# Chern-Simons field theory on the general affine group, $3d$-gravity and the
extension of Cartan connections

S. Capriotti  Departamento de Matemática
Universidad Nacional del Sur and Conicet
8000 Bahía Blanca
Argentina<EMAIL_ADDRESS>
###### Abstract.

The purpose of this article is to study the correspondence between
$3d$-gravity and the Chern-Simons field theory from the perspective of
geometric mechanics, specifically in the case where the structure group is the
general affine group. To accomplish this, the paper discusses a variational
problem of the Chern-Simons type on a principal fiber bundle with this group
as its structure group. The connection to the usual Chern-Simons theory is
established by utilizing a generalization, in the context of Cartan
connections, of the notion of extension and reduction of connections.

##  1\. Integrable field theories and gravity

Chern-Simons field theory is a well-known type of gauge field theory [CS74,
DJT82, Fre95] whose quantization yields to topological field theory [Wit88a,
RSW89]. In its more general setting [Fre95], the fields in Chern-Simons gauge
theory are connections on any $K$-principal bundle with fixed base space. In
this vein, the Chern-Simons action is considered as a function on the
(infinite dimensional) manifold of the $K$-connections on a fixed base space
$M$, and it is evident that the construction of this manifold requires a
precise knowledge of every $K$-principal bundle on $M$. Although this
operation can be performed successfully (getting more complicated as the
dimension of $M$ increases), this scheme is out of range of the geometrical
formulation for field theory [GIM97, Ble81, LSVn15, LMD03, EEMLRR96], because
in this approach the fields should be sections of a definite bundle
(preferably, of finite dimension). A successful formulation for Chern-Simons
field theory fitting in this geometrical scheme is described in [Tej04], using
a variational problem posed by local Lagrangians; a multisymplectic
formulation for Chern-Simons field theory can be found in [GGRR23], where the
structure of the constraints arising from the singular nature of the
Lagrangian is studied. The formulation we will use in this article is inspired
in [Wis+09], and uses Cartan connections as fields.

On the other hand, it is an interesting result [AT86, Wit88] that when gauge
group is the Poincaré group $\text{ISO}\left(2,1\right)$, Chern-Simon field
theory in dimension $3$ can be related with Palatini gravity on a spacetime of
the same dimension. This correspondence is achieved by the splitting

$\mathfrak{iso}\left(2,1\right)=\mathfrak{so}\left(2,1\right)\oplus\mathbb{R}^{3},$

that decomposes the field $A$ in two parts, one living in $\mathbb{R}^{3}$ and
another in $\mathfrak{so}\left(2,1\right)$; the idea is to recognize each of
these fields as a vierbein and a $\mathfrak{so}\left(2,1\right)$-connection
respectively, which can be seen as the basic fields for the Palatini
description of general relativity. In [Wis+09], this scheme is generalized to
pair of algebras $\mathfrak{g}\subset\mathfrak{h}$ defining a Cartan
connection on a $G$-principal bundle $\pi:P\to M$.

A remarkable fact is that, in some of these descriptions of Chern-Simons
theory and its connection with gravity, is usual a certain lack of
definiteness with respect to the nature of the principal bundle to which the
connections of the theory belongs. While the corresponding base space and
structure group are properly set, it is avoided any precision on the
geometrical characteristics of such bundle. Not that it keeps people away from
working with Chern-Simons gauge theory: It is in this context where frameworks
like the one discussed by Freed are fruitful. Moreover, the informed reader
could recall that when dealing with gravity, a metric is available, and it can
be used to select suitable subbundles of the frame bundle. Nevertheless, this
answer should be considered as partial, and in fact might put us in a
paradoxical situation, because the metric is part of the dynamical fields in
gravity, and so it should be necessary to solve the equations of motion of
gravity before constructing the bundle where the fields should live.
Therefore, it seems to appear a tension between the formalism describing field
theory from a geometric viewpoint, and the characteristics that a Chern-Simons
gauge theory must have in order to represent gravity. At the end, the apparent
paradox is solved by invoking the gauge symmetry of the Lagrangian, but it
could be interesting to explore how to deal with this situation from a
geometrical point of view.

These considerations set the aims of the article. Basically, we are looking
for a formulation of the correspondence between Chern-Simons field theory and
gravity (for spacetime dimension $m=3$) where the structure groups of the
involved principal bundle are respectively the affine general group
$A\left(3\right)$ and the general linear group $GL\left(3\right)$. In order to
achieve this objective, we will generalize the scheme posed by Wise. The main
tool used for the generalization is the (as far as I know, novel) formulation
of a concept equivalent to _extension of a connection_ for Cartan connections
(see Section 4 below). The idea for the generalization is that any Cartan
connection can be seen as a principal connection on a suitable bundle (i.e.
Proposition 3); thus the extension of a Cartan connection is the Cartan
connection induced by the extension of this associated principal connection.
As we will see, a drawback of this construction is that, in general, the
connections obtained will be of more general nature than Cartan connections
(are generalized Cartan connections, as defined in [AM95]).

Let us describe in some detail how this scheme will be implemented. In the
formulation of Wise, the basic geometrical data is a $K$-principal bundle
$R_{\zeta}$, and its fields are described by a Cartan connection taking values
in a Lie algebra $\mathfrak{g}$ such that $\mathfrak{k}\subset\mathfrak{g}$.
Let us recall that the Lorentz group $K=SO\left(m-1,1\right)$ becomes a
subgroup of different Lie groups $G$ in the Wise scheme, depending on the sign
of the cosmological constant $\Lambda$; namely, we have that the total group
$G$ is in each case

$G=\begin{cases}SO\left(m,1\right)&\Lambda>0\\\
K\oplus\mathbb{R}^{m}&\Lambda=0\\\ SO\left(m-1,2\right)&\Lambda<0.\end{cases}$

$K$ enters as subgroup in the case $\Lambda<0$ through the immersion

$\iota_{-}\left(A\right)=\begin{bmatrix}A&0\\\ 0&1\end{bmatrix}$

for every $A\in K$, and

$\iota_{+}\left(A\right)=\begin{bmatrix}1&0\\\ 0&A\end{bmatrix}$

in the case $\Lambda>0$. In the present article we will focus in the case
$\Lambda=0$; the other cases will be considered elsewhere.

In order to see how to proceed for the generalization of this scheme to a
principal bundle with structure group $GL\left(m\right)$, let us consider the
following diagram of Lie groups:

${{GL\left(m\right)}}$${{K}{}}$${GL\left(m\right)\oplus\mathbb{R}^{m}}$${{K\oplus\mathbb{R}^{m}}}$

Arrows are induced by canonical inclusions. We want to promote it to a
commutative diagram involving principal bundles with structure groups borrowed
from the nodes of this diagram. Now, what kind of $K$-principal bundle
$R_{\zeta}$ on a manifold $M$ has a Cartan connection with values in the Lie
algebra $\mathfrak{g}=\mathfrak{k}\oplus\mathbb{R}^{m}$? A possible answer can
be found using geometrical considerations. In fact, as it is indicated in
[Sha97, Bar04], whenever the map

$\text{Ad}:K\to
GL\left(\mathfrak{k}\oplus\mathbb{R}^{m}/\mathfrak{k}\right)=GL\left(m\right)$

is injective, $R_{\zeta}$ becomes a $K$-structure, namely, it is a
$K$-subbundle of the frame bundle $LM$; in this case, the previous diagram has
an analogous diagram at principal bundle level

${M}$${LM}$${R_{\zeta}}$${AM}$${LM\left[GL\left(m\right)\oplus\mathbb{R}^{m}\right]}$${R_{\zeta}\left[K\oplus\mathbb{R}^{m}\right]}$$\scriptstyle{\gamma}$$\scriptstyle{\tau}$$\scriptstyle{\beta\circ\tau}$

Here $AM$ indicates the affine frame bundle of the spacetime $M$, maps
$\beta:AM\to LM,\gamma:LM\to AM$ are canonical maps between these bundles (see
Appendix C), and the symbol $P\left[G\right]$, where $\pi:P\to M$ is an
$H$-principal bundle and $G\supset H$ is a Lie group containing $H$, indicates
the extension of $P$ by enlarging its structure group to $G$ (this
construction is detailed in Section 3 below). Therefore, we will restrict
ourselves to Cartan connections describing Palatini gravity on a $K$-structure
$R_{\zeta}$; in this setting we devise a method to extend them to Cartan
connections on $LM$, in order to reproduce the correspondence between Chern-
Simons field theory and Palatini gravity in a case where the structure group
is the general linear group $GL\left(m\right)$.

Let us briefly describe the structure of the article. The geometrical tools
used throughout the article are presented in Section 2. The description of
Cartan connections as sections of a bundle is carried out in Section 3; this
description is necessary due to the type of geometrical formulation adopted
for the variational problems. The operations of extension and reduction for
generalized Cartan connections are developed in Section 4. Although the
author’s knowledge of Cartan connections is far from exhaustive, it seems that
these operations, even though they are a direct consequence of the procedure
of identification between Cartan connections and principal connections on the
extended bundle, have not been described previously in the literature. If this
is indeed the case, this section represents an original contribution of the
present article. The variational problems for Chern-Simons and gravity are
described in Section 5. The main contribution of this section is to find a
global formulation for the Chern-Simons field theory in terms of jet bundles
(Section 5.1); the price that must be paid for this formulation is the
appearance of an additional constraint (see Remark 3). The main result of the
article is discussed in Section 6: A variational problem of the Chern-Simons
type has been found, but with a structure group given by the general affine
group (Section 6.1) such that the extremals of any Chern-Simons theory
(described in terms of the Section 5.1) are in a bijective correspondence with
their extremals through the extension and reduction operations of Section 4.

#### Notations

We are adopting here the notational conventions from [Sau89] when dealing with
bundles and its associated jet spaces. It means that, given a bundle $\pi:P\to
M$, there exists a family of bundles and maps fitting in the following diagram

${\cdots}$${J^{k+1}\pi}$${J^{k}\pi}$${\cdots}$${J^{1}\pi}$${P}$${\cdots}$${\cdots}$${M}$$\scriptstyle{\pi_{k+1,k}}$$\scriptstyle{\pi_{k+1}}$$\scriptstyle{\pi_{k,k-1}}$$\scriptstyle{\pi_{k}}$$\scriptstyle{\pi_{21}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{10}}$$\scriptstyle{\pi}$

Sections of $\pi:P\to M$ will be indicate by the symbol $\Gamma\pi$. The set
of vectors tangent to $P$ in the kernel of $T\pi$ will be represented with the
symbol $V\pi\subset TP$. In this regard, the set of vector fields which are
vertical for a bundle map $\pi:P\to M$ will be indicated by
$\mathfrak{X}^{V\pi}\left(P\right)$. The space of differential $p$-forms,
sections of $\Lambda^{p}(T^{*}Q)\to Q$, will be denoted by $\Omega^{p}(Q)$. We
also write $\Lambda^{\bullet}(Q)=\bigoplus_{j=1}^{\dim Q}\Lambda^{j}(T^{*}Q)$.
If $f\colon P\to Q$ is a smooth map and $\alpha_{x}$ is a $p$-covector on $Q$,
we will sometimes use the notation $\alpha_{f(x)}\circ T_{x}f$ to denote its
pullback $f^{*}\alpha_{x}$. If $P_{1}\to Q$ and $P_{2}\to Q$ are fiber bundles
over the same base $Q$ we will write $P_{1}\times_{Q}P_{2}$ for their fibered
product, or simply $P_{1}\times P_{2}$ if there is no risk of confusion.
Unless explicitly stated, the canonical projections onto its factor will be
indicated by

$\text{pr}_{i}:P_{1}\times P_{2}\to P_{i},\qquad i=1,2.$

Given a manifold $N$ and a Lie group $G$ acting on $N$, the symbol
$\left[n\right]_{G}$ for $n\in N$ will indicate the $G$-orbit in $N$
containing $n$; the canonical projection onto its quotient will be denoted by

$p_{G}^{N}:N\to N/G.$

Also, if $\mathfrak{g}$ is the Lie algebra for the group $G$, the symbol
$\xi_{N}$ will represent the infinitesimal generator for the $G$-action
asssociated to $\xi\in\mathfrak{g}$. Finally, Einstein summation convention
will be used everywhere.

We will be working with several different types of connections, so it will be
necessary to have a notational convention for them. Giving a $H$-principal
bundle $p:Q\to M$, the first jet bundle $J^{1}p$ has a $\mathfrak{h}$-valued
$1$-form that is called _canonical connection_ ; this form will be indicated
by the symbol

$\theta_{J^{1}p}\in\Omega^{1}\left(J^{1}p,\mathfrak{h}\right).$

Because it will become a connection form, its associated curvature form will
be denoted by

$\Theta_{J^{1}p}\in\Omega^{2}\left(J^{1}p,\mathfrak{h}\right).$

On the other hand, two different types of connection will be considered on a
principal bundle $p:Q\to M$. First, it could have a connection form, which
will be indicated as $\omega_{Q}$; its asscoiated curvature form will be
$\Omega_{Q}$. Moreover, if the structure group $H$ is a subgroup of a bigger
Lie group $G\supset H$, then we can have on $Q$ a $\mathfrak{g}$-valued
$1$-form, which will be denoted as $A_{Q}$, and the curvature connection will
become $F_{Q}$.

##  2\. Some geometrical tools

The following section collects geometrical facts regarding principal bundles
and Cartan connections on them.

###  2.1. Geometry of the jet space of a principal bundle

Throughout the article, we will make extensive use of the geometrical tools
related to the jet space associated to a principal bundle, as well as its
connection bundle, as they are discussed in [Gar72, CM01]. So, in order to
proceed, let $p:Q\to N$ be a principal bundle with structure group $H$; then
we can lift the right action of $H$ on $Q$ to a $H$-action on $J^{1}p$, and so
define the bundle

$\overline{p}:C\left(Q\right):=J^{1}p/H\to M$

fitting in the following diagram

${J^{1}p}$${Q}$${C\left(Q\right)}$${M}$$\scriptstyle{p_{10}}$$\scriptstyle{p_{H}^{J^{1}p}}$$\scriptstyle{p}$$\scriptstyle{\overline{p}}$

It can be proved [CM01] that this diagram defines $J^{1}p$ as a pullback,
namely, that

(2.1) $J^{1}p=p^{*}Q=Q\times_{M}C\left(Q\right).$

We know that $J^{1}p$ comes equipped with the _contact structure_ , that can
be described by means of a $Vp$-valued $1$-form

(2.2)
$\left.\theta_{J^{1}p}\right|_{j_{x}^{1}s}:=T_{j_{x}^{1}s}p_{10}-T_{x}s\circ
T_{j_{x}^{1}s}p_{1};$

moreover, because $p:Q\to N$ is a principal bundle, we have the bundle
isomorphism on $Q$

$Vp\simeq Q\times\mathfrak{h}.$

It means that we can consider $\theta$ as a $\mathfrak{h}$-valued $1$-form; in
fact, with respect to the $K$-principal bundle structure

$p_{H}^{J^{1}p}:J^{1}p\to C\left(Q\right),$

the $1$-form $\theta$ becomes a connection form, dubbed _canonical connection
form_. It has the following property.

######  Proposition 1.

Let $\Gamma_{Q}:Q\to J^{1}p$ be a connection on $Q$. Then its connection form
$\omega_{Q}\in\Omega^{1}\left(Q,\mathfrak{h}\right)$ can be constructed from
the canonical connection form through pullback along $\Gamma_{Q}$,

$\omega_{Q}=\Gamma_{Q}^{*}\theta_{J^{1}p}.$

For any manifold $M$ of dimension $m$, the bundle $\tau:LM\to M$ is the _frame
bundle of $M$_, is defined through

(2.3) $LM:=\bigcup_{x\in M}\left\\{u:\mathbb{R}^{m}\to T_{x}M\text{ linear and
with inverse}\right\\}.$

It has a canonical free $GL\left(m,\mathbb{R}\right)$-action, given by the
formula

$u\cdot A:=u\circ A,\qquad u\in LM,A\in GL\left(m,\mathbb{R}\right).$

Let $\eta\in M_{m}\left(\mathbb{R}\right)$ be a non degenerate symmetric
$m\times m$-matrix with real entries; for definiteness, we will fix

$\eta:=\begin{bmatrix}-1&0&\cdots&0\\\ 0&1&&0\\\ \vdots&&\ddots&\vdots\\\
0&\cdots&0&1\end{bmatrix},$

although the constructions we will consider in the present article should work
with any signature. Then we have a Lie group $K\subset
GL\left(m,\mathbb{R}\right)$ defined by

(2.4) $K:=\left\\{A\in M_{m}\left(\mathbb{R}\right):A\eta
A^{T}=\eta\right\\}.$

Then we have an action of $K$ on $LM$; it yields to a bundle

$\tau_{\Sigma}:\Sigma:=LM/K\to M.$

######  Lemma 1.

The bundle $\Sigma$ is the bundle of metrics of $\eta$-signature on $M$.

Let us indicate with $\mathfrak{k}$ the Lie algebra of $K$; then, we have that

$\mathfrak{k}:=\left\\{a\in\mathfrak{gl}\left(m,\mathbb{R}\right):a\eta+\eta
a^{T}=0\right\\}.$

Accordingly, we can define the $K$-invariant subspace

(2.5)
$\mathfrak{p}:=\left\\{a\in\mathfrak{gl}\left(m,\mathbb{R}\right):a\eta-\eta
a^{T}=0\right\\},$

usually called _transvections_ (see [Wis+09]); it follows that

$\mathfrak{gl}\left(m,\mathbb{R}\right)=\mathfrak{k}\oplus\mathfrak{p}.$

We will need this result concerning some natural properties of the canonical
connections.

######  Lemma 2.

Let $p:Q\to N$ be a $H$-principal bundle and
$i_{\zeta}:R_{\zeta}\hookrightarrow Q$ a $L$-principal subbundle, with
projection

$p_{\zeta}:R_{\zeta}\to N.$

We have the following relation between the canonical connections on $J^{1}p$
and $J^{1}p_{\zeta}$, namely

$\left(j^{1}i_{\zeta}\right)^{*}\theta_{J^{1}p}=\theta_{J^{1}p_{\zeta}},$

where $\theta_{J^{1}p}\in\Omega^{1}\left(J^{1}p,\mathfrak{h}\right)$ and
$\theta_{J^{1}p_{\zeta}}\in\Omega^{1}\left(J^{1}p_{\zeta},\mathfrak{l}\right)$
are the corresponding canonical connection forms.

###### Proof.

The lemma follows from the formula

$\left.\theta_{J^{1}p}\right|_{j_{x}^{1}s}\circ
T_{j_{x}^{1}\sigma}\left(j^{1}i_{\zeta}\right)=T_{\sigma\left(x\right)}i_{\zeta}\circ\left.\theta_{J^{1}p_{\zeta}}\right|_{j_{x}^{1}\sigma},$

valid for any $j_{x}^{1}s=j^{1}i_{\zeta}\left(j_{x}^{1}\sigma\right)$ and
$j_{x}^{1}\sigma\in J^{1}p_{\zeta}$. ∎

Assuming that some topological conditions on the manifold $M$ hold111The
existence of a metric with $\left(p,q\right)$-signature is equivalent to the
splitting of the tangent bundle $TM$ in a direct sum of vector subbundles of
rank $p$ and $q$ respectively., a family

$\left\\{O_{\zeta}:\zeta:M\to\Sigma\text{ metric}\right\\}$

of subbundles of $LM$ can be constructed; namely, let us define

$O_{\zeta}:=\left\\{u\in
LM:\zeta\left(u\left(w_{1}\right),u\left(w_{2}\right)\right)=\eta\left(w_{1},w_{2}\right)\text{
for all }w_{1},w_{2}\in\mathbb{R}^{m}\right\\}.$

Here $\eta:\mathbb{R}^{m}\times\mathbb{R}^{m}\to\mathbb{R}$ indicates the
bilinear form associated to the matrix $\eta$.

###  2.2. Introduction to Cartan geometry

Let us recall the definition of Cartan connections on a principal bundle
[Wis10, AM95, Sha97]. Let $\left(G,H\right)$ be a pair of Lie groups such that
$H\subset G$ is a closed subgroup and $G/H$ is connected. Recall that for
every principal bundle $\pi:P\to M$ with structure group $H$ we have a map

$\kappa_{H}:VP\to\mathfrak{h}$

such that

$\kappa_{H}\left(\xi_{P}\left(u\right)\right)=\xi$

for every $\xi\in\mathfrak{h}$ and every $u\in P$.

######  Definition 1 (Cartan geometry).

A _Cartan geometry_ modelled on the pair $\left(G,H\right)$ is an
$H$-principal bundle $\pi:P\to M$ together with a $\mathfrak{g}$-valued
$1$-form $A$ on $P$ such that

  1. (i)

$A_{u}:T_{u}P\to\mathfrak{g}$ is a linear isomorphism for every $u\in P$,

  2. (ii)

$R_{h}^{*}A=\text{Ad}_{h^{-1}}\circ A$ for every $h\in H$, and

  3. (iii)

$\left.A\right|VP=\kappa_{H}$.

The form $A$ is called the _(principal) Cartan connection_ for the given
Cartan geometry.

A Cartan geometry on a manifold provides us with a associated bundle
description of its tangent bundle, as the following result indicates.

######  Proposition 2.

Let $\left(\pi:P\to M,A\right)$ be a Cartan geometry modeled on the pair
$\left(G,H\right)$. Then $TM$ is isomorphic as vector bundle to the associated
bundle

$P\times_{H}\mathfrak{g}/\mathfrak{h},$

where $H$ acts on $\mathfrak{g}/\mathfrak{h}$ through the quotient
representation induced by the $\text{Ad}\left(H\right)$-action on
$\mathfrak{g}$.

###### Proof.

Let $u\in P$; because $A_{u}:T_{u}P\to\mathfrak{g}$ satisfies
$\left.A\right|VP=\kappa_{H}$, it induces a morphism

$\phi_{u}:T_{x}M\to\mathfrak{g}/\mathfrak{h}:v_{x}\mapsto\pi_{\mathfrak{g}/\mathfrak{h}}\left(A_{u}\left(\widehat{v_{x}}\right)\right),$

where $x=\pi\left(u\right)$, $\widehat{v_{x}}\in T_{u}P$ is any element
projecting onto $v_{x}$ via $T_{u}\pi$, and

$\pi_{\mathfrak{g}/\mathfrak{h}}:\mathfrak{g}\to\mathfrak{g}/\mathfrak{h}$

is the canonical projection onto the quotient. It becomes an isomorphism
because $A_{u}$ is also an isomorphism, and given that this map is
$H$-equivariant, we have that

$\phi_{u\cdot h}=\text{Ad}_{h^{-1}}\circ\phi_{u}$

for every $u\in P$ ad $h\in H$. Therefore, we have the isomorphism of bundles
$\overline{q}:P\times_{H}\mathfrak{g}/\mathfrak{h}\to TM$ given by

$\overline{q}\left(\left[u,\left[\xi\right]_{\mathfrak{h}}\right]_{H}\right):=\left(\phi_{u}\right)^{-1}\left(\left[\xi\right]_{\mathfrak{h}}\right),$

that can be proved to be well-defined. ∎

We should stress that a Cartan connection is not a principal connection on
$P$, because it takes values in the larger Lie algebra $\mathfrak{g}$;
nevertheless, there exists a relationship between these concepts [Pan98].
Consider $H$ as a $G$-space through left multiplication, and construct the
associated bundle

$P\left[G\right]:=P\times_{H}G;$

it is a $G$-principal bundle, and we have a canonical inclusion
$\gamma:P\hookrightarrow P\left[G\right]$ given by

$\gamma\left(u\right):=\left[u,e\right]_{H}.$

It is known that every morphism of $G$-principal bundles on a fixed basis $M$
is an isomorphism [Ste99]; this fact gives rise to the following property of
the extension procedure described above.

######  Lemma 3.

Let $P$ be an $H$-principal bundle on $M$ and suppose further that it is a
subbundle of a $G$-principal bundle $Q$ on $M$. Then $Q\simeq
P\left[G\right]$.

###### Proof.

Let $i:P\hookrightarrow Q$ be the immersion of $P$ into $Q$; then we have the
morphism

$\phi:P\left[G\right]\to Q:\left[u,g\right]_{H}\mapsto i\left(u\right)\cdot
g.$

It is a morphism of $G$-principal bundles on $M$; therefore, $Q$ and
$P\left[G\right]$ are isomorphic, as desired. ∎

The extension of a principal bundle can be used to relate Cartan connections
with principal connections [AM95, Sha97, Pan98].

######  Proposition 3.

The Cartan connection $A:TP\to\mathfrak{g}$ induces on $P\left[G\right]$ a
unique principal connection form
$\widetilde{A}:TP\left[G\right]\to\mathfrak{g}$ such that

$\gamma^{*}\widetilde{A}=A.$

Conversely, suppose that $\dim P=\dim G$ and let $\widetilde{A}$ be a
principal connection on $P\left[G\right]$ such that

(2.6) $\ker\widetilde{A}\cap T\gamma\left(TP\right)=\left\\{0\right\\}.$

Then $A:=\gamma^{*}\widetilde{A}$ is a Cartan connection on $P$.

###### Proof.

Because the quotient map $p_{H}^{P\times G}:P\times G\to P\left[G\right]$ is
surjective, any element $W\in T_{\left[u,g\right]_{H}}P\left[G\right]$ can be
represented as

$W=T_{\left(u,g\right)}p_{H}^{P\times G}\left(X_{u},T_{e}L_{g}\zeta\right)$

for $X_{u}\in T_{u}P$ and $\zeta\in\mathfrak{g}$. Then we define

$\left.\widetilde{A}\right|_{\left[u,g\right]_{H}}\left(W\right):=\zeta+\text{Ad}_{g^{-1}}\left.A\right|_{u}\left(X_{u}\right).$

It can be proved that it is well-defined, and defines a principal connection
on $P\left[G\right]$.

Now, let us suppose that we have a principal connection $\widetilde{A}$ on
$P\left[G\right]$ and define

$A:=\gamma^{*}\widetilde{A}:TP\to\mathfrak{g}.$

Let us verify that it is a Cartan connection on $P$. Because of the condition
(2.6), the map

$A_{u}:T_{u}P\to\mathfrak{g}$

is a monomorphism; because $\dim{T_{u}P}=\dim P=\dim G=\dim\mathfrak{g}$, we
have that this map is an isomorphism.

Aditionally, for any $h\in H$

$\displaystyle R_{h}^{*}A$ $\displaystyle=R_{h}^{*}\gamma^{*}\widetilde{A}$
$\displaystyle=\left(\gamma\circ R_{h}\right)^{*}\widetilde{A}$
$\displaystyle=\left(R_{h}\circ\gamma\right)^{*}\widetilde{A}$
$\displaystyle=\gamma^{*}R_{h}^{*}\widetilde{A}$
$\displaystyle=\gamma^{*}\left(\text{Ad}_{h^{-1}}\circ\widetilde{A}\right)$
$\displaystyle=\text{Ad}_{h^{-1}}\circ A.$

Finally, let us take $\xi\in\mathfrak{h}$; because of the identity

$T_{u}\gamma\left(\xi_{P}\left(u\right)\right)=\xi_{P\left[G\right]}\left(\gamma\left(u\right)\right)$

we will have that

$A_{u}\left(\xi_{P}\left(u\right)\right)=\widetilde{A}_{\gamma\left(u\right)}\left(T_{u}\gamma\left(\xi_{P}\left(u\right)\right)\right)=\widetilde{A}_{\gamma\left(u\right)}\left(\xi_{P\left[G\right]}\left(\gamma\left(u\right)\right)\right)=\xi$

for any $u\in P$. ∎

##  3\. Cartan connections and jet bundles

The basic idea for the geometrical interpretation of the correspondence
between Chern-Simons field theory and gravity is due to Wise [Wis+09, Wis10],
and uses a Chern-Simons Lagrangian evaluated on forms that are not principal
connections, but Cartan connections. As we have pointed out before, this
approach is not convenient when you try to understand Chern-Simons field
theory from the viewpoint of geometric mechanics (i.e., the setting described
in [GIM97, GIM04, Got91, Got91a, EEMLRR96]). Therefore, we will devote the
next section to translate the formalism of Cartan connections into the realm
of jet bundles, in order to have at our disposal a language suitable for the
description of Chern-Simons field theory from this viewpoint.

###  3.1. Canonical (generalized) Cartan connection on a jet bundle

Thus, we have that a Cartan connection on an $H$-principal bundle $P$ can be
seen as a principal connection $\widetilde{A}$ on the extended bundle
$P\left[G\right]$, provided that

  1. (a)

$P$ has the same dimension than $G$, and

  2. (b)

the horizontal spaces of this connection are complementary to the tangent
spaces of $P$ (viewed as subspaces of the tangent spaces of
$P\left[G\right]$).

We can reformulate the second condition in terms of jet bundles; in fact, let
us define the set

$U_{\gamma}:=\left\\{j_{x}^{1}s\in
J^{1}\pi_{\left[G\right]}:T_{s\left(x\right)}R_{g^{-1}}\left(T_{x}s\left(T_{x}M\right)\right)\cap
T_{u}\gamma\left(T_{u}P\right)=\left\\{0\right\\}\text{ iff
}s\left(x\right)=\left[u,g\right]\right\\}.$

Using decomposition (2.1), we can see that sections having its images in this
set corresponds exactly with connections on $P\left[G\right]$ that verify
condition b above. It gives us the following characterization for condition
(2.6) in terms of the jet bundle of the extended principal bundle
$P\left[G\right]$.

######  Proposition 4.

A connection $\widetilde{\Gamma}:P\left[G\right]\to J^{1}\pi_{\left[G\right]}$
will satisfy Equation (2.6) if and only if
$\widetilde{\Gamma}\left(\left[u,g\right]\right)\in U_{\gamma}$ for all
$\left[u,g\right]\in P\left[G\right]$.

Now, we have the following result.

######  Lemma 4.

The set $U_{\gamma}$ is open in $J^{1}\pi_{\left[G\right]}$.

###### Proof.

Given $V_{1},V_{2}\subset V$ a pair of subspaces of the (finite-dimensional)
vector space $V$, fix basis $B_{1},B_{2}$ and $B$ for each of them. Thus the
condition

$V_{1}\cap V_{2}=\left\\{0\right\\}$

is equivalent to $F_{k}\not=0$, where $F_{k}$ is the sum of the squares of the
$k\times k$ minors of the matrix formed by the components of the vectors in
$B_{1}\cup B_{2}$ respect to the basis $B$, where $k=\dim V_{1}+\dim V_{2}$.
Consider now

$\displaystyle V$ $\displaystyle=T_{s\left(x\right)}P\left[G\right]$
$\displaystyle V_{1}$
$\displaystyle=T_{s\left(x\right)}R_{g^{-1}}\left(T_{x}s\left(T_{x}M\right)\right)$
$\displaystyle V_{2}$ $\displaystyle=T_{u}\gamma\left(T_{u}P\right);$

in terms of local coordinates on $J^{1}\pi_{\left[G\right]}$, $F_{k}$ gives
rise to a polynomial in the jet variables, and the result follows. ∎

Therefore, instead of considering the variational problem for a Cartan
connection over the entire bundle $J^{1}\pi_{\left[G\right]}$, we can restrict
ourselves to the open set $U_{\gamma}$, thereby abandoning condition b above.
With this consideration in mind, let us generalize the notion of Cartan
connection; to this end we will use the notion of _generalized Cartan
connection_ , as it is defined in [AM95].

######  Definition 2 (Generalized Cartan connection).

A _generalized (principal) $\mathfrak{g}/\mathfrak{h}$-Cartan connection_ on
an $H$-principal bundle $\pi:P\to M$ is a $\mathfrak{g}$-valued $1$-form $A$
on $P$ such that

  1. (i)

$R_{h}^{*}A=\text{Ad}_{h^{-1}}\circ A$ for every $h\in H$, and

  2. (ii)

$\left.A\right|V\pi=\kappa_{H}$.

Let us consider the following diagram

${J^{1}\left(\pi\circ\text{pr}_{1}\right)}$${P\times
G}$${P}$${J^{1}\pi_{\left[G\right]}}$${P\left[G\right]}$${M}$$\scriptstyle{\left(\pi\circ\text{pr}_{1}\right)_{10}}$$\scriptstyle{j^{1}p^{P\times
G}_{H}}$$\scriptstyle{p^{P\times
G}_{H}}$$\scriptstyle{\text{pr}_{1}}$$\scriptstyle{\pi}$$\scriptstyle{\left(\pi_{\left[G\right]}\right)_{10}}$$\scriptstyle{\pi_{\left[G\right]}}$

The bundle $\pi\circ\text{pr}_{1}:P\times G\to M$ together with the action

$\left(u,g\right)\cdot\left(h,g^{\prime}\right):=\left(u\cdot
h,gg^{\prime}\right)$

is an $H\times G$-principal bundle. Thus we have the following result.

######  Lemma 5.

If
$\theta_{J^{1}\left(\pi\circ\text{pr}_{1}\right)}\in\Omega^{1}\left(J^{1}\left(\pi\circ\text{pr}_{1}\right),\mathfrak{h}\times\mathfrak{g}\right)$
denotes the canonical connection on $J^{1}\left(\pi\circ\text{pr}_{1}\right)$
and
$\theta_{J^{1}\pi_{\left[G\right]}}\in\Omega^{1}\left(J^{1}\pi_{\left[G\right]},\mathfrak{g}\right)$
is the canonical connection on $J^{1}\pi_{\left[G\right]}$, then

$\left(j^{1}p_{H}^{P\times
G}\right)^{*}\theta_{J^{1}\pi_{\left[G\right]}}=\text{pr}_{2}^{\mathfrak{h}\times\mathfrak{g}}\circ\theta_{J^{1}\left(\pi\circ\text{pr}_{1}\right)},$

where

$\text{pr}_{2}^{\mathfrak{h}\times\mathfrak{g}}:\mathfrak{h}\times\mathfrak{g}\to\mathfrak{g}$

stands for the projection onto the second factor.

###### Proof.

Recall that the canonical connection on the jet bundle of a principal bundle
$\pi:P\to M$ is nothing but the $V\pi$-valued contact form, which can be seen
as $\mathfrak{g}$-valued through the identification

$V\pi\simeq P\times\mathfrak{g}.$

It means that they have the property

$\left(j^{1}p_{H}^{P\times
G}\right)^{*}\theta_{J^{1}\pi_{\left[G\right]}}=Tp^{P\times
G}_{H}\circ\theta_{J^{1}\left(\pi\circ\text{pr}_{1}\right)},$

and the lemma follows from the identification between the vertical bundles and
the Lie algebras. ∎

###  3.2. Generalized Cartan connections as equivariant sections of a jet
bundle

Recall that a principal connection on $P$ can be represented by an equivariant
map $\Gamma:P\to J^{1}\pi$. The next result gives an analogous representation
for generalized Cartan connections.

######  Proposition 5.

Any generalized Cartan connection $A:TP\to\mathfrak{g}$ gives rise to a
$H$-equivariant bundle map

$\Gamma_{A}:P\to J^{1}\pi_{\left[G\right]}$

covering the immersion $\gamma:P\hookrightarrow P\left[G\right]$. Conversely,
for any $H$-equivariant bundle map $\Gamma:P\to J^{1}\pi_{\left[G\right]}$
making the following diagram

${J^{1}\pi_{\left[G\right]}}$${P}$${P\left[G\right]}$$\scriptstyle{\left(\pi_{\left[G\right]}\right)_{10}}$$\scriptstyle{\Gamma}$$\scriptstyle{\gamma}$

commutative, there exists a generalized Cartan connection
$A_{\Gamma}:TP\to\mathfrak{g}$ such that

$A_{\Gamma}=\Gamma^{*}\theta_{J^{1}\pi_{\left[G\right]}}.$

###### Proof.

The map $\Gamma_{A}:P\to J^{1}\pi_{\left[G\right]}$ is given by

$\Gamma_{A}\left(u\right):v\in T_{x}M\mapsto
T_{\left(u,e\right)}p_{H}^{P\times
G}\left(\widehat{v}_{u},-A_{u}\left(\widehat{v}_{u}\right)\right),$

where $\widehat{v}_{u}\in T_{u}P$ is any lifting of $v\in T_{x}M$ to $T_{u}P$,
and

$p_{H}^{P\times G}:P\times G\to P\left[G\right]$

stands for the canonical projection onto the quotient; it is well-defined
because of the identity

$T_{\left(u,g\right)}p_{H}^{P\times
G}\left(-\zeta_{P}\left(u\right),T_{e}R_{g}\left(\zeta\right)\right)=0$

for every $\left(u,g\right)\in P\times G$ and $\zeta\in\mathfrak{h}$.

Let us now prove that

$\Gamma_{A}^{*}\theta_{J^{1}\pi_{\left[G\right]}}=A$

where
$\theta_{J^{1}\pi_{\left[G\right]}}\in\Omega^{1}\left(J^{1}\pi_{\left[G\right]},\mathfrak{g}\right)$
is the canonical connection form on $J^{1}\pi_{\left[G\right]}$. In order to
accomplish it, let us define the section
$\widetilde{\Gamma}:P\left[G\right]\to J^{1}\pi_{\left[G\right]}$ for
$\left(\pi_{\left[G\right]}\right)_{10}$ such that

$\widetilde{\Gamma}\left(\left[u,g\right]\right):v\in T_{x}M\mapsto
T_{\gamma\left(u\right)}R_{g}\left(\Gamma_{A}\left(u\right)\left(v\right)\right)\in
T_{\left[u,g\right]}P\left[G\right].$

Accordingly, we have the following commutative diagram

${J^{1}\pi_{\left[G\right]}}$${P}$${P\left[G\right]}$$\scriptstyle{\Gamma_{A}}$$\scriptstyle{\gamma}$$\scriptstyle{\widetilde{\Gamma}}$

Moreover, by fixing an auxiliary connection
$\omega_{0}\in\Omega^{1}\left(P,\mathfrak{h}\right)$ for $P$, we can define a
map $\widehat{\Gamma}_{A}^{\omega_{0}}:P\times G\to
J^{1}\left(\pi\circ\text{pr}_{1}\right)$ such that

$\widehat{\Gamma}_{A}^{\omega_{0}}\left(u,g\right)\left(v\right):=\left(v_{u}^{H},-T_{e}R_{g^{-1}}\left(A_{u}\left(v^{H}_{u}\right)\right)\right)$

for every $\left(u,g\right)\in P\times G$ and $v\in T_{x}M$; here $v_{u}^{H}$
indicates the horizontal lifting of $v$ to $T_{u}P$ by means of the connection
$\omega_{0}$. These maps fit in the following diagram

${J^{1}\left(\pi\circ\text{pr}_{1}\right)}$${P\times
G}$${P}$${J^{1}\pi_{\left[G\right]}}$${P\left[G\right]}$$\scriptstyle{j^{1}p^{P\times
G}_{H}}$$\scriptstyle{\widehat{\Gamma}^{\omega_{0}}_{A}}$$\scriptstyle{p^{P\times
G}_{H}}$inc$\scriptstyle{\gamma}$$\scriptstyle{\widetilde{\Gamma}}$

Namely, Cartan connection $\Gamma_{A}$ can be retrieved through the formula

$\Gamma_{A}=\widetilde{\Gamma}\circ\gamma=j^{1}p_{H}^{P\times
G}\circ\widehat{\Gamma}^{\omega_{0}}_{A}\circ\text{inc};$

therefore, because

$\left(\widehat{\Gamma}^{\omega_{0}}_{A}\circ\text{inc}\right)^{*}\theta_{J^{1}\left(\pi\circ\text{pr}_{1}\right)}=\omega_{0}+A,$

Lemma 5 implies that

$\Gamma_{A}^{*}\theta_{J^{1}\pi_{\left[G\right]}}=A$

as required. ∎

We can rephrase Proposition 3 using this correspondence.

######  Corollary 1.

For any map $\Gamma:P\to J^{1}\pi_{\left[G\right]}$ covering $\gamma$ we have
a section $\widetilde{\Gamma}:P\left[G\right]\to J^{1}\pi_{\left[G\right]}$
for $\left(\pi_{\left[G\right]}\right)_{10}$, which is defined through

$\widetilde{\Gamma}\left(\left[u,g\right]\right):v\in T_{x}M\mapsto
T_{\gamma\left(u\right)}R_{g}\left(\Gamma\left(u\right)\left(v\right)\right)\in
T_{\left[u,g\right]}P\left[G\right].$

Conversely, any connection $\widetilde{\Gamma}:P\left[G\right]\to
J^{1}\pi_{\left[G\right]}$ gives rise to a map $\Gamma:P\to
J^{1}\pi_{\left[G\right]}$ covering $\gamma$ by restriction to
$\gamma\left(P\right)\subset P\left[G\right]$.

Recall also the usual extension of principal connections in this context.

######  Proposition 6.

Let $H\subset G$ be a pair of Lie groups. Consider a $G$-principal bundle
$\tau:Q\to M$ and let $\pi:P\to M$ be a $H$-principal subbundle; let
$i:P\hookrightarrow Q$ be the canonical immersion. For every connection

$\Gamma:P\to J^{1}\pi$

the extension $\widehat{\Gamma}:Q\to J^{1}\tau$ of $\Gamma$ is the map

$\widehat{\Gamma}\left(\widehat{u}\right):=j^{1}i\left(j^{1}R_{g}\left(\Gamma\left(u\right)\right)\right)$

if and only if $\widehat{u}=i\left(u\right)g$.

###  3.3. The bundle of (generalized) Cartan connections

We know [Gar72, CM01] that connections on a $H$-principal bundle $\pi:P\to M$
can be seen either as equivariant sections of the map $\pi_{10}:J^{1}\pi\to P$
or as sections of the quotient bundle

$\overline{\pi}:J^{1}\pi/H\to M.$

Both descriptions are related by the following result.

######  Proposition 7.

Let $\pi:P\to M$ be an $H$-principal bundle and suppose that we have a bundle
$p:Q\to P$ together with an $H$-action on $Q$ such that $p$ is an
$H$-equivariant map; consider the induced bundle $\overline{p}:Q/H\to M$.
Then, any section $\sigma:M\to Q/H$ of $\overline{p}$ can be lifted to a
section $\widehat{\sigma}:P\to Q$ of $p$ and viceversa, any section of $p$
induces a section of $\overline{p}$ by quotient.

###### Proof.

Because the $H$-action on $P$ is free, given $\left[q\right]\in Q/H$ and
$u\in\pi^{-1}\left(\overline{p}\left(\left[q\right]\right)\right)$, there
exists a unique element $\widetilde{q}\in\left[q\right]$ such that

$p\left(\widetilde{q}\right)=u.$

Thus, for every $x\in M$, define

$\widehat{\sigma}\left(u\right):=\widetilde{q}$

if and only if $\sigma\left(x\right)=\left[q\right]$ and
$\widetilde{q}\in\left[q\right]$ is such that

$p\left(\widetilde{q}\right)=u.\qed$

In the case of Cartan connections, the description in terms of equivariant
maps is provided by the following corollary of Proposition 5.

######  Corollary 2.

A generalized Cartan connection can be represented as an $H$-equivariant
section of the pullback bundle

${\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)}$${J^{1}\pi_{\left[G\right]}}$${P}$${P\left[G\right]}$$\scriptstyle{\text{pr}_{2}^{\gamma}}$$\scriptstyle{\text{pr}_{1}^{\gamma}}$$\scriptstyle{\left(\pi_{\left[G\right]}\right)_{10}}$$\scriptstyle{\gamma}$

Thus, for $\mathfrak{g}/\mathfrak{h}$-Cartan connections, we have the
following commutative diagram of bundles

${\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)}$${P}$${\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)/H}$${M}$$\scriptstyle{\text{pr}_{1}^{\gamma}}$$\scriptstyle{p_{H}^{\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)}}$$\scriptstyle{\pi}$$\scriptstyle{\overline{\text{pr}}_{1}^{\gamma}}$

where on $\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)$ the $H$-diagonal
action is considered. By proceeding in analogy with the principal connections
case, we obtain the following definition.

######  Definition 3 (Bundle of (generalized)
$\mathfrak{g}/\mathfrak{h}$-Cartan connections).

The _bundle of (generalized) $\mathfrak{g}/\mathfrak{h}$-Cartan connections_
is the bundle

$\overline{\text{pr}}_{1}^{\gamma}:\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)/H\to
M.$

######  Remark 1.

The Cartan connection form is thus recovered from a section

$\sigma:M\to\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)/H$

through the following procedure: we construct the unique $H$-equivariant
section
$\widehat{\sigma}:P\to\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)$, which
can be seen as a map $\Gamma_{\sigma}:P\to J^{1}\pi_{\left[G\right]}$. The
Cartan connection form is then the pullback form

$A_{\sigma}:=\Gamma_{\sigma}^{*}\theta_{J^{1}\pi_{\left[G\right]}}\in\Omega^{1}\left(P,\mathfrak{g}\right).$

Conversely, given a Cartan connection form $A$, we use Proposition 5 in order
to construct an $H$-equivariant map $\Gamma_{A}:P\to
J^{1}\pi_{\left[G\right]}$, and so a section

$\sigma_{A}:M\to\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)/H,$

as required.

##  4\. Extensions and reductions of generalized Cartan connections

We want to describe the connection between gravity and field theory using
generalized connections on principal fiber bundles with the general linear
group as the structure group. To accomplish this task, it will be essential to
have an operation of extension and reduction of connections similar to those
available for principal connections, but that work on generalized Cartan
connections.

Therefore, in the present section we will use the correspondence between
generalized Cartan connections and principal connections on a extended bundle,
as described by Proposition 3, in order to generalize these constructions to
the realm of (generalized) Cartan connection.

###  4.1. How to extend a generalized Cartan connection

Let us consider the following problem: Given the diagram of Lie groups

(4.1) ${G}$${G_{1}}$${G_{2}}$${H}$

such that $G_{1}/H$ and $G/G_{2}$ have the same dimension than $M$, and a
generalized Cartan connection $A:TP\to\mathfrak{g}_{1}$ on a $H$-principal
bundle $\pi:P\to M$, construct a generalized Cartan connection
$A_{\left[G_{2}\right]}:TP\left[G_{2}\right]\to\mathfrak{g}$ in a canonical
way.

In order to properly address this problem, let us establish the following
auxiliary result.

######  Lemma 6.

Let $K\subset G_{1}\subset G$ be a chain of Lie groups, and consider a
$K$-principal bundle $\pi:P\to M$. Then

$P\left[G\right]\simeq\left(P\left[G_{1}\right]\right)\left[G\right].$

###### Proof.

The identification is given by the map

$\left[u,g\right]\in
P\left[G\right]\mapsto\Phi\left(\left[u,g\right]\right):=\left[\left[u,e\right],g\right].$

In fact, for every
$\left[\left[u,h\right],g^{\prime}\right]\in\left(P\left[G_{1}\right]\right)\left[G\right]$,
we have that

$\left[\left[u,h\right],g^{\prime}\right]=\left[\left[u,e\right],hg^{\prime}\right]=\Phi\left(\left[u,hg^{\prime}\right]\right),$

showing that $\Phi$ is surjective. On the other hand, if
$\left[u,g_{1}\right],\left[u,g_{2}\right]\in P\left[G\right]$ are such that

$\Phi\left(\left[u_{1},g_{1}\right]\right)=\Phi\left(\left[u_{2},g_{2}\right]\right),$

we will have that

$\left[\left[u_{1},e\right],g_{1}\right]=\left[\left[u_{2},e\right],g_{2}\right],$

meaning that

$\left[u_{1},h^{-1}\right]=\left[u_{2},e\right],\qquad hg_{1}=g_{2}$

for some $h\in G_{1}$. Therefore

$u_{1}k^{-1}=u_{2},\qquad kh^{-1}=e,\qquad hg_{1}=g_{2}$

and so

$\left[\left[u_{2},e\right],g_{2}\right]=\left[\left[u_{1}k^{-1},e\right],kg_{1}\right]=\left[\left[u_{1},e\right],g_{1}\right].$

This shows that $\Phi$ is also injective. ∎

This lemma allows us to lift Diagram 4.1 to principal bundles level:

(4.2)
${P\left[G\right]}$${P\left[G_{1}\right]}$${P\left[G_{2}\right]}$${P}$$\scriptstyle{\gamma_{G_{1}}}$$\scriptstyle{\gamma_{G_{2}}}$$\scriptstyle{\gamma_{H}^{1}}$$\scriptstyle{\gamma_{H}^{2}}$

######  Remark 2.

It is interesting to note that, in view of Lemma 3, the solution to this
problem described below, will apply to any diagram of principal bundles

${Q}$${P_{1}}$${P_{2}}$${P}$$\scriptstyle{j_{1}}$$\scriptstyle{j_{2}}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$

where $P,P_{1},P_{2},Q$ are $H,G_{1},G_{2},G$-principal bundles respectively,
and the arrows indicate principal bundle immersions.

The idea to extend the $\mathfrak{g}_{1}/\mathfrak{h}$-Cartan connection on
$P$ is to use Proposition 3; with its help, we can find a principal connection
on the bundle $P\left[G_{1}\right]$, and then lift it through the map
$\gamma_{G_{1}}:P\left[G_{1}\right]\hookrightarrow P\left[G\right]$.
Afterwards, we can induce a $\mathfrak{g}/\mathfrak{g}_{2}$-Cartan connection
on $P\left[G_{2}\right]$ using Proposition 3 together with the map
$\gamma_{G_{2}}:P\left[G_{2}\right]\hookrightarrow P\left[G\right]$.

######  Proposition 8.

Given any generalized $\mathfrak{g}_{1}/\mathfrak{h}$-Cartan connection
$A:TP\to\mathfrak{g}_{1}$, there exists a generalized
$\mathfrak{g}/\mathfrak{g}_{2}$-Cartan connection
$B:TP\left[G_{2}\right]\to\mathfrak{g}$ with the following property: If
$\widetilde{B}:TP\left[G\right]\to\mathfrak{g}$ is the principal connection on
$P\left[G\right]$ associated to $B$ and
$\widetilde{A}:TP\left[G_{1}\right]\to\mathfrak{g}_{1}$ is the corresponding
principal connection for $A$, we have that

$\gamma_{G_{1}}^{*}\widetilde{B}=\widetilde{A}.$

In terms of bundle maps, this correspondence proceeds as follows: Given a
Cartan connection

$\Gamma:P\to J^{1}\pi_{\left[G_{1}\right]},$

we can define a principal connection $\widetilde{\Gamma}:P\left[G\right]\to
J^{1}\pi_{\left[G\right]}$ through

$\widetilde{\Gamma}\left(\left[u,g\right]\right):v\in T_{x}M\mapsto
T_{\gamma_{G_{1}}\left(\left[u,e\right]\right)}R_{g}\left(T_{u}\gamma_{G_{1}}\left(\Gamma\left(u\right)\left(v\right)\right)\right).$

Then the induced $\mathfrak{g}/\mathfrak{g}_{2}$-Cartan connection is the map

(4.3) $\Gamma^{\sharp}\left(\left[u,g\right]\right):v\in T_{x}M\mapsto
T_{\gamma_{G_{1}}\left(\left[u,e\right]\right)}R_{g}\left(T_{u}\gamma_{G_{1}}\left(\Gamma\left(u\right)\left(v\right)\right)\right)$

for every $\left[u,g\right]\in
P\left[G_{2}\right]\equiv\gamma_{G_{2}}\left(P\left[G_{2}\right]\right)$.

###  4.2. Reducible Cartan connection

What about the converse of this result? Namely, given a
$\mathfrak{g}/\mathfrak{g}_{2}$-Cartan connection, will it restrict to a
$\mathfrak{g}_{1}/\mathfrak{h}$-Cartan connection on $P$? The problem with
this question is that, because of the way in which the extension is defined,
we have that $\widetilde{B}$ should be $\mathfrak{g}_{1}$-valued, at least
when it is restricted to $P\left[G_{1}\right]$. It poses some restrictions to
the desired converse result.

######  Proposition 9.

Let $\Gamma^{\sharp}:P\left[G_{2}\right]\to J^{1}\pi_{\left[G\right]}$ be a
generalized $\mathfrak{g}/\mathfrak{g}_{2}$-Cartan connection on
$P\left[G_{2}\right]$. If the induced principal connection

$\widetilde{\Gamma}:P\left[G\right]\to J^{1}\pi_{\left[G\right]}$

is such that

$\widetilde{\Gamma}\left(\gamma_{G_{1}}\left(\left[u,g_{1}\right]\right)\right)\in
j^{1}\gamma_{G_{1}}\left(J^{1}\pi_{\left[G_{1}\right]}\right)$

for all $\left[u,g_{1}\right]\in P\left[G_{1}\right]$, then it reduces to a
generalized $\mathfrak{g}_{1}/\mathfrak{h}$-Cartan connection on $P$

$\Gamma:P\to J^{1}\pi_{\left[G_{1}\right]}$

through the formula

$\widetilde{\Gamma}\left(\gamma_{G_{1}}\left(\left[u,e\right]\right)\right)=j^{1}\gamma_{G_{1}}\left(\Gamma\left(u\right)\right).$

A commutative diagram could be clearer in illustrating these matters

(4.4)
${J^{1}\pi_{\left[G\right]}}$${P\left[G\right]}$${P\left[G_{1}\right]}$${P\left[G_{2}\right]}$${J^{1}\pi_{\left[G_{1}\right]}}$${P}$$\scriptstyle{\widetilde{\Gamma}}$$\scriptstyle{\gamma_{G_{1}}}$$\scriptstyle{\gamma_{G_{2}}}$$\scriptstyle{\Gamma^{\sharp}}$$\scriptstyle{j^{1}\gamma_{G_{1}}}$$\scriptstyle{\left(\pi_{\left[G_{1}\right]}\right)_{10}}$$\scriptstyle{\gamma_{H}^{1}}$$\scriptstyle{\gamma_{H}^{2}}$$\scriptstyle{\Gamma}$$\scriptstyle{\gamma_{G}}$

###  4.3. Examples of extensions for (generalized) Cartan connections

Here we will consider some examples for the constructions devised above; the
example concerning $K$-structures will be relevant when we will consider the
relationship between Wise formulation of the correspondence Chern-Simons and
gravity and the formulation of the correspondence in terms of bundle with the
group $GL\left(m\right)$ as structure group (see Section 6.2 below).

####  4.3.1. Extension of principal connections

Let $G$ be a Lie group and $H\subset G$ a closed Lie subgroup. As a first
example, let us consider the pair of diagrams

(4.5) ${Q}$${G}$${P}$${Q}$${H}$${G}$${P}$${H}$

where $\pi:P\to M$ is a $H$-principal bundle and $p:Q\to M$ is a $G$-principal
bundle; the left diagram is the principal bundles diagram, and the right
diagram corresponds to the underlying Lie groups. In this case, Proposition 8
reduces to the usual result on extensions of principal connections (see
Proposition $6.1$ in [KN63]); on the other hand, hypothesis in Proposition 9
is equivalent to the reducibility of the principal connection on $Q$.

####  4.3.2. Induced principal connection for a Cartan connection

Let $H,G,P,Q$ be as in the previous section. We can put Proposition 3 also in
this context; to this end, let us consider the diagrams

${Q}$${G}$${Q}$${Q}$${G}$${G}$${P}$${H}$

comprising Lie groups and their principal bundle counterparts. In this
setting, the additional hypothesis in Proposition 9 is automatically
fulfilled. So, once we realize that $Q\simeq P\left[G\right]$ via Lemma 3,
Proposition 8 and Proposition 9 are nothing but Proposition 3.

####  4.3.3. $K$-structures on space-time

The previous scheme can also be applied to the case in which the original
relationship between Palatini gravity (formulated on a
$SO\left(2,1\right)$-subbundle of $LM$) and Chern-Simons field theory (on a
$SO\left(2,1\right)\ltimes\mathbb{R}^{3}$-principal bundle) is promoted to a
relationship between gravity formulated in the bundle $LM$ and Chern-Simons
theory on a bundle with structure group
$GL\left(3\right)\ltimes\mathbb{R}^{3}$. Diagram (4.1) then becomes

(4.6)
${GL\left(m\right)\ltimes\mathbb{R}^{m}}$${SO\left(p,q\right)\ltimes\mathbb{R}^{m}}$${GL\left(m\right)}$${SO\left(p,q\right)}$

that in terms of principal bundles turns out to be

${AM}$${O_{\zeta}^{\text{aff}}}$${LM}$${O_{\zeta}}$

Here $O_{\zeta}\subset LM$ and $O_{\zeta}^{\text{aff}}\subset AM$ are the
$SO(p,q)$\- and $SO(p,q)\ltimes\mathbb{R}^{m}$-structures respectively,
defined through

$O_{\zeta}:=\bigcup_{x\in M}\left\\{u:\mathbb{R}^{m}\to
T_{x}M:\zeta(u\left(v\right),u\left(w\right))=\eta\left(v,w\right)\text{ for
all }v,w\in\mathbb{R}^{m}\right\\}$

and

$O_{\zeta}^{\text{aff}}:=\bigcup_{x\in M}\left\\{u:\mathbb{R}^{m}\to
A_{x}M:\zeta(\beta\left(u\left(v\right)\right),\beta\left(u\left(w\right)\right))=\eta\left(v,w\right)\text{
for all }v,w\in\mathbb{R}^{m}\right\\}$

for some metric $\zeta:M\to\Sigma:=LM/K$ with $\left(p,q\right)$-signature.

Let us use the local description for the frame bundle and its affine
counterpart (see Appendix C) in order to show how the proposed extension works
in this case. Let us consider $\left(x^{\mu},f^{\nu}_{i}\right)$ the natural
coordinates of an element in $O_{\zeta}$; it means that

$g_{\mu\nu}f^{\mu}_{i}f^{\nu}_{j}=\eta_{ij},$

where $\zeta=g_{\mu\nu}dx^{\mu}\otimes dx^{\nu}$. The map

$\phi:LM\times_{GL\left(m\right)}A\left(m\right)\to AM$

constructed in Proposition 14, Appendix C.2, reads in these coordinates

(4.7)
$\phi\left(\left[\left(x^{\mu},f^{\nu}_{i}\right),\left(a^{i}_{j},0\right)\right]\right)=\left(x^{\mu},a^{j}_{i}f_{j}^{\nu}\right).$

Now, from a local data

(4.8)
$\left(A_{O_{\zeta}}\right)_{U}=\left(\Gamma^{\alpha}_{\gamma\beta}dx^{\beta}\otimes
E_{\alpha}^{\gamma},\sigma^{\alpha}_{\beta}dx^{\beta}\otimes
e_{\alpha}\right)$

for the Cartan connection
$A_{O_{\zeta}}\in\Omega^{1}\left(O_{\zeta},\mathfrak{k}\ltimes\mathbb{R}^{m}\right)$,
using Proposition 5 we obtain a map

$\Gamma_{A_{O_{\zeta}}}:O_{\zeta}\to
J^{1}\left(\left.\left(\beta\circ\tau\right)\right|_{O_{\zeta}^{\text{aff}}}\right)$

that is locally given by

$\Gamma_{A_{O_{\zeta}}}\left(x^{\mu},f^{\nu}_{i}\right)=dx^{\mu}\otimes\left(\frac{\partial}{\partial
x^{\mu}}-f^{\gamma}_{i}\Gamma_{\gamma\mu}^{\alpha}\frac{\partial}{\partial
e^{\alpha}_{i}}-\sigma_{\mu}^{\alpha}\frac{\partial}{\partial
v^{\alpha}}\right)$

As an aside comment, it can be proved that relation (5.9) below implies that

$F_{\mu}:=\frac{\partial}{\partial
x^{\mu}}-f^{\gamma}_{i}\Gamma_{\gamma\mu}^{\alpha}\frac{\partial}{\partial
e^{\alpha}_{i}}$

is a vector field tangent to $O_{\zeta}$, and so

$\Gamma_{A_{O_{\zeta}}}\left(x^{\mu},f^{\nu}_{i}\right)\in
J^{1}\left(\left.\left(\beta\circ\tau\right)\right|_{O_{\zeta}^{\text{aff}}}\right).$

Namely, as expected, the metricity condition implies that
$\Gamma_{A_{O_{\zeta}}}$ takes values in the jet bundle of the affine
subbundle determined by the metric $\zeta$.

Now, using Equation (4.3), we obtain the induced Cartan connection given by
the map

$\Gamma_{A_{O_{\zeta}}}^{\sharp}\left(\left[\left(x^{\mu},f^{\nu}_{i}\right),\left(a^{i}_{j},0\right)\right]\right)=dx^{\mu}\otimes\left(\frac{\partial}{\partial
x^{\mu}}-a_{i}^{j}f^{\gamma}_{j}\Gamma_{\gamma\mu}^{\alpha}\frac{\partial}{\partial
e^{\alpha}_{i}}-\sigma_{\mu}^{\alpha}\frac{\partial}{\partial
v^{\alpha}}\right),$

and using identification (4.7), it will become

$\Gamma_{A_{O_{\zeta}}}^{\sharp}\left(x^{\mu},e^{\nu}_{i}\right)=dx^{\mu}\otimes\left(\frac{\partial}{\partial
x^{\mu}}-e^{\gamma}_{j}\Gamma_{\gamma\mu}^{\alpha}\frac{\partial}{\partial
e^{\alpha}_{i}}-\sigma_{\mu}^{\alpha}\frac{\partial}{\partial
v^{\alpha}}\right).$

This is nothing but the Cartan connection associated to local data (4.8), when
it is considered as providing a
$\mathfrak{a}\left(m\right)/\mathfrak{gl}\left(m\right)$-Cartan connection on
the frame bundle $LM$.

##  5\. Variational problems for Chern-Simons field theory and gravity

We will use the present section to introduce the variational problem for
Chern-Simons field theory used by Wise in [Wis+09, Wis10], and the variational
problem for gravity with basis, both in a form suitable for the purposes of
this article. Concretely, we will try to find a formulation for these
variational problems fitting in the scheme devised by Gotay in the pioneering
works [Got91b, Got91]. It means that we need to find for each of these
descriptions a triple

$\left(\pi:P\to M,\lambda,{\mathcal{I}}\right),$

where $\pi:P\to M$ is a bundle on the base space $M$,
$\lambda\in\Omega^{n}\left(P\right)$ is an $n$-form (where $n=\dim{M}$), and
${\mathcal{I}}\subset\Omega^{\bullet}\left(P\right)$ is a differential ideal
in the exterior algebra of $P$ (a so called _exterior differential system_ ,
see [Bry11, Bry+91, IL03]). As Gotay explained in the article referenced
above, with these data it is possible to formulate a variational problem in
the following way: To find the stationary sections $\sigma:U\subset M\to P$
for the action

$S\left[\sigma\right]:=\int_{U}\sigma^{*}\lambda$

subject to the constraints $\sigma^{*}\alpha=0$ for all
$\alpha\in{\mathcal{I}}$. In the present article we will use the term _Gotay
variational problem_ to refer to this kind of variational problems; in this
secrtion we will describe Chern-Simons field theory and gravity from this
viewpoint.

###  5.1. Wise variational problem

Our first objective is to find a variational problem of this type for the
Chern-Simons field theory, as described in the articles by Wise [Wis+09,
Wis10]. To do this, we will use as a starting point the description of this
field theory given by Tejero Prieto [Tej04], which uses a formulation in terms
of local variational problems. First, we will study how the canonical forms of
the jet space of a principal fiber bundle can be used to globalize this
collection of local variational problems. Then, we will introduce a constraint
that will relate the degrees of freedom of the underlying principal bundle
with the degrees of freedom of the Cartan connection. It should be clarified
that this constraint is absent in Wise’s description because the relevant
degrees of freedom are associated only with the connection, whereas in our
approach, by using the frame bundle as the underlying principal bundle, we
have degrees of freedom that can be used to represent the coframe (see Remark
3 below).

####  5.1.1. Local formulation for Wise variational problem

Let us suppose that we have a Cartan geometry modelled on the pair
$\left(G,H\right)$. The Cartan connection $A$ associated to this geometry can
be described locally by a collection of pairs $\left(U,A_{U}\right)$, where
$U\subset M$ are open sets and $A_{U}\in\Omega^{1}\left(U,\mathfrak{g}\right)$
are $\mathfrak{g}$-valued $1$-forms such that the map

$\pi_{\mathfrak{h}}\circ\left.A_{U}\right|_{x}:T_{x}U\to\mathfrak{g}/\mathfrak{h}$

is a linear isomorphism. These forms are related to the Cartan connection $A$
through a section $s_{U}:U\to P$, via the map

$A_{U}=s_{U}^{*}A.$

Now, according to Freed [Fre95], on every trivializing neighborhood $U\subset
M$ it is possible to define an action for Chern-Simons through

$S_{U}\left[s_{U},A\right]:=\int_{U}s_{U}^{*}Tq\left(A,F\right).$

Modulo some topological assumptions regarding the structure group of the
principal group, it can be proved that this action is independent of the
section $s_{U}$ involved in its definition. Moreover, whenever $\dim{M}=3$ and
the Lie group $H$ is simply connected, any $H$-principal bundle on $M$ is
necessarily trivial, so that this prescription gives rise to a well-defined
variational problem on the whole principal bundle.

On the other hand, Tejero Prieto [Tej04] is able to give global sense to this
collection of local actions by requiring that, whenever the domains intersect,
the associated Euler-Lagrange equations are the same for any of the local
actions. Thus, we will interpret the Wise variational problem as a collection
of local actions

$S_{U}\left[A\right]:=\int_{U}Tq\left(A_{U},F_{U}\right)$

given by the local description $\left(U,A_{U}\right)$ of a Cartan connection,
because its Euler-Lagrange equations will coincide on the intersection of the
corresponding domains.

Using Proposition 3, we consider the equivalent variational problem on the
bundle $\pi_{\gamma}:U_{\gamma}\to M$; therefore, we can see the Wise
variational problem as a problem whose fields are principal connections
$\widetilde{A}$ on $P\left[G\right]$, and the action is given by the formula

$S_{U}\left[\widetilde{A}\right]:=\int_{U}Tq\left(\widetilde{A}_{U},\widetilde{F}_{U}\right),$

where $\widetilde{A}_{U}=\widetilde{s}_{U}^{*}\widetilde{A}$ is the local
description of the principal connection $\widetilde{A}$, with

$\widetilde{s}_{U}\left(x\right)=\gamma\left(s_{U}\left(x\right)\right)=\left[s_{U}\left(x\right),e\right].$

####  5.1.2. Wise variational problem in jet bundle formulation

It is our next aim to formulate Wise variational problem in terms of a
Lagrangian form and a bundle; we are looking for a formulation in which the
integrand $Tq\left(A_{U},F_{U}\right)$ in the action comes from the Lagrangian
form through pullback along a section of a suitable bundle, and this section
is uniquely determined by the Cartan connection. Recalling the discussion
carried out in Section 3.3, we will choose the bundle

$\pi\circ\text{pr}_{1}^{\gamma}:\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)\to
M$

as the bundle whose sections can be put in one-to-one correspondence with
$\mathfrak{g}/\mathfrak{h}$-Cartan connections on $P$.

Our next task is to find the Lagrangian form. To this end, consider a
generalized Cartan connection $\Gamma:P\to J^{1}\pi_{\left[G\right]}$, with
local description $\left\\{A_{U}\right\\}$. Then Remark 1 tells us that

$A_{U}=\left(\Gamma\circ s_{U}\right)^{*}\theta_{J^{1}\pi_{\left[G\right]}},$

where
$\theta_{J^{1}\pi_{\left[G\right]}}\in\Omega^{1}\left(J^{1}\pi_{\left[G\right]},\mathfrak{g}\right)$
is the canonical connection form and $s_{U}:U\to P$ is a local section for the
principal bundle $\pi:P\to M$. Let us define

(5.1)
$\theta_{J^{1}\pi_{\left[G\right]}}^{*}:=\left(\text{pr}_{2}^{\gamma}\right)^{*}\theta_{J^{1}\pi_{\left[G\right]}}$

and

$\Theta_{J^{1}\pi_{\left[G\right]}}^{*}:=\left(\text{pr}_{2}^{\gamma}\right)^{*}\Theta_{J^{1}\pi_{\left[G\right]}}$

where
$\text{pr}_{2}^{\gamma}:\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)\to
J^{1}\pi_{\left[G\right]}$ is the horizontal projection in the pullback
diagram in Corollary 2. Therefore, the Lagrangian form is given by

$\lambda_{CS}:=Tq\left(\theta_{J^{1}\pi_{\left[G\right]}}^{*},\Theta_{J^{1}\pi_{\left[G\right]}}^{*}\right)\in\Omega^{2k-1}\left(\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)\right);$

we have that

$L_{U}:=Tq\left(A_{U},F_{U}\right)=\left(s_{U},\Gamma\circ
s_{U}\right)^{*}\lambda_{CS}.$

Thus, we obtain the following corollary.

######  Corollary 3.

The extremals of the local variational problem determined by the Lagrangian
$L_{U}$ are in one to one correspondence with the local extremals of the
variational problem given by the triple

$\left(\pi\circ\text{pr}_{1}^{\gamma}:\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)\to
M,\lambda_{CS},0\right).$

###### Proof.

The correspondence is given by

$\left(A_{U},s_{U}\right)\mapsto\left(s_{U},\Gamma\circ
s_{U}\right)\in\left(\pi\circ\text{pr}_{1}^{\gamma}\right)^{-1}\left(U\right)\subset\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right).\qed$

####  5.1.3. Wise variational problem for first order geometries

Recall [Sha97, Bar04] that a pair $\left(P,A\right)$, where $\pi:P\to M$ is an
$H$-principal bundle and $A$ is a $\mathfrak{g}/\mathfrak{h}$-Cartan
connection, is called a _first order geometry_ if and only if the
representation

$\text{Ad}:H\to\mathop{\text{GL}}{\left(\mathfrak{g}/\mathfrak{h}\right)}$

is faithful. Then we have the following result.

######  Proposition 10.

$P$ admits a first order geometry if and only if it is an $H$-structure.

###### Proof.

Recall that a Cartan connection $A$ on $P$ induces a family of isomorphisms

$\phi_{u}:T_{x}M\to\mathfrak{g}/\mathfrak{h},\qquad
u\in\pi^{-1}\left(x\right)$

such that $\phi_{uh}=\text{Ad}_{h^{-1}}\circ\phi_{u}$. Thus, the isomorphism
is given by

$u\in
P\mapsto\left(\phi_{u}^{-1}\left(w_{1}\right),\cdots,\phi_{u}^{-1}\left(w_{m}\right)\right),$

where $\left\\{w_{1},\cdots,w_{m}\right\\}$ is a basis for
$\mathfrak{g}/\mathfrak{h}$. ∎

In the present section we will assume that $P$ admits a first order geometry;
according to the previous proposition, it means that $P$ can be considered a
subbundle of the bundle of frames $LM$. Therefore, for every element $v_{x}\in
T_{x}M$ and $u\in\pi^{-1}\left(x\right)$, we have two ways to represent it,
namely

$v_{x}=\phi_{u}^{-1}\left(w_{u}\right)=u\left(c_{u}\right)$

for some elements $w_{u}\in\mathfrak{g}/\mathfrak{h}$ and
$c_{u}\in\mathbb{R}^{m}$; it gives rise to a linear isomorphism

(5.2)
$\kappa_{u}:\mathbb{R}^{m}\to\mathfrak{g}/\mathfrak{h}:c\mapsto\phi_{u}\circ
u\left(c\right).$

######  Example 1 (Affine connections).

Let us calculate this isomorphism for
$\mathfrak{a}\left(m\right)/\mathfrak{gl}\left(m\right)$-Cartan connections on
$LM$, the so called _generalized affine connections_ [KN63]. Assuming that the
local version of a Cartan connection on $LM$ is

$\widetilde{\omega}_{U}=\left(\Gamma^{\alpha}_{\gamma\beta}dx^{\beta}\otimes
E_{\alpha}^{\gamma},\sigma^{\alpha}_{\beta}dx^{\beta}\otimes
e_{\alpha}\right)$

for some local functions $\Gamma^{\alpha}_{\beta\gamma}$ and
$\sigma^{\alpha}_{\beta}$, from Equation (C.5) we obtain

$\phi_{\left(x^{\alpha},e^{\beta}_{i}\right)}=e_{\alpha}^{i}\sigma^{\alpha}_{\beta}dx^{\beta}\otimes
e_{i}.$

Now, recall that in this case
$\mathfrak{a}\left(m\right)/\mathfrak{gl}\left(m\right)=\mathbb{R}^{m}$ in
canonical fashion, so that $\kappa_{\left(x^{\alpha},e^{\beta}_{i}\right)}$ is
a linear endomorphism of $\mathbb{R}^{m}$. Therefore, the matrix for this
morphism in terms of the canonical basis $\left\\{e_{i}\right\\}$ will become

$\left[\kappa_{\left(x^{\alpha},e^{\beta}_{i}\right)}\right]_{j}^{i}=e_{\alpha}^{i}e^{\beta}_{j}\sigma^{\alpha}_{\beta}.$

According to the classical definition [KN63], a Cartan connection on $LM$ is
an _affine connection_ if and only if
$\kappa_{\left(x^{\alpha},e^{\beta}_{i}\right)}$ is the identity on
$\mathbb{R}^{m}$, or equivalently

$e_{\alpha}^{i}e^{\beta}_{j}\sigma^{\alpha}_{\beta}=\delta_{j}^{i}.$

We can see this constraint in terms of the jet bundle description for the
Cartan connection (see Section 3.1). In order to proceed, let us define the
map

(5.3) $j:J^{1}\tau\hookrightarrow
J^{1}\left(\tau\circ\beta\right):j_{x}^{1}s\mapsto j^{1}_{x}\left(\gamma\circ
s\right)-\left.\varphi\right|_{s\left(x\right)},$

where $\varphi$ indicates the canonical $\mathbb{R}^{m}$-valued $1$-form on
$LM$, which can be seen as an element of

$T_{x}^{*}M\otimes\mathbb{R}^{m}\subset
T^{*}_{x}M\otimes\mathfrak{a}\left(m\right)\simeq T_{x}^{*}M\otimes
V_{\gamma\left(s\left(x\right)\right)}\left(AM\right)$

and thus acts on the affine space
$J^{1}_{\gamma\left(s\left(x\right)\right)}\left(\tau\circ\beta\right)$. In
local terms, we have that

$\left.\varphi\right|_{\left(x^{\alpha},e^{\beta}_{i}\right)}=e^{i}_{\beta}dx^{\beta}\otimes
e_{i}$

and if an element $j_{x}^{1}s\in J^{1}\tau$ has coordinates
$\left(x^{\alpha},e_{i}^{\beta},e^{\beta}_{i\gamma}\right)$, then it
represents the map

$j_{x}^{1}s:\frac{\partial}{\partial
x^{\alpha}}\mapsto\frac{\partial}{\partial
x^{\alpha}}+e^{\beta}_{i\alpha}\frac{\partial}{\partial e^{\beta}_{i}};$

additionally, the map $\gamma:LM\to AM$ simply reads

$\gamma\left(x^{\alpha},e_{i}^{\beta}\right)=\left(x^{\alpha},e_{i}^{\beta},0\right).$

Then, from Definition (5.3) we can conclude that

(5.4)
$j\left(x^{\alpha},e_{i}^{\beta},e^{\alpha}_{i\gamma}\right)=\left(x^{\alpha},e_{i}^{\beta},0,e^{\alpha}_{i\gamma},-e^{\alpha}_{i}e^{i}_{\beta}\right),$

as required.

Our next task is to adapt the Equation (5.2) to the description discussed in
Section 3.2, where Cartan connections were considered as sections in a bundle.
The main reason to do that is that, just as we saw in the previous example,
the pullback bundle $\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)$ would
contain degrees of freedom associated to the principal bundle $P$ and the
$\mathfrak{g}/\mathfrak{h}$-part of the Cartan connection. In fact, given an
element
$\overline{u}=\left(u,j_{x}^{1}s\right)\in\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)$,
we can construct the map

${\phi}_{\overline{u}}:T_{x}M\to\mathfrak{g}/\mathfrak{h}$

in the following way: Given a tangent vector $v_{x}\in T_{x}M$, fix a lift
$\widehat{v}_{u}\in T_{u}P$, and use it to construct the tangent vector

$W_{s\left(x\right)}:=T_{u}\gamma\left(\widehat{v}_{u}\right)-T_{x}s\left({v}_{x}\right)\in
T_{s\left(x\right)}\left(P\left[G\right]\right).$

Then we have that

$T_{s\left(x\right)}\pi_{\left[G\right]}\left(T_{u}\gamma\left(\widehat{v}_{u}\right)\right)=v_{x}$

and so $W_{s\left(x\right)}\in
V_{s\left(x\right)}\left(P\left[G\right]\right)$; it means in particular that
there exists an element
$\widetilde{W}\left(\widehat{v}_{u}\right)\in\mathfrak{g}$ such that

$W_{s\left(x\right)}=\left(\widetilde{W}\left(\widehat{v}_{u}\right)\right)_{P\left[G\right]}\left(s\left(x\right)\right).$

Now, changing the lift $\widehat{v}_{u}$ in this definition will produce a
shift in $\widetilde{W}\left(\widehat{v}_{u}\right)$ by an element living in
$\mathfrak{h}$; therefore, we can define the map
${\phi}_{\overline{u}}:T_{x}M\to\mathfrak{g}/\mathfrak{h}$ by projecting into
the quotient by $\mathfrak{h}$, namely

${\phi}_{\overline{u}}\left(v_{x}\right):=\left[\widetilde{W}\left(\widehat{v}_{u}\right)\right]_{\mathfrak{h}},$

where $\left[\cdot\right]_{\mathfrak{h}}$ indicates the equivalence class in
$\mathfrak{g}/\mathfrak{h}$.

Using Example 1 and Equation (5.2), we can define a map

$\kappa_{\overline{u}}:=\phi_{\overline{u}}\circ
u:\mathbb{R}^{m}\to\mathfrak{g}/\mathfrak{h}$

for every
$\overline{u}=\left(u,j_{x}^{1}s\right)\in\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)$.
It will allow us to identify the degrees of freedom mentioned above by means
of the constraint

(5.5) $\kappa_{u}=\kappa_{0}$

for some fixed isomorphism
$\kappa_{0}:\mathbb{R}^{m}\to\mathfrak{g}/\mathfrak{h}$, under the assumption
that $P$ admits a first order geometry. So we are ready to introduce the
following definition.

######  Definition 4 (Wise variational problem for first order geometries).

Let $\pi:P\to M$ be an $H$-principal bundle admitting a first order geometry
associated to the pair $\left(G,H\right)$; fix an isomorphism
$\kappa_{0}:\mathbb{R}^{m}\to\mathfrak{g}/\mathfrak{h}$. The _Wise variational
problem_ is the triple

$\left(\pi\circ\text{pr}_{1}^{\gamma}:\gamma^{*}\left(J^{1}\pi_{\left[G\right]}\right)\to
M,\lambda_{CS},\mathcal{K}_{\kappa_{0}}\right).$

where $\mathcal{K}_{\kappa_{0}}$ is the EDS induced by the constraint (5.5).

######  Remark 3 (On the the constraint $\mathcal{K}_{\kappa_{0}}$).

The introduction of the constraint $\mathcal{K}_{\kappa_{0}}$ implies a
departure from the scheme devised by Wise in the previously cited works, where
the degrees of freedom associated to the $\mathfrak{g}/\mathfrak{h}$-part of
the Cartan connection are the ones devoted to describe the vielbein when the
relationship with gravity is established. In our present description, we are
choosing to use the degrees of freedom of the underlying principal bundle in
order to describe it, and the rôle of the constraint
$\mathcal{K}_{\kappa_{0}}$ is to enforce the identification of the
$\mathfrak{g}/\mathfrak{h}$-part of the Cartan connection with the group
coordinates in the principal bundle $P$. For example, when
$P=LM,H=GL\left(m\right),G=A\left(m\right)$, we can deal with the constraint
$\mathcal{K}_{\kappa_{0}}$ in a straightforward manner. In fact, because of
the Equation (5.3), we have that a section $\sigma:U\subset
M\to\gamma^{*}\left(J^{1}\left(\tau\circ\beta\right)\right)$ is integral for
this EDS (with $\kappa_{0}=\text{id}$, given the identification
$\mathfrak{g}/\mathfrak{h}=\mathbb{R}^{m}$ we have at our disposal is this
case) if and only if there exists a section $s:U\to J^{1}\tau$ such that

$\sigma\left(x\right)=\left(\tau_{10}\left(s\left(x\right)\right),j\left(s\left(x\right)\right)\right).$

Thus we can fulfill this constraint by using this form for the sections we
consider in the variational problem.

###  5.2. Variational problem for gravity with basis

It is usual [Wit88] to describe gravity with a pair
$\left(e^{\mu}_{i},\omega^{i}_{j}\right)$, where $e^{\mu}_{i}$ are the
components of a local basis of the tangent bundle $TM$ respect to some
coordinates $x^{\mu}$, and $\omega^{i}_{j}$ are a set of local $1$-form, the
so called _spin connection forms_. This pair can be represented in terms of
the bundle of frames $LM$ on the space-time: Fixing a common open domain
$U\subset M$, $e_{i}^{\mu}$ gives rise to a local section $\sigma$ of $LM$
through the formula

$x\in
U\mapsto\sigma\left(x\right):=\left(e_{1}^{\mu}\left(x\right)\frac{\partial}{\partial
x^{\mu}},\cdots,e_{m}^{\mu}\left(x\right)\frac{\partial}{\partial
x^{\mu}}\right).$

The forms $\omega_{j}^{i}$ together with the section $\sigma$ can be used in
order to define the Christoffel symbols $\Gamma^{\mu}_{\nu\sigma}$ of the
underlying connection, through the so called _vielbein postulate_

(5.6)
$\omega_{j}^{i}=e_{\mu}^{i}\left(e^{\nu}_{j}\Gamma^{\mu}_{\nu\sigma}+\frac{\partial
e_{j}^{\mu}}{\partial x^{\sigma}}\right)dx^{\sigma}.$

The connection form on $LM$222Or a subbundle… associated to
$\left\\{\omega_{j}^{i}\right\\}$ can be retrieved by the formula [Nak96,
KN63]

$\left.\left(\omega_{LM}\right)^{i}_{j}\right|_{u}=e_{\mu}^{i}e^{\nu}_{j}\Gamma^{\mu}_{\nu\sigma}dx^{\sigma}+e^{i}_{\mu}de^{\mu}_{j}$

where $u\in LM$ is represented by the set of coordinates
$\left(x^{\mu},e^{\mu}_{i}\right)$. This gives rise to a locally defined
action

(5.7)
$S_{\text{PG}}:=\int_{U}\epsilon_{ijkl}\eta^{kp}e^{i}_{\mu}e^{j}_{\nu}dx^{\mu}\wedge
dx^{\nu}\wedge\Omega^{l}_{p},$

where $\Omega_{j}^{i}$ is the local curvature $2$-form associated to
$\omega^{i}_{j}$. A constraint that should be adopted on the set of forms
$\omega^{i}_{j}$ is that the underlying connection form is
$\mathfrak{o}\left(3,1\right)$-valued, namely

(5.8) $\eta^{ij}\omega^{k}_{j}+\eta^{kj}\omega_{j}^{i}=0.$

Before to continue, we have to deal with the constraints given by Eqs. (5.8);
according to formula (5.6), it is equivalent to

(5.9) $\frac{\partial g^{\mu\nu}}{\partial
x^{\sigma}}+g^{\mu\rho}\Gamma^{\nu}_{\rho\sigma}+g^{\nu\rho}\Gamma^{\mu}_{\rho\sigma}=0$

namely, the connection associated to the symbols $\Gamma^{\mu}_{\nu\sigma}$ is
metric with respect to the metric associated to the vielbein,

$g^{\mu\nu}=\eta^{ij}e^{\mu}_{i}e^{\nu}_{j}.$

At jet bundle level it is equivalent to the set of equations

(5.10)
$\left(\eta^{kj}e_{\mu}^{i}+\eta^{ij}e_{\mu}^{k}\right)\left(-e^{\mu}_{j\sigma}+\frac{\partial
e_{j}^{\mu}}{\partial x^{\sigma}}\right)=0.$

Thus, a solution for gravity with basis is a section

$\sigma:U\subset M\to J^{1}\tau$

which is both an extremal for the functional (5.7) and also verifies the
condition (5.10).

On the other hand, there is a formulation for field theory in which the action
is calculated using a Lagrangian density, that is a bundle map

${\mathcal{L}}:J^{1}q\to\wedge^{4}\left(T^{*}M\right),$

where $q:E\to M$ is a bundle on $M$. Namely, for every section $\sigma:M\to E$
of $q$, we have a map

${\mathcal{L}}\circ j^{1}\sigma:M\to\wedge^{4}\left(T^{*}M\right),$

which is a $4$-form on $M$, and so you can integrate,

$S_{PG}\left[\sigma\right]:=\int_{M}{\mathcal{L}}\circ j^{1}\sigma.$

We want to find this kind of formulation for gravity with basis; in order to
carry out this task, it is necessary to identify the bundle $E$, and then to
write down a Lagrangian density on this bundle. Because $e_{i}^{\alpha}$ are
part of the degrees of freedom for this flavor of gravity, the bundle $E$
should include it, namely, we are searching for a bundle of the form

$E=LM\times_{M}\star,$

where $\star$ stands for the degrees of freedom associated to
$\omega_{j}^{i}$. As we said before, these data corresponds to the
specification of a linear connection on $LM$; from a geometrical viewpoint,
the relevant bundle in this regard is the connection bundle
$\overline{\tau}:C\left(LM\right)\to M$, defined in such a way that it fits in
the following diagram

${J^{1}\tau}$${LM}$${C\left(LM\right)}$${M}$$\scriptstyle{\tau_{10}}$$\scriptstyle{p_{GL\left(m\right)}^{J^{1}\tau}}$$\scriptstyle{\tau}$$\scriptstyle{\overline{\tau}}$

Therefore, a section of the connection bundle is equivalent to a equivariant
section of the projection $\tau_{10}$. Now, recall that an element of
$J^{1}\tau$ is a linear map $m:T_{x}M\to T_{u}\left(LM\right)$ such that

$T_{u}\tau\circ m=\text{id}_{T_{x}M}.$

Using the data $\omega_{j}^{i}$, this section can be constructed according to
the formula

$\displaystyle\Gamma\left(u\right)$
$\displaystyle:=dx^{\mu}\otimes\left(\frac{\partial}{\partial
x^{\mu}}\right)_{u}^{H}$
$\displaystyle=dx^{\mu}\otimes\left(\frac{\partial}{\partial
x^{\mu}}-e_{j}^{\rho}\Gamma^{\sigma}_{\rho\mu}\frac{\partial}{\partial
e^{\sigma}_{j}}\right)$

where $u=\left(X_{1},\cdots,X_{m}\right)\in LM$ and the symbols
$\Gamma^{\mu}_{\nu\sigma}$ are calculated through Equation (5.6); here
$\left(\cdot\right)^{H}_{u}$ indicates the horizontal lift of a vector field
to $T_{u}LM$. Thus the bundle describing this flavor of gravity will become

$E=LM\times_{M}C\left(LM\right).$

Moreover, it can be proved [Gar72, CM01] that this bundle is isomorphic as
affine bundle on $LM$ to the first jet bundle $J^{1}\tau$; in terms of the
induced coordinates $\left(x^{\mu},e_{i}^{\nu},e^{\sigma}_{k\rho}\right)$,
this correspondence is given by the formula

(5.11) $\Gamma^{\mu}_{\nu\sigma}=-e^{k}_{\nu}e^{\mu}_{k\sigma}.$

Then, the relevant bundle in the description of gravity with basis will be the
first jet bundle of the frame bundle.

Having identified the basic bundle, we need to write down a Lagrangian density
and to encode the constraint imposed by Eq. (5.8). In order to proceed, let us
define the _canonical connection form_

$\left.\theta_{J^{1}\tau}\right|_{j_{x}^{1}s}:=e^{i}_{\mu}\left(de^{\mu}_{j}-e^{\mu}_{j\rho}dx^{\rho}\right)\otimes
E^{j}_{i}$

where $j_{x}^{1}s=\left(x^{\mu},e^{\nu}_{j},e^{\sigma}_{k\rho}\right)$ are the
induced coordinates on $J^{1}\tau$; it can be proved that this formula defines
a $\mathfrak{gl}\left(m\right)$-valued $1$-form on $J^{1}\tau$, and that it
becomes a connection form on the $GL\left(m\right)$-principal bundle

$p_{GL\left(m\right)}^{J^{1}\tau}:J^{1}\tau\to C\left(LM\right).$

Let

$\rho\left(x\right):=\left(x^{\mu},e^{\mu}_{i}\left(x\right),e^{\nu}_{k\sigma}\left(x\right)\right)$

be a local description for a section $\rho:U\to J^{1}\tau$; then from the
tetrad postulate (5.6) and using Eq. (5.11), we have that

$\rho^{*}\theta_{J^{1}\tau}=\left(\omega_{LM}\right)^{i}_{j}\otimes
E^{j}_{i}.$

Let
$\Omega_{LM}\in\Omega^{2}\left(J^{1}\tau,\mathfrak{gl}\left(m\right)\right)$
be the curvature form associated to $\gamma$; naturatility of the pullback
implies that

$\rho^{*}\Theta_{J^{1}\tau}=\left(\Omega_{LM}\right)^{i}_{j}\otimes E^{j}_{i}$

at curvature forms level. The formula

$\left.\varphi\right|_{j_{x}^{1}s}:=e^{i}_{\mu}dx^{\mu}\otimes e_{j}$

also defines a global $\mathbb{R}^{m}$-valued $1$-form on $J^{1}\tau$; out
from this form and the bilinear map
$\eta:\mathbb{R}^{m}\times\mathbb{R}^{m}\to\mathbb{R}$, we can define a
$GL\left(m\right)$-valued $\left(m-2\right)$-form as follows: First, define
the $\wedge^{m-2}\mathbb{R}^{m}$-valued $\left(m-2\right)$-form given by the
formula

$\varphi^{m-2}:=\overbrace{\varphi\wedge\cdots\wedge\varphi}^{m-2}.$

Then, use the star map
$\star:\wedge^{m-2}\mathbb{R}^{m}\to\wedge^{2}\mathbb{R}^{m}$ determined by
the bilinear form $\eta$ to define a $\wedge^{2}\mathbb{R}^{m}$-valued
$\left(m-2\right)$-form $\star\left(\varphi^{m-2}\right)$; finally, use the
isomorphism $\eta:\mathbb{R}^{m}\to\left(\mathbb{R}^{m}\right)^{*}$ to define
a map

$\eta^{\sharp}:\wedge^{2}\mathbb{R}^{m}\to\left(\mathbb{R}^{m}\right)^{*}\otimes\mathbb{R}^{m}=\mathfrak{gl}\left(m\right)$

giving rise to a $\mathfrak{gl}\left(m\right)$-valued $\left(m-2\right)$-form,
called _Sparkling form_

$\varphi_{m-2}^{\sharp}:=\eta^{\sharp}\left(\varphi^{m-2}\right).$

Additionally, $\eta$ induces a bilinear pairing between
$\mathfrak{gl}\left(m\right)$-forms on $J^{1}\tau$. Using the naturality
properties of the pullback, we can conclude that (in the case $m=4$)

$\rho^{*}\left(\eta\left(\varphi_{2}^{\sharp}\stackrel{{\scriptstyle\wedge}}{{,}}\Gamma\right)\right)=\epsilon_{ijkl}\eta^{kp}e^{i}_{\mu}e^{j}_{\nu}dx^{\mu}\wedge
dx^{\nu}\wedge\Omega^{l}_{p};$

also, for the case $m=3$ it would result

$\rho^{*}\left(\eta\left(\varphi_{1}^{\sharp}\stackrel{{\scriptstyle\wedge}}{{,}}\Gamma\right)\right)=\epsilon_{ijk}\eta^{kp}e^{i}_{\mu}dx^{\mu}\wedge\Omega^{j}_{p}.$

These considerations allows us to define the _Lagrangian form for gravity with
basis_ , that becomes

(5.12)
$\lambda_{\text{PG}}:=\eta\left(\varphi_{m-2}^{\sharp}\stackrel{{\scriptstyle\wedge}}{{,}}\Gamma\right).$

Strictly speaking, $\lambda_{PG}$ is not a $\tau_{1}$-horizontal form on
$J^{1}\tau$, and so it is not associated to a Lagrangian density; if we wanted
to deal with the underlying variational problem as an usual variational
problem, we would have to lift it to the jet bundle $J^{1}\tau_{1}$ (or
perhaps to the subbundle $J^{2}\tau$). Instead, we will treat it as a
Griffiths variational problem, in order to avoid the introduction of
additional variables.

Regarding the constraint (5.8), let

$\mathfrak{gl}\left(m\right)=\mathfrak{p}\oplus\mathfrak{k}$

be the decomposition of the general linear Lie algebra in terms of the $\pm
1$-eigenspaces of the involutive operator

$A\mapsto\eta A^{T}\eta;$

then this constraint is recovered by the equation

(5.13) $\rho^{*}\left(\pi_{\mathfrak{p}}\circ\theta_{J^{1}\tau}\right)=0,$

where $\pi_{\mathfrak{p}}:\mathfrak{gl}\left(m\right)\to\mathfrak{p}$ is the
associated projection onto the factor $\mathfrak{p}$.

######  Definition 5 (Variational problem for gravity with basis).

The _variational problem for gravity with basis_ is the triple

$\left(\tau_{1}:J^{1}\tau\to
M,\lambda_{\text{PG}},{\mathcal{I}}_{\text{PG}}\right)$

where ${\mathcal{I}}_{\text{PG}}$ is the exterior differential system on
$J^{1}\tau$ generated by the forms (5.13).

##  6\. Extension of generalized Cartan connections and Chern-Simons field
theory

In this section we will formulate a Chern-Simons field theory on a principal
bundle with structure group $A\left(m\right)$, and we will relate it with the
usual Chern-Simons field theory on a $K$-structure using the operation of
extension for generalized Cartan connections.

###  6.1. Chern-Simons variational problem with Lie group
$A\left(3,\mathbb{R}\right)$

In the present section we will work with the affine frame bundle on a manifold
of dimension $m=3$. Using Corollary 3 and the geometrical constructions
performed in Appendix C, we will define a Griffiths variational problem on
$\gamma^{*}\left(J^{1}\left(\tau\circ\beta\right)\right)$ in order to
represent Chern-Simons gauge theory. To proceed, let us define the bilinear
form
$\left<\cdot,\cdot\right>:\mathfrak{gl}\left(m\right)\times\mathfrak{gl}\left(m\right)\to\mathbb{R}$
given by

$\left<a,b\right>:=a^{k}_{i}b^{i}_{k}.$

######  Lemma 7.

The bilinear form $\left<\cdot,\cdot\right>$ is non degenerate and
$GL\left(3\right)$-invariant.

Recall that $K\subset GL\left(m,\mathbb{R}\right)$ is the Lorentz group
defined by the matrix $\eta$ (see Equation (2.4)); as always, $\mathfrak{k}$
will indicate its Lie algebra. Then we have the isomorphism [Wis10]

$\mathfrak{k}\simeq\mathbb{R}^{3}$

given by

$\xi=\left(\xi^{i}\right)\mapsto a_{i}^{j}:=\eta^{jk}\epsilon_{ikl}\xi^{l}.$

It allows us to use the prescription

$\left<\left(a,\xi\right),\left(b,\zeta\right)\right>:=\left<a,\zeta\right>+\left<b,\xi\right>$

for the extension of the bilinear form defined above to
$\mathfrak{gl}\left(3\right)\ltimes\mathbb{R}^{3}$. Thus we have a quadratic
form

$q:\mathfrak{a}\left(3\right)\to\mathbb{R}:\left(a,\xi\right)\mapsto\left<\left(a,\xi\right),\left(a,\xi\right)\right>,$

and using Equation (B.2), we obtain the following definition.

######  Definition 6 (Chern-Simons Lagrangian $3$-form).

The _Lagrangian form for Chern-Simons variational problem_ is the $3$-form
${\mathcal{L}}_{CS}\in\Omega^{3}\left(\gamma^{*}\left(J^{1}\left(\tau\circ\beta\right)\right)\right)$
defined through

${\mathcal{L}}_{CS}:=\left<\theta^{*}_{J^{1}\left(\tau\circ\beta\right)}\stackrel{{\scriptstyle\wedge}}{{,}}\Theta^{*}_{J^{1}\left(\tau\circ\beta\right)}\right>-\frac{1}{6}\left<\theta^{*}_{J^{1}\left(\tau\circ\beta\right)}\stackrel{{\scriptstyle\wedge}}{{,}}\left[\theta^{*}_{J^{1}\left(\tau\circ\beta\right)}\stackrel{{\scriptstyle\wedge}}{{,}}\theta^{*}_{J^{1}\left(\tau\circ\beta\right)}\right]\right>,$

where
$\theta^{*}_{J^{1}\left(\tau\circ\beta\right)}\in\Omega^{1}\left(\gamma^{*}\left(J^{1}\left(\tau\circ\beta\right)\right)\right)$
is the pullback to $\gamma^{*}\left(J^{1}\left(\tau\circ\beta\right)\right)$
of the canonical connection on the principal bundle
$p_{A\left(3\right)}^{J^{1}\left(\tau\circ\beta\right)}:J^{1}\left(\tau\circ\beta\right)\to
C\left(AM\right)$ (see Equation (5.1) above).

Because $\theta^{*}_{J^{1}\left(\tau\circ\beta\right)}$ is
$\mathfrak{gl}\left(3\right)\oplus\mathbb{R}^{3}$-valued, we can write

$\theta^{*}_{J^{1}\left(\tau\circ\beta\right)}=o^{*}_{J^{1}\tau}+e$

Using Equation (C.6) we see that, in local terms, this form becomes

$\theta^{*}_{J^{1}\left(\tau\circ\beta\right)}=e_{\beta}^{j}\left(de_{i}^{\beta}-e^{\beta}_{i\alpha}dx^{\alpha}\right)\otimes
E^{i}_{j}-e^{i}_{\beta}v^{\beta}_{\alpha}dx^{\alpha}\otimes e_{i},$

so that

$o^{*}_{J^{1}\left(\tau\circ\beta\right)}=e_{\beta}^{j}\left(de_{i}^{\beta}-e^{\beta}_{i\alpha}dx^{\alpha}\right)\otimes
E^{i}_{j},\qquad e=-e^{i}_{\beta}v^{\beta}_{\alpha}dx^{\alpha}\otimes e_{i}.$

Also,

$\Theta^{*}_{J^{1}\left(\tau\circ\beta\right)}=O^{*}_{J^{1}\left(\tau\circ\beta\right)}+E.$

Now, the splitting
$\mathfrak{a}\left(3\right)=\mathfrak{gl}\left(3\right)\oplus\mathbb{R}^{3}$
has the following properties

$\left[\mathfrak{gl}\left(3\right),\mathfrak{gl}\left(3\right)\right]\subset\mathfrak{gl}\left(3\right),\qquad\left[\mathfrak{gl}\left(3\right),\mathbb{R}^{3}\right]\subset\mathbb{R}^{3},\qquad\left[\mathbb{R}^{3},\mathbb{R}^{3}\right]=0$

and

$\mathfrak{gl}\left(3\right)\perp\mathfrak{gl}\left(3\right),\qquad\mathbb{R}^{3}\perp\mathbb{R}^{3},$

so that we can obtain a result that is the equivalent in this context to
Proposition $1$ in [Wis+09].

######  Proposition 11.

The Chern-Simons Lagrangian can be written as

${\mathcal{L}}_{\text{CS}}=\left<e\stackrel{{\scriptstyle\wedge}}{{,}}O^{*}_{J^{1}\left(\tau\circ\beta\right)}\right>.$

In local coordinates, this Lagrangian becomes

${\mathcal{L}}_{\text{CS}}=\epsilon_{jkl}\eta^{kp}v^{\nu}_{\beta}e^{j}_{\nu}dx^{\beta}\wedge\left(O_{J^{1}\tau}^{*}\right)^{l}_{p}.$

It is interesting to note that, except by the factors $v^{\alpha}_{\beta}$,
this Lagrangian is equivalent to the Lagrangian for $2+1$-gravity with basis;
we need to take care of them, and it will be done through the imposition of
constraints.

So, it is necessary to prescribe the set of constraints that sections
$\sigma:U\subset M\to\gamma^{*}\left(J^{1}\left(\tau\circ\beta\right)\right)$
should obey in order to be evaluated in the action associated to
${\mathcal{L}}_{CS}$. The first set of constraints we need to consider are
those imposed by Equation (5.8) above; in order to achieve it, let us consider
the decomposition

$\mathfrak{a}\left(3\right)=\left(\mathfrak{p}+0\right)\oplus\left(\mathfrak{k}\oplus\mathbb{R}^{3}\right);$

accordingly, let

$\pi_{\mathfrak{p}}:\mathfrak{a}\left(3\right)\to\mathfrak{p}+0,\qquad\pi_{\mathfrak{k}}:\mathfrak{a}\left(3\right)\to\mathfrak{k}\oplus\mathbb{R}^{3}$

be the corresponding projectors. Then we will have that

$\pi_{\mathfrak{p}}\circ\theta^{*}_{J^{1}\left(\tau\circ\beta\right)}=\eta^{ik}\left(o^{*}_{J^{1}\left(\tau\circ\beta\right)}\right)_{k}^{j}+\eta^{jk}\left(o^{*}_{J^{1}\left(\tau\circ\beta\right)}\right)_{k}^{i},$

so that this set of forms is suitable for the incarnation of Equation (5.8) in
this setting.

As we mentioned above, another constraint to be taken into account has to do
with coordinates $v^{\alpha}_{\beta}$; the most natural thing is to use the
map $j:J^{1}\tau\to J^{1}\left(\tau\circ\beta\right)$ discussed in Example 1,
where the classical notion for affine connection was introduced. In
particular, it was proved there (see Equation (5.4)) that the image set of the
map $j$ is described by the equations

$v^{\alpha}=0,\quad v^{\alpha}_{\beta}=-e_{\beta}^{i}e_{i}^{\alpha}.$

Then the constraint in this case deals with the extra degrees of freedom
$v^{\alpha}_{\beta}$ through this map; concretely, we are imposing the form

$x\mapsto\left(\tau_{10}\left(s\left(x\right)\right),j\left(s\left(x\right)\right)\right)\in\gamma^{*}\left(J^{1}\left(\tau\circ\beta\right)\right)$

for the allowed sections in the variational problem with Lagrangian
${\mathcal{L}}_{\text{CS}}$, where $s:U\subset M\to J^{1}\tau$ is a section of
$\tau_{1}:J^{1}\tau\to M$.

######  Definition 7 (Constraints for Chern-Simons variational problem on
$A\left(3,\mathbb{R}\right)$).

We will say that a section $\sigma:U\subset
M\to\gamma^{*}\left(J^{1}\left(\tau\circ\beta\right)\right)$ is _admissible
for the Chern-Simons variational problem_ if and only if

$\sigma\left(x\right)=\left(\tau_{10}\left(s\left(x\right)\right),j\left(s\left(x\right)\right)\right)$

for some section $s:U\subset M\to J^{1}\tau$ and also

$\sigma^{*}\left(\pi_{\mathfrak{p}}\circ\theta^{*}_{J^{1}\left(\tau\circ\beta\right)}\right)=0.$

The admissible section will have the following property.

######  Proposition 12.

Let
$\theta_{J^{1}\left(\tau\circ\beta\right)}\in\Omega^{1}\left(J^{1}\left(\tau\circ\beta\right),\mathfrak{a}\left(m\right)\right)$
be the canonical connection on $J^{1}\left(\tau\circ\beta\right)$ and
$\theta_{J^{1}\tau}\in\Omega^{1}\left(J^{1}\tau,\mathfrak{gl}\left(m\right)\right)$
be the canonical connection form on $J^{1}\tau$. Then

$j^{*}\theta_{J^{1}\left(\tau\circ\beta\right)}=\theta_{J^{1}\tau}+\tau_{10}^{*}\varphi.$

###### Proof.

We will give an argument in local terms. Using Equation (C.6) and the local
expression (5.4), it follows that

$j^{*}\theta_{J^{1}\left(\tau\circ\beta\right)}=e_{\beta}^{j}\left(de_{i}^{\beta}-e^{\beta}_{i\alpha}dx^{\alpha}\right)\otimes
E^{i}_{j}+e^{i}_{\beta}\delta^{\beta}_{\alpha}dx^{\alpha}\otimes
e_{i}=\theta_{J^{1}\tau}+\tau_{10}^{*}\varphi,$

as required. ∎

With these elements at hand, it is immediate to formulate the variational
problem we will use to represent Chern-Simons gauge theory in this context.

######  Definition 8 (Chern-Simons variational problem on
$A\left(3,\mathbb{R}\right)$).

It is the variational problem prescribed by the action

$\sigma\mapsto\int_{U}\sigma^{*}\left({\mathcal{L}}_{CS}\right)$

for $\sigma:U\subset
M\to\gamma^{*}\left(J^{1}\left(\tau\circ\beta\right)\right)$ an admissible
section.

From Proposition 11 and 12 we obtain the correspondence between Chern-Simons
field theory and gravity with basis in this setting.

######  Theorem 1.

The extremals of the Chern-Simons variational problem on
$A\left(3,\mathbb{R}\right)$ are in one-to-one correspondence with the
extremals of the variational problem described in Definition 5.

###### Proof.

The correspondence is determined by the bijective map

$s\mapsto\sigma:=\left(\tau_{10}\circ s,j\circ s\right)$

between a section $s$ of $\tau_{1}:J^{1}\tau\to M$ and an admissible section
$\sigma$ for $\gamma^{*}\left(J^{1}\left(\tau\circ\beta\right)\right)$. ∎

###  6.2. Chern-Simons variational problem on $A\left(3\right)$ as extension
of the Wise variational problem

We will prove in this section that sections of $J^{1}\tau$ that are extensions
of extremals for the Wise variational problem on the jet bundle
$J^{1}\pi_{\zeta}$ associated to any $K$-structure $O_{\zeta}$, are extremals
for the Chern-Simons variational problem on $A\left(3\right)$ and viceversa.
In order to achieve this result, we will need to use the relationship that
connects extensions of Cartan connections with the original connections, as
described in Diagram (4.4). In this case, with the help of the naturality of
the canonical connection established in Lemma 2, the following proposition can
be proven (the notation used is the one employed in Section 4).

######  Proposition 13.

Let

$\Gamma:P\to
J^{1}\pi_{\left[G_{1}\right]}\qquad\Gamma^{\sharp}:P\left[G_{2}\right]\to
J^{1}\pi_{\left[G\right]}$

be a pair of (generalized) Cartan connections such that

$j^{1}\gamma_{G_{1}}\circ\Gamma=\Gamma^{\sharp}\circ\gamma^{2}_{H}.$

Then

$\left(\Gamma^{\sharp}\circ\gamma^{2}_{H}\right)^{*}\theta_{J^{1}\pi_{\left[G\right]}}=\Gamma^{*}\theta_{J^{1}\pi_{\left[G_{1}\right]}}.$

We will use this result to prove that the Chern-Simons variational problem on
$A\left(3\right)$ can be seen as an extension of the Wise variational problem,
namely, that we can establish a one-to-one correspondence between the
extremals of these variational problems through the operations of extension
and reduction of generalized Cartan connections, as defined in Section 4. To
this end, let us apply Proposition 13 to the following diagram

${AM}$${O_{\zeta}^{\text{aff}}}$${LM}$${O_{\zeta}}$$\scriptstyle{\gamma^{\text{aff}}_{\zeta}}$$\scriptstyle{\gamma}$$\scriptstyle{\gamma_{\zeta}}$$\scriptstyle{\gamma_{O_{\zeta}}}$

Now, recall from Section 4.3.3 that the choice of a metric $\zeta:M\to\Sigma$
allows us to select a $K$-structure $O_{\zeta}\subset LM$ and a subbundle
$O_{\zeta}^{\text{aff}}\subset AM$; let us indicate with

$\tau_{\zeta}^{\text{aff}}:O_{\zeta}^{\text{aff}}\to M$

the restriction of the canonical projection $\tau^{\text{aff}}:AM\to M$ to
this subbundle. It follows that if

$\sigma:U\to\gamma^{*}\left(J^{1}\left(\tau\circ\beta\right)\right)$

is a section of the jet space for the affine frame bundle, and

$\sigma_{\zeta}:U\to\gamma_{\zeta}^{*}\left(J^{1}\left(\left.\left(\beta\circ\tau\right)\right|_{O_{\zeta}^{\text{aff}}}\right)\right)$

is a section of the jet space for the restriction of the affine frame bundle
to orthonormal basis respect to the metric $\zeta$, then they will be related
by a relation of reduction or extension if and only if

$\tau_{10}\circ\text{pr}_{1}\circ\sigma=\left(\tau_{\zeta}^{\text{aff}}\right)_{10}\circ\text{pr}_{1}\circ\sigma_{\zeta}$

and

$j^{1}\gamma_{\zeta}^{\text{aff}}\circ\text{pr}_{2}\circ\sigma=\text{pr}_{2}\circ\sigma_{\zeta}\circ\gamma_{O_{\zeta}},$

where $\text{pr}_{i},i=1,2$ are the projections onto the factors in the
cartesian product. Then by Proposition 13 we will have that

$\sigma^{*}{\mathcal{L}}_{\text{CS}}=\sigma_{\zeta}^{*}\lambda_{\text{CS}},$

for sections related by the operations of reduction or extension of
generalized Cartan connections; it means that the correspondences

$\sigma\mapsto\sigma_{\zeta},\qquad\sigma_{\zeta}\mapsto\sigma$

establish a one-to-one correspondence between the Chern-Simons variational
problem on $A\left(3\right)$ and the Wise variational problem, as required.
Thus, we have prove the following result.

######  Theorem 2.

The operations of reduction and extension of generalized Cartan connections
establish a one-to-one correspondence between the extremals of the Wise
variational problem for any first order geometry associated to the pair
$\left(SO\left(2,1\right)\ltimes\mathbb{R}^{3},SO\left(2,1\right)\right)$
(Definition 4) and the Chern-Simons variational problem on $A\left(3\right)$
(as it is described by Definition 8).

##  Appendix A Geometry of principal bundles

###  A.1. Connections and principal bundles

It is a usual to represent a principal connection by a family of locally
defined $\mathfrak{g}$-valued $1$-forms
$A_{U}\in\Omega^{1}\left(U,\mathfrak{g}\right)$, with $U$ belonging to a
covering $\mathcal{C}:=\left\\{U\right\\}$ for $M$; these forms should obey a
_gauge transformation condition_ : For every pair $U,V\in\mathcal{C}$ there
must exists a map

$t_{UV}:U\cap V\to G$

such that

(A.1)
$A_{V}=\text{Ad}_{t_{UV}^{-1}}\left(A_{U}\right)+t_{UV}^{-1}dt_{UV}=\text{Ad}_{t_{UV}^{-1}}\left(A_{U}\right)+t_{UV}^{*}\lambda,$

where $\lambda\in\Omega^{1}\left(G,\mathfrak{g}\right)$ is the (left) Maurer-
Cartan form on $G$. Additionally, these maps must be compatible in the sense
that, for every $U,V,W\in\mathcal{C}$ such that $U\cap V\cap W\not=\emptyset$,
then

$t_{UW}\left(x\right)=t_{UV}\left(x\right)\cdot t_{VW}\left(x\right)$

for every $x\in U\cap V\cap W$. The existence of these maps is equivalent to
have a $G$-principal bundle on $M$ [KN63, p. 51], for which the family
$\mathcal{C}$ becomes a covering of trivializing open sets; as it would seem
obvious, we want to stress the fact that a principal bundle is singled out
when dealing with a connection described in this way.

Before to go on, let us discuss briefly about a notion of equivalence
involving the principal bundle structure. As we know, a gauge transformation
does not change the connection; therefore, a collection of forms

$A_{U}^{\prime}:=\text{Ad}_{g_{U}^{-1}}\left(A_{U}\right)+g_{U}^{*}\lambda$

where $g_{U}:U\to G$ is a family of smooth functions, should describe the same
connection. Accordingly, the structure functions $t_{UV}$ must change obeying
the rule

(A.2) $t_{UV}^{\prime}:U\cap V\to G:x\mapsto
g_{U}\left(x\right)t_{UV}\left(x\right)g_{V}\left(x\right),$

with the underlying principal bundle remaining invariant; therefore, set of
transformations $\left\\{t_{UV}\right\\},\left\\{t_{UV}^{\prime}\right\\}$
related by Equation (A.2) should be considered as equivalent.

Let us look more closely to the principal bundle so constructed. Consider the
triples

$\left(U,x,a\right)\in\mathcal{C}\times M\times G$

such that $x\in U$; we say that $\left(U,x,a\right)$ is equivalent to
$\left(V,y,b\right)$ if and only if $U\cap V\not=\emptyset$, $x=y$ and

$b=t_{VU}\left(x\right)a.$

We will indicate with $\left[U,x,a\right]$ the equivalence class containing
$\left(U,x,a\right)$; the space obtained by quotient by this equivalence
relation becomes a $G$-principal bundle $\pi:P\to M$, where

$\pi\left(\left[U,x,a\right]\right)=x$

and the $G$-action is simply given by

$\left[U,x,a\right]\cdot h=\left[U,x,ah\right].$

The covering $\mathcal{C}$ contains trivializing open sets; in fact, on
$\pi^{-1}\left(U\right)$ we have the trivializing map

$t_{U}:\pi^{-1}\left(U\right)\to U\times
G:\left[U,x,a\right]\mapsto\left(x,a\right).$

This description of the bundle $P$ allows us to construct a family of local
sections for $P$, namely

$s_{U}:U\to\pi^{-1}\left(U\right):x\mapsto\left[U,x,e\right],$

where we have used the symbol $e$ for the unit in $G$; it follows that

(A.3)
$s_{V}\left(x\right)=\left[V,x,e\right]=\left[U,x,t_{UV}\left(x\right)\right]=\left[U,x,e\right]\cdot
t_{UV}\left(x\right)=s_{U}\left(x\right)\cdot t_{UV}\left(x\right)$

for every $x\in U\cap V$. It means that the local sections $s_{U}$ are related
by the gauge transformations associated to the transition functions $t_{UV}$.

Let $U\in\mathcal{C}$ be an open set in the covering; then the pair
$\left(A_{U},s_{U}\right)$ allow us to construct the connection form $\omega$
on $P$ through the formula [Nak96]

(A.4)
$\left.\omega\right|_{u}:=\text{Ad}_{\left(g_{U}\left(u\right)^{-1}\right)}\left(\pi^{*}\left.A_{U}\right|_{x}\right)+g_{U}^{*}\lambda;$

here $x=\pi\left(u\right)$, $g_{U}:\pi^{-1}\left(U\right)\to G$ is defined by

$u=s_{U}\left(x\right)\cdot g_{U}\left(u\right)$

and $\lambda$ is the (left) Maurer-Cartan form on $G$. The transformation
properties of these local data imply that

$\text{Ad}_{\left(g_{U}\left(u\right)^{-1}\right)}\left(\pi^{*}\left.A_{U}\right|_{x}\right)+g_{U}^{*}\lambda=\text{Ad}_{\left(g_{V}\left(u\right)^{-1}\right)}\left(\pi^{*}\left.A_{V}\right|_{x}\right)+g_{V}^{*}\lambda,\qquad
x=\pi\left(u\right)$

for any pair $U,V\in\mathcal{C}$ and $u\in\pi^{-1}\left(U\cap V\right)$;
therefore, this definition is independent of the open set used to calculate it
through Eq. (A.4).

###  A.2. Gauge transformations

Recall that a _gauge transformation of a principal bundle $P$_ is a bundle map
$\phi:P\to P$ over the identity that commutes with the $G$-action, namely

$\phi\left(u\cdot g\right)=\phi\left(u\right)\cdot g.$

Using the trivialization maps $t_{U}:\pi^{-1}\left(U\right)\to U\times G$, a
gauge transformation is locally described by the maps

$\phi_{U}\left(x,a\right):=\left(t_{U}\circ\phi\circ
t_{U}^{-1}\right)\left(x,a\right)=\left(x,h_{U}\left(x\right)a\right),$

where the functions $h_{U}:U\to G$ must have the following transformation
property

$h_{V}\left(x\right)=t_{UV}\left(x\right)h_{U}\left(x\right)$

for every $x\in U\cap V$. In fact, let us define

$\phi\left(\left[U,x,a\right]\right):=\phi_{U}\left(x,a\right)=\left[U,x,h_{U}\left(x\right)a\right]$

for any $\left[U,x,a\right]\in P$; if $x\in U\cap V$, we will have that

$\displaystyle\left[V,x,h_{V}\left(x\right)a\right]$
$\displaystyle=\left[U,x,t_{VU}\left(x\right)h_{V}\left(x\right)a\right]$
$\displaystyle=\left[U,x,h_{U}\left(x\right)a\right],$

showing that this definition is independent of the open set $U$ containing
$x$.

###  A.3. Jet bundle and the bundle of principal connections

Let us suppose that every $U\in\mathcal{C}$ is a coordinate domain; therefore,
we can write

$A_{U}=\zeta^{U}_{i}dx^{i},\qquad\zeta^{U}_{i}:U\to\mathfrak{g}$

and so, locally, we have that a connection can be seen as a section of the
bundle

$\text{pr}_{1}:U\times
G\times\overbrace{\mathfrak{g}\otimes\cdots\otimes\mathfrak{g}}^{m\text{
times}}\to U.$

As with $P$, we can take advantage of the transition functions $t_{UV}$ in
order to glue together these local fibrations. Accordingly, let us consider
the $\left(m+3\right)$-uples

$\left(U,x,a,\xi_{1},\cdots,\xi_{m}\right)\in\mathcal{C}\times M\times
G\times\overbrace{\mathfrak{g}\otimes\cdots\otimes\mathfrak{g}}^{m\text{
times}}$

such that $x\in U$, and define the equivalence relation given by

$\left(U,x,a,\xi_{i}\right)\sim\left(V,y,b,\zeta_{i}\right)$

if and only if $U\cap V\not=\emptyset$, $x=y$, $b=t_{VU}\left(x\right)a$ and

$\zeta_{i}dx^{i}=\xi_{i}dx^{i}+\text{Ad}_{a^{-1}}\left(t_{VU}^{*}\lambda\right).$

The quotient space $J^{1}\pi$ is a manifold, the so called _jet space of the
bundle $\pi:P\to M$_; we have canonical projections

$\pi_{10}:J^{1}\pi\to P:\left[U,x,a,\xi_{i}\right]\mapsto\left[U,x,a\right]$

and

$\pi_{1}:J^{1}\pi\to M:\left[U,x,a,\xi_{i}\right]\mapsto x$

giving it bundle structure on both spaces $P$ and $M$. As before, we have a
local trivialization for $J^{1}\pi$ through the formula

$T_{U}:\pi_{1}^{-1}\left(U\right)\to U\times
G\times\overbrace{\mathfrak{g}\otimes\cdots\otimes\mathfrak{g}}^{m\text{
times}}:\left[U,x,a,\xi_{i}\right]\mapsto\left(x,a,\xi_{i}\right).$

Every element $\left[U,x,g,\xi_{i}\right]\in J^{1}\pi$ is equivalent to a
linear map

$\frac{\partial}{\partial x^{i}}\in T_{x}M\longmapsto
T_{\left[U,x,e\right]}R_{g}\circ T_{x}s_{U}\left(\frac{\partial}{\partial
x^{i}}\right)+\left(\xi_{i}\right)_{P}\left(\left[U,x,g\right]\right)\in
T_{\left[U,x,g\right]}P,$

where $\zeta_{P}$ is the infinitesimal generator of the $G$-action on $P$
corresponding to the element $\zeta\in\mathfrak{g}$. Using the fact that

$t_{U}\circ R_{h}=\left(\text{id}\times R_{h}\right)\circ t_{U}$

for every $h\in G$, we can see that

$Tt_{U}\circ\zeta_{P}=\zeta_{G}^{R}$

for all $\zeta\in\mathfrak{g}$; here $\xi_{G}^{R}$ indicates the infinitesimal
generator on $G$ associated to the right action. Therefore, the map $T_{U}$ is
induced by $Tt_{U}$.

It can be seen that $J^{1}\pi$ is a $G$-space; the action is given by the
formula

$\left[U,x,a,\xi_{i}\right]\cdot
h=\left[U,x,ah,\text{Ad}_{h^{-1}}\xi_{i}\right]$

for all $h\in G$. This action allows us to construct a quotient bundle

$C\left(P\right):=J^{1}\pi/G,$

and the canonical projection gives rise to a new bundle

$p_{G}^{J^{1}\pi}:J^{1}\pi\to C\left(P\right).$

The map $\overline{\pi}:C\left(P\right)\to M$ such that

$\pi_{1}=\overline{\pi}\circ p_{G}^{J^{1}\pi}$

gives $C\left(P\right)$ structure of a bundle on $M$; its elements are the
equivalence classes

$\left[U,x,a,\xi_{i}\right]_{G}:=\left\\{\left[U,x,ah,\text{Ad}_{h^{-1}}\xi_{i}\right]:h\in
G\right\\}.$

For every $U\in\mathcal{C}$, we have a trivialization map

$\phi_{U}:\overline{\pi}^{-1}\left(U\right)\to U\times\mathfrak{g}^{\otimes
m}:\left[U,x,a,\xi_{i}\right]_{G}\mapsto\left(x,\text{Ad}_{a}\xi_{i}\right);$

for every pair $U,V\in\mathcal{C}$ such that $U\cap V\not=\emptyset$, we have
that

$\displaystyle\phi_{V}\circ\phi_{U}^{-1}\left(x,\xi_{i}\right)$
$\displaystyle=\phi_{V}\left(\left[U,x,e,\xi_{i}\right]_{G}\right)$
$\displaystyle=\phi_{V}\left(\left[V,x,t_{VU}\left(x\right),\xi_{i}+\left(t_{VU}^{*}\lambda\right)\left(\frac{\partial}{\partial
x^{i}}\right)\right]_{G}\right)$
$\displaystyle=\left(x,\text{Ad}_{t_{VU}\left(x\right)}\left[\xi_{i}+\left(t_{VU}^{*}\lambda\right)\left(\frac{\partial}{\partial
x^{i}}\right)\right]\right)$
$\displaystyle=\left(x,\text{Ad}_{t_{UV}\left(x\right)^{-1}}\xi_{i}-\left(t_{UV}^{*}\lambda\right)\left(\frac{\partial}{\partial
x^{i}}\right)\right)$

where it was used that $t_{VU}\left(x\right)=t_{UV}\left(x\right)^{-1}$. The
fact that this expression is equivalent to the transformation law (A.1) allows
us to consider $C\left(P\right)$ as the bundle of principal connections for
$P$; namely, we have an identification

(A.5) $\left[U,x,a,\xi_{i}\right]_{G}\longleftrightarrow
A_{U}:=-\text{Ad}_{a}\xi_{i}dx^{i}.$

With this correspondence in mind, the existence of a $\mathfrak{g}$-valued
$1$-form $A_{U}$ for every $U\in\mathcal{C}$ such that compatibility
conditions (A.1) are fulfilled, is equivalent to the existence of a section

$\sigma:M\to C\left(P\right).$

In fact, we have the formula

$\sigma\left(x\right)=\left[U,x,e,-\left.A_{U}\right|_{x}\left(\frac{\partial}{\partial
x^{i}}\right)\right]_{G}$

for every $U\in\mathcal{C}$ and $x\in U$.

###  A.4. The canonical connection form on $J^{1}\pi$

A fundamental geometric structure on $J^{1}\pi$ is the _canonical connection
form_ $A$, which is a $\mathfrak{g}$-valued $1$-form on $J^{1}\pi$ inducing a
connection on the $G$-principal bundle $p_{G}^{J^{1}\pi}:J^{1}\pi\to
C\left(P\right)$; for every $U\in\mathcal{C}$, it is given by the formula

$\left.A\right|_{\left[U,x,a,\xi_{i}\right]}:=\left.\lambda\right|_{a}-\xi_{i}dx^{i}.$

In fact, because $b=t_{VU}\left(x\right)a$, it results that

$\left.\lambda\right|_{b}=\left.\lambda\right|_{a}+\text{Ad}_{a^{-1}}\left(t_{VU}^{*}\lambda\right);$

thus, if $\left[U,x,a,\xi_{i}\right]=\left[V,x,b,\zeta_{i}\right]$ for $x\in
U\cap V$, we will have

$\displaystyle\left.A\right|_{\left[V,x,b,\zeta_{i}\right]}$
$\displaystyle=\left.\lambda\right|_{b}-\zeta_{i}dx^{i}$
$\displaystyle=\left.\lambda\right|_{a}+\text{Ad}_{a^{-1}}\left(t_{VU}^{*}\lambda\right)-\left(\xi_{i}dx^{i}+\text{Ad}_{a^{-1}}\left(t_{VU}^{*}\lambda\right)\right)$
$\displaystyle=\left.\lambda\right|_{a}-\xi_{i}dx^{i}$
$\displaystyle=\left.A\right|_{\left[U,x,a,\xi_{i}\right]},$

and the form $A$ is well-defined.

There is an important property that the canonical form has. In order to
formulate it, let us consider another way to specify a connection, namely,
through an equivariant bundle map

$\Gamma:P\to J^{1}\pi.$

In fact, using the above description of these bundles, and given a local
description $\left\\{A_{U}:U\in\mathcal{C}\right\\}$ for a connection, we can
construct the map

$\Gamma:\left[U,x,a\right]\mapsto\left[U,x,a,-\text{Ad}_{a^{-1}}\left(\left.A_{U}\right|_{x}\left(\frac{\partial}{\partial
x^{i}}\right)\right)\right],$

where identification (A.5) was used. Having the gauge transformation property
(A.1) in mind, we can prove that it is a good definition. Now, given that

$s_{U}\left(x\right)=\left[U,x,e\right]$

and from $u=s_{U}\left(x\right)\cdot g_{U}\left(u\right)$ for every
$u=\left[U,x,a\right]\in\pi^{-1}\left(U\right)$, we obtain the formula

$u=\left[U,x,g_{U}\left(u\right)\right].$

Then

$\Gamma^{*}\left(\left.A\right|_{\Gamma\left(u\right)}\right)=\Gamma^{*}\left(\lambda-\xi_{i}dx^{i}\right)=g_{U}^{*}\lambda+\text{Ad}_{g_{U}\left(u\right)^{-1}}\left.A_{U}\right|_{x}=\left.\omega\right|_{u},$

namely, the connection form can be obtained through pullback along the map
$\Gamma$ of the canonical form.

##  Appendix B Local and global Chern-Simons Lagrangians

Let $K$ be a Lie group and $\pi_{\zeta}:R_{\zeta}\to M$ a $K$-principal bundle
(the notation will be explained later); on its first order jet space

$\left(\pi_{\zeta}\right)_{1}:J^{1}\pi_{\zeta}\to M$

we will define a (global) variational problem, which we will prove to
represent Chern-Simons gauge theory. In order to accomplish this task, it will
be necessary to lift it to $J^{1}\left(\pi_{\zeta}\right)_{1}$, and compare it
with the variational problem defined by local data, which lives on
$J^{1}\overline{\pi}_{\zeta}$.

Let us now suppose that we have an invariant polynomial
$q:\mathfrak{k}\to\mathbb{R}$ of degree $n$. According to the Chern-Simons
theory [CS74, Mor01], the $2n$-form

$\alpha:=q\left(F\right)\in\Omega^{2n}\left(J^{1}\pi_{\zeta}\right)$

is closed. For example, when the Lie algebra comes with an invariant bilinear
form, we can consider the quadratic polynomial

$q\left(F\right):=\left<F\stackrel{{\scriptstyle\wedge}}{{,}}F\right>;$

Additionally, it can be proved that the bundle

$\text{pr}_{1}:J^{1}\pi_{\zeta}\times_{C\left({R}_{\zeta}\right)}J^{1}\pi_{\zeta}\to
J^{1}\pi_{\zeta},$

is a trivial $K$-principal bundle; it means that $\alpha$ is not only closed,
but also exact. Therefore, there exists a $\left(2n-1\right)$-form

(B.1)
$\beta:=Tq\left(A,F\right)\in\Omega^{2n-1}\left(J^{1}\pi_{\zeta}\right),$

such that

$d\beta=\text{pr}_{1}^{*}\alpha;$

the polynomial $Tq$ can be found by a transgression formula [Nak96]. When $q$
is the quadratic form determined by an invariant bilinear form, the
transgression is given by

(B.2)
$Tq\left(A,F\right)=\left<A\stackrel{{\scriptstyle\wedge}}{{,}}F\right>-\frac{1}{6}\left<A\stackrel{{\scriptstyle\wedge}}{{,}}\left[A\stackrel{{\scriptstyle\wedge}}{{,}}A\right]\right>.$

How can these forms be related to the local forms usually used to described
Chern-Simons field theory? In this case, we have another $K$-principal bundle
structure, namely the quotient map

$p_{K}^{J^{1}\pi_{\zeta}}:J^{1}\pi_{\zeta}\to C\left({R}_{\zeta}\right),$

and $q\left(F\right)$ defines a Chern class for it. Accordingly, there exists
a $2n$-form $\gamma$ on $C\left({R}_{\zeta}\right)$ such that

$\left(p_{K}^{J^{1}\pi_{\zeta}}\right)^{*}\gamma=q\left(F\right).$

Moreover, using the canonical $2$-form $F_{2}$ on the bundle of connections
$C\left({R}_{\zeta}\right)$, we can prove that

$\gamma=q\left(F_{2}\right).$

But now, this bundle is not trivial in general; in short, from decomposition

$J^{1}\pi_{\zeta}={R}_{\zeta}\times_{M}C\left({R}_{\zeta}\right),$

we obtain that it is trivial if and only if the bundle

$\pi_{\zeta}:{R}_{\zeta}\to M$

is. In consequence, it is not expected that the $2n$-form $\gamma$, although
closed, should also be exact; it is the reason why, although we have a global
variational problem on $J^{1}\pi_{\zeta}$, it cannot be reproduced on
$C\left({R}_{\zeta}\right)$, even having in mind that the transformations
properties of ${\mathcal{L}}_{CS}$ are telling us that the degrees of freedom
associated to the ${R}_{\zeta}$-factor can be ignored.

Nevertheless, because ${R_{\zeta}}$ admits trivializing open sets, the
previous considerations can be used to associate to every such set $U\subset
M$ a Lagrangian ${\mathcal{L}}_{U}:=Tq\left(A_{2}^{U},F_{2}\right)$, where

$A_{2}^{U}\in\Omega^{1}\left(\left(\overline{\pi}_{\zeta}\right)^{-1}\left(U\right),\mathfrak{k}\right)$

is a $1$-form such that

$\left(p_{K}^{J^{1}\pi_{\zeta}}\right)^{*}A_{2}^{U}=\left.A\right|_{\left(\left(\pi_{\zeta}\right)_{1}\right)^{-1}\left(U\right)}.$

######  Remark 4.

A study of conditions ensuring the existence of global solutions for the local
variational problem for Chern-Simons gauge theory can be found in [PW17]. In
this regard, it is interesting to note that here we have changed a variational
problem determined by local data and whose sections could be globally defined,
by a variational problem described by global data, but whose sections are
forced to have local nature (part of these sections are sections of a
principal bundle).

##  Appendix C Geometry of the affine frame bundle

From now on we will devoted ourselves to particularize this definition to a
very specific principal bundle, the so called _affine frame bundle_ (see
Definition 9 below), and to relate the variational problem so obtained with
Palatini gravity.

###  C.1. The affine general linear group and the affine frame bundle

We have the splitting short exact sequence

${0}$${\mathbb{R}^{m}}$${A\left(m,\mathbb{R}\right)}$${GL\left(m,\mathbb{R}\right)}$${1}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$

where $A\left(m,\mathbb{R}\right)\subset GL\left(m+1,\mathbb{R}\right)$ is the
subgroup of matrices of the form

$B:=\begin{bmatrix}a&\xi\\\ 0&1\end{bmatrix}$

where $a\in GL\left(m,\mathbb{R}\right)$ and $\xi\in\mathbb{R}^{m}$; the maps
in the sequence read

$\alpha\left(\xi\right):=\begin{bmatrix}1&\xi\\\
0&1\end{bmatrix},\qquad\beta\left(\begin{bmatrix}a&\xi\\\
0&1\end{bmatrix}\right):=\begin{bmatrix}a&0\\\ 0&1\end{bmatrix},$

and

$\gamma\left(a\right):=\begin{bmatrix}a&0\\\ 0&1\end{bmatrix}.$

Because this short sequence splits, we can consider

$A\left(m,\mathbb{R}\right)=GL\left(m,\mathbb{R}\right)\oplus\mathbb{R}^{m},$

with the isomorphism of groups given by

$GL\left(m,\mathbb{R}\right)\oplus\mathbb{R}^{m}\to
A\left(m,\mathbb{R}\right):\left(a,\xi\right)\mapsto\gamma\left(a\right)+\alpha\left(\xi\right).$

Now, let $A^{m}$ be the set $\mathbb{R}^{m}$ considered as an affine space; we
can set an isomorphism between $A\left(m,\mathbb{R}\right)$ and the set of
affine maps

$f:A^{m}\to A^{m};$

in fact, given $B=\left(a,\xi\right)$, the associated affine map reads

$f_{B}\left(z\right):=az+\xi$

for every $z\in A^{m}$. Using the expression of an element
$\left(a,\xi\right)$ as a matrix, we have that

$\left(a,\xi\right)^{-1}=\left(a^{-1},-a^{-1}\xi\right)$

and

(C.1)
$\text{Ad}_{\left(a,\xi\right)}\left(b,\zeta\right)=\left(\text{Ad}_{a}b,a\zeta-\left(\text{Ad}_{a}b\right)\xi\right)$

for any $\left(a,\xi\right)\in
A\left(m,\mathbb{R}\right),\left(b,\zeta\right)\in\mathfrak{a}\left(m,\mathbb{R}\right)$.

In the same vein, let $A_{x}\left(M\right)$ be the set $T_{x}M$ considered as
an affine space, for every $x\in M$. As it is well-known [KN63], the set of
affine maps

$u:A^{m}\to A_{x}\left(M\right)$

for every $x\in M$ has structure of $A\left(m,\mathbb{R}\right)$-principal
bundle; the action of an element $B\in A\left(m,\mathbb{R}\right)$ is simply
given by

$u\cdot B:=u\circ B.$

######  Definition 9 (Bundle of affine frames).

The _bundle of affine frames on $M$_ will be the set

$AM:=\bigcup_{x\in M}\left\\{u:A^{m}\to A_{x}\left(M\right)\text{
affine}\right\\}.$

As it follows from the general theory of principal bundles, there exists a
pair of principal bundle morphisms associated to the homomorphisms
$\beta:A\left(m,\mathbb{R}\right)\to GL\left(m,\mathbb{R}\right)$ and
$\gamma:GL\left(m,\mathbb{R}\right)\to A\left(m,\mathbb{R}\right)$

${AM}$${LM}$${M}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$$\scriptstyle{\tau}$

For any $G$-principal bundle $\pi:P\to M$, the affine bundle
$\left(C\left(P\right),\overline{\pi},M\right)$ defined through the diagram

${J^{1}\pi}$${P}$${C\left(P\right):=J^{1}\pi/G}$${M}$$\scriptstyle{\pi_{10}}$$\scriptstyle{p_{G}^{J^{1}\pi}}$$\scriptstyle{\pi}$$\scriptstyle{\overline{\pi}}$

is called the _bundle of connections of the bundle $P$_, and we can establish
a canonical one-to-one correspondence between its sections and principal
connections on $P$. The correspondence is given as follows: Any element
$j_{x}^{1}s\in J^{1}\pi$ is a linear map

$j_{x}^{1}s:T_{x}M\to T_{s\left(x\right)}P$

such that

$T_{s\left(x\right)}\pi\circ j_{x}^{1}s=\text{id}_{T_{x}M},$

and so a $G$-orbit $\left[j_{x}^{1}s\right]_{G}$ can be interpreted as a
linear map

$\left[j_{x}^{1}s\right]_{G}:T_{x}M\to\left(TP/G\right)_{x}.$

Given $u\in P$, there exists a unique $m$-dimensional subspace $H_{u}\subset
T_{u}P$ such that

$\left[j_{x}^{1}s\right]_{G}\left(T_{x}M\right)=p_{G}^{TP}\left(H_{u}\right);$

the assignment $u\mapsto H_{u}$ is the connection associated to
$\left[j_{x}^{1}s\right]_{G}$.

Therefore we have the diagram

${J^{1}\left(\tau\circ\beta\right)}$${C\left(AM\right)}$${J^{1}\tau}$${C\left(LM\right)}$${AM}$${LM}$${M}$$\scriptstyle{p_{A\left(m,\mathbb{R}\right)}^{J^{1}\left(\tau\circ\beta\right)}}$$\scriptstyle{j^{1}\beta}$$\scriptstyle{\left(\tau\circ\beta\right)_{10}}$$\scriptstyle{\left[j^{1}\beta\right]}$$\scriptstyle{p_{GL\left(m,\mathbb{R}\right)}^{J^{1}\tau}}$$\scriptstyle{\tau_{10}}$$\scriptstyle{\overline{\tau}}$$\scriptstyle{\beta}$$\scriptstyle{\tau}$

It is a theorem that any connection $\Gamma:M\to C\left(LM\right)$ gives rise
to a unique connection $\widetilde{\Gamma}:M\to C\left(AM\right)$ such that if

$\omega_{LM}\in\Omega^{1}\left(LM,\mathfrak{gl}\left(m,\mathbb{R}\right)\right)\qquad\text{
and
}\qquad{\omega}_{AM}\in\Omega^{1}\left(AM,\mathfrak{a}\left(m,\mathbb{R}\right)\right)$

are the corresponding connection forms, then

(C.2) $\gamma^{*}{\omega}_{AM}=\omega_{LM}+\varphi,$

where $\varphi\in\Omega^{1}\left(LM,\mathbb{R}^{m}\right)$ is the canonical
solder $1$-form on $LM$.

###  C.2. The affine frame bundle as extension of the frame bundle

It remains to interpret the affine frame bundle as the extension of the frame
bundle using the group immersion $GL\left(m,\mathbb{R}\right)\subset
A\left(m,\mathbb{R}\right)$.

######  Proposition 14.

Let $H=GL\left(m,\mathbb{R}\right)$, $G=A\left(m,\mathbb{R}\right)$ and
$P=LM$. Then

$P\times_{H}G\simeq AM.$

###### Proof.

Let us define the map

$\phi\left(\left[u,\left(h,v\right)\right]_{H}\right):=a\in\left.AM\right|_{x}$

if and only if $x=\tau\left(u\right)$ and

$a:\mathbb{R}^{m}\to T_{x}M:w\mapsto\left(u\circ
h\right)\left(w\right)+u\left(v\right).$

Then, for any $\left(h^{\prime},v^{\prime}\right)\in
A\left(m,\mathbb{R}\right)$, we have that

$\displaystyle\phi\left(\left[u,\left(h,v\right)\right]_{H}\cdot\left(h^{\prime},v^{\prime}\right)\right)$
$\displaystyle=\phi\left(\left[u,\left(h,v\right)\left(h^{\prime},v^{\prime}\right)\right]_{H}\right)$
$\displaystyle=\phi\left(\left[u,\left(hh^{\prime},hv^{\prime}+v\right)\right]_{H}\right)$
$\displaystyle=\left[w\mapsto\left(u\circ h\circ
h^{\prime}\right)\left(w\right)+u\left(hv^{\prime}+v\right)\right]$
$\displaystyle=\left[w\mapsto
u\left(h\left(h^{\prime}w+v^{\prime}\right)\right)+u\left(v\right)\right]$
$\displaystyle=\phi\left(\left[u,\left(h,v\right)\right]_{H}\right)\circ\left(h^{\prime},v^{\prime}\right),$

proving that $\phi$ is a bundle map.

For every $a\in\left.AM\right|_{x}$, we have that

$\phi\left(\left[u,\left(e,v\right)\right]_{H}\right)=a$

if and only if $v=a\left(0\right)$ and

$u\left(w\right)=a\left(w\right)-a\left(0\right)$

for all $w\in\mathbb{R}^{m}$; therefore, $\phi$ es an epimorphism of bundles.

Additionally, if
$\left[u_{1},\left(h_{1},v_{1}\right)\right]_{H},\left[u_{2},\left(h_{2},v_{2}\right)\right]_{H}$
are such that

$\phi\left(\left[u_{1},\left(h_{1},v_{1}\right)\right]_{H}\right)=\phi\left(\left[u_{2},\left(h_{2},v_{2}\right)\right]_{H}\right),$

then

$\left(u_{1}\circ
h_{1}\right)\left(w\right)+u_{1}\left(v_{1}\right)=\left(u_{2}\circ
h_{2}\right)\left(w\right)+u_{2}\left(v_{2}\right)$

for all $w\in\mathbb{R}^{m}$. With $w=0$ it gives us that

(C.3) $u_{1}\left(v_{1}\right)=u_{2}\left(v_{2}\right)$

and so

$u_{2}\circ h_{2}=u_{1}\circ h_{1}\qquad\Longrightarrow\qquad u_{2}=u_{1}\circ
h_{1}\circ h_{2}^{-1}.$

Thus, Equation (C.3) tells us that

$v_{2}=\left(h_{2}\circ h_{1}^{-1}\right)\left(v_{1}\right)$

and it means that

$\left[u_{2},\left(h_{2},v_{2}\right)\right]_{H}=\left[u_{2}\circ h_{1}\circ
h_{2}^{-1},\left(h_{2},\left(h_{2}\circ
h_{1}^{-1}\right)\left(v_{1}\right)\right)\right]_{H}=\left[u_{1},\left(h_{1},v_{1}\right)\right]_{H}.$

Namely, $\phi$ is a monomorphism of bundles. ∎

###  C.3. Local expressions

We will use the constructions developed in Section A in order to find
coordinates for $LM$, $AM$ and its jet bundles. The first thing to note is
that the frame bundle $LM$ can be trivialized on every coordinate chart
$\left(U,\phi\right)$ for $M$; namely, for $u\in\tau^{-1}\left(U\right)$ there
exists a collection $\left(e_{i}^{\alpha}\left(u\right)\right)$ of real
numbers such that

$u\left(c^{1},\cdots,c^{m}\right)=c^{i}e_{i}^{\alpha}\left(u\right)\frac{\partial}{\partial
x^{\alpha}},\qquad\left(c^{1},\cdots,c^{m}\right)\in\mathbb{R}^{m},$

where $\phi=\left(x^{\alpha}\right)$ are the coordinate functions on
$\phi\left(U\right)\subset\mathbb{R}^{m}$. It induces the coordinate chart on
$\tau^{-1}\left(U\right)$ given by

$\Phi_{U}\left(u\right):=\left(x^{\alpha}\left(\tau\left(u\right)\right),e_{i}^{\alpha}\left(u\right)\right).$

In the same vein, given
$\overline{u}\in\left(\tau\circ\beta\right)^{-1}\left(U\right)\subset AM$, we
can find numbers
$\left(e_{i}^{\alpha}\left(\overline{u}\right),v^{\alpha}\left(\overline{u}\right)\right)$
such that

(C.4)
$\overline{u}\left(c^{1},\cdots,c^{m}\right)=\left[c^{i}e_{i}^{\alpha}\left(\overline{u}\right)+v^{\alpha}\left(\overline{u}\right)\right]\frac{\partial}{\partial
x^{\alpha}},\qquad\left(c^{1},\cdots,c^{m}\right)\in\mathbb{R}^{m};$

it defines a coordinate chart on
$\left(\tau\circ\beta\right)^{-1}\left(U\right)$ through the formula

$\overline{\Phi}_{U}\left(\overline{u}\right):=\left(x^{\alpha}\left(\tau\left(\beta\left(\overline{u}\right)\right)\right),e_{i}^{\alpha}\left(\overline{u}\right),v^{\alpha}\left(\overline{u}\right)\right).$

The map $\gamma$ fits nicely with these coordinates; in fact, we have that

$\overline{\Phi}_{U}\circ\gamma\circ\Phi_{U}^{-1}\left(x^{\alpha},e^{\beta}_{i}\right)=\left(x^{\alpha},e^{\beta}_{i},0\right).$

Let us consider now the action of an element $\left(a,w\right)\in
A\left(m,\mathbb{R}\right)=GL\left(m,\mathbb{R}\right)\oplus\mathbb{R}^{m}$ on
$AM$; because of the equation (C.4), we have that

$\displaystyle\left[\overline{u}\cdot\left(a,w\right)\right]\left(c^{1},\cdots,c^{m}\right)$
$\displaystyle=\overline{u}\left(a_{j}^{1}c^{j}+w^{1},\cdots,a_{j}^{m}c^{j}+w^{m}\right)$
$\displaystyle=\left[\left(a_{j}^{i}c^{j}+w^{i}\right)e_{i}^{\alpha}\left(\overline{u}\right)+v^{\alpha}\left(\overline{u}\right)\right]\frac{\partial}{\partial
x^{\alpha}}$
$\displaystyle=\left[c^{i}e_{i}^{\alpha}\left(\overline{u}\right)+\left(v^{\alpha}\left(\overline{u}\right)+c^{i}e_{i}^{\alpha}\left(\overline{u}\right)\right)\right]\frac{\partial}{\partial
x^{\alpha}},$

namely,

$\left(x^{\alpha},e_{i}^{\beta},v^{\rho}\right)\cdot\left(a,w\right)=\left(x^{\alpha},a^{i}_{j}e_{i}^{\beta},v^{\rho}+e^{\rho}_{i}w^{i}\right).$

We are ready to deal with connections on $AM$ in a local fashion; according to
the discussion carried out in Section A.1, the local version of a connection
on $AM$ is an $\mathfrak{a}\left(m,\mathbb{R}\right)$-valued $1$-form, namely

$\widetilde{\omega}_{U}=\left(\Gamma^{\alpha}_{\gamma\beta}dx^{\beta}\otimes
E_{\alpha}^{\gamma},\sigma^{\alpha}_{\beta}dx^{\beta}\otimes
e_{\alpha}\right).$

Accordingly, it can be globalized through formula (A.4); first, recall that in
this case we have

$g_{U}\left(x^{i},e^{\alpha}_{i},v^{\beta}\right)=\left(e^{\alpha}_{i},v^{\beta}\right),$

and so

$g_{U}^{*}\lambda=\left(e^{\alpha}_{i},v^{\beta}\right)^{-1}\left(de^{\alpha}_{i},dv^{\beta}\right)=\left(e^{i}_{\alpha}de^{\alpha}_{j},e^{i}_{\beta}dv^{\beta}\right).$

Additionally, the adjoint action formula (C.1) tells us that

$\text{Ad}_{\left(g_{U}\left(x^{i},e^{\alpha}_{i},v^{\beta}\right)\right)^{-1}}\widetilde{\omega}_{U}=\left(e^{\gamma}_{i}e^{j}_{\alpha}\Gamma^{\alpha}_{\gamma\beta}dx^{\beta}\otimes
E_{j}^{i},e^{i}_{\alpha}\left(\Gamma^{\alpha}_{\gamma\beta}v^{\gamma}+\sigma^{\alpha}_{\beta}\right)dx^{\beta}\otimes
e_{i}\right),$

so that

(C.5)
$\left.\widetilde{\omega}\right|_{\overline{\Phi}_{U}\left(x^{i},e^{\beta}_{j},v^{\gamma}\right)}=\left(e^{j}_{\alpha}\left(de^{\alpha}_{i}+e^{\gamma}_{i}\Gamma^{\alpha}_{\gamma\beta}dx^{\beta}\right)\otimes
E_{j}^{i},e^{i}_{\alpha}\left[dv^{\alpha}+\left(\Gamma^{\alpha}_{\gamma\beta}v^{\gamma}+\sigma^{\alpha}_{\beta}\right)dx^{\beta}\right]\otimes
e_{i}\right).$

Coordinates $\Phi_{U}$ and $\overline{\Phi}_{U}$ induce coordinates on
$J^{1}\tau$ and $J^{1}\left(\tau\circ\beta\right)$ respectively, which will be
indicated as

$\left(x^{\alpha},e^{\beta}_{i},e^{\beta}_{i\gamma}\right)\qquad\text{and}\qquad\left(x^{\alpha},e^{\beta}_{i},v^{\alpha},e^{\beta}_{i\gamma},v^{\alpha}_{\beta}\right).$

The $A\left(m,\mathbb{R}\right)$-action lifts to
$J^{1}\left(\tau\circ\beta\right)$ as follows

$\left(x^{\alpha},e_{i}^{\beta},v^{\rho},e^{\beta}_{i\gamma},v^{\alpha}_{\beta}\right)\cdot\left(a,w\right)=\left(x^{\alpha},a^{i}_{j}e_{i}^{\beta},v^{\rho}+e^{\rho}_{i}w^{i},a_{j}^{i}e^{\beta}_{i\gamma},v^{\alpha}_{\beta}+e^{\alpha}_{i\beta}w^{i}\right).$

using the form of this action, we can consider the projection from
$J^{1}\left(\tau\circ\beta\right)$ to $C\left(AM\right)$; we have that

$p_{A\left(m,\mathbb{R}\right)}^{J^{1}\left(\tau\circ\beta\right)}\left(x^{\alpha},e_{i}^{\beta},v^{\rho},e^{\beta}_{i\gamma},v^{\alpha}_{\beta}\right)=\left(x^{i},e^{j}_{\gamma}e^{\alpha}_{j\beta},v^{\alpha}_{\beta}-e^{i}_{\gamma}e^{\alpha}_{i\beta}v^{\gamma}\right).$

Let us now consider the canonical connection
$\theta_{J^{1}\left(\tau\circ\beta\right)}$ on
$J^{1}\left(\tau\circ\beta\right)$; it results that the contact structure is

$T\left(\tau\circ\beta\right)_{10}-T_{x}s\circ
T\left(\tau\circ\beta\right)_{1}=\left(de_{i}^{\beta}-e^{\beta}_{i\alpha}dx^{\alpha}\right)\otimes\frac{\partial}{\partial
e^{\beta}_{i}}+\left(dv^{\beta}-v^{\beta}_{\alpha}dx^{\alpha}\right)\otimes\frac{\partial}{\partial
v^{\beta}},$

and because the infinitesimal generators for the
$A\left(m,\mathbb{R}\right)$-action on $AM$ are

$\left(E_{i}^{j}\right)_{AM}\left(x^{\alpha},e_{i}^{\beta},v^{\gamma}\right)=e^{\alpha}_{i}\frac{\partial}{\partial
e^{\alpha}_{j}},\qquad\left(e_{i}\right)_{AM}\left(x^{\alpha},e_{i}^{\beta},v^{\gamma}\right)=e^{\alpha}_{i}\frac{\partial}{\partial
v^{\alpha}},$

we obtain that

(C.6)
$\left.\theta_{J^{1}\left(\tau\circ\beta\right)}\right|_{\left(x^{\alpha},e_{i}^{\beta},v^{\rho},e^{\beta}_{i\gamma},v^{\alpha}_{\beta}\right)}=e_{\beta}^{j}\left(de_{i}^{\beta}-e^{\beta}_{i\alpha}dx^{\alpha}\right)\otimes
E^{i}_{j}+e^{i}_{\beta}\left(dv^{\beta}-v^{\beta}_{\alpha}dx^{\alpha}\right)\otimes
e_{i}.$

## References

  * [AM95] D. V. Alekseevsky and P. W. Michor  “Differential geometry of Cartan connections”  In _Publ. Math. Debrecen_ 47, 1995, pp. 349–375
  * [AT86] A. Achúcarro and P.K. Townsend  “A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories”  In _Physics Letters B_ 180.1, 1986, pp. 89 –92  DOI: https://doi.org/10.1016/0370-2693(86)90140-1
  * [Bar04] M. Barakat  “The existence of Cartan connections and geometrizable principle bundles”  In _Archiv der Mathematik_ 83.2 Springer ScienceBusiness Media LLC, 2004  DOI: 10.1007/s00013-004-1048-5
  * [Ble81] D. Bleecker  “Gauge Theory and Variational Principles”  Addison-Wesley, 1981
  * [Bry11] Robert Bryant  “Nine Lectures on Exterior Differential Systems”, 2011  URL: http://maths-people.anu.edu.au/~the/seminar/MSRI_Lectures.pdf
  * [Bry+91] R.L. Bryant et al.  “Exterior differential systems”  Springer-Verlag, 1991
  * [CM01] M. Castrillón López and J. Muñoz Masqué  “The geometry of the bundle of connections”  In _Mathematische Zeitschrift_ 236 Springer Berlin / Heidelberg, 2001, pp. 797–811  URL: http://dx.doi.org/10.1007/PL00004852
  * [CS74] Shiing-Shen Chern and James Simons  “Characteristic Forms and Geometric Invariants”  In _Annals of Mathematics_ 99.1 Annals of Mathematics, 1974, pp. 48–69  URL: http://www.jstor.org/stable/1971013
  * [DJT82] S. Deser, R. Jackiw and S. Templeton  “Three-Dimensional Massive Gauge Theories”  In _Phys. Rev. Lett._ 48 American Physical Society, 1982, pp. 975–978  DOI: 10.1103/PhysRevLett.48.975
  * [EEMLRR96] Arturo Echeverría-Enríquez, Miguel C. Muñoz-Lecanda and Narciso Román-Roy  “Geometry of Lagrangian First-order Classical Field Theories”  In _Fortschritte der Physik/Progress of Physics_ 44.3 WILEY-VCH Verlag, 1996, pp. 235–280  DOI: 10.1002/prop.2190440304
  * [Fre95] Daniel S. Freed  “Classical Chern-Simons theory. Part 1”  In _Adv. Math._ 113, 1995, pp. 237–303  DOI: 10.1006/aima.1995.1039
  * [Gar72] Pedro L. García  “Connections and $1$-jet fiber bundles”  In _Rend. Sem. Mat. Univ. Padova_ 47, 1972, pp. 227–242
  * [GGRR23] Joaquim Gomis, Arnoldo Guerra and Narciso Román-Roy  “Multisymplectic constraint analysis of scalar field theories, Chern-Simons gravity, and bosonic string theory”  In _Nuclear Physics B_ 987, 2023, pp. 116069  DOI: https://doi.org/10.1016/j.nuclphysb.2022.116069
  * [GIM04] M.J. Gotay, J. Isenberg and J.E. Marsden  “Momentum maps and classical relativistic fields. II: Canonical analysis of field theories”, 2004  arXiv:math-ph/0411032
  * [GIM97] M.J. Gotay, J. Isenberg and J.E. Marsden  “Momentum maps and classical relativistic fields. I: Covariant field theory”, 1997  arXiv:physics/9801019
  * [Got91] M.J. Gotay  “A multisymplectic framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism”  In _Mechanics, Analysis and Geometry: 200 Years after Lagrange_ , 1991, pp. 203–235
  * [Got91a] M.J. Gotay  “A multisymplectic framework for classical field theory and the calculus of variations II: space + time decomposition”  In _Differential Geometry and its Applications_ 1.4, 1991, pp. 375 –390  DOI: 10.1016/0926-2245(91)90014-Z
  * [Got91b] M.J. Gotay  “An exterior differential system approach to the Cartan form”  In _Symplectic geometry and mathematical physics. Actes du colloque de géométrie symplectique et physique mathématique en l’honneur de Jean-Marie Souriau, Aix-en-Provence, France, June 11-15, 1990._ Progress in Mathematics. 99. Boston, MA, Birkhäuser, 1991, pp. 160–188
  * [IL03] T. A. Ivey and J. M. Landsberg  “Cartan for beginners: differential geometry via moving frames and exterior differential systems”, Graduate Texts in Mathematics  American Mathematical Society, 2003
  * [KN63] S. Kobayashi and K. Nomizu  “Foundations of Differential Geometry”  Wiley, 1963
  * [LMD03] M. León, J. C. Marrero and D. Martín Diego  “A new geometric setting for classical field theories”  In _Classical and Quantum Integrability_ Institute of Mathematics Polish Academy of Sciences, 2003  DOI: 10.4064/bc59-0-10
  * [LSVn15] Manuel León, Modesto Salgado and Silvia Vilariño  “Methods of Differential Geometry in Classical Field Theories”  World Scientific, 2015  DOI: 10.1142/9693