Adiabatic Limit, Theta Function, and Geometric Quantization
Let π : (, ) → be a non-singular Lagrangian torus fibration on a complete base with prequantum line bundle (, ∇ᴸ) → (, ). Compactness on is not assumed. For a positive integer and a compatible almost complex structure on (, ) invariant along the fiber of π, let be the associated Spinᶜ Dirac op...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2024 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2024
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212355 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Adiabatic Limit, Theta Function, and Geometric Quantization. Takahiko Yoshida. SIGMA 20 (2024), 065, 52 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860280148649050112 |
|---|---|
| author | Yoshida, Takahiko |
| author_facet | Yoshida, Takahiko |
| citation_txt | Adiabatic Limit, Theta Function, and Geometric Quantization. Takahiko Yoshida. SIGMA 20 (2024), 065, 52 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Let π : (, ) → be a non-singular Lagrangian torus fibration on a complete base with prequantum line bundle (, ∇ᴸ) → (, ). Compactness on is not assumed. For a positive integer and a compatible almost complex structure on (, ) invariant along the fiber of π, let be the associated Spinᶜ Dirac operator with coefficients in ⊗ᴺ. First, in the case where is integrable, under certain technical conditions on , we give a complete orthogonal system {ϑb}b ∈ BS of the space of holomorphic ²-sections of ⊗ᴺ indexed by the Bohr-Sommerfeld points BS such that each ϑb converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber π⁻¹(b) by the adiabatic(-type) limit. We also explain the relation of ϑb with Jacobi's theta functions when (, ) is ²ⁿ. Second, in the case where is not integrable, we give an orthogonal family {ϑ~b}b ∈ BS of ²-sections of ⊗ᴺ indexed by BS which has the same property as above, and show that each ϑ~b converges to 0 by the adiabatic(-type) limit with respect to the ²-norm.
|
| first_indexed | 2026-03-17T10:05:36Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 065, 52 pages
Adiabatic Limit, Theta Function,
and Geometric Quantization
Takahiko YOSHIDA
Department of Mathematics, School of Science and Technology, Meiji University,
1-1-1 Higashimita, Tama-ku, Kawasaki, 214-8571, Japan
E-mail: takahiko@meiji.ac.jp
Received March 20, 2023, in final form July 06, 2024; Published online July 19, 2024
https://doi.org/10.3842/SIGMA.2024.065
Abstract. Let π : (M,ω) → B be a non-singular Lagrangian torus fibration on a complete
base B with prequantum line bundle
(
L,∇L
)
→ (M,ω). Compactness onM is not assumed.
For a positive integer N and a compatible almost complex structure J on (M,ω) invariant
along the fiber of π, let D be the associated Spinc Dirac operator with coefficients in L⊗N .
First, in the case where J is integrable, under certain technical condition on J , we give a com-
plete orthogonal system {ϑb}b∈BBS
of the space of holomorphic L2-sections of L⊗N indexed
by the Bohr–Sommerfeld points BBS such that each ϑb converges to a delta-function sec-
tion supported on the corresponding Bohr–Sommerfeld fiber π−1(b) by the adiabatic(-type)
limit. We also explain the relation of ϑb with Jacobi’s theta functions when (M,ω) is T 2n.
Second, in the case where J is not integrable, we give an orthogonal family
{
ϑ̃b
}
b∈BBS
of
L2-sections of L⊗N indexed by BBS which has the same property as above, and show that
each Dϑ̃b converges to 0 by the adiabatic(-type) limit with respect to the L2-norm.
Key words: adiabatic limit; theta function; Lagrangian fibration; geometric quantization
2020 Mathematics Subject Classification: 53D50; 58H15; 58J05
1 Introduction
The purpose of this paper is to investigate the relationship between Spinc quantization and real
quantization from the viewpoint of the adiabatic(-type) limit for Lagrangian torus fibrations on
complete bases. In this paper, a Lagrangian torus fibration is assumed to be non-singular, but
its total space is not assumed to be compact unless otherwise stated.
1.1 Background and motivation
First let us explain the motivation which comes from geometric quantization. For geometric
quantization, see [19, 24, 37, 43]. In physics, quantization is the procedure for building quantum
mechanics starting from classical mechanics. In the mathematical context, it is often thought
of as a representation of the Poisson algebra consisting of certain functions on a symplectic
manifold to some Hilbert space, so called the quantum Hilbert space. When a symplectic
manifold (M,ω) and a prequantum line bundle
(
L,∇L
)
→ (M,ω) are given, geometric quan-
tization provides us with a method to construct the quantum Hilbert space and the represen-
tation from these data geometrically. In the theory of geometric quantization by Kostant and
Souriau [27, 38, 39], we need an additional structure which is called a polarization to obtain the
quantum Hilbert space. By definition, a polarization is an integrable Lagrangian distribution P
of the complexified tangent bundle TM ⊗ C of (M,ω). When a polarization P is given, the
quantum Hilbert space is naively given as the space of L2-sections of
(
L,∇L
)
covariant constant
along P .
mailto:takahiko@meiji.ac.jp
https://doi.org/10.3842/SIGMA.2024.065
2 T. Yoshida
A common example is the Kähler polarization. When (M,ω) is Kähler (i.e., (M,ω) has a com-
patible complex structure) and
(
L,∇L
)
is a holomorphic line bundle with canonical connection,
we can take T 0,1M as a polarization, and the obtained quantum Hilbert space is nothing but
the space of holomorphic L2-sections. This polarization is called the Kähler polarization and
the quantization procedure is called the Kähler quantization. Note that whenM is compact and
the Kodaira vanishing theorem holds, the quantum Hilbert space is H0(M,OL). In particular,
its dimension is equal to the index of the Dolbeault operator with coefficients in L.
Another example is a real polarization. Suppose (M,ω) admits a structure of a Lagrangian
torus fibration π : (M,ω) → B. For each point b ∈ B of the base manifold B, the restriction(
L,∇L
)
|π−1(b) of
(
L,∇L
)
to the fiber π−1(b) is a flat line bundle. Let H0
(
π−1(b);
(
L,∇L
)
|π−1(b)
)
be the space of covariant constant sections of
(
L,∇L
)
|π−1(b). Then, a point b ∈ B is said to be
Bohr–Sommerfeld if H0
(
π−1(b);
(
L,∇L
)
|π−1(b)
)
̸= {0}. It is well known that Bohr–Sommerfeld
points appear discretely. We denote by BBS the set of Bohr–Sommerfeld points. In this case,
we can take TπM ⊗ C, the complexified tangent bundle along the fiber of π as a polarization,
and if M is compact, the quantum Hilbert space is given by ⊕b∈BBS
H0
(
π−1(b);
(
L,∇L
)
|π−1(b)
)
.
See [37] for more details. In this paper, we call this quantization the real quantization.
When (M,ω) has a structure of a Lagrangian torus fibration π : (M,ω) → B as well as
a Kähler structure, it is natural to ask whether Kähler and real quantizations give the same
results. This paper focuses on the quantum Hilbert spaces obtained by both quantizations. It
is easy to see that for any compatible almost complex structure J of (M,ω), (TM, J) admits
a flat connection as a complex vector bundle. So, if (M,ω, J) is closed Kähler, by [29, Ex-
emples 12.5.2 (ii)], as a manifold, M is finitely covered by a torus. A typical example of this
case is an abelian variety, whose geometric quantization is well understood in the context of the
theory of theta functions. For example, see [4]. Moreover, even in the non-compact case, with
an appropriate choice of the quantum Hilbert space for the real quantization, the relationship
between these quantizations has been investigated. For the cotangent bundle of the Lie group
of compact type, these are related by the generalized Segal–Bargmann transform [18].
A completely integrable system can be thought of as a Lagrangian fibration with singular
fibers. As one of such examples, for the moment map of a smooth projective toric variety, Danilov
showed in [9] that H0(M,OL) has the irreducible decomposition H0(M,OL) =
⊕
m∈∆∩t∗Z
Cm as
a compact torus representation, where ∆ is the moment polytope, t∗Z is the weight lattice, and Cm
is the irreducible representation of the torus with weight m. In this case, since singular fibers
of the moment map are isotropic tori, the Bohr–Sommerfeld condition is still meaningful for
singular fibers, and ∆ ∩ t∗Z is identified with the set of Bohr–Sommerfeld points. This implies
the dimensions of the quantum Hilbert spaces obtained by the above two kinds of quantizations
agree with each other. It has been shown that similar results hold for the Gelfand–Cetlin system
on the flag variety [17], the Goldman system on the moduli space of flat SU(2) connections on
a surface [22], and the Kapovich–Millson system on the polygon space [23].
Moreover, for smooth projective toric varieties, in their paper [3], Baier–Florentino–Mourão–
Nunes developed a geometric approach to understand the relationship between Kähler and real
quantizations. Namely, they gave a one-parameter family of complex structures {Jt}t∈[0,∞) and
a basis {smt }m∈∆∩t∗Z of the space of holomorphic sections associated with the complex structure Jt
for each t such that each section smt converges to a delta function section supported on the
corresponding Bohr–Sommerfeld fiber as t goes to ∞. The similar results have been obtained
for flag manifolds [21] and smooth irreducible complex algebraic varieties [20]. But in [21]
and [20] the convergence was showed only for the non-singular Bohr–Sommerfeld fibers whereas
in [3] it was showed for all Bohr–Sommerfeld fibers.
The Kähler quantization can be generalized to a non-integrable compatible almost complex
structure on a closed (M,ω). When a compatible almost complex structure J on (M,ω) is given,
we can consider the associated Spinc Dirac operator D acting on Γ
(
∧•T ∗M0,1 ⊗ L
)
. It is well
Adiabatic Limit, Theta Function, and Geometric Quantization 3
known that D is a formally self-adjoint, first order, elliptic differential operator of degree-one,
and if J is integrable, D agrees with the Dolbeault operator up to constant. If J is not integrable,
T 0,1M is no more polarization. But, even in this case, since D is Fredholm, we can still take
the element of the K-theory of a point
ker(D|∧evenT ∗M0,1⊗L)− ker(D|∧oddT ∗M0,1⊗L) ∈ K(pt)
as a (virtual) quantum Hilbert space. Its virtual dimension is equal to the index of D. We call
this quantization the Spinc quantization. It has been showed in [1, 14, 28] that the equality
between dimensions of two quantum Hilbert spaces still holds by replacing Kähler quantization
with Spinc quantization in terms of the index theory.
1.2 Main theorems
In this paper, we apply the approach taken in [3] to both of Kähler and Spinc quantizations
of Lagrangian torus fibrations. Our setting is as follows. Let π : (M,ω) → B be a Lagrangian
torus fibration on a complete base B with prequantum line bundle
(
L,∇L
)
→ (M,ω). We do
not assume M is compact. Let J be a compatible almost complex structure of (M,ω) invariant
along the fiber of π in the sense of Lemma 3.6. For J , in Section 4.3, we give a one-parameter
family
{
J t
}
t>0
of compatible almost complex structures of (M,ω) with J1 = J so that the fiber
shrinks as t goes to ∞ with respect to the associated Riemannian metrics. We also show that J
is integrable if and only if every J t is integrable. For t > 0 and a positive integer N , let Dt be
the Spinc Dirac operator with coefficients in L⊗N associated with J t.
Firstly, let us consider the case where J is integrable. In this case, we show the following
Theorem, which is obtained by putting Corollary 4.5 and Theorem 4.14 together.
Theorem 1.1. Under the above setting, assume that J is integrable and satisfies certain tech-
nical condition. Then, for each t > 0, there exists a complete orthogonal system
{
ϑtb
}
b∈BBS
of holomorphic L2-sections of L⊗N →
(
M,Nω, J t
)
indexed by the Bohr–Sommerfeld points BBS
such that each ϑtb converges as a delta-function section supported on π−1(b) as t → ∞ in the
following sense, for any L2-section s of L⊗N , we have
lim
t→∞
∫
M
〈
s,
ϑtb
∥ϑtb∥L1
〉
L⊗N
(−1)
n(n−1)
2
ωn
n!
=
∫
π−1(b)
⟨s, δb⟩L⊗N |dy|,
where ⟨ , ⟩L⊗N is the Hermitian metric of L⊗N , δb is the covariant constant section of(
L,∇L
)⊗N |π−1(b) defined by (4.6), and |dy| is the natural one-density on π−1(b).
We give
{
ϑtb
}
b∈BBS
explicitly in Section 4.
One of examples of the total space of a Lagrangian torus fibration with complete base is the
abelian variety. In this case, we show that, under an appropriate choice of J , each ϑb coincides
with Jacobi’s theta function up to function on the base space (see Theorem 4.9). For the theta
functions, see [32, 33].
We remark that there are several works which deal with theta functions from the viewpoint of
geometric quantization of Lagrangian fibrations, for example, [4, 33, 34, 35]. In [7], Borthwick–
Uribe have introduced another approach to generalize the Kähler quantization to non-integrable
almost complex structures by using the metric Laplacian of the connection on the prequantum
line bundle instead of Spinc Dirac operator. Their approach is called the almost Kähler quan-
tization. In the almost Kähler quantization of the Kodaira–Thurston manifold, Kirwin–Uribe
and Egorov separately constructed an analog of the theta function as an element of the quantum
Hilbert space [13, 25]. In [12], Egorov also showed the similar result for Lagrangian T 2-fibrations
on T 2 with zero Euler class.
Secondly, let us consider the case where J is not integrable. Even in this case, we obtain the
following theorem which is a combination of Theorems 5.2 and 5.3.
4 T. Yoshida
Theorem 1.2. Under the above setting, for each t > 0, there exists an orthogonal fam-
ily
{
ϑ̃tb
}
b∈BBS
of L2-sections on L⊗N indexed by BBS such that
(1) each ϑ̃tb converges as a delta-function section supported on π−1(b) as t → ∞ in the above
sense, and
(2) lim
t→∞
∥∥Dtϑ̃tb
∥∥
L2 = 0.
We also give
{
ϑ̃tb
}
b∈BBS
explicitly in Section 5.
WhenM is compact, the index of the Spinc Dirac operator D := D1 can be considered and it
coincides with the number of Bohr–Sommerfeld points. See [14]. Moreover, by the Spinc Dirac
vanishing theorem due to Borthwick–Uribe [7], ker(D|∧oddT ∗M0,1⊗L⊗N ) vanishes for a sufficiently
large N . So, (1.2) in Theorem 1.2 suggests that we can interpret the Hilbert space generated
by
{
ϑ̃tb
}
b∈BBS
as an approximation to the quantum Hilbert space in the Spinc quantization for
a sufficiently large N .
In Kostant–Souriau’s formulation of geometric quantization, there is a systematic method
to associate an operator on the space of L2-sections of L⊗N , called the prequantum operator,
with a smooth function on (M,ω). When (M,ω) is not compact, this induces a nontrivial
representation of the Poisson algebra consisting of certain functions to the quantum Hilbert
space. One of the advantages of our setting is that it enable us to deal with this representation
of the Poisson algebra concretely by using the complete orthogonal system of the quantum
Hilbert space given in this paper, which we will discuss somewhere.
The idea used in this paper is quite simple, and consists of two key facts. The one is
about the symmetry by the fundamental group and the other is about the integrability of al-
most complex structures. Namely, the first key fact is Corollary 2.25 which claims that any
Lagrangian torus fibration π : (M,ω) → B with complete base B and a prequantum line bun-
dle
(
L,∇L
)
→ (M,ω) can be obtained as the quotient of a π1(B)-action on the standard La-
grangian fibration (M0, ω0) := (Rn × Tn,
∑n
i=1 dxi ∧ dyi) → Rn with standard prequantum line
bundle
(
L0,∇L0
)
:=
(
Rn × Tn ×C, d− 2π
√
−1
∑n
i=1 xidyi
)
, where (x1, . . . , xn) and (y1, . . . , yn)
are the coordinates of Rn and Tn, respectively. In particular, any compatible almost complex
structure J on (M,ω) is induced from a π1(B)-equivariant one on (M0, ω0). Since the set of
compatible almost complex structures on (M0, ω0) corresponds one-to-one to the set of smooth
maps from M0 to the Siegel upper half space, it enables us to describe the Spinc Dirac operator
associated with J in terms of the corresponding map. We show that there exists a π1(B)-
invariant compatible almost complex structure whose corresponding map is invariant along the
fiber (see Lemma 3.6). For the Spinc Dirac operator D associated with such an almost complex
structure J , we consider the problem on existence of nontrivial sections of L⊗N
0 contained in
the kernel of D. By taking the Fourier series expansion of a section s of L⊗N
0 with respect to
the fiber coordinates, the equation Ds = 0 can be reduced to a system of partial differential
equations on Rn.
The other key fact is Proposition 3.14 which gives a necessary and sufficient condition in order
that the system of partial differential equations has nontrivial solutions and also shows that it
is equal to the integrability condition for J , i.e., (M0, ω0, J) is Kähler. Moreover, in this case,
we give a family of π1(B)-equivariant solutions of Ds = 0 indexed by the Bohr–Sommerfeld
points, each of which is expressed by the formal Fourier series. If they converge absolutely
and uniformly on any compact set and form square integrable sections, this gives a complete
orthogonal system of the space of holomorphic L2-sections of
(
L,∇L
)⊗N → (M,Nω, J). We
also give a sufficient condition for their convergence and square integrability. Even if J is not
integrable, by considering an approximation of D, we can obtain an orthogonal family of L2-
sections of L⊗N indexed by the Bohr–Sommerfeld points BBS.
The limit used in this paper is slightly different from the adiabatic limit in Riemannian
geometry. When a fiber bundle π : M → B and a Riemannian metric g on M are given, we can
Adiabatic Limit, Theta Function, and Geometric Quantization 5
consider the decomposition (TM, g) = (V, gV ) ⊕ (H, gH), where V is the tangent bundle along
the fiber with fiber metric gV := g|V and H is the orthogonal complement of V with respect
to g with fiber metric gH := g|H . For each t > 0, we deform g by gt := gV ⊕ tgH . Then,
in Riemannian geometry, the adiabatic limit is the procedure for taking the limit of geometric
objects associated with gt as t → ∞. But, since such a deformation 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>0
of compatible almost complex structures on (M,ω) such
that the corresponding one-parameter family of Riemannian metrics is
{
gt = 1
t gV ⊕ tgH
}
t>0
,
and investigate the behavior of ϑtb
(
resp. ϑ̃tb
)
as t goes to ∞.
The paper is organized as follows. In Section 2, we first briefly review some well-known facts
about integral affine geometry and Lagrangian fibrations. Then, by using these, we prove Corol-
lary 2.25. In Section 3, we discuss the π1(B)-equivariant Spinc quantization of (M0, ω0) → Rn
with standard prequantum line bundle
(
L0,∇L0
)
and give a statement of Proposition 3.14. In
Section 4, we prove Theorem 1.1 step by step, and explain the relation between ϑtb and Jacobi’s
classical theta function. Finally, in Section 5, we prove Theorem 1.2. The requirements for
Fourier series are explained in Appendix A. A proof of Proposition 3.14 is given in Appendix B
because it is done by a very long direct computation.
1.3 Notations
For x = t(x1, . . . , xn) and y = t(y1, . . . , yn) ∈ Rn, let us denote the standard inner prod-
uct
∑n
i=1 xiyi by x · y. We use the notation ∂xi for
∂
∂xi
. We also use the following notations:
Tn := (R/Z)n, (M0, ω0) :=
(
Rn × Tn,
n∑
i=1
dxi ∧ dyi
)
,
(
L0,∇L0
)
:=
(
Rn × Tn × C, d− 2π
√
−1
n∑
i=1
xidyi
)
,
where (x1, . . . , xn) and (y1, . . . , yn) are the coordinates of Rn and Tn, respectively. In this paper,
all manifolds and maps are supposed to be smooth unless otherwise stated. When we use the
term “group action”, we mean “left group action” unless otherwise specified.
2 Unfolding Lagrangian fibrations
2.1 Integral affine structures
Let B be a manifold.
Definition 2.1. An integral affine atlas of B is an atlas {(Uα, ϕα)} of B whose coordinate
transformation ϕα ◦ ϕ−1
β on each non-empty overlap Uαβ := Uα ∩ Uβ is an integral affine
transformation. Namely, on each non-empty overlap Uαβ := Uα ∩ Uβ, there exist locally
constant maps Aαβ : Uαβ → GLn(Z) and cαβ : Uαβ → Rn such that ϕα ◦ ϕ−1
β is of the form
ϕα ◦ ϕ−1
β (x) = Aαβx+ cαβ. Two integral affine atlases {(Uα, ϕα)} and {(U ′
β, ϕ
′
β)} of B are said
to be equivalent if on each non-empty overlap Uα∩U ′
β, ϕα ◦ (ϕ′β)−1 is an integral affine transfor-
mation. An integral affine structure on B is an equivalence class of integral affine atlases of B.
A manifold equipped with integral affine structure is called an integral affine manifold.
Example 2.2. An n-dimensional Euclidean space Rn is equipped with a natural integral affine
structure.
Let us give examples of integral affine manifolds obtained from integral affine actions on Rn.
6 T. Yoshida
Example 2.3.
(1) Let v1, . . . , vn ∈ Rn be a linear basis of Rn and C = (v1 · · · vn) ∈ GLn(R) the matrix
whose ith column vector is vi for i = 1, . . . , n. Zn acts on Rn by ργ(x) := x+Cγ for γ ∈ Zn
and x ∈ Rn. Since the action preserves the natural integral affine structure on Rn, the
quotient space, which is topologically Tn, is equipped with an integral affine structure.
(2) Let λ ∈ N be a positive integer and a, b ∈ R>0 positive real numbers. Define the Z2-action
on R2 as follows. First, for the standard basis e1, e2 of Z2, let us define the integral affine
transforms ρe1 , ρe2 by
ρe1(x) := x+
(
a
0
)
, ρe2(x) :=
(
1 λ
0 1
)
x+
(
0
b
)
for x ∈ R2. Since ρe1 and ρe2 are commutative, they form the Z2-action on R2 by
ργ(x) := ργ1e1 ◦ ρ
γ2
e2 (x)
for each γ = t(γ1, γ2) ∈ Z2. By the same manner as in (1), the quotient space is equipped
with an integral affine structure. It is shown in [31, Theorem A] that the quotient space
is topologically T 2, but the induced integral affine structure is not isomorphic to that
obtained in (1) for n = 2 and there are only these two integral affine structures on T 2 up
to isomorphism.
Example 2.4. For γ = t(γ1, γ2, γ3), γ
′ = t(γ′1, γ
′
2, γ
′
3) ∈ Z3, define the product γ ◦ γ′ ∈ Z3 by
γ ◦ γ′ :=
1 0 0
0 0 −1
0 −1 0
γ1
γ′ + γ.
Then, Z3 with product ◦ is a non abelian group
(
Z3, ◦
)
.
(
Z3, ◦
)
acts on R3 by
ργ(x) :=
1 0 0
0 0 −1
0 −1 0
γ1
x+ γ.
The action preserves the natural integral affine structure on R3. Therefore, the quotient
space R3/
(
Z3, ◦
)
is equipped with the integral affine structure induced from that of R3.
Example 2.5. Let n ≥ 2. For γ = t(γ1, . . . , γn), γ
′ = t(γ′1, . . . , γ
′
n) ∈ Zn, define the prod-
uct γ ◦ γ′ ∈ Zn by
γ ◦ γ′ :=
1
(−1)γ1
. . .
(−1)γn−1
γ′ + γ.
Then, Zn with product ◦ is a non abelian group (Zn, ◦). (Zn, ◦) acts on Rn by
ργ(x) :=
1
(−1)γ1
. . .
(−1)γn−1
x+ γ.
The action preserves the natural integral affine structure on Rn. Therefore, the quotient
space Rn/(Zn, ◦) is equipped with the integral affine structure induced from that of Rn. For n=2,
the quotient space is topologically a Klein bottle.
Adiabatic Limit, Theta Function, and Geometric Quantization 7
Example 2.6. Let n ≥ 2 and λ1, . . . , λn−1 ∈ Z. For γ = t(γ1, . . . , γn), γ
′ = t(γ′1, . . . , γ
′
n) ∈ Zn,
define the product γ ◦ γ′ ∈ Zn by
γ ◦ γ′ :=
1 λ1
1 λ2
. . .
. . .
1 λn−1
1
γn
γ′ + γ.
Zn with product ◦ is a group (Zn, ◦), which is non abelian for n ≥ 3. (Zn, ◦) acts on Rn by
ργ(x) :=
1 λ1
1 λ2
. . .
. . .
1 λn−1
1
γn
x+ γ.
Then, the quotient space Rn/(Zn, ◦) is equipped with the integral affine structure induced from
that of Rn. In the case where n = 2 and λ1 > 0, it coincides with the one given in Example 2.3 (2)
with a = b = 1.
Example 2.7. Let Z/4Z ∼=
{
± ( 1 0
0 1 ) ,±
(
0 −1
1 0
)}
act on
(
R2
)n ∖ {0} naturally. Then, the
quotient space is a non-compact manifold and equipped with the integral affine structure induced
from that of
(
R2
)n ∖ {0}.
Let B be an n-dimensional connected integral affine manifold, p : B̃ → B the universal
covering of B. It is clear that B̃ is also equipped with the integral affine structure so that p is
an integral affine map. We set Γ := π1(B). Γ acts on B̃ from the right as a deck transformation.
For each γ ∈ Γ we denote by σγ the inverse of the deck transformation corresponding to γ.
Then, σ : γ 7→ σγ defines a left action σ ∈ Hom
(
Γ,Aut
(
B̃
))
. The following proposition is well
known in affine geometry. See [16, p. 641] for a proof.
Proposition 2.8. There exists an integral affine immersion dev : B̃ → Rn and a homomor-
phism ρ : Γ → GLn(Z)⋉Rn such that the image of dev is an open set of Rn and dev is equivari-
ant with respect to σ and ρ. Such an integral affine immersion is unique up to the composition
of an integral affine transformation on Rn.
We will prove a version of this proposition (see Proposition 2.22) when B is equipped with
a Lagrangian fibration on it in Section 2.3.
Proposition 2.9. Let B, p : B̃ → B, dev : B̃ → Rn, and ρ : Γ → GLn(Z) ⋉ Rn be as in
Proposition 2.8. Suppose that B is compact and the Γ-action ρ on Rn 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 Rn. So, it is
sufficient to show that O is closed in Rn. Since the Γ-action ρ on Rn is properly discontinuous,
the quotient space Rn/Γ becomes a Hausdorff space and the natural projection q : Rn → Rn/Γ is
continuous. O is preserved by the Γ-action ρ on Rn since dev is Γ-equivariant. Then, dev induces
a continuous surjective map dev : B = B̃/Γ → O/Γ. Since B is compact, O/Γ is a compact
subset in the Hausdorff space Rn/Γ. In particular, it is also closed. Hence, O = q−1 (O/Γ) is
also closed in Rn. ■
8 T. Yoshida
Corollary 2.10. Let B, p : B̃ → B, dev : B̃ → Rn, and ρ : Γ → GLn(Z) ⋉ Rn be as in Propo-
sition 2.8 and assume that B compact. If the image of ρ is included in (GLn(Z) ∩ O(n)) ⋉ Rn
and the subgroup ρ(Γ) of (GLn(Z) ∩O(n))⋉Rn is discrete, then dev is surjective.
Proof. It follows from [42, Theorem 3.1.3]. ■
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 of the above examples are complete other than Example 2.7 for n ≥ 2.
Example 2.13. Let B be an n-dimensional compact integral affine manifold B with integral
affine atlas {(Uα, ϕα)} as in Definition 2.1. If on each non-empty overlap Uαβ, the Jacobi matrix
of ϕα ◦ ϕ−1
β is contained in GLn(Z) ∩ O(n), then B has a flat Riemannian metric. Hence, by
Bieberbach’s theorem [5, 6], B is finitely covered by Tn. In particular, B is complete. For flat
Riemannian manifolds, see [42, Chapter 3].
2.2 Lagrangian fibrations
In this section, let us recall Lagrangian fibrations and explain how integral affine structures are
associated with Lagrangian fibrations. After that, let us recall their classification by Duister-
maat. For more details, see [10, 36, 44].
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 Tn = (R/Z)n be an n-dimensional torus. Rn × Tn admits a standard
symplectic structure ω0 =
∑
i dxi ∧ dyi, where (x1, . . . , xn) and (y1, . . . , yn) are the coordinates
of Rn and Tn, respectively. Then, the projection π0 : (Rn×Tn, ω0) → Rn to Rn is a Lagrangian
fibration.
The following theorem shows that Example 2.15 is the local model of Lagrangian fibrations.
Theorem 2.16. Let π : (M,ω) → B be a Lagrangian fibration with compact, path-connected
fibers. Then, for each b ∈ B, there exists a chart (U, ϕ) of B containing b and a symplectomor-
phism φ :
(
π−1(U), ω|π−1(U)
)
→ (ϕ(U)× Tn, ω0) such that the following diagram commutes:(
π−1(U), ω|π−1(U)
)
π
��
φ // (ϕ(U)× Tn, ω0)
π0
��
U
ϕ // ϕ(U).
Proof. In [10, Section 2], Duistermaat showed that any Lagrangian fibration with compact,
path-connected fibers is locally identified with a regular part of a completely integrable Hamil-
tonian system. Theorem 2.16 follows from this fact together with Arnold–Liouville’s theorem
which claims that a regular part of a completely integrable Hamiltonian system is locally iden-
tified with π0 : (Rn × Tn, ω0) → Rn. For Arnold–Liouville’s theorem, see [2, 10, 36, 40]. ■
In particular, Theorem 2.16 says that any Lagrangian fibration with compact, path-connected
fibers has a torus as its fiber.
In this paper, we consider only Lagrangian fibrations with compact, path-connected fibers. In
the rest of this paper, “Lagrangian fibration” refers to a Lagrangian fibration having compact,
path-connected fibers.
Now we investigate automorphisms of the local model. By the direct computation shows the
following lemma. See also [40, Lemma 2.5].
Adiabatic Limit, Theta Function, and Geometric Quantization 9
Lemma 2.17. Let φ : (Rn × Tn, ω0) → (Rn × Tn, ω0) be a fiber-preserving symplectomor-
phism of π0 : (Rn × Tn, ω0) → Rn which covers a map ϕ : Rn → Rn. Then, there exist a ma-
trix A ∈ GLn(Z), a constant c ∈ Rn, and a map u : Rn → Tn with tAJu symmetric such that φ
is written as
φ(x, y) =
(
Ax+ c, tA−1y + u(x)
)
for any (x, y) ∈ Rn × Tn, where Ju 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 at-
las {(Uα, ϕα)}α∈A of B and for each α ∈ A there exists a symplectomorphism
φα :
(
π−1(Uα), ω|π−1(Uα)
)
→ (ϕα(Uα)× Tn, ω0)
such that the following diagram commutes:(
π−1(Uα), ω|π−1(Uα)
)
π
��
φα // (ϕα(Uα)× Tn, ω0)
π0
��
Uα
ϕα // ϕα(Uα).
Moreover, on each non-empty overlap Uαβ there exist locally constant maps Aαβ : Uαβ→GLn(Z),
cαβ : Uαβ → Rn, and a map uαβ : Uαβ → Tn with tAαβJ
(
uαβ ◦ ϕ−1
β
)
symmetric, such that the
overlap map is written as
φα ◦ φ−1
β (x, y) =
(
Aαβx+ cαβ,
tA−1
αβy + uαβ ◦ ϕ−1
β (x)
)
(2.1)
for any (x, y) ∈ ϕβ(Uαβ)× Tn.
Proposition 2.18 implies that the base manifold of a Lagrangian fibration has an integral
affine structure. Conversely, suppose that a manifold B admits an integral affine structure
and let {(Uα, ϕα)}α∈A be an integral affine atlas of B. Then, we can construct a Lagrangian
fibration on B in the following way. For each α ∈ A, let ϕα : T
∗B|Uα → ϕα(Uα) × Rn be the
local trivialization of the cotangent bundle T ∗B induced from (Uα, ϕα). On each nonempty
overlap Uαβ, suppose that ϕα ◦ ϕ−1
β is written by ϕα ◦ ϕ−1
β (x) = Aαβx+ cαβ as in Definition 2.1.
Then, the overlap map is written as
ϕα ◦
(
ϕβ
)−1
(x, y) =
(
Aαβx+ cαβ,
tA−1
αβy
)
. (2.2)
Since Aαβ is in GLn(Z), (2.2) preserves the integer lattice Zn in each fiber Rn. Hence, it
induces the fiber-preserving symplectomorphism from π0 : (ϕβ(Uαβ) × Tn, ω0) → ϕβ(Uαβ) to
π0 : (ϕα(Uαβ)× Tn, ω0) → ϕα(Uαβ) which covers ϕα ◦ ϕ−1
β . By patching {π0 : (ϕα(Uα)× Tn, ω0)
→ ϕα(Uα)}α∈A together by these symplectomorphisms, we obtain a new Lagrangian fibration
πcan : (Mcan, ωcan) → B, namely,
(Mcan, ωcan) :=
∐
α∈A
(ϕα(Uα)× Tn, ω0)/∼
and
πcan([xα, yα]) := ϕ−1
α (xα)
for (xα, yα) ∈ ϕα(Uα) × Tn. This construction does not depend on the choice of an equiv-
alent integral affine atlas and depends only on the integral affine structure on B. We call
πcan : (Mcan, ωcan) → B the canonical model. We summarize the above argument to the follow-
ing proposition.
10 T. Yoshida
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, associated with π : (M,ω) → B,
B has an integral affine atlas {(Uα, ϕα)}α∈A as in Proposition 2.18. Let πcan : (Mcan, ωcan) → B
be the canonical model associated with the integral affine structure on B. On each Uα, let
φα :
(
π−1(Uα), ω|π−1(Uα)
)
→ (ϕα(Uα)× Tn, ω0)
be a local trivialization of π : (M,ω) → B as in Proposition 2.18, and
ϕα :
(
π−1
can(Uα), ωcan
)
→ (ϕα(Uα)× Tn, ω0)
be the local trivialization of πcan : (Mcan, ωcan) → B naturally induced from (Uα, ϕα) as explained
above.1 Then their composition
hα := ϕα
−1 ◦ φα :
(
π−1(Uα), ω|π−1(Uα)
)
→
(
π−1
can(Uα), ωcan
)
gives a local identification between them. On each Uα ∩Uβ, suppose that φα ◦φ−1
β is written as
in (2.1). Then, hα ◦ h−1
β can be written as
hα ◦ h−1
β (p) = ϕα
−1(
Aαβx+ cαβ,
tA−1
αβy + uαβ(π(p))
)
,
where ϕβ(p) = (x, y). uαβ induces the local section ũαβ of πcan : (Mcan, ωcan) → B on Uαβ by
ũαβ(b) := [ϕα(b), uαβ(b)]
for b ∈ Uαβ. It is easy to see that ũαβ satisfies ũ∗αβωcan = 0. A section with this condition is
said to be Lagrangian.
Let S be the sheaf of germs of Lagrangian section of πcan : (Mcan, ωcan) → B. S is the sheaf
of Abelian groups since the fiber of πcan : (Mcan, ωcan) → B has the structure of an Abelian group
by construction. By definition {ũαβ} forms a Čech one-cocycle on B with coefficients in S . The
cohomology class determined by {ũαβ} does not depend on the choice of a specific integral affine
atlas and depends only on π : (M,ω) → B. We denote the cohomology class by u ∈ H1(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 π1 : (M1, ω1) → B and π2 : (M2, ω2) → B
on the same integral affine manifold B, there exists a fiber-preserving symplectomorphism be-
tween them which covers the identity if and only if their Chern classes u1 and u2 agree with each
other. Moreover, if an integral affine manifold B and the cohomology class u ∈ H1(B;S ) are
given, then there exists a Lagrangian fibration π : (M,ω) → B that realizes them.
Remark 2.21. By the construction of u, there exists a fiber-preserving symplectomorphism
between π : (M,ω) → B and πcan : (Mcan, ωcan) → B that covers the identity of B if and only
if u vanishes. In particular, if u vanishes, π : (M,ω) → B possesses a Lagrangian section
since πcan : (Mcan, ωcan) → B has the zero section which is Lagrangian. Conversely, it can be
shown that any Lagrangian fibration with Lagrangian section is identified with the canonical
model.
1Here we use the same notation as the local trivialization of T ∗B because we have no confusion.
Adiabatic Limit, Theta Function, and Geometric Quantization 11
2.3 Lagrangian fibrations with complete bases
Assume that π : (M,ω) → B is a Lagrangian fibration with n-dimensional connected base mani-
fold B, p : B̃ → B the universal covering of B. We denote by π̃ :
(
M̃, ω̃
)
→ B̃ the pullback
of π : (M,ω) → B to B̃. Let Γ be the fundamental group of B and σ ∈ Hom
(
Γ,Aut
(
B̃
))
the action of Γ defined as the inverse of the deck transformation as in Proposition 2.8. By
definition, M̃ admits a natural lift of σ which preserves ω̃. The Γ-action on
(
M̃, ω̃
)
is denoted
by σ̃. By Proposition 2.8, we have a developing map dev : B̃ → Rn and the homomorphism
ρ : Γ → GLn(Z) ⋉ Rn. We denote the image of dev by O. Note that the Γ-action ρ on Rn
preserves O since dev is Γ-equivariant.
Proposition 2.22. There exists a Lagrangian fibration π′ : (M ′, ω′) → O, a fiber-preserving
symplectic immersion d̃ev :
(
M̃, ω̃
)
→ (M ′, ω′) which covers dev, and a lift ρ̃ of the Γ-action ρ
on O to (M ′, ω′) such that d̃ev is Γ-equivariant with respect to σ̃ and ρ̃.
Proof. By Proposition 2.19, B admits an integral affine structure determined by π, and it
also induces the integral affine structure on B̃. Let {(Uα, ϕ
′′
α)} be the integral affine atlas of B̃
and
{(
π̃−1(Uα), ω|π̃−1(Uα), φ
′′
α
)}
the local trivializations of π̃ :
(
M̃, ω̃
)
→ B̃ as in Proposition 2.18
so that on each non-empty overlap Uαβ, there exist locally constant maps Aαβ : Uαβ → GLn(Z)
and c′αβ : Uαβ → Rn, and a map u′αβ : Uαβ → Tn with tAαβJ
(
u′αβ ◦ (ϕ′′β)−1
)
symmetric such
that φ′′
α ◦ (φ′′
β)
−1 is written as in (2.1). Then, Aαβ’s form a Čech one-cocycle {Aαβ} ∈ C1({Uα};
GLn(Z)) and defines a cohomology class [{Aαβ}] ∈ H1
(
B̃; GLn(Z)
)
. It is well known, for ex-
ample, see [30, Appendix A], that H1
(
B̃; GLn(Z)
)
is identified with the moduli space of homo-
morphisms from π1
(
B̃
)
to GLn(Z). Since π1
(
B̃
)
is trivial, there exists a Čech zero-cocycle
{Aα} ∈ C0({Uα}; GLn(Z)) such that Aαβ = AαA
−1
β on each Uαβ. By using the cocycle we mod-
ify the local trivializations
{(
π̃−1(Uα), ω|π̃−1(Uα), φ
′′
α
)}
and the integral affine atlas {(Uα, ϕ
′′
α)}
by replacing φ′′
α, ϕ
′′
α by
φ′
α(p̃) :=
(
A−1
α × tAα
)
◦ ϕ′′α(p̃), ϕ′α := A−1
α ϕ′′α
for each α ∈ A, respectively. Then, on each Uαβ, φ
′
α ◦ (φ′
β)
−1 is written as
φ′
α ◦ (φ′
β)
−1(x̃, y) =
(
x̃+ cαβ, y + uαβ ◦ (ϕ′β)−1(x̃)
)
,
where we set cαβ := A−1
α c′αβ and uαβ := tAαu
′
αβ. Then, cαβ’s form a Čech one-cocycle {cαβ} ∈
C1({Uα};Rn) and defines a cohomology class [{cαβ}] ∈ H1
(
B̃;Rn
)
. By the universal coefficients
theorem, H1
(
B̃;Rn
)
is identified with Hom
(
H1
(
B̃;Z
)
,Rn
)
, which is trivial. So there exists
a Čech zero-cocycle {cα} ∈ C0({Uα};Rn) such that cαβ = cα − cβ on each Uαβ. By using the
cocycle, we again modify
{(
π̃−1(Uα), ω|π̃−1(Uα), φ
′
α
)}
and {(Uα, ϕ
′
α)} by replacing φ′
α, ϕ
′
α by
φα(p̃) := φ′
α(p̃)− (cα, 0), ϕα
(
b̃
)
:= ϕ′α
(
b̃
)
− cα,
respectively for each α ∈ A. Then, on each Uαβ, ϕα coincides with ϕβ and φα ◦φ−1
β is written as
φα ◦ φ−1
β (x̃, y) =
(
x̃, y + uαβ ◦ ϕ−1
β (x̃)
)
.
Now we define the map dev : B̃ → Rn by
dev
(
b̃
)
:= ϕα
(
b̃
)
if b̃ is in Uα. It is well defined, and by construction, it is an integral affine immersion whose
image is ∪α∈Aϕα(Uα). (M
′, ω′) is defined by
(M ′, ω′) :=
∐
α∈A
(ϕα(Uα)× Tn, ω0)/∼,
12 T. Yoshida
where (xα, yα) ∈ ϕα(Uα)×Tn and (xβ, yβ) ∈ ϕβ(Uβ)×Tn are in the relation (xα, yα) ∼ (xβ, yβ)
if they satisfy (xα, yα) = φα ◦ φ−1
β (xβ, yβ), and π′ : (M ′, ω′) → O is defined to be the first
projection. d̃ev :
(
M̃, ω̃
)
→ (M ′, ω′) is defined by
d̃ev(p̃) := [φα(p̃)]
if p̃ is in π̃−1(Uα).
Without loss of generality, we can assume that each Uα is connected,and for each γ ∈ Γ
and α ∈ A there uniquely exists α′ ∈ A such that the deck transformation σγ maps Uα onto Uα′ .
Then, its lift σ̃γ to
(
M̃, ω̃
)
maps π̃−1(Uα) to π̃
−1(Uα′). By Lemma 2.17, ϕα′ ◦ σγ ◦ ϕ−1
α can be
written as
ϕα′ ◦ σγ ◦ ϕ−1
α (x̃) = Aα′α
γ x̃+ cα
′α
γ
for some Aα′α
γ ∈ GLn(Z), cα
′α
γ ∈ Rn. Since ϕα coincides with ϕβ on each overlap Uαβ, ϕα′ ◦ ϕγ ◦
ϕα(x̃) = Aα′α
γ x̃+ cα
′α
γ also agrees with ϕβ′ ◦ ϕγ ◦ ϕβ(x̃) = Aβ′β
γ x̃+ cβ
′β
γ on the overlap ϕα(Uαβ) =
ϕβ(Uαβ). This implies Aα′α
γ ’s and cα
′α
γ ’s do not depend on α and depends only on γ. In fact, for
each γ ∈ Γ and α0 ∈ A, we set
A0 :=
{
α ∈ A | Aα′
0α0
γ = Aα′α
γ and c
α′
0α0
γ = cα
′α
γ
}
.
A0 contains all β ∈ A with Uα0β ̸= ∅. In particular, A0 is not empty since α0 ∈ A0. Then, we
have ( ⋃
α∈A0
Uα
)
∪
( ⋃
α∈A∖A0
Uα
)
= B̃,
( ⋃
α∈A0
Uα
)
∩
( ⋃
α∈A∖A0
Uα
)
= ∅.
If the complement A∖A0 is not empty, this contradicts to the connectedness of B̃. So we denote
them by Aγ and cγ , respectively. Thus, we define the homomorphism ρ : Γ → GLn(Z)⋉Rn by
ργ := (Aγ , cγ).
Γ acts on Rn by ργ(x) = Aγx + cγ for γ ∈ Γ and x ∈ Rn. The lift ρ̃γ of ργ to (M ′, ω′) is
defined by
ρ̃γ([xα, yα]) :=
[
φα′ ◦ σ̃γ ◦ φ−1
α (xα, yα)
]
if (xα, yα) is in ϕα(Uα)× Tn. By construction, ρ̃ is a lift of ρ, and ρ̃ and ρ satisfy d̃ev(σ̃γ(p̃)) =
ρ̃γ
(
d̃ev(p̃)
)
and dev
(
σγ
(
b̃
))
= ργ
(
dev
(
b̃
))
, respectively. ■
Remark 2.23.
(1) By construction, the n-dimensional torus Tn acts freely onM ′ preserving ω′ from the right
so that π′ : M ′ → O is a principal Tn-bundle.
(2) If π : (M,ω) → B admits a Lagrangian section, the restriction of π0 : (Rn × Tn, ω0) → Rn
to O can be taken as π′ : (M ′, ω′) → O. In fact, in this case, since π : (M,ω) → B is identi-
fied with the canonical model, we can take a system of local trivializations
{(
π−1(Uα), φα
)}
with uαβ = 0 on each overlaps Uαβ. By applying the construction of π′ : (M ′, ω′) → O
given in the proof of Proposition 2.22 to such a
{(
π−1(Uα), φα
)}
, we can show the claim.
Suppose that (M,ω) is prequantizable and let
(
L,∇L
)
→ (M,ω) be a prequantum line bundle.
We denote by
(
L̃,∇L̃
)
→
(
M̃, ω̃
)
the pullback of
(
L,∇L
)
→ (M,ω) to
(
M̃, ω̃
)
. By definition,
L̃ admits a natural lift of the Γ-action σ̃ on
(
M̃, ω̃
)
which preserves∇L̃. The Γ-action on
(
L̃,∇L̃
)
is denoted by ˜̃σ. Then, we have the following prequantum version of Proposition 2.22.
Adiabatic Limit, Theta Function, and Geometric Quantization 13
Proposition 2.24. There exists a prequantum line bundle
(
L′,∇L′)→ (M ′, ω′), a bundle im-
mersion ˜̃dev : (L̃,∇L̃
)
→
(
L′,∇L′)
which covers d̃ev, and a lift ˜̃ρ of the Γ-action ρ̃ on (M ′, ω′)
to
(
L′,∇L′)
such that ˜̃dev is equivariant with respect to ˜̃σ and ˜̃ρ.
Proof. Let {(Uα, ϕα)} and
{(
π̃−1(Uα), ω|π̃−1(Uα), φα
)}
be the integral affine atlas of B̃ and the
local trivializations of π̃ :
(
M̃, ω̃
)
→ B̃ obtained in the proof of Proposition 2.22, respectively.
Then, for each α ∈ A there exists a prequantum line bundle(
ϕα(Uα)× Tn × C,∇L̃α
)
→ (ϕα(Uα)× Tn, ω0)
and a bundle isomorphism ψα :
(
L̃,∇L̃
)
|π̃−1(Uα)→
(
ϕα(Uα)× Tn × C,∇L̃α
)
which covers φ. Now
we define
(
L′,∇L′)
by(
L′,∇L′)
:=
∐
α∈A
(
ϕα(Uα)× Tn × C,∇L̃α
)
/∼,
where (xα, yα, zα) ∈ ϕα(Uα) × Tn × C and (xβ, yβ, zβ) ∈ ϕβ(Uβ) × Tn × C are in the relation
(xα, yα, zα) ∼ (xβ, yβ, zβ) if they satisfy (xβ, yβ, zβ) = ψα ◦ ψ−1
β (xβ, yβ, zβ).
˜̃dev : (L̃,∇L̃
)
→(
L′,∇L′)
is defined by
˜̃dev(ṽ) := [ψα(ṽ)]
if ṽ is in
(
L̃,∇L̃
)
|π̃−1(Uα).
Suppose that for each γ ∈ Γ the deck transformation σγ maps each Uα to some Uα′ as before.
Then, ˜̃σγ maps L̃π̃−1(Uα) to L̃π̃−1(Uα′ ). Then, the Γ-action ˜̃ρ is defined by
˜̃ργ(xα, yα, zα) := [ψα′ ◦ ˜̃σγ ◦ ψ−1
α (xα, yα, zα)
]
if (xα, yα, zα) is in ϕα(Uα)× Tn × C. ■
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 : B̃ → B be the univer-
sal covering of B. Let us denote by
(
M̃, ω̃
)
the pullback of (M,ω) to B̃ and denote by
(
L̃,∇L̃
)
the pullback of
(
L,∇L
)
to
(
M̃, ω̃
)
. If B is complete, there exist an integral affine isomor-
phism dev : B̃ → Rn, a fiber-preserving symplectomorphism d̃ev :
(
M̃, ω̃
)
→ (Rn × Tn, ω0), and
a bundle isomorphism ˜̃dev : (L̃,∇L̃
)
→
(
Rn × Tn × C, d− 2π
√
−1x · dy
)
such that d̃ev covers
dev and ˜̃dev covers d̃ev, respectively. Here x · dy denotes
∑n
i=1 xidyi. Moreover, let σ be the
Γ-action on B̃ defined as the inverse of deck transformations, σ̃ the natural lift of σ to
(
M̃, ω̃
)
,
and ˜̃σ the natural lift of σ̃ to
(
L̃,∇L̃
)
, respectively. Then, there exist an integral affine Γ-
action ρ : Γ → GLn(Z)⋉Rn on Rn, its lifts ρ̃ and ˜̃ρ to (Rn × Tn, ω0) and
(
Rn × Tn × C, d −
2π
√
−1x · dy
)
, respectively such that dev, d̃ev, and ˜̃dev are Γ-equivariant.
Proof. By construction of d̃ev given in the proof of Proposition 2.22, if dev is bijective, so is
d̃ev. The argument in [10, p. 696] and Theorem 2.20 also show that π0 : (Rn × Tn, ω0) → Rn is
the unique Lagrangian fibration on Rn up to fiber-preserving symplectomorphism covering the
identity. In particular, π′ : (M ′, ω′) → Rn is identified with π0 : (Rn × Tn, ω0) → Rn.
Concerning the prequantum line bundle, it is sufficient to show that (Rn×Tn, ω0) has a unique
prequantum line bundle
(
Rn × Tn × C, d− 2π
√
−1x · dy
)
up to bundle isomorphism. Since ω0
is exact, any prequantum line bundle on (Rn × Tn, ω0) is trivial as a complex line bundle.
Let
(
Rn × Tn × C, d− 2π
√
−1α
)
be a prequantum line bundle on (Rn × Tn, ω0) with connec-
tion d− 2π
√
−1α. Then, α − x · dy defines a de Rham cohomology class in H1(Rn × Tn;R).
14 T. Yoshida
Since H1(Rn×Tn;R) is isomorphic to H1(Tn;R), in terms of the generators dyi’s of H
1(Tn;R),
α− x · dy can be described as
α− x · dy =
n∑
i=1
τidyi + df
for some τ1, . . . , τn ∈ R and f ∈ C∞(Rn×Tn). Now we define the bundle isomorphism ψ : Rn×
Tn × C → Rn × Tn × C by
ψ(x, y, z) :=
(
x+ (τi), y, e
−2π
√
−1f(x,y)z
)
.
Then, ψ satisfies ψ∗(d− 2π
√
−1x · dy
)
= d− 2π
√
−1α. ■
In the rest of this paper, we use the notations (M0, ω0) :=
(
Rn × Tn,
∑n
i=1 dxi ∧ dyi
)
and
(
L0,∇L0
)
:=
(
Rn × Tn × C, d− 2π
√
−1x · dy
)
for simplicity.
Remark 2.26 (Hermitian metric on
(
L0,∇L0
)
). By Corollary 2.25, any Lagrangian fibra-
tion π : (M,ω) → B on a complete B with prequantum line bundle
(
L,∇L
)
→ (M,ω) is ob-
tained as the quotient space of the Γ-action on π0 : (M0, ω0) → Rn with prequantum line
bundle
(
L0,∇L0
)
→ (M0, ω0). By definition, the prequantum line bundle
(
L,∇L
)
→ (M,ω)
is equipped with a Hermitian metric ⟨·, ·⟩L compatible with ∇L.2 The pull-back of ⟨·, ·⟩L
to
(
L0,∇L0
)
→ (M0, ω0) coincides with the one induced from the standard Hermitian inner
product on C up to constant. In fact, it is easy to see that, up to constant, it is the unique
Hermitian metric on
(
L0,∇L0
)
→ (M0, ω0) compatible with ∇L0 . In the rest of this paper, we
assume that
(
L0,∇L0
)
→ (M0, ω0) is always equipped with the Hermitian metric though we do
not specify it.
2.4 The lifting problem of fiber-preserving symplectomorphisms
to the prequantum line bundle
Let Γ′ be a group, and suppose that Γ′ acts on π0 : (M0, ω0) → Rn as fiber-preserving symplec-
tomorphisms. As in the previous section, we denote by ρ : Γ′ → GLn(Z) ⋉ Rn the Γ′-action
on Rn and also denote by ρ̃ its lift to (M0, ω0). By Lemma 2.17, for each γ ∈ Γ′, there ex-
ist Aγ ∈ GLn(Z), cγ ∈ Rn, and a map uγ : Rn → Tn with tAγJuγ symmetric such that ργ and ρ̃γ
can be described as follows
ργ(x) = Aγx+ cγ , ρ̃γ(x, y) =
(
Aγx+ cγ ,
tA−1
γ y + uγ(x)
)
. (2.3)
Note that since (2.3) is a Γ′-action, Aγ , cγ , and uγ satisfy the following conditions:
Aγ1γ2 = Aγ1Aγ2 , cγ1γ2 = Aγ1cγ2 + cγ1 , uγ1γ2(x) =
tA−1
γ1 uγ2(x) + uγ1(ργ2(x)) (2.4)
for γ1, γ2 ∈ Γ′, and x ∈ Rn. Let ũγ = t
(
ũ1γ , . . . , ũ
n
γ
)
: Rn → Rn be a lift of uγ . For ũγ and
i = 1, . . . , n, we put∫ xi
0
ũγ(x)dxi :=
t
(∫ xi
0
ũ1γ(x)dxi, . . . ,
∫ xi
0
ũnγ (x)dxi
)
and
F i
γ(x) :=
(
tAγ
∫ xi
0
ũγ(x)dxi
)
i
=
n∑
j=1
(
tAγ
)
ij
∫ xi
0
ũjγ(x)dxi.
2A Hermitian metric ⟨·, ·⟩L on L is compatible with ∇L if it satisfies d(⟨s1, s2⟩L) =
〈
∇Ls1, s2
〉
L
+
〈
s1,∇Ls2
〉
L
for all s1, s2 ∈ Γ(L).
Adiabatic Limit, Theta Function, and Geometric Quantization 15
Let N ∈ N be a positive integer. Each ρ̃γ preserves Nω0, hence, Γ
′ also acts on π0 : (M0, Nω0) →
Rn as fiber-preserving symplectomorphisms. Then, we examine in detail the conditions for the
Γ′-action to have a lift to
(
L0,∇L0
)⊗N → (M0, Nω0). The purpose of this subsection is to show
the following lemma which gives the necessary and sufficient condition on the existence of a lift
of the Γ′-action, and which also gives the explicit formula for the lift when this condition is
satisfied.
Lemma 2.27.
(1) For each γ ∈ Γ′, there exists a bundle automorphism ˜̃ργ of
(
L0,∇L0
)⊗N
preserving the
Hermitian metric and the connection such that ˜̃ργ covers ρ̃γ if and only if cγ is contained
in 1
NZn. Moreover, in this case, ˜̃ργ can be described as follows
˜̃ργ(x, y, z) = (ρ̃γ(x, y), gγe2π√−1N{g̃γ(x)+cγ ·(tA−1
γ y)}z
)
(2.5)
for (x, y, z) ∈ L⊗N
0
∼= Rn × Tn × C, where gγ is an arbitrary element in U(1) and
g̃γ(x) := ργ(x) · ũγ(x)− cγ · ũγ(0)−
n∑
i=1
F i
γ(0, . . . , 0, xi, . . . , xn).
The formula (2.5) does not depend on the choice of ũγ.
3
(2) Under the condition given in (1), the map ˜̃ρ : Γ′ → Aut
((
L0,∇L0
)⊗N)
defined by (2.5) is
a homomorphism if and only if the map g : Γ′ ∋ γ 7→ gγ ∈ U(1) is a homomorphism and
for all γ1, γ2 ∈ Γ′ and x ∈ Rn, the following condition holds:{
−cγ1 · uγ1(0) + cγ1 · tA−1
γ1 uγ2(0) + ργ1(cγ2) · uγ1(ργ2(0))
}
−
n∑
i=1
(
tAγ1
∫ (ργ2 (x))i
0
uγ1(0, . . . , 0, τi, (ργ2(x))i+1, . . . , (ργ2(x))n)dτi
)
i
+
n∑
i=1
(
tAγ2
tAγ1
∫ xi
0
uγ1(ργ2(0, . . . 0, τi, xi+1, . . . , xn))dτi
)
i
∈ 1
N
Z.
Proof. For each γ ∈ Γ′, we put
˜̃ργ(x, y, z) = (ρ̃γ(x, y), e2π{g̃Rγ (x,y)+
√
−1g̃Iγ(x,y)}z
)
,
where g̃Rγ and g̃Iγ are real valued functions on M0. By the direct computation, it is easy to see
that ˜̃ργ preserves ∇L⊗N
0 = d− 2π
√
−1Nx · dy if and only if g̃Rγ is constant and g̃Iγ satisfies the
following conditions:
∂xi g̃
I
γ = N(Aγx+ cγ) · ∂xi ũγ , (2.6)
∂yi g̃
I
γ = N
(
A−1
γ cγ
)
i
(2.7)
for i = 1, . . . , n. The conditions for the complete integrability of the system of partial differential
equations (2.6) and (2.7) are as follows:
∂xi∂xj g̃
I
γ = ∂xj∂xi g̃
I
γ , (2.8)
∂xi∂yj g̃
I
γ = ∂yj∂xi g̃
I
γ , (2.9)
∂yi∂yj g̃
I
γ = ∂yj∂yi g̃
I
γ (2.10)
3In the rest of this paper, we often use the notation uγ instead of ũγ .
16 T. Yoshida
for i, j = 1, . . . , n. From (2.6) and (2.7), (2.9) and (2.10) are true because both sides of each
vanish. From (2.6), (2.8) can be expressed as(
tAγ∂xiuγ(x)
)
j
=
(
tAγ∂xjuγ(x)
)
i
(2.11)
for i, j = 1, . . . , n. But, since tAγJuγ is symmetric, (2.11) is also valid. Therefore, we know that
there exists g̃Iγ that satisfies (2.6) and (2.7). In fact, such a g̃Iγ is given by
g̃Iγ(x, y) = g̃Iγ(0, 0) +N
{
ργ(x) · ũγ(x)− cγ · ũγ(0)
−
n∑
i=1
F i
γ(0, . . . , 0, xi, . . . , xn) + cγ · tA−1
γ y
}
. (2.12)
Since y ∈ Tn, g̃Iγ should satisfies e2π
√
−1g̃Iγ(0,ei) = e2π
√
−1g̃Iγ(0,0) for all i = 1, . . . , n and γ ∈ Γ′.
This holds if and only if A−1
γ Ncγ · ei ∈ Z for all i = 1, . . . , n and γ ∈ Γ′. Since Aγ ∈ GLn(Z),
this is equivalent to the condition Ncγ ∈ Zn. In this case, we put gγ := e2π(g̃
R
γ (0,0)+
√
−1g̃Iγ(0,0)).
Since ˜̃ργ preserves the Hermitian metric on
(
L0,∇L0
)
→ (M0, ω0), gγ is contained in U(1). The
formula (2.5) does not depend on the choice of ũγ since the difference of two lifts of uγ is in Zn.
This proves (1).
The map ˜̃ρ defined in (2) is a homomorphism if and only if g̃Iγ(x, y)− g̃Iγ(0, 0) defined by (2.12)
satisfies the cocycle condition. By a direct computation using (2.4), it is equivalent to the ones
given in (2). ■
Example 2.28. Let B be the n-dimensional integral affine torus given in Example 2.3 (1) for
a linear basis v1, . . . , vn ∈ Rn. The product B × Tn admits a symplectic structure ω so that the
trivial torus bundle π : (B × Tn, ω) → B becomes a Lagrangian fibration. This is obtained as
the quotient space of the action of Γ′ := Zn on π0 : (M0, ω0) → Rn which is defined by
ρ̃γ(x, y) = (x+ Cγ, y)
for γ ∈ Γ′ and (x, y) ∈ M0, where C = (v1 · · · vn) ∈ GLn(R). Let N ∈ N be a positive
number. The Γ′-action ρ̃ on (M0, Nω0) has a lift to the prequantum line bundle
(
L0,∇L0
)⊗N →
(M0, Nω0) if and only if all vi’s lie in 1
NZn, and in this case ˜̃ρ is given by
˜̃ργ(x, y, z) = (ρ̃γ(x, y), gγe2π√−1NCγ·yz
)
for γ ∈ Γ′ and (x, y, z) ∈ L⊗N
0
∼= Rn × Tn × C, where g : Γ′ ∋ γ 7→ gγ ∈ U(1) is an arbitrary
homomorphism.
Example 2.29 (the Kodaira–Thurston manifold). Let Γ′ be Z2. Let us consider the Γ′-action
on π0 :
(
R2 × T 2, ω0
)
→ R2 which is defined by
ργ(x) := x+ γ, ρ̃γ(x, y) := (ργ(x), y + uγ(x))
for γ ∈ Γ′ and (x, y) ∈ R2×T 2, where uγ(x) =
t(0, γ1x2). The Lagrangian fibration given by the
quotient of this action is denoted by π : (M,ω) → B. M was first observed by Kodaira in [26] and
Thurston pointed out in [41] that (M,ω) does not admits any Kähler structure. M is nowadays
called the Kodaira–Thurston manifold. Let N ∈ N be a positive number. The Γ′-action ρ̃ on(
R2 × T 2, Nω0
)
has a lift to the prequantum line bundle
(
R2 × T 2 × C, d− 2π
√
−1Nx · dy
)
→(
R2 × T 2, Nω0
)
if and only if N is even, and in this case the lift ˜̃ρ is given by
˜̃ργ(x, y, z) = (ρ̃γ(x, y), gγe2π√−1N{ 1
2
γ1x2
2+γ1γ2x2+γ·y}z
)
for γ ∈ Γ′ and (x, y, z) ∈ R2 × T 2 × C, where g : Γ′ ∋ γ 7→ gγ ∈ U(1) is an arbitrary homomor-
phism.
Adiabatic Limit, Theta Function, and Geometric Quantization 17
Example 2.30. Let B be the n-dimensional integral affine torus given in Example 2.3 (1) for
a linear basis v1, . . . , vn ∈ Rn. When all vi’s are integer vectors, i.e., v1, . . . , vn ∈ Zn, we can
generalize Examples 2.28 and 2.29 in the following way. For i, j = 1, . . . , n, choose uij ∈ Zn
satisfying uij = uji. For each γ ∈ Γ′ := Zn, we define the map uγ : Rn → Tn by
uγ(x) :=
u11 · γ · · · u1n · γ
...
...
un1 · γ · · · unn · γ
x,
and we also define the action of Γ′ on π0 : (M0, ω0) → Rn by
ρ̃γ(x, y) = (x+ Cγ, y + uγ(x)) (2.13)
for γ ∈ Γ′ and (x, y) ∈M0, where C = (v1 · · · vn). Then, the quotient π : (M,ω) → B obtained
as the Γ′-action (2.13) is a Lagrangian fibration on B. Let N ∈ N be a positive number. The
Γ′-action ρ̃ on (M0, Nω0) has a lift to the prequantum line bundle
(
L0,∇L0
)⊗N → (M0, Nω0) if
and only if N
2 vi · Ujvi ∈ Z for all i, j = 1, . . . , n, where
Uj :=
(u11)j · · · (u1n)j
...
...
(un1)j · · · (unn)j
.
And in this case, the lift ˜̃ρ is given by˜̃ργ(x, y, z) = (ρ̃γ(x, y), gγe2π√−1N [ 1
2
{ργ(x)·uγ(ργ(x))−ργ(0)·uγ(ργ(0))}+ργ(0)·y]z
)
for γ ∈ Γ′ and (x, y, z) ∈ L⊗N
0
∼= Rn × Tn × C, where g : Γ′ ∋ γ 7→ gγ ∈ U(1) is an arbitrary
homomorphism.
Example 2.31. Let n ≥ 2 and λ1, . . . , λn−1 ∈ Z. Let Γ′ be the group (Zn, ◦) given in Exam-
ple 2.6. For each γ ∈ Γ′, let Aγ be the matrix
Aγ :=
1 λ1
1 λ2
. . .
. . .
1 λn−1
1
γn
and uγ : Rn → Tn the map defined by
uγ(x) :=
0
...
0
γnxn
.
Let us consider the Γ′-action ρ̃ on π0 : (M0, ω) → Rn which is defined by
ρ̃γ(x, y) :=
(
Aγx+ γ, tA−1
γ y + uγ(x)
)
(2.14)
for γ ∈ Γ′ and (x, y) ∈M0. Then, the quotient π : (M,ω) → B obtained as the Γ′-action (2.14)
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 ρ̃ on (M0, Nω0) has a lift to the prequantum line bun-
dle
(
L0,∇L0
)⊗N → (M0, Nω0) if and only if N is even, and in this case the lift ˜̃ρ is given by˜̃ργ(x, y, z) = (ρ̃γ(x, y), gγe2π√−1N{γnxn(
1
2
xn+γn)+γ·(tA−1
γ y)}z
)
for γ ∈ Γ′ and (x, y, z) ∈ L⊗N
0
∼= Rn × Tn × C, where g : Γ′ ∋ γ 7→ gγ ∈ U(1) is an arbitrary
homomorphism.
18 T. Yoshida
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
of the Lagrangian fibration π0 : (M0, Nω0) → Rn, let D be the associated Spinc Dirac operator
with coefficients in the prequantum line bundle
(
L0,∇L0
)⊗N → (M0, Nω0). An element in the
kernel kerD of D is called a harmonic spinor. In this section, for J which is invariant along the
fiber in the sense of Lemma 3.6, we investigate the condition on the existence of nontrivial degree-
zero harmonic spinors, i.e., nontrivial sections which is contained in kerD. For the construction
and properties of the Spinc Dirac operator, see [11, 30].
3.1 Bohr–Sommerfeld points
Let π : (M,ω) → B be a Lagrangian fibration with prequantum line bundle
(
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
(
L,∇L
)
|π−1(b) admits a non-
trivial covariant constant section. We denote the set of Bohr–Sommerfeld points by BBS.
Let us detect Bohr–Sommerfeld points for π0 : (M0, Nω0) → Rn with prequantum line bun-
dle
(
L0,∇L0
)⊗N → (M0, Nω0).
Proposition 3.2. A point x ∈ Rn is Bohr–Sommerfeld if and only if x is contained in 1
NZn,
i.e., Rn
BS = 1
NZn. Moreover, for a Bohr–Sommerfeld point x ∈ 1
NZn, a covariant constant
section s of
(
L0,∇L0
)⊗N ∣∣
π−1
0 (x)
is of the form s(y) = s(0)e2π
√
−1Nx·y.
Proof. For a fixed x ∈ Rn, a section s of
(
L0,∇L0
)⊗N ∣∣
π−1
0 (x)
→ π−1
0 (x) is covariant constant if
and only if s satisfies
0 = ∇L⊗N
0
∂yi
s = ∂yis− 2π
√
−1Nxis
for i = 1, . . . , n. Hence, any covariant constant section s should be of the form s(y) =
s(0)e2π
√
−1Nx·y. Since π−1
0 (x) is a torus, s is periodic with respect to yi’s. In particular, s satis-
fies s(0) = s(ei) = s(0)e2π
√
−1Nxi for i = 1, . . . , n. This implies that
(
L0,∇L0
)⊗N∣∣
π−1
0 (x)
→ π−1
0 (x)
admits a nontrivial covariant constant section if and only if Nxi ∈ Z for i = 1, . . . , n. ■
Remark 3.3. Suppose that π0 : (M0, Nω0) → Rn is equipped with an action of a group Γ which
preserves all the data, and its lift ˜̃ρ to
(
L0,∇L0
)⊗N
is given by (2.5). Then, by Lemma 2.27 (1),
the Γ-action ρ on Rn preserves Rn
BS. When the Γ-action ρ on Rn is properly discontinuous and
free, let F ⊂ Rn be a fundamental domain of the Γ-action ρ on Rn. Then, the map
Γ×
(
F ∩ 1
N
Zn
)
∋
(
γ,
m
N
)
7→ Nργ
(m
N
)
∈ Zn (3.1)
can be defined and is bijective. In particular, if a Lagrangian fibration π : (M,Nω) → B with
prequantum line bundle
(
L,∇L
)⊗N → (M,Nω) is obtained as the quotient space of the Γ-action,
then F ∩ 1
NZn is identified with BBS.
3.2 Almost complex structures
Let Sn be the Siegel upper half space, namely, the space of n× n symmetric complex matrices
whose imaginary parts are positive definite
Sn :=
{
Z = X +
√
−1Y ∈Mn(C) | X,Y ∈Mn(R), tZ = Z, and Y is positive definite
}
.
Adiabatic Limit, Theta Function, and Geometric Quantization 19
It is well known that Sn is identified with the space of compatible complex structures on the 2n-
dimensional standard symplectic vector space. See [32, Chapter II, Section 4].
For a tangent vector u =
∑n
i=1{(ux)i∂xi + (uy)i∂yi} ∈ T(x,y)M0 at a point (x, y) ∈ M0, we
use the following notation:
u = (∂x1 , . . . , ∂xn , ∂y1 , . . . , ∂yn)
(ux)1
...
(ux)n
(uy)1
...
(uy)n
= (∂x, ∂y)
(
ux
uy
)
,
where
∂x = (∂x1 , . . . , ∂xn), ∂y = (∂y1 , . . . , ∂yn), ux =
(ux)1
...
(ux)n
, uy =
(uy)1
...
(uy)n
.
In terms of the notations of tangent vectors u = (∂x, ∂y) (
ux
uy ) and v = (∂x, ∂y) (
vx
vy ) ∈ T(x,y)M0,
ω0 can be described by
ω0(u, v) =
(
tux,
tuy
)( 0 I
−I 0
)(
vx
vy
)
.
Since the tangent bundle TM0 is trivial, the space of compatible almost complex structures
on (M0, ω0) is identified with the space of C∞ maps from M0 to Sn. For Z = X +
√
−1Y ∈
C∞(M0,Sn), the corresponding almost complex structure JZ is given as follows:
JZu := (∂x, ∂y)
(
XY −1 −Y −XY −1X
Y −1 −Y −1X
)
(x,y)
(
ux
uy
)
(3.2)
for u = (∂x, ∂y) (
ux
uy ) ∈ T(x,y)M0.
4 Then, the Riemannian metric g determined by ω0 and JZ can
be described by
g(u, v) : = ω0(u, Jv) =
(
tux,
tuy
)( 0 I
−I 0
)(
XY −1 −Y −XY −1X
Y −1 −Y −1X
)(
vx
vy
)
=
(
tux,
tuy
)( Y −1 −Y −1X
−XY −1 Y +XY −1X
)(
vx
vy
)
. (3.3)
Let J = JZ be the almost complex structure on (M0, ω0) corresponding to a given Z =
X +
√
−1Y ∈ C∞(M0,Sn). Then, (−J∂y, ∂y) = (−J∂y1 , . . . ,−J∂yn , ∂y1 , . . . , ∂yn) is also a basis
of the tangent space of (M0, ω0). For each tangent vector u ∈ T(x,y)M0, by using this basis, u is
expressed as follows:
u =
∑
i
{(uH)i(−J∂yi) + (uV )i∂yi} = (−J∂y, ∂y)
(
uH
uV
)
.
Then, we have the following transition formula between (∂x, ∂y) and (−J∂y, ∂y):
u = (−J∂y, ∂y)
(
uH
uV
)
= (∂x, ∂y)
((
−XY −1 Y +XY −1X
−Y −1 Y −1X
)(
0
uH
)
+
(
0
uV
))
.
By this formula, we obtain the following lemma.
4
(
XY −1 −Y −XY −1X
Y −1 −Y −1X
)
(x,y)
,
(
XY −1
)
(x,y)
etc. are the values of the maps
(
XY −1 −Y −XY −1X
Y −1 −Y −1X
)
, XY −1 etc. at
(x, y). We will often omit the subscript “(x,y)” for simplicity unless it causes confusion.
20 T. Yoshida
Lemma 3.4. In terms of this notation, the Riemannian metric g defined by (3.3) can be de-
scribed by
g(u, v) =
(
0, tuH
)( Y −1 −Y −1X
−XY −1 Y +XY −1X
)(
0
vH
)
+
(
0, tuV
)( Y −1 −Y −1X
−XY −1 Y +XY −1X
)(
0
vV
)
.
Suppose that a group Γ acts on π0 : (M0, ω0) → Rn and the Γ-actions ρ on Rn and ρ̃
on (M0, ω0) are written as in (2.3). Then, it is easy to see the following lemma.
Lemma 3.5. The Γ-action ρ̃ on (M0, ω0) preserves the almost complex structure J = JZ
on (M0, ω0) corresponding to Z = X +
√
−1Y ∈ C∞(M0,Sn) if and only if the following condi-
tions hold:
Aγ
(
XY −1
)
(x,y)
=
(
XY −1
)
ρ̃γ(x,y)
Aγ −
(
Y +XY −1X
)
ρ̃γ(x,y)
(Juγ)x, (3.4)
Aγ
(
Y +XY −1X
)
(x,y)
=
(
Y +XY −1X
)
ρ̃γ(x,y)
tA−1
γ , (3.5)
(Juγ)x
(
XY −1
)
(x,y)
+ tA−1
γ Y −1
(x,y) = Y −1
ρ̃γ(x,y)
Aγ −
(
Y −1X
)
ρ̃γ(x,y)
(Juγ)x.
Proof. For all γ ∈ Γ and (x, y) ∈ (M0, ω0), the condition
(dρ̃γ)(x,y) ◦ J(x,y) = Jρ̃γ(x,y) ◦ (dρ̃γ)(x,y)
implies above three equalities together with the following equality:
(Juγ)x
(
Y +XY −1X
)
(x,y)
+ tA−1
γ
(
Y −1X
)
(x,y)
=
(
Y −1X
)
ρ̃γ(x,y)
tA−1
γ .
But, this can be obtained from (3.4), (3.5), and t
(
tAγ(Juγ)x
)
= tAγ(Juγ)x. ■
Let π : (M,ω) → B be a Lagrangian fibration with complete n-dimensional base B and
p : B̃ → B the universal covering of B. By Corollary 2.25, the pullback of π : (M,ω) → B to B̃
is identified with π0 : (M0, ω0) → Rn and π : (M,ω) → B can be obtained as the quotient of
the Γ = π1(B)-action on π0 : (M0, ω0) → Rn. In particular, for each compatible almost complex
structure J on (M,ω), there exists a map ZJ = X +
√
−1Y ∈ C∞ (M0,Sn) such that the
pullback p∗J of J to p∗(M,ω) coincides with JZJ
. Then, we have the following lemma.
Lemma 3.6 ([15, Corollary 9.15]). For any Lagrangian fibration π : (M,ω) → B, there exists
a compatible almost complex structure J on (M,ω) such that the corresponding map ZJ does
not depend on y1, . . . , yn. We say such J to be invariant along the fiber.
Proof. Take a Riemannian metric g′ on (M,ω). Then, the pullback p∗g′ is π1(B)-invariant.
Moreover, p∗(M,ω) admits a free Tn-action, and this Tn-action together with the π1(B)-action
forms an action of the semi-direct product π1(B) ⋉ Tn of Tn and π1(B). By averaging p∗g′
over Tn, we obtain a Riemannian metric on p∗M invariant under the π1(B) ⋉ Tn-action. It is
easy to see that p∗ω is also π1(B)⋉Tn-invariant, so by the standard method using the π1(B)⋉Tn-
invariant Riemannian metric and p∗ω, we can obtain a π1(B)⋉ Tn-invariant compatible almost
complex structure on p∗(M,ω). In particular, since the almost complex structure is still invariant
under π1(B)-action, it descends to (M,ω). This is the required almost complex structure. ■
Adiabatic Limit, Theta Function, and Geometric Quantization 21
3.3 A condition on the existence of nontrivial harmonic spinors
of degree-zero
For a map Z = X +
√
−1Y ∈ C∞(M0,Sn), we set
Ω :=
(
Y +XY −1X
)−1
ZY −1. (3.6)
Lemma 3.7. Ω has the following properties:
(1) Ω = Z
−1
, where Z = X −
√
−1Y .
(2) Ω is symmetric, i.e., tΩ = Ω.
Proof. A direct computation shows that ΩZ = I. This proves (1). (2) follows from (1) since Z
is symmetric. ■
Let N ∈ N be a positive integer. Let J = JZ be the compatible almost complex structure
on (M0, Nω0) corresponding to a given Z = X +
√
−1Y ∈ C∞(M0,Sn). Then, the Rieman-
nian metric Ng := Nω0(·, J ·) defines an isomorphism f : T ∗M0
∼= TM0 by τ = Ng (f(τ), ·)
for τ ∈ T ∗M0. For i = 1, . . . , n, let Ωi denote the ith column vector of Ω, and ReΩi and ImΩi
be the real and imaginary parts of Ωi, respectively. Then, we can show the following lemma.
Lemma 3.8. For i = 1, . . . , n,
f(dxi) = − 1
N
J∂yi , f(dyi) = (−J∂y, ∂y)
1
N
ReΩi
1
N
ImΩi
.
Proof. We prove the latter. The former can be proved by the same way. Put f(dyi) =
(−J∂y, ∂y)
(
Y i
H
Y i
V
)
. By definition, for each i, j = 1, . . . , n, we have
dyi(−J∂yj ) = Ng
(
(−J∂y, ∂y)
(
Y i
H
Y i
V
)
, (−J∂y, ∂y)
(
ej
0
))
, (3.7)
dyi(∂yj ) = Ng
(
(−J∂y, ∂y)
(
Y i
H
Y i
V
)
, (−J∂y, ∂y)
(
0
ej
))
. (3.8)
Since −J∂yj is written as
−J∂yj = (∂x, ∂y)
(
−XY −1 Y +XY −1X
−Y −1 Y −1X
)(
0
ej
)
by (3.2), the left-hand side of (3.7) is
(
Y −1X
)
ij
. On the other hand, by Lemma 3.4, the
right-hand side of (3.7) can be described as NY i
H ·
(
Y + XY −1X
)
ej . This implies Y −1X =
N t
(
Y 1
H · · ·Y n
H
)(
Y + XY −1X
)
. Since Y is positive definite, so is Y + XY −1X. In partic-
ular, N
(
Y + XY −1X
)
is invertible. By using tX = X, tY = Y together with this fact,
we can obtain
(
Y 1
H · · ·Y n
H
)
= 1
N
(
Y +XY −1X
)−1
XY −1. By the same way, from (3.8), we ob-
tain I = N t
(
Y 1
V · · ·Y n
V
)(
Y + XY −1X
)
, i.e.,
(
Y 1
V · · ·Y n
V
)
= 1
N
(
Y +XY −1X
)−1
. Hence, 1
NΩ =(
Y 1
H · · ·Y n
H
)
+
√
−1
(
Y 1
V · · ·Y n
V
)
. ■
Define the Hermitian metric on (M0, Nω0, Ng, J) by
h(u, v) := Ng(u, v) +
√
−1Ng(u, Jv) (3.9)
for u, v ∈ T(x,y)M0. Let (W, c) be the Clifford module bundle associated with (Ng, J), i.e., as
a complex vector bundle, W is defined by
W := ∧•(TM0, J)⊗C
(
L⊗N
0
)
.
22 T. Yoshida
W is equipped with the Hermitian metric induced from h and that on L0, and also equipped
with the Hermitian connection, which is denoted by ∇W , induced from the Levi-Civita connec-
tion ∇LC of (M0, Ng) and ∇L0 . c is the Clifford multiplication c : TM0 → EndC(W ) which is
defined by c(u)(τ) := u ∧ τ − u⌞hτ for u ∈ TM0 and τ ∈ W , where ⌞h is the contraction with
respect to the Hermitian metric h on (M0, Nω0, Ng, J). It is well known that W is identified
with ∧•(T ∗M0)
0,1⊗C
(
L0
⊗N
)
as a Clifford module bundle since h induces the isomorphism from
(TM0, J) to (T ∗M0)
0,1 as Hermitian vector bundles. See [11, pp. 12–13] for more details.
Now let us define the Spinc Dirac operator D : Γ(W ) → Γ(W ) by the composition of the
following maps:
D : Γ(W )
∇W
// Γ(T ∗M0 ⊗W )
f⊗idW // Γ(TM0 ⊗W )
c // Γ(W ).
We compute the action of D on degree zero elements in Γ(W ). We identify a section of L0 with
a complex valued function on M0. By using Lemma 3.8, for a section s of L0
⊗N , Ds can be
computed as
Ds = c ◦ (f ⊗ idW ) ◦ ∇W s = c ◦ (f ⊗ idW )
(
ds− 2π
√
−1Nx · dys
)
=
n∑
i=1
{
c(f(dxi))(∂xis) + c(f(dyi))
(
∂yis− 2π
√
−1Nxis
)}
= −
√
−1
N
n∑
i=1
∂yi ⊗C
{
∂xis+
n∑
j=1
Ωij
(
∂yjs− 2π
√
−1Nxjs
)}
.
In particular, the equality Ds = 0 is equivalent to
0 =
∂x1s
...
∂xns
+Ω
∂y1s− 2π
√
−1Nx1s
...
∂yns− 2π
√
−1Nxns
. (3.10)
For a section s of L0
⊗N , let us consider the Fourier series expansion of s with respect to yi’s.
For each x ∈ Rn, as a function of yi’s, s(x, ·) can be expressed as the Fourier series
s(x, y) =
∑
m∈Zn
am(x)e2π
√
−1m·y, (3.11)
where am(x) :=
∫
Tn s(x, y)e
−2π
√
−1m·ydy for m ∈ Zn. Suppose that Z does not depend on
y1, . . . , yn as in Lemma 3.6. Then, by using the Fourier series (3.11), the equation Ds = 0 can
be reduced to the following system of differential equations for am’s with variables x1, . . . , xn.
Lemma 3.9. s satisfies Ds = 0 if and only if am’s satisfy
0 =
∂x1am
...
∂xnam
+ 2π
√
−1amΩ(m−Nx) (3.12)
for all m ∈ Zn.
Proof. By Lemma A.1, the partial derivatives ∂xjs and ∂yjs have the following Fourier series
with respect to yi’s:
∂xjs(x, y) =
∑
m∈Zn
∂xjam(x)e2π
√
−1m·y, (3.13)
∂yjs(x, y) =
∑
m∈Zn
2π
√
−1mjam(x)e2π
√
−1m·y for j = 1, . . . , n. (3.14)
Then, substituting (3.11), (3.13) and (3.14) into (3.10), we can obtain (3.12). ■
Adiabatic Limit, Theta Function, and Geometric Quantization 23
We investigate the equation (3.12).
Lemma 3.10. Let am be a solution of (3.12) for some m ∈ Zn. If there exists p ∈ Rn such
that am(p) = 0, then am(x) = 0 for all x ∈ Rn.
Proof. First, fix the variables x2, . . . , xn to equal p2, . . . , pn, respectively. Then, the first entry
of (3.12), i.e., 0 = ∂x1am+2π
√
−1am(Ω(m−Nx))1 can be thought of as an ordinary differential
equation on x1, and am(x1, p2, . . . , pn) is its solution with initial condition am(p) = 0. On the
other hand, the trivial solution also has the same initial condition. By the uniqueness of the
solution of the ordinary differential equation, am(x1, p2, . . . , pn) = 0 for any x1. Next, by fixing
variables x3, . . . , xn with p3, . . . , pn and fixing x1 with arbitrary value, am(x1, x2, p3, . . . , pn) is
a solution of 0 = ∂x2am +2π
√
−1am(Ω(m−Nx))2 with initial condition am(x1, p2, . . . , pn) = 0.
Then, am(x1, x2, p3, . . . , pn) = 0 for any x1, x2. By repeating the process for x3, . . . , xn, we can
show that am(x) = 0. ■
Lemma 3.11. If am is a nontrivial smooth solution of (3.12) for some m ∈ Zn, then the
condition(
(∂xiΩ)x (m−Nx)
)
j
=
((
∂xjΩ
)
x
(m−Nx)
)
i
(3.15)
holds for all i, j = 1, . . . , n and x ∈ Rn. Conversely, if there exists m ∈ Zn such that (3.15) holds
for all i, j = 1, . . . , n and x ∈ Rn, then (3.12) has a unique nontrivial solution up to constant.
Moreover, in this case, each solution am of (3.12) has the following form:
am(x) = am
(m
N
)
e−2π
√
−1
∑n
i=1 G
i
m(
m1
N
,...,
mi−1
N
,xi,...,xn), (3.16)
where am(mN ) can be taken as an arbitrary constant in C and Gi
m(x) :=
(∫ xi
mi
N
Ω(m−Nx)dxi
)
i
.
Proof. Since am is smooth, am satisfies ∂xi∂xjam = ∂xj∂xiam for all i, j = 1, . . . , n. By differ-
entiating (3.12), we have
∂xi∂xjam = − 2π
√
−1am
{
−2π
√
−1
n∑
k=1
Ωik(mk −Nxk)
n∑
l=1
Ωjl(ml −Nxl)
+
n∑
l=1
(∂xiΩjl) (ml −Nxl)−NΩji
}
for i, j = 1, . . . , n and x ∈ Rn. The condition (3.15) is obtained from this equation.
Conversely, suppose there exists m ∈ Zn such that (3.15) holds for all i, j = 1, . . . , n and
x ∈ Rn. By solving the differential equation appeared as the ith component of (3.12) for
i = 1, . . . , n, we have
am(x) = am
(
x1, . . . , xi−1,
mi
N
, xi+1, . . . , xn
)
e−2π
√
−1Gi
m(x). (3.17)
Using (3.17) repeatedly, we obtain the formula (3.16). By using (3.15), we can show that (3.16)
does not depend on the order of applying (3.17) to xi’s as in the proof of Lemma 2.27. Hence,
(3.16) is well defined. ■
Definition 3.12. We say m ∈ Zn to be integrable if (3.15) holds for all i, j = 1, . . . , n and
x ∈ Rn.
For each m ∈ Zn which is integrable, define the section sm ∈ Γ
(
L0
⊗N
)
by
sm(x, y) := e2π
√
−1{−
∑n
i=1 G
i
m(
m1
N
,...,
mi−1
N
,xi,...,xn)+m·y}. (3.18)
By the elliptic regularity of D and Lemma 3.11, we can obtain the following.
24 T. Yoshida
Proposition 3.13. If
s=
∑
m∈Zn
am(x)e2π
√
−1m·y ∈ Γ
(
L0
⊗N
)
is a nontrivial solution of 0=Ds, then all m ∈ Zn with am ̸= 0 are integrable. Conversely,
suppose that there exists m ∈ Zn such that m is integrable. Then, the section sm defined
by (3.18) satisfies 0 = Dsm.
The following proposition gives a geometric interpretation of the condition (3.15).
Proposition 3.14. The following conditions are equivalent:
(1) All m ∈ Zn are integrable.
(2) ∂xiΩjk = ∂xjΩik for all i, j, k = 1, . . . , n.
(3) ∇LCJ = 0, where ∇LC is the Levi-Civita connection with respect to g.
A proof of Proposition 3.14 is given in Appendix B.
Remark 3.15. When one of (hence, all) the conditions in Proposition 3.14 holds, (M0, ω0, J, g)
is a Kähler manifold and J induces a holomorphic structure on L0 such that ∇L0 is the canonical
connection.
3.4 The Γ-equivariant case
Suppose that π0 : (M0, Nω0, J) → Rn with prequantum line bundle
(
L0,∇L0
)⊗N→(M0, Nω0, J)
is equipped with an action of a group Γ which preserves all the data, and the Γ-actions are
described by (2.3) and (2.5) as before. We assume that the Γ-action ρ on Rn is properly
discontinuous and free. Since the Γ-action preserves all the data, the Spinc Dirac operator D is
Γ-equivariant. In particular, Γ acts on Γ
(
L0
⊗N
)
∩ kerD.
Lemma 3.16. Let s be a section of L0
⊗N with the Fourier series of the form (3.11). Then, s is
Γ-equivariant, i.e., ˜̃ργ ◦s = s◦ ρ̃γ for all γ ∈ Γ if and only if am satisfies the following condition:
aNργ(
m
N
)(ργ(x)) = gγam(x)e2π
√
−1N{g̃γ(x)−ργ(
m
N
)·uγ(x)} (3.19)
for all γ ∈ Γ, m ∈ Zn, and x ∈ Rn. In particular, any Γ-equivariant section of L0
⊗N can be
written as follows:
s(x, y) =
∑
(γ,m
N
)∈Γ×(F∩ 1
N
Zn)
gγam(ργ−1(x))
× e2π
√
−1N{g̃γ(ργ−1 (x))−ργ(
m
N
)·uγ(ργ−1 (x))}e2π
√
−1Nργ(
m
N
)·y. (3.20)
Proof. By computing the both sides separately, we have
˜̃ργ ◦ s(x, y) = gγe
2π
√
−1N{g̃γ(x)+cγ ·tA−1
γ y}
∑
m∈Zn
am(x)e2π
√
−1m·y
= gγ
∑
m∈Zn
am(x)e2π
√
−1Ng̃γ(x)e2π
√
−1Nργ(
m
N
)·tA−1
γ y,
s ◦ ρ̃γ(x, y) =
∑
l∈Zn
al(ργ(x))e
2π
√
−1l·(tA−1
γ y+uγ(x))
=
∑
m∈Zn
aNργ(
m
N
)(ργ(x))e
2π
√
−1Nργ(
m
N
)·uγ(x)e2π
√
−1Nργ(
m
N
)·tA−1
γ y.
Adiabatic Limit, Theta Function, and Geometric Quantization 25
Here, in the last equality, we replace l by Nργ(
m
N ). Note that the map
Zn ∋ m 7→ Nργ
(m
N
)
∈ Zn
is bijective. Then, ˜̃ργ ◦ s = s ◦ ρ̃γ for all γ ∈ Γ implies
gγam(x)e2π
√
−1Ng̃γ(x) = aNργ(
m
N
)(ργ(x))e
2π
√
−1Nργ(
m
N
)·uγ(x)
for all m ∈ Zn. In particular, by (3.1) and (3.19), s can be rewritten as follows:
s(x, y) =
∑
l∈Zn
al(x)e
2π
√
−1l·y (3.1)
=
∑
(γ,m
N
)∈Γ×(F∩ 1
N
Zn)
aNργ(
m
N
)(x)e
2π
√
−1Nργ(
m
N
)·y
(3.19)
=
∑
(γ,m
N
)∈Γ×(F∩ 1
N
Zn)
gγam(ργ−1(x))
× e2π
√
−1N{g̃γ(ργ−1 (x))−ργ(
m
N
)·uγ(ργ−1 (x))}e2π
√
−1Nργ(
m
N
)·y. ■
In the Γ-equivariant case, the condition (3.15) has a symmetry in the following sense.
Lemma 3.17. There exists m0 ∈ Zn with m0
N ∈ F such that m0 is integrable if and only if for
any γ ∈ Γ m = Nργ(
m0
N ) is integrable. Moreover, let am0 be a nontrivial solution of (3.12) for
m0. For each γ ∈ Γ, we define aNργ(
m0
N
) in such a way that it satisfies (3.19). Then, aNργ(
m0
N
)
is a nontrivial solution of (3.12) for m = Nργ(
m0
N ).
Proof. Suppose that there exists m0 ∈ Zn with m0
N ∈ F such that m0 is integrable. By
Lemma 3.11, (3.12) for m0 has a nontrivial solution am0 . Then, for each γ ∈ Γ, define aNργ(
m0
N
)
by (3.19). By Lemma 3.11 again, in order to show this lemma, it is sufficient to prove aNργ(
m0
N
)
is a solution of (3.12) for m = Nργ(
m0
N ). Let us compute the Jacobi matrix of the both sides
of (3.19). The left-hand side is
J(aNργ(
m0
N
) ◦ ργ)x = (JaNργ(
m0
N
))ργ(x)(Jργ)x
= (∂x1aNργ(
m0
N
), . . . , ∂xnaNργ(
m0
N
))ργ(x)Aγ . (3.21)
The right-hand side is
J
(
gγam(x)e2π
√
−1N{g̃γ(x)−ργ(
m
N
)·uγ(x)})
x
= gγe
2π
√
−1N{g̃γ(x)−ργ(
m
N
)·uγ(x)}(Jam)x + gγam(x)J
(
e2π
√
−1N{g̃γ(x)−ργ(
m
N
)·uγ(x)})
(3.12)
= −2π
√
−1gγe
2π
√
−1N{g̃γ(x)−ργ(
m
N
)·uγ(x)}am(x)t(Ωx(m−Nx))
+ 2π
√
−1Ngγam(x)e2π
√
−1N{g̃γ(x)−ργ(
m
N
)·uγ(x)}J
(
g̃γ(x)− ργ
(m
N
)
· uγ(x)
)
(3.19)
= −2π
√
−1aNργ(
m0
N
)(ργ(x))
t
(
ΩxA
−1
γ
(
Nργ
(m
N
)
−Nργ(x)
))
+ 2π
√
−1NaNργ(
m0
N
)(ργ(x))J
(
g̃γ(x)− ργ
(m
N
)
· uγ(x)
)
. (3.22)
For each i = 1, . . . , n, the direct computation shows
∂xi
(
g̃γ(x)− ργ
(m
N
)
· uγ(x)
)
= (∂xiuγ)x ·
(
ργ(x)− ργ
(m
N
))
+
(
tAγuγ(x)
)
i
−
(
tAγuγ(0, . . . , 0, xi, . . . , xn)
)
i
−
∑
j<i
∫ xj
0
(
tAγJuγ
)
ji
(0, . . . , 0, xj , . . . , xn)dxj
26 T. Yoshida
= (∂xiuγ)x ·
(
ργ(x)− ργ
(m
N
))
+
∑
j<i
∫ xj
0
∂xj
(
tAγuγ(0, . . . , 0, xj , . . . , xn)
)
i
dxj
−
∑
j<i
∫ xj
0
(
tAγJuγ
)
ji
(0, . . . , 0, xj , . . . , xn)dxj
= (∂xiuγ)x ·
(
ργ(x)− ργ
(m
N
))
+
∑
j<i
∫ xj
0
(
tAγJuγ
)
ij
(0, . . . , 0, xj , . . . , xn)dxj
−
∑
j<i
∫ xj
0
(
tAγJuγ
)
ji
(0, . . . , 0, xj , . . . , xn)dxj
= −(∂xiuγ)x ·
(
ργ
(m
N
)
− ργ(x)
)
.
In the last equality, we used t
(
tAγJuγ
)
= tAγJuγ . Hence, we have
J
(
g̃γ(x)− ργ
(m
N
)
· uγ(x)
)
= −t
(
ργ
(m
N
)
− ργ(x)
)
(Juγ)x . (3.23)
By (3.21), (3.22) and (3.23), we obtain
tAγ
t(JaNργ(
m0
N
))ργ(x)
= −2π
√
−1aNργ(
m0
N
)(ργ(x))
(
ΩxA
−1
γ + t(Juγ)x
) (
Nργ
(m
N
)
−Nργ(x)
)
.
On the other hand, by (3.4) and (3.5), we have
tAγΩργ(x) = ΩxA
−1
γ + t (Juγ)x . (3.24)
This proves the lemma. ■
Remark 3.18. By Remark 3.3 and Lemma 3.17, all m
N ∈ F ∩ 1
NZn are integrable if and only if
the condition (1), hence all conditions in Proposition 3.14 holds.
4 The integrable case
In this section, we investigate the case where the almost complex structure is integrable in
details. We use the setting and the notations introduced in the previous section.
4.1 Definition and properties of ϑm
N
Let m
N ∈ F ∩ 1
NZn be the point which is integrable, and am the nontrivial solution of (3.12) of the
form (3.16) with am(mN ) = 1. For each γ ∈ Γ, define aNργ(
m
N
) in such a way that it satisfies (3.19).
As we showed in Lemma 3.17, aNργ(
m
N
) is a nontrivial solution of (3.12) for Nργ(
m
N ). Then, we
define the formal Fourier series ϑm
N
by
ϑm
N
(x, y) :=
∑
γ∈Γ
aNργ(
m
N
)(x)e
2π
√
−1Nργ(
m
N
)·y. (4.1)
Proposition 4.1.
(1) ϑm
N
has the following expression:
ϑm
N
(x, y) =
∑
γ∈Γ
gγe
2π
√
−1{Θ(m
N
,γ,x)+Nργ(
m
N
)·y},
Adiabatic Limit, Theta Function, and Geometric Quantization 27
where
Θ
(m
N
, γ, x
)
:= −
n∑
i=1
Gi
m
(m1
N
, . . . ,
mi−1
N
, (ργ−1(x))i, . . . , (ργ−1(x))n
)
+N
{
g̃γ(ργ−1(x))− ργ
(m
N
)
· uγ(ργ−1(x))
}
.
(2) ϑm
N
can be described as ϑm
N
=
∑
γ∈Γ ˜̃ργ ◦ sm ◦ ρ̃γ−1, where sm is the section defined by (3.18).
(3) If Y +XY −1X is constant, then ϑm
N
converges absolutely and uniformly on any compact
set.
Proof. (1) and (2) are obtained by (3.19), (3.16), (2.5) and (3.18). Let us prove (3). By (2.4)
and (3.5), we obtain
tAγ−1
(
Y +XY −1X
)−1
Aγ−1 =
(
Y +XY −1X
)−1
.
By using this formula together with the assumption, the expression in (1) can be rewritten as
ϑm
N
(x, y) =
∑
γ∈Γ
gγe
2π
√
−1[
√
−1N
2
(x−ργ(
m
N
))·(Y+XY −1X)−1(x−ργ(
m
N
))+real part].
Since
(
Y + XY −1X
)−1
is positive definite, there exists a positive constant c > 0 such that(
Y +XY −1X
)−1 ≥ cI. Then,
|gγe2π
√
−1[
√
−1N
2
(x−ργ(
m
N
))·(Y+XY −1X)−1(x−ργ(
m
N
))+real part]|
= e−Nπ(x−ργ(
m
N
))·(Y+XY −1X)−1(x−ργ(
m
N
)) ≤ e−cNπ∥x−ργ(
m
N
)∥2
= e−cNπ∥x− l
N
∥2
(
put l := Nργ
(m
N
))
=
n∏
i=1
e−cNπ(xi−
li
N
)2 .
Hence, the series is dominated by
∏n
i=1
∑
li∈Z e
−cNπ(
li
N
−xi)
2
. Any compact set is contained in
a product of closed intervals I1 × · · · × In, so it is sufficient to show that
∑
l∈Z e
−cNπ( l
N
−x)2
converges uniformly on any closed interval I. Suppose that I is of the form I := [xm, xM ].
Set lM := max
{
l ∈ Z | l
N ∈ I
}
and lm := min
{
l ∈ Z | l
N ∈ I
}
. On I,
∑
−k≤l≤k e
−cNπ( l
N
−x)2
can be estimated as∑
−k≤l≤k
e−cNπ( l
N
−x)2 =
( ∑
−k≤l<lm
+
∑
lm≤l≤lM
+
∑
lM≤l≤k
)
e−cNπ( l
N
−x)2
≤
∑
−k≤l<lm
e
−cπ
N
(l−Nxm)2 + (lM − ln + 1) +
∑
lM<l≤k
e
−cπ
N
(l−NxM )2
≤
∫ lm
−k
e
−cπ
N
(τ−Nxm)2dτ + (lM − ln + 1) +
∫ k
lM
e
−cπ
N
(τ−NxM )2dτ.
It is well known that
∫ lm
−k e
−cπ
N
(τ−Nxm)2dτ and
∫ k
lM
e
−cπ
N
(τ−NxM )2dτ converge as k → +∞. ■
Lemma 4.2. Let s be a section of L0
⊗N with Fourier series of the form (3.11). If s is a nontriv-
ial Γ-equivariant solution of 0 = Ds, then there exists m
N ∈ F ∩ 1
NZn such that m is integrable.
Conversely, suppose that there exists m
N ∈ F ∩ 1
NZn such that m is integrable and ϑm
N
converges
absolutely and uniformly on any compact set. Then, ϑm
N
is a nontrivial Γ-equivariant solution
of 0 = Ds.
28 T. Yoshida
Proof. Since s =
∑
l∈Zn al(x)e
2π
√
−1l·y is nontrivial solution of 0 = Ds, by Proposition 3.13,
there exists l ∈ Zn such that al ̸= 0. On the other hand, as is noticed in Remark 3.3, there
exists
(
γ, mN
)
∈ Γ ×
(
F ∩ 1
NZn
)
such that l = Nργ(
m
N ). Since s is Γ-equivariant, by (3.19),
0 ̸= al = aNργ(
m
N
) implies am ̸= 0. The latter part follows from the definition of ϑm
N
. ■
Let π : (M,ω) → B be a Lagrangian fibration on a complete base B with prequantum line
bundle
(
L,∇L
)
→ (M,ω). By Corollary 2.25, they are obtained as the quotient of an action
of Γ := π1(B) on
(
L0,∇L0
)
→ (M0, ω0). Let J be a compatible almost complex structure
on (M,ω) which is invariant along the fiber in the sense of Lemma 3.6 and DM the associated
Spinc Dirac operator on (M,Nω) with coefficients in L⊗N . We denote by D the Spinc Dirac
operator with coefficients in L0
⊗N associated with the pull-back of J to M0. Since the Γ-action
preserves all the data, Γ
(
L⊗N
)
∩ kerDM is identified with
(
Γ
(
L0
⊗N
)
∩ kerD
)Γ
, the space of
Γ-equivariant elements in Γ
(
L0
⊗N
)
∩kerD. If J is integrable, so is the pull-back of J to M0. In
this case, by Proposition 3.14, all m
N ∈ F ∩ 1
NZn are integrable. So, one can consider ϑm
N
for all
m
N ∈ F ∩ 1
NZn. By Lemma 4.2 and the above identification, if all ϑm
N
’s converge absolutely and
uniformly on any compact set, then they can be thought of as elements of Γ
(
L⊗N
)
∩ kerDM ,
i.e., holomorphic sections of L⊗N indexed by BBS. (As we noticed in Remark 3.3, F ∩ 1
NZn is
identified with BBS.)
We choose the orientation onM so that (−1)
n(n−1)
2
(Nω)n
n! is a positive volume form, and define
the Hermitian inner product on Γ
(
L⊗N
)
by
(s, s′)L2(L⊗N ) :=
∫
M
⟨s, s′⟩L⊗N (−1)
n(n−1)
2
(Nω)n
n!
for s, s′ ∈ Γ
(
L⊗N
)
, where ⟨·, ·⟩L⊗N is the Hermitian metric of L⊗N . For s ∈ Γ
(
L⊗N
)
, we denote
its L2-norm by
∥s∥L2(L⊗N ) := {(s, s)L2(L⊗N )}
1
2
and denote the space of L2-sections of L⊗N by L2
(
L⊗N
)
. Then, we have the following theorem.
Theorem 4.3. Let π : (M,ω) → B be a Lagrangian fibration on a complete base B and(
L,∇L
)
→ (M,ω) a prequantum line bundle. Let J be a compatible integrable almost complex
structure on (M,ω) which is invariant along the fiber in the sense of Lemma 3.6 and DM the
associated Spinc Dirac operator on (M,Nω) with coefficients in L⊗N as above. Assume that ϑm
N
converges absolutely and uniformly on any compact set and is square integrable as a section
of L⊗N for each m
N ∈ F ∩ 1
NZn. Then, L2
(
L⊗N
)
∩kerDM is a Hilbert space and {ϑm
N
}m
N
∈F∩ 1
N
Zn
is a complete orthogonal system of L2
(
L⊗N
)
∩ kerDM indexed by the Bohr–Sommerfeld points.
Proof. By the definition of ϑm
N
and the assumption of Theorem 4.3, {ϑm
N
}m
N
∈F∩ 1
N
Zn is an or-
thogonal system of L2
(
L⊗N
)
. Suppose that l.h.({ϑm
N
}m
N
∈F∩ 1
N
Zn) is the subspace of L2
(
L⊗N
)
generated by {ϑm
N
}m
N
∈F∩ 1
N
Zn , namely,
l.h.({ϑm
N
}m
N
∈F∩ 1
N
Zn) :=
{
k∑
i=1
ciϑmi
N
∣∣∣ k ∈ N, ci ∈ C,
mi
N
∈ F ∩ 1
N
Zn
}
,
and we denote the closure of l.h.({ϑm
N
}m
N
∈F∩ 1
N
Zn) in L2
(
L⊗N
)
by l.h.({ϑm
N
}m
N
∈F∩ 1
N
Zn). Then,
l.h.({ϑm
N
}m
N
∈F∩ 1
N
Zn) is described as
l.h.({ϑm
N
}m
N
∈F∩ 1
N
Zn)
=
{ ∑
m
N
∈F∩ 1
N
Zn
cm
N
ϑm
N
∣∣∣ cm
N
∈ C,
∑
m
N
∈F∩ 1
N
Zn
cm
N
ϑm
N
converges in L2
(
L⊗N
)}
.
Adiabatic Limit, Theta Function, and Geometric Quantization 29
In fact, any φ =
∑
m
N
∈F∩ 1
N
Zn cm
N
ϑm
N
in the right-hand side satisfies
lim
k→∞
∥∥∥∥∥φ−
∑
m
N
∈F∩ 1
N
Zn, |m|≤k
cm
N
ϑm
N
∥∥∥∥∥
L2(L⊗N )
= 0.
This implies φ is contained by the left-hand side. Conversely, since {ϑm
N
}m
N
∈F∩ 1
N
Zn is a com-
pletely orthogonal system of the subspace l.h.({ϑm
N
}m
N
∈F∩ 1
N
Zn), then any φ from the subspace
l.h.({ϑm
N
}m
N
∈F∩ 1
N
Zn) satisfies
lim
k→∞
∥∥∥∥∥φ−
∑
m
N
∈F∩ 1
N
Zn,|m|≤k
(φ, ϑm
N
)L⊗N
∥ϑm
N
∥2
L2(L⊗N )
ϑm
N
∥∥∥∥∥
L2(L⊗N )
= 0.
This implies φ is contained by the right-hand side. We show that
L2
(
L⊗N
)
∩ kerDM = l.h.({ϑm
N
}m
N
∈F∩ 1
N
Zn).
Let s be an element of L2
(
L⊗N
)
∩kerDM . We think of s as an element of
(
Γ
(
L0
⊗N
)
∩ kerD
)Γ
.
By Lemma 3.16, s can be written as in (3.20). Then,
s(x, y)
(3.20)
=
∑
(γ,m
N
)∈Γ×(F∩ 1
N
Zn)
gγam(ργ−1(x))e2π
√
−1N{g̃γ(ργ−1 (x))−ργ(
m
N
)·uγ(ργ−1 (x))+ργ(
m
N
)·y}
(3.16)
=
∑
(γ,m
N
)∈Γ×(F∩ 1
N
Zn)
gγam(
m
N
)e2π
√
−1{Θ(m
N
,γ,x)+Nργ(
m
N
)·y}
=
∑
m
N
∈F∩ 1
N
Zn
am
(m
N
)∑
γ∈Γ
gγe
2π
√
−1{Θ(m
N
,γ,x)+Nργ(
m
N
)·y}
=
∑
m
N
∈F∩ 1
N
Zn
am
(m
N
)
ϑm
N
(x, y).
(Note that it is well known that the Fourier series of s pointwise converges absolutely. In partic-
ular, the order of terms of the Fourier series of s in interchangeable.) This implies s is contained
by l.h.({ϑm
N
}m
N
∈F∩ 1
N
Zn). Conversely, for any s =
∑
m′
N
∈F∩ 1
N
Zn cm′
N
ϑm′
N
in l.h.({ϑm
N
}m
N
∈F∩ 1
N
Zn),
let
s =
∑
(γ,m
N
)∈Γ×(F∩ 1
N
Zn)
bNργ(
m
N
)e
2π
√
−1Nργ(
m
N
)·y
be the Fourier series of s with respect to yi’s. Then, each bNργ(
m
N
) is described by
bNργ(
m
N
) :=
∫
Tn
s(x, y)e−2π
√
−1Nργ(
m
N
)·ydy
=
∫
Tn
∑
m′
N
∈F∩ 1
N
Zn
cm′
N
ϑm′
N
(x, y)e−2π
√
−1Nργ(
m
N
)·ydy
=
∑
m′
N
∈F∩ 1
N
Zn
cm′
N
∫
Tn
ϑm′
N
(x, y)e−2π
√
−1Nργ(
m
N
)·ydy
= cm
N
aNργ(
m
N
)(x).
For each
(
γ, mN
)
∈ Γ×
(
F ∩ 1
NZn
)
, by the definition of ϑm
N
, aNργ(
m
N
)(x) is a nontrivial solution
of (3.12) for Nργ
(
m
N
)
. Hence, so is cm
N
aNργ(
m
N
)(x). This implies s satisfies DMs = 0. ■
30 T. Yoshida
The condition in Proposition 4.1 (3) also gives a sufficient condition on the square integrability
of ϑm
N
as a section of L⊗N . A proof will be given later in more general case. See Lemma 4.13.
Proposition 4.4. If Y +XY −1X is constant, then ϑm
N
is square integrable as a section of L⊗N .
In particular, by Propositions 4.1 and 4.4, and Theorem 4.3, we obtain the following corollary.
Corollary 4.5. Let π : (M,ω) → B be a Lagrangian fibration on a complete base B and(
L,∇L
)
→ (M,ω) a prequantum line bundle. Let J be a compatible almost complex struc-
ture on (M,ω) which is invariant along the fiber in the sense of Lemma 3.6. If J is integrable
and Y +XY −1X is constant, then {ϑm
N
}m
N
∈F∩ 1
N
Zn is a complete orthogonal system of the space
of square integrable holomorphic sections of
(
L,∇L
)⊗N → (M,Nω, J) indexed by the Bohr–
Sommerfeld points.
Let us consider the special case where Γ is trivial. In this case, F = Rn,
(
L,∇L
)
→(M,ω)→B
is
(
L0,∇L0
)
→ (M0, ω0) → Rn, and ϑm
N
is nothing but sm which is defined by (3.18) by
Proposition 4.1 (2). Then, by Proposition 3.13, we have the following corollary.
Corollary 4.6. Let J be a compatible almost complex structure on (M0, ω0) which is invari-
ant along the fiber in the sense of Lemma 3.6 and D the associated Spinc Dirac operator
on (M0, Nω0) with coefficients in L0
⊗N . Assume that J is integrable and sm is in L2
(
L0
⊗N
)
for
all m ∈ Zn. Then, L2
(
L0
⊗N
)
∩ kerD is a Hilbert space and {sm}m∈Zn is a complete orthogonal
system of L2
(
L0
⊗N
)
∩ kerD. The latter assumption holds if Y +XY −1X is constant.
Example 4.7. For Example 2.30, Z = X +
√
−1Y can be chosen so that Y + XY −1X is
a constant map and XY −1 and Y −1 satisfy
(
XY −1
)
x
=
(
Y +XY −1X
)u11 · C
−1x · · · u1n · C−1x
...
...
un1 · C−1x · · · unn · C−1x
,
(
Y −1
)
x
=
u11 · C
−1x · · · u1n · C−1x
...
...
un1 · C−1x · · · unn · C−1x
(Y +XY −1X
)
×
u11 · C
−1x · · · u1n · C−1x
...
...
un1 · C−1x · · · unn · C−1x
+ Y +XY −1X.
In this case, Y +XY −1X is necessarily I and Ω can be written as
Ωx =
u11 · C
−1x · · · u1n · C−1x
...
...
un1 · C−1x · · · unn · C−1x
+
√
−1
(
Y +XY −1X
)−1
,
and the condition (2) in Proposition 3.14 is equivalent to the following condition:(
tC−1ujk
)
i
=
(
tC−1uik
)
j
for all i, j, k = 1, . . . , n.
Assume this condition as well as the condition N
2 vi · Ujvi ∈ Z for all i, j = 1, . . . , n. Then, for
each m
N ∈ F ∩ 1
NZn, ϑm
N
is described by
ϑm
N
(x, y) =
∑
γ∈Γ
gγe
2π
√
−1{Θ(m
N
,γ,x)+Nργ(
m
N
)·y},
Adiabatic Limit, Theta Function, and Geometric Quantization 31
where
Θ
(m
N
, γ, x
)
= N
n∑
i=1
∑
j>i
(
ργ−1(x)−
m
N
)
i
(
ργ−1(x)−
m
N
)
j
(
tC−1uij
)
·
m1
N
...
mi−1
N
1
2
(
ργ−1(x) + m
N
)
i(
ργ−1(x)
)
i+1
...(
ργ−1(x)
)
n
+
N
2
n∑
i=1
(
ργ−1(x)−
m
N
)2
i
(
tC−1uii
)
·
m1
N
...
mi−1
N
1
3
(
2ργ−1(x) + m
N
)
i(
ργ−1(x)
)
i+1
...(
ργ−1(x)
)
n
+
N
2
(
ργ−1(x)−
m
N
)
·
u11 · γ · · · u1n · γ
...
...
un1 · γ · · · unn · γ
+
√
−1
(
Y +XY −1X
)−1
×
(
ργ−1(x)−
m
N
)
− m
N
·
u11 · γ · · · u1n · γ
...
...
un1 · γ · · · unn · γ
m
N
.
By Proposition 4.1 (3), ϑm
N
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 of the Γ := π1(B)-action
on π0 : (M0, ω0) → Rn with prequantum line bundle
(
L0,∇L0
)
→ (M0, ω0). Suppose that the
Γ-actions are described by (2.3) and (2.5) as before. Let J be a compatible almost complex
structure on (M,ω) and Z ∈ C∞(M0,Sn) be the map corresponding to the pull-back of J
to M0. A situation in which (2) in Proposition 3.14 holds occurs when Z is a constant map. In
this subsection, we discuss this case in detail. Note that in this case, Juγ 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 following theorem.
Theorem 4.8.
(1) For each m
N ∈ F ∩ 1
NZn, ϑm
N
can be described as follows:
ϑm
N
(x, y) =
∑
γ∈Γ
gγe
2π
√
−1{Θ(m
N
,γ,x)+Nργ(
m
N
)·y},
where
Θ
(m
N
, γ, x
)
=
N
2
{(
ργ−1(x)−
m
N
)
·
(
Ω+ tAγJuγ
) (
ργ−1(x)−
m
N
)
−m
N
·
(
tAγJuγ
)m
N
}
−Nργ
(m
N
)
· uγ(0).
32 T. Yoshida
(2) For each m
N ∈ F ∩ 1
NZn, ϑm
N
converges absolutely and uniformly on any compact set.
(3) J is integrable and {ϑm
N
}m
N
∈F∩ 1
N
Zn gives a complete orthogonal system of the space of
square integrable holomorphic sections of
(
L,∇L
)⊗N → (M,Nω, J).
Proof. (1) is obtained from Proposition 4.1 (1). (2) is obtained by the assumption and Proposi-
tion 4.1 (3). The first half of (3) holds since J is covariant constant with respect to the associated
Levi-Civita connection. The other half is obtained by Lemma 4.13 later and Corollary 4.5. ■
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 2n-dimensional torus T 2n. In
particular, ϑm
N
’s should be obtained from classical theta functions. So, let us see how ϑm
N
’s
relate with classical theta functions for Example 2.28 with C = I, in which M itself is T 2n.
First, let us briefly recall classical theta functions. For each T ∈ Sn and a, b ∈ Qn, the theta
function with rational characteristics is a holomorphic section on the trivial holomorphic line
bundle Cn × C → Cn which is defined by
ϑ
[
a
b
]
(z, T ) :=
∑
γ∈Zn
eπ
√
−1(γ+a)·T (γ+a)+2π
√
−1(γ+a)·(z+b).
It is well known that ϑ
[
a
b
]
(z, T ) has the following quasi-periodicity:
ϑ
[
a
b
]
(z +m,T ) = e2π
√
−1a·mϑ
[
a
b
]
(z, T ),
ϑ
[
a
b
]
(z + Tm, T ) = e−2π
√
−1b·me−π
√
−1m·Tm−2π
√
−1m·zϑ
[
a
b
]
(z, T )
for m ∈ Zn. For more details, see [32, Chapter II, Section 1] and [33, Section 2]. Here we need
the case where T = NΩ, a = m
N , and b = 0. In this case, define the Z2n = Zn × Zn-action
on Cn × C → Cn by
(γ, γ′) · (z, w) :=
(
z +N(−Ωγ + γ′), e−π
√
−1Nγ·Ωγ+2π
√
−1γ·zw
)
for (γ, γ′) ∈ Z2n and (z, w) ∈ Cn × C. Also define the Z2n-action on the trivial complex line
bundle R2n × C → R2n by
(γ, γ′) · (x, y, w) :=
(
x+ γ, y + γ′, e2π
√
−1Nγ·yw
)
(4.2)
for (γ, γ′) ∈ Z2n and (x, y, w) ∈ R2n × C. Note that by taking the quotient of the latter
Zn-action of (4.2), we can recover Example 2.28 with C = I and gγ = 1. Let F : R2n → Cn
and F̃ : R2n × C → Cn × C be the R-linear isomorphism and the bundle isomorphism covering F
which are defined by
F (x, y) := N(−Ωx+ y), F̃ (x, y, w) :=
(
N(−Ωx+ y), e−π
√
−1Nx·Ωxw
)
.
Then, the direct computation shows the following theorem.
Theorem 4.9.
(1) J√−1I ◦ F = F ◦ (JZ), i.e., F is a C-linear isomorphism from
(
R2n, JZ
)
to the standard
complex vector space (Cn, J√−1I).
(2) F̃ is equivariant with respect to the Z2n-actions defined above.
(3) ϑm
N
satisfies F̃ ◦ ϑm
N
(x, y) = ϑ
[
m
N
0
]
(F (x, y), NΩ), i.e.,
ϑm
N
(x, y) = eπ
√
−1Nx·Ωxϑ
[
m
N
0
]
(N(−Ωx+ y), NΩ) .
Adiabatic Limit, Theta Function, and Geometric Quantization 33
4.3 Adiabatic-type limit
In this subsection let us consider a one parameter family
{(
gt, J t
)}
t>0
of the Riemannian metrics
and the almost complex structures on a Lagrangian fibration so that the fiber shrinks as t goes
to ∞, and investigate the behavior of ϑm
N
defined by (4.1) as t goes to ∞.
Let Z = X +
√
−1Y ∈ C∞(M0,Sn) be the map independent of y1, . . . , yn. Let J = JZ be
the corresponding compatible almost complex structure on (M0, ω0). For each t > 0, we define
the almost complex structure J t by
J tu := (−J∂y, ∂y)
(
0 −1
t
t 0
)(
uH
uV
)
for u = (−J∂y, ∂y) ( uH
uV ) ∈ T(x,y)M0. It is easy to see the following lemma.
Lemma 4.10.
(1) For any t > 0, J t is compatible with ω0. The map Zt ∈ C∞(M0,Sn) corresponding to J t
is described as
Zt =
(
1
t
X +
√
−1Y
)
Y −1
(
Y +XY −1X
)(
tY +
1
t
XY −1X
)−1
Y.
J t can be also written as
J t
(
(∂x, ∂y)
(
ux
uy
))
= (∂x, ∂y)
1
t
(
XY −1 −Y −XY −1X
Y −1
(
t2Y +XY −1X
)(
Y +XY −1X
)−1 −Y −1X
)(
ux
uy
)
.
(2) For any t > 0, let gt be the Riemannian metric corresponding to ω0 and J t. Then, for
u = (−J∂y, ∂y) ( uH
uV ), v = (−J∂y, ∂y) ( vHvV ) ∈ T(x,y)M0, g
t can be written by
gt(u, v) = ω0
(
u, J tv
)
= t
(
0, tuH
)( Y −1 −Y −1X
−XY −1 Y +XY −1X
)(
0
vH
)
+
1
t
(
0, tuV
)( Y −1 −Y −1X
−XY −1 Y +XY −1X
)(
0
vV
)
.
Suppose that a group Γ acts on π0 : (M0, ω0) → Rn and the Γ-actions ρ on Rn and ρ̃
on (M0, ω0) are written as in (2.3).
Lemma 4.11. The Γ-action ρ̃ preserves J t (hence, gt) for all t > 0 if and only if ρ̃ preserves J .
For J t and gt defined as above, the same arguments in Section 3.3 goes well, just by replac-
ing J , g by J t, gt. For each t > 0, let ϑtm
N
be the one defined by (4.1) for J t and gt. Let us
investigate the behavior of ϑtm
N
as t goes to infinity. For t > 0, Ωt defined by (3.6) for Zt can be
described as
Ωt =
(
Y +XY −1X
)−1(
X + t
√
−1Y
)
Y −1. (4.3)
Let Dt be the corresponding Spinc Dirac operator. Then, for a section s of L0
⊗N , Dts can be
described as
Dts = −
√
−1
N
n∑
i=1
∂yi ⊗C
{
∂xis+
n∑
j=1
(
Ωt
)
ij
(
∂yjs− 2π
√
−1Nxjs
)}
.
It is clear that
34 T. Yoshida
Lemma 4.12. For any t > 0, the condition (2) in Proposition 3.14 holds for Ωt if and only if
it holds for Ω = Ω1. In particular, J t is integrable if and only if J is integrable.
Suppose that π0 : (M0, Nω0, J) → Rn with prequantum line bundle
(
L0,∇L0
)⊗N → (M0,
Nω0, J) is equipped with an action of a group Γ which preserves all the data, and the Γ-actions
are described by (2.3) and (2.5) as before. We assume that the Γ-action ρ on Rn is properly
discontinuous and free. Let π : (M,Nω) → B and
(
L,∇L
)⊗N → (M,Nω) be the Lagrangian
fibration and the prequantum line bundle on it obtained by the quotient of the Γ-action. OnM ,
we define the Lp-norm of a section s of L⊗N by
∥s∥Lp(L⊗N ) :=
(∫
M
⟨s, s⟩
p
2
L⊗N (−1)
n(n−1)
2
(Nω)n
n!
) 1
p
,
where ⟨·, ·⟩L⊗N is the Hermitian metric of L⊗N which is induced from the Hermitian met-
ric ⟨·, ·⟩L0
⊗N of L0
⊗N . As noticed in Remark 2.26, there exists a positive constant C such
that ⟨·, ·⟩L0
⊗N can be written as ⟨·, ·⟩L0
⊗N = C⟨·, ·⟩C, where ⟨·, ·⟩C is the standard Hermitian
inner product on C.
For each t > 0 and each point m
N ∈ F ∩ 1
NZn which is integrable, the corresponding ϑtm
N
is
defined by (4.1) for Ωt. We identify F ∩ 1
NZn with BBS the set of Bohr–Sommerfeld points of
π : (M,Nω) → B with prequantum line bundle
(
L,∇L
)⊗N → (M,Nω) and identify ϑtm
N
with
the section of
(
L,∇L
)⊗N → (M,Nω) which is induced from ϑtm
N
. Then, concerning the Lp-norm,
we have the following lemma.
Lemma 4.13. Suppose that Y + XY −1X is constant. Then, as a section of
(
L,∇L
)⊗N →
(M,Nω), the Lp-norm of ϑtm
N
converges and it can be calculated as follows:
∥∥ϑtm
N
∥∥p
Lp(L⊗N )
= C
√
det
(
Y +XY −1X
)(N
pt
)n
2
.
Proof. Let o(B) be the orientation bundle of B which is defined as the quotient bundle
of the trivial real line bundle Rn × R → Rn on the universal cover of B by the Γ-action
ρ′γ(x, r) := (ργ(x), (detAγ)r) for γ ∈ Γ and (x, r) ∈ Rn × R. Then, we have a push-forward
map π∗ : Ω
k(M) → Ωk−n(B, o(B)), where Ω•(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, Section 7].
Then,
∥∥ϑtm
N
∥∥p
Lp(L⊗N )
=
∫
M
〈
ϑtm
N
, ϑtm
N
〉 p
2
L⊗N (−1)
n(n−1)
2
(Nω)n
n!
=
∫
B
π∗
(〈
ϑtm
N
, ϑtm
N
〉 p
2
L⊗N (−1)
n(n−1)
2
(Nω)n
n!
)
= CNn
∑
γ∈Γ
∫
F
e−pNπt(ργ−1 (x)−m
N
)·(Y+XY −1X)−1(ργ−1 (x)−m
N
)|dx|. (4.4)
By changing the coordinates as x′ = ργ−1(x),
(4.4) = CNn
∑
γ∈Γ
∫
ργ−1 (F )
e−pNπt(x′−m
N
)·(Y+XY −1X)−1(x′−m
N
)|dx′|
= CNn
∫
Rn
e−pNπt(x′−m
N
)·(Y+XY −1X)−1(x′−m
N
)|dx′|. (4.5)
Adiabatic Limit, Theta Function, and Geometric Quantization 35
Since Y +XY −1X is positive definite, symmetric, there exists P ∈ O(n) such that
Y +XY −1X = tP
λ1 . . .
λn
P.
Then, we define a positive definite symmetric matrix
√
Y +XY −1X by
√
Y +XY −1X := tP
√
λ1
. . . √
λn
P,
and put τ :=
√(
Y +XY −1X
)−1(
x′ − m
N
)
. Then,
(4.5) = C
√
det
(
Y +XY −1X
)
Nn
∫
Rn
e−pNπt∥τ∥2 |dτ |
= C
√
det
(
Y +XY −1X
)
Nn
n∏
i=1
∫ ∞
−∞
e−pNπtτ2i dτi
= C
√
det
(
Y +XY −1X
)
Nn
(√
1
pNt
)n
. ■
We define the section δm
N
of
(
L,∇L
)⊗N |π−1(m
N
) by
δm
N
(y) :=
1
C
e2π
√
−1m·y. (4.6)
By Proposition 3.2, δm
N
is a covariant constant section of
(
L,∇L
)⊗N |π−1(m
N
). Let T ∗
πM be the
cotangent bundle along the fiber of π. On (∧nT ∗
πM) ⊗ π∗o(B)∗, there exists a natural section,
i.e., a density along the fiber of π, say |dy|, which satisfies
∫
π−1(x)|dy| = 1 on each fiber of π.
Then, we obtain the following theorem.
Theorem 4.14. Suppose that Y +XY −1X is constant. Then, the section
ϑtm
N∥∥ϑtm
N
∥∥
L1(L⊗N )
converges to a delta-function section supported on the fiber π−1(mN ) as t goes to ∞ in the following
sense: for any L2-section s of L⊗N ,
lim
t→∞
(
s,
ϑtm
N∥∥ϑtm
N
∥∥
L1(L⊗N )
)
L2(L⊗N )
=
∫
π−1(m
N
)
⟨s, δm
N
⟩L⊗N |dy|.
Proof. We denote by s̃ the pull-back of s to L0
⊗N →M0. Since s̃ is Γ-equivariant, the Fourier
series of s̃ can be written as in (3.20). Then, by using Proposition 4.1 (1),(
s,
ϑtm
N∥∥ϑtm
N
∥∥
L1(L⊗N )
)
L2(L⊗N )
=
∫
M
〈
s,
ϑtm
N∥∥ϑtm
N
∥∥
L1(L⊗N )
〉
L⊗N
(−1)
n(n−1)
2
(Nω)n
n!
36 T. Yoshida
=
∫
B
π∗
〈s, ϑtm
N∥∥ϑtm
N
∥∥
L1(L⊗N )
〉
L⊗N
(−1)
n(n−1)
2
(Nω)n
n!
=
CNn∥∥ϑtm
N
∥∥
L1(L⊗N )
×
∑
γ∈Γ
∫
F
am(ργ−1(x))e−2π
√
−1
∑n
i=1 G
i
m(
m1
N
,...,
mi−1
N
,(ργ−1 (x))i,...,(ργ−1 (x))n)|dx|. (4.7)
Here, we remark that we can interchange the operations to take infinite sums and integrals by
Lemma A.2. By putting x′ = ργ−1(x), we have
(4.7) =
CNn∥∥ϑtm
N
∥∥
L1(L⊗N )
∑
γ∈Γ
∫
ργ−1 (F )
am(x′)e−2π
√
−1
∑n
i=1 G
i
m(
m1
N
,...,
mi−1
N
,x′
i,...,x
′
n)|dx′|
=
CNn∥∥ϑtm
N
∥∥
L1(L⊗N )
∫
Rn
am(x′)e−2π
√
−1
∑n
i=1 G
i
m(
m1
N
,...,
mi−1
N
,x′
i,...,x
′
n)|dx′|
=
CNn∥∥ϑtm
N
∥∥
L1(L⊗N )
∫
Rn
am(x′)e2π
√
−1
∑n
i=1 ReGi
m(
m1
N
,...,
mi−1
N
,x′
i,...,x
′
n)
× e−πNt(x′−m
N
)·(Y+XY −1X)−1(x′−m
N
)|dx′|. (4.8)
We put
f(x′) := am(x′)e2π
√
−1
∑n
i=1 ReGi
m(
m1
N
,...,
mi−1
N
,x′
i,...,x
′
n)
and τ :=
√(
Y +XY −1X
)−1 (
x′ − m
N
)
. By using Lemma 4.13 for p = 1, (4.8) can be written as
follows:
(4.8) =
CNn∥∥ϑtm
N
∥∥
L1(L⊗N )
∫
Rn
f(x′)e−πNt(x′−m
N
)·(Y+XY −1X)−1(x′−m
N
)|dx′|
=
CNn∥∥ϑtm
N
∥∥
L1(L⊗N )
√
det
(
Y +XY −1X
) ∫
Rn
f
(√
Y +XY −1Xτ +
m
N
)
e−πNt∥τ∥2 |dτ |
= (Nt)
n
2
∫
Rn
f
(√
Y +XY −1Xτ +
m
N
)
e−πNt∥τ∥2 |dτ |. (4.9)
It is well known that limt→∞ (4.9) = f
(
m
N
)
= am
(
m
N
)
. On the other hand, by using the
expression
s̃ =
∑
(γ,m
′
N
)∈Γ×(F∩ 1
N
Zn)
a
Nργ(
m′
N
)
(x)e2π
√
−1Nργ(
m′
N
)·y,
the right-hand side can be computed as∫
π−1(m
N
)
⟨s, δm
N
⟩L⊗N |dy| =
∫
Tn
〈
s̃, δm
N
〉
L0
⊗N |dy|
=
∑
(γ,m
′
N
)∈Γ×(F∩ 1
N
Zn)
a
Nργ(
m′
N
)
(m
N
)∫
Tn
e2π
√
−1(Nργ(
m′
N
)−m)·y|dy|.
The integral
∫
Tn e
2π
√
−1(Nργ(
m′
N
)−m)·y|dy| vanishes unless ργ(m
′
N ) = m
N . Since both m′
N and m
N lie
in the fundamental domain F , this implies γ = e and m′ = m, and in this case,∫
Tn
e2π
√
−1(Nργ(
m′
N
)−m)·y|dy| = 1.
Thus,
∫
π−1(m
N
)⟨s, δm
N
⟩L⊗N |dy| = am
(
m
N
)
. This proves the theorem. ■
Adiabatic Limit, Theta Function, and Geometric Quantization 37
5 The non-integrable case
In this section, let us consider the case where the almost complex structure is not integrable. We
still use the same notations introduced in Section 3. By Lemma 3.11, the equation (3.12) has
no smooth solution for m
N ∈ F ∩ 1
NZn if and only if m is not integrable. For such m
N , instead of
(3.12), let us consider the following equation which is obtained by replacing Ω with its value Ωm
N
at m
N in (3.12)
0 =
∂x1 ãm
...
∂xn ãm
+ 2π
√
−1ãmΩm
N
(m−Nx). (5.1)
The equation (5.1) has a solution of the form
ãm(x) = ãm
(m
N
)
e
π
√
−1N(x−m
N
)·Ωm
N
(x−m
N
)
.
We put the initial condition ãm(mN ) = 1 on the above ãm, and for each γ ∈ Γ, define ãNργ(
m
N
) in
such a way that it satisfies (3.19).
Lemma 5.1. ãNργ(
m
N
) satisfies the following equality:
0 =
∂x1 ãNργ(
m
N
)(x)
...
∂xn ãNργ(
m
N
)(x)
+ 2π
√
−1ãNργ(
m
N
)(x)Ωx
(
Nργ
(m
N
)
−Nx
)
+ 2π
√
−1ãNργ(
m
N
)(x)
tA−1
γ (Ωm
N
− Ωργ−1 (x))A
−1
γ
(
Nργ
(m
N
)
−Nx
)
. (5.2)
Proof. By the same calculation as in the proof of Lemma 3.17, we have
tAγ
∂x1 ãNργ(
m
N
)(ργ(x))
...
∂xn ãNργ(
m
N
)(ργ(x))
= −2π
√
−1ãNργ(
m
N
)(ργ(x))
(
Ωm
N
A−1
γ + t(Juγ)x
) (
Nργ
(m
N
)
−Nργ(x)
)
.
(5.2) can be obtained from this equation and (3.24). ■
By using ãNργ(
m
N
)’s, we define ϑ̃m
N
in the same manner as ϑm
N
, i.e.,
ϑ̃m
N
(x, y) =
∑
γ∈Γ
ãNργ(
m
N
)(x)e
2π
√
−1Nργ(
m
N
)·y.
ϑ̃m
N
converges absolutely and uniformly on any compact set and can be written as ϑ̃m
N
=∑
γ∈Γ ˜̃ργ ◦ s′m ◦ ρ̃γ−1 , where s′m is the section defined by
s′m(x, y) := e
π
√
−1N(x−m
N
)·Ωm
N
(x−m
N
)+2π
√
−1m·y
.
In particular, ϑ̃m
N
defines an Lp-section of L⊗N → M . Moreover,
{
ϑ̃m
N
}
m
N
∈F∩ 1
N
Zn is an or-
thogonal system of the space of L2-sections of L⊗N . These can be proved by the same way as
Proposition 4.1 and Lemma 4.13.
38 T. Yoshida
Next let us consider the one parameter family of J t and gt defined in Section 4.3. Then,
corresponding to J t and gt, we can obtain ϑ̃tm
N
, which can be explicitly described as
ϑ̃tm
N
(x, y) =
∑
γ∈Γ
gγe
2π
√
−1{Θ(m
N
,γ,x)+Nργ(
m
N
)·y},
where
Θ
(m
N
, γ, x
)
=
N
2
(
ργ−1(x)−
m
N
)
· Ωt
m
N
(
ργ−1(x)−
m
N
)
+N
{
g̃γ(ργ−1(x))− ργ
(m
N
)
· uγ(ργ−1(x))
}
and Ωt
m
N
is the value of Ωt given in (4.3) at m
N . Then, ϑ̃tm
N
has the following property. The proof
is same as Theorem 4.14.
Theorem 5.2. For each m
N ∈ F ∩ 1
NZn, the section
ϑ̃tm
N
∥ϑ̃tm
N
∥L1(L⊗N )
converges to a delta-function section supported on the fiber π−1(mN ) as t goes to ∞ in the following
sense: for any L2-section s of L⊗N ,
lim
t→∞
s, ϑ̃tm
N
∥ϑ̃tm
N
∥L1(L⊗N )
L2(L⊗N )
=
∫
π−1(m
N
)
⟨s, δm
N
⟩L⊗N |dy|.
Finally, let us investigate the behavior of Dtϑ̃tm
N
as t goes to ∞. Dtϑ̃tm
N
is a section of(
TM, J t
)
⊗C L
⊗N , and
(
TM, J t
)
⊗CL
⊗N admits a Hermitian metric ⟨·, ·⟩(TM,Jt)⊗CL⊗N induced
by the one parameter version of (3.9) of
(
TM, J t
)
and the Hermitian metric of L. In terms of
this Hermitian metric, the L2-norm is defined as
∥∥Dtϑ̃tm
N
∥∥2
L2((TM,Jt)⊗CL⊗N )
:=
∫
M
〈
Dtϑ̃tm
N
, Dtϑ̃tm
N
〉
(TM,Jt)⊗CL⊗N (−1)
n(n−1)
2
(Nω)n
n!
.
In general, ϑ̃tm
N
is no longer a solution of 0 = Dts, but we can show that ϑ̃tm
N
approximates the
solution of this equation in the following sense:
Theorem 5.3.
lim
t→∞
∥∥Dtϑ̃tm
N
∥∥
L2((TM,Jt)⊗CL⊗N )
= 0.
Proof. For n = 1, it is clear that all m ∈ Z are integrable. Thus, it is sufficient to prove the
theorem for n ≥ 2. By the definition of ϑ̃tm
N
and (5.2), Dtϑ̃tm
N
can be written as
Dtϑ̃tm
N
= −
√
−1
N
n∑
i=1
∂yi ⊗C
∂xi ϑ̃
t
m
N
+
n∑
j=1
(Ωt
x)ij
(
∂yj ϑ̃
t
m
N
− 2π
√
−1Nxjϑ̃
t
m
N
)
= −
√
−1
N
n∑
i=1
∂yi ⊗C
∑
γ∈Γ
{
∂xi ãNργ(
m
N
)(x)
+2π
√
−1ãNργ(
m
N
)(x)
(
Ωt
x
(
Nργ
(m
N
)
−Nx
))
i
}
e2π
√
−1Nργ(
m
N
)·y
Adiabatic Limit, Theta Function, and Geometric Quantization 39
= − 2π
n∑
i=1
∂yi ⊗C
∑
γ∈Γ
ãNργ(
m
N
)(x)
(
B
(m
N
, γ, x, t
))
i
e2π
√
−1Nργ(
m
N
)·y,
where
B
(m
N
, γ, x, t
)
= tA−1
γ
(
Ωt
m
N
− Ωt
ργ−1 (x)
) (m
N
− ργ−1(x)
)
.
Then,〈
Dtϑ̃tm
N
, Dtϑ̃tm
N
〉
(TM,Jt)⊗CL
= (2π)2
∑
γ1,γ2∈Γ
∑
i1,i2
〈
ãNργ1 (
m
N
)(x)e
2π
√
−1Nργ1 (
m
N
)·y, ãNργ2 (
m
N
)(x)e
2π
√
−1Nργ2 (
m
N
)·y〉
L⊗N
×
(
B
(m
N
, γ1, x, t
))
i1
(
B
(m
N
, γ2, x, t
))
i2
Ngt(∂yi1 , ∂yi2 )
= (2π)2
N
t
∑
γ1,γ2∈Γ
〈
ãNργ1 (
m
N
)(x)e
2π
√
−1Nργ1 (
m
N
)·y, ãNργ2 (
m
N
)(x)e
2π
√
−1Nργ2 (
m
N
)·y〉
L⊗N
×B
(m
N
, γ1, x, t
)
·
(
Y +XY −1X
)
x
B
(m
N
, γ2, x, t
)
.
For each x ∈ F and u ∈ Cn, define the norm of u with respect to
(
Y +XY −1X
)
x
by
∥u∥2(Y+XY −1X)x
:= u ·
(
Y +XY −1X
)
x
u.
By (3.5), for each γ ∈ Γ, ∥u∥2(Y+XY −1X)x
satisfies
∥∥tAγu
∥∥2
(Y+XY −1X)x
= ∥u∥2(Y+XY −1X)ργ (x)
.
By using this norm, we obtain∥∥Dtϑ̃tm
N
∥∥2
L2((TM,Jt)⊗CL⊗N )
= (2π)2
CNn+1
t
∑
γ∈Γ
∫
F
e
−2πNt(ργ−1 (x)−m
N
)·(Y+XY −1X)−1
m
N
(ργ−1 (x)−m
N
)
×B
(m
N
, γ, x, t
)
·
(
Y +XY −1X
)
x
B
(m
N
, γ, x, t
)
|dx|
= (2π)2
CNn+1
t
∑
γ∈Γ
∫
F
e
−2πNt(ργ−1 (x)−m
N
)·(Y+XY −1X)−1
m
N
(ργ−1 (x)−m
N
)
×
∥∥∥B (m
N
, γ, x, t
)∥∥∥2
(Y+XY −1X)x
|dx|
= (2π)2
CNn+1
t
∑
γ∈Γ
∫
F
e
−2πNt(ργ−1 (x)−m
N
)·(Y+XY −1X)−1
m
N
(ργ−1 (x)−m
N
)
×
∥∥∥(Ωt
m
N
− Ωt
ργ−1 (x)
) (m
N
− ργ−1(x)
)∥∥∥2
(Y+XY −1X)ρ
γ−1 (x)
|dx|
= (2π)2
CNn+1
t
∑
γ∈Γ
∫
ργ−1 (F )
e
−2πNt(x′−m
N
)·(Y+XY −1X)−1
m
N
(x′−m
N
)
×
∥∥∥(Ωt
m
N
− Ωt
x′
) (m
N
− x′
)∥∥∥2
(Y+XY −1X)x′
|dx′|
(
∵ x′ := ργ−1(x)
)
40 T. Yoshida
= (2π)2
CNn+1
t
∫
Rn
e
−2πNt(x′−m
N
)·(Y+XY −1X)−1
m
N
(x′−m
N
)
×
∥∥∥(Ωt
m
N
− Ωt
x′
) (m
N
− x′
)∥∥∥2
(Y+XY −1X)x′
|dx′|.
Since Ωt can be described as (4.3),∥∥∥(Ωt
m
N
− Ωt
x′
) (m
N
− x′
)∥∥∥2
(Y+XY −1X)x′
=
∥∥∥(Re(Ωm
N
− Ωx′
)) (m
N
− x′
)∥∥∥2
(Y+XY −1X)x′
+ t2
∥∥∥(Im(Ωm
N
− Ωx′
)) (m
N
− x′
)∥∥∥2
(Y+XY −1X)x′
.
We put
R(x′) :=
∥∥∥(Re(Ωm
N
− Ωx′
)) (m
N
− x′
)∥∥∥2
(Y+XY −1X)x′
,
I(x′) :=
∥∥∥(Im(Ωm
N
− Ωx′
)) (m
N
− x′
)∥∥∥2
(Y+XY −1X)x′
.
By changing coordinates as
τ :=
√(
Y +XY −1X
)−1
m
N
(
x′ − m
N
)
,∥∥Dtϑ̃tm
N
∥∥2
L2((TM,Jt)⊗CL)
can be written by∥∥Dtϑ̃tm
N
∥∥2
L2((TM,Jt)⊗CL)
= 22−
n
2 π2CN
n
2
+1
√
det
(
Y +XY −1X
)
m
N
×
{
t−1−n
2
∫
Rn
R
(√(
Y +XY −1X
)
m
N
τ +
m
N
)
(2Nt)
n
2 e−2πNt∥τ∥2 |dτ |
+t1−
n
2
∫
Rn
I
(√(
Y +XY −1X
)
m
N
τ +
m
N
)
(2Nt)
n
2 e−2πNt∥τ∥2 |dτ |
}
.
It is well known that
lim
t→∞
∫
Rn
R
(√(
Y +XY −1X
)
m
N
τ +
m
N
)
(2Nt)
n
2 e−2πNt∥τ∥2 |dτ | = R
(m
N
)
= 0,
lim
t→∞
∫
Rn
I
(√(
Y +XY −1X
)
m
N
τ +
m
N
)
(2Nt)
n
2 e−2πNt∥τ∥2 |dτ | = I
(m
N
)
= 0.
Since n ≥ 2, this proves Theorem 5.2. ■
Example 5.4. For Example 2.29, let us consider the compatible almost complex structure
associated with
Z :=
(
0 0
0 x1
)
+
√
−1
(
1
x2
1+1
0
0 1
)
.
The corresponding Ω is
Ωx =
(√
−1 0
0 x1 +
√
−1
)
.
Adiabatic Limit, Theta Function, and Geometric Quantization 41
This Z does not satisfies (2) in Proposition 3.14, nor the condition (3.15) for any m ∈ Z2. In
fact, for any m ∈ Z2, ((∂x1Ω)(m − Nx))2 = m2 − Nx2 while ((∂x2Ω)(m − Nx))1 = 0. In this
case, ϑ̃tm
N
can be written as
ϑ̃tm
N
(x, y) =
∑
γ∈Z2
gγ exp
{
2π
√
−1N
[1
2
{
t
√
−1
(
x1 − γ1 −
m1
N
)2
+
(m1
N
+ t
√
−1
)(
x2 − γ2 −
m2
N
)2}
+ (x2 − γ2)
{1
2
γ1(x2 + γ2)−
(m2
N
+ γ2
)
γ2
}]}
e2π
√
−1(m+Nγ)·y.
Example 5.5. In the case, where n = 2 of Example 2.31, we can take the compatible almost
complex structure associated with
Z :=
1
x22 + 1
(
λ2x32 λx22
λx22 x2
)
+
√
−1
x22 + 1
((
1 + λ2
)
x22 + 1 λx2
λx2 1
)
.
The corresponding Ω is
Ωx =
( √
−1 −
√
−1λx2
−
√
−1λx2 x2 +
√
−1
(
λ2x22 + 1
)) .
In this case, ∂x2Ω12 = −
√
−1λ and ∂x1Ω22 = 0. So, Z satisfies (2) in Proposition 3.14 if and
only if λ = 0, which is the special case of Example 4.7. Equivalently, Z does not satisfy the
condition (3.15) for any m ∈ Z2 unless λ = 0. In fact, for any m ∈ Z2, ((∂x1Ω)(m−Nx))2 = 0
while ((∂x2Ω)(m−Nx))1 = −
√
−1λ(m2 −Nx2). In this case, ϑ̃tm
N
can be written as
ϑ̃tm
N
(x, y) =
∑
γ∈Γ
gγe
2π
√
−1Θ(m
N
,γ,x)e2π
√
−1{(m1+γ2λm2+Nγ1)y1+(m2+Nγ2)y2},
where
Θ
(m
N
, γ, x
)
= N
[
t
√
−1
2
{
x1 − γ1 − γ2λ(x2 − γ2)−
m1
N
}2
−t
√
−1λ
m2
N
{
x1 − γ1 − γ2λ(x2 − γ2)−
m1
N
}(
x2 − γ2 −
m2
N
)
+
1
2
{
m2
N
+ t
√
−1
(
λ2
m2
2
N2
+ 1
)}(
x2 − γ2 −
m2
N
)2
+
1
2
γ2(x2 − γ2)(x2 + γ2)−
(m2
N
+ γ2
)
γ2(x2 − γ2)
]
.
A Fourier series
Let π : (M,ω) → B be a Lagrangian fibration on a complete n-dimensional B with prequan-
tum line bundle
(
L,∇L
)
→ (M,ω). Then, it is obtained as the quotient of the π1(B)-action
on π0 : (M0, ω0) → Rn with prequantum line bundle
(
L0,∇L0
)
→ (M0, ω0). We take and fix
a fundamental domain F of the π1(B)-action on Rn as before. Let N ∈ N be a positive integer
and s a smooth section of L⊗N . We identify s with a π1(B)-equivariant section of L0
⊗N . Then,
for each x ∈ Rn, s(x, ·) can be expressed as the Fourier series
s(x, y) =
∑
m∈Zn
am(x)e2π
√
−1m·y (A.1)
42 T. Yoshida
in L2
(
L0
⊗N |{x}×Tn
)
, where
am(x) :=
∫
Tn
s(x, y)e−2π
√
−1m·ydy
for m ∈ Zn. Then, we have the following lemma.
Lemma A.1. For j = 1, . . . , n, the partial derivatives ∂xjs and ∂yjs have the following Fourier
series:
∂xjs(x, y) =
∑
m∈Zn
∂xjam(x)e2π
√
−1m·y,
∂yjs(x, y) =
∑
m∈Zn
2π
√
−1mjam(x)e2π
√
−1m·y
in L2
(
L0
⊗N |{x}×Tn
)
.
Proof. Suppose that ∂xjs has the following Fourier series with respect to yi’s:
∂xjs(x, y) =
∑
m∈Zn
bm(x)e2π
√
−1m·y.
Then, bm(x) is computed by
bm(x) :=
∫
Tn
∂xjs(x, y)e
−2π
√
−1m·ydy = ∂xj
∫
Tn
s(x, y)e−2π
√
−1m·ydy = ∂xjam(x).
This proves the first equality. Suppose that ∂yjs has the following Fourier series with respect
to yi’s:
∂yjs(x, y) =
∑
m∈Zn
cm(x)e2π
√
−1m·y.
Then, cm(x) is computed by
cm(x) :=
∫
Tn
∂yjs(x, y)e
−2π
√
−1m·ydy
=
∫
Tn−1
(∫
S1
∂yjs(x, y)e
−2π
√
−1m·ydyj
)
dy1 · · · ˆdyj · · · dyn
=
∫
Tn−1
([
s(x, y)e−2π
√
−1m·y]yj=1
yj=0
−
∫
S1
s(x, y)∂yje
−2π
√
−1m·ydyj
)
dy1 · · · ˆdyj · · · dyn
=
∫
Tn−1
(
2π
√
−1mj
∫
S1
s(x, y)e−2π
√
−1m·ydyj
)
dy1 · · · ˆdyj · · · dyn
= 2π
√
−1mj
∫
Tn
s(x, y)e−2π
√
−1m·ydy
= 2π
√
−1mjam(x).
This proves the second equality. ■
Lemma A.2. If s is in L2
(
L⊗N
)
, then the following formulae hold:
lim
k→∞
∥∥∥∥s− ∑
|m|≤k
am(x)e2π
√
−1m·y
∥∥∥∥
L2(L0
⊗N |F×Tn )
= 0,
Adiabatic Limit, Theta Function, and Geometric Quantization 43
∥s∥2L2(L⊗N ) =
∑
m∈Zn
∥am(x)∥2L2(F ), (A.2)
where |m| := m1 + · · ·+mn for m = (m1, . . . ,mn) ∈ Zn. Namely, the right-hand side of (A.1)
also converges to s in L2
(
L0
⊗N |F×Tn
)
.
Proof. For each x ∈ Rn, we define sx ∈ L2
(
L0
⊗N |{x}×Tn
)
by sx(y) := s(x, y). Then, (A.1) im-
plies
lim
k→∞
∥∥∥∥sx − ∑
|m|≤k
am(x)e2π
√
−1m·y
∥∥∥∥
L2(L0
⊗N |{x}×Tn )
= 0, (A.3)
and sx satisfies
∥sx∥2L2(L0
⊗N |{x}×Tn )
=
∑
m∈Zn
|am(x)|2. (A.4)
By using (A.4) and the monotone convergence theorem, (A.2) can be obtained as follows:
∥s∥2L2(L⊗N ) =
∫
M
⟨s, s⟩L⊗N (−1)
n(n−1)
2
(Nω)n
n!
=
∫
B
π∗
(
⟨s, s⟩L⊗N (−1)
n(n−1)
2
(Nω)n
n!
)
=
∫
F
∥sx∥2L2(L0
⊗N |{x}×Tn )
|dx| =
∫
F
∑
m∈Zn
|am(x)|2|dx|
=
∑
m∈Zn
∫
F
|am(x)|2|dx| =
∑
m∈Zn
∥am(x)∥2L2(F ).
Next, let us prove that∑
m∈Zn
am(x)e2π
√
−1m·y
converges with respect to the norm of L2
(
L0
⊗N |F×Tn
)
. For each k ∈ N, we put
sk(x, y) :=
∑
|m|≤k
am(x)e2π
√
−1m·y.
To prove it, it is sufficient to show {sk}k∈N is a Cauchy sequence in L2
(
L0
⊗N |F×Tn
)
. For k < l
in N,
∥sl − sk∥2L2(L0
⊗N |F×Tn )
=
∥∥∥∥ ∑
|m|≤l
am(x)e2π
√
−1m·y −
∑
|m|≤k
am(x)e2π
√
−1m·y
∥∥∥∥2
L2(L0
⊗N |F×Tn )
=
∥∥∥∥ ∑
k<|m|≤l
am(x)e2π
√
−1m·y
∥∥∥∥2
L2(L0
⊗N |F×Tn )
=
∫
F
∑
k<|m|≤l
|am(x)|2|dx|
=
∣∣∣∣ ∑
|m|≤k
∥am(x)∥2L2(F ) −
∑
|m|≤l
∥am(x)∥2L2(F )
∣∣∣∣. (A.5)
Since s is square integrable, as we showed above,
∑
m∈Zn∥am(x)∥2L2(F ) converges to ∥s∥L2(L⊗N ).
In particular, the sequence
{∑
|m|≤k∥am(x)∥2L2(F )
}
k∈N is a Cauchy sequence. Thus, by (A.5)
lim
k→∞,l→∞
∥sl − sk∥L2(L0
⊗N |F×Tn ) = 0.
Let s̃ ∈ L2
(
L0
⊗N |F×Tn
)
be the limit of {sk}k∈N. Then, {sk}k∈N pointwise converges to s̃. But,
by (A.3), {sk}k∈N also pointwise converges to s. This implies s̃ = s. ■
44 T. Yoshida
Remark A.3. By the continuity of the inner product of the Hilbert space, Lemma A.2 enable
us to interchange operations to take limits and integrals for L2-sections on L⊗N .
B Proof of Proposition 3.14
If all m ∈ Zn are integrable, then by putting m = 0, we have ((∂xiΩ)xx)j = ((∂xjΩ)xx)i.
By substituting this to (3.15), we can see the condition ((∂xiΩ)xm)j = ((∂xjΩ)xm)i holds for
all m ∈ Zn. In particular, by substituting m = ek 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:((
Y +XY −1X
)−1
∂xi
(
XY −1
))
jk
=
((
Y +XY −1X
)−1
∂xj
(
XY −1
))
ik
(B.1)
∂xi
(
Y +XY −1X
)−1
jk
= ∂xj
(
Y +XY −1X
)−1
ik
(B.2)
for i, j, k = 1, . . . , n. For i = 1, . . . , 2n, we set
Γi :=
Γ1
i1 · · · Γ1
i2n
...
...
Γ2n
i1 · · · Γ2n
i2n
,
where Γk
ij is the Christoffel symbol. Then, (3) is equivalent to
0 = ∂iJ + ΓiJ − JΓi, i = 1, . . . , 2n,
where
∂i =
{
∂xi , i = 1, . . . , n,
∂yi−n , i = n+ 1, . . . , 2n.
It is also equivalent to the following conditions:
XY −1
∂x1
(
XY −1
)
1i
· · · ∂xn
(
XY −1
)
1i
...
...
∂x1
(
XY −1
)
ni
· · · ∂xn
(
XY −1
)
ni
−
(
Y +XY −1X
)∂x1
(
Y −1
)
1i
− ∂x1
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
1i
− ∂x1
(
Y −1
)
ni
...
...
∂x1
(
Y −1
)
ni
− ∂xn
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
ni
− ∂xn
(
Y −1
)
ni
=
∂x1
(
XY −1
)
1i
· · · ∂xn
(
XY −1
)
1i
...
...
∂x1
(
XY −1
)
ni
· · · ∂xn
(
XY −1
)
ni
XY −1, (B.3)
Y −1
∂x1
(
XY −1
)
1i
· · · ∂xn
(
XY −1
)
1i
...
...
∂x1
(
XY −1
)
ni
· · · ∂xn
(
XY −1
)
ni
− Y −1X
∂x1
(
Y −1
)
1i
− ∂x1
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
1i
− ∂x1
(
Y −1
)
ni
...
...
∂x1
(
Y −1
)
ni
− ∂xn
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
ni
− ∂xn
(
Y −1
)
ni
Adiabatic Limit, Theta Function, and Geometric Quantization 45
=
∂x1
(
Y −1
)
1i
− ∂x1
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
1i
− ∂x1
(
Y −1
)
ni
...
...
∂x1
(
Y −1
)
ni
− ∂xn
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
ni
− ∂xn
(
Y −1
)
ni
XY −1
+
∂x1
(
Y −1X
)
i1
· · · ∂x1
(
Y −1X
)
in
...
...
∂xn
(
Y −1X
)
i1
· · · ∂xn
(
Y −1X
)
in
Y −1, (B.4)
(
Y +XY −1X
)∂x1
(
Y −1X
)
i1
· · · ∂x1
(
Y −1X
)
in
...
...
∂xn
(
Y −1X
)
i1
· · · ∂xn
(
Y −1X
)
in
=
∂x1
(
XY −1
)
1i
· · · ∂xn
(
XY −1
)
1i
...
...
∂x1
(
XY −1
)
ni
· · · ∂xn
(
XY −1
)
ni
(Y +XY −1X
)
, (B.5)
Y −1X
∂x1
(
Y −1X
)
i1
· · · ∂x1
(
Y −1X
)
in
...
...
∂xn
(
Y −1X
)
i1
· · · ∂xn
(
Y −1X
)
in
=
∂x1
(
Y −1
)
1i
− ∂x1
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
1i
− ∂x1
(
Y −1
)
ni
...
...
∂x1
(
Y −1
)
ni
− ∂xn
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
ni
− ∂xn
(
Y −1
)
ni
(Y +XY −1)
+
∂x1
(
Y −1X
)
i1
· · · ∂x1
(
Y −1X
)
in
...
...
∂xn
(
Y −1X
)
i1
· · · ∂xn
(
Y −1X
)
in
Y −1X, (B.6)
XY −1
∂x1
(
Y +XY −1X
)
1i
· · · ∂xn
(
Y +XY −1X
)
1i
...
...
∂x1
(
Y +XY −1X
)
ni
· · · ∂xn
(
Y +XY −1X
)
ni
+
(
Y +XY −1X
)
×
−∂x1
(
Y −1X
)
1i
+ ∂x1
(
XY −1
)
i1
· · · − ∂xn
(
Y −1X
)
1i
+ ∂x1
(
XY −1
)
in
...
...
−∂x1
(
Y −1X
)
ni
+ ∂xn
(
XY −1
)
i1
· · · − ∂xn
(
Y −1X
)
ni
+ ∂xn
(
XY −1
)
in
=
∂x1
(
Y +XY −1X
)
1i
· · · ∂xn
(
Y +XY −1X
)
1i
...
...
∂x1
(
Y +XY −1X
)
ni
· · · ∂xn
(
Y +XY −1X
)
ni
XY −1, (B.7)
Y −1
∂x1
(
Y +XY −1X
)
1i
· · · ∂xn
(
Y +XY −1X
)
1i
...
...
∂x1
(
Y +XY −1X
)
ni
· · · ∂xn
(
Y +XY −1X
)
ni
+ Y −1X
−∂x1
(
Y −1X
)
1i
+ ∂x1
(
XY −1
)
i1
· · · −∂xn
(
Y −1X
)
1i
+ ∂x1
(
XY −1
)
in
...
...
−∂x1
(
Y −1X
)
ni
+ ∂xn
(
XY −1
)
i1
· · · −∂xn
(
Y −1X
)
ni
+ ∂xn
(
XY −1
)
in
= −
−∂x1
(
Y −1X
)
1i
+ ∂x1
(
XY −1
)
i1
· · · −∂xn
(
Y −1X
)
1i
+ ∂x1
(
XY −1
)
in
...
...
−∂x1
(
Y −1X
)
ni
+ ∂xn
(
XY −1
)
i1
· · · −∂xn
(
Y −1X
)
ni
+ ∂xn
(
XY −1
)
in
XY −1
46 T. Yoshida
+
∂x1
(
Y +XY −1X
)
i1
· · · ∂x1
(
Y +XY −1X
)
in
...
...
∂xn
(
Y +XY −1X
)
i1
· · · ∂xn
(
Y +XY −1X
)
in
Y −1, (B.8)
(
Y +XY −1X
)∂x1
(
Y +XY −1X
)
i1
· · · ∂x1
(
Y +XY −1X
)
in
...
...
∂xn
(
Y +XY −1X
)
i1
· · · ∂xn
(
Y +XY −1X
)
in
=
∂x1
(
Y +XY −1X
)
1i
· · · ∂xn
(
Y +XY −1X
)
1i
...
...
∂x1
(
Y +XY −1X
)
ni
· · · ∂xn
(
Y +XY −1X
)
ni
(Y +XY −1X
)
, (B.9)
Y −1X
∂x1
(
Y +XY −1X
)
i1
· · · ∂x1
(
Y +XY −1X
)
in
...
...
∂xn
(
Y +XY −1X
)
i1
· · · ∂xn
(
Y +XY −1X
)
in
= −
−∂x1
(
Y −1X
)
1i
+ ∂x1
(
XY −1
)
i1
· · · − ∂xn
(
Y −1X
)
1i
+ ∂x1
(
XY −1
)
in
...
...
−∂x1
(
Y −1X
)
ni
+ ∂xn
(
XY −1
)
i1
· · · − ∂xn
(
Y −1X
)
ni
+ ∂xn
(
XY −1
)
in
×
(
Y +XY −1X
)
+
∂x1
(
Y +XY −1X
)
i1
· · · ∂x1
(
Y +XY −1X
)
in
...
...
∂xn
(
Y +XY −1X
)
i1
· · · ∂xn
(
Y +XY −1X
)
in
Y −1X. (B.10)
for i = 1, . . . , n. It is easy to see that (B.6) and (B.10) are obtained by transposing (B.3)
and (B.7), respectively. First, we show that (B.1) is equivalent to (B.5). In fact, (B.1) implies∂x1
(
XY −1
)
1k
· · · ∂x1
(
XY −1
)
nk
...
...
∂xn
(
XY −1
)
1k
· · · ∂xn
(
XY −1
)
nk
(Y +XY −1X
)−1
is symmetric for k = 1, . . . , n. Since X, Y is symmetric, this implies (B.5). Next, we show (B.9)
is equivalent to (B.2). (B.9) is equivalent to∂x1
(
Y +XY −1X
)
i1
· · · ∂x1
(
Y +XY −1X
)
in
...
...
∂xn
(
Y +XY −1X
)
i1
· · · ∂xn
(
Y +XY −1X
)
in
(Y +XY −1X
)−1
=
(
Y +XY −1X
)−1
∂x1
(
Y +XY −1X
)
1i
· · · ∂xn
(
Y +XY −1X
)
1i
...
...
∂x1
(
Y +XY −1X
)
ni
· · · ∂xn
(
Y +XY −1X
)
ni
. (B.11)
By computing the (j, k)-components of the both sides of (B.11), we obtain
n∑
l=1
(
∂xj
(
Y +XY −1X
)−1
kl
)(
Y +XY −1X
)
li
=
n∑
l=1
(
∂xk
(
Y +XY −1X
)−1
jl
)(
Y +XY −1X
)
li
for i, j, k = 1, . . . , n. Here, we used
0 = ∂xj
((
Y +XY −1X
)(
Y +XY −1X
)−1)
=
(
∂xj
(
Y +XY −1X
))(
Y +XY −1X
)−1
+
(
Y +XY −1X
)
∂xj
(
Y +XY −1X
)−1
Adiabatic Limit, Theta Function, and Geometric Quantization 47
and so on. Thus,
∂xj
(
Y +XY −1X
)−1
km
=
n∑
i=1
n∑
l=1
∂xj
((
Y +XY −1X
)−1
kl
)(
Y +XY −1X
)
li
(
Y +XY −1X
)−1
im
=
n∑
i=1
n∑
l=1
(
∂xk
(
Y +XY −1X
)−1
jl
)(
Y +XY −1X
)
li
(
Y +XY −1X
)−1
im
= ∂xk
(
Y +XY −1X
)−1
jm
.
This implies (B.2). In particular, this means (3) ⇒ (2).
We show (B.3), (B.4), (B.7), and (B.8) are obtained from (2). To show (B.7), it is sufficient
to show
0 =
(
Y +XY −1X
)−1
XY −1
∂x1
(
Y +XY −1X
)
1i
· · · ∂xn
(
Y +XY −1X
)
1i
...
...
∂x1
(
Y +XY −1X
)
ni
· · · ∂xn
(
Y +XY −1X
)
ni
−
∂x1
(
Y −1X
)
1i
· · · ∂xn
(
Y −1X
)
1i
...
...
∂x1
(
Y −1X
)
ni
· · · ∂xn
(
Y −1X
)
ni
−
(
Y +XY −1X
)−1
∂x1
(
Y +XY −1X
)
1i
· · · ∂xn
(
Y +XY −1X
)
1i
...
...
∂x1
(
Y +XY −1X
)
ni
· · · ∂xn
(
Y +XY −1X
)
ni
XY −1
+
∂x1
(
XY −1
)
i1
· · · ∂x1
(
XY −1
)
in
...
...
∂xn
(
XY −1
)
i1
· · · ∂xn
(
XY −1
)
in
. (B.12)
Since Ω is symmetric, so is its real part ReΩ =
(
Y +XY −1X
)−1
XY −1. By taking the real part
of (2), we also have
∂xi
((
Y +XY −1X
)−1
XY −1
)
jk
= ∂xj
((
Y +XY −1X
)−1
XY −1
)
ik
.
By using these as well as (B.1) and (B.2), the (j, k)-component of the first two terms of the
right-hand side of (B.12) can be computed as∑
l
((
Y +XY −1X
)−1
XY −1
)
jl
∂xk
(
Y +XY −1X
)
li
− ∂xk
(
Y −1X
)
ji
=
∑
l
(
Y −1X
(
Y +XY −1X
)−1)
jl
∂xk
(
Y +XY −1X
)
li
− ∂xk
(
Y −1X
)
ji
= ∂xk
(∑
l
(
Y −1X
(
Y +XY −1X
)−1)
jl
(
Y +XY −1X
)
li
)
−
∑
l
(
∂xk
(
Y −1X
(
Y +XY −1X
)−1)
jl
)(
Y +XY −1X
)
li
− ∂xk
(
Y −1X
)
ji
= −
∑
l
(
∂xk
(
Y −1X
(
Y +XY −1X
)−1)
jl
)(
Y +XY −1X
)
li
= −
∑
l
(
∂xj
((
Y +XY −1X
)−1
XY −1
)
kl
)(
Y +XY −1X
)
li
.
48 T. Yoshida
On the other hand, the (j, k)-component of the last two terms of the right-hand side of (B.12)
can be computed as
−
∑
m,l
(
Y +XY −1X
)−1
jl
(
∂xm
(
Y +XY −1X
)
li
)(
XY −1
)
mk
+ ∂xj
(
XY −1
)
ik
=
∑
m,l
(
∂xm
(
Y +XY −1X
)−1
jl
)(
Y +XY −1X
)
li
(
XY −1
)
mk
+ ∂xj
(
XY −1
)
ik
=
∑
m,l
(
Y +XY −1X
)
li
(
∂xj
(
Y +XY −1X
)−1
ml
)(
XY −1
)
mk
+
∑
m,l
(
Y +XY −1X
)
li
(
Y +XY −1X
)−1
ml
∂xj
(
XY −1
)
mk
=
∑
l
(
∂xj
((
Y +XY −1X
)−1
XY −1
)
kl
)(
Y +XY −1X
)
li
.
This proves (B.12). We show (B.8). We put
W :=
∂x1
(
Y +XY −1X
)
1i
· · · ∂xn
(
Y +XY −1X
)
1i
...
...
∂x1
(
Y +XY −1X
)
ni
· · · ∂xn
(
Y +XY −1X
)
ni
.
By (B.7) and (B.9), we obtain−∂x1
(
Y −1X
)
1i
+ ∂x1
(
XY −1
)
i1
· · · −∂xn
(
Y −1X
)
1i
+ ∂x1
(
XY −1
)
in
...
...
−∂x1
(
Y −1X
)
ni
+ ∂xn
(
XY −1
)
i1
· · · −∂xn
(
Y −1X
)
ni
+ ∂xn
(
XY −1
)
in
=
(
Y +XY −1X
)−1
WXY −1 −
(
Y +XY −1X
)−1
XY −1W
and (
Y +XY −1X
)
tW =W
(
Y +XY −1X
)
.
In order to show (B.8), it is sufficient to check
0 = Y −1W + Y −1X
(
Y +XY −1X
)−1
WXY −1 − Y −1X
(
Y +XY −1X
)−1
XY −1W
+
(
Y +XY −1X
)−1
WXY −1XY −1
−
(
Y +XY −1X
)−1
XY −1WXY −1 − tWY −1. (B.13)
By using above equalities, the right-hand side of (B.13) can be computed as
Y −1W − Y −1X
(
Y +XY −1X
)−1
XY −1W +
(
Y +XY −1X
)−1
WXY −1XY −1 − tWY −1
= Y −1W −
(
Y +XY −1X
)−1
XY −1XY −1W +
(
Y +XY −1X
)−1
WXY −1XY −1
−
(
Y +XY −1X
)−1
W
(
Y +XY −1X
)
Y −1
= Y −1W −
(
Y +XY −1X
)−1
XY −1XY −1W +
(
Y +XY −1X
)−1
WXY −1XY −1
−
(
Y +XY −1X
)−1
W −
(
Y +XY −1X
)−1
WXY −1XY −1
= Y −1W −
{(
Y +XY −1X
)−1
XY −1X +
(
Y +XY −1X
)−1
Y
}
Y −1W = 0.
This proves (B.8).
Adiabatic Limit, Theta Function, and Geometric Quantization 49
We show (B.3). To see this, we show
0 =
(
Y +XY −1X
)−1
XY −1
∂x1
(
XY −1
)
1i
· · · ∂xn
(
XY −1
)
1i
...
...
∂x1
(
XY −1
)
ni
· · · ∂xn
(
XY −1
)
ni
−
∂x1
(
Y −1
)
1i
− ∂x1
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
1i
− ∂x1
(
Y −1
)
ni
...
...
∂x1
(
Y −1
)
ni
− ∂xn
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
ni
− ∂xn
(
Y −1
)
ni
−
(
Y +XY −1X
)−1
∂x1
(
XY −1
)
1i
· · · ∂xn
(
XY −1
)
1i
...
...
∂x1
(
XY −1
)
ni
· · · ∂xn
(
XY −1
)
ni
XY −1. (B.14)
The (j, k)-component of the right-hand side of (B.14) is∑
l
((
Y +XY −1X
)−1
XY −1
)
jl
∂xk
(
XY −1
)
li
− ∂xk
Y −1
ji + ∂xjY
−1
ki
−
∑
l,m
(
Y +XY −1X
)−1
jm
∂xl
(
XY −1
)
mi
(
XY −1
)
lk
=
((
Y +XY −1X
)−1
XY −1∂xk
(
XY −1
))
ji
− ∂xk
Y −1
ji + ∂xjY
−1
ki
−
∑
l,m
(
Y +XY −1X
)−1
lm
∂xj
(
XY −1
)
mi
(
XY −1
)
lk
=
((
Y +XY −1X
)−1{
∂xk
((
Y +XY −1X
)
Y −1
)
− ∂xk
(
XY −1
)
XY −1
})
ji
− ∂xk
Y −1
ji + ∂xjY
−1
ki −
∑
l,m
(
Y +XY −1X
)−1
ml
(
XY −1
)
lk
∂xj
(
XY −1
)
mi
=
((
Y +XY −1X
)−1(
∂xk
(
Y +XY −1X
))
Y −1 + ∂xk
Y −1
)
ji
−
((
Y +XY −1X
)−1
∂xk
(
XY −1
)
XY −1
)
ji
− ∂xk
Y −1
ji + ∂xjY
−1
ki
−
∑
m
((
Y +XY −1X
)−1
XY −1
)
mk
∂xj
(
XY −1
)
mi
=
((
Y +XY −1X
)−1(
∂xk
(
Y +XY −1X
))
Y −1
)
ji
−
((
Y +XY −1X
)−1
∂xk
(
XY −1
)
XY −1
)
ji
+ ∂xjY
−1
ki
−
∑
m
((
Y +XY −1X
)−1
XY −1
)
km
∂xj
(
XY −1
)
mi
=
((
Y +XY −1X
)−1(
∂xk
(
Y +XY −1X
))
Y −1
)
ji
−
((
Y +XY −1X
)−1
∂xk
(
XY −1
)
XY −1
)
ji
+
(
∂xjY
−1 −
(
Y +XY −1X
)−1
XY −1∂xj
(
XY −1
))
ki
=
(
−
(
∂xk
(
Y +XY −1X
)−1)(
Y +XY −1X
)
Y −1
)
ji
−
∑
l
((
Y +XY −1X
)−1
∂xk
(
XY −1
))
jl
XY −1
li
+
(
∂xjY
−1 −
(
Y +XY −1X
)−1
XY −1∂xj
(
XY −1
))
ki
= −
∑
l
∂xk
(
Y +XY −1X
)−1
jl
((
Y +XY −1X
)
Y −1
)
li
50 T. Yoshida
−
∑
l
((
Y +XY −1X
)−1
∂xj
(
XY −1
))
kl
XY −1
li
+
(
∂xjY
−1 −
(
Y +XY −1X
)−1
XY −1∂xj
(
XY −1
))
ki
= −
∑
l
∂xj
(
Y +XY −1X
)−1
kl
((
Y +XY −1X
)
Y −1
)
li
−
((
Y +XY −1X
)−1
∂xj
(
XY −1
)
XY −1
)
ki
+
(
∂xjY
−1 −
(
Y +XY −1X
)−1
XY −1∂xj
(
XY −1
))
ki
=
(
−
(
∂xj
(
Y +XY −1X
)−1)(
Y +XY −1X
)
Y −1
)
ki
−
((
Y +XY −1X
)−1
∂xj
(
XY −1
)
XY −1
)
ki
+
(
∂xjY
−1 −
(
Y +XY −1X
)−1
XY −1∂xj
(
XY −1
))
ki
=
((
Y +XY −1X
)−1(
∂xj
(
Y +XY −1X
))
Y −1
)
ki
−
((
Y +XY −1X
)−1
∂xj
(
XY −1
)
XY −1
)
ki
+
(
∂xjY
−1 −
(
Y +XY −1X
)−1
XY −1∂xj
(
XY −1
))
ki
=
((
Y +XY −1X
)−1{(
∂xj
(
Y +XY −1X
))
Y −1 +
(
Y +XY −1X
)
∂xjY
−1
})
ki
−
((
Y +XY −1X
)−1{
∂xj
(
XY −1
)
XY −1 +XY −1∂xj
(
XY −1
)})
ki
=
((
Y +XY −1X
)−1{
∂xj (
(
Y +XY −1X
)
Y −1)− ∂xj (XY
−1XY −1)
})
ki
= 0.
This proves (B.3).
Finally, we show (B.4). We put
V :=
∂x1
(
XY −1
)
1i
· · · ∂xn
(
XY −1
)
1i
...
...
∂x1
(
XY −1
)
ni
· · · ∂xn
(
XY −1
)
ni
.
By (B.3) and (B.5), we obtain∂x1
(
Y −1
)
1i
− ∂x1
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
1i
− ∂x1
(
Y −1
)
ni
...
...
∂x1
(
Y −1
)
ni
− ∂xn
(
Y −1
)
1i
· · · ∂xn
(
Y −1
)
ni
− ∂xn
(
Y −1
)
ni
=
(
Y +XY −1X
)−1
XY −1V −
(
Y +XY −1X
)−1
V XY −1
and (
Y +XY −1X
)
tV = V
(
Y +XY −1X
)
.
In order to show (B.4), it is sufficient to check
0 = Y −1V − Y −1X
(
Y +XY −1X
)−1
XY −1V + Y −1X
(
Y +XY −1X
)−1
V XY −1
+
(
Y +XY −1X
)−1
V XY −1XY −1 −
(
Y +XY −1X
)−1
XY −1V XY −1
− tV Y −1. (B.15)
Then, (B.15) can be checked in the same way as (B.13).
Adiabatic Limit, Theta Function, and Geometric Quantization 51
Acknowledgments
The many part of this work was done while the author stayed in McMaster university. The
author would like to thank the department of Mathematics and Statistics, McMaster university
and especially Megumi Harada for their hospitality. The author would also like to express our
sincere gratitude to the referees who carefully read the manuscript and helped him improve it.
This work is supported by Grant-in-Aid for Scientific Research (C) 15K04857 and 19K03479.
References
[1] Andersen J.E., Geometric quantization of symplectic manifolds with respect to reducible non-negative po-
larizations, Comm. Math. Phys. 183 (1997), 401–421.
[2] Arnold V.I., Mathematical methods of classical mechanics, 2nd ed., Grad. Texts Math., Vol. 60, Springer,
New York, 1989.
[3] Baier T., Florentino C., Mourão J.M., Nunes J.P., Toric Kähler metrics seen from infinity, quantization and
compact tropical amoebas, J. Differential Geom. 89 (2011), 411–454, arXiv:0806.0606.
[4] Baier T., Mourão J.M., Nunes J.P., Quantization of abelian varieties: distributional sections and the tran-
sition from Kähler to real polarizations, J. Funct. Anal. 258 (2010), 3388–3412, arXiv:0907.5324.
[5] Bieberbach L., Über die Bewegungsgruppen der Euklidischen Räume, Math. Ann. 70 (1911), 297–336.
[6] Bieberbach L., Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen
mit einem endlichen Fundamentalbereich, Math. Ann. 72 (1912), 400–412.
[7] Borthwick D., Uribe A., Almost complex structures and geometric quantization, Math. Res. Lett. 3 (1996),
845–861, arXiv:dg-ga/9608006.
[8] Bott R., Tu L.W., Differential forms in algebraic topology, Grad. Texts Math., Vol. 82, Springer, New York,
1982.
[9] Danilov V.I., The geometry of toric varieties, Russian Math. Surveys 33 (1978), 97–154.
[10] Duistermaat J.J., On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687–706.
[11] Duistermaat J.J., The heat kernel Lefschetz fixed point formula for the spin-c Dirac operator, Progr. Non-
linear Differential Equations Appl., Vol. 18, Birkhäuser, Boston, MA, 1996.
[12] Egorov D.V., Theta functions on T 2-bundles over T 2 with the zero Euler class, Sib. Math. J. 50 (2009),
647–657, arXiv:1110.2322.
[13] Egorov D.V., Theta functions on the Kodaira–Thurston manifold, Sib. Math. J. 50 (2009), 253–260,
arXiv:0902.2843.
[14] Fujita H., Furuta M., Yoshida T., Torus fibrations and localization of index I, J. Math. Sci. Univ. Tokyo 17
(2010), 1–26, arXiv:0804.3258.
[15] Fujita H., Furuta M., Yoshida T., Torus fibrations and localization of index II: local index for acyclic
compatible system, Comm. Math. Phys. 326 (2014), 585–633, arXiv:0910.0358.
[16] Goldman W., Hirsch M.W., The radiance obstruction and parallel forms on affine manifolds, Trans. Amer.
Math. Soc. 286 (1984), 629–649.
[17] Guillemin V., Sternberg S., The Gelfand–Cetlin system and quantization of the complex flag manifolds,
J. Funct. Anal. 52 (1983), 106–128.
[18] Hall B.C., Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact
type, Comm. Math. Phys. 226 (2002), 233–268, arXiv:quant-ph/0012105.
[19] Hall B.C., Quantum theory for mathematicians, Grad. Texts Math., Vol. 267, Springer, New York, 2013.
[20] Hamilton M.D., Harada M., Kaveh K., Convergence of polarizations, toric degenerations, and Newton–
Okounkov bodies, Comm. Anal. Geom. 29 (2021), 1183–1231, arXiv:1612.08981v2.
[21] Hamilton M.D., Konno H., Convergence of Kähler to real polarizations on flag manifolds via toric degener-
ations, J. Symplectic Geom. 12 (2014), 473–509, arXiv:1105.0741.
[22] Jeffrey L.C., Weitsman J., Bohr–Sommerfeld orbits in the moduli space of flat connections and the Verlinde
dimension formula, Comm. Math. Phys. 150 (1992), 593–630.
https://doi.org/10.1007/BF02506413
https://doi.org/10.1007/978-1-4757-2063-1
https://doi.org/10.4310/jdg/1335207374
https://arxiv.org/abs/0806.0606
https://doi.org/10.1016/j.jfa.2010.01.023
https://arxiv.org/abs/0907.5324
https://doi.org/10.1007/BF01564500
https://doi.org/10.1007/BF01456724
https://doi.org/10.4310/MRL.1996.v3.n6.a12
https://arxiv.org/abs/dg-ga/9608006
https://doi.org/10.1007/978-1-4757-3951-0
https://doi.org/10.1070/RM1978v033n02ABEH002305
https://doi.org/10.1002/cpa.3160330602
https://doi.org/10.1007/978-1-4612-5344-0
https://doi.org/10.1007/s11202-009-0072-x
https://arxiv.org/abs/1110.2322
https://doi.org/10.1007/s11202-009-0029-0
https://arxiv.org/abs/0902.2843
https://arxiv.org/abs/0804.3258
https://doi.org/10.1007/s00220-014-1890-7
https://arxiv.org/abs/0910.0358
https://doi.org/10.2307/1999812
https://doi.org/10.2307/1999812
https://doi.org/10.1016/0022-1236(83)90092-7
https://doi.org/10.1007/s002200200607
https://arxiv.org/abs/quant-ph/0012105
https://doi.org/10.1007/978-1-4614-7116-5
https://doi.org/10.4310/CAG.2021.v29.n5.a6
https://arxiv.org/abs/1612.08981v2
https://doi.org/10.4310/CAG.2021.v29.n5.a6
https://arxiv.org/abs/1105.0741
https://doi.org/10.1007/BF02096964
52 T. Yoshida
[23] Kamiyama Y., The cohomology of spatial polygon spaces with anticanonical sheaf, Int. J. Appl. Math. 3
(2000), 339–343.
[24] Kirillov A.A., Geometric quantization, in Dynamical Systems, IV, Encyclopaedia Math. Sci., Vol. 4, Springer,
Berlin, 2001, 139–176.
[25] Kirwin W.D., Uribe A., Theta functions on the Kodaira–Thurston manifold, Trans. Amer. Math. Soc. 362
(2010), 897–932, arXiv:0712.4016.
[26] Kodaira K., On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964), 751–798.
[27] Kostant B., Quantization and unitary representations, in Lectures in Modern Analysis and Applications, III,
Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, 87–208.
[28] Kubota Y., The joint spectral flow and localization of the indices of elliptic operators, Ann. K-Theory 1
(2016), 43–83, arXiv:1410.5569.
[29] Lascoux A., Berger M., Variétés Kähleriennes compactes, Lecture Notes in Math., Vol. 154, Springer, Berlin,
1970.
[30] Lawson Jr. H.B., Michelsohn M.L., Spin geometry, Princeton Math. Ser., Vol. 38, Princeton University
Press, Princeton, NJ, 1989.
[31] Mishachev K.N., The classification of Lagrangian bundles over surfaces, Differential Geom. Appl. 6 (1996),
301–320.
[32] Mumford D., Tata lectures on theta. I, Mod. Birkhäuser Class., Birkhäuser, Boston, MA, 2007.
[33] Mumford D., Tata lectures on theta. III, Mod. Birkhäuser Class., Birkhäuser, Boston, MA, 2007.
[34] Nohara Y., Projective embeddings and Lagrangian fibrations of abelian varieties, Math. Ann. 333 (2005),
741–757.
[35] Nohara Y., Projective embeddings and Lagrangian fibrations of Kummer varieties, Internat. J. Math. 20
(2009), 557–572, arXiv:math.DG/0604329.
[36] Sepe D., Topological classification of Lagrangian fibrations, J. Geom. Phys. 60 (2010), 341–351,
arXiv:0910.5450.
[37] Śniatycki J., Geometric quantization and quantum mechanics, Appl. Math. Sci., Vol. 30, Springer, New
York, 1980.
[38] Souriau J.-M., Quantification géométrique, Comm. Math. Phys. 1 (1966), 374–398.
[39] Souriau J.-M., Structure of dynamical systems, Progr. Math., Vol. 149, Birkhäuser, Boston, MA, 1997.
[40] Symington M., Four dimensions from two in symplectic topology, in Topology and Geometry of Mani-
folds, Proc. Sympos. Pure Math., Vol. 71, American Mathematical Society, Providence, RI, 2003, 153–208,
arXiv:math.SG/0210033.
[41] Thurston W.P., Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467–468.
[42] Wolf J.A., Spaces of constant curvature, 6th ed., AMS Chelsea Publishing, Providence, RI, 2011.
[43] Woodhouse N.M.J., Geometric quantization, 2nd ed., Oxford Math. Monogr., The Clarendon Press, Oxford
University Press, New York, 1992.
[44] Yoshida T., Local torus actions modeled on the standard representation, Adv. Math. 227 (2011), 1914–1955,
arXiv:0710.2166.
https://doi.org/10.1007/978-3-662-06791-8_2
https://doi.org/10.1090/S0002-9947-09-04852-1
https://arxiv.org/abs/0712.4016
https://doi.org/10.2307/2373157
https://doi.org/10.1007/BFb0079068
https://doi.org/10.2140/akt.2016.1.43
https://arxiv.org/abs/1410.5569
https://doi.org/10.1007/BFb0069331
https://doi.org/10.1016/S0926-2245(96)00024-1
https://doi.org/10.1007/978-0-8176-4577-9
https://doi.org/10.1007/s00208-005-0685-8
https://doi.org/10.1142/S0129167X09005418
https://arxiv.org/abs/math.DG/0604329
https://doi.org/10.1016/j.geomphys.2009.10.004
https://arxiv.org/abs/0910.5450
https://doi.org/10.1007/978-1-4612-6066-0
https://doi.org/10.1090/pspum/071/2024634
https://arxiv.org/abs/math.SG/0210033
https://doi.org/10.2307/2041749
https://doi.org/10.1090/chel/372
https://doi.org/10.1016/j.aim.2011.04.007
https://arxiv.org/abs/0710.2166
1 Introduction
1.1 Background and motivation
1.2 Main theorems
1.3 Notations
2 Unfolding Lagrangian fibrations
2.1 Integral affine structures
2.2 Lagrangian fibrations
2.3 Lagrangian fibrations with complete bases
2.4 The lifting problem of fiber-preserving symplectomorphisms to the prequantum line bundle
3 Degree-zero harmonic spinors and integrability of almost complex structures
3.1 Bohr–Sommerfeld points
3.2 Almost complex structures
3.3 A condition on the existence of nontrivial harmonic spinors of degree-zero
3.4 The Gamma-equivariant case
4 The integrable case
4.1 Definition and properties of vartheta_m/N
4.2 The case when Z is constant
4.3 Adiabatic-type limit
5 The non-integrable case
A Fourier series
B Proof of Proposition 3.14
References
|
| id | nasplib_isofts_kiev_ua-123456789-212355 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T10:05:36Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Yoshida, Takahiko 2026-02-05T09:56:19Z 2024 Adiabatic Limit, Theta Function, and Geometric Quantization. Takahiko Yoshida. SIGMA 20 (2024), 065, 52 pages 1815-0659 2020 Mathematics Subject Classification: 53D50; 58H15; 58J05 arXiv:1904.04076 https://nasplib.isofts.kiev.ua/handle/123456789/212355 https://doi.org/10.3842/SIGMA.2024.065 Let π : (, ) → be a non-singular Lagrangian torus fibration on a complete base with prequantum line bundle (, ∇ᴸ) → (, ). Compactness on is not assumed. For a positive integer and a compatible almost complex structure on (, ) invariant along the fiber of π, let be the associated Spinᶜ Dirac operator with coefficients in ⊗ᴺ. First, in the case where is integrable, under certain technical conditions on , we give a complete orthogonal system {ϑb}b ∈ BS of the space of holomorphic ²-sections of ⊗ᴺ indexed by the Bohr-Sommerfeld points BS such that each ϑb converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber π⁻¹(b) by the adiabatic(-type) limit. We also explain the relation of ϑb with Jacobi's theta functions when (, ) is ²ⁿ. Second, in the case where is not integrable, we give an orthogonal family {ϑ~b}b ∈ BS of ²-sections of ⊗ᴺ indexed by BS which has the same property as above, and show that each ϑ~b converges to 0 by the adiabatic(-type) limit with respect to the ²-norm. The many part of this work were done while the author stayed at McMaster University. The author would like to thank the Department of Mathematics and Statistics, McMaster university and especially Megumi Harada for their hospitality. The author would also like to express our sincere gratitude to the referees who carefully read the manuscript and helped him improve it. This work is supported by Grant-in-Aid for Scientific Research (C) 15K04857 and 19K03479. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Adiabatic Limit, Theta Function, and Geometric Quantization Article published earlier |
| spellingShingle | Adiabatic Limit, Theta Function, and Geometric Quantization Yoshida, Takahiko |
| title | Adiabatic Limit, Theta Function, and Geometric Quantization |
| title_full | Adiabatic Limit, Theta Function, and Geometric Quantization |
| title_fullStr | Adiabatic Limit, Theta Function, and Geometric Quantization |
| title_full_unstemmed | Adiabatic Limit, Theta Function, and Geometric Quantization |
| title_short | Adiabatic Limit, Theta Function, and Geometric Quantization |
| title_sort | adiabatic limit, theta function, and geometric quantization |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212355 |
| work_keys_str_mv | AT yoshidatakahiko adiabaticlimitthetafunctionandgeometricquantization |