# Adiabatic limits, Theta functions, 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
<EMAIL_ADDRESS>
###### Abstract.

Let $\pi\colon(M,\omega)\to B$ be a (non-singular) Lagrangian torus fibration
on a compact, complete base $B$ with prequantum line bundle
$(L,\nabla^{L})\to(M,\omega)$. For a positive integer $N$ and a compatible
almost complex structure $J$ on $(M,\omega)$ invariant along the fiber of
$\pi$, let $D$ be the associated Spinc Dirac operator with coefficients in
$L^{\otimes N}$. Then, we give an orthogonal family
$\\{{\widetilde{\vartheta}}_{b}\\}_{b\in B_{BS}}$ of sections of $L^{\otimes
N}$ indexed by the Bohr-Sommerfeld points $B_{BS}$, and show that each
${\widetilde{\vartheta}}_{b}$ converges to a delta-function section supported
on the corresponding Bohr-Sommerfeld fiber $\pi^{-1}(b)$ and the $L^{2}$-norm
of $D{\widetilde{\vartheta}}_{b}$ converges to $0$ by the adiabatic(-type)
limit. Moreover, if $J$ is integrable, we also give an orthogonal basis
$\\{\vartheta_{b}\\}_{b\in B_{BS}}$ of the space of holomorphic sections of
$L^{\otimes N}$ indexed by $B_{BS}$, and show that each $\vartheta_{b}$
converges to a delta-function section supported on the corresponding Bohr-
Sommerfeld fiber $\pi^{-1}(b)$ by the adiabatic(-type) limit. We also explain
the relation of $\vartheta_{b}$ with Jacobi’s theta functions when
$(M,\omega)$ is $T^{2n}$.

###### Key words and phrases:

adiabatic limit, Theta function, Lagrangian fibration, geometric quantization

###### 2010 Mathematics Subject Classification:

Primary 53D50; Secondary 58H15, 58J05

###### Contents

  1. 1 Motivation and Main Theorems
    1. 1.1 Notations
  2. 2 Developing Lagrangian fibrations
    1. 2.1 Integral affine structures
    2. 2.2 Lagrangian fibrations
    3. 2.3 Lagrangian fibrations with complete bases
    4. 2.4 The lifting problem of the $\Gamma$-action to the prequantum line bundle
  3. 3 Degree-zero harmonic spinors and integrability of almost complex structures
    1. 3.1 Bohr-Sommerfeld points
    2. 3.2 Almost complex structures
    3. 3.3 The existence condition of non-trivial harmonic spinors of degree-zero
    4. 3.4 The $\Gamma$-equivariant case
  4. 4 The integrable case
    1. 4.1 Definition and properties of $\vartheta_{\frac{m}{N}}$
    2. 4.2 The case when $Z$ is constant
    3. 4.3 Adiabatic-type limit
  5. 5 The non-integrable case

##  1\. Motivation and Main Theorems

The purpose of this paper is to show the Spinc quantization converges to the
real quantization by the adiabatic(-type) limit for a Lagrangian torus
fibration on a compact complete base and a compatible almost complex
structures on its total space which is invariant along the fiber. In this
paper, a Lagrangian torus fibration is assumed to be non-singular unless
otherwise stated. First let us explain the motivation which comes from
geometric quantization. For geometric quantization, see [17, 22, 32, 38]. In
physics, quantization is the procedure for building quantum mechanics starting
from classical mechanics. 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, and the geometric quantization gives us the method to construct a
quantum Hilbert space and a representation from a given symplectic manifold
$(M,\omega)$ and a prequantum line bundle $(L,\nabla^{L})\to(M,\omega)$ in the
geometric way. In the theory of geometric quantization by Kostant and Souriau
[25, 34, 33], 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
$TM\otimes\mathbb{C}$ of $(M,\omega)$. For a polarization $P$, the quantum
Hilbert space is naively given as the closure of the space of smooth, square-
integrable sections of $(L,\nabla^{L})$ which are covariant constant along
$P$.

A common example is the Kähler polarization. When $(M,\omega)$ is Kähler and
$(L,\nabla^{L})$ is a holomorphic line bundle with canonical connection, we
can take $T^{0,1}M$ as a polarization, and the obtained quantum Hilbert space
is nothing but the space of holomorphic sections $H^{0}(M,\mathcal{O}_{L})$.
This polarization is called the Kähler polarization and the quantization
procedure is called the Kähler quantization. Note that when $M$ is compact and
the Kodaira vanishing theorem holds, its dimension is equal to the index of
the Dolbeault operator with coefficients in $L$.

Another example is a real polarization. Suppose $(M,\omega)$ admits a
structure of a Lagrangian torus fibration $\pi\colon(M,\omega)\to B$. For each
point $b\in B$ of the base manifold $B$, the restriction
$\left(L,\nabla^{L}\right)|_{\pi^{-1}(b)}$ of $(L,\nabla^{L})$ to the fiber
$\pi^{-1}(b)$ is a flat line bundle. Let
$H^{0}\left(\pi^{-1}(b);\left(L,\nabla^{L}\right)|_{\pi^{-1}(b)}\right)$ be
the space of covariant constant sections of
$\left(L,\nabla^{L}\right)|_{\pi^{-1}(b)}$. Then, an element $b\in B$ is said
to be Bohr-Sommerfeld if
$H^{0}\left(\pi^{-1}(b);\left(L,\nabla^{L}\right)|_{\pi^{-1}(b)}\right)\not=\\{0\\}$.
It is well-known that Bohr-Sommerfeld points appear discretely. In this case,
we can take $T_{\pi}M\otimes\mathbb{C}$, the complexified tangent bundle along
fibers of $\pi$ as a polarization, and if $M$ is compact, the quantum Hilbert
space is defined by $\oplus_{b\in
B_{BS}}H^{0}\left(\pi^{-1}(b);\left(L,\nabla^{L}\right)|_{\pi^{-1}(b)}\right)$,
where the sum is taken over all Bohr-Sommerfeld points. See [32] for more
details. In this paper, we call this quantization the real quantization.

When a Lagrangian torus fibration $\pi\colon(M,\omega)\to B$ with closed total
space $M$ and a prequantum line bundle $(L,\nabla^{L})\to(M,\omega)$ are
given, it is natural to ask whether the quantum Hilbert space obtained by the
Kähler quantization is isomorphic to the one obtained by the real
quantization. A completely integrable system can be thought of as a Lagrangian
fibration with singular fibers. In the case of the the moment map of a
projective toric variety, Danilov has shown in [9] that
$H^{0}(M,\mathcal{O}_{L})$ has the irreducible decomposition
$H^{0}(M,\mathcal{O}_{L})=\oplus_{m\in\Delta\cap\mathfrak{t}^{*}_{\mathbb{Z}}}\mathbb{C}_{m}$
as a compact torus representation, where $\Delta$ is the moment polytope,
$\mathfrak{t}^{*}_{\mathbb{Z}}$ is the weight lattice, and $\mathbb{C}_{m}$ is
the irreducible representation of the torus with weight $m$. Since
$\Delta\cap\mathfrak{t}^{*}_{\mathbb{Z}}$ is identified with the set of Bohr-
Sommerfeld points, this implies the dimensions of the quantum Hilbert spaces
obtained by the above quantizations agree. A similar equality of the
dimensions has been also shown for the Gelfand-Cetlin system on the flag
variety [16], the Goldman system on the moduli space of flat $SU(2)$
connections on a surface [20], and the Kapovich-Millson system on the polygon
space. [21].

Moreover, in the case of smooth projective toric varieties, not only the
numerical equality for the dimensions, but also a geometric correspondence
between the Kähler and the real quantizations has been shown concretely by
Baier-Florentino-Mourão-Nunes in [3]. Namely, they have given a one-parameter
family of complex structures $\\{J_{t}\\}_{t\in[0,\infty)}$ and a basis
$\\{s_{t}^{m}\\}_{m\in\Delta\cap\mathfrak{t}^{*}_{\mathbb{Z}}}$ of the space
of holomorphic sections associated with the complex structure $J_{t}$ for each
$t$ such that each section $s_{t}^{m}$ converges to the delta function section
supported on the corresponding Bohr-Sommerfeld fiber as $t$ goes to $\infty$.
The similar result has been also obtained for flag manifolds in [19] and
smooth irreducible complex algebraic varieties by [18]. But in [19] and [18]
the convergence has been shown only for the non-singular Bohr-Sommerfeld
fibers whereas in [3] it has been shown for all Bohr-Sommerfeld fibers.

The Kähler quantization can be generalized to a non-integrable compatible
almost complex structure on a closed $(M,\omega)$. When a compatible almost
complex structure $J$ on $(M,\omega)$ is given, we can consider the associated
Spinc Dirac operator $D$ acting on
$\Gamma\left(\wedge^{\bullet}T^{*}M^{0,1}\otimes L\right)$. It is well-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, $T^{0,1}M$ 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

(1.1) $\ker\left(D|_{\wedge^{even}T^{*}M^{0,1}\otimes
L}\right)-\ker\left(D|_{\wedge^{odd}T^{*}M^{0,1}\otimes L}\right)\in K(pt)$

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
shown in [1, 13, 26] that the above equality between dimensions of two quantum
Hilbert spaces still holds by replacing the Kähler quantization with the Spinc
quantization by using the index theory.

In this paper, we generalize the approach taken in [3] for the Kähler
quantization to the Spinc quantization of Lagrangian torus fibrations. Let
$\pi\colon(M,\omega)\to B$ be a Lagrangian torus fibration on a compact,
complete base $B$ with prequantum line bundle $(L,\nabla^{L})\to(M,\omega)$.
For a positive integer $N$ and a compatible almost complex structure $J$ of
$(M,\omega)$ invariant along the fiber of $\pi$ (in the sense of Lemma 3.6),
let $D$ be the associated Spinc Dirac operator with coefficients in
$L^{\otimes N}$. Then, the main result is as follows. This theorem is a
combination of Theorem 5.2 and Theorem 5.3.

######  Theorem 1.1.

For the above data, we give one-parameter families of

  * •

$\\{J^{t}\\}_{t>0}$ compatible almost complex structures of $(M,\omega)$ with
$J^{1}=J$

  * •

$\\{{\widetilde{\vartheta}}^{t}_{b}\\}_{b\in B_{BS}}$ sets of sections on
$L^{\otimes N}$ indexed by the Bohr-Sommerfeld points $B_{BS}$

such that

  1. (1)

any pair in $\\{{\widetilde{\vartheta}}^{t}_{b}\\}_{b\in B_{BS}}$ are
orthogonal to each other,

  2. (2)

each ${\widetilde{\vartheta}}^{t}_{b}$ converges as a delta-function section
supported on $\pi^{-1}(b)$ as $t\to\infty$ in the following sense, for any
section $s$ of $L^{\otimes N}$,

$\lim_{t\to\infty}\int_{M}\left\langle
s,\frac{{\widetilde{\vartheta}}^{t}_{b}}{\lVert{\widetilde{\vartheta}}^{t}_{b}\rVert_{L^{1}}}\right\rangle_{L^{\otimes
N}}(-1)^{\frac{n(n-1)}{2}}\dfrac{\omega^{n}}{n!}=\int_{\pi^{-1}\left(b\right)}\left\langle
s,\delta_{b}\right\rangle_{L^{\otimes N}}\lvert dy\rvert,$

where $\langle\,,\,\rangle_{L^{\otimes N}}$ is the Hermitian metric of
$L^{\otimes N}$, $\delta_{b}$ is the covariant constant section of $L^{\otimes
N}|_{\pi^{-1}(b)}$ defined by (4.9), and $\lvert dy\rvert$ is the natural one-
density on $\pi^{-1}\left(b\right)$,

  3. (3)

$\displaystyle\lim_{t\to\infty}\lVert
D^{t}{\widetilde{\vartheta}}^{t}_{b}\rVert_{L^{2}}=0$.

By the Spinc Dirac vanishing theorem due to Borthwick-Uribe [7],
$\ker\left(D|_{\wedge^{odd}T^{*}M^{0,1}\otimes L^{\otimes N}}\right)$ vanishes
for a sufficiently large $N$. So, (3) implies the the complex vector space
spanned by $\\{{\widetilde{\vartheta}}^{t}_{b}\\}_{b\in B_{BS}}$ approximates
the quantum Hilbert space of the Spinc quantization for a sufficiently large
$N$.

If $J$ is integrable, we also give the following refinement of Theorem 1.1,
which is immediately obtained by putting Corollary 4.3 and Theorem 4.12
together.

######  Theorem 1.2.

Under the above setting, assume $J$ is integrable. Then, with a technical
assumption, we give one-parameter families of

  * •

$\\{J^{t}\\}_{t>0}$ compatible complex structures of $(M,\omega)$ with
$J^{1}=J$

  * •

$\\{\vartheta^{t}_{b}\\}_{b\in B_{BS}}$ orthogonal bases of holomorphic
sections of $L^{\otimes N}\to(M,N\omega,J^{t})$ indexed by $B_{BS}$

such that each $\vartheta^{t}_{b}$ converges as a delta-function section
supported on $\pi^{-1}(b)$ as $t\to\infty$ in the following sense, for any
section $s$ of $L^{\otimes N}$,

$\lim_{t\to\infty}\int_{M}\left\langle
s,\frac{\vartheta^{t}_{b}}{\lVert\vartheta^{t}_{b}\rVert_{L^{1}}}\right\rangle_{L^{\otimes
N}}(-1)^{\frac{n(n-1)}{2}}\dfrac{\omega^{n}}{n!}=\int_{\pi^{-1}\left(b\right)}\left\langle
s,\delta_{b}\right\rangle_{L^{\otimes N}}\lvert dy\rvert.$

One of examples of the total space of a Lagrangian torus fibration with
complete base is an abelian variety. In this case, we show that each
$\vartheta_{b}$ coincides with Jacobi’s theta functions up to function on the
base space (Theorem 4.7). For the theta functions, see [28, 29].

We should remark there are several works which deal with theta functions from
the viewpoint of geometric quantization of Lagrangian fibrations, for example,
[29], [4], [30, 31]. 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 quantization. In the almost Kähler quantization of the Kodaira-
Thurston manifold, Kirwin-Uribe and Egorov have constructed an analog of the
theta function as an element of the quantum Hilbert space [23], [12]. In [11],
Egorov has also shown the similar result for Lagrangian $T^{2}$-fibrations on
$T^{2}$ with zero Euler class.

The idea used in this paper is quite simple. One of two key facts is Corollary
2.25 which claims that any Lagrangian torus fibration $\pi\colon(M,\omega)\to
B$ with complete base $B$ and a prequantum line bundle
$\left(L,\nabla^{L}\right)\to\left(M,\omega\right)$ can be obtained as the
quotient of a $\pi_{1}(B)$-action on the standard Lagrangian fibration
$\left({\widetilde{M}},\omega_{0}\right):=\left(\mathbb{R}^{n}\times
T^{n},\sum_{i=1}^{n}dx_{i}\wedge dy_{i}\right)\to\mathbb{R}^{n}$ with the
standard prequantum line bundle. In particular, any compatible almost complex
structure on $(M,\omega)$ is induced from a $\pi_{1}(B)$-equivariant one on
$\left({\widetilde{M}},\omega_{0}\right)$, and the set of compatible almost
complex structures on $\left({\widetilde{M}},\omega_{0}\right)$ corresponds
one-to-one to the set of smooth maps from ${\widetilde{M}}$ to the Siegel
upper half space. We show that there exists a $\pi_{1}(B)$-invariant
compatible almost complex structure $J$ whose corresponding map is invariant
along the fiber (Lemma 3.6). For the Spinc Dirac operator $D$ associated with
such an almost complex structure $J$, we consider the problem on the existence
of non-trivial degree-zero harmonic spinors, i.e., sections of
${\widetilde{L}}^{\otimes N}$ contained in the kernel of $D$. By taking the
Fourier series expansion of a section $s$ of ${\widetilde{L}}^{\otimes N}$
with respect to the fiber coordinates, the equation $Ds=0$ can be reduced to a
system of partial differential equations on $\mathbb{R}^{n}$. The other key
fact is Proposition 3.12 in which we give a necessary and sufficient condition
in order that the system of partial differential equations has non-trivial
solutions and show that it is equal to the integrability condition for $J$,
i.e., $\left({\widetilde{M}},\omega_{0},J\right)$ is Kähler. Moreover, in this
case, we give a family of $\pi_{1}(B)$-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, this gives a linear
basis of the space of holomorphic sections of
$\left(L,\nabla^{L}\right)^{\otimes N}\to\left(M,N\omega,J\right)\to B$. We
also give a sufficient condition for their convergence. Even if $J$ is not
integrable, by considering an approximation of $D$, we can obtain an
orthogonal family of sections of $L^{\otimes N}$ indexed by the Bohr-
Sommerfeld points $B_{BS}$. The limit used in this paper is slightly different
from the adiabatic limit in Riemannian geometry. When a fiber bundle
$\pi\colon M\to B$ and a Riemannian metric $g$ on $M$ are given, we can
consider the decomposition $(TM,g)=(V,g_{V})\oplus(H,g_{H})$, where $V$ is the
tangent bundle along the fiber with fiber metric $g_{V}:=g|_{V}$ and $H$ is
the orthogonal complement of $V$ with respect to $g$ with fiber metric
$g_{H}:=g|_{H}$. For each $t>0$, define the Riemannian metric $g^{t}$ to be
$g^{t}:=g_{V}\oplus tg_{H}$. Then, in Riemannian geometry, the adiabatic limit
is the procedure for taking the limit of geometric objects depending on
$g^{t}$ as $t\to\infty$. 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 $(M,\omega)$ such that the corresponding one-
parameter family of Riemannian metrics is $\\{g^{t}=\frac{1}{t}g_{V}\oplus
tg_{H}\\}_{t>0}$, and investigate the behavior of
${\widetilde{\vartheta}}^{t}_{b}$ (resp. $\vartheta^{t}_{b}$) when $t$ goes to
$\infty$.

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 Corollary 2.25. In Section 3, we discuss the
$\pi_{1}(B)$-equivariant Spinc quantization of $\left(\mathbb{R}^{n}\times
T^{n},\sum_{i=1}^{n}dx_{i}\wedge dy_{i}\right)\to\mathbb{R}^{n}$ with the
standard prequantum line bundle and prove Proposition 3.12. In Section 4, we
prove Theorem 1.2 step by step, and explain the relation between
$\vartheta^{t}_{b}$ and Jacobi’s classical theta function. Finally, in Section
5 we prove Theorem 1.1.

Acknowledgment The most part of this work had been 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. This work is supported by Grant-in-Aid for
Scientific Research (C) 15K04857.

###  1.1. Notations

For $x={}^{t}(x_{1},\ldots,x_{n})$ and
$y={}^{t}(y_{1},\ldots,y_{n})\in\mathbb{R}^{n}$, let us denote the standard
inner product $\sum_{i=1}^{n}x_{i}y_{i}$ by $x\cdot y$. $\partial_{x_{i}}$
denotes $\dfrac{\partial}{\partial x_{i}}$. In this paper, all manifolds and
maps are supposed to be smooth.

##  2\. Developing Lagrangian fibrations

###  2.1. Integral affine structures

Let $B$ be a manifold.

######  Definition 2.1.

An integral affine atlas of $B$ is an atlas $\\{(U_{\alpha},\phi_{\alpha})\\}$
of $B$ on each of whose non-empty overlap $U_{\alpha\beta}$, the transition
function $\phi_{\alpha}\circ\phi_{\beta}^{-1}$ is an integral affine
transformation, namely, $\phi_{\alpha}\circ\phi_{\beta}^{-1}$ is of the form
$\phi_{\alpha}\circ\phi_{\beta}^{-1}(x)=A_{\alpha\beta}x+c_{\alpha\beta}$ for
some locally constant maps $A_{\alpha\beta}\colon
U_{\alpha\beta}\to\operatorname{GL}_{n}(\mathbb{Z})$ and
$c_{\alpha\beta}\colon U_{\alpha\beta}\to\mathbb{R}^{n}$. Two integral affine
atlases $\\{(U_{\alpha},\phi_{\alpha})\\}$ and
$\\{(U^{\prime}_{\beta},\phi^{\prime}_{\beta})\\}$ of $B$ are said to be
equivalent if on each non-empty overlap $U_{\alpha}\cap U^{\prime}_{\beta}$,
the transition function $\phi_{\alpha}\circ(\phi^{\prime}_{\beta})^{-1}$ is an
integral affine transformation. 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 $\mathbb{R}^{n}$ is equipped with a natural
integral affine structure.

Let us give examples of integral affine manifolds obtained from integral
affine actions on $\mathbb{R}^{n}$.

######  Example 2.3.

(1) Let $v_{1},\ldots,v_{n}\in\mathbb{R}^{n}$ be a linear basis of
$\mathbb{R}^{n}$ and $C=(v_{1}\ \cdots\
v_{n})\in\operatorname{GL}_{n}(\mathbb{R})$ the matrix whose $i$th column
vector is $v_{i}$ for $i=1,\ldots,n$. $\mathbb{Z}^{n}$ acts on
$\mathbb{R}^{n}$ by

$\rho_{\gamma}(x):=x+C\gamma$

for $\gamma\in\mathbb{Z}^{n}$ and $x\in\mathbb{R}^{n}$. Since the action
preserves the natural integral affine structure on $\mathbb{R}^{n}$, the
quotient space, which is topologically $T^{n}$, is equipped with an integral
affine structure.
(2) Let $\lambda\in\mathbb{N}$ be a positive integer and
$a,b\in\mathbb{R}_{>0}$ positive real numbers. Define $\mathbb{Z}^{2}$-action
on $\mathbb{R}^{2}$ as follows. First, for the standard basis $e_{1},e_{2}$ of
$\mathbb{Z}^{2}$, let us define the integral affine transform $\rho_{e_{1}}$,
$\rho_{e_{2}}$ by

$\rho_{e_{1}}(x):=x+\begin{pmatrix}a\\\
0\end{pmatrix},\quad\rho_{e_{2}}(x):=\begin{pmatrix}1&\lambda\\\
0&1\end{pmatrix}x+\begin{pmatrix}0\\\ b\end{pmatrix}$

for $x\in\mathbb{R}^{2}$. Since $\rho_{e_{1}}$ and $\rho_{e_{2}}$ are
commutative, they form the $\mathbb{Z}^{2}$-action on $\mathbb{R}^{2}$ by

$\rho_{\gamma}(x):=\rho_{e_{1}}^{\gamma_{1}}\circ\rho_{e_{2}}^{\gamma_{2}}(x)$

for each $\gamma={}^{t}(\gamma_{1},\gamma_{2})\in\mathbb{Z}^{2}$. In the same
manner as in (1), the quotient space is equipped with an integral affine
structure. It is shown in [27, 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
$\gamma={}^{t}(\gamma_{1},\gamma_{2},\gamma_{3}),\gamma^{\prime}={}^{t}(\gamma^{\prime}_{1},\gamma^{\prime}_{2},\gamma^{\prime}_{3})\in\mathbb{Z}^{3}$
define the product $\gamma\circ\gamma^{\prime}\in\mathbb{Z}^{3}$ by

$\gamma\circ\gamma^{\prime}:=\begin{pmatrix}1&0&0\\\ 0&0&-1\\\
0&-1&0\end{pmatrix}^{\gamma_{1}}\gamma^{\prime}+\gamma.$

Then, $\mathbb{Z}^{3}$ with product $\circ$ is a non abelian group
$\left(\mathbb{Z}^{3},\circ\right)$. $\left(\mathbb{Z}^{3},\circ\right)$ acts
on $\mathbb{R}^{3}$ by

$\rho_{\gamma}(x):=\begin{pmatrix}1&0&0\\\ 0&0&-1\\\
0&-1&0\end{pmatrix}^{\gamma_{1}}x+\gamma.$

Then, the quotient space $\mathbb{R}^{3}/\left(\mathbb{Z}^{3},\circ\right)$ is
equipped with the integral affine structure induced from that of
$\mathbb{R}^{3}$.

######  Example 2.5.

Let $n\geq 2$. For
$\gamma={}^{t}(\gamma_{1},\ldots,\gamma_{n}),\gamma^{\prime}={}^{t}(\gamma^{\prime}_{1},\ldots,\gamma^{\prime}_{n})\in\mathbb{Z}^{n}$
define the product $\gamma\circ\gamma^{\prime}\in\mathbb{Z}^{n}$ by

$\gamma\circ\gamma^{\prime}:=\begin{pmatrix}1&&&\\\ &(-1)^{\gamma_{1}}&&\\\
&&\ddots&\\\ &&&(-1)^{\gamma_{n-1}}\end{pmatrix}\gamma^{\prime}+\gamma.$

Then, $\mathbb{Z}^{n}$ with product $\circ$ is a non abelian group
$\left(\mathbb{Z}^{n},\circ\right)$. $\left(\mathbb{Z}^{n},\circ\right)$ acts
on $\mathbb{R}^{n}$ by

$\rho_{\gamma}(x):=\begin{pmatrix}1&&&\\\ &(-1)^{\gamma_{1}}&&\\\ &&\ddots&\\\
&&&(-1)^{\gamma_{n-1}}\end{pmatrix}x+\gamma.$

Then, the quotient space $\mathbb{R}^{n}/\left(\mathbb{Z}^{n},\circ\right)$ is
equipped with the integral affine structure induced from that of
$\mathbb{R}^{n}$. For $n=2$, the quotient space is topologically a Klein
bottle.

######  Example 2.6.

Let $n\geq 2$ and $\lambda_{1},\ldots\lambda_{n-1}\in\mathbb{Z}$. For
$\gamma={}^{t}(\gamma_{1},\ldots,\gamma_{n}),\gamma^{\prime}={}^{t}(\gamma^{\prime}_{1},\ldots,\gamma^{\prime}_{n})\in\mathbb{Z}^{n}$
define the product $\gamma\circ\gamma^{\prime}\in\mathbb{Z}^{n}$ by

$\gamma\circ\gamma^{\prime}:=\begin{pmatrix}1&\lambda_{1}&&&\\\
&1&\lambda_{2}&&\\\ &&\ddots&\ddots&\\\ &&&1&\lambda_{n-1}\\\
&&&&1\end{pmatrix}^{\gamma_{n}}\gamma^{\prime}+\gamma.$

$\mathbb{Z}^{n}$ with product $\circ$ is a group
$\left(\mathbb{Z}^{n},\circ\right)$, which is non abelian for $n\geq 3$.
$\left(\mathbb{Z}^{n},\circ\right)$ acts on $\mathbb{R}^{n}$ by

$\rho_{\gamma}(x):=\begin{pmatrix}1&\lambda_{1}&&&\\\ &1&\lambda_{2}&&\\\
&&\ddots&\ddots&\\\ &&&1&\lambda_{n-1}\\\
&&&&1\end{pmatrix}^{\gamma_{n}}x+\gamma.$

Then, the quotient space $\mathbb{R}^{n}/\left(\mathbb{Z}^{n},\circ\right)$ is
equipped with the integral affine structure induced from that of
$\mathbb{R}^{n}$. In the case where $n=2$ and $\lambda_{1}>0$, it coincides
with the one given in Example 2.3 (2) with $a=b=1$.

######  Example 2.7.

Let $\mathbb{Z}/4\mathbb{Z}\cong\left\\{\pm\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\pm\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}\right\\}$ act on
$(\mathbb{R}^{2})^{n}\smallsetminus\\{0\\}$ naturally. Then, the quotient
space is a non-compact manifold and equipped with the integral affine
structure induced from that of $(\mathbb{R}^{2})^{n}\smallsetminus\\{0\\}$.

As we can guess from above examples, every integral affine manifold is
obtained from a group action. Let $B$ be an $n$-dimensional connected integral
affine manifold, $p\colon\widetilde{B}\to B$ the universal covering of $B$. It
is clear that $\widetilde{B}$ is also equipped with the integral affine
structure so that $p$ is an integral affine map. We set $\Gamma:=\pi_{1}(B)$.
$\Gamma$ acts on $\widetilde{B}$ from the right as a deck transformation. For
each $\gamma\in\Gamma$ we denote by $\sigma_{\gamma}$ the inverse of the deck
transformation corresponding to $\gamma$. Then,
$\sigma\colon\gamma\mapsto\sigma_{\gamma}$ defines a left action
$\sigma\in\operatorname{Hom}\left(\Gamma,\operatorname{Aut}(\widetilde{B})\right)$.

We assume that all the actions considered in this paper are left actions
unless otherwise stated.

The following proposition is well known in affine geometry.

######  Proposition 2.8.

There exists an integral affine immersion
$\operatorname{dev}\colon\widetilde{B}\to\mathbb{R}^{n}$ and a homomorphism
$\rho\colon\Gamma\to GL_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$ such that the
image of $\operatorname{dev}$ is an open set of $\mathbb{R}^{n}$ and
$\operatorname{dev}$ is equivariant with respect to $\sigma$ and $\rho$. Such
an integral affine immersion is unique up to the composition of an integral
affine transformation on $\mathbb{R}^{n}$.

See [15, p.641] for a proof. We will prove a version of this proposition
(Proposition 2.22) when B is equipped with a Lagrangian fibration on it in
Section 2.

######  Proposition 2.9.

Let $B$, $p\colon\widetilde{B}\to B$,
$\operatorname{dev}\colon\widetilde{B}\to\mathbb{R}^{n}$, and
$\rho\colon\Gamma\to GL_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$ be as in
Proposition 2.8. Suppose that $B$ is compact and the $\Gamma$-action $\rho$ on
$\mathbb{R}^{n}$ is properly discontinuous. Then, $\operatorname{dev}$ is
surjective.

###### Proof.

We denote the image of $\operatorname{dev}$ by $O$. By proposition 2.8, $O$ is
an open set in $\mathbb{R}^{n}$. So, it is sufficient to show that $O$ is
closed in $\mathbb{R}^{n}$. Since the $\Gamma$-action $\rho$ on
$\mathbb{R}^{n}$ is properly discontinuous, the quotient space
$\mathbb{R}^{n}/\Gamma$ becomes a Hausdorff space and the natural projection
$q\colon\mathbb{R}^{n}\to\mathbb{R}^{n}/\Gamma$ is continuous. $O$ is
preserved by the $\Gamma$-action $\rho$ on $\mathbb{R}^{n}$ since
$\operatorname{dev}$ is $\Gamma$-equivariant. Then, $\operatorname{dev}$
induces a continuous surjective map ${\overline{\operatorname{dev}}}\colon
B={\widetilde{B}}/\Gamma\to O/\Gamma$. Since $B$ is compact, $O/\Gamma$ is a
compact subset in the Hausdorff space $\mathbb{R}^{n}/\Gamma$. In particular,
it is also closed. Hence, $O=q^{-1}\left(O/\Gamma\right)$ is also closed in
$\mathbb{R}^{n}$. ∎

######  Corollary 2.10.

Let $B$, $p\colon\widetilde{B}\to B$,
$\operatorname{dev}\colon\widetilde{B}\to\mathbb{R}^{n}$, and
$\rho\colon\Gamma\to GL_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$ be as in
Proposition 2.8 and assume that $B$ compact. If the image of $\rho$ lies in
$\left(\operatorname{GL}_{n}(\mathbb{Z})\cap O(n)\right)\ltimes\mathbb{R}^{n}$
and the subgroup $\rho(\Gamma)$ of
$\left(\operatorname{GL}_{n}(\mathbb{Z})\cap O(n)\right)\ltimes\mathbb{R}^{n}$
is discrete, then, $\operatorname{dev}$ is surjective.

###### Proof.

It follows from [37, Theorem 3.1.3]. ∎

######  Definition 2.11.

The integral affine immersion $\operatorname{dev}$ is called a developing map.
$B$ is said to be complete if $\operatorname{dev}$ is bijective. $B$ is said
to be incomplete if $B$ is not complete.

######  Example 2.12.

All the above examples are complete other than Example 2.7 for $n\geq 2$.

######  Example 2.13.

Let $B$ be an $n$-dimensional compact integral affine manifold $B$ with
integral affine atlas $\\{(U_{\alpha},\phi_{\alpha})\\}$ as in Definition 2.1.
If on each non-empty overlap $U_{\alpha\beta}$, the Jacobi matrix of the
coordinate changing map $D\left(\phi_{\alpha}\circ\phi_{\beta}^{-1}\right)$
lies in $GL_{n}(\mathbb{Z})\cap O(n)$, then, $B$ has a flat Riemannian metric.
Hence, by Bieberbach’s theorem [5, 6], $B$ is finitely covered by $T^{n}$. In
particular, $B$ is complete. For flat Riemannian manifolds, see [37, 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 Duistermaat. For more details, see [10, 39].

Let $(M,\omega)$ be a symplectic manifold.

######  Definition 2.14.

A map $\pi\colon(M,\omega)\to B$ from $(M,\omega)$ to a manifold $B$ is called
a Lagrangian fibration if $\pi$ is a fiber bundle whose fiber is a Lagrangian
submanifold of $(M,\omega)$.

######  Example 2.15.

Let $T^{n}=(\mathbb{R}/\mathbb{Z})^{n}$ be an $n$-dimensional torus.
$\mathbb{R}^{n}\times T^{n}$ admits a standard symplectic structure
$\omega_{0}=\sum_{i}dx_{i}\wedge dy_{i}$, where $(x_{1},\ldots,x_{n})$,
$(y_{1},\ldots,y_{n})$ are the coordinates of $\mathbb{R}^{n}$, $T^{n}$,
respectively. Then, the projection $\pi_{0}\colon(\mathbb{R}^{n}\times
T^{n},\omega_{0})\to\mathbb{R}^{n}$ to $\mathbb{R}^{n}$ is a Lagrangian
fibration.

The following theorem shows that Example 2.15 is a local model of Lagrangian
fibration.

######  Theorem 2.16 (Arnold-Liouville’s theorem [2]).

Let $\pi\colon(M,\omega)\to B$ be a Lagrangian fibration with compact, path-
connected fibers. Then, for each $b\in B$, there exists a chart $(U,\phi)$
containing $b$ and a symplectomorphism $\varphi\colon(\pi^{-1}(U)$,
$\omega|_{\pi^{-1}(U)})\to\left(\phi(U)\times T^{n},\omega_{0}\right)$ such
that the following diagram commutes

$\textstyle{\left(\pi^{-1}(U),\omega|_{\pi^{-1}(U)}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\varphi}$$\textstyle{\left(\phi(U)\times
T^{n},\omega_{0}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\phi(U).}$

In the rest of this paper we assume that every Lagrangian fibration has
compact, path-connected fibers.

Now we investigate automorphisms of the local model. By the direct computation
shows the following lemma. See also [35, Lemma 2.5].

######  Lemma 2.17.

Let $\varphi\colon(\mathbb{R}^{n}\times
T^{n},\omega_{0})\to(\mathbb{R}^{n}\times T^{n},\omega_{0})$ be a fiber-
preserving symplectomorphism of $\pi_{0}\colon(\mathbb{R}^{n}\times
T^{n},\omega_{0})\to\mathbb{R}^{n}$ which covers a map $\phi\colon
R^{n}\to\mathbb{R}^{n}$. Then, there exists a matrix
$A\in\operatorname{GL}_{n}(\mathbb{Z})$, a constant $c\in\mathbb{R}^{n}$, and
a map $u\colon\mathbb{R}^{n}\to T^{n}$ with ${}^{t}ADu$ symmetric such that
$\varphi$ is written as

$\varphi(x,y)=\left(Ax+c,{}^{t}A^{-1}y+u(x)\right)$

for any $(x,y)\in\mathbb{R}^{n}\times T^{n}$, where $Du$ is the Jacobi matrix
of $u$.

By Theorem 2.16 and Lemma 2.17 we can obtain the following proposition.

######  Proposition 2.18.

Let $\pi\colon(M,\omega)\to B$ be a Lagrangian fibration. Then, there exists
an atlas $\\{(U_{\alpha},\phi_{\alpha})\\}_{\alpha\in A}$ of $B$ and for each
$\alpha\in A$ there exists a symplectomorphism
$\varphi_{\alpha}\colon\left(\pi^{-1}(U_{\alpha}),\omega|_{\pi^{-1}(U_{\alpha})}\right)\to\left(\phi_{\alpha}(U_{\alpha})\times
T^{n},\omega_{0}\right)$ such that the following diagram commutes

$\textstyle{(\pi^{-1}(U_{\alpha}),\omega|_{\pi^{-1}(U_{\alpha})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\varphi_{\alpha}}$$\textstyle{(\phi_{\alpha}(U_{\alpha})\times
T^{n},\omega_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\textstyle{U_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\alpha}}$$\textstyle{\phi_{\alpha}(U_{\alpha}).}$

Moreover, on each non-empty overlap $U_{\alpha\beta}:=U_{\alpha}\cap
U_{\beta}$ there exist locally constant maps $A_{\alpha\beta}\colon
U_{\alpha\beta}\to\operatorname{GL}_{n}(\mathbb{Z})$, $c_{\alpha\beta}\colon
U_{\alpha\beta}\to\mathbb{R}^{n}$, and a map $u_{\alpha\beta}\colon
U_{\alpha\beta}\to T^{n}$ with
${}^{t}A_{\alpha\beta}D\left(u_{\alpha\beta}\circ\phi_{\beta}^{-1}\right)$
symmetric such that the overlap map is written as

(2.1)
$\varphi_{\alpha}\circ\varphi_{\beta}^{-1}(x,y)=\left(A_{\alpha\beta}x+c_{\alpha\beta},{}^{t}A^{-1}_{\alpha\beta}y+u_{\alpha\beta}\circ\phi_{\beta}^{-1}(x)\right)$

for any $(x,y)\in\phi_{\beta}(U_{\alpha\beta})\times T^{n}$.

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
$\\{(U_{\alpha},\phi_{\alpha})\\}_{\alpha\in A}$ be an integral affine atlas
of $B$. Then, we can construct a Lagrangian fibration on $B$ in the following
way. For each $\alpha\in A$ let $\overline{\phi}_{\alpha}\colon
T^{*}B|_{U_{\alpha}}\to\phi_{\alpha}(U_{\alpha})\times\mathbb{R}^{n}$ be the
local trivialization of the cotangent bundle $T^{*}B$ induced from
$(U_{\alpha},\phi_{\alpha})$. On each nonempty overlap $U_{\alpha\beta}$,
suppose that $\phi_{\alpha}\circ\phi_{\beta}^{-1}$ is written by
$\phi_{\alpha}\circ\phi_{\beta}^{-1}(x)=A_{\alpha\beta}x+c_{\alpha\beta}$ as
in Definition 2.1. Then, the overlap map is written as

(2.2)
$\overline{\phi}_{\alpha}\circ(\overline{\phi}_{\beta})^{-1}(x,y)=(A_{\alpha\beta}x+c_{\alpha\beta},{}^{t}A^{-1}_{\alpha\beta}y).$

Since $A_{\alpha\beta}$ lies in $GL_{n}(\mathbb{Z})$, (2.2) preserves the
integer lattice $\mathbb{Z}^{n}$ of the fiber $\mathbb{R}^{n}$, hence, induces
the fiber-preserving symplectomorphism from
$\pi_{0}\colon\left(\phi_{\beta}(U_{\alpha\beta})\times
T^{n},\omega_{0}\right)\to\phi_{\beta}(U_{\alpha\beta})$ to
$\pi_{0}\colon\left(\phi_{\alpha}(U_{\alpha\beta})\times
T^{n},\omega_{0}\right)\to\phi_{\alpha}(U_{\alpha\beta})$. Then, the
Lagrangian fibrations $\\{\pi_{0}\colon\left(\phi_{\alpha}(U_{\alpha})\times
T^{n},\omega_{0}\right)\to\phi_{\alpha}(U_{\alpha})\\}_{\alpha\in A}$ are
patched together by the symplectomorphisms to form a Lagrangian fibration
$\pi_{T}\colon(T^{*}B_{T},\omega_{T^{*}B_{T}})\to B$, namely

$(T^{*}B_{T},\omega_{T^{*}B_{T}}):=\coprod_{\alpha\in
A}\left(\phi_{\alpha}(U_{\alpha})\times T^{n},\omega_{0}\right)/\sim$

and

$\pi_{T}([x_{\alpha},y_{\alpha}]):=\phi_{\alpha}^{-1}(x_{\alpha})$

for $(x_{\alpha},y_{\alpha})\in\phi_{\alpha}(U_{\alpha})\times T^{n}$. This
construction does not depend on the choice of equivalent integral affine
structures and depends only on the integral affine structure on $B$. We call
$\pi_{T}\colon(T^{*}B_{T},\omega_{T^{*}B_{T}})\to B$ the canonical model.

We summarize the above argument to the following proposition.

######  Proposition 2.19.

Let $B$ be a manifold. $B$ is a base space of a Lagrangian fibration if and
only if $B$ admits an integral affine structure.

Let us give a classification theorem of Lagrangian fibrations in the required
form in this paper. Let $\pi\colon(M,\omega)\to B$ be a Lagrangian fibration.
Then, $B$ has an integral affine structure by Proposition 2.19. We take and
fix an integral affine atlas $\\{(U_{\alpha},\phi_{\alpha})\\}_{\alpha\in A}$
on $B$ and let $\pi_{T}\colon(T^{*}B_{T},\omega_{T^{*}B_{T}})\to B$ be the
canonical model associated with the integral affine structure on $B$. On each
$U_{\alpha}$, let
$\varphi_{\alpha}\colon\left(\pi^{-1}(U_{\alpha}),\omega|_{\pi^{-1}(U_{\alpha})}\right)\to\left(\phi_{\alpha}(U_{\alpha})\times
T^{n},\omega_{0}\right)$ be a local trivialization of $\pi\colon(M,\omega)\to
B$ as in Proposition 2.18, and
$\overline{\phi}_{\alpha}\colon\left(\pi_{T}^{-1}(U_{\alpha}),\omega_{T^{*}B_{T}}\right)\to\left(\phi_{\alpha}(U_{\alpha})\times
T^{n},\omega_{0}\right)$ be the local trivialization of
$\pi_{T}\colon(T^{*}B_{T},\omega_{T^{*}B_{T}})\to B$ naturally induced from
$(U_{\alpha},\phi_{\alpha})$ as explained above.111Here we use the same
notation as the local trivialization of $T^{*}B$ because we have no confusion.
Then their composition

$h_{\alpha}:={\overline{\phi}_{\alpha}}^{-1}\circ\varphi_{\alpha}\colon\left(\pi^{-1}(U_{\alpha}),\omega|_{\pi^{-1}(U_{\alpha})}\right)\to\left(\pi_{T}^{-1}(U_{\alpha}),\omega_{T^{*}B_{T}}\right)$

gives a local identification between them.

On each $U_{\alpha}\cap U_{\beta}$, suppose that
$\varphi_{\alpha}\circ\varphi_{\beta}^{-1}$ is written as in (2.1). Then,
$h_{\alpha}\circ h_{\beta}^{-1}$ can be written as

$h_{\alpha}\circ
h_{\beta}^{-1}(p)={\overline{\phi}_{\alpha}}^{-1}\left(A_{\alpha\beta}x+c_{\alpha\beta},{}^{t}A_{\alpha\beta}^{-1}y+u_{\alpha\beta}(\pi(p))\right),$

where $\overline{\phi}_{\beta}(p)=(x,y)$. $u_{\alpha\beta}$ induces the local
section $\widetilde{u}_{\alpha\beta}$ of
$\pi_{T}\colon(T^{*}B_{T},\omega_{T^{*}B_{T}})\to B$ on $U_{\alpha\beta}$ by

$\widetilde{u}_{\alpha\beta}(b):=\left[\phi_{\alpha}(b),u_{\alpha\beta}(b)\right]$

for $b\in U_{\alpha\beta}$. It is easy to see that
$\widetilde{u}_{\alpha\beta}$ satisfies
$\widetilde{u}_{\alpha\beta}^{*}\omega_{T^{*}B_{T}}=0$. A section with this
condition is said to be Lagrangian.

Let $\mathscr{S}$ be the sheaf of germs of Lagrangian section of
$\pi_{T}\colon(T^{*}B_{T},\omega_{T^{*}B_{T}})\to B$. $\mathscr{S}$ is the
sheaf of Abelian groups since the fiber of
$\pi_{T}\colon(T^{*}B_{T},\omega_{T^{*}B_{T}})\to B$ has the structure of an
Abelian group by construction. By definition
$\\{\widetilde{u}_{\alpha\beta}\\}$ forms a Čech one-cocycle on $B$ with
coefficients in $\mathscr{S}$. The cohomology class determined by
$\\{\widetilde{u}_{\alpha\beta}\\}$ does not depend on the choice of a
specific integral affine structure and depends only on $\pi\colon(M,\omega)\to
B$. We denote the cohomology class by $u\in H^{1}(B;\mathscr{S})$. $u$ is
called the Chern class of $\pi\colon(M,\omega)\to B$ in [10].

Lagrangian fibrations on the same integral affine manifold are classified with
the Chern classes.

######  Theorem 2.20 ([10]).

For two Lagrangian fibrations $\pi_{1}\colon(M_{1},\omega_{1})\to B$ and
$\pi_{2}\colon\\\ (M_{2},\omega_{2})\to B$ on the same integral affine
manifold $B$, there exists a fiber-preserving symplectomorphism between them
which covers the identity if and only if their Chern classes $u_{1}$ and
$u_{2}$ agree with each other. Moreover, if an integral affine manifold $B$
and the cohomology class $u\in H^{1}(B;\mathscr{S})$ are given, then, there
exists a Lagrangian fibration $\pi\colon(M,\omega)\to B$ that realizes them.

######  Remark 2.21.

By the construction of $u$, there exists a fiber-preserving symplectomorphism
between $\pi\colon(M,\omega)\to B$ and
$\pi_{T}\colon(T^{*}B_{T},\omega_{T^{*}B_{T}})\to B$ that covers the identity
of $B$ if and only if $u$ vanishes. Since
$\pi_{T}\colon(T^{*}B_{T},\omega_{T^{*}B_{T}})\to B$ has the zero section
which is Lagrangian, $u$ is the obstruction class in order that
$\pi\colon(M,\omega)\to B$ posses a Lagrangian section. In particular, any
Lagrangian fibration with Lagrangian section is identified with the canonical
model.

###  2.3. Lagrangian fibrations with complete bases

Let $\pi\colon(M,\omega)\to B$ be a Lagrangian fibration with $n$-dimensional
connected base manifold $B$, $p\colon\widetilde{B}\to B$ the universal
covering of $B$. We denote by
$\widetilde{\pi}\colon(\widetilde{M},\widetilde{\omega})\to\widetilde{B}$ the
pullback of $\pi\colon(M,\omega)\to B$ to $\widetilde{B}$. Let $\Gamma$ be the
fundamental group of $B$ and
$\sigma\in\operatorname{Hom}\left(\Gamma,\operatorname{Aut}(\widetilde{B})\right)$
the action of $\Gamma$ defined as the inverse of the deck transformation as in
Proposition 2.8. By definition, $\widetilde{M}$ admits a natural lift of
$\sigma$ which preserves $\widetilde{\omega}$. The $\Gamma$-action on
$(\widetilde{M},\widetilde{\omega})$ is denoted by $\widetilde{\sigma}$. By
Proposition 2.8 we have a developing map
$\operatorname{dev}\colon\widetilde{B}\to\mathbb{R}^{n}$ and the homomorphism
$\rho\colon\Gamma\to\operatorname{GL}_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$.
We denote the image of $\operatorname{dev}$ by $O$. Note that the
$\Gamma$-action $\rho$ on $\mathbb{R}^{n}$ preserves $O$ since
$\operatorname{dev}$ is $\Gamma$-equivariant.

######  Proposition 2.22.

There exists a Lagrangian fibration
$\pi^{\prime}\colon(M^{\prime},\omega^{\prime})\to O$, a fiber-preserving
symplectic immersion
$\widetilde{\operatorname{dev}}\colon(\widetilde{M},\widetilde{\omega})\to(M^{\prime},\omega^{\prime})$
which covers $\operatorname{dev}$, and a lift $\widetilde{\rho}$ of the
$\Gamma$-action $\rho$ on $O$ to $(M^{\prime},\omega^{\prime})$ such that
$\widetilde{\operatorname{dev}}$ is $\Gamma$-equivariant with respect to
$\widetilde{\sigma}$ and $\widetilde{\rho}$.

###### Proof.

By Proposition 2.19 $B$ admits an integral affine structure determined by
$\pi$, and it also induces the integral affine structure on $\widetilde{B}$.
Let $\\{(U_{\alpha},\phi_{\alpha}^{\prime\prime})\\}$ be the integral affine
atlas of $\widetilde{B}$ and
$\\{(\widetilde{\pi}^{-1}(U_{\alpha}),\omega|_{\widetilde{\pi}^{-1}(U_{\alpha})},\varphi^{\prime\prime}_{\alpha})\\}$
the local trivializations of
$\widetilde{\pi}\colon(\widetilde{M},\widetilde{\omega})\to\widetilde{B}$ as
in Proposition 2.18 so that on each non-empty overlap $U_{\alpha\beta}$, there
exist locally constant maps $A_{\alpha\beta}\colon
U_{\alpha\beta}\to\operatorname{GL}_{n}(\mathbb{Z})$ and
$c^{\prime}_{\alpha\beta}\colon U_{\alpha\beta}\to\mathbb{R}^{n}$, and a map
$u^{\prime}_{\alpha\beta}\colon U_{\alpha\beta}\to T^{n}$ with
${}^{t}A_{\alpha\beta}D\left(u^{\prime}_{\alpha\beta}\circ(\phi^{\prime\prime}_{\beta})^{-1}\right)$
symmetric such that
$\varphi^{\prime\prime}_{\alpha}\circ(\varphi^{\prime\prime}_{\beta})^{-1}$ is
written as in (2.1). Then, $A_{\alpha\beta}$’s form a Čech one-cocycle
$\\{A_{\alpha\beta}\\}\in
C^{1}(\\{U_{\alpha}\\};\operatorname{GL}_{n}(\mathbb{Z}))$ and defines a
cohomology class $[\\{A_{\alpha\beta}\\}]\in
H^{1}(\widetilde{B};GL_{n}(\mathbb{Z}))$. It is well known that
$H^{1}(\widetilde{B};GL_{n}(\mathbb{Z}))$ is identified with the moduli space
of homomorphisms from $\pi_{1}(\widetilde{B})$ to $GL_{n}(\mathbb{Z})$. Since
$\pi_{1}(\widetilde{B})$ is trivial, there exists a Čech zero-cocycle
$\\{A_{\alpha}\\}\in
C^{0}(\\{U_{\alpha}\\};\operatorname{GL}_{n}(\mathbb{Z}))$ such that
$A_{\alpha\beta}=A_{\alpha}A_{\beta}^{-1}$ on each $U_{\alpha\beta}$. By using
the cocycle we modify the local trivializations
$\\{(\widetilde{\pi}^{-1}(U_{\alpha}),\omega|_{\widetilde{\pi}^{-1}(U_{\alpha})},\varphi^{\prime\prime}_{\alpha})\\}$
and the integral affine atlas
$\\{(U_{\alpha},\phi_{\alpha}^{\prime\prime})\\}$ by replacing
$\varphi^{\prime\prime}_{\alpha}$, $\phi^{\prime\prime}_{\alpha}$ by

$\varphi^{\prime}_{\alpha}(\widetilde{p}):=(A_{\alpha}^{-1}\times{}^{t}A_{\alpha})\circ\phi_{\alpha}^{\prime\prime}(\widetilde{p}),\quad\phi^{\prime}_{\alpha}:=A_{\alpha}^{-1}\phi^{\prime\prime}_{\alpha}$

for each $\alpha\in A$, respectively. Then, on each $U_{\alpha\beta}$,
$\varphi^{\prime}_{\alpha}\circ(\varphi^{\prime}_{\beta})^{-1}$ is written as

$\varphi^{\prime}_{\alpha}\circ(\varphi^{\prime}_{\beta})^{-1}(\widetilde{x},y)=\left(\widetilde{x}+c_{\alpha\beta},y+u_{\alpha\beta}\circ(\phi^{\prime}_{\beta})^{-1}(\widetilde{x})\right),$

where we set $c_{\alpha\beta}:=A_{\alpha}^{-1}c^{\prime}_{\alpha\beta}$ and
$u_{\alpha\beta}:={}^{t}A_{\alpha}u^{\prime}_{\alpha\beta}$. Then,
$c_{\alpha\beta}$’s form a Čech one-cocycle $\\{c_{\alpha\beta}\\}\in
C^{1}(\\{U_{\alpha}\\};\mathbb{R}^{n})$ and defines a cohomology class
$[\\{c_{\alpha\beta}\\}]\in H^{1}(\widetilde{B};\mathbb{R}^{n})$. By the
universal coefficients theorem, $H^{1}(\widetilde{B};\mathbb{R}^{n})$ is
identified with
$\operatorname{Hom}(H_{1}(\widetilde{B};\mathbb{Z}),\mathbb{R}^{n})$, which is
trivial. So there exists a Čech zero-cocycle $\\{c_{\alpha}\\}\in
C^{0}(\\{U_{\alpha}\\};\mathbb{R}^{n})$ such that
$c_{\alpha\beta}=c_{\alpha}-c_{\beta}$ on each $U_{\alpha\beta}$. By using the
cocycle, we again modify
$\\{(\widetilde{\pi}^{-1}(U_{\alpha}),\omega|_{\widetilde{\pi}^{-1}(U_{\alpha})},\varphi^{\prime}_{\alpha})\\}$
and $\\{(U_{\alpha},\phi^{\prime}_{\alpha})\\}$ by replacing
$\varphi^{\prime}_{\alpha}$, $\phi_{\alpha}^{\prime}$ by

$\varphi_{\alpha}(\widetilde{p}):=\varphi^{\prime}_{\alpha}(\widetilde{p})-(c_{\alpha},0),\quad\phi_{\alpha}(\widetilde{b}):=\phi_{\alpha}^{\prime}(\widetilde{b})-c_{\alpha},$

respectively for each $\alpha\in A$. Then, on each $U_{\alpha\beta}$,
$\phi_{\alpha}$ coincides with $\phi_{\beta}$ and
$\varphi_{\alpha}\circ\varphi_{\beta}^{-1}$ is written as

$\varphi_{\alpha}\circ\varphi_{\beta}^{-1}(\widetilde{x},y)=(\widetilde{x},y+u_{\alpha\beta}\circ\phi_{\beta}^{-1}(\widetilde{x})).$

Now we define the map $\operatorname{dev}\colon\widetilde{B}\to\mathbb{R}^{n}$
by

$\operatorname{dev}(\widetilde{b}):=\phi_{\alpha}(\widetilde{b})$

if $\widetilde{b}$ lies in $U_{\alpha}$. It is well defined, and by
construction, it is an integral affine immersion whose image is
$\displaystyle\cup_{\alpha\in A}\phi_{\alpha}(U_{\alpha})$.
$(M^{\prime},\omega^{\prime})$ is defined by

$(M^{\prime},\omega^{\prime}):=\coprod_{\alpha\in
A}\left(\phi_{\alpha}(U_{\alpha})\times T^{n},\omega_{0}\right)/\sim,$

where $(x_{\alpha},y_{\alpha})\in\phi_{\alpha}(U_{\alpha})\times T^{n}$ and
$(x_{\beta},y_{\beta})\in\phi_{\beta}(U_{\beta})\times T^{n}$ are in the
relation $(x_{\alpha},y_{\alpha})\sim(x_{\beta},y_{\beta})$ if they satisfy
$(x_{\alpha},y_{\alpha})=\varphi_{\alpha}\circ\varphi_{\beta}^{-1}(x_{\beta},y_{\beta})$,
and $\pi^{\prime}\colon(M^{\prime},\omega^{\prime})\to O$ is defined to be the
first projection.
$\widetilde{\operatorname{dev}}\colon(\widetilde{M},\widetilde{\omega})\to(M^{\prime},\omega^{\prime})$
is defined by

$\widetilde{\operatorname{dev}}(\widetilde{p}):=[\varphi_{\alpha}(\widetilde{p})]$

if $\widetilde{p}$ lies in $\widetilde{\pi}^{-1}(U_{\alpha})$.

Without loss of generality, we can assume that each $U_{\alpha}$ is
connected,and for each $\gamma\in\Gamma$ and $\alpha\in A$ there uniquely
exists $\alpha^{\prime}\in A$ such that the deck transformation
$\sigma_{\gamma}$ maps $U_{\alpha}$ onto $U_{\alpha^{\prime}}$. Then, its lift
$\widetilde{\sigma}_{\gamma}$ to $(\widetilde{M},\widetilde{\omega})$ maps
$\widetilde{\pi}^{-1}(U_{\alpha})$ to
$\widetilde{\pi}^{-1}(U_{\alpha^{\prime}})$. By Lemma 2.17,
$\phi_{\alpha^{\prime}}\circ\sigma_{\gamma}\circ\phi_{\alpha}^{-1}$ can be
written as

$\phi_{\alpha^{\prime}}\circ\sigma_{\gamma}\circ\phi_{\alpha}^{-1}(\widetilde{x})=A_{\gamma}^{\alpha^{\prime}\alpha}\widetilde{x}+c_{\gamma}^{\alpha^{\prime}\alpha}$

for some
$A_{\gamma}^{\alpha^{\prime}\alpha}\in\operatorname{GL}_{n}(\mathbb{Z})$,
$c_{\gamma}^{\alpha^{\prime}\alpha}\in\mathbb{R}^{n}$. Since $\phi_{\alpha}$
coincides with $\phi_{\beta}$ on each overlap $U_{\alpha\beta}$,
$\phi_{\alpha^{\prime}}\circ\phi_{\gamma}\circ\phi_{\alpha}(\widetilde{x})=A_{\gamma}^{\alpha^{\prime}\alpha}\widetilde{x}+c_{\gamma}^{\alpha^{\prime}\alpha}$
also agrees with
$\phi_{\beta^{\prime}}\circ\phi_{\gamma}\circ\phi_{\beta}(\widetilde{x})=A_{\gamma}^{\beta^{\prime}\beta}\widetilde{x}+c_{\gamma}^{\beta^{\prime}\beta}$
on the overlap $\phi_{\alpha}(U_{\alpha\beta})=\phi_{\beta}(U_{\alpha\beta})$.
This implies $A_{\gamma}^{\alpha^{\prime}\alpha}$’s and
$c_{\gamma}^{\alpha^{\prime}\alpha}$’s do not depend on $\alpha$ and depends
only on $\gamma$. In fact, for each $\gamma\in\Gamma$ and $\alpha_{0}\in A$,
we set

$A_{0}:=\\{\alpha\in A\mid
A_{\gamma}^{\alpha^{\prime}_{0}\alpha_{0}}=A_{\gamma}^{\alpha^{\prime}\alpha}\text{
and
}c_{\gamma}^{\alpha^{\prime}_{0}\alpha_{0}}=c_{\gamma}^{\alpha^{\prime}\alpha}\\}.$

$A_{0}$ contains all $\beta\in A$ with $U_{\alpha_{0}\beta}\not=\emptyset$. In
particular, $A_{0}$ is not empty since $\alpha_{0}\in A_{0}$. Then, we have

$\left(\cup_{\alpha\in A_{0}}U_{\alpha}\right)\cup\left(\cup_{\alpha\in
A\smallsetminus
A_{0}}U_{\alpha}\right)=\widetilde{B},\quad\left(\cup_{\alpha\in
A_{0}}U_{\alpha}\right)\cap\left(\cup_{\alpha\in A\smallsetminus
A_{0}}U_{\alpha}\right)=\emptyset.$

If the compliment $A\smallsetminus A_{0}$ is not empty, this contradicts to
the connectedness of $\widetilde{B}$. So we denote them by $A_{\gamma}$ and
$c_{\gamma}$, respectively. Thus, we define the homomorphism
$\rho:\Gamma\to\operatorname{GL}_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$ by

$\rho_{\gamma}:=(A_{\gamma},c_{\gamma}).$

$\Gamma$ acts on $\mathbb{R}^{n}$ by $\rho_{\gamma}(x)=A_{\gamma}x+c_{\gamma}$
for $\gamma\in\Gamma$ and $x\in\mathbb{R}^{n}$. The lift
$\widetilde{\rho}_{\gamma}$ of $\rho_{\gamma}$ to
$(M^{\prime},\omega^{\prime})$ is defined by

$\widetilde{\rho}_{\gamma}\left([x_{\alpha},y_{\alpha}]\right):=[\varphi_{\alpha^{\prime}}\circ\widetilde{\sigma}_{\gamma}\circ\varphi_{\alpha}^{-1}(x_{\alpha},y_{\alpha})]$

if $(x_{\alpha},y_{\alpha})$ lies in $\phi_{\alpha}(U_{\alpha})\times T^{n}$.
By construction, $\widetilde{\rho}$ is a lift of $\rho$, and
$\widetilde{\rho}$ and $\rho$ satisfy
$\widetilde{\operatorname{dev}}(\widetilde{\sigma}_{\gamma}(\widetilde{p}))=\widetilde{\rho}_{\gamma}(\widetilde{\operatorname{dev}}(\widetilde{p}))$
and
$\operatorname{dev}(\sigma_{\gamma}(\widetilde{b}))=\rho_{\gamma}(\operatorname{dev}(\widetilde{b}))$,
respectively. ∎

######  Remark 2.23.

(1) By construction, the $n$-dimensional torus $T^{n}$ acts freely on
$M^{\prime}$ preserving $\omega^{\prime}$ from the right so that
$\pi^{\prime}\colon M^{\prime}\to O$ is a principal $T^{n}$-bundle.
(2) When $\pi\colon(M,\omega)\to B$ admits a Lagrangian section, the
restriction of $\pi_{0}\colon(\mathbb{R}^{n}\times
T^{n},\omega_{0})\to\mathbb{R}^{n}$ to $O$ can be taken as
$\pi^{\prime}\colon(M^{\prime},\omega^{\prime})\to O$. In fact, in this case,
since $\pi\colon(M,\omega)\to B$ is identified with the canonical model, we
can take a system of local trivializations
$\\{(\pi^{-1}(U_{\alpha}),\varphi_{\alpha})\\}$ with $u_{\alpha\beta}=0$ on
each overlaps $U_{\alpha\beta}$. By applying the construction of
$\pi^{\prime}\colon(M^{\prime},\omega^{\prime})\to O$ given in the proof of
Proposition 2.22 to such a $\\{(\pi^{-1}(U_{\alpha}),\varphi_{\alpha})\\}$ we
can show the claim.

Suppose that $(M,\omega)$ is prequantizable and let
$(L,\nabla^{L})\to(M,\omega)$ be a prequantum line bundle. We denote by
$(\widetilde{L},\nabla^{\widetilde{L}})\to(\widetilde{M},\widetilde{\omega})$
the pullback of $(L,\nabla^{L})\to(M,\omega)$ to
$(\widetilde{M},\widetilde{\omega})$. By definition, $\widetilde{L}$ admits a
natural lift of the $\Gamma$-action $\widetilde{\sigma}$ on
$(\widetilde{M},\widetilde{\omega})$ which preserves $\nabla^{\widetilde{L}}$.
The $\Gamma$-action on $(\widetilde{L},\nabla^{\widetilde{L}})$ is denoted by
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\sigma}$}}$. Then, we have
the following prequantum version of Proposition 2.22.

######  Proposition 2.24.

There exists a prequantum line bundle
$(L^{\prime},\nabla^{L^{\prime}})\to(M^{\prime},\omega^{\prime})$, a bundle
immersion
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\operatorname{dev}}$}}\colon(\widetilde{L},\nabla^{\widetilde{L}})\to(L^{\prime},\nabla^{L^{\prime}})$
which covers $\widetilde{\operatorname{dev}}$, and a lift
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}$ of the
$\Gamma$-action $\widetilde{\rho}$ on $(M^{\prime},\omega^{\prime})$ to
$(L^{\prime},\nabla^{L^{\prime}})$ such that
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\operatorname{dev}}$}}$ is
equivariant with respect to
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\sigma}$}}$ and
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}$.

###### Proof.

Let $\\{(U_{\alpha},\phi_{\alpha})\\}$ and
$\\{(\widetilde{\pi}^{-1}(U_{\alpha}),\omega|_{\widetilde{\pi}^{-1}(U_{\alpha})},\varphi_{\alpha})\\}$
be the integral affine atlas of $\widetilde{B}$ and the local trivializations
of $\widetilde{\pi}\colon(\widetilde{M},\widetilde{\omega})\to\widetilde{B}$
obtained in the proof of Proposition 2.22, respectively. Then, for each
$\alpha\in A$ there exists a prequantum line bundle
$(\phi_{\alpha}(U_{\alpha})\times
T^{n}\times\mathbb{C},\nabla^{\widetilde{L}_{\alpha}})\to(\phi_{\alpha}(U_{\alpha})\times
T^{n},\omega_{0})$ and a bundle isomorphism
$\psi_{\alpha}\colon(\widetilde{L},\nabla^{\widetilde{L}})|_{\widetilde{\pi}^{-1}(U_{\alpha})}\to(\phi_{\alpha}(U_{\alpha})\times
T^{n}\times\mathbb{C},\nabla^{\widetilde{L}_{\alpha}})$ which covers
$\varphi$. Now we define $(L^{\prime},\nabla^{L^{\prime}})$ by

$(L^{\prime},\nabla^{L^{\prime}}):=\coprod_{\alpha\in
A}\left(\phi_{\alpha}(U_{\alpha})\times
T^{n}\times\mathbb{C},\nabla^{\widetilde{L}_{\alpha}}\right)/\sim,$

where $(x_{\alpha},y_{\alpha},z_{\alpha})\in\phi_{\alpha}(U_{\alpha})\times
T^{n}\times\mathbb{C}$ and
$(x_{\beta},y_{\beta},z_{\beta})\in\phi_{\beta}(U_{\beta})\times
T^{n}\times\mathbb{C}$ are in the relation
$(x_{\alpha},y_{\alpha},z_{\alpha})\sim(x_{\beta},y_{\beta},z_{\beta})$ if
they satisfy
$(x_{\beta},y_{\beta},z_{\beta})=\psi_{\alpha}\circ\psi_{\beta}^{-1}(x_{\beta},y_{\beta},z_{\beta})$.
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\operatorname{dev}}$}}\colon(\widetilde{L},\nabla^{\widetilde{L}})\to(L^{\prime},\nabla^{L^{\prime}})$
is defined by

$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\operatorname{dev}}$}}(\widetilde{v}):=[\psi_{\alpha}(\widetilde{v})]$

if $\widetilde{v}$ lies in
$(\widetilde{L},\nabla^{\widetilde{L}})|_{\widetilde{\pi}^{-1}(U_{\alpha})}$.

Suppose that for each $\gamma\in\Gamma$ the deck transformation
$\sigma_{\gamma}$ maps each $U_{\alpha}$ to some $U_{\alpha^{\prime}}$ as
before. Then,
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\sigma}$}}_{\gamma}$ maps
$\widetilde{L}_{\widetilde{\pi}^{-1}(U_{\alpha})}$ to
$\widetilde{L}_{\widetilde{\pi}^{-1}(U_{\alpha^{\prime}})}$. Then, the
$\Gamma$-action $\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}$ is
defined by

$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}_{\gamma}(x_{\alpha},y_{\alpha},z_{\alpha}):=[\psi_{\alpha^{\prime}}\circ\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\sigma}$}}_{\gamma}\circ\psi_{\alpha}^{-1}(x_{\alpha},y_{\alpha},z_{\alpha})]$

if $(x_{\alpha},y_{\alpha},z_{\alpha})$ lies in
$\phi_{\alpha}(U_{\alpha})\times T^{n}\times\mathbb{C}$. ∎

In the case where $B$ is complete, we obtain the following corollary.

######  Corollary 2.25.

Let $\pi\colon(M,\omega)\to B$ be a Lagrangian fibration with connected
$n$-dimensional base $B$ and $(L,\nabla^{L})\to(M,\omega)$ a prequantum line
bundle on $(M,\omega)$. Let $p\colon\widetilde{B}\to B$ be the universal
covering of $B$. Let us denote by $(\widetilde{M},\widetilde{\omega})$ the
pullback of $(M,\omega)$ to $\widetilde{B}$ and denote by
$(\widetilde{L},\nabla^{\widetilde{L}})$ the pullback of $(L,\nabla^{L})$ to
$(\widetilde{M},\widetilde{\omega})$. If $B$ is complete, there exist an
integral affine isomorphism
$\operatorname{dev}\colon\widetilde{B}\to\mathbb{R}^{n}$, a fiber-preserving
symplectomorphism
$\widetilde{\operatorname{dev}}\colon(\widetilde{M},\widetilde{\omega})\to\left(\mathbb{R}^{n}\times
T^{n},\omega_{0}\right)$, and a bundle isomorphism
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\operatorname{dev}}$}}\colon(\widetilde{L},\nabla^{\widetilde{L}})\to\left(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}x\cdot dy\right)$ such that
$\widetilde{\operatorname{dev}}$ covers $\operatorname{dev}$ and
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\operatorname{dev}}$}}$
covers $\widetilde{\operatorname{dev}}$, respectively. Here $x\cdot dy$
denotes $\sum_{i=1}^{n}x_{i}dy_{i}$. Moreover, let $\sigma$ be the
$\Gamma$-action on $\widetilde{B}$ defined as the inverse of deck
transformations, $\widetilde{\sigma}$ the natural lift of $\sigma$ to
$(\widetilde{M},\widetilde{\omega})$, and
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\sigma}$}}$ the natural lift
of $\widetilde{\sigma}$ to $(\widetilde{L},\nabla^{\widetilde{L}})$,
respectively. Then, there exist an integral affine $\Gamma$-action
$\rho\colon\Gamma\to\operatorname{GL}_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$ on
$\mathbb{R}^{n}$, its lifts $\widetilde{\rho}$ and
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}$ to
$\left(\mathbb{R}^{n}\times T^{n},\omega_{0}\right)$ and
$\left(\mathbb{R}^{n}\times T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}x\cdot
dy\right)$, respectively such that $\operatorname{dev}$,
$\widetilde{\operatorname{dev}}$, and
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\operatorname{dev}}$}}$ are
$\Gamma$-equivariant.

###### Proof.

By construction of $\widetilde{\operatorname{dev}}$ given in the proof of
Proposition 2.22, if $\operatorname{dev}$ is bijective, so is
$\widetilde{\operatorname{dev}}$. The argument in [10, p.696] and Theorem 2.20
also show that $\pi_{0}\colon(\mathbb{R}^{n}\times
T^{n},\omega_{0})\to\mathbb{R}^{n}$ is the unique Lagrangian fibration on
$\mathbb{R}^{n}$ up to fiber-preserving symplectomorphism covering the
identity. In particular,
$\pi^{\prime}\colon(M^{\prime},\omega^{\prime})\to\mathbb{R}^{n}$ is
identified with $\pi_{0}\colon(\mathbb{R}^{n}\times
T^{n},\omega_{0})\to\mathbb{R}^{n}$.

Concerning the prequantum line bundle, it is sufficient to show that
$\left(\mathbb{R}^{n}\times T^{n},\omega_{0}\right)$ has a unique prequantum
line bundle $\left(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}x\cdot dy\right)$ up to bundle
isomorphism. Since $\omega_{0}$ is exact, any prequantum line bundle on
$\left(\mathbb{R}^{n}\times T^{n},\omega_{0}\right)$ is trivial as a complex
line bundle. Let $\left(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}\alpha\right)$ be a prequantum line
bundle on $\left(\mathbb{R}^{n}\times T^{n},\omega_{0}\right)$ with connection
$d-2\pi\sqrt{-1}\alpha$. Then, $\alpha-x\cdot dy$ defines a de Rham cohomology
class in $H^{1}\left(\mathbb{R}^{n}\times T^{n};\mathbb{R}\right)$. Since
$H^{1}\left(\mathbb{R}^{n}\times T^{n};\mathbb{R}\right)$ is isomorphic to
$H^{1}(T^{n};\mathbb{R})$, in terms of the generators $dy_{i}$’s of
$H^{1}(T^{n};\mathbb{R})$, $\alpha-x\cdot dy$ can be described as

$\alpha-x\cdot dy=\sum_{i=1}^{n}\tau_{i}dy_{i}+df$

for some $\tau_{1},\ldots,\tau_{n}\in\mathbb{R}$ and $f\in
C^{\infty}(\mathbb{R}^{n}\times T^{n})$. Now we define the bundle isomorphism
$\psi\colon\mathbb{R}^{n}\times T^{n}\times\mathbb{C}\to\mathbb{R}^{n}\times
T^{n}\times\mathbb{C}$ by

$\psi(x,y,z):=\left(x+(\tau_{i}),y,e^{-2\pi\sqrt{-1}f(x,y)}z\right).$

Then, $\psi$ satisfies $\psi^{*}\left(d-2\pi\sqrt{-1}x\cdot
dy\right)=d-2\pi\sqrt{-1}\alpha$. ∎

######  Remark 2.26.

By Corollary 2.25, any Lagrangian fibration $\pi\colon(M,\omega)\to B$ on a
connected, complete $B$ with prequantum line bundle
$(L,\nabla^{L})\to(M,\omega)$ is obtained as the quotient space of the
$\Gamma$-action on $\pi_{0}\colon(\mathbb{R}^{n}\times
T^{n},\omega_{0})\to\mathbb{R}^{n}$ with prequantum line bundle
$(\mathbb{R}^{n}\times T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}x\cdot
dy)\to(\mathbb{R}^{n}\times T^{n},\omega_{0})$. By definition, the prequantum
line bundle $(L,\nabla^{L})\to(M,\omega)$ is equipped with a Hermitian metric
$\left\langle\cdot,\,\cdot\right\rangle_{L}$ compatible with $\nabla^{L}$.222A
Hermitian metric $\left\langle\cdot,\,\cdot\right\rangle_{L}$ on $L$ is
compatible with $\nabla^{L}$ if it satisfies $d\left(\left\langle
s_{1},s_{2}\right\rangle_{L}\right)=\left\langle\nabla^{L}s_{1},s_{2}\right\rangle_{L}+\left\langle
s_{1},\nabla^{L}s_{2}\right\rangle_{L}$ for all
$s_{1},s_{2}\in\Gamma\left(L\right)$.  The pull-back of
$\left\langle\cdot,\,\cdot\right\rangle_{L}$ to $(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}x\cdot dy)\to(\mathbb{R}^{n}\times
T^{n},\omega_{0})$ coincides with the one induced from the standard Hermitian
inner product on $\mathbb{C}$ up to constant. In fact, it is easy to see that,
up to constant, it is the unique Hermitian metric on $(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}x\cdot dy)\to(\mathbb{R}^{n}\times
T^{n},\omega_{0})$ compatible with $d-2\pi\sqrt{-1}x\cdot dy$. In the rest of
this paper, we assume that $(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}x\cdot dy)\to(\mathbb{R}^{n}\times
T^{n},\omega_{0})$ is always equipped with the Hermitian metric though we do
not specify it.

###  2.4. The lifting problem of the $\Gamma$-action to the prequantum line
bundle

In the rest of this section, we investigate the condition for the
$\Gamma$-action on $\pi_{0}\colon(\mathbb{R}^{n}\times
T^{n},\omega_{0})\to\mathbb{R}^{n}$ to have a lift to $(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},$ $d-2\pi\sqrt{-1}x\cdot dy)\to(\mathbb{R}^{n}\times
T^{n},\omega_{0})$ in detail. Let
$\rho\colon\Gamma\to\operatorname{GL}_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$ be
a $\Gamma$-action on $\mathbb{R}^{n}$ and $\widetilde{\rho}$ its lift to
$\left(\mathbb{R}^{n}\times T^{n},\omega_{0}\right)$. By Lemma 2.17, for each
$\gamma\in\Gamma$, there exist
$A_{\gamma}\in\operatorname{GL}_{n}(\mathbb{Z})$,
$c_{\gamma}\in\mathbb{R}^{n}$, and a map $u_{\gamma}\colon\mathbb{R}^{n}\to
T^{n}$ with ${}^{t}A_{\gamma}Du_{\gamma}$ symmetric such that $\rho_{\gamma}$
and $\widetilde{\rho}_{\gamma}$ can be described as follows

(2.3)
$\rho_{\gamma}(x)=A_{\gamma}x+c_{\gamma},\quad{\widetilde{\rho}}_{\gamma}(x,y)=\left(A_{\gamma}x+c_{\gamma},{}^{t}A_{\gamma}^{-1}y+u_{\gamma}(x)\right).$

Note that since (2.3) is a $\Gamma$-action, $A_{\gamma}$, $c_{\gamma}$, and
$u_{\gamma}$ satisfy the following conditions

(2.4) $\begin{cases}A_{\gamma_{1}\gamma_{2}}&=A_{\gamma_{1}}A_{\gamma_{2}}\\\
c_{\gamma_{1}\gamma_{2}}&=A_{\gamma_{1}}c_{\gamma_{2}}+c_{\gamma_{1}}\\\
u_{\gamma_{1}\gamma_{2}}(x)&={}^{t}A_{\gamma_{1}}^{-1}u_{\gamma_{2}}(x)+u_{\gamma_{1}}\left(\rho_{\gamma_{2}}(x)\right)\end{cases}$

for $\gamma_{1}$, $\gamma_{2}\in\Gamma$, and $x\in\mathbb{R}^{n}$. Let
${\widetilde{u}}_{\gamma}={}^{t}({\widetilde{u}}_{\gamma}^{1},\ldots,{\widetilde{u}}_{\gamma}^{n})\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$
be a lift of $u_{\gamma}$. For ${\widetilde{u}}_{\gamma}$ and $i=1,\ldots,n$,
we put

$\int_{0}^{x_{i}}{\widetilde{u}}_{\gamma}(x)dx_{i}:={}^{t}\left(\int_{0}^{x_{i}}{\widetilde{u}}_{\gamma}^{1}(x)dx_{i},\ldots,\int_{0}^{x_{i}}{\widetilde{u}}_{\gamma}^{n}(x)dx_{i}\right)$

and

$F_{\gamma}^{i}(x):=\left({}^{t}A_{\gamma}\int_{0}^{x_{i}}{\widetilde{u}}_{\gamma}(x)dx_{i}\right)_{i}=\sum_{j=1}^{n}\left({}^{t}A_{\gamma}\right)_{ij}\int_{0}^{x_{i}}{\widetilde{u}}_{\gamma}^{j}(x)dx_{i}.$

Let $N\in\mathbb{N}$ be a positive integer. The $\Gamma$-action
$\widetilde{\rho}$ also preserves $N\omega_{0}$. Then, we can show the
following lemma.

######  Lemma 2.27.

(1) For each $\gamma\in\Gamma$, there exists a bundle automorphism
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}_{\gamma}$ of
$\left(\mathbb{R}^{n}\times T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}Nx\cdot
dy\right)$ which covers $\widetilde{\rho}_{\gamma}$ and preserves the
connection if and only if $c_{\gamma}$ lies in $\frac{1}{N}\mathbb{Z}^{n}$.
Moreover, in this case,
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}_{\gamma}$ can be
described as follows

(2.5)
${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}(x,y,z)=\left({\widetilde{\rho}}_{\gamma}(x,y),\
g_{\gamma}e^{2\pi\sqrt{-1}N\left\\{{\widetilde{g}}_{\gamma}(x)+c_{\gamma}\cdot({}^{t}A_{\gamma}^{-1}y)\right\\}}z\right),$

where $g_{\gamma}$ is an arbitrary element in $U(1)$ and

(2.6)
${\widetilde{g}}_{\gamma}(x):=\rho_{\gamma}(x)\cdot{\widetilde{u}}_{\gamma}(x)-c_{\gamma}\cdot{\widetilde{u}}_{\gamma}(0)-\sum_{i=1}^{n}F_{\gamma}^{i}(0,\ldots,0,x_{i},\ldots,x_{n}).$

The formula (2.5) does not depend on the choice of
${\widetilde{u}}_{\gamma}$.333In the rest of this paper, we often use the
notation $u_{\gamma}$ instead of ${\widetilde{u}}_{\gamma}$.
(2) Under the condition given in (1), the map
$\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}\colon\Gamma\to\operatorname{Aut}\left(\left(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}Nx\cdot dy\right)\right)$ defined by
(2.5) is a homomorphism if and only if the map $g\colon\Gamma\ni\gamma\mapsto
g_{\gamma}\in U(1)$ is a homomorphism and for all
$\gamma_{1},\gamma_{2}\in\Gamma$ and $x\in\mathbb{R}^{n}$, the following
condition holds

$\left\\{-c_{\gamma_{1}}\cdot
u_{\gamma_{1}}(0)+c_{\gamma_{1}}\cdot{}^{t}A_{\gamma_{1}}^{-1}u_{\gamma_{2}}(0)+\rho_{\gamma_{1}}(c_{\gamma_{2}})\cdot
u_{\gamma_{1}}\left(\rho_{\gamma_{2}}(0)\right)\right\\}\\\
-\sum_{i=1}^{n}\left({}^{t}A_{\gamma_{1}}\int_{0}^{(\rho_{\gamma_{2}}(x))_{i}}u_{\gamma_{1}}\left(0,\ldots,0,\tau_{i},(\rho_{\gamma_{2}}(x))_{i+1},\ldots,(\rho_{\gamma_{2}}(x))_{n}\right)d\tau_{i}\right)_{i}\\\
+\sum_{i=1}^{n}\left({}^{t}A_{\gamma_{2}}{}^{t}A_{\gamma_{1}}\int_{0}^{x_{i}}u_{\gamma_{1}}\left(\rho_{\gamma_{2}}(0,\ldots
0,\tau_{i},x_{i+1},\ldots,x_{n})\right)d\tau_{i}\right)_{i}\in\frac{1}{N}\mathbb{Z}.$

###### Proof.

For each $\gamma\in\Gamma$ we put

${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}(x,y,z)=\left({\widetilde{\rho}}_{\gamma}(x,y),e^{2\pi\\{{\widetilde{g}}^{R}_{\gamma}(x,y)+\sqrt{-1}{\widetilde{g}}^{I}_{\gamma}(x,y)\\}}z\right),$

where ${\widetilde{g}}^{R}_{\gamma}$ and ${\widetilde{g}}^{I}_{\gamma}$ are
real valued functions on $\mathbb{R}^{n}\times T^{n}$. By the direct
computation, it is easy to see that
${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}$
preserves $d-2\pi\sqrt{-1}Nx\cdot dy$ if and only if
${\widetilde{g}}^{R}_{\gamma}$ is constant and ${\widetilde{g}}^{I}_{\gamma}$
satisfy the following conditions

(2.7)
$\displaystyle\partial_{x_{i}}{\widetilde{g}}^{I}_{\gamma}=N(A_{\gamma}x+c_{\gamma})\cdot\partial_{x_{i}}{\widetilde{u}}_{\gamma}$
(2.8)
$\displaystyle\partial_{y_{i}}{\widetilde{g}}^{I}_{\gamma}=N(A_{\gamma}^{-1}c_{\gamma})_{i}$

for $i=1,\ldots,n$. From (2.7) we obtain

(2.9)
${\widetilde{g}}^{I}_{\gamma}(x,y)={\widetilde{g}}^{I}_{\gamma}(x_{1},\ldots,x_{i-1},0,x_{i+1},\ldots,x_{n},y)+N\left\\{\left[\rho_{\gamma}(x)\cdot{\widetilde{u}}_{\gamma}(x)\right]_{x_{i}=0}^{x_{i}=x_{i}}-F_{\gamma}^{i}(x)\right\\}.$

Using (2.9) recursively, we obtain

(2.10)
${\widetilde{g}}^{I}_{\gamma}(x,y)={\widetilde{g}}^{I}_{\gamma}(0,y)+N\left\\{\rho_{\gamma}(x)\cdot{\widetilde{u}}_{\gamma}(x)-c_{\gamma}\cdot{\widetilde{u}}_{\gamma}(0)-\sum_{i=1}^{n}F_{\gamma}^{i}\left(0,\ldots,0,x_{i},\ldots,x_{n}\right)\right\\}.$

(2.10) does not depend on the order of applying (2.9) to $x_{i}$’s. In fact,
by applying (2.9) first to $x_{i}$, then next to $x_{j}$, we obtain

$\displaystyle\dfrac{1}{N}{\widetilde{g}}^{I}_{\gamma}(x,y)=$
$\displaystyle\dfrac{1}{N}{\widetilde{g}}^{I}_{\gamma}(x_{1},\ldots,x_{i-1},0,x_{i+1},\ldots,x_{n},y)+\int_{0}^{x_{i}}\left(A_{\gamma}x+c_{\gamma}\right)\cdot\partial_{x_{i}}u_{\gamma}(x)dx_{i}$
$\displaystyle=$
$\displaystyle\dfrac{1}{N}{\widetilde{g}}^{I}_{\gamma}(x_{1},\ldots,x_{i-1},0,x_{i+1},\ldots,x_{j-1},0,x_{j+1},\ldots,x_{n},y)$
$\displaystyle+\int_{0}^{x_{j}}\left(A_{\gamma}x+c_{\gamma}\right)\cdot\partial_{x_{j}}u_{\gamma}(x)dx_{j}\Big{|}_{x_{i}=0}+\int_{0}^{x_{i}}\left(A_{\gamma}x+c_{\gamma}\right)\cdot\partial_{x_{i}}u_{\gamma}(x)dx_{i}.$

So, in order to see this, it is sufficient to show

(2.11)
$\begin{split}\int_{0}^{x_{j}}\left(A_{\gamma}x+c_{\gamma}\right)\cdot\partial_{x_{j}}u_{\gamma}(x)dx_{j}\Big{|}_{x_{i}=0}+\int_{0}^{x_{i}}\left(A_{\gamma}x+c_{\gamma}\right)\cdot\partial_{x_{i}}u_{\gamma}(x)dx_{i}\\\
-\int_{0}^{x_{i}}\left(A_{\gamma}x+c_{\gamma}\right)\cdot\partial_{x_{i}}u_{\gamma}(x)dx_{i}\Big{|}_{x_{j}=0}-\int_{0}^{x_{j}}\left(A_{\gamma}x+c_{\gamma}\right)\cdot\partial_{x_{j}}u_{\gamma}(x)dx_{j}\end{split}$

vanishes. Since ${}^{t}A_{\gamma}Du_{\gamma}$ is symmetric, we have
$\left({}^{t}A_{\gamma}\partial_{x_{i}}u_{\gamma}(x)\right)_{j}=({}^{t}A_{\gamma}Du_{\gamma}(x))_{ji}=({}^{t}A_{\gamma}Du_{\gamma}(x))_{ij}=\left({}^{t}A_{\gamma}\partial_{x_{j}}u_{\gamma}(x)\right)_{i}$
for all $i,j=1,\ldots,n$. By using this, we can show

$\displaystyle\eqref{difference}=$
$\displaystyle\int_{0}^{x_{j}}\partial_{x_{j}}\left(\int_{0}^{x_{i}}\left(A_{\gamma}x+c_{\gamma}\right)\cdot\partial_{x_{i}}u_{\gamma}(x)dx_{i}\right)dx_{j}-\int_{0}^{x_{i}}\partial_{x_{i}}\left(\int_{0}^{x_{j}}\left(A_{\gamma}x+c_{\gamma}\right)\cdot\partial_{x_{j}}u_{\gamma}(x)dx_{j}\right)dx_{i}$
$\displaystyle=\int_{0}^{x_{j}}\int_{0}^{x_{i}}\left(\partial_{x_{j}}\left(A_{\gamma}x+c_{\gamma}\right)\right)\cdot\partial_{x_{i}}u_{\gamma}(x)dx_{i}dx_{j}+\int_{0}^{x_{j}}\int_{0}^{x_{i}}\left(A_{\gamma}x+c_{\gamma}\right)\cdot\partial_{x_{j}}\partial_{x_{i}}u_{\gamma}(x)dx_{i}dx_{j}$
$\displaystyle-\int_{0}^{x_{i}}\int_{0}^{x_{j}}\left(\partial_{x_{i}}\left(A_{\gamma}x+c_{\gamma}\right)\right)\cdot\partial_{x_{j}}u_{\gamma}(x)dx_{j}dx_{i}-\int_{0}^{x_{i}}\int_{0}^{x_{j}}\left(A_{\gamma}x+c_{\gamma}\right)\cdot\partial_{x_{i}}\partial_{x_{j}}u_{\gamma}(x)dx_{j}dx_{i}$
$\displaystyle=\int_{0}^{x_{j}}\int_{0}^{x_{i}}\left({}^{t}A_{\gamma}\partial_{x_{i}}u_{\gamma}(x)\right)_{j}dx_{i}dx_{j}-\int_{0}^{x_{i}}\int_{0}^{x_{j}}\left({}^{t}A_{\gamma}\partial_{x_{j}}u_{\gamma}(x)\right)_{i}dx_{j}dx_{i}$
$\displaystyle=0.$

By the same way, from (3.9) we obtain

(2.12)
${\widetilde{g}}^{I}_{\gamma}(x,y)={\widetilde{g}}^{I}_{\gamma}(x,0)+Nc_{\gamma}\cdot{}^{t}A_{\gamma}^{-1}y.$

Thus, from (2.10) and (2.12) we have

(2.13)
${\widetilde{g}}^{I}_{\gamma}(x,y)={\widetilde{g}}^{I}_{\gamma}(0,0)+N\left\\{\rho_{\gamma}(x)\cdot{\widetilde{u}}_{\gamma}(x)-c_{\gamma}\cdot{\widetilde{u}}_{\gamma}(0)-\sum_{i=1}^{n}F_{\gamma}^{i}(0,\ldots,0,x_{i},\ldots,x_{n})+c_{\gamma}\cdot{}^{t}A_{\gamma}^{-1}y\right\\}.$

Since $y\in T^{n}$, ${\widetilde{g}}^{I}_{\gamma}$ should satisfies
$e^{2\pi\sqrt{-1}{\widetilde{g}}^{I}_{\gamma}(0,e_{i})}=e^{2\pi\sqrt{-1}{\widetilde{g}}^{I}_{\gamma}(0,0)}$
for all $i=1,\ldots,n$ and $\gamma\in\Gamma$. This holds if and only if
$A_{\gamma}^{-1}Nc_{\gamma}\cdot e_{i}\in\mathbb{Z}$ for all $i=1,\ldots,n$
and $\gamma\in\Gamma$. Since $A_{\gamma}\in\operatorname{GL}_{n}(\mathbb{Z})$
this is equivalent to the condition $Nc_{\gamma}\in\mathbb{Z}^{n}$. In this
case, we put
$g_{\gamma}:=e^{2\pi({\widetilde{g}}^{R}_{\gamma}(0,0)+\sqrt{-1}{\widetilde{g}}^{I}_{\gamma}(0,0))}$.
Since ${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}$
preserves the Hermitian metric on $(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}x\cdot dy)\to(\mathbb{R}^{n}\times
T^{n},\omega_{0})$, $g_{\gamma}$ lies in $U(1)$. The formula (2.5) does not
depend on the choice of ${\widetilde{u}}_{\gamma}$ since the difference of two
lifts of $u_{\gamma}$ lies in $\mathbb{Z}^{n}$. This proves $(1)$.

The map ${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}$ defined
in $(2)$ is a homomorphism if and only if
${\widetilde{g}}^{I}_{\gamma}(x,y)-{\widetilde{g}}^{I}_{\gamma}(0,0)$ defined
by (2.13) 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 $v_{1},\ldots,v_{n}\in\mathbb{R}^{n}$. The product $B\times
T^{n}$ admits a symplectic structure $\omega$ so that the trivial torus bundle
$\pi\colon(B\times T^{n},\omega)\to B$ becomes a Lagrangian fibration. This is
obtained as the quotient space of the action of $\Gamma:=\mathbb{Z}^{n}$ on
$\pi_{0}\colon\left(\mathbb{R}^{n}\times
T^{n},\omega_{0}\right)\to\mathbb{R}^{n}$ which is defined by

${\widetilde{\rho}}_{\gamma}(x,y)=\left(x+C\gamma,\ y\right)$

for $\gamma\in\Gamma$ and $(x,y)\in\mathbb{R}^{n}\times T^{n}$, where
$C=(v_{1}\ \cdots\ v_{n})\in\operatorname{GL}_{n}(\mathbb{R})$. Let
$N\in\mathbb{N}$ be a positive number. The $\Gamma$-action
${\widetilde{\rho}}$ on $\left(\mathbb{R}^{n}\times T^{n},N\omega_{0}\right)$
has a lift to the prequantum line bundle $(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}Nx\cdot dy)\to\left(\mathbb{R}^{n}\times
T^{n},N\omega_{0}\right)$ if and only if all $v_{i}$’s lie in
$\frac{1}{N}\mathbb{Z}^{n}$, and in this case
${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}$ is given by

${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}(x,y,z)=\left({\widetilde{\rho}}_{\gamma}(x,y),g_{\gamma}e^{2\pi\sqrt{-1}NC\gamma\cdot
y}z\right)$

for $\gamma\in\Gamma$ and $(x,y,z)\in\mathbb{R}^{n}\times
T^{n}\times\mathbb{C}$, where $g\colon\Gamma\ni\gamma\mapsto g_{\gamma}\in
U(1)$ is an arbitrary homomorphism.

######  Example 2.29 (The Kodaira-Thurston manifold).

Let $\Gamma$ be $\mathbb{Z}^{2}$. Consider the $\Gamma$-action on
$\pi_{0}\colon\left(\mathbb{R}^{2}\times
T^{2},\omega_{0}\right)\to\mathbb{R}^{2}$ which is defined by

$\rho_{\gamma}(x):=x+\gamma,\quad\widetilde{\rho}_{\gamma}(x,y):=\left(\rho_{\gamma}(x),y+u_{\gamma}(x)\right)$

for $\gamma\in\Gamma$ and $(x,y)\in\mathbb{R}^{2}\times T^{2}$, where
$u_{\gamma}(x)={}^{t}(0,\gamma_{1}x_{2})$. The Lagrangian fibration given by
the quotient of this action is denoted by $\pi\colon(M,\omega)\to B$. $M$ was
first observed by Kodaira in [24] and Thurston [36] pointed out in [36] that
$(M,\omega)$ does not admits any Kähler structure. $M$ is nowadays called the
Kodaira-Thurston manifold. Let $N\in\mathbb{N}$ be a positive number. The
$\Gamma$-action ${\widetilde{\rho}}$ on $\left(\mathbb{R}^{2}\times
T^{2},N\omega_{0}\right)$ has a lift to the prequantum line bundle
$(\mathbb{R}^{2}\times T^{2}\times\mathbb{C},d-2\pi\sqrt{-1}Nx\cdot
dy)\to\left(\mathbb{R}^{2}\times T^{2},N\omega_{0}\right)$ if and only if $N$
is even, and in this case the lift
${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}$ is given by

${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}(x,y,z)=\left({\widetilde{\rho}}_{\gamma}(x,y),g_{\gamma}e^{2\pi\sqrt{-1}N\\{\frac{1}{2}\gamma_{1}x_{2}^{2}+\gamma_{1}\gamma_{2}x_{2}+\gamma\cdot
y\\}}z\right)$

for $\gamma\in\Gamma$ and $(x,y,z)\in\mathbb{R}^{n}\times
T^{n}\times\mathbb{C}$, where $g\colon\Gamma\ni\gamma\mapsto g_{\gamma}\in
U(1)$ is an arbitrary homomorphism.

######  Example 2.30.

Let $B$ be the $n$-dimensional integral affine torus given in Example 2.3 (2)
for a linear basis $v_{1},\ldots,v_{n}\in\mathbb{R}^{n}$. When all $v_{i}$’s
are integer vectors, i.e., $v_{1},\ldots,v_{n}\in\mathbb{Z}^{n}$, we can
generalize Example 2.28 and Example 2.29 in the following way. Namely, for
$i,j=1,\ldots,n$, let $u_{ij}$ be an integer vector with $u_{ij}=u_{ji}$. For
each $\gamma\in\Gamma:=\mathbb{Z}^{n}$, define the map
$u_{\gamma}\colon\mathbb{R}^{n}\to T^{n}$ by

$u_{\gamma}(x):=\begin{pmatrix}u_{11}\cdot\gamma&\cdots&u_{1n}\cdot\gamma\\\
\vdots&&\vdots\\\ u_{n1}\cdot\gamma&\cdots&u_{nn}\cdot\gamma\end{pmatrix}x,$

and define the action of $\Gamma$ on $\pi_{0}\colon\left(\mathbb{R}^{n}\times
T^{n},\omega_{0}\right)\to\mathbb{R}^{n}$ by

(2.14) ${\widetilde{\rho}}_{\gamma}(x,y)=\left(x+C\gamma,\
y+u_{\gamma}(x)\right)$

for $\gamma\in\Gamma$ and $(x,y)\in\mathbb{R}^{n}\times T^{n}$, where
$C=(v_{1}\ \cdots\ v_{n})$. Then, the quotient $\pi\colon(M,\omega)\to B$
obtained as the $\Gamma$-action (2.14) is a Lagrangian fibration on $B$. Let
$N\in\mathbb{N}$ be a positive number. The $\Gamma$-action
${\widetilde{\rho}}$ on $\left(\mathbb{R}^{n}\times T^{n},N\omega_{0}\right)$
has a lift to the prequantum line bundle $(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}Nx\cdot dy)\to\left(\mathbb{R}^{n}\times
T^{n},N\omega_{0}\right)$ if and only if $\frac{N}{2}v_{i}\cdot
U_{j}v_{i}\in\mathbb{Z}$ for all $i,j=1,\ldots,n$, where

$U_{j}:=\begin{pmatrix}(u_{11})_{j}&\cdots&(u_{1n})_{j}\\\ \vdots&&\vdots\\\
(u_{n1})_{j}&\cdots&(u_{nn})_{j}\end{pmatrix}.$

And in this case the lift
${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}$ is given by

${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}(x,y,z)=\left({\widetilde{\rho}}_{\gamma}(x,y),\
g_{\gamma}e^{2\pi\sqrt{-1}N\left[\frac{1}{2}\left\\{\rho_{\gamma}(x)\cdot
u_{\gamma}(\rho_{\gamma}(x))-\rho_{\gamma}(0)\cdot
u_{\gamma}(\rho_{\gamma}(0))\right\\}+\rho_{\gamma}(0)\cdot y\right]}z\right)$

for $\gamma\in\Gamma$ and $(x,y,z)\in\mathbb{R}^{n}\times
T^{n}\times\mathbb{C}$, where $g\colon\Gamma\ni\gamma\mapsto g_{\gamma}\in
U(1)$ is an arbitrary homomorphism.

######  Example 2.31.

Let $n\geq 2$ and $\lambda_{1},\ldots\lambda_{n-1}\in\mathbb{Z}$. Let $\Gamma$
be the group $\left(\mathbb{Z}^{n},\circ\right)$ given in Example 2.6. For
each $\gamma\in\Gamma$, let $A_{\gamma}$ be the matrix

$A_{\gamma}:=\begin{pmatrix}1&\lambda_{1}&&&\\\ &1&\lambda_{2}&&\\\
&&\ddots&\ddots&\\\ &&&1&\lambda_{n-1}\\\ &&&&1\end{pmatrix}^{\gamma_{n}}$

and $u_{\gamma}\colon\mathbb{R}^{n}\to T^{n}$ the map defined by

$u_{\gamma}(x):=\begin{pmatrix}0\\\ \vdots\\\ 0\\\
\gamma_{n}x_{n}\end{pmatrix}.$

Consider the $\Gamma$-action ${\widetilde{\rho}}$ on
$\pi_{0}\colon\left(\mathbb{R}^{n}\times T^{n},\omega\right)\to\mathbb{R}^{n}$
which is defined by

(2.15) ${\widetilde{\rho}}_{\gamma}(x,y):=\left(A_{\gamma}x+\gamma,\
{}^{t}A_{\gamma}^{-1}y+u_{\gamma}(x)\right)$

for $\gamma\in\Gamma$ and $(x,y)\in\mathbb{R}^{n}\times T^{n}$, where
$C=(v_{1}\ \cdots\ v_{n})$. Then, the quotient $\pi\colon(M,\omega)\to B$
obtained as the $\Gamma$-action (2.15) is a Lagrangian fibration on the
integral affine manifold $B$ obtained in Example 2.6. Let $N\in\mathbb{N}$ be
a positive number. The $\Gamma$-action ${\widetilde{\rho}}$ on
$\left(\mathbb{R}^{n}\times T^{n},N\omega_{0}\right)$ has a lift to the
prequantum line bundle $(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}Nx\cdot dy)\to\left(\mathbb{R}^{n}\times
T^{n},N\omega_{0}\right)$ if and only if $N$ is even, and in this case the
lift ${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}$ is given by

${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}(x,y,z)=\left({\widetilde{\rho}}_{\gamma}(x,y),\
g_{\gamma}e^{2\pi\sqrt{-1}N\left\\{\gamma_{n}x_{n}\left(\frac{1}{2}x_{n}+\gamma_{n}\right)+\gamma\cdot\left({}^{t}A_{\gamma}^{-1}y\right)\right\\}}z\right)$

for $\gamma\in\Gamma$ and $(x,y,z)\in\mathbb{R}^{n}\times
T^{n}\times\mathbb{C}$, where $g\colon\Gamma\ni\gamma\mapsto g_{\gamma}\in
U(1)$ is an arbitrary homomorphism.

##  3\. Degree-zero harmonic spinors and integrability of almost complex
structures

Let $N\in\mathbb{N}$ be a positive integer. For a compatible almost complex
structure $J$ on the total space of the Lagrangian fibration
$\pi_{0}\colon\left(\mathbb{R}^{n}\times
T^{n},N\omega_{0}\right)\to\mathbb{R}^{n}$, let $D$ be the associated Spinc
Dirac operator with coefficients in the prequantum line bundle
$\left(\mathbb{R}^{n}\times T^{n}\times\mathbb{C},d-2\pi\sqrt{-1}Nx\cdot
dy\right)\to\left(\mathbb{R}^{n}\times T^{n},N\omega_{0}\right)$. An element
in the kernel $\ker D$ of $D$ is called a harmonic spinor. In this section,
for $J$ which is invariant along the fiber, we investigate the existence
condition of non-trivial degree-zero harmonic spinors, i.e., non-trivial
sections which lie in $\ker D$. In the rest of this paper, we put
${\widetilde{M}}:=\mathbb{R}^{n}\times T^{n}$ and
$\left({\widetilde{L}},\nabla^{\widetilde{L}}\right):=(\mathbb{R}^{n}\times
T^{n}\times\mathbb{C},$ $d-2\pi\sqrt{-1}x\cdot dy)$ for simplicity.

###  3.1. Bohr-Sommerfeld points

Let $\pi\colon(M,\omega)\to B$ be a Lagrangian fibration with prequantum line
bundle $(L,\nabla^{L})\to(M,\omega)$. We recall the definition of Bohr-
Sommerfeld points.

######  Definition 3.1.

A point $b\in B$ is said to be Bohr-Sommerfeld if
$\left(L,\nabla^{L}\right)|_{\pi^{-1}(b)}$ admits a non-trivial covariant
constant section. We denote the set of Bohr-Sommerfeld points by $B_{BS}$.

Let us detect Bohr-Sommerfeld points for
$\pi_{0}\colon({\widetilde{M}},N\omega_{0})\to\mathbb{R}^{n}$ with prequantum
line bundle $\left({\widetilde{L}},\nabla^{\widetilde{L}}\right)^{\otimes
N}\to({\widetilde{M}},N\omega_{0})$.

######  Lemma 3.2.

$x\in\mathbb{R}^{n}$ is a Bohr-Sommerfeld if and only if $x$ lies in
$\frac{1}{N}\mathbb{Z}^{n}$, i.e.,
$\mathbb{R}^{n}_{BS}=\frac{1}{N}\mathbb{Z}^{n}$. Moreover, for a Bohr-
Sommerfeld point $x\in\frac{1}{N}\mathbb{Z}^{n}$, a covariant constant section
$s$ of $\left({\widetilde{L}},\nabla^{\widetilde{L}}\right)^{\otimes
N}\Big{|}_{\pi_{0}^{-1}\left(x\right)}$ is of the form
$s(y)=s(0)e^{2\pi\sqrt{-1}Nx\cdot y}$.

###### Proof.

For a fixed $x\in\mathbb{R}^{n}$,
$\left({\widetilde{L}},\nabla^{\widetilde{L}}\right)^{\otimes
N}\Big{|}_{\pi_{0}^{-1}(x)}\to\pi_{0}^{-1}(x)$ admits a non-trivial covariant
constant section $s$ if and only if $s$ satisfies

$0=\nabla^{{\widetilde{L}}^{\otimes
N}}_{\partial_{y_{i}}}s=\partial_{y_{i}}s-2\pi\sqrt{-1}Nx_{i}s$

for $i=1,\ldots,n$. Hence, $s$ should be of the form
$s(y)=s(0)e^{2\pi\sqrt{-1}Nx\cdot y}$. Since $s$ is global,
$s(0)=s(e_{i})=s(0)e^{2\pi\sqrt{-1}Nx_{i}}$. This implies
$Nx_{i}\in\mathbb{Z}$ for $i=1,\ldots,n$. ∎

######  Remark 3.3.

Suppose that $\pi_{0}\colon({\widetilde{M}},N\omega_{0})\to\mathbb{R}^{n}$ is
equipped with an action of a group $\Gamma$ which preserves all the data, and
its lift ${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}$ to
$\left({\widetilde{L}},\nabla^{\widetilde{L}}\right)^{\otimes N}$ is given by
(2.5). Then, by Lemma 2.27 (1), the $\Gamma$-action $\rho$ on $\mathbb{R}^{n}$
preserves $\mathbb{R}^{n}_{BS}$. When the $\Gamma$-action $\rho$ on
$\mathbb{R}^{n}$ is properly discontinuous and free, let
$F\subset\mathbb{R}^{n}$ be a fundamental domain of the $\Gamma$-action $\rho$
on $\mathbb{R}^{n}$. Then, the map

(3.1)
$\Gamma\times\left(F\cap\frac{1}{N}\mathbb{Z}^{n}\right)\ni\left(\gamma,\dfrac{m}{N}\right)\mapsto
N\rho_{\gamma}\left(\dfrac{m}{N}\right)\in\mathbb{Z}^{n}$

can be defined and is bijective. In particular, let $\pi\colon(M,N\omega)\to
B$ be a Lagrangian fibration with prequantum line bundle
$(L,\nabla^{L})^{\otimes N}\to(M,N\omega)$ obtained as the quotient space of
the $\Gamma$-action. Then, $F\cap\frac{1}{N}\mathbb{Z}^{n}$ is identified with
$B_{BS}$.

###  3.2. Almost complex structures

Let ${\mathcal{S}}_{n}$ be the Siegel upper half space, namely, the space of
$n\times n$ symmetric complex matrices whose imaginary parts are positive
definite

${\mathcal{S}}_{n}:=\\{Z=X+\sqrt{-1}Y\in M_{n}(\mathbb{C})\mid X,Y\in
M_{n}(\mathbb{R}),{}^{t}Z=Z,\text{and }Y\text{ is positive definite}\\}.$

It is well known that ${\mathcal{S}}_{n}$ is identified with the space of
compatible complex structures on the $2n$-dimensional standard symplectic
vector space.

For a tangent vector
$u=\sum_{i=1}^{n}\left\\{(u_{x})_{i}\partial_{x_{i}}+(u_{y})_{i}\partial_{y_{i}}\right\\}\in
T_{(x,y)}{\widetilde{M}}$ at a point $(x,y)\in{\widetilde{M}}$ we use the
following notation

$u=\left(\partial_{x_{1}},\ldots,\partial_{x_{n}},\partial_{y_{1}},\ldots,\partial_{y_{n}}\right)\begin{pmatrix}(u_{x})_{1}\\\
\vdots\\\ (u_{x})_{n}\\\ (u_{y})_{1}\\\ \vdots\\\
(u_{y})_{n}\end{pmatrix}=\left(\partial_{x},\partial_{y}\right)\begin{pmatrix}u_{x}\\\
u_{y}\end{pmatrix},$

where

$\partial_{x}=(\partial_{x_{1}},\ldots,\partial_{x_{n}}),\quad\partial_{y}=(\partial_{y_{1}},\ldots,\partial_{y_{n}}),\quad
u_{x}=\begin{pmatrix}(u_{x})_{1}\\\ \vdots\\\ (u_{x})_{n}\end{pmatrix},\quad
u_{y}=\begin{pmatrix}(u_{y})_{1}\\\ \vdots\\\ (u_{y})_{n}\end{pmatrix}.$

In terms of the notations of tangent vectors
$u=\left(\partial_{x},\partial_{y}\right)\begin{pmatrix}u_{x}\\\
u_{y}\end{pmatrix}$ and
$v=\left(\partial_{x},\partial_{y}\right)\begin{pmatrix}v_{x}\\\
v_{y}\end{pmatrix}\in T_{(x,y)}{\widetilde{M}}$, $\omega_{0}$ can be described
by

$\omega_{0}(u,v)=\left({}^{t}u_{x},{}^{t}u_{y}\right)\begin{pmatrix}0&I\\\
-I&0\end{pmatrix}\begin{pmatrix}v_{x}\\\ v_{y}\end{pmatrix}.$

Since the tangent bundle $T{\widetilde{M}}$ is trivial, the space of
$C^{\infty}$ maps from ${\widetilde{M}}$ to ${\mathcal{S}}_{n}$ is identified
with the space of compatible almost complex structures on
$({\widetilde{M}},\omega_{0})$. For $Z=X+\sqrt{-1}Y\in
C^{\infty}\left({\widetilde{M}},{\mathcal{S}}_{n}\right)$, the corresponding
almost complex structure $J_{Z}$ is given as follows

(3.2)
$J_{Z}u:=\left(\partial_{x},\partial_{y}\right)\begin{pmatrix}XY^{-1}&-Y-XY^{-1}X\\\
Y^{-1}&-Y^{-1}X\end{pmatrix}_{(x,y)}\begin{pmatrix}u_{x}\\\
u_{y}\end{pmatrix}$

for $u=\left(\partial_{x},\partial_{y}\right)\begin{pmatrix}u_{x}\\\
u_{y}\end{pmatrix}\in T_{(x,y)}{\widetilde{M}}$.444
$\begin{pmatrix}XY^{-1}&-Y-XY^{-1}X\\\ Y^{-1}&-Y^{-1}X\end{pmatrix}_{(x,y)}$,
$(XY^{-1})_{(x,y)}$ etc. are the values of the maps
$\begin{pmatrix}XY^{-1}&-Y-XY^{-1}X\\\ Y^{-1}&-Y^{-1}X\end{pmatrix}$,
$XY^{-1}$ etc. at $(x,y)$. We will often omit the subscript “(x,y)” for
simplicity unless it causes confusion. Then, the Riemannian metric $g$
determined by $\omega_{0}$ and $J_{Z}$ can be described by

(3.3) $\begin{split}g(u,v):&=\omega_{0}(u,Jv)\\\
&=\left({}^{t}u_{x},{}^{t}u_{y}\right)\begin{pmatrix}0&I\\\
-I&0\end{pmatrix}\begin{pmatrix}XY^{-1}&-Y-XY^{-1}X\\\
Y^{-1}&-Y^{-1}X\end{pmatrix}\begin{pmatrix}v_{x}\\\ v_{y}\end{pmatrix}\\\
&=\left({}^{t}u_{x},{}^{t}u_{y}\right)\begin{pmatrix}Y^{-1}&-Y^{-1}X\\\
-XY^{-1}&Y+XY^{-1}X\end{pmatrix}\begin{pmatrix}v_{x}\\\
v_{y}\end{pmatrix}.\end{split}$

Let $J=J_{Z}$ be the almost complex structure on
$\left({\widetilde{M}},\omega_{0}\right)$ corresponding to a given
$Z=X+\sqrt{-1}Y\in C^{\infty}\left({\widetilde{M}},{\mathcal{S}}_{n}\right)$.
Then,
$\left(-J\partial_{y},\,\partial_{y}\right)=\left(-J\partial_{y_{1}},\ldots,-J\partial_{y_{n}},\partial_{y_{1}},\ldots,\partial_{y_{n}}\right)$
is also a basis of the tangent space of
$\left({\widetilde{M}},\omega_{0}\right)$. With this basis, each tangent
vector $u\in T_{(x,y)}{\widetilde{M}}$ can be written as

$u=\sum_{i}\\{(u_{H})_{i}(-J\partial_{y_{i}})+(u_{V})_{i}\partial_{y_{i}}\\}=\left(-J\partial_{y},\,\partial_{y}\right)\begin{pmatrix}u_{H}\\\
u_{V}\end{pmatrix}.$

Then, we have the following transition formula between
$\left(\partial_{x},\,\partial_{y}\right)$ and
$\left(-J\partial_{y},\,\partial_{y}\right)$

$u=\left(-J\partial_{y},\,\partial_{y}\right)\begin{pmatrix}u_{H}\\\
u_{V}\end{pmatrix}=\left(\partial_{x},\partial_{y}\right)\left(\begin{pmatrix}-XY^{-1}&Y+XY^{-1}X\\\
-Y^{-1}&Y^{-1}X\end{pmatrix}\begin{pmatrix}0\\\
u_{H}\end{pmatrix}+\begin{pmatrix}0\\\ u_{V}\end{pmatrix}\right).$

By this formula, we obtain the following lemma.

######  Lemma 3.4.

In terms of this notation, the Riemannian metric $g$ defined by (3.3) can be
described by

$g(u,v)=(0,{}^{t}u_{H})\begin{pmatrix}Y^{-1}&-Y^{-1}X\\\
-XY^{-1}&Y+XY^{-1}X\end{pmatrix}\begin{pmatrix}0\\\
v_{H}\end{pmatrix}+(0,{}^{t}u_{V})\begin{pmatrix}Y^{-1}&-Y^{-1}X\\\
-XY^{-1}&Y+XY^{-1}X\end{pmatrix}\begin{pmatrix}0\\\ v_{V}\end{pmatrix}.$

Suppose that a group $\Gamma$ acts on
$\pi_{0}\colon({\widetilde{M}},\omega_{0})\to\mathbb{R}^{n}$ and the
$\Gamma$-actions $\rho$ on $\mathbb{R}^{n}$ and $\widetilde{\rho}$ on
$({\widetilde{M}},\omega_{0})$ are written as in (2.3). Then, it is easy to
see the following lemma.

######  Lemma 3.5.

The $\Gamma$-action ${\widetilde{\rho}}$ on $({\widetilde{M}},\omega_{0})$
preserves the almost complex structure $J=J_{Z}$ on
$({\widetilde{M}},\omega_{0})$ corresponding to $Z=X+\sqrt{-1}Y\in
C^{\infty}\left({\widetilde{M}},{\mathcal{S}}_{n}\right)$ if and only if the
following conditions hold

(3.4) $\displaystyle
A_{\gamma}(XY^{-1})_{(x,y)}=(XY^{-1})_{{\widetilde{\rho}}_{\gamma}(x,y)}A_{\gamma}-\left(Y+XY^{-1}X\right)_{{\widetilde{\rho}}_{\gamma}(x,y)}(Du_{\gamma})_{x}$
(3.5) $\displaystyle
A_{\gamma}(Y+XY^{-1}X)_{(x,y)}=\left(Y+XY^{-1}X\right)_{{\widetilde{\rho}}_{\gamma}(x,y)}{}^{t}A_{\gamma}^{-1}$
(3.6)
$\displaystyle(Du_{\gamma})_{x}(XY^{-1})_{(x,y)}+{}^{t}A_{\gamma}^{-1}Y^{-1}_{(x,y)}=Y^{-1}_{{\widetilde{\rho}}_{\gamma}(x,y)}A_{\gamma}-(Y^{-1}X)_{{\widetilde{\rho}}_{\gamma}(x,y)}(Du_{\gamma})_{x}.$

###### Proof.

For all $\gamma\in\Gamma$ and $(x,y)\in({\widetilde{M}},\omega_{0})$, the
condition $\left(d{\widetilde{\rho}}_{\gamma}\right)_{(x,y)}\circ
J_{(x,y)}=J_{{\widetilde{\rho}}_{\gamma}(x,y)}\circ\left(d{\widetilde{\rho}}_{\gamma}\right)_{(x,y)}$
implies above three equalities together with the following equality

$(Du_{\gamma})_{x}(Y+XY^{-1}X)_{(x,y)}+{}^{t}A_{\gamma}^{-1}(Y^{-1}X)_{(x,y)}=(Y^{-1}X)_{{\widetilde{\rho}}_{\gamma}(x,y)}{}^{t}A_{\gamma}^{-1}.$

But, this can be obtained from (3.4), (3.5), and
${}^{t}\left({}^{t}A_{\gamma}(Du_{\gamma})_{x}\right)={}^{t}A_{\gamma}(Du_{\gamma})_{x}$.
∎

Let $\pi\colon(M,\omega)\to B$ be a Lagrangian fibration with connected
$n$-dimensional complete base $B$ and $p\colon\widetilde{B}\to B$ the
universal covering of $B$. By Corollary 2.25, the pullback of
$\pi\colon(M,\omega)\to B$ to $\widetilde{B}$ is identified with
$\pi_{0}\colon({\widetilde{M}},\omega_{0})\to\mathbb{R}^{n}$ and
$\pi\colon(M,\omega)\to B$ can be obtained as the quotient of the
$\Gamma=\pi_{1}(B)$-action on
$\pi_{0}\colon({\widetilde{M}},\omega_{0})\to\mathbb{R}^{n}$. In particular,
for each compatible almost complex structure $J$ on $(M,\omega)$, there exists
a map $Z_{J}=X+\sqrt{-1}Y\in
C^{\infty}\left({\widetilde{M}},{\mathcal{S}}_{n}\right)$ such that the
pullback $p^{*}J$ of $J$ to $p^{*}(M,\omega)$ coincides with $J_{Z_{J}}$.
Then, we have the following lemma.

######  Lemma 3.6 ([14, Corollary 9.15]).

For any Lagrangian fibration $\pi\colon(M,\omega)\to B$, there exists a
compatible almost complex structure $J$ such that the corresponding map
$Z_{J}$ does not depend on $y_{1},\ldots,y_{n}$. We say such $J$ to be
invariant along the fiber.

###### Proof.

Take a Riemannian metric $g^{\prime}$ on $(M,\omega)$. Then, the pullback
$p^{*}g^{\prime}$ is $\pi_{1}(B)$-invariant. Moreover, $p^{*}(M,\omega)$
admits a free $T^{n}$-action, and this $T^{n}$-action together with the
$\pi_{1}(B)$-action forms an action of the semi-direct product
$\pi_{1}(B)\ltimes T^{n}$ of $T^{n}$ and $\pi_{1}(B)$. By averaging
$p^{*}g^{\prime}$ over $T^{n}$, we obtain a Riemannian metric on $p^{*}M$
invariant under the $\pi_{1}(B)\ltimes T^{n}$-action. It is easy to see that
$p^{*}\omega$ is also $\pi_{1}(B)\ltimes T^{n}$-invariant, so by the standard
method using the $\pi_{1}(B)\ltimes T^{n}$-invariant Riemannian metric and
$p^{*}\omega$, we can obtain a $\pi_{1}(B)\ltimes T^{n}$-invariant compatible
almost complex structure on $p^{*}(M,\omega)$. In particular, since the almost
complex structure is still invariant under $\pi_{1}(B)$-action, it descends to
$(M,\omega)$. This is the required almost complex structure. ∎

###  3.3. The existence condition of non-trivial harmonic spinors of degree-
zero

For a map $Z=X+\sqrt{-1}Y\in
C^{\infty}\left({\widetilde{M}},{\mathcal{S}}_{n}\right)$, we set

(3.7) $\Omega:=\left(Y+XY^{-1}X\right)^{-1}ZY^{-1}.$

$\Omega$ has the following properties.

######  Lemma 3.7.

(1) $\Omega={\overline{Z}}^{-1}$, where ${\overline{Z}}=X-\sqrt{-1}Y$.
(2) $\Omega$ is symmetric, i.e., ${}^{t}\Omega=\Omega$.

###### Proof.

A direct computation shows that $\Omega{\overline{Z}}=I$. This proves (1). (2)
follows from (1) since $Z$ is symmetric. ∎

Let $N\in\mathbb{N}$ be a positive integer. Let $J=J_{Z}$ be the compatible
almost complex structure on $\left({\widetilde{M}},N\omega_{0}\right)$
corresponding to a given $Z=X+\sqrt{-1}Y\in
C^{\infty}\left({\widetilde{M}},{\mathcal{S}}_{n}\right)$. Then, the
Riemannian metric $Ng:=N\omega_{0}(\cdot,\,J\cdot)$ defines an isomorphism
$f\colon T^{*}{\widetilde{M}}\cong T{\widetilde{M}}$ by
$\tau=Ng\left(f(\tau),\ \cdot\right)$ for $\tau\in T^{*}{\widetilde{M}}$. For
$i=1,\ldots,n$, let $\Omega_{i}$ denote the $i$th column vector of $\Omega$,
and $\operatorname{Re}\Omega_{i}$ and $\operatorname{Im}\Omega_{i}$ be the
real and imaginary parts of $\Omega_{i}$, respectively. Then, we can show the
following lemma.

######  Lemma 3.8.

For $i=1,\ldots,n$,

$f(dx_{i})=-\frac{1}{N}J\partial_{y_{i}},\quad
f(dy_{i})=\left(-J\partial_{y},\partial_{y}\right)\begin{pmatrix}\frac{1}{N}\operatorname{Re}\Omega_{i}\\\
\frac{1}{N}\operatorname{Im}\Omega_{i}\end{pmatrix}.$

###### Proof.

We prove the latter. The former can be proved by the same way. Put
$f(dy_{i})=\left(-J\partial_{y},\partial_{y}\right)\begin{pmatrix}Y^{i}_{H}\\\
Y^{i}_{V}\end{pmatrix}$. By definition, for each $i,j=1,\ldots,n$, we have

(3.8) $\displaystyle dy_{i}(-J\partial_{y_{j}})$
$\displaystyle=Ng\left(\left(-J\partial_{y},\partial_{y}\right)\begin{pmatrix}Y^{i}_{H}\\\
Y^{i}_{V}\end{pmatrix},\left(-J\partial_{y},\partial_{y}\right)\begin{pmatrix}e_{j}\\\
0\end{pmatrix}\right)$ (3.9) $\displaystyle dy_{i}(\partial_{y_{j}})$
$\displaystyle=Ng\left(\left(-J\partial_{y},\partial_{y}\right)\begin{pmatrix}Y^{i}_{H}\\\
Y^{i}_{V}\end{pmatrix},\left(-J\partial_{y},\partial_{y}\right)\begin{pmatrix}0\\\
e_{j}\end{pmatrix}\right).$

Since $-J\partial_{y_{j}}$ is written as

$-J\partial_{y_{j}}=\left(\partial_{x},\partial_{y}\right)\begin{pmatrix}-XY^{-1}&Y+XY^{-1}X\\\
-Y^{-1}&Y^{-1}X\end{pmatrix}\begin{pmatrix}0\\\ e_{j}\end{pmatrix}$

by (3.2), the left hand side of (3.8) is $\left(Y^{-1}X\right)_{ij}$. On the
other hand, by Lemma 3.4, the right hand side of (3.8) can be described as
$NY^{i}_{H}\cdot(Y+XY^{-1}X)e_{j}$. This implies
$Y^{-1}X=N{}^{t}(Y^{1}_{H}\cdots Y^{n}_{H})(Y+XY^{-1}X)$. Since $Y$ is
positive definite, so is $Y+XY^{-1}X$. In particular, $N(Y+XY^{-1}X)$ is
invertible. By using ${}^{t}X=X$, ${}^{t}Y=Y$ together with this fact, we can
obtain $(Y^{1}_{H}\cdots Y^{n}_{H})=\frac{1}{N}(Y+XY^{-1}X)^{-1}XY^{-1}$. By
the same way, from (3.9), we obtain $I=N{}^{t}(Y^{1}_{V}\cdots
Y^{n}_{V})(Y+XY^{-1}X)$, i.e., $(Y^{1}_{V}\cdots
Y^{n}_{V})=\frac{1}{N}(Y+XY^{-1}X)^{-1}$. Hence,
$\frac{1}{N}\Omega=(Y^{1}_{H}\cdots Y^{n}_{H})+\sqrt{-1}(Y^{1}_{V}\cdots
Y^{n}_{V})$. ∎

Define the Hermitian metric on $({\widetilde{M}},N\omega_{0},Ng,J)$ by

(3.10) $h(u,v):=Ng(u,v)+\sqrt{-1}Ng(u,Jv)$

for $u,v\in T_{(x,y)}{\widetilde{M}}$. Let $(W,c)$ be the Clifford module
bundle associated with $(Ng,J)$, i.e., as a complex vector bundle, $W$ is
defined by

$W:=\wedge^{\bullet}\left(T{\widetilde{M}},J\right)\otimes_{\mathbb{C}}\left({\widetilde{L}}^{\otimes
N}\right).$

$W$ is equipped with the Hermitian metric induced from $h$ and that on
${\widetilde{L}}$, and also equipped with the Hermitian connection, which is
denoted by $\nabla^{W}$, induced from the Levi-Civita connection $\nabla^{LC}$
of $\left({\widetilde{M}},g\right)$ and $\nabla^{\widetilde{L}}$. $c$ is the
Clifford multiplication $c\colon
T{\widetilde{M}}\to\operatorname{End}_{\mathbb{C}}(W)$ defined by

$c(u)(\tau):=u\wedge\tau-u\,\llcorner_{h}\,\tau$

for $u\in T{\widetilde{M}}$ and $\tau\in W$, where $\llcorner_{h}$ is the
contraction with respect to the Hermitian metric $h$ on
$({\widetilde{M}},N\omega_{0},Ng,J)$. It is well known that $W$ is identified
with
$\wedge^{\bullet}(T^{*}{\widetilde{M}})^{0,1}\otimes_{\mathbb{C}}\left({\widetilde{L}}^{\otimes
N}\right)$ as a Clifford module bundle.

Now let us define the Spinc Dirac operator $D\colon\Gamma(W)\to\Gamma(W)$ by
the composition of the following maps

$\textstyle{D\colon\Gamma(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla^{W}}$$\textstyle{\Gamma(T^{*}{\widetilde{M}}\otimes
W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\otimes\mathop{\mathrm{id}}\nolimits_{W}}$$\textstyle{\Gamma(T{\widetilde{M}}\otimes
W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{\Gamma(W).}$

We compute the action of $D$ on a degree zero element in $\Gamma(W)$. We
identify a section of ${\widetilde{L}}$ with a complex valued function on
${\widetilde{M}}$. By using Lemma 3.8, for a section $s$ of
${\widetilde{L}}^{\otimes N}$, $Ds$ can be computed as

$\displaystyle Ds$
$\displaystyle=c\circ(f\otimes\mathop{\mathrm{id}}\nolimits_{W})\circ\nabla^{W}s$
$\displaystyle=c\circ(f\otimes\mathop{\mathrm{id}}\nolimits_{W})(ds-2\pi\sqrt{-1}Nx\cdot
dys)$
$\displaystyle=\sum_{i=1}^{n}\left\\{c\left(f(dx_{i})\right)\left(\partial_{x_{i}}s\right)+c\left(f(dy_{i})\right)\left(\partial_{y_{i}}s-2\pi\sqrt{-1}Nx_{i}s\right)\right\\}$
$\displaystyle=-\frac{\sqrt{-1}}{N}\sum_{i=1}^{n}\partial_{y_{i}}\otimes_{\mathbb{C}}\left\\{\partial_{x_{i}}s+\sum_{j=1}^{n}\Omega_{ij}\left(\partial_{y_{j}}s-2\pi\sqrt{-1}Nx_{j}s\right)\right\\}.$

In particular, the equality $Ds=0$ is equivalent to

(3.11) $0=\begin{pmatrix}\partial_{x_{1}}s\\\ \vdots\\\
\partial_{x_{n}}s\end{pmatrix}+\Omega\begin{pmatrix}\partial_{y_{1}}s-2\pi\sqrt{-1}Nx_{1}s\\\
\vdots\\\ \partial_{y_{n}}s-2\pi\sqrt{-1}Nx_{n}s\end{pmatrix}.$

Suppose that $Z$ does not depend on $y_{1},\ldots,y_{n}$ as in Lemma 3.6.
Then, by substituting a Fourier expansion
$s=\sum_{m\in\mathbb{Z}^{n}}a_{m}(x)e^{2\pi\sqrt{-1}m\cdot y}$ of $s$ with
respect to $y_{i}$’s into (3.11), (3.11) can be reduced to the following
system of differential equations for $a_{m}$’s with variables
$x_{1},\ldots,x_{n}$

(3.12) $0=\begin{pmatrix}\partial_{x_{1}}a_{m}\\\ \vdots\\\
\partial_{x_{n}}a_{m}\end{pmatrix}+2\pi\sqrt{-1}a_{m}\Omega(m-Nx)$

for all $m\in\mathbb{Z}^{n}$.

######  Lemma 3.9.

Let $a_{m}$ be a solution of (3.12) for some $m\in\mathbb{Z}^{n}$. If there
exists $p\in\mathbb{R}^{n}$ such that $a_{m}(p)=0$. Then, $a_{m}(x)=0$ for all
$x\in\mathbb{R}^{n}$.

###### Proof.

First, fix variables $x_{2},\ldots,x_{n}$ with $p_{2},\ldots,p_{n}$. Then, the
first entry of (3.12), i.e.,
$0=\partial_{x_{1}}a_{m}+2\pi\sqrt{-1}a_{m}\left(\Omega(m-Nx)\right)_{1}$ can
be thought of as an ordinary differential equation on $x_{1}$, and
$a_{m}(x_{1},p_{2},\ldots,p_{n})$ is its solution with initial condition
$a_{m}(p)=0$. On the other hand, the trivial solution also has the same
initial condition. By the uniqueness of the solution of the ordinary
differential equation, $a_{m}(x_{1},p_{2},\ldots,p_{n})=0$ for any $x_{1}$.
Next, by fixing variables $x_{3},\ldots,x_{n}$ with $p_{3},\ldots,p_{n}$ and
fixing $x_{1}$ with arbitrary value, $a_{m}(x_{1},x_{2},p_{3},\ldots,p_{n})$
is a solution of
$0=\partial_{x_{2}}a_{m}+2\pi\sqrt{-1}a_{m}\left(\Omega(m-Nx)\right)_{2}$ with
initial condition $a_{m}(x_{1},p_{2},\ldots,p_{n})=0$. Then,
$a_{m}(x_{1},x_{2},p_{3},\ldots,p_{n})=0$ for any $x_{1},x_{2}$. By repeating
the process for $x_{3},\ldots,x_{n}$, we can show that $a_{m}(x)=0$. ∎

######  Lemma 3.10.

If $a_{m}$ is a non trivial smooth solution of (3.12) for some
$m\in\mathbb{Z}^{n}$, then, the condition

(3.13)
$\left(\left(\partial_{x_{i}}\Omega\right)_{x}(m-Nx)\right)_{j}=\left(\left(\partial_{x_{j}}\Omega\right)_{x}(m-Nx)\right)_{i}\
\text{for all }i,j=1,\ldots,n,\text{ and all }x\in\mathbb{R}^{n}$

holds.Conversely, if there exists $m\in\mathbb{Z}^{n}$ such that (3.13) holds,
then, (3.12) has a unique non trivial solution up to constant. Moreover, in
this case, each solution $a_{m}$ of (3.12) has the following form

(3.14)
$a_{m}(x)=a_{m}\left(\frac{m}{N}\right)e^{-2\pi\sqrt{-1}\sum_{i=1}^{n}G_{m}^{i}\left(\frac{m_{1}}{N},\ldots,\frac{m_{i-1}}{N},x_{i},\ldots,x_{n}\right)},$

where $a_{m}\left(\frac{m}{N}\right)$ can be taken as an arbitrary constant in
$\mathbb{C}$ and

$G_{m}^{i}(x):=\left(\int_{\frac{m_{i}}{N}}^{x_{i}}\Omega(m-Nx)dx_{i}\right)_{i}.$

###### Proof.

Since $a_{m}$ is smooth, $a_{m}$ satisfies
$\partial_{x_{i}}\partial_{x_{j}}a_{m}=\partial_{x_{j}}\partial_{x_{i}}a_{m}$
for all $i,j=1,\ldots,n$. By differentiating (3.12), we have

$\displaystyle\partial_{x_{i}}\partial_{x_{j}}a_{m}=-2\pi\sqrt{-1}a_{m}\left\\{-2\pi\sqrt{-1}\sum_{k=1}^{n}\Omega_{ik}(m_{k}-Nx_{k})\sum_{l=1}^{n}\Omega_{jl}(m_{l}-Nx_{l})+\sum_{l=1}^{n}\left(\partial_{x_{i}}\Omega_{jl}\right)(m_{l}-Nx_{l})-N\Omega_{ji}\right\\}$

for $i,j=1,\ldots,n$ and $x\in\mathbb{R}^{n}$. The condition (3.13) is
obtained from this equation.

Conversely, suppose there exists $m\in\mathbb{Z}^{n}$ such that (3.13) holds.
By solving the differential equation appeared as the $i$th component of (3.12)
for $i=1,\ldots,n$, we have

(3.15)
$a_{m}(x)=a_{m}\left(x_{1},\ldots,x_{i-1},\frac{m_{i}}{N},x_{i+1},\ldots,x_{n}\right)e^{-2\pi\sqrt{-1}G_{m}^{i}(x)}.$

Using (3.15) recursively, we obtain the formula (3.14). By using (3.13), we
can show that (3.14) does not depend on the order of applying (3.15) to
$x_{i}$’s as in the proof of Lemma 2.27. Hence, (3.14) is well-defined. ∎

For each $m\in\mathbb{Z}^{n}$ for which the condition (3.13) holds, define the
section $s_{m}\in\Gamma\left({\widetilde{L}}^{\otimes N}\right)$ by

(3.16)
$s_{m}(x,y):=e^{2\pi\sqrt{-1}\left\\{-\sum_{i=1}^{n}G_{m}^{i}\left(\frac{m_{1}}{N},\ldots,\frac{m_{i-1}}{N},x_{i},\ldots,x_{n}\right)+m\cdot
y\right\\}}.$

By the elliptic regularity of $D$ and Lemma 3.10, we can obtain the following.

######  Proposition 3.11.

If $s=\sum_{m\in\mathbb{Z}^{n}}a_{m}(x)e^{2\pi\sqrt{-1}m\cdot
y}\in\Gamma\left({\widetilde{L}}^{\otimes N}\right)$ is a non trivial solution
of $0=Ds$, then, the condition (3.13) holds for all $m\in\mathbb{Z}^{n}$ with
$a_{m}\not=0$. Conversely, suppose that there exists $m\in\mathbb{Z}^{n}$ such
that (3.13) holds. Then, the section $s_{m}$ defined by (3.16) satisfies
$0=Ds_{m}$. In particular, if (3.13) holds for all $m\in\mathbb{Z}^{n}$, then,
$\left\\{s_{m}\right\\}_{m\in\mathbb{Z}^{n}}$ is a linear basis of
$\Gamma\left({\widetilde{L}}^{\otimes N}\right)\cap\ker D$.

The following proposition gives a geometric interpretation of the condition
(3.13).

######  Proposition 3.12.

The following conditions are equivalent:
(1) The condition (3.13) holds for all $m\in\mathbb{Z}^{n}$.
(2) $\partial_{x_{i}}\Omega_{jk}=\partial_{x_{j}}\Omega_{ik}$ for all
$i,j,k=1,\ldots,n$.
(3) $\nabla^{LC}J=0$, where $\nabla^{LC}$ is the Levi-Civita connection with
respect to $g$.

###### Proof.

If (3.13) holds for all $m\in\mathbb{Z}^{n}$, then, by putting $m=0$, we have
$\left(\left(\partial_{x_{i}}\Omega\right)_{x}x\right)_{j}=\left(\left(\partial_{x_{j}}\Omega\right)_{x}x\right)_{i}$.
By substituting this to (3.13), we can see the condition
$\left(\left(\partial_{x_{i}}\Omega\right)_{x}m\right)_{j}=\left(\left(\partial_{x_{j}}\Omega\right)_{x}m\right)_{i}$
holds for all $m\in\mathbb{Z}^{n}$. In particular, by substituting $m=e_{k}$
to this condition for each $k=1,\ldots,n$, we can obtain $(2)$.
$(2)\Rightarrow(1)$ is trivial.

We show $(2)\Leftrightarrow(3)$. $(2)$ is equivalent to the following two
conditions

(3.17)
$\displaystyle\left(\left(Y+XY^{-1}X\right)^{-1}\partial_{x_{i}}\left(XY^{-1}\right)\right)_{jk}=\left(\left(Y+XY^{-1}X\right)^{-1}\partial_{x_{j}}\left(XY^{-1}\right)\right)_{ik}$
(3.18)
$\displaystyle\partial_{x_{i}}\left(Y+XY^{-1}X\right)^{-1}_{jk}=\partial_{x_{j}}\left(Y+XY^{-1}X\right)^{-1}_{ik}$

for $i,j,k=1,\ldots,n$. For $i=1,\ldots,2n$, we set

$\Gamma_{i}:=\begin{pmatrix}\Gamma^{1}_{i\,1}&\cdots&\Gamma^{1}_{i\,2n}\\\
\vdots&&\vdots\\\ \Gamma^{2n}_{i\,1}&\cdots&\Gamma^{2n}_{i\,2n}\end{pmatrix},$

where $\Gamma_{ij}^{k}$ is the Christoffel symbol. Then, $(3)$ is equivalent
to

$0=\partial_{i}J+\Gamma_{i}J-J\Gamma_{i}\quad(i=1,\ldots,2n),$

where

$\partial_{i}=\begin{cases}\partial_{x_{i}}&(i=1,\ldots,n)\\\
\partial_{y_{i-n}}&(i=n+1,\ldots,2n).\end{cases}$

It is also equivalent to the following conditions

(3.19) $\displaystyle
XY^{-1}\begin{pmatrix}\partial_{x_{1}}(XY^{-1})_{1i}&\cdots&\partial_{x_{n}}(XY^{-1})_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(XY^{-1})_{ni}&\cdots&\partial_{x_{n}}(XY^{-1})_{ni}\end{pmatrix}$
$\displaystyle-(Y+XY^{-1}X)\begin{pmatrix}\partial_{x_{1}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{ni}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{ni}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\partial_{x_{1}}(XY^{-1})_{1i}&\cdots&\partial_{x_{n}}(XY^{-1})_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(XY^{-1})_{ni}&\cdots&\partial_{x_{n}}(XY^{-1})_{ni}\end{pmatrix}XY^{-1},$
(3.20) $\displaystyle
Y^{-1}\begin{pmatrix}\partial_{x_{1}}(XY^{-1})_{1i}&\cdots&\partial_{x_{n}}(XY^{-1})_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(XY^{-1})_{ni}&\cdots&\partial_{x_{n}}(XY^{-1})_{ni}\end{pmatrix}$
$\displaystyle-Y^{-1}X\begin{pmatrix}\partial_{x_{1}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{ni}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{ni}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\partial_{x_{1}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{ni}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{ni}\end{pmatrix}XY^{-1}$
$\displaystyle+\begin{pmatrix}\partial_{x_{1}}(Y^{-1}X)_{i1}&\cdots&\partial_{x_{1}}(Y^{-1}X)_{in}\\\
\vdots&&\vdots\\\
\partial_{x_{n}}(Y^{-1}X)_{i1}&\cdots&\partial_{x_{n}}(Y^{-1}X)_{in}\end{pmatrix}Y^{-1},$
(3.21)
$\displaystyle(Y+XY^{-1}X)\begin{pmatrix}\partial_{x_{1}}(Y^{-1}X)_{i1}&\cdots&\partial_{x_{1}}(Y^{-1}X)_{in}\\\
\vdots&&\vdots\\\
\partial_{x_{n}}(Y^{-1}X)_{i1}&\cdots&\partial_{x_{n}}(Y^{-1}X)_{in}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\partial_{x_{1}}(XY^{-1})_{1i}&\cdots&\partial_{x_{n}}(XY^{-1})_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(XY^{-1})_{ni}&\cdots&\partial_{x_{n}}(XY^{-1})_{ni}\end{pmatrix}(Y+XY^{-1}X),$
(3.22) $\displaystyle
Y^{-1}X\begin{pmatrix}\partial_{x_{1}}(Y^{-1}X)_{i1}&\cdots&\partial_{x_{1}}(Y^{-1}X)_{in}\\\
\vdots&&\vdots\\\
\partial_{x_{n}}(Y^{-1}X)_{i1}&\cdots&\partial_{x_{n}}(Y^{-1}X)_{in}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\partial_{x_{1}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{ni}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{ni}\end{pmatrix}(Y+XY^{-1})$
$\displaystyle+\begin{pmatrix}\partial_{x_{1}}(Y^{-1}X)_{i1}&\cdots&\partial_{x_{1}}(Y^{-1}X)_{in}\\\
\vdots&&\vdots\\\
\partial_{x_{n}}(Y^{-1}X)_{i1}&\cdots&\partial_{x_{n}}(Y^{-1}X)_{in}\end{pmatrix}Y^{-1}X,$
(3.23) $\displaystyle
XY^{-1}\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{1i}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y+XY^{-1}X)_{ni}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{ni}\end{pmatrix}$
$\displaystyle+(Y+XY^{-1}X)\begin{pmatrix}-\partial_{x_{1}}(Y^{-1}X)_{1i}+\partial_{x_{1}}(XY^{-1})_{i1}&\cdots&-\partial_{x_{n}}(Y^{-1}X)_{1i}+\partial_{x_{1}}(XY^{-1})_{in}\\\
\vdots&&\vdots\\\
-\partial_{x_{1}}(Y^{-1}X)_{ni}+\partial_{x_{n}}(XY^{-1})_{i1}&\cdots&-\partial_{x_{n}}(Y^{-1}X)_{ni}+\partial_{x_{n}}(XY^{-1})_{in}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{1i}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y+XY^{-1}X)_{ni}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{ni}\end{pmatrix}XY^{-1},$
(3.24) $\displaystyle
Y^{-1}\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{1i}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y+XY^{-1}X)_{ni}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{ni}\end{pmatrix}$
$\displaystyle+Y^{-1}X\begin{pmatrix}-\partial_{x_{1}}(Y^{-1}X)_{1i}+\partial_{x_{1}}(XY^{-1})_{i1}&\cdots&-\partial_{x_{n}}(Y^{-1}X)_{1i}+\partial_{x_{1}}(XY^{-1})_{in}\\\
\vdots&&\vdots\\\
-\partial_{x_{1}}(Y^{-1}X)_{ni}+\partial_{x_{n}}(XY^{-1})_{i1}&\cdots&-\partial_{x_{n}}(Y^{-1}X)_{ni}+\partial_{x_{n}}(XY^{-1})_{in}\end{pmatrix}$
$\displaystyle=-\begin{pmatrix}-\partial_{x_{1}}(Y^{-1}X)_{1i}+\partial_{x_{1}}(XY^{-1})_{i1}&\cdots&-\partial_{x_{n}}(Y^{-1}X)_{1i}+\partial_{x_{1}}(XY^{-1})_{in}\\\
\vdots&&\vdots\\\
-\partial_{x_{1}}(Y^{-1}X)_{ni}+\partial_{x_{n}}(XY^{-1})_{i1}&\cdots&-\partial_{x_{n}}(Y^{-1}X)_{ni}+\partial_{x_{n}}(XY^{-1})_{in}\end{pmatrix}XY^{-1}$
$\displaystyle+\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{i1}&\cdots&\partial_{x_{1}}(Y+XY^{-1}X)_{in}\\\
\vdots&&\vdots\\\
\partial_{x_{n}}(Y+XY^{-1}X)_{i1}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{in}\end{pmatrix}Y^{-1},$
(3.25)
$\displaystyle(Y+XY^{-1}X)\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{i1}&\cdots&\partial_{x_{1}}(Y+XY^{-1}X)_{in}\\\
\vdots&&\vdots\\\
\partial_{x_{n}}(Y+XY^{-1}X)_{i1}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{in}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{1i}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y+XY^{-1}X)_{ni}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{ni}\end{pmatrix}(Y+XY^{-1}X),$
(3.26) $\displaystyle
Y^{-1}X\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{i1}&\cdots&\partial_{x_{1}}(Y+XY^{-1}X)_{in}\\\
\vdots&&\vdots\\\
\partial_{x_{n}}(Y+XY^{-1}X)_{i1}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{in}\end{pmatrix}$
$\displaystyle=-\begin{pmatrix}-\partial_{x_{1}}(Y^{-1}X)_{1i}+\partial_{x_{1}}(XY^{-1})_{i1}&\cdots&-\partial_{x_{n}}(Y^{-1}X)_{1i}+\partial_{x_{1}}(XY^{-1})_{in}\\\
\vdots&&\vdots\\\
-\partial_{x_{1}}(Y^{-1}X)_{ni}+\partial_{x_{n}}(XY^{-1})_{i1}&\cdots&-\partial_{x_{n}}(Y^{-1}X)_{ni}+\partial_{x_{n}}(XY^{-1})_{in}\end{pmatrix}(Y+XY^{-1}X)$
$\displaystyle+\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{i1}&\cdots&\partial_{x_{1}}(Y+XY^{-1}X)_{in}\\\
\vdots&&\vdots\\\
\partial_{x_{n}}(Y+XY^{-1}X)_{i1}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{in}\end{pmatrix}Y^{-1}X.$

for $i=1,\ldots,n$. It is easy to see that (3.22) and (3.26) are obtained by
transposing (3.19) and (3.23), respectively. First, we show that (3.17) is
equivalent to (3.21). In fact, (3.17) implies

$\begin{pmatrix}\partial_{x_{1}}(XY^{-1})_{1k}&\cdots&\partial_{x_{1}}(XY^{-1})_{nk}\\\
\vdots&&\vdots\\\
\partial_{x_{n}}(XY^{-1})_{1k}&\cdots&\partial_{x_{n}}(XY^{-1})_{nk}\end{pmatrix}(Y+XY^{-1}X)^{-1}$

is symmetric for $k=1,\ldots,n$. Since $X$, $Y$ is symmetric, this implies
(3.21). Next, we show (3.25) is equivalent to (3.18). (3.25) is equivalent to

(3.27)
$\displaystyle\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{i1}&\cdots&\partial_{x_{1}}(Y+XY^{-1}X)_{in}\\\
\vdots&&\vdots\\\
\partial_{x_{n}}(Y+XY^{-1}X)_{i1}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{in}\end{pmatrix}(Y+XY^{-1}X)^{-1}$
$\displaystyle=(Y+XY^{-1}X)^{-1}\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{1i}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y+XY^{-1}X)_{ni}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{ni}\end{pmatrix}.$

By computing the $(j,k)$-components of the both sides of (3.27), we obtain

$\sum_{l=1}^{n}\left(\partial_{x_{j}}(Y+XY^{-1}X)^{-1}_{kl}\right)(Y+XY^{-1}X)_{li}=\sum_{l=1}^{n}\left(\partial_{x_{k}}(Y+XY^{-1}X)^{-1}_{jl}\right)(Y+XY^{-1}X)_{li}$

for $i,j,k=1,\ldots,n$. Here, we used

$\displaystyle 0=$
$\displaystyle\partial_{x_{j}}\left(\left(Y+XY^{-1}X\right)\left(Y+XY^{-1}X\right)^{-1}\right)$
$\displaystyle=$
$\displaystyle\left(\partial_{x_{j}}\left(Y+XY^{-1}X\right)\right)\left(Y+XY^{-1}X\right)^{-1}+\left(Y+XY^{-1}X\right)\partial_{x_{j}}\left(Y+XY^{-1}X\right)^{-1}$

and so on. Thus,

$\displaystyle\partial_{x_{j}}(Y+XY^{-1}X)^{-1}_{km}$
$\displaystyle=\sum_{i=1}^{n}\sum_{l=1}^{n}\partial_{x_{j}}\left((Y+XY^{-1}X)^{-1}_{kl}\right)(Y+XY^{-1}X)_{li}(Y+XY^{-1}X)^{-1}_{im}$
$\displaystyle=\sum_{i=1}^{n}\sum_{l=1}^{n}\left(\partial_{x_{k}}(Y+XY^{-1}X)^{-1}_{jl}\right)(Y+XY^{-1}X)_{li}(Y+XY^{-1}X)^{-1}_{im}$
$\displaystyle=\partial_{x_{k}}(Y+XY^{-1}X)^{-1}_{jm}.$

This implies (3.18). In particular, this means $(3)\Rightarrow(2)$.

We show (3.19), (3.20), (3.23), and (3.24) are obtained from $(2)$. To show
(3.23), it is sufficient to show

(3.28) $\displaystyle 0=$
$\displaystyle(Y+XY^{-1}X)^{-1}XY^{-1}\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{1i}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y+XY^{-1}X)_{ni}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{ni}\end{pmatrix}$
$\displaystyle-\begin{pmatrix}\partial_{x_{1}}(Y^{-1}X)_{1i}&\cdots&\partial_{x_{n}}(Y^{-1}X)_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y^{-1}X)_{ni}&\cdots&\partial_{x_{n}}(Y^{-1}X)_{ni}\end{pmatrix}$
$\displaystyle-(Y+XY^{-1}X)^{-1}\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{1i}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y+XY^{-1}X)_{ni}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{ni}\end{pmatrix}XY^{-1}$
$\displaystyle+\begin{pmatrix}\partial_{x_{1}}(XY^{-1})_{i1}&\cdots&\partial_{x_{1}}(XY^{-1})_{in}\\\
\vdots&&\vdots\\\
\partial_{x_{n}}(XY^{-1})_{i1}&\cdots&\partial_{x_{n}}(XY^{-1})_{in}\end{pmatrix}.$

Since $\Omega$ is symmetric, so is its real part
$\operatorname{Re}\Omega=(Y+XY^{-1}X)^{-1}XY^{-1}$. By taking the real part of
$(2)$ we also have

$\partial_{x_{i}}\left((Y+XY^{-1}X)^{-1}XY^{-1}\right)_{jk}=\partial_{x_{j}}\left((Y+XY^{-1}X)^{-1}XY^{-1}\right)_{ik}.$

By using these as well as (3.17) and (3.18), the $(j,k)$-component of the
first two terms of the right hand side of (3.28) can be computed as

$\displaystyle\sum_{l}\left((Y+XY^{-1}X)^{-1}XY^{-1}\right)_{jl}\partial_{x_{k}}(Y+XY^{-1}X)_{li}-\partial_{x_{k}}(Y^{-1}X)_{ji}$
$\displaystyle=$
$\displaystyle\sum_{l}\left(Y^{-1}X(Y+XY^{-1}X)^{-1}\right)_{jl}\partial_{x_{k}}(Y+XY^{-1}X)_{li}-\partial_{x_{k}}(Y^{-1}X)_{ji}$
$\displaystyle=$
$\displaystyle\partial_{x_{k}}\left(\sum_{l}\left(Y^{-1}X(Y+XY^{-1}X)^{-1}\right)_{jl}(Y+XY^{-1}X)_{li}\right)$
$\displaystyle-\sum_{l}\left(\partial_{x_{k}}\left(Y^{-1}X(Y+XY^{-1}X)^{-1}\right)_{jl}\right)(Y+XY^{-1}X)_{li}-\partial_{x_{k}}(Y^{-1}X)_{ji}$
$\displaystyle=$
$\displaystyle-\sum_{l}\left(\partial_{x_{k}}\left(Y^{-1}X(Y+XY^{-1}X)^{-1}\right)_{jl}\right)(Y+XY^{-1}X)_{li}$
$\displaystyle=$
$\displaystyle-\sum_{l}\left(\partial_{x_{j}}\left((Y+XY^{-1}X)^{-1}XY^{-1}\right)_{kl}\right)(Y+XY^{-1}X)_{li}.$

On the other hand, the $(j,k)$-component of the last two terms of the right
hand side of (3.28) can be computed as

$\displaystyle-\sum_{m,l}(Y+XY^{-1}X)^{-1}_{jl}\left(\partial_{x_{m}}(Y+XY^{-1}X)_{li}\right)(XY^{-1})_{mk}+\partial_{x_{j}}(XY^{-1})_{ik}$
$\displaystyle=\sum_{m,l}\left(\partial_{x_{m}}(Y+XY^{-1}X)^{-1}_{jl}\right)(Y+XY^{-1}X)_{li}(XY^{-1})_{mk}+\partial_{x_{j}}(XY^{-1})_{ik}$
$\displaystyle=\sum_{m,l}(Y+XY^{-1}X)_{li}\left(\partial_{x_{j}}(Y+XY^{-1}X)^{-1}_{ml}\right)(XY^{-1})_{mk}+\sum_{m,l}(Y+XY^{-1}X)_{li}(Y+XY^{-1}X)^{-1}_{ml}\partial_{x_{j}}(XY^{-1})_{mk}$
$\displaystyle=\sum_{l}\left(\partial_{x_{j}}\left((Y+XY^{-1}X)^{-1}XY^{-1}\right)_{kl}\right)(Y+XY^{-1}X)_{li}.$

This proves (3.28). We show (3.24). We put

$W:=\begin{pmatrix}\partial_{x_{1}}(Y+XY^{-1}X)_{1i}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y+XY^{-1}X)_{ni}&\cdots&\partial_{x_{n}}(Y+XY^{-1}X)_{ni}\end{pmatrix}.$

By (3.23) and (3.25), we obtain

$\displaystyle\begin{pmatrix}-\partial_{x_{1}}(Y^{-1}X)_{1i}+\partial_{x_{1}}(XY^{-1})_{i1}&\cdots&-\partial_{x_{n}}(Y^{-1}X)_{1i}+\partial_{x_{1}}(XY^{-1})_{in}\\\
\vdots&&\vdots\\\
-\partial_{x_{1}}(Y^{-1}X)_{ni}+\partial_{x_{n}}(XY^{-1})_{i1}&\cdots&-\partial_{x_{n}}(Y^{-1}X)_{ni}+\partial_{x_{n}}(XY^{-1})_{in}\end{pmatrix}$
$\displaystyle=(Y+XY^{-1}X)^{-1}WXY^{-1}-(Y+XY^{-1}X)^{-1}XY^{-1}W$

and

$(Y+XY^{-1}X){}^{t}W=W(Y+XY^{-1}X).$

In order to show (3.24) it is sufficient to check

(3.29) $\displaystyle 0=$ $\displaystyle
Y^{-1}W+Y^{-1}X(Y+XY^{-1}X)^{-1}WXY^{-1}-Y^{-1}X(Y+XY^{-1}X)^{-1}XY^{-1}W$
$\displaystyle+(Y+XY^{-1}X)^{-1}WXY^{-1}XY^{-1}-(Y+XY^{-1}X)^{-1}XY^{-1}WXY^{-1}-{}^{t}WY^{-1}.$

By using above equalities, the right hand side of (3.29) can be computed as

$\displaystyle
Y^{-1}W-Y^{-1}X(Y+XY^{-1}X)^{-1}XY^{-1}W+(Y+XY^{-1}X)^{-1}WXY^{-1}XY^{-1}-{}^{t}WY^{-1}$
$\displaystyle=$ $\displaystyle
Y^{-1}W-(Y+XY^{-1}X)^{-1}XY^{-1}XY^{-1}W+(Y+XY^{-1}X)^{-1}WXY^{-1}XY^{-1}$
$\displaystyle-(Y+XY^{-1}X)^{-1}W(Y+XY^{-1}X)Y^{-1}$ $\displaystyle=$
$\displaystyle
Y^{-1}W-(Y+XY^{-1}X)^{-1}XY^{-1}XY^{-1}W+(Y+XY^{-1}X)^{-1}WXY^{-1}XY^{-1}$
$\displaystyle-(Y+XY^{-1}X)^{-1}W-(Y+XY^{-1}X)^{-1}WXY^{-1}XY^{-1}$
$\displaystyle=$ $\displaystyle
Y^{-1}W-\\{(Y+XY^{-1}X)^{-1}XY^{-1}X+(Y+XY^{-1}X)^{-1}Y\\}Y^{-1}W$
$\displaystyle=$ $\displaystyle 0.$

This proves (3.24).

We show (3.19). To see this, we show

(3.30) $\displaystyle 0=$
$\displaystyle(Y+XY^{-1}X)^{-1}XY^{-1}\begin{pmatrix}\partial_{x_{1}}(XY^{-1})_{1i}&\cdots&\partial_{x_{n}}(XY^{-1})_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(XY^{-1})_{ni}&\cdots&\partial_{x_{n}}(XY^{-1})_{ni}\end{pmatrix}$
$\displaystyle-\begin{pmatrix}\partial_{x_{1}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{ni}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{ni}\end{pmatrix}$
$\displaystyle-(Y+XY^{-1}X)^{-1}\begin{pmatrix}\partial_{x_{1}}(XY^{-1})_{1i}&\cdots&\partial_{x_{n}}(XY^{-1})_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(XY^{-1})_{ni}&\cdots&\partial_{x_{n}}(XY^{-1})_{ni}\end{pmatrix}XY^{-1}.$

The $(j,k)$-component of the right hand side of (3.30) is

$\displaystyle\sum_{l}\left((Y+XY^{-1}X)^{-1}XY^{-1}\right)_{jl}\partial_{x_{k}}(XY^{-1})_{li}-\partial_{x_{k}}Y^{-1}_{ji}+\partial_{x_{j}}Y^{-1}_{ki}-\sum_{l,m}(Y+XY^{-1}X)^{-1}_{jm}\partial_{x_{l}}(XY^{-1})_{mi}(XY^{-1})_{lk}$
$\displaystyle=$
$\displaystyle\left((Y+XY^{-1}X)^{-1}XY^{-1}\partial_{x_{k}}(XY^{-1})\right)_{ji}-\partial_{x_{k}}Y^{-1}_{ji}+\partial_{x_{j}}Y^{-1}_{ki}-\sum_{l,m}(Y+XY^{-1}X)^{-1}_{lm}\partial_{x_{j}}(XY^{-1})_{mi}(XY^{-1})_{lk}$
$\displaystyle=$
$\displaystyle\left((Y+XY^{-1}X)^{-1}\left\\{\partial_{x_{k}}\left((Y+XY^{-1}X)Y^{-1}\right)-\partial_{x_{k}}(XY^{-1})XY^{-1}\right\\}\right)_{ji}-\partial_{x_{k}}Y^{-1}_{ji}+\partial_{x_{j}}Y^{-1}_{ki}$
$\displaystyle-\sum_{l,m}(Y+XY^{-1}X)^{-1}_{ml}(XY^{-1})_{lk}\partial_{x_{j}}(XY^{-1})_{mi}$
$\displaystyle=$
$\displaystyle\left((Y+XY^{-1}X)^{-1}\left(\partial_{x_{k}}(Y+XY^{-1}X)\right)Y^{-1}+\partial_{x_{k}}Y^{-1}-(Y+XY^{-1}X)^{-1}\partial_{x_{k}}(XY^{-1})XY^{-1}\right)_{ji}$
$\displaystyle-\partial_{x_{k}}Y^{-1}_{ji}+\partial_{x_{j}}Y^{-1}_{ki}-\sum_{m}\left((Y+XY^{-1}X)^{-1}XY^{-1}\right)_{mk}\partial_{x_{j}}(XY^{-1})_{mi}$
$\displaystyle=$
$\displaystyle\left((Y+XY^{-1}X)^{-1}\left(\partial_{x_{k}}(Y+XY^{-1}X)\right)Y^{-1}\right)_{ji}-\left((Y+XY^{-1}X)^{-1}\partial_{x_{k}}(XY^{-1})XY^{-1}\right)_{ji}+\partial_{x_{j}}Y^{-1}_{ki}$
$\displaystyle-\sum_{m}\left((Y+XY^{-1}X)^{-1}XY^{-1}\right)_{km}\partial_{x_{j}}(XY^{-1})_{mi}$
$\displaystyle=$
$\displaystyle\left((Y+XY^{-1}X)^{-1}\left(\partial_{x_{k}}(Y+XY^{-1}X)\right)Y^{-1}\right)_{ji}-\left((Y+XY^{-1}X)^{-1}\partial_{x_{k}}(XY^{-1})XY^{-1}\right)_{ji}$
$\displaystyle+\left(\partial_{x_{j}}Y^{-1}-(Y+XY^{-1}X)^{-1}XY^{-1}\partial_{x_{j}}(XY^{-1})\right)_{ki}$
$\displaystyle=$
$\displaystyle\left(-\left(\partial_{x_{k}}(Y+XY^{-1}X)^{-1}\right)(Y+XY^{-1}X)Y^{-1}\right)_{ji}-\sum_{l}\left((Y+XY^{-1}X)^{-1}\partial_{x_{k}}(XY^{-1})\right)_{jl}XY^{-1}_{li}$
$\displaystyle+\left(\partial_{x_{j}}Y^{-1}-(Y+XY^{-1}X)^{-1}XY^{-1}\partial_{x_{j}}(XY^{-1})\right)_{ki}$
$\displaystyle=$
$\displaystyle-\sum_{l}\partial_{x_{k}}(Y+XY^{-1}X)^{-1}_{jl}\left((Y+XY^{-1}X)Y^{-1}\right)_{li}-\sum_{l}\left((Y+XY^{-1}X)^{-1}\partial_{x_{j}}(XY^{-1})\right)_{kl}XY^{-1}_{li}$
$\displaystyle+\left(\partial_{x_{j}}Y^{-1}-(Y+XY^{-1}X)^{-1}XY^{-1}\partial_{x_{j}}(XY^{-1})\right)_{ki}$
$\displaystyle=$
$\displaystyle-\sum_{l}\partial_{x_{j}}(Y+XY^{-1}X)^{-1}_{kl}\left((Y+XY^{-1}X)Y^{-1}\right)_{li}-\left((Y+XY^{-1}X)^{-1}\partial_{x_{j}}(XY^{-1})XY^{-1}\right)_{ki}$
$\displaystyle+\left(\partial_{x_{j}}Y^{-1}-(Y+XY^{-1}X)^{-1}XY^{-1}\partial_{x_{j}}(XY^{-1})\right)_{ki}$
$\displaystyle=$
$\displaystyle\left(-\left(\partial_{x_{j}}(Y+XY^{-1}X)^{-1}\right)(Y+XY^{-1}X)Y^{-1}\right)_{ki}-\left((Y+XY^{-1}X)^{-1}\partial_{x_{j}}(XY^{-1})XY^{-1}\right)_{ki}$
$\displaystyle+\left(\partial_{x_{j}}Y^{-1}-(Y+XY^{-1}X)^{-1}XY^{-1}\partial_{x_{j}}(XY^{-1})\right)_{ki}$
$\displaystyle=$
$\displaystyle\left((Y+XY^{-1}X)^{-1}\left(\partial_{x_{j}}(Y+XY^{-1}X)\right)Y^{-1}\right)_{ki}-\left((Y+XY^{-1}X)^{-1}\partial_{x_{j}}(XY^{-1})XY^{-1}\right)_{ki}$
$\displaystyle+\left(\partial_{x_{j}}Y^{-1}-(Y+XY^{-1}X)^{-1}XY^{-1}\partial_{x_{j}}(XY^{-1})\right)_{ki}$
$\displaystyle=$
$\displaystyle\left((Y+XY^{-1}X)^{-1}\left\\{\left(\partial_{x_{j}}(Y+XY^{-1}X)\right)Y^{-1}+(Y+XY^{-1}X)\partial_{x_{j}}Y^{-1}-\partial_{x_{j}}(XY^{-1})XY^{-1}-XY^{-1}\partial_{x_{j}}(XY^{-1})\right\\}\right)_{ki}$
$\displaystyle=$
$\displaystyle\left((Y+XY^{-1}X)^{-1}\left\\{\partial_{x_{j}}((Y+XY^{-1}X)Y^{-1})-\partial_{x_{j}}(XY^{-1}XY^{-1})\right\\}\right)_{ki}$
$\displaystyle=$ $\displaystyle 0.$

This proves (3.19).

Finally, we show (3.20). We put

$V:=\begin{pmatrix}\partial_{x_{1}}(XY^{-1})_{1i}&\cdots&\partial_{x_{n}}(XY^{-1})_{1i}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(XY^{-1})_{ni}&\cdots&\partial_{x_{n}}(XY^{-1})_{ni}\end{pmatrix}.$

By (3.19) and (3.21), we obtain

$\displaystyle\begin{pmatrix}\partial_{x_{1}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{1i}-\partial_{x_{1}}(Y^{-1})_{ni}\\\
\vdots&&\vdots\\\
\partial_{x_{1}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{1i}&\cdots&\partial_{x_{n}}(Y^{-1})_{ni}-\partial_{x_{n}}(Y^{-1})_{ni}\end{pmatrix}$
$\displaystyle=(Y+XY^{-1}X)^{-1}XY^{-1}V-(Y+XY^{-1}X)^{-1}VXY^{-1}$

and

$(Y+XY^{-1}X){}^{t}V=V(Y+XY^{-1}X).$

In order to show (3.20) it is sufficient to check

(3.31) $\displaystyle 0=$ $\displaystyle
Y^{-1}V-Y^{-1}X(Y+XY^{-1}X)^{-1}XY^{-1}V+Y^{-1}X(Y+XY^{-1}X)^{-1}VXY^{-1}$
$\displaystyle+(Y+XY^{-1}X)^{-1}VXY^{-1}XY^{-1}-(Y+XY^{-1}X)^{-1}XY^{-1}VXY^{-1}-{}^{t}VY^{-1}.$

Then, (3.31) can be checked in the same way as (3.29). ∎

######  Remark 3.13.

When one of (hence, all) the conditions in Proposition 3.12 holds,
$\left({\widetilde{M}},\omega_{0},J,g\right)$ is a Kähler manifold and $J$
induces a natural holomorphic structure on ${\widetilde{L}}$ such that
$\nabla^{\widetilde{L}}$ is the canonical connection.

###  3.4. The $\Gamma$-equivariant case

Suppose that $\pi_{0}\colon({\widetilde{M}},N\omega_{0},J)\to\mathbb{R}^{n}$
with prequantum line bundle
$\left({\widetilde{L}},\nabla^{\widetilde{L}}\right)^{\otimes
N}\to({\widetilde{M}},N\omega_{0},J)$ is equipped with an action of a group
$\Gamma$ which preserves all the data, and the $\Gamma$-actions are described
by (2.3) and (2.5) as before. We assume that the $\Gamma$-action $\rho$ on
$\mathbb{R}^{n}$ is properly discontinuous and free. Since the $\Gamma$-action
preserves all the data, the Spinc Dirac operator $D$ is $\Gamma$-equivariant.
In particular, $\Gamma$ acts on $\Gamma\left({\widetilde{L}}^{\otimes
N}\right)\cap\ker D$.

######  Lemma 3.14.

Let $s=\sum_{m\in\mathbb{Z}^{n}}a_{m}(x)e^{2\pi\sqrt{-1}m\cdot y}$ be a
section of ${\widetilde{L}}^{\otimes N}$. $s$ is $\Gamma$-equivariant, i.e.,
${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}\circ
s=s\circ{\widetilde{\rho}}_{\gamma}$ for all $\gamma\in\Gamma$ if and only if
$a_{m}$ satisfies the following condition

(3.32)
$a_{N\rho_{\gamma}\left(\frac{m}{N}\right)}\left(\rho_{\gamma}(x)\right)=g_{\gamma}a_{m}(x)e^{2\pi\sqrt{-1}N\left\\{{\widetilde{g}}_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)\right\\}}$

for all $\gamma\in\Gamma$, $m\in\mathbb{Z}^{n}$, and $x\in\mathbb{R}^{n}$. In
particular, any $\Gamma$-equivariant section of ${\widetilde{L}}^{\otimes N}$
can be written as follows

(3.33)
$s(x,y)=\sum_{\left(\gamma,\frac{m}{N}\right)\in\Gamma\times\left(F\cap\frac{1}{N}\mathbb{Z}^{n}\right)}g_{\gamma}a_{m}\left(\rho_{\gamma^{-1}}(x)\right)e^{2\pi\sqrt{-1}N\left\\{{\widetilde{g}}_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)\right\\}}e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y}.$

###### Proof.

By computing the both sides separately, we have

$\displaystyle{\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}\circ
s(x,y)$
$\displaystyle=g_{\gamma}e^{2\pi\sqrt{-1}N\left\\{{\widetilde{g}}_{\gamma}(x)+c_{\gamma}\cdot{}^{t}A_{\gamma}^{-1}y\right\\}}\sum_{m\in\mathbb{Z}^{n}}a_{m}(x)e^{2\pi\sqrt{-1}m\cdot
y}$ (3.34)
$\displaystyle=g_{\gamma}\sum_{m\in\mathbb{Z}^{n}}a_{m}(x)e^{2\pi\sqrt{-1}N{\widetilde{g}}_{\gamma}(x)}e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot{}^{t}A_{\gamma}^{-1}y},$
$\displaystyle s\circ{\widetilde{\rho}}_{\gamma}(x,y)$
$\displaystyle=\sum_{l\in\mathbb{Z}^{n}}a_{l}\left(\rho_{\gamma}(x)\right)e^{2\pi\sqrt{-1}l\cdot\left({}^{t}A_{\gamma}^{-1}y+u_{\gamma}(x)\right)}$
(3.35)
$\displaystyle=\sum_{m\in\mathbb{Z}^{n}}a_{N\rho_{\gamma}\left(\frac{m}{N}\right)}\left(\rho_{\gamma}(x)\right)e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)}e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot{}^{t}A_{\gamma}^{-1}y}.$

Here, in the last equality, we replace $l$ with
$N\rho_{\gamma}\left(\frac{m}{N}\right)$. Note that the map $\mathbb{Z}^{n}\ni
m\mapsto N\rho_{\gamma}\left(\frac{m}{N}\right)\in\mathbb{Z}^{n}$ is
bijective. Then,
${\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}\circ
s=s\circ{\widetilde{\rho}}_{\gamma}$ for all $\gamma\in\Gamma$ implies

$g_{\gamma}a_{m}(x)e^{2\pi\sqrt{-1}N{\widetilde{g}}_{\gamma}(x)}=a_{N\rho_{\gamma}\left(\frac{m}{N}\right)}\left(\rho_{\gamma}(x)\right)e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)}$

for all $m\in\mathbb{Z}^{n}$. In particular, by (3.1) and (3.32), $s$ can be
rewritten as follows

$\displaystyle s(x,y)$
$\displaystyle=\sum_{l\in\mathbb{Z}^{n}}a_{l}(x)e^{2\pi\sqrt{-1}l\cdot y}$
$\displaystyle\stackrel{{\scriptstyle\eqref{Gamma times
F}}}{{=}}\sum_{\left(\gamma,\frac{m}{N}\right)\in\Gamma\times\left(F\cap\frac{1}{N}\mathbb{Z}^{n}\right)}a_{N\rho_{\gamma}\left(\frac{m}{N}\right)}(x)e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y}$ $\displaystyle\stackrel{{\scriptstyle\eqref{Gamma-
invariance}}}{{=}}\sum_{\left(\gamma,\frac{m}{N}\right)\in\Gamma\times\left(F\cap\frac{1}{N}\mathbb{Z}^{n}\right)}g_{\gamma}a_{m}\left(\rho_{\gamma^{-1}}(x)\right)e^{2\pi\sqrt{-1}N\left\\{{\widetilde{g}}_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)\right\\}}e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y}.$

∎

In the $\Gamma$-equivariant case, the condition (3.13) has a symmetry in the
following sense.

######  Lemma 3.15.

The condition (3.13) holds for some $m_{0}\in\mathbb{Z}^{n}$ with
$\frac{m_{0}}{N}\in F$ if and only if for any $\gamma\in\Gamma$, (3.13) holds
for $m=N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)$. Moreover, let $a_{m_{0}}$