Adiabatic limits, Theta functions, and geometric quantization
aa r X i v : . [ m a t h . S G ] M a y ADIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION
TAKAHIKO YOSHIDA
Abstract.
Let π : ( M, ω ) → B be a (non-singular) Lagrangian torus fibration on a compact, completebase B with prequantum line bundle ( L, ∇ L ) → ( M, ω ). For a positive integer N and a compatiblealmost complex structure J on ( M, ω ) invariant along the fiber of π , let D be the associated Spin c Diracoperator with coefficients in L ⊗ N . Then, we give an orthogonal family { e ϑ b } b ∈ B BS of sections of L ⊗ N indexed by the Bohr-Sommerfeld points B BS , and show that each e ϑ b converges to a delta-function sectionsupported on the corresponding Bohr-Sommerfeld fiber π − ( b ) and the L -norm of D e ϑ b converges to 0by the adiabatic(-type) limit. Moreover, if J is integrable, we also give an orthogonal basis { ϑ b } b ∈ B BS ofthe space of holomorphic sections of L ⊗ N indexed by B BS , and show that each ϑ b converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber π − ( b ) by the adiabatic(-type)limit. We also explain the relation of ϑ b with Jacobi’s theta functions when ( M, ω ) is T n . Contents
1. Motivation and Main Theorems 11.1. Notations 42. Developing Lagrangian fibrations 42.1. Integral affine structures 42.2. Lagrangian fibrations 72.3. Lagrangian fibrations with complete bases 92.4. The lifting problem of the Γ-action to the prequantum line bundle 123. Degree-zero harmonic spinors and integrability of almost complex structures 153.1. Bohr-Sommerfeld points 153.2. Almost complex structures 163.3. The existence condition of non-trivial harmonic spinors of degree-zero 183.4. The Γ-equivariant case 274. The integrable case 294.1. Definition and properties of ϑ mN Z is constant 324.3. Adiabatic-type limit 335. The non-integrable case 36References 401. Motivation and Main Theorems
The purpose of this paper is to show the Spin c quantization converges to the real quantization bythe adiabatic(-type) limit for a Lagrangian torus fibration on a compact complete base and a compatiblealmost complex structures on its total space which is invariant along the fiber. In this paper, a Lagrangiantorus fibration is assumed to be non-singular unless otherwise stated. First let us explain the motivationwhich comes from geometric quantization. For geometric quantization, see [17, 22, 32, 38]. In physics,quantization is the procedure for building quantum mechanics starting from classical mechanics. Inthe mathematical context, it is often thought of as a representation of the Poisson algebra consistingof certain functions on a symplectic manifold to some Hilbert space, so called the quantum Hilbertspace, and the geometric quantization gives us the method to construct a quantum Hilbert space and a Mathematics Subject Classification.
Primary 53D50; Secondary 58H15, 58J05.
Key words and phrases. adiabatic limit, Theta function, Lagrangian fibration, geometric quantization. representation from a given symplectic manifold (
M, ω ) and a prequantum line bundle ( L, ∇ L ) → ( M, ω )in the geometric way. In the theory of geometric quantization by Kostant and Souriau [25, 34, 33], weneed an additional structure which is called a polarization to obtain the quantum Hilbert space. Bydefinition, a polarization is an integrable Lagrangian distribution P of the complexified tangent bundle T M ⊗ C of ( M, ω ). For a polarization P , the quantum Hilbert space is naively given as the closure ofthe space of smooth, square-integrable sections of ( L, ∇ L ) which are covariant constant along P .A common example is the K¨ahler polarization. When ( M, ω ) is K¨ahler and ( L, ∇ L ) is a holomorphicline bundle with canonical connection, we can take T , M as a polarization, and the obtained quantumHilbert space is nothing but the space of holomorphic sections H ( M, O L ). This polarization is called theK¨ahler polarization and the quantization procedure is called the K¨ahler quantization. Note that when M is compact and the Kodaira vanishing theorem holds, its dimension is equal to the index of the Dolbeaultoperator with coefficients in L .Another example is a real polarization. Suppose ( M, ω ) admits a structure of a Lagrangian torusfibration π : ( M, ω ) → B . For each point b ∈ B of the base manifold B , the restriction (cid:0) L, ∇ L (cid:1) | π − ( b ) of ( L, ∇ L ) to the fiber π − ( b ) is a flat line bundle. Let H (cid:0) π − ( b ); (cid:0) L, ∇ L (cid:1) | π − ( b ) (cid:1) be the space ofcovariant constant sections of (cid:0) L, ∇ L (cid:1) | π − ( b ) . Then, an element b ∈ B is said to be Bohr-Sommerfeldif H (cid:0) π − ( b ); (cid:0) L, ∇ L (cid:1) | π − ( b ) (cid:1) = { } . It is well-known that Bohr-Sommerfeld points appear discretely.In this case, we can take T π M ⊗ C , the complexified tangent bundle along fibers of π as a polarization,and if M is compact, the quantum Hilbert space is defined by ⊕ b ∈ B BS H (cid:0) π − ( b ); (cid:0) L, ∇ L (cid:1) | π − ( b ) (cid:1) , wherethe sum is taken over all Bohr-Sommerfeld points. See [32] for more details. In this paper, we call thisquantization the real quantization.When a Lagrangian torus fibration π : ( M, ω ) → B with closed total space M and a prequantum linebundle ( L, ∇ L ) → ( M, ω ) are given, it is natural to ask whether the quantum Hilbert space obtainedby the K¨ahler quantization is isomorphic to the one obtained by the real quantization. A completelyintegrable system can be thought of as a Lagrangian fibration with singular fibers. In the case of the themoment map of a projective toric variety, Danilov has shown in [9] that H ( M, O L ) has the irreducibledecomposition H ( M, O L ) = ⊕ m ∈ ∆ ∩ t ∗ Z C m as a compact torus representation, where ∆ is the momentpolytope, t ∗ Z is the weight lattice, and C m is the irreducible representation of the torus with weight m .Since ∆ ∩ t ∗ Z is identified with the set of Bohr-Sommerfeld points, this implies the dimensions of thequantum Hilbert spaces obtained by the above quantizations agree. A similar equality of the dimensionshas been also shown for the Gelfand-Cetlin system on the flag variety [16], the Goldman system on themoduli space of flat SU (2) connections on a surface [20], and the Kapovich-Millson system on the polygonspace. [21].Moreover, in the case of smooth projective toric varieties, not only the numerical equality for thedimensions, but also a geometric correspondence between the K¨ahler and the real quantizations has beenshown concretely by Baier-Florentino-Mour˜ao-Nunes in [3]. Namely, they have given a one-parameterfamily of complex structures { J t } t ∈ [0 , ∞ ) and a basis { s mt } m ∈ ∆ ∩ t ∗ Z of the space of holomorphic sectionsassociated with the complex structure J t for each t such that each section s mt converges to the deltafunction section supported on the corresponding Bohr-Sommerfeld fiber as t goes to ∞ . The similarresult has been also obtained for flag manifolds in [19] and smooth irreducible complex algebraic varietiesby [18]. But in [19] and [18] the convergence has been shown only for the non-singular Bohr-Sommerfeldfibers whereas in [3] it has been shown for all Bohr-Sommerfeld fibers.The K¨ahler quantization can be generalized to a non-integrable compatible almost complex structureon a closed ( M, ω ). When a compatible almost complex structure J on ( M, ω ) is given, we can considerthe associated Spin c Dirac operator D acting on Γ (cid:0) ∧ • T ∗ M , ⊗ L (cid:1) . It is well-known that D is a formallyself-adjoint, first order, elliptic differential operator of degree-one, and if J is integrable, D agrees withthe Dolbeault operator up to constant. If J is not integrable, T , M is no more polarization. But, evenin this case, since D is Fredholm, we can still take the element of the K-theory of a point(1.1) ker ( D | ∧ even T ∗ M , ⊗ L ) − ker ( D | ∧ odd T ∗ M , ⊗ L ) ∈ K ( pt )as a (virtual) quantum Hilbert space. Its virtual dimension is equal to the index of D . We call thisquantization the Spin c quantization. It has been shown in [1, 13, 26] that the above equality betweendimensions of two quantum Hilbert spaces still holds by replacing the K¨ahler quantization with the Spin c quantization by using the index theory. DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 3
In this paper, we generalize the approach taken in [3] for the K¨ahler quantization to the Spin c quanti-zation of Lagrangian torus fibrations. Let π : ( M, ω ) → B be a Lagrangian torus fibration on a compact,complete base B with prequantum line bundle ( L, ∇ L ) → ( M, ω ). For a positive integer N and a compat-ible almost complex structure J of ( M, ω ) invariant along the fiber of π (in the sense of Lemma 3.6), let D be the associated Spin c Dirac operator with coefficients in L ⊗ N . Then, the main result is as follows.This theorem is a combination of Theorem 5.2 and Theorem 5.3. Theorem 1.1.
For the above data, we give one-parameter families of • { J t } t> compatible almost complex structures of ( M, ω ) with J = J • { e ϑ tb } b ∈ B BS sets of sections on L ⊗ N indexed by the Bohr-Sommerfeld points B BS such that (1) any pair in { e ϑ tb } b ∈ B BS are orthogonal to each other, (2) each e ϑ tb converges as a delta-function section supported on π − ( b ) as t → ∞ in the followingsense, for any section s of L ⊗ N , lim t →∞ Z M * s, e ϑ tb k e ϑ tb k L + L ⊗ N ( − n ( n − ω n n ! = Z π − ( b ) h s, δ b i L ⊗ N | dy | , where h , i L ⊗ N is the Hermitian metric of L ⊗ N , δ b is the covariant constant section of L ⊗ N | π − ( b ) defined by (4.9) , and | dy | is the natural one-density on π − ( b ) , (3) lim t →∞ k D t e ϑ tb k L = 0 . By the Spin c Dirac vanishing theorem due to Borthwick-Uribe [7], ker ( D | ∧ odd T ∗ M , ⊗ L ⊗ N ) vanishes fora sufficiently large N . So, (3) implies the the complex vector space spanned by { e ϑ tb } b ∈ B BS approximatesthe quantum Hilbert space of the Spin c quantization for a sufficiently large N .If J is integrable, we also give the following refinement of Theorem 1.1, which is immediately obtainedby putting Corollary 4.3 and Theorem 4.12 together. Theorem 1.2.
Under the above setting, assume J is integrable. Then, with a technical assumption, wegive one-parameter families of • { J t } t> compatible complex structures of ( M, ω ) with J = J • { ϑ tb } b ∈ B BS orthogonal bases of holomorphic sections of L ⊗ N → ( M, N ω, J t ) indexed by B BS such that each ϑ tb converges as a delta-function section supported on π − ( b ) as t → ∞ in the followingsense, for any section s of L ⊗ N , lim t →∞ Z M (cid:28) s, ϑ tb k ϑ tb k L (cid:29) L ⊗ N ( − n ( n − ω n n ! = Z π − ( b ) h s, δ b i L ⊗ N | dy | . One of examples of the total space of a Lagrangian torus fibration with complete base is an abelianvariety. In this case, we show that each ϑ b coincides with Jacobi’s theta functions up to function on thebase space (Theorem 4.7). For the theta functions, see [28, 29].We should remark there are several works which deal with theta functions from the viewpoint ofgeometric quantization of Lagrangian fibrations, for example, [29], [4], [30, 31]. In [7], Borthwick-Uribehave introduced another approach to generalize the K¨ahler quantization to non-integrable almost complexstructures by using the metric Laplacian of the connection on the prequantum line bundle instead ofSpin c Dirac operator. Their approach is called the almost K¨ahler quantization. In the almost K¨ahlerquantization of the Kodaira-Thurston manifold, Kirwin-Uribe and Egorov have constructed an analog ofthe theta function as an element of the quantum Hilbert space [23], [12]. In [11], Egorov has also shownthe similar result for Lagrangian T -fibrations on T with zero Euler class.The idea used in this paper is quite simple. One of two key facts is Corollary 2.25 which claimsthat any Lagrangian torus fibration π : ( M, ω ) → B with complete base B and a prequantum line bundle (cid:0) L, ∇ L (cid:1) → ( M, ω ) can be obtained as the quotient of a π ( B )-action on the standard Lagrangian fibration (cid:16) f M , ω (cid:17) := ( R n × T n , P ni =1 dx i ∧ dy i ) → R n with the standard prequantum line bundle. In particular,any compatible almost complex structure on ( M, ω ) is induced from a π ( B )-equivariant one on (cid:16) f M , ω (cid:17) , T. YOSHIDA and the set of compatible almost complex structures on (cid:16) f
M , ω (cid:17) corresponds one-to-one to the set ofsmooth maps from f M to the Siegel upper half space. We show that there exists a π ( B )-invariantcompatible almost complex structure J whose corresponding map is invariant along the fiber (Lemma 3.6).For the Spin c Dirac operator D associated with such an almost complex structure J , we consider theproblem on the existence of non-trivial degree-zero harmonic spinors, i.e., sections of e L ⊗ N containedin the kernel of D . By taking the Fourier series expansion of a section s of e L ⊗ N with respect to thefiber coordinates, the equation Ds = 0 can be reduced to a system of partial differential equationson R n . The other key fact is Proposition 3.12 in which we give a necessary and sufficient conditionin order that the system of partial differential equations has non-trivial solutions and show that it isequal to the integrability condition for J , i.e., (cid:16) f M , ω , J (cid:17) is K¨ahler. Moreover, in this case, we give afamily of π ( B )-equivariant solutions of Ds = 0 indexed by the Bohr-Sommerfeld points, each of whichis expressed by the formal Fourier series. If they converge absolutely and uniformly, this gives a linearbasis of the space of holomorphic sections of (cid:0) L, ∇ L (cid:1) ⊗ N → ( M, N ω, J ) → B . We also give a sufficientcondition for their convergence. Even if J is not integrable, by considering an approximation of D , wecan obtain an orthogonal family of sections of L ⊗ N indexed by the Bohr-Sommerfeld points B BS . Thelimit used in this paper is slightly different from the adiabatic limit in Riemannian geometry. When afiber bundle π : M → B and a Riemannian metric g on M are given, we can consider the decomposition( T M, g ) = (
V, g V ) ⊕ ( H, g H ), where V is the tangent bundle along the fiber with fiber metric g V := g | V and H is the orthogonal complement of V with respect to g with fiber metric g H := g | H . For each t > g t to be g t := g V ⊕ tg H . Then, in Riemannian geometry, the adiabatic limitis the procedure for taking the limit of geometric objects depending on g t as t → ∞ . But, since such adeformation of Riemannian metrics does not fit into our symplectic context, we modify the deformation.Namely, in this paper, we use a one-parameter family { J t } t> of compatible almost complex structures on( M, ω ) such that the corresponding one-parameter family of Riemannian metrics is { g t = t g V ⊕ tg H } t> ,and investigate the behavior of e ϑ tb (resp. ϑ tb ) when t goes to ∞ .The paper is organized as follows. In section 2, we first briefly review some well-known facts aboutintegral affine geometry and Lagrangian fibrations. Then, by using these, we prove Corollary 2.25. InSection 3, we discuss the π ( B )-equivariant Spin c quantization of ( R n × T n , P ni =1 dx i ∧ dy i ) → R n withthe standard prequantum line bundle and prove Proposition 3.12. In Section 4, we prove Theorem 1.2 stepby step, and explain the relation between ϑ tb and Jacobi’s classical theta function. Finally, in Section 5we prove Theorem 1.1. Acknowledgment
The most part of this work had been done while the author stayed in McMasteruniversity. The author would like to thank the department of Mathematics and Statistics, McMasteruniversity and especially Megumi Harada for their hospitality. This work is supported by Grant-in-Aidfor Scientific Research (C) 15K04857.1.1.
Notations.
For x = t ( x , . . . , x n ) and y = t ( y , . . . , y n ) ∈ R n , let us denote the standard innerproduct P ni =1 x i y i by x · y . ∂ x i denotes ∂∂x i . In this paper, all manifolds and maps are supposed to besmooth. 2. Developing Lagrangian fibrations
Integral affine structures.
Let B be a manifold. Definition 2.1. An integral affine atlas of B is an atlas { ( U α , φ α ) } of B on each of whose non-emptyoverlap U αβ , the transition function φ α ◦ φ − β is an integral affine transformation, namely, φ α ◦ φ − β is of theform φ α ◦ φ − β ( x ) = A αβ x + c αβ for some locally constant maps A αβ : U αβ → GL n ( Z ) and c αβ : U αβ → R n .Two integral affine atlases { ( U α , φ α ) } and { ( U ′ β , φ ′ β ) } of B are said to be equivalent if on each non-emptyoverlap U α ∩ U ′ β , the transition function φ α ◦ ( φ ′ β ) − is an integral affine transformation. An integralaffine structure on B is an equivalence class of integral affine atlases of B . A manifold equipped withintegral affine structure is called an integral affine manifold . Example 2.2. An n -dimensional Euclidean space R n is equipped with a natural integral affine structure. DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 5
Let us give examples of integral affine manifolds obtained from integral affine actions on R n . Example 2.3. (1) Let v , . . . , v n ∈ R n be a linear basis of R n and C = ( v · · · v n ) ∈ GL n ( R ) the matrixwhose i th column vector is v i for i = 1 , . . . , n . Z n acts on R n by ρ γ ( x ) := x + Cγ for γ ∈ Z n and x ∈ R n . Since the action preserves the natural integral affine structure on R n , the quotientspace, which is topologically T n , is equipped with an integral affine structure.(2) Let λ ∈ N be a positive integer and a, b ∈ R > positive real numbers. Define Z -action on R asfollows. First, for the standard basis e , e of Z , let us define the integral affine transform ρ e , ρ e by ρ e ( x ) := x + (cid:18) a (cid:19) , ρ e ( x ) := (cid:18) λ (cid:19) x + (cid:18) b (cid:19) for x ∈ R . Since ρ e and ρ e are commutative, they form the Z -action on R by ρ γ ( x ) := ρ γ e ◦ ρ γ e ( x )for each γ = t ( γ , γ ) ∈ Z . In the same manner as in (1), the quotient space is equipped with an integralaffine structure. It is shown in [27, Theorem A] that the quotient space is topologically T , but theinduced integral affine structure is not isomorphic to that obtained in (1) for n = 2 and there are onlythese two integral affine structures on T up to isomorphism. Example 2.4.
For γ = t ( γ , γ , γ ) , γ ′ = t ( γ ′ , γ ′ , γ ′ ) ∈ Z define the product γ ◦ γ ′ ∈ Z by γ ◦ γ ′ := − − γ γ ′ + γ. Then, Z with product ◦ is a non abelian group (cid:0) Z , ◦ (cid:1) . (cid:0) Z , ◦ (cid:1) acts on R by ρ γ ( x ) := − − γ x + γ. Then, the quotient space R / (cid:0) Z , ◦ (cid:1) is equipped with the integral affine structure induced from that of R . Example 2.5.
Let n ≥
2. For γ = t ( γ , . . . , γ n ) , γ ′ = t ( γ ′ , . . . , γ ′ n ) ∈ Z n define the product γ ◦ γ ′ ∈ Z n by γ ◦ γ ′ := − γ . . . ( − γ n − γ ′ + γ. Then, Z n with product ◦ is a non abelian group ( Z n , ◦ ). ( Z n , ◦ ) acts on R n by ρ γ ( x ) := − γ . . . ( − γ n − x + γ. Then, the quotient space R n / ( Z n , ◦ ) is equipped with the integral affine structure induced from that of R n . For n = 2, the quotient space is topologically a Klein bottle. Example 2.6.
Let n ≥ λ , . . . λ n − ∈ Z . For γ = t ( γ , . . . , γ n ) , γ ′ = t ( γ ′ , . . . , γ ′ n ) ∈ Z n define theproduct γ ◦ γ ′ ∈ Z n by γ ◦ γ ′ := λ λ . . . . . .1 λ n − γ n γ ′ + γ. T. YOSHIDA Z n with product ◦ is a group ( Z n , ◦ ), which is non abelian for n ≥
3. ( Z n , ◦ ) acts on R n by ρ γ ( x ) := λ λ . . . . . .1 λ n − γ n x + γ. Then, the quotient space R n / ( Z n , ◦ ) is equipped with the integral affine structure induced from thatof R n . In the case where n = 2 and λ >
0, it coincides with the one given in Example 2.3 (2) with a = b = 1. Example 2.7.
Let Z / Z ∼ = (cid:26) ± (cid:18) (cid:19) , ± (cid:18) −
11 0 (cid:19)(cid:27) act on ( R ) n r { } naturally. Then, the quotientspace is a non-compact manifold and equipped with the integral affine structure induced from that of( R ) n r { } .As we can guess from above examples, every integral affine manifold is obtained from a group action.Let B be an n -dimensional connected integral affine manifold, p : e B → B the universal covering of B .It is clear that e B is also equipped with the integral affine structure so that p is an integral affine map.We set Γ := π ( B ). Γ acts on e B from the right as a deck transformation. For each γ ∈ Γ we denoteby σ γ the inverse of the deck transformation corresponding to γ . Then, σ : γ σ γ defines a left action σ ∈ Hom (cid:16) Γ , Aut( e B ) (cid:17) .We assume that all the actions considered in this paper are left actions unless otherwise stated.The following proposition is well known in affine geometry. Proposition 2.8.
There exists an integral affine immersion dev : e B → R n and a homomorphism ρ : Γ → GL n ( Z ) ⋉ R n such that the image of dev is an open set of R n and dev is equivariant with respect to σ and ρ . Such an integral affine immersion is unique up to the composition of an integral affine transformationon R n . See [15, p.641] for a proof. We will prove a version of this proposition (Proposition 2.22) when B isequipped with a Lagrangian fibration on it in Section 2.
Proposition 2.9.
Let B , p : e B → B , dev : e B → R n , and ρ : Γ → GL n ( Z ) ⋉ R n be as in Proposition 2.8.Suppose that B is compact and the Γ -action ρ on R n is properly discontinuous. Then, dev is surjective.Proof. We denote the image of dev by O . By proposition 2.8, O is an open set in R n . So, it is sufficientto show that O is closed in R n . Since the Γ-action ρ on R n is properly discontinuous, the quotientspace R n / Γ becomes a Hausdorff space and the natural projection q : R n → R n / Γ is continuous. O ispreserved by the Γ-action ρ on R n since dev is Γ-equivariant. Then, dev induces a continuous surjectivemap dev : B = e B/ Γ → O/ Γ. Since B is compact, O/ Γ is a compact subset in the Hausdorff space R n / Γ.In particular, it is also closed. Hence, O = q − ( O/ Γ) is also closed in R n . (cid:3) Corollary 2.10.
Let B , p : e B → B , dev : e B → R n , and ρ : Γ → GL n ( Z ) ⋉ R n be as in Proposition 2.8and assume that B compact. If the image of ρ lies in (GL n ( Z ) ∩ O ( n )) ⋉ R n and the subgroup ρ (Γ) of (GL n ( Z ) ∩ O ( n )) ⋉ R n is discrete, then, dev is surjective.Proof. It follows from [37, Theorem 3.1.3]. (cid:3)
Definition 2.11.
The integral affine immersion dev is called a developing map . B is said to be complete if dev is bijective. B is said to be incomplete if B is not complete. Example 2.12.
All the above examples are complete other than Example 2.7 for n ≥ Example 2.13.
Let B be an n -dimensional compact integral affine manifold B with integral affine atlas { ( U α , φ α ) } as in Definition 2.1. If on each non-empty overlap U αβ , the Jacobi matrix of the coordinatechanging map D (cid:16) φ α ◦ φ − β (cid:17) lies in GL n ( Z ) ∩ O ( n ), then, B has a flat Riemannian metric. Hence, byBieberbach’s theorem [5, 6], B is finitely covered by T n . In particular, B is complete. For flat Riemannianmanifolds, see [37, Chapter 3]. DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 7
Lagrangian fibrations.
In this section let us recall Lagrangian fibrations and explain how integralaffine structures are associated with Lagrangian fibrations. After that let us recall their classification byDuistermaat. For more details, see [10, 39].Let (
M, ω ) be a symplectic manifold.
Definition 2.14.
A map π : ( M, ω ) → B from ( M, ω ) to a manifold B is called a Lagrangian fibration if π is a fiber bundle whose fiber is a Lagrangian submanifold of ( M, ω ). Example 2.15.
Let T n = ( R / Z ) n be an n -dimensional torus. R n × T n admits a standard symplecticstructure ω = P i dx i ∧ dy i , where ( x , . . . , x n ), ( y , . . . , y n ) are the coordinates of R n , T n , respectively.Then, the projection π : ( R n × T n , ω ) → R n to R n is a Lagrangian fibration.The following theorem shows that Example 2.15 is a local model of Lagrangian fibration. Theorem 2.16 (Arnold-Liouville’s theorem [2]) . Let π : ( M, ω ) → B be a Lagrangian fibration withcompact, path-connected fibers. Then, for each b ∈ B , there exists a chart ( U, φ ) containing b and asymplectomorphism ϕ : ( π − ( U ) , ω | π − ( U ) ) → ( φ ( U ) × T n , ω ) such that the following diagram commutes (cid:0) π − ( U ) , ω | π − ( U ) (cid:1) π (cid:15) (cid:15) ϕ / / ( φ ( U ) × T n , ω ) π (cid:15) (cid:15) U φ / / φ ( U ) . In the rest of this paper we assume that every Lagrangian fibration has compact, path-connectedfibers.Now we investigate automorphisms of the local model. By the direct computation shows the followinglemma. See also [35, Lemma 2.5].
Lemma 2.17.
Let ϕ : ( R n × T n , ω ) → ( R n × T n , ω ) be a fiber-preserving symplectomorphism of π : ( R n × T n , ω ) → R n which covers a map φ : R n → R n . Then, there exists a matrix A ∈ GL n ( Z ) , a constant c ∈ R n , and a map u : R n → T n with t ADu symmetric such that ϕ is written as ϕ ( x, y ) = (cid:0) Ax + c, t A − y + u ( x ) (cid:1) for any ( x, y ) ∈ R n × T n , where Du is the Jacobi matrix of u . By Theorem 2.16 and Lemma 2.17 we can obtain the following proposition.
Proposition 2.18.
Let π : ( M, ω ) → B be a Lagrangian fibration. Then, there exists an atlas { ( U α , φ α ) } α ∈ A of B and for each α ∈ A there exists a symplectomorphism ϕ α : (cid:0) π − ( U α ) , ω | π − ( U α ) (cid:1) → ( φ α ( U α ) × T n , ω ) such that the following diagram commutes ( π − ( U α ) , ω | π − ( U α ) ) π (cid:15) (cid:15) ϕ α / / ( φ α ( U α ) × T n , ω ) π (cid:15) (cid:15) U α φ α / / φ α ( U α ) . Moreover, on each non-empty overlap U αβ := U α ∩ U β there exist locally constant maps A αβ : U αβ → GL n ( Z ) , c αβ : U αβ → R n , and a map u αβ : U αβ → T n with t A αβ D (cid:16) u αβ ◦ φ − β (cid:17) symmetric such that theoverlap map is written as (2.1) ϕ α ◦ ϕ − β ( x, y ) = (cid:16) A αβ x + c αβ , t A − αβ y + u αβ ◦ φ − β ( x ) (cid:17) for any ( x, y ) ∈ φ β ( U αβ ) × T n . Proposition 2.18 implies that the base manifold of a Lagrangian fibration has an integral affine struc-ture. Conversely, suppose that a manifold B admits an integral affine structure and let { ( U α , φ α ) } α ∈ A be an integral affine atlas of B . Then, we can construct a Lagrangian fibration on B in the following T. YOSHIDA way. For each α ∈ A let φ α : T ∗ B | U α → φ α ( U α ) × R n be the local trivialization of the cotangent bun-dle T ∗ B induced from ( U α , φ α ). On each nonempty overlap U αβ , suppose that φ α ◦ φ − β is written by φ α ◦ φ − β ( x ) = A αβ x + c αβ as in Definition 2.1. Then, the overlap map is written as(2.2) φ α ◦ ( φ β ) − ( x, y ) = ( A αβ x + c αβ , t A − αβ y ) . Since A αβ lies in GL n ( Z ), (2.2) preserves the integer lattice Z n of the fiber R n , hence, induces the fiber-preserving symplectomorphism from π : ( φ β ( U αβ ) × T n , ω ) → φ β ( U αβ ) to π : ( φ α ( U αβ ) × T n , ω ) → φ α ( U αβ ). Then, the Lagrangian fibrations { π : ( φ α ( U α ) × T n , ω ) → φ α ( U α ) } α ∈ A are patched togetherby the symplectomorphisms to form a Lagrangian fibration π T : ( T ∗ B T , ω T ∗ B T ) → B , namely( T ∗ B T , ω T ∗ B T ) := a α ∈ A ( φ α ( U α ) × T n , ω ) / ∼ and π T ([ x α , y α ]) := φ − α ( x α )for ( x α , y α ) ∈ φ α ( U α ) × T n . This construction does not depend on the choice of equivalent integral affinestructures and depends only on the integral affine structure on B . We call π T : ( T ∗ B T , ω T ∗ B T ) → B the canonical model .We summarize the above argument to the following proposition. Proposition 2.19.
Let B be a manifold. B is a base space of a Lagrangian fibration if and only if B admits an integral affine structure. Let us give a classification theorem of Lagrangian fibrations in the required form in this paper. Let π : ( M, ω ) → B be a Lagrangian fibration. Then, B has an integral affine structure by Proposition 2.19.We take and fix an integral affine atlas { ( U α , φ α ) } α ∈ A on B and let π T : ( T ∗ B T , ω T ∗ B T ) → B be the canon-ical model associated with the integral affine structure on B . On each U α , let ϕ α : (cid:0) π − ( U α ) , ω | π − ( U α ) (cid:1) → ( φ α ( U α ) × T n , ω ) be a local trivialization of π : ( M, ω ) → B as in Proposition 2.18, and φ α : (cid:0) π − T ( U α ) , ω T ∗ B T (cid:1) → ( φ α ( U α ) × T n , ω ) be the local trivialization of π T : ( T ∗ B T , ω T ∗ B T ) → B naturally induced from ( U α , φ α )as explained above. Then their composition h α := φ α − ◦ ϕ α : (cid:0) π − ( U α ) , ω | π − ( U α ) (cid:1) → (cid:0) π − T ( U α ) , ω T ∗ B T (cid:1) gives a local identification between them.On each U α ∩ U β , suppose that ϕ α ◦ ϕ − β is written as in (2.1). Then, h α ◦ h − β can be written as h α ◦ h − β ( p ) = φ α − (cid:16) A αβ x + c αβ , t A − αβ y + u αβ ( π ( p )) (cid:17) , where φ β ( p ) = ( x, y ). u αβ induces the local section e u αβ of π T : ( T ∗ B T , ω T ∗ B T ) → B on U αβ by e u αβ ( b ) := [ φ α ( b ) , u αβ ( b )]for b ∈ U αβ . It is easy to see that e u αβ satisfies e u ∗ αβ ω T ∗ B T = 0. A section with this condition is said to be Lagrangian .Let S be the sheaf of germs of Lagrangian section of π T : ( T ∗ B T , ω T ∗ B T ) → B . S is the sheafof Abelian groups since the fiber of π T : ( T ∗ B T , ω T ∗ B T ) → B has the structure of an Abelian group byconstruction. By definition { e u αβ } forms a ˇCech one-cocycle on B with coefficients in S . The cohomologyclass determined by { e u αβ } does not depend on the choice of a specific integral affine structure and dependsonly on π : ( M, ω ) → B . We denote the cohomology class by u ∈ H ( B ; S ). u is called the Chern class of π : ( M, ω ) → B in [10].Lagrangian fibrations on the same integral affine manifold are classified with the Chern classes. Theorem 2.20 ([10]) . For two Lagrangian fibrations π : ( M , ω ) → B and π :( M , ω ) → B on the same integral affine manifold B , there exists a fiber-preserving symplectomorphismbetween them which covers the identity if and only if their Chern classes u and u agree with each other.Moreover, if an integral affine manifold B and the cohomology class u ∈ H ( B ; S ) are given, then, thereexists a Lagrangian fibration π : ( M, ω ) → B that realizes them. Here we use the same notation as the local trivialization of T ∗ B because we have no confusion. DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 9
Remark 2.21.
By the construction of u , there exists a fiber-preserving symplectomorphism between π : ( M, ω ) → B and π T : ( T ∗ B T , ω T ∗ B T ) → B that covers the identity of B if and only if u vanishes.Since π T : ( T ∗ B T , ω T ∗ B T ) → B has the zero section which is Lagrangian, u is the obstruction class inorder that π : ( M, ω ) → B posses a Lagrangian section. In particular, any Lagrangian fibration withLagrangian section is identified with the canonical model.2.3. Lagrangian fibrations with complete bases.
Let π : ( M, ω ) → B be a Lagrangian fibrationwith n -dimensional connected base manifold B , p : e B → B the universal covering of B . We denote by e π : ( f M , e ω ) → e B the pullback of π : ( M, ω ) → B to e B . Let Γ be the fundamental group of B and σ ∈ Hom (cid:16) Γ , Aut( e B ) (cid:17) the action of Γ defined as the inverse of the deck transformation as in Proposition 2.8.By definition, f M admits a natural lift of σ which preserves e ω . The Γ-action on ( f M , e ω ) is denoted by e σ . ByProposition 2.8 we have a developing map dev : e B → R n and the homomorphism ρ : Γ → GL n ( Z ) ⋉ R n .We denote the image of dev by O . Note that the Γ-action ρ on R n preserves O since dev is Γ-equivariant. Proposition 2.22.
There exists a Lagrangian fibration π ′ : ( M ′ , ω ′ ) → O , a fiber-preserving symplecticimmersion g dev : ( f M , e ω ) → ( M ′ , ω ′ ) which covers dev , and a lift e ρ of the Γ -action ρ on O to ( M ′ , ω ′ ) suchthat g dev is Γ -equivariant with respect to e σ and e ρ .Proof. By Proposition 2.19 B admits an integral affine structure determined by π , and it also induces theintegral affine structure on e B . Let { ( U α , φ ′′ α ) } be the integral affine atlas of e B and { ( e π − ( U α ) , ω | e π − ( U α ) , ϕ ′′ α ) } the local trivializations of e π : ( f M , e ω ) → e B as in Proposition 2.18 so that on each non-empty over-lap U αβ , there exist locally constant maps A αβ : U αβ → GL n ( Z ) and c ′ αβ : U αβ → R n , and a map u ′ αβ : U αβ → T n with t A αβ D (cid:16) u ′ αβ ◦ ( φ ′′ β ) − (cid:17) symmetric such that ϕ ′′ α ◦ ( ϕ ′′ β ) − is written as in (2.1).Then, A αβ ’s form a ˇCech one-cocycle { A αβ } ∈ C ( { U α } ; GL n ( Z )) and defines a cohomology class[ { A αβ } ] ∈ H ( e B ; GL n ( Z )). It is well known that H ( e B ; GL n ( Z )) is identified with the moduli spaceof homomorphisms from π ( e B ) to GL n ( Z ). Since π ( e B ) is trivial, there exists a ˇCech zero-cocycle { A α } ∈ C ( { U α } ; GL n ( Z )) such that A αβ = A α A − β on each U αβ . By using the cocycle we modify thelocal trivializations { ( e π − ( U α ) , ω | e π − ( U α ) , ϕ ′′ α ) } and the integral affine atlas { ( U α , φ ′′ α ) } by replacing ϕ ′′ α , φ ′′ α by ϕ ′ α ( e p ) := ( A − α × t A α ) ◦ φ ′′ α ( e p ) , φ ′ α := A − α φ ′′ α for each α ∈ A , respectively. Then, on each U αβ , ϕ ′ α ◦ ( ϕ ′ β ) − is written as ϕ ′ α ◦ ( ϕ ′ β ) − ( e x, y ) = (cid:0)e x + c αβ , y + u αβ ◦ ( φ ′ β ) − ( e x ) (cid:1) , where we set c αβ := A − α c ′ αβ and u αβ := t A α u ′ αβ . Then, c αβ ’s form a ˇCech one-cocycle { c αβ } ∈ C ( { U α } ; R n ) and defines a cohomology class [ { c αβ } ] ∈ H ( e B ; R n ). By the universal coefficients theorem, H ( e B ; R n ) is identified with Hom( H ( e B ; Z ) , R n ), which is trivial. So there exists a ˇCech zero-cocycle { c α } ∈ C ( { U α } ; R n ) such that c αβ = c α − c β on each U αβ . By using the cocycle, we again modify { ( e π − ( U α ) , ω | e π − ( U α ) , ϕ ′ α ) } and { ( U α , φ ′ α ) } by replacing ϕ ′ α , φ ′ α by ϕ α ( e p ) := ϕ ′ α ( e p ) − ( c α , , φ α ( e b ) := φ ′ α ( e b ) − c α , respectively for each α ∈ A . Then, on each U αβ , φ α coincides with φ β and ϕ α ◦ ϕ − β is written as ϕ α ◦ ϕ − β ( e x, y ) = ( e x, y + u αβ ◦ φ − β ( e x )) . Now we define the map dev : e B → R n by dev( e b ) := φ α ( e b )if e b lies in U α . It is well defined, and by construction, it is an integral affine immersion whose image is ∪ α ∈ A φ α ( U α ). ( M ′ , ω ′ ) is defined by( M ′ , ω ′ ) := a α ∈ A ( φ α ( U α ) × T n , ω ) / ∼ , where ( x α , y α ) ∈ φ α ( U α ) × T n and ( x β , y β ) ∈ φ β ( U β ) × T n are in the relation ( x α , y α ) ∼ ( x β , y β ) ifthey satisfy ( x α , y α ) = ϕ α ◦ ϕ − β ( x β , y β ), and π ′ : ( M ′ , ω ′ ) → O is defined to be the first projection. g dev : ( f M , e ω ) → ( M ′ , ω ′ ) is defined by g dev( e p ) := [ ϕ α ( e p )]if e p lies in e π − ( U α ).Without loss of generality, we can assume that each U α is connected,and for each γ ∈ Γ and α ∈ A there uniquely exists α ′ ∈ A such that the deck transformation σ γ maps U α onto U α ′ . Then, its lift e σ γ to ( f M , e ω ) maps e π − ( U α ) to e π − ( U α ′ ). By Lemma 2.17, φ α ′ ◦ σ γ ◦ φ − α can be written as φ α ′ ◦ σ γ ◦ φ − α ( e x ) = A α ′ αγ e x + c α ′ αγ for some A α ′ αγ ∈ GL n ( Z ), c α ′ αγ ∈ R n . Since φ α coincides with φ β on each overlap U αβ , φ α ′ ◦ φ γ ◦ φ α ( e x ) = A α ′ αγ e x + c α ′ αγ also agrees with φ β ′ ◦ φ γ ◦ φ β ( e x ) = A β ′ βγ e x + c β ′ βγ on the overlap φ α ( U αβ ) = φ β ( U αβ ). Thisimplies A α ′ αγ ’s and c α ′ αγ ’s do not depend on α and depends only on γ . In fact, for each γ ∈ Γ and α ∈ A ,we set A := { α ∈ A | A α ′ α γ = A α ′ αγ and c α ′ α γ = c α ′ αγ } .A contains all β ∈ A with U α β = ∅ . In particular, A is not empty since α ∈ A . Then, we have( ∪ α ∈ A U α ) ∪ ( ∪ α ∈ A r A U α ) = e B, ( ∪ α ∈ A U α ) ∩ ( ∪ α ∈ A r A U α ) = ∅ . If the compliment A r A is not empty, this contradicts to the connectedness of e B . So we denote themby A γ and c γ , respectively. Thus, we define the homomorphism ρ : Γ → GL n ( Z ) ⋉ R n by ρ γ := ( A γ , c γ ) . Γ acts on R n by ρ γ ( x ) = A γ x + c γ for γ ∈ Γ and x ∈ R n . The lift e ρ γ of ρ γ to ( M ′ , ω ′ ) is defined by e ρ γ ([ x α , y α ]) := [ ϕ α ′ ◦ e σ γ ◦ ϕ − α ( x α , y α )]if ( x α , y α ) lies in φ α ( U α ) × T n . By construction, e ρ is a lift of ρ , and e ρ and ρ satisfy g dev( e σ γ ( e p )) = e ρ γ ( g dev( e p ))and dev( σ γ ( e b )) = ρ γ (dev( e b )), respectively. (cid:3) Remark 2.23. (1) By construction, the n -dimensional torus T n acts freely on M ′ preserving ω ′ fromthe right so that π ′ : M ′ → O is a principal T n -bundle.(2) When π : ( M, ω ) → B admits a Lagrangian section, the restriction of π : ( R n × T n , ω ) → R n to O can be taken as π ′ : ( M ′ , ω ′ ) → O . In fact, in this case, since π : ( M, ω ) → B is identified with thecanonical model, we can take a system of local trivializations { ( π − ( U α ) , ϕ α ) } with u αβ = 0 on eachoverlaps U αβ . By applying the construction of π ′ : ( M ′ , ω ′ ) → O given in the proof of Proposition 2.22to such a { ( π − ( U α ) , ϕ α ) } we can show the claim.Suppose that ( M, ω ) is prequantizable and let ( L, ∇ L ) → ( M, ω ) be a prequantum line bundle. Wedenote by ( e L, ∇ e L ) → ( f M , e ω ) the pullback of ( L, ∇ L ) → ( M, ω ) to ( f M , e ω ). By definition, e L admits anatural lift of the Γ-action e σ on ( f M , e ω ) which preserves ∇ e L . The Γ-action on ( e L, ∇ e L ) is denoted by ee σ .Then, we have the following prequantum version of Proposition 2.22. Proposition 2.24.
There exists a prequantum line bundle ( L ′ , ∇ L ′ ) → ( M ′ , ω ′ ) , a bundle immersion gg dev : ( e L, ∇ e L ) → ( L ′ , ∇ L ′ ) which covers g dev , and a lift ee ρ of the Γ -action e ρ on ( M ′ , ω ′ ) to ( L ′ , ∇ L ′ ) suchthat gg dev is equivariant with respect to ee σ and ee ρ .Proof. Let { ( U α , φ α ) } and { ( e π − ( U α ) , ω | e π − ( U α ) , ϕ α ) } be the integral affine atlas of e B and the localtrivializations of e π : ( f M , e ω ) → e B obtained in the proof of Proposition 2.22, respectively. Then, for each α ∈ A there exists a prequantum line bundle ( φ α ( U α ) × T n × C , ∇ e L α ) → ( φ α ( U α ) × T n , ω ) and a bundleisomorphism ψ α : ( e L, ∇ e L ) | e π − ( U α ) → ( φ α ( U α ) × T n × C , ∇ e L α ) which covers ϕ . Now we define ( L ′ , ∇ L ′ )by ( L ′ , ∇ L ′ ) := a α ∈ A (cid:16) φ α ( U α ) × T n × C , ∇ e L α (cid:17) / ∼ , DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 11 where ( x α , y α , z α ) ∈ φ α ( U α ) × T n × C and ( x β , y β , z β ) ∈ φ β ( U β ) × T n × C are in the relation ( x α , y α , z α ) ∼ ( x β , y β , z β ) if they satisfy ( x β , y β , z β ) = ψ α ◦ ψ − β ( x β , y β , z β ). gg dev : ( e L, ∇ e L ) → ( L ′ , ∇ L ′ ) is defined by gg dev( e v ) := [ ψ α ( e v )]if e v lies in ( e L, ∇ e L ) | e π − ( U α ) .Suppose that for each γ ∈ Γ the deck transformation σ γ maps each U α to some U α ′ as before. Then, ee σ γ maps e L e π − ( U α ) to e L e π − ( U α ′ ) . Then, the Γ-action ee ρ is defined by ee ρ γ ( x α , y α , z α ) := [ ψ α ′ ◦ ee σ γ ◦ ψ − α ( x α , y α , z α )]if ( x α , y α , z α ) lies in φ α ( U α ) × T n × C . (cid:3) In the case where B is complete, we obtain the following corollary. Corollary 2.25.
Let π : ( M, ω ) → B be a Lagrangian fibration with connected n -dimensional base B and ( L, ∇ L ) → ( M, ω ) a prequantum line bundle on ( M, ω ) . Let p : e B → B be the universal cover-ing of B . Let us denote by ( f M , e ω ) the pullback of ( M, ω ) to e B and denote by ( e L, ∇ e L ) the pullback of ( L, ∇ L ) to ( f M , e ω ) . If B is complete, there exist an integral affine isomorphism dev : e B → R n , a fiber-preserving symplectomorphism g dev : ( f M , e ω ) → ( R n × T n , ω ) , and a bundle isomorphism gg dev : ( e L, ∇ e L ) → (cid:0) R n × T n × C , d − π √− x · dy (cid:1) such that g dev covers dev and gg dev covers g dev , respectively. Here x · dy denotes P ni =1 x i dy i . Moreover, let σ be the Γ -action on e B defined as the inverse of deck transforma-tions, e σ the natural lift of σ to ( f M , e ω ) , and ee σ the natural lift of e σ to ( e L, ∇ e L ) , respectively. Then, thereexist an integral affine Γ -action ρ : Γ → GL n ( Z ) ⋉ R n on R n , its lifts e ρ and ee ρ to ( R n × T n , ω ) and (cid:0) R n × T n × C , d − π √− x · dy (cid:1) , respectively such that dev , g dev , and gg dev are Γ -equivariant.Proof. By construction of g dev given in the proof of Proposition 2.22, if dev is bijective, so is g dev. Theargument in [10, p.696] and Theorem 2.20 also show that π : ( R n × T n , ω ) → R n is the unique La-grangian fibration on R n up to fiber-preserving symplectomorphism covering the identity. In particular, π ′ : ( M ′ , ω ′ ) → R n is identified with π : ( R n × T n , ω ) → R n .Concerning the prequantum line bundle, it is sufficient to show that ( R n × T n , ω ) has a unique pre-quantum line bundle (cid:0) R n × T n × C , d − π √− x · dy (cid:1) up to bundle isomorphism. Since ω is exact, anyprequantum line bundle on ( R n × T n , ω ) is trivial as a complex line bundle. Let (cid:0) R n × T n × C , d − π √− α (cid:1) be a prequantum line bundle on ( R n × T n , ω ) with connection d − π √− α . Then, α − x · dy definesa de Rham cohomology class in H ( R n × T n ; R ). Since H ( R n × T n ; R ) is isomorphic to H ( T n ; R ), interms of the generators dy i ’s of H ( T n ; R ), α − x · dy can be described as α − x · dy = n X i =1 τ i dy i + df for some τ , . . . , τ n ∈ R and f ∈ C ∞ ( R n × T n ). Now we define the bundle isomorphism ψ : R n × T n × C → R n × T n × C by ψ ( x, y, z ) := (cid:16) x + ( τ i ) , y, e − π √− f ( x,y ) z (cid:17) . Then, ψ satisfies ψ ∗ (cid:0) d − π √− x · dy (cid:1) = d − π √− α . (cid:3) Remark 2.26.
By Corollary 2.25, any Lagrangian fibration π : ( M, ω ) → B on a connected, complete B with prequantum line bundle ( L, ∇ L ) → ( M, ω ) is obtained as the quotient space of the Γ-action on π : ( R n × T n , ω ) → R n with prequantum line bundle ( R n × T n × C , d − π √− x · dy ) → ( R n × T n , ω ).By definition, the prequantum line bundle ( L, ∇ L ) → ( M, ω ) is equipped with a Hermitian metric h· , ·i L compatible with ∇ L . The pull-back of h· , ·i L to ( R n × T n × C , d − π √− x · dy ) → ( R n × T n , ω )coincides with the one induced from the standard Hermitian inner product on C up to constant. Infact, it is easy to see that, up to constant, it is the unique Hermitian metric on ( R n × T n × C , d − π √− x · dy ) → ( R n × T n , ω ) compatible with d − π √− x · dy . In the rest of this paper, we assume A Hermitian metric h· , ·i L on L is compatible with ∇ L if it satisfies d (cid:0) h s , s i L (cid:1) = (cid:10) ∇ L s , s (cid:11) L + (cid:10) s , ∇ L s (cid:11) L for all s , s ∈ Γ ( L ). that ( R n × T n × C , d − π √− x · dy ) → ( R n × T n , ω ) is always equipped with the Hermitian metricthough we do not specify it.2.4. The lifting problem of the Γ -action to the prequantum line bundle. In the rest of thissection, we investigate the condition for the Γ-action on π : ( R n × T n , ω ) → R n to have a lift to( R n × T n × C , d − π √− x · dy ) → ( R n × T n , ω ) in detail. Let ρ : Γ → GL n ( Z ) ⋉ R n be a Γ-action on R n and e ρ its lift to ( R n × T n , ω ). By Lemma 2.17, for each γ ∈ Γ, there exist A γ ∈ GL n ( Z ), c γ ∈ R n ,and a map u γ : R n → T n with t A γ Du γ symmetric such that ρ γ and e ρ γ can be described as follows(2.3) ρ γ ( x ) = A γ x + c γ , e ρ γ ( x, y ) = (cid:0) A γ x + c γ , t A − γ y + u γ ( x ) (cid:1) . Note that since (2.3) is a Γ-action, A γ , c γ , and u γ satisfy the following conditions(2.4) A γ γ = A γ A γ c γ γ = A γ c γ + c γ u γ γ ( x ) = t A − γ u γ ( x ) + u γ ( ρ γ ( x ))for γ , γ ∈ Γ, and x ∈ R n . Let e u γ = t ( e u γ , . . . , e u nγ ) : R n → R n be a lift of u γ . For e u γ and i = 1 , . . . , n , weput Z x i e u γ ( x ) dx i := t (cid:18)Z x i e u γ ( x ) dx i , . . . , Z x i e u nγ ( x ) dx i (cid:19) and F iγ ( x ) := (cid:18) t A γ Z x i e u γ ( x ) dx i (cid:19) i = n X j =1 (cid:0) t A γ (cid:1) ij Z x i e u jγ ( x ) dx i . Let N ∈ N be a positive integer. The Γ-action e ρ also preserves N ω . Then, we can show the followinglemma. Lemma 2.27. (1)
For each γ ∈ Γ , there exists a bundle automorphism ee ρ γ of (cid:0) R n × T n × C , d − π √− N x · dy (cid:1) which covers e ρ γ and preserves the connection if and only if c γ lies in N Z n . Moreover, in this case, ee ρ γ can be described as follows (2.5) ee ρ γ ( x, y, z ) = (cid:16)e ρ γ ( x, y ) , g γ e π √− N { e g γ ( x )+ c γ · ( t A − γ y ) } z (cid:17) , where g γ is an arbitrary element in U (1) and (2.6) e g γ ( x ) := ρ γ ( x ) · e u γ ( x ) − c γ · e u γ (0) − n X i =1 F iγ (0 , . . . , , x i , . . . , x n ) . The formula (2.5) does not depend on the choice of e u γ . (2) Under the condition given in (1) , the map ee ρ : Γ → Aut (cid:0)(cid:0) R n × T n × C , d − π √− N x · dy (cid:1)(cid:1) definedby (2.5) is a homomorphism if and only if the map g : Γ ∋ γ g γ ∈ U (1) is a homomorphism and forall γ , γ ∈ Γ and x ∈ R n , the following condition holds (cid:8) − c γ · u γ (0) + c γ · t A − γ u γ (0) + ρ γ ( c γ ) · u γ ( ρ γ (0)) (cid:9) − n X i =1 t A γ Z ( ρ γ ( x )) i u γ (0 , . . . , , τ i , ( ρ γ ( x )) i +1 , . . . , ( ρ γ ( x )) n ) dτ i ! i + n X i =1 (cid:18) t A γ t A γ Z x i u γ ( ρ γ (0 , . . . , τ i , x i +1 , . . . , x n )) dτ i (cid:19) i ∈ N Z . Proof.
For each γ ∈ Γ we put ee ρ γ ( x, y, z ) = (cid:16)e ρ γ ( x, y ) , e π { e g Rγ ( x,y )+ √− e g Iγ ( x,y ) } z (cid:17) , In the rest of this paper, we often use the notation u γ instead of e u γ . DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 13 where e g Rγ and e g Iγ are real valued functions on R n × T n . By the direct computation, it is easy to see that ee ρ γ preserves d − π √− N x · dy if and only if e g Rγ is constant and e g Iγ satisfy the following conditions ∂ x i e g Iγ = N ( A γ x + c γ ) · ∂ x i e u γ (2.7) ∂ y i e g Iγ = N ( A − γ c γ ) i (2.8)for i = 1 , . . . , n . From (2.7) we obtain(2.9) e g Iγ ( x, y ) = e g Iγ ( x , . . . , x i − , , x i +1 , . . . , x n , y ) + N n [ ρ γ ( x ) · e u γ ( x )] x i = x i x i =0 − F iγ ( x ) o . Using (2.9) recursively, we obtain(2.10) e g Iγ ( x, y ) = e g Iγ (0 , y ) + N ( ρ γ ( x ) · e u γ ( x ) − c γ · e u γ (0) − n X i =1 F iγ (0 , . . . , , x i , . . . , x n ) ) . (2.10) does not depend on the order of applying (2.9) to x i ’s. In fact, by applying (2.9) first to x i , thennext to x j , we obtain1 N e g Iγ ( x, y ) = 1 N e g Iγ ( x , . . . , x i − , , x i +1 , . . . , x n , y ) + Z x i ( A γ x + c γ ) · ∂ x i u γ ( x ) dx i = 1 N e g Iγ ( x , . . . , x i − , , x i +1 , . . . , x j − , , x j +1 , . . . , x n , y )+ Z x j ( A γ x + c γ ) · ∂ x j u γ ( x ) dx j (cid:12)(cid:12)(cid:12) x i =0 + Z x i ( A γ x + c γ ) · ∂ x i u γ ( x ) dx i . So, in order to see this, it is sufficient to show Z x j ( A γ x + c γ ) · ∂ x j u γ ( x ) dx j (cid:12)(cid:12)(cid:12) x i =0 + Z x i ( A γ x + c γ ) · ∂ x i u γ ( x ) dx i − Z x i ( A γ x + c γ ) · ∂ x i u γ ( x ) dx i (cid:12)(cid:12)(cid:12) x j =0 − Z x j ( A γ x + c γ ) · ∂ x j u γ ( x ) dx j (2.11)vanishes. Since t A γ Du γ is symmetric, we have ( t A γ ∂ x i u γ ( x )) j = ( t A γ Du γ ( x )) ji = ( t A γ Du γ ( x )) ij = (cid:0) t A γ ∂ x j u γ ( x ) (cid:1) i for all i, j = 1 , . . . , n . By using this, we can show(2.11) = Z x j ∂ x j (cid:18)Z x i ( A γ x + c γ ) · ∂ x i u γ ( x ) dx i (cid:19) dx j − Z x i ∂ x i (cid:18)Z x j ( A γ x + c γ ) · ∂ x j u γ ( x ) dx j (cid:19) dx i = Z x j Z x i (cid:0) ∂ x j ( A γ x + c γ ) (cid:1) · ∂ x i u γ ( x ) dx i dx j + Z x j Z x i ( A γ x + c γ ) · ∂ x j ∂ x i u γ ( x ) dx i dx j − Z x i Z x j ( ∂ x i ( A γ x + c γ )) · ∂ x j u γ ( x ) dx j dx i − Z x i Z x j ( A γ x + c γ ) · ∂ x i ∂ x j u γ ( x ) dx j dx i = Z x j Z x i (cid:0) t A γ ∂ x i u γ ( x ) (cid:1) j dx i dx j − Z x i Z x j (cid:0) t A γ ∂ x j u γ ( x ) (cid:1) i dx j dx i = 0 . By the same way, from (2.8) we obtain(2.12) e g Iγ ( x, y ) = e g Iγ ( x,
0) +
N c γ · t A − γ y. Thus, from (2.10) and (2.12) we have(2.13) e g Iγ ( x, y ) = e g Iγ (0 , N ( ρ γ ( x ) · e u γ ( x ) − c γ · e u γ (0) − n X i =1 F iγ (0 , . . . , , x i , . . . , x n ) + c γ · t A − γ y ) . Since y ∈ T n , e g Iγ should satisfies e π √− e g Iγ (0 ,e i ) = e π √− e g Iγ (0 , for all i = 1 , . . . , n and γ ∈ Γ. This holdsif and only if A − γ N c γ · e i ∈ Z for all i = 1 , . . . , n and γ ∈ Γ. Since A γ ∈ GL n ( Z ) this is equivalent to thecondition N c γ ∈ Z n . In this case, we put g γ := e π ( e g Rγ (0 , √− e g Iγ (0 , . Since ee ρ γ preserves the Hermitianmetric on ( R n × T n × C , d − π √− x · dy ) → ( R n × T n , ω ), g γ lies in U (1). The formula (2.5) does notdepend on the choice of e u γ since the difference of two lifts of u γ lies in Z n . This proves (1). The map ee ρ defined in (2) is a homomorphism if and only if e g Iγ ( x, y ) − e g Iγ (0 ,
0) defined by (2.13) satisfiesthe cocycle condition. By a direct computation using (2.4), it is equivalent to the ones given in (2). (cid:3)
Example 2.28.
Let B be the n -dimensional integral affine torus given in Example 2.3 (1) for a linearbasis v , . . . , v n ∈ R n . The product B × T n admits a symplectic structure ω so that the trivial torusbundle π : ( B × T n , ω ) → B becomes a Lagrangian fibration. This is obtained as the quotient space ofthe action of Γ := Z n on π : ( R n × T n , ω ) → R n which is defined by e ρ γ ( x, y ) = ( x + Cγ, y )for γ ∈ Γ and ( x, y ) ∈ R n × T n , where C = ( v · · · v n ) ∈ GL n ( R ). Let N ∈ N be a positive number. TheΓ-action e ρ on ( R n × T n , N ω ) has a lift to the prequantum line bundle ( R n × T n × C , d − π √− N x · dy ) → ( R n × T n , N ω ) if and only if all v i ’s lie in N Z n , and in this case ee ρ is given by ee ρ γ ( x, y, z ) = (cid:16)e ρ γ ( x, y ) , g γ e π √− NCγ · y z (cid:17) for γ ∈ Γ and ( x, y, z ) ∈ R n × T n × C , where g : Γ ∋ γ g γ ∈ U (1) is an arbitrary homomorphism. Example 2.29 (The Kodaira-Thurston manifold) . Let Γ be Z . Consider the Γ-action on π : (cid:0) R × T , ω (cid:1) → R which is defined by ρ γ ( x ) := x + γ, e ρ γ ( x, y ) := ( ρ γ ( x ) , y + u γ ( x ))for γ ∈ Γ and ( x, y ) ∈ R × T , where u γ ( x ) = t (0 , γ x ). The Lagrangian fibration given by the quotientof this action is denoted by π : ( M, ω ) → B . M was first observed by Kodaira in [24] and Thurston [36]pointed out in [36] that ( M, ω ) does not admits any K¨ahler structure. M is nowadays called the Kodaira-Thurston manifold. Let N ∈ N be a positive number. The Γ-action e ρ on (cid:0) R × T , N ω (cid:1) has a lift to theprequantum line bundle ( R × T × C , d − π √− N x · dy ) → (cid:0) R × T , N ω (cid:1) if and only if N is even,and in this case the lift ee ρ is given by ee ρ γ ( x, y, z ) = (cid:16)e ρ γ ( x, y ) , g γ e π √− N { γ x + γ γ x + γ · y } z (cid:17) for γ ∈ Γ and ( x, y, z ) ∈ R n × T n × C , where g : Γ ∋ γ g γ ∈ U (1) is an arbitrary homomorphism. Example 2.30.
Let B be the n -dimensional integral affine torus given in Example 2.3 (2) for a linear basis v , . . . , v n ∈ R n . When all v i ’s are integer vectors, i.e., v , . . . , v n ∈ Z n , we can generalize Example 2.28and Example 2.29 in the following way. Namely, for i, j = 1 , . . . , n , let u ij be an integer vector with u ij = u ji . For each γ ∈ Γ := Z n , define the map u γ : R n → T n by u γ ( x ) := u · γ · · · u n · γ ... ... u n · γ · · · u nn · γ x, and define the action of Γ on π : ( R n × T n , ω ) → R n by(2.14) e ρ γ ( x, y ) = ( x + Cγ, y + u γ ( x ))for γ ∈ Γ and ( x, y ) ∈ R n × T n , where C = ( v · · · v n ). Then, the quotient π : ( M, ω ) → B obtainedas the Γ-action (2.14) is a Lagrangian fibration on B . Let N ∈ N be a positive number. The Γ-action e ρ on ( R n × T n , N ω ) has a lift to the prequantum line bundle ( R n × T n × C , d − π √− N x · dy ) → ( R n × T n , N ω ) if and only if N v i · U j v i ∈ Z for all i, j = 1 , . . . , n , where U j := ( u ) j · · · ( u n ) j ... ...( u n ) j · · · ( u nn ) j . And in this case the lift ee ρ is given by ee ρ γ ( x, y, z ) = (cid:16)e ρ γ ( x, y ) , g γ e π √− N [ { ρ γ ( x ) · u γ ( ρ γ ( x )) − ρ γ (0) · u γ ( ρ γ (0)) } + ρ γ (0) · y ] z (cid:17) for γ ∈ Γ and ( x, y, z ) ∈ R n × T n × C , where g : Γ ∋ γ g γ ∈ U (1) is an arbitrary homomorphism. DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 15
Example 2.31.
Let n ≥ λ , . . . λ n − ∈ Z . Let Γ be the group ( Z n , ◦ ) given in Example 2.6. Foreach γ ∈ Γ, let A γ be the matrix A γ := λ λ . . . . . .1 λ n − γ n and u γ : R n → T n the map defined by u γ ( x ) := γ n x n . Consider the Γ-action e ρ on π : ( R n × T n , ω ) → R n which is defined by(2.15) e ρ γ ( x, y ) := (cid:0) A γ x + γ, t A − γ y + u γ ( x ) (cid:1) for γ ∈ Γ and ( x, y ) ∈ R n × T n , where C = ( v · · · v n ). Then, the quotient π : ( M, ω ) → B obtained asthe Γ-action (2.15) is a Lagrangian fibration on the integral affine manifold B obtained in Example 2.6.Let N ∈ N be a positive number. The Γ-action e ρ on ( R n × T n , N ω ) has a lift to the prequantum linebundle ( R n × T n × C , d − π √− N x · dy ) → ( R n × T n , N ω ) if and only if N is even, and in this casethe lift ee ρ is given by ee ρ γ ( x, y, z ) = (cid:16)e ρ γ ( x, y ) , g γ e π √− N { γ n x n ( x n + γ n ) + γ · ( t A − γ y ) } z (cid:17) for γ ∈ Γ and ( x, y, z ) ∈ R n × T n × C , where g : Γ ∋ γ g γ ∈ U (1) is an arbitrary homomorphism.3. Degree-zero harmonic spinors and integrability of almost complex structures
Let N ∈ N be a positive integer. For a compatible almost complex structure J on the total space ofthe Lagrangian fibration π : ( R n × T n , N ω ) → R n , let D be the associated Spin c Dirac operator withcoefficients in the prequantum line bundle (cid:0) R n × T n × C , d − π √− N x · dy (cid:1) → ( R n × T n , N ω ). Anelement in the kernel ker D of D is called a harmonic spinor. In this section, for J which is invariantalong the fiber, we investigate the existence condition of non-trivial degree-zero harmonic spinors, i.e.,non-trivial sections which lie in ker D . In the rest of this paper, we put f M := R n × T n and (cid:16)e L, ∇ e L (cid:17) :=( R n × T n × C , d − π √− x · dy ) for simplicity.3.1. Bohr-Sommerfeld points.
Let π : ( M, ω ) → B be a Lagrangian fibration with prequantum linebundle ( L, ∇ L ) → ( M, ω ). We recall the definition of Bohr-Sommerfeld points.
Definition 3.1.
A point b ∈ B is said to be Bohr-Sommerfeld if (cid:0) L, ∇ L (cid:1) | π − ( b ) admits a non-trivialcovariant constant section. We denote the set of Bohr-Sommerfeld points by B BS .Let us detect Bohr-Sommerfeld points for π : ( f M , N ω ) → R n with prequantum line bundle (cid:16)e L, ∇ e L (cid:17) ⊗ N → ( f M , N ω ). Lemma 3.2. x ∈ R n is a Bohr-Sommerfeld if and only if x lies in N Z n , i.e., R nBS = N Z n . Moreover,for a Bohr-Sommerfeld point x ∈ N Z n , a covariant constant section s of (cid:16)e L, ∇ e L (cid:17) ⊗ N (cid:12)(cid:12)(cid:12) π − ( x ) is of theform s ( y ) = s (0) e π √− Nx · y .Proof. For a fixed x ∈ R n , (cid:16)e L, ∇ e L (cid:17) ⊗ N (cid:12)(cid:12)(cid:12) π − ( x ) → π − ( x ) admits a non-trivial covariant constant section s if and only if s satisfies 0 = ∇ e L ⊗ N ∂ yi s = ∂ y i s − π √− N x i s for i = 1 , . . . , n . Hence, s should be of the form s ( y ) = s (0) e π √− Nx · y . Since s is global, s (0) = s ( e i ) = s (0) e π √− Nx i . This implies N x i ∈ Z for i = 1 , . . . , n . (cid:3) Remark 3.3.
Suppose that π : ( f M , N ω ) → R n is equipped with an action of a group Γ which preservesall the data, and its lift ee ρ to (cid:16)e L, ∇ e L (cid:17) ⊗ N is given by (2.5). Then, by Lemma 2.27 (1), the Γ-action ρ on R n preserves R nBS . When the Γ-action ρ on R n is properly discontinuous and free, let F ⊂ R n be afundamental domain of the Γ-action ρ on R n . Then, the map(3.1) Γ × (cid:18) F ∩ N Z n (cid:19) ∋ (cid:16) γ, mN (cid:17) N ρ γ (cid:16) mN (cid:17) ∈ Z n can be defined and is bijective. In particular, let π : ( M, N ω ) → B be a Lagrangian fibration withprequantum line bundle ( L, ∇ L ) ⊗ N → ( M, N ω ) obtained as the quotient space of the Γ-action. Then, F ∩ N Z n is identified with B BS .3.2. Almost complex structures.
Let S n be the Siegel upper half space, namely, the space of n × n symmetric complex matrices whose imaginary parts are positive definite S n := { Z = X + √− Y ∈ M n ( C ) | X, Y ∈ M n ( R ) , t Z = Z, and Y is positive definite } . It is well known that S n is identified with the space of compatible complex structures on the 2 n -dimensional standard symplectic vector space.For a tangent vector u = P ni =1 { ( u x ) i ∂ x i + ( u y ) i ∂ y i } ∈ T ( x,y ) f M at a point ( x, y ) ∈ f M we use thefollowing notation u = ( ∂ x , . . . , ∂ x n , ∂ y , . . . , ∂ y n ) ( u x ) ...( u x ) n ( u y ) ...( u y ) n = ( ∂ x , ∂ y ) (cid:18) u x u y (cid:19) , where ∂ x = ( ∂ x , . . . , ∂ x n ) , ∂ y = ( ∂ y , . . . , ∂ y n ) , u x = ( u x ) ...( u x ) n , u y = ( u y ) ...( u y ) n . In terms of the notations of tangent vectors u = ( ∂ x , ∂ y ) (cid:18) u x u y (cid:19) and v = ( ∂ x , ∂ y ) (cid:18) v x v y (cid:19) ∈ T ( x,y ) f M , ω canbe described by ω ( u, v ) = (cid:0) t u x , t u y (cid:1) (cid:18) I − I (cid:19) (cid:18) v x v y (cid:19) . Since the tangent bundle T f M is trivial, the space of C ∞ maps from f M to S n is identified with thespace of compatible almost complex structures on ( f M , ω ). For Z = X + √− Y ∈ C ∞ (cid:16) f M , S n (cid:17) , thecorresponding almost complex structure J Z is given as follows(3.2) J Z u := ( ∂ x , ∂ y ) (cid:18) XY − − Y − XY − XY − − Y − X (cid:19) ( x,y ) (cid:18) u x u y (cid:19) DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 17 for u = ( ∂ x , ∂ y ) (cid:18) u x u y (cid:19) ∈ T ( x,y ) f M . Then, the Riemannian metric g determined by ω and J Z can bedescribed by g ( u, v ) : = ω ( u, Jv )= (cid:0) t u x , t u y (cid:1) (cid:18) I − I (cid:19) (cid:18) XY − − Y − XY − XY − − Y − X (cid:19) (cid:18) v x v y (cid:19) = (cid:0) t u x , t u y (cid:1) (cid:18) Y − − Y − X − XY − Y + XY − X (cid:19) (cid:18) v x v y (cid:19) . (3.3)Let J = J Z be the almost complex structure on (cid:16) f M , ω (cid:17) corresponding to a given Z = X + √− Y ∈ C ∞ (cid:16) f M , S n (cid:17) . Then, ( − J∂ y , ∂ y ) = ( − J∂ y , . . . , − J∂ y n , ∂ y , . . . , ∂ y n ) is also a basis of the tangent spaceof (cid:16) f M , ω (cid:17) . With this basis, each tangent vector u ∈ T ( x,y ) f M can be written as u = X i { ( u H ) i ( − J∂ y i ) + ( u V ) i ∂ y i } = ( − J∂ y , ∂ y ) (cid:18) u H u V (cid:19) . Then, we have the following transition formula between ( ∂ x , ∂ y ) and ( − J∂ y , ∂ y ) u = ( − J∂ y , ∂ y ) (cid:18) u H u V (cid:19) = ( ∂ x , ∂ y ) (cid:18)(cid:18) − XY − Y + XY − X − Y − Y − X (cid:19) (cid:18) u H (cid:19) + (cid:18) u V (cid:19)(cid:19) . By this formula, we obtain the following lemma.
Lemma 3.4.
In terms of this notation, the Riemannian metric g defined by (3.3) can be described by g ( u, v ) = (0 , t u H ) (cid:18) Y − − Y − X − XY − Y + XY − X (cid:19) (cid:18) v H (cid:19) + (0 , t u V ) (cid:18) Y − − Y − X − XY − Y + XY − X (cid:19) (cid:18) v V (cid:19) . Suppose that a group Γ acts on π : ( f M , ω ) → R n and the Γ-actions ρ on R n and e ρ on ( f M , ω ) arewritten as in (2.3). Then, it is easy to see the following lemma. Lemma 3.5.
The Γ -action e ρ on ( f M , ω ) preserves the almost complex structure J = J Z on ( f M , ω ) corresponding to Z = X + √− Y ∈ C ∞ (cid:16) f M , S n (cid:17) if and only if the following conditions hold A γ ( XY − ) ( x,y ) = ( XY − ) e ρ γ ( x,y ) A γ − (cid:0) Y + XY − X (cid:1) e ρ γ ( x,y ) ( Du γ ) x (3.4) A γ ( Y + XY − X ) ( x,y ) = (cid:0) Y + XY − X (cid:1) e ρ γ ( x,y ) t A − γ (3.5) ( Du γ ) x ( XY − ) ( x,y ) + t A − γ Y − x,y ) = Y − e ρ γ ( x,y ) A γ − ( Y − X ) e ρ γ ( x,y ) ( Du γ ) x . (3.6) Proof.
For all γ ∈ Γ and ( x, y ) ∈ ( f M , ω ), the condition ( d e ρ γ ) ( x,y ) ◦ J ( x,y ) = J e ρ γ ( x,y ) ◦ ( d e ρ γ ) ( x,y ) impliesabove three equalities together with the following equality( Du γ ) x ( Y + XY − X ) ( x,y ) + t A − γ ( Y − X ) ( x,y ) = ( Y − X ) e ρ γ ( x,y ) t A − γ . But, this can be obtained from (3.4), (3.5), and t ( t A γ ( Du γ ) x ) = t A γ ( Du γ ) x . (cid:3) Let π : ( M, ω ) → B be a Lagrangian fibration with connected n -dimensional complete base B and p : e B → B the universal covering of B . By Corollary 2.25, the pullback of π : ( M, ω ) → B to e B isidentified with π : ( f M , ω ) → R n and π : ( M, ω ) → B can be obtained as the quotient of the Γ = π ( B )-action on π : ( f M , ω ) → R n . In particular, for each compatible almost complex structure J on ( M, ω ),there exists a map Z J = X + √− Y ∈ C ∞ (cid:16) f M , S n (cid:17) such that the pullback p ∗ J of J to p ∗ ( M, ω ) coincideswith J Z J . Then, we have the following lemma. (cid:18) XY − − Y − XY − XY − − Y − X (cid:19) ( x,y ) , ( XY − ) ( x,y ) etc. are the values of the maps (cid:18) XY − − Y − XY − XY − − Y − X (cid:19) , XY − etc.at ( x, y ). We will often omit the subscript “ ( x,y ) ” for simplicity unless it causes confusion. Lemma 3.6 ([14, Corollary 9.15]) . For any Lagrangian fibration π : ( M, ω ) → B , there exists a compatiblealmost complex structure J such that the corresponding map Z J does not depend on y , . . . , y n . We saysuch J to be invariant along the fiber.Proof. Take a Riemannian metric g ′ on ( M, ω ). Then, the pullback p ∗ g ′ is π ( B )-invariant. Moreover, p ∗ ( M, ω ) admits a free T n -action, and this T n -action together with the π ( B )-action forms an action ofthe semi-direct product π ( B ) ⋉ T n of T n and π ( B ). By averaging p ∗ g ′ over T n , we obtain a Riemannianmetric on p ∗ M invariant under the π ( B ) ⋉ T n -action. It is easy to see that p ∗ ω is also π ( B ) ⋉ T n -invariant, so by the standard method using the π ( B ) ⋉ T n -invariant Riemannian metric and p ∗ ω , we canobtain a π ( B ) ⋉ T n -invariant compatible almost complex structure on p ∗ ( M, ω ). In particular, since thealmost complex structure is still invariant under π ( B )-action, it descends to ( M, ω ). This is the requiredalmost complex structure. (cid:3)
The existence condition of non-trivial harmonic spinors of degree-zero.
For a map Z = X + √− Y ∈ C ∞ (cid:16) f M , S n (cid:17) , we set(3.7) Ω := (cid:0) Y + XY − X (cid:1) − ZY − . Ω has the following properties.
Lemma 3.7. (1) Ω = Z − , where Z = X − √− Y . (2) Ω is symmetric, i.e., t Ω = Ω .Proof.
A direct computation shows that Ω Z = I . This proves (1). (2) follows from (1) since Z issymmetric. (cid:3) Let N ∈ N be a positive integer. Let J = J Z be the compatible almost complex structure on (cid:16) f M , N ω (cid:17) corresponding to a given Z = X + √− Y ∈ C ∞ (cid:16) f M , S n (cid:17) . Then, the Riemannian metric N g := N ω ( · , J · )defines an isomorphism f : T ∗ f M ∼ = T f M by τ = N g ( f ( τ ) , · ) for τ ∈ T ∗ f M . For i = 1 , . . . , n , let Ω i denotethe i th column vector of Ω, and Re Ω i and Im Ω i be the real and imaginary parts of Ω i , respectively.Then, we can show the following lemma. Lemma 3.8.
For i = 1 , . . . , n , f ( dx i ) = − N J∂ y i , f ( dy i ) = ( − J∂ y , ∂ y ) (cid:18) N Re Ω i N Im Ω i (cid:19) . Proof.
We prove the latter. The former can be proved by the same way. Put f ( dy i ) = ( − J∂ y , ∂ y ) (cid:18) Y iH Y iV (cid:19) .By definition, for each i, j = 1 , . . . , n , we have dy i ( − J∂ y j ) = N g (cid:18) ( − J∂ y , ∂ y ) (cid:18) Y iH Y iV (cid:19) , ( − J∂ y , ∂ y ) (cid:18) e j (cid:19)(cid:19) (3.8) dy i ( ∂ y j ) = N g (cid:18) ( − J∂ y , ∂ y ) (cid:18) Y iH Y iV (cid:19) , ( − J∂ y , ∂ y ) (cid:18) e j (cid:19)(cid:19) . (3.9)Since − J∂ y j is written as − J∂ y j = ( ∂ x , ∂ y ) (cid:18) − XY − Y + XY − X − Y − Y − X (cid:19) (cid:18) e j (cid:19) by (3.2), the left hand side of (3.8) is (cid:0) Y − X (cid:1) ij . On the other hand, by Lemma 3.4, the right hand sideof (3.8) can be described as N Y iH · ( Y + XY − X ) e j . This implies Y − X = N t ( Y H · · · Y nH )( Y + XY − X ).Since Y is positive definite, so is Y + XY − X . In particular, N ( Y + XY − X ) is invertible. By using t X = X , t Y = Y together with this fact, we can obtain ( Y H · · · Y nH ) = N ( Y + XY − X ) − XY − . By thesame way, from (3.9), we obtain I = N t ( Y V · · · Y nV )( Y + XY − X ), i.e., ( Y V · · · Y nV ) = N ( Y + XY − X ) − .Hence, N Ω = ( Y H · · · Y nH ) + √− Y V · · · Y nV ). (cid:3) DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 19
Define the Hermitian metric on ( f M , N ω , N g, J ) by(3.10) h ( u, v ) := N g ( u, v ) + √− N g ( u, Jv )for u, v ∈ T ( x,y ) f M . Let ( W, c ) be the Clifford module bundle associated with (
N g, J ), i.e., as a complexvector bundle, W is defined by W := ∧ • (cid:16) T f M , J (cid:17) ⊗ C (cid:16)e L ⊗ N (cid:17) .W is equipped with the Hermitian metric induced from h and that on e L , and also equipped with theHermitian connection, which is denoted by ∇ W , induced from the Levi-Civita connection ∇ LC of (cid:16) f M , g (cid:17) and ∇ e L . c is the Clifford multiplication c : T f M → End C ( W ) defined by c ( u )( τ ) := u ∧ τ − u x h τ for u ∈ T f M and τ ∈ W , where x h is the contraction with respect to the Hermitian metric h on( f M , N ω , N g, J ). It is well known that W is identified with ∧ • ( T ∗ f M ) , ⊗ C (cid:16)e L ⊗ N (cid:17) as a Clifford modulebundle.Now let us define the Spin c Dirac operator D : Γ( W ) → Γ( W ) by the composition of the followingmaps D : Γ( W ) ∇ W / / Γ( T ∗ f M ⊗ W ) f ⊗ id W / / Γ( T f M ⊗ W ) c / / Γ( W ) . We compute the action of D on a degree zero element in Γ( W ). We identify a section of e L with a complexvalued function on f M . By using Lemma 3.8, for a section s of e L ⊗ N , Ds can be computed as Ds = c ◦ ( f ⊗ id W ) ◦ ∇ W s = c ◦ ( f ⊗ id W )( ds − π √− N x · dys )= n X i =1 (cid:8) c ( f ( dx i )) ( ∂ x i s ) + c ( f ( dy i )) (cid:0) ∂ y i s − π √− N x i s (cid:1)(cid:9) = − √− N n X i =1 ∂ y i ⊗ C ∂ x i s + n X j =1 Ω ij (cid:0) ∂ y j s − π √− N x j s (cid:1) . In particular, the equality Ds = 0 is equivalent to(3.11) 0 = ∂ x s ... ∂ x n s + Ω ∂ y s − π √− N x s ... ∂ y n s − π √− N x n s . Suppose that Z does not depend on y , . . . , y n as in Lemma 3.6. Then, by substituting a Fourierexpansion s = P m ∈ Z n a m ( x ) e π √− m · y of s with respect to y i ’s into (3.11), (3.11) can be reduced to thefollowing system of differential equations for a m ’s with variables x , . . . , x n (3.12) 0 = ∂ x a m ... ∂ x n a m + 2 π √− a m Ω( m − N x )for all m ∈ Z n . Lemma 3.9.
Let a m be a solution of (3.12) for some m ∈ Z n . If there exists p ∈ R n such that a m ( p ) = 0 .Then, a m ( x ) = 0 for all x ∈ R n .Proof. First, fix variables x , . . . , x n with p , . . . , p n . Then, the first entry of (3.12), i.e., 0 = ∂ x a m +2 π √− a m (Ω( m − N x )) can be thought of as an ordinary differential equation on x , and a m ( x , p , . . . , p n )is its solution with initial condition a m ( p ) = 0. On the other hand, the trivial solution also hasthe same initial condition. By the uniqueness of the solution of the ordinary differential equation, a m ( x , p , . . . , p n ) = 0 for any x . Next, by fixing variables x , . . . , x n with p , . . . , p n and fixing x with arbitrary value, a m ( x , x , p , . . . , p n ) is a solution of 0 = ∂ x a m + 2 π √− a m (Ω( m − N x )) with initial condition a m ( x , p , . . . , p n ) = 0. Then, a m ( x , x , p , . . . , p n ) = 0 for any x , x . By repeating theprocess for x , . . . , x n , we can show that a m ( x ) = 0. (cid:3) Lemma 3.10. If a m is a non trivial smooth solution of (3.12) for some m ∈ Z n , then, the condition (3.13) (( ∂ x i Ω) x ( m − N x )) j = (cid:0)(cid:0) ∂ x j Ω (cid:1) x ( m − N x ) (cid:1) i for all i, j = 1 , . . . , n, and all x ∈ R n holds.Conversely, if there exists m ∈ Z n such that (3.13) holds, then, (3.12) has a unique non trivialsolution up to constant. Moreover, in this case, each solution a m of (3.12) has the following form (3.14) a m ( x ) = a m (cid:16) mN (cid:17) e − π √− P ni =1 G im ( m N ,..., mi − N ,x i ,...,x n ) , where a m (cid:0) mN (cid:1) can be taken as an arbitrary constant in C and G im ( x ) := Z x imiN Ω( m − N x ) dx i ! i . Proof.
Since a m is smooth, a m satisfies ∂ x i ∂ x j a m = ∂ x j ∂ x i a m for all i, j = 1 , . . . , n . By differentiating(3.12), we have ∂ x i ∂ x j a m = − π √− a m ( − π √− n X k =1 Ω ik ( m k − N x k ) n X l =1 Ω jl ( m l − N x l ) + n X l =1 ( ∂ x i Ω jl ) ( m l − N x l ) − N Ω ji ) for i, j = 1 , . . . , n and x ∈ R n . The condition (3.13) is obtained from this equation.Conversely, suppose there exists m ∈ Z n such that (3.13) holds. By solving the differential equationappeared as the i th component of (3.12) for i = 1 , . . . , n , we have(3.15) a m ( x ) = a m (cid:16) x , . . . , x i − , m i N , x i +1 , . . . , x n (cid:17) e − π √− G im ( x ) . Using (3.15) recursively, we obtain the formula (3.14). By using (3.13), we can show that (3.14) doesnot depend on the order of applying (3.15) to x i ’s as in the proof of Lemma 2.27. Hence, (3.14) iswell-defined. (cid:3) For each m ∈ Z n for which the condition (3.13) holds, define the section s m ∈ Γ (cid:16)e L ⊗ N (cid:17) by(3.16) s m ( x, y ) := e π √− { − P ni =1 G im ( m N ,..., mi − N ,x i ,...,x n ) + m · y } . By the elliptic regularity of D and Lemma 3.10, we can obtain the following. Proposition 3.11. If s = P m ∈ Z n a m ( x ) e π √− m · y ∈ Γ (cid:16)e L ⊗ N (cid:17) is a non trivial solution of Ds , then,the condition (3.13) holds for all m ∈ Z n with a m = 0 . Conversely, suppose that there exists m ∈ Z n such that (3.13) holds. Then, the section s m defined by (3.16) satisfies Ds m . In particular, if (3.13) holds for all m ∈ Z n , then, { s m } m ∈ Z n is a linear basis of Γ (cid:16)e L ⊗ N (cid:17) ∩ ker D . The following proposition gives a geometric interpretation of the condition (3.13).
Proposition 3.12.
The following conditions are equivalent: (1)
The condition (3.13) holds for all m ∈ Z n . (2) ∂ x i Ω jk = ∂ x j Ω ik for all i, j, k = 1 , . . . , n . (3) ∇ LC J = 0 , where ∇ LC is the Levi-Civita connection with respect to g .Proof. If (3.13) holds for all m ∈ Z n , then, by putting m = 0, we have (( ∂ x i Ω) x x ) j = (cid:0)(cid:0) ∂ x j Ω (cid:1) x x (cid:1) i . Bysubstituting this to (3.13), we can see the condition (( ∂ x i Ω) x m ) j = (cid:0)(cid:0) ∂ x j Ω (cid:1) x m (cid:1) i holds for all m ∈ Z n .In particular, by substituting m = e k to this condition for each k = 1 , . . . , n , we can obtain (2). (2) ⇒ (1)is trivial.We show (2) ⇔ (3). (2) is equivalent to the following two conditions (cid:16)(cid:0) Y + XY − X (cid:1) − ∂ x i (cid:0) XY − (cid:1)(cid:17) jk = (cid:16)(cid:0) Y + XY − X (cid:1) − ∂ x j (cid:0) XY − (cid:1)(cid:17) ik (3.17) ∂ x i (cid:0) Y + XY − X (cid:1) − jk = ∂ x j (cid:0) Y + XY − X (cid:1) − ik (3.18) DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 21 for i, j, k = 1 , . . . , n . For i = 1 , . . . , n , we setΓ i := Γ i · · · Γ i n ... ...Γ ni · · · Γ ni n , where Γ kij is the Christoffel symbol. Then, (3) is equivalent to0 = ∂ i J + Γ i J − J Γ i ( i = 1 , . . . , n ) , where ∂ i = ( ∂ x i ( i = 1 , . . . , n ) ∂ y i − n ( i = n + 1 , . . . , n ) . It is also equivalent to the following conditions XY − ∂ x ( XY − ) i · · · ∂ x n ( XY − ) i ... ... ∂ x ( XY − ) ni · · · ∂ x n ( XY − ) ni (3.19) − ( Y + XY − X ) ∂ x ( Y − ) i − ∂ x ( Y − ) i · · · ∂ x n ( Y − ) i − ∂ x ( Y − ) ni ... ... ∂ x ( Y − ) ni − ∂ x n ( Y − ) i · · · ∂ x n ( Y − ) ni − ∂ x n ( Y − ) ni = ∂ x ( XY − ) i · · · ∂ x n ( XY − ) i ... ... ∂ x ( XY − ) ni · · · ∂ x n ( XY − ) ni XY − ,Y − ∂ x ( XY − ) i · · · ∂ x n ( XY − ) i ... ... ∂ x ( XY − ) ni · · · ∂ x n ( XY − ) ni (3.20) − Y − X ∂ x ( Y − ) i − ∂ x ( Y − ) i · · · ∂ x n ( Y − ) i − ∂ x ( Y − ) ni ... ... ∂ x ( Y − ) ni − ∂ x n ( Y − ) i · · · ∂ x n ( Y − ) ni − ∂ x n ( Y − ) ni = ∂ x ( Y − ) i − ∂ x ( Y − ) i · · · ∂ x n ( Y − ) i − ∂ x ( Y − ) ni ... ... ∂ x ( Y − ) ni − ∂ x n ( Y − ) i · · · ∂ x n ( Y − ) ni − ∂ x n ( Y − ) ni XY − + ∂ x ( Y − X ) i · · · ∂ x ( Y − X ) in ... ... ∂ x n ( Y − X ) i · · · ∂ x n ( Y − X ) in Y − , ( Y + XY − X ) ∂ x ( Y − X ) i · · · ∂ x ( Y − X ) in ... ... ∂ x n ( Y − X ) i · · · ∂ x n ( Y − X ) in (3.21) = ∂ x ( XY − ) i · · · ∂ x n ( XY − ) i ... ... ∂ x ( XY − ) ni · · · ∂ x n ( XY − ) ni ( Y + XY − X ) , Y − X ∂ x ( Y − X ) i · · · ∂ x ( Y − X ) in ... ... ∂ x n ( Y − X ) i · · · ∂ x n ( Y − X ) in (3.22) = ∂ x ( Y − ) i − ∂ x ( Y − ) i · · · ∂ x n ( Y − ) i − ∂ x ( Y − ) ni ... ... ∂ x ( Y − ) ni − ∂ x n ( Y − ) i · · · ∂ x n ( Y − ) ni − ∂ x n ( Y − ) ni ( Y + XY − )+ ∂ x ( Y − X ) i · · · ∂ x ( Y − X ) in ... ... ∂ x n ( Y − X ) i · · · ∂ x n ( Y − X ) in Y − X,XY − ∂ x ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) i ... ... ∂ x ( Y + XY − X ) ni · · · ∂ x n ( Y + XY − X ) ni (3.23) + ( Y + XY − X ) − ∂ x ( Y − X ) i + ∂ x ( XY − ) i · · · − ∂ x n ( Y − X ) i + ∂ x ( XY − ) in ... ... − ∂ x ( Y − X ) ni + ∂ x n ( XY − ) i · · · − ∂ x n ( Y − X ) ni + ∂ x n ( XY − ) in = ∂ x ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) i ... ... ∂ x ( Y + XY − X ) ni · · · ∂ x n ( Y + XY − X ) ni XY − ,Y − ∂ x ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) i ... ... ∂ x ( Y + XY − X ) ni · · · ∂ x n ( Y + XY − X ) ni (3.24) + Y − X − ∂ x ( Y − X ) i + ∂ x ( XY − ) i · · · − ∂ x n ( Y − X ) i + ∂ x ( XY − ) in ... ... − ∂ x ( Y − X ) ni + ∂ x n ( XY − ) i · · · − ∂ x n ( Y − X ) ni + ∂ x n ( XY − ) in = − − ∂ x ( Y − X ) i + ∂ x ( XY − ) i · · · − ∂ x n ( Y − X ) i + ∂ x ( XY − ) in ... ... − ∂ x ( Y − X ) ni + ∂ x n ( XY − ) i · · · − ∂ x n ( Y − X ) ni + ∂ x n ( XY − ) in XY − + ∂ x ( Y + XY − X ) i · · · ∂ x ( Y + XY − X ) in ... ... ∂ x n ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) in Y − , ( Y + XY − X ) ∂ x ( Y + XY − X ) i · · · ∂ x ( Y + XY − X ) in ... ... ∂ x n ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) in (3.25) = ∂ x ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) i ... ... ∂ x ( Y + XY − X ) ni · · · ∂ x n ( Y + XY − X ) ni ( Y + XY − X ) , DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 23 Y − X ∂ x ( Y + XY − X ) i · · · ∂ x ( Y + XY − X ) in ... ... ∂ x n ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) in (3.26) = − − ∂ x ( Y − X ) i + ∂ x ( XY − ) i · · · − ∂ x n ( Y − X ) i + ∂ x ( XY − ) in ... ... − ∂ x ( Y − X ) ni + ∂ x n ( XY − ) i · · · − ∂ x n ( Y − X ) ni + ∂ x n ( XY − ) in ( Y + XY − X )+ ∂ x ( Y + XY − X ) i · · · ∂ x ( Y + XY − X ) in ... ... ∂ x n ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) in Y − X. for i = 1 , . . . , n . It is easy to see that (3.22) and (3.26) are obtained by transposing (3.19) and (3.23),respectively. First, we show that (3.17) is equivalent to (3.21). In fact, (3.17) implies ∂ x ( XY − ) k · · · ∂ x ( XY − ) nk ... ... ∂ x n ( XY − ) k · · · ∂ x n ( XY − ) nk ( Y + XY − X ) − is symmetric for k = 1 , . . . , n . Since X , Y is symmetric, this implies (3.21). Next, we show (3.25) isequivalent to (3.18). (3.25) is equivalent to ∂ x ( Y + XY − X ) i · · · ∂ x ( Y + XY − X ) in ... ... ∂ x n ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) in ( Y + XY − X ) − (3.27) = ( Y + XY − X ) − ∂ x ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) i ... ... ∂ x ( Y + XY − X ) ni · · · ∂ x n ( Y + XY − X ) ni . By computing the ( j, k )-components of the both sides of (3.27), we obtain n X l =1 (cid:0) ∂ x j ( Y + XY − X ) − kl (cid:1) ( Y + XY − X ) li = n X l =1 (cid:16) ∂ x k ( Y + XY − X ) − jl (cid:17) ( Y + XY − X ) li for i, j, k = 1 , . . . , n . Here, we used0 = ∂ x j (cid:16)(cid:0) Y + XY − X (cid:1) (cid:0) Y + XY − X (cid:1) − (cid:17) = (cid:0) ∂ x j (cid:0) Y + XY − X (cid:1)(cid:1) (cid:0) Y + XY − X (cid:1) − + (cid:0) Y + XY − X (cid:1) ∂ x j (cid:0) Y + XY − X (cid:1) − and so on. Thus, ∂ x j ( Y + XY − X ) − km = n X i =1 n X l =1 ∂ x j (cid:0) ( Y + XY − X ) − kl (cid:1) ( Y + XY − X ) li ( Y + XY − X ) − im = n X i =1 n X l =1 (cid:16) ∂ x k ( Y + XY − X ) − jl (cid:17) ( Y + XY − X ) li ( Y + XY − X ) − im = ∂ x k ( Y + XY − X ) − jm . This implies (3.18). In particular, this means (3) ⇒ (2). We show (3.19), (3.20), (3.23), and (3.24) are obtained from (2). To show (3.23), it is sufficient toshow 0 =( Y + XY − X ) − XY − ∂ x ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) i ... ... ∂ x ( Y + XY − X ) ni · · · ∂ x n ( Y + XY − X ) ni (3.28) − ∂ x ( Y − X ) i · · · ∂ x n ( Y − X ) i ... ... ∂ x ( Y − X ) ni · · · ∂ x n ( Y − X ) ni − ( Y + XY − X ) − ∂ x ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) i ... ... ∂ x ( Y + XY − X ) ni · · · ∂ x n ( Y + XY − X ) ni XY − + ∂ x ( XY − ) i · · · ∂ x ( XY − ) in ... ... ∂ x n ( XY − ) i · · · ∂ x n ( XY − ) in . Since Ω is symmetric, so is its real part Re Ω = ( Y + XY − X ) − XY − . By taking the real part of (2)we also have ∂ x i (cid:0) ( Y + XY − X ) − XY − (cid:1) jk = ∂ x j (cid:0) ( Y + XY − X ) − XY − (cid:1) ik . By using these as well as (3.17) and (3.18), the ( j, k )-component of the first two terms of the right handside of (3.28) can be computed as X l (cid:0) ( Y + XY − X ) − XY − (cid:1) jl ∂ x k ( Y + XY − X ) li − ∂ x k ( Y − X ) ji = X l (cid:0) Y − X ( Y + XY − X ) − (cid:1) jl ∂ x k ( Y + XY − X ) li − ∂ x k ( Y − X ) ji = ∂ x k X l (cid:0) Y − X ( Y + XY − X ) − (cid:1) jl ( Y + XY − X ) li ! − X l (cid:16) ∂ x k (cid:0) Y − X ( Y + XY − X ) − (cid:1) jl (cid:17) ( Y + XY − X ) li − ∂ x k ( Y − X ) ji = − X l (cid:16) ∂ x k (cid:0) Y − X ( Y + XY − X ) − (cid:1) jl (cid:17) ( Y + XY − X ) li = − X l (cid:0) ∂ x j (cid:0) ( Y + XY − X ) − XY − (cid:1) kl (cid:1) ( Y + XY − X ) li . On the other hand, the ( j, k )-component of the last two terms of the right hand side of (3.28) can becomputed as − X m,l ( Y + XY − X ) − jl (cid:0) ∂ x m ( Y + XY − X ) li (cid:1) ( XY − ) mk + ∂ x j ( XY − ) ik = X m,l (cid:16) ∂ x m ( Y + XY − X ) − jl (cid:17) ( Y + XY − X ) li ( XY − ) mk + ∂ x j ( XY − ) ik = X m,l ( Y + XY − X ) li (cid:0) ∂ x j ( Y + XY − X ) − ml (cid:1) ( XY − ) mk + X m,l ( Y + XY − X ) li ( Y + XY − X ) − ml ∂ x j ( XY − ) mk = X l (cid:0) ∂ x j (cid:0) ( Y + XY − X ) − XY − (cid:1) kl (cid:1) ( Y + XY − X ) li . DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 25
This proves (3.28). We show (3.24). We put W := ∂ x ( Y + XY − X ) i · · · ∂ x n ( Y + XY − X ) i ... ... ∂ x ( Y + XY − X ) ni · · · ∂ x n ( Y + XY − X ) ni . By (3.23) and (3.25), we obtain − ∂ x ( Y − X ) i + ∂ x ( XY − ) i · · · − ∂ x n ( Y − X ) i + ∂ x ( XY − ) in ... ... − ∂ x ( Y − X ) ni + ∂ x n ( XY − ) i · · · − ∂ x n ( Y − X ) ni + ∂ x n ( XY − ) in = ( Y + XY − X ) − W XY − − ( Y + XY − X ) − XY − W and ( Y + XY − X ) t W = W ( Y + XY − X ) . In order to show (3.24) it is sufficient to check0 = Y − W + Y − X ( Y + XY − X ) − W XY − − Y − X ( Y + XY − X ) − XY − W (3.29) + ( Y + XY − X ) − W XY − XY − − ( Y + XY − X ) − XY − W XY − − t W Y − . By using above equalities, the right hand side of (3.29) can be computed as Y − W − Y − X ( Y + XY − X ) − XY − W + ( Y + XY − X ) − W XY − XY − − t W Y − = Y − W − ( Y + XY − X ) − XY − XY − W + ( Y + XY − X ) − W XY − XY − − ( Y + XY − X ) − W ( Y + XY − X ) Y − = Y − W − ( Y + XY − X ) − XY − XY − W + ( Y + XY − X ) − W XY − XY − − ( Y + XY − X ) − W − ( Y + XY − X ) − W XY − XY − = Y − W − { ( Y + XY − X ) − XY − X + ( Y + XY − X ) − Y } Y − W =0 . This proves (3.24).We show (3.19). To see this, we show0 =( Y + XY − X ) − XY − ∂ x ( XY − ) i · · · ∂ x n ( XY − ) i ... ... ∂ x ( XY − ) ni · · · ∂ x n ( XY − ) ni (3.30) − ∂ x ( Y − ) i − ∂ x ( Y − ) i · · · ∂ x n ( Y − ) i − ∂ x ( Y − ) ni ... ... ∂ x ( Y − ) ni − ∂ x n ( Y − ) i · · · ∂ x n ( Y − ) ni − ∂ x n ( Y − ) ni − ( Y + XY − X ) − ∂ x ( XY − ) i · · · ∂ x n ( XY − ) i ... ... ∂ x ( XY − ) ni · · · ∂ x n ( XY − ) ni XY − . The ( j, k )-component of the right hand side of (3.30) is X l (cid:0) ( Y + XY − X ) − XY − (cid:1) jl ∂ x k ( XY − ) li − ∂ x k Y − ji + ∂ x j Y − ki − X l,m ( Y + XY − X ) − jm ∂ x l ( XY − ) mi ( XY − ) lk = (cid:0) ( Y + XY − X ) − XY − ∂ x k ( XY − ) (cid:1) ji − ∂ x k Y − ji + ∂ x j Y − ki − X l,m ( Y + XY − X ) − lm ∂ x j ( XY − ) mi ( XY − ) lk = (cid:0) ( Y + XY − X ) − (cid:8) ∂ x k (cid:0) ( Y + XY − X ) Y − (cid:1) − ∂ x k ( XY − ) XY − (cid:9)(cid:1) ji − ∂ x k Y − ji + ∂ x j Y − ki − X l,m ( Y + XY − X ) − ml ( XY − ) lk ∂ x j ( XY − ) mi = (cid:0) ( Y + XY − X ) − (cid:0) ∂ x k ( Y + XY − X ) (cid:1) Y − + ∂ x k Y − − ( Y + XY − X ) − ∂ x k ( XY − ) XY − (cid:1) ji − ∂ x k Y − ji + ∂ x j Y − ki − X m (cid:0) ( Y + XY − X ) − XY − (cid:1) mk ∂ x j ( XY − ) mi = (cid:0) ( Y + XY − X ) − (cid:0) ∂ x k ( Y + XY − X ) (cid:1) Y − (cid:1) ji − (cid:0) ( Y + XY − X ) − ∂ x k ( XY − ) XY − (cid:1) ji + ∂ x j Y − ki − X m (cid:0) ( Y + XY − X ) − XY − (cid:1) km ∂ x j ( XY − ) mi = (cid:0) ( Y + XY − X ) − (cid:0) ∂ x k ( Y + XY − X ) (cid:1) Y − (cid:1) ji − (cid:0) ( Y + XY − X ) − ∂ x k ( XY − ) XY − (cid:1) ji + (cid:0) ∂ x j Y − − ( Y + XY − X ) − XY − ∂ x j ( XY − ) (cid:1) ki = (cid:0) − (cid:0) ∂ x k ( Y + XY − X ) − (cid:1) ( Y + XY − X ) Y − (cid:1) ji − X l (cid:0) ( Y + XY − X ) − ∂ x k ( XY − ) (cid:1) jl XY − li + (cid:0) ∂ x j Y − − ( Y + XY − X ) − XY − ∂ x j ( XY − ) (cid:1) ki = − X l ∂ x k ( Y + XY − X ) − jl (cid:0) ( Y + XY − X ) Y − (cid:1) li − X l (cid:0) ( Y + XY − X ) − ∂ x j ( XY − ) (cid:1) kl XY − li + (cid:0) ∂ x j Y − − ( Y + XY − X ) − XY − ∂ x j ( XY − ) (cid:1) ki = − X l ∂ x j ( Y + XY − X ) − kl (cid:0) ( Y + XY − X ) Y − (cid:1) li − (cid:0) ( Y + XY − X ) − ∂ x j ( XY − ) XY − (cid:1) ki + (cid:0) ∂ x j Y − − ( Y + XY − X ) − XY − ∂ x j ( XY − ) (cid:1) ki = (cid:0) − (cid:0) ∂ x j ( Y + XY − X ) − (cid:1) ( Y + XY − X ) Y − (cid:1) ki − (cid:0) ( Y + XY − X ) − ∂ x j ( XY − ) XY − (cid:1) ki + (cid:0) ∂ x j Y − − ( Y + XY − X ) − XY − ∂ x j ( XY − ) (cid:1) ki = (cid:0) ( Y + XY − X ) − (cid:0) ∂ x j ( Y + XY − X ) (cid:1) Y − (cid:1) ki − (cid:0) ( Y + XY − X ) − ∂ x j ( XY − ) XY − (cid:1) ki + (cid:0) ∂ x j Y − − ( Y + XY − X ) − XY − ∂ x j ( XY − ) (cid:1) ki = (cid:0) ( Y + XY − X ) − (cid:8)(cid:0) ∂ x j ( Y + XY − X ) (cid:1) Y − + ( Y + XY − X ) ∂ x j Y − − ∂ x j ( XY − ) XY − − XY − ∂ x j ( XY − ) (cid:9)(cid:1) ki = (cid:0) ( Y + XY − X ) − (cid:8) ∂ x j (( Y + XY − X ) Y − ) − ∂ x j ( XY − XY − ) (cid:9)(cid:1) ki =0 . This proves (3.19).Finally, we show (3.20). We put V := ∂ x ( XY − ) i · · · ∂ x n ( XY − ) i ... ... ∂ x ( XY − ) ni · · · ∂ x n ( XY − ) ni . By (3.19) and (3.21), we obtain ∂ x ( Y − ) i − ∂ x ( Y − ) i · · · ∂ x n ( Y − ) i − ∂ x ( Y − ) ni ... ... ∂ x ( Y − ) ni − ∂ x n ( Y − ) i · · · ∂ x n ( Y − ) ni − ∂ x n ( Y − ) ni = ( Y + XY − X ) − XY − V − ( Y + XY − X ) − V XY − and ( Y + XY − X ) t V = V ( Y + XY − X ) . In order to show (3.20) it is sufficient to check0 = Y − V − Y − X ( Y + XY − X ) − XY − V + Y − X ( Y + XY − X ) − V XY − (3.31) + ( Y + XY − X ) − V XY − XY − − ( Y + XY − X ) − XY − V XY − − t V Y − . Then, (3.31) can be checked in the same way as (3.29). (cid:3)
Remark 3.13.
When one of (hence, all) the conditions in Proposition 3.12 holds, (cid:16) f
M , ω , J, g (cid:17) is aK¨ahler manifold and J induces a natural holomorphic structure on e L such that ∇ e L is the canonicalconnection.3.4. The Γ -equivariant case. Suppose that π : ( f M , N ω , J ) → R n with prequantum line bundle (cid:16)e L, ∇ e L (cid:17) ⊗ N → ( f M , N ω , J ) is equipped with an action of a group Γ which preserves all the data, andthe Γ-actions are described by (2.3) and (2.5) as before. We assume that the Γ-action ρ on R n is prop-erly discontinuous and free. Since the Γ-action preserves all the data, the Spin c Dirac operator D isΓ-equivariant. In particular, Γ acts on Γ (cid:16)e L ⊗ N (cid:17) ∩ ker D . Lemma 3.14.
Let s = P m ∈ Z n a m ( x ) e π √− m · y be a section of e L ⊗ N . s is Γ -equivariant, i.e., ee ρ γ ◦ s = s ◦ e ρ γ for all γ ∈ Γ if and only if a m satisfies the following condition (3.32) a Nρ γ ( mN ) ( ρ γ ( x )) = g γ a m ( x ) e π √− N { e g γ ( x ) − ρ γ ( mN ) · u γ ( x ) } for all γ ∈ Γ , m ∈ Z n , and x ∈ R n . In particular, any Γ -equivariant section of e L ⊗ N can be written asfollows (3.33) s ( x, y ) = X ( γ, mN ) ∈ Γ × ( F ∩ N Z n ) g γ a m (cid:0) ρ γ − ( x ) (cid:1) e π √− N { e g γ ( ρ γ − ( x ) ) − ρ γ ( mN ) · u γ ( ρ γ − ( x ) ) } e π √− Nρ γ ( mN ) · y . Proof.
By computing the both sides separately, we have ee ρ γ ◦ s ( x, y ) = g γ e π √− N { e g γ ( x )+ c γ · t A − γ y } X m ∈ Z n a m ( x ) e π √− m · y = g γ X m ∈ Z n a m ( x ) e π √− N e g γ ( x ) e π √− Nρ γ ( mN ) · t A − γ y , (3.34) s ◦ e ρ γ ( x, y ) = X l ∈ Z n a l ( ρ γ ( x )) e π √− l · ( t A − γ y + u γ ( x ) )= X m ∈ Z n a Nρ γ ( mN ) ( ρ γ ( x )) e π √− Nρ γ ( mN ) · u γ ( x ) e π √− Nρ γ ( mN ) · t A − γ y . (3.35)Here, in the last equality, we replace l with N ρ γ (cid:0) mN (cid:1) . Note that the map Z n ∋ m N ρ γ (cid:0) mN (cid:1) ∈ Z n isbijective. Then, ee ρ γ ◦ s = s ◦ e ρ γ for all γ ∈ Γ implies g γ a m ( x ) e π √− N e g γ ( x ) = a Nρ γ ( mN ) ( ρ γ ( x )) e π √− Nρ γ ( mN ) · u γ ( x ) for all m ∈ Z n . In particular, by (3.1) and (3.32), s can be rewritten as follows s ( x, y ) = X l ∈ Z n a l ( x ) e π √− l · y (3.1) = X ( γ, mN ) ∈ Γ × ( F ∩ N Z n ) a Nρ γ ( mN )( x ) e π √− Nρ γ ( mN ) · y (3.32) = X ( γ, mN ) ∈ Γ × ( F ∩ N Z n ) g γ a m (cid:0) ρ γ − ( x ) (cid:1) e π √− N { e g γ ( ρ γ − ( x ) ) − ρ γ ( mN ) · u γ ( ρ γ − ( x ) ) } e π √− Nρ γ ( mN ) · y . (cid:3) In the Γ-equivariant case, the condition (3.13) has a symmetry in the following sense.
Lemma 3.15.
The condition (3.13) holds for some m ∈ Z n with m N ∈ F if and only if for any γ ∈ Γ , (3.13) holds for m = N ρ γ (cid:0) m N (cid:1) . Moreover, let a m be a non trivial solution of (3.12) for m . For each γ ∈ Γ , we define a Nρ γ ( m N ) in such a way that it satisfies (3.32) . Then, a Nρ γ ( m N ) is a non trivial solutionof (3.12) for m = N ρ γ (cid:0) m N (cid:1) .Proof. Suppose that there exists m ∈ Z n with m N ∈ F such that (3.13) holds. By Lemma 3.10, (3.12) for m has a non trivial solution a m . Then, for each γ ∈ Γ, define a Nρ γ ( m N ) by (3.32). By Lemma 3.10 again,in order to show this lemma, it is sufficient to prove a Nρ γ ( m N ) is a solution of (3.12) for m = N ρ γ (cid:0) m N (cid:1) .Let us compute the Jacobi matrix of the both sides of (3.32). The left hand side is D (cid:16) a Nρ γ ( m N ) ◦ ρ γ (cid:17) x = (cid:16) Da Nρ γ ( m N ) (cid:17) ρ γ ( x ) ( Dρ γ ) x (3.36) = (cid:16) ∂ x a Nρ γ ( m N ) , . . . , ∂ x n a Nρ γ ( m N ) (cid:17) ρ γ ( x ) A γ . The right hand side is D (cid:16) g γ a m ( x ) e π √− N { e g γ ( x ) − ρ γ ( mN ) · u γ ( x ) } (cid:17) x (3.37) = g γ e π √− N { e g γ ( x ) − ρ γ ( mN ) · u γ ( x ) } ( Da m ) x + g γ a m ( x ) D (cid:16) e π √− N { e g γ ( x ) − ρ γ ( mN ) · u γ ( x ) } (cid:17) (3.12) = − π √− g γ e π √− N { e g γ ( x ) − ρ γ ( mN ) · u γ ( x ) } a m ( x ) t (Ω x ( m − N x ))+ 2 π √− N g γ a m ( x ) e π √− N { e g γ ( x ) − ρ γ ( mN ) · u γ ( x ) } D (cid:16)e g γ ( x ) − ρ γ (cid:16) mN (cid:17) · u γ ( x ) (cid:17) (3.32) = − π √− a Nρ γ ( m N ) ( ρ γ ( x )) t (cid:16) Ω x A − γ (cid:16) N ρ γ (cid:16) mN (cid:17) − N ρ γ ( x ) (cid:17)(cid:17) + 2 π √− N a Nρ γ ( m N ) ( ρ γ ( x )) D (cid:16)e g γ ( x ) − ρ γ (cid:16) mN (cid:17) · u γ ( x ) (cid:17) . For each i = 1 , . . . , n , the direct computation shows ∂ x i (cid:16)e g γ ( x ) − ρ γ (cid:16) mN (cid:17) · u γ ( x ) (cid:17) = ( ∂ x i u γ ) x · (cid:16) ρ γ ( x ) − ρ γ (cid:16) mN (cid:17)(cid:17) + (cid:0) t A γ u γ ( x ) (cid:1) i − (cid:0) t A γ u γ (0 , . . . , , x i , . . . , x n ) (cid:1) i − X j
On the other hand, by (3.4) and (3.5), we have(3.39) t A γ Ω ρ γ ( x ) = Ω x A − γ + t ( Du γ ) x . This proves the lemma. (cid:3)
Remark 3.16.
By Remark 3.3 and Lemma 3.15, the condition (3.13) holds for all mN ∈ F ∩ N Z n if andonly if the condition (1), hence all conditions in Proposition 3.12 holds.4. The integrable case
Definition and properties of ϑ mN . We use the settting and the notations introduced in the previ-ous section. Let mN ∈ F ∩ N Z n be the point for which the condition (3.13) holds, and a m the non trivialsolution of (3.12) of the form (3.14) with a m (cid:0) mN (cid:1) = 1. For each γ ∈ Γ, define a Nρ γ ( mN ) in such a way thatit satisfies (3.32). As we showed in Lemma 3.15, a Nρ γ ( mN ) is a non trivial solution of (3.12) for N ρ γ (cid:0) mN (cid:1) .Then, we can define the formal Fourier series ϑ mN by(4.1) ϑ mN ( x, y ) := X γ ∈ Γ a Nρ γ ( mN )( x ) e π √− Nρ γ ( mN ) · y . Proposition 4.1. (1) ϑ mN has the following expression ϑ mN ( x, y ) = X γ ∈ Γ g γ e π √− h − P ni =1 G im (cid:16) m N ,..., mi − N , ( ρ γ − ( x ) ) i ,..., ( ρ γ − ( x ) ) n (cid:17) + N { e g γ ( ρ γ − ( x ) ) − ρ γ ( mN ) · u γ ( ρ γ − ( x ) ) + ρ γ ( mN ) · y } i . (2) ϑ mN can be described as ϑ mN = P γ ∈ Γ ee ρ γ ◦ s m ◦ e ρ γ − , where s m is the section defined by (3.16) . (3) If Y + XY − X is constant, then, ϑ mN converges absolutely and uniformly on any compact set.Proof. (1) and (2) are obtained by (3.32), (3.14), (2.5), and (3.16). Let us prove (3). By (2.4) and (3.5),we obtain t A γ − (cid:0) Y + XY − X (cid:1) − A γ − = (cid:0) Y + XY − X (cid:1) − . By using this formula together with the assumption, the expression in (1) can be rewritten as ϑ mN ( x, y ) = X γ ∈ Γ g γ e π √− h √− N ( x − ρ γ ( mN )) · ( Y + XY − X ) − ( x − ρ γ ( mN )) +real part i . Since (cid:0) Y + XY − X (cid:1) − is positive definite, there exists a positive constant c > (cid:0) Y + XY − X (cid:1) − ≥ cI . Then, (cid:12)(cid:12)(cid:12)(cid:12) g γ e π √− h √− N ( x − ρ γ ( mN )) · ( Y + XY − X ) − ( x − ρ γ ( mN )) +real part i (cid:12)(cid:12)(cid:12)(cid:12) = e − Nπ ( x − ρ γ ( mN )) · ( Y + XY − X ) − ( x − ρ γ ( mN )) ≤ e − cNπ k x − ρ γ ( mN ) k = e − cNπ k x − lN k (put l := N ρ γ (cid:16) mN (cid:17) )= n Y i =1 e − cNπ (cid:16) x i − liN (cid:17) . Hence, the series is dominated by Q ni =1 P l i ∈ Z e − cNπ ( liN − x i ) . Any compact set is contained in a productof closed intervals I × · · · × I n , so it is sufficient to show that P l ∈ Z e − cNπ ( lN − x ) converges uniformly onany closed interval I . Suppose that I is of the form I := [ x m , x M ]. Set l M := max { l ∈ Z | lN ∈ I } and l m := min { l ∈ Z | lN ∈ I } . On I , P − k ≤ l ≤ k e − cNπ ( lN − x ) can be estimated as X − k ≤ l ≤ k e − cNπ ( lN − x ) = X − k ≤ l By Corollary 2.25, any Lagrangian fibration π : ( M, ω ) → B on a connected complete base B withprequantum line bundle ( L, ∇ L ) → ( M, ω ) are obtained as the quotient of the action of Γ := π ( B ). Let J be a compatible almost complex structure on ( M, ω ) which is invariant along the fiber in the senseof Lemma 3.6 and D M the associated Spin c Dirac operator on ( M, N ω ) with coefficients in L ⊗ N . Sincethe Γ-action preserves all the data, Γ( L ⊗ N ) ∩ ker D M is identified with (cid:16) Γ (cid:16)e L ⊗ N (cid:17) ∩ ker D (cid:17) Γ . Moreover, F ∩ N Z n is identified with B BS as is noticed in Remark 3.3. Thus, we obtain the following corollary. Corollary 4.3. Let π : ( M, ω ) → B be a Lagrangian fibration on a connected complete base B and ( L, ∇ L ) → ( M, ω ) a prequantum line bundle. Let J be a compatible almost complex structure on ( M, ω ) which is invariant along the fiber in the sense of Lemma 3.6. Assume that J is integrable and ϑ mN DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 31 converges absolutely and uniformly for each mN ∈ F ∩ N Z n . Then, (cid:8) ϑ mN (cid:9) mN ∈ F ∩ N Z n gives a basis of thespace of holomorphic sections of ( L, ∇ L ) ⊗ N → ( M, N ω, J ) indexed by the Bohr-Sommerfeld points. Remark 4.4. When M is compact as well as the assumption of Corollary 4.3, we choose the orientationon M so that ( − n ( n − ( Nω ) n n ! is a positive volume form, and define the Hermitian inner product of thespace of sections of L ⊗ N by ( s, s ′ ) L ⊗ L := Z M h s, s ′ i L ⊗ N ( − n ( n − ( N ω ) n n ! , where h· , ·i L ⊗ N is the Hermitian metric of L ⊗ N . Then it is clear that (cid:8) ϑ mN (cid:9) mN ∈ F ∩ N Z n are orthogonalbasis. Example 4.5. For Example 2.30, Z = X + √− Y can be chosen so that Y + XY − X is a constant mapand XY − and Y − satisfy( XY − ) x = ( Y + XY − X ) u · C − x · · · u n · C − x ... ... u n · C − x · · · u nn · C − x , ( Y − ) x = u · C − x · · · u n · C − x ... ... u n · C − x · · · u nn · C − x ( Y + XY − X ) u · C − x · · · u n · C − x ... ... u n · C − x · · · u nn · C − x + Y + XY − X. In this case, Y + XY − X is necessarily I and Ω can be written asΩ x = u · C − x · · · u n · C − x ... ... u n · C − x · · · u nn · C − x + √− Y + XY − X ) − , and the condition (2) in Proposition 3.12 is equivalent to the following condition (cid:0) t C − u jk (cid:1) i = (cid:0) t C − u ik (cid:1) j for all i, j, k = 1 , . . . , n. Assume this condition as well as the condition N v i · U j v i ∈ Z for all i, j = 1 , . . . , n . Then, for each mN ∈ F ∩ N Z n , ϑ mN is described by ϑ mN ( x, y ) = X γ ∈ Γ g γ exp 2 π √− N n X i =1 X j>i (cid:16) ρ γ − ( x ) − mN (cid:17) i (cid:16) ρ γ − ( x ) − mN (cid:17) j (cid:0) t C − u ij (cid:1) · m N ... m i − N (cid:0) ρ γ − ( x ) + mN (cid:1) i (cid:0) ρ γ − ( x ) (cid:1) i +1 ... (cid:0) ρ γ − ( x ) (cid:1) n + 12 n X i =1 (cid:16) ρ γ − ( x ) − mN (cid:17) i (cid:0) t C − u ii (cid:1) · m N ... m i − N (cid:0) ρ γ − ( x ) + mN (cid:1) i (cid:0) ρ γ − ( x ) (cid:1) i +1 ... (cid:0) ρ γ − ( x ) (cid:1) n + 12 (cid:16) ρ γ − ( x ) − mN (cid:17) · u · γ · · · u n · γ ... ... u n · γ · · · u nn · γ + √− (cid:0) Y + XY − X (cid:1) − (cid:16) ρ γ − ( x ) − mN (cid:17) − mN · u · γ · · · u n · γ ... ... u n · γ · · · u nn · γ mN + ρ γ (cid:16) mN (cid:17) · y . By Proposition 4.1 (3), ϑ mN converges absolutely and uniformly on any compact set.4.2. The case when Z is constant. Let π : ( M, ω ) → B be a Lagrangian fibration on a complete n -dimensional B with prequantum line bundle ( L, ∇ L ) → ( M, ω ). Then, it is obtained as the quotient ofthe Γ := π ( B )-action on π : ( f M , ω ) → R n with prequantum line bundle (cid:16)e L, ∇ e L (cid:17) → ( f M , ω ). Supposethat the Γ-actions are described by (2.3) and (2.5) as before. Let J be a compatible almost complexstructure on ( M, ω ) and Z ∈ C ∞ ( f M , S n ) be the map corresponding to the pull-back of J to f M . Asituation in which (2) in Proposition 3.12 holds occurs when Z is a constant map. In this subsection, wediscuss this case in detail. Note that in this case, Du γ is a constant map for each γ ∈ Γ. It is obtained by(3.4). Moreover, as a special case of the setting in the previous subsection, we can obtain the followingtheorem. Theorem 4.6. (1) For each mN ∈ F ∩ N Z n , ϑ mN can be described as follows ϑ mN ( x, y ) = X γ ∈ Γ g γ e π √− N [ { ( ρ γ − ( x ) − mN ) · (Ω+ t A γ Du γ ) ( ρ γ − ( x ) − mN ) − mN · ( t A γ Du γ ) mN } − ρ γ ( mN ) · u γ (0)+ ρ γ ( mN ) · y ] . (2) For each mN ∈ F ∩ N Z n , ϑ mN converges absolutely and uniformly on any compact set. (3) J is integrable and (cid:8) ϑ mN (cid:9) mN ∈ F ∩ N Z n gives a basis of the space of holomorphic sections of ( L, ∇ L ) ⊗ N → ( M, N ω, J ) .Proof. (1) is obtained from Proposition 4.1 (1). (2) is obtained by the assumption and Proposition 4.1 (3).The first half of (3) is true since J is covariant constant with respect to the associated Levi-Civitaconnection. The other half is obtained by Corollary 4.3. (cid:3) When Z is constant, the associated Riemannian metric of M is flat. So, by Bieberbach’s theorem,if M is compact, then, M is finitely covered by the 2 n -dimensional torus T n , hence, ϑ mN ’s should beobtained from classical theta functions. So, let us see how ϑ mN ’s relate with classical theta functions forExample 2.28 with C = I , in which M itself is T n . First, let us briefly recall classical theta functions.For each T ∈ S n and a, b ∈ Q n , the theta function with characteristics is a holomorphic section on thetrivial holomorphic line bundle C n × C → C n which is defined by ϑ (cid:20) ab (cid:21) ( z, T ) := X γ ∈ Z n e π √− γ + a ) · T ( γ + a )+2 π √− γ + a ) · ( z + b ) . It is well-known that ϑ (cid:20) ab (cid:21) ( z, T ) has the following quasi-periodicity ϑ (cid:20) ab (cid:21) ( z + m, T ) = e π √− a · m ϑ (cid:20) ab (cid:21) ( z, T ) ,ϑ (cid:20) ab (cid:21) ( z + T m, T ) = e − π √− b · m e − π √− m · T m − π √− m · z ϑ (cid:20) ab (cid:21) ( z, T )for m ∈ Z n . For more detail, see [28, Chapter II, § 1] and [29, ß2]. Here we need the case when T = N Ω, a = mN , and b = 0. In this case, define the Z n = Z n × Z n -action on C n × C → C n by( γ, γ ′ ) · ( z, w ) := (cid:16) z + N ( − Ω γ + γ ′ ) , e − π √− Nγ · Ω γ +2 π √− γ · z w (cid:17) for ( γ, γ ′ ) ∈ Z n and ( z, w ) ∈ C n × C . Also define the Z n -action on the trivial complex line bundle R n × C → R n by(4.2) ( γ, γ ′ ) · ( x, y, w ) := (cid:16) x + γ, y + γ ′ , e π √− Nγ · y w (cid:17) DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 33 for ( γ, γ ′ ) ∈ Z n and ( x, y, w ) ∈ R n × C . Note that by taking the quotient of the latter Z n -action of(4.2), we can recover Example 2.28 with C = I and g γ = 1. Let F : R n → C n and e F : R n × C → C n × C be the R -linear isomorphism and the bundle isomorphism covering F which are defined by F ( x, y ) := N ( − Ω x + y ) , e F ( x, y, w ) := (cid:16) N ( − Ω x + y ) , e − π √− Nx · Ω x w (cid:17) . Then, the direct computation shows the following theorem. Theorem 4.7. (1) J √− I ◦ F = F ◦ ( J Z ) , i.e., F is a C -linear isomorphism from ( R n , J Z ) to the standardcomplex vector space ( C n , J √− I ) . (2) e F is equivariant with respect to the Z n -actions defined above. (3) ϑ mN satisfies e F ◦ ϑ mN ( x, y ) = ϑ (cid:20) mN (cid:21) ( F ( x, y ) , N Ω) , i.e., ϑ mN ( x, y ) = e π √− Nx · Ω x ϑ (cid:20) mN (cid:21) ( N ( − Ω x + y ) , N Ω) . Adiabatic-type limit. In this subsection let us consider a one parameter family { ( g t , J t ) } t> ofthe Riemannian metrics and the almost complex structures on a Lagrangian fibration so that the fibershrinks as t goes to ∞ , and investigate the behavior of ϑ mN defined by (4.1) when t goes to ∞ . We usethe same notations introduced in the previous sections.Let Z = X + √− Y ∈ C ∞ (cid:16) f M , S n (cid:17) be the map independent of y , . . . , y n . Let J = J Z be thecorresponding compatible almost complex structure on (cid:16) f M , ω (cid:17) . For each t > 0, we define the almostcomplex structure J t by J t u := ( − J∂ y , ∂ y ) (cid:18) − t t (cid:19) (cid:18) u H u V (cid:19) for u = ( − J∂ y , ∂ y ) (cid:18) u H u V (cid:19) ∈ T ( x,y ) f M . It is easy to see the following lemma. Lemma 4.8. (1) For any t > , J t is compatible with ω . The map Z t ∈ C ∞ (cid:16) f M , S n (cid:17) corresponding to J t is described as Z t = (cid:18) t X + √− Y (cid:19) Y − (cid:0) Y + XY − X (cid:1) (cid:18) tY + 1 t XY − X (cid:19) − Y.J t can be also written as J t (cid:18) ( ∂ x , ∂ y ) (cid:18) u x u y (cid:19)(cid:19) = ( ∂ x , ∂ y ) 1 t (cid:18) XY − − Y − XY − XY − (cid:0) t Y + XY − X (cid:1) (cid:0) Y + XY − X (cid:1) − − Y − X (cid:19) (cid:18) u x u y (cid:19) . (2) For any t > , let g t be the Riemannian metric corresponding to ω and J t . Then, for u =( − J∂ y , ∂ y ) (cid:18) u H u V (cid:19) , v = ( − J∂ y , ∂ y ) (cid:18) v H v V (cid:19) ∈ T ( x,y ) f M , g t can be written by g t ( u, v ) = ω (cid:0) u, J t v (cid:1) = t (0 , t u H ) (cid:18) Y − − Y − X − XY − Y + XY − X (cid:19) (cid:18) v H (cid:19) + 1 t (0 , t u V ) (cid:18) Y − − Y − X − XY − Y + XY − X (cid:19) (cid:18) v V (cid:19) . Suppose that a group Γ acts on π : ( f M , ω ) → R n and the Γ-actions ρ on R n and e ρ on ( f M , ω ) arewritten as in (2.3). Lemma 4.9. The Γ -action e ρ preserves J t (hence, g t ) for all t > if and only if e ρ preserves J . For J t and g t defined as above, the same arguments in Section 3.3 goes well, just by replacing J , g by J t , g t . For each t > 0, let ϑ t mN be the one defined by (4.1) for J t and g t . Let us investigate the behaviorof ϑ t mN as t goes to infinity. For t > 0, Ω t defined by (3.7) for Z t can be described as(4.3) Ω t = (cid:0) Y + XY − X (cid:1) − (cid:0) X + t √− Y (cid:1) Y − . Let D t be the corresponding Spin c Dirac operator. Then, for a section s of e L ⊗ N , D t s can be describedas D t s = − √− N n X i =1 ∂ y i ⊗ C ∂ x i s + n X j =1 (cid:0) Ω t (cid:1) ij (cid:0) ∂ y j s − π √− N x j s (cid:1) . (4.4)It is clear that Lemma 4.10. For any t > , the condition (2) in Proposition 3.12 holds for Ω t if and only if it holdsfor Ω = Ω . In particular, J t is integrable if and only if J is integrable. Suppose that π : ( f M , N ω , J ) → R n with prequantum line bundle (cid:16)e L, ∇ e L (cid:17) ⊗ N → ( f M , N ω , J ) isequipped with an action of a group Γ which preserves all the data, and the Γ-actions are describedby (2.3) and (2.5) as before. We assume that the Γ-action ρ on R n is properly discontinuous, free,and cocompact. Let π : ( M, N ω ) → B and ( L, ∇ L ) ⊗ N → ( M, N ω ) be the Lagrangian fibration and theprequantum line bundle on it obtained by the quotient of the Γ-action. On M , we consider the orientationso that ( − n ( n − ( Nω ) n n ! is a positive volume form, and define the L p -norm of a section s of L ⊗ N by k s k L p := (cid:18)Z M h s, s i p L ⊗ N ( − n ( n − ( N ω ) n n ! (cid:19) p , (4.5)where h· , ·i L ⊗ N is the Hermitian metric of L ⊗ N which is induced from the Hermitian metric h· , ·i e L ⊗ N of e L ⊗ N . As noticed in Remark 2.26, there exists a positive constant C such that h· , ·i e L ⊗ N can be writtenas h· , ·i e L ⊗ N = C h· , ·i C , where h· , ·i C is the standard Hermitian inner product on C .For each t > mN ∈ F ∩ N Z n for which the condition (3.13) holds, the corresponding ϑ t mN is defined by (4.1) for Ω t . We identify F ∩ N Z n with B BS the set of Bohr-Sommerfeld points of π : ( M, N ω ) → B with prequantum line bundle ( L, ∇ L ) ⊗ N → ( M, N ω ) and identify ϑ t mN with the sectionof ( L, ∇ L ) ⊗ N → ( M, N ω ) which is induced from ϑ t mN . Then, concerning the L p -norm, we have thefollowing lemma. Lemma 4.11. Suppose that Y + XY − X is constant. Then, the L p -norm of ϑ t mN can be calculated asfollows k ϑ t mN k pL p = C p det ( Y + XY − X ) (cid:18) Npt (cid:19) n . Proof. Let o ( B ) be the orientation bundle of B which is defined as the quotient bundle of the trivial realline bundle R n × R → R n on the universal cover of B by the Γ-action ρ ′ γ ( x, r ) := ( ρ γ ( x ) , (det A γ ) r ) for γ ∈ Γ and ( x, r ) ∈ R n × R . Then, we have a push-forward map π ∗ : Ω k ( M ) → Ω k − n ( B, o ( B )), whereΩ • ( B, o ( B )) is the de Rham complex twisted by o ( B ). B has a natural density which we denote by | dx | .For densities, see [8, Chapter I, § k ϑ t mN k pL p = Z M h ϑ t mN , ϑ t mN i p L ⊗ N ( − n ( n − ( N ω ) n n != Z B π ∗ (cid:18) h ϑ t mN , ϑ t mN i p L ⊗ N ( − n ( n − ( N ω ) n n ! (cid:19) = CN n X γ ∈ Γ Z F e − pNπt ( ρ γ − ( x ) − mN ) · ( Y + XY − X ) − ( ρ γ − ( x ) − mN ) | dx | . (4.6)By changing the coordinates as x ′ = ρ γ − ( x ),(4.6) = CN n X γ ∈ Γ Z ρ γ − ( F ) e − pNπt ( x ′ − mN ) · ( Y + XY − X ) − ( x ′ − mN ) | dx ′ | = CN n Z R n e − pNπt ( x ′ − mN ) · ( Y + XY − X ) − ( x ′ − mN ) | dx ′ | . (4.7) DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 35 Since Y + XY − X is positive definite, symmetric, there exists P ∈ O ( n ) such that Y + XY − X = t P λ . . . λ n P. Then, we define a positive definite symmetric matrix √ Y + XY − X by(4.8) p Y + XY − X := t P √ λ . . . √ λ n P, and put τ := p ( Y + XY − X ) − (cid:0) x ′ − mN (cid:1) . Then,(4.7) = C p det ( Y + XY − X ) N n Z R n e − pNπt k τ k | dτ | = C p det ( Y + XY − X ) N n n Y i =1 Z ∞−∞ e − pNπtτ i dτ i = C p det ( Y + XY − X ) N n (cid:18)r pN t (cid:19) n . (cid:3) We define the section δ mN of (cid:0) L, ∇ L (cid:1) ⊗ N | π − ( mN ) by(4.9) δ mN ( y ) := 1 C e π √− m · y . By Lemma 3.2, δ mN is a covariant constant section of (cid:0) L, ∇ L (cid:1) ⊗ N | π − ( mN ). Let T ∗ π M be the cotangentbundle along the fiber of π . On ( ∧ n T ∗ π M ) ⊗ π ∗ o ( B ) ∗ , there exists a natural section, i.e., a density alongthe fiber of π , say | dy | , which satisfies R π − ( x ) | dy | = 1 on each fiber of π . Then, we obtain the followingtheorem. Theorem 4.12. Suppose that Y + XY − X is constant. Then, the section ϑ t mN k ϑ t mN k L converges to a delta-function section supported on the fiber π − (cid:0) mN (cid:1) as t goes to ∞ in the following sense: for any section s of L ⊗ N , lim t →∞ Z M * s, ϑ t mN k ϑ t mN k L + L ⊗ N ( − n ( n − ( N ω ) n n ! = Z π − ( mN ) (cid:10) s, δ mN (cid:11) L ⊗ N | dy | . Proof. We denote by e s the pull-back of s to e L ⊗ N → f M . Since e s is Γ-equivariant, the Fourier expansionof e s can be written as in (3.33). Then, by using Proposition 4.1 (1), Z M * s, ϑ t mN k ϑ t mN k L + L ⊗ N ( − n ( n − ( N ω ) n n != Z B π ∗ * s, ϑ t mN k ϑ t mN k L + L ⊗ N ( − n ( n − ( N ω ) n n ! ! = CN n k ϑ t mN k L X γ ∈ Γ Z F a m (cid:0) ρ γ − ( x ) (cid:1) e − π √− P ni =1 G im (cid:16) m N ,..., mi − N , ( ρ γ − ( x ) ) i ,..., ( ρ γ − ( x ) ) n (cid:17) | dx | . (4.10) By putting x ′ = ρ γ − ( x ), we have(4.10) = CN n k ϑ t mN k L X γ ∈ Γ Z ρ γ − ( F ) a m ( x ′ ) e − π √− P ni =1 G im ( m N ,..., mi − N ,x ′ i ,...,x ′ n ) | dx ′ | = CN n k ϑ t mN k L Z R n a m ( x ′ ) e − π √− P ni =1 G im ( m N ,..., mi − N ,x ′ i ,...,x ′ n ) | dx ′ | = CN n k ϑ t mN k L Z R n a m ( x ′ ) e π √− P ni =1 Re G im ( m N ,..., mi − N ,x ′ i ,...,x ′ n ) e − πNt ( x ′ − mN ) · ( Y + XY − X ) − ( x ′ − mN ) | dx ′ | . (4.11)We put f ( x ′ ) := a m ( x ′ ) e π √− P ni =1 Re G im ( m N ,..., mi − N ,x ′ i ,...,x ′ n ) and τ := p ( Y + XY − X ) − (cid:0) x ′ − mN (cid:1) . Byusing Lemma 4.11 for p = 1, (4.11) can be written as follows(4.11) = CN n k ϑ t mN k L Z R n f ( x ′ ) e − πNt ( x ′ − mN ) · ( Y + XY − X ) − ( x ′ − mN ) | dx ′ | (4.12) = CN n k ϑ t mN k L p det ( Y + XY − X ) Z R n f (cid:16)p Y + XY − Xτ + mN (cid:17) e − πNt k τ k | dτ | = ( N t ) n Z R n f (cid:16)p Y + XY − Xτ + mN (cid:17) e − πNt k τ k | dτ | . It is well-known that lim t →∞ (4.12) = f (cid:16) mN (cid:17) = a m (cid:16) mN (cid:17) . On the other hand, by using the expression e s = X ( γ, m ′ N ) ∈ Γ × ( F ∩ N Z n ) a Nρ γ ( m ′ N )( x ) e π √− Nρ γ (cid:16) m ′ N (cid:17) · y , the right hand side can be computed as Z π − ( mN ) (cid:10) s, δ mN (cid:11) L ⊗ N | dy | = Z T n (cid:10)e s, δ mN (cid:11) e L ⊗ N | dy | = X ( γ, m ′ N ) ∈ Γ × ( F ∩ N Z n ) a Nρ γ ( m ′ N ) (cid:16) mN (cid:17) Z T n e π √− (cid:16) Nρ γ (cid:16) m ′ N (cid:17) − m (cid:17) · y | dy | . R T n e π √− (cid:16) Nρ γ (cid:16) m ′ N (cid:17) − m (cid:17) · y | dy | vanishes unless ρ γ (cid:16) m ′ N (cid:17) = mN . Since both m ′ N and mN lie in the fundamentaldomain F , this implies γ = e and m ′ = m , and in this case, R T n e π √− (cid:16) Nρ γ (cid:16) m ′ N (cid:17) − m (cid:17) · y | dy | = 1. Thus, Z π − ( mN ) (cid:10) s, δ mN (cid:11) L ⊗ N | dy | = a m (cid:16) mN (cid:17) . This proves the theorem. (cid:3) The non-integrable case We still use the same notations introduced in the previous sections. By Lemma 3.10, the equation(3.12) has no smooth solution for mN ∈ F ∩ N Z n such that (3.13) does not holds. For such mN , instead of(3.12), let us consider the following equation which is obtained from (3.12) by replacing Ω by its valueΩ mN at mN (5.1) 0 = ∂ x e a m ... ∂ x n e a m + 2 π √− e a m Ω mN ( m − N x ) . The equation (5.1) has a solution of the form e a m ( x ) = e a m (cid:16) mN (cid:17) e π √− N ( x − mN ) · Ω mN ( x − mN ) . DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 37 We put the initial condition e a m (cid:0) mN (cid:1) = 1 on the above e a m , and for each γ ∈ Γ, define e a Nρ γ ( mN ) in such away that it satisfies (3.32). Lemma 5.1. e a Nρ γ ( mN ) satisfies the following equality ∂ x e a Nρ γ ( mN )( x ) ... ∂ x n e a Nρ γ ( mN )( x ) + 2 π √− e a Nρ γ ( mN )( x )Ω x (cid:16) N ρ γ (cid:16) mN (cid:17) − N x (cid:17) (5.2) + 2 π √− e a Nρ γ ( mN )( x ) t A − γ (cid:16) Ω mN − Ω ρ γ − ( x ) (cid:17) A − γ (cid:16) N ρ γ (cid:16) mN (cid:17) − N x (cid:17) . Proof. By the same calculation as in the proof of Lemma 3.15, we have t A γ ∂ x e a Nρ γ ( mN ) ( ρ γ ( x ))... ∂ x n e a Nρ γ ( mN ) ( ρ γ ( x )) = − π √− e a Nρ γ ( mN ) ( ρ γ ( x )) (cid:0) Ω mN A − γ + t ( Du γ ) x (cid:1) (cid:16) N ρ γ (cid:16) mN (cid:17) − N ρ γ ( x ) (cid:17) . (5.2) can be obtained from this equation and (3.39). (cid:3) By using e a Nρ γ ( mN )’s, we define e ϑ mN in the same manner as ϑ mN , i.e., e ϑ mN ( x, y ) = X γ ∈ Γ e a Nρ γ ( mN )( x ) e π √− Nρ γ ( mN ) · y . e ϑ mN converges absolutely and uniformly on any compact set and can be written as e ϑ mN = X γ ∈ Γ ee ρ γ ◦ s ′ m ◦ e ρ γ − , where s ′ m is the section defined by s ′ m ( x, y ) := e π √− N ( x − mN ) · Ω mN ( x − mN ) +2 π √− m · y . In particular, when M is compact, it defines a section of L ⊗ N → M . Moreover, these two sections withdifferent mN and m ′ N are orthogonal to each other. These can be proved by the same way as Proposition 4.1.In the rest of this section, we assume that M is compact.Next let us consider the one parameter family of J t and g t defined in Section 4.3. Then, correspondingto J t and g t , we can obtain e ϑ t mN , which can be explicitly describe as e ϑ t mN ( x, y ) = X γ ∈ Γ g γ e π √− N h ( ρ γ − ( x ) − mN ) · Ω tmN ( ρ γ − ( x ) − mN ) + e g γ ( ρ γ − ( x ) ) − ρ γ ( mN ) · u γ ( ρ γ − ( x ) ) i e π √− Nρ γ ( mN ) · y , where Ω t mN is the value of Ω t given in (4.3) at mN . Then, e ϑ t mN has the following property. The proof issame as Theorem 4.12. Theorem 5.2. For each mN ∈ F ∩ N Z n , the section e ϑ t mN k e ϑ t mN k L converges to a delta-function sectionsupported on the fiber π − (cid:0) mN (cid:1) as t goes to ∞ in the following sense: for any section s of L ⊗ N , lim t →∞ Z M * s, e ϑ t mN k e ϑ t mN k L + L ⊗ N ( − n ( n − ( N ω ) n n ! = Z π − ( mN ) (cid:10) s, δ mN (cid:11) L ⊗ N | dy | . e ϑ t mN is not a solution of 0 = D t s , but we can show that e ϑ t mN approximates the solution of this equationin the following sense: Theorem 5.3. lim t →∞ k D t e ϑ t mN k L (( T M,J t ) ⊗ C L ) = 0 . Before proving the theorem, we have to make sure the meaning of L -norm in the left hand side. D t e ϑ t mN is a section of ( T M, J t ) ⊗ C L , and ( T M, J t ) ⊗ C L admits a Hermitian metric h· , ·i ( T M,J t ) ⊗ C L inducedby the one parameter version of (3.10) of ( T M, J t ) and the Hermitian metric of L . In terms of thisHermitian metric, the L -norm is defined as k D t e ϑ t mN k L (( T M,J t ) ⊗ C L ) := Z M h D t e ϑ t mN , D t e ϑ t mN i ( T M,J t ) ⊗ C L ( − n ( n − ( N ω ) n n ! . Proof. For n = 1, it is clear that the condition 3.13 automatically holds for all m ∈ Z . Thus, it is sufficientto prove the theorem for n ≥ 2. By the definition of e ϑ t mN and (5.2), D t e ϑ t mN can be written as D t e ϑ t mN = − √− N n X i =1 ∂ y i ⊗ C ∂ x i e ϑ t mN + n X j =1 (cid:0) Ω tx (cid:1) ij (cid:16) ∂ y j e ϑ t mN − π √− N x j e ϑ t mN (cid:17) = − √− N n X i =1 ∂ y i ⊗ C X γ ∈ Γ n ∂ x i e a Nρ γ ( mN )( x ) + 2 π √− e a Nρ γ ( mN )( x ) (cid:16) Ω tx (cid:16) N ρ γ (cid:16) mN (cid:17) − N x (cid:17)(cid:17) i o e π √− Nρ γ ( mN ) · y = − π n X i =1 ∂ y i ⊗ C X γ ∈ Γ e a Nρ γ ( mN )( x ) (cid:16) t A − γ (cid:16) Ω t mN − Ω tρ γ − ( x ) (cid:17) (cid:16) mN − ρ γ − ( x ) (cid:17)(cid:17) i e π √− Nρ γ ( mN ) · y . Then, h D t e ϑ t mN , D t e ϑ t mN i ( T M,J t ) ⊗ C L = (2 π ) X γ ,γ ∈ Γ X i ,i h e a Nρ γ ( mN )( x ) e π √− Nρ γ ( mN ) · y , e a Nρ γ ( mN )( x ) e π √− Nρ γ ( mN ) · y i L ⊗ N N g t (cid:0) ∂ y i , ∂ y i (cid:1) × (cid:18) t A − γ (cid:18) Ω t mN − Ω tρ γ − ( x ) (cid:19) (cid:16) mN − ρ γ − ( x ) (cid:17)(cid:19) i (cid:18) t A − γ (cid:18) Ω t mN − Ω tρ γ − ( x ) (cid:19) (cid:16) mN − ρ γ − ( x ) (cid:17)(cid:19) i = (2 π ) Nt X γ ,γ ∈ Γ h e a Nρ γ ( mN )( x ) e π √− Nρ γ ( mN ) · y , e a Nρ γ ( mN )( x ) e π √− Nρ γ ( mN ) · y i L ⊗ N × (cid:18) t A − γ (cid:18) Ω t mN − Ω tρ γ − ( x ) (cid:19) (cid:16) mN − ρ γ − ( x ) (cid:17)(cid:19) · (cid:0) Y + XY − X (cid:1) x (cid:18) t A − γ (cid:18) Ω t mN − Ω tρ γ − ( x ) (cid:19) (cid:16) mN − ρ γ − ( x ) (cid:17)(cid:19) . For each x ∈ F and u ∈ C n , define the norm of u with respect to ( Y + XY − X ) x by k u k Y + XY − X ) x := u · ( Y + XY − X ) x u. By (3.5), for each γ ∈ Γ, k u k Y + XY − X ) x satisfies k t A γ u k Y + XY − X ) x = k u k Y + XY − X ) ργ ( x ) . In terms of this norm, we obtain k D t e ϑ t mN k L (( T M,J t ) ⊗ C L ) = (2 π ) CN n +1 t X γ ∈ Γ Z F e − πNt ( ρ γ − ( x ) − mN ) · ( Y + XY − X ) − mN ( ρ γ − ( x ) − mN ) × (cid:16) t A − γ (cid:16) Ω t mN − Ω tρ γ − ( x ) (cid:17) (cid:16) mN − ρ γ − ( x ) (cid:17)(cid:17) · (cid:0) Y + XY − X (cid:1) x (cid:16) t A − γ (cid:16) Ω t mN − Ω tρ γ − ( x ) (cid:17) (cid:16) mN − ρ γ − ( x ) (cid:17)(cid:17) | dx | = (2 π ) CN n +1 t X γ ∈ Γ Z F e − πNt ( ρ γ − ( x ) − mN ) · ( Y + XY − X ) − mN ( ρ γ − ( x ) − mN ) × k t A γ − (cid:16) Ω t mN − Ω tρ γ − ( x ) (cid:17) (cid:16) mN − ρ γ − ( x ) (cid:17) k Y + XY − X ) x | dx | = (2 π ) CN n +1 t X γ ∈ Γ Z F e − πNt ( ρ γ − ( x ) − mN ) · ( Y + XY − X ) − mN ( ρ γ − ( x ) − mN ) DIABATIC LIMITS, THETA FUNCTIONS, AND GEOMETRIC QUANTIZATION 39 × k (cid:16) Ω t mN − Ω tρ γ − ( x ) (cid:17) (cid:16) mN − ρ γ − ( x ) (cid:17) k Y + XY − X ) ργ − x ) | dx | = (2 π ) CN n +1 t X γ ∈ Γ Z ρ γ − ( F ) e − πNt ( x ′ − mN ) · ( Y + XY − X ) − mN ( x ′ − mN ) × k (cid:16) Ω t mN − Ω tx ′ (cid:17) (cid:16) mN − x ′ (cid:17) k Y + XY − X ) x ′ | dx ′ | ( ∵ x ′ := ρ γ − ( x ))= (2 π ) CN n +1 t Z R n k (cid:16) Ω t mN − Ω tx ′ (cid:17) (cid:16) mN − x ′ (cid:17) k Y + XY − X ) x ′ e − πNt ( x ′ − mN ) · ( Y + XY − X ) − mN ( x ′ − mN ) | dx ′ | . Since Ω t can be described as (4.3), k (cid:16) Ω t mN − Ω tx ′ (cid:17) (cid:16) mN − x ′ (cid:17) k Y + XY − X ) x ′ = k (cid:0) Re (cid:0) Ω mN − Ω x ′ (cid:1)(cid:1) (cid:16) mN − x ′ (cid:17) k Y + XY − X ) x ′ + t k (cid:0) Im (cid:0) Ω mN − Ω x ′ (cid:1)(cid:1) (cid:16) mN − x ′ (cid:17) k Y + XY − X ) x ′ . We put R ( x ′ ) := k (cid:0) Re (cid:0) Ω mN − Ω x ′ (cid:1)(cid:1) (cid:16) mN − x ′ (cid:17) k Y + XY − X ) x ′ ,I ( x ′ ) := k (cid:0) Im (cid:0) Ω mN − Ω x ′ (cid:1)(cid:1) (cid:16) mN − x ′ (cid:17) k Y + XY − X ) x ′ . By changing coordinates as τ := q ( Y + XY − X ) − mN (cid:0) x ′ − mN (cid:1) , k D t e ϑ t mN k L (( T M,J t ) ⊗ C L ) can be written by k D t e ϑ t mN k L (( T M,J t ) ⊗ C L ) = 2 − n π CN n +1 q det( Y + XY − X ) mN × (cid:26) t − − n Z R n R (cid:16)q ( Y + XY − X ) mN τ + mN (cid:17) (2 N t ) n e − πNt k τ k | dτ | + t − n Z R n I (cid:16)q ( Y + XY − X ) mN τ + mN (cid:17) (2 N t ) n e − πNt k τ k | dτ | (cid:27) . It is well-known thatlim t →∞ Z R n R (cid:16)q ( Y + XY − X ) mN τ + mN (cid:17) (2 N t ) n e − πNt k τ k | dτ | = R (cid:16) mN (cid:17) = 0 , lim t →∞ Z R n I (cid:16)q ( Y + XY − X ) mN τ + mN (cid:17) (2 N t ) n e − πNt k τ k | dτ | = I (cid:16) mN (cid:17) = 0 . Since n ≥ 2, this proves Theorem 5.2. (cid:3) Example 5.4. For Example 2.29, let us consider the compatible almost complex structure associatedwith Z := (cid:18) x (cid:19) + √− (cid:18) x +1 00 1 (cid:19) . The corresponding Ω is Ω x = (cid:18) √− x + √− (cid:19) . This Z does not satisfies (2) in Proposition 3.12, nor the condition 3.13 for any m ∈ Z . In fact, forany m ∈ Z , (( ∂ x Ω) ( m − N x )) = m − N x while (( ∂ x Ω) ( m − N x )) = 0. In this case, e ϑ t mN can bewritten as e ϑ t mN ( x, y ) = X γ ∈ Z g γ e π √− N [ { t √− x − γ − m N ) + ( m N + t √− ) ( x − γ − m N ) } +( x − γ ) { γ ( x + γ ) − ( m N + γ ) γ } ] e π √− m + Nγ ) · y . Example 5.5. In the case where n = 2 of Example 2.31, we can take the compatible almost complexstructure associated with Z := 1 x + 1 (cid:18) λ x λx λx x (cid:19) + √− x + 1 (cid:18) (1 + λ ) x + 1 λx λx (cid:19) . The corresponding Ω is Ω x = (cid:18) √− −√− λx −√− λx x + √− λ x + 1) (cid:19) . In this case, ∂ x Ω = −√− λ and ∂ x Ω = 0. So, Z satisfies (2) in Proposition 3.12 if and only if λ = 0, which is the special case of Example 4.5. Equivalently, Z does not satisfy the condition 3.13 forany m ∈ Z unless λ = 0. In fact, for any m ∈ Z , (( ∂ x Ω) ( m − N x )) = 0 while (( ∂ x Ω) ( m − N x )) = −√− λ ( m − N x ). In this case, e ϑ t mN can be written as e ϑ t mN ( x, y ) = X γ ∈ Γ g γ e π √− N h t √− { x − γ − γ λ ( x − γ ) − m N } − t √− λ m N { x − γ − γ λ ( x − γ ) − m N } ( x − γ − m N ) + (cid:26) m N + t √− (cid:18) λ m N +1 (cid:19)(cid:27) ( x − γ − m N ) + γ ( x − γ )( x + γ ) − ( m N + γ ) γ ( x − γ ) (cid:21) × e π √− { ( m + γ λm + Nγ ) y +( m + Nγ ) y } . References 1. J. E. 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(2011), no. 5, 1914–1955. Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, 214-8571, Japan E-mail address ::