$\mathrm{Sieg}_{\mathfrak{t}}(M)$ be the full subgroupoid of Siegel bundles of
type $\mathfrak{t}$. Fix a Siegel bundle $P$ of type
$\mathfrak{t}\in\mathrm{Div}^{n}$, whose projection we denote by $\pi$. We
introduce several fiber bundles associated to $P$.

####  3.2.1. The bundle of integral affine symplectic tori defined by $P$

######  Definition 3.3.

A fiber bundle $\mathfrak{A}$ defined on $M$ is called a bundle of integral
affine symplectic tori of rank $n$ if its fibers are $2n$-dimensional tori and
the structure group of $\mathfrak{A}$ reduces to a subgroup of
$\mathrm{Aff}_{\mathfrak{t}}$ for some $\mathfrak{t}\in\mathrm{Div}^{n}$. The
smallest element $\mathfrak{t}_{\mathfrak{A}}\in\mathrm{Div}^{n}$ with this
property is called the type of $\mathfrak{A}$.

Notice that bundles of integral symplectic torus groups coincide with those
bundles of integral affine symplectic tori which admit a smooth global
section. Indeed, such a section gives a further reduction of structure group
from $\mathrm{Aff}_{\mathfrak{t}}$ to
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$. Given a Siegel bundle $P$ of type
$\mathfrak{t}\in\mathrm{Div}^{n}$ defined on $M$, the fiber bundle:

$\mathfrak{A}(P)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}P\times_{\ell_{1}}\mathrm{U}(1)^{2n}$

associated to $P$ through the action (23) is a bundle of integral affine
symplectic tori of type $\mathfrak{t}$. The fibers of the latter admit
integral symplectic forms of type $\mathfrak{t}$ which vary smoothly over $M$.
The group $\mathrm{Sp}(V,\omega)$ acts freely and transitively on the set
$\mathrm{Fr}(V,\omega,\Lambda)$ of integral symplectic bases of any integral
symplectic space $(V,\omega,\Lambda)$. Any bundle $\mathfrak{A}$ of integral
affine symplectic tori of type $\mathfrak{t}$ is associated through the action
$\ell_{1}$ to its Siegel bundle $P(\mathfrak{A})$ of unpointed torus
symplectic frames, which has type $\mathfrak{t}$ and whose fiber at $m\in M$
is defined through:

$P(\mathfrak{A})_{m}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Fr}(H_{1}(\mathfrak{A}_{m},\mathbb{R}),H_{1}(\mathfrak{A}_{m},\mathbb{Z}),\omega_{m})\times\mathfrak{A}_{m}~{}~{}.$

Here $\omega_{m}\stackrel{{\scriptstyle{\rm def.}}}{{=}}[\Omega_{m}]\in
H^{2}(\mathfrak{A}_{m},\mathbb{R})\simeq\wedge^{2}H_{1}(\mathfrak{A}_{m},\mathbb{R})^{\vee}$
is the cohomology class of the symplectic form $\Omega_{m}$ of
$\mathfrak{A}_{m}$, viewed as a symplectic pairing defined on
$H_{1}(\mathfrak{A}_{m},\mathbb{R})$. More precisely, we have:

######  Proposition 3.4.

The correspondences $P\rightarrow\mathfrak{A}(P)$ and $\mathfrak{A}\rightarrow
P(\mathfrak{A})$ extend to mutually quasi-inverse equivalences of groupoids
between $\mathrm{Sieg}(M)$ and the groupoid of bundles of integral affine
symplectic tori and these equivalences preserve type.

This statement parallels a similar correspondence which holds for affine torus
bundles (see [11] as well as Theorem 2.2. and Remark 2.3 in [12]).

####  3.2.2. The Siegel system defined by $P$

Given a Siegel bundle of type $\mathfrak{t}\in\mathrm{Div}^{n}$, consider the
bundle of discrete abelian groups defined through:

$Z(P)_{m}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}H_{1}(\mathfrak{A}(P)_{m},\mathbb{Z})\,,\quad\forall\,\,m\in
M~{}~{}.$

Since torus translations act trivially on
$H_{1}(\mathfrak{A}(P)_{m},\mathbb{Z})$, the structure group of $Z(P)$ reduces
to $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$. Thus $Z(P)$ is a Siegel system
on $M$. Moreover, $Z(P)$ is isomorphic with the bundle of discrete abelian
groups associated to $P$ through the projection morphism
$p_{2}:\mathrm{Aff}_{\mathfrak{t}}\rightarrow\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$,
when the latter is viewed as a left action of $\mathrm{Aff}_{\mathfrak{t}}$
through automorphisms of the group $(\mathbb{Z}^{2n},+)$.

######  Definition 3.5.

$Z(P)$ is called the Siegel system defined by $P$.

Notice that the the monodromy of $P$ at a point $m\in M$ acts through
automorphisms of the integral symplectic space
$(H_{1}(\mathfrak{A}(P)_{m},\mathbb{R}),H_{1}(\mathfrak{A}(P)_{m},\mathbb{Z}),[\Omega_{m}])$.

####  3.2.3. The adjoint bundle and integral duality structure of $P$

The adjoint bundle $\mathrm{ad}(P)$ of $P$ can be identified with the tensor
product $Z(P)\otimes_{\mathbb{Z}}\underline{\mathbb{R}}$, whose fiber at $m\in
M$ is given by:

$\mathrm{ad}(P)_{m}=Z(P)_{m}\otimes_{\mathbb{Z}}\mathbb{R}\simeq
H_{1}(\mathfrak{A}(P)_{m},\mathbb{R})~{}~{}.$

Notice that $\mathrm{ad}(P)$ carries the fiberwise symplectic pairing
$\omega_{P}$ given by $(\omega_{P})_{m}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}[\Omega_{m}]$ for all $m\in M$ (see Lemma 3.1). Since the Lie
algebra of $\mathrm{Aff}_{\mathfrak{t}}$ is abelian, the structure group of
$\mathrm{ad}(P)$ reduces to $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$. The
Siegel system $Z(P)$ is naturally a sub-bundle of $\mathrm{ad}(P)$ whose
fibers $Z(P)_{m}=H_{1}(\mathfrak{A}(P)_{m},\mathbb{Z})$ are full symplectic
lattices with respect to $[\Omega_{m}]$. The monodromy of $Z(P)$ induces a
unique flat connection $\mathcal{D}_{P}$ on $\mathrm{ad}(P)$ whose parallel
transport preserves $Z(P)$. Setting
$\mathcal{S}_{P}\stackrel{{\scriptstyle{\rm def.}}}{{=}}\mathrm{ad}(P)$, it
follows that the system:

${\bm{\Delta}}(P)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\Delta(P),Z(P))~{}~{},~{}~{}\mathrm{where}~{}~{}\Delta(P)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\mathcal{S}_{P},\omega_{P},\mathcal{D}_{P})\,,$

is an integral duality structure of type $\mathfrak{t}$, whose underlying
duality structure is $\Delta(P)$.

######  Proposition 3.6.

There correspondence defined above extends to an essentially surjective
functor:

${\bm{\Delta}}\colon\mathrm{Sieg}(M)\to\mathrm{Dual}_{\mathbb{Z}}(M)\,,$

which associates to every Siegel bundle $P$ of type
$\mathfrak{t}\in\mathrm{Div}^{n}$ defined on $M$ the integral duality
structure ${\bm{\Delta}}(P)$, which has type $\mathfrak{t}$.

###### Proof.

It is clear that the correspondence extends to a functor. Given
${\bm{\Delta}}=(\Delta,\mathcal{L})\in\mathrm{Dual}_{\mathbb{Z}}(M)$, denote
by $Q$ the frame bundle of the Siegel system defined by $\mathcal{L}$, which
is a principal $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$-bundle (see
Proposition 2.12). Let $P$ be the Siegel bundle associated to $Q$ through the
natural left action $l$ of $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ on
$\mathrm{Aff}_{\mathfrak{t}}$:

$P=Q\times_{l}\mathrm{Aff}_{\mathfrak{t}}\,.$

Then ${\bm{\Delta}}(P)={\bm{\Delta}}$, showing that the functor is essentially
surjective. ∎

####  3.2.4. The bundle of integral symplectic torus groups defined by $P$

Consider a Siegel bundle $P$ defined on $M$.

######  Definition 3.7.

The bundle of integral symplectic torus groups defined by $P$ is the bundle:

$\mathcal{A}(P)=\mathcal{A}({\bm{\Delta}}(P))=\mathrm{ad}(P)/Z(P)$

of integral symplectic torus groups defined by the integral duality structure
${\bm{\Delta}}(P)$.

####  3.2.5. Siegel bundles with trivial monodromy

######  Proposition 3.8.

Let $P$ be a Siegel bundle of rank $n$ and type
$\mathfrak{t}\in\mathrm{Div}^{n}$ defined on $M$. Then the following
statements are equivalent:

  1. (a)

The Siegel system $Z(P)$ has trivial monodromy.

  2. (b)

$Z(P)$ is trivial as a bundle of discrete Abelian groups.

  3. (c)

The structure group of $P$ reduces to the torus group $\mathrm{U}(1)^{2n}$.

  4. (d)

The structure group of the bundle of integral symplectic affine tori
$\mathfrak{A}(P)$ reduces to $\mathrm{U}(1)^{2n}$.

  5. (e)

The structure group of the bundle of integral symplectic torus groups
$\mathcal{A}(P)$ is trivial.

  6. (f)

The duality structure $\Delta(P)$ is holonomy-trivial.

In this case, $\mathfrak{A}(P)$ identifies with a principal torus bundle and
$\mathcal{A}(P)$ is a trivial bundle of integral symplectic torus groups.
Moreover, $P$ is isomorphic with the fiber product
$\mathfrak{A}(P)\times_{M}\underline{\Gamma}$, where $\underline{\Gamma}$ is
the trivial $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$-bundle defined on $M$.
Thus $P$ identifies with a countable collection of copies of the principal
torus bundle $\mathfrak{A}(P)$, indexed by elements of
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$.

###### Proof.

The fact that $(a)$ implies $(b)$ follows from the standard characterization
of flat bundles in terms of holonomy representations of the fundamental group
of the underlying manifold. If $Z(P)$ is trivial as a bundle of discrete
groups then the holonomy representation preserves a global frame of $Z(P)$,
which in turn implies, using the explicit form of the adjoint representation
of $\mathrm{Aff}_{\mathfrak{t}}$, that the holonomy representation takes
values in $\mathrm{U}(1)^{2n}\subset\mathrm{Aff}_{\mathfrak{t}}$. The
associated holonomy bundle defines a reduction of $P$ to a principal torus
bundle with structure group $\mathrm{U}(1)^{2n}$. This immediately implies
$(d)$, $(e)$ and $(f)$. Since the flat connection on $\Delta(P)$ is by
definition the real linear extension of the flat connection of $Z(P)$ the
latter has trivial monodromy if and only if $\Delta(P)$ has trivial monodromy,
that is, if and only if $\Delta(P)$ is holonomy-trivial. This proves
$(f)\Rightarrow(a)$. ∎

###  3.3. Classification of Siegel bundles

Let $P$ be a Siegel bundle of type $\mathfrak{t}\in\mathrm{Div}^{n}$ defined
on $M$ and
${\bm{\Delta}}:={\bm{\Delta}}(P)=(\mathcal{S}_{P}=\mathrm{ad}(P),\omega_{P},\mathcal{D}_{P},Z(P))$
be the integral duality structure defined by $P$. The Bockstein isomorphism
(18) reads:

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

It was shown in [11] that $P$ determines a primary characteristic class
$c^{\prime}(P)\in H^{1}(M,\mathcal{C}^{\infty}(\mathcal{A}(P)))$.

######  Definition 3.9.

The twisted Chen class of $P$ is:

$c(P)\stackrel{{\scriptstyle{\rm def.}}}{{=}}\delta(c^{\prime}(P))\in
H^{2}(M,Z(P))~{}~{}.$

Recall from Proposition 3.4 that isomorphism classes of Siegel bundles defined
on $M$ are in bijection with isomorphism classes of bundles of integral affine
symplectic tori. This allows one to classify Siegel bundles by adapting the
classification of affine torus bundles given in [11, Section 2] (see also [12,
Theorem 2.2]). Since the modifications of the argument of loc. cit. are
straightforward, we simply describe the result. Adapting the argument of [11]
we obtain:

######  Theorem 3.10.

Consider the set:

$\Sigma(M)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(Z,c)\,|\,Z\in\mathrm{Ob}[\mathrm{Sg}(M)]\,\,\&\,\,c\in
H^{2}(M,Z)\right\\}/_{\sim},$

where $(Z,c)\sim(Z^{\prime},c^{\prime})$ if and only if there exists an
isomorphism of Siegel systems $\varphi:Z\rightarrow Z^{\prime}$ such that
$\varphi_{\ast}(c)=c^{\prime}$. Then the map:

$P\mapsto(Z(P),c(P))$

induces a bijection between the set of isomorphism classes of Siegel bundles
defined on $M$ and the set $\Sigma(M)$.

A more conceptual explanation of this result is given in [35]. Let
$\rho:\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\rightarrow\mathrm{Aut}(\mathrm{U}(1)^{2n})$
denote the action of $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ on
$\mathrm{U}(1)^{2n}$ and
$\rho_{0}:\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\rightarrow\mathrm{Aut}_{\mathbb{Z}}(\mathbb{Z}^{2n})$
be the corresponding action on $\mathbb{Z}^{2n}$. Then the classifying space
of the $\mathrm{Aff}_{\mathfrak{t}}$ is a twisted Eilenberg-McLane space
$L:=L_{\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})}(\mathbb{Z}^{2n},2)$ in the
sense of [27], which is a sectioned fibration over $\mathrm{B}\Gamma\simeq
K(\Gamma,1)$ whose fibers are homotopy equivalent with
$\mathrm{B}\mathrm{U}(1)^{2n}\simeq
K(\mathbb{Z}^{2n},2)\simeq(\mathbb{C}\mathbb{P}^{\infty})^{\times 2n}$. This
space is a homotopy two-type with:

$\pi_{1}(L)=\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})~{}~{},~{}~{}\pi_{2}(L)=\mathbb{Z}^{2n}~{}~{}.$

The results of [27] are used in [35] to show that isomorphism classes of
principal $\mathrm{Aff}_{\mathfrak{t}}$-bundles $P$ defined on a pointed space
$X$ are in bijection with isomorphism classes of pairs $(\alpha,c)$, where
$\alpha:\pi_{1}(X)\rightarrow\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ is a
morphism of groups and $c\in H^{2}(X,Z_{\alpha})$, where $Z_{\alpha}$ is the
local system with fiber $\mathbb{Z}^{2n}$ and monodromy action at the
basepoint of $X$ given by
$\rho_{0}\circ\alpha:\pi_{1}(X)\rightarrow\mathrm{Aut}_{\mathbb{Z}}(\mathbb{Z}^{2n})$.
When $X=M$ is a manifold, this local system coincides with $Z(P)$, while the
cohomology class $c$ coincides with $c(P)$.

###  3.4. Principal connections on Siegel bundles

Let $P$ be a Siegel bundle of type $\mathfrak{t}\in\mathrm{Div}^{n}$ defined
on a connected manifold $M$, whose projection we denote by $\pi:P\rightarrow
M$. For ease of notation, we set $G\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Aff}_{\mathfrak{t}}$, $\Gamma\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ and
$A\stackrel{{\scriptstyle{\rm def.}}}{{=}}\mathrm{U}(1)^{n}$. We denote the
abelian Lie algebra of $A$ by $\mathfrak{a}\simeq\mathbb{R}^{2n}$. Let
${\bm{\Delta}}=(\Delta,Z)$ be the integral duality structure
${\bm{\Delta}}(P)$ determined by $P$, where
$\Delta=\Delta(P)=(\mathcal{S},\omega,\mathcal{D})$ with
$\mathcal{S}=\mathcal{S}_{P}=\mathrm{ad}(P)$, $\omega=\omega_{P}$ and
$\mathcal{D}=\mathcal{D}_{P}$ and $Z=Z(P)$ is the Siegel system determined by
$P$. Let:

$\mathrm{Conn}(P)=\left\\{\mathcal{A}\in\Omega^{1}(P,\mathfrak{a})\,\,|\,\,r_{a,\gamma}^{\ast}(\mathcal{A})=\gamma^{-1}\mathcal{A}~{}~{}\,\&\,~{}~{}\mathcal{A}_{y}(X^{a}_{y})=a~{}~{}\,\forall
y\in P~{}\,\,\forall\,\,(a,\gamma)\in G\right\\}\,,$

be the set of principal connections on $P$, where $r_{g}$ denotes the right
action of $g\in G$ on $P$ and we used the fact that the adjoint representation
$\mathrm{Ad}:G\rightarrow\mathrm{Aut}(\mathfrak{a})$ is given by (26). Let:

$\Omega_{\mathrm{Ad}}(P,\mathfrak{a})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{\eta\in\Omega(P,\mathfrak{a})\,\,|\,\,r_{a,\gamma}^{\ast}(\eta)=\gamma^{-1}\eta\,\,\&\,\,\iota_{X}\eta=0\,\,\forall(a,\gamma)\in
G~{}\forall X\in V(P)\right\\}\,,$

be the set of equivariant horizontal forms on $P$, where $V(P)$ is the space
of vertical vector fields defined on $P$. Then
$\Omega_{\mathrm{Ad}}(P,\mathfrak{a})$ is naturally isomorphic with
$\Omega(M,\mathcal{S})$. In particular, the curvature
$\Omega_{\mathcal{A}}=\mathrm{d}_{\mathcal{A}}\mathcal{A}\in\Omega^{2}_{\mathrm{Ad}}(P,\mathfrak{a})$
of any principal connection $\mathcal{A}\in\mathrm{Conn}(P)$ identifies with a
$\mathcal{S}$-valued two-form
$\mathcal{V}_{\mathcal{A}}\in\Omega^{2}(M,\mathcal{S})$, which is the adjoint
curvature of $\mathcal{A}$. Since $\mathfrak{a}$ is an abelian Lie algebra, we
are in the situation considered in [35]. Hence the covariant exterior
derivative defined by $\mathcal{A}$ restricts to the ordinary exterior
derivative on the space $\Omega_{\mathrm{Ad}}(P,\mathfrak{a})$:

$\mathrm{d}_{\mathcal{A}}|_{\Omega_{\mathrm{Ad}}(P,\mathfrak{a})}=\mathrm{d}:\Omega_{\mathrm{Ad}}(P,\mathfrak{a})\rightarrow\Omega_{\mathrm{Ad}}(P,\mathfrak{a})~{}~{}.$
(27)

Moreover, the principal curvature of $\mathcal{A}$ is given by:

$\Omega_{\mathcal{A}}=\mathrm{d}\mathcal{A}$

and the Bianchi identity $\mathrm{d}_{\mathcal{A}}\Omega_{\mathcal{A}}=0$
reduces to:

$\mathrm{d}\Omega_{\mathcal{A}}=0~{}~{}.$ (28)

As explained in [35], relation (27) implies that all principal connections on
$P$ induce the same linear connection on the adjoint bundle
$\mathrm{ad}(P)=\mathcal{S}$, which coincides with the flat connection
$\mathcal{D}$ of the duality structure $\Delta$ defined by $P$. Moreover, the
adjoint curvature $\mathcal{V}_{\mathcal{A}}\in\Omega^{2}(M,\mathcal{S})$
satisfies:

$\mathrm{d}_{\mathcal{D}}\mathcal{V}_{\mathcal{A}}=0~{}~{}.$

Let $\mathrm{Sieg}^{c}(M)$ be the groupoid of Siegel bundles with connection,
whose objects are pairs $(P,\mathcal{A})$ where $P$ is a Siegel bundle and
$\mathcal{A}\in\mathrm{Conn}(P)$ and whose morphisms are connection-preserving
based isomorphisms of principal bundles. Let
$\mathrm{Dual}_{\mathbb{Z}}^{c}(M)$ be the groupoid of pairs
$({\bm{\Delta}},\mathcal{V})$, where ${\bm{\Delta}}$ is an integral duality
structure on $M$ and $\mathcal{V}\in\mathrm{Conf}(M,{\bm{\Delta}})$ is a
$\mathrm{d}_{\mathcal{D}}$-closed $\mathcal{S}$-valued 2-form whose
$\mathrm{d}_{\mathcal{D}}$-cohomology class belongs to the charge lattice of
${\bm{\Delta}}$. There exists a natural functor:

${\bm{\Delta}}^{c}\colon\mathrm{Sieg}^{c}(M)\to\mathrm{Dual}_{\mathbb{Z}}^{c}(M)$

which sends $(P,\mathcal{A})$ to the pair
$({\bm{\Delta}}(P),\mathcal{V}_{\mathcal{A}})$. Let
$\mathrm{Sieg}^{c}(M)_{0}\subset\mathrm{Sieg}^{c}(M)$ be the full subgroupoid
consisting of Siegel bundles with flat connection.

######  Theorem 3.11.

There exists a short exact sequence of groupoids:

$1\to\mathrm{Sieg}^{c}(M)_{0}\xrightarrow{\kappa}\mathrm{Sieg}^{c}(M)\xrightarrow{\mathrm{curv}}\mathrm{Dual}_{\mathbb{Z}}^{c}(M)\to
1~{}~{},$

where $\kappa$ is the inclusion and $\mathrm{curv}$ is the curvature map. In
particular, for every integral duality structure ${\bm{\Delta}}$ on $M$ and
every $\mathcal{V}\in\mathrm{Conf}(M,{\bm{\Delta}})$, there exists a Siegel
bundle with connection $(P,\mathcal{A})$ such that:

$\mathcal{V}_{\mathcal{A}}=\mathcal{V}\,,$

and the set of pairs $(\mathcal{A},\mathcal{V}_{\mathcal{A}})$ with this
property is a torsor for $\mathrm{Sieg}^{c}(M)_{0}$.

###### Proof.

It is clear that an object in $\mathrm{Sieg}^{c}(M)$ defines an integral
duality structure and a cohomology class in
$H^{2}_{\mathcal{D}}(M,\mathcal{S})$, whence it defines an object in
$\mathrm{Sieg}^{c}(M)$. Functoriality of this assignment follows from
invariance of the aforementioned cohomology class under gauge transformations.
This is proved in Lemma 4.5. The key ingredient of the proof is now to show
that this cohomology class is in fact _integral_ , that is, belongs to
$j_{\ast}(H^{2}(M,Z))$, where $Z$ is the Siegel system defined by $P$. This is
a technical point which is proved in detail in [35], to which we refer the
reader for more details. Once this is proven, it follows from Theorem 3.10
that the curvature map is surjective onto $\mathrm{Dual}_{\mathbb{Z}}^{c}(M)$
and that its kernel is precisely the pairs of integral duality structures and
flat connections. ∎

The previous theorem shows that _integral_ electromagnetic field strengths can
always be realized as curvatures of principal connections defined on Siegel
bundles, which therefore provide the geometric realization of integral
configurations of abelian gauge theory.

######  Remark 3.12.

Theorem 3.11 can be elaborated to obtain the Dirac quantization of abelian
gauge theory in terms of a certain _twisted_ differential cohomology theory,
though we do not pursue this here. Recall that the DSZ quantization of various
gauge theories using the framework of algebraic quantum field theory and
differentiable cohomology has been considered before in the literature, see
[45, 13] and references therein.

###  3.5. Polarized Siegel bundles and polarized self-dual connections

Let $M$ be a connected manifold.

######  Definition 3.13.

A polarized Siegel bundle is a pair $\mathbf{P}=(P,\mathcal{J})$, where $P$ is
a Siegel bundle and $\mathcal{J}$ is a taming of the duality structure
$\Delta:=\Delta(P)$ defined by $P$.

A polarized Siegel bundle $\mathbf{P}=(P,\mathcal{J})$ determines an integral
electromagnetic structure $\bm{\Xi}_{\mathbf{P}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}({\bm{\Delta}}(P),\mathcal{J})$, where
${\bm{\Delta}}(P)=(\Delta(P),Z(P))$ is the integral duality structure defined
by $P$.

######  Definition 3.14.

Let $\mathbf{P}=(P,\mathcal{J})$ be a polarized Siegel bundle. A principal
connection $\mathcal{A}\in\mathrm{Conn}(P)$ is called _polarized selfdual_ ,
respectively _polarized anti-selfdual_ if its adjoint curvature satisfies:

$\star_{g,\mathcal{J}}\mathcal{V}_{\mathcal{A}}=\mathcal{V}_{\mathcal{A}}\,~{}~{},~{}~{}\mathrm{respectively}~{}~{}\star_{g,\mathcal{J}}\mathcal{V}_{\mathcal{A}}=-\mathcal{V}_{\mathcal{A}}\,.$

Recall the definitions:

$\Omega^{2}_{\pm,\mathcal{J}}(M,\mathcal{S})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{\mathcal{V}\in\Omega^{2}(M,\mathcal{S})\,|\,\star_{g,\mathcal{J}}\mathcal{V}_{\mathcal{A}}=\pm\mathcal{V}_{\mathcal{A}}\right\\}~{}~{},$

where $\Delta(P)=(\mathcal{S},\omega,\mathcal{D})$.

######  Remark 3.15.

The polarized selfduality condition is a first-order partial differential
equation for a connection on a Siegel bundle which, to the best of our
knowledge, has not been studied in the literature on mathematical gauge
theory.

##  4\. Prequantum abelian gauge theory

Let $(M,g)$ be an oriented and connected Lorentzian four-manifold. As
explained in the previous section, imposing the DSZ integrality condition on
an abelian gauge theory allows us to identify its prequantum gauge degrees of
freedom with principal connections on a Siegel bundle. More precisely, let
$\mathbf{P}=(P,\mathcal{J})$ be a polarized Siegel bundle on $(M,g)$ and
${\bm{\Delta}}:={\bm{\Delta}}(P)=(\Delta,Z)$ be the integral duality structure
defined by $P$, where $\Delta:=\Delta(P)=(\mathcal{S},\omega,\mathcal{D})$
(with $\mathcal{S}=\mathrm{ad}(P)$) and $Z:=Z(P)$ are the duality structure
and Siegel system defined by $P$. Let
$\bm{\Xi}:=\bm{\Xi}(P)=({\bm{\Delta}},\mathcal{J})$ be the integral
electromagnetic structure defined by $\mathbf{P}$ and
$\Xi:=\Xi(P)=(\Delta,\mathcal{J})$ be its underlying electromagnetic
structure. By Theorem 3.11, the set of integral field strength configurations
determined by the integral duality structure ${\bm{\Delta}}={\bm{\Delta}}(P)$
coincides with the set of adjoint curvatures of principal connections defined
on $P$:

$\mathrm{Conf}(M,{\bm{\Delta}})=\\{\mathcal{V}_{\mathcal{A}}\,|\,\mathcal{A}\in\mathrm{Conn}(P)\\}~{}~{},$

while the set of integral field strength solutions determined by
$\bm{\Xi}:=\bm{\Xi}(P)=({\bm{\Delta}},\mathcal{J})$ is:

$\mathrm{Sol}(M,g,\bm{\Xi})=\\{\mathcal{V}_{\mathcal{A}}\,|\,\mathcal{A}\in\mathrm{Conn}(P)~{}\&~{}\star_{g,\mathcal{J}}\mathcal{V}_{\mathcal{A}}=\mathcal{V}_{\mathcal{A}}\\}~{}~{}.$

This motivates the following.

######  Definition 4.1.

The set of prequantum gauge configurations determined by $P$ is the affine
set:

$\mathfrak{Conf}(M,P)\stackrel{{\scriptstyle{\rm def.}}}{{=}}\mathrm{Conn}(P)$

of principal connections defined on $P$. The adjoint curvature
$\mathcal{V}_{\mathcal{A}}\in\mathrm{Conf}(M,{\bm{\Delta}})$ of a principal
connection $\mathcal{A}\in\mathrm{Conn}(P)$ is called the integral field
strength configuration defined $\mathcal{A}$. The prequantum abelian gauge
theory defined by $\mathbf{P}$ on $(M,g)$ is described by the condition:

$\star_{g,\mathcal{J}}\mathcal{V}_{\mathcal{A}}=\mathcal{V}_{\mathcal{A}}~{}~{}\mathrm{for}~{}~{}\mathcal{A}\in\mathrm{Conn}(P)\,.$
(29)

The solutions $\mathcal{A}$ of this equation are called gauge potentials or
polarized self-dual connections and form the set:

$\mathfrak{Sol}(M,g,\mathbf{P})=\mathfrak{Sol}(M,g,P,\mathcal{J})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{\mathcal{A}\in\mathrm{Conn}(P)\,|\,\star_{g,\mathcal{J}}\mathcal{V}_{\mathcal{A}}=\mathcal{V}_{\mathcal{A}}\\}\subset\mathfrak{Conf}(M,P)~{}~{}.$

We have:

$\mathrm{Sol}(M,g,\bm{\Xi})=\\{\mathcal{V}_{\mathcal{A}}\,|\,\mathcal{A}\in\mathfrak{Sol}(M,g,\mathbf{P})\\}$

and the adjoint curvature map of $P$ gives surjections:

$\mathfrak{Conf}(M,P)\to\mathrm{Conf}(M,{\bm{\Delta}})~{}~{}\mathrm{and}~{}~{}\mathfrak{Sol}(M,g,\mathbf{P})\to\mathrm{Sol}_{\mathbb{Z}}(M,g,\bm{\Xi})~{}~{}.$

######  Remark 4.2.

Condition (29) reduces locally to a system of first-order differential
equations for $2n$ real-valued one-forms, which describe the local
electromagnetic and magnetoelectric potentials of the theory (see Appendix A).
Also notice that any $\mathcal{A}\in\mathfrak{Sol}(M,g,\mathbf{P})$ satisfies
the following second order equation of Abelian Yang-Mills type:

$\mathrm{d}_{\mathcal{D}}\star_{g,\mathcal{J}}\mathcal{V}_{\mathcal{A}}=0\,.$

###  4.1. The duality hierarchy of prequantum abelian gauge theory

In this subsection we discuss the duality groups of prequantum abelian gauge
theory. Let $P$ be a Siegel bundle of type $\mathfrak{t}\in\mathrm{Div}^{n}$
defined on $M$. For simplicity, we use the notations:

$A\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{U}(1)^{2n}~{}~{},~{}~{}\Gamma\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})~{}~{},~{}~{}G\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Aff}_{\mathfrak{t}}=A\rtimes\Gamma$

and denote the Abelian Lie algebra of $G$ by
$\mathfrak{g}=\mathfrak{aff}_{\mathfrak{t}}\simeq\mathbb{R}^{2n}$. Let
$q:G\rightarrow\Gamma$ be the epimorphism entering the short exact sequence of
groups:

$1\rightarrow A\rightarrow G\xrightarrow{q}\Gamma\rightarrow 1~{}~{},$

which splits from the right.

######  Definition 4.3.

The discrete remnant bundle of $P$ is the principal $\Gamma$-bundle
$\Gamma(P)\stackrel{{\scriptstyle{\rm def.}}}{{=}}P\times_{q}\Gamma$.

We denote the adjoint representation of $G$ by
$\mathrm{Ad}:G\rightarrow\mathrm{Aut}_{\mathbb{R}}(\mathfrak{g})$ and the
adjoint action of $G$ (i.e the action of $G$ on itself by conjugation) by
$\mathrm{Ad}_{G}:G\rightarrow\mathrm{Aut}(G)$. The restriction of the latter
to the normal subgroup $A\subset G$ is denoted by
$\mathrm{Ad}_{G}^{A}:G\rightarrow\mathrm{Aut}(A)$. Since $A$ is Abelian, this
factors through $q$ to the characteristic morphism
$\rho:\Gamma\rightarrow\mathrm{Aut}(A)$:

$\mathrm{Ad}_{G}^{A}=\rho\circ q~{}~{},$

while the adjoint representation factors through $q$ to the reduced adjoint
representation
$\bar{\rho}:\Gamma\rightarrow\mathrm{Aut}_{\mathbb{R}}(\mathfrak{g})$:

$\mathrm{Ad}=\bar{\rho}\circ q~{}~{}.$ (30)

This representation of $\Gamma$ on $\mathfrak{g}$ preserves the canonical
symplectic form on $\mathbb{R}^{2n}\simeq\mathfrak{g}$. The exponential map of
$G$ gives a surjective morphism of Abelian groups
$\exp_{G}:\mathfrak{g}\rightarrow A$ whose kernel is a full symplectic lattice
$\Lambda$ of $\mathfrak{g}$ which identifies with $\Lambda_{\mathfrak{t}}$.
This lattice is preserved by the reduced adjoint representation, which
therefore induces a morphism of groups:

$\rho_{0}:\Gamma\rightarrow\mathrm{Aut}_{\mathbb{Z}}(\Lambda)~{}~{}.$

Accordingly, $\mathrm{Ad}=\bar{\rho}\circ q$ also preserves $\Lambda$ and
hence induces a morphism of groups:

$\mathrm{Ad}_{0}=\rho_{0}\circ
q:G\rightarrow\mathrm{Aut}_{\mathbb{Z}}(\Lambda)~{}~{}.$

Let ${\bm{\Delta}}=(\Delta,Z)$ be the duality structure defined by $P$, where
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ (with $\mathcal{S}=\mathrm{ad}(P)$)
and $Z=Z(P)$ are the duality structure and Siegel system defined by $P$. We
have:

$\mathrm{ad}(P)=P\times_{\mathrm{Ad}}\mathfrak{g}=\Gamma(P)\times_{\rho}\mathfrak{g}~{}~{},~{}~{}Z(P)=P\times_{\mathrm{Ad}_{0}}\Lambda=\Gamma(P)\times_{\rho_{0}}\Lambda~{}~{},~{}~{}\mathcal{A}(P)=P\times_{\mathrm{Ad}_{G}^{A}}A~{}~{},$

where
$\mathcal{A}(P)=\mathcal{A}({\bm{\Delta}}(P))=\mathrm{ad}(P)/Z(P)=\mathcal{S}/Z$
is the bundle of integral symplectic torus groups defined by $P$. As shown in
[35], the connection $\mathcal{D}$ coincides with the flat connection induced
on $\mathcal{S}$ by the monodromy connection of $\Gamma(P)$ and the symplectic
pairing $\omega$ of
$\mathcal{S}=\mathrm{ad}(P)=P\times_{\rho_{0}}\mathfrak{g}$ coincides with
that induced by the canonical symplectic pairing of
$\mathbb{R}^{2n}\simeq\mathfrak{g}$.

Let $\mathrm{Aut}(P)$ be the group of those unbased automorphisms of $P$ which
cover orientation-preserving diffeomorphisms of $M$. We have a short exact
sequence:

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

where $\mathrm{Diff}_{P}(M)\subset\mathrm{Diff}(M)$ is the group formed by
those orientation-preserving diffeomorphisms of $M$ that can be covered by
elements of $\mathrm{Aut}(P)$. Here $\mathrm{Aut}_{b}(P)$ is the group of
based automorphisms of $P$. For any $u\in\mathrm{Aut}(P)$, denote by
$f_{u}\in\mathrm{Diff}(M)$ the orientation-preserving diffeomorphism of $M$
covered by $u$. Every $u\in\mathrm{Aut}(P)$ induces an unbased automorphism
$\mathrm{ad}_{u}\in\mathrm{Aut}(\mathcal{S})$ of the adjoint bundle
$\mathcal{S}=\mathrm{ad}(P)$ defined through:

$\mathrm{ad}_{u}([y,v])\stackrel{{\scriptstyle{\rm
def.}}}{{=}}[u(y),v]\,,\quad\forall\,\,[y,v]\in\mathcal{S}=\mathrm{ad}(P)=P\times_{\mathrm{ad}}\mathfrak{g}\,.$

Notice that $\mathrm{ad}_{u}$ covers $f_{u}$.

######  Proposition 4.4.

For every $u\in\mathrm{Aut}(P)$, the map
$\mathrm{ad}_{u}:\mathcal{S}\rightarrow\mathcal{S}$ is an unbased automorphism
of the integral duality structure ${\bm{\Delta}}$ defined by $P$. Moreover,
the map
$\mathrm{ad}_{P}:\mathrm{Aut}(P)\rightarrow\mathrm{Aut}({\bm{\Delta}})$
defined through:

$\mathrm{ad}_{P}(u)=\mathrm{ad}_{u}~{}~{}\forall u\in\mathrm{Aut}(P)$

is a morphism of groups.

###### Proof.

It is clear from its definition that $\mathrm{ad}_{u}$ preserves $\omega$ and
$Z$. It also preserves $\mathcal{D}$, since the latter is induced by the
monodromy connection of $\Gamma(P)$, which is unique. The fact that
$\mathrm{ad}_{P}$ is a morphism of groups is immediate. ∎

Notice that $\mathrm{ad}_{P}$ restricts to a morphism
$\mathrm{ad}_{P}\colon\mathrm{Aut}_{b}(P)\to\mathrm{Aut}_{b}({\bm{\Delta}})$.
We set:

$g_{u}\stackrel{{\scriptstyle{\rm def.}}}{{=}}(f_{u})_{\ast}(g)~{}~{}\forall
u\in\mathrm{Aut}(P)~{}~{}.$

Let
$\mathbb{A}\colon\mathrm{Aut}(P)\times\mathrm{Conn}(P)\rightarrow\mathrm{Conn}(P)$
be the affine left action of $\mathrm{Aut}(P)$ on $\mathrm{Conn}(P)$ defined
through:

$\mathbb{A}_{u}(\mathcal{A})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}u_{\ast}(\mathcal{A})~{}~{}\forall\mathcal{A}\in\mathrm{Conn}(P)~{}~{}\forall
u\in\mathrm{Aut}(P)~{}~{},$

where
$u_{\ast}\colon\mathcal{C}^{\infty}(P,T^{\ast}P\otimes\mathfrak{g})\to\mathcal{C}^{\infty}(P,T^{\ast}P\otimes\mathfrak{g})$
denotes the push-forward of $u$ extended trivially to $\mathfrak{g}$-valued
forms defined on $P$.

######  Lemma 4.5.

For every $u\in\mathrm{Aut}(P)$, we have a commutative diagram of affine
spaces and affine maps:

$\textstyle{\mathrm{Conn}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{A}_{u}}$$\scriptstyle{\mathcal{V}}$$\textstyle{\mathrm{Conn}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{V}}$$\textstyle{2\pi
j_{0\ast}(c(P))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ad}_{u}}$$\textstyle{2\pi
j_{0\ast}(c(P))}$

where
$\mathcal{V}:\mathrm{Conn}(P)\rightarrow\Omega_{{\mathrm{d}_{\mathcal{D}}\\!\mbox{-}{\mathrm{cl}}}}(M,\mathcal{S})$
is the adjoint curvature map of $P$, $c(P)\in H^{2}(M,Z)$ is the twisted Chern
class of $P$ and the map $j_{0\ast}:H^{2}(M,Z)\rightarrow
H^{2}(M,\mathcal{C}^{\infty}_{\mathrm{flat}}(\mathcal{S}))=H^{2}_{\mathcal{D}}(M,\mathcal{S})$
is induced by the sheaf inclusion
$\mathcal{C}(Z)\hookrightarrow\mathcal{C}^{\infty}_{\mathrm{flat}}(\mathcal{S})$
(see the exact sequence (19)). Here $2\pi j_{0\ast}(c(P))$ is viewed as an
affine subspace of
$\Omega_{{\mathrm{d}_{\mathcal{D}}\\!\mbox{-}{\mathrm{cl}}}}(M,\mathcal{S})$
consisting of $\mathrm{d}_{\mathcal{D}}$-closed $\mathcal{S}$-valued forms
which differ by $\mathrm{d}_{\mathcal{D}}$-exact $\mathcal{S}$-valued forms
and hence as an affine space modeled on the vector space
$\Omega^{2}_{{\mathrm{d}_{\mathcal{D}}\\!\mbox{-}{\mathrm{ex}}}}(M,\mathcal{S})$.

###### Proof.

It is shown in [35] that the curvature map $\mathcal{V}$, which is clearly
affine, takes values in $2\pi j_{\ast}(c(P))$. Therefore, it only remains to
prove that if:

$[\mathcal{V}_{\mathcal{A}}]=2\pi\,j_{\ast}(c(P))\,,$

then:

$[\mathcal{V}_{\mathbb{A}_{u}(\mathcal{A})}]=2\pi\,j_{\ast}(c(P))\,,$

or, equivalently, that
$[\mathcal{V}_{\mathbb{A}_{u}(\mathcal{A})}]=[\mathcal{V}_{\mathcal{A}}]$ for
every $u\in\mathrm{Aut}_{b}(P)$. Since $\mathcal{A}$ is a connection,
$\mathbb{A}_{u}(\mathcal{A})$ is also a connection on $P$, whence there exists
an equivariant and horizontal one-form
$\hat{\tau}\in\Omega^{1}(P,\mathfrak{a})$ such that:

$\mathbb{A}_{u}(\mathcal{A})=\mathcal{A}+\hat{\tau}\,.$

Hence,
$\mathrm{d}\mathbb{A}_{u}(\mathcal{A})=\mathrm{d}\mathcal{A}+\mathrm{d}\hat{\tau}$,
which descends to $M$ as follows:

$\mathcal{V}_{\mathbb{A}_{u}(\mathcal{A})}=\mathcal{V}_{\mathcal{A}}+\mathrm{d}_{\mathcal{D}}\tau\,,$

where $\tau\in\Omega^{1}(M,\mathcal{S}_{P})$ denotes the one-form with values
in $\mathcal{S}_{P}$ defined by $\tau$. On the other hand, considering
$\mathcal{V}_{\mathbb{A}_{u}(\mathcal{A})}\in\Omega^{2}(P,\mathfrak{aff}_{\mathfrak{t}})$
as a two-form on $P$ taking values in $\mathfrak{aff}_{\mathfrak{t}}$ a direct
computation shows that:

$\mathcal{V}_{\mathbb{A}_{u}(\mathcal{A})}=u_{\ast}(\mathcal{V}_{\mathcal{A}})\in\Omega^{2}(P,\mathfrak{aff}_{\mathfrak{t}})\,,$

which, by the equivariance properties of the latter, immediately implies:

$\mathcal{V}_{\mathbb{A}_{u}(\mathcal{A})}=\mathrm{ad}_{u}\cdot\mathcal{V}_{\mathcal{A}}\in\Omega^{2}(M,\Delta)\,,$

where the _dot_ action of an automorphism of $\Delta$ on two- forms taking
values in $\Delta$ was defined in subsection 1.9. ∎

######  Proposition 4.6.

For any $u\in\mathrm{Aut}(P)$, the map
$\mathbb{A}_{u}:\mathrm{Conn}(P)\rightarrow\mathrm{Conn}(P)$ restricts to a
bijection:

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

###### Proof.

Since
$\mathcal{V}_{\mathbb{A}_{u}(\mathcal{A})}=\mathrm{ad}_{u}\cdot\mathcal{V}_{\mathcal{A}}$,
it suffices to prove the relation:

$\star_{g_{u},\mathcal{J}_{u}}\circ\mathrm{ad}_{u}=\mathrm{ad}_{u}\circ\star_{g,\mathcal{J}}~{}~{}.$
(31)

For $\alpha\in\Omega^{k}(M)$ and $\xi\in\mathcal{C}^{\infty}(M,\mathcal{S})$,
we compute:

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

which implies (31) ∎

######  Definition 4.7.

Let $\mathbf{P}=(P,\mathcal{J})$ be a polarized Siegel bundle defined on $M$.

  * •

The group $\mathrm{Aut}(P)$ is the unbased gauge group of $P$. For any
$u\in\mathrm{Aut}(P)$, the map:

$\mathbb{A}_{u}\colon\mathfrak{Sol}(M,g,P,\mathcal{J})\to\mathfrak{Sol}(M,g_{u},P,\mathcal{J}_{u})$

is the unbased gauge transformation induced by $u$.

  * •

The group:

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

is the unbased gauge duality group defined by $P$ and $g$. For any
$u\in\mathrm{Aut}(g,P)$, the map:

$\mathbb{A}_{u}:\mathfrak{Sol}(M,g,P,\mathcal{J})\to\mathfrak{Sol}(M,g,P,\mathcal{J}_{u})$

is the unbased gauge duality transformation induced by $u$.

  * •

The gauge group $\mathrm{Aut}_{b}(P)$ of $P$ is the gauge (electromagnetic)
duality group of the abelian gauge theories with underlying Siegel bundle $P$.
For any $u\in\mathrm{Aut}_{b}(P)$, the map:

$\mathbb{A}_{u}:\mathfrak{Sol}(M,g,P,\mathcal{J})\to\mathfrak{Sol}(M,g,P,\mathcal{J}_{u})$

is called the gauge duality transformation induced by $u$.

Lemma 4.5 implies that for any $u\in\mathrm{Aut}(P)$ we have a commutative
diagram:

$\textstyle{\mathfrak{Sol}(M,g,P,\mathcal{J})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{A}_{u}}$$\scriptstyle{\mathcal{V}}$$\textstyle{\mathfrak{Sol}(M,g_{u},P,\mathcal{J}_{u})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{V}}$$\textstyle{\mathrm{Sol}(M,g,{\bm{\Delta}},\mathcal{J})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ad}_{u}}$$\textstyle{\mathrm{Sol}(M,g_{u},{\bm{\Delta}},\mathcal{J}_{u})}$

and similar diagrams for the other groups in the previous definition. Hence
gauge transformations of $P$ induce integral pseudo-duality and duality
transformations of the abelian gauge theory defined by $(P,\mathcal{J})$.

######  Definition 4.8.

Let $\mathbf{P}=(P,\mathcal{J})$ be a polarized Siegel bundle on $M$.

  * •

The group:

$\mathrm{Aut}(\mathbf{P})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}(P)\,\,|\,\
\mathcal{J}_{u}=\mathcal{J}\right\\}$ (32)

is the unbased unitary gauge group defined by $\mathbf{P}$ on $(M,g)$. For any
$u\in\mathrm{Aut}(\mathbf{P})$, the map:

$\mathbb{A}_{u}:\mathfrak{Sol}(M,g,\mathbf{P})\to\mathfrak{Sol}(M,g_{u},\mathbf{P})$

is the unbased unitary gauge transformation induced by $u$.

  * •

The group:

$\mathrm{Aut}(g,\mathbf{P})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}(P)\,\,|\,\,g_{u}=g~{}~{}\mathrm{and}~{}~{}\mathcal{J}_{u}=\mathcal{J}\right\\}$
(33)

is the unbased unitary gauge duality group defined by $\mathbf{P}$ on $(M,g)$.
For any $u\in\mathrm{Aut}(g,\mathbf{P})$, the map:

$\mathbb{A}_{u}:\mathfrak{Sol}(M,g,\mathbf{P})\to\mathfrak{Sol}(M,g,\mathbf{P})$

is the unbased unitary gauge duality transformation induced by $u$.

  * •

The group:

$\mathrm{Aut}_{b}(\mathbf{P})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}_{b}(P)\,\,|\,\,\mathcal{J}_{u}=\mathcal{J}\right\\}\,.$

is the unitary gauge group defined by $\mathbf{P}$ on $M$. For any
$u\in\mathrm{Aut}_{b}(\mathbf{P})$, the map:

$\mathbb{A}_{u}:\mathfrak{Sol}(M,g,\mathbf{P})\to\mathfrak{Sol}(M,g,\mathbf{P})~{}~{}.$

is the unitary gauge transformation induced by $u$.

Let $\bm{\Xi}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}({\bm{\Delta}},\mathcal{J})$. For any
$u\in\mathrm{Aut}(\mathbf{P})$, we have a commutative diagram:

$\textstyle{\mathfrak{Sol}(M,g,\mathbf{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{A}_{u}}$$\scriptstyle{\mathcal{V}}$$\textstyle{\mathfrak{Sol}(M,g_{u},\mathbf{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{V}}$$\textstyle{\mathrm{Sol}(M,g,\bm{\Xi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ad}_{u}}$$\textstyle{\mathrm{Sol}(M,g_{u},\bm{\Xi})}$

and similar diagrams for the other groups in the previous definition. We have
a short exact sequence:

$1\to\mathrm{Aut}_{b}(P)\rightarrow\mathrm{Aut}(g,P)\rightarrow{\rm
Iso}_{P}(M,g)\to 1\,,$ (34)

where ${\rm Iso}_{P}(M,g)\subset{\rm Iso}(M,g)$ is the group formed by those
orientation-preserving isometries that can be covered by elements of
$\mathrm{Aut}(g,P)$. Similarly, we have an exact sequence:

$1\to\mathrm{Aut}_{b}(\mathbf{P})\rightarrow\mathrm{Aut}(g,\mathbf{P})\rightarrow{\rm
Iso}_{\mathbf{P}}(M,g)\to 1\,,$

where ${\rm Iso}_{\mathbf{P}}(M,g)$ is the group formed by those orientation-
preserving isometries of $M$ which are covered by elements of
$\mathrm{Aut}(g,\mathbf{P})$.

######  Definition 4.9.

The _standard subgroup_ of the unbased gauge group of $P$ is defined through:

$\mathrm{C}(P)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\ker(\mathrm{ad}_{P})\subset\mathrm{Aut}(P)\,.$

When $\dim M>0$ and $\dim A>0$, the group $\mathrm{C}(P)$ is infinite-
dimensional. The classical duality group of a duality structure was shown to
be a finite dimensional Lie group in Section 1. This is no longer true of the
gauge groups introduced above. Instead, they are infinite-dimensional
extensions of the integral duality groups introduced in Section 2.

######  Proposition 4.10.

The gauge group of $P$ fits into the short exact sequence of groups:

$1\to\mathrm{C}(P)\hookrightarrow\mathrm{Aut}_{b}(P)\xrightarrow{\mathrm{ad}_{P}}\mathrm{Aut}_{b}({\bm{\Delta}})\to
1~{}~{}.$ (35)

######  Remark 4.11.

There exist similar short exact sequences for the remaining groups introduced
in Definition 4.7.

###### Proof.

It suffices to prove that
$\mathrm{ad}_{P}(\mathrm{Aut}_{b}(P))=\mathrm{Aut}_{b}({\bm{\Delta}})$. Recall
that
$\mathcal{L}=P\times_{\mathrm{Ad}_{0}}\Lambda=\Gamma(P)\times_{\rho_{0}}\Lambda$
and
$\mathcal{S}=P\times_{\mathrm{Ad}}\mathfrak{g}=\Gamma(P)\times_{\bar{\rho}}\mathfrak{g}$,
where $\Lambda\equiv\Lambda_{\mathfrak{t}}$ and
$\mathfrak{g}\equiv\mathbb{R}^{2n}$. Also recall that the gauge group
$\mathrm{Aut}_{b}(P)$ is naturally isomorphic with the group
$\mathcal{C}^{\infty}(P,G)^{G}$ of $G$-equivariant maps from $P$ to $G$, where
$G=A\times\Gamma=\mathrm{U}(1)^{2n}\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
acts on itself through conjugation. This isomorphism takes
$u\in\mathrm{Aut}_{b}(P)$ to the equivariant map
$f\in\mathcal{C}^{\infty}(P,G)^{G}$ which satisfies:

$u(p)=pf(p)~{}~{}\forall p\in P~{}~{}\forall g\in G~{}~{}.$ (36)

Since the action of $G$ on $P$ is free and we have
$\Gamma\equiv\mathrm{Aut}(\mathbb{R}^{2n},\omega_{2n},\Lambda_{\mathfrak{t}})\simeq\mathrm{Aut}(\mathcal{S}_{m},\omega_{m},\mathcal{L}_{m})$
for all $m\in M$ while the reduced adjoint representation $\bar{\rho}$ is
faithful, every automorphism $\varphi\in\mathrm{Aut}({\bm{\Delta}})$
determines a map $\bar{\varphi}:P\to\Gamma$ which satisfies:

$\varphi([p,v])=[p,\bar{\rho}(\bar{\varphi}(p))(v)]~{}~{}\forall p\in
P~{}~{}\forall v\in\mathfrak{g}$ (37)

as well as:

$\bar{\varphi}(pg)=q(g)^{-1}\bar{\varphi}(p)q(g)~{}~{}\forall p\in
P~{}~{}\forall g\in G~{}~{}.$

The last relation follows from (37) and from the condition
$\varphi([pg,v])=\varphi([p,\bar{\rho}(q(g))(v)])$ of invariance under change
of representative of the equivalence class, where we used (30). Let
$u\in\mathrm{Aut}_{b}(P)$ be the based automorphism of $P$ which corresponds
to the $G$-equivariant map $f:P\rightarrow G$ defined through:

$f(p)\stackrel{{\scriptstyle{\rm def.}}}{{=}}(0_{A},\bar{\varphi}(p))\in
G=A\rtimes\Gamma~{}~{}\forall p\in P~{}~{}.$

For any $p\in P$ and $v\in\mathfrak{g}$, we have:

$\mathrm{ad}_{P}(u)([p,v])=[u(p),v]=[pf(p),v]=[p,\mathrm{Ad}(f(p))(v)]=[p,\bar{\rho}(\bar{\varphi}(p))(p)]=\varphi([p,v])~{}~{},$

where we used (36) and (37). This shows that $\mathrm{ad}_{P}(u)=\varphi$.
Since $\varphi\in\mathrm{Aut}_{b}({\bm{\Delta}})$ is arbitrary, we conclude
that $\mathrm{ad}_{P}(\mathrm{Aut}_{b}(P))=\mathrm{Aut}_{b}({\bm{\Delta}})$. ∎

The previous proposition clarifies the geometric origin of electromagnetic
duality as a ‘discrete remnant’ of gauge symmetry, a notion which is discussed
in more detail in [35]. In particular, $\mathrm{Aut}_{b}(P)$ is an extension
of $\mathrm{Aut}_{b}({\bm{\Delta}})$ by the continuous group $C(P)$.
Intuitively, elements of the latter correspond to the gauge transformations of
a principal torus bundle.

###  4.2. Duality groups for Siegel bundles with trivial monodromy

Let $M$ be a connected and oriented four-manifold.

######  Lemma 4.12.

Let $P$ be a trivial principal $G$-bundle over $M$. Then any trivialization of
$P$ induces an isomorphism of groups:

$\mathrm{Aut}(P)\simeq\mathcal{C}^{\infty}(M,G)\rtimes_{\alpha}\mathrm{Diff}(M)~{}~{},$

where
$\alpha:\mathrm{Diff}(M)\rightarrow\mathrm{Aut}(\mathcal{C}^{\infty}(M,G))$ is
the morphisms of groups defined through:

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

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

$1\rightarrow\mathcal{C}^{\infty}(M,G)\rightarrow\mathrm{Aut}(P)\rightarrow\mathrm{Diff}(M)\rightarrow
1$

which is split from the right.

###### Proof.

Let $\tau:P\stackrel{{\scriptstyle\sim}}{{\rightarrow}}M\times G$ be a
trivialization of $P$. Then the map
$\mathrm{Ad}(\tau):\mathrm{Aut}(P)\rightarrow\mathrm{Aut}(M\times G)$ defined
through:

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

is an isomorphism of groups. Let $f\in\mathrm{Aut}(P)$ be an unbased
automorphism of $P$ which covers the diffeomorphism
$\varphi\in\mathrm{Diff}(M)$. Then $\mathrm{Ad}(\tau)(f)$ is an unbased
automorphism of $M\times G$ which covers $\varphi$ and hence we have:

$\mathrm{Ad}(\tau)(f)(m,g)=(\varphi(m),{\hat{f}}(m,g))\,,\quad\forall\,\,(m,g)\in
M\times G~{}~{},$

where ${\hat{f}}:M\times G\rightarrow G$ is a smooth map which satisfies:

${\hat{f}}(m,g_{1}g_{1})={\hat{f}}(m,g_{1})g_{2}\,,\quad\forall\,\,m\in
M\,,\quad\forall\,\,g_{1},g_{2}\in G~{}~{}.$

The last relation is equivalent with the condition that ${\hat{f}}$ has the
form:

${\hat{f}}(m,g)={\tilde{f}}(m)g\,,\quad\forall\,\,(m,g)\in M\times G~{}~{},$

where ${\tilde{f}}:M\rightarrow G$ is a smooth function which can be recovered
from ${\hat{f}}$ through the relation:

${\tilde{f}}(m)={\hat{f}}(m,1)\,,\quad\forall\,\,m\in M~{}~{}.$

Setting $h\stackrel{{\scriptstyle{\rm
def.}}}{{=}}{\tilde{f}}\circ\varphi^{-1}\in\mathcal{C}^{\infty}(M,G)$, we
have:

$\mathrm{Ad}(\tau)(f)(m,g)=(\varphi(m),h(\varphi(m))g)\,,\quad\forall\,\,(m,g)\in
M\times G\,,$ (38)

and the correspondence $f\rightarrow(h,\varphi)$ gives a bijection between
$\mathrm{Aut}(P)\simeq\mathrm{Aut}(M\times G)$ and the set
$\mathcal{C}^{\infty}(M,G)\times\mathrm{Diff}(M)$. If
$f_{1},f_{2}\in\mathrm{Aut}(P)$ correspond through this map to the pairs
$(h_{1},\varphi_{1}),(h_{2},\varphi_{2})\in\mathcal{C}^{\infty}(M,G)\times\mathrm{Diff}(M)$,
then direct computation using (38) gives:

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

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

Let $P$ be a Siegel bundle or rank $n$ and type
$\mathfrak{t}\in\mathrm{Div}^{n}$ on $(M,g)$ and let
${\bm{\Delta}}=(\Delta,Z)$ be the integral duality structure defined by $P$.
Suppose that $P$ is topologically trivial and that $Z=Z(P)$ has trivial
monodromy, so that $\Delta$ is holonomy trivial. Choosing a trivialization of
$P$ gives:

$P\equiv
M\times\mathrm{Aff}_{\mathfrak{t}}\,,\qquad{\bm{\Delta}}\equiv(M\times\mathbb{R}^{2n},\omega_{2n},\mathrm{d},M\times\Lambda_{\mathfrak{t}})$

and:

$\mathrm{Aut}_{b}(P)\equiv\mathcal{C}^{\infty}(M,\mathrm{Aff}_{\mathfrak{t}})\,,\qquad\mathrm{Aut}(P)\equiv\mathcal{C}^{\infty}(M,\mathrm{Aff}_{\mathfrak{t}})\rtimes_{\alpha}\mathrm{Diff}(M)~{}~{},$

where the last identification follows from Lemma 4.12. Since
$\mathrm{Aff}_{\mathfrak{t}}=\mathrm{U}(1)^{2n}\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
and $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ is discrete, we have:

$\mathcal{C}^{\infty}(M,\mathrm{Aff}_{\mathfrak{t}})=\mathcal{C}^{\infty}(M,\mathrm{U}(1)^{2n})\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})~{}~{}.$

In particular, maps $h\in\mathcal{C}^{\infty}(M,\mathrm{Aff}_{\mathfrak{t}})$
can be identified with pairs $(f,\gamma)$, where
$f\in\mathcal{C}^{\infty}(M,\mathrm{U}(1)^{2n})$ and
$\gamma\in\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$. The unbased gauge
duality group is given by:

$\mathrm{Aut}(g,P)\equiv\mathcal{C}^{\infty}(M,\mathrm{Aff}_{\mathfrak{t}})\rtimes_{\alpha}{\rm
Iso}(M,g)~{}~{}.$

The integral pseudo-duality, relative duality and duality groups of
${\bm{\Delta}}$ are in turn given by:

$\displaystyle\mathrm{Aut}({\bm{\Delta}})\equiv\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\times\mathrm{Diff}(M)\,,$
$\displaystyle\mathrm{Aut}(g,{\bm{\Delta}})\equiv\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\times{\rm
Iso}(M,g)\,,$
$\displaystyle\mathrm{Aut}_{b}({\bm{\Delta}})\equiv\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\,,$

and we have short exact sequences:

$\displaystyle
1\to\mathcal{C}^{\infty}(M,\mathrm{U}(1)^{2n})\to\mathrm{Aut}(P)\to\mathrm{Aut}({\bm{\Delta}})\to
1\,,$ $\displaystyle
1\to\mathcal{C}^{\infty}(M,\mathrm{U}(1)^{2n})\to\mathrm{Aut}(g,P)\to\mathrm{Aut}(g,{\bm{\Delta}})\to
1\,,$ $\displaystyle
1\to\mathcal{C}^{\infty}(M,\mathrm{U}(1)^{2n})\to\mathrm{Aut}_{b}(P)\to\mathrm{Aut}_{b}({\bm{\Delta}})\to
1\,.$

Let us fix a taming of $\Delta$, which we view as a map
$\mathcal{J}\in\mathcal{C}^{\infty}(M,\mathrm{Sp}(2n,\mathbb{R}))$. Then the
unbased unitary gauge group of the tamed Siegel bundle
$\mathbf{P}=(P,\mathcal{J})$ is:

$\mathrm{Aut}(g,\mathbf{P})\equiv\left\\{(f,\gamma,\varphi)\in\left[\mathcal{C}^{\infty}(M,\mathrm{U}(1)^{2n})\times\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\right]\rtimes_{\alpha}{\rm
Iso}(M,g)\,|\,\,\gamma\mathcal{J}\gamma^{-1}=\mathcal{J}\circ\varphi\right\\}~{}~{},$

while its unitary gauge group is:

$\mathrm{Aut}_{b}(\mathbf{P})\equiv\left\\{(f,\gamma)\in\mathcal{C}^{\infty}(M,\mathrm{U}(1)^{2n})\times\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\,|\,\gamma\mathcal{J}\gamma^{-1}=\mathcal{J}\right\\}~{}~{}.$

The integral unbased unitary duality group of the integral electromagnetic
structure $\bm{\Xi}=({\bm{\Delta}},\mathcal{J})$ is:

$\mathrm{Aut}(g,\bm{\Xi})\equiv\left\\{(\gamma,\varphi)\in\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\times{\rm
Iso}(M,g)\,\,|\,\,\gamma\mathcal{J}\gamma^{-1}=\mathcal{J}\circ\varphi\right\\}\,,$

while the integral unitary duality group of $\bm{\Xi}$ is:

$\mathrm{Aut}_{b}(\bm{\Xi})\equiv\left\\{\gamma\in\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\,\,|\,\,\gamma\mathcal{J}\gamma^{-1}=\mathcal{J}\right\\}\subset\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\,.$

We have short exact sequences:

$\displaystyle
1\to\mathcal{C}^{\infty}(M,\mathrm{U}(1)^{2n})\to\mathrm{Aut}(g,\mathbf{P})\to\mathrm{Aut}(g,\bm{\Xi})\to
1\,$ $\displaystyle
1\to\mathcal{C}^{\infty}(M,\mathrm{U}(1)^{2n})\to\mathrm{Aut}_{b}(\mathbf{P})\to\mathrm{Aut}_{b}(\bm{\Xi})\to
1~{}~{}.$

##  5\. Time-like dimensional reduction and polarized Bogomolny equations

This section investigates the time-like dimensional reduction of the equations
of motion of abelian gauge theory on an oriented static space-time $(M,g)$ of
the form:

$(M,g)=(\mathbb{R}\times\Sigma,-\mathrm{d}t^{2}\oplus h)\,,$

where $t$ is the global coordinate on $\mathbb{R}$ and $(\Sigma,h)$ is an
oriented Riemannian three-manifold. We show that the reduction produces an
equation of Bogomolny type, similar to the dimensional reduction of the
ordinary self-duality equation on a Riemannian four-manifold. Unlike that
well-known case, here we reduce the _polarized_ self-duality condition on a
Lorentzian four-manifold.

###  5.1. Preparations

Consider the time-like exact one-form $\theta=\mathrm{d}t\in\Omega^{1}(M)$.
Let $\nu_{h}$ be the volume form of $(\Sigma,h)$ and orient $(M,g)$ such that
its volume form is given by:

$\nu_{g}=\theta\wedge\nu_{h}~{}~{}.$

Let $\ast_{g}$ and $\ast_{h}$ be the Hodge operators of $(M,g)$ and
$(\Sigma,h)$ and $\langle~{},~{}\rangle_{g}$, $\langle~{},~{}\rangle_{h}$ be
the non-degenerate bilinear pairings induced by $g$ and $h$ on
$\Omega^{\ast}(M)$ and $\Omega^{\ast}(\Sigma)$. Let $p:M\rightarrow\Sigma$ be
the projection of $M=\mathbb{R}\times\Sigma$ on the second factor and consider
the distribution $\mathcal{D}=p^{\ast}(T\Sigma)\subset TM$, endowed with the
fiberwise Euclidean pairing given by the bundle pullback $h^{p}$ of $h$. This
distribution is integrable with leaves gives by the spacelike hypersurfaces:

$M_{t}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{t\\}\times\Sigma\,,\quad\forall\,\,t\in\mathbb{R}~{}~{},$

on which $h^{p}$ restricts to the metric induced by $g$. Using $h^{p}$, we
extend $\langle~{},~{}\rangle_{h}$ and $\ast_{h}$ in the obvious manner to the
space
$\mathcal{C}^{\infty}(M,\wedge^{\ast}\mathcal{D}^{\ast})\subset\Omega^{\ast}(M)$.
We have:

$\mathcal{C}^{\infty}(M,\wedge^{\ast}\mathcal{D}^{\ast})=\\{\omega\in\Omega^{\ast}(M)~{}|~{}\iota_{\partial_{t}}\omega=0\\}~{}~{}.$

Since $g$ has signature $(3,1)$ while $h$ has signature $(3,0)$, we have:

$\ast_{g}\circ\ast_{g}=-\pi\,,\quad\ast_{h}\circ\ast_{h}=\mathrm{id}_{\mathcal{C}^{\infty}(M,\wedge\mathcal{D}^{\ast})}~{}~{},$

where $\pi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\oplus_{k=0}^{4}{(-1)^{k}\mathrm{id}_{\Omega^{k}(M)}}$ is the
signature automorphism of the exterior algebra $(\Omega^{\ast}(M),\wedge)$
(see [36]). Moreover, we have $\langle\theta,\theta\rangle_{g}=-1$ and:

$\langle\nu_{g},\nu_{g}\rangle_{g}=-1\,,\quad\langle\nu_{h},\nu_{h}\rangle_{h}=+1~{}~{}.$

Any polyform $\omega\in\Omega^{\ast}(M,g)$ has a unique decomposition:

$\omega=\omega_{\parallel}+\omega_{\perp}~{}~{},$ (39)

such that $\omega_{\parallel},\omega_{\perp}\in\Omega^{\ast}(M)$ satisfy [36]:

$\theta\wedge\omega_{\parallel}=0\,,\quad\iota_{\partial_{t}}\omega_{\perp}=0~{}~{}.$
(40)

The second of these conditions amounts to the requirement that
$\omega_{\perp}\in\mathcal{C}^{\infty}(M,\wedge^{\ast}\mathcal{D}^{\ast})$,
while the first is solved by:

$\omega_{\parallel}=\theta\wedge\omega_{\top}~{}~{}\mathrm{where}~{}~{}\omega_{\top}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}-\iota_{\partial_{t}}\omega\in\mathcal{C}^{\infty}(M,\wedge^{\ast}\mathcal{D}^{\ast})~{}~{}.$

As shown in loc. cit., the map
$\omega\rightarrow(\omega_{\top},\omega_{\perp})$ gives a linear isomorphism
between $\Omega^{\ast}(M)$ and
$\mathcal{C}^{\infty}(M,\wedge^{\ast}\mathcal{D}^{\ast})^{\oplus 2}$. For any
$k=0,\ldots,4$ and any $\omega,\eta\in\Omega^{k}(M)$, we have:

$\langle\omega,\eta\rangle_{g}=-\langle\omega_{\top},\eta_{\top}\rangle_{h}+\langle\omega_{\perp},\eta_{\perp}\rangle_{h}\,,\quad\langle\omega_{\parallel},\eta_{\parallel}\rangle_{g}=-\langle\omega_{\top},\eta_{\top}\rangle_{h}~{}~{}.$

Hence the isomorphism above identifies the quadratic space
$(\Omega^{\ast}(M),\langle~{},~{}\rangle_{h})$ with the direct sum of
quadratic spaces
$(\mathcal{C}^{\infty}(M,\wedge^{\ast}\mathcal{D}^{\ast}),-\langle~{},~{}\rangle_{h})\oplus(\mathcal{C}^{\infty}(M,\wedge^{\ast}\mathcal{D}^{\ast}),\langle~{},~{}\rangle_{h})$.
Notice that $\nu_{h}=\iota_{\partial_{t}}\nu_{g}=-(\nu_{g})_{\top}$. An easy
computation gives:

######  Lemma 5.1.

For any polyform $\omega\in\Omega^{\ast}(M)$, we have:

$(\ast_{g}\omega)_{\top}=\ast_{h}\pi(\omega_{\perp})\,,\quad(\ast_{g}\omega)_{\perp}=-\ast_{h}\omega_{\top}$
(41)

and hence:

$\ast_{g}\omega=-\ast_{h}\omega_{\top}+\theta\wedge\ast_{h}\pi(\omega_{\perp})~{}~{}.$
(42)

Given any vector bundle $V$ defined on $M$, the Hodge operators of $g$ and $h$
extend trivially to operators
$\ast_{g}:\Omega^{\ast}(M,V)\rightarrow\Omega^{\ast}(M,V)$ and
$\ast_{h}:\mathcal{C}^{\infty}(M,\wedge^{\ast}\mathcal{D}^{\ast}\otimes
V)\rightarrow\mathcal{C}^{\infty}(M,\wedge^{\ast}\mathcal{D}^{\ast}\otimes
V)$. The decomposition (39) holds for any $\omega\in\Omega^{\ast}(M,V)$, with
components $\omega_{\parallel},\omega_{\perp}\in\Omega^{\ast}(M,V)$ satisfying
(40). We have
$\omega_{\perp}\in\mathcal{C}^{\infty}(M,\wedge^{\ast}\mathcal{D}^{\ast}\otimes
V)$ and $\omega_{\parallel}=\theta\wedge\omega_{\top}$ with
$\omega_{\top}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}-\iota_{\partial_{t}}\omega\in\mathcal{C}^{\infty}(M,\wedge^{\ast}\mathcal{D}^{\ast}\otimes
V)$. Finally, Lemma 5.1 holds for any $\omega\in\Omega^{\ast}(M,V)$.

###  5.2. Timelike dimensional reduction of abelian gauge theory

Let $P$ be a Siegel bundle of type $\mathfrak{t}\in\mathrm{Div}^{n}$ defined
on $\Sigma$, whose projection we denote by $\pi:P\rightarrow\Sigma$. Let
${\hat{P}}\stackrel{{\scriptstyle{\rm def.}}}{{=}}p^{\ast}(P)$ be the
$p$-pullback of $P$ to $M$, whose projection we denote by $\hat{\pi}$. The map
$\varphi:{\hat{P}}\rightarrow\mathbb{R}\times P$ defined through:

$\varphi(t,\sigma,y)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(t,y)\,,\quad\forall\,\,(t,\sigma)\in\mathbb{R}\times\Sigma\,,\quad\forall\,\,y\in
P_{\sigma}=\pi^{-1}(\sigma)$

allows us to identify ${\hat{P}}$ with the principal
$\mathrm{Aff}_{\mathfrak{t}}$-bundle with total space given by
$\mathbb{R}\times P$, base $M=\mathbb{R}\times\Sigma$ and projection given by
$\mathbb{R}\times P\ni(t,y)\rightarrow(t,\pi(y))\in M$. We make this
identification in what follows. Accordingly, we have:

$\hat{\pi}(t,y)=(t,\pi(y))\,,\quad\forall\,\,(t,y)\in{\hat{P}}\equiv\mathbb{R}\times
P\,.$

Let $\tau:{\hat{P}}\rightarrow P$ be the unbased morphism of principal
$\mathrm{Aff}_{\mathfrak{t}}$-bundles given by projection on the second
factor:

$\tau(t,y)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}y\,,\quad\forall\,\,(t,y)\in{\hat{P}}\,,$

which covers the map $p:M\rightarrow\Sigma$:

$\pi\circ\tau=p\circ\hat{\pi}~{}~{}.$

Consider the action $\rho:\mathbb{R}\rightarrow\mathrm{Aut}({\hat{P}})$ of
$(\mathbb{R},+)$ through unbased automorphisms of ${\hat{P}}$ given by:

$(t,y)\rightarrow\rho(a)(t,y)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(t+a,y)\,,\quad\forall\,\,(t,y)\in{\hat{P}}\equiv\mathbb{R}\times
P\,.$

This covers the action of $\mathbb{R}$ on $M=\mathbb{R}\times\Sigma$ given by
time translations:

$\hat{\pi}(\rho(a)(t,y))=(t+a,\pi(y))\,,\quad\forall\,\,(t,y)\in{\hat{P}}\equiv\mathbb{R}\times
P~{}~{}.$

######  Definition 5.2.

A principal connection on ${\hat{\mathcal{A}}}\in\mathrm{Conn}({\hat{P}})$ is
called time-invariant if it satisfies:

$\rho(a)^{\ast}({\hat{\mathcal{A}}})={\hat{\mathcal{A}}}\,,\quad\forall\,\,a\in\mathbb{R}~{}~{}.$

Notice that time-invariant principal connections defined on ${\hat{P}}$ form
an affine subspace $\mathrm{Conn}^{s}({\hat{P}})$ of
$\mathrm{Conn}({\hat{P}})$. The timelike one-form $\theta\in\Omega^{1}(M)$
pulls back through $\hat{\pi}$ to an exact one-form defined on ${\hat{P}}$
which we denote by $\hat{\theta}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}p^{\ast}(\theta)\in\Omega^{1}({\hat{P}})$.

Let $\Delta=(\mathcal{S},\omega,\mathcal{D})$ be the duality structure defined
by $P$ on $\Sigma$. Then it is easy to see that the duality structure
$\hat{\Delta}=(\hat{\mathcal{S}},\hat{\omega},{\hat{\mathcal{D}}})$ defined by
${\hat{P}}$ on $M$ is given by:

$\hat{\mathcal{S}}=p^{\ast}(\mathcal{S})\,,\quad\hat{\omega}=p^{\ast}(\omega)\,,\quad{\hat{\mathcal{D}}}=p^{\ast}(\mathcal{D})\,.$

######  Lemma 5.3.

A connection ${\hat{\mathcal{A}}}\in\mathrm{Conn}({\hat{P}})$ is time-
invariant if and only if it can be written as:

${\hat{\mathcal{A}}}=-(\Psi^{\pi}\circ\tau)\hat{\theta}+\tau^{\ast}(\mathcal{A})$
(43)

for some $\Psi\in\mathcal{C}^{\infty}(\Sigma,\mathcal{S})$ and some
$\mathcal{A}\in\mathrm{Conn}(P)$. In this case, $\Psi$ and $\mathcal{A}$ are
determined uniquely by ${\hat{\mathcal{A}}}$ and any pair
$(\Psi,\mathcal{A})\in\mathcal{C}^{\infty}(\Sigma,\mathcal{S})\times\mathrm{Conn}(P)$
determines a time-invariant connection on ${\hat{P}}$ though this relation.
Moreover, the curvature of ${\hat{\mathcal{A}}}$ is given by:

$\mathcal{V}_{\hat{\mathcal{A}}}=\theta\wedge
p^{\ast}(\mathrm{d}_{\mathcal{D}}\Psi)+p^{\ast}(\mathcal{V}_{\mathcal{A}})\in\Omega^{2}(\Sigma,\hat{\mathcal{S}})$
(44)

and we have:

$\mathrm{d}_{\mathcal{D}}\mathcal{V}_{\mathcal{A}}=0~{}~{}.$ (45)

######  Remark 5.4.

Relation (44) gives the decomposition (39) of
$\mathcal{V}_{\hat{\mathcal{A}}}$ since
$\iota_{\partial_{t}}p^{\ast}(\mathcal{V}_{A})=\iota_{\partial_{t}}p^{\ast}(\mathrm{d}_{\mathcal{D}}\Psi)=0$.
Thus:

$(\mathcal{V}_{\hat{\mathcal{A}}})_{\top}=p^{\ast}(\mathrm{d}_{\mathcal{D}}\Psi)\,,\quad(\mathcal{V}_{\hat{\mathcal{A}}})_{\perp}=p^{\ast}(\mathcal{V}_{\mathcal{A}})~{}~{}.$
(46)

###### Proof.

Any principal connection
$\mathcal{A}\in\mathrm{Conn}({\hat{P}})\subset\Omega^{1}({\hat{P}},\mathfrak{a})$
decomposes uniquely as:

${\hat{\mathcal{A}}}=-\Phi\hat{\theta}+\mathcal{A}_{\perp}~{}~{},$

where $\Phi\in\Omega^{0}_{\mathrm{Ad}}({\hat{P}},\mathfrak{a})$ and
$\mathcal{A}_{\perp}\in\mathrm{Conn}({\hat{P}},\mathfrak{a})$ satisfies
$\mathcal{A}_{\perp}(\partial_{t})=0$. It is clear that ${\hat{\mathcal{A}}}$
is time-invariant if and only if $\Phi=\Psi^{\prime}\circ\tau$ for some
$\Psi^{\prime}\in\Omega^{0}_{\mathrm{Ad}}(P,\mathfrak{a})$ and
${\hat{\mathcal{A}}}_{\perp}=\tau^{\ast}(\mathcal{A})$ for some
$\mathcal{A}\in\mathrm{Conn}(P)$. Since
$\Omega^{0}_{\mathrm{Ad}}(P,\mathfrak{a})\simeq\mathcal{C}^{\infty}(\Sigma,\mathcal{S})$,
we have $\Psi^{\prime}=\Psi^{\pi}$ for some
$\Psi\in\mathcal{C}^{\infty}(\Sigma,\mathcal{S})$. Since
$\mathrm{d}\hat{\theta}=0$, the principal curvature of ${\hat{\mathcal{A}}}$
reads:

$\Omega_{\hat{\mathcal{A}}}=\mathrm{d}{\hat{\mathcal{A}}}=\mathrm{d}\Phi\wedge\hat{\theta}+\tau^{\ast}(\Omega_{\mathcal{A}})~{}~{},$

which is equivalent with (44). Relation (45) follows from the results of [35].
The remaining statements are immediate. ∎

######  Definition 5.5.

The Bogomolny pair of a time-invariant connection
${\hat{\mathcal{A}}}\in\mathrm{Conn}^{s}({\hat{P}})$ is the pair
$(\Psi,\mathcal{A})\in\mathcal{C}^{\infty}(\Sigma,\mathcal{S})\times\mathrm{Conn}(P)$
defined in Lemma 5.3. The section
$\Psi\in\mathcal{C}^{\infty}(\Sigma,\mathcal{S})$ is called the Higgs field of
the pair.

Let $\mathcal{J}$ be a taming of $\Delta$. Then the $p$-pullback
$\hat{\mathcal{J}}$ of $\mathcal{J}$ defines a time-invariant taming of
$\hat{\mathcal{S}}$, thus $({\hat{P}},\hat{\mathcal{J}})$ is a polarized
Siegel bundle. Let $\star_{h,\mathcal{J}}=\ast_{h}\otimes\mathcal{J}$ be the
polarized Hodge operator defined by $h$ and $\mathcal{J}$. Since
$\mathcal{J}^{2}=-\mathrm{id}_{\mathcal{S}}$ while $\ast_{h}$ squares to the
identity on $\Omega^{\ast}(\Sigma)$, we have:

$\star_{h,\mathcal{J}}\circ\star_{h,\mathcal{J}}=-\mathrm{id}_{\Omega^{\ast}(\Sigma,\mathcal{S})}~{}~{}.$

######  Proposition 5.6.

A time-invariant connection
${\hat{\mathcal{A}}}\in\mathrm{Conn}^{s}({\hat{P}})$ is polarized self-dual
with respect to $\hat{\mathcal{J}}$ if and only if its Bogomolny pair
$(\Psi,\mathcal{A})$ satisfies the polarized Bogomolny equation with respect
to $\mathcal{J}$:

$\star_{h,\mathcal{J}}\mathcal{V}_{\mathcal{A}}=\mathrm{d}_{\mathcal{D}}\Psi\Longleftrightarrow\star_{h,\mathcal{J}}\mathrm{d}_{\mathcal{D}}\Psi=-\mathcal{V}_{\mathcal{A}}~{}~{}.$
(47)

A polarized Bogomolny pair $(\Psi,\mathcal{A})$ which satisfies this equation
is called a polarized abelian dyon relative to $\mathcal{J}$.

###### Proof.

Relations (46) and (42) give:

$\ast_{g}\mathcal{V}_{\hat{\mathcal{A}}}=-p^{\ast}(\ast_{h}\mathrm{d}_{\mathcal{D}}\Psi)+\theta\wedge
p^{\ast}(\ast_{h}\mathcal{V}_{\mathcal{A}})~{}~{},$

which implies:

$\star_{g,\hat{\mathcal{J}}}\mathcal{V}_{\hat{\mathcal{A}}}=-p^{\ast}(\star_{h,\mathcal{J}}\mathrm{d}_{\mathcal{D}}\Psi)+\theta\wedge
p^{\ast}(\star_{h,\mathcal{J}}\mathcal{V}_{\mathcal{A}})~{}~{}.$

Comparing this with (44) shows that the polarized self-duality condition for
$\mathcal{V}_{\hat{\mathcal{A}}}$ amounts to (47), where we used uniqueness of
the decomposition (39). ∎

Let:

$\mathrm{Dyons}(\Sigma,h,\mathbf{P})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{(\Psi,\mathcal{A})\in\mathcal{C}^{\infty}(\Sigma,\mathcal{S})\times\mathrm{Conn}(P)~{}|~{}\star_{h,\mathcal{J}}\mathcal{V}_{\mathcal{A}}=\mathrm{d}_{\mathcal{D}}\Psi\\}$

be the set of all polarized abelian dyons relative to $\mathcal{J}$.

######  Remark 5.7.

Equation (47) is reminiscent of the usual Bogomolny equations obtained by
dimensional reduction of the self-duality equations for a connection on a
principal bundle over a four-dimensional Riemannian manifold. However, it
differs from the latter in two crucial respects:

  * •

The usual Bogomolny equations arise by dimensional reduction of the self-
duality equations (which are first order equations for a connection) in four
_Euclidean_ dimensions. By contrast, equation (47) is the reduction along a
timelike direction of the complete second order equations of motion defining
abelian gauge theory in four _Lorentzian_ dimensions, once these equations
have been re-written as first-order equations by doubling the number of
variables through the inclusion of both electromagnetic and magnetoelectric
gauge potentials. In particular, our reduction yields a system of first-order
differential equations, despite originating in a theory that was initially
defined by local second-order PDEs (see Appendix A).

  * •

Equation (47) is modified by the action of the taming $\mathcal{J}$, which is
absent in the usual Bogomolny equations.

###  5.3. Gauge transformations of polarized abelian dyons

As explained in Subsection 4.1 (see [35] for more detail), the gauge group
$\mathrm{Aut}_{b}(P)$ of the Siegel bundle $P$ over $\Sigma$ has an action:

$\mathrm{ad}_{P}:\mathrm{Aut}_{b}(P)\rightarrow\mathrm{Aut}_{b}(\mathcal{S})$

through based automorphisms of the vector bundle $\mathcal{S}=\mathrm{ad}(P)$.
This action agrees through the adjoint curvature map with the pushforward
action:

$\mathbb{A}\colon\mathrm{Aut}_{b}(P)\rightarrow\mathrm{Aff}(\mathrm{Conn}(P))$

of the gauge group on the space of principal connections defined on $P$. For
any principal connection $\mathcal{A}\in\mathrm{Conn}(P)$ and any
$u\in\mathrm{Aut}_{b}(P)$, we have:

$\mathcal{V}_{\mathbb{A}_{u}(\mathcal{A})}=\mathrm{ad}_{P}(u)(\mathcal{V}_{\mathcal{A}})~{}~{}.$

Similar statements hold for the gauge group of the Siegel bundle ${\hat{P}}$
defined on $M=\mathbb{R}\times\Sigma$.

######  Definition 5.8.

A gauge transformation $\hat{u}\in\mathrm{Aut}_{b}({\hat{P}})$ of ${\hat{P}}$
is called time-invariant if:

$u\circ\rho(a)=\rho(a)\circ u\,,\quad\forall\,\,a\in\mathbb{R}~{}~{}.$

Notice that time-invariant gauge transformations of ${\hat{P}}$ form a
subgroup of $\mathrm{Aut}_{b}({\hat{P}})$, which we denote by
$\mathrm{Aut}_{b}^{s}({\hat{P}})$. Such transformations stabilize the affine
subspace $\mathrm{Conn}^{s}(P)$ of time-invariant principal connections
defined on ${\hat{P}}$. Since ${\hat{P}}=p^{\ast}(P)$, we have
$\mathrm{Ad}_{G}({\hat{P}})=p^{\ast}(\mathrm{Ad}_{P})$. It is easy to see that
$\hat{u}\in\mathrm{Aut}_{b}({\hat{P}})$ is time-invariant if and only if the
corresponding section
$\sigma_{\hat{u}}\in\mathcal{C}^{\infty}(M,\mathrm{Ad}_{G}({\hat{P}}))$ is the
bundle pull-back by $p$ of a section
$\sigma\in\mathcal{C}^{\infty}(\Sigma,\mathrm{Ad}_{G}(P))$. The latter
corresponds to a gauge transformation of $P$ which we denote by
$u\in\mathrm{Aut}_{b}(P)$. We have:

$\sigma_{\hat{u}}=(\sigma_{u})^{p}\,,$ (48)

(where the subscript $p$ denotes bundle pullback by $p$) as well as:

$\tau\circ\hat{u}=u\circ\tau~{}~{}.$ (49)

Conversely, any gauge transformation $u$ of $P$ determines a gauge
transformation $\hat{u}$ of ${\hat{P}}$ by relation (48) and $\hat{u}$
satisfies (49). The correspondence $u\rightarrow\hat{u}$ gives an isomorphism
of groups between $\mathrm{Aut}_{b}(P)$ and $\mathrm{Aut}_{b}^{s}({\hat{P}})$.

The following proposition shows that the map which takes a time-invariant
principal connection defined on ${\hat{P}}$ to its Bogomolny pair intertwines
the action of $\mathrm{Aut}_{b}^{s}({\hat{P}})\simeq\mathrm{Aut}(P)$ on
$\mathrm{Conn}^{s}(P)$ with the action of $\mathrm{Aut}_{b}(P)$ on the set
$\mathcal{C}^{\infty}(\Sigma,\mathcal{S})\times\mathrm{Conn}(P)$ given by:

$\mu\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{ad}_{P}\times\mathbb{A}:\mathrm{Aut}_{b}(P)\rightarrow\mathrm{Aut}_{\mathbb{R}}(\mathcal{C}^{\infty}(\Sigma,\mathcal{S}))\times\mathrm{Aff}(\mathrm{Conn}(P))~{}~{}.$

######  Proposition 5.9.

Let ${\hat{\mathcal{A}}}$ be a time-invariant principal connection on
${\hat{P}}$ and
$(\Psi,\mathcal{A})\in\mathcal{C}^{\infty}(\Sigma,\mathcal{S})\times\mathrm{Conn}(P)$
be the Bogomolny pair defined by ${\hat{\mathcal{A}}}$. For any
$u\in\mathrm{Aut}_{b}(P)$, the Bogomolny pair $(\Psi_{u},\mathcal{A}_{u})$ of
the time-invariant connection $\mathbb{A}_{\hat{u}}({\hat{\mathcal{A}}})$
obtained from ${\hat{\mathcal{A}}}$ by applying the time-invariant gauge
transformation $\hat{u}$ is given by:

$\Psi_{u}=\mathrm{ad}_{P}(u)(\Psi)\,,\quad\mathcal{A}_{u}=\mathbb{A}_{u}(\mathcal{A})\,.$

In particular, we have:

$\mathcal{V}_{\mathcal{A}_{u}}=\mathrm{ad}_{P}(u)(\mathcal{V}_{\mathcal{A}})~{}~{}.$

###### Proof.

We have
$\mathbb{A}_{\hat{u}}({\hat{\mathcal{A}}})=(\hat{u}^{-1})^{\ast}({\hat{\mathcal{A}}})$.
Using Lemma 5.3, this gives:

$\mathbb{A}_{\hat{u}}({\hat{\mathcal{A}}})=-(\Psi^{\pi}\circ
u^{-1}\circ\tau)\hat{\theta}+\tau^{\ast}((u^{-1})^{\ast}(\mathcal{A}))=-(\Psi_{u}^{\pi}\circ\tau)\hat{\theta}+\tau^{\ast}(\mathbb{A}_{u}(\mathcal{A}))~{}~{},$

where we used relation (49) and noticed that
$(\hat{u}^{-1})^{\ast}(\hat{\theta})=\theta$ since
$\hat{\theta}=\hat{\pi}^{\ast}(\theta)$ and
$\hat{\pi}\circ\hat{u}^{-1}=\hat{u}$ because $\hat{u}^{-1}$ is a based
automorphism of ${\hat{P}}$. ∎

Notice that the discrete remnant (see [35]) of any gauge transformation of
${\hat{P}}$ is time-invariant since $\Sigma$ (and hence $M$) is connected and
therefore any discrete gauge transformation of ${\hat{P}}$ is constant on $M$.
In particular, the groups of discrete gauge transformations of ${\hat{P}}$ and
$P$ can be identified. As explained in [35], we have:

$\mathrm{ad}_{P}(u)=\mathrm{ad}_{\Gamma(P)}(\bar{u})\,,\quad\forall\,\,u\in\mathrm{Aut}_{b}(P)~{}~{},$

where $\bar{u}$ is the discrete remnant of $u$. This implies that
$\mathrm{Aut}_{b}(P)$ acts on $\mathcal{S}$ through based automorphism of the
integral duality structure ${\bm{\Delta}}$ determined by $P$:

$\mathrm{ad}_{P}(u)\in\mathrm{Aut}_{b}({\bm{\Delta}})\,,\quad\forall\,\,u\in\mathrm{Aut}_{b}(P)$

and hence induces an integral duality transformation of $\Psi$.

######  Proposition 5.10.

Let $\mathcal{J}$ be a taming of $\mathcal{S}$. Then
$(\Psi,\mathcal{A})\in\mathcal{C}^{\infty}(\Sigma,\mathcal{S})\times\mathrm{Conn}(P)$
is a polarized abelian dyon with respect to $\mathcal{J}$ if and only if
$(\Psi_{u},\mathcal{A}_{u})$ is a polarized abelian dyon with respect to the
taming $\mathcal{J}_{u}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{ad}_{P}(u)\circ\mathcal{J}\circ\mathrm{ad}_{P}(u)^{-1}$.
In particular, the action $\mu$ of $\mathrm{Aut}_{b}(P)$ restricts to
bijections:

$\mu(u):\mathrm{Dyons}(M,g,P,\mathcal{J})\xrightarrow{\sim}\mathrm{Dyons}(\Sigma,g,P,\mathcal{J}_{u})\,,\quad\forall\,\,u\in\mathrm{Aut}_{b}(P)\,.$

###### Proof.

Applying $\mathrm{ad}_{P}(u)$ shows that the polarized Bogomolny equation (47)
relative to $\mathcal{J}$ is equivalent with the polarized Bogomolny equation
relative to $\mathcal{J}_{u}$:

$\star_{h,\mathcal{J}_{u}}\mathcal{V}_{\mathcal{A}_{u}}=\mathrm{d}_{\mathcal{D}}\Psi_{u}~{}~{},$

where we used Proposition 5.9 and the fact that $\mathrm{ad}_{P}(u)$ commutes
with $\mathcal{D}$, which is proved in [35]. ∎

###  5.4. The case when $Z(P)$ has trivial monodromy on $\Sigma$

Suppose that $Z(P)$ has trivial monodromy on $\Sigma$, so the duality
structure $\Delta:=\Delta(P)$ is holonomy-trivial and its Siegel system is
trivial. Then there exists a flat trivialization of $\Delta$ which identifies
the integral duality structure ${\bm{\Delta}}=(\Delta,\mathcal{L})$ of $P$
with
$(\Sigma\times\mathbb{R}^{2n},\omega_{2n},\mathrm{d},\Sigma\times\Lambda_{\mathfrak{t}})$.
Notice that a Higgs field identifies with a smooth map from $\Sigma$ to
$\mathbb{R}^{2n}$, which we decompose into maps
$\Phi,\Upsilon:\Sigma\rightarrow\mathbb{R}^{n}$ according to:

$\Psi=\begin{pmatrix}-\Phi\\\ \Upsilon\end{pmatrix}$ (50)

######  Proposition 5.11.

A pair
$(\Psi,\mathcal{A})\in\mathcal{C}^{\infty}(\Sigma,\mathbb{R}^{2n})\times\mathrm{Conn}(P)$
satisfies the polarized Bogomolny equations iff:

$\mathrm{d}\Psi=\begin{pmatrix}E\\\
-{\mathcal{R}}E-\mathcal{I}\ast_{h}B\end{pmatrix}~{}~{},$ (51)

where $E\in\Omega^{1}(\Sigma,\mathbb{R}^{n})$ and
$B\in\Omega^{2}(\Sigma,\mathbb{R}^{n})$ are determined by
$\mathcal{V}_{\mathcal{A}}$ through the relation:

$\mathcal{V}_{\mathcal{A}}=\begin{pmatrix}B\\\
-{\mathcal{R}}B+\mathcal{I}\ast_{h}E\end{pmatrix}\,.$ (52)

###### Proof.

In the chosen trivialization of $\Delta$, tamings identify with taming maps
$\mathcal{J}:\Sigma\rightarrow\mathrm{GL}(2n,\mathbb{R})$. By Proposition A.7,
the latter have the form:

$\mathcal{J}=\begin{pmatrix}\mathcal{I}^{-1}{\mathcal{R}}&\mathcal{I}^{-1}\\\
-\mathcal{I}-{\mathcal{R}}\mathcal{I}^{-1}{\mathcal{R}}&-{\mathcal{R}}\mathcal{I}^{-1}\end{pmatrix}~{}~{},$

where
$\mathcal{N}={\mathcal{R}}+\mathbf{i}\mathcal{I}:\Sigma\rightarrow\mathbb{S}\mathbb{H}^{n}$
is the corresponding period matrix map. A similar equation relates the taming
$\hat{\mathcal{J}}=\mathcal{J}\circ p$ of $\hat{\Delta}$ to the period matrix
map ${\hat{\mathcal{N}}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{N}\circ
p={\hat{\mathcal{R}}}+\mathbf{i}{\hat{\mathcal{I}}}:M\rightarrow\mathrm{GL}(2n,\mathbb{R})$
(where ${\hat{\mathcal{R}}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}{\mathcal{R}}\circ p$ and
${\hat{\mathcal{I}}}\stackrel{{\scriptstyle{\rm def.}}}{{=}}\mathcal{I}\circ
p$) defined on $M$. By Lemma A.4, the adjoint curvature of $\mathcal{A}$ is
twisted self-dual with respect to $\hat{\mathcal{J}}$ if and only if it has
the form:

$\mathcal{V}_{\hat{\mathcal{A}}}=\begin{pmatrix}{\hat{F}}\\\
G_{g}({\hat{\mathcal{N}}},{\hat{F}})\end{pmatrix}$ (53)

for some ${\hat{F}}\in\Omega^{2}(M,\mathbb{R}^{n})$, where:

$G_{g}({\hat{\mathcal{N}}},{\hat{F}})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}-{\hat{\mathcal{R}}}\,{\hat{F}}-{\hat{\mathcal{I}}}\ast_{g}{\hat{F}}\,.$

Lemma 5.1 gives:

$G_{g}({\hat{\mathcal{N}}},{\hat{F}})_{\top}=-{\hat{\mathcal{R}}}{\hat{F}}_{\top}-{\hat{\mathcal{I}}}\ast_{h}{\hat{F}}_{\perp}\,,\quad
G_{g}({\hat{\mathcal{N}}},{\hat{F}})_{\perp}=-{\hat{\mathcal{R}}}{\hat{F}}_{\perp}+{\hat{\mathcal{I}}}\ast_{h}{\hat{F}}_{\top}\,.$

Thus (53) amounts to:

$(\mathcal{V}_{\hat{\mathcal{A}}})_{\top}=\begin{pmatrix}{\hat{F}}_{\top}\\\
-{\hat{\mathcal{R}}}{\hat{F}}_{\top}-{\hat{\mathcal{I}}}\ast_{h}{\hat{F}}_{\perp}\end{pmatrix}\,,\quad(\mathcal{V}_{\hat{\mathcal{A}}})_{\perp}=\begin{pmatrix}{\hat{F}}_{\perp}\\\
-{\hat{\mathcal{R}}}{\hat{F}}_{\perp}+{\hat{\mathcal{I}}}\ast_{h}{\hat{F}}_{\top}\end{pmatrix}~{}~{}.$
(54)

These relations imply:

${\hat{F}}_{\top}=p^{\ast}(E)\,,\quad{\hat{F}}_{\perp}=p^{\ast}(B)$

for some $E\in\Omega^{1}(\Sigma,\mathbb{R}^{n})$ and
$B\in\Omega^{2}(\Sigma,\mathbb{R}^{n})$. Using (54), we conclude that (46) is
equivalent with:

$\mathrm{d}\Psi=\begin{pmatrix}E\\\
-{\mathcal{R}}E-\mathcal{I}\ast_{h}B\end{pmatrix}\,,\quad\mathcal{V}_{\mathcal{A}}=\begin{pmatrix}B\\\
-{\mathcal{R}}B+\mathcal{I}\ast_{h}E\end{pmatrix}~{}~{}.$

Notice that the pair $(E,B)$ is uniquely determined by
$\mathcal{V}_{\mathcal{A}}$ through the second of these relations. ∎

We will refer to the forms $E\in\Omega^{1}(\Sigma,\mathbb{R}^{n})$ and
$B\in\Omega^{2}(\Sigma,\mathbb{R}^{n})$ as the electrostatic and magnetostatic
field strengths of $\mathcal{A}$. Equation (51) is equivalent to the system:

$\mathrm{d}\Phi=E\,,\quad\mathrm{d}\Upsilon=-{\mathcal{R}}E-\mathcal{I}\ast_{h}B\,,$

which in turn amounts to:

$\vec{E}=-\mathrm{grad}_{h}\Phi\,,\quad\mathcal{I}\vec{B}+{\mathcal{R}}\vec{E}=-\mathrm{grad}_{h}\Upsilon\,,$
(55)

where we defined
$\vec{E},\vec{B}\in\mathcal{C}^{\infty}(\Sigma,T\Sigma)\otimes\mathbb{R}^{n}$
through:

$\vec{E}=E^{\sharp}\,,\quad\vec{B}=(\ast_{h}B)^{\sharp}\,.$

Here $\sharp$ denotes the musical isomorphism given by raising of indices with
the metric $h$. Equation $\mathrm{d}\mathcal{V}_{\mathcal{A}}=0$ amounts to
the system:

$\mathrm{d}B=0\,,\quad\mathrm{d}(-{\mathcal{R}}B+\mathcal{I}\ast_{h}E)=0\Longleftrightarrow\mathrm{div}_{h}\vec{B}=0\,,\quad\mathrm{div}_{h}({\mathcal{R}}\,\vec{B}+\mathcal{I}\,\vec{E})=0\,.$
(56)

###  5.5. Polarized dyons in prequantum electrodynamics

Prequantum electrodynamics defined on $M$ corresponds to setting $n=1$ and
${\mathcal{R}}=\frac{\theta}{2\pi},\mathcal{I}=\frac{4\pi}{g^{2}}$ with
constant $\theta\in\mathbb{R}$ in the previous subsection (see Appendix A).
Then:

$\mathcal{J}=\mathcal{J}_{\theta}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\begin{pmatrix}\frac{g^{2}\theta}{8\pi^{2}}&\frac{g^{2}}{4\pi}\\\
-\frac{4\pi}{g^{2}}-\frac{g^{2}\theta^{2}}{16\pi^{3}}&-\frac{g^{2}\theta}{8\pi^{2}}\end{pmatrix}~{}~{},$

and relation (52) becomes:

$\mathcal{V}_{\mathcal{A}}=\begin{pmatrix}B\\\
-\frac{\theta}{2\pi}B+\frac{4\pi}{g^{2}}\ast_{h}E\end{pmatrix}~{}~{}.$ (57)

In this case, relations (55) reduce to:

$\vec{E}=-\mathrm{grad}_{h}\Phi\,,\quad\frac{4\pi}{g^{2}}\vec{B}+\frac{\theta}{2\pi}\vec{E}=-\mathrm{grad}_{h}\Upsilon\,,$
(58)

which imply
$\vec{B}=-\frac{g^{2}}{4\pi}\,\mathrm{grad}_{h}(\Upsilon-\frac{\theta}{2\pi}\Phi)$
and:

$\mathrm{curl}_{h}\vec{E}=\mathrm{curl}_{h}\vec{B}=0~{}~{}.$ (59)

On the other hand, relations (56) become:

$\mathrm{d}B=\mathrm{d}(\ast_{h}E)=0\Longleftrightarrow\mathrm{div}_{h}\vec{B}=\mathrm{div}_{h}\vec{E}=0~{}~{}.$
(60)

Equations (59) and (60) describe source-free Maxwell electromagnetostatics on
$\Sigma$, where the vector fields $\vec{E},\vec{B}\in\mathcal{A}(\Sigma)$ are
the classical static electric and magnetic fields. Relation (58) shows that
$\Phi$ and $\frac{g^{2}}{4\pi}\Upsilon-\frac{g^{2}\theta}{8\pi^{2}}\Phi$ are
globally-defined classical electrostatic and magnetostatic scalar potentials.
Since $H^{1}(\Sigma,\mathbb{R})$ need not vanish, relation (59) need not imply
(58). Hence polarized dyons describe special potential electromagnetostatic
configurations, i.e. solutions of the static Maxwell equations on $\Sigma$
which admit both a globally-defined electric scalar potential and a globally-
defined magnetic scalar potential. Even though $\vec{E}$ and $\vec{B}$ satisfy
(60), such solutions need not admit a globally-defined vector electric or
vector magnetic potential, since $H^{2}(\Sigma,\mathbb{R})$ may be non-zero.
The condition that the configuration $(\vec{E},\vec{B})$ admits globally-
defined electric and magnetic scalar potentials is a consequence of the fact
that a polarized dyon originates from a principal connection defined on
${\hat{P}}$, which itself is a consequence of the DSZ integrality condition.
We formalize this as follows.

######  Definition 5.12.

An electromagnetostatic configuration defined on $(\Sigma,h)$ is a pair of
vector fields
$(\vec{E},\vec{B})\in\mathcal{A}(\Sigma)\times\mathcal{A}(\Sigma)$ which
satisfies the static source-free Maxwell equations (59) and (60). Such a
configuration is said to be potential if there exist real-valued smooth
functions $\Phi,\Upsilon$ defined on $\Sigma$ such that relations (58) hold.

Let $\mathrm{EMC}(\Sigma,h)$ denote the vector space of all
electromagnetostatic configurations defined on $(\Sigma,h)$ and
$\mathrm{EMC}_{\mathrm{pot}}(\Sigma,h)$ denote the subspace of those
configurations which are potential. Consider the map:

$H_{\theta}:\mathrm{Dyons}(\Sigma,h,P,\mathcal{J}_{\theta})\rightarrow\mathrm{EMC}_{\mathrm{pot}}(\Sigma,h)$

which associates the electromagnetostatic configuration $(\vec{E},\vec{B})$
defined through relation (57) to the polarized dyon $(\Psi,\mathcal{A})$. As
in Subsection 2.10, the chosen trivialization of $\Delta$ induces
isomorphisms:

$H^{2}_{\mathcal{D}}(\Sigma,\mathcal{S})\simeq
H^{2}(M,\mathbb{R}^{2n})\,,\quad H^{2}(\Sigma,\mathcal{L})\simeq
H^{2}(\Sigma,\Lambda_{\mathfrak{t}})\,,$

which allow us to identify the morphism
$j:H^{2}(\Sigma,\mathcal{L})\rightarrow
H^{2}_{\mathcal{D}}(\Sigma,\mathcal{S})$ with the morphism $\iota$ appearing
in the long exact sequence:

$\ldots\rightarrow H^{1}(\Sigma,A)\rightarrow
H^{2}(\Sigma,\Lambda_{\mathfrak{t}})\xrightarrow{i}H^{2}(\Sigma,\mathbb{R}^{2n})\rightarrow
H^{2}(\Sigma,A)\rightarrow\ldots$

induced by the exponential sequence:

$0\rightarrow\Lambda_{\mathfrak{t}}\rightarrow\mathbb{R}^{2n}\rightarrow
A\rightarrow 0~{}~{}.$

The relation $[\mathcal{V}_{\mathcal{A}}]_{\mathcal{D}}=2\pi j_{\ast}(c(P))$
becomes:

$[\mathcal{V}_{\mathcal{A}}]=2\pi i_{\ast}(c(P))\,,$

where $[\omega]$ denotes the de Rham cohomology class of a form
$\omega\in\Omega^{k}(M,S)$. Conversely, any closed two-form
$\mathcal{V}\in\Omega^{2}_{\mathrm{cl}}(\Sigma,\mathbb{R}^{2n})$ which
satisfies $[\mathcal{V}]=2\pi i_{\ast}(c(P))$ identifies with the curvature of
some principal connection $\mathcal{A}$ on $P$. For any
$(\vec{E},\vec{B})\in\mathrm{EMC}(\Sigma,h)$, let:

$\mathcal{V}^{\theta}_{\vec{E},\vec{B}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\begin{pmatrix}B\\\
-\frac{\theta}{2\pi}B+\frac{4\pi}{g^{2}}\ast_{h}E\end{pmatrix}~{}~{},$

where $E\stackrel{{\scriptstyle{\rm def.}}}{{=}}\vec{E}^{\flat}$ and
$B\stackrel{{\scriptstyle{\rm def.}}}{{=}}-\ast_{h}(\vec{B}^{\flat})$.

######  Theorem 5.13.

The image of the map $H_{\theta}$ coincides with the set:

$\mathrm{EMC}_{\mathrm{pot}}(\Sigma,h;\theta,c(P))\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{(\vec{E},\vec{B})\in\mathrm{EMC}_{\mathrm{pot}}(\Sigma,h)~{}|~{}[\mathcal{V}^{\theta}_{\vec{E},\vec{B}}]=2\pi\iota_{\ast}(c(P))\\}~{}~{}.$

Moreover, the $H_{\theta}$-preimage of a configuration
$(\vec{E},\vec{B})\in\mathrm{EMC}_{\mathrm{pot}}(\Sigma,h;\theta,c(P))$ is
given by:

$H_{\theta}^{-1}(\\{(\vec{E},\vec{B})\\})=\Big{\\{}(\Psi,A)\in\mathrm{Dyons}(\Sigma,h,\mathcal{J}_{\theta})~{}|~{}\mathcal{V}_{\mathcal{A}}=\mathcal{V}^{\theta}_{\vec{E},\vec{B}}~{}\&~{}\mathrm{grad}_{h}\Psi=\begin{pmatrix}\vec{E}\\\
-\frac{\theta}{2\pi}\vec{E}-\frac{4\pi}{g^{2}}\vec{B}\end{pmatrix}\Big{\\}}~{}~{}.$

###### Proof.

Let $(\Psi,\mathcal{A})$ be a polarized abelian dyon. Then
$H_{\theta}(\Psi,\mathcal{A})\in\mathrm{EMC}_{\mathrm{pot}}(\Sigma,h)$ and by
equation (57), there exists an electromagnetostatic configuration
$(\vec{E},\vec{B})$ such that:

$\mathcal{V}^{\theta}_{\mathcal{A}}=\begin{pmatrix}B\\\
-\frac{\theta}{2\pi}B+\frac{4\pi}{g^{2}}\ast_{h}E\end{pmatrix}\,.$

Since $\mathcal{V}^{\theta}_{\mathcal{A}}$ is the curvature of a connection
$\mathcal{A}$ on $P$, it satisfies
$[\mathcal{V}^{\theta}_{\mathcal{A}}]_{\mathcal{D}}=2\pi j_{\ast}(c(P))$ and
hence $(\vec{E},\vec{B})\in\mathrm{EMC}_{\mathrm{pot}}(\Sigma,h;\theta,c(P))$.
On the other hand, if
$(\vec{E},\vec{B})\in\mathrm{EMC}_{\mathrm{pot}}(\Sigma,h;\theta,c(P))$,
equation $[\mathcal{V}^{\theta}_{\vec{E},\vec{B}}]=\iota_{\ast}(c(P))$ implies
that there exists a connection $\mathcal{A}$ on $P$ whose curvature is
$\mathcal{V}^{\theta}_{\vec{E},\vec{B}}$ and since $(\vec{E},\vec{B})$ is a
potential electromagnetostatic solution, choosing potentials $\Phi$ and
$\Upsilon$ and defining:

$\Psi=\begin{pmatrix}-\Phi\\\ \Upsilon\end{pmatrix}$ (61)

we conclude that $(\Psi,\mathcal{A})$ is a polarized abelian dyon whence
$(\vec{E},\vec{B})\in\mathrm{Im}(H_{\theta})$. It is now clear that the only
freedom in choosing a preimage of $(\vec{E},\vec{B})$ by $H_{\theta}$ lies
only in choosing the potentials $\Phi$ and $\Upsilon$ as prescribed in the
statement of the theorem and hence we conclude. ∎

###  5.6. Polarized Bogomolny equations on the punctured Euclidean 3-space

In this subsection, we construct families of solutions (which generalize the
dyons of ordinary electromagnetism) on the punctured Euclidean space
$\Sigma=\mathbb{R}^{3}_{0}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathbb{R}^{3}\backslash\left\\{0\right\\}$. Spherical coordinates
give a diffeomorphism:

$\mathbb{R}^{3}_{0}\simeq\mathbb{R}_{>0}\times\mathrm{S}^{2}\,,$

and we denote by $r\in\mathbb{R}_{>0}$ the radial coordinate. The metric $h$
reads:

$h=\mathrm{d}r^{2}+r^{2}h_{\mathrm{S}^{2}}\,,$

where $h_{\mathrm{S}^{2}}$ is the unit round metric of $\mathrm{S}^{2}$. We
have $\nu_{h}=r^{2}\mathrm{d}r\wedge\nu_{\mathrm{S}^{2}}$, where
$\nu_{\mathrm{S}^{2}}$ is the volume form of $\mathrm{S}^{2}$. Let
$q:\mathbb{R}^{3}_{0}\rightarrow\mathrm{S}^{2}$ be the map defined though:

$q(x)\stackrel{{\scriptstyle{\rm def.}}}{{=}}\frac{x}{||x||}~{}~{},$

where $||x||$ is the Euclidean norm of $x$. Up to isomorphism, any Siegel
bundle of rank $n$ and type $\mathfrak{t}\in\mathrm{Div}^{n}$ defined on $P$
on $\mathbb{R}^{3}_{0}$ is the pull-back through $q$ of a Siegel bundle of the
same rank and type defined on $\mathrm{S}^{2}$, which reduces to a principal
torus bundle since $\mathrm{S}^{2}$ is simply connected. Accordingly, the
duality structure $\Delta$ of $P$ is holonomy trivial and we can fix a global
flat trivialization in which $\Delta$ identifies with:

$\Delta\equiv(\Sigma\times\mathbb{R}^{2n},\omega_{2n},\mathrm{d})\,,$ (62)

and the Dirac system $\mathcal{L}$ determined by $P$ identifies with
$\Sigma\times\Lambda_{\mathfrak{t}}$. Hence a Bogomolny pair consists of a map
$\Psi\in\mathcal{C}^{\infty}(\Sigma,\mathbb{R}^{2n})$ and a connection
$\mathcal{A}\in\mathrm{Conn}(P)$ whose curvature can be viewed as a vector-
valued two-form
$\mathcal{V}_{\mathcal{A}}\in\Omega^{2}(\Sigma,\mathbb{R}^{2n})$. Moreover, a
taming of $\Delta$ can be viewed as a map
$\mathcal{J}\in\mathcal{C}^{\infty}(\Sigma,\mathrm{GL}(2n,\mathbb{R}))$ which
squares to minus the identity everywhere and satisfies
$\mathcal{J}^{t}(\sigma){\hat{\omega}}_{2n}\mathcal{J}(\sigma)=\hat{\omega}_{2n}$
for all $\sigma\in\Sigma$.

Let us assume that $\mathcal{J}$ and $\Psi$ depend only on $r$. Then
$\mathrm{d}\Psi=\partial_{r}\Psi\mathrm{d}r$ and
$\ast_{h}(\mathrm{d}r)=\iota_{\partial_{r}}\nu_{h}=r^{2}q^{\ast}(\nu_{\mathrm{S}^{2}})$,
hence the polarized Bogomolny equation reads:

$\mathcal{V}_{\mathcal{A}}=-r^{2}\mathcal{J}(r)\frac{\mathrm{d}\Psi}{\mathrm{d}r}\,q^{\ast}(\nu_{S^{2}})\,,$
(63)

Since $\mathcal{V}_{\mathcal{A}}$ and $\nu_{\mathrm{S}^{2}}$ are closed, a
necessary condition for this equation to admit solutions
$\mathcal{A}\in\mathrm{Conn}(P)$ is:

$\frac{\mathrm{d}}{\mathrm{d}r}(r^{2}\mathcal{J}\frac{\mathrm{d}}{\mathrm{d}r}\Psi\,)=0\,.$

The general solution of this integrability condition is:

$\Psi=-\int\frac{\mathcal{J}v}{2r^{2}}\mathrm{d}r+v^{\prime}\,,$

for constant vectors $v,v^{\prime}\in\mathbb{R}^{2n}$. Using this in (63)
gives:

$\mathcal{V}_{\mathcal{A}}=-\frac{1}{2}v\,q^{\ast}(\nu_{\mathrm{S}^{2}})\,.$

The last relation determines $\mathcal{A}$ up to a transformation of the form:

$\mathcal{A}\rightarrow\mathcal{A}+\omega~{}~{}$

where $\omega\in\Omega^{1}_{\mathrm{Ad}}(P,\mathbb{R}^{2n})$ is closed and
hence corresponds to a closed 1-form
$\omega^{\prime}\in\Omega^{1}(\Sigma,\mathbb{R}^{2n})$ (recall that
$\mathrm{d}_{\mathcal{D}}=\mathrm{d}$ in our trivialization of $\mathcal{S}$).
Notice that $\omega^{\prime}$ need not be exact since
$\Sigma=\mathbb{R}^{3}_{0}$ is not contractible.

####  5.6.1. The integrality condition for $v$

Since $\mathrm{S}^{2}$ is a deformation retract of $\Sigma$, the map
$q^{\ast}:H^{\ast}(\mathrm{S}^{2},\mathbb{R}^{2n}/\Lambda_{\mathfrak{t}})\rightarrow
H^{\ast}(\Sigma,\mathbb{R}^{2n}/\Lambda_{\mathfrak{t}})$ induced by $q$ on
cohomology is an isomorphism of groups for any abelian group of coefficients
$A$. Since $H_{1}(\mathrm{S}^{2},\mathbb{Z})=0$, the universal coefficient
theorem for cohomology gives:

$H^{2}(\mathrm{S}^{2},\Lambda_{\mathfrak{t}})\simeq_{\mathbb{Z}}{\rm
Hom}_{\mathbb{Z}}(H_{2}(\mathrm{S}^{2},\mathbb{Z}),\Lambda_{\mathfrak{t}})\simeq_{\mathbb{Z}}\Lambda_{\mathfrak{t}}~{}~{},$
(64)

where the last isomorphism is given by evaluation on the fundamental class
$[\mathrm{S}^{2}]\in H_{2}(\mathrm{S}^{2},\mathbb{Z})$. This allows us to view
the characteristic class $c(P)=q^{\ast}(c(P_{0}))$ as an element of
$\Lambda_{\mathfrak{t}}$. Moreover, we have isomorphisms of vector spaces:

$H^{2}(\mathrm{S}^{2},\mathbb{R}^{2n})\simeq_{\mathbb{R}}H^{2}(\mathrm{S}^{2},\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{R}^{2n}\simeq_{\mathbb{R}}\mathbb{R}^{2n}~{}~{},$
(65)

the last of which takes $u\otimes_{\mathbb{Z}}x$ to $x$ for all
$x\in\mathbb{R}^{2n}$. Through the isomorphisms (64) and (65), the map
$i_{\ast}:H^{2}(\mathrm{S}^{2},\Lambda_{\mathfrak{t}})\rightarrow
H^{2}(\mathrm{S}^{2},\mathbb{R}^{2n})$ corresponds to the inclusion
$\Lambda_{\mathfrak{t}}\subset\mathbb{R}^{2n}$ and $2\pi c(P)$ identifies with
the de Rham cohomology class of $\mathcal{V}_{\mathcal{A}}$. The free abelian
group $H^{2}(\mathrm{S}^{2},\mathbb{Z})$ is generated by half of the Euler
class $e(\mathrm{S}^{2})$ of $\mathrm{S}^{2}$, which satisfies:

$\int_{\mathrm{S}^{2}}e(\mathrm{S}^{2})=\chi(\mathrm{S}^{2})=1+(-1)^{2}=2~{}~{}.$

On the other hand, the de Rham cohomology class $u\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left[\frac{\nu_{\mathrm{S}^{2}}}{4\pi}\right]\in
H^{2}(\mathrm{S}^{2},\mathbb{R}^{2n})$ satisfies:

$u=i_{\ast}\left(\frac{1}{2}e(\mathrm{S}^{2})\right)~{}~{}.$

We have:

$[\mathcal{V}_{\mathcal{A}}]=2\pi i_{\ast}(c(P))~{}~{},$

which implies $v\in\Lambda_{\mathfrak{t}}$ as well as:

$c(P_{0})=\frac{1}{2}v\,e(\mathrm{S}^{2})~{}~{}\mathrm{and}~{}~{}i_{\ast}(c(P_{0}))=vu~{}~{}.$

Hence we obtain an integrality condition for the integration constant $v$,
which guarantees the existence of a solution and determines through the
previous equation the topological type of the bundle $P$ carrying the
solution.

##  Appendix A Local abelian gauge theory

This appendix discusses the duality-covariant formulation and global
symmetries of source-free $\mathrm{U}(1)^{n}$ abelian gauge theory on a
contractible oriented four-manifold $M$ endowed with an arbitrary metric of
signature $(3,1)$, with the goal of motivating the geometric model introduced
in Section 1 and of making contact with the physics literature. The local
study of electromagnetic duality for Maxwell electrodynamics in four
Lorentzian dimensions has a long history, see [21, 22] as well as [41] and its
references and citations. Let $\ast_{g}$ be the Hodge operator of $(M,g)$.

The prototypical example of local abelian gauge theory is given by classical
electrodynamics, which is defined by the Lagrangian density functional:

$\mathfrak{L}[A]=-\frac{4\pi}{g^{2}}(F_{A})_{ab}(F_{A})^{ab}+\frac{\theta}{2\pi}(F_{A})_{ab}\,(\ast_{g}F_{A})^{ab}\,,\qquad
F=\mathrm{d}A~{}~{}\mathrm{with}~{}~{}A\in\Omega^{1}(M)~{}~{},$

where $\theta\in\mathbb{R}$ is the theta angle. Classical Maxwell theory is
obtained for $\theta=0$. Below, we discuss the more general construction of
local abelian gauge theories, which is motivated by the local structure of
four-dimensional supergravity and supersymmetric field theory and which
involves a finite number of abelian gauge fields.

###  A.1. Lagrangian formulation

Given $n\in\mathbb{Z}_{>0}$ and a Lorentzian metric $g$ on $M$, consider the
following generalization of the previous Lagrangian density, which allows all
couplings to be smooth real-valued functions:

$\mathfrak{L}[A^{1},\ldots,A^{n}]=-\mathcal{I}_{\Lambda\Sigma}\,\langle
F^{\Lambda}_{A},F^{\Sigma}_{A}\rangle_{g}+{\mathcal{R}}_{\Lambda\Sigma}\,\langle
F^{\Lambda}_{A},\ast_{g}F^{\Sigma}_{A}\rangle_{g}~{}~{}.$

Here $\Lambda,\Sigma=1,\ldots,n$ and :

$F^{\Lambda}_{A}=\mathrm{d}A^{\Lambda}\,,\qquad\Lambda=1,\ldots n\,,$

where $A^{\Lambda}\in\Omega^{1}(M)$ and
${\mathcal{R}}_{\Lambda\Sigma},\mathcal{I}_{\Lambda\Sigma}\in\mathcal{C}^{\infty}(M)$
are the components of an $\mathbb{R}^{n}$-valued one-form
$A\in\Omega^{1}(M,\mathbb{R}^{n})$ and of functions
${\mathcal{R}},\mathcal{I}\in\mathcal{C}^{\infty}(M,\mathrm{Mat}_{s}(n,\mathbb{R}))$
valued in the space $\mathrm{Mat}_{s}(n,\mathbb{R})$ of symmetric square
matrices of size $n$ with real entries. To guarantee a positive-definite
kinetic term we must assume that $\mathcal{I}$ is positive-definite
everywhere. Notice that classical electrodynamics corresponds to $n=1$ with
${\mathcal{R}}=\frac{\theta}{2\pi}$ and $\mathcal{I}=\frac{4\pi}{g^{2}}$.
Since $\ast_{g}^{2}F^{\Lambda}=-F^{\Lambda}$, the action is equivalent to:

$\mathcal{S}[A^{1},\ldots,A^{n}]=\int\mathfrak{L}[A^{1},\ldots,A^{n}]\mathrm{vol}_{g}=-\int\left(\mathcal{I}_{\Lambda\Sigma}\,F^{\Lambda}_{A}\wedge\ast_{g}F^{\Sigma}_{A}+{\mathcal{R}}_{\Lambda\Sigma}\,F^{\Lambda}_{A}\wedge
F^{\Sigma}_{A}\right)~{}~{}.$

The partial differential equations obtained as critical points of the
variational problem with respect to compactly supported variations are:

$\mathrm{d}({\mathcal{R}}_{\Lambda\Sigma}F^{\Sigma}_{A}+\mathcal{I}_{\Lambda\Sigma}\ast_{g}F^{\Sigma}_{A})=0\,,$

or, in matrix notation:

$\mathrm{d}({\mathcal{R}}F_{A}+\mathcal{I}\ast_{g}F_{A})=0\,,$

where
$F_{A}=\mathrm{d}A\in\Omega^{2}_{\mathrm{cl}}(M,\mathbb{R}^{n})=\Omega^{2}_{\mathrm{ex}}(M,\mathbb{R}^{n})$
is an $\mathbb{R}^{n}$-valued closed (and hence exact) two-form. These
equations define classical local abelian gauge theory. Since both
${\mathcal{R}}$ and $\mathcal{I}$ are symmetric and $\mathcal{I}$ is positive
definite, the pair $({\mathcal{R}},\mathcal{I})$ is equivalent to a map:

$\mathcal{N}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}{\mathcal{R}}+\mathbf{i}\mathcal{I}\colon M\to\mathbb{SH}_{n}\,,$

where $\mathbb{SH}_{n}$ denotes the Siegel upper half space of symmetric
$n\times n$ complex matrices with positive definite imaginary part.

######  Definition A.1.

A period matrix map of size $n$ on $M$ is a smooth function
$\mathcal{N}\in\mathcal{C}^{\infty}(M,\mathbb{SH}_{n})$ valued in
$\mathbb{SH}_{n}$. We denote the set of such maps by $\mathrm{Per}_{n}(M)$.

When the metric $g$ is fixed, classical local abelian gauge theory is uniquely
determined by a choice of period matrix map.

######  Definition A.2.

Let $\mathcal{N}={\mathcal{R}}+\mathbf{i}\mathcal{I}$ be a period matrix map
of size $n$. The local abelian gauge theory associated to $\mathcal{N}$ is
defined through the following system of equations:

$\mathrm{d}({\mathcal{R}}\,F_{A}+\mathcal{I}\ast_{g}F_{A})=0\,,$ (66)

with unknowns given by the vector valued one-form
$A\in\Omega^{1}(M,\mathbb{R}^{n})$, where $F_{A}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{d}A\in\Omega^{2}_{\mathrm{cl}}(M,\mathbb{R}^{n})$.

######  Remark A.3.

The equations of motion of local abelian gauge theory have a _gauge symmetry_
consisting of transformations of the type:

$A\mapsto
A+\mathrm{d}\alpha\,,\qquad\alpha\in\mathcal{C}^{\infty}(M,\mathbb{R}^{n})\,.$

Variables $A\in\Omega^{1}(M,\mathbb{R}^{n})$ related by such gauge
transformations should be viewed as physically equivalent.

Let us write the equations of motion (66) in a form amenable to geometric
interpretation. Given a period matrix map
$\mathcal{N}={\mathcal{R}}+\mathbf{i}\mathcal{I}$ and a vector of two-forms
$F\in\Omega^{2}(M,\mathbb{R}^{n})$, define:

$G_{g}(\mathcal{N},F)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}-{\mathcal{R}}\,F-\mathcal{I}\ast_{g}F\,.$

Then the condition $\mathrm{d}F=0$ together with the equations of motion (66)
are equivalent with the single equation:

$\mathrm{d}\mathcal{V}=0\,,$ (67)

where the $\mathbb{R}^{2n}$-valued two-form $\mathcal{V}$ is related to $F$
by:

$\mathcal{V}=\begin{pmatrix}F\\\
G_{g}(\mathcal{N},F)\end{pmatrix}\in\Omega^{2}(M,\mathbb{R}^{2n})=\Omega^{2}(M,\mathbb{R}^{n}\oplus\mathbb{R}^{n})\,.$
(68)

As the following lemma shows, not every vector-valued two-form
$\mathcal{V}\in\Omega^{2}(M,\mathbb{R}^{2n})$ can be written as prescribed by
equation (68).

######  Lemma A.4.

Let $\mathcal{N}={\mathcal{R}}+\mathbf{i}\mathcal{I}\in\mathrm{Per}_{n}(M)$ be
a period matrix map of size $n$ on $M$. A vector valued two-form
$\mathcal{V}\in\Omega^{2}(M,\mathbb{R}^{2n})$ can be written as:

$\mathcal{V}=\begin{pmatrix}F\\\ G_{g}(\mathcal{N},F)\end{pmatrix}$

for some $F\in\Omega^{2}(M,\mathbb{R}^{n})$ if and only if:

$\ast_{g}\mathcal{V}=-\mathcal{J}\mathcal{V}\,,$ (69)

where $\mathcal{J}\in\mathcal{C}^{\infty}(M,\mathrm{GL}(2n,\mathbb{R}))$ is
the matrix-valued map defined through:

$\mathcal{J}=\begin{pmatrix}\mathcal{I}^{-1}{\mathcal{R}}&\mathcal{I}^{-1}\\\
-\mathcal{I}-{\mathcal{R}}\mathcal{I}^{-1}{\mathcal{R}}&-{\mathcal{R}}\mathcal{I}^{-1}\end{pmatrix}~{}~{}.$

We have $\mathcal{J}^{2}=-1$. Moreover, $F$ with the property above is
uniquely determined by $\mathcal{V}$.

######  Remark A.5.

Notice that $\mathcal{J}$ can be viewed as a complex structure defined on the
trivial real vector bundle of rank $2n$ over $M$. For classical
electrodynamics, the taming map is constant and given by:

$\mathcal{J}=\begin{pmatrix}\frac{g^{2}\theta}{8\pi^{2}}&\frac{g^{2}}{4\pi}\\\
-\frac{4\pi}{g^{2}}-\frac{g^{2}\theta^{2}}{16\pi^{3}}&-\frac{g^{2}\theta}{8\pi^{2}}\end{pmatrix}~{}~{}.$

In this case, the period matrix map is constant and given by:

$\mathcal{N}=\frac{4\pi}{g^{2}}+\mathbf{i}\frac{\theta}{2\pi}~{}~{},$

being traditionally denoted by $\tau$.

###### Proof.

If:

$\mathcal{V}=\begin{pmatrix}F\\\ G_{g}(\mathcal{N},F)\end{pmatrix}\,,$

then direct computation using the fact that $\ast_{g}^{2}=-1$ on two-forms
shows that $\mathcal{V}$ satisfies
$\ast_{g}\mathcal{V}=-\mathcal{J}\mathcal{V}$. On the other hand, writing
$\mathcal{V}=\begin{pmatrix}F\\\ G\end{pmatrix}$ with
$F,G\in\Omega^{2}(M,\mathbb{R}^{n})$ shows that the equation
$\ast_{g}\mathcal{V}=-\mathcal{J}\mathcal{V}$ is equivalent to:

$\begin{pmatrix}\ast_{g}F\\\
\ast_{g}G\end{pmatrix}=\begin{pmatrix}-\mathcal{I}^{-1}{\mathcal{R}}F-\mathcal{I}^{-1}G\\\
-(\mathcal{I}+{\mathcal{R}}\mathcal{I}^{-1}{\mathcal{R}})F-{\mathcal{R}}\mathcal{I}^{-1}G\end{pmatrix}\,,$

which in turn amounts to $G=G_{g}(\mathcal{N},F)$. ∎

Let $\omega_{2n}$ be the standard symplectic form on $\mathbb{R}^{2n}$, which
in our conventions has the following matrix in the canonical basis
$\mathcal{E}=(e_{1},\ldots,e_{n},f_{1},\ldots,f_{n})$ of the latter:

${\hat{\omega}}_{2n}=\begin{pmatrix}0&I_{n}\\\ -I_{n}&0\end{pmatrix}~{}~{}.$
(70)

Here $I_{n}$ is the identity matrix of size $n$. We have:

$\omega_{2n}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\sum_{a=1}^{n}e_{a}^{\ast}\wedge f_{a}^{\ast}\,,$ (71)

where
$\mathcal{E}^{\ast}=(e_{1}^{\ast},\ldots,e^{\ast}_{n},f^{\ast}_{1},\ldots,f^{\ast}_{n})$
is the basis dual to $\mathcal{E}=(e_{1},\ldots,e_{n},f_{1},\ldots,f_{n})$.
The following result gives a geometric interpretation of the equations of
motion (69). Recall that an almost complex structure $J$ on $\mathbb{R}^{2n}$
is called a taming of the standard symplectic form $\omega_{2n}$ if:

$\omega_{2n}(J\xi_{1},J\xi_{2})=\omega_{2n}(\xi_{1},\xi_{2})\,,\quad\forall\,\,\xi_{1},\xi_{2}\in\mathbb{R}^{2n}\,,$

and:

$\omega_{2n}(J\xi,\xi)>0\,,\quad\forall\,\,\xi\in\mathbb{R}^{2n}\backslash\left\\{0\right\\}\,.$

######  Definition A.6.

A taming map of size $2n$ defined on $M$ is a smooth map
$\mathcal{J}\in\mathcal{C}^{\infty}(M,\mathrm{GL}(2n,\mathbb{R}))$ such that
$\mathcal{J}(m)$ is a taming of $\omega_{2n}$ for every $m\in M$. We denote
the set of all such maps by $\mathfrak{J}_{n}(M)$.

######  Proposition A.7.

A matrix-valued map
$\mathcal{J}\in\mathcal{C}^{\infty}(M,\mathrm{GL}(2n,\mathbb{R}))$ can be
written as:

$\mathcal{J}=\begin{pmatrix}\mathcal{I}^{-1}{\mathcal{R}}&\mathcal{I}^{-1}\\\
-\mathcal{I}-{\mathcal{R}}\mathcal{I}^{-1}{\mathcal{R}}&-{\mathcal{R}}\mathcal{I}^{-1}\,,\end{pmatrix}$
(72)

in terms of a period matrix map
$\mathcal{N}={\mathcal{R}}+\mathbf{i}\mathcal{I}\in\mathrm{Per}_{n}(M)$ iff
$\mathcal{J}\in\mathfrak{J}_{n}(M)$.

###### Proof.

If $\mathcal{J}$ is taken as in equation (72) for a period matrix map
$\mathcal{N}$ then direct computation shows that $\mathcal{J}(m)$ is a taming
of $\omega_{2n}$ for all $m\in M$. For the converse, assume that
$\mathcal{J}(m)\in\mathrm{GL}(2n,\mathbb{R})$ is a taming of $\omega_{2n}$ for
all $m\in M$ (we omit to indicate the evaluation at $m$ for ease of notation).
Let $\mathcal{E}=(e_{1},\ldots,e_{n},f_{1},\ldots,f_{n})$ the canonical basis
of $\mathbb{R}^{2n}$. The vectors $\mathcal{E}_{f}=(f_{1},\ldots,f_{n})$ form
a basis over $\mathbb{C}$ of the complex vector space
$(\mathbb{R}^{2n},\mathcal{J}(m))\simeq\mathbb{C}^{n}$, hence there exists a
unique map $\tau\in\mathcal{C}^{\infty}(M,\mathrm{Mat}(n,\mathbb{C}))$ which
satisfies:

$e_{a}=\tau(m)_{ab}\,f_{b}\,,\quad\forall\,\,a=1,\ldots,n\,,$ (73)

where we use Einstein summation over repeated indices. Thus:

$\delta_{ab}=\omega_{2n}(e_{a},f_{b})=\omega_{2n}(\tau(m)_{ac}\,f_{c},f_{b})=\mathrm{Im}(\tau(m))_{ac}\,\omega_{2n}(\mathcal{J}(m)f_{c},f_{b})~{}~{},$

which implies that $\mathrm{Im}(\tau(m))$ is symmetric and positive-definite.
Using the previous equation and compatibility of $\mathcal{J}(m)$ with
$\omega_{2n}$, we compute:

$0=\omega_{2n}(e_{a},e_{b})=\mathrm{Re}(\tau(m))_{ba}-\mathrm{Im}(\tau(m))_{bc}\,\omega_{2n}(\mathcal{J}(m)(e_{a}),f_{c})=\mathrm{Re}(\tau(m))_{ba}-\mathrm{Re}(\tau(m))_{ab}\,,$

which shows that $\mathrm{Re}(\tau(m))$ is symmetric. Hence the smooth map
$\mathcal{N}\in\mathcal{C}^{\infty}(M,\mathbb{SH}^{n})$ defined through
$\mathcal{N}={\mathcal{R}}+\mathbf{i}\mathcal{I}$, where:

${\mathcal{R}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Re}(\tau)~{}~{},\qquad\mathcal{I}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Im}(\tau)\,,$

is a period matrix map. Equation (73) gives:

$\displaystyle\mathcal{J}(m)(e_{a})={\mathcal{R}}(m)_{ab}\mathcal{I}^{-1}(m)_{bc}\,e_{c}-{\mathcal{R}}(m)_{ab}\mathcal{I}(m)^{-1}_{bc}{\mathcal{R}}(m)_{cd}\,f_{d}-\mathcal{I}(m)_{ad}\,f_{d}$
$\displaystyle\mathcal{J}(m)(f_{a})=\mathcal{I}(m)^{-1}_{ab}\,e_{b}-\mathcal{I}(m)^{-1}_{ab}{\mathcal{R}}(m)_{bc}\,f_{c}\,,$

which is equivalent to (72). ∎

######  Proposition A.8.

The map $\Theta\colon\mathrm{Per}_{n}(M)\to\mathfrak{J}_{n}(M)$ defined
through:

$\mathrm{Per}_{n}(M)\ni\mathcal{N}={\mathcal{R}}+\mathbf{i}\mathcal{I}\mapsto\Theta(\mathcal{N})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\begin{pmatrix}\mathcal{I}^{-1}{\mathcal{R}}&\mathcal{I}^{-1}\\\
-\mathcal{I}-{\mathcal{R}}\mathcal{I}^{-1}{\mathcal{R}}&-{\mathcal{R}}\mathcal{I}^{-1}\end{pmatrix}\in\mathfrak{J}_{n}(M)$

is a bijection between $\mathrm{Per}_{n}(M)$ and $\mathfrak{J}_{n}(M)$.

###### Proof.

Follows directly from the proof of Proposition A.7. The inverse of $\Theta$
takes a taming map $\mathcal{J}\in\mathfrak{J}_{n}(M)$ to the period matrix
$\Theta^{-1}(\mathcal{J})=\mathrm{Re}(\tau)+\mathbf{i}\,\mathrm{Im}(\tau)$,
where, for all $m\in M$, $\tau(m)$ is the complex symmetric matrix of size $n$
uniquely determined by the expansion $e_{a}=\tau_{ab}f_{b}$ of $e_{a}$ over
$\mathbb{C}$ when $\mathbb{R}^{2n}$ is endowed with the complex structure
$\mathcal{J}(m)$. ∎

Since $M$ is contractible, we have
$\Omega^{2}_{\mathrm{cl}}(M,\mathbb{R}^{2n})=\Omega^{2}_{\mathrm{ex}}(M,\mathbb{R}^{2n})$.
By the discussion above, this implies that local abelian gauge theory can be
formulated equivalently as a theory of closed $\mathbb{R}^{2n}$-valued two-
forms $\mathcal{V}\in\Omega^{2}_{\mathrm{cl}}(M,\mathbb{R}^{2n})$ satisfying
the condition:

$\ast_{g}\mathcal{V}=-\mathcal{J}\mathcal{V}$

with respect to a fixed taming map $\mathcal{J}\in\mathfrak{J}_{n}(M)$.
Consequently, the theory is uniquely determined by the choice of taming map.
The condition $\mathrm{d}\mathcal{V}=0$ is equivalent with
$\mathcal{V}=\mathrm{d}\mathcal{A}$, where
$\mathcal{A}\in\Omega^{1}(M,\mathbb{R}^{2n})$ is considered modulo gauge
transformations $\mathcal{A}\mapsto\mathcal{A}+\mathrm{d}\alpha$ with
$\alpha\in\mathcal{C}^{\infty}(M,\mathbb{R}^{2n})$. The map
$[\mathcal{A}]\mapsto\mathrm{d}\mathcal{A}$ gives a well-defined bijection
$\Omega^{1}(M,\mathbb{R}^{2n})/\Omega^{1}_{\mathrm{cl}}(M,\mathbb{R}^{2n})\xrightarrow{\sim}\Omega^{2}_{\mathrm{cl}}(M,\mathbb{R}^{2n})$.
Thus we can formulate classical local abelian gauge theory either in terms of
_electromagnetic gauge potentials_
$\mathcal{A}\in\Omega^{1}(M,\mathbb{R}^{2n})$ taken modulo gauge-equivalence
or in terms of electromagnetic field strengths
$\mathcal{V}\in\Omega^{2}_{\mathrm{cl}}(M,\mathbb{R}^{2n})$.

######  Definition A.9.

Let $\mathcal{J}\in\mathfrak{J}_{n}(M)$ be a taming map. The space of
electromagnetic gauge configurations of the $\mathrm{U}(1)^{n}$ local abelian
gauge is $\Omega^{1}(M,\mathbb{R}^{2n})$.

Two gauge configurations are called gauge equivalent if they differ by an
exact one-form. The theory is defined by the polarized self-duality condition
for $\mathcal{A}\in\Omega^{1}(M,\mathbb{R}^{2n})$:

$\ast_{g}\mathcal{V}_{\mathcal{A}}=-\mathcal{J}\mathcal{V}_{\mathcal{A}}~{}~{},~{}~{}\mathrm{where}\quad\mathcal{V}_{\mathcal{A}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{d}\mathcal{A}\,.$ (74)

The space of electromagnetic gauge fields (or electromagnetic gauge
potentials) of the theory is the linear subspace of
$\Omega^{1}(M,\mathbb{R}^{2n})$ consisting of those elements which satisfy
(74):

$\mathfrak{Sol}_{n}(M,g,\mathcal{J})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{\mathcal{A}\in\Omega^{1}(M,\mathbb{R}^{2n})\,\,|\,\,\ast_{g}\mathcal{V}_{\mathcal{A}}=-\mathcal{J}\mathcal{V}_{\mathcal{A}}\right\\}\,.$

Elements $\mathcal{A}\in\Omega^{1}(M,\mathbb{R}^{2n})$ are called
(electromagnetic) _gauge potentials_ or _gauge fields_. The space of field
strength configurations is the vector space:

$\mathrm{Conf}_{n}(M)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\Omega^{2}_{\mathrm{cl}}(M,\mathbb{R}^{2n})~{}~{},$

while the space of field strengths is defined through:

$\mathrm{Sol}_{n}(M,g,\mathcal{J})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{\mathcal{V}\in\mathrm{Conf}_{n}(M)~{}|~{}\ast_{g}\mathcal{V}=\mathcal{J}\mathcal{V}\\}~{}~{}.$

The map $[\mathcal{A}]\mapsto\mathrm{d}\mathcal{A}$ gives a bijection
$\Omega^{1}(M,\mathbb{R}^{2n})/\Omega^{1}_{\mathrm{cl}}(M,\mathbb{R}^{2n})\xrightarrow{\sim}\mathrm{Conf}_{n}(M)$,
which restricts to a bijection
$\mathfrak{Sol}_{n}(M,g,\mathcal{J})/\Omega^{1}_{\mathrm{cl}}(M,\mathbb{R}^{2n})\xrightarrow{\sim}\mathrm{Sol}_{n}(M,g,\mathcal{J})$.

###  A.2. Duality groups

Let $\mathrm{Diff}(M)$ be the group of orientation-preserving diffeomorphisms
of $M$ and $\mathcal{J}\in\mathfrak{J}_{n}(M)$ be a taming map of rank $2n$
defined on $M$. For
$(\gamma,f)\in\mathrm{GL}(2n,\mathbb{R})\times\mathrm{Diff}(M)$, consider the
linear isomorphism:

$\mathbb{A}_{\gamma,f}\colon\Omega^{k}(M,\mathbb{R}^{2n})\xrightarrow{\sim}\Omega^{k}(M,\mathbb{R}^{2n})\,,\quad\omega\mapsto\gamma(f_{\ast}\omega)\,,$
(75)

where
$f_{\ast}\colon\Omega^{k}(M,\mathbb{R}^{2n})\to\Omega^{k}(M,\mathbb{R}^{2n})$
is the push-forward through the diffeomorphism $f$. This gives a linear action
of $\mathrm{GL}(2n,\mathbb{R})\times\mathrm{Diff}(M)$ on
$\Omega^{k}(M,\mathbb{R}^{2n})$. Since this action commutes with the exterior
derivative, it preserves the space $\mathrm{Conf}_{n}(M)$ of field strength
configurations.

For any $\gamma\in\mathrm{Sp}(2n,\mathbb{R})$, the map:

$\mathcal{J}_{\gamma,f}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\gamma(\mathcal{J}\circ f^{-1})\gamma^{-1}\,,$

is a taming map. This gives an action $\mu$ of
$\mathrm{Sp}(2n,\mathbb{R})\times\mathrm{Diff}(M)$ on $\mathfrak{J}_{n}(M)$
defined through:

$\mu(\gamma,f)(\mathcal{J})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{J}_{\gamma,f}\,,\quad\forall\,\,(\gamma,f)\in\mathrm{Sp}(2n,\mathbb{R})\times\mathrm{Diff}(M)\,.$

######  Proposition A.10.

For every $(\gamma,f)\in\mathrm{Sp}(2n,\mathbb{R})\times\mathrm{Diff}(M)$, the
map $\mathbb{A}_{\gamma,f}$ induces by restriction a linear isomorphism:

$\mathbb{A}_{\gamma,f}\colon\mathrm{Sol}_{n}(M,g,\mathcal{J})\xrightarrow{\sim}\mathrm{Sol}_{n}(M,f_{\ast}(g),\mathcal{J}_{\gamma,f})~{}~{},$

where $f_{\ast}(g)$ denotes the push-forward of $g$ by $f\in\mathrm{Diff}(M)$.

######  Remark A.11.

If we consider a pair
$(\gamma,f)\in\mathrm{GL}(2n,\mathbb{R})\times\mathrm{Diff}(M)$ with
$\gamma\not\in\mathrm{Sp}(2n,\mathbb{R})$, then $\mathcal{J}_{\gamma,f}$ is
not a taming map, so it does not define a local abelian gauge theory. From a
different point of view, such a transformation would not preserve the energy
momentum tensor of the theory and its Lagrangian formulation. See [41] and
references therein for more details about this point.

###### Proof.

For any $\mathcal{V}\in\Omega^{2}_{\mathrm{cl}}(M,\mathbb{R}^{2n})$, we have:

$\ast_{g}\mathcal{V}=-\mathcal{J}\mathcal{V}\Longleftrightarrow\mathbb{A}_{\gamma,f}(\ast_{g}\mathcal{V})=-\mathbb{A}_{\gamma,f}(\mathcal{J}\mathcal{V})\Longleftrightarrow\ast_{f_{\ast}(g)}(\mathbb{A}_{\gamma,f}\mathcal{V})=-\mathcal{J}_{\gamma,f}\mathbb{A}_{\gamma,f}(\mathcal{V})~{}~{}.$

∎

Consider the infinite rank vector bundle with total space:

$\mathrm{Sol}_{n}(M)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\prod_{(g,\mathcal{J})\in\mathrm{Met}_{3,1}(M)\times\mathfrak{J}_{n}(M)}\mathrm{Sol}_{n}(M,g,\mathcal{J})~{}~{},$

and infinite-dimensional base $B_{n}(M)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Met}_{3,1}(M)\times\mathfrak{J}_{n}(M)$, with the natural
projection. Let $\sigma$ be the action of
$\mathrm{Sp}(2n,\mathbb{R})\times\mathrm{Diff}(M)$ on $B_{n}(M)$ defined
though $\sigma=f_{\ast}\times\mu$, i.e.:

$\sigma(\gamma,f)(g,\mathcal{J})=(f_{\ast}(g),\mathcal{J}_{\gamma,f})~{}~{}.$

Then Proposition A.10 shows that the restriction of $\mathbb{A}$ gives a
linearization of $\sigma$ on the vector bundle $\mathrm{Sol}_{n}(M)$. In
particular, each fiber of $\mathrm{Sol}_{n}(M)$ carries a linear
representation of the isotropy group of the corresponding point in the base.
Let ${\rm Iso}(M,g)$ be the group of orientation-preserving isometries of
$(M,g)$. Then:

$\mathrm{Stab}_{\mathrm{Sp}(2n,\mathbb{R})\times\mathrm{Diff}(M)}(g,\mathcal{J})=\\{(\gamma,f)\in\mathrm{Sp}(2n,\mathbb{R})\times{\rm
Iso}(M,g)~{}|~{}\mathcal{J}_{\gamma,f}=\mathcal{J}\\}$

and we have:

######  Corollary A.12.

Let $(\gamma,f)\in\mathrm{Sp}(2n,\mathbb{R})\times{\rm Iso}(M,g)$ such that
$\mathcal{J}_{\gamma,f}=\mathcal{J}$, i.e. $\mathcal{J}\circ
f=\gamma\mathcal{J}\gamma^{-1}$. Then $\mathbb{A}_{\gamma,f}$ is a linear
automorphism of $\mathrm{Sol}_{n}(M,g,\mathcal{J})$.

######  Definition A.13.

Let $\mathcal{J}\in\mathfrak{J}_{n}(M)$ be a taming map and $g$ be a
Lorentzian metric on $M$.

  * •

The group $\mathrm{Sp}(2n,\mathbb{R})\times\mathrm{Diff}(M)$ is called the
unbased pseudo-duality group. The linear isomorphism:

$\mathbb{A}_{\gamma,f}\colon\mathrm{Sol}_{n}(M,g,\mathcal{J})\xrightarrow{\sim}\mathrm{Sol}_{n}(M,f_{\ast}(g),\mathcal{J}_{\gamma,f})\,,$

induced by an element of this group is called a unbased pseudo-duality
transformation.

  * •

The group $\mathrm{Sp}(2n,\mathbb{R})\times{\rm Iso}(M,g)$ is called the
unbased duality group. The linear isomorphism:

$\mathbb{A}_{\gamma,f}\colon\mathrm{Sol}_{n}(M,g,\mathcal{J})\xrightarrow{\sim}\mathrm{Sol}_{n}(M,g,\mathcal{J}_{\gamma,f})\,,$

induced by an element $(\gamma,f)$ of this group is called a unbased duality
transformation.

  * •

The group $\mathrm{Sp}(2n,\mathbb{R})$ is called the duality group. The linear
isomorphism:

$\mathbb{A}_{\gamma,\mathrm{id}_{M}}\colon\mathrm{Sol}_{n}(M,g,\mathcal{J})\xrightarrow{\sim}\mathrm{Sol}_{n}(M,g,\mathcal{J}_{\gamma})\,,$

where $\mathcal{J}_{\gamma}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{J}_{\gamma,\mathrm{id}_{M}}=\gamma\mathcal{J}\gamma^{-1}$,
is called a classical duality transformation.

######  Definition A.14.

Let $\mathcal{J}\in\mathfrak{J}_{n}(M)$ be a taming map and $g$ be a
Lorentzian metric on $M$.

  * •

The stabilizer:

$\mathcal{U}(M,\mathcal{J})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(\gamma,f)\in\mathrm{Diff}(M,g)\times\mathrm{Sp}(2n,\mathbb{R})\,\,|~{}\mathcal{J}\circ
f=\gamma\mathcal{J}\gamma^{-1}\right\\}\,,$ (76)

of $\mathcal{J}$ in $\mathrm{Sp}(2n,\mathbb{R})\times\mathrm{Diff}(M)$ with
respect to the representation $\mu$ is called the unbased unitary pseudo-
duality group. The linear isomorphism:

$\mathbb{A}_{\gamma,f}\colon\mathrm{Sol}_{n}(M,g,\mathcal{J})\xrightarrow{\sim}\mathrm{Sol}_{n}(M,f_{\ast}(g),\mathcal{J})\,,$

induced by an element of this group is called an unbased unitary pseudo-
duality transformation.

  * •

The stabilizer:

$\mathcal{U}(M,g,\mathcal{J})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(\gamma,f)\in\mathrm{Sp}(2n,\mathbb{R})\times{\rm
Iso}(M,g)\,\,|~{}\mathcal{J}\circ f=\gamma\mathcal{J}\gamma^{-1}\right\\}\,,$
(77)

of $\mathcal{J}$ in $\mathrm{Sp}(2n,\mathbb{R})\times{\rm Iso}(M)$ with
respect to the representation $\mu$ is called the unbased unitary duality
group. The linear isomorphism:

$\mathbb{A}_{\gamma,f}\colon\mathrm{Sol}_{n}(M,g,\mathcal{J})\xrightarrow{\sim}\mathrm{Sol}_{n}(M,g,\mathcal{J})\,,$

induced by an element of this group is called an unbased unitary duality
transformation.

  * •

The stabilizer:

$\mathrm{U}_{\mathcal{J}}(n)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{\gamma\in\mathrm{Sp}(2n,\mathbb{R})\,\,|\,\,\gamma\mathcal{J}\gamma^{-1}=\mathcal{J}\right\\}\,.$

of $\mathcal{J}$ in $\mathrm{Sp}(2n,\mathbb{R})$ with respect to the action
$\mathcal{J}\rightarrow\gamma\mathcal{J}\gamma^{-1}$ is called the unitary
duality group. The linear automorphism:

$\mathbb{A}_{\gamma,f}\colon\mathrm{Sol}_{n}(M,g,\mathcal{J})\xrightarrow{\sim}\mathrm{Sol}_{n}(M,g,\mathcal{J})\,$

of $\mathrm{Sol}_{n}(M,g,\mathcal{J})$ induced by an element of this group is
called a unitary duality transformation.

We have inclusions:

$\mathrm{U}_{\mathcal{J}}(n)\subset\mathcal{U}(M,g,\mathcal{J})\subset\mathcal{U}(M,\mathcal{J})$

and short exact sequences:

$\displaystyle
1\to\mathrm{U}_{\mathcal{J}}(n)\to\mathcal{U}(M,g,\mathcal{J})\to{\rm
Iso}_{\mathcal{J}}(M,g)\to 1\,,$ (78) $\displaystyle 1\to{\rm
Iso}(M,g)_{\mathcal{J}}\to\mathcal{U}(M,\mathcal{J})\to\mathrm{Sp}_{\mathcal{J}}(2n,\mathbb{R})\to
1\,,$ (79)

where ${\rm Iso}_{\mathcal{J}}(M,g)$ is the subgroup of those $f\in{\rm
Iso}(M,g)$ for which there exists $\gamma\in\mathrm{Sp}(n,\mathbb{R})$ such
that $\mathcal{J}\circ f=\gamma\mathcal{J}\gamma^{-1}$, while
$\mathrm{Sp}_{\mathcal{J}}(2n,\mathbb{R})$ is the subgroup of those
$\gamma\in\mathrm{Sp}(2n,\mathbb{R})$ for which there exists $f\in{\rm
Iso}(M,g)$ such that $\mathcal{J}\circ f=\gamma\mathcal{J}\gamma^{-1}$.
Finally, the group:

${\rm Iso}(M,g)_{\mathcal{J}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{f\in{\rm Iso}(M,g)~{}|~{}\mathcal{J}\circ f=\mathcal{J}\\}$

is the stabilizer of $\mathcal{J}$ in ${\rm Iso}(M,g)$. In particular, we
have:

######  Corollary A.15.

If $\mathrm{U}_{\mathcal{J}}(n)=1$ then $\mathcal{U}(M,g,\mathcal{J})={\rm
Iso}_{\mathcal{J}}(M,g)$. If ${\rm Iso}(M,g)_{\mathcal{J}}=1$ then
$\mathcal{U}(M,g,\mathcal{J})=\mathrm{Sp}_{\mathcal{J}}(2n,\mathbb{R})$.

###  A.3. Gluing local abelian gauge theories

Let now $M$ be an arbitrary oriented manifold admitting Lorentzian metrics.
Let $\mathcal{U}=(U_{\alpha})_{\alpha\in I}$ be a good open cover of $M$,
where $I$ is an index set. Denote by $g_{\alpha}$ the restriction of $g$ to
$U_{\alpha}$. Roughly speaking, the definition of abelian gauge theory on
$(M,g)$ given in Section 1 is the result of gluing the local
$\mathrm{U}(1)^{n}$ abelian gauge theories defined on the contractible
Lorentzian four-manifolds $(U_{\alpha},g_{\alpha})$ using electromagnetic
dualities. In order to implement this idea, we choose a locally constant
$\mathrm{Sp}(2n,\mathbb{R})$-valued C̆ech cocycle for $\mathcal{U}$:

$u_{\alpha\beta}\colon U_{\alpha}\cap U_{\beta}\to\mathrm{Sp}(2n,\mathbb{R})$

and a family of taming maps $\mathcal{J}_{\alpha}\colon
U_{\alpha}\to\mathbb{R}^{2n}$ for $\omega_{2n}$ such that:

$\mathcal{J}_{\beta}=u_{\alpha\beta}\,\mathcal{J}_{\alpha}u^{-1}_{\alpha\beta}$

on double overlaps. The collection:

$\left\\{(U_{\alpha})_{\alpha\in I},(g_{\alpha})_{\alpha\in
I},(\mathcal{J}_{\alpha})_{\alpha\in I},(u_{\alpha\beta})_{\alpha,\beta\in
I}\right\\}\,,$

is equivalent to a flat symplectic vector bundle
$(\mathcal{S},\omega,\mathcal{D})$ with symplectic form $\omega$ and
symplectic flat connection $\mathcal{D}$, equipped with an almost complex
structure $\mathcal{J}$ which is a taming of $\omega$. A family
$(\mathcal{V}_{\alpha})_{\alpha\in I}$ of solutions of the local abelian gauge
theories defined by $(\mathcal{J}_{\alpha})_{\alpha\in I}$ on
$(U_{\alpha},g_{\alpha})$ which satisfies:

$\mathcal{V}_{\beta}=u_{\alpha\beta}\mathcal{V}_{\alpha}$

corresponds to an $\mathcal{S}$-valued two-form
$\mathcal{V}\in\Omega^{2}(M,\mathcal{S})$ which obeys:

$\mathrm{d}_{\mathcal{D}}\mathcal{V}=0\,,$

where
$\mathrm{d}_{\mathcal{D}}\colon\Omega^{\ast}(M,\mathcal{S})\to\Omega^{\ast}(M,\mathcal{S})$
is the exterior differential twisted by $\mathcal{D}$. This construction
motivates the global geometric model introduced in [32] and further elaborated
in Section 1.

##  Appendix B Integral symplectic spaces and integral symplectic tori

This appendix recalls some notions from the theory of symplectic lattices and
symplectic tori which are used throughout the paper. We also introduce the
notion of integral symplectic torus.

######  Definition B.1.

An integral symplectic space is a triple $(V,\omega,\Lambda)$ such that:

  * •

$(V,\omega)$ is a finite-dimensional symplectic vector space over
$\mathbb{R}$.

  * •

$\Lambda\subset V$ is full lattice in $V$, i.e. a lattice in $V$ such that
$V=\Lambda\otimes_{\mathbb{Z}}\mathbb{R}$.

  * •

$\omega$ is integral with respect to $\Lambda$, i.e. we have
$\omega(\Lambda,\Lambda)\subset\mathbb{Z}$.

An isomorphism of integral symplectic spaces
$f\colon(V_{1},\omega_{1},\Lambda_{1})\to(V_{2},\omega_{2},\Lambda_{2})$ is a
bijective symplectomorphism from $(V_{1},\omega_{1})$ to $(V_{2},\omega_{2})$
which satisfies:

$f(\Lambda_{1})=\Lambda_{2}\,.$

Denote by $\mathrm{Symp}_{\mathbb{Z}}$ the groupoid of integral symplectic
spaces and isomorphisms of such. Let $\mathrm{Aut}(V)$ be the group of linear
automorphisms of the vector space $V$ and
$\mathrm{Sp}(V,\omega)\subset\mathrm{Aut}(V)$ be the subgroup of symplectic
transformations. Then the automorphism group of the integral symplectic space
$(V,\omega,\Lambda)$ is denoted by:

$\mathrm{Sp}(V,\omega,\Lambda)=\left\\{T\in\mathrm{Sp}(V,\omega)\,\,|\,\,T(\Lambda)=\Lambda\right\\}\,.$

######  Definition B.2.

An _integral symplectic basis_ of a $2n$-dimensional integral symplectic space
$(V,\omega,\Lambda)$ is a basis
$\mathcal{E}=(\xi_{1},\ldots,\xi_{n},\zeta_{1},\ldots,\zeta_{n})$ of $\Lambda$
(as a free $\mathbb{Z}$-module) such that:

$\omega(\xi_{i},\xi_{j})=\omega(\zeta_{i},\zeta_{j})=0\,,\qquad\omega(\xi_{i},\zeta_{j})=t_{i}\delta_{ij}~{}~{},~{}~{}\omega(\zeta_{i},\xi_{j})=-t_{i}\delta_{ij}\,,\qquad\forall\,\,i,j=1,\ldots,n\,,$

where $t_{1},\ldots,t_{n}\in\mathbb{Z}$ are strictly positive integers
satisfying the divisibility conditions:

$t_{1}|t_{2}|\ldots|t_{n}\,.$

By the _elementary divisor theorem_ , see [19, Chapter VI], every integral
symplectic space admits an integral symplectic basis and the positive integers
$t_{1},\ldots,t_{n}$ (which are called the elementary divisors of
$(V,\omega,\Lambda)$) do not depend on the choice of such a basis. Define:

$\mathrm{Div}^{n}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(t_{1},\ldots,t_{n})\in\mathbb{Z}_{>0}^{n}\,\,|\,\,t_{1}|t_{2}|\ldots|t_{n}\right\\}\,,$

and:

$\delta(n)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(1,\ldots,1)\in\mathrm{Div}^{n}\,.$

Let $\leq$ be the partial order relation on $\mathrm{Div}^{n}$ defined
through:

$(t_{1},\ldots,t_{n})\leq(t^{\prime}_{1},\ldots,t^{\prime}_{n})~{}~{}\mathrm{iff}~{}~{}t_{i}|t^{\prime}_{i}~{}~{}\forall
i=1,\ldots,n~{}~{}.$

Then $\delta(n)$ is the least element of the ordered set
$(\mathrm{Div}^{n},\leq)$. Notice that this ordered set is directed, since any
two elements $t,t^{\prime}\in\mathrm{Div}^{n}$ have an upper bound given by
$(t_{1}t^{\prime}_{1},\ldots,t_{n}t^{\prime}_{n})$. In fact,
$(\mathrm{Div}^{n},\leq)$ is a lattice with join and meet given by:

$t\vee
t^{\prime}=(\mathrm{lcm}(t_{1},t^{\prime}_{1}),\ldots,\mathrm{lcm}(t_{n},t^{\prime}_{n}))~{}~{},~{}~{}t\wedge
t^{\prime}=(\mathrm{gcd}(t_{1},t^{\prime}_{1}),\ldots,\mathrm{gcd}(t_{n},t^{\prime}_{n}))~{}~{}.$

This lattice is semi-bounded from below with bottom element given by
$\delta(n)$ and it is complete for meets (i.e., it is a complete meet semi-
lattice).

######  Definition B.3.

The _type_ of an integral symplectic space $(V,\omega,\Lambda)$ is the ordered
system of elementary divisors of $(V,\omega,\Lambda)$, which we denote by:

$\mathfrak{t}(V,\omega,\Lambda)=(t_{1},\ldots,t_{n})\in\mathrm{Div}^{n}\,.$

The integral symplectic space $(V,\omega,\Lambda)$ is called _principal_ if:

$\mathfrak{t}(V,\omega,\Lambda)=\delta(n)\in\mathrm{Div}^{n}\,.$

Let $\omega_{2n}$ denotes the standard symplectic pairing on
$\mathbb{R}^{2n}$.

######  Proposition B.4.

Two integral symplectic spaces have the same type if and only if they are
isomorphic. Moreover, every element of $\mathrm{Div}^{n}$ is the type of an
integral symplectic space. Hence the type induces a bijection between the set
of isomorphism classes of integral symplectic spaces and the set
$\mathrm{Div}^{n}$.

###### Proof.

The first statement is obvious. For the second statement, fix
$\mathfrak{t}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(t_{1},\ldots,t_{n})\in\mathrm{Div}^{n}$. Consider the full
lattice $\Lambda_{\mathfrak{t}}\subseteq\mathbb{R}^{2n}$ defined as follows:

$\Lambda_{\mathfrak{t}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(l_{1},\ldots,l_{n},t_{1}l_{n+1},\ldots,t_{n}l_{n+1})\,\,|\,\,l_{1},\ldots,l_{n}\in\mathbb{Z}\right\\}\,.$
(80)

Then $(\mathbb{R}^{2n},\omega_{2n},\Lambda_{\mathfrak{t}})$ is an integral
symplectic space of type $\mathfrak{t}$. ∎

######  Definition B.5.

The lattice $\Lambda_{\mathfrak{t}}$ defined in (80) is called the _standard
symplectic lattice_ of type $\mathfrak{t}$ and
$(\mathbb{R}^{2n},\omega_{2n},\Lambda_{\mathfrak{t}})$ is called the standard
integral symplectic space of type $\mathfrak{t}$.

We have $\Lambda_{\delta(n)}=\mathbb{Z}^{2n}$. Moreover,
$\Lambda_{\mathfrak{t}}$ is a sub-lattice of $\mathbb{Z}^{2n}$ and we have
$\mathbb{Z}^{2n}/\Lambda_{\mathfrak{t}}\simeq\mathbb{Z}_{t_{1}}\times\ldots\times\mathbb{Z}_{t_{n}}$
for all $\mathfrak{t}\in\mathrm{Div}(n)$. For
$\mathfrak{t},\mathfrak{t}^{\prime}\in\mathrm{Div}^{n}$, we have
$\Lambda_{\mathfrak{t}^{\prime}}\subset\Lambda_{\mathfrak{t}}$ if and only if
$\mathfrak{t}\leq\mathfrak{t}^{\prime}$. The lattice $\Lambda_{\mathfrak{t}}$
admits the basis:

$\displaystyle\xi_{1}=e_{1}=(1,0,\ldots,0),\ldots,\xi_{n}=e_{n}=(0,\ldots,0,1,0,\ldots,0)$
$\displaystyle\zeta_{1}=t_{1}f_{1}=(0,\ldots,0,t_{1},0,\ldots,0),\ldots,\zeta_{n}=t_{n}f_{n}=(0,\ldots,0,t_{n})~{}~{},$

in which the standard symplectic form of $\mathbb{R}^{2n}$ has coefficients:

$\displaystyle\omega_{2n}(\xi_{i},\xi_{j})=\omega_{2n}(\zeta_{i},\zeta_{j})=0~{}~{}$
$\displaystyle\omega_{2n}(\xi_{i},\zeta_{j})=t_{i}\delta_{ij}~{}~{},~{}~{}\omega_{2n}(\zeta_{i},\xi_{j})=-t_{i}\delta_{ij}~{}~{}.$

The isomorphism which takes $\xi_{i}$ to $e_{i}$ and $\zeta_{j}$ to $f_{j}$
identifies $(\mathbb{R}^{2n},\omega_{2n},\Lambda_{\mathfrak{t}})$ with the
integral symplectic space
$(\mathbb{R}^{2n},\omega_{\mathfrak{t}},\mathbb{Z}^{2n})$, where
$\omega_{\mathfrak{t}}$ is the symplectic pairing defined on $\mathbb{R}^{2n}$
by:

$\omega_{\mathfrak{t}}(e_{i},e_{j})=\omega_{\mathfrak{t}}(f_{i},f_{j})=0\,,\quad\omega_{\mathfrak{t}}(e_{i},f_{j})=\delta_{ij}\,,\quad\omega_{\mathfrak{t}}(f_{i},e_{j})=-\delta_{ij}\,,\quad\forall\,\,i=1,\ldots,n~{}~{}.$
(81)

Given $\mathfrak{t}=(t_{1},\ldots,t_{n})\in\mathrm{Div}^{n}$, consider the
diagonal $n\times n$ matrix:

$\mathrm{D}_{\mathfrak{t}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{diag}(t_{1},\ldots,t_{n})\in\mathrm{Mat}(n,\mathbb{Z})\,,$

as well as:

$\Gamma_{\mathfrak{t}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\begin{pmatrix}\mathrm{I}_{n}&0\\\
0&\mathrm{D}_{\mathfrak{t}}\end{pmatrix}\in\mathrm{Mat}(2n,\mathbb{Z})\,.$

######  Definition B.6.

The modified Siegel modular group of type $\mathfrak{t}\in\mathrm{Div}^{n}$ is
the subgroup of
$\mathrm{Aut}(\mathbb{R}^{2n},\omega_{2n})\simeq\mathrm{Sp}(2n,\mathbb{R})$
defined through:

$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{T\in\mathrm{Aut}(\mathbb{R}^{2n},\omega_{2n})\,\,|\,\,T(\Lambda_{\mathfrak{t}})=\Lambda_{\mathfrak{t}}\right\\}\simeq\left\\{T\in\mathrm{Sp}(2n,\mathbb{R})\,\,|\,\,\Gamma_{\mathfrak{t}}T\Gamma_{\mathfrak{t}}^{-1}=T\right\\}\,.$

Since
$(\mathbb{R}^{2n},\omega_{2n},\Lambda_{\mathfrak{t}})\simeq(\mathbb{R}^{2n},\omega_{\mathfrak{t}},\mathbb{Z}^{2n})$,
we have
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\simeq\mathrm{Aut}(\mathbb{R}^{2n},\omega_{\mathfrak{t}},\mathbb{Z}^{2n})$.
Hence $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ is a subgroup of
$\mathrm{GL}(2n,\mathbb{Z})$. The remarks above give:

######  Proposition B.7.

[32, Proposition F.12] Let $(V,\omega,\Lambda)$ be an integral symplectic
space of dimension $2n$. Any integral symplectic basis of this space induces
an isomorphism of integral symplectic spaces between $(V,\omega,\Lambda)$ and
$(\mathbb{R}^{2n},\omega_{2n},\Lambda_{\mathfrak{t}})$ as well as an
isomorphism of groups between $\mathrm{Sp}(V,\omega,\Lambda)$ and
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$.

We have $\mathrm{Sp}_{\delta(n)}(2n,\mathbb{Z})=\mathrm{Sp}(2n,\mathbb{Z})$
and
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\subseteq\mathrm{Sp}_{\mathfrak{t}^{\prime}}(2n,\mathbb{Z})$
when $\mathfrak{t}\leq\mathfrak{t}^{\prime}$. Hence
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ forms a direct system of groups
and we have
$\mathrm{Sp}(2n,\mathbb{Z})\subset\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
for all $\mathfrak{t}\in\mathrm{Div}^{n}$. The direct limit
$\varinjlim_{\mathfrak{t}\in\mathrm{Div}^{n}}\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
identifies with the following subgroup of $\mathrm{Sp}(2n,\mathbb{R})$:

$\mathrm{Sp}_{\infty}(2n,\mathbb{Z})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{T\in\mathrm{GL}(2n,\mathbb{R})\,\,|\,\,\exists\,\,\mathfrak{t}_{T}\in\mathrm{Div}^{n}:T\in\mathrm{Sp}_{\mathfrak{t}_{T}}(2n,\mathbb{Z})\\}\,,$

through the isomorphism of groups
$\varphi:\mathrm{Sp}_{\infty}(2n,\mathbb{Z})\rightarrow\varinjlim_{\mathfrak{t}\in\mathrm{Div}^{n}}\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
which sends $T\in\mathrm{Sp}_{\infty}(2n,\mathbb{Z})$ to the equivalence class
$[\alpha(T)]\in\varinjlim_{\mathfrak{t}\in\mathrm{Div}^{n}}\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
of the family
$\alpha(T)\in\sqcup_{\mathfrak{t}\in\mathrm{Div}^{n}}\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
defined through:

$\alpha(T)_{\mathfrak{t}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{\begin{array}[]{ll}T&\mbox{ if
}\mathfrak{t}_{T}\leq\mathfrak{t}\\\ 1&\mbox{ if
}\mathfrak{t}_{T}\nleq\mathfrak{t}\end{array}\right.\,.$

Notice that
$\mathrm{Sp}(2n,\mathbb{Z})=\mathrm{Sp}_{\delta(n)}(2n,\mathbb{Z})$ is a
subgroup of $\mathrm{Sp}_{\infty}(2n,\mathbb{Z})$.

######  Definition B.8.

The type of an element $T\in\mathrm{Sp}_{\infty}(2n,\mathbb{Z})$ is defined as
the greatest lower bound $\mathfrak{t}(T)\in\mathrm{Div}^{n}$ of the finite
set $\\{\mathfrak{t}\in\mathrm{Div}^{n}|\mathfrak{t}|\mathfrak{t}_{T}\\}$,
where $\mathfrak{t}_{T}$ is any element of $\mathrm{Div}^{n}$ such that
$T\in\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$.

Notice that the type of $T$ is well-defined and that we have
$T\in\mathrm{Sp}_{\mathfrak{t}(T)}(2n,\mathbb{Z})$.

###  B.1. Integral symplectic tori

The following definition distinguishes between a few closely related notions.

######  Definition B.9.

A $d$-dimensional torus is a smooth manifold $\mathrm{T}$ diffeomorphic with
the standard $d$-torus $\mathrm{T}^{d}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\mathrm{S}^{1})^{d}$. A $d$-dimensional torus group is a compact
abelian Lie group $A$ which is isomorphic with the standard $d$-dimensional
torus group $\mathrm{U}(1)^{d}$ as a Lie group. A $d$-dimensional affine torus
is a principal homogeneous space $\mathbb{A}$ for a $d$-dimensional torus
group. The standard affine $d$-torus is the $d$-dimensional affine torus
$\mathbb{A}_{d}$ defined by the right action of $\mathrm{U}(1)^{d}$ on itself.

The underlying manifold of the standard $d$-dimensional torus group is the
standard $d$-torus while the underlying manifold of a $d$-dimensional torus
group is a $d$-torus. Moreover, any $d$-dimensional affine torus group is
isomorphic with a standard affine $d$-torus. The transformations of an affine
torus given the right action of its underlying group will be called
translations. Choosing a distinguished point in any affine torus makes it into
a torus group having that point as zero element. Conversely, any torus group
defines an affine torus obtained by ‘forgetting’ its zero element. The
singular homology and cohomology groups of a $d$-torus $\mathrm{T}$ are the
free abelian groups given by: