Shifting Operators in Geometric Quantization

: The original Bohr-Sommerfeld theory of quantization did not give operators of transitions between quantum quantum states. This paper derives these operators, using the ﬁrst principles of geometric quantization.


Introduction
Even though the Bohr-Sommerfeld theory was very successful in predicting some physical results, it was never accepted by physicists as a valid quantum theory in the same class as the Schrödinger theory or the Bargmann-Fock theory. The reason for this was that the original Bohr-Sommerfeld theory did not provide operators of transition between quantum states. The need for such operators in the Bohr-Sommerfeld quantization was already pointed out by Heisenberg [1]. The aim of this paper is to derive operators of transition between quantum states in the Bohr-Sommerfeld theory, which we call shifting operators, from the first principles of geometric quantization.
The first step of geometric quantization of a symplectic manifold (P, ω) is called prequantization. It consists of the construction of a complex line bundle π : L → P with connection whose curvature form satisfies a prequantization condition relating it to the symplectic form ω. A comprehensive study of prequantization, from the point of view of representation theory, was given by Kostant in [2]. The work of Souriau [3] was aimed at quantization of physical systems, and studied a circle bundle over phase space. In Souriau's work, the prequantization condition explicitly involved Planck's constant h. In [4], Blattner combined the approaches of Kostant and Souriau by using the complex line bundle with the prequantization condition involving Planck's constant. Since then, geometric quantization has been an effective tool in quantum theory.
We find it convenient to deal with connection and curvature of complex line bundles using the theory of principal and associated bundles [5]. In this framework, the prequantization condition reads where β is the connection 1-form on the principal C × -bundle π × : L × → P associated to the complex line bundle π : L → P, and C × is the multiplicative group of nonzero complex numbers. The aim of prequantization is to construct a representation of the Poisson algebra (C ∞ (P), { , }, ·) of (P, ω) on the space of sections of the line bundle L. Each Hamiltonian vector field X f on P lifts to a unique C × -invariant vector field Z f on L × that preserves the principal connection β on L × . If the vector field X f is complete, then it generates a 1-parameter group e tX f of symplectomorphisms of (P, ω). Then, the vector field Z f is complete and it generates a 1-parameter group e tZ f of connection preserving diffeomorphisms of the bundle (L × , β), called quantomorphisms, which cover the 1-parameter group e tX f . The term quantomorphism was introduced by Souriau [3] in the context where his Planck's constant h divided by 2π, is a symmetric operator on the Hilbert space H of square integrable sections of L. The operator P f is self adjoint if X f is complete. The whole analysis of prequantization is concerned with global Hamiltonian vector fields. Since every vector field on (P, ω) that preserves the symplectic form is locally Hamiltonian, it is of interest to understand how much of prequantization can be extended to this case. In particular, we are interested in the case where the locally Hamiltonian vector field is the vector field X ϑ of the integer angle variable ϑ that is defined up to an additive term n, where n ∈ Z. For a globally Hamiltonian vector field X f , e tZ f * σ = e −2πi t f /h e t liftX f * σ, (2) where e t liftX f * σ is the horizontal transport of section σ by parameter t along integral curves of X f . Replacing f by a multivalued function ϑ, defined up to an additive n, yields the multivalued expression e tZ ϑ * σ = e −itϑ/h e t liftX ϑ * σ.
We observe that, for t = h, Equation (3) The shifting operator a X ϑ = e h Z ϑ * = e −i ϑ e h liftX ϑ * is an operator on H, which shifts the support of σ ∈ H by h along the integral curve of X ϑ . If the vector field X ϑ is complete, then a n X ϑ = e nhZ ϑ * for every n ∈ Z. Our results provide an answer to Heisenberg's criticism that in Bohr-Sommerfeld theory there are not enough operators to describe transitions between quantum states [1].
Superficially, the shifting operator a X θ = e i θ e h lift X θ * , see Equation (5), appears to be a quantization of an angle θ = 2πϑ. It depends on θ and has the factor e −iθ considered by Dirac [8]. However, the factor e h -liftX θ * , describing the parallel translation by halong integral curves of X θ , makes it nonlocal in the phase space. Therefore, a X θ cannot satisfy local commutation relations with any local quantum variable that is described by a differential operator. Hence, it cannot be the canonical conjugate of the corresponding action operator, or any other operator, which is local in the phase space.
In our earlier papers [9][10][11][12], we followed an algebraic analysis, similar to that used by Dirac [8], supplemented by heuristic guesses about the behaviour of the shifting operators at the points of singularity of the polarization. In particular, we assumed that a X ϑ vanishes on the states concentrated on a set of limit points of e t X ϑ (p) as t → ∞. In the present paper, we derive shifting operators in the framework of geometric quantization, and extend our result to cases with a variable rank polarization.
The second stage in geometric quantization consists of the choice of a polarization, which is an involutive complex Lagrangian distribution F on the phase space. Suppose that P is the cotangent bundle space of the configuration space. In this case, the choice of F containing the vertical directions, leads the quantum mechanics of Schrödinger. If F leads to complex analytic structure on P, we have the Bargmann-Fock theory. If F is spanned by the Hamiltonian vector fields of a completely integrable system, we have Bohr-Sommerfeld theory. Each of these theories have specific structure, which is helpful in formulation and solving problems. In the following, we restrict our investigation to the Bohr-Sommerfeld theory in order to emphasize its membership in the class of quantum theories corresponding to different polarization.
A common problem in arising in quantum theories is occurrence of singularities. Usually, one studies the geometric structure of the theory in the language of differential geometry of smooth manifolds, and then investigates the structure of singularities separately. The theory of differential spaces, introduced by Sikorski [13,14], is a powerful tool in the study of the geometry of spaces with singularities [15]. The main singularity encountered here corresponds to the fact that the polarization F spanned by the Hamiltonian vector fields of a completely integrable system does not have constant rank. This singularity is so well known that we do not have to use the language of differential spaces to get results. It should be noted that the results in [9,11] rely on the theory of differential spaces.
In conclusion, it should be mentioned that the scientists, who used visual presentation of the Bohr-Sommerfeld spectra in terms of dots on the space of the action variables, are familiar with handling shifting operators. The line segments joining two dots corresponding to quantum states represent the shifting operators between these states.
To make the paper more accessible to the reader, we have provided an introductory section with a comprehensive review of geometric quantization. Experts may omit this section and proceed directly to the next section on Bohr-Sommerfeld theory.

Elements of Geometric Quantization
Let (P, ω) be a symplectic manifold. Geometric quantization can be divided into three steps: prequantization, polarization, and unitarization.

Principal Line Bundles with a Connection
We begin with a brief review of connections on complex line bundles. Let C × denote the multiplicative group of nonzero complex numbers. Its Lie algebra c × is isomorphic to the abelian Lie algebra C of complex numbers. Different choices of the isomorphism ι : C → c × lead to different factors in various expressions. Here, to each c ∈ C we associate the 1-parameter subgroup t → e 2πi tc of C × . In other words, we take The prequantization structure for (P, ω) consists of a principal C × bundle π × : L × → P and a c × -valued C × -invariant connection 1-form β satisfying where h is Planck's constant. The prequantization condition requires that the cohomology class [− 1 h ω] is integral, that is, it lies in H 2 (P, Z), otherwise the C × principal bundle π × : L × → P would not exist.
Let Y c be the vector field on L × generating the action of e 2πi tc on L × . In other words, the 1-parameter group e tY c of diffeomorphisms of L × generated by Y c is The connection 1-form β is normalized by the requirement For each c = 0, the vector field, Y c spans the vertical distribution ver TL × tangent to the fibers of π × : L × → P. The horizontal distribution hor TL × on L × is the kernel of the connection 1-form β, that is, hor TL × = ker β.
The vertical and horizontal distributions of L × give rise to the direct sum TL × = ver TL × ⊕ hor TL × , which is used to decompose any vector field Z on L × into its vertical and horizontal components, Z = ver Z + hor Z. Here, the vertical component ver Z has range in ver TL × and the horizontal component has range in hor TL × . If X is a vector field on P, the unique horizontal vector field on L × , which is π × -related to X, is called the horizontal lift of X and is denoted by lift X. In other words, lift X has range in the horizontal distribution hor TL × and satisfies Claim 1. A vector field Z on L × is invariant under the action of C × on L × if and only if the horizontal component of Z is the horizontal lift of its projection X to P, that is, hor Z = lift X and there is a smooth function κ : Proof. Since the direct sum TL × = ver TL × ⊕ hor TL × is invariant under the C × action on L × , it follows that the vector field Z is invariant under the action of C × if and only if hor Z and ver Z are C × -invariant. However, hor Z is C × invariant if Tπ × • hor Z = X • π × for some vector field X on P, that is, hor Z = lift X. However, this holds by definition. On the other hand, the vertical distribution ver TL × is spanned by the vector fields Y c for c ∈ C. Hence, ver Z is C × -invariant if and only if for every fiber L × p the restriction of ver Z to L × p coincides with the restriction of Y c to L × p for some c ∈ C, that is, there is a smooth complex valued function κ on P such that c = κ(p).
Let U be an open subset of P. A local smooth section τ : U ⊆ P → L × of the bundle π × : L × → P gives rise to a diffeomorphism where b ∈ C × is the unique complex number such that × = τ(p)b. In the general theory of principal bundles the structure group of the principal bundle acts on the right. In the theory of C × principal bundles, elements of L × are considered to be one-dimensional frames, which are usually written on the right, see [2]. The diffeomorphism η τ is called a trivialization of L × |U . It intertwines the action of C × on the principal bundle L × with the right action of C × on U × C × , given by multiplication in C × . If a local section σ : U → L of π : L → P is nowhere zero, then it determines a trivialization η τ : L × |U → U × C × . Conversely, a local smooth section τ such that η τ is a trivialization of L × may be considered as a local nowhere zero section of L.
In particular, for every c ∈ C, which is identified with the Lie algebra c × of C × , Equation (7) gives e t Y c • τ = e 2πi tc τ. Differentiating with respect to t and then setting t = 0 gives For every smooth complex valued function κ : P → C, consider the vertical vector field Y κ such that Y κ ( × ) = Y κ(π × ( × )) for every × ∈ L × . The vector field Y κ is complete and the 1-parameter group of diffeomorphisms it generates is e t Y κ : L × → L × : × → × e 2πi tκ(π × ( × )) .
For every smooth section τ of the bundle π × , we have e t Y κ • τ = e 2πi tκ τ so that Let X be a vector field on P and let lift X be its horizontal lift to L × . The local 1-parameter group e t lift X of local diffeomorphisms of L × generated by lift X commutes with the action of C × on L × . For every × , e t lift X ( × ) is called parallel transport of × along the integral curve of X starting at p = π × ( × ). For every p ∈ P the map e t lift X sends the fiber L × p to the fiber L e tX (p) . There are several equivalent definitions of covariant derivative of a smooth section of the bundle π × in the direction of a vector field X on P. We use the following one. The covariant derivative of the smooth section τ of the bundle π × : L × → P in the direction X is Claim 2. The covariant derivative of a smooth local section of the bundle π × : L × → P in the direction X is given by Proof. For every p ∈ P, we have The definition of the connection 1-form β and Equation (13) yield ver (Tτ(X(p)) = Y β|Tτ • X (τ(p)) = 2πi β|Tτ • X τ(p). Hence, which is equivalent to Equation (15).

Associated Line Bundles
The complex line bundle π : L → P associated to the C × principal bundle π × : L × → P is defined in terms of the action of C × on (L × × C) given by Since the action Φ is free and proper, its orbit space L = ( The left action of C × on C gives rise to the left action for every × ∈ L × , every a, b ∈ C × and every c ∈ C. The projection map π × : L × → P induces the projection map

Claim 3.
A local smooth section σ : U → L of the complex line bundle π : L → P corresponds to a unique mapping σ : L × |U → C such that for every p ∈ U and every × ∈ L × Since the action of C × on L × p is free and transitive, it follows that the which satisfies Equation (20). For every b ∈ C × , Equations (18) and (20) imply that If τ : U → L × is a local smooth section of the bundle π × : L × → P, then for every p ∈ P we have σ(p) = [(τ(p), σ (τ(p)))] or σ = [(τ, σ • τ)] suppressing the argument p. The function ψ = σ • τ : U → C is the coordinate representation of the section τ in terms of the trivialization η τ : L × |U → U × C. Let Z be a C × -invariant vector field on L × . Then, Z is π × -related to a vector field X on P, that is, Tπ × • Z = X • π × . We denote by e tX and e tZ the local 1-parameter groups of local diffeomorphisms of P and L × generated by X and Z, respectively. Because the vector fields X and Z are π × -related, we obtain π × • e t Z = e t X • π × . In other words, the flow e t Z of Z covers the flow e t X of X. The local group e t Z of automorphisms of the principal bundle L × act on the associated line bundle L by which holds for all = [( × , c)] for which e t Z ( × ) is defined.
This shows that e t Z covers e t X . Finally, for every = [( × , c)] ∈ L and every b ∈ C × since Z is a C × -invariant vector field on L × . Therefore, This shows that e t Z is a local group of automorphisms of the line bundle π : L → P.
If Z = hor X, then e t lift X ( × ) is parallel transport of × along the integral curve e t X (p) of X starting at is parallel transport of ∈ L along the integral curve e t X (p) of X starting at p. The covariant derivative of a section σ of the bundle π : L → P in the direction of the vector field X on P is Since e −t lift X maps π −1 (e t X ) onto π −1 (p), Equations (22) and (23) are consistent with the definitions in [5].

Theorem 1.
Let σ be a smooth section of the complex line bundle π : L → P and let X be a vector field on P.
Here, L X is the Lie derivative with respect to the vector field X.

Prequantization
Let π : L → P be the complex line bundle associated to the C × principal bundle π × : L × → P. The space S ∞ (L) of smooth sections of π : L → P is the representation space of prequantization. Since C × ⊆ C, we may identify L × with the complement of the zero section in L. With this identification, if σ : U → L is a local smooth section of π : L → P, which is nowhere vanishing, then it is a section of the bundle π × |L × |U : L × |U → U.
where h is Planck's constant.
Proof. The vector field Z on L × preserves the connection 1-form, that is, Since hor Z β = 0, it follows that Z β = ver Z β. The C × -invariance of Z and β imply the C × -invariance of ver Z β. Hence, ver Z β pushes forward to a function π * (ver Z β) ∈ C ∞ (P). Thus, the right hand side of Equation (27) reads By definition Y c β = c, for every c ∈ c. This implies Thus, the left hand side of Equation (27) reads The quantization condition (7) together with (28) and (29) allow us to rewrite Equation (27) in the form Equation (30) shows that X is the Hamiltonian vector field of the smooth function on P. We write X = X f . This implies that We still have to determine the vertical component ver Z of the vector field Z. For each × ∈ L × there is a c ∈ c such that ver Z = Y c . Since Y c is tangent to the fibers of the C × principal bundle π × : L × → P, the element c of c depends only on π × ( × ) = p ∈ P. Therefore, by Equation (31). In other words, for every point × ∈ L × we have ver where p = π × ( × ). Thus, we have shown that Reversing the steps in the above argument proves the converse.
To each f ∈ C ∞ (P), we associate a prequantization operator where e t Z f is the action of e t Z f : L × → L × on L, see (22). Note that the definition of covariant derivative in Equation (23) is defined in terms of the pull back ( e t Z f ) * σ of the section σ by e t Z f , while the prequantization operator in (34) is defined using the push forward ( e t Z f ) * σ of σ by e t Z f .

Theorem 3.
For every f ∈ C ∞ (P) and each σ ∈ S ∞ (L) Proof. Since the horizontal distribution on L × is C × -invariant and the vector field Y c generates multiplication on each fiber of π × by e 2πi c , it follows that and Since Since π • e t Y − f /h = π • id P , where id P is the identity map on P, it follows that Therefore,

Equations (37), (38) and (41) imply Equation (35).
A Hermitian scalar product | on the fibers of L that is invariant under parallel transport gives rise to a Hermitian scalar product on the space S ∞ (L) of smooth sections of L. Since the dimension of (P, ω) is 2k, the scalar product of the smooth sections σ 1 and σ 2 of L is The completion of the space S ∞ c (L) of smooth sections of L with compact support with respect to the norm σ = (σ|σ) is the Hilbert space H of the prequantization representation.

Claim 4.
The prequantization operator P f is a symmetric operator on the Hilbert space H of square integrable sections of the line bundle π : L → P and satisfies Dirac's quantization commutation relations for every f , g ∈ C ∞ (P). Moreover, the operator P f is self adjoint if the vector field X f on (P, ω) is complete.
A complex distribution F ⊆ T C P = C ⊗ TP on a symplectic manifold (P, ω) is Lagrangian if for every p ∈ P, the restriction of the symplectic form ω p to the subspace F p ⊆ T C p P vanishes identically, and rank C F = 1 2 dim P. If F is a complex distribution on P, let F be its complex conjugate. Let A polarization of (P, ω) is an involutive complex Lagrangian distribution F on P such that D and E are involutive distributions on P. Let C ∞ (P) F be the space of smooth complex valued functions of P that are constant along F, that is, The polarization F is strongly admissible if the spaces P/D and P/E of integral manifolds of D and E, respectively, are smooth manifolds and the natural projection P/D → P/E is a submersion. A strongly admissible polarization F is locally spanned by Hamiltonian vector fields of functions in Let F be a strongly admissible polarization on (P, ω). The space S ∞ F (L) of smooth sections of L that are covariantly constant along F is the quantum space of states corresponding to the polarization F.
The space C ∞ F (P) of smooth functions on P, whose Hamiltonian vector field preserves the polarization F, is a Poisson subalgebra of C ∞ (P). Quantization in terms of the polarization F leads to quantization map Q, which is the restriction of the prequantization map In other words, Quantization in terms of positive strongly admissible polarizations such that E ∩ E = {0} lead to unitary representations. For other types of polarizations, unitarity may require additional structure.

Historical Background
Consider the cotangent bundle T * Q of a manifold Q. Let π Q : T * Q → Q be the cotangent bundle projection map. The Liouville 1-form α Q on T * Q is defined as follows. For each q ∈ Q, p ∈ T * q Q and u p ∈ T p (T * Q), The exterior derivative of α Q is the canonical symplectic form dα Q on T * Q.
Assume that the functions H 0 , . . . , H k−1 are independent on a dense open subset of T * Q. For each p ∈ T * Q, let M p be the orbit of the family of Hamiltonian vector fields {X H 0 , . . . , X H k−1 } passing through p. This orbit is the largest connected immersed submanifold in T * Q with tangent space T p (M p ) equal to span R {X H 0 (p ), . . . , X H k−1 (p )}. The integral curve t → e t X H 0 (p) of X H 0 starting at p is contained in M p . Hence, knowledge of the family {M p p ∈ T * Q} of orbits provides information on the evolution of the Hamiltonian system with Hamiltonian H 0 .
Bohr-Sommerfeld theory, see [16,17], asserts that the quantum states of the completely integrable system (H 0 , . . . , H k−1 , T * Q, dα Q ) are concentrated on the orbits M ∈ {M p p ∈ T * Q}, which satisfy the Bohr-Sommerfeld Condition: For every closed loop γ : S 1 → M ⊆ T * Q, there exists an integer n such that where h is Planck's constant.
This theory applied to the bounded states of the relativistic hydrogen atom yields results that agree exactly with the experimental data [17]. Attempts to apply Bohr-Sommerfeld theory to the helium atom, which is not completely integrable, failed to provide useful results. In his 1925 paper [1], Heisenberg criticized Bohr-Sommerfeld theory for not providing transition operators between different states. At present, the Bohr-Sommerfeld theory is remembered by physicists only for its agreement with the quasi-classical limit of Schrödinger theory. Quantum chemists have never stopped using it to describe the spectra of molecules.

Geometric Quantization in a Toric Polarization
To interpret Bohr-Sommerfeld theory in terms of geometric quantization, we consider a set P ⊆ T * Q consisting of points p ∈ T * Q where X H 0 (p), . . . , X H k−1 (p) are linearly independent and the orbit M p of the family {X H 0 , . . . , X H k−1 } of Hamiltonian vector fields on (T * Q, dα T * Q ) is diffeomorphic to the k torus T k = R k /Z k . We assume that P is a 2k-dimensional smooth manifold and that the set B = {M p p ∈ P} is a quotient manifold of P with smooth projection map ρ : P → B. This implies that the symplectic form dα Q on T * Q restricts to a symplectic form on P, which we denote by ω. Let D be the distribution on P spanned by the Hamiltonian vector fields X H 0 , . . . , Since the symplectic form dα Q on T * Q is exact, we may choose a trivial prequantization line bundle Let L × be the restriction of L × T * Q to P and let α be the 1-form on P, which is the restriction of α Q to P, that is, α = α Q |P . Then, L × = C × × P is a principal C × bundle over P with projection map associated to the principal bundle π × is also trivial. Prequantization of this system is obtained by adapting the results of Section 2.
Since integral manifolds of the polarization D are k-tori, we have to determine which of them admit nonzero covariantly constant sections of L. Proof. Suppose that an integral manifold M of D admits a nowhere zero section of L |M . Since σ is nowhere zero, it is a section of L × |M . Let γ : S 1 → M be a loop in M. For each t ∈ S 1 , letγ(t) ∈ T γ(t) M be the tangent vector to γ at t. Since σ is covariantly constant along M, Claim 2 applied to the section for every p ∈ P and every X(p) ∈ T p M. Taking p = γ(t) and X(p) =γ(t) gives Since Hence, Equation (48) is equivalent to If γ bounds a surface Σ ⊆ M, then Stokes' theorem together with Equation (47) and the quantization condition (7) yield because M is a Lagrangian submanifold of (P, ω). Thus, ln b(γ(2π)) = ln b(γ(0)), which implies that the nowhere zero section σ is parallel along γ. If γ does not bound a surface in M, but does satisfy the Bohr-Sommerfeld condition γ * α Q = nh (47) with α Q replaced by its pull back α to P, then Hence, b(γ(2π)) = b(γ(0)) and the nowhere zero section σ is parallel along γ.
Note that the manifolds M that satisfy Bohr-Sommerfeld conditions (47) are k-dimensional toric submanifolds of P. We call them Bohr-Sommerfeld tori. Since Bohr-Sommerfeld tori have dimension k = 1 2 dim P, there is no non-zero smooth section σ 0 : P → L that is covariantly constant along D. For our toral polarization F = D C , the space of smooth functions on P that are constant along F, see Equation (44), is C ∞ F (P) = ρ * (C ∞ (B)), see Lemma A3. For each f ∈ C ∞ F (P), the Hamiltonian vector field X f is in D, that is, ∇ X f σ M = 0 for every basic state σ M ∈ S. Hence, the prequantization and quantization operators act on the basic states σ M ∈ S by multiplication by f , that is, Note that f |M is a constant because f ∈ C ∞ F (P). For a general quantum state σ = ∑ M∈S c M σ M ∈ S, We see that, for every function f ∈ C ∞ (P), each basic quantum state σ M is an eigenstate of Q f corresponding to the eigenvalue f |M . Since eigenstates corresponding to different eigenvalues of the same symmetric operator are mutually orthogonal, it follows that the basis {σ M } of S is orthogonal. This is the only information we have about scalar product in S. Our results do not depend on other details about the scalar product in S.

Shifting Operators
We begin by assuming that P = T * T k with canonical coordinates (p, θ) = (p 1 , ..., p k , θ 1 , ..., θ k ) where, for each i = 1, ..., k, θ i is the canonical angular coordinate on the ith torus and p i is the conjugate momentum. The symplectic form is In this case, action-angle coordinates (j, ϑ) = (j 1 , . . . , j k , ϑ 1 , . . . , ϑ k ) are obtained by rescaling the canonical coordinates so that, for every i = 1, ..., k, we have j i = 2π p i and ϑ i = θ i /2π. Moreover, the rescaled angle coordinate ϑ i : T * T k → T = R/Z is interpreted as a multi-valued real function, the symplectic form and the toric polarization of (P, ω) is given by In terms of action-angle coordinates, the Bohr-Sommerfeld tori in T * T k are given by equation where n = (n 1 , ..., n k ) ∈ Z k . For each n ∈ Z k , we denote by T k n the corresponding Bohr-Sommerfeld form a basis in the space S of quantum states.
For each i = 1, ..., k, the vector field ∂ ∂j i is transverse to D and − ∂ is the Hamiltonian vector field of ϑ i . In the following, we write to describe the actual vector field X i without referring to its relation to the action angle coordinates (j, ϑ). Equation (36) in Section 2.1, for f = ϑ i , is multi-valued because the phase factor is multi-valued, and is well defined.
Let η i be a unique representative of ϑ i|W with values in (a i , b i ). With this notation, The restriction to W of the vector field X ϑ i is the genuinely Hamiltonian vector field of η i , namely, The vector field is well defined. Equation (36) yields e t Z η i = e −2πi η i /h e t lift X η i . Hence, If we make another choice of intervals (a i , b i ) in R such that 0 < b i − a i < 1 and let Since we can cover P by open contractible sets defined in Equation (57), we conclude that e h Z X i is well defined by Equation (56) and depends only on the vector field X i .
Consequently, there exists a connection preserving automorphism A X i : L × → L × such that, if × ∈ L |W × , where W ⊆ P is given by Equation (57), then Claim 6. The connection preserving automorphism A X i : L × → L × , defined by Equation (62) depends only on the vector field X i and not the original choice of the action-angle coordinates.
Proof. If (j 1 , . . . , j k , ϑ 1 , . . . , ϑ k ) is another set of action-angle coordinates then where the matrices A = (a il ) and B = (b il ) lie in Sl(k, Z) and B = (A −1 ) T . In the new coordinates, Clearly, To compare the phase factor entering Equation (55), we consider an open contractible set W ⊆ P. As before, for each i = 1, ..., k, choose a single-valued representative η i of (ϑ i ) |W . Then, where each l j is an integer and thus l = ∑ k j=1 b ij l j is also an integer. Hence, where b i1 , . . . , b ik are integers. Since l is constant, Therefore, which shows that the automorphism A X ϑ i : L × → L × depends on the vector field X i and not on the action angle coordinates in which it is computed.

Claim 7.
For each i = 1, ..., k, the symplectomorphism e hX i : P → P, where h is Planck's constant, preserves the set B of Bohr-Sommerfeld tori in P.
Proof. Since X i is complete, e tX i : P → P is a 1-parameter group of symplectomorphisms of (P, ω). Hence, e h X i : P → P is well defined. By Equation (52), j i |T k n = n i h for every Bohr-Sommerfeld torus T k n , where n = (n 1 , . . . , n k ).
This implies that, for every l = i, (e tX i ) * (j l dϑ l ) = j l dϑ l and (e tX i ) * (j i dϑ i ) = (j i − t)dϑ i . Therefore, if j = nh, then (e h X i ) * j l = j l = n l , if l = i, and (e tX i ) * j i = (j i − h) = (n i − 1)h if = i. This implies that e hX ϑ i (T k n ) is a Bohr-Sommerfeld torus.
We denote by A X i : L → L the action of A X i : L × → L × on L. The automorphism A X i acts on sections of L by pull back and push forward, namely, Since A X i : L × → L × is a connection preserving automorphism, it follows that, if σ satisfies the Bohr-Sommerfeld conditions, then ( A X i ) * σ and ( A X i ) * σ also satisfy the Bohr-Sommerfeld conditions. In other words, ( A X i ) * and ( A X i ) * preserve the space S of quantum states. The shifting operators a X i and b X i , corresponding to X i , are the restrictions to S of ( A X i ) * and ( A X i ) * , respectively. For every n = (n 1 , . . . , n k ) ∈ Z k , Equations (53) and (56) yield In addition, the operators a X i , b X j , for i, j = 1, . . . , k, generate an abelian group A of linear transformations of S into itself, which acts transitively on the space of one-dimensional subspaces of S. Given a non-zero section σ ∈ S supported on a Bohr-Sommerfeld torus, the family of sections {(a n k X k · · · a n 1 X 1 σ) ∈ S n 1 , ...n k ∈ Z} is a linear basis of S, invariant under the action of A. Since A is abelian, there exists a positive, definite Hermitian scalar product · | · on S, which is invariant under the action of A, and such that the basis in (71) is orthonormal. It is defined up to a constant positive factor. The completion of S with respect to this scalar product yields a Hilbert space H of quantum states in the Bohr-Sommerfeld quantization of T * T k . Elements of A extend to unitary operators on H.

General Case of Toral Polarization
Hilbert Space and Operators Let (P, ω) be a symplectic manifold with toroidal polarization D and a covering by domains of action-angle coordinates. If U and U are the domain of the angle-action coordinates (j, ϑ) = (j 1 , . . . , j k , ϑ 1 , . . . , ϑ k ) and (j , ϑ ) = (j 1 , . . . , j k , ϑ 1 , . . . , ϑ k ), respectively, and U ∩ U = ∅, then in U ∩ U we have where the matrices A = (a il ) and B = (b il ) lie in Sl(k, Z) and B = (A −1 ) T . Consider a complete locally Hamiltonian vector field X on (P, ω) such that, for each angle-action coordinates (j, ϑ) with domain U, for some c = (c 1 , ..., c k ) ∈ Z k . Equation (72) shows that in U ∩ U , we have where c i = ∑ k j=1 c j b ji ∈ Z, for i = 1, . . . , k. As in the preceding section, Equation (36) with f = c· ϑ = c 1 ϑ 1 + . . . + c k ϑ k , which is multi-valued, gives e tZ c·ϑ = e −2πi t c·ϑ/h e t liftX , which is multivalued, because the phase factor is multi-valued. As before, if we set t = h, we would get a single-valued expression e h Z c·ϑ = e −2πic·ϑ e h liftX because c 1 , . . . , c k ∈ Z. This would work along all integral curves t → e t X (x) for t ∈ [0, 1], which are contained in U. Now, consider the case when, for x 0 ∈ U, e hX (x) ∈ U and there exists t 1 ∈ (0, h) such that x 1 = e t 1 X (x 0 ) ∈ U ∩ U , where U and U are domains of action-angle variables (j, ϑ) and (j , ϑ ), respectively. Moreover, assume that e tX (x 0 ) ∈ U for t ∈ [0, t 1 ] and e tX (x 1 ) ∈ U for t ∈ [0, h − t 1 ]. Using the multi-index notation, for l × ∈ L × x 0 , we write Let W be a neighborhood of x 1 in P such that U ∩ W and U ∩ W are contractible. For each i = 1, ..., k, let θ i be a single-valued representative of ϑ i as in the proof of Claim 5. Similarly, we denote by η i a single-valued representative of ϑ i . Equation (73) shows that in U ∩ U ∩ W, the functions c 1 η 1 + · · · + c k η k and c 1 η 1 + · · · + c k η k are local Hamiltonians of the vector field X and are constant along the integral curve of X |W . Hence, we have to make the choice of representatives η i and η i so that With this choice, e −2πit 1 (c·ϑ−c ·ϑ )/h = 1, and is well defined. It does not depend on the choice of the intermediate point In the case when m + 1, action-angle coordinate charts with domains U 0 , U 1, ..., U m are needed to reach x m = e h X (x 0 ) ∈ U m from x 0 ∈ U 0 ; we choose . . , x m−1 , we repeat the the argument of the preceding paragraph. We conclude that there is a connection preserving automorphism A X : L × → L × well defined by the procedure given here, and it depends only on the complete locally Hamiltonian vector field X satisfying condition (73). The automorphism A X : L × → L × of the principal bundle L × leads to an automorphism A X of the associated line bundle L. As in Equation (69), the shifting operators corresponded to the complete locally Hamiltonian vector field X are In absence of monodromy, if we have k independent, complete, locally Hamiltonian vector fields X i on (P, ω) that satisfy the conditions leading to Equation (73), then the operators a X i , b X j for i, j = 1, ..., k generate an abelian group A of linear transformations of S. If the local lattice S of Bohr-Sommerfeld tori is regular, then A acts transitively on the space of one-dimensional subspaces of S. This enables us to construct an A-invariant Hermitian scalar product on S, which is unique up to an arbitrary positive constant. The completion of S with respect to this scalar product yields a Hilbert space H of quantum states in the Bohr-Sommerfeld quantization of (P, ω).

Local Lattice Structure
The above discussion does not address the question of labeling the basic sections σ b in H by the quantum numbers n = (n 1 , . . . , n k ) associated to the Bohr-Sommerfeld k-torus T = M b , the support of σ b .
These quantum numbers do depend on the choice of action angle coordinates. If (j , ϑ ) ∈ V × T k is another choice of action angle coordinates in the trivializing chart (U , ψ ), where T ⊆ U , then the quantum numbers n of T in (j , ϑ ) coordinates are related to the quantum numbers n of T in (j, ϑ) coordinates by a matrix A ∈ Gl(k, Z) such that n = A n, because by Claim A2 in Appendix A on U ∩ U the action coordinates j is related to the action coordinate j by a constant matrix A ∈ Gl(k, Z). Let L |U = {n ∈ Z k T n ⊆ U}. Then, L |U is the local lattice structure of the Bohr-Sommerfeld tori T n , which lie in the action angle chart (U, ψ). If (U, ψ) and (U , ψ ) are action angle charts, then the set of Bohr-Sommerfeld tori in U ∩ U are compatible. More precisely, on U ∩ U the local lattices L |U and L |U are compatible if there is a matrix A ∈ Gl(k, Z) such that L |U = A L |U . Let U = {U i } i∈I be a good covering of P, that is, every finite intersection of elements of U is either contractible or empty, such that for each i ∈ I we have a trivializing chart (U i , ψ i ) for action angle coordinates for the toral bundle ρ : P → B. Then, {L U u } i∈I is a collection of pairwise compatible local lattice structures for the collection S of Bohr-Sommerfeld tori on P. We say that S has a local lattice structure.
The next result shows how the operator ( e h Z ϑ i ) * of Section 3.3 affects the quantum numbers of the Bohr-Sommerfeld torus T = T n . Claim 8. Let (U, ψ) be a chart in (P, ω) for action angle coordinates (j, ϑ). For every Bohr-Sommerfeld torus T = T n in U with quantum numbers n = (n 1 , . . . , n k ), the torus e h X ϑ (T) is also a Bohr-Sommerfeld torus T n , where n = (n 1 , . . . , n −1 , n − 1, n +1 , . . . , n k ).

Singularity of Toral Polarization in Completely Integrable Hamiltonian Systems
A completely integrable Hamiltonian system on a symplectic manifold (P, ω) of dimension 2k is given by k functions H 1 , ..., H k ∈ C ∞ (P), which Poisson commute with each other, and are independent on the open dense subset P 0 of P. We assume that, for every i = 1, ..., k, and each x ∈ P 0 , the maximal integral curve of X H i through x is periodic with period T i (x) > 0. The complement P\P 0 of P 0 in P is the set of singular points of the real polarization D = span{X H 1 , ..., X H k } of (P, ω).
Applying the arguments of Section 3.1 and the beginning of Section 3. Let S 0 = {M ∈ S | M ⊂ P 0 } be the set of the Bohr-Sommerfeld tori in P 0 . Then, is the space of quantum states of the system, which are described by the Bohr-Sommerfeld quantization of P 0 . The collection {σ M | M ⊆ P 0 } is a basis of S 0 , and The restriction D |P 0 of D to P 0 is a toral polarization of (P 0 , ω |P 0 ) discussed earlier. The functions H 0 , . . . , H k−1 ∈ C ∞ (P), which define the system, give rise to action-angle coordinates (j, ϑ) on P 0 , where for each i = 0, . . . , k − 1, j i = H i|P 0 |T i|P 0 and ϑ i is the multivalued angle coordinate corresponding to j i . Since we deal with the single set of action-angle coordinates, most of the analysis of Section 3.3.1 applies to this problem. As in Section 3.3.2, Equation (54), for i = 1, . . . , k we introduce the notation Each X i is a locally Hamiltonian vector field on P 0 . However, since P 0 = T * T k , we cannot assume that the vector field X i is complete.
In terms of action-angle coordinates (j, ϑ) on P 0 , the Bohr-Sommerfeld tori in P 0 are given by equation j = (j 1 , . . . , j k ) = (n 1 h, . . . , n k h) = nh, where n = (n 1 , . . . , n k ) ∈ Z k . For n ∈ Z k , denotes the Bohr-Sommerfeld torus in P 0 corresponding to the eigenvalue n of j. If nh is not in the spectrum of j, then M n = ∅. In a trivialization L |P 0 = C × P 0 of the complex line bundle L restricted to P 0 , for each M n = ∅, we can choose σ n : M n → L : (ϑ 1 , . . . , ϑ k ) → (ϑ 1 , . . . , ϑ k ), e −2πi(n 1 ϑ 1 +...+n k ϑ k ) , form a basis in the space S 0 of quantum states in P 0 . Claim 5 implies the following Corollary 1. If, for every x ∈ P 0 and each i = 1, ..., k, Planck constant h is in the domain of the maximal integral curve e tX i (x) of X i starting at x, then e hZ X i = e −2πiϑ i e h lift X i is well defined.
It remains to extend the action of a X i and b X i given above to all states in S. This involves a study of the integral curves of X i on P, which originate or end at points in the singular set P\P 0 .
Suppose we manage to extend the action of the shifting operators to all states in S. Monodromy occurs when, there exist loops in the local lattice of Bohr-Sommerfeld tori such that for some α 1 , . . . , α m ∈ {1, . . . , k} the mapping e hX α m • · · · • e hX α 1 M n : M n → M n need not be the identity on M n . In this case shifting operators are multivalued, and there exists a phase factor e iϕ such that (a X α m • · · · • a X α 1 )σ n = e iϕ σ n .
Given a non-zero section σ ∈ S supported on a Bohr-Sommerfeld torus M. Any maximal family B = {(a n k X k · · · a n 1 X 1 σ) ∈ S n 1 , ...n k ∈ Z} (81) of sections in S, such that no two sections in B are supported on the same Bohr-Sommerfeld torus, is a linear base of S. We can define a scalar product · | · on S as follows. First, assume that basic sections supported on different Bohr-Sommerfeld tori are perpendicular to each other. Then, assume that for every a n k X k · · · a n 1 X 1 σ ∈ B, a n k X k · · · a n 1 X 1 σ | a n k X k · · · a n 1 This definition works even in the presence of monodromy. The completion of S with respect to this scalar product yields a Hilbert space H of quantum states in the Bohr-Sommerfeld quantization of the completely integrable Hamiltonian system under consideration.
Next, we have to consider limits as integral curves of X 1 . Note that the integral curve e tX 1 (x 0 ) of X 1 , originating at x ∈ M 1,m , after time t = h reaches x 1 = e hX 1 (x 0 ) ∈ M 0,m . Moreover, the integral curve e tX 1 (x 0 ) of X 1 originating at x ∈ M n,0 , for n > 0, after time t = h reaches M n−1,0 and after time t = nh reaches the origin M 0,0 . Similarly, the integral curve e −tX 1 (x) of −X 1 originating at x ∈ M n,0 after time t = h reaches M n+1,0 and after time t = kh it reaches M n+k for every k > 0. This argument also applies to X 2 . It enlarges the above table of shifting operators as follows. a X 1 σ 1,m = σ 0,m , provided m ≥ 0, a X 2 σ n,1 = σ n,0 , provided n ≥ 0.
Since X 1 (x) is unbounded as j 1 → 0 + , it is not possible to discuss integral curves of X 1 starting at points in M 0,n . However, for n > 0, b X 1 σ n,m = σ n+1,m and a X 1 σ n+1,m = σ n,m .
Thus, b X 1 shifts in the opposite direction to a X 1 . Similarly, b X 2 shifts in the opposite direction to a X 2 . It is natural to extend these relations to the boundary and assume that b X 1 σ 0,m = σ 1,m , provided m ≥ 0, b X 2 σ n,0 = σ n,1 , provided n ≥ 0.
The actions of the lowering operators a X 1 on states σ 0m and a X 2 on states σ m0 not defined, but they never occur in the theory. Therefore, we may assume that a X 1 σ 0,m = 0, and a X 2 σ n,0 = 0.

Monodromy
Suppose that U = {U i } i∈I is a good covering of P such that for every i ∈ I the chart (U i , ψ i ) is the domain of a local trivialization of the toral bundle ρ : P → B, associated to the fibrating toral polarization D of P, given by the local action angle coordinates We suppose that the set S of Bohr-Sommerfeld tori on P has the local lattice structure {L U i } i∈I of Section 3.3.
Let p and p ∈ P and let γ : [0, 1] → P be a smooth curve joining p to p . We can choose a finite good subcovering {U k } N k=1 of U such that γ([0, 1]) ⊆ ∪ N k=1 U k , where γ(0) ⊆ U 1 and γ(1) ∈ U N . Using the fact that the local lattices {L U k } N k=1 are compatible, we can extend the local action functions j 1 on V 1 = ψ 1 (U 1 ) ⊆ B to a local action function j N on V N ⊆ B. Thus, using the connection E (see Corollary A1), we may parallel transport a Bohr-Sommerfeld torus T n ⊆ U 1 along the curve γ to a Bohr-Sommerfeld torus T n ⊆ U N (see Claim 7). The action function at p , in general depends on the path γ. If the holonomy group of the connection E on the bundle ρ : P → B consists only of the identity element in Gl(k, Z), then this extension process does not depend on the path γ. Thus, we have shown Claim 9. If D is a fibrating toral polarization of (P, ω) with fibration ρ : P → B and B is simply connected, then there are global action angle coordinates on P and the Bohr-Sommerfeld tori T n ∈ S have a unique quantum number n. Thus, the local lattice structure of S is the lattice Z k .
If the holonomy of the connection E on P is not the identity element, then the set S of Bohr-Sommerfeld tori is not a lattice and it is not possible to assign a global labeling by quantum numbers to all the tori in S. This difficulty in assigning quantum numbers to Bohr-Sommerfeld tori has been known to chemists since the early 1920s. Modern papers illustrating it can be found in [18,19]. We give a concrete example where the connection E has nontrivial holonomy, namely, the spherical pendulum.

Example: Spherical Pendulum
The spherical pendulum is a completely integrable Hamiltonian system (H, J, T * S 2 , dα T * S 2 ), where T * S 2 = {(q, p) ∈ T * R 3 q, q = 1 & q, p = 0} is the cotangent bundle of the 2-sphere S 2 with , the Euclidean inner product on R 3 , see [20]. The Hamiltonian is where e T 3 = (0, 0, 1) ∈ R 3 and the e 3 -component of angular momentum is The energy momentum map of the spherical pendulum is E M : T * S 2 → R ⊆ R 2 : (q, p) → H(q, p), J(q, p) .
Here, R is the closure in R 2 of the set R of regular values of the integral map E M. The point (1, 0) ∈ R is an isolated critical value of E M. Thus, the set R has the homotopy type of S 1 and is not simply connected. Every fiber of over a point (h, j) ∈ R is a smooth 2-torus T 2 h,j , see chapter V of [21]. At every point of T * S 2 \ (E M −1 (1, 0) ∪ E M −1 ∂R) there are local action angle coordinates (A 1 , A 2 , ϑ 1 , ϑ 2 ). The actions are while the angles are where t is the time parameter of the integral curves of the vector field X J on the 2-torus T 2 h,j , which are periodic of period 2π, see Section 2.4 of [20]. The action map is a homeomorphism of R \ {(1, 0)} onto (R ≥0 × R) \ {( 4 π , 0)}, which is a real analytic diffeomorphism of R \ {j = 0} onto R >0 × (R \ {0} , see Fact 2.4 in [20]. For every (n, m) ∈ Z ≥0 × Z, the Bohr-Sommerfeld tori are The fibers of E M corresponding to the dark points in Figure 1  which B is an orthogonal basis. The Bohr-Sommerfeld energy momentum spectrum S of the spherical pendulum is the range of the map (n, m) ∈ Z ≥0 × Z are the quantum numbers of the spherical pendulum. In terms of actions A 1 and A 2 , we may write H = H(A 1 , A 2 ). Hence, the quantum operators Q H and Q J act on the basic sections σ m.n as follows Q H σ n,m = H |T 2 n,m σ n,m = h m (n) hσ n,m and Q J σ n,m = J |T 2 n,m σ n,m = m hσ n,m .
In addition, we may assume that a X 1 σ 0,m = 0 and a X 2 σ n,0 = 0.
Since the are no global action angle coordinates, the action function A 1 on R is multi-valued. The holonomy of the connection E is called the monodromy of the fibrating toral polarization D on (P, ω) with fibration ρ : P → B.

Corollary 2.
Let B be the universal covering space of B with covering map Π : B → B. The monodromy map M, which is a nonidentity element holonomy group of the connection E on the bundle ρ sends one sheet of the universal covering space to another sheet.
Proof. Since the universal covering space B of B is simply connected and we can pull back the symplectic manifold (P, ω) and the fibrating toral distribution D by the universal covering map to a symplectic manifold ( P, ω) and a fibrating toral distribution D with associated fibration ρ : P → B.
The connection E on the bundle ρ pulls back to a connection E on the bundle ρ. Let γ be a closed curve on B and let M be the holonomy of the connection E on B along γ. Then, γ lifts to a curve γ on B, which covers γ, that is, ρ • γ = γ. Thus, parallel transport of a k-torus T = R k /Z k , which is an integral manifold of the distribution D, along the curve γ gives a linear map M of the lattice Z k defining the k-torus M( T). The map M is the same as the linear map M of Z k into itself given by parallel transporting T, using the connection E , along the closed γ on B because the connection E is the pull back of the connection E by the covering map ρ. The closed curve γ in B represents an element of the fundamental group of B, which acts as a covering transformation on the universal covering space B that permutes the sheets (= fibers) of the universal covering map Π.
In the spherical pendulum, the universal covering space R of R \ {(1, 0)} is R 2 . If we cut R by the line segment = {(h, 0) ∈ R h > 1}, then R × = R \ is simply connected and hence represents one sheet of the universal covering map of R. For more details on the universal covering map, see [22]. The curve chosen in the example has holonomy M = 1 1 0 1 . It gives a map of R into itself, which sends R × to the adjacent sheet of the covering map. Thus, we have a rule how the labelling of the Bohr-Sommerfeld torus T (n 1 ,n 2 ) , corresponding to (h, j) ∈ R × , changes when we go to an adjacent sheet, which covers R × , namely, we apply the matrix M to the integer vector n 1

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
We return to study the symplectic geometry of a fibrating toral polarization D of the symplectic manifold (P, ω) in order to explain what we mean by its local integral affine structure, see [23].
We assume that the integral manifolds {M p } p∈P of the Lagrangian distribution D on P form a smooth manifold B such that the map ρ : P → B : p → M p is a proper surjective submersion. If the distribution D has these properties we refer to it as a fibrating polarization of (P, ω) with associated fibration ρ : P → B.
Lemma A1. Suppose that D is a fibrating polarization of (P, ω). Then, the associated fibration ρ : P → B has an Ehresmann connection E with parallel translation. Thus, the fibration ρ : P → B is locally trivial bundle.
Proof. We construct the Ehresmann connection as follows. For each p ∈ P let (U, ψ) be a Darboux chart for (P, ω). In other words, (ψ −1 ) * (ω |U ) is the standard symplectic form ω 2k on TV, where V = ψ(U) ⊆ R 2k with ψ(p) = 0. In more detail, for every u ∈ U there is a frame ε(u) of for all w ∈ W. Because the bundle projection map ρ is proper, parallel transport of each fiber of the bundle ρ : P → B by the flow of liftX is defined as long as the flow of X is defined. Because the Ehresmann connection E has parallel transport, the bundle presented by ρ is locally trivial, see pp. 378-379, [21].
Claim A1. If D is a fibrating polarization of the symplectic manifold (P, ω), then for every p ∈ P the integral manifold of D through p is a smooth Lagrangian submanifold of P, which is an k-torus T. In fact T is the fiber over ρ(p) of the associated fibration ρ : P → B.
We say that D is a fibrating toral polarization of (P, ω) if it satisfies the hypotheses of Claim A1. The proof of Claim A1 requires several preparatory arguments.
Let f ∈ C ∞ (B). Then, ρ * f ∈ C ∞ (P). Let X ρ * f be the Hamiltonian vector field on (P, ω) with Hamiltonian ρ * f . We have q. If P q = {0} for some q ∈ P, then the fiber ρ −1 ρ(q) would be diffeomorphic to R k /P q = R k . However, this contradicts the fact the every fiber of the bundle ρ : P → B is compact. Hence, P q = {0} for every q ∈ P. Since R k /P q is diffeomorphic to ρ −1 (b), they have the same dimension, namely, k. Hence, P q is a zero dimensional Lie subgroup of R k . Thus, P q is a rank k lattice Z k . Thus, the fiber ρ −1 (b) is R k /Z k , which is an affine k-torus T k .
We now apply the action angle theorem, see chapter IX of [21], to the fibrating toral Lagrangian polarization D of the symplectic manifold (P, ω) with associated toral bundle ρ : P → B to obtain a more precise description of the Ehresmann connection E constructed in Claim A1. For every p ∈ P there is an open neighborhood U of the fiber ρ −1 ρ(p) in P and a symplectic diffeomorphism u → (j, ϑ) = (j 1 , . . . , j k , ϑ 1 , . . . , ϑ k ) such that is the momentum mapping of the Hamiltonian T k -action on (U, ω |U ). Here, π 1 : V × T k → V : (j, ϑ) → j. Thus, the bundle ρ : P → B is locally a principal T k -bundle. Moreover, we have (ψ −1 ) Corollary A1. Using the chart (U, ψ) for action angle coordinates (j, φ), the Ehresmann connection E |U gives an Ehresmann connection E |V×T n on the bundle π 1 : V × T k → V defined by Proof. This follows because T u ψ ver u = span 1≤i≤k { ∂ ∂ϑ i v=ψ(u) } and T p ψ hor u = span 1≤i≤k { ∂ ∂j i v=ψ(u) }, for every u ∈ U. From the preceding equations for every u ∈ U we have ver u = span 1≤i≤k {X ρ * (j i ) (u)} and hor u = span 1≤i≤k {X (π 2 • ψ) * (−ϑ i ) (u)}. Here, π 2 : V × T k → T k : (j, ϕ) → ϕ.
Corollary A2. The Ehresmann connection E on the locally trivial toral Lagrangian bundle ρ : P → B is flat, that is, ∇ X σ = 0 for every smooth vector field X on B and every local section σ of ρ : P → B.
Proof. In action angle coordinates a local section section σ of the bundle ρ : P → B is given by σ : V → V × T k : j → j, σ(j) . Let X = ∂ ∂j for some 1 ≤ ≤ k with flow e t X . Let liftX be the horizontal lift of X with respect to the Ehresmann connection E V×T k on the bundle π 1 : V × T k → V. Thus, for every j ∈ V we have (∇ X σ)(j) = d dt t=0 e t liftX σ(e −tX (j)) = d dt t=0 e t liftX σ(j(−t)) , where e tX (j) = j(t) = d dt t=0 e t liftX j, σ(j) , since j i for 1 ≤ i ≤ n are integrals of X = d dt t=0 j(t), σ(j(t)) , since π 1 e t liftX (j, σ(j)) = e tX (j) = 0.
This proves the corollary, since every vector field X on W ⊆ B may be written as ∑ k i=1 c i (j) ∂ ∂j i for some c i ∈ C ∞ (W) and the flow {ϕ on V pairwise commute.
Claim A2. Let ρ : P → B be a locally trivial toral Lagrangian bundle, where (P, ω) is a smooth symplectic manifold. Then, the smooth manifold B has an integral affine structure. In other words, there is a good open covering {W i } i∈I of B such that the overlap maps of the coordinate charts (W i , ϕ i ) given by where ϕ i (W i ) = V i , have derivative Dϕ i (v) ∈ Gl(k, Z), which does not depend on v ∈ V i ∩ V .
Proof. Cover P by U = {U i } i∈I , where (U i , ψ i ) is an action angle coordinate chart. Since every open covering of P has a good refinement, we may assume that U is a good covering. Let W i = ρ(U i ). Then, W = {W i } i∈I is a good open covering of B and (W i , ϕ i = π 1 • ψ i ) is a coordinate chart for B. By construction of action angle coordinates, in V i ∩ V the overlap map ϕ i sends the action coordinates j i in V i ∩ V to the action coordinates j in V i ∩ V . The period lattices P ψ −1 i (j i ) and P ψ −1 (j ) are equal since for some p ∈ W i ∩ W we have ψ i (p) = j i and ψ (p) = j . Moreover, these lattices do not depend on the point p. Thus, the derivative Dϕ i (j) sends the lattice Z k spanned by { ∂ Hence, for every j ∈ W i ∩ W the matrix of Dϕ i (j) has integer entries, that is, it lies in Gl(k, Z) and the map j → Dϕ i (j) is continuous. However, Gl(k, Z) is a discrete subgroup of the Lie group Gl(k, R) and W i ∩ W is connected, since W is a good covering. Thus, Dϕ i (j) does not depend on j ∈ W i ∩ W .
Corollary A3. Let γ : [0, 1] → B be a smooth closed curve in B. Let P γ : [0, 1] → P be parallel translation along γ using the Ehresmann connection E on the bundle ρ : P → B. Then, the holonomy group of the k-toral fiber T γ(0) = T k is induced by the group Gl(k, Z) Z k of affine Z-linear maps of Z k into itself.