (∞) ρˆn − ρ 21 ρ˜n = ρ0
sup
ρ−ρ0
sup
ρ−ρ0
+cn −1+2η
147
ρnu (t) − ρnu (∞) 1 sup ρˆn − ρ 21
Eρnu (∞) ρˆn − ρ 21 sup
ρ−ρ0
uˆ n
ρ˜n = ρ0
ρnu (t) − ρnu (∞) 1 = R2 .
(6.3)
The first two inequalities are trivial. In the third inequality we change the expectation from the one with respect to the probability distribution of our data Pρnu (t) to the probability distribution Pρnu (∞) . In doing so, an additional term Pρnu (t) − Pρnu (∞) 1 appears which is bounded from above by ρnu (t) − ρnu (∞) 1 . In the last inequality we can bound
ρˆn − ρ 21 by cn −1+2η for some constant c. Indeed from definitions (5.4) and (5.7) we know that ρˆn − ρ0 1 ≤ c n −1/2+η , and additionally we are under the assumption
ρ − ρ0 1 ≤ n −1/2+ with < η. For the following, recall that all our LAN estimates are valid uniformly around any state ρ 0 = ρ˜ as long as µ − 1/2 ≥ 2 > 0. As we are working with ρ different from the totally mixed state and ρ − ρ
˜ ≤ n −1/2+ , we know that for big enough n, µ−1/2 ˜ ≥ 2 for any possible ρ. ˜ We can then apply the uniform results of the previous sections. The second term in R2 is O(n −5/4+3η+δ , n −3/2+5η+δ ), where δ > 0 can be chosen arbitrarily small. Indeed in the end of Sect. 4 we have proven that after time t ≥ ln n, the following holds: ρnu (t) − ρnu (∞) 1 = O(n −1/4+η+δ , n −1/2+3η+δ ). The contribution to n R(ρ, ρˆn ) brought by this term will not count in the limit, as long as η and are chosen such that 1/12 > η > . We now deal with the first term in R2 . We write ρ in local parametrization around ρ0 = ρ˜ as ρun /√n . We have
ρˆn − ρ 21 = ρu/√n − ρuˆ n /√n 21 =4
(u z − uˆ z )2 −(2µ−1)2 ((u x − uˆ x )2 −(u y − uˆ y )2 ) + O( u − uˆ n 3 n −3/2 ). n (6.4)
The remainder term O( u − uˆ n 3 n −3/2 ) is negligible. It is O(n 3η−3/2 ) which does not contribute to n R(ρ, ρˆn ) for η < 1/6. This is because on the one hand we have asked for ρ˜n − ρ < n −1/2+ , and on the other hand, we have bounded our estimator uˆ n by using (5.4) and (5.7). We now evaluate Eρnu (∞) d(u, uˆ n )2 with the notation (6.5) d(u, v)2 := 4 (u z − vz )2 + (2µ − 1)2 ((u x − vx )2 + (u y − v y )2 ) . Note that the risk of uˆ n is smaller than that of u˜ n (see the discussion below (5.4) and (5.7)). Under the law Pρnu (∞) the estimator u˜ n has a Gaussian distribution as shown in (5.3) and (5.6) with fixed and known variance and unknown expectation. In statistics
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this type of model [49]. Using (5.3) and (5.6), is known as a Gaussian shift experiment we get Eρnu (∞) (u z − uˆ z )2 ≤ µ(1 − µ) and Eρnu (∞) (u i − uˆ i )2 ≤ µ/(2(2µ − 1)2 ) for i = x, y. Substituting these bounds in (6.4), we obtain (6.2). We will now show that the sequence ρˆn is optimal in the local minimax sense: for any ρ0 and any other sequence of estimators ˆ n we have R0 = lim sup
sup
n→∞ ρ−ρ0 1 ≤n −1/2+
n R(ρ, ˆ n ) ≥ 8µ0 − 4µ20 .
We will first prove that the right hand side is the minimax risk Rminimax (µ0 ) for the family of states N u ⊗ φ u which is the limit of the local families ρnu of qubit states centered around ρ0 . We then extend the result to our sequence of quantum statistical models ρnu . The minimax optimality for N u ⊗ φ u can be checked separately for the classical and the quantum part of the experiment. For the quantum part φ u , the optimal measurement is known to be the heterodyne measurement. A proof of this fact can be found in Lemma 7.4 of [22]. For the classical part, which corresponds to the measurement of L z , the optimal estimator is simply the random variable X ∼ N u itself [49]. We now end the proof by using the other direction of LAN. Suppose that there exists a better sequence of estimators ˆ n such that R0 < Rminimax (µ0 ) = 8µ0 − 4µ20 . We will show that this leads to an estimator uˆ of u for the family N u ⊗ φ u whose maximum risk is smaller than the minimax risk Rminimax (µ0 ), which is impossible. By means of a beamsplitter one can divide the state φ u into two independent Gaussian modes, using a thermal state φ := φ 0 as the second input. If r and t are the reflectivity and respective transmitivity of the beamsplitter (r 2 + t 2 = 1), then the transmitted beam u = φ tu and the reflected one φ u = φ r u . By performing a heterodyne meahas state φtr ref surement on the latter, and observing the classical part N u , we can localize u within a big ball around the result u˜ with high probability, in the spirit of Lemma 2.1. More precisely, for any small ˜ > 0 we can find a > 0 big enough such that the risk contribution from ˜ is small unlikely u’s ˜ 2 χ u−u >a E( u − u
) < ˜ . ˜ Summarizing the localization step, we may assume that the parameter u satisfies u < a with an ˜ loss of risk, where a = a(r, ). ˜ Now let n be large enough such that n > a, then the parameter u falls within the domain of convergence of the inverse map Sn of Theorem 3.1 and by (3.9) (with replacing η and δ replacing ) we have
ρntu − S(N tu ⊗ φ tu ) 1 ≤ Cn −1/4++δ , for some constant C. Next we perform the measurement leading to the estimator ˆ n and equivalently to an estimator uˆ n of u. Without loss of risk we can implement the condition u < a into the estimator uˆ n in a similar fashion as in (5.4) and (5.7). The risk of this estimation procedure for φ u is then bounded from above by the sum of three terms: the risk n Rρ (ˆ n )/t 2 coming from the qubit estimation, the error contribution from the map Sn which is a 2 n −1/4++δ , and the localization risk contribution ˜ . This risk bound uses the same technique as the third inequality of (6.3). The second contribution can be made
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arbitrarily small by choosing n large enough, for < 1/4. From our assumption we have R0 < Rminimax (µ0 ) and we can choose t close to one such that R0 /t 2 < Rminimax (µ0 ) and further choose ˜ such that R0 /t 2 + ˜ < Rminimax (µ0 ). In conclusion, we get that the risk for estimating u is asymptotically smaller than the risk of the heterodyne measurement combined with observing the classical part which is known to be minimax [22]. Hence no such sequence ˆ n exists, and ρˆn is optimal. ˆ with Remark. In Theorem 6.1, we have used the risk function R(ρ, ρ) ˆ = E(d 2 (ρ, ρ)), d the L 1 -distance d(ρ, ρ) ˆ = ρ − ρ
ˆ 1 . However, the obtained results can easily be adapted to any distance measure d 2 (ρuˆ , ρu ) which is locally quadratic in uˆ − u, i.e. γαβ (u α − uˆ α )(u β − uˆ β ) + O( u − u
ˆ 3 ). d 2 (ρuˆ , ρu ) = α,β=x,y,z
ˆ ρ) = 1 − F 2 (ρ, ˆ ρ) with the fidelity F(ρ, ˆ ρ) := For instance, one may choose d 2 (ρ, Tr ρρ ˆ ρˆ . For non-pure states, this is easily seen to be locally quadratic with ⎛
(2µ0 − 1)2 0 0 (2µ0 − 1)2 γ =⎝ 0 0
0 0 1 1−(2µ0 −1)2
⎞ ⎠.
For the corresponding risk function R F (ρ, ρˆn ) := E(1 − F 2 (ρ, ρˆn )), this yields lim sup
sup
n→∞ ρ−ρ0 1 ≤n −1/2+
n R F (ρ, ρˆn ) = µ0 + 1/4 ,
with the same asymptotically optimal ρ. ˆ The asymptotic rate R F ∼ earlier in [4], using different methods.
(6.6) 4µ0 +1 4n
was found
7. Conclusions In this article, we have shown two properties of quantum local asymptotic normality (LAN) for qubits. First of all, we have seen that its radius of validity is arbitrarily close to n −1/4 rather than n −1/2 . And secondly, we have seen how LAN can be implemented physically, in a quantum optical setup. We use these properties to construct an asymptotically optimal estimator ρˆn of the qubit state ρ, provided that we are given n identical copies of ρ. Compared with other optimal estimation methods [4,29], our measurement technique makes a significant step in the direction of an experimental implementation. The construction and optimality of ρˆn are shown in three steps. I
II
In the preliminary stage, we perform measurements of σx , σ y and σz on a fraction n˜ = n 1−κ of the n atoms. As shown in Sect. 2, this yields a rough estimate ρ˜n which lies within a distance n −1/2+ of the true state ρ with high probability. In Sect. 3, it is shown that local asymptotic normality holds within a ball of radius n −1/2+η around ρ (η > ). This means that locally, for n → ∞, all statistical problems concerning the n identically prepared qubits are equivalent to statistical problems concerning a Gaussian distribution N u and its quantum analogue, a displaced thermal state φ u of the harmonic oscillator.
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Together, I and II imply that the principle of LAN has been extended to a global setting. It can now be used for a wide range of asymptotic statistical problems, including the global problem of state estimation. Note that this hinges on the rather subtle extension of the range of validity of LAN to neighborhoods of radius larger than n −1/2 . ⊗n√ on III LAN provides an abstract equivalence between the n-qubit states ρu/ n u u the one hand, and on the other hand the Gaussian states N ⊗ φ . In Sect. 4 and 5 it is shown that this abstract equivalence can be implemented physically by two consecutive couplings to the electromagnetic field. For the particular problem of state estimation, homodyne and heterodyne detection on the electromagnetic field then yield the data from which the optimal estimator ρˆn is computed.
Finally, in Sect. 6, it is shown that the estimator ρˆn , constructed above, is optimal in a local minimax sense. Local here means that optimality holds in a ball of radius slightly bigger than n −1/2 around any state ρ0 except the tracial state. That is, even if we had known beforehand that the true state lies within this ball around ρ0 , we would not have been able to construct a better estimator than ρˆn , which is of course independent of ρ0 . For this asymptotically optimal estimator, we have shown that the risk R converges to zero at rate R(ρ, ρˆn ) ∼ we have
8µ0 −4µ20 , n
lim sup
sup
with µ0 > 1/2 an eigenvalue of ρ. More precisely,
n→∞ ρ−ρ0 1 ≤n −1/2+
n R(ρ, ρˆn ) = 8µ0 − 4µ20 .
ˆ where we have chosen d(ρ, ˆ ρ) to be the The risk is defined as R(ρ, ρ) ˆ = E(d 2 (ρ, ρ)), L 1 -distance ρˆ − ρ 1 := Tr(|ρˆ − ρ|). This seems to be a rather natural choice because of its direct physical significance as the worst case difference between the probabilities induced by ρˆ and ρ on a single event. Even still, we emphasize that the same procedure can be applied to a wide range of other risk functions. Due to the local nature of the estimator ρˆn for large n, its rate of convergence in a risk R is only sensitive to the lowest order Taylor expansion of R in local parameters uˆ − u. The procedure can therefore easily be adapted to other risk functions, provided that the distance measure d 2 (ρuˆ , ρu ) is locally quadratic in uˆ − u. Remark. The totally mixed state (µ = 1/2) is a singular point in the parameter space, and Theorem 3.1 does not apply in this case. The effect of the singularity is that the family of states (3.6) collapses to a single degenerate state of infinite temperature. However this phenomenon is only due to our particular parametrisation, which was chosen for its convenience in describing the local neighborhoods around arbitrary states, with the exception of the totally mixed state. Had we chosen a different parametrisation, e.g. in terms of the Bloch vector, we would have found that local asymptotic normality holds for the totally mixed state as well, but the limit experiment is different: it consists of a three dimensional classical Gaussian shift, each independent component corresponding to the local change in the Bloch vector along the three possible directions. Mathematically, the optimal measurement strategy in this case is just to observe the classical variables. However this strategy cannot be implemented by coupling with the field since this coupling becomes singular (see Eq. (4.2)). These issues become more important for higher dimensional systems where the eigenvalues may exhibit more complicated multiplicities, and will be dealt with in that context.
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Acknowledgements. We thank Richard Gill for the many discussions which helped shape up the paper. We thank the School of Mathematics of the University of Nottingham, as well as the Department of Mathematics of the University of Nijmegen, for their warm hospitality during the writing of this paper. M. G. acknowledges the financial support received from the Netherlands Organisation for Scientific Research (NWO).
A. Appendix: Proof of Theorem 3.1 Here we give the technical details of the proof of local asymptotic normality with “slowly growing” local neighborhoods u ≤ n η , with η < 1/4. We start with the map Tn . A.1. Proof of Theorem 3.1; the map Tn . Let us define, for 0 < < (1/4−η) the interval (A.1) Jn = j : (µ − 1/2)n − n 1/2+ ≤ j ≤ (µ − 1/2)n + n 1/2+ . Notice that j ∈ Jn satisfies 2 j ≥ 2 n for all µ − 1/2 ≥ 2 and n big enough, independently of µ. Then Jn contains the relevant values of j, uniformly for µ − 1/2 ≥ 2 : lim pn,u (Jn ) = 1 − O(n −1/2+ ).
n→∞
(A.2)
This is a consequence of Hoeffding’s inequality applied to the binomial distribution, and recalling that pn,u ( j) = B(n/2 + j)(1 + O(n −1/2+ )) for j ∈ Jn . We upper-bound Tn (ρnu ) − N u ⊗ φ u by the sum u u u ∗ u 3 pn, j + N − pn,u ( j)τn, j + sup V j ρ j,n V j − φ 1 . (A.3) j∈ J n j∈Jn j∈Jn 1
The first two terms are “classical” and converge to zero uniformly over u ≤ n η : for the first term, this is (A.2), while the second term converges uniformly on µ − 1/2 ≥ 2 at rate n η−1/2 [37]. The third term can be analyzed as in Proposition 5.1 of [22]: u u ∗ u ∗ u u u V − φ ≤ − V φ V (A.4) V j ρn, ρ j + φ − Pj φ Pj 1 , j j n, j j 1
1
where P j := V j V j∗ is the projection onto the image of V j . We will show that both terms on the right side go to zero uniformly at rate n −1/4+η+ over j ∈ Jn and u ≤ n η . The trick is to note that displaced thermal equilibrium states are Gaussian mixtures of coherent states √ 1 2 2 φu = √ (A.5) e−|z− 2µ−1αu | /2s (|zz|) d 2 z, 2π s 2 where s 2 := (1 − µ)/(4µ − 2). The second term on the left side of (A.4) is bounded from above by √ 1 2 2 e−|z− 2µ−1αu | /2s |zz| − P j |zz|P j 1 d 2 z, √ 2π s 2
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which after some simple computations can be reduced (up to a constant) to 2 2 e−|z| /2s P j⊥ |z + 2µ − 1αu d 2 z.
(A.6)
We now split the integral. The first part is integrating over |z| ≥ n η+δ with 0 < δ < 2(η+δ) /(2s 2 ) 1/4 − η/2. The integral is dominated by the Gaussian and its value is O(e−n ). η+δ η The other part is bounded by the supremum over |z| ≤ 2n (as u ≤ n ) of P j⊥ |z . √ Now P j⊥ |z ≤ |z| j / j! = O(e−n(1/2−η−2δ) ) uniformly on j ∈ Jn , for any µ−1/2 ≥ 2 since then 2 j ≥ 2 n. The same type of estimates apply to the first term u u 0 ∗ u Ad U ρ − V φ V √ ρn, j − V j∗ φ u V j = j j n, j j 1 n 1 u 0 ∗ 0 ∗ 0 ∗ u Ad U V − V ≤ ρn, j − V j φ V j + φ V φ V √ j j j . (A.7) j j 1 n 1 The first term on the right side does not depend on u. From the proof of Lemma 5.4 of [22], we know that p 2 j+1 0 2 j+1 + p ρn, j − V j∗ φ 0 V j ≤ 1 1 − p 2 j+1 with p = (1 − µ)/µ. Now the left side is of the order p 2 j+1 which converges exponentially fast to zero uniformly on µ − 1/2 ≥ 2 and j ∈ Jn . The second term of (A.7) can be bounded again by a Gaussian integral 1 2 2 (A.8) e−|z| /2s (u, z, j) 1 d 2 z, √ 2 2π s where the operator (u, z, j) is given by √ ∗ (u, z, j) := Ad U j u/ n V j |zz|V j −V j∗ |z + 2µ−1αu z+ 2µ−1αu | V j . Again, we split the integral along z ≥ n η+δ . The outer part converges to zero faster than any power of n, as we have already seen. The inner integral, on the other hand, can be bounded uniformly over u ≤ n η , µ − 1/2 ≥ 2 and j ∈ Jn by the supremum of
(u, z, j) 1 over |z| ≤ 2n η+δ , µ√− 1/2 ≥ 2 , j ∈ Jn and u ≤ n η . √ Let z˜ ∈ R2 √ be such that αz˜ = z/ 2µ − 1, and denote ψ(n, j, v) = V j U j (v/ n)| j, j. Then, up to a 2 factor, (u, z, j) 1 is bounded from above by the
ψ(n, j, z˜ ) − |z
+ ψ(n, j, u + z˜ ) − |z + 2µ − 1αu u z˜ u + z˜ | j j − U j √ | j j + U j √ U j √ . n n n
(A.9)
This is obtained by adding and subtracting |ψ(n, j, z˜ )ψ(n, j, z˜ )| and √ |ψ(n, j, u + z˜ )ψ(n, j, u + z˜ )| and using the fact that |ψψ| − |φφ| 1 = 2 ψ − φ for normalized vectors ψ, φ.
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The two first terms are similar; we want to dominate them uniformly: we replace u + z˜ by z˜ with |z| ≤ 2n η+δ . We then write: 2
ψ(n, j, z˜ ) − |z =
∞
|k|ψ(n, j, z˜ ) − k|z|2
k=0
≤
r −1
∞ |k|ψ(n, j, z˜ )|2 + |k|z|2 . |k|ψ(n, j, z˜ ) − k|z| + 2 2
k=0
(A.10)
k=r
If z = |z|eiθ then we have [29] k √ √ 2 j−k 2j sin(|z|/ n)eiθ cos(|z| n) k|ψ(n, j, z˜ ) = , k
k iθ √ e |z| 2µ − 1 (2µ − 1)|z|2 . k|z = exp − √ 2 k! In (A.10) we choose r = n 2η+3 with 3 satisfying the conditions 2δ+2η+ < 2η+3 + < 1/2 and η + 3 < 1/4. Then the tail sums are of the order ∞ k=r ∞
(2n (η+δ) )2n |z|2r ≤ r! (n 2η+3 )!
2η+3
|k|z|2 ≤
|k|ψ(n, j, z˜ )|2 ≤
k=r
k j |z|2 k=r
n
= o exp(−n 2η+3 ) ,
|z|2r (2 j)! ≤n = o exp(−n 2η+3 ) . (2 j − k)!k! r!
For the finite sums we use the following estimates which are uniform over all |z| ≤ 2n η+δ , k ≤ r , j ∈ Jn :
((2µ − 1)n)k/2 (1 + O(n −1/2++2η+3 )), √ k! √ √ (sin(|z|/ n))k = (|z|/ n)k (1 + O(n 4η+3 +2δ−1 )), √ (2µ − 1)|z|2 (1 + O(n 2η−1/2++2δ )), (cos(|z|/ n))2 j−k = exp − 2 2j k
=
where we have used on the last line that (1 + x/n)n = exp(x)(1 + O(n −1/2 x)) for x ≤ n 1/2−4 (cf. [37]). This is enough to show that the finite sum converges uniformly to zero at rate O(n 2η−1/2++3 ) (the worst if 3 is small enough) and thus the first second terms in (A.9) as the square root of this, that is O(n η−1/4+/2+3 /2 ). Notice that the errors terms depend on µ only through j, and that 2 j ≥ n for µ − 1/2 ≥ 2 . Hence they are uniform in µ. We pass now to the third term of (A.9). By direct computation it can be shown that if we consider two general elements exp(i X 1 ) and exp(i X 2 ) of SU (2) with X i selfadjoint elements of M(C2 ) then exp(−i(X 1 +X 2 )) exp(i X 1 ) exp(i X 2 ) exp([X 1 , X 2 ]/2) = 1 + O(X i1 X i2 X i3 ),
(A.11)
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where the O(·) contains only third order terms in X 1 , X 2 . If X 1 , X 2 are in the linear span of σx and σ y then all third order monomials are such linear combinations as well. In particular we get that for z, u ≤ n η+3 : u v u+v U √ U √ exp(i(u x v y − u y vx )σz /n) U (β) := U − √ n n n 1 + O(n −2+4η+43 ) O(n −3/2+3η+33 ) = . (A.12) O(n −3/2+3η+33 ) 1 + O(n −2+4η+43 ) Finally, using the fact that | j, j is an eigenvector of L z , the third term in (A.9) can be written as
| j, j j, j| − U j (β)| j, j j, j|U j (β)∗
and both states are pure, so it suffices to show that the scalar product converges to one uniformly. Using (A.12) and the expression of j|U j (β)| j [29] we get, as j ≤ n, j j, j|U j (β)| j, j = U (β)1,1 = 1 + O(n −1+4η+43 ), which implies that the third term in (A.9) is of order O(n −1+4η+43 ). By choosing 3 and small enough, we obtain that all terms used in bounding (A.8) are uniformly O(n −1/4+η+ ) for any > 0. This ends the proof of convergence (3.8) from the n qubit state to the oscillator.
A.2. Proof of Theorem 3.1; the map Sn . The opposite direction (3.9) does not require much additional estimation, so will only give an outline of the argument. Given the state N u ⊗ φ u , we would like to map it into ρnu or close to this state, by means of a completely positive map Sn . Let X be the classical random variable with probability distribution N u . With X we generate a random j ∈ Z as follows: √ j (X ) = [ n X + n(µ − 1/2)]. This choice is evident from the scaling properties of the probability distribution pnu which we want to reconstruct. Let qnu be the probability distribution of j (X ). By classical local asymptotic normality results we have the convergence sup qnu − pnu 1 = O(n η−1/2 ).
u ≤n η
(A.13)
Now, if the integer j is in the interval Jn then we prepare the n qubits in block diagonal state with the only non-zero block corresponding to the j th irreducible representation of SU (2): 1 u ∗ u ⊥ u τn, := V φ V + Tr(P φ )1 ⊗ . j j j j nj u is trace preserving and completely positive [22]. The transformation φ u → τn, j
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If j ∈ / Jn then we may prepare the qubits in an arbitrary state which we also denote u . The total channel S then acts as follows: by τn, n j Sn : N u ⊗ φ u → τnu :=
n/2
u u qn, j τn, j .
j=0,1/2
We estimate the error ρnu − τnu 1 as u u / Jn ) + sup τn,
ρnu − τnu 1 ≤ qnu − pnu 1 + 2P pnu ( j ∈ j − ρn, j 1 . j∈Jn
The first term on the r.h.s. is O(n η−1/2 ) (see (A.13)), the second term is O(n −1/2 ) (see (A.2)). As for the third term, we use the triangle inequality to write, for j ∈ Jn , u u u ∗ u ∗ ∗ u ∗ u
τn, j − ρn, j 1 ≤ τn, j − V j φ V j 1 + V j φ V j − ρn, j 1 .
The first term is O(e−n(1/2−η−2δ) ), according to the discussion following Eq. (A.6). The second term on the right is O(n −1/4+η+ ) according to equations (A.7) through (A.12). Summarizing, we have Sn (N u ⊗ φ u ) − ρnu 1 = O(n −1/4+η+ ), which establishes the proof in the inverse direction. B. Appendix: Proof of Theorem 4.1 First estimate. We build up the state ρ˜ uj,n by taking linear combinations of number states |m to obtain an approximate coherent state |z, and finally mixing such states with a Gaussian distribution to get an approximate displaced thermal state. Consider the approximate coherent vector Pm˜ |z, for some fixed z ∈ C and m˜ = n γ , with γ to be fixed later. Define the normalized vector n |ψz, j :=
m˜ |z|m 1 √ |m.
Pm˜ |z
m! m=0
(B.1)
We mix the above states to obtain √ 1 2 2 u n n ρ˜ j,n := √ ψz, | d 2 z. e−|z− 2µ−1αu | /2s |ψz, j j 2π s 2 Recall that s 2 = (1 − µ)(4µ − 2), and √ 1 2 2 φu = √ e−|z− 2µ−1αu | /2s (|zz|) d 2 z. 2 2π s n we have From the definition of |ψz, j n
|ψz, j − |z ≤
which implies √
ρ˜ uj,n
2
− φ 1 ≤ √ π s2 u
e
−|z|2 /2s 2
|z +
√ |z|m˜ 2 √ ∧ 2, m! ˜
(B.2)
√ 2µ − 1αu |m˜ √ 2(η+) ), ∧ 2 d 2 z = O(e−n √ m! ˜
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for any > 0, for any γ ≥ 2(η + ). Indeed we can split the integral into two parts. The integral over the domain |z| ≥ n η+ is dominated by the Gaussian factor and is 2(η+) O(e−n ). The integral over the disk |z| ≤ n η+ is bounded by supremum of (B.2) γ since the Gaussian integrates toone, and is O(e−(γ /2−η−)n ). In the last step we use γ √ Stirling’s formula to obtain log (n η+ )n / n γ ! ≈ (η + − γ /2)n γ log n. Note that the estimate is uniform with respect to µ − 1/2 > 2 for any fixed 2 > 0. Second estimate. We now compare the evolved qubits state ρ˜ uj,n (t) and the evolved osciln (t) = U (t) |m ⊗ | be the joint state at time t when the lator state φ u (t). Let |ψm, j,n j initial state of the system is |m corresponding to | j, j − m in the L z basis notation. n (t): We choose the following approximation of |ψm, j n |ξm, j (t) :=
m
cn (m, i)αi (t)|m − i ⊗ |e−1/2u χ[0,t] (u)i ,
i=0
where αi (t) = exp((−m + i)t/2), cn (m, i) := cn (m, i − 1)
f ⊗n
2 j−m+i 2 jn
(B.3)
m−i+1 i
with
cn (m, 0) := 1, and | f n := as defined in (4.1). In particular for µ − 1/2 > 2 and m 2 −1/2+ i ). j ∈ Jn we have cn (m, i) ≤ i (1 + 2 n We apply now the estimate (4.5). By direct computations we get 1 cn (m, i)αi (t)(m − i)|m − i ⊗ |e−1/2u χ[0,t] (u)i dt 2 m
n d|ξm, j (t) = −
i=0
+
m
cn (m, i)αi−1 (t)|m−i ⊗ |e−1/2u χ[0,t] (u)i−1 ⊗s |χ[t,t+dt] , (B.4)
i=1
where f ⊗i ⊗s g :=
i+1
f ⊗ f ⊗ · · · ⊗ g ⊗ · · · ⊗ f.
k=1
From the quantum stochastic differential equation we get n (t) G dt |ξm, j
=−
+
m 2j − m +i + 1 1 cn (m, i)αi (t)(m − i) |m − i ⊗ |e−1/2u χ[0,t] (u)i dt 2 2 jn i=0
m
cn (m, i)αi (t)
i=0
(m − i)(2 j − m + i + 1) |m − i − 1 ⊗ |e−1/2u χ[0,t] (u)i ⊗s |χ[t,t+dt] . 2 jn (i + 1)
(B.5)
j−m+i+1) In the second term of the right side of (B.5) we can replace cn (m, i) (m−i)(2 , 2 jn (i+1) by cn (m, i + 1), and thus we obtain the same sum as in the second term of the left side of (B.4). Thus n n G dt |ξm, j (t) − d|ξm, j (t)
=
m−1 2( jn − j) + m − i − 1 1 cn (m, i)αi (t)(m − i) |m − i ⊗ |e−1/2u χ[0,t] (u)i dt. 2 2 jn i=0
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m −1/2+ )i we get that G ξ n (t) − dξ n (t)
Then using cn (m, i) ≤ dt m, j m, j i (1 + (2/2 )n is bounded from above by ⎤ ⎡ 1/2 m−1 (2( jn − j) + m − i − 1)(m − i) 2 ⎦ 1 ⎣ m −1/2+ −t i −(m−i)t ((1 + n )(1 − e )) e dt. 2 2 jn i i=0
We have (2( jn − j) + m − i − 1)(m − i) = O(m(n −1/2+ + n −1 m)). 2 jn Inside the sum we recognize the binomial terms with the m th term missing. Thus the sum is m m 1 + n −1/2+ − e−t n −1/2+ − (1 − e−t )(1 + n −1/2+ ) ≤ (1 + n −1/2+ )m (1 − (1 − e−t )m ) ≤ (1 + n −1/2+ )m me−t . Then there exists a constant C (independent of µ if µ − 1/2 ≥ 2 ) such that C −t/2 3/2 −1/2+ 2 −1/2+ m/2 n n −1 e
G dt ξm, (t) − dξ (t)
≤ m (n + mn ) 1 + n . j m, j 2 2 By integrating over t we finally obtain 2 −1/2+ m/2 n n 3/2 −1/2+ −1
ψm, j (t) − ξm, j (t) ≤ Cm (n + mn ) 1 + n . (B.6) 2 Note that under the assumption γ < 1/3 − 2/3, the right side converges to zero at rate n 3γ /2−1/2+ for all m ≤ m˜ = n γ . Summarizing, the assumptions which we have made so far over γ are 2η + 2 < γ < 1/3 − 2/3. n as defined in (B.1) and let us denote |ψ n (t) = Now consider the vector |ψz, j z, j n U j,n (t)|ψz, j ⊗ |. Then based on (B.3) we choose the approximate solution
|ξz,n j (t)
=e
−|z|2 /2
m m˜ |z|m cn (m, i)αi (t)|m − i ⊗ |e−1/2u χ[0,t] (u)i . √ m! m=0 i=0
n (t) and |ξ n (t) live in the “k-particle” subspace of H ⊗ Note that the vectors |ψk, j j k, j n (t) and |ξ n (t) with p = k. By F(L 2 (R)) and thus are orthogonal to all vectors |ψ p, j p, j (B.6), the error is ⎛ ⎞ m 1/2 m˜ 2m 2 |z| 2 n n −|z| /2 ⎝
ψz, m 3 (n −1/2+ + mn −1 )2 1 + n −1/2+ ⎠ j (t) − ξz, j (t) ≤ Ce m! 2 m=0
+
|z|2m˜ m! ˜
m/2 ˜ 2 |z|2m˜ . ≤ C m˜ 3/2 (n −1/2+ + mn ˜ −1 ) 1 + n −1/2+ + 2 m! ˜
(B.7)
We now compare the approximate solution ξz,n j (t) with the “limit” solution ψz (t) for the oscillator coupled with the field as described in Sect. 4.2. We can write
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ψz (t) = e−|z|
2 /2
m ∞ m −(m−i)t/2 |z|m e |m − i ⊗ |e−1/2u χ[0,t] (u)i . √ i m! m=0 i=0
Then
ξz,n j (t) − ψz (t) 2 =e
−|z|2
2 ∞ m |z|2m −t i −|z|2 . (1 − e ) + e i m!
m˜ m |z|2m −(m−i)t e cn (m, i) − m!
m=0
m=m˜
i=0
Now cn (m, i) −
2 m m ≤ cn (m, i)2 − i i i % m 2( j − jn ) − m + p 1− ≤ 1+ i 2 jn p=1 m ≤ C2 mn −1/2+ , i
where C2 does not depend on µ as long as µ − 1/2 ≥ 2 (recall that the dependence in µ is hidden in jn = (2µ − 1)n). Thus 2
n (t) − ψ (t) 2 ≤ C n −1/2+ e−|z|
ξz, z 2 j
m˜ |z|2m˜ m|z|2m |z|2m˜ + ≤ C2 n −1/2+ |z|2 + . m! m! ˜ m! ˜
(B.8)
m=0
From (B.7) and (B.8) we get n (t) − ψ (t)
ψz, z j ⎡
& '1/2 ⎤ ˜ 2 −1/2+ m/2 |z|2m˜ |z|2m˜ 3/2 −1/2+ −1 −1/2+ 2 ⎣ ⎦ + C2 n + mn ˜ ) 1+ n + |z| + ≤ 2 ∧ C m˜ (n 2 m! ˜ m! ˜ := E(m, ˜ n, z).
We now integrate the coherent states over the displacements z as we did in the case of local asymptotic normality in order to obtain the thermal states in which we are interested √ 1 2 2 u n n ρ˜ j,n := √ ψz, | d 2 z. e−|z− 2µ−1αu | /2s |ψz, j j 2π s 2 We define the evolved states ρ˜ uj,n (t) := U j,n (t)ρ˜ uj,n U j,n (t)∗ ,
and
Then 1 sup sup ρ˜ uj,n (t) − φ u (t) 1 ≤ sup √ η η π s2
u ≤n j∈Jn u ≤n
φ u (t) := U (t)φ u U (t)∗ , √
e−|z−
2µ−1αu |2 /2s 2
E(m, ˜ n, z) d 2 z.
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Here again we cut the integral in two parts. On |z| ≥ n η+ , the Gaussian dominates, η+ and this outer part is less than e−n . Now the inner part is dominated by sup|z|≤n η+ E(m, ˜ n, z). Now we want m˜ to be not too big for (B.7) to be small, on the other hand, ˜ to go to zero. A choice which satisfies the condition is γ = 2η + 3. we want z2m˜ /m! By renaming we then get E(m, ˜ n, z) = O(n η−1/4+ , n 3η−1/2+ ), for any small enough > 0. Hence we obtain (4.6). References 1. Artiles, L., Gill, R.D., Gu¸ta˘ , M.: An invitation to quantum tomography. J. Royal Statist. Soc. B (Methodological) 67, 109–134 (2005) 2. Bagan, E., Baig, M., Muñoz-Tapia, R.: Optimal Scheme for Estimating a Pure Qubit State via Local Measurements. Phys. Rev. Lett. 89, 277904 (2002) 3. Bagan, E., Baig, M., Muñoz-Tapia, R., Rodriguez, A.: Collective versus local measurements in a qubit mixed-state estimation. Phys. Rev. A 69, 010304(R) (2004) 4. Bagan, E., Ballester, M.A., Gill, R.D., Monras, A., Muñoz-Tapia, R.: Optimal full estimation of qubit mixed states. Phys. Rev. A 73, 032301 (2006) 5. Bagan, E., Monras, A., Muñoz-Tapia, R.: Comprehensive analysis of quantum pure-state estimation for two-level system. Phys. Rev. A 71, 062318 (2005) 6. Barndorff-Nielsen, O.E., Gill, R., Jupp, P.E.: On quantum statistical inference (with discussion). J. R. Statist. Soc. B 65, 775–816 (2003) 7. Barndorff-Nielsen, O.E., Gill, R.D.: Fisher information in quantum statistics. J. Phys. A 33, 1–10 (2000) 8. Bouten, L., Gu¸ta˘ , M., Maassen, H.: Stochastic Schrödinger equations. J. Phys. A 37, 3189–3209 (2004) 9. Butucea, C., Gu¸ta˘ , M., Artiles, L.: Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data. Ann. Stat 35(2), 465–494 (2007) 10. Caves, C.M.: Quantum limits on noise in linear amplifiers. Phys. Rev. D 26, 1817–1839 (1982) 11. Cirac, J.I., Ekert, A.K., Macchiavello, C.: Optimal Purification of Single Qubits. Phys. Rev. Lett. 82, 4344 (1999) 12. D’Ariano, G.M., Leonhardt, U., Paul, H.: Homodyne detection of the density matrix of the radiation field. Phys. Rev. A 52, R1801–R1804 (1995) 13. Dyson, F.J.: General Theory of Spin-Wave Interactions. Phys. Rev. 102, 1217–1230 (1956) 14. Embacher, F., Narnhofer, H.: Strategies to measure a quantum state. Ann. of Phys. (N.Y.) 311, 220 (2004) 15. Fisher, D.G., Kienle, S.H., Freyberger, M.: Quantum-state estimation by self-learning measurements. Phys. Rev. A 61, 032306 (2000) 16. Fujiwara, A.: Strong consistency and asymptotic efficiency for adaptive quantum estimation problems. J. Phys. A 39, 12489–12504 (2006) 17. Fujiwara, A., Nagaoka, H.: Quantum Fisher metric and estimation for pure state models. Phys. Lett. A 201, 119–124 (1995) 18. Gardiner, C.W., Zoller, P.: Quantum Noise. Berlin-Heidelberg-New York: Springer, 2004 19. Geremia, J., Stockton, J.K., Mabuchi, H.: Real-Time Quantum Feedback Control of Atomic Spin-Squeezing. Science 304, 270–273 (2004) 20. Gill, R.D.: Asymptotic information bounds in quantum statistics. http://arxiv.org/abs/math.ST/0512443, 2005, to appear in Annals of Statistics 21. Gill, R.D., Massar, S.: State estimation for large ensembles. Phys. Rev. A 61, 042312 (2000) 22. Gu¸ta˘ , M., Kahn, J.: Local asymptotic normality for qubit states. Phys. Rev. A 73, 052108 (2006) 23. Gu¸ta˘ , M., Jenˇcová, A.: Local asymptotic normality in quantum statistics, preprint quant-ph/0606213, to appear in Commun. Math. Phys 24. Hannemann, T., Reiss, D., Balzer, C., Neuhauser, W., Toschek, P.E., Wunderlich, C.: Self-learning estimation of quantum states. Phys. Rev. A 65, 050303(R) (2002) 25. Hayashi, M.: Presentations at MaPhySto and QUANTOP Workshop on Quantum Measurements and Quantum Stochastics, Aarhus, 2003, and Special Week on Quantum Statistics, Isaac Newton Institute for Mathematical Sciences, Cambridge, 2004 26. Hayashi, M.: Two quantum analogues of Fisher information from a large deviation viewpoint of quantum estimation. J. Phys. A: Math. Gen. 35, 7689–7727 (2002) 27. Hayashi, M.: Quantum estimation and the quantum central limit theorem. Bull. Math. Soc. Japan 55, 368–391 (2003) (in Japanese; Translated into English in quant-ph/0608198)
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Commun. Math. Phys. 277, 161–187 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0368-2
Communications in
Mathematical Physics
Singular Hyperbolic Monopoles Oliver Nash Mathematical Institute, Oxford OX1 3LB, UK. E-mail: [email protected] Received: 20 November 2006 / Accepted: 14 May 2007 Published online: 30 October 2007 – © Springer-Verlag 2007
Abstract: We study the twistor theory of singular hyperbolic SU (2) monopoles following the approach taken by Kronheimer [9] in the Euclidean case. We use our results to show that the moduli space of charge 1 monopoles possesses a natural 2-sphere of scalar flat Kähler metrics. In the zero mass limit, the metrics reduce to a class of metrics first studied by LeBrun in [10]. 1. Overview Singular monopoles were first studied by Kronheimer in [9]. He showed how to extend the twistor theory of non-singular Euclidean monopoles developed in [7] to allow for monopoles with a finite number of prescribed singularities. Furthermore, he then used these techniques to construct the twistor space of the charge 1 moduli space of singular monopoles on R3 and thus found that it carried a natural hyperkähler structure (indeed the moduli space is the smooth part of a quotient of multi–Taub–NUT space). Here we offer a hyperbolic version of Kronheimer’s work. That is, we show how to extend the twistor theory of non-singular hyperbolic monopoles developed in [11] to allow for monopoles on hyperbolic space with a finite number of prescribed singularities. Our main reason for developing this theory is that, just as in the Euclidean case, this allows us to construct the twistor space of the lowest dimensional moduli spaces and hence study their geometry. We find that the moduli space of charge 1 singular hyperbolic monopoles possesses a natural 2-sphere of scalar-flat Kähler metrics all within the same conformal class. The 2-sphere appears naturally as the boundary of hyperbolic space. The motivation for this work is to understand the natural geometry of hyperbolic monopole moduli spaces.1 Since Kronheimer’s work made it possible to see the natural hyperkähler geometry of the charge 1 singular Euclidean monopole moduli spaces in a very explicit way it is natural to ask the same question in the hyperbolic case. 1 Note, see [2], that the natural “L 2 metric”diverges.
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The geometric structure we identify on the charge 1 moduli space perhaps deserves additional interest owing to the fact that a limiting case of it has already been studied in some detail by LeBrun [10]. As we shall see, the spaces studied by LeBrun correspond to the zero mass limit of our monopole moduli spaces. There are also connections between the metrics studied here and a class of metrics introduced in a recent paper of Gibbons and Warnick, see [5]. They study the motion of a single hyperbolic monopole in the presence of a finite number of fixed hyperbolic monopoles. Finally we must point out that singular monopoles have recently been studied by Kapustin and Witten in [8] as part of their work on the Geometric Langlands Programme. Although the context of their work is very different to ours, [8] nevertheless provides a further justification for the study of singular monopoles. 2. Definitions and Elementary Properties The monopoles we are interested in are solutions of the Bogomolny equations with singularities at a fixed set of points in hyperbolic space. The definition below gives the precise behaviour of the solutions at these points. Definition 1. Let { p1 , . . . , pn } ⊂ H3 be n distinct points in hyperbolic space. Let U = H3 \ { p1 , . . . , pn } and let π : E → U be a C ∞ SU (2) vector bundle on U . A singular hyperbolic SU (2) monopole with singularities at p1 , . . . , pn is an SU (2) connection ∇ on E together with a Higgs field Φ ∈ Ω 0 (U, End(E)) such that: (i) Φ and ∇ satisfy the Bogomolny equations: ∇Φ = ∗F∇ (where F∇ ∈ Ω 2 (U, End(E)) is the curvature of ∇). (ii) (∇, Φ) satisfy the boundary conditions BC0, BC1, BC2 defined in [11]. If we fix a point O ∈ H3 then (for SU (2) monopoles) these conditions can be described as follows. Let A B −B ∗ −A be the connection matrix of ∇ in a gauge of unitary eigenvectors of Φ. If ρ is the hyperbolic distance from O then we require – (Φ − m)e2ρ extends smoothly to ∂H3 for some m > 0 – A extends smoothly to ∂H3 – Be2mρ extends smoothly to ∂H3 . (iii) Φ has the following behaviour at singular points: lim (ρi Φ) exists and is (strictly) positive,
ρi →0
d (ρi Φ) is bounded in a neighbourhood of pi , where ρi is the hyperbolic distance from pi . The boundary conditions (iii) of Definition 1 deserve elaboration. To see where they come from, let V : U → R be V =λ+
n i=1
G pi
(1)
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for some λ ≥ 0 and where G p (x) =
1 e2ρ( p,x)
(2)
−1
is the Green’s function for the hyperbolic Laplacian centred at p and normalised so that 1 ∆G p = −2π δ p . Let M be the principal U (1) bundle on U with Chern class 2π [∗d V ]. 1 Let ω be a connection on M with curvature 2π ∗ d V and define the metric g on M by g = V gH3 + V −1 ω ⊗ ω. We give M the orientation defined by volH3 ∧ ω. As shown in [10], we may add in a fixed point pˆ i of the S 1 action on M over each pi ∈ H3 to obtain a smooth manifold ˆ Now if (∇, Φ) is a conMˆ = M ∪ { pˆ 1 , . . . , pˆ n } and the metric extends smoothly to M. nection and Higgs field on U then (suppressing the notation for pull backs) we define a connection ∇ˆ on M according to the correspondence (∇, Φ) → ∇ˆ = ∇ − V −1 Φ ⊗ ω.
(3)
Using the formulae F∇ˆ = F∇ − V −1 Φ ⊗ dω + V −2 Φ ⊗ d V ∧ ω − V −1 ∇Φ ∧ ω and dω = ∗d V, ∗ˆ α = V −1 (∗α) ∧ ω for α ∈ ∧2 T ∗ U , ∗ˆ (α ∧ ω) = V ∗ α for α ∈ T ∗ U , (where ∗ˆ denotes the Hodge ∗-operator on M and ∗ is the Hodge ∗-operator on U ) it follows that (∇, Φ) satisfy the Bogomolny equations on U iff ∇ˆ satisfies the anti-self-dual Yang–Mills equations on M. We thus have a correspondence between solutions of the Bogomolny equations on U and S 1 -invariant solutions of the anti-self-dual Yang–Mills equations on M. The promised elucidation of the aforementioned boundary conditions can now be stated as Lemma 1. In the above notation, (∇, Φ) satisfy the boundary conditions (iii) of Definition 1 iff the corresponding solution of the anti-self-dual Yang-Mills equations on M ˆ extends to a solution on M. Proof. The proof is completely analogous to the corresponding result in [9].
ˆ the fibre of E over Now if we have an S 1 invariant instanton on a bundle E → M, 3 ˆ the point pˆ i ∈ M lying above a singular point pi ∈ H will carry a representation of S 1 . Since the S 1 action is compatible with the SU (2) structure of E, this action must have weights (li , −li ) for some integer li ≥ 0. The question arises of identifying this integer li in terms of the corresponding solution of the Bogomolny equations on U . In fact li = 2 lim (ρ( p, pi )Φ( p)). p→ pi
(4)
To see why, fix a trivialisation of E in a neighbourhood of pˆ i . Let A be the corresponding matrix of 1-forms and let A0 = A(X ), where X is the vector field on Mˆ generated by the
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S 1 action. For each p ∈ H3 near pi choose a gauge transformation g : M p → SU (2) on the corresponding S 1 orbit that takes our fixed trivialisation to an S 1 -invariant one. In view of (3) we thus have −V −1 Φ = g −1 A0 g + g −1 X (g). Now as p → pi , g −1 A0 g = A0 → 0, since X vanishes at pi . Furthermore g −1 X (g) → li since g is approaching the S 1 representation with weights (li , −li ). Equation (4) now follows upon noting that lim V −1 Φ = 2 lim ρi Φ since lim 2ρi V = 1. p→ pi
p→ pi
p→ pi
Definition 2. In the above notation, we define the Abelian charge li of the monopole at n pi by Eq. (4). We also define the total Abelian charge l of the monopole as l = li . i=1
Definition 3. Let O ∈ H3 . From condition (ii) of Definition 1 the limit m=
lim
ρ( p,O)→∞
φ( p) ∈ R
exists and is (strictly) positive. We define m to be the mass of the monopole. Fix a point O ∈ H3 . Since Φ( p) → m > 0 as ρ( p, O) → ∞ we can choose a sphere S in H3 centred at O large enough that the singular points and zeros of Φ all lie inside S. On such a sphere, the bundle E splits as a direct sum of eigenbundles of Φ, E| S = M + ⊕ M − , where M ± is the bundle corresponding to the eigenvalue ±iΦ. (Note that these bundles are interchanged by the quaternionic structure on E.) Definition 4. In the above notation and using the natural orientation of S, we define the total charge N of the monopole by N = c1 (M + )[S]. Definition 5. We define the non-abelian charge k of a monopole to be k = N + l. It is important to address the issue of existence of singular hyperbolic monopoles. As we shall see, the key is a result of LeBrun in [10]. We have seen that given a harmonic function V on U as in (1) we obtain the Riemannian manifold Mˆ of Lemma 1. The function V depends on a choice of λ ≥ 0 and LeBrun [10] shows that for λ = 1, Mˆ has an S 1 -equivariant conformal compactification M c obtained by adding a 2-sphere of fixed points of the S 1 action on Mˆ and gluing along the boundary of H3 (which Mˆ fibres over). Furthermore, after reversing the orientation, M c is diffeomorphic to nCP2 = CP2 # · · · #CP2 and the conformal class (which is self-dual) contains a metric of positive scalar curvature. For n = 0 (i.e. no singularities) this construction is of course the usual observation that round S 4 is an S 1 -equivariant
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conformal compactification of H3 × S 1 which was used very successfully by Atiyah in [1] to study monopoles on H3 . For n = 1 we obtain CP2 with the usual Fubini-Study conformal structure. Now in [11], it is noted that an S 1 -invariant instanton on S 4 corresponds to a solution of the Bogomolny equations on H3 that satisfies the boundary conditions (ii) of Definition 1. Similarly and in view of Lemma 1, an S 1 -invariant anti-self-dual instanton on M c corresponds to a solution of the Bogomolny equations satisfying conditions (ii) and (iii) of 1. This is the same as a self-dual instanton on nCP2 (we don’t have to be careful whether we consider anti-self-dual or self-dual instantons on S 4 since it carries an orientation reversing diffeomorphism). Existence of our singular monopoles then follows from the existence of S 1 -invariant self-dual instantons on the self-dual manifolds nCP2 . Indeed a careful equivariant index calculation can be used to calculate the dimension of the moduli space, 4k − 1. In fact, in view of Buchdahl’s construction [3] of instantons on CP2 , it should even be possible to obtain explicit formulae for singular hyperbolic monopoles just as the same is possible by applying an S 1 -invariant version of the ADHM construction for instantons on S 4 to obtain formulae for non-singular hyperbolic monopoles. 3. The Hitchin–Ward Correspondence The Hitchin–Ward transform is the fundamental theorem that tells us how to interpret solutions of the Bogomolny equations in twistor space. In this section we address the question of what happens to the data in twistor space when the solutions of the Bogomolny equations have singularities as prescribed in Definition 1. Before stating the theorem, we find it convenient to introduce some terminology and make some elementary observations about hyperbolic space. Definition 6. Given an oriented geodesic γ in hyperbolic space and a point O ∈ H3 there exists a unique parameterisation of γ such that γ is parameterised by arc length and γ (0) is the closest point to O on γ . We call this the parameterisation of γ determined by O ∈ H3 . We shall denote the twistor space (i.e. the set of oriented geodesics) of H3 by Q. If x ∈ H3 , we shall denote the corresponding twistor line (the set of all geodesics passing through x) in Q by Px . Finally, if we fix a point O ∈ H3 , then we can identify Q P1 × P1 \∆, where (the so-called anti-diagonal) is ∆ = ( p, τ ( p)), and τ is the usual (anti-podal) real structure on P1 (τ : [z, w] → [−w, z]). The diagonal ∆ ⊂ P1 × P1 \∆ appears as the twistor line of the chosen point O ∈ H3 . Now consider the natural double fibration ν H3
SH3
µ @ R @ Q
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where SH3 ⊂ T H3 is the unit tangent bundle of H3 . Using the hyperbolic metric, we have T H3 T ∗ H3 and so pulling back the natural 1-form on T ∗ H3 , T H3 and hence SH3 carries a natural 1-form θˆ ∈ Ω 1 (SH3 ).
(5)
Thus if fˆ : Q → SH3 is a section of µ we obtain a 1-form θ = fˆ∗ θˆ on Q. Remark 1. A point O ∈ H3 determines a section of µ, namely γ → γ˙ (0) in the parameterisation of γ determined by O. Thus, by the above P1 × P1 \∆ carries a natural 1-form θ . As we shall see later, it is really the (0, 1) component of θ that interests us. Using the coordinates, ([z, 1], [w, 1]) on (an open set of) P1 × P1 \∆, the explicit formula for the (0, 1) component of θ is: d z¯ d w¯ θ 0,1 = (z − w) + . (6) (1 + z z¯ )(1 + z¯ w) (1 + w w)(1 ¯ + z w) ¯ Note that ∂θ 0,1 = 0 and that θ vanishes on the twistor line ∆ (where z = w) in P1 × P1 \∆ corresponding to those geodesics passing through O, and has a singularity along ∆ (where z = −1/w). Lemma 2. Let fˆ be a section of µ above and let f = ν ◦ fˆ : Q → H3 . Let θ = fˆ∗ θˆ and let ω ∈ ∧2 T ∗ H3 . Then ( f ∗ ω)0,2 + iθ 0,1 ∧ ( f ∗ (∗ω))0,1 = 0, where ∗ is the Hodge star on H3 . Proof. This follows from results in [11]. (In particular see Eq. (3.5) of [11]).
Remark 2. The above results also hold with R3 in place of H3 . In the case of R3 if we ∂ use the usual coordinates (η, ζ ) → η ∂ζ on its twistor space T T S 2 , then the formula for its natural 1-form is: θT0,1 =
2η d ζ¯ . (1 + ζ ζ¯ )2
(7)
This 1-form is also the 1-form obtained by using the usual round metric on S 2 to obtain T S 2 T ∗ S 2 and pulling back the natural 1-form on T ∗ S 2 . Definition 7. Let { p1 , . . . , pn } ⊂ H3 be n distinct points in hyperbolic space and let x ∈ H3 . Suppose that there exists a geodesic γ ∈ Px and pi , p j ∈ γ such that x separates pi and p j on γ . Then we say x is geodesically trapped by { p1 , . . . , pn }. We are finally ready to state the Hitchin–Ward correspondence. The proof used here owes most to the proof of the corresponding result in [11]. Theorem 1. Let { p1 , . . . , pn } ⊂ H3 be n distinct points in hyperbolic space. Let U = H3 \ { p1 , . . . , pn }. Let P = P1 ∪ · · · ∪ Pn ⊂ Q (where Pi = Ppi ). Then to each solution of the SU (2) Bogomolny equations on U , there corresponds a pair of rank 2 holomorphic vector bundles (E + , E − ) on Q together with an isomorphism h : E + |Q\P → E − |Q\P of holomorphic vector bundles such that:
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(i) If x ∈ U is a point that is not geodesically trapped by { p1 , . . . , pn }, then there x exists a partition of Px ∩ P into two disjoint sets: Q +x , Q − x such that E is naturally x is the vector bundle isomorphic to the trivial vector bundle with fibre E x , where E over Px obtained by gluing E + | Px\Q +x and E − | Px\Q −x together over Px \ P using h. (ii) E ± carry holomorphic symplectic structures compatible with h. (iii) There exists an anti-holomorphic (anti-linear) map j˜ : E + → E − (covering σ : Q → Q) such that ˜ 2 = −1 (h −1 j) over Q\ P. Furthermore the holomorphic data determines the solution of the Bogomolny equations. Proof. The essence of the theorem is by now standard. We sketch a proof that emphasises the differences that arise because of the singular points pi . Thus, fix a point O ∈ H3 and let R > 0 be large enough that { p1 , . . . , pn } ⊂ B(O, R). Define f : Q → H3 by γ → γ (R), where γ is given the parameterisation determined by O ∈ H3 . Note we have f (Q) ⊂ U ⊂ H3 and so it makes sense to define2 E+ = f ∗ E and ∂ : Ω 0 (Q, E + ) → Ω 0,1 (Q, E + ) by ∂s = (( f ∗ ∇)s − i( f ∗ Φ)(s) ⊗ θ )0,1 , ˆ θˆ is the 1-form of Eq. (5) and fˆ is the map: where θ = fˆ∗ θ, fˆ : Q → SH3 , γ → γ˙ (R). 2 We thus have ∂ = F 0,2 , where F∇ˆ is the curvature of the connection ∇ˆ = f ∗ ∇ − ∇ˆ ∗ i f Φ ⊗ θ . But
F∇ˆ = f ∗ F∇ + iθ ∧ f ∗ (∇Φ) − i f ∗ Φ ⊗ dθ. 2 Although E + depends on R, different values of R give naturally isomorphic bundles.
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Using the Bogomolny equations F∇ = ∗∇Φ and the fact that ∂θ 0,1 = 0 we find 2
∂ = ( f ∗ F∇ )0,2 + iθ 0,1 ∧ ( f ∗ (∗F∇ ))0,1 . 2
In view of Lemma 2 we thus have ∂ = 0 and so we have a holomorphic structure on E + . Define the holomorphic bundle E − using the same construction as for E + but with f ◦ σ in place of f . Note that we thus have E − = σ ∗ E + as complex (but obviously not holomorphic) vector bundles. To define h : E + |Q\P → E − |Q\P , note that if γ ∈ Q then σ (γ ) is just γ parameterised in the opposite direction, i.e. σ (γ ) : t → γ (−t). Thus E γ− E σ+(γ ) E γ (−R) , and so to define h we must define an isomorphism: h γ : E γ (R) E γ (−R) for all γ ∈ Q\ P. Thus fix γ ∈ Q\ P and let v ∈ E γ (R) . Note that γ ⊂ U and so let s be the unique section of E along γ such that s(γ (R)) = v and (∇γ˙ − iΦ)s = 0. We define h γ (v) = s(γ (−R)). It is straightforward to check that h is indeed a holomorphic bijection. To see that condition (i) holds, let x ∈ U be a point that is not geodesically trapped by { p1 , . . . , pn }. Let Q± x = {γ ∈ Px | there exists pi ∈ γ separating x and γ (±R) on γ }. + − Note that Q +x ∪ Q − x = P ∩ Px and Q x ∩ Q x = ∅ since x is not geodesically trapped. We define
ψ + : E + | Px\Q +x → (Px \ Q +x ) × E x as follows. Let γ ∈ Px \ Q +x and let γ + be the closed segment of γ joining x and γ (R). Let v ∈ E γ+ E γ (R) and let s be the unique section of E over γ + ⊂ U such that s(γ (R)) = v and (∇γ˙ − iΦ)s = 0.
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Define ψ + (v) = (γ , s(x)) ∈ (Px \ Q +x ) × E x . Similarly define ψ − : E − | Px\Q −x → (Px \ Q − x ) × Ex by using a section s over the closed segment γ − ⊂ U of γ joining x and γ (−R) such that s(γ (−R)) = v (since E γ− E γ (−R) ). Then define x → Px × E x ψ:E by [v] →
ψ + (v) if v ∈ E +Px\Q + x . ψ − (v) if v ∈ E − − P \Q x
x
It is straightforward to verify that ψ is well defined and is the required trivialisation. To see that condition (ii) holds, note that E carries a symplectic structure χ ∈ C ∞ (U, ∧2 E ∗ ). Using f , we pull this back to a symplectic structure ∗
χ˜ = f ∗ χ ∈ C ∞ (Q, ∧2 E + ). Similarly, E − carries a symplectic structure. It is straightforward to verify that these are holomorphic and compatible with h. Finally for condition (iii) let j : E → E be the quaternionic structure carried by E and σˆ : E − → E + be the bijection induced by σ : Q → Q. We define j˜ : E + → E − by j˜ = σˆ −1 ◦ ( f ∗ j). Again it is straightforward to verify that j˜ has the required properties. We shall omit the proof that the holomorphic data determines the solution to the Bogomolny equations.
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Remark 3. Theorem 1 is of course well known [1,11] in the case n = 0 (i.e. no singularities). In this case, the proof we have offered holds if we take R = 0. Thus h is an isomorphism on all Q and so we really obtain a holomorphic vector bundle on Q rather than a triple (E + , E − , h) as in the singular case. Furthermore, it is clear that the proof could be generalised to groups other than SU (2). In particular we may consider U (1) monopoles with no singularities on H3 . In this case the trivial unit mass U (1) monopole on H3 yields a holomorphic line bundle L over Q. Note that L will carry a canonical anti-holomorphic (anti-linear) bijection: L → L∗
(8)
covering σ : Q → Q since we started with a U (1) bundle. Now it is well known [1,11] that L O(1, −1) however it is worth mentioning that we may recover this result with minimal effort given the approach we have taken. Proposition 1. Let π :L→Q be the holomorphic line bundle corresponding to the unit mass U (1) monopole on H3 . Then L O(1, −1). Proof. From the recipe of Theorem 1, the form defining the ∂-operator of L is θ 0,1 . Using the formula of Eq. (6) we can now read off that L O(1, −1) as required.
Now that we have the Hitchin–Ward correspondence for solutions of the Bogomolny equations over H3 \ { p1 , . . . , pn }, the next step is to work out what consequences the boundary conditions in Definition 1 have for the holomorphic data (E + , E − , h). We first deal with conditions (iii) of Definition 1. Clearly the behaviour of Φ near the singularities pi will be reflected in the behaviour of h near P = P1 ∪ · · · ∪ Pn , where Pi = Ppi (the twistor line that is the set of geodesics passing through pi ). Indeed let p˜ i ∈ H 0 (Q, O(1, 1)) be a section with divisor Pi and let p˜ =
n
p˜ ili ∈ H 0 (Q, O(l, l))
i=1
for some {l1 , . . . , ln } ⊂ N and l =
li . Note that we can regard
h ∈ H 0 (Q\ P, Hom(E + , E − )) and that the singular set of h is exactly the same as the zero set of p. ˜ With this notation in place, we can state Theorem 2. Let (E + , E − , h) be the holomorphic data corresponding to a solution to the Bogomolny equations as in Theorem 1. Let I ⊂ Q be the set of geodesics in H3 that pass through at least two of the singular points pi ∈ H3 . Then the solution of the Bogomolny equations satisfies conditions (iii) of Definition 1 and has Abelian charges l1 , . . . , ln if and only if the section ph ˜ ∈ H 0 (Q\ P, Hom(E + , E − )(l, l)) extends across P to a holomorphic section on all of Q and is non-vanishing on Q\ I .
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Proof. We show that a solution of the Bogomolny equations satisfying conditions (iii) of Definition 1 has the required property and omit the proof of the converse since we do not require it. Now since I is a discrete set of points and Q has complex dimension 2, if ph ˜ extends as a holomorphic section to Q\I then, by Hartog’s theorem, the isolated singularities at the points of I are removable. Let Pi ⊂ Q be the twistor pi ∈ H3 and let
line in Q corresponding to a singularity
x ∈ Pi \ I . Let Ux ⊂ Q\ P j be an open neighbourhood of x in Q\ Pj . j=i
j=i
˜ has a removable Consider p˜ j , 1 ≤ j = i ≤ n. This is non-vanishing on Ux and so ph ˜ is non-vanishing on Ux ∩ Pi iff p˜ ili h is. singularity along Ux ∩ Pi iff p˜ ili h does and ph This means that we can deal with each singularity separately. We just need to prove the result for a single singularity. Identifying Q P1 × P1 \∆, we may take this singularity to be at the point O whose twistor line is the diagonal ∆ ⊂ P1 × P1 \∆. We let l be the Abelian charge. Now, there is a holomorphic trivialisation of O(1, 1) over the open set V of P1×P1\∆ with coordinates ([z, 1], [w, 1]) such that O˜ ∈ H 0 (Q, O(1, 1)) is trivialised as the function (z, w) → z − w. Thus if h has matrix
a1 a2 a3 a4
relative to local trivialisations of E ± , then we need to investigate the behaviour of the functions (z, w) → (z − w)l ai (z, w) as z → w. It is sufficient to do this for each fixed value w = w0 . Furthermore, we may without loss of generality assume w0 = 0 since we may always choose our coordinates to arrange for this. From now on we thus work on a fixed slice w = 0. Now it will follow from our work below that the functions ai (defined on a neighbourhood of 0 in C∗ ) cannot have essential singularities at z = 0. At worst they have poles. Thus there exist unique integers m, n ∈ Z such that n a4 (z) z −a3 (z) and
z
m
−a2 (z) a1 (z)
have removable singularities at 0 which are non-vanishing in a neighbourhood of 0. Using these to define a new trivialisation of E + we find that h has matrix n z 0 . (9) 0 zm
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Since h is compatible with the symplectic structures on E + and E − , it must have a regular determinant and so we must have n = −m. Without loss of generality, we may assume n > 0. Evidently we will be done if we can show that n = l and, in view of (9), to do this it is enough to show that there exist C1 , C2 > 0 and 1 > 1 , 2 > 0 such that for all non-zero z with |z| small enough we have C1 |z|1 < |z|l H (z) < C2 |z|−2 , where H is the matrix of h with respect to local trivialisations of E ± and we are using the l 1 norm on matrices. We will show this by choosing appropriate local trivialisations as detailed below. Thus for any δ > 0 define Γ (δ) = {([z, 1], [0, 1]) ∈ Q | 0 < |z| < δ} ⊂ Q\∆ and Ω(δ) = {γ (t) ∈ H3 | γ ∈ Γ (δ) and |t| < δ} ⊂ H3 \{O}, where we have given γ the parameterisation determined by O ∈ H3 . If δ is small enough, then Φ is non-zero on Ω(δ). Thus, for j = 0, 1, let e j be a unitary eigensection of E|Ω(δ) with eigenvalue (−1) j iΦ. Also let A B −B ∗ −A be the matrix of the monopole connection ∇ with respect to the local trivialisation defined by e0 , e1 . Note that B is bounded in a neighbourhood of O since Φ = 2B, ∇ Φ and it follows easily from conditions (iii) given in Definition 1 that ∇(Φ/Φ) is bounded in a neighbourhood of O. Now if s is a section of E over {γ (t) | |t| < δ} for some γ ∈ Γ (δ) then in terms of the trivialisation of E determined by e0 , e1 the equation (∇γ˙ − iΦ)s = 0 becomes ds Φ + A(γ˙ ) −B ∗ (γ˙ ) s. (10) = B(γ˙ ) −Φ − A(γ˙ ) dt Let H (z, t) be the matrix solution of this equation such that H (z, −δ) = I . Then H (z) = H (z, δ) is the matrix of h in the local trivialisations of E ± determined by e0 , e1 . To proceed with the required analysis of (10) define t G(z, t) = exp − (Φ + A(γ˙ ))ds H (z, t). −δ
Then H solves (10) iff G solves dG 0 −B ∗ (γ˙ ) G. = B(γ˙ ) −2(Φ + A(γ˙ )) dt
(11)
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We estimate the behaviour of solutions of this by regarding the off diagonal terms (for which we have a bound) as a perturbation of the diagonal terms and transforming to an integral equation. Thus let U be the fundamental solution of the diagonal equation, i.e.
1 0 U (t) = . t 0 exp −2 −δ (Φ + A(γ˙ ))ds Note that 1 ≤ U (t) ≤ 2. Define the integral operator T by t 0 −B ∗ (γ˙ ) G(s)ds. U (s)−1 (T G)(t) = U (t) B(γ˙ ) 0 −δ Since B is bounded, let M be a constant such that B < M on Ω(δ). Then for t ∈ [−δ, δ] we have
t 1 0 (T G)(t) ≤ sup GM ds t −δ 0 exp −2 s (Φ + A(γ˙ ))du [−δ,δ] ≤ 4Mδ sup G. [−δ,δ]
Thus by taking δ > 0 small enough, we can have T < for any > 0, where we are using the sup-norm for T . Now G solves (11) iff it solves G = U + T G, and so using the sup-norm for everything, we have G − U = (T + T 2 + · · · )U ≤ (T + T 2 + · · · )U 2 = U . −1 T − 1 Thus provided we take δ > 0 small enough we have G − U < for any > 0 and thus U − < G < U + ⇒ 1 − < G < 2 + , and so finally (1 − ) exp
δ −δ
Φdt
< H (z) < (2 + ) exp
δ
−δ
Φdt .
(12)
δ It only remains to deal with the behaviour of exp −δ Φdt as z → 0. To do this, note that from conditions (iii) of Definition 1 for δ > 0 sufficiently small and 0 < |z| < δ, |t| < δ we have l−
1 1 < 2ρ(γ (t), O)Φ < l + . 2 2
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Next by Pythagoras’s theorem for hyperbolic space (applied to the triangle with vertices O, γ (t), γ (0) which has a right angle at γ (0)) we have cosh ρ(γ (t), O) = ρ(γ (0), O) cosh t and furthermore3 cosh ρ(γ (0), O) =
1 + |z|2
so that we have l−
1 1 < 2 cosh−1 1 + |z|2 cosh t Φ < l + , 2 2
and so l − 1/2 < 2 t 2 + |z|2
cosh−1
1 + |z|2 cosh t
t2
+ |z|2
l + 1/2 Φ < . 2 t 2 + |z|2
Now since the function f (s, t) =
cosh−1
√ 1 + s 2 cosh t , √ t 2 + s2
which is a priori defined on R2 \{0} in fact extends as a continuous function on R2 with value 1 at 0, we have 1 l − 1/2 l + 1/2 1 < Φ < 1 + 2 t 2 + |z|2 1 − 2 t 2 + |z|2 for any > 0, provided δ > 0 is sufficiently small and |t|, |z| < δ. Integrating, we thus have δ l − 1/2 l + 1/2 −1 sinh (δ/|z|) < Φdt < sinh−1 (δ/|z|). 1+ 1 − −δ √ Taking exponentials and remembering that exp(sinh−1 x) = x + x 2 + 1 ∼ 2x as x → ∞, we thus have δ −l+1 < exp Φdt < C2 |z|−l−2 C1 |z| −δ
for constants C1 , C2 > 0 and 1 > 1 , 2 > 0. Combining this with (12) we thus have C1 |z|−l+1 < H (z) < C2 |z|−l+2 for appropriate constants, as required.
3 This a special case of the general formula cosh(ρ) =
√
(1+|z|2 )(1+|w|2 ) . |1+zw|
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Note that the above theorem also applies to h −1 . From now on, when we refer to ph ˜ and ph ˜ −1 it shall be understood that they are regarded as being defined over all of Q. Having dealt with conditions (iii) of Definition 1 we need only note that conditions (ii) have the same effect on E ± as in the non-singular case. Indeed, suppose that (E + , E − , h) is the holomorphic data corresponding to a solution of the Bogomolny equations as in Theorem 1. Suppose also that this solution satisfies conditions (ii) of Definition 1. Define L + ⊂ E + as follows. Let v ∈ E γ+ E γ (R) and let s be the unique section of E over {γ (t) | t ≥ R} such that s(γ (R)) = v and (∇γ˙ − iΦ)s = 0. We define v ∈ L + iff s(γ (t)) → 0 as t → ∞. We then have Theorem 3. In the above notation, L + is a holomorphic line sub-bundle of E + . Furthermore L + L m (0, −N ), where L is the line bundle defined in Proposition 1 and N is the total charge of the monopole. Also, since E + has a holomorphic symplectic structure, we have E + /L + (L + )∗ , and so we can express E + as an extension: 0 → L + → E + → (L + )∗ → 0.
(13)
Proof. This can be proved by modifying a proof in the non-singular case. Since it is only the asymptotic behaviour of the sections of E which matters, the singularities do not cause any complication (note that h does not even enter the statement of the theorem). For a proof of the result in the non-singular case see [11]. Recall now that we have the map j˜ : E + → E − . We thus have a bundle L − = j˜ L + ⊂ E − . It is easy to identify L − in the same way as L + . Let v ∈ E γ− E γ (−R) and let s be the unique section of E over {γ (t) | t ≤ −R} satisfying the initial condition s(γ (−R)) = v and (∇γ˙ − iΦ)s = 0. Then v ∈ L − iff s(γ (t)) → 0 as t → −∞. We thus have a corresponding expression of the bundle E − as an extension: 0 → L − → E − → (L − )∗ → 0.
(14)
Furthermore a choice of isomorphism L + L m (0, −N ) induces an isomorphism L − L −m (−N , 0). In the case of non-singular monopoles, the situation is a little simpler since h is a global isomorphism and so we can work with h −1 L − ⊂ E + .
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4. The Spectral Data We are now in a position to combine the results of the previous section and identify the spectral data which determine a singular hyperbolic SU (2) monopole. Definition 8. Let (E + , E − , h) be the holomorphic data corresponding to a singular hyperbolic SU (2) monopole as in Theorem 1. Define the map ψ as follows: ψ : L + → E + → E − (l, l) → (L − )∗ (l, l),
(15)
where the second arrow is the map ph ˜ of Theorem 2 and the last arrow is formed by tensoring the projection of the exact sequence (14) with the identity map on O(l, l). Note that using an isomorphism L + L m (0, −N ) as in Theorem 3 we can regard ψ ∈ H 0 (Q, O(k, k)), where k is the non-Abelian charge of the monopole. We define the spectral curve S of the monopole to be the divisor of ψ. Remark 4. In the case of non-singular monopoles, the geometric interpretation of the spectral curve is clear. Since S is a subset of twistor space which is the set of all oriented geodesics in H3 and S is preserved by σ , S really defines a set of unoriented lines in H3 . These are known as the spectral lines of the monopole. Chasing through the definitions one finds that a line γ in H3 is a spectral line iff there exists a non-zero section s of E along γ such that (∇γ˙ − iφ)s = 0 and s(γ (t)) → 0 as t → ±∞. For our singular monopoles, the situation is more complicated. The spectral curve S of the singular monopole still defines a set of spectral lines in H3 but it is not as easy to identify them geometrically. For a line γ which does not pass through a singular point, the rule for deciding if γ is a spectral line is the same as for a non-singular monopole. However if γ ∈ S ∩ P this no longer makes sense. Indeed if we return to the definition of S using ph ˜ we see that to define S we had to note that ph ˜ had a removable singularity along P. Obtaining the value of a holomorphic function at a removable singularity requires a limiting process and this means that to decide if a line γ passing through a singular point is a spectral line we will need to examine the behaviour of sections of E along geodesics in a neighbourhood of γ in H3 . For non-singular (Euclidean or hyperbolic) monopoles, the spectral curve determines the monopole. As we shall see, this is almost true for singular monopoles. Except for the special case (which we shall not consider) when S ∩ P is not finite (i.e. when Pi is a connected component of S for some i), exactly one additional piece of spectral data is needed to identify a singular monopole. Let S be the spectral curve of a singular hyperbolic SU (2) monopole for which S ∩ P is finite. Note that by definition of S and the exactness of the sequence 0 → L − (l, l) → E − (l, l) → (L − )∗ (l, l) → 0 it follows that the image of ph ˜ restricted to L + | S is in fact contained in L − (l, l)| S . We − + thus have a map ξ : L | S → L − (l, l)| S . Note that we can regard ξ − ∈ H 0 (S, (L + )∗ L − (l, l)). Similarly, considering
ph ˜ −1
restricted to
L −|
S
(16)
we have
ξ + ∈ H 0 (S, (L − )∗ L + (l, l)).
(17)
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Clearly these two satisfy ξ − ξ + = p˜ 2 | S . Let D be the divisor of ξ + . Using the fact that ˜ 2 = −1 outside P we find that the total map: (h −1 j) j˜
ξ+
j˜
L + | S → L − | S → L + (l, l)| S → L − (l, l)| S is just the map: −ξ − : L + | S → L − (l, l)| S . It thus follows (since j˜ covers σ ) that the divisor of ξ − is σ (D). Definition 9. Using the above notation, we call D the spectral divisor of the monopole. Also since σ (S ∩ P) = S ∩ P we see that S ∩ P really defines a set of unoriented lines in H3 . We call these lines the singular spectral lines of the monopole. We shall refer to (S, D) as the spectral data of a monopole. Note that the support |D| of D is contained in S ∩ P since ph ˜ −1 is an isomorphism − + 2 outside P. Also note that since ξ ξ = p˜ | S we have |D| ∪ σ (|D|) = S ∩ P. Since σ has no fixed points |D| ∩ σ (|D|) = ∅ and so |D| defines a partition of the set of singular spectral lines into disjoint conjugate subsets. Recalling that σ (γ ) is just γ parameterised in the opposite direction, this means that |D| really defines an orientation for each singular spectral line. Finally note that since D + σ (D) = ( p˜ 2 | S ), it follows that that |D| determines D (provided we know p˜ 2 , i.e. the locations of the singularities). As we shall see the spectral data (S, D) determines the monopole and so the spectral data for a singular hyperbolic SU (2) monopole for which S ∩ P is finite may be regarded as the set of spectral lines in H3 together with an orientation for each singular spectral line. We gather together a few important properties of the spectral data for a singular monopole. Proposition 2. Let (S, D) be the spectral data of a singular hyperbolic SU (2) monopole of non-Abelian charge k for which S ∩ P is finite. Then (i) (ii) (iii) (iv)
S is compact. S is real (i.e. preserved by σ ). L 2m+k (0, 2l)| S [D]. If S is non-singular then it has genus (k − 1)2 .
Proof.
(i) Let γ be a geodesic in H3 that does not pass through any of the singular points, pi . We have already noted that the condition for γ to be a spectral line is that there exists a non-zero solution s to (∇γ˙ − iΦ)s = 0 along γ such that s(t) → 0 as t → ±∞. Thus if we fix a point O ∈ H3 , the argument used in [7] to prove compactness of the spectral curve of a non-singular Euclidean monopole shows that there exists M > 0 such that if γ is a spectral line then it
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O. Nash
meets the ball B(O, M) ⊂ H3 of radius M centred about O. Thus if we choose M sufficiently large that we also have { p1 , . . . , pn } ⊂ B(O, M) we find that all spectral lines (i.e. including the singular spectral lines passing through the points pi ) meet B(O, M) from which it follows that S is compact. (ii) This follows since j˜ : L + → L − covers σ . (iii) An isomorphism L + L m (0, −N ) as in Theorem 3 induces an isomorphism (L − )∗ L + (l, l) L 2m+k (0, 2l). Since the spectral divisor D is the divisor of a section of (L − )∗ L + (l, l)| S the result follows. (iv) This follows from the adjunction formula since S is a divisor of O(k, k). The most important of the properties listed in the above proposition is (iii) since it is the only property which differs from the non-singular case. It replaces the condition that L 2m+k | S must be trivial which is what holds in the non-singular case. As we shall see in the next section, it is exactly condition (iii) that means that the spectral curves of k = 1 singular hyperbolic monopoles lift to twistor lines in appropriate twistor spaces. We wish to show that the spectral data determine the monopole. To do this we shall need the following lemma. Lemma 3. Let S be the spectral curve of a singular hyperbolic SU (2) monopole for which S ∩ P is finite and let ξ + be the section of (17). Let a ∈ H 1 (Q, (L + )2 ) be the class representing the extension (13) and let δ : H 0 (S, (L − )∗ L + (l, l)) → H 1 (Q, (L + )2 )
(18)
be the connecting homomorphism associated to the following short exact sequence of sheaves 0 → OQ ((L + )2 ) → OQ ((L − )∗ L + (l, l)) → O S ((L − )∗ L + (l, l)) → 0.
(19)
Then δξ + = a. Proof. By abuse of notation let a ∈ Ω 0,1 (Q, (L + )2 ) be a Dolbeault representative for the extension class of (13). By exactness of the long exact sequence of cohomology groups associated to (19) [a] is in the image of δ if and only if ψa = ∂b
(20)
for some b ∈ Ω 0 (Q, (L − )∗ L + (l, l)), where ψ is the section of Eq. (15). Furthermore, in this case b| S ∈ H 0 (S, (L − )∗ L + (l, l)) is the class mapped to [a] under δ. Now if we fix a smooth splitting E + = L + ⊕ (L + )∗ of (13) on Q then the ∂-operator of E + is ∂ L+ a ∂= 0 ∂ (L + )∗ and from this we can see that (20) holds if and only if we have a meromorphic splitting of (13) with a pole along S. To see this, note that in our fixed smooth splitting, to define
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the meromorphic splitting we only need to define the map (L + )∗ → L + over Q\ S and this is b/ψ : (L + )∗ → L + . To prove the lemma, we thus need a meromorphic splitting α : (L + )∗ → E + of (13) with a pole along S such that if α : (L + )∗ → L + is the induced map (using our fixed smooth splitting) then (ψα )| S = ξ + . However by definition of ψ this is the same as requiring that ξ + is given by the map α
ph ˜
(L + )∗ → E + → E − (l, l) → (L − )∗ (l, l)
(21)
(restricted to S). We thus need only exhibit a meromorphic splitting of (13) that satisfies (21). We thus define α to be ψ −1
ph ˜ −1
(L + )∗ → L − (−l, −l) → E + It is trivial to check that α is indeed a splitting of (13) and satisfies condition (21).
Theorem 4. Let (S, D) be the spectral data of a singular hyperbolic SU (2) monopole of mass m and non-Abelian charge k with Abelian charge li at the singular points pi , i = 1, . . . , n for which S ∩ P is finite. This data determines the monopole. Proof. We must show that we can recover the data (E + , E − , h) of Theorem 1 from the spectral data (S, D). Thus let N = k − l. Let s be a section of L 2m+k (0, 2l) on S with divisor D and such that ss ∗ = 1. Let a = δs ∈ H 1 (Q, L 2m (0, −2N )), where δ : H 0 (S, L 2m+k (0, 2l)) → H 1 (Q, L 2m (0, −2N )) is the connecting homomorphism associated to the short exact sequence of sheaves 0 → OQ (L 2m (0, −2N )) → OQ (L 2m+k (0, 2l)) → O S (L 2m+k (0, 2l)) → 0. Let E + be the bundle on Q defined (up to equivalence) as that extension of L m (0, −N ) by L −m (0, N ) with extension class a . Choose the unique isomorphism L + L m (0, −N ) such that the image of s under the induced isomorphism H 0 (S, L 2m+k (0, 2l)) H 0 (S, (L + )∗ L − (l, l)) is ξ + of (17). Then by Lemma 3 the image of a under the induced isomorphism H 1 (Q, L 2m (0, −2N )) H 1 (Q, (L + )2 ) is the extension class of (13). It follows that we must have E + E + . Similarly, we can recover E − . It remains only to show that h is determined by (S, D). Now we saw in the course of the proof of Lemma 3 that the sequence (13) has a natural splitting on Q\ S, i.e. on Q\ S, we naturally have E + L + ⊕ (L + )∗
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and similarly for E − . A quick review of the relevant definitions reveals that the matrix of h with respect to these splittings is 0 − p/ψ ˜ , ψ/ p˜ 0 and so we see that h is determined by ψ (which is determined by S) and p˜ (which is determined by the points pi ) as required. Remark 5. In the course of the above theorem we made a choice of section s of L 2m+k (0, 2l) on S with divisor D and such that ss ∗ = 1. This choice is unique up to a factor of U (1). Thus the space of all spectral data (S, D) together with a choice of such a section s is naturally a U (1) bundle over the space of spectral data (S, D), i.e. over the monopole moduli space. This natural U (1) bundle on the monopole moduli space is of course the gauged monopole moduli space and is the space whose geometry we are interested in. 5. The Charge 1 Moduli Space We are interested in the natural geometric structure on the moduli space of hyperbolic monopoles4 . The moduli space for k = 1 (non-singular) monopoles on H3 is simply H3 × S 1 but the k = 1 moduli spaces for singular monopoles are more interesting and have essentially already been studied in some detail by LeBrun [10]5 . The goal is to gain some insight into the natural geometric structure of the moduli spaces of hyperbolic monopoles in general by studying this simpler (k = 1) case. The easiest way to identify the moduli space of charge 1 singular SU (2) monopoles is to identify the twistor space. The spectral curves S have genus 0 for k = 1 and the spectral data (S, D) of a monopole can be naturally identified with a twistor line. We are most interested in the local differential geometric structure of the monopole moduli space and so it is enough to work with generic spectral curves. Furthermore, the twistor space naturally carries some additional structure which defines the geometry of the moduli space which is what we are really interested in. In fact, this twistor space arises naturally from the Hitchin–Ward transform for a certain singular U (1) monopole. 5.1. Hitchin–Ward for singular U (1) monopoles. In this subsection, we will review the construction of certain twistor spaces introduced by LeBrun in [10] and show how it can be viewed from the point of view of the Hitchin–Ward transform for singular U (1) monopoles. In the next subsection, we will make use of the observation that a twistor line in this space neatly encodes the spectral data of a charge 1 singular hyperbolic SU (2) monopole to study the geometry of the moduli space of such monopoles. Let λ ∈ [0, ∞) be a non-negative real number, let { p1 , . . . , pn } ⊂ H3 be n distinct points in hyperbolic space and let {l1 , . . . , ln } ⊂ N be n (strictly) positive integers. Let G i = G pi be the Green’s function for the hyperbolic Laplacian introduced in Eq. (2). A solution to the U (1) Bogomolny equations is just a solution to the Laplace equation so that V =λ+ li G i (22) 4 When we talk of the monopole moduli space, we shall always mean the gauged moduli space as discussed in Remark 5. 5 In the notation of (22), LeBrun studied the case with λ = 1 and l = · · · = l = 1. As we shall see for n 1 monopoles of mass m > 0, we have λ = 1 + 2m.
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defines a U (1) monopole on U = H3\{ p1 , . . . , pn }. We can thus apply the U (1) version of Theorem 1 and so obtain the triple (K + , K − , h), where K ± are holomorphic line bundles on Q and h : K + |Q\P → K − |Q\P is a holomorphic isomorphism. We encode this data in a singular 3-dimensional complex space Z as Z = (x, y) ∈ K + ⊕ (K − )∗ | Q\P | h(x)y = 1 . Now we have already seen that if V = 1 then K + = K − = L = O(1, −1) with h the identity map. It follows (for example from the linearity of the U (1) Penrose transform) that if V = λ then K + = K − = L λ with h the identity map. Next, it follows from the work of LeBrun in [10] that V = G i corresponds to K + = O(0, 1), K − = O(−1, 0). Thus, in this case h ∈ H 0 (Q\ P, O(−1, −1)) and if p˜ i ∈ H 0 (Q, O(1, 1)) is a section corresponding to pi ∈ H3 then by the U (1) version of Theorem 2 p˜ i h is a non-vanishing holomorphic function on Q. It is thus constant and we may take this constant to be 1. We thus have h = p˜ i−1 . Combining these observations we find that for V given by Eq. (22) above we have K + L λ (0, l), K − L λ (−l, 0), h = p˜ −1 ∈ H 0 (Q\ P, O(−l, −l)), n n where p˜ = i=1 p˜ ili ∈ H 0 (Q, O(l, l)) and l = i=1 li as in Theorem 2. We can thus explicitly identify Z as ˜ (23) Z (x, y) ∈ L λ (0, l) ⊕ L −λ (l, 0) |Q\P | x y = p(u) (where (x, y) is in the fibre of L λ (0, l) ⊕ L −λ (l, 0) over u ∈ Q.) Since the space Z encodes the data (K + , K − , h) which determines our singular U (1) monopole, the natural question is how to recover the monopole directly from Z . This question has essentially already been answered by LeBrun [10] (using ideas of Hitchin [6].) A natural desingularisation Z of Z is the twistor space for the 4 dimensional real manifold that is the total space of the principal U (1) bundle M of the monopole we started with, endowed with a Gibbons–Hawking type conformal structure (introduced by LeBrun [10]). Since we are following the setup of [10], we shall summarise the details in the language which will be most useful to us, but without providing proofs. Note also that though we can view Z as arising from our version of the Hitchin–Ward transform for singular U (1) monopoles, it is not necessary for the rest of our work. Equation (23) essentially appears in [10] and may be taken as the starting point of our work here. In order to connect the space Z with our SU (2) monopole moduli space, we will need to understand the real structure and the twistor lines. Of course it is LeBrun’s desingularisation Z of Z that is the twistor space, however the real structure and twistor lines are first defined on Z and then lifted to Z so we shall work on Z . This is also consistent with the approach taken by Hitchin in [6]. We define the real structure τˆ : Z→ Z
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by restricting the anti-holomorphic (anti-linear) map L λ (l, 0) → L −λ (0, l) induced by σ on Q. This restricts to Z since p˜ is real. To identify the twistor lines in Z , it is convenient to use the following elementary lemma. Lemma 4. In the above notation, let Pq ⊂ Q be the twistor line of a point q ∈ U . Then we can consistently choose x, y ∈ H 0 (P1 , O(l)) unique up to a factor of U (1) such that x = y ∗ and (π1∗ x)(π2∗ y) = p˜
(24)
on Pq . Proof. Use q to identify Q P1 × P1 \∆ so that Pq corresponds to the diagonal ∆. Let ζ be the natural coordinate on ∆ P1 , then restricted to ∆, p˜ is given by p˜ = (ai ζ 2 + 2bi ζ − a i )li , where ai ∈ C and bi ∈ R. We now simply follow the recipe given in [6]. The discriminant 4(bi2 + |ai |2 ) of the i th quadratic is positive and so we can without ambiguity define ∆i = bi2 + |ai |2 to be the positive square root and the roots αi and βi by −bi + ∆i , ai −bi − ∆i . βi = ai
αi =
We then define x=A
(ζ − αi )li , y = B (ζ − βi )li ,
where A and B satisfy AB = if
aili . x, y now satisfy the required conditions if and only
|A|2 =
(bi − ∆i )li .
We thus have the required factoring of p˜ and the indeterminacy in the argument of A corresponds to the extra U (1) factor. Now we also have a natural holomorphic projection Z → Q\P, compatible with real structures. Composing this with the projection π1 of Q ⊂ P1 × P1 onto the first factor we thus have a holomorphic projection π: Z → P1 . (Note that if we instead used the projection π2 onto the second factor in P1 × P1 , we would just obtain the map τ ◦ π ◦ τˆ so that provided we remember the real structures on Z and P1 , there is no new information.) We can now exhibit the family of twistor lines in Z . Each twistor line will be the image of a holomorphic section of π . As we shall see, this has geometrical significance
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for M. Indeed if π preserved the real structures on Z and P1 , then (see [13]) M would have a hypercomplex structure. This is, of course, not the case here. We define these sections as follows. Pick a point q ∈ U = H3\{ p1 , . . . , pn }. Let Pq be the corresponding twistor line in Q. Note that πi : Pq → P1 is a holomorphic bijection for i = 1, 2. (A geometric reason for this is that Pq is the set of all geodesics through the point q ∈ H3 so if we know the start or end point of the geodesic on sphere at infinity in H3 , then we know the geodesic.) Now choose x, y ∈ H 0 (P1 , O(l)) as in Lemma 4 and let s : Pq → L be a holomorphic trivialisation of L| Pq such that ss ∗ = 1. Evidently s is unique up to a factor of U (1). s defines trivialisations s λ of L λ and s −λ of L −λ over Pq . The pair of sections (s λ π2∗ x, s −λ π1∗ y) ∈ H 0 (Pq , L λ (0, l) ⊕ L −λ (l, 0)) define a real lifting of the twistor line Pq in Q to the required twistor line in Z. With this explicit description of the twistor lines of Z , we are ready to identify them with the spectral data of charge 1 singular hyperbolic SU (2) monopoles. This means that Z is the twistor space of the moduli space M of these monopoles. Furthermore, LeBrun [10] explicitly identified the space which Z is the twistor space of and so by studying its natural geometry we are studying the natural geometry of the monopole moduli space. We summarise here the geometry of M. We saw in Sect. 2 that a harmonic function V as in Eq. (22) defines a Riemannian manifold. The manifold M is the total space of the U (1) bundle on U with Chern class 1 2π [∗d V ] and the metric (which is anti-self-dual with respect to the orientation defined by d x ∧ dy ∧ dz ∧ ω) is Vˆ hˆ + Vˆ −1 ω ⊗ ω,
(25)
where hˆ is the pull back of the hyperbolic metric h to M, Vˆ is the pull back of V to M and ω is a connection on M with curvature ∗d V . According to [10], the twistor space of M with the conformal structure defined by this metric is Z (the natural desingularisation of Z ). We noted above that the fact that the twistor lines were the images of holomorphic sections of the natural holomorphic projection π : Z → P1 would have geometric significance for M. In fact if we fix a point u ∈ P1 then the fibre Σ = π −1 ({u}) is a divisor corresponding to a complex structure on M. Furthermore, as shown in [10], the divisor Σ + Σ represents the line bundle K −1/2 , where K is the canonical bundle of Z . Since [14], holomorphic sections H 0 (Z , K −1/2 ) correspond to scalar-flat Kähler metrics6 in the conformal class on M, Σ defines a Kähler metric on M up to scale. We are fortunate that we may appeal to [10] for an explicit description of these metrics. The details are as follows. 6 Strictly speaking an element of H 0 (Z , K −1/2 ) gives a pair of complex structures J , −J on M and we cannot tell them apart. So the Kähler metric is well defined but the complex structure is only defined up to conjugation. This is not an issue for us however since we have Σ.
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Choose coordinates for H3 so that it is represented in the upper half-space model H3 {(x, y, z) ∈ R3 | z > 0}, d x 2 + dy 2 + dz 2 h= . z2 Suppressing the notation the pull back of data from U to M, note that the 1-forms d x, dy, dz, ω trivialise T ∗ M so that we can define an almost complex structure J on M by J d x = dy, J dz = zV −1 ω. Noting that J (d x + idy) = −i(d x + idy), J (z −1 V dz + iω) = −i(z −1 V dz + iω), d(d x + idy) = 0, ∂V i ∂V 1 ∂V ( −i )dz + dy , d(z −1 V dz + iω) = (d x + idy) ∧ z ∂x ∂y z ∂z we see that J is integrable. Now define the metric g = z 2 (V h + V −1 ω ⊗ ω) on M (which is in the conformal class defined by (25)). We claim that (M, J, g) is a scalar-flat Kähler manifold. If we let Ω = g(J ·, ·) be the associated 2-form then a quick calculation reveals Ω = −(V d x ∧ dy + zdz ∧ ω)
(26)
from which it follows easily (using the fact that d ∗ d V = 0) that dΩ = 0. Thus g is a Kähler metric and we need only note that an anti-self-dual Kähler manifold is scalar-flat (see for example [4]). Finally note that to define this metric on M, we represented H3 in the upper halfspace model. This singles out a point on the conformal 2-sphere at infinity and so we in fact have an entire S 2 of such metrics. This corresponds to the choice of point in P1 giving the divisor Σ above. To see this more clearly, note that the Kähler form (26) can be written as 1 Ω = − (V ∗ d(z 2 ) + d(z 2 ) ∧ ω), 2 where ∗ is the Hodge ∗-operator on H3 . Thus to define the Kähler structure, we only need the function z. Furthermore, z is just a horospherical height function on H3 and as we shall see, any such function will define a Kähler structure. Let us briefly recall some elementary facts about horospherical height functions on H3 . A horospherical height function is the exponential of a Busemann function. To define a Busemann function (see e.g. [15]) on a complete Riemannian manifold with distance function ρ, we choose a geodesic γ parameterised by arc-length and define the associated Busemann function bγ by bγ (x) = lim (t − ρ(x, γ (t))). t→∞
(27)
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Note that bγ (γ (t)) = t and that if we change the parameterisation of γ by t → γ (t + a) for some constant a then bγ is replaced with bγ − a. Let us consider Busemann functions on H3 . We claim that the level sets of bγ are the horospheres passing through γ (∞) ∈ ∂H3 . To see this, we may choose coordinates (x, y, z) so that H3 is represented in the upper half-space model and our geodesic γ appears as γ : t → (0, 0, et ). The horospheres tangential to γ (∞) are given in these coordinates by z = const. Using the formula 2 2 2 2t −1 x + y + z + e ρ((x, y, z), γ (t)) = cosh 2zet it is easy to check from (27) that ebγ ((x,y,z)) = z so that z is the associated horospherical height function as required. Since, if we fix a point p on the boundary, H3 is foliated by the horospheres tangential to ∂H3 at p, we see that to define a Busemann function on H3 , we just need to define a value on each horosphere. Choosing a geodesic such that γ (∞) = p we define the value on each horosphere using bγ (γ (t)) = t. The only remaining degree of freedom is which horosphere contains γ (0). Thus p determines the Busemann function up to the addition of a constant. We have thus seen that a horospherical height function is determined up to a positive scale factor by a point on ∂H3 and that any such function can be represented as the function z in upper half-space coordinates. If we fix a point O ∈ H3 , we thus have a natural 2-sphere of horospherical height functions: for each point on ∂H3 we take the associated horospherical height function q such that O lies in the level set q = 1. In this way, a point in H3 determines a natural 2-sphere of scalar-flat Kähler metrics on M. 5.2. The SU (2) moduli space. In the previous subsection, we reviewed a construction of LeBrun [10] (and also noted how it fits with our singular U (1) Hitchin–Ward transform). The essential point was the geometry of the space M and its twistor space. We now wish to show that M is in fact the moduli space of k = 1 singular SU (2) monopoles. We will do this by showing how the spectral curve S of such a monopole naturally lifts to a twistor line in Z and that this twistor line also encodes the spectral divisor of the monopole. Thus consider a singular SU (2) monopole with non-Abelian charge k = 1 and Abelian charges li at the points pi ∈ H3 , i = 1, . . . , n. Let V = 1 + 2m + 2 li G i . So comparing with (22), we have λ = 1 + 2m and 2li in place of li . As we saw in the previous subsection, associated with V is a twistor space Z (x, y) ∈ L 1+2m (0, 2l) ⊕ L −1−2m (2l, 0) |Q\P | x y = p˜ .
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The key observation is that a twistor line of Z exactly encodes the spectral data (S, D) for a charge 1 singular hyperbolic monopole. We need only observe that D is the divisor of a section ξ + ∈ H 0 (S, (L − )∗ L + (l, l)) and using the isomorphisms L + L m (0, −N ) and L − L −m (−N , 0) we see that we can regard ξ + ∈ H 0 (S, L 1+2m (0, 2l)). Similarly we have ξ − ∈ H 0 (S, L −1−2m (2l, 0)). Thus ξ ± provide a natural lifting of the spectral curve (which is a rational curve) to the twistor space Z . The sections ξ ± are determined up to U (1) factor by the spectral divisor D. This U (1) factor, of course, corresponds to the gauging of the monopole. This is how a monopole corresponds to a twistor line, and hence a point in M. Conversely, by construction, a twistor line in Z projects down to the spectral curve of a singular hyperbolic monopole and this curve obviously comes with a section of L 1+2m (0, 2l) which gives the spectral divisor. To summarise, we thus have Theorem 5. Let m > 0, let { p1 , . . . , pn } ⊂ H3 be n distinct points in hyperbolic space and let {l1 , . . . , ln } ⊂ N be n (strictly) positive integers. Let M be the moduli space of gauged singular hyperbolic SU (2) monopoles of non-Abelian charge 1 with Abelian charges li at pi . Then – M carries a natural self-dual conformal structure. – For each point u ∈ ∂H3 there is a volume form and complex structure J u on M such that the metric determined in the conformal structure makes M together with J u a scalar-flat Kähler manifold. Acknowledgement. This work comprised a part of the author’s D.Phil thesis [12] at Oxford and he finds it a pleasure to thank his supervisor Prof. Nigel Hitchin for all of his help and encouragement.
References 1. Atiyah, M.F.: Magnetic monopoles in hyperbolic spaces. In: Vector bundles on algebraic varieties (Bombay, 1984), Volume 11 of Tata Inst. Fund. Res. Stud. Math., Bombay: Tata Inst. Fund. Res., 1987, pp. 1–33 2. Braam, P.J.: Magnetic monopoles on three-manifolds. J. Differ. Geom. 30(2), 425–464 (1989) 3. Buchdahl, N.P.: Instantons on CP2 . J. Differ. Geom. 24(1), 19–52 (1986) 4. Derdzi´nski, A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compositio Math. 49(3), 405–433 (1983) 5. Gibbons, G.W., Warnick, C.M.: Hidden symmetry of hyperbolic monopole motion. http://arxiv.org/list/ hepth/0609051, 2006 6. Hitchin, N.J.: Polygons and gravitons. Math. Proc. Cambridge Philos. Soc. 85(3), 465–476 (1979) 7. Hitchin, N.J.: Monopoles and geodesics. Commun. Math. Phys. 83(4), 579–602 (1982) 8. Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. http://arxiv. org/list/hepth/0604151, 2006 9. Kronheimer, P.B.: Monopoles and Taub-NUT metrics. Transfer thesis, Oxford University, 1985 10. LeBrun, C.: Explicit self-dual metrics on CP2 # · · · #CP2 . J. Differ. Geom. 34(1), 223–253 (1991) 11. Murray, M., Singer, M.: Spectral curves of non-integral hyperbolic monopoles. Nonlinearity 9(4), 973–997 (1996)
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12. Nash, O.C.: Differential geometry of monopole moduli spaces. Oxford D.Phil. thesis, available at http:// arxiv.org/list/math.DG/0610295, 2006 13. Pedersen, H., Poon, Y.S.: Deformations of hypercomplex structures. J. Reine Angew. Math. 499, 81–99 (1998) 14. Pontecorvo, M.: On twistor spaces of anti-self-dual Hermitian surfaces. Trans. Amer. Math. Soc. 331(2), 653–661 (1992) 15. Shiohama, K.: Topology of complete noncompact manifolds. In: Geometry of geodesics and related topics (Tokyo, 1982), Volume 3 of Adv. Stud. Pure Math., Amsterdam: North-Holland, 1984, pp. 423–450 Communicated by G.W. Gibbons
Commun. Math. Phys. 277, 189–236 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0366-4
Communications in
Mathematical Physics
Optimal Decay Estimates on the Linearized Boltzmann Equation with Time Dependent Force and their Applications Renjun Duan1 , Seiji Ukai1 , Tong Yang1 , Huijiang Zhao2 1 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon,
Hong Kong, P.R. China. E-mail: [email protected]
2 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China
Received: 3 December 2006 / Accepted: 2 April 2007 Published online: 1 November 2007 – © Springer-Verlag 2007
Abstract: Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the linearized Boltzmann equation with a time dependent external force. No time decay assumption is made on the force. The proof is based on both the energy method through the macro-micro decomposition and the L p -L q estimates from the spectral analysis. The decay estimates thus obtained are applied to the study on the global existence of the Cauchy problem to the nonlinear Boltzmann equation with time dependent external force and source. Precisely, for space dimension n ≥ 3, the global existence and decay rates of solutions to the Cauchy problem are obtained under the condition that the force and source decay in time with some rates. This time decay restriction can be removed for space dimension n ≥ 5. Moreover, the existence and asymptotic stability of the time periodic solution are given for space dimension n ≥ 5 when the force and source are time periodic with the same period. Contents 1. 2.
3.
Introduction . . . . . . . . . . . . . . . . . . . . Decay Estimates on the Linearized Equation . . . 2.1 Preliminaries . . . . . . . . . . . . . . . . . 2.2 Estimates on commutators . . . . . . . . . . 2.3 Energy estimates . . . . . . . . . . . . . . . 2.4 Optimal decay rates . . . . . . . . . . . . . . Applications to the Nonlinear Equation . . . . . . 3.1 Basic estimates . . . . . . . . . . . . . . . . 3.2 Global existence for the Cauchy problem . . . 3.3 Existence of time periodic solution . . . . . . 3.4 Asymptotic stability of time periodic solution
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1. Introduction The Boltzmann equation for the hard-sphere gas in n-dimensional space under the influence of an external force and a source takes the form ∂t f + ξ · ∇x f + F · ∇ξ f = Q( f, f ) + S.
(1.1) Rn
Here, the unknown function f = f (t, x, ξ ) with (t, x, ξ ) ∈ R × × is a nonnegative function standing for the number density of gas particles which have position x = (x1 , . . . , xn ) ∈ Rn and velocity ξ = (ξ1 , . . . , ξn ) ∈ Rn at time t ∈ R. Here, the external force field F = F(t, x) and the source term S = S(t, x, ξ ) are assumed to be some given time dependent functions. Q is the usual bilinear collision operator defined by 1 Q( f, g) = ( f g∗ + f ∗ g − f g∗ − f ∗ g)|(ξ − ξ∗ ) · ω|dωdξ∗ , 2 Rn ×S n−1 f = f (t, x, ξ ), f = f (t, x, ξ ), f ∗ = f (t, x, ξ∗ ), f ∗ = f (t, x, ξ∗ ), Rn
ξ = ξ − [(ξ − ξ∗ ) · ω]ω, ξ∗ = ξ∗ + [(ξ − ξ∗ ) · ω]ω, ω ∈ S n−1 ,
and likewise for g. Although the physical space is three dimensional, in this paper, we consider the general space dimension n ≥ 3 to show how the space dimension plays in the decay estimates. Throughout this paper, we consider the perturbative solution near an absolute Maxwellian. Without loss of generality, define the perturbation u = u(t, x, ξ ) by f = M + M1/2 u, where the absolute Maxwellian M=
1 |ξ |2 exp − (2π )n/2 2
is normalized to have zero bulk velocity and unit density and temperature. Then the equation for the perturbation u is:
where
1 S, ∂t u + ξ · ∇x u + F · ∇ξ u − ξ · Fu = Lu + (u) + 2
(1.2)
Lu = M−1/2 Q(M, M1/2 u) + Q(M1/2 u, M) , (u, u) = M−1/2 Q M1/2 u, M1/2 u ,
(1.3)
S = M−1/2 S + M1/2 ξ · F.
(1.4) (1.5)
There are extensive literatures on the existence theory for the Cauchy problem of the Boltzmann equation without external force. The well-known result is the global existence of the renormalized solution with large data proved by DiPerna-Lions [6] where the uniqueness problem remains open. On the other hand, the existence is established in the framework of small perturbation of an absolute Maxwellian [12–14,17,19,21,23,24,29], or an infinite vacuum [2,9,15,16] where uniqueness can be justified. In particular, so far there are two basic methods to deal with solutions near an absolute Maxwellian. One is
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based on the spectral analysis of the linearized Boltzmann equation and the bootstrap argument for the nonlinear equation initiated by Grad and developed by Ukai, cf. [19,23– 25], where the optimal convergence rate to the Maxwellian can be also obtained. Another one is based on the direct energy method for the nonlinear problem through the macromicro decomposition which was initiated by Liu-Yu and developed by Liu-Yang-Yu [17] and Guo [13] independently in two different ways. The former decomposition is around a local Maxwellian while the latter is around an absolute Maxwellian. Here we use the latter decomposition because we are concerned with the decay structure of the linearized equation around the absolute Maxwellian. One of the features of the convergence to the equilibrium for the Boltzmann equation is the coupling of the conservative operator for the free transportation and the degenerate dissipative operator on the velocity variables through the celebrated H-theorem. This property can be found in many kinetic equations and it is now called “hypocoercivity” [32]. For the problems in a torus or in a bounded domain, this property is well investigated where an exponential or almost exponential convergence rate in time to the equilibrium for both space and velocity variables can be obtained, cf. [33] and references therein. However, for problems in the whole space, this property is not yet well understood especially under the influence of some enternal force. And this is one of the motivations of this paper to study the convergence to the equilibrium under the influence of the external force in a general form. To do this, the main part of the paper is concentrated on the decay in time properties of the solution operator for the linearized Boltzmann equation corresponding to (1.2), that is, 1 ∂t u + ξ · ∇x u + F · ∇ξ u − ξ · Fu = Lu. 2 The decay estimates are obtained in some Sobolev space weighted in velocity variables. Our main result is stated in Theorem 2.2 in Sect. 2, where the obtained decay is optimal in the sense that it is equal to the one for the linearized Boltzmann equation without external force. The proof is a combination of the two methods mentioned above for perturbative solutions. In fact, the energy estimate is first carried out for the linearized Boltzmann equation with an error term determined by the space derivative of the macroscopic component in the perturbation. It is then combined with the L p -L q estimates from the spectral analysis to yield the optimal decay in time estimates for the above linear solution operator. The optimal decay estimates on the solution operator to the linearized equation will then be applied to the study on the existence of solutions to the Cauchy problem for the original nonlinear equation. In particular, we will use it to prove the existence and stability of the time periodic solution for some given time periodic force and source. This problem is related to the generation and propagation of sound waves so that it has its physical importance besides its mathematical interest. In fact, for the time periodic solution, the existence and stability have been studied for the Navier-Stokes equaions, cf. [1,10,30,31] and references therein. Recently, some results on this problem are obtained for the nonlinear Boltzmann equation [26–28] in various function spaces when there is a time periodic external source but no external force, for the space dimension n ≥ 3. Thus, it is natural to study the problem under the influence of a time periodic external force. We will show that there exists a time periodic solution if the force is small and time periodic when the space dimension n ≥ 5. The physical case when the space dimension n = 3 is still not known and will be pursued by the authors in the future.
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A lot of work has been done on the convergence rate estimation of the solutions for the Boltzmann equation to the time asymptotic states. For example, the almost exponential decay in time of the solution for the Cauchy problem was given by Desvillettes-Villani [5] for general cutoff potential cases in either torus or smooth bounded domain under the assumption of the existence of smooth global solutions, and also by Strain-Guo [22] for the cutoff soft potentials in the torus for small pertubation of the absolute Maxwellian. Notice that the convergence rate of the perturbative solution for the cutoff hard potentials is exponential in a torus, [23]. For problems in the whole space, the convergence rate should be algebraic and it depends on the space dimension because the low frequency in the Fourier variable dominates the decay estimate, see [24,25]. For the Boltzmann equation with a time independent potential force, the optimal convergence rate of the solution to a local Maxwellian was obtained in [8], where the proof is motivated by the study of the corresponding problems for the Navier-Stokes equations, cf. [7,18,20]. The rest of this paper is arranged as follows. In Sect. 2, we will first present a decomposition of the linearized Boltzmann equation. Then, some basic estimates on the communicators of the linearized collision operator L and the differential operator will be derived. Based on these estimates, the optimal decay in time estimates on the linear solution operator are proved in Theorems 2.1 and 2.2. In Sect. 3, we will apply the estimates obtained in Sect. 2 to prove the global existence and convergence rate of the solution to the Cauchy problem for the nonlinear Boltzmann equation. In addition, the existence and asymptotic stability of the time periodic solution are also given. These existence and stability results are summarized in Theorems 3.1, 3.2 and 3.3. Notation. Throughout this paper, C denotes a general constant. If the dependence needs to be specified, then the notations Ci , i = 1, 2, · · · are used. In addition, c > 0 also denotes a positive constant which may vary from line to line and δ > 0 stands for a small constant. ·, · is the inner product in the space L 2 (Rnx × Rnξ ) with the norm denoted by · . Sometimes, · also denotes the norm of the space L 2 (Rnx ) without any ambiguity. · L p with 1 ≤ p ≤ ∞ denotes the norm in the Lebesgue space L p (Rnx × Rnξ ). The x,ξ
q
norm in the space Z q = L 2ξ (L x ) is defined by u Z q =
Rn
21
2 |u(x, ξ )| d x q
Rn
q
dξ
, u = u(x, ξ ) ∈ Z q .
For the multiple indices α, β, γ with α = (α1 , α2 , . . . , αn ), β = (β1 , β2 , . . . , βn ), and γ β γ β β β γ γ γ = (γ1 , γ2 , . . . , γn ), we adopt the usual notations ∂x ∂ξ = ∂x11 ∂x22 · · · ∂xnn ∂ξ11 ∂ξ22 · · · ∂ξnn , n α = ∂ β ∂ γ when α = β + γ . The length of α is |α| = and in particular ∂x,ξ x ξ i=1 αi . 2. Decay Estimates on the Linearized Equation 2.1. Preliminaries. (i) Linearized equation. In this section, we are concerned with the initial value problem for the linearized Boltzmann equation corresponding to (1.1). More generally, for some initial time s ∈ R, it is in the form ∂t u + ξ · ∇x u + E 1 · ∇ξ u = Lu + ξ · E 2 u, t > s, x ∈ Rn , ξ ∈ Rn , u(t, x, ξ )|t=s = u 0 (x, ξ ), x ∈ Rn , ξ ∈ Rn .
(2.1) (2.2)
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Here u 0 (x, ξ ) is given, denoting the same initial data for different initial time, and E i = E i (t, x), i = 1, 2, are given vector-valued functions for generalization. Formally the solution to the initial value problem (2.1)–(2.2) is written as U (t, s)u 0 , −∞ < s ≤ t < ∞, where U (t, s) is called the solution operator for the linear Eq. (2.1). We shall obtain some basic decay in time estimates on U (t, s) in some Sobolev space weighted with velocity functions H Rnx × Rnξ ; (1 + |ξ |)k d xdξ , ≥ 2, k ≥ 1, which enable us to solve the nonlinear problem by the Duhamel formula and the contraction mapping theorem. (ii) Known properties of the linearized collision operator. For the linearized collision operator L given by (1.3), one has (Lu)(ξ ) = −ν(ξ )u(ξ ) + (K u)(ξ ), ν(ξ ) = |(ξ − ξ∗ ) · ω|M∗ dωdξ∗ , n n−1 R ×S
1 1 1 1 −M 2 u ∗ + (M∗ ) 2 u + (M ) 2 u ∗ |(ξ − ξ∗ ) · ω|M∗2 dωdξ∗ (K u)(ξ ) = n n−1 R ×S = K (ξ, ξ∗ )u(ξ∗ )dξ∗ . Rn
Moreover, the following well-known properties hold; see [3,4,11]. (a) There exists ν0 > 0 such that ν0 (1 + |ξ |) ≤ ν(ξ ) ≤ ν0−1 (1 + |ξ |); (b) K is a self-adjoint compact operator on L 2 (Rnξ ) with a real symmetric integral kernel K (ξ, ξ∗ ) which enjoys the estimate |K (ξ, ξ∗ )|(1 + |ξ∗ |)−β dξ∗ ≤ C(1 + |ξ |)−β−1 , β ≥ 0; (2.3) Rn
(c) the nullspace of the operator L is the space of collision invariants
N = K er L = span M1/2 ; ξi M1/2 , i = 1, 2, . . . , n; |ξ |2 M1/2 ; (d) L is an unbounded, self-adjoint and non-positive operator on L 2 (Rnξ ) with the domain
D(L) = u ∈ L 2 (Rnξ ) ν(ξ )u ∈ L 2 (Rnξ ) . (iii) Macro-micro decomposition. Define P as a velocity projection operator from L 2 (Rnξ ) to N . Then any function u(t, x, ξ ) for any fixed (t, x) can be uniquely decomposed as the sum of the macroscopic component Pu and microscopic component {I −P}u: u(t, x, ξ ) = Pu + {I − P}u.
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With this notion, the linearized collision operator L satisfies − uLu dξ ≥ c0 ν(ξ )({I − P}u)2 dξ, ∀ u ∈ D(L), Rn
Rn
for some constant c0 > 0. Here for simplicity, throughout this section, one sets u 1 = Pu, u 2 = {I − P}u. Equation (2.1) is also decomposed as follows. The microscopic equation for u 2 is obtained by applying the microscopic projection I − P to (2.1): ∂t u 2 − Lu 2 = −{I − P} (ξ · ∇x u) − {I − P} E 1 · ∇ξ u − ξ · E 2 u , or, ∂t u 2 − Lu 2 = −ξ · ∇x u 2 − E 1 · ∇ξ u 2 + ξ · E 2 u 2 − ξ · ∇x u 1 − E 1 · ∇ξ u 1 + ξ · E 2 u 1 + P ξ · ∇x u + E 1 · ∇ξ u − ξ · E 2 u .
(2.4)
In order to write the macroscopic equation, as in [13], one first expands u 1 = Pu as n bi (t, x)ξi + c(t, x)|ξ |2 M1/2 . u 1 = a(t, x) + i=1
Putting this expansion into the following equation: ∂t u 1 + ξ · ∇x u 1 + E 1 · ∇ξ u 1 − ξ · E 2 u 1 = − ∂t u 2 + ξ · ∇x u 2 + E 1 · ∇ξ u 2 − ξ · E 2 u 2 − Lu 2 := , and then collecting the coefficients with respect to the basis M1/2 , ξi M1/2 , |ξi |2 M1/2 , ξi ξ j M1/2 1≤i≤n
1≤i≤n
1≤i< j≤n
(2.5)
, |ξ |2 ξi M1/2
1≤i≤n
,
one has M1/2 : ∂t a + E 1 · b = 0 , ξi M1/2 : ∂t bi + ∂i a − (a E¯ i − 2cE 1i ) = i1 , |ξi |2 M1/2 : ∂t c + ∂i bi − E¯ i bi = i21 , ξi ξ j M
1/2
|ξ | ξi M
1/2
2
: ∂i b j + ∂ j bi − ( E¯ i b j + E¯ j bi ) = : ∂i c − E¯ i c = i3 , ij
ij 22 ,
(2.6) (2.7) (2.8) (2.9) (2.10)
where for simplicity, ∂i = ∂xi , ∂ j = ∂x j , and 0 , i1 , i21 , 22 , i3 with 1 ≤ i = j ≤ n are the corresponding coefficients of with respect to the above basis, and E¯ is defined by 1 E¯ = E 1 + E 2 . 2 Finally we list a basic fact for any function u = u(t, x, ξ ).
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Proposition 2.1. Let m be a non-negative integer and k be any number. Then for any β β β and γ , one has ∂tm ∂x Pu = P∂tm ∂x u with estimates 1 k m β γ ν ∂t ∂x ∂ξ Pu ≤ ∂tm ∂xβ a + ∂tm ∂xβ b + ∂tm ∂xβ c ≤ C ∂tm ∂xβ Pu , C where C > 1 is some constant independent of u. 2.2. Estimates on commutators. In this subsection we study the functional properties of commutators related to L: [L, ξi ], L, ∂ξi , L, ∂ξi , ξ j , L, ∂ξi , ∂ξ j , 1 ≤ i, j ≤ n. Let L denote this kind of commutator. Lemma 2.1. L is a bounded linear operator from L 2 (Rnξ ) to itself, i.e., there is some constant C such that Lu ≤ Cu, for any u = u(ξ ) ∈
(2.11)
L 2 (Rnξ ).
Proof. This lemma is proved by the following steps. Step 1. The explicit expressions of ν and K are available: ν(ξ ) = Cn |ξ − ξ∗ |M(ξ∗ )dξ∗ , Rn
K (ξ, ξ∗ ) = K 1 (ξ, ξ∗ ) + K 2 (ξ, ξ∗ ) |ξ |2 + |ξ∗ |2 , K 1 (ξ, ξ∗ ) = −Cn |ξ − ξ∗ | exp − 4 Cn 1 (|ξ |2 − |ξ∗ |2 )2 |ξ − ξ∗ |2 K 2 (ξ, ξ∗ ) = , exp − − |ξ − ξ∗ |n−2 8 |ξ − ξ∗ |2 8 where for simplicity Cn may be some different positive constants depending only on the space dimension n. The proof for the case n = 3 is given in [11]. The general case n ≥ 3 can be obtained similarly. Step 2. In this step, some preparations are made for the next step. First, from (2.13), one can easily verify that ν(ξ ) is a smooth function of ξ with bounded derivatives of any order. Next, for the integral kernels K 1 and K 2 , set K 1 (ξ, ξ∗ ) = K 11 (|ξ − ξ∗ |)K 12 (ξ, ξ∗ ), K 2 (ξ, ξ∗ ) = K 21 (|ξ − ξ∗ |)K 22 (ξ, ξ∗ ), where K 11 (|ξ − ξ∗ |) = −Cn |ξ − ξ∗ |, Cn |ξ − ξ∗ |2 , exp − K 21 (|ξ − ξ∗ |) = |ξ − ξ∗ |n−2 8 |ξ |2 + |ξ∗ |2 , 4 1 (|ξ |2 − |ξ∗ |2 )2 . K 22 (ξ, ξ∗ ) = exp (V2 ) , V2 = − 8 |ξ − ξ∗ |2
K 12 (ξ, ξ∗ ) = exp (V1 ) , V1 = −
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Finally, for the simplicity of notions, we define velocity differential operators ∂¯i , i = 1, 2, . . . , n by ∂¯i = −{∂ξi + ∂ξi∗ }. Notice that ∂¯i h ≡ 0 for any radial function h = h(|ξ − ξ∗ |), and moreover, ξi + ξi∗ ∂¯i V1 = V1i , V1i = , 2 (ξi − ξi∗ ) (|ξ |2 − |ξ∗ |2 ), ∂¯i V2 = V2i , V2i = 2|ξ − ξ∗ |2 ∂¯ j V1i = ∂¯ j ∂¯i V1 = V1i j , V1i j = −δi j , (ξi − ξi∗ )(ξ j − ξ j∗ ) , ∂¯ j V2i = ∂¯ j ∂¯i V2 = V2i j , V2i j = |ξ − ξ∗ |2 where δi j is Kronecker’s symbol. Then one has ∂¯i K 11 ∂¯i K 12 ∂¯ j (K 12 V1i ) ∂¯ j (K 22 V2i )
= = =
∂¯i K 21 ≡ 0, K 12 V1i , ∂¯i K 22 = K 22 V2i , K 12 V1i V1 j + K 12 V1i j ,
= K 22 V2i V2 j + K 22 V2i j .
Step 3. This step is concerned with the computation of commutators. Set V0i = ξi∗ −ξi ; direct calculations yield [L, ξi ]u = K (ξ, ξ∗ )V0i u(ξ∗ )dξ∗ , Rn L, ∂ξi u = ∂ξi νu + (K 1 V1i + K 2 V2i )u(ξ∗ )dξ∗ , Rn [L, ∂ξi ], ξ j = (K 1 V1i + K 2 V2i )A j u(ξ∗ )dξ∗ , Rn [L, ∂ξi ], ∂ξ j = −∂ξ2i ξ j νu + [K 1 (V1i V1 j + V1i j ) Rn
+K 2 (V2i V2 j + V2i j )]u(ξ∗ )dξ∗ .
Step 4. Write K c (ξ, ξ∗ ) as any one of the following integral kernels: K V0i , K 1 V1i + K 2 V2i , (K 1 V1i + K 2 V2i )V0 j , K 1 (V1i V1 j + V1i j ) + K 2 (V2i V2 j + V2i j ). Direct observations show that K 1 can absorb any finite numbers of velocity functions V0i , V1i and V1i j , while K 2 can absorb any finite number of velocity functions V0i , V2i and V2i j . This means that if one defines |ξ |2 + |ξ∗ |2 , K 1 (ξ, ξ∗ ) = Cn |ξ − ξ∗ | exp − 8 Cn 1 (|ξ |2 − |ξ∗ |2 )2 |ξ − ξ∗ |2 K 2 (ξ, ξ∗ ) = , exp − − |ξ − ξ∗ |n−2 16 |ξ − ξ∗ |2 16 then 1 (ξ, ξ∗ ) + K 2 (ξ, ξ∗ ) := K (ξ, ξ∗ ). |K c (ξ, ξ∗ )| ≤ K
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(ξ, ξ∗ ) satisfies the estimate (2.3) for β = 0 similar to K , it follows that Since K |K c (ξ, ξ∗ )|dξ ≤ C, |K c (ξ, ξ∗ )|dξ∗ ≤ C, Rn
Rn
which implies that
Rn
K c (ξ, ξ∗ )u(ξ∗ )dξ∗ ≤ Cu.
Thus (2.11) is proved. This completes the proof of the lemma.
In general, for any positive integer N , define the iterative commutator L by L = [· · · [[L, X1 ], X2 ] · · · , X N ], where for each k ∈ {1, 2, . . . , N }, Xk denotes the velocity multiplier ξik or the velocity differential operator ∂ξik . Write L as the sum of two parts L I and L I I : L = LI + LI I , L I = [· · · [[−ν(ξ ), X1 ], X2 ] · · · , X N ], L I I = [· · · [[K , X1 ], X2 ] · · · , X N ]. Then L has the same property as in Lemma 2.1. Corollary 2.1. The following properties hold: (i) L I is a bounded linear operator on L 2 (Rnξ ). (ii) L I I is a compact operator on L 2 (Rnξ ) with the integral kernel K c (ξ, ξ∗ ), which satisfies that for any k ≥ 0, there is some constant C depending on k such that ν k L I I u ≤ Cν k−1 u,
(2.12)
for any u = u(ξ ). (iii) L is a bounded linear operator on L 2 (Rnξ ). Proof. It is obvious that (iii) directly follows from (i) and (ii). Thus it suffices to prove (i) and (ii). For the first part L I , in fact it is a velocity multiplier generated by ν(ξ ), given by ⎧ N ⎨ N +1 (−1) Xk ν(ξ ) all Xk are ∂ξik , LI = k=1 ⎩ 0 otherwise. Thus (i) holds from the proof of Lemma 2.1. For the second part L I I , it can be written as (L I I u)(ξ ) = K c (ξ, ξ∗ )u(ξ∗ )dξ∗ , Rn
K c (ξ, ξ∗ ) = K 1 (ξ, ξ∗ )V1 + K 2 (ξ, ξ∗ )V2 , where V1 is the linear combination of products of velocity multipliers V0i , V1i and V1i j , and similarly V2 is the linear combination of products of velocity multipliers V0i , V2i
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and V2i j . Hence, similar to the compact operator K , L I I is also a compact operator on L 2 (Rnξ ) with the integral kernel K c satisfying (2.3). Finally we claim that (2.3) implies (2.12). In fact, for any k ≥ 0 and any u = u(ξ ), 1/2 1/2 |K c (ξ, ξ∗ )|ν −2k (ξ∗ )dξ∗ |K c (ξ, ξ∗ )|ν 2k (ξ∗ )u 2 (ξ∗ )dξ∗ (L I I u)(ξ ) ≤ Rn
≤ Cν
−(2k+1)/2
Rn
(ξ )
Rn
1/2
|K c (ξ, ξ∗ )|ν (ξ∗ )u (ξ∗ )dξ∗ 2k
2
,
which gives 2k 2 2k 2 ν (ξ )(L I I u) (ξ )dξ ≤ C ν (ξ∗ )u (ξ∗ ) |K c (ξ, ξ∗ )|ν −1 (ξ )dξ dξ∗ Rn Rn Rn ≤C ν 2k−2 (ξ∗ )u 2 (ξ∗ )dξ∗ . Rn
That is (2.12). This completes the proof of this lemma.
Finally, Corollary 2.1 directly gives Corollary 2.2. Let γ , k be |γ | ≥ 1 and k ≥ 0. Then there is some constant C such that γ γ [L, ∂ξ ]u ≤ C ∂ξ u, 0≤|γ |≤|γ |−1
γ ν k [K , ∂ξ ]u
≤ Cν k−1 u,
for any u = u(ξ ). 2.3. Energy estimates. From now on, we use the following notation of the index sets for differentiations: Let be any positive integer, 0 (β) 1 (β) 2 (β) i3 (β, γ )
= {0 ≤ |β| ≤ }, = {1 ≤ |β| ≤ }, = {0 ≤ |β| ≤ − 1}, = {|γ | = i, 0 ≤ |β| + |γ | ≤ }, i = 1, 2, . . . , ,
3 (β, γ ) = {|γ | ≥ 1, 0 ≤ |β| + |γ | ≤ } = ∪i=1 i3 (β, γ ), j
4 (β, γ ) = {|γ | = j, 0 ≤ |β| + |γ | ≤ − 1}, j = 1, 2, . . . , − 1,
−1 i 4 (β, γ ) = {|γ | ≥ 1, 0 ≤ |β| + |γ | ≤ − 1} = ∪i=1 4 (β, γ ).
(i) Assumptions and energy inequality. Throughout this subsection, the following assumptions are made: (A1) The integer ≥ 2; (A2) For the functions E 1 and E 2 , there is δ > 0 such that (1 + |x|)∂ β E i (t, x) ∞ + (1 + |x|)∂t ∂ β E i (t, x) ∞ ≤ δ, x x L L 0 (β)
where i = 1, 2.
t,x
2 (β)
t,x
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Under the above assumptions, our final goal of this subsection is to show that if δ > 0 is small enough, then the energy inequality holds: d (2.13) H (t) + cD(t) ≤ C∇x u 1 2 , dt where c > 0 is some positive constant, C is some constant, H (t) is a nonlinear energy functional and D(t) is the corresponding dissipation rate. For the moment, we would not like to expose the precise forms of H (t) and D(t), see Theorem 2.1, but only point out some important characteristics for them: • H (t) contains the microscopic component u 2 and its derivatives with respect to t, x, and ξ up to order of ≥ 2, and also only the derivatives of the macroscopic component u 1 with respect to t and x; • In H (t), for the time derivatives, the differential order of time is at most one, where there is not any weight function, but for others, the velocity function ν is added. • D(t) contains those terms corresponding to H (t) but the power of velocity weight function is higher 1/2. • There is some constant C such that H (t) ≤ C D(t) for any t ≥ 0. (ii) Energy estimates on the microscopic part. Now we turn to the proof of the energy inequality in the form of (2.13). First consider the estimates on some energy functional H1 (t) which is a linear combination of the following terms: β γ 2 β γ 2 ∂ β u 2 , ∂t ∂ β u 2 , ∂ u , ∂ ∂ u u 2 2 , ∂ ∂ . 2 t 2 x x x ξ x ξ 1 (β)
2 (β)
i3 (β,γ )
j
4 (β,γ )
For brevity, define the time dependent linear operator B(t) and D(t) by B(t) = ξ · ∇x + E 1 · ∇ξ − L, D(t) = ξ · ∇x + E 1 · ∇ξ − ξ · E 2 . Using the above notations, (2.1) and (2.4) can be rewritten as ∂t u + B(t)u = ξ · E 2 u,
(2.14)
∂t u 2 + B(t)u 2 = ξ · E 2 u 2 + [P, D(t)]u,
(2.15)
and where [P, D(t)] is the commutator given by [P, D(t)] = PD(t) − D(t)P. In what follows, a series of lemmas are given. The first one is concerned with the L 2x,ξ -estimate on the microscopic component u 2 . For this purpose, from the properties of the linearized Boltzmann operator L, the smallness assumption we imposed on the u1 external forces E 1 , E 2 , and by using the Hardy inequality |x| ≤ C∇x u 1 , we have by applying the standard energy method to (2.15) that Lemma 2.2. If δ > 0 is small enough, then one has 2 d u 2 2 + c ν 1/2 u 2 ≤ C∇x u 1 2 . dt
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The next lemma is on the L 2x,ξ -estimate on ∂x u for β ∈ 1 (β). Lemma 2.3. If δ > 0 is small enough, then one has d 1/2 β 2 β γ 2 ∂ β u 2 + c ∂ β u 1 2 + Cδ ν ∂ ∂ u ≤ Cδ ∂ u . 2 2 x x x x ξ dt 1 (β)
1 (β)
1 (β)
3 (β,γ )
(2.16) β
Proof. Directly applying ∂x with β ∈ 2 (β) to (2.14) gives ∂t (∂xβ u) + B(t)(∂xβ u) = ∂xβ (ξ · E 2 u) + [B(t), ∂xβ ]u.
(2.17)
β
Further multiplying (2.25) by ∂x u and then integrating over Rnx × Rnξ , one has 2 1 d β 2 Ii , ∂x u + c0 ν 1/2 ∂xβ u 2 ≤ 2 dt 2
(2.18)
i=1
where we have used the identity {I − P}∂xβ u = ∂xβ {I − P}u = ∂xβ u 2 , β
and Ii , i = 1, 2, denote the corresponding terms after taking the inner product with ∂x u for ones on the right-hand side of (2.17). Next we estimate I1 and I2 . To this end, from the smallness assumption we imposed on E 1 and E 2 , the Hardy inequality and the Cauchy-Schwarz inequality, we can deduce that 1/2 β 2 β 2 I1 ≤ Cδ ν ∂x u 2 + Cδ ∂x u 1 , 1 (β )
and I2 ≤ Cδ
1 (β )
β 2 ∂x u 1 + Cδ 1 (β )
3 (β ,γ )
2 β γ ∂x ∂ξ u 2 .
Thus taking summation over β ∈ 1 (β) for (2.18) and then collecting all estimates, (2.16) follows if δ > 0 is small enough. This completes the proof of the lemma. γ
For the L 2x,ξ -estimate on ∂t ∂x u(γ ∈ 2 (β)), we have the following result Lemma 2.4. If δ > 0 is small enough, then one has d 1/2 γ 2 ∂t ∂xγ u 2 + c ν ∂t ∂x u 2 dt 2 (β)⎛ 2 (β) ⎞ ≤ Cδ ⎝ ∂xβ u 1 2 + ∂t ∂xβ u 1 2 ⎠ 2 (β) ⎛ 1 (β) ⎞ 2 2 2 1/2 β β γ β γ ⎠ +Cδ ⎝ ν ∂x u 2 + ∂x ∂ξ u 2 + ∂t ∂x ∂ξ u 2 . 1 (β)
3 (β,γ )
4 (β,γ )
(2.19)
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Proof. First it is easy to see that for β ∈ 2 (β), ∂t (∂t ∂xβ u) + B(t)(∂t ∂xβ u) = ξ · ∂t ∂xβ (E 2 u) + [B(t), ∂t ∂xβ ]u, which gives 2 2 1 d ∂t ∂xβ u2 + c0 ν 1/2 ∂t ∂xβ u 2 ≤ Ii . 2 dt
(2.20)
i=1
For I1 , one has
2 2 I1 ≤ δ ν 1/2 ∂t ∂xβ u + Cδ ν 1/2 ∂t ∂xβ (E 2 u) 1/2 β 2 β 2 ≤ Cδ ν ∂t ∂x u 2 + Cδ ∂t ∂x u 1 2 (β )
2 (β ) 1/2 β 2 β 2 +Cδ ν ∂x u 2 + Cδ ∂x u 1 . 1 (β )
1 (β )
For I2 , noticing that
[B(t), ∂t ∂xβ ]u = −
Cβ ∂xβ−β E 1 · ∇ξ ∂t ∂xβ u
0≤|β |≤|β|−1
−
Cβ ∂t ∂xβ−β E 1 · ∇ξ ∂xβ u,
0≤|β |≤|β|
one also has
2 β 2 β 2 I2 ≤ δ ∂t ∂xβ u 2 + Cδ ∂t ∂x u 1 + Cδ ∂x u 1
2 (β ) 1 (β ) β γ 2 β γ 2 +Cδ ∂x ∂ξ u 2 + Cδ ∂t ∂x ∂ξ u 2 . 3 (β,γ )
4 (β,γ )
Thus taking summation over β ∈ 2 (β) for (2.20) and then collecting all estimates, (2.19) follows if δ > 0 is small enough. This completes the proof of the lemma. β γ
As to the L 2x,ξ -estimate on ∂x ∂ξ u 2 for (β, γ ) ∈ i3 (β, γ ), we can conclude that Lemma 2.5. If δ > 0 is small enough, then one has d β γ 2 1/2 β γ 2 ∂x ∂ξ u 2 + c ν ∂x ∂ξ u 2 dt i i 3 (β,γ )
3 (β,γ )
∂ β u 1 2 + C ∂ β u 2 2 ≤C x x 1 (β)
+Ci,i−1
0 (β)
β γ 2 ∂x ∂ξ u 2 + δCi,i+1
i−1 3 (β,γ )
β γ 2 ∂x ∂ξ u 2 ,
(2.21)
i+1 3 (β,γ )
where i = 1, 2, . . . , , and Ci,i−1 , Ci,i+1 are some constants with additional conventions: C1,0 = C , +1 = 0.
(2.22)
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Proof. First apply ∂ξ with |γ | = i = 1, 2, . . . , to (2.15) to get γ
γ
γ
γ
γ
∂t (∂ξ u 2 ) + B(t)∂ξ u 2 = E 2 · ∂ξ (ξ u 2 ) + ∂ξ [P, D(t)]u + [B(t), ∂ξ ]u 2 γ
γ −1
= ξ · E 2 ∂ξ u 2 + eγ · E 2 ∂ξ γ +∂ξ [P, D(t)]u
γ −1
u 2 − eγ · ∇x ∂ξ
u2
γ − [L, ∂ξ ]u 2 ,
(2.23)
where eγ denotes a constant vector, and for simplicity we used the notations γ −1
eγ · E 2 ∂ξ
u2 =
|γ |=1
γ
γ −γ
γ ∂ξ ξ · E 2 ∂ξ
u2 =
0≤|γ |≤|γ |−1
γ −γ
C γ ∂ξ
γ
ξ · E 2 ∂ξ u 2 ,
and γ −1
eγ · ∇x ∂ξ
u2 =
|γ |=1
γ
γ −γ
γ ∂ξ ξ · ∇x ∂ξ
u2 =
0≤|γ |≤|γ |−1
γ −γ
C γ ∂ξ
γ
ξ · ∇ x ∂ξ u 2 .
β
Further apply ∂x with (β, γ ) ∈ i3 (β, γ ) to (2.23) to obtain γ
γ
∂t (∂xβ ∂ξ u 2 ) + B(t)(∂xβ ∂ξ u 2 ) γ = Cβ ξ · ∂xβ−β E 2 ∂xβ ∂ξ u 2 + 0≤|β |≤|β|
−
0≤|β |≤|β|
γ
γ −1
Cβ ∂xβ−β E 1 · ∇ξ ∂xβ ∂ξ u 2 − eγ · ∇x ∂xβ ∂x
0≤|β |≤|β|−1 γ +∂xβ ∂ξ [P, D(t)]u
γ −1
Cβ eγ · ∂xβ−β E 2 ∂xβ ∂ξ
u2
u2
γ
− [L, ∂ξ ]∂xβ u 2 .
(2.24)
β γ
Multiplying (2.24) by ∂x ∂ξ u 2 and integrating it over Rnx × Rnξ , one has 6 2 1 d β γ 2 γ Ii . ∂x ∂ξ u 2 + c0 ν 1/2 {I − P}∂xβ ∂ξ u 2 ≤ 2 dt
(2.25)
i=1
We estimate each term Ii as follows. For I1 , I2 and I3 , one has 2 γ I1 ≤ δ ν 1/2 ∂xβ ∂ξ u 2 + Cδ 2 γ I2 ≤ δ ∂xβ ∂ξ u 2 + δCi,i−1 2 γ I3 ≤ δ ∂xβ ∂ξ u 2 + δCi,i+1
1/2 β γ 2 ν ∂x ∂ξ u 2 ,
i3 (β ,γ )
2 β γ β 2 ∂x ∂ξ u 2 + δCδi1 ∂x u 2 ,
i−1 3 (β ,γ )
2 β γ ∂x ∂ξ u 2 ,
0 (β )
i+1 3 (β ,γ )
where δi1 is the Kroneker symbol and we have set (2.22). In fact, if i = , 3 (β, γ ) γ means β = 0 and |γ | = , i.e. one has taken only the velocity derivative ∂ξ with |γ | = ,
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which implies I3 = 0 for this special case. For I4 , I5 and I6 , similarly it holds that I4 ≤
c0 β γ 2 ∂x ∂ξ u 2 + Ci,i−1 6
1/2 β γ 2 β 2 ν ∂x ∂ξ u 2 + Cδi1 ∂x u 2 ,
0 (β )
i−1 3 (β ,γ )
c0 β 2 β 2 β γ 2 I5 ≤ ∂x u 1 + C ∂x u 2 , ∂x ∂ξ u 2 + C 6 1 (β )
1 (β )
and 2 2 $ c # 0 β γ γ γ γ I6 = − [L, ∂ξ ]∂xβ u 2 , ∂xβ ∂ξ u 2 ≤ ∂x ∂ξ u 2 + C [L, ∂ξ ]∂xβ u 2 6 c0 β γ 2 β 2 β γ 2 ≤ ∂x ∂ξ u 2 + C ∂x u 2 + Ci,i−1 ∂x ∂ξ u 2 , 6 i−1 0 (β )
3 (β ,γ )
where Corollary 2.2 was used. Finally it is noticed that 2 2 2 1/2 γ γ γ ν {I − P}∂xβ ∂ξ u 2 ≥ ν 1/2 ∂xβ ∂ξ u 2 − ν 1/2 P∂xβ ∂ξ u 2 2 β 2 γ ≥ ν 1/2 ∂xβ ∂ξ u 2 − C ∂x u 2 . 0 (β )
Putting all the above estimates into (2.25) and then taking summation over (β, γ ) ∈ i3 (β, γ ) leads to (2.21), provided that δ > 0 is small enough. This completes the proof of the lemma. β γ
j
Finally for the L 2x,ξ -estimate on ∂t ∂x ∂ξ u 2 ((β, γ ) ∈ 4 (β, γ ), j = 1, 2, . . . ,
− 1), we have Lemma 2.6. If δ > 0 is small enough, then one has d dt
β γ 2 1/2 β γ 2 ∂t ∂x ∂ξ u 2 + c ν ∂t ∂x ∂ξ u 2 j
j
4 (β,γ )
4 (β,γ )
∂t ∂ β u 1 2 + C ∂t ∂ β u 2 2 ≤C x x 2 (β)
+C j, j−1
j−1
4
+Cδ
2 (β)
β γ 2 ∂t ∂x ∂ξ u 2 + δC j, j+1
j+1
(β,γ )
4 (β,γ )
β γ 2 ∂ β u 2 2 + Cδ ∂x ∂ξ u 2 , x
0 (β)
β γ 2 ∂t ∂x ∂ξ u 2 (2.26)
3 (β,γ )
where j = 1, 2, . . . , −1, and Ci,i−1 , Ci,i+1 are some constants with additional conventions: C1,0 = C −1, = 0.
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R. Duan, S. Ukai, T. Yang, H. Zhao j
Proof. Notice that (2.24) also holds for (β, γ ) ∈ 4 (β, γ ) with j = 1, 2, . . . , − β γ 1. Then further applying ∂t to it, multiplying the resulting identity by ∂t ∂x ∂ξ u 2 , and integrating the final result over Rn × Rn , we have 7 2 β γ 2 1/2 β γ ∂ ν ∂ ∂ u + c {I − P}∂ ∂ ∂ u ≤ Ii . t x ξ 2 0 t x ξ 2
1 d 2 dt
(2.27)
i=1
First for I1 , I2 and I3 , one has # $ γ γ γ I1 = Cβ ξ · ∂xβ−β E 2 ∂t ∂xβ ∂ξ u 2 + ξ · ∂t ∂xβ−β E 2 ∂xβ ∂ξ u 2 , ∂t ∂xβ ∂ξ u 2 0≤|β |≤|β|
2 γ ≤ δ ν 1/2 ∂t ∂xβ ∂ξ u 2 + Cδ
1/2 β γ 2 ν ∂t ∂x ∂ξ u 2
j
(β ,γ )
4 1/2 β γ 2 ∂ ∂ u ν x ξ 2 ,
+Cδ j
3 (β ,γ )
I2 = ≤
$ # γ −1 γ −1 γ Cβ eγ · ∂xβ−β E 2 ∂t ∂xβ ∂ξ u 2 + ∂t ∂xβ−β E 2 ∂xβ ∂ξ u 2 , ∂t ∂xβ ∂ξ u 2
0≤|β |≤|β| 2 γ δ ∂t ∂xβ ∂ξ u 2
+ δC j, j−1
+δCδ j1
j−1
(β ,γ )
4 β 2 ∂ u ∂t x 2 + δC j, j−1 2 (β )
+δCδ j1
β γ 2 ∂t ∂x ∂ξ u 2 j−1
3
β 2 ∂x u 2 ,
2 β γ ∂x ∂ξ u 2
(β ,γ )
0 (β )
and
# $ γ γ γ Cβ ∂xβ−β E 1 · ∇ξ ∂t ∂xβ ∂ξ u 2 + ∂t ∂xβ−β E 1 · ∇ξ ∂xβ ∂ξ u 2 , ∂t ∂xβ ∂ξ u 2
I3 = −
0≤|β |≤|β|−12 γ ≤ δ ∂t ∂xβ ∂ξ u 2 + δC j, j+1
β γ 2 ∂t ∂x ∂ξ u 2 + Cδ
j+1
4 (β ,γ )
Furthermore, it holds that $ # γ −1 γ I4 = −eγ · ∇x ∂t ∂xβ ∂x u 2 , ∂t ∂xβ ∂ξ u 2 c0 β γ 2 ≤ ∂t ∂x ∂ξ u 2 + C j, j−1 6 j−1 #
4
γ
γ
(β ,γ )
$
2 β γ ∂x ∂ξ u 2 .
j+1
3 (β ,γ )
β γ 2 β 2 ∂t ∂x ∂ξ u 2 + C ∂t ∂x u 2 , 2 (β )
I5 = − ∂t ∂xβ ∂ξ [P, D(t)]u, ∂t ∂xβ ∂ξ u 2 c0 β 2 β 2 β γ 2 ≤ ∂t ∂x u 1 + C ∂t ∂x u 2 , ∂t ∂x ∂ξ u 2 + C 6 2 (β ) 2 (β ) $ # γ γ I6 = − [L, ∂ξ ]∂t ∂xβ u 2 , ∂t ∂xβ ∂ξ u 2
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205
c0 β 2 β γ 2 ∂t ∂x u 2 + C j, j−1 ∂t ∂x ∂ξ u 2 + C 6 2 (β )
β γ 2 ∂t ∂x ∂ξ u 2 .
j−1
4
(β ,γ )
Finally, $ # γ γ I7 = − ∂t E 1 · ∇ξ ∂xβ ∂ξ u 2 , ∂t ∂xβ ∂ξ u 2 2 β γ 2 γ ≤ δ ∂t ∂xβ ∂ξ u 2 + Cδ ∂t ∂x ∂ξ u 2 . j+1
3 (β ,γ )
Inserting all the above estimates into (2.27) and then taking summation over (β, γ ) ∈ j 4 (β, γ ) leads to (2.26), provided that δ > 0 is small enough. This completes the proof of the lemma. Putting all the above estimates together, we can obtain the following elementary energy estimates, which follow directly from a proper linear combination of all the energy inequalities obtained in Lemma 2.2–Lemma 2.6. Corollary 2.3. Under Assumptions (A1)–(A2), if δ > 0 is small enough, then there is an energy functional H1 (t) and a corresponding dissipation rate D1 (t) such that ⎞ ⎛ d (2.28) ∂xβ u 1 2 + ∂t ∂xβ u 1 2 ⎠, H1 (t) + cD1 (t) ≤ C ⎝ dt 1 (β)
2 (β)
where H1 (t) and D1 (t) is defined by H1 (t) ∼ u 2 2 + ∂xβ u2 + ∂t ∂xβ u2
+
1 (β)
3 (β,γ )
D1 (t) ∼ ν +
1/2
u2 +
3 (β,γ )
2 (β)
γ ∂xβ ∂ξ u 2 2 2
+
γ
4 (β,γ )
∂t ∂xβ ∂ξ u 2 2 ,
ν 1/2 ∂xβ u 2 2
1 (β) γ ν 1/2 ∂xβ ∂ξ u 2 2
+
+
ν 1/2 ∂t ∂xβ u 2 2
2 (β)
4 (β,γ )
γ
ν 1/2 ∂t ∂xβ ∂ξ u 2 2 .
(iii) Estimates on the macroscopic part. It should be pointed out that D1 (t) is a lack of the macroscopic dissipation rate. Then it is not true that there is a constant C such that H1 (t) ≤ C D1 (t) for any t ≥ 0. However, except for the first order derivatives of the macroscopic component, the higher order derivatives can be bounded by part of the microscopic dissipation rate D1 (t). Thus a proper further linear combination makes the dissipation rate include the derivatives of the macroscopic component of at least first order. The following estimate is based on the macroscopic equations (2.6)–(2.10) satisfied by a, b, c. Lemma 2.7. Under Assumptions (A1) and (A2), if δ > 0 is small enough, then it holds that
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R. Duan, S. Ukai, T. Yang, H. Zhao
∂ β u 1 2 + ∂t ∂ β u 1 2 x
x
1 (β)
≤C
d dt
2 (β)
& ∂ β 2 , ∂xβ a, ∇x · ∂xβ b + C∇x u 1 2 + C x
%
(2.29)
2 (β)
1≤|β|≤ −1 β
where for any β, ∂x 2 is defined by ∂xβ 2 = ∂xβ 0 2 + ∂xβ 1 2 + ∂xβ 21 2 + ∂xβ 22 2 + ∂xβ 3 2 , β β with ∂x 1 2 = 1≤i≤n ∂x i1 2 , and similarly for other terms. Proof. First consider estimates on the pure space derivatives of a, b, c. We start with b j , which will satisfy a standard elliptic equation. In fact, for any fixed j ∈ {1, 2, . . . , n} and |β| ≥ 0, by (2.8) and (2.9), direct calculations yield ∂ j ∂xβ ( E¯ i bi ) + ∂i ∂xβ ( E¯ i b j + E¯ j bi ) + 2∂ j ∂xβ ( E¯ j b j )
∂xβ b j = −∂ j j ∂xβ b j − −
i = j
∂ j ∂xβ i21
i = j
+
i = j ij ∂i ∂xβ 22
+ ∂ j ∂xβ 21 . j
i = j β
Thus after multiplying by ∂x b j and taking some integrations by part, it holds that ∇x ∂xβ b j 2 + ∂ j ∂xβ b j 2 1 1 β ¯ ∂x ( E ⊗ b)2 + ∂xβ 21 2 + ∂xβ 22 2 ≤ ∇x ∂xβ b j 2 + 2 2 1 ∇x ∂xβ b2 + C ∂xβ 21 2 + ∂xβ 22 2 , ≤ ∇x ∂xβ b j 2 + Cδ 2 2 0≤|β |≤|β|
which implies ∇x ∂xβ b2 ≤ Cδ 2
∇x ∂xβ b2 + C ∂xβ 21 2 + ∂xβ 22 2 .
0≤|β |≤|β|−1
Furthermore, since δ > 0 can be small enough, by iteration, one has that for any |β| ≥ 0, ∇x ∂xβ b2 ≤ C ∂xβ 21 2 + ∂xβ 22 2 , (2.30) 0≤|β |≤|β|
which, after taking summation over 0 ≤ |β| ≤ − 1, gives ∂xβ 21 2 + ∂xβ 22 2 . ∂xβ b2 ≤ C 1 (β)
2 (β)
For the pure space derivatives of c, it follows from (2.10) that for |β| ≥ 0, ¯ 2 + ∂xβ 3 3 ∂xβ ∇x c2 ≤ ∂xβ ( Ec) ≤ Cδ 2 ∂xβ ∇x c2 + ∂xβ 3 2 , 0≤|β |≤|β|
which, with δ > 0 small enough, implies
(2.31)
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207
∂xβ ∇x c2 ≤ C
∂xβ 3 2 .
(2.32)
0≤|β |≤|β|
Then, similar to obtaining (2.31), taking summation for (2.32) over 0 ≤ |β| ≤ − 1 gives ∂xβ c2 ≤ C ∂xβ 3 2 . (2.33) 1 (β)
2 (β)
For the pure space derivatives of a, one has from (2.7) that for any |β| ≥ 0, ∇x ∂xβ a2 =
& % & d % β ∂x a, ∇x · ∂xβ b − ∂xβ ∂t a, ∇x · ∂xβ b dt n # $ + ∂i ∂xβ a, ∂xβ (a E¯ i − 2cE 1 i) + ∂xβ i1 i=1
& 1 d % β 1 1 ∂x a, ∇x · ∂xβ b + ∂xβ ∂t a2 + ∇x ∂xβ b2 + ∇x ∂xβ a2 ≤ dt 2 2 2 1 2 β 2 β 2 β ∇x ∂x a + ∇x ∂x c + ∂x 1 2 . +Cδ (2.34) 2 0≤|β |≤|β|
Notice that (2.6) together with (2.30) gives that for any |β| ≥ 0, ∂xβ ∂t a2 ≤ ∂xβ (E 1 · b)2 + ∂xβ 0 2 ∂xβ 21 2 + ∂xβ 22 2 + ∂xβ 0 2 . ≤ Cδ 2
(2.35)
0≤|β |≤|β|
Putting (2.30), (2.32) and (2.35) into (2.34) and taking summation over 1 ≤ |β| ≤ − 1, one has d β ∂ a, ∇x · ∂xβ b + Cδ 2 ∇x a2 ∇x ∂xβ a2 ≤ C dt x 1≤|β|≤ −1 1≤|β|≤ −1 +C ∂xβ 2 . (2.36) 2 (β) β
Next we estimate ∂t ∂x u 1 with β ∈ 2 (β). It directly follows from (2.35) that ∂t ∂xβ a2 ≤ C ∂xβ 2 . (2.37) 2 (β)
2 (β)
In addition, (2.8) gives that for any |β| ≥ 0,
∂xβ ∂t c2 ≤ C ∇x ∂xβ b2 + ∂xβ ( E¯ · b)2 + ∂xβ 21 2 ≤C ∂xβ 21 2 + ∂xβ 22 2 , 0≤|β |≤|β|
which implies that
2 (β)
∂t ∂xβ c2 ≤ C
2 (β)
∂xβ 2 .
(2.38)
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R. Duan, S. Ukai, T. Yang, H. Zhao
Similarly (2.7) together with (2.33) and (2.36) gives ∂xβ a2 + ∂xβ c2 + C ∂t ∂xβ b2 ≤ C ∂xβ 2 2 (β)
2 (β)
1≤|β|≤
≤C
d β ∂xβ 2 . ∂x a, ∇x · ∂xβ b + C∇x a2 + C dt 2 (β)
1≤|β|≤ −1
Finally, collecting all estimates (2.31), (2.33), (2.36), (2.37), (2.38) and (2.39) yields (2.29). This completes the proof of the lemma. (iv) Combination of estimates on the macro-micro components. As in [13], from the representation (2.5) of , we can prove the following lemma. Lemma 2.8. It holds that ∂ β u 2 2 + C ∂ β ∂t u 2 2 . ∂xβ 2 ≤ C x x 2 (β)
0 (β)
(2.39)
2 (β)
Thus the further linear combination of (2.28), (2.29) and (2.39) gives the following result. Corollary 2.4. Under Assumptions (A1)–(A2), if δ > 0 is small enough, then there is an energy functional H2 (t) and a corresponding dissipation rate D2 (t) such that for any t ≥ 0, d H2 (t) + cD2 (t) ≤ C∇x u 1 2 , dt
(2.40)
and H2 (t) ≤ C D2 (t), where
H2 (t) ∼ u 2 2 +
+
1 (β)
3 (β,γ )
D2 (t) ∼ ν +
1/2
u2 +
3 (β,γ )
+
1 (β)
+
4 (β,γ )
γ ν 1/2 ∂xβ ∂ξ u 2 2
∂xβ u 1 2
+
γ
∂t ∂xβ ∂ξ u 2 2 ,
ν 1/2 ∂xβ u 2 2
1 (β)
∂t ∂xβ u2
2 (β)
γ ∂xβ ∂ξ u 2 2 2
∂xβ u2 +
+
+
ν 1/2 ∂t ∂xβ u 2 2
2 (β)
4 (β,γ )
γ
ν 1/2 ∂t ∂xβ ∂ξ u 2 2 ,
∂t ∂xβ u 1 2 .
2 (β)
(v) Further energy estimates on the microscopic part with velocity weight functions. For later use, we shall make further energy estimates on the microscopic component weighted by velocity functions ν(ξ ). We remark that it is necessary to introduce this velocity weight function to eliminate the time derivatives so that one can make use
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209
of the decay in time estimates for the linearized equation to deal with the nonlinear problems in terms of the contraction mapping theorem. For generality, we shall make the weighted energy estimates on w = w(t, x, ξ ), which is the solution to the following nonhomogeneous linear equation: ∂t w + νw + ξ · ∇x w + E 1 · ∇ξ w = φ + ξ · E 2 w,
(2.41)
where φ = φ(t, x, ξ ) is a given function. Lemma 2.9. Under Assumptions (A1)–(A2), if δ > 0 is small enough, then for any k, the solution w to Eq. (2.41) enjoys the following estimates: d k 2 ν w + cν k+1/2 w2 ≤ Cν k−1/2 φ2 , dt d k β 2 ν ∂x w + c ν k+1/2 ∂xβ w2 dt 1 (β) 1 (β) γ k−1/2 β ≤C ν ∂x φ2 + Cδ ν k−1/2 ∂xβ ∂ξ w2 , 1 (β)
(2.42)
(2.43)
3 (β,γ ) |β|≥1
and d dt
3 (β,γ )
≤C
γ
Cβ,γ ν k ∂xβ ∂ξ w2 + c
3 (β,γ )
γ ν k−1/2 ∂xβ ∂ξ φ2
γ
3 (β,γ )
+C
ν k+1/2 ∂xβ ∂ξ w2
ν k−1/2 ∂xβ w2 ,
(2.44)
0 (β)
where Cβ,γ with (β, γ ) ∈ 3 (β, γ ) are some positive constants, and positive constants c and C may depend on k. Furthermore, it holds that d dt
0≤|α|≤
≤C
α Cα ν k ∂x,ξ w2 + c
α ν k+1/2 ∂x,ξ w2
0≤|α|≤
α ν k−1/2 ∂x,ξ φ2 ,
(2.45)
0≤|α|≤
where Cα are also some positive constants. Proof. For simplicity of presentation, denote the time dependent linear operator A(t) by A(t) = ν + ξ · ∇x + E 1 (t, x) · ∇ξ . Then (2.41) is rewritten as ∂t w + A(t)w = φ + ξ · E 2 w.
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R. Duan, S. Ukai, T. Yang, H. Zhao
Since for each multi-index β and γ , one has γ
γ
∂t (ν k ∂xβ ∂ξ w) + A(t)(ν k ∂xβ ∂ξ w) γ
γ
γ −1
= ν k ∂xβ ∂ξ φ + ν k ξ · ∂xβ ∂ξ (E 2 w) + eγ · ν k ∂xβ ∂ξ γ −γ γ − ∂ξ νν k ∂xβ ∂ξ w − 0≤|γ |≤|γ |−1
γ −1
(E 2 w) − eγ · ν k ∇x ∂xβ ∂ξ β−β
Cβ ∂x
0≤|β |≤|β|−1
w
β
γ E 1 · ν k ∇ ξ ∂ x ∂ξ w
γ
+E 1 · ∇ξ ν k ∂xβ ∂ξ w, and (2.42)–(2.44) can be proved by mimicking the arguments used in the proof of Lemma 2.5. Finally (2.45) follows from the linear combination of (2.42)–(2.44). This completes the proof of the lemma. By applying the above result to the solutions of Eqs. (2.21) and (2.22), one has Corollary 2.5. Under Assumptions (A1)–(A2), if δ > 0 is small enough, then for any k, it holds that 2 d k 2 ν u 2 + c ν k+1/2 u 2 dt 2 (k−1/2)+ −1 2 (k−1/2)+ −1 2 (2.46) ≤ C ∇x u 1 + C ν u 2 + ν ∇x u 2 , d k β 2 ν ∂x u + c ν k+1/2 ∂xβ u2 dt 1 (β) 1 (β) + γ β 2 ≤C ∂x u 1 + C ν (k−1/2) −1 ∂xβ u 2 2 + Cδ ν k−1/2 ∂xβ ∂ξ u 2 2 , 1 (β)
1 (β)
3 (β,γ )
(2.47) and d dt
3 (β,γ )
≤C
γ
Cβ,γ ν k ∂xβ ∂ξ u 2 2 + c
1 (β)
∂xβ u 1 2
+C
3 (β,γ )
ν k−1/2 ∂xβ u 2 2 + C
0 (β)
γ
ν k+1/2 ∂xβ ∂ξ u 2 2 3 (β,γ )
ν (k−1/2)
+ −1
γ
∂xβ ∂ξ u 2 2 , (2.48)
where (·)+ means that (m)+ = m if m ≥ 0 and 0 otherwise. Furthermore, for any k, there is an energy functional H3,k (t) and a corresponding dissipation rate D3,k (t) such that for any t ≥ 0, d − H3,k (t) + cD3,k (t) ≤ C ∂xβ u 1 2 + C ν (k−1/2) −1 ∂xβ u 2 2 dt 1 (β) 0 (β) + γ +C ν (k−1/2) −1 ∂xβ ∂ξ u 2 2 , (2.49) 3 (β,γ )
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211
and H3,k (t) ≤ C D3,k (t),
(2.50)
where H3,k (t) ∼ ν k u 2 2 +
ν k ∂xβ u2 +
1 (β)
D3,k (t) ∼ ν k+1/2 u 2 2 +
3 (β,γ )
γ
ν k ∂xβ ∂ξ u 2 2 ,
ν k+1/2 ∂xβ u2 +
1 (β)
3 (β,γ )
(2.51) γ
ν k+1/2 ∂xβ ∂ξ u 2 2 . (2.52)
Proof. Notice that (2.14) and (2.15) can be rewritten as ∂t u + A(t)u = K u + ξ · E 2 u,
(2.53)
∂t u 2 + A(t)u 2 = K u 2 + [P, D(t)]u + ξ · E 2 u 2 .
(2.54)
and
Thus one can apply the estimate (2.43) to Eq. (2.53) with φ = K u to obtain (2.47), where (2.12) was used. Similarly by applying the estimates (2.42) and (2.44) to Eq. (2.54) with φ = K u 2 + [P, D(t)]u = K u 2 + PD(t)u − D(t)u 1 , one can obtain (2.46) and (2.48). Here we have used the following identities: β
β
β
∂xβ ∂ξ K u 2 = K ∂xβ ∂ξ u 2 − [K , ∂ξ ]∂xβ u 2 , and PD(t)u = PD(t)u 1 + PD(t)ν 1−(k−1/2)
+
+ ν (k−1/2) −1 u 2 .
Finally (2.49) follows from the linear combination of (2.46)–(2.48). It is obvious that (2.50) holds from the equivalent forms (2.51) and (2.52) of H3,k (t) and D3,k (t). This completes the proof of the corollary. So far, based on the energy estimates on the linearized Eq. (2.1) only, we can obtain a standard energy inequality only with the first order derivatives of the macroscopic component u 1 as an error term. In fact, by a proper linear combination of (2.40) and (2.49) with k = 1 yields Theorem 2.1. Under Assumptions (A1)–(A2), if δ > 0 is small enough, then there is an energy functional H (t) and a corresponding dissipation rate D(t) such that for any t ≥ 0, d H (t) + cD(t) ≤ C∇x u 1 2 , dt
(2.55)
H (t) ≤ C D(t),
(2.56)
and
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where
H (t) ∼ νu 2 2 +
+
1 (β)
3 (β,γ )
+
3 (β,γ )
+
+
γ
4 (β,γ )
∂t ∂xβ ∂ξ u 2 2 ,
ν 3/2 ∂xβ u 2 2 +
1 (β)
∂t ∂xβ u2
2 (β)
γ ν∂xβ ∂ξ u 2 2
D(t) ∼ ν 3/2 u 2 2 +
ν∂xβ u2 +
2 (β)
γ ν 3/2 ∂xβ ∂ξ u 2 2
∂xβ u 1 2
ν 1/2 ∂t ∂xβ u 2 2
+
1 (β)
+
γ
4 (β,γ )
ν 1/2 ∂t ∂xβ ∂ξ u 2 2
∂t ∂xβ u 1 2 .
2 (β)
It is noticed that in H (t), the power of the velocity weight function for the time derivatives is one less than that for others. Thus one can eliminate those terms involving the time derivatives by the equation. In fact, at first by u 2 = u − u 1 , it holds that
γ
4 (β,γ )
∂t ∂xβ ∂ξ u 2 2 ≤
γ
4 (β,γ )
∂t ∂xβ ∂ξ u2 +
4 (β,γ )
γ
∂t ∂xβ ∂ξ u 1 2 ,
where it further follows that γ ∂t ∂xβ ∂ξ u 1 2 ≤ ∂t ∂xβ u 1 2 ≤ ∂t ∂xβ u2 . 4 (β,γ )
2 (β)
2 (β)
Then by Eq. (2.1), one has ∂t u = −ξ · ∇x u − E 1 · ∇ξ u − νu 2 + K u 2 + ξ · E 2 u, which implies that 2 (β)
4 (β,γ )
∂t ∂xβ u2 ≤ Cνu 2 2 +
ν∂xβ u2 ,
1 (β)
γ ∂t ∂xβ ∂ξ u2
≤ Cνu 2 + 2
ν∂xβ u2 +
1 (β)
γ
3 (β,γ )
ν∂xβ ∂ξ u 2 2 .
Thus we have proved the following proposition. Proposition 2.2. Under the assumptions of Theorem 2.1, H (t) has the equivalent form: H (t) ∼ νu 2 2 + ∼
1≤|β|≤
ν∂xβ u2 +
1 (β)
∂xβ u 1 2
+
0≤|α|≤
3 (β,γ )
γ
ν∂xβ ∂ξ u 2 2
α ν∂x,ξ u 2 2 .
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2.4. Optimal decay rates. (i) Estimates based on the spectral analysis. Set B = −ξ · ∇x + L. Then from [27], one has Proposition 2.3. The linear operator B generates a semigroup eBt which enjoys the decay in time estimates (2.57) ∇xm eBt g ≤ C(1 + t)−σq,m g Z q + ∇xm g , for any integer m ≥ 0 and any function g = g(x, ξ ), where q ∈ [1, 2] and the decay rate is measured by 1 m n 1 σq,m = − + . (2.58) 2 q 2 2 Note that in terms of the linear operator B, (2.1) can be rewritten as ∂t u = Bu − E 1 · ∇ξ u + ξ · E 2 u. Then the solution to the initial value problem (2.1) and (2.2), with s = 0 for brevity, can be written in the mild form t u(t) = eBt u 0 + eB(t−s) −E 1 · ∇ξ u + ξ · E 2 u (s)ds. (2.59) 0
Based on the above mild form and Proposition 2.3, one has the following lemma. Lemma 2.10. Assume that there is a constant δ > 0 such that (1 + |x|)E i (t, x) L ∞ + |x|E i (t, x) t,x
2q/(2−q) L∞ Lx t
≤ δ,
where i = 1, 2 and 1 ≤ q ≤ 2. Then it holds that −σq,1 ∇x u(t) ≤ Cλ0 (1 + t) t
+Cδ 0
(1 + t − s)−σq,1 (∇x u 1 (s) + ν∇x u 2 (s) + ∇ξ ∇x u 2 (s))ds, (2.60)
where λ0 is given by λ0 = u 0 Z q + ∇x u 0 .
(2.61)
Proof. For simplicity, set G = −E 1 · ∇ξ u + ξ · E 2 u. Then applying (2.57) to (2.59) yields t ∇x u(t) ≤ Cλ0 (1 + t)−σq,1 + Cδ (1 + t − s)−σq,1 G(s) Z q + ∇x G(s) ds. 0
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Furthermore, one has ∇ξ u + Cν |x|E 2 2q/(2−q) u G(s) Z q ≤ |x|E 1 L 2q/(2−q) |x| 2 |x| 2 Lx x Lx L x L 2 ξ ≤ Cδ ∇ξ ∇x u(s) L 2 (L 2 ) + ν∇x u(s) L 2 (L 2 ) x ξ x ξ ≤ Cδ ∇x u 1 (s) + ν∇x u 2 (s) + ∇ξ ∇x u 2 (s) . Similarly it holds that
∇ξ u + E 1 L ∞ ∇x ∇ξ u L 2 ∇x G(s) ≤ |x|∇x E 1 L ∞ x x |x| x L 2x u + Cν E 2 L ∞ ∇x u L 2 +Cν |x|∇x E 2 L ∞ x x x |x| L 2x L 2ξ ≤ Cδ ∇x u 1 (s) + ν∇x u 2 (s) + ∇ξ ∇x u 2 (s) .
Thus (2.60) is proved. This completes the proof of the lemma.
(ii) Optimal decay rates. Combining Theorem 2.1 and Lemma 2.10 gives the optimal decay rates. Lemma 2.11. Assume 2n . (2.62) n+2 Under the assumptions of Theorem 2.1 and Lemma 2.10, if δ > 0 is small enough, then it holds that '
' (2.63) H (t) ≤ C(1 + t)−σq,1 H (0) + u 0 Z q , n ≥ 3, 1 ≤ q <
and u(t) ≤ C(1 + t)−σq,0 Proof. Define M(t) = sup
'
H (0) + u 0 Z q ∩L 2 .
(1 + s)2σq,1 H (s) .
(2.64)
(2.65)
0≤s≤t
Notice that M(t) is non-decreasing and
' ' ∇x u 1 (s) + ν∇x u 2 (s) + ∇ξ ∇x u 2 (s) ≤ C H (s) ≤ C(1 + s)−σq,1 M(t) (2.66)
for any 0 ≤ s ≤ t. Then (2.60) with (2.66) implies that for any t ≥ 0, ∇x u 1 (t) ≤ ∇x u(t)
' (1 + t − s)−σq,1 (1 + s)−σq,0 ds M(t) 0 ' −σq,1 λ0 + δ M(t) , (2.67) ≤ C(1 + t) ≤ Cλ0 (1 + t)
−σq,1
+ Cδ
t
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since σq,1 > 1 from (2.58) and (2.62). On the other hand, by the Gronwall inequality, (2.55) together with (2.56) gives t H (t) ≤ e−ct H (0) + C e−c(t−s) ∇x u 1 (s)2 ds, 0
for some constant c > 0. Then, further using (2.67) yields t H (t) ≤ e−ct H (0) + C e−c(t−s) (1 + s)−2σq,1 ds λ20 + δ 2 M(t) 0 −2σq,1 H (0) + λ20 + δ 2 M(t) . ≤ C(1 + t) Hence for any t ≥ 0, sup 0≤s≤t
i.e.,
(1 + s)2σq,1 H (s) ≤ C H (0) + λ20 + δ 2 M(t) , M(t) ≤ C H (0) + λ20 + δ 2 M(t) .
Then if δ > 0 is small enough, one has
M(t) ≤ C H (0) + λ20 .
(2.68)
Recalling the definitions (2.61) and (2.65) of λ0 and M(t), (2.68) gives (2.63). Finally it follows from (2.57) and (2.63) that t −σq,0 u(t) ≤ C(1 + t) u 0 Z q ∩L 2 + C (1 + t − s)−σq,0 G(s) Z q ∩L 2 ds 0 t ' −σq,0 ≤ C(1 + t) u 0 Z q ∩L 2 + Cδ (1 + t − s)−σq,1 H (s)ds 0
≤ C(1 + t)−σq,0 u 0 Z q ∩L 2 t ' +Cδ (1 + t − s)−σq,0 (1 + s)−σq,1 ds H (0) + u 0 Z q 0 ' H (0) + u 0 Z q ∩L 2 . ≤ C(1 + t)−σq,0 Thus (2.64) is proved. This completes the proof of the lemma.
(iii) Decay estimates on the solution operator U (t, s). For any number k, define a norm [[·]]0,k and a seminorm [[·]]1,k over the Sobolev space H (Rnx × Rnξ ) by α [[u]]0,k = ν k ∂x,ξ u, (2.69) 0≤|α|≤
[[u]]1,k =
1≤|β|≤
∂xβ Pu +
α ν k ∂x,ξ {I − P}u,
(2.70)
0≤|α|≤
where u = u(x, ξ ). Notice that [[u]]0,k ∼ [[u]]1,k + u.
(2.71)
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Theorem 2.2. Suppose that 2n (i) the integers n ≥ 3, ≥ 2 and the number 1 ≤ q < n+2 ; (ii) there a constant δ > 0 such that (1 + |x|)∂ β E i (t, x) ∞ + (1 + |x|)∂t ∂ β E i (t, x) ∞ ≤ δ, x x L L t,x
0≤|β|≤
t,x
0≤|β|≤ −1
and |x|E i (t, x)
2q/(2−q) L∞ Lx t
≤ δ,
where i = 1, 2. Then for any k ≥ 1, there exist constants δ0 > 0 and C0 > 0 such that for any δ ≤ δ0 , the linear solution operator U (t, s), −∞ < s ≤ t < ∞, corresponding to the linear Eq. (2.1) satisfies the decay in time estimates [[U (t, s)u 0 ]]m,k ≤ C0 (1 + t − s)−σq,m ([[u 0 ]]m,k + u 0 Z q ), m = 0, 1, (2.72) for any u 0 = u 0 (x, ξ ), where the constant C0 depends only on n, , q, k and δ0 . Proof. It suffices to consider the case when s = 0. We now prove (2.87) by induction for k ≥ 1. When k = 1, (2.72) follows from Proposition 2.2, Lemma 2.11 and (2.71). Now suppose that (2.72) holds for some k ≥ 1. We claim that it also holds for k + with any 0 ≤ ≤ 3/2. First consider the case of m = 0. Notice that u = U (t, 0)u 0 satisfies ∂t u + νu + ξ · ∇x u + E 1 · ∇ξ u = K u + ξ · E 2 u. Then recalling Eq. (2.41) and then applying the estimate (2.45) with φ = K u, one has d dt
α Cα ν k+ ∂x,ξ u2 + c
0≤|α|≤
≤C
α ν k++1/2 ∂x,ξ u2
0≤|α|≤
α ν k+−1/2 ∂x,ξ K u2 ,
(2.73)
0≤|α|≤
where by Lemma 2.9 and the inductive assumption, it holds that 2 α ν k+−1/2 ∂x,ξ K u2 ≤ C[[u]]20,k ≤ C(1 + t)−2σq,0 [[u 0 ]]0,k + u 0 Z q . 0≤|α|≤
(2.74) Thus by the Gronwall inequality, (2.73) and (2.74) imply (2.72) with m = 0 for k + . Next consider the case of m = 1. Notice that the following equivalent property also holds: k β k k β γ [[u]]1,k ∼ ν ∂x u + ν {I − P}u + ν ∂x ∂ξ {I − P}u . 1 (β)
3 (β,γ )
Thus from Corollary 2.5, similarly (2.72) with m = 1 holds for k + . The details of the proof are omitted for brevity. Hence (2.72) with m = 0 or 1 holds for any k ≥ 1. This completes the proof of the theorem.
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Remark 2.1. In the above theorem, the external force needs not to have time decay. Rather, it may be time independent, time periodic, or even bounded in time, though it should be small. In the case when the force is a small perturbation of some stationary potential force, i.e. in the form F(t, x) = −∇x φ(x) + E(t, x), where φ(x) → 0 as |x| → ∞, we can have the same optimal decay estimates as (2.72) for the linearized equation derived by setting + M1/2 u, f =M where = ρ(x)M, M ˜ ρ(x) ˜ = e−φ(x) . In this case, the linear equation is 1 ∂t u + ξ · ∇x u + F · ∇ξ u − ξ · Fu = ρ(x)Lu. ˜ 2
(2.75)
If the same assumptions of Theorem 2.2 hold for F(t, x) and φ(x) itself is also small in some Sobolev space, then the energy estimate similar to (2.13) still holds. For the estimates on the macroscopic component u 1 , we consider Eq. (2.75) which can be rewritten as 1 ∂t u − Bu = −F · ∇ξ u + ξ · Fu + (ρ˜ − 1)Lu, 2 where the right-hand side can be regarded as a source term. Thus the decay estimate (2.72) is valid for the solution operator corresponding to (2.75) and can be used for the nonlinear problem considered in Sect. 3. 3. Applications to the Nonlinear Equation 3.1. Basic estimates. First from the definition (2.69) of the norm [[·]]0,k , Corollary 2.2 β β β β β β and ∂x ∂ξ K u = K ∂x ∂ξ u − [K , ∂ξ ]∂x u, we have Lemma 3.1. Let k be any number. For any u = u(x, ξ ), it holds that [[K u]]0,k ≤ C[[u]]0,(k−1)+ , where C is some constant. Lemma 3.2. For any u = u(x, ξ ) and v = v(x, ξ ), it holds that (u, v) Z 1 ≤ C (νuv + uνv) , where C is some constant. The proof of the above lemma can be found in [28]. Finally we give a lemma on the estimates on the nonlinear term in the norm [[·]]0,k .
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Lemma 3.3. Let k ≥ 0 and k0 ≤ 1. Suppose that ≥ [n/2]+2. Then for any u = u(x, ξ ) and v = v(x, ξ ), it holds that [[(u, v)]]0,k−k0 ≤ C([[u]]0,k+1−k0 [[v]]0,k + [[u]]0,k [[v]]0,k+1−k0 ),
(3.1)
where C is some constant. Proof. Write (u, v) =
1 {1 (u, v) + 1 (v, u) − 2 (u, v) − 2 (v, u)} , 2
with
1 (u, v) =
2 (u, v) =
|(ξ − ξ∗ ) · ω|M∗ u(ξ )v(ξ∗ )dξ∗ dω, 1/2
Rn ×S n−1
1/2
Rn ×S n−1
|(ξ − ξ∗ ) · ω|M∗ u(ξ )v(ξ∗ )dξ∗ dω.
It is obvious that (3.1) holds if it does for each j , j = 1, 2. First consider 1 . As in [14], after taking a change of variable z = ξ − ξ∗ , 1 can be rewritten as 1 (u, v)(ξ ) = |z · ω|M1/2 (ξ − z)u(ξ )v(z )dzdω, (3.2) Rn ×S n−1
where ξ = ξ − z, z = ξ − z⊥, β γ
α = ∂ ∂ with 0 ≤ |α| ≤ and α = β + γ with z = (z · ω)ω, z ⊥ = z − z . Applying ∂x,ξ x ξ to (3.2) yields β γ α 1 (u, v)(ξ ) = Cβ1 ∂ξ |z · ω|M1/2 (ξ − z)(∂xβ1 u)(ξ )(∂xβ2 v)(z )dzdω ∂x,ξ
Rn ×S n−1
β1 +β2 =β
=
β1 +β2 =β γ1 +γ21 +γ22 =γ
×
γ
Rn ×S n−1
Notice that for any γ1 ,
Then α |∂x,ξ 1 (u, v)(ξ )| ≤ C
β
Cβ1 Cγγ1 Cγγ21−γ1 γ
γ
|z · ω|∂ξ 1 M1/2 (ξ − z)(∂xβ1 ∂ξ 21 u)(ξ )(∂xβ2 ∂ξ 22 v)(z )dzdω.
γ1 1/2 ∂ξ M (ξ − z) ≤ CM1/4 (ξ − z). n n−1 α1 +α2≤α R ×S
α1 α2 |z · ω|M1/4 (ξ −z)|∂x,ξ u(ξ )| |∂x,ξ v(z )|dzdω.
(3.3)
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219
Without loss of generality, suppose |α1 | ≤ |α|/2 in (3.3). Then by integrating (3.3) over Rnx with respect to the space variable and using the Sobolev inequality, one has α ∂x,ξ 1 (u, v)(ξ ) L 2x ≤ C
α1 (ξ ),
(3.4)
|α1 |≤|α|/2
where α1 (ξ ) =
Rn ×S n−1
α1 α2 |z · ω|M1/4 (ξ − z)∇x ∂x,ξ u(ξ ) Hx1 ∂x,ξ v(z ) L 2x dzdω.
Noting that for any k ≥ 0, ν k (ξ )ν k (z ) = ν k (ξ − z )ν k (ξ − z ⊥ ) ≥ Cν k (ξ ),
(3.5)
where the constant C > 0, then for each α1 , one has ν k α1 (ξ ) ≤ C
Rn ×S n−1
α1 α2 |z · ω|M1/4 (ξ − z)ν k ∇x ∂x,ξ u(ξ ) Hx1 ν k ∂x,ξ v(z ) L 2x dzdω
≤C
Rn ×S n−1
×
1/2 |z| M 2
Rn ×S n−1
(ξ − z)dzdω
1/2
2 α1 α2 ν k ∇x ∂x,ξ u(ξ ) Hx1 ν k ∂x,ξ v(z ) L 2x dzdω
≤ Cν(ξ )
1/2
Rn ×S n−1
ν
k
1/2 2 α1 k α2 ∇x ∂x,ξ u(ξ ) Hx1 ν ∂x,ξ v(z ) L 2x dzdω .
Taking further integration over Rnξ with respect to the velocity variable gives ν k−k0 α1 2L 2 ≤ C ξ
Rn ×S n−1
≤C
α1 α2 ν 2−2k0 (ξ )ν k ∇x ∂x,ξ u(ξ )2H 1 ν k ∂x,ξ v(z )2L 2 dξ dzdω
ν 2−2k0 (ξ ) + ν 2−2k0 (z )
x
x
Rn ×S n−1 α1 α2 u(ξ )2H 1 ν k ∂x,ξ v(z )2L 2 dξ dz dω, ×ν k ∇x ∂x,ξ x
x
where we have used the inequality (3.5) since 2 − 2k0 ≥ 0 and taken change of variables (ξ, z) → (ξ , z ), whose Jacobian is unity. Hence 2 k−k0 α1 2 ≤ C [[u]]20,k+1−k0 [[v]]20,k + [[u]]20,k [[v]]20,k+1−k0 . ν Lξ
(3.6)
Thus combining (3.4) and (3.6) implies that (3.1) holds for 1 . Finally it is more straightforward to carry out the estimates on 2 (u, v) in a similar way. The details are omitted. This completes the proof of the lemma.
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3.2. Global existence for the Cauchy problem. In this subsection, we consider the global existence and decay rates of the solution to the Cauchy problem for the nonlinear Boltzmann equation: 1 ∂t u + ξ · ∇x u + F · ∇ξ u − ξ · Fu = Lu + (u) + S, 2 u(t, x, ξ )|t=0 = u 0 (x, ξ ),
(3.7) (3.8)
S is given by (1.5). where u = u(t, x, ξ ), (t, x, ξ ) ∈ R+ × Rn × Rn , and The main result is stated as follows. Theorem 3.1. Suppose that (B1) The integers n ≥ 3, ≥ [n/2] + 2. (B2) The functions F = F(t, x), S = S(t, x, ξ ) and u 0 = u 0 (x, ξ ) satisfy F ∈ Cbi R+t ; H −i (Rnx ) , i = 0, 1, S ∈ Cb0 R+t ; H (Rnx × Rnξ ) , u 0 ∈ H (Rnx × Rnξ ). (B3) There are constants δ > 0, k ≥ 1 and κ > 1 such that F and u 0 are bounded in the sense that (1 + |x|)∂ β F(t, x) ∞ x L t,x
0≤|β|≤
+
(1 + |x|)∂t ∂ β F(t, x) x
L∞ t,x
+ |x|F(t, x) L ∞ 2 ≤ δ, (3.9) t (L x )
0≤|β|≤ −1
[[u 0 ]]0,k+1/2 + u 0 Z 1 ≤ δ,
(3.10)
and moreover, F and S decay in time in the sense that F(t) Hx ∩L 1x ≤ δ(1 + t)−κ , [[M−1/2 S(t)]]0,k−1/2 + M−1/2 S(t)
(3.11) Z1
≤ δ(1 + t)−κ .
(3.12)
Then there are constants δ1 > 0 and C1 > 0 such that for any δ ≤ δ1 , the Cauchy problem (3.7)–(3.8) corresponding to (1.1) has a unique global classical solution u ∈ Cbi R+t ; H −i (Rnx × Rnξ ) , i = 0, 1, (3.13) which satisfies sup(1 + t)2κ0 [[u(t)]]20,k + t≥0
0
∞
[[u(s)]]20,k+1/2 ds ≤ C12 ,
(3.14)
where C1 can be also taken as C1 = C1 δ for another constant C1 independent of δ, and κ0 is given by 1 1 1 2 < κ0 < κ − 2 if σ1,0 ≥ κ − 2 , (3.15) if σ1,0 < κ − 21 . κ0 = σ1,0 Furthermore, it holds that
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221
k−1 α ν ∂t ∂x,ξ u(t) ≤ Cδ(1 + t)−κ0 ,
(3.16)
0≤|α|≤ −1
for some constant C. In order to prove the above theorem, we introduce a function set S(C1 ) by
S(C1 ) = u = u(t, x, ξ ) u ∈ Cb0 R+t ; H (Rnx × Rnξ ) , |||u|||k,κ0 ≤ C1 , where C1 > 0 is some constant to be determined later, and the norm ||| · |||k,κ0 is defined by ∞ |||u|||2k,κ0 = sup(1 + t)2κ0 [[u(t)]]20,k + [[u(s)]]20,k+1/2 ds. t≥0
0
Clearly, S(C1 ) is a complete metric space with the metric induced by the norm ||| · |||k,κ0 . Under some conditions, the solution to (3.7)–(3.8) will be obtained by applying the contraction mapping theorem to find a fixed point in S(C1 ) for some nonlinear mapping , where is defined by t (u) = U (t, 0)u 0 + U (t, s){(u(s), u(s)) + S(s)}ds. (3.17) 0
Thus one has to estimate the time integral in (3.17) in terms of the norm ||| · |||k,κ0 . For this, in what follows, given a function φ = φ(t, x, ξ ), we will first consider the estimate on the general time integral t (Tφ)(t, x, ξ ) = U (t, s)φ(s, x, ξ )ds. 0
This time integral can be written as two parts again by Duhamel’s formula. In fact, define the solution operator U1 (t, s) for any 0 ≤ s ≤ t in the sense that for any v0 = v0 (x, ξ ), v = v(t, x, ξ ) = U1 (t, s)v0 denotes the solution to the following initial value problem: 1 ∂t v + νv + ξ · ∇x v + F · ∇ξ v − ξ · Fv = 0, 2 v(t, x, ξ )|t=s = v0 (x, ξ ). Note that L = −ν + K . Then again by Duhamel’s formula, the solution operator U (t, s) can be rewritten as U (t, s) = U1 (t, s) + U2 (t, s), 0 ≤ s ≤ t, where
t
U2 (t, s) =
U (t, τ )K U1 (τ, s)dτ.
s
Thus we further define
t
(T j φ)(t, x, ξ ) =
U j (t, s)φ(s, x, ξ )ds, j = 1, 2.
0
Then Tφ = T1 φ + T2 φ. The following estimates follow.
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Lemma 3.4. Suppose (3.9). If δ > 0 is small enough, then one has t 2m 2 (1 + s)2m [[T1 φ(s)]]20,k+1/2 ds (1 + t) [[T1 φ(t)]]0,k + 0 t ≤C (1 + s)2m [[φ(s)]]20,k−1/2 ds,
(3.18)
0
for any m ≥ 0 and any k, and t (1 + s)2m T1 φ(s)2Z 1 ds (1 + t)2m T1 φ(t)2Z 1 + 0 t ≤C (1 + s)2m [[φ(s)]]20,k−1/2 + φ(s)2Z 1 ds,
(3.19)
0
for any m ≥ 0 and any k ≥ 1/2. Proof. For simplicity, write w = T1 φ, which by the definitions of T1 and U1 (t, s), satisfies the following Cauchy problem with zero initial data: 1 ∂t w + νw + ξ · ∇x w + F · ∇ξ w − ξ · Fw = φ, 2 w(t, x, ξ )|t=0 = 0.
(3.20) (3.21)
By (2.45), one has the energy inequality d J0,k [w(t)] + c J0,k+1/2 [w(t)] ≤ C[[φ(t)]]20,k−1/2 , dt
(3.22)
for any k, where to the end, the nonlinear functional J0,k [·] is defined by J0,k [w(t)] ∼ [[w(t)]]0,k . After integration, (3.22) implies t t J0,k [w(t)] + J0,k+1/2 [w(s)]ds ≤ C [[φ(s)]]20,k−1/2 ds. 0
(3.23)
(3.24)
0
On the other hand, multiplying (3.22) by (1 + t)2m with m ≥ 0 and further integrating it gives t (1 + t)2m J0,k [w(t)] + c (1 + s)2m J0,k+1/2 [w(s)]ds 0 t t ≤ 2m (1 + s)m−1 J0,k [w(s)]ds + C (1 + s)2m [[φ(s)]]20,k−1/2 ds 0 0 t c t m ≤ (1 + s) J0,k+1/2 [w(s)]ds + C J0,k+1/2 [w(s)]ds 2 0 0 t +C (1 + s)2m [[φ(s)]]20,k−1/2 ds. (3.25) 0
Then (3.25) together with (3.23) and (3.24) yields (3.18).
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Next consider the estimate (3.19) in the norm · Z 1 . It can be based on the explicit form for the solution w from (3.20)–(3.21):
t
w(t, x, ξ ) = 0
e−ν(ξ )(t−s) F · ∇ξ w − ξ/2 · Fw + φ (s, x − (t − s)ξ, ξ )ds,
which implies w(t, ξ ) L 1 (Rnx ) ≤ C
t 0
e−ν0 (t−s) ∇ξ ∇x w(s, ξ ) L 2 (Rnx )
+ν∇x w(s, ξ ) L 2 (Rnx ) + φ(s, ξ ) L 1 (Rnx ) ds.
Further taking the norm · L 2 (Rn ) gives ξ
w(t) Z 1 ≤ C
t
e−ν0 (t−s) G(s)ds,
(3.26)
0
where for simplicity, we used the notion G(s) = ∇ξ ∇x w(s) + ν∇x w(s) + φ(s) Z 1 .
(3.27)
From (3.26), we claim that for any m ≥ 0, (1 + t)
2m
w(t)2Z 1
t
+
(1 + s)
2m
0
w(s)2Z 1 ds
≤C
t
(1 + s)2m G(s)2 ds.
(3.28)
0
In fact, on one hand, by the Hölder inequality, it is easy to see from (3.26) that
t e−2ν0 (t−s) (1 + s)−2m ds (1 + s)2m G(s)2 ds 0 0 t −2m 2m 2 ≤ C(1 + t) (1 + s) G(s) ds.
w(t)2Z 1 ≤ C
t
(3.29)
0
On the other hand, again by (3.26), one has
t 0
(1 + s)2m w(s)2Z 1 ds ≤
t 0
(
s
(1 + s)2m
e−ν0 (s−τ ) G(τ )dτ
)2
0
By the Schwarz inequality, it holds that (
s
−ν0 (s−τ )
e s
)2 G(τ )dτ
s e−ν0 (s−τ ) (1 + τ )−2m dτ e−ν0 (s−τ ) (1 + τ )2m G(τ )2 dτ 0 0 s ≤ C(1 + s)−2m e−ν0 (s−τ ) (1 + τ )2m G(τ )2 dτ, 0
≤
0
ds.
(3.30)
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which together with (3.30) gives
t 0
t
s
e−ν0 (s−τ ) (1 + τ )2m G(τ )2 dτ ds 0 0 t t =C dτ (1 + τ )2m G(τ )2 e−ν0 (s−τ ) ds 0 τ t 2m 2 (1 + τ ) G(τ ) dτ. ≤C
(1 + s)2m w(s)2Z 1 ds ≤ C
(3.31)
0
Thus (3.28) follows from (3.29) and (3.31). Furthermore, notice from (3.27) and k ≥ 1/2 that G(s)2 ≤ C ∇ξ ∇x w(s)2 + ν∇x w(s)2 + φ(s)2Z 1 ≤ C [[w(t)]]20,k+1/2 + φ(s)2Z 1 , which by (3.18), implies
t
(1 + s)2m G(s)2 ds ≤ C
0
t
0
≤C
0
t
(1 + s)2m [[w(t)]]20,k+1/2 + φ(s)2Z 1 ds (1 + s)2m [[φ(t)]]20,k−1/2 + φ(s)2Z 1 ds.
(3.32)
With the notion w = T1 φ, combining (3.28) and (3.32) leads to (3.19). This completes the proof of the lemma. Lemma 3.5. Suppose (3.9). If δ > 0 is small enough, then one has t (1 + t)2m [[T2 φ(t)]]20,k + [[T2 φ(s)]]20,k+1/2 ds 0 t ≤C (1 + s)2m [[φ(s)]]20,k−1/2 + φ(s)2Z 1 ds,
(3.33)
0
for any 1/2 < m ≤ σ1,0 and any k ≥ 1. Proof. First fix some m and k with 1/2 < m ≤ σ1,0 and k ≥ 1. Set z = T2 φ for simplicity. By the definitions of Ti and Ui (t, s), i = 1, 2, note that
t
z(t) = T2 φ(t) = 0
t
U2 (t, s)φ(s)ds = 0
U (t, s)K T1 φ(s)ds.
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225
Then by Theorem 2.2 and Lemma 3.4, it holds that t 2 2 −σ1,0 [[z(t)]]0,k ≤ C (1 + t − s) [[K T1 φ(s)]]0,k + K T1 φ(s) Z 1 ds 0
t 2 ≤ C (1 + t − s)−σ1,0 ([[T1 φ(s)]]0,k−1 + T1 φ(s) Z 1 )ds 0 t ≤C (1 + t − s)−2σ1,0 (1 + s)−2m ds 0 t 2 × (1 + s)2m [[T1 φ(s)]]20,k+1/2 + T1 φ(s) Z 1 ds 0 t ≤ C(1 + t)−2m (1 + s)2m [[φ(s)]]20,k−1/2 + φ(s)2Z 1 ds.
(3.34)
0
On the other hand, z = z(t, x, ξ ) is the solution to the following initial value problem with zero initial data: 1 ∂t z + νz + ξ · ∇x z + F · ∇ξ z − ξ · F z = K z + K T1 φ, 2 z(t, x, ξ )|t=0 = 0. This means that z = T1 (K z + K T1 φ). Use (3.18) with m = 0 to deduce t t 2 [[z(s)]]0,k+1/2 ds ≤ C [[K z + K T1 φ]]20,k−1/2 ds 0 0 t t 2 ≤C [[z(s)]]0,k−3/2 ds + C [[T1 φ(s)]]20,k−3/2 ds, 0
0
where further, it holds from (3.34) that t t 2 [[z(s)]]0,k−3/2 ds ≤ [[z(s)]]20,k ds 0 0 t s −2m ≤C (1 + s) ds sup (1 + τ )2m [[φ(τ )]]20,k−1/2 + φ(τ )2Z 1 dτ 0≤s≤t
0
≤C
0
t
0
(1 + τ )2m [[φ(τ )]]20,k−1/2 + φ(τ )2Z 1 dτ,
and again from (3.18) with m = 0 that t t t [[T1 φ(s)]]20,k−3/2 ds ≤ [[T1 φ(s)]]20,k+1/2 ds ≤ C [[φ(s)]]20,k−1/2 ds. 0
Then,
0
t 0
0
[[z(s)]]20,k+1/2 ds ≤ C
0
t
(1 + s)2m [[φ(s)]]20,k−1/2 + φ(s)2Z 1 ds. (3.35)
Thus (3.33) follows from (3.34) and (3.35). This completes the proof of the lemma.
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Corollary 3.1. Suppose (3.9). If δ > 0 is small enough, then one has t [[Tφ(s)]]20,k+1/2 ds (1 + t)2m [[Tφ(t)]]20,k + 0 t 2m ≤C [[φ(s)]]20,k−1/2 + φ(s)2Z 1 ds, (1 + s) 0
for any 1/2 < m ≤ σ1,0 and any k ≥ 1. Now we are in a position to prove the global existence of the solution to the Cauchy problem for the nonlinear Boltzmann equation. Proof of Theorem 3.1. First we prove that there is a proper constant C1 > 0 such that is a contraction mapping from S(C1 ) to itself, and thus it has a fixed point in S(C1 ) which is a unique solution to the Cauchy problem (3.7)–(3.8). For this purpose, we start with a claim that there is a constant C such that for any u, v ∈ S(C1 ), |||(u)|||k,κ0 ≤ Cδ + C|||u|||2k,κ0 , |||(u) − (v)|||k,κ0 ≤ C|||u + v|||k,κ0 |||u − v|||k,κ0 .
(3.36) (3.37)
In fact, recall the definition (3.17) of , and then it is straightforward to compute |||U (t, 0)u 0 |||2k,κ0
≤ sup(1 + t)
2κ0
t≥0
≤ C sup(1 + t)
[[U (t, 0)u 0 ]]20,k
2κ0 −2σ1,0
t≥0
[[u 0 ]]20,k
∞
+ 0
[[U (s, 0)u 0 ]]20,k+1/2 ds
+C 0
∞
(1 + s)−2σ1,0 ds[[u 0 ]]20,k+1/2
≤ C[[u 0 ]]20,k+1/2 ≤ Cδ 2 ,
(3.38)
where we used (3.10), and the inequalities κ0 ≤ σ1,0 and 2σ1,0 > 1 since n ≥ 3. Furthermore, noticing from (3.15) and n ≥ 3 that 1/2 < κ0 ≤ σ1,0 , one can apply Corollary 3.1 with m = κ0 to obtain t 2 U (t, s)(u(s), u(s))ds 0 k ∞ 2κ0 [[(u(s), u(s))]]20,k−1/2 + (u(s), u(s))2Z 1 ds ≤C (1 + s) 0 ∞ ≤C (1 + s)2κ0 [[u(s)]]20,k+1/2 [[u(s)]]20,k ds 0 ∞ [[u(s)]]20,k+1/2 ds sup(1 + s)2κ0 [[u(s)]]20,k ≤C s≥0
0
≤
C|||u|||2k,κ0 ,
where Lemma 3.3 was used. Since (3.11) and (3.12) together with (1.5) imply S(s) Z 1 ≤ Cδ(1 + s)−κ , [[ S(s)]]0,k−1/2 +
(3.39)
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227
similarly applying Corollary 3.1 with m = κ0 yields t 2 U (t, s) S(s)ds ≤ C 0
k
≤ Cδ
∞ 0
2
S(s)]]20,k−1/2 + (1 + s)2κ0 [[ S(s)2Z 1 ds ∞
(1 + s)2κ0 −2κ ds
0
≤ Cδ 2 ,
(3.40)
where by (3.15), κ0 < κ − 1/2 was used. Thus by (3.17), combining (3.38), (3.39) and (3.40) proves (3.36). For (3.37), notice that since is bilinear, (u, u) − (v, v) = (u + v, u − v). Then it holds that (u) − (v) =
t
U (t, s)(u + v, u − v)(s)ds,
0
which similar to the proof of (3.39), implies (3.37). Now suppose u, v ∈ S(C1 ). Then based on (3.36) and (3.37), it is easy to see that (u), (v) ∈ Cb0 R+t ; H (Rnx ) , with estimates |||(u)|||k,κ0 ≤ Cδ + CC12 , |||(u) − (v)|||k,κ0 ≤ 2CC1 |||u − v|||k,κ0 . If δ ≤ δ1 with δ1 > 0 small enough, then there is a constant C1 > 0 depending only on δ1 and C such that Cδ + CC12 ≤ C1 , 2CC1 < 1. Thus (u), (v) ∈ S(C1 ) and |||(u) − (v)|||k,κ0 ≤ µ|||u − v|||k,κ0 , µ = 2CC1 < 1. Therefore is a contraction mapping over S(C1 ). Thus there is a unique fixed point u in S(C1 ) as a mild solution to the Cauchy problem (3.7)–(3.8). Then (3.13) with i = 0 and (3.14) are proved. In addition, it is obvious that C1 can be also taken as C1 = C1 δ for another constant C1 independent of δ. Finally the time-differentiability (3.13) with i = 1 of the solution u and the estimate (3.16) directly follow from the equation. This completes the proof of the theorem.
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3.3. Existence of time periodic solution. In this subsection, we are concerned with the existence of the time periodic solution to the nonlinear Boltzmann equation 1 ∂t u + ξ · ∇x u + F · ∇ξ u − ξ · Fu = Lu + (u) + S, 2
(3.41)
where u = u(t, x, ξ ), (t, x, ξ ) ∈ R × Rn × Rn , and S is given by (1.5). Roughly speaking, our goal is to show that if the time dependent external force F and source S are time periodic with period T , then Eq. (3.41) should have a time periodic solution with the same period under some additional assumptions. When the space dimension n ≥ 5, this can be achieved by making use of the decay in time property of the linearized equation which is established in Sect. 2. Precisely, the main result is stated as follows. Theorem 3.2. Suppose that (C1) the integers n ≥ 5, ≥ [n/2] + 2; (C2) the functions F = F(t, x) and S = S(t, x, ξ ) are time periodic with period T , satisfying F ∈ Cbi Rt ; H −i (Rnx ) , i = 0, 1, S ∈ Cb0 Rt ; H (Rnx × Rnξ ) ; (C3) there are constants δ > 0 and k ≥ 1 such that F and S are bounded in the sense that (1 + |x|)∂ β F(t, x) ∞ x L t,x
0≤|β|≤
+
(1 + |x|)∂t ∂ β F(t, x) x
0≤|β|≤ −1
sup F(t) Hx ∩L 1x + [[M t∈R
−1/2
L∞ t,x
+ |x|F(t, x) L ∞ 2 ≤ δ, (3.42) t (L x )
−1/2 ≤ δ. (3.43) S(t)]]0,k−1/2 + M S(t) Z1
Then there are constants δ2 > 0 and C2 > 0 such that for any δ ≤ δ2 , Eq. (3.41) corresponding to (1.1) has a unique time periodic solution u ∗ ∈ Cbi Rt ; H −i (Rnx × Rnξ ) , i = 0, 1, with the same period T , which satisfies T ∗ 2 sup [[u (t)]]0,k + [[u ∗ (t)]]20,k+1 dt ≤ C22 , 0≤t≤T
(3.44)
0
where precisely, C2 can be chosen as C2 = C2 δ with C2 independent of δ. Furthermore, it holds that k−1 α ∗ sup [[u ∗ (t)]]0,k+1/2 + sup ν ∂t ∂x,ξ u (t) ≤ Cδ, (3.45) 0≤t≤T
for some constant C.
0≤t≤T 0≤|α|≤ −1
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229
In order to prove Theorem 3.2, we shall use the arguments developed in [26] to deal with the existence of the periodic solution. Define (u) =
t
−∞
U (t, s){(u(s), u(s)) + S(s)}ds.
Suppose that has a unique fixed point u(t). ¯ Then if S(t) is time periodic with period T , so is u(t) ¯ as in [26]. Furthermore, u(t) ¯ is a desired time periodic solution provided that it is differentiable with respect to time t. Thus it suffices to find the fixed point of in a proper complete metric space. We choose it as S(C2 ) defined by u is time periodic with period T , S(C2 ) = u = u(t, x, ξ ) , u ∈ Cb0 Rt ; H (Rnx × Rnξ ) , |||u|||k,∗ ≤ C2 where C2 > 0 is some constant to be determined later, and |||u|||2k,∗ = sup [[u(t)]]20,k + 0≤t≤T
T 0
[[u(s)]]20,k+1 ds.
As before, we first consider some general estimates on a linear operator T∗ given by T∗ φ(t) =
t
−∞
U (t, s)φ(s)ds,
for any φ = φ(t, x, ξ ). Lemma 3.6. Suppose that φ is time periodic with period T and
T
φ0 = 0
[[φ(t)]]20,k + φ(s)2Z 1 dt < ∞.
Under the assumptions of Theorem 3.2, if δ > 0 is small enough, then T∗ φ is welldefined, time periodic with the same period T , and the following estimate holds sup [[T∗ φ(t)]]20,k+1/2 +
0≤t≤T
T 0
[[T∗ φ(t)]]20,k+1 dt ≤ Cφ0 .
(3.46)
Proof. For simplicity, set w = T∗ φ. By Theorem 2.2, it holds that [[w(t)]]0,k ≤ C
t −∞
(1 + t − s)−σ1,0 G(s)ds = C
∞
I j (t),
(3.47)
j=0
where G(s) = [[φ(s)]]0,k + φ(s) Z 1 , t− j T (1 + t − s)−σ1,0 G(s)ds. I j (t) = t−( j+1)T
(3.48) (3.49)
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Since φ is time periodic with period T and so is G(s), one has from (3.49) that t− j T t− j T 2 −2σ1,0 I j (t) ≤ (1 + t − s) ds G 2 (s)ds =
t−( j+1)T T
t−( j+1)T T
(1 + ( j + 1)T − r )−2σ1,0 dr
0
G 2 (r )dr
0
≤ C(1 + j T )−2σ1,0 G2L 2 (0,T ) , which implies ∞
I j (t) ≤ C
j=0
∞ (1 + j T )−σ1,0 G L 2 (0,T ) ≤ CG L 2 (0,T ) ,
(3.50)
j=0
where σ1,0 = n/4 > 1 was used because n ≥ 5. Then it follows from (3.47), (3.48) and (3.50) that T [[φ(t)]]20,k + φ(s)2Z 1 dt ≤ Cφ0 . (3.51) [[w(t)]]20,k ≤ CG2L 2 (0,T ) ≤ C 0
Next, the periodicity of w directly follows from t+T w(t + T ) = U (t + T, s)φ(s)ds −∞ t U (t + T, s + T )φ(s + T )ds = −∞ t U (t, s)φ(s)ds, = −∞
where we have used that for any −∞ < s ≤ t < ∞, φ(s + T ) = φ(s), U (t + T, s + T ) = U (t, s). Finally consider the estimate (3.46). Notice that w satisfies the initial value problem 1 ∂t + νw + ξ · ∇x w + F · ∇ξ w − ξ · Fw = K w + φ, 2 w(t, x, ξ )|t=0 = 0. Recalling Eq. (2.41) and the corresponding estimate (2.45), one has T T [[w(t)]]20,k+1/2 + c [[w(t)]]20,k+1 dt ≤ C [[K w(t) + φ(t)]]20,k dt 0
T
≤C 0
0
[[K w(t)]]20,k dt + Cφ0 ,
where further by Lemma 3.1 and (3.51), it holds that T T 2 [[K w(t)]]0,k dt ≤ [[w(t)]]20,k−1 dt ≤ C T sup [[w(t)]]20,k−1 ≤ Cφ0 . 0
0
0≤t≤T
Thus (3.46) holds. This completes the proof of the lemma.
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231
Proof of Theorem 3.2. Similar to the proof of Theorem 3.1, we first prove that there is a constant C such that for any u, v ∈ S(C2 ) with some constant C2 to be determined later, |||(u)|||k,∗ ≤ Cδ + C|||u|||2k,∗ , |||(u) − (v)|||k,∗ ≤ C|||u + v|||k,∗ |||u − v|||k,∗ .
(3.52) (3.53)
Notice that (3.43) implies [[ S(t)]]0,k + S(t) Z 1 ≤ δ, for any t ∈ R. Thus based on Lemma 3.6, (3.52) and (3.53) are proved similarly as before and the details are omitted for brevity. Hence the contraction mapping theorem can be applied over the complete metric space S(C2 ) for a proper constant C2 > 0, provided that δ ≤ δ2 with δ2 > 0 small enough. Then there is a unique fixed point u ∗ in S(C2 ) for the nonlinear mapping . Notice that it is obvious that C2 can be also chosen as C2 δ for some constant C2 independent of δ. Finally by u ∗ = (u ∗ ), it follows from (3.46) and (3.52) that sup [[u ∗ (t)]]0,k+1/2 ≤ Cδ + C(C2 δ)2 ≤ Cδ,
0≤t≤T
since δ ≤ δ2 with δ2 small enough. Further by the equation, the estimate (3.44) holds. Thus this complete the proof of the theorem.
3.4. Asymptotic stability of time periodic solution. In order to study the stability of the time periodic solution u ∗ , we shall consider the Cauchy problem 1 S, ∂t u + ξ · ∇x u + F · ∇ξ u − ξ · Fu = Lu + (u) + 2 u(t, x, ξ )|t=t0 = u 0 (x, ξ ),
(3.54) (3.55)
for some t0 ∈ R, where u = u(t, x, ξ ), (t, x, ξ ) ∈ (t0 , ∞) × Rn × Rn . It it noticed that the initial time t0 can be chosen arbitrarily. By putting v = u − u∗, the initial value problem (3.54) and (3.55) can be rewritten as 1 ∂t v + ξ · ∇x v + F · ∇ξ v − ξ · Fv = Lv + (v, v) + 2(u ∗ , v), 2 v(t, x, ξ )|t=t0 = v0 (x, ξ ), where v0 (x, ξ ) ≡ u 0 (x, ξ ) − u ∗ (t0 , x, ξ ). Then we have the following result.
(3.56) (3.57)
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Theorem 3.3. Let all assumptions in Theorem 3.2 hold and u ∗ be the corresponding time periodic solution obtained. Moreover, suppose that u 0 ∈ H (Rnx × Rnξ ) and there are constants δ > 0 and k ≥ 2 such that [[v0 ]]0,k + v0 Z 1 ≤ δ. Then there are constants δ3 > 0 and C3 > 0 such that for any δ ≤ δ3 , the Cauchy problem (3.56)–(3.57) has a unique global solution v ∈ Cbi [t0 , ∞); H −i (Rnx × Rnξ ) , i = 0, 1, (3.58) with bounds
sup(1 + t − t0 )2κ1 [[v(t)]]20,k +
t≥t0
∞ t0
(1 + s)2κ1 [[v(s)]]20,k+1/2 ds ≤ C32 ,
(3.59)
where κ1 is some constant with σ1,0 /2 ≤ κ1 < σ1,0 − 1/2,
(3.60)
and C3 can be also chosen as C2 = C3 δ with C3 independent of δ. Furthermore it holds that [[v(t)]]0,k ≤ Cδ(1 + t − t0 )−σ1,0 ,
(3.61)
for some constant C. To prove the above theorem, as before we first consider the decay in time estimates on (t, t0 ), −∞ < t0 ≤ t < ∞ corresponding to the nonlinear the linear solution operator U (t, t0 ) is defined in the sense that for any w0 = w0 (x, ξ ), then equation (3.56). Here U (t, t0 )w0 denotes the solution to the following initial value problem: w=U 1 ∂t w + ξ · ∇x w + F · ∇ξ w − ξ · Fw = Lw + 2(u ∗ , w), 2 w(t, x, ξ )|t=t0 = w0 (x, ξ ).
(3.62) (3.63)
Lemma 3.7. Let all assumptions in Theorem 3.2 hold and u ∗ be the corresponding time periodic solution obtained. Moreover, let k ≥ 2. Then there exist constants δ4 > 0 and (t, t0 ), −∞ < t0 ≤ t < ∞ C4 such that for any δ ≤ δ4 , the linear solution operator U satisfies the following decay estimates: (t, t0 )w0 ]]0,k ≤ C4 (1 + t − t0 )−σ1,0 [[w0 ]]0,k + w0 Z 1 , [[U (3.64) for any w0 = w0 (x, ξ ), where the constant C4 depends only on n, , k and δ4 . Proof. Without loss of generality, it suffices to prove this lemma for t0 = 0. By (2.45) and (3.45), for Eq. (3.62) one has d J0,k [w(t)] + c J0,k+1/2 [w(t)] ≤ C[[K w(t) + 2(u ∗ (t), w(t))]]0,k−1/2 dt ≤ C[[w(t)]]20,k−3/2 + C[[u ∗ (t)]]20,k+1/2 [[w(t)]]20,k+1/2 ≤ C[[w(t)]]20,k−1 + Cδ 2 J0,k+1/2 [w(t)],
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233
where the nonlinear functional J0,k [·] is given by (3.23). Thus if δ > 0 is small enough, then d J0,k [w(t)] + c J0,k+1/2 [w(t)] ≤ C[[w(t)]]20,k−1 . (3.65) dt On the other hand, by the Duhamel’s principle, w can be written as the mild form t w(t) = U (t, 0)w0 + U (t, s){2(u ∗ (s), w(s))}ds, 0
which from Theorem 2.2, (3.45) and k ≥ 2, implies [[w(t)]]0,k−1 ≤ C [[w0 ]]0,k−1 + w0 Z 1 (1 + t)−σ1,0 t +Cδ (1 + t − s)−σ1,0 [[w(s)]]k ds.
(3.66)
0
Since σ1,0 > 1 from n ≥ 5, then similar to the proof of Lemma 2.11, combining (3.65) and (3.66) yields (3.64) with t0 = 0. This completes the proof of the lemma. Furthermore, define the linear mapping T by t (t, s)φ(s)ds, Tφ(t) = U (3.67) 0
for any φ = φ(t, x, ξ ). Then similar to Corollary 3.1, we have the following estimates. Lemma 3.8. Under the assumptions of Lemma 3.7, if further δ > 0 is small enough, then one has t Tφ(t)]]20,k + (1 + s)2m [[ Tφ(s)]]20,k+1/2 ds (1 + t)2m [[ 0 t ≤ (3.68) (1 + s)2m [[φ(s)]]20,k−1/2 + φ(s)2Z 1 ds, 0
for any 0 ≤ m < σ1,0 − 1/2. Proof. For simplicity, set z(t) = Tφ(t). Fix some 0 ≤ m < σ1,0 − 1/2. Then similar to the proof of (3.65) in Lemma 3.7, one has d J0,k [z(t)] + c J0,k+1/2 [z(t)] ≤ C[[z(t)]]20,k−1/2 + C[[φ(t)]]20,k−1/2 . (3.69) dt Further applying Lemma 3.7 to (3.67) gives t [[z(t)]]0,k−1/2 ≤ C (1 + t − s)−σ1,0 [[φ(s)]]0,k−1/2 + φ(s) Z 1 ds. (3.70) 0
Since σ1,0>1 and 0 ≤ m < σ1,0 − 1/2, then similar to the proof of (3.28), it follows from (3.70) that t 2m 2 (1 + t)2m [[z(t)]]20,k−1/2 (1 + t) [[z(t)]]0,k−1/2 + 0 t (3.71) ≤C (1 + s)2m [[φ(s)]]20,k−1/2 + φ(s)2Z 1 ds. 0
Finally similar to the proof of (3.25), combining (3.69) and (3.71) gives (3.68). This completes the proof of the lemma.
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Now we are in a position to prove the asymptotical stability of the time periodic solution. Proof of Theorem 3.3. The proof is almost the same as that for Theorem 3.1. In fact, Without loss of generality, it suffices to prove Theorem 3.3 for t0 = 0. The corresponding integral equation to solve is v(t) = ϒ(v)(t) for any t ≥ 0, where the nonlinear mapping ϒ is given by t (t, 0)v0 + (t, s)(v(s), v(s))ds. ϒ(v)(t) = U U 0
By the contraction mapping theorem, the solution v will be obtained as a fixed point of ϒ on the complete metric space
S(C3 ) = v = v(t, x, ξ )|v ∈ Cb0 R+t ; H (Rnx × Rnξ ) , |||v|||k,κ1 ≤ C3 , where κ1 is given by (3.60) and the norm ||| · |||k,κ1 is defined by ∞ |||v|||k,κ1 = sup(1 + t)2κ1 [[v(t)]]20,k + (1 + s)2κ1 [[v(s)]]20,k+1/2 ds. t≥0
0
In fact, based on Lemma 3.7 and Lemma 3.8 with m = κ1 , as before it is easy to show that there is a constant C such that for any u, v ∈ S(C3 ) with some constant C3 to be determined later, |||ϒ(u)|||k,κ1 ≤ Cδ + C|||u|||2k,κ1 , |||ϒ(u) − ϒ(v)|||k,κ1 ≤ C|||u + v|||k,κ1 |||u − v|||k,κ1 , where κ1 < σ1,0 − 1/2 was used. Thus if δ ≤ δ3 with δ3 > 0 small enough and C3 is chosen properly, the unique fixed point v in S(C3 ) as a solution is found. Hence (3.58) with i = 0 and (3.59) are proved. In addition, it is easy to see that the constant C3 can be chosen as C3 δ for another constant C3 , and (3.58), and i = 1 follows from the equation. Finally we consider the improved decay rate (3.61). From the mild form v = ϒ(v) of the solution v, it follows that t −σ1,0 +C (1 + t − s)−σ1,0 [[v(s)]]0,k+1/2 [[v(s)]]0,k−1/2 ds [[v(t)]]0,k−1/2 ≤ Cδ(1 + t) 0
≤ Cδ(1 + t)−σ1,0 + C
t
(1 + t − s)−2σ1,0 (1 + s)−4κ1 ds
0
t
×
(1 + s)
2κ1
0
≤ Cδ(1 + t)
−σ1,0
[[v(s)]]20,k+1/2 ds
1/2
1/2
sup(1 + s)κ1 [[v(s)]]0,k s≥0
,
since 4κ1 ≥ 2σ1,0 > 1. Furthermore, in terms of Eq. (3.56) satisfied by v, then similar to the proof of (3.69), one has d J0,k [v(t)] + c[[v(t)]]20,k+1/2 ≤ C[[v(t)]]20,k−1/2 + C[[(v(t), v(t))]]20,k−1/2 dt ≤ Cδ 2 (1 + t)−2σ1,0 + C[[v(t)]]2k+1/2 [[v(t)]]2k−1/2 ≤ Cδ 2 (1 + t)−2σ1,0 + Cδ 2 [[v(t)]]2k+1/2 ,
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which implies d J0,k [v(t)] + c J0,k+1/2 [v(t)] ≤ Cδ 2 (1 + t)−2σ1,0 , dt since δ ≤ δ3 with δ3 > 0 small enough. Thus by the Gronwall’s inequality, it holds that [[v(t)]]20,k ≤ C J0,k [v(t)] ≤ Cδ 2 (1 + t)−2σ1,0 . Hence (3.61) is proved. This completes the proof of the theorem. Acknowledgement. Firstly, the authors would like to thank the referee for the helpful comments on revising the manuscript. The research of Seiji Ukai was supported by Liu Bie Ju Center for Mathematical Sciences and Department of Mathematics of the City University of Hong Kong. He would like to thank them for their invitation and hospitality. The research of Tong Yang was supported by the RGC Competitive Earmarked Research Grant of Hong Kong, CityU #102805, and the Changjiang Scholar Program of Chinese Educational Ministry in Shanghai Jiao Tong University. The research of Huijiang Zhao was supported by a grant from the National Natural Science Foundation of China under contract 10431060. The research was also supported in part by the National Natural Science Foundation of China under contract 10329101.
References 1. Beir˜ao da Veiga, H.: Time periodic solutions of the Navier-Stokes equations in unbounded cylindrical domains—Leray’s problem for periodic flows. Arch. Rat. Mech. Anal. 178, 301–325 (2005) 2. Bellomo, N., Toscani, G.: On the Cauchy problem for the nonlinear Boltzmann equation: global existence, uniqueness and asymptotic behaviour. J. Math. Phys. 26, 334–338 (1985) 3. Bouchut, F., Golse, F., Pulvirenti, M.: Kinetic Equations and Asymptotic Theory, Edited by Perthame, B., Desvillettes, L., Series in Applied Mathematics 4, Paris: Gauthier-Villars, 2000 4. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences 106. New York: Springer-Verlag, 1994. viii+347 pp. 5. Desvillettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation. Invent. Math. 159(2), 245–316 (2005) 6. DiPerna, R.J., Lions, P.L.: On the Cauchy problem for Boltzmann equation: global existence and weak stability. Ann. Math. 130, 321–366 (1989) 7. Duan, R.J., Ukai, S., Yang, T., Zhao, H.J.: Optimal convergence rates for the compressible Navier-Stokes equations with potential forces. Math. Mod. Meth. Appl. Sci. 17(5), 737–758 (2007) 8. Duan, R.J., Ukai, S., Yang, T., Zhao, H.J.: Optimal convergence rates to the stationary solutions for the Boltzmann equation with potential force. Preprint, 2006 9. Duan, R.J., Yang, T., Zhu, C.J.: Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete and Continuous Dynamical Systems 16, 253–277 (2006) 10. Feireisal, E., Matuš˙u-Neˇcasovà, Š., Petzeltovà, H., Straškraba, I.: On the motion of a viscous compressible fluid driven by a time periodic external force. Arch. Rat. Mech. Anal. 149, 69–96 (1999) 11. Glassey, R.: The Cauchy Problem in Kinetic Theory. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1996. xii+241 pp. 12. Grad, H.: Asymptotic Theory of the Boltzmann Equation II. In: Rarefied Gas Dynamics, J.A. Laurmann, ed., Vol. 1, New York: Academic Press, 1963 26–59 13. Guo, Y.: The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53, 1081–1094 (2004) 14. Guo, Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55(9), 1104–1135 (2002) 15. Guo, Y.: The Vlasov-Poisson-Boltzmann system near vacuum. Commun. Math. Phys. 218(2), 293–313 (2001) 16. Illner, R., Shinbrot, M.: Global existence for a rare gas in an infinite vacuum. Commun. Math. Phys. 95, 217–226 (1984) 17. Liu, T.-P., Yang, T., Yu, S.-H.: Energy method for the Boltzmann equation. Physica D 188(3–4), 178–192 (2004) 18. Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A 55, 337–342 (1979) 19. Nishida, T., Imai, K.: Global solutions to the initial value problem for the nonlinear Boltzmann equation. Publ. Res. Inst. Math. Sci. 12, 229–239 (1976/77)
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20. Shibata, Y., Tanaka, K.: Rate of convergence of non-stationary flow to the steady flow of compressible viscous fluid. Comput. Math. Appl. 53, 605–623 (2007) 21. Strain, R.M.: The Vlasov–Maxwell–Boltzmann System in the Whole Space. Commun. Math. Phys. 268,2, 543–567 (2006) 22. Strain, R.M., Guo, Y.: Almost exponential decay near Maxwellian. Commun. Par. Differ. Eqs. 31(3), 417–429 (2006) 23. Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proceedings of the Japan Academy 50, 179–184 (1974) 24. Ukai, S.: Les solutions globales de l’équation de Boltzmann dans l’espace tout entier et dans le demiespace. C. R. Acad. Sci. Paris 282A, 317–320 (1976) 25. Ukai, S.: Solutions of the Boltzmann equation. In: Pattern and Waves-Qualitive Analysis of Nonlinear Differential Equations, Mimura, M., Nishida, T., eds., Studies of Mathematics and Its Applications, Vol. 18, Tokyo: Kinokuniya-North-Holland, 1986, pp. 37–96 26. Ukai, S.: Time-periodic solutions of the Boltzmann equation. Discrete Cont. Dyn. Syst. 14A, 579–596 (2006) 27. Ukai, S., Yang, T.: Mathematical Theory of Boltzmann Equation. Lecture Notes Series-No. 8, Hong Kong: Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong, March 2006 28. Ukai, S., Yang, T.: The Boltzmann equation in the space L 2 ∩ L ∞ β : Global and time-periodic solutions. Anal. Appl. 4, 263–310 (2006) 29. Yang, T., Zhao, H.J.: Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 268(3), 569–605 (2006) 30. Valli, A.: Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10, 607–647 (1983) 31. Valli, A., Zajaczkowski, W.M.: Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103, 259–296 (1986) 32. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of mathematical fluid dynamics, Vol. I, Amsterdam: North-Holland, 2002, pp. 71–305 33. Villani, C.: Hypocoercive diffusion operators. Proceedings of the International Congress of Mathematicians, Madrid (2006) Communicated by P. Constantin
Commun. Math. Phys. 277, 237–285 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0323-2
Communications in
Mathematical Physics
On the Structure of Framed Vertex Operator Algebras and Their Pointwise Frame Stabilizers Ching Hung Lam1, , Hiroshi Yamauchi2, 1 Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan 2 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku,
Tokyo 153-8914, Japan. E-mail: [email protected] Received: 13 February 2007 / Accepted: 30 March 2007 Published online: 28 August 2007 – © Springer-Verlag 2007
Dedicated to Professor Koichiro Harada on his 65th birthday Abstract: In this paper, we study the structure of a general framed vertex operator algebra (VOA). We show that the structure codes (C, D) of a framed VOA V satisfy certain duality conditions. As a consequence, we prove that every framed VOA is a simple current extension of the associated binary code VOA VC . This result suggests the feasibility of classifying framed vertex operator algebras, at least if the central charge is small. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in the pointwise stabilizer, which are of order 1, 2 or 4. The 4A-twisted sector and the 4A-twisted orbifold theory of the famous moonshine VOA V are also constructed explicitly. We verify that the top module of this twisted sector is of dimension 1 and of weight 3/4 and the VOA obtained by 4A-twisted orbifold construction of V is isomorphic to V itself. Contents 1. 2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . . . . . Preliminaries on Simple Current Extensions . Ising Frame and Framed VOA . . . . . . . . . Representation of Code VOAs . . . . . . . . . Structure of Framed VOAs . . . . . . . . . . . Frame Stabilizers and Order Four Symmetries 4A-Twisted Orbifold Construction . . . . . .
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1. Introduction A framed vertex operator algebra V is a simple vertex operator algebra (VOA) which contains a sub VOA F called a Virasoro frame isomorphic to a tensor product of Partially supported by NSC grant 94-2115-M-006-001 of Taiwan, R.O.C.
Supported by JSPS Research Fellowships for Young Scientists.
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n-copies of the simple Virasoro VOA L(1/2, 0) such that the conformal element of F is the same as the conformal element of V . There are many important examples such as the moonshine VOA V and the Leech lattice VOA. In [DGH], a basic theory of framed VOAs was established. A general structure theory about the automorphism group and the frame stabilizer, the subgroup which stabilizes F setwise, was also included. Moreover, Miyamoto [M3] showed that if V = ⊕n∈Z Vn is a framed VOA over R such that V has a positive definite invariant bilinear form and V1 = 0, then the full automorphism group Aut(V ) is finite (see also [M1, M2]). Hence, the theory of framed VOA is very useful in studying certain finite groups such as the Monster. It is well-known (cf. [DMZ, DGH, M3]) that for any framed VOA V with a frame F, one can associate two binary codes C and D to V as follows. Since F L(1/2, 0)⊗n is a rational vertex operator algebra, V is completely reducible as an F-module. That is, V = m h 1 ,...,h n L(1/2, h 1 ) ⊗ · · · ⊗ L(1/2, h n ), h i ∈{0,1/2,1/16}
where the non-negative integer m h 1 ,...,hr is the multiplicity of the F-module L(1/2, h 1 ) ⊗ · · · ⊗ L(1/2, h n ) in V . In particular, all the multiplicities are finite and m h 1 ,...,hr is at most 1 if all h i are different from 1/16. Let M = L(1/2, h 1 ) ⊗ · · · ⊗ L(1/2, h n ) be an irreducible module over F. The 1/16word (or τ -word) τ (M) of M is a binary codeword β = (β1 , . . . , βn ) ∈ Zn2 such that 0 if h i = 0 or 1/2, (1.1) βi = 1 if h i = 1/16. For any α ∈ Zn2 , define V α as the sum of all irreducible submodules M of V such that τ (M) = α. Denote D := {α ∈ Zn2 | V α = 0}. Then D is an even linear subcode of Zn2 and we obtain a D-graded structure on V = ⊕α∈D V α such that V α · V β = V α+β . In particular, V 0 itself is a subalgebra and V can be viewed as a D-graded extension of V 0 . For any γ = (γ1 , . . . , γn ) ∈ Zn2 , denote V (γ ) := L(1/2, h 1 ) ⊗ · · · ⊗ L(1/2, h n ), where h i = 1/2 if γi = 1 and h i = 0 elsewhere. Set C := {γ ∈ Zn2 | m γ1 /2,...,γn /2 = 0}. Then V (0) = F and V 0 = ⊕γ ∈C V (γ ). The sub VOA V 0 forms a C-graded simple current extension of F which has a unique simple VOA structure [M2]. A VOA of the form V 0 = ⊕γ ∈C V (γ ) is often referred to as a code VOA associated to C. The codes C and D are very important parameters for V and we shall call them the structure codes of V with respect to the frame F. One of the main purposes of this paper is to study the precise relations between the structure codes C and D. As our main result, we shall show in Theorem 6 that for any α ∈ D, the subcode Cα := {β ∈ C | supp(β) ⊂ supp(α)} contains a doubly even subcode which is self-dual with respect to α. From this we can prove that every framed VOA forms a D-graded simple current extension of a code VOA associated to C in Theorem 7. This shows that one can obtain any framed VOA by performing simple current extensions in two steps: first extend F to a code VOA VC associated to C, then form a D-graded simple current extension of VC by adjoining suitable irreducible VC -modules. The structure and representation theory of simple current extensions is well-developed by many authors [DM1, M2, L3, Y1, Y2]. It is known that a simple current extension has a unique structure of a simple vertex
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operator algebra. Since F is rational, this implies there exist only finitely many inequivalent framed VOAs with a given central charge. Therefore, together with the conditions on (C, D) in Theorem 6, our results provide a method for determining all framed VOAs with a fixed central charge, at least if the central charge is small. It is well-known that the structure codes (C, D) of a holomorphic framed VOA must satisfy C = D ⊥ (cf. [DGH, M3]). In this case, we shall describe some necessary and sufficient conditions which C has to satisfy. Namely, we shall show in Theorem 10 that there exists a holomorphic framed VOA with structure codes (C, C ⊥ ) if and only if C satisfies the following: (1) The length of C is divisible by 16. (2) C is even, every codeword of C ⊥ has a weight divisible by 8, and C ⊥ ⊂ C. (3) For any α ∈ C ⊥ , the subcode Cα of C contains a doubly even subcode which is self-dual with respect to α. We shall call such a code an F-admissible code. Since the conditions above provide quite strong restrictions on a code C, it is possible to classify all the codes satisfying these conditions if the length is small. Once the classification of the F-admissible codes of a fixed length is done, one can consider the classification of holomorphic framed VOAs with the corresponding central charge since a holomorphic framed VOA is always a simple current extension of a code VOA. Based on the results of the present paper, one can also characterize the moonshine vertex operator algebra as the unique holomorphic framed vertex operator algebra of central charge 24 with trivial weight one subspace (cf. [LY], see also Remark 8). It is a special case of the famous uniqueness conjecture of Frenkel-Lepowsky-Meurman [FLM]. In our argument, doubly even self-dual codes play an important role in prescribing structures of framed VOAs, and it is also revealed that if we omit the doubly even property, then we lose the self-duality of certain summands V α of V which will give rise to an involutive symmetry analogous to the lift of the (−1)-isometry on a lattice VOA VL . By the standard notation as in [FLM], a lattice VOA has a form VL = Mh(α), (1.2) α∈L
where Mh(α) denotes the irreducible highest weight representation over the free bosonic vertex operator algebra Mh(0) associated to the vector space h = C ⊗Z L with highest weight α ∈ h∗ = h. Since the fusion algebra associated to Mh(0) is canonically isomorphic to the group algebra C[h], one has a duality relation Mh(α)∗ Mh(−α). This shows that there exists an order two symmetry inside the decomposition (1.2), namely, we can define an involution θ ∈ Aut(VL ) such that θ Mh(α) = Mh(−α), which is an extension of an involution on Mh(0). However, since a framed VOA V has a decomposition V = ⊕α∈D V α graded by an elementary abelian 2-group D, one cannot see the analogous symmetry directly from the decomposition. We shall show that by breaking the doubly even property in (C, D), we can find a pair of structure subcodes (C 0 , D 0 ) with [C : C 0 ] = [D : D 0 ] = 2 such that one can obtain a decomposition ⎛ ⎞ ⎛ ⎞ ⎝ V =⎝ V α+ ⊕ V α− ⎠ V α+ ⊕ V α− ⎠ (1.3) α∈D 0
α∈D 1
which forms a (D 0 ⊕ Z4 )-graded simple current extension of a code VOA associated to C 0 , where D 1 is the complement of D 0 in D. Actually, the main motivation
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of the present work is to obtain the decomposition above. In the study of McKay’s E 8 -observation on the Monster simple group [LYY1, LYY2], the authors found that McKay’s E 8 -observation is related to the conjectural Z p -orbifold construction of the moonshine VOA from the Leech lattice VOA for p > 2, where the case p = 2 is solved in [FLM, Y3]. Based on the decomposition (1.3), we can perform a Z4 -twisted orbifold construction on V . The order four symmetry defined by the decomposition in (1.3) can be found as an automorphism fixing F pointwise. The group of automorphisms which fixes F pointwise is referred to as the pointwise frame stabilizer of V . We shall show that the pointwise frame stabilizer only has elements of order 1, 2 or 4 and it is completely determined by the structure codes (C, D). As an example, we compute the pointwise stabilizer of the Moonshine VOA V associated with a frame given in [DGH, M3]. A 4A-element of the Monster is described as an element of the pointwise frame stabilizer and the associated McKay-Thompson series is computed in the proof of Theorem 14. In addition, the 4A-twisted sector and the 4A-twisted orbifold theory of V are constructed. We shall verify that the lowest degree subspace of this twisted sector is of dimension 1 and of weight 3/4, and the VOA obtained by the 4A-twisted orbifold construction of V is isomorphic to V itself. Notation and Terminology. In this article, N, Z and C denote the set of non-negative integers, integers, and the complex numbers, respectively. For disjoint subsets A and B of a set X , we use A B to denote the disjoint union. Every vertex operator algebra is defined over the complex number field C unless otherwise stated. A VOA V is called of CFT-type if it has the grading V = ⊕n≥0 Vn with V0 = C1. For a VOA structure (V, Y (·, z), 1, ω) on V , the vector ω is called the conformal vector1 of V . For simplicity, we often use (V, ω) to denote the structure (V, Y (·, z), 1, ω). The vertex operator Y (a, z) of a ∈ V is expanded as Y (a, z) = n∈Z a(n) z −n−1 . For subsets A ⊂ V and B ⊂ M of a V -module M, we set A · B := SpanC {a(n) v | a ∈ A, v ∈ B, n ∈ Z}. If M has an L(0)-weight space decomposition M = ⊕∞ n=0 Mn+h with Mh = 0, we call Mh the top level or top module of M and h the top weight of M. The top level and top weight of a twisted module can be defined similarly. For c, h ∈ C, let L(c, h) be the irreducible highest weight module over the Virasoro algebra with central charge c and highest weight h. It is well-known that L(c, 0) has a simple VOA structure. An element e ∈ V is referred to as a Virasoro vector with central charge ce ∈ C if e ∈ V2 and it satisfies e(1) e = 2e and e(3) e = (1/2)ce 1. It is well-known that by setting L e (n) := e(n+1) , n ∈ Z, we obtain a representation of the Virasoro algebra on V (cf. [M1]), i.e., [L e (m), L e (n)] = (m − n)L e (m + n) + δm+n,0
m3 − m ce . 12
Therefore, a Virasoro vector together with the vacuum element generates a Virasoro VOA inside V . We shall denote this subalgebra by Vir(e). In this paper, we define a sub VOA of V to be a pair (U, e) such that U is a subalgebra of V containing the vacuum element 1 and e is the conformal vector of U . Note that 1 We have changed the definition of the conformal vector and the Virasoro vector. In our past works, their definitions are opposite.
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(U, e) inherits the grading of V , that is, U = ⊕n≥0 Un with Un = Vn ∩ U , but e may not be the conformal vector of V . In the case that e is also the conformal vector of V , we shall call the sub VOA (U, e) a full sub VOA.2 For a positive definite even lattice L, we shall denote the lattice VOA associated to L by VL (cf. [FLM]). We adopt the standard notation for VL as in [FLM]. In particular, VL+ denotes the fixed point subalgebra of VL by a lift of (−1)-isometry on L. The letter
always denotes the Leech lattice, the unique even unimodular lattice of rank 24 without roots. Given an automorphism group G of V , we denote by V G the fixed point subalgebra of G in V . The subalgebra V G is called the G-orbifold of V in the literature. For a σ (a, z) := Y (σ a, z) for a ∈ V . V -module (M, Y M (·, z)) and σ ∈ Aut(V ), we set Y M M σ (·, z)). σ Then the σ -conjugate module M of M is defined to be the module (M, Y M We denote the ring Z/ pZ by Z p with p ∈ Z and often identify the integers 0, 1, . . . , p − 1 with their images in Z p . An additive subgroup C of Zn2 together with the standard Z2 -bilinear form is called a linear code. For a codeword α = (α1 , . . . , αn ) ∈ C, we define the support of α by supp(α) := {i | αi = 1} and the weight by wt(α) := |supp(α)|. For a subset A of C, we define supp(A) := ∪α∈A supp(α). For a binary codeword γ ∈ Zn2 and for any linear code C ⊂ Zn2 , we denote Cγ := {α ∈ C | supp(α) ⊂ supp(γ )} and C ⊥γ := {β ∈ C ⊥ | supp(β) ⊂ supp(γ )}, where C ⊥ = {δ ∈ Zn2 | α, δ = 0 for all α ∈ C}. A subcode H of C is called self-dual with respect to β ∈ C if supp(H ) = supp(β) and H ⊥β = H (see also Definition 3). The all-one vector is a codeword (11 . . . 1) ∈ Zn2 . For α = (α1 , . . . , αn ) and β ∈ (β1 , . . . , βn ) ∈ Zn2 , we define α · β := (α1 β1 , . . . , αn βn ) ∈ Zn2 . That is, the product α · β is taken in the ring Zn2 . Note that α · β ∈ (Zn2 )α ∩ (Zn2 )β . 2. Preliminaries on Simple Current Extensions We shall present some basic facts on simple current extensions of a rational C2 -cofinite vertex operator algebra of CFT-type. 2.1. Fusion algebras. We recall the notion of the fusion algebra associated to a rational VOA V . It is known that a rational VOA V has finitely many inequivalent irreducible modules (cf. [DLM2]). Let Irr(V ) = {X i | 1 ≤ i ≤ r } be the set of inequivalent irreducible V -modules. It is shown in [HL] that the fusion product X i V X j exists for a rational VOA V . The irreducible decomposition of X i V X j is referred to as the fusion rule r Xi X j = Nikj X k , (2.1) V
k=1
∈ Z denotes the multiplicity of X k in the fusion product, and where the integer is called the fusion coefficient which is also the dimension of the space of all V -inter k twining operators of type X i × X j → X k . We shall denote by X iX X j V the space of V -intertwining operators of type X i × X j → X k . We define the fusion algebra (or the Verlinde algebra) associated to V by the linear space V(V ) = ⊕ri=1 CX i spanned by Nikj
2 It is also called a conformal sub VOA in the literature.
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a formal basis {X i | 1 ≤ i ≤ r } equipped with a product defined by the fusion rule (2.1). By the symmetry of fusion coefficients, the fusion algebra V(V ) is commutative (cf. [FHL]). Moreover, it is shown in [H3] that if V is rational, C2 -cofinite and of CFT-type, then V(V ) is associative. In this subsection, we assume that V is rational, C2 -cofinite and of CFT-type. A V -module M is called a simple current if for any irreducible V -module X , the fusion product M V X is also irreducible. In other words, a simple current V -module M induces a permutation on Irr(V ) via X → M V X for X ∈ Irr(V ). Note that V itself is a simple current V -module. Next we shall recall the notion of the dual module. For a graded V -module M = ∗ , ⊕n∈N Mn+h such that dim Mn+h < ∞, define its restricted dual by M ∗ = ⊕n∈N Mn+h ∗ where Mn+h := HomC (Mn+h , C) is the dual space of Mh+n . Let Y M (·, z) be the vertex ∗ (·, z) on M ∗ operator on M. We can introduce the contragredient vertex operator Y M defined by ∗ (a, z)x, v := x, Y M (e z L(1) (−z −2 ) L(0) a, z −1 )v (2.2) Y M ∗ (·, z)) is called the ∗ for a ∈ V , x ∈ M and v ∈ M (cf. [FHL]). The module (M ∗ , Y M dual (or contragredient) module of M. Note that if the dual module M ∗ of M is isomorphic to N , there exists a V -isomorphism f ∈ Hom V (N , M ∗ ). Then f induces a V -intertwining operator of type ∗ V × N → M ∗ . This implies that MV N V = 0 or equivalently M V N ⊃ V ∗ . A V -module M is called self-dual if M ∗ M. It is obvious that the space of V -invariant bilinear forms on an irreducible self-dual V -module is one-dimensional. Lemma 1. ([Y2]) Let U, W be V -modules such that U V W = V in the fusion algebra. Then both U and W are simple current V -modules. Proof. First, we show that U V M = 0 for any V -module M. We may assume that M is irreducible as V is rational. Since the fusion product is commutative and associative, we have (U V M) V W = (U V W ) V M = V V M = M. This shows that U V M = 0. Similarly, W V M = 0. Now assume that U V M = M 1 ⊕ M 2 for V -submodules M 1 and M 2 . Then W V (U V M) = (W V M 1 ) ⊕ (W V M 2 ). On the other hand, W (U M) = (W U ) M = V M = M. V
V
V
V
V
Therefore, (W V ⊕ (W V = M. Since W V M i = 0 if M i = 0, we see that U V M is irreducible if M is. This shows that U , and also W , are simple current V -modules. M 1)
M 2)
Corollary 1. Assume that V is simple, rational, C2 -cofinite, of CFT-type and self-dual. Then the following hold. (1) Every simple current V -module is irreducible. (2) A V -module U is simple current if and only if U V U ∗ = V . (3) The set of simple current V -modules forms a multiplicative abelian group in V(V ) under the fusion product. Proof. Let U be a simple current V -module. Then U = V V U is irreducible as V ∗ is simple. By the symmetry of fusion rules VUU V UUV V UVU ∗ V (cf. [FHL]) and the assumption V ∗ V , we have U V U ∗ ⊃ V . Since U and U ∗ are irreducible, we have U V U ∗ = V . This shows (1) and (2). Now let A be the subset of V(V ) consisting of all the (inequivalent) simple current V -modules. Since a fusion product of simple current modules is again a simple current, A is closed under the fusion product.
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Clearly V ∈ A so that A contains a unit element. Finally, if U ∈ A, then U V U ∗ = V so that the inverse U ∗ ∈ A by (2). This completes the proof. 2.2. Simple current extensions. We review some basic results about simple current extensions from [L3, Y1]. Let V 0 be a simple rational C2 -cofinite VOA of CFT type and let {V α | α ∈ D} be a set of inequivalent irreducible V 0 -modules indexed by an abelian group D. A simple VOA VD = ⊕α∈D V α is called a D-graded extension of V 0 if V 0 is a full sub VOA of VD and VD carries a D-grading, i.e., V α · V β ⊂ V α+β for α, β ∈ D. In this case, the dual group D ∗ of D acts naturally and faithfully on VD . If all V α , α ∈ D, are simple current V 0 -modules, then VD is referred to as a D-graded simple current extension of V 0 . The abelian group D is automatically finite since V 0 is rational (cf. [DLM2]). Proposition 1. ([ABD, DM2, L3, Y1]) Let V 0 be a simple rational C2 -cofinite VOA of CFT type. Let VD = ⊕α∈D V α be a D-graded simple current extension of V 0 . Then (1) VD is rational and C2 -cofinite. (2) If V˜ D = ⊕α∈D V˜ α is another D-graded simple current extension of V 0 such that V˜ α V α as V 0 -modules for all α ∈ D, then VD and V˜ D are isomorphic VOAs over C. (3) For any subgroup E of D, a subalgebra VE := ⊕α∈E V α is an E-graded simple current extension of V 0 . Moreover, VD is a D/E-graded simple current extension of VE . A representation theory of simple current extensions is developed in [L3, Y1]. It is shown that each irreducible module over a simple current extension corresponds to an irreducible module over a finite dimensional semisimple associative algebra. Moreover, it is also proved that any V 0 -module can be extended to certain twisted modules over VD . Let M be an irreducible VD -module. Since V 0 is rational, we can take an irreducible V 0 -submodule W of M. Define DW := {α ∈ D | V α V 0 W V 0 W }. Then DW is a subgroup of D. Note that the subgroup DW is independent of the choice of the irreducible V 0 -module W in M. In other words, DW = DW for any irreducible V 0 -submodules W and W of M. We call M D-stable if DW = 0. In this case, V α V 0 W V 0 V β V 0 W if and only if α = β and by setting M α := V α V 0 W , we have a D-graded isotypical decomposition M = ⊕α∈D M α ( V V 0 W ) as a V 0 -module. Theorem 1. ([L3, Y1]) Let W be an irreducible V 0 -module. Then there exists a unique χW ∈ D ∗ ⊂ Aut(VD ) such that W can be extended to an irreducible χW -twisted VD -module. If DW = 0, then the extension of W to an irreducible χW -twisted V D -module is unique and D-stable. Moreover, the extension of W is given by VD V 0 W as a V 0 -module. One can easily compute fusion rules among irreducible D-stable modules. Proposition 2. ([SY, Y1]) Let VD be a D-graded simple current extension of a simple rational C2 -cofinite VOA V 0 of CFT-type. Let M i , i = 1, 2, 3 be irreducible D-stable VD -modules. Denote by M i = ⊕α∈D (M i )α a D-graded isotypical decomposition of M i . Then the following linear isomorphism holds:
M3 (M 3 )γ , M 1 M 2 VD (M 1 )α (M 2 )β V 0 where α, β, γ ∈ D are arbitrary.
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We shall need the following result on Z2 -graded simple current extensions. Proposition 3. Let V 0 be a simple rational C2 -cofinite self-dual VOA of CFT-type. Let V 1 be a simple current V 0 -module not isomorphic to V 0 such that V 1 V 0 V 1 = V 0 . Assume that V 1 has an integral top weight and the invariant bilinear form on V 1 is symmetric. Then there exists a unique simple VOA structure on V = V 0 ⊕ V 1 as a Z2 -graded simple current extension of V 0 . Proof. For a, b ∈ V 0 and u, v ∈ V 1 , define a vertex operator Y (·, z) as follows: Y (a, z)b := YV 0 (a, z)b, Y (a, z)u := YV 1 (a, z)u, Y (u, z)a := e z L(−1) Y (a, −z)u, and Y (u, z)v is defined by means of the matrix coefficients Y (u, z)v, aV 0 = v, Y (e z L(1) (−z −2 ) L(0) u, z −1 )aV 1 , where ·, ·V i denotes the invariant bilinear form on V i , i = 0, 1. Since the invariant bilinear form on V 1 is symmetric, we have the skew-symmetry Y (u, z)v = e z L(−1) Y (v, −z)u for any u, v ∈ V 1 by Proposition 5.6.1 of [FHL]. It is also shown in [FHL, Li2] that (V 0 ⊕ V 1 , Y (·, z)) forms a Z2 -graded simple vertex operator algebra if and only if we have a locality for any three elements in V 1 , that is, for any u, v ∈ V 1 , there exists N ∈ N such that for any w ∈ V 1 we have (z 1 − z 2 ) N Y (u, z 1 )Y (v, z 2 )w = (z 1 − z 2 ) N Y (v, z 2 )Y (u, z 1 )w. By Huang [H3, Theorem 3.5] (see also Theorem 3.2 and 3.5 of [H1]), it is shown that there exists λ ∈ C∗ such that for any u, v, w ∈ V 1 and sufficiently large k ∈ Z, we have (z 0 + z 2 )k Y (u, z 0 + z 2 )Y (v, z 2 )w = λ(z 2 + z 0 )k Y (Y (u, z 0 )v, z 2 )w.
(2.3)
We shall show that the associativity above leads to the locality. The idea of the following argument comes from [R]. Let N ∈ Z such that z N Y (u, z)v ∈ V 0 [[z]]. Take sufficiently large s, t ∈ Z such that z s Y (v, z)w ∈ V 0 [[z]] and (2.3) holds for (u, v, w) and (v, u, w) with k = t, s. Then z 1t z 2s (z 1 − z 2 ) N Y (u, z 1 )Y (v, z 2 )w
= e−z 2 ∂z1 (z 1 + z 2 )t z 2s z 1N Y (u, z 1 + z 2 )Y (v, z 2 )w
= λe−z 2 ∂z1 (z 2 + z 1 )t z 2s z 1N Y (Y (u, z 1 )v, z 2 )w
= λe−z 2 ∂z1 (z 2 + z 1 )t z 2s z 1N Y (e z 1 L(−1) Y (v, −z 1 )u, z 2 )w
= λe−z 2 ∂z1 e z 1 ∂z2 z 2t (z 2 − z 1 )s z 1N Y (Y (v, −z 1 )u, z 2 )w . Define p(z 1 , z 2 ) := z 2t (z 2 − z 1 )s z 1N Y (Y (v, −z 1 )u, z 2 )w. The equations above show that p(z, w) ∈ V 1 [[z 1 , z 2 ]]. On the other hand, z 1t z 2s (−z 2 + z 1 ) N Y (v, z 2 )Y (u, z 1 )w
= e−z 1 ∂z2 z 1t (z 2 + z 1 )s (−z 2 ) N Y (v, z 2 + z 1 )Y (u, z 1 )w
= λe−z 1 ∂z2 z 1t (z 1 + z 2 )s (−z 2 ) N Y (Y (v, z 2 )u, z 1 )w = λe−z 1 ∂z2 p(−z 2 , z 1 ).
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Thus the locality follows from e−w∂z e z∂w p(z, w) = e−w∂z p(z, w + z) = e−w∂z p(z, z + w) = p(z − w, z) and e−z∂w p(−w, z) = p(−w + z, z) = p(z − w, z). The uniqueness has already been shown in [DM2] in a general fashion.
Later, we shall consider a construction of framed VOAs. The following extension property will be used frequently. Theorem 2. (Extension property [Y2, Theorem 4.6.1]) Let V (0,0) be a simple rational C2 -cofinite VOA of CFT-type, and let D1 , D2 be finite abelian groups. Assume that we have a set of inequivalent irreducible simple current V (0,0) -modules {V (α,β) | (α, β) ∈ D1 ⊕ D2 } with D1 ⊕ D2 -graded fusion rules V (α1 ,β1 ) V (0,0) V (α2 ,β2 ) = V (α1 +α2 ,β1 +β2 ) for any (α1 , β1 ), (α2 , β2 ) ∈ D1 ⊕ D2 . Further assume that all V (α,β) , (α, β) ∈ D1 ⊕ D2 , have integral top weights and we have D1 - and D2 -graded simple current extensions VD1 = ⊕α∈D1 V (α,0) and VD2 = ⊕β∈D2 V (0,β) . Then VD1 ⊕D2 := ⊕(α,β)∈D1 ⊕D2 V (α,β) possesses a unique structure of a simple vertex operator algebra as a (D1 ⊕ D2 )-graded simple current extension of V (0,0) . 3. Ising Frame and Framed VOA We shall review the notion of an Ising frame and a framed vertex operator algebra.
3.1. Miyamoto involutions. We begin by the definition of an Ising vector. Definition 1. A Virasoro vector e is called an Ising vector if Vir(e) L(1/2, 0). Two Virasoro vectors u, v ∈ V are called orthogonal if [Y (u, z 1 ), Y (v, z 2 )] = 0. A decomposition ω = e1 + · · · + en of the conformal vector ω of V is called orthogonal if ei are mutually orthogonal Virasoro vectors. Let e ∈ V be an Ising vector. By definition, Vir(e) L(1/2, 0). It is well-known that L(1/2, 0) is rational, C2 -cofinite and has three irreducible modules L(1/2, 0), L(1/2, 1/2) and L(1/2, 1/16). The fusion rules of L(1/2, 0)-modules are computed in [DMZ]: L(1/2, 1/2) L(1/2, 1/2) = L(1/2, 0),
L(1/2, 1/2) L(1/2, 1/16) = L(1/2, 1/16),
L(1/2, 1/16) L(1/2, 1/16) = L(1/2, 0) ⊕ L(1/2, 1/2).
(3.1)
By (3.1), one can define some involutions in the following way. Let Ve (h) be the sum of all irreducible Vir(e)-submodules of V isomorphic to L(1/2, h) for h = 0, 1/2, 1/16. Then one has the isotypical decomposition V = Ve (0) ⊕ Ve (1/2) ⊕ Ve (1/16). Define a linear automorphism τe on V by 1 on Ve (0) ⊕ Ve (1/2), τe = −1 on Ve (1/16).
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Then by the fusion rules in (3.1), τe defines an automorphism on the VOA V (cf. [M1]). On the fixed point subalgebra V τe = Ve (0) ⊕ Ve (1/2), one can define another linear automorphism σe by 1 on Ve (0), σe = −1 on Ve (1/2). Then σe also defines an automorphism on V τe (cf. [M1]). The automorphisms τe ∈ Aut(V ) and σe ∈ Aut(V τe ) are often called Miyamoto involutions. 3.2. Framed VOAs and their structure codes. Let us define the notion of a framed VOA. Definition 2. ([DGH, M3]) A simple vertex operator algebra (V, ω) is called framed if there exists a set {e1 , . . . , en } of Ising vectors of V such that ω = e1 + · · · + en is an orthogonal decomposition. The full sub VOA F generated by e1 , . . . , en is called an Ising frame or simply a frame of V . By abuse of notation, we sometimes call the set of Ising vectors {e1 , . . . , en } a frame, also. Let (V, ω) be a framed VOA with an Ising frame F. Then F Vir(e1 ) ⊗ · · · ⊗ Vir(ei ) L(1/2, 0)⊗ n n and V is a direct sum of irreducible F-submodules ⊗i=1 L(1/2, h i ) with h i ∈ n {0, 1/2, 1/16}. For each irreducible F-module W = ⊗i=1 L(1/2, h i ), we define its binary 1/16-word (or τ -word) τ (W ) = (α1 , . . . , αn ) ∈ Zn2 by αi = 1 if and only if h i = 1/16. For α ∈ Zn2 , denote by V α the sum of all irreducible F-submodules of V whose 1/16words are equal to α. Define a linear code D ⊂ Zn2 by D = {α ∈ Zn2 | V α = 0}. Then we have the 1/16-word decomposition V = ⊕α∈D V α . By the fusion rules of L(1/2, 0)modules, it is easy to see that V α · V β ⊂ V α+β . Hence, the dual group D ∗ of D acts on V . In fact, the action of D ∗ coincides with the action of the elementary abelian 2-group generated by Miyamoto involutions {τei | 1 ≤ i ≤ n}. Therefore, all V α , α ∈ D, are irreducible V 0 -modules by [DM1]. Since there is no L(1/2, 1/16)-component in V 0 , the ∗ fixed point subalgebra V D = V 0 has the following shape: V0 = m h 1 ,...,h n L(1/2, h 1 ) ⊗ · · · ⊗ L(1/2, h n ), h i ∈{0,1/2}
where m h 1 ,...,h n ∈ N denotes the multiplicity. On V 0 we can define Miyamoto involutions σei for i = 1, . . . , n. Denote by Q the elementary abelian 2-subgroup of Aut(V 0 ) generated by {σei | 1 ≤ i ≤ n}. Then the fixed point subalgebra (V 0 ) Q = F and each m h 1 ,...,h n L(1/2, h 1 ) ⊗ . . . ⊗ L(1/2, h n ) is an irreducible F-submodule again by [DM1]. Thus m h 1 ,...,h n ∈ {0, 1} and we obtain an even linear code C := {(2h 1 , . . . , 2h n ) ∈ Zn2 | h i ∈ {0, 1/2}, m h 1 ,...,h n = 0}, namely, V0 = L(1/2, α1 /2) ⊗ · · · ⊗ L(1/2, αn /2). (3.2) α=(α1 ,...,αn )∈C
Since L(1/2, 0) and L(1/2, 1/2) are simple current L(1/2, 0)-modules, V 0 is a C-graded simple current extension of F. By Proposition 1, the simple VOA structure on V 0 is
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unique. The simple VOA V 0 of the form (3.2) is called the code VOA associated to C and denoted by VC . It is clear that VC is simple, rational, C2 -cofinite and of CFT-type. Since L(1)(VC )1 = 0 by its F-module structure, VC has a non-zero invariant form and thus is self-dual as a VC -module by [Li1]. Similarly, we also have L(1)V1 = 0 and V is self-dual as a V -module. Summarizing, there exists a pair (C, D) of even linear codes such that V is an D-graded extension of a code VOA VC associated to C. We call the pair (C, D) the structure codes of a framed VOA V associated with the frame F. Since the powers of z in an L(1/2, 0)-intertwining operator of type L(1/2, 1/2) × L(1/2, 1/2) → L(1/2, 1/16) are half-integral, the structure codes (C, D) satisfy C ⊂ D ⊥ . Notation. Let V be a framed VOA with the structure codes (C, D), where C, D ⊂ Zn2 . For a binary codeword β ∈ Zn2 , we define: σβ :=
σei ∈ Aut(V 0 )
and
τβ :=
i∈supp(β)
τei ∈ Aut(V ).
(3.3)
i∈supp(β)
Namely, by associating Miyamoto involutions to a codeword of Zn2 , σ : Zn2 → Aut(V 0 ) and τ : Zn2 → Aut(V ) define group homomorphisms. It is also clear that ker σ = C ⊥ and ker τ = D ⊥ . 4. Representation of Code VOAs Since every framed VOA is an extension of its code sub VOA, it is quite natural to study a framed VOA as a module over its code sub VOA. Let us first review a structure theory for the irreducible modules over a code VOA. 4.1. Central extension of codes. Let ν 1 = (10 . . . 0), ν 2 = (010 . . . 0), . . . , ν n = (0 . . . 01) ∈ Zn2 . Define ε : Zn2 × Zn2 → C∗ by ε(ν i , ν j ) := −1
if i > j and 1
otherwise,
(4.1)
and extend to Zn2 linearly. Then ε defines a 2-cocycle in Z 2 (Zn2 , C∗ ). By definition, ε(α, β)ε(β, α) = (−1)α,β+wt(α)wt(β) and ε(α, α) = (−1)wt(α)(wt(α)−1)/2
(4.2)
for all α, β ∈ Zn2 . In particular, ε(α, α) = (−1)wt(α)/2 and ε(α, β)ε(β, α) = (−1)α,β if α, β ∈ Zn2 are even. Let G be the central extension of Zn2 by C∗ with associated 2-cocycle ε. Recall that G = Zn2 × C∗ as a set, but the group operation is given by (α, u)(β, v) = (αβ, ε(α, β)uv) for all α, β ∈ Zn2 and u, v ∈ C∗ . Let C be a binary even linear code of Zn2 . Since ε takes values in {±1}, we can take a subgroup C˜ = {(α, u) ∈ G | α ∈ C, u ∈ {±1}} of G so that C˜ forms a central extension of C by {±1}: π
1 −→ {±1} −→ C˜ −→ C −→ 1.
(4.3)
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We shall note that the radical of the standard bilinear form ·, · on C is given by ˜ Take R = C ∩ C ⊥ and thus by (4.2), the preimage R˜ = π −1 (C ∩ C ⊥ ) is the center of C. a subgroup D of C such that C = R ⊕ D. Then the form ·, · is non-degenerate on D. It follows from (4.2) that the preimage D˜ := π −1 (D) is an extra-special 2-subgroup of ˜ The central extension C˜ is then isomorphic3 to the central product of D˜ and R˜ over C. ˜ {±1} ⊂ C∗ which we shall denote by D˜ ∗{±1} R. We identify the multiplicative group C∗ with the central subgroup (0, C∗ ) = {(0, u) ∈ G | u ∈ C∗ } of G, and let C∗ C˜ = {(α, u) ∈ G | α ∈ C, u ∈ C∗ } be the subgroup of G ˜ Then we have the exact sequence: generated by C∗ = (0, C∗ ) and C. πC ∗
1 −→ C∗ −→ C∗ C˜ −→ C −→ 1. C∗
Since is injective in the category of abelian groups, the preimage of C splits and one has an isomorphism
∩ C⊥
(4.4) in C∗ C˜
˜ C∗ C˜ (C ∩ C ⊥ ) × (C∗ ∗{±1} D). Now let ψ : C → End(V ) be a ε-projective representation of C on V , that is, ψ(α)ψ(β) = ε(α, β)ψ(α + β) for α, β ∈ C. Then one defines a linear representation ψ˜ ˜ of C∗ C˜ via ψ(α, u) := uψ(α) ∈ End(V ) for α ∈ C and u ∈ C∗ . Since C∗ C˜ is isomor˜ ψ˜ is a tensor product of a linear phic to a direct product of R = C ∩ C ⊥ and C∗ ∗{±1} D, character of R and an irreducible non-linear character of D˜ if ψ˜ is irreducible. Since D˜ is an extra-special 2 group, D˜ has only one non-linear irreducible character up to isomorphisms (cf. [Go] and [FLM, Theorem 5.5.1]). Therefore, the number of inequivalent irreducible ε-projective representations of C is equal to the order of R = C ∩ C ⊥ . Let us review the structure of the irreducible non-linear character of D˜ in more detail. Let H be a maximal self-orthogonal subcode of D. Then by (4.2) the preimage πC−1∗ (H ) of H in C∗ C˜ splits. Hence, there exists a map ι : H → C∗ such that ε(α, β) = (∂ι)(α, β) = ι(α)ι(β)/ι(α + β) for all α, β ∈ H . Since ε(α, β) ∈ {±1}, one also has ε(α, β) = ε(α, β)−1 = ι(α + β)/ι(α)ι(β). Then the section map H α → (α, ι(α)) ∈ πC−1∗ (H ) is a group homomorphism. Let χ be a linear character of H and define a linear character χ˜ of πC−1∗ (H ) by χ˜ (α, ι(α)u) = uχ (α) for α ∈ H and u ∈ C∗ . Since the preimage H˜ := π −1 (H ) is a subgroup of πC−1∗ (H ), we may view χ˜ as a linear character of H˜ . Then the irreducible non-linear character of D˜ is realized by the induced ˜ module Ind D˜ χ˜ (cf. Theorem 5.5.1 of [FLM]). Summarizing, we have: H
Proposition 4. (Theorem 5.5.1 of [FLM]) Let ψ be an irreducible ε-projective representation of C. Then the associated linear representation ψ˜ of C∗ C˜ is of the form ˜ ˜ where λ is a linear character of C ∩ C ⊥ , H˜ is the preimage of a maxiλ ⊗C Ind D˜ χ, H ˜ and χ˜ is a linear character of H˜ such that mal self-orthogonal subcode H of D in C, ˜ χ(0, ˜ −1) = −1. In particular, ψ is induced from a linear character of a maximal abelian ˜ subgroup of C. 3 Note that the isomorphism type of D ˜ is determined by the dimension of maximal isotropic subspaces of D with respect to the quadratic form q(α) = ε(α, α) (cf. [Go] and [FLM, Sect. 5.3]), which depends on the choice of the complement D if R is not doubly even. For example, we can take C = SpanZ2 {(11000), (00110), (00101)}. Then the radical R = {(00000), (11000)}. Set D = {(00000), (00110), (00101), (00011)} and D = {(00000), (11110), (00101), (11011)}. Then both D and D are complements of R in C but D˜ D˜ , for D˜ is a quaternion group whereas D˜ is a dihedral group of order 8. Nevertheless, the central product D˜ ∗{±1} R˜ is still uniquely determined by C up to isomorphisms.
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4.2. Structure of modules. Let C be an even linear code of Zn2 . For a codeword α = (α1 , . . . , αn ) ∈ C, we set V (α) := L(1/2, α1 /2) ⊗ · · · ⊗ L(1/2, αn /2). Let VC = ⊕α∈C V (α) be the code VOA associated to C. Since V (0) = L(1/2, 0)⊗ n is a rational full sub VOA of VC , every VC -module is completely reducible as a V (0)module. We shall review the structure theory of irreducible VC -modules from [M2, L3, Y1, Y2]. Let M be an irreducible VC -module. Take an irreducible V (0)-submodule W of M, which is possible as V (0) is rational. Let τ (W ) ∈ Zn2 be the binary 1/16-word of W as defined in (1.1) (see also Sect. 3.2). Then it follows from the fusion rules of L(1/2, 0)-modules that τ (W ) ∈ C ⊥ and τ (W ) = τ (W ) for any irreducible V (0)submodule W of M. Set C W := {α ∈ C | V (α) V (0) W W }. Then C W = {α ∈ C | supp(α) ⊂ supp(τ (W ))} and C W = C W for any irreducible V (0)-submodule W of M. Let {αi | 1 ≤ i ≤ r } be the coset representatives for C W in C. By the definition of C W , it follows V (αi ) V (0) W V (α j ) V (0) W if i = j, because if V (β) V (0) W = V (γ ) V (0) W in the fusion algebra then W = V (β) V (0) V (γ ) V (0) W = V (β + γ ) V (0) W for β, γ ∈ C. Note that the fusion algebra associated to V (0) is associative and V (β) V (0) V (γ ) = V (β + γ ). For simplicity, we set W i := V (αi ) V (0) W . Then we have the following isotypical decomposition: M=
r
W i ⊗ Hom V (0) (W i , M).
i=1
In the decomposition above, each homogeneous component W i ⊗ Hom V (0) (W i , M) of M forms an irreducible VC W -submodule, where VC W is the code VOA associated to C W . Let U := Hom V (0) (W, M). It is shown in [M2, L3, Y2] that U is an irreducible ε-projective representation of C W so that U is also an irreducible C∗ C˜ W -module. Moreover, the VC -module structure on M is uniquely determined by the C∗ C˜ W -module structure on U . Theorem 3. ([M2, L3, Y2]) Let C be an even linear code and VC = ⊕α∈C V (α) the associated code VOA. Let W be an irreducible V (0)-module such that τ (W ) ∈ C ⊥ . Then there is a one to one correspondence between the isomorphism classes of irreducible ε-projective representations of C W and the isomorphism classes of irreducible VC -modules containing W as a V 0 -submodule. In the following, we shall give an explicit construction of irreducible VC -modules from irreducible ε-projective C W -modules. An explicit construction. Let W be an irreducible V (0)-module such that the 1/16word τ (W ) ∈ C ⊥ . Let H be a maximal self-orthogonal subcode of C W = {α ∈ C | supp(α) ⊂ supp(τ (W ))}. Since the preimage πC−1∗ (H ) of H in (4.4) splits, there is a map ι : H → C∗ such that (α, ι(α))(β, ι(β)) = (α + β, ι(α + β)) for all α, β ∈ H . Let χ be a linear character of H . Then we can define a linear character χ˜ ι of πC−1∗ (H ) by χ˜ ι (α, ι(α)u) = uχ (α)
for α ∈ H, u ∈ C∗ .
(4.5)
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In this case, χ˜ ι is also a linear character on the preimage H˜ = π −1 (H ) of H in (4.3). Let Cε [C] be the twisted group algebra associated to the 2-cocycle ε ∈ Z 2 (C, C∗ ) defined in (4.1). That means Cε [C] = SpanC {eα | α ∈ C} as a linear space and eα eβ = ε(α, β)eα+β . By (4.2), we have eα eβ = (−1)α,β eβ eα .
(4.6)
It is clear that Cε [C W ] = ⊕α∈C W Ceα and Cε [H ] = ⊕β∈H Ceβ are subalgebras of Cε [C]. Moreover, Cε [H ] C[H ] as C-algebras. Let {α1 , . . . , αr } be a set of coset representatives for C W in C and let {β1 , . . . , βs } be a set of coset representatives for H ˜ in C W . Consider an induced module IndC˜ χ˜ ι . As a linear space, it is defined by H
˜
IndCH˜ χ˜ ι =
r s
C eαi +β j ⊗ vχ˜ι , Cε [ H˜ ]
i=1 j=1
where Cvχ˜ι is a Cε [H ]-module affording the character χ˜ ι , that is, ι(α)eα · vχ˜ι = χ (α)vχ˜ι for all α ∈ H . Note also that the components U i :=
s
Ceαi +β j ⊗ vχ˜ι , 1 ≤ i ≤ r, Cε [ H˜ ]
j=1
are irreducible Cε [C W ]-modules. Set W i := V (αi ) V (0) W for 1 ≤ i ≤ r . Let I α,i (·, z) be a V (0)-intertwining operator of type V (α) × W i → V (α) V (0) W i . Since all V (α), α ∈ C, are simple current V (0)-modules, I α,i (·, z) are unique up to scalars. It is possible to choose these intertwining operators such that
(z 0 + z 2 )m I α, j (x α , z 0 + z 2 )I β, j (x β , z 2 )w j = ε(α, β)(z 2 + z 0 )m I α+β, j (YVC (x α , z 0 )x β , z 2 )w j for x α ∈ V α , x β ∈ V β , w j ∈ W j , α j + C W = β + α j + C W and m 0 (cf. [M2, Y2]). We can also choose I 0,i (·, z) so that I 0,i (1, z) = id W i . Now put M = Ind VVCH (W, χ˜ ι ) :=
r i=1
Wi ⊗ Ui, C
and define a vertex operator Y (·, z) : VC × M → M((z)) by Y (x α , z)wi ⊗ u i := I α,i (x α , z)wi ⊗(eα · u i ) C
C
for x α ∈ V α , wi ∈ W i and u i ∈ U i . Theorem 4. ([M2, L3, Y1]) The induced module Ind VVC0 (W, χ˜ ι ) equipped with the vertex operator defined above is an irreducible VC -module. Moreover, every irreducible VC -module is isomorphic to an induced module. Remark 1. Even if τ (W ) ∈ C ⊥ , one can still define an irreducible Z2 -twisted VC -module structure on Ind VVCH (W, χ˜ ) (cf. [L1, Y1]).
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Parameterization by a pair of binary codewords. The irreducible VC -modules can also be parameterized by a pair of binary codewords. For given β ∈ C ⊥ and γ ∈ Zn2 , we define a weight vector h β,γ = (h 1β,γ , . . . , h nβ,γ ), h iβ,γ ∈ {0, 1/2, 1/16} by
h iβ,γ
⎧ 1 ⎪ ⎨ := 16 ⎪ ⎩ γi 2
if βi = 1, if βi = 0.
Let L(h β,γ ) := L(1/2, h 1β,γ ) ⊗ · · · ⊗ L(1/2, h nβ,γ ) be the irreducible L(1/2, 0)⊗ n -module with the weight h β,γ . Set Cβ := {α ∈ C | supp(α) ⊂ supp(β)} and let R β = Cβ ∩ (Cβ )⊥ be the radical of Cβ . Fix a map ι : R β → C∗ such that the section map R β α → (α, ι(α)) ∈ πC−1∗ (R β ) is a group homomorphism. Take a maximal self-orthogonal subcode H of Cβ . Then R β ⊂ H and we can extend ι to H such that the section map H α → (α, ι(α)) ∈ πC−1∗ (H ) is a group homomorphism. For, there exists a map j : H → C∗ such that H α → (α, j (α)) ∈ πC−1∗ (H ) is a group homomorphism. Then µ = ι/j restricted on R β is a character since ∂µ = ∂ι/∂j = ε/ε = 1 on R β . Take a complement K such that H = R β ⊕ K and extend µ to H by letting µ(K ) = 1. Then µj coincides with ι on R β as desired. Define a character χγ of H by χγ (α) := (−1)γ ,α for α ∈ H and extend to a character χ˜ γ ;ι of πC−1∗ (H ) by χ˜ γ ;ι (α, ι(α)u) := χγ (α)u for α ∈ H and u ∈ C∗ . Note that χ˜ γ ;ι also defines a linear character on H˜ . Moreover, every character ϕ of H˜ such that ϕ(0, −1) = −1 is of the form χ˜ γ ;ι for some γ ∈ Zn2 . Then by Theorem 4, the pair (β, γ ) determines an irreducible VC -module MC (β, γ ; ι) := Ind VVCH (L(h β,γ ), χ˜ γ ;ι ). Note that if C is self-orthogonal and supp(C) ⊂ supp(β), then MC (β, γ ; ι) L(h β,γ ) as a V (0)-module. If β = 0, then H = 0 and ι is trivial. We shall simply denote MC (0, γ ; ι) by MC (0, γ ). It is also obvious that MC (0, γ ) =
L(1/2, α1 /2) ⊗ · · · ⊗ L(1/2, αn /2).
α=(α1 ,...,αn )∈C+γ
This module is called a coset module in [M2]4 . We sometimes denote MC (0, γ ) by VC+γ , also. We shall review some basic properties of MC (β, γ ; ι). Lemma 2. ([DGL]) The module structure of MC (β, γ ; ι) is independent of the choice of the maximal self-orthogonal subcode H of Cβ and the choice of the extension of ι from R β to H . 4 This name has nothing to do with so-called the coset construction (cf. [FZ, GKO]) of a commutant subalgebra.
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Proof. Let H be a maximal self-orthogonal subcode of Cβ and ι : H → C∗ an extension of ι to H such that the section map H α → (α, ι (α)) ∈ πC−1∗ (H ) is a group homomorphism. Define a linear character χ˜ γ ;ι of πC−1∗ (H ) by χ˜ γ ;ι (α, ι (α)u) := (−1)γ ,α u for α ∈ H and u ∈ C∗ . Then χ˜ γ ;ι is also a linear character of the preimage H˜ of H in (4.3). We shall show that Ind VVC (L(h β,γ ), χ˜ γ ;ι ) Ind VVCH (L(h β,γ ), χ˜ γ ,ι ). H
C˜
C˜
For this, it suffices to show that Ind ˜β χ˜ γ ;ι Ind ˜β χ˜ γ ;ι by Theorem 3 and the construcH H tion of induced modules. By definition, it is clear that χ˜ γ ;ι | R β = χ˜ γ ;ι | R β . For simplicity, we denote it by λ. Let K be a compliment of R β in H + H and take D be a complement of R β in Cβ ˜ such that K ⊂ D. Then H = R β ⊕(H ∩ K ), H = R β ⊕(H ∩ K ) and C˜ β R˜ β ∗{±1} D. It is obvious that both H1 = H ∩ K and H2 = H ∩ K are maximal self-orthogonal subcodes of D. Therefore, by Proposition 4, we have Ind and Ind
C˜ β ˜ χ H˜ γ ;ι
C˜ β χ˜ H˜ γ ;ι
˜ H1
λ ⊗C Ind D˜ χ˜ γ ;ι | H˜ 1
˜ H2
λ ⊗C Ind D˜ χ˜ γ ;ι | H˜ 2 . Since there is only one linear representation of ˜
C ˜ ˜ D˜ such that (0, −1) acts non-trivially, we have Ind D˜ χ˜ γ ;ι Ind D˜ χ˜ γ ;ι and Ind ˜β χ˜ γ ;ι
Ind
C˜ β χ˜ H˜ γ ;ι
H1
as desired.
H2
H
Remark 2. If we choose another map ι : R β → C∗ such that the section map R β α → (α, ι (α)) ∈ πC−1∗ (R β ) is a group homomorphism, then ι/ι is a linear character of R β . Thus, there exists ξ ∈ (Zn2 )β such that ι(α)/ι (α) = (−1)α,ξ for α ∈ R β . Hence, for any α ∈ R β and u ∈ C∗ , we have χ˜ γ ;ι (α, ι(α)u) = χ˜ γ ;ι (α, (−1)α,ξ ι (α)u) = (−1)α,ξ · (−1)α,γ u = (−1)α,γ +ξ u = χ˜ γ +ξ ;ι (α, ι(α)u). Hence, χ˜ γ ;ι = χ˜ γ +ξ ;ι on R˜ β and we have MC (β, γ ; ι ) MC (β, γ + ξ ; ι). Similarly, one can show the following by considering linear characters of H˜ . Lemma 3. ([DGL]) Let β, β ∈ C ⊥ and γ , γ ∈ Zn2 . Then the irreducible VC -modules MC (β, γ ; ι) and MC (β , γ ; ι) are isomorphic if and only if β = β and γ + γ ∈ C + H ⊥β , where H is a maximal self-orthogonal subcode of Cβ and H ⊥β = {α ∈ Zn2 | supp(α) ⊂ supp(β) and α, δ = 0 for all δ ∈ H }. Proof. Assume that MC (β, γ ; ι) MC (β , γ ; ι). Then clearly β = β by 1/16-word decompositions. It is also obvious from the definition of MC (β, γ ; ι) that MC (β, γ ; ι) MC (β, γ + δ; ι) for any δ ∈ H ⊥β . Let {α1 , . . . , αr } and {δ1 , . . . , δs } be transversals for Cβ in C and H in Cβ , respectively. Then by definition we have a decomposition
r s
αi +δ j V (αi ) L(h β,γ ) ⊗ e MC (β, γ ; ι) = ⊗ χ˜ γ ;ι . i=1 j=1
V (0)
C
Cε [H ]
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It follows from (4.6) that
V (αi ) L(h β,γ ) ⊗ eαi +δ j ⊗ χ˜ γ ;ι M H (β, γ + αi + δ j ; ι) V (0)
Cε [H ]
C
(4.7)
as a VH -submodule. Therefore, we have the following decompositions:
MC (β, γ ; ι) =
M H (β, γ + δ; ι),
δ+H ∈C/H
MC (β, γ ; ι) =
M H (β, γ + δ; ι).
δ+H ∈C/H
Since H = Cβ ∩ H ⊥β by the maximality of H , all M H (β, γ + δ; ι), δ ∈ C/H , are inequivalent irreducible VH -submodules. Thus, if MC (β, γ ; ι) MC (β, γ ; ι), then χ˜ γ ;ι = χ˜ γ +δ;ι for some δ ∈ C. This is possible if and only if γ + γ ∈ C + H ⊥β . Conversely, if γ +γ ∈ C + H ⊥β , then MC (β, γ ; ι) and MC (β, γ ; ι) contain isomorphic irreducible VH -submodules. Since VC -module structures on MC (β, γ ; ι) and MC (β, γ ; ι) are uniquely determined by their VH -module structures, they are isomorphic. In the proof above, we have shown the following useful fact. Corollary 2. Let MC (β, γ ; ι) be an irreducible VC -module. Let H be a maximal self-orthogonal subcode of Cβ . Then as a VH -module,
MC (β, γ ; ι) =
M H (β, γ + δ; ι).
δ+H ∈C/H
In particular, every irreducible VH -submodule of MC (β, γ ; ι) is multiplicity-free. Lemma 4. Let R be the radical of Cβ with respect to the standard bilinear form and H a maximal self-orthogonal subcode of Cβ . Then Cβ + H ⊥β = R ⊥β and hence the code C + H ⊥β = C + R ⊥β is again independent of the choice of the maximal self-orthogonal subcode H . Proof. Since R ⊂ H , we have H ⊥β ⊂ R ⊥β . By definition, it is also clear that Cβ ⊂ R ⊥β and we have Cβ + H ⊥β ⊂ R ⊥β . Since the bilinear form ·, · restricted on the quotient space Cβ /R is non-degenerate and H/R is a maximal self-orthogonal subspace, we have dim Cβ /R = 2 dim H/R and hence dim Cβ = 2 dim H − dim R. On the other hand, dim(Cβ + H ⊥β ) = dim Cβ + dim H ⊥β − dim(Cβ ∩ H ⊥β ) = dim Cβ + wt β − 2 dim H = wt β − dim R = dim R ⊥β . Thus, we have Cβ + H ⊥β = R ⊥β
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4.3. Dual module. We shall determine the structure of the dual module of a VC -module MC (β, γ ; ι). Recall that the dual module of a V -module M = ⊕n∈N Mn+h is defined to ∗ equipped with a vertex operator Y ∗ (·, z) defined be its restricted dual M ∗ = ⊕n∈N Mn+h M by (2.2). First, we consider the case when the code is self-orthogonal. Let H be a self-orthogonal code of Zn2 . In this case, one can define a character ϕ of H by ϕ(α) = (−1)wt(α)/2 for α ∈ H . So there exists a codeword κ ∈ Zn2 such that ϕ(α) = (−1)κ,α for all α ∈ H . Lemma 5. Let H ⊂ Zn2 be a self-orthogonal code. For any γ ∈ Zn2 , the dual module of M H ((1n ), γ ; ι) is isomorphic to M H ((1n ), γ + κ; ι), where κ ∈ Zn2 is given by (−1)κ,α = (−1)wt(α)/2 for all α ∈ H . Proof. By assumption, M H ((1n ), γ ; ι) = L(1/2, 1/16)⊗ n ⊗ χ˜ γ ;ι is an irreducible V (0) = L(1/2, 0)⊗ n -module. Therefore, M H ((1n ), γ ; ι)∗ M H ((1n ), γ ; ι)
for some γ ∈ Zn2 .
Since L(1/2, 1/16)⊗ n is a self-dual V (0)-module, we have a V (0)- isomorphism f : M → M ∗ . Let Y (·, z) and Y ∗ (·, z) be the vertex operators on M and M ∗ , respectively. For α ∈ H , let x α ∈ V (α) be a non-zero highest weight vector of weight wt(α)/2. Then Y ∗ (x α , z) f and f Y (x α , z) are V (0)-intertwining operators of type V (α) × M H ((1n ), γ ; ι) → M H ((1n ), γ ; ι)∗ . Since the space of V (0)-intertwining operators of this type is one-dimensional, there exists a scalar λα ∈ C∗ such that Y ∗ (x α , z) f = λα f Y (x α , z). Let v be a non-zero highest weight vector of M H ((1n ), γ ; ι). Then by (2.2), one has Y ∗ (x α , z) f v, v = (−1)wt(α)/2 z −wt(α) f v, Y (x α , z −1 )v α v. = (−1)wt(α)/2 z −wt(α)/2 f v, x(wt(α)/2−1)
On the other hand, Y ∗ (x α , z) f v, v = λα f Y (x α , z)v, v α = λα z −wt(α)/2 f x(wt(α)/2−1) v, v. α v = tv for some t ∈ C∗ and f v, v = 0, we have λα = (−1)wt(α)/2 = Since x(wt(α)/2−1) (−1)κ,α . Therefore, by considering the linear character associated to M H ((1n ), γ ; ι)∗ , we see that M H ((1n ), γ ; ι)∗ M H ((1n ), γ + κ; ι).
Proposition 5. Let C be an even linear code, β ∈ C ⊥ and γ ∈ Zn2 . Let H be a maximal self-orthogonal subcode of Cβ . Then the dual module MC (β, γ ; ι)∗ is isomorphic to MC (β, γ + κ H ; ι), where κ H ∈ Zn2 is such that supp(κ H ) ⊂ supp(β) and (−1)κ H ,α = (−1)wt(α)/2 for all α ∈ H . Proof. By Corollary 2, MC (β, γ ; ι) contains a VH -submodule M H (β, γ ; ι) L(h β,γ ) ⊗ χ˜ γ ;ι . By the previous lemma, the dual module MC (β, γ ; ι)∗ contains a VH -submodule isomorphic to M H (β, γ ; ι)∗ M H (β, γ + κ H ; ι) = L(h β,γ ) ⊗ χ˜ γ +κ H ;ι .
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Therefore, by the structure of irreducible VC -modules, MC (β, γ ; ι)∗ Ind VVCH (L(h β,γ ), χ˜ γ +κ H ;ι ), and hence MC (β, γ ; ι)∗ MC (β, γ + κ H ; ι).
As an immediate corollary, we have: Corollary 3. With reference to the proposition above, MC (β, γ ; ι) is self-dual if and only if κ H ∈ C. In particular, MC (0, γ ) is self-dual for all γ ∈ Zn2 . 4.4. Fusion rules. We shall compute the fusion rules among some irreducible VC modules. First, we recall a result from [M2] which is a direct consequence of Proposition 2. Lemma 6. ([M2]) For α, β ∈ Zn2 , MC (0, α) VC MC (0, β) = MC (0, α + β). By the lemma above, we see that MC (0, α) VC MC (0, α) = MC (0, 0) VC . Therefore, all MC (0, α), α ∈ Zn2 , are simple current VC -modules by Corollary 1. It also follows that MC (0, α) VC MC (β, γ ; ι) is an irreducible VC -module with the 1/16word β. The corresponding fusion rules are also computed by Miyamoto [M3] in the case supp(α) ⊂ supp(β). Lemma 7. ([M3]) Let α, β, γ ∈ Zn2 with β ∈ C ⊥ . Then MC (0, α) MC (β, γ ; ι) = MC (β, α + γ ; ι). VC
Moreover, the difference of the top weight of MC (β, γ ; ι) and the top weight of MC (β, α+ γ ; ι) is congruent to α, α + β/2 modulo Z. Proof. The assertion is proved in Lemma 3.27 of [M3] in the case supp(α) ⊂ supp(β). We generalize his argument to obtain the desired fusion rule. Since MC (0, α) is a simple current VC -module, we know that there exists γ ∈ Zn2 such that MC (0, α) VC MC (β, γ ; ι) MC (β, γ ; ι). Therefore, if we can construct a non-zero VC -intertwining operator of type MC (0, α) × MC (β, γ ; ι) → MC (β, α + γ ; ι), then we are done. To do this, we have to extend VC to a larger algebra. The case α ∈ C is trivial so that we assume that α ∈ C. Set C = C (C + α). We can define a simple vertex operator (super)algebra structure on the space VC = VC ⊕ VC+α = MC (0, 0) ⊕ MC (0, α). This is a VOA if α is even, and an SVOA if α is odd. Set H := Cβ ∩ H ⊥ , which is the unique maximal subcode of Cβ containing H such that its preimage H˜ is a maximal abelian subgroup of C˜ in the extension (4.3). β
We can take j : H → C∗ such that j | H = ι and the section map H δ → (δ, j (δ)) ∈ πC−1∗ (H ) defines a group homomorphism. In the definition of the induced module MC (β, γ ; ι) = Ind VVCH L(h β,γ , χ˜ γ ;ι ), if we use Cε [C ] instead of Cε [C] and replace ι by j , then we obtain an irreducible VC -module V
MC (β, γ ; j ) := Ind VC (L(h β,γ , χ˜ γ ,j ) H
which contains MC (β, γ ; ι) = Ind VVCH (L(h β,γ , χ˜ γ ,j | H ) as a VC -submodule. The induced module MC (β, γ ; j ) is an untwisted VC -module if α, β = 0 and otherwise it is a
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Z2 -twisted VC -module (cf. [Y2]). Nevertheless, the subspace M = VC+α · MC (β, γ ; ι) of MC (β, γ ; j ) is an irreducible VC -submodule. It follows from (4.6) and (4.7) that there exists an irreducible VH -submodule of M isomorphic to M H (β, α + γ ; j | H ). Then M MC (β, α + γ ; j | H ) by the structure of an irreducible VC -module. Since M MC (0, α) VC MC (β, γ ; ι), we obtain the desired fusion rule. Since the L(1/2, 0)-intertwining operators of types L(1/2, h) × L(1/2, 1/16) → L(1/2, 1/16), h ∈ {0, 1/2}, keep the top weights but the L(1/2, 0)-intertwining operators of type L(1/2, 1/2) × L(1/2, 0) → L(1/2, 1/2) and L(1/2, 1/2) × L(1/2, 1/2) → L(1/2, 0) change the top weights by 1/2 or −1/2, the difference of top weights is as in the assertion. By this lemma, we can compute the following fusion rule. Proposition 6. Let β ∈ C ⊥ and γ ∈ Zn2 . Let H be a maximal self-orthogonal subcode of Cβ . Then MC (β, γ ; ι) MC (β, γ ; ι)∗ = VC
MC (0, δ),
⊥ δ+C∈C+H β
where δ ∈ Zn2 runs over a transversal for C in C + H ⊥β . Proof. It follows from the 1/16-word decomposition that the fusion product MC (β, γ ; ι) M (β, γ ; ι)∗ contains only modules of type MC (0, δ). Now assume that VC CMC (0,δ) = 0. Then by the symmetry of fusion rules, we have MC (β,γ ;ι) MC (β,γ ;ι)∗ VC
MC (0, δ) MC (β, γ ; ι) MC (β, γ ; ι)∗
VC
MC (β, γ ; ι) MC (β, γ ; ι) MC (0, δ)
= 0. VC
Since MC (β, γ ; ι) VC MC (0, δ) = MC (β, γ + δ; ι) by the previous lemma, this is possible if and only if δ ∈ C + H ⊥β by Lemma 3. Therefore, we have the fusion rule as stated. By the lemma above, we introduce the following definition. Definition 3. Let β ∈ Zn2 and H be a subcode with supp(H ) ⊂ supp(β). H is said to be self-dual with respect to β if H = H ⊥β . Remark 3. Note that if H is a self-dual subcode of Cβ w.r.t. β then C + H ⊥β = C. By Corollary 1 and Proposition 6, we have Corollary 4. MC (β, γ ; ι) is a simple current module if and only if Cβ contains a selfdual subcode w.r.t. β. p Remark 4. Now suppose MC (β, 0; ι) VC MC (β, 0; ι)∗ = i=1 MC (0, δi ). Let H be a maximal self-orthogonal subcode of Cβ , and let κ H ∈ (Zn2 )β such that κ H , α = α, α/2 mod 2 for all α ∈ H as in Proposition 5. Then MC (β, 0; ι)∗ = MC (β, κ H ; ι) = MC (0, κ H ) MC (β, 0; ι), VC
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and thus MC (β, 0; ι) = MC (0, κ H ) VC MC (β, 0; ι)∗ . Using this, we can compute the following fusion rule: MC (β, γ1 ; ι) MC (β, γ2 ; ι) VC = MC (0, γ1 ) MC (β, 0; ι) MC (0, γ2 ) MC (β, 0; ι) VC
VC
VC
= MC (0, γ1 + γ2 ) MC (β, 0; ι) MC (β, 0; ι) VC VC = MC (0, γ1 + γ2 ) MC (β, 0; ι) MC (0, κ H ) MC (β, 0; ι)∗ VC
VC
VC
= MC (0, γ1 + γ2 + κ H ) MC (β, 0; ι) MC (β, 0; ι)∗ VC VC p = MC (0, γ1 + γ2 + κ H ) MC (0, δi ) VC
=
p
i=1
MC (0, γ1 + γ2 + κ H + δi ).
i=1
5. Structure of Framed VOAs We shall prove that every framed VOA is a simple current extension of a code VOA. This result has many fruitful consequences. For example, the irreducible representations of a framed VOA can be determined by a notion of induced modules. Another interesting result is the conditions on possible structure codes of holomorphic framed VOAs, namely we obtain a necessary and sufficient condition for a pair of codes (C, D) to be the structure codes of some holomorphic framed VOAs in Theorem 10.
5.1. Simple current structure. In this subsection we discuss how a code VOA can be extended to a framed VOA. First, we give a construction of a non-trivial simple current extension. Lemma 8. Let C be an even linear subcode of Zn2 and β ∈ C ⊥ a non-zero codeword. Let γ ∈ Zn2 be a binary codeword such that the irreducible VC -module MC (β, γ ; ι) has an integral top weight. If Cβ contains a doubly even self-dual subcode w.r.t. β, then there exists a unique structure of a framed VOA on VC ⊕ MC (β, γ ; ι) which forms a Z2 -graded simple current extension of VC . Proof. Let H be a doubly even self-dual subcode of Cβ w.r.t. β. By Proposition 5, MC (β, γ ; ι) is self-dual, and by Corollary 4, MC (β, γ ; ι) is a simple current VC - module. By Corollary 2, MC (β, γ ; ι) has a VH -module structure MC (β, γ ; ι) = M H (β, γ + δ; ι), δ+H ∈C/H
where all irreducible VH -submodules M H (β, γ + δ; ι) are self-dual by Proposition 5. It is clear that a VC -invariant bilinear form on MC (β, γ ; ι) induces a non-degenerate
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VH -invariant bilinear form on M H (β, γ + δ; ι). It is shown in [Li1] that a VC -invariant bilinear form on MC (β, γ ; ι) is either symmetric or skew-symmetric. Since the top level of M H (β, γ + δ; ι) is one-dimensional, the VC -invariant bilinear form on MC (β, γ ; ι) must be symmetric. Therefore, VC ⊕ MC (β, γ ; ι) forms a Z2 -graded simple current extension of VC by Proposition 3. Lemma 9. Let β ∈ C ⊥ and γ ∈ Zn2 with β = 0. Assume that V = VC ⊕ MC (β, γ ; ι) forms a framed VOA. Then there exists a maximal self-orthogonal subcode K of Cβ which is doubly even. Proof. Let H be a maximal self-orthogonal subcode of Cβ . If H is doubly even, then we are done. So we assume that H contains a codeword whose weight is congruent to 2 modulo 4. Since V = VC ⊕ MC (β, γ ; ι) forms a simple VOA, MC (β, γ ; ι) is a selfdual VC -module. Therefore, by Corollary 3, there exists a codeword κ H ∈ Cβ such that (−1)κ H ,α = (−1)wt(α)/2 for all α ∈ H . By Corollary 2, MC (β, γ ; ι) has the following decomposition as a VH -module: MC (β, γ ; ι) = M H (β, γ + δ; ι). δ+H ∈C/H
By our choice of H , κ H ∈ H ⊥β so that M H (β, γ ; ι) and its dual M H (β, γ + κ H ; ι) are inequivalent irreducible VH -submodules of V . We consider a sub VOA U generated by M H (β, γ ; ι) ⊕ M H (β, γ + κ H ; ι). By the fusion rule given in Proposition 6, U has the following shape as a VH -module: U = M H (0, 0) ⊕ M H (0, κ H ) ⊕ M H (β, γ ; ι) ⊕ M H (β, γ + κ H ; ι).
(5.1)
Note that H = C ∩ H ⊥β by the maximality of H . Set H := H (H + κ H ), H0 := H ∩ κ H ⊥ and take any α ∈ H \H0 . Then H = H0 (H0 + α ) (H0 + κ H ) (H0 + α + κ H ). We set K := H0 (H0 + κ H ). It is clear that U also possesses a symmetric invariant bilinear form which we shall denote by ·, ·U . Since M H (β, γ ; ι) and M H (β, γ + κ H ; ι) are dual to each other, we have M H (β, γ ; ι), M H (β, γ ; ι)U = M H (β, γ + κ H ; ι), M H (β, γ + κ H ; ι)U = 0. By Lemma 3, M H (β, γ ; ι) and M H (β, γ + κ H ; ι) are isomorphic irreducible VH0 -modules and there exists a VH0 -isomorphism ϕ : M H (β, γ ; ι) → M H (β, γ +κ H ; ι). Let u be a non-zero highest weight vector of M H (β, γ ; ι). Since the top level of M H (β, γ ; ι) is one-dimensional, we may assume that u, ϕ(u)U = 1. Now consider the decomposition (5.1) of U with respect to a series of sub VOAs VH0 ⊂ VK ⊂ VH of U . It is clear that M H (0, 0) ⊕ M H (0, κ H ) = M K (0, 0) ⊕ M K (0, α ). Therefore, there exists a decomposition of U as a VK -module U = M K (0, 0) ⊕ M K (0, α ) ⊕ W with W = M H (β, γ ; ι) ⊕ M H (β, γ + κ H ; ι). Let W 0 be an irreducible VK -submodule of W . Since K is a self-orthogonal subcode of Cβ , the top level of W 0 is one-dimensional. Let v ∈ W 0 be a non-zero highest weight vector. As we mentioned, M H (β, γ ; ι) and M H (β, γ + κ H ; ι) are isomorphic VH0 -submodules. But M H (β, γ ; ι) and M H (β, γ +
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κ H ; ι) cannot form VK -submodules by the fusion rule of VH -modules. Therefore, we can write v = c1 u + c2 ϕ(u) with c1 , c2 = 0. This shows that v, vU = 2c1 c2 = 0 so that W 0 is a self-dual VK -submodule. Then K is a doubly even code by Corollary 3. Since |K | = |H | = 2|H0 |, K is a maximal self-orthogonal subcode of Cβ . Therefore, Cβ contains the desired subcode K . We recall the following fact from the coding theory. Theorem 5. ([McST]) Let n be divisible by 8 and H a doubly even code of Zn2 containing the all-one vector (11 . . . 1) ∈ Zn2 . Then there exists a doubly even self-dual code H such that H ⊂ H . Now we begin to prove that every framed VOA is a simple current extension of a code VOA. For this, it suffices to show the following proposition. Proposition 7. Let β ∈ C ⊥ and γ ∈ Zn2 . Assume that V = VC ⊕ MC (β, γ ; ι) forms a framed VOA. Then Cβ contains a doubly even self-dual subcode w.r.t. β. Proof. By Lemma 9, Cβ contains a maximal self-orthogonal subcode H which is doubly even. By Corollary 2, V has a decomposition V = M H (0, δ) ⊕ M H (β, γ + δ; ι) δ+H ∈C/H
as a VH -module. By the fusion rule in Proposition 6, the subspace U := VH ⊕ M H (β, γ ; ι) forms a sub VOA of V , since H = Cβ ∩ H ⊥β . If H is not self-dual, then there exists a doubly even self-dual subcode H of Zn2 w.r.t. β such that H ∪ (H + β) ⊂ H by Theorem 5. Note that the weight of β is divisible by 8 since MC (β, γ ; ι) has an integral top weight. Let us consider the code VOA VH associated to H . Since H ⊂ H , it is clear that VH contains VH as a sub VOA. We can also take a map j: H → C∗ such that j | H = ι and the section map H α → (α, j (α)) ∈ πC−1∗ (H ) is a group homomorphism. By the structure theory in Theorem 4, we can define an irreducible VH -module M H (β, γ ; j ) such that M H (β, γ ; j )|VH M H (β, γ ; ι) as a VH -module. For simplicity, we shall denote M H (β, γ ; j ) by W . Since the top level of W is one-dimensional, the VH -invariant bilinear form on W is symmetric. Therefore, by Proposition 2, we can define a framed VOA structure on U := VH ⊕ W. We denote the vertex operator on U by Y (·, z). Now suppose H is a proper subcode of H . Then VH = Vδ+H , Vδ+H = M H (0, δ), δ+H ∈H /H
as a VH -module. Let πδ+H : VH → Vδ+H be the projection map. Then for u, v ∈ W , we have Y (u, z)v = πδ+H Y (u, z)v. δ+H ∈H /H
Since the simple VOA structure is unique on VH ⊕W , we may assume that π H Y (u, z)v = YV (u, z)v. Take any α ∈ H \H and set K := H (H + α). We shall show the following claim:
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Claim. For u, v ∈ W , there exists N = N (u, v) ∈ N such that (z 1 − z 2 ) N Y (u, z 1 )π H +α Y (v, z 2 )w = (z 1 − z 2 ) N Y (v, z 2 )π H +α Y (u, z 1 )w
(5.2)
for any w ∈ W . Take any a ∈ VH +α . Since U = VH ⊕ M H (β, γ ; ι) forms a framed VOA by assumption, there exists N = N (u, v) ∈ N such that (z 1 − z 2 ) N Y (u, z 1 )π H Y (v, z 2 )w = (z 1 − z 2 ) N Y (v, z 2 )π H Y (u, z 1 )w.
(5.3)
Take a sufficiently large k ∈ N. Then one has (z 1 − z 2 ) N (z 0 − z 1 )k (z 0 − z 2 )k Y (u, z 1 )π H +α Y (v, z 2 )Y (a, z 0 )w = (z 1 − z 2 ) N (z 0 − z 1 )k (z 0 − z 2 )k Y (a, z 0 )Y (u, z 1 )π H Y (v, z 2 )w = (z 1 − z 2 ) N (z 0 − z 1 )k (z 0 − z 2 )k Y (a, z 0 )Y (v, z 2 )π H Y (u, z 1 )w = (z 1 − z 2 ) N (z 0 − z 1 )k (z 0 − z 2 )k Y (v, z 2 )π H +α Y (u, z 1 )Y (a, z 0 )w. Since the expansions of both sides of the equations have only finitely many negative powers of z 0 , we get (z 1 − z 2 ) N Y (u, z 1 )π H +α Y (v, z 2 )w = (z 1 − z 2 ) N Y (v, z 2 )π H +α Y (u, z 1 )w. Note that Y (a, z)π H = π H +α Y (a, z) on VH and W = VH +α · W . By the Claim above, we can introduce a framed VOA structure on X := VK ⊕ M K (β, γ ; j | K ) as follows. Since W as a VK -module is isomorphic to M K (β, γ ; j | K ), we can identify these structures. For a, b ∈ VK and u, v ∈ M K (β, γ ; j | K ), we define the vertex operator map Y X (·, z) by Y X (a, z)b := Y (a, z)b, Y X (a, z)u := Y (a, z)u, Y X (u, z)a := Y (u, z)a, and Y X (u, z)v := π H Y (u, z)v + π H +α Y (u, z)v. Let ·, ·U be an non-zero invariant bilinear form on U , which is unique up to scalar multiples. Since VK is a subalgebra of U , we can define an invariant bilinear form ·, ·VK on VK by a, bVK := a, bU . In addition, since W as a VK -module is isomorphic to M K (β, γ ; j | K ), we may view ·, ·U restricted on W as a VK -invariant bilinear form on M K (β, γ ; j | K ). Then a, Y X (u, z)vVK = a, π H Y (u, z)vVK + a, π H +α Y (u, z)vVK = a, π H Y (u, z)vU + a, π H +α Y (u, z)vU = a, Y (u, z)vU = Y (e z L(1) (−z −2 ) L(0) u, z −1 )a, vU . By the equality above, it follows from Sect. 5.6 of [FHL] and [Li2] that Y X (·, z) satisfies the Jacobi identity if and only if we have a locality for any three elements in M K (β, γ ; j | K ), which follows from (5.2) and (5.3). Therefore, (X, Y X (·, z)) is also a
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framed VOA. In fact, one can define a framed VOA structure on VE ⊕ M E (β, γ ; j | E ) for any subcode E of H containing H in a similar way. We shall deduce a contradiction from this observation. Let W 1 and W 2 be VH -modules isomorphic to M H (β, γ ; ι). Since W i and M K (β, γ ; j | K ) are isomorphic VH -modules, we have VH -isomorphisms ϕi : W i → M K (β, γ ; j | K ) ⊂ X for i = 1, 2. Set X := VH ⊕ VH +α ⊕ W 1 ⊕ W 2 . We shall define a vertex operator Y X on X as follows. For a 0 , b0 ∈ VH , a 1 , b1 ∈ VH +α , u 1 , v 1 ∈ W 1 and u 2 , v 2 ∈ W 2 , define ⎤ ⎡ 0 0 0 Y X (a 0 , z) ⎥ ⎢ 0 0 0 Y X (a 0 , z) ⎥, Y X (a 0 , z) := ⎢ −1 0 ⎦ ⎣ 0 0 0 ϕ1 Y X (a , z)ϕ1 −1 0 0 0 0 ϕ2 Y X (a , z)ϕ2 ⎤ ⎡ 1 0 0 0 Y X (a , z) ⎥ ⎢Y X (a 1 , z) 0 0 0 ⎥, Y X (a 1 , z) := ⎢ −1 1 ⎣ 0 0 0 ϕ1 Y X (a , z)ϕ2 ⎦ 0 0 ϕ2−1 Y X (a 1 , z)ϕ1 0 Y X (u 1 , z) ⎡
⎤ 0 0 0 π H Y X (ϕ1 u 1 , z)ϕ1 ⎢ 0 0 0 π H +α Y X (ϕ1 u 1 , z)ϕ2 ⎥ ⎥, := ⎢ ⎦ ⎣ϕ −1 Y X (ϕ1 u 1 , z) 0 0 0 1 −1 1 0 0 0 ϕ2 Y X (ϕ1 u , z)
Y X (u 2 , z) ⎡
⎤ 0 0 0 π H Y X (ϕ2 u 2 , z)ϕ2 ⎥ ⎢ 0 0 π H +α Y X (ϕ2 u 2 , z)ϕ1 0 ⎥, := ⎢ −1 2 ⎦ ⎣ 0 0 0 ϕ1 Y X (ϕ2 u , z) −1 2 0 0 0 ϕ2 Y X (ϕ2 u , z)
on t [b0 , b1 , v 1 , v 2 ] ∈ VH ⊕ VH +α ⊕ W 1 ⊕ W 2 . Note that Y X (·, z) is considered as a VH intertwining operator and π H : X → VH and πα+H : X → Vα+H are VH -projections on VH and Vα+H , respectively. By (5.2) and (5.3), it is straightforward to check that Y X (·, z) satisfies the locality and hence (X , Y X (·, z)) itself forms a VOA. In fact, we have defined a VOA structure on X such that V 1 · W 1 = W 2 , V 1 · W 2 = W 1 , W 1 · W 1 = W 2 · W 2 = V 0 and W 1 · W 2 = V 2 based on the framed VOA structure on X . Now take a subspace Z := {a 0 + b0 + u 1 + ϕ2−1 ϕ1 u 1 ∈ X | a 0 ∈ VH , b0 ∈ VH +α , u 1 ∈ W 1 }. Then it follows from the definition of Y X (·, z) that Z is a subalgebra of X and the linear isomorphism √ ψ: a 0 + b0 + u 1 + ϕ2−1 ϕ1 u 1 −→ a 0 + b0 + 2ϕ1 u 1 , a 0 ∈ VH , b0 ∈ VH +α , u 1 ∈ W 1 ,
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defines a vertex operator algebra isomorphism between Z and X . Since X is a framed VOA and every framed VOA is rational, X is a completely reducible Z -module. However, since the quotient X /Z has no 1/16-word component corresponding to a codeword 0, we obtain ψ −1 (M K (β, γ ; j | K )) · (X /Z ) = 0 which is a contradiction by Proposition 11.9 of [DL]. This contradiction comes from the assumption that H = H . Hence, H = H as desired. Now we present the main theorem of this paper: Theorem 6. Let V = ⊕α∈D V α be a framed VOA with structure codes (C, D). Then (1) For every non-zero α ∈ D, the subcode Cα of C contains a doubly even self-dual subcode w.r.t. α. (2) C is even, every codeword of D has a weight divisible by 8, and D ⊂ C ⊂ D ⊥ . Proof. (1) follows from Proposition 7 since V 0 ⊕ V α is a framed sub VOA of V for any non-zero α ∈ D. (2) follows from (1), since a self-dual subcode of Cα w.r.t. α always contains the codeword α. As a corollary, we can also prove the following theorem. Theorem 7. Let V = ⊕α∈D V α be a framed VOA with structure codes (C, D). Then V = ⊕α∈D V α is a D-graded simple current extension of the code VOA V 0 = VC . Proof. The assertion follows immediately from Theorem 6 and Corollary 4.
There are many applications of Theorems 6 and 7. Corollary 5. For a positive integer n, the number of isomorphism classes of framed VOAs with a fixed central charge n/2 is finite. Proof. By Theorem 7, every framed VOA is a simple current extension of a code VOA. A code VOA is uniquely determined by its structure code by Proposition 1, and it has finitely many irreducible representations as it is rational. In particular, there are finitely many inequivalent simple current modules over a code VOA. Therefore, the number of isomorphism classes of framed VOAs of given central charge is finite by the uniqueness of a simple current extension in Proposition 1. By Theorem 7, we can immediately classify all irreducible (both untwisted and Z2 -twisted) modules over a framed VOA. Corollary 6. Let V = ⊕α∈D V α be a framed VOA with structure codes (C, D). Let W be an irreducible V 0 -module. Then there exists η ∈ Zn2 , which is unique modulo D ⊥ , such that W can be uniquely extended to an irreducible τη -twisted V -module which is given by V V 0 W as a V 0 -module. In particular, every irreducible untwisted V -module is D-stable. Proof. Let β ∈ C ⊥ be the 1/16-word of W . Since all V α , α ∈ D, are simple current V 0 -submodules, the fusion product W α := V α V 0 W is again irreducible. It is clear that the binary 1/16-word of W α is α + β so that all W α , α ∈ D, are inequivalent V 0 -modules. Therefore, there exists a unique untwisted or Z2 -twisted V -module structure on Ind VV 0 W = V V 0 W = ⊕α∈D W α by Theorem 1. Since any element in the dual group D ∗ Zn2 /D ⊥ is realized as a Miyamoto involution τη associated to a codeword η ∈ Zn2 , the induced module Ind VV 0 W is indeed a τη -twisted V -module.
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Remark 5. By the corollary above and Proposition 2, we can compute the fusion rules of V -modules from those of VC -modules, Corollary 7. ([DGH, M3]) A framed VOA V with structure codes (C, D) is holomorphic if and only if C = D ⊥ . Proof. That a framed VOA having a structure code (D ⊥ , D) is holomorphic is proved in [M3] by showing that every module contains a vacuum-like vector (cf. [Li1]). The converse is also proved in [DGH] by using modular forms. Here we give another, rather representation-theoretical proof. Let V be a holomorphic framed VOA with structure codes (C, D) and the 1/16-word decomposition V = ⊕α∈D V α . Take any codeword δ ∈ D ⊥ . By the previous corollary, a VC -module MC (0, δ) can be uniquely extended to either an untwisted or Z2 -twisted V -module. As a V 0 -module, it is given by an induced module V MC (0, δ) = V α MC (0, δ). VC
α∈D
VC
By Lemma 7, the top weight of V α and that of V α VC MC (0, δ) are congruent modulo Z for all α ∈ D. Therefore, the induced module V VC MC (0, δ) is an irreducible untwisted V -module and thus isomorphic to V itself, as V is holomorphic. Then by considering the 1/16-word decomposition we see that MC (0, δ) = V 0 = MC (0, 0). Therefore, δ ∈ C by Lemma 3 and hence D ⊥ = C. 5.2. Construction of a framed VOA. In [M3, Y2], certain constructions of a framed VOA are discussed. Assume the following: (1) (C, D) is a pair of even linear codes of Zn2 such that
(1-i) C ⊂ D ⊥ , (1-ii) for each α ∈ D, there is a subcode E α ⊂ Cα such that E α is a direct sum of the [8,4,4]-Hamming codes. 0 (2) V is a code VOA associated to C. (3) {V α | α ∈ D} is a set of irreducible V 0 -modules such that (3-i) the 1/16-word of V α is α, (3-ii) all V α , α ∈ D, have integral top weights, (3-iii) the fusion product V α V 0 V β contains V α+β for all α, β ∈ D. Then it is shown in [M3, Y2] that the space V := ⊕α∈D V α forms a framed VOA with structure codes (C, D). Instead of the condition (1-ii), assume that (1-iii) for each α ∈ D, Cα contains a doubly even self-dual subcode w.r.t. α. Then we have already shown in Lemma 8 that V 0 ⊕ V α forms a framed VOA. So by the extension property of simple current extensions in Theorem 2, we can again show that V = ⊕α∈D V α forms a framed VOA with structure codes (C, D) under the other conditions. The key idea in [M3, Y2] is to use a special symmetry of the code VOA associated to the [8,4,4]-Hamming code to form a minimal Z2 -graded extension V 0 ⊕ V α . Thanks to Lemma 8, we can transcend this step without the [8,4,4]-Hamming code. Theorem 8. With reference to the conditions (1)–(3) above, assume the condition (1-iii) instead of (1-ii). Then V = ⊕α∈D V α forms a framed VOA with structure codes (C, D).
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Proof. Let {α i | 1 ≤ i ≤ r } be a linear basis of D and set D ( j) := SpanZ2 {α j | 1 ≤ j ≤ i}. By induction on i, we show that the space V [i] = ⊕α∈D (i) V α forms a framed VOA with structure codes (C, D (i) ). The case i = 0 is trivial, and the case i = 1 is done in Lemma 8. Now assume that V [i] forms a framed VOA for i ≥ 1. Then we have two i+1 simple current extensions V [i] = ⊕α∈D (i) V α and V 0 ⊕ V α . Applying Theorem 2 to a α (i+1) 0 set {V | α ∈ D } of simple current V -modules, we obtain a D (i+1) -graded simple current extension V [i + 1] = ⊕α∈D (i+1) V α of V 0 . Repeating this, finally we shall obtain the desired framed VOA structure on V [r ] = ⊕α∈D V α . We can also generalize Theorem 7.4.9 of [Y2] as follows: Theorem 9. Let V = ⊕α∈D V α be a framed VOA with structure codes (C, D). For any even subcode E such that C ⊂ E ⊂ D ⊥ , the space IndCE V := Ind VVCE V α = VE V α α∈D
α∈D
VC
forms a framed VOA with structure codes (E, D). Proof. The idea of the proof is almost the same as that of Theorem 7.4.9 of [Y2]. Let {γ i + C | 1 ≤ i ≤ r } be a transversal for C in E. It is clear that VE = ⊕ri=1 VC+γ i is an E/C-graded simple current extension of VC by Proposition 1. First, we show that V α is uniquely extended to an untwisted VE -module. For this, it suffices to show that VC+γ i VC V α , 1 ≤ i ≤ r , are inequivalent VC -modules. Assume VC+γ VC V α VC+δ VC V α . It follows from a given framed VOA structure and Theorem 6 that V α VC V α V 0 VC . Since the fusion product is commutative and associative, by multiplying both sides of VC+γ VC V α VC+δ VC V α by V α with respect to the fusion product, we have VC+γ VC+γ V α V α VC+δ V α V α VC+δ . VC
VC
VC
VC
Thus, γ ≡ δ mod C and hence all VC+γ i VC V α , 1 ≤ i ≤ r , are inequivalent VC -modules. Since E ⊂ D ⊥ , the top weight of V α and that of VC+γ i VC V α are congruent modulo Z by Lemma 7. Therefore, V α is uniquely extended to an irreducible untwisted VE -module Ind VVCE V α = VE VC V α by Theorem 1. Since all Ind VVCE V α , α ∈ D, are E/C-stable VE -modules, we have the fusion rule
Ind VVCE V α Ind VVCE V β Ind VVCE V α V β Ind VVCE V α+β VE
VC
by Proposition 2. Therefore, the space Ind VVCE V =
α∈D
Ind VVCE V α
forms a framed VOA with structure codes (E, D) by Theorem 8.
By Theorem 6, its corollaries and Theorem 8, a pair of structure codes (C, C ⊥ ) of a holomorphic framed VOA satisfies the following conditions: Condition 1 (F-admissible condition).
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(1) The length of C is divisible by 16. (2) C is even, every codeword of C ⊥ has a weight divisible by 8, and C ⊥ ⊂ C. (3) For any α ∈ C ⊥ , the subcode Cα of C contains a doubly even self-dual subcode w.r.t. α. For simplicity, we will call a code C F-admissible if it satisfies Condition 1. Indeed, we can construct a holomorphic framed VOA starting from an F-admissible code. Remark 6. A linear code C is F-admissible if and only if its dual C ⊥ satisfies the following three conditions: (i) the length of C ⊥ is divisible by 16, (ii) C ⊥ contains the all-one vector, (iii) C ⊥ is triply even, that is, wt(α) is divisible by 8 for any α ∈ C ⊥ . Let D satisfy the conditions (i), (ii) and (iii) above. Then for any α, β ∈ D, the weight of their intersection α · β is divisible by 4 and so α · D is doubly even. Then there exists a doubly even code E containing α · D such that E is self-dual w.r.t. α by Theorem 5. For any δ ∈ (α · D)⊥α , we have δ, D = δ · α, D = δ, α · D = 0, showing E ⊂ (α · D)⊥α ⊂ (D ⊥ )α . Therefore, D ⊥ is F-admissible. Let C be an F-admissible code. Then the all-one vector 1 = (11 . . . 1) is contained in C ⊥ . Since n = wt(1) is divisible by 16, all irreducible VC -modules with the 1/16-word 1 have integral top weights. Let V 1 be an irreducible VC -module with the 1/16-word 1. Then V 1 is a self-dual simple current VC -module. Lemma 10. Let C be an F-admissible code. For α ∈ C ⊥ , α ∈ {0, 1}, there exists an irreducible VC -module W α such that τ (W α ) = α and both W α and the fusion product V 1 VC W α have integral top weights. Proof. Clearly, we can find an irreducible VC -module X such that τ (X ) = α and X has an integral top weight. The 1/16-word of the fusion product V 1 VC X is then 1 + α and its weight is divisible by 8. Thus, the top weight of V 1 VC X is in either Z or Z + 1/2. In the former case, we just set W α = X . If the top weight is in Z + 1/2, we take a codeword δ ∈ (Zn2 )α such that δ is odd. By Lemma 7, the VC -module (V 1 VC X ) VC MC (0, δ) has an integral top weight. Set W α = X VC MC (0, δ). Since supp(δ) ⊂ supp(α), the top weight of X is congruent to that of W α modulo Z by Lemma 7, which is integral. Moreover, V 1 VC W α V 1 VC (X VC MC (0, δ)) (V 1 VC X ) VC MC (0, δ) also has an integral top weight as desired. Proposition 8. Let C be an F-admissible code and D a proper subcode of C ⊥ containing 1. Suppose that we have a framed VOA V = ⊕α∈D V α with structure codes (C, D). Then for β ∈ C ⊥ \D, there exists a self-dual simple current V -module W such that W has the 1/16-word decomposition W = ⊕α∈D W α+β and V˜ = V ⊕ W forms a framed VOA with structure codes (C, D + β). Proof. By the previous lemma, we can take an irreducible VC -module W β such that τ (W β ) = β and both W β and V 1 VC W β are of integral weights. Since β ∈ C ⊥ , it follows from (3) of Condition 1 that W β is a self-dual simple current VC -module. Then the induced module Ind VV 0 W β = ⊕α∈D V α VC W β is an irreducible τη -twisted V -module for some η ∈ Zn2 by Corollary 6. If η ∈ D ⊥ , then the space V ⊕ Ind VV 0 W β forms a framed VOA with structure codes (C, D + β) by Theorem 8.
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If η ∈ D ⊥ , then τη is not trivial. Set D + := {α ∈ D | α, η = 0} and D − := {α ∈ D | α, η = 1}. Then D = D + D − and D ± = ∅. By our choice of W β , the all-one vector 1 is in D + so that η is an even codeword. We set V ± := V α , W ± := V α Wβ. α∈D ±
α∈D ±
VC
Then all V ± , W ± are irreducible V + -modules. The top weight of W + is integral but the top weight of W − is in Z + 1/2. We shall deform W − so that it has an integral top weight also. Since [(D + )⊥ ∩ β⊥ , D ⊥ ∩ β⊥ ] = 2, there exists a codeword γ ∈ (D + )⊥ ∩ β⊥ such that γ , D − = 1 mod 2. Then it follows from Corollary 6 that W˜ ± := VC+γ W ± VC
are irreducible untwisted V + -modules. Moreover, by our choice of γ , both of W˜ ± have integral top weights since the top weight of W˜ + is congruent to γ , γ + β/2 modulo Z, whereas the top weight of W˜ − is congruent to γ , β + γ + D − /2 + 1/2 modulo Z. Therefore, by Theorem 8, we have a framed VOA structure on V˜ := V + ⊕ V − ⊕ W˜ + ⊕ W˜ − with structure codes (C, D + β). Now setting W := W˜ + ⊕ W˜ − , we have the desired extension of V . This completes the proof. Remark 7. In the proof above, we can construct another extension of V + which also has the structure codes (C, D + β) in the following way. Take a codeword γ ∈ D ⊥ with γ , β = 1, which is possible as [D ⊥ : D ⊥ ∩β⊥ ] = 2, and set V˜ − = VC+γ VC V − and W˜ − = VC+γ VC W − . Then one can similarly verify that the space V + ⊕ V˜ − ⊕W + ⊕ W˜ − also forms a framed VOA with structure codes (C, D + β). Theorem 10. There exists a holomorphic framed VOA with structure codes (C, C ⊥ ) if and only if C is F-admissible, i.e., C satisfies Condition 1. Proof. Let {α1 , . . . , αr } be a linear basis of C ⊥ with α1 = 1 and set D[i] := SpanZ2 {α j | 1 ≤ j ≤ i} for 1 ≤ i ≤ r . By Lemma 8 we can construct a framed V [1] := VC ⊕ V 1 with structure codes (C, D[1]). By Proposition 8, we can construct a framed VOA V [2] with structure codes (C, D[2]) which is a Z2 -graded simple current extension of V [1]. Recursively, we can construct a Z2 -graded simple current extension V [i + 1] of V [i] which has structure codes (C, D[i + 1]) and we shall obtain a holomorphic framed VOA V [r ] with structure codes (C, D[r ]) = (C, C ⊥ ). Remark 8. Condition 1, especially (2) and (3), give quite strong restrictions on a code C. Roughly speaking, C must be much bigger than its dual C ⊥ by (3) of Condition 1. In addition, if we assume that the minimum weight of C is greater than 2, then the corresponding framed VOA may have a finite full automorphism group (cf. [LSY, Corollary 3.9]). It seems possible to classify all F-admissible codes C if the length is small. It suggests a possibility for classifying all holomorphic framed VOAs of small central charge. The most interesting (and the first non-trivial) case would be the classification of c = 24 holomorphic framed VOAs. In fact, one can prove that the moonshine vertex
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operator algebra V is the unique holomorphic framed VOA of central charge 24 whose weight one subspace is trivial, which is a variant of the famous uniqueness conjecture of the moonshine vertex operator algebra proposed in [FLM] (see also [DGL]). The key point is that the structure codes of V (or any holomorphic framed VOA V of central charge 24 and V1 = 0) are closely related to those of the Leech lattice VOA V . If V = ⊕α∈C ⊥ V α with V 0 VC is a holomorphic framed VOA of central charge 24 and V1 = 0, then the minimal weight of C is greater than or equal to 4. In this case, for any δ ∈ Z48 2 of weight 2, the τδ -twisted orbifold construction yields a VOA
V α ⊕ VC+δ V α , D = {α ∈ C ⊥ | α, δ = 0}, V (τδ ) = VC
α∈D
which is isomorphic to the Leech lattice V and a pair (C (δ + C), D) will be the structure codes of V . Note that the weight one subspace of V (τδ ) forms an abelian Lie algebra with respect to the bracket [a, b] = a(0) b. We shall give more details on this point in our next work [LY]. 6. Frame Stabilizers and Order Four Symmetries In Sect. 5, we have seen that structure codes (C, D) of a framed VOA V = ⊕α∈D V α satisfy certain duality conditions. The main property is that for any α ∈ D, the subcode Cα contains a doubly even self-dual subcode w.r.t. α and V α is a simple current V 0 -module. However, it is shown in Corollary 4 that V α is a simple current module without the assumption on the doubly even property. In this section, we shall discuss the role of the doubly even property. It turns out that by relaxing the doubly even property, we can obtain a refinement of the 1/16-word decomposition and define an automorphism of order four in the pointwise frame stabilizer. We begin by defining the frame stabilizer and the pointwise frame stabilizer of a framed VOA. Definition 4. Let V be a framed VOA with a frame F = Vir(e1 ) ⊗ · · · ⊗ Vir(en ). The frame stabilizer of F is the subgroup of all automorphisms of V which stabilizes the frame F setwise. The pointwise frame stabilizer is the subgroup of Aut(V ) which fixes F pointwise. The frame stabilizer and the pointwise frame stabilizer of F are denoted pt by StabV (F) and StabV (F), respectively. Let (C, D) be the structure code of V with respect to F, i.e., V = V α , τ (V α ) = α and V 0 = VC . α∈D pt For any θ ∈ StabV (F), it is easy to see that τei = τθei = θ τei θ −1 and thus θ centralizes τe1 , . . . , τen . Therefore, the group τ (Zn2 ) = τe1 , . . . , τen generated by the pt τ -involutions is a central subgroup of StabV (F) isomorphic to Zn2 /D ⊥ . In addition, we have θ V α = V α for all α ∈ D and hence θ |V 0 is an automorphism of V 0 .
The following results can be proved easily using the fusion rules. Lemma 11. Let V = ⊕α∈D V α be a framed VOA. (1) Let φ ∈ Aut(V 0 ) such that φ| F = id F . Then φ ∈ σ (Zn2 ) = σe1 , . . . , σen . (2) Let g ∈ Aut(V ) such that g|V 0 = id V 0 . Then g ∈ τ (Zn2 ) = τe1 , . . . , τen .
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Proof. (1) Consider the 1/2-word decomposition V0 = L(1/2, β1 /2) ⊗ · · · ⊗ L(1/2, βn /2) β=(β1 ,...,βn )∈C
of V 0 as an F = Vir(e1 ) ⊗ · · · ⊗ Vir(en )-module. Since φ| F = id F , it follows from Schur’s lemma that φ|V 0 acts on L(1/2, β1 /2) ⊗ · · · ⊗ L(1/2, βn /2) by a non-zero scalar aα for each α ∈ C. Moreover, it follows from the fusion rules of L(1/2, 0)-modules in (3.1) that aα aβ = aα+β for all α, β ∈ C. Thus the association C α → aα ∈ C defines a character of C and hence there is a codeword ξ ∈ Zn2 such that aα = (−1)ξ,α . Now it is easy to see that φ|V 0 is realizable as a product of σei , 1 ≤ ei ≤ n, that is, φ|V 0 = σξ . (2) Since each V α , α ∈ D, is an irreducible V 0 -module, it follows from Schur’s lemma that g acts on V α by a non-zero scalar tα ∈ C. Then again by the fusion rules of L(1/2, 0)-modules in (3.1) we have tα tβ = tα+β so that the map α → tα defines a character of D. Therefore, there exists a codeword η ∈ Zn2 such that g = τη ∈ τe1 , . . . , τen . As a corollary, we have the following theorem. Theorem 11. Let V be a framed VOA with a frame F = Vir(e1 ) ⊗ · · · ⊗ Vir(en ). For pt any θ ∈ StabV (F), there exist ξ and η ∈ Zn2 such that θ |V 0 = σξ
and
θ 2 = τη .
In particular, we have θ 4 = 1. pt
Let θ ∈ StabV (F). Then θ |V 0 = σξ for some ξ ∈ Zn2 . That means θ is an extension of a σ -involution on V 0 to the whole framed VOA V . In this section, we shall give a necessary and sufficient condition on whether a σ -involution σξ can be extended to the whole V . Our argument is based on the representation theory of code VOAs developed in Sect. 4 and 5. pt First let us consider θ ∈ StabV (F) such that θ |V 0 = σξ = id V 0 , i.e., ξ ∈ / C ⊥ . Set 0 1 0 C := {α ∈ C | ξ, α = 0} and C := {α ∈ C | ξ, α = 1}. Then C is a subcode of C, [C : C 0 ] = 2 and C = C 0 C 1 . Note also that VC 0 is fixed by θ and θ acts by −1 on VC 1 . In other words, V 0 = VC 0 ⊕ VC 1 is the eigenspace decomposition of θ on V 0 . Now assume that θ 2 = τη for some η ∈ Zn2 . For each non-zero α ∈ D, it is clear that V 0 ⊕ V α is a subalgebra of V and θ stabilizes V 0 ⊕ V α . If α ∈ D ∩ η⊥ , then θ 2 acts as an identity on V 0 ⊕ V α and the eigenvalues of θ on V α are ±1. Let V α+ and V α− be the eigenspaces of θ with eigenvalues +1 and −1, respectively. Note that both V α± are non-zero inequivalent irreducible V 0+ -submodules. For if V α+ = 0, then V 0− · V α− = V α+ = 0, which contradicts Proposition 11.9 of [DL]. Since the subalgebra V 0 ⊕ V α = V 0+ ⊕ V 0− ⊕ V α+ ⊕ V α− affords a faithful action of a group Z2 × Z2 of order 4, V α± are inequivalent irreducible V 0+ -submodules by the quantum Galois theory [DM1]. √ If α ∈ D\η⊥ , then θ 2 = −1 on V α and the eigenvalues of θ on V α are ± −1. Let √ V α± be the eigenspace of θ of eigenvalues of ± −1 on V α . Then V α = V α+ ⊕ V α− . V α± are again non-zero inequivalent irreducible V 0+ -submodules. The argument above actually shows that θ is of order 2 if and only if η, D = 0, i.e., η ∈ D ⊥ . By the observation above, we have
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Lemma 12. For any α ∈ D, let V α± be defined as above. Then the dual V 0+ -module (V α± )∗ is isomorphic to V α± if and only if α ∈ η⊥ . Otherwise, (V α± )∗ is isomorphic to V α∓ . Proof. Since any framed VOA is self-dual, the sub VOA V 0 ⊕ V α of V is also self-dual. Since V α± · V α± = V 0+ if and only if α ∈ η⊥ , the duality is as in the assertion. pt
We have shown that for any θ ∈ StabV (F)\τ (Zn2 ), |θ | = 2 if and only if all irreducible V 0+ -submodules of V are self-dual, and otherwise |θ | = 4. We rewrite this condition in terms of the structure codes as follows. Lemma 13. Let θ ∈ StabV (F) be such that θ |V 0 = σξ and θ 2 = τη for some ξ ∈ Zn2 \C ⊥ and η ∈ Zn2 . Set C 0 = {α ∈ C | ξ, α = 0} and C 1 = {α ∈ C | ξ, α = 1}. pt
(1) For α ∈ D ∩ η⊥ , (C 0 )α contains a doubly even self-dual subcode w.r.t. α. (2) For α ∈ D\η⊥ , (C 0 )α contains a self-dual subcode w.r.t. α, but (C 0 )α does not contain any doubly even self-dual subcode w.r.t. α. Proof. (1) For α ∈ D ∩ η⊥ , let V α = V α+ ⊕ V α− be the eigenspace decomposition such that θ acts on V α± by ±1. In this case the subspace V 0+ ⊕ V α+ forms a framed sub VOA of V . By Proposition 7, (C 0 )α contains a doubly even self-dual subcode w.r.t. α. (2) For α ∈ D\η⊥ ,√ let V α = V α+ ⊕ V α− be the eigenspace decomposition such α± that θ acts on V by ± −1. In this case the restriction of θ on the sub VOA V 0 ⊕ V α = V 0+ ⊕ V 0− ⊕ V α+ ⊕ V α− is of order 4. By the quantum Galois theory [DM1], V α+ and V α− are inequivalent irreducible V 0+ = VC 0 -modules. By Lemma 12, V α+ and V α− are dual to each other. Therefore, by Proposition 5, any maximal self-orthogonal subcode of (C 0 )α is not doubly even. Let H be a doubly even self-dual subcode of Cα w.r.t. α and H 0 a maximal self-orthogonal subcode of (C 0 )α . Since (C 0 )α does not contain a doubly even self-dual subcode w.r.t. α, (C 0 )α is a proper subgroup of Cα so that [Cα : (C 0 )α ] = 2. It follows from Theorem 4 that V α is a direct sum of [C : Cα ] inequivalent V (0)-submodules with the multiplicity [Cα : H ]. Similarly, each of V α± is a direct sum of [C 0 : (C 0 )α ] inequivalent irreducible F-submodules with the multiplicity [(C 0 )α : H 0 ]. Since V α+ and V α− are dual to each other, they are isomorphic as F-modules. Therefore, by counting multiplicity of irreducible F-submodules of V α and V α± , one has [Cα : H ] = 2[(C 0 )α : H 0 ]. Combining with |Cα | = 2|(C 0 )α |, we obtain |H | = |H 0 | = 2wt(α)/2 . Therefore, H 0 is a self-dual subcode of (C 0 )α w.r.t. α. Lemma 14. Let C be an even code and β ∈ C ⊥ . Assume that V = VC ⊕ MC (β, γ ; ι) forms a framed VOA and C contains a subcode E with index two such that E contains a self-dual subcode w.r.t. β. Then V decomposes into a direct sum of four inequivalent simple current VE -submodules V = M E (0, 0) ⊕ M E (0, δ) ⊕ M E (β, γ ; j ) ⊕ M E (β, γ + δ; j ), where δ is an element of C such that C = E (E + δ) and j : E ∩ E ⊥ → C∗ is a map such that j | E∩C ⊥ = ι and (α, j (α)) · (β, j (β)) = (α + β, j (α + β)) ∈ πC−1∗ (E) for all α, β ∈ E ∩ E ⊥ . Moreover, all irreducible VE -submodules of V are self-dual if and only if E contains a doubly even self-dual subcode w.r.t. β.
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Proof. Let δ ∈ C be such that C = E (E + δ). Then the decomposition VC = M E (0, 0) ⊕ M E (0, δ) is obvious. Let H be a self dual subcode of E β w.r.t. β. Then H is still a maximal self-orthogonal subcode of Cβ . Let j : H → C∗ be an extension of ι : C ∩ C ⊥ → C∗ such that (α, j (α)) · (β, j (β)) = (α + β, j (α + β)) for all α, β ∈ H . Then j | E∩C ⊥ = ι and the decomposition MC (β, γ ; ι) = M E (β, γ ; j )⊕ M E (β, γ +δ; j ) follows from Corollary 2. That all irreducible VE -submodules are simple currents follows from Corollary 4. If E contains a doubly even self-dual subcode w.r.t. β, then all irreducible VE -submodules of V are self-dual by Proposition 5. Conversely, if all irreducible VE -submodules of V are self-dual, then VE ⊕ M E (β, γ ; j ) forms a sub VOA of V so that E contains a doubly even self-dual subcode w.r.t. β by Proposition 7. This completes the proof. Theorem 12. Let V be a framed VOA with a frame F = Vir(e1 ) ⊗ · · · ⊗ Vir(en ) and let (C, D) be the corresponding structure codes. For a codeword ξ ∈ Zn2 \C ⊥ , there exists pt θ ∈ StabV (F) such that θ |V 0 = σξ if and only if α · ξ ∈ C for all α ∈ D. Moreover, |θ | = 2 if and only if wt(α · ξ ) ≡ 0 mod 4 for all α ∈ D, and otherwise |θ | = 4. Proof. Let θ ∈ StabV (F) be such that θ |V 0 = σξ with ξ ∈ Zn2 \C ⊥ . Set C 0 := C ∩ ξ ⊥ and C 1 := C\C 0 . Then by Lemma 13, (C 0 )α contains a self-dual subcode w.r.t. α for any α ∈ D. Since pt
(C 0 )α = {β ∈ C | β, ξ = 0 and supp(β) ⊂ supp(α)} and β, ξ = β, α ·ξ = 0 for all β ∈ (C 0 )α , α ·ξ ∈ ((C 0 )α )⊥ for all α ∈ D. Therefore, α · ξ is contained in all self-dual subcodes of (C 0 )α w.r.t. α and hence α · ξ ∈ (C 0 )α ⊂ C as claimed. Conversely, assume that a codeword ξ ∈ Zn2 \C ⊥ satisfies α · ξ ∈ C for all α ∈ D. Then C 0 = C ∩ ξ ⊥ is a proper subcode of C with index 2. By definition, α · ξ ∈ ((C 0 )α )⊥ for all α ∈ D. Therefore, any maximal self-orthogonal subcode of (C 0 )α contains α · ξ . Set D 0 := {α ∈ D | wt(α · ξ ) ≡ 0 mod 4} and D 1 := {α ∈ D | wt(α · ξ ) ≡ 2 mod 4}. It is clear that D = D 0 D 1 . If α ∈ D 0 , then there exists a doubly even self-dual subcode of (C 0 )α w.r.t. α. Let H be a doubly even self-dual subcode of Cα , which exists by Proposition 7. If H is contained in (C 0 )α , then we are done. If not, then α · ξ ∈ H and H ∩ (C 0 )α = H ∩ ξ ⊥ is a subcode of H with index 2 so that (H ∩ ξ ⊥ ) (H ∩ ξ ⊥ + α · ξ ) gives a doubly even self-dual subcode of (C 0 )α w.r.t. α. Similarly, we can show that (C 0 )α contains a self-dual subcode w.r.t. α for any α ∈ D 1 . But in this case any selfdual subcode of (C 0 )α w.r.t. α is not doubly even, as it always contains α · ξ . We have shown that for each α ∈ D, (C 0 )α contains a self-dual subcode w.r.t. α so that one has a VC 0 -module decomposition V α = V α,1 ⊕ V α,2 and V α, p , p = 1, 2, are simple current VC 0 -submodules by Lemma 14. Let {α 1 , . . . , αr } be a linear basis of D 0 . For each i, 1 ≤ i ≤ r , choose an irreducible i i VC 0 -submodule U α of V α arbitrary. Then for α = α i1 + · · · + α ik ∈ D 0 , set U α := U α
i1
· · · Uα . ik
VC 0
VC 0
(6.1)
Since all U α , 1 ≤ i ≤ r , are simple current self-dual VC 0 -modules, U α is uniquely defined by (6.1) for all α ∈ D 0 . Note that U 0 = VC 0 . Since ⊕α∈D 0 V α is a sub VOA i
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of V , U α are irreducible VC 0 -submodules of V α for all α ∈ D 0 . Therefore, we obtain a framed sub VOA U := ⊕α∈D 0 U α of V with structure codes (C 0 , D 0 ). It is easy to see that V α = U α ⊕ (VC 1 VC 0 U α ) for α ∈ D 0 by Lemma 14. If D = D 0 , then we have V = U ⊕ (VC 1 VC 0 U ) as a VC 0 -module. In this case we define a linear automorphism θξ on V by θξ :=
1 −1
on U, on VC 1 VC 0 U. pt
Then it follows from Lemma 14 and Proposition 6 that θξ ∈ StabV (F) and θξ |V 0 = σξ . Therefore, σξ can be extended to an involution on V . If D = D 0 , then D = D 0 D 1 with D 1 = ∅. In this case, take one β ∈ D 1 and an irreducible VC 0 -submodule W β of V β . Since W β and all U α , α ∈ C 0 , are simple current VC 0 -modules, we have a VC 0 -module decomposition V α+β = (U α W β ) ⊕ (VC 1 U α W β ) VC 0
VC 0
VC 0
of V α+β for all α ∈ C 0 by Lemma 14. Since (C 0 )α+β contains no doubly even self-dual subcode w.r.t. α + β, the decomposition V 0 ⊕ V α+β = VC 0 ⊕ VC 1 ⊕ (U α W β ) ⊕ (VC 1 U α W β ) VC 0
VC 0
VC 0
induces an order four automorphism on a sub VOA V 0 ⊕ V α+β of V by Lemma 14 and Proposition 6. Set W := Uα Wβ. VC 0
α∈C 0
Then we have obtained the following decomposition of V as a VC 0 -module: V = U ⊕ (VC 1 U ) ⊕ W ⊕ (VC 1 W ). VC 0
We define a linear automorphism θξ on V ⎧ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨−1 θξ := √ ⎪ ⎪ −1 ⎪ ⎪ ⎪ ⎪ ⎩−√−1
VC 0
by on U, on VC 1 VC 0 U, on W, on VC 1 VC 0 W. pt
Then it follows from the argument above that θξ ∈ StabV (F) and θξ |V 0 = σξ . Therefore, σξ gives rise to an automorphism of order 4. pt Summarizing, we have shown that there exists θ ∈ StabV (F) such that θ |V 0 = σξ if and only if α · ξ ∈ C for all α ∈ D. It remains to show that for such θ , |θ | = 2 if and only if wt(α · ξ ) ≡ 0 mod 4. But this is almost obvious by the preceding argument.
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Remark 9. Let V = ⊕α∈D V α be a framed VOA with structure codes (C, D). It was conjectured in [M3, Conjecture 1] that for any codeword β ∈ C, the σ -type involution σβ ∈ Aut(V 0 ) can be extended to an automorphism of V , that means there exists pt g ∈ StabV (F) such that g|V 0 = σβ . By the theorem above, we know that this is not correct; we have to take a codeword β such that α · β ∈ C for all α ∈ D. Motivated by Theorem 12, we define P := {ξ ∈ Zn2 | α · ξ ∈ C for all α ∈ D}. It is clear that P is a linear subcode of C. Moreover, we have Lemma 15. C ⊥ ⊂ P. Proof. Let δ ∈ C ⊥ . For α ∈ D, one has δ, Cα = 0 by definition and hence α·δ, Cα = 0. Since Cα contains a self-dual subcode w.r.t. α by Theorem 6, α · δ ∈ Cα ⊂ C. pt
For each codeword ξ ∈ P, there exists θξ ∈ StabV (F) such that θξ |V 0 = σξ by Theorem 12. However, the construction of θξ in the proof of Theorem 12 is not unique since we have to choose a linear basis of D 0 and irreducible VC 0 -submodules. Nevertheless, the following holds. pt
Lemma 16. Let θ, φ ∈ StabV (F) such that θ |V 0 = φ|V 0 = σξ . Then φ = θ τη for some η ∈ Zn2 . Proof. Since θ |V 0 = φ|V 0 , we have θ −1 φ|V 0 = id V 0 . By Lemma 11, there exists η ∈ Zn2 such that θ −1 φ = τη and hence φ = θ τη as desired. In other words, θξ is only determined modulo τ -involutions. We have also seen in Lemma 11 that θξ ∈ τ (Zn2 ) if and only if ξ ∈ C ⊥ . Since C ⊥ ⊂ P by Lemma 15, the association ξ + C ⊥ → θξ τ (Zn2 ) defines a group isomorphism between P/C ⊥ and pt StabV (F)/τ (Zn2 ). Therefore, we have the following central extension: pt
pt
1 −→ τ (Zn2 ) −→ StabV (F) −→ StabV (F)/τ (Zn2 ) −→ 1 ↓
||
↓
1 −→ Zn2 /D ⊥ −→ StabV (F) −→
P/C ⊥
pt
−→ 1
pt
The commutator relation in StabV (F) can also be described as follows. pt
Theorem 13. For ξ 1 , ξ 2 ∈ P, let θξ i , i = 1, 2, be extensions of σξ i to StabV (F). Then [θξ 1 , θξ 2 ] = 1 if and only if α · ξ 1 , α · ξ 2 = 0 for all α ∈ D. Proof. Since the case θξ 1 ∈ θξ 2 τ (Zn2 ) is trivial, we assume that σξ 1 = σξ 2 . For i = 1, 2, i
1
2
i
set C 0,ξ := {α ∈ C | α, ξ i = 0} and E := C 0,ξ ∩ C 0,ξ . Then C 0,ξ are subcodes of C with index 2 and E is a subcode of C with index 4. Let δ 1 , δ 2 ∈ C be such that i C 0,ξ = E (E + δ i ). By definition, α · ξ 1 , α · ξ 2 ∈ (E α )⊥ for all α ∈ D so that E α contains a self-dual subcode w.r.t. α if and only if α · ξ 1 , α · ξ 2 = 0. We have seen that θξ i acts semisimply on each V α , α ∈ D, with two eigenvalues, and these eigenspaces are inequivalent irreducible VC 0,ξ i -submodules. For an irreducible VE -submodule W of V α , the subspace W + (VE+δi · W ) forms a VC 0,ξ i -submodule so that θξ i acts on W by an eigenvalue. Therefore, θξ 1 commutes with θξ 2 if and only if V α splits into a direct
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sum of 4 irreducible VE -submodules for all α ∈ D. Let m α be the number of irreducible VE -submodules of V α . For α ∈ D, let Hα and Hα0 be maximal self-orthogonal subcodes of Cα and E α , respectively. By the structure of irreducible modules over a code VOA shown in Theorem 4, V α is a direct sum of [C : H ] irreducible F-submodules. Moreover, again by Theorem 4 any irreducible VE -submodule of V α is a direct sum of [E : Hα0 ] irreducible F-submodules. By counting the number of irreducible F-submodules of V α , we have m α |Hα | = 4|Hα0 | as [C : E] = 4. Thus, E α contains a self-dual subcode w.r.t. α if and only if m α = 4. Hence, θξ 1 commutes with θξ 2 if and only if α · ξ 1 , α · ξ 2 = 0. pt
We have shown that the structure of StabV (F) is determined by Theorems 12 and 13 only in terms of the structure codes (C, D). Remark 10. In [Y2, Y3], one of the authors has shown that for any Ising vector e ∈ V , we have no automorphism g ∈ Aut(V ) such that g restricted on (V )τe is equal to σe . Thanks to Theorem 12, we can give a simpler proof of this. As shown in [DMZ], the moonshine VOA V is framed. Take any Ising frame F = Vir(e1 ) ⊗ · · · ⊗ Vir(e48 ) of V pt and set ξ = (1047 ) ∈ Z48 2 . Since ξ is odd, there is no extension of σξ = σe1 to Stab V (F) by Theorem 12. Since all the Ising vectors of V are conjugate under Aut(V ) = M (cf. [C, M1]), e and e1 are conjugate. Therefore, there is no extension of σe to V . At the end of this section, we give a brief description of the frame stabilizer StabV (F). pt Its structure is also discussed in [DGH]. It is clear that StabV (F) is a normal subgroup of StabV (F). Let g ∈ StabV (F). Then g induces a permutation µg ∈ Sn on the set of Ising vectors {e1 , . . . , en } of F, namely gei = eµg (i) . Since g preserves the 1/16-word decomposition V = ⊕α∈D V α , it follows that gV α = V µg (α) with µg (α) = (αµg (1) , . . . , αµg (n) ). In particular, g restricted on V 0 defines an element of Aut(V 0 ) = Aut(VC ) which is a lift of Aut(C). Therefore, every element of StabV (F) is a lift of Aut(C) ∩ Aut(D). Conversely, we know that for any µ ∈ Aut(C), there exists µ˜ ∈ Aut(VC ) such that µe ˜ i = eµ(i) for 1 ≤ i ≤ n by Theorem 3.3 of [Sh]. It is shown in Lemma 3.15 of [SY] that if µ˜ lifts to an element of Aut(V ) then {(V α )µ˜ | α ∈ D} coincides with {V α | α ∈ D} as a set of inequivalent irreducible VC -modules. Therefore, there exists a lift of µ˜ ∈ Aut(VC ) = Aut(V 0 ) to an element of Aut(V ) if and only if the subgroup {V α | α ∈ D} of the group formed by all the simple current VC -module in the fusion algebra is invariant under the conjugation action of µ. ˜ If such a lift of µ˜ exists, it is pt unique modulo StabV (F). If µ˜ and µ˜ are two lifts of µ, µ˜ −1 µ˜ fixes F pointwise, showpt pt ing µ˜ −1 µ˜ ∈ StabV (F). Thus, the factor group StabV (F)/StabV (F) is isomorphic to a subgroup of Aut(C) ∩ Aut(D) which gives a slight refinement of (3) of Theorem 2.8 in [DGH]. The VC -module structure of (V α )µ˜ involves some extra information other than C and D, namely, if V α MC (α, γ ; ια ) for α ∈ D then (V α )µ˜ MC (µ−1 α, γ ; ιµ−1 α ) for some codeword γ ∈ Zn2 , and this γ depends not only on C and D but also on γ , ια and ιµ−1 α . We do not have a general result for the lifting property of Aut(C) ∩ Aut(D) at present. 7. 4A-Twisted Orbifold Construction Let V be the moonshine VOA constructed in [FLM]. In this section, we shall apply Theorem 12 to define a 4A-element of the Monster M = Aut(V ) and exhibit that the 4A-twisted orbifold construction of the moonshine VOA V will be V itself.
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By [DGH, M3], we can take an Ising frame F = Vir(e1 ) ⊗ · · · ⊗ Vir(e48 ) of V such that the associated structure codes (C, D) are as follows: C = D⊥ , D = SpanZ2 {(116 032 ), (032 116 ), (α, α, α) | α ∈ RM(1, 4)}, where RM(1, 4) ⊂ Z16 2 is the first order matrix ⎡ 1111 1111 ⎢1111 1111 ⎢ ⎢1111 0000 ⎣1100 1100 1010 1010
Reed-Muller code defined by the generator 1111 0000 1111 1100 1010
⎤ 1111 0000⎥ ⎥ 0000⎥ . 1100⎦ 1010
Note that 16 C = {(α, β, γ ) ∈ Z48 2 | α, β, γ ∈ Z2 are even and α + β + γ ∈ RM(2, 4)}.
(7.1)
Remark 11. The weight enumerator of RM(1, 4) is X 16 + 30X 8 Y 8 + Y 16 . Let V = ⊕α∈D (V )α be the 1/16-word decomposition. Set P := {ξ ∈ Z48 2 | α · ξ ∈ C for all α ∈ D}. pt
Then for each ξ ∈ P, one can define an automorphism θξ ∈ StabV (F) such that θξ |V 0 = σξ by Theorem 12. Note also that D = C ⊥ < P < C and σξ 1 = σξ 2 if and only if ξ 1 + ξ 2 ∈ C ⊥ = D. Lemma 17. Let C, D and P be defined as above. Then P = {(α, β, γ ) ∈ Z48 2 | α, β, γ ∈ RM(2, 4) and α + β + γ ∈ RM(1, 4)}, where RM(2, 4) = RM(1, 4)⊥ ⊂ Z16 2 is the second order Reed-Muller code of length 16. Proof. Set E := {(α, β, γ ) | α, β, γ ∈ RM(2, 4) and α + β + γ ∈ RM(1, 4)}. We shall first show that E ⊂ P. It is clear that if d1 ·ξ , d2 ·ξ ∈ C then (d1 +d2 )·ξ = d1 ·ξ +d2 ·ξ ∈ C. Thus, we only need to show that d · ξ ∈ C for ξ ∈ E and d in a generating set of D. 1 2 3 Let ξ = (ξ 1 , ξ 2 , ξ 3 ) ∈ E with each ξ i ∈ Z16 2 . Then ξ , ξ , ξ ∈ RM(2, 4). Hence, α · ξ ∈ C for α = (116 032 ), (016 116 016 ) and (032 116 ) ∈ D. Take any two codewords β, γ ∈ RM(1, 4) with weight 8. Then the weight of β · γ is either 0, 4 or 8 by Remark 11. By the definition of E, we have ξ 1 + ξ 2 + ξ 3 ∈ RM(1, 4). Therefore, β, γ · (ξ 1 + ξ 2 + ξ 3 ) ≡ wt(β · γ · (ξ 1 + ξ 2 + ξ 3 )) ≡ 0 mod 2, and hence γ · (ξ 1 + ξ 2 + ξ 3 ) ∈ RM(1, 4)⊥ = RM(2, 4). Since ξ i ∈ RM(2, 4) = RM(1, 4)⊥ , all γ · ξ i , i = 1, 2, 3, are even codewords. Therefore, (γ , γ , γ ) · ξ = (γ · ξ 1 , γ · ξ 2 , γ · ξ 3 ) ∈ C for all γ ∈ RM(1, 4). As D is generated by the elements of the form: (116 032 ), (016 116 016 ), (032 116 ) and (γ , γ , γ ) with γ ∈ RM(1, 4), we have E ⊂ P. 1 2 3 Conversely, assume (α 1 , α 2 , α 3 ) ∈ P with α i ∈ Z16 2 . Then one has (α , α , α ) · β ∈ 16 32 16 16 16 32 16 i C for β = (1 0 ), (0 1 0 ) and (0 1 ) ∈ D so that α ∈ RM(2, 4) for i = 1, 2, 3. Moreover, for (γ , γ , γ ) ∈ D with γ ∈ RM(1, 4), (α 1 , α 2 , α 3 ) · (γ , γ , γ ) ∈ C is an even codeword. Then it follows from (7.1) that (α 1 + α 2 + α 3 ) · γ ∈ RM(2, 4) and thus α 1 + α 2 + α 3 ∈ RM(1, 4). Hence E = P.
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Take ξ = (1100 0000 1100 0000 0110 0000 0110 0000 1010 0000 1010 0000) ∈ Z48 2 . (7.2) Then ξ ∈ P by Lemma 17. Set D0 := {α ∈ D | wt(α · ξ ) ≡ 0 mod 4} and D1 := {α ∈ D | wt(α · ξ ) ≡ 2 mod 4}. It is easy to see that α = (116 ), ({14 04 }2 ), D0 = SpanZ2 (116 032 ), (032 116 ), (α, α, α) ({12 02 }4 ) or ({10}8 ) and D1 = ({18 08 }3 ) + D0 . Therefore, the index [D : D0 ] is 2 and in this case the involupt tion σξ ∈ Aut((V )0 ) can be extended to an automorphism θξ ∈ StabV (F) of order 4 by Theorem 12. We also set C 0 := {α ∈ C | α, ξ = 0} and C 1 := {α ∈ C | α, ξ = 1}. Let us consider a subgroup of D × C∗ defined by √ D˜ := (D0 × {±1}) D1 × ± −1 . (7.3) For (α, u) ∈ D × C∗ , set (V )(α,u) := {x ∈ (V )α | θξ x = ux}. Then we have a ˜ D-graded decomposition (V )(α,u) , (V )(0,1) = VC 0 , (V )(0,−1) = VC 1 , (7.4) V = (α,u)∈D˜
where VC 0 denotes the code VOA associated to C 0 and VC 1 is its module. Since (C 0 )α ˜ are contains a self-dual subcode w.r.t. α ∈ D by Lemma 13, all (V )(α,u) , (α, u) ∈ D, ˜ simple current VC 0 -modules by Corollary 4. Therefore, V is a D-graded simple current extension of a code VOA (V )(0,1) = VC 0 . By direct computation, it is not difficult to obtain the following lemma. Lemma 18. For any non-zero α ∈ D0 , the subset (C 1 )α is not empty. In other words, [Cα , (C 0 )α ] = 2. Remark 12. For a general framed VOA with structure codes (C, D), it is possible that the set (C 1 )α is empty for some non-zero α ∈ D. For example, we can take D = {(016 ), (18 08 ), (08 18 ), (116 )}, C = D ⊥ , ξ = (12 014 ) and α = (08 18 ). Then, Cα = (C 0 )α and (C 1 )α is empty. Note that (C, D) can be realized as the structure codes of the lattice VOA VE 8 (cf. [DGH]). Theorem 14. θξ is a 4A-element of the Monster. Proof. We shall compute the McKay-Thompson series of θξ defined by Tθξ (z) := tr V θξ q L(0)−1 , q = e2π
√
−1 z
.
Recall the notion of the conformal character of a module M = ⊕n≥0 Mn+h over a VOA V: ch M (q) := tr M q L(0)−c/24 =
∞ n=0
It is clear that
dimC Mn+h q n+h−c/24 .
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Tθξ (z) =
ch(V )(α,1) (q) −
α∈D 0
ch(V )(α,−1) (q)
α∈D 0
√ √ + −1 ch(V )(α,√−1) (q) − −1 ch(V )(α,−√−1) (q).
α∈D 1 α∈D 1 √ Let α ∈ D1 . Since (V )(α,± −1) are dual to each other, their conformal characters are the same. Let α ∈ D0 be a non-zero codeword. By Lemma 18, there exists a codeword in (C 1 )α . Then by Corollary 2 one sees that (V )(α,1) and (V )(α,−1) are isomorphic
F-modules. Therefore, they have the same conformal characters. Then Tθξ (z) = ch(V )(0,1) (q) − ch(V )(0,−1) (q) = ch VC0 (q) − ch VC1 (q) = 2ch VC0 (q) − ch VC (q). The conformal character of a code VOA can be easily computed. The following conformal characters are well-known (cf. [FFR]): ∞ ch L(1/2,0) (q) ± ch L(1/2,1/2) (q) = q −1/48 (1 ± q n+1/2 ). n=0
D⊥ and C 0
(D+ξ )⊥ , the weight enumerators of these codes are calculated
Since C = = by the MacWilliams identity [McS]:
1 WD (x + y, x − y), |D| 1 WD+ξ (x + y, x − y), WC 0 (x, y) = |D + ξ | WC (x, y) =
where WD (x, y) = x 48 + 3x 32 y 16 + 120x 24 y 24 + 3x 16 y 32 + y 48 , WD+ξ (x, y) = x 48 + 2x 36 y 12 + 3x 32 y 16 + 30x 28 y 20 + 184x 24 y 24 + 30x 20 y 28 +3x 16 y 32 + 2x 12 y 36 + y 48 . Now set f (x, y) := WD+ξ (x, y) − WD (x, y) = 2x 36 y 12 + 30x 28 y 20 + 64x 24 y 24 + 30x 20 y 28 + 2x 12 y 36 . Then one has Tθξ (z) = 2ch VC0 (q) − ch VC (q) = 2WC 0 (x, y) − WC (x, y)
! x=ch L(1/2,0) (q), y=ch L(1/2,1/2) (q)
! 1 = 7 WD+ξ (x + y, x − y) − WD (x + y, x − y) 2
∞ ∞ 1 −1 n+1/2 n+1/2 = 7q f (1 + q ), (1 − q ) 2 n=0 n=0 = q −1 + 276q + 2048q 2 + · · · . Therefore, θξ is a 4A-element of the Monster by [ATLAS].
x=ch L(1/2,0) (q), y=ch L(1/2,1/2) (q)
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Next, we shall construct the irreducible 4A-twisted V -module. Let us consider irreducible VC 0 -modules whose 1/16-word is ξ defined in (7.2). Lemma 19. (1) (C 0 )ξ is a self-dual subcode w.r.t. ξ . (2) For any γ ∈ Zn2 , the dual of MC 0 (ξ, γ ; ι) is isomorphic to MC 0 (ξ, γ + κ; ι) with κ = ({107 }2 032 ) ∈ Z48 2 . Proof. By a direct computation, one can show that Cξ = (C 0 )ξ is generated by the following generator matrix: ⎤ ⎡ 11000000 11000000 00000000 00000000 00000000 00000000 ⎢00000000 00000000 01100000 01100000 00000000 00000000⎥ ⎥ ⎢ ⎢00000000 00000000 00000000 00000000 10100000 10100000⎥ ⎢10000000 10000000 01000000 01000000 00000000 00000000⎥ ⎥ ⎢ ⎣10000000 10000000 00000000 00000000 10000000 10000000⎦ 11000000 00000000 01100000 00000000 10100000 00000000 From this, it is easy to see that (C 0 )ξ is a self-dual code w.r.t. ξ . If we set κ = (1000 0000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000) ∈ Z48 2 , (7.5) then supp(κ) ⊂ supp(ξ ) and we have (−1)α,κ = (−1)wt(α)/2 for all α ∈ (C 0 )ξ . Therefore, the dual of MC 0 (ξ, γ ; ι)∗ is isomorphic to MC 0 (ξ, γ + κ; ι) by Proposition 5. By the lemma above, we know that (C 0 )ξ is self-dual w.r.t. ξ and thus (C 0 )ξ equals its own radical. From now on, we shall fix a map ι : (C 0 )ξ → C∗ such that the section map (C 0 )ξ α → (α, ι(α)) ∈ πC−1∗ (C 0 ) is a group homomorphism. We shall simply denote MC 0 (ξ, γ ; ι) by MC 0 (ξ, γ ) and set W := MC 0 (ξ, 0). ˜ are inequivalent irreducible VC 0 -modules. Lemma 20. All (V )(α,u) VC0 W , (α, u) ∈ D, ˜ Since Proof. Suppose (V )(α,u) VC0 W (V )(β,v) VC0 W with (α, u), (β, v) ∈ D.
((V )(α,u) )∗ (V )(α,u)
−1
= (V )(α,u
−1 )
, we have
W = (V )(α,u
−1 )
VC0 (V )(α,u) VC0 W
= (V )(α,u
−1 )
VC0 (V )(β,v) VC0 W
= (V )(α+β,u
−1 v)
VC0 W
in the fusion algebra. By considering 1/16-word decompositions, one has α = β and −1 u −1 v ∈ {±1}. Let δ ∈ C be such that C 1 = C 0 + δ. If u −1 v = −1, then (V )(α+β,u v) = (V )(0,−1) = VC 1 = MC 0 (0, δ) so that by Lemma 7 one has W = (V )(α+β,u
−1 v)
W = MC 0 (0, δ) MC 0 (ξ, 0) = MC 0 (ξ, δ). V
VC 0
C0
It is shown in Lemma 19 that (C 0 )ξ contains a self-dual subcode w.r.t. ξ . Then W is not isomorphic to MC 0 (ξ, δ) by Lemma 3 and we obtain a contradiction. Therefore, u −1 v = 1 and hence (α, u) = (β, v). This completes the proof.
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By Theorem 1 and the lemma above, the space (V )(α,u) V (θξ ) := V W = VC 0
(α,u)∈D˜
(V )(0,1)
MC 0 (ξ, 0)
(7.6)
carries a unique structure of an irreducible χW -twisted V -module for some χW ∈ ˜ ∗ ⊂ M. It is clear that χW ∈ Stabpt (F). Since the top weight of W is 3/4, V (θξ ) is (D) V neither a 2A-twisted nor a 2B-twisted V -module so that χW is an element of M of order 2 | 4. Hence, there exists ξ ∈ P such that χW (V )0 = σξ . By the construction of V (θξ ), 0 we know that C ⊂ {α ∈ C | α, ξ = 0} C so that σξ = σξ and χW = θξ · τη for some η ∈ Z48 2 . Since the definition of θξ is only unique up to a product of τ -involutions, we can take θξ = χW . Note that the definition of V (θξ ) in (7.6) depends only on the decomposition of V as a VC 0 -module. Replacing θξ by θξ τβ with β ∈ Z48 2 will only change the labeling of the VC 0 -modules in (7.4) and does not affect the isotypical decomposition of V as a VC 0 -module. Thus, we have constructed the irreducible θξ -twisted V -module. Theorem 15. V (θξ ) defined in (7.6) is an irreducible 4A-twisted module over V . Remark 13. By Lemma 19 and Corollary 2, we see that the top weight of the 4A-twisted module is 3/4 and the dimension of the top level is 1. Let us consider the dual module W ∗ of W . It is clear that W and W ∗ are inequivalent ˜ are inequivalent VC 0 -modules, and so since κ ∈ C 0 . All (V )(α,u) VC0 W ∗ , (α, u) ∈ D, the space V (θξ3 ) := V W ∗ = (V )(α,u) MC 0 (ξ, κ) (7.7) VC 0
(α,u)∈D˜
(V )(0,1)
has a unique irreducible χW ∗ -twisted V -module structure with a linear character χW ∗ ∈ D˜ ∗ (cf. Theorem 1). It is also shown [DLM2] that the dual of χW -twisted module forms a −1 χW -twisted module. The dual V -module of V (θξ ) contains W ∗ as a VC 0 -submodule, and V (θξ3 ) is uniquely determined by W ∗ , so χW ∗ is actually equal to θξ3 = θξ−1 by our choice of θξ . Therefore, V (θξ3 ) is the irreducible θξ3 -twisted V -module. In order to perform the 4A-twisted orbifold construction of V , we classify the irreducible representations of (V )θξ . By (7.4), the fixed point subalgebra (V )θξ is a framed VOA with structure codes (C 0 , D0 ). Proposition 9. There are 16 inequivalent irreducible (V )θξ -modules. Every irreducible (V )θξ -module is a submodule of an irreducible θξi -twisted V -module for 0 ≤ i ≤ 3. Among them, 8 irreducible modules have integral top weights. Proof. Since V and (V )θξ are simple current extensions of the code VOA VC 0 , V is a Z4 -graded simple current extension of (V )θξ by Proposition 1. Then by Theorem 1, every irreducible (V )θξ -module is a submodule of a θξi -twisted module for 0 ≤ i ≤ 3. It is shown in [DLM3] that V has a unique irreducible θξi -twisted module for 0 ≤ i ≤ 3 as V is holomorphic. Therefore, we only have to show that each irreducible θξi -twisted V -module decomposes into a direct sum of 4 inequivalent irreducible (V )θξ -submodules. Since V (θξ ) and V (θξ3 ) are dual to each other (cf. [DLM2]),
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we know each of them has four inequivalent irreducible (V )θξ -submodules. So we shall consider the θξ2 -twisted module. Let κ ∈ Z48 2 be defined as in (7.5). Then one can easily verify that θξ2 = τκ . We consider an irreducible VC 0 -module MC 0 (0, κ). We claim ˜ are inequivalent irreducible VC 0 -modules. that (V )(α,u) V 0 MC 0 (0, κ), (α, u) ∈ D, C
The irreducibility of (V )(α,u) VC0 MC 0 (0, κ) is clear since all (V )(α,u) are simple current VC 0 -modules. If (V )(α,u) VC0 MC 0 (0, κ) (V )(β,v) VC0 MC 0 (0, κ), then by the 1/16-word decomposition, we have α = β and u −1 v ∈ {±1}. Let δ ∈ Z48 2 such that −1 C 1 = C 0 + δ. If u −1 v = −1, then (V )(α+β,u v) = MC 0 (0, δ) and one has MC 0 (0, κ) = (V )(α,u
−1 )
= (V )(α+β,u
VC0 (V )(β,v) VC0 MC 0 (0, κ)
−1 v)
VC0 MC 0 (0, κ)
= MC 0 (0, δ) VC0 MC 0 (0, κ) = MC 0 (0, κ + δ). But this is a contradiction by Lemma 3. Therefore, all (V )(α,u) VC0 MC 0 (0, κ), (α, u) ∈ ˜ are inequivalent irreducible VC 0 -modules. Now by Theorem 1 and Lemma 7 the space D, V (θξ2 ) :=
(α,u)∈D˜
(V )(α,u) MC 0 (0, κ) VC 0
(7.8)
forms an irreducible τκ = θξ2 -twisted V -module. Thus V (θξ2 ) splits into four irreducible (V )θξ -submodules as follows: ⎧ ⎫ ⎧ ⎫ ⎨ ⎬⎨ ⎬ (V )(α,1) MC 0 (0, κ) (V )(α,−1) MC 0 (0, κ) V (θξ2 ) = ⎩ ⎭ ⎩ ⎭ VC 0 VC 0 α∈D 0 α∈D 0 ⎫ ⎧ ⎫ ⎧ ⎬⎨ ⎬ √ √ ⎨ (V )(α, −1) MC 0 (0, κ) (V )(α,− −1) MC 0 (0, κ) . ⎭ ⎩ ⎭ ⎩ VC 0 VC 0 1 1 α∈D
α∈D
Therefore, all irreducible θξi -twisted V -modules are direct sums of 4 inequivalent irreducible (V )θξ -submodules and we have obtained 16 irreducible (V )θξ -modules. It remains to show that these 16 modules are inequivalent. Since every irreducible (V )θξ module can be uniquely extended to a θξi -twisted V -module by Theorem 1, these 16 irreducible modules are actually inequivalent. Among these 16 irreducible (V )θξ -modules, we have 4 modules having integral top weights from V , 2 from θξ2 -twisted V -module, 1 from θξ -twisted and 1 from θξ3 -twisted modules, respectively. This completes the proof. We have also shown that every irreducible θξi -twisted V -module has a Z4 -grading which agrees with the action of θξ on V . By this fact, we adopt the following notation. For u ∈ C∗ satisfying u 4 = 1, we set V (1, u) := {a ∈ V | θξ a = ua}. For i = 1 or 3, we define V (θξi , 1) to be the unique irreducible (V )θξ -submodule of V (θξi ) which
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has an integral top weight. They can be defined explicitly as follows: √ % V (θξ , 1) := α∈D1 (V )(α,− −1) VC0 MC 0 (ξ, 0), √ % V (θξ3 , 1) := α∈D1 (V )(α, −1) VC0 MC 0 (ξ, κ).
For i = 2, there are two irreducible (V )θξ -submodules in V (θξ2 ) having integral top weights. We shall define % V (θξ2 , 1) := α∈D0 (V )(α,−1) VC0 MC 0 (0, κ), % (7.9) V (θξ2 , −1) := α∈D0 (V )(α,1) VC0 MC 0 (0, κ). In addition, we define V (θξi , u) := V (1, u)
(V )θξ
V (θξi , 1) for 1 ≤ i ≤ 3.
(7.10)
Set G := θξ × {u ∈ C∗ | u 4 = 1}. Then {V (g, u)| (g, u) ∈ G} is the set of all inequivalent irreducible (V )θξ -modules. Proposition 10. The fusion algebra associated to (V )θξ is isomorphic to the group algebra of G. The isomorphism is given by V (g, u) → (g, u). Proof. Since the structure codes of (V )θξ is (C 0 , D0 ), (V )θξ is a D0 -graded simple current extension of VC 0 . So we have the following fusion rules: V (1, u)
(V )θξ
V (1, v) = V (1, uv) for u, v ∈ C∗ , u 4 = v 4 = 1.
Since all V (g, u), (g, u) ∈ G, are D0 -stable, we can use Proposition 2. The following fusion rules of VC 0 -modules are already known: MC 0 (0, κ) VC0 MC 0 (0, κ) = MC 0 (0, 0), MC 0 (0, κ) VC0 MC 0 (ξ, 0) = MC 0 (ξ, κ), MC 0 (ξ, 0) VC0 MC 0 (ξ, 0) = MC 0 (0, κ), MC 0 (ξ, 0) VC0 MC 0 (ξ, κ) = MC 0 (0, 0). Therefore, we have the following fusion rules of (V )θξ -modules: V (θ 2 , 1) (V )θξ V (θ 2 , 1) = V (1, 1), V (θ 2 , 1) (V )θξ V (θ, 1) = V (θ 3 , 1), V (θ, 1) (V )θξ V (θ, 1) = V (θ 2 , 1), V (θ, 1) (V )θξ V (θ 3 , 1) = V (1, 1). Since the fusion algebra is commutative and associative, the remaining fusion rules are deduced from the above and we can establish the isomorphism. A θξ -twisted orbifold construction of V refers to a construction of a Z4 -graded (simple current) extension of the θξ -fixed point subalgebra (V )θξ by using the irreducible submodules of V (θξi ) with integral weights. By Proposition 9, such modules are denoted by √ V (1, ±1), V (1, ± −1), V (θξ2 , ±1), V (θξ , 1) and V (θξ3 , 1).
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By the fusion rules in Proposition 10, there are three possible extensions of (V )θξ , namely, √ √ V = V (1, 1) ⊕ V (1, −1) ⊕ V (1, −1) ⊕ V (1, − −1), V2B = V (1, 1) ⊕ V (1, −1) ⊕ V (θξ2 , 1) ⊕ V (θξ2 , −1),
(7.11)
V4 A = V (1, 1) ⊕ V (θξ , 1) ⊕ V (θξ2 , 1) ⊕ V (θξ3 , 1). Consider the fixed point subalgebra (V )θξ = V (1, 1) ⊕ V (1, −1). Since θξ is a 4Aelement of M, its square θξ2 belongs to the 2B conjugacy class of M [ATLAS]. Therefore, 2
by the original construction of the moonshine VOA in [FLM], the subalgebra (V )θξ is isomorphic to the fixed point subalgebra V + of the Leech lattice VOA V . It is shown in [D] that V + has four inequivalent irreducible modules which are denoted by V ± and V T ± in [FLM]. Since the top weights of irreducible V + -modules belong to Z/2, the 2
inequivalent irreducible (V )θξ -modules are given by the list below: √ √ V (1, 1) ⊕ V (1, −1), V (1, −1) ⊕ V (1, − −1), √ √ V (θξ2 , 1) ⊕ V (θξ2 , −1), V (θξ2 , −1) ⊕ V (θξ2 , − −1). 2
(7.12)
It is shown in [H2] that there are two inequivalent simple extensions of V + ; one is the moonshine VOA V = V + ⊕ V T + and the other is V = V + ⊕ V − . So we have the following isomorphisms: √ √ (7.13) V + V (1, 1) ⊕ V (1, −1), V T + V (1, −1) ⊕ V (1, − −1). We shall prove that θξ2 -twisted orbifold construction V2B in (7.11) is isomorphic to the Leech lattice VOA V . For this, it is enough to show the following. Lemma 21. The top weight of V (θξ2 , −1) is 1 and the dimension of the top level is 24. Proof. Recall the 1/16-word decomposition (7.9) of V (θξ2 , −1) as a module over VC 0 . It contains a VC 0 -submodule (V )(0,1) MC 0 (0, κ) = MC 0 (0, κ) = VC 0 +κ . V C
By a straightforward computation we see that there are exactly 24 weight two codewords in the coset C 0 + κ and the support of each is one of the following: {1, 9}, {2, 10}, {3, 11}, {4, 12}, {5, 13}, {6, 14}, {7, 15}, {8, 16}, {17, 25}, {18, 26}, {19, 27}, {20, 28}, {21, 29}, {22, 30}, {23, 31}, {24, 32}, {33, 41}, {34, 42}, {35, 43}, {36, 44}, {37, 45}, {38, 46}, {39, 47}, {40, 48}.
(7.14)
Therefore, the top weight of V (θ 2 , −1) is 1. Then by the list of irreducible V + -modules in (7.12), the dimension of the top level of V (θ 2 , −1) must be 24. Or, one can directly check that all (V )(α,1) VC0 MC 0 (0, κ) has the top weight greater than 1 for any non-zero α ∈ D0 by considering their F-module structures.
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Since the top weights of V − and V T − are 1 and 3/2, we have the remaining isomorphisms as follows. √ √ V − V (θξ2 , 1) ⊕ V (θξ2 , −1), V T − V (θξ2 , −1) ⊕ V (θξ2 , − −1). (7.15) Therefore, the θξ2 -twisted orbifold V2B is isomorphic to V . Remark 14. One can identify V2B with the Leech lattice VOA without the isomorphisms in (7.13) and (7.15). For, V2B can be defined as a framed VOA with structure codes (C 0 C 1 (C 0 + κ) (C 1 + κ), D0 ) by Theorem 8, which is holomorphic by Corollary 7. It follows from (7.14) that V2B contains a free bosonic VOA associated to a vector space of rank 24. Then V2B is isomorphic to a lattice VOA associated to an even unimodular lattice by [LiX]. Since the weight one subspace of V2B is 24-dimensional, V2B is actually isomorphic to the lattice VOA associated to the Leech lattice. Similarly, one can also show that (V )θξ = V (1, 1) is isomorphic to a Z2 -orbifold VL+ of a lattice VOA VL for a certain sublattice L of . For we know that V (1, 1) ⊕ V (θξ2 , −1) forms a subVOA of V2B isomorphic to a lattice VOA again by [LiX]. Since V (1, 1) is a Z2 -fixed point subalgebra of V (1, 1) ⊕ V (θξ2 , −1) under an involution acting on V (θξ2 , −1) by −1, we have the isomorphism as claimed. This isomorphism is first pointed out by Shimakura from a different viewpoint. Next we consider the proper θξ -twisted orbifold construction V4 A in (7.11). Let α ∈ D˜ 1 be arbitrary. We can find the following VC 0 -submodule in V4 A : √
U := VC 0 ⊕ VC 1 +κ ⊕ {(V )(α,−
−1)
√
MC 0 (ξ, 0)} ⊕ {(V )(α,
−1)
VC 0
MC 0 (ξ, κ)}.
VC 0
Lemma 22. There exists a unique structure of a framed VOA on U . Proof. Set U 0 := VC 0 ⊕ VC 1 +κ and √
U 1 := {(V )(α,−
−1)
√
MC 0 (ξ, 0)} ⊕ {(V )(α, V
−1)
C0
MC 0 (ξ, κ)}.
VC 0
Then U 0 is a code VOA associated to C 0 (C 1 + κ) and U 1 is an irreducible U 0 -module with the 1/16-word α + ξ . Clearly the top weight of U 1 is integral. The dual code of C 0 (C 1 + κ) is given by D0 (D1 + ξ ) and it is straightforward to verify that D0 (D1 + ξ ) is triply even, i.e., wt(α) is divisible by 8 for any α ∈ D0 (D1 + ξ ). Therefore, (C 0 (C 1 + κ))α+ξ contains a doubly even self-dual subcode w.r.t. α + ξ by Remark 6. Then by Lemma 8, U = U 0 ⊕ U 1 forms a framed VOA. By the lemma above, we can apply the extension property of simple current extensions in Theorem 2 to define a framed VOA structure on V4 A with structure codes (C 0 (C 1 + κ), D0 (D1 + ξ )). We know that V4 A is holomorphic by Corollary 7. We shall prove that V4 A is isomorphic to the moonshine VOA V . On V2B = V (1, 1) ⊕ V (1, −1) ⊕ V (θξ2 , 1) ⊕ V (θξ2 , −1), define ψ1 , ψ2 ∈ Aut(V2B ) by ψ1 :=
1 on V (1, 1) ⊕ V (1, −1), −1 on V (θξ2 , 1) ⊕ V (θξ2 , −1),
Framed Vertex Operator Algebras and Pointwise Frame Stabilizers
and ψ2 :=
283
⎧ ⎨ 1 on V (1, 1) ⊕ V (θξ2 , 1), ⎩−1 on V (1, −1) ⊕ V (θ 2 , −1). ξ
Then both of ψ1 , ψ2 are involutions on V2B by the fusion rules in Proposition 10. ψ
ψ
Lemma 23. The fixed point subalgebras V2B 1 and V2B 2 are isomorphic to the Z2 -orbifold subalgebra V + of the Leech lattice VOA. Proof. We have shown that V2B is isomorphic to the Leech lattice VOA V . By (7.13), ψ the fixed point subalgebra V2B 1 is isomorphic to the Z2 -orbifold V + . So it remains to ψ ψ prove that V2B 2 is isomorphic to V2B 1 . Since the weight one subspace (V2B )1 of V2B is a subspace of V (θ 2 , −1) by Lemma 21, both ψ1 and ψ2 acts as −1 on (V2B )1 . The weight one subspace of V2B generates a sub-VOA isomorphic to the free bosonic VOA MC (0) associated to the linear space C = C ⊗Z . Since ψ1 ψ2−1 trivially acts on the weight one subspace of V2B , ψ1 ψ2−1 commute with the action of MC (0) on V2B . Therefore, √ ψ1 ψ2−1 is a linear character ρh = exp(2π −1 h (0) ) ∈ Aut(V2B ) induced by a weight one vector h ∈ (V2B )1 . Since ψ1 = ψ2 = −1 on the weight one subspace, we have −1 ψi ρh = ρ−h ψi = ρh−1 ψi for i = 0, 1. Then ψ1 = ρh ψ2 = ρh/2 ρh/2 ψ2 = ρh/2 ψ2 ρh/2 so that ψ1 and ψ2 are conjugate in Aut(V2B ). From this we have the desired isomorphism ∼ ρh/2 : (V2B )ψ2 → (V2B )ψ1 . Corollary 8. There exists ρ ∈ Aut((V )θξ ) such that V (1, −1)ρ V (θξ2 , 1). Proof. Since V (1, 1) ⊂ ρh/2 (V2B )ψ2 ∩ (V2B )ψ2 = (V2B )ψ1 ∩ (V2B )ψ2 , ρh/2 keeps (V )θξ = V (1, 1) invariant. Thus the restriction of ρh/2 on (V )θξ is the desired automorphism. By the classification of irreducible modules over V + and the fusion rules in Proposition 10, the irreducible untwisted (V2B )ψ2 -modules are as follows: V + V (1, 1) ⊕ V (θξ2 , 1),
V T + V (θξ , 1) ⊕ V (θξ3 , 1),
V − V (1, −1) ⊕ V (θξ2 , −1), V T − V (θξ , −1) ⊕ V (θξ3 , −1). Actually, the isomorphisms above are induced by ρ ∈ Aut((V )θξ ) defined in Corollary 8. By the isomorphisms above, the space V4 A = V (1, 1) ⊕ V (θξ , 1) ⊕ V (θξ2 , 1) ⊕ V (θξ3 , 1) is a Z4 -graded simple current extension of (V )θξ and isomorphic to V = V + ⊕ V T + as a (V )θξ -module. Since both V4 A and V are simple current extensions of (V )θξ , these two VOA structures are isomorphic. Therefore, we have obtained our main result in this section. Theorem 16. The VOA V4 A obtained by the 4A-twisted orbifold construction of V is isomorphic to V .
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Acknowledgement. The authors thank Masahiko Miyamoto for discussions and valuable comments on the proof of Theorem 6. They also thank Masaaki Kitazume and Hiroki Shimakura for discussions on binary codes. The second-named author wishes to thank Markus Rosellen for information on the associativity and the locality of vertex operators. Part of the work was done when the second author was visiting the National Center for Theoretical Sciences, Taiwan in February 2006. He thanks the staff of the center for their hospitality.
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[H3] [HL] [L1]
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: ATLAS of finite groups. Oxford: Clarendon Press, 1985 Abe, T., Buhl, G., Dong, C.: Rationality, regularity, and C2 -cofiniteness. Trans. Amer. Math. Soc. 356, 3391–3402 (2004) Conway, J.H.: A simple construction for the Fischer-Griess monster group. Invent. Math. 79, 513–540 (1985) Dong, C.: Representations of the moonshine module vertex operator algebra. In: Mathematical aspects of conformal and topological field theories and quantum groups. Proc. Joint Summer Research Conference, Mount Holyoke, 1992, edited by P. Sally, M. Flato, J. Lepowsky, N. Reshetikhin, G. Zuckerman, Contemporary Math. 175, Providence, RI: Amer. Math. Soc., 1994, pp. 27–36 Dong, C., Griess, R.L., Höhn, G.: Framed vertex operator algebras, codes and the moonshine module. Commun. Math. Phys. 19, 407–448 (1998) Dong, C., Griess, R.L., Lam, C.H.: Uniqueness results for the moonshine vertex operator algebra. Amer. J. Math. 139(2), 583–609 (2007) Dong, C., Lepowsky, J.: Generalized vertex algebras and relative vertex operators. Progress in Math. 112, Boston: Birkhäuser, 1993 Dong, C., Li, H., Mason, G.: Some twisted sectors for the moonshine module. In: Moonshine, the Monster, and related topics (South Hadley, MA, 1994), Contemp. Math. 193, Providence, RI: Amer. Math. Soc., 1996, pp. 25–43 Dong, C., Li, H., Mason, G.: Twisted representations of vertex operator algebras. Math. Ann. 310, 571–600 (1998) Dong, C., Li, H., Mason, G.: Modular-invariance of trace functions in orbifold theory and generalized moonshine. Commun. Math. Phys. 214, 1–56 (2000) Dong, C., Mason, G.: On quantum Galois theory. Duke Math. J. 86, 305–321 (1997) Dong, C., Mason, G.: Rational vertex operator algebras and the effective central charge. Internat. Math. Res. Notices 56, 2989–3008 (2004) Dong, C., Mason, G., Zhu, Y.: Discrete series of the Virasoro algebra and the moonshine module. Proc. Symp. Pure. Math., 56, II Providence, RI: American Math. Soc. 1994, pp. 295–316 Feingold, A.J., Frenkel, I.B., Ries, J.F.X.: Spinor construction of vertex operator algebras, triality, (1) and E 8 . Contemp. Math. 121, Providence, RI: Amer. Math. Soc., 1991 Frenkel, I., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs Amer. Math. Soc. 104, 494, Providence, RI: Amer. Math. Soc., 1993 Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. New York: Academic Press, 1988 Frenkel, I.B., Zhu, Y.: Vertex operator algebras associated to representation of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992) Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys. 103, 105–119 (1986) Gorenstein, D.: Finite groups. New York-London: Harper & Row, Publishers, 1968 Huang, Y.-Z.: Virasoro vertex operator algebras, (non-meromorphic) operator product expansion and the tensor product theory. J. Algebra 182, 201–234 (1996) Huang, Y.-Z.: A nonmeromorphic extension of the moonshine module vertex operator algebra. In: “Moonshine, the Monster and Related Topics, Proc. Joint Summer Research Conference, Mount Holyoke, 1994” C. Dong, G. Mason, eds., Contemporary Math. 193, Providence, RI: Amer. Math. Soc., 1996, pp. 123–148 Huang, Y.-Z.: Differential equations and intertwining operators. Comm. Contemp. Math. 7, 375–400 (2005) Huang, Y.-Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra, III. J. Pure Appl. Alg. 100, 141–171 (1995) Lam, C.H.: Twisted representations of code vertex operator algebras. J. Algebra 217, 275–299 (1999)
Framed Vertex Operator Algebras and Pointwise Frame Stabilizers
[L2] [L3] [LSY] [LY] [LYY1] [LYY2] [Li1] [Li2] [LiX] [McS] [McST] [M1] [M2] [M3] [R] [SY] [Sh] [Y1] [Y2] [Y3] [Z]
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Lam, C.H.: Some twisted module for framed vertex operator algebras. J. Algebra 231, 331–341 (2000) Lam, C.H.: Induced modules for orbifold vertex operator algebras. J. Math. Soc. Japan 53, 541–557 (2001) Lam, C.H., Sakuma, S., Yamauchi, H.: Ising vectors and automorphism groups of commutant subalgebras related to root systems. Math. Z. 255, 597–626 (2007) Lam, C.H., Yamauchi, H.: A characterization of the Moonshine vertex operator algebra by means of Virasoro frames. Internat. Math. Res. Notices 2007 article ID:mm003, p. 9 (2007) Lam, C.H., Yamada, H., Yamauchi, H.: Vertex operator algebras, extended E 8 -diagram, and McKay’s observation on the Monster simple group. Trans. Amer. Math. Soc. 359, 4107–4123 (2007) Lam, C.H., Yamada, H., Yamauchi, H.: Mckay’s observation and vertex operator algebras generated by two conformal vectors of central charge 1/2. Internat. Math. Res. Papers 3, 117–181 (2005) Li, H.: Symmetric invariant bilinear forms on vertex operator algebras. J. Pure Appl. Algebra 96, 279–297 (1994) Li, H.: Local systems of vertex operators, vertex superalgebras and modules. J. Pure Appl. Algebra 109, 143–195 (1996) Li, H., Xu, X.: A characterization of vertex algebras associated to even lattices. J. Algebra 173, 253–270 (1995) MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. Amsterdam: North-Holland, 1998 MacWilliams, F.J., Sloane, N.J.A., Thompson, J.G.: Good self-dual codes exist. Discrete Math. 3, 153–162 (1972) Miyamoto, M.: Griess algebras and conformal vectors in vertex operator algebras. J. Algebra 179, 528–548 (1996) Miyamoto, M.: Representation theory of code vertex operator algebras. J. Algebra 201, 115–150 (1998) Miyamoto, M.: A new construction of the moonshine vertex operator algebra over the real number field. Ann. of Math 159, 535–596 (2004) Rosellen, M.: A course in Vertex Algebra. http://arxiv.org/abs/math/0607270, 2006 Sakuma, S., Yamauchi, H.: Vertex operator algebra with two miyamoto involutions generating S3 . J. Algebra 267, 272–297 (2003) Shimakura, H.: Automorphism group of the vertex operator algebra VL+ for an even lattice L without roots. J. Algebra 280, 29–57 (2004) Yamauchi, H.: Module categories of simple current extensions of vertex operator algebras. J. Pure Appl. Algebra 189, 315–328 (2004) Yamauchi, H.: A theory of simple current extensions of vertex operator algebras and applications to the moonshine vertex operator algebra. Ph.D. thesis, University of Tsukuba, 2004.; available on the author’s web site: http://auemath.aichi-edu.ac.jp/∼yamauchi/math/phd/myphd.pdf Yamauchi, H.: 2A-orbifold construction and the baby-monster vertex operator superalgebra. J. Algebra 284, 645–668 (2005) Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996)
Communicated by Y. Kawahigashi
Commun. Math. Phys. 277, 287 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0363-7
Communications in
Mathematical Physics
Erratum
Kraichnan Turbulence via Finite Time Averages C. Foias1,2 , M. S. Jolly2 1 Department of Mathematics, Texas A&M University, College Station, TX 77843, USA 2 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA.
E-mail: [email protected] Received: 26 June 2007 / Accepted: 23 August 2007 Published online: 26 October 2007 – © Springer-Verlag 2007 Commun. Math. Phys. 255, 329–361 (2005)
To our regret, a number of misprints have occurred in [FJM]. The argument is correct, but the following typographical corrections should be made: in (4.24) ∼ should be , and the exponent on the logarithm should be −3/2. Two lines down, > should be , and also the final κ˜ η should be η. ˜ At the top of page 352, it is for (4.21) that we cannot provide a converse, and for (4.22) that we do find a weak converse. Accordingly, in the statement of Proposition 4.10, should be . Consistent with this, should be in (4.29), (4.30), (4.31), and in the last relation in the proof of Proposition 4.10. Missing in the right-hand side of (4.29) is the factor 1/ψ(κ). In the last two relations on page 353, 1/3 κ˜ η should be η˜ 1/3 . There are other minor misprints, such as · where there should be ˜ which can readily be corrected from the context. ·, Reference [FJM] Foias, C., Jolly, M.S., Manley, O.P.: Kraichnan turbulence via finite time averages. Commun. Math. Phys. 255, 329–361 (2005) Communicated by P. Constantin
The online version of the original article can be found under doi:10.1007/s00220-004-1274-5. This work was partially supported by NSF grant number DMS-0511533.
Commun. Math. Phys. 277, 289–304 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0362-8
Communications in
Mathematical Physics
Robustness of Quantum Markov Chains Ben Ibinson, Noah Linden, Andreas Winter Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK. E-mail: [email protected]; [email protected]; [email protected] Received: 8 November 2006 / Accepted: 1 April 2007 Published online: 20 October 2007 – © Springer-Verlag 2007
Abstract If the conditional information of a classical probability distribution of three random variables is zero, then it obeys a Markov chain condition. If the conditional information is close to zero, then it is known that the distance (minimum relative entropy) of the distribution to the nearest Markov chain distribution is precisely the conditional information. We prove here that this simple situation does not obtain for quantum conditional information. We show that for tri-partite quantum states the quantum conditional information is always a lower bound for the minimum relative entropy distance to a quantum Markov chain state, but the distance can be much greater; indeed the two quantities can be of different asymptotic order and may even differ by a dimensional factor. I. Introduction From the point of view of information theory, as well as physics, it is very interesting to know when entropy or, more generally, information inequalities are saturated. For example, the basic quantities von Neumann entropy S(A) = S(ρ A ) = − Tr ρ A log ρ A , quantum mutual information I (A : B) = S(A)+ S(B)− S(AB) for a bipartite state ρ AB , and conditional mutual information I (A : C|B) = S(AB) + S(BC) − S(B) − S(ABC) for a tripartite state ρ ABC are all non-negative; for the latter two this is known as the subadditivity and strong subadditivity of the entropy, respectively [17]. The entropy is 0 if and only if the state is pure, and the mutual information is 0 if and only if the state ρ AB is a product state, ρ AB = ρ A ⊗ ρ B . However, in many applications it is not the case or not known that the state is exactly pure or a product, only that it is very close to being so. In such situations, there are continuity bounds on entropic quantities that one can use to quantify how small the entropy or mutual information is. Fannes’ inequality √ [11] states that if ρ A − σ A 1 ≤ ≤ 1/e (with the trace norm X 1 := Tr |X | = Tr X ∗ X ), then |S(ρ) − S(σ )| ≤ − log + log d A ,
(1)
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where d A is the dimension of the Hilbert space supporting the states. (“log” in this paper is always the binary logarithm; the natural logarithm is denoted “ln”.) In particular, if ρ has trace distance ≤ 1/e to a pure state, then S(ρ) ≤ − log + log d A . Note that by the monotonicity of the trace norm under quantum channels, and non-decrease of the entropy under conditional expectations, the general quantum case reduces to the classical (i.e. commuting density operators). Recently, Alicki and Fannes [2] proved an extension of the Fannes inequality to quantum conditional entropy S(A|B) = S(AB) − S(B) for bipartite states ρ AB and σ AB : if ρ A − σ A 1 ≤ ≤ 1, then |S(A|B)ρ − S(A|B)σ | ≤ −2 log − 2(1 − ) log(1 − ) + 4 log d A .
(2)
The crucial observation here is that the bound only depends on and d A , not d B as the bound yielded by a naive application of the original Fannes inequality. This gives an upper bound on the mutual information for a state that is at trace distance from a product state (using convexity of the trace distance, and (1) and (2) together with the triangle inequality). Conversely, one may ask, if say the entropy of a state is small, S(ρ) ≤ , is it close to being pure? Indeed yes, as the following argument shows. Fix a diagonalisation d A of ρ, ρ = i=1 λi |ei ei | with eigenvalues λi arranged in decreasing order. Then, as −x log x ≥ x for 0 ≤ x ≤ 1/2, ≥ S(ρ) =
dA
−λi log λi ≥
i=1
dA
λi = 1 − λ1 .
(3)
i=2
Hence, ρ − |e1 e1 |1 = 2(1 − λ1 ) ≤ 2.
(4)
Note however that this bound and Fannes’ inequality are not “inverse” to each other; plugging the 2 into the Fannes bound yields something much larger than order . Similarly, what can we say about the state when I (A : B) ≤ ? Here, a new quantity, the relative entropy D(ρσ ) = Tr ρ(log ρ − log σ ), comes into play, when we observe that I (A : B)ρ = D(ρ AB ρ A ⊗ ρ B ). Invoking another inequality between distance measures for states, namely Pinsker’s inequality, see [19], Thm. 5.5, D(ρσ ) ≥
1 ρ − σ 21 , 2
(5)
√ we conclude that ρ AB − ρ A ⊗ ρ B 1 ≤ 2. (Again, the general quantum case follows from the classical, if one invokes monotonicity of the relative entropy under quantum channels.) Note that in both examples discussed, we found an explicit candidate for the closest pure/product state to the given state (as can be checked), and that the bound on the trace distance depends only on , not on dimensions as in the converse Fannes-style inequalities, and third, that the relative entropy gives even tighter control on the distance due to Pinsker’s inequality. In this paper we study the quantum conditional information. If the quantum conditional information of a tri-partite state ρ vanishes, then ρ obeys a quantum Markov chain condition (see Sect. III, Eq. (24) below for the technical statement). Here we analyze what can be said if ρ has small quantum conditional information; in particular we investigate how close it is to a Markov chain state. The motivation is partly
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classical (e.g. cryptographic [14]), but in the quantum case a strong motivation comes from considerations of new entropy inequalities: in [18] a so-called constrained inequality for the von Neumann entropies of subsystems was found, namely a relation which is valid provided three quantum conditional mutual informations are zero. The desire to turn this into an unconstrained, universal inequality lead to speculations that if one understood the near-vanishing of these constraints, then perhaps a trade-off between the constraints and the new inequality solely in terms of entropies might be established. In Sect. II we review, as a model, the classical case, where it turns out that the conditional mutual information is exactly the minimum relative entropy distance between the distribution and the closest Markov chain distribution. In Sect. III we formulate the analogous quantum problem, which we analyse in the rest of the paper: Sect. IV presents several simplifications of the question – we prove continuity of the minimum relative entropy, and that it is lower bounded by the quantum conditional mutual information, and some useful formulas for later numerical and analytical evaluation of the quantity. Then, in Sect. V, we specialise to pure states: we relate the minimum relative entropy to the so-called entanglement of purification, and for a large family of states show upper and lower bounds of matching order. These results are then used in Sect. VI to provide examples of states for which the minimum relative entropy is much larger than the quantum conditional mutual information, and also ones where the dimension enters explicitly, showing that the classical and the quantum case are very different indeed. II. Classical Case In the classical case, Markov chain distributions are used to define the conditional mutual information. A classical distribution PX 1 X 2 ...X n (x1 , x2 , . . . , xn ) forms a Markov chain denoted as X 1 → X 2 → X 3 → · · · → X n if the distribution can be written as PX 1 X 2 ...X n (x1 , x2 , . . . , x3 ) = PX 1 X 2 (x1 , x2 )PX 3 |X 2 (x3 |x2 ) . . . PX n |X n−1 (xn |xn−1 ).
(6)
If we take any three linked variables in the above Markov chain, i.e. X α−1 ,X α ,X α+1 , then the conditional distribution of PX α+1 |X α ...X 1 (xα+1 |xα . . . x1 ) depends only on X α , and X α+1 is conditionally independent of X α−1 , given X α . Consider three random variables X, Y and Z which form a Markov chain X → Y → Z . The probability distribution for this system is PX Y Z (x yz) = PX Y (x y)PZ |Y (z|y) = PY (y)PX |Y (x|y)PZ |Y (z|y).
(7)
Aside, from this we define the conditional mutual information as I (X : Z |Y ) =
x,y,z
PX Y Z (x yz) log
PX Z |Y (x z|y) , PX |Y (x|y)PZ |Y (z|y)
(8)
which can be written in terms of joint entropies entirely: I (X : Z |Y ) = H (X Y ) + H (Y Z ) − H (X Y Z ) H (Z ). Note that throughout this section we use the convention 0 log 0 = 0. (This is justified by looking at the behavior of x log x as x → 0.) This conditional mutual information is equal to zero if and only if for all x, y and z, PX Z |Y (x z|y) = 1. PX |Y (x|y)PZ |Y (z|y)
(9)
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Therefore, PX Z |Y (x z|y) = PX |Y (x|y)PZ |Y (z|y), PX Y Z (x yz) = PY (y)PX |Y (x|y)PZ |Y (z|y).
(10)
Hence a classical Markov chain distribution is characterized by zero conditional mutual information. The classical case is characterized by an exact correspondence between the conditional mutual information and the relative entropy distance to the set of Markov chains: for any joint distribution PX Y Z of three random variables X, Y, Z [14], I (X : Z |Y ) = min {D(PQ) : Q Markov} .
(11)
It can be shown that the Markov chain required to minimise this quantity is Q X Y Z (x yz) = PY (y)PX |Y (x|y)PZ |X (z|x).
(12)
To see this, imagine a joint probability distribution Q X Y Z that also forms a general Markov chain: Q X Y Z (x yz) = Q Y (y)Q X |Y (x|y)Q Z |Y (z|y).
(13)
We can write the probability distribution of PX Y Z as follows PX Y Z (x yz) = PX Y Z (x yz) = PY (y)PZ |Y (z|y)PX |Y Z (x|yz),
(14)
therefore the relative entropy between the two probability distributions is D(PQ) =
x yz
PX Y Z (x yz) log
PY (y)PZ |Y (z|y)PX |Y Z (x|yz) . Q Y (y)Q Z |Y (z|y)Q X |Y (x|y)
(15)
Since we have a product of logarithms we can represent the relative entropy as such, PZ |Y (z|y) PX |Y Z (x|yz) PY (y) D(PQ) = +log +log . (16) PX Y Z (x yz) log Q Y (y) Q Z |Y (z|y) Q X |Y (x|y) x yz On inspection of the final term we can use the following equivalence: PX |Y Z (x|yz) PZ |X Y (z|x y) PX |Y (x|y) PX Y Z (x yz) = = . Q X |Y (x|y) PY Z (yz)Q X |Y (x|y) PZ |Y (z|y) Q X |Y (x|y)
(17)
Observing that the first two terms of Eq. (16) are relative entropy terms, we have D(PQ) = D (PY (y)Q Y (y)) + D PZ |Y (z|y)Q Z |Y (z|y) PZ |X Y (z|x y) . (18) +D PX |Y (x|y)Q X |Y (x|y) + PX Y Z (x yz) log PZ |Y (z|y) x yz Note that the only terms that depend on the distribution of Q are the first three relative entropy terms. Since relative entropy is non-negative and D(ST ) = 0 if and only if S = T , the Markov chain that provides the minimum relative entropy between P and Q can be achieved by setting these terms to zero which gives the required distribution in (12). This concludes the proof.
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From this result it is simple to show that conditional mutual information can be achieved. Since we know the relative entropy terms in Eq. (18) are zero if we use Q as the minimising Markov chain: D(PQ) =
PX Y Z (x yz) log
x yz
PZ |X Y (z|x y) . PZ |Y (z|y)
(19)
Using the following equivalence: PZ |X Y (z|x y) =
PX Z |Y (x z|y) PX Y Z (x yz) PY (y) PX Y Z (x yz) = = . PX Y (x y) PY (y) PX Y (x y) PX |Y (x|y)
(20)
We can substitute this into (19) to produce the final result D(PQ) =
PX Z |Y (x z|y) = I (X : Z |Y ). PX |Y (x|y)PZ |Y (z|y)
PX Y Z (x yz) log
x yz
(21)
III. Quantum Analogue A quantum analogue of (short) Markov chains, i.e. quantum states of some tripartite system ABC with a suitably defined Markov property, was first proposed by Accardi and Frigerio [1]. In finite Hilbert space dimension, which will be the case we will consider in the present paper, this property reads as follows: µ ABC is a quantum Markov state if there exists a quantum channel, i.e. a completely positive and trace preserving (c.p.t.p.) map T : B(B) −→ B(B) ⊗ B(C) such that µ ABC = (id A ⊗ T )µ AB , with µ AB = Tr C µ ABC . In [20] it was shown that this Markov condition is equivalent to vanishing conditional mutual information, I (A : C|B)µ := S(µ AB ) + S(µ BC ) − S(µ ABC ) − S(µ B ) = 0,
(22)
just as in the classical case, and in [13] the most general form of such states was given, as follows: system B has a direct sum decomposition into tensor products, B= b Lj ⊗ b Rj , (23) j
such that µ ABC =
j
( j)
( j)
p j µ Ab L ⊗ µb R C . j
(24)
j
Note that this precisely generalises Eq. (7). We introduce the notation δ for the direct sum decomposition of B. Note that we can always think of H B as being a subspace of a larger Hilbert space H B (for which inclusion we use the shorthand B → B ). This doesn’t change the fact that a state is a Markov chain state or not, but it leads to more possibilities of decomposing the ambient Hilbert space as a sum of products as in Eq. (23). In other words, in a larger space there is a larger set of quantum Markov chains. This latter is evidently going to be relevant when comparing a given state ρ ABC to the class of Markov chain states: in general we will have to admit that all three systems A, B, C are subspaces of larger Hilbert spaces, and we have to take into account the Markov states on the extended system.
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Now we go on to develop some formalism to deal with these embeddings: If we
have the embedded system B → B then we define δ ≡ B → B= j B j as both the isometric embedding and the orthogonal decomposition of the embedding system. For a specific such direct sum decomposition δ we introduce τ = τδ for the family of tensor product decompositions of the B j , which are, w.l.o.g., embeddings τδ ≡
τ j : B j → b Lj ⊗ b Rj
j
. Note that this latter only gives us increased flexibility:
we could as well demand that each τ j is actually a unitary isomorphism between B j and b Lj ⊗ b Rj , because one can always blow up the spaces B j , extending the isometry to a unitary. This brings us to the main question of this paper: for given state ρ ABC , to find (ρ) := inf {D(ρµ) : µ Markov} ,
(25)
and to compare it to I (A : C|B)ρ . The remainder of this paper will be devoted to a study of the properties of this function. To be precise, we would like to consider µ to be with A ⊂ A, B⊂ (with ρ a Markov state on a tripartite system A B C, B and C ⊂ C understood to be also a state on A B C via these embeddings), which is why above we is unbounded. This appears to be have to use the infimum, since the dimension of A BC necessary for the reason that the decompositions as in Eq. (24) depend on the dimension of B. and C = C, and We will show below (in the next section) that w.l.o.g. A = A 4 dim B ≤ d B , so that the infimum is actually a minimum. We also show lower bounds on comparing it to I (A : C|B), in particular exhibiting examples of states ρ with (ρ) I (A : C|B)ρ . IV. General Properties of Here we show that the problem of determining the minimum relative entropy to a Markov state is really only a minimisation over decompositions of the type (23) for B. For given dimensions of the quantum system, there is only a finite number of decomposition types. Therefore we need to perform a finite-dimensional optimisation for each decomposition (some of which we can perform explicitly) and choose the global minimum. Proposition 1. The optimal state for given direct sum and tensor decomposition we denote ω[δ, τ ], describing the specific direct sum as δ and the chosen tensor decomposition for that direct sum τ = τδ . We obtain ω[δ, τ ] by the following procedure: first, with the subspace projections P j onto b Lj ⊗ b Rj ⊂ B, let ( j) ω[δ] := (1 AC ⊗ P j )ρ(1 AC ⊗ P j ) = q j ω Ab L b R C , (26) j
j
j
j
where for each part j of the direct sum for the given decomposition, we project system B ( j) via the corresponding projections P j to produce ω Ab L b R C with corresponding probability j
( j)
j
( j)
( j)
( j)
q j . Then, form the reduced states σ Ab L = Tr b R C ω Ab L b R C and χb R C = Tr Ab L ω Ab L b R C , j
j
and let ω[δ, τ ] :=
j
j
( j)
j
j
( j)
q j σ Ab L ⊗ χb R C . j
j
j
j
j
(27)
Robustness of Quantum Markov Chains
295
With these definitions, among all Markov states with decomposition (23) of B, ω[δ, τ ] is the one with smallest relative entropy, given by ( j) ( j) (28) q j S σ Ab L + S χb R C . D (ρω[δ, τ ]) = −S(ρ) + H (q) + j
j
j
Proof. The relative entropy between a general state ρ ABC and a general quantum Markov state µ ABC is, with given decompositions δ and τ , D(ρ ABC µ ABC ) = −S(ρ ABC ) − Tr(ρ ABC log µ ABC ),
(29)
where the general quantum Markov state is defined as ( j) ( j) µ ABC = p j µ Ab L ⊗ µb R C . j
j
(30)
j
Therefore we can calculate the logarithm of µ ABC , ( j) ( j) log p j (1 AC ⊗ P j ) + log(µ Ab L ⊗ µb R C ) =: log µ ABC = L j, j
j
j
(31)
j
B onto b Lj b Rj . For clarity we assume ρ where P j are the subspace projections from indicates the state over all parties unless otherwise indicated: Tr ρ ABC log µ ABC = (32) Tr (1 AC ⊗ P j )ρ(1 AC ⊗ P j )L j . j ( j)
( j)
Now (1 AC ⊗ P j )ρ(1 AC ⊗ P j ) = q j ω Ab L b R C with Tr ω Ab L b R C = 1. Therefore, j
Tr ρ ABC log µ ABC =
j
j
( j)
j
( j)
( j)
( j)
Tr q j ω Ab L b R C log(µ Ab L ⊗ µb R C ) j
j
=
j
Tr q j ω Ab L b R C log p j (1 AC ⊗ P j )
j
+
j
q j log p j +
j
j
j
(33)
j
( j)
( j)
( j)
q j Tr ω Ab L b R C log(µ Ab L ⊗ µb R C ) (34) j
j
j
j
j
= −H (q) − D(q p) ( j) ( j) ( j) ( j) − q j S(σ Ab L ) + S(χb R C ) + D(σ Ab L µ Ab L ) j
j ( j)
( j)
j
j
j
j
j
+D(χb R C µb R C ) , ( j)
( j)
(35)
( j)
( j)
where σ Ab L = Tr b R C ω Ab L b R C and χb R C = Tr Ab L ω Ab L b R C . For a given decomposij
j
j
j
j
j
j
j
( j) tion of system B, the subspace projections P j and hence q j and ω Ab L b R C are fixed. j
j
Since we want to minimise the relative entropy, we want to maximise the quantity Tr ρ ABC log µ ABC . Therefore to maximise the first relative entropy term we set p j = q j .
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For the sum we consider each i individually and only have freedom of setting the last two relative entropy terms to zero. Therefore we can maximise this expression by setting ( j) ( j) ( j) ( j) µ Ab L = σ Ab L and µb R C = χb R C . This concludes the proof.
j
j
j
j
A nice little observation is that the relative entropy of interest can be decomposed into two relative entropies, as follows: D (ρω[δ, τ ]) = D (ρω[δ]) + D (ω[δ]ω[δ, τ ]) = D (ρω[δ]) + q j I (Ab Lj : b Rj C)ω( j) .
(36)
j
(This is really a special case of the general conditional expectation property of the relative entropy.) Note that this result has an important consequence for the infimum defining (ρ): we only need to worry about embedding B into a larger system B; A and C can, w.l.o.g., stay the same.
k≤d 2 Theorem 2. The infimum of Eq. (25) is achieved on a decomposition B = i=1 B b Lj ⊗ b Rj , with dim b Lj , dim b Rj ≤ d B . In particular, because it is one of a continuous function over a compact domain, the infimum is actually a minimum. Also, it means that (ρ), as the minimum of a continuous function over a compact domain, is itself a continuous function of its argument ρ. The reader may wish to skip the rather lengthy and somewhat technical proof of this theorem; note however that in it some notation is introduced which is referred to later. Proof. The proof has two parts – first, that the direct sum decomposition δ may be taken to have only d B2 terms, and second, that each direct summand may be embedded into a space of not more than d B × d B dimensions. These two arguments are quite independent of each other; we start with the first. 1. For given embedding B → B and decomposition of B, we have ω[δ] = (1 AC ⊗ P j )ρ ABC (1 AC ⊗ P j ) = (1 AC ⊗ P j P)ρ ABC (1 AC ⊗ P P j ), j
j
(37) with the projector P of B onto B. Note that the operators P j P form a complete Kraus system: (P j P)† (P j P) = P Pj P = P = 1B . (38) j
j
Hence the operators M j = P P j P form a POVM on B, and introducing an auxiliary register J with orthogonal states | j to reflect the direct sum, ω[δ] is equivalent, up to local isometries, to the state = 1 AC ⊗ M j ρ ABC 1 AC ⊗ M j ⊗ | j j| J . (39) j
At the same time, the embedding τ j can be reinterpreted as a family of isometries τ j : B → b L ⊗ b R controlled by the content j of the J -register (note that we may,
Robustness of Quantum Markov Chains
297
w.l.o.g., assume that the τ j all map into the same tensor product space), so that the state after the action of τ is 1 AC ⊗ τ j M j ρ ABC 1 AC ⊗ M j τ †j ⊗ | j j| J . Ab L b R C J = (40) j
In this notation, our formula (28) can be rewritten as D(ρω[δ, τ ]) = −S(ρ) + S(J ) + S(Ab L |J ) + S(b R C|J ).
(41)
Now, to reduce the number of POVM elements (i.e., entries of the J -register with nonzero probability amplitude), we invoke a theorem of Davies [10] on extremal POVMs: One looks at all real vectors (λ j ) j such that the operators λ j M j form a POVM, i.e., j λ j M j = 1 B . It is clear that the all-ones vector is eligible, and that this set is compact and convex – in fact, it is a polytope, and Davies’ theorem states that its extremal points have at most d B2 non-zero entries (actually, this is just a special case of Caratheodory’s lemma). On the other hand, the all-ones vector can be convex-decomposed into extremal ones, i.e., (k) ∀j Mj = rk λ j M j , (42) k
(k) with extremal vectors (λ j ) j and positive reals rk with k rk = 1. In operational terms, the POVM (M j ) is equivalent to choosing K = k with probability rk and then measuring the POVM (λ(k) j M j ). This means that we can extend the state above to
(k) (k) † rk 1 AC ⊗ τ j λ j M j ρ ABC 1 AC ⊗ λ j M j τ j Ab L b R C J K = jk
⊗| j j| J ⊗ |kk| K ,
(43)
of which it can be readily verified that tracing over K gives Eq. (40). Then, by the concavity of the von Neumann entropy, S(J ) ≥ S(J |K ) and by the way we constructed the POVMs, S(Ab L |J ) = S(Ab L |J K ),
S(b R C|J ) = S(b R C|J K ).
(44)
Hence, Eq. (41) is lower bounded by − S(ρ) + S(J |K ) + S(Ab L |J K ) + S(b R C|J K ),
(45)
and there exists a value k of K for which D(ρω[δ, τ ]) ≥ −S(ρ) + S(J |K = k) + S(Ab L |J K = k) + S(b R C|J K = k).
(46)
But for each K = k, the right-hand side is a relative entropy with a Markov state referring to the POVM (λ(k) j M j ); it can be lifted, by Naimark’s theorem, to an orthogonal measurement on a larger space B. It is clear that w.l.o.g. B j has dimension at most d B : ( j) the state ω AB j C is supported in B j on a subspace of dimension at most d B . 2. Now for the second part: looking at Eq. (28), we see that once δ is fixed, we have ( j) states ω AB j C and we need to find, for each j individually, a decomposition/embedding
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B. Ibinson, N. Linden, A. Winter
( j) ( j) τ j : B j → b Lj ⊗ b Rj that minimises the term S(σ Ab L ) + S(χb R C ) in Eq. (28). Dropping j
j
the index j for now, since we will keep it fixed, let us introduce a purification |φ ABC D of ω ABC ; then, with the isometric embedding τ : B → b L b R implicit and the slight abuse of notation |φ Ab L b R C D := (1 AC D ⊗ τ )|φ ABC D ,
(47)
our task is to minimise, over all choices of τ , S(Ab L ) + S(b R C) = S(Ab L ) + S(ADb L ).
(48)
Now notice that the latter quantity refers only to subsystems AD and b L , and that hence we can describe it entirely by the state TrC φ ABC D =: ϑ AB D and the completely positive and trace preserving map T := Tr b R ◦τ mapping density operators on B to density operators on b L – by Stinespring’s theorem, conversely every such quantum channel can be lifted to an isometric dilation τ : B → b L ⊗b R (the system b R would be called the environment of the channel). For a fixed output system b L the set of these quantum channels is convex and the state (1 AD ⊗ T )ϑ AB D is a linear function of the map. Hence, by the concavity of the von Neumann entropy S, the smallest sum of entropies (48) is attained for extremal channels, which by a theorem of Choi [6] have at most d B operator terms in the Kraus decomposition – which translates into a dimension of at most d B of b R . The dimensionality of the subsystem covered by the output of that channel in b L is thus at most d B2 . But now we can run the same argument for b R instead – the whole setup is symmetric, so the channel from B to b R is also w.l.o.g. extremal, entailing dim b L ≤ d B (note that we fix the output dimension here to ≤ d B from the previous argument).
There is a special case of the second part of the above proof in the literature that has inspired the present argument: that is the dimension bounds in the so-called entanglement of purification [21]. There it was shown that in the problem of, for a pure state ω ABC , minimising the entropy S(AE) =
1 (S(AE) + S(C F)) , 2
(49)
over all isometric embeddings B → E F, one may restrict to a priori bounded dimensions dim E = d B and dim F = d B2 , or, vice versa, dim E = d B2 and dim F = d B . What is noticed above is that, apart from the generalisation to mixed states, one can apply the argument of the extremal channels twice, to get the same bound d B on the dimensions of both E and F: Corollary 3. The entanglement of purification, E P (ρ AC ) =
inf
B→E F
S(AE),
(50)
the entropy understood with respect to the state φ AE FC , is attained at an embedding with dimensions dim E, dim F ≤ d B = rank ρ AC . Theorem 4. For any state ρ ABC , the quantity (ρ ABC ) has the following lower bound: (ρ) ≥ I (A : C|B)ρ .
(51)
Robustness of Quantum Markov Chains
299
Proof. Indeed, it is sufficient to show, for any ρ ABC and decomposition of B as in Eq. (12) with accompanying state ω[δ, τ ], that D (ρω[δ, τ ]) ≥ I (A : C|B)ρ ,
(52)
which, by Eq. (28), is equivalent to H (q) +
j
qj
( j) ( j) S(σ Ab L ) + S(χb R C ) ≥ S(B)ρ + S(A|B)ρ + S(C|B)ρ . j
(53)
j
It turns out to be convenient to introduce the following state of five registers to represent the entropic quantities in the above: J Ab L b R C =
j
( j)
q j | j j| J ⊗ ω Ab L b R C , j
(54)
j
observing that we may think of all b Lj , b Rj as subspaces of one b L , b R , respectively. Then the inequality we need to prove reads S(J ) + S(Ab L |J ) + S(b R C|J ) ≥ S(B)ρ + S(A|B)ρ + S(C|B)ρ .
(55)
This is done by invoking standard inequalities as follows: S(J ) + S(Ab L |J ) + S(b R C|J ) = S(J ) + S(A|b L J ) + S(b L |J ) +S(C|b R J ) + S(b R |J ) ≥ S(J ) + S(b L b R |J ) + S(A|b L J ) +S(C|b R J ) = S(J b L b R ) + S(A|b L J ) + S(C|b R J ) ≥ S(B)ρ + S(A|B)ρ + S(C|B)ρ , (56) where in the second line we have used ordinary subadditivity of entropy, and in the fourth line the fact that is obtained from ρ by a unital c.p.t.p. map on B; it can only increase the entropy, and, since it induces c.p.t.p. maps from B to J b L and J b R , we can use the non-decrease of the conditional entropy under processing of the condition (that’s basically strong subadditivity). That means, for given dimensions d A , d B , dC , we may define the continuous and monotonic real function (t; d A , d B , dC ) := max (ρ ABC ) : I (A : C|B)ρ ≤ t ,
(57)
which has the property (t; d A , d B , dC ) = 0 if and only if t = 0 and (t; d A , d B , dC ) ≥ t (for not too large t, i.e. t ≤ 2 log min {d A , dC }).
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V. Pure States Here we give some results when ρ is a pure state. The entropy of the density matrix of a pure state is zero, and the minimum over τ j of the j th von Neumann entropy term in ( j) the sum in Eq. (28) is the entanglement of purification of ρ AC , [21]. Thus, we arrive at the formula ( j) min D (ψω[δ, τ ]) = H (q) + 2 (58) q j E P ρ AC . τ
j
Using the calculation of E P for symmetric and antisymmetric states in [8], we can now show: Theorem 5. Let A and C be systems of the same dimension d. For any pure state ψ ABC such that ρ AC = Tr B ψ ABC is supported either on the symmetric or on the antisymmetric subspace of AC, we have S(ρ A ) ≤ (ψ) ≤ 2S(ρ A ).
(59)
Proof. The upper bound can be simply derived by considering a single decomposition of system B and calculating the value of D(ψω[δ, τ ]). Since (ρ) is a minimum over all possible decompositions of system B, choosing one will immediately give an upper bound. Consider the following decomposition: B = b L ⊗ b R := B ⊗ C.
(60)
This gives a single term of tensor products leading to the following density matrix: ω[δ, τ ] = ρ AB ⊗ ρC ,
(61)
D(ψω[δ, τ ]) = 2E P (ρ AC ).
(62)
therefore we have,
A property of the entanglement of purification [21] is that if a two-party state ρ AC is completely supported either on the symmetric or antisymmetric subspace of AC then the entanglement of purification is simply the entropy of reduced state of one of the parties [8], E P (ρ AC ) = S(ρ A ) = S(ρC ).
(63)
Hence we prove the upper bound. The lower bound is a consequence of strong subadditivity of quantum entropy. We know from Eq. (63) that ( j) ( j) (ρ) = H (q) + 2 q j S(ρ A ) ≥ H (q) + q j S(ρ A ). (64) j
j
Note, however that H (q) +
j
⎛ ( j)
q j S(ρ A ) ≥ S ⎝
⎞ ( j) q j ρ A ⎠ = S(ρ A ).
(65)
j
Hence we have shown the lower bound and this concludes the proof.
Robustness of Quantum Markov Chains
301
VI. Examples In this section we examine families of states which we can use to numerically illustrate the bounds on (ρ). We look at two families of examples: first, on three qubits, Example 6. Consider the following family of three qubit states: 1 |ψ(x) ABC := √ (|ϕx A |0 B |ϕx C + |ϕ−x A |1 B |ϕ−x C ) , (66) 2 √ where |ϕx := 1 − x 2 |0 + x|1, and x is a real parameter. Using the notation y = √ 1 − x 2 so that y 2 + x 2 = 1, we can calculate the following reduced density matrices for this pure state: 2 0 y , (67) ρ A = ρC = 0 x2 ρB =
1 2
1 (y 2 − x 2 )2
(y 2 − x 2 )2 1
.
(68)
Therefore we can calculate the entropy of each single party density matrix. S(ρ A ) = S(ρC ) = −y 2 log y 2 − x 2 log x 2 = H2 (x 2 ),
(69)
S(ρ B ) = −(y 4 + x 4 ) log (y 4 + x 4 ) − 2x 2 y 2 log 2x 2 y 2 .
(70)
From Theorem 5 we know that for totally symmetric or totally anti-symmetric states, S(ρ A ) ≤ (ρ) ≤ 2S(ρ A ). Note also that for this state I (A : C|B)ψ(x) = S(AB) + S(BC) − S(B) = 2S(A) − S(B). Thus, we wish to understand the ratio S(ρ A ) . 2S(ρ A ) − S(ρ B )
(71)
If we look at the leading order terms of the single party entropies, since 0 < x < 1, we know that x 2 log x and x 2 are of lower order than x 4 , x 6 , etc. Thus, only taking x 2 log x and x 2 terms, S(ρ A ) = −(1 − x 2 ) log (1 − x 2 ) − 2x 2 log x x2 2x 2 ln x + + O(x 4 ), =− ln 2 ln 2 S(ρ B ) = −(1 − 2x 2 + 2x 4 ) log(1 − 2x 2 + 2x 4 ) −2x 2 (1 − x 2 )[1 + 2 log x + log(1 − x 2 )] 4x 2 ln x 2x 2 + − 2x 2 + O(x 4 ). =− ln 2 ln 2
(72)
(73)
Inserting these expressions, we find ln 2 (ρ) S(ρ A ) =− + O(1) as x → 0. ≥ I (A : C|B)ρ 2S(ρ A ) − S(ρ B ) ln x
(74)
Therefore for this state we can make this quantity approach +∞ as the value of x decreases, marking a striking deviation from the classical case.
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Example 7. Another use of Theorem 5 is for the pure states |ζ (d) ABC on systems A and C of dimension d and B of dimension d(d + 1)/2: namely, |ζ (d) is the purification of the completely mixed state on the symmetric subspace of AC, i.e. ζ AC = Tr B ζ ABC is proportional to the symmetric subspace projector, of rank d(d + 1)/2. For this family of states, we have I (A : C|B)ζ (d) = 1 + log
d < 1, d +1
(75)
while according to our theorem, (ζ (d)) ≥ S(A) = log d.
(76)
This example shows that not only must any bound on depend nonlinearly on I (A : C|B), but that a log-dimensional factor is also necessary. Example 8. Consider the class of states p j | j j| A ⊗ |ψ j ψ j | B ⊗ | j j|C , ρ ABC =
(77)
j
characterised by an ensemble of pure states { p j , |ψ j } on B – the states of A and C are meant to be mutually orthogonal states. For a POVM (Mk ) on B, and using the previous notation of B = K b L b R , the optimal state is given by p j | j j| A ⊗ Mk |ψ j ψ j | Mk L R ⊗ |kk| K ⊗ | j j|C . (78) ω[δ, τ ] A BC = jk
b b
We can calculate the following using formula (41), using the fact that the system is symmetric in systems A and C and S(ρ) = S(A), D(ρω[δ, τ ]) = −S(A) + S(K ) + S(Ab L |K ) + S(Ab R |K ).
(79)
It is fairly clear from the formula since S(Ab L |K ) + S(Ab R |K ) ≥ 2S(A|K ) that the optimal choice of b L b R is to make one trivial, the other B, so that D(ρω[δ, τ ]) = −S(A) + S(K ) + 2S(A|K ) = S(A|K ) + S(K |A),
(80)
all entropies relative to the state ω. Note that A and K are essentially classical registers, so that the above is really a classical probabilistic/entropic formula for the relative entropy. It is also quite amusing to see a quantity appearing that is known as information-distance in other contexts (see e.g. [5]). VII. Conclusions We have investigated the relation between the quantum conditional mutual information of a three-party state, and its relative entropy distance from the set of all (short) quantum Markov chains. While the latter is always larger or equal than the former, with equality in the classical case, in general the relative entropy distance can be much larger than the conditional mutual information. We showed this by developing tools to lower bound the relative entropy distance, in particular for pure states of a special symmetric form. In the process we found many useful properties of the minimum relative entropy distance
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from Markov states. Our findings indicate that the characterisation of quantum Markov chains in terms of vanishing quantum conditional mutual information is not robust, or at least not at all like the classical case, or the (quantum and classical) case of ordinary mutual information. Since these lower bounds are additive for tensor products of states, this surprising and perhaps displeasing behaviour will not go away in an asymptotic limit of many copies of the state. What we haven’t found is an upper bound of the relative entropy distance in terms of the conditional mutual information I (A : C|B); our examples above show that such a bound has to depend nonlinearly on I and it has to contain a factor proportional to the logarithm of one or more of the local dimensions. Note that if there were a bound of the form (ρ) ≤ f (I ) log(d A dC ) – in particular not depending on the dimension of B –, then this would settle a question left open in [7]: namely, it would imply that the “squashed entanglement” E sq (ρ AB ) of a bipartite state ρ AB is zero if and only if the state is separable. (We are grateful to Paweł Horodecki for pointing this out to us.) We close by pointing out that our results cast doubts on earlier ideas of two of the present authors (NL and AW), reported in [18], on how to prove a non-standard inequality for the von Neumann entropy. The heuristics given there don’t seem to bear out, in the light of the present paper; of course, the conjectured entropy inequality itself may well still be true. Acknowledgements. It is our pleasure to acknowledge discussions on the topics of this paper with Michał and Paweł Horodecki. BI, NL and AW acknowledge support from the U.K. Engineering and Physical Sciences Research Council through “QIP IRC”; NL and AW furthermore were supported through the EC project QAP (contract IST-2005-15848), and AW gratefully acknowledges support via a University of Bristol Research Fellowship.
Note added in proof: Anna Jenˇcová has kindly pointed out to us a very simple proof of Theorem 4 as well as other properties of . Namely, using the framework of Proposition 1, the following holds. Let ρABC be the state of a tripartite system and let µ ABC be a Markov state. Then the conditional mutual information satisfies I (A : C|B)ρ = D(ρ ABC µ ABC ) + D(ρ B µ B ) − D(ρ AB µ AB ) − D(ρ BC µ BC ). It follows immediately that (ρ) ≥ I (A : C|B)ρ . Further, it follows that (ρ) = I (A : C|B)ρ if and only if there exists a Markov state µ, such that ρ AB = µ AB and ρ BC = µ BC . (This is always satisfied in the classical case.) References 1. Accardi, L., Frigerio, A.: Markovian cocycles. Proc. Roy. Irish Acad. 83(2), 251–263 (1983) 2. Alicki, R., Fannes, M.: Continuity of quantum conditional information. J. Phys. A: Math. Gen. 37, L55–L57 (2004) 3. Bennett, C.H., Wiesner, S.: Communication via one- and two-particle operators on Einstein-PodolskyRosen states. Phys. Rev. Lett. 69, 2881–2884 (1992) 4. Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70(13), 1895–1899 (1993) 5. Bennett, C.H., Gács, P., Li, M., Vitányi, P.M.B., Zurek, W.H.: Information Distance. IEEE Trans. Inf. Theory 44(4), 1407–1423 (1998) 6. Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Algebra and Appl. 10, 285–290 (1975) 7. Christandl, M., Winter, A.: Squashed entanglement: An additive entanglement measure. J. Math. Phys. 45(3), 829–840 (2004)
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8. Christandl, M., Winter, A.: Uncertainty, Monogamy, and Locking of Quantum Correlations. IEEE Trans. Inf. Theory 51(9), 3159–3165 (2005) 9. Cover, T.M., Thomas, J.A.: Elements of Information Theory. New York: John Wiley & Sons, Inc. 1991 10. Davies, E.B.: Information and Quantum Measurement. IEEE Trans. Inf. Theory. 24, 596–599 (1978) 11. Fannes, M.: A continuity property of the entropy density for spin lattice systems. Commun. Math. Phys. 31, 291–294 (1973) 12. Fuchs, C.A., van de Graaf, J.: Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inf. Theory 45(4), 1216–1227 (1999) 13. Hayden, P., Jozsa, R., Petz, D., Winter, A.: Structure of states which satisfy strong subadditivity of quantum entropy with equality. Commun. Math. Phys. 246(2), 359–374 (2004) 14. Horodecki, K., Horodecki, M., Horodecki, P., Oppenheim, J.: Information Theories with Adversaries, Intrinsic Information, and Entanglement. Found. Physics 35(12), 2027–2040 (2005) 15. Ibinson, B., Linden, N., Winter, A.: All inequalities for the relative entropy. Commun. Math. Phys. 269(1), 223–238 (2007) 16. Ibinson, B., Linden, N., Winter, A.: in preparation, 2006 17. Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14, 1938–1941 (1973) 18. Linden, N., Winter, A.: A new inequality for the von Neumann entropy. Commun. Math. Phys. 259(1), 129–138 (2005) 19. Ohya, M., Petz, D.: Quantum Entropy and Its Use. 2nd edition, Berlin: Springer Verlag, 2004 20. Petz, D.: Sufficiency of channels over von Neumann algebras. Quart. J. Math. Oxford Ser. (2) 39(153), 97–108 (1988) 21. Terhal, B.M., Horodecki, M., Leung, D.W., DiVincenzo, D.P.: The entanglement of purification. J. Math. Phys. 43(9), 4286–4298 (2002) Communicated by M.B. Ruskai
Commun. Math. Phys. 277, 305–321 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0360-x
Communications in
Mathematical Physics
Improved Estimates for Correlations in Billiards N. Chernov1 , H.-K. Zhang2 1 Department of Mathematics, University of Alabama at Birmingham, Birmingham, USA.
E-mail: [email protected]
2 Department of Mathematics and Physics, North China Electric Power University, Baoding,
Hebei, 071003, China. E-mail: [email protected] Received: 15 November 2006 / Accepted: 14 April 2007 Published online: 18 October 2007 – © Springer-Verlag 2007
Abstract: We consider several classes of chaotic billiards with slow (polynomial) mixing rates, which include Bunimovich’s stadium and dispersing billiards with cusps. In recent papers by Markarian and the present authors, estimates on the decay of correlations were obtained that were sub-optimal (they contained a redundant logarithmic factor). We sharpen those estimates by removing that factor.
1. Introduction Here we sharpen estimates on mixing rates (i.e. the decay of correlations) for several classes of chaotic billiards, including the celebrated stadium introduced by Bunimovich [4,5] and dispersing tables with cusps studied by Machta [15,16]. In all our models the billiard map is known to be hyperbolic, as well as ergodic and Bernoulli, but its hyperbolicity is very non-uniform and consequently its mixing rates are slow (polynomial). Physicists described this phenomenon as “intermittent chaos”. If you watch a typical trajectory of a billiard with polynomial mixing rates, then you observe that periods of truly chaotic behavior alternate with long regular-looking cycles when the orbit remains confined to a small and special part of phase space. That part acts as a ‘trap’ and that trap will play an important role in our analysis. The study of mixing rates in intermittent chaotic systems is more difficult than that of truly chaotic ones, and the resulting estimates may depend on delicate details of the dynamics in the traps. Our two main models are stadia. Bunimovich’s original stadium is a convex billiard table bounded by two equal semicircles and two parallel straight lines, see Fig. 1. Due to its simplicity and geometric appeal it became popular in many theoretical and experimental studies, see for example [2,8,22,26] and [25, Sect. 5.3]. We also discuss a ‘skewed’ stadium shown on Fig. 2, which is a convex domain bounded by two unequal circular arcs and two non-parallel lines; we call it ‘drivebelt’ due to its shape. Note that both types of stadia have C 1 , but not C 2 , boundary.
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Fig. 1. Bunimovich’s (straight) stadium
Fig. 2. Skew stadium (‘drivebelt’ table)
Another interesting class of billiards with slow decay of correlations is made by tables with concave (dispersing) boundary that includes cusps (corner points with zero interior angle). For example, Fig. 3 shows a table studied by Machta [15,16] made by three identical circular arcs. We also consider billiards in a square with a small fixed circular obstacle removed (Fig. 4). Such models are known as semi-dispersing billiards. Let denote the boundary of a billiard table and M = ×[−π/2, π/2] the standard collision space whose canonical coordinates are r, ϕ, where r is the arc length parameter on and ϕ ∈ [−π/2, π/2] the angle of reflection, see Fig. 1. The collision map F : M → M taking a collision point to the next collision, see Fig. 1, preserves smooth measure dµ = c cos ϕ dr dϕ on M, here c = (2||)−1 is normalizing constant. Let f, g ∈ L 2µ (M) be two functions. Correlations are defined by n ( f ◦ F ) g dµ − f dµ g dµ. (1.1) Cn ( f, g, F, µ) = M
M
M
It is well known that F : M → M is mixing if and only if lim Cn ( f, g, F, µ) = 0
n→∞
∀ f, g ∈ L 2µ (M).
(1.2)
The rate of mixing of F is characterized by the speed of convergence in (1.2) for smooth enough functions f and g. We will always assume that f and g are Hölder continuous or piecewise Hölder continuous with singularities that coincide with those of the map F k for some k. For example, the free path between successive reflections is one such function. Bunimovich proved that under very general conditions (that easily hold for both types of stadia) billiards bounded by circular arcs and straight lines are hyperbolic, ergodic and K-mixing [3–5]. Due to other general results, these systems are also Bernoulli
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Fig. 3. Machta’s table with three cusps
Fig. 4. Semi-dispersing billiard
[10,18]. Similar results were proved for semi-dispersing billiards and dispersing tables with cusps by Sinai [20], Reháˇcek [19] and others. It has been long expected based on heuristic analysis [22,15] that correlations decay as O(1/n), but rigorous estimates were obtained only recently: Markarian [17] proved that |Cn ( f, g, F, µ)| ≤ const · (ln n)2 /n
(1.3)
for the ‘straight’ Bunimovich stadia; the present authors [11] extended this result to the drivebelt stadia and semi-dispersing billiards, and lastly one of us with Markarian [13] derived the same bound on correlations for tables with cusps. It was clear to all of us [17,11,13] that the logarithmic factor (ln n)2 was just an artifact of our method. Here we refine the method and remove that factor: Theorem 1.1. For both types of stadia, semi-dispersing billiards, and dispersing tables with cusps the correlations (1.1) for the billiard map F : M → M and piecewise Hölder continuous functions f, g on M decay as |Cn ( f, g, F, µ)| ≤ const/n.
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We note that Bálint and Gouëzel very recently obtained a lower bound on correlations for stadia: they proved that |Cn ( f, g, F, µ)| ≥ const/n for infinitely many n’s, see details in [1, Corollary 1.3]. Thus the speed of the decay of correlations for stadia is now completely determined. 2. General Scheme We start by briefly repeating the general scheme for the analysis of correlations for nonuniformly hyperbolic maps developed in [17,11]. The first step is to localize places in the phase space M where the hyperbolicity of the map F : M → M deteriorates (becomes non-uniform). In those places long sequences of iterations of the map F occur without exponential divergence of nearby trajectories. For example, in stadia, see [3–5] and [12, Chap. 8], hyperbolicity is ensured by the ‘defocusing mechanism’, which works when the billiard particle moves from one circular arc to the other, hence during long series of consecutive collisions with the same arc or bounces between two flat sides, the hyperbolicity weakens. This suggests to us to reduce the space M to a subset M ⊂ M on which the induced (first return) map F : M → M would be uniformly hyperbolic. For stadia [11] we define M ⊂ M as a set consisting of first (initial) collisions with every circular arc, that is, M = {x ∈ M : x lies on an arc C ⊂ and F −1 x ∈ / C}. It is proven in [11] that indeed the first return map F : M → M is uniformly hyperbolic (more precisely, we proved uniform expansion and contraction of, respectively, unstable and stable tangent vectors). A similar reduction of the collision space was constructed for semi-dispersing billiards and tables with cusps [13]. It is also proven in [11,13] that in all these cases the reduced map F : M → M enjoys exponential decay of correlations. This was done by constructing Young’s tower [23] in M. That tower plays the role of a Markov partition of M; its full description is fairly complicated, but we only need two elements of it here. The first element is the ‘base of the tower’ 0 ⊂ M, which Young calls a ‘horseshoe with hyperbolic structure’. For us, its structure is irrelevant, we can just think of 0 as a subset of M of positive measure. Then for a.e. point x ∈ M, its orbit {F n x} makes infinitely many returns to 0 , according to the Poincaré theorem. Young only counts ‘proper returns’ (or ‘Markov returns’), as she defines in [23], but for us the exact meaning of proper returns is not relevant. Young proved that a.e. point x ∈ M properly returns to 0 infinitely many times. Let R(x; F, 0 ) denote the time of the first proper return of x to 0 (under F). The second important element of Young’s tower construction is the exponential tail bound µ (x ∈ M : R(x; F, 0 ) > n) ≤ const · θ n
∀n ≥ 1,
(2.1)
where θ < 1 is a constant. Next we turn back to the map F : M → M. The tower in M can be easily (and naturally) extended to M, thus we get a bigger tower (with the same base 0 ⊂ M); and a.e. point x ∈ M again properly returns to 0 (now under F) infinitely many times. For every x ∈ M, let R(x; F, 0 ) denote the time of the first proper return of x to 0 (under F). Since M is larger than M, it takes typical points longer to return to 0 . In fact, only a polynomial tail bound on return times presumably holds: µ (x ∈ M : R(x; F, 0 ) > n) ≤ const · n −1
∀n ≥ 1.
(2.2)
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Assuming that (2.2) holds, the bound on correlations in Theorem 1.1 immediately follows from Young’s general result [24]. Our goal is to prove (2.2). Consider the return times to M under F, i.e. R(x; F, M) = min{r ≥ 1 : F r (x) ∈ M}
(2.3)
for x ∈ M. The following estimate is standard for systems under consideration [7,13, 11,17,22]: µ(x ∈ M : R(x; F, M) > n) ≤ const · n −1
∀n ≥ 1.
(2.4)
µ(x ∈ M : R(x; F, M) > n) ≤ const · n −2
∀n ≥ 1.
(2.5)
Equivalently,
The equivalence of (2.4) and (2.5) is proven in [11]. Now for every n ≥ 1 and x ∈ M denote r (x; n, M) = #{1 ≤ i ≤ n : F i (x) ∈ M} and An = {x ∈ M : R(x; F, 0 ) > n}, Bn,b = {x ∈ M : r (x; n, M) > b ln n}, where b > 0 is a constant to be chosen shortly. By (2.1), µ(An ∩ Bn,b ) ≤ const · n θ b ln n . Let us choose and fix b > 0 large enough so that n θ b ln n < n −1 . It remains to prove the following: Proposition 2.1. µ(An \Bn,b ) ≤ const · n −1 . This proposition constitutes the main result of our paper and will be proven in the next sections. Proposition 2.1 concludes the proof of (2.2). Now Theorem 1.1 readily follows from Young’s general result [24]. To conclude this section, we recall how µ(An \Bn,b ) is estimated in [17,11], this should clarify our ideas. Since points x ∈ An \Bn,b return to M at most b ln n times during the first n iterates of F, it is observed in [17,11] that there are ≤ b ln n time intervals between successive returns to M, and hence the longest such interval, we call it I , has length ≥ n/(b ln n). Applying the bound (2.5) to the interval I gives µ(An \Bn,b ) ≤ const · n (ln n)2 n −2
(2.6)
(the extra factor of n must be included because the interval I may appear anywhere within the longer interval [1, n], and the measure µ is invariant). This gives us a weaker version of (2.2): µ(x ∈ : R(x; F, 0 ) > n) ≤ const · (ln n)2 n −1
∀n ≥ 1.
Now Young’s general result [24] implies the sub-optimal correlation bound (1.3), which is the main result of [17,11]. But it is clear that the estimate (2.6) can only be sharp if most of the intervals between returns to M have length ∼ n/ ln n. This is, however, the ‘worst case scenario’, which is extremely unlikely due to a special character of the dynamics between returns to M. We explore these special features to improve the estimate on µ(An \Bn,b ) in this paper.
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3. Analysis of the Set An \Bn,b In this section we develop a strategy of the proof of Proposition 2.1 in the case of stadia. (Semi)dispersing billiards require a different approach that will be discussed in Sect. 5. The set An \Bn,b consists of points x ∈ M whose images under n iterations of the map F satisfy two conditions: (i) they never return to the ‘base’ 0 of Young’s tower and (ii) they return to M at most b ln n times. Our goal is to show that µ(An \Bn,b ) = O(n −1 ). The following subsets of M: Mm = {x ∈ M : R(x; F, M) = m + 1} are called m-cells, m ≥ 0. Here M0 constitutes the ‘bulk’ of the space M and M1 , M2 , . . . are (usually) small regions for which we make the following assumptions. First, their measures decrease polynomially: µ(Mm ) ≤ C/m r ,
(3.1)
where r ≥ 3 and C > 0 are constants. (In all our models, r = 3, see [11,13], but we adopt a more general assumption here.) Second, if x ∈ Mm then F(x) ∈ Mk with β −1 m − C ≤ k ≤ βm + C.
(3.2)
Here β > 1 is another constant. We will denote by C > 0 various constants whose exact values are not important. Our next assumption concerns transition probabilities ‘from cells to other cells’: pk,m 1 ,...,m t = µ(Mk /F Mm 1 ∩ F 2 Mm 2 ∩ · · · ∩ F t Mm t ),
(3.3)
where k, m 1 , . . . , m t ≥ 2 are indices and µ(A/B) = µ(A ∩ B)/µ(B) denotes the conditional measure. If we fix a sequence m 1 , . . . , m t , then k = km 1 ,...,m t becomes a random variable with probability distribution { pk,m 1 ,...,m t }. We will also use the random variable ξm 1 ,...,m t = ln(km 1 ,...,m t /m 1 ).
(3.4)
Note that |ξm 1 ,...,m t | ≤ ln β + O(1/k), due to (3.2). It is known that in the billiards under consideration the distribution of ξm 1 ,...,m t weakly converges to a fixed probability distribution on the interval [− ln β, ln β], as m 1 , . . . , m t → ∞. The reason is that the map F on the cells Mm with high indices m can be well approximated by a stationary random walk; this fact was already explored in [1, Sect. 4]. We only need a somewhat weaker property here, we state it as an assumption below and prove it in the next section. We assume that for any m ≥ 2 there exists a subset M˜ m ⊂ Mm such that µ(Mm \ M˜ m )/µ(Mm ) ≤ C/m d
(3.5)
for some d > 0, and for any t ≥ 1 and m 1 , . . . , m t the transition probabilities µ(Mk /F M˜ m 1 ∩ F 2 M˜ m 2 ∩ · · · ∩ F t M˜ m t ) define a random variable k = k˜m 1 ,...,m t so that the logarithmic variable ξ˜m 1 ,...,m t = ln k˜m 1 ,...,m t /m 1
(3.6)
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satisfies ξ˜m 1 ,...,m t ≤ η,
(3.7)
where η is a random variable supported on the interval [− ln β, ln β + 1] and having a negative mean value η¯ = E(η) < 0.
(3.8)
We stress that the distribution of η is fixed (independent of m 1 , . . . , m t ). Note that our approximation of (3.3) by (3.6) is only good as long as all m 1 , . . . , m t are large, due to (3.5). Proposition 3.1. Under the above assumptions µ(An \Bn,b ) = O(n −r +2 ) The rest of this section is devoted to the proof of this proposition. For every point x ∈ An \Bn,b we consider all its returns to M within the first n iterations of F, i.e. all 0 ≤ i 1 , . . . , i J∗ ≤ n such that F i j (x) ∈ M. Note that the sequence {i 1 , . . . , i J∗ } and its length J∗ depend on x and recall that J∗ ≤ b ln n. For every i j we have F i j (x) ∈ Mm j for some m j ≥ 1. We fix a small q > 0 and say that m j is large if m j ≥ n q . We call an island a subinterval I ⊂ [0, n] such that for all k ∈ I the point F k (x) either belongs to M\M or lies in an m-cell Mm with a large m, i.e. m ≥ n q . Each island terminates at 0 or n or at a point k satisfying F k (x) ∈ Mm with some ‘small’ m < n q . Let Imax ⊂ [0, n] be the longest island (note that Imax depends on x). Lemma 3.2. There is a constant κ = κ(b, β, q) > 0 such that for every x ∈ An \Bn,b we have |Imax | ≥ κn. Proof. The point x will be fixed throughout the proof. Consider an arbitrary island I . If it does not terminate at 0 or n, then due to (3.2) we have #{k ∈ I : F k (x) ∈ M} ≥ t, where t is the smallest integer satisfying |I | ≤ n q (1 + β + · · · + β t ), hence t≥
ln |I | − ln n q + O(1) . ln β
Now it is easy to see that ln |Imax | − ln n q + O(1) n × ≤ b ln n. ln β |Imax | Suppose q < 1/2, then for large n we have |Imax | ≥
n ≥ n 2q , b ln n
thus ln |Imax | − ln n q ≥ q ln n, which implies the lemma with any choice of κ < q/(b ln β).
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Our further analysis is done within the maximal island Imax = [K 0 , K 1 ]. Given x ∈ An \Bn,b , we call a subinterval J ⊂ Imax a run if F k (x) ∈ / M for every k ∈ J . Let Jmax = [n 0 , n 1 ] be the longest run within Imax . As the number of runs does not exceed b ln n, we have |Jmax | ≥
κn |Imax | ≥ . b ln n b ln n
Without loss of generality, we assume that K 1 − n 1 ≥ n 0 − K 0 , i.e. the right subinterval of Imax \Jmax is at least as long as the left one (because the time reversibility of the billiard dynamics allows us to turn the time backwards). Before going into further detail, we describe the idea of the proof. If |Jmax | ∼ n, then due to (3.1) the measure of the corresponding points x is O(n −r ). Summing over all possible n 0 and n 1 gives the measure bound O(n −r +2 ) as claimed by Proposition 3.1. The problem may arise when |Jmax | = o(n), because the measure of such points x will be O(|Jmax |−r ) n −r . But then we will show that, due to (3.8), for typical points y ∈ M|Jmax | we have F t (y) ∈ Mm t , where m t decreases exponentially fast. Then if we add up all runs covered by the trajectory F k (x) of x during the interval n 1 ≤ k ≤ K 1 , we get O(|Jmax |). On the other hand, K 1 − n 0 ≥ κn/2. This will prove that |Jmax | ≥ cn for some c > 0, taking us back to the case |Jmax | ∼ n, which is handled already. Let n 1 < · · · < n s ≤ K 1 be all the moments such that F n t (x) ∈ M. We need to estimate the measure of points x such that (a) F n 0 (x) ∈ M|Jmax | ; (b) F n t (x) ∈ Mm t for n q ≤ m t ≤ m 0 = |Jmax | for all t = 0, . . . , s; (c) m 0 + · · · + m s ≥ κn/2 and s < b ln n. We will use the assumptions (3.5)–(3.8). At each iteration of F we incur relative losses bounded by O(1/m dt ) = O(1/n qd ) due to (3.5), thus the total losses are bounded by r µ(M Jmax )/n qd = O(n −r −qd ln n), which is n −r . Next our assumptions (3.5)–(3.8) allow us to estimate the cell index m t from above m t ≤ m˜ t = m 0 eη1 +···+ηt ,
(3.9)
where η1 , . . . , ηt are independent random variables having the same distribution as η. The probability distribution here is induced by the measure on M˜ m 0 , so we denote it by Pm 0 . Of course it is possible that the random variable m˜ t defined by (3.9) will exceed |Jmax | or fall below n q , thus violating the above restriction (b) on m t . But this only means that our probabilistic estimates will exceed the actual measure of points satisfying (a)–(c). Since we are estimating our measures from above, this approach is logically consistent. Lemma 3.3. For every ε > 0 there are C > 0 and γ ∈ (0, 1) such that for all t ≥ 1 we have the following probability estimate: Pm 0 (η1 + · · · + ηt > (η¯ + ε)t) ≤ Cγ t . Proof. This follows from the classical theorem on large deviations, see e.g. [14, Theorem 2.2.3].
¯ We fix ε > 0 such that η¯ + ε < 0, i.e. θ : = eη+ε < 1 and then fix the corresponding γ < 1. Thus with an overwhelming probability we should have m˜ t ≤ m 0 θ t for all large enough t. This idea lies behind the following lemma.
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Lemma 3.4. There is C > 0 such that n
µ(Mm 0 ) Pm 0 m˜ t > m 0 θ t for some t > κn/(10m 0 ) ≤ Cn −r +1 .
m 0 =n/(b ln n)
Proof. Due to the previous lemma, our probability is bounded by n
µ(Mm 0 )
m 0 =κn/(b ln n)
Cγ κn/10m 0 . 1−γ
Since µ(Mm 0 ) = O(m −r 0 ), we can obtain an upper bound by integral estimation: n m 0 =κn/(b ln n)
γ κn/(10m 0 ) ≤ const m r0 =
const nr −1
n
κn
x −r γ 10x d x
1
n
y r −2 γ κ y/10 dy,
1
where we used the change of variables y = n/x. It remains to note that < ∞ since γ < 1.
∞ 1
y r −2 γ κ y/10 dy
Adding over n 0 gives an upper bound O(n −r +2 ) on the measure as required by Proposition 3.1. Finally, if m t ≤ m 0 θ t for all t > κn/(10m 0 ), then due to the above restrictions (b) and (c) we have ∞
κn m0 m 0 κn κn ≤ m 0 + · · · + mr ≤ + , + m0θ t ≤ 2 10m 0 10 1 − θ t=1
hence m 0 ≥ (1 − θ )κn/3 and µ(Mm 0 ) = O(n −r ). This completes the proof of Proposition 3.1. 4. Dynamics in Cells for Stadia It remains to prove our assumptions (3.5)–(3.8). They essentially follow from the Markovian character of the map F restricted to cells Mm with high indices m. First we do this for Bunimovich’s ‘straight’ stadia, for which the structure of m-cells is well known [6,7,11,17,22] and the Markovian character of the map F is already explored in [1]. Here we just recall relevant facts. First of all, the m-cell Mm for any large m is a union of domains of two types: one consists of points whose trajectories experience a long series of consecutive collisions at the same semicircle, and the other is made of points whose trajectories bounce between the two parallel flat lines. The first type domains are small, they have measure ∼ m −4 , which is negligible. The second type domains have measure ∼ m −3 and we only consider them. Slightly abusing notation we will call those domains m-cells. The set M ⊂ M is the union of two identical parallelograms in the r, ϕ coordinates, each one is constructed on one semicircle C ⊂ . Figure 5 shows one of them (C E D B), and the m-cells make a nested structure of self-similar domains converging to the vertices C and D as m → ∞.
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Fig. 5. One half of the set M
Each m cell consists of two identical parts, one near C and the other near D. Due to the obvious symmetry, we only describe the domain of Mm near C. It, in turn, is a union of two strips, one (Mm1 ) below the line C F and the other (Mm2 ) above the line C F. The domain Mm1 contains points mapped (by F) directly to a straight side of the stadium; the domain Mm2 contains points that are mapped by F onto the same arc (but to its almost diametrically opposite point) and then to a straight side. Figure 6 shows the image F(Mm ) of an m-cell. These images, too, make a nested structure of self-similar domains converging to the vertices C and D as m → ∞. The intersection of k-cells with the image F(Mm ) is depicted on Fig. 6. Each F(Mm ) intersects Mk with 1 3m
− O(1) ≤ k ≤ 3m + O(1).
(4.1)
Most of the intersections F(Mm ) ∩ Mk are parallelogram-looking domains, but a few (≤ const) intersections near the ends of the strip F(Mm ) look like less regular polygons with 3, 4 or 5 sides. To clarify our ideas, let us assume for a moment that the domains Mm1 and Mm2 are exact trapezoids which shrink homotetically as m growth. Also let the strip F(Mm ) be a perfect trapezoid that scales with m, i.e. shrinks homotetically as m → ∞. Let the measure µ have constant density and the map F be linear within every m-cell. Lastly, let us ignore the irregular intersections F(Mm ) ∩ Mk , i.e. assume that all of these intersections are parallelograms, and (4.1) holds without the O(1) terms. Under these ideal conditions, it is easy to see that the cells Mm would make a Markov partition, so the action of F would be equivalent to a discrete Markov chain. Moreover, the random variables (3.4) would be almost identically distributed, i.e. their distribution would not depend on m 1 , . . . , m t and t, modulo a O(1/k) error term accounting for their discrete character (and of course, as long as all the indices are ≥ 3). Effectively, the transitions between cells could be described by a sequence of independent random variables. It is known, see [1, Eq. (35)], that
µ (Mk /F(Mm )) =
1 3m + O 8k 2 m2
(4.2)
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Fig. 6. The domains Mk1 and Mk2 and the image F(Mm ) near the vertex C. The vertex C has coordinates r = 0 and ϕ = 0; other coordinates are shown, to the leading order
for k ∈ [m/3 − C, 3m + C]. Under our ideal assumptions, the same estimate would hold for multistep transition probabilities (3.3). Observe that 3m 3m 5 3m 3m k x ln ∼ ln d x = 1 − ln 3 < 0. 2 8k 2 m 8x m 4 m/3
(4.3)
k=m/3
Thus one can easily find a random variable η satisfying (3.7) and (3.8). Now in reality cell boundaries are curvilinear, the density of µ is not constant, and the map F is nonlinear. We just need to estimate the effect of nonlinearity in order to prove our assumptions (3.5)–(3.8). This was in fact done in [1, Sect. 4], we only outline the argument here. For one-step transition probabilities, i.e. for t = 1 in (3.6), this can be done by a direct and fairly elementary analysis. The measure µ has density cos ϕ, and |ϕ| = O(m −1 ) on Mm , so the density variation over Mm is just O(m −2 ). The derivative of the map F : (r, ϕ) → (r1 , ϕ1 ) is given by (see [12, Chap. 2]) −1 R −1 τ + cos ϕ τ DF = , cos ϕ1 R −2 τ + R −1 cos ϕ1 + R −1 cos ϕ τ R −1 + cos ϕ1 where R is the radius of the semicircles bounding the stadium (Fig. 1). Within the m-cell, we have
τ = 4R 2 m 2 + L 2 + O(m −2 ), where L is the length of the flat lines bounding the stadium. Thus the derivative of F 1 varies by O(m −2 ) within every m-cell. Lastly, the curve separating Mm1 from Mm−1 has equation r = Rϕ + R sin−1
R sin ϕ (2m
− 1)2 R 2
+
L2
+ R tan−1
L , (2m − 1)R
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2 i.e. its slope is dr/dϕ = R + O(m −1 ). The curve separating Mm2 from Mm−1 has similar equation
R sin ϕ L , r = 3Rϕ + R sin−1
+ R tan−1 2 2 2 (2m − 1)R (2m − 1) R + L i.e. its slope is dr/dϕ = 3R + O(m −1 ). So the curves separating neighboring m-cells can be approximated in the C 1 metric by parallel straight lines, up to some O(1/m) error terms. It is now easy to check that the width of the domain Mmi , i = 1, 2, is ci m −2 + O(m −3 ), where c1 , c2 > 0 are constants. Thus our map, measure, and cells admit good linear approximations: the non-linearity only affects higher-order terms in all relevant parameters. Also, the irregular intersections F(Mm ) ∩ Mk , see above, have to be thrown out (of Mm ), but their relative measure is O(1/k). We conclude that (3.5)–(3.8) hold for t = 1, we can even afford a luxury to set d = 1 in (3.5). For multi-step transition probabilities, i.e. for t ≥ 2, such a direct analysis is hardly feasible, so one needs a more sophisticated argument. In [1, pp. 488–490], multi-step transition probabilities are estimated by foliating each m-cell Mm by unstable curves {W u } which stretch completely across Mm , i.e. terminate on its ‘long’ sides (separating Mm from Mm−1 and Mm+1 ). Such a foliation can be chosen smooth enough so that conditional measures on the fibers W u are nearly uniform. To analyze multistep transition probabilities (3.3), we would like our foliations to be F-invariant under relevant iterations of F, i.e. until the image of a fiber either falls into an irregular intersection F(Mm ) ∩ Mk , see above, or lands in the ‘bulk’ M0 . Such a ‘limited’ invariance can be ensured with a little extra work [1, pp. 488–490]. We describe here an alternative approach that gives the invariance of the foliation ‘for free’. Let us foliate m-cells by unstable manifolds of the map F, so that in each cell Mm only unstable manifolds that stretch across Mm completely (terminating on its long sides) are used. The invariance of this foliation under F is then automatic. A little price to pay for this convenience is to deal with ‘gaps’ between unstable manifolds. Indeed, since arbitrarily short unstable manifolds are dense in M, our foliation is ‘holey’ (it covers a Cantor-like set), there infinitely many gaps in Mm where unstable manifolds fail to reach one of the two long sides of Mm . We need to estimate the relative measure of gaps in Mm . For x ∈ M, let r u (x) denote the distance from x to the nearer endpoint of the unstable manifold passing through x (if none exists, we put r u (x) = 0). Then for any stable curve W ⊂ M and ε > 0 we have m(x ∈ W : r u (x) < ε) < Cε, where C > 0 is a constant (independent of W ) and m denotes the Lebesgue measure on W . (For the proof, see [12]: it is shown in [12, Sect. 5.12] that this estimate follows from the so-called first growth lemma in the case of dispersing billiards, and the argument applies to stadia without change; and the first growth lemma for stadia is proved in [12, Sect. 8.14].) Thus the union of all gaps has relative measure O(1/m) in Mm , so it can be simply excluded from Mm . After all bad parts of Mm are removed, as described above, we obtain the desired subset M˜ m ⊂ Mm . The conditional measure on each unstable manifold W is smooth and its density ρ satisfies d C ln ρ(x) ≤ , dx |W |1/2
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where C > 0 is a constant, see [12, Sect. 8.12]. Hence the fluctuations of the density on our fibers in F(Mm ) are bounded by maxW ρ − min W ρ ≤ C|W |1/2 ≤ Cm −1/2 . min W ρ The distortions of unstable manifolds under the map F are also estimated in [12, Sect. 8.12]: if W is an unstable manifold on which F n is smooth, then maxW JW F n − min W JW F n ≤ C|W |1/2 ≤ Cm −1/2 , min W JW F n where JW F n is the Jacobian of the map F n restricted to W , and the constant C > 0 is independent of W and n. These facts imply that the transformation of the conditional measures on unstable manifolds can be tightly approximated by a Markov chain as non-linearity only affects higher order terms. Now in order to handle multistep transition probabilities (3.6) we note that F M˜ m 1 ∩ 2 F M˜ m 2 ∩ · · · ∩ F t M˜ m t is a union of unstable manifolds in F M˜ m 1 , each stretching completely across the strip F M˜ m 1 . Thus it is enough to show that for any two such unstable manifolds W1 , W2 ⊂ F Mm 1 and the conditional measures νW1 , νW2 on them we have νW (Mk ) − νW (Mk ) ≤ Cm −1/2 , 1 2 1 for every k ∈ [m 1 /3 + C, 3m 1 − C]. This readily follows from standard estimates on the Jacobian of the holonomy map, see [12, Sect. 8.13]. In summary, we obtain (3.5)–(3.8) with d = 1/2. This concludes our analysis of the ‘straight’ Bunimovich stadium. The dynamics in the ‘drivebelt’ billiard table is studied in [11, Sect. 9]. The m-cells Mm there consist only of points experiencing long series of consecutive collisions at the same arc (as the number of possible consecutive bounces off the two nonparallel flat sides is limited). But there are two very different types of series of collisions with the same arc. First, there are ‘sliding’ trajectories (where ϕ ≈ ±π/2), just like in the straight stadium, which make a small set of measure ∼ m −4 , and so they are negligible. Second, there are trajectories bouncing off within the bigger circle almost orthogonally to its boundary (i.e. with ϕ ≈ 0), see Fig. 2. It was shown in [11, Sect. 9] that they make a set of measure ∼ m −3 , thus they are of interest to us. The structure of m-cells in the drivebelt stadium are described in [11, Sect. 9]. It is a sequence of self-similar domains accumulating at two corner points of the space M (which again consists of two parallelograms in the r ϕ coordinates). The images of the m-cells are also self-similar domains accumulating at the same corner points of M. More precisely, each F(Mm ) intersects Mk with 1 7m
− O(1) ≤ k ≤ 7m + O(1),
(4.4)
which is similar to (4.1), but now β = 7 instead of β = 3. Accordingly, the transition probabilities are µ (Mk /F Mm ) =
1 7m + O 48k 2 m2
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for k ∈ [m/7−C, 7m +C], which is an analogue of (4.2). Here 7/48 is just a normalizing factor resulting from the requirement k µ (Mk /F Mm ) = 1. A simple calculation analogous to (4.3) gives 7m k=m/7
7 k ln ∼ 2 48k m
7m m/7
50 7m x ln 7 < 0. ln d x = 1 − 2 48x m 48
Thus we get a necessary Markov approximation in this case, too. We note that the billiard dynamics in the drivebelt region is very similar to that in Bunimovich’s stadia, but because the latter has been popular for a long time, its dynamical properties have been investigated in great detail (see, for example in [12, Chap. 8]). These include sharp estimates on distortion bounds, conditional densities on unstable manifolds, and the Jacobian of the holonomy map. For the drivebelt region, such estimates are obtained, in a weaker form, in [7, Appendix 1.5]. We plan to publish separately a detailed investigation of the drivebelt stadia along the lines of [12, Chap. 8]. 5. Cell Dynamics for (Semi-)Dispersing Tables Lastly we turn to the semi-dispersing billiards (Fig. 4) and dispersing billiards with cusps (Fig. 3). They turn out to be much easier than the stadia. For semi-dispersing billiards, we define M to consist of all collisions with the circular obstacle. Now the return map F : M → M is equivalent to the well studied Lorentz gas billiard map without horizon [11]. Then m-cells are made of points colliding with the sides of the square exactly m times before returning to the obstacle. The structure of m cells in the semi-dispersing billiards is described in the literature [6,7,12]. In particular, we still have µ(Mm ) = O(m −3 ), as for the stadia. But there is a crucial difference: the bound (3.2) fails, and instead the image F(Mm ) of the m-cell intersects other cells Mk with O(m 1/2 ) < k < O(m 2 ).
(5.1)
Moreover, typical points x ∈ Mm land in cells Mk with k m, in fact the average value of k is E(k) = O(m 1/2 ). Thus the majority of points x ∈ Mm ‘escape’ from ‘high cells’ into M0 much faster than they do in the case of stadia. This is good, but our method used for the stadia (based on Lemma 3.2) will no longer apply, so a different strategy must be employed. A crucial estimate is proved in [21, Lemma 16]: Lemma 5.1 ([21]). There are constants p, q > 0 such that for any large b > 0 there is a subset M˜ m ⊂ Mm such that µ(Mm \ M˜ m ) ≤ Cm − p µ(Mm ), where C = C(b) > 0 is a constant, and for every x ∈ M˜ m the images F i (x) for i = 1, . . . , b ln m never appear in cells Mk with k > m 1−q . Lemma 16 in [21] is stated and proved for the iterations of T −1 , but the time reversibility of the billiard dynamics makes it applicable to T as well. For the sake of completeness,
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we outline the proof here. Examining the cell structure it is easy to see that the one-step transition probabilities are µ (Mk /F(Mm )) (m + k)/k 3 ,
(5.2)
where A B means that 0 < c1 < A/B < c2 for two constants c1 , c2 and k satisfies (5.1). Thus for any small e > 0, 2 2e µ ∪m = O 1/m . (5.3) M /M k m 1/2+e k=m 1
Hence we can neglect points x ∈ Mm such that F(x) ∈ Mk with k > m 2 +e and those for 1 which F 2 (x) ∈ M j with j > m 4 +3e . It remains to estimate the probability that points 1 y ∈ M j with j ≤ m 4 +3e will come up to Mi , i ≥ m 1−q , within O(ln m) iterations of F. The key observation is that the cells M j are long (their length is l j ∼ j −1/2 ) and the cells Mi are very short (because li ∼ i −1/2 = O(l 3j )). Actually we need to deal with homogeneous sections of m-cells, in which distortion bounds can be enforced [12, Chap. 5]. Every cell M j has length ∼ j 1/2 , and it is divided into homogeneous sections of length k −3 for k ≥ j 1/4 . We will only 1/4+e , as the union of the rest has measure keep homogeneous with k ≤ j sections −3−4e 4e O( j ) = O µ(M j )/j , which is negligible. Just as we indicated in the previous section, we can foliate each homogeneous section of Mm by smooth unstable curves, then their images in other cells will be homogeneous unstable curves stretching completely across homogeneous sections in those cells (with negligible exceptions caused by irregular intersections at the ends of homogeneous sections). Then we consider an arbitrary homogeneous unstable curve W ⊂ M j , 1 j ≤ m 4 +3e , in the k th section, where k ≤ j 1/4+e . Its length is mW (W ) ∼ k −3 ≥ 3 2 j −3/4−3e ≥ m − 16 −3e−9e , where mW denotes the Lebesgue measure on W . Next we use the key estimate of the ‘growth lemma’ [9, Theorem 3.1], which says that there are constants α ∈ (0, 1) and β > 0 such that for every homogeneous unstable curve W , every n ≥ 1 and ε > 0, mW (rn < ε) ≤ (α)n mW (r0 < ε/n ) + βεmW (W ).
(5.4)
Here > 1 is the hyperbolicity constant and rn (x) is a function on W equal to the distance from F n (x) to the nearest endpoint of the component of F n (W ) that contains x, we refer to [9] for details. A crucial observation is that if F n (x) ∈ Mk , then because the length of the largest homogeneous section in Mk is O(k −3/4 ), we have rn (x) = O(k −3/4 ). So applying (5.4) with ε = m −3(1−q)/4 (note that ε mW (W )) completes the proof of the lemma. We now turn back to our analysis of the set An \Bn,b in the end of Sect. 2. Let Cn = {x ∈ An \Bn,b : |I | > n/10}, where I is again the longest time interval, within [1, n], between successive returns to M. It is immediate that µ(Cn ) = O(n −1 ), because µ(Mm ) = O(m −3 ), just like in the proof of Proposition 3.1. On the other hand, the above lemma implies that µ(An \Bn,b \Cn ) is even smaller: it can be bounded, say, by n −1−q/2 . Finally we deal with dispersing billiards with cusps. Here the hyperbolicity of the map F is weak during long series of reflections deep in a cusp, see [13] and Fig. 3. We
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define M ⊂ M to consist of points that do not belong to series of N ≥ N0 reflections is a cusp, where N0 is a large constant. A cell Mm is then made by points whose trajectories enter a cusp and come out of it after m bounces. The cell structure is described in [13] in detail, and it is surprisingly similar to the cell structure of semi-dispersing billiards treated above. In particular, the bound (5.1) holds. Lemma 5.1 carries over, and its proof is essentially the same. For example, Eq. (5.2) takes form µ (Mk /F(Mm )) ∼ m 2/3 /k 7/3 , hence for any small e > 0, 2 4e/3 = O 1/m , M /M µ ∪m m k=m 1/2+e k which is similar to (5.3). The rest of the argument goes word for word, requiring only changes in the values of some constants. Acknowledgement. We thank P. Bálint and D. Dolgopyat for useful discussions and the anonymous referee for helpful remarks. N. Chernov was partially supported by NSF grant DMS-0354775. H.-K. Zhang was partially supported by SRF for ROCS, SEM.
References 1. Bálint, P., Gouëzel, S.: Limit theorems in the stadium billiard. Commun. Math. Phys. 263, 461–512 (2006) 2. Benettin, G., Strelcyn, J.-M.: Numerical experiments on the free motion of a point in a plane convex region: stochastic transition and entropy. Phys. Rev. A 17, 773–785 (1978) 3. Bunimovich, L.A.: On billiards close to dispersing. Math. USSR. Sb. 23, 45–67 (1974) 4. Bunimovich, L.A.: The ergodic properties of certain billiards. Funk. Anal. Prilozh. 8, 73–74 (1974) 5. Bunimovich, L.A.: On ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65, 295–312 (1979) 6. Bunimovich, L.A., Sinai, Ya.G., Chernov, N.I.: Markov partitions for two-dimensional billiards. Russ. Math. Surv. 45(3), 105–152 (1990) 7. Bunimovich, L.A., Sinai, Ya.G., Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46(4), 47–106 (1991) 8. Chernov, N.: A new proof of Sinai’s formula for entropy of hyperbolic billiards. Its Application to Lorentz Gas and Stadium. Funct. Anal. Appl. 25, 204–219 (1991) 9. Chernov, N.: Decay of correlations and dispersing billiards. J. Stat. Phys. 94, 513–556 (1999) 10. Chernov, N.I., Haskell, C.: Nonuniformly hyperbolic K-systems are Bernoulli. Ergod. Th. Dynam. Sys. 16, 19–44 (1996) 11. Chernov, N., Zhang, H.-K.: Billiards with polynomial mixing rates. Nonlinearity 18, 1527–1553 (2005) 12. Chernov, N., Markarian, R.: Chaotic Billiards, Mathematical Surveys and Monographs, 127, Providence, RI: Amer. Math. Soc., 2006 13. Chernov, N., Markarian, R.: Dispersing billiards with cusps: slow decay of correlations. Commun. Math. Phys. 270, 727–758 (2007) 14. Dembo, A., Zeitouni, O.: Large Deviations, Techniques and Applications. NY: Springer, 1998 15. Machta, J.: Power law decay of correlations in a billiard problem. J. Stat. Phys. 32, 555–564 (1983) 16. Machta, J., Reinhold, B.: Decay of correlations in the regular Lorentz gas. J. Stat. Phys. 42, 949–959 (1986) 17. Markarian, R.: Billiards with polynomial decay of correlations. Ergod. Th. Dynam. Syst. 24, 177–197 (2004) 18. Ornstein, D., Weiss, B.: On the Bernoulli nature of systems with some hyperbolic structure. Ergod. Th. Dynam. Sys. 18, 441–456 (1998) 19. Reháˇcek, J.: On the ergodicity of dispersing billiards. Rand. Comput. Dynam. 3, 35–55 (1995) 20. Sinai, Ya.G.: Dynamical systems with elastic reflections. Ergodic Properties of Dispersing Billiards. Russ. Math. Surv. 25, 137–189 (1970) 21. Szász, D., Varjú, T.: Limit laws and recurrence for the planar Lorentz process with infinite horizon. Manuscript, available at www.ma.utexas.edu, archive 06-274, 2006
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22. Vivaldi, F., Casati, G., Guarneri, I.: Origin of long-time tails in strongly chaotic systems. Phys. Rev. Let. 51, 727–730 (1983) 23. Young, L.-S.: Statistical properties of systems with some hyperbolicity including certain billiards. Ann. Math. 147, 585–650 (1998) 24. Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999) 25. Zaslavsky, G.: Stochastisity in quantum systems. Phys. Rep. 80, 157–250 (1981) 26. Zheng, W.-M.: Symbolic dynamics of the stadium billiard. Phys. Rev. E. 56, 1556–1560 (1997) Communicated by G. Gallavotti
Commun. Math. Phys. 277, 323–367 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0369-1
Communications in
Mathematical Physics
Higgs Bundles, Gauge Theories and Quantum Groups Anton A. Gerasimov1,2,3 , Samson L. Shatashvili2,3,4 1 2 3 4
Institute for Theoretical and Experimental Physics, Moscow, 117259, Russia School of Mathematics, Trinity College, Dublin 2, Ireland. E-mail: [email protected] The Hamilton Mathematics Institute TCD, Dublin 2, Ireland IHES, 35 route de Chartres, Bures-sur-Yvette, France
Received: 30 November 2006 / Accepted: 12 March 2007 Published online: 13 November 2007 – © Springer-Verlag 2007
Abstract: The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U (N ) gauge theory reproduce quantum wave functions of the Nonlinear Schrödinger equation in the N -particle sector. This implies the full equivalence between the above gauge theory and the N -particle sub-sector of the quantum theory of the Nonlinear Schrödinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine Hecke algebra. We propose a similar construction based on the G/G gauged WZW model leading to the representation theory of the double affine Hecke algebra. 1. Introduction In [1] a relation between a certain type of two-dimensional Yang-Mills theory and the Bethe Ansatz equations for the quantum theory of the Nonlinear Schröinger equation was uncovered. The topological Yang-Mills-Higgs theory considered in [1] captures the hyperkähler geometry of the moduli space of Higgs bundles introduced in [2] by Hitchin. It was shown that the path integral in this theory can be localized on the disconnected set whose components are naturally enumerated by the solutions of a system of the Bethe Ansatz equations. The conceptual explanation of the appearance of the Bethe Ansatz equations as a result of the localization in the topological Yang-Mills-Higgs theory, as well as potential consequences were missing. To elucidate the structure of the theory we consider the space of wave functions of the two-dimensional gauge theory introduced in [1]. We argue that this space can be identified with the space of wave-functions in the N -particle sector of the quantum theory of the Nonlinear Schrödinger equation constructed in the framework of the coordinate Bethe Ansatz (see e.g. [3,4]). This implies the equivalence between two seemingly different quantum field theories. Taking into account the interpretation of the coordinate
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Bethe Ansatz in the Nonlinear Schrödinger theory via the Quantum Inverse Scattering Method (algebraic Bethe Ansatz) [5] this also provides a direct correspondence between gauge theory based on the moduli problem for Higgs bundles and the representation theory of quantum groups. The partition function of the theory considered in [1] depends on an additional parameter playing the role of the first Chern class of the tautological line bundle on the classifying space BU (1) of the U (1) group in the description of the equivariant cohomology of the Hitchin moduli space. In the Nonlinear Schrödinger theory the same parameter plays the role of the coupling constant. In addition we show that the considerations of [1] can be generalized from the YangMills-Higgs theory to the G/G gauged WZW model. The corresponding partition function is expressed in terms of solutions of Bethe Ansatz equations similar to the ones for XXZ spin chains. Thus presumably the wave functions for G = U (N ) can be identified with the wave functions of the spin chains. One can suspect that the relations discussed in this paper are more general and other examples considered in [1] have similar interpretation in terms of the representation theory of quantum groups1 . The case of the instanton moduli space in the four-dimensional Yang-Mills theory studied in [1] looks especially interesting in this regard and will be considered elsewhere. Let us finally note that the revealed correspondence between topological quantum field theories and integrable structures captured by the Bethe Ansatz equations obviously imply some relation with the old standing wish to unify three-dimensional hyperbolic geometry, two dimensional conformal/integrable theories and (some fragments of) algebraic K-theory [8–11]. The plan of the paper is as follows. In Sect. 2 we recall the standard facts from the two-dimensional Yang-Mills theory and the G/G gauged WZW model. This provides a template for further considerations in the Yang-Mills-Higgs theory. This part can be skipped by the reader familiar with the subject. In Sect. 3, following [1], the topological Yang-Mills-Higgs theory is introduced and the application of the cohomological localization technique is discussed. We also provide the explicit description of the two important limiting cases: c → ∞ and c → 0 - these limiting cases are instructive in establishing the correspondence between answers computed in this model and representation theory. In Sect. 4 we recall the description of exact N -particle wave functions in the quantum theory of the Nonlinear Schrödinger equation emphasizing the role of the degenerate double affine Hecke algebra. We also consider the same limiting cases: c → ∞ and c → 0. In addition, we stress the fact that the quantum wave functions of the Nonlinear Schrödinger equation are p → 1 limits of the (generalized) spherical functions for G L(N , Q p ) with p - prime given by Hall-Littlewood polynomials. In Sect. 5 we propose the explicit expressions for the exact wave functions in the Yang-Mills-Higgs theory and identify the bases of wave functions with the bases of eigenfunctions of the Hamiltonian in the finite-particle sector for the Nonlinear Schrödinger equation. This provides the conceptual explanation of the appearance of the Bethe Ansatz equations in [1]. In Sect. 6 we give the equivariant cohomology description of the Hilbert space in the topological Yang-Mills-Higgs theory. In Sects. 7 we discuss the connection between the gauge theory and the quantum Nonlinear Schrödinger integrable system using the Nahm duality. In Sect. 8 the relevant generalizations of G/G gauge Wess-Zumino-Witten is proposed and the partition function is derived. We conclude with the discussion of the possible general framework. 1 See in this respect [6,7] where the role of the moduli spaces of monopoles in the representation theory of infinite-dimensional quantum groups was studied.
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We include two relevant topics in the Appendices. In Appendix A we provide the twistor type description for the Yang-Mills-Higgs theory and Appendix B is devoted to the topic of quantization of a singular manifold relevant to the construction of the wave function in the main part of the paper.
2. Two-Dimensional Gauge Theories with Compact Group There is an interesting class of two-dimensional gauge theories that are exactly solvable on an arbitrary Riemann surface. The simplest examples are given by the Yang-Mills theory on a Riemann surface with a gauge group G = Map(, G) with G - a compact Lie group [12–17], and more generally by the G/G gauged WZW theory [18,19]. The Lagrangian of the G/G gauged WZW model depends on an integer number k and the two-dimensional Yang-Mills theory can be recovered in the limit k → ∞. In all these cases the space of the classical solutions of the theory is naturally described in terms of the moduli spaces associated with the underlying Riemann surface and the partition function can be represented as a sum over the set of irreducible representations of some algebraic object. For the Yang-Mills theory it is the set of finite-dimensional representations of the group G and for the G/G gauged WZW model it is the set of finitedimensional irreducible representations of the quantum group Uq g with g = Lie(G) and q = ex p(2πi/(k + cv )), cv - the dual Coxeter number. In this section we recall the standard facts about these two-dimensional gauge theories.
2.1. Two-dimensional Yang-Mills theory. We start with a two-dimensional Yang-Mills theory for a compact group G on a Riemann surface (we mostly follow [16]). The partition function is given by: 1 1 1 2 2 2 Z Y M () = Dϕ D A Dψ e 2π d z (iTr ϕ F(A)+ 2 Tr ψ∧ψ−gY M Tr ϕ vol ) , Vol(G ) (2.1) where Vol(G ) is a volume of the gauge group G = Map(, G) and vol is a volume form on . Here A is a connection on a principal G-bundle over , ϕ ∈ A0 (, adg) is a section of the vector bundle adg and ψ ∈ A1 (, adg) is an odd one-form taking values in adg, g = Lie(G). The measure D A Dψ is a canonical flat measure and the measure Dϕ is defined using the standard normalization of the Killing form on g. We also imply the sum over all topological classes of the principal G-bundle over h . The Feynman path integral (2.1) is invariant under the action of the following odd and even vector fields: Q A = iψ, Q ψ = −(dϕ + [A, ϕ]), Q ϕ = 0, Lϕ A = (dϕ + [A, ϕ]), Lϕ ψ = −[ϕ, ψ], Lϕ ϕ = 0,
(2.2) (2.3)
such that Q 2 = −iLϕ . The invariance of the path integral under transformations (2.2), (2.3) allows to solve the theory exactly. We will be mostly interested in the (generalizations of) the two-dimensional Yang-Mills theory with gY2 M = 0. The theory for gY2 M = 0 will be called topological Yang-Mills theory because its correlation functions depend only on the topology of the underlying surface.
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The observables in the theory are constructed using the standard descent procedure. Given any Ad G -invariant function f ∈ Fun(g) the corresponding local observable is given by: O(0) f = Tr f (ϕ),
(2.4)
where trace is taken in the adjoint representation. The corresponding nonlocal observables are given by the solutions of the descent equations: dO(i) = −i Q(O(i+1) ).
(2.5)
Thus we have: (1) Of
(2) Of
=
1 = 2
dz
dim( g) a=1
2
d z
∂ f (ϕ) a ψ , ∂ϕ a
dim( g) a,b=1
∂ 2 f (ϕ) a ψ ∧ ψb + i ∂ϕ a ∂ϕ b
(2.6)
d2z
dim( g) a=1
∂ f (ϕ) F(A)a , (2.7) ∂ϕ a
dim(g) where ϕ = a=1 ϕ a ta , {t a } is a basis in g and ∈ is a closed curve on a twodimensional surface . To describe the Hilbert space of the Yang-Mills theory consider the theory on a two dimensional torus = T 2 supplied with the flat coordinates (t, σ ) ∼ (t +2π m, σ +2π n), n, m ∈ Z. The action has the form: 1 S(ϕ, A) = dtdσ Tr (ϕ∂t Aσ + At (∂σ ϕ + [Aσ , ϕ])). (2.8) 2π T 2 Here we have integrated out fermionic degrees of freedom using the appropriate choice of the measure in the path integral. Due to the gauge invariance generated by the first-class constraint: ∂σ ϕ + [Aσ , ϕ] = 0,
(2.9)
the phase space of the theory can be reduced to the finite-dimensional space - the phase space is an orbifold: MG = (T ∗ H )/W.
(2.10)
Its non-singular open part is given by: (0)
MG = (T ∗ H0 )/W,
(2.11)
where H0 = H ∩ G r eg is an intersection of the Cartan torus H ⊂ G with a subset G r eg of regular elements of G (the set of the elements of G such that its centralizer has the dimension equal to the rank of g) and W - is Weyl group. Note that Cartan torus is given by H = h/Q ∨ , where Q ∨ is a coroot lattice of g. In the case of G = U (N ) the set rank(g) H/H0 = ∪ j
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The Hilbert space of the theory can be realized as a space of Ad G -invariant functions on G. The pairing is defined by the integration with a bi-invariant normalized Haar measure: 1 , 2 = dg 1 (g) 2 (g). (2.12) G
Being restricted to the subspace of AdG -invariant functions it descends to the integral over Cartan torus H : 1 1 , 2 = d x 2G (e2πi x ) 1 (x) 2 (x), (2.13) |W | H where the Jacobian is given by: 2G (e2πi x ) =
2 eiπ α(x) − e−iπ α(x) ,
(2.14)
α∈R +
and R + is a set of positive roots of g. The set of invariant operators descending onto M includes a commuting family of the operators given by Ad G -invariant polynomials of ϕ. In the case of G = U (N ) one can take: (0)
Ok (ϕ) =
1 Tr ϕ k , (2π )k
(2.15)
where trace is taken in the N -dimensional representation. Define the bases of wavefunctions by the condition: Ok(0) (ϕ)λ (x1 , . . . , x N ) = pk (λ)λ (x1 , . . . , x N ), λ (xw(1) , . . . , xw(N ) ) = λ (x1 , . . . , x N ),
w ∈ W,
(2.16)
where λ = (λ1 , . . . , λ N ) are elements of the weight lattice P of g and pk ∈ C[h∗ ]W is the basis of invariant polynomials on the dual h∗ to Cartan subalgebra h. It is useful to redefine wave functions by multiplying them on: eiπ α(x) − e−iπ α(x) , (2.17) G (e2πi x ) = α∈R +
so that the integration measure becomes a flat measure on H : 1 1 , 2 = d x 1 (x) 2 (x), |W | H
(2.18)
where i (x) = G (e2πi x )i (x). Then the action of the operators ck on the wavefunctions has a simple form: (0)
Ok =
N 1 ∂k . (2πi)k ∂ xik i=1
(2.19)
Note that after the redefinition the wave functions are skew-symmetric with respect to the action of W . To get the symmetric wave functions one can use a slightly different redefinition i (x) = | G (e2πi x )|i (x).
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The space of states of the theory allows simple description in terms of the representation theory of G. The bases of W -skew-invariant eigenfunctions can be expressed through the characters chµ (g) = Tr Vµ g of the finite-dimensional irreducible representations of G as: µ+ρ (g) = G (e2πi x ) chµ (g),
µ ∈ P++ ,
where P++ is a subset of the dominant weights of G, ρ = 21 positive roots and λ = µ + ρ in (2.16). Explicitly we have: µ+ρ (x) =
α∈R +
(2.20) α is a half-sum of
(−1)l(w) e2πi w(µ+ρ)(x) ,
(2.21)
w∈W
where l(w) is a length of a reduced decomposition of w ∈ W . The partition function of the topological Yang-Mills theory on a Riemann surface h of a genus h can be expressed as a sum over unitary irreducible representations of G: Z Y M (h ) =
2h−2
Vol(G) (2π )dim(G)
(dim Vµ )2−2h ,
(2.22)
µ∈P++
where: dim Vµ =
(µ + ρ, α) , (ρ, α)
(2.23)
α∈R+
is a dimension of the irreducible representation Vµ given by the Weyl formula. Here (α, β) is an invariant symmetric pairing on h∗ . Thus for g = u N we have:
dim Vµ =
1≤i< j≤N
(µi − µ j + j − i) , ( j − i)
(2.24)
where µi ∈ Z+ and µ1 ≥ µ2 ≥ · · · ≥ µ N . Note that for h = 0, 1 the partition function (2.22) is divergent. It can be made finite by taking gY M = 0 in (2.1). We use more general regularization compatible with symmetries of the theory - we consider the following deformation of the action: SY M = −
∞ k=1
tk
h
(0)
d 2 z Ok (ϕ) volh ,
(2.25)
where Ok(0) (ϕ) is a basis of Adg-invariant polynomials on ϕ and the number of tk = 0 is finite. We also impose the additional condition on tk such that the functional integral (0) is well defined. For g = u N one can take Ok (ϕ) = (2π1 )k Tr ϕ k . Then the partition function is given by: Z Y M (h ) =
Vol(G) (2π )dim(G)
2h−2 µ∈P++
(dim Vµ )2−2h e−
∞
k=1 tk
pk (µ+ρ)
, (2.26)
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where pk ∈ R[h∗ ]W . Note that pk (µ + ρ) are equal to the eigenvalues of particular combinations of the (higher) Casimir operator ck acting on Vµ . For example in the case of the quadratic Casimir operator we have: c2 |Vµ =
1 1 1 1 1 (µ + 2ρ, µ) = (µ + ρ, µ + ρ) − (ρ, ρ) = p2 (µ + ρ) − (ρ, ρ). 2 2 2 2 2
In fact, not only the dimensions of the irreducible representations show up in the 2d Yang-Mills theory, but also the characters of the irreducible representations enter the explicit expressions for correlation functions. To illustrate this let us consider the (0) correlation function of the operator Ok = tr ϕ k inserted at the center of the disk D. Boundary conditions are defined by fixing the holonomy of the connection over the boundary. The explicit representation is given by: On(0) D (x) =
Vol(G) (2π )dim(G)
−1
pn (µ + ρ) e−
∞
k=1 tk
pk (µ+ρ)
dim Vµ µ+ρ (x),
µ∈P++
where the holonomy of the boundary connection is g = exp(2πi x) ∈ H and µ+ρ (x) = G (e2πi x )chµ (e2πi x ). In particular for n = 0 and tk = 0 we have: Z Y M (D) ≡ 1 D (g) =
Vol(G) (2π )dim(G)
−1
dim Vµ µ+ρ (g) = δe(G) (g),
µ∈P++
(2.27) G)
where we lift the expression to Ad G -invariant functions on G. The delta-function δe (g) on the group G has a support at the unit element e ∈ G and is a vacuum wave function corresponding to a disk. Let us remark that δe(G) (g) can be considered as a character of the regular representation of G and (2.27) is a decomposition of the regular representation over the right action of the group G. Note that the factor dim Vµ enters in the power of the Euler characteristic of the disk χ (D) = 1, naturally generalizing the representation (2.26) for the compact surface. Finally note that (2.27) is obviously compatible with (2.22) for h = 0 if we represent a sphere as glued from two disks: Z Y M (0 ) = dg1 D (g) 1 D (g −1 ) G
=
Vol(G) (2π )dim(G)
−2
(dim Vµ )2 e−
∞
k=1 tk
pk (µ+ρ)
.
(2.28)
µ∈P++
2.2. One-dimensional reduction. Consider the dimensional reduction of the twodimensional Yang-Mills theory to one dimension (quantum mechanics). We have for the partition function: 1 1 2 2 Z Q M () = Dϕ Da DbDηDζ e 2 dt (iTr ϕ (∂t a+[b,a])+η ζ −gY M Tr ϕ vol ) , Vol(G ) (2.29)
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where is a trivalent graph supplied with the volume form vol on its edges, Vol(G ) is a volume of the gauge group G = Map(, G). Here (a, b) and (η, ζ ) are 1d reductions of two-dimensional fields (Aσ , At ) and (ψσ , ψt ). The sewing conditions for the fields on different edges of the graph are chosen in such way that the gauge invariance of (2.29) holds. As in the case of the Yang-Mills theory we will be mostly interested in (the generalizations of) the theory with gY2 M = 0. The path integral (2.29) is invariant under the action of the odd and even vector fields: Q a = iη, Q b = iζ, Q η = −[a, ϕ], Q ζ = −(∂t ϕ + [b, ϕ]), Q ϕ = 0, Lϕ a = −[ϕ, a], Lϕ b = −(∂t ϕ + [ϕ, b]), Lϕ η = −[ϕ, η], Lϕ ζ = −[ϕ, ζ ], Lϕ ϕ = 0, (2.30) such that Q 2 = iLϕ . Consider the simplest case of the Yang-Mills theory for G = U (N ) on the circle = S 1 . The bosonic part of the action has the form: 1 S(ϕ, a, b) = dt Tr (ϕ∂t a + b [a, ϕ]). (2.31) 2 S1 Using the invariance with respect to the gauge transformations generated by the firstclass constraint: [a, ϕ] = 0,
(2.32)
the phase space of the theory can be reduced to the finite-dimensional space. The (open part of the) phase space is given by: M(0) = (T ∗ h0 )/W,
(2.33)
where h0 = h ∩ gr eg is an intersection of the Cartan subalgebra h ⊂ g with the subset gr eg of regular elements of g and W - is Weyl group. For G = U (N ) the set h/h0 = ∪ j
On Ad G -invariant functions it descends to the integral over Cartan subalgebra h: 1 1 , 2 = d x 2g(x) 1 (x) 2 (x), (2.35) |W | h where the corresponding Jacobian is given by: 2g(x) = α(x)2 .
(2.36)
α∈R +
The same set of invariant operators (2.15) descends onto M(0) and we fix the bases of wave functions by the conditions (2.16), where (λ1 , . . . , λ N ) now take values in a
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rank(g)-dimensional vector space P(R) = P ⊗ R. Similar to the case of wave-function of the Yang-Mills theory it is useful to redefine wave functions by multiplying them on: g(x) = α(x), (2.37) α∈R +
so that the integration measure becomes flat: 1 1 , 2 = d x 1 (x) 2 (x), |W | h
(2.38)
where i (x) = g(x)i (x). In the case of the dimensionally reduced theory the space of states does not have a direct connection with the representation theory of G. However as a basis of W skew-invariant eigenfunctions one can use the renormalized characters of the finitedimensional irreducible representations continued to the arbitrary weights (λ1 , . . . , λ N ) in the positive Weyl chamber P+ (R). Explicitly we have: λ (x) = (−1)l(w) e2πiw(λ)(x) . (2.39) w∈W
The partition function of the dimensionally reduced topological Yang-Mills theory on a graph h can be expressed as a formal integral over the cone PR+ : d N λ dλ2−2h , (2.40) Z Y M (h ) = λ∈P+ (R)
where: dλ =
(λ, α),
(2.41)
α∈R+
is a continuation of the renormalized Weyl expression for the dimension of the irreducible representation Vλ from P++ to P+ (R). 2.3. G/G gauged Wess-Zumino-Witten model. The Yang-Mills theory in two dimensions allows the following nontrivial generalization to the G/G gauged WZW model. The partition function for the G/G gauged WZW model on h is given by the following path integral: 1 Z GW Z W (h ) = (2.42) Dg D A Dψ ek S(g,A,ψ) , Vol(Gh ) 1 S(g, A, ψ) = SW Z W (g)− d 2 z Tr (A z g −1 ∂¯ z¯ g+g∂z g−1 A z¯ +g A z g−1 A z¯ − A z A z¯ ) 2π h 1 + d 2 z Tr(ψ ∧ ψ), 4π h where SW Z W (g) is an action functional for Wess-Zumino-Witten model: 1 SW Z W = − d 2 z Tr(g −1 ∂z g · g −1 ∂z¯ g) − i(g), 8π h 1 (g) = d 3 y i jk Tr g −1 ∂i g · g −1 ∂ j g · g −1 ∂k g. 12π B
(2.43)
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Here k is a positive integer and ∂ B = h . The gauged WZW model allows even and odd symmetries extending those for the Yang-Mills theory [20]: −1
Q A = iψ, Q ψ (1,0) = i(A g )(1,0) − i A(1,0) , Q ψ (0,1) = −i(A g )(0,1) +i A(0,1) , Q g = 0, (0,1)
Lg A(1,0) = (A g )(1,0) − A(1,0) Lg A A
−1
= −(A g )(0,1) + A(0,1) ,
(2.44)
Lg ψ (1,0) = −gψ (1,0) g −1 + ψ (1,0) , Lg ψ (0,1) = g −1 ψ (0,1) g − ψ (0,1) , Lg g = 0. (2.45) Here A g = g −1 dg + g −1 Ag is a gauge transformation. We have the following relation Q 2 = Lg . It is useful to compare these generators with those in the pure Yang-Mills theory. Generators (2.3) and (2.2) realize infinitesimal symmetries (from the Lie algebra) of the action functional. In contrast the transformations (2.45) and (2.44) realize the finite (from the gauge group) symmetries of the action. Note that in the limit g → 1 + iϕ0 , → 0 (2.45) and (2.44) are reduced to (2.3) and (2.2). Consider a deformation of the theory by: S = tµ d 2 z Tr Vµ g volh . (2.46) h
µ∈P++
We take tµ = 0 for all but a finite subset of P++ to make the path integral well defined. Similarly to (2.22) the partition function can be represented in the following form: 1 dim MG (h ) 2 2h−2 k + cv Z GW Z W (h ) = |Z (G)| Volq (G)2h−2 2 4π 2πi λˆ k tµ ch Vµ e 2−2h − µ∈P++ × (dimq Vµ ) e , (2.47) k µ∈P++
where dim MG (h ) = dim G(2h − 2) is the dimension of the moduli space of flat G-bundles on h , |Z (G)| is a dimension of the center of G and: dimq Vµ = Tr Vµ q
−ρˆ
=
(q 21 (µ+ρ,α) − q − 12 (µ+ρ,α) )
, 1 1 (q 2 (ρ,α) − q − 2 (ρ,α) ) −1 1 1 1 q 2 (ρ,α) − q − 2 (ρ,α) , Volq (G) = (2π )dimG (k + cv )− 2 (dim G−dim H )
(2.48)
α∈R+
(2.49)
α∈R+ k of integrable representations of the affine group
LG k at and the sum is over the set P++ the level k. The same set also enumerates irreducible representations of Uq (g) for q = ˆ exp(2πi/(k + cv )). We define exp(2πi λˆ ) = exp(2πi j λ j e j ) ∈ H and ch Vµ (e2πi λ ) is a character of the element exp(2πi λˆ ) taken in the representation Vµ . The expressions dimq Vµ are known as quantum dimensions of the representations of the quantum group. For example in the case of g = gl N we have:
dimq Vµ =
1 1 N (q 2 (µi −µ j + j−i) − q 2 (µ j −µi +i− j) ) 1
i< j
1
(q 2 ( j−i) − q 2 (i− j) )
,
(2.50)
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and µ1 ≥ µ2 ≥ · · · ≥ µ N , µi ∈ Z+ . The q-analog of the character is given by: µ (x) = chµ q −ρ+x ,
(2.51)
and corresponding q-generalizations of (2.27) holds. The representation (2.47) for the partition function of the gauged WZW model can be derived using the cohomological localization technique [20] (see also [21] for slightly different approach). The choice of the regularization scheme leading to a particular normalization of the partition function (2.47) is compatible with the interpretation of (2.47) as number of the conformal blocks in WZW theory on h . This interpretation follows from the fact that the partition function in the G/G gauged WZW model coincides with partition function in 3d Chern-Simons theory for three-dimensional manifold h × S 1 . 3. Topological Yang-Mills-Higgs Theory In [1] a two dimensional gauge theory was proposed such that the space of classical solutions of the theory on a Riemann surface h is closely related with the cotangent space to the space of solutions of the dimensionally reduced (to 2d) four-dimensional selfdual Yang-Mills equations studied by Hitchin [2]. Below we present a way to calculate the partition function of the theory following [1]. Given a principal G-bundle PG over a Riemann surface supplied with a complex structure, we then have an associated vector bundle adg = (PG × g)/G with the fiber g = Lie(G) supplied with a coadjoint action of G. Consider the pairs (A, ) where A is a connection on PG and is a one-form taking values in adg. Then Hitchin equations are given by: F(A) − ∧ = 0,
(1,0)
∇A
(0,1) = 0,
(0,1)
∇A
(1,0) = 0.
(3.1)
The space of the solutions has a natural hyperkähler structure and admits compatible U (1) action. The correlation functions in the theory introduced in [1] can be described by the integrals of the products of U (1) × G-equivariant cohomology classes over the moduli space of solutions of (3.1). Note that the U (1)-equivariance makes the path integral well-defined. The field content of the theory introduced in [1] can be described as follows. In addition to the triplet (A, ψ A , ϕ0 ) of the topological Yang-Mills theory one has: ( , ψ ) : (ϕ± , χ± ) :
∈ A1 (, adg), ϕ± ∈
A0 (, ad
g),
ψ ∈ A1 (, adg), χ± ∈
A0 (, ad
g),
(3.2) (3.3)
where , ϕ± are even and ψ , χ± are odd fields. We will use also another notation ϕ± = ϕ1 ± iϕ2 . The theory is described by the following path integral: 1 Z Y M H (h ) = Dϕ0 Dϕ± D A D Dψ A Dψ Dχ± e S(ϕ0 ,ϕ± ,A, ,ψ A ,ψ ,χ± ) , Vol(Gh ) (3.4) where S = S0 + S1 with: S0 (ϕ0 , ϕ± , A, , ψ A , ψ , χ± ) =
1 2π
d 2 z Tr(iϕ0 (F(A) − ∧ ) − c ∧ ∗ )
h (1,0) +ϕ+ ∇ A (0,1)
(0,1)
+ ϕ− ∇ A
(1,0)
(3.5)
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and
1 1 1 2 S1 (ϕ0 , ϕ± , A, , ψ A , ψ , χ± ) = d z Tr ψ A ∧ ψ A + ψ ∧ ψ 2π h 2 2 (1,0) (0,1) (1,0) (0,1) (0,1) (1,0) (0,1) (1,0) + χ− ψ A , + χ+ ∇ A ψ + χ− ∇ A ψ , +χ+ ψ A , (3.6)
(1,0) (0,1) where the decompositions = (1,0) + (0,1) and ψ = ψ + ψ correspond to the decomposition of the space of one-forms A1 (h ) = A(1,0) (h ) ⊕ A(0,1) (h ) defined in terms of a fixed complex structure on h . The theory is invariant under the action of the following even vector field: (1,0) (1,0) Lv (1,0) = + (1,0) , Lv (0,1) = − (1,0) , Lv ψ = +ψ , (0,1) Lv ψ
(3.7)
(0,1) −ψ
= Lv ϕ± = ∓ϕ± , Lv χ± = ±χ± , Lϕ0 A = −∇ A ϕ0 , Lϕ0 ψ A = −[ϕ0 , ψ A ], Lϕ0 = −[ϕ0 , ], Lϕ0 ψ = −[ϕ0 , ψ ], (3.8) Lϕ0 ϕ0 = 0, Lϕ0 ϕ± = −[ϕ0 , ϕ± ], Lϕ0 χ± = −[ϕ0 , χ± ], and an odd vector field generated by the BRST operator: Q = iψ ,
Q A = iψ A , Qψ A = −∇ A ϕ0 , Qϕ0 = 0,
(1,0) (0,1) Qψ = − ϕ0 , (1,0) +c (1,0) , Qψ = − ϕ0 , (0,1) −c (0,1) , (3.9) Qχ± = iϕ± , Qϕ± = −[ϕ0 , χ± ] ± cχ± .
We have Q 2 = iLϕ0 + cLv and Q can be considered as a BRST operator on the space of Lϕ0 and Lv -invariant functionals. The action functional of the topological Yang-MillsHiggs theory can be represented as a sum of the action functional of the topological pure Yang-Mills theory (written in terms of fields φ0 , A, ψ A ) and an additional part which can be represented as a Q-anti-commutator: 1 (1,0) (0,1) SY M H = SY M + Q, ∧ ψ + ϕ+ ∇ A (0,1) + ϕ− ∇ A (1,0) d 2 z Tr . 2 h + (3.10) The theory given by (3.4) is a quantum field theory whose correlation functions are given by the intersection pairings of the equivariant cohomology classes on the moduli spaces of Higgs bundles. To simplify the calculations it is useful to consider the more general action given by: 1 2 SY M H = SY M + Q, ∧ ψ d z Tr 2 h (1,0) (0,1) (0,1) (1,0) . + τ2 (χ+ ϕ− + χ− ϕ+ ) volh +τ1 ϕ+ ∇ A + ϕ− ∇ A (3.11) Cohomological localization of the functional integral takes the simplest form for τ1 = 0, τ2 = 0. Note that it is not obvious that the theory for τ1 = 0, τ2 = 0 is equivalent to
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that for τ1 = 0, τ2 = 0. However taking into account that the action functionals in these two cases differ on the equivariantly exact form and for c = 0 the space of fields is essentially compact, one can expect that the theories are equivalent. For τ1 = 0 the path integrals over , ϕ± and χ± is quadratic. Thus we have for the partition function the following formal representation:
=
1 Vol(Gh )
Z Y M H (h ) = D A Dϕ0 Dψ A e
1 2π
h
d 2 z Tr (iϕ0 F(A)+ 21 ψ A ∧ψ A )
Sdet V (adϕ0 + ic), (3.12)
where the super-determinant is taken over the super-space: V = Veven ⊕ Vodd = A0 (h , adg) ⊕ A(1,0) (h , adg),
(3.13)
and should be properly understood using a regularization compatible with Q-symmetry of the path integral (e.g. τ1 = 0). Thus the Yang-Mills-Higgs theory can be considered as a pure Yang-Mills theory deformed by a non-local gauge invariant observable. Let us stress that there are two interesting limiting cases c → ∞ and c → 0 for the theory (3.4), obvious from the definition of the corresponding Lagrangian. Note that the dependence on c is through the mass term for the field in (3.5) and (3.6). Thus in the limit c → ∞ the field (and corresponding fermions) drops out and we get the 2d Yang-Mills theory with a compact group G. Therefore, in this limit we have to recover the known answers from 2d Yang-Mills theory. On the other hand for c = 0 the topological Yang-Mills-Higgs theory for group G is equivalent to 2d topological Yang-Mills theory for complex group G c . This might be considered as a manifestation of the general relation between the hyperkähler quotient over a compact group and a Kähler quotient over its complexification. In the case of (3.4) the gauge symmetry group is G = Map(h , G) while in the complexified Yang-Mills theory one would have G c = Map(h , G c ) as a gauge group. Thus in the limit c → 0 one might expect the relation with the representation theory of the complexified group G c . Let us demonstrate the relation of c = 0 Yang-Mills-Higgs theory with the YangMills theory for the complexified gauge group explicitly. The Feynman path integral representation for Yang-Mills-Higgs theory at c = 0 in (3.4) can be considered as a result of a partial gauge fixing of the symmetry in the complex Yang-Mills theory. Consider a complex gauge field: ∇A = d + Ac = d + A + i ,
(3.14)
where A and are skew-hermitian one forms. The corresponding curvature is naturally decomposed into the skew-hermitian and hermitian parts: F(Ac ) = (d A + A ∧ A − ∧ ) + i(d + A ∧ ).
(3.15)
Define two-dimensional topological Yang-Mills theory for the complex group G c as: Z
Y Mc
1 (h ) = c ) Vol(G h
Dϕ0 Dϕ− D A D e SY M c (ϕ± ,A, ) ,
(3.16)
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where:
1 d 2 z Tr (ϕc F(Ac ) + ϕ c F(Ac )) SY M c (ϕ0 , ϕ1 , A, ) = 2π h 1 = d 2 z Tr(ϕ0 (d A + A ∧ A− ∧ )+ϕ1 (d + A ∧ )), 2π h (3.17)
and ϕc = ϕ0 + iϕ1 . This theory has an infinitesimal gauge symmetry: A → d + [, A] − [η, ], A0 (
→ dη + [η, A] + [, ],
(3.18)
with the gauge parameter ( + iη) ∈ h partially fix the gauge freedom generated by the η-dependent part of (3.18) by adding: d 2 z Tr(ϕ2 ∇ A (∗ )) S = h 1 1,0 (0,1) + ∇ A χ+ + i (1,0) , χ+ ∇ 0,1 d 2 z Tr , χ− , A χ− + i 2 h (3.19)
, gc ). To make contact with (3.4) one should
where the last term is the ghost-antighost contribution. Note that this term is invariant with respect to -symmetry if the field ϕ− takes values in the coadjoint representation of the group. Taking c = 0 and integrating over ψ A and ψ in (3.4) one can see that the theory (3.4) is equivalent to this, partially gauge fixed, complex Yang-Mills theory. The partition function (3.4) of the Yang-Mills-Higgs theory on a compact Riemann surface can be calculated using the standard methods of the cohomological localization [17]. As in the case of Yang-Mills theory we consider the deformation of the action of the theory: ∞ SY M H = − tk d 2 z Tr ϕ0k volh , (3.20) h
k=1
where we impose the condition that tk = 0 for all but a finite set of indexes. The path integral with the action (3.11) at τ1 = 0 and τ2 = 1 is easily reduced to the integral over abelian gauge fields. The contribution of the additional nonlocal observable in (3.12) can be calculated as follows. The purely bosonic part of the nonlocal observable after reduction to abelian fields can be easily evaluated using any suitable regularization (i.e. zeta function regularization) and the result is the change of the bosonic part of the abelian action d 2 z(ϕ0 )i F i (A) by: S =
2
h
+
1 2
d z
N i, j=1
h
d2z
(ϕ0 )i − (ϕ0 ) j + ic log (ϕ0 )i − (ϕ0 ) j − ic
N
F(A)i
√ log((ϕ0 )i − (ϕ0 ) j + ic)R (2) g;
(3.21)
i, j=1
where F(A)i is the i th component of the curvature of the abelian connection A and R (2) (g) is the curvature on h for 2d metric g used to regularize non-local observable.
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We will use the notation (ϕ0 )i = λi in the remaining part of the paper. This leads to the unique Q-closed completion. The completion of the term in (3.21) containing the curvature of the gauge field is given by the two-observable O(2) f corresponding to the descendent of the following function on the Cartan subalgebra isomorphic to R N : f (diag(λ1 , . . . , λ N )) =
N
λ j −λk
arctg λ/c dλ.
(3.22)
k, j=1 0
Thus, the abelianized action is defined by a two-observable descending from: I (λ) =
N 1
2
j=1
λi2
− 2π n j λ j
+
N
λ j −λk
arctg λ/c dλ,
(3.23)
k, j=1 0
according to the formula (2.7). On the other hand the term containing the metric curvature R in (3.21) is Q-closed and thus does not need any completion. It can be considered as an integral of the zero observable: O(0) =
N
log((ϕ0 )i − (ϕ0 ) j + ic)
(3.24)
i, j=1
over h weighted by the half of the metric curvature. Note that the function I (λ) plays an important role in Nonlinear Schrödinger theory which we explain in the next section. After integrating out the fermionic partners of abelian connection A the standard localization procedure leads to the following final finite-dimensional integral representation for the partition function [1]: N e(1−h)a(c) Z Y M H (h ) = d N λ µ(λ)h e2πi m=1 λm n m |W | RN (n 1 ,...,n N )∈Z N ∞ × (λk − λ j )n k −n j +1−h (λk − λ j − ic)n k −n j +1−h e− k=1 tk pk (λ) , k= j
k, j
(3.25) where µ(λ) = det
∂ 2 I (λ) , ∂λi ∂λ j
(3.26)
and pk (λ) are S N -invariant polynomial functions of degree k on R N and a(c) is an h-independent constant defined by the appropriate choice of the regularization of the functional integral. One can write the n i -dependent parts of the products in (3.25) as the exponent of the sum: e(1−h)a(c) d N λ µ(λ)h e2πi j n j α j (λ) Z Y M H (h ) = |W | RN (n 1 ,...,n N )∈Z N ∞ × (λk − λ j )1−h (λk − λ j − ic)1−h e− k=1 tk pk (λ) , (3.27) k= j
k, j
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with notation: e2πiα j (λ) = F j (λ) ≡ e2πiλ j
λk − λ j − ic . λk − λ j + ic
(3.28)
k= j
After taking the sum over (n 1 , . . . , n N ) ∈ Z N using: e2πi j n j αi (λ) = µ(λ) µ(λ) (n 1 ,...,n N )∈Z N
(λ∗1 ,...,λ∗N )∈R N
j
=
(m 1 ,...,m N )∈Z N
δ(α j (λ) − m j )
j
δ(λ j − λ∗j )
(3.29)
(see definition of R N below) and the integral over (λ1 , . . . , λ N ) ∈ R N , we see that only α j (λ) ∈ Z, or the same - F j (λ) = 1, contribute to the partition function which now can be written in the form similar to (2.22), (2.47) for g = u N : ∞ Dλ2−2h e− k=1 tk pk (λ) , (3.30) Z Y M H (h ) = e(1−h)a(c) λ∈R N
where: Dλ = µ(λ)−1/2
(λi − λ j )(c2 + (λi − λ j )2 )1/2 ,
(3.31)
i< j
and the R N in (3.29) and (3.30) denotes a set of the solutions of the Bethe Ansatz equations F j (λ) = 1: e2πiλ j
λk − λ j + ic = 1, k = 1, . . . , N , λk − λ j − ic
(3.32)
k= j
for the N -particle sector of the quantum theory of Nonlinear Schrödinger equation (see e.g. [3,4]). Note that the sum in (3.30) is taken over the classes of the solutions up to the action of the symmetric group on λi . This set can be enumerated by the multiplets of the integer numbers ( p1 , . . . , p N ) ∈ Z N such that p1 ≥ p2 ≥ · · · ≥ p N , pi ∈ Z. Thus, the sum in (3.30) is the sum over the same set of partitions as in 2d Yang-Mills theory. The structure of the representation (3.30) for the partition function of YangMills-Higgs theory is very similar to the analogous representation (2.22), (2.47) of the partition functions for Yang-Mills and gauged Wess-Zumino-Witten theories. 3.1. Reduction of Yang-Mills-Higgs theory to one dimension. Let us use the following notations for the one-dimensional reduction of the non-scalar fields entering the description of Yang-Mills-Higgs theory: A → (a, b), → (φ, ρ), ψ A → (ηa , ζb ), ψ → (η , ζ ).
(3.33)
Reduction of topological Yang-Mills-Higgs theory to one dimension (Yang-Mills-Higgs Quantum Mechanics) is described by the following path integral: Z Y M H Q M (h ) = D(ϕ0 , ϕ± , a, b, φ, ρ, ηa , ζb , ηφ , ζρ , χ± ) ×e S(ϕ0 ,ϕ± ,a,b,φ,ρ,ηa ,ζb ,ηφ ,ζρ ,χ± ) ,
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where the action is given by S = S0 + S1 : 1 S0 = (3.34) dt Tr (iϕ0 (∂t a + [b, a] + [φ, ρ]) − c(φ 2 + ρ 2 ) 4π +ϕ1 (∂t φ + [b, φ] − [a, ρ]) + ϕ2 (∂t ρ + [b, ρ] + [a, φ])), 1 1 1 ηa ζb + ηφ ζρ + χ1 ([ηa , φ] + [ζb , ρ]) + χ2 ([ζb , φ] − [ηa , ρ]) S1 = dt Tr 2π 2 2 +χ1 (∂t η + [b, η ] − [a, ζ ]) + χ2 (∂t ζ + [b, ζ ] + [a, η ]) . (3.35) The theory is invariant under the action of the vector fields: Lv φ = +ρ, Lv ρ = −φ, Lv ζ = +η Lv ϕ± = ∓ϕ± ,
Lv η = −ζ , Lv χ± = ±χ± ,
Lϕ0 a = −[a, ϕ0 , ], Lϕ0 b = −(∂t ϕ0 +[b, ϕ0 , ]), Lϕ0 ηa = −[ϕ0 , ηa ], Lϕ0 ζb = −[ϕ0 , ζb ], (3.36) Lϕ0 φ = −[ϕ0 , φ], Lϕ0 ρ = −[ϕ0 , ρ], Lϕ0 η = −[ϕ0 , η ], Lϕ0 ζ = −[ϕ0 , ζ ], Lϕ0 ϕ0 = 0,
Lϕ0 χ± = −[ϕ0 , χ± ],
and a fermionic symmetry generated by the BRST operator: Qa = iηa , Qb = iζb , Qηa = −[a, ϕ0 ], Qζb = −(∂t b + [b, ϕ0 ]), Qϕ0 = 0, (3.37) Qφ = iη , Qρ = iζ , Q η = −[ϕ0 , φ] + cρ, Q ζ = −[ϕ0 , ρ] − cφ, (3.38) Qχ± = iϕ± , Qϕ± = −[ϕ0 , χ± ] ± cχ± . (3.39) We have Q 2 = iLϕ0 + cLv and Q can be considered as a BRST operator on the space of Lϕ0 and Lv -invariant functionals. The partition function on the graph h for g = u N after localization is given by: e(1−h)a(c) dNλ (λk − λ j )1−h (λk − λ j − ic)1−h Z Y M H Q M (h ) = N |W | R /S N k= j k= j (1−h)a(c) e = d N λ Dλ2−2h , (3.40) |W | R N /S N where: Dλ =
(λi − λ j )(c2 + (λi − λ j )2 )1/2 .
(3.41)
i< j
In contrast with the two-dimensional case we have the integral over R N /S N instead of the sum over the solutions of Bethe Ansatz equations. Note also that the factor µ(λ) is a constant for the dimensionally reduced theory and is included into the proper normalization of the partition function. If we compare (3.27) and (3.40) we see that the only difference is that in (3.27) we have an additional insertion under the integral over λ’s of the sum over integers (n 1 , . . . , n N ) with the exponential factor that reduces the integral to the sum over the zeros of the exponent, αi (λ), which is the same as a reduction to
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those λ’s that solve the Bethe Ansatz equation. This simple fact will be important later in computation of wave functions for Yang-Mills-Higgs theory. The obvious similarities between (3.30) and (2.22), (2.47) suggests that there should be the full analog of the results discussed in Sect. 2, including the interpretation of Dλ as a (formal) dimension of the representation of some algebraic structure together (0) with the identification of the eigenfunctions of the operators Ok = (2π1 )k Tr ϕ0k in the appropriate polarization with the corresponding characters. The obvious candidate for the replacement of (2.21) in the Yang-Mills-Higgs theory is a set of wave-functions in the N -particle sector of the Nonlinear Schrödinger theory. The basis in this space is defined in terms of the eigenfunctions of the quantum Hamiltonian operator of the Nonlinear Schrödinger equation and has an interpretation in terms of the representation theory of the degenerate (double) affine Hecke algebra. Before we consider this proposal in detail let us discuss two limiting cases of the representation (3.30), (3.31) that have connections with representation theory of the classical Lie groups. 3.2. c → ∞. In the limit c → ∞ the c-dependent term in the action (3.34) transforms into the delta-function of the fields and the path integral reduces effectively to the path integral of the two-dimensional Yang-Mills theory discussed in Sect. 2. One should expect that the representation for the partition function (3.30) reproduces (2.22) in this limit. Indeed in the limit c → ∞ we have µ(λ) → 1 and : lim Dλ = g(λ) = (λ j − λk ), (3.42) c→∞
1≤ j
and λi are the solutions of the limiting Bethe Ansatz equations: (−1) N −1 e2πiλk = 1.
(3.43)
Thus λi = m i + (N 2−1) , m i ∈ Z and one can identify m i = µi − i in the expressions (2.24) and (3.42). 3.3. c → 0. In the opposite limit c → 0 we again have µ(λ) → 1 and : N (N −1) lim c 2 Dλ = g(λ)2 = (λ j − λk )2 , c→0
(3.44)
1≤ j
and λi are solutions of the limiting Bethe Ansatz equation: e2πiλk = 1.
(3.45)
The interpretation of the limit is not so obvious because the localization technique is not straightforwardly applicable for c = 0. However let us recall that in the case of c = 0 the theory (3.4) is equivalent to the two-dimensional Yang-Mills theory with complex gauge group G c . Thus one might expect that in the limit c → 0 one gets an answer with at least some interpretation in terms of the representation theory of the complexified group G c . In the Yang-Mills theory for G c one expects to have a sum (more exactly an integral and a sum) over the set of unitary representations arising in the decomposition of the regular representation of G in L 2 (G), i.e. over the principal series of unitary representations. In order to compare this with the limit c → 0 let us first recall standard
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facts in the representation theory of complex groups (see e.g. [22,23]). For simplicity we discuss only the case G = G L(N , C). Principal series of unitary representations of G L(N , C) can be studied by inducing them from a Borel subgroup B ⊂ G L(N , C) using the character of B: χ (b) =
N
m
|b j j |iρ j −m j b j j j
b ∈ B, ρ j ∈ R, m j ∈ Z.
(3.46)
j=1
For complex groups all unitary representations are infinite-dimensional and thus the definitions of the character of the representation and the dimension of the representation deserve some care. The character of the representation π : G → End(V ) is defined as follows. Let f (g) be a smooth function with compact support. Then define the trace of f as: Tr V f ≡ Tr( dg f (g)π(g)). (3.47) Under some conditions (3.47) is well defined (i.e. the operator (3.47) is of trace class) and one calls the generalized function ch V on G c a character if Tr V f = ch V , f .
(3.48)
It was shown by Harich-Chandra that a thus defined generalized function is an ordinary function and therefore ch V can be considered as a generalization of the characters of finite dimensional representations. The simplest example is a representation of G L(N , C) obtained by quantization of the regular coadjoint orbit generalizing a two-sheet hyperboloid for G L(2, C). The corresponding character is given by: chλ (e x ) =
2πi N λ x 1 j=1 w( j) j , e x | G (e )|
(3.49)
w∈S N
where λ j = m j + iρ j and S N is a Weyl group of G L(N , C). In the case of the finite-dimensional representations the dimension of the representation is given by the value of the corresponding character at the unit element of the group. However in the case of the infinite-dimensional representations this relation can not be used to define the notion of dimension even formally. In particular the value of the corresponding character at the unit element can be infinite. For example (3.49) tends to infinity when x → 0, which is a manifestation of the fact that the corresponding representation is infinite-dimensional. The correct definition of the dimension Dλ of the principal series unitary representations is provided by the decomposition: δe(G) (g) = Dλ chλ (g), (3.50) λ∈G (G)
where δe (g) is a delta-function with the support at the unit element e ∈ G of the group, is a unitary dual to G (i.e. the set of isomorphism classes chλ (g) is a character and G of the unitary representations entering the decomposition of the regular representation). The dimension Dλ defined in such way (known as a formal degree of the representation)
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coincides with the ratio of the Plancherel measure arising in the decomposition of the regular representation and the flat measure on the space of characters (see e.g. [22,23]). Explicitly we have for Dλ : Dλ = |λi − λ j |2 . (3.51) 1≤ j
Comparing (3.51) with (3.44) one infers that in the limit c → 0 one obtains the subset of the principal series of representations corresponding to λk = m k ∈ Z (i.e. ρk = 0). It is reasonable to guess that in the limit c → 0 the only information that remains is a class of functions and a particular class of representations that arise in the spectral decomposition corresponds to this class of functions2 . This is natural because the localization demands a compactification of the configuration space and the c = 0 term just provides effectively this compactification. In the limit c → 0 not all elements of L 2 (G) arise in the description of Hilbert space of the theory and we get a subset of the representations. Thus the wave-functions of Yang-Mills-Higgs theory for c = 0 should interpolate between characters of finite-dimensional representations of G and characters of a class of infinite-dimensional representations of G c . As we will demonstrate in the next section the wave-functions in the N -particle sector of the Nonlinear Schrödinger theory provide exactly this interpolation. 4. N-particle Wave Functions in Nonlinear Schrödinger Theory The appearance of a particular form of Bethe Ansatz equations (3.32) strongly suggests the relevance of quantum integrable theories in the description of wave-functions in topological Yang-Mills-Higgs theory. Precisely this form of Bethe Ansatz equations (3.32) arises in the description of the N -particle wave functions for the quantum Nonlinear Schrödinger theory with the coupling constant c = 0 [24–27]. In this section we recall the standard facts about the construction of these wave-functions using the coordinate Bethe Ansatz. We also discuss the relation with the representation theory of the degenerate (double) affine Hecke algebras and the representation theory of the Lie groups over complex and p-adic numbers. For the application of the quantum inverse scattering method to Nonlinear Schrödinger theory see [28,29]. One can also recommend [30] as a quite readable introduction into the Bethe Ansatz machinery. The Hamiltonian of the Nonlinear Schrödinger theory with a coupling constant c is given by: 1 ∂φ ∗ (x) ∂φ(x) H2 = d x (4.1) + c(φ ∗ (x)φ(x))2 , 2 ∂x ∂x with the following Poisson structure for bosonic fields: {φ ∗ (x), φ(x )} = δ(x − x ). The operator of the number of particles: H0 = d x φ ∗ (x)φ(x),
(4.2)
(4.3)
2 This may be compared with the Bogomolony limit of the monopole equations where the only information on the potential that survives in the limit is encoded in the asymptotic behaviour of the solutions.
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commutes with the Hamiltonian H2 and thus one can solve the eigenfunction problem in the sub-sector for a given number of particles H0 = N . We will consider both the theory on the infinite interval (x ∈ R) and its periodic version x ∈ S 1 . The equation for eigenfunctions in the N -particle sector has the following form: ⎛ ⎞ N N 2 1 ∂ 2 2 ⎝− ⎠ +c δ(xi − x j ) λ (x) = 2π λi λ (x) i = 1, . . . , N . 2 ∂ xi2 1≤i< j≤N i=1 i=1 (4.4) This equation is obviously symmetric with respect to the action of symmetric group S N on the coordinates xi . Thus the solutions are classified according to the representations of S N . Quantum integrability of the Nonlinear Schrödinger theory implies the existence of the complete set of the commuting Hamiltonian operators. The corresponding eigenvalues are given by the symmetric polynomials pk (λ). Finite-particle sub-sectors of the Nonlinear Schrödinger theory can be described in terms of the representation theory of a particular kind of Hecke algebra [31–34]. Let R = {α1 , . . . , αl } be a root system, W - the corresponding Weyl group and P - a weight lattice. The degenerate affine Hecke algebra H R,c associated to R is defined as an algebra with the basis Sw , w ∈ W and {Dλ , λ ∈ P} such that Sw w ∈ W generate subalgebra isomorphic to the group algebra C[W ] and the elements Dλ , λ ∈ P generate the group algebra C[P] of the weight lattice P. In addition one has the relations: Ssi Dλ − Dsi (λ) Ssi = c
2(λ, αi ) , i = 1, . . . , n. (αi , αi )
(4.5)
Here si are the generators of the Weyl algebra corresponding to the reflection with respect to the simple roots αi . The center of H R,c is isomorphic to the algebra of W -invariant polynomial functions on R ⊗ C. The degenerate affine Hecke algebras were introduced by Drinfeld [35] and independently by Lusztig [36]. Below we consider only the case of the gl N root system and thus we have W = S N . Let us introduce the following differential operators (Dunkle operators [37]): Di = −i
N ∂ c +i ((xi − x j ) + 1)si j . ∂ xi 2
(4.6)
j=i+1
Here (x) is a sign-function and si j ∈ S N is a transposition (i j). These operators together with the action of the symmetric group (4.6) provide a representation of the degenerate affine algebra H N ,c for g = gl(N ): Ssi → si , Di → Di , i = 1, . . . , N .
(4.7)
The image of the quadratic element of the center is given by: 1 ∂2 1 2 Di = − +c 2 2 ∂ xi2 N
N
i=1
i=1
1≤i< j≤N
δ(xi − x j ),
(4.8)
and thus coincides with the restriction of the quantum Hamiltonian on the N -particle sector of the Nonlinear Schrödinger theory on the infinite interval.
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We are interested in S N -invariant solutions of (4.4). They play the role of spherical vectors (with respect to the spherical subalgebra C[W ] ∈ H N ,c ) in the representation theory of the degenerate affine Hecke algebra. The eigenvalue problem (4.4) allows the equivalent reformulation as an eigenvalue problem in the domain x1 ≤ x2 ≤ · · · ≤ x N for the differential operator:
1 ∂2 − 2 ∂ xi2 N
λ (x) = 2π 2
N
i=1
λi2 λ (x)
i = 1, . . . , N ,
(4.9)
i=1
with the boundary conditions: (∂xi+1 λ (x) − ∂xi λ (x))xi+1 −xi =+0 = 4π c λ (x)xi+1 −xi =0 .
(4.10)
The solution is given by: (0) λ (x)
=
w∈W 1≤i< j≤N
λw(i) − λw( j) + ic exp 2πi λw(k) xk , (4.11) λw(i) − λw( j) k
or equivalently: (0) λ (x) =
1 (−1)l(w) g(λ) w∈W
(λw(i) −λw( j) + ic) exp 2πi
1≤i< j≤N
λw(k) xk ,
k
(4.12)
where Fg (λ) = 1≤i< j≤N (λi − λ j ). Note that the wave-function is explicitly symmetric under the action of symmetric group S N on λ = (λ1 , . . . , λ N ). This solution can be also constructed using the representation theory of degenerate affine Hecke algebra H N ,c (see [31–34]). Given a solution of Eq. (4.9) with boundary conditions (4.10) S N -symmetric solutions of (4.4) on R N can be represented in the following form: ⎛ ⎞ − λ + ic(x − x ) λ i j w(i) w( j) (0) ⎝ exp 2πi λ (x) = λw(k) xk ⎠ , λw(i) − λw( j) w∈W
i< j
k
(4.13) where (x) is a sign-function. This gives the full set of solutions of (4.4) for (λ1 ≤ · · · ≤ λ N ) ∈ R N satisfying the orthogonality condition with respect to the natural pairing: < λ , µ >=
1 N!
d x1 . . . d x N λ (x) µ (x) = G(λ)
N
δ(λi − µi ), (4.14)
i=1
where: G(λ) =
1≤i< j≤N
(λi − λ j )2 + c2 . (λi − λ j )2
(4.15)
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Therefore the normalized wave functions are given by: λ (x) =
(−1)l(w)
w∈W
1 λw(i) − λw( j) + ic(xi − x j ) 2 exp 2πi λw(k) xk . λw(i) − λw( j) − ic(xi − x j )
i< j
k
(4.16) The eigenvalue problem for the periodic N -particle Hamiltonian of the Nonlinear Schrödinger theory can be reformulated in the following way. Consider the eigenfunction problem for the differential operator: ⎞ ⎛ N N 2 ∂ 1 ⎝− +c δ(xi − x j + n)⎠ λ (x) = 2π 2 λi2 λ (x) 2 2 ∂ x i i=1 i=1 n∈Z 1≤i< j≤N i = 1, . . . , N .
(4.17)
The wave function of the periodic Nonlinear Schrödinger equation is the eigenfunction of (4.17) satisfying the following invariance conditions: λ (x1 , . . . , x j + 1, . . . , x N ) = λ (x1 , . . . , x N ), j = 1, . . . , N , λ (xw(1) , . . . , xw(N ) ) = λ (x1 , . . . , x N ), w ∈ S N .
(4.18)
These are the conditions of invariance under the action of the affine Weyl group on the space of wave functions. The solutions can be obtained imposing the additional periodicity conditions on the wave functions (4.16). This leads to the following set of the Bethe Ansatz equations for (λ1 , . . . , λ N ): F j (λ) ≡ e2πiλ j
λk − λ j − ic = 1, λk − λ j + ic
j = 1, . . . , N .
(4.19)
k= j
The set of solutions of these equations can be enumerated by sets of integer numbers ( p1 ≥ · · · ≥ p N ) - for each ordered set of these integers there is exactly one solution to Bethe Ansatz equations [27]. Let us remark that there is the following equivalent representation for the periodic wave-functions: 1 λw(i) − λw( j) + ic 2 +[xi −x j ] l(w) λ (x) = (−1) exp 2πi λw(k) xk , λw(i) − λw( j) − ic w∈W
i< j
k
(4.20) where [x] is an integer part of x defined by the conditions: [x] = 0 for 0 ≤ x < 1 and [x + n] = [x] + n. It easy to see that these wave functions are periodic and descend to the wave functions (4.16) if λ = (λ1 , . . . , λ N ) satisfy (4.19). The normalized wave functions in the periodic case are given by: ∂ log F j (λ) −1/2 nor m λ (x) = det λ (x) = µ(λ)−1/2 λ (x). (4.21) ∂λk Note that the normalization factor is closely related to the factor (3.26) arising in the representation of the partition function of Yang-Mills-Higgs theory. Indeed the function
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I (λ) introduced in (3.26) is known in the theory of Nonlinear Schrödinger equations as the Yang function [27]; critical points of the Yang function are in one to one correspondence with the solutions of Bethe Ansatz equations: α j (λ) = log F j (λ) =
∂ I (λ) = n j. ∂λ j
(4.22)
Below we will see that this is not accidental. Finally note that the periodic Nonlinear Schrödinger theory has an interpretation in terms of the representation theory of the degenerate double affine Hecke algebras introduced by Cherednik [38]. For the details in this regard see [34]. 4.1. c → ∞: Representation theory of compact Lie groups. The limit c → ∞ corresponds to the case of impenetrable bosons and the correlation functions are naturally represented in terms of free fermions. Indeed in the limit c → ∞ Bethe Ansatz equations are reduced to the condition: λj =
N −1 + mi , 2
m i ∈ Z,
(4.23)
and the wave-function is given by the wave-function of free fermions (up to a simple sign factor): c=∞ (x) = λ
N | (e x )| | (e x )| iλk x j det e | = (−1)l(w) e2πi k=1 λw(k) xk . (4.24) x x (e ) (e ) w∈W
Note that we have a simple relation with the characters of the finite-dimensional representations of U (N ): chλ (x) =
1 c=∞ (x). | (e x )| λ
(4.25)
Thus in the limit c → ∞ the wave function λ (x) in Nonlinear Schrödinger theory can be considered as a wave-function in two-dimensional Yang-Mills theory renormalized according to i (x) = | G (e x )|i (x) (see Sect. 2). 4.2. c → 0: Representation theory of complex Lie groups. In the limit c → 0 Bethe Ansatz equations are reduced to the condition: λi = m i ,
m i ∈ Z,
and the wave-functions are given by: N (−1)l(w) e2πi k=1 λw(k) xk . c=0 λ (x) =
(4.26)
(4.27)
w∈W
Note that wave functions (4.27) are normalized with respect to the standard scalar product. Now we have a simple relation with the characters (3.49) of the infinite-dimensional representations of G L(N , C): chλ (x) =
1 c=0 (x). | (e x )| λ
(4.28)
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4.3. c = 0, ∞: Representation theory of p-adic Lie groups. Wave functions of Nonlinear Schrödinger theory for c = 0, ∞ have also a connection with the representation theory of Lie groups due to the relation between the representation theory of the degenerate affine Hecke algebras H N ,c and the representation theory of G L(N , Q p ), where Q p is a field of p-adic numbers. More specifically the wave functions of Nonlinear Schrödinger theory can be obtained as a limit of Hall-Littlewood polynomials that can be considered as generalized zonal spherical functions for G L(N , Q p ). The limit is a kind of p → 1 limit3 . For the detailed discussion of p-adic zonal spherical functions see Macdonald [40] and for the relation with Nonlinear Schrödinger theory through representation theory of degenerate Hecke algebras see [33]. Let us start with the definition of Hall-Littlewood polynomials (see e.g. [41]). Let {i }, i = 1, . . . , N be a set of formal variables and µ = (µ1 , . . . , µ N ) be a partition of length N . Then the Hall-Littlewood polynomial depending on the additional formal variable t is defined as: ⎛ ⎞ i − j t 1 µ µ ⎠ w ⎝1 1 . . . NN Pµ (1 , . . . , N |t) = vµ (t) i − j =
1 vµ (t) ()
w∈S N
i< j
(−1)l(w) w (1X 1 . . . NX N
w∈S N
(i − j t)),
(4.29)
i< j
where for the partition µ = (1m 1 , 2m 2 , . . . , r m r , . . .): vµ =
mj N 1 − ti j=1 i=1
1−t
,
() =
(i − j ).
(4.30)
i< j
Hall-Littlewood polynomials enter the explicit formulas for the zonal spherical functions for p-adic Lie groups. The zonal spherical functions are defined as follows. Given a p-adic Lie group G and its maximal compact subgroup K ⊂ G, the zonal spherical function ω(g) on G is a continuous complex-valued function satisfying the following conditions: (1) the function is invariant with respect to the left and right action of the compact subgroup ω(kgk ) = ω(g), k, k ∈ K , (2) the normalization condition ω(1) = 1, (3) the function is an eigenfunction for the convolution with any function with compact support on G satisfying (1). The spherical functions for G = G L(N , Q p ), K = G L(N , Z p ) (here Z p is a ring of p-adic integers) have the following representation in terms HallLittlewood polynomials. Note that the set of the representatives of the double-coset K \G/K can be identified with the elements of the form ( p µ1 , . . . , p µ N ) ∈ K \G/K , where µ = (µ1 ≥ µ2 ≥ · · · ≥ µ N ) is a partition. Then for the zonal spherical functions we have: ωs ( p µ1 , . . . , p µ N ) = p −
N
i=1 (n−i)µi
vµ ( p −1 ) Pµ ( p −s1 , . . . , p −s N | p −1 ), (4.31) v N ( p −1 )
where s = (s1 , . . . , s N ) ∈ Z N and v N (t) =
N 1 − ti i=1
1−t
.
3 For another example where the same limit p → 1 is relevant in string theory see [39].
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Note that the “spectral” indexes in the spherical function and Hall-Littlewood polynomial are interchanged. There is a generalization of the notion of the spherical function which is similar to the multivariable hypergeometric functions for general root systems introduced by Heckman and Opdam (see e.g. [42]). For the case of the p-adic Lie groups the generalized spherical function depends on the additional integer parameter k ∈ Z and is given by: ωs(k) ( p µ1 , . . . , p µ N ) = p −
N
i=1 (n−i)µi
vµ ( p −k ) Pµ ( p −s1 , . . . , p −s N | p −k ). v N ( p −k )
(4.32)
Now one can see how in a particular limit the wave functions of the N -particle sector of the Nonlinear Schródinger theory on R obtained using the coordinate Bethe Ansatz arise. Taking µi = −1 xi , i = exp(2π λi ) and t = e2πic , while → 0 (we use analytical continuation over parameters here) we have for (4.29): ⎛ ⎞ N λk − λ j + ic vµ (t) 1 1 ⎠. w ⎝e2πi k=1 xk λk Pµ (1 , . . . , N ; t) → v N (t) (c) N N ! λk − λ j w∈S N
k< j
(4.33) This expression coincides with the restriction of the Bethe wave function on the subspace x1 < x2 < · · · < x N . Thus taking into account (4.32) one can conclude that the generalized zonal spherical functions in the formal limit → 0 while p = e2πi , µi = 1 xi are given by the wave functions for the N -particle sector of the Nonlinear Schrödinger equation with c = k. It is known that Hall-Littlewood polynomials are a special case of the Macdonald polynomials and one can expect that Hall-Littlewood polynomials before degeneration and more general Macdonald polynomials should be related with the quantum integrable/topological field theories these along the line discussed in this section. Below in Sect. 8 we propose the generalization of the topological Yang-Mills-Higgs theory that provides a realization of these more general polynomials. 5. Wave-Function in Topological Yang-Mills-Higgs Theory In this section we provide the evidence for the identification of a basis of the wave functions of topological Yang-Mills-Higgs theory for G = U (N ) (given by a path integral on a disk with the insertion of observables in the center) with the eigenfunctions of the N -particle Hamiltonian operator of Nonlinear Schrödinger theory. First by counting the observables of the theory we show that the phase space of the Yang-Mills-Higgs theory can be considered as a deformation of the phase space of Yang-Mills theory. This implies that the bases of wave functions in Yang-Mills-Higgs theory can be obtained by a deformation of the bases of wave functions in Yang-Mills theory. Next, using the explicit representation of the partition function on the two-dimensional torus we derive the transformation properties of the wave functions under large gauge transformations. They are in agreement with the known explicit transformation properties of the wave function in Nonlinear Schrödinger theory. Finally, we compute the cylinder path integral (Green function) and torus partition function in Nonlinear Schrödinger theory (with all higher Hamiltonians) and show that the latter coincides with the torus partition function in Yang-Mills-Higgs theory (with arbitrary observables turned on). Taking into account that the constructed wave functions in the gauge theory are the eigenfunctions of the full
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set of the Hamiltonian operators these considerations fix the set of wave functions up to a unitary transformation.
5.1. Local Q-cohomology. We start with a description of the Hilbert space of the YangMills-Higgs theory using the operator-state correspondence. In the simplest form the operator-state correspondence is as follows. Each operator, by acting on the vacuum state, defines a state in the Hilbert space. In turn for each state there is an operator, creating the state from the vacuum state. Moreover, for the maximal commutative subalgebra of the operators this correspondence should be one to one. For example, the space of local gauge-invariant Q-cohomology classes in topological U (N ) Yang-Mills theory is spanned, linearly, by the operators: (0)
Ok =
1 Tr ϕ k , (2πi)k
(5.1)
and thus this space coincides with the space of Ad G -invariant regular functions on the Lie algebra u N . This is in accordance with the description of the Hilbert space of the theory given in Sect. 2. We would like to apply the same reasoning to the topological Yang-Mills-Higgs theory. To get an economical description of the Hilbert space of the theory one should find a maximal (Poisson) commutative subalgebra of local Q-cohomology classes (where Q given by (3.37) acts on the space of functions invariant under the symmetries generated by (3.7), (2.3)). Obviously the operators (5.1) provide non-trivial cohomology classes. One can show that these operators provide a maximal commutative subalgebra for c = 0 and therefore the reduced phase space in the Yang-Mill-Higgs system can be identified with a phase space of pure Yang-Mills theory. Thus the Hilbert space of Yang-MillsHiggs theory (c = 0) can be naturally identified with the Hilbert space of Yang-Mills theory (identified with c → ∞). The fact that the Hilbert space of Yang-Mills-Higgs theory is the same for all c = 0 implies that the bases of wave-functions for c = 0 should be a deformation of the bases for c = ∞. One should stress that this reasoning is not applicable to the case c = 04 . The local cohomology for c = 0 contains additional operators. For example, the following operators provide non-trivial cohomology classes for arbitrary t ∈ C and c = 0: (0)
Ok (t) =
1 t 2 k χ Tr ϕ + tϕ − . 0 + (2πi)k 2 +
(5.2)
This is a manifestation of the fact that c = 0 theory is a Yang-Mills theory for the complexified group G c and thus its phase space is given by MC = T ∗ H c /W . The identification of the Hilbert spaces of Yang-Mills-Higgs theory and of pure YangMills theory supports the idea to use wave functions in Nonlinear Schrödinger theory discussed in the previous section as a basis in the Hilbert space of Yang-Mills-Higgs theory. Below we provide further evidence for this identification. 4 We distinguish the case corresponding to Yang-Mills theory for a complex group (c = 0) and c → 0 in the Yang-Mills-Higgs theory. In the latter case, as we explained before, the Hilbert space is the same as for Yang-Mills-Higgs theory with c = 0 and the corresponding bases of wave functions is given by the formal characters (3.49) for a subset of the unitary representations of the complex group.
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5.2. Gauge transformations of wave function. The discreteness of the spectrum of the N -particle Hamiltonian operator in periodic Nonlinear Schrödinger theory arises due to periodicity condition on wave functions. Thus the eigenfunctions in the periodic case are given by a subset of eigenfunctions on R N descending to S N -invariant functions on (S 1 ) N . The eigenfunction (4.16) of the Hamiltonian operator on R N is represented as the sum over elements of the symmetric group of simple wave functions. For generic eigenvalues each term of the sum is multiplied by some function under the shift xi → xi + n i , n i ∈ Z of the coordinates. Below we will show how these multiplicative factors arising in Nonlinear Schrödinger theory can be derived in Yang-Mills-Higgs gauge theory. We start with a simple case of Yang-Mills theory. The partition function of U (N ) Yang-Mills theory on the torus 1 is given by: N ∞ Z Y M (1 ) = dNλ e2πi m=1 λm n m e− k=1 tk pk (λ) R N /S N
=
(n 1 ,...,n N )∈Z N − ∞ k=1 tk pk (m+ρ)
e
,
(5.3)
(m 1 ,...,m N )∈P++
where P++ is a set of the dominant weights of U (N ). The sum over (n 1 , . . . , n N ) ∈ Z N has a meaning of the sum over topological classes of U (1) N -principle bundles on the torus 1 . It results in the replacement of the integration over λ by a sum over a discrete subset. This should be compared with the partition function of dimensionally reduced U (N ) Yang-Mills theory on S 1 : ∞ Z Q M (S 1 ) = d N λ e− k=1 tk pk (λ) . (5.4) R N /S N
Contrary to the two-dimensional Yang-Mills theory in the last case we do not have any additional restriction on the spectrum (λ1 , . . . , λ N ) ∈ R N /S N . The appearance of the additional sum in (3.40) can be traced back to the difference between the Hilbert spaces of dimensionally reduced and non-reduced theories. The mechanism of the spectrum restriction via the sum over the topological sectors can be explained in terms of the structure of the Hilbert space of the theory as follows. In the Hamiltonian formalism the partition function on a torus is given by the trace of the evolution operator over the Hilbert space of the theory. Let us consider first the dimensionally reduced U (N ) Yang-Mills theory. The phase space of the theory is T ∗ R N /S N , where we divide over the Weyl group W = S N . To construct the Hilbert space we quantize the phase space using the following polarization. Consider Lagrangian projection π : T ∗ R N → R N supplied with a section. We choose the coordinates on the base as position variables and the coordinates on the fibers as the corresponding momenta. Thus the Hilbert space in this polarization is realized as a space of S N (skew)invariant functions on the base R N of the projection. Now consider two-dimensional Yang-Mills theory. For the phase space we have T ∗ H/S N , where H is the Cartan subgroup. We use similar polarization associated with the projection π : T ∗ H → H . Thus the wave functions are S N invariant functions on a torus H or equivalently the functions on R N invariant under action of the semidirect product of the lattice P0 = π1 (H ) and Weyl W = S N group (i.e. under the action of the affine Weyl group W a f f ). The lattice P0 can be interpreted as a lattice of the R N -valued constant connections on S 1 which are gauge equivalent to the zero connection. The
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corresponding gauge transformations act on the wave functions by the shifts x j → x j + n j , n j ∈ Z of the argument of the wave functions in the chosen polarization and the wave functions in two-dimensional Yang-Mills theory can be obtained by the averaging over this gauge transformation and global gauge transformations by the nontrivial elements of the normalizer of Cartan torus W = N (H )/H . It is possible to relate the averaging over the topologically non-trivial transformations with the sum over topological classes of the H bundle on the torus. Note that the maps of S 1 to the gauge group H are topologically classified by π1 (H ) = Z N . Consider a 1 such connection A = (A1 , . . . , A N ) on an H bundle over a cylinder L, ∂ L = S+1 ∪ S− 1 1 that the holonomies along the boundaries S+ and S− are in the different topological classes [(m 1 , . . . , m N )] ∈ π1 (H ) and [(m 1 + n 1 , . . . , m N + n N )] ∈ π1 (H ). Gluing boundaries of the cylinder L we obtain a torus supplied with a connection ∇ A such that the first Chern classes of the bundles corresponding to each U (1)-factor are given 1 by c1 (∇ Ai ) = 2πi L F(A j ) = n j , j = 1, . . . , N . Thus we see that the sum over the topologically non-trivial gauge transformations on S 1 can be translated into the sum over topological classes of the H -bundles on the torus. Let us re-derive the partition function of Yang-Mills theory on the torus (5.3) using the averaging procedure. We start with the dimensionally reduced theory. Let us choose a basis in the Hilbert space of the dimensionally reduced Yang-Mills theory given by (0) the S N skew-invariant eigenfunctions of the quadratic operator H2 = trϕ 2 . In the polarization discussed above we have: ⎞ ⎛ N N 2 ∂ ⎠ 1 (0) 2 H2 ψλ (x) = − ⎝ (x) = 2π λi2 ψλ (x), (5.5) ψ λ 2 2 ∂ x j j=1 j=1 where (x1 , . . . , x N ) ∈ R N . The set of normalized skew-invariant eigenfunctions is given by: ⎛ ⎞ N ψλ (x) = (−1)l(w) exp ⎝2πi λw( j) x j ⎠ , (λ1 , . . . , λ N ) ∈ R N /W, (5.6) w∈S N
j=1
N 1 d N x ψλ (x) ψλ (x) = (2π ) N (−1)l(w) δ(λw( j) − λj ) N ! RN w∈S N j=1
(5.7)
= δ (SN ) (λ − λ ). The integral kernel of the identity operator acting on the skew-symmetric functions can be represented (due to translation invaraince it is a function of the difference x − x ) as: K 0 (x, x ) = K 0 (x − x ) = δ (SN ) (x − x ) = d N λ ψλ (x) ψλ (x ). (5.8) R N /S N
The partition function of the dimensionally reduced Yang-Mills theory on S 1 is given by the trace of an evolution operator and can be written explicitly as: (0) 1 −t2 H2 ( p, ˆ q) ˆ Z Q M (S ) = T r e = d N x d N λ ψλ (x) e−t2 H2 (i∂x ,x) ψλ (x) (R N ×R N )/S N = d N x d N λ e−t2 p2 (λ) . (5.9) (R N ×R N )/S N
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The Green function of the theory is: (0) G 0 (x, x ) = d N λ ψλ (x ) e−t2 H2 (i∂x ,x) ψλ (x) R N /S N = d N λ ψλ (x )e−t2 p2 (λ) ψλ (x). R N /S N
(5.10)
Up to the infinite factor given by the integral over x = (x1 , . . . , x N ) ∈ R N the integral in (5.14) coincides with the expression (5.4) for the partition function for ti=2 = 0. Now consider two-dimensional Yang-Mills theory. In this case we have the periodic eigenvalue problem for (5.5). Then for the normalized eigenfunctions of H2 we have: ψn (x) =
(−1)l(w) exp(2πi
w∈S N
1 N!
N
(n w( j) + ρw( j) )x j ), (n 1 , . . . , n N ) ∈ P++ ,
j=1
(5.11)
(S 1 ) N
d N x ψn (x) ψn (x) =
(−1)l(w)
w∈S N
N
SN δn w( j) ,n j = δn,n .
(5.12)
j=1
Here ρ = (ρ1 , . . . , ρ N ) is a half-sum of the positive roots of u N . The integral kernel of the identity operator can be represented as: K (x, x ) = K (x − x ) = δ(SN ) (x − x ) = ψn (x) ψn (x ). (5.13) n∈P++
The partition function of the Yang-Mills theory on a torus 1 is given by the trace of a evolution operator and: (0) −t2 H2 ( p, ˆ q) ˆ Z Y M (1 ) = T r e = d N x ψn (x) e−t2 H2 (i∂x ,x) ψn (x). (5.14) n∈P++
(S 1 ) N
The kernel for the periodic case can be obviously represented as a matrix element of the projection operator as follows: K (x, x ) = d N λ ψλ (x) P(λ) ψλ (x ), (5.15) R N /S N
where the wave-functions ψλ (x) are given by (5.6) and: N
P(λ) =
δ(λ j − m j ) =
m∈Z N j=1
Equivalently we have:
K (x, x ) =
k∈Z N
=
k∈Z N
R N /S N
R N /S N
e2πi
N
j=1 λ j k j
.
(5.16)
k∈Z N
d N λ ψλ (x) e2πi
N
j=1 λ j k j
d N λ ψλ (x) ψλ (x + k).
ψλ (x ) (5.17)
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We conclude that the Green function G(x, x ) (the path integral on the cylinder with insertion of exp (−t2 H2(0) )) is represented as: N G Y M (x, x ) = d N λ ψλ (x) e2πi j=1 λ j n j e−t2 p2 (λ) ψλ (x ) (5.18) n∈Z N
λ∈R N /S N
or equivalently as: G Y M (x, x ) =
k∈Z N
λ∈R N /S
d N λ ψλ (x)e−t2 p2 (λ) ψλ (x + k).
(5.19)
N
Let us note that the identities in (5.17), (5.19) are based on the following transformation property of the complete set of skew-symmetric normalized wave-functions on R N : N N ψλ (x + k) = (−1)l(w) e2πi j=1 λw( j) k j e2πi j=1 λw( j) x j . (5.20) w∈S N
Thus each elementary term in the averaging over S N is multiplied on the simple exponent factor entering the description of the projector (5.16). Let us also note that the shift transformations in (5.20) can be interpreted as large gauge transformations in YangMills theory discussed above. The representation (5.19) can be written in the following form: G Y M (x, x ) = G 0 (x, x + k). (5.21) k∈Z N
If we set the coupling t2 to zero, t2 = 0, we recover the formula (5.17) for K (x, x ): K 0 (x, x + k). (5.22) K (x, x ) = k∈Z N
For the partition function of Yang-Mills theory on a torus we get (after setting x = x above and integrating over x): N N Z Y M (1 ) = d λ d N x ψλ (x) e2πi j=1 λ j n j e−t2 p2 (λ) ψλ (x) n∈Z N
=
λ∈R N /S N
e−t2 p2 (m+ρ) ,
(S 1 ) N
(5.23)
m∈P++
and this coincides with the representation (5.3). Note the obvious relations between (5.17), (5.21), (5.22) and the averaging over the topologically non-trivial gauge transformations discussed above. Let us remark that the averaging procedure represented by (5.21), (5.22) is a standard tool in construction of Green functions on non-simply connected spaces. At the first step one computes the Green function on the universal covering space and then averages with respect to the action of the action of π1 of the underlying non-simply connected space. For example this procedure has been used in a similar problem of the quantization of the coadjoint orbits of compact Lie groups in [43]. We will apply this procedure to the Yang-Mills-Higgs theory in a fashion described above for 2d Yang-Mills theory.
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Now we are finally ready to consider the case of Yang-Mills-Higgs theory. As it was conjectured above one can choose as a basis of the wave-functions the basis of eigenfunctions of the set of the Hamiltonian operators in the N -particle subsector of Nonlinear Schrödinger theory. Below we construct the Green function and partition function in Nonlinear Schrödinger theory and demonstrate that identifying the Hamil(0) tonian operator with quadratic observable O2 = (2π1 )2 Trϕ02 in Yang-Mills-Higgs theory we reproduce the partition function of Yang-Mills-Higgs theory on a torus. Let us start with the construction of the kernel of the unit operator in the bases of the N -particle eigenfunctions of the Nonlinear Schrödinger theory. The representation for the kernel (5.17) can be straightforwardly generalized to this case: m (x) nor m (x ) = (x, x ) = nor d N λ λ (x) P(λ) λ (x ), K λ λ R N /S N
(λ1 ,...,λ N )∈R N
(5.24) where λ (x) are normalized skew-invariant eigenfunctions on R N given by (4.16), m (x) are normalized periodic eigenfunctions given by (4.21) and the sum goes over nor λ the set R N of the solutions of Baxter Ansatz equations. The projector here is given by: P(λ) = µ(λ)
N
(λ∗1 ,...,λ∗N )∈R N
j
δ(α j (λ) − m j ) =
m∈Z N j=1
where α j (λ) are defined as follows (compare with (3.28)): λk − λ j − ic 1 . α j (λ) = λ j + log 2πi λk − λ j + ic
δ(λ j − λ∗j ), (5.25)
(5.26)
k= j
Then we have: (x, x ) = K n∈Z N
=
d λ µ(λ) λ (x) e N
N
m=1 λm n m
R N /S N
k∈Z N
2πi
R N /S N
λl − λ j − ic n j λ (x ) λl − λ j + ic
l= j
d N λ λ (x) λ (x + n).
(5.27)
The last equality follows from the following property of the eigenfunctions (4.20) of the N -particle Hamiltonian in Nonlinear Schrödinger theory: λw(i) − λw( j) + ic n i l(w) λ (x + n) = (−1) exp 2πi λw(m) n m λw(i) − λw( j) − ic m w∈W i< j 1 λw(i) − λw( j) + ic 2 +[xi −x j ] × exp 2πi λw(k) xk . (5.28) λw(i) − λw( j) − ic i< j
k
These wave functions are periodic and descend to the wave functions (4.16) if λ = (λ1 , . . . , λ N ) satisfy (4.19). The representation for the kernel (5.27) leads to the following representation for the Green function (cylinder path integral) and torus partition function for U (N ) YangMills-Higgs theory for ti=2 = 0:
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n∈Z N
R N /S N
355
G Y M H (x, x ) (5.29) N λl − λ j − ic n j d N λ µ(λ) λ (x)e2πi m=1 λm n m e−t2 p2 (λ) λ (x ), λl − λ j + ic
or the same: G Y M H (x, x ) =
l= j
G 0Y M H (x, x + k) =
k∈Z N
k∈Z N
R N /S N
d N λ λ (x)e−t2 p2 (λ) λ (x + k).
0 (x, x + k), since the kernel is a Green = k∈Z N K Similarly for the kernel: function at t2 = 0. Integrating over x after setting x = x we obtain the representation for the partition function on the torus: (x, x ) K
=
R N /S N
d N λ µ(λ)
(n 1 ,...,n N )∈Z N
Z Y M H (1 ) e2πi
N
m=1 λm n m
λl − λ j − ic n j e−t2 p2 (λ) . λl − λ j + ic
l= j
(5.30) This is in complete agreement with a representation for the partition function of U (N ) Yang-Mills-Higgs theory on a torus discussed in Sect. 3, formula (3.30). All the above expressions have the property to recover corresponding well-known answers of 2d YM theory in the limit c → ∞, as they should from the general arguments presented before. Note that one can repeat the same arguments for all observables and higher differential operators of Nonlinear Schrödinger theory, traces of higher powers of the Dunkle operator from Sect. 4, by simply turning on all other couplings tk . The identification of the representation of the partition function of the Nonlinear Schrödinger operator and Yang-Mills-Higgs theory on the torus strongly suggests the full equivalence of the theories. It would be very desirable to obtain the same answer for the Green function and the wave-function (cylinder path integral) directly from the path integral for cylinder topology using the cohomology localization technique. Finally let us comment on the explicit form of the wave function (5.28) from the gauge theory point of view. The appearance of an integer part [x] in (5.28) is not quite an unexpected phenomena from the point of view of the proposed identification of the wave functions in Yang-Mills-Higgs theory with the wave functions in Nonlinear Schrödinger theory. Let us remark that the interpretation of the topologically non-trivial bundles as an interpolation between topologically nontrivial gauge transformations naturally arises in the discussion of the spectral flow of the eigenfunctions of the gauge invariant operators on the boundary (see [44,45] for the details and examples). To make closer contact with this interpretation let us recall the simplest instance of the Atiyah–Patodi–Singer index theorem on an two-dimensional manifold with non-empty boundary [46]. Given an even-dimensional Riemann spin manifold M with a boundary ∂ M, and a vector bundle E supplied with a connection ∇ A , consider the Dirac operator on a vector bundle E ⊗ S, where S is a spinor bundle supplied with a connection ∇ S . The definition of the index of the Dirac operator on a non-compact manifold M relies on the correct treatment of the boundary conditions. In [46] specific non-local boundary conditions were defined corresponding to a vacuum state in the Hilbert space of the Dirac fermions. We would like to apply the index theorem to the two-dimensional cylinder L, 1 with a flat metric and U (1) bundles supplied with a connection ∇ . The ∂ L = S+1 ∪ S− A spinor bundle on L can be identified with 0,0 (L) ⊕ 0,1 (L) and the Dirac operator is given by D = ∂¯ A + ∂¯ A+ . We have the following expression for the index of D which
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coincides with the index of ∂¯ A :
IndexD =
1 1 c1 (∇) + η(S+1 ∪ S− ), 2 L
(5.31)
1 ) is defined in terms of the spectrum of the restriction where the η-invariant η(S+1 ∪ S− of D to the boundary : η(∂ L) = lim sign(λi ) |λi |s . (5.32) s→0
Taking into account the relation
L
λi ∈Spec(D |∂ L )
F(A) =
S+1
A−
1 S−
A we have:
i i 1 1 η(S+ ) − A − η(S− ) − A = IndexD ∈ Z. 2π S+1 2π S−1
(5.33)
An easy calculation shows that the η-invariant for a constant connection ∇ A = ∂t + x, on S 1 is given by: 1 1 − η(S 1 ) = x − [x] − , 2 2 where [x] is an integer part of x. Thus: i 1 1 1 A − η(S 1 ) = −x + (x − [x] − ) = [x] + . 2π S 1 2 2 2
(5.34)
(5.35)
This makes the appearance of integer values in (5.28) less mysterious. One should stress however that the proper derivation of the wave function (5.28) using this reasoning should use a more refined form of the η-invariant also introduced in [46]. Let B be an operator commuting with D. Then the character-valued η-invariant is given by: η B (∂ M) = lim sign(λi ) |λi |s tr Vλi B, (5.36) s→0
λi ∈Spec(D |∂ M )
where Vλi is an eigenspace of D corresponding to the eigenvalue λi . Using an appropriate operator B one can reproduce the phase factor in (5.28). Finally note that η-invariant is defined using the vacuum boundary condition on the quantum fields [46]. Thus the appearance of η-invariant in the explicit expression for a wave function can be traced back to the fact that the additional fields ( , ψ , ϕ± , χ± ) entering the description of the Yang-Mills-Higgs theory are in the vacuum state. Thus the only contribution to the total wave function is a phase factor. 6. On Equivariant Cohomology Description of the Hilbert Space The realization of the representations of the degenerate affine Hecke algebras Hg,c in the space of the S 1 × G-equivariant cohomology of the flag spaces for the Lie group G, such that Lie(G) = g considered in [47,48] (see also [49,50]), bears an obvious resemblance with the constructions discussed in this paper. Below we make some preliminary remarks regarding this relation. The detailed consideration will be postponed for another occasion.
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We start with the simplest case of the dimensionally reduced Yang-Mills theory. The one-dimensional Yang-Mills theory in the Hamiltonian formulation is a system with a first class constraint and thus the Hilbert space is naturally described as a cohomology of the corresponding BRST operator. The standard approach is to realize a Hilbert space as the cohomology of the BRST operator acting in some extended space including ghost variables. One considers a pair of canonically conjugate ghost-antighost fields (b, c) of ghost numbers (−1, 1) such that the BRST operator acts on the space of functions of ϕ and c, and is given by: 1 ∂ ∂ Q B R ST = Tr c ϕ, − Tr [c, c], . (6.1) ∂ϕ 2 ∂c The Hilbert space H = H Q∗ B R ST of the theory is naturally graded by the ghost number. The relevant cohomology can be interpreted as a Lie algebra cohomology of g = Lie(G) with coefficients in the space of functions on g considered as a g-module with respect to the adjoint action: H = H Q∗ B R ST = H ∗ (g, Fun(g)).
(6.2)
Note that the higher cohomologies are non-trivial and the result differs from the naive expectation of finding the space of the gauge invariant function on g as a realization of the Hilbert space. Let us remark that (6.2) is close to HG (G)∗ . One can try to use a more economical way to quantize the theory. Let us consider the same set of fields but use a modified BRST operator: 1 ∂ ∂ ∂ − T r [c, c], +ϕ . Q B R ST = T r c ϕ, (6.3) ∂ϕ 2 ∂c ∂c The cohomology of this BRST operator provides a BRST model for G-equivariant cohomology of the point (see [51] for discussion of various models for equivariant cohomology): H = HG∗ ( pt) ≡ H ∗ (BG) = C[G]G .
(6.4)
We see that in this formulation we get the correct Hilbert space. The interpretation of the Hilbert space of the dimensionally reduced Yang-Mills theory in terms of the equivariant cohomology is very natural. Let us recall that according to [52] the Hilbert space of the G/G gauged Wess-Zumino-Witten theory at the level k can be described in terms of twisted equivariant cohomology K G (G)∗+k+cv +dim(G) . In the limit k → ∞ this provides a description of the states in Yang-Mills theory in terms of equivariant cohomologies HG∗ (G). After the reduction to one dimension one gets the equivariant cohomology HG∗ (G) ⊗ C with coefficients in C which provide a model for the Hilbert space of Yang-Mills theory. 7. Relation with Nahm Transform Taking into account previous considerations it is natural to look for more direct correspondence between two-dimensional U (N ) Yang-Mills-Higgs theory and quantum nonrelativistic Nonlinear Schrödinger theory associated with U (2) ( Nonlinear Schrödinger theory exists for any group, see e. g. [29]; here we only utilized the U (2) version of it). In this section we briefly comment on this issue leaving the detailed considerations for future work.
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Let us consider a covariant description of the phase space of the theory as a space of solutions of the equations of motion (see [53] for discussion of the general formalism). Applying the general construction to the two-dimensional theory one can identify the phase space with a space of classical solutions on S 1 × R. Consider first the Yang-Mills theory. Equations of motion for ϕ lead to the flatness condition F(A) = 0 and thus the moduli space of the classical solutions has a natural projection onto the space of unitary flat G-connections on S 1 × R. Under appropriate boundary conditions this space can be identified with G/Ad G = H/W . The fiber of the projection is given by the space of covariantly constant sections ∇ A ϕ = 0, and thus the total phase space can be identified with the cotangent bundle to a moduli space of flat connections. Thus, indeed the covariant phase space coincides with the phase space M = T ∗ H/W described in Sect. 2. Consider now Yang-Mills-Higgs theory for c = 0. Equations of motion obtained by the variation over (ϕ0 , ϕ+ , ϕ− ) are given by: F(A) − ∧ = 0,
(1,0)
∇A
(0,1) = 0,
(0,1)
∇A
(1,0) = 0,
(7.1)
and those given by the variation over (A, ) are: (1,0)
∇A
(0,1)
∇A
ϕ+ = c (1,0) + [ (1,0) , ϕ0 ],
ϕ− = −c (0,1) + [ (0,1) , ϕ0 ],
∇ A ϕ0 = [
(1,0)
, ϕ+ ] − [
(0,1)
(7.2)
, ϕ− ].
Let us start with the space M H of the solutions of the first set of equations. The equations: F(A) − ∧ = 0,
∇ A = 0,
(7.3)
are equivalent to a flatness condition of the modified connection: (d + A + i )2 = 0,
(7.4)
and have a simple solution on C∗ = S 1 × R: Ac = A + i = gc−1 dgc + gc−1 AcD gc ,
(7.5)
where AcD is a constant one form taking values in the diagonal matrices. It is useful to represent the complex matrix gc as gc = bg, where b ∈ G L(N , C)/U (N ) is a Hermitian matrix and g ∈ U (N ) (Cartan decomposition). Then g can be gauged away and we have: A = AbD − (AbD )+ ,
= −i(AbD + (AbD )+ ),
(7.6)
where AbD = b−1 db+b−1 AcD b. The third equation from the first set provides a constraint on b which fixes it up to a holomorphic map C∗ = S 1 × R → G (harmonicity condition). A variant of Narasimhan-Seshadri and Ramanathan arguments [54,55] allows to describe, in holomorphic terms, the moduli space of flat G-bundles for a compact complex curve. Consider the equation ∇ (0,1) 1,0 = 0. It describes a holomorphic section of A the holomorphic bundle. The rest of the equations define the unitary structure and have unique solutions on the compact surface. In the non-compact case the same arguments work for appropriate boundary conditions. Thus, we can think of the phase space of Yang-Mills-Higgs theory as a moduli space of Hitchin equations on C∗ = S 1 × R. Now we can try to apply Nahm duality to characterize this moduli space in other terms. Indeed, the solutions of Hitchin equations on S 1 × R can be considered as solutions of
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four-dimensional (anti)self-dual YM equations on R×S 1R1 ×S 1R2 ×S 1R2 when R2 → 0. By the Nahm duality [56,57] the moduli space of (anti)self-dual gauge fields on S 1R0 × S 1R1 × S 1R2 × S 1R2 for a gauge group U (N ) is equivalent to the moduli space of (anti)self-dual 1 1 1 1 × S1/R × S1/R × S1/R for another gauge group U (M), where M is gauge fields on S1/R 0 1 2 2 the second Chern class of the gauge field (instanton number). Therefore taking R0 → ∞, R2 → 0 we get an equivalence of the moduli space of solutions of Hitchin equations for G = U (N ) with the moduli space of periodic monopoles on S 1R1 × R × R with the gauge group U (M), where M is an appropriately defined topological characteristic of the solutions with fixed boundary conditions. Taking into account that the cylinder has two boundaries, the simplest nontrivial boundary conditions lead to M = 2. Thus, one can expect that the moduli space for U (N )-gauge theory will be equivalent to the N -monopole solution in a U (2) gauge theory. These considerations correspond to the case c = 0. One can hope that considering the S 1 -equivariant version of the Nahm correspondence one obtains the analogous relations for c = 0. This would provide a hint for the direct connection between Yang-Mills-Higgs theory for G = U (N ) and Nonlinear Schrödinger theory associated with U (2). 8. Generalization of G/ G Gauged WZW Model It is natural to expect that the story presented in previous sections extends from the YangMills-Higgs theory to the certain generalization of G/G gauged WZW theory. In this section we describe such a construction and show that the partition function can be represented as a sum over solutions of a certain generalization of the Bethe Ansatz equation. We start with the definition of the set of fields and the action of the odd and even symmetries in the spirit of (3.7), (2.3), (3.37). Let us note that the gauged Wess-ZuminoWitten model can be obtained from the topological Yang-Mills theory by using the group-valued field g instead of algebra-valued field ϕ. Correspondingly, we replace the generators of the Lie algebra actions with the parameter ϕ (2.2), (2.3) by the generators of the Lie group action (2.45), (2.44) with the parameter g. Thus it is natural to introduce the set of fields (A, ψ A , , ψ , χ± , ϕ± , g) and t ∈ R∗ with the following action of the odd and even symmetries: (0,1)
L(g,t) A(1,0) = (A g )(1,0) − A(1,0) , L(g,t) A A
−1
= −(A g )(0,1) + A(0,1) , (8.1)
L(g,t) ψ A(1,0) = −gψ A(1,0) g −1 + ψ A(1,0) , L(g,t) ψ A(0,1) = g −1 ψ A(0,1) g − ψ A(0,1) , L(g,t) g = 0, L(g,t) (1,0) = tg (1,0) g −1 − (1,0) , (1,0)
L(g,t) ψ
(1,0) −1 g
= tgψ
L(g,t) χ+ = tgχ+ L(g,t) ϕ+ = QA
(1,0)
− ψ
g −1
(0,1)
,
− χ+ ,
t −1 gϕ+ g −1
L(g,t) (0,1) = −t −1 g −1 (0,1) g + (0,1) , L(g,t) ψ
L(g,t) χ− =
(0,1)
= −t −1 g −1 ψ
−t −1 g −1 χ
− ϕ+ ,
(0,1)
g + ψ
,
− g + χ− , −1 −tg ϕ+ g + ϕ+ ,
L(g,t) ϕ+ = (1,0) = iψ A , Q ψ A = i(A g )(1,0) − i A(1,0) , −1 (0,1) Q ψ A = −i(A g )(0,1) + i A(0,1) ,
(8.2)
Q g = 0,
Q = iψ ,
(1,0) Qψ = tg (1,0) g −1
(0,1)
− (1,0) , Qψ
Qχ± = iϕ± , Qϕ+ = tgχ+ g −1 − χ+ ,
= −t −1 g −1 (0,1) g − (0,1) ,
Qϕ− = −t −1 g −1 χ− g + χ− .
(8.3)
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We have Q 2 = L(g,t) and Q can be considered as a BRST operator on the space of L(g,t) -invariant functionals. We define the action of the theory in analogy with the construction of the action for Yang-Mills-Higgs theory as follows: 1 2 S = SGW Z W + Q, d z Tr ∧ ψ 2 h (1,0) (0,1) (0,1) (1,0) + τ2 (χ+ ϕ− + χ− ϕ+ ) volh +τ1 ϕ+ ∇ A + ϕ− ∇ A . (8.4) +
Taking τ1 = 0, τ2 = 1 and applying the standard localization technique to this theory we obtain for the partition function: N e(1−h)a(t) Z GW Z W H (h ) = d N λ µq (λ)h e2πi m=1 λm n m (k+cv ) |W | H (n 1 ,...,n N )∈Z N × (e2πi(λ j −λk ) − 1)n j −n k +1−h (te2πi(λ j −λk ) − 1)n j −n k +1−h , j=k
j,k
(8.5) where a(t) is an h-independent constant, the integral goes over the Cartan torus H = (S 1 ) N and ∂β j (λ) (8.6) µq (λ) = det ∂λ , k with: e2πiβ j (λ) = e2πiλ j (k+cv )
te2πi(λ j −λk ) − 1 . te2πi(λk −λ j ) − 1 k= j
(8.7)
We can rewrite this formula in the form similar to (3.27): N e(1−h)a(t) Z GW Z W H (h ) = d N λ µq (λ)h e2πi m=1 βm (λ)n m |W | H (n 1 ,...,n N )∈Z N iπ(λ j −λk ) iπ(λk −λ j ) 2−2h × (e −e ) j
×
|teiπ(λ j −λk ) − eiπ(λk −λ j ) |2−2h .
(8.8)
j
Summation over integers in (8.5) leads to the following restriction on the integration parameters: e2πiλ j (k+cv )
te2πi(λ j −λk ) − 1 = 1, 2πi(λk −λ j ) − 1 te k= j
i = 1, . . . , N .
(8.9)
It is useful to rewrite Eq. (8.9) in the standard form of the Bethe Ansatz equations: sin(iπ(λ j − λk + ic)) = 1, i = 1, . . . , N . (8.10) e2πiλ j (k+cv ) sin(iπ(λ j − λk − ic)) k= j
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This clearly shows that we are dealing with a kind of XXZ quantum integrable chain. The particular form (8.10) can be obtained by taking the limit s → −i∞ in the following Bethe equations: sin(iπ(λ j − isc)) (k+cv ) sin(iπ(λ j − λk + ic)) = 1, i = 1, . . . , N , (8.11) sin(iπ(λ j + isc)) sin(iπ(λ j − λk − ic)) k= j
corresponding to formal limit of the infinite spin s of XXZ chain. The partition function is the generalization of Yang-Mills-Higgs theory, discussed above, and can be written in the following form: q Z GW Z W H (h ) = (Dλ )2−2h , (8.12) λi ∈Rq
where Rq is a set of the solutions of (8.9) and: 1 1 1 1 q (q 2 (λi −λ j ) − q 2 (λ j −λi ) ) |tq 2 (λi −λ j ) − q 2 (λ j −λi ) |, (8.13) Dλ = µq (λ)−1/2 i< j
i< j
where we use the standard parametrization q = exp(2πi/(k + cv )). Note that in the limit t → ∞ Eq. (8.9) and the expression for the partition function (8.13) up to an overall scaling factor become the corresponding expressions for a gauged Wess-Zumino-Witten model. Finally note that the form of (8.9) and the explicit expressions for the q-Casimir operators, playing the role of the Hamiltonians, strongly imply the description of the wave functions of the theory in terms of the wave functions in a particular X X Z finite spin chain. This proposition will be discussed in detail elsewhere. 9. Conclusion Let us put the results of this paper in a more general perspective. Any two-dimensional topological theory satisfying the appropriate cutting/gluing relations [58] can be described by a commutative Frobenius algebra. Generally this Frobenius algebra comes from the chiral ring R of some Conformal Field Theory (CFT). For example the gauged WZW model leads to finite-dimensional Frobenius algebra associated with the representation theory of the finite-dimensional quantum groups. The corresponding CFT is a WZW model. Similarly, the two-dimensional topological Yang-Mills theory is related to the infinite-dimensional Frobenius algebra constructed in terms of the representation theory of finite-dimensional Lie groups. The associated CFT is a particular degeneration of the WZW model. Thus, one should expect that the topological Yang-Mills-Higgs theory introduced in [1] and its generalizations defined in this paper correspond to some interesting classes of two-dimensional CFT. One can speculate that such CFT should be constructed by an appropriate deformation of the WZW model for complex gauge groups. Taking into account the relation between WZW models for complex groups and the (generalizations) two-dimensional quantum gravity this might be elaborated in precise form. In this paper we mostly restrict ourselves to the case G = U (N ). This restriction is not essential. One can study these theories for an arbitrary semisimple Lie group (for the construction of finite-particle wave functions for arbitrary G see e.g. [3]). Let us stress that the simple expression for the partition function as a sum over the solutions of Bethe Ansatz equations can be considered as a kind of nonlinear Fourier
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transform of the similar, but more conventional, representation as a sum over a set of C∗ -fixed point components of the Higgs bundle moduli spaces (see [1] for details). The equality of the two dual representations of the partition function can be considered as an example of the Arthur-Selberg trace formula if one takes into account the interpretation of C∗ -fixed points as variations of the Hodge structures on the underlying curve. Note also that the existence of the two dual representations leads to sum rules for the solutions of the Bethe Ansatz equations and the corresponding nonlinear Fourier transform looks very similar to the Quantum Inverse Scattering Method for quantum integrable systems [5]. It would be interesting to make this analogy more precise. Non-linear Fourier transforms become standard tools in the study of topological sigma models and mirror symmetry and here we see another important appearance. Let us note that it would be also interesting to establish the relation between results of the present paper and the other type of connection between 2d Yang-Mills theory and many-body systems described in [59]. Given a U (N ) gauge theory it is natural to consider its ’t Hooft limit N → ∞. In this limit one expects to find a dual description of the theory in terms of stringy expansion. The known results for the dual string description of two-dimensional Yang-Mills theory [60–62] implies that the similar description of the N → ∞ limit of Yang-Mills-Higgs theory and its generalizations can be very instructive. In this respect let us note that recently the same program has been completed for q-deformed two-dimensional YangMills theory [63]. The q-deformed theories are closely related to the gauged WZW deformations of two-dimensional Yang-Mills theory. Thus the deformation of the YangMills-Higgs theory proposed in Sect. 10 can provide a dual description of interesting string backgrounds. In this respect it is interesting to note that the proper quasi-classical expansion of the Bethe Ansatz solutions, given via the double scaling limit c → 0, N → ∞ with N c-fixed, is known to lead to Riemann surfaces and genus expansion [64]. The gauge theories considered in this paper seem to provide a proper framework for the quantum field theory version of a Kazhdan-Lusztig type construction of the representations of double affine Hecke algebras (DAHA). Indeed the essential role played by S 1 × G-equivariant cohomology groups and K -groups both in the Kazhdan-Lusztig constructions and in the quantum gauge theory construction implies a deep relation between these two subjects. In the topological gauge theories considered here the natural objects are gauge invariant and thus reproduce only the properties of the center of DAHA. One can expect that the consideration of the corresponding CFT discussed above should provide a deeper connection with the representation theory of DAHA. This seems compatible with the ideas regarding the relation between the representation theory of the affine Hecke algebra and hyperkähler geometry of Higgs bundles advanced in [65]. Note that the Hecke algebras are important ingredients of the explicit construction of the Langlands correspondence (see [66,67] for modern introductions into the subject). Thus it is tempting to suggest that the relation between gauge theories and double affine Hecke algebras considered in this paper can lead to a better understanding of this correspondence. The latter might be related to recent studies in [68] where an interpretation of the geometric Langlands correspondence in terms of quantum field theory was described. Acknowledgements. We are grateful to J. Bernstein, M. Kontsevich, W. Nahm, N. Nekrasov, F. Smirnov and L. Takhtajan for discussions. The research of the first author is partly supported by the grant RFBR 04-01-00646 and Enterprise Ireland Basic Research Grant; that of the second author is supported by Enterprise Ireland Basic Research Grant, SFI Research Frontiers Programme and Marie Curie RTN ForcesUniverse from EU.
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A. Twistor Type Description In this appendix we give a Chern-Simons type representation of the bosonic part of the Yang-Mills-Higgs action using a twistor approach. Consider the one-dimensional family of flat connections: D = d + A(λ) = d +
1 (1,0) + A + λ (0,1) . λ
(A.1)
The flatness condition for D: F(A) ≡ D2 = 0
(A.2)
is equivalent to the Hitchin equations for the pair (A, ). Then we have the following representation for the bosonic part of the action S0 (3.34): ∂ S(A, ) = Resλ=0 d 2 z T r (ϕ(z, z¯ , λ)F(A(z, z¯ , λ)) + c A(z, z¯ , λ) A(z, z¯ , λ)). ∂λ (A.3) Here ϕ(z, z¯ , λ) = ϕ+ (z, z¯ ) + λ−1 ϕ0 (z, z¯ ) + λ−2 ϕ− (z, z¯ ). Let us note that the elements of the gauge group are considered to be independent of λ. In this representation it is natural to denote ϕ(z, z¯ , λ) ≡ Aλ (z, z¯ , λ) to get, formally, a three-dimensional action. Note however that there are severe constraints on the dependence of the fields on λ. Thus, it might be considered as a reduction of three dimensional Chern-Simons theory. B. On Lagrangian Geometry of the Singular Manifold In this appendix we discuss the classical gauge theory counterpart of the c-dependent boundary conditions (4.10) used for the construction of the eigenfunctions in Nonlinear Schrödinger theory. We will consider only the case of G = U (2) in the gauge theory reduced to one dimension. The phase space in this case is M = T ∗ R2 /Z2 . Choose the coordinates (x1 , x2 , π1 , π2 ) on T ∗ R2 such that (x1 , x2 ) are coordinates on R2 and (π1 , π2 ) are coordinates on the fibers of the projection T ∗ R2 → R2 . Let X = x2 − x1 , Y = π2 − π1 and the action of Z2 be given by: w0 : (X, Y ) → (−X, −Y ).
(B.1)
The space of functions on the phase space M is generated by the invariant functions a1 = 21 X 2 , a2 = 21 Y 2 and a3 = X Y satisfying: F(a) = 4a1 a2 − a32 = 0.
(B.2)
One has a1 ≥ 0, a2 ≥ 0 and the fiber of the projection on the subspace a1 > 0, a2 > 0 consists of two points while the fiber over (a1 = 0) ∪ (a2 = 0) consists of one point. The point a1 = a2 = 0 is singular (i.e. F = 0, ∂i F = 0 at this point). The quantization of this space in the standard representation Xˆ = x, Yˆ = ∂x is given by the representation of the algebra sl(2): [aˆ 2 , aˆ 1 ] = aˆ 3 , [aˆ 3 , aˆ 1 ] = 2aˆ 1 , [aˆ 3 , aˆ 2 ] = −2aˆ 2 ,
(B.3)
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A. A. Gerasimov, S. L. Shatashvili
where: aˆ 1 =
1 2 1 x , aˆ 2 = ∂x2 , 2 2
1 (x∂x + ∂x x). 2
aˆ 3 =
(B.4)
The equation 4a1 a2 − a32 = 0 translates into the restriction on the value of the second Casimir operator in this representation. Thus we are dealing here with the representations associated with the nilpotent orbits of sl(2). The representations can be described in terms of the space of regular functions on the Lagrangian submanifold L0 defined by the equation a2 = 0. The functions on L0 are the functions of a1 (a3 = 0 on L0 ) and the action of aˆ i is given by: aˆ 1 f (a1 ) = a1 f (a1 ), aˆ 2 f (a1 ) =
∂ f (a1 ) ∂a1
(B.5)
2 + 4a1 ∂∂a 2f 1
aˆ 3 f (a1 ) = 4a1 ∂ f∂a(a11 ) +
1 2
,
(B.6)
f (a1 ).
(B.7)
Regular functions depending on x 2 can be characterized by the following conditions: δ(x)∂x (x −1 ∂x )n f (x) = 0,
n ≥ 0.
(B.8)
It easy to see that these conditions are invariant with respect to the action of Z2 and compatible with the action of sl(2). The first condition can be written in a more usual form as: ∂ X f (X )| X =0 = 0, which is the boundary condition discussed above. Now consider the case c = 0. The change of the boundary condition can be understood in geometric terms as follows. As it was discussed previously the phase space is defined by the equations: a1 a2 − a32 = 0, a1 ≥ 0, a2 ≥ 0.
(B.9)
In the case of the Yang-Mills theory we quantize the cone (B.9) using the Lagrangian submanifold L0 defined by the equation a2 = 0. This Lagrangian submanifold is rather special. Note that the associated representation of sl(2) is irreducible. Its naive deformation Lc defined by the equation a2 = c2 provides a reducible representation. To c of Lc to (x, y)-plane. The lift of L0 is given understand it better consider the lift L c=0 of Lc=0 consists by the connected submanifold x = 0. On the other hand the lift L of two components y = ±c. This means that the representation associated with Lc=0 can be reducible. Indeed one can choose as a Z2 -invariant Lagrangian submanifold the submanifold Lˆ +c=0 defined by the equations: +c=0 = (y = +c, x > 0) ∪ (y = −c, x < 0). L
(B.10)
Taking the factor over Z2 we obtain: L+c=0 = (a1 > 0, a3 > 0, a2 = c2 ).
(B.11)
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Let us describe the corresponding spaces of functions and the action of the generators c , where c = 0. As functions of x it is of sl(2). We start with the space of functions on L given by a pair of functions with the action of yˆ by the following differential operator: yˆ
∂ +c 0 f + (x) f + (x) = x , f − (x) 0 ∂x − c f − (x)
(B.12)
and the action of the generator w0 of Z2 is given by: w0
f − (−x) f + (x) = . f − (x) f + (−x)
(B.13)
Equivalently we have: yˆ
f + (x)ecx f − (x)e−cx
f + (x)ecx ∂x 0 = , 0 ∂x f − (x)e−cx
(B.14)
and w0
f + (x)ecx f − (x)e−cx
=
f − (−x)ecx . f + (−x)e−cx
(B.15)
Thus the symmetry condition is reduced to f + (x) = f − (−x). Note that we automatically have f + (0) = f − (0). Now consider the functions defined on the lift of the Lagrangian submanifold L+c . They can be described as follows: f (x) = f + (x)ecx , x > 0, f (x) = f + (−x)e−cx , x < 0.
(B.16) (B.17)
The symmetry condition implies that f + (x) is a symmetric function. Note that: f (x)|x→+0 = f (x)|x→−0 and ∂x f (x)|x→+0 = c f (0), ∂x f (x)x→−0 = −c f (0). Thus we have the deformed boundary conditions. As an example consider the function that is a solution of the equation ∂x2 f (x) = λ2 f (x) for x = 0 and satisfies the boundary conditions at x = 0: f λ (x) = (λ − ic) eiλx + (λ + ic) e−iλx , f λ (x) = (λ + ic) eiλx + (λ − ic) e−iλx ,
x > 0, x < 0.
(B.18) (B.19)
This function can be easily represented in the form discussed above: f λ (x) = (λ − ic) ei(λ+ic)x + (λ + ic) e−i(λ−ic)x ecx , x > 0, f λ (x) = (λ + ic) ei(λ−ic)x + (λ − ic) e−i(λ+ic)x e−cx , x < 0.
(B.20) (B.21)
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Commun. Math. Phys. 277, 369–384 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0370-8
Communications in
Mathematical Physics
Tensor Invariants of the Poisson Brackets of Hydrodynamic Type Oleg I. Bogoyavlenskij Department of Mathematics, Queen’s University, Kingston, K7L 3N6, Canada. E-mail: [email protected] Received: 6 December 2006 / Accepted: 28 May 2007 Published online: 30 October 2007 – © Springer-Verlag 2007
Abstract: Form-invariant solutions for the Poisson brackets of hydrodynamic type on a manifold M n with (2,0)-tensor g i j (u) of rank m ≤ n are derived. Tensor invariants of the Poisson brackets are introduced that include a vector field V (or dynamical system V ) on M n , the Lie derivative L V g i j and symmetric (k, 0)-tensors h i j··· . Several scalar invariants of the Poisson brackets are defined. A nilpotent Lie algebra structure is disclosed in the space of 1-forms Au ⊂ Tu∗ (M n ) that annihilate the (2,0)-tensor g i j (u). Applications to the one-dimensional gas dynamics are presented. 1. Introduction I. We study quasi-linear systems of partial differential equations [1–5] u it =
n
j
Aij (u 1 , . . . , u n )u x .
(1.1)
j=1
Here u 1 , . . . , u n form a chart of local coordinates on a manifold M n and u i (t, x) are unknown functions. Properties of system (1.1) are determined by the geometry of the (1,1)-tensor Aij (u) [6]. For a Hamiltonian system with a local [7] or non-local [8] structure of Poisson brackets, the (1,1)-tensor Aij (u) has the form Aij (u) = g iα (u)
∂ 2 f (u) ∂ f (u) + biα + K f (u)δ ij . j (u) ∂u α ∂u j ∂u α
(1.2)
Here f (u) is the density of the Hamiltonian functional and K = const. The symmetric (2,0)-tensor g i j (u) and the non-tensorial coefficients bik j (u) satisfy certain nonlinear equations that follow from the Jacobi identity. As is known [7], the equations for the non-degenerate (2,0)-tensor g i j (u) for K = 0 mean that the covariant metric gi j (u) has
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zero curvature. The corresponding Poisson brackets are always degenerate since they have n Casimirs. II. In Sect. 2, we introduce the form-invariant solutions for the Poisson brackets of hydrodynamic type with rank g i j (u) = m ≤ n and K = 0: g i j (u) =
m α,β=1
cαβ Uαi (u)Uβ (u), bk (u) = Bk (u) = j
ij
ij
m α,β=1
j
cαβ Uαi (u)
∂Uβ (u) ∂u k
. (1.3)
Here U1 (u), . . . , Um (u) are arbitrary commuting vector fields on the manifold M n . The ij main property of these exact solutions is that in spite of the coefficients Bk (u) do not form a tensor, they have the same invariant form (1.3) in all systems of coordinates. Solutions (1.3) define all Poisson brackets with non-degenerate (2,0)-tensor g i j (u) and K = 0. For 1 ≤ rank g i j (u) ≤ n − 1, the general solutions for K = 0 have ij ij ij ij the form bk (u) = Bk (u) + Tk (u), where Tk (u) is a certain skew-symmetric (2,1)tensor. The Hamiltonian systems (1.1)–(1.2) corresponding to the Poisson brackets with coefficients (1.3) have the form m ∂ ∂ f (u) i αβ i Uβ (u) c Uα (u) ut = ∂x ∂u α,β=1
and are systems of conservation laws. III. The Poisson brackets of hydrodynamic type form a special class in the space of all possible Poisson brackets in the space of functions u j (x). For the non-degenerate (2,0)-tensor g i j (u) and K = 0, the covariant metric gi j (u) has zero curvature and hence ij is constant in some coordinates where bk (u) ≡ 0 [7]. Evidently this is an analogue of the Darboux theorem for the finite-dimensional non-degenerate Poisson structures. The result of [8] for K = 0 states that metric gi j (u) has constant curvature K . Hence all its tensor invariants (notably the Riemann tensor Ri jk ) are polynomials of the metric components and all scalar invariants are constant. In this paper we show that the analogy with the Darboux theorem fails for the Poisson brackets of hydrodynamic type with rank g i j (u) ≤ n − 1. Indeed,the finite-dimensional Poisson structures P i j have only one local invariant - the rank P i j that evidently is locally constant. However, the Poisson brackets of hydrodynamic type with a degenerate (2,0)-tensor g i j (u) possess several scalar and tensor invariants that are not locally constant and depend on the coordinates u 1 , . . . , u n . IV. In Sects. 4–6, we present the locally non-constant invariants for the Poisson brackets with rank g i j (u) = m ≤ n − 1, and for any values of K . We demonstrate that in spite ij of the coefficients bk (u) do not form a tensor, there are the following tensor invariants ij constructed from the bk (u) and g i j (u): (a) A vector field V (u) (or a dynamical system) on M n . (b) A (1,1)-tensor L ij (u) that is invariantly defined on the leaves of the foliation F m [10]; its characteristic polynomial yields an invariant smooth mapping F : M n −→ Rm . (c) A symmetric (2,0)-tensor h i j (u) of rank ≤ m and symmetric (k, 0)-tensors h i j... (u). (d) A structure of nilpotent Lie algebra in the space of 1-forms Au ⊂ Tu∗ (M n ) that annihilate the (2,0)-tensor g i j (u). The simplest of such Lie algebras is the Heisenberg Lie algebra H 3 .
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2. Form-Invariant Solutions for the Poisson Brackets I. The Poisson brackets structure for the Hamiltonian systems (1.1) is defined by the operator I i j (u) for the local [7] and non-local case K = 0 [8]: I i j (u) = g i j (u)
∂ ij + bk (u)u kx + K u ix ∂x
∂ ∂x
−1
j
ux .
(2.1)
The corresponding Hamiltonian systems (1.1) have the form u it = I iα (u)
∂ f (u) j = Aij (u 1 , . . . , u n )u x , ∂u α
(2.2)
with the (1,1)-tensors Aij (u) (1.2), where f (u) is the density of the Hamiltonian functional. The operator I i j (u) (2.1) defines a structure of Poisson brackets ∞ δ F1 (u) i j δ F2 (u) ˆ ˆ I (u) j dx (2.3) { F1 (u), F2 (u)} = i (x) δu δu (x) −∞ ∞ for the local functionals Fˆγ = −∞ Fγ (u, u x , u x x , . . .)dx. The coefficients g i j (u) and ij
bk (u) (2.1) satisfy certain equations [7,9] that follow from the conditions that the Poisson brackets (2.3) are skew and the Jacobi identity holds. The skew-symmetricity condition for the Poisson brackets (2.3) is equivalent to the relations g i j = g ji ,
∂g i j ij ji = bk + bk . ∂u k
(2.4)
The Jacobi identity for the Poisson brackets (2.3) for K = 0 is equivalent to the equations [9]
σ,τ
⎛ ⎝
jr
jr ∂b ∂bα − kα ∂u k ∂u
σ (i)σ ( j)
∂bτ (k)
∂u α
bαik g α j = bαjk g αi ,
αj j g αi + bαi j bkαr − bαir bk + K g i j δkr − g ir δk = 0,
(2.5) (2.6)
⎞ σ (i)σ ( j)
∂bα ⎠ bασ () + K bσ ()σ (i) − bσ (i)σ () δ σ ( j) = 0. (2.7) − τ (s) τ (s) τ (k) τ (k) ∂u τ (k)
Here summation is taken with respect to the index α and in (2.7) with respect to the three cyclic permutations σ of indices i, j, and two transpositions τ of indices k and s. II. Any degenerate (2, 0)-tensor g i j (u) defines a distribution L u ⊂ Tu (M n ) that consists of all tangent vectors v i = g iα (u)ωα , where ωα are arbitrary covectors. We have dim L u = rank g i j (u). If the functions g i j (u) are smooth, or C ∞ , the manifold M n is n ij the closure of the union n−1 k=0 Ok of subsets Ok ⊂ M , where rank g (u) = k; some of i j Ok can be empty. If the functions g (u) are analytic then there is only one open domain Om ; it is everywhere dense in M n and its complement is a set of measure zero. For each non-empty domain Om , the subspaces L u form a smooth distribution of the constant dimension m.
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Any two vectors v i and w j belonging to the distribution L u have the form v i = and w j = g jβ (u)ηβ . Here the covectors ωα and ηβ are defined up to any covectors ωα and ηβ satisfying the equations g iα (u)ωα = 0, g jβ (u)ηβ = 0. In [10], we defined the following bilinear form: g iα (u)ωα
v, w = g αβ (u)ωα ηβ = v β ηβ = w α ωα .
(2.8)
The formulae (2.8) yield that the form v, w is correctly defined. It is evidently bilinear and symmetric. The form (2.8) is non-degenerate because v, w = v β ηβ and for any non-zero vector v β there exists a covector ηβ satisfying v β ηβ = 0, hence v, w = 0. In paper [10], we proved the following theorem for rank g i j (u) = m ≤ n − 1 and any K : (a) In the open domain Om , the distribution L u is involutive and defines an m-dimensional foliation F m . (b) Equations (2.4)–(2.7) imply that metric v, w (2.8) on the leaves of F m has constant curvature K . (c) The distribution L u is invariant under any (1,1)-tensor Aij (u) (1.2) and operators biα j (u) for α = const. III. The invariance of the Poisson bracket (2.1), (2.3) with respect to the changes of local coordinates v i = v i (u 1 , . . . , u n ) yields the transforms gr (v) =
∂vr ∂v i j ∂vr ∂v ∂u k i j ∂vr ∂ 2 v ∂u q r ij g (u), b (v) = b (u) + g (u) . p ∂u i ∂u j ∂u i ∂u j ∂v p k ∂u i ∂u j ∂u q ∂v p (2.9) ij
Formulae (2.9) show that the object bk (u) is not a tensor if g i j (u) = 0. However, we conij struct below certain tensors from the coefficients bk (u) provided that 1 ≤ rank g i j (u) ≤ n − 1. ij
Remark 1. For the non-degenerate (2,0)-tensor g i j (u), the coefficients bk (u) are connected with the Christoffel symbols ∂gkp ∂g pq ∂gqk 1 j (2.10) + − pk = g jq 2 ∂u k ∂u p ∂u q ij
j
by the formulae bk = −g iα αk [7]. Using formula (2.10), we find js is 1 ∂g i j ij ir ∂g jr ∂g . + gks g −g bk = 2 ∂u k ∂u r ∂u r
(2.11)
For this case, formula (2.9) follows from the classical transform [15] i
k (v) =
∂v i ∂u β ∂u γ α ∂ 2 v i ∂u β ∂u γ (u) − . ∂u α ∂v k ∂v βγ ∂u β ∂u γ ∂v k ∂v
Proposition 1. (a) Any Poisson bracket with rank g i j (u) = 1 and any constant K ij uniquely defines a (2,1)-tensor Tk (u) that satisfies the equations ij
ji
Tk = −Tk ,
Tαik g α j = 0.
(2.12)
(b) Any smooth 1-dimensional foliation X 1 on M n is realizable as the foliation F 1 for a Poisson bracket with rank g i j (u) = 1 and K = 0.
Tensor Invariants of the Poisson Brackets of Hydrodynamic Type
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Proof. (a) Any (2,1)-tensor g i j (u) of rank 1 has the form g i j (u) = cU i (u)U j (u), c = ±1,
(2.13)
where vector field U i (u) is tangent to the invariant foliation F 1 and is defined up to a factor µ = ±1. We introduce the non-tensorial coefficients ij
n k (u) = cU i (u)
∂U j (u) , ∂u k
(2.14)
that are defined by the tensor g i j (u) uniquely since they are unchanged after the substitution U i −→ µU i . The coefficients (2.14) are form-invariant and are transformed according to the formulae nrp (v) =
∂vr ∂v ∂u k i j ∂vr ∂ 2 v ∂u q ij n (u) + g (u) , ∂u i ∂u j ∂v p k ∂u i ∂u j ∂u q ∂v p
(2.15)
ij
that coincide with (2.9). The coefficients g i j (u) and n k (u) (2.13)–(2.14) satisfy all Eqs. (2.4)–(2.7) for K = 0. Indeed, Eqs. (2.4)–(2.6) are verified directly. Equations (2.7) after substitution of (2.14) take the form n σ () ∂U σ (i) ∂U σ ( j) ∂U σ (i) ∂U σ ( j) γ ∂U − = 0. (2.16) U ∂u γ ∂u τ (k) ∂u τ (k) ∂u γ ∂u τ (s) σ,τ γ =1
After summation with respect to the three cyclic permutations σ and two transpositions τ , all terms in (2.16) are cancelled out. ij For any Poisson bracket with coefficients bk (u), rank g i j (u) = 1 and any constant K , we define ij
ij
ij
Tk (u) = bk (u) − n k (u).
(2.17)
The formulae (2.9) and (2.15) yield the law of transformation r
T p (v) =
∂vr ∂v ∂u k i j T (u). ∂u i ∂u j ∂v p k
ij
(2.18) ij
ij
Hence the coefficients Tk (u) (2.17) form a (2,1)-tensor. Since both bk (u) and n k (u) ij ji satisfy Eqs. (2.4)–(2.5) that do not contain K , we get from Eqs. (2.4) Tk + Tk = 0 and jk from Eqs. (2.5) Tαik g α j = Tα g αi . Using these equations, we find Tαik g α j = −Tαki g α j = −Tαji g αk = Tαi j g αk = Tαk j g αi = −Tαjk g αi = −Tαik g α j . Hence Tαik g α j = 0. (b) Let g i j (u) be some Riemannian metric on M n . For any 1-dimensional foliation X 1 on M n , we define at each point u two vectors U i (u) and −U i (u) tangent to the foliation X 1 and having unit length in g i j (u). We introduce a global Poisson structure with the ij
(2,0)-tensor g i j (u) of rank 1 and the coefficients bk (u) that are uniquely defined by ij the formulae g i j (u) = U i (u)U j (u) and bk (u) = U i (u)∂U j (u)/∂u k (2.14) that do not depend on a local choice of the vector field U i (u) or −U i (u). Since Eqs. (2.4)–(2.7) hold for K = 0, we obtain a global Poisson structure. The corresponding foliation F 1 evidently coincides with the X 1 .
374
O. I. Bogoyavlenskij
Proposition 2. (a) Any solution to Eqs. (2.4)–(2.7) with a non-degenerate (2,0)-tensor g i j (u) and K = 0 locally has the form g (u) = ij
n α,β=1
c
αβ
j Uαi (u)Uβ (u),
ij bk (u)
=
n
j
c
αβ
α,β=1
Uαi (u)
∂Uβ (u) ∂u k
,
(2.19)
where cαβ are constant and U1 (u), . . . , Un (u) are mutually commuting vector fields on M n . (b) For m = rank g i j (u) ≤ n − 1, any solution to Eqs. (2.4)–(2.7) for K = 0 locally has the form g i j (u) =
m α,β=1
cαβ Uαi (u)Uβ (u), bk (u) = j
j
m
ij
α,β=1
cαβ Uαi (u)
∂Uβ (u) ∂u k
ij
+ Tk (u), (2.20)
where U1 (u), . . . , Um (u) are commuting vector fields on M n . The coefficients ij Tk (u) form a certain (2,1)-tensor that satisfies Eqs. (2.12). Proof. (a) As is known [7], any Poisson structure (2.1), (2.3) with K = 0 in some local coordinates v 1 , . . . , v n has constant metric g i j (v) = ci j . Let U1 , . . . , Un be the corresponding coordinate vectors fields, Uαi = δαi . Hence the metric g i j (u) in any coordinates ij u 1 , . . . , u n has the form (2.19). Calculating the coefficients bk (u) by the formulae (2.11) and using three times the commutativity relations (α, β = 1, . . . , n), j
[Uα , Uβ ] = j
Uα
∂Uβ ∂u
− Uβ
j
∂Uα = 0, ∂u
(2.21)
we arrive at the second formula (2.19). (b) Let g i j (u) be the non-degenerate m × m metric (2.8) v, w = g i j v i w j ,
(2.22)
induced on the leaves of the foliation F m . Here g i j is a non-degenerate symmetric m ×m matrix and i, j ∈ {1, . . . , m}. Applying Theorem 1 of paper [10] for K = 0, we find that metric g i j (u) has zero curvature. Hence there exist (at least locally) mutually commuting vector fields U1i (u), . . . , Umi (u) tangent to the leaves of F m that are constant in some local coordinates v 1 , . . . , v n and have constant scalar products in the metric g i j (v) [15,16]. Hence the (2,0)-tensor g i j (u) has the form (2.20) where cαβ are constant. We define the form-invariant coefficients ij Bk (u)
=
m α,β=1
j
c
αβ
Uαi (u)
∂Uβ (u) ∂u k
.
(2.23)
It is easy to verify that the coefficients (2.23) are transformed from one system of coordinates to another according to the formulae (2.15) that coincide with (2.9). The ij coefficients Bk (v) (2.23) vanish in the local coordinates v 1 , . . . , v n because the vector i fields U1 (v), . . . , Umi (v) are constant there, together with the (2,0)-tensor g i j (v) (2.20).
Tensor Invariants of the Poisson Brackets of Hydrodynamic Type ij
375
ij
Therefore the g i j (v) and bk (v) = Bk (v) satisfy Eqs. (2.4)–(2.7) in the coordinates v 1 , . . . , v n and thus define a Poisson bracket (2.3). Hence the (2,0)-tensor g i j (u) (2.20) ij ij and the coefficients bk (u) = Bk (u) (2.23) satisfy Eqs. (2.4)–(2.7) in any coordinates u 1 , . . . , u n . It is easy to verify this fact also by a direct substitution using the commutativity relations (2.21). ij ij ij ij ij We define Tk (u) = bk (u) − Bk (u). Since the coefficients bk (u) and Bk (u) (2.23) ij are transformed by the same formulae (2.9), we obtain that the coefficients Tk (u) are subject to the law of transformation (2.18) and therefore form a (2,1)-tensor. Since ij ij the bk (u) and Bk (u) both satisfy Eqs. (2.4)–(2.7) for K = 0, we get from Eqs. (2.4) ij ji jk Tk (u)+Tk (u) = 0 and from Eqs. (2.5) Tαik g α j = Tα g αi . Hence equations Tαik g α j = 0 follow as in Proposition 1.
IV. Remark 2. For any 1 ≤ m ≤ n and any m commuting vector fields U1 (u), . . . , ij Um (u) on a manifold M n , the formulae (2.20) with Tk (u) = 0 define exact forminvariant solutions to Eqs. (2.4)–(2.7) for K = 0. Such commuting vector fields can be obtained from the Liouville-integrable Hamiltonian systems on the symplectic manifolds M 2k , or from dynamical systems integrable in the broad sense of paper [17]. The m-dimensional foliation F m defined by the (2,0)-tensor g i j (u) (2.20) is generated by the commuting vector fields U1 (u), . . . , Um (u). ij
Remark 3. For the Poisson structures defined by the coefficients (2.20) with Tk (u) = 0, the operator I i j (u) (2.1) has the form I i j (u) =
m α,β=1
cαβ Uαi (u) ·
∂ j · Uβ (u). ∂x
The corresponding Hamiltonian systems (1.1)–(1.2) are m ∂ ∂ f (u) i αβ i . ut = Uβ (u) c Uα (u) ∂x ∂u
(2.24)
α,β=1
The system (2.24) locally is a system of conservation laws. Indeed, in the local coordinates v 1 , . . . , v n , where the commuting vector fields Uα are constant, system (2.24) takes the form vti = (h i (v))x ,
m
h i (v) =
α,β=1
cαβ Uαi Uβ
∂ f (v) . ∂v
Remark 4. It is evident that the commuting vector fields U1 (u), . . . , Um (u) are defined non-uniquely by the (2,0)-tensor g i j (u) with 2 ≤ rank g i j (u) ≤ n − 1. Indeed, let γ Aα (u) be any m × m matrix function that is constant along the leaves of the foliation γ F m and satisfies the equation Aα (u)cαβ Aδβ (u) = cγ δ , that means At (u) = c−1 A−1 (u)c. γ Then the vector fields Vαi (u) = Aα (u)Uγi (u) mutually commute and define the same i j (2,0)-tensor g (2.20). However, the corresponding non-tensorial coefficients ij B k (u)
=
m α,β=1
j
c
αβ
Vαi (u)
∂ Vβ (u) ∂u k
ij
= Bk (u) +
m α,β,γ ,δ=1
cαβ Aγα (u)
∂ Aδβ (u) ∂u k
j
Uγi (u)Uδ (u)
376
O. I. Bogoyavlenskij ij
ij
are different from the coefficients Bk (u) (2.23). Hence the (2,1)-tensors Tk (u) (2.17) ij are defined non-uniquely by the coefficients g i j (u) and bk (u) if rank g i j (u) ≥ 2. However, below we introduce certain uniquely defined invariants. 3. The Key Lemma ∗ n Let L ⊥ u ⊂ Tu (R ) be the dual distribution of the 1-forms ω that are annihilated by the i j (2,0)-tensor g (u): ωi g i j (u) = 0. Evidently dim L ⊥ u = n − m. For any operator A that ∗ (R n ) invariant, we denote as Tr A| ⊂ T leaves the subspace L ⊥ L ⊥ the trace of A on u u . L⊥ u
Lemma 1. (a) For any Poisson bracket (2.1), (2.3), and any 1-form ω ∈ L ⊥ the object j αj (Aω )k = ωα bk is a (1,1)-tensor satisfying Aω (Tu∗ ) ⊂ L ⊥ u,
Aω (L u ) = 0.
(3.1)
ij
(b) The coefficients bk (u) invariantly define the bilinear product αβ
∗ ⊥ L⊥ u × Tu −→ L u ,
b(ω, η)i = ωα bi ηβ ,
(3.2)
∗ where ω ∈ L ⊥ u and η ∈ Tu . (c) For any 1-form η, the object
(Bη )ik =
n
iβ
bk ηβ
(3.3)
β=1 ⊥ ⊥ ⊥ is an invariantly defined operator on the subspaces L ⊥ u : Bη (L u ) ⊂ L u . For ω ∈ L u and any 1-forms η, ζ we have
Bη (ω) = Aω (η), Bζ (Aω η) = Bη (Aω ζ ).
(3.4)
(d) All operators Bη , Bν on L ⊥ u commute with each other. (e) For the 1-forms ν, θ ∈ L ⊥ , the operators Bν , Bθ on L ⊥ u are nilpotent and satisfy the identities Bν Bθ = 0,
TrBν | L ⊥ = 0.
(3.5)
ij
Proof. (a) Let ωi , η j be differential 1-forms and g i j (u), bk (u) be the coefficients defining the Poisson brackets (2.1), (2.3) in local coordinates u 1 , . . . , u n . Let ωr , η , gr (v) r and b p (v) be their components in another local coordinates v 1 , . . . , v n . For the 1-forms we have ωr =
∂u α ωα , ∂vr
η =
∂u β ηβ . ∂v
Using formulae (2.9), we get r
ωr b p (v) = ωα
α r ∂ 2 v ∂u q ∂u α ∂vr ∂v ∂u k i j i j ∂u ∂v b + ω g α ∂vr ∂u i ∂u j ∂v p k ∂vr ∂u i ∂u j ∂u q ∂v p αj
= ωα bk
2 ∂u q ∂v ∂u k αj ∂ v + ω g . α ∂u j ∂v p ∂u j ∂u q ∂v p
(3.6)
Tensor Invariants of the Poisson Brackets of Hydrodynamic Type j
377 αj
Equation (3.6) implies that the object (Aω )k = ωα bk is a (1,1)-tensor provided that the 1-form ωi is annihilated by the (2,0)-tensor g i j (u), ωα g α j = 0. Indeed, for this case formula (3.6) reads r
αj
(Aω )p = ωr b p (v) = ωα bk
k ∂v ∂u k j ∂v ∂u = (A ) . ω k ∂u j ∂v p ∂u j ∂v p
(3.7)
Using Eq. (2.5) and equation ωi g ki = 0, we derive for any 1-form θ j : (Aω θ )k g k = αj j (ωα bk θ j )g k = ωα bk θ j g kα = 0. Hence Aω (Tu∗ ) ⊂ L ⊥ u. Any vector v i ∈ L u has the form v i = g iγ ηγ . Applying Eq. (2.5), we find (Aω v)i = γi ωα bβαi g βγ ηγ = ωα bβ g βα ηγ = 0 since ωα g βα = 0. Hence (3.1) follows. (b)–(c) Formula (3.7) yields the transformation r
αj
ωr b p (v)η = ωα bk
∂v ∂u k ∂u β ∂u k αβ ηβ = ωα bk ηβ . j p ∂u ∂v ∂v ∂v p
αβ
Hence the object ωα bk ηβ is an invariantly defined 1-form provided that the 1-form ωi is annihilated by the (2,0)-tensor g i j (u) that means ω ∈ L ⊥ , ωα g α j = 0. From (3.1), αβ ⊥ Aω (Tu∗ ) ⊂ L ⊥ u , we obtain ωα bk ηβ ∈ L u and (3.2) follows. ij For any 1-form ω ∈ L ⊥ , formula (3.3) yields Bη (ω) = ωi bk η j = Aω (η). Hence we get that operators Bη are correctly defined on the subspaces L ⊥ u because Aω is a (1,1)-tensor for ω ∈ L ⊥ . Thus Bη (ω) = Aω (η) ∈ L ⊥ . u Contracting Eq. (2.6) with ωi η j ζr for ω ∈ L ⊥ and arbitrary 1-forms η and ζ and using g αi ωi = 0, we find αj
ωi bαi j η j bkαr ζr = ωi bαir ζr bk η j .
(3.8)
The equality reads Bζ (Aω η) = Bη (Aω ζ ) and (3.4) follows. (d) Equation (3.8) means also that Bζ Bη ω = Bη Bζ ω.
(3.9)
Hence all operators Bζ , Bη commute on L ⊥ u. (e) For 1-forms θ, ν ∈ L ⊥ , using Eqs. (2.4) we find ij
ji
(Bθ ν + Bν θ )k = (bk + bk )νi θ j =
∂(g i j νi θ j ) ∂θ j ∂νi ∂g i j ν θ = − g i j k θ j − g i j νi k = 0. i j ∂u k ∂u k ∂u ∂u
Hence we get Bθ ν = −Bν θ.
(3.10)
Using Eqs. (3.9) and (3.10) for η, ζ, ω ∈ L ⊥ , we derive Bζ Bη ω = −Bζ Bω η = −Bω Bζ η = Bω Bη ζ = Bη Bω ζ = −Bη Bζ ω.
(3.11)
Equations (3.9) and (3.11) yield for η, ζ ∈ L ⊥ , Bζ Bη = 0. Hence the operators Bη for η ∈ L ⊥ are nilpotent and TrBη | L ⊥ = 0.
(3.12)
378
O. I. Bogoyavlenskij
Remark 5. Lemma 1 shows that the operators Bη (3.3) for all 1-forms η ∈ Tu∗ (M n ) form an invariant algebra of commuting linear transforms in the dual distribution L ⊥ u ⊂ define a nilpotent Tu∗ (M n ). In Sect. 7, we demonstrate that the operators Bη for η ∈ L ⊥ u Lie algebra structure in L ⊥ u . Also we prove that all (1,1)-tensors Aω are nilpotent and satisfy the identity A3ω = 0. j αβ i Remark 6. For the metric g i j (u) = m α,β=1 c Uα (u)Uβ (u) (2.20), the 1-forms ωα ∈ i L⊥ u annihilate all vectors Uα (u): ωi Uα (u) = 0. Hence the second formula (2.20) gives ∗ for any µβ ∈ Tu : iβ
iβ
ωi bk µβ = ωi Tk µβ .
(3.13) ij
Therefore in spite of their non-uniqueness, all (2,1)-tensors Tk (2.17) define the same iβ bilinear product (3.13). Hence for a given 1-form µ ∈ Tu∗ all (1,1)-tensors Tk µβ on the iβ ∗ invariant subspaces L ⊥ u ⊂ Tu coincide with operator Bµ = bk µβ . 4. Invariant Vector Field Applying Lemma 1, we introduce the following invariants: Definition 1. (a) A vector field V i (u) on M n is defined by the formula V i µi = TrBµ | L ⊥
(4.1)
for any 1-form µ. Vector field V i (u) leads to the invariant dynamical system du i = V i (u 1 , . . . , u n ) dt
(4.2)
on the manifold M n . (b) The symmetric (2,0)-tensor h i j and (k, 0)-tensors h i j... are defined by h i j ηi ζ j = Tr(Bη Bζ )| L ⊥ , h i j... ηi ζ j . . . θ = Tr(Bη Bζ . . . Bθ )| L ⊥ ,
(4.3)
for any 1-forms η, ζ, . . . , θ . Proposition 3. For rank g i j (u) = n − 1, the vector field V i (u) (4.1) is defined also by the formula V =
1 Aω U, ω(U )
(4.4)
where ω ∈ L ⊥ u and U ∈ Tu is an arbitrary vector satisfying ω(U ) = 0. The vector field V (u) (4.4) belongs to the distribution L u . Proof. We have dim L u = n − 1 and dim L ⊥ u = 1. Hence the subspace u = Aω (Tu ) is 1-dimensional because Aω (L u ) = 0 by Lemma 1. In the coordinates u 1 , . . . , u n , where the distribution L u has equation δu n = 0, we have for ω ∈ L ⊥ u : ω = (0, . . . , 0, ωn ). For any vector (U 1 , . . . , U n ) ∈ Tu using Aω (L u ) = 0, we find nj
(Aω U ) j µ j = ωn bn U n µ j = ω(U ) Tr Bµ | L ⊥ .
Tensor Invariants of the Poisson Brackets of Hydrodynamic Type
379
Hence Eq. (4.1) yields (Aω U ) j = ω(U )V j and formula (4.4) follows. For the 1-form ω ∈ L⊥ u we have ω(V ) =
1 1 1 αβ ω(Aω U ) = ωα bk ωβ U k = (Bω ω)(U ). ω(U ) ω(U ) ω(U )
Equation (3.10) yields Bω ω = 0. Hence ω(V ) = 0 or V (u) ∈ L u .
For K = 0, some statements concerning Poisson brackets with degenerate (2,0)tensor g i j (u) were announced in 1985 without any proofs in a short communications [14]. Since 1985, no proofs of these statements had appeared in the literature; we have discussed this announcement in [10]. 5. Invariant (1,1)-Tensor and Mapping F Let g i j (u) be the non-degenerate m × m matrix (2.20) of the induced metric v, w (2.8) in some coordinates on the leaves of the foliation F m . The Lie derivative with respect to the dynamical system (4.2) (L V g)i j = V α
∂ gi j ∂u α
+ g iα
∂V α ∂V α + g , 1 ≤ i, j, α ≤ m α j ∂u j ∂u i
(5.1)
is another (0,2)-tensor on F m . The repeated Lie derivatives L kV give the higher (0,2)tensors L kV g i j . Theorem 1. (a) Vector field V i (u) is tangent to the foliation F m and defines a functional invariant h(u) = V, V (u).
(5.2)
(b) An invariant (1,1)-tensor L ij (u) is defined on the m-dimensional leaves of the foliation F m : L ij (u) =
m
(L V g(u)) jα (g −1 (u))αi ,
(5.3)
α=1
where i, j, α = 1, . . . , m. The characteristic polynomial B(λ, u) = λm + bm−1 (u)λm−1 + . . . + b0 (u) = det ||λδ ij − L ij (u)||
(5.4)
defines a smooth mapping F : M n −→ R m , u −→ (bm−1 (u), . . . , b0 (u))
(5.5)
into the m-dimensional Euclidean space R m . (c) rank h i j ≤ m; the expressions h i j... ηi ζ j . . . θ = Tr(Bη Bζ . . . Bθ )| L ⊥ (4.3) vanish if at least one of the 1-forms η, ζ, . . . , θ belongs to the distribution L ⊥ . (d) The roots µ of the equation rank ||µg i j (u) − h i j (u)|| ≤ m − 1 form m invariants µ1 (u), . . . , µm (u) of the Poisson bracket (2.1), (2.3).
(5.6)
380
O. I. Bogoyavlenskij
Proof. (a) For any 1-form η ∈ L ⊥ , Eq. (3.5) gives TrBη | L ⊥ = 0. Hence (4.1) implies V i ηi = 0. Therefore vector field V belongs to the distribution L u ⊂ Tu∗ (M n ). Hence its norm (5.2) in the induced on L u metric (2.8) is well defined. (b) In another basis of local coordinates on the foliation F m the two (0,2)-tensors g i j (u) and L V g i j (u) have the form Qg Q t and Q L V g Q t with some non-degenerate matrix Q(u). Hence since det ||g(u)|| = 0, the (1,1)-tensor L ij (u) (5.3), its characteristic polynomial (5.4) and the mapping F (5.5) are invariantly defined. (c)–(d) For a 1-form η ∈ L ⊥ and any 1-form ζ , we have on L ⊥ : (Bη Bζ )2 = Bη2 Bζ2 in view of (3.9). Equation (3.12) yields Bη2 = 0. Hence operator Bη Bζ is nilpotent, (Bη Bζ )2 = 0, and thus h i j ηi ζ j = Tr(Bη Bζ )| L ⊥ = 0. Hence we have h i j ηi = 0 for any i j... . 1-form η ∈ L ⊥ u , analogously for the (k, 0)-tensors h i j Since the (2,0)-tensor h and the roots µ of (5.6) are invariant, we consider them in the special coordinates u 1 , . . . , u n , where the integrable distribution L u has the form δu m+1 = 0, . . . , δu n = 0. In these coordinates we have g i j = 0 for i > m or j > m and vector field V i ∈ L u has components V m+1 = 0, . . . , V n = 0. The subspace L ⊥ u consists of all 1-forms η with η1 = 0, . . . , ηm = 0. Since h i j ηi = 0 for η ∈ L ⊥ u , we find that h i j = 0 for i > m or j > m. Hence rank h i j ≤ m and the roots µ of Eq. (5.6) coincide with the m roots of the characteristic polynomial ij
Ph (λ, u) = det ||µg i j (u) − h (u)||, ij
where h (u) and g i j (u) are the corresponding m×m blocks in the coordinates u 1 , . . . , u n .
Remark 7. Using the repeated Lie derivatives (5.1), we define the invariant (1,1)-tensors (L k )ij (u) on the m-dimensional leaves of the foliation F m : (L k )ij (u) =
m
(L kV g(u)) jα (g −1 (u))αi ,
α=1
where i, j, α = 1, . . . , m. The operators (L k )ij (u) satisfy the equations L k g = gL tk . 6. Nilpotent Lie Algebraic Invariants Proposition 4. For rank g i j (u) = m ≤ n − 1, the formula [η, ω]k =
n
ij
bk ωi η j
(6.1)
i, j=1
defines an invariant structure of the nilpotent Lie algebra Au in the space of 1-forms that annihilate the foliation F m . The derivative Lie subalgebra Au relates to the center Cu of Au : Au ⊂ Cu .
(6.2)
The invariants Au belong to the moduli space Mn−m of nilpotent Lie algebras satisfying relation (6.2).
Tensor Invariants of the Poisson Brackets of Hydrodynamic Type
381
∗ n Proof. Applying Lemma 1 for the 1-forms ω, η ∈ L ⊥ u ⊂ Tu (R ), we get that (6.1) is invariantly defined, since [η, ω] = Aω (η) = Bη (ω), see (3.4). Equation (3.10) proves that bracket (6.1) is skew-symmetric: [ω, η] = −[η, ω]. Equation (3.5), Bν Bη (ω) = 0, takes the form
[ν, [η, ω]] = 0.
(6.3)
The latter evidently implies the Jacobi identity [[ω, η], ν] + [[η, ν], ω] + [[ν, ω], η] = 0.
(6.4)
Hence the operation (6.1) defines structure of Lie algebra in L ⊥ u . The identity (6.3) implies adν adη = 0, where adη = Bη . Hence the Lie algebra Au in L ⊥ u is nilpotent and belongs to the moduli space Mn−m . The identity (6.3) is evidently equivalent to the relation (6.2).
Remark 8. The invariants Au exist when rank g i j (u) ≤ n − 3, since the nilpotent Lie algebras occur in dimensions ≥ 3. For rank g i j (u) = n − 3, the non-zero invariant is the Heisenberg’s nilpotent Lie algebra H 3 with the commutator relations [e1 , e2 ] = e3 , [e2 , e3 ] = 0, [e3 , e1 ] = 0. Example 1. For n = m + p + q, we define in R n a (2,0)-tensor g i j (u) of rank m that has a ij ji constant m ×m block and is zero for i > m or j > m. Let bk (u) = −bk (u) be non-zero only for m + p + 1 ≤ k ≤ n and for m + 1 ≤ i, j ≤ m + p, and for these indices be arbitrary smooth functions of variables u m+ p+1 , . . . , u m+ p+q . Equations (2.4) hold since ij bk (u) are skew and g i j (u) = const. Equations (2.5) evidently hold. Equations (2.6) for ij αj K = 0 are satisfied because bα bkαr = 0 and bαir bk = 0. Equations (2.7) for K = 0 σ (i)σ ( j) hold because ∂bτ (k) /∂u α = 0 for α = 1, . . . , m + p and hence all products in (2.7) are zero. The m-dimensional distribution L u ⊂ Tu (R n ) corresponds to the coordinates ∗ n and the dual distribution L ⊥ u ⊂ Tu (R ) corresponds to the coordinates m+1 n , . . . , u and has dimension p + q. u For the corresponding Lie algebras Au in the dual distribution L ⊥ u , we have in the basis of the coordinate 1-forms em+1 , . . . , en :
u1, . . . , um
ij
ij
[ei , e j ] = bm+ p+1 (u)em+ p+1 + . . . + bn (u)en
(6.5)
for m + 1 ≤ i, j ≤ m + p and [ei , e j ] = 0 otherwise. Equation (6.5) implies the identity [[ei , e j ], ek ] = 0 (6.3). Hence (6.5) defines the nilpotent Lie algebras Au that satisfy the relation Au ⊂ Cu (6.2). Thus we get a Poisson bracket of hydrodynamic type for p+q K = 0 with deformations of non-isomorphic Lie algebraic structures Au in L ⊥ u = R for p + q ≥ 4. For p + q = 3 there is only one nilpotent Lie algebra - the Heisenberg’s H 3. Proposition 5. For any 1-forms σ, θ, ω ∈ L ⊥ u , the (1,1)-tensors Aσ , Aθ and Aω are nilpotent and satisfy the equation Aσ Aθ Aω = 0.
(6.6)
382
O. I. Bogoyavlenskij
∗ Proof. Applying Eq. (3.1) of Lemma 1, we find Aω (η) ∈ L ⊥ u for any 1-form η ∈ Tu . ⊥ For any 1-forms ψ, ζ ∈ L u , formula (6.1) yields Aζ (ψ) = [ψ, ζ ]. Hence we get Aθ (Aω (η)) = [Aω (η), θ ]. Therefore using identity (6.3) we find
Aσ (Aθ (Aω (η))) = [[Aω (η), θ ], σ ] = −[σ, [Aω (η), θ ]] = 0.
(6.7)
Since the 1-form η is arbitrary, the identity (6.6) follows from (6.7). The identity (6.6) evidently implies A3ω = 0.
7. Examples. Gas Dynamics I. Example 2. Hamiltonian formulation of 3-dimensional non-isentropic gas dynamics was introduced in [11]. Hamiltonian formulation of 1-dimensional gas dynamics (that is a reduction of the 3-dimensional case) was published later in [12,13]. Equations of 1-dimensional gas dynamics are pρ ∂ρ ∂v ps ∂s ∂v = −v − − , ∂t ∂x ρ ∂x ρ ∂x
∂ρ ∂ρv =− , ∂t ∂x
∂s ∂s = −v , ∂t ∂x
(7.1)
where u 1 = v is the gas velocity, u 2 = ρ is the mass density, u 3 = s is the entropy density and p(ρ, s) is the gas pressure. Let us use formulae of [13], where the (2,0)-tensor ij g i j (u) and the only non-zero coefficients bk (u) are ⎛ ⎞ 0 −1 0 (7.2) g i j = ⎝ −1 0 0 ⎠ , b313 (u) = ρ −1 , b331 (u) = −ρ −1 . 0 0 0 A direct verification proves that Eqs. (2.4)–(2.7) for K = 0 hold. Equations (7.1) have Hamiltonian form (2.1)–(2.2) with coefficients (7.2) and Hamiltonian function f (u) = 21 ρv 2 +g(ρ, s). Here function g(ρ, s) is connected with pressure p(ρ, s) by the compatible system of equations [13]: ρgρρ = pρ , ρgρs − gs = ps . The (2,0)-tensor (7.2) defines the 2-dimensional distribution L u ∈ Tu (R 3 ) consisting ∗ 3 of vectors (u 1 , u 2 , 0). The dual distribution L ⊥ u ∈ Tu (R ) is generated by the 1-form 2 ω = (0, 0, ω3 ). Hence the leaves of the foliation F are the planes u 3 = const and the induced flat metric v, w on F 2 is the 2 × 2 block of matrix (7.2). For any 1-form (µ1 , µ2 , µ3 ), the invariant operator Bµ (3.3) on the dual distribu−1 i tion L ⊥ u has the form Bµ (ω) = −ρ µ1 ω. Equation (4.1) yields V µi = Tr Bµ | L ⊥ = −1 i −1 −ρ µ1 . Hence vector field V has components (−ρ , 0, 0) and the corresponding dynamical system (4.2) is dv = −ρ −1 , dt
dρ = 0, dt
ds = 0. dt
(7.3)
The 1-forms ω ∈ L ⊥ have components (0, 0, ω3 ). In view of (7.2), the (1,1)-tensor j αj j j (Aω )k = ωα bk is (Aω )k = −ρ −1 ω3 δ1 δk3 and is evidently nilpotent, A2w = 0. Vector field V = Aω U/ω(U ) (4.4) is (−ρ −1 , 0, 0). Its invariant norm (5.2) is h(u) = V, V = gi j V i V j = 0. The (2,0)-tensor h i j and the (k, 0)-tensors h i j... (4.3) have the form h i j = V i V j , h i j... = V i V j . . . V . For the Lie derivative (5.1) we find L V g i j = −2ρ −2 δi2 δ 2j , L kV g i j = 0, k ≥ 2. The mapping F (5.5) is constant: (v, ρ, s) −→ (0, 0); the invariants µi are zeros.
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The coefficients (7.2) have the form (2.20), where the two commuting vector fields ij U1 , U2 and the skew (2,1)-tensor Tk in the coordinates u 1 = v, u 2 = ρ, u 3 = s have the form U1 = (1, 0, 0), U2 = (0, 1, 0), T313 (u) = ρ −1 , T331 (u) = −ρ −1 , all other components Tk are zeros. The coefficients cαβ are c12 = c21 = −1, c11 = c22 = 0. Evidently Eqs. (2.12) Tαik g α j = 0 hold. Hamiltonian structures for some systems of two hyperbolic conservation laws are presented in [18]. ij
Example 3. Let us consider in R n the (2,0)-tensor g i j = q i δ ij , where q n = 0, q i =
ci−1 = const = 0 for i = 1, . . . , n − 1. Let the only non-zero coefficients b be nj jn bn = −bn that are smooth functions of the coordinates u 1 , . . . , u n−1 . Equations (2.4) and (2.5) evidently hold. Equations (2.6) for K = 0 reduce to the system ij
nj
∂bn nj = −ci bnni bn , ∂u i
(7.4)
where there is no summation with respect to index i. The overdetermined system of nj nj partial differential Eqs. (7.4) evidently implies bn (u)/bnnk (u) = const. Hence bn (u) = j p ψ(u) and Eq. (7.4) are reduced to ∂ψ(u) = −ci pi ψ 2 (u). ∂u i Therefore all solutions to the system (7.4) have the form bn = p j ψ(u), ψ(u) = (c1 p 1 u 1 + . . . + cn−1 p n−1 u n−1 − d)−1 , nj
(7.5)
where p j and d are arbitrary constants. Equations (2.7) for coefficients (7.5) are satisij fied identically. Hence the coefficients g i j and b define the Poisson bracket (2.1), (2.3) with rank g i j = n − 1. The corresponding invariant foliation F n−1 has plane leaves u n = const and the induced metric is gi j = ci δ ij for i, j ≤ n − 1. The dual distribution L⊥ u is 1-dimensional and is generated by a 1-form ω = (0, . . . , 0, ωn ). The operators 1 n−1 µ Bµ (3.3) on L ⊥ n−1 )ψ(u)ω. Hence u have the form Bµ ω = ( p µ1 + . . . + p Tr Bµ | L ⊥ = ( p 1 µ1 + . . . + p n−1 µn−1 )ψ(u) and vector field V (4.1) has the components V 1 = p 1 ψ(u), . . . , V n−1 = p n−1 ψ(u), V n = 0. The corresponding dynamical system (4.2) is du 1 = p 1 ψ(u), . . . , dt j
αj
du n−1 = p n−1 ψ(u), dt j
du n = 0. dt
(7.6)
The (1,1)-tensor (Aω )k = ωα bk is (Aω )k = p j ψ(u)ωn δkn and evidently is nilpotent, A2ω = 0. The vector field V = Aω U/ω(U ) (4.4) has the above components V j = p j ψ(u), V n = 0.
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The invariant norm (5.2) of the vector field V is h(u) = V, V = gi j V i V j = ci ( pi )2 ψ 2 (u). The symmetric (2,0)-tensor h i j has the form h i j = V i V j . Hence the invariant roots of Eq. (5.6) are µ1 (u) = h(u), µ j = 0 for 2 ≤ j ≤ n − 1. The symmetric (k, 0)-tensors (4.3) have the components h i j... = V i V j . . . V . For the Lie derivatives (5.1), we find L V g i j = −2ci c j pi p j ψ 2 (u),
L 2V g i j = 8(cα p α )ci c j pi p j ψ 3 (u).
The characteristic polynomial (5.4) is B(λ, u) = λn−1 + 2ci ( pi )2 ψ 2 (u)λn−2 . Hence the mapping (5.5) is u −→ (2h(u), 0, . . . , 0). Acknowledgements. The author thanks the referee for useful comments.
References 1. Courant, R., Hilbert, D.: Methods of Mathematical Physics, II, New York: Interscience Publishers, 1962 2. Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm. Pure Appl. Math. 7, 159–193 (1954) 3. Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 697–715 (1965) 4. Ablowitz, M.J., Benney, D.J.: The evolution of multiphase modes for nonlinear dispersive waves. Studies Appl. Math. 49(N3), 225–238 (1970) 5. Flaschka, H., Forest, M.G., McLaughlin, D.W.: Multiphase averaging and the inverse spectral solution. Commun. Pure Appl. Math. 33(N6), 739–784 (1980) 6. Nijenhuis, A.: X n−1 -forming sets of eigenvectors. Proc. Kon. Ned. Akad. Amsterdam 54, 200–212 (1951) 7. Dubrovin, B.A., Novikov, S.P.: Hamiltonian formalism of one-dimensional hydrodynamic type systems and Bogoliubov - Whitham averaging method. Soviet Math. Dokl. 27, 665–669 (1983) 8. Mokhov, O.I., Ferapontov, E.V.: Non-local Hamiltonian operators of hydrodynamic type related to metrices of constant curvature. Russ. Math. Surv. 45(N2), 218–219 (1990) 9. Mokhov, O.I.: Hamiltonian systems of hydrodynamic type and constant curvature metrics. Phys. Lett. A 166, 215–216 (1992) 10. Bogoyavlenskij, O.I.: Invariant foliations for the Poisson brackets of hydrodynamic type. Phys. Lett. A 360, 539–544 (2007) 11. Morrison, P.J., Greene, J.M.: Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics. Phys. Rev. Lett. 45(N10), 790–794 (1980) 12. Novikov, S.P.: The Hamiltonian formalism and a many-valued analogue of Morse theory. Russ. Math. Surv. 37(N5), 1–56 (1982) 13. Verosky, J.M.: First-order conserved densities for gas dynamics. J. Math. Phys. 27(N12), 3061–3063 (1986) 14. Grinberg, N.I.: On Poisson brackets of hydrodynamic type with a degenerate metric. Russ. Math. Surv. 40(N4), 231–232 (1985) 15. Eisenhart, L.P.: Riemannian geometry. Princeton, NJ : Princeton University Press, 1964 16. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. New York: Springer Verlag, 1999 17. Bogoyavlenskij, O.I.: Extended integrability and bi-Hamiltonian systems. Commun. Math. Phys. 196, 19–51 (1998) 18. Olver, P.J., Nutku, Y.: Hamiltonian structures for systems of hyperbolic conservation laws. J. Math. Phys. 29(N7), 1610–1619 (1988) Communicated by L. Takhtajan
Commun. Math. Phys. 277, 385–421 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0371-7
Communications in
Mathematical Physics
A Duality Theorem for Ergodic Actions of Compact Quantum Groups on C ∗ -Algebras Claudia Pinzari1 , John E. Roberts2 1 Dipartimento di Matematica, Università di Roma La Sapienza, 00185, Roma, Italy.
E-mail: [email protected]
2 Dipartimento di Matematica, Università di Roma Tor Vergata, 00133, Roma, Italy
Received: 12 December 2006 / Accepted: 16 May 2007 Published online: 10 November 2007 – © Springer-Verlag 2007
Abstract: The spectral functor of an ergodic action of a compact quantum group G on a unital C ∗ -algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor ∗ -functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C ∗ -algebra. As an application, we associate an ergodic G-action on a unital C ∗ -algebra to an inclusion of Rep(G) into an abstract tensor C ∗ -category T. If the inclusion arises from a quantum subgroup K of G, the associated G-system is just the quotient space K \G. If G is a group and T has permutation symmetry, the associated G-system is commutative, and therefore isomorphic to the classical quotient space by a subgroup of G. If a tensor C ∗ -category has a Hecke symmetry making an object ρ of dimension d and µ-determinant 1, then there is an ergodic action of Sµ U (d) on a unital C ∗ -algebra having the (ι, ρ r ) as its spectral subspaces. The special case of Sµ U (2) is discussed.
1. Introduction A theorem in [5] asserts that any abstract tensor C ∗ -category with conjugates and permutation symmetry is the representation category of a unique compact group, thus generalizing the classical Tannaka–Krein duality theorem, where one starts from a subcategory of the category of Hilbert spaces (see, e.g., [12]). This paper fits into the program of generalizing the abstract duality theorem of [5] to tensor C ∗ -categories without permutation symmetry. Our interest in this problem is motivated by the tensor C ∗ -categories with conjugates, but also with a unitary symmetry of the braid group, arising from low dimensional QFT [9].
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In [25]−[27] Woronowicz introduced compact quantum groups, generalizing the classical representation theory of compact groups. He proved a Tannaka–Krein duality theorem, asserting that the representation categories of compact quantum groups are precisely the tensor ∗ -subcategories of categories of Hilbert spaces where every object has a conjugate. This theorem allowed him to construct the quantum deformations Sµ U (d) of the classical SU (d) groups by a real parameter µ. Therefore if a tensor C ∗ -category T with conjugates can be embedded into a tensor category of Hilbert spaces, T is necessarily the representation category of a compact quantum group. In [14] the first named author characterized the representation category of Sµ U (d) among braided tensor C ∗ -categories T with conjugates. Consequently, if an object ρ has a symmetry of the Hecke algebra type H∞ (µ2 ) making ρ of dimension d with µ-determinant one, there is a faithful tensor ∗ -functor Rep(Sµ U (d)) → T. The notion of subgroup of a compact quantum group G was given by Podles in [17], who computed all the subgroups of the quantum SU (2) and S O(3) groups. In the same paper he introduced quantum quotient spaces, showing that these spaces have an action of G spliting into a direct sum of irreducibles, with multiplicity bounded above by the Hilbert space dimension. Later Wang proved in [22] that compact quantum group actions on quotient spaces are ergodic, as in the classical case, and found an example of an ergodic action on a commutative C ∗ -algebra which is not a quotient action. Classifying all the ergodic actions of Sµ U (2) is an open problem. Tomatsu described all those arising from the invariant C ∗ -subalgebras of Sµ U (2) [19]. A compact quantum subgroup K of G gives rise to an inclusion of tensor C ∗ -categories Rep(G) → Rep(K ), and, by Tannaka–Krein duality, every tensor ∗ -inclusion of Rep(G) into a category of Hilbert spaces, taking a representation u to its Hilbert space and acting trivially on the arrows, is of this form. When a tensor ∗ -inclusion ρ : Rep(G) → T into an abstract tensor C ∗ -category is given, it is natural to look for a tensor ∗ -functor of T into the category of Hilbert spaces, acting as the embedding functor on Rep(G). This amounts to asking whether ρ arises as an inclusion associated with a quantum subgroup of G. The aim of this paper is twofold. Assume given a tensor ∗ -inclusion ρ : Rep(G) → T. Then one can construct a unital C ∗ -algebra B associated with the inclusion, and an ergodic action of G on B whose spectral subspaces are the spaces (ι, ρu ), where u varies over the set of unitary irreducible representations of G. The relevance of this to abstract duality for compact quantum groups will be discussed elsewhere [6]. We exhibit two interesting particular cases of this construction. If T = Rep(K ), with K a compact quantum subgroup of G, the ergodic system thus obtained is just the quotient space K \G (Theorem 10.1). If instead G is a group and T has a permutation symmetry, B turns out to be commutative, and therefore identifiable with the C ∗ -algebra of continuous functions over a quotient of G by a point stabilizer subgroup (Theorem 9.2). We apply this to an abstract tensor C ∗ -category T and an object of dimension d and µ-determinant one finding ergodic C ∗ -actions of Sµ U (d) (Theorem 9.3). We also discuss the particular case d = 2 (Cor. 9.5). Our second aim is to characterize, among all ergodic actions, those which are isomorphic to the quotient spaces by some quantum subgroup. By a well known theorem of Høegh–Krohn, Landstad and Størmer [10], an irreducible representation of a compact group G appears in the spectrum of an ergodic action of G on a unital C ∗ -algebra with a multiplicity bounded above by its dimension.
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Generalizing to compact quantum groups, Boca shows that the multiplicity of a unitary irreducible representation of the quantum group in the action is, instead, bounded above by its quantum dimension [3]. In [2], Bichon, De Rijdt and Vaes construct examples of ergodic actions of Sµ U (2) where the multiplicity of the fundamental representation can be any integer n with 2 ≤ n ≤ µ + µ1 . Therefore these actions are not on quotient spaces of Sµ U (2). They also give a simpler proof of Boca’s result by introducting a new invariant, the quantum multiplicity of a spectral representation, showing it to be bounded below by the multiplicity and above by the quantum dimension. They construct examples using a duality theorem between ergodic quantum actions where the quantum multiplicity equals the quantum dimension, and certain tensor ∗ -functors of RepG into tensor categories of Hilbert spaces. When the quantum multiplicity is not maximal, our main tool is the spectral functor associated with a generic ergodic action (Sect. 6). This covariant ∗ -functor associates to any unitary, finite dimensional representation u of G, the dual space L u of the space of all multiplets in B transforming like u under the action η. By ergodicity, this space L u , becomes a Hilbert space in a natural way. We stress that the functor L satisfies two crucial properties: first L u⊗v naturally contains a copy of L u ⊗ L v , in such a way that the copy of L u ⊗ L v ⊗ L z contained in both L u⊗v ⊗ L z and L u ⊗ L v⊗z is the same. Secondly the projection from L u⊗v⊗z to L u⊗v ⊗ L z actually takes L u ⊗ L v⊗z onto L u ⊗ L v ⊗ L z (Theorem 6.3). We call any functor F from a generic tensor C ∗ -category T to the category of Hilbert spaces satisfying the above properties, quasitensor (Sect. 3). We show that quasitensor functors, like the tensor ones, have the property that if ρ is a conjugate of ρ then the Hilbert space F(ρ) must be a conjugate of F(ρ), although this conjugate must be found in the image category of F enriched with the projection maps from the spaces F(ρ ⊗ σ ) onto F(ρ) ⊗ F(σ ) (Theorem 3.7). It follows that F(ρ) is automatically finite dimensional and endowed with an intrinsic dimension, in the sense of [13], bounded below by the Hilbert space dimension of F(ρ) and above by the intrinsic dimension of ρ (Cor. 3.8). This result applied to the spectral functor L allows us to identify the quantum multiplicity of a spectral irreducible representation of an ergodic action of [2], with the intrinsic dimension of L u , and to recover the multiplicity bound theorems of [3] and [2]. Maximal quantum multiplicity is characterized by L preserving conjugates. (see Cor. 6.5 for a precise statement). The spectral functor determines uniquely the ∗ -algebra structure of the dense ∗ -subalgebra of spectral elements. Our main result is a duality theorem for ergodic C ∗ -actions of compact quantum groups, showing that any quasitensor ∗ -functor F from the representation category of a compact quantum group G to the category of Hilbert spaces is the spectral functor of an ergodic G-action on a unital C ∗ -algebra (Theorem 8.1). Thus the problem of classifying the ergodic actions of Sµ U (2) becomes equivalent to the problem of classifying the quasitensor ∗ -functors of Sµ U (2). The Duality Theorem 8.1 is applied to inclusions of tensor C ∗ -categories. In fact, quasitensor ∗ -functors defined on the representation category of a compact quantum group G arise very naturally from tensor ∗ -functors ρ : Rep(G) → T: just map a unitary Grepresentation u onto the Hilbert space (ι, ρu ), and let an intertwiner T ∈ (u, v) act on (ι, ρu ) by left composition with ρ(T ) (Example 3.5). Therefore for any tensor ∗ -functor ρ : Rep(G) → T, our duality theorem yields an ergodic G-action over a unital C ∗ -algebra, having (ι, ρu ) as its spectral subspaces (Theorem 9.1).
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Quasitensor functors arising from quantum quotient spaces share the property of being subobjects of the embedding functor H taking u to its Hilbert space Hu . For these functors, there is a natural unitary transformation identifying the spectral subspace L u with the subspace K u of Hu of all vectors invariant under the restriction of u to the subgroup (Theorem 6.7). For maximal compact quantum groups, we characterize algebraically the spaces of invariant vectors K u for a unique maximal subgroup K (Theorem 5.5). From this we derive our second main result. A maximal ergodic action (B, η) is isomorphic to a compact quantum quotient space, if and only if the corresponding quasitensor functor is concrete (or embedded) and satisfies suitable properties (Theorem 10.3). After the first version of our paper was written, the referee informed us that Tomatsu had in the meantime characterized compact quantum quotient spaces of a compact quantum group of G, which are embedded in the Hopf C ∗ -algebra of G [20]. We discuss the relationship between his result and ours in the remark at the end of Sect. 10. 2. Preliminaries 2.1. Compact quantum groups and their representations. In this paper G = (A, ) will always denote a compact quantum group in the sense of [27]: a unital C ∗ -algebra A with a unital ∗ -homomorphism (the coproduct) : A → A ⊗ A such that a) ⊗ ι ◦ = ι ⊗ ◦ , with ι the identity map on A, b) the sets {b ⊗ I (c), b, c ∈ A} {I ⊗ b(c), b, c ∈ A} both span dense subspaces of A ⊗ A. By a unitary representation of G we mean a unitary finite dimensional representation u : Hu → Hu ⊗ A on the Hilbert space Hu [27]. If u is a unitary representation, the matrix coefficients of u are the elements of A: u φ,ψ := ∗φ ◦ u(ψ), ψ, φ ∈ Hu , with φ : A → H ⊗ A the operator between the indicated right Hilbert A–modules which tensors on the left by φ. Let u : Hu → Hu ⊗ A be any linear map, and let (φi ) be an orthonormal basis of Hu . Consider the matrix (u i j ) with entries in A, where u i j := u φi ,φ j . Then u is a unitary representation if and only if the matrix (u i j ) is unitary and (u i j ) = k u ik ⊗ u k j . The linear span A∞ of all the matrix coefficients is known to be a unital dense ∗ -subalgebra of A, and a Hopf ∗ -algebra G ∞ = (A∞ , ) with the restricted coproduct [27]. The category Rep(G) of unitary representations of G with arrows, the spaces (u, v) of linear maps T : Hu → Hv intertwining u and v: T ⊗ I ◦u = v◦T is a tensor C ∗ -category with conjugates [26]. Recall that the tensor product u ⊗ v of two representations u and v and the conjugate representation u have Hilbert spaces Hu ⊗ Hv and Hu and coefficients, (u ⊗ v)φ⊗φ ,ψ⊗ψ := u φ,ψ vφ ,ψ , u jφ, j ∗ −1 ψ := (u φ,ψ )∗ ,
φ, ψ ∈ Hu , φ , ψ ∈ Hv , φ, ψ ∈ Hu ,
respectively, where j : Hu → Hu is an antilinear invertible intertwiner. 2.2. Spectra of Hopf ∗ -algebra actions. Let C be a unital ∗ -algebra and G = (A, ) a compact quantum group. Consider the dense Hopf ∗ -subalgebra G ∞ = (A∞ , ) of G and a unital ∗ -homomorphism η : C → C A∞ such that η ⊗ ι ◦ η = ι ⊗ ◦ η.
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We define the spectrum sp(η) of η, to be the set of all unitary G-representations u : Hu → Hu ⊗ A with a faithful linear map T : Hu → C intertwining u and η: η ◦ T = T ⊗ ι ◦ u. Representing u as a matrix u = (u i j ) with respect to some orthonormal basis of Hu , u ∈ sp(η) if and only if there is a multiplet (c1 , . . . , cd ), with d the dimension of u, of linearly independent elements of C, with η(ci ) = j c j ⊗ u ji . For compact quantum groups this notion was introduced by Podles in [17], as a generalization of the classical notion of an action of a compact group on a C ∗ -algebra [8]. The spectrum has the following simple properties. Proposition 2.1. a) If u ∈ sp(η) and if z is a unitary representation of G such that (z, u) contains an isometry, then z ∈ sp(η). b) If u ∈ sp(η) and u is a unitary representation equivalent to the complex conjugate u ∗ then u ∈ sp(η). Here u ∗ denotes the representation, in general not unitary, whose matrix elements are the adjoints of those of u. We denote the linear span of all spectral multiplets by Csp . Part a) of the previous proposition tells us that Csp is generated, as a linear space, by those nonzero multiplets transforming according to unitary irreducible G-representations (such multiplets are automatically linearly independent by irreducibility). Proposition 2.2. a) If T : Hu → C is any linear map satisfying η ◦ T = T ⊗ ι ◦ u then the image of T lies in Csp . b) Csp is a unital ∗ -subalgebra of C invariant under the G ∞ -action: η(Csp ) ⊂ Csp A∞ . Consider an action η : B → B ⊗ A of G = (A, ) on a unital C ∗ -algebra B (with ⊗ the minimal tensor product): a unital ∗ -homomorphism with η ⊗ ι ◦ η = ι ⊗ ◦ η. In the C ∗ -algebraic case, one can similarly define the spectrum of the action, sp(η), and the spectral ∗ -subalgebra Bsp , a unital ∗ -subalgebra invariant under the action of G ∞ . When B = A and η = , Asp = A∞ . We call an action η : B → B⊗A of G = (A, ) on a unital C ∗ -algebra B nondegenerate if η(B)I ⊗A is dense in B⊗A. In [17] Podles shows that if η is nondegenerate Bsp is dense and generated, as a linear space, by the subspaces Wu containing any spectral multiplet transforming like u, where u ranges over the irreducibles of G. Furthermore each Wu is the direct sum of subspaces Wui , i ∈ Iu , each of them corresponding to u. The cardinality of the set Iu is called the multiplicity of the irreducible u in η, and denoted mult(u). 2.3. Quantum subgroups and quotient spaces. A compact quantum subgroup K = (A , ) of G, as introduced in [17], is a compact quantum group with a surjective ∗ -homomorphism π : A → A such that π ⊗ π ◦ = ◦ π. A closed bi–ideal I of A is a norm closed two–sided ideal of A such that (I) ⊂ A ⊗ I + I ⊗ A. There is a surjective correspondence between quantum subgroups of G and closed bi–ideals of A, which associates and a subgroup defined by the surjection π , the kernel of π . Subgroups defined by surjections with the same kernel are isomorphic as Hopf C ∗ -algebras [15]. The subgroup K acts (on the left) on the C ∗ -algebra A via δ := π ⊗ ι ◦ : A → A ⊗ A. The fixed point algebra Aδ := {T ∈ A : δ(T ) = I ⊗ T } is defined to be the quantum quotient space of right cosets. This algebra has a natural right action of G:
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η K := Aδ : Aδ → Aδ ⊗ A, known to be nondegenerate and ergodic [22], see also [15]. We set: K \G := (Aδ , η K ). Similarly the action of K on the right on A: ρ := ι ⊗ π ◦ : A → A ⊗ A , yields the quantum quotient space of left cosets G/K over Aρ . As in the group case, unitary representations of G can be restricted to unitary representations of K on the same Hilbert space: u K := ι ⊗ π ◦ u : Hu → Hu ⊗ A . Proposition 2.3. [15]. If T ∈ (u, v) then T ∈ (u K , v K ) as well. So the map Rep(G) → Rep(K ), taking u → u K , and acting trivially on the arrows defines a faithful tensor ∗ -functor. The smallest full tensor ∗ -subcategory of Rep(K ) with subobjects and direct sums containing the u K , for u ∈ Rep(G), is Rep(K ). We shall consider the subspace K u of Hu of K –invariant vectors for the restricted representation u K : this is the set of all k ∈ Hu with u K (k) = k ⊗ I . The dimension of K u is the multiplicity of the trivial representation ι K of K in u K . Proposition 2.4. If K is a compact quantum subgroup of G defined by the surjection π then for any representation u of G and vectors ψ ∈ Hu , k ∈ K u , the elements u ψ,k − (ψ, k)I and u k,ψ − (k, ψ)I belong to the closed bi–ideal kerπ . As in the group case, elements of the coset spaces arise from invariant vectors for the subgroup: for any representation u of G, if we pick vectors k ∈ K u , ψ ∈ Hu , the coefficient u k,ψ lies in Aδ and u ψ,k lies in Aρ . We only show the former: if (ψ j ) is an orthonormal basis of Hu : i
δ(u k,ψ j ) = π ⊗ ι ◦ (u k,ψ j ) = π(u k,ψi ) ⊗ u i j = I ⊗ (k, ψi )u i j = I ⊗ u k,ψ j . i
Fix a complete set Gˆ of unitary irreducible representations of G, and set Gˆ K := {u ∈ Gˆ : K u = 0}. Proposition 2.5. The linear space generated by the matrix coefficients {u k,ψ }, (resp. ρ {u ψ,k }, ) as u ∈ Gˆ K , k ∈ K u , ψ ∈ Hu , vary, coincides with Aδsp (resp. Asp )). Consequently, the above subspace is a unital ∗ -algebra endowed with the restricted action of the Hopf ∗ -algebra G ∞ := (A∞ , ), still denoted by η K : Aδsp → Aδsp A∞ . 3. Quasitensor Functors and a Finiteness Theorem Definition 3.1. Let T and R be strict tensor C ∗ -categories [5,4]. We shall always assume that tensor units are irreducible: (ι, ι) = C. A (covariant) ∗ -functor F : T → R will be called quasitensor if
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F(ι) = ι,
(3.1)
if for objects ρ, σ ∈ T there is an isometry Sρ,σ : F(ρ) ⊗ F(σ ) → F(ρ ⊗ σ )
(3.2)
such that Sρ⊗σ,τ ◦ Sρ,σ
Sρ,ι = Sι,ρ = 1F(ρ) , ⊗ 1F(τ ) = Sρ,σ ⊗τ ◦ 1F(ρ) ⊗ Sσ,τ =: Sρ,σ,τ , E ρ⊗σ,τ ◦ E ρ,σ ⊗τ ≤ E ρ,σ,τ ,
(3.3) (3.4) (3.5)
with E ρ,σ ∈ (F(ρ ⊗ σ ), F(ρ ⊗ σ )) the range projection of Sρ,σ and E ρ,σ,τ ∈ (F(ρ ⊗ σ ⊗ τ ), F(ρ ⊗ σ ⊗ τ )) the range projection of Sρ,σ,τ , and if F(S ⊗ T ) ◦ Sρ,σ = Sρ ,σ ◦ F(S) ⊗ F(T ), for any other pair of objects
ρ,
σ
and arrows S ∈
(ρ, ρ ),
T ∈
(3.6) (σ, σ ).
Here we treat the case where R is the tensor C ∗ -category H with objects Hilbert spaces and arrows the set (H, H ) of all bounded linear mappings from H to H . We shall assume that H is strictly tensor, i.e. that the tensor product between objects is strictly associative. To economize on brackets, we evaluate tensor products of arrows before composition. Moreover in the sequel, for simplicity, we shall occasionally identify, with a less precise but lighter notation, F(ρ) ⊗ F(σ ) with a subspace of F(ρ ⊗ σ ), the image of Sρ,σ . Equations (3.2)–(3.6) shall then be written: F(ρ) ⊗ F(σ ) ⊂ F(ρ ⊗ σ ), E ρ,ι = E ι,ρ = 1F(ρ) , E ρ,σ ⊗ 1F(τ ) ◦ E ρ⊗σ,τ = 1F(ρ) ⊗ E σ,τ ◦ E ρ,σ ⊗τ =: E ρ,σ,τ , E ρ⊗σ,τ (F(ρ) ⊗ F(σ ⊗ τ )) ⊂ F(ρ) ⊗ F(σ ) ⊗ F(τ ), F(S ⊗ T ) F(ρ)⊗F(σ ) = F(S) ⊗ F(T ).
(3.7) (3.8) (3.9) (3.10) (3.11)
Equation (3.10) combined with (3.9) requires the projection onto F(ρ ⊗ σ ) ⊗ F(τ ) to take the subspace F(ρ) ⊗ F(σ ⊗ τ ) onto F(ρ) ⊗ F(σ ) ⊗ F(τ ). Therefore we must have E ρ,σ,τ = E ρ⊗σ,τ ◦ E ρ,σ ⊗τ = E ρ,σ ⊗τ ◦ E ρ⊗σ,τ .
(3.12)
Notice that any tensor ∗ -functor from T to H is quasitensor. Example 3.2. Assume that T = Rep(G), the representation category of a compact quantum group G. Then the embedding functor H : Rep(G) → H associating to each representation u its Hilbert space Hu and acting trivially on the arrows, is tensor, and therefore quasitensor. Quasitensor ∗ -functors arise naturally in abstract tensor C ∗ -categories. Proposition 3.3. Let T be a tensor C ∗ -category with (ι, ι) = C1ι . For any object ρ of T, consider the Hilbert space ρˆ := (ι, ρ), with inner product (φ, φ )1ι := φ ∗ ◦ φ , φ, φ ∈ (ι, ρ). For T ∈ (ρ, σ ) define a bounded linear map Tˆ : ρˆ → σˆ by Tˆ (φ) = T ◦ φ. This is a quasitensor ∗ -functor.
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∗ = Tˆ ∗ . Proof. It is easy to check that, for T ∈ (ρ, σ ), S ∈ (σ, τ ), S ◦ T = Sˆ ◦ Tˆ and T ∗ Thus we have a -functor with ιˆ = (ι, ι) = C, and (3.1) is satisfied. For φ ∈ ρˆ = (ι, ρ), ψ ∈ σˆ = (ι, σ ), the map φ, ψ → φ ⊗ ψ = φ ⊗ 1σ ◦ ψ ∈ ρ ⊗ σ = (ι, ρ ⊗ σ ) defines an isometric map from ρˆ ⊗ σˆ to ρ ⊗ σ . The copy of ρˆ ⊗ σˆ ⊗ τˆ sitting inside ρ ⊗ σ ⊗ τˆ is the subspace generated by elements φ ⊗ 1σ ⊗τ ◦ ψ ⊗ 1τ ◦ η, for φ ∈ ρ, ˆ ψ ∈ σˆ , η ∈ τˆ , and coincides with the copy of the same Hilbert space in ρˆ ⊗ σ ⊗ τ , hence (3.9) holds. We now check (3.10). First note that if φ is an orthonormal basis of σˆ then every i finite sum finite 1ρ ⊗ (φi ◦ φi∗ ) is an element of (ρ ⊗ σ, ρ ⊗ σ ), defining a projection map from ρ ⊗ σ onto a subspace of ρˆ ⊗ σˆ by composition. The strong limit of this net converges, in the strong topology defined by ρ ⊗ σ , to the projection map E ρ,σ . This shows that the projection E ρ⊗σ,τ is the strong limit of i 1ρ⊗σ ⊗ ψi ◦ ψi∗ , with ψi ∈ τˆ ⊗ τ into ρˆ ⊗ σˆ ⊗ τˆ , completing the an orthonormal basis. Thus E ρ⊗σ,τ takes ρˆ ⊗ σ proof of (3.10).
The following proposition helps to construct more examples. Proposition 3.4. Let S, T be tensor C ∗ -categories. If G : S → T is a tensor ∗ -functor and F : T → H is quasitensor then F ◦ G is quasitensor. An analogous statement holds if F is tensor and G is quasitensor. Example 3.5. Let G be a compact quantum group, T be a tensor C ∗ -category and ρ : Rep(G) → T a tensor ∗ -functor. Composing ρ with the quasitensor ∗ -functor T → H associated with T as in Prop. 3.3, we obtain an interesting ‘abstract’ quasitensor ∗ -functor F : Rep(G) → H with Hilbert spaces F (u) = (ι, ρ ) and defined on arrows ρ ρ u by ˆ )φ = ρ(T ) ◦ φ ∈ (ι, ρv ), φ ∈ (ι, ρu ), T ∈ (u, v). Fρ (T )φ := ρ(T Example 3.6. Apply the above construction described to the tensor C ∗ -category T = Rep(K ), where K is a compact quantum subgroup of G, with ρ given by the canonical tensor ∗ -functor Rep(G) → Rep(K ) described in Prop. 2.3. We now obtain a ‘concrete’ quasitensor ∗ -functor, that we shall denote, with abuse of notation, by K : Rep(G) → H, where K u = (ι K , u K ). K will be called the invariant vectors functor. In Sect. 6 we shall exhibit examples of quasitensor ∗ -functors associated with ergodic actions of compact quantum groups. The rest of this section is devoted to showing the following theorem. Theorem 3.7. Let T be a strict tensor C ∗ -category and F : T → H a quasitensor If ρ has a conjugate ρ in T then F(ρ) is finite dimensional and
∗ -functor.
dimF(ρ) = dimF(ρ). Furthermore if F(ρ) = 0 and if R ∈ (ι, ρ ⊗ ρ) and R ∈ (ι, ρ ⊗ ρ) is a solution ∗ ◦ F(R) ∈ F(ρ) ⊗ F(ρ) and of the conjugate equations for ρ in T then Rˆ := Sρ,ρ Rˆ := S ∗ ◦ F(R) ∈ F(ρ) ⊗ F(ρ) is a solution of the conjugate equations for F(ρ) in
H.
ρ,ρ
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Proof. For simplicity, we shall use the simplified notation (3.7)–(3.11), replacing Sρ,σ ∗ by E by the identity and Sρ,σ ρ,σ . Apply the functor F to the relation ∗
R ⊗ 1ρ ◦ 1ρ ⊗ R = 1ρ , and get F(R ⊗ 1ρ )∗ ◦ F(1ρ ⊗ R) = 1F(ρ) . Using successively (3.11), (3.8), (3.9), (3.10), we get F(1ρ ⊗ R)(ψ) = ψ ⊗ F(R), ψ ∈ F(ρ), F(R ⊗ 1ρ )(φ) = F(R) ⊗ φ, φ ∈ F(ρ). So (φ, ψ) = (φ, F(R ⊗ 1ρ )∗ ◦ F(1ρ ⊗ R)ψ) = (F(R) ⊗ φ, ψ ⊗ F(R)) = (F(R) ⊗ φ, E ρ⊗ρ,ρ (ψ ⊗ F(R)) = (F(R) ⊗ φ, E ρ,ρ,ρ ◦ E ρ⊗ρ,ρ (ψ ⊗ F(R)) = (E ρ,ρ ◦ F(R) ⊗ φ, ψ ⊗ E ρ,ρ ◦ F(R)), so ∗ Rˆ ⊗ 1F(ρ) ◦ 1F(ρ) ⊗ Rˆ = 1F(ρ) .
At this point we can start with ρ and obtain the relation Rˆ ∗ ⊗ 1F(ρ) ◦ 1F(ρ) ⊗ Rˆ = 1F(ρ) ,
implying that F(ρ) is finite dimensional, too, with the same dimension as F(ρ). In a tensor C ∗ -category with conjugates, the infimum of all the d R,R (ρ) := RR is the intrinsic dimension of ρ, denoted d(ρ) [13]. Corollary 3.8. If F : T → H is a quasitensor ∗ -functor and ρ an object of T with a conjugate defined by R and R then dim(F(ρ)) ≤ d ˆ ˆ (F(ρ)) ≤ d R,R (ρ). R, R
Furthermore d ˆ ˆ (F(ρ)) = d R,R (ρ) if and only if F(R) ∈ ImageSρ,ρ and F(R) ∈ R, R ImageSρ,ρ . ˆ ≤ R Proof. Note that F(R)2 = F(R)∗ F(R) = F(R ∗ R) = R2 , so R ˆ and, similarly, R ≤ R, proving the last inequality. Thus d ˆ ˆ (F(ρ)) = d R,R (ρ) if R, R ˆ = F(R), and the last statement follows. ˆ = F(R) and R and only if R
ˆ Rˆ by J ψ = r ∗ ◦ R. ˆ Let J : F(ρ) → F(ρ) be the antilinear invertible associated to R, ψ Then d 2 ˆ (F(ρ)) = Trace(J J ∗ )Trace((J J ∗ )−1 ). ˆ R R,
The first inequality follows as n 2 ≤ Trace(Q)Trace(Q −1 ), for any positive invertible matrix Q ∈ Mn .
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In Sect. 6 we shall relate this result to the work of [2]. Let F : S → T be a quasitensor functor and R, R¯ solve the conjugate equations for ˆ ¯ solve an object ρ of S. Then, as we have seen, Rˆ := E ρ,ρ ¯ ◦ F(R) and R¯ := E ρ.ρ¯ ◦ F( R) the conjugate equations for F(ρ). This construction has certain functorality properties. Given T ∈ (ρ, ρ ) define T • by T • ⊗ 1ρ ◦ Rρ := 1ρ¯ ⊗ T ∗ ◦ Rρ . Then F(T • ⊗ 1ρ ) ◦ F(Rρ ) = F(1ρ¯ ⊗ T ∗ ) ◦ F(Rρ ), so E ρ¯ ,ρ ◦ F(T • ⊗ 1ρ ) ◦ F(Rρ ) = E ρ¯ ,ρ ◦ F(1ρ¯ ⊗ T ∗ ) ◦ F(Rρ ), and then F(T • ) ⊗ 1F(ρ) ◦ Rˆ ρ = 1F(ρ¯ ) ⊗ F(T ∗ ) ◦ Rˆ ρ . It follows that F(T • ) ⊗ 1F(ρ) ◦ Rˆ ρ = F(T )• ⊗ 1F(ρ) ◦ Rˆ ρ , giving F(T )• = F(T • ). Now suppose that R, R¯ is a standard solution of the conjugate equations. Then R = i W¯ i ⊗ Wi ◦ Ri , where Ri ∈ (ι, ρ¯i ρi ) and R¯ i ∈ (ι, ρi ρ¯i ) are normalized solutions of the conjugate equations for the irreducible ρi . Thus F(R) = i F(W¯ i ⊗ Wi ) ◦ F(Ri ) ˆ¯ F(W ) ⊗ ¯ ˆ giving Rˆ = E ρ,ρ ¯ ◦ F(R) = i i F( Wi ) ⊗ F(Wi ) ◦ Ri . We similarly get R = i ˆ Rˆ¯ is a standard solution of F(W¯ ) ◦ R¯ˆ . Consequently, if the F(ρ ) are irreducible, R, i
i
i
the conjugate equations for F(ρ). For future use we note that Sρ,σ is still a natural transformation when antilinear intertwiners are allowed, as follows from the next result.
Lemma 3.9. If F : T → H is a quasitensor functor, the antilinear operators J associated ˆ Rˆ¯ of the conjugate equations satisfy with solutions R, Jρ⊗σ Sρ,σ = Sσ¯ ,ρ¯ Jσ ⊗ Jρ ◦ θρ,σ , with θρ,σ the flip map, provided Rρ⊗σ := 1σ¯ ⊗ Rρ ⊗ 1σ ◦ Rρ . Remark. Antilinear arrows are discussed in [16], where it is pointed out that Jσ ⊗ Jρ ◦ θρ,σ is the natural tensor product.
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Proof. Identifying F(ρ) ⊗ F(σ ) with a subspace of F(ρ ⊗ σ ), we must show that Jρ⊗σ ψ ⊗ φ = Jσ φ ⊗ Jρ ψ, ψ ∈ F(ρ), φ ∈ F(σ ). If χ ∈ F(ρ ⊗ σ ) then rχ∗ ◦ Rˆ ρ⊗σ = Jρ⊗σ χ . Hence
∗ rψ⊗φ
∗ rψ⊗φ
∗ Jρ⊗σ ψ ⊗ φ = rψ⊗φ ◦ Rˆ ρ⊗σ =
◦ E σ¯ ⊗ρ,ρ⊗σ F(1σ¯ ⊗ Rρ ⊗ 1σ ◦ Rσ ) = ¯
◦ 1F(σ¯ ⊗ρ) F(1σ¯ ⊗ Rρ ⊗ 1σ ◦ Rσ ). ¯ ¯ ⊗ E ρ,σ ◦ E σ¯ ⊗ρ,ρ⊗σ
On the other hand by (3.9), 1F(σ¯ ⊗ρ) = E σ¯ ⊗ρ,ρ , ¯ ¯ ⊗ 1F(σ ) ◦ E σ¯ ⊗ρ⊗ρ,σ ¯ ¯ ⊗ E ρ,σ ◦ E σ¯ ⊗ρ,ρ⊗σ hence the last term above equals ∗ rψ⊗φ ◦ E σ¯ ⊗ρ,ρ ◦ F(1σ¯ ⊗ Rρ ⊗ 1σ ) ◦ F(Rσ ). ¯ ⊗ 1F(σ ) ◦ E σ¯ ⊗ρ⊗ρ,σ ¯
(3.13)
Now by (3.11), F(1σ¯ ⊗ Rρ∗ ⊗ 1σ ) ◦ E σ¯ ⊗ρ⊗ρ,σ = F(1σ¯ ⊗ Rρ∗ ) ⊗ 1F(σ ) ◦ E σ¯ ⊗ρ⊗ρ,σ = ¯ ¯ E σ¯ ,σ ◦ F(1σ¯ ⊗ Rρ∗ ) ⊗ 1F(σ ) ◦ E σ¯ ⊗ρ⊗ρ,σ , ¯
hence taking the adjoint, E σ¯ ⊗ρ⊗ρ,σ ◦ F(1σ¯ ⊗ Rρ ⊗ 1σ ) = F(1σ¯ ⊗ Rρ ) ⊗ 1F(σ ) ◦ E σ¯ ,σ ¯ = 1F(σ¯ ) ⊗ F(Rρ ) ⊗ 1F(σ ) ◦ E σ¯ ,σ , and (3.13) becomes ∗ rψ⊗φ ◦ E σ¯ ⊗ρ,ρ ¯ ⊗ 1F(σ ) ◦ 1F(σ¯ ) ⊗ F(Rρ ) ⊗ 1F(σ ) ◦ E σ¯ ,σ ◦ F(Rσ ).
Now E σ¯ ⊗ρ,ρ = 1F(σ¯ ) ⊗ E ρ,ρ ¯ ◦ E σ¯ .ρ⊗ρ ¯ ¯ ◦ E σ¯ ,ρ⊗ρ ¯ and E σ¯ ,ρ⊗ρ ◦ 1F(σ¯ ) ⊗ F(Rρ ) = 1F(σ¯ ) ⊗ F(Rρ ), ¯ so (3.14) equals ∗ rψ⊗φ ◦ 1F(σ¯ ) ⊗ Rˆρ ⊗ 1F(σ ) ◦ Rˆσ = Jσ φ ⊗ Jρ ψ
as required.
(3.14)
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4. Concrete Quasitensor Functors Let G be a compact quantum group. We construct quasitensor ∗ -functors Rep(G) → H associating a subspace K u of the representation Hilbert space Hu with the representation u, and generalizing the invariant vectors functor of a compact quantum subgroup K of a compact quantum group G = (A, ) described in Example 3.6. They are functors more general than the invariant vectors functors, and characterize the C ∗ -subalgebras of A invariant under the coproduct (see Theorem 10.4). To each unitary representation u of G, we assign a subspace K u of the representation Hilbert space Hu , imposing conditions on the projection maps E u : Hu → K u . Lemma 4.1. For u ∈ Rep(G), let E u : Hu → Hu be an orthogonal projection such that E ι = 1C , T E u = E v T, T ∈ (u, v), E u ⊗ E v = I ⊗ E v ◦ E u⊗v .
(4.1) (4.2) (4.3)
Then K u := E u Hu ,
K T := T K u ∈ (K u , K v ),
is quasitensor. The notion of quasitensor natural transformation describes the isometric inclusion map K u ⊗ K v ⊂ K u⊗v well. Definition 4.2. Let F, G be quasitensor ∗ -functors from a tensor C ∗ -category T to the F and S G be the defining set of isometensor C ∗ -category of Hilbert spaces H. Let Sρ,σ ρ,σ tries for F and G respectively. A natural transformation η : F → G is quasitensor if for objects ρ, σ ∈ T, ηι : F(ι) = C → G(ι) = C is the identity map F = SG ◦ η ⊗ η . ηρ⊗σ ◦ Sρ,σ ρ σ ρ,σ
Proposition 4.3. Let K : Rep(G) → H be the quasitensor ∗ -functor obtained from projections (E u ) satisfying properties (4.1)–(4.3). Then the inclusion map Wu : K u → Hu defines a quasitensor natural transformation from the functor K to the embedding functor H : Rep(G) → H. Remark. The invariant vectors functors associated with quantum subgroups fit this description. Indeed, let K be a compact quantum subgroup of G with Haar measure h , and, for a unitary representation u of G, let E uK : Hu → Hu project onto the subspace of u K –fixed vectors obtained averaging over the K -action: E uK (ψ) = ι ⊗ h ◦ u K (ψ). If u is unitary, E uK is easily seen to be selfadjoint. A straightforward computation shows that for any pair of f.d. representations u and v, and for any T ∈ (u K , v K ), and a fortiori for any T ∈ (u, v), T ◦ E uK = E vK ◦ T. We are left to check property (4.3). Since K , the tensor product of K –invariant vectors is K –invariant, we have: E uK ⊗ E vK ≤ E u⊗v K K , i.e. to the implication = I ⊗ E vK ◦ E u⊗v and (4.3) is equivalent to E uK ⊗ E vK E u⊗v ψ ∈ K v , η ∈ K u⊗v implies rψ∗ (η) ∈ K u . Now the restriction to K of a tensor product representation is the tensor product of the restrictions, so η ∈ K u⊗v = (ι, (u ⊗ v) K ) = (ι, u K ⊗v K ).
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Using ψ ∈ K v = (ι, v K ), too, gives, rψ∗ (η) = 1u K ⊗ ψ ∗ ◦ η ∈ (ι, u K ) = K u , the operations being understood in Rep(K ). Remark. A similar argument shows that the projections E uK also satisfy K . E uK ⊗ E vK = E uK ⊗ I ◦ E u⊗v
(4.4)
Remark. If G is a maximal quantum group (i.e. obtained from its smooth part A∞ by completing in the maximal C ∗ -seminorm) and K is the trivial subgroup (corresponding to the counit e : A → C of G) then K is just the embedding tensor functor H : Rep G → H. At the other extreme, setting K = G, K u = (ι, u), does not give a tensor functor, as K u = K u = 0, for example, if u is irreducible, but K u⊗u = (ι, u ⊗ u) = 0. We next construct certain maps used later to describe multiplicities of spectral representations in quantum quotient spaces. Given a unitary representation u : Hu → Hu ⊗A, we set u K := E uK ⊗ I ◦ u : Hu → K u ⊗ A. For any pair of vectors φ, ψ ∈ Hu , the coefficients of u K : K := ∗φ ◦ u K (ψ) = ∗E K (φ) ◦ u(ψ) = u E uK (φ),ψ , u φ,ψ u
belong to Aδsp . So u K has its range in K u Aδsp : u K : Hu → K u Aδsp . The map u → u K satisfies the following properties. Proposition 4.4. Let u and v be unitary representations of G. a) K T ⊗ I ◦ u K = v K ◦ T, for T ∈ (u, v), K K K b) (u ⊗ v)φ⊗φ ,ψ⊗ψ = u φ,ψ u φ ,ψ , for φ ∈ K u , φ ∈ K v , ψ ∈ Hu , ψ ∈ Hv . 5. Characterizing the Invariant Vectors Functor We characterize the invariant vectors functors among the concrete quasitensor functors considered in the previous section. For each u ∈ Rep(G), let E u be an orthogonal projection on the representation Hilbert space Hu satisfying properties (4.1)–(4.3). In terms of the Hilbert spaces K u = E u Hu , these conditions can be written K ι = Hι = C, T K u ⊂ K v . T ∈ (u, v), rk∗ K u⊗v ⊂ K u , k ∈ K v , K u ⊗ K v ⊂ K u⊗v .
(5.1) (5.2) (5.3) (5.4)
(ι, u ⊗ v) ⊂ K u⊗v .
(5.5)
Consequently,
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In fact, pick T ∈ (ι, u ⊗ v). Properties (5.1) and (5.2) show that for any λ ∈ K ι = C, T λ ∈ K u⊗v , so T = T 1 ∈ K u⊗v . In particular, thanks to (5.3), rk∗ T ∈ K u , T ∈ (ι, u ⊗ v), k ∈ K v .
(5.6)
Now let u be a conjugate of u defined by R ∈ (ι, u ⊗u), R ∈ (ι, u ⊗u), and j : Hu → Hu the associated antilinear invertible intertwiner defined by: jφi ⊗ φi , R= j −1 ψ j ⊗ ψ j , R= i
j
with (φi ) and (ψ j ) orthonormal bases of Hu and Hu respectively, chosen to complete the orthonormal bases of K u and K u . Property (5.6) applied to R and R shows that j Ku = Ku .
(5.7)
When G is a group, conditions (5.1), (5.2), (5.4) and (5.7) are known to characterize a subgroup of G having each K u as the invariant subspace of u, restricted to the subgroup [18]. The question naturally arises of whether, given a compact quantum group G and subspaces K u ⊂ Hu for each finite–dimensional unitary representation u of G, conditions (5.1), (5.2), (5.4) and (5.7) still suffice for the existence of a unique compact quantum subgroup of G whose subspaces of invariant vectors are the K u . It is well known that for a compact quantum group, the reduced representation, defined by its Haar measure, and the universal representation, defined by its maximal C ∗ -norm, give rise to the same Hopf ∗ -algebras, and therefore to the same representation categories. Therefore uniqueness for compact quantum groups should mean that the corresponding underlying Hopf ∗ -algebras of smooth elements are isomorphic. This means restricting attention to maximal compact quantum groups. Concerning existence, we start with the following result. Theorem 5.1. Let G = (A, ) be a compact quantum group, and, for each u ∈ Rep(G), let K u be a subspace of the representation Hilbert space Hu satisfying conditions (5.1), (5.2), (5.4) and (5.7). Then there is a compact quantum subgroup K of G such that K u ⊂ (ι K , u K ) for u ∈ Rep(G) and such that the linear span of {u k,φ , k ∈ K u , φ ∈ Hu , u ∈ Rep(G)} is a unital ∗ -subalgebra of the quantum quotient space Aδ . Proof. Consider a complete set A of irreducible representations u of G with K u = 0. Let M denote the linear span of the set u {xφ,k := u φ,k − (φ, k)I, u ∈ A, k ∈ K u , φ ∈ Hu },
in the Hopf C ∗ -algebra A. It is easy to check that u u )= u φ,φi ⊗ xφui ,k + xφ,k ⊗ I, (xφ,k i
with (φi ) an orthonormal basis of Hu . Therefore (M) ⊂ A∞ M + M CI . Let J be the closed two–sided ideal of A generated by M. Then (J) ⊂ A ⊗ J + J ⊗ A, so J is a closed bi–ideal. Consider the associated compact quantum subgroup K = (A/J, )
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of G with coproduct (q(a)) = q ⊗ q ◦ (a), where q : A → A/J is the canonical surjection. We show that K u ⊂ (ι K , u K ). For k ∈ K u : φi ⊗ u φi ,k ) = u K (k) = ι ⊗ q ◦ u(k) = ι ⊗ q(
φi ⊗ q(u φi ,k ) =
i
i
φi ⊗ (φi , k)I = k ⊗ I.
i
We are left to show that the linear span V of all the u k,φ is a unital ∗ -subalgebra of Aδ . Since K u ⊂ (ι, u K ), V is contained in Aδ , and I ∈ V , as K ι = C. Therefore it suffices to show that V is a ∗ -subalgebra of A. On the other hand the ∗ -algebra structure of A recalled at the end of Subsect. 2.1 and properties (5.4) and (5.7) show that V is a ∗ -subalgebra.
Remark. Under conditions (5.1), (5.2), (5.4) and (5.7) alone, one can not, in general, identify the subspace K u with the space of all the invariant vectors (ι K , u K ) for some quantum subgroup K of G. In fact, Wang [22] provides an ergodic action δ on a commutative C ∗ -algebra C which is not a quotient action. Now C being commutative, there is a faithful embedding of (Csp , δ) into a quantum quotient space by a quantum subgroup and subspaces K u satisfying the above equations can be constructed (see Theorem 10.4). Such an embedding, though, does not extend to an isomorphism of C and therefore the K u can not be the spaces of all invariant vectors. This phenomenon occurs for Sµ U (2), too [19]. We can give a positive answer if we replace (5.4) by a stronger, still necessary condition, motivated as follows. In the group case, the representation category of G contains the permutation symmetry among its intertwiners: the operators θu,v ∈ (u ⊗ v, v ⊗ u) permute the order of factors in the tensor product. Consequently, for u, v, z ∈ Rep(G), 1u ⊗ φ ⊗ 1z ◦ k ∈ K u⊗v⊗z , k ∈ K u⊗z , φ ∈ K v .
(5.8)
In fact, by (5.4), φ ⊗ k ∈ K v⊗u⊗z , so 1u ⊗ φ ⊗ 1z ◦ k = (ϑv,u ⊗ 1z )φ ⊗ k, an element of K u⊗v⊗z thanks to (5.2). On the other hand, the argument in the remark following Prop. 4.3, showing the necessity (4.3) also shows (5.8) to be still necessary for the K u to be the invariant subspaces of the restriction of u to a quantum subgroup. Therefore, in the quantum group case, it seems natural to replace (5.4) by the stronger condition (5.8). Given a functor K associating to any representation u ∈ Rep(G) a subspace K u ⊂ Hu satisfying (5.1), (5.2), (5.7) and (5.8), consider, for u, v ∈ Rep(G), the subspace of (Hu , Hv ) defined by ∗
< Hu , Hv >:= {R ⊗ 1v ◦ 1u ⊗ φ, φ ∈ K u⊗v }, where R ∈ (ι, u ⊗ u) is an intertwiner arising from a standard solution of the conjugate equations for u in Rep(G).
Proposition 5.2. The space < Hu , Hv > is independent of u and R. Proof. If R ∈ (ι, u ⊗ u) ˜ arises from another standard solution of the conjugate equations then there is a unitary U ∈ (u, u) ˜ such that R = 1u ⊗ U ◦ R [13]. Since U ⊗ 1v ∈ (u ⊗ v, u˜ ⊗ v), U ⊗ 1v K u⊗v = K u⊗v by (5.2). Therefore any φ ∈ K u⊗v is of the form ˜ ˜ ∗
U ⊗ 1v φ with φ ∈ K u⊗v . This implies R ∗ ⊗ 1v ◦ 1u ⊗ φ = R ⊗ 1v ◦ 1u ⊗ φ.
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Proposition 5.3. For u, v ∈ Rep(G), (u, v) ⊂< Hu , Hv >. Proof. Fix a pair R ∈ (ι, u ⊗ u) R ∈ (ι, u ⊗ u) solving the conjugate equations. Pick T ∈ (u, v) and set φT := 1u ⊗ T ◦ R, So φT ∈ (ι, u ⊗ v) ⊂ K u⊗v . Since ∗
∗
R ⊗ 1v ◦ 1u ⊗ φT = R ⊗ 1v ◦ 1u⊗u ⊗ T ◦ 1u ⊗ R = ∗
T ◦ R ⊗ 1u ◦ 1u ⊗ R = T, we can conclude that T ∈< Hu , Hv >.
Proposition 5.4. The Hilbert spaces Hu and the subspaces < Hu , Hv >, as u and v vary in Rep(G), form, respectively, the objects and the arrows of a tensor ∗ -subcategory of the category of Hilbert spaces. This category contains the image of Rep(G) under the embedding functor H , therefore it has conjugates. Proof. Since (u, v) ⊂< Hu , Hv >, 1u ∈< Hu , Hu > for u ∈ Rep(G). We show that if T ∈< Hu , Hv > and S ∈< Hv , Hz > then S ◦ T ∈< Hu , Hz >. In fact, writing ∗ ∗ T = R u ⊗ 1v ◦ 1u ⊗ φ and S = R v ⊗ 1z ◦ 1v ⊗ ψ then ∗
∗
S ◦ T = R v ⊗ 1z ◦ 1v ⊗ ψ ◦ R u ⊗ 1v ◦ 1u ⊗ φ = ∗
∗
R v ⊗ 1z ◦ R u ⊗ 1v⊗v⊗z ◦ 1u⊗u⊗v ⊗ ψ ◦ 1u ⊗ φ = ∗
∗
∗
R u ⊗ 1z ◦ 1u⊗u ⊗ R v ⊗ 1z ◦ 1u ⊗ (φ ⊗ ψ) = ∗
R u ⊗ 1z ◦ 1u ⊗ (1u ⊗ R v ⊗ 1z (φ ⊗ ψ)) ∈ as φ ⊗ ψ ∈ K u⊗v⊗v⊗z and (1u ⊗ R v ⊗ 1z )∗ (φ ⊗ ψ) ∈ K u⊗z by (5.2). ∗ We next show that ∗ =. Pick φ ∈ K u⊗v and set T = R u ⊗ 1v ◦ ∗ ∗ 1u ⊗ φ. Then T = 1u ⊗ φ ◦ R u ⊗ 1v . We use the solution Ru⊗v := 1v ⊗ R u ⊗ 1v ◦ Rv ∈ (ι, v ⊗ u ⊗ u ⊗ v) and R u⊗v := 1u ⊗ R v ⊗ 1u ◦ Ru ∈ (ι, u ⊗ v ⊗ v ⊗ u) of the conjugate equations for u ⊗ v. Thanks to (5.7), ψ := 1v⊗u ⊗ φ ∗ Ru⊗v = jφ ∈ K v⊗u . We have: ∗
∗
R v ⊗ 1u ◦ 1v ⊗ ψ = R v ⊗ 1u ◦ 1v⊗v⊗u ⊗ φ ∗ ◦ 1v⊗v ⊗ R u ⊗ 1v ◦ 1v ⊗ Rv = ∗
1u ⊗ φ ∗ ◦ R u ⊗ 1v ◦ R v ⊗ 1v ◦ 1v ⊗ Rv = 1u ⊗ φ ∗ ◦ R u ⊗ 1v = T ∗ . Therefore T ∗ ∈< Hv , Hu >. ∗ ∗ We are left to show that if S = R u ⊗ 1u ◦ 1u ⊗ φ ∈, T = R v ⊗ 1v ◦ 1v ⊗ ψ ∈< Hv , Hv > then S ⊗ T ∈< Hu⊗v , Hu ⊗v >. Consider the following solution of the conjugate equations for u ⊗v: Ru⊗v = 1v ⊗ Ru ⊗1v ◦ Rv , R u⊗v = 1u ⊗ R v ⊗1u ◦ R u . ∗ Since η := 1v ⊗ φ ⊗ 1v ◦ ψ ∈ K v⊗u⊗u ⊗v by (5.8) then R u⊗v ⊗ 1u ⊗v ◦ 1u⊗v ⊗ η ∈< Hu⊗v , Hu ⊗v >. On the other hand, ∗
R u⊗v ⊗ 1u ⊗v ◦ 1u⊗v ⊗ η =
∗
∗
R u ⊗ 1u ⊗v ◦ 1u ⊗ R v ⊗ 1u⊗u ⊗v ◦ 1u⊗v⊗v ⊗ φ ⊗ 1v ◦ 1u⊗v ⊗ ψ = ∗
∗
R u ⊗ 1u ⊗v ◦ 1u ⊗ φ ⊗ 1v ◦ 1u ⊗ R v ⊗ 1v ◦ 1u⊗v ⊗ ψ = S ⊗ 1v ◦ 1u ⊗ T = S ⊗ T.
A compact quantum subgroup of G will be called maximal if it is maximal as a compact quantum group (meaning that the norm on the dense Hopf ∗ -subalgebra coincides with the maximal C ∗ -seminorm). The above results lead to the following theorem.
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Theorem 5.5. Consider the tensor C ∗ -category of finite dimensional unitary representations of a maximal compact quantum group G. Suppose for each representation u of G on a Hilbert space Hu there is a subspace K u ⊂ Hu satisfying (5.1), (5.2), (5.7) and (5.8). Then the K u are the subspaces of invariant vectors for a unique maximal compact quantum subgroup of G. Proof. Consider the category T obtained completing the spaces , for u, v ∈ Rep(G), with respect to subobjects and direct sums. This is still a tensor ∗ -category of Hilbert spaces with conjugates, now with subobjects and direct sums. By Woronowicz’s Tannaka–Krein Theorem, we can construct a Hopf ∗ -algebra (C, ). Since Rep(G) ⊂ T, this Hopf ∗ -algebra is a model, in the sense of [26], for the Hopf ∗ -algebra constructed from Rep(G), which, in turn, is isomorphic to (A∞ , ). Therefore we can find a ∗ -epimorphism π : A ∞ → C with ◦ π = π ⊗ π ◦ . Set J := ker(π ). This is ∗ obviously a -ideal, but also an algebraic coideal, as if a ∈ J then b := ι ⊗ π((a)) must belong to the kernel of π ⊗ιC. Since ker(π ⊗ιC) = J C = Image(ιJ ⊗π ), we can find c ∈ J A such that b = ιJ ⊗ π(c). Set d := (a) − c, which is easily checked to lie in ker(ιA ⊗ π ), that in turn, equals A ⊗ J. Then (a) = c + d ∈ J A + A J. Consider the completion A of C in the maximal C ∗ -seminorm, so extends to the completion. This yields a compact quantum group (A , ) and a ∗ -homomorphism π˜ : A∞ → A intertwining and with the same kernel as π , as the inclusion C ⊂ A is faithful. Therefore we can extend π˜ to a ∗ -homomorphism q from A to A interwining the coproducts. This ∗ -homomorphism is a surjection, as its range contains C, which is dense in A . Thus (A , ) is a maximal compact quantum subgroup of G, that we denote, by abuse of notation, by K . Since q(A∞ ) = C, and q(A∞ ) = A∞ for any quantum subgroup, we deduce that C = A∞ . Therefore the representation category of K is the category we started from: Rep(K ) = T. In particular, for any representation u of G, (ι K , u K ) == K u . If K 1 = (A1 , 1 ) is another maximal quantum subgroup of G with spaces of invariant vectors given by the K u , then by Frobenius reciprocity, for any pair of representations u and v of G, (u K 1 , v K 1 ) is canonically linearly isomorphic to (ι K , u K 1 ⊗v K 1 ) = (ι K , (u ⊗ v) K 1 ) = K u⊗v . But (u K , v K ) is also linearly isomorphic, according to the same isomorphism, to K u⊗v , so (u K 1 , v K 1 ) = (u K , v K ). Hence Rep(K 1 ) = Rep(K ). It follows that there is a ∗ -isomorphism η : A1∞ → A∞ intertwining the corresponding coproducts, which extends to a ∗ -isomorphism of the completions.
We now give two constructions of subspaces K u ⊂ Hu satisfying our equations making no mention of a compact quantum group, and therefore not motivated by invariant subspaces. Let T be a tensor C ∗ -subcategory with conjugates of a tensor category of Hilbert spaces. The objects of T will be denoted by u, v, . . . as in the category of finite dimensional representations of a compact quantum group whilst the arrows will be written for example as T ∈ (u, v). We suppose that T comes equipped with a permutation symmetry ε. We now define K u = {ϕ ∈ Hu : ε(u, v)ϕ ⊗ 1v = 1v ⊗ ϕ for all objects v}. Note that the defining condition can be replaced by ε(v, u)1v ⊗ ϕ = ϕ ⊗ 1v for all objects v. But more is true: we have ε(u, v)ϕ ⊗ ψ = ψ ⊗ ϕ provided either ϕ ∈ K u and ψ ∈ Hv or ϕ ∈ Hu and ψ ∈ K v . We check the validity of our Eqs. (5.1), (5.2), (5.7), (5.4) for this definition of K . Equation (5.1) is true by definition. If T ∈ (u, v) and ϕ ∈ K u then
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ε(v, w)T ϕ ⊗ 1w = 1w ⊗ T ε(u, w)ϕ ⊗ 1w = 1w ⊗ T ϕ. Thus (u, v)K u ⊂ K v , which is (5.2). If ϕ ∈ K u and ψ ∈ K v then ε(u ⊗ v, w)ϕ ⊗ ψ ⊗ 1w = ε(u, w) ⊗ 1v 1u ⊗ ε(v, w)ϕ ⊗ ψ ⊗ 1w = 1w ⊗ ϕ ⊗ ψ. ¯ Thus K u ⊗ K v ⊂ K u⊗v , which is (5.4). Let R ∈ (ι, u¯ ⊗ u) and R ∈ (ι, u ⊗ u) solve the conjugate equations and define the corresponding antilinear operator j by: jϕ := rϕ∗ ◦ R = 1u¯ ⊗ ϕ ∗ R. If ϕ ∈ K u then jϕ ⊗ 1v = 1u¯ ⊗ ϕ ∗ ⊗ 1v ◦ R ⊗ 1v = 1u¯ ⊗ ϕ ∗ ⊗ 1v ◦ ε(v, u¯ ⊗ u) ◦ 1v ⊗ R = 1u¯ ⊗ ϕ ∗ ⊗ 1v ◦ 1u¯ ⊗ ε(v, u) ◦ε(v, u) ¯ ⊗ 1u ◦ 1v ⊗ R = 1u¯ ⊗ 1v ⊗ ϕ ∗ ◦ ε(v, u) ¯ ⊗ 1u ◦ 1v ⊗ R. Acting on the left by ε(u, ¯ v), we get ¯ v) ⊗ 1u ◦ ε(v, u) ¯ ⊗ 1u ◦ 1v ⊗ R = 1v ⊗ jϕ. ε(u, ¯ v) jϕ ⊗ 1v = 1v ⊗ 1u¯ ⊗ ϕ ∗ ◦ ε(u, This proves (5.7). A further property, (5.8), is valid here. 1u ⊗ φ ⊗ 1z ◦ k ∈ K u⊗v⊗z , k ∈ K u⊗z , φ ∈ K v . In fact, ε(u ⊗ v ⊗ z, w)(1u ⊗ φ ⊗ 1z ◦ k) ⊗ 1w = ε(u ⊗ v, w) ⊗ 1z ◦ 1u⊗v ⊗ ε(z, w) ◦ 1u ⊗ φ ⊗ 1z ⊗ 1w ◦ k ⊗ 1w = ε(u ⊗ v, w) ⊗ 1z ◦ 1u ⊗ φ ⊗ 1w ⊗ 1z ◦ 1u ⊗ ε(z, w) ◦ k ⊗ 1w = 1w ⊗ 1u ⊗ φ ⊗ 1z ◦ ε(u, w) ⊗ 1z ◦ 1u ⊗ ε(z, w) ◦ k ⊗ 1w = 1w ⊗ 1u ⊗ φ ⊗ 1z ◦ ε(u ⊗ z, w)k ⊗ 1w = 1w ⊗ (1u ⊗ φ ⊗ 1z ◦ k), as required. A second example of defining subspaces K u ⊂ Hu is the following. Consider an inclusion A ⊂ F of C ∗ –algebras with unit and suppose that T is a tensor C ∗ –category of endomorphisms with conjugates of A, where each object ρ of the category is induced by a Hilbert space Hρ of support I in F. We suppose that A ∩ F = C then this Hilbert space is unique. We thus have an embedding of T in a category of Hilbert spaces in F and T can be regarded as a category of endomorphism of F. If B is a C ∗ –algebra with A ⊂ B ⊂ F and invariant under the objects of T, set K ρ := Hρ ∩ B. The K ρ are Hilbert spaces in B. In this case (5.1), (5.2) and (5.4) are obvious. Let R, R solve the conjugate equations for ρ. Note that jρ (ψ) := ρ(ψ)∗ R ∈ Hρ , where ψ ∈ Hρ . The inverse jρ of jρ is defined by jρ ψ := ρ(ψ)∗ R. If ψ ∈ K ρ then jρ ψ = ρ(ψ)∗ R ∈ B and (5.7) follows. If φ ∈ K σ and χ ∈ K ρ⊗σ , then ρ(φ ∗ )χ A = ρ(φ ∗ )ρσ (A)χ = ρ(A)ρ(φ ∗ )χ , for A ∈ A. Thus ρ(φ ∗ )χ ∈ Hρ ∩ B = K ρ , verifying (5.3). It follows similarly that ψ ∈ K ρ implies ψ ∗ χ ∈ K σ and that ξ ∈ K ρ⊗τ and φ ∈ K σ implies ρ(φ)ξ ∈ K ρ⊗σ ⊗τ which is (5.8). Finally, we make K into a functor from T to the category of Hilbert spaces by setting K T := T K ρ for T ∈ (ρ, σ ).
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In the last part of this section we set up a Galois correspondence between functors K : u → K u satisfying (5.1), (5.2), (5.7) and (5.8) above and closed bi-ideals J. Given K let K ⊥ be the closed ideal generated by the (u i j − δi j I ) for all i and all j ≤ m u and all corepresentations u, where m u is the Hilbert space dimension of K u , and we have chosen an orthonormal basis of Hu whose first m u vectors are an orthonormal basis of K u . Obviously, if K 1 ⊂ K 2 , then K 1⊥ ⊂ K 2⊥ . Given J, we define the functor J⊥ by ⊥ ⊥ ϕ ∈ J⊥ u if and only if (u − ι(u))ϕ ∈ Hu ⊗ J. Again, if J1 ⊂ J2 , then J1,u ⊂ J2,u . Given K and u, if j ≤ m u then (u − ι(u))ϕ j =
ϕi ⊗ (u i j − δi j I ) ∈ Hu ⊗ K ⊥ .
i
Thus K u ⊂ K u⊥⊥ . Given J, then if J ∈ J⊥⊥ , J is in the closed ideal generated by the (u i j − δi j I ) for all i, j ≤ m u and all u. But if j ≤ m u then uϕ j = i ϕi ⊗ (u i j − δi j I ) ∈ Hu ⊗J, so (u i j −δi j I ) ∈ J and J⊥⊥ ⊂ J. From these relations, the usual closure relations follow: K ⊥ = K ⊥⊥⊥ and J⊥ = J⊥⊥⊥ . Up to this point, relations (5.1), (5.2), (5.7) and (5.8) have not been used, but we now want to show that K u = K u⊥⊥ when, by Theorem 5.5, the K u are the spaces of invariant vectors for a unique maximal compact quantum subgroup G of G, where G = (A/J, ). We let π : A → A/J denote the canonical surjection. Now, by definition, ϕ ∈ J⊥ u if (u − ι(u))ϕ ∈ Hu ⊗ J. But then 1 ⊗ π u(ϕ) = ϕ ⊗ I so ϕ ∈ K u . Conversely, if ϕ ∈ K u , then 1 ⊗ π uϕ = 1 ⊗ π ι(u)ϕ. But ker(1 ⊗ π ) = Hu ⊗ J so ⊥ ⊥⊥ = J⊥⊥⊥ = J⊥ = K as required. ϕ ∈ J⊥ u u . Thus K u = Ju and K u u u 6. Functors Arising from Ergodic Actions Consider a nondegenerate action η : B → B ⊗ A of a compact quantum group G = (A, ) on a unital C ∗ -algebra B. If η is ergodic (Bη = CI ), much can be said about the spectrum of the action. Boca showed that any irreducible representation in the spectrum of η has a finite multiplicity mult(u) bounded above by the quantum dimension q–dim(u) of u [3]. [2] showed that mult(u) can be bigger than the Hilbert space dimension of u. For any unitary (not necessarily irreducible) representation u of G of finite dimension, consider the vector space L u of linear maps T : Hu → B intertwining u with η. For S, T ∈ L u , T (ei )S(ei )∗ is independent of the choice of orthonormal basis (ei ) of Hu , and defines an element of the fixed point algebra, which, though, reduces to the complex numbers. Thus L u is a Hilbert space with inner product: (S, T )I :=
T (ei )S(ei )∗ .
Proposition 6.1. The Hilbert space L u is non-trivial if and only if u contains a subrepresentation v ∈ sp(η). In particular, if u is irreducible then u ∈ sp(η) if and only if L u = 0. For any intertwiner A ∈ (u, v), let L A : L v → L u be the linear map defined by: L A (T ) = T ◦ A.
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It is easy to check that L is a contravariant ∗ -functor. We obtain a covariant functor passing to the dual space. Consider, for any u ∈ sp(η) another Hilbert space, L u , which, as a vector space, is the complex conjugate of L u , with inner product: (S, T ) :=
S(ei )T (ei )∗ = (T, S),
where (ei ) is an orthonormal basis of Hu . Identify the complex conjugate of L u with the dual of L u , and set, for A ∈ (u, v), L A : φ ∈ Lu → φ ◦ L A ∈ Lv. This is now a covariant ∗ -functor. The functors L and L are related by: L A (T ) = L A∗ (T ),
A ∈ (v, u), T ∈ L v .
Definition 6.2. The functor L is called the spectral functor associated with the ergodic action η : B → B⊗A, and L u the spectral subspace corresponding to the representation u. Theorem 6.3. The ∗ -functor L : Rep(G) → H is quasitensor. Therefore for any u ∈ Rep(G), L u is finite dimensional and dim(L u ) = dim(L u ). Proof. The space L ι is, by definition, the fixed point algebra, reducing to the complex numbers, so L ι = C = L ι , and (3.1) follows. If S ∈ L u , T ∈ L v then the map S ⊗ T : ψ ⊗ φ ∈ Hu ⊗ Hv → S(ψ)T (φ) intertwines u ⊗ v with η, so S ⊗ T ∈ L u⊗v . If S ∈ L u , T ∈ L v is another pair, and (ei ) and ( f j ) are orthonormal bases of Hu and Hv respectively, then (S ⊗ T , S ⊗ T )I =
S(ei )T ( f j )(S (ei )T ( f j ))∗ = (S , S)(T , T )I,
i, j
so the linear span of all S ⊗ T is just a copy of L u ⊗ L v sitting inside L u⊗v . Moreover, a straightforward computation shows that if A ∈ (u , u), B ∈ (v , v), S ∈ L u , T ∈ L v , then L A⊗B (S ⊗ T ) = L A (S) ⊗ L B (T ). Hence L A⊗B L u ⊗L v = L A ⊗ L B . Next we define an inclusion Su,v : L u ⊗ L v → L u⊗v by taking S ⊗ T to S ⊗ T , for S ∈ L u , T ∈ L v and hence S ⊗ T ∈ L u⊗v . As for L, one can check that this inclusion is an isometry from the tensor product Hilbert space L u ⊗ L v to the Hilbert space L u⊗v , and (3.2) follows. The relation between L and L on arrows shows that L A⊗B L ⊗L = L A ⊗ L B , and this shows (3.6). Equation (3.3) u
v
is obvious for L and L as well. Equation (3.4) follows as if R ∈ L u , S ∈ L v , T ∈ L z then (R ⊗ S) ⊗ T = R ⊗ (S ⊗ T ). It remains to show (3.5). We define, for any S ∈ L v , ˜ ) : Hu → B by T ∈ L u⊗v , a linear map S(T ˜ )(ψ) = S(T
k
T (ψ ⊗ f k )S( f k )∗ ,
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˜ ) ∈ Lu. independent of the choice of orthonormal basis ( f k ) of Hv . We check that S(T In fact, on an orthonormal basis (ψi ) of Hu , ˜ )(ψi ) = η( η ◦ S(T T (ψi ⊗ f k )S( f k )∗ ) = k
∗
η(T (ψi ⊗ f k ))η(S( f k )) =
k
(T (ψr ⊗ f s ) ⊗ u ri vsk )(S( f p ) ⊗ v pk )∗ =
r,s,k, p
T (ψr ⊗ f s )S( f p )∗ ⊗ u ri vsk v ∗pk =
r,s,k, p
T (ψr ⊗ f s )S( f s )∗ ⊗ u ri =
r,s
˜ )(ψr ) ⊗ u ri . S(T
r
We have thus defined a linear map: S˜ : L u⊗v → L u .
(6.1)
˜ ) ∈ L u . Consider the operator Conjugating S˜ gives a linear map Sˆ : T ∈ L u⊗v → S(T r (S) : L u → L u ⊗ L v of tensoring on the right by S and the previously defined isometric inclusion map Su,v : L u ⊗ L v → L u⊗v . We claim that Sˆ ∗ = Su,v ◦ r (S). In fact, for T ∈ L u⊗v , T ∈ L u , ˆ , T ) = (T , Sˆ ∗ T ) = ( ST ˜ ), T ) = ( S(T T (ψi ⊗ f j )S( f j )∗ T (ψi )∗ = i, j
(T , T
⊗ S) = (T , Su,v T ⊗ S) = (T , Su,v ◦ r (S)T ),
∗ , which implies and the claim is proved. Taking the adjoint, Sˆ = r (S)∗ ◦ Su,v ∗ ˆ Su,v = p r (S p ) S p , with (S p ) an orthonormal basis of L v . Replace u by a tensor product representation u ⊗ z, and compute, for T ∈ L u , T ∈ L z⊗v , ∗ ◦ Su,z⊗v (T ⊗ T ) = p r (S p ) ◦ Sˆp (T ⊗ T ) = Su⊗z,v ˜ p r (S p )( S p (T ⊗ T )).
Now for ψ ∈ Hu , φ ∈ Hz ,
S˜p (T ⊗ T )(ψ ⊗ φ) = k (T ⊗ T )(ψ ⊗ φ ⊗ f k )S p ( f k )∗ = T (ψ) k T (φ ⊗ f k )S p ( f k )∗ = T ⊗ S˜p (T )(ψ ⊗ φ),
so E u⊗z,v ◦ Su,z⊗v (T ⊗ T ) =
T ⊗ S˜p (T ) ⊗ S p ,
p
and therefore E u⊗z,v (ImageSu,z⊗v ) ⊂ ImageSu,z,v and the proof is complete.
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Let u be a conjugate of u defined by R ∈ (ι, u ⊗ u) and R ∈ (ι, u ⊗ u) solving the ˆ Rˆ of the conjugate equations for L conjugate equations. We shall identify the solution R, u
constructed in Theorem 3.7 relating it to the notion of quantum multiplicity introduced in [2]. We can write R = i jei ⊗ ei and R = k j −1 f k ⊗ f k , with j : Hu → Hu an antilinear invertible map and (ei ) and ( f k ) orthonormal bases of Hu and Hu , respectively. For T ∈ L u , the multiplet T (ei )∗ transforms like the complex conjugate invertible representation u ∗ , so the linear map T˜ := ψ ∈ Hu → T (ψ)∗ ∈ B, ψ ∈ Hu intertwines u ∗ with η. Consider the linear map Q : Hu ∗ = Hu → Hu obtained composing the (antiunitary) complex conjugation Hu → Hu with j ∗ −1 . Regard Hu as the Hilbert space for u ∗ . The condition that R be an intertwiner can be written: Q ∈ (u ∗ , u). Thus T˜ ◦ Q −1 : Hu → B intertwines u with η. We have therefore defined an antilinear invertible map J : L u → L u by J (T )(ψ) = T˜ ◦ Q −1 (ψ) = T ( j ∗ (ψ))∗ , with inverse J −1 : L u → L u , J −1 (S)(φ) = S( j ∗ −1 (φ))∗ , φ ∈ Hu . If we conjugate J with the antilinear invertible map T ∈ L u → T ∈ L u , we obtain an antilinear invertible J : L u → L u . If u is irreducible, the quantum multiplicity of u is defined in [2] by ∗
∗
q–mult(u)2 := Trace(J J )Trace((J J )−1 ).
Theorem 6.4. If u is irreducible, Rˆ =
J Ti ⊗ Ti ,
Rˆ =
J
−1
Sj ⊗ Sj,
where (Ti ) and (S j ) are orthonormal bases of L u and L u , respectively. Therefore q–mult(u) = d ˆ ˆ (L u ). R, R
∗ L , Proof. We need to show that for T ∈ L u , r (T )∗ ◦ Rˆ = J T . Recall that Rˆ = Su,u R so, with the notation of the previous theorem, ∗ ◦ L R = Tˆ ◦ L R = Tˆ (L R ). r (T )∗ ◦ Rˆ = r (T )∗ ◦ Su,u
In the last equation we have identified L R ∈ (L ι , L u⊗u ) with the element L R (1), with 1 ∈ L ι the intertwiner 1 ∈ Hι → I ∈ B. Now L R (1) = L R ∗ (1) = 1 ◦ R ∗ ,
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so Tˆ (L R ) = T˜ (1 ◦ R ∗ ). But 1 ◦ R ∗ ∈ L u⊗u takes ξ ∈ Hu ⊗ Hu to (R, ξ )I , hence T˜ (1 ◦ R ∗ ) ∈ L u is the map taking ψ ∈ Hu to (1 ◦ R ∗ )(ψ ⊗ ei )T (ei )∗ = (R, ψ ⊗ ei )T (ei )∗ =
So T˜ (1 ◦
i
i
( jei , ψ)T (ei ) = ( j ∗ ψ, ei )T (ei )∗ = T ( j ∗ ψ)∗ = J (T )(ψ). ∗
i
i
R∗)
= J (T ), and the proof is complete.
Consequently, we get the following result, proved in [2], in turn extending Boca’s result. Corollary 6.5. Let η be an ergodic nondegenerate action of a compact quantum group G on a unital C ∗ -algebra B. Then, for any irreducible representation u of G, mult(u) ≤ q–mult(u) ≤ q–dim(u). Furthermore q–mult(u) = q–dim(u) if and only if L R ∈ ImageSu,u and L R ∈ ImageSu,u for some (and hence any) solution (R, R) of the conjugate equations for u. Proof. Thanks to the previous theorem, we are stating that dim(L u ) ≤ d ˆ ˆ (L u ) ≤ d R,R (u), R, R
as follows from Corollary 3.8 applied to the functor L. In the last part of this section we relate the functor L associated with the G-action on a quantum quotient space to the functor K of invariant vectors.
Proposition 6.6. Let K be a compact quantum subgroup of a compact quantum group G. A unitary G-representation u contains a subrepresentation of sp(η K ) if and only if K u = 0. In particular, if u is irreducible, then u ∈ sp(η K ) if and only if K u = 0. In this case, mult(u) = dim(K u ). Proof. Proposition 2.5 shows that any unitary irreducible G-representation v for which K v = 0 lies in sp(η K ). If u is any representation with K u = 0 then u contains an irreducible subrepresentation v such that K v = 0. So v ∈ sp(η K ). Conversely, if u contains a spectral subrepresentation u and if z is an irreducible component of u , then by Proposition 2.1, z ∈ sp(η K ). By Prop. 2.5 K z = 0, so K u = 0.
Any ψ ∈ K u , yields a map Tψ : Hu → Aδsp defined by Tψ (φ) = u ψ,φ , which actually lies in L u , as on an orthonormal basis of Hu : η K ◦ Tψ (ei ) = (u ψ,ei ) = u ψ,ek ⊗ u ek ,ei = Tψ ⊗ ι ◦ u(ei ). k
Theorem 6.7. Let K be a compact quantum subgroup of G. The map Uu : ψ ∈ K u → Tψ ∈ L u is a quasitensor natural transformation from the functor u → K u to the functor u → L u and Uu is unitary whenever K u = 0.
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7. Multiplicity Maps in Ergodic Actions Let η : B → B ⊗ A be a nondegenerate ergodic action of a compact quantum group G = (A, ) on a unital C ∗ -algebra B. In this section we introduce multiplicity maps, studied in the group case, by A. Wassermann [23]. Consider the Hilbert space L u associated with the unitary representation u, and an orthonormal basis (Tk ) in L u . Let cu : Hu → L u ⊗ B denote the linear map defined by cu (φ) =
Tk ⊗ Tk (φ).
k
This map does not depend on the choice of orthonormal basis of L u . We shall call the map cu the multiplicity map of u in η. Proposition 7.1. The map cu is nonzero if and only if L u is nonzero, i.e. if and only if u contains a spectral subrepresentation. Proof. We need to show that cu = 0 implies L u = 0. Indeed, if L u were = 0 then for any orthonormal basis (T k ) of L u , Tk = 0. Since (Tk ) is a linear basis of L u , we must have L u = 0. A contradiction.
We can also represent cu as the rectangular matrix, still denoted by cu , whose k th row is (Tk (e1 ), . . . , Tk (ed )), with (ei ) an orthonormal basis of Hu , d = dim(u), k = 1, . . . , dim(L u ). Proposition 7.2. Let u contain a subrepresentation in the spectrum of η. Then the matrix cu satisfies a) cu cu∗ = I , b) each row of cu transforms like u, c) for any multiplet c transforming like u, c Pu = c, with Pu the domain projection of cu . Conversely, if cu ∈ M p,dim(u) (B) satisfies a)–c) then p = dim(L u ) and cu is of the form cu . Proof. Property b) is obviously satisfied, a) and c) follow as (Tk ) is an orthonormal basis of L u . We show uniqueness. If cu is as in the statement then the rows of cu lie in L u by b). Furthermore these rows must be an orthonormal basis of L u by a) and c). In particular, p = dim(L u ).
We next investigate the relationship between u → cu and the functor u → L u . Define general coefficients of cu : for φ ∈ L u , ψ ∈ Hu , set u cφ,ψ := ∗φ cu (ψ) ∈ B.
Proposition 7.3. The map cu : Hu → L u ⊗ B satisfies a) L A ⊗ I ◦ cu = cv ◦ A, A ∈ (u, v), u⊗v u u b) for φ ∈ L u , φ ∈ L v , ψ ∈ Hu , ψ ∈ Hv , cφ⊗φ ,ψ⊗ψ = cφ,ψ cφ ,ψ .
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Let K be a compact quantum subgroup of G and η K : Aδ → Aδ ⊗A the ergodic action defining the quantum quotient space K \G. We can then associate to any representation u of G the multiplicity map cuK . In the last part of Sect. 4 we have also defined, for any representation u, the map u K = E uK ⊗ I ◦ u : Hu → K u Aδsp . Now consider the natural unitary transformation Uu : K u → L u between the functors u → K u and u → L u defined in the previous section. Proposition 7.4. If U : K → L is the unitary natural transformation defined in Theorem 6.7, then for any representation u, Uu ⊗ ι ◦ u K = cuK . Proof. If (ψk ) is an orthonormal basis of K u , we can write u K = rest of the proof is now clear.
k
ψk ⊗ Tψk . The
Corollary 7.5. If (ψk ) and (e j ) are orthonormal bases of K u and Hu respectively, the matrix (u ψKk ,e j ) = (∗ψk u(e j )) satisfies properties a)–c) of Prop. 7.2. We next show the linear independence of the coefficients of the cu ’s for inequivalent irreducibles. The proof generalizes a result of Woronowicz [25] stating the linear independence of matrix coefficients of inequivalent irreducible representations of a compact matrix pseudogroup. Proposition 7.6. Let η : C → C A∞ be an action of the Hopf ∗ -algebra G ∞ on a unital ∗ -algebra C, and let S be a set of unitary, irreducible, pairwise inequivalent, representations of G in the spectrum of η. For each u ∈ S, let cu = (ciuj ) ∈ M pu ,dim(u) (C) satisfy a) and b) of Prop. 7.2. Then the set of matrix coefficients {ciuj , i = 1, . . . , pu , j = 1, . . . , dim(u), u ∈ S} is linearly independent. Proof. Let F be a finite subset of S. Consider the linear subspace of ⊕u∈F M pu ,dim(u) , M := {⊕u∈F cρu := ⊕u∈F (ρ(ciuj )), ρ ∈ C }, with C the dual of C as a vector space and also the subspace of ⊕u∈F Mdim(u) , B := {⊕u∈F u σ := ⊕u∈F (σ (u pq )), σ ∈ A∞ }. u , where ρ ∗ σ := Notice that M B ⊂ M, since for ρ ∈ C , σ ∈ A∞ , u ∈ S, cρu u σ = cρ∗σ ρ ⊗ σ ◦ η. On the other hand, by Lemma 4.8 in [25], B = ⊕u∈F Mdu . Thus, for each u0 , fixed u 0 ∈ F, choosing σ such that u σ is zero for u = u 0 and u 0σ is a matrix unit θh,k shows that for every ρ ∈ C , every u 0 ∈ F and every h, k = 1, . . . , dim(u 0 ) there 0 exists ρ ∈ C such that cρu is zero if u = u 0 and cρu is the matrix with identically 0
zero columns except for the k th one, which coincides with the h th column of cρu . Now assume that a finite linear combination in C vanishes: i, j,u λiuj ciuj = 0. Then for all ρ ∈ C , λiuj ρ (ciuj ) = 0. Choose F finite and large enough so that this sum runs over F. By the previous arguments, for every ρ ∈ C , every u 0 ∈ S and every choice of h, k, m u 0 u0 i=1 λi,k ρ(ci,h ) = 0. Thus 0 0 u u λi,k ci,h = 0, h, k = 1, . . . , du 0 . i
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Multiplying on the right by cu
0∗
h, p ,
0
summing up over h and using cu 0 cu∗0 = I gives
λup,k = 0 for all u 0 ∈ S, p = 1, . . . , pu 0 , k = 1, . . . , dim(u 0 ). Theorem 7.7. Let η : B → B ⊗ A be a nondegenerate, ergodic G-action of a compact quantum group on a unital C ∗ -algebra B. Let ηˆ be a complete set of unitary irreducible elements of sp(η). Associate with any u ∈ ηˆ a corresponding multiplicity map cu . Then the set of all matrix coefficients ˆ (Ti ) o.n.b. of L u , (e j ) o.n.b. of Hu } {∗T cu (e j ) = Ti (e j ), u ∈ η, i
is a linear basis for the dense spectral ∗ -subalgebra Bsp . 8. A Duality Theorem for Ergodic C ∗-Actions We have seen in Sect. 6 that an ergodic nondegenerate action η of a compact quantum group G on a unital C ∗ -algebra has an associated quasitensor ∗ -functor L : Rep(G) → H to the category of, necessarily finite dimensional, Hilbert spaces. In this section we shall conversely construct from a quasitensor ∗ -functor F an ergodic nondegenerate action having F as its spectral functor. Theorem 8.1. Let G = (A, ) be a compact quantum group and F : Rep(G) → H be a quasitensor ∗ -functor. Then there is a unital C ∗ -algebra BF with an ergodic nondegenerate G-action ηF : BF → BF ⊗ A and a quasitensor natural unitary transformation from the associated spectral functor L to F. The proof of the above theorem is inspired by the proof of the Tannaka–Krein duality theorem given by Woronowicz in [26]. A similar result is obtained in [2] for unitary fibre functors. Proof. We start by considering a complete set Fˆ of inequivalent, unitary, irreducible representations u ∈ Gˆ such that F(u) = 0, and form the algebraic direct sum CF = ⊕u∈Fˆ F(u) ⊗ Hu . ˆ orthonormal bases We shall make CF into a unital ∗ -algebra. Consider, for each u ∈ F, (Tk ) and (ei ) of F(u) and Hu respectively, form the orthonormal basis (Tk ⊗ ei ) of F(u) ⊗ Hu . The linear map Tk ⊗ (Tk ⊗ φ) ∈ F(u) ⊗ CF cu : φ ∈ Hu → k
is independent of the choice of orthonormal basis. Extend the definition of cu to all objects of Rep(G): first set cu = 0 if u ∈ Gˆ but F(u) = 0. For u ∈ Rep(G), define cu uniquely by: ˆ cu ◦ A = F(A) ⊗ I cv , v ∈ G,
A ∈ (v, u).
(8.1)
ˆ T ⊗φ ∈ Relation (8.1) can be easily shown to hold for u, v ∈ Rep(G). For u, v ∈ F, F(u) ⊗ Hu , T ⊗ φ ∈ F(v) ⊗ Hv , set (T ⊗ φ)(T ⊗ φ ) := ∗T ⊗T ◦ cu⊗v (φ ⊗ φ ).
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In this way CF becomes an associative algebra with identity I = 1 ⊗ 1 ∈ F(ι) ⊗ Hι . We next define the ∗ -involution on CF using the conjugate representation defined, up to unitary equivalence, by solutions R ∈ (ι, u ⊗ u) and R ∈ (ι, u ⊗ u) of the conjugate ˆ of the conjugate equations for ˆ R) equations. Recall that (R, R) defines a solution ( R, F(u) as in Theorem 3.7. Writing R=
jei ⊗ ei ,
Rˆ =
J Tp ⊗ Tp ,
p
i
ˆ we set, for u ∈ F, ∗
(∗T cu (φ))∗ = ∗J T cu ( j −1 φ),
(8.2)
ˆ Since u is irreducible, the spaces where u denotes the unique conjugate of u in F. (ι, u ⊗ u) and (ι, u ⊗ u) are one dimensional, so any other solution of the conjugate equations is of the form (λR, µR) with λ, µ ∈ C and µλ = 1. Therefore the adjoint is independent of the choice of (R, R). Replacing u by u, and therefore R by R and Rˆ by ˆ and j and J in turn by their inverses, shows that R, (∗T cu (φ))∗∗ = ∗T cu (φ). We have thus defined an antilinear involutive map ∗ : CF → CF. Now let u ∈ Rep(G) be any representation, and u a conjugate of u defined by the pair (R, R), corresponding to j. If u 1 is another conjugate defined by R1 = X ⊗ 1u ◦ R, R1 = 1u ⊗ X ∗ −1 ◦ R, with X ∈ (u, u 1 ), then j becomes j1 = X j and, by (3.6), J becomes J1 = F(X )J . Therefore for φ ∈ Hu , T ∈ F(u), ∗
∗
∗J1 T cu 1 ( j1−1 φ) = ∗J T cu ( j −1 φ) by (8.1) applied to u and u 1 . Computing the adjoint of ∗T cu (φ) using (8.1) and the definition (8.2) of the adjoint in an irreducible representation, we find, see Lemma 8.2, that formula (8.2) still holds independently of the choice of conjugate of u. We next ˆ φ ∈ Hu , T ∈ F(u), φ ∈ Hv , T ∈ F(v), show that for u, v ∈ F, ((T¯ ⊗ φ)(T¯ ⊗ φ ))∗ = (T¯ ⊗ φ )∗ (T¯ ⊗ φ)∗ . The left hand side is the adjoint of ∗T ⊗T cu⊗v (φ ⊗ φ ), which equals ∗ −1 φ ⊗ φ ), ∗Ju⊗v T ⊗T cu⊗v ( ju⊗v
where ju⊗v defines a conjugate of u ⊗ v. If we choose a tensor product solution then u ⊗ v = v ⊗ u, ∗ −1 ju⊗v φ ⊗ φ = jv∗ −1 φ ⊗ ju∗ −1 φ
and, by Lemma 3.9, Ju⊗v T ⊗ T = Jv T ⊗ Ju T,
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and the antimultiplicativity of the adjoint follows. We now have a unital ∗ -algebra CF. Consider the linear map ηF : CF → CF A∞ ,
ˆ ηF(T ⊗ φ) = T ⊗ u(φ), T ∈ F(u), φ ∈ Hu , u ∈ F. It is easy to check that ηF is multiplicative and that ηF ⊗ ι ◦ ηF = ι ⊗ ◦ ηF . We show that ηF preserves the involutions. Recall that R ∈ (ι, u ⊗ u) is equivalent to Q ∈ (u ∗ , u), where, Q : ψ ∈ Hu → j ∗ −1 ψ ∈ Hu so ηF((∗T cu (ei ))∗ ) = ηF(∗J T cu (Qei )) = J T ⊗ u(Qei ) = J T ⊗ j ∗ −1 ek ⊗ u ∗ki = ηF(∗T cu (ei ))∗ . k
Consider the linear functional h on CF defined by h(I ) = 1 and ˆ u = ι. h(T ⊗ φ) = 0 T ∈ F(u), φ ∈ Hu , u ∈ F, We claim that h is a positive faithful state on CF, so CF has a C ∗ -norm. Consider the conditional expectation E onto the fixed point ∗ -subalgebra obtained averaging over the group action. Since the Haar measure of G annihilates the coefficients of all the irreducible representations u, except for u = ι, we see that E = h, and can conclude that ηF is an ergodic algebraic action. Evidently the maps, γT : ψ ∈ Hu → T ⊗ ψ ∈ CF, for T ∈ F(u) and u ∈ Fˆ span the Hilbert space L u associated with the algebraic ergodic ˆ space. Thus we have surjective linear maps, for u ∈ F, Vu : T ∈ F(u) → γT ∈ L u , ˆ We show that Vu is an isometry, and hence unitary. For S, T ∈ F(u), and an for u ∈ F. orthonormal basis (ψi ) of Hu , γT (ψi )γ S (ψi )∗ = (Vu (T ), Vu (S))I = i
(T ⊗ ψi )(S ⊗ ψi )∗ = (T ⊗ ψi )(J S ⊗ j ∗ −1 ψi ). i
i
Consider pairwise inequivalent irreducible representations u 1 , . . . u N of G and, for α = 1, . . . , N , isometries sα, j ∈ (u α , u ⊗ u), with ranges adding up to the identity. Then the last term above becomes ∗ ∗ −1 F(sα, j )∗ (T ⊗ J S) ⊗ sα, ψi ) = j (ψi ⊗ j i
α
j
α
j
∗ F(sα, j )∗ (T ⊗ J S) ⊗ sα, j Ru .
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∗ R ∈ (ι, u ), hence s ∗ R = 0 unless u = ι. In this case N = 1 and s Now sα, α α α, j j u α, j u −1 can be chosen to coincide with R u R u . So the last term above equals ∗ ∗ R u −2 F(R u )∗ (T ⊗ J S) ⊗ R u R u = Rˆu T ⊗ J S ⊗ 1ι =
(S, T )1ι ⊗ 1ι = (T, S)Iι . We extend V to a quasitensor natural transformation from F to L. First set Vu = 0 if u is irreducible but F(u) = 0. Then for any representation u ∈ Rep(G), consider pairwise inequivalent irreducible representations u α ∈ Fˆ and isometries sα, j ∈ (u α , u) decomposing u. Define a unitary map Vu : F(u) → L u by L sα, j Vu α F(sα, j )∗ . Vu = α
j
Vu is a natural transformation, obviously independent of the choice of isometries and as a consequence of the definition of multiplication in CF, Vu⊗v F(u)⊗F(v) = Vu ⊗ Vu , ˆ Routine computations show that V is a quasitensor unitary natural transfor u, v ∈ F. formation from F to L. Recalling the definition of the maps cu at the beginning of the proof, Vu ⊗ I cu (ψ) = γTk ⊗ γTk (ψ), k
ˆ Thus cu is the multiplicity map of the functor L in the representation for ψ ∈ Hu , u ∈ F. u. So CF is linearly spanned by entries of coisometries in matrix algebras over CF, the maps cu , and the maximal C ∗ -seminorm on CF is finite. Completing CF in the maximal C ∗ -seminorm yields a unital C ∗ -algebra BF, a nondegenerate ergodic G-action ηF : BF → BF ⊗ A,
with spectral functor L, and a natural unitary transformation from F to L. We now show the claimed statements Lemma 8.2. If S ∈ (u, v) and if ju : Hu → Hu , jv : Hv → Hv are antilinear invertible maps defining conjugates of u and v respectively, with corresponding antilinear invertible maps Ju : F(u) → F(u) Jv : F(v) → F(v), in the sense of Theorem 3.7, then F( jv S ju−1 ) = Jv F(S)Ju−1 . Proof. Let Ru , R u be the solution of the conjugate equations corresponding to ju , and, similarly, Rv , R v the solution corresponding to jv . We need to show that ∗ ∗ F(1v ⊗ R u ◦ 1v ⊗ S ∗ ⊗ 1u ◦ Rv ⊗ 1u ) = 1F(v) ⊗ Rˆu ◦ 1F(v) ⊗ F(S)∗ ⊗ 1F(u) ◦ Rˆv ⊗ 1F(u) .
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∗ by We shall use the simplified notation (3.7)–(3.11), replacing Su,v by the identity, Su,v E u,v , Rˆ by E u,u ◦ R and Rˆ by E u,u ◦ R. Properties (3.11) and (3.8) show that, regarding F(Rv ) as an element of F(v ⊗ v),
F(Rv ⊗ 1u ) = F(Rv ) ⊗ 1F(u) . We next show that ∗
F(1v ⊗ R u ) = 1F(v) ⊗ F(Ru )∗ ◦ E v,u⊗u . In fact, if ξ ∈ F(v ⊗ u ⊗ u) and η ∈ F(v), ∗
(η, F(1v ⊗ R u )ξ ) = (η, F(1v ⊗ R u )∗ ξ ) = (F(1v ⊗ R u )η, ξ ) = (η ⊗ F(R u ), ξ ) = (η ⊗ F(R u ), E v,u⊗u ξ ) = (η, 1F(v) ⊗ F(R u )∗ ◦ E v,u⊗u ξ ). Pick a vector ζ ∈ F(u). Then ∗
F(1v ⊗ R u ◦ 1v ⊗ S ∗ ⊗ 1u ◦ Rv ⊗ 1u )ζ = ∗
1F(v) ⊗ F(R u ) ◦ E v,u⊗u (F(1v ⊗ S)∗ (F(Rv )) ⊗ ζ ).
(8.3)
But thanks to (3.12), E v,u⊗u (F(1v ⊗ S)∗ (F(Rv )) ⊗ ζ ) = E v,u,u (F(1v ⊗ S)∗ (F(Rv )) ⊗ ζ ) = E v,u (F(1v ⊗ S)∗ F(Rv )) ⊗ ζ = E v,u ◦ F(1v ⊗ S)∗ (F(Rv )) ⊗ ζ = 1F(v) ⊗ F(S)∗ (E v,v ◦ F(Rv )) ⊗ ζ = (1F(v) ⊗ F(S)∗ ( Rˆv )) ⊗ ζ = 1F(v) ⊗ F(S)∗ ⊗ 1F(u) ◦ Rˆv ⊗ 1F(u) (ζ ) = 1F(v) ⊗ E u,u ◦ 1F(v) ⊗ F(S)∗ ⊗ 1F(u) ◦ Rˆv ⊗ 1F(u) (ζ ). Substituting back in (8.3) gives ∗ 1F(v) ⊗ Rˆ u ◦ 1F(v) ⊗ F(S)∗ ⊗ 1F(u) ◦ Rˆv ⊗ 1F(u) (ζ ).
Lemma 8.3. The linear functional h is a faithful state on the ∗ -algebra CF. ˆ pick T ∈ F(u), S ∈ F(v), φ ∈ Hu , ψ ∈ Hv . Choose isometries Proof. For u, v ∈ F, sα ∈ (u α , u ⊗ v), with u α irreducible and orthogonal ranges summing up to the identity. Then (T ⊗ φ)∗ (S ⊗ ψ) = ∗Ju T ⊗S ◦ cu⊗v ( ju∗ −1 φ ⊗ ψ) = ∗Ju T ⊗S ◦ F(sα ) ⊗ I ◦ cu α (sα∗ ju∗ −1 φ ⊗ ψ). α
Let A be the subset of all α with u α = ι. Then for α ∈ A, (ι, u ⊗ v) is always zero unless u = v. In this case, there is, up to a phase, a single sα , of the form λu Ru , with λu = Ru −1 , as (ι, u ⊗ u) is one–dimensional. Thus h((T ⊗ φ)∗ (S ⊗ ψ)) = δu,v |λu |2 (F(Ru )∗ Ju T ⊗ S)∗ Ru∗ ju∗ −1 φ ⊗ ψ) = ∗ δu,v |λu |2 ( Rˆu Ju T ⊗ S)∗ (Ru∗ ju∗ −1 φ ⊗ ψ)) = δu,v |λu |2 (Ju T, Ju S)(φ, ψ).
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If a ∈ CF is written in the form a = u∈F,i, j µiuj Tiu ⊗φ uj with F a finite set, Tiu ∈ F(u), φ uj ∈ Hu orthonormal bases, the previous computation gives h(a ∗ a) = Trace( ju∗ ju )−1 µiuj µruj (Ju Tiu , Ju Tru ) ≥ 0. If
h(a ∗ a)
=0
u∈F i,r, j
u u i µi j Ju Ti
= 0, for all j, u, hence µiuj = 0 for all i, j, u, so a = 0.
Remark. The above computations F(u) ⊗ Hu , up show that h acts, on each subspace to normalization, as h(a ∗ a) = i ϕu (θξi ), for a = ξi ⊗ φiu , where θξi is the rank one operator on F(u) defined by ξi and ϕu is the positive linear functional of B(F(u)) ˆ defined by ϕu (T ) = Rˆ ∗ 1F(u) ⊗ T R. Natural transformations induce ∗ -isomorphisms from (BF, ηF) to (BG, ηG). Proposition 8.4. Let F, G : Rep(G) → H be two quasitensor ∗ -functors and let U : F → G be a unitary quasitensor natural transformation. Then there is a unique ∗ -isomorphism α : B → B intertwining the corresponding G-actions, such that U F G ˆ αU (T ⊗ φ) = Uu T ⊗ φ, T ∈ F(u), φ ∈ Hu , u ∈ F. Proof. The formula obviously defines a linear multiplicative map αU commuting with the actions and one can check that G F Uu r T∗ ◦ E u,u = rU∗ u T ◦ E u,u ◦ Uu⊗u ,
for T ∈ F(u). The ∗ -invariance of αU follows from Uu JuF(T ) = JuGUu T, an easy consequence of unitarity, quasitensoriality and naturality of U . Thus αU extends to the completions in the maximal C ∗ -seminorms.
9. Applications to Abstract Duality Theory As a first application of Theorem 8.1, consider a tensor C ∗ -category T and a faithful tensor ∗ -functor ρ : Rep(G) → T from the representation category of a compact quantum group G to T. Then, as shown in Example 3.5, one has a quasitensor ∗ -functor Fρ : Rep(G) → H associating to any representation u of G the Hilbert space (ι, ρu ). Applying Theorem 8.1 to Fρ we obtain an ergodic G–space canonically associated with the inclusion ρ. We summarize this in the following theorem. Theorem 9.1. Let ρ : Rep(G) → T be a tensor ∗ -functor from the representation category of a compact quantum group G to an abstract tensor C ∗ -category T. Then there is a canonically associated ergodic nondegenerate action of G on a unital C ∗ -algebra B whose spectral functor can be identified with Fρ . The above G–space plays a central role in abstract duality theory for compact quantum groups. This matter will be developed elsewhere [6]. Here we note some consequences of the previous theorem. The notion of permutation symmetry for an abstract tensor C ∗ -category has been introduced in [5]. This is defined assigning a unitary intertwiner ε(ρ, σ ) ∈ (ρ ⊗σ, σ ⊗ρ) for any pair of objects ρ, σ in T satisfying suitable properties. If G is a compact group, Rep(G) has permutation symmetry defined by the intertwiners ϑu,v ∈ (u ⊗ v, v ⊗ u) exchanging the order of factors in the tensor product. The previous theorem allows us to recover the following result.
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Theorem 9.2. If G is a compact group, T a tensor C ∗ -category with permutation symmetry ε and ρ : Rep(G) → T a tensor ∗ -functor with ρ(ϑu,v ) = ε(ρu , ρv ) then there is a compact subgroup K of G, unique up to conjugation, and an isomorphism of the G–ergodic system associated with ρ and the ergodic C ∗ -system induced by the homogeneous space K \G over G. Proof. We show that the C ∗ -algebra associated with the quasitensor ∗ -functor u → (ι, ρu ) is commutative. This suffices as any ergodic action of a compact group on a commutative C ∗ -algebra arises from the transitive G-action on a quotient space K \G by a point stabilizer subgroup, unique up to conjugation. We need to show that if u and v are irreducible then for k ∈ (ι, ρu ), k ∈ (ι, ρv ), ψ ∈ Hu , ψ ∈ Hv , then k ⊗ ψ and k ⊗ ψ commute, as these elements span a dense ∗ -subalgebra. Choose inequivalent representations (u α ) of G and isometries sα, j ∈ (u α , u ⊗ v) irreducible ∗ =1 such that α j sα, j sα, u⊗v . Thus j (k ⊗ ψ)(k ⊗ ψ ) =
α
∗ ∗ ρ(s α, j )(k ⊗ 1ρv ◦ k ) ⊗ sα, j (ψ ⊗ ψ ).
(9.1)
j
Now ϑv,u ψ ⊗ ψ = ψ ⊗ ψ and ε(ρv , ρu )(k ⊗ 1ρu ◦ k) = ε(ρv , ρu )(k ⊗ k) = k ⊗ k = k ⊗ 1ρv ◦ k . Thus the right-hand side of (9.1) can be written α
∗ ∗ ρ(s α, j )(ε(ρv , ρu )k ⊗ 1ρu ◦ k) ⊗ sα, j ϑv,u (ψ ⊗ ψ).
(9.2)
j
On the other hand ε(ρu , ρv ) ◦ ρ(s α, j ) = ρ(ϑu,v ) ◦ ρ(s α, j ), and this is the map taking an element ξ ∈ (ι, ρα ) to the element ρ(ϑu,v ) ◦ ρ(sα, j ) ◦ ξ = ρ(ϑu,v ◦ sα, j ) ◦ ξ ∈ (ι, ρv ⊗ ρu ). Thus ε(ρu , ρv ) ◦ ρ(s ◦ sα, j ). Set tα, j := ϑu,v ◦ sα, j ∈ (u α , v ⊗ u). Then α, j ) = ρ(ϑu,v (9.2) equals α
∗ ∗ ρ(t α, j )(k ⊗ 1ρu ◦ k) ⊗ tα, j (ψ ⊗ ψ).
j
Since the isometries tα, j := ϑu,v ◦ sα, j ∈ (u α , v ⊗ u) give an orthogonal decomposition of v ⊗ u into irreducibles, the last term above equals (k ⊗ ψ )(k ⊗ ψ), and the proof is complete.
Recall that a q–Hecke symmetry for an object ρ in a tensor C ∗ -category T is given by representations εn : Hn (q) → (ρ ⊗n, ρ ⊗n ) of the Hecke algebras Hn (q) interchanging the homomorphism σ : Hn (q) → Hn+1 (q) that takes gi of Hn (q) to gi+1 with left tensoring by 1ρ [14]. Also recall that an object ρ
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of T is called µ–special of dimension d if there is a µ2 –Hecke symmetry for ρ and an intertwiner R ∈ (ι, ρ ⊗d ) for some d ≥ 2, satisfying R ∗ R = d!q , R∗
− 1)!q (−µ)d−1 1ρ ,
⊗ 1ρ ◦ 1ρ ⊗ R = (d R R ∗ = εd (Ad ), ε(g1 . . . gd )R ⊗ 1ρ = −(−µ)d−1 1ρ ⊗ R,
(9.3) (9.4) (9.5) (9.6)
where q := µ2 , g1 , . . . , gn−1 are the generators of the Hecke algebra Hn (q) and Ad is the totally symmetric projection, as defined in [14]. If T admits a special object of dimension d for some µ > 0, Theorem 6.2 in [14] then assures the existence of a tensor ∗ -functor Rep(Sµ U (d)) → T taking the fundamental representation u of Sµ U (d) to ρ and the canonical intertwiner S ∈ (ι, u ⊗d ) to R, where u is the fundamental representation of Sµ U (d). Under this isomorphism the Jimbo representation of the Hecke symmetry [11] corresponds to ε(ρ, ρ). We are thus in a position to apply Theorem 9.1. Theorem 9.3. Let ρ be a µ–special object of a tensor C ∗ -category T with dimension d ≥ 2 and parameter µ > 0. Then there is an ergodic nondegenerate action of Sµ U (d) on a unital C ∗ -algebra B whose spectral functor can be identified on the objects with u ⊗r → (ι, ρ r ), where u is the fundamental representation of Sµ U (d). When d = 2 the notion of a µ–special object of dimension 2 simplifies considerably. Proposition 9.4. If an object ρ of a tensor C ∗ -category T admits an intertwiner R ∈ (ι, ρ ⊗2 ) satisfying relations R ∗ ◦ R = (1 + q)1ι , R ∗ ⊗ 1ρ ◦ 1ρ ⊗ R = −µ1ρ ,
(9.7) (9.8)
with µ > 0 and q = µ2 , then ρ can be made uniquely into a µ–special object of dimension 2 with intertwiner R. Proof. If ε is any µ2 –Hecke symmetry for ρ making ρ into a µ–special object of dimension 2 with intertwiner R, then (9.5) shows that R R ∗ = ε2 (A2 ). Since A2 = 1+g1 , ε2 (g1 ) = R R ∗ − 1ρ ⊗2 . Therefore for all n, i = 1, . . . , n − 1, 1ρ ⊗i−1
εn (gi ) = εn (σ i−1 (g1 )) = ⊗ ε2 (g1 ) = 1ρ ⊗i−1 ⊗ (R R ∗ − 1ρ ⊗2 ),
and the symmetry is uniquely determined. The existence of a symmetry follows from that 1 formula: the orthogonal projection e = 1+q R R ∗ ∈ (ρ 2 , ρ 2 ) satisfies the Temperley–
Lieb relations [7] with parameter (q + q1 +2)−1 , thanks to (9.8). Since the Temperley–Lieb
algebra T L n ((q + q1 + 2)−1 ) is the quotient of the Hecke algebra Hn (q) by the ideal generated by A3 (see [7]), there is a Hecke symmetry for ρ with ε2 (g1 ) = (q + 1)e − 1ρ ⊗2 = R R ∗ − 1ρ ⊗2 . Since A2 = 1 + g1 , ε2 (A2 ) = (q + 1)e = R ◦ R ∗ , so (9.5) follows. We show (9.6) for d = 2: ε3 (g2 ) ◦ R ⊗ 1ρ = (1ρ ⊗ (R R ∗ − 1ρ ⊗2 )) ◦ R ⊗ 1ρ = −µ1ρ ⊗ R − R ⊗ 1ρ ,
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hence ε3 (g1 g2 ) ◦ R ⊗ 1ρ = ((R R ∗ − 1ρ ⊗2 ) ⊗ 1ρ ) ◦ (−µ1ρ ⊗ R − R ⊗ 1ρ ) =
µ2 R ⊗ 1ρ − (q + 1)R ⊗ 1ρ + µ1ρ ⊗ R + R ⊗ 1ρ = µ1ρ ⊗ R. Remark. There is a canonical isomorphism from Rep(Sµ U (2)) to Rep(Ao (F)). Relations (9.7) and (9.8) can be implemented in Hilbert spaces of dimension ≥ 2 as follows. Let j be any antilinear invertible map on a finite dimensional Hilbert space H , and set R = i jei ⊗ ei ∈ H ⊗2 . Then, for this R, (9.7) and (9.8) become, Trace( j ∗ j) = 1 + q,
j 2 = −µ.
Consider the involutive antiunitary map c of H acting trivially on the orthonormal basis ei , and set F = jc and F = cFc. Then the above conditions can be equivalently written Trace(F ∗ F) = 1 + q,
F F = −µ.
The maximal compact quantum group with representation category generated by R is the universal quantum group Ao (F) defined in [21]. Therefore the fundamental representation of Ao (F) is a µ–special object of dimension 2 in Rep(Ao (F)). Theorem 6.2 in [14] then shows that there is a unique isomorphism of tensor C ∗ -categories Rep(Sµ U (2)) → Rep(Ao (F)) taking the fundamental representation u of Sµ U (2) to the fundamental representation ofAo (F) and the quantum determinant S = ψ1 ⊗ ψ2 − µψ2 ⊗ ψ1 ∈ (ι, u ⊗2 ) to R = jei ⊗ ei . For related results, see Cor. 5.4 in [2], where the authors find similar necessary and sufficient conditions for the existence of a monoidal equivalence between a generic pair of universal compact quantum groups, and the result of Banica [1], where it is shown that the fusion rules of Ao (F) are the same as those of SU (2). Corollary 9.5. Let ρ be an object of a tensor C ∗ -category T with an intertwiner R ∈ (ι, ρ ⊗2 ) satisfying conditions (9.7) and (9.8). Then the unique tensor ∗ -functor Rep(Sµ U (2)) → T taking the fundamental representation u to ρ and the quantum determinant S = ψ1 ⊗ψ2 −µψ2 ⊗ψ1 ∈ (ι, u ⊗2 ) to R gives rise to an ergodic nondegenerate action of Sµ U (2) on a unital C ∗ -algebra B with spectral subspaces L u ⊗r = (ι, ρ r ). 10. Actions Embeddable into Quantum Quotient Spaces As a second application of the duality Theorem 8.1, consider the invariant vectors functor K associated with a compact quantum subgroup K of G. We know that this is just a copy of the spectral functor of the quotient space K \G. Theorem 10.1. Let K be a compact quantum subgroup of a maximal compact quantum group G. Then the nondegenerate ergodic G-system associated with the invariant vectors functor K is isomorphic to the quotient G–space K \G. Proof. The construction of the dense Hopf ∗ -algebra C K in Proposition 2.5 shows that C K is ∗ -isomorphic to the dense spectral subalgebra of Aδ , its G-action corresponding to the right G-action on the right coset space. Therefore we are left to show that the maximal C ∗ -seminorm ˙1 on C K = Aδsp coincides with the restriction of the maximal C ∗ -seminorm ˙ 2 on A∞ . Any Hilbert space representation of A∞ restricts to a
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Hilbert space representation on Aδsp , so ˙ 2 ≤ ˙ 1 . Conversely, if π is a Hilbert space representation of Aδsp , we can induce it up to a Hilbert space representation π˜ of A∞ via the conditional expectation m : A∞ → Aδsp obtained averaging over the action δ of the subgroup. There is an isometry V from the Hilbert space of π to the Hilbert space of π˜ such that V ∗ π˜ (a)V = π(m(a)), for a ∈ A∞ . In particular, if a ∈ Aδsp ,
π(a) = V ∗ π˜ (a)V , so π(a) ≤ π˜ (a) ≤ a2 , and a1 ≤ a2 . Definition 10.2. An ergodic G-action η : B → B ⊗ A will be called maximal if G is a maximal compact quantum group and if B is obtained completing the dense spectral ∗ -subalgebra with respect to the maximal C ∗ -seminorm. Combining Theorem 10.1, Theorem 5.5 and Prop. 8.4, gives the following characterization of maximal ergodic systems isomorphic to quotient spaces. Theorem 10.3. Let (B, η) be a maximal, nondegenerate, ergodic G-action. Given, for each unitary representation u of G, a subspace K u ⊂ Hu satisfying (5.1), (5.2), (5.7), (5.8) and a quasitensor natural unitary transformation from the spectral functor L associated with (B, η) to the functor K , then there exists a unique maximal compact quantum subgroup K of G such that (B, η) K \G. The last application concerns a functor K satisfying conditions (5.1)–(5.4), which are weaker than the conditions describing the invariant vectors functor. Theorem 10.4. Let G = (A, ) be a compact quantum group, and ζ : C → C ⊗ A a nondegenerate ergodic G-action on a unital C ∗ -algebra C with associated spectral functor L. Then the following properties are equivalent: a) Csp has a ∗ -character, b) there is a subfunctor K of the embedding functor H satisfying properties (5.1)–(5.4) and a quasitensor unitary natural transformation from L to K , c) there is a compact quantum subgroup K of G and a faithful ∗ -homomorphism φ : Csp → Aδ intertwining ζ with the G-action on the compact quantum quotient space K \G, d) there is a faithful ∗ -homomorphism φ : Csp → A intertwining ζ with the coproduct . Proof. We first show that a) implies b). Let χ be a ∗ -character of Csp , and define, for u ∈ Rep(G), the map ηu : L u → Hu by ηu (T ) =
χ (T (ei )∗ )ei ,
i
with ei an orthonormal basis of Hu . This map is an isometry, as (ηu (T ), ηu (T )) =
i
χ (T (ei )∗ )χ (T (ei )∗ ) = χ (
T (ei )T (ei )∗ ) = (T , T ).
i
Actually η is a quasitensor natural transformation from L to H , as for A ∈ (u, v), T ∈ Lu,
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ηv (L A (T )) = ηv (T A∗ ) =
χ (T (ei )∗ )( f j , Aei ) f j =
i, j
χ ((T A∗ ( f j ))∗ ) f j =
j
χ (T (ei )∗ )Aei = H A (ηu (T )),
i
and, for T ∈ L u , T ∈ L v , i, j
ηu⊗v (T ⊗ T ) = ηu⊗v (T ⊗ T ) = χ (T ⊗ T (ei ⊗ f j )∗ )ei ⊗ f j = χ ((T (ei )T ( f j ))∗ )ei ⊗ f j = i
∗
χ (T (ei ) )ei ⊗
i, j
χ (T ( f j )∗ ) f j = ηu (T ) ⊗ ηv (T ).
j
It follows that the functor K u := ηu (L u ), K A := A K u , for A ∈ (u, v) and u, v ∈ Rep(G) is a quasitensor ∗ -subfunctor of H . Therefore K satisfies properties (5.1), (5.2) and (5.4). We are left to show that (5.3) holds as well. Consider, for S ∈ L v , the map Sˆ : L u⊗v → L u defined in (6.1). A straightforward computation shows that ηu ◦ Sˆ = rη∗ (S) ◦ ηu⊗v . Therefore r K∗ v K u⊗v ⊂ ηu (L u ) = K u , and this is (5.3). We next show that v b) implies c). Thanks to Lemma 4.1, K is a quasitensor subfunctor of H . By Prop. 8.4 there is a faithful ∗ -homomorphism φ : C L → C K intertwining the G-actions. Now, C K , regarded as the ∗ -algebraic G-system defined by K , is ∗ -isomorphic to the ∗ -algebraic G-system defined by the linear span of the coefficients u k,ϕ , with k ∈ K u , ϕ ∈ Hu , u ∈ Rep(G), with the restricted G-action, thanks to the ∗ -algebraic structure of A recalled at the end of Subsect. 2.1. This is in turn a ∗ -subsystem of some quantum quotient space K \G, by Theorem 5.1. On the other hand the system (C L , η L ) is in turn ∗ -isomorphic to (C , ζ ), and the proof is now complete. The implication from c) to d) is sp obvious. We are left to show that d) implies a). The range of φ must be contained in the spectral ∗ -subalgebra of A because of the intertwining relation between the G-actions. Since Asp is the dense ∗ -subalgebra of A generated by the matrix coefficients, u ϕ,ψ , we can define a ∗ -character χ on Csp simply by composing φ with the counit e of G.
Remark. Under the assumptions of Theorem 10.4, if one has an everywhere defined ∗ -character on C and if the action ζ and the Haar measure of G are faithful, there is a faithful embedding of (C, ζ ) into a compact quantum quotient space K \G, as shown in Theorem 6.4 in [15]. Remark. Tomatsu has shown in [20] that, among the right invariant C ∗ -subalgebras of the Hopf C ∗ -algebra (A, ) of a coamenable compact quantum group G, quotient spaces are characterized by: 1) the property of being the range of a Haar invariant conditional expectation; 2) the property of being invariant under a natural action β of the discrete quantum group dual to G. On the other hand our results lead to a similar characterization, but from a categorical viewpoint: Theorem 10.4 shows that concrete quasitensor functors characterize the right invariant C ∗ -subalgebras of (A, ), and Theorem 10.3 shows that among them, quotient spaces are characterized by the extra condition (5.8). The two characterizations are of course related: assume that we start with a concrete quasitensor functor defined by projections E u : Hu → K u , with associated ergodic system (C, δ). Its dense ∗ -subsystem is naturally embedded in A (see Theorem 5.1). Then one has a densely defined, Haar invariant projection map E : Asp → Csp by E(u φ,ψ ) = u E u φ,ψ .
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Furthermore if we spell out Tomatsu’s property 2) in detail, it amounts to asking that ∗ v cv ∈ C for all c ∈ C, and all irreducible unitary representations (v pq ) of G. Now is i ij if we specify this condition on the generators of C, c = k ⊗ ψ, with ψ ∈ Hu , k ∈ K u , we find that Tomatsu’s condition 2) reduces precisely to 1u ⊗ k ⊗ 1u ◦ Rv ∈ K u⊗v⊗u . This property clearly follows from (5.8), and is in fact all we need to prove Theorem 5.5. Acknowledgements. The authors are very grateful to S. Doplicher for many discussions during the preparation of the manuscript. C.P. would also like to thank S. Vaes for drawing our attention to [2]. This research was supported by the European network ‘Quantum Spaces - Noncommutative Geometry’ HPRN-CT-2002-00280.
References 1. Banica, T.: Le groupe quantique compact libre U (n). Commun. Math. Phys. 190, 143–172 (1997) 2. Bichon, J., De Rijdt, A., Vaes, S.: Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups. Commun. Math. Phys. 262, 703–728 (2006) 3. Boca, F.: Ergodic actions of compact matrix pseudgroups on C ∗ -algebras. Asterisque 232, 93–109 (1995) 4. Doplicher, S., Roberts, J.E.: Duals of compact Lie groups realized in the Cuntz algebras and their actions on C ∗ -algebras. J. Funct. Anal. 74, 96–120 (1987) 5. Doplicher, S., Roberts, J.E.: A new duality theory for compact groups. Invent. Math. 98, 157–218 (1989) 6. Doplicher, S., Pinzari, C., Roberts, J.E.: Work in progress 7. Goodman, F.M., de la Harpe, P., Jones, V.F.R.: Coxeter graphs and towers of algebras. MSRI Publications 14, New York: Springer-Verlag, 1989 8. Ghez, P., Lima, R., Roberts, J.E.: The spectral category and the Connes invariant . J. Operator Th. 14, 129–146 (1985) 9. Haag, R.: Local quantum physics. Fields, particles, algebras. Second edition. Berlin: Springer-Verlag, 1996 10. Høegh–Krohn, R., Landstad, M., Størmer, E.: Compact ergodic groups of automorphisms. Ann. of Math. 114, 137–149 (1981) 11. Jimbo, M.: A q–analogue of U (gl(N + 1)), Hecke algebras and the Yang–Baxter equation. Lett. Math. Phys. 10, 63–69 (1985) 12. Kirillov, A.A.: Elements of the theory of representations. Berlin–New York: Springer-Verlag, 1976 13. Longo, R., Roberts, J.E.: A theory of dimension. K –Theor 11, 103–159 (1997) 14. Pinzari, C.: The representation category of the Woronowicz compact quantum group Sµ U (d) as a braided tensor C ∗ -category. Int. J. Math. 18, 113–136 (2007) 15. Pinzari, C.: Embedding ergodic actions of compact quantum groups on C ∗ -algebras into quotient spaces. Int. J. Math. 18, 137–164 (2007) 16. Pinzari, C., Roberts, J.E.: Regular objects, multiplicative unitaries and conjugation. Int. J. Math. 13, 625–665 (2002) 17. Podles, P.: Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU (2) and S O(3) groups. Commun. Math. Phys. 170, 1–20 (1995) 18. Roberts, J.E.: Spontaneously broken gauge symmetries and superselection rules. In: International school of mathematical physics. Proceedings. Università di Camerino, 1974, G. Gallavotti (ed.), Camerino: Università di Camerino, 1976 19. Tomatsu, R.: Compact quantum ergodic systems. http://arxiv.org/list/math.OA/0412012, 2004 20. Tomatsu, R.: A characterization of right coideals of quotient type and its application to classification of Poisson boundaries. http://arxiv.org/list/math.OA/0611327, 2006 21. Van Daele, A., Wang, S.: Universal quantum groups. Int. J. Math. 7, 255–263 (1996) 22. Wang, S.: Ergodic actions of universal quantum groups on operator algebras. Commun. Math. Phys. 203, 481–498 (1999) 23. Wassermann, A.: Ergodic actions of compact groups on operator algebras. Ann. of Math. 130, 273–319 (1989) 24. Wenzl, H.: Hecke algebras of type An and subfactors. Invent. Math 92, 349–383 (1988) 25. Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111, 613–665 (1987) 26. Woronowicz, S.L.: Tannaka–Krein duality for compact matrix pseudogroups. Twisted SU (N ) groups. Invent. Math. 93, 35–76 (1988) 27. Woronowicz, S.L.: Compact quantum groups, Les Houches, 1995, Amsterdam: North Holland, 1998, pp 845–884 Communicated by Y. Kawahigashi
Commun. Math. Phys. 277, 423–437 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0367-3
Communications in
Mathematical Physics
Some Geometric Calculations on Wasserstein Space John Lott Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA. E-mail: [email protected] Received: 5 January 2007 / Accepted: 9 April 2007 Published online: 7 November 2007 – © Springer-Verlag 2007
Abstract: We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold. 1. Introduction If M is a smooth compact Riemannian manifold then the Wasserstein space P2 (M) is the space of Borel probability measures on M, equipped with the Wasserstein metric W2 . We refer to [21] for background information on Wasserstein spaces. The Wasserstein space originated in the study of optimal transport. It has had applications to PDE theory [16], metric geometry [8,19,20] and functional inequalities [9,17]. Otto showed that the heat flow on measures can be considered as a gradient flow on Wasserstein space [16]. In order to do this, he introduced a certain formal Riemannian metric on the Wasserstein space. This Riemannian metric has some remarkable properties. Using O’Neill’s theorem, Otto gave a formal argument that P2 (Rn ) has nonnegative sectional curvature. This was made rigorous in [8, Theorem A.8] and [19, Prop. 2.10] in the following sense: M has nonnegative sectional curvature if and only if the length space P2 (M) has nonnegative Alexandrov curvature. In this paper we study the Riemannian geometry of the Wasserstein space. In order to write meaningful expressions, we restrict ourselves to the subspace P ∞ (M) of absolutely continuous measures with a smooth positive density function. The space P ∞ (M) is a smooth infinite-dimensional manifold in the sense, for example, of [7]. The formal calculations that we perform can be considered as rigorous calculations on this smooth manifold, although we do not emphasize this point. In Sect. 3 we show that if c is a smooth immersed curve in P ∞ (M) then its length in P2 (M), in the sense of metric geometry, equals its Riemannian length as computed with Otto’s metric. In Sect. 4 we compute the Levi-Civita connection on P ∞ (M). We use it to derive the equation for parallel transport and the geodesic equation. This research was partially supported by NSF grant DMS-0604829.
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In Sect. 5 we compute the Riemannian curvature of P ∞ (M). The answer is relatively simple. As an application, if M has sectional curvatures bounded below by r ∈ R, one can ask whether P ∞ (M) necessarily has sectional curvatures bounded below by r . This turns out to be the case if and only if r = 0. There has been recent interest in doing Hamiltonian mechanics on the Wasserstein space of a symplectic manifold [1,4,5]. In Sect. 6 we briefly describe the Poisson geometry of P ∞ (M). We show that if M is a Poisson manifold then P ∞ (M) has a natural Poisson structure. We also show that if M is symplectic then the symplectic leaves of the Poisson structure on P ∞ (M) are the orbits of the group of Hamiltonian diffeomorphisms, thereby making contact with [1,5]. This approach is not really new; closely related results, with applications to PDEs, were obtained quite a while ago by Alan Weinstein and collaborators [10,11,22]. However, it may be worth advertising this viewpoint.
2. Manifolds of Measures In what follows, we use the Einstein summation convention freely. Let M be a smooth connected closed Riemannian manifold of positive dimension. We denote the Riemannian density by dvol M . Let P2 (M) denote the space of Borel probability measures on M, equipped with the Wasserstein metric W2 . For relevant results about optimal transport and the Wasserstein metric, we refer to [8, Sects. 1 and 2] and references therein. Put P ∞ (M) = {ρ dvol M : ρ ∈ C ∞ (M), ρ > 0, ρ dvol M = 1}. (2.1) M
Then P ∞ (M) is a dense subset of P2 (M), as is the complement of P ∞ (M) in P2 (M). We do not claim that P ∞ (M) is necessarily a totally convex subset of P2 (M), i.e. that if µ0 , µ1 ∈ P ∞ (M) then the minimizing geodesic in P2 (M) joining them necessarily lies in P ∞ (M). However, the absolutely continuous probability measures on M do form a totally convex subset of P2 (M) [12]. For the purposes of this paper, we give P ∞ (M) the smooth topology. (This differs from the subspace topology on P ∞ (M) coming from its inclusion in P2 (M).) Then P ∞ (M) has the structure of an infinite-dimensional smooth manifold in the sense of [7]. The formal calculations in this paper can be rigorously justified as being calculations on the smooth manifold P ∞ (M). However, we will not belabor this point. Given φ ∈ C ∞ (M), define Fφ ∈ C ∞ (P ∞ (M)) by Fφ (ρ dvol M ) =
φ ρ dvol M .
(2.2)
M
This gives an injection P ∞ (M) → (C ∞ (M))∗ , i.e. the functions Fφ separate points in P ∞ (M). We will think of the functions Fφ as “coordinates” on P ∞ (M). Given φ ∈ C ∞ (M), define a vector field Vφ on P ∞ (M) by saying that for F ∈ C ∞ (P ∞ (M)), (Vφ F)(ρ dvol M ) =
d =0 F ρ dvol M − ∇ i (ρ∇i φ) dvol M . d
(2.3)
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The map φ → Vφ passes to an isomorphism C ∞ (M)/R → Tρ dvol M P ∞ (M). This parametrization of Tρ dvol M P ∞ (M) goes back to Otto’s paper [16]; see [2] for further discussion. Otto’s Riemannian metric on P ∞ (M) is given [16] by Vφ1 , Vφ2 (ρ dvol M ) = ∇φ1 , ∇φ2 ρ dvol M M = − φ1 ∇ i (ρ∇i φ2 ) dvol M . (2.4) M
In view of (2.3), we write δVφ ρ = Then Vφ1 , Vφ2 (ρ dvol M ) = φ1 δVφ2 ρ dvol M = φ2 δVφ1 ρ dvol M . − ∇ i (ρ∇i φ).
M
(2.5)
M
In terms of the weighted L 2 -spaces L 2 (M, ρ dvol M ) and 1L 2 (M, ρ dvol M ), let d be the usual differential on functions and let dρ∗ be its formal adjoint. Then (2.4) can be written as dφ1 , dφ2 ρ dvol M = φ1 dρ∗ dφ2 ρ dvol M . (2.6) Vφ1 , Vφ2 (ρ dvol M ) = M
M
We now relate the function Fφ and the vector field Vφ . Lemma 1. The gradient of Fφ is Vφ . Proof. Letting ∇ Fφ denote the gradient of Fφ , for all φ ∈ C ∞ (M) we have φ ∇ i (ρ∇i φ ) dvol M ∇ Fφ , Vφ (ρ dvol M ) = (Vφ Fφ )(ρ dvol M ) = − M
= Vφ , Vφ (ρ dvol M ). This proves the lemma.
(2.7)
3. Lengths of Curves In this section we relate the Riemannian metric (2.4) to the Wasserstein metric. One such relation was given in [17], where it was heuristically shown that the geodesic distance coming from (2.4) equals the Wasserstein metric. To give a rigorous relation, we recall that a curve c : [0, 1] → P2 (M) has a length given by L(c) = sup
J
sup
J ∈N 0=t0 ≤t1 ≤...≤t J =1 j=1
W2 c(t j−1 ), c(t j ) .
(3.1)
From the triangle inequality, the expression Jj=1 W2 c(t j−1 ), c(t j ) is nondecreasing under a refinement of the partition 0 = t0 ≤ t1 ≤ . . . ≤ t J = 1. If c : [0, 1] → P ∞ (M) is a smooth curve in P ∞ (M) then we write c(t) = ρ(t) dvol M ∂ρ i and let φ(t) satisfy dt = − ∇ (ρ∇i φ), where we normalize φ by requiring for example that M φ ρ dvol M = 0. If c is immersed then ∇φ(t) = 0. The Riemannian length of c, as computed using (2.4), is 1 1 1 2 1 2 2 c (t), c (t) dt = |∇φ(t)| (m) ρ(t) dvol M dt. (3.2) 0
0
M
The next proposition says that this equals the length of c in the metric sense.
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Proposition 1. If c : [0, 1] → P ∞ (M) is a smooth immersed curve then its length L(c) in the Wasserstein space P2 (M) satisfies 1 1 L(c) = c (t), c (t) 2 dt. (3.3) 0
Proof. We can parametrize c so that M |∇φ(t)|2 ρ(t) dvol M is a constant C > 0 with respect to t. Let {St }t∈[0,1] be the one-parameter family of diffeomorphisms of M given by ∂ St (m) = (∇φ(t))(St (m)) ∂t
(3.4)
with S0 (m) = m. Then c(t) = (St )∗ (ρ(0) dvol M ). Given a partition 0 = t0 ≤ t1 ≤ . . . ≤ t J = 1 of [0, 1], a particular transference plan . Then from c(t j−1 ) to c(t j ) comes from the Monge transport St j ◦ St−1 j−1 2 d(m, St j (St−1 (m)))2 ρ(t j−1 ) dvol M W2 c(t j−1 ), c(t j ) ≤ j−1 M = d(St j−1 (m), St j (m))2 ρ(0) dvol M M
≤ M
2
tj
|∇φ(t)|(St (m)) dt
t j−1
≤ (t j − t j−1 )
tj
|∇φ(t)|2 (St (m)) dt ρ(0) dvol M
M t j−1 tj
= (t j − t j−1 )
t j−1
ρ(0) dvol M
|∇φ(t)|2 (m) ρ(t) dvol M dt,
(3.5)
M
so 1 W2 c(t j−1 ), c(t j ) ≤ (t j − t j−1 ) 2
tj
t j−1
= (t j − t j−1 ) M
1
2
|∇φ(t)| (m) ρ(t) dvol M dt 2
M
|∇φ(t j )|2 (m) ρ(t j ) dvol M
1 2
(3.6)
for some t j ∈ [t j−1 , t j ]. It follows that
1
L(c) ≤
1
c (t), c (t) 2 dt.
(3.7)
0
Next, from [8, Lemma A.1], 2 (t j − t j−1 ) φ(t j−1 ) ρ(t j ) dvol M − φ(t j−1 ) ρ(t j−1 ) dvol M M M tj 2 ≤ W2 (c(t j−1 ), c(t j )) |∇φ(t j−1 )|2 dµt dt, t j−1
M
(3.8)
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where {µt }t∈[t j−1 ,t j ] is the Wasserstein geodesic between c(t j−1 ) and c(t j ). Now φ(t j−1 ) ρ(t j ) dvol M − φ(t j−1 ) ρ(t j−1 ) dvol M M M tj =− φ(t j−1 ) ∇ i (ρ(t)∇i φ(t)) dt dvol M =
M tj
t j−1
∇φ(t j−1 ), ∇φ(t) ρ(t) dvol M dt,
t j−1
(3.9)
M
so (3.8) becomes (t j − t j−1 )
2
tj
∇φ(t j−1 ), ∇φ(t) ρ(t) dvol M dt
t j−1
M
≤ W2 (c(t j−1 ), c(t j ))2
tj
t j−1
|∇φ(t j−1 )|2 dµt dt.
(3.10)
M
Thus tj
L(c) ≥
J j=1
t j−1
M ∇φ(t j−1 ),∇φ(t) ρ(t)
1 t j −t j−1
tj t j−1
dvol M dt
t j −t j−1 2 M |∇φ(t j−1 )| dµt dt tj t
(t j − t j−1 ).
M ∇φ(t j−1 ),∇φ(t) ρ(t)
(3.11)
dvol M dt
As the partition of [0, 1] becomes finer, the term j−1 t j −t j−1 uniformly approaches the constant C. The Wasserstein geodesic {µt }t∈[t j−1 ,t j ] has the form µt = (Ft )∗ µt j−1 for measurable maps Ft : M → M with Ft j−1 = Id [12]. Then tj 1 2 |∇φ(t j−1 )| dµt dt − C t j − t j−1 t j−1 M
tj 1 = |∇φ(t j−1 )|2 dµt − |∇φ(t j−1 )|2 dµt j−1 dt t j − t j−1 t j−1 M M tj 1 = |∇φ(t j−1 )|2 ◦ Ft − |∇φ(t j−1 )|2 dµt j−1 dt t j − t j−1 t j−1 M tj 1 ≤ ∇|∇φ(t j−1 )|2 ∞ d(m, Ft (m)) dµt j−1 (m) dt t j − t j−1 t j−1 M t j 1 ≤ ∇|∇φ(t j−1 )|2 ∞ d(m, Ft (m))2 dµt j−1 (m) dt t j − t j−1 t j−1 M tj 1 = ∇|∇φ(t j−1 )|2 ∞ W2 (µt j−1 , µt ) dt t j − t j−1 t j−1 ≤ ∇|∇φ(t j−1 )|2 ∞ W2 (c(t j−1 ), c(t j )).
(3.12)
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Now continuity of a 1-parameter family of smooth measures in the smooth topology implies continuity in the weak-∗ topology, which is metricized by W2 (as M is compact). It tj 2 follows that as the partition of [0, 1] becomes finer, the term t j −t1 j−1 t j−1 M |∇φ(t j−1 )| dµt dt uniformly approaches the constant C. Thus from (3.11), L(c) ≥
1 √ 1 C = c (t), c (t) 2 dt.
(3.13)
0
This proves the proposition.
Remark 1. Let X be a finite-dimensional Alexandrov space and let R be its set of nonsingular points. There is a continuous Riemannian metric g on R so that lengths of curves in R can be computed using g [15]. (Note that in general, R and X − R are dense in X .) This is somewhat similar to the situation for P ∞ (M) ⊂ P2 (M). In fact, there is an open dense subset O ⊂ X with a Lipschitz manifold structure and a Riemannian metric of bounded variation that extends g [18]. We do not know if there is a Riemannian manifold structure, in some appropriate sense, on an open dense subset of P2 (M). Other approaches to geometrizing P2 (M), with a view toward gradient flow, are in [2,3]; see also [14].
4. Levi-Civita Connection, Parallel Transport and Geodesics In this section we compute the Levi-Civita connection of P ∞ (M). We derive the formula for parallel transport in P ∞ (M) and the geodesic equation for P ∞ (M). We first compute commutators of our canonical vector fields {Vφ }φ∈C ∞ (M) . Lemma 2. Given φ1 , φ2 ∈ C ∞ (M), the commutator [Vφ1 , Vφ2 ] is given by [Vφ1 , Vφ2 ]F (ρ dvol M ) d = =0 F ρ dvol M − ∇i ρ (∇ i ∇ j φ2 )∇ j φ1 − (∇ i ∇ j φ1 )∇ j φ2 dvol M d (4.1) for F ∈ C ∞ (P ∞ (M)). Proof. We have [Vφ1 , Vφ2 ]F (ρ dvol M ) = Vφ1 (Vφ2 F) (ρ dvol M ) − Vφ2 (Vφ1 F) (ρ dvol M ) d = 1 =0 (Vφ2 F) ρ dvol M − 1 ∇ i (ρ∇i φ1 ) dvol M d1 d − 2 =0 (Vφ1 F) ρ dvol M − 2 ∇ i (ρ∇i φ2 ) dvol M d2 d d = 2 =0 F (ρ − 1 ∇ i (ρ∇i φ1 )) dvol M 1 =0 d1 d2 − 2 ∇ j ((ρ − 1 ∇ i (ρ∇i φ1 ))∇ j φ2 ) dvol M
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d d i 2 =0 1 =0 F (ρ − 2 ∇ (ρ∇i φ2 )) dvol M d2 d1 −1 ∇ j ((ρ − 2 ∇ i (ρ∇i φ2 ))∇ j φ1 ) dvol M d = =0 F ρ dvol M +∇ j (∇ i (ρ∇i φ1 )∇ j φ2 ) dvol M −∇ j (∇ i (ρ∇i φ2 )∇ j φ1 ) dvol M . d (4.2) One can check that ∇ j (∇ i (ρ∇i φ1 )∇ j φ2 ) − ∇ j (∇ i (ρ∇i φ2 )∇ j φ1 ) = − ∇i ρ (∇ i ∇ j φ2 )∇ j φ1 − (∇ i ∇ j φ1 )∇ j φ2 , from which the lemma follows.
(4.3)
We now compute the Levi-Civita connection. Proposition 2. The Levi-Civita connection ∇ of P ∞ (M) is given by d ((∇ Vφ1 Vφ2 )F)(ρ dvol M ) = =0 F ρ dvol M − ∇i ρ ∇ j φ1 ∇ i ∇ j φ2 dvol M d (4.4) for F ∈ C ∞ (P ∞ (M)). Proof. Define a vector field DVφ1 Vφ2 by d ((DVφ1 Vφ2 )F)(ρ dvol M ) = =0 F ρ dvol M − ∇i ρ ∇ j φ1 ∇ i ∇ j φ2 dvol M d (4.5) for F ∈ C ∞ (P ∞ (M)). We also write (4.6) δ DVφ Vφ2 ρ = − ∇i ρ ∇ j φ1 ∇ i ∇ j φ2 . 1
It is clear from Lemma 2 that DVφ1 Vφ2 − DVφ2 Vφ1 = [Vφ1 , Vφ2 ].
(4.7)
Next, Vφ1 Vφ2 , Vφ3 (ρ dvol M ) = − ∇ i φ2 ∇i φ3 ∇ j (ρ∇ j φ1 ) dvol M M = ∇ j φ1 ∇ i ∇ j φ2 ∇i φ3 ρ dvol M M + ∇ j φ1 ∇ i ∇ j φ3 ∇i φ2 ρ dvol M M = − φ3 ∇i (ρ ∇ j φ1 ∇ i ∇ j φ2 ) dvol M M − φ2 ∇i (ρ ∇ j φ1 ∇ i ∇ j φ3 ) dvol M M = φ3 δ DVφ Vφ2 ρ dvol M + φ2 δ DVφ M
1
M
1
Vφ3 ρ
dvol M
=DVφ1 Vφ2 , Vφ3 (ρ dvol M ) + Vφ2 , DVφ1 Vφ3 (ρ dvol M ). (4.8)
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J. Lott
Vφ1 Vφ2 , Vφ3 = DVφ1 Vφ2 , Vφ3 + Vφ2 , DVφ1 Vφ3 .
(4.9)
As 2∇ Vφ1 Vφ2 , Vφ3 = Vφ1 Vφ2 , Vφ3 + Vφ2 Vφ3 , Vφ1 − Vφ3 Vφ1 , Vφ2 + Vφ3 , [Vφ1 , Vφ2 ] − Vφ2 , [Vφ1 , Vφ3 ] − Vφ1 , [Vφ2 , Vφ3 ], (4.10) substituting (4.7) and (4.9) into the right-hand side of (4.10) shows that ∇ Vφ1 Vφ2 , Vφ3 = DVφ1 Vφ2 , Vφ3 for all φ3 ∈
C ∞ (M).
The proposition follows.
(4.11)
Lemma 3. The connection coefficients at ρ dvol M are given by ∇ Vφ1 Vφ2 , Vφ3 = ∇i φ1 ∇ j φ3 ∇ i ∇ j φ2 ρ dvol M .
(4.12)
M
Proof. This follows from (2.5) and (4.4).
Let G ρ be the Green’s operator for dρ∗ d on L 2 (M, ρ dvol M ). (More explicitly, if f ρ dvol M = 0 and φ = G ρ f then φ satisfies − ρ1 ∇ i (ρ∇i φ) = f and M M φ ρ dvol M = 0, while G ρ 1 = 0.) Let ρ denote orthogonal projection onto Im(d) ⊂ 1L 2 (M, ρ dvol M ). Lemma 4. At ρ dvol M , we have ∇ Vφ1 Vφ2 = Vφ , where φ = G ρ dρ∗ (∇i ∇ j φ2 ∇ j φ1 d x i ). Proof. Given φ3 ∈ C ∞ (M), we have Vφ3 , Vφ (ρ dvol M ) = dφ3 , dG ρ dρ∗ (∇i ∇ j φ2 ∇ j φ1 d x i ) ρ dvol M M = dφ3 , ρ (∇i ∇ j φ2 ∇ j φ1 d x i ) ρ dvol M M = dφ3 , ∇i ∇ j φ2 ∇ j φ1 d x i ρ dvol M M
= Vφ3 , ∇ Vφ1 Vφ2 (ρ dvol M ). The lemma follows.
(4.13)
To derive the equation for parallel transport, let c : (a, b) → P ∞ (M) be a smooth curve. As before, we write c(t) = ρ(t) dvol M and define φ(t) ∈ C ∞ (M), up to a ∞ constant, by dc dt = Vφ(t) . Let Vη(t) be a vector field along c, with η(t) ∈ C (M). If ∞ ∞ ∞ ∞ {φα }α=1 is a basis for C (M)/R then {Vφα }α=1 is a global basis for T P (M) and we can write η(t) = α ηα (t) Vφα c(t) . The condition for Vη to be parallel along c is dηα Vφα c(t) + ηα (t) ∇ Vφ(t) Vηα = 0, (4.14) dt α
or
α
V ∂η + ∇ Vφ(t) Vη(t) = 0. ∂t
c(t)
(4.15)
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Proposition 3. The equation for Vη to be parallel along c is
i ∂η i j ∇i ρ ∇ + ∇jφ ∇ ∇ η = 0. ∂t Proof. This follows from (2.3), (4.4) and (4.15).
(4.16)
As a check on Eq. (4.16), we show that parallel transport along c preserves the inner product. Lemma 5. If Vη1 and Vη2 are parallel vector fields along c then M ∇η1 , ∇η2 ρ dvol M is constant in t. Proof. We have ∂η1 ∂η2 d ∇i η2 ρ dvol M + ρ dvol M ∇η1 , ∇η2 ρ dvol M = ∇i ∇i η1 ∇ i dt M ∂t dt M M − ∇i η1 ∇ i η2 ∇ j (ρ∇ j φ) dvol M M ∂η1 ∂η2 = ∇i ∇i η1 ∇ i ∇i η2 ρ dvol M + ρ dvol M ∂t dt M M ∇ i ∇ j η1 ∇i η2 + ∇i η1 ∇ i ∇ j η2 ∇ j φ ρ dvol M +
M i ∂η1 i j dvol M + ∇ j φ ∇ ∇ η1 = − η2 ∇i ρ ∇ ∂t M
∂η2 dvol M + ∇ j φ ∇ i ∇ j η2 − η1 ∇i ρ ∇ i ∂t M = 0. (4.17) This proves the lemma.
Finally, we derive the geodesic equation. Proposition 4. The geodesic equation for c is 1 ∂φ + |∇φ|2 = 0, ∂t 2
(4.18)
modulo the addition of a spatially-constant function to φ. Proof. Taking η = φ in (4.16) gives
∂φ 1 ∇i ρ ∇ i = 0. + |∇φ|2 ∂t 2
(4.19)
1 2 Thus ∂φ ∂t + 2 |∇φ| is spatially constant. Redefining φ by adding to it a function of t alone, we can assume that (4.18) holds.
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Remark 2. Equation (4.18) has been known for a while, at least in the case of Rn , to be the formal equation for Wasserstein geodesics. For general Riemannian manifolds M, it was formally derived as the Wasserstein geodesic equation in [17] by minimizing lengths of curves. For t > 0, it has the Hopf-Lax solution
φ(t, m) = inf
m ∈M
φ(0, m ) +
d(m, m )2 2t
.
(4.20)
Given µ0 , µ1 ∈ P ∞ (M), it is known that there is a unique minimizing Wasserstein geodesic {µt }t∈[0,1] joining them. It is of the form µt = (Ft )∗ µ0 , where Ft ∈ Diff(M) is given by Ft (m) = expm (−t∇m φ0 ) for an appropriate Lipschitz function φ0 [12]. If φ0 happens to be smooth then defining ρ(t) by µt = ρ(t) dvol M and defining φ(t) ∈ C ∞ (M)/R as above, it is known that φ satisfies (4.18), with φ(0) = φ0 [21, Sect. 5.4.7]. In this way, (4.18) rigorously describes certain geodesics in the Wasserstein space P2 (M). 5. Curvature In this section we compute the Riemannian curvature tensor of P ∞ (M). Given φ, φ ∈ C ∞ (M), define Tφφ ∈ 1L 2 (M) by Tφφ = (I − ρ ) ∇ i φ ∇i ∇ j φ d x j .
(5.1)
(The left-hand side depends on ρ, but we suppress this for simplicity of notation.) Lemma 6. Tφφ + Tφ φ = 0. Proof. As ∇ i φ ∇i ∇ j φ d x j + ∇ i φ ∇i ∇ j φ d x j = d∇φ, ∇φ , and I − ρ projects away from Im(d), the lemma follows.
(5.2)
Theorem 1. Given φ1 , φ2 , φ3 , φ4 ∈ C ∞ (M), the Riemannian curvature operator R of P ∞ (M) is given by R(Vφ1 , Vφ2 )Vφ3 , Vφ4 = R(∇φ1 , ∇φ2 )∇φ3 , ∇φ4 ρ dvol M − 2Tφ1 φ2 , Tφ3 φ4 M
+ Tφ2 φ3 , Tφ1 φ4 − Tφ1 φ3 , Tφ2 φ4 ,
(5.3)
where both sides are evaluated at ρ dvol M ∈ P ∞ (M). Proof. We use the formula R(Vφ1 , Vφ2 )Vφ3 , Vφ4 = Vφ1 ∇ Vφ2 Vφ3 , Vφ4 − ∇ Vφ2 Vφ3 , ∇ Vφ1 Vφ4 − Vφ2 ∇ Vφ1 Vφ3 , Vφ4 + ∇ Vφ1 Vφ3 , ∇ Vφ2 Vφ4 − ∇ [Vφ1 ,Vφ2 ] Vφ3 , Vφ4 .
(5.4)
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First, from (2.3) and (3),
Vφ1 ∇ Vφ2 Vφ3 , Vφ4 = − ∇i φ2 ∇ j φ4 ∇ i ∇ j φ3 ∇ k (ρ∇k φ1 ) dvol M M = ∇ k ∇i φ2 ∇ j φ4 ∇ i ∇ j φ3 ∇k φ1 ρ dvol M M + ∇i φ2 ∇ k ∇ j φ4 ∇ i ∇ j φ3 ∇k φ1 ρ dvol M M + ∇i φ2 ∇ j φ4 ∇ k ∇ i ∇ j φ3 ∇k φ1 ρ dvol M .
(5.5)
M
Similarly, Vφ2 ∇ Vφ1 Vφ3 , Vφ4 =
∇ k ∇i φ1 ∇ j φ4 ∇ i ∇ j φ3 ∇k φ2 ρ dvol M
M
+ M
∇i φ1 ∇ k ∇ j φ4 ∇ i ∇ j φ3 ∇k φ2 ρ dvol M ∇i φ1 ∇ j φ4 ∇ k ∇ i ∇ j φ3 ∇k φ2 ρ dvol M .
+
(5.6)
M
Next, using (2.4), Lemma 4 and (5.1), ∇ Vφ2 Vφ3 , ∇ Vφ1 Vφ4 = dG ρ dρ∗ (∇i ∇ j φ3 ∇ j φ2 d x i ), dG ρ dρ∗ (∇k ∇l φ4 ∇ l φ1 d x k ) L 2 = ρ (∇i ∇ j φ3 ∇ j φ2 d x i ), ρ (∇k ∇l φ4 ∇ l φ1 d x k ) L 2 = ∇i ∇ j φ3 ∇ j φ2 d x i , ∇k ∇l φ4 ∇ l φ1 d x k L 2 − Tφ2 φ3 , Tφ1 φ4 = ∇i ∇ j φ3 ∇ j φ2 ∇ i ∇l φ4 ∇ l φ1 ρ dvol M − Tφ2 φ3 , Tφ1 φ4 . M
(5.7) Similarly, ∇ Vφ1 Vφ3 , ∇ Vφ2 Vφ4 =
M
∇i ∇ j φ3 ∇ j φ1 ∇ i ∇l φ4 ∇ l φ2 ρ dvol M −Tφ1 φ3 , Tφ2 φ4 . (5.8)
Finally, we compute ∇ [Vφ1 ,Vφ2 ] Vφ3 , Vφ4 . From (4.1), we can write [Vφ1 , Vφ2 ] = Vφ , where (5.9) φ = G ρ dρ∗ ∇i ∇ j φ2 ∇ j φ1 d x i − ∇i ∇ j φ1 ∇ j φ2 d x i . Then from (4.12),
∇ [Vφ1 ,Vφ2 ] Vφ3 , Vφ4 =
M
∇i φ ∇ j φ4 ∇ i ∇ j φ3 ρ dvol M = dφ, ∇ j φ4 ∇i ∇ j φ3 d x i L 2
= dG ρ dρ∗ (∇i ∇ j φ2 ∇ j φ1 d x i − ∇i ∇ j φ1 ∇ j φ2 d x i ), ∇ j φ4 ∇i ∇ j φ3 d x i L 2 = ρ ∇i ∇ j φ2 ∇ j φ1 d x i − ∇i ∇ j φ1 ∇ j φ2 d x i , ρ ∇ j φ4 ∇i ∇ j φ3 d x i L 2
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=
∇i ∇ j φ2 ∇ j φ1 − ∇i ∇ j φ1 ∇ j φ2 ∇k φ4 ∇ i ∇ k φ3 ρ dvol M M
− Tφ φ , Tφ4 φ3 + Tφ2 φ1 , Tφ4 φ3 1 2 = ∇i ∇ j φ2 ∇ j φ1 − ∇i ∇ j φ1 ∇ j φ2 ∇k φ4 ∇ i ∇ k φ3 ρ dvol M M
+ 2 Tφ1 φ2 , Tφ3 φ4 .
(5.10)
The theorem follows from combining Eqs. (5.4)-(5.10).
∞ 2 2 Corollary 1. Suppose that φ1 , φ2 ∈ C (M) satisfy M |∇φ1 | ρ dvol M = M |∇φ2 | ρ dvol M = 1 and M ∇φ1 , ∇φ2 ρ dvol M = 0. Then the sectional curvature at ρ dvol M ∈ P ∞ (M) of the 2-plane spanned by Vφ1 and Vφ2 is K (Vφ1 , Vφ2 ) = K (∇φ1 , ∇φ2 ) |∇φ1 |2 |∇φ2 |2 −∇φ1 , ∇φ2 2 ρ dvol M + 3|Tφ1 φ2 |2, M
(5.11) where K (∇φ1 , ∇φ2 ) denotes the sectional curvature of the 2-plane spanned by ∇φ1 and ∇φ2 . Corollary 2. If M has nonnegative sectional curvature then P ∞ (M) has nonnegative sectional curvature. Remark 3. One can ask whether the condition of M having sectional curvature bounded below by r ∈ R implies that P ∞ (M) has sectional curvature bounded below by r . This is not the case unless r = 0. The reason is one of normalizations. The normalizations on φ1 and φ2 are M |∇φ1 |2 ρ dvol M = M |∇φ2 |2 ρ dvol M = 1 and M ∇φ1 , ∇φ2 ρ dvol M = 0. One cannot conclude from this that M |∇φ1 |2 |∇φ2 |2 − ∇φ1 , ∇φ2 2 ρ dvol M is ≥ 1 or ≤ 1. More generally, if M has nonnegative sectional curvature then P2 (M) is an Alexandrov space with nonnegative curvature [8, Theorem A.8], [19, Prop. 2.10(iv)]. On the other hand, if M does not have nonnegative sectional curvature then one sees by an explicit construction that P2 (M) is not an Alexandrov space with curvature bounded below [19, Prop. 2.10(iv)]. Remark 4. The formula (5.3) has the structure of the O’Neill formula for the sectional curvature of the base space of a Riemannian submersion. In the case M = Rn , Otto argued that P ∞ (Rn ) is formally the quotient space of Diff(Rn ), with an L 2 -metric, by the subgroup that preserves a fixed volume form [16]. As Diff(Rn ) is formally flat, it followed that P ∞ (Rn ) formally had nonnegative sectional curvature. 6. Poisson Structure Let M be a smooth connected closed manifold. We do not give it a Riemannian metric. In this section we describe a natural Poisson structure on P ∞ (M) arising from a Poisson structure on M. If M is a symplectic manifold then we show that the symplectic leaves in P ∞ (M) are orbits of the action of the group Ham(M) of Hamiltonian diffeomorphisms acting on P ∞ (M). We recover the symplectic structure on the orbits that was considered in [1,5]. Let M be a smooth manifold and let p ∈ C ∞ (∧2 T M) be a skew bivector field. Given f 1 , f 2 ∈ C ∞ (M), one defines the Poisson bracket { f 1 , f 2 } ∈ C ∞ (M) by { f 1 , f 2 } =
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435
p(d f 1 ⊗ d f 2 ). There is a skew trivector field ∂ p ∈ C ∞ (∧3 T M) so that for f 1 , f 2 , f 3 ∈ C ∞ (M), (∂ p)(d f 1 , d f 2 , d f 3 ) = {{ f 1 , f 2 }, f 3 } + {{ f 2 , f 3 }, f 1 } + {{ f 3 , f 1 }, f 2 }.
(6.1)
One says that p defines a Poisson structure on M if ∂ p = 0. We assume hereafter that p is a Poisson structure on M. Definition 1 Define a skew bivector field P ∈ C ∞ (∧2 T P ∞ (M)) by saying that its Poisson bracket is {Fφ1 , Fφ2 } = F{φ1 ,φ2 } , i.e. {Fφ1 , Fφ2 }(µ) = {φ1 , φ2 } dµ (6.2) M
for µ ∈
P ∞ (M).
The map φ → d Fφ µ passes to an isomorphism C ∞ (M)/R → Tµ∗ P ∞ (M). As the right-hand side of (6.2) vanishes if φ1 or φ2 is constant, Eq. (6.2) does define an element of C ∞ (∧2 T P ∞ (M)). Proposition 5. P is a Poisson structure on P ∞ (M). Proof. It suffices to show that ∂ P vanishes. This follows from the equation (∂ P)(d Fφ1 , d Fφ2 , d Fφ3 ) = {{Fφ1 , Fφ2 }, Fφ3 }+{{Fφ2 , Fφ3 }, Fφ1 }+{{Fφ3 , Fφ1 }, Fφ2 } = F{{φ1 ,φ2 },φ3 } + {{φ2 ,φ3 },φ1 } + {{φ3 ,φ1 },φ2 } = 0.
(6.3)
A finite-dimensional Poisson manifold has a (possibly singular) foliation with symplectic leaves [6]. The leafwise tangent vector fields are spanned by the vector fields W f defined by W f h = { f, h}. The symplectic form on a leaf is given by saying that (W f , Wg ) = { f, g}. Suppose now that (M, ω) is a closed 2n-dimensional symplectic manifold. Let Ham(M) be the group of Hamiltonian symplectomorphisms of M [13, Chap. 3.1]. Proposition 6. The symplectic leaves of P ∞ (M) are the orbits of the action of Ham(M) φ1 , H φ2 ∈ Tµ P ∞ (M) be on P ∞ (M). Given µ ∈ P ∞ (M) and φ1 , φ2 ∈ C ∞ (M), let H the infinitesimal motions of µ under the flows generated by the Hamiltonian vector fields φ1 , H φ2 ) = Hφ1 , Hφ2 on M. Then ( H {φ , φ } dµ. 1 2 M d − {φ, ρ} ωn ) |=0 F(µ Proof. Write µ = ρ ωn . We claim that (W Fφ F)(µ) = d ∈ C ∞ (P ∞ (M)). To show this, it is enough to check it for each F = Fφ , with for F φ ∈ C ∞ (M). But (W Fφ Fφ )(µ) = F{φ,φ } (µ) = {φ, φ } ρ ωn = − φ {φ, ρ} ωn , (6.4) M
φ . from which the claim follows. This shows that W Fφ = H Next, at µ ∈ P ∞ (M) we have φ1 , H φ2 ) = (W Fφ , W Fφ ) = {Fφ1 , Fφ2 }(µ) = ( H 1 2 This proves the proposition.
M
{φ1 , φ2 } dµ. M
(6.5)
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φ2 Remark 5. As a check on Proposition 6, suppose that φ2 ∈ C ∞ (M) is such that H vanishes at µ = ρ ωn . Then {φ2 , ρ} = 0, so by our formula we have φ2 ) = φ1 , H {φ1 , φ2 } dµ = {φ1 , φ2 } ρ ωn = φ1 {φ2 , ρ} ωn = 0. (6.6) ( H M
M
M
Remark 6. The Poisson structure on P ∞ (M) is the restriction of the Poisson structure on (C ∞ (M))∗ considered in [10,11,22]. Here the Poisson structure on (C ∞ (M))∗ comes from the general construction of a Poisson structure on the dual of a Lie algebra, considering C ∞ (M) to be a Lie algebra with respect to the Poisson bracket on C ∞ (M). The cited papers use the Poisson structure on (C ∞ (M))∗ to show that certain PDE’s are Hamiltonian flows. Acknowledgements. I thank Wilfrid Gangbo, Tommaso Pacini and Alan Weinstein for telling me of their work. I thank Cédric Villani for helpful discussions and the referee for helpful remarks.
References 1. Ambrosio, L., Gangbo, W.: Hamiltonian ODE’s in the Wasserstein space of probability measures. to appear, Comm. Pure Applied Math., DOI: 10.1002/cpa.20188 http://www.math.gatech.edu/~gangbo/ publications/ 2. Ambrosio, L., Gigli, N., Savarè, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, Basel: Birkhaüser, 2005 3. Carrillo, J.A., McCann, R.J., Villani, C.: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rat. Mech. Anal. 179, 217–263 (2006) 4. Gangbo, W., Nguyen, T., Tudorascu, A.: Euler-Poisson systems as action-minimizing paths in the Wasserstein space. Preprint, 2006 5. Gangbo, W., Pacini, T.: Infinite dimensional Hamiltonian systems in terms of the Wasserstein distance. Work in progress 6. Kirillov, A.: Local Lie algebras. Usp Mat. Nauk 31, 57–76 (1976) 7. Kriegl, A., Michor, P.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs 53, Providence, RI: Amer. Math. Soc., 1997 8. Lott, J., Villani, C.: Ricci curvature for metric-measure space via optimal transport. To appear, Ann. of Math., available at http://www.arxiv.org/abs/math.DG/0412127, 2004 9. Lott, J., Villani, C.: Weak curvature conditions and functional inequalities. J. of Funct. Anal. 245, 311–333 (2007) 10. Marsden, J., Weinstein, A.: The Hamiltonian structure of the Maxwell-Vlasov equations. Phys. D 4, 394–406 (1982) 11. Marsden, J., Ratiu, T., Schmid, R., Spencer, R., Weinstein, A.: Hamiltonian systems with symmetry, coadjoint orbits and plasma physics. In: Proceedings of the IUTAM-ISIMM symposium on modern developments in analytical mechanics, vol. I (Torino, 1982), Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 117, suppl. 1, 289–340 (1983) 12. McCann, R.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11, 589–608 (2001) 13. McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Second edition, Oxford Mathematical Monographs, New York: The Clarendon Press, Oxford University Press, 1998 14. Ohta, S.: Gradient flows on Wasserstein spaces over compact Alexandrov spaces. Preprint, http://www. math.kyoto-u.ac.jp/~sohta/, 2006 15. Otsu, Y., Shioya, T.: The Riemannian structure of Alexandrov spaces. J. Diff. Geom. 39, 629–658 (1994) 16. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26, 101–174 (2001) 17. Otto, F., Villani, C.: Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000) 18. Perelman, G.: DC structure on Alexandrov space. Unpublished preprint 19. Sturm, K.-T.: On the geometry of metric measure spaces I. Acta Math. 196, 65–131 (2006) 20. Sturm, K.-T.: On the geometry of metric measure spaces II. Acta Math. 196, 133–177 (2006)
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21. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics 58, Providence, RI: Amer. Math. Soc., 2003 22. Weinstein, A.: Hamiltonian structure for drift waves and geostrophic flow. Phys. Fluids 26, 388–390 (1983) Communicated by P. Constantin
Commun. Math. Phys. 277, 439–458 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0374-4
Communications in
Mathematical Physics
Uniqueness and Examples of Compact Toric Sasaki-Einstein Metrics Koji Cho1 , Akito Futaki2 , Hajime Ono2 1 Department of Mathematics, Kyushu University, 6-10-1, Hakozaki, Higashiku, Fukuoka-City,
Fukuoka 812-8581, Japan. E-mail: [email protected]
2 Department of Mathematics, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro,
Tokyo 152-8551, Japan. E-mail: [email protected]; [email protected] Received: 11 January 2007 / Accepted: 15 May 2007 Published online: 7 November 2007 – © Springer-Verlag 2007
Abstract: In [11] it was proved that, given a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on S 5 k(S 2 × S 3 ) for each positive integer k. 1. Introduction In [11] the existence of an Einstein metric is proved on a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle D; these two conditions will be denoted by c1B > 0 and c1 (D) = 0. The purposes of the present paper are firstly to prove the uniqueness of Sasaki-Einstein metrics up to a connected Lie group action and secondly to clarify the meaning of the assumptions c1B > and c1 (D) = 0 in relation with toric diagrams. A Sasaki manifold is a Riemannian manifold (S, g) whose cone manifold (C(S), g) with C(S) ∼ = S × R+ and g = dr 2 + r 2 g is Kähler where r is the standard coordinate + on R . From this definition S is odd-dimensional and we put dim S = 2m + 1, and thus dimC C(S) = m + 1. A Sasaki manifold (S, g) is said to be toric if the Kähler cone manifold C(S) is toric, namely (m + 1)-dimensional torus G acts on (C(S), g) effectively as holomorphic isometries. Note that C(S) does not contain the apex. Then S is a contact manifold with the contact form η = (i(∂ − ∂) log r )|r =1 ,
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where S is identified with the submanifold {r = 1} ⊂ C(S), and has the Reeb vector field ξ with the defining properties i(ξ )η = 1 and i(ξ )dη = 0, where i(ξ ) denotes the inner product. The Reeb field ξ is a Killing vector field on S and also lifts to a Killing vector field on C(S), and thus ξ is contained in the Lie algebra g of G since G already has the maximal dimension of possible torus actions on C(S). The Reeb vector field ξ generates a 1-dimensional foliation, called the Reeb foliation, on S. Since ξ naturally lifts to a holomorphic vector field on C(S) in the form ∂ ) the Reeb foliation shares common local leaf spaces with the ξ − i J ξ with ξ = J (r ∂r holomorphic flow generated by ξ − i J ξ on C(S). Thus the local leaf spaces give the Reeb foliation a transverse holomorphic structure. The contact structure of S determines a Kähler structure on the transverse holomorphic structure, which we call the transverse Kähler structure. Recall that a smooth differential form α on S is basic if i(ξ )α = 0 and Lξ α = 0, where Lξ denotes the Lie derivative by ξ . The basic forms are preserved by the exterior derivative d which decomposes into d = ∂ B + ∂ B , and we can define basic cohomology groups and basic Dolbeault cohomology groups. We also have the transverse ChernWeil theory and can define basic Chern classes for complex vector bundles with basic transition functions. The Sasaki manifold is said to have positive basic first Chern class if the first Chern class of the normal bundle of the Reeb foliation is represented by a positive basic (1, 1)-form; as mentioned above this condition is denoted by c1B > 0. This is a necessary condition for the existence of Sasaki-Einstein metric or equivalently the existence of positive transverse Kähler-Einstein metric. There is another necessary condition c1 (D) = 0 as a de Rham cohomology class, where D = Ker η is the toric bundle. Conversely if c1B > 0 and c1 (D) = 0 then c1B = τ [dη] for some positive constant τ . See Proposition 4.3 in [11] for more details. Given a Sasaki manifold (S, g), we say that another Sasaki metric g is compatible with the Sasaki structure of (S, g) if g and g have the same Reeb vector field and thus define the same transverse holomorphic structure. The automorphism group of the transverse holomorphic structure is the group of all biholomorphic automorphisms of C(S) which commute with the holomorphic flow generated by ξ − i J ξ . Such automorphisms descend to an action on S preserving the transverse holomorphic structure of the Reeb foliation, see Sect. 2 for more detail. In this paper we first prove the uniqueness theorem of compatible Sasaki-Einstein metrics modulo connected group actions of automorphisms for the transverse holomorphic structure. Theorem 1.1. Let (S, g) be a compact toric Sasaki manifold with c1B > 0 and c1 (D) = 0. Then the identity component of the automorphism group for the transverse holomorphic structure acts transitively on the space of all Sasaki-Einstein metrics compatible with g. In order to make clear which Sasaki manifolds the result of [11] applies to, we wish to explain the conditions c1B > 0 and c1 (D) = 0. Since a three dimensional Einstein manifold of positive scalar curvature is finitely covered by the standard three sphere we may restrict ourselves to the case when the dimension of S is bigger than or equal to five.
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Theorem 1.2. Let S be a compact toric Sasaki manifold with dim S ≥ 5. Then the following three conditions are equivalent. (a) c1B > 0 and c1 (D) = 0. (b) The Sasaki manifold S is obtained from a toric diagram with height for some positive integer defined by λ1 , · · · , λd ∈ g and γ ∈ g∗ (cf. Definition 3.1 and 3.2) and the Reeb field ξ ∈ g satisfies γ , ξ = −m − 1 and y, ξ > 0 for all y ∈ C, where C = {y ∈ g∗ |y, λ j ≥ 0, j = 1, · · · , d}. ⊗ (c) For some positive integer , the th power K C(S) of the canonical line bundle K C(S) is trivial. Remark 1.3. We denote by C(S) the closure of the cone C(S), that is C(S) plus the apex, and consider it as an affine toric variety. It is a known fact that the condition of toric diagram with height is equivalent to the apex being a Q-Gorenstein singularity, that is the th power K⊗ of the canonical sheaf KC(S) is invertible, see [2]. C(S)
In the literature there are toric Sasaki manifolds denoted by Y p,q ([12,19]), L p,q,r ([9,19]), X p,q ([14]) and Z p,q ([24,3]) which are constructed from toric diagrams with height 1. They are all of positive basic first Chern class by Theorem 1.2, and thus admit a Sasaki-Einstein metric by [11]. Combining the existence result of [11] with Theorem 1.1 and Theorem 1.2 we get the following corollary. Corollary 1.4. Given a toric diagram, there is a unique Sasaki structure whose cone is the one obtained from the toric diagram by Delzant construction and on which there exist compatible Einstein metrics. Moreover the identity component of the automorphism group of the transverse holomorphic structure acts transitively on the set of all compatible Einstein metrics. Thus the Sasaki-Einstein metrics constructed in [11] on Y p,q coincide with those which have been known in the literature [12,19]. Using diagrams we show that compact connected toric Sasaki manifolds associated with toric diagrams of height bigger than 1 are not simply connected and that the converse is not true by giving an example. We will also show the following. Theorem 1.5. For each positive integer k there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on the k-fold connected sum S 5 k(S 2 × S 3 ) of S 2 × S 3 with S 5 . The existence of (possibly non-toric) Sasaki-Einstein metrics on S 5 k(S 2 × S 3 ) has been known by the works of Boyer, Galicki, Nakamaye and Kollár ([5,4,15]), and that the existence of toric Sasaki-Einstein metrics for all odd k’s has been known by van Coevering ([26]). Hence our result is new in that we obtain toric constructions for all even k’s. Moreover most of our examples should be irregular while the previous ones are all quasi-regular.
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2. Uniqueness of Compatible Sasaki-Einstein Metrics In Kähler geometry a well known method of proving uniqueness of constant scalar curvature metrics is to use geodesics on the space of all Kähler metrics in a fixed Kähler class ([18,10,8]). This idea becomes substantially simpler when the Kähler manifold under consideration is toric because the geodesic becomes a line segment expressed by the symplectic potentials ([13]). To prove Theorem 1.1 we wish to use the same idea, but have to consider geodesics both on the space of transverse Kähler metrics on S and on the space of Kähler metrics on C(S). We therefore give an outline of the idea in the case of compact Kähler manifolds first and then explain how we modify it in the Sasakian case. Let V be a compact Kähler manifold and H the space of Kähler potentials in a fixed Kähler class [ω0 ]: √ H = {ϕ ∈ C ∞ (V ) | ωϕ = ω0 + −1∂∂ϕ > 0}. The tangent space Tϕ H at ϕ ∈ H is identified with the set C ∞ (V ) of all real smooth functions via d (ϕ + sψ) = ψ ∈ C ∞ (V ). ds |s=0 We have a natural Riemannian metric on H, (ψ1 , ψ2 ) := ψ1 ψ2 ωϕn , M
where n = dimC V and ψ1 , ψ2 ∈ Tϕ H ∼ = C ∞ (V ). For a smooth path ϕ = {ϕt | a ≤ t ≤ b} in H, let ψ = {ψt | a ≤ t ≤ b} be a vector field along ϕ, considered as ψt =
d (ϕt + sψt ) ∈ Tϕt H, ds |s=0
a ≤ t ≤ b.
Then the covariant derivative by Levi-Civita connection is expressed as D 1 ψ = ψ˙t − Re(∂ ϕ˙t , ∂ψt )ωt = ψ˙t − (d ϕ˙t , dψt )ωt , ∂t 2
(1)
where ωt = ωϕt . Thus the equation of geodesics is given by ϕ¨t − |∂ ϕ˙t |2ωt = 0. The K-energy, or Mabuchi energy, is defined by 1 µ(ωϕ ) := M(ωϕ , ω0 ) = − (σωt − σ )ϕ˙ ωtn dt, 0
V
where ϕt := tϕ, 0 ≤ t ≤ 1, ωt = ωϕt and nc1 (V )ωn−1 σ = V n 0 . V ω0 The fundamental facts are the following:
(2)
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• • •
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ωϕ is a critical point of µ if and only if ωϕ is a Kähler metric of constant scalar curvature. The Hessian of µ is positive semi-definite, so µ is a convex function. We have for any smooth path {ϕt }0≤t≤1 , d 2 µ(ωt ) 2 n = |∂Y | ω − (ϕ¨t − |∂ ϕ˙t |2ωt )ωtn , t ωt t dt 2 V V where Yt = ωt−1 (∂ ϕ˙t ). In particular, for geodesics ϕt we have d2 µ(ω ) = |∂Yt |2ωt ωtn ≥ 0. t dt 2 V
(3)
Now we can prove the uniqueness of constant scalar curvature metrics modulo the action of the identity component of the group of biholomorphic automorphisms of V provided we have a geodesic joining two such metrics as follows. Suppose that both ω0 and ω1 are Kähler forms with constant scalar curvature and that we have a geodesic ωt , 0 ≤ t ≤ 1, joining them. Then it follows from the above facts that d 2 µ(ωt ) ≥ 0, dt 2
dµ(ωt ) = 0, dt |t=0
dµ(ωt ) = 0. dt |t=1
t) These imply dµ(ω = 0 for all t ∈ [0, 1]. But Eq. (3) shows that Yt is a holomorphic dt vector field and ωt is the pull-back of ω0 by an automorphism of V . The problem is then whether we can find a geodesic in the space of Kähler metrics. The geodesic equation is reduced to a degenerate Monge-Ampère equation ([25]). The existence of C 1,1 -solution was proved by X.X. Chen [8], but C 1,1 geodesics are not enough to prove uniqueness and Chen used ε-approximations of the solutions. In the toric case, however, geodesics are obtained as a line segment of symplectic potentials as shown by Guan [13], which is explained next. Let V be a toric Kähler manifold. Then V is a completion of (C∗ )n with coordinates j 1 w , · · · , w n . Put w j = e z and z j = x j + iθ j . Let F(x) be the Kähler potential of a T n -invariant Kähler metric so that
i ∂2 F dz j ∧ dz k . 4 ∂x j ∂xk
ω = ig jk dz j ∧ dz k =
The symplectic potential G is the Legendre transform of F: G(y) =
n
xj
j=1
with y j =
∂F ∂x j
∂F −F ∂x j
. There is a symmetrical relation xj =
Thus as matrices
∂2 F ∂ xi ∂ x j
∂G , ∂yj
=
∂ yi ∂x j
F(x) =
n
yj
j=1
=
∂xi ∂yj
∂G − G. ∂yj
−1
=
∂2G ∂ yi ∂ y j
−1 .
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If {ωt } is a curve in the space of Kähler forms and Ft is the corresponding Kähler potential then we have the t-dependent coordinates yt =
∂ Ft ∂x
on the image of the moment map. Conversely if we start from a curve G t of symplectic potential with t-independent coordinates y we have t-dependent coordinates xt =
∂G t ∂y
on Rn . To understand the geodesic equation better it is convenient to consider Ft (xt ) in terms of t-dependent coordinates xt and G t (yt ) in terms of yt with the relations yt j =
∂ Ft
, j
∂ xt
G t (yt ) =
n
j
xt
j=1
∂ Ft j
∂ xt
− Ft .
(4)
We suppress t for the notational convenience. First of all ∂G ∂G(y) ˙ ˙ = G(y) + y˙ j = G(y) + x j y˙ j , ∂t ∂yj n
n
j=1
j=1
and ⎛ ∂ ⎝ ∂t
n j=1
⎞
xj
n 2 ˙ ∂F ∂F j j ∂F j ∂F j ∂ F k ⎠= x ˙ − F˙ − F + x + x x ˙ − x ˙ ∂x j ∂x j ∂x j ∂x j ∂xk ∂x j j=1
=
n
˙ x j y˙ j − F(x).
j=1
Thus ˙ ˙ G(y) = − F(x).
(5)
Taking the derivative of (5) we get F¨ +
n n ∂ F˙ j ∂ G˙ j ¨ + x ˙ = − G y˙ . ∂x j ∂y j j=1
(6)
j=1
In what follows we omit obvious indices and sum notations. Taking the derivative of y = ∂∂ Fx we have y˙ =
∂ F˙ ∂ y + x. ˙ ∂x ∂x
(7)
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Proposition 2.1. Let M be a toric Kähler manifold. (a) Let Ft (xt ) and G t (yt ) be Kähler and symplectic potentials of t-dependent coordinates xt and yt satisfying the relations (4). Then the geodesic equations are insensitive to t-dependent coordinates in that 1 ˙ 2 ¨ F − |d F|t (xt ) = 0 2 if and only if ¨ t ) = 0. G(y (b) For any two Kähler potentials there exists a unique geodesic joining them. In the coordinates y, θ with fixed standard symplectic form ω =
n action-angle i i=1 dyi ∧ dθ , the geodesic can be expressed as t G 1 (y) + (1 − t)G 0 (y). Proof. (a)
(b)
It follows from (6), (7) and (5) that 1 ˙ 2 ∂ x ∂ F˙ ∂ F˙ ¨ F − |d F|t (xt ) = F¨ − 2 ∂y ∂x ∂x ∂ F˙ ∂ x ∂ F˙ ∂ F˙ ∂ G˙ y˙ − x˙ − = −G¨ − ∂y ∂x ∂y ∂x ∂x ˙ ˙ ˙ ∂ G ∂ F ∂x ∂F = −G¨ − y˙ − x˙ − y˙ − x˙ ∂y ∂x ∂y ∂x ˙ ˙ ∂F ∂G y˙ − y˙ = −G¨ − ∂y ∂y ¨ t ). = −G(y
The existence of a geodesic joining two Kähler potentials can be shown as follows. We first fix t-independent coordinates x on Rn and y on the convex polytope. Let F0 (x) and F1 (x) be two Kähler potentials, and y0 , y1 , G 0 and G 1 be defined by y0 =
∂ F0 , G 0 = x y0 − F0 ; ∂x
y1 =
∂ F1 , G 1 = x y1 − F1 . ∂x
Put yt = t y1 +(1−t)y0 . Then yt is the moment map of the Kähler potential t F1 +(1−t)F0 and thus gives coordinates on the image of the moment map. Put G t (y) = t G 1 (yt ) + ¨ t ) = 0, so the corresponding Legendre transform (1 − t)G 0 (yt ). Then obviously G(y ∂G t − G t (yt ) ∂ yt ˙ 2t (xt ) = 0. Thus we get a geodesic Ft (x) satisfies the geodesic equation F¨ − 21 |d F| joining F0 and F1 . Notice that we inserted t-independent coordinates x in Ft so that Ft (x) becomes a geodesic in the original sense. We could perturb yt and get a different xt , but Ft (x) does not change because of the uniqueness of the geodesic proved by X.-X. Chen [8]. One can argue as in [13] to show that Ft defines metrics on the whole Kähler manifold. Taking the Legendre transform of the geodesic Ft obtained in this way one sees that in the action-angle coordinates on the polytope (see [1]) G t is expressed as t G 1 (y) + (1 − t)G 0 (y). Ft (xt ) = yt
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Now we consider the case of transverse Kähler structure of compact Sasaki manifolds of positive basic Chern class. We begin with the study of the automorphisms of transverse holomorphic structure. Proposition 2.2. Let S be a compact Sasaki manifold. Then the Lie algebra of the automorphism group of transverse holomorphic structure is the Lie algebra of all Hamiltonian holomorphic vector fields in the sense of Definition 4.4 of [11]. Proof. Since the Reeb foliation has transverse holomorphic structure we can choose local transverse holomorphic coordinates z 1 , · · · , z m . They are used as part of local holomorphic coordinates as well as local coordinates on S. A local holomorphic vector field of the form X i ∂z∂ i is considered as a local vector field on C(S) as well as one on S. We will denote by X the former and by X the latter. Note that along {r = 1}, X is the tangential part to S ∼ X . = {r = 1} of
If a vector field X generates a one-parameter group of automorphisms of C(S) which commutes with the holomorphic flow generated by ξ − i J ξ , then [ X , ξ − i J ξ ] = 0. If we set X = Y − i J Y with Y the real part of X then [ξ, Y ] = 0. From this one sees [Y, J ξ ] = J [Y, ξ ] = 0. These mean that the holomorphic flow descends to S and local leaf spaces, and that X descends to a holomorphic vector field on each local leaf space. This local vector field can be regarded as a local vector field X on C(S) as well as X on S. Recall from [11] that the contact form η on S lifts to C(S) as η = 2d c log r = i(∂ − ∂) log r, where we use the same letter η by the abuse of notation. We then have η( X ) = η(X ). This is because if p : C(S) = S × R+ → S is the projection then η on C(S) is p ∗ and X = p∗ X . Then X can be expressed as
X )(ξ − i J ξ )). X = η( X )(ξ − i J ξ ) + ( X − η(
(8)
Note that the right-hand side is an orthogonal splitting. Taking ∂ of both sides of (8) we get ∂η( X ) = ∂η( X ) = ∂ B η(X ).
(9)
Taking the tangential component to S of X we obtain X := η( X )ξ + X − η(X )ξ. Then since η(X ) = η( X ), X may be written as X = η(X )ξ + X − η(X )ξ. Since η is of the form η = dt − i∂ B f + i∂ B f, where t is the leaf coordinate with ξ t = 1 and f is the Kähler potential for the transverse Kähler form 21 dη we have dη = 2i∂ B ∂ B f.
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Hence we get i(X )dη = i(X )dη = 2i(X )∂ B η = −2∂ B (η(X )) = −2∂ B (η(X )). Hence X is a Hamiltonian holomorphic vector field in the sense of Definition 4.4 of [11]. It is easy to see that the Lie algebra consisting of all X is isomorphic to the Lie algebra consisting of all Hamiltonian holomorphic vector fields X . Recall that a basic function ϕ is a smooth function on S such that ξ ϕ = 0, where ξ is the Reeb field. The transverse Kähler form ω T is given by ωT =
1 dη, 2
where η = 2d c log r |{r =1}∼ =S = i(∂ − ∂) log r |{r =1}∼ =S and the transverse Kähler deformation is given by ω T + i∂ B ∂ B ϕ for some basic function ϕ where ∂ B and ∂ B are basic ∂ and ∂-operators. The tangent space to transverse Kähler metrics is therefore the set of all basic functions ϕ. We may define geodesics in the space of transverse Kähler metrics by the equation ϕ¨ − |∂ B ϕ| ˙ 2t = 0. We can derive the similar conclusion that if one can always find a geodesic joining two Kähler potentials one can show that the identity component of the automorphism group of the transverse holomorphic structure acts transitively on the space of transverse Kähler metrics of constant scalar curvature by using the principle stated in the Appendix of [11]. In fact the corresponding equation to (3) shows that the geodesic joining two transverse Kähler metrics of constant scalar curvature is tangent to the Hamiltonian function of a Hamiltonian holomorphic vector field. Of course since transverse KählerEinstein metrics have constant scalar curvature these arguments give the uniqueness of transverse Kähler-Einstein metrics modulo the action of the identity component of the automorphism group of the transverse holomorphic structure. Suppose now that the compact Sasaki manifold S is toric so that the cone C(S) is a toric Kähler manifold. We may also define the covariant derivative and geodesic equation by (1) and (2). By the above arguments we can always find a geodesic joining two Kähler potentials on C(S). The Kähler form on C(S) is given by 1 1 1 d(r 2 η) = idd c r 2 = i∂∂r 2 . 2 2 2 A function on S can be lifted to C(S) = S × R+ and we use the same notation for a function on S and its lift to C(S). The transverse Kähler deformation is given using a basic function ϕ by ω=
η = η + 2d Bc ϕ = 2d c log(r exp ϕ), and hence the Kähler form ω on C(S) is deformed by
ω=
1 1 d(˜r 2 η) = i∂∂(r 2 exp(2ϕ)). 2 2
(10)
Let K be the space of all Kähler metrics on C(S) of the form i∂∂ H for some real smooth function H on C(S), and Kω be the submanifold consisting of Kähler metrics obtained by transverse Kähler deformations of the form (10).
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Lemma 2.3. Kω is a totally geodesic submanifold in K. Proof. By (10) a curve in Kω is of the form 21 r 2 exp(2ϕt ) so that its tangent vector is rt2 ϕ˙t where we put rt = r exp ϕt . Similarly a vector field along 21 r 2 exp(2ϕt ) is of the form rt2 ψt for a curve ψt of basic functions. The covariant derivative of rt2 ψt along 1 2 2 r exp(2ϕt ) is computed by 1 D 2 (rt ψt ) = 2rt2 ϕ˙ t ψt + rt2 ψ˙ t − (d(rt2 ϕ˙t ), d(rt2 ψt )) dt 2 1 2 = rt ψ˙ t − (d θ˙t , dψt ) 2 D = rt2 ψt , dt
(11)
where the covariant derivative in the last term is the one for the transverse Kähler structure. This shows that the covariant derivative of a vector field in the tangent spaces of Kω along a curve in Kω is tangent to Kω . Thus Kω is a totally geodesic submanifold. Proposition 2.4. A curve i∂∂( 21 r 2 exp(2ϕt )) in Kω is a geodesic if and only if ω T + i∂ B ∂ B ϕt is a geodesic in the space of the transverse Kähler metrics. Moreover, for any given two transverse Kähler metrics with the same Reeb field ξ corresponding to toric Kähler cone metrics there exists a unique geodesic joining them. Proof. The first statement follows immediately from (11). Let ω0T and ω1T be the transverse Kähler metrics with the common Reeb field ξ corresponding to toric Kähler metrics ω0 and ω1 on C(S). Let G 0 and G 1 be the corresponding symplectic potentials. We use the action-angle coordinates yi , θ i . Since G 0 and G 1 have common Reeb field ξ , g = G 1 − G 0 satisfies ⎞ ⎛ m+1 ∂ ⎠ ∂g = 0 ⎝ yj ∂ y j ∂ yi j=1
by (2.40) in [21]. Thus the geodesic t G 1 (y)+(1−t)G 0 (y) = G 0 +tg joining G 0 and G 1 has the same Reeb field ξ , and thus the corresponding Kähler potentials define the same transverse holomorphic structure. From the first statement it follows that the geodesic Ft = 21 rt2 = 21 r 2 exp(2ϕt ) with ϕt basic smooth functions descends to a geodesic in the space of transverse Kähler metrics. Proof of Theorem (1.1). Let ω0T and ω1T be the transverse Kähler metrics corresponding to two Sasaki-Einstein metrics. As proved in [22] and [7] the Lichnerowicz-Matsushima theorem for compact Kähler manifolds of constant scalar curvature extends to compact Sasaki manifolds of constant transverse scalar curvature. Thus, we may assume that both ω0T and ω1T are invariant under the maximal compact subgroup of the group of automorphisms of the transverse holomorphic structure. In particular we may assume that they are invariant under the maximal torus G, and thus we only need to consider the toric Sasaki-Einstein metrics. In [21] and [11] it is shown that the volume functional of Sasakian structures depends only on the Reeb fields, that there is a unique critical Reeb field ξ which minimizes the
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volume functional and that only for the critical point ξ the obstruction to the existence of transverse Kähler-Einstein metric vanishes. Thus ω0T and ω1T must have a common Reeb field ξ . They can be joined by a geodesic by Proposition 2.4. We then apply the standard method known in Kähler geometry as explained above, and it follows from (3) that the geodesic is tangent to the Hamiltonian function of a Hamiltonian holomorphic vector field. This completes the proof. 3. Toric Diagrams We begin with the definition of a good rational polyhedral cone. Definition 3.1 (cf. [16]). Let g∗ be the dual of the Lie algebra g of the (m + 1) dimensional torus G. Let Zg be the integral lattice of g, that is the kernel of the exponential map exp : g → G. A subset C ⊂ g∗ is a rational polyhedral cone if there exists a finite set of vectors λi ∈ Zg, 1 ≤ i ≤ d, such that C = {y ∈ g∗ | y, λi ≥ 0 for i = 1, · · · , d}. We assume that the set λi is minimal in that for any j, C = {y ∈ g∗ | y, λi ≥ 0 for all i = j}, and that each λi is primitive, i.e. λi is not of the form λi = aµ for an integer a ≥ 2 and µ ∈ Zg. (Thus d is the number of codimension 1 faces if C has non-empty interior.) Under these two assumptions a rational polyhedral cone C with nonempty interior is good if the following condition holds. If {y ∈ C | y, λi j = 0 for all j = 1, · · · , k} is a non-empty face of C for some {i 1 , · · · , i k } ⊂ {1, · · · , d}, then λi1 , · · · , λik are linearly independent over Z and ⎧ ⎫ ⎧ ⎫ k k ⎨ ⎬ ⎨ ⎬ a j λi j | a j ∈ R ∩ Zg = m j λi j | m j ∈ Z . (12) ⎩ ⎭ ⎩ ⎭ j=1
j=1
Let M be a 2m + 1-dimensional compact connected contact toric manifold with the contact form η. Namely there is an effective action of the (m + 1)-dimensional torus G which preserves η. Then the moment map µ : M → g∗ is defined by µ( p), X = (η(X M ))( p), where X M denotes the vector field on M induced by X ∈ g. We assume dim M = 2m + 1 ≥ 5. It is well-known ([16]) that if the action of G is not free then the image of the moment map is a good rational polyhedral cone. Definition 3.2. An (m + 1)-dimensional toric diagram with height is a collection of λi ∈ Zm+1 ∼ = Zg satisfying (12) and γ ∈ Qm+1 ∼ = (Qg)∗ such that (1) is a positive integer such that γ is a primitive element of the integer lattice Zm+1 ∼ = Z∗g. (2) γ , λi = −1.
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We say that a good rational polyhedral cone C is associated with a toric diagram of height if there exists a rational vector γ satisfying (1) and (2) above. The reason why we use the terminology “height ” is because of the following proposition. Proposition 3.3. Using a transformation by an element of S L(m + 1, Z) we may assume that ⎛ 1⎞ − ⎜0 ⎟ ⎜ ⎟ γ = ⎜. ⎟ ⎝ .. ⎠ 0 and the first component of λi is equal to for each i. Proof. By elementary group theory there is an element A of S L(m + 1, Z) which sends the primitive vector γ in Zm+1 to t (−1, 0, · · · , 0) where the left upper t denotes the transpose. Then Aγ = t (− 1 , 0, · · · , 0). By transforming g by t A−1 , the transpose of A−1 , we get Aγ , t A−1 λi = γ , λi = −1. This implies the first component of t A−1 λi is .
Before we give a proof of Theorem 1.2 we outline the proof of the following fact (cf. [16,20]). Proposition 3.4. For each pair of a good rational polyhedral cone C and an element ξ ∈ C0∗ , where C0∗ = {ξ ∈ g | v, ξ > 0 for all v ∈ C}, there is a compact connected toric Sasaki manifold S whose moment map image is equal to C\{0} and whose Reeb vector field is generated by ξ . Outline of the proof. The construction of a contact manifold from a good rational polyhedral cone is the so-called Delzant construction. Let e1 , · · · , ed be the canonical basis of Rd . Of course they generate the lattice Zd . Let βZ : Zd → Zg ∼ = Zm+1 be the homomorphism defined by βZ (ei ) = λi , and βR : Rd → g ∼ = Rm+1 be the natural linear map induced by βZ . Since C has non-empty interior then βR is surjective, i.e. there is a subset {i 1 , · · · , i m+1 } such that λi1 , · · · , λim+1 are linearly independent over R. Then βZ and βR naturally induce a homomorphism βT : T d → G ∼ = T m+1 of the tori. Let K be the kernel of βT . We write [a] ∈ T d for the image of a ∈ Rd . Then d K = [a] | ai λi ∈ Zg . i=1
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It is a compact abelian subgroup of T d and its Lie algebra is ker βR . Note that K is not connected in general. Let us consider the standard action of T d on Cd with the Kähler
d dv i ∧ dv j by form 2i i=1 [a] · (v 1 , · · · , v d ) = (e2πia1 v 1 , · · · , e2πiad v d ). Consider the action of K on Cd obtained as the restriction of the T d -action and the moment map µ K : Cd → k∗ . The Kähler cone manifold C(S) is obtained as the Kähler quotient C(S) = (µ−1 K (0)\{0})/K . See [16] for more detail. The closure C(S) is obtained as C(S) = µ−1 K (0)/K which is realized also as a normal complex analytic space via the standard method using fans in algebraic geometry ([23]). A Sasaki manifold S is obtained as 2d−1 S = (µ−1 )/K , K (0) ∩ S
where S 2d−1 is the standard (2d − 1)-sphere in Cd . This Sasaki metric is often called the canonical Sasaki metric, and the symplectic potential on C(S) and the Reeb field are respectively given by 1 li (y) log li (y), 2 d
G can =
i=1
ξ can =
d
λi ,
i=1
where li (y) = λi , y. For a general Reeb field ξ ∈ g a symplectic potential G can on ξ C(S) is given by 1 1 1 li (y) log li (y) + lξ (y) log lξ (y) − l∞ (y) log l∞ (y), 2 2 2 d
G can ξ (y) =
i=1
where lξ (y) = ξ, y and l∞ = ξ can , y, see [20] for more detail. The corresponding Kähler potential Fξcan , computed by the Legendre transform, is given by Fξcan =
1 lξ (y), 2
see (61) in [11]. Since the Kähler potential is equal to 21 r 2 , then r 2 = lξ (y) and the Sasakian structure is determined via the identification S ∼ = {lξ (y) = 1} ⊂ C(S).
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Proof of Theorem 1.2. First we prove that (a) implies (b). Suppose c1B > 0 and c1 (D) = 0. By our assumption the (m + 1)-dimensional torus G acts on S preserving the Sasakian structure. By Proposition 4.3 in [11] we can then choose a G-invariant transverse Kähler form ω T such that c1B = (2m + 2)[ω T ]. Let ρ T be the Ricci form of ω T . Note that ω T and ρ T are defined on each local leaf space of the Reeb foliation, but they can be lifted to S to define global 2-forms on S. There exists a basic G-invariant smooth function h on S such that ρ T = (2m + 2)ω T + i∂ B ∂ B h
(13)
on S. By an elementary curvature computation in Sasakian geometry the equation (13) is equivalent to ρ = −i∂∂ log det(Fi j ) = i∂∂h
(14)
on C(S), where h is pulled back to C(S) ∼ = R+ × S so that h satisfies r
r h = ξ h = 0, ∂r
and where F is the Kähler potential on C(S), (e x of G C ∼ = (C∗ )m+1 and Fi j =
0 +iθ 0
(15) , · · · , ex
m +iθ m
) is the coordinate
∂2 F . ∂ xi ∂ x j
Note that since F is G-invariant it is independent of θ i ’s. Since any G-invariant pluriharmonic function on C(S) is an affine function then there exists a γ ∈ g∗ such that log det(Fi j ) = −2
m
γi x i − h
(16)
i=0
by replacing h + constant by h. Using the Legendre transform G of F we get log det(G i j ) = 2
m
γi G i + h.
(17)
i=0
Using Abreu-Guillemin arguments about the boundary behavior of G it is shown in [20] that λ j , γ = −1 for j = 1, · · · , d.
(18)
Since the moment map image has non-empty interior there are (m + 1) vectors λ j1 , · · · , λ jm+1 linearly independent over R. Hence one can consider γ as a solution to the linear equations λ ji , γ = −1 for i = 1, · · · , m + 1 and sees that γ ∈ Qm+1 g∗ . Choosing a positive integer such that γ is a primitive element of the integer lattice. Since η(ξ ) = 1 and the moment map on C(S) is given by 21 r 2 η
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we have y, ξ > 0 for all y ∈ C. It is also shown in [20] that γ , ξ = −m − 1. This proves that (a) implies (b). Next we prove that (b) implies (c). Return to the Delzant construction in the proof of Proposition 3.4. One sees that µ−1 K (0) is given by d −1 d 2 d µ K (0) = v ∈ C | bi |vi | = 0 for all b ∈ k ⊂ R . i=1
By Proposition 3.3 we may assume that the first component of λi is for each i. Recall that d
ai λi ∈ Zm+1
i=1
for all [a] ∈ K . Looking at the first component we get (a1 + · · · + ad ) ∈ Z for all [a] ∈ K . Thus (e2πi(a1 +···+ad ) dv1 ∧ · · · ∧ dvd )⊗ = (dv1 ∧ · · · ∧ dvd )⊗ . Let b1 , · · · , bd−m−1 be a basis of k, and put bi = (bi1 , · · · , bid ) and d
Xi =
j=1
bi j
∂ . ∂v j
Then (i(X 1 ) · · · i(X d−m−1 )dv1 ∧ · · · ∧ dvd )⊗
(19)
⊗ ⊗ descends to a nowhere zero section of K C(S) . Hence K C(S) is a trivial line bundle. This proves that (d) implies (c). Note that this proof shows the section (19) extends to the apex of C(S), as a token of Q-Gorenstein property (cf. Remark 1.3). We now prove that (c) implies (a). Suppose we are given a G-invariant Sasakian struc⊗ ture with Reeb field ξ and with trivial line bundle K C(S) . Thus we have a G-invariant ⊗ . Let h 1 be Kähler metric ω and a nowhere vanishing holomorphic section 1 of K C(S) defined by
h1 =
1 log ||1 ||2 ,
where the norm of 1 is taken with respect to ω. Then the Ricci form ρ of ω is written as ρ=
1 ∂∂h 1 . 2π
(20)
Let h be the average of h 1 by the action of G. Since ρ is G-invariant we see from (20) that ρ=
i ∂∂h. 2π
(21)
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Starting from (14) which is identical to (21) we get (16) and (17) (though we do not have (15)). Then it is shown in [20] that ξ, γ = −(m + 1).
(22)
Equation (16) says that eh det(Fi j ) is a flat metric on C(S). Consider the (m + 1)-form written as = e−i = e−
m
i=0 γi θ
m
i=0 γi
zi
i
h
1
e 2 (det(Fi j )) 2 dz 0 ∧ · · · ∧ dz m
(23)
dz 0 ∧ · · · ∧ dz m ,
where we used (16). Then is multi-valued if γ is not integral but only rational. Let 1 be the positive integer such that 1 γ is a primitive element of the integer lattice. Then ⊗1 ⊗1 is a holomorphic section of K C(S over the open set corresponding to the interior of ⊗ the moment map image. But since || 1 || = 1 we see that ⊗1 extends to the whole C(S). We further have
and
Lξ = (m + 1)i
(24)
m+1 i 1 ωm+1 . (−1)m(m+1)/2 ∧ = exp(h) 2 (m + 1)!
(25)
Since ξ is decomposed into the holomorphic and the anti-holomorphic parts ξ=
1 1 (ξ − i J ξ ) + (ξ + i J ξ ) 2 2
∂ with J ξ = −r ∂r we have
Lξ = L 1 (ξ +ir 2
=
∂ ∂r
)
i (m + 1)i + Lr ∂ . 2 2 ∂r
From this and (24) it follows that Lr ∂ = (m + 1) ∂r
and Lr ∂ ( ∧ ) = 2(m + 1) ∧ . ∂r
On the other hand since ω = i∂∂(r 2 /2) we have Lr ∂ ωm+1 = 2(m + 1)ωm+1 . ∂r
∂ we get Taking the Lie derivative of both sides of (25) by r ∂r
r
∂h = 0. ∂r
Since h is G-invariant we also have ξ h = 0. Hence (25) implies [ρ T ] = (2m + 2)[ω T ] as basic cohomology classes, from which we get c1B > 0 and c1 (D) = 0 by Proposition 4.3 in [11]. This completes the proof of Theorem 1.2.
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4. Examples and Remarks on the Fundamental Groups As is mentioned in the Introduction there are examples of 3 dimensional toric diagrams of height 1 denoted by X p,q , Y p,q , Z p,q and L p,q,r known in the physics literature and all the corresponding Sasaki manifolds have Sasaki-Einstein metrics by the existence result of [11]. To check that these toric diagrams satisfy the goodness condition of Definition 3.2 the following proposition is useful. Proposition 4.1. Let C be a convex polyhedral cone in R3 given by C = {y ∈ R3 | y, λi ≥ 0, j = 1, · · · , d} with
⎛
⎛ ⎞ ⎞ 1 1 λ1 = ⎝ p 1 ⎠ , · · · , λ d = ⎝ p d ⎠ . q1 qd
Then C is good in the sense of Definition 3.2 if and only if either (i) | pi+1 − pi | = 1 or |qi+1 − qi | = 1 or (ii)
pi+1 − pi and qi+1 − qi are relatively prime non-zero integers
for i = 1, · · · , d where we have put λd+1 = λ1 . Further the area of the 2-dimensional convex polytope formed by pd p1 p1 ,··· , , q1 qd q1 is an invariant of the equivalent classes given by the action of some element of S L(3, Z) on the set of all such diagrams. Proof. Let a1 and a2 be real numbers such that a1 λi + a2 λi+1 ∈ Z3 . Then we have a1 + a2 ∈ Z, pi a1 + pi+1 a2 ∈ Z, qi a1 + qi+1 a2 ∈ Z.
(26)
It follows from these that ( pi+1 − pi )a2 ∈ Z, (qi+1 − qi )a2 ∈ Z.
(27)
If ( pi+1 − pi ) and (qi+1 − qi ) satisfy (i) or (ii) then there exist s, t ∈ Z such that s( pi+1 − pi ) + t (qi+1 − qi ) = 1. Then from (27) we get a2 ∈ Z. From (26) we also have a1 ∈ Z. Conversely if a2 satisfying (27) is always in Z then ( pi+1 − pi ) and (qi+1 − qi ) are relatively prime. If a diagram of the first variable 1 is transformed to another by an element of S L(3, Z) then the volume of the 3-dimensional truncated cone {a1 λ1 + · · · + ad λd | 0 ≤ a1 ≤ 1, · · · , 0 ≤ ad ≤ 1, 0 ≤ a1 + · · · + ad ≤ 1} is invariant. But this is equal to one third of the area described in the statement of the proposition. This completes the proof of Proposition 4.1.
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We give a simplest toric diagram of height . Let C be the convex polyhedral cone defined by C = {y ∈ R3 | y, λi ≥ 0, j = 1, 2, 3} with
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 1 λ1 = ⎝ 0 ⎠ , λ 2 = ⎝ 1 ⎠ , λ 3 = ⎝ 1 ⎠ . 0 0
Then this is a good cone and defines a smooth Sasaki manifold. One can show that ⎛ ⎞ −1 γ = ⎝ −1 ⎠ . 1
Taking ⎛
⎞ 0 0 −1 A = ⎝ −1 1 0 ⎠ 10 we have
⎛
⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − 1 Aγ = ⎝ 0 ⎠ , t A−1 λ1 = ⎝ 0 ⎠ , t A−1 λ2 = ⎝ 1 ⎠ , t A−1 λ3 = ⎝ 1 ⎠ . 1 1 2 0
By following the Delzant construction one sees that the resulting Sasaki manifold is the Lens space S 5 /Z . Next let C be the convex polyhedral cone defined by C = {y ∈ R3 | y, λi ≥ 0, j = 1, 2, 3, 4} with
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 1 1 ⎠. λ1 = ⎝ 0 ⎠ , λ2 = ⎝ 1 ⎠ , λ3 = ⎝ 1 ⎠ , λ 4 = ⎝ 1 0 0 −1
Then this is a good cone and defines a smooth Sasaki manifold. The resulting Sasaki manifold does not satisfy the conditions of Theorem 1.2 because there is no γ with γ , λ j = −1 for j = 1, 2, 3, 4. One can show that if > 1 then the resulting Sasaki manifold S is not simply connected. This follows from a result of Lerman [17] which is stated as follows. Let L be the subgroup of Zg generated by λ1 , · · · , λd . Then π1 (S) ∼ = Zg/L. Obviously Zg/L is not trivial if > 1. Thus we proved the following. Proposition 4.2. Let S be a compact connected toric Sasaki manifold associated with a toric diagram of height > 1. Then S is not simply connected.
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Note that the converse is not true as the following example shows. Consider the toric diagram with height 1 defined by the three normal vectors ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 1 1 λ1 = ⎝ 0 ⎠ , λ2 = ⎝ 2 ⎠ , λ3 = ⎝ 3 ⎠ . 0 1 4 The resulting Sasaki manifold is the lens space with a different Z5 -action from the above example with = 5. Note also, for example, Y p,q is not simply connected unless p and q are relatively prime. Proof of Theorem 1.5. Put n = k + 3. We construct diagrams of height 1 with either | pi+1 − pi | = 1 or |qi+1 − qi | = 1 such that ( p1 , q1 ), · · · , ( pn , qn ), ( p1 , q1 ) form a convex polytope with n vertices and that they generate Z2 . Then S is simply connected since L = Z3 . By another theorem of Lerman [17] we know that b2 (S) = n − 3 = k. It follows from the classification of five dimensional simply connected spin manifolds with T 3 -action ([6]) that S = S 5 k(S 2 × S 5 ). There are many ways to construct such examples. For instance if k = 2r so that n = 2r + 3 then take 0 p1 1 p0 = , = ,..., q0 q1 0 1 r r +1 pr pr +1 , = r (r +1) , = (r +1)(r +2) qr qr +1 +s 2 2 r pr +2 ,..., = (r +1)(r +2) qr +2 +s−1 2 1 p2r +1 p2r +2 0 . = (r +1)(r +2) = r (r +1) , q 1 q2r +1 + s − 2r +2 2 2 For different values of s they give inequivalent toric diagrams because they have different areas. If k = 2r − 1 so that n = 2r + 2 then take r −1 p0 p1 p2 pr −1 0 1 2 , , ,..., = = = = (r −1)r , 0 1 3 q0 q1 q2 qr −1 2 r 0 pr pr +1 , , = r (r +1) = (r )(r +1) qr qr +1 +s +s+1 2 2 −r −(r − 1) pr +3 pr +4 , = r (r +1) = (r −1)r ,..., qr +3 q + s r +4 2 2 p2r +1 p2r −2 −1 . = = 3 0 q2r q2r +1 Then again different values of s give inequivalent diagrams. This completes the proof of Theorem 1.5. Acknowledgement. We are grateful to Charles Boyer for pointing out our careless statement of the results without the condition c1 (D) = 0 in the first version of the paper.
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References 1. Abreu, M.: Kähler geometry of toric manifolds in symplectic coordinates. Fields Institute Comm. 35, Providence, RI: Amer. Math. Soc., 2003, pp. 1–24 2. Altman, K.: Toric Q-Gorenstein singularities. Prepint, 1994 3. Argurio, R., Bertolini, M., Closset, C., Cremonesi, S.: On Stable Non-Supersymmetric Vacua at the Bottom of Cascading Theories. JHEP 0609, 030(2006) 4. Boyer, C.P., Galicki, K., Kollár, J.: Einstein metrics on spheres. Ann. of Math. 162(1), 557–580 (2005) 5. Boyer, C.P., Galicki, K., Nakamaye, M.: On the geometry of Sasakian-Einstein 5-manifolds. Math. Ann. 325(3), 485–524 (2003) 6. Boyer, C.P., Galicki, K., Ornea, L.: Constructions in Sasakian geometry. http://arxiv.org/list/math.DG/ 0602233, 2006, to appear in math.zeit 7. Boyer, C.P., Galicki, K., Simanca, S.R.: Canonical Sasakian metrics. http://arxiv.org/list/math.DG/ 0604325, 2006 8. Chen, X.X.: The space of Kähler metrics. J. Diff. Geom. 56, 189–234 (2000) 9. Cvetiˇc, M., Lü, H., Page, D.N., Pope, C.N.: New Einstein-Sasaki Spaces in Five and Higher Dimensions. Phys. Rev. Lett. 95, 071101 (2005) 10. Donaldson, S.K.: Symmetric spaces, Kähler geometry and Hamiltonian dynamics. In: Northern California Symplectic Geometry Seminar’ (Eliashberg et al eds.), Providence, RI: Amer. Math. Soc. 1999, pp. 13–33 11. Futaki, A., Ono, H., Wang, G.: Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. http://arxiv.org/list/math.DG/0607586, 2006 12. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Sasaki-Einstein Metrics on S 2 × S 3. Adv. Theor. Math. Phys. 8, 711–734 (2004) 13. Guan, D.: On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles. Math. Res. Letters 6, 547–555 (1999) 14. Hanany, A., Kazakopoulos, P., Wecht, B.: A New Infinite Class of Quiver Gauge Theories. JHEP 0508, 054(2005) 15. Kollár, J.: Einstein metrics on connected sums of S 2 × S 3 . http://arxiv.org/list/math.DG/0402141, (2004) 16. Lerman, E.: Contact toric manifolds. J. Symplectic Geom. 1(4), 785–828 (2003) 17. Lerman, E.: Homotopy groups of K-contact toric manifolds. Trans. Amer. Math. Soc. 356(4), 4075–4084 (2004) 18. Mabuchi, T.: Some Sympletic geometry on compact Kähler manifolds. I. Osaka J. Math. 24, 227–252 (1987) 19. Martelli, D., Sparks, J.: Toric Sasaki-Einstein metrics on S 2 × S 3 . Phys. Lett. B 621, 208–212 (2005) 20. Martelli, D., Sparks, J., Yau, S.-T.: The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds. Commun. Math. Phys. 268, 39–65 (2006) 21. Martelli, D., Sparks, J., Yau, S.-T.: Sasaki-Einstein manifolds and volume minimization. http://arxiv.org/ list/hepth/0603021, 2006 22. Nishikawa, S., Tondeur, P.: Transversal infinitesimal automorphisms for harmonic Kähler foliation. Tohoku Math. J. 40, 599–611 (1988) 23. Oda, T.: Convex bodies and algebraic geometry - An introduction to the theory of toric varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete 15. Berlin: Springer-Verlag, 1988 24. Oota, T., Yasui, Y.: New Example of Infinite Family of Quiver Gauge Theories. Nucl. Phys. B 762, 377–391 (2007) 25. Semmes, S.: Complex Monge-Ampère and symplectic manifolds. Amer. J. Math. 114, 495–550 (1992) 26. van Coevering, C.: Toric surfaces and Sasakian-Einstein 5-manifolds. http://arxiv.org/list/math.DG/ 0607721, 2006 Communicated by G.W. Gibbons
Commun. Math. Phys. 277, 459–496 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0379-z
Communications in
Mathematical Physics
Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi-Periodic Perturbations W.-M. Wang1,2, 1 Departement de Mathematique, Universite Paris Sud, 91405 Orsay Cedex, France 2 Department of Mathematics, University of Massachussetts, Amherst, Ma 01003, USA.
E-mail: [email protected]; [email protected] Received: 11 January 2007 / Accepted: 6 June 2007 Published online: 7 November 2007 – © Springer-Verlag 2007
Abstract: We prove that the 1-d quantum harmonic oscillator is stable under spatially localized, time quasi-periodic perturbations on a set of Diophantine frequencies of positive measure. This proves a conjecture raised by Enss-Veselic in their 1983 paper [EV] in the general quasi-periodic setting.The motivation of the present paper also comes from construction of quasi-periodic solutions for the corresponding nonlinear equation. Contents 1. 2. 3. 4. 5.
Introduction and Statement of the Theorem . . . . . . . . . . . . . The Floquet Hamiltonian in the Hermite-Fourier Basis . . . . . . . Exponential Decay of Green’s Functions at Fixed E: Estimates in θ Frequency Estimates and the Elimination of E . . . . . . . . . . . Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . .
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459 468 471 487 491
1. Introduction and Statement of the Theorem The stability of the quantum harmonic oscillator is a long standing problem since the establishment of quantum mechanics. The Schrödinger equation for the harmonic oscillator in Rn (in appropriate coordinates) is the following: n 1 ∂2 ∂ 2 − 2 + xi ψ, (1.1) −i ψ = ∂t 2 ∂ xi i=1 where we assume ψ ∈ C 1 (R, L 2 (Rn )) Partially supported by NSF grant DMS-05-03563.
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for the moment. We start from the 1 dimensional case, n = 1. (1.1) then reduces to −i
∂ 1 ∂2 ψ= − 2 + x 2 ψ. ∂t 2 ∂x
(1.2)
The Schrödinger operator d2 2 H= − 2 +x dx def
(1.3)
is the 1-d harmonic oscillator. Since H is independent of t, it is amenable to a spectral analysis. It is well known that H has pure point spectrum with eigenvalues λn = 2n + 1, n = 0, 1..,
(1.4)
and eigenfunctions (the Hermite functions) Hn (x) −x 2 /2 e h n (x) = √ , n = 0, 1 . . . , 2n n!
(1.5)
where Hn (x) is the n th Hermite polynomial, relative to the weight e−x (H0 (x) = 1) and ∞ 2 e−x Hm (x)Hn (x)d x −∞ √ = 2n n! π δmn . (1.6) 2
Using (1.4–1.6), the normalized L 2 solutions to (1.1) are all of the form ψ(x, t) =
∞
an h n (x)ei
λn 2
t
|an |2 = 1 ,
(1.7)
n=0
corresponding to the initial condition ψ(x, 0) =
∞
an h n (x)
|an |2 = 1 .
(1.8)
n=0
The functions in (1.7) are almost-periodic (in fact periodic here) in time with frequencies λn /4π, n = 0, 1 . . . . Equation (1.2) generates a unitary propagator U (t, s) = U (t − s, 0) on L 2 (R). Since the spectrum of H is pure point, ∀u ∈ L 2 (R), ∀, ∃R, such that inf U (t, 0)u L 2 (|x|≤R) ≥ (1 − )u t
(1.9)
by using eigenfunction (Hermite function) expansions. The harmonic oscillator (1.3) is an integrable system. The above results are classical. It is natural to ask how much of the above picture remains under perturbation, when the system is no longer integrable. In this paper, we investigate stability of the 1-d harmonic
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oscillator under time quasi-periodic, spatially localized perturbations. To simplify the exposition, we study the following “model” equation: ν ∂ 1 ∂2 2 −i ψ = cos(ωk t + φk )ψ, − 2 + x ψ + δ|h 0 (x)|2 ∂t 2 ∂x
(1.10)
k=1
on C 1 (R, L 2 (R)), where 0 < δ 1, ω = {ωk }νk=1 ∈ [0, 2π )ν , φ = {φk }νk=1 ∈ [0, 2π )ν , h 0 (x) = e−x
2 /2
(1.11)
is the 0th Hermite function.
In particular, we shall study the validity of (1.9) for solutions to (1.10), when U is the propagator for (1.10). The method used here can be generalized to treat the equation 1 ∂2 ∂ 2 − 2 + x ψ + δV (t, x), −i ψ = ∂t 2 ∂x where V is C0∞ in x and analytic, quasi-periodic in t. The perturbation term, O(δ) term in (1.10) is motivated by the nonlinear equation: −i
∂ 1 ∂2 ψ= − 2 + x 2 ψ + Mψ + δ|ψ|2 ψ (0 < δ 1), ∂t 2 ∂x
(1.12)
where M is a Hermite multiplier, i.e., in the Hermite function basis, M = diag (Mn ), Mu =
∞
Mn ∈ R,
Mn (h n , u)h n , for all u ∈ L 2 (R).
n=0
Specifically, (1.10) is motivated by the construction of time quasi-periodic solutions to (1.12) for appropriate initial conditions such as ψ(x, 0) =
ν
cki h ki (x).
(1.13)
i=1
In (1.10), for computational simplicity, we take the spatial dependence to be |h 0 (x)|2 as it already captures the essence of the perturbation in view of (1.12, 1.13, 1.5). The various computations and the theorem extend immediately to more general finite combinations of h k (x).
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The Floquet Hamiltonian and formulation of stability. It follows from [Y2,Y3] that (1.10) generates a unique unitary propagator U (t, s), t, s ∈ R on L 2 (R), so that for every s ∈ R and u 0 ∈ B 2 = { f ∈ L 2 (R)| f 2B 2 = x α ∂xβ f 2L 2 < ∞}, |α+β|≤2
u(·) = U (·, s)u 0 ∈ C 1 (R, L 2 (R)) ∩ C 0 (R, B 2 )
(1.14)
is a unique solution of (1.10) in L 2 (R) satisfying u(s) = u 0 . When ν = 1, (1.10) is time periodic with period T = 2π/ω. The 1-period propagator U (T +s, s) is called the Floquet operator. The long time behavior of the solutions to (1.10) can be characterized by means of the spectral properties of U (T + s, s) [EV,Ho,YK]. Furthermore the nature of the spectrum of U is the same (apart from multiplicity) as that of the Floquet Hamiltonian K [Y1]: ∂ 1 ∂2 2 K = iω + − 2 + x ψ + δ|h 0 (x)|2 cos φ ∂φ 2 ∂x on L 2 (R) ⊗ L 2 (T), where L 2 (T) is L 2 [0, 2π ) with periodic boundary conditions. Decompose L 2 (R) into the pure point H pp and continuous Hc spectral subspaces of the Floquet operator U (T + s, s): L 2 (R) = H pp ⊕ Hc . We have the following equivalence relations [EV,YK]: u ∈ H pp (U (T + s, s)) if and only if ∀ > 0, ∃R > 0, such that inf t U (t, s)u L 2 (|x|≤R) ≥ (1 − )u; and u ∈ Hc (U (T + s, s)) if and only if ∀R > 0, 1 t limt→±∞ dt U (t , s)u2L 2 (|x|≤R) = 0. t 0 (Needless to say, the above statements hold for general time periodic Schrödinger equations.) When ν ≥ 2, (1.10) is time quasi-periodic. The above constructions extend for small δ, cf. [Be,E,JL] leading to the Floquet Hamiltonian K : ν ν ∂2 ∂ 1 2 K =i − 2 + x ψ + δ|h 0 (x)|2 ωk + cos φk (1.15) ∂φk 2 ∂x k=1
k=1
L 2 (R) ⊗ L 2 (Tν ), cf. [BW1]. This is related to the so called reducibility of skew prod-
on uct flows in dynamical systems, cf. [E]. We note that the Hermite-Fourier functions: e−in·φ h j (x), n ∈ Zν , φ ∈ Tν ,
j ∈ {0, 1 . . .}
(1.16)
provide a basis for L 2 (R) ⊗ L 2 (Tν ). We say that the harmonic oscillator H is stable if K has pure point spectrum. Let s ∈ R. This implies (by expansion using eigenfunctions of K ) that given any u ∈ L 2 (R), ∀ > 0, ∃R > 0, such that inf t U (t, s)u L 2 (|x|≤R) ≥ (1 − )u, a.e. φ, cf. [BW1,JL]. So (1.9) remains valid and we have dynamical stability. We now state the main results pertaining to (1.10).
(1.17)
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
463
Theorem. There exists δ0 > 0, such that for all 0 < δ < δ0 , there exists ⊂ [0, 2π )ν of positive measure, asymptotically full measure: mes → (2π )ν as δ → 0, such that for all ω ∈ , the Floquet Hamiltonian K defined in (1.15) has pure point spectrum: σ (K ) = σpp . Moreover the Fourier-Hermite coefficients of the eigenfunctions of K have subexponential decay. As an immediate consequence, we have Corollary. Assume that is as in the theorem. Let s ∈ R. For all ω ∈ , all u ∈ L 2 (R), all > 0, there exists R > 0, such that inf t U (t, s, φ)u L 2 (|x|≤R) ≥ (1 − )u, a.e. φ,
(1.18)
where U is the unitary propagator for (1.10). We note that this good set of ω is a subset of Diophantine frequencies. This is typical for a KAM type of persistence theorem. Stability under time quasi-periodic perturbations as in (1.10) is, generally speaking a precursor for stability under nonlinear perturbation as in (1.12) (cf. [BW1,BW2]), where M plays the role of ω and varies the tangential frequencies. The above theorem resolves the Enss-Veselic conjecture dated from their 1983 paper [EV] in a general quasi-periodic setting. A sketch of the proof of the theorem. Instead of working with K defined on L 2 (R) ⊗ L 2 (Tν ) directly, it is more convenient to work with its unitary equivalent H on 2 (Zν × {0, 1 . . .}), using the Hermite-Fourier basis in (1.16). We have δ 1 + W ⊗ (1.19) H = diag n · ω + j + 2 2 on 2 (Zν × {0, 1 . . .}), where W acts on the j indices, j = 0, 1, 2 . . ., W j j ∼
1 j + j
2
e
− (2(j−j+j j) )
for j + j 1;
(1.20)
acts on the n indices, n ∈ Zν , nn = 1, |n − n |1 = 1, = 0, otherwise.
(1.21)
The reduction from (1.15) to (1.19–1.21) is performed in Sect. 2. The main work is to compute W , which involves integrals of products of Hermite functions. We will explain shortly this computation, which is independent from the main thread of construction. √ The principal new feature here is that W is long range. The j th row has width O( j) about the diagonal element W j j . It is not and cannot be approximated by a convolution matrix. The potential x 2 breaks translational invariance. The annihilation and creation operators of the harmonic oscillator a = √1 ( ddx + x), a ∗ = √1 (− ddx + x), satisfying 2 2 [a, a ∗ ] = 1, are generators of the Heisenberg group. So (1.19) presents a new class of problems distinct from that considered in [B1,B2,B3,BW1,BW2,EK,Ku1,KP].
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The proof of pure point spectrum of H is via proving pointwise decay as |x − y| → ∞ of the finite volume Green’s functions: (H − E)−1 (x, y), where are finite subsets of Zν × {0, 1 . . .} and Zν × {0, 1 . . .}. We need decay of the Green’s functions at all scales, as assuming E an eigenvalue, a priori we do not have information on the center and support of its eigenfunction ψ. The regions where (H − E)−1 has pointwise decay is precisely where we establish later that ψ is small there. For the initial scales, the estimates on G (E) = (H − E)−1 are obtained by direct perturbation theory in δ for 0 < δ 1. For subsequent scales, the proof is a multiscale induction process using the resolvent equation. Assume we have estimates on G for cubes at scale L and is a cube at a larger scale L, L L . Intuitively, if we could establish that for most of ⊂ , G (E) has pointwise decay, then assuming we have some a priori estimates on G (E), we should be able to prove that G (E) also has pointwise decay. There are “two” directions in the problem, the higher harmonics direction n and the spatial direction j. The off-diagonal part of H is Toeplitz in the n direction, corresponding to the discrete Laplacian . Since the frequency ω is in general a vector (if ν ≥ 2), n · ω does not necessarily → ∞ as |n| → ∞. So the n direction is non-perturbative. We use estimates on G and semi-algebraic techniques as in [BGS,BW1] to control the number of resonant , where G is large, in . In the j direction, we do analysis, i.e., perturbation theory. This is the new feature. From (1.19) and Schur’s lemma, W ⊗ = O(1). So the 2 norm of the perturbation does not decay (relative to eigenvalue spacing) in j. However when δ = 0, H is diagonal with eigenvalues n · ω + j and eigenfunctions δn, j , the canonical basis for 2 (Zν × {0, 1 . . .}). We have 1 ( j ≥ 1), [W ⊗ ]δn, j = O j 1/4 which decays in j. This is intuitively reasonable, as W stems from a spatially localized perturbation from (1.10). As j increases, the Hermite functions h j become more extended, cf. (1.5). So the effect of the spatial perturbation should decrease as j increases. Assuming ω is Diophantine: n · ωT ≥
c (c > 0, n = 0, α > 2ν), |n|α
where · T is the distance to the nearest integer; this enables us to preserve local eigenvalue spacing for which are appropriately proportioned in n, j. This in turn leads to decay of Green’s functions. Combining the estimates in the n and j directions, we obtain estimates on the Green’s function at the larger scale L.
Integrals of products of Hermite functions. From (1.15, 1.19), computation of W involves computing the following integrals: ∞ h 20 (x)h m (x)h n (x)d x −∞ ∞ 1 2 =√ e−2x H02 (x)Hm (x)Hn (x)d x, m, n = 0, 1 . . . , (1.22) n+m 2 m!n! −∞
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
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where Hm , Hn are respectively the m th , n th Hermite polynomial, H0 (x) = 1. Let ∞ 2 e−2x H02 (x)Hm (x)Hn (x)d x. I = −∞
√ 2 The idea is to view e H02 (x) as e−x H0 ( 2x), i.e., the 0th Hermite function relative to 2 the weight e−2x and to use the generating function of Hermite polynomials to reexpress −x 2
Hm (x)Hn (x) =
m+n
√ a H ( 2x).
(1.23)
=0
We then have
I = a0
√ 2 [H0 ( 2x)]2 e−2x d x
= a0 π/2 using (1.6). This computation is carried out in Sect. 2, (2.7–2.10), recovering an apparently classical result, which could be found in e.g., [GR,PBM]. More generally, we are interested in computing ∞ 2 I = e−2x H p (x)Hq (x)Hm (x)Hn (x)d x, p, q, m, n = 0, 1 . . . , (1.24) −∞
which are needed for the nonlinear equation or if we consider more general perturbations of the harmonic oscillator. Following the same line of arguments, we decompose H p (x)Hq (x) into H p (x)Hq (x) =
p+q
√ b H ( 2x).
(1.25)
=0
Combining (1.23) with (1.25), and assuming (without loss of generality), p + q ≤ m + n, we then have I =
p+q
a b c ,
=0
√
∞ 2 where c = −∞ [H ( 2x)]2 e−2x d x. The computation for general p, q is technically more involved and is carried out in [W]. Unlike the special case p = q = 0, we did not find the corresponding result for general p, q in the existing literature. The computation of I in (1.24) is exact (see (2.10)), reflecting the integrable nature of the quantum harmonic oscillator. The proof of the theorem is, however, general. It is applicable as soon as the kernel W satisfies (1.20). Following the precedent discussion on I for general p, q, and using properties of the Hermite series (cf. [T] and references therein), one should be able to extend the theorem to V , which are C0∞ in x and analytic quasi-periodic in t, leading to perturbation kernels in the Hermite-Fourier basis satisfying conditions similar to (1.20) in the j direction and the exponential decay condition in the n direction.
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When the perturbation V is independent of time and is a 0th order symbol, satisfying |∂ α V | ≤ Cα (1 + |x|)−α , α = 0, 1 . . .
(1.26)
the corresponding Schrödinger equation has been studied in e.g., [BBL,KRY,Z], where it was shown that certain properties of the harmonic oscillator equation extend to the perturbed equation. The spectral property needed for the construction here is more detailed and stringent. Hence it is reasonable to believe that the set of potentials V will be more restrictive than that in (1.26) Some perspectives on the theorem. The theorem shows that for small δ, there is a subset ⊂ [0, 2π )ν of Diophantine frequencies of positive measure, such that if ω ∈ , then (1.9) is satisfied. Hence spatially localized solutions remain localized for all time. It is natural to ask what happens if the forcing frequencies ω are in the complement set, ω ∈ c . If ω is rational, the perturbing potential V is bounded and has sufficiently fast decay at infinity, it is known from general compactness argument [EV] that the Floquet Hamiltonian has pure point spectrum. In our example, this can be seen as follows. In (1.10) restricting to periodic perturbation (ν = 1), it is easy to see that for ∀ω, A = (nω + j + z)−1 W ⊗ , where z = 1 is compact.
(1.27)
Assume ω is rational: ω = p/q, (q = 0). Since H0 = nω+ j has pure point spectrum (with infinite degeneracy) and the spacing between different eigenvalues is 1/q, (1.27) implies that H has pure point spectrum. When ω is irrational, H0 typically has dense spectrum. No conclusion can be drawn from (1.27). It is worth remarking that (1.27) holds for all scalar ω. In the quasi-periodic case, ω is a vector, the compactness argument breaks down. The proof of Lemma 3.5 in the present paper is a replacement. If V is unbounded, we have a different situation. The results in [HLS,GY] combined with [YK] show that for the following unbounded time periodically perturbed harmonic oscillator: ∂u 1 i = (− + x 2 )u + 2(sin t)x1 u + µV (t, x)u, x = (x1 , . . . xn ) ∈ Rn , (1.28) ∂t 2 where V (t, x) is a real valued smooth function of (t, x), satisfying V (t + 2π, x) = V (t), |V (t, x)| |x| as x → ∞, |∂xα V (t, x)| ≤ Cα , |α| ≥ 1, the solutions diffuse to infinity as t → ∞. More precisely, for all u 0 ∈ L 2 (Rn )∩ H 2 (Rn ), for any R > 0, the solution u t satisfies 1 T lim dtu t L 2 (|x|≤R) = 0. (1.29) T →±∞ T 0 In (1.28), ν = 1 (periodic), ω = 1 , ω ∈ c , (1.29) is an opposite of (1.9). However the perturbation is unbounded. Moreover the proof in [GY] uses in an essential way that the potential is linear at infinity, hence positivity of the commutator: [ ddx1 , x1 ] = 1. In the exactly solvable case where the time periodic perturbation is quadratic in the spatial coordinates, it is known that the Floquet Hamiltonian exhibits a transition between pure point and continuous spectrum as the frequency is varied [Co1]. The perturbation there is again unbounded.
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Some related results. To our knowledge, when ω ∈ is nonresonant, there were no results in the literature on the perturbed harmonic oscillator equation of type (1.10), even in the time periodic case, i.e., ω ∈ [0, 2π ). The main difficulties encountered by the traditional KAM method seem to be (i) the eigenvalue spacing for the unperturbed operator does not grow, λk+1 − λk = 1, (ii) the perturbation W in the Hermite basis has slow decay (1.20). When the eigenvalue spacing for the unperturbed operator grows: |E j+1 − E j | > j β (β > 0), which corresponds to a potential growing faster than quadratically at infinity, and when the perturbation is periodic in time, related stability results were proven in [DS]. In [Co2], under time periodic perturbation and replacing W in (1.20) by a faster decaying kernel, hence decaying norm in j, which no longer corresponds to the physical case of harmonic oscillator under time periodic, spatially localized perturbation, stability results were also proven. Both papers used some modified KAM method. Motivation for studying (1.10). As mentioned earlier, the motivation for analyzing (1.10) partly comes from the nonlinear equation (1.12). In [B1,B2,B3,EK], time quasi-periodic solutions were constructed for the nonlinear Schrödinger equation in Rd with Dirichlet or periodic boundary condition i
∂ ψ = (− + M)ψ + δ|ψ|2 p ψ, ∂t
( p ∈ N+ ; 0 < δ 1),
(1.30)
where M is a Fourier multiplier; see [Ku1,KP] for the Dirichlet case in R with a potential in place of M. In [BW2], time quasi-periodic solutions were constructed for the nonlinear random Schrödinger equation in Zd , i
∂ ψ = (− + V )ψ + δ|ψ|2 p u, ∂t
( p ∈ N+ ; , 0 < δ 1),
(1.31)
where V = {v j } j∈Zd is a family of random variables. The proofs in [B1,B2,B3,BW2] use an operator method, which traces its origin to the study of Anderson localization [FS]. This method was first applied in the context of Hamiltonian PDE in [CW]. The proofs in [EK,Ku1,KP] use a KAM type of method. In (1.30) (specializing to 1-d), the eigenvalues of the linear operator are n 2 , so E n+1 − E n ∼ n, the eigenfunctions einx , however, are extended: |einx | = 1 for all x. Let us call this case A, where there is eigenvalue separation. In (1.31), the eigenvalues of the linear operator form a dense set, the eigenfunctions, on the other hand are not only localized but localized about different points in Zd from Anderson localization theory, see e.g., [GB,GK]. This is case B, where there is eigenfunction separation. The existence of time quasi-periodic solutions, i.e., KAM type of solutions in A is a consequence of eigenvalue separation; while in B, eigenfunction separation. Equation (1.10) and its nonlinear counterpart 1 ∂2 ∂ 2 − 2 + x ψ + Mψ + δ|ψ|2 p ψ, −i ψ = ( p ∈ N+ ; 0 < δ 1), ∂t 2 ∂x (1.32) where M is a Hermite multiplier, stand apart from both (1.30, 1.31). It is neither A, nor B. There is eigenvalue spacing, but it is a constant: λn+1 − λn = 1. In particular, it does not grow with n. The eigenfunctions (Hermite functions) h n are “localized” about the
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origin. But they become more extended as n increases because of the presence of the Hermite polynomials, cf. (1.5). This in turn leads to the long range kernel W in (1.20) and long range nonlinearity in (1.32) in the Hermite function basis, cf. [W]. From the KAM perspective a la Kuksin, this is a borderline case, where Theorem 1.1 in [Ku2] does not apply. The more recent KAM type of theorem in [EK] does not apply either, because W is long range and not close to a Toeplitz matrix (cf. (1.20)) for the reasons stated earlier. These are the features which make (1.10, 1.32) interesting from a mathematics point of view, aside from its apparent relevance to physics. 2. The Floquet Hamiltonian in the Hermite-Fourier Basis Recall from Sect. 1, the Floquet Hamiltonian K =i
ν k=1
ν ∂ 1 ∂2 2 ωk + cos φk − 2 + x + δ|h 0 (x)|2 ∂φk 2 ∂x
(2.1)
k=1
on L 2 (R) ⊗ L 2 (Tν ), where 0 < δ 1, ωk ∈ [0, 2π ), k = 1, . . . , ν φk ∈ [0, 2π ), k = 1, . . . , ν h 0 (x) = e−x
2 /2
(2.2)
.
As mentioned in Sect. 1, h 0 (x) is the 0th Hermite function, 0th eigenfunction of the 1-d harmonic oscillator and more generally, d2 2 − 2 + x h n = λn h n , dx (2.3) λn = 2n + 1, n = 0, 1 . . . , Hn (x) −x 2 /2 h n (x) = √ e , n = 0, 1 . . . , 2n n! where Hn (x) is the n th Hermite polynomial, relative to the weight e−x (H0 (x) = 1) and ∞ √ 2 e−x Hm (x)Hn (x)d x = 2n n! π δmn . (2.4) 2
−∞
Integral of products of Hermite functions. We express (2.1) in the Hermite function basis for small δ and compute the integral: ∞ h 20 (x)h m (x)h n (x)d x −∞ (2.5) ∞ 1 2 =√ e−2x H02 (x)Hm (x)Hn (x)d x, m, n = 0, 1 . . . . 2n+m m!n! −∞
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
The integral of the more general product ∞ 2 e−2x H p (x)Hq (x)Hm (x)Hn (x)d x −∞
469
(2.6)
= (H p (x)Hq (x)e−x , Hm (x)Hn (x)e−x ), 2
2
p, q, m, n = 0, 1 . . .
is done in [W]. We use generating functions of Hermite polynomials to find a of (1.23) as follows. Since 2
∞ n t
2
n=0 ∞
e2t x−t = e2sx−s =
Hn (x),
(2.7)
sm Hm (x), m!
(2.8)
n!
m=0
which can be found in any mathematics handbook (cf. [CFKS,T] for connections with the Mehler formula), multiplying (2.7, 2.8), we obtain t n sm 2 2 e2(t+s)x−(t +s ) = Hn (x)Hm (x) n!m! n,m √ √ ) 2x−( t+s √ )2 2( t+s
1
2 =e 2 · e− 2 (t−s) t+s ∞ ∞ ( √ ) √ (t − s)2 p 2 = H ( 2x) · (−1) p · . ! 2 p p!
=0
2
(2.9)
p=0
√ From (1.23), we are only interested in the coefficient in front of H0 ( 2x). So we set = 0. To obtain a0 , we equate the coefficient in front of t n s m . Comparing the LHS with the RHS of (2.9), n, m must have the same parity, otherwise it is 0. We deduce a0 =
(−1) 2
m+n 2
n−m 2
( m+n 2 )!
· (n + m)!,
n, m same parity,
=0
(2.10)
otherwise,
by taking 2 p = n + m, which is the only contributing term. Taking into account the normalization factors in the third equation of (2.3), we then obtain Lemma 2.1. def
Wmn =
=
∞ −∞
h 20 (x)h m (x)h n (x)d x n−m 2
(−1) √
2m+n =0
(m + n)! m+n m!n! ( 2 )! ·
π 2
m, n same parity
(2.11)
otherwise.
Let N=
n−m n+m ,k= , 2 2
(2.12)
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W.-M. Wang
assuming n ≥ m, without loss. When N 1, n−m (−1) 2 N ! 1 , Wmn = 1 + O √ N 2N (N + k)!(N − k)! 1 2 |Wmn | ≤ √ e−k /2N . N
(2.13) (2.14)
Proof. We only need to obtain the asymptotics in (2.13, 2.14). This is an exercise in Stirling’s formula: n n √ 1 1 n! = + 2π n 1 + + ··· (2.15) e 12n 288n 2 or its log version √ 1 log n! = n + log n − n + log 2π + . . . . 2
(2.16)
Here it is more convenient to use the latter. Using (2.12, 2.16), log
(m + n)! 2m+n ( m+n 2 )!
= log
= N log N − N + So
(2N )! − log 22N N!
1 log 2 + O(N −1 ). 2
(2.17)
(m + n)! 1 N! = 1 + O , √ m+n m+n 2 ( 2 )! N πN
(2.18)
using (2.17). Hence Wmn
n−m (−1) 2 N ! 1 , = 1+O √ N 2N (N + k)!(N − k)!
N 1
which is (2.13). Using the fact that n! =
√
2π n
n n e
eλn
with 1 1 < λn < , for all n ≥ 1, 12n + 1 12n and applying the inequalities (with x = k/N ): def
φ(x) = (1 + x) log(1 + x) + (1 − x) log(1 − x) ≥ x 2 for all x ∈ [0, 1) and φ(x) ≥ ax 2 with a > 1 for x ∈ [7/10, 1), we obtain (2.14). (When x = k/N = 1, (2.14) follows by a direct computation using Stirling’s formula.)
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
From (2.14), the matrix element Wmn has subexponential decay |m−n|2 m+n 1 1 N= |Wmn | ≤ √ e− 2N 2 N √ √ when |m − n| > N . When |m − n| ≤ N , we only have the estimate 1 |Wmn | ≤ √ . N
471
(2.19)
(2.20)
√ Hence W is a matrix with a slowly enlarging region of size O( N ) around the prin2 2 ∞ cipal diagonal √ Wnn , the norm of W is of O(1), but the local to norm is of order O(1/ N ). These new features will need to be taken into account when we do the analysis in Sects. 3, 4. (2.1) in the Hermite-Fourier basis. In the Hermite-Fourier basis, e−in·φ h j (x), n ∈ Zν , φ ∈ Tν , j ∈ {0, 1 . . .}, the Floquet Hamiltonian, which is unitarily equivalent to the K defined in (1.15) is then δ 1 + W ⊗ H = diag n · ω + j + (2.21) 2 2 on 2 (Zν × {0, 1 . . .}), where W is the matrix operator defined in (2.11), acting on the j indices, j = 0, 1, 2 . . ., acts on the n indices, n ∈ Zν , nn = 1, |n − n |1 = 1, = 0, otherwise.
(2.22)
˜ δ. We then have Let H˜ = H − 1/2 and rename H˜ , H ; let δ˜ = δ/2 and rename δ, def
H = diag(n · ω + j) + δW ⊗
(2.23)
on 2 (Zν × {0, 1 . . .}), with W , defined in (2.11, 2.22). 3. Exponential Decay of Green’s Functions at Fixed E: Estimates in θ Let H be the operator defined in (2.23), i.e., H = diag (n · ω + j) + δW ⊗
(3.1)
on 2 (Zν × {0, 1, . . .}), where n ∈ Zν , j ∈ {0, 1, . . .}, 0 < δ 1.
(3.2)
W acts on the j indices, j = 0, 1, 2 . . ., j− j
W j j
(−1) 2 ( j + j )! = · j+ j , 2 j+ j j! j ! ( 2 )!
j, j same parity,
=0
otherwise.
(3.3)
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W.-M. Wang
Write J=
j + j j − j ,k= , (assume j ≥ j ). 2 2
When J 1, J! J (J + k)!(J − k)! 1 − k2 ≤ √ e 2J J
|W j j | ≤ √
(3.4)
from (2.14). acts on the n indices, n ∈ Zν , nn = 1, |n − n |1 = 1, = 0, otherwise.
(3.5)
In view of the theorem, our aim is to prove that on a good set of ω, H has pure point spectrum. To achieve that goal, we add a parameter θ (θ ∈ R) to H : H (θ ) = H + θ = diag (n · ω + θ + j) + δW ⊗ .
(3.6)
We consider a sequence of finite volume Green’s functions at fixed E: G (θ, E) = (H (θ ) − E)−1 ,
(3.7)
where are cubes in Zν × {0, 1 . . .}, Zν × {0, 1 . . .} in an appropriate way, H (θ ) are H (θ ) restricted to . In this section, E ∈ R, ω ∈ [0, 2π )ν are fixed and we assume that ω is a Diophantine frequency. We do estimates in θ . Specifically, we prove (inductively) that for any large enough, away from a set of θ of small measure in R, G (θ, E) is bounded and |G (θ, E)(x, y)| has subexponential decay for |x − y| ∼ linear scale of . (For a precise statement, see Proposition 3.10.) As mentioned in Sect. 1, the proof is a combination of a non-perturbative part in the n direction and a perturbatuve part in the j direction. We note that this set of bad θ depends on E, ω; the estimate on the measure is, however, uniform in E and ω for Diophantine ω. In the next section (Sect. 4), we eliminate the E dependence by excluding double resonances and converting the estimate in θ into estimates in ω in the process. The conversion is possible because ω, θ appear in (3.6) in the form n · ω + θ . Below we start the induction process. To simplify notations, we extend H to a linear operator on 2 (Zν+1 ) with W j j as in (3.3) for j, j ∈ {0, 1 . . .} and W j j = 0 otherwise. 3.1. The initial estimate (0th step). For any subset ⊂ Zν+1 , we define H (θ )(n, j; n , j ) = H (θ )(n, j; n , j ), (n, j; n , j ) ∈ × , = 0, otherwise.
(3.8)
Let 0 = [−J, J ]ν+1 for some J > 0 to be determined. For a fixed E, we study the Green’s function G 0 (θ, E) = (H0 (θ ) − E)−1 by doing perturbation theory in δ. We have
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
473
Lemma 3.1. Assume 0 < δ 1. For any fixed σ , 0 < σ < 1/4, there exists J ∈ N such that the following statement is satisfied. Let 0 = [−J, J ]ν+1 . There exists a set B(0 , E) in R, with mes B(0 , E) ≤ e−J
σ/2
,
(3.9)
such that if θ ∈ R\B(0 , E), then σ
G 0 (θ, E) ≤ e J ,
(3.10)
|G 0 (θ, E)(n, j; n j )| ≤ e
−|(n, j)−(n , j )|1/4
,
|(n, j) − (n , j )| > J/10. (3.11)
Proof. Since this is the initial estimate, we do perturbation theory in δ. Let κ > 0. Let B(0 , E) be the set such that if θ ∈ B(0 , E), then |n · ω + j + θ − E| ≤ 2κ
(3.12)
for some (n, j) ∈ 0 . Clearly mes B(0 , E) ≤ 4|0 |κ = 4(2J + 1)ν+1 κ.
(3.13)
Let def
D0 (θ ) = diag(n · ω + j + θ ), (n, j) ∈ 0
(3.14)
be the unperturbed diagonal operator. Since δW ⊗ 2 ≤ O(δ), if θ ∈ / B(0 , E), then G 0 (θ, E) = (D0 (θ ) − E + δW ⊗ )−1 ≤ κ −1
(3.15)
for κ δ. From the resolvent equation G 0 (θ, E)(n, j; n j ) = (n · ω + j + θ − E)−1 [(δW ⊗ )G 0 (θ, E)](n, j; n j ) (3.16) for (n, j) = (n , j ). Hence |G 0 (θ, E)(n, j; n j )| ≤ O(1)δκ −2 , if |(n, j) − (n , j )| > J/10.
(3.17)
J = | log δ| (hence δ = e−J ),
(3.18)
Let
κ=e
−J σ
.
Equations (3.13–3.17) then imply (3.9–3.11).
(3.19)
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W.-M. Wang
3.2. A Wegner estimate in θ for all scales. We now state an apriori estimate in θ for (H (θ ) − E)−1 valid for all finite subsets ⊂ Zν+1 and all δ. This estimate will be useful in the induction process. Following the Anderson localization tradition, we call it a Wegner estimate. Proposition 3.2. For any E ∈ R, and any finite subset in Zν+1 , the following estimate is satisfied for all κ > 0: mes {θ | dist (E, σ (H (θ ))) ≤ κ} ≤ 2||κ.
(3.20)
Proof. Let λk , k = 1, . . . , || be eigenvalues of H (θ = 0). Then ||
{θ | dist (E, σ (H (θ ))) ≤ κ} = ∪k=1 {θ ||E − θ − λk | ≤ κ}. It follows that the measure of the left side is bounded by 2||κ.
(3.21)
Our goal now is to obtain inductively the equivalent of estimates (3.9–3.11) for larger subsets , Zν+1 ⊃ ⊃ 0 . We note that in proving Lemma 3.1, we did perturbation theory about the diagonal operator D0 (θ ) = diag (n · ω + j + θ )|(n, j)∈0
(3.22)
using the smallness of δ. This was sufficient for one initial scale. For subsequent scales, however, we need more detailed information on the spectrum of H . 3.3. Local spectral property of H . We assume that ω is Diophantine, i.e., ∃ c > 0, α > 2ν, such that |n · ω + j| ≥
c |n|α
(3.23)
for all n ∈ Zν \{0}, all j ∈ Z. In this subsection, we make statements which hold for any fixed θ . We look at finite subsets ⊂ Zν+1 , such that j = 0, if (n, j) ∈ . When j 1, this is the perturbative region. Proposition 3.3. Assume ω satisfies (3.23). Let 0 < β < 1/5α and 3/4 < β < 1. Let be a rectangle centered at (N , 2L) ∈ Zν × Z, where Zν is identified with Zν × {0}:
= (N , 2L) + [−L β , L β ]ν × [−L β , L β ] ⊂ Zν+1 .
(3.24)
|L| 1.
(3.25)
Assume
For any fixed θ ∈ R, the eigenvalues λn, j (θ ) of H (θ ) satisfy |λn, j (θ ) − λn , j (θ )| >
1 L 1/5
(n, j) = (n j );
(3.26)
the eigenfunctions φn, j may be chosen such that φn, j − δn, j <
1 . L 1/20
(3.27)
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
475
Remark. It is crucial to note that the estimates (3.26, 3.27) are independent of θ and the specific ω satisfying (3.23). In Lemma 3.5, we exploit further the consequences of (3.26). Proof of Proposition 3.3. Let (n, j), (n , j ) ∈ , (n, j) = (n , j ). Then for all θ , the difference of the diagonal elements |n · ω + j + θ − (n · ω + j + θ )| = |(n − n ) · ω + ( j − j )| c 1 ≥ β α > 1/5 for 0 < β < 1/5α, L 1, L L (3.28) from (3.23). Use as approximate eigenfunctions δn, j with approximate eigenvalues λ˜ n, j (θ ) = n · ω + j + θ , (n, j) ∈ , and let (H − λ˜ n, j )δn, j = ψ.
(3.29)
ψ(n , j ) = δW j j , |n − n | = 1, (n , j ) ∈ = 0, otherwise,
(3.30)
ψ2 = O(|L|−1/4 )
(3.31)
Then from (3.7),
and
from (3.4). Equations (3.28–3.31) imply that λ˜ n, j (θ ) = n ·ω + j +θ is an approximate eigenvalue of H to O(|L|−1/4 ). This can be seen as follows. Let λ = λ˜ n, j (θ ). Assume dist (λ, σ (H (θ ))) > O(|L|−1/4 ). Take any f , f 2 () = 1, f = of H . Then
cm ψm ,
(H − λ) f =
|cm |2 = 1, where ψm are eigenfunctions
(E m − λ)cm ψm ,
where E m is the corresponding eigenvalue for ψm . Hence (H − λ) f =
(E m − λ)2 |cm |2 > O(|L|−1/4 ).
(3.32)
This is a contradiction if f = δn, j . Hence (3.28) gives (3.26). Equation (3.27) follows from (3.26, 3.31) and standard perturbation theory, see e.g., [Ka].
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3.4. The first iteration (1st step). We now increase the scale from J to J C , where C > 1 (independent of δ) is the geometric expansion factor, which will be specified in Sect. 3.5. Recall that J is the initial scale, large enough so that (3.18) holds. Hence on 0 = [−J, J ]ν+1 , (3.9–3.11) hold. Let =[−J C , J C ]ν+1 . def
(3.33)
Our aim is to prove the analogue of (3.9–3.11) when 0 is replaced by . The general strategy in going from scale J to scale J C is to distinguish the region near j = 0, where we use the estimate from the previous scale, here (3.9–3.11) and non-perturbative arguments and the region away from j = 0, where we use Proposition 3.3 and perturbation theory. The general iteration strategy here is similar to that in [BW1]. Toward that end, we define def
T ={(n, j) ∈ | | j| ≤ 2J − 1}.
(3.34)
0 (n, j) = (n, j) + [−J, J ]ν+1 , (n, j) ∈ T
(3.35)
Let
be cubes of the previous scale. Let
∗ (n, 2 j) = (n, 2 j) + [− j β , j β ]ν ×[− j β , j β ] (0 < β < 1/5α, 3/4 < β < 1), (n, 2 j) ∈ \T , (3.36) be cubes of type (3.24). We cover with 0 , ∗ cubes, i.e., T with 0 , \T with ∗ . G are then obtained by using the resolvent equation, (3.9–3.11), Propositions 3.2 and 3.3. We implement this strategy in our first iteration. This iteration is special, as in the tube region T , we use smallness of δ, cf. (3.18). We need the following notion of pairwise disjointness. Let Sk , k = 1, . . . , K be finite K . If S ∩ S = ∅, ∀k = k , then we say there is sets , Sk = Sk if k = k . Let S = {Sk }k=1 k k 1 pairwise disjoint set in S. More generally, if ∃I1 , I2 , . . . , I P , I p ∩ I p = ∅, if p = p , {I p } Pp=1 = {1, 2, . . . , K } such that Sk ∩ Sk = ∅ if and only if k, k ∈ I p for some p. Then we say there are P pairwise disjoint sets in S. Lemma 3.4. Let 0 be a covering of T with 0 cubes defined in (3.35). Assume ω is Diophantine satisfying (3.23). Fix E, σ (0 < σ < 1/4) as in Lemma 3.1. For all θ , there ˜ ∈ 0 , such that exists at most 1 pairwise disjoint σ
dist (E, σ (H˜ (θ ))) ≤ e−J .
(3.37)
˜ = ∅, Moreover if 0 ∈ 0 and 0 ∩ dist (E, σ (H0 (θ ))) ≥
c . J Cα
(3.38)
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
477
σ
Proof. Let 0 ∈ 0 . If |λn − E| < e−J for a λn ∈ σ (H0 (θ ), then since δW ⊗ ≤ Cδ = Ce−J , σ
|n · ω + j + θ − E| ≤ e−J + Ce−J ≤ 2e−J
σ
˜ satisfy (3.37), then for some (n, j) ∈ 0 . Thus if both 0 and |(n − n ) · ω + j − j | ≤ |n · ω + j + θ − E| + |n · ω + j + θ − E| ≤ 4e−J
σ
(3.39) ˜ This implies 0 ∩ ˜ = ∅ for large J , since for some (n, j) ∈ 0 and (n , j ) ∈ . otherwise the left side is larger than c/(2ν J C )α by the Diophantine condition (3.23), which is a contradiction if C J σ / log J . ˜ ∈ 0 , then for any (n, j) ∈ 0 and for some If 0 ∈ 0 is disjoint from ˜ (n , j ) ∈ , |n · ω + j + θ − E| ≥ |(n − n ) · ω + j − j | − |n · ω + j + θ − E| c σ − 2e−J . ≥ C α (2ν J ) Since δW ⊗ ≤ Cδ = Ce−J , (3.38) follows.
(3.40)
˜ = ∅, then (3.10, 3.11) are available. Let ∗ be a covering of \T . For If 0 ∩ ∗ ∈ ∗ , we need Lemma 3.5. Let 0 < β < 1/5α and 3/4 < β < 1. Let ∗ be the rectangle:
∗ (n, 2 j) = (n, 2 j) + [− j β , j β ]ν × [− j β , j β ], (n, 2 j) ∈ \T . For a fixed E ∈ R, there exists W ⊂ R, with mes W ≤ C|∗ |e−J θ ∈ R\W , J 1, |[H∗ (θ ) − E]−1 (n , j ; n , j )| ≤ e
β
− 21 [|n −n |+ | j
β
(|n − n | > j /10, or | j − j | ≥ j /10), [H∗ (θ ) − E]
−1
≤e
Jβ
/2
.
β /2
− j | ] j 1/2
(3.41)
such that for
, (3.42) (3.43)
Proof. For any fixed θ , E, there exists at most 1 bad site b = (n , j ) ∈ ∗ , such that |n · ω + j + θ − E| ≤
1 . j 1/5
This is because if there were (n , j ), (n , j ) ∈ ∗ , (n , j ) = (n , j ) such that |n · ω + j + θ − E| ≤ |n · ω + j + θ − E| ≤
1 j 1/5
,
1 , j 1/5
then |(n − n ) · ω + ( j − j )| ≤
2 j 1/5
,
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W.-M. Wang
which contradicts the Diophantine condition (3.23) on ω: (n − n ) · ωT ≥
c c 2 ≥ β α 1/5 |n − n |α j j
(3.44)
for j ≥ J 1, using (3.41). To obtain (3.42, 3.43), we first consider H∗ \b (θ ) and make estimates on [H∗ \b (θ )− E]−1 (n , j ; n , j ). We then use Proposition 3.2 and the resolvent equation to obtain (3.42). We prove below that [H∗ \b (θ ) − E]−1 2 →2 < 2 j 1/5 ,
(3.45)
for j 1. To obtain estimates on the matrix elements, we use weighted 2 space and show that [H∗ \b (θ ) − E]−1 remain bounded. Equation (3.45) is then the special case when the weight equals to 1. Toward that end, for any p ∈ Zν+1 , we define | p|ν = |( p1 , . . . , pν , 0)| and | p|1 = |(0, . . . , 0, pν+1 )|. Let ρ = {ρa }a∈Zν+1 be a family of weights, such that ρa ( p) = e
| p−a|ν +
| p−a|1 j 1/2
(a ∈ Zν+1 ), ∀ p ∈ Zν+1 ,
(3.46)
where j is as in (3.41). Hence ∀ p, q ∈ Zν+1 , ρa−1 ( p)ρa (q) = e ≤e
−| p−a|ν − | p−q|ν +
| p−a|1 j 1/2
| p−q|1 j 1/2
·e
|q−a|ν +
|q−a|1 j 1/2
,
(3.47)
for all ρa ∈ ρ. Let D be the diagonal part in (3.8). To arrive at (3.42), we consider the deformed operator H˜ ∗ \b (θ ): H˜ ∗ \b (θ )( p, q) =def [ρa−1 H∗ \b (θ )ρa ]( p, q) = D pq + δρa−1 ( p)ρa (q)(W ⊗ ) pq , ˜ pq , = D pq + δ(W˜ ⊗ )
(3.48)
where p, q ∈ ∗ \b and ˜ pq def (W˜ ⊗ ) = ρa−1 ( p)ρa (q)(W ⊗ ) pq .
(3.49)
To prove boundedness of [ H˜ ∗ \b (θ ) − E]−1 , we use resolvent series and perturb about the diagonal. We have formally ˜ [ H˜ ∗ \b (θ ) − E]−1 = (D − E)−1 + δ(D − E)−1 (W˜ ⊗ )(D − E)−1 ˜ ˜ +δ 2 (D − E)−1 (W˜ ⊗ )(D − E)−1 (W˜ ⊗ )(D − E)−1 +....
(3.50)
Let ˜ W =(W˜ ⊗ )(D − E)−1 . def
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
479
Then ˜ n n (n · ω + j + θ − E)−1 , |n − n | = 1 W(n , j ; n , j ) = W˜ j j (3.51) = 0 otherwise. We bound W using Schur’s lemma [Ka]: ⎛ ⎞1/2 ⎛ ⎞1/2 W ≤ ⎝ sup |W(n , j ; n , j )|⎠ ⎝ sup |W(n , j ; n , j )|⎠ , n , j n , j
n , j n , j
(3.52) where (n , j ), (n , j ) ∈ ∗ \b, |W(n , j ; n , j )| = n , j
˜ n n (n · ω + j + θ − E)−1 |, |W˜ j j
|n −n |=1 | j −2 j|≤ j β
(n , j ) ∈ ∗ \b. From (3.47, 3.4),
( j − j )2 | j − j | e ˜ n n | ≤ √ e − 2( j + j ) + j 1/2 |W˜ j j j j )2 200 e −(j − 10 j ≤ √ e , j
(3.53)
where we used j + j ≤ 4 j + 2 j β ≤ 9 j/2 for j , j ∈ ∗ \b. So
n , j
e200 |W(n , j ; n , j )| ≤ √ j
|n · ω + j + θ − E|−1
|n −n |=1 | j −2 j|≤ j β
O(1) ≤ √ ( j 1/5 + log j). j def
(3.54)
Let a j = n · ω + j + θ − E. To arrive at (3.54), we used the fact that |a j | = |n · ω + j + θ − E| > j −1/5 for all (n , j ) ∈ ∗ \b and that a p − aq = p − q, for all p, q. Using (3.53), we have |W(n , j ; n , j )| ≤ O(1) j 1/5 . (3.55) n , j
Substituting (3.54, 3.55) into (3.52), we have 1/2 j 2/5 W ≤ O(1) j 1/2 < O(1) j −1/20 ( j 1). So the Neumann series in (3.50) is norm convergent: [ H˜ ∗ \b (θ ) − E]−1 ≤ (D − E)−1 (1 + δW + δ 2 W2 + . . .) < 2(D − E)−1 < 2 j 1/5 ,
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W.-M. Wang
which is equivalent to ρa (H∗ \b (θ ) − E)−1 ρa−1 < 2 j 1/5 .
(3.56)
For each pair p, q ∈ ∗ \b, we can then always choose a so that | j − j | −1 1/5 − |n −n |+ j 1/2 |[H∗ \b (θ ) − E] (n , j ; n , j )| ≤ C j e . Write G 0 for [H∗ \b (θ ) −
E]−1 ,
G for [H∗ (θ ) −
E]−1 .
(3.57)
From the resolvent equation:
G = G 0 + G 0 Hb G 0 + G 0 Hb G Hb G 0 , where Hb = H∗ − H∗ \b . The matrix element Hb ( p, q) = 0, unless p = b or q = b. Using this, (3.57), the Wegner estimate (3.20) on G with κ = e−J and self-adjointness, we obtain (3.42, 3.43)
β /2
(0 < β < 1/5α)
We now write the estimate at scale J C . Assume δ, J satisfying (3.18), so that Lemma 3.1 holds. Let J1 = J C , C > 1, the same geometric expansion factor as before. Let = [−J1 , J1 ]ν+1 . We have Lemma 3.6. Assume ω is Diophantine satisfying (3.23) and 0 < δ 1 is the same as in Lemma 3.1. For any fixed σ , 0 < σ < 1/5α, there exists B(, E) in R, with σ/2
mes B(, E) ≤ e−J1 ,
(3.58)
such that if θ ∈ R\B(, E), then σ
G (θ, E) ≤ e J1 ,
(3.59) −|(n, j)−(n , j )|1/4
, |G (θ, E)(n, j; n j )| ≤ e for all (n, j), (n j ) such that |(n, j) − (n , j )| > J1 /10,
(3.60)
provided the expansion factor C satisfies 1 < C < β /σ , 0 < β < 1/5α is as in (3.41). So we have the same estimate as in Lemma 3.1 at the larger scale J1 = J C (C > 1). Proof. This is similar to the proof of Lemma 2.4 in [BW1]. So we summarize the main steps. We prove (3.59’, 3.60) using the resolvent equation and cover with cubes of types 0 , ∗ defined in (3.35, 3.36), i.e., T , defined in (3.34), with a covering 0 of 0 ’s and \T , a covering ∗ of ∗ ’s. From Lemma 3.4, for all fixed E, all θ , there exists at most 1 pairwise disjoint 0 ∈ 0 on which (3.10, 3.11) do not hold. We use Proposition 3.2 on this 0 , (3.10, 3.11) on all other 0 . For a given ∗ , let W∗ be the set such that (3.42, 3.43) hold if θ ∈ R\W∗ . Let W = ∪W∗ , where the union is over all possible ∗ with centers in \T , 2(ν+1) −J β
mes W ≤ O(1)J1
e
/2
β 2C
≤ e−J1
2(ν+1)
· J1
,
where the first term is an upper bound on the number of possible ∗ with centers in \T multiplied by the volume of ∗ . For θ ∈ R\W, we use (3.42, 3.43) on ∗ . Using the resolvent equation, combining (3.10, 3.11, 3.42, 3.43) and (3.20) with σ κ = e−J1 (0 < σ < 1/5α) on the only bad 0 , we obtain (3.59’, 3.60). (For more general iterations using the resolvent equation, cf. proof of Lemma 3.8, in particular σ (3.72).) Combining the measure estimate from (3.20) with κ = e−J1 and the above measure estimate on W, we obtain (3.58), provided C < β /σ .
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
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For the induction process to follow, it is convenient to define the following. For any fixed σ , 0 < σ < 1/5α and any given box ⊂ Zν+1 of side length 2J + 1, we say G (θ, E) at fixed (θ , E) is good if σ
G (θ, E) ≤ e J ,
1/4
|G (θ, E)(n, j; n j )| ≤ e−|(n, j)−(n , j )| , ∀(n, j), (n j ) such that |(n, j) − (n , j )| > J/10.
(3.61)
Otherwise, it is bad. We also define def
W() =
W ∗ ,
(3.62)
where the union is over all possible ∗ of the form (3.36) with centers in \T and T is as defined in (3.34), so that for θ ∈ R\W(), (3.42, 3.43) are valid for all ∗ with centers in \T . We say ∗ is good, if (3.42, 3.43) hold. Otherwise ∗ is bad. 3.5. A large deviation estimate in θ for the Green’s functions at fixed E at all scales. We now increase the scale from J1 to J1C (1 < C < β /σ , the geometric expansion factor will be determined here). Our task is again to derive estimates (3.58–3.60) for the cube [−J1C , J1C ]ν+1 starting from the estimates (3.58–3.60) for the cube [−J1 , J1 ]ν+1 . For simplicity of notation, we rename J1 , J and [−J1C , J1C ]ν+1 , in this section. As in the first iteration, we distinguish the tube region T , defined as in (3.34) with the new J . We cover T with 0 ’s defined in (3.35) with the new J , and \T , ∗ defined in (3.36). We note that 0 are at scale J with centers in T , while ∗ are at scales from J β to J Cβ (0 < β < 1/5α, 3/4 < β < 1) with centers away from T . As in the first iteration, we use the resolvent equation to obtain estimates on G from estimates on G 0 and G ∗ . Let 0 be a covering of T and ∗ of \T . For θ ∈ R\W(), (3.42, 3.43) are valid on all ∗ ∈ ∗ . So for any fixed θ ∈ R\W(), we only need to control the number of pairwise disjoint bad 0 boxes on which estimate (3.61) is not available. In particular, we need the number of such bad boxes to be J C , the linear scale of the box . (This is intuitively clear, as otherwise without further detail on the location of the bad boxes, we could not accumulate decay at the linear scale as in (3.60).) Recall that for the first iteration, there is at most 1 such (pairwise disjoint) bad box. Lemma 3.7. Assume ω is Diophantine, satisfying (3.23) and Lemma 3.6 is valid on cubes 0 (0, j) = (0, j) + [−J, J ]ν+1 , ∀ j ∈ [−(2J − 1), 2J − 1].
(3.63)
Then for all fixed θ , #{(n, j) ∈ T |0 (n, j) is a bad box } ≤ O(1)J 5(ν+1) = (J C )1− J C , by choosing 5(ν + 1) < C J σ/2 (0 < σ < 1/5α).
(3.64)
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W.-M. Wang σ
Proof. Write 0 for 0 (0, j). We first replace the estimate G 0 (θ, E)2 →2 ≤ e J1 in (3.59’) by the estimate on the Hilbert-Schmidt norm: σ
G 0 (θ, E)HS ≤ e J1 .
(3.59’)
This leaves the measure estimate in (3.58) unchanged for J1 1 (cf. proof of Lemma 3.6). Define def B(0 (0, j)). (3.65) A= j∈[−(2J −1), 2J −1]
Since the conditions on the Green’s function in (3.59’, 3.60) can be rewritten as polynomial inequalities in θ by using Cramer’s rule, A is semi-algebraic of total degree less than (2J + 1)2(ν+1) · (2J + 1)2(ν+1) · (4J + 1) ≤ Oν (1)J 5(ν+1) ,
(3.66)
where the first factor is an upper bound of the degree of polynomial for each entry of the matrix G 0 (θ, E), the second is an upperbound on the # of entries of each G 0 (θ, E) plus the one for the Hilbert-Schmidt norm, the third is the # of different matrices G 0 ’s. For more details, cf. the proof of Lemma 2.6 in [BW1]. A is therefore the union of at most Oν (1)J 5(ν+1) intervals in R by using Theorem 1 in [Ba] (see also [BGS], where the special case we need is restated as Theorem 7.3). For any fixed θ ∈ R, let (3.67) I = {n ∈ [−J C , J C ]ν n · ω + θ ∈ A}. Then 0 (n, j) is a bad box if and only if n ∈ I and, therefore, n is in one of the intervals of A. But each interval does not contain two such n if ω satisfies (3.23) and |I | ≤ Oν (1)J 5(ν+1) by virtue of (3.58). This is because for Diophantine ω satisfying (3.23), if there exist n, n ∈ [−J C , J C ]ν , n = n , then c σ/2 |(n − n ) · ω ≥ e−J . (3.68) (2J C )α Hence each interval can contain at most 1 integer point in [−J C , J C ]ν . We therefore obtain (3.64). For any fixed θ ∈ R\W(), W() defined as in (3.62), the only bad boxes are of type 0 . Lemma 3.7 shows that there are only few (of order ((J C )1− ) bad 0 boxes in . The following iteration lemma will enable us to obtain estimates (3.58–3.60) for G at scale J C . Lemma 3.8. Fix b ∈ (0, 1/8) and assume τ satisfies 3/4 + b < τ < 1 − b. Suppose M, N are integers satisfying N τ ≤ M ≤ 2N τ .
(3.69)
¯ ⊆ with diameter L, the Green’s function Let = [−N , N ]ν+1 . Assume for all −1 G ¯ (E) = (H¯ − E) at energy E satisfies b
G ¯ ≤ e L .
(3.69’)
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
483
Let be cubes of side length 2M. We say that is good if in addition to (3.69), the Green’s function exhibits off-diagonal decay: |G (E)(x, y)| ≤ e−|x−y|
1/4
for all x, y ∈ satisfying |x − y| > M/10. Otherwise is bad. Assume for any family F of pair-wise disjoint bad cubes in , #F ≤ N b . Under these assumptions, one has |G (E)(x, y)| ≤ e−|x−y|
1/4
for all x, y ∈ satisfying |x − y| ≥ N /10, provided N is sufficiently large, i.e., N ≥ N0 (b, τ ). Proof. The proof is similar to the proof of Lemma 2.4 in [BGS]. As we will see from (3.73, 3.74), because of the conditions on b, τ , it only needs a one step iteration. To estimate G (E)(x, y), x, y ∈ , |x − y| ≥ N /10, let Q be cubes of side length 4M, we make an exhaustion {Si (x)}i=0 of of width 2M centered at x as follows: def
S−1 (x) =∅, S0 (x) = (x) ∩ , def
def
Si (x) = ∪ y∈Si−1 Q(y) ∩ , for 1 ≤ i ≤ , where is maximal such that S (x) = . We say an annulus Ai = Si (x)\Si−1 (x) is good if Ai ∩ F = ∅. Let Ai (x), Ai+1 (x), . . . , Ai+s (x) be an adjacent good annuli and define U = ∪i+s k=i Ak (x). Let ∂∗ S−1 (x) = {x} and ∂∗ S j (x) = {y ∈ S j (x)|∃z ∈ \S j (x), |y − z| = 1} for j ≥ 0. By construction dist (∂∗ Si−1 , ∂∗ Si+s ) = 2M(s + 1) (s ≥ 0). For any subset B ⊆ , let H B be defined as in (3.8). Let def
B = H − (H B ⊕ H\B ). Assume x ∈ B, from the resolvent equation, G (E)(x, y) = G B (E)(x, y) G B (E)(x, z) B (z, z )G (E)(z , y). +
(3.70)
z∈B z ∈ \B
The proof follows the same line of arguments as in the proof of Lemma 2.4 in [BGS] by iterating (3.70). There are two modifications:
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W.-M. Wang
• In view of B , with matrix elements B (z, z ) = δW ( j, j )(n, n ), z = (n, j) ∈ B, z = (n , j ) ∈ \B, where (n, n ) as defined in (2.22) and |W ( j, j )| ≤ e
2
− (2(j−j+j j) )
≤ e−
( j− j )2 4N
from (2.14) for 1 j + j ≤ 2N , we define (1)
∂ Si (x) = {y ∈ Si (x)|dist(y, ∂∗ Si (x)) ≤ N 11/16 }, (2)
∂ Si (x) = {y ∈ \Si (x)|dist(y, ∂∗ Si (x)) ≤ N 11/16 }. (1)
(2)
We note that for y ∈ ∂ Si (x) ∪ ∂ Si (x), dist(y, ∂∗ Si (x)) ≤ N 11/16 M, the size of cubes, and for j, j such that | j − j | > N 11/16 , |W ( j, j )| e−N
1/4
.
(3.71)
• For all x ∈ U , with dist(x, ∂∗ Si−1 ) ≥ M/4, there exists x ∈ U such that (x ) ⊂ U (2) (1) and dist(x, ∂∗ (x )) ≥ M/5. We estimate G U (E)(x, y) with x ∈ ∂ Si−1 , y ∈ ∂ Si+s using (x ) ⊂ U . To obtain subexponential decay of off-diagonal elements G (E)(x, y) we proceed (1) as in [BGS]. This entails to estimate iteratively G Sni (E)(x, z), where z ∈ ∂ Sn i and An i = Sn i \Sn i −1 is a good annulus. Let n i < m i < n i+1 , so that all the annulus in Sm i \Sn i are bad and all the annulus in Sn i+1 \Sm i = U are good. (1) Using the resolvent equation to relate G Sni+1 (E)(x, z), z ∈ ∂ Sn i+1 with G Sni (E)(x, y), y ∈ ∂ Sn(1) i , we have
G Sni+1 (E)(x, z) =
G U (E)(z, w)U (w, w )G Sni+1 (E)(w , x)
w∈U w ∈ Sn i+1 \U
=
G U (E)(z, w)U (w, w )G Sni+1 (E)(w , x) + o(e−N
1/4
)
(2) w ∈ ∂ Sm i (1) w ∈ ∂ Sm i
=
G U (E)(z, w)U (w, w )G Sni (E)(y, x)
(2) (1) w ∈ ∂ Sm w ∈ ∂ Sm i i (1) (2) y ∈ ∂ Sn y ∈ ∂ Sn i i
× Sni (y, y )G Sni+1 (E)(y , w ) + o(e−N
1/4
),
(3.72)
where we used (3.71, 2.22). This is the analogue of (2.25, 2.26) in [BGS]. Iterating (3.72) and taking the log log, lead to the conditions log log[(e N ) N /M ] < log log e N b
log log e
[M 1/4 (N /M−N b )]
1/4
,
> log log e
(3.73) N 1/4
,
(3.74)
for N 1, where (3.73) originates from the estimates on G Sni+1 and (3.74) from the decay estimates on G U . Equations(3.73, 3.74) in turn lead to 3/4 + b < τ < 1 − b for M, N satisfying (3.69) (cf. [BGS]).
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
485
In order to apply the above lemma, we need to convert the covering of = [−J C , which is of diverging scales from J β (0 < β < 1/5α) to J Cβ (3/4 < β < 1) Cβ to a covering of a single scale J . Let be cubes at scale J Cβ (3/4 < β < 1), i.e., J C ]ν+1
(n, j) = (n, j) + [−J Cβ , J Cβ ]ν+1 , (n, j) ∈ Zν+1 .
(3.75)
Let W be as in (3.62). We define ˜ = ∅, where ˜ is a bad 0 box, i.e., (3.61) bad (for a fixed θ ∈ R\W()), if ∩ is violated. (3.76) Let be a covering of with boxes defined in (3.75). Lemma 3.7 gives Lemma 3.9. Fix σ ∈ (0, 1/5α) as in Lemma 3.6. For any fixed θ ∈ R\W(), has at most O(1)J 5(ν+1) = (J C )1− J C (5(ν + 1) < C J σ/2 ) pairwise disjoint bad boxes. On the good box, we have G (θ ) ≤ e(J
Cβ )σ
,
1/4
|G (θ ; n, j; n j )| ≤ e−|(n, j)−(n , j )| , ∀(n, j), (n j ) such that |(n, j) − (n , j )| > J Cβ /10,
(3.77)
provided β β max 5(ν + 1), 0 be arbitrary. We have G (E + i)(x, y) = G U (x) (E + i)(x, y) + G U (x) (E + i)(x, z)U (x) (z, z )G (E + i)(z , y), z ∈ U (x) z ∈ \U (x)
where U (x) is as defined above (3.70), U (x) is either ∗ or 0 and x ∈ U (x). Write U for U (x). For y ∈ ∂∗ U = {y ∈ U |∃z ∈ \U, |z − y | = 1}, x and y satisfy the distance condition in (3.42) if U = ∗ and the condition in (3.11) if U = 0 . It is easy to see that for all x ∈ , such a U (x) exists. Summing over y, we have |G (E + i)(x, y)| ≤ G U y
y∈U (x)
+ 2ν+1 J C(ν+1) e−
Jβ 11
δ sup w
y
|G (E + i)(w, y)|,
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W.-M. Wang
where we used (2.14, 2.22, 3.10, 3.11, 3.42, 3.43) and the conditions σ, β < 1/5α, α > 2ν. Taking the supremum over x and U (x), we obtain the bound on G (θ ) in (3.77), provided C > β /2βσ , J 1. Applying the resolvent equation one more time, similar to the proof of Lemma 3.8, and using the bound on G (θ ), we obtain the off-diagonal decay in (3.77). Let 0 < β < 1/5α, (α as in (3.23)), 3/4 < β < 1. Choose σ ∈ (0, 1/5α) satisfying 5(ν + 1) < β /σ , which leads to 0<σ <
β 1 < . 5(ν + 1) 25α(ν + 1)
Choosing appropriate C large enough so that both Lemme 3.8 and 3.9 are available, we then arrive at the following estimate for G (θ ) = (H (θ ) − E)−1 valid for all = [−J, J ]ν+1 , with J large enough and any fixed E. Proposition 3.10. Assume ω is Diophantine, satisfying (3.23). For any 0 < β < 1/5α, fix 0<σ <
β . 20(ν + 1)
Then for all 0 < δ 1, there exists J0 such that the following statement is satisfied for all = [−J, J ]ν+1 , with J ≥ J0 : There exists B(, E) in R, with mes B(, E) ≤ e−J
σ/2
,
(3.78)
such that if θ ∈ R\B(, E), then σ
G (θ, E) ≤ e J ,
(3.79) −|(n, j)−(n , j )|1/4
, |G (θ, E)(n, j; n j )| ≤ e for all (n, j), (n j ) such that |(n, j) − (n , j )| > J/10.
(3.80)
Proof. Choose C > 40(ν + 1) such that β < Cσ < β < 1. 2β 1
Assume 0 < δ 1. Let J0 = | log δ| C . For the scales J0 to J0C , we use perturbation theory in δ as in Lemma 3.1, with a further lowering of δ if necessary in order that (3.9–3.11) hold for all scales in [J0 , J0C ]. For the scales ≥ J0C , Lemme 3.8 and 3.9 are available, in view of the choice of C. Assume (3.78–3.80) hold at some scale J 1. Using Lemma 3.9 in Lemma 3.8, we obtain the corresponding estimate at scale J C . Hence the proposition holds by induction.
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
487
4. Frequency Estimates and the Elimination of E We now convert the estimate in θ in (3.78–3.80) for fixed E, ω into estimates in ω for fixed θ (θ = 0), eliminating the E dependence in the process by excluding double resonances. This is the key estimate leading to the proof of the theorem in Sect. 5. We will first define a double resonant set in (ω, θ ) ∈ Tν × R. We use 2 scales N , N¯ with N < N¯ , log log N¯ log N . Let N ( j) = [−N , N ]ν+1 + (0, j), j ∈ Z, N¯ = [− N¯ , N¯ ]ν+1 . Assume N¯ is resonant at θ = 0 for some E, i.e., E is close to some eigenvalue of H N¯ (see (4.1) below). We prove in Lemma 4.1 that with further reductions in the frequency set, the N ( j) boxes with | j| ≤ N¯ and are at appropriate distances from N¯ are all nonresonant (at θ = 0). Toward that end, let DC( N¯ ) be the Diophantine condition to order N¯ , i.e., if ω ∈ DC( N¯ ), then n · ωT ≥
c (α > 2ν), ∀n ∈ [− N¯ , N¯ ]ν \{0}, |n|α
where as before · T is the distance to the nearest integer. Denote H N¯ by H N¯ . We define D(N , N¯ ) ⊂ Tν × R as D(N , N¯ ) = {(ω, θ ) ∈ Tν × R|∃ E, such that { j∈Z| N ( j)∩ N¯ =∅}
−1 2 H N¯ (ω, 0) − E ≥ e N and θ ∈ B ( N ( j), E)},
and for fixed 0 < a < 1 and C > 1, D(N , N¯ ) (DC( N¯ ) × R) , S(N ) =
(4.1)
(4.2)
N¯ N C
where N¯ N C means a N C ≤ N¯ ≤ N C /a. We note that a is kept fixed in this section. The set S(N ) is the double resonant set in (ω, θ ) restricted to ω ∈ DC( N¯ ), S(N ) ⊂ ˜ ), S(N ˜ ) ⊂ Tν × R × R, i.e., S(N ) = Tν × R. It is the projection of the set S(N ˜ )) with Proj Tν ×R ( S(N ˜ , N¯ ) (DC( N¯ ) × R × R) , ˜ )= S(N D(N (4.3) N¯ N C
where
˜ , N¯ ) = (ω, θ, E) ∈ Tν × R × R| H ¯ (ω, 0) − E −1 ≥ e N 2 D(N N and θ ∈ B ( N ( j), E) for some N ( j) ∩ N¯ = ∅ .
(4.4)
Lemma 4.1. Let N ∈ N be sufficiently large, C > 1 and δ > 0 sufficiently small. Let N be the union of ∗N¯ over N¯ N C , where ∗N¯ ⊂ DC( N¯ ) is the set of ω, such that • there exists E ∈ R such that −1 2 H N¯ (ω, 0) − E ≥ eN ;
(4.5)
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W.-M. Wang
• there exist j ∈ Z, N ( j) ∩ N¯ = ∅, and ∈ Zν , || exp[(log N )2 ] = K such that 1/4
|(H N ( j) − E)−1 (ω, · ω)(m, m )| > e−|m−m | m, m
for some ∈ N ( j), |m Then the set N satisfies
− m|
(4.6)
> N /10.
1 (log N )2 . mes N ≤ exp − 100
(4.7)
The proof of Lemma 4.1 uses the following decomposition lemma [B3, Lemma 9.9]. Lemma 4.2. Let S ⊂ [0, 1]2ν be a semi-algebraic set of degree B and mes2ν S < η, log B log 1/η. Denote by (x, y) ∈ [0, 1]ν × [0, 1]ν the product variable. Fix > η1/2ν . Then there is a decomposition S = S1 S2 , with S1 satisfying |Projx S1 | < B C
(4.8)
and S2 satisfying the transversality property mesν (S2 ∩ L) < B C −1 η1/2ν ,
(4.9)
for any ν-dimensional hyperplane L such that max |Proj L (e j )| <
1≤ j≤ν
1 , 100
(4.10)
where e j are the basis vectors for the x-coordinates. For our usage, the variable (x, y) ∈ R2ν in the lemma will be (ω, θ ) after identifying θ with (θ, 0) ∈ Rν . Proof of Lemma 4.1. Let (ω, θ ) ∈ S(N ), then θ ∈ B ( N ( j), E k ) for some N ( j) and 2 some eigenvalue E k of H N¯ satisfying |E k − E| < e−N . This is because the perturbation 2 e−N essentially preserves the condition of the definition of a bad set, see (3.8–3.10) (cf., proof of Lemma 4.1 [BW1]). By the Diophantine restrictions on the frequencies mes S(N ) ≤ O(1)e−N
σ/2
N C(ν+2)
(4.11)
from Proposition 3.10, where the third factor in the RHS is an upper bound on the number of N ( j) and the number of eigenvalues of H N¯ . ˜ ) is semi-algebraic with total degree at most N C1 ν (C1 > 1), cf., proof of Since S(N Lemma 3.7, S(N ) its projection onto Tν × R is also semi-algebraic with degree at most N C2 ν (C2 > C1 ) [B3, Chap 9]. Equation (4.8) and Lemma 4.2 then conclude the proof by taking exp[−(log N )2 ] (cf., also [BW2]). Here we also used the fact that the sum of the measure estimate in (4.9) over all L corrsponding to the hyperplanes (ω, · ω) with || exp[(log N )2 ] is much smaller than exp[−(log N )2 ] for the corresponding B, η and > 0.
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489
Lemma 4.1 shows that the boxes N (, j) = (, j) + [−N , N ]ν+1 , where || exp[(log N )2 ], j ∈ [−N C , N C ] are non-resonant with the box N¯ = [− N¯ , N¯ ]ν+1 , N¯ N C . Let T be the tube region, T = (n, j) ∈ Zν+1 | j ∈ [−N C , N C ] . (4.12) The boxes N (, j) have centers in T . We now exclude resonances of N¯ with boxes with centers in Zd \T . Recall from Sect. 3, that boxes with centers in Zd \T are in fact rectangles:
∗ (n, 2 j) = (n, 2 j) + [− j β , j β ]ν ×[− j β , j β ] (0 < β < 1/5α, 3/4 < β < 1), (n, 2 j) ∈ Zd \T . (4.13) To exclude resonances of boxes ∗ (n, 2 j) where max(|n|, |2 j|) exp[(log N )2 ] with N¯ , we use direct perturbation. We have Lemma 4.3. mes {ω ∈ [0, 2π )ν | dist (σ (H N¯ ), σ (H∗ )) ≤ κ, ω satisfies (3.23)} 1 ≤ Cκ| N¯ ||∗ |, (κ exp[− (log N )2 ]) 5
(4.14)
for any ∗ = ∗ (n, 2 j) in (4.13) satisfying |n| > exp[ 21 (log N )2 ] and max(|n|, |2 j|) exp[(log N )2 ], provided 0 < β < min(1/20, 1/5α). Proof. Let λm,k , φm,k be eigenvalues and eigenfunctions of H∗ :
Write φm,k = λm,k =
H∗ φm,k = λm,k φm,k . |am ,k |2 = 1. Then (m ,k )∈∗ am ,k δm ,k , with
|am ,k |2 (m · ω + k ) + δ
(m ,k )
am ,k a¯ m ,k m m Wk k .
(m ,k ), (m ,k )
From (3.27) of Proposition 3.3,
|am ,k |2 = O
(m ,k )=(m,k)
1 j 1/20
.
The first order eigenvalue variation: m·
∂ λm,k = |am ,k |2 m · m ∂ω (m ,k ) = |am ,k |2 m · m + (m ,k )
(m ,k ), m =m
O(1) ≥ |m|2 − 1/20 |m| · j β , j
|am ,k |2 m · (m − m)
(4.15)
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where we used (4.15) and the fact that |m − m | ≤ O(1) j β in ∗ . Lowering β to satisfy β < 1/20 if necessary, we obtain m·
1 ∂ λm,k ≥ |m|2 m N C ∂ω 2
(4.16)
for |m| > exp[ 21 (log N )2 ] N C . Let µ,i (ω) be the eigenvalues of H N¯ . Then µ,i (ω) are piecewise holomorphic in each ω p , p = 1, . . . , ν and Lipshitz in ω: µ,i Lip ≤ C N¯ N C . Using this and (4.16), we have that mes {ω ∈ [0, 2π )ν | min min |λm,k (ω) − µ,i (ω)| ≤ κ} ,i m,k
≤ Cκ| N¯ ||∗ |. Using Lemma 4.3, we arrive at Lemma 4.4. Let N ∈ N be sufficiently large, C > 2/β , 0 < β < min(1/20, 1/5α) as in Lemma 4.3, δ sufficiently small. Let N be a subset of the Diophantine set defined in (3.23) with the properties: • there are N¯ N C and E such that −1 2 ≥ eN ; (4.17) H N¯ (ω, 0) − E • there is a ˜ (n, 2 j) = (n, 2 j) + [− j β , j β ]ν+1 , (3/4 < β < 1) with | j| ≥ N C /2 and max(|n|, |2 j|) exp[(log N )2 ] such that 1/4
−1 −|m−m | |(H(n,2 ˜ j) − E) (ω, 0)(m, m )| > e
(4.18)
˜ satisfying |m − m | > j β /10. for some m, m ∈ Then mes N ≤ e−N .
(4.19)
˜ with ∗ of the form (4.13). We use a resolvent expansion similar Proof. We cover to the proof of Lemma 3.9 to obtain the opposite of (4.18). For that purpose we need to take away an additional set of ω so that (3.42, 3.43) hold on all ∗ of the form (4.13) at θ = 0. For any ∗ (n, 2 j) such that |n| ≥ exp[ 21 (log N )2 ], max(|n|, |2 j|) exp[(log N )2 ], 2 (4.14) is available. Take one such ∗ and let κ = e−2N e−N for N large. Assume 2 (4.17) holds, so |E − E k | ≤ e−N e−2N for some eigenvalue E k of H N¯ . Lemma 4.3 then says that (E − H∗ )−1 ≤ e2N (1 + 2e−N ) < 2e2N 2
by taking away a set in ω of measure ≤ Cκ| N¯ ||∗ |.
(4.20)
Pure Point Spectrum of the Floquet Hamiltonian for Quantum Harmonic Oscillator
491
Comparing (4.12) with (3.34) J = N C here. So (E − H∗ )−1 < 2e2N < e J
β /2
= eN
Cβ /2
,
provided Cβ /2 > 1 or C > 2/β . Therefore (3.42, 3.43) hold on this ∗ . Multiplying (4.20) by the number of all possible ∗ (n, 2 j) such that |n| ≥ exp [ 21 (log N )2 ], max(|n|, |2 j|) exp[(log N )2 ], we have that (3.42, 3.43) are available on all such ∗ after taking away an additional set in ω of measure ≤ e−N . For ∗ (n, 2 j) such that |n| < exp[ 21 (log N )2 ], since max(|n|, |2 j|) exp[(log N )2 ], | j| exp[(log N )2 ]. So dist E, σ (H∗ ) exp[(log N )2 ], ∀ω ∈ [0, 2π )ν , if E satisfies (4.17). This is because H N¯ ≤ O( N¯ ) = N C exp[(log N )2 ]. The proof now proceeds as the proof of Lemma 3.9 and we arrive at the conclusion. 5. Proof of the Theorem We use Lemme 4.1, 4.4 to prove that the Floquet Hamiltonian K in (1.15), or rather its unitary equivalent H in (1.19) has pure point spectrum. Let N , N be the frequency ˜ to be sets as in Lemme 4.1, 4.4, N ∈ N, sufficiently large. Define ˜ = Tν \ ( N ∪ N ), N0 1 (depending on δ), (5.1) N ≥N0
˜ → (2π )ν as δ → 0. then mes ˜ From the Schnol-Simon theorem [CFKS,S], to prove H in (1.19) has Fix ω ∈ . pure point spectrum, it suffices to prove that the generalized eigenfunctions have fast decay, hence are in 2 . More precisely, let ψ be a non-zero function on Zν+1 satisfying (H − E)ψ = 0, |ψ(m)| ≤ 1 + |m|c0 for all m ∈ Zν+1 ,
(5.2)
where E is arbitrary and c0 > 0 is some constant. We will prove using Lemme 4.1, 4.4 that ψ has subexponential decay and hence is in 2 (Zν+1 ). We first verify that (4.5) is satisfied. This implies that (4.17) is also satisfied as they are the same condition. So we need to show that there is some box N¯ centered at 0, N¯ = [− N¯ , N¯ ]ν+1 , for some N¯ N C (C > 2/β , 0 < β < min(1/20, 1/5α) as in Lemme 4.1, 4.4), such that −1 2 H N¯ − E ≥ eN .
(5.3)
For this we let
T = {(n, j) ∈ N¯ | j ∈ [−N C , N C ]}, C > 1 to be determined from (5.8–5.11), C chosen to be > C . Let N C be boxes of side length 2N C . We cover T with N C , N¯ \T with ∗ of the form (4.13).
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From Proposition 3.10, Lemma 3.7, there are at most N¯ 1−τ0 (τ0 > 0) pairwise disjoint bad N C boxes in T by taking C large enough. In N¯ \T , there is at most 1 ˜ set as follows. pairwise disjoint bad ∗ box by a further reduction in the Assume ∃(n, j) ∈ ∗ , ∃(m, k) ∈ ∗ , ∗ ∩ ∗ = ∅ such that |λn, j (ω) − λm,k (ω)| ≤ κ,
(5.4)
where λn, j , λm,k are eigenvalues of H∗ , H∗ . From Proposition 3.3, λn, j = n · ω + j + O( j −1/4 ), λm,k = m · ω + k + O(k −1/4 ). So |λn, j − λm,k | ≥ 1/2, if n = m, since (n, j) = (m, k). We only need to look at the case n = m. This is similar to the proof of Lemma 4.3, except we look at the first order eigenvalue variation in the (n − m) direction. Let a n , j δn , j φn, j = (n , j )∈∗
be the eigenfunction with eigenvalue λn, j : H∗ φn, j = λn, j φn, j ; and ψm,k =
bm ,k δm ,k
(m ,k )∈∗
be the eigenfunction with eigenvalue λm,k : H∗ ψm,k = λm,k ψm,k . We have for the first order variation: ∂ [λn, j − λm,k ] (n − m) · ∂ω ⎡
⎤ ∂ ⎣ = (n − m) · |an , j |2 n · ω − |bm ,k |2 m · ω⎦ ∂ω (n , j )∈∗ (m ,k )∈∗ ⎡ ∂ ⎣ = (n − m) · |an , j |2 (n − n) · ω (n − m) · ω + ∂ω (n , j )∈∗ , n=n ⎤ − |bm ,k |2 (m − m) · ω⎦ (m ,k )∈∗ , m=m
= |n − m| + O 2
1 |n − m|2 2 1 ≥ , 2
1 j 1/20
β
· j |n − m| + O
1 k 1/20
k β |n − m|
>
(5.5)
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493
where we used (3.27) of Proposition 3.3 and the definitions of ∗ , ∗ in (4.13). So mes {ω||λn, j (ω) − λm,k (ω)| ≤ κ, ω satisfies (3.23)} ≤ Cκ. Take κ = e−(log N ) and denote the set N such that ∃∗ ⊂ N¯ , ∗ ⊂ N¯ , ∗ ∩ ∗ = ∅, ∃(n, j) ∈ ∗ , ∃(m, k) ∈ ∗ , such that (5.4) holds. Then 2
1
mes N ≤ e− 2 (log N ) , provided 2/β < C log N . Let ˜ = \ def
2
(5.6)
N ,
(5.7)
N ≥N0
mes → (2π )ν as δ → 0. We now assume ω ∈ . Then in N¯ , N¯ N C , there are at most N¯ 1−τ0 (0 < τ0 < 1) bad N C boxes with centers in T and 1 bad ∗ box with center in N¯ \T . Since N¯ N 5(ν+1)C
> N¯ τ0
(5.8)
for C large enough, there has to be an annulus A at a distance N C to the origin of thickness 10N C in T and of thickness N C /4 in N¯ \T (as there is at most 1 bad ∗ box) devoid of bad points. Since (H − E)ψ = 0 from (5.2), let (H(c) − E)ψ = ξ˜ , ˜
(5.9)
˜ where (c) is a box of type = N C or ∗ centered in c. In view of (1.20, 5.2), the restriction on β for ∗ in (3.36) ˜
|ξ˜ (m)| |m|c0 e−c [ dist (m,∂ )] , 1/2
(5.10)
˜ (1) ) > |m − c|1 , where if dist(m, ∂ ˜ ˜ |z − y|1 = 1} ˜ (1) = {y ∈ |∃z ∈ Zd \, ∂ and | |1 is as defined above (3.46). ˜ ˜ Let c ∈ A, (c) ⊂ A. So (c) is a good box of type N C with centers in T or ∗ of the form (4.13) with centers in \T . Estimates (3.42, 3.80) are available. From (5.9) ψ = (H(c) − E)−1 ξ˜ . ˜ Using (5.10), (3.42) or (3.80), we have that |ψ(c)| < e−N , 2
(5.11)
˜ provided C > 2/β for all c ∈ A, such that ∃(c) ⊂ A. We now choose 2/β < C < C log N , so that (5.8) is satisfied. Equation(5.11) implies immediately that |ψ(m)| < e−N
2
(5.12)
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for all m ∈ A ⊂ A, where A is a smaller annulus contained in A of thickness 6N C in T and thickness N C /5 in N¯ \T . Since dist A to the origin N C , one can always find a square N¯ centered at the origin, N¯ N C , such that ∂ N¯ ⊂ A ,
dist (∂ N¯ , A ∩ T ) > 2N C ,
dist (∂ N¯ , A ∩ T ) > N /10. c
C
(5.13) (5.14)
˜ Using (5.9) with N¯ in place of (c), we have ψ = (H N¯ − E)−1 ξ N¯ . Using (5.12–5.14), we have ξ N¯ < e−N by slightly increasing C if necessary. Since we may always assume ψ(0) = 1, (5.10, 5.11) give that 2
(H N¯ − E)−1 > e N . 2
Subexponential decay of eigenfunctions of H . Let K = exp[(log N )2 ]. Lemme 4.1, 4.4 then imply (via a proof similar to the proofs of Lemme 3.8 and 3.9) that the Green’s function of the set def
U = 2K (0)\ K (0), where L (0) = [−L , L]ν+1 , exhibits off-diagonal decay, i.e., |G U (E, m, n)| ≤ e−|m−n|
1/4
m, n ∈ U, |m − n| > K /10.
˜ Fix m ∈ U . Assume dist (m, ∂U ) ≥ Let ξU be defined as in (5.9) with U replacing . K /4. For n such that dist (n, ∂U ) > K /10, we use the bound in (5.10) with ξU replacing ξ˜ . Otherwise we use the fact that ξU is polynomially bounded since ψ is polynomially bounded. The equality ψ(m) =
G U (E, m, n)ξ(n),
n
then implies that |ψ(m)| ≤ e−|m| provide K and thus |m| is large enough.
κ
(0 < κ < 1/4),
Acknowledgement. I am deeply indebted to the referee for a thorough reading of the paper and for many helpful comments.
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References [Ba] [Be] [B1] [B2] [B3] [BGS] [BW1] [BW2] [BBL] [Co1] [Co2] [CW] [CFKS] [DS] [E] [EK] [EV] [FS] [GB] [GK] [GR] [GY] [HLS] [Ho] [JL] [KRY] [Ka] [Ku1] [Ku2] [KP]
Basu, S.: On bounding the betti numbers and computing the euler characteristic of semi-algebraic sets. Discrete Comput. Geom. 22, 1–18 (1999) Bellissard, J.: Stability and chaotic behavior of quantum rotators in stochastic process in classical and quantum systems. Lecture Notes in Physics, vol. 262. Springer, Berlin (1986) Bourgain, J.: Construction of quasi-periodic solutions for hamiltonian perturbations of linear equations and applications to nonlinear pde. IMRN 11, 475–497 (1994) Bourgain, J.: Quasi-periodic solutions of hamiltonian perturbations of 2d linear schrödinger equations. Ann. Math 148, 363–439 (1998) Bourgain, J.:Green’s function estimates for latttice Schrödinger operators and applications. Princeton, NJ: Princeton University Press, 2005 Bourgain, J., Goldstein, M., Schlag, W.: Anderson localization for schrödinger operators on Z2 with quasi-periodic potential. Acta Math. 188, 41–86 (2002) Bourgain, J., Wang, W.-M.: Anderson localization for time quasi-periodic random schrödinger and wave equations. Commun. Math. Phys. 248, 429–466 (2004) Bourgain, J., Wang, W.-M.: Quasi-periodic solutions for nonlinear random schrödinger. J. European Math. Society 10(1), 1–45 (2008) Boutel de Monvel-Berthier, A., Boutet de Monvel, L., Lebeau, G.: Sur les valeurs propres d’un oscillateur harmonique perturbe. J. d’Anal. Math. 58, 39–60 (1992) Combescure, M.: A quantum particle in a quadrupole radio-frequency trap. Ann. Inst. Henri. Poincare 44, 293–314 (1986) Combescure, M.: The quantum stability problem for time-periodic perturbation of the harmonic oscillator. Ann. Inst. Henri. Poincare 47, 63–83, 451–454 (1987) Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equation. Commun. Pure Appl. Math. 46, 1409–1498 (1993) Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators. Berlin-HeidelbergNew York: Springer-Verlag, 1987 Duclos, P., Stovicek, P.: Floquet hamiltonians with pure point spectrum. Commun. Math. Phys. 177, 327–347 (1996) Eliasson, L.H.: Almost reducibility of linear quasi-periodic systems. Proceedings of Symp in Pure Math 69, 679–705 (2001) Eliasson, L.H., Kuksin, S.B.: KAM for the nonlinear Schrödinger equation. available at http://theory.physics.unige.ch/mp_arc/c/06/06-144pdf, 2006 Enss, V., Veselic, K.: Bound states and propagating states for time-dependent hamiltonians. Ann IHP 39(2), 159–191 (1983) Fröhlich, J., Spencer, T.: Absence of diffusion in the anderson tight binding model for large disorder. Commun. Math. Phys. 88, 151–184 (1983) Germinet, F., de Bievre, S.: Dynamical localization for discrete and continuous random schrödinger operators. Commun. Math. Phys. 194, 323–341 (1998) Germinet, F., Klein, A.: Bootstrap multiscale analysis and localization in random media. Commun. Math. Phys. 222, 415–448 (2001) Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products (6th ed). LondonNew York; Academic Press, 2000 Graffi, S., Yajima, K.: Absolute continuity of the floquet spectrum for a nonlinearly forced harmonic oscillator. Commun. Math. Phys. 215, 245–250 (2000) Hagedorn, G., Loss, M., Slawny, J.: Non stochasticity of time-dependent quadratic hamiltonians and the spectra of canonical transformations. J. Phys. A 19(4), 521–531 (1986) Howland, J.S.: Stationary scattering theory for time-dependent hamiltonians. Math. Ann. 207, 315–335 (1974) Jauslin, H.R., Lebowitz, J.L.: Spectral and stability aspects of quantum chaos. Chaos 1, 114–121 (1991) Kapitanski, L., Rodnianski, I., Yajima, K.: On the fundamental solution of a perturbed harmonic oscillator. Topo. Methods Nonlinear Anal. 9(1), 77–106 (1997) Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: SpringerVerlag, 1980 Kuksin, S.: Hamiltonian perturbation of infinite-dimensional linear systems. Funts. Anal. Prilozh. 21, no. 3, 22–37 (1987); English translation in Funct. Anal. Appl. 21, 192–205 (1987) Kuksin, S.: Nearly Integrable Infinite-Dimensional Hamiltonian Systems. Lect. Notes. Math. 1556, Berlin-Heidelberg-New York: Springer-Verlag, 1993 Kuksin, S., Pöschel, J.: Invariant cantor manifolds of quasi-periodic osillations for a nonlinear schrödinger equation. Ann. Math. 143, 149–179 (1996)
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Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series, Vol I. New York: Gordon and Breach Science Publishers, 1986 Schnol, I.: On the behaviour of the Schrödinger equation. Mat. Sb. (Russian) 273–286, (1957) Thangavelu, S.: Lectures on Hermite and Laguerre Expansions. Math. Notes 42, Princeton, NJ: Princeton University Press, 1993 Wang, W.-M.: Quasi-periodic solutions of nonlinearly perturbed quantum harmonic oscillator. In preparation Yajima, K.: Resonances for the ac-stark effect. Commun. Math. Phys. 78, 331–352 (1982) Yajima, K.: On smoothing property of Schrödinger propagators. Lect. Notes Math. 1450, BerlingHeidelberg-New York: Springer, 1989, pp. 20–35 Yajima, K.: Schrödinger evolution equations with magnetic fields. J. d’Anal. Math. 56, 29–76 (1991) Yajima, K., Kitada, H.: Bound states and scattering states for time periodic hamiltonians. Ann. IHP, A 39, 145–157 (1983) Zelditch, S.: Reconstruction of singularities for solutions of schrödinger’s equation. Commun. Math. Phys. 90, 1–26 (1983)
Communicated by B. Simon
Commun. Math. Phys. 277, 497–529 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0375-3
Communications in
Mathematical Physics
Logarithmic Intertwining Operators and Vertex Operators Antun Milas Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY 12222, USA. E-mail: [email protected]; [email protected] Received: 9 February 2007 / Accepted: 30 May 2007 Published online: 15 November 2007 – © Springer-Verlag 2007
Abstract: This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg and lattice vertex algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra M(1)a , of central charge 1 − 12a 2 . We classify these operators in terms of depth and provide explicit constructions in all cases. Our intertwining operators resemble puncture operators appearing in quantum Liouville field theory. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)0 and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1, 0)-module.
0. Introduction The theory of vertex algebras continues to be very effective in proving rigorous results in two-dimensional Conformal Field Theory (CFT) (for some recent breakthrough see [H1,H2,Le]). In 1993, Gurarie [Gu] studied a CFT-like structure with two features absent in the ordinary CFT: logarithmic behavior of matrix coefficients and appearance of indecomposable representations of the Virasoro algebra underlying the theory. There are several additional examples of “logarithmic” models that have been discovered since then (see for instance [GK1,F1,G1], and especially [FFHST,F2,G2] and references therein). By now, a structure that involves a family of modules for the chiral algebra, closed under the “fusion”, with logarithmic terms in the operator product expansion is usually called a logarithmic conformal field theory (LCFT). The most important examples of LCFTs are the so-called rational LCFTs. These involve only finitely many inequivalent irreducible representations, but also some indecomposable logarithmic representations so that the
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modular invariance is preserved (e.g., the triplet model [GK1,GK2,GK3,BF], etc.). We stress here that neither LCFT nor rational LCFT are mathematically precise notions. In [M1] we proposed a purely algebraic approach to LCFT based on the notion of logarithmic modules and logarithmic intertwining operators. The key idea is to introduce a deformation parameter log(x) and to define logarithmic intertwining operators as expressions involving intertwining like operators multiplied with appropriate powers of log(x), such that the translation invariance is preserved. These operators can be used to explain appearance of logarithms in correlation functions. In our setup we do not require an extension of the space of “states” for the underlying vertex algebra. There are other proposals in the literature [FFHST] where log(x) is also viewed as a deformation parameter, but with an important difference that log(x) is also part of an extended chiral algebra (or an OPE algebra). Even though the construction in [FFHST] has been shown successful in explaining various logarithmic behaviors of CFTs, it is unclear to us if the approach in [FFHST] can be used to address the problem of fusion. On the other hand, logarithmic intertwining operators have been used by Huang, Lepowsky and Zhang in [HLZ] as a convenient tool to develop a generalization of Huang-Lepowsky’s tensor product theory [HL] to non-semisimple tensor categories. Another important contribution is Miyamoto’s generalization of Zhu’s modular invariance theorem for vertex operator algebras satisfying the C2 -condition [My1], which possibly involve logarithmic modules. In view of [My1] we tend to believe that rational LCFT give rise from vertex algebras satisfying the C2 -cofiniteness condition. From everything being said it appears that several important aspects of LCFTs can be studied in the framework of vertex (operator) algebras. This paper continues naturally [M1] and [M2]. In the present work we focus on a simple, yet interesting class of vertex operator algebras those associated with Heisenberg Lie algebras and the Virasoro algebra. The aim here is to construct a family of logarithmic intertwining operators associated with certain weak M(1)a -modules, which can be used for building logarithmic intertwining operators among indecomposable representations of the Virasoro algebra and various W-algebras ( [AM,M3]). The most interesting part of our work is an explicit construction of the so-called hidden logarithmic intertwining operators, i.e., those which intertwine a pair of ordinary and one logarithmic module. We should stress here that our logarithmic intertwining operators are closely related to puncture operators appearing in quantum Liouville field theory [Se,ZZ]. Let us elaborate the construction on an example. Consider the Feigin-Fuchs module 2 M(1, λ)a of central charge c = 1 − 12a 2 and lowest conformal weight λ2 − aλ. Tensor the module M(1, λ)a with a two-dimensional space , where h(0) acts on (in some basis) as λ 1 0 λ so we obtain a weak M(1)a -module M(1, λ)a ⊗. Now it is easy to see that M(1, λ)a ⊗ is an ordinary module if and only if λ = a, so that M(1, a)a ⊗ is of lowest conformal 2 weight − a2 . Our main results (cf. Theorem 5.5 and Theorem 7.5) then provides us with a genuine logarithmic intertwining operator of type W , M(1, a)a ⊗ M(1, a)a ⊗ where W is a logarithmic module. Notice that for special a this result is in “agreement” with some results in the physics literature. For instance, for c = −2 and a = 21 , the
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−1 8 . This model is known to be logarithmic 2 after [Gu]. More generally, one takes c = 1 − 6( p−1) , p ≥ 2 (cf. [AM,M3]). p We should say here that Feigin-Fuchs modules M(1, a)a with the lowest conformal 2 weight − a2 and central charge 1 − 12a 2 are indeed special from at least two points of
lowest conformal weight of M(1, 21 ) 1 ⊗ is 2
view. These modules are self-dual (cf. (3.9)) and in addition we have h−
c −1 = , 24 24
so that tr| M(1,a)a q L(0)−c/24 =
1 , η(τ )
∞ where η(τ ) = q 1/24 i=1 (1 − q i ) is the Dedekind η-function. In fact these are the only Feigin-Fuchs modules whose modified graded dimensions are modular functions. Let us briefly outline the content of the paper. In Sect. 2 we recall the notions of logarithmic module and of logarithmic intertwining operator. In Sects. 3 and 4 we prove some standard results about extended Heisenberg Lie algebras, the corresponding vertex operator algebras and associated logarithmic intertwining operators. In Sect. 5 we derive a logarithmic version of the Li-Tsuchiya-Kanie’s “Nuclear Democracy Theorem” for logarithmic intertwining operators needed for our construction. In Sects. 5 and 7 we provide a classification of logarithmic intertwining operators among a triple of logarithmic M(1)a -modules. Finally, in Sects. 9 and 10 we use certain restriction theorems for construction of logarithmic intertwining operators among triples of logarithmic L(1, 0)modules. In particular, we obtain a family of hidden intertwining operators from a pair of ordinary L(1, 0)-modules. Sect. 11 is an introduction to [AM] and [M3]. This paper is based on a talk the author gave at the AMS Special Session on Representations of Infinite Dimensional Lie Algebras and Mathematical Physics in Bloomington, IN, April 2003. Because of the recent increasing interest in LCFT we decided to make our paper public. 1. Logarithmic Modules and Logarithmic Intertwining Operators In this section notation and definitions are mostly from [DLM,HLZ,LL]and[M1]. For an accessible introduction to vertex (operator) algebras and their representations we refer the reader to [LL]. Definition 1.1. Let (V, Y, 1, ω), V = Vi , be a vertex operator algebra. A logarithmic i∈Z
V -module M is a weak V -module [DLM] which admits a decomposition M= Mh , h i ∈ C, for some k ∈ N, h∈{h i +Z+ , i=1,...,k}
Mh = {v ∈ M : (L(0) − h I )m v = 0, for some m}, dim(Mh ) < +∞. Here Mh denotes the generalized homogeneous subspace of M of generalized weight h.
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It is important to mention that we use the definition of weak module as in [DLM], so that M carries an action of the full Virasoro algebra and not only the L(−1)-operator. Hence, for every v ∈ Vi and m ∈ Z we have vm Mh ⊆ Mh+i−m−1 . The category of logarithmic V -modules LOG is essentially just a different name for the category O introduced in [DLM]. In particular every logarithmic module is admissible [DLM]. We say that a weak V -module is a generalized logarithmic module if it decomposes into (not necessarily finite-dimensional) generalized L(0)-eigenspaces. Furthermore, we will say that a logarithmic V -module M is genuine if under the action of L(0) it admits at least one Jordan block of size 2 or more. We recall the definition of logarithmic intertwining operators from [M1]. More precisely, here we are using a slightly more general definition following [HLZ]. In what follows log(x) is just a formal variable satisfying ddx log(x) = x1 . Definition 1.2. A logarithmic intertwining operator among a triple of logarithmic V -modules W1 , W2 and W3 is a linear map Y(·, x)· : W1 ⊗ W2 −→ W3 {x}[log(x)], (n) Y(w1 , x)w2 = logn (x)x −α−1 (w1 )(n) α w2 , (w1 )α ∈ End(W2 , W3 ),
(1.1)
n∈N α∈C
satisfying: (i) (Truncation condition) For every w1 ∈ W1 , w2 ∈ W2 and α ∈ C, (w1 )(k) α+m w2 = 0, for m ∈ N, m >> 0. (ii) (Translation invariance) For every w1 ∈ W1 , [L(−1), Y(w1 , x)] =
d Y(w1 , x). dx
(iii) (Jacobi identity) For every wi ∈ Wi , i = 1, 2 and v ∈ V , we have x1 − x2 Y (v, x1 )Y(w1 , x2 )w2 x0−1 δ x0 −x2 + x1 Y(w1 , x2 )Y (v, x1 )w2 −x0−1 δ x0 x1 − x0 Y(Y (v, x0 )w1 , x2 )w2 . = x2−1 δ x2
(1.2)
We will denote the vector space of logarithmic intertwining operators among W1 , W2 and W3 by W3 . I W1 W2 (k)
We say that a logarithmic intertwining operator is genuine if (w1 )α is nonzero for k ≥ 1 and some w1 ∈ W1 . A genuine logarithmic intertwining operator of type some W3 W1 W2 is called hidden if two modules Wi and W j are ordinary V -modules, where {i, j} ⊂ {1, 2, 3}.
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Definition 1.3. A logarithmic intertwining operator Y of type if
W3 W1 W2 is said to be strong
depth(Y) := sup{k : (w1 )(k) α w2 = 0, α ∈ C, w1 ∈ W1 , w2 ∈ W2 , k ∈ N}
(1.3)
is finite. In other words, for strong logarithmic intertwining operators the powers of log(x) are globally bounded. Definition 1.4. A logarithmic intertwining operator Y is said to be locally strong if Y(w1 , x) ∈ Hom(W2 , W3 ){x}[log(x)], for every w1 ∈ W1 . The previous definition was used in [M1] as the definition of logarithmic intertwining operators. In this paper we shall only study strong intertwining operators. From the previous definitions we clearly have a chain of embeddings W3 W3 W3 W3 ⊆ Ist ⊆ Ilst ⊆I , (1.4) Ior d W1 W2 W1 W2 W1 W2 W1 W2 where Ist , Ilst and Ior d stand for the vector space of strong, locally strong and ordinary intertwining operators, respectively. 3 Proposition 1.5. (i) Every strong logarithmic intertwining operator Y of type WW 1 W2 defines an ordinary intertwining operator of the same type. (ii) Every locally strong logarithmic intertwining operator Y among a triple of finitely generated modules is strong. Proof. Let k = depth(Y). From Y(w, x) =
k
Y (i) (w, x)logi (x)
i=0
it is clear that the truncation condition and Jacobi identity hold for Y (k) . From d k k k−1 (x) it follows that d x log (x) = x log [L(−1), Y (k) (w, x)] =
d (k) Y (w, x). dx
To prove (ii) it suffices to assume that W1 is finitely generated. Let {w1,1 , . . . , w1,m } be a generating set of W1 so that for every w1 ∈ W1 , there exist vi ∈ V and n i ∈ Z such
k (vi )n i w1,i . Let that w1 = i=1 Y(w1,i , x) =
ri
Y (i) (w1,i , x)logi (x), ri ∈ N.
i=1
The Jacobi identity now gives
x1 − x Y (vi , x1 )Y(w1,i , x) Y(Y (vi , x0 )w1,i , x) = Resx1 x0−1 δ x0 −x + x1 Y(w1,i , x)Y (vi , x1 ). −x0−1 δ x0
(1.5)
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After we take Coeff x −ni −1 in (1.5) we see that the powers of log(x) in Y((vi )n i w1,i , x) 0
are bounded by the highest power of log(x) in Y(w1,i , x). Because of the finiteness of the generating set W1 , the powers of log(x) in Y are globally bounded.
Remark 1. In the previous proposition we considered a canonical map W3 W3 −→ Ior d , Ist W1 W2 W1 W2 sending a strong logarithmic inertwining operator to its “top” (or the depth) component. This map is clearly surjective, but it is far from being injective 3 (see Lemma 5.4). Therefore, in general, the logarithmic fusion rules (i.e., dim I WW ) are not the same as 1 W2 W3 the nonlogarithmic fusion rules (i.e., dim Ior d W1 W2 ). Remark 2. It is tempting to relax the condition in (1.1) and assume instead that Y(·, x)· : W1 ⊗ W2 −→ W3 {x}[[log(x)]].
(1.6)
We will show (cf. Sect. 8) that there is a downside for doing that. One could even put further restriction on (1.1) and force a freshman calculus “formula” elog(x) = x.
(1.7)
This identity, in our formal variable setup, is unnatural and it should be avoided. The main problem is that elog(x) − x is a unit in the formal ring C[[x, log(x)]]. 2. An Extended Heisenberg Lie Algebra Let h be a finite-dimensional complex abelian Lie algebra with an inner product (·, ·). ˆ generated by Consider the central extension of the affinization of h, denoted by h, h(n) := h ⊗ t n , n ∈ Z, h ∈ h, with the bracket relations: [a(m), b(n)] = m(a, b)δm+n,0 C, [C, h(m)] = 0, where a, b ∈ h, m, n ∈ Z and C is the central element. For purposes of this paper we will assume that dim(h) = 1 with a fixed unit vector h, so that [h(m), h(n)] = mδm+n,0 C. The Lie algebra hˆ is an example of an extended Heisenberg Lie algebra (of course, the Heisenberg Lie algebra associated with h does not involve h(0)). Let hˆ = hˆ <0 ⊕ hˆ >0 ⊕ Ch(0) ⊕ CC, where hˆ <0 and hˆ >0 are defined as usual. Denote by Ck the category of restricted Z-graded, ˆ Wn , h–modules of level k, i.e., the category of Z-graded modules W = n∈Z
h(n)Wm ⊆ Wm−n , for every m, n ∈ Z,
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503
such that there exists N ∈ N, so that Wn = 0 for n < N , and the central element C acts as the multiplication with k. Let us denote by M(k) ≡ U(hˆ <0 ) · 1 essentially unique irreducible lowest weight module of level k, where h(n) · 1 = 0, for n ≥ 0 and the grading is the obvious one (see [FLM]). Then we have a version of the Stone-Von Neumann theorem for the extended ˆ Heisenberg algebra h: ˆ Lemma 2.1. Let W be a restricted h-module of level k. Then ∼ W = M(k) ⊗ (W ),
(2.8)
where (W ) = {w ∈ W : h(n)w = 0, n > 0} (the vacuum space of W ) is h-stable. Proof. It is known (see Theorem 1.7.3 in [FLM], for instance) that every restricted module for the Heisenberg algebra h˜ = hˆ \ Ch(0) admits a decomposition (2.8), where (W ) is the vacuum space of W . Now, h(0) commutes with h(n) for every n, hence it preserves the vacuum space (W ).
3. Logarithmic M(1)a -Modules The Heisenberg vertex operator algebra is omnipresent in conformal field theory and representation theory. Despite of its simplicity, it is the main tool for building (more interesting) rational vertex operator algebras (e.g., lattice VOAs [D,FLM]). Moreover, Heisenberg vertex operator algebras also contain many interesting subalgebras (e.g., W -algebras [FKRW]). It is well-known (cf. [FrB,FLM,K2]) that (M(1), Y, 1, ωa ) has a vertex operator algebra structure of central charge c = 1 − 12a 2 , where h(−1)2 1 + ah(−2)1. 2 This VOA will be denoted by M(1)a . It is known that M(1)a has infinitely many inequivalent irreducible modules, which can be easily classified [LL]. For every highest weight irreducible M(1)a -module W there exists λ ∈ C such that W ∼ = M(1)a ⊗ λ , where λ is one-dimensional and h(0) acts as the multiplication with λ. Such a module ˆ will be denoted by M(1, λ)a . The restricted h-modules are essentially logarithmic ˆ M(1)a and Virasoro algebra moM(1)a -modules. Since M(1)a is at the same time h, dule, we stress some differences when undertaking the contragradient module. If we ˆ of M(1, λ), under the standard anti-involution denote by M(1, λ)∗ the dual h-module h(n) → −h(−n), then we have ∼ M(1, −λ), M(1, λ)∗ = ωa =
ˆ On the other hand, if we denote by M(1, λ) the contragradient M(1)a viewed as h. a module (or the dual Virasoro algebra module) of M(1, λ)a (cf. [FHL]), then we have (3.9) M(1, λ) ∼ = M(1, 2a − λ)a . a
Let us also mention the automorphism τ of M(1)a , uniquely determined by τ (h(−n 1 ) · · · h(−n k )1) = (−1)k h(−n 1 ) · · · h(−n k )1. This map does not preserve the Virasoro element ωa , unless a = 0. As in Lemma 2.1 we now describe all logarithmic modules for the vertex operator algebra M(1)a .
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Proposition 3.1. Suppose that W is a logarithmic M(1)a -module. Then, viewed as an ˆ h-module, W ∼ = M(1)a ⊗ (W ). Proof. Every weak M(1)a -module carries a level one representation of hˆ via the expansion h(n)x −n−1 . Y (h(−1)1, x) = n∈Z
ˆ Because W is logarithmic, it is also a restricted Z-graded h-module. An application of
Lemma 2.1 yields W = M(1)a ⊗ (W ), for some vacuum space (W ). Corollary 3.2. Suppose that ( h 2(0) − ah(0))| admits a Jordan block of size at least 2. Then M(1)a ⊗ is a genuine logarithmic M(1)a -module. 2
In the case of irreducible M(1)-modules it possible to classify all intertwining operators and the corresponding fusion rules. The following result seems to be known (see [FrB] for instance). Proposition 3.3. Let M(1, λ)a , M(1, τ )a and M(1, ν)a be three (ordinary) M(1)a M(1,ν)a modules. Then the vector space of ordinary intertwining operators Ior d M(1,λ) a M(1,τ )a M(1,λ+τ )a is nontrivial if and only if ν = λ + τ . If so, Ior d M(1,λ) is one-dimensional. a M(1,τ )a In Sect. 5 we will substantially generalize this result. 4. Generalized Logarithmic Intertwining Operators Here we show that certain spaces of “logarithmic operators” give rise to (generalized) logarithmic modules. In particular, this construction will be useful for classification of logarithmic intertwining operators among certain triples of logarithmic M(1)a -modules. We will closely follow Li’s work [Li1] (see also [Li2]). The following definition is a logarithmic version of Definition 6.1.1 in [Li2] (see also [Li1]). Definition 4.1. Let V be a vertex operator algebra and W1 and W2 a pair of logarithmic V –modules. An operator valued formal series φ(x) = φα(n) logn (x)x −α−1 ∈ Hom(W1 , W2 ){x}[[log(x)]] α∈C n∈N
is called a generalized logarithmic intertwining operator of generalized weight h if it satisfies the following conditions: (i) For every w ∈ W1 we have φ(x)w ∈ W2 {x}[log(x)], and for every i (i)
φα+m w = 0, for m large enough. (ii) [L(−1), φ(x)] =
d d x φ(x).
Logarithmic Operators and Vertex Operators
505
(iii) For every v ∈ V , there is a positive integer n v such that (x1 − x2 )n v Y (v, x1 )φ(x2 ) = (x1 − x2 )n v φ(x2 )Y (v, x1 ). (iv) There exists k ∈ N such that k d ad L(0) − x − h φ(x) = 0, dx where ad L(0) φ(x) = [L(0), φ(x)]. We will denote by G h (W1 , W2 ) the vector space of generalized logarithmic intertwining operators of generalized weight h and by G log (W1 , W2 ) = ⊕h∈C G h (W1 , W2 ) the vector space of generalized logarithmic intertwining operators. Proposition 4.2. The vector space G log (W1 , W2 ) has a generalized logarithmic V -module structure. Proof. We will closely follow Sect. 6 in [Li2] (or Theorem 7.1.6 in [Li1]), with appropriate modifications due to logarithms. As in Definition 6.1.2 in [Li2] we let x1 − x2 Y (u, x0 ) ◦ φ(x2 ) = Resx1 x0−1 δ Y (u, x1 )φ(x2 ) x0 −x2 + x1 φ(x2 )Y (u, x1 ) . −x0−1 δ x0 The L(−1)-property is then proven as in [Li2]. Let a ∈ V be a homogeneous vector of weight wt(a) and φ(x) a generalized logarithmic intertwining operator of generalized weight h, so that (ad L(0) − x
d − h)k φ(x) = 0, dx
for some k ∈ N. We claim that ∂ ∂ k ad L(0) − h − wt(a) − x0 − x2 Y (a, x0 ) ◦ φ(x2 ) = 0. ∂ x0 ∂ x2 By using Lemma 7.1.4 in [Li1] we get the identity ∂ ∂ [L(0), Y (a, x0 ) ◦ φ(x2 )] = wt(a) + x0 Y (a, x0 ) ◦ φ(x2 ) + x2 ∂ x0 ∂ x2 ∂ φ(x2 ) . +Y (a, x0 ) ◦ ad L(0) φ(x2 ) − x2 ∂ x2 Thus
∂ ∂ ad L(0) − wt(a) − h − x0 Y (a, x0 ) ◦ φ(x2 ) − x2 ∂ x0 ∂ x2 ∂ − h φ(x2 ). = Y (a, x0 ) ◦ ad L(0) − x2 ∂ x2
From the previous formula and property (iv) in Definition 4.1 the claim now follows. Finally, the Jacobi identity is the verbatim repetition of the proof of Theorem 6.1.7 in [Li2].
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In parallel with the ordinary case, generalized logarithmic intertwining operators are closely related to logarithmic intertwining operators. Lemma 4.3. Let W be a logarithmic V -module and φ ∈ Hom V (W, G log (W1 , W2 )). Define Iφ (·, x) : W −→ Hom(W1 , W2 ){x}[[log(x)]], Iφ (w, x) = φ(w)(x). Then Iφ (·, x) is a logarithmic intertwining operator of type WWW2 1 . Proof. Firstly, Iφ (w, x)w1 ∈ W2 {x}[log(x)]. Now we have to check properties (i)-(iii) in Definition 1.2. Let Iφ (w, x) = wα(n) log(x)n x −α−1 . α∈C n≥0
Then the truncation property (i) wα+m w1 = 0 for m >> 0
is just a consequence of Definition 4.1,(i). Similarly, the L(−1)-property holds. The Jacobi identity is then proven as in formula (6.2.3) in [Li2].
W2 The map between Hom V (W, G log (W1 , W2 )) and I W W1 defined in the previous lemma injective. Conversely, to every logarithmic intertwining operator Y of is clearly type WWW2 1 we associate a map ψ ∈ Hom V (W, G log (W1 , W2 )), via ψ(w) = Y(w, x), w ∈ W. Combined together we obtain: Theorem 4.4. Let W , W1 and W2 be logarithmic V -modules. Then the vector space I WWW2 1 is naturally isomorphic to Hom V (W, G log (W1 , W2 )). The next result is a logarithmic analogue of Li’s Theorem 7.3.1 [Li1] (after Tsuchiya and Kanie who proved an important spacial case V = L sl2 (k, 0), k ∈ N). We do not need this theorem in full generality so we just focus on a special case V = M(1)a . Theorem 4.5. Let W1 and W2 be two logarithmic M(1)a -modules. Let be a finitedimensional h-module and Y(·, x) a linear map from to Hom(W1 , W2 )[[log(x)]]{x} satisfying the truncation condition, the L(−1)-property and (x1 − x2 )wt(a)−1 Y (a, x1 )Y(w, x2 ) − (−x2 + x1 )wt(a)−1 Y(w, x2 )Y (a, x1 ) x2 Y(awt(a)−1 · w, x), (4.10) = x1−1 δ x1 for every homogeneous a ∈ V andw ∈ . Then Y extends uniquely to an intertwining W2 . operator of type M(1)a ⊗ W1
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507
Proof. From (4.10) it follows that Y(u, x) is a generalized intertwining operator for every u ∈ . Moreover, is also an A(M(1)a )-module (where Zhu’s algebra A(M(1)a ) is just the polynomial ring in one variable). Now we may proceed as in [Li1]. The linear map Y extends to an intertwining map from M(1)a ⊗ . (The universal Verma M(1)a -module ¯ M() appearing in Li’s theorem is M(1)a ⊗ .) If there is another intertwining operator Y extending Y| , then Y − Y would be trivial on . But M(1)a ⊗ is generated by , so Y = Y .
5. Logarithmic Intertwining Operators Among M(1)a -Modules In this section we will give a sharp upper bound on the dimension of the vector space of strong logarithmic intertwining operators among certain logarithmic M(1)a -modules. From now on we shall assume that every logarithmic M(1)a -module is of the form M(1)a ⊗ , where is a finite-dimensional h-module such that (h(0) − λ)n | = 0 for some λ and n large enough. Then Proposition 1.10 in [M1] implies that every intertwining operator among a triple of such modules is strong. If we remove the finite-dimensionality condition on , it is not hard to construct logarithmic intertwining operators that are neither strong nor locally strong. We prove a few lemmas first. Lemma 5.1. Suppose that (h(0) − λ)m 1 |1 = 0 and (h(0) − ν)m 2 |2 = 0 for some W . Then for every w1 ∈ m i ∈ N, i = 1, 2. Let Y ∈ I M(1)a ⊗ 1 M(1)a ⊗ 2 M(1)a ⊗ 1 and w2 ∈ M(1)a ⊗ 2 we have (h(0) − λ − ν)m 1 +m 2 −1 Y(w1 , x)w2 = 0. are no genuine Moreover, if 1 and 2 are one-dimensional h-modules, then there W . logarithmic intertwining operators of type M(1)a ⊗ 1 M(1)a ⊗ 2 Proof. Let w1 ∈ M(1)a ⊗ 1 and w2 ∈ M(1)a ⊗ 2 so that (h(0) − λ)m 1 · w1 = (h(0) − ν)m 2 · w2 = 0. From the Jacobi identity it follows that (h(0) − λ − ν)m 1 +m 2 −1 Y(w1 , x)w2 m1 + m2 − 1 Y((h(0) − λ)i1 w1 , x)(h(0) − ν)i2 w2 . = i 1 i ≥ 0, i ≥ 0 1 2 i1 + i2 = m 1 + m 2 − 1
Now, it is easy to see that every term on the right-hand side is zero. If m 1 = m 2 = 1 then m 1 + m 2 − 1 = 1, so h(0) is diagonalizable on the image of Y, but so is L(0) = 21 h(0)2 − ah(0) + n>0 h(−n)h(n). Now, apply Proposition 1.10 in [M1].
Let us recall (cf. that for every wi ∈ Wi of generalized weight h i , i = 1, 2 [M1,HLZ]) 3 , the vector and every Y ∈ I WW 1 W2 (w1 )(k) α w2 is of generalized weight h 1 + h 2 − α − 1 (independently of k).
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m1 m2 Lemma 5.2. Suppose that (h(0) − λ) |1 = (h(0) − ν) |2 = 0 and let W . Then Y∈I M(1)a ⊗ 1 M(1)a ⊗ 2
Y(w1 , x)w2 ∈ x µλ W ((x)) ⊕ x µλ log(x)W ((x)) ⊕ · · · ⊕ x µλ log(x)m 1 +m 2 −2 W ((x)), for every wi ∈ M(1) ⊗ i , i = 1, 2. Equivalently, depth(Y) ≤ m 1 + m 2 − 2. Proof. The proof goes by induction on m 1 + m 2 ≥ 2. For m 1 + m 2 = 2 the statement holds by the previous lemma. From the same lemma and L(0)|1 =
1 h(0)2 − ah(0) 2
(5.11)
it follows that L(0)| M(1)a ⊗1 (resp. L(0)| M(1)a ⊗2 ) does not admit a Jordan block of size larger than m 1 (resp. m 2 ). For the induction step we apply the bracket relation between L(0) and Y(w1 , x) and an elementary ODE argument as in Proposition 1.10, [M1].
Lemma 5.3. Let Y and i be as in Lemma 5.2. Suppose further that λ = 0. Then for every wi ∈ M(1)a ⊗ i , i = 1, 2 we have Y(w1 , x)w2 ∈ W ((x)) ⊕ log(x)W ((x)) ⊕ · · · ⊕ logm 1 −1 (x)W ((x)).
(5.12)
If ν = 0, then (5.12) holds with m 1 replaced by m 2 . W Proof. By using the isomorphism I W1WW2 ∼ = I W2 W1 [M1], it is sufficient to consider the λ = 0 case, so that h(0)|1 is a nilpotent operator. We prove the formula (5.12) by induction on m 1 . Firstly, let m 1 = 1, so M(1)a ⊗ 1 ∼ = M(1)a . In this case we have to show that there are no genuine logarithmic intertwining operators (i.e., Y(w1 , x)w2 ∈ W ((x))). From the L(−1)-property and L(−1)1 = 0, it follows that w3 , Y(L(−1)1, x)w2 =
d w , Y(1, x)w2 = 0, w3 ∈ W3 . dx 3
Thus Y(1, x) is a constant term (operator) and it does not involve powers or log(x). Similarly, from the Jacobi identity, it follows that Y(w, x)w1 does not involve nonzero powers of log(x) for every w ∈ M(1)a and w1 ∈ M(1)a ⊗ 2 . Now, suppose that (5.12) holds for every m < m 1 . For w1 ∈ 1 , and w2 ∈ M(1)a ⊗ 2 and w3 ∈ (W ) = (W ) we clearly have w3 , Y(L(−1)w1 , x)w2 =
d w , Y(w1 , x)w2 . dx 3
On the other hand, the Jacobi identity gives Y(L(−1)w1 , x) = Y(h(−1)h(0)w1 , x) = •• h(x)Y(h(0)w1 , x) •• , so that d w , Y(w1 , x)w2 = w3 , Y(h(0)w1 , x)h(0)x −1 w2 dx 3
Logarithmic Operators and Vertex Operators
or x
509
d w , Y(w1 , x)w2 = w3 , Y(h(0)w1 , x)h(0)w2 . dx 3
(5.13)
If we denote ¯ 1 ) = {h(0)w1 : w1 ∈ 1 }, (W then h(0)m 1 −1 |(W ¯ 1 ) = 0 so by induction hypothesis the right-hand side in (5.13) is of the form P(log(x)), where deg(P) ≤ m 1 − 2. Therefore w3 , Y(w1 , x)w2 is a polynomial in log(x) of degree at most m 1 − 1. Now, the Jacobi identity implies that in w3 , Y(w1 , x)w2 the powers of log(x) are bounded by m 1 − 1 for every w1 ∈ M(1)a ⊗ 1 .
3 be a strong logarithmic intertwining operator of depth Lemma 5.4. Let Y ∈ I WW 1 W2 k among an arbitrary triple of logarithmic V -modules. Then Y−1 (·, x) :=
k−1 (i + 1)Y (i+1) (·, x)logi (x), i=0
defines a strong intertwining logarithmic operator of depth k − 1. Proof. The truncation condition and the Jacobi identity clearly hold for Y−1 (·, x). It remains to prove the L(−1)-property. If we distribute the L(−1)-property for Y(·, x) among Y (i) (·, x) we get [L(−1), Y (k) (w, x)] =
d (k) Y (w, x), dx
and [L(−1), Y (i) (w, x)] =
d (i) i + 1 (i+1) Y (w, x) + Y (w, x), dx x
for i ≤ k − 1. Thus [L(−1), Y−1 (w, x)] =
k−1 (i + 1)[L(−1), Y (i+1) (w, x)]logi (x) i=0
d (k) Y (w, x) logk−1 (x) dx k−2 d (i+1) i + 2 (i+2) Y Y + (i + 1) (w, x) + (w, x) logi (x) dx x
=k
i=0
=
k−1
(i + 1)
i=0
d (i+1) Y (w, x)logi (x) . dx
The following result gives a sharp upper bound on the depth of (strong) logarithmic intertwining operators among a triple of logarithmic M(1)a -modules.
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Theorem 5.5. Let h(0)|1 and h(0)|2 such that (h(0) − λ)m 1 |1 = (h(0) − ν)m 2 |2 = 0, and (h(0) − λ)m 1 −1 |1 = 0, (h(0) − ν)m 2 −1 |2 = 0, for some λ, ν ∈ C. Then we have W , (i) For every Y ∈ I M(1)a ⊗ 1 M(1)a ⊗ 2 ⎧ m1 + m2 − 2 ⎪ ⎨ m1 − 1 0 ≤ depth(Y) ≤ k = m ⎪ 2−1 ⎩ min(m 1 − 1, m 2 − 1)
for λν = 0, for λ = 0 and ν = 0 . (5.14) for λ = 0 and ν = 0 for λ=ν=0
(ii) There exists a canonical embedding W k → Homh(1 ⊗ 2 , (W ))⊕ , (5.15) I M(1)a ⊗ 1 M(1)a ⊗ 2 where k is as in (5.14). (iii) The range of depth(Y) in (5.14) is the best possible. More precisely, for every nonnegative integer m ≤ k there exists a logarithmic intertwining operator of depth exactly m. Proof. Here we prove (i) and (ii) only. We will complete the proof of (iii) in Sect. 7. The assertion (i) followsfrom Lemma 5.2 and 5.3. Let Y ∈ I W . Then Y admits a canonical expansion M(1)a ⊗ 1 M(1)a ⊗ 2 Y(w1 , x) =
m1 +m 2 −2
Y (i) (w1 , x)logi (x),
i=0
for every w1 ∈ M(1)a ⊗ 1 . By Lemma 5.2, for every i, Y (i) (w1 , x)w2 ∈ x λν W ((x)). Also, for w1 and w2 satisfying (L(0) − h 1 )m 1 w1 = (L(0) − h 2 )m 2 w2 = 0, the vector (i) (w1 )α w2 is homogeneous of generalized weight h 1 + h 2 − α − 1 for every i. Suppose 2 2 that w1 ∈ 1 and w2 ∈ 2 are of generalized weight λ2 − aλ and ν2 − aν, respectively. Then for every i, (i)
w1 ⊗ w2 → (w1 )−λν−1 (w2 ) defines an h-module map (i)
FY : 1 ⊗ 2 −→ (W ).
Logarithmic Operators and Vertex Operators
We claim that F : I
511
W M(1)a ⊗ 1 M(1)a ⊗ 2 (0)
(m 1 +m 2 −2)
F(Y) = (FY , . . . , FY
−→ Homh(1 ⊗ 2 , (W ))⊕
k
),
(i)
defines an embedding. Suppose that FY = 0 for every i. We have to show that Y ≡ 0. Firstly, we prove that Y(w1 , x)w2 = 0 for every w1 ∈ 1 and w2 ∈ 2 . In order to prove that we observe first that the contragradient (or dual) module of W , denoted by W , is again a logarithmic module generated by (W ) = (W ) . By the assumption we have w3 , Y(w1 , x)w2 = 0, w3 ∈ (W ). Furthermore, for every w3 ∈ (W ) and n ≥ 1 we have h(−n) · w3 , Y(w1 , x)w2 = −w3 , x −n Y(h(0)w1 , x)w2 = 0. Since W is generated by h(n), n ≤ −1 from (W ), the previous formula gives w3 , Y(w1 , x)w2 = 0, for every w3 ∈ W . Hence, Y(w1 , x)w2 = 0 for every w1 ∈ 1 and w2 ∈ 2 . Finally, M(1)a ⊗ i is generated by i , so Y(w1 , x)w2 = 0 for every wi ∈ M(1)a ⊗ i , i = 1, 2. This proves the injectivity. In Sect. 7 we will complete the proof of (iii). Now, let us assume that there exists Y and W such that depth(Y) = k, where k is as in (5.14) (i.e., depth(Y) reaches its upper bound). Then to construct Yi with depth(Yi ) = k − i, for i = 1, . . . , k we simply apply Lemma 5.4.
6. The Operator “x h(0) ” Let us recall that every endomorphism of a finite-dimensional complex vector space admits a unique decomposition h(0) = h s (0) + h n (0), where h s (0) and h n (0) are the semisimple and nilpotent part of h(0), respectively such that h s (0) and h n (0) commute. The following elementary fact will be of use in the next section: Let be a finite dimensional vector space and h(0) an endomorphism of . Then elog(x)h n (0) x h s (0) ,
(6.16)
is a solution of the ODE x
d A(x) = h(0)A(x), dx
where A(x) : V −→ V {x}[log(x)], is a linear map. The operator valued expression (6.16) is a replacement for x h(0) that (in the case when h(0) is semisimple) appears in many places in the literature. Clearly, the “operator” x h(0) for h(0) nonsemisimple is not well-defined.
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7. Logarithmic Intertwining Operators for M(1)a : The Proof of the Existence In this section to every T13,2 ∈ Homh(1 ⊗ 2 , 3 ) we associate a linear map Y(·, x)· : 1 ⊗ W2 −→ W3 {x}[log(x)], Wi = M(1)a ⊗ i , i = 2, 3, with the following properties: [h(n), Y(w, x)] = x n Y(h(0) · w, x), n ∈ Z, (7.17) d Y(w, x), (7.18) [L(−1), Y(w, x)] = dx for every w ∈ 1 . We will construct this map explicitly. Our construction mimics the well-known formulas when i are all one-dimensional, which will be a special case of our construction. Firstly, we fix a basis B = {w1 , . . . , wn } for 1 in which h(0)|1 admits a Jordan form consisting of a single Jordan block so that h(0) · w1 = λw1 , h(0) · wi = λwi + wi−1 , for 2 ≤ i ≤ n. Let
+
h(x) = h(0)log(x) +
(7.19)
h(m)x −m , −m
m>0
−
h(m)x −m . h(x) = −m m<0
Formal differentiation yields + d h(x) + dx Let
−
h(x) =
h(n)x −n−1 .
n∈Z
λh(m)x −m E (λ, x) = exp , −m m>0 λh(m)x −m − E (λ, x) = exp . −m +
m<0
Here we use a slightly different notation compared with [LL]; E ± (λ, x) are usually denoted by E ± (−λ, x). The following lemma is easy to prove (see for instance [FLM, K1,LL], etc.). Lemma 7.1. [L(−1), E − (λ, x)] = (
λh(n)x −n−1 )E − (λ, x),
n≤−2
[L(−1), E + (λ, x)] = E + (λ, x)(
n≥0
λh(n)x −n−1 ).
(7.20)
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513
Similarly, [h(n), E − (λ, x)E + (λ, x)] = x n λE − (λ, x)E + (λ, x), for n = 0, [h(0), E − (λ, x)E + (λ, x)] = 0. Also, [h(0), T23 (w)] = T23 (h(0) · w), w ∈ 1 , so that [h(0), E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) ] = E − (λ, x)E + (λ, x)T23 (h(0) · w)elog(x)λh n (0) x λh s (0) .
(7.21)
Lemma 7.2. For every w ∈ 1 we have [L(−1), E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) ] ⎞ ⎛ λh(n)x −n−1 ⎠ E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) =⎝ n≤−2
⎛ ⎞ +E − (λ, x)E + (λ, x)T 3 (w)elog(x)λh n (0) x λh s (0) ⎝ λh(n)x −n−1 ⎠ 2
n≥0 −
+h(−1)E (λ, x)E
+
(λ, x)T23 (h(0) · w)elog(x)λh n (0) x λh s (0) .
(7.22)
In particular if h(0) · w = λw, then [L(−1), E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) ]
d − = E (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) . dx Proof. Let us recall that L(−1) = on a. For simplicity let
1 2
n∈Z
• •
(7.23)
h(−n − 1)h(n) •• , which does not depend
A(λ, w, x) = E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) . By using [L(−1), T23 (w)] = h(−1)T23 (h(0) · w),
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A. Milas
we have [L(−1), A(λ, w, x)] ⎞ ⎛ λh(n)x −n−1 ⎠ E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) =⎝ n≤−2
⎛
+E − (λ, x)E + (λ, x) ⎝
⎞ λh(n)x −n−1 ⎠ T23 (w)elog(x)λh n (0) x λh s (0)
n≥0
+E (λ, x)E (λ, x)h(−1)T23 (h(0) · w)elog(x)λh n (0) x λh s (0) ⎛ ⎞ =⎝ λh(n)x −n−1 ⎠ E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) −
+
n≤−2
⎛
+E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) ⎝
⎞ λh(n)x −n−1 ⎠
n≥0
+λx E (λ, x)E (λ, x)T23 (h(0) · w)elog(x)λh n (0) x λh s (0) −x −1 λE − (λ, x)E + (λ, x)T23 (h(0) · w)elog(x)λh n (0) x λh s (0) +h(−1)E − (λ, x)E + (λ, x)T23 (h(0) · w)elog(x)λh n (0) x λh s (0) −1
⎛
=⎝
−
+
⎞
λh(n)x −n−1 ⎠ E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0)
n≤−2
⎛
+E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) ⎝
⎞ λh(n)x −n−1 ⎠
n≥0 −
+h(−1)E (λ, x)E
+
(λ, x)T23 (h(0) · w)elog(x)λh n (0) x λh s (0) .
Lemma 7.3. [h(n), E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) ] = λx n E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) , for n = 0, [h(0), E − (λ, x)E + (λ, x)T23 (w)elog(x)λh n (0) x λh s (0) ] = E − (λ, x)E + (λ, x)T23 (h(0) · w)elog(x)λh n (0) x λh s (0) . The following theorem gives a solution to Eqs. (7.17) and (7.18) for a Jordan block of length n. Theorem 7.4. Assume that wi satisfy (7.19). Let Y(w1 , x) = E − (λ, x)E + (λ, x)T23 (w1 )elog(x)λh n (0) x λh s (0) ,
(7.24)
Logarithmic Operators and Vertex Operators
515
and for 2 ≤ i ≤ n, let ⎛ i i−l − ( h(x)) j − ⎝ E (λ, x)E + (λ, x)T23 (wl ) Y(wi , x) = j! l=1 j=0 + h(x))i−l− j log(x)λh n (0) λh s (0) ( ×e x . (i − l − j)!
(7.25)
Then d Y(wi , x), dx n [h(n), Y(wi , x)] = x Y(h(0) · wi , x).
[L(−1), Y(wi , x)] =
(7.26)
Proof. We already proved the formula in the i = 1 case. So we may assume i ≥ 2. For every j ≥ 1 we have −
+ h(x)) j − h(x))i−l− j 3 + log(x)λh n (0) λh s (0) ( E (λ, x)E (λ, x)T2 (w1 )e ] x [L(−1), j! (i − l − j)! ⎛ ⎞ − ( h(x)) j−1 − E (λ, x)E + (λ, x)T23 (w1 ) =⎝ h(n)x −n−1 ⎠ ( j − 1)! n≤−2 + h(x))i−l− j log(x)λh n (0) λh s (0) ( ×e x (i − l − j)! −
( + h(x))i−l− j j ( h(x)) d E − (λ, x)E + (λ, x)T23 (w1 )elog(x)λh n (0) x λh s (0) + ( j)! dx (i − l − j)! −
j ( h(x)) d E − (λ, x)E + (λ, x)T23 (w1 )elog(x)λh n (0) x λh s (0) + ( j)! dx ⎞ ⎛ + ( h(x))i−l− j−1 ⎝ × h(n)x −n−1 ⎠ (i − l − j − 1)! n≥0 − + h(x))i−l− j d ( h(x)) j − 3 + log(x)λh n (0) λh s (0) ( E (λ, x)E (λ, x)T2 (w1 )e = x dx j! (i − l − j)! − ( h(x)) j−1 − E (λ, x)E + (λ, x)T23 (w1 ) −h(−1) ( j − 1)! + h(x))i−l− j log(x)λh n (0) λh s (0) ( ×e x . (i − l − j)! (
For 2 ≤ l ≤ i − 1, by using Lemma 7.2 we have [L(−1),
(
−
+ h(x)) j − h(x))i−l− j 3 + log(x)λh n (0) λh s (0) ( x E (λ, x)E (λ, x)T2 (wl )e ] j! (i − l − j)!
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A. Milas
⎛
⎞ − ( h(x)) j−1 − E (λ, x)E + (λ, x)T23 (wl ) =⎝ h(n)x −n−1 ⎠ ( j − 1)! n≤−2 + h(x))i−l− j log(x)λh n (0) λh s (0) ( ×e x (i − l − j)! ⎧⎛ ⎞ − ( h(x)) j ⎨ ⎝ + λh(n)x −n−1⎠ E − (λ, x)E + (λ, x)T23 (wl )elog(x)λh n (0) x λh s (0) ⎩ j! n≤−2 ⎛ ⎞ +E − (λ, x)E + (λ, x)T23 (wl )elog(x)λh n (0) x λh s (0) ⎝ λh(n)x −n−1⎠
n≥0
+ h(−1)E − (λ, x)E + (λ, x)T23 (h(0) · wl )elog(x)λh n (0) x λh s (0)
⎫ ⎬ ( + h(x))i−l− j
⎭ (i − l − j)!
+ h(x))i−l− j−1 ( h(x)) j − 3 + log(x)λh n (0) λh s (0) ( E (λ, x)E (λ, x)T2 (wl )e x + ( j)! (i − l − j − 1)! ⎛ ⎞ ×⎝ h(n)x −n−1 ⎠ −
n≥0
⎛
=⎝
n≤−2 +
⎞ − ( h(x)) j−1 − E (λ, x)E + (λ, x)T23 (wl )elog(x)λh n (0) x λh s (0) h(n)x −n−1 ⎠ ( j − 1)!
(
h(x))i−l− j (i − l − j)! ⎧⎛ ⎞ − j ( h(x)) ⎨ ⎝ + λh(n)x −n−1⎠ E − (λ, x)E + (λ, x)T23 (wl )elog(x)λh n (0) x λh s (0) ⎩ j! n≤−1 ⎛ ⎞ 3 − + log(x)λh n (0) λh s (0) ⎝ −n−1 ⎠ +E (λ, x)E (λ, x)T2 (w)e x λh(n)x
×
n≥0
+ h(−1)E − (λ, x)E + (λ, x)T23 (wl−1 )elog(x)λh n (0) x λh s (0)
⎫ ⎬ ( + h(x))i−l− j
⎭ (i − l − j)! −
( + h(x))i−l− j−1 ( h(x)) j − E (λ, x)E + (λ, x)T23 (wl )elog(x)λh n (0) x λh s (0) + ( j)! (i − l − j − 1)! ⎛ ⎞ ×⎝ h(n)x −n−1 ⎠ n≥0
d = dx
− + h(x))i−l− j ( h(x)) j − 3 + log(x)λh n (0) λh s (0) ( E (λ, x)E (λ, x)T2 (wl )e x j! (i − l − j)!
Logarithmic Operators and Vertex Operators
517
−
+ h(x)) j−1 − h(x))i−l− j 3 + log(x)λh n (0) λh s (0) ( −h(−1) x E (λ, x)E (λ, x)T2 (wl )e ( j − 1)! (i − l − j)! − + ( h(x)) j − ( h(x))i−l− j E (λ, x)E + (λ, x)T23 (wl−1 )elog(x)λh n (0) x λh s (0) . +h(−1) j! (i − l − j)! (
Also, by Lemma 7.2 we have [L(−1), E − (λ, x)E + (λ, x)T23 (wi )elog(x)λh n (0) x λh s (0) ]
d − = E (λ, x)E + (λ, x)T23 (wi )elog(x)λh n (0) x λh s (0) dx +h(−1)E − (λ, x)E + (λ, x)T23 (wi−1 )elog(x)λh n (0) . By combining the previous three formulas we obtain i−l i
( + h(x))i−l− j ( − h(x)) j − E (λ, x)E + (λ, x)T 3 (wl )elog(x)λh n (0) x λh s (0) ] 2 j! (i − l − j)!
[L(−1),
l=1 j=0 i−l i d = dx l=1 j=0
−h(−1)
( + h(x))i−l− j ( − h(x)) j − 3 + log(x)λh (0) λh (0) n s E (λ, x)E (λ, x)T (wl )e x + 2 j! (i − l − j)! ( + h(x))i−l− j ( − h(x)) j−1 − 3 + log(x)λh (0) λh (0) n s E (λ, x)E (λ, x)T (wl )e x 2 ( j − 1)! (i − l − j)!
i−1 i−l l=1 j=1
+h(−1)
i−l i l=2 j=0
( − h(x)) j − ( + h(x))i−l− j 3 + log(x)λh (0) λh (0) n s E (λ, x)E (λ, x)T (wl−1 )e x 2 j! (i − l − j)!
d Y(wi , x) = dx −h(−1)
( + h(x))i−l− j ( − h(x)) j−1 − 3 + log(x)λh (0) λh (0) n s E (λ, x)E (λ, x)T (wl )e x 2 ( j − 1)! (i − l − j)!
i−1 i−l l=1 j=1
+h(−1)
i−l i l=2 j=0
( + h(x))i−l− j ( − h(x)) j − 3 + log(x)λh (0) λh (0) n s E (λ, x)E (λ, x)T (wl−1 )e x 2 j! (i − l − j)!
d Y(wi , x). = dx
Assume now that n > 0. The formula (7.26) certainly holds for i = 1. Then for i ≥ 2, by Lemma 7.3, we get [h(n),
i l=1
⎛
⎛ ⎝
⎞ i−l − ( h(x)) j − ( + h(x))i−l− j 3 + log(x)λh (0) λh (0) n s ⎠] E (λ, x)E (λ, x)T (wl )e x 2 j! (i − l − j)! j=0
⎞ h(x)) j−1 − ( + h(x))i−l− j 3 + log(x)λh (0) λh (0) n s ⎠ E (λ, x)E (λ, x)T (wl )e x = 2 ( j − 1)! (i − l − j)! l=1 j=1 ⎛ ⎞ + i i−l i−l− j ( − h(x)) j − ( h(x)) n + log(x)λh (0) λh (0) 3 n ⎝ ⎠ + E (λ, x)E (λ, x)T (wl )e λx x s 2 j! (i − l − j)! l=1 j=0 ⎛ ⎞ + i−1 i−l−1 i−l− j−1 ( − h(x)) j − ( h(x)) n + log(x)λh (0) λh (0) n ⎝ ⎠ E (λ.x)E (λ, x)T 3 (wl )e x x s = 2 j! (i − l − j − 1)! i−1
l=1
⎝
i−l
−
( xn
j=0
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A. Milas
+
i l=1
⎛ ⎝
i−l j=0
−
( λx n
⎞ h(x)) j − ( + h(x))i−l− j 3 + log(x)λh (0) λh (0) n s ⎠ E (λ, x)E (λ, x)T (wl )e x 2 j! (i − l − j)!
= x n Y(wi−1 , x) + x n λY(wi , x).
Consequently, [h(n), Y(wi , x)] = x n Y(h(0) · wi , x). It is easy to see that the previous formula holds for n < 0 as well. Finally, by Lemma 7.3, [h(0), Y(wi , x)] = Y(h(0) · wi , x).
Remark 3. Even though the intertwining operator Y(w, x) associated with T13,2 was defined for a single Jordan block only, it is now straightforward to define Y(w, x) for an arbitrary finite-dimensional h-module 1 . Finally, we have a description of logarithmic intertwining operators among a triple of logarithmic modules with finite-dimensional vacuum spaces. Theorem 7.5. Let W1 = M(1)a ⊗ 1 , W2 = M(1)a ⊗ 2 and W3 = M(1)a ⊗ 3 as above and T13,2 ∈ Homh(1 , Hom(2 , 3 )), then Y(w, x) associated with T13,2 as in Theorem 7.4, and defined for w ∈ 1 only, 3 extends uniquely to a logarithmic intertwining operator of type WW . 1 W2 Proof. Because of Theorem 7.4 and Theorem 4.5, it only remains to show (4.10). The formula (4.10) clearly holds in the case when v = h(−1)1 by Theorem 7.4. But the vertex operator algebra is generated by the vector h(−1)1 so (4.10) holds for every homogeneous v ∈ M(1)a (cf. Proposition 7.1.5 and 7.3.3 in [Li1] ).
⊗2 = Id ∈ Proof of Theorem 5.5. (iii). We choose (W ) = 1 ⊗ 2 , such that T11, 2 M(1)a ⊗ 1 ⊗ 2 as constructed in Homh(1 ⊗ 2 , 1 ⊗ 2 ). Let Y ∈ I M(1)a ⊗ 1 M(1)a ⊗ 2 Theorem 7.5 and Theorem 7.4. It is not hard to see that for such Y and λν = 0, we have depth(Y) = m 1 + m 2 − 2, because of (7.24) and (7.25). Similarly for λ = 0 or ν = 0.
8. Mock Logarithmic Intertwiners Among Ordinary Modules In this section (which is completely independent from the rest of the paper) we construct certain operators related to logarithmic intertwining operators studied in earlier sections. These operators involve logarithms but operate among ordinary M(1)a -modules. The next result has already been proven. Proposition 8.1. The only logarithmic intertwining operators among M(1, λ)a , M(1, ν)a and M(1, λ + ν)a are the ordinary intertwining operators.
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519
Proposition 8.2. Suppose that in Definition 1.2 we allow Y(·, x)· to satisfy (1.6). Under this new definition, there exists a nontrivial (mock) logarithmic intertwining operator among every triple of ordinary M(1)a -modules M(1, λ)a , M(1, ν)a and M(1, λ + ν)a , provided that λν = 0. M(1,λ+ν)a Proof. Let Y ∈ I M(1,λ) . It is easy to see that the operator a M(1,ν)a ¯ x) = Y(·, x)x −λh(0) Y(·, satisfies the Jacobi identity, but it doesn’t satisfy the L(−1)-property. For the same reason Y(·, x)x −λh(0) h(0)k log(x)k satisfies the Jacobi identity (notice that h(0)k acting on M(1, ν)a is merely ν k ), but again it does not satisfy the L(−1)-property. Finally, we consider Ylog (·, x) = Y(·, x)x −λh(0) eλh(0)log(x) , where eλh(0)log(x) =
∞ λn h(0)n log(x)n n=0
n!
.
Now, as before Ylog (·, x) satisfies the Jacobi identity but also the L(−1)-property [L(−1), Ylog (w, x)] =
d Ylog (w, x), dx
which follows from L(−1)-property for Y(·, x) and the formula d −λh(0) λh(0)log(x) (x e ) = 0. dx
9. Indecomposable and Logarithmic Representations of the Virasoro Algebra of Central Charge c = 1 In Sect. 7 we described logarithmic intertwining operators associated with logarithmic M(1)a -modules. Here we restrict our construction to an important vertex operator subalgebra L(1, 0) = U(V ir ) · 1 ⊂ M(1)0 , where L(c, h), (c, h) ∈ C2 denote the irreducible lowest weight irreducible module for the Virasoro algebra of central charge c and lowest conformal weight h [KR]. For simplicity, we shall also use M(1) instead of M(1)0 . Results from this section form logarithmic extensions of several results from [M2]. Let us introduce some notation. Let W be a V ir -module. By • we will denote a lowest weight vector inside W such that U (V ir≤0 ) · • is an irreducible lowest weight module (every • is of course a singular vector in W ). By a we will denote a vector that becomes a lowest weight vector in the quotient of W by moding out the submodule generated by all lowest weight vectors. These vectors will be called subsingular vectors. Similarly will denote a vector that becomes a lowest vector after quotienting with the submodule of W generated by all lowest weight vectors and all subsingular vectors. Such a vector is called a sub-subsingular vector. One can continue in this manner and introduce vectors
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A. Milas
that become lowest weight vectors after quotienting with the submodule generated by the lowest weight, subsingular and sub-subsingular vectors, but we shall not need those in the paper. An arrow → • indicates that • is contained in the submodule generated by , etc. We also recall here that cosingular vectors are those vectors that are being mapped to singular vectors in the contragradient module. There is also a pairing between singular vectors and equivalence classes of cosingular vectors in a module. Let us illustrate these definitions with an example: Let W be a V ir -module such that 0 → L(1, m 2 ) → W → L(1, (m + 1)2 ) → 0,
(9.27)
is a nonsplit extension, so that W is generated by a subsingular vector of weight (m +1)2 . This extension may be visualized as follows: • _? ?? ?? ??
where the arrow pointing up indicates that the conformal weight of • is smaller than that of . More complicated diagrams will appear later. In the previous example we implicitly assumed the following result (see [M2]). Proposition 9.1. For every k, m ∈ Z, we have Ext1V ir,L(0) (L(1, k 2 ), L(1, m 2 )) = C, if and only if |k − m| = 1. In all other cases Ext 1V ir,L(0) is trivial. Our aim is to determine the Virasoro submodule structure of M(1) ⊗ (viewed as a V ir -module) for some special h(0)| . The first part in the following result is well-known (see for instance [KR]). For the second part see [DG] or [M2]. Theorem 9.2. Viewed as a Virasoro module M(1) decomposes as a direct sum of irreducible Virasoro modules ∞ M(1) = L(1, m 2 ). (9.28) m=0
If u m denotes the lowest weight vector (unique up to a nonzero scalar) of L(1, m 2 ), then the Virasoro module generated by Y (u n , x)u m decomposes as L(1, (m + n)2 ) ⊕ L(1, (m + n − 2)2 ) ⊕ · · · ⊕ L(1, (m − n)2 ). The previous theorem can be used for construction of some intertwining operators among irreducible L(1, 0)-modules. This construction L(1,1)relies on non-vanishing of certain 3 jsymbols [DG,M2]. For instance, dim I L(1,1) L(1,1) = 1, but this “fusion rule” is not covered by Theorem 9.2. Here is a useful consequence of Theorem 9.2
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521
Corollary 9.3. The Virasoro module generated by {h(−n)u m : n ∈ Z} is isomorphic to
L(1, (m − 1)2 ) ⊕ L(1, (m + 1)2 ),
for m ≥ 1. If m = 0, the first summand in (9.29) is trivial. Proof. We just have to observe that h(−1)1 = u 1 , so that Y (u 1 , x) = Now, apply the previous theorem.
(9.29)
n∈Z h(n)x
−n−1 .
Lemma 9.4. Let h(0)| be a nilpotent operator. Then M(1) ⊗ is a self-dual M(1)module, i.e., (M(1) ⊗ ) ∼ = M(1) ⊗ . Clearly, the same is true if M(1) ⊗ is viewed as a Virasoro algebra module. Proof. Every logarithmic M(1)-module W is uniquely determined by the h(0)-action on (W ). Thus, two logarithmic M(1)-modules M(1) ⊗ 1 and M(1) ⊗ 2 are equivalent if and only if there exists : 1 −→ 2 such that h(0)|1 = −1 h(0)|2 . The module (M(1) ⊗ ) is isomorphic to M(1) ⊗ , where the action of h(0) on the dual space is given by h(0) · w , w = −w , h(0) · w, w ∈ , w ∈ , so that h(0)| = −h ∗ (0), where h ∗ (0) is the dual map. The operator h(0) is nilpotent, thus −h ∗ (0) and consequently h(0)| are nilpotent as well. But −h(0)∗ and h(0) admit the same Jordan form, so there exists with wanted properties.
Theorem 9.5. Let be a two-dimensional space and h(0)| = 0, h(0)2 | = 0. Then viewed as a Virasoro module, M(1) ⊗ is generated by a sequence of subsingular1 vectors as on the following diagram: / • aD DD zz DD z z zz DDD }zz / • aD DD zz DzDz zz DDD z }z / • aD DD zz DzDz zz DDD z }z / • aD DD zz DzDz zz DDD }zz / • bFF FF yyy Fy yyFF yy FF | y ... ...
(9.30)
where the s th • and , counting from the top, have conformal weight (s −1)2 , s ≥ 1. Here dotted arrows indicate the action of the transpose of h(0), which uniquely determines every . 1 These subsingular vectors are also cosingular.
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A. Milas
Proof. Firstly, we may and will choose a basis {w1 , w2 } of such that h(0) · w1 = 0, h(0) · w2 = w1 , so that h(0)T · w1 = w2 and h(0)T · w2 = 0. Since h(0)2 | = 0, the module M(1) ⊗ is L(0)-diagonalizable. From the Virasoro algebra embedding ˆ M(1) → M(1)⊗, where we use the identification M(1) = U(h)·w 1 , and Theorem 9.2, it is clear that M(1) ⊗ contains a sequence of singular vectors of weight m 2 for every m ≥ 0. These singular vectors, displayed in the left column of (9.30) by • are determined up to a constant. Let u m = Pm (h)w1 denote such a vector of weight m 2 , where Pm (h) is a polynomial in h(−i), i ≥ 1, of degree m 2 . As we already mentioned there are also vectors in M(1)⊗ that become singular after quotienting with M(1). These vectors are uniquely determined if we assume that every is obtained from • by applying h(0)T to u m . These vectors will be denoted by u 2,m , m ≥ 0, so that u 2,m = h(0)T u m = Pm (h)·w2 . It is clear that M(1) ⊗ is generated by S = {u m : m ≥ 0} ∪ {u 2,m : m ≥ 0}. It remains to prove that we can reduce the generating set S down to {u 2,m : m ≥ 0}. The short exact sequence of Vir-modules π
0 −→ M(1) −→ M(1) ⊗ −→ M(1) −→ 0, together with Theorem 9.2 gives π
0 −→ ⊕m≥0 L(1, m 2 ) −→ M(1) ⊗ −→ ⊕m≥0 L(1, m 2 ) −→ 0. Thus M(1) ⊗ , which is L(0)-diagonalizable, gives a nonzero element in Ext 1V ir,L(0) (⊕m≥0 L(1, m 2 ), ⊕n≥0 L(1, n 2 )) ∼ Ext 1V ir,L(0) (L(1, m 2 ), L(1, n 2 )). = m
n
(9.31) Now, Proposition 9.1 implies that
Ext1V ir,L(0) (⊕m≥0 L(1, m 2 ), ⊕n≥0 L(1, n 2 )) ∼ =
Ext 1V ir,L(0) (L(1, m 2 ), L(1, n 2 )).
|m−n|=1
(9.32) Already from the previous formula it is clear that there could be at most two arrows exiting from u 2,m . Now, we determine these arrows for every m. For m = 0, there is precisely one outgoing arrow from u 2,0 pointing to u 1 . This follows from L(−1)w2 = h(−1)h(0)w2 = h(−1)w1 = u 1 , where u 1 is the lowest weight vector of L(1, 1) ⊂ M(1). Claim. For m ≥ 1, from each u 2,m there are precisely two outgoing arrows; one pointing to u m−1 and the other pointing to u m+1 . To see that we write the generator L(n), n ∈ Z as h(0)h(n) +
1 2
• •
h(k)h(l) •• .
k+l=n,kl=0
Now, L(n)u 2,m = L(n)Pm (h)w2 ⎛ 1 = ⎝h(0)h(n) + 2
⎞ • •
h(k)h(l) •• ⎠ Pm (h)w2
k+l=n,kl=0
¯ = h(n)Pm (h)w1 + L(n)P m (h)w2 ,
Logarithmic Operators and Vertex Operators
where
523
⎛ 1 ¯ L(n) = ⎝ 2
⎞ • •
h(k)h(l) •• ⎠ .
k+l=n,kl=0
By using Corollary 9.3, we have h(n)Pm (h)w1 ∈ L(1, (m − 1)2 ) ⊕ L(1, (m + 1)2 ), n ∈ Z. 2,m = 0, n ≥ 1, it follows that there exists a + ∈ U(V ir ), such ¯ Combined with L(n)u >0 that
a + · u 2,m = u m−1 . This proves that there is an arrow pointing to u m−1 . Let us recall that the duality functor interchanges singular and cosingular vectors. In addition, orientations of arrows are reversed. Now, Lemma 9.4 yields an isomorphism between M(1) ⊗ and its dual, which maps singular vectors u m to cosingular vectors w 2,m and cosingular vectors u 2,m to singular vectors w m such that w m and w 2,m form a Jordan block with respect to h(0) (i.e., h(0) · w m = 0, h(0) · w 2,m = w m ). Clearly, w m generates the lowest weight module L(1, m 2 ). Thus, there will be an arrow pointing down from w 2,m−1 to w m . Now, by using the same argument as before we argue that there is an arrow pointing up from every w 2,m to w m−1 .
Here is a useful consequence of the previous theorem. For simplicity we only consider with dim() = 3. Corollary 9.6. Let be a three dimensional h-module such that h(0)3 | = 0 and h(0)2 | = 0. Then, viewed as a V ir -module, M(1) ⊗ is generated by a sequence of sub-subsingular vectors (denoted by ) as on the following diagram • aC CC {{ aCCC {{ CC{{ C { C{ {C {{ CCC {{{ CCC }{ }{{ • aC CC {{ aCCC {{ CC{{ C { C { {C {{ CCC {{{ CCC }{ }{{ • aC CC {{ aCCC {{ CC{{ C { C { {C {{ CCC {{{ CCC }{ }{{ • aC CC {{ aCCC {{ CC{{ C { C { {C {{ CCC {{{ CCC }{ }{{ • bEE bE EE yyy EEE yyy E E y y E E y E yy E yy EE |yyy EE ... . . .|y ...
(9.33)
where every (resp. ) is obtained from • (resp. ) of the same generalized weight by applying the transpose of h(0). For simplicity we do not display dotted arrows and arrows obtained by the “addition of arrows” rule.
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Proof. From an embedding M(1) ⊗ 2 → M(1) ⊗ 3 it is clear that there will be arrows connecting u 2,m and u m±1 as in Theorem 9.5. Let u 3,n = h T (0)u 2,n , so u 3,n is represented by a . By arguing as before, from each u 3,m there will be arrows pointing to u 2,m+1 and u 2,m−1 . So we only have to show that there are no additional arrows from u 3,m except those displayed on (9.33). From the formula L(0)u 3,n = n 2 u 3,n + 21 u n it follows that u n and u 3,n form a Jordan block with respect to L(0). But u n can be also reached from u 3,n via an oriented path u 3,n → u 2,n+1 → u n so there is no need to display an arrow from u 3,n to u n , because the submodule generated by u 3,n contains u n , and similarly with u 3,n and u n+2 .
10. Hidden Logarithmic Intertwining Operators
01 , and 00
Suppose that h(0)|2 in some basis {w1 , w2 } for 2 is represented by ⎡ ⎤ 010 h(0)|3 , in some basis {w˜ 1 , w˜ 2 , w˜ 3 } for 3 , is represented by ⎣ 0 0 1 ⎦. Then a surjective 000 map T23,2 ∈ Hom(2 ⊗ 2 , 3 ), defined by w2 ⊗ w2 → w˜ 3 , w2 ⊗ w1 →
w˜ 2 w˜ 2 w˜ 1 , w1 ⊗ w2 → , w1 ⊗ w1 → 2 2 2
(10.34)
commutes with h(0). Let us denote by W2 (1, m 2 ) ⊂ M(1) ⊗ 2 a cyclic V ir -module generated by u 2,m of weight m 2 . For m > 0, W2 (1, m 2 ) can be visualized as a “wedge” in (9.30) • _? ?? ?? ??
•
or a single arrow
•
in the m = 0 case. Similarly, we denote by W3 (1, m 2 ) ⊂ M(1) ⊗ 3 (cf. Corollary 9.6) the module generated by , of generalized weight m 2 . For every m > 1 this module can be visualized as
Logarithmic Operators and Vertex Operators
525
• _? ?? ?? ??
`@ @@@ @@ @ • _? ?? ~~ ?? ~ ?? ~~~ ~~ • Similarly, W3 (1, 0) may be visualized as • _? ?? ~ ~ ?? ~ ?? ~~~ ~~ • Again, we shall assume that (resp. ) is obtained from (resp. •) of the same generalized weight by applying h(0)T . The following lemma is just a consequence of L(0)u 3,m = m 2 u 3,m + 21 u m , so we omit the proof. Lemma 10.1. For every m ≥ 0 the module W3 (1, m 2 ) is a genuine logarithmic module. The module W3 (1, m 2 ) is a nonsplit extension of L(1, m 2 ) by W2 (1, (m − 1)2 ) + W2 (1, (m + 1)2 ), where if m = 0 the first summand is trivial. Now we have a logarithmic version of Theorem 9.2. Theorem 10.2. Let 2 and 3 be as above. Then there exists a nontrivial Y ∈ I M(1) ⊗ 3 , such that Y projects down to a hidden logarithmic intertM(1) ⊗ 2 M(1) ⊗ 2 wining operator Y¯ ∈ I of depth one, where W =
W W2 (1, m 2 ) W2 (1, n 2 )
|m − n| ≤ k ≤ m + n k ≡ m + n mod 2
This sum is not direct, whenever mn = 0.
W3 (1, k 2 ).
(10.35)
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A. Milas
Proof. Let Y be as in Theorem 7.5, with T23,2 as in (10.34). Let ¯ , x)· = Y(· , x) · |W (1,m 2 )⊗W (1,n 2 ) . Y(· 2 2 We recall (cf. Sect. 7) that in this case Y(u 0 , x) = Y(w1 , x) = T23,2 (w1 ), − h(x)T23,2 (w1 ) + T23,2 (w1 ) Y(u 2,0 , x) = Y(w2 , x) = + h(x) + T23,2 (w2 ). ×
(10.36)
Y(u 2,m , x) = Y(Pm (h)u 2,0 , x).
(10.37)
Also, We need a more precise information about the image of Y. Since W2 (1, m 2 ) is cyclic, and ¯ denoted by Wˆ , is actually the Virasoro submodule gegenerated by u 2,m , the image of Y, ¯ 2,m , x)u 2,n , Y(u ¯ m±1 , x)u 2,n , Y(u ¯ 2,m , x)u n±1 nerated by the Fourier coefficients of Y(u ¯ m±1 , x)u n±1 . As before, let u 3,k denote a generator of W3 (1, k 2 ) of generalized and Y(u weight k 2 . Since L(1, (m − 1)2 ) ⊕ L(1, (m + 1)2 ) ⊂ W2 (1, m 2 ), then Theorem 9.2, ¯ m±1 , x)u n±1 is (10.36) and (10.37) imply that the submodule of Wˆ generated by Y(u precisely L(1, (m − n − 2)2 ) ⊕ L(1, (m − n)2 ) ⊕ · · · ⊕ L(1, (m + n + 2)2 ).
(10.38)
Now we move “one step higher” or “deeper” in the filtration and determine the Virasoro ¯ 2,m , x)u n . From the formula submodule generated by Y(u h(0)Y(u 2,m , x)u n = Y(u m , x)u n , and the previous discussion it follows that the submodule generated by Y(u 2,m , x)u n±1 is contained inside W2 (1, (m − n − 1)2 ) + · · · + W2 (1, (m + n + 1)2 ) and possibly some L(1, k 2 ), k 2 ∈ / {(m − n − 2)2 , . . . , (m + n + 2)2 }. But having in the image such L(1, k 2 ) ¯ m±1 , x)u 2,n would contradict Theorem 9.2. Similarly, the submodule generated by Y(u 2 2 lies again inside the sum W2 (1, (m − n − 1) ) + · · · + W2 (1, (m + n − 1) ). Furthermore, from h(0)Y(u 2,m , x)u 2,n = Y(u m , x)u 2,n + Y(u 2,m , x)u n , ¯ 2,m , x)u 2,n is it follows that the Virasoro module generated by the coefficients of Y(u 2 2 contained inside W3 (1, (m − n) ) + · · · + W3 (1, (m + n) ) and possibly some irreducible module L(1, k 2 ) not included in (10.38). But this would again contradict Theorem 9.2. Thus, we have shown that the image Wˆ is contained inside W (cf. 10.35). In fact, it is not hard to show that u 3,k ∈ Wˆ for k 2 ∈ {(m − n)2 , . . . , (m + n)2 }, which would imply Wˆ = W . Finally, from (10.36) and (10.37) it is clear that Y¯ is a genuine logarithmic intertwining operator of depth one.
Logarithmic Operators and Vertex Operators
527
The module W in the previous theorem has the following diagram representation
~ @@@ ~ @@ ~ @@ ~~ ~~ ~ } AAA } } AA AA }}} ~} • •
... || | | || ~|| ... BB BB BB BB .! . .
. . .B BB BB BB B ... || | || || } | ...
~~ ~~ ~ ~~ ~ AA AA AA A
@ @@ @@ @@
•
}} }} } }~ }
? ?? ?? ??
•
Remark 4. It is not hard to generalize the results from this section to logarithmic modules with Jordan blocks of arbitrary size.
11. Hidden Logarithmic Intertwining Operators Among Feigin-Fuchs Modules at c = 1 − 12a2 Let us recall that M(1, λ)a is a M(1)a -module of lowest conformal weight λ2 − aλ. As we have already mentioned in the introduction, the λ = a case (e.g., λ = 0 for a = 0) is indeed very special. Here is a consequence of Corollary 3.2. 2
Lemma 11.1. Let M(1)a ⊗ , where is two-dimensional and h(0)| is represented λ1 by , in some basis. Then M(1)a ⊗ is an ordinary M(1)a -module if and only if 0λ λ = a, in which case the Feigin-Fuchs module M(1, a)a is of lowest conformal weight −a 2 2 . Let be as in the lemma. It is easy to see that M(1)a ⊗ ( ⊗ ) is a genuine logarithmic M(1)a -module. Now, we have a consequence of Theorem 7.5. Corollary 11.2. Let M(1)a ⊗ be as in Lemma 11.1. Then there exists a genuine M(1)a ⊗ ( ⊗ ) . logarithmic intertwining operator of type M(1)a ⊗ M(1)a ⊗ Example. For a = 21 the vertex operator algebra M(1) 1 has central charge c = −2. 2 Then M(1) 1 -module M(1) 1 ⊗ , where h(0)| is represented by 2
2
1 2
0
1
1 2
is an ordinary M(1) 1 -module with the lowest conformal weight − 18 . The intertwining 2 operator constructed in Corollary 11.2 is closely related to logarithmic operators studied in [Gu]. Acknowledgements. We are grateful to anonymous referees for several useful comments.
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References Adamovi´c, D., Milas, A.: Logarithmic intertwining operators and W(2, 2P −1)-algebras. J. Math. Phys. 48(7), 073504 (2007) [BF] Bredthauer, A., Flohr, M.: Boundary states in c = −2 logarithmic conformal field theory. Nucl. Phys. B 639, 450–470 (2002) [D] Dong, C.: Vertex algebras associated with even lattices. J. Alg. 161(1), 245–265 (1993) [DG] Dong, C., Griess, R.: Rank one lattice type vertex operator algebras and their automorphism groups. J. Alg. 208, 262–275 (1998) [DLM] Dong, C., Li, H., Mason, G.: Regularity of rational vertex operator algebras. Adv. Math. 132, 148–166 (1997) [FFHST] Fjelstad, J., Fuchs, J., Hwang, S., Semikhatov, A.M., Yu, I.: Tipunin, logarithmic conformal field theories via logarithmic deformations. Nuclear Phys. B 633, 379–413 (2002) [F1] Flohr, M.: On modular invariant partition functions of conformal field theories with logarithmic operators. Int. J. Mod. Phys. A 11, 4147–4172 (1996) [F2] Flohr, M.: Singular vectors in logarithmic conformal field theory. Nucl. Phys. B 514, 523–552 (1998) [F3] Flohr, M.: Bits and pieces in logarithmic conformal field theory. proceedings of the school and workshop on logarithmic conformal field theory and its applications (Tehran, 2001). Int. J. Mod. Phys. A 18, 4497–4591 (2003) [FrB] Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves. Mathematical Surveys and Monographs, 88, Providence, RI: Amer. Math. Soc., 2001 [FKRW] Frenkel, E., Kac, V., Radul, A., Wang, W.: W1+∞ and W(glN ) with central charge N . Commun. Math. Phys. 170, 337–357 (1995) [FHL] Frenkel, I.B., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs Amer. Math. Soc. 104, Providence, RI: Amer. Math. Soc., 1993 [FLM] Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster. Pure and Appl. Math., 134, New York: Academic Press, 1988 [G1] Gaberdiel, M.: Fusion rules and logarithmic representations of a wzw model at fractional level. Nucl. Phys. B 618, 407–436 (2001) [G2] Gaberdiel, M.: An algebraic approach to logarithmic conformal field theory. In: Proceedings of the School and Workshop on Logarithmic Conformal Field Theory and its Applications (Tehran, 2001). Int. J. Mod. Phys. A 18, 4593–4638 (2003) [GK1] Gaberdiel, M., Kausch, H.G.: A rational logarithmic conformal field theory. Phy. Lett. B 386, 131–137 (1996) [GK2] Gaberdiel, M., Kausch, H.G.: Indecomposable fusion products. Nucl. Phy. B 477, 293–318 (1996) [GK3] Gaberdiel, M., Kausch, H.G.: A local logarithmic conformal field theory. Nucl. Phys. B 538, 631–658 (1999) [Gu] Gurarie, V.: Logarithmic operators in conformal field theory. Nucl. Phys. B 410, 535–549 (1993) [H1] Huang, Y.-Z.: Vertex operator algebras and the Verlinde conjecture. http://arxiv.org/list/math.QA/ 0406291, 2004 [H2] Huang, Y.-Z.: Vertex operator algebras, the verlinde conjecture, and modular tensor categories. Proc. Nat. Acad. Sci. 102(15), 5352–5356 (2005) [HL] Huang, Y.-Z., Lepowsky, J.: Tensor products of modules for a vertex operator algebra and vertex tensor categories. In: Lie Theory and Geometry, in honor of Bertram Kostant, ed. R. Brylinski, J.-L. Brylinski, V. Guillemin, V. Kac, Boston: Birkhäuser, 1994, pp. 349–383 [HLZ] Huang, Y.-Z., Lepowsky, J., Zhang, L.: A logarithmic generalization of tensor product theory for modules for a vertex operator algebra. Int. J. Math. 17(8), 975–1012 (2006) [K1] Kac, V.: Infinite-dimensional Lie algebras. Third edition, Cambridge: Cambridge University Press, 1990 [K2] Kac, V.: Vertex algebras for beginners. University Lectures Series, Vol. 10, Providence, RI: Amer. Math. Soc., 1998 [KR] Kac, V., Raina, A.: Bombay lectures on highest weight representations of infinite-dimensional Lie algebras. In: Advanced Series in Mathematical Physics, Vol 2, River Edge, NJ: World Scientific, 1987 [Le] Lepowsky, J.: From the representation theory of vertex operator algebras to modular tensor categories in conformal field theory. Proc. Nat. Acad. Sci. 102(15), 5304–5305 (2005) [LL] Lepowsky, J., Li, H.: Introduction to Vertex Operator Algebras and Their Representations. Progress in Mathematics, 227, Boston: Birkhäuser, 2003 [Li1] Li, H.: Representation theory and a tensor product theory for vertex operator algebras. PhD thesis, Rutgers University, 1994 [AM]
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Communicated by Y. Kawahigashi
Commun. Math. Phys. 277, 531–553 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0385-1
Communications in
Mathematical Physics
On the Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model Fanghua Lin2 , Ping Zhang1,2 , Zhifei Zhang3 1 Academy of Mathematics & Systems Science, The Chinese Academy of Sciences, 100080, Beijing, China.
E-mail: [email protected]
2 Courant Institute, New York University, 10012, New York, NY, USA. E-mail: [email protected] 3 School of Mathematical Science, Peking University, 100871, Beijing, China.
E-mail: [email protected] Received: 22 February 2007 / Accepted: 17 May 2007 Published online: 10 November 2007 – © Springer-Verlag 2007
Abstract: In two space dimension, we prove the global existence of smooth solutions to a coupled microscopic-macroscopic co-rotational FENE dumbbell model which arises from the kinetic theory of diluted solutions of polymeric liquids with noninteracting polymer chains.
1. Introduction In this paper, we consider the global existence of smooth solutions to a coupled microscopic-macroscopic model which arises from the kinetic theory of diluted solutions of polymeric liquids with noninteracting polymer chains. More precisely, we consider in two space dimension the co-rotational FENE Dumbbell Model, which models polymers by nonlinear springs, and which takes into account the finite extensibility of the polymer chains. Mathematically, this system reads (in a non-dimensional form): ⎧ ε 1−ε u = −∇ p + ReW x ∈ R2 , ⎨ u t + u · ∇u − Re e div τ, 2 div u = 0, x ∈ R , ⎩ f t + u · ∇x f + ∇q · (S(u)q f ) = 2W1eN q f + 2W1 e divq (∇q U f ), (x, q) ∈ R2 ×D, (1.1) with S(u), the potential U (s) and the extra-stress tensor τ being given by 2 b ε ∇u − ∇u t |q| |q|2 = − log 1 − , τ= ∇q U ⊗q f dq, S(u) = , U 2 2 2 b We D (1.2) √ √ where D = B(0, b), the ball with center 0 and radius b, the Reynolds number Re > 0, the Weissenberg number W e > 0, ε ∈ (0, 1) and N > 0 are the non-dimensional parameters in the system. We should point out that in general S(u) = ∇u in (1.2), and in the simpler co-rotational case, S(u) is given by the anti-symmetric part of
532
F. Lin, P. Zhang, Z. Zhang
∇u. The physical and mechanical background of (1.1) can be found in the following textbooks: [4,11]. The known mathematical results for micro-macro models of polymeric fluids are usually limited to the small-time existence and uniqueness of strong solutions of the corresponding PDE systems. In the setting when S(u) = ∇u and the last equation of (1.1) is replaced by a stochastic PDE, one may find interesting studies in [14] for FENE models with b > 2 and sometimes b > 6, in [13], an additional polynomial force term is added. See also discussions in [24] for FENE models with b > 76. Concerning the general coupled PDE systems, some preliminary studies were made in the earlier work [22]. It did not contain the FENE models however. In a recent work [18], the authors studied the global (in time) existence of smooth solutions to micro-macro models of polymeric fluids near equilibrium. The pure marcoscopic model corresponding to these micro-marco models studied in [18] is in fact that of the Olyroyd B-model, for which similar mathematical issues were studied in an earlier work [17] by the same authors. For the FENE models, the global in time existence of smooth solutions near the equilibrium were also established by the first two authors in a recent preprint [16]. When the special dimension is two, the authors [9] proved a global existence of smooth solutions to a coupled nonlinear Fokker-Planck and Navier-Stokes system. The advantage of the systems considered in [9] is that it avoids the difficulty of unboundedness of the polymer chains, q, as that in [18], and that it also avoids the singularity of the potential due to finite extendibility of polymer chains in the FENE models. In another direction, see [3], an existence result for global weak solutions of (1.1) with b ≥ 10 was proved with, however, a technical smoothing procedure applied to the velocity field and the stress tensor in (1.1). On the other hand, one can, as in [15], investigate the long-time behavior of both Hookean models and FENE models in various special flows in a bounded domain with suitable boundary conditions. Comparing with the result proved in [9], we made the following two improvements. First, we do not need to replace the advection velocity in the last equation of (1.1) by 1 u(t, ¯ x) = τ
t
(t−τ )+
u(s, x) ds
for some τ > 0. (After this paper was submitted, the authors heard that the above assumption was removed recently in [10] as well.) Secondly, the potential U given by (1.2) has natural singularity at the boundary of D. But the price to pay is that here we need S(u) to be an antisymmetric part of ∇u. Actually a similar assumption was made by the authors in [19] in order to prove the global existence of a weak solution to an Oldroyd model. We point out that the global existence of smooth solutions to the model in [19] remains unknown. One of the main technical devices used in this paper is the so called estimations on losing derivatives in [2,8]. Compared to the method in [2,8] and [21], our presentation for the last step presented in this paper seems to be more explicit and easy to understand. For simplicity, we will take the following values for the non-dimensional parameters: Re = 21 , W e = 1, N = 1, ε = 21 . And we assume f satisfies the natural flux boundary condition:
1 1 ∇q U f + ∇q f − S(u)q f 2 2
·
q |∂ D = 0. |q|
(1.3)
Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model
533
In order to get around the singularity of ∇ Q U in (1.3), as in [12], we introduce the following transformation:
f (t, x, q) = e− so that as long as b > 4 and
U
|q|2 2 2
g(t, x, q),
g|∂ D = 0,
(1.4)
the flux boundary condition (1.3) is automatically satisfied. Furthermore, the system (1.1) is now reduced to the following new system for (u, g): ⎧ u t + u · ∇u − u = −∇ p + div τ, x ∈ R2 , ⎪ ⎪ ⎨ div u = 0, x ∈ R2 , (1.5) g + u · ∇x g + ∇q · (S(u)qg) − 21 ∇q U · (S(u)q)g ⎪ ⎪ ⎩ t 1 1 1 = 2 q g + 4 (q U − 2 |∇q U |2 )g, (x, q) ∈ R2 ×D, with the extra-stress tensor τ given by
∇q U ⊗ qe−
τ=
U
|q|2 2 2
g(t, x, q)dq,
(1.6)
D
together with the initial conditions: u|t=0 = u 0 ,
g|t=0 = g0 , for (x, q) ∈ R2 ×D.
(1.7)
Now we can state our main result as follows. Theorem 1.1. Let s ≥ 3 be an integer and b > 12, let (u 0 , g0 ) ∈ H s (R2 ) ⊗ H s (R2 ; H01 (D)). Then (1.4–1.7) has a unique global solution (u, g) such that for any T > 0, there holds u ∈ C([0, +∞); H s (R2 )) ∩ L 2 ((0, T ); H s+1 (R2 )), g ∈ C([0,+∞); H s (R2 ; H01 (D))), and g q g, ∈ L 2 ((0, T ); H s (R2 ; L 2 (D))). (b−|q|2 )2 Remark 1.1. We should point out that the term ∇q U · (S(u)q)g which appears in the g equation of (1.5) vanishes when S(u) is given by the anti-symmetric part of ∇u. This property is only used to prove the global existence result as in Theorem 1.1. Therefore, Theorem 3.1 holds even for S(u) = ∇u. The structure of the proof of Theorem 1.1 is as follows. In Sect. 2, we will recall some basic facts from Littlewod-Paley theory; in Sect. 3, we will prove the local existence of solutions to (1.4–1.7) and its blow-up criterion; in Sect. 4, we shall prove that if the maximal time T ∗ of existence for the solution constructed in Sect. 3 is finite, then we can give an estimate of u(t) L ∞ (C σ (R2 )) for T0 , T1 close enough to T ∗ and for [T0 ,T ]
1 < σ < 2. The last conclusion contradicts with the blow-up criterion established in Sect. 3, and this will complete the proof of Theorem 1.1. Let us end the introduction by introducing some notations that will be used in the subsequent sections.
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Notations. We will denote by ∂ α the derivatives with respect to x variables, and
φ L p =
1
p
|φ(x, q)| d x dq p
R2 ×D
⎛ , |g|s = ⎝
R2 ×D |α|≤s
⎞1 2
|∂ g| d x dq ⎠ , α
2
and |u| H s the standard Sobolev norm of u when u depends only x variables. We shall
use the convention ( f, g) to stand for both the inner product on R2 , R2 f g d x, and on
R2 × D, R2 ×D f g d x dq. And we will denote C··· a bounded positive constant which depends on the listed variables, but may change from line to line. 2. Littlewood-Paley Analysis In this section, we are going to recall some basic facts on Littlewood-Paley theory, one may check [5,6] for more details. def
def
Let B = {ξ ∈ R2 , |ξ | ≤ 43 } and C = {ξ ∈ R2 , 43 ≤ |ξ | ≤ 83 }. Let χ ∈ Cc∞ (B) and ϕ ∈ Cc∞ (C) which satisfy
χ (ξ ) + ϕ(2− j ξ ) = 1, ξ ∈ R2 . j≥0
def def We denote h = F −1 ϕ and h˜ = F −1 χ . Then the dyadic operators j and S j can be defined as follows:
j f = ϕ(2− j D) f = 22 j h(2 j y) f (x − y)dy, for j ≥ 0, R2
−j 2j ˜ j y) f (x − y)dy, and S j f = χ (2 D) f =
k f = 2 h(2 −1≤k≤ j−1
R2
−1 f = S0 f.
(2.1)
With the introduction of j and S j , let us recall the definition of the inhomogenous Besov space from [23]: Definition 2.1. Let s ∈ R, 1 ≤ p, q ≤ ∞; the inhomogenous Besov space B sp,q (R2 ) is defined by B sp,q (R2 ) = { f ∈ S (R2 ); f B sp,q < ∞}, where
f B sp,q
⎧⎛ ⎞1 ⎪ q ∞ ⎪
⎪ ⎪ q ⎠ jsq ⎨⎝ 2
j f L p , for q < ∞, = j=−1 ⎪ ⎪ js ⎪ ⎪ ⎩ sup 2
j f L p , for q = ∞. j≥−1
s is the usual Sobolev space H s and that B s Let us point out that that B2,2 ∞,∞ is the usual Hölder space C s . We refer to [23] for more details. For convenience, let us also recall the following lemmas from [6,7]:
Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model
535
Lemma 2.1. (Bernstein’s inequality). Let 1 ≤ p ≤ q ≤ ∞. Assume that f ∈ L p , then there exists a constant C independent of f , j such that 1
1
j|α|+2 j ( p − q ) supp fˆ ⊂ {|ξ | ≤ C2 j } ⇒ ∂ α f L q ≤ C2
f L p , 1 j 2 ≤ |ξ | ≤ C2 j ⇒ f L p ≤ C2− j|α| sup ∂ β f L p . supp fˆ ⊂ C |β|=|α|
Lemma 2.2. Let φ be a smooth function supported in the annulus {ξ ∈ R2 : 1 ≤ |ξ | ≤ 2}. Then there exist two positive constants c and C depending only φ such that for any 1 ≤ p ≤ ∞ and λ > 0, we have
φ(λ−1 D)et f L p ≤ Ce−ctλ φ(λ−1 D) f L p . 2
In the sequel, we will constantly use Bony’s decomposition from [5] that uv = Tu v + Tv u + R(u, v),
(2.2)
where Tu v =
S j−1 u j v and R(u, v) =
j u j v.
| j− j |≤1
j≥−1
Let us conclude this section by recalling a result due to Chemin and Masmoudi [8] concerning 2-D Navier-Stokes equations. We first recall the following definition, see [7]. p Definition 2.2. Let p ∈ [1, ∞], s ∈ R . The space L [T0 ,T ] (C s ) is the space of the distributions u such that p
u L
[T0 ,T ] (C
s)
= sup 2 js
j u L p ([T0 ,T ];L ∞ ) < ∞. j≥−1
Proposition 2.1. (Theorem 3.3 of [8]). Assume that u 0 ∈ L 2 (R2 ) and f ∈ L 1[0,T ] 2 2 2 2 −1 −1 ∞ 2 2 1 (C (R )) ∩ L [0,T ] ( H˙ (R )). Let u ∈ L ([0, T ]; L (R )) ∩ L ((0, T ); H (R2 )) be a solution of the 2-D Navier-Stokes equations on [0, T ], ⎧ ⎨ u t + u · ∇u − u = −∇ p + f, div u = 0, ⎩ u(0, x) = u (x). 0 Then for any ε > 0, there exists T0 ∈ (0, T ) such that
∇u L1
[T0 ,T ] (C
0)
≤ ε.
(2.3)
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F. Lin, P. Zhang, Z. Zhang
3. Local Existence and Blow-up Criterion In this section, we will prove the local wellposedness of (1.4–1.7), and give a Beal-KatoMajda [1] type Blow-up criterion for thus obtained solutions. More precisely, Theorem 3.1. Let s ≥ 3 be an integer and b > 12. Let (u 0 , g0 ) ∈ H s (R2 ) ∩ H s (R2 ; H01 (D)). Then there exists T > 0 such that (1.4–1.7) has a unique solution (u, g) on [0, T ] with u ∈ C([0, T ); H s (R2 )) ∩ L 2 ((0, T ); H s+1 (R2 )), g ∈ C([0, T ); H s (R2 ; H01 (D))), and g q g, ∈ L 2 ((0, T ); H s (R2 ; L 2 (D))). (b − |q|2 )2 Furthermore, let T ∗ be the maximal time of existence, then T∗ ∗ if T < ∞ =⇒
∇u(t) 2L ∞ dt = +∞.
(3.1)
0
Remark 3.1. Here we present only the local wellposedness of (1.4–1.7) in two dimensions, a similar result holds for general space dimension as well. We start the proof of Theorem 3.1 by the following two lemmas: Lemma 3.1. Let g ∈ H s (R2 ; L 2 (D)) and u ∈ H s (R2 ) with div u = 0, then for any |α| ≤ s, there hold |ug|s ≤ C( u L ∞ |g|s + |u| H s g L ∞ 2 ), x (L q )
[∂ α , u] · ∇x g L 2 ≤ C( ∇u L ∞ |g|s + |u| H s+1 g L ∞ 2 ). x (L q ) 2 Lemma 3.2. Let U |q|2 be given by (1.2) with b > 4, and g ∈ H01 (D). Then there holds 1 2 2 |∇q g + ∇q U g| dq ≥ |∇q g| dq − C |g|2 dq. 2 D D D Proof. We first get by integrating by parts that 1 1 1 2 2 2 |∇q U | − q U g 2 dq. (3.2) |∇q g + ∇q U g| dq = |∇q g| dq + 2 2 D D D 4 √ Taking a such that 2 < a < b, then we split the second integral on the right-hand of (3.2) as: 1 1 1 1 2 |∇q U |2 − q U g 2 dq + |∇ U | − U g 2 dq, q q √ 2 2 |q|≤a 4 a<|q|< b 4 while thanks to (1.2), it is easy to calculate that 2 b2 |q|4 − 1 1 1 , |∇q U |2 − q U = 4 2 (b − |q|2 )2
Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model
537
from which, we deduce that
1 |∇q g + ∇q U g|2 dq ≥ 2 D
|∇q g|2 dq −
D
≥
2
|q|
|∇q g| dq − C
|g|2 dq.
2
D
This completes the proof of Lemma 3.2.
b2 (1 − |q|4 ) 2 g dq (b − |q|2 )2
D
Proof of Theorem 3.1. As the existence of solutions to (1.4–1.7) follows from Galerkin’s method and the a priori estimates for the approximate solutions, here we only need to present the a priori estimate to smooth enough solutions of (1.4–1.7). Step 1. The estimate of |u(t)| H s and |g(t)|s . A standard energy estimate applied to the first equation of (1.5) gives
1 d |u|2H s + |∇u|2H s = −(∂ α (u · ∇u) − u · ∇∂ α u, ∂ α u) + (∂ α divτ, ∂ α u) 2 dt |α|≤s
≤ C( ∇u L ∞ |u|2H s + |g|s |∇u| H s ), where we used (1.6) such that |τ |
Hs
(3.3)
≤C
|g(t, x, q)| H s dq ≤ C|g|s . D
Similarly, we apply ∂ α with |α| ≤ s to the third equation of (1.5), and take the L 2 (R2 ×D) inner product of the resulting equation with ∂ α g to get ⎡ ⎤
1 1 1 1 d |g|2 + ⎣ |∇q g|2s − q U − |∇q U |2 ∂ α g, ∂ α g ⎦ 2 dt s 2 4 2 |α|≤s
(∂ α (u · ∇x g) − u · ∇x ∂ α g, ∂ α g) + (∂ α (S(u)q · ∇q g), ∂ α g) =− |α|≤s
1 − (∇q U · ∂ α (S(u)qg), ∂ α g) . 2
Lemma 3.1 can be applied, and it gives |(∂ α (u · ∇x g) − u · ∇x ∂ α g, ∂ α g)| ≤ C( ∇u L ∞ |g|2s + g L ∞ and 2 |g|s |∇u| H s ), x (L q ) |(∂ α (S(u)q · ∇q g), ∂ α g)| ≤ C( ∇u L ∞ |g|s |∇q g|s + ∇q g L ∞ 2 |∇u| H s |g|s ). x (L q ) bq Note that ∇q U = b−|q| 2 and g|∂ D = 0; we get by using Lemma 3.1 and the Hardy inequality [20] that
|(∇q U · ∂ α (S(u)qg), ∂ α g)| ≤ C( ∇u L ∞ |g|s |∇q g|s + ∇q g L ∞ 2 |∇u| H s |g|s ). x (L q ) And again as g|∂ D = 0, we get by integrating by parts that 2
1 1 1 1 1 2 2 α α |∇q g|s + q U − |∇q U | ∂ g, ∂ g = ∇q g + ∇q U g . 2 4 2 2 2 s |α|≤s
538
F. Lin, P. Zhang, Z. Zhang
Therefore, thanks to Lemma 3.2, (3.3), and the Poincaré inequality, we obtain 1 2 d 2 |u| H s + |g|2s + |∇u|2H s + |∇q g|2s ≤ C 1 + ∇u L ∞ + ∇q g L ∞ 2) (L x q dt 4 × |u|2H s + |g|2s .
(3.4)
g Step 2. The estimate of ∇q g L ∞
∞ 2 2 . It turns out that we need to estimate
b−|q|2 L x (L q ) x (L q ) before estimating ∇q g L ∞ . In order to do so, let us take any ε > 0 and multiply 2 x (L q ) g the g equation in (1.5) by (b+ε−|q| 2 )2 , then we get by integrating the resulting equation over D that
1 ∂t 2
D
g 2 (t) 1 dq + u · ∇ 2 2 (b + ε − |q| ) 2
D
g 2 (t) dq (b + ε − |q|2 )2
1 1 1 g [ q g + (q U − |∇q U |2 )g] dq 4 2 (b + ε − |q|2 )2 D 2 g 1 g =− ∇q · (S(u)qg) dq + ∇q U · (S(u)q)g dq. 2 2 (b + ε−|q| ) 2 D (b + ε−|q|2 )2 D −
Note that divu = 0 implies that divq (S(u)q) = 0; we get by integrating by parts that ∇q · (S(u)qg) D
g 1 dq = − 2 2 (b + ε − |q| ) 2
g 2 S(u)q · ∇q D
1 dq = 0, (b + ε − |q|2 )2
where we used the fact that S(u) is an antisymmetric matrix. While it is easy to notice that g ∇q U · (S(u)q)g dq 2 2 (b + ε − |q| ) D
≤ C ∇u L ∞ D
g2 dq (b + ε − |q|2 )2
21 D
g2 dq 2 (b − |q| )(b + ε − |q|2 )2
and we get by integration by parts,
g − q g dq = (b + ε − |q|2 )2 D
|∇q g|2 dq 2 2 D (b + ε − |q| ) 1 1 2 dq, − g q 2 D (b + ε − |q|2 )2
which together with the fact that q
1 (b + ε − |q|2 )2
=
8 24|q|2 + (b + ε − |q|2 )3 (b + ε − |q|2 )4
21
,
Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model
539
implies that 1 − 2
|∇q g|2 dq − 4 2 2 D (b + ε − |q| ) g2 dq. −12 2 4 D (b + ε − |q| )
g 1 q g dq = 2 2 (b + ε − |q| ) 2 D
D
g2 dq (b + ε − |q|2 )3
Thanks to (1.2), one gets by an explicit calculation that g 1 2 q U − |∇q U | g dq 2 (b + ε − |q|2 )2 D |q|2 2 1 − g2 4 b =− dq. 2 D (b − |q|2 )2 (b + ε − |q|2 )2
1 − 4
Then as in the proof of Lemma 3.2 for b > 12, one has
1 g 1 1 q g + q U − |∇q U |2 g dq 4 2 (b + ε − |q|2 )2 D 2 |∇q g|2 1 g2 ≥ dq + c dq 0 2 2 2 2 2 D (b + ε − |q|2 )2 D (b − |q| ) (b + ε − |q| ) g2 dq. −C 2 2 D (b + ε − |q| )
−
(3.5)
Therefore, we obtain g 2 (t) g 2 (t) dq + u · ∇ dq 2 2 2 2 D (b + ε − |q| ) D (b + ε − |q| ) |∇q g|2 g2 1 c0 + dq + dq 4 D (b + ε − |q|2 )2 2 D (b − |q|2 )2 (b + ε − |q|2 )2 g 2 (t) ≤ C(1 + ∇u L ∞ ) dq, 2 2 D (b + ε − |q| )
∂t
from which we deduce
t |∇q g(t , t (x), q)|2 1 g 2 (t, t (x), q) dq + dq 2 2 4 D (b + ε − |q|2 )2 D (b + ε − |q| ) 0 g 2 (t , t (x), q) c0 + dq dt 2 D (b − |q|2 )2 (b + ε − |q|2 )2 t g02 (x, q) g 2 (t , t (x), q) ∞) ≤ dq + C (1 +
∇u(t )
dq dt , L 2 2 2 2 D (b + ε − |q| ) D (b + ε − |q| ) 0 (3.6)
540
F. Lin, P. Zhang, Z. Zhang
where t (x) is determined by
d dt t (x) = u(t, t (x)),
t (x)|t=0 = x.
Notice that g|∂ D = 0; we get by using the Hardy inequality [20] and taking ε to 0 in (3.6) that t |∇q g(t , t (x), q)|2 1 g 2 (t, t (x), q) dq + dq (b − |q|2 )2 4 D (b − |q|2 )2 D 0 g 2 (t , t (x), q) c0 + dq dt 2 D (b − |q|2 )4 t g 2 (t , t (x), q) ≤ |∇q g0 |2 dq + C (1 + ∇u(t ) L ∞ ) dq dt . (3.7) (b − |q|2 )2 D D 0
Now let us turn to the estimate D |∇q g(t)|2 dq. Multiplying the equation for g in (1.5) by q g and integrating the resulting equation over D, we obtain
1 1 |∇q g|2 dq + u · ∇ |∇q g|2 dq + |q g|2 dq 2 2 D D D 1 1 2 q U − |∇q U | gq g dq + ∇q · (S(u)qg)q g dq =− 4 D 2 D 1 − ∇q U · (S(u)qg)q g dq, 2 D
1 ∂t 2
from which, we get by repeating the proof of (3.7) that
1 t |q g(t , t (x), q)|2 dq dt 2 0 D D t g 2 (t , t (x), q) ≤ |∇q g02 (x, q) dq + C dq dt 2 )4 (b − |q| 0 D D t 2
∇u(t ) L ∞ |∇q g(t , t (x), q)|2 dq dt . +C |∇q g(t, t (x), q)|2 dq +
0
(3.8)
D
Thanks to (3.7) and (3.8), we obtain g 2 (t) 2 2 ∞ dq
+ η
|∇ g(t)| dq L q 2 ∞ ≤ C ∇q g0 L ∞ x 2 2 x (L ) D (b − |q| ) D Lx t g 2 (t ) ∞ +C (1 + ∇u(t ) L ∞ )2 [
dq
+ η
|∇q g(t )|2 dq L ∞ ] dt , Lx x 2 )2 (b − |q| D D 0 (3.9) for η small. Step 3. Closing estimates for |u(t)| H s , |g(t)|s and ∇q g L ∞ 2 . Equations (3.4) and x (L q ) (3.9) ensure a maximal time T ∗ > 0 such that (1.4)–(1.7) has a unique solution (u, g)
Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model
541
on [0, T ∗ ) with |u(t)|2H s
+ |g(t)|2s
g(t) 2 + (b − |q|2 ) ∞
L x (L q2 )
+ ∇q g(t) 2L ∞ (L 2 ) x
q
t |∇u(t )|2H s + |∇q g(t )|2s dt + 0
≤ C(T ∗ , |u 0 | H s , |g0 |s , |∇q g0 |s ).
(3.10)
Furthermore, by (3.4), we get by using the Gronwall inequality that
≤ (|u 0 |2H s
1 4
t
(|∇u(t )|2H s + |∇q g(t )|2s )dt 0 t 2 2 , + |g0 |s ) exp C (1 + ∇u(t ) L ∞ + ∇q g(t ) L ∞ ) dt 2 x (L q )
|u(t)|2H s + |g(t)|2s +
0
while Gronwall’s inequality applied to (3.9) gives t |∇q g(t)|2 dq ≤ C|∇q g0 |s exp C (1 + ∇u(t ) L ∞ )2 dt . D
0
T∗
Therefore if < ∞, (3.1) holds. g(t) Step 4. The estimate of |∇q g(t)|s . As in Step 2, here we need to estimate | b−|q| 2 |s before dealing with |∇q g(t)|s . For any ε > 0 and |α| ≤ s, we apply ∂ α to the third equation of ∂α g (1.5) and take the L 2 (R2 ×D) inner product of the resulting equation with (b+ε−|q| 2 )2 to get 2 g 1 1 1 ∂αg α 2 α − ∂ |∇ ∂ g+ U − U | g, q q q (b+ε−|q|2 ) 4 2 (b + ε−|q|2 )2 s |α|≤s 2
∂αg α = − ∇q · ∂ (S(u)qg), (b + ε − |q|2 )2 |α|≤s 1 ∂αg α + ∇q U · ∂ (S(u)qg), 2 (b + ε − |q|2 )2
∂αg α α −(∂ (u · ∇x g) − u · ∇x ∂ g, ) = Iα + I Iα + I I Iα . (3.11) 2 2 (b + ε − |q| )
1 d 2 dt
|α|≤s
Lemma 3.1 can be applied, and it gives ∇q g g , |Iα | ≤ C|∇u| H s 2 2 (b + ε − |q| ) s (b + ε − |q| ) s g g , |I Iα | ≤ C|∇u| H s 2 2 2 (b + ε − |q| ) s (b − |q| )(b + ε − |q| ) s 2 g , |I I Iα | ≤ C|∇u| H s 2 (b + ε − |q| ) s
542
F. Lin, P. Zhang, Z. Zhang
while the same argument as that used in the proof of (3.8) ensures −
1 1 1 ∂αg q ∂ α g + q U − |∇q U |2 ∂ α g, 2 4 2 (b + ε − |q|2 )2 |α|≤s 2 2 2 ∇q g g g 1 . ≥ + c0 −C 2 2 2 2 2 (b + ε − |q| ) s (b − |q| )(b + ε − |q| ) s (b + ε − |q| ) s
Therefore, by (3.11), we obtain 2 2 2 ∇q g g g d +1 + c0 dt (b + ε − |q|2 ) s 4 (b + ε − |q|2 ) s 2 (b − |q|2 )(b + ε − |q|2 ) s 2 g , ≤ C(1 + |∇u| H s )2 2 (b + ε − |q| ) s
g0 while as g0 |∂ D = 0, the Hardy inequality [20] ensures that | (b−|q| 2 ) |s < ∞, from which, we deduce from Gronwall’s inequality and Fatou’s Lemma that
g(t) 2 (b−|q|2 ) + s
t 0
∇q g(t ) 2 g(t ) 2 + (b−|q|2 ) (b − |q|2 )2 dt ≤ C(T, |u 0 | H s , |∇q g0 |s ). s
s
(3.12) On the other hand, we get as in the proof of (3.11) that
1 1 d 1 1 2 2 2 α α − |∇q g|s + |q g|s = q U − |∇q U | ∂ g, ∂ q g 2 dt 2 4 2 |α|≤s
1 +(∇q · ∂ α (S(u)qg), ∂ α q g)− (∇q U · ∂ α (S(u)qg), ∂ α q g) 2 +(∂ α (u · ∇x g) − u · ∇x ∂ α g, ∂ α q g) ,
from which, we can repeat the proof of (3.12) to obtain 2 g d 1 2 2 2 2 2 , |∇q g|s + |q g|s ≤ C|∇u| H s |∇q g|s + |g|s + C dt 4 (b − |q|2 )2 s which together with the Gronwall inequality, (3.10) and (3.12) gives |∇q g(t)|2s
t
+ 0
|q g(t )|2s dt ≤ C(T, |u 0 | H s , |g0 |s , |∇q g0 |s ).
This completes the proof of Theorem 3.1.
(3.13)
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4. The proof of Theorem 1.1 The main aim of this section is to prove the following theorem: Theorem 4.1. Let (u, g) be the unique solution of (1.4–1.7) constructed in Theorem 3.1. Then if T ∗ < ∞, for any T0 < T < T ∗ , and T ∗ − T0 << 1, there holds
u(t) L ∞
[T0 ,T ] (C
σ (R2 ))
≤ C(|u(T0 )| H σ +1 , |g(T0 )|σ )
(4.1)
for any 1 < σ < 2. Proof. Step 1. The estimate of the stress τ (t, x). Multiplying the equation for g in (1.5) by g, then integrating the resulting equation over D, one obtains 1 1 1 2 2 ∂t |g| dq + u · ∇x |g| dq + ∇q · (S(u)qg)gdq − ∇q U · (S(u)q)g 2 dq 2 D 2 2 D D D 1 1 1 2 2 q U − |∇q U | g 2 dq. |∇q g| dq + =− 2 D 4 D 2 Note that divu = 0 implies that divq (S(u)q) = 0; we get by integrating by parts that 1 1 ∇q · (S(u)qg)gdq = − (S(u)q) · ∇q g 2 dq = divq (S(u)q)g 2 dq = 0, 2 2 D D D and as S(u) is an antisymmetric matrix, we have bqi 2 ∇q U · (S(u)q)g dq = S(u)i, j q j g 2 dq b − |q|2 D D bq j S(u) j,i qi g 2 dq = 0. =− 2 b − |q| D On the other hand, we get, by integration by parts, that 2 1 1 1 1 ∇q g + 1 ∇q U g dq. q U − |∇q U |2 g 2 dq = − − |∇q g|2 dq + 2 D 4 D 2 2 D 2 Therefore, we obtain ∂t |g|2 dq + u · ∇x |g|2 dq + D
D
2 ∇q g + 1 ∇q U g dq = 0. 2 D
In particular, this implies that 1 1 2 2 2 2 ∂t |g| dq + u · ∇x |g| dq ≤ 0, D
D
from which and divu = 0, we deduce 1 1 2 2 |g(t, x, q)|2 dq ≤ |g0 (x, q)|2 dq D p D L
, for p = 2, ∞. (4.2) Lp
On the other hand, thanks to (1.2) and (1.6), we have ∇q U =
bq , b−|q|2
1
|g(t, x, q)|dq ≤ C
|τ (t, x)| ≤ C D
and
|g(t, x, q)| dq 2
D
2
.
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F. Lin, P. Zhang, Z. Zhang
The latter estimate together with (4.2) ensures that for any T < T ∗ , 1 2 ≤C |g0 (x, q)|2 dq D
sup τ (t, x) L p 0≤t≤T
, for p = 2, ∞.
(4.3)
Lp
Step 2. The losing estimate for u. Given any T0 < T < T ∗ , we are going to show that for any λ > 0 and σ ≤ s − 1, there holds σ Mλ,[T (u) 0 ,T ]
≤ C u(T0 ) C σ + C def
σ where Mλ,[T (u) = 0 ,T ]
sup
1 σ −1 σ + u 4 (u) + C Mλ,[T (τ ), Mλ,[T 1 0 ,T ] 0 ,T ] L [T ,T ] (H 2 ) λ 0 (4.4) jσ − λ (t) ∞ 2
j u(t) L with
t∈[T0 ,T ], j≥−1
λ (t, t ) = λ
t
t
∇u(t )
L∞
t 2 dt + λ |g(t )| dq
D
t
λ (t) = λ (t, T0 ).
L∞
dt ,
Since we will prove later that
u(t) L 2 ≤ u 0 L 2 +
t 0
τ (t ) L 2 dt ,
σ by (4.3), there is thus no need to consider indices j ≤ 4 in the definition of Mλ,[T (u). 0 ,T ] Instead we shall always assume that j ≥ 5 in all subsequent discussions. Applying the Leray projector P to the first equation of (1.5), then we rewrite it in the equivalent integral form
u(t) = e
(t−T0 )
u(T0 ) −
t
e
(t−t )
P(u · ∇u)(t )dt +
T0
t
e(t−t ) P(divτ )(t )dt .
T0
By Lemma 2.2 and Lemma 2.1, we get
j u(t) L ∞ ≤ Ce
−ct22 j
j u(T0 ) L ∞ + C
t
e−c(t−t )2
j (u · ∇u)(t ) L ∞ dt 2j
T0 t
+C
e
−c(t−t )22 j
2 j
j τ (t ) L ∞ dt ,
(4.5)
T0
where we used the fact that
P j u L p ≤ C p
j u L p , ∀ 1 ≤ p ≤ ∞. Let us assume for the time being that 3
σ (u)( ∇u(t) L ∞ + 2 2 j u(t)
2 jσ − λ (t)
j (u · ∇u)(t) L ∞ ≤ C Mλ,[T 0 ,T ]
1
H2
). (4.6)
Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model
545
Then multiplying by 2 jσ − λ (t) on both sides of (4.5) and using (4.6), we obtain t σ σ σ + CM (u) ≤ C u(T )
(u) sup 2− λ (t,t ) ∇u(t ) L ∞ dt Mλ,[T 0 C λ,[T0 ,T ] 0 ,T ] t∈[T0 ,T ] T0
σ −1 σ +C Mλ,[T (τ )+C Mλ,[T (u) sup 0 ,T ] 0 ,T ]
t
t∈[T0 ,T ] T0
σ −1 ≤ C u(T0 ) C σ + C Mλ,[T (τ ) + C 0 ,T ]
3
e−c(t−t )2 2 2 j |u(t )| 2j
1
H2
dt
1 σ + u 4 (u). Mλ,[T 1 0 ,T ] L [T ,T ] (H 2 ) λ 0
This proves (4.4). Now let us turn to the proof of (4.6). In order to do so, we first use Bony’s decomposition (2.2) to get
j (u · ∇u) = j (Tul ∂l u) + j (T∂l u u l ) + R(u l , ∂l u). Considering the support of the Fourier transform of the term Tul ∂l u, we have
j (Tul ∂l u) =
j (S j −1 u · ∇ j u). | j − j|≤5
Lemma 2.1 can be applied, and it gives 2 jσ − λ (t)
j (Tul ∂l u) L ∞ ≤ C2 jσ − λ (t)
2 j S j −1 u L ∞
j u L ∞
| j − j|≤5
≤ C2
3 2
j
|u(t)|
and also 2 jσ − λ (t)
j (T∂l u u l ) L ∞ ≤ C2 jσ − λ (t) ≤ Since divu = 0, we have R(u l , ∂l u) =
1
H2
σ Mλ,[T (u), 0 ,T ]
j u L ∞ S j −1 ∇u L ∞ | j − j|≤5 σ C ∇u(t) L ∞ Mλ,[T (u). 0 ,T ]
(4.7)
(4.8)
j div( i u ⊗ i u),
|i−i |≤1 i,i j
where i j stands for i ≥ j − N for some fixed positive integer N . Then we get by using Lemma 2.1 that
2 jσ − λ (t) R(u l , ∂l u) L ∞ ≤ C2 jσ − λ (t) 2 j
i u L ∞
i u L ∞
≤C
|i−i |≤1 i,i j s 2( j−i)(σ +1)
i ∇u L ∞ Mλ,[T (u) 0 ,T ]
ij σ ≤ C ∇u(t) L ∞ Mλ,[T (u), 0 ,T ]
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which together with (4.7)–(4.8) proves (4.6). Step 3. The losing estimate for g. As a convention in this step, let us fix σ ∈ (1, 2). We observe first, by (1.6), that 1 2 σ −1 j (σ −1)− λ (t) 2 Mλ,[T (τ ) ≤ C sup 2 |
g(t, x, q)| dq . j ,T ] 0 ∞ D j≥−1,t∈[T0 ,T ] L
(4.9) def
This reduces the estimate of τ to that of g. To proceed further, let us set g j = j g. Then we deduce, by applying j to the equation for g in (1.5), that 1 1 1 2 q U − |∇q U | g j ∂t g j + u · ∇x g j − q g j − 2 4 2 1 = − j ∇q · (S(u)qg) + j ∇q U · (S(u)q)g − j (u · ∇g) − u · ∇x g j . 2 Multiplying the above equation by g j , and integrating the result over D, we get 2 1 1 ∇q g j + 1 ∇q U g j dq |g j |2 dq + u · ∇x |g j |2 dq + 2 2 D 2 D D 1 =−
j ∇q · (S(u)qg) g j dq +
j ∇q U · (S(u)q)g g j dq 2 D D def −
j (u · ∇g) − u · ∇x g j g j dq = I + I I + I I I. (4.10)
1 ∂t 2
D
Now we are going to estimate each term on the right-hand side of the above equation. def Step 3.1. The estimate of I I I. Let G j (t, x) = D |g j (t, x, q)|2 dq; we get by using the Hölder inequality that |I I I | ≤ G j (t)
1 2
1 | j (u · ∇g) − u · ∇x g j | dq 2
2
.
(4.11)
D
On the other hand, by using Bony’s decomposition (2.2), one deduces that 2
j (u · ∇g) − u · ∇x g j = [ j , Tul ]∂l g + j (T∂l g u l ) l=1
+ j R(u l , ∂l g) − T∂l g j u l − R(u l , ∂l g j ) . In what follows, we will frequently use the fact that: for any ψ ∈ L 1 (R2 ), we have
1 |ψ ∗x g(t, ·, q)|2 dq D
2
1 2 |ψ(x − y)| |g(t, y, q)|2 dq dy R2 D 1 2 2 ≤ |ψ(x)|d x |g(t, x, q)| dq . 2 ∞ R D
≤
L
(4.12)
Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model
547
Thanks to (2.2), we get by using (4.12) that
1 D
2
| j (T∂l g u )| dq l
2
≤
| j ( j u · ∇ S j −1 g)| dq
D
| j − j|≤5
≤C
1 2
2
1 2 2 2
j u L ∞ |g| dq D j
| j − j|≤5
, L∞
and
1 2
| j R(u , ∂l g)| dq l
2
≤
D
|i−i |≤1 i,i j
≤ C2
j
ij
1 | j div( i u i g)| dq 2
2
D
1 2 2
i u L ∞ |g| dq D
. L∞
Examining the support of the Fourier transform of g j and thanks to (2.2), we have
T∂l g j u l =
j u · ∇ S j −1 g j . j j
Thus (4.12) can be applied, and it gives
1 |T∂l g j u | dq l 2
D
2
≤ C2
j
j j
1 2 2
j u L ∞ |g| dq D
. L∞
Similarly, we have
1 |R(u , ∂l g j )| dq l
2
2
≤ C2
j
D
|i− j|≤5
1 2 2
i u ∞ |g| dq D
. L∞
Finally let us estimate the terms [ j , Tul ]∂l g. By (2.2), we have
[ j , Tul ]∂l g = [ j , S j −1 u] · ∇ j g. | j − j|≤5
With the definition of j , we have [ j , S j −1 u] · ∇ j g = 2
2j R2
h(2 j (x − y))[S j −1 u(y) − S j −1 u(x)]
·∇ j g(t, y, q)dy, from which, it follows that |[ j , S j −1 u] · ∇ j g| ≤ C2− j ∇u L ∞ 22 j (2 j | · | × |h(2 j ·)| ∗ |∇ j g|)(x).
548
F. Lin, P. Zhang, Z. Zhang
Applying (4.12), one has
1 2
|[ j , Tul ]∂l g| dq 2
D
≤ C ∇u L ∞
1
G j (t) 2 .
| j − j|≤5
Therefore, we obtain
⎡
1 2 2 |I I I | ≤ C G j (t) ⎣2
j u L ∞ |g| dq ∞ D L j j ⎤
1 + ∇u L ∞ G j (t) 2 ⎦ . 1 2
j
(4.13)
| j − j|≤5
Step 3.2. The estimate of I. Note that D ∇q · (S(u)qg j )g j dq = 0, one has
j (∇q · (S(u)qg)) − ∇q · (S(u)qg j ) g j dq I =− D =
j (S(u)qg) − (S(u)qg j ) · ∇q g j dq, D
and hence 1
| j (S(u)qg) − (S(u)qg j )| dq
|I | ≤
2
D
1
2
2
|∇q g j | dq 2
,
D
while thanks to (2.2), we decompose j (S(u)qg) − (S(u)qg j ) as def [ j , TS(u)q ]g+ j (Tg S(u)q)+ j R(S(u)q, g)−Tg j S(u)q − R(S(u)q, g j ) = A+ B. Repeating the argument in Step 3.1, we obtain
1 |B| dq 2
2
≤C
D
j j
1 2 2 2
j u L ∞ |g| dq D j
. L∞
Next, we use (2.1) to deduce
2j A= 2 h(2 j (x − y))[S j −1 S(u)(y) − S j −1 S(u)(x)]q · j g(t, y, q)dy, | j − j|≤5
R2
from which it follows that
S j −1 ∇ 2 u L ∞ 22 j (2 j | · | × |h(2 j ·)| ∗ | j g|)(x). |A| ≤ C2− j | j − j|≤5
The latter together with (4.12) and Lemma 2.1 lead to
1 |A| dq 2
D
2
≤ C2
−j
j j
2
2 j
1 2 2
j u L ∞ |g| dq D
. L∞
Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model
549
Therefore, we conclude 1 1 2 2 2 |g| dq |∇q g j |2 dq |I | ≤ C ∞ D D L ⎡ ⎤
×⎣ 2 j
j u L ∞ + 2− j 22 j
j u L ∞ ⎦ .
Step 3.3. The estimate of I I. Note that 1 II = 2
(4.14)
j j
j j 1 2
D
∇q U · (S(u)q)g j g j dq = 0; we have
j (∇q U · (S(u)q)g) − ∇q U · (S(u)q)g j g j dq, D
from which, we get by Hardy inequality [20] that
1
21 |g j |2 |I I | ≤ C | j (S(u)qg) − (S(u)qg j )| dq dq 2 2 D D (b − |q| ) 1 1 2 2 ≤C | j (S(u)qg) − (S(u)qg j )|2 dq |∇q g j |2 dq . 2
2
D
D
Then the argument in Step 3.2 can be applied, and one has 1 1 2 2 2 2 |I I | ≤ C |g| dq |∇q g j | dq D D ∞ ⎤ ⎡
×⎣ 2 j
j u L ∞ + 2− j 22 j
j u L ∞ ⎦ .
(4.15)
j j
j j
By summing up (4.10), (4.13), (4.14)-(4.15), we obtain
∂t
1 |g j | dq + u · ∇x |g j | dq + |∇q g j + ∇q U g j |2 dq 2 D D D ⎡ 1
2 1 j 2 ⎣ ≤ C G j (t) 2 2
j u L ∞ |g| dq + ∇u L ∞ ∞ D 2
2
j j
2 +C |g| dq D
+
1 2
⎣ L∞
|∇q g j |2 dq. D
⎡
L
j j
j
2
j u L ∞ + 2− j
j j
| j − j|≤5
⎤2
2
2 j
j u L ∞ ⎦
⎤ G j (t) ⎦ 1 2
550
F. Lin, P. Zhang, Z. Zhang
The above together with Lemma 3.2 gives ∂t |g j |2 dq + u · ∇x |g j |2 dq ≤ C |g j |2 dq D D D ⎡ 1
2 1 j 2 +C G j (t) 2 ⎣2
j u L ∞ |g| dq D j j
2 +C |g| dq D
⎡ ⎣
L∞
j
2
j u L ∞ + 2− j
+ ∇u L ∞
2
G j (t) ⎦
| j − j|≤5
L∞
⎤ 1 2
⎤2
2 j
j u L ∞ ⎦ .
(4.16)
j j
j j
Let us denote 2(σ −1) (G) = Mλ,[T 0 ,T ]
sup
j≥−1,t∈[T0 ,T ]
2 22 j (σ −1)−2 λ (t) |g | dq j D
L∞
.
Notice that 1 < σ < 2; we get by using the standard L p estimate for the transport equations and then multiplying (4.16) by 22 j (σ −1)−2 λ (t) that 2(σ −1)
2(σ −1)
Mλ,[T0 ,T ] (G) ≤ sup 22 j (σ −1) G j (T0 ) L ∞ + C(T − T0 )Mλ,[T0 ,T ] (G) j≥−1
σ +C[Mλ,[T (u)]2 sup 0 ,T ]
t
2 2−2 λ (t,t ) |g(t )| dq
t∈[T0 ,T ] T0 t −2 λ (t,t )
2(σ −1)
+C Mλ,[T0 ,T ] (G) sup
t∈[T0 ,T ] T0
2
2 ≤ sup 22(σ −1) |g (T )| dq j 0 D
j≥−1
L∞
D
L∞
dt
∇u(t ) L ∞ dt
1 2(σ −1) Mλ,[T0 ,T ] (G) + C T − T0 + λ
C σ (u)]2 . + [Mλ,[T 0 ,T ] λ From the above estimate we deduce that: for T − T0 sufficiently small and λ sufficiently large, there holds C 2(σ −1) 2 j (σ −1) 2 σ 2 Mλ,[T0 ,T ] (G) ≤ C sup 2 |g j (T0 )| dq ∞ + λ [Mλ,[T0 ,T ] (u)] . D j≥−1 L By (4.9), we finally arrive at σ −1 Mλ,[T (τ ) 0 ,T ]
≤ C sup 2 j≥−1
1 2 2 |g j (T0 )| dq D
j (σ −1)
+ L∞
C λ
1 2
σ Mλ,[T (u) 0 ,T ]
(4.17) for λ 1 and T − T0 1. Step 4. The estimate of the Hölder norm of u. Thanks to (4.3), we get, by using the standard energy estimate for the first equation of (1.5), that t t
u(t) 2L 2 +
∇u(t ) 2L 2 dt ≤ u 0 2L 2 +
τ (t ) 2L 2 dt ≤ C T ∗ , u 0 L 2 , g0 L 2 , 0
0
(4.18)
Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model
551
which together with Proposition 2.1 implies that for any ε > 0, there exists T0 ∈ (0, T ) such that
∇u (4.19) L1 (C 0 ) < ε. [T0 ,T ]
On the other hand, by combining (4.4) and (4.17), one has that for T − T0 1, there holds 1 2 σ j (σ −1) 2 σ + C sup 2 Mλ,[T (u) ≤ C u(T )
|
g (x, q)| dq 0 C j 0 ,T ] 0 ∞ D j≥−1 L 1 σ +C + u 4 (u). (4.20) Mλ,[T 1 1 0 ,T ] 2) L (H 2 [T ,T ] λ 0 Note that from (4.18) and the standard interpolation, we have
u
1
1 L 4[T ,T ] (H 2 ) 0
≤ u L2 ∞
[T0
1
2 ,T ] (L )
u L2 2
[T0 ,T ] (H
1)
< ∞.
Thus if we take λ 1 and T − T0 1, then 1 1 C + u 4 ≤ , 1 1 2 L [T ,T ] H 2 λ2 0 which together with (4.20) implies that σ Mλ,[T (u) 0 ,T ]
≤ C u(T0 ) C σ + C sup 2
1 2 2 |g j (T0 )| dq D
j (σ −1)
j≥−1
. L∞
Therefore, there exists A < ∞, which depends on u(T0 ) H σ +1 , |g(T0 )|σ such that for t ∈ [T0 , T ], we have t
∇u(t ) L ∞ dt .
u(t) C σ ≤ A exp λ T0
Consequently, one has that log(e + u(t) C σ ) ≤ log(e + A) + λ
t
∇u(t ) L ∞ dt .
(4.21)
T0
On the other hand, according to Littlewood-Paley theory, for any integer N , which will be determined later on, we can decompose u into three parts: low frequency, median frequency, and high frequency: u = −1 u +
N
j=0
ju +
j u.
j>N
Then we apply Lemma 2.1 to deduce
∇u L ∞ ≤ C u L 2 +
N
j=0
j ∇u L ∞ + C2−N (σ −1) u C σ ,
552
F. Lin, P. Zhang, Z. Zhang
and hence,
t T0
∇u(t ) L ∞ dt ≤ C u L 1
[T0 ,t] (L
2)
+ (N + 1) ∇u L1
[T0 ,t] (C
0)
σ . +C(T − T0 )2−N (σ −1) u L ∞ [T ,t] (C )
(4.22)
0
σ ≤ 1, i.e. Now we choose N such that 2−N (σ −1) u L ∞ [T ,t] (C ) 0
N≥
σ ) log(e + u L ∞ [T ,t] (C ) 0
(σ − 1) log 2
.
Substituting (4.22) into (4.21), one obtains that σ ) ≤ log(e + A) + C u 1 log(e + u L ∞ L [T ,T ] (C )
[T0 ,T ] (L
0
+C ∇u L1
[T0 ,T ] (C
0)
2)
+ C(T − T0 )
σ ). log(e + u L ∞ [T ,T ] (C ) 0
At this stage, if we choose ε small enough, we obtain by (4.19) that σ ) ≤ 2 log(e + A) + C u 1 log(e + u L ∞ L [T ,T ] (C )
[T0 ,T ] (L
0
2)
+ C(T − T0 ).
The latter implies (4.1), and this completes the proof of Theorem 4.1.
Now we are in a position to conclude the proof of Theorem 1.1. Proof of Theorem 1.1. Thanks to Theorem 3.1, we only need to prove that T ∗ = ∞. Otherwise, if T ∗ < ∞, we deduce from (4.1) that for any 1 < σ < 2, lim ∇u(t) L ∞ ≤ lim∗ u(t) C σ < ∞,
t→T ∗
t→T
which contradicts with (3.1), and therefore T ∗ = ∞. This completes the proof of Theorem 1.1. Acknowledgements. This work was done while Ping Zhang and Zhifei Zhang were visiting Courant Institute of New York University. We are thankful for the hospitality and support of the Institute. Fanghua Lin is partially supported by NSF Grant DMS-0201443; Ping Zhang is partially supported by NSF of China under Grant 10525101 and 10421101, National 973 project and the innovation grant from Chinese Academy of Sciences; and Zhifei Zhang is supported by NSF of China under Grant 10601002.
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Commun. Math. Phys. 277, 555–571 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0324-1
Communications in
Mathematical Physics
Restricting Positive Energy Representations of Diff + (S1 ) to the Stabilizer of n Points Mihály Weiner Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, I-00133, Roma, Italy. E-mail: [email protected] Received: 2 March 2007 / Accepted: 3 March 2007 Published online: 28 August 2007 – © Springer-Verlag 2007
Abstract: Let G n ⊂ Diff + (S 1 ) be the stabilizer of n given points of S 1 . How much information do we lose if we restrict a positive energy representation Uhc associated to an admissible pair (c, h) of the central charge and lowest energy, to the subgroup G n ? The question, and a part of the answer originate in chiral conformal QFT. The value of c can be easily “recovered” from such a restriction; the hard question concerns the value of h. If c ≤ 1, then there is no loss of information, and accordingly, all of these restrictions are irreducible. In this work it is shown that Uhc |G n is always irreducible for n = 1 and, if h = 0, it is irreducible at least up to n ≤ 3. Moreover, an example is given for c > 2 and certain values of h = h˜ such that Uhc |G 1 U ˜c |G 1 . It is also concluded h that for these values Uhc |G n cannot be irreducible for n ≥ 2. For further values of c, h and n, the question is left open. Nevertheless, the example already shows that, on the circle, there are conformal QFT models in which local and global intertwiners are not equivalent. 1. Introduction This paper concerns a purely mathematical problem regarding the representation theory of infinite dimensional Lie groups, and it is intended to be largely self-contained. The actual proofs, apart from those of Prop. 4.2 and Corollary 6.5, will not require any knowledge of chiral conformal QFT. Nevertheless, at least in this introductory section, we shall shortly discuss the physical motivations. A chiral component of a conformal QFT “lives” on a lightline, but it is often extended to the compactified lightline, that is, to the circle. For several reasons, it is more convenient to study such a theoretical model on the circle than on the lightline. However, Supported by MIUR, GNAMPA-INdAM, EU networks “Noncommutative Geometry” (MRTN-CT-2006031962) and “Quantum Spaces – Noncommutative Geometry” (HPRN-CT-2002-00280), and by the “Deutsche Forschungsgemeinschaft”.
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keeping in mind the physical motivation, one should always clear the relation between the properties that a model has on the lightline and the properties that it has on the circle. One may adjust the Doplicher-Haag-Roberts (DHR) theory to describe charged sectors of a model A given on the circle in the setting of Haag-Kastler nets; see e.g. [Fre, GL]. In particular, one may introduce the universal C ∗ algebra of A, and to each sector of A one can associate a localized endomorphism ρ of this algebra. (The endomorphism ρ is localized in the open proper interval I ⊂ S 1 , iff ρ(A(I )) ⊂ A(I ) and ρ(A) = A for all A ∈ A(I c ), where I c is the interior of S 1 \ I .) It is well known that a model, when restricted to the lightline, may admit new sectors that cannot be obtained by restrictions. These sectors are usually called solitonic. However, so far one may have thought that the restriction from the circle to the lightline is at least injective: each sector restricts to a sector (i.e. to something irreducible, and not to a sum of sectors), and different sectors restrict to different sectors. In fact, under the assumption of strong additivity, this is indeed true. However, there are interesting (i.e. not pathological) models, in which strong additivity fails; most prominently, the Virasoro net with central charge c > 1. (The Virasoro nets are fundamental, because each chiral conformal model contains a Virasoro net as a subnet in an irreducible way.) As is known, see e.g. the book [KR], for certain values of the central charge c and lowest energy h, there exists a unitary lowest energy representation of the Virasoro algebra. By [GW], each of these representations gives rise to a projective unitary representation Uhc of the group Diff + (S 1 ), that is, of the group of orientation-preserving smooth diffeomorphisms of the circle. These representations are all irreducible, and every positive energy irreducible representation of Diff + (S 1 ) is equivalent with Uhc for a certain admissible pair (c, h). Moreover, two of these representations are equivalent if and only if both their central charges and their lowest energies coincide. A representation Uhc with lowest energy h = 0 gives rise to a conformal net (in its vacuum representation) on the circle. This is the so-called Virasoro net at central charge c, and it is denoted by AVirc . Every charged sector of AVirc arises from a positive energy representation of Diff + (S 1 ) with (the same) central charge c. Two charged sectors are equivalent if and only if they arise from equivalent positive energy representations of Diff + (S 1 ). Viewing the circle as the compactified lightline, i.e. the lightline together with the “infinite” point, one has that a diffeomorphism of the circle restricts to a diffeomorphism of the lightline if and only if it stabilizes the chosen infinite point. So by what was roughly explained, we are motivated to ask the following questions. Let G n ⊂ Diff + (S 1 ) be the stabilizer subgroup of n given points of S 1 . Then – is the restriction of Uhc to G 1 irreducible? ˜ do we have U c |G 1 U c˜ |G 1 ? – for what values of (c, h) and (c, ˜ h) h ˜ h
Actually, with G 1 replaced with G n , there are reasons to consider these questions not only for n = 1, but in general. (Note that though the actual elements of G n depend on the choice of the n points, different choices result in conjugate subgroups: thus all of these questions are well-posed.) In fact, the (possible) irreducibility of Uhc |G n for h = 0, is directly related to the (possible) n-regularity of AVirc . (See [GLW] for more on the notion of n-regularity.) The other reason is the relation between the answers regarding different values of n. Of course, we have some trivial relations, since G n may be considered to be a subgroup of G m whenever n ≥ m. However, as it will be proved at Prop. 4.1, we have the further relation: Uhc |G n+1 is irred.
⇒
˜ Uhc |G n Uh˜c˜ |G n if and only if (c, h) = (c, ˜ h).
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As it was mentioned, both the questions, and a part of their answers originate in chiral conformal QFT. For example, Haag-duality is known [BGL, FG] to hold in the vacuum sector of any chiral conformal net on the circle. This could be used to conclude that U0c |G 2 is irreducible for all values of the central charge c. However, we shall not enter into details of this argument, because in any case we shall prove some stronger statements regarding irreducibility. In particular, by considering the problem at the Lie algebra level, it will be shown that the representation Uhc |G n for n = 1, is always irreducible (Corollary 3.6), and for h = 0, we have irreducibility at least up to n ≤ 3 (Prop. 4.3). Apart from general statements regarding conformal nets, by now we have a detailed knowledge, in particular, of Virasoro nets. For example, it is known that for c ≤ 1 they are strongly additive, [KL, Xu]. This permits us to conclude (Prop. 4.2), that whenever c ≤ 1, the representation Uhc |G n is irreducible for any positive integer n, and accordingly, ˜ U c |G n U c˜ |G n if and only if (c, h) = (c, ˜ h). h
h˜
Thus for c ≤ 1, our questions are answered. Let us discuss now the region c > 1. It is easy to show that — in general — Uhc |G n U ˜c˜ |G n implies c = c˜ (Corollary 3.3). h Hence the real problem is to “recover” the value of the lowest energy. The main result of this paper is an example, showing that already for n = 1, the value of the lowest energy cannot be always determined by the restriction, since in particular c ˜ c for h = 32 , h = 32 + 21 and c > 2 we have Uhc |G 1 U ˜c˜ |G 1 . h It follows (Corollary 6.5) that the Virasoro net with c > 2 is an example for a local net on the circle, in which local and global intertwiners are not equivalent. In particular, the Virasoro net AVirc with c > 2 has two (globally) inequivalent localized endomorphisms ρ1 and ρ2 , localized in a common open proper interval I ⊂ S 1 , which are locally equivalent; i.e.there exists a unitary operator U ∈ AVirc (I ) such that ρ2 (A) = Uρ1 (A)U ∗ for all A ∈ AVirc (I ). (Note that for strongly additive nets, local equivalence implies global equivalence, too.) The values here exhibited may have more to do with the actual construction than with the problem itself. (For example, it could turn out that when c > 1, all of the representation Uhc |G 1 with h varying over the positive numbers, are equivalent.) The method of showing equivalence is obtained by a combined use of two known tricks: – the realization (appearing e.g. in [BS]) of the lightline-restriction of the Virasoro net at c > 1 as a subnet of the so called U (1)-current, – the observation (appearing e.g. in [LX]) that for any k = 1, 2, . . ., the map ln → 1 C 1 k lkn + 24 (k − k )δn,0 gives an endomorphism of the Virasoro algebra. 2. Preliminaries The Virasoro algebra (Vir) is spanned by the elements {ln : n ∈ Z} together with the central element C obeying the commutation relations [ln , lm ] = (n − m)ln+m + [ln , C] = 0.
C 3 (n − n)δ−n,m , 12 (1)
For a representation π of Vir on a complex vector space V , set L n ≡ π(ln ). An eigenvalue of L 0 is usually referred to as a value of the energy, and the corresponding eigenspace as the energy level associated to that value. If L 0 = λ, then by use of the commutation
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relations L 0 (L n ) = (λ − n), i.e.the operator L n decreases the value of the energy by n. We say that π , with representation space V = 0, is a lowest energy representation with central charge c ∈ C and lowest energy h ∈ C, iff (1) h is an eigenvalue of L 0 , and if Re(s) < Re(h) then s is not an eigenvalue of L 0 (i.e. h is the “lowest energy”), (2) π(C) = c1, (3) V is spanned by the orbit (under π ) of a vector of Ker(L 0 − h1). In this case Ker(L 0 − h1) is one-dimensional, so up to a multiplicative constant there exists a unique lowest energy vector hc and L n hc = 0 for all n > 0. Moreover, V = ⊕∞ n=0 V(h+n) , where V(h+n) = Ker(L 0 − (h + n)1) for n = 0, 1, . . . and actually the dimension of V(h+n) is smaller than or equal to the number of partitions of n, as in fact V(h+n) = Span{L −n 1 . . . L −n j hc | j ∈ N, n 1 ≥ . . . ≥ n j > 0,
j
nl = n},
(2)
l=0
where j = 0 means that no operator is applied to hc . A unitary representation of the Virasoro algebra is a representation π of Vir on a complex vector space V endowed with a (skew symmetric, positive definite) scalar product ·,· satisfying the condition π(ln )1 , 2 = 1 , π(l−n )2
(1 , 2 ∈ V, n ∈ Z),
(3)
or in short, that π(ln )+ ≡ π(ln )∗ |V = π(l−n ). (We use the symbol “+ ”, keeping “∗ ” exclusively for the adjoint defined in the von Neumann sense on a Hilbert space.) Note that the formula θ (ln ) = l−n defines a unique antilinear involution with the property that [θ (x), θ (y)] = θ ([y, x]), and that unitarity means that π(x)+ = π(θ (x)) for every x ∈ Vir. A pair (c, h) is called admissible, if there exists a unitary lowest energy representation with central charge c and lowest energy h. If (c, h) is admissible, then up to equivalence there exists a unique unitary lowest energy representation with central charge c and lowest energy h. In this paper this unique representation will be denoted by πhc , the corresponding representation space by Vhc , and the (up-to-phase unique) normalized lowest energy vector by hc . As is known, this representation is irreducible (in the algebraic sense) and two such representations are equivalent (in the algebraic sense) if and only if their central charges, as well as their lowest energies, coincide. Of course (c, h) = (0, 0) is an admissible pair and the corresponding representation is trivial, but a pair (c, h) = (0, 0), as is known (see e.g.the book [KR] for further explanations), is admissible if and only if it belongs either to the continuous part [1, ∞)×[0, ∞) or to the discrete part {(c(m), h p,q (m))|m ∈ N, p = 1, . . . , m + 1; q = 1, . . . , p}, where c(m) = 1 −
6 , (m + 2)(m + 3)
h p,q (m) =
((m + 3) p − (m + 2)q)2 − 1 . 4(m + 2)(m + 3)
(4)
Let us see now what all this has to do with the so-called positive energy representations of Diff + (S 1 ), where by the symbol “Diff + (S 1 )” we mean the group of orientation preserving (smooth) diffeomorphisms of the unit circle S 1 ≡ {z ∈ C| z = 1}. We shall always consider Diff + (S 1 ) as a continuous group with the usual C ∞ topology.
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We shall often think of a smooth function f ∈ C ∞ (S 1 , R) as the vector field d symbolically written as z = eiθ → f (eiϑ ) dϑ . The corresponding one-parameter group of diffeomorphisms will be denoted by t → Exp(t f ). We shall denote by U(H) the group of unitary operators of a Hilbert space H. A projective unitary operator on H is an element of the quotient group U(H)/{z1|z ∈ S 1 }. A (strongly continuous) projective representation of a (continuous) group G is a (strongly continuous) homomorphism from G to U(H)/{z1|z ∈ S 1 }. We shall often think of a projective unitary operator Z as a unitary operator. Although there is more than one way of fixing phases, note that expressions like Ad(Z ) or Z ∈ M for a von Neumann algebra M ⊂ B(H) are unambiguous. Note also that the self-adjoint generator of a one-parameter group of strongly continuous projective unitaries t → Z (t) is well defined up to a real additive constant: there exists a self-adjoint operator A such that Ad(Z (t)) = Ad(ei At ) for all t ∈ R, and if A is another self-adjoint with the same property then A = A + r 1 for some r ∈ R. Let now (c, h) be an admissible pair for Vir, and denote by Hhc the Hilbert space obtained by the completion of the representation space Vhc of πhc . The operator L n = πhc (ln ) may be viewed as a densely defined operator on this space. By the unitarity of πhc , we have that L ∗n ⊃ L −n (i.e. L ∗n is an extension of L −n ) and hence L n is closable. If f : S 1 → C is a smooth function with Fourier coefficients 2π 1 e−inθ f (eiθ )dθ (n ∈ Z), (5) fˆn ≡ 2π 0 then the sum n∈Z fˆn L n is strongly convergent on Vhc , and the operator given by the sum is closable. Denoting by Thc ( f ) the corresponding closed operator, by use of Nelson’s commutator theorem [Ne, Prop. 2], one has that Thc ( f )∗ = Thc ( f ) and so in particular that Thc ( f ) is self-adjoint whenever f is a real function. By the main result of [GW], there exists a unique projective unitary representation Uhc of Diff + (S 1 ) on H such that Uhc (Exp( f )) = ei Th ( f ) c
(6)
for every f ∈ C ∞ (S 1 , R). This representation is strongly continuous, and moreover, it is irreducible. Note that this latter property does not follow immediately from the fact that πhc is irreducible (in the algebraic sense). For example, Uhc could have a nontrivial invariant closed subspace which has a trivial intersection with the dense subspace Vhc ; see also the related remark after Prop. 3.1. However, this is not so. Indeed, if W is bounded operator in the commutant of Uhc , then, in particular W L 0 ⊂ L 0 W and hence W preserves each eigenspace of L 0 . But by assumption one can form a complete orthonormed system consisting of eigenvectors of L 0 , and so the eigenspaces of L 0 are exactly the eigenspaces of L 0 . Thus W preserves each energy space and so also the dense subspace ˜ V c . Similar arguments show that U c ≡ U c˜ if and only if c = c˜ and h = h. h
h
h˜
A positive energy representation U of Diff + (S 1 ) on H is a strongly continuous homomorphism from Diff + (S 1 ) to U(H)/{z1|z ∈ S 1 } such that the self-adjoint generator of the anticlockwise rotations is bounded from below. (Note that although the generator is defined only up to a real additive constant, the fact whether it is bounded from below is unambiguous.) Diffeomorphisms of S 1 of the form z → az+b with a, b ∈ C, |a|2 − |b|2 = 1 bz+a
are called Möbius-transformations. The subgroup Möb ⊂ Diff + (S 1 ) formed by these
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transformations is isomorphic to PSL(2, R), and it is generated by the (real combinations of the complex) vector fields z → z ±1 and z → 1. Note that in the representation Uhc , the three listed complex vector fields correspond to the three operators L ±1 and L 0 . A strongly continuous projective representation of Möb always lifts to a unique strongly continuous unitary representation of the universal covering group M öb ≡ R) of Möb. Through restriction and this lifting, one may fix the additive conPSL(2, stant in the definition of the self-adjoint generator of anticlockwise rotations and define the conformal Hamiltonian L 0 of a strongly continuous projective representation of Diff + (S 1 ). As is well known, L 0 is bounded from below (i.e.the representation is of positive energy type) if and only if L 0 is actually bounded by 0. Moreover, each irreducible positive energy representation of Diff + (S 1 ) is equivalent to Uhc for a certain admissible pair (c,h); see [Ca, Theorem A.2]. 3. The Passage to the Lie Algebra Level One could view G n as a Lie subgroup of Diff + (S 1 ), with the corresponding Lie algebra consisting of those vector fields that vanish at the given n points. Without any loss of 2π generality, let us assume that the given points of the unit circle are ei n k for k = 1, . . . , n. Then the mentioned (complexified) Lie subalgebra can be identified with the set of func2π tions Gn ≡ { f ∈ C ∞ (S 1 , C) : f (e n k ) = 0 for k = 1, . . . , n}. To find a suitable base, consider the function defined by the formula e j,r (z) ≡ z j − z r + j = z j (1 − z r ),
(7)
where r ∈ Z and j ∈ N. Then {e j,r n : r ∈ Z, j = 0, . . . , n − 1} is a set of linearly independent elements of Gn whose span is dense in Gn , where the latter is considered with the usual C ∞ topology. Omitting the indices of central charge and lowest energy, we have that T (e j,r ) = π(l j − lr + j ). So let us set k j,r ≡ l j − lr + j .
(8)
We shall often use k0,r . To shorten formulae, we shall set kr ≡ k0,r . By direct calculation we find that [kr , km ] = r kr − mkm − (r − m)kr +m +
C 3 (r − r )δ−r,m , 12
(9)
implying that the elements {kr : r ∈ Z} together with the central element C span a Lie subalgebra of the Virasoro algebra, which we shall denote by K. In fact, by a similar straightforward calculation one has that Kn ≡ Span {k j,r n : r ∈ Z, j = 0, . . . , n − 1} ∪ {C} (10) is a Lie subalgebra of Vir for any positive integer n. (Note that for n = 1 we get back K, i.e. K = K1 .) Intuitively, viewing the Virasoro algebra from the point of view of vector fields on the circle, Kn corresponds to the algebra of Laurent-polynomial (polynomial in z and z −1 ) vector fields, that are zero at the chosen n points of S 1 . In what follows, and throughout the rest of this paper, for a densely defined operator A we shall denote its closure by A.
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˜ be two admissible pairs for the Virasoro algebra, Proposition 3.1. Let (c, h) and (c, ˜ h) and assume that Uhc |G n U ˜c˜ |G n . Then there exists a unitary operator V : Hhc → Hc˜˜ h h and a linear functional φ : Kn → C with Ker(φ) ⊃ [Kn , Kn ] such that for all x ∈ Kn we have V πhc (x)V ∗ = π c˜˜ (x) + φ(x)1. h
Proof. If the real vector field f belongs to Gn , then Exp(t f ) ∈ G n for every t ∈ R. Using that both the real part and the imaginary part of x is in Gn , the fact that the finite energy vectors form a core for all operators of the form T ( f ), and some standard arguments, one can easily show that if the two representations are equivalent, then there exists a unitary V such that V πhc (x)V ∗ = π c˜˜ (x) + an additive constant, that may depend h (linearly) on x; say φ(x)1. (Recall that we are dealing with projective representations.) As π is a Lie algebra representation, we have that [π(x), π(y)] = π([x, y]). Actually, as π(x) = π(θ (x))∗ , we have that [π(x), π(y)] = [π(θ (x))∗ , π(θ (y))∗ ] ⊂ [π(θ (y)), π(θ (x))]∗ = π(θ ([x, y]))∗ = π([x, y]).
(11)
Thus it follows that V πhc ([x, y])V ∗ = V [πhc (x), πhc (y)]V ∗ = [V πhc (x)V ∗ , V πhc (y)V ∗ ] ⊂ [(π c˜˜ (x) + φ(x)1), (π c˜˜ (y) + φ(y)1)] = [π c˜˜ (x), π c˜˜ (y)] h
⊂
π c˜˜ ([x, h
h
h
h
y]),
(12)
and hence φ([x, y]) = 0, which concludes our proof.
Remark. Note that even if φ = 0, the unitary operator V appearing in the above proposition does not necessarily make an equivalence between πhc |Kn and π c˜˜ |Kn , since it may h
not take the dense subspace Vhc into V ˜c˜ . h This possibility is not something which is specific to infinite dimensional Lie groups. In fact, consider two of the unitary lowest energy irreducible representations (with lowest energy different from zero), say η1 and η2 of the Lie algebra sl(2, R). Moreover, consider the base e+ , e− and h (in the complexified) Lie algebra satisfying the usual commutation relations [h, e± ] = ∓e± and [e− , e+ ] = 2h. The two elements t = 2h − (e− + e+ ) and s = i(e− − e+ ) span a two-dimensional Lie subalgebra of sl(2, R), and it is easy to prove, that if η1 and η2 are inequivalent, then also their restrictions to this subalgebra are inequivalent (in the algebraic sense). However, the corresponding representations of the corresponding Lie subgroups are in fact equivalent; see for example the remarks in the proof of [GLW, Theorem 2.1]. Proposition 3.2. C, kr n ∈ [Kn , Kn ] for every r ∈ Z and positive integer n. In particular, [K, K] = K. Proof. Let φ : Kn → C be a linear functional such that Ker(φ) = [Kn , Kn ]. Our aim is to show that φ(C) = φ(kr n ) = 0. To shorten notations, we shall set φr ≡ φ(kr n ). Then by Eq. (9) one finds that r φr − mφm − (r − m)φr +m +
φ(C) 2 3 (n r − r )δ−r,m = 0 12
(13)
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for all r, m ∈ Z. Let us now analyze the above relation (together with the fact that φ0 = φ(k0 ) = φ(0) = 0). If r > 1 and m = 1, then by substituting into (13) we obtain the recursive relation r φr + φ1 − (r − 1)φr −1 = 0. Resolving the recursive relation we get that for r > 1 we have φr = (r − 1)(φ2 − φ1 ) + φ1 .
(14)
Similarly, letting r < −1 and m = −1 and resolving the resulting recursive relation we get that φr is a (possibly different) first order polynomial of r for the region r < −1, too. Then letting m = −r and using that φ0 = 0, we find by substitution that r φr + r φ−r +
φ(C) 2 3 (n r − r ) = 0. 12
(15)
The expression on the left-hand side — by what was just explained — for the region r > 1, is a polynomial of r . Thus each coefficient of this polynomial must be zero, and hence, by what was just obtained about degrees, we find that φ(C) = 0 which then by the above equation further implies that φr = −φ−r for every r ∈ Z. Moreover, returning to (13), we have that r φr − mφm − (r − m)φr +m = 0,
(16)
and also, by exchanging m with −m and using that φ−m = −φm , we have that r φr − mφm − (r + m)φr −m = 0.
(17)
Taking the difference of these two equations and setting m = r − 1, we find that φ2r −1 = (2r − 1)φ1 . Restricting our attention to the region r > 1, and confronting what we have just obtained with (14), we get that φ1 = φ2 = 0, and hence again by (14) that φr = 0 for all r ≥ 0 and so actually for all r ∈ Z, which concludes our proof. ˜ be two admissible pairs for the Virasoro algebra, Corollary 3.3. Let (c, h) and (c, ˜ h) and assume that Uhc |G n U ˜c˜ |G n . Then c = c. ˜ h
Proof. It follows trivially from Prop. 3.1 and 3.2.
We shall now formulate a useful condition of irreducibility. Fix an admissible pair (c, h) of the Virasoro algebra, and a positive integer n. Recall that we have denoted by hc the (up to phase) unique normalized lowest energy vector of the representation πhc . It is clear that the subset of Kn , Och,n ≡ {x ∈ Kn : π(x)hc = λx hc for some λx ∈ C},
(18)
is in fact a Lie subalgebra. Note that θ (Kn ) = Kn , but θ (Och,n ) = Och,n . Proposition 3.4. Suppose that Vhc is spanned by vectors of the form π(θ (x1 )) . . . π(θ (x j ))hc , where x1 , . . . , x j ∈ Och,n and j ∈ N (with j = 0 meaning the vector hc itself). Then Uhc |G n is irreducible.
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Proof. Simple arguments (similar to those appearing in the proof of Prop. 3.1) show that if V is a unitary operator commuting with Uhc (G n ), then for all x ∈ Kn we have V πhc (x) = πhc (x)V . Let B ≡ V − hc , V hc 1; then for every j ∈ N and x1 , . . . , x j ∈ Och,n , we have πhc (θ (x1 )) . . . πhc (θ (x j ))hc , Bhc = π(x1 )∗ . . . π(x j )∗ hc , Bhc = hc , Bπhc (x j ) . . . πhc (x1 )hc = multiple of Bhc , hc = 0,
(19)
and hence by the condition of the proposition Bhc = 0. In turn this implies that Bπhc (θ (x1 )) . . . πhc (θ (x j ))hc = πhc (θ (x1 )) . . . πhc (θ (x j ))Bhc = 0 and hence that B = 0; i.e. that V = hc , V hc 1. In order to use the above proposition, let us fix a certain admissible value of c and h. To simplify notations, we shall set L n ≡ πhc (ln ) and K n ≡ πhc (kn ) = L 0 − L n . Note that K 0 = 0 and that πhc (C) = c1. Lemma 3.5. The vectors of the form K −n 1 K −n 2 . . . K −n k hc , where k ∈ N, n j ∈ N+ ( j = 1 . . . k) and n 1 ≥ n 2 ≥ . . . ≥ n k (and where k = 0 means the vector hc in itself, without any operator acting on it) span the representation space Vhc . Proof. The statement with “K ” everywhere replaced by “L” is true by definition. On the other hand, L −n 1 hc = (h1 − L 0 + L −n 1 )hc = −K −n 1 hc + hhc .
(20)
L −n 1 L −n 2 hc = ((h + n 2 )1 − L 0 + L −n 1 )(h1 − L 0 + L −n 2 )hc = ((h + n 2 )1 − K −n 1 )(h1 − K −n 2 ) = K −n 1 K −n 2 h − (h + n 2 )K −n 2 hc + hhc ,
(21)
Similarly,
and it is not too difficult to generalize the above argument, by induction, to show that L −n 1 L −n 2 . . . L −n k hc is a linear combination of vectors of the discussed form. The lowest energy vector hc , though (in general) it is not annihilated by the operators K n (n > 0), is still a common eigenvector for them: ∀n > 0 : K n hc = hhc .
(22)
Hence θ (k−n ) = kn ∈ Och,1 and thus by Lemma 3.5 and Prop. 3.4 we can draw the following conclusion. Corollary 3.6. Let (c, h) be any admissible pair. If n = 1 then Uhc |G n is irreducible. By Prop. 3.1 and 3.2, if V is a unitary operator making an equivalence between and U ˜c˜ |G 1 , then it also makes an equivalence between πhc |K and π c˜˜ |K. To show h h the converse, one needs to overcome the following difficulty: we do not know whether the subgroup of G n generated by the exponentials is dense in G n . In what follows we shall denote this subgroup by G˜ n .
Uhc |G 1
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Proposition 3.7. Let B be a bounded operator from Hhc to Hc˜˜ . Then B intertwines Uhc |G 1 with U ˜c˜ |G 1 if and only if it intertwines Uhc |G˜ 1 with U ˜c˜ |G˜ 1 . h
h
h
Proof. Clearly, we have never used in our proof of irreducibility the whole group G 1 , but only the subgroup G˜ 1 generated by the exponentials. Hence we have that also U ˜c˜ |G˜ 1 h
and U ˜c˜ |G˜ 1 are irreducible representations. h
If B intertwines Uhc |G 1 with U ˜c˜ |G 1 then of course it also intertwines Uhc |G˜ 1 with h
U ˜c˜ |G˜ 1 . So let us assume that B is an intertwiner of the latter two. Then, since these h representations are unitary and irreducible, it follows that B is a multiple of the unitary operator. So we may assume that B is unitary. Then, using the intertwining property and the fact that the conjugate of an exponential in G 1 is still an exponential, it is easy to show ˜ = U ˜c˜ (g gg ˜ −1 ) for all g ∈ G 1 and g˜ ∈ G˜ 1 . Thus by the that Ad(BUhc (g)B ∗ )(U ˜c˜ (g)) h
h
irreducibility of U ˜c˜ |G˜ 1 it follows that Ad(BUhc (g)B ∗ ) = Ad(U ˜c˜ (g)) and so that in the h
h
projective sense BUhc (g)B ∗ = U ˜c˜ (g), which finishes our proof. h
˜ be two admissible pairs for the Virasoro algeCorollary 3.8. Let (c, h) and (c, ˜ h) c c ˜ bra. Then Uh |G 1 U ˜ |G 1 if and only if there exists a unitary operator V such that h
V πhc (x)V ∗ = π c˜˜ (x) for all x ∈ K. h
Proof. The “only if” part follows from Prop. 3.1 and 3.2. As for the “if” part: it is clear that if the two representations are equivalent at the Lie algebra level, then they are also equivalent on the subgroup generated by the exponentials. Hence the “if” part follows directly from the previous proposition. 4. Further Observations ˜ be two admissible pairs, n a positive integer, and Proposition 4.1. Let (c, h) and (c, ˜ h) ˜ suppose that Uhc |G n+1 is irreducible. Then Uhc |G n U ˜c˜ |G n if and only if (c, h) = (c, ˜ h). h
Proof. The “if” part is trivial; we only need to prove the “only if” part. So suppose the two representations of G n in question are equivalent. In fact, assume that they actually coincide. (Clearly, we can safely do so.) So we shall fix n (different) points p1 , . . . , pn on the circle, we shall think of G n as their stabilizer, and we shall assume that Uhc |G n = U ˜c˜ |G n h
(so in particular we assume that the two representations of Diff + (S 1 ) are given on the same Hilbert space). To simplify notations, for the rest of the proof we shall further set U ≡ Uhc and U˜ ≡ U ˜c˜ . h
Suppose ξ ∈ Diff + (S 1 ) is such that it preserves all but one of our n fixed points. Let this point be p j . Set q ≡ ξ( p j ), and let us think of G n+1 as the stabilizer of the points p1 , . . . , pn and q; then ξ −1 G n+1 ξ ⊂ G n . Accordingly, we have that for all ϕ ∈ G n+1 , (23) Ad U (ξ )U˜ (ξ −1 ) (U (ϕ)) = U (ϕ).
However, we cannot immediately conclude that U (ξ )U˜ (ξ −1 ) commutes with U (ϕ), since the above equation is meant in the sense of projective unitaries. Nevertheless, it
Restricting Positive Energy Representations of Diff + (S 1 )
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follows that there exists a complex unit number λ(ξ ), such that in the sense of unitary operators (i.e. not only in the projective sense) we have U (ϕ)∗ Ad U (ξ )U˜ (ξ −1 ) (U (ϕ)) = λ(ξ )1. (24) Clearly, the value of λ(ξ ) is independent of the chosen phase of U (ϕ), and moreover, it is largely independent from the diffeomorphism ξ . Indeed, if ξ is another diffeomorphism such that ξ ( pk ) = pk for k = j and ξ ( p j ) = q, then ξ = ξ ◦ β, where β ≡ ξ −1 ◦ ξ ∈ G n and thus U (β) = U˜ (β) and so in the projective sense U (ξ )U˜ (ξ −1 ) = U (ξ )U˜ (β) U˜ (ξ −1 ) = U (ξ )U˜ (ξ −1 ),
(25)
implying that λ(ξ ) = λ(ξ ). However, the map ξ → λ(ξ ) is clearly continuous, so the above argument actually shows that λ(ξ ) = 1; i.e. that U (ξ )U˜ (ξ −1 ) commutes with U (ϕ). Hence we have shown that U (ξ )U˜ (ξ −1 ) is in the commutant of U (G n ) and so — by the condition of irreducibility — it follows that U (ξ ) = U˜ (ξ ). This concludes our proof, since Diff + (S 1 ) is evidently generated by the diffeomorphisms that preserve all but one of the points p1 , . . . , pn . At this point it is natural to ask: what are the admissible pairs (c, h), for which we can prove the irreducibility of Uhc |G n for some n > 1? (Recall that for n = 1 we have already obtained irreducibility, but in order to use the above proposition, we need n > 1.) Here we shall prove irreducibility for two (overlapping) regions: for c ≤ 1, and for h = 0 (the latter only for n ≤ 3). The irreducibility in the first of them is an evident consequence of the known properties of the Virasoro nets. Nevertheless, it is worth stating it. Proposition 4.2. Let (c, h) be an admissible pair with c ≤ 1. Then Uhc |G n is irreducible ˜ for every positive integer n. Moreover, Uhc |G n U ˜c |G n if and only if h = h. h
Proof. As it is known, [KL, Xu], the Virasoro net with c ≤ 1 is strongly additive. Moreover, it is also known, if (c, h) is an admissible pair with c ≤ 1, then the representation Uhc gives rise to a locally normal representation of the conformal net AVirc ; see the discussion before [Ca, Prop.2.1] explaining for which values of c and h it is known (and from where) that Uhc gives a locally normal representation of AVirc . This clearly shows that Uhc |G n is irreducible. The rest of the proposition follows from irreducibility and the previous proposition. Proposition 4.3. Let (c, h = 0) be an admissible pair and n ≤ 3. Then U0c |G n is irreducible. Proof. It is enough to show the statement for n = 3. As usual in case of h = 0, we shall omit the index of the lowest energy, and we shall denote the lowest energy vector by (the “vacuum vector”) instead of 0 . Moreover, we shall set L k ≡ π0c (lk ) (k ∈ Z). The proof relies on the simple fact that in the case of h = 0, the equality L k = L +k = 0 is satisfied for 3 different values of k; namely for k = 0, ±1. Using this, we shall show that each energy level of V0c is in S ≡ Span{A+1 . . . A+j | j ∈ N, A1 , . . . A j ∈ π0c (K3 ), A1 = . . . = A j = 0} (26) (where j = 0 means the vector itself). This is enough; then the statement follows by Prop. 3.4.
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We shall argue by induction on the energy level. The zero energy level (V0c )(0) is in S, since (V0c )(0) = C. So suppose that (V0c )(k) ⊂ S for all k ≤ m, and consider the case k = m + 1. Of course, the energy level (Vhc )(m+1) is spanned by vectors of the form L −n 1 . . . L −n j , where j and n 1 ≤ n 2 . . . ≤ n j are positive integers such that n 1 +. . .+n j = m+1. However, as it is well known, for h = 0, these vectors are not independent, and (V0c )(m+1) is already spanned by the vectors of the above form with the further condition that 2 ≤ n1 ≤ n2 . . . ≤ n j . So consider one of these vectors, and let r be the number in {0, ±1} such that r ≡ n 1 modulo 3. Then setting A ≡ L n 1 − L r we have that A ∈ π0c (K3 ) and A = 0. It follows that A+ S ⊂ S. Moreover, L −n 1 . . . L −n j = A+ (L −n 2 . . . L −n j ) + L −r (L −n 2 . . . L −n j ),
(27)
and of course by the inductive condition both the vector L −n 2 . . . L −n j and the vector L −r L −n 2 . . . L −n j is in S (as r < 2 ≤ n 1 , the energies of both vectors are smaller than m + 1). Thus by the above equation L −n 1 . . . L −n j ∈ S and so (Vhc )(m+1) ⊂ S, which concludes the inductive argument and our proof. 5. Constructing Representations of K Recall that θ (kn ) = k−n and θ (C) = C and hence θ (K) = K. A representation η of K on complex scalar product space V satisfying η(θ (x)) = η(x)+ for every x ∈ K will be said to be unitary. So far, as a concrete example for such representation, we only had the representations πhc |K obtained by restriction. We shall now exhibit more examples. Let us now begin our list of constructions with an abstract one. Suppose that we have a γ : K → K endomorphism that commutes with the antilinear involution θ . Then it is clear that for any unitary representation η, the composition η ◦ γ is still a unitary representation. For example, following similar constructions for the Virasoro algebra, cf. [LX], for any r ∈ N+ consider the linear map γr given by 1 1 C kr n + (r − ) (n ∈ Z \ {0}), r 24 r C → rC.
kn →
(28)
By a straightforward calculation using the commutation relations of the algebra K, we have that γr is an endomorphism and it is clear that it commutes with θ . Thus for any unitary representation we can construct a family of new unitary representations by taking compositions with γr . Just as in the case of the Virasoro algebra, we can also get some interesting constructions considering the U (1) current algebra. As it is well-known, for every q ∈ R there exists a linear space Vq with positive scalar product, a unit vector q ∈ Vq and a set of operators {Jn ∈ End(Vq )|n ∈ Z} satisfying the following properties. • • • •
[Jn , Jm ] = n δ−n,m 1 and J−n = Jn+ . Jn q = 0 for all n > 0. J0 = q1. Vq is the smallest invariant subspace for {Jn |n ∈ Z}, containing q .
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We shall call this representation of the U (1) current algebra the representation with charge q ∈ R. The formally infinite sum of the normal product of the current with itself : J 2 :n ≡
Jn−k Jk +
k>n−k
Jk Jn−k
(29)
k≤n−k
becomes finite on each vector of Vq , thus giving a well-defined linear operator. Setting L n ≡ 21 : J 2 :n , one finds that the map ln → L n extends to a unitary representation of the Virasoro algebra with the central charge represented by 1. Moreover, one finds that • [L n , Jm ] = −m Jn+m , • L n q = 0 for all n > 0, • L 0 q = 21 q 2 q . We shall now give a new construction for some unitary representation of K. The next proposition — although it can be understood and justified even without knowing anything more than what was so far listed — needs some “explanations”. Without making explicit definitions and rigourous arguments, let us mention the following. (In any case, the precise statement and its proof will make no explicit use of this.) The main idea of [BS] is the fact, that — using their settings1 and notations, but changing the singular point from −1 to 1 — on the punctured plane C \ {1},
z+1 Tα (z) = T (z) + α J (z) + i J (z) z−1
(30)
follows the commutation relations of a stress-energy tensor at central charge c = 1+12α 2 . z+1 However, the function z → i z−1 has a singularity at point z = 1, and the power series expansion of Tα will depend on the chosen region. As a consequence, the operators appearing in the expansion will give rise to a representation of the Virasoro algebra, which — on the full representation space V0 — will not satisfy the unitarity condition. On the other hand, with h(z) ≡ 1 − z n , the product hTα will have an unambiguous expansion, as in fact z+1 h(z) = −(z + 1)(1 + z + z 2 . . . z n−1 ) = 1 + z n − 2 zk z−1 n
(31)
k=0
is a polynomial. This suggests the following statement. Proposition 5.1. For any fixed α ∈ R, setting ⎛
⎞ max(0,n) 1 K nα ≡ (L 0 − L n ) + inα ⎝ Jn + (J0 + Jn − 2 Jk )⎠ |n| k=min(0,n)
the assignment kn → K nα (n ∈ Z \ {0}), C → c(α) = 1 + 12α 2 gives a unitary representation of K. 1 After a Cayley transformation, the formula on the real lines simplifies to “stress-energy tensor + α-times the (real-line) derivative of the current”.
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Unitarity is manifest, and the rest of the proposition may be justified by a long, but straightforward calculation using what was previously listed about the U (1) current. Note that here the formula is given in a compact way, which is a fine thing for a proposition, but not necessarily the best for actual calculations. (For example, there is a hidden sign factor, appearing as n/|n|.) The reader is encouraged to really check the commutation relations. It might seem something tedious (and boring), but — according to the author’s personal opinion — in fact it is interesting to observe how the apparent contradictions disappear by some “miraculous cancellations” of the terms. 6. Equivalence with h1 = h2 Let (c, h) be an admissible pair and consider the representation πhc on the representation space Vhc . It is clear that for any ∈ Vhc we have that πhc (kn ) = (πhc (l0 ) − πhc (ln )) which, for n sufficiently large, is further equal to πhc (l0 ). This shows two things: first, that up to phase = hc is the unique normalized vector in Vhc such that πhc (kn ) = h for all n > 0; second, that by knowing the restriction πhc |K one can “recover” the operator πhc (l0 ) and hence the whole representation πhc . It follows that πhc |K is equivalent to ˜ However, as π c˜˜ |K if and only if πhc is equivalent to π c˜˜ ; i.e.if and only if (c, h) = (c, ˜ h). h h it was already explained, what we are interested in is not the equivalence of πhc |K and π c˜˜ |K but the equivalence of πhc |K and π c˜˜ |K. In order to investigate the second kind of h h equivalence, first we shall consider different ways of exhibiting the representation πhc |K. Let η be a representation of K on a complex scalar product space V . Recall that θ (kn ) = k−n , θ (C) = C and so θ (K) = K. Assume that η satisfies the following properties: (A) η is unitary: η(θ (x)) = η(x)+ for all x ∈ K, (B) η(C) = c1, (C) up to phase there exists a unique normalized vector with the property η(kn ) = h for all n > 0, (D) V is the smallest invariant space for η containing . Using the commutation relations and the listed properties it is an exercise to show that • The value of the scalar product η(kn 1 )η(kn 2 ) . . . η(knr ), η(km 1 )η(km 2 ) . . . η(km s ) is “universal”: it is completely determined by the values of c, h and the integers n 1 , . . . , nr and m 1 , . . . , m s . That is, the scalar product can be calculated by knowing these values; even without having the actual form of the representation η or knowing anything more (than just the required properties) about it. • The representation space is spanned by the vectors of the form appearing in Lemma 3.5. These two consequences imply that the representation, up to equivalence, is uniquely determined by the pair (c, h). It is worth to state this in the form of a statement. Corollary 6.1. Let everything be as it was explained. Then the map η(kn 1 )η(kn 2 ) . . . η(knr ) → πhc (kn 1 )πhc (kn 2 ) . . . πhc (knr )hc extends to a unique unitary operator which establishes an isomorphism between η and πhc |K.
Restricting Positive Energy Representations of Diff + (S 1 )
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We shall now get to the “main trick” of this paper, which is a combination of the two constructions discussed in the previous section. So consider the unitary representation of K given by Proposition 5.1 for q = 0, and compose it with the endomorphism γ2 (α,2) given by (28). We get that for every α ∈ R, the map kn → K n (n ∈ (Z \ {0})), where 1 (L 0 − L r n ) 2 ⎛
K n(α,2) ≡
+ inα ⎝ J2n
⎞ max(0,2n) 1 (J2n − 2 + Jk )⎠ 2|n| k=min(0,2n)
+
1 + 12α 2 16
1
(32)
extends to a unitary representation ρ(α,2) of K with central charge c(α, 2) = 2(1 + 12α 2 ) (i.e. the element C is represented by c(α, 2)1). Note that in the above formula we omitted J0 , since we are in the vacuum representation of the U (1) current (i.e. q = 0 and so J0 = 0, too). Moreover, as it is usual in the vacuum representation, we shall denote the lowest energy vector — corresponding to the “true conformal energy” L 0 — by , rather than by 0 , and we shall call it the vacuum vector. 16α Lemma 6.2. Let ≡ J−1 + i 1+12α 2 . Then for every n > 0 we have
1 + 12α 2 9 + 12α 2 , K n(α,2) = . 16 16 Proof. Anything which lowers the energy (in the sense of L 0 ) by more than 1, annihilates both the vector and . Thus Jk = L k = Jk = L k = 0 for every k > 1 and (α,2) hence one finds that the operator K n , for every n > 0, acts exactly like the operator K n(α,2) =
1 1 + 12α 2 L 0 − iα J1 + 1 2 16 on the mentioned vectors. The rest is trivial calculation.
(33)
c c By the previous lemma and by Corollary 6.1, for c > 2, h 1 = 32 and h 2 = 21 + 32 , the representations πhc1 |K and πhc2 |K appear as subrepresentations of a common, non irreducible representation, namely the representation ρ(α,2) with α = c/2−1 12 . Let Vh 1 be the minimal invariant subspace for ρ(α,2) containing , and Vh 2 the one containing the previously given vector . These are the subspaces on which the representation is isomorphic to πhc1 |K and πhc2 |K, respectively, since — as we have seen — the vectors and behave like “lowest energy vectors” for the representation ρ(α,2) of K. The important observation is that these two vectors, since α = 0, are not orthogonal. Thus, neither the two subspaces Vh 1 and Vh 2 can be so. It should follow therefore, that the corresponding irreducible representations (consisting of closed operators) cannot be inequivalent. This argument however, is not completely rigourous as we deal with (unbounded) operators rather than unitary representations of groups. So in what follows, we shall find a way to deal with the technical difficulties.
Lemma 6.3. Let Q 1 and Q 2 be the orthogonal projections onto V h 1 and V h 2 . Then for every x ∈ K such that θ (x) = x, we have that ρ(α,2) (x) is self-adjoint and
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Q j eit ρ(α,2) (x) = eit ρ(α,2) (x) Q j (t ∈ R, j = 1, 2). Proof. The finite energy vectors are analytic for every operator which is a finite sum of the “L” operators. Indeed, let X = | j|≤k λ j L j and ∈ Ker(L 0 − m1). Then by the well-known linear energy bounds (see e.g. [We, Eq. (3.45)] and the discussion before), there exists an r > 0 such that X η ≤ r (1 + L 0 )η for all vectors η. Moreover, (1 + L 0 )X l ≤ (1 + m + lk)X l as X l ∈ ⊕ j≤m+lk Ker(L 0 − j1), and thus X l+1 ≤ r (1 + L 0 )X l ≤ r (1 + m + lk)X l ,
(34)
from where X l ≤ r l k l (2 + m)l l! and hence is analytic for X in the strip |t| < 1/(r k(2 + m)). By the above, in particular, the finite energy vectors are analytic for ρ(α,2) (x), too. This shows that if θ (x) = x then ρ(α,2) (x) is essentially self-adjoint. Moreover, as the subspaces Vh j ( j = 1, 2) are invariant for ρ(α,2) (x), the analyticity property also shows that the subspaces V h j ( j = 1, 2) are invariant for eit ρ(α,2) (x) (t ∈ R). c and h 2 = Corollary 6.4. Let c > 2, h 1 = 32 c and Uh 2 |G 1 are unitary equivalent.
1 2
+
c 32 .
Then the representations Uhc1 |G 1
Proof. By Corollary 3.8, the restrictions of ρ(α,2) onto Vh 1 and Vh 2 give rise to two unitary representations (in that corollary, all elements of K appear in the condition, but it is clear that the hermitian ones, i.e. the elements invariant under θ , are sufficient for us) of G 1 on V h 1 and V h 1 , respectively, with the first one being unitary equivalent to Uhc1 |G 1 , while the second one to Uhc2 |G 1 . By Corollary 3.6, these representations are irreducible, and by the previous lemma and Prop.3.7, the restriction of Q 1 is an intertwiner between them. Hence if Q 1 , as a map from V h 2 to V h 1 , is not zero, then the two representations 16iα are equivalent. This is indeed the case, as , = 1+12α 2 = 0, and ∈ Vh 1 whereas ∈ Vh 2 . Thus we have managed to give examples for values h = h˜ such that Uhc |G 1 U ˜c |G 1 . h More examples could be generated by i) taking tensor products (and then restrictions), ii) using the endomorphism γr with r different from 2 (which we have used so far). However, at the moment our aim was just to find some examples. Corollary 6.5. The Virasoro net on the circle AVirc with c > 2, admits sectors that are (globally) inequivalent but locally equivalent: there exist two endomorphisms ρ1 , ρ2 of the universal C ∗ algebra of AVirc such that ρ j (AVirc (I )) ⊂ AVirc (I )
and
ρ j |AVirc (I c ) = id|AVirc (I c ) ( j = 1, 2)
(that is, ρ1 and ρ2 are both locallized in I ), ρ1 and ρ2 give rise to two inequivalent sectors, and yet there exists an U ∈ AVirc (I ) such that ρ2 |AVirc (I ) = Ad(U ) ◦ ρ1 |AVirc (I ) . c c and h˜ = 21 + 32 give rise to two Proof. The representations Uhc and U ˜c , where h = 32 h locally normal, irreducible representations of the conformal net AVirc ; see the discussion before [Ca, Prop. 2.1] explaining for which values of c and h it is known (and from where) that Uhc gives a sector of AVirc .
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In any locally normal irreducible representation of AVirc , there is a unique strongly continuous unitary representation of the universal covering group of the Möbius group, which implements the Möbius symmetry in the given locally normal irreducible representation of the net; see [DFK] for the details. This shows that the value of the lowest energy in any locally normal irreducible representation of AVirc is well-determined and hence, globally, the sectors given by Uhc and U ˜c are inequivalent. However, considering h
the point of S 1 kept fixed by G 1 to be one of the endpoints of I , by the previous result it follows evidently that they are locally equivalent. Acknowledgements. The author would like to thank Roberto Longo for suggesting the problem. The main construction of this paper, providing a counter-example for the equivalence of local and global intertwiners, was found in March 2006, while the author stayed at the “Institute für Theoretische Physik” in Göttingen (Germany) and was supported in part by the EU network program “Quantum Spaces – Noncommutative Geometry” and by the “Deutsche Forschungsgemeinschaft”. The author would like to thank the Institute for hospitality and in particular appreciated the hospitality of Karl-Henning Rehren and moreover, the useful discussions with him that contributed in a significant way to finding the main construction. The author would also like to thank Sebastiano Carpi for further useful discussions. Finally, the author would like to thank the organizers of the conference “Recent Advances in Operator Algebras” held in Rome, November 8–11, 2006 (on the occasion of the 60th birthday of László Zsidó), where the result presented here was first announced.
References [DFK] [BGL] [BS] [Ca] [Fre] [FG] [GW] [GL] [GLW] [KR] [KL] [LX] [Ne] [Xu] [We]
D’Antoni, C., Fredenhagen, K., Köster, S.: Implementation of conformal covariance by diffeomorphism symmetry. Lett. Math. Phys. 67, 239–247 (2004) Brunetti, R., Guido, D., Longo, R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156, 201–219 (1993) Buchholz, D., Schulz-Mirbach, H.: Haag duality in conformal quantum field theory. Rev. Math. Phys. 2, 105–125 (1990) Carpi, S.: On the representation theory of virasoro nets. Commun. Math. Phys. 244, 261–284 (2004) Fredenhagen, K.: Generalization of the theory of superselection sectors. In: The algebraic theory of superselection sectors, edited by D. Kastler, Singapore: World Scientific, 1990 Gabbiani, F., Fröhlich, J.: Operator algebras and conformal field theory. Commun. Math. Phys. 155, 569–640 (1993) Goodman, R., Wallach, N.R.: Projective unitary positive-energy representations of diff(s 1 ). J. Funct. Anal. 63, 299–321 (1985) Guido, D., Longo, R.: Relativistic invariance and charge conjugation in quantum field theory. Comm. Math. Phys. 148, 521–551 (1992) Guido, D., Longo, R., Wiesbrock, H.-W.: Extensions of conformal nets and superselection structures. Comm. Math. Phys. 192, 217–244 (1998) Kac, V.G., Raina, A.K.: Bombay lectures on highest weight representations of infinite-dimensional Lie algebras. Advanced Series in Mathematical Physics, 2. Singapore: World Scientific Publishing Co., 1987 Kawahigashi, Y., Longo, R.: Classification of local conformal nets. case c < 1. Ann. of Math. 160, 493–522 (2004) Longo, R., Xu, F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251, 321–364 (2004) Nelson, E.: Time-ordered operator product of sharp-time quadratic forms. J. Funct. Anal. 11, 211–219 (1972) Xu, F.: Strong additivity and conformal nets. Pacific J. Math. 221(1), 167–199 (2005) Weiner, M.: Conformal covariance and related properties of chiral QFT. Phd Thesis (2005), Dipartimento di Matematica, Università di Roma “Tor Vergata”. http://arxiv.org/list/math/0703336, 2007
Communicated by Y. Kawahigashi
Commun. Math. Phys. 277, 573–576 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0365-5
Communications in
Mathematical Physics
Erratum
Ground State of N Coupled Nonlinear Schrodinger Equations in Rn , n ≤ 3 Tai-Chia Lin1,2 , Juncheng Wei3 1 Department of Mathematics, National Taiwan University, Taipei 106, Taiwan 2 National Center of Theoretical Sciences, National Tsing Hua University, Hsinchu, Taiwan.
E-mail: [email protected]
3 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong.
E-mail: [email protected] Received: 4 May 2006 / Accepted: 23 August 2007 Published online: 7 November 2007 – © Springer-Verlag 2007 Commun. Math. Phys. 255(3), 629–653 (2005)
Certain statements in [1] need to be reformulated. The reason is that the infimum in Lemma 3 on p. 636 and the infimum c on p. 642 may not be finite. Throughout the whole paper [1] due to physical considerations, the coupling constants βi j ’s satisfy βi j = β ji , for i = j. In [1], Theorem 2 should be restated as follows: Theorem 2. There exists β0 > 0 depending on λ j ’s, µ j ’s, n and N such that if 0 < βi j < β0 , βi j = β ji , ∀i = j and the matrix (defined at (1.9) of [1]) is positively definite, then there exists a ground state solution (u 01 , . . . , u 0N ). All u 0j ’s are positive, radially symmetric and strictly decreasing. Theorem 3 of [1] should also be restated as follows: Theorem 3. There exists β0 > 0 depending on λ j ’s, µ j ’s, n and N such that if the matrix is positively definite, βi j = β ji , ∀i = j and βi0 j < 0, ∀ j = i 0 , and 0 < βi j < β0 , ∀i = i 0 , j ∈ {i, i 0 }, for some i 0 ∈ {1, . . . , N }, then the ground state solution to (1.2) doesn’t exist. The reason for this correction is that the current form of Lemma 3 is incorrect. We now modify the statement by setting 1 E λ1 [u] = (|∇u|2 + λu 2 ). (0.1) 4 Rn Then the revised Lemma 3 can be stated as follows: The online version of the original article can be found under doi:10.1007/s00220-005-1313-x.
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Lemma 3. inf u∈N
λ,µ
E λ1 [u] is attained only by wλ,µ .
The proof is similar by noting that (|∇u 0 |2 + λu 20 ) = 2<∇ E λ1 [u 0 ], u 0 >. Rn
For the proof of Theorem 1, we replace Iλ j ,µ j [u j ] by E λ1 j [u j ] and note that if c is
attained by some (u 01 , . . . , u 0N ) ∈ N, then (u 01 , . . . , u 0N ) satisfies (1.2). In fact, let G j [u] = |∇u j |2 + λ j u 2j − µ j u 4j − βi j u i2 u 2j . Rn
i= j
Rn
Then there are Lagrange multipliers α1 , . . . , α N such that ∇E +
N
α j ∇G j = 0,
j=1
which implies that N
α j βi j
j=1
Rn
(u i0 )2 (u 0j )2 = 0.
Since (u 01 , . . . , u 0N ) ∈ N, we have |βi j | (u i0 )2 (u 0j )2 < i= j
Rn
Rn
(0.2)
β j j (u 0j )4 ,
which implies that the matrix ( R n βi j (u i0 )2 (u 0j )2 ) is diagonally dominant, and hence from (0.2), we deduce that α1 = · · · = α N = 0. The rest is the same as in [1]. For the proof of Theorem 2, we remark that c = inf E[u] = inf E 1 [u] ≥ inf E 1 [u] := c u∈N
u∈N
(0.3)
u∈N
and replace E[u 1 , . . . , u N ] by E 1 [u 1 , . . . , u N ] in the rest of the proof, where E 1 is defined by N 1 E 1 [u] = (|∇u j |2 + λ j u 2j ). (0.4) 4 Rn j=1
As in our paper, we can show that a minimizer (u 1 , . . . , u N ) of c exists. Since βi j < β0 , by the same proof as those of Lemma 2.1 of [2], we infer that C1 ≤ u 4j ≤ C2 , j = 1, . . . , N , (0.5) Rn
where C1 and C2 are positive constants depending on n, N , λ j , βi j .
Erratum
575
√ √ We now claim that (u 1 √ , . . . , u N )√∈ N. To this end, let ( t1 u 1 , . . . , t N u N ) ∈ N , where each t j > 0. Then ( t1 , . . . , t N ) satisfies
Rn
(|∇u j |2 + λ j u 2j ) = t j
µ j u 4j +
Rn
i=1 i = j
Consequently, N j=1
Rn
(|∇u j |2 + λ j u 2j ) =
N
N
⎛ ⎜ tj ⎝
j=1
Rn
Rn
ti βi j u i2 u 2j ,
µ j u 4j +
⎞
N i=1 i = j
j = 1, . . . , N .
Rn
⎟ βi j u i2 u 2j ⎠ .
(0.6)
Here we have used the fact that βi j = β ji . √ √ Due to ( t1 u 1 , . . . , t N u N ) ∈ N ⊂ N , we have √ √ c ≤ E 1 [ t1 u 1 , . . . , t N u N ], and hence N
(t j − 1)
j=1
i.e.
N j=1
Rn
(|∇u j |
2
Rn
(|∇u j |2 + λ j u 2j ) ≥ 0,
+ λ j u 2j )
≤
N
tj
Rn
j=1
(|∇u j |2 + λ j u 2j ).
(0.7)
Substituting (0.6) into the left-hand side of (0.7), and regrouping all the terms, we obtain ⎤ ⎡ N tj ⎣ u 4j + βi j u i2 u 2j − (|∇u j |2 + λ j u 2j )⎦ ≤ 0. j=1
Rn
i= j
Rn
Rn
Each of the terms above are nonnegative. Since (u 1 , . . . , u N ) ∈ N and each t j > 0, we obtain that 4 2 2 uj + βi j u i u j = (|∇u j |2 + λ j u 2j ) , ∀ j = 1, . . . , N . Rn
Rn
i= j
Therefore, (u 1 , . . . , u N ) ∈ N and hence (u 1 , . . . , u N ) also attains c. By the same proof of Lemma 2.2 of [2], (u 1 , . . . , u N ) is a critical point of E[u]. The rest of proof then follows. (It is remarkable that this argument has been used in the proof of Lemma 2.2 in [2].) Actually, we have shown that inf E[u] = inf E 1 [u] = inf E 1 [u].
u∈N
u∈N
(0.8)
u∈N
The main idea for the proof of Theorem 3 remains unchanged. Here we modify the proof of Theorem 3 as follows: By (0.8), (6.6) can be replaced by E ∗1 [u 2 , . . . , u N ] ≥
inf
(u 2 ,...,u N )∈N1
E ∗1 [u 2 , . . . , u N ] =
inf
(u 2 ,...,u N )∈N1
E [u 2 , dots, u N ] = c1 , (0.9)
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T.-C. Lin, J. Wei
where 1 4 N
E ∗1 [u 2 , · · · , u N ] =
j=2
Rn
(|∇u j |2 + λ j u 2j ).
Besides, the revised Lemma 3 may imply E λ11 [u 1 ] ≥ E λ11 [wλ1 ,µ1 ] .
(0.10)
inf E[u] = inf E 1 [u] ≥ E λ11 [wλ1 ,µ1 ] + c1 .
(0.11)
Thus by (0.8)–(0.10), we have u∈N
u∈N
However, by (6.10), inf E[u] ≤ Iλ1 ,µ1 [wλ1 ,µ1 ] + c1 < E λ11 [wλ1 ,µ1 ] + c1 ,
u∈N
which may contradict (0.11). Therefore, we may complete the proof of Theorem 3. Acknowledgements. The authors want to express their sincere thanks to B. Sirakov for the comments on our previous paper [1].
References 1. Lin, T.C., Wei, J.C.: Ground state of N coupled nonlinear Schrödinger equations in Rn , n ≤ 3. Commun. Math. Phys. 255(3), 629–653 (2005) 2. Lin, T.C., Wei, J.: Spikes in two coupled nonlinear Schrödinger equations. Ann. I. H. Poincaré Analyse. Non. 22(4), 403–439 (2005) Communicated by M. Aizenman
Commun. Math. Phys. 277, 577–625 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0399-8
Communications in
Mathematical Physics
Fusion of Symmetric D-Branes and Verlinde Rings Alan L. Carey, Bai-Ling Wang Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia. E-mail: [email protected]; [email protected] Received: 22 May 2005 / Accepted: 14 September 2007 Published online: 1 December 2007 – © Springer-Verlag 2007
Abstract: We explain how multiplicative bundle gerbes over a compact, connected and simple Lie group G lead to a certain fusion category of equivariant bundle gerbe modules given by pre-quantizable Hamiltonian LG-manifolds arising from AlekseevMalkin-Meinrenken’s quasi-Hamiltonian G-spaces. The motivation comes from string theory namely, by generalising the notion of D-branes in G to allow subsets of G that are the image of a G-valued moment map we can define a ‘fusion of D-branes’ and a map to the Verlinde ring of the loop group of G which preserves the product structure. The idea is suggested by the theorem of Freed-Hopkins-Teleman. The case where G is not simply connected is studied carefully in terms of equivariant bundle gerbe modules for multiplicative bundle gerbes. Contents 1. 2. 3.
4.
5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Some physics background . . . . . . . . . . . . . . . . . . 1.2 Mathematical summary . . . . . . . . . . . . . . . . . . . Bundle Gerbe D-Branes . . . . . . . . . . . . . . . . . . . . . 2.1 Equivariant bundle gerbe D-branes . . . . . . . . . . . . . Bundle Gerbe D-Branes from Group-Valued Moment Maps . . 3.1 Equivariant bundle gerbes over G . . . . . . . . . . . . . . 3.2 Quasi-Hamiltonian G-spaces and Hamiltonian LG-spaces 3.3 Equivariant bundle gerbe modules . . . . . . . . . . . . . Moduli Spaces of Flat Connections on Riemann Surfaces . . . 4.1 Relationship with Chern-Simons . . . . . . . . . . . . . . 4.2 Recasting the Segal-Witten reciprocity law . . . . . . . . . 4.3 The multiplicative structure of Gk . . . . . . . . . . . . . . Spin c Quantization and Fusion of D-Branes . . . . . . . . . . The Non-Simply Connected Case . . . . . . . . . . . . . . . .
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578 578 581 583 585 587 588 589 591 593 596 598 600 600 605
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6.1 Some background about the center of G˜ and the loop group 6.2 Multiplicative bundle gerbes . . . . . . . . . . . . . . . . 6.3 G-equivariant bundle gerbe modules . . . . . . . . . . . . 6.4 The fusion category of bundle gerbe modules . . . . . . . 6.5 An example for G = S O(3) . . . . . . . . . . . . . . . . 6.6 An example for G = SU (3)/Z3 . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LG . . . . . . . . . . . . . . . . . .
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606 612 613 617 619 621 624
1. Introduction It was shown in [10] that there is an additive group structure on equivalence classes of bundle gerbe modules, for a bundle gerbe over a manifold M with Dixmier-Douady class [H ] ∈ H 3 (M, Z), such that the resulting group is isomorphic to the twisted K-theory of M twisted by [H ], denoted K [H ] (M). On the other hand the theorem in [25] is that the Verlinde ring of positive energy representations of the loop group of a compact Lie group (with fusion ∗ as the product and denoted (Rk (LG), ∗)) is isomorphic to the dimG (G), where h ∨ is the dual Coxeter number. Here equivariant twisted K-theory K G,k+h ∨ ∨ k + h is viewed as the level of the twisting class in HG3 (G) with the G-action on G given by conjugation. The aim of this paper is to answer the natural question: is there is a fusion product which can be constructed using bundle gerbe modules and a direct map to (Rk (LG), ∗) preserving the fusion product structure. For simplicity, we only deal with a bundle gerbe over a compact, connected and simply-connected simple Lie group in this introduction. We have some results on the general situation in Sect. 6. Our approach depends on a second circle of ideas. First, (Rk (LG), ∗) provides the quantization of classical Wess-Zumino-Witten models with target G. Second, bundle gerbes over G provide a differential geometric way to approach Wess-Zumino-Witten models [16]. Third, the main result of [15] shows that classical Wess-Zumino-Witten models which arise by transgression from Chern-Simons gauge theories have the property that their associated bundle gerbe has internal extra structure termed ‘multiplicative’. In fact we showed more, namely that a bundle gerbe G over G is multiplicative (see Theorem 5.8 in [15]), if and only if its Dixmier-Douady class is transgressive, that is, lies in the image of the transgression map τ : H 4 (BG, Z) −→ H 3 (G, Z). We will see that the multiplicative property gives a fusion product for bundle gerbe modules. For a simplyconnected, compact simple Lie group G, we know that H 4 (BG, Z) ∼ = H 3 (G, Z) ∼ = Z, 3 ∼ and so for any integer k ∈ H (G, Z) = Z there is a corresponding multiplicative bundle gerbe Gk . To assist the reader we review the key notions from [15] in Subsect. 4.1. While it is not necessary in order to understand the mathematical results of this paper, our motivation comes from string theory considerations. Namely we expand the notion of a D-brane (as there is no fusion product on the space of D-branes) to include the image of G-equivariant smooth maps from a manifold to G (cf. Definition 2.2). We are able, using the multiplicative property, to define fusion for these generalised D-branes.
1.1. Some physics background. This subsection is not really needed for our results but we use it to introduce some notation and for the interested reader we include some background which amplifies the preceding remarks. First we recall that the background
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Kalb-Ramond field (the so-called “B-field”) is a 2-form potential for an invariant 3form on the target group manifold G. In Type II string theory with non-trivial “B-field”, twisted K-theory is believed to classify those D-branes with Chan-Paton fields ([54]). The quantized Wess-Zumino-Witten model of level k (a positive integer) for closed strings moving on a group manifold G is determined by a closed string Hilbert space HkW Z W =
H∗λ ⊗ Hλ ,
λ∈∗k
where Hλ is the positive energy irreducible projective representation of the loop group LG at level k with dominant weight λ in the space of level k dominant weights ∗k , and ∗λ is the dominant weight of the irreducible representation of G complex conjugate to the one with weight λ. In addition there is an assignment of trace class operators Z k () : HkW Z W ⊗ · · · ⊗ HkW Z W −→ HkW Z W ⊗ · · · ⊗ HkW Z W m times
n times
to any Riemann surface with analytically parametrised boundaries divided into m incoming boundaries and n outgoing boundaries, such that the operators Z k () satisfy certain gluing formulae under composition of surfaces (see [48] and the references therein). We call such Riemann surfaces ‘extended’. With primary fields determined by a dominant weight of level k inserted at each boundary of , the space of correlations or conformal blocks is given by the multiplicity space of the modular functor from the category of extended Riemann surfaces with conformal structure to the category of positive energy irreducible projective representations of the loop group LG at level k: H =
− → − → λ in , λ out
− → − → → . → ⊗ H− Vk (∗ λ in , λ out ) ⊗ H∗− λ λ out
in
− → − → The space of conformal blocks Vk (∗ λ in , λ out ) also satisfies certain gluing formulae, the well-known Verlinde factorization formulae. Varying the conformal structure on , the space of conformal blocks forms a holomorphic vector bundle over the moduli space of conformal structure, equipped with a canonical projective flat connection (the Knizhinik-Zamolodchikove connection). For 0,3 , the genus 0 surface with 3 boundary components, two incoming boundary circles labelled by the weight λ, µ and the third outgoing boundary circle labelled by the weight ν, the dimension of the space of conformal blocks ν Nλ,µ = dimVk 0,3 (∗λ, ∗µ, ν)
(1.1)
is given by the Verlinde fusion coefficient ([51]). Another definition of Verlinde coeffiν is given by (cf. [8]): cients Nλµ ν = dim Nλ,µ
⎧ ⎨ ⎩
u ∈ H om G Vλ ⊗ Vµ , Vν |u
( p) Vλ
⊗ Vµ(q) ⊆
p+q+r ≤k
Vν(r )
⎫ ⎬ ⎭
,
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A. L. Carey, B.-L. Wang
where Vλ , Vµ and Vν denote the representation of G with highest weight λ, µ and ν respectively. The highest root ϑ determines a copy of SU (2) with respect to which Vλ admits a decomposition Vλ =
k/2
(i)
Vλ ,
i=0 (i)
where Vλ (i = 0, 1/2, 1, . . . , k/2) are the spin i isotypic components. In boundary conformal field theory, the D-brane is described by a boundary state in the closed string Hilbert space, which is a linear combination of the so-called Ishibashi states. The coefficients should satisfy the Cardy condition and some sewing relations (cf. [14]). For the Wess-Zumino-Witten model of conformal field theory, the stringy geometry can be studied via bundle gerbes ([31,32]) and embedded submanifolds Q of G, the D-branes. For the boundary Wess-Zumino-Witten theory on a simply connected group manifold G (cf. [32]), symmetry preserving boundary conditions are labelled by λ ∈ ∗k and the open string Hilbert space labelled by λ1 and λ2 admits the following decomposition: open Hλ1 ,λ2 ∼ =
µ∈∗k
Wλλ12µ ⊗ Hµ .
In order to get a consistent quantum conformal field theory, the multiplicity space Wλλ12µ is identified with the space of conformal blocks Vk 0,3 (∗λ1 , ∗µ, λ2 ). In particular, the dimension of the multiplicity space dim Wλλ12µ = Nλλ12,µ is also given by the Verlinde fusion coefficients. For general boundary conditions, the consistency condition implies that µ → (dim Wλλ12µ ) realises a representation of the Verlinde algebra. See also [2,11,22,27,29] for some earlier discussion of D-branes on group manifolds. Geometrically, for a simply-connected Lie group G, D-branes on G are classified into symmetric D-branes: 2πiλ Cλ = g · ex p · g −1 |g ∈ G k for λ ∈ ∗k ; twisted D-branes 2πiλ · G (g)−1 |g ∈ G , Cλ = g · ex p k where G is an outer automorphism of G, and λ ∈ ∗k is a fixed point of G , and the coset D-branes such as D-branes in N = 2 coset models in [35,36,45,46]. In this paper, we will study these symmetric D-branes from the equivariant bundle gerbe module viewpoint.
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1.2. Mathematical summary. We now look at the mathematical content of this paper. We introduce a new notion, ‘generalized rank n bundle gerbe D-branes’ for a bundle gerbe G over G. These are smooth manifolds Q with a smooth map µ : Q → G such that the pull-back bundle gerbe µ∗ (G) admits a rank n bundle gerbe module (Definition 2.2). There is also a corresponding notion for G-equivariant bundle gerbes G over a G-manifold M. When the compact simple Lie group G is simply-connected then we can construct a G-equivariant bundle gerbe Gk over G whose Dixmier-Douady class is represented by a multiple by a positive integer k of the canonical bi-invariant 3-form on G (see Prop. 3.1 in Sect. 3). A particularly interesting example of a generalized G-equivariant bundle gerbe D-brane is provided by a quasi-Hamiltonian manifold (M, ω, µ) (see Definition 3.2) where M is a G-manifold, ω is an invariant 2-form and µ : M → G is a group-valued moment map. Quasi-hamiltonian manifolds are extensively studied by Alekseev-MalkinMeinrenken in [1] whose results are reviewed in Sect. 3. We focus on the correspondence between quasi-Hamiltonian manifolds and Hamiltonian LG-manifolds at level k as illustrated by the following diagram: Mˆ
µˆ
π
M
/ Lg∗
(1.2)
H ol
µ
/ G,
where µˆ : Mˆ → Lg∗ is the moment map for the Hamiltonian LG-action at level k and the vertical arrows define G-principal bundles. The quasi-Hamiltonian manifold, when “pre-quantizable”, is naturally a generalized rank 1 bundle gerbe D-brane (cf. Theorem 3.5) of the bundle gerbe over G. When G is semisimple and simply connected, any bundle gerbe Gk is multiplicative [15] and so in Sect. 4, applying Theorem 3.5 to the moduli spaces of flat connections on Riemann surfaces, we relate these generalized bundle gerbe D-branes to the ChernSimons bundle 2-gerbe of [15] over the classifying space BG . A corollary is that the Segal-Witten reciprocity law is explained in the set-up of multiplicative bundle gerbes and their generalized bundle gerbe D-branes. In Sect. 5, we define the fusion category of generalized bundle gerbe D-branes of Gk to be the category of pre-quantizable quasi-Hamiltonian manifolds with fusion product k ∗ ∗¯ (M1 , ω1 , µ1 ) (M2 , ω2 , µ2 ) = M1 × M2 , ω1 + ω2 + < µ1 θ, µ2 θ >, µ1 · µ2 , 2 where the G-action on M1 × M2 is via the diagonal embedding G → G × G, θ , θ¯ are the left and right Maurer-Cartan forms on G, and µ1 · µ2 (x1 , x2 ) = µ1 (x1 ) · µ2 (x2 ). This fusion product and the corresponding fusion product on Hamiltonian LG-manifolds were studied in [40]. Denote by (QG,k , ) the fusion category of bundle gerbe D-branes of Gk . Let Rk (LG) be the free group over Z generated by the isomorphism classes of positive energy, irreducible, projective representations of LG at level k. The central extension of LG at level k we write as LG. The positive energy representation labelled by λ ∈ ∗k acts on Hλ and the Kac-Peterson character of Hλ is c
χk,λ (τ ) = T rHλ e2πiτ (L 0 − 24 ) ,
(1.3)
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kdimG k + h∨ c is the Virasoro central charge. We mention that e2πiτ (L 0 − 24 ) is a trace class operator (cf. Theorem 6.1 in [33] and Lemma 2.3 in [21]) for τ ∈ C with I m(τ ) > 0. Equipped with the fusion ring structure: ν χλ,k ∗ χµ,k = Nλ,µ χν,k , where τ ∈ C with I m(τ ) > 0, L 0 is the energy operator on Hλ (cf.[43]), and c =
ν∈∗k
ν is the Verlinde fusion coefficient (1.1), we obtain (R (LG), ∗), the Verlinde where Nλ,µ k ring. Motivated by Guillemin-Sternberg’s “quantization commutes with reduction” philosophy, we define a quantization functor on the fusion category of generalized bundle gerbe D-branes of Gk using Spin c quantization of the reduced spaces (see Definition 5.2):
χk,G : QG,k −→ Rk (LG). Note that for a quasi-Hamiltonian manifold M obtained from a pre-quantizable Hamiltonian G-manifold, χk,G (M) is the equivariant index of the Spin c Dirac operator twisted by the pre-quantization line bundle. Main Theorem. The quantization functor χk,G : (QG,k , ) −→ (Rk (LG), ∗) satisfies χk,G (M1 M2 ) = χk,G (M1 ) ∗ χk,G (M2 ),
where the product ∗ on the right-hand side denotes the fusion ring structure on the Verlinde ring (Rk (G), ∗). The fusion product on Hamiltonian LG-manifolds at level k involves the moduli space of flat connections on a canonical pre-quantization line bundle over the ‘trousers’ 0,3 . The multiplicative property of the bundle gerbe Gk over G is essential for this part of the construction. In Sect. 6, we discuss various subtle issues concerning the non-simply connected case. ˜ Given a compact, connected, non-simply connected simple Lie group G = G/Z for a ˜ ˜ subgroup Z in the center Z (G) of the universal cover G, we construct a G-equivariant bundle gerbe G(k,χ ),G associated to a multiplicative level k and a char˜ Z) and k is the acter χ ∈ H om(Z , U (1)), where the so-called level lies in H 4 (B G, 4 4 ˜ multiplicative if it is transgressed from H (BG, Z) to H (B G, Z). The G-equivariant bundle gerbe G(k,χ ),G is obtained from the central extension of LG in [50], 1 → U (1) −→ LG χ −→ LG → 1, associated to (k, χ ). We classify all irreducible positive energy representations of LG χ following the work of Toledano Laredo in [50]. Let Rk,χ (LG) be the Abelian group generated by the positive energy, irreducible representations of LG χ . We define the category Q(k,χ ),G of G-equivariant bundle gerbe modules of G(k,χ ).G . The quantization functor χ(k,χ ),G : Q(k,χ ),G −→ Rk,χ (LG) can also be established (see Definition 6.13). When k is multiplicative and χ is the trivial homomorphism 1, then Q(k,1),G admits a natural fusion product structure whose resulting category is denoted by (Q(k,1),G , ). Then χ(k,1),G induces a ring structure on Rk,1 (LG) (cf. Theorem 6.16).
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2. Bundle Gerbe D-Branes It is now known that the “B-fields” on a manifold M can be described by a bundle gerbe with connection and curving, and topologically classified by the degree 2 Deligne cohomology H 2 (M, D2 ). (This was understood in [10] using [28].) Explicitly, choose a good covering {Ui } of M. Denote double intersections Ui ∩ U j by Ui j and extend this notation in the obvious way to n-intersections. Then a degree 2 Deligne cohomology class is given by an equivalence class of triples (gi jk , Ai j , Bi ), where gi jk ∈ C ∞ (Ui jk , U (1)), Ai j ∈ 1 (Ui j , iR) and Bi ∈ 2 (Ui , iR) satisfy the following cocycle condition −1 gi jk gi−1 jl gikl g jkl = 1,
on Ui jkl ,
Ai j + A jk + Aki = gi−1 jk dgi jk , on Ui jk , Bi − B j = d Ai j
on Ui j .
The equivalence relation is given by adding a coboundary term (h i j h jk h ki , A j − Ai − h i−1 j dh i j , 0) for h i j ∈ C ∞ (Ui j , U (1)) and Ai ∈ 1 (Ui , iR). Differential geometrically, a degree 2 Deligne cohomology class can be realized by a bundle gerbe with connection and curving over M [42]. A bundle gerbe G over M consists of a quadruple (G, m; Y, M), where Y is a smooth manifold with a surjective submersion π : Y → M, and a principal U (1)-bundle (also denoted by G) over the fibre product Y [2] = Y ×π Y together with a groupoid multiplication m on G, which is compatible with the natural groupoid multiplication on Y [2] . We represent a bundle gerbe G = (G, m; Y, M) by the following diagram: G Y [2]
(2.1) π1 π2
//
Y M
with the bundle gerbe product m given by an isomorphism m : p1∗ G ⊗ p3∗ G → p2∗ G
(2.2)
of principal U (1)-bundles over Y [3] = Y ×π Y ×π Y , and pi , i = 1, 2, 3 are the three natural projections from Y [3] to Y [2] obtained by omitting the entry in position i for pi . The maps π j , j = 1, 2 are the projections onto the first and second factors in Y [2] . A bundle gerbe with connection and curving G over M is given by (2.1), together with a U (1)-connection A on the principal U (1)-bundle G over Y [2] which is compatible with the bundle gerbe product m: m ∗ ( p2∗ A) = p1∗ A + p3∗ A,
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A. L. Carey, B.-L. Wang
and a 2-form B on Y , such that the curvature FA of A, satisfies the relation FA = π1∗ (B) − π2∗ (B). The connection A is called a bundle gerbe connection and B is called the curving of A. Then there exists a closed 3-form H called the bundle gerbe curvature of (G, A), such that d B = π ∗ H . (The notation is chosen to match with the corresponding objects in the Deligne point of view.) The characteristic class of the Deligne class [(gi jk , Ai j , Bi )] is given by the class of ˇ the Cech cocycle {gi jk }, in H 2 (M, U (1)) ∼ = H 3 (M, Z). The corresponding class in H 3 (M, Z) for any realizing bundle gerbe G is called the Dixmier-Douady class of G. In [10], bundle gerbe modules are defined to study twisted K-theory. Given a bundle gerbe G = (G, m; Y, M) over M, a rank n bundle gerbe module of G is a rank n Hermitian vector bundle E over Y , associated to a U (n)-principal bundle P over Y for which there is an isomorphism of principal bundles over Y [2] , ρ : G ⊗ π2∗ P ∼ = π1∗ P,
(2.3)
which is compatible with the bundle gerbe product: ρ ◦ (m ⊗ id) = ρ ◦ (id ⊗ ρ). Note that a bundle gerbe G admits a rank n bundle gerbe module (E, ρ) if and only if the Dixmier-Douady class of G is a torsion class in H 3 (M, Z). Remark 2.1. The next definition is motivated by the following stringy considerations. In Type II superstring theory with non-trivial B-field on a 10-dimensional oriented, spin manifold M, a D-brane Q, as defined in [28,54], is given by a smooth oriented submanifold ι : Q → M such that ι∗ z B + W3 (Q) = 0, where z B ∈ H 3 (M, Z) is the characteristic class of the B-field, and W3 (Q) is the third integral Stieffel-Whitney class, the obstruction to the existence of a Spin c structure on Q. Now there is a torsion bundle gerbe GW3 (with the Dixmier-Douady class W3 (Q)) called the lifting bundle gerbe. It arises [42] from the central extension 1 → U (1) −→ Spin c −→ S O → 1. Denote by G B the bundle gerbe determined by the B-field on M. Now if W3 (Q) = 0 (that is, Q is a Spin c manifold), then ι∗ z B = 0 so that ι∗ G B admits a trivialization. Definition 2.2. Let G be a bundle gerbe over a manifold M equipped with a bundle gerbe connection and curving. (1) A rank 1 bundle gerbe D-brane of a bundle gerbe G over M is a smooth oriented submanifold ι : Q → M such that ι∗ G admits a trivialization. Given a bundle gerbe connection and curving on G, a twisted gauge field on the D-brane is a trivialization of the corresponding Deligne class.
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(2) A rank n bundle gerbe D-brane of a bundle gerbe G over M is a smooth oriented submanifold ι : Q → M such that ι∗ G admits a rank n bundle gerbe module. A twisted gauge field on the D-brane is a bundle gerbe module connection on the bundle gerbe module. (3) A generalized rank n bundle gerbe D-brane is a smooth manifold Q with a smooth map µ : Q → M such that µ∗ G admits a rank n bundle gerbe module. A twisted gauge field on the D-brane is a bundle gerbe module connection on the bundle gerbe module. 2.1. Equivariant bundle gerbe D-branes. Now we recall the definition of an equivariant bundle gerbe from [38] (see also [37]). Let M be a smooth G-manifold, acted on by G from the left. A G-equivariant bundle gerbe over a G-manifold M is a bundle gerbe (G, m; Y, M), where Y is a smooth G-manifold with a G-equivariant surjective submersion π : Y → M, and a G-equivariant principal U (1)-bundle (also denoted by G) over the fibre product Y [2] = Y ×π Y together with a G-equivariant groupoid multiplication m on G. Note that the diagonal embedding of G into G p defines an action of G on Y [ p] = Y × M Y × M · · · × M Y . p times
A G-equivariant bundle gerbe G defines a bundle gerbe (E G ×G G, m, E G ×G Y, E G ×G M) over E G ×G M whose Dixmier-Douady class defines an element in HG3 (M, Z) = H 3 (E G ×G M, Z), called the equivariant Dixmier-Douady class of G. Conversely, given an element of HG3 (M, Z), Sect. 6 of [5] associates to it a G-equivariant stable projective bundle over M whose structure group is PU (H) with the norm topology (or the compact-open topology), satisfying some mild local conditions. That is, there is a bundle of projective spaces P with G-action, mapping Px to Pg·x by a projective isomorphism for any x ∈ M and g ∈ G, and satisfying ∼ P ⊗ L 2 (G); (1) P is stable, i.e., P = (2) each point x ∈ M with isotropy group G x has a G x -invariant neighbourhood Ux such that there is an isomorphism of bundles with G x -action P|Ux ∼ = Ux × P(Hx ) for some projective Hilbert space P(Hx ) with G x -action; (3) the transition functions between two trivializations are given by maps Ux ∩ U y −→ I som(Hx , H y ) which are continuous in the compact-open topology. As shown in [37], given such a G-equivariant stable projective bundle P over M, the lifting bundle gerbe ([42]) associated to the corresponding principal bundle and the central extension 1 → U (1) −→ U (H) −→ PU (H) → 0, is a G-equivariant bundle gerbe. The following construction is, in a sense, a special case of this more general approach. With LG being the smooth loop group C ∞ (S 1 , G) and G the based loop group, we have LG = G × G. There is a G-action on G given by conjugation.
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Proposition 2.3. Given an LG-manifold Mˆ with a free G-action such that the quotient map ˆ G π : Mˆ → M := M/ defines a locally trivial principal G-bundle, then the lifting bundle gerbe over M arising from a central extension 1 → U (1) −→
G −→ G → 1 is a G-equivariant bundle gerbe over M. Proof. It is easy to see that the quotient map π : Mˆ → M is a G-equivariant surjective submersion. The lifting bundle gerbe is represented by the diagram gˆ ∗
G Mˆ [2]
π1 π2
// ˆ M M
where gˆ : Mˆ [2] → G is determined by x2 = g(x ˆ 1 , x2 ) · x1 for (x1 , x2 ) ∈ Mˆ [2] , and satisfies g(x ˆ 2 , x3 ) · g(x ˆ 1 , x2 ) = g(x ˆ 1 , x3 ) for (x1 , x2 , x3 ) ∈ Mˆ [3] . The bundle gerbe product is given by m : gˆ ∗ × gˆ ∗ → gˆ ∗
G
G
G (x1 ,x2 )
which maps
(x2 ,x3 )
(x1 ,x3 )
x1 , x2 , γˆ1 , x2 , x3 , γˆ2 → x1 , x3 , γˆ1 · γˆ2
ˆ ,x ) . Under the conjugation action of G on G,
G)g(x for γˆ1 ∈ ( ˆ 1 ,x2 ) and γˆ2 ∈ ( G)g(x 2 2 we see that the central extension 1 → U (1) −→
G −→ G → 1 is G-equivariant. This implies that gˆ ∗
G is G-equivariant. It remains to show that the bundle gerbe product m is G-equivariant. Using the fact that the G-action on Mˆ is free and a direct calculation from the definition of g, ˆ we obtain, for g ∈ G, g(g ˆ · x1 , g · x2 ) = g · g(x ˆ 1 , x2 ) · g −1 . From this equation we deduce the following commutative diagram:
m / x1 , x3 , γˆ1 γˆ2 x1 , x2 , γˆ1 , x2 , x3 , γˆ2 _ _ g
gx1 , gx2 , Adg (γˆ1 ) , gx2 , gx3 , Adg (γˆ2 )
g
/ gx1 , gx3 , Adg (γˆ1 γˆ2 ) ,
i.e., m is G-equivariant. Hence the lifting bundle gˆ ∗
G over M is a G-equivariant bundle gerbe.
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Given any positive energy projective representation of G acting on H of level determined by the central extension of G, we see that Mˆ × G P(H) −→ M is a G-equivariant stable projective bundle over M whose invariant (cf. [5]) in HG3 (M, Z) agrees with the equivariant Dixmier-Douady class of the lifting bundle gerbe defined in Proposition 2.3. Given a G-equivariant bundle gerbe G = (G, m; Y, M) over M, a rank n G-equivariant bundle gerbe module of G is a bundle gerbe module (E, ρ), such that E is a G-equivariant Hermitian vector bundle over Y , and the bundle gerbe action ρ in (2.3) is G-equivariant. Definition 2.4. We call a generalized rank n bundle gerbe D-brane (Q, µ) of a Gequivariant bundle gerbe G equivariant if Q is a G-manifold and µ is G-equivariant with respect to the conjugate action of G on itself such that µ∗ (G) admits a rank n G-equivariant bundle gerbe module. Following [32] for the Wess-Zumino-Witten model on a group manifold G, the conjugacy classes of G give so-called symmetric D-branes. We will see that they provide many examples of rank 1G-equivariant bundle gerbe D-branes in G and in fact that any quasi-Hamiltonian G-manifold corresponding to a pre-quantizable Hamiltonian LGmanifold at level k is a generalized rank 1G-equivariant bundle gerbe D-brane. 3. Bundle Gerbe D-Branes from Group-Valued Moment Maps Until the end of Sect. 5, G will denote a compact, connected and simply-connected simple Lie group with Lie algebra g. We fix a smooth infinite dimensional model of BG by embedding G into U (N ) and letting E G be the Stiefel manifold of N orthonormal vectors in a separable complex Hilbert space. Let ·, · be the normalized invariant inner product on g such that the highest co-root with respect to a basis of the root system for a fixed maximal torus in G has norm 2. Then k·, · defines an element in H 4 (BG, Z) ∼ = H 3 (G, Z) ∼ = Z, which in turn determines a central extension of LG at level k ([43]): 1 → U (1) −→ LG −→ LG → 1. There is a technical issue, namely we need to complete the smooth loop loop group LG in an appropriate Sobolev norm for the ensuing discussion. None of the constructions in the previous section are changed by using this completion. Thus in the above exact sequence we let LG consist of maps of a fixed Sobolev class L 2p ( p > 3/2). The based loop groups will continue to be denoted by G. The Lie algebra of LG is the space of maps Lg = Map(S 1 , g) of Sobolev class L 2p−1 . Denote by Lg∗ = 1 (S 1 , g) whose elements are of Sobolev class p − 1. Note that Lg∗ ⊂ (Lg)∗ , via the natural pairing of Lg∗ and Lg, (a, ξ ) = a, ξ . S1
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We can view Lg∗ = 1 (S 1 , g) as the affine space of L 2p−1 -connections on the trivial bundle S 1 × G, with an LG-action by gauge transformations (the affine coadjoint action at level k): ¯ γ · A = Adγ A − kγ ∗ θ.
(3.1)
Then there is a well-defined holonomy map H ol = H ol1 :
Lg∗ −→ G
defined by solving the differential equation H ols (a)−1
∂ H ols (a) = k −1 a, ∂t
H ol0 (a) = e,
where s is the coordinate of R, and S 1 = R/Z. The holonomy map H ol is equivariant with respect to the evaluation homomorphism LG → G, γ → γ (1), and the conjugate action of G on itself. We remark that the holonomy map H ol : Lg∗ −→ G also defines the universal
G-principal bundle over G, and for a ∈ Lg∗ = 1 (S 1 , g), the stabilizer of a for the LG-action, denoted by (LG)a , is diffeomorphic to G H ol(a) , the centralizer of H ol(a) in G. 3.1. Equivariant bundle gerbes over G. Denote by θ, θ¯ ∈ 1 (G, g) the left- and rightinvariant Maurer-Cartan forms. In a faithful matrix representation ρ of G, θ = ρ −1 dρ and θ¯ = dρρ −1 . Let k ∈ 3 (G) be the canonical closed bi-invariant 3-form on G: k =
k k θ, [θ, θ ] = θ¯ , [θ¯ , θ¯ ]. 12 12
Then k represents an integral de Rham cohomology class of G in H 3 (G, R) defined by k ∈ H 4 (BG, Z) ∼ = Z. The lifting bundle gerbe construction of [42] starts from the universal G-principal bundle H ol : Lg∗ −→ G and the central extension 1 → U (1) −→
G −→ G → 1 determined by the element k ∈ H 4 (BG, Z) ∼
G, where gˆ : = Z. Then set Gk = gˆ ∗ ˆ 1 , ξ2 ) · ξ1 for (ξ1 , ξ2 ) ∈ (Lg∗ )[2] . (Lg∗ )[2] → G is defined by ξ2 = g(ξ Proposition 3.1. The lifting bundle gerbe Gk is a G-equivariant bundle gerbe over G, whose equivariant Dixmier-Douady class is the class in HG3 (G, Z) ∼ = Z represented by k . Proof. Under the identification Lg∗ ∼ = Z, = 1 (S 1 , g) determined by k ∈ H 4 (BG, Z) ∼ the LG-action on Lg∗ makes the holonomy map H ol : Lg∗ −→ Lg∗ / G ∼ =G a G-equivariant principal G-bundle with the conjugation action of G on G. Then from Proposition 2.3 and the discussion after the proof of Proposition 2.3, we see that Gk is a G-equivariant bundle gerbe over G, and the Dixmier-Douady class agrees with the non-equivariant Dixmier-Douady class of Gk under the isomorphisms HG3 (G, Z) ∼ =Z∼ = H 3 (G, Z), for any connected, compact, simply-connected simple Lie group G.
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3.2. Quasi-Hamiltonian G-spaces and Hamiltonian LG-spaces. We begin with a review from [1] of the definition of a group-valued moment map for a quasi-Hamiltonian G-space. Definition 3.2. A quasi-Hamiltonian G-space is a G-manifold with an invariant 2-form ω ∈ 2 (M)G and an equivariant map µ ∈ C ∞ (M, G)G such that (1) The differential of ω satisfies dω = µ∗ k . k ¯ ξ , where vξ is the fundamental vector (2) The map µ satisfies ι(vξ ) = µ∗ θ + θ, 2 field on M generated by ξ ∈ g. (3) At each x ∈ M, the kernel of ωx is given by ker ωx = {vξ | ξ ∈ ker (Adµ(x)+1 )}. The map µ is called the Lie group valued moment map of the quasi-Hamiltonian G-space M. Basic examples of quasi-Hamiltonian G-spaces are provided by conjugacy classes C ⊂ G as in [1], where the one-to-one correspondence between Hamiltonian loop group manifolds with proper moment map and quasi-Hamiltonian G-manifold is established. ˆ ω, Definition 3.3. A Hamiltonian LG-manifold at level k is a triple ( M, ˆ µ), ˆ consisting of a Banach manifold Mˆ with a smooth LG-action, an invariant weakly symplectic (that is, closed and weakly nondegenerate) 2-form ωˆ and an equivariant moment map µˆ : Mˆ → Lg∗ : kµ, ˆ ξ . ι(vξ )ωˆ = d S1
Remark 3.4. It will be important later to observe that a Hamiltonian LG-manifold at level k is a Hamiltonian LG-manifold Mˆ with an LG-equivariant moment map µˆ : Mˆ −→ Lg∗ = Lg∗ × {k} → Lg∗ ⊕ R, where LG acts on Lg∗ × {k} by the conjugation action. This conjugation action defines an affine coadjoint action of LG at level k. ˆ ω, A Hamiltonian LG-manifold ( M, ˆ µ) ˆ at level k is pre-quantizable if Mˆ has an ˆ with an invariant connection ∇ whose LG-equivariant Hermitian line bundle L → M, curvature is given by −2πi ωˆ and 2π kiµ, ˆ ξ = V er t (ξL ). Here ξL denotes the fundamental vector field on L and V er t : T L → T L is the vertical projection defined by the connection ∇. We call (L, ∇) the pre-quantisation line ˆ ω, bundle for ( M, ˆ µ). ˆ ˆ ω, Given a Hamiltonian LG-manifold ( M, ˆ µ) ˆ at level k with a proper moment map, ˆ G is a compact, smooth then the G-action on Mˆ is free and the quotient space M/
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ˆ G, manifold of finite dimension. We define the holonomy manifold of Mˆ as M = M/ then the following diagram commutes: Mˆ
µˆ
π
M
/ Lg∗
(3.2)
H ol
µ
/ G.
There exists a unique invariant 2-form ω such that (M, ω, µ) is a quasi-Hamiltonian G-manifold with Lie group valued moment map µ. Conversely, given a quasi-Hamiltonian G-manifold (M, ω, µ), there exists a unique ˆ ω, ˆ G, and the comHamiltonian LG-manifold ( M, ˆ µ) ˆ at level k such that M = M/ ∗ ˆ mutative diagram (3.2) holds. In fact, M = M ×G Lg is a principal G-bundle over M and the LG-invariant weakly symplectic 2-form is given by ωˆ = π ∗ ω + µˆ ∗ , where is the following 2-form on Lg∗ : 1 ∂ ¯ ¯ H ols∗ (θ), = dsH ols∗ (θ), 2 S1 ∂s satisfying H ol ∗ k = −d (for details, see Theorem 8.3 in [1]). Example 3.1. (cf. Prop. 3.1 in [1]) Choose a maximal torus T in G with its Lie algebra t. The integral lattice r∨ ⊂ t (the co-root lattice) is the kernel of the exponential map ex p : t → T . Let R and R ∨ be the root system and the co-root system of G. The root and co-root lattices r ⊂ t∗ and r∨ ⊂ t are the lattices spanned by R and R ∨ with their Z-basis given by simple roots and simple co-roots = {α1 , . . . , αn }, and ∨ = {α1∨ , . . . , αn∨ } 2α . The weight and co-weight lattices w ⊂ t∗ and α, α ∨ ∨ w ⊂ t are the lattices dual to r and r . Then affine coadjoint LG-orbits at level k are labelled by respectively, where α ∨ =
Uk = {λ ∈ t∗ |λ, α ∨j ≥ 0 for any simple co-root α j , λ, ϑ ≤ k}, where ϑ is the highest root in R with respect to . We denote by Oλ the affine coadjoint LG-orbit at level k through λ ∈ Uk . The conjugacy classes in G are labelled by elements 2πiλ ex p k under g ∼ = g∗ defined by k·, ·. A conjugacy class Cλ (for λ ∈ Uk ) in G 2πiλ g = g0 · ex p · g0−1 |g0 ∈ G k has a canonical 2-form ω(g) =
k θ, (1 − Adg )−1 (θ ) 2
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such that dω = k |Cλ and (C, ω) is a quasi-Hamiltonian G-manifold with moment map the embedding C → G. The corresponding Hamiltonian LG-manifold at level k is given by the affine coadjoint LG-orbit Oλ . Then Oλ as a Hamiltonian LG-manifold at level k is pre-quantizable if and only if λ ∈ ∗k := w ∩ Uk , whose elements are called the dominant weights at level k. The pre-quantization line bundle over Oλ is given by LG × LG λ C(∗λ,1) , where LG λ acts on C(∗λ,1) with weight (∗λ, 1), ∗λ is the dominant weight of the irreducible representation of G complex conjugate to the one with weight λ. The geometric quantization on Oλ by the Borel-Weil construction as in [43] gives rise to the irreducible positive energy representation of LG with the highest weight (λ, k). 3.3. Equivariant bundle gerbe modules. Our main theorem in this section is the following existence result for generalized G-equivariant bundle gerbe modules in terms of pre-quantizable Hamiltonian LG-manifolds at level k. Theorem 3.5. Given a quasi-Hamiltonian G-manifold (M, ω, µ) such that the correˆ ω, sponding Hamiltonian LG-manifold ( M, ˆ µ) ˆ at level k is pre-quantizable and the moment map µˆ is proper, then the pull-back of the bundle gerbe Gk over G, µ∗ Gk , admits a canonical G-equivariant trivialization µ∗ Gk ∼ = δ(L Mˆ ) ˆ M [2]
π1 π2
// ˆ M M
ˆ and δ(L ˆ ) = π ∗ L ˆ ⊗ π ∗ L−1 . where L Mˆ is the pre-quantization line bundle over M, 2 M 1 M ˆ M
Proof. The pull-back of the bundle gerbe Gk is determined by the following diagram: µ∗ Gk Mˆ [2]
π1 π2
// ˆ M M
Specifically µ∗ Gk is the pullback to Mˆ [2] of the U (1) bundle determined by the central extension U (1) →
G → G corresponding to k under the map gˆM : Mˆ [2] → G
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defined by x2 = gˆM (x1 , x2 ) · x1 for (x1 , x2 ) ∈ Mˆ [2] . For a pre-quantizable ˆ ω, Hamiltonian LG-manifold ( M, ˆ µ) ˆ at level k, the pre-quantization line bundle L Mˆ carries an G-action such that the following diagram commutes:
G × L Mˆ
G × Mˆ
/L
Mˆ
/ M. ˆ
This implies that, for (x1 , x2 ) ∈ Mˆ [2] , (µ∗ Gk )(x1 ,x2 ) ⊗ (L Mˆ )x1 ∼ = (L Mˆ )x2 . The associativity of the
G-action ensures that L Mˆ is a rank one bundle gerbe module of µ∗ Gk . As a rank one bundle gerbe module of µ∗ Gk , we know that µ∗ Gk ∼ . = π2∗ L Mˆ ⊗ π1∗ L−1 Mˆ As the pre-quantization line bundle L Mˆ is actually an LG-equivariant line bundle, ˆ is G-equivariant we immediately know that the rank one bundle gerbe module (L Mˆ , M) ˆ in the sense that L Mˆ is a G-equivariant line bundle over M and the bundle gerbe action µ∗ Gk ⊗ π1∗ L Mˆ ∼ = π2∗ L Mˆ is G-equivariant.
Remark 3.6. In the set-up of differentiable stacks and their presenting Lie groupoids, a more general moment map theory is developed in [56]. Results analogous to Theorem 3.5 in terms of pre-quantizations of quasi-symplectic groupoids and the compatible pre-quantizations of their quasi-Hamiltonian spaces are also discussed in [34] for nonequivariant cases. From this theorem, we can deduce easily the following existence result for equivariant bundle gerbe D-branes of the bundle gerbe Gk over G. Corollary 3.1. Given a quasi-Hamiltonian G-manifold (M, ω, µ) such that the correˆ ω, sponding Hamiltonian LG-manifold ( M, ˆ µ) ˆ at level k is pre-quantizable and proper, then (M, ω, µ) is a generalized rank one G-equivariant bundle gerbe D-brane of Gk . For the quasi-Hamiltonian G-manifolds from conjugacy classes Cλ in G, we obtain the corresponding symmetric D-brane in G, which is pre-quantizable if and only if λ is a dominant weight at level k. ˆ ω, Remark 3.7. Relax the pre-quantizable condition in Corollary 3.1 on ( M, ˆ µ) ˆ to allow ˆ ˆ ˆ M to have an LG-equivariant Hermitian vector bundle E → M of rank n, with an invariant connection ∇ whose curvature is given by −2πi ωˆ ⊗ I d. Then the same proof implies that there exists a G-equivariant bundle gerbe action µ∗ Gk ⊗ π1∗ E −→ π2∗ E, that is to say, (M, ω, µ) is a generalized G-equivariant bundle gerbe D-brane of rank n.
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4. Moduli Spaces of Flat Connections on Riemann Surfaces In this section, we recall a particular class of quasi-Hamiltonian G-manifolds, given by the moduli spaces of flat connections on Riemann surfaces with boundaries. Denote by A S 1 the space of L 2p G-connections on the trivial principal G-bundle over S 1 . The holonomy map H ol : A S 1 −→ G defines a principal G-bundle over G, where G is the based loop group, identified with the based gauge transformation group under a choice of parametrization of S 1 . With a fixed trivial connection, we can identify A S 1 with 1 (S 1 , g). Using the isomorphism g ∼ = g∗ defined by <, >, we have ∗ ∼ A S 1 = Lg . We know that H ol : A S 1 → G agrees with the universal G-bundle over G constructed in Proposition 3.2. The bundle gerbe Gk over G is the lifting bundle gerbe (cf. [42]) associated to the principal G-bundle A S 1 → G: Gk A[2] S1
(4.1) π1 π2
// A 1 S G
G , where
G is the central extension U (1) →
G → G such that Gk = gˆ ∗ determined by φ ∈ H 4 (BG, Z) (cf. [43]), and gˆ : A[2] → G is defined by A2 = S1
A1 · g(A ˆ 1 , A2 ) for (A1 , A2 ) ∈ A[2] . The full gauge group (identified with LG) action S1 and Proposition 2.3 tell us that Gk is a G-equivariant bundle gerbe over G with equivariant Dixmier-Douady class k ∈ Z ∼ = HG3 (G, Z). The classifying map for the universal G-bundle over G, a homotopy equivalence between (BG) and B( G) and the evaluation map from S 1 × (BG) to BG define a homotopy class of maps, formally denoted by [ev]: [ev] : S 1 × G ∼ S 1 × B G ∼ S 1 × (BG) → BG, such that the Dixmier-Douady class of Gk is given by the cohomology class [ωk ] = ◦[ev](φ), S1
where φ ∈ H 4 (BG, Z) is defined by <, >. Now, given a Riemann surface with one boundary component which is pointed by fixing a base point on the boundary, denote by M , {flat L 2
p− 21
{g : L 2
p+ 12
G-connections on × G}
(, G)|g(base point) = I d ∈ G}
,
the based moduli space of flat G-connections on (that is, the space of flat G-connections on × G modulo gauge transformations which are the identity at the base point). Note that the based moduli space M is a finite dimensional smooth manifold and the boundary holonomy map defines a group valued moment map (cf. [1]) µ : M −→ G. There
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is a canonical principal G-bundle with connection over × M given by choosing a classifying map Ev :
× M −→ BG,
(4.2)
such that ev, the restriction of Ev to ∂ × M = S 1 × M , represents a map in the homotopy class of maps: S 1 × M
I d×µ
/ S1 × G
[ev]
/ BG.
i Now we can represent φ by a differential form FA on BG, where is the 2π corresponding G-invariant degree two polynomial on the Lie algebra g of G, determined by the inner product ·, ·, on g. Then from direct calculation, we see that i i ∗ ∗ ∗ d = µ ev FA FA . Ev 2π 2π S1 Hence µ∗ ωk is exact so the Dixmier-Douady class of the pullback bundle gerbe µ∗ Gk over M is trivial. Proposition 4.1. The quasi-Hamiltonian G-space (M , µ ) determines a unique Hamiltonian LG-space at level k which is diffeomorphic to ˆ , µ∗ A S 1 = M ×G A S 1 ∼ =M ˆ is the moduli space of flat connections modulo gauge transformations which with M ˆ is preare the identity on the boundary. Moreover the Hamiltonian LG-manifold M quantizable and admits a proper moment map, hence (M , µ ) is a generalized rank 1 G-equivariant bundle gerbe D-brane of Gk . ˆ is an infinite dimensional symplectic manifold Proof. By results in [4] and [20], M admitting a residual Hamiltonian action of LG at level k, whose moment map is given by the pullback of connections to the boundary ˆ −→ A S 1 ∼ µˆ : M = Lg∗ . ˆ → M is the ˆ such that the quotient map M The induced G-action is free on M induced principal G-bundle and the following diagram is commutative: ˆ M
µˆ
π
M
/ Lg∗
(4.3)
H ol
µ
/ G.
ˆ , and the moment map µˆ is proper. which confirms µ∗ A S 1 ∼ =M Now we give a construction of a pre-quantization line bundle following [39] and [4]. f lat Denote by A the space of flat G-connections on × G, and denote by G0 () and
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G0 (∂) the based gauge transformation groups on and ∂ respectively. Let G(, ∂) be the kernel of the restriction map to the boundary G0 () −→ G0 (∂) ∼ = G,
∂:
(4.4)
then G(, ∂) consists of those gauge transformations which are the identity on the whole boundary. Since G is simply connected, we have the following exact sequence: 1 → G(, ∂) −→ G0 () −→ G0 (∂) → 1, ˆ −→ M is induced by the residual action of the and the principal G-bundle M gauge group G0 (). The pullback of the central extension 1 → U (1) −→
G −→ G → 1 under the map ∂ (4.4) defines a central extension G0 () of G0 () whose 2-cocycle is given by −1 −1 c(g1 , g2 ) = ex p 2πi g1 dg1 , dg2 g2 . (4.5)
It is known that this extension has a canonical trivialisation over G(, ∂) ⊂ G0 (). f lat f lat The pre-quantization line bundle over A is given by the trivial line bundle A × C with connection 1-form f lat
θ A : T A A
∼ = {α ∈ 1 (, g)|dα = 0} → R α → α, A.
(4.6)
Here 1 (, g) is the space of the Lie algebra g valued 1-form on . This pre-quantization line bundle admits a connection-preserving action of G0 () via the local action (g, z) · (A, w) = g · A, ex p −2πi g −1 dg, A zw ,
whose quotient under the G(, ∂)-action f lat
L = (A
× C)/G(, ∂)
ˆ . is the pre-quantization line bundle over M We claim that µ : M → G is a generalized rank 1 bundle gerbe D-brane, with the canonical trivialisation of µ∗ Gk given by µ∗ Gk ∼ = δ(L ) ˆ [2] M
(4.7) π1 π2
// ˆ M M
where L is the pre-quantization line bundle over Mˆ in [39,53].
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As L carries an action of LG, it is straightforward to show that L is a G-equivariant bundle gerbe module of µ∗ Gk (see the proof of Theorem 3.5), therefore, we have f lat shown that µ∗ Gk ∼ = δ(L ) as in (4.7). The connection θ (4.6) on A × C descends to a bundle gerbe module connection on L . Hence, (M , µ ) is a generalized rank 1 G-equivariant bundle gerbe D-brane of Gk .
4.1. Relationship with Chern-Simons. In this subsection we summarise some observations about the present situation which may be deduced from [15]. In that paper the universal Chern-Simons bundle 2-gerbe Qφ associated to φ ∈ H 4 (BG, Z) is defined to be a bundle 2-gerbe (Qφ , E G [2] ; E G, BG) with connection illustrated by the following diagram: Gτ (φ)
(4.8)
;w G w gˆ www w w w ww π1 // [2] EG EG Qφ
π2
π
BG
The way to read this diagram is that Qφ is obtained as the pull-back of a multiplicative bundle gerbe Gτ (φ) over G, with connection and curving, whose bundle gerbe curvature τ (φ) ∈ H 3 (G, Z) is determined by φ and where τ : H 4 (BG, Z) → H 3 (G, Z) is the usual transgression map. The technicalities in [15] are handled by recognising that transformations between stable isomorphisms of bundle 1-gerbes provide 2-morphisms making the category BGrb M of bundle 1-gerbes over a manifold M and stable isomorphisms between bundle 1-gerbes into a bi-category. The space of 2-morphisms between two stable isomorphisms is in one-to-one correspondence with the space of line bundles over M. We also recall from [15] the definition of a multiplicative bundle gerbe on a compact semi-simple Lie group G. Let BG • be the following simplicial manifold: BG • = {BG n = G × · · · × G (n copies)} (where n = 0, 1, 2, . . .), endowed with face operators πi 0, 1, . . . , n + 1), ⎧ ⎪ ⎨(g1 , . . . , gn ), πi (g0 , . . . , gn ) = (g1 , . . . , gi−1 gi , gi+1 , . . . , gn ), ⎪ ⎩(g , . . . , g ), 0 n−1
: G n+1 → G n , (i = i = 0, 1 ≤ i ≤ n, i = n + 1.
In particular, the face operators from G × G → G consist of π0 (g1 , g2 ) = g2 , π1 (g1 , g2 ) = g1 g2 and π2 (g1 , g2 ) = g1 for (g1 , g2 ) ∈ G × G.
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The face operator πi : G n+1 → G n defines a bi-functor πi∗ : BGrbG n −→ BGrbG n+1 sending objects, stable isomorphisms and 2-morphisms to the pull-backs by πi . Definition 4.2 (cf. [15]). A multiplicative bundle gerbe on G is a bundle gerbe G over G together with a stable isomorphism m : π0∗ G ⊗ π2∗ G → π1∗ G over G × G, where πi∗ G is the pull-back bundle gerbe over G × G, such that, the stable isomorphism m is associative up to a 2-morphism in BGrbG×G×G : ϕ : π2∗ m ◦ (π0∗ m ⊗ I d) =⇒ π1∗ m ◦ (I d ⊗ π3∗ m), for which the corresponding line bundle Lϕ over G × G × G is trivial. Moreover, there is a canonical isomorphism between two trivial line bundles over G 4 with their induced trivializing sections: π1∗ Lϕ ⊗ π3∗ Lϕ ⊗ π5∗ Lϕ ∼ = π2∗ Lϕ ⊗ π4∗ Lϕ . The main result of [15] is that a bundle gerbe G over G is multiplicative if and only if the corresponding Dixmier-Douady class lies in the image of the transgression map τ : H 4 (BG, Z) → H 3 (G, Z). The relation between the Chern-Simons bundle gerbe and the moduli space of flat G-connections is given by the following proposition. Proposition 4.3. The transgression of our universal Chern-Simons bundle 2-gerbe Qφ to the moduli space of flat G-connections on a closed Riemann surface is the Chern-Simons line bundle over the moduli space. Proof. Given a closed Riemann surface with a base point, we cut along a separating simple curve through the base point so that = 1 ∪ S 1 2 and i is a Riemann surface with one pointed boundary. It is easy to see that the based moduli space of flat G-connections on × G is given by the fiber product of the group valued moment maps for M1 and M2 , M ∼ = M1 ×G M2 ∼ ˆ 2 // G ˆ 1 × M = M ∼ ˆ 2 / G, ˆ 1 ×A M = M S1
(4.9)
with the induced map µ : M → G. Here we use the notation “//” for the symplectic reduction by the diagonal G-action with respect to the moment map µˆ 1 − µˆ 2 , as 0 is a regular value ([4]), ˆ 2 , ˆ 1 ×A M (µˆ 1 − µˆ 2 )−1 (0) = M S1 and the action of G is free. The pull-back bundle gerbe µ∗ Gk over M from Gk over G now has two trivialisations from the canonical trivialisations of the pull-back bundle gerbes over M1 and M2 (see Proposition 4.1). These two trivialisations give a line bundle L over M
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which we refer to as the transgression of Qφ . In [15], there is a natural bundle 2-gerbe connection on Qφ . The Chern-Simons bundle 2-gerbe connection induces canonical bundle gerbe module connections on the bundle gerbe modules L1 and L2 . These bundle gerbe module connections define a canonical connection on the line bundle L . The curvature of this canonical connection is given by i i i ∗ ∗ ∗ FA − FA = FA , Ev1 Ev2 Ev 2π 1 2π 2 2π 1 −2 where Ev is the classifying map for the canonical G-bundle over × M with connection Ev ∗ A, and is the corresponding G-invariant degree two polynomial on the Lie algebra g of G. This agrees with the curvature formula in [4,44] for the Chern-Simons line bundle over M . 4.2. Recasting the Segal-Witten reciprocity law. We now interpret the Segal-Witten reciprocity law (cf. [12]) from the viewpoint of bundle gerbes over G and their bundle gerbe D-branes. Let G C be the complexification of a connected, compact and semi-simple Lie group (not necessarily simply-connected) G. Let LG C denote the smooth loop group. A central extension of LG C by C∗ , 1 → C∗ −→ LG C −→ LG C → 1, has the reciprocity property if, for an extended Riemann surface whose boundary ∂ is a disjoint union of parametrized circles, the extension of C ∞ (∂, G C ) induced by the Baer product of the extension of boundary components G C ) −→ C ∞ (∂, G C ) → 1 1 → C∗ −→ C ∞ (∂, splits canonically over the subgroup of holomorphic maps (denoted by H ol(, G C )) from to G C . G C ) to denote the canonical section of the We use s : H ol(, G C ) → H ol(, splitting of the induced extension G C ) −→ H ol(, G C ) → 1. 1 → C∗ −→ H ol(, Given a central extension LG C with the reciprocity property, LG C satisfies the glueing property if whenever an extended Riemann surface is obtained by glueing two extended Riemann surfaces 1 and 2 along some boundary components with the obvious restriction maps ρi : H ol(, G C ) −→ H ol(i , G C ), G C ) and ρ ∗ H ol(1 , G C ) ⊗ then there is a canonical isomorphism between H ol(, 1 ρ2∗ H ol(2 , G C ) carrying the section s to ρ1∗ s1 ⊗ ρ2∗ s2 . G C ) satisfies the The Segal-Witten reciprocity law claims that an extension H ol(, reciprocity and glueing properties if the characteristic class of the extension lies in the image of the transgression map τ : H 4 (BG, Z) −→ H 3 (G, Z). In [12], the converse of the Segal-Witten reciprocity law is established: any extension of LG C satisfying the reciprocity and glueing properties must lie in the image of τ .
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In the light of the properties of multiplicative bundle gerbes, their bundle gerbe modules and their relationship with the Chern-Simons bundle 2-gerbes, we can recast the Segal-Witten reciprocity law as in the following proposition for a simply-connected simple Lie group G. Proposition 4.4. For an extended Riemann surface = g,n of genus g with n pointed and parametrized boundary components, the transgression of the Chern-Simons bundle 2-gerbe Qφ (4.8) provides a canonical G-equivariant trivialization of the pull-back equivariant bundle gerbe associated to the group-valued moment map µ : M −→ G n . Proof.Given Riemann surface = g,n of genus g with n pointed boundary compon nents i=1 Si , the boundary holonomy map defines the group valued moment map for the based moduli space M , µ : M −→ G n . We can identify as before µ∗ (AnS 1 ) with ˆ = {flat G-connections on × G} , M {g : → G| g|∂ = I d ∈ G} which is an infinite dimensional symplectic manifold carrying a Hamiltonian action of (LG)n . The corresponding moment map is given by the restriction map to the boundˆ −→ (Lg∗ )n defining the following commutative diagram ary components µˆ : M analogous to (4.3): ˆ M
µˆ
π
M
µ
/ (Lg∗ )n
(4.10)
H ol
/ Gn .
Therefore, the pull-back bundle gerbe n pi∗ Gk ) µ∗ (⊗i=1
over M , where pi : G n → G is the projection on its ith factor, has a canonical ˆ by a construction trivialisation given by the pre-quantization line bundle L over M analogous to that in the proof of Proposition 4.1. Hence, (M , µ ) gives rise to a generalized equivariant bundle gerbe D-brane in G n . Remark 4.5. To see precisely the relationship between the Segal-Witten reciprocity law and our Proposition 4.4, we remind the reader of the following two observations: (1) Proposition 4.4 holds for all Sobolev classes L 2p and the corresponding moduli space ˆ contains a dense set C ∞ (∂, G C )/H ol(, G C ) which is a Frechet manifold. M (2) The pre-quantization line bundle L is given by the extension of G C )/H ol(, G C ) ×C∗ C, C ∞ (∂, where the canonical splitting over H ol(, G C ) enters naturally. It is not hard to see that these canonical trivialisations obtained from the transgression of our universal Chern-Simons bundle 2-gerbe Qφ satisfy natural glueing properties under cutting and pasting of Riemann surfaces.
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4.3. The multiplicative structure of Gk . We assume that the classifying map for the principal G-bundle over 0,n is given by a smooth map 0,k → BG such that base points on the boundary components are mapped to a base point in BG. Then the based moduli space of flat G-connections on 0,n , still denoted by M0,n , satisfies n n gi = 1 . M ∼ = (g1 , . . . , gn ) ∈ G | 0,n
i=1
For a sphere with three holes 0,3 , the pull-back bundle gerbe over M0,3 is isomorphic to the bundle gerbe δ(Gk ) = p0∗ (Gk ) ⊗ p1∗ (Gk∗ ) ⊗ p2∗ (Gk ) over G × G, where πi : G × G → G is given by π0 (g1 , g2 ) = g2 , π1 (g1 , g2 ) = g1 g2 and π2 (g1 , g2 ) = g1 for (g1 , g2 ) ∈ G. Then the induced canonical trivialisation of δ(Gk ) defines the multiplicative structure on Gk (see [15]). The associator for the multiplicative structure is given by the canonical trivialisation of the pull-back bundle gerbe over the based moduli space M0,4 ∼ =G×G×G of flat G-connections on a sphere with four holes, 0,4 , and the induced trivialisations from two ways of decomposing the four holed sphere into three holed spheres. The cocycle condition for the associator is given by the canonical trivialisation of the pullback bundle gerbe over the based moduli space M0,5 ∼ =G×G×G×G of flat G-connections on a sphere with five holes, 0,5 , and various ways of decomposing the 5-holed sphere. 5. Spi n c Quantization and Fusion of D-Branes Fix a quasi-Hamiltonian G-manifold (M, ω, µ) with corresponding pre-quantizable ˆ ω, Hamiltonian LG-manifold ( M, ˆ µ) ˆ at level k, as illustrated in the following diagram: ˆ ω) ( M, ˆ
µˆ
π
(M, ω)
/ Lg∗
(5.1)
H ol
µ
/ (G, k ).
ˆ together with the pre-quantization line bundle From Theorem 3.5, we know that M, L = L Mˆ , defines a generalized rank one G-equivariant bundle gerbe module of Gk over G. Equivalently, (M, ω, µ) is a generalized G-equivariant bundle gerbe D-brane of Gk . In this section, we will define a quantization procedure which gives rise to an eleˆ ω, ˆ µ) ˆ at level ment in Rk (LG) for any pre-quantizable Hamiltonian LG-manifold ( M, k. We will apply the fusion product defined in [40] for pre-quantizable Hamiltonian LG-manifolds at level k to show that our quantization functor from the category of
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generalized G-equivariant bundle gerbe modules of Gk to Rk (LG) commutes with these fusion products. Suppose that λ is a quasi-regular value of µ. ˆ The Hamiltonian reduction at the dominant weight λ of level k, given by Mλ := µˆ −1 (λ)/(LG)λ ∼ = µˆ −1 (Oλ )/LG, is a symplectic orbifold with the reduced symplectic form ωλ and a pre-quantization line bundle with a connection ∇λ , C(∗λ,1) , Lλ := L|µˆ −1 (λ) ×( LG) λ
where ∗λ is the dominant weight of the irreducible representation of G dual to the one λ acts on C(∗λ,1) with weight (∗λ, 1) (cf. [3] and [55]). Choose an with weight λ, (LG) almost complex structure J , compatible with ωλ , which defines a canonical Spin c structure S := S ± J . Twisted by the pre-quantization line bundle Lλ , a Hermitian connection on T Mλ defines a Spin c Dirac operator ∂/λ : L 2 (S + ⊗ Lλ ) −→ L 2 (S − ⊗ Lλ ). k
k−1
The index of ∂/λ , denoted by I ndex(/ ∂ λ , Mλ ), is independent of the choice of the almost complex structure and the Hermitian connection. The symplectic invariant defined as ∂ λ ), is a rational number in general (an integer if Mˆ λ is a smooth the index of ∂/λ , I nd(/ symplectic manifold). Remark 5.1. (1) If Mλ is Kähler and (L λ , ∇λ ) is holomorphic, then the canonical Spin c bundle and the Spin c Dirac operator are given by √ S ± = 0,even/odd (T ∗ Mλ ), ∂/λ = 2(∂¯∇λ + ∂¯∇∗ λ ). Hence, we have I ndex(/ ∂ λ , Mλ ) = χ (Mλ , Lλ ), the Euler characteristic for the sheaf of holomorphic sections of Lλ . ˆ 0,3 (∗λ, ∗µ, ν) be the Hamiltonian reduction of M ˆ 0,3 at (∗λ, ∗µ, ν) ∈ (2) Let M ∗ 3 c ˆ (k ) , then the index of the Spin Dirac operator on M0,3 (∗λ, ∗µ, ν), see [8,39] ˆ 0,3 (∗λ, ∗µ, ν)) = N ν , I ndex(/ ∂, M λ,µ the fusion coefficient determined by the Verlinde factorization formula ([51]). The ν agrees vanishing theorems for higher cohomology groups in [49] imply that Nλ,µ with the dimension of the space of holomorphic sections of the pre-quantization ˆ 0,3 (∗λ, ∗µ, ν). line bundle over the reduced space M Definition 5.2. The quantization of a pre-quantizable quasi-Hamiltonian G-manifold (M, ω, µ) at level k is defined to be ˆ = χk,G (M) := χk,G ( M) I ndex(/ ∂ λ , Mλ ) · χλ,k ∈ Rk (LG), λ∈∗k
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where χλ,k is the character of the irreducible representation of LG with highest weight (λ, k), and Rk (LG) is the Abelian group generated by the isomorphism classes of irreducible positive energy representations of LG at level k. Equipped with the fusion product ν χλ,k ∗ χµ,k = Nλ,µ χν,k . ν∈∗k
(Rk (LG), ∗) becomes the Verlinde ring. It was shown in [25] that (Rk (G), ∗) can be identified with the G-equivariant twisted dimG (G) for the conjugacy action of G on itself, where h ∨ is the dual Coxeter K -group K G,h ∨ number. This motivates the following definition. Definition 5.3. The category of equivariant bundle gerbe D-branes of the equivariant bundle gerbe Gk over G is given by the category of pre-quantizable quasi-Hamiltonian G-manifolds whose objects are (M, ω, µ) and the morphism between (M1 , ω1 , µ1 ) and (M2 , ω2 , µ2 ) is given by a G-equivariant map f : M1 → M2 such that ω1 = f ∗ ω2 ,
µ1 = µ2 ◦ f.
Equivalently, we say that the category of equivariant bundle gerbe modules of Gk is given by the category of pre-quantizable Hamiltonian LG-manifolds at level k with proper moment maps. We denote this category by QG,k . On the category of pre-quantizable Hamiltonian LG-manifolds at level k with proper moment maps, there is a product structure, called the fusion product of Hamiltonian LG-manifolds in [40]. Recall from Sect. 4 that M0,3 is the based moduli space of flat G-connections on 0,3 (the genus 0 surface with 3 pointed boundary components). Note that M0,3 is a quasi-Hamiltonian (G × G × G)-manifold, and the corresponding Hamiltonian (LG)3 ˆ 0,3 : manifold at level k is denoted by M ˆ 0,3 M π
M0,3
µˆ 0,3
µ
/ (Lg∗ )3
H ol
/ G3.
ˆ at level k, Given a Hamiltonian LG × LG-manifold M ˆ M
M0,3
µˆ
/ Lg∗ ⊕ Lg∗ H ol
µ
/ G×G
ˆ ×M ˆ 0,3 is a Hamiltonian (LG)5 -manifold at level k. The diagonal embedding then M LG × LG −→ (LG × LG) × (LG × LG × LG),
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mapping (γ1 , γ2 ) → (γ1 , γ2 )×(γ¯1 , γ¯2 , e) (where γ¯i (θ ) = γi (−θ ) : R/Z → G), defines a Hamiltonian LG × LG action on Mˆ × Mˆ 0,3 with moment map ˆ ×M ˆ 0,3 −→ Lg∗ ⊕ Lg∗ µˆ diag : M (x, [A]) → µ(x) ˆ − pr12 ◦ µˆ 0,3 ([A]), where pr12 denotes the projection from Lg∗ ⊕ Lg∗ ⊕ Lg∗ to the first two factors. As 0 is a regular value of pr12 ◦ µˆ 0,3 , we can define the Hamiltonian quotient, denoted by ˆ ×M ˆ 0,3 //diag(LG)2 , M as the symplectic reduction µˆ −1 diag (0)/(LG × LG). Note that
∼ ˆ 0,3 /(LG × LG), ˆ × L g∗ ⊕L g∗ M µˆ −1 (0)/(LG × LG) M = diag
ˆ → Lg∗ ⊕ Lg∗ and which is an LG × LG quotient of the fiber product of µˆ : M ˆ 0,3 → Lg∗ ⊕ Lg∗ . The remaining LG-action on M ˆ 0,3 descends to a pr12 ◦ µˆ 0,3 : M 2 ˆ ˆ Hamiltonian LG-action on Mγ × M0,3 //diag(LG) , with the natural moment map induced from the map ˆ ×M ˆ 0,3 −→ Lg∗ ⊕ Lg∗ pr3 ◦ µˆ 0,3 M (x, [A]) → pr3 ◦ µˆ 0,3 ([A]), where pr3 is the projection from Lg∗ ⊕ Lg∗ ⊕ Lg∗ to the third factor. As any λ ∈ Lg∗ ˆ × is a regular value of pr3 ◦ µˆ 0,3 , we can define the symplectic reduction of M 2 ˆ M0,3 //diag(LG) at λ as ˆ ×M ˆ 0,3 (·, ·, λ) /diag(LG)2 , ( pr3 ◦ µˆ 0,3 )−1 (0)/(LG)λ ∼ = M ˆ 0,3 (·, ·, λ) is given by the subset of M ˆ 0,3 with holonomy around the outgoing where M boundary component in the conjugacy class Cλ of G through exp(2πiλ/k). Definition 5.4. The fusion product on the category QG,k is defined as follows: given two pre-quantizable Hamiltonian LG-manifolds ( Mˆ 1 , ωˆ 1 , µˆ 1 ) and ( Mˆ 2 , ωˆ 2 , µˆ 2 ) at level k with proper moment maps, the fusion of product Mˆ 1 Mˆ 2 is the Hamiltonian LGmanifold at level k obtained as the Hamiltonian quotient ˆ 0,3 //diag(LG)2 , Mˆ 1 Mˆ 2 := ( Mˆ 1 × Mˆ 2 ) × M with the resulting moment map denoted by µˆ 1 µˆ 2 . For two pre-quantizable quasi-Hamiltonian G-manifolds (M1 , ω1 , µ1 ) and (M2 , ω2 , µ2 ), the corresponding fusion product is given by k ∗ ∗¯ M1 M2 = M1 × M2 , ω1 + ω2 + µ1 θ, µ2 θ, µ1 · µ2 . 2
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Remark 5.5. By direct calculation, it is shown in [40] that ˆ g ,n M ˆ g ,n = M ˆ g +g ,n +n −1 , M 1 1 2 2 1 2 1 2 and, for three level k dominant weights λ, µ and ν in ∗k , ˆ 0,3 (∗λ, ∗µ, ν), (Oλ Oµ )ν = M where Oλ denotes the coadjoint orbit of the affine LG action on Lg∗ at level k with the corresponding quasi-Hamiltonian G-manifold given by 2πiλ Cλ = g · ex p · g −1 |g ∈ G . k The fusion product on QG,k is well-defined in the sense that given two pre-quantizable quasi-Hamiltonian G-manifolds (M1 , ω1 , µ1 ) and (M2 , ω2 , µ2 ) whose Hamiltonian LG-manifolds at level k are denoted by Mˆ 1 and Mˆ 2 , then the corresponding Hamiltonian LG-manifold at level k for M1 M2 is given by ˆ 1 Mˆ 2 , M 1 M2 = M which is also pre-quantizable. See [1] for a proof of this claim. Moreover, modulo LGequivariant symplectomorphisms, QG,k is a monoidal tensor category: (1) For any Hamiltonian LG-manifold Mˆ at level k, there is an LG-equivariant sympˆ that is, G is the unit object. lectomorphism Mˆ G ∼ = M, ˆ ˆ ˆ (2) Let M1 , M2 , M3 be Hamiltonian LG-manifolds at level k with proper moment maps. There exist LG-equivariant symplectomorphisms ∼ Mˆ 2 Mˆ 1 , Mˆ 1 Mˆ 2 = ˆ ˆ ˆ ( M1 M2 ) M3 ∼ = Mˆ 1 ( Mˆ 2 Mˆ 3 ). The category QG,k together with the fusion product is called the fusion category of equivariant bundle gerbe D-branes (QG,k , ). Our main theorem about the structure of the category QG,k is the following result on “quantization commutes with fusion”, which gives a geometric way to think of the ring structure on equivariant twisted K -theory (cf [25]). Theorem 5.6. The quantization functor defined by the Spin c quantization as in Definition 5.2 χk,G : (QG,k , ) −→ (Rk (LG), ∗) satisfies χk,G ( Mˆ 1 Mˆ 2 ) = χk,G ( Mˆ 1 ) ∗ χk,G ( Mˆ 2 ),
(5.2)
where the product ∗ on the right-hand side denotes the fusion ring structure on the Verlinde ring Rk (LG). Proof. By definition, we see that χk,G ( Mˆ 1 Mˆ 2 ) =
ν∈∗k
I ndex ∂/ν , ( Mˆ 1 Mˆ 2 )ν · χν,k ,
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and
χk,G ( Mˆ 1 ) ∗ χk,G ( Mˆ 2 ) =
λ,µ,ν∈∗k
605
ν ·χ . I ndex ∂/λ , ( Mˆ 1 )λ · I ndex ∂/µ , ( Mˆ 2 )µ · Nλ,µ ν,k
Note that for ν ∈ ∗k ,
ˆ 0,3 (ν) //diag(LG)2 . ( Mˆ 1 Mˆ 2 )ν = ( Mˆ 1 × Mˆ 2 ) × M
Applying Theorem 2.1 in [39] to the Hamiltonian L(G × G × G × G)-manifold at level k ˆ 0,3 (ν), ( Mˆ 1 × Mˆ 2 ) × M we obtain
I ndex ∂/ν , ( Mˆ 1 Mˆ 2 )ν ˆ 0,3 (ν))λ,µ,∗λ,∗µ = λ,µ∈∗ I ndex ∂/, (( Mˆ 1 × Mˆ 2 ) × M k ˆ 0,3 (∗λ, ∗µ, ν) = λ,µ∈∗ I ndex ∂/, ( Mˆ 1 × Mˆ 2 )λ,µ × M k = λ,µ∈∗ I ndex ∂/λ , ( Mˆ 1 )λ · I ndex ∂/µ , ( Mˆ 2 )µ k ˆ 0,3 (∗λ, ∗µ, ν) ·I ndex ∂/, M ν · I ndex ∂ /λ , ( Mˆ 1 )λ · I ndex ∂/µ , ( Mˆ 2 )µ , = λ,µ∈∗ Nλ,µ k
which leads to (5.2) by direct calculation.
For a dominant weight λ of level k in ∗k , the pre-quantizable quasi-Hamiltonian G-manifold given by the conjugacy class Cλ defines an object in Qk,G . Then it is easy to see that χk,G (Cλ ) = χk,G (Oλ ) = χλ,k ∈ Rk (LG). Hence, the quantization functor χk,G : Qk,G → Rk (LG) is surjective. 6. The Non-Simply Connected Case For a compact, connected, non-simply connected simple Lie group G, there are a few subtleties in the construction of the fusion category of bundle gerbe modules. These issues also surface in the ring structure on the equivariant twisted K-theory of G, see for example [18]. ˜ , where Z is a subgroup of the center Let G˜ be the universal cover of G, G = G/Z ˜ ˜ Z (G) of G, 1 → Z −→ G˜ −→ G → 1, and the covering map π : G˜ → G identifies Z with the fundamental group π1 (G). For a compact, connected and simply connected simple Lie group with non-trivial center, G˜
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is one of the Cartan series SU (n + 1), Spin(2n + 1), Sp(n), Spin(4n), Spin(4n + 2), E 6 and E 7 with the center given by Zn+1 , Z2 , Z2 , Z2 × Z2 , Z4 , Z3 and Z2 respectively. The first subtlety comes from the non-connectness of the loop group LG, in fact its connected components are given by the fundamental group π1 (G). Let L 0 G be the identity component of LG, and 0 G the identity component of G, then we have the following exact sequences: 1 → L 0 G −→ LG −→ Z → 1,
1 → 0 G −→ G −→ Z → 1.
We now note the following difficulties: (1) The central extension of LG, 1 → U (1) −→ LG −→ LG → 1, is not uniquely determined by a class σ ∈ H 3 (G, Z), rather by a class in H 3 (G, Z) ⊕ H om(Z , U (1)) ([43] and [50]). (2) The fusion object for the category of bundle gerbe modules for a non-simply connected G, as a moduli space of flat connections on 0,3 modulo those gauge transformations which are trivial on boundary components, is actually a Hamiltonian L 0 G × L 0 G × L 0 G-manifold at level k, not a Hamiltonian LG × LG × LG manifold. (3) The pre-quantization condition for Hamiltonian L 0 G-manifolds at level k, such as those from the moduli spaces of flat connections on a Riemann surface, only holds when the level k is transgressive for G. The second subtlety comes from the fact that we have to restrict to bundle gerbes Pσ whose Dixmier-Douady class σ lies in the image of τ : H 4 (BG, Z) → H 3 (G, Z) in order that Pσ is multiplicative. Remark 6.1. (1) For G = S O(3), a Dixmier-Douady class σ is transgressive if it is an even class in H 3 (S O(3), Z), equivalently, a multiple of 4 under the map H 3 (S O(3), Z) −→ H 3 (SU (2), Z) ∼ = Z. (2) For a general compact, connected, simple Lie group G, we have the following commutative diagram: / H 3 (G, Z)
H 4 (BG, Z) ˜ Z) H 4 (B G,
∼ =
(6.1)
/ H 3 (G, ˜ Z),
˜ Z) ∼ ˜ Z) ∼ where H 4 (B G, = H 3 (G, = Z and the generator is determined by the basic inner product ·, · on g uniquely specified by requiring that the highest co-root of G˜ ˜ Z) is specified by its “level” k coming has norm 2. Then each element in H 4 (B G, from identifying the induced inner product on g as being given by k·, ·. 6.1. Some background about the center of G˜ and the loop group LG. We first review some basic properties about the center of the universal cover G˜ of G following [50] where irreducible positive energy projective representations of LG are classified. ˜ can be characterized as follows: choose a maximal torus T˜ of G˜ The center Z (G) ˜ The with Lie algebra t, let R and R ∨ be the root system and the co-root system of G.
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root and co-root lattices r ⊂ t∗ and r∨ ⊂ t are the lattices spanned by R and R ∨ with their Z-basis given by = {α1 , . . . , αn }, and ∨ = {α1∨ , . . . , αn∨ } 2α . The weight and co-weight lattices w ⊂ t∗ and α, α ∨ ∨ w ⊂ t are the lattices dual to r and r with their Z-basis given by the fundamental weights λi and the fundamental co-weights λi∨ such that respectively, where α ∨ =
λi , α ∨j = λi∨ , α j = δi j . Under the identification t ∼ = t∗ defined by the basic inner product, r ⊂ w ⊂ t∗ ∪ ∪ r∨ ⊂ ∨ w ⊂ t, from which we know that: ∨ ∼ ˜ (1) the map h ∨ → ex pT˜ (2πi h ∨ ) induces an isomorphism ∨ w /r = Z (G), where ex pT˜ denotes the exponential map for T˜ ; ∨ (2) the map sends µ ∈ w to the pairing µ(ex pT˜ (2π h ∨ )) = e2πiµ,h induces an ˜ U (1)). isomorphism w /r ∼ = H om(Z (G),
˜ in terms of special roots (cf. Lemma 2.3 in There is another characterisation of Z (G) ˜ [50]): non-trivial elements in Z (G) correspond one-to-one to the special fundamental co-weights {λi(z) }z∈Z (G) ˜ such that the corresponding αi(z) ∈ carries the coefficient 1 in the expression for the highest root ϑ. For each special root αi(z) , there exists a unique Weyl group element ωi(z) (cf. Prop. 4.1.2 in [50]) which preserves ∪ {−ϑ} and sends −ϑ to αi(z) , such that ωi(z 1 ) ωi(z 2 ) = ωi(z 1 ·z 2 ) , ˜ where z 1 · z 2 denotes the group multiplication in Z (G). ˜ The dominant weight of G at level k is given by ∗k = {λ ∈ w |λ, α ∨ ≥ 0, λ, θ ≤ k},
(6.2)
˜ is isomorphic to the which is non-empty only if k is a positive integer. Note that Z (G) group of automorphisms of the extended Dynkin diagram of G˜ which induces an action ˜ on ∗ as given by Proposition 4.1.4 in [50], where the explicit action for all of Z (G) k ˜ is given by the affine classical groups is explained. Geometrically, this action of Z (G) Weyl group element ([50]) ∨ z → τ (kλi(z) )ωi(z) , ∨ ) denotes the translation by kλ∨ in the affine Weyl group. where τ (kλi(z) i(z)
(6.3)
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˜ the integral lattice of G = G/Z ˜ , ∨ = H om(U (1), Given a subgroup Z ⊂ Z (G), Z ∨ ∨ ∼ T˜ /Z ), then r∨ ⊂ ∨Z ⊂ ∨ w , and Z /r = Z . The basic level b of G is the smallest integer k such that the restriction of k·, · to ∨Z is integral. Introduce the group of ‘discontinuous loops’ ˜ (t + 2π )γ (t)−1 ∈ Z }. L Z G˜ = {γ ∈ C ∞ (R, G)|γ Then we have the following commutative diagram with all rows and columns being exact: 1 Z
1
1
(6.4)
1 ∼ =
/Z
/ L G˜
/ L G˜ Z
/Z
/ L0G
/ LG
/Z
1
1
/1 ∼ =
/1
As Z ∼ = ∨Z /r∨ , we can associate to µ∨ ∈ ∨Z , the discontinuous loop ˜ ζµ∨ (t) = ex pT˜ (2πitµ∨ ) ∈ L Z (G).
(6.5)
Notice that if µ∨ ∈ r∨ , then ζµ∨ (t) ∈ L T˜ . In particular, for each z ∈ Z , fix a representative wz ∈ G˜ for the unique Weyl group element ωi(z) , then z → ζz := ζλ∨ wz i(z)
(6.6)
˜ to each z, we call ζz the distinguished discontinassigns a discontinuous loop in L Z (G) uous loop associated to z. According to Theorem 4.3.3 in [50], the conjugation action of ζz on L G˜ induces an action of Z on ∗k which agrees with the induced action of ˜ on ∗ as from (6.3). We denote by z · λ the action of z ∈ Z on λ ∈ ∗ . Z ⊂ Z (G) k k ˜ is the smallest positive integer such Definition 6.2. The basic level b of G = G/Z that the restriction of ·, · to ∨Z is integral. It was shown in Proposition 3.5.1 of [50], the level for which LG admits a central extension is a multiple of the basic level b . Now we review the construction of the central extension of LG from [50]. ˜ of L Z G˜ (see Given a level k ∈ b Z, there is a canonical central extension L ZG Proposition 3.5.1 and Theorem 3.2.1 in [50]) at level k and the trivial extension of ∨Z where we regard ∨Z as a subgroup of L Z G˜ through the discontinuous loop ζµ∨ given
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by (6.5) for µ∨ ∈ ∨Z . The central extension of ∨Z is classified by its commutator map defined by ˜ −1 ω(λ∨ , µ∨ ) = ζ˜λ∨ ζ˜µ∨ ζ˜λ−1 ∨ ζµ∨ ,
(6.7)
˜ are arbitrary lifts of ζ˜λ∨ , ζ˜µ∨ . Note that there is a necessary and where ζ˜λ∨ , ζ˜µ∨ ∈ L ZG sufficient compatibility condition (cf. Proposition 3.3.1) ∨ ,µ∨
ω(λ∨ , µ∨ ) = (−1)kλ ˜ whenever λ∨ ∈ r∨ . for the existence of L Z G,
˜ to G, ˜ is canonically As G˜ is simple and simply-connected, the restriction of L ZG trivial, hence, restricted to Z , Z admits a canonical section s : Z → Z . Given a character χ : Z → U (1), following [50], we can construct a canonical central extension of LG associated to a level k ∈ b Z and χ ∈ H om(Z , U (1)), defined by ˜ )χ −1 (γ )|γ ∈ Z }. LG χ := L Z G/{s(γ
(6.8)
We denote this central extension of LG by 1 → U (1) −→ LG χ −→ LG → 1.
(6.9)
Remark 6.3. For all compact, connected simple Lie groups G except P S O(4n) =
Spin(4n) , Z2 × Z2
we have H 3 (G, Z) ∼ = Z. Then from (6.8) we obtain all central extensions of LG labelled by (k, χ ). For G = Spin(4n)/Z with Z = Z2 × Z2 , H 3 (G, Z) ∼ = Z ⊕ Z2 , ˜ the canonical one where Z2 corresponds to two inequivalent central extensions of L Z G: for LG χ and the other one with the commutator ω (6.7) defined by the pull-back of the non-trivial, skew-symmetric form on Z2 × Z2 . We denote the corresponding central extension of LG with respect to this non-trivial commutator ω by 1 → U (1) −→ LG χ ,− −→ LG → 1. LG χ ,− for G = The following discussion for LG χ can be extended to the case of P S O(4n) with some minor modifications; we shall point out the difference for this latter case. We are now able to classify all irreducible positive energy representations of LG χ . We assume from now on that the level k ∈ b Z. The induced central extensions of L 0 G ˜ are denoted by and L G˜ from LG χ and L L G˜ respectively. Then we have L 0 G χ and ZG
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the following commutative diagrams relating various exact sequences and their central extensions: 1
/ L0Gχ
/ LG χ
/Z
/ LG
/Z
/1
(6.10)
∼ =
/ 1,
1
/ L0G
1
/Z
/ ˜ LG
/ L0Gχ
/1
1
/Z
/ L G˜
/ L0G
/ 1,
1
/Z
/ ˜ LZG
/ LG χ
/1
1
/Z
/ L G˜ Z
/ LG
/ 1.
(6.11)
and (6.12)
The key observation is the following proposition which characterizes the irreducible positive energy representation of LG χ and Theorem 4.3.3 of [50]. The proof of Proposition 6.4 follows directly from the definition of LG χ in (6.8). Proposition 6.4. For any irreducible positive energy representation of LG χ , the pull ˜ back representation to L Z G˜ is an irreducible positive energy representation of L ZG (as classified in [50]) such that the center Z , as a subset of L G˜ under the canonical section s : Z → Zˆ , acts as multiplication by the character χ .
Z
Proposition 6.5 (Theorem 4.3.3 of [50]). Let (Hλ , π ) be an irreducible positive energy representation of L G˜ of level k and highest weight λ ∈ ∗k . For the distinguished discontinuous loop ζz associated to z ∈ Z , the conjugated representation γ → π(ζz−1 γ ζz ) of L G˜ on Hλ is an irreducible positive energy representation of L G˜ of level k and the ˜ highest weight z · λ ∈ ∗k , where z · λ denotes Z ⊂ Z (G)-action as in (6.3). Given a distinguished discontinuous loop ζz ∈ L Z G˜ associated to z ∈ Z , from Proposition 6.5, we know that the conjugated representation of (Hλ , π ) by ζz , denoted by ˜ (ζz )∗ Hλ , is equivalent to the irreducible positive energy module Hz·λ of L G. There is a character map ∗k −→ H om(Z , U (1)) given by λ → e2πiλ,· , where e2πiλ,· is the character on Z : h ∨ → e2πiλ,h
(6.13) ∨
for h ∨ ∈ ∨Z .
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Lemma 6.6 (cf. Lemma 7.1 in [50]). For any positive energy irreducible representation L G˜ at level k, the unique lift of Z ⊂ G˜ acts on Hλ by the character Hλ of h ∨ → e2πiλ,h
∨
for [h ∨ ] ∈ ∨Z /r∨ ∼ = Z. Denote by ∗k,χ the pre-image of χ ∈ H om(Z , U (1)) for the character map (6.13). Then ∗k is partitioned into different sectors labelled by χ ∈ H om(Z , U (1)): ∗k = ∗k,χ . χ ∈H om(Z ,U (1))
Note that the character map (6.13) factors through ∗k /Z , the orbit space of the Z -action on ∗k (this follows from Lemma 7.2 and the proof of Corollary 7.3 in [50]). We fix a choice of a representative λ for each Z -orbit Z · λ. The irreducible positive energy representations of LG χ are labelled by elements in the space of orbits for ∗k,χ /Z (cf. Theorem 6.1 and Corollary 7.3 in [50]), together with a character ρ ∈ H om(Z λ , U (1)) for an orbit Z · λ with a non-trivial stabilizer Z λ = {z ∈ Z |z · λ = λ}. Given an orbit Z · λ ∈ ∗k,χ /Z , ρ
and ρ ∈ H om(Z λ , U (1)), denote by H Z ·λ the irreducible positive energy module of ˜ LG χ . The pull-back representation, as a L G-module, through the compositions of maps L G˜
/ ˜ LZG
L0Gχ
/ LG χ
admits the following decomposition: ρ H Z ·λ ∼ =
Hλ ⊗ Cm λ ,
(6.14)
λ ∈Z ·λ
where Hλ is the irreducible positive energy module of L G˜ at level k with highest weight λ and m λ = 1 except for L(P S O(4n))χ ,− (cf. Remark 6.3) with Z · λ = {λ}, k is even in which case m λ = 2. Moreover, the group of discontinuous loops corresponding ρ ρ to elements in Z λ acts on H Z ·λ via the character ρ. Note that, if Z λ is non-trivial, H Z ·λ (ρ ∈ H om(Z λ , U (1))) have the same Virasoro character χk,λ (τ ). λ ∈Z ·λ
The appearance of the character ρ ∈ H om(Z λ , U (1)) in the representation of LG χ should be understood in terms of the Borel-Weil theory for loop groups (cf. [43]).
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Denote by Rk,χ (LG) the Abelian group generated by the irreducible positive energy ρ representations of LG χ . Denote by χk,Z λ the Kac-Peterson character corresponding ρ to the irreducible positive energy representation H Z ·λ of LG χ for Z · λ ∈ ∗k,χ , and ρ ρ ∈ H om(Z λ , U (1)). Then Rk,χ (LG) is an Abelian group generated by those χk,Z λ . 6.2. Multiplicative bundle gerbes. Multiplicative bundle gerbes on G have transgressive Dixmier-Douady class ([15]). For a compact, connected and simple Lie group G, ˜ Z), a level H 4 (BG, Z) ∼ = Z, in terms of the generators of H 4 (BG, Z) and H 4 (B G, 4 (B G, ˜ ˜ Z) is transgressive for G = G/Z , if and only if (cf. [19] and [41]) k∈Z∼ H = k λi(z) , λi(z) ∈ Z, 2
(6.15)
where {λi(z) }z∈Z are those special fundamental co-weights corresponding to elements in ˜ Z . Let m be the smallest positive integer such that all transgressive levels for G = G/Z is a multiple of m . We call m the multiplicative level of G. We let f be the smallest positive integer lying in the image of H 3 (G, Z) → ˜ Z); it is called the fundamental level of G in [50], see also [24]. Note that H 3 (G, the fundamental level for S O(3) is 2, and the multiplicative level of S O(3) is 4. We need to construct a G-equivariant bundle gerbe over G for a multiplicative level k ∈ m Z. Note that the multiplicative level m is always a multiple of the basic level b . Given k ∈ m Z and χ ∈ H om(Z , U (1)), there exists a canonical central extension of LG given by (6.8). Let P G be the space of smooth maps f : R → G such that θ → f (θ + 2π ) f (θ )−1 is constant. The map P G → G given by f → f (2π ) f (0)−1 defines a principal LGbundle over G. Then we can follow the construction in the proof of Proposition 2.3 to define a G-equivariant bundle gerbe over G. Proposition 6.7. Given a level k ∈ b Z and χ ∈ H om(Z , U (1)), the lifting bundle gerbe associated to the central extension LG χ as in (6.9) and the principal LG-bundle P G over G is a G-equivariant bundle gerbe over G, denoted by G(k,χ ),G , whose equivariant Dixmier-Douady class is determined by (k, χ ). Proof. It is easy to show that the lifting bundle gerbe associated to the central extension LG χ as in (6.9) and the principal LG-bundle P G over G is a G-equivariant bundle ˜ we get a central extension of L Z G˜ which is classified gerbe. Pull-back LG χ to L Z G, by the level k, and a cocycle in H 2 (Z , U (1)). From the exact sequence, 0 → E xt (G, U (1)) −→ HG3 (G, Z) −→ H 3 (G, Z), we know that the equivariant Dixmier-Douady class of G(k,χ ),G , as a class in HG3 (G, Z), is canonically determined by (k, χ ). We keep the notation Gk = Gk,G˜ for the bundle gerbe of level k on the simply connected Lie group G˜ whose equivariant Dixmier-Douady class is k times the generator ˜ Z). in H 3˜ (G, G
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6.3. G-equivariant bundle gerbe modules. Let k be the closed bi-invariant 3-form on G, k =
k k θ, [θ, θ ] = θ¯ , [θ¯ , θ¯ ], 12 12
where θ, θ¯ ∈ 1 (G, g) the left- and right- invariant Maurer-Cartan forms. Note that k has to be a multiple of the fundamental level f in order that k defines an integral cohomology class in H 3 (G, Z). Here we require that k is a multiple of the basic level b , note that b is a multiple of f (cf. [50]). Group-valued moment maps for quasi-Hamiltonian G-manifolds and the corresponding Hamiltonian LG-manifolds at level k have been studied also for compact, semi-simple, non-simply connected Lie groups ([1]). A quasi-Hamiltonian manifold (M, ω, µ) ˆ ω, and its Hamiltonian LG-manifold ( M, ˆ µ) ˆ at level k give rise to the following diagram: Mˆ
µˆ
π
M
/ Lg∗
(6.16)
H ol
µ
/ G,
where µˆ : Mˆ → Lg∗ is the moment map for the Hamiltonian LG-action. To be precise, ˆ ω, LG χ -equivariant moment map ( M, ˆ µ) ˆ is actually a Hamiltonian LG χ -manifold with µˆ : Mˆ −→ Lg∗ = Lg∗ × {k} → Lg∗ ⊕ R, and Lg∗ is again identified as the L 2p -connections on the principal G-bundle over S 1 . The proof of the following proposition is straightforward, see the proof of Theorem 3.5. Proposition 6.8. Given a level k ∈ b Z and χ ∈ H om(Z , U (1)), an LG χ -equivariant ˆ vector bundle E over a Hamiltonian LG-manifold M defines a G-equivariant bundle gerbe module of the G-equivariant bundle gerbe G(k,χ ),G . That means the corresponding quasi-Hamiltonian G-manifold is a generalized equivariant bundle gerbe D-brane of G(k,χ ),G . Let Q(k,χ ),G be the category of G-equivariant bundle gerbe modules of G(k,χ ),G . Then the quantization functor defined as in Definition 5.2 can be carried over to Q(k,χ ),G . The coadjoint orbits of the affine LG-action on Lg∗ through λ ∈ ∗k,χ provide examples in Q(k,χ ),G . Remark 6.9. Given a Riemann surface g,1 of genus g with only one boundary compoˆ g,1 nent (which is pointed by fixing a base point on the boundary), the moduli space M of flat connections on g,1 × G modulo those gauge transformations which are trivial on the boundary is only a Hamiltonian L 0 G-manifold at level k, for a transgressive level k. The holonomy map ˆ g,1 / 0 G M ˆ g,1 does not admit a LG-action. defines a quasi-Hamiltonian G manifold. But M
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A. L. Carey, B.-L. Wang
Given a Hamiltonian L 0 G-manifold Mˆ at level k with a free 0 G-action such that ˆ M/
0 G is a quasi-Hamiltonian G-manifold, we need to construct in a canonical way a Hamiltonian LG-manifold Mˆ # at level k, and an associated principal G-bundle over M, i.e. fill in the diagram, Mˆ #
µˆ #
G
M
/ Lg∗
G
/ G.
µ
Lemma 6.10. Given a Hamiltonian L 0 G-manifold Mˆ at level k with its quasi-Hamiltonian G-manifold µ : M → G, then the fiber product M˜ = M ×G G˜ is a quasi˜ ˜ Hamiltonian G-manifold and the corresponding Hamiltonian L G-manifold Mˆ # is also a Hamiltonian LG-manifold. ˜ Proof. Define G-action on M ×G G˜ via ˜ · m, g˜ g˜1 g˜ −1 ), g˜ · (m, g˜1 ) = (π(g) where π : G˜ → G is the covering map. Then the projection µ˜ : M˜ = M ×G G˜ → G˜ is ˜ ˜ µ) a G-equivariant map. M is a quasi-Hamiltonian G-manifold; it is easy to see that ( M, ˜ ˜ is a quasi-Hamiltonian G-manifold and the following diagram commutes: M˜ M
µ˜
/ G˜
(6.17)
π
/ G.
µ
˜ As G˜ is simply-connected, the corresponding Hamiltonian L G-manifold is given by the # ∗ ˜ ˆ fiber product M = M ×G˜ Lg : Mˆ # ˜ M
µˆ #
/ Lg∗
(6.18)
H ol µ˜
/ G, ˜
˜ ˜ Composing Diagram (6.17) where H ol : Lg∗ → G˜ is a universal G-bundle over G. and Diagram (6.18), we see that Mˆ # = M˜ ×G˜ Lg∗ is a Hamiltonian LG-manifold with LG-action given by the affine coadjoint action of LG on Lg∗ at level k: Mˆ #
µˆ #
G
G
M
/ Lg∗
µ
/ G.
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˜ and Z ⊂ L G, ˜ as conRemark 6.11. (1) Note that the center Z ⊂ G˜ acts trivially on M, stant loops, acts trivially on Mˆ # . Using the surjection L G˜ → L 0 G, a Hamiltonian ˜ at level k. LG-manifold Mˆ # at level k admits a Hamiltonian L G-action (2) Suppose Mˆ # is pre-quantizable as a Hamiltonian LG-manifold with a LG χ -equi# ˆ variant pre-quantization line L Mˆ # , then M is also pre-quantizable as a Hamiltonian ˜ ˜ L G-manifold and the L G-equivariant line bundle L Mˆ # on which the center Z ⊂ L G˜ acts via the character χ . This defines a natural map D πk,χ : Q(k,χ ),G −→ Qk,G˜ .
(6.19)
Notice that the coadjoint orbit of the affine coadjoint LG-action on Lg∗ consists of the {ζz |z ∈ Z }-orbit, where the distinguished discontinuous loops {ζz } become smooth loops in LG, the affine coadjoint action of these smooth loops on Lg∗ is exactly the Z -action defined by (6.3). We fix a representative λ in each Z -orbit Z · λ ⊂ ∗k . Given a Hamiltonian LG-manifold ( Mˆ # , µˆ # ) at level k with a LG χ -equivariant prequantization line L Mˆ # , from Lemma 6.10 and Remark 6.11, we know that a Hamiltonian ˜ LG-manifold ( Mˆ # , µˆ # ) at level k is also a Hamiltonian L G-manifold at level k and its ˜ ˜ pre-quantization line bundle L Mˆ # is L G-equivariant. The Hamiltonian L G-reduction at λ # ˜ λ, Mˆ λ, := (µˆ # )−1 (L G˜ · λ)/L G˜ ∼ = (µˆ # )−1 (λ)/(L G) G˜
(6.20)
with its pre-quantization line bundle is given by Lλ,G˜ := L Mˆ # |(µˆ # )−1 (λ) ×( ˜ C(∗λ,1) , L G) λ
where the action of L G˜ λ on C(∗λ,1) ∼ = C is given by the weight (λ, 1); notice that for ∗ L G˜ λ . λ ∈ k,χ , the weight (+λ, 1) agrees with the character χ when restricted to Z ⊂ ∗ The Hamiltonian LG-reduction at λ ∈ k,χ , # := (µˆ # )−1 (LG · λ)/LG ∼ Mˆ λ,G = (µˆ # )−1 (λ)/(LG)λ ,
doesn’t depend on the choice of λ in its Z -orbit. Here (LG)λ denotes the isotropic group of LG-action at λ. The pre-quantization line bundle over Mˆ λ# depends on a choice of a character ρ ∈ H om(Z λ , U (1)) if Z λ = {z ∈ Z |z · λ = λ} is non-trivial. For each character ρ ∈ H om(Z λ , U (1)), the corresponding pre-quantiza# is given by tion line bundle over Mˆ λ,G ρ
Lλ,G := L Mˆ # |(µˆ # )−1 (λ) ×( LG)λ C(∗λ,ρ −1 ,1) , where ∗λ is the dominant weight of the irreducible representation of G dual to the one with weight λ, the action of LG λ on C(∗λ,ρ −1 ,1) ∼ = C is determined by the action of ( L 0 G)λ on C via the weight (∗λ, 1) and the character ρ −1 through the exact sequence 1 → ( L 0 G)λ −→ ( LG)λ −→ Z λ → 1,
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A. L. Carey, B.-L. Wang
where the quotient group LG λ / L 0 G λ is identified with the group of distinguished loops ρ defined by (6.6) corresponding to elements in the stabilizer Z λ . We denote by ∂/λ,G the ρ Spin c Dirac operator associated to the line bundle Lλ . It is understood that when Z λ is trivial, then ρ
ρ
Lλ,G = L1λ,G and ∂/λ,G = ∂/1λ,G . ρ
The following proposition identifies the spaces of sections of the line bundles Lλ,G ρ and Lλ,G˜ , denoted by (Lλ,G ) and (Lλ,G˜ ) respectively. ρ
Proposition 6.12. Given λ ∈ ∗k,χ , the space of sections of the line bundle Lλ,G consists of sections for the line bundle L Mˆ # |(µˆ # )−1 (λ) over (µˆ # )−1 (λ) with weight (λ, ρ, 1) for the action of ( LG)λ ; and the space of sections of the line bundle L ˜ consists of sections of λ,G
the line bundle L Mˆ # |(µˆ # )−1 (λ) Moverover,
˜ over (µˆ # )−1 (λ) with weight (λ, 1) for the action of ( L G)
(Lλ,G˜ ) ∼ =
ρ∈H om(Z λ ,U (1))
λ.
ρ
(Lλ,G ).
Proof. We know that Mˆ #
has |Z λ |-components, on which Z λ acts transitively via the group of distinguished discontinuous loops. Each component of Mˆ # is diffeomorphic λ,G˜
λ,G˜
# . to Mˆ λ,G
˜ λ -equivariant. From The line bundle L Mˆ # |(µˆ # )−1 (λ) is ( LG)λ -equivariant and ( L G) the definition of Lλ,G˜ , we can see that the space of sections of the line bundle Lλ,G˜ consists of sections of the line bundle L Mˆ # |(µˆ # )−1 (λ) over (µˆ # )−1 (λ) with weight (λ, 1) for ˜ λ ; and similarly the space of sections of the line bundle Lρ consists the action of ( L G) λ,G
of sections for the line bundle L Mˆ # |(µˆ # )−1 (λ) over (µˆ # )−1 (λ) with weight (λ, ρ, 1) for the action of ( LG)λ . There is a Z λ -covering map # # −→ Mˆ λ,G , π : Mˆ λ, G˜ ρ
ρ
and there is a bundle isomorphism between Lλ,G˜ and π ∗ Lλ,G . For a section s ∈ (Lλ,G ), ρ the linear map s → π ∗ s identifies (Lλ,G ) with a subspace of (Lλ,G˜ ) such that
ρ (Lλ,G ). (Lλ,G˜ ) ∼ = ρ∈H om(Z λ ,U (1))
Definition 6.13. Given a G-equivariant bundle gerbe module ( Mˆ # , E) ∈ Q(k,χ ),G , we define the quantization of ( Mˆ # , E) to be ˆ E) = χ(k,χ ),G ( M,
Z ·λ∈∗k,χ /Z
ρ∈H om(Z λ ,U (1))
ρ
ρ
# )χ I ndex(/ ∂ λ,G ⊗ E, Mˆ λ,G k,Z ·λ ∈ Rk,χ (LG).
This gives rise to a quantization functor χ(k,χ ),G : Q(k,χ ),G → Rk,χ (LG).
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6.4. The fusion category of bundle gerbe modules. Now recall that our fusion product in the simply connected case uses a pre-quantizable line bundle over the moduli space ˆ 0,3 . For a non-simply connected Lie group, the moduli space M ˆ 0,3 is quantizable M for any transgressive level k. Proposition 6.14. The G-equivariant bundle gerbe G(k,χ ),G over G is a G-equivariant multiplicative bundle gerbe if k is transgressive and χ is the trivial homomorphism. Proof. The first statement holds from the main result of [15]. To be G-equivariant and multiplicative, the central extension LG χ of LG has to be G-equivariant as a principal U (1)-bundle over LG = G × G with G-action on G given by conjugation, and under the face operators from πi : G × G → G, where π0 (g1 , g2 ) = g2 , π1 (g1 , g2 ) = g1 g2 and π2 (g1 , g2 ) = g1 for (g1 , g2 ) ∈ G × G, there is a G-equivariant stable isomorphism π0∗ G(k,χ ),G ⊗ π2∗ G(k,χ ),G −→ π1∗ G(k,χ ),G . These conditions hold if χ ∈ H om(Z , U (1)) is the trivial homomorphism.
This proposition determines the conditions under which we may obtain the fusion category of bundle gerbe modules in this non-simply connected situation. From now on in this subsection, we assume that the level k is multiplicative for G, i.e., k ∈ m Z; this excludes the case of Q(k,χ ,−),G for G = P S O(4n), as χ has to be non-trivial for LG χ ,− . We can define the fusion of two quasi-Hamiltonian G-manifolds µi : (Mi , ωi ) → G (i = 1, 2) as (M1 , ω1 , µ1 ) (M2 , ω2 , µ2 ) = (M1 × M2 , µ1 · µ2 ),
(6.21)
1 with the 2-form ω1 + ω2 + µ∗1 θ, µ∗2 θ¯ . Then (M1 × M2 , µ1 · µ2 ) is again a quasi2 Hamiltonian G-manifold, and the corresponding Hamiltonian LG-manifold at level k is given by ∗ M 1 M2 = (M1 × M2 ) ×G Lg ,
(6.22)
˜ Lg∗ → G˜ → G. where the universal principal G-bundle over G factors through G: ˆ g,1 at any multiplicative level k, the proof For the Hamiltonian L 0 G-manifold M ˆ # admits a LG 1 pre-quantization of Proposition 4.1 can be adapted to show that M g,1 line bundle. The key point in the proof is the Segal-Witten reciprocity property for transgressive central extensions (we omit the details). The category of G-equivariant bundle gerbe modules for G(k,1),G , when k is transgressive for G, admits the fusion object given by the fiber product ˆ # = M ˆ 0,3 ×G 3 G˜ 3 , M 0,3 3
1 -equivariant pre-quanwhich is a Hamiltonian (LG)3 -manifold at level k with a (LG) tization line bundle L0,3 . The fact can be verified that M 1 M2 defined by (6.22) is diffeomorphic to the Hamiltonian quotient ˆ # //diag(LG × LG). ( Mˆ 1 × Mˆ 2 ) × M 0,3
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Moreover, if Mˆ 1 and Mˆ 2 admit LG 1 -equivariant pre-quantization line bundles L1 and L2 respectively, then
2 (L1 × L2 ) × L0,3 //diag(LG) 1 defines a LG 1 -equivariant pre-quantization line bundle over M 1 M2 . Hence, the fusion category of rank one G-equivariant bundle gerbe modules of G(k,1),G is well defined. We denote this fusion category by (Q(k,1),G , G ). We can apply the quantization functor χ(k,1),G : (Q(k,1),G , G ) −→ Rk,1 (LG) to define a fusion product on Rk,1 (LG). Note that irreducible positive energy representations in Rk,1 (LG) are labelled by {(Z · λ, ρλ )|Z · λ ⊂ ∗k,1 , ρλ ∈ H om(Z λ , U (1))}. Definition 6.15. We define the fusion coefficient for Rk,1 (LG) as: (ρλ ,ρµ ,ρν ) (Z ·ν,ρ ) ˆ # (G, ∗λ, ∗µ, ν) , ,M N(Z ·λ,ρλν ),(Z ·µ,ρµ ) := I ndex ∂/(∗λ,∗µ,ν),G 0,3
(6.23)
ˆ# ˆ # (G, ∗λ, ∗µ, ν) denotes the Hamiltonian (LG)3 -reduction of M where M 0,3 0,3 at (ρ ,ρ ,ρ )
λ µ ν is the corresponding Spin c Dirac operator. (∗λ, ∗µ, ν), and ∂/(∗λ,∗µ,ν),G
ρµ ρλ (Z ·ν,ρν ) ρν Theorem 6.16. χk,Z (Z ·ν,ρν ) N(Z ·λ,ρλ ),(Z ·µ,ρµ ) χk,Z ·ν defines a fusion ring ·λ ∗χk,Z ·µ = structure on Rk,1 (LG) with the unit given by χk,Z ·0 , the representation corresponding to the Z -orbit through 0. Proof. The fusion product defined on the Q(k,1),G is commutative and associative modulo LG-equivariant symplectomorphisms and equivalence of LG 1 -equivariant line bundles. This implies that the fusion product on Rk,1 (LG) is commutative and associative.
G 1 ×U (1) C, The unit in Q(k,1),G is given by G with its pre-quantization line bundle the quantization functor χ(k,1),G : Q(k,1),G −→ Rk,1 (LG) sends G to χk,Z ·0 .
Proposition 6.17. If Z · λ, Z · µ and Z · ν are free Z -orbits, then z·ν ·ν Nλ,µ . N ZZ·λ,Z ·µ = z∈Z ν } for R (L G) ˜ satisfy the following symProof. Note that the Verlinde coefficients {Nλ,µ k metry under the action of Z : z 2 ·ν ν Nzz11·λ,z = Nλ,µ 2 ·µ
for any z 1 , z 2 ∈ Z . This is due to the fact that the moduli spaces for calculating the ν and N z 1 z 2 ·ν ˜ Verlinde coefficients Nλ,µ z 1 ·λ,z 2 ·µ for Rk (L G) are identical: ˆ # (G, ˆ # (G, ˜ ∗λ, ∗µ, ν) ∼ ˜ ∗(z 1 · λ), ∗(z 2 · µ), z 1 z 2 · ν) M =M 0,3 0,3
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from the holonomy descriptions of these moduli spaces. These facts imply that ˆ # (G, ˆ # (G, ∗λ, ∗µ, ν) ∼ ˜ ∗λ, ∗µ, z · ν), M M = 0,3 0,3 z∈Z
as symplectic manifolds, and their corresponding pre-quantization line bundles are also equivalent for free Z -actions on Z · λ, Z · µ and Z · ν. Hence, we have z·ν ·ν N ZZ·λ,Z Nλ,µ . ·µ = z∈Z
Remark 6.18. The fusion category (Q(k,1),G , G ) is actually a braided tensor category, see [7] for a definition of a braided tensor category, where the braiding isomorphism for two Hamiltonian LG-manifolds Mˆ 1 and Mˆ 2 , Mˆ 1 Mˆ 2 −→ Mˆ 2 Mˆ 1 , is induced by a diffeomorphism of 0,3 exchanging the two incoming boundaries. Applying the conformal model for 0,3 Pw,q,q = {z ∈ C||q| ≤ |z| ≤ 1, |z − w| ≥ |q|} with boundary points 1, q and w + q, where 0 < |w| < 1, and 0 < |q| < |w| − |q| < 1 − 2|q|. Then the conformal model for 0,3 with two incoming boundaries exchanged is given by P−w,q,q . Note that Pw,q,q and P−w,q,q are connected by the path Peiθ w,q,q for θ ∈ [0, π ]. The Pentagon axiom, Triangle axiom and Hexagon axioms follow from the multiplicative property of the equivariant bundle gerbe G(k,1),G . This braiding isomorphism is important to determine the fusion coefficients for Rk,1 (LG) involving Z -orbits with non-trivial stabilizer. To illustrate our result, we end this section by a detailed study for G = S O(3) and G = SU (3)/Z3 . 6.5. An example for G = S O(3). Note that S O(3) = SU (2)/Z2 , the basic level is 2, and the level is transgressive if and only if it is a multiple of 4. Given a class (k, χ ) ∈ 2Z ⊕ (Z2 , U (1)), where χ = ±1 ∈ Z2 , we have the corresponding equivariant bundle gerbes Gk,±1 for k = 4n or k = 4n + 2 (n > 0). We first give a complete classification of all irreducible positive energy representations of L S O(3)χ at level k ∈ 2Z and χ = ±1. For k = 4n and χ = +1, the irreducible positive energy representations of L S O(3)+1 are labelled by Z2 = Z (SU (2))-orbits in the space of level k dominant weights. We instead use the half-integers (half weights) j = 0, 1/2, 1, 3/2, . . . , 2n −1/2, 2n to label level k dominant weights of L SU (2). Denote by H0 , H1/2 , H1 , . . . , H2n−1/2 , H2n the corresponding irreducible positive energy representations of L SU (2) at level 4n. These representations of L SU (2) can be obtained by (geometric) quantization of equivariant bundle gerbe D-branes given by conjugacy classes labelled by those half-integer
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representations of SU (2), or equivalently, quantization of equivariant bundle gerbe modules of the corresponding affine coadjoint L SU (2)-orbits at level 4n. Then the irreducible positive energy representations of L S O(3)+1 , as L SU (2)-modules, are given by H0 ⊕ H2n , H1 ⊕ H2n−1 , H2 ⊕ H2n−2 , .. . Hn−1 ⊕ Hn+1 , Hn± , where the spin n is the fixed point of the Z2 -action: j → 2n − j, and Hn± ∼ = Hn as a L SU (2)-module, with the group of discontinuous loops corresponding to Z2 acting via the character ±1. These representations can be thought of as quantization of the projection of Z2 -orbits of those conjugacy classes in SU (2) with integer spin weights. It is straightforward to verify that R4n,+1 (L S O(3)) admits a fusion product, from which we obtain the non-diagonal modular invariant (diagonal modular invariant for an extension of the chiral algebra by Z , Z4n,+1 =
k/4−1
|χk, j + χk,k/2− j |2 + 2|χk,n |2 .
j=0
This agrees with the formula from fixed point resolution for simple current extensions in [13,47]. For k = 4n and χ = −1, the irreducible positive energy representations of L S O(3)−1 , as L SU (2)-modules, are given by H1/2 ⊕ H2n−1/2 , H3/2 ⊕ H2n−3/2 , H5/2 ⊕ H2n−5/2 , .. .
Hn−1/2 ⊕ Hn+1/2 . Similarly, for k = 4n + 2 and χ = +1, the irreducible positive energy representations of L S O(3)+1 , as L SU (2)-modules, are given by H0 ⊕ H2n+1 , H1 ⊕ H2n , H2 ⊕ H2n−1 , .. . Hn−1 ⊕ Hn+2 , Hn ⊕ Hn+1 .
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For k = 4n+2 and χ = −1, the irreducible positive energy representations of L S O(3)−1 , as L SU (2)-modules, are given by H1/2 ⊕ H2n+1/2 , H3/2 ⊕ H2n−1/2 , H5/2 ⊕ H2n−3/2 , .. . Hn−1/2 ⊕ Hn+3/2 , ± Hn+1/2 , ± ∼ where the spin n+1/2 is the fixed point of the Z2 -action: j → 2n+1− j, Hn+1/2 = Hn+1/2 as a L SU (2)-module, with the group of discontinuous loops corresponding to Z2 acting via the character ±1. For each k ∈ 2Z and χ ∈ Z2 ∼ = H om(Z2 , U (1)), the quantization functor
χ(k,χ ),S O(3) : Q(k,χ ),S O(3) −→ Rk,χ (L S O(3)), from the category Q(k,χ ),S O(3) of S O(3)-equivariant generalized bundle gerbe modules of Gk,χ over S O(3) to Rk,χ (L S O(3)), is surjective. Among the four cases discussed above, Q(k,χ ),S O(3) admits a fusion product structure if k = 4n and χ = +1, and the corresponding quantization functor χ(4n,+1),S O(3) : Q(4n,+1),S O(3) → R4n,+1 (L S O(3)) preserves the fusion products. 6.6. An example for G = SU (3)/Z3 . Denote by {α1 , α2 }, and {α1∨ , α2∨ } the simple roots and the simple co-roots of SU (3) respectively. The fundamental weights and co-weights are denoted by ∨ {λ1 , λ2 }, and {λ∨ 1 , λ2 }
respectively. Then we know that the highest root is given by α1 + α2 and ∨ ∨ α1 + α2∨ = λ∨ 1 + λ2 , ∨ α1∨ − α2∨ = 3(λ∨ 1 − λ2 ),
which gives the isomorphism ∨ ∼ ∨ ∨ ∨ ∨ ∼ ∨ w /r = Z(λ1 − λ2 )/3Z(λ1 − λ2 ) = Z3 .
The set of dominant weights at level ∗k is given by {k1 λ1 + k2 λ2 |ki ≥ 0, k1 + k2 ≤ k},
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with the action of Z3 generated by k1 λ1 + k2 λ2 → (k − k1 − k2 )λ1 + k1 λ2 . It is easy to see that the character map ∗k → H om(Z3 , U (1)) is induced by k1 λ1 + k2 λ2 → (k1 − k2 )(mod 3), and Z3 -action admits a fixed point if and only if k ∈ 3Z with the fixed point given by k k 3 λ1 + 3 λ2 . For G = SU (3)/Z3 , as the multiplicative level for SU (3)/Z3 is 3, we know that the transgressive level for SU (3)/Z3 is given by k ∈ 3Z, we know that the multiplicative bundle gerbes over G are classified by their Dixmier-Douady classes: the level k ∈ 3Z. We note that b = 1 for SU (3)/Z3 . Hence, the equivariant bundle gerbe G(k,χ ),SU (3)/Z3 exists for any level k ∈ Z and χ ∈ H om(Z3 , U (1)). Here we only consider the transgressive levels. Given k ∈ 3Z and χ ∈ Z3 ∼ = H om(Z3 , U (1)), we have the corresponding quantization functor: χ(k/3,χ ),SU (3)/Z3 : Q(k/3,χ ),SU (3)/Z3 −→ Rk,χ (L(SU (3)/Z3 )). SU (3) of the highDenote by H(k1 ,k2 ) the positive energy irreducible representation of L est weight k1 λ1 + k2 λ2 ∈ ∗k . Then Rk,χ (L(SU (3)/Z3 )) is generated by H(k1 ,k2 ) ⊕ H(k−k1 −k2 ,k1 ) ⊕ H(k2 ,k−k1 −k2 ) , k1 −k2
2πi 3 ∨ for k1 λ1 +k2 λ2 with trivial stabilizer. for k1 λ1 +k2 λ2 ∈ ∗k and χ ([λ∨ 1 −λ2 ]) = e For χ = 1 and k1 = k2 = k/3, there are three additional representations: ρ
ρ
ρ
0 1 2 H(k/3,k/3) , H(k/3,k/3) , H(k/3,k/3) ,
SU (3)-modules, with the group of discontinuous which are equivalent to H(k/3,k/3) as L loops corresponding to Z3 acting via the character ρi ∈ H om(Z3 , U (1)). We give a complete list for k = 3 and k = 6 as follows: ∨ (1) For k = 3, if χ ([λ∨ 1 − λ2 ]) = 1, then R3,χ (L(SU (3)/Z3 )) is generated by
H(0,0) ⊕ H(3,0) ⊕ H(0,3) , ρ
ρ
ρ
0 1 2 H(1,1) , H(1,1) , H(1,1) ,
ρi ∼ as L SU (3)-modules, where (1, 1) is the fixed point of the Z3 -action, H(1,1) = H(1,1) as a L SU (3)-module, with the group of discontinuous loops corresponding to ∨ 2πi/3 , then R Z2 acting via the character ρi ; if χ ([λ∨ 3,χ (L(SU (3)/Z3 )) 1 − λ2 ]) = e is generated by
H(1,0) ⊕ H(2,1) ⊕ H(0,2) , ∨ 4πi/3 , then R as L SU (3)-modules; if χ ([λ∨ 3,χ (L(SU (3)/Z3 )) is gen1 − λ2 ]) = e erated by
H(2,0) ⊕ H(1,2) ⊕ H(0,1) ,
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as L SU (3)-modules. Note that the Verlinde ring R3,+1 (L(SU (3)/Z3 )) gives rise to a modular invariant for SU (3)/Z3 at level 3 (cf. [9]): Z(3,+1),SU (3)/Z3 = |χ3,(0,0) + χ3,(3,0) + χ3,(0,3) |2 + 3|χ3,(1,1) |2 . ∨ (2) For k = 6, if χ ([λ∨ 1 − λ2 ]) = 1, then R6,χ (L(SU (3)/Z3 )) is generated by
H(0,0) ⊕ H(6,0) ⊕ H(0,6) , H(1,1) ⊕ H(4,1) ⊕ H(1,4) , ρ
ρ
ρ
0 1 2 , H(2,2) , H(2,2) , H(2,2)
H(3,3) ⊕ H(0,3) ⊕ H(3,0) , ρi ∼ as L SU (3)-modules, where (2, 2) is the fixed point of the Z3 -action, H(2,2) = H(2,2) as a L SU (3)-module, with the group of discontinuous loops corresponding to ∨ 2πi/3 , then R Z2 acting via the character ρi ; if χ ([λ∨ 6,χ (L(SU (3)/Z3 )) 1 − λ2 ]) = e is generated by
H(1,0) ⊕ H(5,1) ⊕ H(0,5) , H(2,1) ⊕ H(3,2) ⊕ H(1,3) , H(4,0) ⊕ H(2,4) ⊕ H(0,2) , ∨ 4πi/3 , then R as L SU (3)-modules; if χ ([λ∨ 6,χ (L(SU (3)/Z3 )) is gen1 − λ2 ]) = e erated by
H(2,0) ⊕ H(4,2) ⊕ H(0,4) , H(3,1) ⊕ H(2,4) ⊕ H(1,2) , H(5,0) ⊕ H(1,5) ⊕ H(0,1) , as L SU (3)-modules. Note that only R6,+1 (L(SU (3)/Z3 )) admits a ring structure, which gives rise to a modular invariant for SU (3)/Z3 at level 6 (cf. [6]) Z(6,+1),SU (3)/Z3 = |χ6,(0,0) + χ6,(6,0) + χ6,(0,6) |2 + |χ6,(1,1) + χ6,(4,1) + χ6,(1,4) |2 +|χ6,(3,3) + χ6,(0,3) + χ6,(3,0) |2 + 3|χ6,(2,2) |2 . Acknowledgement. The authors acknowledge the support of the Australian Research Council. BLW thanks Valerio Toledano Laredo for stimulating e-mail correspondence. ALC thanks the Erwin Schrodinger Institute for their support.
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Commun. Math. Phys. 277, 627–641 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0392-2
Communications in
Mathematical Physics
Renormalization: A Number Theoretical Model Bertfried Fauser Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22-26, D-04103 Leipzig, Germany. E-mail: [email protected] Received: 23 June 2006 / Accepted: 29 June 2007 Published online: 24 November 2007 – © Springer-Verlag 2007
Abstract: We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals. Contents 1.
Dirichlet Convolution Ring of Arithmetic Functions . . . . . . 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Multiplicativity versus complete multiplicativity . . . . . . 1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Products and Coproducts Related to Dirichlet Convolution . . . 2.1 Multiplicativity of the coproducts . . . . . . . . . . . . . . 3. Hopf Gebra: Multiplicativity Versus Complete Multiplicativity 3.1 Plan A: The modified crossing . . . . . . . . . . . . . . . 3.2 Plan B: Unrenormalization . . . . . . . . . . . . . . . . . 3.3 The co-ring structure . . . . . . . . . . . . . . . . . . . . 3.4 Coping with overcounting : renormalization . . . . . . . . 4. Taming Multiplicativity . . . . . . . . . . . . . . . . . . . . . Appendix A. Some Facts about Dirichlet and Bell Series . . . . . . A.1 Characterizations of complete multiplicativity . . . . . . . A.2 Groups and subgroups of Dirichlet convolution . . . . . . Appendix B. Densities of Generators . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Dirichlet Convolution Ring of Arithmetic Functions 1.1. Definitions. In this section we recall a few well know facts about formal Dirichlet series and the associated convolution ring of Dirichlet functions [1,4]. An arithmetic function is a map f : N → C. Equivalently we can consider integer indexed sequences of complex numbers. It is convenient to introduce formal generating functions to encode this information in a more compact form f (s) :=
f (n) , ns n≥1
s = σ + i t ∈ C,
(1-1)
where the formal complex parameter is traditionally written as s. No confusion should arise between the series elements f (n) and the generating function f (s) formally denoted in the same way. A ring structure is imposed in the obvious manner: Definition 1.1. The Dirichlet convolution ring of arithmetic functions is defined on the set of arithmetic functions as ( f + g)(s) :=
f (n) + g(n) , ns n≥1
f (d) · g(n/d) , ( f g)(s) := ns
(1-2)
n≥1 d|n
where
d|n
is the sum over all divisors d of n.
The component-wise addition imposes a module structure on the arithmetic functions, and the convolution product is actually the point-wise product of the generating functions f (s) · g(s) as is easily seen. Furthermore the product is commutative, associative, and unital with unit u(s) := n≥1 δn,1 n −s , where δn,1 is the Kronecker delta symbol. If f (1) = 0, then a unique inverse Dirichlet generating function exists w.r.t. the convolution product f f −1 = u = f −1 f, n = 1 : f −1 (1) = 1/ f (1), 1 n n > 1 : f −1 (n) = −1 f −1 ( ) f (d). f (1) d|n d
(1-3)
d
The invertible arithmetic functions form a group: f u = f = u f, ( f g)−1 = g −1 f −1 , due to the associativity of the convolution.
(1-4)
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1.2. Multiplicativity versus complete multiplicativity. Two integers n, m are called relatively prime if their greatest common divisor gcd(n, m) = (n, m) is 1, hence if they have no prime factor in common. Many important number theoretical functions enjoy a weakened homomorphism property, called multiplicativity. Definition 1.2. An arithmetic function f is called complete multiplicative if f (n · m) = f (n) · f (m) ∀n, m.
(1-5)
An arithmetic function f is called multiplicative if f (n · m) = f (n) · f (m) ∀n, m with (n, m) = 1.
(1-6)
Hence multiplicative functions fail in general to be homomorphisms of the multiplicative structure of the natural numbers iff the product has a nontrivial common prime number content (n, m) = k, such that n = k · n and m = k · m . We may call k the overlapping or meet part of n, m. Actually gcd and lcm form a distributive lattice on the integers.
1.3. Examples. We give some examples of arithmetic functions, among them multiplicative, complete multiplicative and non-multiplicative ones, which all play important roles in number theory. Let n = piri , m = pisi and define ν to be the function ν(n) = 2 ri . One has ν(n · m) = ν piri +si = 2 (ri +si ) = 2( ri )+( si ) =2
ri
2
si
= ν(n) ν(m).
Hence ν is a homomorphism or complete multiplicative function. The Möbius function is defined as ⎧ n=1 ⎨ 1 0 n contains a square µ(n) = . k ⎩ (−1)k n = i=1 pi , k distinct primes
(1-7)
(1-8)
The sequence of integer values of the Möbius function is a random-looking list of ±1, 0 entries: n 1 2 3 4 5 6 7 8 ... . µ(n) 1 −1 −1 0 −1 1 −1 0 . . .
(1-9)
Another interesting arithmetic function is the Euler totient function, which counts the number of relative prime numbers d having (d, n) = 1 smaller than n. Using # for cardinality it reads φ = #{d ∈ N; d < n, (d, n) = 1}: n 1 2 3 4 5 6 7 8 ... . φ(n) 1 1 2 2 4 2 6 4 . . .
(1-10)
Introducing the arithmetic function N (n) = n, ∀n one finds φ(n) = (µ N )(n) = n p|n (1 − 1p ). The Möbius and Euler totient functions are multiplicative, but not complete multiplicative.
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A further example of a non multiplicative function is the von Mangoldt function:
log p if n = p m , m ≥ 1, p a prime (n) = . (1-11) 0 otherwise (including 1) Tabulated this reads n 1 2 3 4 5 6 7 8 ... . (n) 0 log 2 log 3 log 2 log 5 0 log 7 log 2 . . .
(1-12)
The importance of the von Mangoldt function stems from the fact that it encodes the derivation with respect to the formal parameter s of a Dirichlet generating function in terms of the convolution product. We use d|n (d) = log n to show this: ∂ ∂ f (n) n −s = f (n)(− log n)n −s . f (s) = ∂s ∂s n≥1
(1-13)
n≥1
In particular one obtains for the Riemann zeta function ζ −1 = µ, ζ (n) = 1 ∀n the formula −
∂ ζ (s) = − log ζ (s) = (s). ζ (s) ∂s
(1-14)
The von Mangoldt function appears in the Selberg formula [23], which allows one to embark on an ‘elementary’, that is nonanalytic, proof of the prime number theorem. 2. Products and Coproducts Related to Dirichlet Convolution In previous work, we studied extensively the Dirichlet Hopf algebra of arithmetics [12]. We extract from that work the two coproducts needed for the present purpose. We dualize the (semi)ring structure1 of the natural numbers (N, +, ·) using the Kronecker duality written as a scalar product | : N × N → Z2 , n | m = δn,m . Definition 2.3. The coproduct of addition is defined as + (n) := n1 ⊕ n2 n 1 +n 2 =n
= n (1) ⊕ n (2) ,
(2-1)
and the coproduct of multiplication is defined as n · (n) := n1 × n2 = d× d n ·n =n 1
2
= n [1] × n [2] .
d|n
(2-2)
We introduced Sweedler indices and the Brouder-Schmitt convention [3] to denote Sweedler indices of different coproducts by different parentheses. 1 We will later on always complete the natural numbers à la Grothendieck to a group, the integers, hence ‘ring’ will be a posteriori justified.
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Our coproducts allow us to define convolution algebras on the (set of coefficients of) arithmetic functions f : N → C together with the product µ in C. If the codomain of such a function is in the nonnegative integers, it is an endomorphism of C under a suitable identification of N in C. The process of extending the convolution of (set) endomorphisms on N to (set) homomorphisms hom(N, C) is subtle, since ring extensions have to be considered. In our formal treatment we do not care about this. Definition 2.4. A convolution algebra Conv(µ, ) is defined on homomorphisms f, g ∈ hom(N, C) as ( f + g)(n) = ‘ + ( f × g) + (n) = f (n) + g(n), n f (n) · g ( f g)(n) = µ( f × g) (n) = d
(2-3)
d|n
for the addition ‘+ the product µ and coproducts + respectively · . It is easy to show the following Proposition 2.5. Conv(+, + ) is biassociative, biunital, bicommutative with antipode S+ : N → Z given by S+ (n) = −n Note that the antipode is Z valued forcing us to extend the codomain of the homomorphisms at least to Z. We introduce the Hadamard product . : hom(N, C) × hom(N, C) → hom(N, C), that is the coefficient-wise product of Dirichlet series, as ( f.g)(s) = n≥1 f (n) · g(n)n −1 to be able to state the Proposition 2.6. Conv(·, · ) is biassociative, biunital, bicommutative with antipode S· : N → Z given by S· (n) = (N .µ)(n) = n · µ(n) or alternatively written as generating function S· (s) = µ(s − 1). While the first statements are almost trivial, the antipode can be derived as a group inverse using a recursion argument. Tabulated it reads n 1 2 3 4 5 6 7 8 ... , S· (n) 1 −2 −3 0 −5 6 −7 0 . . .
(2-4)
which should be compared with the table (1-9). The coproduct of multiplication models exactly split arguments in the Dirichlet convolution. In this case the Hopf algebraic version acts directly on the elements of the series representation of the arithmetic functions. The remarkable fact is that this coproduct can be obtained from an almost trivial dualization of multiplication of integers. The coproduct of addition will come into play later. We want to make this duality explicit, using the Kronecker pairing n | m = δn,m , n + m | k = n ⊕ m | + (k) = n | k(1) m | k(2) k1 × k2 = k(1) ⊕ k(2) , ⇔ + (k) =
(2-5)
k1 +k2 =k
and for the coproduct of multiplication one has n · m | k = n × m | · (k) = n | k[1] m | k[2] n ⇔ · (k) = k1 × k2 = d × = k[1] × k[2] . d k1 ·k2 =k
d|n
(2-6)
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B. Fauser
We close this discussion by exhibiting the primitive elements with respect to coaddition and comultiplication. A (1, 1)-primitive element (or simply primitive element) p is defined satisfying the relation ( p) = p ⊗ 1 + 1 ⊗ p. Using the particular monoidal structure, i.e. direct sums for addition, cartesian product for multiplication, and the respective units for addition and multiplication, we find as an easy consequence of the definitions: Corollary 2.7. With respect to the coproduct of addition + , 1 is the only primitive element and N is additively generated by 1. ∞ the set of all Corollary 2.8. With respect to the coproduct of multiplication · , { pi }i=1 prime numbers represents all primitive elements and N is multiplicatively generated by these primes.
This poses the opportunity to introduce two gradings on N turning the integers into a graded set, first by setting N = ⊕n∈N 1n , where every number represents its own grade. Addition is a graded map (binary ‘product’) under this grading. Now let P be the set of all prime numbers and Pk the set of all integers having exactly k prime factors (including multiplicities). Let P0 = 1. The grading suggested by the multiplicative structures is defined as: N = ⊕i≥0 Pi .
(2-7)
This regrouping will have a great influence on how densities or the asymptotic behaviour of Dirichlet arithmetic functions have to be considered, see Appendix B. For a detailed discussion of the algebraic aspects, including Hopf algebra cohomology, see [12]. 2.1. Multiplicativity of the coproducts. A remarkable fact is the following Proposition 2.9. The coproduct of multiplication · is a multiplicative function. Proof. First consider relative prime numbers pr , q s , pr · q s . · ( pr · q s ) = d× d r s
(2-8)
d| p ·q
Since pr | q s = 1, from which follows d | pr · q s = a | pr · b | q s , we obtain · ( pr · q s ) =
a| pr b|q s
=
pr · q s = pl q k × pr −l q s−k a·b r
a·b×
r
s
l=0 k=0
pl × pr −l
l=0 · r
s
q k × q s−k =
k=0 ·
c| pr
c×
pr qs d× c d s d|q
= ( p ) (q ). ·
s
(2-9)
is not complete multiplicative due to · (4) = 1 × 4 + 2 × 2 + 4 × 1 · (2) · (2) = (1 × 2 + 2 × 1)2 = 1 × 4 + 2 × 2 + 2 × 2 + 4 × 1,
which completes the proof.
(2-10)
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The lack of complete multiplicativity of the coproduct map spoils a major axiom of Hopf algebra theory, namely the homomorphism axiom µ(n × m) = (µ ⊗ µ)(Id ⊗ sw ⊗ Id)( ⊗ )(n × m),
(2-11)
which fails to hold(!) in the present case, but is only true as a multiplicative relation for (n, m) = 1. The multiplicative convolution, despite being bicommutative, biassociative, biunital, and having a nice antipode, is alas not a Hopf algebra. 3. Hopf Gebra: Multiplicativity Versus Complete Multiplicativity The fact that the convolution Conv(·, · ) is not a Hopf algebra spoils the idea of employing a vast amount of standard machinery. To distinguish the presently studied antipodal convolution from a proper Hopf algebra we give it a new name. Definition 3.10. A biassociative, biunital, antipodal convolution Conv(µ, , S) is called a Hopf gebra (HG). If the product is a comultiplicative map and if the coproduct is a multiplicative map fulfilling Eq. (2-11) then the Hopf gebra is called multiplicative.2 3.1. Plan A: The modified crossing. To be able to deal with the multiplicative, or even the general case, one has to introduce new technical devices. For definitions etc. see [15,10]. A first attempt at a cure would be to ask if there could be a deformed crossing or switch cV,U : V ⊗ U → U ⊗ V so that the homomorphism axiom Eq. (2-11) could be reestablished in a complete multiplicative fashion. This hope is nourished by the following: Theorem 3.11. [20]: Every biassociative antipodal convolution has a unique crossing cV,U , such that is a monoid homomorphism and µ is a comonoid homomorphism
cV,U =S
S
(3-1)
If cV,U is a braid, i.e. (cV,W ⊗ IdU )(IdV ⊗ cU,W )(cU,V ⊗ IdW ) = (IdW ⊗ cU,V )(cU,W ⊗ IdV )(IdU ⊗ cV,W ), on U ⊗ V ⊗ W , then the Hopf gebra is a braided Hopf algebra. If cU,V is a (graded) switch the Hopf gebra becomes a (graded) Hopf algebra. The further route of such studies involves the possible classifications of crossings obtained this way, and to detect if they are braided, compute their minimal polynomial and so on. Such research is quite tedious, as was shown in [15]. The difficulties are so large that in fact plan A has to be disregarded. 2 This notion is in the Bourbaki tradition [2] and was used in [10] but originally coined by Oziewicz [20], however, with a different connotation.
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B. Fauser
3.2. Plan B: Unrenormalization. We need to come up with a new strategy. The idea is to reestablish a Hopf algebra structure as close as possible to the given multiplicative Hopf gebra in question. Then use the nicely behaved Hopf algebra for computations, and try to find a transformation back to the Hopf gebra formulation. That there is actually hope to do so, stems from the fact that we are going to establish a Hopf algebra which is isomorphic to the multiplicative Hopf gebra on all relatively prime inputs and differs only on common ‘overlapping’ prime factors. To comply with the usage of the term ‘renormalization’ in physics, we need to call such a map assigning to a multiplicative Hopf gebra a Hopf algebra an ‘unrenormalization’ map. Definition 3.12. The unrenormalized coproduct of multiplication · related to the (renormalized) coproduct of multiplication · is recursively defined as i) · ( p) = · ( p) = p × 1 + 1 × p on primes ii) · (n · m) = · (n) · · (m) ∀n, m
(3-2)
forcing complete multiplicativity. In this way the homomorphism axiom (2-11) holds automatically on all pairs n, m of (non-negative) integers. It is important to note that this is a minimal alteration of the coproduct in the sense that the unrenormalized coproduct differs only on the diagonal (on the gcd’s) from the original coproduct. While the counit still remains as the counit of the unrenormalized coproduct, unrenormalization has, however, serious impacts, for example · Corollary ri3.13. The unrenormalized antipode is given as S (n) = (−) n = i pi .
ri n,
where
This result shows that the antipode is just the grade involution with respect to the grading of the natural numbers by prime number content. This is a natural map in Hopf algebra theory, but far from being an interesting number theoretic arithmetic function, like the Möbius function, which was related to the renormalized antipode. We can now wonder which duality connects multiplication and the new unrenormalized co-product. Corollary 3.14. Let n =
i
piri and m = (n | m) =
s
j
p j j . The pairing ( | ) defined by
δri ,si ri ! = z n
(3-3)
i
dualizes the multiplication · into · . Proof. (Sketch). Use Laplace expansion demanding that · and · are Milnor-Moore dual w.r.t. ( | ). For details see [12]. Note that also in this case all alterations are just scalings: (n | m) = z n n | m, which is up to a rescaling by z n , the Kronecker delta again.
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3.3. The co-ring structure. Before we try to set up the number theoretical model of renormalization, we want to exhibit the co-ring structure. This implies a relation between the coproduct of addition and the coproduct of multiplication in analogy with a ring structure. Such relations were used in [14] to investigate new group branchings. Let us introduce a further group-like coproduct δ : P → P × P, δ( p) = p × p. Coaddition and comultiplication are related as (n = i piri ) · (n) =
d×
d|n
n d
= δ (n) = +
( pi × pi )
i
=
ri +ri =ri
r pi i
+ (r
×
i
i)
r pi i
,
(3-4)
i
where the notion · = δ should be taken as a mnemonic only. The unrenormalized case follows along the same lines, and actually can be used to define the unrenormalized coproduct of addition: n + n1 × n2. (n) := (3-5) n1 n +n =n +
1
Let n =
i
2
piri , unrenormalized addition and unrenormalized multiplication relate as: · (n) = δ (n) r + r r r i i pi i × pi i . = r i +
ri +ri =ri
i
(3-6)
i
The appearance of the binomial factors is well known from calculations in quantum field theories, describing the coproduct of scalar fields for example.
3.4. Coping with overcounting : renormalization. Our paradigm is that the number theoretically interesting structure is the renormalized one, which is only multiplicative, and hence forms a multiplicative Hopf gebra (HG) only. To use nice algebraic machinery, we associate to it an unrenormalized Hopf algebra (HA) which differs only on common prime content, hence in a minimal way. The relation of the HG and HA can be summarized in the following commutative diagram: H G(·, · )
unrenormalization
diff. comp. /NT H G(·, · )
H A(·, · ) alg. manip. /pQFT
renormalization
H A(·, · ) (3-7)
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B. Fauser
Number theoretical (NT) computations in the Dirichlet ring of arithmetic functions, i.e. in the convolution ring over the Hopf gebra (HG), are performed along the left down arrow and are usually involved and complex. Perturbative quantum field theory (pQFT) starts with a Hopf algebra structure assuming an algebra structure on the duals of the fields, either explicitly or implicitly. Then algebraic calculations are performed explicitly or implicitly using the underlying Hopf algebra (HA) structure. However, the final formal expressions are plagued by infinities, which are removed by a rescaling technique called renormalization. Our point is that this rescaling ends up in a Hopf gebra in analogy to number theory. The first step, the unrenormalization, is not seen in physics, since the modelling is done by assuming a Hopf algebra structure or equivalently a compatible algebra structure of the fields and their duals, which vice versa implies a comultiplication. The technique of renormalization hence copes with overcounting on the diagonals (gcd generalized to common maximal ideals). In pQFT these overcountings are infinite, since summations are replaced by integrations which in general diverge. In number theory one obtains finite overcountings, and a Hopf algebra approach would have just failed to work by producing wrong results. However, after having established the relation of the diagram Eq. (3-7) one attempted to try to unrenormalize problems in number theory and to use methods from QFT to handle them and ‘renormalize’ the formal result. Our approach opens at least two new possibilities: a: pQFT starts with a HA structure, the unrenormalization is hence superfluous. Due to scalings by counterterms renormalization takes care of ‘overcounting the diagonal’. An enlargement of modelling to start with unrenormalized quantities would possibly allow to introduce number theoretic machinery, i.e. celebrated theorems and particular techniques, to solve problems in physics. b: Via the unrenormalization, there may arise new possibilities to deal with hard number theoretic problems in the ring of arithmetic functions by using methods from quantum field theory. Hence renormalization should be understood as a sort of rewriting rule, allowing insights to be moved from one side to the other. There are several approaches to the theory of renormalization, discussed for example in the topical review [7]. However, from our point of view, the approach proposed by Brouder-Schmitt [3], based on Epstein-Glaser renormalization [8], seems to be more natural and we have adopted it in our work [12]. Therein it was shown for the example of occupation number representations that the ordering process which we introduced in [9] also applies for QM and used both algebraic structures, the unrenormalized and renormalized ones. Since the same process of deformation, but on another level of complexity, produces the renormalization mechanism, we argue that the ‘ordering’ or ‘deformation’ if done on the higher level of complexity –multiplication versus addition, or composition versus multiplication– enters at least in a twofold manner, the more complex one giving rise to the renormalization map. In terms of symmetric functions this leads to the Hopf algebra of plethysm [13]. The crucial fact is that addition can be obtained as iteration of the successor map, multiplication as the iteration of the addition, and exponentiation as the iteration of multiplication. Further generalization fails, since the iteration functor needs a transposition, which is equivalent to demanding a commutative binary underlying operation [17]. In that sense, our number theoretic model needs to be enlarged to include exponentiation to actually parallel the ‘renormalization’ encountered in pQFT.
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4. Taming Multiplicativity A complete multiplicative function g is defined if its values are known on all primes, i.e. on P. Let n = i piri , then complete multiplicativity allows to write r i g(n) = g pi = g( pi )ri . (4-1) i
i
However, a multiplicative function f needs to be specified on all prime powers { pik }, ∀i, k r i f (n) = f pi = f ( piri ). (4-2) i
i
While both sets have the same cardinality it is awkward that a multiplicative function is not well defined by its values on generators, here the primes in P. In what follows, we want show how one might tame multiplicativity by giving data only on primes, and controlling the values on f ( p n ) by a recursion involving a complete multiplicative function. This idea is based on the analogy that the expectation values of powers of quantum fields 0 | (ψ(x))2 | 0 should be computable from a function of the expectation values 0 | ψ(x)ψ(y) | 0 in a suitable limit y → x. The device we want to use is that of Bell series. These are series encoding an arithmetic function on all prime powers of a given prime p. Definition 4.15. A Bell series of an arithmetic function f for a fixed prime p is given as an ordinary power series f ( p n )x n , (4-3) f p (x) = n≥0
employing a formal indeterminate x. Corollary 4.16. If f is complete multiplicative its Bell series reads f p (x) =
f ( p)n x n =
1 . 1 − f ( p)x
(4-4)
The Bell series of the Möbius function and the Euler totient function read µ p (x) = 1 − x, 1−x , φ p (x) = 1 − px
(4-5)
showing that they are not complete multiplicative. The most important fact about Bell series for us is that the Dirichlet convolution product of arithmetic functions is transformed into the Cauchy product of Bell series. Let h = f g, then h p (x) = f p (x)g p (x), reducing the complexity of the operation dramatically.
(4-6)
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B. Fauser
We use an example from Apostol [1] to demonstrate how this might be used to model the process of renormalization in number theory in analogy to renormalization in pQFT, by adding counterterms or modifying the pole structure of the ‘propagator’. Let g be complete multiplicative, and recall that then g(1) = 1. We define a recursion for a multiplicative function f so that all values of f on prime powers are determined. In terms of coefficients a particular recursion reads f ( p n+1 ) = f ( p) f ( p n ) − g( p) f ( p n−1 ).
(4-7)
This allows to compute the Bell series f p (x) =
1 . 1 − f ( p)x + g( p)x 2
(4-8)
It can be shown that Eq. (4-8) follows from Eq. (4-7) and vice versa. Using this recursion, it is now possible to establish the following product formula: m·n . (4-9) f (m · n) = f (m) f (n) − d|gcd(n,m) g(d) f d2 1
Together with Eq. (4-8) this establishes a number theoretic analog of renormalization the1 ory. g( p) would serve as an additive renormalization of the ‘propagator’ 1− f ( p)x+g( p)x 2 and the sum in the right hand side of Eq. (4-9) constitutes counterterms. Acknowledgements. It is a pleasure to thank Peter Jarvis for many helpful discussions and for ongoing collaboration on this subject. Part of this work was done in Hobart during a visit supported by the ARC research grant DP0208808, and the Alexander von Humboldt Foundation.
Appendix A. Some Facts about Dirichlet and Bell Series A.1. Characterizations of complete multiplicativity. Since multiplicativity versus complete multiplicativity plays a major role in our argumentation we want to recall useful characterizations of multiplicativity. Lambek [16] proved that an arithmetical function f is completely multiplicative iff its Hadamard product distributes over every Dirichlet product: f.(g h) = ( f.g) ( f.h) for all arithmetical functions g, h. In terms of coefficients this reads f (n) g(d)h(n/d) = f (d)g(d) f (n/d)h(n/d). d|n
(A-1)
(A-2)
d|n
This can be rephrased saying that the convolution is a Laplace pairing [11] for the Hadamard product. Carlitz [5] posed the problem to characterize complete multiplicativity by distributivity over particular Dirichlet convolutions. Let τ = ζ ζ be the number of positive divisors function. f is complete multiplicative iff f.τ = ( f.ζ ) ( f.ζ ) = f f.
(A-3)
A nice way to generalize such notions is by using Möbius categories C [18,19]. These are categories defined to generalize and unify the theory of Möbius inversions. In terms
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of morphisms one investigates incidence functions f, g forming an incidence algebra A(C) by employing the product f (α )g(α ). (A-4) ( f g)(α) = α α =α
An incidence function is complete multiplicative iff f (α) = f (α ) f (α )
(A-5)
with α = α α the composition of morphisms. Now, Lambek’s characterization generalizes to this case, while Carlitz’ characterization has to be altered [22]. It is, however, nice to have a generalization to this general setting allowing to export the concept of multiplicativity to incidence (or functionals on operator) algebras. This way of generalization is needed on the way to establish our analogy between number theory and pQFT in a more concrete way. A.2. Groups and subgroups of Dirichlet convolution. This section follows the exposition of Dehaye [6]. Let F0 be the set of multiplicative functions different from the zero function 0(n) = 0 for all n. This amounts to have f (1) = 1 for all f in F0 . The pair (F0 , ) is an abelian group with Dirichlet convolution as product. For any prime p we define F p = { f ∈ F0 | f (n) = 0 for every n s.t. p n}. That is, an f ∈ F p has support on prime powers p k only. For every prime p, (F p , ) is a subgroup of F0 . Furthermore, there exists an isomorphism between F p and the group of upper-triangular non-zero infinite matrices M1 , M1 = {m ∈ M | m(a, a) = 1, ∀a ∈ N, and m(a, b) = 0, ∀a, b ∈ N s.t. a > b}. (A-6) M1 is a group with the infinite unit F p → M1 is given by ⎛ 1 ⎜0 ⎜ 0 φ( f ) = ⎜ ⎜0 ⎝ .. .
matrix as identity element. The isomorphism φ : ⎞ f ( p) f ( p 2 ) f ( p 3 ) · · · 1 f ( p) f ( p 2 ) · · · ⎟ ⎟ 0 1 f ( p) · · · ⎟ 0 0 1 ···⎟ ⎠ .. .. .. . . . . . .
(A-7)
and φ( f ) · φ(g) = φ( f g),
(A-8)
where in the l.h.s. the product is matrix multiplication. This isomorphism shows that the Bell series are particular Dirichlet series or restrictions of Dirichlet series to the subgroup F p . These groups are isomorphic for every pair of primes pi , p j . It is possible to consider F0 as a complete (or Cartesian) direct product of the subgroups F pi for all primes F0 = F pi . (A-9) i∈N
It can be shown [6] that
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B. Fauser
a) The group F0 is torsion free (i.e. has no element of finite order). b) F p is a subgroup of F0 for every prime p, all such subgroups are pairwise isomorphic and are isomorphic to infinite upper-triangular non-zero matrices or to Bell series. c) F0 is isomorphic to the complete direct product of the subgroups F p . d) F0 is divisible and has a natural structure of a vector space over Q.
Appendix B. Densities of Generators We want to emphasise another point which connects our work with renormalization of quantum fields. As our discussion here and in [12] demonstrated, one can grade the natural numbers in two canonical ways attached to addition and multiplication. If we generate the natural numbers additively, we have but one generator, the one 1, which is the target of the successor map, and all numbers are generated as successors of the zero 0. The successor map is assumed to have no torsion and composition is associative. From this construction it is evident that the density of natural numbers in the natural numbers d(n, n 0 ) with respect to the 1 as generator is constant. That is, in every neighbourhood, that is an interval containing n 0 , of a natural number n 0 , one finds the same density of natural numbers. The second way to grade the natural numbers was induced by the multiplicative structure and the primitive elements, i.e. generators, were shown to be the set of prime numbers { pi }. However, the density of prime numbers in the natural numbers is a nontrivial function. The celebrated prime number theorem states that the number of prime numbers below n 0 is logn 0n 0 for n 0 → ∞. This renders it obvious that a multiplicative construction of the integers behave quite differently with respect to the densities of generators in the natural numbers. A. Petermann showed in a remarkable paper [21] that a renormalization group analysis provides a proof for the prime number theorem. This supports our claim that the present simplified model of renormalization is actually rich enough to contain main features of renormalization in quantum field theory. It is possible to iterate this process by asking what kind of ‘primitive elements’ occur if one looks for exponentiation as an iteration of multiplication. This question leads into the realm of modular forms and one obtains higher order corrections in the densities of ‘higher primitive elements’ along the same lines as one obtains higher order loop corrections, and divergencies in perturbative quantum field theory. This will be explored elsewhere. References 1. Apostol, T.M.: Introduction to Analytic Number Theory. New York: Springer-Verlag, 1979 [fouth printing 1995] 2. Bourbaki, N.: Elements of Mathematics: Algebra I Chapters 1–3. Berlin: Springer-Verlag, 1989 3. Brouder, C., Schmitt, W.: Renormalization as a functor on bialgebra. J. Pure Appl. Alg. 209, 477–495 (2007) 4. Brüdern, J.: Einführung in die analytische Zahlentheorie. Berlin: Springer-Verlag, 1995 5. Carlitz, L.: Problem E 2268. Amer. Math. Monthly 78, 1140 (1971) 6. Dehaye, P.-O.: On the structure of the group of multiplicative arithmetical functions. Bull. Belg. Math. Soc. Simon Stevin 9(1), 15–21 (2002) 7. Ebrahimi-Fard, K., Kreimer, D.: The Hopf algebra approach to Feynman diagram calculations. J. Phys. A: Math. Gen. 38, R385–R407 (2005) 8. Epstein, H., Glaser, V.: The role of locality in perturbation theory. Ann. Inst. Henri Poincaré 19, 211–295 (1973)
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9. Fauser, B.: On the Hopf-algebraic origin of Wick normal-ordering. J. Phys. A: Math. Gen. 34, 105–115 (2001) 10. Fauser, B.: A Treatise on Quantum Clifford Algebras. Konstanz, 2002, Habilitationsschrift, available at http://arxiv.org/list/math.QA/0202059, 2002 11. Fauser, B., Jarvis, P.D.: A Hopf laboratory for symmetric functiuons. J. Phys. A: Math. Gen. 37(5), 1633–1663 (2004) 12. Fauser, B., Jarvis, P.D.: The Dirichlet Hopf algebra of arithmetics. J. Knot Theor. and Its Ramif. 16(4), 379–438 (2007) 13. Fauser, B., Jarvis, P.D.: The Hopf algebra of plethysms. Work in progress, 2007 14. Fauser, B., Jarvis, P.D., King, R.C., Wybourne, B.G.: New branching rules induced by plethysm. J. Phys A: Math. Gen. 39, 2611–2655 (2006) 15. Fauser, B., Oziewicz, Z.: Clifford Hopf gebra for two dimensional space. Misc. Alg. 2(1), 31–42 (2001) 16. Lambek, J.: Arithmetical functions and distributivity. Amer. Math. Monthly 73, 969–973 (1966) 17. Lawvere, F.W., Rosebrugh, R.: Sets for Mathematics. Cambridge: Cambridge Univ. Press, 2003 18. Leroux, P.: Les Catégories Möbius. Cahiers Top. Géom. Différ. Catég. 16, 280–282 (1975) 19. Leroux, P.: Reduced matrices and q-log-concavity properties of q-Stirling numbers. J. Combin. Theory Ser. A 54, 64–84 (1990) 20. Oziewicz, Z.: Clifford Hopf gebra and biuniversal Hopf gebra. Czech. J. Phys. 47(12), 1267–1274 (1997) 21. Petermann, A.: The so-called Renormalization Group method applied to the specific prime numbers logarithmic decrease Eur. Phys. J. C 17, 367–369 (2000) 22. Schwab, E.D.: Characterizations of Lambek-Carlitz type. Arch. Math. (Brno) 40, 295–300 (2004) 23. Selberg, A.: An elementary proof of the prime number theorem. Ann. Math. 50, 305–313 (1949) Communicated by A. Connes
Commun. Math. Phys. 277, 643–706 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0396-y
Communications in
Mathematical Physics
D-Branes, RR-Fields and Duality on Noncommutative Manifolds Jacek Brodzki1 , Varghese Mathai2 , Jonathan Rosenberg3 , Richard J. Szabo4 1 School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK.
E-mail: [email protected]
2 Department of Pure Mathematics, University of Adelaide, Adelaide 5005, Australia.
E-mail: [email protected]
3 Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
E-mail: [email protected]
4 Department of Mathematics and Maxwell Institute for Mathematical Sciences,
Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK. E-mail: [email protected] Received: 5 July 2006 / Accepted: 19 June 2007 Published online: 5 December 2007 – © Springer-Verlag 2007
Abstract: We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of T-duality as well as establishing a very general formula for D-brane charges. This formula is closely related to a noncommutative Grothendieck-Riemann-Roch theorem that is proved here. Our approach relies on a very general form of Poincaré duality, which is studied here in detail. Among the technical tools employed are calculations with iterated products in bivariant K-theory and cyclic theory, which are simplified using a novel diagram calculus reminiscent of Feynman diagrams.
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. D-Branes and Ramond-Ramond Charges . . . . . . . . . 1.1 Flat D-branes . . . . . . . . . . . . . . . . . . . . . 1.2 Ramond-Ramond fields . . . . . . . . . . . . . . . . 1.3 Noncommutative D-branes . . . . . . . . . . . . . . 1.4 Twisted D-branes . . . . . . . . . . . . . . . . . . . 2. Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . 2.1 Exterior products in K-theory . . . . . . . . . . . . . 2.2 KK-theory . . . . . . . . . . . . . . . . . . . . . . . 2.3 Strong Poincaré duality . . . . . . . . . . . . . . . . 2.4 Duality groups . . . . . . . . . . . . . . . . . . . . . 2.5 Spectral triples . . . . . . . . . . . . . . . . . . . . 2.6 Twisted group algebra completions of surface groups 2.7 Other notions of Poincaré duality . . . . . . . . . . . 3. KK-Equivalence . . . . . . . . . . . . . . . . . . . . . . 3.1 Strong KK-equivalence . . . . . . . . . . . . . . . . 3.2 Other notions of KK-equivalence . . . . . . . . . . .
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3.3 Universal coefficient theorem . . . . . . . . . . . . . . . 3.4 Deformations . . . . . . . . . . . . . . . . . . . . . . . 3.5 Homotopy equivalence . . . . . . . . . . . . . . . . . . 4. Cyclic Theory . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Formal properties of cyclic homology theories . . . . . . 4.2 Local cyclic theory . . . . . . . . . . . . . . . . . . . . 5. Duality in Bivariant Cyclic Cohomology . . . . . . . . . . . 5.1 Poincaré duality . . . . . . . . . . . . . . . . . . . . . . 5.2 HL-Equivalence . . . . . . . . . . . . . . . . . . . . . . 5.3 Spectral triples . . . . . . . . . . . . . . . . . . . . . . 6. T-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Duality for crossed products . . . . . . . . . . . . . . . 6.2 T-duality and KK-equivalence . . . . . . . . . . . . . . 6.3 T-duality and HL-equivalence . . . . . . . . . . . . . . . 7. Todd Classes and Gysin Maps . . . . . . . . . . . . . . . . . 7.1 The Todd class . . . . . . . . . . . . . . . . . . . . . . 7.2 Gysin homomorphisms . . . . . . . . . . . . . . . . . . 7.3 Strongly K-oriented maps . . . . . . . . . . . . . . . . . 7.4 Weakly K-oriented maps . . . . . . . . . . . . . . . . . 7.5 Grothendieck-Riemann-Roch formulas: the Strong case . 7.6 Grothendieck-Riemann-Roch formulas: the Weak case . 8. Noncommutative D-Brane Charges . . . . . . . . . . . . . . 8.1 Poincaré pairings . . . . . . . . . . . . . . . . . . . . . 8.2 D-Brane charge formula for noncommutative spacetimes Appendix A. The Kasparov Product . . . . . . . . . . . . . . . . Appendix B. A Diagram Calculus for the Kasparov Product . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction As proposed by [63] and elaborated in [37,44,64,66,86], D-brane charges and RR-fields in string theory are classified by the K-theory of spacetime X , or equivalently by the K-theory of the C ∗ -algebra C0 (X ) of continuous functions on X vanishing at infinity. Recently, in a far-sighted suggestion at KITP, I.M. Singer suggested working out string theory and duality on spacetimes that are general noncommutative C ∗ -algebras, with some minimal assumptions. This paper can be viewed as a preliminary step towards this goal. Some of our main results are a formula for the charges of D-branes in noncommutative spacetime and a fairly complete treatment of a general framework for T-duality. The main technical tools are a study of Poincaré duality in both KK-theory and bivariant cyclic theories, a definition of Gysin (“wrong-way”) maps, and a version of the Grothendieck-Riemann-Roch theorem. Previous work ([9,59], among many other references) already showed that a good formulation of T-duality requires the use of noncommutative algebras. We develop a formalism for dealing with T-duality in the context of general separable C ∗ -algebras and in Sect. 6 we give an axiomatic definition. This includes the requirement that the RR-fields and D-brane charges of A should be in bijective correspondence with the RRfields and D-brane charges of the T-dual T(A), and that T-duality applied twice yields a C ∗ -algebra which is physically equivalent to the C ∗ -algebra that we started out with.
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This general T-duality formalism can be viewed as a noncommutative version of the [topological aspects of the] Fourier-Mukai transform. In the classical case, D-brane charges are expressed in terms of the Chern character in K-homology (see [75]), as formulated topologically by the Baum-Douglas construction of [6]. In formulating the notions of D-brane charges and RR-fields on arbitrary C ∗ algebras, one is faced with the problem of developing Poincaré duality and constructing characteristic classes in this general setting. In [21], Connes initiated this study, pointing out that the analogue of a spinc structure for a C ∗ -algebra A is a fundamental class for its K-theory, whereas the analogue of a spin structure is a fundamental class for its KO-theory. In [65], Moscovici gives an elegant application of Poincaré duality, deriving an analogue of the Vafa-Witten inequalities for spectral triples that implement Poincaré duality, under a finite topological type hypothesis. One of the goals of this paper is to define the Todd class and Todd genus for a spinc C ∗ -algebra A, which generalize the notion of the classical Todd class and Todd genus of a compact spinc manifold X . If is a fundamental class for the K-theory of the spinc C ∗ -algebra A and is a fundamental class in bivariant cyclic homology of A (which is the analogue of an orientation for a smooth manifold), then we define in Sect. 7.1, the Todd class of A to be the invertible element Todd(A) = ∨ ⊗Ao ch() in bivariant cyclic homology of the algebra A. (The notations are explained below; ∨ is the dual fundamental class to and Ao is the opposite algebra to A.) In the special case when A is a spin C ∗ -algebra and the K-theory fundamental class comes from a fundamental class in KO-theory, the characteristic class as defined above is called the Atiyah-Hirzebruch class A(A). One of our main results, Theorem 7.10, shows that the Todd class as defined above is exactly the correction factor needed in the noncommutative Grothendieck-Riemann-Roch formula. Our final main result, the D-brane charge formula of Sect. 8.2, is a noncommutative analogue of the well-known formula (1.1) in [63] (cf. [44,64,66,86]). It takes the familiar form, Qξ = ch( f ! (ξ )) ⊗A Todd(A), for a D-brane B in a noncommutative spacetime A with given weakly K-oriented morphism f : A → B and Chan-Paton bundle ξ ∈ K• (B), where f ! denotes the Gysin map associated to f . With this modification of the Chern character, one obtains an isometry between the natural intersection pairings in K-theory and cyclic theory of A. There is also a similar dual formula for the charge of a D-brane given by a Fredholm module, representing the Chern-Simons coupling of D-branes with RR-fields. The central mathematical technique of the paper is the development of a novel diagram calculus for KK theory and the analogous diagram calculus for bivariant cyclic theory, in Appendix B. The rules of this diagram calculus are reminiscent of those for the calculus of Feynman diagrams, and are likely to become an important tool for simplifying iterated sequences of intersection products in KK-theory and in cyclic theory, and for establishing identities in these theories. 1. D-Branes and Ramond-Ramond Charges In this section we give a detailed mathematical description of brane charges in the language of topological K-homology and singular cohomology. Our aim later on is then
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to generalize these constructions to analytic K-homology and cyclic cohomology suitable to generic noncommutative settings, some examples of which we describe below. For a description of D-branes in terms of K-theory see [63,64,66,86], and in terms of K-homology see [1,75,84]. 1.1. Flat D-branes. Let X be a spin manifold of dimension d = 10 with metric. In Type II superstring theory, X is called the spacetime. If X is non-compact, appropriate compact support conditions are always implicitly understood throughout. In our later applications we can typically relax some of these requirements and only assume that X is a finite-dimensional Hausdorff space which has the homotopy type of a finite CW-complex. Definition 1.1. A flat D-brane in X is a triple (W, E, φ), where φ : W → X is a closed, embedded spinc submanifold and E ∈ K0 (W ). The submanifold W is called the worldvolume and the class E the Chan-Paton bundle of the D-brane. When E is the stable isomorphism class of a complex vector bundle over W , we assume that it is equipped with a connection and refer to the triple as a brane system. When E is only a virtual bundle, say E = E + − E − , we can loosely regard it as the class of a complex Z2 -graded bundle E + ⊕ E − equipped with a superconnection and the triple is called a brane-antibrane system. The requirement that a D-brane (W, E, φ) be invariant under processes involving brane-antibrane creation and annihilation is the statement of stable isomorphism of Chan-Paton bundles. Physical quantities which are invariant under deformations of E thereby depend only on its K-theory class in K0 (W ) [86]. Deformation invariance, gauge symmetry enhancement and the possibility of branes within branes then imply that any D-brane (W, E, φ) should be subjected to the usual equivalence relations of topological K-homology [6], i.e., bordism, direct sum and vector bundle modification, respectively [75]. We will not distinguish between a D-brane and its K-homology class in K• (X ), nor between the Chan-Paton bundle and its isomorphism class in K0 (W ). To define the charge of a flat D-brane in the spacetime manifold X , we begin by introducing a natural bilinear pairing on the K-theory of X , index(D / (−) )
⊗
−, − : K0 (X ) × K0 (X ) −→ K0 (X ) −−−−−−→ Z,
(1.1)
where D / N : C ∞ (X, S+X ⊗ N ) → C ∞ (X, S− X ⊗ N ) is the twisted Dirac operator on X , with respect to a chosen connection on the complex vector bundle N → X , and S± X → X are the two half-spinor bundles over X . When tensored over Q the pairing (1.1) is nondegenerate, which is equivalent to Poincaré duality in rational K-theory. In the topological setting, Poincaré duality is generically determined by the bilinear cap product ∩
K0 (X ) ⊗ K• (X ) −→ K• (X )
(1.2)
defined for any complex vector bundle F → X and any D-brane (W, E, φ) in X by F ∩ (W, E, φ) = (W, E ⊗ φ ∗ F, φ).
(1.3)
The index pairing K0 (X ) ⊗ K• (X ) → Z is then provided by the Dirac operator on W as F ⊗ (W, E, φ) −→ index(D / E⊗φ ∗ F ) .
(1.4)
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On the other hand, in cohomology the natural bilinear pairing is given by the intersection form (−)[X ]
∪
(−, −) : H p (X, Z) × Hd− p (X, Z) −→ Hd (X, Z) −−−−→ Z.
(1.5)
Again nondegeneracy of this pairing over Q is equivalent to the Poincaré duality theorem of classical rational cohomology theory [7, p. 44]. For a compact oriented manifold X , the pairing between cohomology groups of complementary degrees leads to the duality ∨ H p (X, Q) ∼ (1.6) = Hd− p (X, Q) ∼ = Hd− p (X, Q). It is important to realize that this pairing is determined purely in terms of the topology of X , while the index pairing between K-theory and K-homology uses both the knowledge of the topology of X and the analysis of the Dirac operator D / . This difference will become important when we compare the two pairings using the Chern character below. The statement of cohomological Poincaré duality in the non-oriented case requires the use of twisted coefficients, while in K-theory the Poincaré pairing involves twisting whenever X is not spinc . This links very importantly with twisted K-theory [3]. ) of the manifold X is invertible with Recall that the Atiyah-Hirzebruch class A(X respect to the cup product on cohomology [55, p. 257]. An application of the AtiyahSinger index theorem (recalling that X is spin) then immediately gives the following fundamental result. Proposition 1.2. The modified Chern isomorphism Ch : K0 (X ) ⊗ Q −→ Heven (X, Q) =
H2n (X, Q)
(1.7)
n≥0
defined by )1/2 Ch(N ) = ch(N ) ∪ A(X is an isometry with respect to the natural inner products (1.1) and (1.5), N , N = Ch(N ), Ch(N ) .
(1.8)
(1.9)
Note that the ordinary Chern character ch preserves the addition and multiplication on K-theory and cohomology, but not the bilinear forms. The modified Chern character Ch preserves addition but not the cup products. A similar is also true for the Chern
statement 2n+1 (X, Q). However, because character on K−1 (X ) ⊗ Q → Hodd (X, Q) = n≥0 H of the suspension isomorphism K−1 (X ) ∼ = K0 (X × R) it will suffice to work explicitly 0 with K groups alone in the following. In string theory terms this means that we work only with Type IIB D-branes, the analogous results for Type IIA branes being obtainable by T-duality (see Sect. 6). There is an elementary but useful alternative interpretation of Proposition 1.2. Since the A-class is an even degree (inhomogeneous) class in the cohomology ring Heven (X , )1/2 is also invertible. It follows Q), with non-zero constant term, its square root A(X that taking products with this class produces an isomorphism h : Heven (X, Q) → Heven (X, Q) which is given explicitly by )1/2 . ω −→ ω ∪ A(X
(1.10)
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When we combine this isomorphism with the pairing given by Poincaré duality, we obtain a new nondegenerate pairing (−, −)h : Heven (X, Q) × Heven (X, Q) → R defined by )1/2 , α ∪ A(X )1/2 α, α h := α ∪ A(X )1/2 ∪ α ∪ A(X )1/2 [X ] = α ∪ A(X ) [X ], = α ∪ α ∪ A(X (1.11) where we have used commutativity of the cup product. It is now easy to see that the classical Chern character is an isometry with respect to the two pairings (1.1) and (1.11), N , N = ch(N ), ch(N ) h . (1.12) From this point of view the isomorphism h transforms the purely topological pairing (−, −) to the “index” pairing (−, −)h, where the latter contains the information about the extra piece of index machinery given by the Atiyah-Hirzebruch class. For any closed oriented embedding φ : W → X of dimension p, we denote by [W ] its orientation cycle in H p (X, Z), by Pd X (W ) = PdW →X = ([X ] ∩ )−1 [W ] its Poincaré dual in Hd− p (X, Z), and by φ! : K• (W ) → K•+d− p (X ) the corresponding K-theoretic Gysin homomorphism. Recall that on cohomology, the Gysin map is given • •+d− p (X, Z). explicitly by φ! = Pd X ◦ φ∗ ◦ Pd−1 W : H (W, Z) → H Definition 1.3. The Ramond-Ramond charge (RR-charge for short) of a D-brane (W, E, φ) in X is the modified Chern characteristic class Ch(φ! E) ∈ H• (X, Q). If (W , E , φ ) is any other D-brane in X , then the (W , E , φ )-charge of (W, E, φ) is the integer Q W ,E ,φ (W, E, φ) = Pd X (W ), Ch(φ! E) = φ ∗ Ch(φ! E)[W ]. (1.13)
When (W , E , φ ) = (W, E, φ), we write simply Q W,E,φ = Q W,E,φ (W, E, φ) for the charge of the D-brane (W, E, φ) itself. Note that this charge formula for a D-brane is written entirely in terms of spacetime quantities. Let us momentarily assume, for simplicity, that the spacetime manifold X is compact. Let C(X ) denote the C ∗ -algebra of continuous complex-valued functions on X . A standard construction in K-homology then provides the following result. Proposition 1.4. There is a one-to-one correspondence between flat D-branes in X , modulo Baum-Douglas equivalence, and stable homotopy classes of Fredholm modules over the algebra C(X ). Proof. Consider a D-brane (W, E, φ) such that dim(W ) is odd. The worldvolume W inherits a metric from X and its Chan-Paton bundle E is equipped with a (super)connection ∇. Let SW → W be the spinor bundle over W , and consider the usual twisted Dirac operator D / E : C ∞ (W, SW ⊗ E) → C ∞ (W, SW ⊗ E) with respect to the chosen connection ∇. Using the metric we can complete the vector space of smooth sections C ∞ (W, SW ⊗ E) of the twisted spinor bundle and view D / E : H → H as an unbounded self-adjoint Fredholm operator on the separable Hilbert space H = L 2 (W, SW ⊗ E).
(1.14)
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Let us now define a unital algebra ∗-homomorphism ρ : C(X ) → B(H) as pointwise multiplication on H via ρ( f ) = m f ◦φ ⊗ 11SW ⊗E , ∀ f ∈ C(X ),
(1.15)
where m g : C(W ) → C(W ) is the pointwise multiplication operator h → g h. Since the Dirac operator is closable, we can thereby form an odd Fredholm module (H, ρ, F) over the algebra C(X ), where F = D / E /|D / E | is the partial isometry in the polar decomposition of D / E . Similarly, when W is even-dimensional, we can form an even Fredholm module (H, ρ, F), where the Z2 -grading H = H+ ⊕ H− on the Hilbert space (1.14) is given by H± = L 2 (W, S± W ⊗ E)
(1.16)
with S± W → W the two half-spinor bundles over W , the odd bounded Fredholm operator F constructed as above from the corresponding twisted Dirac operator D /E : C ∞ (W, S+W ⊗ E) → C ∞ (W, S− ⊗ E), and the even ∗-homomorphism ρ : C(X )→ W B(H± ) defined as in (1.15). The clases of the Fredholm modules built in this way are independent of the choice of metric on X and connection ∇ on E. Conversely, allow arbitrary coefficient classes in K-theory (this requires certain care with the defining equivalence relations [48,75]). Then the K-homology class of the cycle (X, E, id X ) is the Poincaré dual of E, which can be any class in K• (X ). We conclude that all classes in the K-homology K• (A) = KK• (A, C) of the algebra A = C(X ) can be obtained by using an appropriate D-brane. Proposition 1.4 of course simply establishes the equivalence between the analytic and topological descriptions of K-homology. Any Fredholm module over the C ∗ -algebra C(X ) is therefore a flat D-brane in the spacetime X . The usefulness of this point of view is that it can be extended to more general brane configurations (that we describe in the following) which are represented by noncommutative algebras. Namely, a D-brane may be generically regarded as the homotopy class of a suitable Fredholm module over an algebra A. In what follows we will reformulate the description of D-brane charge in the language of cyclic cocycles. This will require, in particular, an analytic reformulation of the natural pairings introduced above. More precisely, one of our main goals in this paper is to provide a generic, noncommutative version of the result (1.12). 1.2. Ramond-Ramond fields. Closely related to the definition of D-brane charge given above is the notion of a Ramond-Ramond field. In what follows we use the cup product ∪ when multiplying together cohomology classes, and exterior products ∧ when multiplying arbitrary differential forms. In a similar vein to what we have done before, we will not distinguish between (co)homology classes and their explicit representatives. Let Fred = Fred(H) be the space of Fredholm operators on a separable Hilbert space H. Then Fred is a classifying space for K-theory of X and any vector bundle E → X can be obtained as the index bundle of a map into Fred. Let c be a choice of cocycle representative for the universal Chern character. If f E : X → Fred is the classifying map of a bundle E → X , then ch(E) = f E∗ c ∈ Heven (X, R). Consider triples ( f, C, ω), where f ∈ [X, Fred], ω is an inhomogeneous form of even degree, and C is an inhomogeneous cochain of odd degree satisfying dC = ω − f ∗ c.
(1.17)
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The collection of all such triples is denoted K0 (X ). Two elements ( f 0 , C0 , ω0 ) and ( f 1 , C1 , ω1 ) of K0 (X ) are called equivalent if there is a triple ( f, C, ω) on X × [0, 1], with ω constant on {x} × [0, 1] for each x ∈ X , such that ( f, C, ω)| X ×{0} = ( f 0 , C0 , ω0 ) and ( f, C, ω)| X ×{1} = ( f 1 , C1 , ω1 ). The set of equivalence classes forms an abelian group under addition of triples called the differential K-theory group K˘ 0 (X ), cf. §4 in [43]. It fits into the short exact sequence 0 −→ K−1 (X ) ⊗ R/Z −→ K˘ 0 (X ) −→ A0 (X ) −→ 0, where
A0 (X )
(1.18)
is defined by the pullback square / even cl (X ) =
A0 (X ) K0 (X )
ch
n≥0
2n cl (X )
(1.19)
/ Heven (X, R)
with 2n cl (X ) the space of closed 2n-forms on X . The RR-fields (of Type IIA superstring theory) are closed even degree forms associated to elements of K0 (X ) [36,64]. Definition 1.5. The Ramond-Ramond field (RR-field for short) G associated to an element of K0 (X ) which maps to (E, ω) ∈ A0 (X ) under (1.18) is the closed differential form )1/2 . G(E, ω) = ω ∧ A(X
(1.20)
The topological equivalence class of the RR-field is the D-brane charge regarded as an element of the appropriate K-theory group. The D-branes “couple” to RR-fields, and another way to define D-brane charge is through the pairings of their characteristic classes with these differential forms. Definition 1.6. The Chern-Simons coupling of a D-brane (W, E, φ) to an RR-field corresponding to the element ( f, C, ω) ∈ K0 (X ) is the spacetime integral (1.21) SCS (W, E, φ|C) = C ∧ Ch(φ! E). X
Given this notion, we can now formulate an alternative homological definition of D-brane charge. Definition 1.7. The dual Ramond-Ramond charge (dual RR-charge for short) of a D-brane (W, E, φ) in X is the rational homology class Ch(W, E, φ) ∈ H• (X, Q) such that SCS (W, E, φ|C) = C (1.22) Ch(W,E,φ)
for all RR-fields on X .
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Evidently, the natural framework for the Chern-Simons couplings of D-branes is K-homology. The Chern character in topological K-homology is the isomorphism ch : K• (X ) ⊗ Q −→ H• (X, Q)
(1.23)
ch(W, E, φ) = φ∗ ◦ Pd−1 W (ch(E) ∪ Todd(W ))
(1.24)
defined by
for any D-brane (W, E, φ) in X . The Todd class is related to the Atiyah-Hirzebruch class by ), Todd(W ) = e −d(W ) ∪ A(W
(1.25)
where d(W ) ∈ H2 (W, Z) is a characteristic class whose reduction modulo 2 is the second Stiefel-Whitney class w2 (W ) ∈ H2 (W, Z2 ). This specifies the spinc structure on the brane worldvolume W as follows. The spinc groups Spin c (n) = Spin(n) ×Z2 U (1) fit into a commutative diagram (1.26)
1
1
U (1) L LLL LzLL→ z 2 j LLL % l c / Spin (n) / U (1)
ı / Spin(n) MMM MMM MM λ λ MMM & S O(n)
/ 1
1 whose row and column are exact sequences. The map λ : Spin(n) → S O(n) is the universal cover of the group S O(n), while j : U (1) → Spin c (n) and ı : Spin(n) → Spin c (n) are natural inclusions. The homomorphism l : Spin c (n) → U (1) is defined by (g, z) → z 2 . It induces a map H1 (W, Spin c (n)) → H1 (W, U (1)) and thus we may associate a complex line bundle L → W with the worldvolume W . The corresponding Chern class is the characteristic class d(W ) := c1 (L). The homological Chern character preserves sums, as well as the cap product in the sense that ch (F ∩ (W, E, φ)) = ch(F) ∩ ch(W, E, φ)
(1.27)
for any complex vector bundle F → X . This follows from its definition (1.24), the multiplicativity of the cohomological Chern character, the index theorem, and the Atiyah-Hirzebruch version of the Riemann-Roch theorem, ), φ! (ch(E) ∪ Todd(W )) = ch(φ! E) ∪ A(X
(1.28)
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which together give ch(F) ch(W, E, φ) = ch(F) ∪ φ! (ch(E) ∪ Todd(W )) [X ] )[X ] = ch(F ⊗ φ! E) ∪ A(X = index(D / F⊗φ! E )
(1.29)
with : H• (X, Z) × H• (X, Z) the pairing between cohomology and homology. This is just the index pairing (1.4). As the notation suggests, the dual charge of a D-brane is a modification of the homological Chern character analogous to the modification in the case of cohomology. Proposition 1.8. The dual RR-charge Ch(W, E, φ) ∈ H• (X, Q) of a D-brane (W, E, φ) in X can be represented by −1/2 Pd . Ch(W, E, φ) = Pd−1 ◦ ch(W, E, φ) ∪ A(X ) X X
(1.30)
Proof. We use (1.28) along with (1.8) to rewrite the D-brane charge as )−1/2 . Ch(φ! E) = φ! (ch(E) ∪ Todd(W )) ∪ A(X
(1.31)
Along with the definition (1.24), we can use (1.31) to rewrite the Chern-Simons coupling (1.21) in the form SCS (W, E, φ|C) =
)−1/2 . C ∧ Pd X ◦ ch(W, E, φ) ∪ A(X
(1.32)
X
By comparing this with the definition (1.22) of the dual charge, (1.30) follows.
1.3. Noncommutative D-branes. There are many sorts of noncommutative D-branes, i.e., D-branes modeled as Fredholm modules over a noncommutative algebra, and here we will discuss only a few special instances. To motivate the first generalization of our definition of a D-brane given above, we look at an alternative way of regarding the embedding φ : W → X of a flat D-brane into spacetime. Consider a tubular neighbourhood W of W in X . For any point u ∈ W , there is an isomorphism Tu X ∼ = Tu W ⊕ Nu (X/W ), where N (X/W ) → W is the normal bundle, which can be identified with φ ∗ (T X )/T W , in terms of the proper differentiable map φ : W → X . Let : W → N (X/W ) be the diffeomorphism which identifies the normal bundle N (X/W ) with the tubular neigh := ◦ φ is the zero section of N (X/W ) → W , and in this way bourhood W . Then φ we may identify the embedding of the worldvolume into spacetime as a smooth section ∈ C ∞ (W, N (X/W )). of the corresponding normal bundle, φ ), where φ : Definition 1.9. A flat nonabelian D-brane in X is a quadruple (W, E, φ, φ W → X is a closed, embedded spinc submanifold, E ∈ K0 (W ), and ∈ C ∞ (W, N (X/W ) ⊗ End(E)). φ
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= φ and in this case the nonabelian When E is a complex line bundle, we identify φ D-brane is the same object that we defined above. Nonabelian D-branes are classified by the same K-theory as abelian ones. In general, the algebra A = C(X )⊗C ∞ (W, End(E)) acts on the Hilbert spaces (1.14) or (1.16), and so one can formulate this definition in the language of Fredholm modules over an algebra which is Morita equivalent to C(X ). A related notion arises within the framework of Fredholm modules when one replaces the algebra of functions on spacetime with an appropriate noncommutative algebra. Definition 1.10. A flat noncommutative D-brane in X is a Fredholm module over a deformation Aθ of the algebra A = C(X ). For the most part, noncommutative D-branes are classified by the same K-theory as commutative ones. However, this assumes that K-theory is preserved under deformation [79], which is not always the case. See [56] for an interesting counterexample. Example 1.11. Consider X = R2n (with compactly-supported cohomology groups), and let S(R2n ) be the space of complex Schwartz functions on R2n . Let θ = (θ i j ) be a real, invertible skew-symmetric 2n × 2n matrix. For f, g ∈ S(R2n ), we define the corresponding twisted product −2n f x − 21 θ u g (x + v) e − i u·v d2n u d2n v, (1.33) f θ g(x) := (2π ) where d2n u is the Lebesgue measure on R2n . The deformed algebra Aθ is then defined as (1.34) Aθ = S(R2n ), θ . This is an associative Fréchet algebra which defines a noncommutative space that is often called the Moyal n-plane or noncommutative Euclidean space. D-branes may be constructed analogously to the commutative case. For instance, for f ∈ Aθ let m θf : Aθ → n Aθ denote the left multiplication operator g → f θ g, and let H = L 2 (R2n ) ⊗ C2 be the Hilbert space of ordinary square-integrable spinors on R2n . Let D / be the ordinary Euclidean Dirac operator, and define a ∗-representation ρ θ : Aθ → B(H) by ρ θ ( f ) = m θf ⊗ 112n . Then (H, ρ θ , F), with F = D / /|D / |, is a Fredholm module over the algebra (1.34). Example 1.12. Let X be a closed Riemannian spin manifold equipped with a smooth isometric action of a 2n-torus T2n . The periodic action of T2n on X induces by pullback an action of T2n by automorphisms τ on the algebra A = C ∞ (X ) of smooth functions on X . The orbits on which T2n acts freely determine maps σs : C ∞ (X ) → C ∞ (T2n ). Let Tθ2n := (C ∞ (T2n ), θ ) be the noncommutative torus defined as the algebra of smooth functions on the ordinary torus endowed with the periodised version of the twisted product (1.33). Pulling back this deformation by the maps σs gives rise to an algebra Aθ := (C ∞ (X ), ×θ ). This defines a broad class of noncommutative spaces known as toric noncommutative manifolds. The product f ×θ g is given by a periodic twisted product just like (1.33), with the non-periodic translations replaced by the periodic T2n ˆ Tθ2n action. Alternatively, Aθ may be defined as the fixed point subalgebra of C ∞ (X ) ⊗ ˆ the projective tensor product of under the action of the automorphism τ ⊗ τ −1 (with ⊗ Fréchet algebras). The construction of D-branes in these cases again parallels that of the commutative case and Example 1.11 above.
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The special classes of noncommutative branes given by Examples 1.11 and 1.12 above will be referred to as isospectral deformations of flat D-branes. Other interesting examples may be found in [23] and [22]. 1.4. Twisted D-branes. A very important instance in which noncommutative D-branes arise is through the formulation of the notion of a curved D-brane. These arise when the spacetime manifold X carries certain topologically non-trivial characteristics in the following sense. Recall that a gerbe over X is an infinite rank principal bundle over X with projective unitary structure group and characteristic class H . A gerbe connection is a Deligne cohomology class on X with top form H . Definition 1.13. A B-field (X, H ) is a gerbe with one-connection over X and characteristic class H ∈ H3 (X, Z) called an NS–NS H -flux. For any oriented submanifold W ⊂ X , we denote by W3 (W ) ∈ H3 (W, Z) the third integer Stiefel-Whitney class of its normal bundle N (X/W ). It is the obstruction to a spinc structure on W . Definition 1.14. A curved or twisted D-brane in a B-field (X, H ) is a triple (W, E, φ), where φ : W → X is a closed, embedded oriented submanifold with φ ∗ H = W3 (W ), and E ∈ K0 (W ). The condition on the brane embedding is required to cancel the global Freed-Witten anomalies [37] arising in the worldsheet functional integral. Suitable equivalence classes of curved D-branes take values in the twisted topological K-homology K• (X, H ) [84]. For H = 0, the worldvolume W is spinc and the definition reduces to that of the flat case. One should also require that the brane worldvolume W carry a certain projective structure that reduces for H = 0 to the usual characteristic class d(W ) ∈ H2 (W, Z) specifying a spinc structure on W . A B-field can be realized by a bundle of algebras over X whose sections define a noncommutative C ∗ -algebra. When H ∈ Tor(H3 (X, Z)) is a torsion class, this is known as an Azumaya algebra bundle [13]. Via the Sen-Witten construction, D-branes in (X, H ) may then be realized in terms of n D9 brane-antibrane pairs carrying a principal SU (n)/Zn = U (n)/U (1) Chan-Paton bundle. Cancellation of anomalies then requires n H = 0. To accommodate non-torsion characteristic classes, one must consider a certain n → ∞ limit which can be realized as follows. Let us fix a separable Hilbert space H, and denote by PU (H) = U (H)/U (1) the group of projective unitary automorphisms of H. Let K(H) be the C ∗ -algebra of compact operators on H. For any g ∈ U (H), the map Ad g : K(H) → K(H) defined by Ad g (T ) = g T g −1 is an automorphism. The assignment g → Ad g defines a continuous epimorphism Ad : U (H) → Aut(K(H)) with respect to the strong operator topology on U (H) and the point-norm topology on Aut(K(H)) with ker(Ad) = U (1). It follows that one can identify the group PU (H) with Aut(K(H)) under this homomorphism. The exact sequence of sheaves of germs of continuous functions on X given by 1 −→ U (1) X −→ U (H) X −→ PU (H) X −→ 1
(1.35)
induces a long exact sequence of sheaf cohomology groups as δ 1 −→ H1 X, U (H) X −→ H1 X, PU (H) X −→ H2 X, U (1) X −→ . (1.36)
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Since the unitary group U (H) is contractible with respect to the strong operator topology, the sheaf U (H) X is soft and so H j (X, U (H) X ) = 0 for all j ≥ 1. It follows that the map δ1 is an isomorphism. From the exact sequence of groups 1 −→ Z −→ R −→ U (1) −→ 1
(1.37)
we obtain the long exact cohomology sequence δ 2 −→ H2 X, R X −→ H2 X, U (1) X −→ H3 (X, Z) −→ H3 X, R X −→ . Again, since R X is a fine sheaf, one has H j (X, R X ) = 0 for all j ≥ 1 and so the map δ2 is an isomorphism. The map δ X = δ2 ◦ δ1 : H1 X, PU (H) X −→ H3 (X, Z) (1.38) is thus an isomorphism on stable equivalence classes of principal PU (H)-bundles over the spacetime X . If P → X is a PU (H)-bundle and [P] ∈ H1 (X, PU (H) X ) is its isomorphism class, then δ X (P) := δ X ([P]) ∈ H3 (X, Z) is called the DixmierDouady invariant of P [13,15,61]. The set of isomorphism classes of locally trivial bundles over X with structure group Aut(K(H)) and fibre K(H) form a group Br ∞ (X ) under tensor product called the infinite Brauer group of X . Using the identification PU (H) ∼ = Aut(K(H)), it follows that such algebra bundles are also classified by H3 (X, Z). If E is a bundle of this kind, then the corresponding element of H3 (X, Z) is also called the Dixmier-Douady invariant of E [61] and denoted δ X (E). Given a B-field (X, H ), there corresponds a unique, locally trivial C ∗ -algebra bundle E H → X with fibre K(H) and structure group PU (H) whose Dixmier-Douady invariant is δ X (E H ) = H.
(1.39)
Let C0 (X, E H ) be the C ∗ -algebra of continuous sections, vanishing at infinity, of this algebra bundle. The twisted K-theory K• (X, H ) = K• (C0 (X, E H )) [3,78] may then be computed as the set of stable homotopy classes of sections of an associated algebra bundle PH × PU (H) Fred(H), where PH is a principal PU (H)-bundle over X and Fred(H) is the algebra of (self-adjoint) Fredholm operators on H with PU (H) acting by conjugation. On the other hand, one can define Dixmier-Douady classes over any D-brane worldvolume W in complete analogy with (1.38) and show that [69] W3 (W ) = δW (Cliff(N (X/W ))) ,
(1.40)
where Cliff(N (X/W )) → W is the Clifford algebra bundle of the normal bundle N (X/W ). The Dixmier-Douady class δW (Cliff(N (X/W ))) is the global obstruction to existence of a spinor bundle SW with Cliff p (N (X/W )) ∼ = End p (SW )
(1.41)
for p ∈ W . This observation leads to the following result. Proposition 1.15. There is a one-to-one correspondence between twisted D-branes in (X, H ) and stable homotopy classes of Fredholm modules over the algebra C0 (X, E H ).
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The proof of Proposition 1.15 fixes the appropriate equivalence relations required for twisted topological K-homology. One of these equivalence relations (in addition to the appropriate twisted analogs of bordism, direct sum and vector bundle modification) is based on the observation [57] that while a triple (W , E , φ ) may violate the embedding condition of a curved D-brane, one can still cancel the Freed-Witten anomalies on the submanifold W by adding a “source” in W corresponding to a twisted D-brane. This D-brane can be unstable and decay due to the configuration (W , E , φ ). This physical process can be stated more precisely as follows. Lemma 1.16 (Stabilization). Let (W, E, φ) be a twisted D-brane in (X, H ) whose orientation cycle [W ] is non-trivial in H• (X, Z). Suppose that there exists a closed, embedded oriented submanifold φ : W → X such that W is a codimension 3 submanifold of W and its Poincaré dual PdW →W satisfies the equation φ ∗ H = W3 (W ) + PdW →W
(1.42)
in H3 (W , Z). Then (W, E, φ) is trivial in K• (X, H ) (up to twisted vector bundle modification). The structure of D-branes in torsion B-fields simplifies drastically. When H ∈ Tor(H3 (X, Z)) the algebra C0 (X, E H ) is Morita equivalent to an Azumaya algebra bundle over X , i.e., a bundle whose fibres are Azumaya algebras with local trivializations reducing them to n × n matrix algebras Mn (C). Two Azumaya bundles E, F over X are called equivalent if there are vector bundles E, F over X such that E ⊗ End(E) is isomorphic to F ⊗ End(F). The set of equivalence classes is a group Br(X ) under tensor product called the Brauer group of X . There is also a notion of Dixmier-Douady invariant δ X for Azumaya bundles over X , which is constructed using the same local description as above but now with H a finite-dimensional complex vector space. By Serre’s theorem one has Br(X ) ∼ = Tor(H3 (X, Z)). This gives two descriptions of Tor(H3 (X, Z)), one in terms of locally trivial bundles over X with fibre K(H) and structure group Aut(K(H)), and the other in terms of Azumaya bundles. They are related by the following result from [61]. Proposition 1.17. If X is a compact manifold and E is a locally trivial bundle over X with fibre K(H) and structure group Aut(K(H)), then the algebra C(X, E) is stably unital if and only if its Dixmier-Douady invariant is a torsion element in H3 (X, Z). These constructions allow us to describe the K-theory of the noncommutative C ∗ -algebra C(X, E H ) [61]. Morita equivalence induces an isomorphism between the K-theories of C0 (X, E H ) and C0 (X, A H ), where A H is an Azumaya bundle associated to E H via the Dixmier-Douady invariant [13]. A geometric description of this K-theory is provided by the notion of projective vector bundle [61], while in the infinite-dimensional setting of a non-torsion B-field one needs to introduce the notions of bundle gerbes and bundle gerbe modules [8]. 2. Poincaré Duality A crucial point of our construction of flat D-brane charges in Sect. 1.1 was the role played by Poincaré duality. With an eye to generalizing the construction to the more general settings described above, in this section we will explore how and to what extent this classical notion of topology can be generalized to generic C ∗ -algebras in the context
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of KK-theory. We will describe various criteria which guarantee the duality. There are several natural inequivalent versions of Poincaré duality, which we define and study, giving many purely noncommutative examples. Our examples range from those of classical spaces to noncommutative deformations of spinc manifolds, and also the more general examples of Poincaré duality spaces such as those arising in the case of the free group acting on its boundary or, more generally, for hyperbolic groups acting on their Gromov boundaries.
2.1. Exterior products in K-theory. To describe Poincaré duality generically in K-theory, we first need to make some important remarks concerning the product structure. Let A1 and A2 be unital C ∗ -algebras. If p1 ∈ Mk (A1 ) is a projection representing a Murrayvon Neumann equivalence class in K0 (A1 ) and a projection p2 ∈ Ml (A2 ) represents a class in K0 (A2 ), then the tensor product p1 ⊗ p2 is a projection in Mk (A1 ) ⊗ Ml (A2 ) ∼ = Mk l (A1 ⊗ A2 ) for any C ∗ -tensor product and so it represents a class in K0 (A1 ⊗ A2 ) (in this section we will work mostly with the maximal tensor product). In this way we obtain a map K0 (A1 ) × K0 (A2 ) −→ K0 (A1 ⊗ A2 ).
(2.1)
This definition extends to non-unital algebras in a standard way [41, p. 104]. In the special case A1 = A2 = A we obtain a map K0 (A) × K0 (A) −→ K0 (A ⊗ A).
(2.2)
It is important to note that, in contrast to the topological case, it is not possible in general to make K0 (A) into a ring. We recall that for a compact topological space X there is an exterior product map K0 (X ) × K0 (X ) −→ K0 (X × X )
(2.3)
which is defined using the exterior tensor product of vector bundles. The diagonal map X → X × X induces a natural transformation K0 (X × X ) → K0 (X ). The composition of the two maps thereby leads to the product K0 (X ) × K0 (X ) −→ K0 (X ).
(2.4)
If A = C(X ) is the algebra of continuous functions on X , then the diagonal map translates into the product map m : A ⊗ A −→ A
(2.5)
on the algebra A given by m(a ⊗ b) = a b for all a, b ∈ A. Since A is commutative, the multiplication (2.5) is an algebra homomorphism and there is an induced map m ∗ : K0 (A ⊗ A) −→ K0 (A).
(2.6)
For a noncommutative C ∗ -algebra the multiplication map is not an algebra homomorphism and so we cannot expect that in general the map (2.6) will be defined. Recall that the suspension of a generic C ∗ -algebra A is the C ∗ -algebra (A) := C0 (R) ⊗ A. By definition one has K p (A) = K0 ( p (A)) = K0 (C0 (R p ) ⊗ A). Bott
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periodicity ensures that up to isomorphism there are only two distinct K-theory groups K0 (A) and K1 (A). One has k (A1 ) ⊗ l (A2 ) = C0 (Rk ) ⊗ A1 ⊗ C0 (Rl ) ⊗ A2 ∼ = C0 (Rk+l ) ⊗ (A1 ⊗ A2 ) = k+l (A1 ⊗ A2 ).
(2.7)
If we combine this formula with the exterior product (2.1) defined for K0 -groups then we obtain the general exterior product Kk (A1 ) × Kl (A2 ) −→ Kk+l (A1 ⊗ A2 ).
(2.8)
See [41, §4.7] for more details and examples. Many important statements in K-theory admit a concise formulation in terms of the product structure. For example, there exists a canonical class β ∈ K2 (C) = K0 (C0 (R2 )), called the Bott generator, such that the exterior product with β defines a map ⊗β
K0 (A) −−→ K2 (A ⊗ C) = K2 (A).
(2.9)
This provides the isomorphism required by the Bott periodicity theorem [41, §4.9]. These observations all find their most natural generalisation in Kasparov’s KK-theory [51], which we now proceed to describe. (See [5, Ch. VIII] for a more detailed exposition.) 2.2. KK-theory. Let B be a C ∗ -algebra. A Hilbert B-module H is a module over B equipped with a B-valued inner product H × H −→ B, (ζ, ζ ) −→ ζ | ζ ∈ B
(2.10)
which satisfies similar properties to those of an inner product with values in C [54]. We denote by L(H) the algebra of linear operators on H which admit an adjoint with respect to this inner product. The closed subalgebra generated by all rank 1 operators of the form θζ,ζ : ξ → ζ ζ | ξ is denoted K(H) and called the algebra of compact operators on H. The algebra K(H) is a closed ideal in L(H). Definition 2.1. Let A and B be C ∗ -algebras. An odd A–B Kasparov bimodule is a triple (H, ρ, F), where H is a countably generated Hilbert B-module, the map ρ : A → L(H) is a ∗-homomorphism, and F ∈ L(H) is a self-adjoint operator such that for each a ∈ A one has ρ(a) idH − F 2 ∈ K(H) and F ρ(a) − ρ(a) F ∈ K(H). (2.11) An even A–B Kasparov bimodule is a triple (H, ρ, F) where H = H+ ⊕ H− is Z2 -graded, φ is an even degree map, F is an odd map, and the compactness conditions (2.11) are satisfied. In both cases a triple is called degenerate if the operators ρ(a) (idH − F 2 ) and F ρ(a) − ρ(a) F are zero for all a ∈ A.
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We denote by E0 (A, B) and E1 (A, B) the sets of isomorphism classes of even and odd Kasparov bimodules, respectively. These two sets are made into semi-groups using the direct sum of Kasparov bimodules. Two triples (H, ρ, F) and (H, ρ , F ) are regarded as equivalent if (by adding a degenerate triple to both if necessary) F can be obtained from F via operator homotopy. Imposing these equivalence relations on E0 (A, B) and E1 (A, B) yields two abelian groups KK0 (A, B) and KK1 (A, B). The functor KK• (A, B) is homotopy invariant and satisfies excision in both variables with respect to C-split exact sequences of C ∗ -algebras. The special case where B = C is important. A Hilbert C-module H is just a Hilbert space, and the algebra L(H) is in this case the C ∗ -algebra of bounded linear operators on H. The compactness conditions (2.11) provide an abstraction of the essential properties of elliptic operators [2], and a Kasparov bimodule in this case is just a Fredholm module. Thus we can define the K-homology of the C ∗ -algebra A as K• (A) = KK• (A, C).
(2.12)
One can also show that KK• (C, A) is isomorphic to the K-theory K• (A) of the algebra A. The key property of the bivariant functor KK• (A, B) is the existence of an associative product ⊗B : KKi (A, B) × KK j (B, C) −→ KKi+ j (A, C)
(2.13)
induced by the composition of bimodules, which is additive in both variables. This product is called the composition or intersection product and it is compatible with algebra homomorphisms in the following sense. There is a functor from the category of separable C ∗ -algebras to an additive category KK whose objects are separable C ∗ -algebras and whose morphisms A → B are precisely the elements of KK• (A, B). An algebra homomorphism φ : A → B thus defines an element KK(φ) ∈ KK0 (A, B), and if ψ : B → C is another homomorphism then KK(ψ ◦ φ) = KK(φ) ⊗B KK(ψ) ∈ KK0 (A, C).
(2.14)
The intersection product makes KK• (A, A) into a Z2 -graded ring whose unit element is 1A := KK(idA), the element of KK0 (A, A) determined by the identity map idA : A → A. The operation of taking the composition product by a fixed element α ∈ KK0 (A, B) gives a map KKi (C, A) −→ KKi (C, B),
(2.15)
i.e., a homomorphism α∗ : Ki (A) → Ki (B) in K-theory, and also a map KKi (B, C) −→ KKi (A, C),
(2.16)
i.e., a homomorphism of K-homology groups α ∗ : Ki (B) → Ki (A). If α is the class of a bimodule (H, ρ, F), then (2.15) is the index map index F : Ki (A) → Ki (B). In general, we will say that the element α is invertible if there exists β ∈ KK0 (B, A) such that α ⊗B β = 1A ∈ KK0 (A, A) and β ⊗A α = 1B ∈ KK0 (B, B). We call β the inverse of α and write β = α −1 . An invertible element of KK0 (A, B) gives an isomorphism Ki (A) ∼ = Ki (B) of K-theory groups and of K-homology groups Ki (A) ∼ = Ki (B). This construction will be generalized in Sect. 3.1. The composition product (2.13), along with the natural map KKi (A, B) → KKi (A⊗ C, B⊗C) given by α → α⊗1C, imply the existence of a more general associative product
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in KK-theory called the Kasparov product. For any collection of separable C ∗ -algebras A1 , B1 , A2 , B2 and D there is a bilinear map KKi (A1 , B1 ⊗ D) ⊗D KK j (D ⊗ A2 , B2 ) −→ KKi+ j (A1 ⊗ A2 , B1 ⊗ B2 ). (2.17) This product can be thought of as a mixture between the usual cup and cap products. For D = C it specializes to the exterior product (also written ×): KKi (A1 , B1 ) ⊗ KK j (A2 , B2 ) −→ KKi+ j (A1 ⊗ A2 , B1 ⊗ B2 ).
(2.18)
When A1 = A2 = C, (2.18) is just the exterior product on K-theory that we discussed in Sect. 2.1 above. On the other hand, when we put B1 = C and A1 = C in (2.17) we recover the original composition product (2.13) in the form KKi (A1 , D) ⊗D KK j (D, B2 ) −→ KKi+ j (A1 , B2 ).
(2.19)
Various technical details of the Kasparov product, useful for explicit computations, are collected in Appendix A, and a pictorial method for keeping track of these is given in Appendix B. 2.3. Strong Poincaré duality. Poincaré duality for C ∗ -algebras was defined by Connes [20,21] in the context of real KK-theory as a means of defining noncommutative spinc manifolds. It was subsequently extended to more general situations by Kaminker and Putnam [50], Emerson [32,33], amongst others. These latter works motivate our first definition of the duality in the context of complex KK-theory. Definition 2.2 (Strong Poincaré Duality). A pair of separable C ∗ -algebras (A, B) is said to be a strong Poincaré dual pair (strong PD pair for short) if there exists a class ∈ KKd (A ⊗ B, C) = Kd (A ⊗ B) in the K-homology of A ⊗ B and a class ∨ ∈ KK−d (C, A ⊗ B) = K−d (A ⊗ B) in the K-theory of A ⊗ B with the properties ∨ ⊗B = 1A ∈ KK0 (A, A) and ∨ ⊗A = (−1)d 1B ∈ KK0 (B, B). (2.20) The element is called a fundamental K-homology class for the pair (A, B) and ∨ is called its inverse. A separable C ∗ -algebra A is said to be a strong Poincaré duality algebra (strong PD algebra for short) if (A, Ao ) is a strong PD pair, where Ao denotes the opposite algebra of A, i.e., the algebra with the same underlying vector space as A but with the product reversed. Remark 2.3. The use of the opposite algebra in this definition is to describe A-bimodules as (A ⊗ Ao )-modules. We will see this explicitly in Sect. 2.5 below. Let us indicate how Definition 2.2 is used to implement Poincaré duality. First of all, we note that the tensor product algebra A ⊗ B is canonically isomorphic to the algebra B⊗A through the “flip” map A⊗B → B⊗A which interchanges the two factors. Thus Kd (A ⊗ B) ∼ = Kd (B ⊗ A) and K−d (A ⊗ B) ∼ = K−d (B ⊗ A). With this observation, we can use the Kasparov product (2.17) to induce a map ⊗A : KKd (A ⊗ B, C) ⊗ Ki (A) ∼ = KKd (B ⊗ A, C) ⊗ KKi (C, A) −→ KKd+i (B, C) = Kd+i (B).
(2.21)
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Thus taking the product on the right with the element ∈ KKd (A ⊗ B, C) produces a map ⊗A
Ki (A) −−−→ Ki+d (B)
(2.22)
from the K-theory Ki (A) of the algebra A to the K-homology Ki+d (B) of the algebra B. Since the element has an inverse ∨ ∈ KK−d (C, A ⊗ B), using the exterior product again we can define a map ⊗B : KK−d (C, A ⊗ B) ⊗ KKi (B, C) −→ KK−d+i (C, A) = K−d+i (A). (2.23) Thus multiplying on the left by the element ∨ establishes a map ∨ ⊗B
Ki (B) −−−−→ Ki−d (A) from the K-homology of B to the K-theory of A. Since and ∨ are inverses to each other, for any x ∈ Ki (A) one has ∨ ⊗B (x ⊗A ) = ∨ ⊗B ⊗A x = 1A ⊗A x = x.
(2.24)
(2.25)
As a consequence the two maps (2.22) and (2.24) are inverse to each other, up to the sign given in (2.20) which results from graded commutativity of the exterior product (2.18). (See [20, p. 588] and [32, §3] for further details.) Thus when (A, B) is a strong PD pair, the elements and ∨ establish isomorphisms Ki (A) ∼ = Ki+d (B)
and
Ki (B) ∼ = Ki−d (A),
(2.26)
which is the fundamental property of any form of Poincaré duality. More generally, for any pair of separable C ∗ -algebras (C, D), the maps ⊗A : KKi (C, A ⊗ D) −→ KKi+d (C ⊗ B, D), ∨ ⊗B : KKi (C, B ⊗ D) −→ KKi−d (C ⊗ A, D)
(2.27)
are also isomorphisms, showing that Poincaré duality with arbitrary coefficients holds in this case (Compare [20]). By setting C, D equal to various choices from the collection of algebras C, A, B, we may infer from (2.27) that the four maps ⊗A : KKi (A, A) ∨ ⊗B : KKi (B, B) ⊗A : KKi (C, A ⊗ B) ∨ ⊗B : KKi (C, B ⊗ A)
−→ −→ −→ −→
KKi+d (A ⊗ B, C), KKi−d (B ⊗ A, C), KKi+d (B, B), KKi−d (A, A)
(2.28)
are all isomorphisms. It follows that if (A, B) is a strong PD pair, then a fundamental class for (A, B) induces isomorphisms Ki+d (A ⊗ B) ∼ = KKi (A, A) ∼ = KKi (B, B) ∼ = Ki+d (A ⊗ B). Proposition 2.4. Let A and B be separable C ∗ -algebras. Then: (1) A is a strong PD algebra if and only if Ao is a strong PD algebra; and
(2.29)
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(2) If A is Morita equivalent to B, then A is a strong PD algebra if and only if B is a strong PD algebra. Proof. (1) follows easily since is a fundamental class for A if and only if it is a fundamental class for Ao , where we identify the KK-groups of A ⊗ Ao with those of the flip Ao ⊗ A. The proof of (2) is in [52, §4 Theorem 7]. Example 2.5. Let X be a complete oriented manifold. Then the two pairs of C ∗ -algebras (C0 (X ), C0 (T ∗ X )) and (C0 (X ), C0 (X, Cliff(T ∗ X ))) are both strong PD pairs, where Cliff(T ∗ X ) is the Clifford algebra bundle of the cotangent bundle T ∗ X . If in addition X is spinc then C0 (Cliff(T ∗ X )) and C0 (X ) are Morita equivalent, and so C0 (X ) is a strong PD algebra. If moreover X is compact with boundary ∂ X equipped with the induced spinc structure, then the K-homology connecting homomorphism takes the fundamental class of X to the fundamental class of ∂ X . It follows that (C(X ), C(X, ∂ X )) is a strong PD pair, where C(X, ∂ X ) is the C ∗ -algebra of continuous functions on X which vanish at the boundary ∂ X . Example 2.6. Let be a K-amenable, torsion-free discrete group whose classifying space B is a smooth oriented manifold. Suppose that has the Dirac-dual Dirac property, i.e., for any proper –C ∗ -algebra A, a Dirac element α ∈ KK0 (A, C) in the -equivariant K-homology of A and a dual Dirac element β ∈ KK0 (C, A) in the -equivariant K-theory of A satisfy the conditions α ⊗C β = 1A ∈ KK0 (A, A) and β ⊗A α = 1C ∈ KK0 (C, C).
(2.30)
The Dirac element α is constructed using a spinc Dirac operator. Recall that a multiplier σ on the group is a normalized, U (1)-valued group 2-cocycle on . Its Dixmier-Douady invariant δ (σ ) ∈ H3 (, Z) is induced in the usual way via the short exact sequence of coefficients in (1.37). Given σ we consider the reduced twisted group C ∗ -algebra Cr∗ (, σ ). Then (C0 (T ∗ B), Cr∗ (, σ )) is a strong PD pair for every multiplier σ on with trivial Dixmier-Douady invariant (we have used the fact that the Baum-Connes conjecture holds for in this case). In particular, this holds whenever is: • A torsion-free, discrete subgroup of S O(n, 1) or of SU (n, 1); or • A torsion-free, amenable group. If moreover B is spinc , then C0 (T ∗ B) is a strong PD algebra and hence Cr∗ (, σ ) is a strong PD algebra for every multiplier σ on with trivial Dixmier-Douady invariant. In particular, we conclude that the noncommutative torus Tθ2n is a strong PD algebra, g and more generally the noncommutative higher genus Riemann surface Rθ is a strong PD algebra. We will consider these latter examples in more detail in Sect. 2.6.
2.4. Duality groups. We will now determine how many fundamental classes a given strong PD pair admits. Proposition 2.7. Let (A, B) be a strong PD pair, and let ∈ Kd (A ⊗ B) be a fundamental class with inverse ∨ ∈ K−d (A ⊗ B). Let ∈ KK0 (A, A) be an invertible element. Then ⊗A ∈ Kd (A ⊗ B) is another fundamental class, with inverse ∨ ⊗A −1 ∈ K−d (A ⊗ B).
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A
C
∆∨
663 −1
A
A
A
◦
◦
B
1B
∆
C
B
Fig. 2.1. Diagram representing the proof of Proposition 2.7
Proof. Using associativity of the Kasparov product along with the flip isomorphism A ⊗ B → B ⊗ A, we compute ∨ ⊗A −1 ⊗A ( ⊗A ) = ∨ ⊗A −1 ⊗A ⊗A = ∨ ⊗A (1A ⊗A ) = ∨ ⊗ A = (−1)d 1B.
(2.31)
The calculation in the other direction is similar, but slightly trickier because of notational quirks in the way the Kasparov product is written. The calculation goes as follows: ∨ ⊗A −1 ⊗B ( ⊗A ) = ⊗A ∨ ⊗B ⊗A −1 = ⊗A 1A ⊗A −1 = ⊗A −1 = 1A.
(2.32)
It would appear that we needed to reorder many of the factors in the product, but in fact, all we are really using is the associativity of the product, in the form discussed in Appendix B. In terms of the diagram calculus discussed there, we simply need to consider the diagram depicted in Fig. 2.1. This result has a converse. Proposition 2.8. Let (A, B) be a strong PD pair, and let 1 , 2 ∈ Kd (A ⊗ B) be fun∨ ∨ damental classes with inverses ∨ 1 , 2 ∈ K−d (A⊗B). Then 1 ⊗B 2 is an invertible ∨ d element in KK0 (A, A), with inverse given by (−1) 2 ⊗B 1 ∈ KK0 (A, A). Proof. As above we compute ∨ ∨ ∨ 1 ⊗B 2 ⊗A ∨ 2 ⊗ B 1 = 1 ⊗ B 2 ⊗ A 2 ⊗ B 1 d = ∨ 1 ⊗B (−1) 1B ⊗B 1
= (−1)d ∨ 1 ⊗ B 1 = (−1)d 1A, and similarly with 1 and 2 interchanged.
(2.33)
As an immediate consequence of Propositions 2.7 and 2.8 above, we deduce the following.
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Corollary 2.9. Let (A, B) be a strong PD pair. Then the moduli space of fundamental classes for (A, B) is isomorphic to the group of invertible elements in the ring KK0 (A, A). We now make some clarifying comments concerning the corollary above. In the physics literature, K-orientations on a smooth manifold X are generally linked to spinc structures, which (once an orientation has been fixed) are an affine space modeled on the second integral cohomology of X , H2 (X, Z), i.e., the isomorphism classes of line bundles over X . However, the space of K-orientations of X is in general much larger, being a principal homogeneous space for the abelian group of units in the ring K• (X ), consisting of stable isomorphism classes of virtual vector bundles over X of virtual rank equal to 1, and with group operation given by tensor product. The space of all fundamental classes of X in KK-theory is in general still larger, being a principal homogeneous space for the abelian group of units of KK0 (C(X ), C(X )), which in turn by [80] is an extension of Aut K• (X ) by Ext Z (K• (X ), K•+1 (X )). Recall that in the situation of Corollary 2.9 there is an isomorphism KK0 (A, A) ∼ = KK0 (B, B). This moduli space is called the duality group of the pair (A, B) and is denoted KK0 (A, A)−1 . It can be computed explicitly using (2.29) from either the K-theory or the K-homology of the algebra A ⊗ B. We will now describe two illustrative and broad classes of examples of strong PD algebras.
2.5. Spectral triples. There is a natural object that encodes the geometry of a D-brane whose construction can be motivated by the observation that bounded Kasparov modules are not the most useful ones in practical applications of KK-theory, especially when it comes to defining the Kasparov product (2.17) [4]. A zeroth order elliptic pseudodifferential operator F on a smooth closed manifold X determines a class in the K-homology of X , i.e., a class in KK• (C(X ), C). However, the product of two such operators need not be a pseudodifferential operator. The Kasparov product is handled better when one uses first order operators instead. But a first order elliptic pseudodifferential operator D will not in general extend to a bounded operator on L 2 (X ) and so will not generally provide a class in KK• (C(X ), C). The trick here, due to Kasparov and reformulated by Baaj and Julg [4], is to replace D by the operator D (1 + D 2 )−1/2 which gives rise to a bounded Fredholm module and so produces a class in KK• (C(X ), C). The Baaj-Julg construction generalises to noncommutative C ∗ -algebras. Definition 2.10. A spectral triple over a unital C ∗ -algebra A is a triple (A, H, D), where the algebra A is represented faithfully on a Hilbert space H and D is an unbounded self-adjoint operator on H with compact resolvent such that the commutator [D, a] is bounded for all a ∈ A. In the even case we assume that the Hilbert space H is Z2 -graded and that D is an odd operator with respect to this grading, i.e., there is an involution γ on H which implements the grading and which anticommutes with D. One can prove [4] that all classes in the K-homology of A are obtained from such spectral triples, which in this context are also called unbounded K-cycles. Together with Connes’ axioms [21] such a K-cycle defines a noncommutative spinc manifold. We will not enter into a detailed account of this latter characterization but refer to [21] and [39, Chap. 10] for a thorough discussion. In the example of a flat D-brane (W, E, φ) in the spacetime X , the spectral triple is (C(X ), L 2 (W, SW ⊗ E), D / E ) as specified in the proof of Proposition 1.4. This definition can be generalized to provide unbounded A–B
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bimodules, which have the property that every element of KK• (A, B) arises from such an unbounded bimodule. Let (A, H, D) be an unbounded spectral triple. This fixes our noncommutative spacetime. Let Ao be the opposite algebra of the algebra A. The action of the algebra A ⊗ Ao on the Hilbert space H is described by using commuting actions of the algebras A and Ao , making H into a bimodule over the algebra A. The action of A is provided by the representation of A on H which is part of the data given by the spectral triple (A, H, D). The algebra Ao is assumed to act by means of operators bo for b ∈ A. We assume that the two actions commute, i.e., [a, bo ] = 0 for all a, b ∈ A, and that [D, bo ] is bounded for all b ∈ A. Definition 2.11. The index class of the noncommutative spacetime (A, H, D) is the class D ∈ Kd (A ⊗ Ao ) given by the data (A ⊗ Ao , H, D). If the index class is also a fundamental class for A, then its inverse ∨D ∈ K−d (A ⊗ Ao ) is called the Bott class of A. One advantage of the spectral triple formulation is that under suitable circumstances it enables the straightforward construction of a smooth subalgebra A∞ of a C ∗ -algebra A. This is a dense ∗-subalgebra of A which is stable under holomorphic functional calculus, i.e., if a ∈ A∞ with a = a ∗ and a > 0, then f (a) ∈ A∞ for every holomorphic function f on a neighbourhood of the spectrum Spec(a). By the Karoubi density theorem (see for example [19]), the inclusion homomorphism ι : A∞ → A then induces an isomorphism in K-theory. Given a spectral triple (A, H, D), we assume that the smooth domain H∞ = (2.34) Dom D k k∈N
of the operator D is an A∞ -bimodule for some smooth subalgebra A∞ ⊂ A, or equivalently an A∞ ⊗ (A∞ )o -module. We further assume that A∞ is a Fréchet algebra for the family of semi-norms
qk (a) = δ k (a)
(2.35)
provided by the derivation δ : a → [ |D|, a] on A. The usage of smooth subalgebras, and in particular topological algebras, will be important later on when we start employing cyclic theory. Example 2.12. Let X be a compact spinc manifold of dimension d. Let A∞ = C ∞ (X ) be the Fréchet algebra of smooth functions on X . It acts by pointwise multiplication on the Hilbert space H = L 2 (X, S X ) of square integrable spinors on X . This Hilbert space is Z2 -graded when d is even, with the usual grading operator γ defining the split S X = S+X ⊕ S− X into irreducible half-spinor bundles, and ungraded in the odd case. For the operator D we take the usual spinc Dirac operator D = D / acting on H. Then (A∞ , H, D / ) defines a cycle [D / ] in Kd (A) [39, Theorem 9.20]. This data determines the index class in Kd (A ⊗ Ao ). Because the algebra A is commutative in this case, one has A = Ao and the multiplication map (2.5) is an algebra homomorphism. Since the K-homology functor is contravariant, there is a map m ∗ : Kd (A) −→ Kd (A ⊗ A)
(2.36)
induced by (2.5). The image of the class [D / ] under this homomorphism is the index class D/ [39, p. 488].
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2.6. Twisted group algebra completions of surface groups. The noncommutative twotorus Tθ2 provides the original example of noncommutative Poincaré duality which was described by Connes [21] in the context of real spectral triples. It is the specialization to genus one of the example that we present here. Let g be the fundamental group of a compact, oriented Riemann surface g of genus g ≥ 1. It has the presentation g [U j , V j ] = 1 , (2.37) g = U j , V j , j = 1, . . . , g j=1
and Bg = g is a smooth spin manifold. Since H2 (g , U (1)) ∼ = R/Z, for each θ ∈ [0, 1) we can identify a unique multiplier σθ on g up to isomorphism. Let C(g , σθ ) be the σθ -twisted convolution algebra of finitely supported maps g → C, which is spanned over C by a set of formal letters δγ , γ ∈ g satisfying δγ δµ = σθ (γ , µ) δγ µ . Let f denote the operator norm of the operator on 2 (g ) given by left convolution with f ∈ g . Then the completion of C(g , σθ ) with respect to this norm is the reduced twisted group C ∗ -algebra Cr∗ (g , σθ ). It can also be viewed as the C ∗ -algebra generated by unitaries U j and V j satisfying the commutation relation g
[U j , V j ] = exp(2π i θ ).
(2.38)
j=1
On Cr∗ (g , σθ ) there is a canonical trace τ defined by evaluation at the identity element of g . Let D be the operator defined by Dδγ = (γ ) δγ ,
(2.39)
where (γ ) ∈ [0, ∞) is the word length of γ ∈ g . Let δ = ad(D) denote the commutator [D, −]. Then δ is an unbounded closed derivation on the reduced twisted group C ∗ -algebra Cr∗ (g , σθ ). Consider the smooth subalgebra R∞ (g , σθ ) := (2.40) Dom δ k . k∈N
Since R∞ (g , σθ ) contains δγ ∀γ ∈ g , it contains C(g , σθ ). Hence it is dense in Cr∗ (g , σθ ). Since R∞ (g , σθ ) is defined as a domain of derivations, it is closed under holomorphic functional calculus. Because g is a surface group, it follows from a variant of a result by Jolissaint [49] that there exists k ∈ N and a positive constant C such that for all f ∈ C(g , σθ ) one has the Haagerup inequality, f ≤ C νk ( f ),
(2.41)
where ⎛ νk ( f ) = ⎝
⎞1/2 (1 + (γ ))2k | f (γ )|2 ⎠
.
(2.42)
γ ∈g
Using this, it is routine to show that R∞ (g , σθ ) is a Fréchet algebra, complete in the semi-norms (2.42) induced by δ k .
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667 g
To define the fundamental class of the noncommutative Riemann surface Rθ := ∗ g , σθ ), we recall from §8 [16], the K-theory of Cr (g , σθ ). For any θ , the multiplier σθ has trivial Dixmier-Douady invariant, as δg (σθ ) ∈ H3 (g , Z) = 0, and so (via the Baum-Connes assembly map) K0 (Cr∗ (g , σθ )) ∼ = K0 (g ) = Z2 . For irrational θ , the algebras Cr∗ (g , σθ ) are distinguished for different values of θ by the image of the trace map induced on K-theory by the trace τ . In a basis e0 , e1 of K0 (Cr∗ (g , σθ )) the trace is given by R∞ (
τ (n e0 + m e1 ) = n + m θ.
(2.43)
We choose e0 = [1], the class of the identity element, and e1 such that τ (e1 ) = θ . Another result in §8 [16] is K1 (Cr∗ (g , σθ )) ∼ = K1 (g ) = Z2g . Moreover, the unitaries ∗ U j and V j form a basis for K1 (Cr (g , σθ )), §6 [58]. Then the inverse fundamental class of Cr∗ (g , σθ ) is given by ∨ = e0 ⊗ e1o − e1 ⊗ e0o +
g U j ⊗ V jo − V j ⊗ U oj .
(2.44)
j=1
The trace τ also leads to an inner product on Rθ defined by (a, b) = τ (b∗ a) for g g g a, b ∈ Rθ . Let L 2 (Rθ ) denote the completion of Rθ with respect to this inner product, g g and define H := L 2 (Rθ ) ⊕ L 2 (Rθ ). Then the element (2.44) is the Bott class of the g g spectral triple (Rθ , H, D), with Rθ acting diagonally on H by left multiplication and D odd with respect to the canonical Z2 -grading γ on H. g
2.7. Other notions of Poincaré duality. We now return to the general theory and introduce some alternative weaker forms of the duality described in Sect. 2.3 above, all of which imply the fundamental property (2.26). We start with a “pointwise” version of Definition 2.2. Definition 2.13 (Weak Poincaré Duality). A pair of separable C ∗ -algebras (A, B) is said to be a weak Poincaré duality pair (weak PD pair for short) if there exists a class ∈ KKd (A ⊗ B, C) = Kd (A ⊗ B) in the K-homology of A ⊗ B and a class ∨ ∈ KK−d (C, A ⊗ B) = K−d (A ⊗ B) in the K-theory of A ⊗ B with the properties ∨ ⊗B ⊗A x = x ∀ x ∈ KK0 (C, A) (2.45) and ∨ ⊗A ⊗B y = (−1)d y
∀ y ∈ KK0 (C, B).
(2.46)
A separable C ∗ -algebra A is said to be a weak Poincaré duality algebra (weak PD algebra for short) if (A, Ao ) is a weak PD pair. Example 2.14. Let be a torsion-free, discrete group having the Dirac-dual Dirac property such that B is a smooth oriented manifold. Then (C0 (T ∗ B), Cr∗ (, σ )) is a weak PD pair for every multiplier σ on with trivial Dixmier-Douady invariant. If moreover B is spinc , then C0 (T ∗ B) is a weak PD algebra and so Cr∗ (, σ ) is a weak PD algebra for every multiplier σ on with trivial Dixmier-Douady invariant. In particular, this holds whenever is:
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• A torsion-free, word hyperbolic group; • A torsion-free, cocompact lattice in a product of a finite number of groups among Lie or p-adic groups of rank one, or in S L 3 (F) with F a local field, H or E 6(−26) ; or • A torsion-free lattice in a reductive Lie group or in reductive groups over non-archimedean local fields. The duality of Definition 2.13 can also be weakened to hold only modulo torsion elements of the K0 -groups of the algebras involved. Definition 2.15 (Rational Poincaré Duality). A pair of separable C ∗ -algebras (A, B) is said to be a rational Poincaré duality pair (Q–PD pair for short) if there exists a class ∈ KKd (A ⊗ B, C) = Kd (A ⊗ B) in the K-homology of A ⊗ B and a class ∨ ∈ KK−d (C, A ⊗ B) = K−d (A ⊗ B) in the K-theory of A ⊗ B with the properties ∨ ⊗B ⊗A x = x ∀ x ∈ KK0 (C, A) ⊗ Q (2.47) and ∨ ⊗A ⊗B y = (−1)d y
∀ y ∈ KK0 (C, B) ⊗ Q.
(2.48)
A separable C ∗ -algebra A is said to be a rational Poincaré duality algebra (Q–PD algebra for short) if (A, Ao ) is a Q–PD pair. Example 2.16. Let X be an oriented rational homology manifold, such as the quotient of a manifold by an orientation-preserving action of a finite group. Then C0 (X ) is a Q–PD algebra. Example 2.17. Let be a discrete group with the Dirac-dual Dirac property and with a torsion-free subgroup 0 of finite index such that B0 is a smooth oriented manifold. Then (C0 (T ∗ B), Cr∗ (, σ )) is a Q–PD pair for every multiplier σ on . (Note that the Dixmier-Douady invariant in this case is always torsion.) If moreover B0 is spinc , then C0 (T ∗ B) is a Q–PD algebra and hence Cr∗ (, σ ) is a Q–PD algebra for every multiplier σ on . In particular, this holds whenever is: • A word hyperbolic group; or • A cocompact lattice in a product of a finite number of groups among Lie or p-adic groups of rank one, or in S L 3 (F) with F a local field, H or E 6(−26) . Finally, we can take the fundamental property (2.26) itself as the weakest form of the duality. Definition 2.18 (Poincaré Duality). A pair of separable C ∗ -algebras (A, B) is said to be a Poincaré duality pair (PD pair for short) if there exist isomorphisms Ki (A) ∼ = Ki+d (B)
and
Ki (B) ∼ = Ki−d (A).
(2.49)
A separable C ∗ -algebra A is said to be a Poincaré duality algebra (PD algebra for short) if (A, Ao ) is a PD pair. Remark 2.19. For any C ∗ -algebra A, the K-theory groups of A and Ao are isomorphic. This follows easily from the fact that if p is a projection in A and p o is the corresponding element of Ao , then p o is also a projection. Similarly for a unitary u ∈ A, the corresponding element u o of Ao is also unitary.
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In fact, there is a more general statement, which we will need later. The additive category KK with separable C ∗ -algebras as objects and with KK• (A, B) as the morphisms from A to B may be viewed as a certain completion of the stable homotopy category of separable C ∗ -algebras [5, §22]. As such, it has an involution o induced by the involution f → f o sending f : A → B to the ∗-homomorphism f o : Ao → Bo which sends a o to ( f (a))o . The above isomorphism from K• (A) to K• (Ao ) is simply this involution KK• (C, A) ∼ = KK• (Co = C, Ao ). 3. KK-Equivalence In this section we introduce the notion of KK-equivalence and describe its intimate connection to Poincaré duality and Morita equivalence for C ∗ -algebras. As in the previous section, we will describe various criteria for the equivalence and describe several natural inequivalent versions of it, giving illustrative commutative and noncommutative examples.
3.1. Strong KK-equivalence. Our first notion of equivalence in KK-theory is a generalization of the standard definition that was essentially already described in Sect. 2.2. Definition 3.1 (Strong KK-Equivalence). A pair of separable C ∗ -algebras (A, B) are said to be strongly KK-equivalent if there are elements α ∈ KKn (A, B)
and
β ∈ KK−n (B, A)
(3.1)
α ⊗B β = 1A ∈ KK0 (A, A)
and
β ⊗A α = 1B ∈ KK0 (B, B).
(3.2)
such that
The significance of this definition stems from the following results. Lemma 3.2. Suppose that the pair of separable C ∗ -algebras (A, B) are strongly KKequivalent. Then the maps α⊗B : Ki (B) −→ Ki+n (A), ⊗Bα : Ki (B) −→ Ki+n (A),
β⊗A : Ki (A) −→ Ki−n (B), ⊗Aβ : Ki (A) −→ Ki−n (B)
(3.3) (3.4)
are all isomorphisms. Proof. By the associativity property of the Kasparov product, the maps in (3.3) satisfy β ⊗A (α ⊗B x) = (β ⊗A α) ⊗B x = 1B ⊗B x =x ∀ x ∈ Ki (B), α ⊗B (β ⊗A y) = (α ⊗B β) ⊗A y = 1A ⊗A y =y ∀ y ∈ Ki (A),
(3.5)
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and are therefore all isomorphisms. Again by the associativity property of the Kasparov product, the maps in (3.4) satisfy (z ⊗B α) ⊗A β = = = (w ⊗A β) ⊗B α = = = and thus are also isomorphisms.
z ⊗B (α ⊗A β) z ⊗B 1B z ∀ z ∈ Ki (B), w ⊗A (β ⊗B α) w ⊗A 1A w ∀ w ∈ Ki (A),
(3.6)
Remark 3.3. As in (2.27), one can generalize the isomorphisms of Lemma 3.2 above to arbitrary coefficients. For any pair of separable C ∗ -algebras (C, D), the maps α⊗B β⊗A ⊗Bα ⊗Aβ
: : : :
KKi (C ⊗ B, D) KKi (C ⊗ A, D) KKi (C, B ⊗ D) KKi (C, A ⊗ D)
−→ KKi+n (C ⊗ A, D), −→ KKi−n (C ⊗ B, D), −→ KKi+n (C, A ⊗ D), −→ KKi−n (C, B ⊗ D)
(3.7)
are all isomorphisms. Lemma 3.4. Suppose that the pair of separable C ∗ -algebras (A, B) are strongly KKequivalent. Then A is a PD algebra (resp., strong PD algebra, weak PD algebra) if and only if B is a PD algebra (resp., strong PD algebra, weak PD algebra). Proof. If A is a PD algebra, then there are isomorphisms Ki (A) ∼ = Ki+d (A). Combining this with the isomorphisms given in Lemma 3.2, we deduce that there are isomorphisms Ki (B) ∼ = Ki+d (B), showing that B is also a PD algebra. The argument is symmetric, proving the result. We will now investigate some circumstances under which KK-equivalence holds. Let A be a C ∗ -algebra, and let H be a Hilbert A-module. Recall from Sect. 2.2 that the norm closure of the linear span of the set ζ | ζ | ζ, ζ ∈ H is the algebra K(H) of compact operators on H. The module H is said to be full if K(H) is equal to A. Definition 3.5 (Strong Morita equivalence). Two C ∗ -algebras A and B are said to be strongly Morita equivalent if there is a full Hilbert A-module H such that K(H) ∼ = B. Upon identifying B with K(H), we define a Hilbert (A, B)-bimodule H∨ as follows. As sets (or real vector spaces), one has H∨ = H. Let ζ → ζ ∨ denote the identity map on H → H∨ . Since λ ζ ∨ = ( λ ζ )∨ for λ ∈ C and ζ ∈ H, it follows that the identity map is conjugate linear. For ζ1 , ζ2 ∈ H∨ , a ∈ A and b ∈ B we set a ζ1∨ = (ζ1 a ∗ )∨ , ζ1∨ b = (b∗ ζ1 )∨ , and ζ1∨ , ζ2∨ = ζ1 ζ2 , −.
(3.8)
Then H∨ is a Hilbert B-module which is full by definition. Moreover, the map ζ1∨ ζ2∨ , − → ζ1 , ζ2 identifies K(H∨ ) with A. From the point of view of the present paper, the importance of this notion stems from the fact that Morita equivalent algebras encode the same physics. The following well-known lemma relates strong Morita equivalence to strong KK-equivalence.
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Lemma 3.6. Let A and B be separable C ∗ -algebras. If A is strongly Morita equivalent to B, then A is strongly KK-equivalent to B. Proof. Let H be a full Hilbert A-module implementing the Morita equivalence between A and B. Define elements β ∈ KK0 (B, A) by the equivalence class of (H, i, 0), where i : B → K(H) is the identity, and α ∈ KK0 (A, B) by the equivalence class of (H∨ , i ∨ , 0) in the notation above. Then the map ζ1 ⊗ ζ2∨ → ζ1 ζ2 , − identifies the B-bimodule H ⊗A H∨ with B, and hence β ⊗A α = 1B. Similarly, the map ζ1∨ ⊗ ζ2 → ζ1∨ ζ2∨ , − identifies the A-bimodule H∨ ⊗B H with A. Therefore α ⊗B β = 1A, proving that (A, B) is a strongly KK-equivalent pair. There are many examples of strongly KK-equivalent algebras that are not strongly Morita equivalent. For example, by [80] any two type I separable C ∗ -algebras with the same K0 and K1 groups are automatically strongly KK-equivalent. Another famous example concerns the two-dimensional noncommutative tori Tθ2 . We recall [68,77] that Tθ2 is Morita equivalent to Tθ2 if and only if θ and θ belong to the same G L 2 (Z) orbit. On the other hand, the algebras Tθ2 and C(T2 ) are strongly KK-equivalent for all θ [68]. The following lemma from [52] gives us more examples of strongly KK-equivalent algebras. Lemma 3.7. The Thom isomorphism for an oriented real vector bundle E → X gives a natural strong KK-equivalence between the algebra C0 (E) of continuous functions on E vanishing at infinity and the algebra C0 (X, Cliff(E)) of continuous sections, vanishing at infinity, of the Clifford algebra bundle Cliff(E) of E. Remark 3.8. If E → X is a spinc vector bundle then δ X (Cliff(E)) = 0, and the C ∗ -algebras C0 (Cliff(E)) and C0 (X ) are strongly Morita equivalent. In this case the space of sections C0 (X, Cliff(E)) can be replaced by C0 (X ) in Lemma 3.7. 3.2. Other notions of KK-equivalence. We now introduce variants of the concept of strong KK-equivalence. Definition 3.9 (Weak KK-Equivalence). A pair of separable C ∗ -algebras (A, B) are said to be weakly KK-equivalent if there are elements α ∈ KKn (A, B)
and
β ∈ KK−n (B, A)
(3.9)
such that (α ⊗B β) ⊗A y = y ∀ y ∈ Ki (A), z ⊗B (α ⊗A β) = z ∀ z ∈ Ki (B) (3.10) and (β ⊗A α) ⊗B x = x ∀ x ∈ Ki (B), w ⊗A (β ⊗B α) = w ∀ w ∈ Ki (A).(3.11) Definition 3.10 (Rational KK-Equivalence). A pair of separable C ∗ -algebras (A, B) are said to be rationally KK-equivalent if there are elements α ∈ KKn (A, B)
and
β ∈ KK−n (B, A)
(3.12)
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such that (α ⊗B β) ⊗A y = y ∀ y ∈ Ki (A) ⊗ Q, z ⊗B (α ⊗A β) = z ∀ z ∈ Ki (B) ⊗ Q (3.13) and (β ⊗A α) ⊗B x = x ∀ x ∈ Ki (B) ⊗ Q, w ⊗A (β ⊗B α) = w ∀ w ∈ Ki (A) ⊗ Q. (3.14) Definition 3.11 (K-Equivalence). A pair of separable C ∗ -algebras (A, B) are said to be K-equivalent if there are isomorphisms Ki (A) ∼ = Ki−n (B)
and
Ki (A) ∼ = Ki−n (B).
(3.15)
With a proof along the lines of Lemma 3.4, one can prove the following. Lemma 3.12. Suppose that the pair of separable C ∗ -algebras (A, B) are weakly KKequivalent (resp., rationally KK-equivalent, K-equivalent). Then A is a weak PD algebra (resp., Q–PD algebra, PD algebra) if and only if B is a weak PD algebra (resp., Q–PD algebra, PD algebra). In the remainder of this section we will give some classes of examples illustrating the various notions of KK-equivalence introduced above.
3.3. Universal coefficient theorem. To understand the relation between weak and strong KK-equivalence, we appeal to the universal coefficient theorem of Rosenberg and Schochet [80]. It holds precisely for the class N of C ∗ -algebras which are KK-equivalent to commutative C ∗ -algebras. For every pair of C ∗ -algebras (A, B) with A ∈ N and B separable, there is a split short exact sequence of abelian groups given by 0 → Ext Z (K•+1 (A), K• (B)) → KK• (A, B) → HomZ (K• (A), K• (B)) → 0. (3.16) Since there are many examples of C ∗ -algebras which are not in N [83], the notion of K-equivalence may strictly contain that of strong KK-equivalence. More precisely, suppose that A is not in N. If A satisfies the universal coefficient theorem for one-variable K-homology but not for KK-theory, then A is K-equivalent to a commutative C ∗ -algebra but not strongly KK-equivalent to such an algebra. (We do not know if such algebras exist, but this is a possibility.) Remark 3.13. From Remark 2.19 and the universal coefficient theorem (3.16) it follows that if A lies in the category N of C ∗ -algebras discussed above, then A and Ao are strongly KK-equivalent. (It is easy to construct examples, however, where they are not Morita equivalent.) We are not sure if A and Ao are always strongly KK-equivalent, without any hypotheses on A.
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3.4. Deformations. Let A and B be C ∗ -algebras. A deformation of A into B is a continuous field of C ∗ -algebras over a half-open interval [0, ε), locally trivial over the open interval (0, ε), whose fibre over 0 is A and whose fibres over ∈ (0, ε) are all isomorphic to B. A deformation gives rise to an extension of C ∗ -algebras 0 −→ C0 ((0, ε), B) −→ C −→ A −→ 0.
(3.17)
Connes and Higson observed that any deformation from A into B defines a morphism K• (A) → K• (B), which is simply the connecting homomorphism ∂ in the six-term exact K-theory sequence associated to (3.17). Moreover, when A is nuclear, the extension (3.17) has a completely positive cross-section and thus defines an element in KK0 (A, B) which induces the map on K-theory groups. 3.5. Homotopy equivalence. Let A and B be C ∗ -algebras. Two algebra homomorphisms φ0 , φ1 : A → B are homotopic if there is a path γt , t ∈ [0, 1] of homomorphisms γt : A → B such that t → γt (a) is a norm continuous path in B for every a ∈ A and such that γ0 = φ0 , γ1 = φ1 . The algebras A and B are said to be homotopy equivalent if there exist morphisms φ : A → B and η : B → A whose compositions η ◦ φ and φ ◦ η are homotopic to the identity maps on A and B, respectively. The algebra A is called contractible if it is homotopy equivalent to the zero algebra. Lemma 3.14. If A and B are homotopy equivalent C ∗ -algebras, then the pair (A, B) are strongly KK-equivalent. Proof. Suppose that φ : A → B and η : B → A are ∗-homomorphisms which are homotopy inverses to one another. Then they define classes KK(φ) ∈ KK0 (A, B) and KK(η) ∈ KK0 (B, A) whose Kasparov products are simply KK(η ◦ φ) ∈ KK0 (A, A) and KK(φ ◦ η) ∈ KK0 (B, B). Since η ◦ φ is homotopic to idA, one has KK(η ◦ φ) = 1A by homotopy invariance of KK-theory, and similarly KK(φ ◦ η) = 1B. 4. Cyclic Theory As was crucial in the definition of D-brane charge given in Sect. 1.1, the topological K-theory of a spacetime X is connected to its cohomology through the rational isomorphism provided by the Chern character ch : K• (X ) ⊗ Q → H• (X, Q). While this works well in the case of flat D-branes, in the more general settings described before we need a more general cohomological framework in which to express the D-brane charge, particularly when the branes are described by a noncommutative algebra A. The appropriate receptacle for the Chern character in analytic K-theory is a suitable version of the cyclic cohomology of the given algebra A. In this section we will present an overview of the general aspects of cyclic homology and cohomology. As we will see later on, this general formulation provides a nice intrinsic definition of the D-brane charge even in the flat commutative case, which moreover has a suitable extension to the noncommutative situations. 4.1. Formal properties of cyclic homology theories. Cyclic cohomology of a complex algebra, from its introduction by Connes, was developed in parallel with K-theory as a noncommutative analogue of the de Rham cohomology of a differentiable manifold. One
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of the main features of that theory, which made it a very useful tool to aid in computations of K-theory, is the existence of the Chern character from K-theory of the algebra to the cyclic cohomology of the algebra. Compared to K-theory, cyclic type homology theories exhibit a major weakness: they are all defined using a suitably chosen deformation of the tensor algebra. In the case of a topological algebra A, there are many ways to topologize the tensor algebra T A: this makes cyclic type homology theories very sensitive to the choice of topology. In this section we introduce the properties that we shall require of the cohomology theory to make it suitable for our purposes. We shall then briefly outline the main points in the construction of cyclic type homology theories that will satisfy those properties. Let A and B be topological algebras, whose topology will be specified shortly. We shall denote by HLi (A, B), i = 0, 1, a bivariant cyclic theory associated with A and B that has the following formal properties: (1) HLi (A, B) is covariant in the second variable and contravariant in the first variable; (2) For any three algebras A, B and C there is a natural composition product ⊗B : HLi (A, B) × HL j (B, C) → HLi+ j (A, C) ; (3) The functor HLi (−, −) is split exact and satisfies excision in each variable; (4) HLi (−, −) is homotopy invariant; (5) For any algebras A1 , A2 , B1 , and B2 , there is an exterior product HLi (A1 , B1 ) × HL j (A2 , B2 ) → HLi+ j (A1 ⊗ A2 , B1 ⊗ B2 ) compatible with the composition product; and (6) When A and B are C ∗ -algebras, there is a natural transformation of functors, the Chern character, ch : KKi (A, B) → HLi (A, B) which is compatible with the product on KK and the composition product on HL. Often we suppress the subscript i when i = 0. Axiom (2) ensures that HL• (A, A) is a Z2 -graded ring. There are now many definitions of bivariant cyclic theories of this kind, each suited to a specific category of algebras. (See [73] for a survey of these theories, as well as the relationships among them.) With every choice of a class of algebras we need to specify the notions of homotopy and stability which are suitable for the given category. In many cases, for example when A and B are multiplicatively convex (m-convex) algebras, KK in property (6) needs to be replaced by a different form of bivariant K-theory, for example, Cuntz’s kk [26], which is defined on a class of m-convex algebras. Cuntz’s kk is much easier to define than KK, but it is harder to compute, and at present, the precise relation between kk and KK is unclear. This is why we are led to consider Puschnigg’s local bivariant cyclic cohomology, developed primarily in [73], which we shall denote HLi (−, −).1 This theory, which can be defined on a class of C ∗ -algebras which is suitable for our purposes, is closest to Kasparov’s KK-theory. Furthermore we need to point out that the correct notion of tensor product in property (5) depends on the theory. When working with nuclear C ∗ -algebras, the usual C ∗ -tensor product is appropriate, but when working with Fréchet algebras, one might need the projective tensor product. When we do not assume that A and B are topological algebras, 1 Puschnigg calls it HCloc or HEloc , but as this is a bit cumbersome, we have chosen to simplify the notation.
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then the natural cyclic type homology theory to consider is the bivariant periodic cyclic homology HPi (A, B) of Cuntz and Quillen [30]. This theory is closest to the cyclic homology and cohomology defined by Connes. We have that HPi (A, C) is the same as the periodic cyclic cohomology of A, HPi (A), while HPi (C, A) coincides with the periodic cyclic homology, HPi (A). When the algebra A is equipped with a topology, Meyer’s work indicates [62] that in the construction of a reasonable cyclic type theory one should consider bounded rather than continuous maps. More precisely, this means the following. As is well known, a map of topological vector spaces f : E → F is bounded if and only if it sends bounded sets in E to bounded sets in F. If E and F are locally convex, then a reasonable definition of a bounded set states that a subset of E is bounded if and only if it is absorbed by every open neighbourhood of zero. Since the choice of open neighbourhoods in this definition is dictated by topology, the class of bounded sets in E is fixed by the choice of topology. The class of bounded sets in a topological space is called a bornology B; the bornology associated with the topology of a space is called the von Neumann bornology. A space equipped with a chosen family of bounded sets is called a bornological space. A bornology on a vector space E is a class B of subsets of E, which have the properties that {e} ∈ B for all e ∈ E; if S ∈ B and T ⊂ S, then T ∈ B (a subset of a bounded set is bounded); if S, T ∈ B, then S ∪ T ∈ B (the union of two bounded sets is bounded); c · S ∈ B if S ∈ B and c ∈ C (a scalar multiple of a bounded set is bounded); and if S, T ∈ B then S + T ∈ B (vector addition in E is a bounded map). A vector space equipped with a bornology is called a bornological vector space. A map f : E → F of bornological spaces is called bounded if and only if it sends elements of the bornology in E (i.e., the ‘bounded’ sets in E) to elements of the bornology in F. In the study of bornological spaces it became clear that it is useful to consider the choice of bornology to be independent from the choice of topology. This observation lies at the basis of Meyer’s construction of his analytic cyclic homology HAi (−, −), which is a bivariant functor defined on a class of bornological algebras. A bornological vector space A is a bornological algebra if and only if it is equipped with a multiplication m : A × A → A which is bounded in the bornological sense. Meyer’s analytic theory is very flexible and can be used in a variety of contexts. For example, it contains the Cuntz-Quillen bivariant cyclic theory HPi (A, B) of [30]. Moreover, it can be defined for Fréchet algebras and, in particular, for Banach algebras. Meyer showed in his thesis that for a suitable choice of bornology on a locally convex algebra A his analytic cyclic cohomology HAi (A, C) = HAi (A) is isomorphic to Connes’ entire cyclic cohomology HEi (A) [62, Thm. 3.47]. A very important example of an entire cyclic cohomology class is given by the JLO cocycle, which provides the character of a θ -summable Fredholm module [47]. 4.2. Local cyclic theory. We shall now outline, in broad terms, the construction and main properties of Puschnigg’s local cyclic theory. For any algebra A, unital or not, the (non-unital) tensor algebra T A of A is defined by T A = A ⊕ (A ⊗ A) ⊕ (A ⊗ A ⊗ A) ⊕ . . . . In applications, A will be assumed to be complete with respect to some additional structure. For example, A may be a Fréchet or Banach algebra, or in the bornological case, A will be assumed to be a complete bornological algebra [62, 2.2]. In each case the definition of T A will require a choice of a completed tensor product which is relevant
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to the given situation. For example, for Banach algebras a reasonable choice is the comˆ π while for bornological algebras we need to take the pleted projective tensor product ⊗ completed bornological tensor product [62, 2.2.3]. The tensor algebra is closely related to the algebra of noncommutative differential forms A, which is defined (as a vector space) by n A = A⊗n+1 ⊕ A⊗n , n > 0. We put 0 A = A.2 One defines a differential d on A of degree +1, which for n ≥ 1 is given by the two-by-two matrix 0 0 d= , 1 0 while in degree zero we put d = (0, 1) : A → A⊗2 ⊕ A. One can then show that, for n > 0, n A ∼ = Span{a˜ 0 da1 . . . dan }, where a˜ 0 is an element of the unitization A˜ of A, and ai ∈ A. The multiplication on A is uniquely determined by the requirements that d be a derivation (satisfying the Leibnitz rule) and that (da1 . . . dan ) · (dan+1 . . . dan+m ) = da1 . . . dan+m , (a˜ 0 da1 . . . dan ) · (dan+1 . . . dan+m ) = a˜ 0 da1 . . . dan+m . A key point in the construction of any cyclic type homology theory is the choice of a suitable completion (depending on whether A is considered to be a topological or a bornological algebra) of A. To retain the important universal property of the tensor algebra, this completion is also usefully described as a deformation of the tensor algebra denoted X (T A). This is a Z2 graded complex defined very simply as follows: b
1 (T A)/[ 1 (T A), 1 (T A)] − → 0 (T A), where [ 1 (T A), 1 (T A)], the commutator space of 1 (T A), is spanned by the set of all commutators [ω, η] with ω, η ∈ 1 (T A). The map b is given by ω0 dω1 → [ω0 , ω1 ] for any two ω0 , ω1 ∈ T A. There is a differential going the other way, which is the composition of the differential d : T A → 1 (T A) with the quotient map 1 (T A) → 1 (T A)/[ 1 (T A), 1 (T A)]. Let A and B be two complete bornological algebras and let X (T A)c be the Puschnigg completion of the X (T A) (see [72, §5], [28, §23]). There is a Z2 -graded complex of bounded maps HomC (X (T A)c , X (T B)c ). We define the bivariant local cyclic homology by HLi (A, B) = Hi (HomC (X (T A)c , X (T B)c )), where i = 0, 1 [28]. This homology theory coincides with other theories discussed there under suitable conditions. For example, when B is a Fréchet algebra whose bornology is specified by the family of pre-compact sets (or is nuclear) then HA• (B) = HL• (B) and there is a natural map HA• (A, B) → HL• (A, B). We recall the notion of a smooth subalgebra of a complete bornological algebra. 2 Caution: In [73], Puschnigg forgets to mention this, i.e., to mention that the definition of n A is different when n = 0.
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Definition 4.1. [28, 23.3]. Let A be a complete bornological algebra with bornology B(A), which is a dense subalgebra of a Banach algebra B with closed unit ball U . Then A is a smooth subalgebra of B if and only if for every element S ∈ B(A) such that S ⊂ r U , for some r < 1, the set S ∞ = n S n is an element of B(A). Smooth subalgebras of Banach algebras are closed under the holomorphic functional calculus. The following result will be important to us. Theorem 4.2. [28, 23.4]. Let B be a Banach algebra with the metric approximation property. Let A be a smooth subalgebra of B. Then A and B are HL-equivalent, that is the inclusion map A → B induces an invertible element of HL0 (A, B). Note by [17] that all nuclear C ∗ -algebras have the metric approximation property. Some, but not all, non-nuclear C ∗ -algebras have it as well. Example 4.3. Let X be a compact manifold. Then the Fréchet algebra C ∞ (X ) is a smooth subalgebra of the algebra C(X ) of continuous functions on X . Furthermore, the inclusion C ∞ (X ) → C(X ) is an invertible element in HL(C ∞ (X ), C(X )) by Theorem 4.2 above. In particular, both the local homology and cohomology of these two algebras are isomorphic. Puschnigg also proves that HL• (C ∞ (X )) ∼ = HP• (C ∞ (X )), and so, in this case, Puschnigg’s local cyclic theory coincides with the standard periodic cyclic homology. The following fundamental result of Connes makes it possible to establish contact between Puschnigg’s local cyclic theory of C(X ) and the de Rham cohomology of X . Theorem 4.4. For X a compact manifold, the periodic cyclic homology HP• (C ∞ (X )) is canonically isomorphic to the periodic de Rham cohomology: HP0 C ∞ (X ) ∼ and HP1 C ∞ (X ) ∼ (4.1) = Heven (X ) = Hodd (X ) . dR
dR
It is in the sense of this theorem that we may regard cyclic homology as a generalization of de Rham cohomology to other (possibly noncommutative) settings. The local cyclic theory HL admits a Chern character with the required properties. Theorem 4.5. [28, 23.5]. Let A and B be separable C ∗ -algebras. Then there exists a natural bivariant Chern character ch : KK• (A, B) → HL• (A, B) which has the following properties: (1) ch is multiplicative, i.e., if α ∈ KKi (A, B) and β ∈ KK j (B, C) then ch(α ⊗B β) = ch(α) ⊗B ch(β);
(4.2)
(2) ch is compatible with the exterior product; and (3) ch(KK(φ)) = HL(φ) for any algebra homomorphism φ : A → B. The last property implies that the Chern character sends invertible elements of KK-theory to invertible elements of bivariant local cyclic cohomology. Moreover, if A and B are in the class N of C ∗ -algebras for which the Universal Coefficient Theorem holds in KK-theory, then HL• (A, B) ∼ = HomC (K• (A) ⊗Z C, K• (B) ⊗Z C). If K• (A) is finitely generated, this is also equal to KK• (A, B) ⊗Z C.
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5. Duality in Bivariant Cyclic Cohomology In this section we shall formulate and analyse Poincaré duality in the context of bivariant cyclic cohomology of generic noncommutative algebras. Our analysis of Poincaré duality in KK-theory from Sect. 2 and of KK-equivalence in Sect. 3 indicates that it is possible to define analogous notions in any bivariant theory that has the same formal properties as KK-theory. An important example of such a situation is provided by the bivariant local cyclic theory as introduced in Sect. 4.2, where we have the additional tool of the bivariant Chern character from KK. Rather than repeating all the details, we shall simply state the main points. Duality in cyclic homology and periodic cyclic homology has also been considered by Gorokhovsky [38, §5.2]. 5.1. Poincaré duality. We will now develop the periodic cyclic theory analogues of the versions of Poincaré duality introduced in Sect. 2. Because we want to link everything with KK and not with kk or its variants, we will work throughout with HL and not with HP, even though the latter is probably more familiar to most readers. Definition 5.1. Two complete bornological algebras A, B are a strong cyclic Poincaré dual pair (strong C-PD pair for short) if there exists a class ∈ HLd (A ⊗ B, C) = HLd (A⊗B) in the local cyclic cohomology of A⊗B and a class ∨ ∈ HLd (C, A⊗B) = HLd (A ⊗ B) in the local cyclic homology of A ⊗ B with the properties ∨ ⊗B = 1A ∈ HL0 (A, A) and ∨ ⊗A = (−1)d 1B ∈ HL0 (B, B). The class is called a fundamental cyclic cohomology class for the pair (A, B) and ∨ is called its inverse. A complete bornological algebra A is a strong cyclic Poincaré duality algebra ( strong C-PD algebra for short) if (A, Ao ) is a strong C-PD pair. As in the case of KK-theory, these hypotheses establish an isomorphism between the periodic cyclic homology and cohomology of the algebras A and B as HL• (A) ∼ = HL•+d (B)
HL• (B) ∼ = HL•+d (A).
and
(5.1)
One also has the isomorphisms HL•+d (A ⊗ B) ∼ = HL• (A, A) ∼ = HL• (B, B) ∼ = HL•+d (A ⊗ B).
(5.2)
The moduli space of fundamental cyclic cohomology classes for the pair (A, B) is the cyclic duality group HL0 (A, A)−1 of invertible elements of the ring HL0 (A, A) ∼ = HL0 (B, B). Similarly to Sect. 2.7, one has the alternative notions of weak C-PD pairs and of cyclic Poincaré duality. Example 5.2. Let A = C ∞ (X ) be the algebra of smooth functions on a compact oriented manifold X of dimension d. Then the image of the class [ϕ X ] of the cyclic d-cocycle 1 0 1 d f0 d f1 ∧ ··· ∧ d fd (5.3) ϕX ( f , f , . . . , f ) = d! X
∼ HL• (C(X )) → HL• (C(X ) ⊗ for ∈ A, under the homomorphism : HP• (A) = C(X )) induced by the product map (2.5), is the fundamental class ∈ HLd (C(X ) ⊗ C(X )) of X in cyclic cohomology. Thus in this case corresponds to the orientation fi
m∗
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cycle [X ] and our notion of Poincaré duality agrees with the classical one. More generally, if X is non-orientable, i.e., w1 (X ) = 0, we choose the local coefficient system C X → X associated to w1 (X ) whose fibres are each (non-canonically) isomorphic to Z. Then (C(X ), C(X, C X ⊗ C)) is a strong C-PD pair. Thanks to the existence of a universal, multiplicative Chern character which maps the bivariant KK-theory to bivariant local cyclic cohomology, we can show that Definitions 2.20 and 5.1 are compatible. Let (A, B) be a strong PD pair of algebras in KK-theory with fundamental class ∈ Kd (A ⊗ B) and inverse ∨ ∈ K−d (A ⊗ B). Then there is a commutative diagram ⊗A
K• (A) ch
HL• (A)
/ K•+d (B)
ch()⊗A
(5.4)
ch
/ HL•+d (B).
Since the Chern character is a unital homomorphism the cocycle ch() is an invertible class in HLd (A ⊗ B) with inverse ch(∨ ) ∈ HLd (A ⊗ B), and so it establishes Poincaré duality in local cyclic cohomology, i.e., Poincaré duality in KK-theory implies Poincaré duality in cyclic theory. However, the converse is not true, since the cyclic theories constructed in Sect. 4 give vector spaces over C and are thus insensitive to torsion. A simple example is provided by any compact oriented manifold X for which W3 (X ) = 0. Then the algebra A = C(X ) is a strong C-PD algebra but not a PD algebra. In the cases where the Chern characters chA and chB are both isomorphisms after tensoring the K-groups with C, Poincaré duality in cyclic theory implies rational Poincaré duality in K-theory. The commutative diagram (5.4) allows us to transport the structure of Poincaré duality in KK-theory to local cyclic cohomology. In particular, all examples that we presented in Sect. 2 in the context of KK-theory also apply to local cyclic cohomology. Note, however, that if a strong PD pair of algebras (A, B) are equipped with their own fundamental cyclic cohomology class ∈ HLd (A ⊗ B), then generically ch() = . We will see an example of this in Sect. 5.3 below. In fact, this will be the crux of our construction of D-brane charge cycles in Sect. 8. The choice ch() = has certain special properties which will be discussed in Sect. 7.1.
5.2. HL-Equivalence. Exactly as we did above, it is possible to define the analogous notion of KK-equivalence from Sect. 3 in the bivariant local cyclic cohomology. We now briefly discuss how this works. Definition 5.3. Two complete bornological algebras A and B are said to be strongly HL-equivalent if there are elements ξ ∈ HLn (A, B)
and
η ∈ HLn (B, A)
(5.5)
ξ ⊗B η = 1A ∈ HL0 (A, A)
and
η ⊗A ξ = 1B ∈ HL0 (B, B).
(5.6)
such that
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As in the case of KK-theory, these hypotheses induce isomorphisms between the local cyclic homology and cohomology groups of the algebras A and B as HL• (A) ∼ = HL•+n (B)
and
HL• (A) ∼ = HL•+n (B).
(5.7)
One similarly has the notions of weak equivalence and of HL-equivalence. The Chern character again allows us to transport results of Sect. 3 to cyclic theory. If (A, B) is a pair of strongly KK-equivalent algebras, with the equivalence implemented by classes α ∈ KKn (A, B) and β ∈ KK−n (B, A), then there are commutative diagrams K• (B) o ch
HL• (B) o
α⊗B
/
β⊗A ch(α)⊗B ch(β)⊗A
/
K•+n (A)
(5.8)
ch
HL•+n (A)
and K• (B) o ch
HL• (B) o
⊗B α ⊗A β ⊗B ch(α) ⊗A ch(β)
/K
•+n (A)
/ HL
(5.9)
ch
•+n (A).
Thus KK-equivalence implies HL-equivalence, but not conversely. Remark 5.4. In the various versions of cyclic theory, one needs different notions of MoL1 ) in rita equivalence. For example, one has an isomorphism HP• (A) ∼ = HP• (A⊗ periodic cyclic homology, where L1 is the algebra of trace-class operators on a separable Hilbert space. Fortunately, since HL is well behaved for C ∗ -algebras, which are our main examples of interest, we will usually not have to worry about this point. 5.3. Spectral triples. Let us model a D-brane by an even spectral triple (A, H, D) as prescribed in Sect. 2.5. Assume that the resolvent of the operator D is of order p, i.e., its eigenvalues µk decay as k −1/ p . This is the situation, for example, for the case when D is the canonical Dirac operator over a finite-dimensional spinc manifold. We further make the following regularity assumption on the spectral triple. Let us assume that both A and [D, A] are contained in k>0 Dom(δ k ), where δ = ad(|D|). Let ⊂ C denote the set of all singularities of the spectral zeta-functions ζ P (z) = Tr H(P |D|−z ), where z ∈ C and P is an element of the algebra generated by δ k (A) and δ k ([D, A]). The set is called the dimension spectrum of the spectral triple (A, H, D). We will assume that (A, H, D) has discrete and simple dimension spectrum , i.e., that ζ P , ∀P can be extended as a meromorphic function to C \ with simple poles in . Such a spectral triple is said to be regular. Under these circumstances, the residue formula (5.10) − P = Resz=0 Tr H P |D|−2z
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defines a trace on the algebra generated by A, [D, A] and |D|z , z ∈ C. Using it we may CM ) define the Connes-Moscovici cocycle ϕ CM = (ϕ2n n≥0 in the (b, B) bi-complex of the [k] algebra A [25]. For this, we denote by a the k th iterated commutator of a ∈ A with the operator D 2 , ## # a [k] := D 2 , D 2 , · · · D 2 , a · · · . ! "
(5.11)
k times
For n = 0 and a0 ∈ A we set ϕ0CM (a0 ) = Tr H γ a0
ker(D)
+ Resz=0
1 z
Tr H γ a0 |D|−2z ,
(5.12)
where ker(D) is the orthogonal projection onto the kernel of the operator D on H. For n > 0 and ai ∈ A, we define CM ϕ2n (a0 , a1 , . . . , a2n ) (5.13) $ % +n−1 ! 2n (−1)|k| |k| = [D, a j ][k j ] |D|−2|k|−2n , − γ a0 2n j=1 k 2 k! (k1 + · · · + k j + j) j=1
:= k1 + · · · + k2n where the sum runs through all multi-indices k = (k1 , . . . , k2n ) with |k| and k! := k1 ! · · · k2n !. It can be shown that this formula has only a finite number of nonCM ] ∈ HE0 (A) is called the (even) cyclic cohomology zero terms. The class ch(D) = [ϕ2n Chern character and it may be regarded as a map ch : K0 (A) −→ HE0 (A).
(5.14)
It is instructive again to look at the case where X is a compact, smooth spin manifold of even dimension d. For the spectral triple we then take A = C ∞ (X ), H± = L 2 (X, S± X ), ), the usual (untwisted) Dirac operator. Then and D = D / : C ∞ (X, S+X ) → C ∞ (X, S− X the dimension spectrum consists of relative integers < d, and is simple. (Multiplicities would arise in the case that the spacetime X is a singular orbifold, for example.) We can thereby apply the Connes-Moscovici cocycle construction to this situation. One finds and hence its components are given that the contributions to (5.14) vanish unless k = 0, explicitly by [70] 1 CM 0 1 2n ) ϕ2n ( f , f , . . . , f ) = f 0 d f 1 ∧ · · · ∧ d f 2n ∧ A(X (5.15) (2n)! X
with f i ∈ C ∞ (X ). In this case, the entire cyclic cohomology HE• (A) is naturally isomorphic to the local cyclic cohomology HL• (A) and to the periodic cyclic cohomology HP• (A) [72]; the resulting entire cocycle is cohomologous to an explicit periodic cyclic cocycle given in terms of the spectral triple. This result implies an important characterization that will be crucial to the construction of brane charges as cyclic classes.
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Theorem 5.5. Let X be a compact, smooth spin manifold of even dimension. Then the cyclic cohomology Chern character of the spectral triple (C ∞ (X ), L 2 (X, S X ), D / ) coincides with the Atiyah-Hirzebruch class of X in even de Rham homology, ) . ch(D / ) = Pd−1 A(X (5.16) X Remark 5.6. In some cases the p-summability requirements are not met, most notably when the spectral triple is infinite-dimensional. In such instances we can still compute the cyclic cohomology Chern character if the spectral triple (A, H, D) is θ -summable, i.e., [D, a] is bounded for all a ∈ A, and the eigenvalues µk of the resolvent of D grow no faster than log(k). This implies that the corresponding heat kernel is trace-class, 2 Tr H( e −t D ) < ∞ ∀t > 0. Within this framework, we can then represent the Chern character of the even spectral triple (A, H, D) in the entire cyclic cohomology of A by JLO ) using the JLO cocycle ϕ JLO = (ϕ2n n≥0 [47]. With a0 , a1 . . . , a2n ∈ A, it is defined by the formula JLO (a0 , a1 . . . , a2n ) ϕ2n $ $ %% 2n −t0 D 2 −t j D 2 = dt0 dt1 · · · dt2n Tr H γ a0 e [D, a j ] e , 2n
(5.17)
j=1
& where n = {(t0 , t1 , . . . , tn ) | ti ≥ 0, i ti = 1} denotes the standard n-simplex in Rn+1 . This entire cyclic cocycle is cohomologous to the Chern character. Once again, consider the example of the canonical triple (C ∞ (X ), L 2 (X, S X ), D / ) over a spin manifold X , and replace D / everywhere in the formula (5.17) by s D / with s > 0. By using asymptotic symbol calculus, one can then show [24,74] that the character (5.17) retracts as s → 0 to the Connes-Moscovici cocycle (5.15). 6. T-Duality In this section we will show that there is a strong link between KK-equivalence for crossed product algebras and T-duality in string theory. This will lead to a putative axiomatic characterization of T-duality for C ∗ -algebras. We also describe an analogous characterization in local cyclic cohomology. 6.1. Duality for crossed products. We begin with some general results regarding KKequivalence and Poincaré duality for crossed product algebras, and then use them to give some more examples of PD algebras. Let G be a locally compact, connected Lie group. Recall [18] that G is said to satisfy the Haagerup property if it has a metrically proper isometric action on some Hilbert space. Examples are S O(n, 1), SU (n, 1) and locally compact, connected, amenable Lie groups. An amenable Lie group is one that has an invariant mean, examples of which include abelian Lie groups, nilpotent Lie groups and solvable Lie groups. Let K be a maximal compact subgroup of G. Let V denote the cotangent space to the symmetric space G/K at the point (K ). Let Cliff(V ) be the Clifford algebra of V with respect to some positive definite inner product on V . We start by recalling a theorem of Higson-Kasparov [40] and Tu [85], generalizing a theorem of Kasparov [52, §6 Theorem 2].
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Theorem 6.1. Let A be a G-C ∗ -algebra, where G is a locally compact, connected Lie group satisfying the Haagerup property. Then in the notation above, the pair of crossed product C ∗ -algebras (A G, (A ⊗ Cliff(V )) K ) are strongly KK-equivalent. If in addition the coadjoint action of K on V is spin then the pair (AG, (A K )⊗C0 (Rd )) are strongly KK-equivalent, where d = dim(G/K ). A special case of Theorem 6.1, which was proved earlier by Fack and Skandalis [34] generalising an argument of Connes [19], is as follows. Corollary 6.2. Let A be a G-C ∗ -algebra, where G is a simply connected, locally compact, solvable Lie group of dimension k. Then the pair of C ∗ -algebras (A ⊗ C0 (Rk ), A G) are strongly KK-equivalent. As an immediate consequence of Theorem 6.1 and Lemma 3.4 we obtain the following. Corollary 6.3. Let A be a G-C ∗ -algebra, where G is a locally compact, connected Lie group satisfying the Haagerup property. Then in the notation of Theorem 6.1, (A ⊗ Cliff(V )) K is a (strong, weak) PD algebra if and only if A G is a (strong, weak) PD algebra. If in addition the coadjoint action of K on V is spin, then A K is a (strong, weak) PD algebra if and only if A G is a (strong, weak) PD algebra. In addition, an immediate consequence of Corollary 6.2 and Lemma 3.4 is as follows. Corollary 6.4. Let A be a G-C ∗ -algebra, where G is a simply connected, locally compact, solvable Lie group. Then A is a (strong, weak) PD algebra if and only if A G is a (strong, weak) PD algebra. Example 6.5. Let be a torsion-free, discrete subgroup of a connected semisimple Lie group G with finite center. Let P be a minimal parabolic subgroup of G and K a maximal compact subgroup of G. Then G/P is the Furstenberg boundary (at infinity) of the symmetric space G/K . By Green’s theorem, C(G/P) is strongly Morita equivalent to C0 (\G) P. By Lemmas 3.4 and 3.6 it follows that C(G/P) is a strong PD algebra if and only if C0 (\G) P is a strong PD algebra. By Corollary 6.3 above, C0 (\G) P is a strong PD algebra if and only if C0 (\G) K is a strong PD algebra, i.e., if and only if \G/K is a spinc manifold. We conclude that C(G/P) is a strong PD algebra if and only if \G/K is a spinc manifold. In particular, C(S1 ) g is a strong PD algebra whenever g is the fundamental group of a compact, oriented Riemann surface of genus g ≥ 1. There is a deep variant of this example, analysed in detail by Emerson [32], dealing with the crossed product C ∗ -algebra C(∂) for a hyperbolic group with Gromov boundary ∂. Another important property of such crossed products is Takai duality. If G is a locally compact, abelian group we denote by G˜ its Pontrjagin dual, i.e., the set of characters of G, which is also a locally compact, abelian group. Pontrjagin duality G˜˜ ∼ = G follows ˜ n = Zn , and Z˜ n = Tn . If A is a ˜ n = Rn , T by Fourier transformation. For example, R ˜ G-C ∗ -algebra, then the crossed product A G carries a G-action. Theorem 6.6 (Takai Duality). Let A be a G-C ∗ -algebra, where G is a locally compact, abelian Lie group. Then there is an isomorphism of C ∗ -algebras (6.1) (A G) G˜ ∼ = A ⊗ K L 2 (G) .
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In other words, the algebras A and (A G) G˜ are strongly Morita equivalent. If we interpret the crossed product A G as the noncommutative analogue of an abelian orbifold spacetime X/G, then Takai duality asserts that “orbifolding twice” gives back a spacetime which is physically equivalent to the original spacetime X . The essential ˜ physical phenomenon is that the states which were projected out by G are restored by G. 6.2. T-duality and KK-equivalence. We next explain how KK-equivalence of crossed products is related to T-duality. Throughout X will be assumed to be a locally compact, finite-dimensional, homotopically finite space. Consider first the simplest case of flat Dbranes in Type II superstring theory on a spacetime X = M × Tn which is compactified on an n-torus Tn = Vn /!n , where !n is a lattice of maximal rank in an n-dimensional, real vector space Vn . As shown in [45], T-duality in this instance is explained by using the correspondence Tn M × Tn ×J J t JJ tt JJ tt JJ tt t JJ t tp J t p JJJ tt t JJ t JJ tt % ytt n M ×T M × Tn
(6.2)
where Tn = (Vn )∨ /(!n )∨ denotes the dual torus, with (!n )∨ the dual lattice in the dual vector space (Vn )∨ . This gives rise to an isomorphism of K-theory groups ≈ (6.3) Tn T! : K• (M × Tn ) −−−−→ K•+n M × given by p! T! (−) =
p ! (−) ⊗ P ,
(6.4)
Tn pulled back to M × Tn × Tn where P is the Poincaré line bundle over the torus Tn × n n n n via the projection map pr 1 : M × T × T → T × T . Thus T-duality can be viewed in this case as a smooth analog of the Fourier-Mukai transform. If G is the metric of the torus Tn inherited from the non-degenerate bilinear form of the lattice !n , then the dual torus Tn has metric G −1 inherited from the dual lattice (!n )∨ . As argued by [63], detailed in [37,44,64,66,86], and discussed in Sect. 1, RR-fields are classified by K1 -groups and RR-charges by K0 -groups of the spacetime X in Type IIB string theory, whereas RR-fields are classified by K0 -groups and RR-charges by K1 groups in Type IIA string theory. Thus if spacetime X = M × Tn is compactified on a torus of rank n, then the isomorphism (6.3) is consistent with the fact that T-duality is an equivalence between the Type IIA and Type IIB string theories if n is odd, while if n is even it is a self-duality for both string theories. Given this compelling fact, we will take this isomorphism to mean the equivalence itself here (although in string theory the duality is much more complicated and involves many more ingredients). As was observed in [59], all of this can be reformulated in terms of the C ∗ -algebra C0 (M × Tn ). The locally compact, abelian vector Lie group Vn ∼ = Rn acts on Tn = Vn /!n via left translations, and consider the crossed product algebra C0 (M × Tn ) Vn with Vn acting trivially on M. By Rieffel’s imprimitivity theorem [76], there is a strong Morita equivalence C0 (M × Tn ) Vn ∼ C0 (M) !n ,
(6.5)
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where the discrete group !n acts trivially on C0 (M). One therefore has C0 (M × Tn ) Vn ∼ C0 (M) ⊗ C ∗ (!n ). C ∗ -algebra
(6.6)
By Fourier transformation the group of !n can be identified as C( Tn ), and as a consequence there is a strong Morita equivalence n C0 M × Tn Vn ∼ C0 (M) ⊗ C Tn . T ∼ = C0 M × C ∗ -algebras
C ∗ (!n )
∼ =
(6.7)
Tn ),
By Lemma 3.6 and Corollary 6.2, the pair of (C0 (M × C0 (M × Tn ) Vn ∼ C0 (M × Tn )) are strongly KK-equivalent (with a degree shift of n mod 2). Thus by Lemma 3.2 there are isomorphisms ≈ T! : K• C0 (M × Tn ) −→ K•+n C0 (M × Tn ) , ≈ T ! : K• C0 (M × Tn ) −→ K•+n C0 (M × Tn ) . (6.8) The upshot of this analysis is that the Fourier-Mukai transform, or equivalently T-duality for flat D-branes in Type II string theory, on a spacetime X that is compactified on a torus Tn , can be interpreted as taking a crossed product with the natural action of Vn ∼ = Rn ∗ on the C -algebra C0 (X ). This point of view was generalized in a series of papers [9–12,59,60] to twisted D-branes in Type II superstring theory in a B-field (X, H ) and for a spacetime X which is a possibly non-trivial principal torus bundle π : X → M of rank n. As described in Sect. 1.4, in this case the type I, separable C ∗ -algebra in question is the algebra C0 (X, E H ) of continuous sections vanishing at infinity of a locally trivial C ∗ -algebra bundle E H → X with fibre K(H) and Dixmier-Douady invariant δ X (E H ) = H ∈ Br ∞ (X ) ∼ = H3 (X, Z). This is a stable, continuous trace algebra with spectrum X . It is a fundamental theorem of Dixmier and Douady [31] that H is trivial in cohomology if and only if C0 (X, E H ) is strongly Morita equivalent to C0 (X ) (in fact C0 (X, E0 ) ∼ = C0 (X, K(H))), consistent with the above discussion. If in addition X is a Calabi-Yau threefold, then T-duality in these instances coincides with mirror symmetry. As above, the abelian Lie group Vn acts on X via left translations of the fibres Tn . In [60] the following fundamental technical theorem was proven. Theorem 6.7. In the notation above, the natural Vn -action on X lifts to a Vn -action on the total space E H , and hence to a Vn -action on C0 (X, E H ), if and only if the restriction of H to the fibres of X is trivial in cohomology. This is a non-trivial obstruction if and only if the fibres of X are of rank n ≥ 3. The T-dual is then defined as the crossed product algebra C0 (X, E H ) Vn . This algebra is a continuous trace algebra if and only if π∗ (H ) = 0 in H1 (M, H2 (Tn , Z)). In the general case, the crossed product C0 (X, E H ) Vn is not of type I but is rather a continuous field of (stabilized) rank n noncommutative tori fibred over M. The fibre over the point m ∈ M is isomorphic to T nf (m) ⊗ K(H), where π∗ (H ) = [ f ] is the Mackey obstrucn tion class with f : M → H2 (!n , U (1)) ∼ = (R/Z)k , k = 2 a continuous map. This obstruction is due to the presence of discrete torsion in the fibres of the string background, represented by multipliers f (m) on the discrete group !n , which is essentially due to the presence of non-trivial global B-fields along the fibres of X . Corollary 6.8. In the notation of Theorem 6.7, suppose that the restriction of H to the fibres of X is trivial in cohomology. Then the T-dual C0 (X, E H ) Vn is a strong PD algebra if and only if X is a spinc manifold.
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Proof. By a theorem of Parker [67], the continuous trace algebra C0 (X, E H ) is a strong PD algebra if and only if X is a spinc manifold. The result now follows from Theorem 6.7 and Corollary 6.4. The Takai duality theorem in these examples implies that (C0 (X, E H ) Vn ) Vn is strongly Morita equivalent to C0 (X, E H ), i.e., T-duality applied twice returns the original string theory. We can now combine all of these observations to formulate a generically noncommutative version of T-duality for C ∗ -algebras in very general settings. Definition 6.9 (K-Theoretic T-Duality). Let T be a suitable category of separable C ∗ algebras, possibly equipped with some extra structure (such as the Rn -action above). Elements of T are called T-dualizable algebras, with the following properties: (1) There is a covariant functor T:T → T which sends an algebra A to an algebra T(A) called its T-dual; (2) There is a functorial map A → γA ∈ KKn (A, T(A)) such that γA is a KK-equivalence; and (3) The pair (A, T(T(A))) are Morita equivalent, and the Kasparov product γA ⊗T(A) γT(A) is the KK-equivalence associated to this Morita equivalence. 6.3. T-duality and HL-equivalence. As we have mentioned, the isomorphisms (6.8) are only part of the story behind T-duality, as they only dictate how topological charges behave under the duality. In particular, the isomorphism T! on K-theory bijectively relates the RR-fields in T-dual spacetimes, while the bijection T ! relates the RR-charges themselves. As explained in Sects. 1.1 and 1.2 in the case of flat D-branes, the RR-fields are represented by closed differential forms on the spacetime X while the branes themselves are associated to non-trivial (worldvolume) cycles of X . It is therefore natural to attempt to realize our characterization of T-duality above in the language of cyclic theory, in order to provide the bijections between the analogues of these (and other) geometric structures. We begin with the following observation. Theorem 6.10. Let A be separable C ∗ -algebra, and suppose that A admits an action by a locally compact, real, abelian vector Lie group Vn of dimension n. Then there is a commutative diagram K• (A) ch
T∗
(6.9)
ch
HL• (A)
/ K•+n (A Vn )
T∗
/ HL•+n (A Vn )
whose horizontal arrows are isomorphisms. Proof. The isomorphism K• (A) ∼ = K•+n (A Rn ) in the top row is the Connes-Thom isomorphism [19] (cf. Corollary 6.2 above), while the isomorphism in the bottom row comes from transporting this isomorphism to local cyclic homology, as in [82]. (See also the review of [82] in MathSciNet, MR2117221 (2005j:46041), for more of an explanation.)
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The isomorphism in the bottom row of (6.9) is the local cyclic homology version of T-duality. Theorem 6.10 shows that the T-duality isomorphisms in K-theory descend to isomorphisms of cyclic cohomology, giving the mappings at the level of RR-field representatives in HL• (A). This motivates the local cyclic cohomology version of the axioms spelled out in Definition 6.9. Definition 6.11 (Cohomological T-Duality). Let A be a complete bornological algebra. A cyclic T-dual of A is a complete bornological algebra THL (A) which satisfies the following three axioms: (1) The map A → THL (A) is a covariant functor on an appropriate category of algebras; (2) The pair (A, THL (A)) are HL-equivalent; and (3) The pair (A, THL (THL (A))) are topologically Morita equivalent. As in Definition 6.9, there should be an explicit functorial HL-equivalence in (2) compatible with the Morita equivalence in (3). From these definitions it follows that K-theoretic T-duality for a separable C ∗ -algebra A implies cohomological T-duality for the same algebra, but the converse need not necessarily be true (because of torsion in K-theory, for example). Remark 6.12. There are competing points of view concerning T-duality in the nonclassical case, that is, in the case when the T-dual of a spacetime X , which is a principal torus bundle with nontrivial H -flux, is not another principal torus bundle. Unlike the approach discussed in this section, where the T-dual is a globally defined but possibly noncommutative algebra, the T-dual in the competing points of view is not globally defined. For example, in [46], Hitchin’s generalized complex geometry is used to construct a T-dual which is a purely local object, that does not patch together to give a global object, and is referred to as a T-fold. See also [81] for a related point of view. 7. Todd Classes and Gysin Maps In this section we apply the concept of Poincaré duality in KK-theory and bivariant cyclic cohomology to define the notion of a Todd class for a very general class of C ∗ -algebras. As follows from the discussion of Sect. 1.1, this will be one of the main building blocks of our definitions of generalized D-brane charges. Another crucial ingredient in these definitions is the application of Poincaré duality to the construction of Gysin maps (or “wrong way” maps) in KK-theory and in cyclic theory, which also came up in our discussion of T-duality above. These general constructions combine together to yield a generalization of the Grothendieck-Riemann-Roch theorem for an appropriate class of C ∗ -algebras, which in turn yields another perspective on the concept of a T-dual C ∗ -algebra that was introduced in the previous section.
7.1. The Todd class. Our general definition of Todd classes is motivated by Theorem 5.5 and Proposition 2.8. We begin with the following observation. Lemma 7.1. Let A, B1 , B2 be separable C ∗ -algebras such that (A, B1 ) and (A, B2 ) are both strong PD pairs. Then the pair of algebras (B1 , B2 ) are strongly KK-equivalent.
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Proof. Let 1 ∈ Kd (A ⊗ B1 ) and 2 ∈ Kd (A ⊗ B2 ) be the respective fundamental classes. Then with a proof along the lines of Proposition 2.8, one shows that the clasd ∨ ses α := ∨ 1 ⊗A 2 ∈ KK 0 (B1 , B2 ) and β := (−1) 2 ⊗A 1 ∈ KK 0 (B2 , B1 ) implement the required equivalence. Let D denote the class of all separable C ∗ -algebras A for which there exists another separable C ∗ -algebra B such that (A, B) is a strong PD pair. For any such A, we fix a ˜ In general there is representative of the KK-equivalence class of B and denote it by A. ˜ no canonical choice for A. If A is a strong PD algebra, the canonical choice A˜ := Ao will always be made. ˜ be a fundamental K-homology class for Definition 7.2. Let A ∈ D, let ∈ Kd (A ⊗ A) d ˜ ˜ the pair (A, A) and let ∈ HL (A ⊗ A) be a fundamental cyclic cohomology class. Then the Todd class of A is defined to be the class (7.1) Todd (A) = Todd, A, A˜ := ∨ ⊗A˜ ch () in the ring HL0 (A, A). ˜ ∼ Recall that the map ∨ ⊗A˜ (−) implements an isomorphism HLd (A⊗ A) = HL0 (A, A). The element (7.1) is invertible with inverse given by (7.2) Todd (A)−1 = (−1)d ch ∨ ⊗A˜ . Remark 7.3. Observe that the Todd class of an algebra A ∈ D is trivial, i.e., Todd(A) = ˜ 1A in HL0 (A, A), if and only if ch() = in HLd (A ⊗ A). The Todd class depends on a number of choices, but this dependence can be described by “covariant” actions on the classes. Theorem 7.4. In the notation above, the Todd class of an algebra A ∈ D has the following properties: ˜ with (1) Suppose and are fundamental classes for the strong PD pair (A, A), inverse fundamental classes ∨ and ∨ . If there are KK-equivalences α ∈ KK0 (A, ˜ A˜ 1 ), then (A1 , A˜ 1 ) is a strong PD pair, with fundamental A1 ) and β ∈ KK0 (A, classes 1 = (α −1 × β −1 ) ⊗A⊗A˜ ,
1 = (ch(α)−1 × ch(β)−1 ) ⊗A⊗A˜
(where × denotes the exterior product) and inverse fundamental classes ∨ ∨ ˜ (α × β), 1 = ⊗A⊗A
∨ ∨ ˜ (ch(α) × ch(β)). 1 = ⊗A⊗A
Furthermore, ˜ ⊗A ch(α). Todd1 ,1 (A1 , A˜ 1 ) = ch(α)−1 ⊗A Todd, (A, A)
(7.3)
(2) If (, HL ) is an element of the duality group KK0 (A, A)−1 × HL0 (A, A)−1 , then Todd⊗A , HL ⊗A A, A˜ = ch () ⊗A Todd, A, A˜ ⊗A −1 HL . (7.4)
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Proof. The fact that (A1 , A˜ 1 ) is a strong PD pair with fundamental classes 1 and 1 is routine and quite similar to the calculation in Proposition 2.7. We proceed to compute the Todd class. In terms of the diagram calculus of Appendix B, the picture is:
C
∨ 1
@A =
C
∨
A1 @ @@ @@ @@ @
A1 ~? ~ ~~ ~~ ~~ ◦? ?? ?? ?? ? A˜ 1
◦= == == == =
A˜
ch(α)
ch(β)
/ A˜ 1
1
/ A1
/ A˜ 1
A1
1
◦
ch(1 )
/C
ch(α −1 )
/A
−1
/ A˜ ch(β) / A˜ 1
>> >> >> >>
◦
ch()
/C
(7.5) This yields formula (7.3) via associativity in the formulation of Appendix B. The proof of (2) is done with a very similar diagram. Corollary 7.5. Suppose A is a C ∗ -algebra that is strongly KK-equivalent to C(X ), where X is an even-dimensional compact spinc manifold. Let Todd(X ) be the usual Todd class of X , but viewed as a bivariant cyclic homology class as above. If α ∈ KK(A, C(X )) and β ∈ KK(C(X ), A) are explicit KK-equivalences inverse to one another, then Todd(A) = ch(α) ⊗C(X ) Todd(X ) ⊗C(X ) ch(β).
(7.6)
Proof. Immediate from Theorem 7.4, since D / X ×X is a KK fundamental class for C(X ) and the usual homology fundamental class provides another fundamental class in cyclic homology. Remark 7.6. There is a more subtle real version of Kasparov’s KK-theory defined for complex C ∗ -algebras with involution [39, Definition 9.18]. For any separable C ∗ -algebra A, the algebra A ⊗ Ao may be equipped with a canonical involution τ defined by τ (a ⊗ bo ) = b∗ ⊗ (a ∗ )o
(7.7) •
for all a, b ∈ A. The corresponding real K-homology groups are denoted KR (A ⊗ Ao ). There is a forgetful map f : KR• (A ⊗ Ao ) −→ K• (A ⊗ Ao )
(7.8)
from the real to the complex K-homology of the algebra A ⊗ Ao . Suppose that A is a strong PD algebra which admits a fundamental KR-homology class R ∈ KRd (A⊗Ao ). Let ∈ HLd (A ⊗ Ao ) be a fundamental cyclic cohomology class for A. The image of
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R under the homomorphism (7.8) is a fundamental K-homology class for A and the corresponding Todd class (7.1) in HL0 (A, A) is called the Atiyah-Hirzebruch class of A, denoted (A) := ∨ ⊗Ao ch (f(R )) . A
(7.9)
It satisfies the same basic properties as the Todd class above, and may be related to (7.1) for any other fundamental K-homology class by using the action of the duality group KK0 (A, A)−1 in (7.4). 7.2. Gysin homomorphisms. Let f : A → B be a morphism of separable C ∗ -algebras. It induces morphisms in K-theory, f ∗ : K• (A) −→ K• (B),
(7.10)
and morphisms in K-homology, f ∗ : K• (B) −→ K• (A).
(7.11)
We will now describe how to construct Gysin maps (or “wrong way” maps) on these groups. If both A and B are PD algebras, then they are easily constructed as analogues of the classical “Umkehrhomomorphismus”. In this case, there are isomorphisms ≈
≈
PdA : K• (A) −→ K•−dA (Ao ) and PdB : K• (B) −→ K•−dB (Bo ). (7.12) We can then define the Gysin map in K-theory, f ! : K• (B) −→ K•+d (A),
(7.13)
(where d = dA − dB) as the composition PdB
( f o )∗
Pd−1 A
f ! : K• (B) −−→ K•−dB (Bo ) −−−→ K•−dB (Ao ) −−−→ K•+d (A).
(7.14)
Under the same hypotheses, we can similarly define the Gysin map in K-homology, f ! : K• (A) −→ K•+d (B),
(7.15)
as the composition Pd−1 A
f ∗o
PdB
f ! : K• (A) −−−→ K•+dA (Ao ) −→ K•+dA (Bo ) −−→ K•+d (B).
(7.16)
However, this relies on the fact that A and B are PD algebras, which is in general too stringent a requirement. We will therefore proceed to some more general constructions.
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B
A
C
∆∨ A
◦
◦ fo
Ao
∆B
C
Bo
Fig. 7.2. Diagram representing the construction of f !. The “free ends” are on the top line and concatenation is done on the bottom line.
7.3. Strongly K-oriented maps. We will consider a subcategory C of the category of separable C ∗ -algebras and morphisms of C ∗ -algebras, consisting of strongly K-oriented morphisms. C comes equipped with a contravariant functor ! : C → KK, sending f
C " (A −→ B) −→ f ! ∈ KKd (B, A) and having the following properties: (1) For any C ∗ -algebra A, the identity morphism idA : A → A is strongly K-oriented with (idA)! = 1A, and the 0-morphism 0A : A → 0 is strongly K-oriented with (0A)! = 0 ∈ KK(0, A); fo
f
(2) If (A −→ B) ∈ C, then (Ao −→ Bo ) ∈ C and moreover ( f !)o = ( f o )!; f
(3) If A and B are strong PD algebras, then any morphism (A −→ B) ∈ C, and f ! is determined as follows: o f ! = (−1)dA ∨ A ⊗Ao [ f ] ⊗Bo B,
where for the rest of the paper, [ f ] = KK( f ) denotes the class in KK(A, B) of the f
morphism (A −→ B) and [ f o ] is defined similarly. As Kasparov products like this are rather hard to visualize when written this way, it is useful to use the diagram calculus developed in Appendix B. In these terms, f ! is represented by the picture depicted by Fig. 7.2. Actually, it is not immediately obvious that property (3) above is compatible with the required functoriality. However, consistency of the definition follows from the following: Lemma 7.7 (Functoriality of the Gysin map). If A, B and C are strong PD algebras, and if f : A → B, g : B → C are morphisms of C ∗ -algebras, then o dB ∨ o (−1)dA ∨ A ⊗Ao [ f ] ⊗Bo B ⊗B (−1) B ⊗Bo [g ] ⊗Co C o o [(g ◦ f ) ] ⊗Co C . = (−1)dA ∨ ⊗ A A Proof. Note that, by associativity of the Kasparov product, ∨ o A ⊗Ao [ f o ] ⊗Bo B ⊗B ∨ B ⊗Bo [g ] ⊗Co C o ∨ o = ∨ A ⊗Ao [ f ] ⊗Bo B ⊗B B ⊗Bo [g ] ⊗Co C.
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But o o ∨ o [ f o ] ⊗Bo B ⊗B ∨ B ⊗Bo [g ] = [ f ] ⊗Bo (B ⊗B B) ⊗Bo [g ] = [ f o ] ⊗Bo (−1)dB 1Bo ⊗Bo [g o ]
= (−1)dB [ f o ] ⊗Bo [g o ] = (−1)dB [(g ◦ f )o ], and the result follows.
We now exhibit more examples of elements in this category C. In the following example, the C ∗ -algebras are not strong PD algebras, but yet we can get an element in this category C. Suppose that we are given oriented manifolds X and Y , and classes H X ∈ H3 (X, Z) and HY ∈ H3 (Y, Z). A smooth map f : X → Y defines a morphism f ∗ : C0 (Y, E HY ) −→ C0 (X, E H X ) if f ∗ HY = H X . Since X and Y are oriented, then by Example 2.5, the pair (C0 (X ), C0 (X, Cliff(T X ))) is a strong PD pair, that is, there is a fundamental class X ∈ KK(C0 (X ) ⊗ C0 (X, Cliff(T X ), C). Since E H X ⊗ EoH X is stably isomorphic to the trivial bundle X × K, it follows that C0 (X ) ⊗ C0 (X, Cliff(T X ) is stably isomorphic to C0 (X, E H X ) ⊗ C0 (X, EoH X ⊗ Cliff(T X ). Therefore KK(C0 (X ) ⊗ C0 (X, Cliff(T X ), C) ∼ = KK(C0 (X, E H X ) ⊗ C0 (X, EoH X ⊗ Cliff(T X ), C), giving rise to a fundamental class in KK(C0 (X, E H X )⊗C0 (X, EoH X ⊗Cliff(T X ), C), and showing that (C0 (X, E H X ), C0 (X, EoH X ⊗Cliff(T X ))) is a strong PD pair. The analogous statement is true for Y . Finally, if f ∗ W3 (Y ) = W3 (X ), where W3 (X ) ∈ H3 (X, Z) is the third integral Stiefel-Whitney class of X , then we get the commutative diagram, K• (C0 (Y, EoHY ⊗ Cliff(T Y )))
f∗
/ K• (C0 (X, EoH ⊗ Cliff(T X ))) X
PdY
K• (C0 (Y, E HY ))
Pd X
f∗
/ K• (C0 (X, E H X )),
(7.17)
where the vertical arrows are isomorphisms. Then ( f ∗ )! ∈ KK(C0 (X, E H X ), C0 (Y, E HY )) ∗ is defined as the Kasparov product (−1)dim Y ∨ Y ⊗ [ f ] ⊗ X . This is a special case of the more general situation given as follows. Let (Ai , Bi ), i = 1, 2 be strong PD pairs with fundamental classes i , i = 1, 2 respectively, and let f : A1 → A2 be a morphism. Then f ! ∈ KK(B2 , B1 ) is defined using the diagram calculus in Appendix B as (−1)d1 ∨ 1 ⊗A1 [ f ] ⊗A2 2 . There are also many interesting examples of strongly K-oriented maps between noncommutative foliation C ∗ -algebras constructed in [42].
7.4. Weakly K-oriented maps. We will consider a subcategory Cw of the category of separable C ∗ -algebras and morphisms of C ∗ -algebras, consisting of “weakly K-oriented morphisms”. Cw comes equipped with a contravariant functor ! : Cw → Ab (Ab denotes the category of Z2 -graded abelian groups) sending f
Cw " (A −→ B) −→ f ! ∈ HomZ (K• (B), K•+d (A)) and having the following properties:
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(1) For any C ∗ -algebra A, the identity morphism idA : A → A is weakly K-oriented; f
fo
(2) If (A −→ B) ∈ Cw , then (Ao −→ Bo ) ∈ Cw and moreover ( f ! )o = ( f o )! ; f
(3) If A and B are weak PD algebras, then any morphism (A −→ B) ∈ Cw , and f ! is determined as follows: o ∗ f ! = Pd−1 A ◦ ( f ) ◦ PdB,
where ( f o )∗ denotes the morphism in HomZ (K• (Ao ), K• (Bo )). This definition generalizes the one in the previous subsection in the following sense: Proposition 7.8. C can be taken to be a subcategory of Cw . In other words, if f : A → B is a morphism in C, then f ! = ( f !)∗ : K• (B) → K•+d (A) satisfies the above requirements. Proof. Functoriality is obvious since we are merely composing the functor from C to KK with the functor from KK to Ab that sends A → K• (A), KK(A, B) " x → x∗ ∈ HomZ (K• (A), K• (B)). We need to check property (3). In other words, suppose A and B are strong PD algeo o o bras and f ! ∈ KKd (B, A) is defined to be (−1)dA ∨ A ⊗A [ f ] ⊗B B. We want to −1 o ∗ show that the induced map on K• is PdA ◦ ( f ) ◦ PdB. However, this is obvious, since the Kasparov product with B ∈ KKdB (B ⊗ Bo , C) is PdB : K• (B) → K•−dB (Bo ) −1 o • and the Kasparov product with (−1)dA ∨ A ∈ KK dA (C, A ⊗ A ) is PdA : K (A) → o K•+dA (A ). Remark 7.9. The Gysin maps in K-homology f ! ∈ HomZ (K• (A), K•+d (B)) can also be defined with completely analogous properties. There are also the obvious HL-theory analogues, used in the next subsection.
7.5. Grothendieck-Riemann-Roch formulas: the Strong case. The Grothendieck-Riemann-Roch formula compares the two bivariant cyclic classes ch( f !) and f HL !. Theorem 7.10. Suppose A and B are strong PD algebras with given HL fundamental classes. Then one has the Grothendieck-Riemann-Roch formula, ch( f !) = (−1)dB Todd(B) ⊗B f HL ! ⊗A Todd(A)−1 .
(7.18)
Proof. We will write out the right-hand side of (7.18) and simplify. In the notation of Definition 7.2, the Todd class of B is the class Todd (B) = ∨ ˜ ch (B) ∈ HL0 (B, B), B ⊗B and the inverse of the Todd class of A is the class Todd (A)−1 = (−1)dA ch ∨ ˜ A ∈ HL0 (A, A). A ⊗A Since A and B are strong PD algebras, then f HL ! is determined as follows: HL o f HL ! = (−1)dA ∨ ) ] ⊗B ˜ [( f ˜ B, A ⊗A
(7.19)
(7.20)
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where [ f HL ] = HL( f ) denotes the class in HL(A, B) of the morphism (A −→ B) and [( f HL )o ] is defined similarly. Therefore the right-hand side of (7.18) is equal to ∨ ∨ HL o (−1)dB ∨ ) ] ⊗B ˜ ch (B) ⊗B A ⊗A ˜ [( f ˜ B ⊗A ch A ⊗A ˜ A , B ⊗B which by the associativity of the intersection product, or equivalently by the diagram calculus of Appendix B (there it is worked out for KK, but it works the same way for HL), is equal to ∨ ∨ HL o B ⊗B ) ]⊗B (−1)dB ∨ ˜ A ⊗A ˜ [( f ˜ ˜ ch (B) ⊗B B . A ⊗A ch A ⊗A On the other hand, o f ! = (−1)dA ∨ ˜ [ f ] ⊗B ˜ B. A ⊗A
Therefore the left-hand side of (7.18) is equal to o (−1)dA ch(∨ ˜ ch[ f ] ⊗B ˜ ch(B). A) ⊗A
By the functorial properties of the bivariant Chern character, one has ch[ f o ] = [( f HL )o ].
(7.21)
In order to prove the theorem, it therefore suffices to prove that
dB ∨ ch(B), ˜ ch (B) ⊗B B = (−1) B ⊗B
(7.22)
∨ dA ∨ ch(∨ ˜ A = (−1) A ⊗A ch A ⊗A A).
(7.23)
and
But both of these equalities also follow easily from the diagram calculus: ∨ (∨ ˜ ch (B)) ⊗B B = (B ⊗B B) ⊗B ˜ ch (B) B ⊗B
= (−1)dB 1B ˜ ⊗B ˜ ch (B) = (−1)
dB
(7.24)
ch (B)
and ∨ ∨ ∨ ∨ ˜ A = ch A ⊗A ˜ A ⊗A A A ⊗A ch A ⊗A dA = ch ∨ ˜ (−1) 1A ˜ A ⊗A ∨ dA = (−1) ch A .
(7.25)
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7.6. Grothendieck-Riemann-Roch formulas: the Weak case. For (A −→ B), weakly K-oriented, the Grothendieck-Riemann-Roch formula repairs the noncommutativity of the diagram, f!
K• (B) ch
HL• (B)
/ K•+d (A)
f !HL
ch
(7.26)
/ HL•+d (A).
The following can be proved in an analogous way to the strong case of the GrothendieckRiemann-Roch formula, Theorem 7.10, so we will omit the proof. Theorem 7.11. If f : A → B is a morphism in Cw , and ξ ∈ K• (B), then one has the Grothendieck-Riemann-Roch formula, ch( f ! ξ ) ⊗A Todd(A) = (−1)dB f !HL (ch(ξ ) ⊗B Todd(B)) .
(7.27)
Remark 7.12. Let A be a unital PD algebra having an even degree fundamental class in K-theory. Then there is a canonical morphism, λ : C → A, given by C " z → z · 1 ∈ A, where 1 denotes the unit in A. Observe that λ is always weakly K-oriented, since C is a PD algebra, and the Gysin map λ! : K0 (A) → Z is the analog of the topological index morphism (for compact manifolds). Theorem 7.11 above applied to this situation says that, λ! (ξ ) = λHL ! (ch(ξ ) ⊗A Todd(A)) , : HL0 (A) → C is the associated Gysin morphism in cyclic theory. In the where λHL ! case where A = C(X ), X a compact spinc manifold, this is just the usual Atiyah-Singer index theorem. Indeed, λ! (ξ ) = index Pd X (ξ ) = index(D / ξ ), while the other side of the HL index formula is λ! (ch(ξ ) ⊗A Todd(A)) = (Todd(X ) ∪ ch(ξ )) [X ]. Note in particular that when ξ is the canonical rank one free module over A, then we obtain a numerical invariant which we call the Todd genus of A, a characteristic number of the algebra. 8. Noncommutative D-Brane Charges In this final section we will come to the main motivation for the present work, the “Dbrane charge formula” for very general noncommutative spacetimes. The crux of the definition of D-brane charge in Sect. 1 relied upon the introduction of natural pairings in K-theory and singular cohomology, which in turn arose as a consequence of Poincaré duality. We are now ready to describe the analogs of the natural pairings in appropriate noncommutative cases. The key point is that the multiplication map m : A ⊗ A → A is an algebra homomorphism only in the commutative case and one needs to replace its role with some new construct. This is where the formalism of KK-theory plays a crucial role. Mathematically, the problem is concerned with taking the square root of the Todd class of a noncommutative spacetime, under mild hypotheses. This then enables one to “correct” the Chern character so that the index pairing in K-theory and the given pairing in HL-theory agree.
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8.1. Poincaré pairings. In the notation of Sect. 7.1, let A ∈ D and α ∈ Ki (A), ˜ Then there is a pairing β ∈ K−d−i (A). (α, β) −→ α, β = (α × β) ⊗A⊗A˜ ∈ KK0 (C, C) = Z.
(8.1)
In the case where A = A˜ = C(X ) is the algebra of continuous functions on a spinc manifold and the fundamental class comes from the Dirac operator, this is the same as the pairing (1.1) introduced earlier, and is the K-theory analogue of the cup-product pairing (1.5). Indeed, in this case, α, β = Pd X (α) ⊗C(X ) β = D / α ⊗C(X ) β = index(D / α⊗β ). If A and A˜ have finitely generated K-theory and satisfy the Universal Coefficient Theorem (UCT), then the pairing (8.1) is nondegenerate modulo torsion. In the case of a strong PD algebra, since we have A˜ = Ao , whose K-theory is canonically isomorphic to that of A itself (by Remark 2.19), the pairing (8.1) can be viewed as a pairing of K• (A) with itself. Then we are led to consider the following additional condition. Definition 8.1. A fundamental class of a strong PD algebra A is said to be symmetric if σ ()o = ∈ Kd (A ⊗ Ao ), where σ : A ⊗ Ao −→ Ao ⊗ A
(8.2)
yo ⊗ x
→ and σ also denotes the induced map on K-homology. is the involution x In terms of the diagram calculus of Appendix B, being symmetric implies that ⊗ yo
A
Ao
x
yo
/A
/ Ao
@@ @@ @@ @@ ~~ ~~ ~ ~ ~~
A ◦
/C
y
/A
=
Ao
xo
/ Ao
@@ @@ @@ @@ ~~ ~~ ~ ~ ~~
◦
/C
for all x and y. Symmetry is a natural condition to consider, since the intersection pairing on an even-dimensional manifold is symmetric. Proposition 8.2. For any strong Poincaré duality algebra A there exists a bilinear pairing on K-theory: −, − : Ki (A) × Kd−i (A) −→ Z defined by α, β = (α × β o ) ⊗A⊗Ao ∈ KK0 (C, C) = Z.
(8.3)
Moreover, if the fundamental class is symmetric, then the bilinear pairing (8.3) on K-theory is symmetric. If A satisfies the UCT in K-theory and has finitely generated K-theory, then the pairing (8.3) is nondegenerate modulo torsion.
D-Branes on Noncommutative Manifolds
Proof. Immediate from the remarks above.
697
If A is a strong C-PD algebra, then the local cyclic homology and cohomology of A are isomorphic. This is equivalent to saying that the canonical pairing (−, −) : HLi (A) ⊗C HLd−i (A) −→ C
(8.4)
on cyclic homology, given by (x, y) = (x × y o ) ⊗A⊗Ao
(8.5)
for x ∈ HLi (A) and y ∈ HLd−i (A), is non-degenerate, since the pairing between HL• (A) and HL• (A) is always non-degenerate for any algebra, at least if the universal coefficient theorem holds. In the commutative case, this pairing coincides with the intersection form (1.5). If A is a strong PD algebra, then one can also define a bilinear form on cyclic homology determined by the class ch() as (−, −)h : HLi (A) ⊗C HLd−i (A) −→ C
(8.6)
(x, y)h = (x × y o ) ⊗A⊗Ao ch()
(8.7)
by setting
for x ∈ HLi (A) and y ∈ HLd−i (A). A fundamental class in HL-theory is said to be symmetric if σ ()o = ∈ HLd (A ⊗ o A ), where σ is the involution defined earlier in (8.2) and σ also denotes the induced map on HL-theory.
8.2. D-Brane charge formula for noncommutative spacetimes. If A is a strong PD algebra, then we have defined in the previous subsection two pairings, one given by the formula (8.7), and the other by the formula (8.5). These two pairings will a priori be different. Comparing them is the crux of our definition of D-brane charge. Let us begin with the following observation. Proposition 8.3. If A is a strong PD algebra, then the Chern character ch : K• (A) → HL• (A) is an isometry with respect to the inner products given in Eqs. (8.3) and (8.7), p, q = (ch( p), ch(q))h .
(8.8)
Proof. Using multiplicativity of the Chern character, one has (ch( p), ch(q))h = ch ( p × q o ) ⊗A⊗Ao .
(8.9)
Now use the fact that ch is a unital homomorphism (this is essentially the index theorem, i.e., the statement that the index pairing (8.3) coincides with the canonical pairing between the corresponding Chern characters in local cyclic homology).
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From this proposition it follows that the bilinear form (8.7) is the analogue of the twisted inner product defined in (1.11). Finding the appropriate modified Chern character which maps (8.5) onto (8.7) will thereby yield the formula for D-brane charge that we are looking for. The technical problem that we are faced with is to take the square root of the Todd class Todd(A) in HL(A, A). To do this, we will assume that the Universal Coefficient Theorem holds for the noncommutative spacetime A. Then HL(A, A) = End(HL• (A)). In addition, we will assume that dimC HL• (A) is finite, say equal to n. Then since Todd(A) is in G L(HL• (A)) ∼ a square root (use the = G L n (C) and every matrix in G L n (C) has √ Jordan√canonical form to prove this!), we can take a square root, Todd(A). Using the UCT, Todd(A) can again be considered as an element in HL(A, A). The square root is not unique, but we fix a choice. In some cases, the Todd class may be self-adjoint and positive with respect to a suitable inner product on HL• (A), which might help to pin down a more canonical choice. In any event, we have the following theorem. Theorem 8.4 (Isometric pairing formula). Suppose that the noncommutative spacetime A satisfies the UCT for local cyclic homology, and that HL• (A) is a finite dimensional vector space. If A has symmetric (even-dimensional) fundamental classes both in K-theory and in cyclic theory, then the modified Chern character ch ⊗A
Todd(A) : K• (A) → HL• (A)
(8.10)
is an isometry with respect to the inner products (8.1) and (8.5), p, q = ch( p) ⊗A Todd(A), ch(q) ⊗A Todd(A) .
(8.11)
Proof. To prove the theorem, we use Proposition 8.3 and observe that it’s enough to show that the right-hand sides of Eqs. (8.11) and (8.9) agree. For this we use the diagram calculus of Appendix B, A
Todd /
√
A; ;; ;; ;;
Ao √
A
Todd/
A
A ◦
/C
Todd/
√
A
=
Ao
1A
/A
= √ o Todd/ o o A A
Todd/
1A o
√
;; ;; ;; ;
A ◦
/C
=
A; ;; ;; ;;
/ Ao
Todd/
◦
A; ;; ;; ;;
√ o Todd/ o o A A
◦
/C
/C .
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Note that the symmetry of is used here in a crucial way. This computation shows that √ √ (ch( p) ⊗A Todd(A), ch( p) ⊗A Todd(A)) = (ch( p) ⊗ ch(q)) ⊗A⊗Ao (Todd(A) ⊗A ). Since, by definition, Todd(A) = ∨ ⊗Ao ch(), a similar computation shows that Todd(A) ⊗A = (∨ ⊗A ) ⊗Ao ch() = ch() and so √ √ (ch( p) ⊗A Todd(A), ch( p) ⊗A Todd(A)) = (ch( p) ⊗ ch(q)) ⊗A⊗Ao ch() = ch(( p ⊗ q) ⊗A⊗Ao ) = ch(( p, q)). Finally, the Chern character Z = KK(C, C) → HL(C, C) = C is injective, which gives the desired result. Corollary 8.5 (D-brane charge formula for noncommutative spacetimes). Suppose that the noncommutative spacetime A satisfies the hypotheses of Theorem 8.4 above. Then there is a noncommutative analogue of the well-known formula (1.1) in [63] for the charge associated to a D-brane B in a noncommutative spacetime A with given weakly K-oriented morphism f : A → B and Chan-Paton bundle ξ ∈ K• (B), Qξ = ch( f ! (ξ )) ⊗A Todd(A). (8.12) This is still not quite the most general situation. Corollary 8.5 deals with charges coming from a (weakly) K-oriented morphism f : A → B, when a Chan-Paton bundle, i.e., a K-theory class, is given on B. This is the obvious translation of the situation coming from a flat D-brane in the commutative case, but one can imagine more general noncommutative D-branes, where the algebra B is missing, i.e., one simply has a Fredholm module for A representing a class in K• (A). (In Corollary 8.5, the associated class in K• (A) is PdA( f ! (ξ )) = f ∗ (PdB(ξ )).) The final version of the charge formula is the following: Proposition 8.6 (D-brane charge formula, dual version). Suppose that the noncommutative spacetime A satisfies the hypotheses of Theorem 8.4 above. Then there is a noncommutative analogue of formula (1.30) of Proposition 1.8 above for the dual charge associated to a D-brane in the noncommutative spacetime A represented by a class µ ∈ K• (A): −1 Qµ = Todd(A) ⊗A ch(µ). (8.13) This formula satisfies the isometry rule: ∨ ⊗A⊗Ao Qµ × Qoν = ∨ ⊗A⊗Ao (µ × ν o ).
(8.14)
Proof. We need to check (8.14). By multiplicativity of the Chern character, the righthand side is equal to ch(∨ ) ⊗A⊗Ao (ch(µ) × ch(ν)o ). The left-hand side is ∨ ⊗A⊗Ao Qµ × Qoν −1 −1 ⊗A⊗Ao ch(µ) × ch(ν)o Todd(A) × Todd(A)o = ∨ ⊗A⊗Ao = (by symmetry of ∨ as in the proof of Theorem 8.4) ∨ ⊗A⊗Ao Todd(A) −1 × 1Ao ⊗A⊗Ao ch(µ) × ch(ν)o .
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But ch(∨ ) = ∨ ⊗A Todd(A) −1 , since (by the diagram calculus) ∨ ⊗A Todd(A) −1 = ∨ ⊗A ch(∨ ) ⊗Ao = ch(∨ )⊗Ao ∨ ⊗A = ch(∨ ). Remark 8.7. Although our noncommutative formulas for D-brane charges have been derived under the assumption that A is a strong PD algebra, they hold more generally for any algebra A belonging to the class D introduced in Sect. 7.1. This allows us to write down charge formulas in a variety of very general situations. For instance, one can in this way obtain a bilinear pairing on twisted K-theory, K• (X, H ) × K• (X, −H ) → Z, and hence an isometric pairing between twisted K-theory and twisted cohomology, recovering the charge formula (1.13) of [8] for twisted D-branes.
Appendix A. The Kasparov Product In this appendix, we will summarize the main properties of the intersection product. If A is a separable algebra then the exterior (or cup) product exists and defines a bilinear pairing, [53, Thm 2.11]: KKi (A, B1 ) ⊗B1 KK j (B1 , B2 ) → KKi+ j (A, B2 ).
(A.1)
In [53, Def. 2.12], Kasparov also defines the intersection product (which he calls the cap-cup product) KKi (A1 , B1 ⊗ D) ⊗D KK j (D ⊗ A2 , B2 ) → KKi+ j (A1 ⊗ A2 , B1 ⊗ B2 ) by the formula x1 ⊗D x2 = (x1 ⊗ 1A2 ) ⊗B1 ⊗D⊗A2 (x2 ⊗ 1B1 ). The exterior (or cup) product is obtained when D = C. This exterior product has the following properties [53, Thm. 2.14]. Theorem A.1. Let A1 and A2 be separable algebras. Then the intersection (cup-cap) product exists and is: (1) (2) (3) (4)
bilinear; contravariant in A1 and A2 ; covariant in B1 and B2 ; functorial in D: for any morphism f : D1 → C one has f (x1 ) ⊗D2 x2 = x1 ⊗D1 f (x2 ) ;
(5) associative: for any x1 ∈ KKi (A1 , B1 ⊗ D1 ), x2 ∈ KK j (D1 ⊗ A2 , B2 ⊗ D2 ) and x3 ∈ KKk (D2 ⊗ A3 , B3 ), where A1 , A2 , A3 and D1 are assumed separable, the following formula holds: (x1 ⊗D1 x2 ) ⊗D2 x3 = x1 ⊗D1 (x2 ⊗D2 x3 ); (6) For any x1 ∈ KKi (A1 , B1 ⊗ D1 ⊗ D), x2 ∈ KK j (D ⊗ D2 ⊗ A2 , B2 ), where A1 , A2 , D2 are separable and D1 is σ -unital, the following formula holds: x1 ⊗D x2 = (x1 ⊗ 1D2 ) ⊗D1 ⊗D⊗D2 (1D1 ⊗ x2 );
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B
C
◦
A
◦
D
Fig. B.1. Diagram representing an element of KK(B ⊗ A, C ⊗ D)
(7) For x1 ∈ KKi (A1 , B1 ⊗D), x2 ∈ KK j (D⊗A2 , B2 ), for separable algebras A1 , A2 and D, the following formula holds: (x1 ⊗D x2 ) ⊗C 1D1 = (x1 ⊗ 1D1 ) ⊗D⊗D1 (x2 ⊗ 1D1 ); (8) the cup product is commutative (over C): x1 ⊗C x2 = x2 ⊗C x1; and (9) the element 1C ∈ KK0 (C, C) is a unit for this product: 1C ⊗C x = x ⊗C 1C = x for all x ∈ KKi (A, B), where A is assumed to be separable.
Appendix B. A Diagram Calculus for the Kasparov Product Keeping track of Kasparov products and the associativity formulae in the general case described above in Appendix A can be quite complicated. In this appendix we describe a pictorial calculus for keeping track of these things, which one of us (J.R.) has often found useful as a guide to calculations. In this appendix we will not write degree labels explicitly on KK-groups for the sake of notational convenience — in the most important case, all elements lie in KK0 anyway. The idea is to represent an element of a KK group by a diagram (which we read from left to right), with one “input” for each tensor factor in the first argument of KK, and one “output” for each tensor factor in the second argument of KK. For convenience, we can also add arrowheads pointing toward the outputs. Thus, for example, an element of KK(B ⊗ A, C ⊗ D) would be represented by a diagram like the one in Fig. B.1. Note that an element of KK(A ⊗ B, C ⊗ D) would be represented by an almost identical diagram, having the two input terminals switched. The basic rule is that permutation of the input or output terminals may involve at most the switch of a sign. The Kasparov product corresponds to concatenation of diagrams, except that one is only allowed to attach an input to a matching output. For example, in Fig. B.1, there are input terminals corresponding to both B and A, so one can take the product over a Kasparov class having a B or A as an output terminal. For example, a class in KK(E, A) would be represented by a diagram like Fig. B.2, and we can concatenate the diagrams as shown in Fig. B.3 to obtain the product (over A) in KK(B ⊗ E, C ⊗ D) shown in Fig. B.4.
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E
A
Fig. B.2. Diagram representing an element of KK(E, A)
B
C
◦
E
◦
A
D
Fig. B.3. Diagram representing the intersection product, ⊗A : KK(E, A) ⊗ KK(B⊗ A, C⊗ D) → KK(B⊗ E, C ⊗ D)
C
B
◦
◦
E
D
Fig. B.4. Diagram representing an element of KK(B ⊗ E, C ⊗ D)
E
z
B
◦
D
y
x
C
A
Fig. B.5. Diagram showing that z ⊗B (y ⊗A x) = ±y ⊗A (z ⊗B x)
The associativity of the Kasparov product corresponds to the principle that if one has multiple concatenations to do, the concatenations can be done in any order, except perhaps for keeping track of signs. For example, if x ∈ KK(B ⊗ A, C),
y ∈ KK(D, A), and
z ∈ KK(E, B),
then the associativity of the product gives z ⊗B (y ⊗A x) = ±y ⊗A (z ⊗B x), even though when written this way, it seems to be somewhat counter-intuitive. But one can “prove” this graphically with the picture in Fig. B.5. Of course, a picture by itself is not a rigorous proof, but it can be made into one as follows. Here × is used to denote the “exterior” Kasparov product, and for simplicity we assume that all elements lie in KK0 , so that we don’t have to worry about sign changes
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(which all have to do with conventions about orientation of the Bott element). On the one hand, we have z ⊗B (y ⊗A x) := (z × 1D) ⊗B⊗D (y ⊗A x) = (z × 1D) ⊗B⊗D (1B × y) ⊗B⊗A x # = (z × 1D) ⊗B⊗D (1B × y) ⊗B⊗A x.
(B.1)
But on the other hand we have y ⊗A (z ⊗B x) := (1E × y) ⊗E⊗A (z ⊗B x) = (1E × y) ⊗E⊗A (z × 1A) ⊗B⊗A x # = (1E × y) ⊗E⊗A (z × 1A) ⊗B⊗A x.
(B.2)
So to prove the associativity formula, it suffices to observe that (z × 1D) ⊗B⊗D (1B × y) = z × y = (1E × y) ⊗E⊗A (z × 1A).
(B.3)
We should mention incidentally that essentially everything we said about the Kasparov product applies equally well to products in bivariant cyclic homology, whose formal properties are exactly the same. Acknowledgements We would like to thank P. Bouwknegt, K. Hannabuss, J. Kaminker, R. Plymen, R. Reis and I.M. Singer for helpful discussions. J.B. and R.J.S. were supported in part by the London Mathematical Society. V.M. was supported by the Australian Research Council. J.R. was supported in part by the USA National Science Foundation, grant number DMS-0504212. R.J.S. was supported in part by PPARC Grant PPA/G/S/2002/00478 and by the EU-RTN Network Grant MRTN-CT-2004-005104. J.B., V.M., and J.R. all thank the Erwin Schrödinger International Institute for Mathematical Physics for its hospitality under the auspices of the programme in Gerbes, Groupoids, and Quantum Field Theory, which made part of this collaboration possible.
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Commun. Math. Phys. 277, 707–714 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0372-6
Communications in
Mathematical Physics
KdV Preserves White Noise Jeremy Quastel, Benedek Valkó Departments of Mathematics and Statistics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S2E4. E-mail: [email protected]; [email protected] Received: 19 December 2006 / Accepted: 21 May 2007 Published online: 8 November 2007 – © Springer-Verlag 2007
Abstract: It is shown that white noise is an invariant measure for the KortewegdeVries equation on T. This is a consequence of recent results of Kappeler and Topalov establishing the well-posedness of the equation on appropriate negative Sobolev spaces, together with a result of Cambronero and McKean that white noise is the image under the Miura transform (Ricatti map) of the (weighted) Gibbs measure for the modified KdV equation, proven to be invariant for that equation by Bourgain. 1. KdV on H −1 (T) and White Noise The Korteweg-deVries equation (KdV) on T = R/Z, u t − 6uu x + u x x x = 0,
u(0) = f
(1.1)
defines nonlinear evolution operators St f = u(t),
(1.2)
−∞ < t < ∞ on smooth functions f : T → R. Theorem 1.1 (Kappeler and Topalov [KT1]). St extends to a continuous group of nonlinear evolution operators S¯t : H −1 (T) → H −1 (T).
(1.3)
In concrete terms, take f ∈ H −1 (T) and let f N be smooth functions on T with f N − f H−1 (T) → 0 as N → ∞. Let u N (t) be the (smooth) solutions of (1.1) with initial data f N . Then there is a unique u(t) ∈ H −1 (T) which we call u(t) = S¯t f with u N (t) − u(t) H −1 (T) → 0.
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White noise on T is the unique probability measure Q on the space D(T) of distributions on T satisfying 1 2 (1.4) eiλ,u d Q(u) = e− 2 λ2 for any smooth function λ on T, where · 22 = ·, · are the L 2 (T, d x) norm and inner product (see [H]). Let {en }n=0,1,2,... be an orthonormal of smooth functions in L 2 (T) with e0 = 1. basis ∞ White noise is represented as u = n=0 x n en , where x n are independent Gaussian random variables, each with mean 0 and variance 1. Hence Q is supported in H −α (T) for any α > 1/2. Mean zero white noise Q 0 on T is the probability measure on distributions u with T u = 0 satisfying 1 2 (1.5) eiλ,u d Q 0 (u) = e− 2 λ2 for any mean zero smooth function λ on T. It is represented as u = ∞ n=1 x n en . Recall that if f : X 1 → X 2 is a measurable map between metric spaces and Q is a probability measure on (X 1 , B(X 1 )), then the pushforward f ∗ Q is the measure on X 2 given by f ∗ Q(A) = Q({x : f (x) ∈ A}) for any Borel set A ∈ B(X 2 ). Our main result is: Theorem 1.2. White noise Q 0 is invariant under KdV; for any t ∈ R, S¯t∗ Q 0 = Q 0 .
(1.6)
Remarks. 1. In terms of classical solutions of KdV, the meaning of Theorem 1.2 is as follows. Let f N , N = 1, 2, . . . be a sequence of smooth mean zero random initial data approximating mean zero white noise. For example, one could take N f N (ω) = n=1 xn (ω)en , where xn and en are as above. Solve the KdV equation for each ω up to a fixed time t to obtain St f N . The limit in N exists [KT1] in H −1 (T), for almost every value of ω , and is again a white noise. 2. It follows immediately that Sˆt : L 2 (Q 0 ) → L 2 (Q 0 ), (Sˆt )( f ) = (S¯t f )
(1.7)
L 2 (Q
3.
are a group of unitary transformations of 0 ), defining a continuous Markov process u(t), t ∈ (−∞, ∞) on H −1 (T) with Gaussian white noise one dimensional marginals, invariant under time+space inversions. The correlation functions S(x, t) = f (0)S¯t f (x)d Q 0 may have an interesting structure. Q 0 is certainly not measure for KdV. The Gibbs measure formally the only invariant written as Z −1 1 T u 2 ≤ K e−H2 , where 1 (1.8) H2 (u) = − u 3 − u 2x 2 T is known to be invariant [Bo]. Note that (after subtraction of the mean) this Gibbs measure is supported on a set of Q 0 -measure 0. Q 0 is also a Gibbs measure, corresponding to the Hamiltonian u2. (1.9) H1 (u) = T
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709
The existence of two Gibbs measures corresponds to the bihamiltonian structure of KdV: It can be written u˙ = Ji
4.
δHi , δu
i = 1, 2
(1.10)
with symplectic forms J1 = ∂x3 + 4u∂x + 2∂x u and J2 = ∂x . Because of all the conservation laws of KdV, there are many other invariant measures as well. We were led to Theorem 1.2 after noticing that the discretization of KdV used by Kruskal and Zabusky in the numerical investigation of solitons, u˙ i = (u i+1 + u i + u i−1 )(u i+1 − u i−1 ) − (u i+2 − 2u i+1 + 2u i−1 − u i−2 ),
(1.11)
preserves discrete white noise (independent Gaussians mean 0 and variance σ 2 > 0). The invariance follows from two simple properties of the special discretization (1.11). First of all u˙ i = bi preserves Lebesgue measure whenever ∇ ·b = i ∂i bi = 0, and (1.11) is of this form. Furthermore, it is easy to check (though something 2 − 1 2 i u i2 −1 du i is of a miracle) that i u i is invariant under (1.11). Hence Z e 2σ also invariant. Note that the discretization (1.11) is not completely integrable, and we are not aware of a completely integrable discretization which does conserve discrete white noise. For example, consider the following family of completely integrable discretizations of KdV, depending on a real parameter α [AL]: u˙ i = (1 − αu i ){−αu i−1 (u i−2 − u i ) − α(u i−1 + 2u i + u i+1 )(u i−1 − u i+1 ) −αu i+1 (u i − u i+2 ) + u i−2 − 2u i−1 + 2u i+1 − u i+2 }. (1.12) They conserve Lebesgue measure by the Liouville theorem. We want α = 0; otherwise the quadratic term of KdV is not represented. In that case the conserved quantity analogous to T u 2 is u i2 + 2u i u i+1 . (1.13) Q= i
5.
But Q is non-definite, and hence the corresponding measure e−Q i du i cannot be normalized to make a probability measure. At a completely formal level the proof proceeds as follows. Note first of all that the flow generated by u t = u x x x is easily solved and seen to preserve white noise. So consider the Burgers’ flow u t = 2uu x , 2 2 2 δf δf ∂t f (u(t))e− u = , u t e− u = , (u 2 )x e− u δu δu δ 2 − u2 =− f (u )x e (1.14) δu and
2 δ 2 − u2 = (2u x − (u 2 )x 2u)e− u . (u )x e δu
(1.15)
The last term vanishes because (u 2 )x 2u = 23 (u 3 )x and because of periodic boundary 1 conditions any exact derivative integrates to zero: f x = 0 f = 0.
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Such an argument is known in physics [S]. Note that the problem is subtle, and requires an appropriate interpretation. In fact the result is not correct for the standard mathematical interpretation of the Burgers’ flow as the limit as ↓ 0 of u t = 2u u x + u x x , as can be checked with the Lax-Oleinik formula. On the other hand, the argument is rigorous for (1.11). 2. Invariant Measures for mKdV on T
Let P0 denote Wiener measure on φ ∈ C(T) conditioned to have T φ = 0. It can be derived from the standard circular Brownian motion P on C(T) defined as follows: Condition a standard Brownian motion β(t), t ∈ [0, 1] starting at β(0) = x to have β(1) = x as well, and now distribute x on the real line according to Lebesgue measure. P0 is obtained from P by conditioning on T φ = 0. (4) Define P0 to be the measure absolutely continuous to P0 given by 4 1 (4) −1 J (φ)e− 2 T φ d P0 , (2.1) P0 (B) = Z B
for Borel sets B ⊂ C(T), where Z is the normalizing factor to make P0(4) a probability measure and 1
J (φ) = (2π )−1/2 K (φ)K (−φ)e 2 where
1
K (φ) =
φ2
2
,
(2.2)
e2(x) d x
(2.3)
φ(y)dy.
(2.4)
0
and
(x) =
x
0
For smooth g and −∞ < t < ∞, let φ(t) = Mt g denote the (smooth) solution of the modified KdV (mKdV) equation, φt − 6φ 2 φx + φx x x = 0,
φ(0) = g.
(2.5)
Theorem 2.1 (Kappeler and Topalov [KT3]). Mt extends to a continuous group of nonlinear evolution operators ¯ t : L 2 (T) → L 2 (T). M
(2.6)
1 φ 4 + φx2 . 2 T
(2.7)
Let H (φ) =
mKdV can be written in Hamiltonian form, φt = ∂ x (4)
P0
δH . δφ
gives rigorous meaning to the weighted Gibbs measure J (φ)e−H (φ) on
(2.8)
φ = 0.
KdV Preserves White Noise
711 (4)
Theorem 2.2 (Bourgain [Bo]). P0
is invariant for mKdV,
¯ ∗t P (4) = P (4) . M 0 0
(2.9) 1
Proof In fact what is proven in [Bo] is that Z −1 e− 2 T φ d P is invariant for mKdV. The main obstacle at the time was a lack of well-posedness for mKdV on the support H 1/2− of the measure. This statement follows with less work once one has the results of Kappeler and Topalov proving well-posedness on a larger set (Theorem 2.1). We have in addition to show that J (φ) is a conserved quantity for mKdV. It is well known that T φ 2 is preserved. So the problem is reduced to showing that K (φ) and K (−φ) are conserved. Let φ(t) be a smooth solution of mKdV. Note that 4
∂t = 2φ 3 − φx x . Hence
1
∂t K = 2
(2.10)
(2φ 3 − φx x )e2(x) d x.
(2.11)
0
But integrating by parts we have, since φ is periodic and x = φ, 1 1 1 2(x) 2(x) 2 2(x) φx x e dx = − 2φx φe d x = − (φ )x e dx = 0
0
0
1
2φ 3 e2(x) d x.
0
(2.12)
Therefore ∂t K (φ(t)) = 0. One can easily check with the analogous integration by parts that ∂t K (−φ(t)) = 0. ¯ t φ = φ(t) with φ ∈ L 2 (T). From Theorem 2.1 we have smooth φn Now suppose M with φn → φ and φn (t) → φ(t) in L 2 (T), K (φ(t)) − K (φ) = [K (φ(t)) − K (φn (t))] − [K (φn ) − K (φ)],
(2.13)
¯ t. so if K is a continuous function on L 2 (T) then K (φ) and K (−φ) are conserved by M To prove that K is continuous simply note that 1 x x 2φ L 2 (T) 2ψ−φ L 2 (T) |K (φ) − K (ψ)| = | e2 0 φ [e2 0 ψ−φ − 1]d x| ≤ e [e − 1]. 0
(2.14)
3. The Miura Transform on L 20 (T) The Miura transform φ → φx +φ 2 maps smooth solutions of mKdV to smooth solutions of KdV. It is basically two to one, and not onto. But this is mostly a matter of the mean T φ. Since the mean is conserved in both mKdV and KdV, it is more natural to consider the map corrected by subtracting the mean. The corrected Miura transform is defined for smooth φ by, 2 µ(φ) = φx + φ − φ2. (3.1) T
Let
L 20 (T)
and
H0−1 (T)
denote the subspaces of L 2 (T) and H −1 (T) with T φ = 0.
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Theorem 3.1 (Kappeler and Topalov [KT2]). The corrected Miura transform µ extends to a continuous map µ¯ : L 20 → H0−1
(3.2)
which is one to one and onto. µ¯ takes solutions φ of mKdV (2.5) on L 2 (T), to solutions u = µ(φ) of KdV (1.1) on H −1 (T); ¯ t. S¯t µ¯ = µ¯ M
(3.3)
Remark The Ricatti map is given by r (φ, λ) = φx + φ 2 + λ.
(3.4)
Note that Kappeler and Topalov use the term Ricatti map for µ = r (φ, − T φ 2 ). 4. The Miura Transform on Wiener Space Theorem 4.1 (Cambronero and McKean [CM]). The corrected Miura transform µ¯ maps (4) P0 into mean zero white noise Q 0 ; (4)
µ¯ ∗ P0 (4) Proof Let Pˆ0 be given by
1 Pˆ (4) (B) = √ 2π
= Q0.
(4.1)
1
(φ,λ)∈B
K (φ)K (−φ)e− 2
T (φ
2 +λ)2
d Pdλ,
(4.2)
where B is a Borel subset of C(T) × R. Let rˆ = (r, T φ). Let Qˆ on C(T) × R be given by Qˆ = Q× Lebesgue measure. What is actually proved in [CM] is that ˆ rˆ ∗ Pˆ (4) = Q.
(4.3)
Equation (4.1) is obtained by conditioning on λ = − T φ 2 and T φ = 0.
Remark There is a simple heuristic argument explaining (4.1). Formally (4)
d P0 = Z 1−1 K (φ)K (−φ)e− 2
1
1 0
1 (φ 2 +φ − 0 φ 2 )2
d F(φ),
d Q 0 = Z 2−1 e− 2 1
1 0
u2
d F(u), (4.4)
1 (4) where F is the (mythical) flat measure on 0 φ = 0. Note that in the exponent of d P0 1 2 we have assumed that integration by parts gives 0 φ φ = 0. Since the corrected 1 Miura transform u = φ 2 + φ − 0 φ 2 the only mystery is the form of the Jacobian C K (φ)K (−φ). Let D be the map D f = f and φ stand for the map of multiplication by 1 φ with a subtraction to make the result mean zero, φ f = φ · f − 0 φ · f . The Jacobian is then, f (φ) = det(1 + 2φ D −1 ) = exp{Tr log(1 + 2φ D −1 )}.
(4.5)
KdV Preserves White Noise
713
∂ For fixed x, y ∈ T let ∂x y = ∂(φ(y)−φ(x)) , i.e. the Gâteaux derivative in the direction −1 δ y − δx , ∂x y F(φ) = lim→0 (F(φ + (δ y − δx )) − F(φ). We have
∂x y log f (φ) = ∂x y Tr log(1 + 2ϕ D −1 ) = Tr[{∂x y (1 + 2φ D −1 )}{1 + 2φ D −1 )−1 }]. (4.6) If we let G(x, y) denote the Green function of D + 2φ this gives ∂x y log f (φ) = 2[G(y, y) − G(x, x)]. It is not hard to compute the Green function with the result that y y 2 x e−2 2 x e−2 ∂x y log f (φ) = 1 − 1 . (4.7) −2 −2 0 e 0 e The argument is completed by a straightforward verification that this is satisfied by f (φ) = K (φ)K (−φ). The heuristic argument can be made rigorous by taking finite dimensional approximations where this set of equations actually identifies the determinant. Since the computations become exactly those of [CM], we do not repeat them here. 5. Proof of Theorem 1.2 S¯t∗ Q 0
4.1 = S¯t∗ µ¯ ∗ P0(4)
Thm
3.1 ¯ ∗t P (4) Thm=2.2 µ¯ ∗ P (4) Thm=4.1 Q 0 . = µ¯ ∗ M 0 0
Thm
(5.1)
Acknowledgement. Thanks to K. Khanin, M. Goldstein and J. Colliander for enlightening conversations. J. Q. and B. V. are supported by the Natural Sciences and Engineering Research Council of Canada, B. V. is partially supported by the Hungarian Scientific Research Fund grant K60708.
References [AL] [Bi] [Bo] [CKSTT] [CM] [H] [KPV] [KT1] [KMT] [KT2] [KT3]
Ablowitz, M.J., Ladik, J.F.: On the solution of a class of nonlinear partial difference equations. Studies in Appl. Math. 57(1), 1–12 (1976/77) Billingsley, P.: Convergence of Probability Measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. New York: John Wiley & Sons, Inc., 1999 Bourgain, J.: Periodic nonlinear Schrödinger equation and invariant measures. Commun. Math. Phys. 166(1), 1–26 (1994) Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Sharp global well-posedness for KdV and modified KdV on R and T. J. Amer. Math. Soc. 16(3), 705–749 (2003) Cambronero, S., McKean, H.P.: The ground state eigenvalue of Hill’s equation with white noise potential. Comm. Pure Appl. Math. 52(10), 1277–1294 (1999) Hida, T.: Brownian motion. Applications of Mathematics 11. New York-Berlin: Springer-Verlag, 1980 Kenig, C., Ponce, G., Vega, L.: A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9, 573–603 (1996) Kappeler, T., Topalov, P.: Well-posedness of KdV on H −1 (T). Mathematisches Institut, GeorgAugust-Universität Göttingen: Seminars 2003/2004, Göttingen: Universitätsdrucke Göttingen, 2004, pp. 151–155 Kappeler, T., Möhr, C., Topalov, P.: Birkhoff coordinates for KdV on phase spaces of distributions. Selecta Math. (N.S.) 11(1), 37–98 (2005) Kappeler, T., Topalov, P.: Riccati map on L 20 (T) and its applications. J. Math. Anal. Appl. 309(2), 544–566 (2005) Kappeler, T., Topalov, P.: Global well-posedness of mKdV in L 2 (T, R). Comm. Partial Diff. Eqs. 30(1–3), 435–449 (2005)
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[S]
Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics, Berlin-Heidelberg-New York: Springer-Verlag 1991, p. 267 Takaoka, H., Tsutsumi, Y.: Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition. Int. Math. Res. Not. 56, 3009–3040 (2004)
[TT]
Communicated by A. Kupiainen
Commun. Math. Phys. 277, 715–727 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0378-0
Communications in
Mathematical Physics
Torsion Cycles as Non-Local Magnetic Sources in Non-Orientable Spaces Marcos Alvarez Centre for Mathematical Science, City University, Northampton Square, London EC1V 0HB, UK. E-mail: [email protected] Received: 8 January 2007 / Accepted: 19 April 2007 Published online: 7 November 2007 – © Springer-Verlag 2007
Abstract: Non-orientable spaces are known to be able to support net magnetic fluxes through closed surfaces, even in the absence of magnetic sources. To an observer for whom the non-orientability of space appears to be confined to a certain finite region of space, such a magnetic flux appears as a magnetic charge quasi-localised in that region. In this paper it is shown that this effect is a physical manifestation of the existence of torsion cycles of codimension one in the homology of space. 1. Introduction The idea that electromagnetic charge may be a manifestation of the topology of space has a very long history [Whe62, MW57, Lub63, Sor77]. The basic configuration considered by Wheeler and Misner was a space that contained a handle-like region, often called a “wormhole”. This space is traversed by sourceless electric lines of force that enter the wormhole radially through one mouth and exit through the other. For an observer situated sufficiently far from the wormhole, space appears approximately flat, but not sourceless: the two mouths look like two pointlike electric charges. A “dual” configuration can be constructed in which the lines of force are oriented circles that surround the wormhole mouths and extend to its interior in a continuous way, as sketched in Fig. 1. From a distance, the two mouths appear to carry dual charges of opposite signs. As a matter of terminology, we will refer to the dual lines of force, and the charges they correspond to, as magnetic, although the motivation for doing that will only become clear later on. Serious objections can be raised against Misner and Wheeler’s wormhole as a realistic model of electromagnetic charge, on the grounds that wormholes are not satisfactory solutions in General Relativity. Under fairly general assumptions, the null energy condition has to be violated at some points near the wormhole throat [HV97]. This indicates that wormhole solutions require the coupling of gravity to some type of exotic matter. Even if the right type of matter and coupling is found, known wormhole solutions are
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Fig. 1. An orientable handle
very unstable. Charged wormholes also exist in General Relativity in the presence of scalar fields non-minimally coupled to gravity. The scalar field provides a good example of the type of exotic matter required, but these charged wormholes are still unstable under spherically symmetric perturbations [BG03]. Another unsatisfactory feature of the Misner-Wheeler construction is that it is unable to accommodate individual isolated charges. Besides, the throat that joins the two apparent charges establishes a correlation between them that is difficult to reconcile with quantum mechanics [Sor77]. For these reasons, the Misner-Wheeler wormhole cannot be taken seriously as a realistic model of charged particles. Some time later, Sorkin [Sor77] constructed a non-orientable version of the WheelerMisner wormhole, and showed that the effect of the non-orientability was that, to a distant observer, the non-orientable handle appears to be a source of net flux. Figure 2 gives a two-dimensional illustration of this idea. A non-orientable surface is crossed by magnetic lines of force, and the non-orientability causes the lines of force to emerge with the same orientation from both ends. A distant observer would assign the same magnetic charge to both of them. In contrast, electric flux lines would have behaved very much as in Fig. 1, and each end would have seemed to still carry opposite electric charges. This example suggests that the non-orientability of space affects only one kind of charge, namely the one we called dual or magnetic. The construction proposed in [Sor77] covers a large class of non-orientable manifolds, not limited to handles, and provides a model of individual, isolated charges that is based purely upon the topology of space. These ideas were further elaborated and extended in [DH99], who generalised Sorkin’s proposal and provided new illustrations of it. The ideas that will be discussed in the main body of this paper are similar to those in [Sor77], although the point of view will be rather different. None of the objections pointed out earlier will be answered, and it will not be claimed that these models can provide an alternative model of electromagnetic charge. The goal will be to show that a different connection between magnetic charge and non-orientability can be made, in which the topological properties of non-orientable spaces appear more explicitly. 2. First Notions We shall be working in an (m + 1)-dimensional spacetime of the form R × K, where the real line R is the time direction. By construction, this spacetime is time-orientable,
Torsion Cycles as Non-Local Magnetic Sources in Non-Orientable Spaces
717
Fig. 2. A non-orientable handle
so it will be orientable if and only if the space manifold K is. In this spacetime there is defined a closed (m − 1)-form F, the electromagnetic field strength. The presence of magnetic or electric sources is expressed by the dual currents µm and µe , which are related to the field strength F and its Hodge dual ∗F by the equations, d F = µm , d∗F = µe .
(1)
The forms µm and µe are taken to be purely spatial, i.e., are dual to timelike forms. This is a physical assumption based on the fact that the world-volumes of localised charged sources are always timelike. The dual current, being transversal to the worldvolume, must then be always spacelike. Assuming that the Stokes theorem for differential forms can be applied, (1) relate the magnetic and electric charges to the fluxes of F and ∗F respectively. Suppose that N is an m-dimensional region of K whose frontier is a smoothly embedded compact (m − 1)-manifold S. Then the magnetic charge contained in N , Q m (N ) = µm , (2) N
equals the flux of F through S,
Q m (N ) =
µm =
N
dF =
N
F. S
Similarly, if ν is a 3-dimensional region of K bounded by σ , the electric charge contained in ν, defined as Q e (ν) = µe , (3) ν
equals the flux of ∗F through σ ,
Q e (ν) =
µe =
ν
d∗F =
ν
∗F. σ
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Although charges can be defined directly in terms of the currents, as in (2) and (3), we will only be concerned with those charges that can be expressed as fluxes. The general question of whether charges are always fluxes or not depends on certain relationships between the homologies of the world-volumes of the sources and of the spacetime manifold in which they propagate. For a recent analysis of this question in compact orientable manifolds with boundary, see [AO06]. 3. Global Considerations Equations (1), which hold locally at every point in space, leave open the possibility that F or ∗F, or the dual currents µm and µe , may not be globally defined [Kis78]. We shall assume that the (m − 1)-form F is global, i.e., single-valued. In our model, the dual magnetic current µm is an m-form on the m-manifold K. Consequently, it must be proportional to ΩK , the volume form of K, µm = C(w)ΩK ,
(4)
in which C(w) is a distribution localised on w, the world-line of the magnetic charge. Equation (4) only makes sense globally if K is orientable. If it is not, there exists a loop λ along which an ΩK chosen at one point gets transported back into −ΩK , which would be interpreted as the magnetic charge having changed sign after traversing λ. Because we are assuming that F is single-valued, the same must be true of µm . In what follows, K will be non-orientable, and so, as discussed, µm can only be defined globally up to a sign. This leaves µm = 0 as the only possibility compatible with our assumptions. That makes F a closed form, i.e., the first equation in (1) reduces to d F = 0. No nonzero single-valued magnetic charge, as defined in (2), can exist in K. Another implication of K being non-orientable is that, because there is no global volume element with which to define the Hodge star, the dual field strength ∗F cannot exist globally. It will reverse sign whenever ΩK does, as was the case when it was transported around the loop λ. Therefore µe will also have to change sign upon traversing λ. In conclusion, within the framework of the equations of motion (1), spacetimes of the form R × K, where the spatial manifold K is non-orientable cannot support nonzero single-valued electric or magnetic currents, and hence neither electric nor magnetic charges in the sense of (2) and (3). It will be shown below that, despite the exclusion of magnetic sources implied by the condition d F = 0, it is still possible to define a different kind of magnetic charge, understood as a flux of F through a cycle in K, which owes its existence precisely to the non-orientability of the space manifold K. 4. Fluxes, Orientations, and Charges Given the closed differential (m − 1)-form F and a smoothly embedded orientable (m − 1)-cycle S in K, the magnetic flux Φ(S) is defined as Φ(S) = F. (5) S
The assumption of orientability of S is needed in order to integrate the differential form F on S [AMR88, BT82]. Let us now add the extra assumption that S separates the
Torsion Cycles as Non-Local Magnetic Sources in Non-Orientable Spaces
719
space manifold K into two parts that we call M and N , with M non-compact and orientable, and N compact, possibly non-orientable, and has S as its only frontier. An N non-orientable will be interpreted below as a quasi-localised region of space whose topology is capable of displaying magnetic charge, as first suggested by Sorkin [Sor77]. Configurations of this type can be constructed by taking the connected sum of two m-manifolds, one non-compact and orientable and the other closed and non-orientable. The connected sum consists in removing the interior of an embedded m-ball from each manifold, and pasting the remainders together by means of an homeomorphism on the boundary spheres of these balls. The identification of the two boundary spheres gives the cycle S. If there is net magnetic flux Φ(S), we will say that N carries F-charge. We now proceed to relate Φ(S) to the behaviour of F in the interior of N by means of the Stokes theorem. In fact, there are two versions of the Stokes theorem, depending on whether we are working with differential forms, or with densities [AMR88, BT82]. Integrating differential forms requires an orientation with which to turn the volume element into a measure. Hence, the Stokes theorem for differential forms is only applicable to orientable manifolds, and in particular we need N to be orientable. If so, then dF = F. (6) N
S
Remembering that F is closed, it follows that an orientable region N never carries F-charge, N orientable
=⇒
Φ(S) = 0.
(7)
The question that will occupy us in the rest of this paper is whether the F-charge can be non-zero if N is non-orientable, and what its values can be. The Stokes theorem for differential forms (6) is inapplicable when N is non-orientable, so no conclusions relevant to our question can be drawn from it. Nevertheless, we will find a way to make sense of (6) for N non-orientable by means of a cell decomposition, as will be explained below. This procedure will relate the values of Φ(S) to certain topological quantities intrinsic to N , independent of the choice of cell decomposition. 5. Two Notions of Boundary In point-set topology, an m-manifold with boundary is locally mapped to R+ × R m−1 , and the boundary is the submanifold that gets mapped into 0 × R m−1 . That conforms to the ordinary idea of boundary as a frontier where the manifold ends, and in fact we have been referring to this notion as “frontier” in previous paragraphs. From the point of view of homology, an m-manifold can be regarded as a sum of m-cells, each cell being homeomorphic to an m-ball. The m-cells can be visualised as m-dimensional polyhedra that fill out the entire manifold without overlapping. The boundary of an m-cell, defined as in point-set topology, i.e., as its frontier, is now regarded as the sum of its faces. These faces now are the (m − 1)-cells in our cell decomposition of the manifold, and so on. The collection of all k-cells is known as the k-skeleton of the manifold, where k runs from 0 to m. We will adopt the notation Vak for the elements of the k-skeleton, where a is an appropriate index that labels its elements. Because the cells Vak are homeomorphic to k-balls, the boundaries ∂ Vak are homeomorphic to (k − 1)-spheres.
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The precise way in which the boundaries of the (r + 1)-cells fit into the r -skeleton is r +1,r summarised in the incidence matrix Mab , defined by ∂ Var +1 =
r +1,r r Mab Vb .
(8)
b
The entries of these matrices can only be 0 or ±1. As a matter of terminology, we will r +1,r say that Var +1 incides upon Vbr whenever Mab = 0. The operation of taking the boundary of a cell is required to be nilpotent, that is, it must always be the case that ∂ 2 = 0. At the level of cells, this requirement follows from the earlier statement that ∂ Vam is homeomorphic to the sphere S m−1 , and hence boundaryless. In order to reflect this nilpotency, the incidence matrices are required to satisfy the relationship r +1,r r,r −1 Mab Mbc = 0. (9) b
In the definition of the m-dimensional cells Vam , there is implied a choice of orientation for each one of them. Recall that the m-cells are homeomorphic to m-balls, and for these the notion of orientation is straightforward. Then, just as for an m-ball, there are two choices possible for the orientation of each cell. Although there is no preferred way to choose all the orientations, some choices are better than others in a sense to be explained below. For the time being, we will assume that every m-cell has been given a definite orientation. This induces an orientation on its boundary in the usual way. Once this is done, the integration of differential forms is defined both in the interior of the cells and on their boundaries. Now, we identify the manifold with the sum of all the m-cells, N = a Vam . When we combine this identification between the manifold and its cell decomposition with the definition (8) of the incidence matrix, the boundary of N can be given as m,m−1 ∂N = ∂ Vam = Mab Vbm−1 . (10) a
a
b
Every (m − 1)-cell in the cell decomposition of a manifold is the face of one or two m-cells. In contrast, cells of dimension m − 2 and lower can be shared by any number of higher-dimensional cells, depending on the details of the cell decomposition. It follows m,m−1 can be of three types: that the colums of the incidence matrix Mab m−1 ⎛ Va, type II ⎞
m−1
V ⎛ a, type I ⎞
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0 . . . 0 1 0 . . . 0
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
or
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0 . . . 0 1 0 . . . 0 −1 0 . . . 0
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
m−1 ⎛ Va, type III ⎞
or
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0 . . . 0 1 0 . . . 0 1 0 . . . 0
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Torsion Cycles as Non-Local Magnetic Sources in Non-Orientable Spaces
721
up to overall signs that can be arranged to be as shown. The first type corresponds to cells Va,m−1 type I that lie entirely on the frontier of N . The second and third types correspond to m−1 cells Va,m−1 type II or Va, type III that are shared by two different m-dimensional cells. The difference between these two types becomes apparent when we use (10) to calculate ∂N . All the type II cells cancel out, and we are left with ∂N = Va,m−1 Va,m−1 (11) type I + 2 type III . a
a
This formula illustrates the difference between the notions of boundary used in homology theory and point-set topology. In our notation, the latter comprises all the type I cells, and nothing else; as a subspace of N , it coincides with the frontier S. The former includes also an (m − 1) cycle made of all the type III cells, multiplied by two. It will be explained below that the appearence of this cycle is due to the existence of a torsion subgroup in the homology group Hm−1 (N , S; Z) that is cyclic of order two whenever N is non-orientable. Lacking a standard terminology, we will use the name “free boundary” for the pointset boundary, and the name “torsion boundary” for the contribution of the torsion (m−1)cycles. The “homology boundary” is the sum of both. The symbol ∂N will be reserved for the homology boundary of N , and so we write ∂N = Va,m−1 Va,m−1 (12) type I + 2 type III = (∂N )free + (∂N )torsion . a
a
As a chain, the free boundary is the sum of all type I cells, and so it manifestly depends upon the choice of cell decomposition. As a subspace of N , however, it does not, because, as has already been pointed out, it coincides with the frontier S. In contrast, the torsion boundary does depend on that choice, even as a subspace of N . It will now be shown that, if N is orientable, then there always exists a cell decomposition for which there is no torsion boundary. 6. Orientability and Cell Decompositions When we calculate ∂N using an arbitrary cell decomposition, the result is generally of the form ∂N = (∂N )free + 2γ ,
(13)
where γ is made entirely of the type III cells of dimension m − 1. The torsion boundary is the term 2γ , and γ is the torsion cycle. Formula (13) shows that the double cycle 2γ is homologous to minus the free boundary of N . In the language of relative homology, γ is a relative torsion (m − 1)-cycle. It is known that the torsion subgroup of Hm−1 (N , S; Z) is zero if N is orientable (see e.g. [Mas91, Bre93]). In that case, γ itself must be a relative boundary, say γ = ∂α − β for some chains α ⊂ N and β ⊂ S. Then, ∂(N − 2α) = (∂N )free − 2β. This shows that our cell decomposition had the wrong signs in the regions α and β. Reversing those signs provides a corrected cell decomposition that is more natural in the sense that the ∂N calculated with its help does not introduce any spurious torsion
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M. Alvarez
B1 s1
C1
s2
C2
s2
γ
s1 B2
Fig. 3. A cell decomposition of the Möbius band
boundaries. These are the “better” choices of orientation we mentioned earlier in relation to the cell decomposition of N . The situation for non-orientable manifolds is different. No choice of cell decomposition can eliminate the torsion boundary, although it can change it by an absolute homology. We are interested in the case where the frontier of the non-orientable manifold is orientable (the (m − 1)-cycle S). Then we can begin by orientating the frontier, that is, the type I (m − 1)-cells, and then provide the adjoining m-cells with matching orientations. There is no natural prescription for orientating the rest of the cells. These ideas are best illustrated by means of an example. The Möbius band is a non-orientable 2-manifold with boundary. It can be obtained from the 2-dimensional projective plane by cutting out an open disk, which leaves behind a one-dimensional frontier. A cell decomposition of the Möbius band that only uses two 2-cells is shown in Fig. 6. An orientation was first chosen for the frontier, and then the cells C1 and C2 were orientated accordingly. In the notation shown in the figure, the boundaries of the two cells are ∂C1 = s1 + s2 + B1 + γ , ∂C2 = −s1 − s2 + B2 + γ .
(14)
This can be rewritten in terms of an incidence matrix as follows: ⎛
⎞ s1 ⎜s ⎟
1 1 1 0 1 ⎜ 2⎟ C1 B1 ⎟ . = ∂ −1 −1 0 1 1 ⎜ C2 ⎝B ⎠ 2 γ From the incidence matrix we see that B1 and B2 (columns 3 and 4) are type I cells, s1 and s2 (columns 1 and 2) are type II, and γ (column 5) is type III. Then, B1 and B2 form a cell decomposition of the frontier of the Möbius band and γ is a torsion cycle. The boundary of the Möbius band is given by ∂N = ∂(C1 + C2 ) = B1 + B2 + 2γ ,
(15)
which is of the general form (13), with B1 + B2 the free boundary S, and 2γ the torsion boundary.
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7. Stokes Theorem for Cell Decompositions We are now in a position to apply the Stokes theorem for differential forms (6) to a closed (m − 1)-form F that exists in a compact m-manifold N , not necessarily orientable, with non-empty frontier S. We start by introducing a cell decomposition of the manifold. Because each m-cell has been provided with an orientation, (6) can be applied to it, dF = F. (16) ∂ Vam
Vam
Recalling that F is closed, this shows that F = 0.
(17)
∂ Vam
Let us consider what will happen when the contribution of all the m-cells is added. Depending on whether the orientations of two adjoining m-cells agree or not, the (m −1)cells that separate them will be type II or III. The orientations induced on a type II cell by those of the two m-cells that coincide on it will be opposite. Therefore, F is integrated twice on every type II cell, but with opposite orientations. The two integrals will cancel out, and so we find that type II cells make no contribution to the sum. In contrast, the induced orientations will agree on all type III cells, so that the two fluxes will add up instead of cancelling out. Finally, the contribution of type I cells is straightforward: it is the flux of F on the free boundary of N , that is, on its frontier S. In fact, the way the fluxes of F combine when all the contributions from the cells are added reproduces the calculation of the homology boundary of N , as defined in (10), F= F. a
∂ Vam
∂N
When this is combined with (17), the result is what we regard as the Stokes theorem for closed differential forms in a format that is applicable whether N is orientable or not, F = 0. (18) ∂N
It has already been explained that, if N is orientable, then ∂N coincides with its frontier S, in which case (18) is a repetition of (7). On the other hand, if N is non-orientable, ∂N consists of two distinct parts, namely the free and torsion boundaries, as in (13), and then F = −2 F. (19) S
γ
The left-hand side of this is Φ(S), the F-charge contained inside S. The result (19) shows that the F-charge need not be zero, as we now discuss.
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M. Alvarez
S
P
F
Q
γ
Q
P
Fig. 4. A torsion cycle acting as a magnetic source
8. Homological Aspects of F-Charge The result (19) shows that, if N is non orientable, it may contain a non-zero F-charge, Φ(S) = −2
F.
(20)
γ
It is the presence of the torsion cycle γ that allows the flux of F through S to be nonzero. Because the torsion subgrop of Hm−1 (N , S; Z) is cyclic of order two, a γ will always exist and be unique up to homology. In fact, (20) depends only on the absolute homology class of γ ,
F=
dF =
F+ γ
γ +∂C
F. γ
C
Therefore it is the absolute homology class [γ ]abs , rather than any particular choice of γ , that acts as a magnetic source. In contrast with a magnetic current, which is localised at the world-line of a magnetic charge, [γ ]abs is not a localised object, and for that reason it is not meaningful to ask for a spacetime picture of how exactly the flux appears in N and then crosses the sphere S. A rough representation of the process can be given if a choice of γ is made; Fig. 4 illustrates this point for a Möbius strip. In that figure, the frontier of the Möbius strip is denoted S to match our notation for bounding spheres. Let us recall that γ was originally a representative of Hm−1 (N , S; Z). Let [γ ]rel be the relative homology class of γ in N . As explained, the F-charge (20) is invariant only under absolute homologies of γ . Therefore the absolute homology class [γ ]abs should be understood as the preimage of [γ ]rel under the map j∗ in the homology exact sequence of the pair (N , S), i∗
j∗
∂
i∗
· · · −→ Hm−1 (N ; Z) −→ Hm−1 (N , S; Z) −→ Hm−2 (S; Z) −→ · · · . For the exceptional case m = 2 we use reduced homology groups in dimension zero, and the sequence is still exact. Then the preimage of [γ ]rel in Hm−1 (N ; Z) always exists for m ≥ 2 because then Hm−2 (S; Z) (or its reduced version) is zero.
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9. Existence of Manifolds Supporting Non-Zero F-Charge The existence of a torsion (m − 1)-cycle γ is a necessary condition for the F-charge not to be identically zero, but it is not sufficient. We also need the space manifold K to support closed (m − 1)-forms (such as F) with non-zero periods over γ . Thus we need to analyse the cohomology of the class of non-orientable manifolds that we have been considering in this paper, and not just the homology. A quick way to find the cohomology groups of a manifold given its homology groups is to apply Poincaré duality [Mas]. Unfortunately, Poincaré duality for non-orientable manifolds requires the use of Z2 coefficients, in which case F cannot be a differential form, unless it is zero. For real (or integer) coefficients, there is no straightforward relationship between cohomology and homology of a non-orientable manifold. We now proceed to show that the non-orientable manifolds we have been considering do in fact possess the correct cohomology structure to support non-zero F-charges, at least when M is Rm minus a solid m-ball and S the boundary of that ball. Then M has the same de Rham cohomology as S, and in particular it admits a globally defined closed (m − 1)-form with non-zero periods on S. That (m − 1)-form is a candidate for the field strength F, bearing in mind that F should be globally defined in the entire space manifold K and not just in M. Then we must find out whether this candidate for F, which we shall call FS , extends to N , the non-orientable piece of space bounded by S. The following argument, based on the Mayer-Vietoris exact sequence of the triad (X ; N ; B ), shows that it does and hence FS extends to a globally defined (m − 1)-form F in K, as required. Recall that N is constructed by removing an open m-ball B from some compact nonorientable m-manifold X without frontier. Let B be a slightly larger open ball so that X = (interion N ) ∪ (interion B ). The Mayer-Vietoris sequence of (X ; N ; B ) contains the section · · · ← H m (X ) ← H m−1 (N ∩ B ) ← H m−1 (N ) ⊕ H m−1 (B ) ← H m−1 (X ) ← · · · (21) (see for example [BT, Mas]). The subspace N ∩ B is a small neighbourhood of the sphere S so that H m−1 (N ∩ B ) = H m−1 (S). Then FS will fit into (21) as a representative of H m−1 (S). Because FS is a differential form we need only consider the free parts of the cohomology groups, and in particular we can put H m (X ) = 0. Noting also that H m−1 (B ) = 0, (21) reduces to 0 ← H m−1 (S) ← H m−1 (N ) ← H m−1 (X ) ← · · · .
(22)
Therefore every differential form in H m−1 (S) lifts to a differential form in N , and in particular FS lifts to the form F that we identify as the field strength. In conclusion, the non-orientable manifolds discussed in this paper possess the correct homology and cohomology structure to support non-zero F-charges. Before discussing examples, we shall show that the (m − 1)st relative homology group of N is isomorphic to the absolute homology group of X of the same dimension. Then the problem of identifying the torsion cycle in N is reduced to finding the torsion cycle in the original boundaryless manifold X . The argument is based on the relative homology sequence of the triad (X , N , B ) [Mas91], which contains the section Hm (X , X ) −→ Hm−1 (N , S) −→ Hm−1 (X , B ) −→ Hm−1 (X , X ).
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It follows that Hm−1 (N , S) ≈ Hm−1 (X , B ) because the first and last homology groups shown are always zero. On the other hand the homology sequence of the pair (X , B ) contains the section i∗
j∗
∂
Hm−1 (B ) −→ Hm−1 (X ) −→ Hm−1 (X , B ) −→ Hm−2 (B ) in which the first and last homology groups are zero (taking reduced homology in dimension zero if m = 2). Then Hm−1 (X ) ≈ Hm−1 (X , B ). In combination with the previous result, this shows that Hm−1 (N , S) ≈ Hm−1 (X ), as required. The real projective plane RP2 and the Klein bottle K are the basic examples for m = 2. When a 2-ball is removed from RP2 the result is a Möbius strip, an example that has already been discussed. The first homology group of the Klein bottle with integer coefficients is Z ⊕ Z2 , and the cycle γ whose homology class acts as a non-local magnetic source after a ball is removed from K is the generator of the torsion part Z2 . This example is more generic than the Möbius band in that the homology group contains both free and torsion parts, but the discussion in entirely analogous. Easy examples with m = 3 are RP2 × S 1 and K× S 1 . The two-cycle whose homology class acts as a magnetic source is now γ × S 1 , in which γ is the one-cycle in RP2 or K S 2 , the non-trivial S 2 bundle over S 1 described above. A less obvious example is S 1 × [Ste51]. It is constructed by identifying the two spheres at the ends of S 2 × I after an S 2 ) = Z2 , pure torsion, generated orientation-reversing transformation. Now H2 (S 1 × by a two-cycle that can be identified with the fibre S 2 at any point in the base S 1 . As usual, the absolute homology class of that cycle is the source of magnetic flux after a three-ball is removed from the manifold. S 2 has been discussed previously in the literature [FM82] in relaThe manifold S 1 × tion to the idea that sourceless electric charges can be defined in a non-orientable manifold. The work of [FM82] is based on a geometric construction that requires the manifold to possess a rotational Killing pseudovector. An “electric field” is then defined in terms of covariant derivatives of the Killing vector and non-zero fluxes through a sphere are then obtained, in the absence of electric charges. Our construction is purely topological and therefore independent of such considerations. Note also that our identification of the non-zero charge as magnetic was fixed by the convention that the field strength (m − 1)form F was globally defined. The opposite convention, that the 2-form ∗F is globally defined, would have yielded non-zero electric fluxes by an analogous construction, but only in the particular case m = 3 where torsion cycles are 2-dimensional. Acknowledgements. I would like to thank Rafael Sorkin, Paul Martin, Anton Cox, Mark Hadley and Allen Hatcher for stimulating discussions on different aspects of this work, and especially Esperanza Gómez for her constant encouragement. This research was supported by PPARC through the Advanced Fellowship PPA/A/S/1999/00486.
References [AMR88] [AO06] [BG03]
Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Number 75 in Applied Mathematical Sciences. New York: Springer-Verlag, Second edition, 1988 Alvarez, M., Olive, D.I.: Charges and fluxes in maxwell theory on compact manifolds with boundary. Commun. Math. Phys. 267, 279–305 (2006) Bronnikov, K., Grinyok, S.: Charged wormholes with non-minimally coupled scalar fields: existence and stability. In: Festschrift in honor of Mario Novello’s 60th birthday. France: Frontier, 2003
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[Bre93] [BT82] [DH99] [FM82] [HV97] [Kis78] [Lub63] [Mas91] [MW57] [Sor77] [Ste51] [Whe62]
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Bredon, G.E.: Topology and Geometry. Number 139 in Graduate Texts in Mathematics. New York: Springer-Verlag, 1993 Bott, R., Tu, L.W.: Differential forms in algebraic topology. New York: Springer-Verlag, 1982 Diemer, T., Hadley, M.: Charge and the topology of spacetime. Class. Quant. Grav. 16(11), 3567–3577 (1999) Friedman, J.L., Mayer, S.: Vacuum handles carrying angular momentum; electrovac handles carrying net charge. J. Math. Phys. 23(1), 109–115 (1982) Hochberg, D., Visser, M.: Geometric structure of the generic static traversable wormhole throat. Phys. Rev. D56, 4745 (1997) Kiskis, J.: Disconnected gauge groups and the global violation of charge conservation. Phys. Rev. D17, 3196 (1978) Lubkin, E.: Geometric definition of gauge invariance. Ann. Phys. 23, 233–283 (1963) Massey, W.S.: A Basic Course in Algebraic Topology. Number 127 in Graduate Texts in Mathematics. New York: Springer-Verlag, 1991 Misner, C.W., Wheeler, J.A.: Classical physics as geometry: gravitation, electromagnetism, unquantized charge, and mass as properties of curved empty space. Ann. Phys. 2(6), 525–603 (1957) Sorkin, R.: On the relation between charge and topology. J. Phys. A 10, 717–725 (1977) Steenrod, N.: The Topology of Fibre Bundles. Princeton, NJ: Princeton University Press, 1951 Wheeler, J.A.: Geometrodynamics. New York: Academic Press, 1962
Communicated by G.W. Gibbons
Commun. Math. Phys. 277, 729–758 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0373-5
Communications in
Mathematical Physics
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules Felix Otto1 , Athanasios E. Tzavaras2,3 1 Institute of Applied Mathematics, University of Bonn, Bonn, Germany 2 Department of Mathematics, University of Maryland, College Park, USA.
E-mail: [email protected]
3 Institute for Applied and Computational Mathematics, FORTH, Crete, Greece.
E-mail: [email protected] Received: 13 January 2007 / Accepted: 18 May 2007 Published online: 22 November 2007 – © Springer-Verlag 2007
Abstract: We investigate the Doi model for suspensions of rod–like molecules in the dilute regime. For certain parameter values, the velocity gradient vs. stress relation defined by the stationary and homogeneous flow is not rank–one monotone. We then consider the evolution of possibly large perturbations of stationary flows. We prove that, even in the absence of a microscopic cut–off, discontinuities in the velocity gradient cannot occur in finite time. The proof relies on a novel type of estimate for the Smoluchowski equation. 1. Introduction 1.1. Summary. We consider the Doi model for suspensions of rod–like molecules in the dilute regime. This kinetic model couples a microscopic to a macroscopic equation. The macroscopic one is the Stokes equation for the fluid velocity, the microscopic equation is a Fokker–Planck (Smoluchowski) equation for the probability distribution of rod orientations in every point of physical space. Velocity gradients distort the isotropic equilibrium concentration; these deviations from isotropy in turn generate an additional macroscopic stress, which is elastic in nature and entropic in origin. The model is characterized by two non–dimensional parameters: The Deborah number which relates the externally imposed time scale to the intrinsic relaxation time, and a non-dimensional measure of concentration which quantifies the relative importance of elastic vs. viscous stress. For sufficiently large values of these parameters, the strain rate vs. stress relation defined by the stationary and homogeneous flow is not rank–one monotone. This non–monotonicity has been related to the occurrence of transition layers in velocity gradients. We consider the evolution of flows that are (possibly large) perturbations of stationary homogeneous flows. We prove that even in the absence of a microscopic cut–off these discontinuities in the velocity gradients cannot occur in finite time. This is a confirmation of Doi model. The proof relies on a novel type of estimate for the Smoluchowski equation.
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u
L
.
X
n
b Fig. 1. Rod–like molecule
1.2. The Doi model. As a first approximation, we think of the identical liquid crystal molecules as inflexible rods of a thickness b which is much smaller than their length L, as illustrated in Fig. 1. Let ν denote their constant number density. Following [5], we distinguish three regimes for the solution: • Dilute regime. The rods are well separated, as expressed by ν L −3 . • Concentrated regime. In this regime, the excluded volume effects reduce the entropy substantially. The theory by Onsager shows that this happens for ν b−1 L −2 . For a critical value of the dimensionless number ν b L 2 , this leads to the isotropic nematic–phase transition [3, Sect. 2.2], [5, Sect. 10.2]. • Semi–dilute regime. On one hand, there is the kinetic effect that rods hinder themselves in their rotational movement. On the other hand, there is not yet an entropic effect: L −3 ν b−1 L −2 . We will focus on the dilute regime. We are interested in creeping flows, where the inertia of solvent (and rods) can be neglected. Doi [4] introduced the model we will consider, see also [5, Chap. 8]. The system is described by a local probability distribution f (x, t, n) dn. It gives the time–dependent probability that a rod with center of mass at x has an axis n in the area element dn. The evolution of f is given by the Smoluchowski equation: ∂t f = − u · ∇x f − ∇n · (Pn ⊥ ∇x u n f ) + D x f + Dr n f.
(1) (2)
The two terms in (1) describe advection of the centers of mass by the velocity u respectively the rotation of the axes due to velocity gradients ∇x u. Here and in the sequel ∇n , ∇n · and n denote the gradient, divergence and Laplacian on S 2 . Finally, Pn ⊥ ∇x u n = ∇x u n − (n · ∇x u n) n denotes the projection of the vector ∇x u n on the tangent space in n. The two terms in (2) describe the Brownian effects: translational diffusion, respectively rotational diffusion. The Kirkwood theory [5, App. 8.1] derives asymptotic expres-
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
731
sions for the diffusivities D and Dr from a microscopic theory. They scale as D ∼
kB T ηs L
kB T ηs L 3
and Dr ∼
(3)
(up to a logarithmic correction in Lb ), where ηs denotes the viscosity of the solvent. The Kirkwood theory [5, Sect. 8.3, App. 8.1] predicts that the longitudinal and transversal translational diffusion differ by an O(1)–factor. This difference is neglected here and the longitudinal and transversal diffusivities are taken equal; it will then turn out that the effect of translational diffusion is negligible. In the semi–dilute regime, the rotational diffusion would be hindered by the neighboring rods. This effect can be modeled by a mean–field ansatz on the level of the one–point statistics f (t, x, n) dn; it leads to a substantially reduced diffusivity Dr (t, x, n). We also neglect this effect. Diffusion can be seen as a gradient flow of the entropy functional E[ f ] := ν k B T f ln f dn d x. (4) system S 2 In the concentrated regime, the excluded volume effect would become important. Within a mean–field ansatz, this can be done on the level of the one–point statistics f (t, x, n) dn; it leads to the Onsager Potential. As we focus on the dilute regime, we neglect this term. As can be seen from (1), a velocity gradient ∇x u distorts an isotropic distribution f which leads to an increase in entropy. Thermodynamic consistency [5, Sect. 8.6] requires that this is balanced by a stress tensor σ (t, x) given by σ (t, x) := ν k B T (3 n ⊗ n − id) f (t, x, n) dn. S2
Notice that E plays the role of a stored energy functional and σ that of an elastic stress. The presence of the rod–like molecules gives also rise to a viscous stress which modifies the solvent viscosity. In the dilute and semi-dilute regimes, this additional viscous stress can be neglected. Hence the averaged continuity and momentum equations are given by ∇x · u = 0 and ∇x · ηs (∇x + ∇xt ) u − p id + σ = 0. (5) Notice the coupling of the Smoluchowski equation (1) & (2) and the macroscopic equation (5) via the drift terms and the stress tensor σ . Together, they define an evolution for f. We want to mimic a simple flow situation. The Doi model admits a special class of solutions that correspond to stationary flows driven by an externally imposed velocity gradient ∇u ext , and we consider perturbations of such flows. For such flows there is a characteristic externally imposed time scale |∇x u1 ext | , and a macroscopic length scale L ext related to the size of the perturbation. This evolution is a gradient flow of the entropy (4) and this will play a role in the analysis. 1.3. Non–dimensionalization. The problem has three characteristic time scales: • The time scale related to rotational diffusion: • A visco–elastic time scale k BηTs ν . • An externally imposed time scale: |∇x u1 ext | .
1 Dr
.
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F. Otto, A. E. Tzavaras
We non–dimensionalize based on the visco–elastic time scale ηs t = tˆ. kB T ν length2 Since the translational diffusion has units of time , this gives rise to three length scales. In addition, there is the external length scale, which we use for non–dimensionalization: x = L ext x. ˆ This imposes the following non–dimensionalization of velocity, strain and stresses: kB T ν kB T ν u, ˆ ∇x u = ∇xˆ u, ˆ ηs ηs ˆ σ = k B T ν σˆ , p = k B T ν p. u = L ext
We are left with three non–dimensional parameters: (3) ηs ∼ (L 3 ν)−1 , kB T ν (3) ηs L −2 ∼ (L L 2ext ν)−1 , Dˆ = D k B T ν ext ηs ∇ ∇x u ext . x u ext = kB T ν Dˆ r = Dr
ˆr = Sometimes, it is more convenient to think in terms of the Deborah number ∇ x u ext / D ∇x u ext /Dr , which relates the externally imposed time scale to the rotational relaxation time. We collect the nondimensionalized equations (dropping the hats): ∂t f + ∇x f · u + ∇n · (Pn⊥ ∇x u n f ) − Dr n f − Dx f = (3 n ⊗ n − id) f dn = S2 ∇x · (∇x u + ∇xt u) − p id + σ = ∇x · u =
0,
(6)
σ,
(7)
0, 0.
(8) (9)
These form a system consisting of the transport equation (6) coupled with the Stokes system (8)–(9). The coupling is effected via (7) that determines the viscoelastic stresses as moments of the probability distribution f . The function f (t, x, n) is a probability density on S 2 , f ≥ 0, f (t, x, n) dn = 1 . (10) S2
This requirement is consistent with the evolution (6). Our system is supplemented with initial conditions f (0, x, n) = f 0 (x, n)
(11)
and one checks that property (10) is propagated from the initial data to solutions of (6).
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
733
The model (6)–(9) admits a special class of stationary steady states: Let ∇x u ext be a given traceless tensor, tr∇x u ext = 0, then u ext (x) = (∇x u ext ) x
(12)
gives rise to an incompressible vector field. Define f eq (n) to be the unique solution of the stationary Fokker-Planck equation (13) ∇n · Pn ⊥ ∇x u ext n f − Dr ∇n f = 0 satisfying f eq (n) ≥ 0 and S 2 f eq (n)dn = 1. Notice that ( f eq (n), u ext (x)) is a stationary, steady solution of (6)–(9) associated to a constant pressure and with σeq = (3n ⊗ n − id) f eq dn . S2
This class plays an important role in our analysis. It will be used as a building block for constructing non-monotone spatially varying steady states, and we will study the evolution of (large) perturbations of ( f eq (n), u ext (x)). 1.4. Non-monotonicity of steady states. Let ( f eq (n), u ext (x)) be as defined in (12) & (13) and σeq be the associated moment in (7). By varying parametrically the imposed velocity gradient κ = ∇x u ext we define a mapping End(R3 ) κ → σκ ∈ Sym(R3 ),
(14)
taking strain-rates to elastic stresses and defined by (7). A necessary condition for structural stability of the homogeneous flow κ x is that the mapping from deformation–rates to total stresses End(R3 ) κ → (κ + κ t ) + σκ ∈ Sym(R3 )
(15)
be rank–one monotone at the κ under consideration. After appropriate rescaling with Dr , (14) is universal (see Definition 1). We will argue in Sect. 3.2 that it fails to be monotone along the shear direction (Lemmas 4 and 5). This implies that (15) fails to be monotone along the shear direction for sufficiently small Dr . One effect of this non–monotonicity in the shear direction is that there exist spatially discontinuous solutions ( f (x, n), u(x)) for vanishing translational diffusivity (D = 0), i. e. solutions of ∇x · ( f u) + ∇n · (Pn⊥ ∇x u n f ) − Dr n f (3 n ⊗ n − id) f dn S2 ∇x · (∇x u + ∇xt u) − p id + σ ∇x · u
= 0,
(16)
= σ,
(17)
= 0, = 0.
(18) (19)
More precisely, we will show: Theorem 1. There exist Dr > 0 such that (16)–(19) admits a solution (in the distributional sense) with discontinuous ∇x u.
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F. Otto, A. E. Tzavaras
This failure of ellipticity for (16)–(19) has been frequently seen as a deficiency of the Doi model. On the contrary, other schools have advocated the failure of ellipticity of steady states as the cause of the onset of instabilities in viscoelastic flows, and in particular as an explanation of the phenomenon of spurt. Spurt refers to a sudden increase of the volumetric flow rate at a critical stress which has been observed experimentally, see for instance [18]. Spurt has been connected in [11] to the non–monotonicity of the map (15), which allows for jumps in the steady strain rate when the driving pressure gradient exceeds a critical value. Such jumps can account for the sudden increase in the flow rate observed in experiments. This explanation of spurt motivated analytical results regarding existence of discontinuous steady states and their stability properties, accomplished for macroscopic models in the absence of translational diffusion D and for a 1–d geometry. The macroscopic model (Oldroyd B or Johnson–Segalman) can be seen as an exact closure of a kinetic model with Hookean springs instead of rigid rods. It has been shown that discontinuities in the strain rate ∇x u form in infinite time, see [13,12].
1.5. Continuity of velocity gradients. The main goal of this paper is to investigate on which time scale these near–discontinuities occur. We want to study a forced problem, and to this end we consider a solution that is a perturbation of the stationary steady state ( f eq (n), u ext (x)) in (12)–(13). Our analysis is valid even for large perturbations and we find that the time scale can be bounded by below independently of the translational diffusion D, just in terms of the non–dimensional parameter Dr , and ∇x u ext , see Theorem 2. We now outline our strategy. Qualitatively speaking, we want to control the modulus of continuity of ∇x u. By Sobolev’s embedding, this is a consequence of control of |∇x2 u| p d x R3
for some fixed 3 < p < ∞. In view of (8)&(9) and standard L p –regularity theory for the Stokes system, this is a consequence of |∇x σ | p d x, control of R3
see Lemma 2. In view of (7), which yields (3 n ⊗ n − id) ∂xi f dn, ∂xi σ =
(20)
S2
this requires control of ∇x f . This control has to be the L p –norm with respect to x but can be a weak norm with respect to n, for instance an H −1 (S 2 )–norm. Recall that the H −1 (S 2 )–norm of ∇x f can be defined as
∇x f H −1 (S 2 ) :=
1/2 S2
|∇n φ|2 dn
,
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
735
where the potential φ = (φ1 , φ2 , φ3 ) is the solution of the Poisson problem for the Laplace operator ∂xi f − n φi = 0. For reasons intrinsic to the Fokker–Planck–like equation (16) (see the proof of Proposition 1), our choice is an f –dependent version of an H −1 (S 2 )–norm of ∇x f : 1/2 2 |∇n φ| f dn ,
∇x f H −1 (S 2 ) := f
S2
where the potential φ = (φ1 , φ2 , φ3 ) is the solution of the following elliptic problem on S2: ∂xi f − ∇n · ( f ∇n φi ) = 0.
(21)
(Note that, for any tensor g, |g|2 denotes the sum of the squares of the entries with respect to orthonormal bases.) This norm comes from a natural Riemannian structure of the space of probability densities f which was introduced in [14], see also [15, Sect. 3]. Accordingly, we define the quantities w(t, x) = |∇n φ|2 f dn (22) S2
and
W (t) :=
R3
w p/2 d x =
R3
p/2 S2
|∇n φ|2 f dn
dx ,
and seek to establish control of W (t). A second ingredient is an identity for the relative entropy density f ln f dn, e(t, x) := f eq S2 see Lemma 1 and [10], that yields differential control for the relative entropy f ln f dnd x . E(t) := e(t, x) d x = f eq R3 R3 S 2
(23)
(24)
(25)
We prove: Theorem 2. Let ( f, u, p) be a solution of (6)–(9) (with D = 0 allowed), let ( f eq , u ext ) be as in (12)–(13) and assume that the data f 0 satisfy f0 ln f 0 dn d x < +∞ . E(0) = f eq R3 S 2 There exists a constant C only depending on p ∈ (3, ∞) and a constant K only depending on |∇x u ext |/Dr such that dE ≤ K E, dt 1 dW ≤ −Dr W + C (1 + |∇x u ext | + ln E(t) + ln W ) W. p dt
(26) (27)
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F. Otto, A. E. Tzavaras
Remark 1. Integrating the differential inequalities we obtain the bounds E(t) ≤ E(0)e K t , W (t) ≤ exp (ln W (0) + K ) e K t . Moreover, the following estimate: |∇x (u − u ext )|2 d x ≤ R3
≤C
R3
R3
(28) (29)
|σ − σeq |2 d x
2
S2
| f − f eq |dn
d x < E(0) e K t
(30)
is derived in Proposition 3 as a byproduct of the proof. Let us comment only on the most pertinent mathematical literature: In [8], a purely macroscopic viscoelastic model is considered (Oldroyd–B). It can be interpreted as an exact closure of a kinetic model for Hookean springs instead of rigid rods. The existence of weak solutions is established by “propagation of compactness”. This approach can be extended to our kinetic model [9]. Theorem 2 might be seen as a quantification of the more qualitative approach in [8]. In [6], a kinetic model for nonlinear springs is investigated (FENE). Among other things, sufficient conditions on the asymptotic stability of the homogeneous flow ∇x u ≡ ∇x u ext are given. A more careful analysis, to appear in [10], reveals that
|∇x u ext | dE |∇x u ext | 2 −1 − C Dr exp −C ≤ C E. dt Dr Dr Hence also Theorem 2, in this extended version, yields a stability result in the regime of sufficiently small concentration Dr 1 and sufficiently small Deborah number |∇x u ext | Dr (provided the initial perturbation W (0) is sufficiently small). Finally, we refer to [2] for a recent global existence result, which is also valid in the concentrated regime, for flows that are asymptotically at rest at infinity driven by a body force. For comparison purposes, the present flow lies in the dilute regime but approaches any constant gradient flow at infinity. 2. Proof of Theorem 2 Theorem 2 is based on the following ingredients: an identity for the transport of the relative entropy density e(t, x) defined in (24), a transport inequality for the norm w(t, x) defined in (22), an L ∞ estimation for the Stokes system, and the derivation of differential inequalities for the quantities E(t) and W (t) in (25) and (23) respectively. 2.1. A relative entropy identity. Let (u ext , f eq (n)) be a stationary steady state as in (12)– (13) and let ( f, u, p) be a solution of (6)–(9) which approaches at infinity (u ext , f eq (n)). The relative entropy density f e(t, x) := ln f dn, (31) f eq S2 serves as a measure of the distance between f eq and f and satisfies the following identity.
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
737
Lemma 1. Let ( f, u, p) satisfy (6)–(9), then (∂t + u · ∇x − Dx ) e f f 2 | f dn + Dr |2 f dn |∇x ln |∇n ln +D f eq f eq S2 S2 = ∇x (u − u ext ) : (σ − σeq ) − ∇n ln f eq ⊗ n ( f − f eq ) dn . S2
Proof of Lemma 1. Using the property that f eq is independent of x and t, we derive from the Smoluchowski equation (6) the formula f 2 + D f ∇x ln (∂t + u · ∇x − Dx ) f ln f eq f eq f −∇n · (Pn ⊥ ∇x u n f ) + Dr n f = 1 + ln f eq f ∇n · Pn ⊥ ∇x u n f − Dr ∇n f , = − 1 + ln f eq
f
which after an integration over S 2 gives by integration by parts 2 ∇x ln f f dn (∂t + u · ∇x − Dx ) e + D f eq S2 f = ∇n ln · Pn ⊥ (∇x u − ∇x u ext ) n f dn 2 f eq S f ∇n ln · Pn ⊥ ∇x u ext n f − Dr ∇n f dn f eq S2 =: J1 + J2 .
(32)
We first treat J2 , by a classical formula for drift–diffusion equations. We write Pn ⊥ ∇x u ext n f − Dr ∇n f f f = −Dr f eq ∇n Pn ⊥ ∇x u ext n f eq − Dr ∇n f eq + f eq f eq f f = −Dr f ∇n ln Pn ⊥ ∇x u ext n f eq − Dr ∇n f eq , + f eq f eq so that
· Pn ⊥ ∇x u ext n f − Dr ∇n f dn f |2 f dn = −Dr |∇n ln f eq S2 f f · Pn ⊥ ∇x u ext n f eq − Dr ∇n f eq dn. + ∇n ln f eq S 2 f eq
J2 =
S2
∇n ln
f
f eq
(33)
738
F. Otto, A. E. Tzavaras
The last term in (33) vanishes by definition (13) of f eq : f f · Pn ⊥ ∇x u ext n f eq − Dr ∇n f eq dn ∇n ln f eq S 2 f eq f · Pn ⊥ ∇x u ext n f eq − Dr ∇n f eq dn = ∇n f eq S2 f = − ∇n · Pn ⊥ ∇x u ext n f eq − Dr ∇n f eq dn = 0. S 2 f eq Hence we have
J2 = −Dr
f |∇n ln |2 f dn f eq S2
(34)
We now turn to J1 : f J1 = ∇n ln · ∇x (u − u ext ) n f dn f eq S2 f = ∇x (u − u ext ) : ∇n ln ⊗ n f dn 2 f eq S = ∇x (u − u ext ) : ∇n f ⊗ n dn − ∇n f eq ⊗ n dn 2 S2 S (∇n ln f eq ⊗ n) ( f − f eq ) dn . −
(35)
S2
According to formula (95) in Appendix II and the definition (7) of σ we have ∇n f ⊗ n dn − ∇n f eq ⊗ n dn S2 S2 = (3 n ⊗ n − id) f dn − (3 n ⊗ n − id) f eq dn S2
S2
= σ − σeq . Hence J1 can be rewritten as
J1 = ∇x (u − u ext ) : (σ − σeq ) − (∇n ln f eq ⊗ n)( f − f eq ) dn .
(36)
S2
Lemma 1 follows from a combining of (32) with (34) and (36).
−1 2.2. Transport inequality for the H −1 f -norm. We introduce the H f -norm as defined by (22) and (21) and proceed to derive a differential inequality for w.
Proposition 1. For any solution of (6) we have the partial differential inequality: ∂t ( 21 w) + ∇x ( 21 w) · u − D x ( 21 w) ≤ −Dr w + |∇x u + ∇xt u| w + |∇x2 u| w 1/2 .
(37)
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
739
Proof of Proposition 1. We start by differentiating the defining Eq. (21) with respect to t: ∂xi ∂t f − ∇n · ( f ∇n ∂t φi ) − ∇n · (∂t f ∇n φi ) = 0. Thus we have d 1 |∇n φi |2 f dn dt S 2 2
∂t f 21 |∇n φi |2 + f ∇n φi · ∇n ∂t φi dn = 2 S
∂t f 21 |∇n φi |2 − ∇n · ( f ∇n ∂t φi ) φi dn = S2
(38) ∂t f 21 |∇n φi |2 + ∇n · (∂t f ∇n φi ) φi − ∂xi ∂t f φi dn = 2 S
−∂t f 21 |∇n φi |2 − ∂xi ∂t f φi dn. =
(38)
(39)
S2
The contributions of the terms ∂t f and ∂xi ∂t f in (39) are calculated by invoking (6). We start with the contribution of the rotational diffusion term. It is given by
−n f 21 |∇n φi |2 − ∂xi n f φi dn S2
−n f 21 |∇n φi |2 − ∂xi f n φi dn = S2
(21) −n f 21 |∇n φi |2 − ∇n · ( f ∇n φi ) n φi dn = 2 S
−n ( 21 |∇n φi |2 ) + ∇n φi · ∇n n φi f dn. = S2
We now appeal to Bochner’s formula n ( 21 |∇n φi |2 ) = ∇n φi · ∇n n φi + tr(Hessn φi Hesstn φi ) + ∇n φi · Ric ∇n φi , where Hessn φi denotes the Hessian (a covariant notion) and Ric the Ricci curvature tensor. We refer for instance to [16, Prop. 3.3, p. 175]. On S 2 , Ric is just the metric tensor. Hence we obtain −n ( 21 |∇n φi |2 ) + ∇n φi · ∇n n φi ≤ −|∇n φi |2 , and the contribution of rotational diffusion is
−n f 21 |∇n φi |2 − n ∂xi f φi dn ≤ − i
S2
S2
|∇n φ|2 f dn.
We now treat the term coming from the translational diffusion. In view of (39), it is given by
−x f 21 |∇n φi |2 − ∂xi x f φi dn S2
(21) −x f 21 |∇n φi |2 − x ∇n · ( f ∇n φi ) φi dn = 2 S
= −x f 21 |∇n φi |2 + x ( f ∇n φi ) · ∇n φi dn. S2
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F. Otto, A. E. Tzavaras
The identities −x f 21 |∇n φi |2 + x ( f ∇n φi ) · ∇n φi = x f 21 |∇n φi |2 + 2 ∂x j f ∂x j ∇n φi · ∇n φi + f x (∇n φi ) · ∇n φi j
= x f +
2 1 2 |∇n φi |
j
= x = x
+2
∂x j f ∂x j 21 |∇n φi |2
j
f ∂x j ∂x j (∇n φi ) · ∇n φi − f ∂x j ∇n φi · ∂x j ∇n φi
f 21 |∇n φi |2 − f ∂x j ∇n φi · ∂x j ∇n φi
f
2 1 2 |∇n φi |
j 2 − f |∇x,n φi |2
show that the contribution is given by x
S2
2 1 2 |∇n φ|
f dn −
S2
2 |∇x,n φ|2 f dn.
For the inequality (37), we drop the non positive second term. We now treat the contribution from the advection term in x. It splits into two parts
∇x f · u 21 |∇n φi |2 + ∂xi (∇x f · u) φi dn S2
= ∇x f · u 21 |∇n φi |2 + (∇x ∂xi f · u + ∇x f · ∂xi u) φi dn S2
∂x j f 21 |∇n φi |2 + ∂x j ∂xi f φi dn u j = S2
j
+
j
∂xi u j
S2
∂x j f φi dn .
For the first term we observe
∂x j f 21 |∇n φi |2 + ∂x j ∂xi f φi dn S2
(21) ∂x j f 21 |∇n φi |2 + ∂x j ∇n · ( f ∇n φi ) φi dn = 2 S
∂x j f 21 |∇n φi |2 − ∂x j ( f ∇n φi ) · ∇n φi dn = S2 = − ∂x j ( f 21 |∇n φi |2 ) dn S2 2 1 = −∂x j |∇ φ | f dn . n i 2 S2
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
For the second term we notice (21) ∂xi u j ∂x j f φi dn = ∂xi u j S2
j
j
= −
S2
741
∇n · ( f ∇n φ j ) φi dn
∂xi u j
j
S2
∇n φ j · ∇n φi f dn.
Hence the contribution from advection in x is
∇x f · u 21 |∇n φi |2 + ∂xi (∇x f · u) φi dn i
S2
= −
i, j
∂x j
− 21
i, j
≤ −∇x
S2
= −∇x
S2
S2
S2
2 1 2 |∇n φi |
2 1 2 |∇n φ|
f dn
uj −
i, j
f dn · u
S2
∇n φ j · ∇n φi f dn ∂xi u j
∇n φ j · ∇n φi f dn (∂xi u j + ∂x j u i ) 2 1 2 |∇n φ|
f dn · u +
1 2
S2
|∇n φ|2 f dn |∇x u + ∇xt u|.
We finally come to the contribution from the drift term in n. We introduce the notation b = Pn ⊥ ∇x u n for the drift term:
∇n · (b f ) 21 |∇φi |2 + ∂xi ∇n · (b f ) φi dn S2
∇n · (b f ) 21 |∇φi |2 + ∇n · (b ∂xi f + ∂xi b f ) φi dn = S2 (21) ∇n · (b f ) 21 |∇φi |2 + ∇n · b ∇n · ( f ∇n φi ) + ∂xi b f φi dn = 2 S
−b · ∇n ( 21 |∇n φi |2 ) + ∇n (b · ∇n φi ) · ∇n φi − ∂xi b · ∇n φi f dn. (40) = S2
We now use the formula −b · ∇n ( 21 |∇n φi |2 ) + ∇n φi · ∇n (b · ∇n φi ) = −∇n φi · Hessn φi b + (b · Hessn φi ∇n φi + ∇n φi · Dn b ∇n φi ) = ∇n φi · Dn b ∇n φi , where Dn b denotes the covariant derivative of b on S 2 . Since b = Pn ⊥ ∇x u n = ∇x u n − (n · ∇x u n) n , we obtain for the component-wise derivative in a tangential direction τ ∈ n ⊥ , ∇n b τ = ∇x u τ − (τ · ∇x u n) n − (n · ∇x u τ ) n − (n · ∇x u n) τ,
(41)
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F. Otto, A. E. Tzavaras
and thus for the covariant derivative Dn b τ = Pn ⊥ ∇n b τ =
Pn ⊥ ∇x u − (n · ∇x u n) id τ.
(42)
Furthermore we have ∂xi b = Pn ⊥ ∂xi ∇x u n.
(43)
Inserting (43), (42) into (41) and (40), and since ∇n φi is on the tangent space of the sphere, we obtain
∇n · (b f ) 21 |∇φi |2 + ∂xi ∇n · (b f ) φi dn S2 = ∇n φi · (∇x u − (n · ∇x u n) id) ∇n φi f dn S2 ∇n φi · ∂xi ∇x u n f dn. − S2
Thus the contribution from the drift term in n is
∇n · (b f ) 21 |∇φi |2 + ∂xi ∇n · (b f ) φi dn S2
i
=
1 2
i
−
S2
S2
∇n φi · (∇x u + ∇xt u) − (n · (∇x u + ∇xt u) n) id ∇n φi f dn
∇n φi · ∂xi ∇x u n f dn
≤
1 2
|∇n φ| f dn |∇x u 2
S2
+ ∇xt u|
1/2 |∇n φ| f dn 2
+ S2
|∇x2 u|.
2.3. Bounds on the Stokes system. Consider the Stokes system (8)&(9), ∇x · (∇x u + ∇xt u) − p id + σ = 0, ∇x · u = 0,
(44) (45)
in R3 . We need a standard and a not so standard regularity result. Lemma 2. There exists a constant C depending only on p ∈ (1, ∞) with 2 p |∇x u| d x ≤ C |∇x σ | p d x. R3
R3
Proposition 2. There exists a constant C only depending on p ∈ (3, ∞) such that ⎡ ⎛ 3 1 ⎞⎤ 2 d x 3 (1− p ) p d x 1/ p |σ | |∇ σ | 3 x R3 ⎠ ⎦ sup |σ |. sup |∇x u| ≤ C ⎣1 + ln ⎝1 + R 2 (1− 3 )+1 x x 3 p supx |σ | Lemma 2 follows from standard regularity theory for the Stokes system; see [17, ChI, Prop 2.2]. Results of the type of Proposition 2 go back to Weigant & Kazhikov [19], see also [7, App. F]. We present a proof in Appendix I which is not based on a fundamental solution but on a dyadic decomposition in Fourier space.
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
743
2.4. Bound of the relative entropy. Given a stationary steady state (u ext , f eq (n)) and a solution ( f, u, p) of (6)–(9) which is a (possibly large) perturbation of (u ext , f eq (n)), the relative entropy is defined by f E(t) := ln f dnd x . (46) f eq R3 S 2 Proposition 3. Let (u ext , f eq ) be as in (12)–(13) and ( f, u, p) be a solution of (6)–(9) with data satisfying f0 ln f 0 dn d x < +∞ . E0 = (47) f eq R3 S 2 There exists a constant K = K (|∇x u ext |/Dr ) such that for t ∈ (0, ∞), dE ≤ K E(t) dt and
(48)
|∇x (u − u ext )| d x ≤ 2
R3
R3
|σ − σeq |2 d x ≤ C E 0 e K t ,
(49)
where C denotes a universal constant. A far more detailed estimation along the lines of Proposition 3 is derived in [10] and is used to study the stability of equilibria. Proof. The functions u − u ext and σ − σeq satisfy x (u − u ext ) − ∇x p + ∇x · (σ − σeq ) = 0, ∇x · (u − u ext ) = 0 . Therefore, we multiply by (u − u ext ) and integrate by parts to obtain − ∇x (u − u ext ) : (σ − σeq ) d x = |∇x (u − u ext )|2 d x.
(50)
Combine next Lemma 1 with (25), (9) and (50) to obtain d e dx + |∇x (u − u ext )|2 d x dt R3 R3 = − ∇x (u − u ext ) : (∇n ln f eq ⊗ n)( f − f eq ) dn d x R3 S2 |∇x (u − u ext )| | f − f eq | dn d x, ≤ K1
(51)
R3
R3
R3
S2
where K 1 = supn |∇n ln f eq ⊗ n| is a constant that depends only on the quotient |∇x u ext |/Dr . Next, we use the Kullback–Csiszar inequality, i. e. 2 f ln f dn (52) | f − f eq | dn ≤ 8 f eq S2 S2
744
F. Otto, A. E. Tzavaras
together with Young’s inequality to obtain dE ≤ K E, dt and thus
E(t) =
R3
e(x, t) d x ≤ E 0 e K t .
Observe next that (7) and (52) imply |σ − σeq |2 d x =
2 (3n ⊗ n − id)( f − f eq )dn d x R3 S 2 2 ≤C | f − f eq | dn d x 3 2 R S ≤C e(t, x) d x ,
R3
R3
hence, (49) follows from (50) and (48).
2.5. Derivation of the differential inequality. Let w be defined in (22)–(21) and W be as in (23). We derive a differential inequality for W . Proposition 4. Let W be defined as in (23). There exists a constant C only depending on p ∈ (3, ∞) such that |∇x2 u| p d x ≤ C W, (53) R3
sup |∇x u − ∇x u ext | ≤ C (1 + ln E(t) + ln W ) ,
(54)
x
and W satisfies the differential inequality: 1 dW ≤ −Dr W + C 1 + sup |∇x u ext | + ln E(t) + ln W W . p dt x
(55)
Proof. We evoke Proposition 1. With (37) as a starting point, we obtain for p ≥ 2 the differential inequality ∂t (w p/2 ) + ∇x · (uw p/2 ) − D x (w p/2 ) ≤ − p Dr w p/2 + p|∇x u + ∇xt u| w p/2 + p|∇x2 u| w
p−1 2
.
(56)
Let us address the three terms on the right side of (56). The first term gives rise to w p/2 d x = −Dr W. −Dr R3
For the second term we notice w p/2 |∇x u + ∇xt u| d x ≤ sup |∇x u + ∇xt u| W. R3
R3
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
745
Finally, the last term is estimated with help of Hölder’s inequality 1/ p p−1 w 2 |∇x2 u| d x ≤ |∇x2 u| p d x W 1−1/ p . R3
R3
Combining these together gives 1/ p 1 dW t 2 p |∇x u| d x W 1−1/ p . ≤ −Dr + sup |∇x u + ∇x u| W + p dt x R3 We next observe that there exists a universal constant C such that p/2 p 2 |∇x σ | d x ≤ C |∇n φ| f dn d x = C W. R3
R3
(57)
S2
Indeed, the starting point for deriving (57) is (20), written component-wise: ∂xi σkl = (3 n k nl − δkl ) ∂xi f dn. S2
According to definition (21), we obtain ∇n (n k nl ) · ∇n φi f dn, ∂xi σkl = −3 S2
and thus
|∂xi σkl |2 ≤ 9 sup |∇n (n k nl )|2 n
S2
|∇n φi |2 f dn.
It remains to sum over all i, k, l, raise to power p/2 and integrate over R3 . This establishes (57). Estimate (53) now follows from Lemma 2 and, in turn, provides the differential inequality 1 dW ≤ −Dr W + 2 sup |∇x u| W + C W . p dt x Next, we observe that σ is uniformly bounded: 1 2 2 |n ⊗ n − id| f dn ≤ 6 f dn = 6. |σ | ≤ 9 3 S2 S2
(58)
(59)
We also recall that (u − u ext , σ − σeq ) satisfies the Stokes system (44) &(45), and evoke Proposition 2. This implies p 2 sup |∇x u − ∇x u ext | ≤ C 1 + ln |∇x σ | d x + ln |σ − σeq | d x x R3 R3 (57),(49) ≤ C 1 + ln W + ln e(t, x) d x ≤
R3
C (1 + ln W + ln E(t)) ,
which with (58) gives (55) and completes the proof.
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F. Otto, A. E. Tzavaras
3. General Properties of the Doi Model We list here certain properties of the Doi model: the invariance under rotations of Eqs. (6)–(9), and the non-monotonicity of steady states for steady shear flows (16)– (19).
3.1. Invariances of the Doi model. We consider the model (6)–(9) and will show that the system is invariant under rotations. Proposition 5. Let ( f, u, p), with f = f (t, x, n), u = u(t, x) and p = p(t, x), satisfy (6)–(9). Then ( fˆ, u, ˆ p), ˆ defined by fˆ(t, x, n) = f (t, Qx, Qn) u(t, ˆ x) = Q t u(t, Qx), p(t, ˆ x) = p(t, Qx)
Q ∈ O(3),
(60)
satisfies (6)–(9). Moreover, σˆ (t, x) = Q t σ (t, Qx)Q. The proof is based on invariance properties of the transport equation ∂t f + ∇x f · u + ∇n · (Pn⊥ ∇x u n f ) − Dr n f − Dx f = 0
(61)
in conjunction with well known invariances of the Stokes system. We use the notation f u,∇x u = f u(t,x),∇x u(t,x) (t, x, n) for the solution of (61) generated by the fields u = u(t, x) and ∇x u = ∇x u(t.x). Lemma 3. Let f u,∇x u satisfy the transport equation (61). Then, f˜u,Q t ∇x u Q (t, x, n) := f u,∇x u (t, x, Qn) ,
f¯R t u(Rx),(∇x u)(Rx) (t, x, n) := f u,∇x u (t, Rx, n) ,
Q ∈ O(3),
(62)
R ∈ O(3),
(63)
fˆR t u(Rx),Q t ∇x u(Rx)Q (t, x, n) := f u,∇x u (t, Rx, Qn) , Q, R ∈ O(3),
(64)
satisfy transport equations (61) with velocity and velocity-gradient fields as stated in (62), (63), (64). . Proof of Lemma 3. This is a symmetry consideration. Let f = f u,∇x u satisfy (61) with fields u and ∇x u, Q ∈ O(3), and define f˜(n) := f (Q n). We then have ∇ f˜ = Q t ∇ f (Qn) and ∇n f˜(n) = Pn ⊥ ∇ f˜ = ∇ f˜ − (n · ∇ f˜)n = Q t (∇ f (Qn) − (Qn · ∇ f (Qn)) Qn) = Q t ∇n f (Qn)
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
747
and Pn ⊥ (Q t ∇x u Q) n = (Q t ∇x u Q) n − n · (Q t ∇x u Q) n n = Q t (∇x u (Q n) − (Q n) · ∇x u (Q n) (Q n)) = Q t Pn ⊥ ∇x u n (Q n), so that
−∇n f˜ + Pn ⊥ (Q t ∇x u Q) n f˜ (n) = Q t −∇n f + Pn ⊥ ∇x u n f (Q n). We infer
−n f˜ + ∇n · Pn ⊥ (Q t ∇x u Q) n f˜ (n)
= ∇n · −∇n f˜ + Pn ⊥ (Q t ∇x u Q) n f˜ (n) = ∇n · −∇n f + Pn ⊥ ∇x u n f (Q n) = −n f + ∇n · Pn ⊥ ∇x u n f (Q n), and thus
∂t f˜ + u · ∇x f˜ + ∇n · Pn ⊥ (Q t ∇x u Q) n f˜ − Dr n f˜ − Dx f˜ = ∂t f + u · ∇x f + ∇n · Pn ⊥ ∇x u n f − Dr n f − Dx f (t, x, Qn) = 0. Next, again with f = f u,∇x u and R ∈ O(3), set f¯ = f (Rx). Observe that ∇x f¯ = R t (∇x f )(Rx) and that x f¯ = (x f )(Rx). We infer ∂t f¯ + R t u(Rx) · ∇x f¯ + ∇n · Pn ⊥ (∇x u)(Rx) n f¯ − Dr n f¯ − Dx f¯ = ∂t f + u · ∇x f + ∇n · Pn ⊥ ∇x u n f − Dr n f − Dx f (t, Rx, n) = 0. The last statement follows by combining the first two. Proof of Proposition 5. Consider the function ( fˆ, u, ˆ p) ˆ defined by (60), and let Q ∈ O(3). We then have uˆ = Q t u(t, Qx), ∇x uˆ = Q t (∇x u)(t, Qx) Q , and, according to Lemma 3,
∂t fˆ + uˆ · ∇x fˆ + ∇n · Pn ⊥ ∇x uˆ n fˆ − Dr n fˆ − Dx fˆ = ∂t f + u · ∇x f + ∇n · Pn ⊥ ∇x u n f − Dr n f − Dx f (t, Qx, Qn) =0 .
748
F. Otto, A. E. Tzavaras
The transformation of viscoelastic stresses can be seen from (7). We have (3 n ⊗ n − id) fˆ dn σˆ (t, x) = 2 S (3 n ⊗ n − id) f (t, Qx, Qn) dn = 2 S (3 Q t n ⊗ Q t n − id) f (t, Qx, n) dn = S2 t
= Q σ (t, Qx)Q . Moreover, ∇x · uˆ = (∇x · u)(t, Qx) = 0 and
ˆ − pˆ id + σˆ ∇x · (∇x uˆ + ∇xt u) = Q t ∇x · (∇x u + ∇xt u) − p id + σ (t, Qx) = 0,
that is ( fˆ, u, ˆ p) ˆ satisfy (6)–(9). 3.2. Non–monotonicity and discontinuous solutions. In this section we prove Theorem 1. The proof is based on the properties of the normalized strain–rate to elastic stress mapping which we now define. Definition 1. The map
: End(R3 ) κ → σ ∈ Sym(R3 ) is defined via
σ =
S2
(3 n ⊗ n − id) f κ dn,
where f κ is the unique solution of with f ≥ 0 and
∇n · (Pn ⊥ κ n f ) − n f = 0
S2
(65)
f dn = 1.
We denote by κs the gradient of a normalized shear flow, i. e. ⎛ ⎞ 010 κs = ⎝ 0 0 0 ⎠ . 000 Hence x1 is the flow direction, x2 the shear direction and x3 the vorticity direction. We will need the following three properties for (γ˙ κs ): Lemma 4. d
12 (γ˙ κs ) > 0. d γ˙ |γ˙ =0
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
749
Lemma 5. lim 12 (γ˙ κs ) = 0.
γ˙ →∞
Lemma 6.
23 (γ˙ κs ) = 0. Proof of Lemma 4. We start with remarking that the components of 3n ⊗ n − id, i. e. 3 n i n j − δi j , are surface spherical harmonics of order 2. This means that they are harmonic polynomials on R3 of order 2, restricted to S 2 . It is well known that surface spherical harmonics are eigenfunctions of the Laplacian on S 2 . Their eigenvalue is − ( + 1), where is the order [1, App. E]. Hence n (3 n i n j − δi j ) = −6 (3 n i n j − δi j ). This observation yields an alternative representation of the map : (3n ⊗ n − id) f κ dn
(κ) = S2 n n (66) 1 ⊗ − id f κ dn n 3 = − 6 S2 |n| |n| 1 = − n ⊗ n n f κ dn 2 S2 (65) 1 n ⊗ n ∇n · (Pn ⊥ κn f κ ) dn. = − 2 S2 According to (67), we have in particular γ˙
12 (γ˙ κs ) = − n 1 n 2 ∇n · (Pn ⊥ κs n f γ˙ κs ) dn. 2 S2 Hence we obtain d 1
12 (γ˙ κs ) = − n 1 n 2 ∇n · (Pn ⊥ κs n f 0 ) dn d γ˙ |γ˙ =0 2 S2 1 =− n 1 n 2 ∇n · (Pn ⊥ κs n) dn 8π S 2 1 = ∇n (n 1 n 2 ) · Pn ⊥ κs n dn 8π S 2 ⎛ ⎞ ⎛ ⎞ n2 n2 1 Pn ⊥ ⎝ n 1 ⎠ · Pn ⊥ ⎝ 0 ⎠ dn = 8π S 2 0 0 ⎛ ⎞ ⎛ ⎞ n2 n2 1 = Pn ⊥ ⎝ n 1 ⎠ · ⎝ 0 ⎠ dn 8π S 2 0 0 1 (1 − 2 n 21 ) n 22 dn. = 8π S 2
(66)
(67)
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F. Otto, A. E. Tzavaras
By symmetry we have (1 − 2 n 21 )n 22 dn = (1 − n 21 − n 23 ) n 22 dn = S2
S2
S2
n 42 dn,
so that the above turns into d 1
12 (γ˙ κs ) = n 4 dn > 0. d γ˙ |γ˙ =0 8π S 2 2 Proof of Lemma 5. According to the definition of , we have to show n 1 n 2 f γ˙ κs dn = 0. lim γ˙ ↑∞ S 2
Because of Jensen
S
3 n 1 n 2 f γ˙ κs dn ≤ 2
S2
|n 1 n 2 |3 f γ˙ κs dn
and the inequality |n 1 n 2 |3 ≤ |n 2 |3 ≤ (n 22 + n 23 ) |n 2 | = (1 − n 21 ) |n 2 |, it suffices to show
lim
γ˙ ↑∞ S 2
(1 − n 21 ) |n 2 | f γ˙ κs dn = 0.
(68)
We now argue in favor of (68). According to (65), we have for any test function ζ ,
∇n ζ · Pn ⊥ κs n + γ˙ −1 n ζ f γ˙ κs dn = 0. (69) S2
We now make a special ansatz for ζ . We fix a smooth function ϕ(nˆ 2 ) with ϕ(nˆ 2 ) = 1 for nˆ 2 ≥ 1 and ϕ(nˆ 2 ) = −1 for nˆ 2 ≤ −1. For given λ > 0 to be optimized later, we consider n 2 , ζλ (n) = n 1 ϕ λ which we think of as an approximation of n 1 sign(n 2 ) for λ 1. On one hand we have 1 λ2 with a universal generic constant C < ∞. On the other hand, we have ⎛ ⎛ ⎞ ⎞ ϕ nλ2 n2 ∇n ζλ · Pn ⊥ κs n = Pn ⊥ ⎝ nλ1 ϕ nλ2 ⎠ · Pn ⊥ ⎝ 0 ⎠ 0 0 ⎛ ⎞ n2 ⎞ ⎛ 2 ϕ λ n2 − n1 n2 = ⎝ nλ1 ϕ nλ2 ⎠ · ⎝ −n 1 n 22 ⎠ 0 −n 1 n 2 n 3 n n2 n2 2 = (1 − n 21 ) n 2 ϕ − n 21 2 ϕ . λ λ λ |n ζλ | ≤ C
(70)
(71)
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
Since
751
(1 − n 2 ) n 2 ϕ n 2 − (1 − n 2 ) |n 2 | ≤ λ | n 2 | |ϕ n 2 − sign n 2 | ≤ C λ, 1 1 λ λ λ λ 2 n 2 n n 2 ϕ 2 ≤ λ n 2 2 ϕ n 2 ≤ C λ, 1 λ λ λ λ
(71) yields
∇n ζλ · Pn ⊥ κs n − (1 − n 21 ) |n 2 | ≤ C λ.
(72)
From (70) and (72) we obtain 1 . ∇n ζλ · Pn ⊥ κs n + γ˙ −1 n ζλ − (1 − n 21 ) |n 2 | ≤ C λ + γ˙ λ2 With the choice of λ = γ˙ −1/3 this turns into ∇n ζλ · Pn ⊥ κs n + γ˙ −1 n ζλ − (1 − n 21 ) |n 2 | ≤ C γ˙ −1/3 . In view of (69) this yields (1 − n 2 ) |n 2 | f γ˙ κ dn ≤ C γ˙ −1/3 , s 1 2 S
which is a quantitative version of (68). Proof of Lemma 6. This is an outcome of symmetry considerations. We notice that for any Q ∈ O(3), f Q t κ Q (n) = f κ (Q n).
(73)
Indeed, consider the transformation f˜(n) := f (Q n). Proceeding as in the proof of Lemma 3 we obtain
−n f˜ + ∇n · Pn ⊥ (Q t κ Q) n f˜ (n) = −n f + ∇n · Pn ⊥ κ n f (Q n). This identity implies (73) by uniqueness of (65). We now notice that ⎛ ⎞ 10 0 Q := ⎝ 0 1 0 ⎠ ∈ O(3) and Q t (γ˙ ks ) Q = γ˙ ks . 0 0 −1 Hence by (73) we have f γ˙ κs (n 1 , n 2 , −n 3 ) = f γ˙ κs (n 1 , n 2 , n 3 ), which in turn yields
23 (γ˙ κs ) = =
S2 S2
=−
n 2 n 3 f γ˙ κs (n 1 , n 2 , n 3 ) dn n 2 n 3 f γ˙ κs (n 1 , n 2 , −n 3 ) dn
S2
n 2 n 3 f γ˙ κs (n 1 , n 2 , n 3 ) dn
= − 23 (γ˙ κs ).
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F. Otto, A. E. Tzavaras
Proof of Theorem 1. According to Lemmas 4 and 5, R γ˙ → 12 (γ˙ κs ) is not monotone. Hence for sufficiently small Dr , also
R γ˙ → γ˙ + 12 We fix such a Dr and select γ˙± with
γ˙+ = γ˙− and γ˙+ + 12 We introduce
f (x, n) :=
γ˙ κs Dr
γ˙+ κs Dr
f γ˙+ κs (n) f γ˙− κs (n)
is not monotone.
= γ˙− + 12
γ˙− κs Dr
.
(74)
for x2 > 0 , for x2 < 0
γ˙+ (x2 , 0, 0) u(x) := γ˙− (x2 , 0, 0) ⎧
⎨ 22 γ˙+ κs Dr p(x) := ⎩ 22 γ˙− κs Dr
for x2 > 0 , for x2 < 0 ⎫ for x2 > 0 ⎬ . for x2 < 0 ⎭
We notice that u is continuous with weak gradient γ˙+ κs for x2 > 0 , ∇x u(x) = γ˙− κs for x2 < 0
(75)
(76)
that ∇x u is discontinuous and (19) is satisfied (in the weak sense). We now argue that (18), which in view of (19) can be rewritten as ∇x · (∇x u − p id + σ ) = 0,
(77)
holds in the weak sense. Indeed, because of (76), (75) and Definition 1 we have ⎧ ⎫
⎨ γ˙+ κs − 22 γ˙+ κs id + γ˙+ κs ⎬ for x > 0 2 D D r r ∇x u − p id + σ = . ⎩ γ˙− κs − 22 γ˙− κs id + γ˙− κs for x2 < 0 ⎭ Dr Dr Since this tensor is piecewise constant, it remains to show that u i,2 − p δi2 + σi2 is continuous across {x2 = 0} for i = 1, 2, 3 in order to conclude (77). For the flow direction i = 1 we have ⎧ ⎫
⎨ γ˙+ + 12 γ˙+ κs ⎬ for x > 0 2 Dr u 1,2 − p δ12 + σ12 = , γ ˙ κ ⎩ γ˙− + 12 − s ⎭ for x < 0 2 Dr so that (78) follows from (74). For the shear direction i = 2 we notice that u 2,2 − p δ22 + σ22 = 0
(78)
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
753
due to the definition (75) of the pressure. For the vorticity direction i = 3 we remark ⎧ ⎫
⎨ 23 γ˙+ κs ⎬ for x > 0 2 Dr u 3,2 − p δ32 + σ32 = , γ ˙ κ ⎩ 23 − s ⎭ for x < 0 2 Dr which vanishes according to Lemma 6. Hence (78) is established. The Smoluchowski equation (16) itself is satisfied by definition (65) of f γ˙± κs and because of ∇x · ( f u) = 0 distributionally, since u has only a u 1 –component and f u depends on x only through x2 . 4. Appendix I. Proof of Proposition 2. We select a ϕ in S(R3 ), the Schwartz space, such that its Fourier transform satisfies ϕ(k) ˆ = The constant is chosen such that
1 for |k| ≤ 1. (2π )3/2
(79)
R3
ϕ d x = 1.
(80)
We recall that u is periodic and that the Fourier symbol which relates σ to ∇x u via the Stokes system (44) & (45) is given by kj ki k km − δi σˆ m (k), uˆ i, j (k) = (81) |k| |k| |k| |k| where we sum over repeated indices. Thanks to (79), kj km ki k k ˆ − ϕ(k) ˆ − δi ψi j m (k) = ϕˆ 2 |k| |k| |k| |k| defines a ψi j m ∈ S(R3 ). We introduce the dyadically rescaled version of these Schwartz functions: (ν)
ϕ (ν) (x) = (2ν )3 ϕ(2ν x), ψi j m (x) = (2ν )3 ψi j m (2ν x) for ν ∈ {0, 1, · · · }, and recall that ϕ (ν) (k) = ϕˆ 2kν . We now fix an N ∈ N which we will choose at the end. The decomposition of the right-hand side σ into σ m = σ m − ϕ (N ) ∗ σ m +(ϕ (N ) − ϕ (N −1) ) ∗ σ m + · · · + (ϕ (1) − ϕ (0) ) ∗ σ m +ϕ (0) ∗ σ m
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F. Otto, A. E. Tzavaras
translates by definition of ψi j m into u i, j = u i, j − ϕ (N ) ∗ u i, j −1) + ψi(N j m (0)
(82)
∗ σ m + · · · + ψi(0) j m
∗ σ m
(83)
∗ u i, j .
+ϕ
(84)
Each of the terms in line (83) is easily estimated as follows (ν) (ν) sup |ψi j m ∗ σ m | ≤ |ψi j m | dz sup |σ m | R3
x∈R3
x∈R3
=
R3
|ψi j m | d zˆ sup |σ m | x∈R3
≤ C sup |σ |,
(85)
x
where C denotes a generic constant only depending on p. For the term in line (84) we obtain (0) 2 (0) 2 2 |ϕ ∗ u i, j (x)| ≤ |ϕ (x − y)| dy |u i, j | dy R3 R3 ≤C |σ |2 dy . R3
(86)
We now address the term in line (82). We recall the Sobolev embedding theorem for functions in W 1, p (R3 ), |u i, j (x) − u i, j (y)| ≤ C |x − y|
1/ p
1−3/ p
|∇u i, j | d x
,
p
R3
(87)
and Lemma 2 with the L p (R3 )–estimate for the Stokes operator:
1/ p |∇ u| d x 2
R3
≤ C
p
1/ p |∇σ | d x
.
p
R3
(88)
This allows to tackle the term in line (82): |(u i, j − ϕ (N ) ∗ u i, j )(x)| (80) (N ) ϕ (x − y) (u i, j (x) − u i, j (y)) dy = R3 1/ p (87,88) ≤ C |ϕ (N ) (z)| |z|1−3/ p dz |∇u i, j | p d x R3
= C (2
−N 1−3/ p
)
= C 2−N (1−3/ p)
R3
R3
|ϕ(ˆz )| |ˆz |
1−3/ p
R3
d zˆ
1/ p |∇σ | p d x
.
1/ p |∇σ | d x p
R3
(89)
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
Combining (85), (86) & (89), we gather sup |∇u| ≤ C
2
−N (1−3/ p)
x∈R3
755
1/ p |∇σ | d x p
+N sup |σ | +
R3
1/2
|σ | d x
.
2
R3
x∈R3
(90)
The Stokes system (44)–(45) in R3 is invariant under the rescaling u (x) =
1 1 u( x) , σ (x) = σ ( x) , 2
p (x) =
1 p( x).
We apply (90) to the rescaled functions u , σ and use the identities − 3p
5
σ L 2 (R3 ) = − 2 σ L 2 (R3 ) , ∇x σ L p (R3 ) = to obtain
∇x σ L p (R3 ) ,
1− 3p
2−N (1−3/ p)
sup |∇u| ≤ C
x∈R3
∇x σ L p (R3 )
− 23
+N sup |σ | + x∈R3
σ L 2 (R3 ) .
(91)
The interpolation estimate (91) depends on two parameters N and . We proceed to optimize their selection. First choose N ∈ N such that 2(N −1)(1−3/ p) ≤ 1 + so that
!
1−3/ p ∇x σ L p (R3 ) supx |σ |
N ≤ C 1 + ln 1 + Then (91) turns into !! sup |∇u| ≤ C
1 + ln 1 +
1−3/ p ∇x σ L p (R3 ) supx |σ |
1−3/ p ∇x σ L p (R3 ) supx |σ |
x∈R3
≤ 2 N (1−3/ p) ,
" .
"
" −3/2
sup |σ | + x
σ L 2 (R3 ) . (92)
Next we select =
σ L 2 (R3 ) supx |σ |
in (92) and complete the proof of Proposition 2.
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F. Otto, A. E. Tzavaras
5. Appendix II. The operator ∇n satisfies certain elementary properties that are extensively used in this article: Let F be a vector-valued function and f , g be scalar-valued functions, then (∇n · F) f dn = − F · (∇n f − 2n f )dn, (93) 2 2 S S (∇n · ∇n f )gdn = (∇n · ∇n g) f dn, (94) 2 S2 S n ⊗ ∇n f dn = ∇n f ⊗ n dn = (3n ⊗ n − id) f dn. (95) S2
S2
S2
A convenient way to prove such formulas is by expressing them to spherical coordinates, see [1, App. A.6 and E.6]. The change of variables for a point P with Cartesian coordinates (n x , n y , n z ) to spherical coordinates is n x = r sin θ cos ϕ , n y = r sin θ sin ϕ , n z = r cos θ, where 0 < θ < π , 0 ≤ ϕ < 2π . Let er , eθ , eϕ be the orthonormal coordinate system associated to spherical coordinates and attached at P. It satisfies the derivative formulas ∂er = 0, ∂r ∂eθ = 0, ∂r ∂eϕ = 0, ∂r
∂er = eθ , ∂θ ∂eθ = −er , ∂θ ∂eϕ = 0, ∂θ
∂er = eϕ sin θ, ∂ϕ ∂eθ = eϕ cos θ, ∂ϕ ∂eϕ = −er sin θ − eθ cos θ. ∂ϕ
(96)
We visualize the sphere S 2 as embedded in the Euclidean space. The operator ∇n is related to the gradient operator ∇ through ∇n = r (id − n ⊗ n) · ∇ = eθ
∂ 1 ∂ + eϕ . ∂θ sin θ ∂ϕ
For a scalar-valued function f , ∂f 1 ∂f + eϕ , ∂θ sin θ ∂ϕ ∂f 1 ∂2 f 1 ∂ n f = ∇n · ∇n f = sin θ + . sin θ ∂θ ∂θ sin2 θ ∂ϕ 2 ∇n f = eθ
For a vector-valued function F, expressed in spherical coordinates in the form F = Fr er + Fθ eθ + Fϕ eϕ , we compute ∂ 1 ∂ + eϕ · Fr er + Fθ eθ + Fϕ eϕ eθ ∂θ sin θ ∂ϕ ∂ 1 1 ∂ Fϕ (96) + 2Fr . = (sin θ Fθ ) + sin θ ∂θ sin θ ∂ϕ
∇n · F =
Continuity of Velocity Gradients in Suspensions of Rod–like Molecules
757
Observe next that 1 ∂ 1 ∂ Fϕ (∇n · F) f dn = + 2Fr f sin θ dθ dϕ (sin θ Fθ ) + sin θ ∂θ sin θ ∂ϕ S2 1 ∂f ∂f + Fϕ sin θ dθ dϕ −2Fr f + Fθ =− ∂θ sin θ ∂ϕ =− F · (∇n f − 2n f )dn S2
gives (93). Formula (94) follows by applying (93) twice: (∇n · ∇n f )gdn = − ∇n f · (∇n g − 2ng) dn 2 S2 S ∇n f · ∇n g dn =− S2 f (∇n · ∇n g)dn . = S2
Finally, using integration by parts, we have the chain of identities n ⊗ ∇n f dn S2 ∂f 1 ∂f + eϕ sin θ dθ dϕ = er ⊗ eθ ∂θ sin θ ∂ϕ $ # ∂ cos θ 1 ∂ (er ⊗ eθ ) + er ⊗ eθ + (er ⊗ eϕ ) f sin θ dθ dϕ =− ∂θ sin θ sin θ ∂ϕ (96) = − eθ ⊗ eθ + eϕ ⊗ eϕ − 2er ⊗ er f sin θ dθ dϕ = (3n ⊗ n − id) f dn. (97) S2
Since (a ⊗ b)t = b ⊗ a and the final equation in (97) is a symmetric tensor, we deduce (95). Acknowledgements. FO thanks Claude Le Bris and his group for inspiring discussions. AET partially supported by the National Science Foundation. FO partially supported by the German Science Foundation through SFB 611.
References 1. Bird, R.B., Curtiss, Ch.F., Armstrong, R.C., Hassager, O.: Dynamics of Polymeric Liquids, Vol. 2, Kinetic Theory. New York: Wiley Interscience, 1987 2. Constantin, P.: Nonlinear Fokker-Planck Navier-Stokes systems. Comm. Math. Sci. 3, 531–544 (2005) 3. De Gennes, P.G., Prost, J.: The physics of liquid crystals. Oxford: Oxford Univ. Press, 1993 4. Doi, M.: Molecular-dynamics and rheological properties of concentrated–solutions of rodlike polymers in isotropic and liquid–crystalline phases. J. Polym. Sci. Polym. Phys. Ed. 19, 229–243 (1981) 5. Doi, M., Edwards, S.F.: The theory of polymer dynamics. Oxford: Oxford Univ. Press, 1986 6. Lelièvre, T.: PhD thesis, CERMICS, Ecole Nationale des Ponts et Chaussées, 2004 7. Lions, P.-L.: Mathematical topics in fluid mechanics, Vol 2. Oxford: Oxford Univ. Press, 1998
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8. Lions, P.-L., Masmoudi, N.: Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann. Math. B 21(2), 131–146 (2000) 9. Masmoudi, N.: Private communication, 2004 10. Loeschcke, Ch., Otto, F., Wachsmuth, J.: Suspensions in rod-like molecules: Nonlinear stability of homogeneous flows. Preprint 11. Malkus, D.S., Nohel, J.A., Plohr, B.J.: Analysis of the spurt phenomena for a non-Newtonian fluid. In: Problems Involving Change of type, K. Kirchgässuer ed., Lecture Notes in Physics 359, Berlin New York: Springer, 1990, pp. 112–132 12. Nohel, J.A., Pego, R.L.: On the generation of discontinuous shearing motions of a non–Newtonian fluid. Arch. Rat. Mech. Anal. 139, 355–376 (1997) 13. Nohel, J.A., Pego, R.L., Tzavaras, A.E.: Stability of discontinuous steady states in shearing motions of a non–Newtonian fluid. Proc. Roy. Soc. Edinburgh 115 A, 39–59 (1990) 14. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Diff. Equations 26, 101–174 (2001) 15. Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000) 16. Petersen, P.: Riemannian Geometry, Berlin-Heidelberg-New York: Springer, 1991 17. Temam, R.: Navier-Stokes Equations. Amsterdam: North-Holland, 1984 18. Vinogradov, G., Malkin, A., Yanovskii, Y., Borisenkova, E., Yarlykov, B., Berezhnaya, G.: Viscoelastic properties and flow of narrow distribution polybutadienes and polyisoprenes. J. Polymer Sci. Part A-2 10, 1061–1084 (1972) 19. Weigant, V.A., Kazhikhov, A.V.: Stability “in the large” of an initial–boundary value problem for equations of the potential flows of a compressible viscous fluid at low Reynolds numbers (Russian). Dokl. Akad. Nauk 340(4), 460–462 (1995) Communicated by P. Constantin
Commun. Math. Phys. 277, 759–769 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0376-2
Communications in
Mathematical Physics
Gaussian Thermostats as Geodesic Flows of Nonsymmetric Linear Connections Piotr Przytycki1 , Maciej P. Wojtkowski2 1 Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-956 Warszawa, Poland.
E-mail: [email protected]
2 Department of Mathematics and Informatics, University of Warmia and Mazury, ul. Zolnierska 14,
10-561 Olsztyn, Poland. E-mail: [email protected] Received: 30 January 2007 / Accepted: 25 May 2007 Published online: 4 December 2007 – © P. Przytycki and M. P. Wojtkowski 2007
Abstract: We establish that Gaussian thermostats are geodesic flows of special metric connections. We give sufficient conditions for hyperbolicity of geodesic flows of metric connections in terms of their curvature and torsion. 1. Introduction Let M be a compact manifold with a Riemannian metric g, whose scalar product will be denoted by ·, ·. Denote by ∇ the Levi–Civita connection of the metric g. Let E be a vector field on M. A Gaussian thermostat is the dynamical system on T M defined by the equations du Dv E, vv = v, =E− , dt dt v, v
(1)
D where u(t) ∈ M, v(t) ∈ Tu(t) M, and dt = ∇v is the covariant derivative, [G-R]. 2 Since v is a first integral of the system we can restrict our attention to one level set. Although the dynamics is quite different for different values of v 2 , there is no loss of generality in considering the Gaussian thermostat on the unit sphere bundle S M = {v ∈ T M : |v| = 1}. Indeed the change in the value of v 2 is equivalent, up to parameterization, to the rescaling of E. On S M we can write Eqs. (1) as equations of a spray, Dv du = v, = v 2 E − E, vv. (2) dt dt Every spray can be viewed as a geodesic flow of a canonical symmetric linear connection ∇ s , [A-P-S], defined in this case as
1 1 ∇ Xs Y = ∇ X Y − X, Y E + X, EY + Y, EX. 2 2 Reproduction of the entire article for non-commercial purposes is permitted without charge.
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It is not the only linear connection that can be used for that purpose. We want to argue that there are indeed two more useful linear connections with the same geodesics up to parameterization. First of all the trajectories of a Gaussian thermostat are geodesics of the Weyl connection, [W1], X Y = ∇ X Y − X, Y E + X, EY + Y, EX. ∇ The advantage of the Weyl connection over the spray connection ∇ s is that its parallel transport is conformal. However the parameterization of the trajectories of (1) are unrelated to the Weyl connection. Moreover the geodesic flow of a Weyl connection on T M is not in general complete. The geodesics on M can be extended indefinitely but their velocity may go to infinity in finite time. Dynamical systems obtained from geodesic flows of Weyl connections by the parameterization with the arclength of a background riemannian metric were called W -flows in [W1]. The starting point of the paper was that Gaussian thermostats are W -flows. In this paper we propose to consider the equations of a Gaussian thermostat as the geodesic flow of the linear connection ∇, X Y = ∇ X Y − X, Y E + Y, EX. ∇ This connection is nonsymmetric but it has isometric parallel transport, i.e., it is a metric is connection. The torsion of ∇ (X, Y ) = ∇ Y X − [X, Y ] = Y, EX − X, EY. X Y − ∇ T Metric connections are uniquely determined by their torsions. We prove the following theorem. is the only Theorem 1. For a Gaussian thermostat system (1) on S M the connection ∇ linear connection on T M satisfying the following properties: i) the trajectories of the system are geodesics for ∇, ii) parallel transport defined by ∇ is isometric, (X, Y ) of the connection has values in span{X, Y }. iii) the torsion T rather than the Weyl connection ∇ will allow us to obtain in a simpler, The use of ∇ more transparent way the basic results of [W1, W2] on hyperbolic properties of Gaussian thermostats. The linearization of geodesic flows is provided by Jacobi equations. For nonsym metric connections the Jacobi equations involve both the curvature tensor R(X, Y) = . With a chosen metric g we introduce the X ∇ Y ∇ [X,Y ] and the torsion T Y − ∇ X − ∇ ∇ () and the sectional torsion T () of a connection in the direction sectional curvature K of a plane by the formulas 2 1 () = R(X, K Y )Y, X , T˜ () = T (X, Y ) , 4 where (X, Y ) is an orthonormal frame in the plane . We get the following generalization of a result from [W1]. Theorem 2. If for every plane the sum of the sectional curvature and the sectional torsion of a metric connection is negative then the geodesic flow is Anosov. This theorem is formulated for an arbitrary metric connection. For special metric connec(X, Y ) = ϕ(Y )X − ϕ(X )Y , for some tions related to Gaussian thermostats, where T linear form ϕ, the following theorem was proven in [W1].
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Theorem 3. If all the sectional curvatures of the metric connection are negative then the geodesic flow has a dominated splitting with exponential growth/decay of volumes. The dominated splitting (also called the exponential dichotomy) is the property of the linearized equations to have two subspaces of solutions E + and E − such that the exponential rates of growth in E + dominate those in E − . We refer the reader to [M] and [W3] for detailed formulations and discussions. It is a much weaker property than Anosov. In our case we have the additional property that there is uniform growth in E + , and uniform decay in E − , of volumes. In particular in the case of two dimensional manifolds M the subspaces E + and E − are one dimensional, the volume becomes length, and we obtain the Anosov property. This theorem applies to Gaussian thermostats studied by Bonetto, Gentile and Mastropietro, [B-G-M], and it gives Anosov property for electrical fields E of any strength. Dairbekov and Paternain, [D-P], showed recently that on two dimensional manifolds, if a Gaussian thermostat is Anosov then its SRB measure is singular, except when E has a global potential (i.e., ϕ is exact). It remains an open problem to decide if in higher dimensions one gets the Anosov property from negative sectional curvatures alone, either in the general case of a metric connection or in the Gaussian thermostat case. The plan of the paper is the following. In Sect. 2 we introduce the Jacobi equations and prove Theorems 2 and 3. of a metric connection ∇ In Sect. 3 we study the class of linear connections on M whose geodesics coincide with the trajectories of the system (2), which leads us to the special role played by ∇ and the proof of Theorem 1. In Sect. 4 we discuss the interpretation of Gaussian thermostats as geodesic flows and explore the role of the conformal class of the background metric. appears again in the Finally in Sect. 5 we show that the antisymmetric tensor T interpretation of Gaussian thermostats as generalized hamiltonian systems, obtained in [W-L] and [W4]. 2. The Curvature, Torsion and Hyperbolic Properties of Geodesic Flows of Linear Metric Connections We consider the geodesic flow t : S M → S M of a metric connection ∇. and let u(s, a) ∈ M be a family Let u(s) be a fixed geodesic of the connection ∇, of geodesics, u(s) = u(s, 0), where s is the arclength parameter on a geodesic, and a is some real parameter, taken from a small interval around 0. Define the unit velocity field v = v(s) and the Jacobi field ξ = ξ(s) along the geodesic u(s) by v=
du ∂u , ξ= ∈ Tu M. ds ∂a a=0
The velocity field v is a special Jacobi field. Jacobi fields form a vector space which can be naturally identified with the tangent spaces of S M along the chosen geodesic. ξ v ∈ Tu M we get the following Jacobi equations If we introduce χ = ∇ (v, ξ ), v ξ = χ + T ∇ ξ )v, v χ = R(v, ∇ and T are the curvature and the torsion tensors respectively. These equations where R are completely general and they follow immediately from the definitions of the tensors and R. Indeed T
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(v, ξ ) = χ + T (v, ξ ), ξ v + T v ξ = ∇ ∇ ξ )v. v v = R(v, ∇v χ = ∇v ∇ξ v = R(v, ξ )v + ∇ξ ∇ Due to the restriction of the geodesic flow to S M we have that χ , v = 0. The dynamical significance of the Jacobi equations is that they provide a convenient, and geometrically meaningful, linearization of the geodesic flow. More specifically a Jacobi field ξ is uniquely determined by the Cauchy data (ξ(s), χ (s)) for the Jacobi equations. Hence the pair (ξ(s), χ (s)) can be thought of as a tangent vector to the phase space S M, that is we have that (ξ(s), χ (s)) ∈ Tv (S M). With this identification we get that D s ((ξ(0), χ (0)) = (ξ(s), χ (s)). Thus the Jacobi equations give us a way to study the hyperbolic properties of the geodesic flow. In particular its Lyapunov exponents are the exponential rates of growth of the Jacobi fields. We introduce a quadratic form J on S M by (v, ξ ). v ξ − ξ, T J (ξ ) = ξ, χ = ξ, ∇ For a fixed Jacobi field ξ the evaluation of J on ξ along the geodesic becomes the function of the arclength parameter s, namely J (ξ )(s) = ξ(s), χ (s). Following [W3] we introduce the definition Definition 1. We say that the geodesic flow s is a) strictly J -monotone if for every Jacobi field ξ which is not colinear with the velocity d field v we have ds J (ξ )(s) > 0, b) strictly J -separated if for any Jacobi field ξ which is not colinear with the velocity d field v, and for which J (ξ )(0) = 0, we have that ds J (ξ )(s)|s=0 > 0. Clearly if a geodesic flow is strictly J -monotone then it is strictly J -separated. The interest in this definition comes from the following Theorem 4 ([W3]). If a flow is strictly J -separated then it has a dominated splitting. If a flow is strictly J -monotone then it is Anosov. is metric we get Proof of Theorem 2. Using the fact that the connection ∇ d d v ξ, χ + ξ, ∇ v χ . J = ξ, χ = ∇ ds ds The Jacobi equations give us that the last expression is equal to (v, ξ ), χ + ξ, R(v, ξ )v χ 2 + T 2 1 v)v − = χ + T (v, ξ ) − ξ, R(ξ, 2
2 1 T (v, ξ ) . 4
Theorem 2 follows now from Theorem 4 and the second part of 3.
(3)
In the special case of Weyl connections the equivalent of Theorem 2 was formulated in [W1].
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In the Riemannian case there is a codimension one subspace of Jacobi fields which are orthogonal to the velocity field v. That is not the case in general, unless the values (v, ·) are orthogonal to v. For lack of invariant subspaces transversal to the flow we of T need to consider the quotient of the tangent space to S M by the velocity field. One way to do it is to consider the projection of the Jacobi field on the subspace perpendicular to the velocity, that is for a given Jacobi field ξ we consider the field ζ = ξ − ξ, vv. In particular the form J factors to the quotient space, that is its value depends only on ζ . In the rest of the section we restrict ourselves to the special case of the torsion (X, Y ) = ϕ(Y )X − ϕ(X )Y , for some 1-form ϕ. In this case we get from the Jacobi T equations the following quotient equations for the field ζ : v ζ = χ − ϕ(v)ζ, ∇ ζ )v. v χ = R(v, ∇
(4)
The proof of Theorem 3 can be extracted from [W1]. For completeness we provide it here in detail in the new setup. we get T (v, ξ ), χ = ϕ(ξ )v, χ − Proof of Theorem 3. For our special form of T ϕ(v)ξ, χ . We have always that v, χ = 0, because v, v = 1. Under the assumption that J (ξ )(0) = ξ(0), χ (0) = 0 we get from the first part of 3, d v)v. J (ξ )(s)s=0 = χ 2 − ξ, R(ξ, ds We get that the geodesic flow is strictly J -separated and the first part of our theorem follows from Theorem 4. To prove the second part we choose an orthonormal frame at an initial point on our geodesic such that the first vector of the frame is the velocity vector. We transport the With these frames fixed we frame parallely along the geodesic using the connection ∇. n−1 can consider ζ (s) and χ (s) as vectors in R and the quotient equations (4) become d ζ = χ − ϕ(v)ζ, ds d ζ )v. χ = R(v, ds
(5)
The dominated splitting property gives us two invariant subspaces E + and E − . We will establish exponential growth of volume on E + . The exponential decay of volume on E − follows then from the reversibility of the geodesic flow. To prove the exponential growth we will introduce a special volume element in E + . We represent the subspace E + ⊂ Rn−1 ×Rn−1 as a graph of an operator U : Rn−1 → Rn−1 , that is E + = {(ζ, U ζ ) | ζ ∈ Rn−1 }. The evolution of U follows from (5) and it is described by the following Riccatti equation: d v)v. U = ϕ(v)U − U 2 − R, where Rζ = R(ζ, (6) ds Since by the construction of E + the quadratic form J = ζ, χ is positive definite on E + , we get that the symmetric part of U is positive definite. In contrast to the Riemannian case we are not guaranteed that U itself is symmetric, because the operator R is not in general symmetric. Let us split the operators U = Us + Ua and R = Rs + Ra into symmetric and antisymmetric parts respectively. By the assumption of negative sectional curvatures everywhere we get that −Rs is positive definite. (Using the formula for the
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curvature tensor from [W1] it can be calculated that Ra ζ, η = have that Ua2 is negative semidefinite. We get from (6),
1 2 dϕ(ζ, η).)
We also
d Us = ϕ(v)Us − Us2 − Ua2 − Rs . ds
(7)
We introduce new linear coordinates κ ∈ Rn−1 by the formula κ = Us ζ . We will show that in these coordinates the standard volume has uniform exponential growth. Indeed we get from (5) and (7), d κ = −Us + Us UUs−1 + (−Ua2 − Rs )Us−1 κ. ds Since tr (Us UUs−1 ) = tr Us we get that the trace of the operator in the right hand side of the equation is equal to tr (−Ua2 − Rs )Us−1 . It is positive since a product of two symmetric positive definite operators has positive trace. It follows that the standard volume in the coordinates κ is uniformly exponentially expanded. 3. Linear Connections Determined by the Family of Geodesics Let us recall that two linear connections on a manifold differ by a tensor. We consider two such connections ∇ 1 and ∇ 2 , ∇ X2 Y = ∇ X1 Y + A(X, Y ) + B(X, Y ), where A is a symmetric and B an antisymmetric tensor. Clearly the equations of geodesics are not effected by the antisymmetric tensor B. The following proposition is the classical theorem of H. Weyl. Proposition 1. The linear connections ∇ 1 and ∇ 2 have the same geodesics up to parameterizations if and only if A(X, Y ) = α(X )Y + α(Y )X for some linear form α. In the proof we will need the following elementary lemma. Lemma 1. For a bilinear map C : Rn × Rn → Rn the following are equivalent (a) C(X, Y ) ∈ span{X, Y }, for every X, Y ∈ Rn , (b) there are linear forms α and β such that C(X, Y ) = α(X )Y + β(Y )X . Proof. If C satisfies (a) then both its symmetric and antisymmetric parts satisfy (a). Hence it is enough to establish (b) separately for symmetric and antisymmetric maps C. We give the proof here only for the symmetric case. The antisymmetric case is somewhat more involved and we leave it to the reader. Let us assume that a symmetric bilinear map C satisfies (a). Then C(X, X ) must be colinear with X . Let (x 1 , x 2 , . . . , x n ) be linear coordinates in Rn . If x 1 = 0 then we get that also the first coordinate of C(X, X ) has to vanish. It follows that the first coordinate of C(X, X ) is a quadratic form all of whose terms must contain x 1 . The same can be repeated for other coordinates. Hence we obtain that C(X, X ) = (α1 (X )x 1 , . . . , αn (X )x n ) for some linear forms αk , k = 1, . . . , n. Since C(X, X ) must be colinear with X it follows immediately that the linear forms must coincide, which gives us (b). Proof of Proposition 1. Let γ (t) be a geodesic of ∇ 1 , and let us assume that there is a change of time t = t (u) such that γ (t (u)) is a geodesic of ∇ 2 . We have dγ dγ = v, ∇v1 v = 0 and = w, ∇w2 w = 0. dt du
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Further w = t v and 0 = ∇w2 w = t v + (t )2 ∇v2 v = t v + (t )2 A(v, v). It follows that A(v, v) ∈ span{v}. Using Lemma 1 we obtain the desired conclusion. The converse is straightforward. As a corollary of Proposition 1 we obtain that the trajectories of a Gaussian thermostat 2 2 can be obtained by integrating the system ddtx = v, Dv dt = v E, in which v is not preserved. This may be simpler than the integration of the original equation as in the following example. Example 1. Let M be the two dimensional flat torus with coordinates (x1 , x2 ) and E = (1, 0) be a constant vector field. The trajectories of the Gaussian thermostat satisfy dv1 2 2 dv2 dt = v1 + v2 , dt = 0. Integrating the first equation for a constant v2 = 0 we get v1 = v2 tan(v2 t + c) which yields trajectories x1 = − ln cos(x2 + c1 ) + c2 . The remaining trajectories are horizontal lines (v2 = 0). Proposition 2. The linear connections ∇ 1 and ∇ 2 define the same parallel transport up to dilation if and only if ∇ X2 Y − ∇ X1 Y = α(X )Y for some linear form α. Proof. As we observed before C(X, Y ) = ∇ X2 Y − ∇ X1 Y is a tensor. Let us assume that the parallel transports of the two connections differ only by dilations. Let Y be a vector field parallel in direction X with respect to ∇ 2 and let f be a positive function such that f Y is parallel in the same direction with respect to ∇ 1 . We have C(X, f Y ) = ∇ X2 ( f Y ) − ∇ X1 ( f Y ) = d f (X )Y. In view of the arbitrariness of X and Y the claim follows now from Lemma 1. The proof in the other direction is straightforward. In view of Proposition 1 we consider the family of all linear connections that share the same geodesics up to parameterization, X Y = ∇ X Y − X, Y E + α(X )Y + α(Y )X + B(X, Y ), ∇
(8)
where α is a linear form and B(X, Y ) is an antisymmetric tensor. We will say that a linear connection from this family is compatible with the conformal class (of the Riemannian metric) if the parallel transport along a geodesic of vectors perpendicular to the geodesic results in perpendicular vectors. Proposition 3. A linear connection from the family (8) is compatible with the conformal class if and only if there is a linear form β such that B(X, Y ) = β(X )Y − β(Y )X + B1 (X, Y ), B1 (X, Y ) is perpendicular to span{X, Y }, and α(Y ) − β(Y ) = E, Y . be a linear connection from the family (8) compatible with the conformal Proof. Let ∇ class and let γ (t) be one of its geodesics. We have for v = dγ dt , v v = ∇v v − v 2 E + 2α(v)v. 0=∇ Let further Y be a parallel vector field along γ (t) so that v Y = ∇v Y − v, Y E + α(v)Y + α(Y )v + B(v, Y ). 0=∇
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If Y is perpendicular to the geodesic then 0=
d v, Y = ∇v v, Y + v, ∇v Y = v 2 E, Y − v 2 α(Y ) − B(v, Y ), v dt
We get B(v, Y ), v = (E, Y − α(Y ))v 2 = −β(Y )v 2 ,
(9)
where β(Y ) = α(Y ) − E, Y is a linear form. We conclude that if the parallel transport takes perpendicular vectors into perpendicular vectors then (9) holds for any orthogonal vectors v, Y . Let us consider the antisymmetric tensor B1 (X, Y ) = B(X, Y ) − β(X )Y + β(Y )X . It follows from (9) that if X and Y are orthogonal then B1 (X, Y ), X = 0, and B1 (X, Y ), Y = −B1 (Y, X ), Y = 0. It follows that B1 (X, Y ) is orthogonal to span{X, Y } for any X and Y .The “only if” part of the proposition is proven. The other implication is straightforward. Guided by Proposition 3 we will restrict our attention to the family of linear connections X Y = ∇ X Y − X, Y E + E, Y X + E, X Y − γ (X )Y, ∇
(10)
where γ is a linear form. The fact that we dropped the antisymmetric tensor B1 comes from our inability to make an advantageous choice different from zero. In view of Proposition 2 all of these connections share the same parallel transport up X ∇ Y ∇ [X,Y ] have Y − ∇ X − ∇ to dilation and hence their curvature tensors R(X, Y) = ∇ the same antisymmetric part. Let us note that we are using a Riemannian metric to describe the family of connections. Let us examine the role of the conformal class of the metric. For a linear connecX Y = 0, that is tion (10) and a parallel field Y defined along a path we have ∇ ∇ X Y = X, Y E − E, Y X − E, X Y + γ (X )Y . This leads us to d Y, Y = 2∇ X Y, Y = 2 (γ (X ) − E, X ) Y, Y . (11) dt We have established that the parallel transport with respect to any of the connections X Y = ∇ X Y = ∇ X Y − is conformal. For γ = 0 we obtain the symmetric connection ∇ X, Y E + E, Y X + E, X Y . A linear symmetric connection with conformal parallel transport is called a Weyl connection [F]. The only connection in the family (10) with isometric parallel transport is obtained for the resulting metric connection. for γ (X ) = E, X . We will reserve the notation ∇ We are ready to prove Theorem 1. satisfies i). The property ii) Proof of Theorem 1. By Proposition 1 the connection ∇ follows from (11) and the property iii) is checked by direct calculation. To prove the converse we need to invoke again Proposition 1 to get the restriction to connections of the form (8). The torsion of the connection (8) is equal to 2B(X, Y ). If the connection is metric, that is with isometric parallel transport, then it is clearly compatible with the conformal class and hence is covered by Proposition 3. Now the property iii) implies that the tensor B1 from Proposition 3 vanishes, so that our connection belongs to the family (10). Finally by (11) the only connection of that family with isometric parallel transport is the one with the form γ dual to the vector field E.
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4. Gaussian Thermostats as Geodesic Flows and the Role of the Conformal Class of the Metric We have two ways of interpreting the Gaussian thermostat system (2) defined by a background metric g and a tangent vector field E. One way is to introduce the Weyl connection ∇. From the point of view of a Weyl connection there is nothing special about the background metric used in the definition of a Gaussian thermostat. Weyl connections are defined in terms of the conformal class of metrics, [F]. For a chosen metric g in the is uniquely determined by the linear form ϕ, conformal class the Weyl connection ∇ ϕ(X ) = E, X , i.e., ϕ is the dual of the vector field E. The defining property of the Weyl X g = −2ϕ(X )g. If we change the metric g to e−2U g we get that the connection is that ∇ Weyl connection defined by the pair (g, ϕ) is also defined by the pair (e−2U g, ϕ + dU ). as we change the metric and use it to parameterize the For a fixed Weyl connection ∇ Weyl geodesics by the arc length we get a family of Gaussian thermostats which are flow equivalent by the obvious identification of the respective unit tangent bundles, via the rescaling. Certain hyperbolic properties, e.g. Anosov property or dominated splitting, are shared by flow equivalent systems. Our new point of view is that the Gaussian thermostat defined by a pair (g, ϕ) is the (X, Y ) = ϕ(X )Y − with the torsion T geodesic flow of the unique metric connection ∇ ϕ(Y )X . The pairs (g, ϕ) and (e−2U g, ϕ + dU ) define flow equivalent systems. Indeed changes, so does the connection ∇ X Y = ∇ X Y − ϕ(X )Y . However by as the torsion T Proposition 1 all of these connections share the same geodesics, albeit parameterized differently by the respective arclength parameters. and any ∇ differ by dilations Moreover by Proposition 2 the parallel transports of ∇ only. The curvature operator of a linear connection represents infinitesimal parallel is equal to the antisymmetric part of the transport. Hence the curvature tensor of ∇ curvature tensor of the Weyl connection. In particular it does not change when we change the background metric g, in the conformal class, to e−2U g. However the sectional (). curvature of the new metric connection does change; it is obviously equal to e2U K We see that the negativity of the sectional curvature is the property of the Weyl connection alone and holds simultaneously for all the metric connections. It is consistent with Theorem 3 and the fact that the presence of a dominated splitting is not destroyed by the change of time in a flow. (X, Y ) = ϕ(X )Y − ϕ(Y )X we get the sectional torsion For the special torsion T 2 1 () = ϕ| . Hence as the metric g and the form ϕ change to e−2U g and ϕ + dU T 4 respectively, the sectional torsion changes to e2U
2 1 (ϕ + dU )| , 4
where the norm | · | is determined by the old Riemannian metric. These formulas allow the optimization of the sufficient condition for the geodesic flow to be Anosov from Theorem 3. Again the Anosov property is not affected by the parameterization of geodesics so it is enough to establish it for a conveniently chosen metric. As we change the metric in the conformal class the sum of the sectional curvature and the sectional torsion is equal to () + 1 (ϕ + dU )| 2 . e2U K 4
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2 Hence we would like to minimize (ϕ + dU )| over all possible functions U . One way to fine tune the function U is to use the Hodge theory. It allows the minimization of the L 2 norm of ϕ + dU over the whole manifold by the orthogonal projection of the form ϕ on the subspace of “divergence free” forms. If our original vector field E has zero divergence this optimization is void; we already have the optimal metric. Let us note that in general the resulting optimal form ϕ + dU does not have zero divergence with respect to the new Riemannian metric. That brings us to another way of optimization by requesting that the new form is divergence free with respect to the new metric. By the result of Gauduchon ([G]), it can be achieved on a compact manifold and there is a unique way to do it. Applications of Theorem 3 hinge on the understanding of sectional curvatures of Although the difference between the curvature tensors of the the metric connection ∇. metric connection and the Levi-Civita connection of the underlying metric contains many terms, the difference between the respective sectional curvatures simplifies to the following transparent formula in terms of the vector field E ([W1]), 2 () = K () − (E 2 − E K ) − ∇ X E, X − ∇Y E, Y ,
where for any orthonormal basis (X, Y ) of the plane the vector E = X, EX + Y, EY is the orthogonal projection of E on the plane and K () denotes the Gaussian sectional curvature of the background metric. We have also 2 (X, Y )2 = (E, Y X − E, X Y )2 = E . T
In particular we see that in dimension 2 the curvature of a metric connection is negative if and only if the respective Gauduchon metric has negative curvature. These formulas allow to obtain the Anosov property for the Gaussian thermostats with divergence free fields E on surfaces of negative curvature, studied in [B-G-M]. They were also used to study Weyl connections with nonpositive sectional curvatures on tori, [W4]. 5. The Torsion and the Hamiltonian Formulation The torsion comes in an interesting way into the generalized hamiltonian formulation of the problem, [W-L, W4]. Using the background metric g we will identify a tangent space to T M at (u, v) with Tu M ⊕ Tu M. Namely for a tangent vector defined by a parameterized curve (u(a), v(a)), u(a) ∈ M, v(a) ∈ Tu(a) M, |a| < , we use (ξ, η) ∈ Tu(0) M ⊕ Tu(0) M, ξ=
du , η = ∇ξ v|a=0 , da |a=0
as coordinates. 2 Take the hamiltonian function H = v2 on T M and the symplectic form on T M given by
ω (ξ1 , η1 ), (ξ2 , η2 ) = η1 , ξ2 − η2 , ξ1 . We are looking for an antisymmetric 2-form γ on T M, such that the Gaussian thermostat (2) becomes the generalized hamiltonian flow with respect to the form = ω−γ . More precisely we want the vector field − → H (u, v) = v, v 2 E − E, vv
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to satisfy − → ( H , ·) = −d H (·). It turns out that γ is given by
(ξ1 , ξ2 ), v . γ (ξ1 , η1 ), (ξ2 , η2 ) = T Indeed: −d H (ξ, η) = −η, v and − → − → − → ( H , (ξ, η)) = ω( H , (ξ, η)) − γ ( H , (ξ, η))
= v 2 E − E, vv, ξ − η, v − E, ξ v − E, vξ, v = −η, v. The form is always nondegenerate. It was shown in [W-L] that it is conformally symplectic if ϕ is closed. Acknowledgements. The second author acknowledges very useful conversations with Tomasz Maszczyk, and the hospitality of IMPAN in Warsaw, Poland.
References [A-P-S] Ambrose, W., Palais, R.S., Singer, I.M.: Sprays. Acad. Brasileira de Ciencas 32, 163–178 (1960) [B-G-M] Bonetto, F., Gentile, G., Mastropietro, V.: Electric fields on a surface of constant negative curvature. Erg. Th. Dynam. Sys. 20(3), 681–696 (2000) [D-P] Dairbekov, N.S., Paternain, G.P.: Entropy production in gaussian thermostats. Commun. Math. Phys. 269(2), 533–543 (2007) [F] Folland, G.B.: Weyl manifolds. J. Diff. Geom. 4, 145–153 (1970) [G] Gauduchon, P.: Fibrés hermitiens avec endomorphisme de Ricci nonnegatif. Bull. Soc. Math. France 105, 113–140 (1970) [G-R] Gallavotti, G., Ruelle, D.: SRB states and nonequilibrium statistical mechanics close to equilibrium. Commun. Math. Phys. 190, 279–285 (1997) [M] Mañé, R.: Oseledec’s theorem from the generic viewpoint. Proc. ICM Warsaw 1983, Warsaw:PWN, 1984, pp. 1269–1276 [W-L] Wojtkowski, M.P., Liverani, C.: Conformally symplectic dynamics and symmetry of the lyapunov spectrum. Commun. Math. Phys. 194, 47–60 (1998) [W1] Wojtkowski, M.P.: W-flows on weyl manifolds and gaussian thermostats. J. Math. Pures Appl. 79(10), 953–974 (2000) [W2] Wojtkowski, M.P.: Weyl manifolds and gausssian thermostats. In: Proc. ICM Beijing 2002, Beijing: Higher Education Press, 2003, pp. 3511–3523 [W3] Wojtkowski, M.P.: Monotonicity, J -algebra of Potapov and lyapunov exponents. In: Smooth Ergodic Theory and its Applications, Proc. Symp. Pure Math. 69, Providence, RI: Amer. Math. Soc., 2001, pp 499–521 [W4] Wojtkowski, M.P.: Magnetic flows and gaussian thermostats. Fund. Math. 163, 177–191 (2000) [W5] Wojtkowski, M.P.: Rigidity of some weyl manifolds with nonpositive sectional curvature. PAMS 133, 3395–3402 (2005) Communicated by G. Gallavotti
Commun. Math. Phys. 277, 771–819 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0383-3
Communications in
Mathematical Physics
Topological Strings and (Almost) Modular Forms Mina Aganagic1 , Vincent Bouchard2 , Albrecht Klemm3 1 Department of Mathematics, University of California, Berkeley, CA 94720, USA.
E-mail: [email protected]
2 Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720, USA 3 Department of Physics, University of Wisconsin, Madison, WI 53706, USA
Received: 12 February 2007 / Accepted: 8 June 2007 Published online: 22 November 2007 – © Springer-Verlag 2007
Abstract: The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group , generated by monodromies of the periods of X . This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H 3 (X ). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under . Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP2 and IP1 × IP1 . As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold C3 /ZZ3 . 1. Introduction Topological string theory has led to many insights in both physics and mathematics. Physically, it computes non-perturbative F-terms of effective supersymmetric gauge and gravity theories in string compactifications. Moreover, many dualities of superstring theory are better understood in terms of topological strings. Mathematically, the A-model explores the symplectic geometry and can be written in terms of GromovWitten, Donaldson-Thomas or Gopakumar-Vafa invariants, while the mirror B-model depends on the complex structure deformations and usually provides a more effective tool for calculations. The topological string is well understood for non-compact toric Calabi-Yau manifolds. For example, the B-model on all non-compact toric Calabi-Yau manifolds was solved to all genera in [1] using the W∞ symmetries of the theory. Geometrically, the W∞ symmetries are the ω-preserving diffeomorphisms of the Calabi-Yau manifold, where ω is the (3, 0) holomorphic volume form. By contrast, for compact Calabi-Yau manifolds the genus expansion of the topological string is much harder to compute and so far only
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known up to genus four in certain cases, for instance for the quintic Calabi-Yau threefold. It is natural to think that understanding quantum symmetries of the theory may hold the key in the compact case as well. In this paper, we will not deal with the full diffeomorphism group, but we will ask how does the finite subgroup of large, ω-preserving diffeomorphisms, constrain the amplitudes. In other words, we ask: what can we learn from the study of the group of symmetries generated by monodromies of the periods of the Calabi-Yau? For this, we need to know how acts in the quantum theory. The remarkable fact about the topolog 2g−2 ical string is that its partition function Z = exp( g gs Fg ) is a wave function in a Hilbert space obtained by quantizing H3 (X ), where gs2 plays the role of h¯ .1 Classically, acts on H3 (X ) as a discrete subgroup of the group Sp(2n, ZZ) of symmetries that preserve the symplectic form, where n = 21 b3 (X ). This has a natural lift to the quantum theory. The answer turns out to be beautiful. Namely, the Fg ’s turn out to be (almost) modular forms of . By “(almost) modular form” we mean one of two things: a form which is holomorphic, but quasi-modular (i.e. it transforms with shifts), or a form which is modular, but not quite holomorphic. By studying monodromy transformations of the topological string partition function in “real polarization”, where Z depends holomorphically on the moduli space, we find that it is a quasi-modular form of of weight 0. The symmetry transformations under imply that the genus g partition function Fg is fixed r ecur sively in terms of lower genus data, up to the addition of a holomorphic modular form. Thus, modular invariance constrains the wave function, but does not determine it uniquely. The holomorphic modular form that is picked out by the topological string can be deduced (at least in principle) by its behavior at the boundaries on the moduli space. On the other hand, if we consider the topological string partition function in “holomorphic polarization”, this turns out to be a modular form of weight 0, which is not holomorphic on the moduli space. While it fails to be holomorphic, it turns out to be “almost holomorphic” in a precise sense. Moreover, it is again determined recursively, up to the holomorphic modular form. Thus, the price to pay for insisting on holomorphicity is that the Fg ’s fail to be precisely modular, and the price of modularity is failure of holomorphicity! The recursive relations we obtain contain exactly the same information as what was extracted in [6] from the holomorphic anomaly equation. In [6], through a beautiful study of topological sigma models coupled to gravity, the authors extracted a set of equations that the genus g partition function Fg satisfies, expressing an anomaly in holomorphicity of Fg . The equations turn out to fix Fg in terms of lower genus data, up to an holomorphic function with a finite set of undetermined coefficients. Here, we have formulated the solutions to the holomorphic anomaly equation by exploiting the underlying symmetry of the theory. In the context of [6], solving the equations was laborious, the particularly difficult part being the construction of certain “propagators”. From our perspective, the propagators are simply the “generators” of (almost) modular forms, that is the analogues of the second Eisenstein series of S L(2, ZZ) and its non-holomorphic counterpart! That a reinterpretation of [6] in the language of (almost) modular forms should exist was anticipated by R. Dijkgraaf in [13]. For local Calabi-Yau manifolds, the relevant modular forms are Siegel modular forms. In the compact Calabi-Yau manifold case, our formalism seems to predict the existence of a new theory of modular forms of 1 This fact was also recently explored in [18,14,31,39].
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(subgroups of) Sp(2n, ZZ), defined on spaces with Lorentzian signature (instead of the usual Siegel upper half-space). The paper is structured as follows. In Sect. 2, we describe the B-model topological string theory, from a wave function perspective, for both compact and non-compact target spaces. In Sect. 3, we take a first look at how the topological string wave function behaves under the symmetry group generated by the monodromies. Then, we give a more precise analysis of the resulting constraints on the wave function in Sect. 4. We also explain the close relationship between the topological string amplitudes and (almost) modular forms in this section. In the remaining sections we give examples of our formalism: in Sect. 5 we study SU (N ) Seiberg-Witten theory, in Sect. 6 local IP2 — where we also use the wave function formalism to extract the Gromov-Witten invariants of the orbifold C3 /ZZ3 , aand in Sect. 7 local IP1 × IP1 . To conclude our work, in Sect. 8 we present some open questions, speculations and ideas for future research. Finally, Appendix A and B are devoted to a review of essential facts and conventions about modular forms, quasi-modular forms and Siegel modular forms. 2. B-model and the Quantum Geometry of H 3 (X,C) The B-model topological string on a Calabi-Yau manifold X can be obtained by a particular topological twisting of the “physical” string theory, two-dimensional (2, 2) supersymmetric sigma model on X coupled to gravity. The genus zero partition function of the B-model F0 is determined by the variations of complex structures on X . The higher genus amplitudes Fg>0 can be thought of as quantizing this. When X has a mirror Y , this is dual to the A-model topological string, which is the Gromov-Witten theory of Y , obtained by an A-type twist of the physical theory on Y . As is often the case, many properties of the theory become transparent when the moduli of X and Y are allowed to vary, and the global structure of the fibration of the theory over its moduli space is considered. This is quite hard to do in the A-model directly, but the mirror B-model is ideally suited for these types of questions. 2.1. real Polarization. Let us first recall the classical geometry of H 3 (X,C) = H 3 (X, ZZ) ⊗ C. In the following, we will assume that X is a compact Calabi-Yau manifold, and later explain the modifications that ensue in the non-compact, local case. Choose a complex structure on X by picking a particular 3-form ω in H 3 (X,C). Any other 3-form differing from this by a multiplication by a non-zero complex number determines the same complex structure. The set of (3, 0)-forms is a line bundle L over the moduli space M of complex structures. Given a symplectic basis of H3 (X, ZZ), A I ∩ B J = δ JI , where I, J = 1, . . . , n, and n = 21 b3 (X ), we can parameterize the choices of complex structures by the periods xI = ω, pI = ω. AI
BI
The periods are not independent, but satisfy the special geometry relation: p I (x) =
∂ F0 (x). ∂x I
(2.1)
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As is well known, F0 turns out to be given in terms of the classical, genus zero, free energy of the topological strings on X . In the above, we picked a symplectic basis of H3 . Different choices of symplectic basis differ by Sp(2n, ZZ) transformations: p˜ I = A I J p J + B I J x J , x˜ I = C I J p J + D I J x J , where
B ∈ Sp(2n, ZZ). D
A C
M=
(2.2)
For future reference, note that the period matrix τ , defined by τI J =
∂ pI ∂x J
transforms as τ˜ = (Aτ + B)(Cτ + D)−1 .
(2.3)
For a discrete subgroup ⊂ Sp(2n, ZZ), the changes of basis can be undone by picking a different 3-form ω. Conversely, we should identify the choices of complex structure that are related by changes of basis of H3 (X, ZZ). The x’s can be viewed as projective coordinates on the Teichmuller space T of X , on which acts as the mapping class group. Consequently, the space of inequivalent complex structures is M = T / . Generically, the moduli space M has singularities in complex codimension one, and is generated by monodromies around the singular loci. It is natural to think of H 3 (X, ZZ) as a classical phase space, with symplectic form, d x I ∧ dp I , and (2.1) as giving a lagrangian inside it. In fact, the analogy is precise. As shown in [40], in the quantum theory x I and p J become canonically conjugate operators [ p I , x J ] = gs2 δ IJ ,
(2.4)
where gs2 plays the role of h¯ , and the topological string partition function χ
Z (x I ) = gs24
−1
exp [
∞
2g−2
gs
Fg (x I )],
(2.5)
g=0
where Fg is the genus g free energy of the topological string, becomes a wave function. More precisely, the B-model topological string theory determines a par ticular state |Z in the Hilbert space obtained by quantizing H 3 (X, ZZ). The wave function, x I |Z = Z (x I )
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describes the topological string partition function in one, “real” polarization2 of H 3 (X ). The semi-classical, genus zero approximation to the topological string wave function is determined by the classical geometry of X , and the lagrangian (2.1): p I Z (x) = gs2
∂ ∂ Z (x) ∼ ( I F0 ) Z (x). ∂x I ∂x
The lagrangian does not determine the full quantum wave function. In general, there are normal ordering ambiguities, and to resolve them, the full topological B-model string theory is needed.3 The partition function Z implicitly depends on the choice of symplectic basis. Classically, changes of basis ( p, x) → ( p, ˜ x) ˜ which preserve the symplectic form are canonical transformations of the phase space. For the transformation in (2.2), the corresponding generating function S(x, x) ˜ that satisfies d S = p I d x I − p˜ I d x˜ I
(2.6)
1 1 S(x, x) ˜ = − (C −1 D) J K x J x K + (C −1 ) J K x J x˜ K − (AC −1 ) J K x˜ J x˜ K . 2 2
(2.7)
is given by4
This has an unambiguous lift to the quantum theory, with the wave function transforming as5 ˜ s2 Z (x). (2.8) Z˜ (x) ˜ = d x e−S(x,x)/g We should specify the contour used to define (2.8); however, as long as we work with the perturbative gs2 expansion of Z (x), the choice of contour does not enter. To make sense of (2.8) then, consider the saddle point expansion of the integral. Given x˜ I , the saddle point of the integral x I = xclI solves the classical special geometry relations that follow from (2.2): ∂S |x = p I (xcl ). ∂ x I cl Expanding around the saddle point, and putting x I = xclI + y I , we can compute the integral over y by summing Feynman diagrams where I J = −(τ + C −1 D) I J
(2.9)
2 For us, ω naturally lives in the complexification H 3 (X,C) =C ⊗ H 3 (X, IR), so “real” polarization is a bit of a misnomer. 3 Note that due to (2.4), g is a section of L, so that F is a section of L2−2g . The full partition function s g χ
is a section of L 24 −1 , where χ is the Euler characteristic of the Calabi-Yau, due to the prefactor. 4 Note that (2.6) only defines S up to an addition of a constant on the moduli space. This ambiguity can be absorbed in F1 , since only derivatives of it are physical anyhow. 5 It is important to note that this makes sense only on the large phase space, where the integral is over the n-dimensional space spanned by the x I ’s. In particular, the choice of section of L does not enter.
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Fig. 1. Pictorial representation of the Feynman expansion at genus 2 in terms of degenerations of Riemann surfaces
is the inverse propagator, and derivatives of Fg , ∂ I1 . . . ∂ In Fg (xcl ),
(2.10)
the vertices. As a short hand we summarize the saddle point expansion by F˜ g = Fg + g ( I J , ∂ I1 . . . ∂ In Fr
1 log det(−), 2
where by we mean the propagator I J in matrix form. At genus two one has 1 1 IJ IJ ∂ I ∂ J F1 + ∂ I F1 ∂ J F1 2 , ∂ I1 . . . ∂ In Fr <2 = 2 2 1 1 IJ KL ∂ I F1 ∂ J ∂ K ∂ L F0 + ∂ I ∂ J ∂ K ∂ L F0 + 2 8 1 ∂ I ∂ J ∂ K F0 ∂ L ∂ M ∂ N F0 + I J K L M N 8 1 (2.11) ∂ I ∂ K ∂ M F0 ∂ J ∂ L ∂ N F0 , + 12 where we suppressed the argument xcl for clarity. It is easy to see from the path integral that this describes all possible degenerations of a Riemann surface of genus g to “stable” curves of lower genera, with I J being the corresponding contact term, as shown in the figure below. Stable here means that the conformal Killing vectors were removed by adding punctures, so that every genus zero component has at least three punctures, and every genus one curve, one puncture.6 6 Note that in particular this implies that at each genus, the equations are independent of the choice of section of L we made, the left and the right-hand side transforming in the same way.
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Mirror symmetry and Gromov-Witten theory picks out the real polarization which is natural at large radius where instanton corrections are suppressed, and where the classical geometry makes sense. However, also by mirror symmetry, there is a larger family of topological A-model theories which exist, though they may not have an interpretation as counting curves. For a generic element M of Sp(2n, ZZ), (2.8) simply takes one polarization into another. However, for M in the mapping class group ⊂ Sp(2n, ZZ), the transformation (2.8) should translate into a constraint on Fg , since is a group of symmetries of the theory. We will explore the consequences of this in the rest of this paper. 2.2. Holomorphic polarization. Instead of picking a symplectic basis of H3 (X ) to parameterize the variations of complex structure on X , we can choose a fixed background complex structure ∈ H 3 (X,C), and use it to define the Hodge decomposition of H 3 (X,C): H 3 = H 3,0 ⊕ H 2,1 ⊕ H 1,2 ⊕ H 0,3 . Here is the unique H 3,0 form and the Di ’s span the space of H 2,1 forms, where ¯ This implies that: Di = ∂i − ∂i K and K is the Kähler potential K = log[i X ∧ ]. ¯ + ϕ¯ , ¯ ω = ϕ + z i Di + z¯ i D¯ i
(2.12)
where (ϕ, z i ), and (ϕ, ¯ z¯ i ) become coordinates on the phase space.7 Correspondingly we can express |Z as a wave function in holomorphic polarization z i , ϕ|Z = Z (z i , ϕ). The topological string partition function Z (z i , ϕ) depends on the choice of background , and this dependence is not holomorphic. This is the holomorphic anomaly of [6]. One way to see this is through geometric quantization of H 3 (X ) in this polarization [40] . We will take a different route, and exhibit this by exploring the canonical transformation from real to holomorphic polarizations. Using special geometry relations it is easy to see that I x = ω = z I + c.c, AI pI = ω = τ I J z J + c.c, BI
where we defined z I = ϕ X I + z i Di X I in terms of
X =
I
AI
,
PI =
, BI
7 Since ω for us does not live in H 3 (X, IR), but rather in H 3 (X,C), ϕ¯ and z¯ i are not honest complex conjugates of ϕ, z i .
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and where τI J =
∂ PJ . ∂XI
From this it easily follows that dp I ∧ d x I = (τ − τ¯ ) I J dz I ∧ d z¯ J and hence the canonical transformation from (x I , p I ) to (z I , z¯ I ) is generated by ˆ d S(x, z) = p I d x I + (τ − τ¯ ) I J z¯ I dz J . This corresponds to 1 1 ˆ S(x, z) = τ¯I J x I x J + x I (τ − τ¯ ) I J z J − z I (τ − τ¯ ) I J z J + c, 2 2 where c is a constant, but which can now depend on the background. In the quantum theory, this implies that the topological string partition function in the holomorphic polarization is related to that in real polarization by: 2 ˆ Zˆ (z; t, t¯) = d x e− S(x,z)/gs Z (x), (2.13) where t i are local coordinates on the moduli space, parameterizing the choice of background, i.e. X I = X I (t). Note that all the background dependence of Zˆ (z) comes from ˆ 8 Let the kernel of S. χ 1 − 1 log(gs ), (2.14) c(X, X¯ ) = −F1 (X ) − log[det(τ − τ¯ )](X, X¯ ) − 2 24 where χ is the Euler characteristic of the Calabi-Yau. Consider now the perturbative expansion of the integral. For simplicity, let us pick ϕ = 1,
z i = 0,
so that z I = X I . The saddle point equation, which can be written as9 (τ¯ (X ) − τ (xcl )) I J xclJ + (τ (X ) − τ¯ ( X¯ )) I J z J = 0, has then a simple solution, xclI = X I . Expanding around this solution,10 we can compute the integral by summing Feynman diagrams where − (τ (X ) − τ¯ ( X¯ )) I J
(2.15)
8 In what follows, we will use hats to label quantities which are not holomorphic. 9 We used here the special geometry relation p = τ x J . I IJ 10 It should now be clear why (2.14) is natural. The above normalization of the integral ensures that Zˆ con-
tains no one loop term without insertions (the vanishing of genus zero terms with zero, one and two insertions is automatic in the saddle point expansion).
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is the inverse propagator, and derivatives of Fg , ∂ I1 . . . ∂ In Fg (X ), the vertices. That is, we get I J Fˆ g (t, t¯) = Fg (X ) + g − (τ − τ¯ )−1 , ∂ I1 . . . ∂ In Fr
(2.16)
where the properties of the functionals g obtained by the Feynman graph expansion have been discussed in the previous section. Finally, one can show [39] that Zˆ satisfies the holomorphic anomaly equations of [6]. Differentiating the left and the right-hand side of (2.13) with respect to t¯ we get ∂ ˆ ∂ ˆ gs2 ¯ jk ∂ 2 ] Z. + G i¯ j z j Z = [ Ci¯ ¯ i j 2 ∂z ∂z ∂ϕ ∂ t¯i In the above equation, Ci jk is the amplitude at genus zero with three punctures, G i¯ j is ¯ ¯ jk the Kähler metric, and C¯ = e2K C¯ ¯ ¯ ¯ G j j G kk . It also satisfies the second holomorphic anomaly equation11 [
∂ ∂t i¯
+ ∂i K (z j
i¯
i jk
∂ ∂ ∂ χ 1 − ϕ )] Zˆ = [ϕ i − ∂i Fˆ 1 − ( − 1)∂i K − 2 Ci jk z j z k ] Zˆ . j ∂z ∂ϕ ∂z 24 2gs
The second anomaly equation implies that Zˆ has the form Zˆ (ϕ, z; t, t¯) = exp(
1 2g−2 (n) χ Fˆ g;i1 ,...in z i1 . . . z in ϕ 2−2g−n − ( − 1) log ϕ), gs n! 24 g,n
(n) where Fˆ g;i = Di1 . . . Din Fˆ g for 2g − 2 + n > 0, and zero otherwise, for some Fˆ g ’s, 1 ,...i n a fact that we will need later. The holomorphic polarization, as explained in [6,40] is the natural polarization of the topological string theory, in the following sense. The topological string is obtained by twisting a physical string on the Calabi-Yau at some point in the moduli space. The physical string theory naturally depends not only on X , but also on X¯ , so the space of physical theories is labeled by (X, X¯ ). After twisting, it is natural to deform by purely topological observables which are in one-to-one correspondence with the h 2,1 moduli— we have parameterized the resulting deformations by z i above. While one would naively expect the topological theory to depend only on z, this fails and the theory depends on the background (X, X¯ ) that we used to define it as well. 11 We used here the explicit form of Fˆ from [6], from which follows that ∂ Fˆ + ( χ − 1)∂ K = ∂ F − 1 i 1 i i 1 24
1 2 ∂i log(τ − τ¯ ).
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2.3. Local Calabi-Yau manifolds. In the previous subsections we assumed that the Calabi-Yau X is compact. In this subsection we explain the modifications required in the local case. We can derive the results of this section by viewing the B-model on a local Calabi-Yau simply as a limit of the compact one. This is the perspective that was taken in [10,23]. Since today, there is now far more known about the topological string in the local than in the compact case, it is natural to work directly in the language of local Calabi-Yau manifolds. For a string theory on a non-compact Calabi-Yau manifold, gravity decouples. As a consequence, the moduli space is governed by rigid special geometry, and not local special geometry as in the compact Calabi-Yau case. The partition functions are no longer sections of powers of line bundle L; the latter disappears altogether. Consider the local Calabi-Yau manifold given by the equation X : uw = H (y, z)
(2.17)
in C4 . This has a holomorphic three-form ω given by ω=
du ∧ dy ∧ dz. u
(2.18)
The Calabi-Yau can be viewed as a C∗ fibration over the y − z plane where a generic fiber is given by uw = const. It is easy to see that the 3-cycles on X descend to 1-cycles on a Riemann surface given by
: 0 = H (y, z), and, moreover, that the periods of the holomorphic three-form ω on X descend to the periods of a meromorphic 1-form λ on , ω= λ, 3−cycle
1−cycle
where λ = ydz. On a genus g Riemann surface there are 2g compact 1-cycles that form a symplectic basis,12 i = 1, . . . , g, Ai ∩ B j = δ ij . Let
x =
i
Ai
λ,
pi =
λ; Bi
the x i ’s are the nor mali zable moduli of the Calabi-Yau manifold. However, since the Calabi-Yau is non-compact, H (y, z) may depend on additional parameters which are 12 This is a slight over-simplification. Since the Riemann surface is non-compact, it can happen that one cannot find compact representatives of the homology satisfying this, and that instead one has to work with Ai ∩ B j = n ij , with n ij integral. We will see examples of this in the later sections. Since it is very easy to see how this modifies the discussion of this section, we will not do this explicitly, but assume the simpler case for clarity of presentation.
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non-normalizable complex structure moduli s α . Corresponding to these, there are compact 3-cycles Cα in H3 (X ) and 1-cycles on such that λ. sα = Cα
But, since the homology dual cycles to the C α are non-compact, the metric on the moduli space along the corresponding directions will not be normalizable. As a consequence, the s α are parameters of the model, not moduli. This implies that the monodromy group corresponds to elements of the form p˜ i = Ai j p j + Bi j x j + E iα s α , x˜ i = C i j p j + D i j x j + F i α s α ,
(2.19)
where s α , being parameters which do not vary, are monodromy invariant. Since preserves the symplectic form d x i ∧ dpi , we have that
A C
B ∈ Sp(2g, ZZ). D
Note that, while pi and x j transform in a somewhat unconventional way, the period matrix τi j =
∂ pi ∂x j
transforms as usual: τ˜ = (Aτ + B)(Cτ + D)−1 . The corresponding generator of canonical transformations is easily found to be 1 1 S(x, x) ˜ = − (C −1 D) jk x j x k + (C −1 ) jk x j x˜ k − (AC −1 ) jk x˜ j x˜ k 2 2 j i α i α + Ci−1 x F s − E x ˜ s . iα α j
(2.20)
In the quantum theory, once again x i and p j are promoted to operators with canonical commutation relations [x i , p j ] = gs2 δ ij . The B-model determines a state |Z , and a wave function Z (x i ) = x i |Z . The wave function depends on the choice of real polarization, the different polarization choices being related in the usual way: ˜ s2 Z˜ (x) ˜ = d x e−S(x,x)/g Z (x). (2.21)
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Computing the path integral, in the saddle point expansion around (2.19), we find that i j = −(τ + C −1 D)i j
(2.22)
is the inverse propagator, and derivatives of Fg , ∂i1 . . . ∂in Fg (xcl ),
(2.23)
the vertices. This implies that F˜ g = Fg + g (i j , ∂i1 . . . ∂in Fr
, Pi =
, s =
. Ai
Bi
Cα
It is easy to see that d x i ∧ dpi = (τi j − τ¯i j )d Z¯ i ∧ d Z j , where τi j (X ) = ∂ Pi /∂ X j depends on the background and we put Zi = z j∂j Xi. The corresponding canonical transformation is easily found: 1 1 ˆ S(x, z) = τ¯i j (x − X )i (x − X ) j + (τ − τ¯ )i j Z i (x − X ) j − (τ − τ¯ )i j Z i Z j + Pi x i . 2 2 The wave functions in holomorphic and real polarizations are now simply related by 2 ˆ ˆ (2.24) Z (z) = d x e− S(x,z)/gs Z (x) The saddle point equation reads τ¯ ( X¯ )i j (xcl − X ) j + (τ (X ) − τ¯ ( X¯ ))i j Z j − ( p − P)i = 0,
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and if we put z i = 0, which corresponds to Z vanishing, it has a simple solution: xcli = X i . Expanding around this, we get a Feynman graph expansion with inverse propagator −(τ (X ) − τ¯ ( X¯ ))i j and derivatives of Fg (X ) as vertices. This gives the by now familiar expansion relating the partition functions in holomorphic and real polarizations: i j Fˆ g (t, t¯) = Fg (t) + g − (τ − τ¯ )−1 , ∂i1 . . . ∂in Fr
(2.25)
Before we go on, it is worth noting that the wave function in holomorphic polarization satisfies a set of differential equations, expressing the dependence of Zˆ on the background — the local holomor phic anomaly equations. These can be derived easily by differentiating both the left and the right-hand side of (2.24) with respect to t¯ (here, t i is the local coordinate parameterizing the choice of background, X = X (t)). This is straightforward, we state here only the answer: ∂2 ˆ ∂ ˆ 1 jk Z = gs2 C¯ i¯ Z, ¯ 2 ∂z j ∂z k ∂ t¯i
(2.26)
¯
where indices are raised by the inverse g i j of the Kähler metric on the moduli space gi j¯ = ∂i X k (τ − τ¯ )k ∂¯ j¯ X¯ . In summary, apart from a few subtleties, the quantum mechanics of the compact and local Calabi-Yau manifolds are analogous. In the following section we will use the language of the compact theory, but everything we will say will go over, without modifications, to the non-compact case as well.
3. A First Look at the Action In this section we take a first look at how topological string amplitudes behave under monodromies. On general grounds, is a group of symmetries of the physical string theory. This implies that the state |Z in the Hilbert space that the topological string partition function determines should be invariant under monodromies. The associated wave functions, however, need not be. By definition, the wave function in real polarization requires a choice of symplectic basis of H3 on which acts nontrivially; thus, it cannot be monodromy invariant. By contrast, the wave function in the holomorphic polarization is the physical partition function. It is a well defined function13 all over the moduli space; however, it is not holomorphic. 13 We are assuming a definite choice of gauge, throughout. Of course, changing the gauge, the amplitudes transform as sections of the apropriate powers of L.
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3.1. The wave function in real polarization. Given a symplectic basis {A I , B I }, I = 1, . . . , n of H3 (X, ZZ), with n = 21 b3 , a pick a definite 3-form ω in H 3 (X,C). The topological string partition function determines a wave function Z (x I ) = x I |Z , where
xI =
AI
ω,
and a corresponding state |Z in the Hilbert space obtained by quantizing H 3 (X,C). Having picked a definite section ω of the line bundle L, x I ’s and Z (x I ) are at least locally, functions on the moduli space x I = x I (ψ), where the n −1 variables ψ i are some arbitrary local coordinates on M. For definiteness, we take here the Calabi-Yau manifold to be compact, but everything carries over to the non-compact space as well, the only real modification being that there the moduli space would have dimension n, instead. The moduli space M has singular loci in complex codimension 1 around which the cycles A I , B J undergo monodromies in . As one goes around the singular locus, by sending ψ ψ → γ · ψ, for γ an element of , the periods transform as pI pI pI (ψ) → (γ · ψ) = Mγ (ψ), xI xI xI where Mγ is a symplectic matrix corresponding to γ . What happens in the quantum theory? The monodromy group is a symmetry of the theory, so the state |Z determined by the topological string partition function should be invariant under it: |Z → |Z . The state x(ψ)|, by contrast, is not invariant. There are two ways to express what happens to x| under monodromies. On the one hand, x I is a function of ψ, so we get a purely classical variation of the ket vector, x(ψ)| → x(γ · ψ)|. But on the other hand, we have seen in Sect. 2 that any element Mγ ∈ Sp(2n, ZZ) acting classically on the period vector has a unique lift to the quantum theory as an operator Uγ that acts on the Hilbert space. In particular, x(γ · ψ)| = x(ψ)| Uγ . Putting these facts together implies that x(γ · ψ)|Z = x(ψ)| Uγ |Z ,
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or, schematically in terms of wave functions, Z (x(γ · ψ)) = e Sγ Z (x(ψ)),
(3.1)
where exp(Sγ ) computes the corresponding matrix element of Uγ . There is one such equation for each monodromy transformation g and its corresponding element Mγ ∈ . Thus, the symmetry group imposes the constraints (3.1) on Z , one for each generator. Using the results of Sect. 2, Eq. (3.1) implies constraints on the free energy, genus by genus. For example, (3.1) implies that the free energies satisfy14 IJ (3.2) Fg (x(γ · ψ)) = Fg (x(ψ)) + g M , ∂ . . . ∂ F I I r
Mγ =
A C
B . D
(3.3)
(3.4)
To summarize, non-trivial monodromy (with det(C) = 0) around a point in the moduli space corresponds to choosing A-cycles which are not well defined there, but instead transform by x I → C I J pJ + D I J x J . This leads to an obstruction to analytic continuation of the amplitudes all over the moduli space. It also lead us to the notion of “good variables” in the moduli space, which are implicit in Gromov-Witten computations: near a point in the moduli space, the “good” variables are those with no non-trivial monodromy, meaning that C I J = 0. 3.2. Another perspective. Consider instead the wave function in holomorphic polarization. Pick a background complex structure , and write ω as in (2.12), ¯ + ϕ¯ . ¯ ω = ϕ + z i Di + z¯ i D¯ i Using ϕ and z i as coordinates, we can write |Z as a wave function in holomorphic polarization Zˆ (ϕ, z i ) = ϕ, z i |Z . Note that z i are coordinates on M, centered at . How does Z (ϕ, z i ) transform under ? In real polarization, the non-trivial transformation law of the wave function came about from having to pick a basis of periods x I |, which were not invariant under . In writing down the wave function in holomorphic polarization, that is in defining ϕ, z i |, we made no reference to the periods, so Zˆ (ϕ, z i ) 14 It is important to emphasize that this does not depend on the choice of section either. We could have written here simply x I (ψ) = x I and x I (γ · ψ) = C I J p J (x) + D IJ x J .
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has to be invariant. There is another, independent reason why this has to be so. Namely, Z (ϕ, z i ) is the physical wave function everywhere on M and as such, it better be well defined everywhere! We have seen above that the wave function in real polarization has rather complicated monodromy transformations under , while the wave function in holomorphic polarization is invariant. Since the two polarizations are related in a simple way, we could have derived the transformation properties of one from that of the other. Consider for example the genus two amplitudes in (2.16) for a compact Calabi-Yau, and in (2.25) for a non-compact one. While on the left-hand side Fˆ 2 is manifestly invariant under , on the right-hand side all the ingredients have non-trivial monodromy transformations. In fact, we have
(τ − τ¯ )−1
I J
K L → (Cτ + D) I K (Cτ + D) J L (τ − τ¯ )−1 − C I L (Cτ + D) J L , (3.5)
where C, D enter M as in (3.4), and analogously in the local case. These quasi-modular transformations of (τ − τ¯ )−1 must precisely cancel the transformations of the genus zero, one and two amplitudes in real polarization. We will come back to this in the next section. 4. Topological Strings and Modular Forms In the previous section we took a first look at how the topological string partition functions transform under . In this section we give a simple and precise description of how, and to which extent, the discrete symmetry group can constrain the topological string amplitudes. Along the way, we will discover a close relationship of topological string partition functions and modular forms. On the one hand, we have seen in the previous sections that the partition function in holomorphic polarization satisfies i. Fˆ g (x, x) ¯ is invariant under —that is, it is a modular form of of weight zero. ˆ ii. Fg (x, x) ¯ is “almost” holomorphic—its anti-holomorphic dependence can be summarized in a finite power series in (τ − τ¯ )−1 . On the other hand, the topological string partition function in r eal polarization satisfies iii. Fg (x) is holomorphic, but not modular in the usual sense. iv. Fg (x) is the constant part of the series expansion of Fˆ g (x, x) ¯ in (τ − τ¯ )−1 . Forms of this type were considered by Kaneko and Zagier [23].15 In [23] forms satisfying i. and ii. (with arbitrary weight) are called almost holomorphic modular forms of . Moreover, for every almost holomorphic modular form, [23] defines the associated quasi-modular form as that satisfying iii. and iv. These are holomorphic forms which are not modular in the usual sense. This suggests that the genus g amplitudes are in fact naturally (almost) modular functions of τ (and τ¯ in holomorphic polarization), which can be extended from functions on the moduli space M of complex structures to the space H X parameterized by the period matrix τ I J on X modulo . In the following, we 15 To be precise, [23] considers only modular forms of S L(2, Z Z). However, this has an obvious generalizaZ). tion, at least in principle, to (subgroups of) Sp(2n, Z
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will mainly study this in the local Calabi-Yau examples, and show that this indeed is the case, leaving compact Calabi-Yau manifolds for future work. Now, take a holomorphic, quasi-modular form E I J (τ ) of , such that I J (4.1) Eˆ I J (τ, τ¯ ) = E I J (τ ) + (τ − τ¯ )−1 is a modular form, albeit an almost holomorphic one. Since (τ − τ¯ )−1 transforms under as in (3.5), for Eˆ I J to be modular, E I J must transform as E I J (τ ) → (Cτ + D) I K (Cτ + D) J L E K L (τ ) + C I L (Cτ + D) J L .
(4.2)
Then Eˆ transforms simply as Eˆ I J (τ, τ¯ ) → (Cτ + D) I K (Cτ + D) J L Eˆ K L (τ, τ¯ ).
(4.3)
Of course, E I J and Eˆ I J are just ⊂ Sp(2n, ZZ) analogues (up to normalization) of the second Eisenstein series E 2 (τ ) of S L(2, ZZ), and its modular but non-holomorphic counterpart E 2∗ (τ, τ¯ ) — see Appendix A. It is important to note that the transformation properties given above do not define E and Eˆ uniquely: shifting E I J by any holomorphic modular form e I J of , E I J (τ ) → E I J (τ ) + e I J (τ ) with e I J (τ ) transforming as e I J (τ ) → (Cτ + D) I K (Cτ + D) J L e K L (τ ), we still get a solution of (4.2). With this in hand, one can reorganize each Fg as a finite power series in E with coefficients that are strictly holomorphic modular forms [23]. In particular, the free energy at genus g in holomorphic polarization can be written as (1) ˆIJ Fˆ g (τ, τ¯ ) = h (0) g (τ ) + (h g ) I J E (τ, τ¯ ) + . . . (3g−3)
+(h g
) I1 ...I6g−6 Eˆ I1 I2 (τ, τ¯ ) . . . Eˆ I6g−7 I6g−6 (τ, τ¯ ),
(4.4)
where h (k) g (τ ) are holomorphic modular forms of in the usual sense. Moreover, taking Fˆ g (τ, τ¯ ) and sending τ¯ to infinity,16 Fg (τ ) = lim Fˆ g (τ, τ¯ ), τ¯ →∞
we recover the modular expansion of the partition function in real polarization: (3g−3)
(1) IJ Fg (τ ) = h (0) g (τ ) + (h g ) I J E (τ ) + . . . + (h g
) I1 ...I6g−6 E I1 I2 (τ ) . . . E I6g−7 I6g−6 (τ ).
This gives us a way to construct modular invariant quantities out of the free energy and correlation functions. For example, it is easy to see that the highest order term in the (τ − τ¯ )−1 expansion of Fˆ g is always modular. It is constructed solely out of genus zero 16 By sending τ¯ to infinity what we really mean is keeping the constant term in the finite power series in (τ − τ¯ )−1 . For S L(2, Z Z), this is simply the isomorphism between the rings of almost holomorphic modular Z). forms and quasi-modular forms described in [23], which can be easily generalized to Sp(2n, Z
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amplitudes, as it corresponds to the most degenerate genus g Riemann surface that breaks up into (2g − 2) genus zero components with three punctures each. Moreover, it follows that ∂ I ∂ J ∂ K F0 is itself modular and corresponds to an irreducible representation—a third rank symmetric tensor: I J K ∂ I ∂ J ∂ K F0 → (Cτ + D)−1 (Cτ + D)−1 (Cτ + D)−1 ∂ I ∂ J ∂ K F0 , I
J
K
(4.5) which can be verified directly as well. (0) From h g , we get a modular form of weight zero, constructed out of Fg and lower genus amplitudes via IJ (h (0) g )(τ ) = Fg (τ ) + g (E (τ ), ∂ I1 . . . ∂ I N Fr
(4.6)
where g is the functional introduced in the previous sections. While none of the terms on the right-hand side is modular on its own, added together we get a modular invariant of . We can turn this around and read this equation as follows: given the genus r < g amplitudes and the propagator E I J , the free energy Fg (τ ) is fixed, up to the addition (0) of a precisely modular holomorphic form h (0) g ! In practice, this means that h g is a meromorphic function on the moduli space.17 We can write this compactly as follows. Let H(τ ) =
∞
h (0) g (τ ) gs
2g−2
g=1
be the generating functional of weight zero modular forms, and define the generating function of correlation functions 1 2g−2 ∂ I1 . . . ∂ In Fg (x) y I1 . . . y In gs , W(y, x) = n! g,n where the sum over n runs from zero to infinity, except at genus zero and one, where it starts at n = 3 and n = 1, respectively. Then, the above can be summarized by writing 1 exp( H(x)) = dy exp( − 2 E I J y I y J ) exp( W(y, x)), 2gs where E I J is the inverse of E I J , E I K E K J = δ JI . This follows directly from the path integral of Sect. 2 relating the wave functions in the real and holomorphic polarizations, which can be written as 1 ˆ I J y I y J ) exp( W(y, x)), Zˆ (x, x) ¯ = dy(− 2 (E − E) 2gs where one views Eˆ as a perturbation. 17 As stated in Sect. 2, throughout we assumed a definite choice of a gauge, and picked a 3-form ω as a (0) (0) definite section of L. Like Fg ’s, h g depend on this choice – they are sections of L2−2g , so h g is more pre2−2g . On a non-compact Calabi-Yau, however, it is simply a meromorphic cisely a meromorphic section of L function.
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Furthermore, one can show that similar equations hold when F and E are replaced by their non-holomorphic counterparts. To see this, note that the inverse of (2.13) is 2 ˆ Z (x) = dz e S(x,z)/gs Zˆ (z; X, X¯ ), (4.7) with all the quantities as defined in Sect. 2. If we choose the background X I = x I , this has a saddle point at z I = x I . Expanding around it, by putting z I = x I + y I , where y I = −ϕx I + z i Di x I , and integrating over y, we get 1 ˆ Z (x) = dy exp(− 2 (τ − τ¯ ) I J y I y I ) exp( W(y; x, x) ¯ ), gs where 2g−2 ˆ Fˆ g ((1 − ϕ)x + z i Di x, x) gs ¯ W(y; x, x) ¯ = g
1 2g−2 gs (1 − ϕ)2−2g−n z i1 . . . z in Di1 . . . Din Fˆ g (x, x) ¯ = n! n,g
χ − 1) log(1 − ϕ). 24 From this, and thinking about Z (x) in terms of a power series in E, it follows immediately that 1 ˆ exp(H(x)) = dy exp( − 2 Eˆ I J (x, x) ¯ y I y J ) exp( W(y, x, x) ¯ ). (4.8) 2gs −(
Equation (4.8) has appeared before. In the seminal paper [6] the authors derived a set of equations that the physical free energies Fˆ g must satisfy, through analysis of the worldsheet theory. These equations were interpreted in [40] as saying that the topological string partition function is a wave function in the Hilbert space obtained from the geometric quantization of H 3 (X,C), the fact that we used repeatedly here. Holomorphic anomaly equations (and modular invariance) constrain what the topological string amplitudes can be. Here we described the solutions to the equations using symmetry ˆ which was the guts of the method of [6] alone. The construction of the propagators E, for solving the equations, was quite complicated. The answers were messy, with ambiguities that had no clear interpretation. Now, the meaning of the propagators Eˆ I J and E I J is simple and beautiful—they are simply generators of (almost) modular forms of the symmetry group ! The only remaining thing to show is that the propagators of our expansion and of [6] agree. In [6] the authors gave a set of relations that the inverse propagators satisfy (p. 103 of [6]). It is easily shown that our propagators (4.1) satisfy these relations (for any holomorphic form E I J ). Let Eˆ ϕϕ = Eˆ I J x I x J ,
Eˆ ϕi = Eˆ I J x I Di x J ,
Eˆ i j = Eˆ I J Di x I D j x J ,
where Di is the Kähler covariant derivative Di = ∂i − ∂i K and K is the Kähler form of the special geometry of X . Then, with a bit of algebra it follows that these satisfy ∂¯i¯ Eˆ jk = C¯ i¯mn Eˆ m j Eˆ nk + G i¯ j Eˆ ϕk + G ik ¯ Eˆ ϕ j , ∂¯i¯ Eˆ jϕ = C¯ i¯mn Eˆ m j Eˆ nϕ + G i¯ j Eˆ ϕϕ , ∂¯i¯ Eˆ ϕϕ =
C¯ i¯mn Eˆ mϕ Eˆ nϕ ,
(4.9)
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where G i¯ j = ∂¯i¯ ∂ j K , C¯ i¯mn = e−2K G m m¯ G n n¯ C¯ i¯m¯ n¯ , C¯ i¯m¯ n¯ = C¯ I J K D¯ i¯ x¯ I D¯ j¯ x¯ J D¯ k¯ x¯ K . Equations (4.9) are exactly the equations of [6] with obvious substitutions.
4.1. A mathematical subtlety. As we have shown in the previous sections, our results are completely general and apply to both non-compact and compact Calabi-Yau threefolds. However, to make contact with the theory of modular forms in mathematics there is an important subtlety that we have not mentioned yet. In the theory of modular forms, the period matrix τ I J acquires a crucial role. A modular form is defined to be a holomorphic function f : Hk → C satisfying certain transformation properties, where Hk is the Siegel upper half-space: Hk = {τ ∈ Mat k×k (C)| τ T = τ, τ − τ¯ > 0}, which is the space of k × k symmetric matrices with positive definite imaginary part. The period matrix is the τ in the definition of the Siegel upper half-space. Note that strictly speaking, this defines Siegel modular forms; proper modular forms are obtained for k = 1.18 For the non-compact case, the mirror symmetric geometry reduces to a family of Riemann surfaces of a certain genus. Thus, it is clear that the period matrix τ I J has positive definite imaginary part. Therefore, in this case our results should be interpreted mathematically as Siegel modular forms, where k depends on the genus g of the Riemann surface. In particular, if the mirror geometry is a family of elliptic curves, k = 1, and we recover proper modular forms. However, in the compact case the situation changes slightly. The period matrix τ I J does not have positive definite imaginary part anymore; it has signature (h 2,1 , 1), as explained for instance in [15]. Thus, in this case the Siegel upper half-space is not the relevant object anymore, and we cannot make contact directly with Siegel modular forms. This seems to call for a new theory of modular forms defined on spaces with indefinite signature. It would be very interesting to develop this mathematically. Another possibility, in order to make contact with already known mathematical concepts in the compact case, is to replace the period matrix τ I J by a different but related matrix N I J —see for instance [15] for a definition—which has positive definite imaginary part, but is not holomorphic. This is usually done in the context of supergravity. Roughly speaking, it amounts to replacing the intersection pairing by the Hodge star pairing. In that way perhaps we can come back into the realm of Siegel modular forms, perhaps along the lines of what was done in [15] in a related context. In the following sections we will give applications of the modular approach we have developed so far, for local Calabi-Yau threefolds. 5. Seiberg-Witten Theory As is well known, type II string theory compactified on local Calabi-Yau manifolds gives rise to N = 2 gauge theories in four dimensions. The topological string theory on these 18 See Appendix A and B for definitions and conventions.
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manifolds computes topological terms in the effective action of N = 2 Seiberg-Witten theory with gauge group G [6,24]. These terms are summarized in a partition function Z SW = exp(λ2g−2 Fg (a)),
(5.1)
where Fg coincides with the genus g topological string free energy, and the a’s are local parameters in the vacuum manifold of the gauge theory. Each term in (5.1) has a physical meaning in the effective action of the N = 2 gauge theory. The genus zero topological string amplitude yields the exact gauge coupling τi j =
∂ 2 F0 , ∂ai ∂a j
(5.2)
with i, j = 1, . . . r , where r = rank(G), while the higher genus topological string amplitudes yield the gravitational coupling of the self-dual part of the curvature R+ to 2g−2 the self-dual part of the graviphoton field strength dx 4 Fg R+2 F+ . The Fg (a)’s for g > 1 were in fact extensively studied in the weak electric coupling limit [32]. The corresponding Calabi-Yau manifold is given by an equation of the form (2.17) with an appropriate H (y, z) depending on the theory. For example, for G = SU (n) without matter, H (y, z) = y 2 − (Pn (z))2 + 1,
(5.3)
where Pn (z) = + u2 + . . . u n , and the holomorphic 3-form is given by (2.18). The parameters u i are complex coordinates on the moduli space of the Calabi-Yau. In the gauge theory, they correspond to the expectation values of the gauge invariant observables zn
uk =
z n−2
1 Trφ k + products of lower order Casimirs, k
(5.4)
where φ is the adjoint valued Higgs field. The family of Riemann surfaces obtained by setting
g :
H (y, z) = 0
is the Seiberg-Witten curve of the gauge theory. The genus g of the Riemann surface is the rank of the gauge group r . The gauge coupling constant Im(τi j ) is the period matrix of the Riemann surface. Alternatively, τi j is the complex structure of the Jacobian of the Riemann surface g , which is an abelian variety. The abelian variety is spanned by the periods pi a Di Bi λ , (5.5) = = ai xi Ai λ with i = 1, . . . r , and where the A- and B-cycles generate the symplectic integer basis of H1 ( g , ZZ). Here λ is a meromorphic differential, which is part of the data of the theory. As explained in Sect. 2, in the string theory context, λ comes from the reduction of the holomorphic 3-form of the parent Calabi-Yau threefold to a one-form on g . For theories with matter, there can be additional periods on g —λ then has poles whose residues correspond to the mass parameters. The monodromy group of the curve g , which is naturally a subgroup of Sp(2r, ZZ), played the central role in [35]. It is generated by the BPS particles going massless at
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a codimension one loci in the moduli space and captures the non-perturbative duality symmetries of the N = 2 gauge theory, since it acts non-trivially on the coupling constant τi j . From the monodromies of the periods around the perturbative limits in the moduli space, [35] showed that one can deduce the periods themselves everywhere in the moduli space—this is the Riemann-Hilbert problem—and hence also τi j and F0 . It is then very natural to ask: what does the group of symmetries imply about the full partition function Z SW ? In fact, this question, and the close relation of Seiberg-Witten theory and topological strings in general, is what motivated this paper. The topological string partition function is a wave function for both compact Calabi-Yau threefolds, studied in [6], and non-compact Calabi-Yau threefolds, as we have seen in Sect. 2. This implies that the Seiberg-Witten partition function [40] Z SW is a wave function, arising by geometric quantization of H1 ( g )—see [21]. In particular, in holomorphic polarization, it satisfies the local holomorphic anomaly equation (2.26). In fact, it would be very interesting to derive this directly from the N = 2 gauge theory. Since the partition function Z SW is known, this gives us a testing ground for exploring the restrictions that follow from the duality symmetries generated by , but now acting on the full quantum wave function Z SW .19
5.1. Seiberg-Witten theory and modular forms. One crucial property of the abelian variety is that Im(τi j ) > 0, which ensures positivity of the kinetic terms of the vector multiplet. Thus, in this case the period matrix τi j can be used to define the Siegel upper half space Hr as Hr = {τ ∈ Matr ×r (C)|τ T = τ, Im(τ ) > 0}.
(5.6)
The monodromy group ⊂ Sp(2r, ZZ) of the family of Riemann surfaces g acts on τi j as −1
τ → (Aτ + B)(Cτ + D)
for
A C
B ∈ . D
Thus, in principle, we should be able to give explicit expressions for the Seiberg-Witten higher genus amplitudes in terms of Siegel modular forms under the corresponding subgroup ⊂ Sp(2r, ZZ) (see Appendix B for a brief review of Siegel modular forms). To start with, however, let us consider SU (2) gauge theory, where the modular group ⊂ S L(2, ZZ), and correspondingly standard modular forms suffice. i.SU (2) Seiberg-Witten theory. The curve of the SU (2) gauge theory can be written as20 y 2 = (x 2 − 1)(x − u).
(5.7)
19 The observation that duality transformations imply quasi-modular properties of the F ’s has been made g earlier in [12]. However, their results are different from ours in that their partition function Z = exp F does not transform like a wave function; rather, it transforms by Legendre transformations of F . 20 As explained in [36] there are two curves corresponding to this gauge theory, differing by a factor of 2 in the normalization of the A-period and electric charge. The curve at hand has #(A ∩ B) = 2 between the generators of H1 ( , Z Z). The curve which is the n = 2 specialization of (5.23) has the A-period A = A/2. Correspondingly, the modular groups will differ: in the second case we would get the 0 (4) subgroup of Z) instead of (2). S L(2, Z
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There are three singular points in the moduli space, corresponding to u = ±1, ∞ with monodromies −1 2 1 0 −1 2 M∞ = , M1 = , M−1 = (5.8) 0 −1 −2 1 −2 3 acting on =
λ p , = B x Aλ
where #(A ∩ B) = 2.
(5.9)
The monodromies (5.8) generate the (2) subgroup of S L(2, ZZ); that is, the subgroup of 2 × 2 matrices congruent to the identity matrix, modulo 2. The x = a, p = a D are by now canonical variables of Seiberg-Witten theory [35], so we will mainly use that notation. The periods a, a D solve the Picard-Fuchs equation L = 0, ∂ where L = θ (θ − 1) − u 2 (θ − 21 )2 and θ = u ∂u . From the previous sections, we can predict that the genus g amplitudes Fg of this theory are (almost) modular forms of (2), with definite transformation properties. Since the higher genus amplitudes are known from [35,31], they will provide a direct check of our predictions. The parameter τ of the modular curve is defined by τ = ∂∂ xp , or in usual Seiberg-Witten notation
τ=
∂a D ∂2 = 2 2 F0 (a). ∂a ∂a
(5.10)
Solving the Picard-Fuchs equation for the periods, we can obtain τ as a function of u. Alternatively, we can proceed as follows. Recall that the j-function of the elliptic curve, which has the q-expansion j (τ ) =
1 + 744 + 196884q + . . . , q
where q = e2πiτ , provides a coordinate independent way of characterizing the curve. Roughly speaking, elliptic curves are the same if their j-functions are equal. Bringing Eq. (5.7) of the family of elliptic curves in Weierstrass form y 2 = 4x 3 − g2 x − g3 ,
(5.11)
the j function can be computed as g23
.
(5.12)
64(3 + u 2 )3 . (u 2 − 1)2
(5.13)
j = 1728
g23 − 27g32
For the family of elliptic curves (5.7), this gives j (τ ) =
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Then, using the q-expansion of the j-function, we get a q-expansion for u, in the large u limit: u=
1 −1/2 5 1/2 31 3/2 q + q − q + 27q 5/2 + O(q 7/2 ). 8 2 4
However, what we want is an expression of u in terms of τ which is valid everywhere in the moduli space, not just a q-expansion when u is large; in other words, we want to find the modular form of (2) which has the above q-expansion. Since u is a good coordinate on the moduli space, which is the quotient of the Teichmuller space by (2), it has to be invariant under (2); i.e., it must be a modular form of weight zero. For a brief review of modular forms of (2), see Appendix A. The modular forms of (2) are generated by the following θ -constants: b(τ ) := θ24 (τ ),
c(τ ) := θ34 (τ ),
d(τ ) := θ44 (τ )
which all have weight 2. These are not independent, but satisfy the relation c = b + d. It is easy to show that [21] u(τ ) =
c+d (τ ), b
which is modular invariant, as claimed. The genus one amplitude [30] ∂a 1 1 F1 = − log det − log(u 2 − 1) 2 ∂u 12
(5.14)
(5.15)
can be rewritten, using the results we have obtained so far, as [26] F1 (τ ) = − log η(τ ),
(5.16)
where η(τ ) is the Dedekind η-function. Note that this transforms under modular transformation in (2) exactly as predicted in Sect. 2, namely Aτ + B 1 1 = F1 (τ ) + log F1 Cτ + D 2 τ + C −1 D (up to a constant that is irrelevant, as only ∂F1 is well defined).21 3 Next, from Sect. 4, we expect that ∂∂aF30 = 21 ∂τ ∂a is a modular form of weight −3. ∂ ∂u ∂ Using the fact that ∂a = ∂a ∂u , the modular expression for u (5.14) and the modular expression for ∂u ∂a obtained by combining (5.26) and (5.16), we get √ b ∂3 (5.17) F0 (a) = − 3 ∂a cd which indeed transforms as expected. 21 In this case, F transforms in this way under the whole S L(2, Z Z), but this is an accident of the model. In 1 particular, had we worked with 0 (4) (and hence with τ = τ/2), F1 would transform like this under 0 (4), Z). but not under the full S L(2, Z
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Now, consider the genus two amplitude. In [21] it was shown that this can be written as (0)
(1)
(2)
(3)
F2 (τ ) = h 2 (τ ) + h 2 (τ ) E(τ ) + h 2 (τ ) (E(τ ))2 + h 2 (τ ) (E(τ ))3 ,
(5.18)
where the propagator E(τ ) is given in terms of the second Eisenstein series E(τ ) =
2πi E 2 (τ ), 6
and the modular coefficients are (0)
1 (c + d)(16b2 + 19cd) X, 30 6 (b2 + cd) X, = −2 2πi 6 2 =3 (c + d) X, 2πi 6 3 5 =− X, 3 2πi
h2 = h (1) 2 (2)
h2
(3)
h2
(5.19)
where we defined X=
1 b . 2 1728 c d 2
We will now see that this is exactly as predicted in Sect. 4! First, consider how this transforms under modular transformations in . Note that (k) the coefficients h 2 are modular forms of of weight (−3k): (k)
(k)
h 2 ((Aτ + B)/(Cτ + D)) = (Cτ + D)−3k h 2 (τ ). Moreover, k ranges from zero to 3g − 3, where g = 2 in this case. On the other hand E(τ ) transforms as a quasi-modular form: E((Aτ + B)/(Cτ + D)) = (Cτ + D)2 E(τ ) + 2 C(Cτ + D);
(5.20)
in other words it is a holomorphic form, modular up to shifts (cf. (4.2)). The fact that F2 is a finite power series in E(τ ), with coefficients that are strictly modular forms of (2) means that F2 is itself a quasi-modular form of (2), per definition. Note that the propagator in (5.20) transforms by a factor of 2 relative to (4.2). This factor of two is a consequence of the fact that the intersection number of the A and the B periods of the curve is twice bigger than the conventional one (5.9). It is very easy to derive this from Sects. 2 and 3 (see footnote 6). Moreover, it is easy to check, starting from (5.16), (5.17) and (5.18) (with the help of some standard modular formulae given in Appendix A), that F2 transforms under modular transformations exactly as predicted in Sect. 3. To do so, note that, looping around u = 1 for example, simply acts on τ by the (2) ⊂ S L(2, ZZ) transformation M1 given in (5.8). Using the usual transformation properties of modular forms and the expression (5.18) for F2 in terms of modular forms of (2), it is then easy to work out the transformation property of F2 under M1 .
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Furthermore, while the Fg and the vertices ∂i1 , . . . , ∂in Fg are not quite modular, the combinations Fg (τ ) + g (E(τ ), ∂i1 . . . ∂in Fr
(5.21)
are exactly invariant under modular transformations and agree with h (0) g (τ ), as expected from Sect. 4. We can trade quasi-modular forms for almost holomorphic forms by replacing E(τ ) in all formulae by its modular, but not holomorphic counterpart ˆ τ¯ ) = E(τ ) + E(τ,
2 τ − τ¯
which transforms as ˆ ˆ τ¯ ). E((Aτ + B)/(Cτ + D), (Aτ¯ + B)/(C τ¯ + D)) = (Cτ + D)2 E(τ, Also, note that F1 can be made exactly modular by writing 1 Fˆ1 (τ, τ¯ ) = − log((τ − τ¯ ) 2 |η(τ ))|2 ).
This is exactly the one-loop amplitude of the local Calabi-Yau in holomorphic polari∂ ∂ ˆ zation. More precisely, it is only the holomorphic derivatives ∂a F1 that are F1 , and ∂a physical, but this is the natural way to write it. ˆ τ¯ ) is exactly the propagator of [6] ! One has that Finally, E(τ, ˆ τ¯ ), ∂i1 , . . . , ∂in Fr
(5.22)
is strictly holomorphic, with the same modular form h (0) g (τ ) as in (5.21). In the next subsection, we consider gauge groups of higher rank, corresponding to Riemann surfaces of genus higher than one. 5.2. The SU (n), n > 2 Seiberg-Witten theory. As mentioned earlier, the Riemann surface corresponding to SU (n) Seiberg Witten theory is a genus g = n − 1 curve y 2 − (Pn (z))2 + 2n = 0,
(5.23)
where Pn (z) = z n + u 2 z n−2 + · · · u n+1 . The singular loci in the moduli space correspond to the zeroes of the discriminant = (ei (u) − e j (u))2 , (5.24) i< j
where ei (u) are roots of Pn (z, u)2 − 2n . That is, at the values of the moduli u for which any pair of roots come together ei (u) → e j (u), the curve becomes singular. There is a natural basis of (n − 1) A-cycles corresponding to pairs of branch points that pair up as
goes to zero. This corresponds to points where the non-abelian gauge bosons become
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797
massless in the classical theory. The monodromy group ⊂ Sp(2g, ZZ) of the quantum theory can be determined [25], by following the exchange paths of the branch points. We will leave the detailed analysis of this and the corresponding implications for the structure of the topological string amplitudes as an interesting exercise, and only consider briefly the one-loop amplitude. On general grounds [5,6], the one-loop amplitude in the topological string theory has the universal form 1 1 log(). F1 (τ ) = − log(det(Di X )) − 2 12
(5.25)
This result was also derived in a purely gauge theory context in [29,30]. There, the authors computed the one-loop amplitude of the (twisted) N = 2 gauge theory on a curved four-manifold, namely the coefficients of the R 2 term in the effective action. Restricting the curvature to be anti-self dual, R− = 0, this is precisely the term that the topological string computes.22 This gives 1 ∂ai 1 − log(), (5.26) F1 (τ ) = − log det 2 ∂u k 12 where is the discriminant of the Seiberg-Witten curve. For example, for G = SU (n) with curve given by (5.23), is (5.24). Note that the u’s are necessarily modular invariants of , as they are just parameters entering into the algebraic definition of the curve, and hence they do not ‘talk’ to its periods. On the other hand, is simply a rational function of u, so also necessarily a Siegel modular form of of weight zero. To write the full amplitude in terms of modular forms, note that from [29,30] we have 1
2 1 ∂a j 0 8 (0, τ ), (5.27) det =θ δ ∂u i
where δ = 21 , . . . , 21 and we defined the ‘generalized’ θ -functions with characteristic in Appendix B. As a consequence we can write 1 0 (0, τ ) + log(). F1 (τ ) = − log θ δ 24 This is consistent with the transformation properties of F1 , since θ 0δ is a scalar Siegel modular form of weight 1/2. 6. Local IP2 We now study the local IP2 , from the mirror B-model point of view. In this case the mirror is a family of elliptic curves with monodromy group (3). The Gromov-Witten theory of the local IP2 at large radius was solved in [3,4]. Using those results, we can show 22 Practically, in terms of [29,30] this corresponds to setting the signature σ of the four-manifold equal to σ = − 23 χ , where χ is its Euler character. One way to see this is that it holds exactly for the K 3, for example, where the curvature is anti-self dual.
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explicitly that the predictions for modular properties of the topological string amplitudes are satisfied. Another interesting point in the moduli space of the local IP2 is the C3 /ZZ3 orbifold point. One can in principle formulate the Gromov-Witten theory of the orbifold point as well, however the amplitudes are not yet available [7,34]. We now have a simple prescription to carry over the large radius results to other points in the moduli space, the orbifold point in particular, so we can make new predictions there. 6.1. Mirror family of elliptic curves. The mirror data is a family of elliptic curves , given by the equation 3
xi3 − 3ψ
i=1
3
xi = 0
(6.1)
i=1
in IP3 , and a meromorphic 1-form λ = log(x2 /x3 )d x1 /x1 . This has an obvious ZZ3 symmetry α = e2πi/3 ,
ψ → αψ,
since it can be undone by a coordinate transformation x1 → α −1 x1 that affects neither
nor λ. The discriminant of the curve is = (1 − ψ 3 ). This vanishes at the three singular points ψ 3 = 1, corresponding to conifold singularities. To make contact with standard elliptic functions and their modular properties we make a P G L(3,C) transform to bring the equation of the curve to its Weierstrass form y 2 = 4x 3 − g2 x − g3 with g2 =
α(8 + ψ 3 ) , 2(2/3) 24ψ 3
g3 =
8 + 20ψ 3 − ψ 6 , 864ψ 6
so that its j-function is given by j (τ ) = −
27ψ 3 (8 + ψ 3 )3 . (1 − ψ 3 )3
(6.2)
As usual, ∂p (6.3) ∂x is the standard complex structure modulus of the family of elliptic curves, where we view as a quotient of a complex plane by a lattice generated by 1 and τ . Here23 p= λ(ψ), x= λ(ψ), τ=
B
A
23 We use x to denote both the coordinate on the Riemann surface and the period of λ. It should be clear
from the context which meaning we assign to x.
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where λ(ψ) = log(x)dy/y. Our j-function is normalized to j=
1 + 744 + 196884q + O(q 2 ), q
(6.4)
where q = exp(2πiτ ). Combining the two expressions for the j-function, we find a series expansion for ψ(q) in the large ψ limit: 3ψ =
1 q
1 3
2
5
8
+ 5q 3 − 7q 3 + O(q 3 ).
(6.5)
Alternatively, we can obtain the same expansion by first using the Picard-Fuchs equations to find the periods x(ψ), p(ψ), and then computing τ (ψ) directly using the definition (6.3). We will study in more detail the Picard-Fuchs equation and its solutions in the next subsection. For now, we only note one interesting aspect to this. Namely, as discussed in Sect. 2.3, due to the non-compactness of the Calabi-Yau, it may not be possible to find a basis of periods that are normalized canonically. This occurs in the present example: the compact B period satisfies #(A ∩ B) = −3.
(6.6)
One way to see this is in the mirror A-model: the compact parts of H4 and H2 of the manifold are generated by the IP2 , which we take to be mirror to the B-period, and the IP1 line inside it, mirror to the A period. In the Calabi-Yau, these do intersect, but the intersection number is −3. Correspondingly, if we put x = t, p = −3
∂ F0 (t), dt
and therefore τ = −3 dt∂ 2 F0 (t). The above expression for ψ(τ ) is valid for Im(τ ) → ∞. In the next subsection, we will show that the local IP2 is governed by a (3) subgroup of S L(2, Z ). This will allow us to give a globally well defined expression for ψ in terms of modular forms under (3). 2
6.2. The monodromy group. The meromorphic 1-form λ turns out to have a nonvanishing residue: in addition to the usual A and B periods—by this we mean the periods associated to the A and B cycles—of the genus one Riemann surface, it has an additional period, which we will call C. As discussed in Sect. 2.3, the extra period does not correspond to a modulus of the Riemann surface, but to an auxiliary parameter. While the monodromies mix up all the periods, the monodromy action on the extra period C should be highly constrained. To derive the monodromy action on the full period vector ⎛ ⎞ B λ = ⎝ A λ⎠ Cλ we will solve the Picard-Fuchs (PF) differential equations that satisfies L = 0,
(6.7)
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ever ywher e in the moduli space. A certain linear combination of the solutions to Eq. (6.7) will have the property that its monodromies are integral, and that gives . Before doing that, note that, since the additional period C is just an auxiliary parameter, the modular properties of the topological string amplitudes should be governed by the monodromy group of the family of elliptic curves . It is well known that this is a (3) subgroup of S L(2, ZZ), when viewed as a fibration over the punctured ψ plane. We will see below that this is indeed the case. Now let us come back to the study of the full Picard-Fuchs equation. It is convenient to work in the coordinate z, centered at large radius: z=−
1 . (3ψ)3
(6.8)
There are three special points in the z plane. In addition to the large radius point at z = 0, there is also the conifold point, coming from ψ 3 = 1, and the orbifold point z = ∞, with ZZ3 monodromy around it. In this coordinate the Picard-Fuchs differential operator L has the form L = θz3 + 3z(3θz + 2)(3θz + 1)θ. This has three independent solutions, one of which is a constant, corresponding to the period of λ around the C−cycle. The corresponding new cycle C encircles the residue of λ(ψ). The solutions near large radius (z = 0) can be found by the Frobenius method from the generating function ω(z, s) :=
∞ n=1
z s+n , (−3(n + s) + 1) 3 (n + s + 1)
with Lω(z, s) = 0. This gives 3 independent solutions, di 1 ωi = ω(z, s) , i i (2πi) d s s=0 i.e. ω0 = 1, ω1 =
1 2πi (log(z)
+ σ1 (z)) and ω2 =
1 (log(z)2 + 2σ1 log(z) + σ2 (z)), (2πi)2 2 + 45 z 2 + O(z 3 ) and σ2 = −18 z + 4232 z + O(z 3 ).
where the first orders are σ1 = −6 z Linear combinations of these solutions will give the periods over cycles in integer cohomology. This requires analytic continuation to all singular points. The result is ⎞ ⎞ ⎛1 ⎛ 1 1 −3∂t F0 2 ω2 − 2 ω1 − 4 ⎠. ⎠=⎝ (6.9) =⎝ t ω1 1 1 The factor of −3 in the above equation comes from (6.6) as we explained earlier. From above, we can read off the mirror map, giving the A-period in terms of the coordinates on the moduli space, and its inverse: z(Q) = Q + 6 Q 2 + 9 Q 3 + 56 Q 4 + O(Q 5 ),
(6.10)
Topological Strings and (Almost) Modular Forms
801
where Q = e2πit , and z is defined in (6.8).24 From this, we can also read off the monodromy of the periods around large radius, i.e. around z = 0 (or ψ = ∞). From (6.10) it follows that this is equivalent to shifting t by one, and, since −3∂F0 = 21 t 2 − 2t − 41 + O(eπit ), this gives ⎛
M∞
⎞ 110 = ⎝0 1 1⎠ . 001
(6.11)
Expanding the periods at the conifold point ψ 3 = 1, one finds the monodromy ⎛ ⎞ 1 00 M1 = ⎝−3 1 0⎠ . (6.12) 0 01 This is the Picard-Lefshetz monodromy around the shrinking B-cycle with intersection form (6.6). The C-period corresponds to an auxiliary parameter, and correspondingly the C-cycle does not intersect the A and B cycles. From M∞ and M1 , we can recover the monodromy around the orbifold point M0 , as holomorphy requires M0 M1 M∞ = 1, ⎛
⎞ −2 −1 1 M0 = ⎝ 3 1 −1⎠ . 0 0 1
(6.13)
This satisfies (M0 )3 = 1, as it should, since the monodromy is of third order. Note that in all three cases, the monodromies act trivially on the C-period, which is consistent with the fact that this corresponds simply to a parameter. Moreover, the monodromy action on the A and the B periods generates the 0 (3) subgroup of S L(2, ZZ). If instead of the z-plane, we choose to work with the ψ-plane, then ψ = 0 is a regular point, with trivial monodromy around it, but instead we have three conifold singularities, 2πi at ψ = 1, α, α 2 , with α = e 3 . The monodromies M˜ in the ψ-plane can be derived from the expressions above. For example, M˜ 1 = M1 ,
M˜ α = M0 M1 M0−1 ,
M˜ α 2 = M02 M1 M0−2
with monodromy at infinity given by M˜ ∞ = M˜ 1 M˜ α M˜ α 2 . These turn out to generate the (3) subgroup of S L(2, ZZ). Below, we will choose to work with modular forms of (3), in terms of which both ψ and z will be given by exactly modular forms. 24 For the genus zero partition function this gives
∂t F 0 = −
1 45 Q 2 244 Q 3 12333 Q 4 t2 t + + +3Q− + − + O(Q 5 ), 6 6 12 4 3 16
which agrees with the Gromow-Witten large radius expansion. Using this, and the definition of τ we can explicitly check (6.5).
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6.3. Topological strings on local IP2 and modular forms. To get modular expressions for the topological string amplitudes we need to know a bit about modular forms of the subgroup (3) of S L(2, ZZ). Essential facts about them are reviewed in Appendix A; for a detailed study of modular forms of (3), see [17]. The set of θ -constants that generate modular forms of (3) is: a := θ 3
1 6 1 6
, b := θ 3
1 6 1 2
, c := θ 3
1 6 5 6
, d := θ 3
1 2 1 6
,
which all have weight 3/2. They satisfy the relations [17] c = b − a,
d = a + αb,
i abcd. To begin with, note that since and the Dedekind η-function is given by η12 = 33/2 ψ is a coordinate on the moduli space, it has to be a weight zero modular form of (3). Indeed, we find that
ψ(τ ) =
a−c−d . d
(6.14)
From [5] we know that the genus one free energy is given by ∂t 1 1 F1 = − log − log(1 − ψ 3 ). 2 ∂ψ 12 It is easy to show, using the Q-expansion of z around z = 0, that √ d ∂t =− 3 , ∂ψ η
(6.15)
and that, on the other hand, = 1 − ψ 3 = −33
η12 . d4
Combining these three expressions, we get F1 (τ ) = − log(η(τ )) +
1 1 log() = − log(dη3 ), 24 6
up to an irrelevant constant term. This transforms under as − log(η) does, since the discriminant is invariant, which is exactly what we predicted. As a consistency check, if we use the Q-expansion of q and the modular expression for F1 we get the expansion F1 = −
Q 3Q 2 23Q 3 1 log Q + − − + O(Q 4 ), 12 4 8 3
which is precisely the genus 1 amplitude of local IP2 .
Topological Strings and (Almost) Modular Forms
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6.4. Higher genus amplitudes. To find the higher genus amplitudes, we need the mod3 ular expression for the Yukawa coupling Cttt = ∂t∂ 3 F0 . We know that Cttt = −
1 ∂ψ ∂τ 1 ∂τ =− . 3 ∂t 3 ∂t ∂ψ
∂t Using the modular expressions for ψ (6.14), for ∂ψ (6.15), and the formulae for logarithmic derivatives derived in Appendix A, we get
Cttt = −
1 d . 35/2 η9
(6.16)
Another useful object is the (3)-invariant Yukawa coupling, expressed in terms of the globally defined variable ψ. We obtain Cψψψ =
∂t ∂ψ
3 Cttt = −
9 .
(6.17)
Using the results of the previous subsection, we can now find a modular expression for higher genus amplitudes, through their Feynman expansions. The propagator E(τ ) must transform under modular transformations as in (4.2), E((Aτ + B)/(Cτ + D)) = (Cτ + D)2 E(τ ) − 3 C (Cτ + D); the factor of −3 comes from the intersection numbers (6.6). For example, we can take E =−
2πi E 2 (τ ). 4
∂ We could have worked with the full E = 6 ∂τ F1 as well, since the propagator is defined (0) up to a modular invariant piece; it would have only changed the modular invariant h 2 . We obtain that the general form of the higher genus amplitudes reads
Fg = X g−1
3(g−1)
(3g−3−k)
E 2k h g
(K 2 , K 4 , K 6 ),
(6.18)
k=0
where we defined the weight −6 object X=
d2 1 C2 = 29 34 η18 1536 ttt (d)
and the ring of modular forms of (3) generating the weight 2d forms h g is given by K 2 = −α 2
(a − αc)2 , η2
K4 =
1 ac(a + c)(α 2 a − c) , α2 − 1 η4
K6 =
(ac)2 (a + c)2 . η6
The coefficients of E 2 are fixed by the Feynman graph expansion and we obtain for example
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1 (0) h 2 = F2 − X 5E 23 + E 22 K 2 + E 2 K 22 , 3 (0)
h 3 = F3 − X 2 (180E 26 + 240E 25 K 2 + 4E 24 (145K 22 − 1008K 4 ) 32 + E 23 (199K 23 − 1908K 2 K 4 + 648K 6 ) 9 4 2 + E 2 (563K 24 − 7936K 22 K 4 + 26496K 42 ) 5 16 + E 2 (149K 25 − 2536K 23 K 4 + 11952K 2 K 42 − 3456K 4 K 6 )). 15
(6.19)
Now, using known results for Fg in the large radius limit (obtained for instance (0) through the topological vertex formalism), we can find the h g ’s explicitly—this corresponds to fixing the holomorphic ambiguity in the BCOV formalism. For instance, we obtain (0)
1 1 11 + − , 69120 34560 76802 (6.20) 269 19393 17 337 373 + − = + − . 6289280 46448640 2786918402 22118403 41287684
h2 = h (0) 3
6.5. The C3 /ZZ3 orbifold point. In this section we explain how to extract the GromovWitten generating functions of the orbifold C3 /ZZ3 from the large radius amplitudes, through the wave function formalism. Let us first discuss this theory from the mirror A-model point of view. The target space X is an X =C3 /ZZ3 orbifold, with ZZ3 acting on the three coordinates z i , i = 1, 2, 3 by z i → αz i ,
α=e
2πi 3
.
In quantizing string theory on X , the Hilbert space splits into 3 twisted sectors, corresponding to strings closed up to α k , k = 0, 1, 2 (and projecting onto ZZ3 invariant states). The supersymmetric ground states in the k th sector correspond to the cohomology of the fixed point set of α k . This has an interpretation in terms of the cohomology of X as well. In the case at hand, the ground states in the sector twisted by α k correspond to the generators of H k,k (X ). Namely, the contribution to the cohomology of X is determined by the U (1) L × U (1) R charges of the states, where the charge ( pi , qi ) corresponds to H pi ,qi . In the twisted sectors, however, these receive a zero-point shift: in the sector twisted by z i → e2πiki z i with 0 ≤ ki < 1 the shift is ( i ki , i ki ). As there is precisely one such state for each k, the stringy cohomology of the orbifold agrees with the cohomology of the smooth resolution of X , i.e. the O(−3) → IP2 , as is generally true (see however [38]). As explained in [37], the orbifold theories have discrete quantum symmetries. In the present case, this is the ZZ3 symmetry which sends a state in the k th twisted sector to itself times α k . This is respected by interactions, so it is a well defined symmetry of the quantum theory. This implies that the only non-vanishing correlation functions are those that have net charge zero (mod 3). In particular, if we consider correlation functions of n insertions of topological observables Oσ corresponding to the generator of H 1,1 (X ), Oσ Oσ . . . Oσ g n
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at any genus g, this does not vanish only if n = 0 (mod 3). We will describe in this section how to compute the generating functions of correlation functions at genus g, Fgorb (σ ) =
1 (Oσ )n σ n , n! n
and show that this is indeed the case. By Fgorb here, we mean the generating function at the orbifold point—in this section, we will denote the generating function in the large radius limit by Fg∞ to avoid confusion. From what we explained in Sect. 3, the expectation is the following. The good coordinate in one region of the moduli space generally fails to be good at other regions of the moduli space. The good variable at large radius is t, as the corresponding monodromy is trivial (6.11), according to our criterion in Sect. 3. However, the monodromy of the period t is not trivial around the orbifold point, being given by (6.13), as 3 = 0. Correspondingly, even though we know the topological string amplitudes near the large radius point, we cannot simply analytically continue them to the orbifold point—the resulting objects would have bad singularities. Changing to good variables at the orbifold point involves a wave function transform that mixes up the genera. What is the good variable at the orbifold point? Clearly, it is the mirror B-model realization of the parameter σ that enters the orbifold Gromov-Witten partition functions in the A-model language and corresponds to H 1,1 (X ). The dual variable σ D , σ D = −3
∂ orb F , ∂σ 0
corresponds to H 2,2 (X ). To identify them in the B-model, note that, on the one hand, under the quantum symmetry ZZ3 symmetry σ and σ D transform as (1, σ, σ D ) → (1, α σ, α 2 σ D ). On the other hand, the symmetry acts in the mirror theory by [37] ψ → α ψ. The fixed point of this, ψ = 0, corresponds to the elliptic curve with ZZ3 symmetry, which is mirror to the C3 /ZZ3 orbifold. We can easily find the solutions to the Picard-Fuchs equations with these properties. A basis of solutions is given by the hypergeometric system 3 F2 , k 3 k ∞ (−1) 3 3 n k 3n (3ψ) Bk (ψ) = 3 k+i ψ , k i=1 n=0
3
(6.21)
n
for k = 1, 2, where we defined the Pochhammer symbols [a]n := (a+n) (a) . We also set B0 (ψ) = 1. The B’s diagonalize the monodromy around the orbifold point, namely ψ → αψ takes (B0 , B1 , B2 ) → (B0 , α B1 , α 2 B2 ). Consequently, we can identify (1, σ, σ D ) = (B0 , B1 , B2 ).
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The relative normalization of σ and σ D can be fixed using σ D = −3 ∂σ0 and hence 3 ∂ψ ∂ 2σD ∂ τ˜ , since ψ is globally defined. ∂σ = ∂σ 2 = −3C ψψψ ∂σ We can already make a prediction for the genus zero free energy at the orbifold point, up to an overall constant. By integrating σ D = 3 F0orb (σ ) =
∂ F0orb ∂σ ,
∞ N orb g=0,n n=1
(3n)!
we get
σ 3n
where, for example 1 1 1 1093 orb orb orb orb = , N0,2 = − 3 , N0,3 = 2 , N0,4 =− 6 , N0,1 3 3 3 3 119401 27428707 orb orb N0,5 = , N0,6 = − ,... . 37 38 Let us now turn to higher genus amplitudes. The analytic continuation from the point at infinity to the orbifold point can be done with the Barnes integral, as also explained in [8]. This relates ⎛ 1 ⎞⎛ ⎞ α σD − 1−α c2 1−α c1 13 = ⎝ c2 (6.22) c1 0 ⎠ ⎝ σ ⎠ 1 0 0 1 with the coefficients
i 13 c1 = , 2π 2 23
i 23 c2 = − , 2π 2 13
(6.23)
which are not integers. This is because the natural basis (σ, σ D ) diagonalizes the monodromy around the orbifold point, and this cannot be done in S L(2, ZZ).25 Note that c1 c2 = α(α−1) ; correspondingly the change of basis does not preserve the sym(2πi)3 plectic form, we have rather that dp ∧ d x =
1 dσ D ∧ dσ β
where β = −(2πi)3 . 25 We could have derived the change of basis in another way. There is another natural basis at the orbifold, (C0 , C1 , C2 ), corresponding to the 3 fractional branes. This basis has monodromy around the orbifold point, which is the cyclic Z Z3 permutations of the branes, ⎛ ⎞ ⎛ ⎞⎛ ⎞ C0 010 C0 ⎝C1 ⎠ → ⎝0 0 1⎠ ⎝C1 ⎠ . 100 C2 C2
The fractional brane basis is related to the large radius basis by an integral transformation—respecting the integrality of the D-brane charges—and the symplectic form. On the other hand, it is known [16] how the fractional branes couple to the twisted sectors: in particular, the i th twisted sector corresponds to j α i j C j . This reproduces (6.22).
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Because of this fact, the analysis of Sect. 2 goes through, but one has to be careful with normalizations. More precisely, it implies that the effective string coupling at the orbifold (gsorb )2 is renormalized relative to the large radius gs2 by (gsorb )2 = βgs2 . Then, knowing the Gromov-Witten amplitudes at large radius, we can predict them at the orbifold: β g−1 Fgorb = Fg∞ + g (, ∂i1 . . . ∂in Fr∞
(6.24)
where the coefficient β comes from the renormalization of the string coupling, and 3 . τ + C −1 D
=
The coefficient 3 above comes from (6.6). The coefficients C and D are computed from (the inverse of) (6.22) as before, which gives C −1 D =
1 . 1−α
(6.25)
In order to extract the σ -expansion of Fgorb such as we presented for F0orb , we compute the right-hand side of (6.24) in terms of the period t, and then use the relation between σ and t given in (6.22) to get expansions around σ = 0. D Since τ˜ = ∂σ ∂σ vanishes at the orbifold point σ = 0, it follows from (6.22) that τ (σ = 0) =
α . 1−α
(6.26)
− √π
Numerically, this corresponds to q(σ = 0) = −e 3 ∼ −0.16; at this value, the qexpansion of the modular expression (6.18) still converges rapidly. Indeed, since the coefficients of the σ -expansion of the topological string amplitude at the orbifold point are rational numbers, they can be easily recovered from their convergent q-expansion. At genus 1, we get F1orb (σ )
=
∞ N orb g=1,n
(3n)!
n=1
σ 3n ,
where, for instance, orb N1,1 = 0, orb = N1,4
orb N1,2 =
13007 , 38
1 , 35
orb N1,3 =−
orb N1,5 =−
14 , 35
8354164 ,... . 310
It is good to note that simply expanding Fg∞ (τ ) near τ (σ = 0), that is, doing only the analytic continuation of the amplitudes, would lead to non-rational coefficients in the σ -expansion. Instead of (6.24), it is faster to use the recursion relations at the orbifold point directly in terms of the modular ambiguity (6.20) and the corresponding propagator, E orb (τ ) =
lim
τ¯ →τ¯ (σ =0)
ˆ τ¯ ), E(τ,
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where τ¯ (σ = 0) = −C −1 D is just the complex conjugate of (6.26). This follows from the fact that Fˆ g , on the one hand, satisfies the same recursion relations as Fg∞ with E’s and Fr∞ ’s replaced by their hatted counterparts, and on the other hand Fˆ g (τ, τ¯ ) at τ¯ set to τ¯ = −C −1 D gives Fgorb . In fact, the right-hand side of (6.24) can be interpreted as computing just that. Either way, for Fgorb , we find that Fgorb (σ ) =
∞ orb N g,n n=0
(3n)!
σ 3n
orb , with the numbers N g,n≥1
g
n=1
0
1 3
1
0
2
1 24 ·34 ·5 − 5 31 2 35 5·7 313 27 39 52 − 9 519961 2 311 52 7·11 14609730607 212 313 53 72 11
3 4 5 6
2
3
4
5
1 119401 − 13 − 1093 3 32 36 37 1 13007 − 145 − 8354164 8 5 3 310 3 3 31429111 13 20693 12803923 − 4 6 − 4 10 2 ·3 24 ·38 ·5 2 ·3 ·5 24 310 11569 871749323 2429003 1520045984887 − − 24 311 5·7 25 39 5·7 25 310 5·7 25 313 5·7 115647179 29321809247 22766570703031 − 1889 − 27 39 26 313 52 28 312 52 27 315 5 339157983781 196898123 78658947782147 − 1057430723091383537 − 29 312 52 7·11 29 314 52 7·11 29 316 5·7 29 317 52 7·11 2453678654644313 − 40015774193969601803 5342470197951654213739 − 258703053013 212 314 53 72 11 211 318 53 72 11 212 319 5·72 11 210 315 51 72 11
where we also included the genus 0 and 1 numbers obtained earlier for completeness. The n = 0 numbers, corresponding to untwisted maps for g ≥ 2 (these are not well-defined for g = 0, 1), read −1 1 χ χ orb = + , N3,0 − , 2160 5760 544320 1451520 χ 7 + , =− 41990400 87091200 χ 3161 − , = 77598259200 2554675200 691χ 6261257 + ,... , =− 317764871424000 31384184832000
orb N2,0 = orb N4,0 orb N5,0 orb N6,0
where χ is the “Euler number" of local IP2 . The natural value of χ is 3. Generally in Gromov-Witten theory the denominators come from dividing by the finite automorphisms of the moduli space Mg,n . In the ZZ3 orbifold case there are obviously various automorphisms of order 3, corresponding to the powers of 3 in the denominators. We note that all other prime factors in the denominators do not exceed |B2g B2g−2 | the prime factors in 2g(2g−2)(2g−2)! . Automorphism groups of this order arise already for the constant map Gromov-Witten invariant.
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7. Local IP1 × IP1 Our last example is the Gromov-Witten theory of the Calabi-Yau Y which is the total space of the canonical bundle over IP1 × IP1 . We will study this using modularity of the B-model topological string on the mirror manifold X . To start with, let us review elementary facts about Y . Let A1 and A2 denote the classes of the two IP1 ’s in H2 (Y ). There is also one compact four cycle – the IP1 × IP1 itself, and denote by B the corresponding class in H4 (Y ). The intersection numbers of the cycles on Y are #(A1 ∩ B) = −2 = #(A2 ∩ B). The class C = A1 − A2 does not have a dual cycle in H4 (Y ), as it does not intersect B. From our discussion in Sect. 2, C will correspond to a non-normalizable modulus of the theory. For the normalizable modulus A we can take A2 , for example, so let us define A = A2 , C = A1 − A2 . The mirror manifold is a family of elliptic curves , which is given by the following equation [10,20] in IP1 × IP1 : x02 y02 + z 1 x12 y02 + x02 y12 + z 2 x12 y12 + x0 x1 y0 y1 = 0,
(7.1)
where [x0 : x1 ] and [y0 : y1 ] are homogeneous coordinates of the two IP1 ’s. The large radius point corresponds to z 1 = 0 = z 2 . Let t1 and t2 denote the periods of the one form λ around the 1-cycles mirror dual to A1 and A2 (which we also denote by A1 and A2 ): λ, t2 = λ, t1 = A1
A2
and let t D be the period around the 1-cycle mirror dual to B: λ. tD = B
The periods t1 and t2 compute the physical Kähler parameters, i.e. the masses of BPS D2-branes wrapping the two IP1 ’s.26 At large radius the complex structure parameters z 1 and z 2 are related to the Kähler parameters t1 , t2 of Y by z 1,2 ∼ e2πit1,2 . More specifically, we can find the periods ti in terms of the parameters z i as the solutions of the Picard-Fuchs equations of X , L1 = 21 − 2z 1 (1 + 2 )(1 + 21 + 22 ), L2 = 22 − 2z 2 (1 + 2 )(1 + 21 + 22 ),
(7.2)
26 The IP1 ’s of the embedding space of the mirror will hopefully not be confused with the two IP1 ’s generating H2 (Y ) on the A-model side.
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where i = z i ∂z∂ i for i = 1, 2. The solutions around the large radius point z 1 = 0 = z 2 can be determined by the Frobenius method from ω(z 1 , z 2 , r1 , r2 ) :=
∞ m,n=1
z r11 +m z r22 +n (−2(m + r1 ) − 2(n + r2 ) + 1) 2 (m + r1 + 1) 2 (n + r2 + 1)
as d 1 ti = ω(z 1 , z 2 , r1 , r2 ) . (2πi) dri r1,2 =0 Thus t1 (z 1 , z 2 ) = log(z 1 ) + 2z 1 + 2z 2 + 3z 12 + 12z 1 z 2 + 3z 22 + · · · , and similarly for t2 with z 1 and z 2 exchanged. By inverting the above, we get the mirror maps: z 1 = q1 − 2(q1 + q1 q2 ) + 3(q13 + q1 q22 ) − 4(q14 + q13 q2 + q12 q22 + q1 q23 ) + . . . , z 2 = q2 − 2(q2 + q1 q2 ) + 3(q23 + q2 q12 ) − 4(q24 + q23 q1 + q12 q22 + q2 q13 ) + . . . , (7.3) where qi = exp(2πiti ) for i = 1, 2. In addition to this there are two other solutions to the Picard-Fuchs equations. First, there is a double logarithmic solution, which is the period t D introduced previously. Second, there is a constant solution, corresponding to the period mirror to the D0 brane in the A-model. This constant period, together with m = t1 − t 2 =
λ, C
where C is the 1-cycle of the curve mirror dual to the class C of Y (again we use the same letter to denote mirror dual objects), should be regarded as constant parameters that enter in specifying the model. In fact, it is easy to see that the period m does not receive instanton corrections, i.e. qm = exp(2πim) satisfies qm = q1 /q2 = z 1 /z 2 , which is consistent with the interpretation of m as an auxiliary parameter. In the following we will denote the physical modulus by T , T = t2 =
λ, A
and define Q = exp(2πi T ). In order to find the modularity properties of the amplitudes, we now study in more detail the family of elliptic curves .
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7.1. The family of elliptic curves. The family of elliptic curves in (7.1) can be brought into Weierstrass form,27 y 2 = 4x 3 − g2 x − g3 with 22/3 (16z 12 + (1 − 4z 2 )2 + 8z 1 (−1 + 28z 2 )), 3 2 g3 = (64z 13 + (−1 + 4z 2 )3 − 48z 12 (1 + 44z 2 ) + z 1 (12 + 480z 2 − 2112z 22 ). 27
g2 =
Its j-function reads j (τ ) =
(16z 12 + (1 − 4z 2 )2 + 8z 1 (−1 + 28z 2 ))3 z 1 z 2 (16z 12 + (1 − 4z 2 )2 − 8z 1 (1 + 4z 2 ))2
.
(7.6)
As usual, by j (τ ) we mean that the j-function is a function of the standard complex parameter τ of the family of elliptic curves =C/(ZZ ⊕ τ ZZ). As it turns out, we have met this curve before! Recall that the j-function of the (2) modular curve, the SU (2) Seiberg-Witten curve, is (7.7) j (τ ) =
64(3 + u 2 )3 . (u 2 − 1)2
(7.7)
If we make the substitution −1/2
u=
1 1/2 qm −1/2 − (qm + qm ) 8z 2 2
(7.8)
in (7.7), we get exactly the j-function (7.6), using the fact that qm = z 1 /z 2 . Since the j-function captures all the coordinate-invariant data of the elliptic curve, the curves in the family mirror to local IP1 × IP1 are in fact isomorphic to the curves in the SU (2) Seiberg-Witten family, through reparameterization of the moduli space as in (7.8). In particular, it follows immediately that the curves in the family have monodromy group (2). 27 To do so, we first use the Segre embedding of IP1 × IP1 into IP3 given by the map
([x0 : x1 ], [y0 : y1 ]) → [X 0 : X 1 : X 2 : X 3 ] = [x0 y0 , x1 y0 , x0 y1 , x1 y1 ], where [x0 : x1 ] and [y0 : y1 ] are homogeneous coordinates of the two IP1 ’s and X i , i = 0, . . . , 3 are homogeneous coordinates of IP3 . Then IP1 × IP1 is given by the hypersurface X0 X3 − X1 X2 = 0
(7.4)
in IP3 . The family of elliptic curves is now given by the complete intersection of (7.4) and the hypersurface defined by X 02 + z 1 X 12 + X 22 + z 2 X 32 + X 0 X 3 = 0.
(7.5)
After a linear change of variable, (7.5) becomes linear in X 3 , so X 3 can be eliminated from (7.5) and (7.4) to get a cubic equation in IP2 . Then, given any cubic in IP2 we can use Nagell’s algorithm [9,11] to transform it into its Weierstrass form.
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We could also have found the monodromy transformations of the periods directly from the Picard-Fuchs equations, as we did for local IP2 , but it requires more work. The j-function approach, when the mirror geometry is a family of elliptic curves, provides a simpler way to determine the monodromy group, at least the part of it restricted to the physical periods. Fortunately, this is all that is relevant for our purposes. Using this result, we can borrow heavily the results from the SU (2) theory. In particular, using the expression for u in terms of modular forms of (2) in (5.14) and relating z 2 to the period T , we find28 −1/2 1/2
Q(qm , q) = qm
q
−3/2
− (2 + 2qm−1 ) q + qm
(5 − 4qm + 5qm2 ) q 3/2 + · · · , (7.9)
where q = e2πiτ , qm = e2π m and Q = e2πi T . From this expansion, we see that the period T does not only depend on τ ; the coefficients of the power series in q depend explicitly on the auxiliary parameter m (or qm ). 7.2. Genus 0, 1 and Yukawa coupling. Let us start by finding the partition function at genus 1. Recall that F1 is fixed by its modular properties and its behavior at the discriminant of the family of elliptic curves . In the local IP1 × IP1 case, we can show that F1 = − log η(τ )
(7.10)
transforms as required and has precisely the good behavior at the discriminant—this is the same expression as in SU (2) Seiberg-Witten theory. As a consistency check, if we expand (7.10) using thee expansion of q in terms of qm and Q we get 1 1 1 log(qm Q 2 ) − (1 + qm )Q − (1 + 4qm + qm2 )Q 2 24 6 12 1 2 3 3 − (1 + 9qm + 9qm + qm )Q + O(Q 4 ), 18
F1 = −
which reproduces precisely the genus one partition function of local IP1 × IP1 . Now consider the Yukawa coupling, i.e. the third derivative of F0 (m, T ) with respect to T , which we will need to compute higher genus amplitudes. Using ∂3 1 ∂ τ (m, T ) F0 (m, T ) = − 3 ∂T 2 ∂T and the expansion for τ in terms of qm and Q we get the following expansion: ∂3 F0 (m, T ) = −1 − 2(1 + qm )Q − 2(1 + 16qm + qm2 )Q 2 + O(Q 3 ). ∂T 3
(7.11)
However, what we would like to obtain is a modular expression for ∂∂T 3 F0 defined globally over the moduli space of complex structures, such as our expression (7.10) for F1 , not just an expansion in the large complex structure limit. To identify the modular form we make use of the change of variable (7.8), which relates the usual (2) curve to our curve with the auxiliary parameter qm . Through this change of variable, we identify the j-functions of the two curves, and correspondingly 3
28 Note that we could invert the series because q is just a parameter, i.e. it must be τ -independent. m
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the parameter τ , via the q-expansion of the j-function. In particular, this implies a relation between the periods a = a(T, m), where a is the usual Seiberg-Witten period, coming from the identification of the jfunctions, which we write schematically as j (a) = j (τ ) = j (T, m). As a result, acting on any function of τ (at fixed m), we get ∂a ∂ ∂ = . ∂T ∂ T ∂a For instance, we can write ∂ 3 F0 1 ∂a ∂τ 1 ∂τ =− . =− 3 ∂T 2 ∂T 2 ∂ T ∂a Now, we saw in Sect. 5 that √ ∂τ b = −2 (τ ), ∂a cd and we can compute that 1/2 1 ∂a d +c 1/2 −1/2 =− qm + qm (τ ) f := +2 , ∂T 2 b
(7.12)
using (7.11) and (7.9). In the above equations we used the modular forms b, c and d as defined in the (2) part of Appendix A. Putting all this together, we get √ 1/2 ∂ 3 F0 1 b d +c 1/2 −1/2 qm + qm (τ ) =− +2 ∂T 3 2 cd b which is a modular form of (2) of weight (−3), as expected. Note that f itself has weight zero. To summarize, given the function f = ∂∂aT in (7.12), which relates the a-period of the (2) curve to the T and m periods of the IP1 × IP1 curve, we directly obtain modular expressions for the higher genus amplitudes in terms of the modular expressions already obtained for SU (2) Seiberg-Witten theory. 7.3. Higher genus amplitudes. First, we can take the propagator to be E(τ ) = −
2πi E 2 (τ ), 6
which is the same propagator as in SU (2) Seiberg-Witten theory, up to a sign (see Sect. 5). The sign comes from the different conventions for the relative orientation of the A and the B-cycles. To get higher genus amplitudes, we use the by now familiar Feynman expansions with the above propagator. To relate the expansions to the SU (2) Seiberg-Witten expansions,
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we simply use the chain rule for derivatives: whenever we need to take derivatives with respect to T in the Feynman expansions, we use the function f given in (7.12) to write ∂ ∂ = f . ∂T ∂a This relates the amplitudes on the local IP1 × IP1 to those in the SU (2) SeibergWitten theory, up to an exactly modular form. Plugging all these results in the Feynman expansion for the genus 2 partition function F2 we get the nice and simple expression (0) for the modular function h 2 in terms of the partition functions FgSW , g ≤ 2 of SU (2) Seiberg-Witten theory: 1 E 22 1 SW 1/2 c+d −1/2 h (0) q F − . = F + + q + 2 2 m m 2 4 2 b 576 cd This is an interesting result. Through our modular formalism, we can express higher genus amplitudes of local Calabi-Yau manifolds in a very simple way in terms of higher genus amplitudes of the corresponding theory with no auxiliary parameters—in this case SU (2) Seiberg-Witten theory. More precisely, given two theories governed by elliptic curves with j-functions related by a change of variables (that generically also involves the auxiliary parameters), all one needs to do is to determine the function f = ∂∂aT relating the physical periods, and everything else follows from the formalism. Finally, by plugging in the known expansion for F2 (obtained for instance through (0) the topological vertex formalism) we could determine h 2 , and show that it is a modular 2 form of weight 0, as we did for local IP . We could also go to higher genera, and relate the expressions to the Seiberg-Witten expressions; we will not present the explicit formulae here, but it is straightforward to calculate them.
7.4. Seiberg-Witten limit. Let us end this section by showing that the double scaling limit to recover SU (2) Seiberg-Witten theory from the local IP1 × IP1 topological string amplitude is consistent with our results above. Since we know the j-function of the mirror family of elliptic curves in terms of the complex moduli z 1 and z 2 , we first express the limit in these parameters, and then show that taking the limit gives the j-function of the SU (2) Seiberg-Witten curve. The double scaling limit was explained in detail in [22,24]. Define first new parameters x and y satisfying z 1 = 1/4x 2 and z 2 = y/4, and then parameters x1 and x2 such that √ y . x1 = (1 − x), x2 = 1−x The double scaling limit is given by letting x1 = 2 u and x2 = 1/u, and then sending → 0. Taking this limit in our j-function (7.6) for the elliptic curve mirror to local IP1 × IP1 , we get j (τ ) =
64(3 + u 2 )3 , (u 2 − 1)2
which is indeed exactly the j-function of the SU (2) Seiberg-Witten curve.
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8. Open Questions and Speculations In this paper we showed how to use symmetries to constrain the topological string amplitudes. As a result, we obtained nice expressions for the amplitudes in terms of (almost) holomorphic modular forms. However, various open questions remained, and new ideas for future research emerged. i. Compact case. Our formalism is completely general, and applies to both compact and non-compact Calabi-Yau threefolds. However, all the examples that we worked out explicitly consisted in non-compact target spaces. As explained in Sect. 4.1, the reason is that in the compact case the period matrix τ I J does not have positive definite imaginary part. It would be interesting to understand how to get modular expressions in this case, perhaps using the closely related matrix N I J , as also explained in Sect. 4.1. ii. Full group of symmetries.In this paper, we considered the group of symmetries of the topological string generated by monodromies of the periods. However, as explained in the introduction, this is just a subgroup of the full group of symmetries, which consists in the group of ω-preserving diffeomorphisms. In the local case, the ω preserving diffeomorphisms were used in [1] to solve completely the topological string. It would be very interesting to see if this generalizes to compact Calabi-Yau manifolds. iii. Away from the weak coupling. In this work we obtained nice modular expressions for the topological string amplitudes genus by genus. However, the main object of study was the topological string wave function Z (gs , x), which should make sense at any value of the string coupling. It would be interesting to use the symmetries to constrain the topological string amplitude for all values of the string coupling. This would correspond to solving the Eqs. (3.1) away from the weak coupling regime. However, this may be hard, as one has to pick the correct non-perturbative definition of (3.1). Acknowledgements. We would like to thank Jim Bryan, Tom Coates, Robbert Dijkgraaf, Chuck Doran, Thomas Grimm, Minxin Huang, Marcos Mariño, Andrew Neitzke, Nikita Nekrasov, Yong-Bin Ruan, Albert Schwarz, Jan Stienstra, Cumrun Vafa and Don Zagier for useful discussions. We would also like to thank Ruza Markov for pointing out a few misprints in the first version of the paper. The research of M.A. is supported in part by a DOI OJI Award, the Alfred P. Sloan Fellowship, and the NSF grant PHY-0457317. A.K. is supported in part by the DOE-FG02-95ER40896 grant. V.B. is supported by an MSRI postdoctoral fellowship for the “New topological structures in physics” program, and by an NSERC postdoctoral fellowship.
Appendix A. Modular Forms and Quasi-Modular Forms In this appendix we review essential facts in the theory of modular forms and quasimodular forms, mainly in order to fix our conventions. Denote by H = {τ ∈ C|Im(τ ) > 0} the complex upper half-plane, and let ⊂ S L(2, ZZ) be a subgroup of finite index. The action of the modular group on H is given by Aτ + B A B ∈ . τ → , for γ = C D Cτ + D A modular form of weight k on is a holomorphic function f : H →C satisfying A B ∈ , f (γ τ ) = (Cτ + D)k f (τ ) for all γ = C D
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and growing at most polynomially in 1/Im(τ ) as Im(τ ) → 0. We can also define an almost holomorphic modular form of weight k on as a function fˆ : H →C satisfying the same transformation property and growth condition as above, but with the form fˆ(τ, τ¯ ) =
M
f m (τ )Im(τ )−m ,
m=0
for some integer M ≥ 0, where the functions f m (τ )’s are holomorphic. The constant term in the series, f 0 (τ ), is a quasi-modular form of weight k; it is holomorphic, but not quite modular. It has the form f 0 (τ ) =
M
h m (τ )E 2 (τ )m ,
m=0
where the h m (τ )’s are holomorphic modular forms and we defined the second Eisenstein series ∞ nq n , E 2 (τ ) = 1 − 24 (1 − q n ) n=1
which is itself quasi-modular of weight 2. Its almost holomorphic counterpart is defined as 3 . E 2∗ (τ, τ¯ ) = E 2 (τ ) − π Im(τ ) Note that there is an isomorphism between the ring of almost holomorphic modular forms and the ring of quasi-modular forms. A.1. Modular forms of (2). Our conventions for the theta functions with characteristics are as follows: a 1 2 θ (z, τ ) = q 2 (n+a) e2πi(n+a)(b+z) . b n As usual, we denote the (2) theta constants by
1 0 2 θ2 = θ (0|τ ), (0|τ ), θ3 = θ 0 0
θ4 = θ
0 1 2
(0|τ ).
We also define the fourth powers b := θ24 (τ ),
c := θ34 (τ ),
d := θ44 (τ ),
which satisfy the identity c = b + d. Also, η12 = 2−4 bcd, where η is the Dedekind η-function. Here are some useful formulae involving derivatives of modular forms: d 24q dq log(η) = E 2 , d 6q dq log(d) = E 2 − b − c, d 6q dq log(c) = E 2 + b − d, d 6q dq log(b) = E 2 + c + d.
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A.2. Modular forms of (3). For the congruence subgroup (3), the relevant theta constants (taking their third powers) are29 a := θ 3
1 6 1 6
(0, τ ), b := θ 3
1 6 1 2
(0, τ ), c := θ 3
1 6 5 6
(0, τ ), d := θ 3
1 2 1 6
(0, τ ),
satisfying the identities b = a + c,
d = a + αb,
2πi
i abcd. with α = e 3 . Moreover, the Dedekind η-function is given by η12 = 33/2 We need derivative formulae for these theta constants as well. Let us first define the six following modular forms of weight 2:
ac ab bc , t2 = 2 , t3 = 2 , η2 η η bd ad cd t4 = 2 , t 5 = 2 , t 6 = 2 . η η η t1 =
Then we found the relations: d log a dq d log b 8q dq d log c 8q dq d log d 8q dq 8q
1 τ +1 2 E2 = E 2 (τ ) − (t4 + t6 + αt3 ), 3 3 3 1 τ 2 = E2 = E 2 (τ ) + (t1 − t5 + t6 ), 3 3 3 τ +2 1 2 = E2 = E 2 (τ ) + (t4 + t5 − α 2 t2 ), 3 3 3 2 = 3E 2 (3τ ) = E 2 (τ ) + (−t1 + α 2 t2 + αt3 ). 3 =
Note that the second equality in each line are ‘triple’ analogs of the doubling identities for the Eisenstein series E 2 (τ ). Appendix B. Siegel Modular Forms A good reference on Siegel modular forms is Ghitza’s elementary introduction [19] and the more complete textbook [28]. Let be a subgroup of finite index of the symplectic group Sp(2r, ZZ) defined by A B T T T T T T Sp(2r, ZZ) = ∈ G L(2r, ZZ)|A C = C A, B D = D B, A D − C B = I , C D where I is the r × r identity matrix. Define the Siegel upper half space Hr = {τ ∈ Mat r×r (C)|τ T = τ, Im(τ ) > 0}; 29 We use the same variables to denote the fourth powers of the (2) theta constants and the third powers of the (3) theta constants, but it should always be clear from the context which subgroup we are considering.
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this is the space of r × r symmetric matrices with positive definite imaginary part. The action of on Hr is given by A B τ → (Aτ + B)(Cτ + D)−1 for γ = ∈ . C D A weight k (scalar-valued) Siegel modular form of is a holomorphic function f : Hr →C satisfying A B f (γ τ ) = det(Cτ + D)k f (τ ) for all γ = ∈ . C D Note that for r > 1 we do not need to impose the condition of holomorphicity at infinity in the definition of a modular form, as was the case for r = 1. Moreover, for r > 1 one can define more general objects, which transform under irreducible representations of G L(r,C). Given such a representation ρ : G L(r,C) → G L(V ), where V is a finite-dimensional vector space, we say that a function transforming under ρ is a Siegel modular form of weight ρ—see for instance [19]. We can also defined ‘generalized’ theta functions as ⎛ ⎞ a θ (z i , τ ) = exp ⎝πi (n i + a i )τi j (n j + a j ) + 2πi (z i + bi )n i ⎠ , b r n∈ZZ
ij
i
where a, b and z are vectors of length r . References 1. Aganagic, M., Dijkgraaf, R., Klemm, A., Mariño, M., Vafa, C.: Topological strings and integrable hierarchies. Commun. Math. Phys. 261, 451 (2006) 2. Aganagic, M., Klemm, A., Mariño, M., Vafa, C.: Matrix model as a mirror of Chern-Simons theory. JHEP 0402, 010 (2004) 3. Aganagic, M., Klemm, A., Mariño, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254, 425 (2005) 4. Aganagic, M., Mariño, M., Vafa, C.: All loop topological string amplitudes from Chern-Simons theory. Commun. Math. Phys. 247, 467 (2004) 5. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Holomorphic anomalies in topological field theories. Nucl. Phys. B 405, 279 (1993) 6. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994) 7. Bryan, J. et al.: Work in progress 8. Candelas, P., DeLa Ossa, X.C., Green, P.S., Parkes, L.: A Pair Of Calabi-Yau Manifolds As An Exactly Soluble Superconformal Theory. Nucl. Phys. B 359, 21 (1991) 9. Cassels, J.W.S.: Lectures on Elliptic Curves, London Mathematical Society Student Texts 24, Cambridge: Cambridge University Press, 1991 10. Chiang, T.M., Klemm, A., Yau, S.T., Zaslow, E.: Local mirror symmetry: Calculations and interpretations. Adv. Theor. Math. Phys. 3, 495 (1999) 11. Connell, I.: Elliptic curve handbook, Montreal: McGill University, 1996, available at http://www.math. mcgill.ca/connell/public/ECH1/, 1996 12. de Wit, B.: N = 2 electric-magnetic duality in a chiral background. Nucl. Phys. Proc. Suppl. 49, 191 (1996) ; de Wit, B.: N = 2 symplectic reparametrizations in a chiral background. Fortsch. Phys. 44, 529 (1996); de Wit, B., Lopes Cardoso, G., Lust, D., Mohaupt, T., Rey, S.J.: Higher-order gravitational couplings and modular forms in N = 2, D = 4 heterotic string compactifications. Nucl. Phys. B 481, 353 (1996) 13. Dijkgraaf, R.: Mirror symmetry and elliptic curves. In: The moduli space of curves (Texel Island, 1994), Boston MA: Birkhuser Boston, 1995, pp. 149–163
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14. Dijkgraaf, R., Gukov, S., Neitzke, A., Vafa, C.: Topological M-theory as unification of form theories of gravity. Adv. Theor. Math. Phys. 9, 593 (2005) 15. Dijkgraaf, R., Verlinde, E.P., Vonk, M.: On the partition sum of the NS five-brane. http://arXiv.org/list/ hepth/0205281, 2002 16. Douglas, M.R., Moore, G.W.: D-branes, Quivers, and ALE Instantons. http://arXiv.org/list/hepth/ 9603167, 1996 17. Farkas, H., Kra, I.: Theta Constants, Riemann Surfaces and the Modular Group. Providence, RI: Amer. Math. Soc., 2001 18. Gerasimov, A.A., Shatashvili, S.L.: Towards integrability of topological strings. I: Three-forms on Calabi-Yau manifolds. JHEP 0411, 074 (2004) 19. Ghitza, A.: An elementary introduction to Siegel modular forms. Talk given at the Number Theory Seminar, University of Illinois at Urbana-Champaign, http://www.math.mcgill.ca/~ghitza/uiuc-siegel1.pdf 20. Hori, K., Vafa, C.: Mirror symmetry. arXiv:hepth/0002222 21. Huang, M.X., Klemm, A.: Holomorphic anomaly in gauge theories and matrix models. http://arXiv.org/ list/hepth/0605195, 2006 22. Kachru, S., Klemm, A., Lerche, W., Mayr, P., Vafa, C.: Nonperturbative results on the point particle limit of N = 2 heterotic string compactifications. Nucl. Phys. B 459, 537 (1996) 23. Kaneko, M., Zagier, D.B.: A generalized Jacobi theta function and quasimodular forms. In: The moduli space of curves. Progr. Math. 129, Boston, MA:Birkhauser, 1995, pp. 165–172 24. Katz, S., Klemm, A., Vafa, C.: Geometric engineering of quantum field theories. Nucl. Phys. B 497, 173 (1997) 25. Klemm, A., Lerche, W., Theisen, S.: Nonperturbative effective actions of N = 2 supersymmetric gauge theories. Int. J. Mod. Phys. A 11, 1929 (1996) 26. Klemm, A., Mariño, M., Theisen, S.: Gravitational corrections in supersymmetric gauge theory and matrix models. JHEP 0303, 051 (2003) 27. Klemm, A., Zaslow, E.: Local mirror symmetry at higher genus. http://arXiv.org/list/hepth/9906046, 1999 28. Klingen, H.: Introductory lectures on Siegel modular forms. Vol. 20 of Cambridge Studies in Advanced Mathematics, Cambridge:Cambridge University Press, 1990 29. Mariño, M., Moore, G.W.: The Donaldson-Witten function for gauge groups of rank larger than one. Commun. Math. Phys. 199, 25 (1998); Mariño, M.: The uses of Whitham hierarchies. Prog. Theor. Phys. Suppl. 135, 29 (1999) 30. Moore, G.W., Witten, E.: Integration over the u-plane in Donaldson theory. Adv. Theor. Math. Phys. 1, 298 (1998) 31. Nekrasov, N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831 (2004) 32. Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. http://arXiv.org/list/hepth/ 0306238, 2003 33. Nekrasov, N.: Z-theory: Chasing M-F-Theory. Comptes Rendus Physique 6, 261 (2005) 34. Ruan, Y.-B. et al.: Work in progress 35. Seiberg, N., Witten, E.: Electric - magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19 (1994) [Erratum-ibid. B 430, 485 (1994)] 36. Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nucl. Phys. B 431, 484 (1994) 37. Vafa, C.: Quantum Symmetries Of String Vacua. Mod. Phys. Lett. A 4, 1615 (1989) 38. Vafa, C., Witten, E.: On orbifolds with discrete torsion. J. Geom. Phys. 15, 189 (1995) 39. Verlinde, E.P.: Attractors and the holomorphic anomaly. http://arXiv.org/list/hepth/0412139, 2004 40. Witten, E.: Quantum background independence in string theory. http://arXiv.org/list/hepth/9306122, 1993 Communicated by N.A. Nekrasov
Commun. Math. Phys. 277, 821–860 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0381-5
Communications in
Mathematical Physics
Construction of Quantum Field Theories with Factorizing S-Matrices Gandalf Lechner Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria. E-mail: [email protected] Received: 13 February 2007 / Accepted: 29 March 2007 Published online: 27 November 2007 – © Springer-Verlag 2007
Abstract: A new approach to the construction of interacting quantum field theories on two-dimensional Minkowski space is discussed. In this program, models are obtained from a prescribed factorizing S-matrix in two steps. At first, quantum fields which are localized in infinitely extended, wedge-shaped regions of Minkowski space are constructed explicitly. In the second step, local observables are analyzed with operatoralgebraic techniques, in particular by using the modular nuclearity condition of Buchholz, d’Antoni and Longo. Besides a model-independent result regarding the Reeh–Schlieder property of the vacuum in this framework, an infinite class of quantum field theoretic models with non-trivial interaction is constructed. This construction completes a program initiated by Schroer in a large family of theories, a particular example being the Sinh-Gordon model. The crucial problem of establishing the existence of local observables in these models is solved by verifying the modular nuclearity condition, which here amounts to a condition on analytic properties of form factors of observables localized in wedge regions. It is shown that the constructed models solve the inverse scattering problem for the considered class of S-matrices. Moreover, a proof of asymptotic completeness is obtained by explicitly computing total sets of scattering states. The structure of these collision states is found to be in agreement with the heuristic formulae underlying the Zamolodchikov-Faddeev algebra.
1. Introduction In relativistic quantum field theory, the rigorous construction of models with non-trivial interaction is still a largely open problem. Apart from the well-known results of Glimm and Jaffe [33], most interacting quantum field theories are treated only perturbatively, usually without any control over the perturbation series.
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The main difficulties in the construction of interacting quantum field theories arise from the principle of Einstein causality, demanding that physical observables must be strictly local, i.e. represented by commuting operators when spacelike separated. In view of this problem, several authors [17,18,55,58,48,50] have proposed to construct models by first considering easier manageable, non-local theories, and then passing to a local formulation in a second step. A particular example of such a constructive scheme is the program initiated by Schroer [58,55,56], which deals with quantum field theories on two-dimensional Minkowski space. Here the interaction of the models to be constructed is not formulated in terms of classical Lagrangians, but rather by prescribed S-matrices, i.e. the inverse scattering problem is considered. The S-matrices are assumed to be factorizing [36], i.e. of the type found in completely integrable models such as the Sinh-Gordon-, O(N ) Sigma-, or Thirring model [1,25]. Adopting the idea of constructing local theories by first considering non-local auxiliary quantities, one starts in this program from a given factorizing S-matrix S and defines two non-local quantum fields φ, φ depending on S [42]. Although these fields are not local, they are relatively wedge-local to each other in the following sense. Consider the so-called right wedge W R := {x ∈ R2 : x1 > |x0 |}, (1.1) and its causal complement W L := W R = −W R , the left wedge. Then φ(x) and φ (y) commute (in a suitable sense) if the wedges W L + x and W R + y are spacelike separated. Hence φ(x) is not, as usual, localized at the spacetime point x, but rather spread out over the infinitely extended spacetime region W L + x. The advantage of these non-local field operators is that they can be cast into a very simple form in momentum space. In fact, the only difference to free fields are the deformed commutation relations of their creation and annihilation parts, which form a representation of the Zamolodchikov-Faddeev algebra [63]. In particular, φ and φ create only single particle states from the vacuum, without accompanying vacuum polarization clouds. In view of this special property, such operators have been termed polarizationfree generators [56], see [14] for a model-independent discussion of this concept. The first step of the inverse scattering construction of models with factorizing S-matrices, i.e. the construction and analysis of their wedge-local fields, has by now been completed for a large class of underlying scattering operators [42,58,17]. It is the aim of the present article to accomplish the second step of the construction, i.e. the passage from the wedge-local fields to theories complying with the principle of locality, for the family of S-matrices considered in [42]. Usually the task of classifying and constructing quantum field theories with factorizing S-matrices is taken up in the so-called form factor program [59,6]. In that approach, one studies local fields A in terms of their matrix elements in scattering states (form factors). The n-point functions of A are then represented as infinite series of integrals over form factors. These form factors have been calculated for a multitude of models [4,30,7,5], at least for the lowest particle numbers. But in almost all cases1 one is still lacking control over the series representing the n-point functions [6]. This is due to the complicated form factor functions, and can be understood as a consequence of the complicated momentum space structure a local quantum field must have in the presence of interaction. So at present, the existence of models with prescribed S-matrices cannot be decided within the form factor program. 1 The only non-trivial example of a proof of convergence of a form factor expansion which is known to us is the case of a two-point function in the Yang-Lee model (F. A. Smirnov, private communication).
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If a given collision operator is expected to be related to a classical Lagrangian, there is also the possibility of applying the Euclidean techniques of constructive quantum field theory. Along these lines, the existence of the Sine-Gordon model and the non-triviality of its S-matrix have been established by Fröhlich [31,32]. But the hard task of explicitly computing the scattering operator and making contact with the form factor approach is still an open problem in this framework. In the context of the wedge-local fields φ, φ , local observables can be characterized by commutation relations with φ and φ [58]. Solving these relations amounts to solving the form factor program. We will follow here a different approach, motivated by the observation that for the analysis of basic questions, such as the existence of theories with certain properties, it is not necessary to have explicit expressions for strictly local quantities. The problem to decide if a given factorizing S-matrix is realized as the collision operator of a well-defined quantum field theory can be solved by considering only the structure of observables localized in wedge regions. To accomplish this task, one has to answer the question whether the wedge-local models defined by the fields φ, φ contain also observables localized in bounded spacetime regions. A strategy how to solve this existence problem was proposed in [17]. The main idea is to consider the algebras generated by bounded functions of φ, φ rather than the fields themselves. One proceeds to a net of wedge algebras, i.e. a collection of von Neumann algebras A(W ), where W runs through the family {W R + x, W L + x : x ∈ R2 } of all wedges in two-dimensional Minkowski space. In this formulation, powerful operator-algebraic techniques become available for the solution of the existence problem, which have not been employed in other approaches. It has been shown in [17] that non-trivial observables localized in a double cone region of the form W R ∩ (W L + x), x ∈ W R (cf. Fig. 2.1, p. 825) do exist if the so-called modular nuclearity condition [15] holds, i.e. if the map Ξ (x) : A(W R ) −→ H,
Ξ (x)A := ∆1/4 U (x)AΩ,
(1.2)
is nuclear. Here ∆ denotes the modular operator [39] of (A(W R ), Ω). So the existence of local observables can be established by estimates on wedge-local quantities. In the context of theories with factorizing S-matrices, the crucial question arises whether the modular nuclearity condition holds in such models. We will show here that this condition takes a very concrete form in these models, and can be solved by analyzing analytic continuations of form factors of observables localized in wedges. As our main result, we will give a proof of the nuclearity condition for a large class of underlying S-matrices, thereby establishing the existence of the corresponding models as well-defined, local quantum field theories. Moreover, once the modular nuclearity condition has been established, it is possible to apply the usual methods of scattering theory. Doing so, we will compute total sets of n-particle collision states in these models, and prove that the construction solves the inverse scattering problem. This article is organized as follows. In Sect. 2, we extend the analysis of [17] and derive further consequences of the modular nuclearity condition in a model-independent, operator-algebraic framework. It is shown there that the Reeh–Schlieder property of the vacuum [61], which is a prerequisite for doing scattering theory [2], follows from the nuclearity condition (Theorem 2.5). For the sake of self-containedness, the basic definitions and results regarding the models based on the fields φ, φ are recalled in Sect. 3. Also the classes of factorizing S-matrices which we consider are defined there (Definitions 3.1 and 3.3).
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In Sect. 4, analytic continuations of form factors of observables localized in wedges are studied. It is then shown in Sect. 5 how these analytic properties can be used to verify the modular nuclearity condition. We obtain two proofs of different generality (Theorems 5.6 and 5.8). Section 6 is devoted to a study of the collision states of the constructed models. It is shown that our construction solves the inverse scattering problem for the considered class of S-matrices (Theorem 6.3), and a proof of asymptotic completeness is given (Proposition 6.2). The paper closes in Sect. 7 with our conclusions, and the technical proof of a lemma needed in Sect. 4 can be found in the Appendix. This article is based on the PhD thesis of the author [45].
2. Construction of Local Nets from a Wedge Algebra In this section we discuss some model-independent aspects of the construction procedure. In contrast to the following chapters, we will here base our analysis only on assumptions which are satisfied in a wide class of quantum field theories, and do not use the special structure of the integrable models to be studied later. As explained in the Introduction, it is our aim to construct strictly local quantum field theories, but use auxiliary objects which are localized only in wedge regions during the construction. We therefore consider an algebra M modelling the observables localized in the reference wedge W R (1.1), and a representation U of the two-dimensional translation group (R2 , +). Given these data, we will construct a corresponding quantum field theory by specifying, for arbitrary regions O in two-dimensional Minkowski space, the algebras A(O) containing all its observables localized in O, and show that they have the right physical properties. The “wedge algebra” M is taken to be a von Neumann algebra acting on a Hilbert space H, which in order to exclude trivialities we assume to satisfy dimH > 1. Moreover, we require A1) U is strongly continuous and unitary. The joint spectrum of the generators P0 , P1 of U (R2 ) is contained in the forward light cone { p ∈ R2 : p0 ≥ | p1 |}. There is an up to a phase unique unit vector Ω ∈ H which is invariant under the action of U. A2) Ω is cyclic and separating for M. A3) For each x ∈ W R , the adjoint action of the translation U (x) induces endomorphisms on M, M(x) := U (x)MU (x)−1 ⊂ M,
x ∈ WR .
(2.1)
Assumption A1) is standard in quantum field theory [61,34], and identifies Ω as the vacuum vector. A2) and A3) are abstract characterizations of M as an algebra of observables localized in W R [11,17]. It should be mentioned that Assumptions A1)–A3) put strict constraints on the algebraic structure of M. The following result has been found in [47,26]. Lemma 2.1. Consider a triple (M, U, H) satisfying Assumptions A1)–A3). Then M is a type III1 factor according to the classification of Connes [22].
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Fig. 2.1. The double cone O x,y (2.2) and its causal complement O x,y = (W L + x) ∪ (W R + y)
Keeping this structure of M in mind, let us recall how a net O → A(O) of local algebras can be constructed from the data (M, U, H) [11,17]. As the operators in M(x) (2.1) are interpreted as observables localized in the translated right wedge W R + x, the elements of the commutant M(x) correspond to observables in its causal complement W L + x, where W L := W R = −W R is the left wedge. For an operator A representing an observable in the double cone region Ox,y with vertices x, y, Ox,y := (W R + x) ∩ (W L + y),
y − x ∈ WR ,
(2.2)
Einstein causality demands that A must commute with both algebras, M(x) and M(y) (see Fig. 2.1). The maximal von Neumann algebra of operators A ∈ B(H) compatible with this condition is A(Ox,y ) := M(x) ∩ M(y) .
(2.3)
Denoting the set of all double cones in R2 by O := {Ox,y : y − x ∈ W R }, this definition is extended to arbitrary regions R ⊂ R2 by additivity, A(R) := A(O). (2.4) R⊃O∈O
This prescription determines in particular the locally generated subalgebras A(W R ) ⊂ M, A(W L ) ⊂ M associated to the right and left wedges, and the von Neumann algebra A(R2 ) ⊂ B(H) of all local observables. It is straightforward to verify that the so defined algebras A(O) comply with the basic principles of isotony, locality and covariance [34], i.e. they fulfill [11, Sect. III], O, O1 , O2 ⊂ R2 , Isotony: A(O1 ) ⊂ A(O2 )
for O1 ⊂ O2 .
(2.5)
A(O1 ) ⊂ A(O2 )
for O1 ⊂ O2 .
(2.6)
Locality:
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Translation Covariance: U (x)A(O)U (x)−1 = A(O + x),
x ∈ R2 .
(2.7)
As a consequence of A2), the modular theory of Tomita and Takesaki (cf., for example, [39]) applies to the pair (M, Ω). It has been shown by Borchers [11] that in the present situation, the modular unitaries and modular group of (M, Ω) can be used to extend U to a representation of the proper Poincaré group, under which the net O → A(O) also transforms covariantly. But this fact will not be needed in our subsequent considerations. The three properties of the algebras A(O) mentioned above allow to interpret the elements of A(O) as observables which are localized in O ⊂ R2 . However, two important properties of these algebras are still missing. First of all, it is not clear if our definition contains any non-trivial observables localized in bounded spacetime regions, i.e. the intersections (2.3) could be trivial in the sense that A(O) = C·1. Since a quantum field theory should contain local observables, at least in spacetime regions above some minimal size, a condition implying the non-triviality of the algebras (2.3) is necessary. Thinking of applications to the explicit construction of models in an inverse scattering approach, one would also like to implement the postulate that the models defined by the observable algebras A(O) have a well-defined S-matrix. In the framework of algebraic quantum field theory, collision states and the S-matrix can be calculated with the help of Haag-Ruelle scattering theory [2]. For this method to be applicable, however, two additional conditions have to be satisfied. Firstly, more detailed information about the energy momentum spectrum encoded in U is needed [2]. As we can choose U from the outset, this requirement poses no difficulties here. But besides these spectral properties, scattering theory relies on the notion of quasi-localized excitations of the vacuum, which can only be constructed if the Reeh–Schlieder property [61,2] holds, i.e. if the vacuum vector Ω is cyclic for the local algebras A(O). In [17], the following additional assumption, known as the modular nuclearity condition in the literature [15,16], was made to exclude the case of a quantum field theory without local observables. A4) Let ∆ denote the modular operator of (M, Ω). Then for x ∈ W R , the maps Ξ (x) : M → H,
Ξ (x)A := ∆1/4 U (x)AΩ
(2.8)
are assumed to be nuclear.2 The modular nuclearity condition A4) is known [15] to imply the split property [24] for the inclusion M(x) ⊂ M, x ∈ W R , i.e. the existence of a type I factor Nx such that M(x) ⊂ Nx ⊂ M.
(2.9)
Moreover, it has been shown in [17] that this inclusion is even a standard split inclusion in the terminology of [24], i.e. there exist vectors in H which are cyclic and separating for the three algebras M, M(x) and M ∩ M(x) . Since M (and hence M(x), too) is a factor (Lemma 2.1), the standard split property of M(x) ⊂ M is equivalent to the following condition [23,24]: 2 The definition of a nuclear map between two Banach spaces is recalled in Sect. 5.
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A4 ) For x ∈ W R , there exists a unitary Vx : H → H ⊗ H such that3 Vx M ∨ M(x) Vx∗ = M ⊗ M(x), Vx M
N Vx∗
= M ⊗ N,
M ∈ M , N ∈ M(x).
(2.10) (2.11)
Given A1)–A3), Condition A4) implies A4 ), but A4 ) is slightly weaker than A4) [15]. In the present section, devoted to a model-independent analysis of the algebraic structure, A4 ) turns out to be the more convenient condition to work with. However, the somewhat stronger modular nuclearity condition A4) has the advantage that it can be checked more easily in concrete applications. We therefore formulate also the results of this section in terms of the latter condition, and begin by recalling the non-triviality result of [17]. Theorem 2.2 [17]. Consider a triple (M, U, H) satisfying Assumptions A1)–A4). Then the double cone algebras A(O) (2.3), O ∈ O, are isomorphic to the hyperfinite type III1 factor. As type III algebras, the double cone algebras are far from trivial, and therefore local observables exist in abundance if the modular nuclearity condition is satisfied. Moreover, it follows that the set of vectors which are cyclic for a given A(O), O ∈ O, is dense G δ in H [17]. In the remainder of this section, we will show that it can also be deduced in the general situation described by Assumptions A1)–A4) that Ω lies in this set of cyclic vectors for arbitrary O ∈ O. In a particular example of a triple (M, U, H), the cyclicity of the vacuum has already been established by Buchholz and Summers by explicit calculation of local observables [20]. In the following lemma, we start our analysis by comparing M to the locally generated wedge algebra A(W R ) ⊂ M (2.4). Lemma 2.3. Consider a triple (M, U, H) satisfying Assumptions A1)–A4). Then M is locally generated, i.e. A(W R ) = M. Proof. Let x ∈ W R be fixed and consider the sequence of double cones On := O0,nx = W R ∩ (W L + nx),
n ∈ N.
(2.12)
As x ∈ W R , this sequence is increasing in the sense that On ⊂ On+1 , n ∈ N. Moreover, it exhausts all of W R , i.e. every bounded subset of W R lies in some On . Hence n A(On ) = A(W R ). The left vertex of each On is the origin, and the algebras A(On ) are according to the definition (2.3) given by A(On ) = M ∩ M(nx) . The split property (2.10) provides us with a unitary Vx : H → H ⊗ H implementing an isomorphism between A(O1 ) = M ∨ M(x) and M ⊗ M(x). In view of (2.11), we find by restriction to M , Vx M Vx∗ = M ⊗ 1, and since M(nx) ⊂ M(x) (A3), also Vx A(On ) Vx∗ = Vx M ∨ M(nx) Vx∗ = M ⊗ M(nx),
(2.13)
n ∈ N.
3 We write M ∨ M(x) to denote the von Neumann algebra generated by M and M(x).
(2.14)
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We thus obtain
A(W R ) =
A(On ) = Vx∗
n∈N
= Vx∗ M ⊗
M ⊗ M(nx) Vx n∈N
M(nx)
Vx .
(2.15)
n∈N
In the last step, we used the commutation theorem for tensor products of von Neumann algebras (or rather, a consequence thereof, see [62, Cor. IV.5.10]). Now consider M∞ := n M(nx). By construction, this algebra is stable under translations, U (y)M∞ U (y)−1 ⊂ M∞ , y ∈ R2 . The same is true for its commutant M∞ , which furthermore has Ω as a cyclic vector since it contains M , and Ω is cyclic for M (A2). But as Ω is (up to multiples) the only translation invariant vector and the spectrum condition holds (A1), it follows by standard arguments (cf., for example, [12]) that M∞ = B(H), i.e. M∞ = C · 1. Inserting this equality into (2.15) yields A(W R ) = Vx∗ (M ⊗ 1)Vx , which in view of (2.13) equals M . Hence the claim A(W R ) = M follows. To go on, we need another lemma, which is due to Müger [49, Lemma 2.7]. Lemma 2.4 (Müger). Let (M, U, H) satisfy Assumptions A1)–A4), and consider three points x, y, z ∈ R2 such that y − x ∈ W R and z − y ∈ W R . Then A(Ox,y ) ∨ A(O y,z ) = A(Ox,z ). Geometrically speaking, this lemma states that the algebras of two double cones O1 , O2 having one of their (left or right) vertices in common generate the algebra of the smallest double cone containing O1 and O2 . It thus establishes a relation between the algebras of double cones of different sizes, and enables us to derive the Reeh–Schlieder property. In the following theorem, we give a proof of this property and some related consequences of Lemma 2.3 and 2.4. Theorem 2.5. Consider a triple (M, U, H) satisfying A1)–A4) and the local net O → A(O) defined by (2.3,2.4). Then a) The Reeh–Schlieder property holds, i.e. Ω is cyclic and separating for each double cone algebra A(O), O ∈ O. b) Haag duality holds, i.e. A(O) = A(O ) for any double cone or wedge O. c) Weak additivity holds, i.e. for any open region O ⊂ R2 , A(O + x) = A(R2 ) = B(H). (2.16) x∈R2
Proof. a) Given any two double cones O, O˜ ∈ O, there exists n ∈ N and translations x1 , . . . , xn ∈ R2 such that O˜ ⊂ ((O + x1 ) ∪ . . . ∪ (O + xn )) ,
(2.17)
and (O + xk ), (O + xk+1 ) have one vertex in common, k = 1, . . . , n − 1. So, by iterated application of Lemma 2.4, it follows that ˜ = A(R2 ). A(O + x) = A( O) (2.18) x∈R2
˜ O O∈
Construction of Quantum Field Theories with Factorizing S-Matrices
829
Hence we can use the standard Reeh–Schlieder argument making use of the spectrum condition of U (cf., for example, [2]) to show that Ω is cyclic for A(O) if and only if it is cyclic for A(R2 ). But in view of Lemma 2.3, M = A(W R ) is contained in A(R2 ). Since Ω is cyclic for M, it is also cyclic for A(R2 ), and hence for A(O). It has been shown by Borchers [11] that the modular conjugation J of (M, Ω) acts as the total spacetime reflection, J A(O)J = A(−O). Since W L = −W R , this implies together with Lemma 2.3 A(W L ) = J MJ = M , i.e. wedge duality holds. Taking into account the translation covariance of the net, we furthermore find A(Ox,y ) = M(x) ∨ M(y) = A(W L + x) ∨ A(W R + y) = A(Ox,y ),
(2.19)
showing the Haag duality of the net (b). According to the above remarks, A(R2 ) contains M and M . Since M is a factor, also A(R2 ) = B(H) follows, as the last claim to be proven. Besides the properties of the net O → A(O) mentioned in Theorem 2.5, further additional features like the split property for double cones, the time slice property and n-regularity can be derived, as has been shown by Müger [49]. The strong results of Theorems 2.2 and 2.5 open up a new perspective on the construction of quantum field theories on two-dimensional Minkowski space, emphasizing the role of the local observable algebras. Each triple (M, U, H) satisfying Assumptions A1)–A4) gives rise to a net of local algebras, which can be interpreted as the (non-trivial) observable algebras of a well-defined quantum field theory satisfying the Reeh–Schlieder property. Hence model theories can be constructed by finding examples of such triples. In the following sections, we will construct these objects, verify Assumptions A1)–A4), and discuss the properties of the corresponding model theories. 3. A Class of Models with Factorizing S-Matrices We now turn to the concrete construction of interacting quantum field theories on twodimensional Minkowski space. For simplicity, we consider here models containing only a single species of particles4 of mass m > 0. We will use the rapidity θ to parametrize the (one-dimensional) upper mass shell according to p(θ ) := m(cosh θ, sinh θ ). Our approach is that of inverse scattering theory, i.e. a given S-matrix S is the input in the construction. The family of theories we will study is characterized by the condition that S is factorizing. (For an introduction to factorizing S-matrices, see for example the review [25].) This term derives from the fact that in a model with a factorizing S-matrix, all scattering amplitudes are products of delta distributions and a single function [36], the so-called scattering function S2 . On rapidity wavefunctions Ψnin (θ1 , . . . , θn ) of n incoming particles, the S-matrix S therefore acts as a multiplication operator, S2 (|θl − θk |) · Ψnin (θ1 , . . . , θn ). (3.1) (SΨnin )(θ1 , . . . , θn ) = 1≤l
In particular, the particle number is a conserved quantity in collision processes governed by a factorizing S-matrix. This feature is typical for completely integrable models, which provide a rich class of examples for such scattering operators [1]. 4 The extension of the program to models with a richer particle spectrum is currently under investigation.
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G. Lechner
Basic properties of S, like unitarity, crossing symmetry and its analytic properties, imply corresponding properties of the scattering function S2 [1,25,6], which we take as a definition. Here and in the following, we write S(a, b) := {ζ ∈ C : a < Im ζ < b}
(3.2)
for strips in the complex plane. Definition 3.1 (Scattering functions). A scattering function is a bounded and continuous function S2 : S(0, π ) → C which is analytic in the interior of this strip and satisfies, θ ∈ R, S2 (θ ) = S2 (θ )−1 = S2 (θ + iπ ) = S2 (−θ ). (3.3) The set of all scattering functions is denoted S. A special scattering function is S2free (θ ) = 1, which belongs to the interaction-free S-matrix S free = id (3.1). A simple example with non-trivial interaction is given by the Sinh-Gordon model. By comparison with perturbation theory, the scattering function of this model is expected to be [3] S2ShG (θ ) =
sinh θ − i sin b , sinh θ + i sin b
(3.4)
where the parameter b is related to the coupling constant g of the Sinh-Gordon Lagrangian by b = πg 2 (4π + g 2 )−1 . Fixing S2 ∈ S, we now recall the construction of an associated triple (M, U, H) S2 consisting of a Hilbert space H, a representation U of the translations on H, and a “wedge algebra” M ⊂ B(H) of the type studied in Sect. 2. For details, we refer the reader to [42,45,55]. To describe the Hilbert space, we introduce on L 2 (Rn ) an S2 -dependent representation Dn of the group Sn of permutations of n letters. Given ρ ∈ Sn , we put (Dn (ρ) f n ) (θ1 , . . . , θn ) = S ρ (θ1 , . . . , θn ) · f n (θρ(1) , . . . , θρ(n) ), S2 (θρ(l) − θρ(k) ). S ρ (θ1 , . . . , θn ) :=
(3.5) (3.6)
1≤lρ(k)
In particular, the transpositions τ j , j = 1, . . . , n − 1, are represented as (Dn (τ j ) f n )(θ1 , . . . , θn ) = S2 (θ j+1 − θ j ) · f n (θ1 , . . . , θ j+1 , θ j , . . . , θn ).
(3.7)
Using the properties (3.3) of S2 , it has been shown in [45] (see also [46,42]) that Dn is a unitary representation of Sn on L 2 (Rn ), and that the mean over Dn , 1
Dn (ρ), (3.8) Pn := n! ρ∈Sn is the orthogonal projection onto the Dn -invariant functions in L 2 (Rn ). With the help of the “S2 -symmetrization” Pn , we define the Hilbert space H of the model with scattering function S2 as H0 := C,
Hn := Pn L 2 (Rn ), n ≥ 1,
H :=
∞ n=0
Hn .
(3.9)
Construction of Quantum Field Theories with Factorizing S-Matrices
831
The vectors in H are sequences Ψ = (Ψ0 , Ψ1 , . . . ), Ψn ∈ Hn , such that the norm corresponding to the scalar product Ψ, Φ := Ψ0 Φ0 + ∞ n=1 Ψn , Φn L 2 (Rn ) is finite. Due to its invariance under Dn (τ j ) (3.7), a function Ψn ∈ Hn has the symmetry property Ψn (θ1 , . . . , θ j+1 , θ j , . . . θn ) = S2 (θ j − θ j+1 ) · Ψn (θ1 , . . . , θ j , θ j+1 , . . . , θn ).
(3.10)
Note that for S2 = 1, this construction yields precisely the Bose Fock space over H1 . For generic S2 ∈ S, we refer to H as the S2 -symmetric Fock space. As a domain for some unbounded operators on H, we also introduce the dense subspace of terminating sequences (Ψ0 , Ψ1 , . . . , Ψn , 0, 0, . . . ), Ψk ∈ Hk , which will be denoted D. For example, the particle number operator N , (N Ψ )n := n · Ψn , is well defined on D. On H, there acts a representation U of the proper Poincaré group5 P+ . The proper ↑ orthochronous Poincaré transformations (x, λ) ∈ P+ consisting of a boost with rapidity parameter λ ∈ R and a subsequent translation along x ∈ R2 are represented as, Ψ ∈ H, n
p(θk ) · x · Ψn (θ1 − λ, . . . , θn − λ), (3.11) (U (x, λ)Ψ )n (θ1 , . . . , θn ) := exp i k=1
and the reflection j (x) := −x as (U ( j)Ψ )n (θ1 , . . . , θn ) := Ψn (θn , . . . , θ1 ).
(3.12)
For the proof that U is an (anti-) unitary representation of P+ on H, see [42]. By inspection of U , it follows that Ω := (1, 0, 0, . . . ) is up to multiples the only U -invariant vector in H. The representation of the translations required in the discussion of the previous section will be identified with the restriction of U to the translation subgroup, and we will also employ the shorthand notation U (x) := U (x, 0). Clearly, U satisfies the spectrum condition and is strongly continuous, i.e. Assumption A1) of Sect. 2 is satisfied. Having specified the Hilbert space H and the representation U , we now turn to the construction of the “wedge algebra” M ⊂ B(H). On the S2 -symmetric Fock space, there acts an algebra of creation and annihilation operators z † (ψ), z(ψ), ψ ∈ H1 . These are unbounded operators, in general, but always contain D in their domains. They are defined by, Φ ∈ D, (z † (ψ)Φ)n :=
√
n Pn (ψ ⊗ Φn−1 ),
z(ψ) := z † (ψ)∗ ,
ψ ∈ H1 .
(3.13)
Since z(ψ)Ω = 0, z † (ψ)Ω = ψ, we call z and z † annihilation and creation operators, respectively. The following bounds with respect to the particle number operator N hold [17]: z(ψ)Φ ≤ ψN 1/2 Φ, z † (ψ)Φ ≤ ψ(N + 1)1/2 Φ, Φ ∈ D.
(3.14)
† From time to time, we will also work with the distributions z(θ are related
), z (θ ), which # to the above operators by the formal integrals z (ψ) = dθ ψ(θ )z # (θ ), z # = z, z † . 5 Actually, this representation extends to the whole Poincaré group P, but this fact will not be needed here.
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G. Lechner
These distributions satisfy the relations of the Zamolodchikov-Faddeev algebra [63,59], z # (θ1 )z # (θ2 ) = S2 (θ1 − θ2 ) z # (θ2 )z # (θ1 ), z(θ1 )z (θ2 ) = S2 (θ2 − θ1 ) z (θ2 )z(θ1 ) + δ(θ1 − θ2 ) · 1, †
†
(3.15a) (3.15b)
where 1 denotes the identity in B(H). For later reference, we mention here that in view of the definition (3.13), there holds in particular √ z † (θ1 ) · · · z † (θn )Ω, Ψ = n! Ψn (θ1 , . . . , θn ), Ψ ∈ H. (3.16) Following Schroer [55,56,58], the Zamolodchikov operators can be combined to define a quantum field φ. For Schwartz test functions f ∈ S (R2 ), we put 1 † + − ± f (θ ) := (3.17) φ( f ) := z ( f ) + z( f ), d 2 x f (x)e±i p(θ)·x . 2π This field has a number of interesting properties [42]. To begin with, it transforms covariantly under the adjoint action of the proper orthochronous Poincaré transformations U (x, λ), and it has the Reeh–Schlieder property. Moreover, φ is a solution of the Klein-Gordon equation since it creates single particle states from the vacuum. Convenient mathematical properties of φ( f ) are its essential self-adjointness for real f , and the fact that f → φ( f )Ψ , Ψ ∈ D, is a vector valued tempered distribution. As is familiar from free field theory, φ has well-defined time zero fields ϕ, π , which are given by ϕ( f ) = z † ( fˆ) + z( fˆ− ), π( f ) = i z † (ω fˆ) − z(ω fˆ− ) ,
fˆ(θ ) := f (m sinh θ ), fˆ− (θ ) := fˆ(−θ ).
(3.18a) (3.18b)
Here the single particle Hamiltonian ω = m cosh θ acts as a multiplication operator on its domain in H1 . However, as a consequence of the commutation relations (3.15), φ is not local, in general. Locality holds if and only if S2 = 1, and in this case φ coincides with the free scalar field of mass m. For generic scattering function S2 ∈ S, it was discovered by Schroer [55] that although φ is not strictly local, it is not completely delocalized either. The localization properties of φ are most easily understood by introducing a second field operator φ [42], φ ( f ) := U ( j)φ( f j )U ( j), f j (x) := f (−x). (3.19) The two fields φ, φ are relatively wedge-local in the following sense: For (real) test functions f, g with supp f ⊂ W L , supp g ⊂ W R , the selfadjoint closures of φ( f ) and φ (g) commute, i.e. [eiφ( f ) , eiφ (g) ] = 0. Therefore φ(x) can be consistently interpreted as being localized in the left wedge W L + x, and φ (y) is localized in the right wedge W R + y. Switching to the algebraic formulation, we consider the “wedge algebra” M := eiφ( f ) : f ∈ SR (W L ) = eiφ ( f ) : f ∈ SR (W R ) . (3.20) Here the first equality defines M, and the second is a result of [17].
Construction of Quantum Field Theories with Factorizing S-Matrices
833
For f ∈ S (R2 ) with supp f ⊂ W L , the field operator φ( f ) satisfies [42] Ψ, [φ( f ), A] Φ = 0,
A ∈ M, Ψ, Φ ∈ D.
(3.21)
Analogously, one can show that for testfunctions h ∈ S (R), supp h ⊂ R− , Ψ, [ϕ(h), A] Φ = 0, Ψ, [π(h), A] Φ = 0
A ∈ M, Ψ, Φ ∈ D.
(3.22)
The definition of M completes the data (M, U, H) S2 . Assumptions A2) and A3), regarding the cyclicity and separating property of Ω for M, and the isotony of this algebra under translations inside W R , can be deduced from the Reeh–Schlieder property and the translation covariance of φ and φ [42,45]. We note down these facts as a theorem. Theorem 3.2 [42]. Let S2 ∈ S. Then the triple (M, U, H) S2 defined through (3.9), (3.11) and (3.20) satisfies Assumptions A1)–A3) of Sect. 2. With A1)–A3) fulfilled, we can apply the construction of Sect. 2 to define a net O → A(O) of local observable algebras on R2 . As discussed there, the crucial question in this context is whether also the modular nuclearity condition A4) holds. If it holds, the existence of observables, the Reeh–Schlieder property, and, as we shall see in Sect. 6, the anticipated form (3.1) of the S-matrix follow. If, on the other hand, condition A4) fails, the status of all these important properties is unclear. It is thus doubtful if the inverse scattering problem has a solution, i.e. if there exists a local quantum theory with the considered S-matrix, in that case. It has been shown in [17] and [43] that the modular nuclearity condition is satisfied for the constant scattering functions S2 = 1 and S2 = −1, respectively. In the following, we will analyze this condition for the class of models with regular scattering functions, defined below. Definition 3.3 (Regular scattering functions). A scattering function S2 ∈ S is called regular if there exists κ > 0 such that S2 continues to a bounded analytic function on the strip S(−κ, π + κ). In this case, we put κ(S2 ) := min{κ, π2 } and define (3.23) S2 := sup |S2 (ζ )| : ζ ∈ S(−κ(S2 ), π + κ(S2 )) < ∞. The family of all regular scattering functions is denoted S0 . The two regularity assumptions made in this definition can be understood as follows. As a consequence of the relations S2 (−θ ) = S2 (θ )−1 = S2 (θ + iπ ) (3.3), S2 can be continued to a meromorphic function on all of C. A pole ζ in the “unphysical sheet” −π < Im ζ < 0 is usually interpreted as evidence for an unstable particle with a finite lifetime [27], and the lifetime of such a resonance becomes arbitrarily long if the corresponding pole lies sufficiently close to the real axis. A scattering function with a sequence of poles in S(−π, 0) which approach the real axis might therefore have infinitely many almost stable resonances with “masses” such that the thermodynamical partition function diverges. But the modular nuclearity condition is closely related [16] to the thermodynamically motivated energy nuclearity condition of Buchholz and Wichmann [21], and the latter condition might well be violated for the previously described distribution of resonances. We therefore expect the modular nuclearity condition to fail in this situation (although there might still exist local observables). To exclude such models, we require all singu-
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G. Lechner
larities of a regular scattering function S2 to lie a finite distance off the real axis, i.e. S2 to continue analytically to a strip of the form S(−κ, π + κ), κ > 0. The second requirement, postulating S2 to stay bounded also on the enlarged strip, amounts to a condition on the phase shift of S2 (cf. [40] for a similar assumption). All scattering functions known from Lagrangian models, like the Sinh-Gordon model, satisfy the regularity assumptions of Definition 3.3. Particular examples for S2 ∈ S0 are S2 (θ ) = ±
N sinh βk − sinh θ , sinh βk + sinh θ
0 < Im β1 , . . . , Im β N < π,
(3.24)
k=1
where with each βk , also −βk is required to be included in the set {β1 , . . . , β N }. In the following, it is our aim to prove the modular nuclearity condition A4) for the models with scattering functions S2 ∈ S0 , i.e. to show that the maps Ξ (x) : M → H,
Ξ (x)A := ∆1/4 U (x)AΩ,
x ∈ WR ,
(3.25)
are nuclear maps between the Banach spaces (M, · B(H) ) and (H, · ). This task is facilitated by the fact that in the models at hand, the modular data J, ∆ of (M, Ω) are known to act geometrically “correct”, i.e. as expected from the Bisognano-Wichmann theorem [8,9]. More precisely, the modular conjugation coincides with the TCP operator, J = U ( j) (3.12), and the modular unitaries are given by the boost transformations ∆it = U (0, −2π t) [17]. In particular, ∆1/4 U (x) commutes with the projection Pn onto Hn (3.8), such that the n-particle restrictions Ξn (x) of Ξ (x) take the form Ξn (x) : M → Hn ,
Ξn (x)A := Pn Ξ (x)A = ∆1/4 U (x)(AΩ)n .
(3.26)
We now consider a purely spatial translation x = (0, s) =: s, s > 0, and write Ξn (s) instead of Ξn (s). The notation s = (0, s) will be used throughout the following sections, and the parameter s > 0 will be referred to as the splitting distance. As ∆1/4 acts as the boost with imaginary rapidity parameter iπ 2 , and since i p(θ − iπ 2 ) · (0, s) = −ms cosh θ , the maps Ξn (s) are explicitly given by, A ∈ M, (Ξn (s)A)n (θ1 , . . . , θn ) =
n
e−ms cosh θk · (AΩ)n (θ1 −
iπ 2 , . . . , θn
−
iπ 2 ).
(3.27)
k=1
The right-hand side has to be understood in terms of analytic continuation, and suggests to study the analytic properties of (AΩ)n for the proof of the nuclearity condition. This is done in the subsequent Sect. 4. In Sect. 5, the nuclearity of the maps (3.27) will then be established. 4. Analytic Properties of Wedge-Local Form Factors In this section, we consider a regular scattering function S2 ∈ S0 and a fixed operator A in the associated wedge algebra M (3.20). We will study analyticity and boundedness properties of the n-particle rapidity functions (AΩ)n = Pn AΩ. These functions are precisely the form factors of A (3.16), 1 (AΩ)n (θ1 , . . . , θn ) = √ z † (θ1 ) · · · z † (θn )Ω, AΩ. n!
(4.1)
Construction of Quantum Field Theories with Factorizing S-Matrices
835
Our notation will be as follows. Vectors in Rn and Cn are denoted by boldface letters λ, θ , ζ , and their components by λk , θk , ζk . As multidimensional generalizations of the strip regions S(a, b) (3.2), we will consider tubes of the form T := Rn + i C ⊂ Cn , where the base C is an open convex domain in Rn . For functions F : T → C, we introduce the notation Fλ (θ ) := F(θ + iλ), λ ∈ C. The main result of the present section is Proposition 4.4, stating that (AΩ)n is the boundary value of a function analytic in some tube in Cn , and that this function is bounded on Rn + i C, where C ⊂ Rn is a neighborhood of the point (− π2 , . . . , − π2 ) corresponding to the action of the modular operator in (3.27). Lemma 4.1. Let A ∈ M, n 1 , n 2 ∈ N0 , Ψn 1 ∈ Hn 1 , Φn 2 ∈ Hn 2 . There exists a function K : S(−π, 0) → C which is analytic in the interior of this strip and whose boundary values satisfy, θ ∈ R, K (θ ) = Ψn 1 , [z(θ ), A] Φn 2 ,
K (θ − iπ ) = −Ψn 1 , [z † (θ ), A] Φn 2 , (4.2) √ √ in the sense of distributions. Moreover, with c(n 1 , n 2 ) := n 1 + 1+ n 2 + 1, there holds the bound 1/2 ≤ c(n 1 , n 2 )Ψn 1 Φn 2 A, 0 ≤ λ ≤ π. (4.3) dθ |K (θ − iλ)|2 Proof. Consider the distributions K # : S (R) → C, K # ( fˆ) := Ψn 1 , [z # ( fˆ), A] Φn 2 ,
z # = z, z † .
(4.4)
In view of the bounds (3.14), there holds |K ( fˆ)| ≤ z † ( fˆ)Ψn 1 AΦn 2 + A∗ Ψn 1 z( fˆ)Φn 2 √ n 1 + 1 + n 2 Ψn 1 Φn 2 A · fˆ, ≤ √ |K † ( fˆ)| ≤ n 1 + n 2 + 1 Ψn 1 Φn 2 A · fˆ.
(4.5) (4.6) (4.7)
By application of Riesz’ Lemma, it follows that both distributions, K and K † , are given by integration against functions in L 2 (R) (denoted by the same symbols) with norms K # 2 ≤ c(n 1 , n 2 )Ψn 1 Φn 2 A.
(4.8)
To obtain the analytic continuation of K , we consider the time zero fields ϕ, π of φ (3.18), and the corresponding expectation values k± : S (R) → C, k− ( f ) := Ψn 1 , [ϕ( f ), A] Φn 2 ,
k+ ( f ) := Ψn 1 , [π( f ), A] Φn 2 .
(4.9)
As a consequence of the localization of φ in the left wedge and A in the right wedge, k± ( f ) vanishes for test functions with support on the left half line R− (3.22). Hence the Fourier transforms of k± are the boundary values of functions p → k˜± ( p) analytic in the lower half plane, which satisfy polynomial bounds at the boundary and at infinity [52, Thm. IX.16]. Since sinh(·) maps S(−π, 0) to the lower half plane, the functions K + (θ ) := k˜+ (m sinh θ ),
K − (θ ) := m cosh θ · k˜− (m sinh θ ),
(4.10)
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G. Lechner
are analytic in the strip S(−π, 0). The relation between K ± and K # is found by expressing z # in terms of the time zero fields ϕ, π of φ (3.18), 1 (ϕ( f ) − iπ(ω−1 f )), 2 1 z( fˆ) = (ϕ( f − ) + iπ(ω−1 f − )), 2
z † ( fˆ) =
fˆ(θ ) = f (m sinh θ ),
(4.11)
f − (x) = f (−x).
(4.12)
For the annihilation operator this yields, f ∈ S (R), 1 dθ K (θ ) fˆ(θ ) = Ψn 1 , [z( fˆ), A]Φn 2 = (k− ( f − ) + i k+ (ω−1 f − )) 2 ˜ 1 ( p) i k + = f ( p) dp k˜− ( p) + 2 p2 + m 2 1 dθ (K − (θ ) + i K + (θ )) fˆ(θ ). = 2 Similarly, one obtains for the creation operator 1 dθ K † (θ ) fˆ(θ ) = dθ (K − (−θ ) − i K + (−θ )) fˆ(θ ). 2 It follows from these equations that the boundary values of K ± exist as square integrable functions, and we have the identities K (θ ) =
1 (K − (θ ) + i K + (θ )), 2
K † (θ ) =
1 (K − (−θ ) − i K + (−θ )). 2
(4.13)
Hence K is analytic in S(0, π ), too, and since K ± (θ −iπ ) = ±K ± (−θ ) holds for θ ∈ R (4.10), also the claimed relation K (θ − iπ ) = −K † (θ ) (4.2) follows. It remains to prove the L 2 -bound (4.3). Consider the “shifted” function K (s) (ζ ) := −ims sinh ζ · K (ζ ), s > 0, e (s) 1 −ms sin λ cosh θ |K − (θ − iλ) + i K + (θ − iλ)| . (4.14) K −λ (θ ) = e 2 As θ → K ± (θ − iλ) are bounded by polynomials in cosh θ for |θ | → ∞, 0 < λ < π, (s) we have K −λ ∈ L 2 (R) for all λ ∈ [0, π ], s > 0. In view of the previous estimates (4.8) on the L 2 -norms of the boundary values of K , the three lines theorem can be applied and we conclude (s)
K −λ 2 ≤ c(n 1 , n 2 )Ψn 1 Φn 2 A,
0 ≤ λ ≤ π.
(4.15)
But as (4.14) is monotonically increasing as s → 0, this uniform bound holds also for (0) K −λ = K −λ , 0 ≤ λ ≤ π . Lemma 4.1 is our basic tool for deriving analytic properties of the functions (AΩ)n . In the following, we study matrix elements of the form z † (θk+1 ) · · · z † (θn )Ω, A z † (θk ) · · · z † (θ1 )Ω, with certain contractions between the rapidity variables θk+1 , . . . , θn in the left and θ1 , . . . , θk in the right argument of the scalar product.
Construction of Quantum Field Theories with Factorizing S-Matrices
837
Some notation needs to be introduced. Given two integers 0 ≤ k ≤ n, we define a contraction C to be a set of pairs, C = {(l1 , r1 ), . . . , (l N , r N )}, with pairwise different “left indices” l1 , . . . , l N ∈ {k + 1, . . . , n} and “right indices” r1 , . . . , r N ∈ {1, . . . , k}. The set of all such contractions is denoted Cn,k . The number N of pairs (l j , r j ) in a given C ∈ Cn,k will be called the length of C, and notated as |C| := N ≤ min{k, n − k}. We also write l C := {l1 , . . . , l N } and r C := {r1 , . . . , r N } for the sets of left and right indices of the contraction C. With these notations, a contracted matrix element of A is defined as † l C | A |r C n,k := z k+1 · · · zl†1 · · · zl†|C| · · z n† Ω, A z k† · · · zr†1 · · · zr†|C| · ·z 1† Ω,
(4.16)
where z a† := z † (θa ) is considered as an operator-valued distribution in θa and the hats indicate omission of the corresponding creation operators. Given the particle number bounds (3.14) and the boundedness of A, we can apply the nuclear theorem to conclude that these contracted matrix elements are well-defined tempered distributions on S (Rn−2|C| ). For square-integrable functions FL ∈ L 2 (Rn−k−|C| ) and FR ∈ L 2 (Rk−|C| ) depending on {θk+1 , . . . , θn }\{θl1 , . . . , θl|C| } and {θ1 , . . . , θk }\{θr1 , . . . , θr|C| }, respectively, there hold the bounds (cf. (3.16)) l C | A |r C n,k (FL ⊗ FR ) ≤ (n − k − |C|)! (k − |C|)! FL FR A. (4.17) Employing the shorthand notations δl,r := δ(θl − θr ) and Sb,a ; a ≤ k < b or b ≤ k < a (k) Sa,b := S2 (θa − θb ), Sa,b := , Sa,b ; otherwise
(4.18)
we associate with each contraction C = {(l1 , r1 ), . . . , (l|C| , r|C| )} ∈ Cn,k the following (k) distribution δC and function SC : δC := (−1)
|C|
|C|
δl j ,r j ,
(k) SC
:=
|C|
l j −1
j=1 m j =r j +1
j=1
Sm(k)j ,r j ·
ri
(k)
Sr j ,li .
(4.19)
In the following, the main objects of interest are the completely contracted matrix elements of A, defined as
Acon δC · SC(k) · l C | A |r C n,k . (4.20) n,k := C∈Cn,k (k)
The product δC · SC · l C | A |r C n,k is defined in the sense of distributions. Note that the product of δC and l C | A |r C n,k is well-defined because these distributions act on different variables. Since S2 ∈ S0 can be continued to a bounded analytic function on a strip containing the real axis (cf. Definition 3.3), the functions SC(k) are smooth, and all their derivatives are bounded on Rn . Hence (4.20) exists as a tempered distribution on S (Rn ). To discuss the analytic properties of Acon n,k , it is convenient to represent this distribution by two alternative formulae, stated below.
838
G. Lechner
Lemma 4.2. Let Cˆn,k ⊂ Cn,k denote the subset of those contractions C ∈ Cn,k which do not contract k + 1, i.e. fulfill k + 1 ∈ / l C . Then
(k) δC SC l C ∪ {k + 1}| [z k+1 , A] |r C n,k , (4.21) Acon n,k = C∈Cˆn,k
Acon n,k+1 =
† δC SC(k+1) l C ∪ {k + 1}| [A, z k+1 ] |r C n,k .
(4.22)
C∈Cˆn,k
The proof of Lemma 4.2 is based on the exchange relations of the ZamolodchikovFaddeev algebra (3.15); it can be found in the Appendix. The analyticity and boundedness properties of the contracted matrix elements Acon n,k are explained in the following lemma. Lemma 4.3. In a model with scattering function S2 ∈ S0 , let A ∈ M. a) Acon n,k has an analytic continuation in the variable θk+1 to the strip S(−π, 0), k ≤ n − 1. Its distributional boundary value at Im θk+1 = −π is given by con Acon n,k (θ1 , . . . , θk+1 − iπ, . . . , θn ) = An,k+1 (θ1 , . . . , θk+1 , . . . , θn ).
(4.23)
b) There holds the bound, f 1 , . . . , f n ∈ S (R), 0 ≤ λ ≤ π , n n √ ≤ 2n n! A d n θ Acon (θ1 , . . . , θk+1 − iλ, . . . , θn ) f (θ ) f j 2 . j j n,k j=1 j=1 (4.24) Proof. a) Consider the distribution Acon n,k , expressed as in (4.21). As k + 1 is not contracted in C ∈ Cˆn,k , the delta distribution δC does not depend on θk+1 . The function
SC(k) depends on θk+1 only via m j = k + 1 in Sm(k)j ,r j in (4.19) because li , r j = k + 1.
(k) Since S2 is analytic in S(0, π ), the factor Sk+1,r = Sr j ,k+1 has an analytic continuation j in θk+1 to the strip S(−π, 0), with the crossing-symmetric boundary value S2 (θr j − (k+1)
(θk+1 − iπ )) = S2 (θk+1 − θr j ) = S2 (k)
(k)
(θk+1 − θr j ). All other factors in SC are of (k)
(k+1)
(k)
the form Sa,b , a, b = k + 1, and therefore satisfy Sa,b = Sa,b . Thus θk+1 → SC (θ), with θ1 , . . . , θk , θk+2 , . . . , θn ∈ R fixed, can be analytically continued to S(−π, 0), with (k+1) boundary value SC at R − iπ . According to Lemma 4.1, also l C ∪ {k + 1}| [z k+1 , A] |r C n,k has an analytic continuation in θk+1 ∈ S(−π, 0), and its boundary value at Im θk+1 = −π is obtained by † ]. Hence Acon exchanging [z k+1 , A] with [A, z k+1 n,k has an analytic continuation to the strip S(−π, 0), and its boundary value at Im θk+1 = −π is (in the sense of distributions)
(k+1) † Acon δC SC l C ∪ {k + 1}| [A, z k+1 ] |r C n,k . n,k (θ1 , . . . , θk+1 − iπ, . . . , θn ) = C∈Cˆn,k
Taking into account the formula (4.22) for Acon n,k , this shows that the boundary value con con of An,k at Im θk+1 = −π is An,k+1 . (k) b) Let C ∈ Cˆn,k and put θ r := (θr1 , . . . , θr|C| ). In the product SC (4.19), at least one of the two variables θa , θb of each factor Sa,b is contracted, i.e. either a ∈ l C ∪ r C or
Construction of Quantum Field Theories with Factorizing S-Matrices
839
b ∈ l C ∪ r C . After the multiplication with δC , the variables θl j and θr j are identified. We (k)
(k)
can therefore split δC SC into a product of three factors, δC SC = δC SCL SCM SCR , where SCL depends on {θk+2 , . . . , θn }\{θl1 , . . . , θl|C| } and θ r , and SCR depends on {θ1 , . . . , θk }. |C| Only SCM = j=1 Sr j ,k+1 depends on θk+1 . For f 1 , . . . , f n ∈ S (R), let FθLr := SCL · f k+2 ⊗ . . . ⊗ f l1 ⊗ . . . ⊗ fl|C| ⊗ . . . ⊗ f n , FθRr := SCR · f k ⊗ . . . ⊗ f r1 ⊗ . . . ⊗ fr|C| ⊗ . . . ⊗ f 1 ,
(4.25) (4.26)
where the hats indicate omission of the corresponding factors. FθLr and FθRr are considered as functions of the n − k − 1 − |C| variables {θk+2 , . . . , θn }\{θl1 , . . . , θl|C| } and the k − |C| variables {θ1 , . . . , θk }\{θr1 , . . . , θr|C| }, respectively, and the dependence of these functions on θ r ∈ R|C| is treated as a parameter. In view of |S2 (θ )| = 1, θ ∈ R, the L/R L 2 -norms of Fθ r are FθLr =
n
f j 2 ,
FθRr =
j=k+2 j∈ /lC
k
θ r ∈ R|C| .
f j 2 ,
(4.27)
j=1 j∈ /rC
Let Acon n,k ( f ; θk+1 ) :=
Acon n,k (θ1 , . . . , θk+1 , . . . , θn )
n
f j (θ j ) dθ j .
j=1 j =k+1
After analytic continuation in θk+1 , and after carrying out the integration over the delta distributions in (4.21), we find Acon n,k ( f ; θk+1 − iλ) =
(−1)|C|
d |C| θ r IθCr ,λ (θk+1 )
C∈Cˆn,k
IθCr ,λ (θk+1 ) =
|C|
|C|
fl j (θr j ) fr j (θr j ),
j=1
S2 (θr j − θk+1 + iλ)
j=1
× l C ∪ {k + 1}| [z(θk+1 − iλ), A] |r C n,k (FθLr ⊗ FθRr ). Putting the bounds |S2 (ζ )| ≤ 1, ζ ∈ S(0, π ), (4.3), (4.17) and (4.27) together, we arrive at, 0 ≤ λ ≤ π , dθk+1 f k+1 (θk+1 )I C (θk+1 ) ≤ γ C · f j 2 · A, (4.28) θ r ,λ n,k R
C γn,k
:=
n − k − |C| + k − |C| + 1
j=l C ∪ r C
√ 2 n! . (n − k − 1 − |C|)!(k − |C|)! ≤ |C|!
840
G. Lechner
C follows from the inequality a!b! ≤ (a + b)!. From (4.28) we The given estimate on γn,k conclude n
A √ dθk+1 f k+1 (θk+1 )Acon ( f ; θk+1 − iλ) ≤ 2 n! f j 2 . (4.29) n,k |C|! C∈Cˆn,k
j=1
It remains the combinatorial problem to find a bound on the sum over Cˆn,k . Note that the number of all contractions C ∈ Cˆn,k with fixed length |C| = N is N ! Nk n−k−1 , since N each such contraction is given by two N -element subsets {r1 , . . . , r N } ⊂ {1, . . . , k} and {l1 , . . . , l N } ⊂ {k + 2, . . . , n}, and a permutation of {1, . . . , N } to determine which element of {l1 , . . . , l N } is contracted with which element of {r1 , . . . , r N }. Using |C| ≤ min{k, n − k − 1}, we find
C∈Cˆn,k
1 = |C|! ≤
min{k,n−k−1}
N =0
k N
n−k−1 N
k n−k−1
k n − k − 1 N =0 M=0
N
M
= 2n−1 .
In combination with (4.29), this implies the desired bound (4.24).
The analyticity and boundedness properties of the contracted matrix elements Acon n,k imply corresponding properties of the n-particle form factors (AΩ)n . In order not to overburden our notation, we will denote the analytic continuation of (AΩ)n by the same symbol. In the following, more specific information on the underlying regular scattering function is needed. We will exploit the fact that each S2 ∈ S0 can be continued to the enlarged strip S(−κ(S2 ), π + κ(S2 )), and is bounded by S2 < ∞ on this domain. Also recall that κ(S2 ) ≤ π2 by definition. The regions which are relevant in this context are, κ > 0, (4.30) n := λ ∈ Rn : π > λ1 > λ2 > . . . > λn > 0 , n Bn (κ) := λ ∈ R : 0 < λ1 , . . . , λn < π, λk − λl < κ, 1 ≤ l < k ≤ n , (4.31) π κ κ ×n π Cn (κ) := − , . (4.32) λ0 := − , . . . , − , 2 2 2 2 Note that Cn (κ) + λ0 ⊂ Bn (κ) (cf. Fig. 4.1 for the case n = 2). The tubes based on these sets are denoted Tn := Rn − in , Tn (κ) := Rn + i (λ0 + Cn (κ)).
(4.33) (4.34)
Proposition 4.4 (Analyticity and boundedness properties of ( AΩ)n ). In a model with regular scattering function, let A ∈ M. a) (A)n is analytic in the tube Rn − iBn (κ(S2 )). b) Let 0 < κ < κ(S2 ). There holds the bound, n S2 8 |(A)n (ζ )| ≤ · A, √ π κ(S2 ) − κ
ζ ∈ Tn (κ).
(4.35)
Construction of Quantum Field Theories with Factorizing S-Matrices
841
Fig. 4.1. The two-dimensional bases −Λ2 (triangle on the left), −B2 (κ) (pentagon on the right) and λ0 +C2 (κ) (square on the right), inscribed in the square [−π, 0] × [−π, 0]
Proof. a) Let f ∈ S (Rn ). As the first statement to be proven, we claim that the convolution (θ1 , . . . , θk ) → ((A)n ∗ f )(θ1 , . . . , θn ), considered as a function of θ1 , . . . , θk , with θk+1 , . . . , θn ∈ R fixed, is analytic in the tube Rk − ik and continuous on its closure. Our proof is based on induction in k √ ∈ {1, . . . , n}. For k = 1, note that (AΩ)n = Acon / n!, since Cn,0 contains only the empty n,0 contraction (4.20). But according to Lemma 4.3 a), Acon n,0 is the boundary value of a 1 function analytic in S(−π, 0) = R − i1 . Thus the claim for k = 1 follows. So assume analyticity of (θ1 , . . . , θk ) −→ ((A)n ∗ f )(θ1 , . . . , θn ) in Rk − ik . In view of Lemma √ 4.3 a), the boundary value at Im θ1 = . . . = Im θk = −π is given ∗ f / n!, which in turn has an analytic continuation in θk+1 ∈ S(−π, 0). by Acon n,k By application of the Malgrange Zerner (“flat tube”) theorem (cf., for example, [29]), it follows that (A)n ∗ f , considered as a function of the first k + 1 variables, has an analytic continuation to the convex closure of the set Rk+1 − i ({(λ1 , . . . , λk , 0) : (λ1 , . . . , λk ) ∈ k } ∪ {(π, . . . , π, λk+1 ) : π > λk+1 > 0}), which coincides with Rk+1 − ik+1 . Hence our claim follows. Since f was arbitrary, we conclude that (A)n is the boundary value in the sense of distributions of a function (denoted by the same symbol) analytic in Tn = Rn − iΛn (4.30). Now let Sn denote the group of permutations of n objects and consider the “permuted form factors” (A)ρn (θ ) := (A)n (θρ(1) , . . . , θρ(n) ), ρ
ρ ∈ Sn , ρ
which are analytic in the “permuted tubes” Tn := Rn − in , ρn := λ ∈ Rn : π > λρ(1) > . . . > λρ(n) > 0 . Recall that (AΩ)n ∈ Hn is invariant under the representation Dn of Sn (3.5), (A)n (θ) = (Dn (ρ)(AΩ)n )(θ ) = S ρ (θ ) · (A)ρn (θ), S2 (θρ(l) − θρ(k) ). S ρ (θ ) = 1≤lρ(k)
(4.36) (4.37)
842
G. Lechner
As S2 ∈ S0 is analytic in S(−κ(S2 ), π + κ(S2 )), all the functions S ρ , ρ ∈ Sn , are analytic in the tube Rn + i Bn (κ(S2 )) with base Bn (κ(S2 )) := λ ∈ Rn : −κ(S2 ) < λk − λl < π + κ(S2 ), 1 ≤ l < k ≤ n . Hence the right-hand side of (4.36) can be analytically continued to the tube based on ρ Bn (κ(S2 )) ∩ (−n ). But the left-hand side of (4.36) is analytic in Rn − in , and both sides converge in the sense of distributions to the same boundary values on Rn . So we may apply Epstein’s generalization of the Edge of the Wedge Theorem [28] to conclude that (A)n has an analytic continuation to the tube whose base is the convex closure of Bn (κ(S2 )) ∩ (−ρn ). ρ∈Sn ρ
Since the convex closure of ρ n is the cube (0, π )×n , it follows that (A)n is analytic in the tube based on (−π, 0)×n ∩ Bn (κ(S2 )) = −Bn (κ(S2 )), and the proof of part a) is finished. b) We first derive an estimate on |S ρ (ζ )| (4.37), ζ ∈ Tn (κ(S2 )). Clearly, S ρ is bounded on Tn (κ(S2 )), because each factor S2 (ζρ(l) − ζρ(k) ) is bounded. By the multidimensional analogue of the three lines theorem [10], the supremum of S ρ over this tube is attained on a subspace of the form Rn + iλ0 + iξ , where ξ is a vertex of Cn (κ(S2 )), i.e. |S ρ (ζ )| ≤ sup |S2 (θρ(l) − θρ(k) + i(ξρ(l) − ξρ(k) ))|, ζ ∈ Tn (κ(S2 )). θ ∈Rn 1≤l
ρ(l)>ρ(k)
Since ξ is a vertex of Cn (κ(S2 )), there holds ξρ(l) − ξρ(k) ∈ {0, κ(S2 ), −κ(S2 )}. We have sup |S2 (θ )| = 1,
θ∈R
sup |S2 (θ + iκ(S2 ))| ≤ 1,
sup |S2 (θ − iκ(S2 ))| ≤ S2 .
θ∈R
θ∈R
As at most (n − 1) of the differences ξρ(l) − ξρ(k) can equal −κ(S2 ) simultaneously, and since S2 ≥ 1, we conclude |S ρ (ζ )| ≤ S2 n−1 ≤ S2 n ,
ζ ∈ Tn (κ(S2 )).
(4.38)
Now let f 1 , . . . , f n ∈ S (R) and put f := f 1 ⊗ . . . ⊗ f n . Lemma 4.3 b) implies that at points θ − iλ ∈ Tn with λ = (π, . . . , π, λk+1 , 0, . . . , 0), 0 ≤ λk+1 ≤ π , there holds the bound |((A)n ∗ f )(θ − iλ)| ≤ 2n A
n
f j 2 .
(4.39)
j=1
By a standard argument (cf., for example [44, Lemma A.2]), this bound can be seen to hold for arbitrary λ ∈ n . Moreover, taking into account the S2 -symmetry of (AΩ)n (4.36) and the bound (4.39), we find |((A)n ∗ f )(ζ )| ≤ (2S2 )n A
n
f j 2 ,
ζ ∈ Tn (κ(S2 )),
j=1
and this inequality extends to f 1 , . . . , f n ∈ L 2 (R) by continuity.
(4.40)
Construction of Quantum Field Theories with Factorizing S-Matrices
843
To proceed to the desired bound (4.35), we fix arbitrary θ ∈ Rn , λ ∈ λ0 + Cn (κ), and 0 < κ < κ(S2 ), put θ − iλ =: ζ ∈ Tn (κ), and consider the disc Dr (ζk ) ⊂ C of radius r := 21 (κ(S2 ) − κ) and center ζk . For this value of the radius, the polydisc Dr (ζ1 ) × . . . × Dr (ζn ) is contained in Tn (κ(S! 2 )). Denoting the characteristic function of
the interval [−r (λk ), r (λk )], with r (λk ) := r 2 − (λk )2 , by χλk , we can use the mean value property for analytic functions as follows. 2 −n dθ1 dλ1 · · · dθn dλn (AΩ)n (θ + iλ ) (AΩ)n (ζ ) = (πr ) Dr (ζ1 )
2 −n
= (πr )
Dr (ζn ) r(λ1 )
r(λn )
−r (λ1 )
−r (λn )
n
dθ1 · · ·
d λ
[−r,r ]×n
= (πr 2 )−n
dθn (AΩ)n (θ + θ − iλ + iλ )
d n λ (AΩ)n ∗ (χλ1 ⊗ . . . ⊗ χλn ) (θ − iλ + iλ ).
[−r,r ]×n
Since θ − ! iλ + iλ ∈ Tn (κ(S2 )), we can apply the estimate (4.40). Taking into account √ χλk 2 = 2r (λk ) ≤ 2r and r = 21 (κ(S2 ) − κ), we get |(AΩ)n (ζ )| ≤ (πr 2 )−n · (2r )n (2S2 )n A · (2r )n/2 = Since ζ ∈ Tn (κ) was arbitrary, the proof is finished.
S2 8 √ π κ(S2 ) − κ
n
A.
5. Proof of the Nuclearity Condition With the help of the results of the previous section, we can now proceed to the proof of the modular nuclearity condition. We begin by recalling the definition of a nuclear map [38,51]. Definition 5.1 (Nuclear maps). Let X and Y be two Banach spaces. A linear map T : X −→ Y is said to be nuclear if there exists a sequence of vectors {Ψk }k ⊂ Y and a sequence of linear functionals {ηk }k ⊂ X∗ such that, X ∈ X , T (X ) =
∞
∞
ηk (X ) Ψk ,
k=1
ηk X∗ Ψk Y < ∞.
(5.1)
k=1
The nuclear norm of such a mapping is defined as T 1 := inf
ηk ,Ψk
∞
ηk X∗ Ψk Y ,
(5.2)
k=1
where the infimum is taken over all sequences {Ψk }k ⊂ Y, {ηk }k ⊂ X∗ complying with the above conditions. The sets of all bounded, respectively nuclear, maps between X and Y will be denoted B(X , Y) and N (X , Y), respectively. We will use the following well-known facts about nuclear maps, mostly without further mention.
844
G. Lechner
Lemma 5.2 (Properties of nuclear maps). Let X , X1 , Y, Y1 be Banach spaces. a) Let A1 ∈ B(X , X1 ), T ∈ N (X1 , Y1 ), A2 ∈ B(Y1 , Y). Then A2 T A1 ∈ N (X , Y), and A2 T A1 1 ≤ A2 · T 1 · A1 . (5.3) b) (N (X , Y), · 1 ) is a Banach space. c) Let H be a separable Hilbert space. Then N (H, H) coincides with the set of trace class operators on H, and T 1 = Tr |T |,
T ∈ N (H, H).
(5.4)
For a proof of this lemma, see for example [38]. We also have to recall the notion of Hardy spaces on tubes. Definition 5.3 (Hardy spaces on tube domains). Let C ⊂ Rn be open. The Hardy space H 2 (T ) on the tube T = Rn + i C is the space of all analytic functions F : T → C for which Fλ is an element of L 2 (Rn ) for each λ ∈ C, and which have finite Hardy norm 1/2 |||F||| := sup Fλ 2 = sup d n θ |F(θ + iλ)|2 < ∞. (5.5) λ∈C
Rn
λ∈C
(H 2 (T ), ||| · |||) is a Banach space [60]. As in the preceding section, we choose κ in 0 < κ < κ(S2 ) ≤ π2 , and consider the tube π κ κ ×n π ,..., , Cn (κ) := − , λ0 := − . Tn (κ) := λ0 + i Cn (κ), 2 2 2 2 Having set up our notation, we now turn to the analysis of the properties of the concrete mappings Ξn (s) (3.27) appearing in the modular nuclearity condition, Ξn (s) : M −→ Hn ⊂ L 2 (Rn ), (Ξn (s)A)n (θ1 , . . . , θn ) =
n
s > 0,
e−ms cosh θk · (AΩ)n (θ1 −
iπ 2 , . . . , θn
−
iπ 2 ).
(5.6)
k=1
We decompose Ξn (s) as depicted in the following diagram:
M Ξ
n(
Σn (s, κ) H 2 (Tn (κ))
s)
∆n (s, κ)
L2 (Rn )
The two maps Σn (s, κ) : M → H 2 (Tn (κ)) and ∆n (s, κ) : H 2 (Tn (κ)) → L 2 (Rn ) appearing here are defined as Σn (s, κ)A := (A( 21 s)Ω)n , (∆n (s, κ)F)(θ) :=
n k=1
e−
ms 2
cosh θk
s := (0, s),
(5.7)
· F(θ + iλ0 ).
(5.8)
Construction of Quantum Field Theories with Factorizing S-Matrices
845
In view of (5.6), the above diagram commutes, i.e. there holds Ξn (s)A = ∆n (s, κ)Σn (s, κ)A,
A ∈ M.
(5.9)
Σn (s, κ) and ∆n (s, κ) are investigated in the following two lemmas. Lemma 5.4. Let S2 ∈ S0 and 0 < κ < κ(S2 ). The map Σn (s, κ), s > 0, is a bounded operator between the Banach spaces (M, · B(H) ) and (H 2 (Tn (κ)), ||| · |||). Its operator norm satisfies Σn (s, κ) ≤ σ (s, κ)n .
(5.10)
For fixed κ, the function s → σ (s, κ) is monotonously decreasing, with the limits σ (s, κ) → 0 for s → ∞ and σ (s, κ) → ∞ for s → 0. Proof. Given the translation invariance of Ω and the form of U (3.11), we have (Σn (s, κ)A)(ζ ) = (A( 21 s)Ω)n (ζ ) = u n,s (ζ ) · (AΩ)n (ζ ), u n,s (ζ ) =
n
e−
ims 2
sinh ζk
.
k=1
Since u n,s is entire, the analyticity of (AΩ)n (Proposition 4.4) carries over to Σn (s, κ)A. Moreover, it follows from a straightforward calculation that u n,s is an element of H 2 (Tn (κ)), with Hardy norm n/2 −ms cos κ cosh θ |||u n,s ||| = dθ e , (5.11) R
and this integral converges since s > 0 and 0 < κ < π2 . In view of the uniform bound (4.35) on (AΩ)n (ζ ), ζ ∈ Tn (κ), it follows that also u n,s · (AΩ)n lies in the Hardy space H 2 (Tn (κ)), with norm bounded by 1/2 n 8 S2 |||Σn (s, κ)A||| −ms cos κ cosh θ ≤ · dθ e . (5.12) √ A π κ(S2 ) − κ R The claimed behaviour of Σn (s, κ) with respect to s can be directly read off from this formula. Lemma 5.5. Let s > 0, κ > 0, and ∆n (s, κ) be defined as in (5.8). a) ∆n (s, κ) is a nuclear map between the Banach spaces (H 2 (Tn (κ)), ||| · |||) and (L 2 (Rn ), · 2 ). b) Let Ts,κ be the integral operator on L 2 (R, dθ ) with kernel Ts,κ (θ, θ ) =
e−
ms 2
cosh θ
iπ (θ − θ −
iκ 2)
.
(5.13)
Ts,κ is of trace class, and there holds the bound ∆n (s, κ)1 ≤ Ts,κ 1n < ∞.
(5.14)
Proof. Let F ∈ H 2 (Tn (κ)), and pick θ ∈ Rn and a polydisc Dn (θ + iλ0 ) ⊂ Tn (κ) with center θ + iλ0 . By virtue of Cauchy’s integral formula, we can represent F(θ + iλ0 ) as a contour integral over Dn (θ + iλ0 ),
846
G. Lechner
F(θ + iλ0 ) =
1 (2πi)n
" Dn (θ +i λ0 )
d n ζ n
F(ζ )
k=1 (ζk
− θk +
iπ 2 )
.
(5.15)
As a consequence of the mean value property, the Hardy space function F is uniformly bounded on the subtubes Tn (κ ) ⊂ Tn (κ), κ < κ (cf. the line of argument at the end of the proof of Proposition 4.4 b). Moreover, F ∈ H 2 (Tn (κ)) can be continued to the boundary of Tn (κ) as follows: The map Cn (κ) λ → Fλ ∈ L 2 (Rn ) extends continuously (in the norm topology of L 2 (Rn )) to the closed cube λ0 + [− κ2 , κ2 ]×n [60, Ch. III, Cor. 2.9]. Taking advantage of these two properties of F, we can deform the contour of integration in (5.15) to the boundary of Tn (κ). After multiplication with the exponential factor (5.8) we arrive at ms n εk e− 2 cosh θk 1
n (∆n (s, κ)F)(θ) = d θ · Fλ0 − κ2 ε (θ ), − θ − iεk κ (2πi)n ε θ k 2 k=1 k n R
where the summation runs over ε = (ε1 , . . . , εn ), ε1 , . . . , εn = ±1. Expressed in terms of the integral operator Ts,κ , this equation reads
∆n (s, κ)F = 2−n ε1 · · · εn (Ts,ε1 κ ⊗ . . . ⊗ Ts,εn κ )Fλ0 − κ2 ε . (5.16) ε
The integral operators Ts,± κ are of trace class on L 2 (R), as can be shown by a standard argument [53, Thm. XI.21]. Hence Ts,ε1 κ ⊗ . . . ⊗ Ts,εn κ is a trace class operator on L 2 (Rn ), for any ε1 , . . . , εn = ±1. Note that since Ts,κ and Ts,−κ are related by C Ts,κ C ∗ = −Ts,−κ , (C f )(θ ) := f (−θ ), they have the same nuclear norm. Hence Ts,ε1 κ ⊗ . . . ⊗ Ts,εn κ 1 = Ts,κ n1 . Moreover, it follows from the L 2 -convergence of F to its boundary values that the maps F −→ Fλ0 − κ2 ε are bounded as operators from H 2 (Tn (κ)) to L 2 (Rn ) for any ε, with norm not exceeding one. According to Lemma 5.2, this implies the nuclearity of ∆n (s, κ) (5.16). Since the sum in (5.16) runs over 2n terms, we also obtain the claimed bound ∆n (s, κ)1 ≤ Ts,κ n1 . Lemma 5.5 implies our first nuclearity result for the maps Ξ (s) (3.25). Theorem 5.6 (Nuclearity for sufficiently large splitting distances). In each model theory with regular scattering function, there exists a splitting distance smin < ∞ such that Ξ (s) is nuclear for all s > smin . Hence in these models, for each double cone Oa,b = (W R + a) ∩ (W L + b) with 2 , the corresponding observable algebra A(O ) = b − a ∈ W R and −(b − a)2 > smin a,b M(a) ∩ M(b) (2.3) has Ω as a cyclic vector. Proof. Let κ ∈ (0, κ(S2 )). We have Ξn (s) = ∆n (s, κ)Σn (s, κ), and in view of the previously established results, Ξn (s) is nuclear, with nuclear norm bounded by (5.10, 5.14), n Ξn (s)1 ≤ Σn (s, κ) · ∆n (s, κ)1 ≤ σ (s, κ) · Ts,κ 1 . (5.17) ∞ To obtain nuclearity for Ξ (s) = n=0 Ξn (s) (3.26), note that for s → ∞, Ts,κ 1 and σ (s, κ) converge monotonously to zero (cf. Lemma 5.4 and (5.13)). So there exists smin < ∞ such that σ (s, κ)Ts,κ 1 < 1 for all s > smin . But for these values of s, there holds
Construction of Quantum Field Theories with Factorizing S-Matrices ∞
n=0
Ξn (s)1 ≤
∞
n σ (s, κ) Ts,κ 1 < ∞,
847
(5.18)
n=0
and the series ∞ n=0 Ξn (s) converges in nuclear norm to Ξ (s). Since the set of nuclear operators between two Banach spaces is closed with respect to convergence in · 1 , the nuclearity of Ξ (s) follows. The Reeh–Schlieder property for the double cone algebras A(O0,s ) is a consequence of the nuclearity of Ξ (s) (Theorem 2.5). 2 , there exists a rapidity parameter For a region of the form O0,x , x ∈ W R , −x 2 > smin λ such that 0 cosh λ sinh λ , s > smin . x= (5.19) s sinh λ cosh λ Since the modular operator of (M, Ω) commutes with the boosts U (0, λ), and U (0, λ) Ω = Ω, it follows that Ξ (x) and Ξ (s) are related by Ξ (x) = U (0, λ)Ξ (s)αλ−1 ,
αλ (A) := U (0, λ)A U (0, λ)−1 .
(5.20)
So the invariance of M under the modular group αλ and the unitarity of U (0, λ) imply that Ξ (x) is nuclear, too, with Ξ (x)1 = Ξ (s)1 . The corresponding statement for double cone regions Oa,b with b − a ∈ W R , −(b − 2 , follows by translation covariance. a)2 > smin Theorem 5.6 establishes the Reeh–Schlieder property (and all the other consequences of the modular nuclearity condition discussed in Sect. 2) for double cones having a minimal “relativistic size”. This size is measured by the length smin and depends on the scattering function S2 and the mass m. For example, if we consider a scattering function of the form (3.24) with N = 1 and β1 = iπ 4 , one can derive the estimate smin < lC , where lC is the Compton wavelength corresponding to the mass m. Whereas the occurrence of a minimal localization length in theories describing quantum effects of gravity is expected for physical reasons, we conjecture that the minimal length smin appearing here is an artifact of our estimates. This conjecture is supported by a second theorem, stated below, which improves the previous one under an additional assumption on the underlying scattering function. The set S0 of regular scattering functions can be divided into a “Bosonic” and a “Fermionic” class according to S0± := {S2 ∈ S0 : S2 (0) = ±1},
S0 = S0+ ∪ S0− .
(5.21)
We emphasize that, independently of the scattering function, all the models under consideration describe Bosons in the sense that their scattering states are completely symmetric (see Sect. 6). However, as will be shown below, there exist certain distinguished unitaries Y ± mapping a model with S2 ∈ S0± onto the Hilbert space H± corresponding to the special model with the constant scattering function S2 = ±1. In order to distinguish between the different scattering functions involved, we adopt the convention that the usual notations z, z † , Dn , Pn , Hn , H refer to the generic S2 ∈ S0 under consideration. All objects corresponding to the constant scattering functions † , Dn± , Pn± , Hn± , H± . S2 = ±1 are tagged with an index “±”, i.e. we write z ± , z ± In preparation for the construction of the unitaries Y ± : H → H± , note that each S2 ∈ S0 is analytic and nonvanishing in the strip S(−κ(S2 ), κ(S2 )), since zeros and
848
G. Lechner
poles are related by S2 (−ζ ) = S2 (ζ )−1 (cf. (3.3) and Definition 3.3). So there exists an analytic function δ : S(−κ(S2 ), κ(S2 )) → C (the phase shift) such that S2 (ζ ) = S2 (0) e2iδ(ζ ) ,
ζ ∈ S(−κ(S2 ), κ(S2 )).
(5.22)
Since S2 has modulus one on the real line, δ takes real values on R, and we fix it uniquely by the choice δ(0) = 0. Note that in view of S2 (−θ ) = S2 (θ ), θ ∈ R, δ is odd. Lemma 5.7. Let S2 ∈ S0± and δ : S(−κ(S2 ), κ(S2 )) −→ C be defined as above. Consider the functions Y0± = 1, ±eiδ(ζk −ζl ) , n ≥ 2, (5.23) Y1± (ζ ) = 1, Yn± (ζ ) := 1≤k
and the corresponding multiplication operators (denoted by the same symbol Yn± ). a) Viewed as an operator on H 2 (Tn (κ(S2 ))), Yn± is a bounded map with operator norm Yn± ≤ S2 n/2 . b) Viewed as an operator on L 2 (Rn ), Yn± is a unitary intertwining the representations Dn and Dn± of Sn , and hence mapping the S2 -symmetric subspace Hn ⊂ L 2 (Rn ) onto the totally (anti-) symmetric subspace Hn± ⊂ L 2 (Rn ). Proof. a) Since δ is analytic in S(−κ(S2 ), κ(S2 )), so is the function Yn± in S(− 21 κ(S2 ), 1 ×n ± 2 κ(S2 )) . Depending only on differences ζk − ζl of rapidities, Yn is also analytic in 1 1 ×n the tube Tn (κ(S2 )) = S(− 2 κ(S2 ), 2 κ(S2 )) + iλ0 . In view of (4.38), it follows that ± Y (ζ ) ≤ S2 n/2 , ζ ∈ Tn (κ(S2 )). (5.24) n n
Hence Yn± maps H 2 (Tn (κ(S2 ))) into itself, and the bound |||Yn± F||| ≤ S2 2 |||F|||, F ∈ H 2 (Tn (κ(S2 ))), proves a). b) Considered as a multiplication operator on L 2 (Rn ), Yn± multiplies with a phase and is hence unitary. Let τ j ∈ Sn denote the transposition exchanging j and j + 1, j ∈ {1, . . . , n}, and pick arbitrary f n ∈ L 2 (Rn ), θ ∈ Rn , ±eiδ(θk −θl ) f n (θ1 , . . . , θ j+1 , θ j , . . . , θn ) (Dn± (τ j )Yn± f n )(θ ) = eiδ(θ j+1 −θ j ) 1≤k
=
±eiδ(θk −θl ) · S2 (θ j+1 − θ j ) f n (θ1 , . . . , θ j+1 , θ j , . . . , θn ) 1≤k
= (Yn± Dn (τ j ) f n )(θ). As the transpositions τ j generate Sn , this calculation shows that Yn± intertwines Dn± and Dn . In particular, Yn± restricts to a unitary mapping Hn onto Hn± . The operator Y ± :=
∞
Yn± : H −→ H±
(5.25)
n=0
will be used to improve the estimate on Ξn (s)1 underlying Theorem 5.6. In a model theory with scattering function S2 ∈ S0± , we consider the maps Ξn± (s) := Yn± Ξn (s) : M −→ Hn± ,
Ξ ± (s) := Y ± Ξ (s).
Construction of Quantum Field Theories with Factorizing S-Matrices
849
Since Y ± : H → H± is unitary, Ξ (s) is nuclear if and only if Ξ ± (s) is, and in this case Ξ (s)1 = Ξ ± (s)1 . Moreover, as Yn± acts by multiplication with a function depending only on differences of rapidities, this operator commutes with the translation U (s) and the modular operator ∆, i.e. ± Ξn± (s)A = ∆1/4 U ( 21 s)Yn± (A( 21 s))n =: ∆± n (s, κ) Yn Σn (s, κ) A. Here Σn (s, κ) is defined as in (5.7) and ∆± n (s, κ) acts as ∆n (s, κ) (5.8), but is now considered as a map from the subspace H±2 (Tn (κ)) ⊂ H 2 (Tn (κ)), consisting of the totally (anti-) symmetric functions in H 2 (Tn (κ)), to Hn± . Lemma 5.4 and Lemma 5.7 a) imply that Yn± Σn (s, κ) is a bounded linear map from M to H±2 (Tn (κ)), κ ∈ (0, κ(S2 )), with norm n Yn± Σn (s, κ) ≤ S2 1/2 · σ (s, κ) . (5.26) In the case S2 ∈ S0− , the Pauli principle effectively reduces the size of the image of ∆− n (s, κ), which results in an improved estimate on Ξ (s)1 , implying the following theorem. In the case S2 (0) = +1, the Pauli principle does not apply and the subsequent argument cannot be used to obtain nuclearity for arbitrarily small splitting distances. It should be mentioned, however, that the scattering functions of all models known from Lagrangian formulations belong to the class S0− [4]. Theorem 5.8 (Proof of the modular nuclearity condition). In a model theory with scattering function S2 ∈ S0− (5.21), the maps Ξ (s) are nuclear for every splitting distance s > 0. In particular, in these models there exist observables localized in arbitrarily small open regions O ⊂ R2 , and the Reeh–Schlieder property holds without restriction. Proof. Proceeding along the same lines as in the proof of Lemma 5.5, we infer that ∆− n (s, κ) is nuclear and can be represented as in (5.16). With the notations used there, ε = (ε1 , . . . , εn ), εk = ±1, there holds for F − ∈ H−2 (Tn (κ)),
− −n ε1 · · · εn (Ts,ε1 κ ⊗ . . . ⊗ Ts,εn κ )Fλ−− κ ε . (5.27) ∆− n (s, κ)F = 2 0
ε
2
Choosing an orthonormal basis {ψk }k of L 2 (R), the vectors √ † † Ψk− := z − (ψk1 ) · · · z − (ψkn )Ω = n! Pn− (ψk1 ⊗ . . . ⊗ ψkn ) 1
= √ sign(ρ) ψρ(k1 ) ⊗ . . . ⊗ ψρ(kn ) n! ρ∈Sn
(5.28)
form an orthonormal basis of Hn− if k = (k1 , . . . , kn ) varies over k1 < k2 < . . . < kn , k1 , . . . , kn ∈ N, as a consequence of the Pauli principle. Expanding the right-hand side of (5.27) in this basis, we find
− −n ∆− ε1 · · · εn Ψk− , (Ts,ε1 κ ⊗ . . . ⊗ Ts,εn κ )Fλ−− κ ε Ψk− n (s, κ)F = 2 ε
0
k1 <...
2
2−n
=√ ε1 · · · εn sign(ρ)× n! ε ρ∈Sn
∗ ∗ × Ts,ε ψ ⊗ . . . ⊗ Ts,ε ψ , Fλ−− κ ε Ψk− . n κ ρ(kn ) 1 κ ρ(k1 ) k1 <...
0
2
850
G. Lechner
This is an example of a nuclear decomposition (5.1) of ∆− n (s, κ), with the functionals ηk from Definition 5.1 being given by ψ
ηε,ρ,k (F − ) :=
ε1 · · · εn sign(ρ) ∗ ∗ Ts,ε1 κ ψρ(k1 ) ⊗ . . . ⊗ Ts,ε ψ , Fλ−− κ ε . (5.29) √ n κ ρ(kn ) 0 2 n 2 n! ψ
To obtain a good bound on ηε,ρ,k , we have to choose the basis {ψk }k in an appropriate ∗ |2 + |T ∗ |2 )1/2 , which is of trace way. Consider the positive operator Tˆs,κ := (|Ts,κ s,−κ class on L 2 (R) and satisfies Tˆs,κ 1 ≤ 2 Ts,κ 1 [41]. We choose {ψk }k as normalized eigenvectors of Tˆs,κ , with eigenvalues tk ≥ 0. − ∗ ψ ≤ Tˆ − Noting Ts,±κ kj s,±κ ψk j = tk j and Fλ0 − κ ε ≤ |||F |||, we can estimate 2 √ ψ (5.29) according to ηε,ρ,k (F − ) ≤ tk1 · · · tkn /(2n n!) · |||F − |||. Since Ψk− = 1 and n ε ,ρ 1 = 2 · n!, this yields ∆− n (s, κ)1 ≤
√
n!
k1 <...
∞
Ts,κ n 1 tk 1 · · · tk n ≤ √ tk 1 · · · tk n = √ 1 . n! k ,...,k =1 n! 1
In view of the bound (5.26) on Yn− Σn (s, κ), we arrive at − − nuclear norm of Ξ − (s) = ∞ n=0 ∆n (s, κ)Yn Σn (s, κ): Ξ − (s)1 ≤
n
the following estimate for the
n ∞
σ (s, κ) S2 1/2 Ts,κ 1 < ∞. √ n! n=0
(5.30)
This series converges for arbitrary values of σ (s, κ)S2 1/2 Ts,κ 1 , i.e. for arbitrary splitting distances s > 0. 6. Collision States and Reconstruction of the S-Matrix The theorems of the preceding section establish the existence of a class of quantum field theories. In this section, we investigate the collision states of these models and prove that they provide the solution of the inverse scattering problem for the considered class of S-matrices. More precisely, we will show that the function S2 , which entered as a parameter into the construction, is related to the S-matrix of the model as in (3.1) (Theorem 6.3). Moreover, we will find explicit formulae for n-particle scattering states and give a proof of asymptotic completeness (Proposition 6.2). To compute n-particle collision states, it is sufficient to restrict to the family S0 of regular scattering functions (Definition 3.3), as Theorem 5.6 ensures that in this case there exist compactly localized observables satisfying the Reeh–Schlieder property, at least in double cones above some minimal size. Since any number of double cones of any size can be spacelike separated by translation, it is possible to apply the usual methods of collision theory in this class of theories – localization with arbitrarily high precision is not needed. The method to be used for the calculation of the S-matrix is Haag-Ruelle scattering theory [2, Ch. 5] in the same form as in [14], where scattering properties of polarizationfree generators have been analyzed. As usual in this approach, we consider quasilocal operators of the form, A ∈ A(O), A( f t ) = d 2 x f t (x)A(x), A(x) = U (x)A U (x)−1 . (6.1)
Construction of Quantum Field Theories with Factorizing S-Matrices
851
The functions f t , t ∈ R, are defined in terms of momentum space wavefunctions f by 1/2 1 f ( p0 , p1 ) ei( p0 −ω p )t e−i p·x , ω p := m 2 + p12 . (6.2) f t (x) := d2 p 2π Here f is taken to be a Schwartz test function, such that the integral (6.1) converges in operator norm. For the construction of collision states, the asymptotic properties as t → ±∞ of these functions are important. We introduce the velocity support of f as V( f ) := (1, p1 · ω−1 ) : ( p , p ) ∈ supp f , ω p := ( p12 + m 2 )1/2 . (6.3) 0 1 p Recall that the support of f t is essentially contained in t V( f ) for asymptotic times t [35]. More precisely, let χ be a smooth function which is equal to 1 on V( f ) and vanishes in the complement of a slightly larger region. Then the asymptotically dominant part of f t is fˆt (x) := χ (x/t) f t (x), and for any N ∈ N, the difference |t| N ( f t − fˆt ) converges to zero in the topology of S (R2 ) as t → ±∞ ([35], see also [53, Cor. to Thm. XI.14]). We also adopt the notation from [14] to write f ≺ g if V(g) − V( f ) ⊂ {0} × (0, ∞). This notation will be used for single particle wavefunctions as well: Given smooth, compactly supported θ → ψ1 (θ ), θ → ψ2 (θ ), we write ψ1 ≺ ψ2 if supp ψ2 − supp ψ1 ⊂ (0, ∞). It is straightforward to show that in this situation, there exist testfunctions f 1 , f 2 ∈ S (R2 ) such that f 1+ = ψ1 , f 2+ = ψ2 , and f 1 ≺ f 2 in the previously defined sense. If the support of f is concentrated around a point (ω p , p1 ) on the upper mass shell and does not intersect the energy momentum spectrum elsewhere, A( f t )Ω ∈ H1 is a single particle state which does not depend on the time parameter t. Furthermore, there exist the following (strong) limits: lim A( f t )Ψ = A( f ) out Ψ,
t→±∞
in
lim A( f t )∗ Ψ = A( f )∗out Ψ,
t→±∞
(6.4)
in
to the asymptotic creation and annihilation operators A( f )out/in and A( f )∗out/in , respectively. These limits are known to hold for all scattering states Ψ of compact energy momentum support, in particular, for all single particle states of the form φ( f )Ω = f + , where f + has compact support [14]. The creation and annihilation operators A( f )ex (∗) , ex = in/out, are related to the Zamolodchikov operators z +† , z + with the constant scattering function S2 = 1, acting on the totally symmetric Bose Fock H+ space over H. This relation is implemented by the Møller operators Vex : H+ → H, (6.5) A( f )ex = Vex z +† (A( f )Ω) Vex ∗ , A( f )ex ∗ = Vex z + A( f )Ω Vex ∗ . Having recalled these basic facts of scattering theory, we now fix a regular scattering function and compute n-particle collision states in the corresponding model theory. Using the standard notation for scattering states, we find the following lemma. Lemma 6.1. (Calculation of n-particle collision states). Consider testfunctions f1 , . . . , f n ∈ S (R2 ) having pairwise disjoint compact supports concentrated around points on the upper mass shell such that f 1 ≺ . . . ≺ f n . Then √ ( f 1+ × . . . × f n+ )out = φ( f 1 ) · · · φ( f n )Ω = n! Pn ( f 1+ ⊗ . . . ⊗ f n+ ), (6.6) √ + + + + (6.7) ( f 1 × . . . × f n )in = φ( f n ) · · · φ( f 1 )Ω = n! Pn ( f n ⊗ . . . ⊗ f 1 ).
852
G. Lechner
Proof. Since the supports of the f k do not intersect the lower mass shell, the annihilation parts of the fields φ( f k ) vanish, φ( f k ) = z † ( f k+ ). So the second identity in (6.6) and (6.7) follows from (3.13). The proof of the first identity in (6.6) and (6.7) is based on induction in the particle number n. For n = 1, we have φ( f 1 )Ω = f 1+ = ( f 1+ )out = ( f 1+ )in ,
(6.8)
since f 1+ is a single particle state. For the step from n to (n + 1) particles, consider operators A1 , . . . , An ∈ A(O) localized in a double cone O large enough for Ω to be cyclic for A(O). We want to establish commutation relations between φ( f ) and the creation operators Ak (gk )out , where f ≺ g1 ≺ . . . ≺ gn and the test functions f, g1 , . . . , gn have the same support properties as the f 1 , . . . , f n . As the support of f intersects the energy momentum spectrum only in the upper mass shell, it readily follows from the definitions (3.17) of f ± and (6.2) of f t , that f t+ = f + , f t− = 0, t ∈ R. Since fˆt − f t converges to zero in S (R2 ) for t → ∞, and since f → φ( f )Ψ , Ψ ∈ D is a vector valued tempered distribution, this implies φ( f )Ψ = φ( f t )Ψ = lim φ( fˆt )Ψ, t→∞
Ψ ∈ D.
(6.9)
Since A(gˆ k,t ) − A(gk,t ) ≤ Agˆ k,t − gk,t 1 → 0 for t → ∞, we can use the strong convergence Ak (gk,t ) → Ak (gk )out and the hermiticity of φ to obtain, Ψ ∈ D, φ( f )∗ Ψ, (A1 (g1 )Ω × . . . × An (gn )Ω)out = lim φ( fˆt )∗ Ψ, A1 (gˆ 1,t ) · · · An (gˆ n,t )Ω. t→∞
For large t, the functions fˆt and gˆ k,t have supports in small neighborhoods of t V( f ) (t) and t V(gk ), respectively. Hence φ( fˆt )∗ is localized in a wedge W L slightly larger than W L +t V( f ), and Ak (gˆ k,t ) is localized in a neighborhood of O +t V(gk ). For large enough t > 0, these regions are spacelike separated since f ≺ gk . As φ( fˆt )∗ is affiliated with (t) A(W L ), it follows that this operator commutes with Ak (gˆ k,t ), k = 1, . . . , n. Thus φ( f )∗ Ψ, (A1 (g1 )Ω × . . . × An (gn )Ω)out = lim Ψ, A1 (gˆ 1,t ) · · · An (gˆ n,t )φ( fˆt )Ω t→∞
= lim Ψ, A1 (gˆ 1,t ) · · · An (gˆ n,t ) fˆt+ . t→∞
A straightforward estimate yields A1 (gˆ 1,t ) · · · An (gˆ n,t ) ≤ c t 2n with a constant c > 0. But since t 2n ( fˆt − f t ) converges to zero in the topology of S (R2 ), it follows that also t 2n fˆt+ − f + 2 → 0. So we may replace fˆt+ in the above equation by f + , and use the strong convergence Ak (gˆ k,t ) → Ak (gk )out on this single particle state to conclude φ( f )∗ Ψ, (A1 (g1 )Ω × . . . × An (gn )Ω)out = Ψ, A1 (g1 )out · · · An (gn )out f + = Ψ, (A1 (g1 )Ω × . . . × An (gn )Ω × f + )out = Ψ, ( f + × A1 (g1 )Ω × . . . × An (gn )Ω)out , where in the last step we used the Bose symmetry of the scattering states. In view of the Reeh–Schlieder property of A(O), we can approximate the single particle state f k+ by Ak (gk )Ω. Given any ε > 0, there exist local operators A1 , . . . , An ∈
Construction of Quantum Field Theories with Factorizing S-Matrices
853
A(O) and functions g1 , . . . , gn , with gk having support in an arbitrarily small neighborhood of the support of f k , such that f k+ − Ak (gk )Ω < ε. As the left and right-hand side of the above equation are continuous in the Ak (gk )Ω, this implies φ( f )∗ Ψ, ( f 1+ × . . . × f n+ )out = Ψ, ( f + × f 1+ × . . . × f n+ )out .
(6.10)
Since Ψ ∈ D was arbitrary and D ⊂ H is dense, we can use the induction hypothesis and obtain φ( f )φ( f 1 ) · · · φ( f n )Ω = φ( f )( f 1+ × . . . × f n+ )out = ( f + × f 1+ × . . . × f n+ )out , proving (6.6). For incoming n-particle states, the order of the velocity supports of f 1 , . . . , f n has to be reversed, since W L + t V( f 1 ) is spacelike separated from O + t V( f k ) for t → −∞ if f f k . Apart from this modification, the same argument can be used to derive formula (6.7). Given smooth, compactly supported single particle functions ψ1 , . . . , ψn ∈ H1 with supports ordered according to ψ1 ≺ . . . ≺ ψn , there exist testfunctions f 1 , . . . , f n ∈ S (R2 ) such that f k+ = ψk , f k− = 0, k = 1, . . . , n, and f 1 ≺ . . . ≺ f n . Hence for these ψk , √ (ψ1 × . . . × ψn )out = n! Pn (ψ1 ⊗ . . . ⊗ ψn ), ψ1 ≺ . . . ≺ ψn , (6.11) √ ψ1 ≺ . . . ≺ ψn . (6.12) (ψ1 × . . . × ψn )in = n! Pn (ψn ⊗ . . . ⊗ ψ1 ), In terms of improper n-particle states with sharp rapidities, we have thus shown that z † (θ1 ) · · · z † (θn )Ω = | θ1 , . . . , θn out ,
θ1 < . . . < θn ,
(6.13a)
z (θ1 ) · · · z (θn )Ω = | θ1 , . . . , θn in ,
θ1 > . . . > θn ,
(6.13b)
†
†
are asymptotic collision states in the sense of the Haag-Ruelle scattering theory. The identification of incoming and outgoing n-particle states with n-fold products of such creation operators acting on the vacuum, arranged in order of decreasing, respectively increasing, rapidities, is one of the basic assumptions in the framework of the form factor program. In fact, it has motivated the very definition of the ZamolodchikovFaddeev algebra [63]. It is therefore gratifying that with the help of the approach presented here, the heuristic picture underlying the relations of this algebra can be rigorously justified. The outgoing and incoming scattering states (6.11,6.12) form total sets in the Hilbert space H. To prove this, note that the functions θ → nk=1 ψk (θk ) form a total set in the space L 2 (E n ) of all square integrable functions on the simplex E n := {(θ1 , . . . , θn ) ∈ Rn : θ1 ≤ . . . ≤ θn } when the ψk are varied within the limitations specified above. But the S2 -symmetrization Pn is a linear and continuous map from L 2 (E n ) to Hn , with dense range. Hence the totality of the constructed outgoing n-particle collision states in Hn follows. Analogously, one can show that also the incoming n-particle states form a total set in Hn . Taking linear combinations of states of different particle number, it also follows that the spaces Hout and Hin spanned by all outgoing and incoming scattering states are dense in H. So we arrive at the following proposition. Proposition 6.2 (Asymptotic completeness). All model theories with regular scattering functions are asymptotically complete.
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This result seems to be the first proof of asymptotic completeness in an interacting relativistic quantum field theory [19]. We finish this section by computing the Møller operators#Vin , Vout and the S-matrix S. + 2 The asymptotic states span the Bosonic Fock space H+ = ∞ n=0 Hn over H1 = L (R). + + Denoting the orthogonal projection onto Hn by Pn , we infer from the form (6.11, 6.12) of the collision states that the Møller operators are given by Vout Pn+ (ψ1 ⊗ . . . ⊗ ψn ) = Pn (ψ1 ⊗ . . . ⊗ ψn ), Vin Pn+ (ψn
⊗ . . . ⊗ ψ1 ) = Pn (ψn ⊗ . . . ⊗ ψ1 ),
ψ1 ≺ . . . ≺ ψn ,
(6.14)
ψ1 ≺ . . . ≺ ψn .
(6.15)
In view of the ordering of the supports of the ψk , these equations determine two welldefined linear operators Vin/out with dense domains and ranges, and since Pn+ (ψ1 ⊗ . . . ⊗ ψn ) = n!−1/2 ψ1 · · · ψn = Pn (ψ1 ⊗ . . . ⊗ ψn ),
(6.16)
Vin and Vout continue to unitaries mapping H+ onto H. The S-matrix is the product of the Møller operators, S := Vout ∗ Vin : H+ → H+ . (6.17) Theorem 6.3 (Calculation of the S-matrix). The model with scattering function S2 ∈ S0 solves the inverse scattering problem for the corresponding S-matrix, i.e. its scattering operator (6.17) is, Ψ + ∈ H+ , S2 (|θl − θk |) · Ψn+ (θ1 , . . . , θn ). (6.18) (SΨ + )n (θ1 , . . . , θn ) = 1≤l
Proof. Recall that the S2 -symmetrization operator Pn has the form (3.5) 1 ρ Sn (θ1 , . . . , θn ) · Ψn (θρ(1) , . . . , θρ(n) ), n! ρ∈Sn Snρ (θ1 , . . . , θn ) = S2 (θρ(l) − θρ(k) ).
(Pn Ψn )(θ1 , . . . , θn ) =
(6.19) (6.20)
1≤lρ(k)
Consider ψ1 , . . . , ψn ∈ C0∞ (R), ψ1 ≺ · · · ≺ ψn and a point θ ∈ Rn such that θπ(1) < · · · < θπ(n) for some permutation π ∈ Sn . In this situation, there holds (Pn (ψ1 ⊗ . . . ⊗ ψn ))(θ) =
1 π S (θ) · ψ1 (θπ(1) ) · · · ψn (θπ(n) ), n! n
(6.21)
and Vout (6.14) is seen to act on n-particle states by multiplication with the function θ → {Snπ (θ ) : θπ(1) ≤ . . . ≤ θπ(n) }. Similarly, Vin (6.15) acts by multiplication with θ → {Snπ ι (θ ) : θπ(1) ≤ . . . ≤ θπ(n) }, where ι ∈ Sn is the total inversion permutation, ι(k) := n − k + 1. This implies that the n-particle S-matrix is the multiplication operator Ψ + ∈ H+ , (SΨ + )n (θ) = Sˆn (θ ) · Ψn+ (θ ), Sˆn (θ) := {Snπ (θ)−1 Snπ ι (θ) : θπ(1) ≤ . . . ≤ θπ(n) }.
(6.22) (6.23)
Since Dn (3.5) is a representation of Sn , there holds Snπ ι (θ1 , . . . , θn ) = Snπ (θ1 , . . . , θn )Snι (θπ(1) , . . . , θπ(n) ),
θ1 , . . . , θn ∈ R.
(6.24)
Construction of Quantum Field Theories with Factorizing S-Matrices
855
Hence, for θπ(1) < . . . < θπ(n) , Sˆn (θ1 , . . . , θn ) = Snι (θπ(1) , . . . , θπ(n) ) = =
1≤l
S2 (θπ ι(l) − θπ ι(k) )
1≤l
S2 (|θιπ(l) − θιπ(k) |) =
S2 (|θl − θk |).
(6.25) (6.26)
1≤l
As all reference to the permutation π has been eliminated, this formula is valid for arbitrary θ1 , . . . , θn ∈ R, and finishes the proof of the claimed expression (6.18) for the S-matrix. 7. Conclusions In the present article, the construction of a large class of quantum field theories with factorizing S-matrices has been completed. The starting point of this construction is a pair of wedge-local quantum fields associated with a given S-matrix S, and the observation that the structure of the local observables corresponding to S are fixed by commutation relations with these fields. By employing operator-algebraic techniques, basic problems such as the existence of models with a prescribed S-matrix were solved without having to specify explicit formulae for local interacting quantum fields. It is interesting to notice that, at least in the class of models considered here, the rather abstract modular nuclearity condition needed to prove the existence of local observables amounts to very explicit conditions of analyticity and boundedness properties of matrix elements of observables localized in wedges. So these form factors play an important role also in the construction presented here, although in a manner quite different from their use in the form factor program. For a complete understanding of these models, both, the algebraic approach presented here and the form factor program, are relevant. Structural properties like asymptotic completeness (which enters into the form factor program as an assumption) can be more conveniently analyzed in the algebraic framework. Furthermore, it is possible to discuss large classes of models at the same time in this approach. In comparison, the form factor program is better suited for deriving approximate formulae for local quantities such as n-point Wightman functions. Although the convergence of the form factor expansion is not under control yet, one might speculate that this situation can be improved in view of the now established existence theorem, just as the heuristic motivation of the relations of the Zamolodchikov-Faddeev algebra were rigorously justified in Haag–Ruelle scattering theory. In addition to properties of the scattering states, also something about the thermodynamics of models with a factorizing S-matrix can be learned from our analysis. By a slight generalization of our arguments, and following the reasoning in [16], it can be shown that the maps Θβ (s) : A(Os ) → H, Θβ (s)A := e−β H AΩ, where H denotes the Hamiltonian and Os = W R ∩ (W L + (0, s)), s > 0, are nuclear if Ξ (s) is. An estimate on the nuclear norms Θβ (s)1 can be calculated. As the quantity Θβ (s)1 is to be interpreted as the partition function of the restriction of the considered theory to the “relativistic box” Os at inverse temperature β [21], such estimates provide information about gross thermodynamical properties of the system. In the present paper, we restricted ourselves to models describing a single species of neutral, scalar particles. There also exist many integrable quantum field theories with
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richer particle spectra, containing bound states and solitons. The generalization of the construction procedure presented here to this larger class of models is currently under investigation.6 Before a generalization to models with bound states can be realized, one probably needs to develop an operator-algebraic understanding of the singularity structure of the corresponding S-matrices [1,6], just as the crossing symmetry of factorizing S-matrices is now known to be linked to the wedge-locality of its associated polarization-free generators [57]. Besides these more specific aspects of models with factorizing S-matrices, we note in conclusion that the general idea of constructing interacting model theories by first considering nets of wedge algebras and then analyzing their relative commutants is applicable to higher-dimensional spacetimes as well. However, the modular nuclearity condition cannot be satisfied if the spacetime dimension is larger than two. Finding an adequate condition, applicable in physical spacetime and ensuring the non-triviality of intersections of wedge algebras, might therefore lead to considerable progress in the construction of interacting quantum field theories. Acknowledgements. Since this article is the result of a rather long investigation, I have reason to thank many people and institutions. Many discussions with my PhD advisor D. Buchholz have been important for this work. Regarding the theory of complex analysis, I had the opportunity to learn a lot from H.-J. Borchers and J. Bros. G. Garrigos pointed out reference [60] to me, and M. Karowski, F. A. Smirnov and A. Fring informed me about the status of the form factor program. I also benefitted from several conversations with B. Schroer, J. Mund, M. Müger, K.-H. Rehren and R. Verch during different stages of this work. I wish to thank them all. Financial support by the Deutsche Forschungsgemeinschaft DFG, and travel grants by the universities of Gainesville and São Paulo, the DFG, the Daniel Iagolnitzer foundation and the Oberwolfach Institute are thankfully acknowledged. Last but not least, my thanks go to J. Yngvason for inviting me to the Erwin Schrödinger Institute in Vienna, where the final version of this article was written up.
A. Proof of Lemma 4.2 In this appendix, we prove the two formulae (4.21) and (4.22) for the completely contracted matrix elements Acon n,k (4.20). Recall that a contraction C ∈ Cn,k is a set of pairs, C = {(l1 , r1 ), . . . , (l|C| , r|C| )},
(A.1)
with |C| ≤ min{k, n − k}. The “right indices” satisfy r1 , . . . , r|C| ∈ {1, . . . , k}, and the “left indices” l1 , . . . , l|C| ∈ {k + 1, . . . , n}. As before, we write l C and r C for the sets {l1 , . . . , l|C| } and {r1 , . . . , r|C| }, respectively. We will need to distinguish between those contractions C ∈ Cn,k which do not contract k + 1, i.e. fulfill k + 1 ∈ / l C , and those contractions which have k + 1 ∈ l C as a left index. The former set will be denoted Cˆn,k , and the latter Cˇn,k . The set of all contractions is the disjoint union Cn,k = Cˆn,k Cˇn,k . Also recall the shorthand notations δl,r := δ(θl − θr ), Sa,b := S2 (θa − θb ) and the (k) definitions of δC and SC , δC := (−1)
|C|
|C|
δl j ,r j ,
SC(k)
j=1 (k)
Sa,b :=
:=
|C|
j=1 m j =r j +1
Sb,a ; a ≤ k < b or b ≤ k < a . Sa,b ; otherwise
6 H. Grosse and G. Lechner, work in progress.
l j −1
Sm(k)j ,r j ·
ri
Sr(k) , j ,li
(A.2)
(A.3)
Construction of Quantum Field Theories with Factorizing S-Matrices
857
Note that a contraction C ∈ Cˇn,k is always a union C = C ∪ {(k + 1, r )}, where C ∈ Cˆn,k has length |C| = |C | − 1, and r ∈ / r C . In this situation, there holds δC = −δk+1,r · δC , (k)
SC =
|C|
(A.4)
l j −1
j=1 m j =r j +1 k
= SC(k) ·
Sm(k)j ,r j ·
k
(k) Sm,r ·
ri
m=r +1
Sm,r ·
m=r +1
(k)
Sr j ,li ·
ri
(k)
Sr,li ·
r
Sk+1,r j ,
(k)
Sr j ,k+1
(A.5)
r
since l1 , . . . , l|C| > k + 1. Taking into account Sa,b = Sb,a −1 (3.3), we get δ
C
·
(k) SC
= −δC ·
(k) SC
k
· δk+1,r ·
Sm,k+1 .
(A.6)
m=r +1 m =r j for r j >r
Similarly, contractions C ∈ Cˇn,k+1 contracting k + 1 (as a right index) are unions of the form C = {(l, k + 1)} ∪ C, with C ∈ Cˆn,k and l ∈ / l C . By a computation analogous to the one above one finds in this situation (k+1)
δC = −δC · δl,k+1 ,
SC
(k+1)
= SC
·
l−1
Sk+1,m .
(A.7)
m=k+2 m =li for li
Now consider some contraction C ∈ Cˆn,k . Repeated application of the relations of Zamolodchikov’s algebra (3.15) yields (cf. (4.16)) † · · · zl†1 · · · zl†|C| · · · z n† Ω, z k+1 A z k† · · · zr†1 · · · zr†|C| · · · z 1† Ω l C | A |r C n,k = z k+2 = l C ∪ {k + 1}| [z k+1 , A] |r C n,k +
k
r =1 r∈ /rC
δk+1,r
k
Sm,k+1 · l C ∪ {k + 1}| A |r C ∪ {r }n,k . (A.8)
m=r +1 m =r j for r j >r
(k) Consider the last line, multiplied with δC SC and summed over all C ∈ Cˆn,k . Taking into account the remarks made at the beginning of the proof, there holds rk=1,r ∈/ r C C∈Cˆ = n,k being related by C = C ∪ {(k + 1, r )}. More, with the contractions C and C ˇ C ∈Cn,k over, the delta distributions and scattering functions appearing in (A.8) are the same as in (A.6). So we conclude
δC SC(k) l C | A |r C n,k = δC SC(k) l C ∪ {k + 1}| [z k+1 , A] |r C n,k C∈Cˆn,k
C∈Cˆn,k
−
C ∈Cˇn,k
(k)
δC SC l C | A |r C n,k ,
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G. Lechner
and as Cn,k = Cˆn,k Cˇn,k ,
δC SC(k) l C ∪ {k + 1}| [z k+1 , A] |r C n,k = δC SC(k) l C | A |r C n,k . C∈Cn,k
C∈Cˆn,k
Since the right-hand side coincides with Acon n,k (4.20), this proves the first formula (4.21) of Lemma 4.2. For the second formula (4.22), we argue in a similar manner. Considering a contraction C ∈ Cˆn,k , the relations (3.15) of Zamolodchikov’s algebra and Sa,b = Sb,a (3.3) imply † ] |r C n,k = l C | A |r C n,k+1 l C ∪{k + 1}| [A, z k+1
−
n
δl,k+1
l=k+2 l∈ /lC
l−1
Sk+1,m · l C ∪ {l}| A |r C ∪ {k + 1}n,k+1 .
(A.9)
m=k+2 m =li for li
According to the remarks made at the beginning of the proof, all contractions in Cˇn,k+1 are of the form C := C ∪ {(l, k + 1)}, C ∈ Cˆn,k , l ∈ / l C , i.e. we have the equality of n sums l=k+2,l ∈/ l C C∈Cˆ = C ∈Cˇ . Taking into account the relations (A.7), it n,k
n,k+1
(k+1)
follows that the second term on the right hand-side in (A.9), multiplied with δC SC (k+1) and summed over C ∈ Cˆn,k , gives C ∈Cˇ δC SC l C | A |r C n,k+1 . As the first n,k+1 term in (A.9) yields the sum over all C ∈ Cˆn,k , and since Cn,k+1 = Cˆn,k Cˇn,k+1 , we arrive at
† δC SC(k+1) l C ∪ {k + 1}| [A, z k+1 ] |r C n,k C∈Cˆn,k
=
(k+1)
δC SC
l C | A |r C n,k+1 = Acon n,k+1 .
C∈Cn,k+1
This is the desired Eq. (4.22).
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Commun. Math. Phys. 277, 861–863 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0364-6
Communications in
Mathematical Physics
Erratum
2-Matrix versus Complex Matrix Model, Integrals over the Unitary Group as Triangular Integrals B. Eynard1 , A. Prats Ferrer2 1 Service de Physique Théorique de Saday, CEA/DSM/SPhT-CNRS/SPM/URA 2306,
91191 Gif-sur-Yuette Cedex, France. E-mail: [email protected]
2 Departamenta d’Estructura i constituents de la Matèria, Universitad Barcelona, Av. Diagonal 647,
08028 Barcelona, Spain. E-mail: [email protected] Received: 5 September 2006 / Accepted: 11 September 2007 Published online: 15 November 2007 – © Springer-Verlag 2007 Commun. Math. Phys. 264, 115–144 (2006)
In the above article Theorem 3.4 contains an error which is easily resolved through a slight reformulation of the statement, including in it part of Theorem 4.1. We keep the conventions and definitions given in the paper. Original Version This is Theorem 3.4 as it appears in the published article. Theorem 3.4. For any polynomial invariant function F(A, B), one has: α1 α2 J˜n2 2 2 d X dY e− 2 Tr X e− 2 Tr Y W˜ F (X, Y ) 2 n! Z H (n, γ , α1 , α2 ) Dn(R)×Dn (R) 2 α1 α2 Jn 2 = d X e− 2 Tr X e− 2 Tr X W F (X, X ). (0-1) n! Z C (n, γ , α1 , α2 ) Dn (C) Proof. Start from Theorem 3.3, diagonalize M1 and M2 on the hermitian side, and jordanize Z on the complex side, α1 α2 J˜n2 2 2 F H = 2 d X dY e− 2 Tr X e− 2 Tr Y W˜ F (X, Y ) n! Z H (n, γ , α1 , α2 ) Dn (R)×Dn (R) 2 α1 α2 Jn 2 = FC = d X e− 2 Tr X e− 2 Tr X e−γ Tr X X ω F (X, X ) n! Z C (n, γ , α1 , α2 ) Dn (C)
The online version of the original article can be found under doi:10.1007/s00220-006-1541-8.
862
B. Eynard, A. Prats Ferrer
2 α1 α2 Jn 2 d X e− 2 Tr X e− 2 Tr X e−γ Tr X σ X τ ω F (X σ , X τ ) n! Z C (n, γ , α1 , α2 ) Dn (C) 2 α1 α2 Jn 2 d X e− 2 Tr X e− 2 Tr X W F (X σ , X τ ). (0-2) = n! Z C (n, γ , α1 , α2 ) Dn (C)
=
The equality in the first line is obtained by diagonalizing M1 and M2 (with Jacobian given in Eq.2-17), the equality in the second line is obtained by Jordanizing Z (with Jacobian given in Eq.2-19), the equality between the second and third line holds for any pair of permutations σ and τ (it can be proven with Lemma A.1 given in the Appendix), and the equality of the last line comes from the definition of W F . The main mistake appears from second to third line, since at this point it is false that permuting the two matrices X and X to X σ and X τ independently does not change the integral. However, a small rearrangement of Theorems 3.4 and 4.1 yields the desired result. Corrected Version To fix the mistake, we reformulate Theorem 3.4 as follows: Theorem 3.4. For any polynomial invariant function F(A, B), one has: α1 α2 J˜n2 2 2 d X dY e− 2 Tr X e− 2 Tr Y W˜ F (X, Y ) 2 n! Z H (n, γ , α1 , α2 ) Dn(R)×Dn (R) α1 α2 Jn 2 2 = d X dY e− 2 Tr X e− 2 Tr Y W F (X, Y ). n! Z C (n, γ , α1 , α2 ) Dn (R)×Dn (R) (0-3) Proof. α1 α2 J˜n2 2 2 F H = 2 d X dY e− 2 Tr X e− 2 Tr Y W˜ F (X, Y ) n! Z H (n, γ , α1 , α2 ) Dn (R)×Dn (R) 2 α1 α2 Jn 2 d X e− 2 Tr X e− 2 Tr X e−γ Tr X X ω F (X, X ) = FC = n! Z C (n, γ , α1 , α2 ) Dn (C) n Jn √π−δ α1 2 d X dY e− 2 Tr X = n 2π Dn (R)×Dn (R) n! Z C (n, γ , α1 , α2 ) √ ×e =
−
α2 2
δ
Tr Y 2 −γ Tr X Y e n Jn √π−δ
n! Z C (n, γ , α1 , α2 ) ×
ω F (X, Y ) 2π √ δ
Dn (R)×Dn (R) n Jn √π−δ
n
1 n!2
d X dY e−
α1 2
Tr X 2 −
α2 2
Tr Y 2 −γ Tr X σ Yτ
α1 2
Tr X 2 −
α2 2
Tr Y 2
e
e
ω F (X σ , Yτ )
σ,τ
=
n! Z C (n, γ , α1 , α2 ) × σ,τ
2π √ δ
Dn (R)×Dn (R)
n
1 n!2
d X dY e−
e
W F (X, Y ).
(0-4)
Erratum
863
The equality in the first line is obtained by diagonalizing M1 and M2 (with Jacobian given in Eq.2-17), the equality in the second line is obtained by Jordanizing Z (with Jacobian given in Eq.2-19), the equality between the second and third line follows from Lemma A.1 given in the Appendix, the equality between the third and fourth line can be done thanks to the symmetry of the measure. It is convenient to do so in order to compare integrands later, since the integrand in the first line is symmetric by the action of the unitary integral inside W˜ F (X, Y ). Finally the equality between the fourth and fifth line comes from the definition of W F . Additional minor modification. Due to the inclusion of part of Theorem 4.1 in the corrected Theorem 3.4, we must rewrite the proof of the former in the following terms Theorem 4.1. For any invariant function F(A, B) one has: † dU F(X, U Y U † ) e−γ Tr XU Y U U (n) σ τ −γ Tr X σ Yτ † −γ Tr T T † dT cn σ τ (−1) (−1) e T (n) F(X σ + T, Yτ + T ) e , = n! (X )(Y ) (0-5) where
n−1 cn =
k=0
(−2π )
k!
n(n−1) 2
,
(0-6)
i.e. W˜ F (X, Y ) = n! cn W F (X, Y ). Proof. From Theorem 3.4 we have α1 α2 2 d X dY e− 2 Tr X e− 2 0= Dn (R)×Dn (R)
Tr Y 2
(n! cn W F (X, Y ) − W˜ F (X, Y )).
(0-7)
(0-8)
Notice that if f (A) and g(B) are invariant functions, i.e. f (U AU −1 ) = f (A) for all A and U (resp. g(U BU −1 ) = g(B) for all B and U ), one has: W f (X )g(Y )F(X,Y ) (X, Y ) = f (X )g(Y )W F (X, Y ), W˜ f (X )g(Y )F(X,Y ) (X, Y ) = f (X )g(Y )W˜ F (X, Y ).
(0-9)
Thus, for any symmetric polynomials f (x1, . . . , xn) and g(y1, . . . , yn), one has: α1 α2 2 2 d X dY e− 2 Tr X e− 2 Tr Y f (X )g(Y ) 0= Dn (R)×Dn (R)
×(n! cn W F (X, Y ) − W˜ F (X, Y )). α − 21
α X 2 − 22
(0-10)
Tr Tr Y e (n! cn W F (X, Y ) − W˜ F (X, Y )) is a continuous function Notice that e quickly decreasing at ∞ in all variables, and it is symmetric in the x’s and in the y’s. Using the Stone-Weierstrass Theorem (polynomials are dense in the space of continuous functions), one gets that (n! cn W F (X, Y ) − W˜ F (X, Y )) = 0.
Communicated by L. Takhtajan
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