# The duality covariant geometry and DSZ quantization of abelian gauge theory

C. Lazaroiu  Department of Theoretical Physics, Horia Hulubei National
Institute for Physics and Nuclear Engineering, Bucharest-Magurele, Romania
<EMAIL_ADDRESS>and  C. S. Shahbazi  Department of Mathematics,
University of Hamburg, Germany<EMAIL_ADDRESS>
###### Abstract.

We develop the Dirac-Schwinger-Zwanziger (DSZ) quantization of classical
abelian gauge theories with general duality structure on oriented and
connected Lorentzian four-manifolds $(M,g)$ of arbitrary topology, obtaining
as a result the duality-covariant geometric formulation of such theories
through connections on principal bundles. We implement the DSZ condition by
restricting the field strengths of the theory to those which define classes
originating in the degree-two cohomology of a local system valued in the
groupoid of integral symplectic spaces. We prove that such field strengths are
curvatures of connections $\mathcal{A}$ defined on principal bundles $P$ whose
structure group $G$ is the disconnected non-abelian group of automorphisms of
an integral affine symplectic torus. The connected component of the identity
of $G$ is a torus group, while its group of connected components is a modified
Siegel modular group which coincides with the group of local duality
transformations of the theory. This formulation includes electromagnetic and
magnetoelectric gauge potentials on an equal footing and describes the
equations of motion through a first-order polarized self-duality condition for
the curvature of $\mathcal{A}$. The condition involves a combination of the
Hodge operator of $(M,g)$ with a taming of the duality structure determined by
$P$, whose choice encodes the self-couplings of the theory. This description
is reminiscent of the theory of four-dimensional euclidean instantons, even
though we consider a two-derivative theory in Lorentzian signature. We use
this formulation to characterize the hierarchy of duality groups of abelian
gauge theory, providing a gauge-theoretic description of the electromagnetic
duality group as the discrete remnant of the gauge group of $P$. We also
perform the time-like reduction of the polarized self-duality condition to a
Riemannian three-manifold, obtaining a new type of Bogomolny equation which is
modified by the given taming and duality structure induced by $P$. We give
explicit examples of such solutions, which we call _polarized dyons_.

###### Key words and phrases:

Abelian gauge theory, Electromagnetic duality, symplectic vector bundles,
character varieties, symplectic lattices

2010 MSC. Primary: 53C80. Secondary: 53C07.

###### Contents

  1. 1 Classical abelian gauge theory
  2. 2 The Dirac-Schwinger-Zwanziger condition
  3. 3 Siegel bundles and connections
  4. 4 Prequantum abelian gauge theory
  5. 5 Time-like dimensional reduction and polarized Bogomolny equations
  6. A Local abelian gauge theory
  7. B Integral symplectic spaces and integral symplectic tori

## Introduction

Abelian gauge theory on Lorentzian four-manifolds is a natural extension of
Maxwell electrodynamics, which locally describes a finite number of abelian
gauge fields interacting through couplings which are allowed to vary over
space-time. Such theories occur frequently in high energy physics. For
instance, the low energy limit of a non-abelian gauge theory coupled to scalar
fields which can be maximally higgsed contains an abelian gauge theory sector;
this occurs in particular on the Coulomb branch of supersymmetric non-abelian
gauge theories. Moreover, the universal bosonic sector of four-dimensional
ungauged supergravity involves a fixed number of abelian gauge fields
interacting with each other through couplings which can vary over space-time
due to their dependence on the scalars of the theory. This sector of four-
dimensional supergravity can be described by pulling back an abelian gauge
theory defined on the target space of the sigma model of the scalar fields
[32].

The local behavior of abelian gauge theories (including their supersymmetric
extensions) was studied intensively in the physics literature, where the
subject has achieved the level of textbook material [16]. Despite intense
activity, a global geometric formulation of such theories on arbitrary
spacetimes is still missing. At the level of field strengths, such a
description was given in [32] (see [34] for a summary) in the wider context of
the geometric description of the universal sector of classical supergravity in
four dimensions. As explained there and recalled in Section 1, the global
formulation requires the specification of a “duality structure”, defined as a
flat symplectic vector bundle $\Delta=(\mathcal{S},\omega,\mathcal{D})$ on the
spacetime manifold $M$. The even rank $2n$ of $\mathcal{S}$ equals the number
of field strengths, where both electromagnetic and magnetoelectric fields are
included. When the spacetime is not simply connected, such a bundle need not
be trivial and it ‘twists’ the local formulation in such a way that the
combination of all electromagnetic and magnetoelectric field strengths can be
described globally by a $\mathrm{d}_{\mathcal{D}}$-closed two-form
$\mathcal{V}$ defined on $M$ and valued in $\mathcal{S}$, where
$\mathrm{d}_{\mathcal{D}}$ is the differential induced by the flat connection
$\mathcal{D}$ of the duality structure. A classical electromagnetic field
strength configuration is a $\mathrm{d}_{\mathcal{D}}$-closed two-form
$\mathcal{V}$. The classical equations of motion are encoded by the condition
that $\mathcal{V}$ be self-dual with respect to a ‘polarized Hodge operator’
obtained by tensoring the Hodge operator $\ast_{g}$ defined by the Lorentzian
spacetime metric $g$ with a (generally non-flat) taming $\mathcal{J}$ of the
symplectic bundle $(\mathcal{S},\omega)$, an object which encodes all
couplings and theta angles in a fully geometric manner. A classical field
strength solution is a field strength configuration which obeys the polarized
self-duality condition. This provides a global formulation of the theory on
oriented Lorentzian four manifolds of arbitrary topology, which is manifestly
covariant with respect to electromagnetic duality. In this formulation,
classical duality transformations are described by (based) flat symplectic
automorphisms of $\Delta$. Such theories admits global solutions which
correspond to ‘classical electromagnetic U-folds’ – a notion which had been
used previously in the physics literature without being given a mathematically
clear definition.

While the treatment found in the physics literature discusses local gauge
potentials and local gauge transformations (which are described using
differential forms defined locally on spacetime), a fully general and
manifestly duality-covariant geometric formulation of such theories in terms
of connections on an appropriate principal bundle has not yet been given. Such
a formulation is required by the Aharonov-Bohm effect [1] and by Dirac-
Schwinger-Zwanziger (DSZ) “quantization” [23, 42, 47], which force the field
strengths to obey an _integrality condition_ implied by the requirement of a
consistent coupling between classical gauge fields and quantum charge
carriers. Imposing this condition restricts the set of allowed field
strengths, defining a so-called prequantum abelian gauge theory As pointed out
in [32], the general formulation of this condition involves the choice of a
Dirac system $\mathcal{L}$ for $\Delta$, defined as a $\mathcal{D}$-flat fiber
sub-bundle of $\mathcal{S}$ whose fibers are full symplectic lattices inside
the fibers of $\mathcal{S}$. Every Dirac system has a type
$\mathfrak{t}=(t_{1},\ldots,t_{n})$, where $t_{1},\ldots,t_{n}$ are positive
integers such that $t_{1}|t_{2}|\ldots|t_{n}$. A duality structure is called
semiclassical if it admits a Dirac system. The choice of a Dirac system
$\mathcal{L}$ refines a semiclassical duality structure $\Delta$ to an
integral duality structure
${\bm{\Delta}}=(\mathcal{S},\omega,\mathcal{D},\mathcal{L})$ and reduces the
group of duality transformations to a discrete group, which generalizes the
arithmetic duality group known from the local formulation found in the physics
literature. In the global setting, the Dirac system replaces the “Dirac
lattice” of the local approach and makes the DSZ condition is rather subtle
since it requires the use of cohomology with local coefficients (see [28, 43,
48, 44]). In particular, the bundle-valued two-form $\mathcal{V}$ (which
describes all electromagnetic and magnetoelectric field strengths
simultaneously) cannot in general be the curvature of a connection defined on
a principal torus bundle. Indeed, the vector bundle $\mathcal{S}$ is generally
non-trivial, while the adjoint bundle of any principal torus bundle is
trivial.

In the present paper, we ‘solve’ the DSZ integrality condition determined by a
Dirac system $\mathcal{L}$ by giving the geometric formulation of prequantum
abelian gauge theory in terms of connections defined on an adequate principal
bundle $P$. More precisely, we show that the combined field strength
$\mathcal{V}$ is the curvature of a connection defined on a principal bundle
with non-abelian and disconnected structure group $G$, whose connected
component of the identity is a torus group and whose group of connected
components is a modified Siegel modular group. This shows that the manifestly
duality-covariant formulation of such gauge theories is not truly abelian,
since it involves a non-abelian structure group. Instead, the structure group
$G$ is weakly abelian, in the sense that only its Lie algebra is abelian. As a
consequence, the gauge group of $P$ has a “discrete remnant” which encodes
equivariance of the theory under electromagnetic duality transformations.
Using this framework, we show that the electromagnetic duality transformations
of the theory are given by (non-infinitesimal) gauge transformations, a fact
that provides a geometric interpretation for the former. Principal connections
$\mathcal{A}$ on $P$ describe the combined electromagnetic and magnetoelectric
gauge potentials of the theory. The polarized self-duality condition becomes a
first-order differential equation for $\mathcal{A}$ which is reminiscent of
the instanton equations, though the signature of our spacetime is Lorentzian
and the theory is of second order. These results provide a global geometric
formulation of prequantum abelian gauge theory as a theory of principal
connections, in a manner that is manifestly covariant under electromagnetic
duality.

To extract the principal bundle description, we proceed as follows. We first
show that integral duality structures of rank $2n$ and type $\mathfrak{t}$ are
associated to Siegel systems of rank $2n$, which we define to be local systems
$Z$ of free abelian groups of rank $2n$ whose monodromy is contained in the
modified Siegel modular group
$\Gamma=\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ of type $\mathfrak{t}$. The
latter is defined as the group of automorphisms of a full symplectic lattice
of rank $2n$ and type $\mathfrak{t}$ and is a subgroup of
$\mathrm{Sp}(2n,\mathbb{R})$ which contains the usual Siegel modular group
$\mathrm{Sp}(2n,\mathbb{Z})$, to which it reduces when $\mathfrak{t}$
coincides with the principal type $\delta=(1,\ldots,1)$. The vector bundle of
the duality structure is given by
$\mathcal{S}=Z\otimes_{\mathbb{Z}}\mathbb{R}$ while the flat connection
$\mathcal{D}$ is induced by the monodromy connection of $Z$. A classical field
strength configuration $\mathcal{V}$ is called integral if it satisfies the
DSZ quantization condition, which states that the cohomology class
$[\mathcal{V}]_{\mathcal{D}}\in H^{2}_{\mathcal{D}}(M,\mathcal{S})$ of
$\mathcal{V}$ with respect to $\mathrm{d}_{\mathcal{D}}$ lies in the image of
the local coefficient cohomology group $H^{2}(M,Z)$ through the natural
morphism $H^{2}(M,Z)\rightarrow H^{2}_{\mathcal{D}}(M,\mathcal{S})$, where
$H^{2}_{\mathcal{D}}(M,\mathcal{S})$ is the the second cohomology of the
complex $(\Omega^{\ast}(M,\mathcal{S}),\mathrm{d}_{\mathcal{D}})$. We then
show that $\mathcal{V}$ is integral if and only if it coincides with the
adjoint curvature $\mathcal{V}_{\mathcal{A}}$ of a connection $\mathcal{A}$
defined on a principal bundle $P$ (called a Siegel bundle) whose structure
group is the group $G=\mathrm{Aff}_{\mathfrak{t}}$ of automorphisms of an
integral affine symplectic torus and whose adjoint bundle identifies with
$\mathcal{S}$. The group $\mathrm{Aff}_{\mathfrak{t}}$ is a semidirect product
$A\rtimes\Gamma$ of the torus group $A=\mathbb{R}^{2n}/\mathbb{Z}^{2n}$ with
the modified Siegel modular group
$\Gamma=\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ and hence has a countable
group of components which is isomorphic with $\Gamma$. The
$\mathrm{d}_{\mathcal{D}}$-cohomology class of $\mathcal{V}_{\mathcal{A}}$
coincides with the image in $H^{2}_{\mathcal{D}}(M,\mathcal{S})$ of the
twisted Chern class $c(P)\in H^{2}(M,Z)$ of $P$. The Siegel system $Z$ (and
hence the duality structure $\Delta$) is uniquely determined by the Siegel
bundle $P$ and Siegel bundles are determined up to isomorphism by the pair
$(Z,c)$. The classifying space of such principal bundles is a twisted
Eilenberg MacLane space [27] of type $2$, namely a
$K(\mathbb{Z}^{2n},2)$-fibration over
$K(\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z}),1)$ whose $\kappa$-invariant is
trivial and whose monodromy is induced by the fundamental action of
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ on $\mathbb{Z}^{2n}$.

There exists a large group of classical ‘pseudo-duality’ transformations which
identifies the spaces of classical field strength configurations and solutions
of _different_ abelian gauge theories. Locally, such transformations involve
matrices $T\in\mathrm{Sp}(2n,\mathbb{R})$ acting on the field strengths and
the DSZ integrality condition with respect to a Dirac lattice of type
$\mathfrak{t}$ restricts $T$ to lie in the arithmetic group
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$. As already pointed out in [32],
the global theory of such duality transformations is much richer. We develop
this theory from scratch, defining a hierarchy of duality groups and providing
short exact sequences to compute them. We exploit this geometric framework to
show that the electromagnetic duality transformations of abelian gauge theory
correspond to gauge transformations of Siegel bundles, elucidating the
geometric origin of electromagnetic duality. In particular, we emphasize the
role played by the _type_ $\mathfrak{t}$ of the underlying Dirac system and by
the monodromy representation of the Siegel system $Z$ in the correct
definition and computation of the discrete duality group of the prequantum
theory. The fact that a symplectic lattice need not be of principal type is
well-known in the theory of symplectic tori as well as in that of Abelian
varieties, where non-principal types correspond to non-principal
polarizations. The physical implications of non-principal types have been
systematically explored only recently in the context of supersymmetric field
theories, see for instance [6, 7, 4, 5, 8, 14, 15] and references therein.

The construction of the present paper produces a class of geometric gauge
models which is amenable to the methods of mathematical gauge theory [24]. In
particular, it allows for the study of moduli spaces of solutions (which, as
shown in [32], can be viewed as ‘electromagnetic U-folds’) using techniques
borrowed from the theory of instantons. In this spirit, we perform the time-
like reduction of the polarized self-duality equations, obtaining a novel
system of Bogomolny-like equations. Solutions of these equations define
polarized abelian dyons, of which we describe a few examples.

The paper is organized as follows. Section 1 recalls the description of
classical abelian gauge theories with arbitrary duality structure in terms of
combined electromagnetic and magnetoelectric field strengths, following [32].
In the same section, we describe the hierarchy of duality groups of such
theories and give a few short exact sequences to characterize them. Section 2
discusses the DSZ integrality condition for general duality structures,
relating the notion of Dirac system (which appears in its formulation) to
various equivalent objects. Section 3 discusses Siegel bundles and connections
showing in particular that a Siegel induces a canonical integral duality
structure. In Section 4, we give the formulation of “prequantum” abelian gauge
theory (defined as classical abelian gauge theory supplemented by the DSZ
integrality condition) as a theory of principal connections on a Siegel
bundle. Section 5 discusses the time-like dimensional reduction of the
polarized self-duality equations, which leads to the notion of polarized
abelian dyon. In the same section, we construct examples of polarized abelian
dyons on the punctured affine 3-space. Appendix A recalls the duality-
covariant formulation of abelian gauge theory on contractible Lorentzian four-
manifolds, starting from the local treatment found in the physics literature.
Appendix B discusses integral symplectic spaces and integral symplectic tori,
introducing certain notions used in the main text.

###  0.1. Notations and conventions

All manifolds and fiber bundles considered in the paper are smooth, Hausdorff
and paracompact. The manifold denoted by $M$ is assumed to be connected. In
our convention, a Lorentzian four-manifold $(M,g)$ has “mostly plus” signature
$(3,1)$. If $E$ is a fiber bundle defined on a manifold $M$, we denote by
$\mathcal{C}^{\infty}(M,E)$ the set of globally-defined smooth sections of $E$
and by $\mathcal{C}^{\infty}(E)$ the sheaf of smooth sections of $E$.

######  Acknowledgements.

We thank V. Cortés for useful comments and discussions. C.S.S would like to
thank P. C. Argyres, A. Bourget, M. Martone and R. Szabo for useful comments
and suggestions. Part of the work of C. I. L. on this project was supported by
grant IBS-R003-S1. The work of C.S.S. is supported by the Germany Excellence
Strategy _Quantum Universe_ \- 390833306.

##  1\. Classical abelian gauge theory

In this section we introduce the configuration space and equations of motion
defining classical abelian gauge theory on an oriented Lorentzian four-
manifold $(M,g)$ and discuss its global dualities and symmetries. Appendix A
gives the description of this theory for the special case when $M$ is
contractible, which recovers the local treatment found in the physics
literature. The global formulation presented in this section was proposed in a
wider context in reference [32], to which we refer the reader for certain
details. The definition of classical abelian gauge theory on $(M,g)$ is given
in terms of field strengths and relies on the choice of a _duality structure_
(defined as a flat symplectic vector bundle
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ on $M$) equipped with a taming
$\mathcal{J}$ of $(\mathcal{S},\omega)$, which encodes the gauge-kinetic
functions (coupling constants and theta angles) in a globally-correct and
frame-independent manner. The equations of motion of the theory are encoded by
the $\mathcal{J}$-polarized self-duality condition for the combined
electromagnetic and magnetoelectric field strengths, which are modeled
mathematically by a $\mathcal{D}$-flat two-form valued in the underlying
vector bundle $\mathcal{S}$ of the duality structure.

###  1.1. Duality structures

Let $M$ be a connected smooth manifold and $(\mathcal{S},\omega)$ be a
symplectic vector bundle defined on $M$ with symplectic structure $\omega$.
Since $\mathrm{Sp}(2n,\mathbb{R})$ and $\mathrm{GL}(n,\mathbb{C})$ are
homotopy equivalent to their common maximal compact subgroup $\mathrm{U}(n)$,
the classification of symplectic, complex and Hermitian vector bundles defined
on $M$ are equivalent. In particular, any complex vector bundle admits a
Hermitian pairing and any symplectic vector bundle admits a complex structure
which is compatible with its symplectic pairing and makes it into a Hermitian
vector bundle. Thus a real vector bundle of even rank admits a symplectic
pairing if and only if it admits a complex structure. The classifying spaces
$\mathrm{B}\mathrm{Sp}(2n,\mathbb{R})$ and $\mathrm{B}\mathrm{U}(n)$ are
homotopy equivalent, hence the fundamental characteristic classes of a
symplectic vector bundle $(\mathcal{S},\omega)$ are Chern classes, which we
denote by $c_{k}(\mathcal{S},\omega)$.

######  Remark 1.1.

Suppose that $\dim M=4$ and let $\mathcal{S}$ be an oriented real vector
bundle of rank $2n$ defined on $M$, thus $w_{1}(\mathcal{S})=0$. If
$\mathcal{S}$ admits a complex structure $\mathcal{J}$ inducing its
orientation, then its third Stiefel-Whitney class $w_{3}(\mathcal{S})$ must
vanish and its even Stiefel-Whitney classes $w_{2}(\mathcal{S})$ and
$w_{4}(\mathcal{S})$ must coincide with the $\mathrm{mod}~{}2$ reduction of
the Chern classes $c_{1}(\mathcal{S},\mathcal{J})$ and
$c_{2}(\mathcal{S},\mathcal{J})$ of the complex rank $n$ vector bundle defined
by $\mathcal{S}$ and $\mathcal{J}$. In particular, the third integral Stiefel-
Whitney class $W_{3}(\mathcal{S})\in H^{3}(M,\mathbb{Z})$ must vanish, i.e.
$\mathcal{S}$ must admit a $\mathrm{Spin}^{c}$ structure (notice that
$W_{5}(\mathcal{S})$ vanishes for dimension reasons). These conditions are not
always sufficient. To state the necessary and sufficient conditions (see
[37]), we distinguish the cases:

  * •

$n=1$, i.e. ${\rm rk}\mathcal{S}=2$. Then $\mathcal{S}$ always admits a
complex structure (equivalently, a symplectic pairing) which induces its
orientation (since $\mathrm{SO}(2)=\mathrm{U}(1)$).

  * •

$n=2$, i.e. ${\rm rk}\mathcal{S}=4$. In this case, $\mathcal{S}$ admits a
complex structure (equivalently, a symplectic pairing) which induces its
orientation if and only if it satisfies $W_{3}(\mathcal{S})=0$ and
$\mathfrak{c}^{4}(\mathcal{S})=0$, where $\mathfrak{c}^{4}(\mathcal{S})\in
H^{4}(M,\mathbb{Z})$ is an integral obstruction class described in [37,
Theorem II].

  * •

$n\geq 3$, i.e. ${\rm rk}\mathcal{S}\geq 6$. Then $\mathcal{S}$ admits a
complex structure (equivalently, a symplectic pairing $\omega$) which induces
its orientation if and only if $W_{3}(\mathcal{S})=0$.

Notice that an oriented real vector bundle $\mathcal{S}$ of rank four on a
four-manifold $M$ is determined up to isomorphism by its first Pontryaghin
class $p_{1}(\mathcal{S})\in H^{4}(M,\mathbb{Z})$, its second Stiefel-Whitney
class $w_{2}(\mathcal{S})\in H^{2}(M,\mathbb{Z}_{2})$ and its Euler class
$e(\mathcal{S})\in H^{4}(M,\mathbb{Z})$.

######  Definition 1.2.

A duality structure $\Delta\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\mathcal{S},\omega,\mathcal{D})$ on $M$ is a flat symplectic
vector bundle $(\mathcal{S},\omega)$ over $M$ equipped with a flat connection
$\mathcal{D}:\mathcal{C}^{\infty}(M,\mathcal{S})\rightarrow\Omega^{1}(M,\mathcal{S})$
which preserves $\omega$. The rank of $\Delta$ is the rank of the vector
bundle $\mathcal{S}$, which is necessarily even.

Notice that the Chern classes $c_{1}(\mathcal{S},\omega)$ and
$c_{2}(\mathcal{S},\omega)$ of the underlying symplectic vector bundle of a
duality structure must be torsion classes.

######  Definition 1.3.

A based isomorphism of duality structures from
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ to
$\Delta^{\prime}=(\mathcal{S}^{\prime},\omega^{\prime},\mathcal{D}^{\prime})$
is a based isomorphism of vector bundles
$f:\mathcal{S}\xrightarrow{\sim}\mathcal{S}^{\prime}$ which satisfies the
conditions $\omega^{\prime}\circ(f\otimes f)=\omega$ and
$\mathcal{D}^{\prime}\circ f=(\mathrm{id}_{T^{\ast}M}\otimes
f)\circ\mathcal{D}$.

Here and below, we let based morphisms of vector bundles act on sections in
the natural manner. We denote by $\mathrm{Dual}(M)$ be the groupoid of duality
structures defined on $M$ and based isomorphisms of such.

The group $\mathrm{Aut}_{b}(\mathcal{S},\omega)$ of based automorphisms of a
symplectic vector bundle $(\mathcal{S},\omega)$ is called its group of
symplectic gauge transformations. Such transformations $\varphi$ act on the
set of linear connections $\mathcal{D}$ defined on $\mathcal{S}$ through:

$\mathcal{D}\rightarrow(\mathrm{id}_{T^{\ast}M}\otimes\varphi)\circ\mathcal{D}\circ\varphi^{-1}~{}~{}$

and preserve the set of flat symplectic connections. The group
$\mathrm{Aut}_{b}(\Delta)$ of based automorphism of a duality structure
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ coincides with the stabilizer of
$\mathcal{D}$ in $\mathrm{Aut}_{b}(\mathcal{S},\omega)$. For any such duality
structure, we have:

$c_{1}(\mathcal{S},\omega)=\delta({\hat{c}}_{1}(\mathcal{D}))\,,\quad
c_{2}(\mathcal{S},\omega)=\delta({\hat{c}}_{2}(\mathcal{D}))~{}~{},$

where ${\hat{c}}_{1}(\mathcal{D})\in H^{1}(M,\mathrm{U}(1))$ and
${\hat{c}}_{2}(\mathcal{D})\in H^{3}(M,\mathrm{U}(1))$ are the Cheeger-Chern-
Simons invariants of the flat connection $\mathcal{D}$ and
$\delta:H^{i}(M,\mathbb{C}/\mathbb{Z})\rightarrow H^{i+1}(M,\mathbb{Z})$ are
the Bockstein morphisms in the long exact sequence:

$\ldots\rightarrow H^{i}(M,\mathbb{Z})\rightarrow
H^{i}(M,\mathbb{R})\stackrel{{\scriptstyle\exp_{\ast}}}{{\rightarrow}}H^{i}(M,\mathrm{U}(1))\stackrel{{\scriptstyle\delta}}{{\rightarrow}}H^{i+1}(M,\mathbb{Z})\rightarrow
H^{i+1}(M,\mathbb{R})\rightarrow\ldots$

induced by the exponential sequence:

$0\rightarrow\mathbb{Z}\rightarrow\mathbb{R}\rightarrow\mathrm{U}(1)\rightarrow
0~{}~{}.$

The Cheeger-Chern-Simons invariants depend only on the gauge equivalence class
of $\mathcal{D}$.

###  1.2. The twisted de Rham complex of a duality structure

Given a duality structure $\Delta=(\mathcal{S},\omega,\mathcal{D})$ on a
connected manifold $M$, let
$\mathrm{d}_{\mathcal{D}}:\Omega(M,\mathcal{S})\rightarrow\Omega(M,\mathcal{S})$
be the exterior covariant derivative twisted by $\mathcal{D}$ (notice that
$\mathrm{d}_{\mathcal{D}}|_{\Omega^{0}(M,\mathcal{S})}=\mathcal{D}$). This
defines a cochain complex:

$0\to\mathcal{C}^{\infty}(M,\mathcal{S})\xrightarrow{\mathcal{D}}\Omega^{1}(M,\mathcal{S})\xrightarrow{\mathrm{d}_{\mathcal{D}}}\Omega^{2}(M,\mathcal{S})\xrightarrow{\mathrm{d}_{\mathcal{D}}}\Omega^{3}(M,\mathcal{S})\xrightarrow{\mathrm{d}_{\mathcal{D}}}\Omega^{4}(M,\mathcal{S})\to
0\,,$ (1)

whose total cohomology group (viewed as a $\mathbb{Z}$-graded abelian group)
we denote by $H_{\mathcal{D}}^{\ast}(M,\mathcal{S})$.

######  Definition 1.4.

The vector spaces $H_{\mathcal{D}}^{k}(M,\mathcal{S})$ (where $k=0,\ldots,4$)
are called the twisted de Rham cohomology spaces of the duality structure
$\Delta$.

Let $\Omega^{k}_{\mathrm{flat}}(\mathcal{S})$ be the locally-constant sheaf of
$\mathcal{D}$-flat $\mathcal{S}$-valued $k$-forms. Then a straightforward
modification of the Poincaré lemma shows that the complex of sheaves:

$0\to\mathcal{C}^{\infty}_{\mathrm{flat}}(\mathcal{S})\hookrightarrow\Omega^{0}(\mathcal{S})\xrightarrow{\mathcal{D}}\Omega^{1}(\mathcal{S})\xrightarrow{\mathrm{d}_{\mathcal{D}}}\Omega^{2}(\mathcal{S})\xrightarrow{\mathrm{d}_{\mathcal{D}}}\Omega^{3}(\mathcal{S})\xrightarrow{\mathrm{d}_{\mathcal{D}}}\Omega^{4}(\mathcal{S})\to
0$

is exact and hence provides a resolution of the locally-constant sheaf
$\mathcal{C}^{\infty}_{\mathrm{flat}}(\mathcal{S})$. Since $M$ is paracompact,
each of the sheaves $\Omega^{k}(\mathcal{S})$ is acyclic. Thus the sheaf
cohomology of $\Omega^{0}_{\mathrm{flat}}(\mathcal{S})$ can be computed as the
cohomology of the complex (1). This gives a natural isomorphism of graded
vector spaces:

$H^{\ast}_{\mathcal{D}}(M,\mathcal{S})\simeq
H^{\ast}(M,\mathcal{C}^{\infty}_{\mathrm{flat}}(\mathcal{S}))~{}~{},$

where the right hand side denotes sheaf cohomology.

###  1.3. Flat systems of symplectic vector spaces

Let $\Pi_{1}(M)$ be the fundamental groupoid of a connected manifold $M$,
whose objects are the points of $M$ and whose set of morphisms
$\Pi_{1}(m,m^{\prime})$ from $m$ to $m^{\prime}$ is the set of homotopy
classes of piecewise-smooth curves starting at $m$ and ending at $m^{\prime}$.
Let $\mathrm{Symp}$ be the groupoid of finite-dimensional symplectic vector
spaces (see Appendix B). The functor category $[\Pi_{1}(M),\mathrm{Symp}]$ is
a groupoid since all its morphisms (which are natural transformations) are
invertible.

######  Definition 1.5.

A flat system of symplectic vector spaces (or $\mathrm{Symp}$-valued local
system) on $M$ is a functor $F:\Pi_{1}(M)\rightarrow\mathrm{Symp}$, i.e. an
object of the groupoid $[\Pi_{1}(M),\mathrm{Symp}]$. An isomorphism of such
systems is an isomorphism in this groupoid. A flat system $F$ of symplectic
vector spaces has rank111The rank is constant on $M$ since $M$ is connected.
$2n$ if $\dim F(m)=2n$ for all $m\in M$.

Recall that a symplectic representation of a group $G$ is a representation
$\rho:G\rightarrow\mathrm{Aut}(V,\omega)$ of $G$ through automorphisms of a
finite-dimensional symplectic vector space $(V,\omega)$. An equivalence (or
isomorphism) of symplectic representations from
$\rho:G\rightarrow\mathrm{Aut}(V,\omega)$ to
$\rho^{\prime}:G\rightarrow\mathrm{Aut}(V^{\prime},\omega^{\prime})$ is an
isomorphism of symplectic vector spaces
$\varphi:(V,\omega)\xrightarrow{\sim}(V^{\prime},\omega^{\prime})$ such that
$\varphi\circ\rho(g)=\rho^{\prime}(g)\circ\varphi^{\prime}$ for all $g\in G$.
We denote by $\mathrm{SympRep}(G)$ the groupoid of symplectic representations
of $G$ equipped with this notion of isomorphism.

######  Definition 1.6.

Let $F$ be a flat system of symplectic vector spaces on $M$. The holonomy
representation of $F$ at a point $m\in M$ is the morphism of groups
$\mathrm{hol}_{m}(F):\pi_{1}(M,m)=\Pi_{1}(m,m)\rightarrow\mathrm{Aut}(F(m))=\mathrm{Symp}(F(m),F(m))$
defined through:

$\mathrm{hol}_{m}(F)(\mathbf{c})=F(\mathbf{c})\,,\quad\forall\,\,\mathbf{c}\in\pi_{1}(M,m)~{}~{}.$

The holonomy group of $F$ at $m$ is the subgroup of $\mathrm{Aut}(F(m))$
defined through:

$\mathrm{Hol}_{m}(F)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{im}(\mathrm{hol}_{m}(F))~{}~{}.$

Notice that $\mathrm{hol}_{m}(F)$ is a symplectic representation of
$\pi_{1}(M,m)$ on $F(m)$. Any homotopy class
$\bm{\gamma}\in\Pi_{1}(m,m^{\prime})$ of paths from $m$ to $m^{\prime}$
induces an isomorphism of symplectic vector spaces
$F(\bm{\gamma}):F(m)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}F(m^{\prime})$
which intertwines the holonomy representations of $F$ at the points $m$ and
$m^{\prime}$:

$F(\bm{\gamma})\circ\mathrm{hol}_{m}(F)(\mathbf{c})=\mathrm{hol}_{m^{\prime}}(F)(\bm{\gamma}\,\mathbf{c}\,\bm{\gamma}^{-1})\circ
F(\bm{\gamma})\,,\quad\forall\,\,\mathbf{c}\in\pi_{1}(M,m)~{}~{}.$

Since $M$ is connected, it follows that the holonomy representation of $F$ at
a fixed basepoint $m_{0}\in M$ determines its holonomy representation at any
other point of $M$. Moreover, the isomorphism class of the holonomy group of
$F$ at $m_{0}$ does not depend on the choice of $m_{0}\in M$.

######  Proposition 1.7.

Let $m_{0}\in M$ be any point of $M$. Then the map $\mathrm{hol}_{m_{0}}$
which sends a $\mathrm{Symp}$-valued local system $F$ defined on $M$ to its
holonomy representation $\mathrm{hol}_{m_{0}}(F)$ at $m_{0}$ and sends an
isomorphism $f:F\xrightarrow{\sim}F^{\prime}$ of $\mathrm{Symp}$-valued local
systems to the equivalence of representations
$\mathrm{hol}_{m_{0}}(f)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}f(m_{0}):\mathrm{hol}_{m_{0}}(F)\xrightarrow{\sim}\mathrm{hol}_{m_{0}}(F^{\prime})$
defines an equivalence of groupoids from $[\Pi_{1}(M),\mathrm{Symp}]$ to
$\mathrm{SympRep}(\pi_{1}(M,m_{0}))$. In particular, isomorphism classes of
flat systems of symplectic vector spaces of rank $2n$ defined on $M$ are in
bijection with the points of the character variety:

$\mathfrak{R}(\pi_{1}(M,m_{0}),\mathrm{Sp}(2n,\mathbb{R}))\stackrel{{\scriptstyle{\rm
def.}}}{{=}}{\rm
Hom}(\pi_{1}(M,m_{0}),\mathrm{Sp}(2n,\mathbb{R}))/\mathrm{Sp}(2n,\mathbb{R})\,,$
(2)

where $\mathrm{Sp}(2n,\mathbb{R})$ acts on ${\rm
Hom}(\pi_{1}(M),\mathrm{Sp}(2n,\mathbb{R}))$ through its adjoint
representation.

###### Proof.

An isomorphism $f:F\xrightarrow{\sim}F^{\prime}$ of $\mathrm{Symp}$-valued
local systems is a collection of isomorphisms of symplectic vector spaces
$f(m):F(m)\rightarrow F^{\prime}(m)$ for all $m\in M$ which satisfies:

$f(m)\circ F(\bm{\gamma})=F^{\prime}(\bm{\gamma})\circ
f(m)\,,\quad\forall\,\,\bm{\gamma}\in\Pi_{1}(m,m^{\prime})\,,\quad\forall\,\,m,m^{\prime}\in
M~{}~{}.$

Taking $m^{\prime}=m=m_{0}$ in this relation shows that $f(m)$ is an
equivalence between the holonomy representations of $F$ and $F^{\prime}$ at
$m$:

$\mathrm{hol}_{m_{0}}(F^{\prime})(\mathbf{c})=f(m)\circ\mathrm{hol}_{m_{0}}(F)(\mathbf{c})\circ
f(m_{0})^{-1}\,,\quad\forall\,\,\mathbf{c}\in\pi_{1}(M,m_{0})~{}~{}.$

Conversely, it is easy to see that any such equivalence of representations
extends to an isomorphism from $F$ to $F^{\prime}$. ∎

###  1.4. The flat system of symplectic vector spaces defined by a duality
structure

######  Definition 1.8.

The parallel transport functor
$\mathcal{T}_{\Delta}\in[\Pi_{1}(M),\mathrm{Symp}]$ of a duality structure
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ is the functor which associates to
each point $m\in M$ the symplectic vector space
$\mathcal{T}_{\Delta}(m)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\mathcal{S}_{m},\omega_{m})$ and to the homotopy class (with
fixed endpoints) $\mathbf{c}\in\Pi_{1}(M)$ of any piecewise-smooth curve
$c\colon[0,1]\to M$ the isomorphism of symplectic vector spaces:

$\mathcal{T}_{\Delta}(\mathbf{c}):(\mathcal{S}_{c(0)},\omega_{c(0)})\xrightarrow{\sim}(\mathcal{S}_{c(1)},\omega_{c(1)})$

given by the parallel transport of $\mathcal{D}$.

Notice that $\mathcal{T}_{\Delta}:\Pi_{1}(M)\rightarrow\mathrm{Symp}$ is a
flat system of symplectic vector spaces. The map which sends $\Delta$ to
$\mathcal{T}_{\Delta}$ extends in an obvious manner to an equivalence of
groupoids:

$\mathcal{T}:\mathrm{Dual}(M)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}[\Pi_{1}(M),\mathrm{Symp}]~{}~{},$

which sends a based isomorphism
$f:\Delta=(\mathcal{S},\omega,\mathcal{D})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Delta^{\prime}=(\mathcal{S}^{\prime},\omega^{\prime},\mathcal{D}^{\prime})$
of duality structures to the invertible natural transformation
$\mathcal{T}(f):\mathcal{T}_{\Delta}\xrightarrow{\sim}\mathcal{T}_{\Delta^{\prime}}$
given by the isomorphisms of symplectic vector spaces:

$\mathcal{T}(f)(m)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}f_{m}:(\mathcal{S}_{m},\omega_{m})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathcal{S}^{\prime}_{m},\omega_{m})\,,\quad\forall\,\,m\in
M\,.$

These isomorphisms intertwine $\mathcal{T}^{\Delta}(\mathbf{c})$ and
$\mathcal{T}^{\Delta^{\prime}}(\mathbf{c})$ for any
$\mathbf{c}\in\Pi_{1}(M)(m,m^{\prime})$ since $f$ satisfies
$\mathcal{D}^{\prime}\circ f=(\mathrm{id}_{T^{\ast}M}\otimes
f)\circ\mathcal{D}$. Hence one can identify duality structures and systems of
flat symplectic vector spaces defined on $M$. For any duality structure
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ on $M$, the holonomy representation
$\mathrm{hol}_{m}(\mathcal{D})$ of the flat connection $\mathcal{D}$ at $m\in
M$ coincides with the holonomy representation of the flat system of symplectic
vector spaces defined by $\Delta$:

$\mathrm{hol}_{m}(\mathcal{D})=\mathrm{hol}_{m}(\mathcal{T}_{\Delta})~{}~{}.$

In particular, the holonomy groups of $\mathcal{D}$ and $\mathcal{T}_{\Delta}$
at $m\in M$ coincide:

$\mathrm{Hol}_{m}(\mathcal{D})=\mathrm{Hol}_{m}(\mathcal{T}_{\Delta})~{}~{}.$

Since $M$ is connected, Proposition 1.7 implies that isomorphism classes of
duality structures defined on $M$ are in bijection with the character variety
(2).

###  1.5. Trivial duality structures

######  Definition 1.9.

Let $\Delta=(\mathcal{S},\omega,\mathcal{D})$ be a duality structure defined
on $M$. We say that $\Delta$ is:

  1. (1)

topologically trivial, if the vector bundle $\mathcal{S}$ is trivializable,
i.e. if it admits a global frame.

  2. (2)

_symplectically trivial_ if the symplectic vector bundle
$(\mathcal{S},\omega)$ is isomorphic to the trivial symplectic vector bundle,
i.e. if it admits a global symplectic frame (a global frame in which $\omega$
has the standard form).

  3. (3)

trivial (or _holonomy trivial_), if $\Delta$ admits a global flat symplectic
frame, i.e. if the holonomy group of $\mathcal{D}$ is the trivial group.

A symplectically trivial duality structure is automatically topologically
trivial, while a holonomy trivial duality structure is symplectically trivial.
If $M$ is simply-connected then every duality structure is holonomy trivial. A
global flat symplectic frame of a holonomy-trivial duality structure
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ of rank $2n$ has the form:

$\mathcal{E}=(e_{1},\ldots,e_{n},f_{1},\ldots,f_{n})~{}~{},$

where $e_{i},f_{j}$ are $\mathcal{D}$-flat sections of $\mathcal{S}$ such
that:

$\omega(e_{i},e_{j})=\omega(f_{i},f_{j})=0~{}~{},~{}~{}\omega(e_{i},f_{j})=-\omega(f_{i},e_{j})=\delta_{ij}\,,\quad\forall\,\,i,j=1,\ldots,n~{}~{}.$

Any choice of such a frame induces a trivialization isomorphism
$\tau_{\mathcal{E}}:\Delta\xrightarrow{\sim}\Delta_{n}$ between $\Delta$ and
the canonical trivial duality structure:

$\Delta_{n}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\underline{\mathbb{R}}^{2n},\underline{\omega}_{2n},\mathrm{d})\,,$
(3)

of rank $2n$, where $\underline{\omega}_{2n}$ is the constant symplectic
pairing induced on the trivial vector bundle
$\underline{\mathbb{R}}^{2n}=M\times\mathbb{R}^{2n}$ by the canonical
symplectic pairing $\omega_{2n}$ of $\mathbb{R}^{2n}$ and
$\mathrm{d}\colon\mathcal{C}^{\infty}(M,\underline{\mathbb{R}}^{2n})\rightarrow\Omega^{1}(M,\underline{\mathbb{R}}^{2n})$
is the ordinary differential. Any two flat symplectic frames $\mathcal{E}$ and
$\mathcal{E}^{\prime}$ of $\Delta$ are related by a based automorphism
$T\in\mathrm{Aut}_{b}(\Delta)$:

$\mathcal{E}^{\prime}=T\mathcal{E}\,,$

which corresponds to the constant automorphism
$\tau_{\mathcal{E}^{\prime}}\circ\tau_{\mathcal{E}}^{-1}\in\mathrm{Aut}_{b}(\Delta_{n})$
of $\Delta_{n}$ induced by an element
${\hat{T}}\in\mathrm{Sp}(2n,\mathbb{R})$.

###  1.6. Electromagnetic structures

As before, let $M$ be a connected manifold.

######  Definition 1.10.

An electromagnetic structure defined on $M$ is a pair
$\Xi=(\Delta,\mathcal{J})$, where $\Delta=(\mathcal{S},\omega,\mathcal{D})$ is
a duality structure on $M$ and $\mathcal{J}\in\mathrm{End}(\mathcal{S})$ is a
taming of the symplectic vector bundle $(\mathcal{S},\omega)$, i.e. a complex
structure on $\mathcal{S}$ which:

  * •

is compatible with $\omega$, i.e. it satisfies:

$\omega(\mathcal{J}x,\mathcal{J}y)=\omega(x,y)~{}~{}\forall(x,y)\in\mathcal{S}\times_{M}\mathcal{S}$

  * •

has the property that the symmetric bilinear pairing $Q_{\mathcal{J},\omega}$
defined on $\mathcal{S}$ through:

$Q_{\mathcal{J},\omega}(x,y)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\omega(\mathcal{J}x,y)~{}~{}\forall(x,y)\in\mathcal{S}\times_{M}\mathcal{S}$
(4)

is positive-definite.

The rank of $\Xi$ is the rank of $\Delta$.

Notice that the taming $\mathcal{J}$ is not required to be flat with respect
to $\mathcal{D}$. The bundle-valued one-form:

$\Psi_{\Xi}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{D}^{\mathrm{ad}}(\mathcal{J})=\mathcal{D}\circ\mathcal{J}-(\mathrm{id}_{T^{\ast}M}\otimes\mathcal{J})\circ\mathcal{D}\in\Omega^{1}(M,\mathrm{End}(\mathcal{S}))$

is called the fundamental form of $\Xi$ and measures the failure of
$\mathcal{D}$ to preserve $\mathcal{J}$. The electromagnetic structure $\Xi$
is called unitary if $\Psi_{\Xi}=0$. We refer the reader to [32] for more
detail on the fundamental form.

######  Definition 1.11.

Let $\Xi=(\Delta,\mathcal{J})$ and
$\Xi^{\prime}=(\Delta^{\prime},\mathcal{J}^{\prime})$ be two electromagnetic
structures defined on $M$. A based isomorphism of electromagnetic structures
from $\Xi$ to $\Xi^{\prime}$ is a based isomorphism of duality structures
$f:\Delta\xrightarrow{\sim}\Delta^{\prime}$ such that
$\mathcal{J}^{\prime}=f\circ\mathcal{J}\circ f^{-1}$.

######  Remark 1.12.

A taming $\mathcal{J}$ of a duality structure $\Delta$ can also be described
using a _positive complex polarization_. Let
$\Delta_{\mathbb{C}}=(\mathcal{S}_{\mathbb{C}},\omega_{\mathbb{C}},\mathcal{D}_{\mathbb{C}})$
be the complexification of $\Delta$. Then $\mathcal{J}$ is equivalent to a
complex Lagrangian sub-bundle $L\subset\mathcal{S}_{\mathbb{C}}$ such that:

$\mathbf{i}\omega(x,\bar{x})>0\,,\quad\forall\,\,x\in\dot{\mathcal{S}}_{\mathbb{C}}\,,$

where $\dot{\mathcal{S}}_{\mathbb{C}}$ is the complement of the zero section
in $\mathcal{S}_{\mathbb{C}}$. By definition, such a Lagrangian sub-bundle is
a positive complex polarization of the symplectic vector bundle
$(\mathcal{S},\omega)$. A detailed description of this correspondence can be
found in [32, Appendix A]. Note that the physics literature sometimes uses a
local version of positive polarizations when discussing Abelian gauge theory.
In our global framework this requires the supplementary step of complexifying
the vector bundle $\mathcal{S}$.

######  Definition 1.13.

A taming map of size $2n$ defined on $M$ is a smooth map
$\mathcal{J}:M\rightarrow\mathrm{Mat}(2n,\mathbb{R})$ such that
$\mathcal{J}(m)$ is a taming of the standard symplectic form $\omega_{2n}$ of
$\mathbb{R}^{2n}$ for all $m\in M$ (in particular, we have
$\mathcal{J}(m)\in\mathrm{Sp}(2n,\mathbb{R})$ for all $m$).

We denote by $\mathfrak{J}_{n}(M)$ the set of all taming maps of size $2n$.
Let $\Xi=(\Delta,\mathcal{J})$ be an electromagnetic structure of rank $2n$
defined on $M$ whose underlying duality structure
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ is holonomy trivial. Choosing a flat
global symplectic frame $\mathcal{E}$, we can identify $\Delta$ with the
canonical trivial duality structure (3) through an isomorphism
$\tau_{\mathcal{E}}:\Delta\xrightarrow{\sim}\Delta_{n}$. This identifies
$\mathcal{J}$ with the section
$\tau_{\mathcal{E}}\circ\mathcal{J}\circ\tau_{\mathcal{E}}^{-1}$ of the
trivial vector bundle $\underline{\mathbb{R}}^{2n}$, which in turn can be
viewed as a smooth map from $M$ to $\mathbb{R}^{2n}$. Since
$\tau_{\mathcal{E}}$ transports $\omega$ to the constant symplectic form
induced on $\underline{\mathbb{R}}^{2n}$ by $\omega_{2n}$, it is easy to see
that this map is a taming map of size $2n$ defined on $M$. Hence any choice of
global flat symplectic frame identifies the set of tamings of $\Delta$ with
$\mathfrak{J}_{n}(M)$.

###  1.7. The polarized Hodge operator

Let $(M,g)$ be an oriented Lorentzian four-manifold. Let
$\Xi=(\Delta,\mathcal{J})$ be an electromagnetic structure of rank $2n$
defined on $M$ and let $Q:=Q_{\mathcal{J},\omega}$ be the Euclidean scalar
product induced by $\mathcal{J}$ and $\omega$ on $\mathcal{S}$ as in equation
(4). The Hodge operator $\ast_{g}$ of $(M,g)$ extends trivially to an
automorphism of the vector bundle
$\wedge^{\ast}(M,\mathcal{S})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\wedge^{\ast}T^{\ast}M\otimes\mathcal{S}$, which we denote by the
same symbol.

######  Definition 1.14.

The $\mathcal{J}$-polarized Hodge operator of $(M,g,\Xi)$ is the automorphism
$\star_{g,\mathcal{J}}$ of the vector bundle $\wedge^{\ast}(M,\mathcal{S})$
defined through:

$\star_{g,\mathcal{J}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\ast_{g}\otimes\mathcal{J}=\mathcal{J}\otimes\ast_{g}\,.$

Let $(\cdot,\cdot)_{g}$ be the pseudo-Euclidean scalar product induced by $g$
on the total exterior bundle $\wedge^{\ast}(M)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\wedge^{\ast}T^{\ast}M$. Together with $Q$, this pairing induces a
pseudo-Euclidean scalar product $(\cdot,\cdot)_{g,Q}$ on the vector bundle
$\Lambda^{\ast}(M,\mathcal{S})$, which is uniquely determined by the
condition:

$(\rho_{1}\otimes\xi_{1},\rho_{2}\otimes\xi_{2})_{g,Q}=\delta_{k_{1},k_{2}}(-1)^{k_{1}}Q(\xi_{1},\xi_{2})(\rho_{1},\rho_{2})_{g}\,$

for all $\rho_{1}\in\Omega^{k_{1}}(M)$, $\rho_{2}\in\Omega^{k_{2}}(M)$ and
$\xi_{1},\xi_{2}\in\mathcal{C}^{\infty}(M,\mathcal{S})$. The polarized Hodge
operator $\star_{g,\mathcal{J}}$ induces an involutive automorphism of the
space $\Omega^{2}(M,\mathcal{S})$. We have a $(\cdot,\cdot)_{g,Q}$-orthogonal
splitting:

$\wedge^{2}(M,\mathcal{S})=\wedge^{2}_{+}(M,\mathcal{S})\oplus\wedge^{2}_{-}(M,\mathcal{S})$

into eigenbundles of $\star_{g,\mathcal{J}}$ with eigenvalues $+1$ and $-1$
for the polarized Hodge operator. The sections of
$\wedge^{2}_{+}(M,\mathcal{S})$ and $\wedge^{2}_{-}(M,\mathcal{S})$ are called
respectively polarized self-dual and polarized anti-self-dual
$\mathcal{S}$-valued two-forms. Notice that the definition of these notions
does not require complexification of $\mathcal{S}$.

###  1.8. Classical abelian gauge theory

Let $(M,g)$ be an oriented Lorentzian four-manifold.

######  Definition 1.15.

The classical configuration space defined by the duality structure
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ on $M$ is the vector space of two-
forms valued in $\mathcal{S}$ which are closed with respect to
$\mathrm{d}_{\mathcal{D}}$:

$\mathrm{Conf}(M,\Delta)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{\mathcal{V}\in\Omega^{2}(M,\mathcal{S})\,\,|\,\,\mathrm{d}_{\mathcal{D}}\mathcal{V}=0\right\\}\,.$

######  Definition 1.16.

The classical abelian gauge theory determined by an electromagnetic structure
$\Xi=(\Delta,\mathcal{J})$ on $(M,g)$ is defined by the polarized self-duality
condition:

$\star_{g,\mathcal{J}}\mathcal{V}=\mathcal{V}$

for $\mathcal{V}\in\mathrm{Conf}(M,\Delta)$. The solutions of this equation
are called classical electromagnetic field strengths and form the vector
space:

$\mathrm{Sol}(M,g,\Xi)=\mathrm{Sol}(M,g,\Delta,\mathcal{J})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{\mathcal{V}\in\mathrm{Conf}(M,\Delta)\,|\,\star_{g,\mathcal{J}}\mathcal{V}=\mathcal{V}\right\\}\,.$

Notice that classical abelian gauge theory is formulated in terms of field
strengths. In later sections, we will formulate the pre-quantum version of
this theory (which is obtained by imposing an appropriate DSZ quantization
condition) in terms of connections defined on certain principal bundles called
_Siegel bundles_ (see Definition 3.2). The classical theory simplifies when
$M$ is contractible, since in this case any duality structure defined on $M$
is holonomy-trivial. This case corresponds to the local abelian gauge theory
discussed in Appendix A, which makes contact with the formulation used in the
physics literature. Notice that the traditional physics formulation (which is
valid only locally or for holonomy-trivial duality structures) relies on
gauge-kinetic functions, leading to complicated formulas which obscure the
geometric structure displayed by Definition 1.16. It was shown in [32] that
elements of $\mathrm{Sol}(M,g,\Xi)$ correspond to classical electromagnetic
U-folds when the duality structure $\Delta$ is not holonomy trivial.

###  1.9. Classical duality groups

Let $(M,g)$ be an oriented Lorentzian four-manifold and $\mathcal{S}$ be a
vector bundle defined on $M$. Let $\mathrm{Aut}(\mathcal{S})$ be the group of
those unbased vector bundle automorphisms of $\mathcal{S}$ which cover
orientation-preserving diffeomorphisms of $M$ and let
$\mathrm{Aut}_{b}(\mathcal{S})\subset\mathrm{Aut}(\mathcal{S})$ the subgroup
of based automorphisms, i.e. those automorphisms which cover the identity of
$M$. Let $\mathrm{Diff}(M)$ be the group of orientation-preserving
diffeomorphisms of $M$. Given $u\in\mathrm{Aut}(\mathcal{S})$, let
$f_{u}\in\mathrm{Diff}(M)$ be the orientation-preserving diffeomorphism of $M$
covered by $u$. This defines a morphism of groups:

$\mathrm{Aut}(\mathcal{S})\ni u\rightarrow f_{u}\in\mathrm{Diff}(M)~{}~{}.$

The group $\mathrm{Aut}(\mathcal{S})$ admits an $\mathbb{R}$-linear
representation:

$\mathrm{Aut}(\mathcal{S})\ni
u\rightarrow\mathbb{A}_{u}\in\mathrm{Aut}(\Omega^{\ast}(M,\mathcal{S}))$

given by push-forward; we will occasionally also use the notation:
$u\cdot(-)\stackrel{{\scriptstyle{\rm def.}}}{{=}}\mathbb{A}_{u}(-)$. When
$f_{u}$ is not the identity the push-forward action of
$u\in\mathrm{Aut}(\mathcal{S})$ must be handled with care; we refer the reader
to [32, Appendix D] for a detailed description of this operation and its
properties. On decomposable elements of
$\Omega^{\ast}(M,\mathcal{S})=\Omega^{\ast}(M)\otimes\mathcal{C}^{\infty}(M,\mathcal{S})$,
it is given by:

$\mathbb{A}_{u}(\alpha\otimes\xi)=u\cdot(\alpha\otimes\xi)=(f_{u\ast}\alpha)\otimes(u\cdot\xi)=(f_{u\ast}\alpha)\otimes(u\circ\xi\circ
f^{-1}_{u})\,,$

where $f_{u\ast}\alpha$ is the push-forward of $\alpha$ on $M$ as defined in
[32, Appendix D]. For instance, if $\alpha\in\Omega^{1}(M)$ we have:

$(f_{u\ast}\alpha)(v)=\alpha(f_{u\ast}^{-1}(v))\circ
f^{-1}_{u}=\alpha(\mathrm{d}f_{u}^{-1}(v)\circ f_{u})\circ f^{-1}_{u}\in
C^{\infty}(M)\,,\quad\forall\,\,v\in\mathfrak{X}(M)\,.$

In particular, restricting to a given point $m\in M$ we obtain the familiar
formula:

$(f_{u\ast}\alpha)(v)_{m}=\alpha(f_{u\ast}^{-1}(v))_{f^{-1}_{u}(m)}=\alpha_{f^{-1}_{u}(m)}(\mathrm{d}f_{u}^{-1}|_{m}(v_{m}))\,.$

Recall that $f_{u\ast}^{-1}(v)\in\mathfrak{X}(M)$ is again a vector field on
$M$ whereas $\mathrm{d}f_{u}^{-1}(v)\colon M\to TM$ is a vector field _along_
$f_{u}$. For any duality structure $\Delta=(\mathcal{S},\omega,\mathcal{D})$
defined on $M$, let:

$\mathrm{Aut}(\Delta)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}(\mathcal{S})\,\,|\,\,\omega_{u}=\omega\,,\,\,\mathcal{D}_{u}=\mathcal{D}\right\\}\,,$

be the group of unbased automorphisms of $\Delta$, defined as the stabilizer
of the pair $(\omega,\mathcal{D})$ in $\mathrm{Aut}(\mathcal{S})$. Here
$\mathcal{D}_{u}$ is the push-forward of the connection $\mathcal{D}$ by $u$,
which is defined through:

$(\mathcal{D}_{u})_{v}(s)=u\cdot\mathcal{D}_{f_{u\ast}^{-1}(v)}(u^{-1}\cdot
s)=u\circ(\mathcal{D}_{f_{u\ast}^{-1}(v)}(u^{-1}\circ s\circ f_{u}))\circ
f^{-1}_{u}\,,\quad\forall\,\,s\in\Gamma(\mathcal{S})\quad\forall\,\,v\in\mathfrak{X}(M)\,,$

where $f_{u\ast}^{-1}\cdot v=\mathrm{d}f^{-1}_{u}(v)\circ f_{u}$. Note that if
$s\in\Gamma(S)$ is a parallel section of $\mathcal{S}$ with respect to
$\mathcal{D}$, then $u\cdot s\in\Gamma(S)$ is parallel with respecto
$\mathcal{D}_{u}$ for all $u\in\mathrm{Aut}(\Delta)$. We have a short exact
sequence:

$1\to\mathrm{Aut}_{b}(\Delta)\to\mathrm{Aut}(\Delta)\to\mathrm{Diff}_{\Delta}(M)\to
1\,,$

where $\mathrm{Diff}_{\Delta}(M)\subset\mathrm{Diff}(M)$ is the subgroup
formed by those orientation-preserving diffeomorphisms of $M$ that can be
covered by elements of $\mathrm{Aut}(\Delta)$. Given a taming
$\mathcal{J}\in\mathrm{End}(\mathcal{S})$ of $(\mathcal{S},\omega)$, we
define:

$\mathcal{J}_{u}\stackrel{{\scriptstyle{\rm def.}}}{{=}}u\circ\mathcal{J}\circ
u^{-1}\in\mathrm{End}(\mathcal{S})~{}~{},$

which is a taming of the duality structure
$\Delta_{u}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\mathcal{S},\omega_{u},\mathcal{D}_{u})$. On the other hand, if
$g$ is a Lorentzian metric on $M$, we set:

$g_{u}\stackrel{{\scriptstyle{\rm def.}}}{{=}}(f_{u})_{\ast}(g)~{}~{}.$

Finally, we define $\Xi_{u}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\Delta_{u},\mathcal{J}_{u})$ and we denote by ${\rm Iso}(M,g)$
the group of orientation-preserving isometries of $(M,g)$.

######  Definition 1.17.

  * Let $\Xi=(\Delta,\mathcal{J})$ be an electromagnetic structure defined on $M$.

  * •

The group $\mathrm{Aut}(\Delta)$ of unbased automorphisms of $\Delta$ is
called the unbased pseudo-duality group defined by $\Delta$. The unbased
pseudo-duality transformation defined by $u\in\mathrm{Aut}(\Delta)$ is the
linear isomorphism:

$\mathbb{A}_{u}\colon\mathrm{Conf}(M,\Delta)\xrightarrow{\sim}\mathrm{Conf}(M,\Delta)~{}~{},$
(5)

which restricts to an isomorphism:

$\mathbb{A}_{u}\colon\mathrm{Sol}(M,g,\Delta,\mathcal{J})\xrightarrow{\sim}\mathrm{Sol}(M,g_{u},\Delta,\mathcal{J}_{u})~{}~{}.$

  * •

The unbased duality group $\mathrm{Aut}(g,\Delta)$ of the pair $(g,\Delta)$ is
the stabilizer of $g$ in $\mathrm{Aut}(\Delta)$:

$\mathrm{Aut}(g,\Delta)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Stab}_{\mathrm{Aut}(\Delta)}(g)=\left\\{u\in\mathrm{Aut}(\Delta)\,\,|\,\,f_{u}\in{\rm
Iso}(M,g)\right\\}\,.$

The unbased duality transformation defined by $u\in\mathrm{Aut}(g,\Delta)$ is
the linear isomorphism (5), which restricts to an isomorphism:

$\mathbb{A}_{u}\colon\mathrm{Sol}(M,g,\Delta,\mathcal{J})\xrightarrow{\sim}\mathrm{Sol}(M,g,\Delta,\mathcal{J}_{u})~{}~{}.$

  * •

The group $\mathrm{Aut}_{b}(\Delta)$ is called the (electromagnetic) duality
group defined by $\Delta$. The duality transformation defined by
$u\in\mathrm{Aut}_{b}(\Delta)$ is the linear isomorphism (5), which restricts
to an isomorphism:

$\mathbb{A}_{u}\colon\mathrm{Sol}(M,g,\Delta,\mathcal{J})\to\mathrm{Sol}(M,g,\Delta,\mathcal{J}_{u})~{}~{}.$

######  Remark 1.18.

The fact that $\mathbb{A}_{u}$ restricts as stated above follows from a direct
yet subtle computation. Since the conditions involved are linear, it is enough
to verify it on an homogeneous element. If
$\mathcal{V}=\alpha\otimes\xi\in\mathrm{Sol}(M,\Delta)$,
$u\in\mathrm{Aut}(\Delta)$ and $v\in\mathfrak{X}(M)$ we compute:

$\displaystyle\mathrm{d}_{(\mathcal{D}_{u})_{v}}\mathbb{A}_{u}(\mathcal{V})=\mathrm{d}_{(\mathcal{D}_{u})_{v}}((f_{u\ast}\alpha)\otimes(u\cdot\xi))=(\iota_{v}f_{u\ast}\mathrm{d}\alpha)\otimes(u\cdot\xi)+(f_{u\ast}\alpha)\otimes((\mathcal{D}_{u})_{v}u\cdot\xi)$
$\displaystyle=(f_{u\ast}\iota_{f_{u\ast}^{-1}(v)}\mathrm{d}\alpha)\otimes(u\cdot\xi)+(f_{u\ast}\alpha)\otimes(u\cdot\mathcal{D}_{f_{u\ast}^{-1}(v)}\xi)=\mathbb{A}_{u}((\iota_{f_{u\ast}^{-1}(v)}\mathrm{d}\alpha)\otimes(\xi)+\alpha\otimes(\mathcal{D}_{f_{u\ast}^{-1}(v)}\xi))$
$\displaystyle=\mathbb{A}_{u}(\mathrm{d}_{\mathcal{D}_{f_{u\ast}^{-1}(v)}}\mathcal{V})=0\,.$

On the other hand, if $\mathcal{V}=\alpha\otimes\xi\in\mathrm{Sol}(M,g,\Xi)$
we compute:

$\star_{g_{u},\mathcal{J}_{u}}\mathbb{A}_{u}(\mathcal{V})=(\ast_{g_{u}}f_{u\ast}\alpha)\otimes(\mathcal{J}_{u}\circ
u\cdot\xi)=f_{u\ast}(\ast_{g}\alpha)\otimes u\circ\mathcal{J}(\xi)\circ
f_{u}^{-1}=\mathbb{A}_{u}(\star_{g_{,}\mathcal{J}}\mathcal{V})=\mathbb{A}_{u}(\mathcal{V})\,,$

implying that the restrictions of $\mathbb{A}_{u}$ to $\mathrm{Sol}(M,\Delta)$
and $\mathrm{Conf}(M,g,\Xi)$ are well-defined as stated above.

We have obvious inclusions:

$\mathrm{Aut}_{b}(\Delta)\subset\mathrm{Aut}(g,\Delta)\subset\mathrm{Aut}(\Delta)$
(6)

and a short exact sequence:

$1\to\mathrm{Aut}_{b}(\Delta)\to\mathrm{Aut}(g,\Delta)\to{\rm
Iso}_{\Delta}(M,g)\to 1\,,$ (7)

where ${\rm Iso}_{\Delta}(M,g)\subset{\rm Iso}(M,g)$ denotes the group formed
by those orientation-preserving isometries of $(M,g)$ that can be covered by
elements of $\mathrm{Aut}(g,\Delta)$.

The classical duality group $\mathrm{Aut}_{b}(\Delta)$ consists of all based
automorphisms of $\mathcal{S}$ which preserve both $\omega$ and $\mathcal{D}$.
Therefore (see [24, Lemma 4.2.8]) fixing a point $m_{0}\in M$ it can be
realized as the centralizer $C_{m_{0}}(\Delta)$ of the holonomy group
$\mathrm{Hol}_{m_{0}}(\mathcal{D})$ of $\mathcal{D}$ at $m_{0}$ inside the
group
$\mathrm{Aut}(\mathcal{S}_{m_{0}},\omega_{m_{0}})\simeq\mathrm{Sp}(2n,\mathbb{R})$:

$\mathrm{Aut}_{b}(\Delta)\simeq C_{m_{0}}(\Delta)~{}~{}.$

In particular, $\mathrm{Aut}_{b}(\Delta)$ is a closed subgroup of
$\mathrm{Sp}(2n,\mathbb{R})$ and hence it is a finite-dimensional Lie group.
The same holds for ${\rm Iso}_{\Delta}(M,g)$, which is a closed subgroup of
the finite-dimensional222It is well-known that the isometry group of any
pseudo-Riemannian manifold is a finite-dimensional Lie group. See for example
[31, Theorem 5.1, p. 22]. Lie group ${\rm Iso}(M,g)$. The sequence (7) implies
that $\mathrm{Aut}(g,\Delta)$ is also a finite-dimensional Lie group. In
general, the groups defined above differ markedly from their local
counterparts described in Appendix A.2. We stress that the latter are not the
adequate groups to consider when dealing with electromagnetic U-folds (since
in that case the duality structure is not holonomy trivial).

######  Definition 1.19.

Let $\Xi=(\Delta,\mathcal{J})$ be an electromagnetic structure defined on $M$.

  * •

The group:

$\mathrm{Aut}(\Xi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{u\in\mathrm{Aut}(\Delta)~{}|~{}\mathcal{J}_{u}=\mathcal{J}\\}$

of unbased automorphisms of $\Xi$ is called the unbased unitary pseudo-duality
group defined by $\Xi$. The unbased unitary pseudo-duality transformation
defined by $u\in\mathrm{Aut}(\Xi)$ is the linear isomorphism:

$\mathbb{A}_{u}\colon\mathrm{Sol}(M,g,\Xi)\xrightarrow{\sim}\mathrm{Sol}(M,g_{u},\Xi)\,.$

  * •

The unbased unitary duality group of the pair $(g,\Xi)$ is the stabilizer of
$g$ in $\mathrm{Aut}(\Xi)$:

$\mathrm{Aut}(g,\Xi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Stab}_{\mathrm{Aut}(\Xi)}(g)=\mathrm{Stab}_{\mathrm{Aut}(g,\Delta)}(\mathcal{J})=\left\\{u\in\mathrm{Aut}(\Delta)\,\,|\,\,\mathcal{J}_{u}=\mathcal{J}~{}\&~{}f_{u}\in{\rm
Iso}(M,g)\right\\}~{}~{}.$ (8)

The unbased unitary duality transformation defined by
$u\in\mathrm{Aut}(g,\Xi)$ is the linear automorphism:

$\mathbb{A}_{u}\colon\mathrm{Sol}(M,g,\Xi)\xrightarrow{\sim}\mathrm{Sol}(M,g,\Xi)~{}~{}.$

  * •

The group $\mathrm{Aut}_{b}(\Xi)$ of based automorphisms of $\Xi$ is called
the classical unitary duality group defined by $\Xi$:

$\mathrm{Aut}_{b}(\Xi)=\mathrm{Stab}_{\mathrm{Aut}_{b}(\Delta)}(\mathcal{J})=\left\\{u\in\mathrm{Aut}_{b}(\Delta)\,\,|\,\,\mathcal{J}_{u}=\mathcal{J}\right\\}\,.$

The classical unitary duality transformation defined by
$u\in\mathrm{Aut}_{b}(\Xi)$ is the linear automorphism:

$\mathbb{A}_{u}\colon\mathrm{Sol}(M,g,\Xi)\xrightarrow{\sim}\mathrm{Sol}(M,g,\Xi)$

We have obvious inclusions:

$\mathrm{Aut}_{b}(\Xi)\subset\mathrm{Aut}(g,\Xi)\subset\mathrm{Aut}(\Xi)$

and a short exact sequence:

$1\to\mathrm{Aut}_{b}(\Xi)\to\mathrm{Aut}(g,\Xi)\to{\rm Iso}_{\Xi}(M,g)\to
1\,,$ (9)

where ${\rm Iso}_{\Xi}(M,g)$ is the group formed by those orientation-
preserving isometries of $(M,g)$ which are covered by elements of
$\mathrm{Aut}(g,\Xi)$. Arguments similar to those above show that
$\mathrm{Aut}_{b}(\Xi)$, $\mathrm{Aut}(g,\Xi)$ and ${\rm Iso}_{\Xi}(M,g)$ are
finite-dimensional Lie groups. The previous definitions give global
mathematically rigorous descriptions of several types of _duality groups_
associated to abelian gauge theory on Lorentzian four-manifolds. In general,
these can differ markedly from their “local” counterparts described in
Appendix A.2, which are considered traditionally in the physics literature.

###  1.10. The case of trivial duality structure

Let $(M,g)$ be an oriented Lorentzian four-manifold. For any $n\geq 0$, the
set $\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))$ of smooth
$\mathrm{Sp}(2n,\mathbb{R})$-valued functions defined on $M$ is a group under
pointwise multiplication, whose group of automorphisms we denote by
$\mathrm{Aut}(\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R})))$.

######  Lemma 1.20.

Let $(\mathcal{S},\omega)$ be a symplectic vector bundle of rank $2n$ defined
on $M$ which is symplectically trivializable. Then any symplectic
trivialization of $(\mathcal{S},\omega)$ induces an isomorphism of groups:

$\mathrm{Aut}(\mathcal{S},\omega)\simeq\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))\rtimes_{\alpha}\mathrm{Diff}(M)~{}~{},$

where
$\alpha:\mathrm{Diff}(M)\rightarrow\mathrm{Aut}(\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R})))$
is the morphism of groups defined through:

$\alpha(\varphi)(f)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}f\circ\varphi^{-1}\,,\quad\forall\,\,\varphi\in\mathrm{Diff}(M)\,,\,\,\forall\,\,f\in\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))~{}~{}.$

In particular, we have a short exact sequence of groups:

$1\rightarrow\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))\rightarrow\mathrm{Aut}(\mathcal{S},\omega)\rightarrow\mathrm{Diff}(M)\rightarrow
1$

which splits from the right.

###### Proof.

Let
$\tau:\mathcal{S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}M\times\mathbb{R}^{2n}$
be a symplectic trivialization of $(\mathcal{S},\omega)$. Then the map
$\mathrm{Ad}(\tau):\mathrm{Aut}(\mathcal{S},\omega)\rightarrow\mathrm{Aut}(M\times\mathbb{R}^{2n},\omega_{2n})$
defined through:

$\mathrm{Ad}(\tau)(f)\stackrel{{\scriptstyle{\rm def.}}}{{=}}\tau\circ
f\circ\tau^{-1}\,,\quad\forall\,\,f\in\mathrm{Aut}(\mathcal{S},\omega)\,,$

is an isomorphism of groups. Let $f\in\mathrm{Aut}(\mathcal{S},\omega)$ be an
unbased automorphism of $(\mathcal{S},\omega)$ which covers the diffeomorphism
$\varphi\in\mathrm{Diff}(M)$. Then $\mathrm{Ad}(\tau)(f)$ is an unbased
automorphism of $M\times\mathbb{R}^{2n}$ which covers $\varphi$ and hence we
have:

$\mathrm{Ad}(\tau)(f)(m,x)=(\varphi(m),{\tilde{f}}(m)(x))\,,\quad\forall\,\,(m,x)\in
M\times\mathbb{R}^{2n}~{}~{},$

where ${\tilde{f}}:M\rightarrow\mathrm{Sp}(2n,\mathbb{R})$ is a smooth map.
Setting $h\stackrel{{\scriptstyle{\rm
def.}}}{{=}}{\tilde{f}}\circ\varphi^{-1}\in\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))$,
we have:

$\mathrm{Ad}(\tau)(f)(m,x)=(\varphi(m),h(\varphi(m))(x))\,,\quad\forall\,\,(m,x)\in
M\times\mathbb{R}^{2n}$ (10)

and the correspondence $f\rightarrow(h,\varphi)$ gives a bijection between
$\mathrm{Aut}(\mathcal{S},\omega)$ and the set
$\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))\times\mathrm{Diff}(M)$. If
$f_{1},f_{2}\in\mathrm{Aut}(\mathcal{S},\omega)$ correspond through this map
to the pairs:

$(h_{1},\varphi_{1}),(h_{2},\varphi_{2})\in\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))\times\mathrm{Diff}(M)\,,$

then direct computation using (10) gives:

$\mathrm{Ad}(\tau)(f_{1}\circ
f_{2})(m,x)=((\varphi_{1}\circ\varphi_{2})(m),h_{1}(m)(h_{2}\circ\varphi_{1}^{-1})(m)(x))~{}~{},$

showing that $f_{1}\circ f_{2}$ corresponds to the pair
$(h_{1}\cdot\alpha(\varphi_{1})(h_{2}),\varphi_{1}\circ\varphi_{2})$. ∎

######  Corollary 1.21.

Let $\Delta=(\mathcal{S},\omega,\mathcal{D})$ be a holonomy trivial duality
structure defined on $M$. Then any trivialization of $\Delta$ induces an
isomorphism of groups:

$\mathrm{Aut}(\Delta)\simeq\mathrm{Sp}(2n,\mathbb{R})\times\mathrm{Diff}(M)~{}~{}.$

###### Proof.

Follows from Lemma 1.20 by noticing that the action $\alpha$ of
$\mathrm{Diff}(M)$ on $\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))$
restricts to the trivial action on the subgroup:

$\\{f\in\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))~{}|~{}\mathrm{d}f=0\\}\simeq\mathrm{Sp}(2n,\mathbb{R})$

of constant $\mathrm{Sp}(2n,\mathbb{R})$-valued functions defined on $M$. ∎

Fix an electromagnetic structure $\Xi=(\Delta,\mathcal{J})$ of rank $2n$
defined on $M$ with holonomy-trivial underlying duality structure
$\Delta=(\mathcal{S},\omega,\mathcal{D})$. Choosing a global flat symplectic
frame $\mathcal{E}$ of $\mathcal{S}$, we identify $\Delta$ with the canonical
trivial duality structure (3) and $\mathcal{J}$ with a taming map of size
$2n$. Then $\mathcal{D}$ identifies with the trivial connection and
$\mathrm{d}_{\mathcal{D}}$ identifies with the exterior derivative
$\mathrm{d}\colon\Omega(M,\mathbb{R}^{2n})\to\Omega(M,\mathbb{R}^{2n})$
extended trivially to vector-valued forms. Using Lemma 1.20 and its obvious
adaptation, we obtain:

$\displaystyle\mathrm{Aut}_{b}(\mathcal{S})\equiv\mathcal{C}^{\infty}(M,\mathrm{GL}(2n,\mathbb{R}))\,,\quad\mathrm{Aut}(\mathcal{S})\equiv\mathcal{C}^{\infty}(M,\mathrm{GL}(2n,\mathbb{R}))\rtimes_{\alpha}\mathrm{Diff}(M)\,,$
$\displaystyle\mathrm{Aut}_{b}(\mathcal{S},\omega)\equiv\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))\,,\quad\mathrm{Aut}(\mathcal{S},\omega)\equiv\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))\rtimes_{\alpha}\mathrm{Diff}(M)\,.$

On the other hand, Corollary 1.21 gives:

$\displaystyle\mathrm{Aut}_{b}(\Delta)\equiv\mathrm{Sp}(2n,\mathbb{R})\,,\quad\mathrm{Aut}(\Delta)\equiv\mathrm{Sp}(2n,\mathbb{R})\times\mathrm{Diff}(M)\,,$
$\displaystyle\mathrm{Aut}(g,\Delta)=\mathrm{Sp}(2n,\mathbb{R})\times{\rm
Iso}(M,g)\,.$

Moreover, we have:

$\displaystyle\mathrm{Aut}_{b}(\Xi)\equiv\mathrm{U}_{\mathcal{J}}(n)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{\gamma\in\mathrm{Sp}(2n,\mathbb{R})\,\,|\,\,\gamma\mathcal{J}\gamma^{-1}=\mathcal{J}\\}\,,$
$\displaystyle\mathrm{Aut}(\Xi)\equiv\\{(\gamma,f)\in\mathrm{Sp}(2n,\mathbb{R})\times\mathrm{Diff}(M)\,\,|\,\,\gamma\mathcal{J}\gamma^{-1}=\mathcal{J}\circ
f\\}\,,$

as well as:

$\mathrm{Aut}(g,\Xi)\equiv\\{(\gamma,f)\in\mathrm{Sp}(2n,\mathbb{R})\times{\rm
Iso}(M)\,\,|\,\,\gamma\mathcal{J}\gamma^{-1}=\mathcal{J}\circ f\\}~{}~{}.$

Hence we recover the local formulas obtained in Appendix A.2 for the duality
groups of local abelian gauge theory. Notice that $\mathrm{Aut}_{b}(\Xi)$ is
isomorphic with the unitary group $\mathrm{U}(n)$ when the electromagnetic
structure $\Xi$ is unitary, which amounts to the map $\mathcal{J}$ being
constant.

##  2\. The Dirac-Schwinger-Zwanziger condition

The previous section introduced classical abelian gauge theory as a theory of
_field strengths_ , i.e. a theory of $\mathrm{d}_{\mathcal{D}}$-closed two-
forms valued in the flat symplectic vector bundle $\mathcal{S}$ with equation
of motion given by the polarized self-duality condition. Well-known arguments
originally due to Dirac as well as the Aharonov-Bohm effect [1] imply that a
consistent coupling of the theory to quantum charge carriers imposes an
integrality condition on field strength configurations. This is traditionally
called the DSZ “quantization” condition, even though it constrains classical
field strength configurations – in fact, only particles which carry the
corresponding charges are quantized in such arguments, but not the gauge
fields themselves. To avoid confusion, we prefer to call it the DSZ
integrality condition. For local abelian gauge theories of rank $2n$ (which
are discussed in Appendix A), this condition can be implemented using a full
symplectic lattice in the standard symplectic vector space
$(\mathbb{R}^{2n},\omega_{2n})$, as usually done in the physics literature
[42, 47]. For abelian gauge theories with non-trivial electromagnetic
structure defined on an arbitrary Lorentzian four-manifold, we shall implement
this condition using a Dirac system, as originally proposed in [32]. We begin
with some preliminaries.

###  2.1. Principal bundles with discrete structure group

Let $\Gamma$ be a discrete group and $Q$ be a principal bundle with structure
group $\Gamma$ and projection $p:Q\rightarrow M$. Then the total space of $Q$
is a (generally disconnected) covering space of $M$. Let
$U_{Q}:\Pi_{1}(M)\rightarrow\Phi_{0}(Q)$ be the monodromy transport of the
covering map $p:Q\rightarrow M$, where $\Phi_{0}(Q)$ is the bare fiber
groupoid of $Q$. By definition, the objects of $\Phi_{0}(Q)$ are the fibers of
$Q$ while its morphisms are arbitrary bijections between the latter. By
definition, the functor $U_{Q}$ associates to the homotopy class
$\mathbf{c}\in\Pi_{1}(M)(m,m^{\prime})$ of any curve $c:[0,1]\rightarrow M$
with $c(0)=m$ and $c(1)=m^{\prime}$ the bijection
$U(\mathbf{c}):Q_{m}\xrightarrow{\sim}Q_{m^{\prime}}$ given by
$U_{Q}(\mathbf{c})(x)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}{\tilde{c}}_{x}(1)\in Q_{m^{\prime}}$, where ${\tilde{c}}_{x}$ is
the the unique lift of $c$ to $Q$ through the point $x\in Q$ (thus
$\tilde{c}_{x}(0)=x$). Notice that $x$ and $U(\mathbf{c})(x)$ lie on the same
connected component of $Q$ and hence the diffeomorphism
$U_{Q}(\mathbf{c}):Q\xrightarrow{\sim}Q$ induces the trivial permutation of
$\pi_{0}(Q)$. For any $\gamma\in\Gamma$, the curve ${\tilde{c}}^{\gamma}_{x}$
defined through ${\tilde{c}}_{x}^{\gamma}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}{\tilde{c}}(t)\gamma$ for all $t\in[0,1]$ is a lift of $c$ through
the point $x\gamma$. The homotopy lifting property of $p$ implies that
${\tilde{c}}^{\gamma}_{x}$ and ${\tilde{c}}_{x\gamma}$ are homotopic and hence
$U_{Q}(\mathbf{c})(x\gamma)=U(\mathbf{c})(x)\gamma$. This shows that $U_{Q}$
acts through isomorphisms of $\Gamma$-spaces and hence it is in fact a
functor:

$U_{Q}:\Pi_{1}(M)\rightarrow\Phi(Q)~{}~{},$

where $\Phi(Q)$ is the principal fiber groupoid of $Q$ (whose objects coincide
with those of $\Phi_{0}(Q)$ but whose morphisms are isomorphisms of
$\Gamma$-spaces). This implies that $U_{Q}$ is the parallel transport of a
flat principal connection defined on $Q$, which we shall call the monodromy
connection of $Q$. The holonomy morphism:

$\alpha_{m}(Q):\pi_{1}(M,m)\rightarrow\mathrm{Aut}_{\Gamma}(Q_{m})$

of this connection at a point $m\in M$ will be called the monodromy morphism
of $Q$ at $m$, while its image:

$\mathrm{Hol}_{m}(Q)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{im}(\alpha_{m}(Q))\subset\mathrm{Aut}_{\Gamma}(Q_{m})$

will be called the monodromy group of $Q$ at $m$. The monodromy morphism at a
fixed point $m_{0}\in M$ induces a bijection between the set of isomorphism
classes of principal $\Gamma$-bundles and the character variety:

$\mathfrak{R}(\pi_{1}(M,m_{0}),\Gamma)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}{\rm Hom}(\pi_{1}(M,m),\Gamma)/\Gamma~{}~{}.$

######  Remark 2.1.

For the purposes of this work, the most important class of principal bundles
with discrete structure group are principal
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ bundles, which are naturally
associated to Siegel bundles (see Section 3).

###  2.2. Bundles of finitely-generated free abelian groups

Let $\mathcal{F}$ be a bundle of free abelian groups of rank $r$ defined on
$M$. Then $\mathcal{F}$ is isomorphic with the bundle of groups with fiber
$\mathbb{Z}^{r}$ associated to a principal $\mathrm{GL}(r,\mathbb{Z})$-bundle
$Q$ through the left action
$\ell:\mathrm{GL}(r,\mathbb{Z})\rightarrow\mathrm{Aut}_{\mathbb{Z}}(\mathbb{Z}^{r})$:

$\mathcal{F}\simeq\mathcal{F}_{r}(Q)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}Q\times_{\ell}\mathbb{Z}^{r}\simeq\hat{M}\times_{\ell\circ\alpha_{m}(Q)}\mathbb{Z}^{r}~{}~{},$

where $\alpha_{m}(Q):\pi_{1}(M,m)\rightarrow\mathrm{GL}(r,\mathbb{Z})$ is the
monodromy morphism of $Q$ at $m$. The monodromy connection of $Q$ induces a
flat Ehresmann connection which we shall call the monodromy connection of
$\mathcal{F}$ and whose parallel transport:

$U_{\mathcal{F}}:\Pi_{1}(M)\rightarrow\Phi(\mathcal{F})$

acts by isomorphisms of groups between the fibers of $\mathcal{F}$. The
holonomy morphism:

$\sigma_{m}(\mathcal{F}):\pi_{1}(M,m)\rightarrow\mathrm{Aut}_{\mathbb{Z}}(\mathcal{F}_{m})$

at $m\in M$ can be identified with the morphism
$\ell\circ\alpha_{m}(P):\pi_{1}(M,m)\rightarrow\mathrm{Aut}_{\mathbb{Z}}(\mathbb{Z}^{r})$
upon choosing a basis of $\mathcal{F}_{m}$. The holonomy group:

$\mathrm{Hol}_{m}(\mathcal{F})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{im}(\sigma_{m}(\mathcal{F}))\subset\mathrm{Aut}_{\mathbb{Z}}(\mathcal{F}_{m})\simeq\mathrm{GL}(r,\mathbb{Z})$

is called the monodromy group of $\mathcal{F}$ at $m$ and identifies with a
subgroup of $\mathrm{GL}(r,\mathbb{Z})$ upon choosing an appropriate basis of
$\mathcal{F}_{m}$.

Conversely, let $\mathrm{Fr}(\mathbb{Z}^{r})$ be the set of all bases of the
free $\mathbb{Z}$-module $\mathbb{Z}^{r}$. Then $\mathrm{GL}(r,\mathbb{Z})$
has a natural free and transitive left action $\mu$ on this set. Taking the
set of bases of each of fiber gives the bundle of frames
$\mathrm{Fr}(\mathcal{F})$ of $\mathcal{F}$. This is a principal
$\mathrm{GL}(r,\mathbb{Z})$-bundle whose monodromy morphism coincides with
that of $\mathcal{F}$. This gives the following result.

######  Proposition 2.2.

The correspondences $Q\mapsto\mathcal{F}_{r}(Q)$ and
$\mathcal{F}\mapsto\mathrm{Fr}(\mathcal{F})$ extend to mutually quasi-inverse
equivalences between the groupoid of bundles of free abelian groups of rank
$r$ defined on $M$ and the groupoid of principal
$\mathrm{GL}(r,\mathbb{Z})$-bundles defined on $M$.

###  2.3. Dirac systems and integral duality structures

######  Definition 2.3.

Let $\Delta=(\mathcal{S},\omega,\mathcal{D})$ be a duality structure defined
on $M$. A Dirac system for $\Delta$ is a smooth fiber sub-bundle
$\mathcal{L}\subset\mathcal{S}$ of full symplectic lattices in
$(\mathcal{S},\omega)$ which is preserved by the parallel transport
$\mathcal{T}_{\Delta}$ of $\mathcal{D}$. That is, for any piece-wise smooth
path $\gamma:[0,1]\rightarrow M$ we have:

$\mathcal{T}_{\Delta}(\gamma)(\mathcal{L}_{\gamma(0)})=\mathcal{L}_{\gamma(1)}\,.$

A pair:

${\bm{\Delta}}\stackrel{{\scriptstyle{\rm def.}}}{{=}}(\Delta,\mathcal{L})\,$

consisting of a duality structure $\Delta$ and a choice of Dirac system
$\mathcal{L}$ for $\Delta$ is called an integral duality structure.

Let $\mathrm{Dual}_{\mathbb{Z}}(M)$ be the groupoid of integral duality
structures defined on $M$, with the obvious notion of isomorphism. For every
$m\in M$, the fiber $(\mathcal{S}_{m},\omega_{m},\mathcal{L}_{m})$ of an
integral duality structure ${\bm{\Delta}}=(\Delta,\mathcal{L})$ of rank $2n$
is an _integral symplectic space_ of dimension $2n$ (see Appendix B for
details). Each such space defines an integral vector (called its _type_)
belonging to a certain subset $\mathrm{Div}^{n}$ of $\mathbb{Z}_{>0}^{n}$
endowed with a partial order relation $\leq$ which makes it into a complete
meet semi-lattice. The type of an integral symplectic space depends only on
its isomorphism class (which it determines uniquely) and every element of
$\mathrm{Div}^{n}$ is realized as a type. Moreover, the group of automorphisms
of an integral symplectic space of type $\mathfrak{t}\in\mathrm{Div}^{n}$ is
isomorphic with the modified Siegel modular group
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ of type $\mathfrak{t}$ (see
Definition B.6). This is a discrete subgroup of $\mathrm{Sp}(2n,\mathbb{R})$
which contains the Siegel modular group $\mathrm{Sp}(2n,\mathbb{Z})$, to which
it reduces when $\mathfrak{t}$ equals the principal type
$\delta=(1,\ldots,1)$. If $\mathfrak{t}$ and $\mathfrak{t}^{\prime}$ are
elements of $\mathrm{Div}^{n}$ such that
$\mathfrak{t}\leq\mathfrak{t}^{\prime}$, then the lattice $\Lambda$ of any
integral symplectic space $(V,\omega,\Lambda)$ of type $\mathfrak{t}$ admits a
full rank sublattice $\Lambda^{\prime}$ such that
$(V,\omega,\Lambda^{\prime})$ is an integral symplectic space of type
$\mathfrak{t}^{\prime}$. In this case, we have
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\subset\mathrm{Sp}_{\mathfrak{t}^{\prime}}(2n,\mathbb{Z})$.
Since we assume that $M$ is connected and that $\mathcal{D}$ preserves
$\mathcal{L}$, the integral symplectic spaces
$(\mathcal{S}_{m},\omega_{m},\mathcal{L}_{m})$ are isomorphic to each other
through the parallel transport of $\mathcal{D}$, hence their type does not
depend on the base-point $m\in M$.

######  Definition 2.4.

The type $\mathfrak{t}_{{\bm{\Delta}}}\in\mathrm{Div}^{n}$ of an integral
duality structure ${\bm{\Delta}}=(\mathcal{S},\omega,\mathcal{D},\mathcal{L})$
(and of the corresponding Dirac system $\mathcal{L}$) is the common type of
the integral symplectic spaces $(\mathcal{S}_{m},\omega_{m},\mathcal{L}_{m})$,
where $m\in M$.

Let $\mathrm{Dual}_{\mathbb{Z}}^{\mathfrak{t}}(M)$ be the full subgroupoid of
$\mathrm{Dual}_{\mathbb{Z}}(M)$ consisting of integral duality structures of
type $\mathfrak{t}$.

######  Definition 2.5.

A duality structure $\Delta$ is called semiclassical if it admits a Dirac
system.

Not every duality structure is semiclassical, as the following proposition
shows.

######  Proposition 2.6.

A duality structure $\Delta=(\mathcal{S},\omega,\mathcal{D})$ admits a Dirac
system of type $\mathfrak{t}\in\mathrm{Div}^{n}$ if and only if the holonomy
representation of $\mathcal{D}$ at some point point (equivalently, at any
point) $m\in M$:

$\mathcal{T}_{\Delta}|_{\pi_{1}(M,m)}\colon\pi_{1}(M,m)\to\mathrm{Sp}(\mathcal{S}_{m},\omega_{m})\simeq\mathrm{Sp}(2n,\mathbb{R})$

can be conjugated so that its image lies inside the modified Siegel modular
group:

$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\subset\mathrm{Sp}(2n,\mathbb{R})$

of type $\mathfrak{t}$. In this case, $\Delta$ is semiclassical and the
greatest lower bound of those $\mathfrak{t}\in\mathrm{Div}^{n}$ with this
property is called the type of $\Delta$ and denoted by
$\mathfrak{t}_{\Delta}$.

###### Proof.

Assume $\Delta=(\mathcal{S},\omega,\mathcal{D})$ admits a Dirac system
$\mathcal{L}$ of type $\mathfrak{t}$. Then (as explained in Appendix B) the
automorphism group of every fiber
$(\mathcal{S}_{m},\omega_{m},\mathcal{L}_{m})$ is isomorphic to
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$, which is the automorphism group
of the standard integral symplectic space
$(\mathbb{R}^{2n},\omega_{2n},\Lambda_{\mathfrak{t}})$ of type $\mathfrak{t}$.
Since $\mathcal{L}$ is preserved by the parallel transport of $\mathcal{D}$,
it follows that we have
$\mathcal{T}_{\Delta}(\pi_{1}(M,m))\subset\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
after identifying $(\mathcal{S}_{m},\omega_{m},\mathcal{L}_{m})$ with
$(\mathbb{R}^{2n},\omega_{2n},\Lambda_{\mathfrak{t}})$ and hence
$\mathrm{Sp}(\mathcal{S}_{m},\omega_{m},\mathcal{L}_{m})$ with
$\mathrm{Sp}(2n,\mathbb{R})$. The converse follows immediately from the
associated bundle construction. ∎

######  Remark 2.7.

A duality structure $\Delta=(\mathcal{S},\omega,\mathcal{D})$ of rank $2n$
admits a Dirac system $\mathcal{L}$ of type
$\mathfrak{t}=(t_{1},\ldots,t_{n})\in\mathrm{Div}^{n}$ if and only if $M$
admits an open cover $\mathcal{U}=(U_{\alpha})_{\alpha\in I}$ such that for
each $\alpha\in I$ there exists a $\mathcal{D}$-flat frame
$(e_{1}^{(\alpha)},\ldots,e_{2n}^{(\alpha)})$ of $\mathcal{S}|_{U_{\alpha}}$
with the property:

$\omega(e_{i},e_{j})=\omega(e_{n+i},e_{n+j})=0\,,\,\,\omega(e_{i},e_{n+j})=t_{i}\delta_{ij}\,,\,\,\omega(e_{n+i},e_{j})=-t_{i}\delta_{ij}\,,\quad\forall\,\,i,j=1,\ldots,n\,.$
(11)

For each $\alpha,\beta\in I$ with $U_{\alpha}\cap U_{\beta}\neq\emptyset$ we
have:

$e_{k}^{(\beta)}=\sum_{l=1}^{2n}T^{(\alpha\beta)}_{lk}e^{(\alpha)}_{l}\,,\quad\forall\,\,k=1,\ldots,2n~{}~{}\mathrm{on}~{}~{}U_{\alpha}\cap
U_{\beta}\,,$

where $T^{(\alpha\beta)}_{lk}\in\mathbb{Z}$ for all $k,l=1,\ldots,2n$.
Furthermore:

$\oplus_{k=1}^{2n}\mathbb{Z}e^{(\alpha)}_{k}(m)=\mathcal{L}_{m}\,,\quad\forall\,\,\alpha\in
I\,,\quad\forall\,\,m\in U_{\alpha}\,,$

and the matrices $T^{(\alpha\beta)}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(T^{(\alpha\beta)}_{kl})_{k,l=1,\ldots,2n}$ belong to
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$.

Every integral duality structure
${\bm{\Delta}}=(\mathcal{S},\omega,\mathcal{D},\mathcal{L})$ on $M$ defines a
parallel transport functor:

$\mathcal{T}_{{\bm{\Delta}}}\colon\Pi_{1}(M)\to\mathrm{Symp}_{\mathbb{Z}}\,,$

where $\mathrm{Symp}_{\mathbb{Z}}$ is the groupoid of integral symplectic
spaces defined in Appendix B. This functor associates the integral symplectic
vector space $(\mathcal{S}_{m},\omega_{m},\mathcal{L}_{m})$ to every point
$m\in M$ and the isomorphism of symplectic vector spaces
$\mathcal{T}_{{\bm{\Delta}}}(\mathbf{c})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{T}_{\Delta}(\mathbf{c})$ to every homotopy class
$\mathbf{c}\in\Pi_{1}(M)(m,m^{\prime})$ of curves from $m\in M$ to
$m^{\prime}\in M$. The functor $\mathcal{T}_{{\bm{\Delta}}}$ defines a flat
system of integral symplectic vector spaces (that is, a
$\mathrm{Symp}_{\mathbb{Z}}$-valued local system) on $M$. As in Section 1, the
correspondence ${\bm{\Delta}}\mapsto\mathcal{T}_{{\bm{\Delta}}}$ extends to an
equivalence of groupoids:

$\mathcal{T}:\mathrm{Dual}_{\mathbb{Z}}(M)\xrightarrow{\sim}[\Pi_{1}(M),\mathrm{Symp}_{\mathbb{Z}}]$

between $\mathrm{Dual}_{\mathbb{Z}}(M)$ and the functor groupoid
$[\Pi_{1}(M),\mathrm{Symp}_{\mathbb{Z}}]$. Thus one can identify integral
duality structures with $\mathrm{Symp}_{\mathbb{Z}}$-valued local systems
defined on $M$. This implies the following result, whose proof is similar to
that of Proposition 1.7.

######  Proposition 2.8.

For any $m_{0}\in M$, the set of isomorphism classes of integral duality
structures of type $\mathfrak{t}$ defined on $M$ is in bijection with the
character variety:

$\mathfrak{R}(\pi_{1}(M,m_{0}),\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z}))\stackrel{{\scriptstyle{\rm
def.}}}{{=}}{\rm
Hom}(\pi_{1}(M,m_{0}),\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z}))/\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})~{}~{}.$
(12)

For later reference, we introduce the following:

######  Definition 2.9.

An _integral electromagnetic structure_ defined on $M$ is a pair:

$\bm{\Xi}\stackrel{{\scriptstyle{\rm def.}}}{{=}}(\Xi,\mathcal{L})\,,$

where $\Xi=(\mathcal{S},\omega,\mathcal{D},\mathcal{J})$ is an electromagnetic
structure on $M$ and $\mathcal{L}$ is a Dirac system for the duality structure
$\Delta=(\mathcal{S},\omega,\mathcal{D})$.

###  2.4. Siegel systems

Integral duality structures are associated to certain local systems of free
abelian groups of even rank defined on $M$.

######  Definition 2.10.

Let $n\in\mathbb{Z}_{>0}$. A Siegel system of rank $2n$ on $M$ is a bundle $Z$
of free abelian groups of rank $2n$ defined on $M$ equipped with a reduction
of its structure group from $\mathrm{GL}(2n,\mathbb{Z})$ to a subgroup of some
modified Siegel modular group $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$,
where $\mathfrak{t}\in\mathrm{Div}^{n}$. The greatest lower bound of the set
of those $\mathfrak{t}\in\mathrm{Div}^{n}$ with this property is called the
type of $Z$ and is denoted by $\mathfrak{t}_{Z}$.

Let $\mathrm{Sg}(M)$ be the groupoid of Siegel systems on $M$ and
$\mathrm{Sg}_{\mathfrak{t}}(M)$ be the full sub-groupoid of Siegel systems of
type $\mathfrak{t}$.

######  Remark 2.11.

Let $\underline{\mathbb{R}}$ be the trivial real line bundle on $M$ and
$U_{0}:\Pi_{1}(M)\rightarrow\Phi(\underline{\mathbb{R}})$ be the transport
functor induced by its trivial flat connection. The following statements are
equivalent for a bundle $Z$ of free abelian groups of rank $2n$ defined on
$M$:

  1. (a)

$Z$ is a Siegel system of type $\mathfrak{t}$ defined on $M$.

  2. (b)

The vector bundle $\mathcal{S}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}Z\otimes_{\mathbb{Z}}\underline{\mathbb{R}}$ carries a symplectic
pairing $\omega$ which is invariant under the parallel transport
$U_{Z}\otimes_{\mathbb{Z}}U_{0}$ of the flat connection induced from $Z$ and
which makes the triplet $(\mathcal{S}_{m},\omega_{m},Z_{m})$ into an integral
symplectic space of type $\mathfrak{t}$ for any $m\in M$.

  3. (c)

For any $m\in M$, the $2n$-dimensional vector space
$\mathcal{S}_{m}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}Z_{m}\otimes_{\mathbb{Z}}\mathbb{R}$ carries a symplectic form
$\omega_{m}$ which makes the triplet $(\mathcal{S}_{m},\omega_{m},Z_{m})$ into
an integral symplectic space of type $\mathfrak{t}$ and we have
$\mathrm{Hol}_{m}(Z)=\mathrm{Aut}(\mathcal{S}_{m},\omega_{m},Z_{m})$.

By definition, any Siegel system $Z$ of type $\mathfrak{t}$ is isomorphic with
the bundle of groups with fiber $\mathbb{Z}^{2n}$ associated to a principal
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$-bundle $Q$ through the left action
$\ell:\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\rightarrow\mathrm{Aut}_{\mathbb{Z}}(\mathbb{Z}^{2n})$
of $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ on $\mathbb{Z}^{2n}$:

$Z\simeq Z(Q)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}Q\times_{\ell}\mathbb{Z}^{2n}\simeq\hat{M}\times_{\ell_{\mathfrak{t}}\circ\alpha_{m}(Q)}\mathbb{Z}^{2n}~{}~{},$

where
$\alpha_{m}(Q):\pi_{1}(M,m)\rightarrow\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
is the monodromy morphism of $Q$ at $m$ and ${\hat{M}}$ is the universal cover
of $M$. The monodromy morphism $\sigma_{m}(Z)$ of $Z$ at $m$ identifies with
$\ell\circ\alpha_{m}(Q)$ upon choosing an integral symplectic basis of the
integral symplectic space $(\mathcal{S}_{m},\omega_{m},Z_{m})$. This also
identifies the monodromy group
$\mathrm{Hol}_{m}(Z)\subset\mathrm{Aut}_{\mathbb{Z}}(Z_{m})$ with
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$. Conversely, let
$\mathrm{Fr}_{\mathfrak{t}}(\mathbb{Z}^{2n})$ be the set of those bases of the
free $\mathbb{Z}$-module $\mathbb{Z}^{2n}$ in which the standard symplectic
form $\omega_{2n}$ of
$\mathbb{R}^{2n}=\mathbb{Z}^{2n}\otimes_{\mathbb{Z}}\mathbb{R}$ takes the form
$\omega_{\mathfrak{t}}$ (see Appendix B). Then
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ has a natural free and transitive
left action $\mu_{\mathfrak{t}}$ on this set. Taking the set of bases of each
of fiber gives the bundle of frames $\mathrm{Fr}(Z)$ of the Siegel system $Z$,
which is a principal $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$-bundle. The
previous discussion implies the following result.

######  Proposition 2.12.

The correspondences $Q\mapsto Z(Q)$ and $Z\mapsto\mathrm{Fr}(Z)$ extend to
mutually quasi-inverse equivalence of groupoids between
$\mathrm{Prin}_{\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})}(M)$ and
$\mathrm{Sg}_{\mathfrak{t}}(M)$.

In particular, the set of isomorphism classes of Siegel systems of type
$\mathfrak{t}$ defined on $M$ is in bijection with the character variety
$\mathfrak{R}(\pi_{1}(M,m_{0}),\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z}))$ of
equation (12). Let $\underline{1}$ be the unit section of the trivial real
line bundle $\underline{\mathbb{R}}=M\times\mathbb{R}$.

######  Proposition 2.13.

Let $Z$ be a Siegel system defined on $M$. Then there exists a unique integral
duality structure $\Delta=(\mathcal{S},\omega,\mathcal{D},\mathcal{L})$ such
that $\mathcal{S}=Z\otimes_{\mathbb{Z}}\underline{\mathbb{R}}$ and
$\mathcal{L}=Z\otimes_{\mathbb{Z}}\mathbb{Z}$ and this duality structure has
the same type as $Z$. Moreover, the parallel transport
$U_{\mathcal{D}}:\Pi_{1}(M)\rightarrow\Phi(\mathcal{S})$ of $\mathcal{D}$ is
given by:

$U_{\mathcal{D}}(\mathbf{c})=U_{Z}(\mathbf{c})\otimes_{\mathbb{Z}}U_{0}(m,m^{\prime})\,,\quad\forall\,\,\mathbf{c}\in\Pi_{1}(M)(m,m^{\prime})\,,\quad\forall\,\,m,m^{\prime}\in
M~{}~{},$ (13)

where $U_{Z}:\Pi_{1}(M)\rightarrow\Phi(Z)$ is the monodromy transport of $Z$
and $U_{0}:\Pi_{1}(M)\rightarrow\Phi(\underline{\mathbb{R}})$ is the trivial
transport of $\underline{\mathbb{R}}$.

######  Remark 2.14.

The fiber of $\mathcal{L}$ at $m\in M$ is given by:

$\mathcal{L}_{m}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{z\otimes_{\mathbb{Z}}1~{}|~{}z\in Z_{m}\\}\equiv Z_{m}~{}~{},$
(14)

where $1$ is the unit element of the field $\mathbb{R}$. It is clear that the
transport $U_{\mathcal{D}}$ defined by (13) gives bijections from
$\mathcal{L}_{m}$ to $\mathcal{L}_{m^{\prime}}$ and hence preserves
$\mathcal{L}$. Any locally-constant frame $(s_{1},\ldots,s_{2n})$ of $Z$
defined above a non-empty open set $V\subset M$ determines a local flat
symplectic frame $(e_{1},\ldots,e_{2n})$ of $\Delta$ defined above $V$ given
by:

$e_{i}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}s_{i}\otimes_{\mathbb{Z}}1\,,\quad\forall\,\,i=1,\ldots,2n~{}~{}$
(15)

and the matrix of $\omega$ with respect to this frame is integer-valued.

###### Proof.

The restriction of $U_{\mathcal{D}}$ to $\mathcal{L}$ gives isomorphisms of
groups between the fibers (14) of $\mathcal{L}$ and hence it must agree with
the monodromy transport $U_{Z}$ of $Z$ in the sense that:

$U_{\mathcal{D}}(\mathbf{c})(z\otimes_{\mathbb{Z}}1)=U_{Z}(\mathbf{c})(z)\otimes_{\mathbb{Z}}1\,,\quad\forall\,\,\mathbf{c}\in\Pi_{1}(M)(m,m^{\prime})\,,\quad\forall\,\,m,m^{\prime}\in
M~{}~{}.$

This implies (13) since $U_{\mathcal{D}}$ is $\mathbb{R}$-linear and
$\mathcal{L}_{m}$ are full lattices in
$\mathcal{S}_{m}=Z_{m}\otimes_{\mathbb{Z}}\mathbb{R}$. Remark 2.14 gives a
$\mathcal{D}$-flat symplectic pairing $\omega_{Z}$ on $\mathcal{S}_{Z}$ such
that the integral symplectic spaces $(\mathcal{S}_{m},\omega,\mathcal{L}_{m})$
have type $\mathfrak{t}_{Z}$ and such that $\mathcal{L}$ is preserved by the
parallel transport of $\mathcal{D}$. ∎

######  Remark 2.15.

Notice that $\mathcal{S}$ identifies with the vector bundle associated to the
frame bundle $\mathrm{Fr}(Z)$ of $Z$ through the linear representation
$q=\varphi\circ\iota:\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\rightarrow\mathrm{Aut}_{\mathbb{R}}(\mathbb{R}^{2n})$
defined by the inclusion morphism:

$\iota:\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\hookrightarrow\mathrm{Sp}(2n,\mathbb{R})~{}~{},$

where $\mathrm{Sp}(2n,\mathbb{R})$ acts on $\mathbb{R}^{2n}$ through the
fundamental representation
$\varphi:\mathrm{Sp}(2n,\mathbb{R})\rightarrow\mathrm{Aut}_{\mathbb{R}}(\mathbb{R}^{2n})$.
The representation $q$ preserves the canonical symplectic form
$\omega_{\mathfrak{t}}$ of $\mathbb{R}^{2n}$ and the latter induces the
symplectic pairing $\omega$ of $\mathcal{S}$.

We denote by ${\bm{\Delta}}(Z)$ the integral duality structure defined by $Z$
as in Proposition 2.13. Conversely, any integral duality structure
${\bm{\Delta}}=(\mathcal{S},\omega,\mathcal{L},\mathcal{D})$ defines a Siegel
system $Z({\bm{\Delta}})$ upon setting:

$Z({\bm{\Delta}})\stackrel{{\scriptstyle{\rm def.}}}{{=}}\mathcal{L}$

and it is easy to see that ${\bm{\Delta}}$ is isomorphic with
${\bm{\Delta}}(\mathcal{L})$. Therefore, we obtain the following result:

######  Proposition 2.16.

The correspondences $Z\mapsto{\bm{\Delta}}(Z)$ and ${\bm{\Delta}}\rightarrow
Z({\bm{\Delta}})$ extend to mutually quasi-inverse equivalences of groupoids
between $\mathrm{Sg}_{\mathfrak{t}}(M)$ and
$\mathrm{Dual}^{\mathfrak{t}}_{\mathbb{Z}}(M)$.

###  2.5. Bundles of integral symplectic torus groups

In the following we use the notions of integral symplectic torus group
discussed in Appendix B.

######  Definition 2.17.

A bundle of integral symplectic torus groups of rank $2n$ is a bundle
$\mathcal{A}$ of $2n$-dimensional torus groups defined on $M$ whose structure
group reduces from $\mathrm{GL}(2n,\mathbb{Z})$ to a subgroup of some modified
Siegel modular group $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$, where
$\mathfrak{t}\in\mathrm{Div}^{n}$. The greatest lower bound
$\mathfrak{t}_{\mathcal{A}}$ of the set of elements
$\mathfrak{t}\in\mathrm{Div}^{n}$ with this property is called the type of
$\mathcal{A}$.

Let $\mathcal{A}$ be a bundle of integral symplectic torus groups of type
$\mathfrak{t}$. Then the zero elements of the fibers determine a section
$s_{0}\in\mathcal{C}^{\infty}(M,\mathcal{A})$. The structure group
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ acts on $\mathcal{A}_{m}$
preserving the distinguished point $s_{0}(m)$ and the abelian group structure
of each fiber. Since such a bundle is associated to a principal
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$-bundle, it carries an induced flat
Ehresmann connection whose holonomy group is a subgroup of
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ and whose holonomy representation
at $m\in M$ we denote by:

$\rho_{m}(\mathcal{A}):\pi_{1}(M,m)\rightarrow\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\subset\mathrm{GL}(2n,\mathbb{Z})~{}~{}.$
(16)

The parallel transport of this connection preserves the image of the section
$s_{0}$ as well as a fiberwise symplectic structure which makes each fiber
into an integral symplectic torus group of type $\mathfrak{t}$ in the sense of
Appendix B. We have:

$\mathcal{A}\simeq{\hat{M}}\times_{\rho}\left(\mathbb{R}^{2n}/\mathbb{Z}^{2n}\right)~{}~{}.$

######  Remark 2.18.

When $M$ is compact, the fiber bundle $\mathcal{A}$ is virtually trivial by
[17, Theorem 1.1.], i.e. the pull-back of $\mathcal{A}$ to some finite
covering space of $M$ is topologically trivial.

It follows from the above that bundles of integral symplectic torus groups of
type $\mathfrak{t}$ defined on $M$ are classified by group morphisms (16),
i.e. the set of isomorphism classes of such bundles is in bijection with the
character variety
$\mathfrak{R}(\pi_{1}(M,m_{0}),\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$,
which also classifies integral duality structures. This also follows from the
results below.

######  Proposition 2.19.

Let ${\bm{\Delta}}=(\mathcal{S},\omega,\mathcal{D},\mathcal{L})$ be an
integral duality structure of type $\mathfrak{t}$ defined on $M$. Then the
fiberwise quotient:

$\mathcal{A}({\bm{\Delta}})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{S}/\mathcal{L}\,,$

is a bundle of integral symplectic torus groups of type $\mathfrak{t}$ defined
on $M$.

###### Proof.

It is clear that $\mathcal{A}({\bm{\Delta}})$ is a fiber bundle of even-
dimensional torus groups, whose zero section $s_{0}$ is inherited from the
zero section of $\mathcal{S}$. The fiberwise symplectic pairing $\omega$ of
$\mathcal{S}$ descends to a translation-invariant collection of symplectic
forms on the fibers of $\mathcal{A}({\bm{\Delta}})$, making the latter into
integral symplectic torus groups of type $\mathfrak{t}$. Since the parallel
transport of $\mathcal{D}$ preserves both $\mathcal{L}$ and $\omega$, this
bundle of torus groups inherits a flat Ehresmann connection which preserves
both its symplectic structure and the image of the section $s_{0}$ and whose
holonomy group reduces to $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$. In
particular, the structure group of $\mathcal{A}({\bm{\Delta}})$ reduces to
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$. ∎

As explained in Appendix B, any integral symplectic torus group
$\mathbf{A}=(A,\Omega)$ of type $\mathfrak{t}$ determines an integral
symplectic space $(H_{1}(A,\mathbb{R}),\omega,H_{1}(A,\mathbb{Z}))$, where
$\omega$ is the cohomology class of $\Omega$, viewed as a symplectic pairing
on the vector space $H_{1}(A,\mathbb{R})$.

######  Proposition 2.20.

Given a bundle $\mathcal{A}$ of integral symplectic torus groups of type
$\mathfrak{t}$ defined on $M$, let $\mathcal{S}_{\mathcal{A}}$ be the vector
bundle with fiber at $m\in M$ given by $H_{1}(\mathcal{A}_{m},\mathbb{R})$ and
$\mathcal{L}_{\mathcal{A}}$ be the bundle of discrete Abelian groups with
fiber at $m\in M$ given by $H_{1}(\mathcal{A}_{m},\mathbb{Z})$. Moreover, let
$\omega_{\mathcal{A}}$ be the fiberwise symplectic pairing defined on
$\mathcal{S}_{\mathcal{A}}$ through:

$\omega_{\mathcal{A},m}=\omega_{m}\,,\quad\forall\,\,m\in M~{}~{},$

where $\omega_{m}$ is the cohomology class of the symplectic form $\Omega_{m}$
of the fiber $\mathcal{A}_{m}$. Then the flat Ehresmann connection of
$\mathcal{A}$ induces a flat linear connection $\mathcal{D}_{\mathcal{A}}$ on
$\mathcal{S}_{\mathcal{A}}$ which makes the quadruplet:

${\bm{\Delta}}(\mathcal{A})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\mathcal{S}_{\mathcal{A}},\omega_{\mathcal{A}},\mathcal{D}_{\mathcal{A}},\mathcal{L}_{\mathcal{A}})\,,$

into an integral duality structure of type $\mathfrak{t}$ defined on $M$.

###### Proof.

$\mathcal{D}_{\mathcal{A}}$ is the connection induced by the flat Ehresmann
connection of $\mathcal{D}$ on the bundle of first homology groups of the
fibers, which preserves the bundle $\mathcal{L}_{\mathcal{A}}$ of integral
first homology groups of these fibers. The remaining statements are immediate.
∎

The two propositions above imply the following statement.

######  Proposition 2.21.

The correspondences ${\bm{\Delta}}\mapsto\mathcal{A}({\bm{\Delta}})$ and
$\mathcal{A}\mapsto{\bm{\Delta}}(\mathcal{A})$ extend to mutually quasi-
inverse equivalences between the groupoid
$\mathrm{Dual}^{\mathfrak{t}}_{\mathbb{Z}}(M)$ of integral duality structures
of type $\mathfrak{t}$ defined on $M$ and the groupoid
$\mathfrak{T}_{\mathfrak{t}}(M)$ of bundles of integral symplectic torus
groups of type $\mathfrak{t}$ defined on $M$.

Combining everything, we have a chain of equivalences of groupoids:

$\mathrm{Prin}_{\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})}(M)\simeq\mathrm{Sg}_{\mathfrak{t}}(M)\simeq\mathfrak{T}_{\mathfrak{t}}(M)\simeq\mathrm{Dual}^{\mathfrak{t}}_{\mathbb{Z}}(M)~{}~{}.$

###  2.6. The exponential sheaf sequence defined by a Siegel system

Let $Z$ be a Siegel system of type $\mathfrak{t}\in\mathrm{Div}^{n}$ on $M$.
Let $\mathcal{S}_{Z}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}Z\otimes_{\mathbb{Z}}\underline{\mathbb{R}}$ be the underlying
vector bundle of the integral duality structure ${\bm{\Delta}}(Z)$ defined by
$Z$ and $\mathcal{A}_{Z}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{S}_{Z}/Z$ be the associated bundle of integral symplectic
torus groups. The exponential sequence the torus group
$\mathbb{R}^{2n}/\Lambda_{\mathfrak{t}}\simeq\mathbb{R}^{2n}/\mathbb{Z}^{2n}$
(where the canonical symplectic lattice $\Lambda_{\mathfrak{t}}$ of type
$\mathfrak{t}$ is defined in Appendix B) induces a short exact sequence of
bundles of abelian groups (where $j$ is the inclusion):

$0\rightarrow Z\stackrel{{\scriptstyle
j}}{{\rightarrow}}\mathcal{S}_{Z}\stackrel{{\scriptstyle\exp}}{{\rightarrow}}\mathcal{A}_{Z}\rightarrow
0~{}~{}.$

In turn, this induces an exact sequence of sheaves of abelian groups:

$0\rightarrow\mathcal{C}(Z)\stackrel{{\scriptstyle
j}}{{\rightarrow}}\mathcal{C}^{\infty}(\mathcal{S}_{Z})\stackrel{{\scriptstyle\exp}}{{\rightarrow}}\mathcal{C}^{\infty}(\mathcal{A}_{Z})\rightarrow
0~{}~{},$

where $\mathcal{C}(Z)$ is the sheaf of continuous (and hence locally constant)
sections of $Z$. This induces a long exact sequence in sheaf cohomology, of
which we are interested in the following piece:

$H^{1}(M,\mathcal{C}(Z))\stackrel{{\scriptstyle
j_{\ast}}}{{\rightarrow}}H^{1}(M,\mathcal{C}^{\infty}(\mathcal{S}_{Z}))\stackrel{{\scriptstyle\exp_{\ast}}}{{\rightarrow}}H^{1}(M,\mathcal{C}^{\infty}(\mathcal{A}_{Z}))\stackrel{{\scriptstyle\delta}}{{\rightarrow}}H^{2}(M,\mathcal{C}(Z))\rightarrow
H^{2}(M,\mathcal{C}^{\infty}(\mathcal{S}_{Z}))\,,$ (17)

where $\delta$ is the Bockstein morphism. The sheaf
$\mathcal{C}^{\infty}(\mathcal{S}_{Z})$ is fine and hence acylic since $M$ is
paracompact and $\mathcal{S}_{Z}$ is a vector bundle on $M$. Thus:

$H^{j}(M,\mathcal{S}_{Z})=0\,,\quad\forall j>0\,,$

which by the long sequence above implies that $\delta$ is an isomorphism of
groups. We also have $H^{\ast}(M,\mathcal{C}(Z))=H^{\ast}(M,Z)$, where in the
right hand side $Z$ is viewed as a local system of $\mathbb{Z}^{2n}$
coefficients. Hence we can view $\delta$ as an isomorphism of abelian groups:

$\delta:H^{1}(M,\mathcal{C}^{\infty}(\mathcal{A}_{Z}))\xrightarrow{\sim}H^{2}(M,Z)~{}~{}.$
(18)

###  2.7. The lattice of charges of an integral duality structure

For every integral duality structure ${\bm{\Delta}}=(\Delta,\mathcal{L})$ on
$M$, the sheaf $\mathcal{C}_{\mathrm{flat}}^{\infty}(\mathcal{A})$ of flat
smooth local sections of the bundle $\mathcal{A}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{A}({\bm{\Delta}})=\mathcal{S}/\mathcal{L}$ of integral
symplectic torus groups defined by ${\bm{\Delta}}$ fits into the short exact
sequence of sheaves of abelian groups:

$1\to\mathcal{C}(\mathcal{L})\xrightarrow{j_{0}}\mathcal{C}^{\infty}_{\mathrm{flat}}(\mathcal{S})\xrightarrow{\exp}\mathcal{C}^{\infty}_{\mathrm{flat}}(\mathcal{A})\to
1\,,$

where $j$ is the inclusion. This induces a long exact sequence in sheaf
cohomology, of which we are interested in the following terms:

$\ldots\rightarrow
H^{1}(M,\mathcal{C}^{\infty}_{\mathrm{flat}}(\mathcal{A}))\xrightarrow{\delta_{0}}H^{2}(M,\mathcal{C}(\mathcal{L}))\xrightarrow{j_{0\ast}}H^{2}(M,\mathcal{C}^{\infty}_{\mathrm{flat}}(\mathcal{S}))\xrightarrow{\exp_{\ast}}H^{2}(M,\mathcal{C}^{\infty}_{\mathrm{flat}}(\mathcal{A}))\rightarrow\ldots~{}~{},$

where $\delta_{0}$ is the Bockstein morphism. Notice that
$H^{\ast}(M,\mathcal{C}(\mathcal{L}))\simeq H^{\ast}(M,Z)$, where
$Z=\mathcal{L}$ is the Siegel system defined by $\mathcal{L}$, which we view
of a local system of $\mathbb{Z}^{2n}$ coefficients. Moreover, we have
$H^{2}(M,\mathcal{C}^{\infty}_{\mathrm{flat}}(\mathcal{A}))=H^{2}(M,\mathcal{A}_{\mathrm{disc}})$,
the right hand side being the cohomology with coefficients in the local system
defined by $\mathcal{A}$ when the fibers of the latter are endowed with the
discrete topology. Since
$H^{\ast}(M,\mathcal{C}_{\mathrm{flat}}^{\infty}(\mathcal{S}))=H^{\ast}_{\mathcal{D}}(M,\mathcal{S})$,
the sequence above can be written as:

$\ldots\rightarrow
H^{1}(M,\mathcal{A}_{\mathrm{disc}})\xrightarrow{\delta_{0}}H^{2}(M,Z)\xrightarrow{j_{0\ast}}H^{2}_{\mathcal{D}}(M,\mathcal{S})\xrightarrow{\exp_{\ast}}H^{2}(M,\mathcal{A}_{\mathrm{disc}})\rightarrow\ldots$
(19)

Denote by $H^{2}(M,Z)^{\mathrm{tf}}\subset H^{2}(M,Z)$ the torsion free part
of $H^{2}(M,Z)$.

######  Definition 2.22.

The lattice:

$L_{\bm{\Delta}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}j_{0\ast}(H^{2}(M,Z))=j_{0\ast}(H^{2}(M,Z)^{\mathrm{tf}})\subset
H^{2}_{\mathcal{D}}(M,\mathcal{S})$

is called the lattice of charges defined the integral duality structure
${\bm{\Delta}}$. Elements of this lattice are called _integral cohomology
classes_ or charges.

######  Proposition 2.23.

There exists a natural isomorphism:

$H^{k}_{\mathcal{D}}(M,\mathcal{S})\simeq
H^{k}(M,Z)\otimes_{\mathbb{Z}[\bm{\pi}]}\mathbb{R}\simeq
H^{k}(M,Z)^{\mathrm{tf}}\otimes_{\mathbb{Z}[\bm{\pi}]}\mathbb{R}~{}~{},$ (20)

for all $k$. In particular, the kernel of $j_{0\ast}$ coincides with the
torsion part of $H^{2}(M,Z)$ and $j_{\ast}(H^{2}(M,Z))$ is a full lattice in
$H^{2}_{\mathcal{D}}(M,\mathcal{S})$.

###### Proof.

Let $\bm{\pi}\stackrel{{\scriptstyle{\rm def.}}}{{=}}\pi_{1}(M,m)$ and
$\mathbb{Z}[\bm{\pi}]$ be the group ring of $\bm{\pi}$. The universal
coefficient theorem for cohomology with local coefficients of [29] gives a
short exact sequence:

$0\to H^{k}(M,Z)\otimes_{\mathbb{Z}[\bm{\pi}]}\mathbb{R}\to
H^{k}_{\mathcal{D}}(M,\mathcal{S})\to\mathrm{Tor}_{\mathbb{Z}[\bm{\pi}]}(H^{k+1}(M,Z),\mathbb{R})\to
0\,,$

where $\mathbb{R}$ is the $\mathbb{Z}[\bm{\pi}]$-module corresponding to the
trivial representation of $\bm{\pi}$ in $\mathbb{R}$. Since the latter module
is free, we have
$\mathrm{Tor}_{\mathbb{Z}[\bm{\pi}]}(H^{k+1}(M,Z),\mathbb{R})=0$ and the
sequence above gives the natural isomorphism (20). ∎

######  Remark 2.24.

We have a commutative diagram with exact rows:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}(Z)~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces~{}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{j_{0}}$$\textstyle{~{}\mathcal{C}_{\mathrm{flat}}^{\infty}(\mathcal{S}_{Z})~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{~{}~{}\exp}$$\scriptstyle{\tau}$$\textstyle{\mathcal{C}_{\mathrm{flat}}^{\infty}(\mathcal{A}_{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces~{}}$$\textstyle{~{}\mathcal{C}(Z)~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{~{}\mathcal{C}^{\infty}(\mathcal{S}_{Z})~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{~{}~{}\exp}$$\textstyle{\mathcal{C}^{\infty}(\mathcal{A}_{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $j_{0},j$ and $\tau,\iota$ are inclusions. In turn, this induces a
commutative diagram with exact rows:

$\textstyle{H^{1}(M,Z)~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces~{}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{j_{0\ast}}$$\textstyle{~{}H^{1}_{\mathcal{D}_{P}}(M,\mathcal{S}_{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{\ast}}$$\scriptstyle{\exp_{\ast}}$$\textstyle{H^{1}(M,\mathcal{A}_{Z,\mathrm{disc}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\ast}}$$\scriptstyle{\delta_{0}}$$\textstyle{H^{2}(M,Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{j_{0\ast}}$$\textstyle{~{}H^{2}_{\mathcal{D}_{P}}(M,\mathcal{S}_{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{\ast}}$$\textstyle{~{}H^{1}(M,Z)~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\ast}}$$\textstyle{~{}0~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp_{\ast}}$$\textstyle{H^{1}(M,\mathcal{C}^{\infty}(\mathcal{A}_{Z}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H^{2}(M,Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\ast}}$$\textstyle{0}$

In particular, we have $\delta_{0}=\delta\circ\iota_{\ast}$.

###  2.8. The DSZ integrality condition

Let $(M,g)$ be an oriented and connected Lorentzian four-manifold. Given an
integral duality structure ${\bm{\Delta}}=(\Delta,\mathcal{L})$, we implement
the DSZ condition by restricting the configuration space
$\mathrm{Conf}(M,\Delta)$ to a subsets determined by the charge lattice
$L_{\bm{\Delta}}$. We will show in later sections that integral field
strengths are adjoint curvatures of connections defined on a certain principal
bundle.

######  Definition 2.25.

Let ${\bm{\Delta}}=(\Delta,\mathcal{L})$ be an integral duality structure on
$(M,g)$ with underlying duality structure
$\Delta=(\mathcal{S},\omega,\mathcal{D})$. The set of integral electromagnetic
field strength configurations defined by ${\bm{\Delta}}$ on $M$ is the
following subset of $\mathrm{Conf}(M,\Delta)$:

$\mathrm{Conf}(M,{\bm{\Delta}})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{\mathcal{V}\in\mathrm{Conf}(M,\Delta)\,\,|\,\,2\pi[\mathcal{V}]_{\mathcal{D}}\in
L_{\bm{\Delta}}\right\\}\,,$

where $[\mathcal{V}]_{\mathcal{D}}\in H^{2}_{\mathcal{D}}(M,\mathcal{S})$ is
the $\mathrm{d}_{\mathcal{D}}$-cohomology class of the $\mathcal{S}$-valued
two-form $\mathcal{V}\in\mathrm{Conf}(M,\Delta)$ and $L_{\bm{\Delta}}\subset
H^{2}_{\mathcal{D}}(M,\mathcal{S})$ is the lattice of charges defined by
${\bm{\Delta}}$.

######  Definition 2.26.

Let $\bm{\Xi}=({\bm{\Delta}},\mathcal{J})$ be an integral electromagnetic
structure defined on $(M,g)$ with electromagnetic structure
$\Xi=(\Delta,\mathcal{J})$ and integral duality structure
${\bm{\Delta}}=(\Delta,\mathcal{L})$. The set of integral field strength
solutions defined by $\bm{\Xi}$ on $(M,g)$ is the subset of
$\mathrm{Sol}(M,g,\Xi)$ defined through:

$\mathrm{Sol}(M,g,\bm{\Xi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Sol}(M,g,\Xi)\cap\mathrm{Conf}(M,{\bm{\Delta}})$

and hence consists of those elements of $\mathrm{Conf}(M,{\bm{\Delta}})$ which
satisfy the equations of motion (i.e. the polarized self-duality condition) of
the classical abelian gauge theory defined by $\Xi$.

###  2.9. Integral duality groups

The DSZ integrality condition restricts the classical duality groups of
Subsection 1.9 to certain subgroups.

######  Definition 2.27.

Fix an integral duality structure ${\bm{\Delta}}$ on $(M,g)$.

  * •

The integral unbased pseudo-duality group defined by ${\bm{\Delta}}$ is the
group $\mathrm{Aut}({\bm{\Delta}})\subset\mathrm{Aut}(\Delta)$ formed by those
elements $u\in\mathrm{Aut}(\Delta)$ which satisfy
$u(\mathcal{L})=\mathcal{L}$.

  * •

The integral unbased duality group defined by ${\bm{\Delta}}$ is the subgroup
$\mathrm{Aut}(g,{\bm{\Delta}})$ of $\mathrm{Aut}({\bm{\Delta}})$ which covers
${\rm Iso}(M,g)$.

  * •

The integral duality group defined by ${\bm{\Delta}}$ is the subgroup
$\mathrm{Aut}_{b}({\bm{\Delta}})$ of $\mathrm{Aut}({\bm{\Delta}})$ consisting
of those elements which cover the identity of $M$.

######  Definition 2.28.

Fix an integral electromagnetic structure
$\bm{\Xi}=({\bm{\Delta}},\mathcal{J})$ on $(M,g)$.

  * •

The integral unbased unitary pseudo-duality group defined by $\bm{\Xi}$ is the
group:

$\mathrm{Aut}(\bm{\Xi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{u\in\mathrm{Aut}({\bm{\Delta}})~{}|~{}\mathcal{J}_{u}=\mathcal{J}\\}$

  * •

The integral unbased unitary duality group defined by $\bm{\Xi}$ is:

$\mathrm{Aut}(g,\bm{\Xi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}(\bm{\Xi})\,\,|\,\,g_{u}=g\right\\}~{}~{}.$

  * •

The integral unitary duality group defined by $\bm{\Xi}$ is the subgroup
$\mathrm{Aut}_{b}(\bm{\Xi})$ of $\mathrm{Aut}(\bm{\Xi})$ consisting of those
elements which cover the identity of $M$.

It is easy to check that $\mathbb{A}_{u}$ with $u$ belonging to the groups
defined above restrict to transformations similar to those of Subsection 1.9
between the sets of integral configurations and solutions. The discrete
duality groups introduced above are the global counterparts of the discrete
duality group considered in the physics literature on local abelian gauge
theory. The latter is usually taken to be $\mathrm{Sp}(2n,\mathbb{Z})$ due to
the fact that the symplectic lattice of charges appearing in the local
treatment of abelian gauge theory is traditionally assumed to have principal
type $\mathfrak{t}=\delta=(1,\ldots,1)$. As explained in [32] and recalled in
Appendix A, $\mathrm{Sp}(2n,\mathbb{Z})$ is not always the correct duality
group even in the local case, since the local lattice of charges need not be
principal. In Section 4.1, we consider a natural gauge-theoretic extension of
the discrete duality groups defined above, which clarifies the geometric
origin of electromagnetic duality.

###  2.10. Trivial integral duality structures

Let $Z$ be a trivializable Siegel system of type
$\mathfrak{t}\in\mathrm{Div}^{n}$ and
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ be the associated duality structure,
where $\mathcal{S}=Z\otimes_{\mathbb{Z}}\underline{\mathbb{R}}$. Pick a flat
trivialization $\tau:\mathcal{S}\xrightarrow{\sim}M\times S$ of $\mathcal{S}$,
where $S\simeq\mathbb{R}^{2n}$. This takes $\omega$ into a symplectic pairing
$\omega_{S}$ on the vector space $S$ and restricts to an isomorphism
$\tau_{0}:Z\xrightarrow{\sim}M\times\Lambda$ between $Z$ and $M\times\Lambda$,
where $\Lambda$ is a full symplectic lattice in $(S,\omega_{S})$. Let
$A\stackrel{{\scriptstyle{\rm def.}}}{{=}}S/\Lambda$ be the torus group
defined by $(S,\Lambda)$ and $\mathcal{A}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{S}/Z$ be the bundle of torus groups defined by
$(\mathcal{S},Z)$. Then $\tau$ induces a trivialization
$\bar{\tau}:\mathcal{A}\xrightarrow{\sim}M\times A$ of $\mathcal{A}$, which
fits into a commutative diagram of fiber bundles:

$\textstyle{Z~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces~{}}$$\scriptstyle{\tau_{0}}$$\scriptstyle{j}$$\textstyle{~{}\mathcal{S}~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\tau}}$$\textstyle{M\times\Lambda~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{~{}M\times
S~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\times
A}$

where $i$ and $j$ are inclusions. Since $\tau$ identifies $\mathcal{D}$ with
the trivial connection on $M\times S$, it induces an isomorphism of graded
vector spaces:

$\tau_{\ast}:H^{\ast}_{\mathcal{D}}(M,\mathcal{S})\xrightarrow{\sim}H^{\ast}(M,S)$

whose restriction coincides with the isomorphism of graded abelian groups:

$\tau_{0\ast}:H^{\ast}(M,Z)\xrightarrow{\sim}H^{\ast}(M,\Lambda)$

induced by $\tau_{0}$. Moreover, $\bar{\tau}$ induces an isomorphism of graded
abelian groups
$\bar{\tau}_{\ast}:H^{\ast}(M,\mathcal{A})\xrightarrow{\sim}H^{\ast}(M,A)$.
Hence the diagram above induces an isomorphism of long exact sequences of
abelian groups:

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Since $\Lambda$ is free while $\mathcal{S}$ is a vector space, we have
isomorphisms of abelian groups:

$H^{\ast}(M,S)\simeq
H^{\ast}(M,\mathbb{R})\otimes_{\mathbb{R}}S~{}~{},~{}~{}H^{\ast}(M,\Lambda)\simeq_{\mathbb{Z}}H^{\ast}(M,\mathbb{Z})\otimes_{\mathbb{Z}}\Lambda~{}~{}.$

and:

$H^{\ast}(M,S)\simeq H^{\ast}(M,\Lambda)\otimes_{\mathbb{Z}}\mathbb{R}\simeq
H^{\ast}(M,\Lambda)^{\mathrm{tf}}\otimes_{\mathbb{Z}}\mathbb{R}~{}~{}.$

The latter agrees with the isomorphism (20) through the maps $\tau_{0\ast}$
and $\tau_{\ast}$. The map $i_{\ast}:H^{k}(M,\Lambda)\rightarrow H^{k}(M,S)$
is obtained by tensoring the map $H^{k}(M,\mathbb{Z})\rightarrow
H^{k}(M,\mathbb{R})$ with the inclusion $\Lambda\subset S$, while its
restriction $i_{\ast}^{\mathrm{tf}}:H^{k}(M,\Lambda)^{\mathrm{tf}}\rightarrow
H^{k}(M,S)$ is obtained by tensoring the inclusion $\Lambda\subset S$ with the
map $H^{l}(M,\mathbb{Z})^{\mathrm{tf}}\rightarrow H^{k}(M,\mathbb{R})$. Since
the latter is injective, it follows that $i_{\ast}^{\mathrm{tf}}$ is injective
and hence $H^{l}(M,\Lambda)^{\mathrm{tf}}$ identifies with a full lattice in
$H^{k}(M,S)$. Since $A$ and $(S,+)$ are divisible groups while
$H_{0}(M,\mathbb{Z})=\mathbb{Z}$ and $\Lambda$ are free, the universal
coefficient sequence for cohomology gives isomorphisms:

$\displaystyle H^{k}(M,S)\simeq{\rm
Hom}_{\mathbb{Z}}(H_{k}(M,\mathbb{Z}),S)={\rm
Hom}_{\mathbb{Z}}(H_{k}(M,\mathbb{Z})^{\mathrm{tf}},S)\,,$ $\displaystyle
H^{k}(M,\Lambda)\simeq{\rm Hom}_{\mathbb{Z}}(H_{k}(M,\mathbb{Z}),\Lambda)={\rm
Hom}_{\mathbb{Z}}(H_{k}(M,\mathbb{Z})^{\mathrm{tf}},\Lambda)$ (21)

and:

$H^{k}(M,A)\simeq{\rm Hom}_{\mathbb{Z}}(H_{k}(M,\mathbb{Z}),A)\simeq{\rm
Hom}_{\mathbb{Z}}(H_{k}(M,\mathbb{Z})^{\mathrm{tf}},A)$

for all $k$. The first of these is the period isomorphism:

$\mathrm{per}(\omega)(c):=\mathrm{per}_{c}(\omega)=c\cap\omega=\int_{c}\omega\,,\quad\forall\,\,\omega\in
H^{k}(M,S)\,,\quad\forall\,\,c\in H_{k}(M,\mathbb{Z})\,.$

The map $i_{\ast}:H^{k}(M,\Lambda)\rightarrow H^{k}(M,\mathcal{S})$ agrees
with the injective map induced by the inclusion $\Lambda\hookrightarrow S$
though the isomorphisms (2.10). Hence:

$H^{k}(M,\Lambda)\simeq\mathrm{per}^{-1}({\rm
Hom}_{\mathbb{Z}}(H_{k}(M,\mathbb{Z}),\Lambda))=i_{\ast}(H^{k}(M,\Lambda))=\\{\omega\in
H^{k}(M,S)~{}|~{}\mathrm{per}_{c}(\omega)\in\Lambda\,\&\,c\in
H_{k}(M,\mathbb{Z})\\}~{}~{}.$

##  3\. Siegel bundles and connections

In this section we use the notion of integral affine symplectic torus, for
which we refer the reader to Appendix B.

###  3.1. Automorphisms of integral affine symplectic tori

We denote by $\mathrm{Aff}_{\mathfrak{t}}$ the group of affine
symplectomorphisms of the integral affine symplectic torus
$\mathbbold{A}_{\mathfrak{t}}=(\mathbb{A},\Omega_{\mathfrak{t}})$ of type
$\mathfrak{t}\in\mathrm{Div}^{n}$. Here $\mathbb{A}$ is the underlying
$2n$-dimensional affine torus (which is a principal homogeneous space for the
torus group $\mathrm{U}(1)^{2n}\simeq\mathbb{R}^{2n}/\mathbb{Z}^{2n}$), while
$\Omega_{\mathfrak{t}}$ is the integral symplectic form of type $\mathfrak{t}$
on $\mathbb{A}$, which is translationally-invariant. As explained in Appendix
B, $\mathrm{Aff}_{\mathfrak{t}}$ is a non-compact disconnected Lie group whose
connected component of the identity is the $2n$-dimensional torus group
$\mathrm{U}(1)^{2n}$. We have
$\pi_{0}(\mathrm{Aff}_{\mathfrak{t}})\simeq\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
and:

$\mathrm{Aff}_{\mathfrak{t}}\simeq\mathrm{U}(1)^{2n}\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\,,$
(22)

where $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ acts on $\mathrm{U}(1)^{2n}$
through the restriction of the group morphism defined in equation (82) of
Appendix B, an action which we denote by juxtaposition. Thus
$\mathrm{Aff}_{\mathfrak{t}}$ identifies with the set
$\mathrm{U}(1)^{2n}\times\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$, endowed
with the composition rule:

$(a_{1},\gamma_{1})\,(a_{2},\gamma_{2})=(a_{1}+\gamma_{1}a_{2},\gamma_{1}\gamma_{2})\,,\quad\forall\,\,a_{1},a_{2}\in\mathrm{U}(1)^{2n}\,,\quad\forall\,\,\gamma_{1},\gamma_{2}\in\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\,.$

Let
$\ell:\mathrm{Aff}(t)\rightarrow\mathrm{Diff}(\mathrm{Aff}_{\mathfrak{t}})$ be
the left action of $\mathrm{Aff}_{\mathfrak{t}}$ on itself:

$\ell(g)(g^{\prime})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}gg^{\prime}\,,\quad\forall\,\,g,g^{\prime}\in\mathrm{Aff}_{\mathfrak{t}}\,,$

and let
$\mathrm{pr}_{1}:\mathrm{Aff}_{\mathfrak{t}}\rightarrow\mathrm{U}(1)^{2n}$ and
$\mathrm{pr}_{2}:\mathrm{Aff}_{\mathfrak{t}}\rightarrow\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
be the projections of the set-theoretic decomposition
$\mathrm{Aff}_{\mathfrak{t}}=\mathrm{U}(1)^{2n}\times\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$.
Notice that $\mathrm{pr}_{2}$ is a morphism of groups. Define left actions
$\ell_{1}$ and $\ell_{2}$ of $\mathrm{Aff}_{\mathfrak{t}}$ on
$\mathrm{U}(1)^{2n}$ and $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ through:

$\ell_{1}(g)(a)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{pr}_{1}(\ell(g)(a,1))\,,\quad\ell_{2}(g)(\gamma)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{pr}_{2}(\ell(g)(0,\gamma))\,~{},~{}\forall
g\in\mathrm{Aff}_{\mathfrak{t}}~{},~{}\forall
a\in\mathrm{U}(1)^{2n}~{},~{}\forall\gamma\in\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})~{}~{}.$

Then $\ell_{1}$ is given by:

$\ell_{1}(a,\gamma)(a^{\prime})=a+\gamma
a^{\prime}\,,\quad\forall\,\,\gamma\in\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\,,\quad\forall\,\,a,a^{\prime}\in\mathrm{U}(1)^{2n}~{}~{}.$
(23)

This action is transitive with stabilizer isomorphic with
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$. On the other hand, $\ell_{2}$ is
given by:

$\ell_{2}(a,\gamma)(\gamma^{\prime})=\gamma\gamma^{\prime}=p_{2}(a,\gamma)\gamma^{\prime}\,,\quad\forall\,\,\gamma,\gamma^{\prime}\in\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\,,\quad\forall\,\,a\in\mathrm{U}(1)^{2n}\,$
(24)

and is transitive with stabilizer isomorphic to $\mathrm{U}(1)^{2n}$. This
gives the right-split short exact sequence:

$0\rightarrow\mathrm{U}(1)^{2n}\rightarrow\mathrm{Aff}_{\mathfrak{t}}\rightarrow\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\rightarrow
1~{}~{}.$

Notice that $\ell=\ell_{1}\times\ell_{2}$. The Lie algebra
$\mathfrak{aff}_{\mathfrak{t}}$ of $\mathrm{Aff}_{\mathfrak{t}}$ is abelian
and coincides with the Lie algebra of $\mathrm{U}(1)^{2n}$:

$\mathfrak{aff}_{\mathfrak{t}}=\mathbb{R}^{2n}\simeq
H_{1}(\mathbb{A}_{\mathfrak{t}},\mathbb{R})~{}~{}.$

The exponential map
$\exp:\mathfrak{aff}_{\mathfrak{t}}\rightarrow\mathrm{Aff}_{\mathfrak{t}}$ has
kernel $\Lambda_{\mathfrak{t}}\simeq
H_{1}(\mathbb{A}_{\mathfrak{t}},\mathbb{Z})$ and image $A$, giving the
exponential sequence:

$0\rightarrow\Lambda_{\mathfrak{t}}\rightarrow\mathfrak{aff}_{\mathfrak{t}}\stackrel{{\scriptstyle\exp}}{{\rightarrow}}\mathrm{U}(1)^{2n}\rightarrow
0~{}~{}.$ (25)

######  Lemma 3.1.

The adjoint representation
$\mathrm{Ad}\colon\mathrm{Aff}_{\mathfrak{t}}\to\mathrm{GL}(2n,\mathbb{R})$ of
$\mathrm{Aff}_{\mathfrak{t}}$ coincides with its fundamental linear
representation, that is:

$\mathrm{Ad}(a,\gamma)(v)=\gamma(v)\,,\quad\forall\,\,(a,\gamma)\in\mathrm{Aff}_{\mathfrak{t}}\,,\quad\forall\,\,v\in\mathbb{R}^{2n}\,.$
(26)

In particular, we have $\mathrm{Ad}=j\circ\mathrm{pr}_{2}$, where
$j:\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\rightarrow\mathrm{GL}(2n,\mathbb{R})$
is the fundamental representation of
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$.

###### Proof.

Let $\alpha=(\gamma,1)\colon
I\to\mathrm{Aff}(\mathfrak{t})=\mathrm{U}(1)^{2n}\times\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
be a smooth path on $\mathrm{Aff}(\mathfrak{t})$ such that
$\alpha(0)=\mathrm{Id}$. Set:

$\frac{\mathrm{d}}{\mathrm{d}t}\alpha(t)|_{t=0}=v\in\mathbb{R}^{2n}\,.$

For every
$x=(x_{1},x_{2})\in\mathrm{Aff}(\mathfrak{t})=\mathrm{U}(1)^{2n}\times\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
we have:

$x\,\alpha(t)\,x^{-1}=(x_{1},x_{2})\,(\gamma,1)\,(-x^{-1}_{2}x_{1},x_{2}^{-1})=(x_{2}\gamma,1)\,.$

Hence:

$\frac{\mathrm{d}}{\mathrm{d}t}(x\,\alpha(t)\,x^{-1})|_{t=0}=x_{2}(v)\,,$

which immediately implies:

$\mathrm{Ad}(x)=j\circ\mathrm{pr}_{2}(x)\,,$

for every $x\in\mathrm{Aff}(\mathfrak{t})$ and hence we conclude. ∎

###  3.2. Siegel bundles

Let $M$ be a connected manifold.

######  Definition 3.2.

A Siegel bundle $P$ of rank $n$ and type $\mathfrak{t}\in\mathrm{Div}^{n}$ is
a principal bundle on $M$ with structure group $\mathrm{Aff}_{\mathfrak{t}}$.
An isomorphism of Siegel bundles is a based isomorphism of principal bundles.

Let $\mathrm{Sieg}(M)$ be the groupoid of Siegel bundles defined on $M$ and