Geometric quantization via cotangent models

Analysis and Mathematical Physics (2021) 11:118 https://doi.org/10.1007/s13324-021-00559-4 Geometric quantization via cotangent models Pau Mir1·Eva Miranda2,3 Received: 8 May 2021 / Revised: 25 May 2021 / Accepted: 27 May 2021 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 Abstract In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the cotangent models in [13] by both consideringsingularorbitsandaddingtothecotangentmodelsamodelfortheprequantum line bundle. These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hyperbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on T∗(SO(3)) to a sphere. Our geometric quantization formulation coincides with the models given in [11] and [21] away from the singularities and corrects former models for hyperbolic and focus-focus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [4]. Our cotangent models obey a local-to-global principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general. BEva Miranda e[email protected] Pau Mir pau.mir[email protected] 1Laboratory of Geometry and Dynamical Systems, Department of Mathematics, Universitat Politècnica de Catalunya and BGSMath, Av. Dr. Marañón 44-50, 08028 Barcelona, Spain 2Laboratory of Geometry and Dynamical Systems Department of Mathematics & Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech), Universitat Politècnica de Catalunya & Centre de Recerca Matemàtica, Av. Dr. Marañón 44-50, 08028 Barcelona, Spain 3IMCCE, CNRS-UMR8028, Observatoire de Paris, PSL University, Sorbonne Université, 77 Avenue Denfert-Rochereau, 75014 Paris, France 0123456789().: V,-vol 118 Page 2 of 35 P. Mir, E. Miranda 1 Introduction Quantization is a mathematical procedure which seeks to associate a quantum system to a classical Hamiltonian system by replacing functions by operators and Poisson brackets of functions by brackets of operators. Several paths have been traced for this passionate journey from geometry and analysis into Physics: geometric quantization, formal quantization, BRST quantization and semi-classical quantization, to cite a few. All of them supply Taylor-made master formulas to the day-dreamer mathematicians who are looking into the quantum world. Inthisarticlewefocusonthegeometricquantizationapproachandweprovideanew model which corrects former models and brings us closer to the role of quantization as a mathematical tamer of quantum physics. Some almost metaphysical questions still waft through the air: Can this be achieved? Do these methods depend on additional data? Can we find a universal model? One of the virtues of our model is that it takes the cotangent bundle as a general setup for our systems. The connection between a Hamiltonian system and the cotangent bundle is given by the cotangent lift and provides a unified approach to former attempts in the literature. On the other hand, one of the downfalls of our model is that, unlike other quantization models like Kähler quantization, it depends on choices (in our case, on the choice of a real polarization given by an integrable system) as it usually happens in the standard geometric quantization. One of the advantages of our method is that geometric quantization of integrable systems can be computed even if global classification of integrable systems with non-degenerate singularities is unknown in general (not even semi-local classification), as its recipe is based on gluing local models. So the "from local to global" principle prevails here. Geometric quantization and integrable systems are common mathematical objects on the interface of Geometry and Physics. Integrable systems represent a class of Hamiltonian systems which can be associated to an extra set of functions called first integrals, and are ubiquitous in Physics. Many known systems, such as any two dimensional system, or more complicated systems, such as the coupled harmonic oscillators or the spherical pendulum, are integrable. Other classical systems defined by attracting or repelling particles, such as Toda systems, are integrable. The geometric quantization procedure meets integrable systems when these are used as data attached to the geometric quantization process, in particular as providers of (real) polarizations. In this article, we contemplate the quantization problem considering precisely the real polarization associated to an integrable system. The geometric quantization procedure starts with a prequantum complex line bundle L, which is naturally associated to a symplectic manifold (M2n,ω)of integral class, and an attached connection ∇with curvature ω.Aflat section s of the line bundle is a solution to the equation ∇Xs=0, where the derivation takes place along the direction Xof a polarization, which in this article is considered to be real and given by the integrable system. Flat sections form a sheaf, from which one constructs a cohomology that eventually gives the quantization. Because of the maximum principle, an integrable system defined via smooth functions on a compact phase space must have singularities. Then, polarizations given by these systems are singular too. In this article, we analyze the contribution of singularities to the geometric quantization of singular integrable systems. Geometric quantization via cotangent models Page 3 of 35 118 The simplest type of singularities of smooth functions are Morse-type singularities, which admit a Morse or a Morse-Bott normal form. For integrable systems on symplectic manifolds one might also demand a normal form in a neighbourhood of their singularities. That is to say, one might assume that there exist local coordinates such that the functions defining the system are simultaneously of Morse type and such that the symplectic form is Darboux. Those singularities were initially considered by Eliasson [6] and later by Miranda [17] and Miranda-Zung [22]. In former works by Hamilton [10], Hamilton-Miranda [11] and Miranda-PresasSolha [21], the authors analyze the contributions of non-degenerate singularities of integrable systems to quantization. They find no contribution from elliptic points and infinite dimensional contributions for hyperbolic and focus-focus type singularities. Those infinite dimensional models clash with the initial expectations of obtaining a finite dimensional representation space as the quantization space (and thus, representation space) of a system defined on a compact manifold. In this article, we work out "cotangent models" for integrable systems with nondegenerate singularities which can be of elliptic, hyperbolic and focus-focus type. This is a first step towards understanding the cotangent models of the pairs given by the polarization associated to such integrable systems and the prequantum line bundle. Those singularities naturally appear in polarizations on compact manifolds given by integrable systems of Morse-Bott type. In particular, any semitoric system (such as the ones studied in [23,24]) gives rise to singularities of this type). These structures also show up naturally in algebraic geometry, for instance in the study of the K3 surface1, which can be viewed as a semitoric system. When it comes to considering their quantization, several models have been proposed. However, none of them can compete with the model of Kähler quantization in terms of independence of the polarization and in terms of the principle quantization commutes with reduction, or simply [Q,R]=0(see[7–9]). Notwithstanding, Kähler quantization cannot always be applied since the conditions to have a polarization of Kähler type are not always fulfilled. For regular integrable systems (without singularities) action-angle coordinates (the classicalArnold-Liouville-Mineurtheorem)providecotangent models as aneighbourhood of the Liouville torus can be symplectically interpreted as its cotangent bundle, T∗(Tn). This canonical identification gives a way to relate the choice of the Liouville 1-form of the cotangent bundle with the connection 1-form of the prequantum line bundle. In other words, the Liouville 1-form of the cotangent bundle yields a canonical choice of the connection 1-form. This connects the cotangent model to quantization in the regular case (see for instance [20] and [25]). With the ambition of extending these ideas to the singular set-up, we analyze the cotangent lift technique for different types of non-degenerate singularities (in the sense of Eliasson-Williamson) and provide brand-new cotangent models for the pair given by the polarization and the connection one-form. Additionally, the existence of a local model of cotangent type allows to capture symmetry and is compatible with the [Q,R]=0 principle. The cotangent models 1A K3 surface is an example of a hyperkähler manifold with three compatible complex structures i,j,k. The denomination K3 comes from Kummer, Kähler and Kodaira and, according to André Weil, it is a reminiscence of the beautiful mountain K2 in Kashmir. 118 Page 4 of 35 P. Mir, E. Miranda used to define the new proposal for geometric quantization for non-degenerate singularities also allows to obtain a unique universal cotangent model. In contrast to the former models of geometric quantization for real polarizations endowed with nondegenerate singularities in [11] and [21], our new models provide finite dimensional representations for systems on compact manifolds which match the physical models. One interesting advantage of our models is that they fit well with the sheaftheoretical geometric quantization kit provided in [20]. In particular, the Künneth formula and Mayer-Vietoris recipe which were established there can be used to patch the cotangent models to provide global quantization on a compact manifold even if the global symplectic classification of non-degenerate integrable systems is still unknown in some cases. In other words, our cotangent models can be seen as building pieces of the geometric quantization puzzle as partitions of unity in Differential geometry allow us to invoke a local-to-global principle. Organization of this article: InSection2werevisetherudimentsofintegrablesystems, the main features of the theory of non-degenerate singularities of such systems, the cotangent lift technique and the basics of geometric quantization. In Section 3 as a novel result we present the local normal form theorem for integrable systems with non-degenerate singularities described as a product of lower dimensional (2 or 4) cotangent models. we introduce the notion of discrete cotangent lift and show that it generates the set of Bohr-Sommerfeld leaves. In Section 4we connect cotangent models of Section 3with the connection 1-form of geometric quantization. We prove that an integrable system with only non-degenerate singularities of hyperbolic and elliptic type can be realized as a cotangent lift. In Section 5we propose a new local model, redefining the previous quantizations for the hyperbolic and focus-focus singularities. We unify the quantization procedure for non-degenerate singularities. In Section 6we discuss some applications to the quantization of K3 surfaces and the advantages of the new model. Finally, in Appendix A, for the sake of completeness we compute explicitly the sheaf cohomology of the cylinder. We obtain the expression of the flat sections, the cochains and the cocycles by hand, a procedure that may help a non-familiar reader to understand the techniques applied in the proofs of Section 5. 2 Background. Integrable systems, the cotangent lift, quantization We provide in this section the basic concepts that will be necessary for the rest of the article, dividing them into three parts. First, we review some important results on integrable systems defined on symplectic manifolds and on classification of singularities. Then, we introduce the cotangent lift, a tool which extends group actions to the cotangent bundle. Finally, we give a complete scheme of geometric quantization. 2.1 Moment maps, Hamiltonian systems and singularities Hamiltonian actions and the moment map are the absolute key concepts in the link between symplectic geometry and integrable systems. In this section we give a brief Geometric quantization via cotangent models Page 5 of 35 118 review on them, giving special attention to integrable systems with non-degenerate singularities. Definition 2.1 Let H∈C∞(M2n)be a smooth function on a symplectic manifold (M2n,ω).TheHamiltonian vector field XHassociated to His defined as the only solution of ιXHω=−dH. Remark 2.2 In Physics, the Hamiltonian represents a function of the total energy of a system. Definition 2.3 An integrable system on a symplectic manifold (M2n,ω)is given by a smooth map f=(f1,..., fn):M2n→Rnsuch that {fi,fj}=ω(Xfi,Xfj)=0 for all 1 ≤fi,fj≤nand rank f=nalmost everywhere. Definition 2.4 Let Gbe a Lie group and gits Lie algebra. Consider also g∗, the dual of g. Suppose ψ:G→Diff(M)is an action on a symplectic manifold (M,ω).Itis called a Hamiltonian action if there exists a map μ:M→g∗which satisfies: – For each X∈g,dμX=ιX#ω, i.e., μXis a Hamiltonian function for the vector field X#, where –μX:p−→  μ(p), X:M−→ Ris the component of μalong X, –X#is the vector field on Mgenerated by the one-parameter subgroup {exp tX | t∈R}⊂G. –Themapμis equivariant with respect to the given action ψon Mand the coadjoint action: μ◦ψg=Ad∗ g◦μ, for all g∈G. Then, (M,ω,G,μ)is called a Hamiltonian G-space and μis called the moment map. The normal form of a completely integrable system around a whole leaf of a regular point is well-known by the Arnold-Liouville-Mineur theorem (see [1] and [16]). Theorem 2.5 (Arnold-Liouville-Mineur, [1]) Let (M2n,ω)be a symplectic manifold. Let {f1,..., fn}be a set of n functions on M which are functionally independent (d f1∧···∧df n= 0on a dense set and pairwise in involution). Suppose that m is a regular point of F =(f1,..., fn)and that the level set of F through m, which we denote by Fm, is compact and connected. Then, Fmis a torus and on a neighbourhood U of Fmthere exist R-valued smooth functions (p1,...,pn)and R/Z-valued smooth functions (θ1,...,θ n)such that the following holds: 1. The functions (θ1,...,θ n,p1,...,pn)define a diffeomorphism U ≃Tn×Bn. 2. The symplectic structure can be written in terms of these coordinates as ω= n  i=1 dθi∧dpi. 3. The leaves of the surjective submersion F =(f1,..., fs)are given by the projection onto the second component Tn×Bn, in particular, the functions f1,..., fs depend only on p1,...,pn. 118 Page 6 of 35 P. Mir, E. Miranda We call the piaction coordinates and the θiangle coordinates. For singular points, where Arnold-Liouville-Mineur theorem does not apply, it is necessary to explore the moment map there at a local or semi-local level in order to obtain the topology of the whole singular leaf. It can be really difficult to understand both the geometry and the dynamics of the system depending on the degeneracy of dF =(df 1,...,df n)but some results already do this work in the case of the simplest singularities, the non-degenerate singularities. For this type of singularities powerful classification results have been obtained and, for instance, we have local normal forms. Definition 2.6 A point m∈M2nis singular of an integrable Hamiltonian system given by F=(f1,..., fn)if the rank of dF =(df 1,...,df n)at mis less than n. Definition 2.7 Let (M2n,ω)be a symplectic manifold with an integrable Hamiltonian system of nindependent and commuting first integrals f1,..., fn. Consider a singular point p∈Mof rank 0, i.e. (df i)p=0 for all i. It is called a non-degenerate singular point if the operators ω−1d2f1,...,ω −1d2fnform a Cartan subalgebra in the symplectic Lie algebra sp(2n,R)=sp(TpM,ω). The classification of non-degenerate singular points in the real case is equivalent to the classification of Cartan subalgebras and was obtained by Williamson [30]. Theorem 2.8 (Williamson, [30]) For a Cartan subalgebra Cof sp(2n,R), there exists a symplectic system of coordinates (x1,...,xn,y1,...,yn)in R2nand a basis of n functions f1,..., fnof Csuch that each of the quadratic polynomials fiis one of the following: fi=x2 i+y2 ifor 1 ≤i≤ke fi=xiyifor ke+1≤i≤ke+kh fi=xiyi+1−xi+1yi fi+1=xiyi+xi+1yi+1 for i=ke+kh+2j−1,1≤j≤kf The three types are called elliptic, hyperbolic and focus-focus, respectively. The triple (ke,kh,kf)at a singular point of rank k=n−ke−kh−2kf, called Williamson type of the singularity, is an invariant of the point and an invariant of the orbit of the integrable system through the point [31]. The following result of Eliasson [6] and Miranda and Zung ([17,18,22]) extends the classification to Hamiltonian systems in symplectic manifolds. Theorem 2.9 (Eliasson, Miranda, Zung, [6,17,18,22]) Let F be an smooth integrable Hamiltonian system of degree n on a symplectic manifold (M2n,ω). The Liouville foliation in a neighborhood of a non-degenerate singular point m of rank k and Williamson type (ke,kh,kf)is locally symplectomorphic to the foliation defined by the basis functions of Theorem 2.8 plus regular functions fi=xifor i =ke+kh+2kf+1 to n. Geometric quantization via cotangent models Page 7 of 35 118 This theorem can be extended to an orbit of the integrable system via the following two Theorems. Theorem 2.10 (Model in a covering) The symplectic manifold (M,ω) can be represented, locally at a non-degenerate singularity of rank k and Williamson type (ke,kh,kf), as the direct product k   Mreg ×···×Mreg × ke   Mell ×···×Mell × kh   Mhyp ×···×Mhyp × kf   Mfoc ×···×Mfoc Where: –M reg is a regular block, representing the regular moment map given by fr=x, –M ell is an elliptic block, representing the elliptic singularity given by fe=x2+y2, –M hyp is an hyperbolic block, representing the hyperbolic singularity given by fh=xy, –M foc is a focus-focus block, representing the focus-focus singularity given by ff=(f1,f2)=(x1y2−x2y1,x1y1+x2y2). For the first three types of blocks the symplectic form is ω=dx ∧dy, while for the focus-focus block it is ω=dx1∧dy1+dx2∧dy2. In the case of a smooth system (defined by a smooth moment map), a similar result was proved and described by Miranda and Zung in [22]. It summarizes some former results proved independently and fixes the case where there are hyperbolic components (kh= 0), because in this case the result is slightly different and it has to betakenthesemi-directproductin the decomposition. Contrary to thecase where there are only elliptic and focus-focus singularities, in which the base of the fibration of the neighbourhood is an open disk, if there are hyperbolic components the topology of the fiber can become complicated. The reason is essentially that for the smooth case a level set of the form {xiyi=ε}is not connected but consists of two components. Precisely, because of these two components, one cotangent model gives raise to two different local models whenever there is an hyperbolic singularity. To overcome this duplicity, one takes a quotient by a finite group Γ(typically Z2). This is why an equivariant version in the presence of symmetries (Theorem 2.11) yields the total classification. Theorem 2.11 (Miranda-Zung, [22]) Let V =Dk×Tk×D2(n−k)with the following coordinates: (p1, ..., pk)for Dk,(q1(mod1), ..., qk(mod1)) for Tk, and 118 Page 8 of 35 P. Mir, E. Miranda (x1,y1, ..., xn−k,yn−k)for D2(n−k)be a symplectic manifold with the standard symplectic form dpi∧dqi+dxj∧dyj. Let F be the moment map corresponding to a singularity of rank k with Williamson type (ke,kh,kf). There exists a finite group Γ, a linear system on the symplectic manifold V /Γ and a smooth Lagrangian-fibrationpreserving symplectomorphism φfrom a neighborhood of O into V /Γ , which sends Otothetorus{pi=xi=yi=0}. The smooth symplectomorphism φcan be chosen so that via φ, the system-preserving action of a compact group G near O becomes a linear system-preserving action of G on V /Γ . If the moment map F is real analytic and the action of G near O is analytic, then the symplectomorphism φcan also be chosen to be real analytic. If the system depends smoothly (resp., analytically) on a local parameter (i.e. we have a local family of systems), then φcan also be chosen to depend smoothly (resp., analytically) on that parameter. In summary, given an integrable system, there is a naturally associated Lagrangian foliation given by a distribution generated by the Hamiltonian. This result does not only classify integrable systems but also classifies Lagrangian foliations [18]. 2.2 The cotangent lift The cotangent bundle of a smooth manifold can be naturally equipped with a symplectic structure in the following way. Let Mbe a differential manifold and consider its cotangent bundle T∗M. There is an intrinsic canonical linear form λon T∗Mdefined pointwise by λp,v=p,dπpv,p=(m,ξ)∈T∗M,v ∈Tp(T∗M), where dπp:Tp(T∗M)−→ TmMis the differential of the canonical projection at p. In local coordinates (qi,pi), the form is written as λ=ipidqiand is called the Liouville 1-form. Its differential ω=dλ=idpi∧dqiis a symplectic form on T∗M. Definition 2.12 Let ρ:G×M−→ Mbe a group action of a Lie group Gon a smooth manifold M. For each g∈G, there is an induced diffeomorphism ρg:M−→ M. The cotangent lift of ρg, denoted by ˆρg, is the diffeomorphism on T∗Mgiven by ˆρg(q,p):= (ρg(q), ((dρg)∗ q)−1(p)), (q,p)∈T∗M which makes the following diagram commute: T∗MT ∗M MM π ˆρg ρg π Geometric quantization via cotangent models Page 9 of 35 118 Given a difeomorphism ρ:M−→ M, its cotangent lift preserves the canonical form λ(see computations in [19]). Then, the canonical 1-form is preserved by ˆρ. As a consequence: ˆρ∗(ω) =ˆρ∗(dλ) =d(ˆρ∗λ) =dλ=ω. Meaning that the cotangent lift ˆρgpreserves the Liouville form and the symplectic form of T∗M. Example 2.13 Let ρ:(R3,+)×R3→R3be the Lie group action corresponding to a space translation defined by ρx(q)=q+x. Write (q,p)for an element of the cotangent bundle T∗R3∼ =R6. By definition, ˆρx, the cotangent lift of ρxis ˆρx(q,p)=(ρx(q), ((dρx)∗ q)−1(p)) =(q+x,((Id∗)−1(p)) =(q+x,p). (2.1) Example 2.14 Let ρ:SO(3,R)×R3→R3be a Lie group action defined by ρA(q)= Aq. Write (q,p)for an element of T∗ qR3. By definition, ˆρA, the cotangent lift of ρA is ˆρA(q,p)=(ρA(q), ((dρA)∗ q)−1(p)) =(Aq,((A∗)−1(p)) =(Aq,Ap), where the last equality holds because Ais orthogonal. Like any cotangent lift, since the induced action in the cotangent bundle is Hamiltonian, it has an associated momentum map which, in this case, corresponds to the classical quantity q∧p. 2.3 Overview on geometric quantization As a general principle, quantization consists in associating a Hilbert space Qto a symplectic manifold (M,ω). In geometric quantization, this Hilbert space is constructed using the sections of a complex line bundle L. Normally, one declares as representation space the space formed by the flat sections of this bundle in some direction (given by a polarization). Such sections are not always defined globally along the leaves of the polarization. This is why it is convenient to use the sheaf-theoretic language (with sheaf meaning the sheaf of flat sections of the bundle) to surmount this difficulty. Kostant introduced the main ideas of geometric quantization in the 70s [14] and, today, they remain useful and have applications in representation theory, a big variety of physical problems and many other fields. One of the main characters of the theory of geometric quantization are Bohr-Sommerfeld leaves. Kostant’s model goes through the cohomology associated to the sheaf of flat sections of Land is well-adapted for real polarizations given by integrable systems and toric manifolds, which are symplectic manifolds endowed with an effective Hamiltonian action of a torus whose rank is half of the dimension of the manifold [18]. An important result of Delzant, which connects quantization and moment maps, states the existence of a one-to-one correspondence between closed toric manifolds in dimension 2nand the Delzant polytope on Rn[5]. The Delzant polytope gives the real 118 Page 16 of 35 P. Mir, E. Miranda Therefore, t0has to satisfy 1 =e2πit0and we get the condition t0∈Z. Then, the Bohr-Sommerfeld leaves are the leaves of Pof the form {k}×S1with m∈Z. The space of global covariant constant sections over one leaf is one-dimensional, i.e., there is freedom in the choice of the value of a∈Cin σ=aeit0θ. To compute the sheaf cohomology of Mwe can determine the cohomology groups Hkapplying directly ´ Sniatycki Theorem. In this case, we obtain that, for any open interval I⊂Rand U=I×S1⊂M, H1(U,J)∼ = m∈Z∩I C,Hk(U,J)=0,k= 1, where Jis the sheaf of flat sections σ. But we can also compute the sheaf cohomology of the cylinder explicitly, as we do in Appendix A. 3 Integrable systems as cotangent lifts and the discrete cotangent lift Cotangent lifts arise naturally in physics problems, and the link between integrable systems and cotangent models is clear in view of the following Kiesenhofer and Miranda result [13], which restates Theorem 2.5 to reveal that at a semi-local level the regular leaves are equivalent to a completely toric cotangent lift model. Theorem 3.1 (Kiesenhofer-Miranda, [13]) Let F =(f1,..., fn)be an integrable system on a symplectic manifold (M,ω). Then, semi-locally around a regular Liouville torus, the system is equivalent to the cotangent model (T∗Tn)can restricted to a neighbourhood of the zero section (T∗Tn)0of T ∗Tn. In the classical models of the harmonic oscillator, the simple pendulum and the spherical pendulum one already finds the three different types of non-degenerate singularities in its lowest dimensional case. A simple elliptic singularity is appears in the harmonic oscillator, a simple hyperbolic singularity shows up in the simple pendulum and a simple focus-focus singularity arises in the spherical pendulum. The Hamiltonian vector fields associated each of the three non-degenerate singularities can be obtained from the cotangent lift of a Lie group action. Definition 3.2 We denote by ρhthe following Lie group action: ρh:R×R−→ R (t,x)−→ e−tx, by ρethe following Lie group action: ρe:R×C−→ C (t,z)−→ eitz, Geometric quantization via cotangent models Page 17 of 35 118 and by ρfthe following Lie group action: ρf:(S1×R)×R2−→ R2 (θ, t), x1 x2−→ e−t0 0e−tcos θsin θ −sin θcos θx1 x2. Lemma 3.3 The infinitesimal generators of the cotangent lift of the Lie group actions ρh,ρe,ρfcoincide, respectively, with the vector fields corresponding to the hyperbolic, elliptic and focus-focus singularities. Proof For the hyperbolic singularity, take coordinates (x,y)on R2such that the symplectic form is ω=dx ∧dy and the moment map is f=xy. The Hamiltonian vector field associated to fis X=(−x,y). Consider the action of Ron Rgiven by: ρh:R×R−→ R (t,x)−→ e−tx. Then, ((dρh t)∗ x)−1acts as y−→ ety. The cotangent lift ˆρh tassociated to the group action ρh t, in coordinates (x,y)of T∗R,is: ˆρh:T∗R−→ T∗R x y−→ e−tx ety. Deriving the last vector with respect to tand evaluating at t=0, we obtain X= (−x,y), the vector field associated to the hyperbolic singularity. For the elliptic singularity, consider R2with real coordinates (x,y)and define the complex conjugate coordinates (z,¯z)=(x+iy,x−iy)such that the symplectic form is ω=i 2dz ∧d¯z. The moment map corresponding to the elliptic singularity is f=1 2x2+y2=1 2z¯z. The Hamiltonian vector field associated to fin the complex setting is X=(iz,−i¯z). Consider the following action of Ron C, which corresponds to a rotation of zof angle t: ρe:R×C−→ C (t,z)−→ eitz. Then, ((dρe t)∗ z)−1acts as ¯z−→ e−it ¯z, and the cotangent lift ˆρe tassociated to the group action ρe t, in coordinates (z,¯z)of T∗Cis: ˆρe:T∗C−→ T∗C z ¯z−→ eitz e−it ¯z. 118 Page 18 of 35 P. Mir, E. Miranda Deriving the last vector with respect to tand evaluating at t=0 we obtain X= (iz,−i¯z), the vector field associated to the elliptic singularity. The cotangent lift in the elliptic case uses a complex moment map which is not holomorphic. It is a formal development and holomorphicity is not assumed. For the focus-focus singularity, take coordinates (x1,x2,y1,y2)in R4in such a way that the symplectic form is ω=dx1∧dy1+dx2∧dy2and the moment map is F=(f1,f2)=(x1y2−x2y1,x1y1+x2y2). TheHamiltonian vectorfieldsassociatedto f1and f2are X1=(x2,−x1,y2,−y1), X2=(−x1,−x2,y1,y2). Let G=S1×Rand M=R2. Consider the action of a rotation and a radial dilation of R2given by ρf:(S1×R)×R2−→ R2 (θ, t), x1 x2−→ e−t0 0e−tcos θsin θ −sin θcos θx1 x2. Then, the cotangent lift ˆρfassociated to the group action is: ˆρf:T∗R2−→ T∗R2 ⎛ ⎜ ⎜ ⎝ x1 x2 y1 y2 ⎞ ⎟ ⎟ ⎠−→ ⎛ ⎜ ⎜ ⎝ e−t(x1cos θ+x2sin θ) e−t(−x1sin θ+x2cos θ) et(y1cos θ+y2sin θ) et(−y1sin θ+y2cos θ) ⎞ ⎟ ⎟ ⎠ . Deriving the vector with respect to θand evaluating at 0 we obtain X1= (x2,−x1,y2,−y1). While deriving the vector with respect to tand evaluating at 0 we obtain X2=(−x1,−x2,y1,y2). In view of this realization of the three non-degenerate singularities as cotangent lifts of Lie group actions, we can automatically state a block form result, which is the cotangent analogous to Theorem 2.10. Definition 3.4 Let pbe a non-degenerate singularity of rank kand Williamson type (ke,kh,kf)in a symplectic manifold (M,ω).Letρr,ρe,ρh,ρfbe the Lie group actions of the regular case and the non-degenerate singular cases in Definition 3.2.We defineρpasthefollowingcompositionofLiegroupactionsactinginthecorresponding molecules: ρp=(ρr:R×T→R)k×(ρe:R×C→C)ke ×(ρh:R×R→R)kh×(ρ f:(S1×R)×R2→R2)kf. In detail, the map ρpacts on the product manifold (R×T)k×(R×C)ke×(R× R)kh×((S1×R)×R2)kfas the following composition: ρp= k   ρr◦···◦ρr◦ ke   ρe◦···◦ρe◦ kh   ρh◦···◦ρh◦ kf   ρf◦···◦ρf, Geometric quantization via cotangent models Page 19 of 35 118 with ρracting on R×T,ρeacting on R×C,ρhacting on R×Rand ρfacting on (S1×R)×R2and all of them acting as the identity in the respective other components of the product of manifolds. Remark 3.5 The cotangent lift ˆρpof ρpis now naturally defined as the following action: (ˆρr:R×T∗T→T∗T)k×(ˆρe:R×T∗C→T∗C)ke ×(ˆρh:R×T∗R→T∗R)kh×(ˆρf:(S1×R)×T∗R2→T∗R2)kf Theorem 3.6 (Cotangent model in a covering) Take the definitions and notation of the section, let (M,ω) be a symplectic manifold. In a neighbourhood U of a nondegenerate singularity p of rank k and Williamson type (ke,kh,kf), the manifold can be represented as the integral manifold of the infinitesimal generators of ˆρp.Each component of the cotangent lift of ρp, when the Lie group element is fixed, acts on the corresponding blocks as presented in Theorem 2.10, i.e., ˆρracts on Mreg ∼ =T∗T,ˆρe acts on Mell ∼ =T∗C,ˆρhacts on Mhyp ∼ =T∗Rand ˆρfacts on Mfoc ∼ =T∗R2, and all of them acting as the identity in the respective other components of the product of manifolds. Proof The cotangent lift commutes with the product of manifolds and, hence, with the composition of functions in each block. The block decomposition form around a non-degenerate singularity of Theorem 2.10 is then compatible with the cotangent models for each singularity in Lemma 3.3, proving the cotangent lift block form.  Remark 3.7 This theorem says that at each non-degenerate singularity, the symplectic manifoldcanberealizedasacotangentmodelthankstothenaturalsymplecticstructure of the cotangent bundle and the existence of a cotangent lift model for each type of non-degenerate singularity. The associated moment map is: F=( k   fr,..., fr, ke   fe,..., fe, kh   fh,..., fh, kf   ff,..., ff), where fr,fe,fh,ffare the elementary forms of the regular, elliptic, hyperbolic and focus-focus focus singularities respectively. 3.1 The discrete cotangent lift As a discrete analog of the the classical cotangent lift of a Lie group action, we define the discrete cotangent lift, a tool which connects the cotangent models with the geometric quantization and allows to see Bohr-Sommerfeld leaves as a cotangent lift in the classical sense. On an integrable system of dimension 2n, Bohr-Sommerfeld orbits correspond to the integer points in the interior of the Delzant polytope. Since they are all of them n-dimensional tori, they can be seen as the orbit of a discrete translation action of a single torus. 118 Page 20 of 35 P. Mir, E. Miranda Consider the integer points in the Delzant polytope of an integrable system, written in coordinates as (x1+m1,...,xn+mn)∈Rnfor some fixed (x1,...,xn)∈Rn and a set N={(m1,...,mn)∈Zn}of n-tuples. Here, Rnis the ambient space of the image of the moment map. Consider the action α:Zn×Rn−→ Rn (m1,...,mn), (x1,...,xn)−→ (x1+m1,...,xn+mn), which can be seen as the restriction to integer n-tuples in Nof the associate continuous translation ˜α:Rn×Rn−→ Rn (t1,...,tn), (x1,...,xn)−→ (x1+t1,...,xn+tn). The orbit of (x1,...,xn)∈Rnby ˜αrestricted to Nis the set of integer points in the interior of the Delzant polytope. Definition 3.8 Consider the actions αand ˜αas defined before. The discrete cotangent lift ˆα:Zn×T∗Rn→T∗Rnof αis the restriction to Zn×T∗Rnof the classical cotangent lift of the action ˜α. In practice, the discrete cotangent lift ˆαextends αto the cotangent bundle of Rn. Lemma 3.9 The discrete cotangent lift coincides with the classical cotangent lift of ˜α at the integer points of the basis. Proof It is direct from the definition, since it is constructed as its restriction to the integer points.  The discrete cotangent lifted action ˆαacts on the pairs of points in the interior of the Delzant polytope and tori in the manifold. Its orbit is not only the whole set of integer points in the Delzant polytope but the pairs of integer points and their associated Bohr-Sommerfeld leaves (see Figure 2). The action ˆαis well defined on the cotangent bundle of a vector space. Not only this, but it is also compatible with the symplectic structure of the cotangent bundle. To see this, consider the coordinates (x1,...,xn)on the base and the symplectic dual coordinates (y1,...,yn)on the fiber of T∗Rn. Since the natural pairing of each coordinate xiwith its dual yicoincides with the symplectic conjugation (reflected in theLiouville1-formλ=yidxiandinthesymplecticformω=dλ=dyi∧dxi), the lift of the discrete action αpreserves the symplectic structure. Having introduced this language, we can reformulate Theorems 2.26 and 2.27, since in both ´ Sniatycki’s and Hamilton’s cases there is a toric fibration. Theorem 3.10 Let (F,M,ω) be a toric integrable system in a compact symplectic manifold with a prequantization line bundle L, equipped with a locally toric Lagrangian fibration. Let Δbe the Delzant polytope of the momentum map of F. Then, the set of Bohr-Sommerfeld leaves coincides with the intersection of Δwith the image of the discrete cotangent lift of the action α:Zn×Rn−→ Rn (m1,...,mn), (x1,...,xn)−→ (x1+m1,...,xn+mn), Geometric quantization via cotangent models Page 21 of 35 118 Fig. 2 The set of tori corresponding to the integer points in the interior of Delzant’s polytope is the orbit of the action ˆα where (m1,...,mn)is n-tuple of integers corresponding to any of the BohrSommerfeld leaves and (x1,...,xn)can be transported via a symplectomorphism φto the base coordinates of the Lagrangian fibration of M. Proof First, recall that in Theorem 3.6 we proved that the neighbourhood of every regular point and every non-degenerate singularity can be seen as a cotangent lift. Then,sincebyLemma 3.9thediscretecotangentlift canbethoughtastherestriction of the classical cotangent lift and its image is provided by the integer n-tuples. Finally, Theorems 2.26 and 2.27 prove that Bohr-Sommerfeld fibers are isolated compact tori which are separated by integer values which form a lattice in the interior of the Delzant polytope. Then, in a toric manifold (with only elliptic singularities), the Bohr-Sommerfeld fibers, which correspond to Lagrangian tori, are the image of the discrete cotangent lift of α. The set of regular Bohr-Sommerfeld leaves in the regular and toric case both coincides with the discrete cotangent lift of a lattice of the basis. 4 Cotangent models for quantization In both the geometric quantization procedure and the cotangent lift technique there appears a 1-form. In the first case it is the connection 1-form Θand in the second case it is the Liouville 1-form λ, and in both cases their differential is the symplectic form ω. It seems that there is a sort of freedom of choice for a 1-form to satisfy that its differential is ω, but the fact is that, in both cases, it is determined by precise conditions. In the prequantization of the elliptic case (the cylinder of Section 2.6) one can already get an intuition of how the 1-form is determined and which conditions it has 118 Page 22 of 35 P. Mir, E. Miranda to fulfill, apart from the obvious of having the symplectic form as its differential. In both the elliptic and the hyperbolic cases the conditions are essentially the same. If we go back to the cotangent models in Section 3, we see that it is natural to present the cotangent models associating them not only to a set of a symplectic manifold (M2n,ω)and an integrable system Fwith non-degenerate singularities but also to a connection ∇with a connection 1-form Θ. We just proved that the connection 1-form is a Liouville 1-form of the type λ=pdq, where pis the moment map "coordinate" (it could correspond to the singular one) and qis the symplectic orthogonal to p. Theorem 4.1 Let F be an integrable system with only non-degenerate singularities of elliptic and hyperbolic type defined in the prequantum line bundle of a symplectic manifold (M2n,ω). Let p be a singular point of rank k and Williamson type (ke,kh,0). Then, in a neighbourhood of each point, there exists a unique product-like cotangent model such that the connection 1-form Θcoincides with the Liouville 1-form λ=pidqi, where the piare the moment map coordinates and the qiare the symplectic orthogonal to pi. The connection 1-form Θ(q1,...,qn,p1,...,pn)of the prequantization of F is determined up to a constant by the following conditions: –dΘ=ω, –ι∂ ∂pi Θ=Θ∂ ∂pi=0,fori=1,...,n –ι∂ ∂qi Θ=Θ∂ ∂qi=F(pi),fori=1,...,n. Proof We prove that it is true for regular and simple elliptic and hyperbolic blocks on a 2-manifold (M2,ω). Since any singularity of Williamson type (ke,kh,0)can be written as a product of these types of 2-dimensional blocks, the 1-form Θwill simply be the sum of the form at each block. Around a regular point, the moment map writes simply as f=xin Cartesian coordinates (x,y).Wetakeω=dx∧dy and the connection 1-form will be written as Θ(x,y)=α(x,y)dx +β(x,y)dy for some smooth functions αand β. The second condition implies that α(x,y)=0. Then, the third condition implies β=β(x)and, finally, by the first condition we obtain that Θr=xdy is the 1-form we were looking for. The moment map can be written as fe=1 2(x2+y2)around an elliptic singularity in Cartesian coordinates; whilst in polar coordinates (r,θ)it can be written as fe=1 2r2. The symplectic form can be written as ω=dx ∧dy =rdr ∧dθ. Since we want to work with the coordinate of the moment map fe, which is a regular function of the rpolar coordinate, we change to elliptic coordinates (s,θ), which are essentially polar coordinates where with the transformation s=1 2r2. Then, ω=ds ∧dθand a connection 1-form is written as Θ(s,θ) =α(s,θ)ds +β(s,θ)dθfor some smooth functions αand β. Since Θhas to satisfy Θ∂ ∂s=0, αneeds to be 0. The condition Θ∂ ∂θ =fe(s) implies that β=β(s). Finally, since we want that dΘ=ω=ds ∧dθ, we need to require that ∂β ∂s=1. Then, Θe=sdθis the unique 1-form satisfying the conditions. The moment map around an hyperbolic singularity can be written as f=xy in Cartesian coordinates, with symplectic form ω=dx ∧dy.Inhyperbolic coordinates Geometric quantization via cotangent models Page 23 of 35 118 (h,b), where: h=xy, b=− 1 2ln  x y, the connection 1-form can be written as Θ=α(h,b)dh+β(h,b)db. Again, the three conditions together imply Θh=hdb. The definition of the Liouville 1-form on a cotangent bundle of a smooth manifold is precisely built from the fibration position-momentum, and it finishes the proof. The 1-form is, precisely: Θ= k  i=1 Θr+ k+ke  i=k+1 Θe+ k+ke+kh  i=k+ke+1 Θh, with each Θjacting on the corresponding block in the decomposition of Min 2-blocks given by Theorem 2.10. Remark 4.2 In both cases we can interpret the contraction conditions as a symplectic orthogonality, which determines a unique choice of the 1-form. In particular, the contraction of the 1-form with the vector field parallel to the foliation of the moment map of the singularity gives precisely the moment map function, while its contraction with a vector field which is symplectic orthogonal to this foliation vanishes. With the notation of Section 3, we can state this result as a cotangent model as follows. Theorem 4.3 Consider a triple ((M,ω),F=(f1,..., fn), Θ), where (M,ω) is a symplectic manifold, F is an integrable system with only non-degenerate singularities of rank k and Williamson type (ke,kh,0)and Θis a connection 1-form. Then, locally at any singularity p, the system is equivalent to a triple where Θis determined by the Liouville 1-form of the cotangent bundle corresponding to the cotangent lift model of the singularity. Corollary 4.4 The triple ((M,ω),F=(f1,..., fn), Θ) with the same properties as in Theorem 4.3 locally decomposes as products of two dimensional triples. Explicitly, it decomposes as: ((Mreg,ω), fr,Θ r)k×((Mell,ω), fe,Θ e)ke×((Mhyp,ω), fh,Θ h)kh. Proof By Theorem 2.10, in a neighbourhood of a non-degenerate singularity, the manifold decomposes as the direct product of blocks which are all of them 2-dimensional if there are no focus-focus type components. By Theorem 3.6, the block decomposition model is a cotangent model, so the triple ((M,ω),F=(f1,..., fn), Θ) decomposes as products of two dimensional triples.  Remark 4.5 By Theorem 4.3, the integrable system with only elliptic and hyperbolic singularities is automatically a cotangent lift. 118 Page 24 of 35 P. Mir, E. Miranda 5 Proposal of the new model of geometric quantization Jacques Vey proved in [29] that there was a unique complex model for the linearization of analytical systems. Indeed, he proved the following theorem in the holomorphic set-up. Theorem 5.1 (Vey, [29]) Let (M2n,ω) be an analytic complex symplectic manifold. Let A be a Liouville algebra of critical function germs at a point p ∈M. Then, there exist holomorphic coordinates (p1,...,pn,q1,...,qn)in a neighbourhood of p on M such that ω=idpi∧dqiand such that A is the analytic algebra generated by the n functions hi=piqi. From his idea of a unique model for a non-degenerate singularity in the complexes, wepresentanewmodelforgeometricquantization.Thismodelunifiesthecomputation for the regular case and the three types of (real) non-degenerate singularities in the Williamson sense, which are equivalent in the complexes. 5.1 Sheaf surgery for non-degenerate singularities Theorem 2.27 already unifies the geometric quantization of elliptic singularities with the regular case. We introduce some sheaf surgery that will allow us to redefine the quantization of the hyperbolic and the focus-focus singularity to also bring together their geometric quantization. To set up for the hyperbolic case, recall from [11] the following result Proposition 5.2 (Hamilton-Miranda, [11])Let Z be the neighbourhood of a hyperbolic singular point. If σ:Z→Lis a smooth leaf-wise flat section defined over Z, then σ is Taylor flat at the singular point. That is, ∂j+kσ ∂jx∂ky(0,0) =0for all j,k Corollary 5.3 The only leaf-wise flat analytic section in a semi-local neighbourhood of a hyperbolic singularity is the zero section. Proof By Theorem 5.2, leaf-wise flat analytic sections in a neighbourhood of a singularity are the zero sections. Since the analytic extension of the zero section is zero, the only leaf-wise flat analytic section in a semi-local neighbourhood of a hyperbolic singularity is the zero section.  Now, we construct a special sheaf by changing the sheaf of smooth flat sections in a neighbourhood on a hyperbolic singularity. Lemma 5.4 Let p ∈(M,ω,F)be a non-degenerate singular point of hyperbolic type on an integrable system defined on a symplectic manifold. Suppose that it is equipped with a complex line bundle Land a connection ∇whose curvature is ω. Then, the sheaf of leaf-wise smooth flat sections is still a sheaf when a in a neighbourhood of p smooth sections are required to be analytic. We denote this sheaf by Jh. Geometric quantization via cotangent models Page 25 of 35 118 Proof Analytic sections are a subclass of smooth sections. Observe that the process consists of two steps. Fisrt, thanks to the system linearization around a singularity of Eliasson [6], the flat sections equation ∇Xσ=0 is formed by analytic data (the field X=x∂ ∂x−y∂ ∂yin dimension 2) and the solutions sections are analytic. Requiring smooth sections to be analytic in a small neighbourhood does not change the fact that theintersectionworks well.Thisisbecauseweareextendingapiecewherethe sections are required to be zero and by Corollary 5.3) this is compatible, since analyticity is a local condition. Therefore, conditions for being a sheaf (recall Definitions 2.20 and 2.21) are satisfied.  To set up for the focus-focus case, we need another kind of sheaf manipulation. In this case, we define a new sheaf by replacing a section in a neighbourhood. Lemma 5.5 Let p ∈(M,ω,F)be a non-degenerate singular point of focus-focus type on an integrable system defined on a symplectic manifold. Suppose that it is equipped with a complex line bundle Land a connection ∇whose curvature is ω. Then, the sheaf cohomology of leaf-wise flat smooth sections is still a sheaf cohomology when the cohomology of a neighbourhood of p is replaced by the sheaf cohomology of a cylinder in that neighbourhood. We denote this sheaf by Jf. Proof Consider a semi-local neighbourhood Uof the singularity of focus-focus type p∈(M,ω,F). Apart from the singular focus-focus leaf Lcontaining p, the rest of the leaves of Uare regular tori. In the intersection of Land a local neighbourhood Vof pchange the cohomology of the focus-focus leaf presented in [21] by the sheaf cohomology of the torus. Explicitly, take ω=dt ∧dθand take as flat sections the complex functions of the form σ=a(t)eitθ. For the complete construction of the cohomology, see Section 2.6 and especially Appendix A. The definition of these flat sections in the singular fiber automatically sets the cohomology of all the fibers in the entire Uinto the torus cohomology. By continuity, this is a well-defined cohomology in U. The cohomology glues back well because the focus-focus singularity is isolated and the topology of the complementary is glued back to a cylinder. So the pieces of the puzzle will glue back normally as in the computation of the sheaf cohomology in a neighbourhood of the torus. This is done using the Mayer-Vietoris formula in Section 4 of [20]. If this torus lies over an integer point of the lattice this would add a Bohr-Sommerfeld leaf to the computation.  Remark 5.6 Observe that this argument works whenever there are several pinched nodes in the singular focus-focus fiber. In this case, the total count does not depend on the number of nodes of the multiple pinched torus. At each node of the pinched torus we redo the puzzle argument above to conclude. In practice, we do an interchange of a single cohomology piece, a focus-focus cohomology piece by a torus cohomology piece which is so close that the substitution can be done smoothly (see Figure 3). This allows to define the geometric quantization of the focus-focus singularity as the quantization of the regular case. We can think of the sheaf-theoretical approach as a puzzle construction where the manifold comes endowed with an adapted covering admitting local data (sections and 118 Page 32 of 35 P. Mir, E. Miranda We want to identify the ˇ Cech 0-cochains and 1-cochains with respect to the cover A. A 0-cochain αis an assignment of a flat section over A,Band Cto that same subsets. Then, αassigns Ato the section aA(t)eitθ,Bto aB(t)eitθand Cto aC(t)eitθ. The angular coordinate θcan not be defined on all of S1so a branch of θhas to be fixed on each rectangle. We choose the branches so that θA=θBon A∩B,θB=θC on B∩C, and θC=θA+2πon A∩C. The coboundary operator δacts on αas (δα)ij =ηj−ηi=aj(t)eitθj−ai(t)eitθi, for i,j∈{A,B,C}. We impose that, at the three intersections, δα is 0, obtaining the following three equations: 0=aB(t)eitθB−aA(t)eitθAon A∩B(A.1) 0=aC(t)eitθC−aB(t)eitθBon B∩C(A.2) 0=aA(t)eitθA−aC(t)eitθCon C∩A(A.3) Then, αis a cocycle if and only if the following three equations simultaneously: aB(t)=aA(t)on A∩B(A.4) aC(t)=aB(t)on B∩C(A.5) aA(t)=aC(t)e2πit on C∩A(A.6) which is not possible since e2πit can not equal 1 in an entire interval of values of t. We conclude that are no 0-cocycles and that H0=0. Now, a 1-cochain βis an assignment of a flat section over A∩B,B∩Cand C∩Ato that same subsets. Then, βassigns A∩Bto the section bAB(t)eitθ,B∩Cto bBC(t)eitθand C∩Ato bCA(t)eitθ. There only possible triple intersection in the cover Ais empty and 2-cochains do not exist, implying that every 1-cochain is a cocycle. Since the 1-cochain is determined essentially by the three smooth functions bij(t)on I, the space of 1-cocycles is isomorphic to C∞(I)3. A 1-cochain βis a coboundary if there exists a 0-cochain α={aA(t)eitθA,aB(t) eitθB,aC(t)eitθC}with δα =β. Or, equivalently, if these three equations are satisfied: βAB =αB−αAon A∩B(A.7) βBC =αC−αBon B∩C(A.8) βCA =αA−αCon C∩A(A.9) Giving the sections explicitly, these equations transform to: bAB(t)eitθA=aB(t)eitθB−aA(t)eitθAon A∩B(A.10) bBC(t)eitθB=aC(t)eitθC−aB(t)eitθBon B∩C(A.11) bCA(t)eitθC=aA(t)eitθA−aC(t)eitθCon C∩A(A.12) Geometric quantization via cotangent models Page 33 of 35 118 Notice that, on each ordered intersection of two sets E∩F,weusetheθcoordinate from E. In each equation all the θcoordinates coincide except in Equation A.12, where they differ by a factor of 2π. Then, we obtain a system of equations in the three unknown functions aA,aB, and aCwhich has to be true for each value of tin Iand which has, as a matrix of coefficients, the following: ⎛ ⎝ −110 0−11 e−2πit 0−1⎞ ⎠(A.13) We observe that the matrix has rank 3 (and therefore the system has a unique solution) when e−2πit = 1. Then if e−2πit is never 1 on U, every cocycle is a coboundary, and Uhas the zero cohomology. Otherwise, if e−2πit =1 somewhere in I, which happens if and only if Icontains an integer m, the system only has a solution if the matrix ⎛ ⎝ −110bAB(m) 0−11bBC(m) e−2πit 0−1bCA(m)⎞ ⎠(A.14) has rank 2, i.e., if βsatisfies the condition bAB(m)+bBC(m)+bCA(m)=0.(A.15) Then, we have the following result: Proposition A.1 the cohomology group H1of U is precisely H1=C∞(I)3/{bAB(m)+bBC(m)+bCA(m)=0},(A.16) whichisisomorphictoC. Observe that for k>1 there are no (k+1)-fold intersections in the cover A. Therefore, all the cohomology groups Hk Aare zero for k>1. The condition e2πit =1 is satisfied exactly at the Bohr-Sommerfeld leaves, so we conclude that if Uis a band on the cylinder, the sheaf cohomology of Uwith respect to the cover Aby the three rectangles is trivial if Udoes not contain a Bohr-Sommerfeld leaf, and it is: Hk A(U;J)∼ =Ck=1 0k= 1 One can see that if another cover Bof Uis made by krectangles instead of 3, the cohomology calculated with respect to Bis the same as that calculated with respect to A. 118 Page 34 of 35 P. Mir, E. Miranda References 1. Arnold, Vladimir: Mathematical methods of classical mechanics, volume 60. Graduate Texts in Mathematics, (1974) 2. Bolsinov, Alexey, Izosimov, Anton: Smooth invariants of focus-focus singularities and obstructions to product decomposition. J. Symplectic Geom. 17(6), 1613–1648 (2019) 3. Castejon-Diaz, Hector: Berezin-Toeplitz Quantization on K3 Surfaces and Hyperkähler BerezinToeplitz Quantization. University of Luxembourg 9, (2016) 4. 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