Bohr-Sommerfeld quantization of b-symplectic toric manifolds

BOHR-SOMMERFELD QUANTIZATION OF b-SYMPLECTIC TORIC MANIFOLDS PAU MIR, EVA MIRANDA, AND JONATHAN WEITSMAN Abstract. We define the Bohr-Sommerfeld quantization via T-modules for a b-symplectic toric manifold and show that it coincides with the formal geometric quantization of [GMW18b]. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the toric action on the manifold. Dedicated to Victor Guillemin 1. Introduction Singular symplectic manifolds appear in the investigation of geometrical and dynamical facets of non-compact manifolds as their natural compactifications. This is exemplified in the work undertaken by [KMS16] and [MO21] on the restricted three-body problem. They also appear in the realm of quantization, where new procedures are required to extend the classic quantization scheme to the compactification of symplectic manifolds. For compact manifolds coming from a physical system, one of the minimal requirements for a quantization model is that it is finite-dimensional, something that has long been sought after. This quantization challenge was tackled, for instance, by Guillemin, Miranda and Weitsman using the formal geometric quantization of [Wei01] and [Par09], and the surprising result they proved in [GMW18b] is that the formal geometric quantization of a b-symplectic manifold a is indeed a finitedimensional vector space. This raised the natural question of whether there is a true geometric quantization of such a space. An answer was given in the affirmative in [BLS21] and [LLSS21], where virtual modules agreeing with the formal geometric quantization of [GMW18b] were constructed analytically using index theory. The purpose of this paper is to revisit the question in the context of Bohr-Sommerfeld quantization, as it was done in [GS83], but restricting ourselves to the case of symplectic and b-symplectic toric manifolds. P. Mir is funded in part by the Doctoral INPhINIT - RETAINING grant LCF/BQ/DR21/11880025 of “la Caixa” Foundation. P. Mir and E. Miranda are partially supported by the AEI grant PID2019-103849GB-I00 of MCIN/ AEI /10.13039/501100011033. E. Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prizes 2016 and 2021 and by the Spanish State Research Agency, through the Severo Ochoa and Mar´ıa de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M). J. Weitsman was supported in part by a Simons collaboration grant. 1 2 PAU MIR, EVA MIRANDA, AND JONATHAN WEITSMAN We start revisiting a result of Guillemin and Sternberg [GS83], which identifies the Bohr- Sommerfeld leaves of a symplectic manifold endowed with an integrable system with the integer points in the image of its moment map. This allows to read the geometric quantization of a symplectic toric manifold from its Delzant polytope. Then, we prove that the same can be done for b-symplectic toric manifolds. Next, we prove our main result, Theorem 5.2. It states that, for any integral b-symplectic toric manifold, the Bohr-Sommerfeld quantization with sign agrees with the formal geometric quantization of [GMW18b]. For this, we need to introduce the definition of Bohr-Sommerfeld quantization with sign, which we do via T-modules. The article is organized as follows. In Section 2 we recall the necessary preliminaries on bsymplectic manifolds, Bohr-Sommerfeld and formal geometric quantization. In Section 3, following the idea of [GS83], we prove that the Bohr-Sommerfeld leaves of an integral b-symplectic toric manifold can be obtained from the image of the moment map of the toric action. In Section 4 we introduce the Bohr-Sommerfeld quantization with sign via T-modules. In Section 5 we prove the equivalence between Bohr-Sommerfeld quantization and formal geometric quantization both for integral symplectic and b-symplectic toric manifolds. 2. Preliminaries. b-Symplectic manifolds, Bohr-Sommerfeld quantization and formal geometric quantization In this preliminaries section we review the definitions of b-symplectic manifolds, Bohr-Sommerfeld quantization and formal geometric quantization. We also recall the results that we use in the other sections. 2.1. b-Symplectic manifolds. b-Symplectic geometry is a generalization of symplectic geometry that furnishes manifolds with boundary with a Poisson structure which lowers rank at a singular hypersurface. It is possible to associate a tangent and a cotangent bundle to the class of b-symplectic manifolds and to expand there the classical symplectic tools. This singular model, presented in [GMP11], [GMP14] and [GMPS15], reveals to be useful for many families of physical problems for which symplectic manifolds are not enough. We proceed to briefly summon the necessary definitions in b-symplectic geometry and we refer to the aforementioned articles for a complete overview. Recall that a b-manifold is a pair (M, Z) where Zis a hypersurface in a manifold Mand a b-map is a map f: (M1, Z1)−→ (M2, Z2) between b-manifolds with fis transverse to Z2and Z1=f−1(Z2). Definition 2.1 (b-vector field).A vector field on a b-manifold (M, Z) is called a b-vector field if it is tangent to Zat every point p∈Z. Let (Mn, Z) be a b-manifold. If xis a local defining function for Zon an open set U⊂Mand (x, y1, . . . , yn−1) is a chart on U, then the set of b-vector fields on Uis a free C∞(M)-module with basis (x∂ ∂x,∂ ∂y1 ,..., ∂ ∂yn ). There exists a vector bundle associated to this module called the b-tangent bundle and denoted by bTM. The b-cotangent bundle bT∗Mof Mis defined to be the vector bundle dual to bT M. BOHR-SOMMERFELD QUANTIZATION OF b-SYMPLECTIC TORIC MANIFOLDS 3 For each k > 0, let bΩk(M) denote the space of sections of the vector bundle Λk(bT∗M), called b-de Rham k-forms. For any defining function fof Z, every b-de Rham k-form can be written as (1) ω=α∧df f+β, with α∈Ωk−1(M) and β∈Ωk(M). This decomposition enables us to extend the exterior operator dto bΩ(M) by setting dω =dα ∧df f+dβ. The right hand side agrees with the usual exterior operator don M\Zand extends smoothly over Mas a section of Λk+1(bT∗M). The fact that d2= 0 allows us to define a complex of bforms, the b-de Rham complex. The cohomology associated to this complex is the b-cohomology and it is denoted by bH∗(M). The elements of bΩ0(M) are also called b-functions and the following definition characterizes them. Definition 2.2 (b-function).The set of b-functions bC∞(M) consists of functions with values in R∪ {∞} of the form clog|f|+g, where c∈R,fis a defining function for Zand gis a smooth function on M. The differential operator dis defined as: d(clog|f|+g) := c df f+dg ∈bΩ1(M), where dg and df are the standard de Rham derivatives. A special class of closed 2-forms of the complex of b-forms is the class of b-symplectic forms as defined in [GMP14]. It contains forms with singularities and can be introduced formally for b-symplectic manifolds, making it possible to extend the symplectic structure from M\Zto the whole manifold M. Definition 2.3 (b-symplectic manifold).Let (M2n, Z) be a b-manifold and ω∈bΩ2(M) a closed b-form. We say that ωis b-symplectic if ωpis of maximal rank as an element of Λ2(bT∗ pM) for all p∈M. The triple (M, Z, ω) is called a b-symplectic manifold. The Mazzeo-Melrose Theorem describes the relationship between b-cohomology and de Rham cohomology and makes it natural to talk of integrality of b-forms. Theorem 2.4 (Mazzeo-Melrose).The b-cohomology groups of (M2n, Z)satisfy bH∗(M)∼ =H∗(M)⊕H∗−1(Z). Remark 2.5.The integrality of a b-form ωin the sense of [GMW18b] implies the integrality of the form on M\Z. Since we will work with a line bundle on Mwhose Chern class is given by the projection of [ω] to H∗(M), its restriction to M\Zhas Chern class [ωM\Z]. 4 PAU MIR, EVA MIRANDA, AND JONATHAN WEITSMAN 2.2. Symplectic and b-symplectic toric manifolds. A classical theorem on the classification of symplectic toric manifolds is Delzant’s Theorem. It characterizes any such manifold via its Delzant polytope, the convex polytope corresponding to the entire image of the moment map of the toric action. Theorem 2.6 (Delzant, [Del88]).Let (M2n, ω, µ)be a toric manifold. Then, there is a bijective correspondence between the following two sets, which is given by the image of the moment map µ: {toric manifolds} −→ {Delzant polytopes} (M2n, ω, µ)−→ µ(M) In a symplectic toric manifold (M2n, ω, µ), the singularities of the moment map µcan only be of elliptic type, in the sense of Williamson [Wil36]. In fact, the singular leaves of the toric foliation are in correspondence with the points in the boundary of Delzant’s polytope, as it is detailed in the following observation. Remark 2.7.Let (M2n, ω, µ) be a symplectic toric manifold and ∆ its Delzant’s polytope. For k= 1, . . . , n, the points in the intersection of k≤nfacets of ∆ correspond to the leaves of M where µhas ksingular elliptic components. In particular, the vertices of ∆ correspond to the fixed points of µ. On the other hand, and in the appropriate coordinates, the elliptic singular components of the moment map at any singular leaf can be written as a sum of squares [Eli90]. The classification of b-symplectic toric surfaces was established by Guillemin, Miranda, Pires and Scott. Their result is that any b-symplectic surface is either a b-symplectic sphere or a b-symplectic torus. Theorem 2.8 (Guillemin-Miranda-Pires-Scott, [GMPS15]).Ab-symplectic surface with a toric S1-action is equivariantly b-symplectomorphic to either (S2, Z)or (T2, Z), where Zis a collection of latitude circles (in the T2case, an even number of such circles), the action is the standard rotation, and the b-symplectic form is determined by the modular periods of the critical curves and the regularized Liouville volume. The theorem above, in fact, boils down from the following result on the classification of higherdimensional b-symplectic toric manifolds also carried out in [GMPS15]. Proposition 2.9 (Guillemin-Miranda-Pires-Scott, Remark 38 in [GMPS15]).Every b-symplectic toric manifold is either the product of a b-symplectic T2with a classic symplectic toric manifold, or it can be obtained from the product of a b-symplectic S2with a classic symplectic toric manifold by a sequence of symplectic cuts performed at the north and south “polar caps”, away from the critical hypersurface Z. The image of the moment map of a b-symplectic toric manifold is a b-Delzant polytope and the classification of Proposition 2.9 is a consequence of the b-Delzant theorem (Theorem 35 of [GMPS15]). The main skeleton of the proof builds up from the proposition below (Proposition 18 in [GMPS15]). Proposition 2.10. Let (M2n, Z, ω, µ)be a b-symplectic toric manifold, La leaf of its symplectic foliation and vZthe modular weight of Z. Pick a lattice element X∈tthat represents a generator BOHR-SOMMERFELD QUANTIZATION OF b-SYMPLECTIC TORIC MANIFOLDS 5 of t/tZand pairs positively with vZ. Then, there is a neighbourhood L×S1×(−ε, ε)∼ =U⊆Mof Zsuch that the Tn-action on U\Zhas moment map µU\Z:L×S1×((−ε, ε)\ {0})→t∗∼ =t∗ Z×R,(ℓ, ρ, t)7→ (µL(ℓ), c log |t|), where cis the modular period of Z, the map µL:L→t∗ Zis a moment map for the Tn−1 Z-action on L, and the isomorphism t∗∼ =t∗ Z×Ris induced by the splitting t∼ =tZ⊕ ⟨X⟩. Since the moment map of a group action over a b-symplectic manifold is a b-function (see Definition 2.2), it can be unbounded due to the logarithm term. Hence, its image is neither convex (in the sense of classical analysis, for a more sophisticated notion of convexity confer [GMPS17]) in general (see Figure 1) nor bounded. This is the main handicap when one tries to obtain a finite Bohr-Sommerfeld quantization of a b-symplectic toric manifold and the reason why we introduce the Bohr-Sommerfeld quantization with sign in Section 4. Zµ=−log |h| µ Figure 1. The moment map of the rotation action over the canonical b-symplectic sphere is unbounded in any neighbourhood of Z. 2.3. Bohr-Sommerfeld quantization. Let us recall the Bohr-Sommerfeld quantization of a compact symplectic toric manifold using the classical definitions of Kostant [Kos70] and Guillemin- Sternberg [GS82]. Let (M, ω) be an integral symplectic manifold and let Lbe a complex line bundle with connection ∇whose curvature is ω. Geometric quantization is a process which associates to the quadruple (M, ω, L,∇) a Hilbert space Q(M) and to each f∈C∞(M) a self-adjoint operator Q(f) on that space. Following [Kos70], we define the quantization using the additional data given by a real polarization of M. That is, a foliation of Mby Lagrangian submanifolds. This foliation may be given by the fibres of a map π:M→Band, in this case, the quantization is given by sections s∈Γ(L) satisfying (2) ∇Xs= 0, for any Xtangent to the fibres of π. If (M, ω, µ) a toric manifold, a natural foliation is given by the fibres of the moment map µ:M→t∗. When Mis compact, there are no smooth sections satisfying equation (2) defined globally on all M. Instead, such leafwise constant sections or flat sections are concentrated on the fibres π−1(b) such that b∈Im(π) and L|π−1(b)is a trivial bundle. The quantization space is defined as 6 PAU MIR, EVA MIRANDA, AND JONATHAN WEITSMAN (3) Q(M) = M b∈BBS C⟨sb⟩, where BBS is the Bohr-Sommerfeld set, namely, BBS ={b∈Im(π)⊂B:L|π−1(b)is trivial}, and sbis the corresponding flat section of L|π−1(b). This definition of quantization is based on the identification of the Bohr-Sommerfeld leaves and we call it Bohr-Sommerfeld quantization. It coincides with the classical definition of geometric quantization of a symplectic manifold equipped with a completely integrable system that provides a real polarization. Some authors use sheaves to compute the Bohr-Sommerfeld leaves and obtain the same quantization (see [´ S77] or [Ham08]). 2.4. Formal geometric quantization. We take the basic definitions of formal geometric quantization of Hamiltonian T-spaces from [GMW18b]. For further reading on this quantization see [Wei01], [HM16] and [Par09]. Suppose (M, ω) is a compact symplectic manifold and let (L,∇) be a complex line bundle with connection of curvature ω. By twisting the spin-CDirac operator on Mby L, we obtain an elliptic operator ¯ ∂L. The geometric quantization Q(M) of Mis defined by Q(M) = ind(¯ ∂L), and it is a virtual vector space. If (M, ω) is a compact integral symplectic manifold, one can always find a complex line bundle Lwith connection ∇of curvature ωand the quantization Q(M) is independent of this choice. If Mis equipped with a Hamiltonian action of a torus T, the action can be lifted to Land the almost complex structure of Lcan be chosen to be T-invariant. In this case, the quantization Q(M) is a finite-dimensional virtual T-module. For ξ∈t∗,denote by M//ξTthe reduced space of Mat ξ. For αa weight of Tand Va virtual T-module, denote by Vαthe sub-module of Vof weight α. The following result states that the component of weight αof the quantization of Mequals the quantization of the reduced space of Mat α. Theorem 2.11 (Quantization commutes with reduction, [Mei96]).Let (M, ω)is a compact integral symplectic manifold. Suppose Mis equipped with a Hamiltonian action of a torus Tand let αbe a weight of T. Then (4) Q(M)α=Q(M//αT). In other words, (5) Q(M) = M α Q(M//αT)α. BOHR-SOMMERFELD QUANTIZATION OF b-SYMPLECTIC TORIC MANIFOLDS 7 Theorem 2.11 and equation (5) are valid only for regular values of the moment map of the Hamiltonian T-action. In the case where αis a singular value of the moment map, the singular quotient must be replaced by a slightly different construction using a shift of α[Mei96]. A similar caution applies in the case of Hamiltonian T-spaces which are non-compact and in the case of b-symplectic manifolds. 2.4.1. Formal geometric quantization of non-compact Hamiltonian T-spaces. In the case where M is non-compact, equation (4) is used to define the quantization of such Hamiltonian T-spaces. Definition 2.12 (Weitsman, [Wei01]).Let Mbe a Hamiltonian T-space with integral symplectic form. Suppose the moment map for the T-action is proper. Let Vbe an infinite-dimensional virtual T-module with finite multiplicities. We say V=Q(M) if for any compact Hamiltonian T-space Nwith integral symplectic form, we have (6) (V⊗Q(N))T=Q((M×N)//0T). In other words, as in (5), Q(M) = M α Q(M//αT)α, where the sum is taken over all weights αof T. The fact that the moment map is proper implies that the reduced space (M×N)//0Tis compact for any compact Hamiltonian T-space N, so that the right hand side of equation (6) is well-defined. 2.4.2. Formal geometric quantization of b-symplectic manifolds. Suppose now that (M, ω) is compact, connected, oriented and not symplectic but b-symplectic. Suppose that it is equipped with a Hamiltonian action of a torus T, with nonzero leading modular weight. Let Lbe a complex line bundle on Mwith connection ∇on L|M\Zwhose curvature is ω|M\Z. In [GMW21], the formal geometric quantization Q(M) is defined as the following virtual T- module. Definition 2.13. Let Vbe a virtual T-module with finite multiplicities. We say V=Q(M) if for any compact Hamiltonian T-space Nwith integral symplectic form, we have (7) (V⊗Q(N))T=εQ((M×N)//0T), where Q(N) denotes the standard geometric quantization of N,Q(M×N)//0Tis the geometric quantization of the compact integral symplectic manifold (M×N)//0T, and εis +1 if the symplectic orientation on the symplectic quotient (M×N)//0Tagrees with the orientation inherited from M×Nand −1 otherwise. This means that Q(M) = Q(M\Z) = ⊕iεiQ((M\Z)i), where the (M\Z)iare the connected components of M\Z,Q(M\Z) is the formal geometric quantization of the non-compact Hamiltonian T-space M\Z, and the εi∈ {±1}are determined by the relative orientations of the symplectic 8 PAU MIR, EVA MIRANDA, AND JONATHAN WEITSMAN forms on the components of M\Zand the overall orientation of M. Alternatively, Q(M) = M α ε(α)Q(M//αT)α, where Q(M//αT) must be defined using the shifting trick if αis not a regular value of the moment map, and each ε(α)∈ {±1}is determined by the relative orientations of Mand M//αT. In this b-symplectic case, the condition that the modular weight is non-zero guarantees that the reduced space (M×N)//0Tis compact and symplectic for any compact Hamiltonian T-space N, so that Q((M×N)//0T) is well-defined. 3. Bohr-Sommerfeld leaves via the moment map In this section we prove that, in an integral symplectic toric manifold, the Bohr-Sommerfeld set coincides with the set of integer points in the image of the moment map of the toric action. We prove that the same result also holds for integral b-symplectic toric manifolds. First, let us recall a result from Guillemin and Sternberg [GS83] that identifies the Bohr- Sommerfeld leaves in a symplectic manifold using the moment map of an integrable system. In particular, it proves that the count of Bohr-Sommerfeld leaves of an integral symplectic manifold equals the count of the integer points in the image of the moment map. Theorem 3.1 (Guillemin-Sternberg, Theorem 2.4 in [GS83]).Let (M, ω)be a 2n-dimensional symplectic manifold endowed with an integrable system with moment map µ:M→B. Let pand qbe two distinct points of Bcontained in an open simply connected subset B0of B. Then: •There exists a globally defined system of action coordinates f1, . . . , fnon B0such that f1(p) = · · · =fn(p)=0. •If p∈Bis in the Bohr-Sommerfeld set, q∈Bis in the Bohr-Sommerfeld set if and only if f1(q), . . . , fn(q)are integers. In Theorem 3.1, the correspondence between the Bohr-Sommerfeld leaves and the integer points of the moment map is established after the election of a globally defined system of action coordinates. Once a Bohr-Sommerfeld leaf is identified at a point p∈B, the 0 of all the action coordinates is set there and the other Bohr-Sommerfeld leaves correspond to the integer points in these coordinates. As a consequence, the integer condition that Bohr-Sommerfeld leaves have to satisfy can be shifted by an additive constant as long as it is the same constant for all the leaves, since the essential implication of Theorem 3.1 is that the difference between the action variables at any two Bohr-Sommerfeld leaves is an integer. In view of this, a value of the moment map has to be fixed at some point (and leaf) of Mor, equivalently, a choice of the constant in the moment map has to be made. We will prove that this choice of a constant in the moment map is also equivalent to the choice of the connection 1-form Θ such that dΘ = ω. BOHR-SOMMERFELD QUANTIZATION OF b-SYMPLECTIC TORIC MANIFOLDS 9 3.1. Dependence on the connection. In the following statements we show that we can always find a connection 1-form Θ with dΘ = ωsuch that the Bohr-Sommerfeld set coincides with the integer points in the image of the moment map in the appropriate coordinates. Lemma 3.2. Let (M, ω, µ)be a toric symplectic manifold. Let Lbe a complex line bundle over M with connection ∇whose curvature is ω. Given an arbitrary connection 1-form Θ, we can always produce a connection ˜ Θthat is invariant under the toric action. Proof. Since the group G=Tnacting on Mis compact, there exists a Haar measure dg such that RGdg = 1. Then, the averaging of a form Θ via RGL∗ gΘdg provides a G-invariant form ˜ Θ. □ Proposition 3.3. Let (M, ω, µ)be a symplectic toric manifold. Let Lbe a complex line bundle over Mwith connection ∇whose curvature is ω. If Θ1and Θ2are two invariant connection 1-forms, the function ⟨Θ1−Θ2, X⟩is constant when Xis a vector field tangent to the polarization of Mby µ. Proof. By definition, the connection 1-forms Θisatisfy µ= Θi(X) [Kos70], where µis a moment map of the toric action and Xis a vector field tangent to the polarization given by µ. Take αsuch that π∗α= Θ1−Θ2. By Lemma 3.2, Θ1and Θ2can be chosen invariant so that LXα= 0, since αis invariant under X. Then, by Cartan’s magic formula, we have that dα(X) = diXα=LXα−iXdα. If we have two invariant connection 1-forms Θ1and Θ2, we know that their difference Θ1−Θ2= π∗αis a constant. Then π∗dα =dΘ1−dΘ2=ω−ω= 0 and dα = 0 is zero. Finally, if d(α(X)) = 0, α(X) is a constant. □ Remark 3.4.In view of Proposition 3.3, α(Xi) is a constant for any component Xiof X. On the other hand, for any Θ1,Θ2, we have that Θ1= Θ2+π∗αand µ1=µ2+π∗α(X). Then, fixing α(Xi) is equivalent to make a choice of the connection 1-form and to fix the constant in the moment map. In other words, the choice of a constant, which has to be made at some point in the process of quantization, can be made either by selecting a specific connection 1-form or, equivalently, by setting at 0 the coordinates of the moment map at a particular Bohr-Sommerfeld leaf. 3.2. The Bohr-Sommerfeld set in the image of the moment map. Observe that any toric action on a symplectic manifold defines an integrable system. Then, in a symplectic toric manifold (M, ω, µ), we can apply Theorem 3.1 to identify the Bohr-Sommerfeld leaves of Mwith the integer points in the image of the moment map. We want to extend the correspondence between Bohr-Sommerfeld leaves and integer points in the image of the moment map to b-symplectic toric manifolds. For this reason, we prove first Theorem 3.5, which is a particular case of Theorem 3.1 for symplectic toric manifolds. Then, we obtain Corollary 3.6, the b-symplectic toric version of Theorem 3.5. Theorem 3.5. Let (M, ω, µ)be an integral symplectic toric manifold. Then, the Bohr-Sommerfeld set coincides with the integer points in the image of µ. 16 PAU MIR, EVA MIRANDA, AND JONATHAN WEITSMAN h Z1Z2Z3Z4Z5 −1hZ1hZ2hZ3hZ4hZ51 µ Figure 4. The moment map of the b-sphere of Figure 3. Blue dots are the Bohr- Sommerfeld leaves in positively oriented components of S2\Z. Red dots are the Bohr-Sommerfeld leaves in negatively oriented components of S2\Z. White-filled dots represent Bohr-Sommerfeld leaves at the neighbourhood of each Ziwhose associated quantization spaces symmetrically cancel. a positively-oriented connected component and as ϵ(b) = −1 if πµ−1(b)belongs to a negativelyoriented one. We call ϵ(b) the sign of b. Definition 4.11 (Bohr-Sommerfeld quantization with sign of (M2n, Z, ω, µ)).Let BBS be the Bohr-Sommerfeld set of (M2n, Z, ω, µ). The quantization with sign of (M2n, Z, ω, µ) is ˜ Q(M2n) = M b∈BBS ϵ(b)C(b). Lemma 4.12. ˜ Q(M2n)is a finite-dimensional vector space. Proof. Take a symmetric neighbourhood U⊂M2n\Z, which always exists by Proposition 2.10. Project Uby πto bS2or bT2and use the same argument in the proof of Lemmas 4.6 and 4.9 to obtain that ˜ Q(M2n) is finite-dimensional. □ Remark 4.13.Observe that in the definition of the Bohr-Sommerfeld quantization with sign and, specifically, in the sum Lb∈BBS ϵ(b)C(b), we are using the fact that we have a group Tacting with weights with finite multiplicity, thus each C(b) is finite dimensional. Then, the infinite sum Lb∈BBS ϵ(b)C(b) is of finite dimension. To illustrate the behaviour of this sum, consider the abelian case where all representations C(w) are one-dimensional and correspond to a weight w, so that A=⊕wmwC(w) and B=⊕wnwC(w), where mwand nware integers. Then, one can consider A−Bto be ⊕w(mw−nw)C(w), which is a virtual T-module with finite multiplicities. The dimension of A−Bis d=Pw(mw−nw) if this sum is finite and, then, one can talk about a finite-dimensional virtual module of dimension d. 5. The final count. Bohr-Sommerfeld quantization equals formal geometric quantization In this section we compare Bohr-Sommerfeld quantization with formal geometric quantization. We prove that they are equivalent because they both equal the count of integer points in the image BOHR-SOMMERFELD QUANTIZATION OF b-SYMPLECTIC TORIC MANIFOLDS 17 of the moment map. We do it both for the symplectic case, in Theorem 5.1, and for the b-symplectic case, in Theorem 5.2. Theorem 5.1. Let (M2, ω, µ)be a symplectic toric manifold. Then, the formal geometric quantization of Mcoincides with the Bohr-Sommerfeld quantization. Proof. We compute the formal geometric quantization of a symplectic toric manifold and then we count the Bohr-Sommerfeld leaves. We see that they are both the same and, in particular, they coincide with the count of the integer points in the image of the moment map (with sign) of the toric action. In view of Theorem 2.11 (Quantization commutes with reduction), the formal geometric quantization of a symplectic toric manifold (M2n, ω, u) is given by (13) Q(M) = M α∈Zn Q(M//αT)α. Notice that the sum is taken over all weights αof T. Suppose µ:M→tis the moment map of the toric action. Then, the reduced spaces M//αT are either empty if αis not in µ(M) or a point if it is. Since the quantization of each single point is given by C, we have that (14) Q(M) = M α∈Zn∩µ(M) C(α). Then, the formal geometric quantization of Mis given by as many copies of Cas integer points in the image of the moment map. On the other hand, by Theorem 3.5, the Bohr-Sommerfeld quantization is given by the count of Bohr-Sommerfeld leaves of M, which coincides with the integer points in the image of the moment map. □ Theorem 5.1 also holds when the manifold is b-symplectic. Theorem 5.2. Let (M2n, Z, ω, µ)be a b-symplectic toric manifold such that the image of the moment map is simply connected. Then, the formal geometric quantization of Mcoincides with the Bohr-Sommerfeld quantization with sign. Proof. Recall from [GMW18a] that for any b-symplectic toric manifold (M, Z, ω, µ), the quantization space Q(M) is defined as the vector space such that the following equality holds (15) (Q(M)⊗Q(N))α=ε(α)Q((M×N)//αT) for any compact symplectic manifold Nand any weight αof T, where Tis the torus generating the action with moment map µ[GMW18b]. Since we are considering the case of a toric action, the reduced spaces are empty or just points. Observe that the right hand side of equation 15 is symplectic. Therefore we can apply the quantization commutes with reduction scheme. 18 PAU MIR, EVA MIRANDA, AND JONATHAN WEITSMAN We can take Nto be the coadjoint orbit of this Tn-action of integral weight α, which is in the integral lattice by definition. Applying equation 15 to equation 4 from Theorem 2.11, we see that the formal geometric quantization of a b-symplectic manifold is given by the direct sum of the reduced spaces Q(M//αT) at each α. The weights αare precisely the points in the integer lattice of the moment map of the toric action, meaning that {α}=Zn∩µ(M). On the other hand, Then, the quantization of Mis (16) Q(M) = M α ε(α)Q(M//αT)α=M α∈Zn∩µ(M) ε(α)C(α). On the other hand, by Corollary 3.6 the Bohr-Sommerfeld set of Mcoincides with the lattice of integer points in the image of µ. Therefore, by Definition 4.11, the Bohr-Sommerfeld quantization with sign of Mis (17) ˜ Q(M) = M b∈BBS ϵ(b)C(b) = M b∈Zn∩µ(M) ϵ(b)C(b). Finally, for any point pin the Bohr-Sommerfeld set BBS, the sign ϵ(p) coincides with the sign ε(p). By definition, both of them are +1 if the relative orientations of the symplectic forms on the component of µ−1(p) of M\Zand the overall orientation of Magree and −1 otherwise. 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Non-abelian symplectic cuts and the geometric quantization of noncompact manifolds. Lett. Math. Phys., 56(1):31–40, 2001. EuroConf´erence Mosh´e Flato 2000, Part I (Dijon). [Wil36] John Williamson. On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems. Amer. J. Math., 58(1):141–163, 1936. Pau Mir, Laboratory of Geometry and Dynamical Systems, Universitat Polit` ecnica de Catalunya, Avinguda del Doctor Mara˜ non 44-50, 08028, Barcelona Email address:[email protected] Eva Miranda, Laboratory of Geometry and Dynamical Systems &Institut de Matem` atiques de la UPC-BarcelonaTech (IMTech), Universitat Polit` ecnica de Catalunya, Avinguda del Doctor Mara˜ non 44-50, 08028, Barcelona, CRM Centre de Recerca Matem` atica, Campus de Bellaterra Edifici C, 08193 Bellaterra, Barcelona Email address:[email protected] Jonathan Weitsman, Department of Mathematics, Northeastern University, Boston, MA 02115 Email address:[email protected]