GGEOMETRIC QUANTIZATION VIA COTANGENT MODELS
PAU MIR AND EVA MIRANDA
Abstract.
In this article we give a universal model for geometric quantization associated to a real polar-ization given by an integrable system with non-degenerate singularities. This universal model is inspired bythe cotangent bundle, as any neighbourhood of a regular compact orbit (Liouville torus) of an integrablesystem can be symplectically described as its cotangent bundle, T ∗ ( T n ). As any integrable system on acompact manifold will certainly have singularities we need to make the recipe work in some singular situa-tions. Our recipe works for real polarizations given by integrable systems with non-degenerate singularities.These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hy-perbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoricand almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, thespherical 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 [HM10] and [MPS20] away fromthe singularities and corrects the former models for hyperbolic and focus-focus singularities cancelling outthe infinite dimensional contributions obtained by former approaches. The geometric quantization mod-els provided here match the classical physical methods for mechanical systems, for instance the sphericalpendulum as presented in [CD88]. Introduction
However, on the second timeround, she came upon a lowcurtain she had not noticed before,and behind it was a little doorabout fifteen inches high: she triedthe little golden key in the lock,and to her great delight it fitted!
Lewis Carrol, Alice in Wonderland.
The quantization seeks to associate a quantum system to a classical Hamiltonian system replacing func-tions by operators and Poisson brackets of functions by brackets of operators. Several paths have been tracedfor this passionate journey from geometry and analysis into Physics: geometric quantization, formal quan-tization, BRST quantization and semi-classical quantization to cite a few. All of them supply Taylor-mademaster formulas to the day dreamer mathematicians who are looking into the quantum world through theirquantization mirror.In this article we focus on the geometric quantization approach and we provide a new model proposalwhich corrects former models and brings us closer to the role of quantization as a mathematical tamer ofquantum physics. The almost metaphysical questions still waft through the air: Can this be achieved? Dothese methods depend on additional data? Can we find a universal model?
Pau Mir is supported by an FPI-UPC grant. Eva Miranda is supported by the Catalan Institution for Research and AdvancedStudies via an ICREA Academia Prize 2016. Both authors are supported by the grants reference number MTM2015-69135-P (MINECO/FEDER) and reference number 2017SGR932 (AGAUR) and the project PID2019-103849GB-I00 with referencenumber AEI/10.13039/501100011033. a r X i v : . [ m a t h . S G ] F e b PAU MIR AND EVA MIRANDA
One of the virtues of our model is that it takes the cotangent bundle as a general set-up for our systems.The connection between the Hamiltonian system and the cotangent bundle is given by the idea of cotangentlift. This provides a unified approach to former attempts in the literature. One of the downfalls of our modelis that, unlike other models like the one of K¨ahler quantization, it depends on choices (in our case, on thechoice of a real polarization given by an integrable system).Geometric quantization and integrable systems are common mathematical objects on the interface ofGeometry and Physics. Integrable systems represent a class of Hamiltonian systems which can be associatedto an extra number of functions (first integrals). And geometric quantization meets integrable systems whenthese systems are used as data attached to the geometric quantization process (a polarization).A prequantum line bundle is naturally associated to a symplectic manifold of integral class, and a con-nection ∇ with curvature ω (the symplectic form) can be attached. A flat section of the bundle is a solutionto the equation ∇ X s = 0, where the derivation takes place along the direction of a polarization. An inte-grable system naturally provides examples of (real) polarizations. In this article, we will contemplate thequantization problem considering as polarizations the real polarizations associated to an integrable system.Integrable systems 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, areintegrable. Other classical systems defined by attracting or repelling particles, such as the Toda systems,are integrable.Because of the maximum principle, when the phase space is compact the integrable system defined viasmooth functions must have singularities. In this article we analyze the contribution of these singularities inthe study of their geometric quantization. The simplest type of singularities of smooth functions are Morse-type singularities, which admit a Morse or Morse-Bott normal form. For integrable systems, one mightdemand a similar behaviour in a neighbourhood of their singularities. That is to say, we assume that thereexist local coordinates such that the functions are simultaneously of Morse type and the symplectic form isDarboux. Those singularities were initially considered by Eliasson [Eli90] and later by Miranda [Mir03] andMiranda-Zung [MZ04].In former works by Hamilton [Ham10], Hamilton-Miranda [HM10] and Miranda-Presas-Solha [MPS20],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 afinite dimensional representation space as quantization space of a system defined on a compact manifold.In this article, we work out cotangent lift models for integrable systems with non-degenerate singularitieswhich can be of elliptic, hyperbolic and focus-focus type. Those singularities naturally appear in polarizationson compact manifolds given by integrable systems. In particular, any semitoric system (as the ones studiedin [PVuN09, PVuN11]) gives rise to singularities of this type). These structures also show up in algebraicgeometry naturally, for instance in the K3 surface , which can be viewed as a semitoric system. When itcomes to considering their quantization, several models have been proposed. Nevertheless, none of them cancompete with the model of K¨ahler quantization (which cannot always be applied since the polarization needsto be of K¨ahler type) in terms of independence of the polarization and in terms of the principle ”quantizationcommutes with reduction [ Q, R ] = 0” (see [GS83], [GS82a], [GS82b]).For regular integrable systems (without singularities) action-angle coordinates (the classical Arnold-Liouville theorem) provide cotangent models because, semilocally, the manifold is symplectically interpretedas the cotangent bundle of a Liouville torus, T ∗ ( T n ). This canonical identification gives a way to relate thechoice of the Liouville one-form of the cotangent bundle with the connection 1-form of the prequantum line A K3 surface is an example of a hyperk¨ahler manifold with three compatible complex structures i, j, k . The denominationK3 comes from Kummer, K¨ahler and Kodaira and, according to Andr´e Weil, it is a reminiscent of the beautiful mountain K2in Kashmir.
EOMETRIC QUANTIZATION VIA COTANGENT MODELS 3 bundle. In other words, Liouville one-forms of the cotangent bundle yield a canonical choice of the connection1-form. This connects the cotangent model to quantization in the regular case (see for instance [MP15] and[´S77]). With the ambition of extending these ideas to the singular set-up, we analyze the cotangent lift tech-nique for different types of non-degenerate singularities (in the sense of Eliasson-Williamson). Additionally,the existence of a local model of cotangent type allows to capture symmetry and makes them compatiblewith the principle of quantization commutes with reduction. We use the cotangent models to define a newproposal for geometric quantization for non-degenerate singularities. By complexifying these system, we ob-tain a unique universal cotangent model which allows us to give a unified model of complex elliptic type. Incontrast to the former models of geometric quantization for real polarizations endowed with non-degeneratesingularities in [HM10] and [MPS20] our new models provide finite dimensional representations for systemson compact manifolds.
Organization of this article:
In section 2 we revise the rudiments of geometric quantization. In Section3 the main features of the theory of non-degenerate singularities of integrable systems are revised. In Section4 we revise the basics of cotangent lifts and introduce the notion of discrete cotangent lift. In section 5we attach cotangent models to integrable systems in a neighbourhood of a compact regular orbit and inthe neighbourhood of an elliptic and hyperbolic points and study their complexification. In Section 6 wepropose a new local model and glue it globally for almost toric systems giving new finite dimensional recipesfor focus-focus singularities. We end up this section with some applications to the quantization of K3 surfacesand the spherical pendulum. Section 7 contains the conclusions of this article.2.
A crash course on geometric quantization
As a general principle, quantization consists in associating a Hilbert space Q to a symplectic manifold( M, ω ). In geometric quantization, this Hilbert space is constructed using the sections of a complex linebundle L . Normally, one declares as representation space flat sections of this bundle in some direction (givenby a polarization). Such sections are not always defined globally along the leaves of the polarization. Thisis why it is convenient to use the sheaf-theoretic language (with sheaf meaning the sheaf of flat sections ofthe bundle) to surmount this difficulty. Kostant introduced the main ideas of geometric quantization in the70s [Kos70] and, today, they remain useful and have applications in representation theory, a big variety ofphysical problems and many other fields. One of the main characters of the theory of geometric quantizationare Bohr-Sommerfeld leaves. Kostant’s model goes through the cohomology associated to the sheaf of flatsections of L and is well-adapted for real polarizations given by integrable systems and toric manifolds, whichare symplectic manifolds endowed with an effective Hamiltonian action of a torus whose rank is half of thedimension of the manifold [Mir14].An important result of Delzant, which connects quantization and moment maps, states the existence ofa one-to-one correspondence between closed toric manifolds in dimension 2 m and the Delzant polytope on R m [Del88]. The Delzant polytope gives the real geometric quantization of closed toric manifolds [Ham10]because, given a toric manifold, its real geometric quantization is completely determined by the count ofinteger points in the interior of its associated Delzant polytope.Segal also proposed a way to quantize a Hamiltonian system consisting essentially in associating to thephase space a real Hilbert space ( F , ( , )) of the states of one particle [Seg67], bringing a symplectic structure ω and a complex structure J such that the complexification H of F under J has a complex scalar product( , ) C defined as ( , ) C = ( , ) + Jω and the Hamiltonian evolution of the system is expressed by a unitary flow.Although Segal quantization is quite useful for many purposes, in this article we focus on Kostant’sviewpoint, since it is a more convenient geometric quantization to deal with cotangent models. In particular, PAU MIR AND EVA MIRANDA as we will provide cotangent models for regular leaves and non-degenerate singularities of integrable systems,it will be the right approach for this article.
Definition 2.1.
Let (
M, ω ) be a symplectic manifold. A prequantization line bundle is a complex line bundle L over M , equipped with a connection ∇ whose curvature is ω . Definition 2.2. A real polarization of a symplectic manifold ( M, ω ) is a foliation of M into Lagrangiansubmanifolds.We will usually want to compute the quantization of a compact completely integrable system ( M, ω, F ),using the singular real polarization given by the singular foliation by level sets of F , which are genericallyLagrangian tori. Definition 2.3.
A section σ of L is flat along the leaves or leafwise flat if it is covariant constant along thefibres of F , with respect to the prequantization connection ∇ . Namely, if ∇ X σ = 0 for any vector field X tangent to fibres of F . The sheaf of sections which are flat along the leaves is denoted by J . Definition 2.4.
With (
M, ω, F ), L , and J as above, the quantization of M is Q ( M ) = (cid:77) k ≥ H k ( M ; J ) . Definition 2.5.
A leaf (cid:96) of the (singular) foliation is a
Bohr-Sommerfeld leaf if there is a leafwise flatsection σ defined over all of (cid:96) .Although leafwise flat sections always exist locally (because by construction the curvature of ∇ is ω ,which is zero when restricted to a leaf), we are requiring global existence, which is a strong condition. Theset of Bohr-Sommerfeld leaves is discrete in the leaf space and a leaf is Bohr-Sommerfeld if and only if itsholonomy is trivial around all the loops contained in the leaf.The main result about quantization using real polarizations is a theorem of ´Sniatycki [´S77] which statesthat, if the leaf space B n is a manifold and the map π : M n → B n is a fibration with compact fibres, thenall of the cohomology groups are zero except in degree n . Furthermore, H n can be expressed in terms of theBohr-Sommerfeld leaves and the dimension of H n is exactly the number of Bohr-Sommerfeld leaves.This result implies that, for a toric manifold foliated by fibres of the moment map, the Bohr-Sommerfeldleaves correspond to the integer lattice points in the interior of moment polytope, excluding the ones on theboundary (see Figure 1). To prove that the Bohr-Sommerfeld leaves on C are precisely the discrete circleswhich are function of an integer, one goes through the following computations.Suppose L is the trivial line bundle C × C . Take polar coordinates ( r, φ ), so that the symplectic form is ω = rdr ∧ dφ = d ( r dφ ), so it is exact. It is not defined at r = 0, but it extends smoothly there.We equip the plane with a singular real polarization given by the distribution P = span { ∂∂φ } , which isintegrable and has as integral manifolds the circles of constant r (and it is a singular foliation). Then, theconnection defined in the canonical trivialization of L by ∇ X σ = dσ ( X ) − σi r dφ ( X ) , where σ : C → C is a prequantization connection.A section of L is given by σ . If we impose the flatness condition along the leaves, we get0 = ∇ X σ = X ( σ ) − σ i r dφ ( X ) ∀ X ∈ P, which is equivalent to 0 = ∇ ∂∂φ σ = ∂σ∂φ − σi r , EOMETRIC QUANTIZATION VIA COTANGENT MODELS 5
Figure 1.
The integer points in the interior of Delzant’s polytope correspond to the Bohr-Sommerfeld leaves, which are regular tori.and ∂σ∂φ = i r σ. Then, the sections which are flat along the leaves are those of the form σ ( r, φ ) = a ( r ) · e i r φ , for arbitrarysmooth functions a ( r ).This calculation applies anywhere except at the origin ( r = 0), but this is enough to determine the valueof σ at 0 [Ham10]. Now, in order for e isφ to be defined on an entire leaf, that is, the entire range of φ from0 to 2 π , it is necessary for s to be an integer. So the Bohr-Sommerfeld leaves on C are precisely the circles { r = k } k ∈ N .One of the key points of this computation is that the connection one-form is canonically associated to thechoice of a Liouville one form adapted to the singular foliation. This can be related to the cotangent lift aswe will see later.In [HM10], Hamilton and Miranda prove the following key theorem for compact two-dimensional integrablesystems with non-degenerate singularities. Theorem 2.6.
Let ( M, ω, F ) be a two-dimensional, compact, completely integrable system, whose momentmap has only non-degenerate singularities. Suppose M has a prequantum line bundle L , and let J be thesheaf of sections of L flat along the leaves. The cohomology H ( M, J ) has two contributions of the form C N for each hyperbolic singularity, each one corresponding to a space of Taylor series in one complex variable.It also has one C term for each non-singular Bohr-Sommerfeld leaf. That is, (2.1) H ( M ; J ) ∼ = (cid:77) p ∈H (cid:0) C N ⊕ C N (cid:1) ⊕ (cid:77) b ∈ BS C b . The cohomology in other degrees is zero. Thus, the quantization of M is given by (2.1) . Two more important theorems were proved by Miranda, Presas and Solha in [MPS20], concerning thegeometric quantization of focus-focus fibers.
Theorem 2.7.
The geometric quantization of a saturated neighborhood of a focus-focus fiber with n nodesis: • if the singular fiber is not Bohr–Sommerfeld. • isomorphic to ( C ∞ ( R ; C )) n f , PAU MIR AND EVA MIRANDA if the singular fiber is Bohr–Sommerfeld, where n f = n (for compact fibers) and n f = n − otherwise. Theorem 2.8.
For a -dimensional closed almost toric manifold M , with n r regular Bohr–Sommerfeld fibersand n f focus-focus Bohr–Sommerfeld fibers: Q ( M ) ∼ = C n r ⊕ (cid:77) j ∈{ ,...,n f } ( C ∞ ( R ; C )) n ( j ) , with n ( j ) the number of nodes on the j -th focus-focus Bohr–Sommerfeld fiber. These results allow to compute the quantization of some particular systems such as K H in some specific cases is also done in [MPS20]. Consider M = T ∗ I × ( I s × S ) with ω = dx ∧ dy + dx ∧ dy , endowed with a trivial pre-quantum line bundle with connection ∇ = d − i ( x dy + x dy ), and P generated by ∂∂y and ∂∂y , where x is the coordinate function along thefibers of T ∗ I , y is the coordinate function along the open interval I ⊂ R , x is the coordinate function alongthe open interval I s ⊂ R , and y is the periodic coordinate function along S . Proposition 2.9.
The geometric quantization of M = T ∗ I × ( I s × S ) , with the extra structures describedabove, is given by H ( T ∗ I × ( I s × S ); J ) ∼ = ( H ( T ∗ I ; J )) n ∼ = ( C ∞ ( R ; C )) n , where n is the number of integers inside I s . Also, there exist isomorphisms from H ( T ∗ I × ( I s × S ); J ) to flat sections of trivial pre-quantum linebundles L over T ∗ I , for a given open interval I ⊂ R . These flat sections are all of the form T ∗ I (cid:51) ( x, y ) (cid:55)→ h ( x ) e ixy s ( x, y ) ∈ L | ( x,y ) ∼ = C where h ∈ C ∞ ( R ; C ) and s ∈ Γ( L ) is a unitary section of L with potential 1-form α = − xdy . For example H ( T ∗ I × ( I s × S ); J ) ∼ = H ( T ∗ I ; J ) ⊗ H ( I s × S ; J ) ∼ = C ∞ ( R ; C ) ⊗ C n . Although the geometric quantization of the regular case is quite clear and for specific systems with non-degenerate singularities of hyperbolic and focus-focus type there do exist results, they are not so general.3.
Moment maps and Hamiltonian systems
Hamiltonian actions and moment maps are the absolute key concepts in the link between symplecticgeometry and integrable systems. In this section we give a brief review on them, giving special attention tointegrable systems with non-degenerate singularities.
Definition 3.1.
An integrable system on M is given by a smooth map f = ( f , . . . , f n ) : M → R n suchthat { f i , f j } = 0 for all 1 ≤ f i , f j ≤ n and rank f = n almost everywhere. If the Hamiltonian vectorfields provided by each f i (i.e. the X i satisfying ι X i ω = − df i ), which have commuting flows φ t , . . . , φ nt , arecomplete, then the system induces an R n action on M , called the joint flow : ρ : R n × M −→ M (3.1) ( t , . . . , t n , p ) (cid:55)−→ φ t ◦ · · · ◦ φ nt ( p )(3.2) Definition 3.2.
Let H ∈ C ∞ be a smooth function on a symplectic manifold ( M, ω ) (in Physics, a Hamil-tonian, the function of total energy). The
Hamiltonian vector field X H associated to H is defined as theonly solution of ι X H ω = − dH . EOMETRIC QUANTIZATION VIA COTANGENT MODELS 7
Definition 3.3.
Let G be a Lie group and g its Lie algebra. Consider also g ∗ , the dual of g . Suppose ψ : G → Diff( M ) is an action on a symplectic manifold ( M, ω ). It is called a
Hamiltonian action if thereexists a map µ : M → g ∗ which satisfies: • For each X ∈ g , dµ X = ι X ω , i.e., µ X is a Hamiltonian function for the vector field X , where – µ X : p (cid:55)−→ (cid:104) µ ( p ) , X (cid:105) : M −→ R is the component of µ along X , – X is the vector field on M generated by the one-parameter subgroup { exp tX | t ∈ R } ⊂ G . • The map µ is equivariant with respect to the given action ψ on M and the coadjoint action: µ ◦ ψ g =Ad ∗ g ◦ µ , for all g ∈ G .Then, ( M, ω, G, µ ) is called a
Hamiltonian G -space and µ is called the moment map .The normal form of a completely integrable system around a whole leaf of a regular point is well-knownby the Arnold-Liouville-Mineur theorem. Theorem 3.4 (Arnold-Liouville-Mineur) . Let ( M n , ω ) be a symplectic manifold. Let f , . . . , f n functionson M which are functionally independent (i.e. df ∧ · · · ∧ df n (cid:54) = 0 ) on a dense set and which are pairwisein involution. Assume that m is a regular point of F = ( f , . . . , f n ) and that the level set of F through m ,which we denote by F m , is compact and connected.Then, F m is a torus and on a neighbourhood U of F m there exist R -valued smooth functions ( p , . . . , p n ) and R / Z -valued smooth functions ( θ , . . . , θ n ) such that:(1) The functions ( θ , . . . , θ n , p , . . . , p n ) define a diffeomorphism U (cid:39) T n × B n .(2) The symplectic structure can be written in terms of these coordinates as ω = n (cid:88) i =1 dθ i ∧ dp i . (3) The leaves of the surjective submersion F = ( f , . . . , f s ) are given by the projection onto the secondcomponent T n × B n , in particular, the functions f , . . . , f s depend only on p , . . . , p n .The functions p i are called action coordinates; the functions θ i are called angle coordinates. For singularities, and refined results have been obtained for local normal forms.At singularities, one has to dig deeper because it can be very difficult to understand both the geometry andthe dynamics of the system depending on the degeneracy of dF . For the case of non-degenerate singularitiespowerful results hav been obtained and, for instance, we have local normal forms. Definition 3.5.
A point p ∈ M n is a singular point (or a singularity ) of an integrable Hamiltonian systemgiven by F = ( f , . . . , f n ) if the rank of dF = ( df , . . . , df n ) at p is less than n . The singular point p has rank k and corank of n − k if rank( dF ) p = rank (( df ) p , . . . , ( df n ) p ) = k . Definition 3.6.
Let g be a Lie algebra. A Cartan subalgebra h is a nilpotent subalgebra of g that is self-normalizing, i.e., if [ X, Y ] ∈ h for all X ∈ h , then Y ∈ h . If g is finite-dimensional and semisimple overan algebraically closed field of characteristic zero, a Cartan subalgebra is a maximal abelian subalgebra (asubalgebra consisting of semisimple elements). Definition 3.7.
Let ( M n , ω ) be a symplectic manifold with an integrable Hamiltonian system of n inde-pendent and commuting first integrals f , . . . , f n . Consider a singular point p ∈ M of rank 0, i.e. ( df i ) p = 0for all i . It is called a non-degenerate singular point if the operators ω − d f , . . . , ω − d f n form a Cartansubalgebra in the symplectic Lie algebra sp (2 n, R ) = sp ( T p M, ω ). Remark . The operators ω − d f i , where df i is the Hessian of f i , associate a function to the Hessian byvisualizing the Hessian as a quadratic form H ( u, v ) and taking the symplectic dual of the function obtained.A good reference for details of the algebraic construction of the Cartan subalgebra is [BF04]. PAU MIR AND EVA MIRANDA
The classification of non-degenerate critical points of the moment map in the real case was obtained byWilliamson [Wil36]. In the complex case, all the Cartan subalgebras are conjugate and hence there is onlyone model for non-degenerate critical points of the moment map.
Theorem 3.9 (Williamson) . For any Cartan subalgebra C of sp (2 n, R ) , there exists a symplectic systemof coordinates ( x , . . . , x n , y , . . . , y n ) in R n and a basis f , . . . , f n of C such that each of the quadraticpolynomials f i is one of the following: f i = x i + y i for 1 ≤ i ≤ k e f i = x i y i for k e + 1 ≤ i ≤ k e + k h (cid:40) f i = x i y i +1 − x i +1 y i f i +1 = x i y i + x i +1 y i +1 for i = k e + k h + 2 j − , ≤ j ≤ k f The three types are called elliptic, hyperbolic and focus-focus, respectively.Remark . Notice that the focus-focus components always go by pairs. Because of theorem 3.9, the triple( k e , k h , k f ) at a singular point it is an invariant. It is also an invariant of the orbit of the integrable systemthrough the point [Zun96].If p is a non-degenerate singularity of the moment map F , it is characterized by four integer numbers, therank k of the singularity and the triple ( k e , k h , k f ). By the way they are defined, they satisfy k + k e + k h +2 k f = n , where n is the number of degrees of freedom of the integrable system.The following is a result of Eliasson [Eli90] and Miranda and Zung ([Mir03], [Mir14], [MZ04]). Theorem 3.11 (Smooth local linearization) . Given an smooth integrable Hamiltonian system with n degreesof freedom on a symplectic manifold ( M n , ω ) , the Liouville foliation in a neighborhood of a non-degeneratesingular point of rank k and Williamson type ( k e , k h , k f ) is locally symplectomorphic to the model Liouvillefoliation, which is the foliation defined by the basis functions of Theorem 3.9 plus ”coordinate functions” f i = x i for i = k e + k h + 2 j + 1 to n .Remark . The theorem states the existence of a semilocal symplectomorphism between foliations witha non degenerate singularity of rank k and the same parameters ( k e , k h , k f ). One could think that functionsare also preserved via a symplectomorphism, but it is not possible to guarantee this statement when h k (cid:54) = 0as one can add up analytically flat terms on different connected components (see counterexample in [Mir03]).In general one needs more information about the topology of the leaf to conclude. Remark . Because of Theorem 3.11, if one considers the Taylor expansions of F = ( f , . . . , f n ) at thenon-degenerate singular point in a canonical coordinate system and removes all terms except for linear andquadratic, the functions obtained remain commuting and define a Liouville foliation that can be consideredas the linearization of the initial foliation F given by f , . . . , f n , to which it is symplectomorphic.The description of non-degenerate singularities at the semilocal level is completed with the following tworesults. Theorem 3.14 (Model in a covering) . The manifold can be represented, locally at a non-degenerate singu-larity of rank k and Williamson type ( k e , k h , k f ) , as the direct product M reg × k · · · × M reg × M ell × k e · · · × M ell × M hyp × k h · · · × M hyp × M foc × k f · · · × M foc Where: • M reg is a ”regular block”, given by f = x, EOMETRIC QUANTIZATION VIA COTANGENT MODELS 9 • M ell is an ”elliptic block”, representing the elliptic singularity given by f = x + y , • M hyp is an ”hyperbolic block”, representing the hyperbolic singularity given by f = xy, • M foc is a ”focus-focus block”, representing the focus-focus singularity given by (cid:40) f = x y − x y f = x y + x y . For the first three types of blocks the symplectic form is ω = dx ∧ dy , while for the focus-focus block it is ω = dx ∧ dy + dx ∧ dy . In the case of a smooth system (defined by a smooth moment map), a similar result was proved anddescribed by Miranda and Zung in [MZ04]. It summarizes some previously results proved independently andfixes the case where there are hyperbolic components ( k h (cid:54) = 0), because in this case the result is slightlydifferent and it has to be taken the semi-direct product in the decomposition. As opposite to the case wherethere are only elliptic and focus-focus singularities, in which the base of the fibration of the neighbourhoodis an open disk, if there are hyperbolic components the topology of the fiber can become complicated. Thereason is essentially that for the smooth case a level set of the form { x i y i = ε } is not connected but consistsof two components. Theorem 3.15 (Miranda-Zung) . [MZ04] Let V = D k × T k × D n − k ) with the following coordinates: ( p , ..., p k ) for D k , ( q ( mod , ..., q k ( mod for T k , and ( x , y , ..., x n − k , y n − k ) for D n − k ) be a symplecticmanifold with the standard symplectic form (cid:80) dp i ∧ dq i + (cid:80) dx j ∧ dy j . Let F be the moment map correspond-ing to a singularity of rank k with Williamson type ( k e , k h , k f ) . There exists a finite group Γ , a linear systemon the symplectic manifold V / Γ and a smooth Lagrangian-fibration-preserving symplectomorphism φ from aneighborhood of O into V / Γ , which sends O to the torus { p i = x i = y i = 0 } . The smooth symplectomorphism φ can be chosen so that via φ , the system-preserving action of a compact group G near O becomes a linearsystem-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 φ can also be chosen to be real analytic. If the system dependssmoothly (resp., analytically) on a local parameter (i.e. we have a local family of systems), then φ can alsobe chosen to depend smoothly (resp., analytically) on that parameter. Theorem 3.16 (Miranda, [Mir03]) . Let ω be a symplectic form defined in a neighbourhood of the singularityat p for which the foliation F is Lagrangian. Then, there exists a local diffeomorphism φ : ( U, p ) −→ ( φ ( U ) , p ) such that φ preserves the foliation and φ ∗ ( (cid:80) i dx i ∧ dy i ) = ω , where x i , y i are local coordinates on ( φ ( U ) , p ) . The cotangent lift
The cotangent bundle of a smooth manifold can be naturally equipped with a symplectic structure inthe following way. Let M be a differential manifold and consider its cotangent bundle T ∗ M . There is anintrinsic canonical linear form λ on T ∗ M defined pointwise by (cid:104) λ p , v (cid:105) = (cid:104) p, dπ p v (cid:105) , p = ( m, ξ ) ∈ T ∗ M, v ∈ T p ( T ∗ M ) , where dπ p : T p ( T ∗ M ) −→ T m M is the differential of the canonical projection at p . In local coordinates( q i , p i ), the form is written as λ = (cid:80) i p i dq i and is called the Liouville 1-form . Its differential ω = dλ = (cid:80) i dp i ∧ dq i is a symplectic form on T ∗ M . Definition 4.1.
Let ρ : G × M −→ M be a group action of a Lie group G on a smooth manifold M . Foreach g ∈ G , there is an induced diffeomorphism ρ g : M −→ M . The cotangent lift of ρ g , denoted by ˆ ρ g , isthe diffeomorphism on T ∗ M given byˆ ρ g ( q, p ) := ( ρ g ( q ) , (( dρ g ) ∗ q ) − ( p )) , ( q, p ) ∈ T ∗ M which makes the following diagram commute: T ∗ M T ∗ MM Mπ ˆ ρ g ρ g π Given a difeomorphism ρ : M −→ M , its cotangent lift preserves the canonical form λ as the simplefollowing computation shows. At a point p = ( m, ξ ) ∈ T ∗ M : λ p = ( dπ ) ∗ p ξ == ( dπ ) ∗ p ( dρ ) ∗ m (( dρ ) ∗ m ) − ξ == ( d ( ρ ◦ π )) ∗ p (( dρ ) ∗ m ) − ξ == ( d ( π ◦ ˆ ρ )) ∗ p (( dρ ) ∗ m ) − ξ == ( d ˆ ρ ) ∗ p ( dπ ) ∗ ˆ ρ ( p ) (( dρ ) ∗ m ) − ξ == ( d ˆ ρ ) ∗ p λ ˆ ρ ( p ) , where we have used the definitions of the Liouville 1-form and the cotangent lift and the fact that ρ ◦ π = π ◦ ˆ ρ .Then, the canonical 1-form is preserved by ˆ ρ .As a consequence: ˆ ρ ∗ ( ω ) = ˆ ρ ∗ ( dλ ) = d (ˆ ρ ∗ λ ) = dλ = ω. Meaning that the cotangent lift ˆ ρ g preserves the Liouville form and the symplectic form of T ∗ M and weconclude the following standard result in the theory of cotangent lifts: Example . Let ρ : ( R , +) × R → R be the Lie group action corresponding to a space translation definedby ρ x ( q ) = q + x . Write ( q, p ) for an element of the cotangent bundle T ∗ R ∼ = R .By definition, ˆ ρ x , the cotangent lift of ρ x isˆ ρ x ( q, p ) = ( ρ x ( q ) , (( dρ x ) ∗ q ) − ( p )) == ( q + x, (( Id ∗ ) − ( p )) = ( q + x, p ) Example . Let ρ : SO (3 , R ) × R → R be a Lie group action defined by ρ A ( q ) = Aq . Write ( q, p ) for anelement of T ∗ q R . By definition, ˆ ρ A , the cotangent lift of ρ A isˆ ρ A ( q, p ) = ( ρ A ( q ) , (( dρ A ) ∗ q ) − ( p )) = ( Aq, (( A ∗ ) − ( p )) = ( Aq, Ap ) , where the last equality holds because A is orthogonal. Like any cotangent lift, since the induced action inthe cotangent bundle is Hamiltonian, it has an associated momentum map which, in this case, correspondsto the classical quantity q ∧ p . EOMETRIC QUANTIZATION VIA COTANGENT MODELS 11
The discrete cotangent lift.
As a discrete analog of the the classical cotangent lift of a Lie groupaction, we define the discrete cotangent lift , a tool which allows to see Bohr-Sommerfeld leaves as a cotangentlift in the classical sense.In an integrable system of dimension 2 n , Bohr-Sommerfeld orbits correspond to the integer points in theinterior of the Delzant polytope. Since they are all of them n -dimensional tori, they can be seen as the orbitof a discrete translation action of a single torus.Consider the integer points in the Delzant polytope of an integrable system, written in coordinates as( x + m , . . . , x n + m n ) ∈ R n for some fixed ( x , . . . , x n ) ∈ R n and a set N = { ( m , . . . , m n ) ∈ Z n } of n -tuples. Consider the action α : Z n × R n −→ R n (cid:0) ( m , . . . , m n ) , ( x , . . . , x n ) (cid:1) (cid:55)−→ ( x + m , . . . , x n + m n ) , which can be seen as the restriction to integer n -tuples in N of the associate continuous translation˜ α : R n × R n −→ R n (cid:0) ( t , . . . , t n ) , ( x , . . . , x n ) (cid:1) (cid:55)−→ ( x + t , . . . , x n + t n ) . The orbit of ( x , . . . , x n ) ∈ R n by ˜ α restricted to N is the set of integer points in the interior of theDelzant polytope. Definition 4.4.
Consider the actions α and ˜ α as defined before. The discrete cotangent lift of α is the actionˆ α : Z n × T ∗ R n → T ∗ R n which extends α to the cotangent bundle of R n and coincides with the classicalcotangent lift of ˜ α at the integer points.The discrete cotangent lifted action ˆ α acts on the pairs of points in the interior of the Delzant polytopeand tori in the manifold. Its orbit is not only the whole set of integer points in the Delzant polytope butthe pairs of integer points and their associated Bohr-Sommerfeld leaves (see Figure 2).The action ˆ α is well defined on the cotangent bundle of a vector space. Not only this, but it is alsocompatible with the symplectic structure of the cotangent bundle. To see this, consider the coordinates( x , . . . , x n ) on the base and the symplectic dual coordinates ( y , . . . , y n ) on the fiber of T ∗ R n . Since thenatural pairing of each coordinate x i with its dual y i coincides with the symplectic conjugation (reflected inthe Liouville 1-form λ = (cid:80) y i dx i and in the symplectic form ω = dλ = (cid:80) dy i ∧ dx i ), the lift of the discreteaction α preserves the symplectic structure. α ˆ α Figure 2.
The set of tori corresponding to the integer points in the interior of Delzant’spolytope is the orbit of the action ˆ α . Integrable systems as cotangent lifts
The regular case as a cotangent lift.
Cotangent lifts arise naturally in physics problems, and thelink between integrable systems and cotangent models is clear in view of the following Kiesenhofer andMiranda result [KM17], which restates Theorem 3.4 to reveal that at a semilocal level the regular leaves areequivalent to a completely toric cotangent lift model.
Theorem 5.1.
Let F = ( f , . . . , f n ) be an integrable system on a symplectic manifold ( M, ω ) . Then,semilocally around a regular Liouville torus, the system is equivalent to the cotangent model ( T ∗ T n ) can restricted to a neighbourhood of the zero section ( T ∗ T n ) of T ∗ T n . Non-degenerate singularities as cotangent lifts.
In the classical models of the harmonic oscillator,the simple pendulum and the spherical pendulum one already finds the three different types of non-degeneratesingularities 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 arisesin the spherical pendulum.5.3.
The hyperbolic singularity as a cotangent lift.
Take coordinates ( x, y ) on T ∗ R such that thesymplectic form is ω = dx ∧ dy and the moment map is f = xy .The Hamiltonian vector field associated to f is(5.1) X = ( − x, y ) . Consider the action of R on R given by: ρ : R × R −→ R ( t, x ) (cid:55)−→ e − t x . Then, (( dρ t ) ∗ x ) − acts as y (cid:55)−→ e t y , and the cotangent lift ˆ ρ t associated to the group action ρ t , incoordinates ( x, y ) of T ∗ R is exactly: ˆ ρ : T ∗ R −→ T ∗ R (cid:18) xy (cid:19) (cid:55)−→ (cid:18) e − t xe t y (cid:19) . Deriving the last vector with respect to t and evaluating at t = 0, we obtain exactly X = ( − x, y ), thevector field associated to the hyperbolic singularity.5.4. The elliptic singularity as a cotangent lift.
The cotangent lift in the elliptic case uses a complexmoment map which is not holomorphic. It is a formal development and by no means holomorphicity isassumed.Take complex coordinates ( z, ¯ z ) = ( x + iy, x − iy ) such that the symplectic form is ω = i dz ∧ d ¯ z . Themoment map corresponding to the elliptic singularity is f = (cid:0) x + y (cid:1) = z ¯ z .The Hamilton’s equations in this complex setting are: ι X ω = − df ⇐⇒⇐⇒ ι ( α ∂∂z + β ∂∂ ¯ z ) (cid:18) i dz ∧ d ¯ z (cid:19) = − ∂f∂z dz − ∂f∂ ¯ z d ¯ z ⇐⇒⇐⇒ iα d ¯ z − iβ dz = −
12 ¯ zdz − zd ¯ z ⇐⇒⇐⇒ (cid:40) α = izβ = − i ¯ z EOMETRIC QUANTIZATION VIA COTANGENT MODELS 13
Then, the Hamiltonian vector field associated to f is(5.2) X = ( iz, − i ¯ z ) . Now, consider the following action of R on C , which corresponds to a rotation of z of angle t : ρ : R × C −→ C ( t, z ) (cid:55)−→ e it z . Then, (( dρ t ) ∗ z ) − acts as ¯ z (cid:55)−→ e − it ¯ z , and the cotangent lift ˆ ρ t associated to the group action ρ t , incoordinates ( z, ¯ z ) of T ∗ C is: ˆ ρ : T ∗ C −→ T ∗ C (cid:18) z ¯ z (cid:19) (cid:55)−→ (cid:18) e it ze − it ¯ z (cid:19) . Deriving the last vector with respect to t and evaluating at t = 0 we obtain X = ( iz, − i ¯ z ), the vectorfield associated to the elliptic singularity.5.5. The focus-focus singularity as a cotangent lift.
In a singularity of focus-focus type in a manifoldof dimension 4, we can take coordinates ( x , x , y , y ) in a way that the symplectic form is ω = dx ∧ dy + dx ∧ dy and the moment map is F = ( f , f ) = ( x y − x y , x y + x y ).The Hamiltonian vector fields associated to f and f are X = ( x , − x , y , − y ) , X = ( − x , − x , y , y ) . Let G = S × R and M = R . Consider the action of a rotation and a radial dilation of R given by ρ : ( S × R ) × R → R (( θ, t ) , (cid:32) x x (cid:33) ) (cid:55)→ (cid:18) e − t e − t (cid:19) (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) x x (cid:19) Then, the cotangent lift ˆ ρ associated to the group action is exactlyˆ ρ : T ∗ R → T ∗ R (5.3) x x y y (cid:55)→ e − t ( x cos θ + x sin θ ) e − t ( − x sin θ + x cos θ ) e t ( y cos θ + y sin θ ) e t ( − y sin θ + y cos θ ) (5.4)Deriving the vector with respect to θ and evaluating at 0 we obtain exactly X = ( x , − x , y , − y ). Whilederiving the vector with respect to t and evaluating at 0 we obtain exactly X = ( − x , − x , y , y ).5.6. The focus-focus singularity as a complexification.
On another level, the focus-focus singularitycan be seen as a complexification, that is, its moment map can be locally given by two complex coordinates.Take F = ( f , f ) = ( x y − x y , x y + x y ), the moment map associated to the focus-focus singularity insymplectic coordinates ( x , x , y , y ) and define f = f + if , a complex moment map. Define two complexvariables z and z the following way: z = x + ix , z = y + iy . In these new coordinates ( z , z ), in which ω = (cid:60) ( d ¯ z ∧ dz ) (and λ = (cid:60) ¯ z dz )), we have f = ¯ z z , andwe can see that this model resembles the moment map of the hyperbolic singularity. In the same way that,in coordinates ( x, y ), we can see the hyperbolic singularity either given as the product xy or given as the difference of squares s − t via the definition of s = ( x + y ) / , t = ( x − y ) /
2, we can see f either as given asthe product ¯ z z or given as a sum of squares if we apply the following complex linear change of variables ζ = ¯ z + z , ζ = ¯ z − z i . This change of variables is equivalent to define directly ζ = 12 ( x + y + i ( − x + y ))(5.5) ζ = 12 ( − x − y + i ( − x + y ))(5.6)In the new coordinates ζ , ζ , in which ω = (cid:60) (2 idζ ∧ dζ ) (and λ = (cid:60) (2 iζ dζ )), we have f = ζ + ζ , whichis really convenient to consider the quantization of the focus-focus singularity.The conclusion we derive is that the focus-focus singularity can be seen as the cotangent lift of a non-compact action and as a complexification, but both approaches can not be made compatible with each other.We would say that the cotangent lift and the complexification do not commute. In fact, if it was possibleto do it, then the focus-focus singularity as seen in Theorem 3.14 would not be a 4-dimensional elementaryblock but the product of two 2-dimensional elementary blocks, which is not the case (by Theorem 3.9, whichproves that the focus-focus singularity is elementary).A way to avoid this lack of smoothness in the case of the focus-focus singularity is to use its complexifiedmodel as defined in Section 5.6. In the complexified version in variables ( ζ , ζ ), the complex moment mapof the focus-focus singularity writes locally as f = ζ + ζ , offering a way to compute the quantization whichis analogous to the case of the elliptic singularity, which has an equivalent real moment map.The moment map f ( ζ , ζ ) takes values in C ∼ = R ⊕ i R . If we consider the points ( ζ , ζ ) ∈ C satisfying f ( ζ , ζ ) ∈ Z ⊕ i Z , we obtain a submanifold of C .The flat section σ of the quantization, which satisfies ∇ z σ = 0, has to solve(5.7) X ( σ ) − i < X, λ > ( σ ) = 0 , where λ = (cid:60) (2 iζ dζ ) is the symplectic 1-form (on the complexes). The solution of the differential equation5.7 has to be searched with respect to the complex variables ζ and ζ .5.7. Computation of complex flat sections.
Suppose L is the trivial line bundle C × C . Take complexcoordinates ( ζ , ζ ), which are defined from 5.5 and 5.6. The symplectic form ω is ω = (cid:60) (2 idζ ∧ dζ ) = d (cid:60) (2 iζ dζ )), so it is exact.Recall that the definition of ζ and ζ is ζ = 12 ( x + y + i ( − x + y ))(5.8) ζ = 12 ( − x − y + i ( − x + y ))(5.9)At this point, we would want to equip C with a singular real polarization given by some distribution P such that the vector fields generating it are the ones tangent to the foliation given by the equation f = ζ + ζ = a + bi, a, b ∈ R We know that the equation is equivalent, in the ( x , x , y , y ) coordinates to(5.10) (cid:40) x y + x y = ax y − x y = b , a, b ∈ R And for a = b = 0, we know that the topology of the solutions of 5.10 is that of a pinched torus. And theneighbouring leaves are T by Arnold-Liouville-Mineur theorem.The corresponding Hamiltonian vector fields are EOMETRIC QUANTIZATION VIA COTANGENT MODELS 15 (5.11) ˙ x = − x ˙ x = − x ˙ y = y ˙ y = y , ˙ x = x ˙ x = − x ˙ y = y ˙ y = − y Suppose that we can write ζ in polar coordinates as ζ = r e iφ and ζ = r e iφ . Then, we could try towrite the polarization as P = span { ∂∂φ , ∂∂φ } , but nothing leads us suspect that ζ + ζ = a + bi is equivalent to ∂r ∂t = ∂r ∂t = 0, so this way does not allowa direct computation.The expression of the flat sections, nevertheless, is known (see [Sol15]). If we consider the connection1-form(5.12) Θ = 12 ( x dy − y dx + x dy − y dx ) , which corresponds to a Liouville 1-form, we can check that the unitary section is s = e i ( x y + x y + x y − x y ) .That is the reason why wee look at the focus-focus singularity as an elliptic singularity on the complexes,neglecting completely the associated flat sections equations that would lead to exactly the same solutionsobtained in [Sol15, p. 21]. Namely, we can give them explicitly, without having to compute them throughthese equations.5.8. The focus-focus singularity as a complexified model.
We have just seen that the complexificationof S ∼ = SO (2 , R ) gives SO (2 , C ) ∼ = S × R . On the other hand, in Section 5.5 we showed how the focus-focussingularity can be presented as a cotangent model, a model which precisely starts from the cotangent lift ofthe action of the non-compact Lie group S × R . Our discussion at the end of that section, though, alreadymade it clear that it is not possible to make the cotangent lift compatible with the complexification. If it waspossible to find an action of a compact group (like S ) which, after a cotangent lift and a complexification,provided the infinitesimal generator of the singularity, one would be able to apply ideas of rigidity used in[MM20] to the focus-focus singularity. But it is not expected to find any action of this type because thefocus-focus singularity can not be displayed as a product of two simpler singularities.Notice that this conclusion is coherent with the results of Bolsinov and Izosimov. In [BI19] a particularintegrable system with non-degenerate singularities is built. This system is homeomorphic to the directproduct of a rank 0 focus-focus singularity and a trivial fibration, but is not diffeomorphic to it.Consider a family F t of double-pinched symplectic focus-focus singularities on ( M , ω ) that depend on aparameter t ∈ ( a, b ) ⊂ R . This family can generate a Lagrangian fibration on M := M × ( a, b ) × S , whichis endowed with the symplectic structure ω + dt ∧ dφ , where φ ∈ S . Its moment map ˜ F : M → R is givenby ˜ F ( x, t, φ ) = ( F , t ). This fibration has a focus-focus singularity of rank 1 with two critical circles on eachfiber and, by construction, is homeomorphic to the direct product of a rank 0 double-pinched focus-focussingularity and a regular foliation of an annulus by concentric circles.Bolsinov and Izosimov show that there exists an invariant which should be a constant function on fordirect-product-type singularities and also constant for almost direct products. And, on the other hand, theyprove that this invariant, in the case of the M system, is a non-trivial function. Therefore, they provethat the singularity is not diffeomorphic to any almost direct product of the elementary non-degeneratesingularities. Proposal of the new model
The model we propose for geometric quantization, which is simultaneously convenient for the three typesof non-degenerate singularities can be stated as follows.This model is a local model in a neighbourhood of the singularity and identifies the local normal forms ofthe singularity with the complexified model (which is the only local normal form). Building up on this ideawe define:
Definition 6.1.
The geometric quantization of non-degenerate integrable systems is locally zero in a neigh-bourhood of a non-degenerate singular point.As a consequence, we obtain:
Proposition 6.2.
The geometric quantization of an integrable system with only non-degenerate singularpoints is locally given by the number of regular Bohr-Sommerfeld leaves in the neighbourhood of any singularpoint.Remark . Locally, in a neighbourhood of a singular point, this quantization coincides with the count ofinteger points in the interior of the Delzant polytope close to an elliptic vertex.6.1.
Extending the new model from local to global: The almost toric case.
A system is calledalmost toric if it admits only singularities with elliptic and focus-focus components. These include toricmanifolds and semitoric manifolds (which admit a global circle action). Let us describe how to globalize ournew geometric quantization model.In order to make this construction global, we think of the sheaf-theoretical approach as a puzzle con-struction where the manifold comes endowed with an adapted covering admitting local data (sections andpolarization).We just need to replace the neighbourhood of a singularity by a piece of a regular torus which is so closethat the substitution can be done smoothly (see Figure 3).
Figure 3.
At practice, the model replaces a neighbourhood of the singular point in thesingular fiber by another neighbourhood, like pieces of a puzzle. From left to right, thepieces represent a single focus-focus singularity, a regular torus, a hyperbolic singularityand a double focus-focus singularity.
EOMETRIC QUANTIZATION VIA COTANGENT MODELS 17
The attachment of a new piece keeps the global smoothness of the fiber, its topology only changes in asmall neighbourhood of the singularity (see Figure 4).
Figure 4.
Examples of 4 possible attachments.This piece interchange achieves the desingularization. It is followed by a small perturbation of the entirefiber (see Figure 5), which moves the image of moment map out of the integer value so that it is does notcorrespond to a Bohr-Sommerfeld leaf any more.
Figure 5.
Artistic representation of the perturbation of the desingularized fiber.Understanding the sheaf cohomology computation in [MPS20] under the computation kit provided bytwo of the same authors in [MP15] we can simply reconstruct the sheaf computation in the focus-focus casereplacing the ”puzzle piece” containing the focus-focus singularities by an elliptic singularity. In practice,this will behave as replacing the puzzle by a red piece (regular cylinder) and gluing it back to the sheaf. Inthe picture above other possibilities are considered: Taking a yellow piece would add an extra focus-focussingularity to the leaf so it is of no interest. Replacing the original focus-focus piece by the green piece would yield a sphere as leaf and this would imply, in particular, that the singularity is degenerate as it happens inthe degenerate Lagrangian spheres appearing in some Gelfand-Ceitlin systems (see [BMZ18]). This situationhas to be discussed, as here we are considering systems with only non-degenerate singularities.Why does cohomology glue back well? The focus-focus singularity is isolated and the topology of thecomplementary is glued back to a cylinder. So the pieces of the puzzle will glue back normally as in thecomputation of the sheaf cohomology in a neighbourhood of the torus. This is done using the Mayer-Vietorisformulae in [MP15]. If this torus lies over an integer point of the lattice this would add a Bohr-Sommerfeldleaf to the computation. To make the computation compatible with the elliptic result we should perturbslightly the torus so it is not aligned over an integer point of the lattice as pictured above.A way to avoid doing the perturbation of the torus would be to adopt a more sophisticated (albeit standardin the quantization jargon) model by using the metaplectic correction. This would add back the points tothe computation.Applying this mantra to the set-up of [MPS20] we obtain a simplification of the quantization formulae.Theorem 2.7 now becomes:
Theorem 6.4.
The new geometric quantization of a saturated neighborhood of a focus-focus fiber with n nodes is . This computation can be made global using the Mayer-Vietoris formulae in [MPS20] to obtain a newquantization model for almost toric manifolds. Theorem 2.8 becomes:
Theorem 6.5.
For a -dimensional closed almost toric manifold M , with n r regular Bohr–Sommerfeld fibersand n f focus-focus Bohr–Sommerfeld fibers: Q ( M ) ∼ = C n r . Examples.
Example . The quantization of a K3 surface.
In [MPS20] the authors apply Kostant’s model to singularities of focus-focus type to compute the coho-mology groups associated to the real geometric quantization of a neighborhood of a focus-focus fiber of a4-dimensional semitoric integrable system. They conclude that the first cohomology group is trivial but thesecond is not, since it is infinite dimensional if the singular fiber is Bohr–Sommerfeld (Theorem 2.7). As anapplication of Theorem 2.8 in a particular physical example, they analyze the effect of nodal trades [LS10] inthe real quantization of a toy model of the spin-spin system of [SZ99], which has the product of two spheresand is a toric manifold, and a in a K3 surface, which is an almost toric manifold.The spin-spin system can be represented by its Delzant polytope, see Figure 6, and when a nodal tradeis performed in one of its vertices, one obtains a semitoric system. (cid:115) (cid:115)(cid:115)(cid:115)(cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:0)(cid:0)(cid:9) (cid:45) (cid:113)(cid:115) (cid:115)(cid:115) (cid:115)(cid:113) (cid:113) (cid:113)(cid:113) (cid:113) (cid:113)(cid:113) (cid:113)(cid:101)
Figure 6.
Nodal trade on S × S .In the other example, a K3 surface is built from two copies of a symplectic and toric blowup of the complexprojective plane at 9 different points, also applying nodal trades to all of their elliptic-elliptic singular fibersand taking their symplectic sum along the symplectic tori corresponding to the preimage of the boundary oftheir respective bases. This construction provides a K3 surface with up to 24 Bohr–Sommerfeld focus-focusfibers (see Figure 7). EOMETRIC QUANTIZATION VIA COTANGENT MODELS 19
The construction is fully detailed in [MPS20], but we remark some of its more noticeable points. For theK3 surface with up to 24 Bohr–Sommerfeld focus-focus fibers, one obtains that its quantization, with theprevious model, is Q ( K ∼ = C ⊕ (cid:77) j ∈{ ,..., } C ∞ ( R ; C ) . Then, the real geometric quantization of the K3 surface is essentially different from the K¨ahler case,which is always finite dimensional. In the particular example we are considering, the dimension of its K¨ahlerquantization is 12 c ( L ) + 2 = 12 (2 ·
24 + 2 ·
24) + 2 = 50 . But, on the other hand, even when the real geometric quantization is finite dimensional, for this symplectic K Q ( K ∼ = C , which is still different from C . This difference is due to how real andK¨ahler quantization behave with respect to singular Bohr–Sommerfeld elliptic fibers of a toric manifold.In the real case, those fibers (lying on the the boundary of the Delzant polytope) do not contribute tothe geometric quantization [Ham10], while they do contribute in the K¨ahler case. The advantage of thequantization model presented in [MPS20] is that its finite part coincides with K¨ahler quantization. Figure 7.
K3 surface as a singular fiber bundle over the sphere. Mayuko Yamashita ob-served that the preimage of the equator of the sphere contains 12 regular Bohr-Sommerfeldfibers, obtained from the 12 elliptic singular Bohr-Sommerfeld fibers which become regularafter the symplectic sum.We observe that the nodal trading, which transforms toric into semitoric manifolds, makes the modelsprovided in [MPS20] not compatible with the functorality of the quantization. By functorality, we mean asmooth application, which is not achieved for the K Example . The quantization of the spherical pendulum
The classical physical model of the spherical pendulum is an integrable system on the cotangent bundle ofthe sphere which contains a focus-focus singular fiber. Its natural phase space is the cotangent bundle T ∗ S and, while spherical coordinates are the convenient setting to study the dynamics, Cartesian coordinatesare more appropriated to analyze the focus-focus singularity. In coordinates q = ( q , q , q ) for position and p = ( p , p , p ) for momentum, and normalizing the invariant physical quantities, the Hamiltonian of thesystem is(6.1) H ( q, p ) = (cid:104) p, p (cid:105) q The angular momentum in the q direction, defined as L = q p − q p is the other first integral of thesystem, i.e., satisfies { H, L } = 0 and both are independent almost everywhere. There are two singularities in the pendulum system, corresponding to q A = (0 , , −
1) and q B = (0 , , − q B , one can see that the lower terms of the Hamiltonian are of the form ˜ H = ( p q + p q ) and the angularmomentum is still ˜ L = q p − q p . Therefore, and in view of Theorem 3.9, the singularity is of focus-focustype.The quantization of the spherical pendulum (see Figure 8) is based on two global actions A , A . The firstis a continuous function of the energy H and the angular momentum L , and the latter is directly A = L . TheBohr-Sommerfeld conditions A = 2 πn (cid:126) and A = 2 πm (cid:126) , with n and m integers, give a quantization of theaction functions A , A . The pairs ( n, m ) of integers label the basic states of a basis { σ n,m } of distributionsections of the prequantization line bundle of the space of quantum states H of the system. h(cid:96) Figure 8.
Integer points in the image of the moment map in the case of the spherical pendulum.7.
Conclusions
The definition of the geometric quantization of the focus-focus singular leaf as the number of Bohr-Sommerfeld orbits has advantages with respect to other definitions. From the physical point of view, themost important advantage is that it associates a discrete set to a model which is semilocally compact. Thisis much more natural than to associate to it a continuous set. On the other hand, since the complexificationsof the real, elliptic and hyperbolic singularities are equivalent, it is reasonable that the quantizations ofthe focus-focus and hyperbolic cases essentially coincide with the quantization of the elliptic cases. Thenodal trade, which can be thought as a smooth operator, allows the compatibility of this definition with theproperties of the Delzant polytope. Our definition, also, can be used to explicitly compute the quantizationin particular examples, such as the K π (cid:126) = 8 /n ,where n is the first component of the lattice in the image of the moment map and takes some integer values.This is a really strong condition on Planck’s constant which is not likely to be satisfied in physics, since EOMETRIC QUANTIZATION VIA COTANGENT MODELS 21
Figure 9.
This model replaces a neighbourhood of the focus-focus singularity by neigh-bourhood of a regular torus. It can be done smoothly by a desingularization and a pertur-bation.Planck’s constant cannot equal a set of different values. Although for some computations Planck’s constantis normalized in the sense that 2 π (cid:126) = 1 (see [HM10] or [Ham10]), normalization would still not avoid thisphysical impossibility. In consequence, the Bohr-Sommerfeld conditions are not satisfied in the focus-focuspoint and it makes sense that they do not contribute to the quantization. References [BF04] A. V. Bolsinov and A. T. Fomenko.
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Laboratory of Geometry and Dynamical Systems, Department of Mathematics, Universitat Polit`ecnica deCatalunya and BGSMath, Barcelona, Spain
Email address : [email protected] Laboratory of Geometry and Dynamical Systems Department of Mathematics & Institut de Matem`atiques dela UPC-BarcelonaTech (IMTech) ), Universitat Polit`ecnica de Catalunya & Centre de Recerca Matem`atica,Barcelona, Spain, and, IMCCE, CNRS-UMR8028, Observatoire de Paris, PSL University, Sorbonne Universit´e, 77Avenue Denfert-Rochereau, 75014 Paris, France
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