# The geometry and DSZ quantization of four-dimensional supergravity

C. I. Lazaroiu  Center for Geometry and Physics, Institute for Basic Science,
Pohang, Republic of Korea<EMAIL_ADDRESS>and  C. S. Shahbazi  Department of
Mathematics, University of Hamburg, Germany<EMAIL_ADDRESS>
###### Abstract.

We develop the Dirac-Schwinger-Zwanziger (DSZ) quantization of four-
dimensional bosonic ungauged supergravity on an oriented four-manifold $M$ of
arbitrary topology and use it to obtain its manifestly duality-covariant
gauge-theoretic geometric formulation. Classical bosonic supergravity is
completely determined by a submersion $\pi$ over $M$ equipped with a complete
Ehresmann connection, a vertical euclidean metric and a vertically-polarized
flat symplectic vector bundle $\Xi$. We implement the Dirac-Schwinger-
Zwanziger quantization condition in the aforementioned classical supergravity
through the choice of an element in the degree-two sheaf cohomology group with
coefficients in a locally constant sheaf $\mathcal{L}\subset\Xi$ valued in the
groupoid of integral symplectic spaces. We show that this data determines a
Siegel principal bundle $P_{\mathfrak{t}}$ of fixed type
$\mathfrak{t}\in\mathbb{Z}$ whose connections provide the global geometric
description of the electromagnetic guage potentials of the theory. The Maxwell
gauge equations of the theory reduce to the polarized self-duality condition
determined by $\Xi$ on the connections of $P_{\mathfrak{t}}$. We investigate
the continuous and discrete U-duality groups of the theory, characterizing
them through short exact sequences and realizing the latter through the gauge
group of $P_{\mathfrak{t}}$ acting on its adjoint bundle. This elucidates the
geometric origin of U-duality, which we explore in several examples,
illustrating its dependence on the topology of the fiber bundles $\pi$ and
$P_{\mathfrak{t}}$ as well as on the isomorphism type of $\mathcal{L}$.

###### Key words and phrases:

Mathematical supergravity, abelian gauge theory, electromagnetic duality,
symplectic vector bundles

2010 MSC. Primary: 53C80. Secondary: 83E50.

##  1\. Introduction

The goal of this article is to construct the geometric, gauge-theoretic and
duality-covariant global formulation of the universal bosonic sector of four-
dimensional (ungauged) supergravity and study its U-duality group. In order to
do so, we develop the Dirac-Schwinger-Zwanziger (DSZ) quantization [15, 32,
35] of the gauge sector of the theory and interpret the result geometrically.

The local formulation of classical four-dimensional supergravity theories has
been studied intensively in the physics literature, see for instance the
seminal references [2, 3, 7, 8, 10, 11, 13, 14] and the reviews and books [4,
18, 20, 29, 31]. Such theories share a universal bosonic sector, which is
subject to increasingly stringent constraints according to the number of
supersymmetry generators of the theory. The global classical geometric
formulation of the universal bosonic sector of supergravity was obtained in
[24, 25], see also [28] for a mathematically rigorous approach to supergravity
based on supergeometry. In [24, 25] it was found that the local structure of
supergravity does not suffice to determine the theory on spacetimes which are
not simply-connected. As show in loc. cit., the global formulation of the
classical universal bosonic sector of ungauged supergravity on an oriented
four-manifold $M$ is a Generalized Einstein-Section-Maxwell theory, which is
determined by the following data:

  * •

A _scalar bundle_ $(\pi,\mathcal{H},\mathcal{G})$, where $\pi\colon X\to M$ is
a submersion equipped with a complete flat Ehresmann connection $\mathcal{H}$
and a Euclidean vertical metric $\mathcal{G}$, i.e. a Euclidean metric on the
vertical bundle of $\pi$ which is invariant under the parallel transport of
$\mathcal{H}$.

  * •

A _duality bundle_ $\Delta=(\mathcal{S},\omega,\mathcal{D})$ i.e. a flat
symplectic vector bundle over the total space $X$ of $\pi$.

  * •

A vertical polarization $\mathcal{J}$ on $\Delta$, i.e. a taming of
$(\mathcal{S},\omega)$ which is invariant under the extended parallel
transport induced by $\mathcal{H}$ and $\mathcal{D}$.

The configuration space of the bosonic supergravity theory determined by a
tuple $(\pi,\mathcal{H},\mathcal{G},\Delta,\mathcal{J})$ as introduced above
is the set of triples $(g,s,\mathcal{F})$, where $g$ is a Lorentzian metric,
$s$ is a global section of $\pi$ and
$\mathcal{F}\in\Omega^{2}(M,\mathcal{S}^{s})$ is a field strength, that is, a
two-form on $M$ taking values in the pull-back $\mathcal{S}^{s}$ of
$\mathcal{S}$ by $s$ which is flat with respect to the pullback connection
$\mathcal{D}^{s}$. The classical equations of motion of the theory were given
in terms of global geometric structures in [25]. In particular, the Maxwell
equations (i.e. the equations of motion for $\mathcal{F}$) correspond to the
polarized self-duality condition determined by the taming $\mathcal{J}$.

The description of the gauge sector in terms of field strengths $\mathcal{F}$
is unsatisfactory when coupling the theory to quantized charged particles.
Indeed, the Aharonov-Bohm effect [1] implies that this sector should admit a
global description in terms of gauge potentials, which are expected to be
modeled by connections $\mathcal{A}$ on an appropriate principal bundle $P$
defined on $M$. To determine this bundle, we develop the geometric formulation
of the Dirac-Schwinger-Zwanziger (DSZ) quantization condition of the gauge
sector. We then show that this condition implies that $P$ is a Siegel bundle
in the sense of [26], i.e. a principal bundle whose structure group is the
automorphism group of an integral symplectic affine torus. The latter is
isomorphic to a certain semidirect product of an even-dimensional torus group
with a modified Siegel modular group. This process parallels the DSZ
quantization of Abelian gauge theories with manifest electromagnetic duality,
developed in [26], which depends on a Siegel system $Z$ on $X$. The latter was
defined in [26] as a local system of finitely-generated free Abelian groups
whose structure group reduces to a modified Siegel modular group and which is
isomorphic to $\Delta$ upon tensorization with $\mathbb{R}$ over $\mathbb{Z}$.
We define a classical configuration $(g,s,\mathcal{F})$ to be integral if the
cohomology class of $\mathcal{F}$ with respect to the de Rham differential
twisted by $\mathcal{D}^{s}$ belongs to the second cohomology group of $M$
with coefficients in $Z^{s}$, the pull-back of $Z$ by $s$. By the results of
[26], any element of $H^{2}(M,Z)$ is the twisted Chern class of a Siegel
bundle defined on the total space of $\pi$. Using this fact, we show that the
DSZ quantization of classical bosonic supergravity is determined by the
following data:

  * •

A _scalar bundle_ $(\pi,\mathcal{H},\mathcal{G})$.

  * •

A Siegel bundle $P_{\mathfrak{t}}$ (of type $\mathfrak{t}$) defined on the
total space $X$ of $\pi$.

  * •

A vertical polarization $\mathcal{J}$ on the adjoint bundle
$\mathrm{ad}(P_{\mathfrak{t}})$ of $P_{\mathfrak{t}}$.

The configuration space of the DSZ quantization of bosonic supergravity
determined by $(\pi,\mathcal{H},\mathcal{G})$ and
$(P_{\mathfrak{t}},\mathcal{J})$ is given by triples $(g,s,\mathcal{A})$,
where $g$ and $s$ are as defined above and $\mathcal{A}$ is a connection on
$P^{s}_{\mathfrak{t}}$, which describes both the electric and magnetic
potentials of the theory. The Maxwell equations become a first-order condition
on the connections of $P_{\mathfrak{t}}$ which depends on the Hodge operator
of $g$ and the polarization $\mathcal{J}$. In particular, we obtain the global
and duality-covariant equations of motion of the universal bosonic sector of
four-dimensional ungauged supergravity on $(P_{\mathfrak{t}},\mathcal{J})$ in
terms of the variables $(g,s,\mathcal{A})$.

Using this geometric and gauge-theoretic formulation of bosonic supergravity,
we study its group of continuous and discrete U-duality transformations, which
we characterize through short exact sequences involving the group of
automorphisms of $P_{\mathfrak{t}}$ and its adjoint bundle. In general, these
groups can differ markedly from their local counterparts considered in the
physics literature [22]. In this regard, we emphasize the dependence of
U-duality groups on the _type_ 111This is a classical notion in the theory of
symplectic lattices see for instance [12] or [26, Appendix B] for more
details. of the corresponding Siegel modular group and of the isomorphism
class of the Siegel system $Z$, a point which does not appear to have been
noticed in the supergravity literature. In particular, the explicit
computation of the discrete U-duality groups becomes a hard arithmetic problem
in the theory of automorphisms of local systems. As an application of the
framework that we develop, we show that group of discrete electromagnetic
duality transformations is the _discrete remnant_ of the unbased group of
automorphisms of $P_{\mathfrak{t}}$. This clarifies the geometric origin of
U-duality in supergravity.

The geometric formulation described in this paper provides the basis of the
mathematical framework necessary to investigate the differential-geometric
problems arising in four-dimensional supergravity. In particular, it allows
for a mathematically rigorous formulation of the geometric constraints imposed
by supersymmetry through the corresponding Killing spinor equations, in the
spirit of [9], thus opening the way for developing the mathematical theory of
supergravity supersymmetric solutions and moduli spaces of such.

######  Acknowledgements.

We thank Vicente Cortés and Tomás Ortín for useful comments and discussions.
The work of C. I. L. was supported by grant IBS-R003-S1. The work of C.S.S. is
supported by the Germany Excellence Strategy _Quantum Universe_ \- 390833306.

##  2\. Classical bosonic supergravity

In this section we recall the construction of generalized Einstein-Section-
Maxwell theories on an oriented four-manifold $M$ given in [24, 25]. These
give the global geometric formulation of the universal bosonic sector of four-
dimensional supergravity (also called _classical geometric bosonic
supergravity_ or classical bosonic supergravity for short, see [9]).

###  2.1. Preparations

Let $M$ be an oriented and connected four-manifold. We start by introducing
the geometric data needed to formulate classical bosonic supergravity on $M$,
a detailed account of which was given in [25].

######  Definition 2.1.

A _scalar bundle_ of rank $n_{s}$ on $M$ is a triple
$(\pi,\mathcal{H},\mathcal{G})$ consisting of:

  * •

A smooth submersion $\pi\colon X\to M$, where $X$ is a connected and oriented
differentiable manifold of dimension $n_{s}+4$.

  * •

A complete Ehresmann connection $\mathcal{H}\subset TX$ on $\pi$.

  * •

A vertical Euclidean metric, i.e. a Euclidean metric $\mathcal{G}$ defined on
the vertical bundle $\mathcal{V}\subset TX$ of $\pi$ which is preserved by the
parallel transport of $\mathcal{H}$. Recall that the vertical bundle
$\mathcal{V}\subset TX$ is defined as the kernel of the differential map
$\mathrm{d}\pi\colon TX\to TM$.

We say that a scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ is _flat_ if
$\mathcal{H}$ is Frobenius integrable.

In the following we let $\mathcal{O}_{m}$ denote the restriction to the fiber
$X_{m}$ of $\pi$ at $m\in M$ of any geometric structure $\mathcal{O}$ defined
on $X$.

######  Remark 2.2.

Since the Ehresmann connection $\mathcal{H}$ of a scalar bundle is complete
and its parallel transport preserves the vertical metric $\mathcal{G}$, the
fibers $(X_{m},\mathcal{G}_{m})$ of $\pi$ are isomorphic to each other as
Riemannian manifolds. By the results of [17], it follows that $\pi$ is a fiber
bundle associated to a principal bundle with structure group given by the
isometry group of $(X_{m},\mathcal{G}_{m})$. Notice that $X_{m}$ is typically
non-compact in physics applications.

######  Definition 2.3.

A scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ is:

  * •

_topologically trivial_ if $\pi$ is topologically trivial as fiber bundle,
i.e. $X$ is diffeomorphic with $M\times\mathcal{M}$ for some manifold
$\mathcal{M}$ and $\pi$ identifies with the projection
$\mathrm{pr}:M\times\mathcal{M}\rightarrow M$ on the first factor.

  * •

_holonomy-trivial_ if the holonomy of $\mathcal{H}$ is trivial.

######  Remark 2.4.

Every holonomy-trivial scalar bundle is topologically trivial. Moreover, its
Ehresmann connection identifies with the pull-back of $TM$ through the
projection $\mathrm{pr}:M\times\mathcal{M}\rightarrow M$.

Let $\mathcal{P}(M)$ and $\mathcal{P}(X)$ be the sets of piece-wise smooth
paths in $M$ and $X$ defined on the unit interval. Given a scalar bundle
$(\pi,\mathcal{H},\mathcal{G})$, let $T$ be the parallel transport defined by
$\mathcal{H}$, which associates to a path $\gamma\in\mathcal{P}(M)$ the
diffeomorphism:

$T_{\gamma}\colon X_{\gamma(0)}\xrightarrow{\sim}X_{\gamma(1)}\,,$

obtained by parallel transport along $\mathcal{H}$.

######  Definition 2.5.

A _duality bundle_ $\Delta=(\mathcal{S},\omega,\mathcal{D})$ over $\pi$ is a
triple $(\mathcal{S},\omega,\mathcal{D})$ where $\mathcal{S}$ is a vector
bundle on $X$, $\omega$ is a symplectic structure on $\mathcal{S}$ and
$\mathcal{D}$ is a flat connection on $\mathcal{S}$ preserving $\omega$. We
denote the rank of $\mathcal{S}$ by $2n_{v}$.

######  Remark 2.6.

As shown in [25], a duality bundle of rank $2n_{v}$ corresponds locally to a
supergravity theory coupled to $n_{v}$ _vector multiplets_.

Let $(\pi,\mathcal{H},\mathcal{G})$ be a scalar bundle over $M$ and
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ be a duality bundle on $\pi$, which
we shall also call a duality bundle over $(\pi,\mathcal{H},\mathcal{G})$. For
any $m\in M$, let $(\mathcal{S}_{m},\omega_{m},\mathcal{D}_{m})$ be the
restriction of $(\mathcal{S},\omega,\mathcal{D})$ to the fiber
$X_{m}=\pi^{-1}(m)$. This is a flat symplectic vector bundle on $X_{m}$ and
hence a duality structure on the latter as defined in [24]. For any path
$\Gamma\in\mathcal{P}(X)$ in the total space $X$ of $\pi$, we denote by
$\mathfrak{U}_{\Gamma}:\mathcal{S}_{\Gamma(0)}\rightarrow\mathcal{S}_{\Gamma(1)}$
the parallel transport of $\mathcal{D}$ along $\Gamma$. Since $\mathcal{D}$ is
a symplectic connection, $\mathfrak{U}_{\Gamma}$ is a symplectomorphism
between the symplectic vector spaces
$(\mathcal{S}_{\Gamma(0)},\omega_{\Gamma(0)})$ and
$(\mathcal{S}_{\Gamma(1)},\omega_{\Gamma(1)})$. For any
$\gamma\in\mathcal{P}(M)$, let $\bar{\gamma}_{x}\in\mathcal{P}(X)$ be the
horizontal lift of $\gamma$ starting at the point $x\in X_{\gamma(0)}$. By the
definition of $T$, we have $\bar{\gamma}_{x}(1)=T_{\gamma}(x)$.

######  Definition 2.7.

The extended horizontal transport along a path $\gamma\in\mathcal{P}(M)$ is
the unbased isomorphism of flat symplectic vector bundles
$\mathcal{T}_{\gamma}:\mathcal{S}_{\gamma(0)}\rightarrow\mathcal{S}_{\gamma(1)}$
defined by:

$\mathcal{T}_{\gamma}(x)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathfrak{U}_{\bar{\gamma}_{x}}\colon\mathcal{S}_{x}\rightarrow\mathcal{S}_{T_{\gamma}(x)}\,,\quad\forall\,x\in
X_{\gamma(0)}\,,$

which linearizes the Ehresmann transport $T_{\gamma}:X_{\gamma(0)}\rightarrow
X_{\gamma(1)}$ along $\gamma$.

Given a duality bundle $\Delta=(\mathcal{S},\omega,\mathcal{D})$, a
(compatible) _taming_ $\mathcal{J}\in\mathrm{Aut}_{b}(\mathcal{S})$ on
$\Delta$ is a complex structure on $\mathcal{S}$ which tames the symplectic
pairing $\omega$, i.e. it satisfies the compatibility condition:

$\omega(\mathcal{J}\xi_{1},\mathcal{J}\xi_{2})=\omega(\xi_{1},\xi_{2})\,,\qquad\forall\,\,(\xi_{1},\xi_{2})\in\mathcal{S}\times_{X}\mathcal{S},,$

and the positivity condition:

$\omega(\mathcal{J}\xi,\xi)>0\,,\qquad\forall\,\,\xi\in\dot{\mathcal{S}}~{}~{},$

where $\dot{\mathcal{S}}$ is the complement of the image of the zero section
in $\mathcal{S}$.

######  Definition 2.8.

A taming on the duality bundle $\Delta$ over $(\pi,\mathcal{H},\mathcal{G})$
is called _vertical_ if it is preserved by the extended horizontal transport
$\mathcal{T}$, i.e. if
$\mathcal{T}_{\gamma}\colon(\mathcal{S}_{\gamma(0)},\omega_{\gamma(0)},\mathcal{J}_{\gamma(0)})\rightarrow(\mathcal{S}_{\gamma(1)},\omega_{\gamma(1)},\mathcal{J}_{\gamma(1)})$
an isomorphism of tamed symplectic vector bundles for all
$\gamma\in\mathcal{P}(M)$.

######  Remark 2.9.

As explained in [24], a vertical taming on $\Delta$ is equivalent to a
_positive_ Lagrangian sub-bundle of the complexification of
$(\mathcal{S},\omega)$ which is preserved by the complexified extended
horizontal transport.

As in [24, 25], we refer to the pair $\Xi=(\Delta,\mathcal{J})$ consisting of
a duality bundle $\Delta$ defined on $(\pi,\mathcal{H},\mathcal{G})$ and a
vertical taming $\mathcal{J}$ as an _electromagnetic bundle_ on
$(\pi,\mathcal{H},\mathcal{G})$. We will refer to a choice scalar bundle
$(\pi,\mathcal{H},\mathcal{G})$ together with a choice of electromagnetic
bundle $\Xi$ as a scalar-electromagnetic bundle $\Phi$, that is:

$\Phi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\pi,\mathcal{H},\mathcal{G},\Xi)\,.$

As shown in [24, 25], the universal bosonic sector of supergravity defined on
$M$ is determined by the choice of a scalar-electromagnetic bundle. Morphisms
of duality and electromagnetic bundles are defined in the natural way (see
[25]). Note that standard bundle theory implies that isomorphism classes of
duality bundles over a fixed submersion $\pi:X\rightarrow M$ are in one to one
correspondence with the character variety:

$\mathfrak{M}_{d}(X)\stackrel{{\scriptstyle{\rm def.}}}{{=}}{\rm
Hom}(\pi_{1}(X),\mathrm{Sp}(2n_{v},\mathbb{R}))/\mathrm{Sp}(2n_{v},\mathbb{R})\,.$

######  Remark 2.10.

In general, the character variety above has positive dimension, giving a
moduli space of inequivalent duality bundles. This implies [25] that one can
construct an uncountable infinity of globally inequivalent bosonic geometric
supergravities which are however all locally equivalent.

######  Remark 2.11.

A duality bundle $\Delta=(\mathcal{S},\omega,\mathcal{D})$ is called
topologically trivial if the vector bundle $\mathcal{S}$ is trivial, i.e. if
it admits a global frame. It is called _symplectically trivial_ if the
$(\mathcal{S},\omega)\in\Delta$ is symplectically trivial, i.e. if
$\mathcal{S}$ admits a global symplectic frame. Finally, we say that $\Delta$
is _holonomy trivial_ if the holonomy of $\mathcal{D}$ is the trivial group.
Holonomy-triviality implies symplectic triviality, which in turn implies
topological triviality. If $X$ is simply connected then every duality bundle
is holonomy trivial.

Smooth sections of the submersion $\pi:X\rightarrow M$ are called _scalar
sections_. For every scalar section $s\colon M\to X$ we use a superscript $s$
to denote the bundle pull-back by $s$ and the subscript $s$ to denote push-
forward by $s$ in the appropriate category. For instance,
$\Delta^{s}=(\mathcal{S}^{s},\omega^{s},\mathcal{D}^{s})$ denotes the bundle
pull-back of $\Delta=(\mathcal{S},\omega,\mathcal{S})$ by $s$, which is a flat
symplectic vector bundle over $M$. Similarly,
$\Xi^{s}=(\Delta^{s},\mathcal{J}^{s})$ denotes the pull-back of
$\Xi=(\Delta,\mathcal{J})$ by $s$, which is an electromagnetic structure on
$M$ in the sense of [24]. Let $\Phi$ be a scalar-electromagnetic bundle on
$M$. For every Lorentzian metric $g$ on $M$ and every scalar section
$s\in\Gamma(\pi)$, consider the isomorphism of vector bundles:

$\star_{g,\mathcal{J}^{s}}\colon\wedge
T^{\ast}M\otimes\mathcal{S}^{s}\to\wedge T^{\ast}M\otimes\mathcal{S}^{s}\,,$

defined through $\star_{g,\mathcal{J}^{s}}=\ast_{g}\otimes\mathcal{J}^{s}$.
Since both $\ast_{g}$ and $\mathcal{J}^{s}$ square to minus the identity, this
restricts to an involutive automorphism:

$\star_{g,\mathcal{J}^{s}}\colon\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s}\to\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s}$

which gives a direct sum decomposition into eigenbundles corresponding to the
eigenvalues $+1$ and $-1$:

$\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s}=(\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s})_{+}\oplus(\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s})_{-}\,.$

Here the subscript denotes the sign of the corresponding eigenvalue of
$\star_{g,\mathcal{J}^{s}}$. The spaces of smooth global sections of these
sub-bundles are denoted by $\Omega^{2}_{\pm}(M,\mathcal{S}^{s})$ and their
elements are called polarized (anti)-selfdual $\mathcal{S}^{s}$-valued 2-forms
with respect to $\mathcal{J}^{s}$. We have:

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

The flat symplectic connection $\mathcal{D}^{s}$ of $\Delta^{s}$ defines an
exterior covariant derivative acting on $\mathcal{S}^{s}$-valued forms defined
on $M$, which we denote by:

$\mathrm{d}_{\mathcal{D}^{s}}\colon\Omega(M,\mathcal{S}^{s})\to\Omega(M,\mathcal{S}^{s})\,.$

This operator squares to zero since $\mathcal{D}^{s}$ is flat. We denote its
cohomology groups by $H^{k}(M,\Delta^{s})$ and the corresponding total
cohomology by $H(M,\Delta^{s})$. For every scalar section $s\in\Gamma(\pi)$,
we denote by $\mathfrak{G}^{s}_{\Delta}$ the sheaf of flat sections of
$\Delta^{s}$, defined as follows:

$\mathfrak{G}^{s}_{\Delta}(U)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{\xi\in\Gamma(U,\mathcal{S}^{s})\,\,|\,\,\mathcal{D}^{s}\xi=0\right\\}~{}~{},$

for any open set $U\subset M$. This is a locally-constant sheaf of symplectic
vector spaces of rank $2n_{v}$, whose stalk is isomorphic to the typical fiber
of $\Delta$. Since the sheaf of smooth $\mathcal{S}^{s}$-valued forms is
acyclic, there exists a natural isomorphism of graded vector spaces:

$H(M,\Delta^{s})\simeq H(M,\mathfrak{G}^{s}_{\Delta})\,,$

where $H(M,\mathfrak{G}^{s}_{\Delta})$ is the sheaf cohomology of
$\mathfrak{G}^{s}_{\Delta}$. Note that the definition of an electromagnetic
bundle $\Xi=(\Delta,\mathcal{J})$ does not require $\mathcal{D}\in\Delta$ to
be compatible with $\mathcal{J}$, a fact which is crucial for recovering the
correct local description of bosonic geometric supergravity. The failure of
$\mathcal{D}$ to be compatible with $\mathcal{J}$ is measured by the
_fundamental form_ of an electromagnetic bundle.

######  Definition 2.12.

Let $\Phi$ be an scalar-electromagnetic bundle. The fundamental form $\Psi$ of
$\Xi$ is the following $End(\mathcal{S})$-valued one-form defined on $X$:

$\Psi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{D}\mathcal{J}\in\Omega^{1}(X,End(\mathcal{S}))\,.$

######  Remark 2.13.

For every $v\in\Gamma(TX)$, the endomorphism
$\Psi(v)\in\mathrm{End}(\mathcal{S})=\Gamma(End(\mathcal{S}))$ is an
$\mathcal{J}$-antilinear $Q$-symmetric, where the scalar product $Q$ is the
Euclidean metric induced by $\omega$ and $\mathcal{J}$ on $\mathcal{S}$ as
follows:

$Q(\xi_{1},\xi_{2})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\omega(\xi_{1},\mathcal{J}\xi_{2})~{}~{}\forall(\xi_{1},\xi_{2})\in\mathcal{S}\times_{X}\mathcal{S}\,,$

See [24, 25] for more details.

######  Definition 2.14.

An electromagnetic bundle $\Xi$ is called unitary if $\Psi=0$.

To describe the universal bosonic sector of 4d supergravity, we introduce
three natural operations which are determined by a choice of a scalar-
electromagnetic bundle $\Phi$, a Lorentzian metric $g$ on $M$ and a scalar
section $s\in\Gamma(\pi)$.

######  Definition 2.15.

The twisted exterior pairing $(\cdot,\cdot)_{g,Q^{s}}$ is the unique pseudo-
Euclidean scalar product on $\wedge T^{\ast}M\otimes\mathcal{S}^{s}$ which
satisfies:

$(\rho_{1}\otimes\xi^{s}_{1},\rho_{2}\otimes\xi^{s}_{2})_{g,Q^{s}}=(\rho_{1},\rho_{2})_{g}Q^{s}(\xi^{s}_{1},\xi^{s}_{2})=(\rho_{1},\rho_{2})_{g}Q^{s}(\xi_{1},\xi_{2})$

for all $\rho_{1},\rho_{2}\in\Omega(M)$ and all
$\xi_{1},\xi_{2}\in\Gamma(\mathcal{S}^{s})$.

Given any vector bundle $W$ on $M$, we extend this trivially to a $W$-valued
pairing (which for simplicity we denote by the same symbol) between the
bundles $W\otimes\wedge T^{\ast}M\otimes\mathcal{S}^{s}$ and $\wedge
T^{\ast}M\otimes\mathcal{S}^{s}$. Thus:

$(w\otimes\eta_{1},\eta_{2})_{g,Q^{s}}=w\otimes(\eta_{1},\eta_{2})_{g,Q^{s}}\,,\quad\forall\,\,w\in\Gamma(W)\,,\quad\forall\,\,\eta_{1},\eta_{2}\in\Omega(M,\mathcal{S}^{s})~{}~{}.$

######  Definition 2.16.

The inner $g$-contraction of (2,0)-tensors is the bundle morphism
$\oslash_{g}:(\otimes^{2}T^{\ast}M)^{\otimes
2}\rightarrow\otimes^{2}T^{\ast}M$ uniquely determined by the condition:

$(\alpha_{1}\otimes\alpha_{2})\oslash_{g}(\alpha_{3}\otimes\alpha_{4})=(\alpha_{2},\alpha_{4})_{g}\alpha_{1}\otimes\alpha_{3}\,,\quad\forall\,\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in
T^{\ast}M\,.$

We define the _inner $g$-contraction of two-forms_ to be the restriction of
$\oslash_{g}$ to
$\wedge^{2}T^{\ast}M\otimes\wedge^{2}T^{\ast}M\subset(\otimes^{2}T^{\ast}M)^{\otimes
2}$.

######  Definition 2.17.

The twisted inner contraction of $\mathcal{S}^{s}$-valued two-forms is the
unique morphism of vector bundles:

$\oslash_{Q^{s}}\colon\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s}\times_{M}\wedge^{2}T^{\ast}M\otimes\mathcal{S}^{s}\rightarrow\otimes^{2}(T^{\ast}M)$

which satisfies:

$(\rho_{1}\otimes s_{1})\oslash_{Q^{s}}(\rho_{2}\otimes
s_{2})=Q^{s}(s_{1},s_{2})\rho_{1}\oslash_{g}\rho_{2}\,,$

for all $\rho_{1},\rho_{2}\in\Omega^{2}(M)$ and all
$s_{1},s_{2}\in\Gamma(\mathcal{S}^{s})$.

###  2.2. The configuration space and equations of motion

We are ready to give the geometric formulation of the universal bosonic sector
of 4d classical supergravity, whose global solutions can be interpreted as
locally geometric classical supergravity U-folds [25].

######  Definition 2.18.

Let $\Phi=(\pi,\mathcal{H},\mathcal{G},\mathcal{S},\omega,\mathcal{D})$ be a
scalar-electromagnetic bundle on an oriented four-manifold $M$. The
configuration space of the universal bosonic sector determined by $(M,\Phi)$
is the set:

$\mathrm{Conf}(\Phi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(g,s,\mathcal{F})\,\,|\,\,g\in\mathrm{Lor}(M)\,,\,\,s\in\Gamma(\pi)\,,\,\,\mathcal{F}\in\Omega^{2}_{\mathrm{d}_{\mathcal{D}^{s}}\\!\mbox{-}\mathrm{cl}}(M,\mathcal{S}^{s})\right\\}\,,$

where $\mathrm{Lor}(M)$ denotes the set of Lorentzian metrics on $M$ and
$\Omega^{2}_{\mathrm{d}_{\mathcal{D}^{s}}\\!\mbox{-}\mathrm{cl}}(M,\mathcal{S}^{s})$
denotes the set of $\mathcal{D}^{s}$-closed 2-forms on $M$ valued in
$\mathcal{S}^{s}$.

######  Remark 2.19.

In general, the isomorphism class of $\mathcal{S}^{s}$ depends on the scalar
section $s\in\Gamma(\pi)$.

Given a scalar bundle $(\pi,\mathcal{H},\mathcal{G})$, the complete Ehresmann
connection $\mathcal{H}$ can be described equivalently by a one-form
$\mathcal{C}\in\Omega^{1}(X,\mathcal{V})={\rm Hom}(TX,\mathcal{V})$ which
satisfies the condition $\mathcal{C}\circ\mathcal{C}=\mathcal{C}$. Thus
$\mathcal{C}\colon TX\to\mathcal{V}$ is a projection of the tangent bundle of
$X$ onto $\mathcal{V}$. The horizontal distribution $\mathcal{H}$ is recovered
as the kernel of $\mathcal{C}$. Given
$(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi)$, we define the vertical first
fundamental form $(s^{\ast}_{\mathcal{C}}\mathcal{G})\in\Gamma(T^{\ast}M\odot
T^{\ast}M)$ through [33, 34]:

$(s^{\ast}_{\mathcal{C}}\mathcal{G})(v_{1},v_{2})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{G}(\mathcal{C}\circ\mathrm{d}s(v_{1}),\mathcal{C}\circ\mathrm{d}s(v_{2}))\,,\quad(v_{1},v_{2})\in
TM\times_{M}TM\,;$

which depends explicitly on $\mathcal{C}$ or, equivalently, $\mathcal{H}$. The
trace of this tensor with respect to $g$ is called the _vertical tension_ of
the section $s\in\Gamma(\pi)$. For ease of notation, we define
$\mathrm{d}^{\mathcal{C}}s(v)=\mathcal{C}\circ\mathrm{d}s(v)\in\mathcal{V}$,
where $v\in TM$. Notice that
$\mathrm{d}^{\mathcal{C}}s\in\Omega^{1}(M,\mathcal{V}^{s})$. Denote by
$\mathrm{Lor}(X)$ the set of Lorentzian metrics on $X$. Every Lorentzian
metric $g$ on $M$ can be lifted to $\mathcal{H}$ using the isomorphism of
vector bundles
$(\mathrm{d}\pi|_{\mathcal{H}})\colon\mathcal{H}\xrightarrow{\sim}TM$ given by
the restriction of $\mathrm{d}\pi$ to $\mathcal{H}$. Thus there exists a
natural map (see [25]):

$h\colon\mathrm{Conf}(\Phi)\to\mathrm{Lor}(X)\,,\quad(g,s,\mathcal{F})\mapsto
h(g)\stackrel{{\scriptstyle{\rm def.}}}{{=}}\pi^{\ast}g+\mathcal{G}\,,$

where $h(g)$ is written using the direct sum decomposition
$TX=\mathcal{H}\oplus\mathcal{V}$. The Lorentzian metric $h(g)$ enters the
equations of motion of bosonic supergravity, as explained below. Note that,
equipped with the lifted metric $h_{g}$, $\pi\colon(X,h_{g})\to(M,g)$ becomes
a Lorentzian submersion. Given $(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi)$, we
denote by $\nabla^{h(g)}$ the Levi-Civita connection defined by $h(g)$ on $X$.
For every scalar section $s\colon M\to X$, we denote by $\nabla^{\Phi(g,s)}$
the connection on $TM\otimes\mathcal{V}^{s}$ given by the tensor product of
the Levi-Civita connection $\nabla^{g}$ of $g$ with the the vertical
projection of the pull-back by $s$ of the connection $\nabla^{h(g)}$. We then
have:

$\nabla^{\Phi(g,s)}\mathrm{d}^{\mathcal{C}}s\in\Gamma(T^{\ast}M\otimes
T^{\ast}M\otimes\mathcal{V}^{s})\,,$

as explained in [33, 34].

######  Definition 2.20.

Let $\Phi$ be a scalar-electromagnetic bundle on $M$. The universal bosonic
sector defined by $\Phi$ on $M$ is described by following system of partial
differential equations for triples $(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi)$:

  * •

The Einstein equations:

$\mathrm{Ric}^{g}-\frac{g}{2}\mathrm{R}^{g}=\frac{1}{2}\mathrm{Tr}_{g}(s^{\ast}_{\mathcal{C}}\mathcal{G})\,g-s^{\ast}_{\mathcal{C}}\mathcal{G}+2\mathcal{F}\oslash_{Q^{s}}\mathcal{F}\,,$
(1)

where $\mathrm{Ric}^{g}$ and $\mathrm{R}^{g}$ are respectively the Ricci
tensor and Ricci scalar of $g$, while $\mathrm{Tr}_{g}$ denotes trace with
respect to $g$.

  * •

The scalar equations:

$\nabla^{\Phi(g,s)}\mathrm{d}^{\mathcal{C}}s=\frac{1}{2}(\ast\mathcal{F},\Psi^{s}\mathcal{F})_{g,Q^{s}}\,.$
(2)

  * •

The Maxwell equations:

$\star_{g,\mathcal{J}^{s}}\mathcal{F}=\mathcal{F}\,.$ (3)

We denote by $\mathrm{Sol}(\Phi)\subset\mathrm{Conf}(\Phi)$ the set of
solutions to these equations.

######  Remark 2.21.

The configuration space $\mathrm{Conf}(\Phi)$ is formulated using the _field
strength_ two-forms instead of the appropriate notion of gauge potential, as
required by the Aharonov-Bohm effect [1]. The latter suggests that the gauge
potentials of the theory should be described by connections on an appropriate
principal bundle. To identify this bundle, we must impose an appropriate DSZ
quantization condition on the field strength $\mathcal{F}$. We consider this
condition and its geometric interpretation in Section 3.

######  Remark 2.22.

The fact that the formulation given above reduces locally to the usual
formulas of local bosonic supergravity found in the physics literature was
proved in detail in references [24, 25], to which we refer the reader for
further details. It is not known if this theory can be supersymmetrized when
$\mathcal{H}$ is not flat, although the Killing spinor equations can be
formulated exactly as in the case when $\mathcal{H}$ is flat.

###  2.3. The classical U-duality group

In this section we characterize the _global_ U-duality group of the bosonic
supergravity associated to a fixed scalar electromagnetic bundle
$\Phi=(\pi,\mathcal{H},\mathcal{G},\Xi)$. Given a duality bundle
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ let $\mathrm{Aut}(\mathcal{S})$
denote the group of all _unbased_ automorphisms of the vector bundle
$\mathcal{S}$. Let $f_{u}\in\mathrm{Diff}(X)$ be the diffeomorphism covered by
$u\in\mathrm{Aut}(\mathcal{S})$. Moreover, let $\mathrm{Aut}(\Delta)$ be the
group of those unbased automorphisms of $\mathcal{S}$ which preserve both
$\omega$ and $\mathcal{D}$:

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

Let $\mathrm{Aut}_{\pi}(\Delta)$ be the subgroup consisting of all elements of
$\mathrm{Aut}(\Delta)$ which cover based automorphisms of the fiber bundle
$\pi$:

$\mathrm{Aut}_{\pi}(\Delta)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}(\Delta)\,\,|\,\,f_{u}\in\mathrm{Aut}_{b}(\pi)\right\\}=\left\\{u\in\mathrm{Aut}(\Delta)\,\,|\,\,\pi\circ
f_{u}=\pi\right\\}\,.$

We have a short exact sequence of groups:

$1\to\mathrm{Aut}_{b}(\Delta)\to\mathrm{Aut}_{\pi}(\Delta)\to\mathrm{Aut}_{b}^{0}(\pi)\to
1\,,$ (4)

where $\mathrm{Aut}_{b}^{0}(\pi)\subset\mathrm{Aut}_{b}(\pi)$ is the subgroup
of those automorphisms of $\pi$ which are covered by elements of
$\mathrm{Aut}(\Delta)$.

Given a scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ and an element
$u\in\mathrm{Aut}_{\pi}(\Delta)$, the fiber bundle automorphism
$f_{u}\in\mathrm{Aut}_{b}(\pi)$ covered by $u$ acts as a gauge transformation
on $\mathcal{H}$ through push-forward
$\mathcal{H}_{u}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(f_{u})_{\ast}\mathcal{H}$. Similarly, since $f_{u}$ is an
automorphism of $\pi$ covering the identity, the push-forward of $\mathcal{G}$
by $f_{u}$ defines a new vertical Riemannian metric
$\mathcal{G}_{u}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(f_{u})_{\ast}\mathcal{G}$ on $\pi$ such that
$(X_{m},\mathcal{G}_{m})$ is isometric to $(X_{m},(\mathcal{G}_{u})_{m})$ for
all $m\in M$. Given an electromagnetic bundle $\Xi=(\Delta,\mathcal{J})$,
push-forward by $f_{u}$ produces another electromagnetic bundle which we
denote by $(\Delta_{u},\mathcal{J}_{u})$. Given a scalar-electromagnetic
bundle $\Phi=(\pi,\mathcal{H},\mathcal{G},\Delta,\mathcal{J})$, the system:

$\Phi_{u}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\pi,\mathcal{H}_{u},\mathcal{G}_{u},\Delta_{u},\mathcal{J}_{u})$

is a scalar-electromagnetic bundle with the same underlying submersion
$\pi\colon X\to M$. If $\mathcal{C}\in\Omega^{1}(X,\mathcal{V})$ is the
connection one-form associated to $\mathcal{H}$ then the natural push-forward
$f_{u\ast}\mathcal{C}\in\Omega^{1}(X,\mathcal{V})$ is the connection one-form
associated to $\mathcal{H}_{u}$.

######  Remark 2.23.

Since elements of $\mathrm{Aut}(\mathcal{S})$ may cover non-trivial
diffeomorphisms of $X$, the pull-back or push-forward operations must be dealt
with care (see [24]). Explicitly, define the following action of
$\mathrm{Aut}(\mathcal{S})$ on sections of $\mathcal{S}$:

$u\cdot\xi=u\circ\xi\circ f_{u}^{-1}\colon M\to\mathcal{S}\,,\qquad
u\in\mathrm{Aut}(\mathcal{S})\,,\qquad\xi\in\Gamma(\mathcal{S})\,.$

This gives an isomorphism of real vector spaces
$u\colon\Gamma(\mathcal{S})\to\Gamma(\mathcal{S})$ for every element
$u\in\mathrm{Aut}(\mathcal{S})$. We have $\omega^{u}=\omega$ if and only if:

$(\omega^{u})(\xi_{1},\xi_{2})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\omega(u\cdot\xi_{1},u\cdot\xi_{2})\circ
f_{u}=\omega(\xi_{1},\xi_{2})\,,\qquad\forall\,\,\xi_{1},\xi_{2}\in\Gamma(\mathcal{S})\,.$

Likewise, we have $\mathcal{D}^{u}=\mathcal{D}$ if and only if:

$\mathcal{D}^{u}_{v}(\xi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}u^{-1}\cdot\mathcal{D}_{f_{u\ast}\cdot
v}(u\cdot\xi)=\mathcal{D}_{v}(\xi)\,,\quad\forall\,\,\xi\in\Gamma(\mathcal{S})\,,\quad\forall\,\,v\in\Gamma(TX)\,,$

where $f_{u\ast}\cdot v=\mathrm{d}f_{u}(v)\circ f^{-1}_{u}$ and
$\mathrm{d}f_{u}\colon TX\to TX$ is the ordinary differential of
$f_{u}\in\mathrm{Diff}(X)$. Recall that if $v\in\Gamma(TX)$ then
$\mathrm{d}f_{u}(v)$ is not a vector field on $X$ but a section of $TX$ along
$f_{u}$, whereas $f_{u\ast}\cdot v\in\Gamma(TX)$ is again a vector field on
$X$. To illustrate the inner workings of the pull-backed connection $D^{u}$ we
verify that it satisfies the Leibniz identity:

$\displaystyle\mathcal{D}^{u}_{v}(\kappa\,\xi)=u^{-1}\cdot\mathcal{D}_{f_{u\ast}\cdot
v}(u\cdot(\kappa\,\xi))=u^{-1}\cdot\mathcal{D}_{f_{u\ast}\cdot v}((\kappa\circ
f_{u}^{-1})\,u\cdot\xi)=u^{-1}\cdot(\mathrm{d}(\kappa\circ
f_{u}^{-1})(f_{u\ast}\cdot v)\,u\cdot\xi)$
$\displaystyle+u^{-1}\cdot(\kappa\circ f_{u}^{-1}\,\mathcal{D}_{f_{u\ast}\cdot
v}(u\cdot\xi))=u^{-1}\cdot(\mathrm{d}\kappa(v\circ
f_{u}^{-1})\,u\cdot\xi)+u^{-1}\cdot(\kappa\circ
f_{u}^{-1}\,\mathcal{D}_{v}(u\cdot\xi))=\mathrm{d}\kappa(v)\,\xi+\kappa\,\mathcal{D}^{u}_{v}(\xi)\,,$

where $\kappa\in C^{\infty}(X)$ is a function on $X$. On the other hand, the
push-forward of $\mathcal{J}$ by $u\in\mathrm{Aut}(\mathcal{S})$ is given by:

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

for every $\xi\in\Gamma(\mathcal{S})$.

Given a duality bundle $\Delta$ over the scalar bundle
$(\pi,\mathcal{H},\mathcal{G})$, every element $u\in\mathrm{Aut}(\Delta)$ maps
a triplet of the form:

$(g,s,\mathcal{F})\in\mathrm{Lor}(M)\times\Gamma(\pi)\times\Omega^{2}(M,\mathcal{S}^{s})\,,$

to a triplet of the form:

$\mathbb{A}_{u}(g,s,\mathcal{F})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(g,f_{u}\circ
s,u\cdot\mathcal{F})\in\mathrm{Lor}(M)\times\Gamma(\pi)\times\Omega^{2}(M,\mathcal{S}^{f_{u}(s)})\,,$

where _dot_ denotes the natural action of $\mathrm{Aut}(\mathcal{S})$ on
$\mathcal{S}^{s}$-valued forms.

######  Remark 2.24.

Recall that
$\mathcal{F}_{m}\in\wedge^{2}T^{\ast}_{m}M\otimes\mathcal{S}^{s}_{m}$ or,
equivalently:

$\mathcal{F}_{m}\in\wedge^{2}T^{\ast}_{m}M\otimes\mathcal{S}_{s(m)}\,.$

The push-forward $u\cdot\mathcal{F}\in\Omega^{2}(M,\mathcal{S}^{f_{u}(s)})$ of
$\mathcal{F}\in\Omega^{2}(M,\mathcal{S}^{s})$ by $u$ produces a
$\mathcal{S}^{f_{u}(s)}$-valued two-form on $M$ whose value at $m\in M$ is
given by:

$(u\cdot\mathcal{F})_{m}=u_{s(m)}(\mathcal{F}_{m})\in\wedge^{2}T^{\ast}_{m}M\otimes\mathcal{S}_{f_{u}(s(m))}\,,$

where $u_{s(m)}\colon\mathcal{S}_{s(m)}\to\mathcal{S}_{f_{u}(s(m))}$ acts
trivially on the two-form components of $\mathcal{F}$.

Given $u\in\mathrm{Aut}(\Delta)$, the map $\mathbb{A}_{u}$ defined above need
not preserve the configuration space $\mathrm{Conf}(\Phi)$ defined by a fixed
scalar-electromagnetic bundle $\Phi=(\pi,\mathcal{H},\mathcal{G},\Xi)$.
Instead we have the following result.

######  Theorem 2.25.

Let $\pi\colon X\to M$ be a smooth submersion. For every connection
$\mathcal{H}$, vertical metric $\mathcal{G}$ and electromagnetic bundle $\Xi$
on $\pi$, an element $u\in\mathrm{Aut}_{\pi}(\Delta)$ defines a bijection:

$\mathbb{A}_{u}\colon\mathrm{Conf}(\Phi)\xrightarrow{\sim}\mathrm{Conf}(\Phi_{u})\,,\quad(g,s,\mathcal{F})\mapsto(g,f_{u}\circ
s,u\cdot\mathcal{F})\,,$

which restricts to a bijection:

$\mathbb{A}_{u}\colon\mathrm{Sol}(\Phi)\xrightarrow{\sim}\mathrm{Sol}(\Phi_{u})\,,$

between the solution spaces of the bosonic supergravities associated to $\Phi$
and $\Phi_{u}$ on $(\pi,\mathcal{H},\mathcal{G})$.

###### Proof.

Assume that $(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi)$, where
$\Phi=(\pi,\mathcal{H},\mathcal{G},\Delta,\mathcal{J})$ and
$u\in\mathrm{Aut}_{\pi}(\Delta)$ covers $f_{u}\in\mathrm{Aut}_{b}(X)$.
Clearly, $f_{u}\circ s\colon M\to X$ is again a section of $\pi$ since
$f_{u}\colon X\to X$ is covers the identity over $M$. On the other hand,
$u\cdot\mathcal{F}$ is by construction a two-form on $M$ taking values in
$\mathcal{S}^{f_{u}(s)}$ whence $(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi_{u})$.
The fact that this map takes solutions to solutions follows by a computation
that involves several different pull-backs through unbased automorphisms of
fiber bundles. The reader is referred to [24, Appendix D] for a detailed
account of the operations involved. Assume that
$(g,s,\mathcal{F})\in\mathrm{Sol}(\Phi)$. For the Einstein equation (1), we
compute:

$s^{\ast}_{\mathcal{C}}\mathcal{G}=(f_{u}^{-1}\circ f_{u}\circ
s)^{\ast}_{\mathcal{C}}\mathcal{G}=(f_{u}\circ
s)^{\ast}((f_{u}^{-1})^{\ast}_{\mathcal{C}}\mathcal{G})=(f_{u}\circ
s)^{\ast}_{f_{u\ast}\mathcal{C}}\mathcal{G}\,.$ (5)

On the other hand, we have:

$\mathcal{F}\oslash_{Q^{s}}\mathcal{F}=(u^{-1}\cdot
u\cdot\mathcal{F})\oslash_{Q^{s}}(u^{-1}\cdot
u\cdot\mathcal{F})=(u\cdot\mathcal{F})\oslash_{Q^{f_{u}(s)}_{u}}(u\cdot\mathcal{F})\,,$
(6)

where $Q^{f_{u}(s)}_{u}$ denotes the bilinear form on $\mathcal{S}^{s}$
defined as follows:

$Q^{f_{u}(s)}_{u}(\xi^{f_{u}(s)},\xi^{f_{u}(s)})=\omega(\xi(f_{u}(s)),\mathcal{J}(f_{u}(s))\xi(f_{u}(s)))\,,$

and where $\xi^{f_{u}(s)}\in\mathcal{S}^{f_{u}(s)}$ for every
$\xi\in\mathcal{S}$. Combining equations (5) and (6) together with the fact
that the left hand side of Equation (1) is invariant under $u$ we obtain that
$(g,f_{u}\circ s,u\cdot\mathcal{F})$ satisfies the Einstein equations with
respect to the scalar-electromagnetic structure
$(\pi,\mathcal{G},\mathcal{H}_{u},\Delta_{u},\mathcal{J}_{u})$. For the scalar
equation (2), we compute:

$\nabla^{\Phi(g,s)}\mathrm{d}^{\mathcal{C}}s=\nabla^{\Phi(g,s)}\mathrm{d}^{\mathcal{C}}(f_{u}^{-1}\circ
f_{u}\circ
s)=\nabla^{\Phi_{u}(g_{u},f_{u}(s))}\mathrm{d}^{f_{u\ast}\mathcal{C}}(f_{u}\circ
s)\,.$

Similarly, using the fact that $u\in\mathrm{Aut}(\Delta)$ preserves the flat
connection $\mathcal{D}$ determined by $\Delta$ together with equation (6), we
obtain:

$(\ast\mathcal{F},\Psi^{s}\mathcal{F})_{g,Q^{s}}=(\ast(u\cdot\mathcal{F}),\Psi^{f_{u}(s)}_{u}(u\cdot\mathcal{F}))_{g,Q^{f_{u}(s)}}\,,$

where we have defined:

$\Psi^{f_{u}(s)}_{u}=(\mathcal{D}\mathcal{J}_{u})^{f_{u}(s)}\,.$

Hence $(g,f_{u}\circ s,u\cdot\mathcal{F})$ satisfies the scalar equations
associated to the scalar-electromagnetic structure
$\Phi_{u}=(\pi,\mathcal{G},\mathcal{H}_{u},\Delta_{u},\mathcal{J}_{u})$. Since
the flatness condition for $\mathcal{F}$ is linear in $\mathcal{F}$ it is
enough to verify it on an homogenous element of the form
$\mathcal{F}=\alpha\otimes\xi^{s}$, where $\alpha\in\Omega^{2}(M)$ and
$\xi^{s}$ is the pull-back by $s$ of a section $\xi\in\Gamma(\mathcal{S})$. We
compute:

$\displaystyle\mathrm{d}_{\mathcal{D}^{f_{u}(s)}}(u\cdot\mathcal{F})=\mathrm{d}_{\mathcal{D}^{f_{u}(s)}}(\alpha\otimes
u\cdot\xi^{s})=\mathrm{d}_{\mathcal{D}^{f_{u}(s)}}(\alpha\otimes(u\circ\xi\circ
f^{-1}_{u}\circ f_{u}(s)))=\mathrm{d}\alpha\otimes(u\cdot\xi^{s})$
$\displaystyle+\alpha\otimes\mathcal{D}^{f_{u}(s)}(u\circ\xi\circ
f^{-1}_{u})^{f_{u}(s)}=\mathrm{d}\alpha\otimes(u\cdot\xi^{s})+\alpha\otimes(\mathcal{D}(u\circ\xi\circ
f^{-1}_{u}))^{f_{u}(s)}=\mathrm{d}\alpha\otimes(u\cdot\xi^{s})$
$\displaystyle+\alpha\otimes(u\circ\mathcal{D}\xi\circ
f^{-1}_{u})^{f_{u}(s)}=\mathrm{d}\alpha\otimes(u\cdot\xi^{s})+\alpha\otimes
u\cdot\mathcal{D}^{s}\xi^{s}=u\cdot\mathrm{d}_{\mathcal{D}^{s}}\mathcal{F}=0\,,$

where we have usted that $u\in\mathrm{Aut}_{\pi}(\Delta)$ preserves the
symplectic connection $\mathcal{D}$. Whence $u\cdot\mathcal{F}$ is flat with
respect to $\mathcal{D}^{f_{u}(s)}$. On the other hand, the Maxwell equation
is also linear in $\mathcal{F}$ hence it is enough to verify it on an
homogenous element $\mathcal{F}=\alpha\otimes\xi^{s}$. We obtain:

$\displaystyle\star_{g,\mathcal{J}^{f_{u}(s)}_{u}}(u\cdot\mathcal{F})=\ast_{g}\alpha\otimes\mathcal{J}^{f_{u}(s)}_{u}(u\cdot\xi^{s})=\ast_{g}\alpha\otimes\mathcal{J}^{f_{u}(s)}_{u}(u\circ\xi\circ
f_{u}^{-1}\circ f_{u}(s))$
$\displaystyle=\ast_{g}\alpha\otimes(\mathcal{J}_{u}(u\circ\xi\circ
f_{u}^{-1}))^{f_{u}(s)}=\ast_{g}\alpha\otimes(u\circ\mathcal{J}(\xi)\circ
f_{u}^{-1}))^{f_{u}(s)}=\ast_{g}\alpha\otimes
u\cdot\mathcal{J}^{s}(\xi^{s})=u\cdot\star_{g,\mathcal{J}^{s}}\mathcal{F}=u\cdot\mathcal{F}\,,$

whence $(g,f_{u}\circ s,u\cdot\mathcal{F})$ also satisfies the Maxwell
equations associated to
$(\pi,\mathcal{G},\mathcal{H}_{u},\Delta_{u},\mathcal{J}_{u})$. Thus
$(g,f_{u}\circ
s,u\cdot\mathcal{F})\in\mathrm{Sol}(\pi,\mathcal{H}_{u},\mathcal{G}_{u},\Delta_{u},\mathcal{J}_{u})$
and reversing the previous relations it is easy to see that the map
$(g,s,\mathcal{F})\mapsto(g,f_{u}\circ s,u\cdot\mathcal{F})$ is a bijection. ∎

######  Remark 2.26.

The group $\mathrm{Aut}_{\pi}(\Delta)$ is the global counterpart of the so-
called _pseudo-duality group_ introduced in [23] as the direct product of the
symplectic group and the diffeomorphism group of the simply connected open set
on which the theory is considered. When both $\Delta$ and $\pi$ are non-
trivial, the group $\mathrm{Aut}_{\pi}(\Delta)$ can differ markedly from the
local pseudo-duality group of loc. cit.

The following statement follows from [16, Lemma 4.2.8].

######  Lemma 2.27.

Let $\Delta$ be a duality bundle over the submersion $\pi:X\rightarrow M$ and
consider a point $x\in X$. Then there exists a canonical isomorphism:

$\mathrm{Aut}_{b}(\Delta)=\mathrm{C}(\mathrm{Hol}_{x}(\mathcal{D}),\mathrm{Aut}(S_{x},\omega_{x}))\,,$

where $\mathrm{Hol}_{x}(\mathcal{D})$ is the holonomy group of $\mathcal{D}$
at $x$, $\mathrm{Aut}(S_{x},\omega_{x})\simeq\mathrm{Sp}(2n_{v},\mathbb{R})$
is the automorphism group of the fiber $(S_{x},\omega_{x})=(S,\omega)|_{x}$
and
$\mathrm{C}(\mathrm{Hol}_{x}(\mathcal{D}),\mathrm{Aut}(S_{x},\omega_{x})))$
denotes the centralizer of $\mathrm{Hol}_{x}(\mathcal{D})$ in
$\mathrm{Aut}(S_{x},\omega_{x})$.

Fixing $x\in X$, this shows that (4) is isomorphic with the exact sequence:

$1\to\mathrm{C}(\mathrm{Hol}_{x}(\mathcal{D}),\mathrm{Aut}(S_{x},\omega_{x})))\to\mathrm{Aut}_{\pi}(\Delta)\to\mathrm{Aut}^{0}_{b}(\pi)\to
1\,.$

Given a scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ and an electromagnetic
bundle $\Xi=(\Delta,\mathcal{J})$ we next introduce a subgroup of
$\mathrm{Aut}_{\pi}(\Delta)$ which preserves the Ehresmann connection
$\mathcal{H}$, the vertical metric $\mathcal{G}$ and the vertical taming
$\mathcal{J}$. This subgroup gives the global counterpart of the group of
_continuous_ U-dualities studied traditionally in the supergravity literature.

######  Definition 2.28.

Let $\Phi=(\pi,\mathcal{H},\mathcal{G},\Delta,\mathcal{J})$ be a scalar-
electromagnetic bundle on $M$. The classical U-duality group of $\Phi$ is the
subgroup $\mathrm{U}(\Phi)$ of $\mathrm{Aut}_{\pi}(\Delta)$ consisting of
those elements which preserve the Ehresmann connection $\mathcal{H}$, the
vertical metric $\mathcal{G}$ and the vertical taming $\mathcal{J}$:

$\mathrm{U}(\Phi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}_{\pi}(\Delta)\,\,|\,\,\mathcal{H}_{u}=\mathcal{H}\,,\,\,\mathcal{G}_{u}=\mathcal{G}\,,\,\,\mathcal{J}_{u}=\mathcal{J}\right\\}\,.$

Let $\mathrm{Aut}_{b}(\Xi)\subset\mathrm{Aut}_{b}(\Delta)$ be the group based
automorphisms of $\Xi$, which consists of those vector bundle automorphisms of
$\mathcal{S}$ which cover the identity and preserve $\omega$, $\mathcal{D}$
and $\mathcal{J}$. The U-duality group fits into a short exact sequence:

$1\to\mathrm{Aut}_{b}(\Xi)\to\mathrm{U}(\Phi)\to\mathrm{Aut}_{b}^{0}(\pi,\mathcal{H},\mathcal{G})\to
1\,,$

where
$\mathrm{Aut}_{b}^{0}(\pi,\mathcal{H},\mathcal{G}))\subset\mathrm{Aut}_{b}(\pi)$
denotes the subgroup of those based automorphisms of $\pi$ that can be covered
by elements of $\mathrm{U}(\Phi)$ and preserve both the Ehresmann connection
$\mathcal{H}$ and the vertical metric $\mathcal{G}$. If the scalar bundle
$(\pi,\mathcal{H},\mathcal{G})$ is flat then the group
$\mathrm{Aut}_{b}(\pi(\mathcal{H},\mathcal{G}))$ is finite-dimensional by
Lemma 2.27, which in turn implies that $\mathrm{U}(\Phi)$ is a finite-
dimensional Lie group. In general, this group is markedly different from the
U-duality group traditionally considered in the local formulation of the
theory. The main feature of the latter is that maps solutions to solutions and
hence it can be used as a solution generating mechanism. This key property
also holds for $\mathrm{U}(\Phi)$ as a consequence of Theorem 2.25.

######  Corollary 2.29.

The action $\mathbb{A}$ of the U-duality group $\mathrm{U}(\Phi)$ preserves
both $\mathrm{Conf}(\Phi)$ and $\mathrm{Sol}(\Phi)$. i.e. it maps
configurations to configurations and solutions to solutions. Moreover, if the
scalar bundle $(\pi,\mathcal{H},\mathcal{G})\in\Phi$ is flat then
$\mathrm{U}(\Phi)$ is a finite-dimensional Lie group.

For further reference we introduce the following definition.

######  Definition 2.30.

The _classical U-duality transformation_ defined by an element
$u\in\mathrm{U}(\Phi)$ is the bijection
$\mathbb{A}_{u}\colon\mathrm{Sol}(\Phi)\to\mathrm{Sol}(\Phi)$.

##  3\. The Dirac-Schwinger-Zwanziger integrality condition

This section discusses the geometric model obtained by imposing the DSZ
quantization condition on the universal bosonic sector of four-dimensional
supergravity defined by a fixed scalar-electromagnetic bundle. This condition
depends on the choice of a Dirac system for the underlying duality bundle
$\Delta$ and of the choice of an _integral_ cohomology class in
$H^{2}(M,\Delta)$, where integrality is defined relative to that Dirac system.

###  3.1. The vector space of integral field strengths

The DSZ quantization condition of local supergravity is implemented using a
full symplectic lattice. Similarly, we implement the DSZ quantization of the
universal bosonic sector defined by a scalar-electromagnetic bundle $\Phi$ in
terms of a smoothly varying fiber-wise choice of full symplectic lattices for
the underlying duality bundle $\Delta$, as proposed in [24]. Recall that a
full lattice $\Lambda$ in a $2n$-dimensional symplectic vector space
$(V,\omega)$ is called symplectic if the restriction of the symplectic pairing
$\omega$ to $\Lambda$ takes integer values. Such lattices are characterized up
to symplectomorphism by their type $\mathfrak{t}\in\mathrm{Div}^{n}$ (see [12,
Proposition 1.1]), where:

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

Any full symplectic lattice of type $\mathfrak{t}\in\mathrm{Div}^{n}$ in
$(V,\omega)$ admits a basis
$\lambda_{1},\ldots,\lambda_{n},\mu_{1},\ldots,\mu_{n}$ such that:

$\omega(\lambda_{i},\mu_{j})=t_{j}\delta_{ij}~{}~{},~{}~{}\omega(\lambda_{i},\lambda_{j})=\omega(\mu_{i},\mu_{j})=0~{}~{}\forall
i,j=1,\ldots,n~{}~{}$

and any element $\mathfrak{t}\in\mathrm{Div}^{n}$ is realized as the type of
some full symplectic lattice. The symplectic lattice $\Lambda$ is called
principal if $\mathfrak{t}=\delta_{n}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(1,\ldots,1)$. The modified Siegel modular group of type
$\mathfrak{t}\in\mathrm{Div}^{n}$ is the subgroup
$\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})\subset\mathrm{Sp}(2n,\mathbb{R})\simeq\mathrm{Aut}(V,\omega)$
consisting of those symplectic transformations which preserve a symplectic
lattice of type $\mathfrak{t}$. We have
$\mathrm{Sp}_{\delta_{n}}(2n,\mathbb{Z})=\mathrm{Sp}(2n,\mathbb{Z})$ and
$\mathrm{Sp}(2n,\mathbb{Z})\subset\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$
for all $\mathfrak{t}\in\mathrm{Div}^{n}$.

######  Definition 3.1.

Let $\Delta=(\mathcal{S},\omega,\mathcal{D})$ be a duality bundle on the
scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ with submersion
$\pi:X\rightarrow M$. A Dirac system on $\Delta$ is a smooth fiber sub-bundle
$j\colon\mathcal{L}\hookrightarrow\mathcal{S}$ of full symplectic lattices in
$(\mathcal{S},\omega)$ which is preserved by the parallel transport $T$ of the
flat connection $\mathcal{D}$ in the sense that we have:

$T_{\gamma}(\mathcal{L}|_{\gamma(0)})=\mathcal{L}|_{\gamma(1)}$

for any piece-wise smooth path $\gamma\in\mathcal{P}(X)$. The common type of
these fiberwise symplectic lattices is called the type of $\mathcal{L}$. A
pair:

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

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

For every $x\in X$, the fiber $(\mathcal{S}_{x},\omega_{x},\mathcal{L}_{x})$
of an integral duality bundle ${\bm{\Delta}}=(\Delta,\mathcal{L})$ with
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ is an _integral symplectic space_ as
defined in [26, Appendix B]. All fibers of ${\bm{\Delta}}$ are isomorphic as
integral symplectic spaces, hence their type does not depend on $x\in X$ since
we assume that $X$ is connected.

######  Remark 3.2.

The existence of a Dirac system is obstructed. A duality bundle
$\Delta=(\mathcal{S},\omega,\mathcal{D})$ of rank $2n_{v}$ admits a Dirac
system of type $\mathfrak{t}\in\mathrm{Div}^{n}$ if and only if the structure
group of $\mathcal{S}$ can be reduced from $\mathrm{Sp}(2n,\mathbb{R})$ to
$\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})$. We say that $\Delta$ is
_semiclassical_ if it admits a Dirac system.

######  Definition 3.3.

Let ${\bm{\Delta}}_{1}=(\Delta_{1},\mathcal{L}_{1})$ and
${\bm{\Delta}}_{2}=(\Delta_{2},\mathcal{L}_{2})$ be two integral duality
bundles on $M$. A morphism of integral duality bundles from
${\bm{\Delta}}_{1}$ to ${\bm{\Delta}}_{2}$ is a morphism of duality bundles
$f\colon\Delta_{1}\to\Delta_{2}$ such that
$f(\mathcal{L}_{1})=\mathcal{L}_{2}$.

######  Remark 3.4.

Given a Dirac system $\mathcal{L}$ for a duality bundle
$\Delta=(\mathcal{S},\omega,\mathcal{D})$, let
$\mathfrak{S}_{{\bm{\Delta}}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{C}(\mathcal{L})$ be the locally-constant sheaf of
continuous sections of the discrete fiber bundle $\mathcal{L}$. This is a
subsheaf of the sheaf $\mathfrak{S}_{{\bm{\Delta}}}$ of flat sections of
$(\mathcal{S},\mathcal{D})$ whose stalk at $x\in X$ identifies with the
symplectic lattice $\mathcal{L}_{x}\subset\mathcal{S}_{x}$.

For every scalar section $s\in\Gamma(\pi)$, let
$\mathfrak{S}^{s}_{{\bm{\Delta}}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}s^{\ast}(\mathfrak{S}_{{\bm{\Delta}}})$ be the locally constant
sheaf on $M$ obtained as the pullback of $\mathfrak{S}_{\bm{\Delta}}$ through
$s$. The sheaf cohomology groups $H^{k}(M,\mathfrak{S}^{s}_{{\bm{\Delta}}})$
are naturally isomorphic with the cohomology groups $H^{k}(M,\mathcal{L}^{s})$
of $M$ with coefficients in the local system
$\mathcal{L}^{s}=s^{\ast}(\mathcal{L})$ and play a crucial role in what
follows.

######  Definition 3.5.

An _integral electromagnetic bundle_ is a pair:

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

where $\Xi$ is an electromagnetic bundle on $M$ and $\mathcal{L}$ is Dirac
system for the duality bundle of $\Xi$. An _integral scalar-electromagnetic
bundle_ on $M$ is a pair:

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

where $\Phi$ is a scalar-electromagnetic bundle on $M$ and $\mathcal{L}$ is a
Dirac system for the duality bundle of $\Phi$.

Given an integral electromagnetic bundle ${\bm{\Xi}}=(\Xi,\mathcal{L})$ with
integral duality structure
${\bm{\Delta}}=(\mathcal{S},\omega,\mathcal{D},\mathcal{L})$ over a submersion
$\pi:X\rightarrow M$, the quotient:

$\mathcal{X}_{{\bm{\Delta}}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{S}/\mathcal{L}$

is a flat fibration over $X$ by symplectic torus groups. The taming
$\mathcal{J}$ of $\Delta$ makes this into a fibration by polarized Abelian
varieties which however need not be flat since $\mathcal{J}$ is not flat
unless the underlying electromagnetic bundle $\Xi$ is unitary. The sheaf
$\mathfrak{S}_{\mathcal{X}_{\bm{\Delta}}}$ of smooth flat sections of
$\mathcal{X}_{{\bm{\Delta}}}$ fits into a short exact sequence of sheaves of
Abelian groups defined on $X$:

$0\to\mathfrak{S}_{{\bm{\Delta}}}\xrightarrow{j}\mathfrak{S}_{\Delta}\to\mathfrak{S}_{\mathcal{X}_{{\bm{\Delta}}}}\to
0\,.$

which pulls-back to a short exact sequence of sheaves of Abelian groups
defined on $M$:

$0\to\mathfrak{S}^{s}_{{\bm{\Delta}}}\xrightarrow{j^{s}}\mathfrak{S}^{s}_{\Delta}\to\mathfrak{S}^{s}_{\mathcal{X}_{\bm{\Delta}}}\to
0\,.$

The latter induces a long exact sequence in sheaf cohomology, of which we are
interested in the following portion:

$\ldots\rightarrow H^{1}(M,\mathfrak{S}^{s}_{\mathcal{X}_{{\bm{\Delta}}}})\to
H^{2}(M,\mathfrak{S}^{s}_{{\bm{\Delta}}})\xrightarrow{j^{s}_{\ast}}H^{2}(M,\mathfrak{S}^{s}_{\Delta})\to
H^{2}(M,\mathfrak{S}^{s}_{\mathcal{X}_{{\bm{\Delta}}}})\rightarrow\ldots\,.$

######  Definition 3.6.

The charge lattice of the integral scalar-electromagnetic structure
${\bm{\Xi}}=(\Xi,\mathcal{L})$ relative to the scalar section
$s\in\Gamma(\pi)$ is the lattice:

$L_{{\bm{\Xi}}}^{s}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}j^{s}_{\ast}(H^{2}(M,\mathfrak{S}^{s}_{{\bm{\Delta}}}))\subset
H^{2}(M,\mathfrak{S}^{s}_{\Delta})\,,$

Elements of this lattice are called _integral cohomology classes_.

It can be shown that $L_{{\bm{\Xi}}}^{s}$ is a full lattice in
$H^{2}(M,\mathfrak{S}^{s}_{\Delta})$ (see [26, Proposition 2.24]). Given an
integral scalar-electromagnetic bundle $\bm{\Phi}$, we implement DSZ
quantization by restricting the configuration space $\mathrm{Conf}(\Phi)$ to a
subset $\mathrm{Conf}(\bm{\Phi})\subset\mathrm{Conf}(\Phi)$ obtained by
imposing an _integrality condition_ on the elements of $\mathrm{Conf}(\Phi)$.
This is the appropriate implementation of the DSZ quantization condition in
our situation.

######  Definition 3.7.

Let $\bm{\Phi}$ be an integral scalar-electromagnetic bundle. The integral
configuration space $\mathrm{Conf}(\bm{\Phi})$ of defined by $\bm{\Phi}$ is
the set:

$\mathrm{Conf}(\bm{\Phi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(g,s,\mathcal{F})\in\mathrm{Conf}(M,\Phi)\,\,|\,\,[\mathcal{F}]\in
2\pi L^{s}_{\Xi}\right\\}\,.$

The integral solution space
$\mathrm{Sol}(\bm{\Phi})\subset\mathrm{Conf}(\Phi)$ defined by $\bm{\Phi}$ is
the the set:

$\mathrm{Sol}(\bm{\Phi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Sol}(\Phi)\cap\mathrm{Conf}(\bm{\Phi})~{}~{}.$

For further reference we introduce a refinement of the previous definition.

######  Definition 3.8.

Let $\bm{\Phi}$ be an integral scalar-electromagnetic bundle and let
$\mathfrak{V}\in H^{2}(X,\mathfrak{S}_{{\bm{\Delta}}})$. The framed integral
configuration space $\mathrm{Conf}(\mathfrak{V},\bm{\Phi})$ with _framing_
$\mathfrak{V}$ of the classical geometric supergravity theory associated to
$\bm{\Phi}$ is defined as the following subset of $\mathrm{Conf}(\bm{\Phi})$:

$\mathrm{Conf}(\mathfrak{V},\bm{\Phi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(g,s,\mathcal{F})\in\mathrm{Conf}(\Phi)\,\,|\,\,[\mathcal{F}]=2\pi
j^{s}_{\ast}(\mathfrak{V}^{s})\right\\}\,,$

The framed integral solution space
$\mathrm{Sol}(\mathfrak{V},\bm{\Phi})\subset\mathrm{Conf}(\mathfrak{V},\Phi)$
is the set:

$\mathrm{Sol}(\mathfrak{V},\bm{\Phi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Sol}(\Phi)\cap\mathrm{Conf}(\mathfrak{V},\bm{\Phi})~{}~{}.$

######  Definition 3.9.

The _arithmetic U-duality group_ of an integral scalar-electromagnetic
structure $\bm{\Phi}=(\Phi,\mathcal{L})$ is the subgroup of $\mathrm{U}(\Phi)$
defined through:

$\mathrm{U}(\bm{\Phi})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{U}(\Phi)\,\,|\,\,u(\mathcal{L})=\mathcal{L}\right\\}\,.$

The arithmetic U-duality group $\mathrm{U}(\bm{\Phi})$ is the global
counterpart of the arithmetic U-duality group of local supergravity normally
considered in the physics literature [22, 30]. We remark that the supergravity
literature seems to have considered thus far only holonomy trivial Dirac
systems $\mathcal{L}$ of principal type, though there is a priori no physical
or mathematical reason to make that assumption. We will consider some simple
examples of arithmetic U-duality groups in Section 5. More elaborated examples
will be considered in a separate publication.

##  4\. The DSZ quantization of 4d bosonic supergravity

In this section we describe the geometric and gauge-theoretic formulation of
the universal bosonic sector of 4d supergravity implied by the DSZ
quantization condition. This formulation can be constructed through a step-by-
step process as done in [26] for Abelian gauge theory. Instead of going
through the details of that process, which are similar to those in [26], we
give the description of the theory in its final form, verifying then that it
satisfies the appropriate DSZ quantization (see Theorem 4.5). The key
ingredient occurring in the construction is a _Siegel bundle_ , a special kind
of principal bundle which was introduced and discussed in detail in [26,
Section 3] and forms a particular case of the more general notion of principal
bundle with weakly-Abelian structure group studied in [27], to which we refer
the reader for background and further details. Given
$\mathfrak{t}\in\mathrm{Div}^{n_{v}}$, we define the following disconnected
Lie group:

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

where $\mathrm{U}(1)^{2n_{v}}\simeq\mathbb{R}^{2n_{v}}/\mathbb{Z}^{2n_{v}}$ is
an affine torus group of dimension $2n_{v}$. The group
$\mathrm{Aff}_{\mathfrak{t}}$ identifies with the set
$\mathrm{U}(1)^{2n_{v}}\times\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})$
equipped with the multiplication rule:

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

The modified Siegel modular group
$\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})$ coincides with the
automorphism group of the standard integral symplectic space
$(\mathbb{R}^{2n_{v}},\omega_{n_{v}},\wedge_{\mathfrak{t}})$ of type
$\mathfrak{t}$, where $\omega_{n_{v}}$ is the standard symplectic form on
$\mathbb{R}^{2n_{v}}$ and:

$\Lambda_{\mathfrak{t}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathbb{Z}^{n_{v}}\oplus\oplus_{i=1}^{n_{v}}{t_{i}\mathbb{Z}}\subset\mathbb{R}^{2n_{v}}$

is the standard symplectic lattice of type $\mathfrak{t}$ (see [26, Appendix
B]). Moreover, $\mathrm{Aff}_{\mathfrak{t}}$ coincides with the group of
affine symplectomorphisms of the $2n_{v}$-dimensional symplectic torus
$(\mathbb{R}^{2n_{v}}/\Lambda_{\mathfrak{t}},\Omega_{\mathfrak{t}})$, whose
symplectic form $\Omega_{\mathfrak{t}}$ is induced by $\omega_{n_{v}}$. The
connected component of the identity in $\mathrm{Aff}_{\mathfrak{t}}$ is the
torus group $\mathrm{U}(1)^{2n_{v}}$, while the group of components of
$\mathrm{Aff}_{\mathfrak{t}}$ is the discrete group
$\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})$, which is infinite and non-
Abelian when $n_{v}>0$.

######  Definition 4.1.

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

Let $(\pi,\mathcal{H},\mathcal{G})$ be a scalar bundle with submersion
$\pi:X\rightarrow M$ and consider a Siegel bundle $P_{\mathfrak{t}}$ of rank
$n_{v}$ and type $\mathfrak{t}\in\mathrm{Div}^{n_{v}}$ over $X$. As shown in
[26], the adjoint bundle of $P_{\mathfrak{t}}$ admits a natural structure of
integral duality bundle of type $\mathfrak{t}$ which we denote by
${\bm{\Delta}}(P_{\mathfrak{t}})$. By definition, a vertical taming
$\mathcal{J}$ of $P_{\mathfrak{t}}$ is a vertical taming of
${\bm{\Delta}}(P_{\mathfrak{t}})$. Given such a taming, the pair
$(P_{\mathfrak{t}},\mathcal{J})$ is called a (positively) polarized Siegel
bundle (cf. [26, 27]). The integral electromagnetic bundle
${\bm{\Xi}}(P_{\mathfrak{t}},\mathcal{J})$ determined by
$(P_{\mathfrak{t}},\mathcal{J})$ is defined through:

${\bm{\Xi}}(P_{\mathfrak{t}},\mathcal{J})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}({\bm{\Delta}}(P_{\mathfrak{t}}),\mathcal{J})\,.$

Given a scalar section $s\in\Gamma(\pi)$, we denote by $P^{s}_{\mathfrak{t}}$
the pullback of $P_{\mathfrak{t}}$ by $s$, which becomes a Siegel bundle over
$M$. Similarly, we denote by ${\bm{\Delta}}(P^{s}_{\mathfrak{t}})$ and
${\bm{\Xi}}(P^{s}_{\mathfrak{t}},\mathcal{J}^{s})$ the integral duality and
integral electromagnetic bundles defined by $P^{s}$ and $\mathcal{J}^{s}$,
which coincide with the $s$-pullbacks of the corresponding bundles defined by
$P$ and $\mathcal{J}$ on $X$. When necessary, we will write:

${\bm{\Delta}}(P^{s}_{\mathfrak{t}})=(\mathcal{S}^{s},\omega^{s},\mathcal{D}^{s})\,.$

Let $\mathrm{Conn}(P^{s}_{\mathfrak{t}})$ be the affine space of connections
on $P^{s}_{\mathfrak{t}}$. Elements of this space are invariant one-forms on
$P_{\mathfrak{t}}$ mapping the fundamental vector fields of $P_{\mathfrak{t}}$
to their generators in $\mathrm{aff}_{\mathfrak{t}}$, where
$\mathrm{aff}_{\mathfrak{t}}\simeq\mathbb{R}^{2n_{v}}$ is the Lie algebra of
$\mathrm{Aff}_{\mathfrak{t}}$, which has trivial Lie bracket. The adjoint
curvature of a connection $\mathcal{A}\in\mathrm{Conn}(P^{s}_{\mathfrak{t}})$
will be denoted by
$\mathcal{F}_{\mathcal{A}}\in\Omega^{2}(M,\mathcal{S}^{s})$. This bundle-
valued 2-form is $\mathrm{d}_{\mathcal{D}^{s}}$-closed by the Bianchi identity
since (by the results of [26, 27]) all connections on $P^{s}$ induce the same
connection on $\mathcal{S}^{s}$, which coincides with the connection induced
by $\mathcal{D}^{s}$ on the adjoint bundle of $P^{s}_{\mathfrak{t}}$. Thus:

$\mathrm{d}_{\mathcal{D}}^{s}\mathcal{F}_{\mathcal{A}}=0\,.$

######  Definition 4.2.

A scalar-Siegel bundle of rank $n_{v}$ and type
$\mathfrak{t}\in\mathrm{Div}^{n_{v}}$ over $M$ is a system
$\zeta\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}})$, where
$(\pi,\mathcal{H},\mathcal{G})$ is a scalar bundle over $M$ with submersion
$\pi:X\rightarrow M$ and $P_{\mathfrak{t}}$ is a Siegel bundle of rank $n_{v}$
and type $\mathfrak{t}\in\mathrm{Div}^{n_{v}}$ defined on $X$. Given a
vertical taming $\mathcal{J}$ of $\Delta(P_{\mathfrak{t}})$, the system
$\bm{\upzeta}\stackrel{{\scriptstyle{\rm def.}}}{{=}}(\Psi,\mathcal{J})$ is
called a polarized scalar-Siegel bundle of rank $n_{v}$ and type
$\mathfrak{t}$ over $M$.

######  Definition 4.3.

Let $\bm{\upzeta}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}},\mathcal{J})$ be a
polarized scalar-Siegel bundle over $M$. The configuration space of the
bosonic supergravity defined by $\bm{\upzeta}$ is the set:

$\mathfrak{Conf}(\bm{\upzeta})=\left\\{(g,s,\mathcal{A})\,\,|\,\,g\in\mathrm{Lor}(M)\,,\,\,s\in\Gamma(\pi)\,,\,\,\mathcal{A}\in\mathrm{Conn}(P^{s}_{\mathfrak{t}})\right\\}\,.$

The universal bosonic sector of four-dimensional supergravity determined on
$M$ by $\bm{\upzeta}$ is defined through the following system of partial
differential equations for triples
$(g,s,\mathcal{A})\in\mathfrak{Conf}(\bm{\upzeta})$:

  * •

The Einstein equations:

$\mathrm{Ric}^{g}-\frac{g}{2}\mathrm{R}^{g}=\frac{1}{2}\mathrm{Tr}_{g}(s^{\ast}_{\mathcal{C}}\mathcal{G})\,g-s^{\ast}_{\mathcal{C}}\mathcal{G}+2\mathcal{F}_{\mathcal{A}}\oslash_{Q^{s}}\mathcal{F}_{\mathcal{A}}\,.$
(7)

  * •

The scalar equations:

$\nabla^{\Phi(g,s)}\mathrm{d}^{\mathcal{C}}s=\frac{1}{2}(\ast\mathcal{F}_{\mathcal{A}},\Psi^{s}\mathcal{F}_{\mathcal{A}})_{g,Q^{s}}\,.$
(8)

  * •

The Maxwell equations:

$\star_{g,\mathcal{J}^{s}}\mathcal{F}_{\mathcal{A}}=\mathcal{F}_{\mathcal{A}}\,,$
(9)

whose set of solutions we denote by
$\mathfrak{Sol}(\bm{\upzeta})\subset\mathfrak{Conf}(\bm{\upzeta})$.

Connections $\mathcal{A}$ satisfying equation (9) will be called polarized
self-dual, following the terminology introduced in [26] in the context of
Abelian gauge theory.

######  Remark 4.4.

The Maxwell equations of the bosonic gauge sector of local supergravity are
given by a system of second-order partial differential equations for a number
$n_{v}$ of _electromagnetic_ local gauge potentials whose curvatures satisfy a
generalization of the Maxwell equations. This is locally equivalent with the
description given by the first order global equation (9), which reduces
locally to a system of first-order partial differential equations for $2n_{v}$
local gauge fields, both _electric_ and _magnetic_ (considered up to gauge
transformations of the principal bundle $P^{s}_{\mathfrak{t}}$).

The Bianchi identity and polarized self-duality condition imply that the gauge
potential of any solution $(g,s,\mathcal{A})\in\mathrm{Sol}(\bm{\upzeta})$
automatically satisfies the following second order differential equation of
Yang-Mills type:

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

These differ from the usual Yang-Mills equations since
$\mathcal{F}_{\mathcal{A}}$ involves both electric and magnetic degrees of
freedom while the equations themselves involve the pulled-back taming
$\mathcal{J}^{s}$.

######  Theorem 4.5.

Let $\bm{\Phi}=(\pi,\mathcal{H},\mathcal{G},{\bm{\Delta}},\mathcal{J})$ be an
integral scalar-electromagnetic bundle of type $\mathfrak{t}$. For every
framed integral configuration space $\mathrm{Conf}(\mathfrak{V},\Phi)$ there
exists a vertically polarized Siegel bundle $(P_{\mathfrak{t}},\mathcal{J})$
on $(\pi,\mathcal{H},\mathcal{G})$ such that
${\bm{\Delta}}={\bm{\Delta}}(P_{\mathfrak{t}})$ and the twisted Chern
class222See [26, 27] for its precise definition. $c(P_{\mathfrak{t}})$ of
$P_{\mathfrak{t}}$ satisfies $c(P_{\mathfrak{t}})=\mathfrak{V}$. Moreover, the
map:

$\mathfrak{Sol}(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}},\mathcal{J})\to\mathrm{Sol}(\mathfrak{V},\bm{\Phi})\,,\qquad(g,s,\mathcal{A})\mapsto(g,s,\mathcal{F}_{\mathcal{A}})\,,$

is surjective.

###### Proof.

Given ${\bm{\Delta}}$ and $\mathfrak{V}$, it follows from the results of [5,
6] (see also [27]) that there exists a Siegel bundle $P_{\mathfrak{t}}$ of
type $\mathfrak{t}$ (unique up to isomorphism) whose twisted Chern class
$c(P_{\mathfrak{t}})$ equals $\mathfrak{V}$ and whose adjoint bundle is
isomorphic to ${\bm{\Delta}}$ as an integral duality bundle. The vertical
taming $\mathcal{J}$ in $\bm{\Phi}$ makes $P_{\mathfrak{t}}$ into a polarized
Siegel bundle. On the other hand, the curvature of any connection
$\mathcal{A}\in\mathrm{Conn}(P^{s}_{\mathfrak{t}})$ defines a
$\mathrm{d}_{\mathcal{D}^{s}}$-cohomology class
$[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}$ in
$H^{2}(M,\mathfrak{S}_{\Delta^{s}})$ since, as remarked earlier:

$\mathrm{d}_{\mathcal{D}^{s}}\mathcal{F}_{\mathcal{A}}=\mathrm{d}_{\mathcal{A}}\mathcal{F}_{\mathcal{A}}=0\,.$

Given any other connection $\mathcal{A}^{\prime}$ on $P_{\mathfrak{t}}$ we
have:

$\mathcal{A}^{\prime}=\mathcal{A}+\bar{\tau}\,,$

for a unique horizontal and invariant one-form
$\bar{\tau}\in\Omega^{1}(P_{\mathfrak{t}},\mathfrak{aff}_{\mathfrak{t}})$.
Therefore, the curvatures of $\mathcal{F}_{\mathcal{A}}$ and
$\mathcal{F}_{\mathcal{A}^{\prime}}$ are related as follows:

$\mathcal{F}_{\mathcal{A}^{\prime}}=\mathcal{F}_{\mathcal{A}}+\mathrm{d}_{\mathcal{D}^{s}}\tau\in\Omega^{2}(M,\mathcal{S}^{s})\,,$

where $\tau\in\Omega^{1}(M,\mathcal{S}^{s})$ is uniquely determined by
$\bar{\tau}\in\Omega^{1}(P_{\mathfrak{t}},\mathfrak{aff}_{\mathfrak{t}})$.
This implies that the cohomology class
$[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}\in
H^{2}(M,\mathfrak{S}_{\Delta}^{s})$ does not depend on the connection
$\mathcal{A}$. A similar argument shows that the cohomology class
$[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}$ is invariant under
automorphisms of $P_{\mathfrak{t}}$ and therefore only depends on the
isomorphism class of the latter. This is further elaborated in [27] to show
that $[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}$ is equal to the _real_
twisted Chern class of $P_{\mathfrak{t}}$ as follows:

$[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}=2\pi
j^{s}_{\ast}(c(P^{s}_{\mathfrak{t}}))\in L^{s}_{\bm{\Delta}}\,.$

Since $c(P_{\mathfrak{t}})=\mathfrak{V}$ by construction, we immediately
conclude that:

$[\mathcal{F}_{\mathcal{A}}]_{\mathcal{D}^{s}}=2\pi
j^{s}_{\ast}(\mathfrak{V}^{s})\in L^{s}_{\bm{\Delta}}\,.$

Hence, $(g,s,\mathcal{F}_{\mathcal{A}})$ belongs to
$\mathrm{Sol}(\mathfrak{V},\bm{\Phi})$ for all
$(g,s,\mathcal{A})\in\mathrm{Conf}(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}},\mathcal{J})$.
Furthermore, every element in $\mathrm{Sol}(\mathfrak{V},\bm{\Phi})$ is of the
form $(g,s,\mathcal{F}_{\mathcal{A}})$ for some
$(g,s,\mathcal{A})\in\mathfrak{Sol}(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}},\mathcal{J})$.
An explicit way to prove this is to use a good open cover
$M\subset\left\\{U_{a}\right\\}_{a\in I}$ of $M$. Then, given
$(g,s,\mathcal{F})\in\mathrm{Sol}(\mathfrak{V},\bm{\Phi})$, the restriction:

$\mathcal{F}_{a}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{F}|_{U_{a}}=\mathrm{d}\mathcal{A}_{a}\,,\qquad\mathcal{A}_{a}\in\Omega^{1}(U_{a},\mathbb{R}^{2n_{v}})\qquad
a\in I\,,$

is $\mathrm{d}_{\mathcal{D}^{s}}$-exact and hence $\mathrm{d}$-exact, since we
can trivialize $\Delta$ over $U_{a}$ as the latter is simply connected. The
family of one-forms $\left\\{\mathcal{A}_{a}\right\\}$ taking values in
$\mathbb{R}^{2n_{v}}$ can be shown to define a connection on
$P^{s}_{\mathfrak{t}}$ whose curvature is precisely $\mathcal{F}$ and hence we
conclude. ∎

The previous theorem shows that Definition 4.3 realizes geometrically the DSZ
quantization of the universal bosonic supergravity sector defined by $\Phi$
since it shows that, given a Dirac system for the duality structure of $\Phi$,
every element in the solution space $\mathrm{Sol}(\Phi,\mathcal{L})$ can be
realized through a Lorentz metric on $M$, a section of $\pi$ and a gauge
potential $\mathcal{A}\in\mathrm{Conn}(P^{s}_{\mathfrak{t}})$ for some Siegel
bundle $P^{s}_{\mathfrak{t}}$ on $X$. The latter is the novel geometric object
attached to the DSZ quantization condition.

##  5\. The electromagnetic U-duality group

In this section we investigate the gauge U-duality group of the DSZ
quantization of bosonic supergravity, which yields a natural extension of its
arithmetic U-duality group and provides the geometric interpretation of
U-duality transformations as gauge transformations.

Fix a vertically polarized Siegel bundle $(P_{\mathfrak{t}},\mathcal{J})$ on
the total space $X$ of a scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ and let
$\mathrm{Aut}(P_{\mathfrak{t}})$ be the automorphism group of
$P_{\mathfrak{t}}$. For every $u\in\mathrm{Aut}(P_{\mathfrak{t}})$, denote by
$\mathfrak{ad}_{u}\colon{\bm{\Delta}}(P_{\mathfrak{t}})\to{\bm{\Delta}}(P_{\mathfrak{t}})$
the automorphism of the integral duality structure
${\bm{\Delta}}(P_{\mathfrak{t}})$ defined by $u$. Let
$\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})\subset\mathrm{Aut}(P_{\mathfrak{t}})$ be
the subgroup formed by all elements of $\mathrm{Aut}(P_{\mathfrak{t}})$ which
cover based automorphisms of the fiber bundle $\pi$, that is:

$\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}(P_{\mathfrak{t}})\,\,|\,\,f_{u}\in\mathrm{Aut}_{b}(\pi)\right\\}=\left\\{u\in\mathrm{Aut}(P_{\mathfrak{t}})\,\,|\,\,\pi\circ
f_{u}=\pi\right\\}\,.$

We have the a short exact sequence of groups:

$1\to\mathrm{Aut}_{b}(P_{\mathfrak{t}})\to\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})\to\mathrm{Aut}^{0}_{b}(\pi)\to
1\,,$

where $\mathrm{Aut}^{0}_{b}(\pi)$ is the subgroup of $\mathrm{Aut}_{b}(\pi)$
formed by those based automorphisms of $\pi$ that can be covered by elements
of $\mathrm{Aut}(P_{\mathfrak{t}})$.

######  Definition 5.1.

Let $(P_{\mathfrak{t}},\mathcal{J})$ be a vertically polarized Siegel bundle
over the scalar-bundle $(\pi,\mathcal{H},\mathcal{G})$. The gauge U-duality
group $\mathrm{U}(\bm{\upzeta})$ of the polarized scalar-Siegel bundle
$\bm{\upzeta}=(\pi,\mathcal{H},\mathcal{G},P_{\mathfrak{t}},\mathcal{J})$ is
the subgroup of $\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})$ consisting of those
elements which preserve the Ehresmann connection $\mathcal{H}$, the metric
$\mathcal{G}$ and the vertical taming $\mathcal{J}$:

$\mathrm{U}(\bm{\upzeta})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{u\in\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})\,\,|\,\,\mathcal{H}_{u}=\mathcal{H}\,,\,\,\mathcal{G}_{u}=\mathcal{G}\,,\,\,\mathcal{J}_{u}=\mathcal{J}\right\\}\,.$

Let $\mathrm{Aut}_{b}(P_{\mathfrak{t}},\mathcal{J})$ be the subgroup of
$\mathrm{Aut}_{b}(P_{\mathfrak{t}})$ consisting of those based automorphisms
of $P_{\mathfrak{t}}$ which preserve $\mathcal{J}$. The gauge U-duality group
fits into a short exact sequence:

$1\to\mathrm{Aut}_{b}(P_{\mathfrak{t}},\mathcal{J})\to\mathrm{U}(\bm{\upzeta})\to\mathrm{Aut}_{b}^{0}(\pi,\mathcal{H},\mathcal{G})\to
1\,,$

where
$\mathrm{Aut}^{0}_{b}(\pi,\mathcal{H},\mathcal{G})\subset\mathrm{Aut}^{0}_{b}(\pi)$
is the subgroup consisting of those based automorphisms of $\pi$ that can be
covered by elements of $\mathrm{U}(\bm{\upzeta})$ and preserve the Ehresmann
connection $\mathcal{H}$ and the metric $\mathcal{G}$.

The main feature of the local U-duality group of a local supergravity theory
is that maps solutions to solutions and thus can be used as a solution
generating mechanism. This key property also holds for
$\mathrm{U}(\bm{\upzeta})$ as we show below. Let:

$(g,s,\mathcal{A})\in\mathfrak{Sol}(\bm{\upzeta})~{}~{}.$

Recall that $\mathcal{A}\in\mathrm{Conn}(P^{s}_{\mathfrak{t}})$ is a
connection on the pull-back of $P_{\mathfrak{t}}$ through $s\in\Gamma(\pi)$.
An element $u\in\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})$ which covers
$f_{u}\in\mathrm{Aut}_{b}(\pi)$ acts on $(g,s,\mathcal{A})$ through:

$u\cdot(g,s,\mathcal{A})=(g,f_{u}(s),\mathcal{A}_{u})\,,$

where $f_{u}(s)=f_{u}\circ s\in\Gamma(\pi)$ and $\mathcal{A}_{u}$ is the push-
forward of $\mathcal{A}$ by the based isomorphism of $P^{s}_{\mathfrak{t}}$
naturally associated to $u$ as follows:

$u_{m}\colon(P^{s}_{\mathfrak{t}})_{m}=(P_{\mathfrak{t}})_{s(m)}\to(P^{f_{u}(s)}_{\mathfrak{t}})_{m}=(P_{\mathfrak{t}})_{f_{u}(s(m))}\,,\quad
p\mapsto u_{s(m)}(p)\,$

for all $m\in M$. Notice that $\mathcal{A}_{u}$ is a connection on the bundle
$P^{f_{u}(s)}_{\mathfrak{t}}$, where the latter denotes the pull-back of
$P_{\mathfrak{t}}$ by the section $f_{u}(s)\in\Gamma(s)$. Denote by:

$\bm{\upzeta}_{u}=(\pi,\mathcal{H}_{u},\mathcal{G}_{u},P_{\mathfrak{t}},\mathcal{J}_{u})$

the push-forward of $\bm{\upzeta}_{u}$ of $\bm{\upzeta}$ by
$u\in\mathrm{Aut}_{\pi}(P_{\mathfrak{t}})$.

######  Corollary 5.2.

Let $\bm{\upzeta}=(\pi,\mathcal{H},\mathcal{G},P,\mathcal{J})$ be a polarized
scalar-Siegel bundle with submersion $\pi\colon X\to M$. Every element
$u\in\mathrm{Aut}(P)$ defines a bijection of sets:

$\mathbb{A}_{u}\colon\mathfrak{Conf}(\bm{\upzeta})\to\mathfrak{Conf}(\bm{\upzeta}_{u})\,,\quad(g,s,\mathcal{A})\mapsto(g,f_{u}(s),\mathcal{A}_{u})\,,$

which restricts to a bijection:

$\mathbb{A}_{u}\colon\mathfrak{Sol}(\bm{\upzeta})\to\mathfrak{Sol}(\bm{\upzeta}_{u})~{}~{}.$

In particular, if $u\in\mathrm{U}(\bm{\upzeta})$ then
$\mathbb{A}_{u}\colon\mathfrak{Sol}(\bm{\upzeta})\to\mathfrak{Sol}(\bm{\upzeta})$
preserves the solution space of the given supergravity theory.

###### Proof.

The result follows directly from Theorem 2.25 upon noticing that:

$\mathcal{F}_{\mathcal{A}_{u}}=u\cdot\mathcal{F}_{\mathcal{A}}\,~{}~{},$

which shows that $\mathcal{F}_{A}$ transforms as the field strength
$\mathcal{F}$ considered in Section 2 in the classical formulation of the
theory. ∎

For further reference we introduce the following definition.

######  Definition 5.3.

The gauge _U-duality transformation_ induced by $u\in\mathrm{U}(\bm{\upzeta})$
is the bijection
$\mathbb{A}_{u}\colon\mathrm{Sol}(\bm{\upzeta})\to\mathrm{Sol}(\bm{\upzeta})$.

We have a canonical morphism of groups:

$\mathfrak{ad}\colon\mathrm{U}(\bm{\upzeta})\to\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))\,,\quad
u\mapsto\mathfrak{ad}_{u}\,,$

where $\bm{\Phi}(\bm{\upzeta})$ is the integral scalar-electromagnetic bundle
determined by the polarized scalar-Siegel bundle $\bm{\upzeta}$. This morphism
associates to $u$ the automorphism of the adjoint bundle of $P_{\mathfrak{t}}$
defined canonically by the latter.

######  Definition 5.4.

The _continuous subgroup_ of the gauge U-duality group
$\mathrm{U}(\bm{\upzeta})$ is:

$\mathrm{C}(\bm{\upzeta})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\ker(\mathfrak{ad})\subset\mathrm{U}(\bm{\upzeta})\,.$

The classical U-duality group was shown to be a finite-dimensional Lie group
in Section 2.3 when the scalar bundle is flat. This is no longer true for the
gauge U-duality group. Instead, if the rank of
$P_{\mathfrak{t}}\in\mathrm{U}(\bm{\upzeta})$ is positive
$\mathrm{U}(\bm{\upzeta})$ is an extension of the arithmetic duality group
$\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))$ by an _infinite-dimensional_ abelian
group, a fact that allows us to pinpoint the geometric origin of U-duality.

######  Proposition 5.5.

The gauge U-duality group $\mathrm{U}(\bm{\upzeta})$ fits into a short exact
sequence:

$1\to\mathrm{C}(\bm{\upzeta})\hookrightarrow\mathrm{U}(\bm{\upzeta})\xrightarrow{\mathfrak{ad}}\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))\to
1\,,$ (10)

where $\bm{\Phi}(\bm{\upzeta})$ is the polarized integral scalar-
electromagnetic bundle determined by $\bm{\upzeta}$.

###### Proof.

Since it is clear that the natural inclusion
$\mathrm{C}(\bm{\upzeta})\hookrightarrow\mathrm{U}(\bm{\upzeta})$ is injective
and the map $\mathfrak{ad}$ is a homomorphism, it suffices to prove that
$\mathfrak{ad}$ is surjective. Write the Dirac system $\mathcal{L}$ in
$\bm{\upzeta}$ as an associated bundle
$\mathcal{L}=P_{\mathfrak{t}}\times_{\ell}\mathbb{Z}^{2n}$ to
$P_{\mathfrak{t}}$ through the natural representation $\ell$ of
$\mathrm{Aff}_{\mathfrak{t}}$ on $\mathbb{Z}^{2n}$. Let
$\phi\in\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))$ be an automorphism of the scalar-
electromagnetic bundle $\bm{\Phi}(\bm{\upzeta})$ associated to $\bm{\upzeta}$
and covering the diffeomorphism $f_{\phi}\in\mathrm{Diff}(M)$. Since the
latter is a diffeomorphism, the principal bundles $P_{\mathfrak{t}}$ and
$P_{\mathfrak{t}}^{f_{\phi}}$, where the latter denotes the pull-back of
$P_{\mathfrak{t}}$ by $f_{\phi}$, are isomorphic. Fix such an isomorphism,
which is equivalent to fixing an automorphism $u^{\prime}_{\phi}\colon
P_{\mathfrak{t}}\to P_{\mathfrak{t}}$ covering $f_{\phi}$. Then, for every
$[p,v]\in\mathcal{L}$ there exists a unique map $\Scr
B_{p}\in\mathbb{Z}^{2n}\to\mathbb{Z}^{2n}$ such that:

$\phi([p,v])=[u^{\prime}_{\phi}(p),\Scr B_{p}(v)]\,.$

Since $\phi$ is a linear automorphism of the integral duality structure
determined by $\bm{\upzeta}$ it follows that the map $\Scr
B_{p}\in\mathbb{Z}^{2n}\to\mathbb{Z}^{2n}$ is a linear automorphism of the
standard symplectic lattice of type $\mathfrak{t}\in\mathbb{Z}$ and therefore
belongs to $\mathrm{Aff}_{\mathfrak{t}}$. Furthermore, independence of the
representative in $[p,v]\in\mathcal{L}$ in the definition of $\Scr B_{p}$
implies:

$\Scr B_{px}=x^{-1}\circ\Scr B_{p}\circ x\,,$

for every $x\in\mathrm{Aff}_{\mathfrak{t}}$. Hence, the assignment
$p\mapsto\Scr B_{p}$ defines a smooth map $\Scr B\colon
P_{\mathfrak{t}}\to\mathrm{Aff}_{\mathfrak{t}}$ which is equivariant with
respect to the adjoint action. Therefore, we have:

$\phi([p,v])=[u^{\prime}_{\phi}(p)\Scr B_{p},(v)]\,,$

and the automorphism $u_{\phi}\colon P_{\mathfrak{t}}\to P_{\mathfrak{t}}$
defined as follows:

$u_{\phi}(p)\stackrel{{\scriptstyle{\rm def.}}}{{=}}u^{\prime}_{\phi}(p)\Scr
B_{p}\,,\quad p\in P_{\mathfrak{t}}\,,$

covers $f_{\phi}\in\mathrm{Diff}(M)$ and satisfies
$\mathfrak{ad}(u_{\phi})=\phi$ by construction. Hence $\mathfrak{ad}$ is
surjective and thus we conclude. ∎

It is clear that $\mathfrak{ad}_{u}$ is trivial when
$u\in\mathrm{C}(\bm{\upzeta})$. Intuitively speaking, elements in
$\mathrm{C}_{\pi}(P_{\mathfrak{t}},\mathcal{J})$ behave as gauge
transformations on a principal torus bundle and therefore act trivially on the
curvature of any connection. In fact, the arithmetic U-duality group
$\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))$ identifies with the _discrete remnant_
(in the sense of [27]) of the gauge group $\mathrm{Aut}(P_{\mathfrak{t}})$,
which shows that U-dualities in supergravity are but gauge transformations of
the underlying Siegel bundle, a fact that elucidates their geometric origin.
We discuss next a few examples. An in-depth study of the gauge U-duality group
will be presented in a separate publication.

###  5.1. Rank-zero Siegel bundle

Let $(\pi,\mathcal{H},\mathcal{G})$ be a scalar bundle over $M$ with
submersion $\pi:X\rightarrow M$ and consider the rank zero Siegel bundle
$P_{0}=(\mathrm{id}_{X}:X\rightarrow X)$ on $X$ (which is necessarily
trivial). In this case $\mathrm{Aff}_{\mathfrak{t}}$ is the trivial group and
we have:

$\mathrm{Aut}(P_{0})=\mathrm{Diff}(X)\,,\qquad\mathrm{Aut}_{\pi}(P_{0})=\mathrm{Aut}_{b}(\pi)\,,\qquad\mathrm{Aut}_{b}(P_{0})=\\{\mathrm{id}_{X}\\}~{}~{},$

as well as:

${\bm{\Delta}}(P_{0})=X\times\left\\{0\right\\}\,.$

Let $\bm{\upzeta}_{0}=(\pi,\mathcal{H},\mathcal{G},P_{0},\mathcal{J}_{0})$,
where $\mathcal{J}_{0}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{id}_{{\bm{\Delta}}(P_{0})}$. Then:

$\mathrm{U}(\bm{\upzeta}_{0})=\mathrm{Aut}^{0}_{b}(\pi,\mathcal{H},\mathcal{G})=\left\\{u\in\mathrm{Aut}_{b}(\pi)\,\,|\,\,\mathcal{H}_{u}=\mathcal{H}\,,\,\,\mathcal{G}_{u}=\mathcal{G}\right\\}\,.$

Lemma 2.27 shows that $\mathrm{U}(\bm{\upzeta}_{0})$ is isomorphic to the
commmutant of the holonomy group of $\mathcal{H}$ inside the isometry group of
the typical fiber of $(\mathcal{M},\mathcal{G})$ of $(\pi,\mathcal{G})$. When
the holonomy of $\mathcal{H}$ is trivial, $\mathrm{U}(\bm{\upzeta}_{0})$
reduces to the orientation-preserving isometry group ${\rm
Iso}(\mathcal{M},\mathcal{G})$ of the scalar manifold but is in general
different. In particular, when the holonomy of $\mathcal{H}$ is full, that is,
equal to ${\rm Iso}(\mathcal{M},\mathcal{G})$, then
$\mathrm{U}(\bm{\upzeta}_{0})$ is isomorphic to the center of ${\rm
Iso}(\mathcal{M},\mathcal{G})$ and hence possibly trivial. This gives a simple
and explicit example illustrating how the supergravity duality group may
differ from its local counterpart considered in the literature, which in this
case would correspond always with ${\rm Iso}(\mathcal{M},\mathcal{G})$.

###  5.2. Rank zero scalar bundle

Let $(\pi,\mathcal{H},\mathcal{G})$ be the rank zero scalar bundle, that is,
$X=M$, $\pi\colon M\to M$ is the identity map, $\mathcal{H}=TM$ is canonically
identified with the tangent bundle of $M$ and $\mathcal{G}$ is the trivial
metric on the rank zero vector bundle over $M$. Then, the isometry group of
the typical fiber of $(\pi,\mathcal{H},\mathcal{G})$ is the trivial group
whence the short exact sequence:

$1\to\mathrm{Aut}_{b}(P_{\mathfrak{t}},\mathcal{J})\to\mathrm{U}(\bm{\upzeta})\to\mathrm{Aut}_{b}^{0}(\pi,\mathcal{H},\mathcal{G})\to
1\,,$

reduces to an isomorphism of groups
$\mathrm{U}(\bm{\upzeta})=\mathrm{Aut}_{b}(P_{\mathfrak{t}},\mathcal{J})$
where $(P_{\mathfrak{t}},\mathcal{J})$ is a polarized Siegel bundle over $M$.
Therefore, the gauge U-duality group reduces to the gauge group of the Siegel
bundle underlying the given bosonic supergravity. This corresponds, in fact,
with the electromagnetic gauge duality group of the abelian gauge theory
determined by $(P_{\mathfrak{t}},\mathcal{J})$ as explained in detail in [26].

###  5.3. Holonomy-trivial scalar bundle

Consider a holonomy-trivial scalar bundle $(\pi,\mathcal{H},\mathcal{G})$ in
the presentation:

$X=M\times\mathcal{M}\,,\qquad\mathcal{H}=TM^{\mathrm{pr_{1}}}\,,$

where $\mathcal{M}$ is an oriented $n_{s}$-dimensional manifold and
$\mathrm{pr}_{1}\colon M\times\mathcal{M}\to M$ is the canonical projection
onto the first factor. In this situation
$\mathcal{V}=T\mathcal{M}^{\mathrm{pr_{2}}}$, where $\mathrm{pr}_{2}\colon
M\times\mathcal{M}\to\mathcal{M}$ is the canonical projection onto the second
factor, and the vertical metric $\mathcal{G}$ descends to a Riemannian metric
on $\mathcal{M}$ which we denote by the same symbol for ease of notation.
Furthermore, consider the vertically polarized Siegel bundle
$(P_{\mathfrak{t}},\mathcal{J})$ obtained by pull-back through
$\mathrm{pr}_{2}\colon M\times\mathcal{M}\to\mathcal{M}$ of a vertically
polarized Siegel bundle on $(\mathcal{M},\mathcal{G})$, which we denote again
by $(P_{\mathfrak{t}},\mathcal{J})$ for ease of notation. Then:

$\mathrm{Aut}_{b}(\pi)=\mathrm{Diff}(\mathcal{M})\,,$

where $\mathrm{Diff}(\mathcal{M})$ the group of oriented diffeomorphisms of
$\mathcal{M}$. In particular, we obtain the following short exact sequence:

$1\to\mathrm{Aut}_{b}(P_{\mathfrak{t}})\to\mathrm{Aut}(P_{\mathfrak{t}})\to\mathrm{Diff}_{0}(\mathcal{M})\to
1\,,$

where $\mathrm{Diff}_{0}(\mathcal{M})$ denotes the subgroup of
$\mathrm{Diff}(\mathcal{M})$ that can be covered by elements in
$\mathrm{Aut}(P_{\mathfrak{t}})$. Here $P_{\mathfrak{t}}$ is considered as a
Siegel bundle over $\mathcal{M}$. In this case, the gauge U-duality group is:

$\mathrm{U}(\bm{\upzeta})=\left\\{u\in\mathrm{Aut}(P_{\mathfrak{t}})\,\,|\,\,\mathcal{G}_{u}=\mathcal{G}\,,\,\,\mathcal{J}_{u}=\mathcal{J}\right\\}\,,$

and fits into the short exact sequence:

$1\to\mathrm{Aut}_{b}(P_{\mathfrak{t}},\mathcal{J})\to\mathrm{U}(\bm{\upzeta})\to{\rm
Iso}_{0}(\mathcal{M},\mathcal{G})\to 1\,,$

where ${\rm Iso}_{0}(\mathcal{M},\mathcal{G})$ is group of those isometries of
$(\mathcal{M},\mathcal{G})$ which can be covered by elements in
$\mathrm{U}(\bm{\upzeta})$. Assume in addition that $P_{\mathfrak{t}}$ is
topologically trivial and write:

$P_{\mathfrak{t}}=\mathcal{M}\times\mathrm{Aff}_{\mathfrak{t}}=\mathcal{M}\times\left[\mathrm{U}(1)^{2n_{v}}\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})\right]\,.$

in a fixed trivialization. We have:

$\displaystyle\mathrm{Aut}_{b}(P_{\mathfrak{t}})=\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}}\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z}))\,,$
$\displaystyle\mathrm{Aut}(P_{\mathfrak{t}})=\mathrm{Diff}(\mathcal{M})\ltimes\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}}\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z}))\,.$

Since $\mathrm{Sp}_{\mathfrak{t}}(2n,\mathbb{Z})$ is discrete, we find:

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

as well as:

$\mathrm{U}(\bm{\upzeta})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(f,u_{T},\mathfrak{U})\in{\rm
Iso}(\mathcal{M},\mathcal{G})\ltimes(\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}})\rtimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z}))\,\,|\,\,\mathfrak{U}\mathcal{J}\mathfrak{U}^{-1}=\mathcal{J}\circ
f\right\\}\,.$

In particular, the morphism
$\mathfrak{ad}\colon\mathrm{U}(\bm{\upzeta})\to\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))$
is given explicitly by:

$\mathfrak{ad}((f,u_{T},\mathfrak{U}))=(f,\mathfrak{U})\in{\rm
Iso}(\mathcal{M},\mathcal{G})\ltimes\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})\,,$

The short exact sequence:

$1\to\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}})\hookrightarrow\mathrm{U}(\bm{\upzeta})\xrightarrow{\mathfrak{ad}}\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))\to
1\,,$

shows how
$\mathrm{C}(\bm{\upzeta})=\mathcal{C}^{\infty}(\mathcal{M},\mathrm{U}(1)^{2n_{v}})$
captures the _non-discrete_ gauge transformations in
$\mathrm{U}(\bm{\upzeta})$, which act trivially on the adjoint bundle of $P$.
Consequently, we have:

$\mathrm{U}(\bm{\Phi}(\bm{\upzeta}))\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\left\\{(f,\mathfrak{U})\in{\rm
Iso}(\mathcal{M},\mathcal{G})\times\mathrm{Sp}_{\mathfrak{t}}(2n_{v},\mathbb{Z})\,\,|\,\,\mathfrak{U}\mathcal{J}\mathfrak{U}^{-1}=\mathcal{J}\circ
f\right\\}\,,$

which illustrates the explicit dependence of the arithmetic U-duality group on
the type $\mathfrak{t}\in\mathrm{Div}^{n}$ of its underlying Siegel bundle
$P_{\mathfrak{t}}$.

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