# The Tulczyjew triple for gauge field theories

Marcin Zajac

###### Abstract

In this article we construct and discuss a new rigorous geometric formalism
for gauge field theories. The basis of our work is the notion of the Tulczyjew
triple, a geometric structure which successfully solved numerous problems in
mathematical description of mechanics and classical field theory. In
particular, we construct a Tulczyjew triple for gauge field theories and
reduce it for systems that depend only on the value of a connection and
curvature instead of the entire first jet of the gauge field. We also
introduce new geometric structures such as the vector-affine product of
bundles and analyse the connection bundle from a new perspective. Finally, we
apply the derived formalism to Yang-Mills theory.

Department of Mathematical Methods in Physics,

Faculty of Physics. University of Warsaw,

ul. Pasteura 5, 02-093 Warsaw, Poland.

<EMAIL_ADDRESS>

###### Contents

  1. 1 Introduction
  2. 2 Jet bundles and principal bundles
    1. 2.1 First order jet bundles
    2. 2.2 Higher order jet bundles
    3. 2.3 Principal bundles and adjoint bundles
    4. 2.4 Connection in a principal bundle
    5. 2.5 Exterior covariant derivative
  3. 3 Tulczyjew triple in mechanics and field theory
    1. 3.1 Tulczyjew triple in mechanics
    2. 3.2 Tulczyjew triple in field theory
  4. 4 Vector-affine products of bundles
    1. 4.1 Vector-affine product
    2. 4.2 Tangent and cotangent bundle of vector-affine product
  5. 5 The bundle of connections
    1. 5.1 The bundle of connections
    2. 5.2 Local picture of the connection bundle
    3. 5.3 The configuration bundle for gauge theories
    4. 5.4 The phase bundle
  6. 6 Lagrangian description
    1. 6.1 The structure of iterated bundles
    2. 6.2 The structure of the bundle $\mathsf{V}^{*}\mathsf{J}^{1}C$
    3. 6.3 The Tulczyjew map for gauge theories
  7. 7 Hamiltonian description
    1. 7.1 The dual side of vector-affine product
    2. 7.2 Reduced Hamiltonian formalism
  8. 8 Tulczyjew triple for gauge theories
  9. 9 Example: Yang-Mills theory
  10. 10 Acknowledgments

##  1 Introduction

The Tulczyjew triples. Geometrical tools of the classical field theory were
developed as a generalisation of ideas coming from classical mechanics. In
most textbooks and papers, analytical mechanics is based on variational
calculus and its main objective is Euler- Lagrange equation. Analogously,
using variational principle, one may derive field equations in classical field
theories. However, if one wants to describe field theory geometrically, it
turns out that there is a surprisingly large variety of approaches. The most
common one is to extend the symplectic formalism from mechanics to field
theory, which leads to polysymplectic structures [19], $k$-symplectic
structures [2, 31, 28], $k$-almost cotangent structures [26, 27] and
multisymplectic structures [21, 8, 9, 10]. However, each of these approaches
has significant limits, which immediately appear when one wants to consider
nonregular systems, reduction with respect to symmetries or inclusion of
contraints [1].

In 70’s W. M. Tulczyjew suggested a new understanding of variational calculus
in mechanics and field theories. In his numerous works (e.g [35, 36, 37, 38])
he presented a formalism in which, in contrast to most textbooks, the crucial
object is the phase dynamics instead of Euler-Lagrange equation. This
formalism was later called a Tulczyjew triple. It has been recently recognised
by many theoretical physicist and mathematicians. It provides more general and
complete description of a mechanical system and is much simpler on the
conceptual level than the traditional one.

Tulczyjew triple is a very useful commutative diagram built on maps that are
essential in Lagrangian and Hamiltonian description of physical systems. In
fact the name Tulczyjew triple refers not to one diagram but a collection of
diagrams adapted to various physical situations. The very first triple
introduced by Tulczyjew in his numerous works (e.g [37, 38, 40]) served for
autonomous analytical mechanics. It was then adapted and generalised for time-
dependent mechanics, mechanics on algebroids [12, 14, 16], for field theory
[11], for higher order systems [15] etc. The concept of Tulczyjew triple also
points to the certain philosophy of interpreting concepts of variational
calculus within physical theories.

The main advantage of the approach developed by Tulczyjew and his
collaborators is its generality. For example, using the Tulczyjew triple for
autonomous mechanics we can derive the phase equations for systems with
singular Lagrangians and understand properly the Hamiltonian description of
such systems. One can even discuss systems with more general generating
objects than just a Lagrangian function, e.g. systems described by family of
Lagrangians or a Lagrangian function defined on a submanifold. Another
advantage of the Tulczyjew’s approach is its flexibility. Being based on well-
defined general principles, it can be easily adapted to different settings
[11, 12, 15, 25]. Finally, we do not postulate ad hoc the ingredients of the
theory, but obtain them as unavoidable consequences of the variational
calculus [11].

Gauge field theories. Within our work we will be interested in gauge field
theories. From the mathematical point of view gauge field is represented by a
connection in a principal bundle $\pi:P\to M$ with the structure group $G$.
Therefore, gauge theories may be, to some extent, reduced to the geometry of
principal bundles. A gauge transformation of the principal bundle $P$ is an
equivariant diffeomorphism $\Phi:P\to P$ preserving the projection on the base
manifold $M$. By equivariance we mean that the condition $\Phi(pg)=\Phi(p)g$
is satisfied for each $p\in P$, $g\in G$. The set of gauge transformations
form an infinite-dimensional group with a composition of maps as a group
multiplication. We denote this group by $\mathcal{G}(P)$. We say that
$\mathcal{G}(P)$ is a symmetry group of a given theory if its action
functional is invariant under the transformations given by $\mathcal{G}(P)$
[7, 32].

The first historically discovered gauge theory was classical electrodynamics
formulated by J. C. Maxwell [30]. The existence of a gauge symmetry within
Maxwell equations initially did not appear to be a fact of huge importance.
Only the papers of Weyl on the unification of electrodynamics with general
relativity consciously introduced the notion of a local symmetry to
theoretical physics [42]. In his seminal paper [42] Weyl introduced a term
,,gauge transformation”, and in particular gauge invariance (ger.
”eichinvarianz”). However, perhaps the greatest success of gauge theories came
in 1954, when C. H. Yang and R. Mills introduced non-abelian gauge theories to
describe strong interaction confining nucleons in atomic nucleus [43]. These
theories came to be known in the literature as Yang-Mills theories, after
their inventors. Ever since then Yang-Mills theories and gauge theories in
general became one of the main objects of study in theoretical physics [3, 4].
Arguably, the most important discovery within this field was the emergence of
the Standard Model of particle physics.

Main goals and results. The first goal of our paper is to find a mathematical
formalism, which will allow us to describe in a simple way the dynamics of
gauge fields both on the Lagrangian and Hamiltonian side. In this part of our
work we mainly rely on the formalism of the Tulczyjew triple, which is
described in detail in a section 3. One of the main advantages of the
Tulczyjew description is that it does not require any regularity of the
Lagrangian to find the dynamics of the system and to pass to the Hamiltonian
side. This feature is particularly important in the context of field theory.
While in mechanics regular sytems constitute a vast majority of the physically
interesting models, in field theory (especially gauge theories) most of the
systems is nonregular. Whatsmore, many of these teories are naturally theories
with constraints. Since the Tulczyjew formalism turned out to be very
efficient in application to both constrained systems and nonregular systems,
it seems to be a perfect tool to analyse such problems.

In the next step we want to reduce the derived Tulczyjew triple with respect
to internal symmetries appearing in gauge theories. Let us stress here that in
the current paper we do not consider the problem of gauge symmetries
reduction. We plan to work on this issue in a separate paper that is aimed as
the continuation of this one. In particular, we will be interested now in a
following problem. The general formalism of the classical field theory is
based on the geometry of jet bundles. The space of fields is represented by a
fiber bundle $E\to M$, where $E$ is the space of values of the field. To find
the dynamics of a system one usually takes the Lagrangian, which is a map

$L:\mathsf{J}^{1}E\to\Omega^{m},$

where $\Omega^{m}$ is the space of $m$-covectors on the $m$-dimensional
manifold $M$. However, in gauge theories a Lagrangian usually does not depend
on the entire first jet of a connection but only on the value of the field and
on its curvature. This is the case for instance of such important theories as
classical electrodynamics or Yang-Mills and Yang-Mills-Higgs theories.
Therefore, a natural question arises, how to describe geometrically the
projection associated with the passage from the first jet of the connection to
the curvature of this connection in a given point. This kind of projection
should imply a reduction of the entire structure of gauge theories depending
only on the value of the field and its curvature. In this paper we discuss
this reduction in detail and, as a result, we obtain a reduced Tulczyjew
triple for gauge fields.

An additional result of the paper is the further development of geometric
structures associated with gauge theories. In our formulation the space of
gauge fields is a bundle $C\to M$, where $C=\mathsf{J}^{1}P/G$. In section 5
we provide a meticulous analysis of the connection bundle in the realm of jet
bundles and group actions. In order to do that we introduce a new kind of
product of bundles in section 4, which we call a vector-affine product. The
main result of this part of our work is Theorem 1 in subsection 5.3, which
shows that $\mathsf{J}^{1}C$ has a natural structure of the vector-affine
product. Let us mention that although the jet bundle geometry is a natural
area to develop classical field theory, the literature concerning gauge
theories in the jet bundle setting is extremely modest [32, 33]. In
particular, in [32] Sardanashvily mentions a result similar to Theorem 1 but
provides a wrong formula for it. We will discuss this issue in detail in
section 5.3.

Outline of the paper. The paper is organized as follows. Section 2 contains a
brief recalling of the basic tools used in the geometric description of
classical field theory of first order and gauge theories. The main notions in
this context are jet bundles and principal bundles. In section 3 we recall the
concept of a Tulczyjew triple in mechanics and classical field theory of the
first order. In section 4 we introduce a notion of the vector-affine bundles
and we consider the tangent and cotangent bundle of such products. Section 5
contains the detailed discussion of the connection bundle in a bit different
language than the one presented in most textbooks. Sections 6 and 7 are main
results of the paper and they contain a reduction of the Tulczyjew formalism
for gauge theories. The results of this reduction are summarised in section 8.
In section 9 we apply the reduced Tulczyjew triple to Yang-Mills theory.

##  2 Jet bundles and principal bundles

###  2.1 First order jet bundles

Let us briefly recall the notion of first order jet spaces and their duals. We
will follow the notation from [11]. For a more detailed discussion of the jet
bundle geometry see e.g. [34, 6].

Let $\pi:E\to M$ be a bundle with the total space of dimension
$\mathrm{dim}E=n+m$. We introduce in a domain $U\in M$ a local coordinate
system $(x^{i})^{n}_{i=0}$ on $M$. In field theory fields are represented by
sections of a fibration $\pi$. The total space is the space of values of the
field e.g vector field is a section of $\pi$ being a vector bundle, scalar
field is a section of a trivial bundle $E=M\times\mathbb{R}$ or
$E=M\times\mathbb{C}$, etc. On an open subset $V\subset E$ such that
$\pi(V)=U$ we introduce local coordinates $(x^{i},u\indices{{}^{\alpha}})$
adapted to the structure of the bundle.

In $\mathsf{T}E$ we have a vector subbundle $\mathsf{V}E\to E$ consisting of
those tangent vectors that are vertical with respect to the projection $\pi$,
i.e. $\mathsf{T}\pi(v_{p})=0$ for $v_{p}\in\mathsf{V}_{p}E$. We will also need
its dual vector bundle $\mathsf{V}^{*}E\to E$.

The space of first jets of sections of the bundle $\pi$ will be denoted by
$\mathsf{J}^{1}E$. By definition, the first jet $\mathsf{j}^{1}_{m}\phi$ of a
section $\phi$ at the point $m\in M$ is an equivalence class of sections
having the same value at the point $m$ and such that the spaces tangent to the
graphs of the sections at the point $\phi(m)$ coincide. Therefore, there is a
natural projection $\pi_{1,0}$ from the space $\mathsf{J}^{1}E$ onto the
manifold $E$

$\pi_{1,0}:\mathsf{J}^{1}E\to
E:\quad\mathsf{j}^{1}_{m}\phi\longmapsto\phi(m).$

Moreover, every jet $\mathsf{j}^{1}_{m}\phi$ may be identified with a linear
map $\mathsf{T}\phi:\mathsf{T}_{m}M\to\mathsf{T}_{\phi(m)}E$. Linear maps
coming from jets at the point $m$ form an affine subspace in a vector space
$Lin(\mathsf{T}_{m}M,\mathsf{T}_{e}E)$ of all linear maps from
$\mathsf{T}_{m}M$ to $\mathsf{T}_{e}E$. A map belongs to this subspace if
composed with $\mathsf{T}\pi$ gives identity. In a tensorial representation we
have an inclusion

$\mathsf{J}^{1}_{e}E\subset\mathsf{T}^{*}_{m}M\otimes\mathsf{T}_{e}E.$

It is easy to check that the affine space $\mathsf{J}^{1}_{e}E$ is modelled on
the vector space $\mathsf{T}_{m}^{*}M\otimes\mathsf{V}_{e}E$. Summarising, the
bundle $\mathsf{J}^{1}E\to E$ is an affine bundle modelled on the vector
bundle

$\pi^{*}(\mathsf{T}^{*}M)\otimes_{E}\mathsf{V}E\to E.$

In the following we will omit the symbol of the pullback in
$\pi^{*}(\mathsf{T}^{*}M)$ writing simply
$\mathsf{T}^{*}M\otimes_{E}\mathsf{T}E$ and
$\mathsf{T}^{*}M\otimes_{E}\mathsf{V}E$.

Using the adapted coordinates $(x^{i},u\indices{{}^{\alpha}})$ in $V\subset
E$, we can construct the induced coordinate system
$(x^{i},u\indices{{}^{\alpha}},u\indices{{}^{\beta}_{j}})$ on
$\pi_{1,0}^{-1}(V)$ such that for any section $\phi$ given by $n$ functions
$\phi^{a}(x^{i})$ we have

$u\indices{{}^{\beta}_{j}}(\phi^{\alpha}(x^{i}))=\frac{\partial\phi^{\beta}}{\partial
x^{j}}(x^{i}(m)).$

In the tensorial representation the first jet $\mathsf{j}^{1}_{m}\phi$ may be
written as

$\mathrm{d}x^{i}\otimes\frac{\partial}{\partial
x^{i}}+\frac{\partial\phi^{\alpha}}{\partial
x^{j}}(x^{i}(m))\mathrm{d}x^{j}\otimes\frac{\partial}{\partial
u\indices{{}^{\alpha}}},$

where we have used local bases of sections of $\mathsf{T}^{*}M$ and
$\mathsf{T}E$ coming from the chosen coordinates.

We will introduce now the bundle which is dual to $\mathsf{J}^{1}E\to E$. Let
us recall that each fiber $\mathsf{J}^{1}_{e}E$ is an affine space. We can
consider a set of affine maps $\mathsf{J}^{1}_{e}E\to\mathbb{R}$, say
$\mathsf{Aff}(\mathsf{J}^{1}_{e}E,\mathbb{R})$, for each $e\in E$. Collecting
$\mathsf{Aff}(\mathsf{J}^{1}_{e}E,\mathbb{R})$ point by point we obtain the
bundle of affine maps on $\mathsf{J}^{1}E$, namely
$\mathsf{Aff}(\mathsf{J}^{1}E,\mathbb{R})\to E$. From now on we will use the
notation $\mathsf{J}^{\dagger}E:=\mathsf{Aff}(\mathsf{J}^{1}E,\mathbb{R})$. It
is a vector bundle over $E$. If
$(x^{i},u\indices{{}^{\alpha}},u\indices{{}^{\beta}_{j}})$ are coordinates in
$\mathsf{J}^{1}E$, then we introduce coordinates
$(x^{i},u\indices{{}^{\alpha}},r,\varphi\indices{{}^{b}_{j}})$ in
$\mathsf{J}^{\dagger}E$. The evaluation between $\mathsf{J}^{1}E$ and
$\mathsf{J}^{\dagger}E$ in coordinates reads

$\mathsf{J}^{\dagger}E\times_{E}\mathsf{J}^{1}E\to\mathbb{R},\qquad\langle
T_{e},\mathsf{j}_{e}\psi\rangle=r+\varphi\indices{{}^{b}_{j}}y\indices{{}^{j}_{b}}.$

Let us recall that each affine map has the associated linear part. Since
$\mathsf{J}^{1}_{e}E$ is modelled on the vector bundle
$\mathsf{T}^{*}_{\pi(e)}M\otimes_{E}\mathsf{V}_{e}E$ the linear part of an
affine map $T_{e}:\mathsf{J}^{1}_{e}E\to\mathbb{R}$ is an element of
$\mathsf{T}_{\pi(e)}M\otimes_{E}\mathsf{V}^{*}_{e}E$. The bundle

$\mu:\mathsf{J}^{\dagger}E\to\mathsf{T}M\otimes_{E}\mathsf{V}^{*}E$

is an affine bundle, which projects an affine map onto its linear part. The
model bundle of $\mu$ is a trivial bundle
$\mathsf{T}M\otimes_{E}\mathsf{V}^{*}E\times\mathbb{R}\to\mathsf{T}M\otimes_{E}\mathsf{V}^{*}E$.

###  2.2 Higher order jet bundles

Consider the bundle $\pi_{1}:\mathsf{J}^{1}E\to M$. The bundle of first jets
of sections of $\pi_{1}$ is by definition an affine bundle

$(\pi_{1})_{1,0}:\mathsf{J}^{1}\mathsf{J}^{1}E\to\mathsf{J}^{1}E,\quad\mathsf{j}^{1}_{m}\psi\longmapsto\psi(m),$

where $\mathsf{j}^{1}_{m}\psi$ is the first jet of a section
$\psi:M\to\mathsf{J}^{1}E$ at the point $m$. It turns out that the projection

$\mathsf{j}^{1}\pi_{1,0}:\mathsf{J}^{1}\mathsf{J}^{1}E\to\mathsf{J}^{1}E,\quad\mathsf{j}^{1}_{m}\psi\longmapsto\mathsf{j}^{1}_{m}(\pi_{1,0}\circ\psi)$

is an affine bundle as well. The bundle $\mathsf{J}^{1}\mathsf{J}^{1}E$ has
the structure of a double affine bundle [17] represented by the diagram

$\textstyle{\mathsf{J}^{1}\mathsf{J}^{1}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\pi_{1})_{1,0}}$$\scriptstyle{\mathsf{j}^{1}\pi_{1,0}}$$\textstyle{\mathsf{J}^{1}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1,0}}$$\textstyle{\mathsf{J}^{1}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1,0}}$$\textstyle{E}$

In the bundle $\mathsf{J}^{1}\mathsf{J}^{1}E$ we have a subbundle
$\widehat{\mathsf{J}^{2}E}$ consisting of elements having the same projection
on both sides, i.e.

$\widehat{\mathsf{J}^{2}E}:=\\{\mathsf{j}^{1}_{m}\psi\in\mathsf{J}^{1}\mathsf{J}^{1}E,\quad\mathsf{j}^{1}\pi_{1,0}(\mathsf{j}^{1}_{m}\psi)=(\pi_{1})_{1,0}(\mathsf{j}^{1}_{m}\psi)\\}.$

The bundle $\widehat{\mathsf{J}^{2}E}\to\mathsf{J}^{1}E$ is called the bundle
of semiholonomic jets [34]. It is an affine bundle modelled on the vector
bundle
$\otimes^{2}\mathsf{T}^{*}M\otimes_{\mathsf{J}^{1}E}\mathsf{V}E\to\mathsf{J}^{1}E$.
If
$(q^{i},u\indices{{}^{\alpha}},u\indices{{}^{\beta}_{j}},z\indices{{}^{\gamma}_{k}},w\indices{{}^{\delta}_{l}{}_{m}})$
are coordinates in $\mathsf{J}^{1}\mathsf{J}^{1}E$ then the subbundle
$\widehat{\mathsf{J}^{1}E}$ is given by the condition
$u\indices{{}^{\beta}_{j}}=z\indices{{}^{\beta}_{j}}$. Therefore, we have
natural coodinates
$(q^{i},u\indices{{}^{\alpha}},u\indices{{}^{\beta}_{j}},u\indices{{}^{\beta}_{j}{}_{k}})$
in $\widehat{\mathsf{J}^{1}E}$ induced from $\mathsf{J}^{1}\mathsf{J}^{1}E$.

In $\widehat{\mathsf{J}^{2}E}$ there exists a subbundle
$\mathsf{J}^{2}E\to\mathsf{J}^{1}E$, such that

$\mathsf{J}^{2}E:=\\{\mathsf{j}^{1}_{m}\psi\in\mathsf{J}^{1}\mathsf{J}^{1}E,\quad\psi=\mathsf{j}^{1}\phi\\}.$

The bundle $\mathsf{J}^{2}E\to\mathsf{J}^{1}E$ is called the bundle of second
jets. The elements of $\mathsf{J}^{2}E$ are called second jets of the section
$\phi$ or holonomic jets. The bundle $\mathsf{J}^{2}E\to\mathsf{J}^{1}E$ is an
affine bundle modelled on the vector bundle
$\vee^{2}\mathsf{T}^{*}M\otimes_{\mathsf{J}^{1}E}\mathsf{V}E\to\mathsf{J}^{1}E$,
where $\vee^{2}\mathsf{T}^{*}M$ is the subbundle of symmetric tensors in
$\mathsf{T}^{*}M\otimes_{M}\mathsf{T}^{*}M$. In $\mathsf{J}^{2}E$ we have
induced coordinates
$(q^{i},u\indices{{}^{\alpha}},u\indices{{}^{\beta}_{j}},u\indices{{}^{\beta}_{j}{}_{k}})$,
such that $u\indices{{}^{\beta}_{j}{}_{k}}=u\indices{{}^{\beta}_{k}{}_{j}}$.
There are natural inclusions

$\mathsf{J}^{2}E\subset\widehat{\mathsf{J}^{2}E}\subset\mathsf{J}^{1}\mathsf{J}^{1}E.$

The structure of the bundle $\widehat{\mathsf{J}^{2}E}$ will play a
particularly important role in the following sections.

Let us mention that the bundle of second jets may be defined independently
from its embedding in $\mathsf{J}^{1}\mathsf{J}^{1}E$. We say that two
sections $\phi,\phi^{\prime}:M\to E$ are in relation $\sim$ if and only if

$\displaystyle\phi(m)$ $\displaystyle=$ $\displaystyle\phi^{\prime}(m),$ (1)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}_{|t=0}(\phi\circ\gamma)$
$\displaystyle=$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}_{|t=0}(\phi^{\prime}\circ\gamma),$
(2)
$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}_{|t=0}(\phi\circ\gamma)$
$\displaystyle=$
$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}_{|t=0}(\phi^{\prime}\circ\gamma),$
(3)

where $\gamma:\mathbb{R}\to M$ is any smooth curve in $M$ such that
$\gamma(0)=m$. The second jet of a section $\phi$ at the point $m\in M$ is the
equivalence class of sections with respect to the above relation and we denote
it by $\mathsf{j}^{2}_{m}\phi$. The total space $\mathsf{J}^{2}E$ is by
definition a set of equivalence classes $\mathsf{j}^{2}_{m}\phi$ at each point
in $M$. Similarly, extending the condition of equality of derivatives up to
the $k$-th order, we can define the bundle of $k$-jets denoted by
$\mathsf{J}^{k}E$.

###  2.3 Principal bundles and adjoint bundles

In this and subsequent subsection we will briefly recall the fundamentals of
principal bundles with a particular focus on the notion of a connection in
this class of bundles. We will also fix the notation necessary for our
subsequent work. The reader is referred, e.g. to [23, 5] for the standard
exposition of the principal bundle geometry. Our introduction concerning this
topic is in large based on [7].

Let $G$ be a Lie group with the Lie algebra $\mathfrak{g}$. We denote by $P$ a
smooth manifold such that $G$ acts on it from the right in a smooth, free and
proper way. Then, the space $M:=P/G$ of orbits of the action of $G$ on $P$ is
a smooth manifold as well. The bundle $\pi:P\to M$ is called a principal
bundle with a structure group $G$. It is locally isomorphic to $M\times G$.
Let $U_{\alpha}\subset M$ be an open subset in $M$. A local trivialisation of
$P$ is a $G$-equivariant diffeomorphism

$\Psi_{\alpha}:\pi^{-1}(U_{\alpha})\to U_{\alpha}\times
G,\qquad\Psi_{\alpha}(p)=(\pi(p),g_{\alpha}(p)),$

where $g_{\alpha}:\pi^{-1}(U_{\alpha})\to G$ is a $G$-valued function
associated with the map $\Psi_{\alpha}$. The equivariance condition means that
$\Psi_{\alpha}(pg)=\Psi_{\alpha}(p)g$, which implies that $g_{\alpha}$ is also
$G$-equivariant, says $g_{\alpha}(pg)=g_{\alpha}(p)g$. Notice that the
function $g_{\alpha}$ uniquely defines a local trivialisation of $P$. The
transition between trivialisations $g_{\alpha}$ and $g_{\beta}$ defined on
$\pi^{-1}(U_{\alpha}\cap U_{\beta})$ is realised by the function

$g_{\alpha\beta}:U_{\alpha}\cap U_{\beta}\to G,\qquad
g_{\alpha\beta}(\pi(p))=g_{\alpha}(p)g_{\beta}(p)^{-1}.$

Let $F$ be a smooth manifold and let $G$ act on $F$ from the left. We
introduce the action of $G$ on product $P\times F$ given by

$g(p,f)=(pg,g^{-1}f).$

We denote by $N:=(P\times F)/G$ the space of orbits of this action. The bundle

$\xi:N\to M,\quad[(p,f)]\to\pi(p),$

where $[(p,f)]$ is the orbit of the element $(p,f)\in P\times F$, is called an
associated bundle of $P$ with a typical fiber $F$. Notice that the above
projection does not depend on the choice of a representative in $[(p,f)]$,
therefore it is well-defined. For $F$ being a vector space, the associated
bundle is a linear bundle over $M$. The most important examples of associated
bundles of $P$ in the context of our work are the bundles with fibers
$F=\mathfrak{g}$ and $F=G$, i.e. $N=(P\times\mathfrak{g})/G$ and $N=(P\times
G)/G$. From now on, we will use the notation

$\mathrm{ad}(P):=(P\times\mathfrak{g})/G\quad\qquad\textrm{and}\quad\qquad\mathrm{Ad}(P):=(P\times
G)/G.$

The action of $G$ on $\mathfrak{g}$ and the action of $G$ on $G$ is given by
the adjoint map, namely

$\mathrm{Ad}:G\times\mathfrak{g}\to\mathfrak{g},\quad(g,X)\longmapsto\mathrm{Ad}_{g}(X),$
$\mathrm{Ad}:G\times G\to G,\quad(g,h)\longmapsto\mathrm{Ad}_{g}(h),$

respectively. For the sake of the simplicity of our notation we have denoted
both actions by the same symbol $\mathrm{Ad}$.

Denote by $\Omega^{k}(M,\mathfrak{g})$ the bundle of $\mathfrak{g}$-valued
$k$-forms on $M$. Let $\\{U_{\alpha}\\}$ be an open covering of $M$ and let
$\\{\xi_{\alpha}\\}$ be a family of local $k$-forms on $M$ such that
$\xi_{\alpha}\in\Omega^{k}(U_{\alpha},\mathfrak{g})$ for each $\alpha\in
I\subset\mathbb{R}$. We also require that for each overlapping $U_{\alpha}\cap
U_{\beta}$ the condition

$\xi_{\alpha}(m)=\mathrm{Ad}_{g_{\alpha\beta}(m)}\circ\xi_{\beta}(m),\qquad
m\in U_{\alpha\beta},\quad g_{\alpha\beta}:U_{\alpha\beta}\to G$ (4)

is satisfied. We claim that the family of $k$-forms $\\{\xi_{\alpha}\\}$
defines a $k$-form on $M$ with values in $\mathrm{ad}P$. The space of
$\mathrm{ad}P$-valued $k$-forms on $M$ will be denoted by
$\Omega^{k}(M,\mathrm{ad}P)$.

###  2.4 Connection in a principal bundle

Let $\mathsf{V}P$ be the vertical subbundle in $\mathsf{T}P$, i.e. the
subbundle of tangent vectors, which are tangent to the fibers of $\pi:P\to M$.
A connection in $\pi:P\to M$ is a $G$-invariant distribution $\mathcal{D}$ in
$\mathsf{T}P$, which is complementary to $\mathsf{V}P$ at each point $p\in P$.
By definition we have

$\mathsf{T}_{p}P=\mathsf{V}_{p}P\oplus\mathcal{D}_{p},\qquad p\in P$ (5)

and

$\mathcal{D}_{p}g=\mathcal{D}_{pg},\quad g\in G.$ (6)

The above definition is very elegant and general, however, when it comes to
applications, it is more convenient to represent a connection in a different
way. We will start with introducing some basic mathematical tools. Let $X$ be
an element of $\mathfrak{g}$. The group action of $G$ on $P$ defines the
vertical vector field $\sigma_{X}$ on $P$ associated with the element $X$,
namely

$\sigma_{X}(p):=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}p\exp(tX).$

The field $\sigma_{X}$ is called the fundamental vector field corresponding to
the element $X$. The fundamental vector field is equivariant in the sense that

$\sigma_{X}(pg)=\sigma_{\mathrm{Ad}_{g^{-1}}(X)}(p).$

A connection form in $P$ is a $G$-equivariant, $\mathfrak{g}$-valued one-form
$\omega$

$\omega:\mathsf{T}P\to\mathfrak{g},$

such that $\omega(\sigma_{X}(p))=X$ for each $p\in P$ and $X\in\mathfrak{g}$.
The $G$-equivariance means that

$R^{*}_{g}\omega(p)=\mathrm{Ad}_{g^{-1}}\circ\omega(p).$

It is easy to check that the distribution $\mathcal{D}_{p}:=\ker\omega(p)$
defines a connection in $P$. Since the connection form is an identity on
vertical vectors, the difference of two connections is a horizontal form. It
follows that the space of connections is the space of sections of an affine
subbundle $\mathcal{A}\subset\mathsf{T}^{*}P\otimes\mathfrak{g}$ modelled on
the vector bundle of $\mathfrak{g}$-valued, $G$-equivariant horizontal one-
forms on $P$. It turns out, that the space of such horizontal forms may be
identified with the space of sections of the bundle
$\mathsf{T}^{*}M\otimes\mathrm{ad}P\to M$.

The connection one-form may be equivalently defined by means of a family of
$\mathfrak{g}$-valued one-forms on $M$. This approach is widely used by
physicists working in classical field theory. Let $s_{\alpha}:M\supset
U_{\alpha}\to P$ be a local section. A pull-back of the connection form
$\omega$ defines a $\mathfrak{g}$-valued one-form
$A_{\alpha}:=s_{\alpha}^{*}\omega$. In local coordinates it reads

$A_{\alpha}(m)=A_{\alpha}^{a}(m)\otimes
e_{a}=A\indices{{}^{a}_{j}}(m)\mathsf{d}q^{j}\otimes e_{a}\quad m\in M,$

where $\\{e_{a}\\}$ is a basis of $\mathfrak{g}$. In order to simplify the
notation we have skipped the index $\alpha$ in $A\indices{{}^{a}_{j}}$. One
can show that two one-forms $A_{\alpha}$ and $A_{\beta}$ must satisfy the
gluing condition

$A_{\alpha}=\mathrm{Ad}_{g_{\alpha\beta}}\circ
A_{\beta}+g^{*}_{\beta\alpha}\theta$

on the overlapping $U_{\alpha\beta}$. In the above formula we have used the
Maurer-Cartan form

$\theta:G\to\mathsf{T}^{*}G\otimes\mathfrak{g},\quad\theta(g)=\mathsf{T}L_{g^{-1}}.$

On the other hand, once we have a family of forms $\\{A_{\alpha}\\}$ we can
restore a connection form $\omega$ on $P$. The restriction of $\omega$ to
$\pi^{-1}(U_{\alpha})$ is given by

$\omega_{\alpha}(p)=\mathrm{Ad}_{{g_{\alpha}(p)}^{-1}}\circ\pi^{*}A_{\alpha}(p)+g^{*}_{\alpha}\theta(p).$
(7)

The proof of the above statement is rather technical so we will skip it here.

The curvature of the connection is the two-form
$\Omega_{\omega}:=(\mathrm{d}\omega)^{h}$, where $(\mathrm{d}\omega)^{h}$ is
the horizontal part of $\mathrm{d}\omega$. After some computations one can
show that

$\Omega_{\omega}=\mathrm{d}\omega+\frac{1}{2}[\omega\wedge\omega],$ (8)

where $[\omega\wedge\omega]$ is the bracket of forms on $P$ with values in
$\mathfrak{g}$. The curvature form, same as the connection form, is
equivariant in the sense that
$R_{g}^{*}\Omega_{\omega}=\mathrm{Ad}_{g^{-1}}\circ\Omega_{\omega}$. The
curvature form is horizontal and $G$-equivariant therefore it defines the
$\mathrm{ad}P$-valued two-form

$F_{\omega}:M\to\wedge^{2}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P.$ (9)

From now on we will use the following notation

$\displaystyle\mathcal{V}$
$\displaystyle:=\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P,$ (10)
$\displaystyle\mathcal{F}$
$\displaystyle:=\wedge^{2}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P.$ (11)

Section 9 can be also constructed from the local picture of $\Omega$. One can
use $s_{\alpha}$ to pull-back the curvature form and obtain a
$\mathfrak{g}$-valued two-form

$F_{\alpha}:=s^{*}_{\alpha}\Omega,\qquad
F_{\alpha}=dA_{\alpha}+\frac{1}{2}[A_{\alpha}\wedge A_{\alpha}].$

It is an easy exercise to check that each $F_{\alpha}$ and $F_{\beta}$ satisfy
the condition (4) on the overlapping $U_{\alpha\beta}$. Therefore, we claim
that the family of forms $\\{F_{\alpha}\\}$ define a section

$F:M\to\wedge^{2}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P.$

It turns out that the curvature of the connection $\omega$ may be equally
represented by the global $\mathfrak{g}$-valued two-form $\Omega$ on $P$ or by
the global $\mathrm{ad}P$-valued two-form $F$ on $M$.

###  2.5 Exterior covariant derivative

Let $\Omega_{G}^{k}(P,V)$ be a subspace in $\Omega^{k}(P,V)$, which consists
of $V$-valued $k$-forms that are both horizontal and equivariant. The exterior
covariant derivative with respect to a connection $\omega$ and a
representation $\rho:G\to GL(V)$ is the operator

$D_{\omega}:\Omega_{G}^{k}(P,V)\to\Omega_{G}^{k+1}(P,V),\quad\zeta\longmapsto(\mathrm{d}\zeta)^{h}.$

One can show that

$D_{\omega}\zeta=\mathrm{d}\zeta+\rho^{\prime}(\omega)\wedge\zeta,$ (12)

where $\rho^{\prime}$ is given by

$\rho^{\prime}:\mathfrak{g}\to
gl(V),\quad\rho^{\prime}(X)v=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\rho(\exp(tX))v$
(13)

and $\rho^{\prime}(\omega)\wedge\zeta$ reads

$\rho^{\prime}(\omega)\wedge\zeta(v_{1},..,v_{1+k}):=\frac{1}{k!}\sum_{\sigma\in
S_{1+k}}sgn(\sigma)\rho^{\prime}\Big{(}\omega(v_{\sigma(1)}\Big{)}\zeta(v_{\sigma(2)},..,v_{\sigma(1+k)}).$
(14)

Introducing a basis $(e_{i})$ in $V$ we can write

$\omega=\omega^{i}\otimes e_{i},\quad\zeta=\zeta^{j}\otimes e_{j},$
$\rho^{\prime}(\omega)\wedge\zeta=\omega^{i}\wedge\zeta^{j}\otimes\rho(e_{i})e_{j}.$

Since $\Omega_{G}^{k}(P,V)\simeq\Omega^{k}(M,P\times_{G}V)$ we have that the
exterior covariant derivative in $\Omega_{G}^{k}(P,V)$ defines also an
exterior covariant derivative in $\Omega^{k}(M,P\times_{G}V)$.

Consider forms on $P$ with values in the dual space $V^{*}$. The
representation $\rho$ on $V$ defines the contragredient representation
$\rho^{\\#}$ on $V^{*}$. By definition we have

$\rho^{\\#}:G\to
End(V^{*}),\quad\rho^{\\#}(g):={\rho(g)^{*}}^{(-1)}=\rho(g^{-1})^{*}.$

One can check that the action of $\mathfrak{g}$ on $V^{*}$ reads

$\rho^{\\#}:\mathfrak{g}\to
End(V^{*}),\quad{\rho^{\\#}}^{\prime}(X):=-\rho^{\prime}(X)^{*}.$

Therefore, the covariant derivative $D_{\omega}$ in $\Omega_{G}^{k}(P,V)$
defines also a covariant derivative, say $D^{\\#}_{\omega}$, in
$\Omega_{G}^{k}(P,V^{*})$ and $\Omega^{k}(M,P\times_{G}V^{*})$. The total
space $P\times_{G}V^{*}$ is the space of orbits of the action

$G:P\times V^{*}\to P\times V^{*},\quad g(p,f)=(pg,\rho(g^{-1})^{\\#}f).$

For $\xi\in\Omega_{G}^{k}(P,V^{*})$ we have

$D^{\\#}_{\omega}:\Omega_{G}^{k}(P,V^{*})\to\Omega_{G}^{k+1}(P,V^{*}),\quad\xi\longmapsto(\mathrm{d}\xi)^{h},$
$D^{\\#}_{\omega}\xi=\mathrm{d}\xi+{\rho^{\\#}}^{\prime}(\omega)\wedge\xi=\mathrm{d}\xi-\rho^{\prime}(\omega)^{*}\wedge\xi.$

In the end lets us recall that the space $\Omega(P,\mathfrak{g})$ is equipped
with the canonical graded bracket

$[\cdot\wedge\cdot]:\Omega^{k}(P,\mathfrak{g})\times\Omega^{l}(P,\mathfrak{g})\to\Omega^{k+l}(P,\mathfrak{g}),\quad(\alpha,\beta)\to[\alpha\wedge\beta],$
(15)

such that

$[\alpha\wedge\beta](v_{1},..,v_{k+l}):=\frac{1}{k!l!}\sum_{\sigma\in
S_{k+l}}sgn(\sigma)[\alpha(v_{\sigma(1)},..,v_{\sigma(k)}),\beta(v_{\sigma(k+1)},..,v_{\sigma(k+l)})],$

where $[\cdot,\cdot]$ is the Lie bracket in $\mathfrak{g}$.

##  3 Tulczyjew triple in mechanics and field theory

###  3.1 Tulczyjew triple in mechanics

We will present now a point of view on describing a mechanical system, which
is alternative to the one present in most textbooks [22]. Its essence lies in
the so-called Tulczyjew triple. Tulczyjew triple enables us to describe
systems in both Lagrangian and Hamiltonian approach and shows relation between
the two. It is important to notice, that the triple is based only on canonical
structures of proper bundles. We will not derive here the whole formalism
whose origin lies in the reinterpretation of variational description of
statical systems. One can find the thorough analysis with all the details in
numeorus Tulczyjew papers e.g. [35, 36, 37, 38, 39, 40].

The Tulczyjew triple is a geometrical structure presented in the followig
diagram

$\textstyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathsf{T}}^{*}{\mathsf{T}}^{*}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathsf{T}^{*}M}}$$\textstyle{{\mathsf{T}}{\mathsf{T}}^{*}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{M}}$$\scriptstyle{\alpha_{M}}$$\scriptstyle{\tau_{\mathsf{T}^{*}M}}$$\scriptstyle{\mathsf{T}\pi_{M}}$$\textstyle{{\mathsf{T}}^{*}{\mathsf{T}}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathsf{T}M}}$$\textstyle{{\mathsf{T}}^{*}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}X_{H}}}$$\scriptstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\mathsf{d}}H}}$$\textstyle{{\mathsf{T}}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{{\mathsf{d}}L}}$$\textstyle{M}$
(16)

The right-hand side of the diagram is related to Lagrangian formalism while
the left-hand side to Hamiltonian one. Both formalisms are based on the same
scheme and the only difference is in generating objects on both sides. We will
discuss now more precisely Lagrangian and Hamiltonian description in the
language of the Tulczyjew triple. For simplicity, we will consider only
autonomous systems i.e. with no external forces.

Let $M$ be the configuration manifold of the system and
$L:\mathsf{T}M\longrightarrow\mathbb{R}$ its Lagrangian. The dynamics of the
system is a subset

$\mathcal{D}:=\alpha_{M}^{-1}\circ\mathrm{d}L(\mathsf{T}M)$ (17)

of $\mathsf{T}\mathsf{T}^{*}M$. Since $\mathsf{d}L(\mathsf{T}M)$ is a
Lagrangian submanifold in $\mathsf{T}^{*}\mathsf{T}M$, and $\alpha_{M}$ is a
symplectomorphism, the dynamics is a Lagrangian submanifold in
$\mathsf{T}\mathsf{T}^{*}M$. From the physical point of view it is a (possibly
implicit) first order differential equation for a trajectory in the phase
space. A curve $\eta:\mathbb{R}\supset I\rightarrow\mathsf{T}^{\ast}M$ is a
phase trajectory if $\dot{\eta}(t)\in\mathcal{D}$ for $t\in I$.

Now let us assume that Hamiltonian $H:\mathsf{T}^{\ast}M\rightarrow\mathbb{R}$
of the system does exist. The dynamics of the system is then the image of the
Hamiltonian vector field $X_{H}$ i.e.

$\mathcal{D}=X_{H}(\mathsf{T}^{\ast}M)=\beta_{M}^{-1}(\mathrm{d}H(\mathsf{T}^{\ast}M)).$
(18)

Phase space trajectories are integral curves of the field $X_{H}$.

The dynamics of the system may be projected on
$\mathsf{T}M\times\mathsf{T}^{*}M$. The projection
$\Lambda=\mathsf{T}\pi_{M}\times\tau_{\mathsf{T}^{\ast}M}(\mathcal{D})$ is a
subset of Cartesian product of $\mathsf{T}M$ and $\mathsf{T}^{*}M$ therefore
it can be understood as a relation between these two manifolds. If the
dynamics comes from Lagrangian it is a graph of the Legendre map
$\lambda=\zeta_{M}\circ\mathrm{d}L$. If the dynamics comes from Hamiltonian it
is a graph of a map ‘in the opposite direction’. In general, it can be a
relation that is not a map at all. In such a case in place of Lagrangian and
Hamiltonian we have to consider more general objects generating Lagrangian
submanifolds than functions on manifolds [29, 41].

###  3.2 Tulczyjew triple in field theory

The Tulczyjew triple for mechanics may be generalised to the classical field
theory. The very general Tulczyjew triple for the first order field theory has
already been derived in [11]. We will briefly present results obtained in this
paper.

Let $\mathsf{V}E\to E$ be the bundle of vectors vertical with respect to the
projection $\pi:E\to M$, and $\mathsf{V}^{*}E\to E$ its dual bundle. Again,
fields are represented by sections of the fibre bundle $\pi:E\to M$ and the
space of infinitesimal configuration is $\mathsf{J}^{1}E$. As a natural
consequence of the variational calculus we obtain that the phase space of the
system is the space $\mathcal{P}:=\mathsf{V}^{*}E\otimes\Omega^{m-1}$. The
bundle $\mathcal{P}\to E$ is a vector bundle. Let us introduce coordinates
$(q^{i},y^{a},p^{b}_{j})$ in $\mathcal{P}$. The field analogue of the
Tulczyjew triple is a diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.19583pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathsf{P}\mathsf{J}^{\dagger}E}$}}}}}}}{\hbox{\kern
38.19583pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathsf{J}^{1}\mathcal{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.80435pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\quad\beta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.19585pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
67.65483pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha\quad\quad}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 82.9403pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
82.9403pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathsf{V}^{*}\mathsf{J}^{1}E\otimes\Omega^{m}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

The bundle $\mathsf{P}\mathsf{J}^{\dagger}E$ is the so-called ‘affine phase
bundle’, which is an affine analog of the cotangent bundle [13]. Again, the
left-hand side of the diagram refers to the Hamiltonian description of the
sytem, while the right-hand side to the Lagrangian one. The main elements in
the triple are maps $\alpha$ and $\beta$ which constitute the Lagrangian and
Hamiltonian description, respectively. Both maps are morphisms of double
vector-affine bundles [24, 17].

Let $L:\mathsf{J}^{1}E\to\Omega^{m}$ be the Lagrangian of a system. We denote
by $\mathrm{d}^{\mathsf{v}}L$ the vertical differential of $L$, i.e. the
restricton of $\mathrm{d}L$ to
$\mathsf{V}\mathsf{J}^{1}E\subset\mathsf{T}\mathsf{J}^{1}E$. The phase
dynamics of the system is a subset

$\mathsf{J}^{1}\mathcal{P}\supset\mathcal{D}:=\alpha^{-1}\circ\mathrm{d}L(\mathsf{J}^{1}E).$
(19)

From the mathematical pont of view the dynamics is a first-order partial
differential equation for phase trajectories in $\mathcal{P}$. We say that a
section $p:M\to\mathcal{P}$ is a solution of the Lagrange field equations if

$\mathsf{j}^{1}_{m}p\in\mathcal{D}.$

Let us introduce coordinates $(q^{i},y^{a},p^{b}_{j},y^{c}_{k},p^{l}_{dm})$ in
$\mathsf{J}^{1}\mathcal{P}$. Then, the Lagrange field equations read

$\displaystyle\sum_{j}\frac{\partial p^{j}_{b}}{\partial q^{j}}=\frac{\partial
L}{\partial y^{b}},\qquad\displaystyle p^{j}_{b}=\frac{\partial L}{\partial
y^{b}_{j}}.$

The Euler-Lagrange equations for field theory are consequence of the Lagrange
field equations. Let us also mention that, same as in the case of mechanics,
we do not require any regularity of the Lagrangian and the definition of the
momenta comes directly from the theory.

On the other hand, a Hamiltonian is not a function and not even a map with
values in forms, but a section of the bundle
$\mathsf{J}^{\dagger}E\to\mathcal{P}$, where $\mathsf{J}^{\dagger}E$ is the
affine dual of $\mathsf{J}^{\dagger}E$. The affine differential $\mathsf{d}H$
can be viewed as a map
$\mathsf{d}H:\mathcal{P}\to\mathsf{P}\mathsf{J}^{\dagger}E$. It defines the
phase dynamics $\mathcal{D}=\beta^{-1}(\mathsf{d}H(\mathcal{P}))$. All the
rigorous results concerning a very general Tulczyjew triple for field theory
may be found in [11].

Let us briefly recall the construction of the maps $\alpha$ and $\beta$. We
will start with the map $\alpha$. We will use the symbol ${\bf
t}:=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}$ to denote the operation of taking a
tangent vector to a curve in $t=0$, and the functor $\mathsf{V}$ will always
mean verticality with respect to the projection on $M$. Let
$\mathsf{J}^{1}\mathsf{V}E$ be the first jet bundle of $\mathsf{V}E\to M$ and
$\mathsf{V}\mathsf{J}^{1}E$ the subbundle of vertical vectors in
$\mathsf{T}\mathsf{J}^{1}E$. Each element of $\mathsf{J}^{1}\mathsf{V}E$ and
$\mathsf{V}\mathsf{J}^{1}E$ is represented by a map

$\chi:M\times\mathbb{R}\to E$

preserving the projection on $M$. The elements of $\mathsf{V}\mathsf{J}^{1}E$
and $\mathsf{J}^{1}\mathsf{V}E$ have a form ${\bf t}\mathsf{j}^{1}_{m}\chi$
and $\mathsf{j}^{1}_{m}{\bf t}\chi$, respectively. There is a canonical
isomorphism

$\kappa:\mathsf{V}\mathsf{J}^{1}E\to\mathsf{J}^{1}\mathsf{V}E,\quad\quad{\bf
t}\mathsf{j}^{1}_{m}\chi\longmapsto\mathsf{j}^{1}_{m}{\bf t}\chi.$ (20)

The above map is a generalisation of the canonical involution
$\kappa_{M}:\mathsf{T}\mathsf{T}M\to\mathsf{T}\mathsf{T}M$ known in mechanics.
One can check that $\kappa$ is well-defined, i.e. it does not depend on the
choice of the representative $\chi$.

The map $\alpha$ is a dual map to $\kappa$. The bundle
$\mathsf{V}^{*}\mathsf{J}^{1}E\to\mathsf{J}^{1}E$ is the dual bundle of
$\mathsf{V}\mathsf{J}^{1}E\to\mathsf{J}^{1}E$. It turns out that there exists
also a canonical evaluation (over $\mathsf{J}^{1}E$) between
$\mathsf{J}^{1}\mathcal{P}$ and $\mathsf{J}^{1}\mathsf{V}E$ given by

$\mathsf{J}^{1}\mathcal{P}\times_{\mathsf{J}^{1}E}\mathsf{J}^{1}\mathsf{V}E\to\Omega^{m},\quad(\mathsf{j}^{1}_{m}p,\mathsf{j}^{1}_{m}\delta\sigma)\longmapsto\mathrm{d}\langle
p,\delta\sigma\rangle.$ (21)

We will denote the above evaluation by
$\langle\\!\langle\mathsf{j}^{1}_{m}p,\mathsf{j}^{1}_{m}\delta\sigma\rangle\\!\rangle$.
The map $\alpha$, which is a generalisation of the Tulczyjew isomorphism in
mechanics [40, 11], is a dual map to $\kappa$, understood as a morphism of the
vector bundles $\mathsf{J}^{1}\mathsf{V}E\to\mathsf{J}^{1}E$ and
$\mathsf{V}\mathsf{J}^{1}E\to\mathsf{J}^{1}E$. The duality is taken with
respect to the canonical evaluation between $\mathsf{V}\mathsf{J}^{1}E$ and
$\mathsf{V}^{*}\mathsf{J}^{1}E\otimes_{\mathsf{J}^{1}E}\Omega^{m}$ and
evaluation (21). However, it is not a duality in the ordinary sense, which
comes from the fact that (21) is degenerated. By definition we have

$\alpha:\mathsf{J}^{1}\mathcal{P}\to\mathsf{V}^{*}\mathsf{J}^{1}E\otimes_{\mathsf{J}^{1}E}\Omega^{m},\quad\langle\alpha(u),w\rangle=\langle\\!\langle
u,\kappa(w)\rangle\\!\rangle.$

In coordinates the map $\alpha$ has a form

$\alpha(q^{i},y^{a},p\indices{{}^{j}_{b}},y\indices{{}^{c}_{k}},p\indices{{}^{l}_{d}{}_{m}})=(q^{i},y^{a},y\indices{{}^{c}_{k}},\sum_{l}p\indices{{}^{l}_{d}{}_{l}},p\indices{{}^{j}_{b}}).$
(22)

Notice that, in a contrast to mechanics, the map $\alpha$ is not an
isomorphism of bundles.

On the Hamiltonian side we have a canonical diffeomorphism

$R_{\mathsf{J}^{1}E}:\mathsf{V}^{*}\mathsf{J}^{1}E\otimes_{M}\Omega^{m}\to\mathsf{P}\mathsf{J}^{\dagger}E.$
(23)

We define $\beta$ as a composition

$\beta:=R_{\mathsf{J}^{1}E}\circ\alpha,\quad\beta:\mathsf{J}^{1}\mathcal{P}\to\mathsf{P}\mathsf{J}^{\dagger}E.$

The map $\beta$ is a generalisation of the map $\beta_{M}$ in 16 and it
constitutes the basis of the Hamiltonian description of field theory. In
coordinates we have

$\beta(q^{i},y^{a},p\indices{{}^{j}_{b}},y\indices{{}^{c}_{k}},p\indices{{}^{l}_{d}{}_{s}})=(q^{i},y^{a},p\indices{{}^{j}_{b}},\sum
p\indices{{}^{k}_{c}{}_{k}},y\indices{{}^{c}_{k}}).$ (24)

A Hamiltonian in classical field theory is not a function on the phase space
with values in $\Omega^{m}$ but a family of sections of the affine bundle

$\Sigma:\mathsf{J}^{1}E\times_{E}\mathcal{P}\to\mathsf{J}^{\dagger}E$

parametrised by points in $\mathsf{J}^{1}E$. Notice that the fibers of the
bundle $\mathsf{J}^{\dagger}E\to\mathsf{J}^{1}E\times_{E}\mathcal{P}$ are one-
dimensional. The above family of sections corresponds to the family of
functions

$H:\mathsf{J}^{1}E\times_{E}\mathsf{J}^{\dagger}E\to\Omega^{m}$ (25)

with values in $\Omega^{m}$. In some cases the family of sections and the
corresponding family of functions reduces to the single generating section and
function

$\Sigma:\mathcal{P}\to\mathsf{J}^{\dagger}E,$
$H:\mathsf{J}^{\dagger}E\to\Omega^{m}.$

The dynamics is given then by the formula

$\mathcal{D}=\beta^{-1}(\mathrm{d}H(\mathcal{P})).$

##  4 Vector-affine products of bundles

###  4.1 Vector-affine product

We will discuss now a particular case of an affine bundle, which can be
decomposed on a properly defined sum of its affine subbundle and certain
vector bundle. Before we will move to fibrations let us analyse this problem
in a simpler case. Let $A$ be an affine space modelled on a vector space $V$
and let $B\subset A$ be an affine subspace in $A$ modelled on a vector
subspace $V_{1}\subset V$. Moreover, we will assume that there exists a
subspace $V_{2}\subset V$ such that $V$ is a direct sum $V=V_{1}\oplus V_{2}$.
Let us choose an element $b_{0}\in B$. Then, any element $a\in A$ can be
written in the form

$a=b_{0}+v=b_{0}+v_{1}+v_{2},$

where the decomposition $v=v_{1}+v_{2}$ on $v_{1}\in V_{1}$ and $v_{2}\in
V_{2}$ is unique. We introduce the element $b=b_{0}+v_{1}$, so that
$a=b+v_{2}$. Notice that $b$ depends only on $a$, but it does not depend on
$b_{0}$. Indeed, if $b^{\prime}_{0}=b_{0}+u$ for a certain $u\in V_{1}$ and

$a=b^{\prime}_{0}+v^{\prime}=b^{\prime}_{0}+v^{\prime}_{1}+v^{\prime}_{2}=b^{\prime}+v^{\prime}_{2}$

then $v=u+v^{\prime}$. From the uniqueness of the decomposition $V=V_{1}\oplus
V_{2}$ we obtain that $v_{1}=u+v^{\prime}_{1}$ and $v_{2}=v^{\prime}_{2}$, so
that

$b^{\prime}=b^{\prime}_{0}+v^{\prime}_{1}=b_{0}+u+v_{1}-u=b_{0}+v_{1}=b.$

Therefore, the map

$A\to B\times V_{2},\quad a\longmapsto(b,v_{2})$

is well-defined. Above decomposition will be called a vector-affine sum of $B$
and $V_{2}$ and we will denote it by $A=B\bar{\oplus}V_{2}$.

The above construction may be generalised to affine bundles over a manifold.
Let $A\to N$ be an affine bundle modeled on a vector bundle $V\to N$, and let
$V$ be a Whitney sum of vector bundles $V_{1}\to N$ and $V_{2}\to N$. If
$B\subset A$ is an affine subbundle in $A$ modelled on a vector subbundle
$V_{1}$, then we have a decomposition

$A=B\bar{\oplus}_{N}V_{2}$ (26)

over $N$. By definition, each fiber of $B\bar{\oplus}_{N}V_{2}\to N$ over a
point $n\in N$ is a vector-affine sum $B_{n}\bar{\oplus}V_{2n}$, where $B_{n}$
and $V_{2n}$ are fibers of $B\to N$ and $V_{2}\to N$, respectively.

As an example of the above construction let us consider

$\displaystyle A$ $\displaystyle=$ $\displaystyle\widehat{\mathsf{J}^{2}E},$
(27) $\displaystyle B$ $\displaystyle=$ $\displaystyle\mathsf{J}^{2}E,$ (28)
$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\mathsf{J}^{1}E\times_{E}(\vee^{2}\mathsf{T}^{*}M\otimes_{M}\mathsf{V}E),$
(29) $\displaystyle V_{2}$ $\displaystyle=$
$\displaystyle\mathsf{J}^{1}E\times_{E}(\wedge^{2}\mathsf{T}^{*}M\otimes_{M}\mathsf{V}E),$
(30) $\displaystyle N$ $\displaystyle=$ $\displaystyle\mathsf{J}^{1}E,$ (31)

where $\vee^{2}\mathsf{T}^{*}M$ is the space of symmetric tensors on
$\mathsf{T}^{*}M$. By means of (26) we have

$\widehat{\mathsf{J}^{2}E}=\mathsf{J}^{2}E\bar{\oplus}_{\mathsf{J}^{1}E}(\mathsf{J}^{1}E\times_{E}\wedge^{2}\mathsf{T}^{*}M\otimes_{M}\mathsf{V}E).$
(32)

Informally, we can say that each semiholonomic jet consists of the second jet
of a certain section of a bundle $E\to M$ and a suitable antisymmetric two-
vector on $M$ with values in $\mathsf{V}E$.

###  4.2 Tangent and cotangent bundle of vector-affine product

Let us consider the tangent and cotangent bundle of a vector-affine product of
bundles. We will start with the tangent bundle. Let $A\to N$ be an affine
bundle modelled on a vector bundle $V\to N$. The tangent bundle $\mathsf{T}A$
is a double affine-vector bundle [17, 24] represented by the diagram

$\textstyle{\mathsf{T}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{A}}$$\textstyle{\mathsf{T}N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{N}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N}$

The bundle $\mathsf{T}A\to\mathsf{T}N$ is an affine bundle modeled on the
vector bundle $\mathsf{T}V\to\mathsf{T}N$. The addition of elements
$v\in\mathsf{T}A$ and $w\in\mathsf{T}V$ having the same projection on
$\mathsf{T}N$ is defined in a following way. Let $\gamma_{v},\gamma_{w}$ be
curves representing $v$ and $w$, respectively, and having the same projection
on $N$. Then, we can construct a curve

$(\gamma_{v}+\gamma_{w})(t):=\gamma_{v}(t)+\gamma_{w}(t)$

and define the element $v\dot{+}w$ as a tangent vector of the above curve in
$t=0$. This vector is fixed in $\tau_{A}(v)+\tau_{V}(w)$ and it has the same
projection on $\mathsf{T}N$ as $v$ and $w$. The affine structure of
$\mathsf{T}A\to\mathsf{T}N$ is therefore a result of the application of the
tangent functor to the structure of $A\to N$. In a similar way this functor
acts on the vector-affine sum. Let us notice, that each representative
$\gamma_{v}:\mathbb{R}\to A$ of the tangent vector $v\in\mathsf{T}_{a}A$ may
be written in the form $\gamma_{v}(t)=\gamma_{B}(t)+\gamma_{V_{2}}(t)$, where
$\gamma_{B}:\mathbb{R}\to B$ and $\gamma_{V_{2}}:\mathbb{R}\to V_{2}$ are
unique curves determined by the decomposition $A=B\bar{\oplus}V_{2}$ and
having the same projection on $N$. Therefore, we can write

$v=v_{B}\dot{+}w_{2},\qquad$
$v_{B}=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\gamma_{B}\in\mathsf{T}_{b}B,\quad
w_{2}=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\gamma_{V_{2}}\in\mathsf{T}_{p_{2}}V_{2},$

where $a=(b,p_{2})$. Notice that $v,v_{B}$ and $w_{2}$ have the same
projection on $\mathsf{T}N$. Finally, we obtain an isomorphism
$\mathsf{T}A\simeq\mathsf{T}B\oplus_{\mathsf{T}N}\mathsf{T}V_{2}$ and the
diagram

$\textstyle{\mathsf{T}B\bar{\oplus}_{\mathsf{T}N}\mathsf{T}V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{T}N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{N}}$$\textstyle{B\bar{\oplus}_{N}V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N}$

Let us consider now the cotangent bundle
$\mathsf{T}^{*}A=\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})$. Same as
$\mathsf{T}A$, it is a double affine-vector bundle represented by a diagram

$\textstyle{\mathsf{T}^{*}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}}$$\textstyle{V^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{N}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N}$

The bundle $\mathsf{T}^{*}A\to V^{*}$ is an affine bundle, in which the
projection is constructed in a similar way as the projection
$\mathsf{T}^{*}E\to E^{*}$ for $E\to N$ being a vector bundle [17, 24]. In
short, each covector $\varphi\in\mathsf{T}_{a}^{*}A$ may be restricted to the
subspace $\mathsf{V}_{a}A\simeq V_{x}$, and this restriction is identified
with an element of $V_{x}^{*}$.

Let us recall a general fact from linear algebra that if $V$ is a vector space
and $W\subset V$ is a subspace then $W^{*}\simeq V^{*}/W^{o}$, where $W^{o}\in
V^{*}$ is an annihilator of $W$ in $V$. It means that there exists a
projection $V^{*}\to W^{*}$ but in general there is no canonical embedding of
$W^{*}$ in $V^{*}$. On the level of fiber bundles this fact has the following
implications. If $V_{1}\to N$ and $V_{2}\to N$ are vector bundles, then in an
obvious way we have

$\mathsf{T}(V_{1}\times
V_{2})\simeq\mathsf{T}V_{1}\times\mathsf{T}V_{2},\qquad\mathsf{T}^{*}(V_{1}\times
V_{2})\simeq\mathsf{T}^{*}V_{1}\times\mathsf{T}^{*}V_{2}.$

The bundle $V_{1}\oplus_{N}V_{2}$ is a vector subbundle in $V_{1}\times
V_{2}$, which means that
$\mathsf{T}(V_{1}\oplus_{N}V_{2})=\mathsf{T}V_{1}\oplus_{\mathsf{T}N}\mathsf{T}V_{2}$
and there exists a projection

$\mathsf{T}^{*}(V_{1}\times
V_{2})\rightarrowtriangle\mathsf{T}^{*}(V_{1}\oplus_{N}V_{2}),$

which is a symplectic relation. This relation may be obtained by choosing a
subbundle
$\mathsf{T}^{*}(V_{1}\times
V_{2})\supset\mathsf{T}_{V_{1}\times_{N}V_{2}}^{*}(V_{1}\times V_{2})$, where
$\mathsf{T}_{V_{1}\times_{N}V_{2}}^{*}(V_{1}\times V_{2})$ is the set of those
covectors in $\mathsf{T}^{*}(V_{1}\times V_{2})$, which are fixed at
$V_{1}\times_{N}V_{2}$, and subsequently dividing it by the annihilator
$\mathsf{T}(V_{1}\oplus_{N}V_{2})^{o}$, so that

$\mathsf{T}^{*}(V_{1}\oplus_{N}V_{2})\simeq\mathsf{T}_{V_{1}\times_{N}V_{2}}^{*}(V_{1}\times
V_{2})/\mathsf{T}(V_{1}\oplus_{N}V_{2})^{o}.$

A similar situation occurs in the case of vector-affine sum, i.e. when we
replace $V_{1}$ by $B$. The corresponding relation reads

$\mathsf{T}^{*}B\times\mathsf{T}^{*}V_{2}\rightarrowtriangle\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2}).$

The bundle $\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})$ has the structure of a
double affine-vector bundle represented by the diagram

$\textstyle{\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{1}^{*}\oplus_{N}V_{2}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\bar{\oplus}_{N}V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N}$

The bundle $\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})\to
V_{1}^{*}\oplus_{N}V_{2}^{*}$ is an affine bundle modelled on the vector
bundle $\mathsf{T}^{*}V\to V^{*}$. Since now we will denote it by

$\rho:\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})\to V_{1}^{*}\oplus_{N}V_{2}^{*}.$

Let us stress, that in a sharp contrast to the tangent bundle
$\mathsf{T}(B\bar{\oplus}_{N}V_{2})$, the bundle
$\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})$ is not a direct product of bundles.

Let us consider now the set
$\rho^{-1}(0\oplus_{N}V_{2}^{*})\subset\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})$.
By definition it consists of differentials of functions
$f:B\bar{\oplus}_{N}V_{2}^{*}\to\mathbb{R}$, which are constant in the
direction of $B$. The set $\rho^{-1}(0\oplus_{N}V_{2}^{*})$ may be subject to
symplectic reduction

$\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})\supset\rho^{-1}(0\oplus_{N}V_{2}^{*})\to\mathsf{T}^{*}V_{2}.$

The above reduction will play an important role in the Lagrangian description
of gauge field theories.

##  5 The bundle of connections

###  5.1 The bundle of connections

Let $E\to M$ be a fiber bundle and let
$\mathsf{j}^{1}_{m}\phi\in\mathsf{J}^{1}E$ be the first jet of a section
$\phi:M\to E$ at the point $m\in M$. The tangent space to $\phi(M)$ at the
point $\phi(m)$ defines the decomposition of the tangent space
$\mathsf{T}_{\phi(m)}E$ on the direct sum of the vertical subspace and the
subspace, which is tangent to the image of the section $\phi$, i.e.

$\mathsf{T}_{\phi(m)}E=\mathsf{V}_{\phi(m)}E\oplus\mathsf{T}\phi(\mathsf{T}_{m}M).$

Let us notice that $\mathsf{T}\phi(\mathsf{T}_{m}M)$ does not depend on the
choice of a representative in $\mathsf{j}^{1}_{m}\phi$. By taking a collection
of first jets at each point $e\in E$ we obtain the decomposition of each
tangent space $\mathsf{T}_{e}E$. We conclude that a connection in $E\to M$ may
be defined as a smooth section $\Gamma:E\to\mathsf{J}^{1}E$.

From the above considerations we obtain that a connection in a principal
bundle $P$ can be given by a global section of the bundle $\mathsf{J}^{1}P\to
P$. The connection has to be compatible with the group action of $G$. Notice
that the right action $R_{g}$ of $G$ on $P$ may be lifted to the action on
$\mathsf{J}^{1}P$ given by

$\mathsf{J}^{1}R_{g}:\mathsf{J}^{1}P\to\mathsf{J}^{1}P:\quad\mathsf{j}^{1}_{m}\phi\to\mathsf{j}^{1}_{m}(\phi
g),$

where $\phi g$ is a section given by the formula $(\phi g)(m):=\phi(m)g$. From
the above definition we obtain that the diagram

$\textstyle{\mathsf{J}^{1}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{J}^{1}R_{g}}$$\scriptstyle{\pi_{1,0}}$$\textstyle{\mathsf{J}^{1}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1,0}}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R_{g}}$$\textstyle{P}$

is commutative. From now on we will say that a section
$\Gamma:P\to\mathsf{J}^{1}P$ is invariant with respect to the action of $G$ if
$\Gamma(pg)=\Gamma(p)g$ for each $p\in P,g\in G$. Principal connections in $P$
are in a one-to-one correspondence with invariant sections of
$\mathsf{J}^{1}P\to P$. The invariance of $\Gamma$ is equivalent to the
invariance of the horizontal distribution in $\mathsf{T}P$ defined by
$\Gamma$. Let us notice that each invariant section $\Gamma$ defines a unique
section

$\omega:M\to C,\quad C:=\mathsf{J}^{1}P/G.$

The bundle $C\to M$ turns out to be the bundle of principal connections in
$P$. Each section of $C$ represents a gauge field of a given theory. Since
$\mathsf{J}^{1}P\to P$ is an affine bundle modelled on the vector bundle
$\mathsf{T}^{*}M\otimes_{P}\mathsf{V}P\to P$, we obtain that $C\to M$ is an
affine bundle as well and it is modelled on the vector bundle
$\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P\to M$.

###  5.2 Local picture of the connection bundle

Let $g_{\alpha}:\pi^{-1}(U_{\alpha})\to U_{\alpha}\times G$ be a local
trivialisation of $P$. It induces the local trivialisation of
$\mathsf{J}^{1}P$ given by

$\Psi_{\alpha}:\mathsf{J}^{1}P\supset\pi_{1}^{-1}(U_{\alpha})\to
G\times(\mathsf{T}^{*}M\otimes\mathfrak{g}),\qquad(\mathsf{j}^{1}_{m}\phi)\longmapsto\Big{(}g_{\alpha}(\phi(m)),(g_{\alpha}\circ\phi)^{*}\theta\Big{)},$
(33)

where

$(g_{\alpha}\circ\phi)^{*}\theta(m):=-\mathrm{Ad}_{{g_{\alpha}(p)}^{-1}}\circ
A_{\alpha}(m),\quad p=\phi(m).$ (34)

In the above formula we have denoted by $\theta$ the Maurer-Cartan form on
$G$. For two different trivialisations $\Psi_{\alpha}$ and $\Psi_{\beta}$ we
have the transition function

$\Psi_{\beta}\circ\Psi_{\alpha}^{-1}:G\times\mathsf{T}^{*}M\otimes\mathfrak{g}\to
G\times\mathsf{T}^{*}M\otimes\mathfrak{g},$
$\Big{(}g,B(m)\Big{)}\longmapsto\Big{(}g_{\beta\alpha}(m)g,B(m)+\mathrm{Ad}_{g^{-1}}\circ
g_{\beta\alpha}^{*}\theta(m)\Big{)}.$ (35)

Using (33) we can also write the action of the group $G$ on $\mathsf{J}^{1}P$,
i.e.

$\mathsf{J}^{1}R_{h}:G\times(\mathsf{T}^{*}M\otimes\mathfrak{g})\to
G\times(\mathsf{T}^{*}M\otimes\mathfrak{g}),$
$(g,j_{\alpha})\longmapsto(gh,\mathrm{Ad}_{h^{-1}}\circ j_{\alpha}),\qquad
j_{\alpha}\in\mathsf{T}^{*}M\otimes\mathfrak{g},$

where $R_{h}$ is the right action of an element $h\in G$ on $P$. To simplify
the notation we have denoted by the same symbol $\mathsf{J}^{1}R_{h}$ both the
first jet prolongation of $R_{h}$ and its trivialised version.

Each trivialisation of $C$ comes from the trivialisation of $\mathsf{J}^{1}P$
divided by the action of $G$. Let $\mathsf{j}^{1}_{m}\phi$ be a first jet,
which in trivialisation $g_{\alpha}$ reads

$\mathsf{j}^{1}_{m}\phi=\Big{(}g_{\alpha}(p),-\mathrm{Ad}_{g_{\alpha}(p)^{-1}}\circ
A_{\alpha}\Big{)},\quad p=\phi(m).$ (36)

In our notation we will represent the equivalence class
$[\mathsf{j}^{1}_{m}\phi]$ by its representative in the neutral element of
$G$. For instance, the equivalence class of
$\Big{(}g_{\alpha}(p),-\mathrm{Ad}_{g_{\alpha}(p)^{-1}}\circ
A_{\alpha}\Big{)}$ will be represented by the element $(e,-A_{\alpha})$. It
implies that the trivialisation of $C$ reads

$C\to\mathsf{T}^{*}M\otimes\mathfrak{g},\qquad[\mathsf{j}^{1}_{m}\phi]\longmapsto-
A_{\alpha}.$

From the above formula we can see that sections of the bundle $C$ locally look
like gauge fields. The transition formulas for two different trivialisations
of $C$ coming from $g_{\alpha}$ and $g_{\beta}$ are the same as for
$A_{\alpha}$, i.e.

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{\beta}}$$\scriptstyle{\Psi_{\alpha}}$$\textstyle{[\mathsf{j}^{1}_{m}\phi]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{\beta}}$$\scriptstyle{\Psi_{\alpha}}$$\textstyle{\mathsf{T}^{*}M\otimes\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{\alpha}^{-1}\circ\Psi_{\beta}}$$\textstyle{\mathsf{T}^{*}M\otimes\mathfrak{g}}$$\textstyle{(j_{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{\alpha}^{-1}\circ\Psi_{\beta}\quad\qquad}$$\textstyle{(\mathrm{Ad}_{g_{\beta\alpha}(m)}\circ
j_{\alpha}+g_{\alpha\beta}^{*}\theta)}$

###  5.3 The configuration bundle for gauge theories

The bundle of infinitesimal configurations for a gauge field theory described
by $C\to M$ is the bundle $\mathsf{J}^{1}C$. It is an affine bundle over $C$
modelled on the vector bundle

$\mathsf{T}^{*}M\otimes_{M}\mathsf{V}C\simeq
C\times_{M}\mathsf{T}^{*}M\otimes_{M}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P.$

In the above formula we have used the isomorphism $\mathsf{V}C\simeq
C\times_{M}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P$, which comes from the fact
that $C\to M$ is an affine bundle modelled on the vector bundle
$\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P\to M$. Therefore, the model bundle of
$\mathsf{J}^{1}C\to C$ can be written as

$C\times_{M}\otimes^{2}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P\to C.$

The affine structure of $C\to M$ implies that $\mathsf{J}^{1}C\to M$ is also
an affine bundle modelled on the vector bundle
$\mathsf{J}^{1}(\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P)\to M$. In the
following we will analyse in detail the internal structure of
$\mathsf{J}^{1}C$. Since by definition $C=\mathsf{J}^{1}P/G$ we expect that
there exists some relation between $\mathsf{J}^{1}C$ and the space of
semiholonomic jets $\widehat{\mathsf{J}^{2}P}$. In particular, we will show
that there exists an isomorphism
$\mathsf{J}^{1}C\simeq\widehat{\mathsf{J}^{2}P}/G$.

Let us start with the brief recalling of the space $\widehat{\mathsf{J}^{2}E}$
(see Section 2.2) in the case $E=P$. The bundle
$\widehat{\mathsf{J}^{2}P}\to\mathsf{J}^{1}P$ is an affine bundle modelled on
the vector bundle
${\mathsf{J}^{1}P}\times_{P}\otimes^{2}\mathsf{T}^{*}M\otimes_{P}\mathsf{V}P\to\mathsf{J}^{1}P$.
Since $\mathsf{V}P\simeq P\times\mathfrak{g}$ we obtain a natural
identification

${\mathsf{J}^{1}P}\times_{P}\otimes^{2}\mathsf{T}^{*}M\otimes_{P}\mathsf{V}P\simeq{\mathsf{J}^{1}P}\times_{M}(\otimes^{2}\mathsf{T}^{*}M\otimes\mathfrak{g}).$

From (32) we know that the bundle of semiholonomic jets
$\widehat{\mathsf{J}^{2}P}$ has a natural decomposition

$\widehat{\mathsf{J}^{2}P}=\mathsf{J}^{2}P\bar{\oplus}_{\mathsf{J}^{1}P}(\mathsf{J}^{1}P\times_{P}\wedge^{2}\mathsf{T}^{*}M\otimes_{M}\mathsf{V}P).$

Dividing the above product by $G$ we obtain

$\widehat{\mathsf{J}^{2}P}/G=\mathsf{J}^{2}P/G\bar{\oplus}_{C}(C\times_{M}\wedge^{2}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P).$
(37)

The bundle $\widehat{\mathsf{J}^{2}P}/G\to C$ is an affine bundle modelled on
the vector bundle
$C\times_{M}\mathsf{T}^{*}M\otimes_{M}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P$.
The model bundle has a natural decomposition

$C\times_{M}\mathsf{T}^{*}M\otimes_{M}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P=\Big{(}C\times_{M}\vee^{2}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P\Big{)}\oplus_{C}\Big{(}C\times_{M}\wedge^{2}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P\Big{)},$

where $\vee^{2}\mathsf{T}^{*}M$ is the subspace of symmetric tensors in
$\otimes^{2}\mathsf{T}^{*}M$. Notice that $\mathsf{J}^{2}P/G\to C$ is modelled
on $C\times_{M}\vee^{2}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P\to C$.

Our purpose now is to show that there exists isomorphism
$\widehat{\mathsf{J}^{2}P}/G\simeq\mathsf{J}^{1}C$. This isomorphism together
with the decomposition (32) will allow us to ”extract” geometrically the
curvature from the first jet of a connection. In order to do that we will
prove first two preparatory lemmas. Let us recall that each section
$\omega:M\to C$ uniquely defines an invariant section
$\bar{\omega}:P\to\mathsf{J}^{1}P$. Notice, that each jet
$\bar{\omega}(p)\in\mathsf{J}^{1}_{p}P$ is represented by the equivalence
class of sections $\phi_{\bar{\omega}}:M\to P$, such that
$\phi_{\bar{\omega}}(m)=p$. The composition
$\bar{\omega}\circ\phi_{\bar{\omega}}$ defines a local section of the bundle
$\mathsf{J}^{1}P\to M$ in a neighborhood of $m\in M$. We have the following
lemma

######  Lemma 1.

Let $\bar{\omega}:P\to\mathsf{J}^{1}P$ be an invariant section and let
$\phi_{\bar{\omega}}:M\to P$ be a representative of
$\bar{\omega}(p)\in\mathsf{J}^{1}_{p}P$ at a point $m\in M$, i.e.
$\bar{\omega}(p)=\mathsf{j}^{1}_{m}\phi_{\bar{\omega}}$. Then, we have

$\mathsf{j}^{1}_{m}\Big{(}\bar{\omega}\circ\phi_{\bar{\omega}}\Big{)}\in\widehat{\mathsf{J}^{2}P}.$

Proof:
We will show first that
$\mathsf{j}^{1}_{m}\Big{(}\bar{\omega}\circ\phi_{\bar{\omega}}\Big{)}$ does
not depend on the choice of a representative $\phi_{\bar{\omega}}$. The first
jet $\mathsf{j}^{1}_{m}\Big{(}\bar{\omega}\circ\phi_{\bar{\omega}}\Big{)}$ is
uniquely defined by the tangent map
$\mathsf{T}(\bar{\omega}\circ\phi_{\bar{\omega}}):\mathsf{T}_{m}M\to\mathsf{T}_{\bar{\omega}(p)}\mathsf{J}^{1}P$.
From the composition rule for tangent maps we have

$\mathsf{T}_{m}(\bar{\omega}\circ\phi_{\bar{\omega}})=\mathsf{T}_{p}\bar{\omega}\circ\mathsf{T}_{m}\phi_{\bar{\omega}}.$

Since $\mathsf{T}_{m}\phi_{\bar{\omega}}$ does not depend on the choice of a
representative in $\bar{\omega}(p)$, we have that the above map does not
depend on the choice of $\phi_{\bar{\omega}}$ neither. Applying the
projections $(\pi_{1})_{1,0}$ and $\mathsf{j}^{1}\pi_{1,0}$ to
$\mathsf{j}^{1}_{m}(\bar{\omega}\circ\phi_{\bar{\omega}})$ we obtain

$(\pi_{1})_{1,0}\Big{(}\mathsf{j}^{1}_{m}(\bar{\omega}\circ\phi_{\bar{\omega}})\Big{)}=(\bar{\omega}\circ\phi_{\bar{\omega}})(m)=\bar{\omega}(p)$
$\mathsf{j}^{1}\pi_{1,0}\Big{(}\mathsf{j}^{1}_{m}(\bar{\omega}\circ\phi_{\bar{\omega}})\Big{)}=\mathsf{j}^{1}_{m}\Big{(}\pi_{1,0}\circ(\bar{\omega}\circ\phi_{\bar{\omega}})\Big{)}=\mathsf{j}^{1}_{m}\phi_{\bar{\omega}}=\bar{\omega}(p),$

which proves that indeed
$\mathsf{j}^{1}_{m}\Big{(}\bar{\omega}\circ\phi_{\bar{\omega}}\Big{)}\in\widehat{\mathsf{J}^{2}P}$.

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\blacksquare$

From now on we will denote by $[\mathsf{j}^{1}_{m}\psi]$ the equivalence class
of a first jet $\mathsf{j}^{1}_{m}\psi\in\widehat{\mathsf{J}^{2}P}$ with
respect to the group action of $G$.

In the next step we will be interested in the following construction. Let us
consider an invariant section $\bar{\omega}:P\to\mathsf{J}^{1}P$. By
definition, at each point $p\in P$ we have
$\bar{\omega}(p)=\mathsf{j}^{1}_{m}\varphi$, where $\varphi:M\to P$ is a
representative of the suitable equivalence class of sections. Notice that the
first jet of the section $\bar{\omega}\circ\varphi$ at the point $m=\pi(p)$ is
independent on the choice of a representative $\varphi$ in
$\mathsf{j}^{1}_{m}\varphi$. In particular, for each representative $\varphi$
we have $\varphi(m)=p$. In the following it will be important to write down
the first jet of $\bar{\omega}\circ\varphi$ in a given trivialisation
$\mathsf{J}^{1}P\simeq G\times\mathsf{T}^{*}M\otimes\mathfrak{g}$. Using Lemma
1 and the constructions from chapter 5.2 we will prove a following lemma,
which will simplify the calculations in the main theorem of this section.

######  Lemma 2.

Let

$\eta:M\to\mathsf{T}^{*}M\otimes\mathfrak{g},\quad
x\longmapsto\mathrm{Ad}_{\varphi_{\alpha}(x)^{-1}}B_{\alpha}(x)$

be a section of the bundle $\mathsf{T}^{*}M\otimes\mathfrak{g}\to M$, where
$B_{\alpha}:M\to\mathsf{T}^{*}M\otimes\mathfrak{g}$ is a one-form with values
in $\mathfrak{g}$ and $\varphi_{\alpha}:M\to G$ is a $G$-valued function, such
that

$A_{\alpha}(x):=-\mathrm{Ad}_{{\varphi_{\alpha}(x)}}\varphi_{\alpha}^{*}\theta(x)$

(see 34) and $B_{\alpha}(m)=A_{\alpha}(m)$ in a given point $m\in M$. Then,
the first jet of the section $\eta$ at the point $m\in M$ may be written as

$\mathsf{j}^{1}_{m}\eta=\Big{(}\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}(\mathsf{j}^{1}_{m}B_{\alpha})+\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\frac{1}{2}[B_{\alpha}(m)\wedge
B_{\alpha}(m)]\Big{)}\quad\in\mathsf{J}^{1}(\mathsf{T}^{*}M\otimes\mathfrak{g}),$

where

$\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\mathsf{j}^{1}_{m}B_{\alpha}:=\mathsf{j}^{1}_{m}\Big{(}\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}B_{\alpha}\Big{)}.$

Proof:
The first jet $\mathsf{j}^{1}_{m}\eta$ is uniquely defined by the tangent map
$\mathsf{T}\eta$. Let $v\in\mathsf{T}_{m}M$ be a tangent vector represented by
a curve, say $m(t)$, and such that $m(0)=m$. Applying $\mathsf{T}\eta$ to $v$
we obtain

$\mathsf{T}\eta(v)=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\eta\circ\gamma_{v}(t)=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\Big{(}\mathrm{Ad}_{\varphi_{\alpha}(m(t))^{-1}}B_{\alpha}(m(t))\Big{)}=$
$=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\mathrm{Ad}_{\varphi_{\alpha}(m(t))^{-1}}B_{\alpha}(m)+\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}B_{\alpha}(m(t)).$
(38)

The last equality comes from the fact that

$\mathrm{Ad}_{\varphi_{\alpha}(m(t))^{-1}}B_{\alpha}(m(t))=\varphi_{\alpha}(m(t))^{-1}B_{\alpha}(m(t))\varphi_{\alpha}(m(t)).$

From the second element in (38) we obtain

$\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}B_{\alpha}(m(t))=\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\mathsf{T}B_{\alpha}(v),$

which means that we can identify it with
$\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\mathsf{j}^{1}_{m}B_{\alpha}(v)$. Let
us consider now the first element in (38), which is a bit more complicated. We
will introduce a following notation. The tangent vector
$\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\varphi_{\alpha}(m(t))$ is by definition
an element of $\mathsf{T}G$, which in trivialisation $\mathsf{T}G\simeq
G\times\mathfrak{g}$ reads $(\varphi_{\alpha}(m),X)$, where

$X:=\varphi_{\alpha}(m)^{-1}\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\varphi_{\alpha}(m(t))\in\mathfrak{g}.$

Notice that using (36) we can write $X$ as

$X=-\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}A_{\alpha}(v),$

where the relation between $\varphi_{\alpha}$ and $A_{\alpha}$ is the same as
in (34) (recall that $\varphi_{\alpha}=g_{\alpha}\circ\phi$). From the
assumptions of the lemma we know that at the point $m$ we have
$B_{\alpha}(m)=A_{\alpha}(m)$, so we can replace $A_{\alpha}(m)$ by
$B_{\alpha}(m)$. Then, in a neighborhood of $t=0$ we have

$\varphi_{\alpha}(m(t))\sim\varphi_{\alpha}(m)\exp(tX)=\varphi_{\alpha}(m)\exp\Big{(}-t\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}B_{\alpha}(m)(v)\Big{)},$

where $\varphi_{\alpha}(m(t))\sim\varphi_{\alpha}(m)\exp(tX)$ means that the
tangent vectors to $\varphi_{\alpha}(m(t))$ and $\varphi_{\alpha}(m)\exp(tX)$
at $t=0$ are the same. Notice that there is a relation

$\mathrm{Ad}_{\varphi_{\alpha}(m(t))^{-1}}=\mathrm{Ad}_{\exp(-tX)}\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}},$

where we have used the formula $(\exp(tX))^{-1}=\exp(-tX)$.

Let us come back to the first element in (38). Using the above conclusions we
can write

$\Big{(}\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\mathrm{Ad}_{\varphi_{\alpha}(m(t))^{-1}}\Big{)}B_{\alpha}(m)=\Big{(}\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\mathrm{Ad}_{\exp(-tX)}\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\Big{)}B_{\alpha}(m)=$
$=\Big{(}\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\mathrm{Ad}_{\exp(-tX)}\Big{)}\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}B_{\alpha}(m)=[-X,\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}B_{\alpha}(m)]=$
$[\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}B_{\alpha}(m)(v),\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}B_{\alpha}(m)]=\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}[B_{\alpha}(m)(v),B_{\alpha}(m)]=$
$=\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\frac{1}{2}[B_{\alpha}(m)\wedge
B_{\alpha}(m)](v,\cdot).$

where $[\cdot,\cdot]$ is the Lie bracket in $\mathfrak{g}$ and
$[\cdot\wedge\cdot]$ is the bracket of forms on $M$ with values in
$\mathfrak{g}$. Using the identification
$\mathsf{T}_{m}\eta\sim\mathsf{j}^{1}_{m}\eta$ and the calculations associated
with (38) we can finally write

$\mathsf{j}^{1}_{m}\eta=\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\mathsf{j}^{1}_{m}B_{\alpha}+\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\frac{1}{2}[B_{\alpha}(m)\wedge
B_{\alpha}(m)].$

Notice that
$\mathsf{j}^{1}_{m}B_{\alpha}\in\mathsf{J}^{1}(\mathsf{T}^{*}M\otimes\mathfrak{g})$
and $[B_{\alpha}(m)\wedge
B_{\alpha}(m)]\in\mathsf{T}^{*}M\otimes\mathsf{T}^{*}M\otimes\mathfrak{g}$,
which means that the above addition is well-defined. $\blacksquare$

Using the above lemmas we will prove now the crucial theorem of this section.

######  Theorem 1.

The map

$\gamma:\mathsf{J}^{1}C\to\widehat{\mathsf{J}^{2}P}/G,\quad\mathsf{j}^{1}_{m}\omega\longmapsto[\mathsf{j}^{1}_{m}\Big{(}\bar{\omega}\circ\phi_{\bar{\omega}}\Big{)}]$

is an isomorphism of affine bundles over $C$ and its linear part is the
identity on
$\mathsf{T}^{*}M\otimes_{M}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P$.

Proof:
We will start with the derivation of $\gamma$ in a trivialisation. Let
$g_{\alpha}:P\to G$ be a local trivialisation of $P$, which induces
trivialisations $\mathsf{J}^{1}P\simeq
G\times\mathsf{T}^{*}M\otimes\mathfrak{g}$,
$C\simeq\mathsf{T}^{*}M\otimes\mathfrak{g}$ and
$\mathsf{J}^{1}C\simeq\mathsf{J}^{1}(\mathsf{T}^{*}M\otimes\mathfrak{g})$. Let
$\mathsf{j}^{1}_{m}\omega\in\mathsf{J}^{1}_{m}C$ be a first jet, which in the
above trivialisation has a form $\mathsf{j}^{1}_{m}B_{\alpha}$, where
$B_{\alpha}$ is, by definition, a representative of the equivalence class of
sections

$B_{\alpha}:M\to\mathsf{T}^{*}M\otimes\mathfrak{g},$

such that the tangent maps at the point $m$ given by each representative are
the same. A family of sections $B_{\alpha}$ defines an invariant section
$\bar{\omega}:P\to\mathsf{J}^{1}P$, which in our trivialisation reads

$\bar{\omega}:M\times G\to
G\times\mathsf{T}^{*}M\otimes\mathfrak{g},\quad\bar{\omega}(m,g_{\alpha}(p))=(g_{\alpha}(p),\mathrm{Ad}_{g_{\alpha}(p)^{-1}}B_{\alpha}(m)).$

To simplify the notation we have denoted by the same symbol $\bar{\omega}$ a
section of the bundle $\mathsf{J}^{1}P\to P$ and its trivialised form.

Let us choose now a point $p\in P$, such that $\pi(p)=m$. Let $\phi:M\to P$ be
a representative of $\bar{\omega}(p)=\mathsf{j}^{1}_{m}\phi$, where
$\phi(m)=p$. In the trivialisation we obtain

$\bar{\omega}(m,g_{\alpha}\circ\phi(m))=(\varphi_{\alpha}(m),\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}B_{\alpha}(m)),$

where $\varphi_{\alpha}:=g_{\alpha}\circ\phi$. Notice that
$\bar{\omega}(p)=\mathsf{j}^{1}_{m}\phi$ implies that
$m\longmapsto\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}B_{\alpha}(m)$ satisfies
assumptions of Lemma 2. Therefore, the first jet
$\mathsf{j}^{1}_{m}(\bar{\omega}\circ\phi)$ has a form

$\mathsf{j}^{1}_{m}(\bar{\omega}\circ\phi)\in(G\times\mathsf{T}^{*}M\otimes\mathfrak{g})\times_{\mathsf{T}^{*}M\otimes\mathfrak{g}}\mathsf{J}^{1}(\mathsf{T}^{*}M\otimes\mathfrak{g}),$
$\mathsf{j}^{1}_{m}(\bar{\omega}\circ\phi)=\Big{(}\varphi_{\alpha}(m),\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}B_{\alpha}(m),\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}(\mathsf{j}^{1}_{m}B_{\alpha})+\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\frac{1}{2}[B_{\alpha}\wedge
B_{\alpha}](m)\Big{)}.$

Since both projections on $\mathsf{J}^{1}P$ are the same we can write

$\mathsf{j}^{1}_{m}(\bar{\omega}\circ\phi)\in
G\times\mathsf{J}^{1}(\mathsf{T}^{*}M\otimes\mathfrak{g}),$
$\mathsf{j}^{1}_{m}(\bar{\omega}\circ\phi)=\Big{(}\varphi_{\alpha}(m),\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}(\mathsf{j}^{1}_{m}B_{\alpha})+\mathrm{Ad}_{\varphi_{\alpha}(m)^{-1}}\frac{1}{2}[B_{\alpha}\wedge
B_{\alpha}](m)\Big{)}.$

Taking the equivalence class of the above jet with respect to the group action
we obtain

$[\mathsf{j}^{1}_{m}(\bar{\omega}\circ\phi)]=\Big{(}\mathsf{j}^{1}_{m}B_{\alpha}+[B_{\alpha},B_{\alpha}](m)\Big{)}=\Big{(}\mathsf{j}^{1}_{m}B_{\alpha}+\frac{1}{2}[B_{\alpha}\wedge
B_{\alpha}](m)\Big{)}\in\mathsf{J}^{1}(\mathsf{T}^{*}M\otimes\mathfrak{g}),$

where $[B_{\alpha}\wedge B_{\alpha}]$ is a product of $\mathfrak{g}$-valued
forms on $M$. Let us notice, that the above result is independent on the
choice of a representative $\phi$ and, as a consequence, on the previously
chosen point $p$. Furthermore, it does not depend on the choice of a
representative $\omega$ in $\mathsf{j}^{1}_{m}\omega$. Indeed, if
$\omega,\omega^{\prime}$ are two representatives of
$\mathsf{j}^{1}_{m}\omega$, which in trivialisation read $B_{\alpha}$ and
$B^{\prime}_{\alpha}$, respectively, then by definition
$B_{\alpha}(m)=B^{\prime}_{\alpha}(m)$ and
$\mathsf{j}^{1}_{m}B_{\alpha}=\mathsf{j}^{1}_{m}B^{\prime}_{\alpha}$. Finally,
the map $\gamma$ reads

$\gamma:\mathsf{j}^{1}_{m}B_{\alpha}\longmapsto\mathsf{j}^{1}_{m}B_{\alpha}+\frac{1}{2}[B_{\alpha}(m)\wedge
B_{\alpha}(m)].$

Using the above formula we can easily show that $\gamma$ is an affine map
whose linear part is the identity. Let
$\mathsf{j}^{1}_{m}B_{\alpha}\in\mathsf{J}^{1}(\mathsf{T}^{*}M\otimes\mathfrak{g})$
and
$\mathsf{j}^{1}_{m}D_{\alpha}\in\mathsf{J}^{1}(\mathsf{T}^{*}M\otimes\mathfrak{g})$
be elements of $\mathsf{J}^{1}_{e}C$. We have

$\gamma(\mathsf{j}^{1}_{m}B_{\alpha})-\gamma(\mathsf{j}^{1}_{m}D_{\alpha})=\mathsf{j}^{1}_{m}B_{\alpha}+\frac{1}{2}[B_{\alpha}(m)\wedge
B_{\alpha}(m)]-\mathsf{j}^{1}_{m}D_{\alpha}-\frac{1}{2}[D_{\alpha}(m)\wedge
D_{\alpha}(m)]=$ $=\mathsf{j}^{1}_{m}B_{\alpha}-\mathsf{j}^{1}_{m}D_{\alpha},$

where in the last equality we have used the fact that
$B_{\alpha}(m)=D_{\alpha}(m)$. From the above formula one can see that the
linear part of $\gamma$ is the identity on
$\mathsf{T}^{*}M\otimes_{M}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P$.
$\blacksquare$

Using the above theorem and (37) we obtain the decomposition

$\mathsf{J}^{1}C=\mathsf{J}^{2}P/G\bar{\oplus}_{C}(C\times_{M}\wedge^{2}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P),$
(39)
$\mathsf{j}^{1}_{m}\omega=\Big{(}[\mathsf{j}^{2}_{m}\bar{\omega}],\omega(m),F_{\omega}(m)\Big{)}$

over $C$. In the above formula we have denoted by
$[\mathsf{j}^{2}_{m}\bar{\omega}]$ the equivalence class of the second jet of
the section $\bar{\omega}$ with respect to the group action and
$F_{\omega}(m)$ is the value of the curvature form (9) associated with
$\mathsf{j}^{1}_{m}\omega$.

Notice that a similar result to Theorem 1 was briefly discussed by
Sardanashvily in [32]. In the formula (6.6) in his book he presents the
decomposition

$\mathsf{J}^{1}C=\mathsf{J}^{2}P/G\bar{\oplus}_{C}(\wedge^{2}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P),$
$k_{\mu\lambda}^{m}=\frac{1}{2}(k_{\mu\lambda}^{m}+k_{\lambda\mu}^{m}+c^{m}_{nl}k_{\lambda}^{n}k_{\mu}^{l})+\frac{1}{2}(k_{\mu\lambda}^{m}-k_{\lambda\mu}^{m}-c^{m}_{nl}k_{\lambda}^{n}k_{\mu}^{l}).$

where $(x^{\mu},k^{m}_{\mu},k_{\mu\lambda}^{m})$ are coordinates in
$\mathsf{J}^{1}C$. However, his result is not true. First of all, we have
$c^{m}_{nl}=-c^{m}_{ln}$, which implies that the element
$\frac{1}{2}(k_{\mu\lambda}^{m}+k_{\lambda\mu}^{m}+c^{m}_{nl}k_{\lambda}^{n}k_{\mu}^{l})$
is not symmetric with respect to $\mu\lambda$, therefore it can not be part of
$\mathsf{J}^{2}P/G$. Secondly, [32] does not contain any calculations that
would support his statements.

###  5.4 The phase bundle

In subsection 3.2 it was said that the phase bundle in field theory is
represented by the bundle
$\mathcal{P}=\mathsf{V}^{*}E\otimes_{E}\Omega^{m-1}$, where $E\to M$ is the
space of fields. From now on we will use the notation
$\mathcal{V}:=\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P$. In gauge field theories
the phase space is given by

$\mathcal{P}=\mathsf{V}^{*}C\otimes_{C}\Omega^{m-1}=C\times_{M}\mathcal{V}^{*}\otimes_{M}\Omega^{m-1},$

where $\mathcal{V}^{*}$ is the dual bundle of $\mathcal{V}$. Using isomorphism
$\mathsf{T}M\otimes_{M}\Omega^{m}\simeq\Omega^{m-1}$ we can write

$\mathcal{V}^{*}\otimes_{M}\Omega^{m-1}\simeq\mathsf{T}M\otimes_{M}\mathrm{ad}^{*}P\otimes_{M}\mathsf{T}M\otimes_{M}\Omega^{m}=\mathsf{T}M\otimes_{M}\mathsf{T}M\otimes_{M}\mathrm{ad}^{*}P\otimes_{M}\Omega^{m}.$

Notice that the tensor bundle $\mathsf{T}M\otimes_{M}\mathsf{T}M$ has the
canonical decomposition

$\mathsf{T}M\otimes_{M}\mathsf{T}M=\wedge^{2}\mathsf{T}M\oplus_{M}\vee^{2}\mathsf{T}M.$

Using the above formula we obtain

$\mathsf{T}M\otimes_{M}\mathsf{T}M\otimes_{M}\mathrm{ad}^{*}P\otimes_{M}\Omega^{m}=(\wedge^{2}\mathsf{T}M\oplus_{M}\vee^{2}\mathsf{T}M)\otimes_{M}\Omega^{m}\otimes_{M}\mathrm{ad}^{*}P$
$=(\wedge^{2}\mathsf{T}M\otimes_{M}\Omega^{m}\otimes_{M}\mathrm{ad}^{*}P)\oplus_{M}(\vee^{2}\mathsf{T}M\otimes_{M}\Omega^{m}\otimes_{M}\mathrm{ad}^{*}P)=$
$=(\Omega^{m-2}\otimes_{M}\mathrm{ad}^{*}P)\oplus_{M}(\vee^{2}\mathsf{T}M\otimes_{M}\Omega^{m}\otimes_{M}\mathrm{ad}^{*}P).$
(40)

The phase bundle $\mathcal{P}\to C$ turns out to be the Whitney sum of vector
bundles

$\displaystyle C\times_{M}(\Omega^{m-2}\otimes_{M}\mathrm{ad}^{*}P)\to C,$
(41) $\displaystyle
C\times_{M}(\vee^{2}\mathsf{T}M\otimes_{M}\Omega^{m}\otimes\mathrm{ad}^{*}P)\to
C.$ (42)

For the clarity of the presentation we will introduce the notation

$\overline{\mathcal{P}}:=\Omega^{m-2}\otimes_{M}\mathrm{ad}^{*}P,$
$\mathcal{S}:=\vee^{2}\mathsf{T}M\otimes_{M}\Omega^{m}\otimes\mathrm{ad}^{*}P.$

The bundle $C\times_{M}\overline{\mathcal{P}}\to C$ will be called the reduced
phase bundle and it represents the reduced phase space of the system. Notice
that (40) implies that $\mathsf{J}^{1}\mathcal{P}$ is the product of bundles

$\mathsf{J}^{1}\mathcal{P}=\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}\overline{\mathcal{P}}\oplus_{M}\mathsf{J}^{1}\mathcal{S}.$
(43)

The decompositions (39) and (43) will be crucial in the following part of our
work.

##  6 Lagrangian description

In sections 5 we have carefully analysed the geometric structure of the bundle
of connections. Now we will focus on the dynamics of gauge fields. The
starting point for our considerations will be the Tulczyjew triple presented
in section 3. In the previous section we have shown that the bundles
$\mathsf{J}^{1}C$ and $\mathsf{J}^{1}\mathcal{P}$ have the natural structure
of a product of bundles. In particular, we have derived the decomposition 39
of the bundle $\mathsf{J}^{1}C$, which allows to project the first jet of a
connection in a given point onto its value and curvature. As we mentioned
before, in most of the physically interesting field theories Lagrangian does
not depend on the entire first jet but only on some informations contained in
it and associated with its antisymmetric part. Each description of the gauge
fields dynamics should be therefore independent of the symmetry associated
with the addition of the symmetric tensor to the first jet of a connection. In
this section we will reduce the Tulczyjew formalism with respect to that
symmetry.

###  6.1 The structure of iterated bundles

Let us consider the map $\kappa$ on a bundle $C$. To simplify the notation we
will use the symbol ${\bf t}:=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}$ from the
subsection 3.2. Notice that each element of $\mathsf{V}\mathsf{J}^{1}C$ has a
form ${\bf t}\mathsf{j}^{1}_{x}\chi$, where $\chi$ is a map

$\chi:\mathbb{R}\times M\to C$ (44)

preserving the projection on $M$. By definition 20 the map $\kappa$ on $C$
reads

$\kappa:\mathsf{V}\mathsf{J}^{1}C\to\mathsf{J}^{1}\mathsf{V}C,\quad{\bf
t}\mathsf{j}^{1}_{x}\chi\longmapsto\mathsf{j}^{1}_{x}{\bf t}\chi.$ (45)

The affine structure of the bundle $C$ allows to write the above equation in a
simpler way. From the fact that $C\to M$ is an affine bundle modelled on the
vector bundle $\mathcal{V}=\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P\to M$, we
obtain that the bundle $\mathsf{J}^{1}C\to M$ is an affine bundle as well and
it is modelled on the vector bundle $\mathsf{J}^{1}\mathcal{V}\to M$. We have
isomorphisms

$\displaystyle\mathsf{V}C$ $\displaystyle\simeq C\times_{M}\mathcal{V},$ (46)
$\displaystyle\mathsf{V}\mathsf{J}^{1}C$
$\displaystyle\simeq\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}\mathcal{V},$ (47)
$\displaystyle\mathsf{J}^{1}\mathsf{V}C$
$\displaystyle\simeq\mathsf{J}^{1}(C\times_{M}\mathcal{V})\simeq\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}\mathcal{V},$
(48)

where for simplicity we have used the notation (10) and (11).

The map $\kappa$ can be expressed in terms of the above identifications. Since
the bundle of connections is an affine bundle, each element ${\bf
t}\mathsf{j}^{1}_{x}\chi$ has the representative of the form

$\chi(t,x)=\omega(x)+t\tau(x),\qquad\omega:M\to C,\quad\tau:M\to\mathcal{V}.$

By taking the first jet of the above map in $x\in M$ we obtain a vertical
curve in $\mathsf{J}^{1}_{x}C$

$\mathsf{j}^{1}_{x}\chi(\cdot):\mathbb{R}\to\mathsf{J}^{1}C,\quad
t\longmapsto\mathsf{j}^{1}_{x}\omega+t\mathsf{j}^{1}_{x}\tau.$

We can calculate the derivative of the above curve over $t$ in $t=0$ and
obtain

${\bf
t}\mathsf{j}^{1}_{x}\chi(t)=(\mathsf{j}^{1}_{x}\omega,\mathsf{j}^{1}_{x}\tau),$

where we have used the identfication (47). If we now repeat the above
procedure in the opposite order, i.e. we calculate the derivative of $\chi$
over $t$ in $t=0$ and then take the first jet, we will obtain

$\mathsf{j}^{1}_{x}{\bf
t}\chi=(\mathsf{j}^{1}_{x}\omega,\mathsf{j}^{1}_{x}\tau),$

where we have used isomorphism (48). In the above identifications the map
$\kappa$ turns out to be the identity

$\kappa:\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}\mathcal{V}\to\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}\mathcal{V},\qquad(\mathsf{j}^{1}_{x}\omega,\mathsf{j}^{1}_{x}\tau)\longmapsto(\mathsf{j}^{1}_{x}\omega,\mathsf{j}^{1}_{x}\tau).$

Let us fix now an element $\mathsf{j}^{1}_{x}\omega\in\mathsf{J}^{1}_{x}C$.
The first jet $\mathsf{j}^{1}_{x}\omega$ has a canonical projection on the
value $\omega(x)$, which defines a covariant derivative $D_{\omega}$ in
$\mathcal{V}_{x}$. Then, we have in $\mathsf{J}^{1}_{x}\mathcal{V}$ a subspace

$S_{x}:=\\{\mathsf{j}^{1}_{x}\tau_{0}\in\mathsf{J}^{1}_{x}\mathcal{V},\quad(D_{\omega}\tau_{0})(x)=0\\}.$

Notice that each first jet
$\mathsf{j}^{1}_{x}\tau\in\mathsf{J}^{1}_{x}\mathcal{V}$ can be decomposed on
a pair $\Big{(}\mathsf{j}^{1}_{x}\tau_{0},D_{\omega}\tau(x)\Big{)}$, where
$\tau_{0}:M\to\mathcal{V}$ is a section of the bundle $\mathcal{V}$, such that
$\tau_{0}(x)=\tau(x)$ and $\mathsf{j}^{1}_{x}\tau_{0}\in S_{x}$, and
$D_{\omega}\tau(x)$ is the covariant derivative of the section $\tau$ in the
point $x\in M$. Notice that $D_{\omega}\tau(x)$ does not depend on the choice
of the representative in $\mathsf{j}^{1}_{x}\tau$. Therefore, we obtain a
decomposition

$\mathsf{J}^{1}_{x}\mathcal{V}=S_{x}\times\mathcal{F}_{x},\quad\mathsf{j}^{1}_{x}\tau\longmapsto\Big{(}\mathsf{j}^{1}_{x}\tau_{0},D_{\omega}\tau(x)\Big{)}.$
(49)

Using the above identification we can project $\mathsf{j}^{1}_{x}\tau$ on
$D_{\omega}\tau(x)$. Furthermore, there is the canonical projection
$\mathsf{J}^{1}_{x}\mathcal{V}\to\mathcal{V}$ so that we obtain a map

$\mathsf{J}^{1}\mathcal{V}\to\mathcal{V}\times_{M}\mathcal{F},$
$\mathsf{j}^{1}_{x}\tau\longmapsto\Big{(}\tau(x),D_{\omega}\tau(x)\Big{)}.$
(50)

Finally, the maps (39) and (50) define a projection

$\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}\mathcal{V}\to(C\times_{M}\mathcal{F})\times_{M}\mathcal{V}\times_{M}\mathcal{F},\quad(\mathsf{j}^{1}_{x}\omega,\mathsf{j}^{1}_{x}\tau)\longmapsto\Big{(}\omega(x),F_{\omega}(x),\tau(x),D_{\omega}\tau(x)\Big{)}.$
(51)

Notice that isomorphisms (47) and (48) imply that the bundles
$\mathsf{J}^{1}\mathsf{V}C$ and $\mathsf{V}\mathsf{J}^{1}C$ may be reduced by
means of (51). Therefore, we can use (51) to reduce the map $\kappa$ given by
(45). The reduction of $\kappa$ is presented on the diagram

$\textstyle{\mathsf{V}\mathsf{J}^{1}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{rzut12})}$$\scriptstyle{\kappa}$$\textstyle{\mathsf{J}^{1}\mathsf{V}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{rzut12})}$$\textstyle{(C\times_{M}\mathcal{F})\times_{M}\mathcal{V}\times_{M}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\kappa}}$$\textstyle{(C\times_{M}\mathcal{V})\times_{M}\mathcal{F}\times_{M}\mathcal{F}}$

and

$\textstyle{(\mathsf{j}^{1}_{m}\omega,\mathsf{j}^{1}_{m}\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{rzut12})}$$\scriptstyle{\kappa}$$\textstyle{(\mathsf{j}^{1}_{m}\omega,\mathsf{j}^{1}_{m}\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{rzut12})}$$\textstyle{\Big{(}\omega(x),F_{\omega}(x),\tau(x),D_{\omega}\tau(x)\Big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\kappa}}$$\textstyle{\Big{(}\omega(x),\tau(x),F_{\omega}(x),D_{\omega}\tau(x)\Big{)}}$

From the above diagram we see that the reduced map $\overline{\kappa}$ has a
form

$\overline{\kappa}:\Big{(}\omega(x),F_{\omega}(x),\tau(x),D_{\omega}\tau(x)\Big{)}\longmapsto\Big{(}\omega(x),\tau(x),F_{\omega}(x),D_{\omega}\tau(x)\Big{)}.$
(52)

###  6.2 The structure of the bundle $\mathsf{V}^{*}\mathsf{J}^{1}C$

Let $\mathsf{V}^{*}\mathsf{J}^{1}C\to\mathsf{J}^{1}C$ be the dual bundle of
$\mathsf{V}\mathsf{J}^{1}C\to\mathsf{J}^{1}C$. Since $\mathsf{J}^{1}C\to M$ is
an affine bundle modelled on the vector bundle $\mathsf{J}^{1}\mathcal{V}\to
M$ we have a canonical isomorphism
$\mathsf{V}^{*}\mathsf{J}^{1}C\simeq\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}{{}^{*}\mathcal{V}}$,
where $\mathsf{J}^{1}{{}^{*}\mathcal{V}}\to M$ is the dual bundle of
$\mathsf{J}^{1}\mathcal{V}\to M$.

In section (4.2) we have considered an affine bundle $A\to N$ with an affine
subbundle $B\to N$ modelled on a vector bundle $V_{2}\to N$. In particular we
have obtained the fibration

$\rho:\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})\to V_{1}^{*}\oplus_{N}V_{2}^{*}$

and the set
$\rho^{-1}(0\oplus_{N}V_{2}^{*})\subset\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})$.
By definition it consists of the differentials of functions
$f:B\bar{\oplus}_{N}V_{2}\to\mathbb{R}$, which are constant in the direction
of $B$. The set $\rho^{-1}(0\oplus_{N}V_{2}^{*})$ is a coisotropic
submanifold, which means that we can perform the symplectic reduction

$\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})\supset\rho^{-1}(0\oplus_{N}V_{2}^{*})\to\mathsf{T}^{*}V_{2}.$

Let us apply the above construction to the case

$\displaystyle A$ $\displaystyle=$ $\displaystyle\mathsf{J}^{1}_{x}C,$
$\displaystyle B$ $\displaystyle=$ $\displaystyle(\mathsf{J}^{2}P/G)_{x},$
$\displaystyle V_{1}$ $\displaystyle=$ $\displaystyle
C_{x}\times(\vee^{2}\mathsf{T}_{x}^{*}M\otimes_{M}\mathrm{ad}_{x}P),$
$\displaystyle V_{2}$ $\displaystyle=$ $\displaystyle
C_{x}\times\mathcal{F}_{x},$ $\displaystyle N$ $\displaystyle=$ $\displaystyle
C_{x}.$

We introduce the notation

$\mathsf{V}^{*}\mathsf{J}^{1}C\supset\mathcal{K}:=\rho^{-1}(0\oplus_{C}C\times_{M}\mathcal{F}^{*}).$
(53)

Using isomorphisms
$\mathsf{T}^{*}\mathsf{J}^{1}_{x}C\simeq\mathsf{V}_{x}^{*}\mathsf{J}^{1}C$ and
$\mathsf{T}^{*}(C_{x}\times\mathcal{F}_{x})\simeq\mathsf{V}_{x}^{*}(C\times_{M}\mathcal{F})$
we can perform the symplectic reduction

$\mathsf{V}_{x}^{*}\mathsf{J}^{1}C\supset\mathcal{K}_{x}\to\mathsf{V}_{x}^{*}(C\times\mathcal{F}).$

If we take the above map point by point in $M$ we will obtain a reduction

$\mathsf{V}^{*}\mathsf{J}^{1}C\supset\mathcal{K}\to\mathsf{V}^{*}(C\times_{M}\mathcal{F})\simeq
C\times_{M}\mathcal{F}\times_{M}\mathcal{V}^{*}\times_{M}\mathcal{F}^{*}$ (54)

over $C\times_{M}\mathcal{F}$. In the following, we will be rather interested
in the bundle
$\mathsf{V}^{*}\mathsf{J}^{1}C\otimes_{\mathsf{J}^{1}C}\Omega^{m}$ than
$\mathsf{V}^{*}\mathsf{J}^{1}C$ itself. The suitable reduction for
$\mathsf{V}^{*}\mathsf{J}^{1}C\otimes_{\mathsf{J}^{1}C}\Omega^{m}$ reads

$\mathsf{V}^{*}\mathsf{J}^{1}C\otimes_{\mathsf{J}^{1}C}\Omega^{m}\supset\mathcal{K}\to
C\times_{M}\mathcal{F}\times_{M}\mathcal{V}^{*}\otimes_{M}\Omega^{m}\times_{M}\mathcal{F}^{*}\otimes_{M}\Omega^{m}\simeq$
(55)
$\simeq(C\times_{M}\mathcal{F})\times_{M}(\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P)\times_{M}\overline{\mathcal{P}}.$

###  6.3 The Tulczyjew map for gauge theories

According to (22) the Tulczyjew map for gauge theories reads

$\alpha:\mathsf{J}^{1}\mathcal{P}\to\mathsf{V}^{*}\mathsf{J}^{1}C\otimes\Omega^{m},$
$(q^{i},A\indices{{}^{a}_{i}},X\indices{{}_{a}^{i}{}^{k}},A\indices{{}^{a}_{i}{}_{j}},X\indices{{}_{a}^{i}{}^{k}_{s}})\longmapsto\Big{(}q^{i},A\indices{{}^{a}_{i}},A\indices{{}^{b}_{i}{}_{j}},\sum_{k}X\indices{{}_{a}^{i}{}^{k}_{k}},X\indices{{}_{a}^{i}{}^{k}}\Big{)}.$

Using the decomposition of $\mathsf{J}^{1}\mathcal{P}$ described in (43) and
the isomorphism
$\mathsf{V}^{*}\mathsf{J}^{1}C\simeq\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}{{}^{*}\mathcal{V}}$,
where $\mathsf{J}^{1}{{}^{*}\mathcal{V}}\to M$ is a dual bundle of
$\mathsf{J}^{1}\mathcal{V}\to M$, we can write the Tulczyjew map in the form

$\alpha:\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}\bar{\mathcal{P}}\times_{M}\mathsf{J}^{1}\mathcal{S}\to\mathsf{J}^{1}C\times_{M}\mathsf{J}^{*}\mathcal{V}\otimes_{M}\Omega^{m}.$
(56)

From (56) we can see that the structure of the Tulczyjew map is encoded in two
maps

$\alpha_{1}:\mathsf{J}^{1}C\to\mathsf{J}^{1}C,$

being an identity, and

$\alpha_{2}:\mathsf{J}^{1}\bar{\mathcal{P}}\times_{M}\mathsf{J}^{1}\mathcal{S}\to\mathsf{J}^{*}\mathcal{V}\otimes_{M}\Omega^{m}.$

We will derive now the reduced Tulczyjew map, which forms the basis of the
Lagrangian description of gauge theories, and which is one of the main results
of this paper. In subsection 3.2 we have defined $\alpha$ as a map, which is
dual to $\kappa$ with respect to the evaluation (21). The construction of the
Tulczyjew map for field theory is therefore based on the evaluation

$\mathsf{J}^{1}\mathcal{P}\times_{\mathsf{J}^{1}C}\mathsf{J}^{1}\mathsf{V}C\to\Omega^{m},\qquad\langle\mathsf{j}^{1}_{x}p,\mathsf{j}^{1}_{x}\delta\tau\rangle=\mathrm{d}\langle
p,\delta\tau\rangle(x),$ (57)

where $x$ is a point in $M$. Let us recall that the bundles appearing in the
above evaluation have the natural identifications

$\mathsf{J}^{1}\mathcal{P}=\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}\bar{\mathcal{P}}\times_{M}\mathsf{J}^{1}\mathcal{S},\qquad\mathsf{J}^{1}\mathsf{V}C\simeq\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}\mathcal{V}.$

Therefore, the evaluation (57) is in fact the evaluation between elements of
$\mathsf{J}^{1}\bar{\mathcal{P}}\times_{M}\mathsf{J}^{1}\mathcal{S}$ and
$\mathsf{J}^{1}\mathcal{V}$. Notice that in $\mathsf{J}^{1}\mathcal{P}$ there
exists a subbundle
$\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}\overline{\mathcal{P}}$, which can be
identified with

$\mathsf{J}^{1}C\times\mathsf{J}^{1}\overline{\mathcal{P}}\simeq\mathsf{J}^{1}\mathcal{P}/\mathsf{J}^{1}\mathcal{S}.$

Furthermore, we can project $\mathsf{J}^{1}C$ onto the second factor in (39)
and obtain the reduction of the phase bundle

$\mathsf{J}^{1}\mathcal{P}\to
C\times_{M}\mathcal{F}\times_{M}\mathsf{J}^{1}\overline{\mathcal{P}}.$

On the other hand, in
$\mathsf{J}^{1}C\times_{M}\mathsf{J}^{*}\mathcal{V}\otimes\Omega^{m}$ there
exists a subbundle $\mathcal{K}$ (53), which can be subject to the symplectic
reduction (54). The idea of the reduction of the Tulczyjew map is represented
by the diagram

$\textstyle{\mathsf{J}^{1}C\times_{M}\mathsf{J}^{1}\overline{\mathcal{P}}\times_{M}\mathsf{J}^{1}\mathcal{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathsf{J}^{1}C\times_{M}\mathsf{J}^{*}\mathcal{V}\otimes\Omega^{m}}$$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(C\times_{M}\mathcal{F})\times_{M}\mathsf{J}^{1}\overline{\mathcal{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\alpha}\qquad\qquad}$$\textstyle{(C\times_{M}\mathcal{F})\times_{M}(\Omega^{m-1}\otimes\mathrm{ad}^{*}P)\times_{M}\overline{\mathcal{P}}}$

The left arrow represents the division of the phase bundle by
$\mathsf{J}^{1}\mathcal{S}$ composed with the projection $\mathsf{J}^{1}C\to
C\times_{M}\mathcal{F}$. The right-hand side of the diagram represents the
symplectic reduction of the submanifold $\mathcal{K}$ composed with the
projection $\mathsf{J}^{1}C\to C\times_{M}\mathcal{F}$.

Let us move to the calculations. We start with elements

$(\mathsf{j}^{1}_{x}\omega,\mathsf{j}^{1}_{x}p)\in\mathsf{J}^{1}_{x}C\times\mathsf{J}^{1}_{x}\overline{\mathcal{P}},\qquad(\mathsf{j}^{1}_{x}\omega,\mathsf{j}^{1}_{x}\tau)\in\mathsf{J}^{1}_{x}C\times\mathsf{J}^{1}_{x}\mathcal{V}.$

In the above identifications $\mathsf{j}^{1}_{x}\omega$ is a point on the base
so by means of (57) we can write the evaluation between $\mathsf{j}^{1}_{x}p$
and $\mathsf{j}^{1}_{x}\tau$

$\langle\mathsf{j}^{1}_{x}p,\mathsf{j}^{1}_{x}\tau\rangle=\mathrm{d}\langle
p,\tau\rangle(x).$

From now on we will abuse our notation and we will denote by $p$ a section of
the bundle $\overline{\mathcal{P}}$ instead of $\mathcal{P}$. Let us introduce
a basis $(e_{i})$ in $\mathrm{ad}_{x}P$ and the dual basis $(e_{*}^{i})$ in
$\mathrm{ad}_{x}^{*}P$. We have

$p=p\indices{{}_{a}}\otimes e_{*}^{a},\quad p_{a}\in\Omega^{m-2},$
$\tau=\tau^{a}\otimes e_{a},\quad\tau^{a}\in\Omega^{1}.$

The evaluation between those elements reads

$\mathrm{d}\langle
p,\tau\rangle(x)=\mathrm{d}(p\indices{{}_{a}}\wedge\tau^{a})=\mathrm{d}p\indices{{}_{a}}\wedge\tau^{a}+(-1)^{m-2}p\indices{{}_{a}}\wedge\mathrm{d}\tau^{a}=\langle\mathrm{d}p,\tau\rangle+(-1)^{m}\langle
p,\mathrm{d}\tau\rangle.$

Since $(\mathsf{j}^{1}_{x}\omega,\mathsf{j}^{1}_{x}p)$ contains information
about the value $\omega(x)$, we can decompose above expression in a following
way

$\langle\mathrm{d}p,\tau\rangle+(-1)^{m}\langle
p,\mathrm{d}\tau\rangle=\langle\mathrm{d}p,\tau\rangle+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(-1)^{m}\langle
p,[\omega\wedge\tau]\rangle}+(-1)^{m}\langle
p,\mathrm{d}\tau\rangle-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(-1)^{m}\langle
p,[\omega\wedge\tau]\rangle}.$

Furthermore, we have

$\langle p,\mathrm{d}\tau\rangle+\langle p,[\omega\wedge\tau]\rangle=\langle
p,\mathrm{d}\tau+[\omega\wedge\tau]\rangle=\langle p,D_{\omega}\tau\rangle,$
$\langle\mathrm{d}p,\tau\rangle-(-1)^{m}\langle
p,[\omega\wedge\tau]\rangle=\langle\mathrm{d}p,\tau\rangle-\langle\mathrm{ad}^{*}(\omega)\wedge
p,\tau\rangle=\langle D^{\\#}_{\omega}p,\tau\rangle,$

where in the last equality we have used the identity

$\langle p,[\omega\wedge\tau]\rangle=\langle
p,\mathrm{ad}(\omega)\wedge\tau\rangle=\langle p_{c}\otimes
e_{*}^{c},\omega^{a}\wedge\tau^{b}\otimes\mathrm{ad}(e_{a})(e_{b})\rangle=p_{c}\wedge\omega^{a}\wedge\tau^{b}\otimes\langle
e_{*}^{c},\mathrm{ad}(e_{a})(e_{b})\rangle=$
$=p_{c}\wedge\omega^{a}\wedge\tau^{b}\otimes\langle\mathrm{ad}^{*}(e_{a})e_{*}^{c},(e_{b})\rangle=(-1)^{m-2}\omega^{a}\wedge
p_{c}\wedge\tau^{b}\otimes\langle\mathrm{ad}^{*}(e_{a})e_{*}^{c},(e_{b})\rangle=$
$=(-1)^{m}\langle\omega^{a}\wedge
p_{c}\otimes\mathrm{ad}^{*}(e_{a})e_{*}^{c},\tau^{b}\otimes
e_{b}\rangle=(-1)^{m}\langle\mathrm{ad}^{*}(\omega)\wedge p,\tau\rangle.$

In the above formula $\mathrm{ad}(\omega)\wedge\tau$ is the particular case of
the expression (14) for $\rho$ being the adjoint representation. Notice that
$\omega$ and $\tau$ can be understood as $\mathfrak{g}$-valued forms on $P$,
so that the expression $\mathrm{ad}(\omega)\wedge\tau$ in this identification
is the exterior product of the $\mathfrak{g}$-valued forms on $P$ given by 15.
Finally, we obtain

$\langle\mathsf{j}^{1}_{x}p,\mathsf{j}^{1}_{x}\tau\rangle=\mathrm{d}\langle
p,\tau\rangle(x)=\langle D^{\\#}_{\omega}p,\tau\rangle+(-1)^{m}\langle
p,D_{\omega}\tau\rangle.$ (58)

Notice that the above formula does not depend on the entire
$\mathsf{j}^{1}_{x}\tau$ but only on its projection onto
$\mathcal{V}\times_{M}\mathcal{F}$. We interpret (58) as the evaluation of the
pair $(D^{\\#}_{\omega}p,(-1)^{m}p)$ with $(\tau,D_{\omega}\tau)$. Combined
with the fact that the map $\overline{\kappa}$ has form 52 we obtain that the
reduced $\alpha$ reads

$\overline{\alpha}:(C\times_{M}\mathcal{F})\times_{M}\mathsf{J}^{1}\overline{\mathcal{P}}\to(C\times_{M}\mathcal{F})\times(\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P)\times_{M}\overline{\mathcal{P}},$
$\Big{(}\omega,F_{\omega},\mathsf{j}^{1}_{x}p\Big{)}\longmapsto\Big{(}\omega,F_{\omega},D^{\\#}_{\omega}p,(-1)^{m}p\Big{)}.$
(59)

In the above formula we have used isomorphism

$\mathsf{V}^{*}(C\times_{M}\mathcal{F})\otimes_{M}\Omega^{m}\simeq(C\times_{M}\mathcal{F})\times(\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P)\times_{M}\overline{\mathcal{P}}.$

##  7 Hamiltonian description

In this section we will subject the Hamiltonian formalism to a similar
reduction as the one performed in section 6. The Hamiltonian side of gauge
theories is strongly related to the dual bundle of $A=B\bar{\oplus}_{N}V_{2}$.
Therefore, it is an important question in this context what is the picture of
the vector-affine structure on the dual side. Before we can analyse
Hamiltonian formalism we have to consider this issue in detail.

Since the constructions presented here are quite complicated from the
technical point of view, for the clarity of presentation we will consider
first some preparatory constructions concerning general vector-affine products
and then we will apply them to the bundle $\mathsf{J}^{1}C$. More precisely,
in section 7.1 we will consider the dual side of $A=B\bar{\oplus}_{N}V_{2}$
and in 7.2 we will apply these results to $\mathsf{J}^{1}C$ and reduce the
Hamiltonian formalism.

###  7.1 The dual side of vector-affine product

In section (4.1) we were considering an affine bundle $A\to N$ modelled on a
vector bundle $V\to N$, which is the direct sum of bundles $V_{1}\to N$ and
$V_{2}\to N$. Furthermore, we assumed that there exists in $A$ an affine
subbundle $B\to N$ modelled on $V_{1}\to N$. Then, we have the decomposition
26, namely

$A=B\bar{\oplus}_{N}V_{2}.$ (60)

Let us consider affine maps on $A\to N$ having the structure (60). We denote
by $U\to N$ a one-dimensional vector bundle over $N$ and by
$\mathsf{Aff}(A,U)$ the set of all affine maps $T:A\to U$. Recall that
$\mathsf{Aff}(A,U)$ is a vector bundle over $N$ and an affine bundle over
$V^{*}\otimes_{N}U$ with the model bundle $V^{*}\otimes_{N}U\times U\to
V^{*}\otimes_{N}U$ [11]. Let

$T:A\to U$

be an affine map and

$\bar{T}:V\to U$

the linear part of $T$. The bundle $V$ is the direct sum of the bundles
$V_{1}$ and $V_{2}$, which means that $\bar{T}$ is the direct sum of two
linear maps

$\bar{T}=\bar{T}_{1}+\bar{T}_{2},\quad\bar{T}_{1}:=\bar{T}|_{V_{1}},\quad\bar{T}_{2}:=\bar{T}|_{V_{2}}.$

On the other hand, using the decomposition (60) for $a=(b,w_{2})$ we obtain

$T(a)=T(b+w_{2})=T(b)+\bar{T}(w_{2}).$

It turns out that each affine map can be decomposed on two maps. The first one
is an affine map

$T_{1}:B\to U,\quad T_{1}:=T|_{B},$

with the linear part $\bar{T}_{1}$. The second one is a linear map

$\bar{T}_{2}:V_{2}\to U.$

From the above relations we obtain the following decomposition of the bundle
$\mathsf{Aff}(A,U)\to N$

$\mathsf{Aff}(A,U)=\mathsf{Aff}(B,U)\oplus_{N}(V_{2}^{*}\otimes_{N}U).$ (61)

The bundle $\mathsf{Aff}(B,U)\to V_{1}^{*}\otimes_{N}U$ is an affine bundle
modelled on the trivial bundle $(V_{1}^{*}\otimes_{N}U)\times U\to
V_{1}^{*}\otimes_{N}U$. The addition of the elements from $\mathsf{Aff}(A,U)$
and the elements of the model bundle in the decomposition (61) reads

$\qquad\qquad\qquad\qquad T+u=(T_{1}+u,\bar{T}_{2}),\qquad
T\in\mathsf{Aff}(A,U),\quad u\in(V_{1}^{*}\otimes_{N}U)\times U.$ (62)

Our goal now will be the reduction of the map $R_{A}$ presented in [11] with
respect to (61). Recall that the map $R_{A}$ reads

$R_{A}:\mathsf{T}^{*}A\otimes_{A}U\to\mathsf{P}\mathsf{Aff}(A,U),\quad(x^{i},f^{\alpha},\sigma_{i},\psi_{\alpha})\to(x^{i},\psi_{\alpha},\sigma_{i},-f^{\alpha})$
(63)

and it comes from the map $\widetilde{R}_{A}$

$\widetilde{R}_{A}:\mathsf{T}^{*}\mathsf{Aff}(A,U)\otimes_{\mathsf{Aff}(A,U)}U\to\mathsf{T}^{*}A\otimes_{A}U,\quad(x^{i},\varphi_{\alpha},r,\sigma_{i},f^{\alpha},-1)\to(x^{i},-f^{\alpha},\sigma_{i},\varphi_{\alpha}),$
(64)

which can be constructed by means of the evaluation

$A\times_{N}\mathsf{Aff}(A,U)\to U,$ $(a,T)\longmapsto T(a).$

Notice that for the bundle $V_{2}\to N$ there exists a canonical isomorphism
[18, 24]

$R_{V_{2}}:\mathsf{T}^{*}(V_{2}^{*}\otimes_{N}U)\otimes_{(V_{2}^{*}\otimes_{N}U)}U\to\mathsf{T}^{*}V_{2}\otimes_{V_{2}}U,$

which comes from the evaluation

$V_{2}\times_{N}V_{2}^{*}\otimes_{N}U\to U.$

We will show now that the reduction of $\widetilde{R}_{A}$ with respect to
(61) naturally leads to $R_{V_{2}}$. From the definition of
$\widetilde{R}_{A}$ we have

$\widetilde{R}_{A}:\mathsf{T}^{*}(\mathsf{Aff}(B,U)\oplus_{N}(V_{2}^{*}\otimes_{N}U))\otimes_{\mathsf{Aff}(A,U)}U\to\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})\otimes_{A}U.$
(65)

Let us introduce coordinates $(x^{i},b^{a})$ in $B$ and $(x^{i},v^{\alpha})$
in $V_{2}$. Then, we have induced coordinates $(x^{i},b^{a},v^{\alpha})$ in
$B\bar{\oplus}_{N}V_{2}$.

In $\mathsf{T}(B\bar{\oplus}_{N}V_{2})$ we have the subbundle of vectors
tangent to the fibers of the bundle $B\to N$. We will denote this subbundle by
$\Delta B$ and in coordinates it reads
$(x^{i},b^{a},v^{\alpha},0,\dot{b}^{a},0)$. Consider now the annihilator of
$\Delta B$ in $\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})$, which will be denoted
by $\Delta B^{o}$. In coordinates $\Delta B^{o}$ has a form
$(x^{i},b^{a},v^{\alpha},\sigma_{i},0,\phi_{j}^{V})$. The annihilator $\Delta
B^{o}$ by definition consists of those covectors in
$\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})$, which come from functions constant
along the fibers of $B$. These covectors may be identified with elements of
$\mathsf{T}^{*}V_{2}$, which means that there exists a natural projection

$\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})\supset\Delta
B^{o}\to\mathsf{T}^{*}V_{2},\qquad(x^{i},b^{a},v^{\alpha},\sigma_{i},0,\phi_{j}^{V})\longmapsto(x^{i},v^{\alpha},\sigma_{i},\phi_{j}^{V}).$

A similar construction may be applied to $\mathsf{T}\mathsf{Aff}(A,U)$. From
61 we have

$\mathsf{T}\mathsf{Aff}(A,U)=\mathsf{T}(\mathsf{Aff}(B,U)\oplus_{N}(V_{2}^{*}\otimes_{N}U)).$

Let us introduce coordinates $(x^{i},\phi_{j}^{B},r)$ in $\mathsf{Aff}(B,U)$,
$(x^{i},\phi_{j}^{V})$ in $V^{*}_{2}\otimes_{N}U$ and
$(x^{i},\phi_{j}^{B},\phi_{k}^{V},r)$ in
$\mathsf{Aff}(B,U)\bar{\oplus}_{N}V_{2}$. Now we can proceed similarly as in
the case of $\mathsf{T}(B\bar{\oplus}_{N}V_{2})$. In
$\mathsf{T}\mathsf{Aff}(A,U)$ we have the subbundle of vectors tangent to the
fibers of the bundle $\mathsf{Aff}(B,U)\to N$, which we denote by
$\Delta\mathsf{Aff}(B,U)$, and its annihilator in
$\mathsf{T}^{*}\mathsf{Aff}(A,U)$ denoted by $\Delta\mathsf{Aff}(B,U)^{o}$.
The subbundle $\Delta\mathsf{Aff}(B,U)^{o}$ can be projected onto
$\mathsf{T}^{*}(V_{2}\otimes_{N}U)$ by means of the map

$\mathsf{T}^{*}(\mathsf{Aff}(B,U)\oplus_{N}(V_{2}^{*}\otimes_{N}U))\supset\Delta\mathsf{Aff}(B,U)^{o}\to\mathsf{T}^{*}(V^{*}_{2}\otimes_{N}U),$
$(x^{i},\phi_{j}^{B},\phi_{k}^{V},r,\sigma_{i},0,v^{\alpha},0)\longmapsto(x^{i},\phi_{k}^{V},\sigma_{i},v^{\alpha}).$

Using both projections we can perform the reduction of the map $R_{A}$, which
is presented on the diagram

$\textstyle{\mathsf{T}^{*}(\mathsf{Aff}(B,U)\oplus_{N}(V_{2}^{*}\otimes_{N}U))\otimes_{\mathsf{Aff}(A,U)}U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad\qquad\quad\widetilde{R}_{A}}$$\textstyle{\mathsf{T}^{*}(B\bar{\oplus}_{N}V_{2})\otimes_{A}U}$$\textstyle{\Delta\mathsf{Aff}(B,U)^{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta
B^{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{T}^{*}(V_{2}^{*}\otimes_{N}U)\otimes_{(V_{2}^{*}\otimes_{N}U)}U}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{T}^{*}V_{2}\otimes_{V_{2}}U}$$\scriptstyle{\qquad\quad
R_{V_{2}}}$ (66)

From the above diagram we obtain the reduced version of the map $R_{A}$

$R_{V_{2}}:\mathsf{T}^{*}V_{2}\otimes_{V_{2}}U\to\mathsf{T}^{*}(V_{2}^{*}\otimes_{N}U)\otimes_{(V_{2}^{*}\otimes_{N}U)}U,\quad(x^{i},v^{\alpha},\sigma_{i},\phi_{j}^{V})\to(x^{i},\phi_{j}^{V},\sigma_{i},-v^{\alpha}).$
(67)

Recall that according to [11] the Hamiltonian in classical field theory is
represented by the family of functions

$H:A\times_{N}\mathsf{Aff}(A,U)\to U$ (68)

parametrised by points in $A$. As a result of the reduction described above
the bundle $\mathsf{Aff}(A,U)$ has been replaced by the bundle
$V^{*}_{2}\otimes_{N}U$, which is a vector bundle. It implies that the derived
Hamiltonian side, in a sharp contrast to the general case, is rather linear in
its nature than an affine like. In the reduced description the Hamiltonian 68
reduces to the family of functions

$H:V_{2}\times_{N}V^{*}_{2}\otimes_{N}U\to U,$ (69)

parametrised by points in $V_{2}$.

###  7.2 Reduced Hamiltonian formalism

Recall that according to (23) the map $R_{\mathsf{J}^{1}C}$ reads

$R_{\mathsf{J}^{1}C}:\mathsf{V}^{*}\mathsf{J}^{1}C\otimes_{M}\Omega^{m}\to\mathsf{P}\mathsf{J}^{\dagger}C.$

We will use now the constructions described in the previous subsection to
reduce the above map and, subsequently, to reduce the entire Hamiltonian
description of gauge fields. We will be interested in the case

$\displaystyle A$ $\displaystyle=$ $\displaystyle\mathsf{J}^{1}_{x}C,$
$\displaystyle B$ $\displaystyle=$ $\displaystyle(\mathsf{J}^{2}P/G)_{x},$
$\displaystyle V_{1}$ $\displaystyle=$ $\displaystyle
C_{x}\times(\vee^{2}\mathsf{T}_{x}^{*}M\otimes\mathrm{ad}_{x}P),$
$\displaystyle V_{2}$ $\displaystyle=$ $\displaystyle
C_{x}\times\mathcal{F}_{x},$ $\displaystyle N$ $\displaystyle=$ $\displaystyle
C_{x},$ $\displaystyle U$ $\displaystyle=$ $\displaystyle\Omega^{m}_{x}.$

From the above we obtain immediately

$\displaystyle\mathsf{Aff}(A,U)$ $\displaystyle=$
$\displaystyle\mathsf{J}^{\dagger}_{x}C,$ $\displaystyle\mathsf{Aff}(B,U)$
$\displaystyle=$ $\displaystyle(\mathsf{J}{{}^{2}}P/G)^{\dagger}_{x},$
$\displaystyle V_{1}^{*}\otimes_{N}U$ $\displaystyle=$ $\displaystyle
C_{x}\times(\vee^{2}\mathsf{T}_{x}M\otimes\mathrm{ad}^{*}_{x}P)\otimes\Omega_{x}^{m},$
$\displaystyle V_{2}^{*}\otimes_{N}U$ $\displaystyle=$ $\displaystyle
C_{x}\times\mathcal{F}^{*}_{x}\otimes\Omega_{x}^{m}\simeq
C_{x}\times\overline{\mathcal{P}}_{x},$

where by $(\mathsf{J}{{}^{2}}P/G)^{\dagger}_{x}$ we have denoted the set of
affine maps on $(\mathsf{J}{{}^{2}}P/G)_{x}$ with values in $\Omega^{m}$.

From (67) we have that the map $R_{V_{2}}$ reads

$R_{V_{2}}:\mathsf{T}^{*}(C_{x}\times\mathcal{F}_{x})\otimes\Omega_{x}\to\mathsf{T}^{*}(C_{x}\times\overline{\mathcal{P}}_{x})\otimes\Omega_{x}.$

Notice that we have isomorphisms
$\mathsf{T}^{*}(C_{x}\times\mathcal{F}_{x})\simeq\mathsf{V}^{*}_{x}(C\times\mathcal{F})$
and
$\mathsf{T}^{*}(C_{x}\times\overline{\mathcal{P}}_{x})\simeq\mathsf{V}^{*}_{x}(C\times\overline{\mathcal{P}})$.
Taking above identifications point by point in $M$ we obtain a map

$R_{V_{2}}:\mathsf{V}^{*}(C\times_{M}\mathcal{F})\otimes\Omega^{m}\to\mathsf{V}^{*}(C\times_{M}\overline{\mathcal{P}})\otimes_{M}\Omega^{m}.$

Notice that the relations

$\displaystyle\mathsf{V}^{*}C$ $\displaystyle=$ $\displaystyle
C\times_{M}\mathsf{T}M\otimes_{M}\mathrm{ad}^{*}P,$
$\displaystyle\mathsf{V}^{*}(C\times_{M}\mathcal{F})$ $\displaystyle=$
$\displaystyle
C\times_{M}\mathcal{F}\times_{M}\mathsf{T}M\otimes_{M}\mathrm{ad}^{*}P\times_{M}\mathcal{F}^{*},$
$\displaystyle\mathsf{V}(C\times_{M}\overline{\mathcal{P}})$ $\displaystyle=$
$\displaystyle
C\times_{M}\overline{\mathcal{P}}\times_{M}\mathsf{T}^{*}M\otimes_{M}\mathrm{ad}P\times_{M}\overline{\mathcal{P}}^{*}$

imply the following identifications for the domain

$\mathsf{V}^{*}\Big{(}C\times_{M}\mathcal{F}\Big{)}\otimes_{M}\Omega^{m}\simeq\Big{(}C\times_{M}\mathcal{F}\Big{)}\times_{M}\Big{(}\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P\Big{)}\times_{M}\overline{\mathcal{P}}$

and pre-domain

$\mathsf{V}^{*}\Big{(}C\times_{M}\overline{\mathcal{P}}\Big{)}\otimes\Omega^{m}\simeq\Big{(}C\times_{M}\overline{\mathcal{P}}\Big{)}\times_{M}\Big{(}\mathsf{T}M\otimes_{M}\mathrm{ad}^{*}P\otimes\Omega^{m}\Big{)}\times_{M}\Big{(}\mathfrak{X}^{m-2}\otimes_{M}\mathrm{ad}P\otimes\Omega^{m}\Big{)}\simeq$
$\simeq\Big{(}C\times_{M}\overline{\mathcal{P}}\Big{)}\times_{M}\Big{(}\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P\Big{)}\times_{M}\mathcal{F}$

of the map $R_{V_{2}}$. In the above formula we have denoted by
$\mathfrak{X}^{m-2}$ the space of $(m-2)$-tangent vectors to $M$.

From now on we will use the notation $\overline{R}:=R_{V_{2}}$. Let us
introduce coordinates $(x^{i},A\indices{{}^{a}_{j}})$ in $C$,
$(x^{i},F\indices{{}^{a}_{j}})$ in $\mathcal{F}$,
$(x^{i},\sigma\indices{{}^{a}_{j}})$ in $\Omega^{m-1}\otimes_{M}\mathrm{ad}P$
and $(x^{i},p\indices{{}^{a}_{j}})$ in $\overline{\mathcal{P}}$. Then, the map
$\overline{R}$ reads

$\overline{R}:\Big{(}C\times_{M}\mathcal{F}\Big{)}\times_{M}\Big{(}\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P\Big{)}\times_{M}\overline{\mathcal{P}}\to\Big{(}C\times_{M}\overline{\mathcal{P}}\Big{)}\times\Big{(}\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P\Big{)}\times\mathcal{F},$
$(x^{i},A\indices{{}^{a}_{j}},F\indices{{}^{b}_{k}},\sigma\indices{{}^{a}_{j}},p\indices{{}^{a}_{j}})\longmapsto(x^{i},A\indices{{}^{a}_{j}},p\indices{{}^{a}_{j}},\sigma\indices{{}^{a}_{j}},-F\indices{{}^{b}_{k}}).$

From the above formula we can recognise the geometric version of
$\overline{R}$, which reads

$\overline{R}:\Big{(}C\times_{M}\mathcal{F}\Big{)}\times_{M}\Big{(}\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P\Big{)}\times_{M}\overline{\mathcal{P}}\to\Big{(}C\times_{M}\overline{\mathcal{P}}\Big{)}\times\Big{(}\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P\Big{)}\times\mathcal{F},$
$(\omega,F,b,X)\longmapsto(\omega,X,b,-F).$ (70)

The reduced map $\beta$ will be denoted by $\overline{\beta}$ and we define it
as the composition of maps (59) and (70), i.e.
$\overline{\beta}=\overline{R}\circ\overline{\alpha}$. After some
straightforward calculations we obtain

$\overline{\beta}:(C\times_{M}\mathcal{F})\times\mathsf{J}^{1}\overline{\mathcal{P}}\to\Big{(}C\times_{M}\overline{\mathcal{P}}\Big{)}\times\Big{(}\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P\times\mathcal{F}\Big{)},$
(71)
$\Big{(}\omega,F,\mathsf{j}^{1}p\Big{)}\longmapsto\Big{(}\omega,(-1)^{m}p,D_{\omega}^{\\#}p,-F\Big{)}.$

Let us move to the description of the dynamics. From (69) we have that the
reduced Hamiltonian in gauge theories is the family of $\Omega^{m}$-valued
functions

$\bar{H}:C\times_{M}(\mathcal{F}\times_{M}\overline{\mathcal{P}})\to\Omega^{m},\quad\bar{H}(\omega,F,p)=\bar{L}(\omega,F)-\langle
F,p\rangle$ (72)

parametrised by elements in $\mathcal{F}$. In some cases this family can be
reduced to a single function

$\bar{H}:C\times_{M}\overline{\mathcal{P}}\to\Omega^{m}.$

The dynamics on the Hamiltonian side is given by the formula

$\mathcal{D}=\overline{\beta}^{-1}(\mathrm{d}\bar{H}(\overline{\mathcal{P}})).$

##  8 Tulczyjew triple for gauge theories

The reduced Lagrangian and Hamiltonian descriptions constitute together the
reduced Tulczyjew triple, which is presented on the diagram

$\textstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(C\times_{M}\mathcal{F})\times_{M}\mathsf{J}^{1}\overline{\mathcal{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad
pr_{C\times_{M}\mathcal{F}}}$$\scriptstyle{\overline{\alpha}\qquad\qquad}$$\textstyle{(C\times_{M}\mathcal{F})\times_{M}(\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P)\times_{M}\overline{\mathcal{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{C\times_{M}\mathcal{F}}\qquad}$$\textstyle{C\times_{M}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\mathsf{d}}L}}$
(73)

for the Lagrangian side and on the diagram

$\textstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Big{(}C\times_{M}\overline{\mathcal{P}}\Big{)}\times_{M}\Big{(}\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P\times_{M}\mathcal{F}\Big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{C\times_{M}\overline{\mathcal{P}}}}$$\textstyle{(C\times_{M}\mathcal{F})\times_{M}\mathsf{J}^{1}\overline{\mathcal{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad\qquad\overline{\beta}}$$\scriptstyle{pr_{C\times_{M}\overline{\mathcal{P}}}}$$\textstyle{C\times_{M}\overline{\mathcal{P}}}$
(74)

for the Hamiltonian side.

The reduced Tulczyjew triple allows to describe the dynamics of gauge
theories, with Lagrangian and Hamiltonian that does not depend on the entire
first jet of a gauge field but only on the value of the connection and
curvature in a given point. The dynamics is a subset in the space
$(C\times_{M}\mathcal{F})\times_{M}\mathsf{J}^{1}\overline{\mathcal{P}}$ and
its solutions are sections of the reduced phase bundle
$C\times_{M}\overline{\mathcal{P}}\to C$.

Notice that the process of the dynamics generation is in our description
extremely simple from the conceptual point of view. Let us stress that this
description is also independent on the regularity of the system, which is a
very important feature in physical applications of gauge field theories. We
recall that most of the physically interesting field theories belong to the
class of nonregular systems, including such important examples like
electrodynamics or Yang-Mills theories, which play the fundamental role in the
theory of elementary interactions. On the Lagrangian side the dynamics is
given by the formula

$D:=\overline{\alpha}^{-1}\circ\mathrm{d}^{\mathsf{v}}L(C\times_{M}\mathcal{F}).$

On the Hamiltonian side, when Hamiltonian reduces to a single function, the
dynamics reads

$\mathcal{D}=\overline{\beta}^{-1}(\mathrm{d}\bar{H}(\overline{\mathcal{P}})).$

The relation between Lagrangian and Hamiltonian of the system is given by
formula (72). From the diagram of the reduced Tulczyjew triple we can easily
derive the Legendre map for gauge theories. The vertical differential of the
Lagrangian is a map

$\mathrm{d}^{\mathsf{v}}L:C\times_{M}\mathcal{F}\to(C\times_{M}\mathcal{F})\times_{M}(\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P)\times_{M}\overline{\mathcal{P}}.$

Composing above map with the projection on $\overline{\mathcal{P}}$ in the
last element we obtain a map

$\lambda:\mathcal{F}\to\overline{\mathcal{P}},\qquad F\longmapsto
pr_{\overline{\mathcal{P}}}\circ\mathrm{d}^{\mathsf{v}}L(F),$

which we call the Legendre map for gauge field theories.

##  9 Example: Yang-Mills theory

We will present now the example of application of the reduced Tulczyjew
triple, which is the dynamics of Yang-Mills theory [43, 20]. Let us notice
that in the physical literature this class of theories is usually considered
on the flat Minkowski space. Our approach is more general, whereas it allows
to formulate Yang-Mills theory on any smooth manifold equipped with a metric
$g$. Lagrangian of a free Yang-Mills field reads

$\bar{L}:C\times_{M}\mathcal{F}\to\Omega^{m},\quad\bar{L}(\omega,F)=\frac{1}{2}K_{ab}F^{a}\wedge\star
F^{b},$

where $F=F^{a}\otimes
e_{a}=F\indices{{}^{a}_{i}{}_{j}}\mathrm{d}q^{i}\wedge\mathrm{d}q^{j}\otimes
e_{a}$, $K$ is a scalar product on $\mathrm{ad}P$ and $\star$ is the Hodge
star on $M$ coming from the metric $g$. We denote by $\Xi$ a projection on the
second factor in (39), i.e.

$\Xi:\mathsf{J}^{1}C\to C\times_{M}\mathcal{F}.$

By means of $\Xi$ we can construct the Lagrangian

$L:\mathsf{J}^{1}C\to\Omega^{m},\quad L=\bar{L}\circ\Xi.$

We start with calculating the vertical differentials

$\mathrm{d}^{\mathsf{v}}L:\mathsf{J}^{1}C\to\mathsf{V}^{*}\mathsf{J}^{1}C\otimes\Omega^{m},$
$\mathrm{d}^{\mathsf{v}}\bar{L}:C\times_{M}\mathcal{F}\to\mathsf{V}^{*}(C\times_{M}\mathcal{F})\otimes\Omega^{m}.$

Above bundles may be written in the form

$\mathsf{V}^{*}(C\times_{M}\mathcal{F})=(C\times_{M}\mathcal{F})\times_{M}\mathcal{V}^{*}\times_{M}\mathcal{F},$
$\mathsf{V}^{*}(C\times_{M}\mathcal{F})\otimes\Omega^{m}=(C\times_{M}\mathcal{F})\times_{M}(\Omega^{m-1}\otimes_{M}\mathrm{ad}^{*}P)\times_{M}(\Omega^{m-2}\otimes_{M}\mathrm{ad}^{*}P).$

Let

$\gamma:\mathbb{R}\to\mathsf{J}^{1}_{m}C,\qquad
t\longmapsto\mathsf{j}^{1}_{m}\omega+t\mathsf{j}^{1}_{m}\tau,\qquad\textrm{where}\qquad\mathsf{j}^{1}_{m}\omega\in\mathsf{J}^{1}_{m}C,\quad\mathsf{j}^{1}_{m}\tau\in\mathsf{J}^{1}_{m}\mathcal{V}$

be a curve representing a vertical tangent vector
$(\mathsf{j}^{1}_{m}\omega,\mathsf{j}^{1}_{m}\tau)$ from
$\mathsf{V}_{m}\mathsf{J}^{1}C\simeq\mathsf{J}^{1}_{m}C\times\mathsf{J}^{1}_{m}\mathcal{V}$.
Acting by $\mathrm{d}^{\mathsf{v}}L$ on that vector we obtain

$\mathrm{d}^{\mathsf{v}}L(\mathsf{j}^{1}_{m}\omega)(\mathsf{j}^{1}_{m}\tau)=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}L(\mathsf{j}^{1}_{m}\omega+t\mathsf{j}^{1}_{m}\tau)=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\bar{L}(\omega+t\tau,F_{\omega+t\tau})=$
$=\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}\Big{(}\frac{1}{2}K_{ab}F_{\omega+t\tau}^{a}\wedge\star
F_{\omega+t\tau}^{b}\Big{)}=\frac{1}{2}K_{ab}\Big{(}\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}F_{\omega+t\tau}^{a}\Big{)}\wedge\star
F_{\omega}^{b}+\frac{1}{2}K_{ab}F_{\omega}^{a}\wedge\star\Big{(}\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}F_{\omega+t\tau}^{b}\Big{)}$
$=K_{ab}\Big{(}\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}F_{\omega+t\tau}^{a}\Big{)}\wedge\star
F_{\omega}^{b}.$

The curve $F_{\omega+t\tau}$ reads

$F_{\omega+t\tau}=\mathrm{d}(\omega+t\tau)+\frac{1}{2}[(\omega+t\tau)\wedge(\omega+t\tau)]=F_{\omega}+td\tau+t[\omega\wedge\tau]+t^{2}\frac{1}{2}[\tau\wedge\tau]=$
$=F_{\omega}+tD_{\omega}\tau+t^{2}\frac{1}{2}[\tau\wedge\tau]$

and by taking its derivative in $t=0$ we obtain

$\frac{\mathsf{d}}{\mathsf{d}t}_{|t=0}F_{\omega+t\tau}=D_{\omega}\tau,$

where $D_{\omega}\tau=d\tau+[\omega\wedge\tau]$ is the covariant derivative of
the one-form $\tau$. Finally, we obtain

$\mathrm{d}^{\mathsf{v}}L(\mathsf{j}^{1}_{m}\omega)(\mathsf{j}^{1}_{m}\tau)=K_{ab}(D\tau)^{a}\wedge\star
F_{\omega}^{b}.$

Notice that

$K_{ab}(D\tau)^{a}\wedge\star F_{\omega}^{b}=\langle D\tau|\tilde{K}(\star
F_{\omega})\rangle,$

which implies

$\mathrm{d}^{\mathsf{v}}\bar{L}(\omega,F)=\Big{(}\omega,F,0,\star\tilde{K}(F)\Big{)}.$
(75)

Now we can apply the reduced Tulczyjew map. For simplicity let us assume that
$M$ represents a four-dimensional spacetime so that $\dim M=4$. Then, the
coefficient $(-1)^{m}$ in (59) disappears and we obtain

$\overline{\alpha}:\Big{(}\omega(m),F(m),\mathsf{j}^{1}_{m}p\Big{)}\longmapsto\Big{(}\omega(m),F(m),D^{\\#}_{\omega}p(m),p(m)\Big{)}.$

From (75) we obtain that the equations describing the dynamics
$\mathcal{D}=\overline{\alpha}^{-1}\mathrm{d}^{\mathsf{v}}\bar{L}(C\times_{M}\mathcal{F})$
read

$\displaystyle p$ $\displaystyle=$ $\displaystyle\star\tilde{K}(F),$ (76)
$\displaystyle D_{\omega}^{\\#}p$ $\displaystyle=$ $\displaystyle 0.$ (77)

Let us notice that the above relations imply an equation

$D_{\omega}^{\\#}\star\tilde{K}(F)=0,$

which is a generalised Yang-Mills equation. For $M=\mathbb{R}^{4}$ equipped
with the Minkowski metric we obtain the traditional Yang-Mills equation

$D_{\omega}\star F=0.$

##  10 Acknowledgments

We gratefully acknowledge prof. Katarzyna Grabowska for the fruitful
conversations and valuable comments, which definitely make this paper more
correct and easy to read.

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