††institutetext: Departamento de Física, Universidade Federal de São Carlos,
Caixa Postal 676, CEP 13565-905, São Carlos-SP, Brasil

# A generalized 4d Chern-Simons theory

David M<EMAIL_ADDRESS>
###### Abstract

A generalization of the 4d Chern-Simons theory action introduced by Costello
and Yamazaki is presented. We apply general arguments from symplectic geometry
concerning the Hamiltonian action of a symmetry group on the space of gauge
connections defined on a 4d manifold and construct an action functional that
is quadratic in the moment map associated to the group action. The
generalization relies on the use of contact 1-forms defined on non-trivial
circle bundles over Riemann surfaces and mimics closely the approach used by
Beasley and Witten to reformulate conventional 3d Chern-Simons theories on
Seifert manifolds. We also show that the path integral of the generalized
theory associated to integrable field theories of the PCM type, takes the
canonical form of a symplectic integral over a subspace of the space of gauge
connections, turning it a potential candidate for using the method of non-
Abelian localization. Alternatively, this new quadratic completion of the 4d
Chern-Simons theory can also be deduced in an intuitive way from manipulations
similar to those used in T-duality. Further details on how to recover the
original 4d Chern-Simons theory data, from the point of view of the
Hamiltonian formalism applied to the generalized theory, are included as well.

Keywords: Chern-Simon theories, non-Abelian localization, string sigma models,
integrable deformations.

##  1 Introduction

A relatively new approach to integrable lattice models and 2-dimensional
integrable field theories that, as a main tool, uses a gauge theory of the
Chern-Simons (CS) type, has attracted a great deal of attention due to its
potential in offering novel insights into the quantum integrable structure of
these systems and their general properties. Such a gauge theory, known as the
4-dimensional semi-holomorphic Chern-Simons theory, or 4d CS theory for short,
introduced in C1 ; C2  and studied in detail in CWY1 ; CWY2 ; CY  has, in the
last few years, triggered several new interesting results in the fields of
integrable systems and string theory.

The 4d CS theory under consideration is defined by an action functional of the
form

$S=ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\omega\wedge
CS\left(\mathbb{A}\right),$ (1.1)

where $c$ is a constant222Usually taken to be $1/2\pi$ or $1/4\pi$.,
$\mathbb{M}=\Sigma\times C$ is a 4-dimensional manifold constructed out of a
cylinder $\Sigma=\mathbb{R}\times S^{1}$ with Minkowskian signature and a
Riemann surface $C$, $\omega$ is a meromorphic 1-form defined on $C$ and
$CS(\mathbb{A})$ is the usual Chern-Simons 3-form for a connection
$\mathbb{A}$ on $\mathbb{M}$. The theory is topological along $\Sigma$ and
holomorphic along $C$, hence the name semi-holomorphic. For an introduction to
the theory and some of its properties, see the nice review Lacroix .

What is important for the narrative of this work, is the fact that if we
consider an atlas $\mathcal{T}$ covering the Riemann surface $C$, we can
interpret the 4-dimensional manifold $\mathbb{M}$ as the total space of a
trivial bundle over the base space $C$, with a typical fiber $\Sigma$. This
observation motivates us to relax such a condition in favor of a more general
one, namely, that the manifold $\mathbb{M}$ indeed looks like the product
$\Sigma\times C$ but only locally over any chart
$\mathcal{U}\subset\mathcal{T}$. This causes some changes into the structure
of the theory because, besides the 1-form $\omega$ defined on $\mathbb{M}$,
which is now seen as the pullback of a 1-form $\omega_{C}$ defined on $C$,
another interesting differential 1-form $\alpha$ (to be identified later on as
a contact 1-form) defined on $\mathbb{M}$ can also be introduced via the
pullback from $C$ to $\mathbb{M}$ of a certain symplectic 2-form $\sigma_{C}$
defined on $C$, if the fiber bundle is taken to be inherently non-trivial.
This allows to generalize the 4d CS theory action (1.1) into a new one
depending on the former dynamical field $\mathbb{A}$, two new non-dynamical
1-forms $\Omega$, $\kappa$ (to be specified below) constructed out of
$\alpha$, $\omega$ and some parameters.

The time direction in $\mathbb{M}$ is considered, from physical grounds, as
globally defined and oriented. This means we can restrict the manifold
$\mathbb{M}$ a bit more and consider instead a 4-dimensional space of the form
$\mathbb{M}=\mathbb{R}\times\text{M}$, hence the assumed non-triviality of the
fiber bundle now resides entirely on the manifold M. In this work we will
consider a Minkowskian cylinder $\Sigma$ and identify the time direction with
the $\mathbb{R}$ factor in the decomposition $\Sigma=\mathbb{R}\times S^{1}$,
thus the 3-dimensional manifold M now becomes the total space of a non-trivial
circle bundle over the base space $C$. Fortunately, the structure of circle
bundles of this sort was exploited heavily in the seminal paper NA loc CS ,
devoted to a new formulation of the conventional 3-dimensional Chern-Simons
theories on Seifert manifolds which are, roughly speaking, total spaces with a
circle bundle structure plus a certain technical condition over the action of
an Abelian group on the total space Orlik . This ultimately allowed to compute
the partition function of some CS theories on these manifolds by using the
method of non-Abelian localization of symplectic integrals Witten revisited .
See also Wilson NA loc , for the inclusion of Wilson loops within the
formalism.

It is then natural to ask if the partition function of the theory defined by
the action (1.1) is a candidate for using the non-Abelian localization method
as well, due to the structure of the 4-dimensional manifold
$\mathbb{M}=\mathbb{R}\times\text{M}$ that we are considering. After all, the
only difference between $\mathbb{M}$ and M is a trivial $\mathbb{R}$ factor.
To be more precise NA loc CS ; Witten revisited , if we want to show that this
is the case, one first need to put the action (1.1) in the quadratic form

$S=ic(\mu,\mu)$ (1.2)

and second, one has to show that the partition function of the theory is
canonical, in the sense that it can be written as a symplectic integral of the
form

$Z(\epsilon)\sim\mathop{\displaystyle\int}\nolimits_{X}\text{exp}\left[\hat{\underline{\Omega}}-\frac{1}{2\epsilon}(\mu,\mu)\right].$
(1.3)

Here, $X$ is a symplectic manifold with symplectic form
$\hat{\underline{\Omega}}$ constructed out of the space $\mathcal{A}$ of gauge
connections $\mathbb{A}$ defined on $\mathbb{M}$. We assume that a Lie group
$\mathcal{H}$ acts on $X$ in a Hamiltonian fashion, with moment map
$\mu:X\rightarrow\mathfrak{h}^{*}$, where $\mathfrak{h}^{*}$ is the dual of
the Lie algebra $\mathfrak{h}$ of $\mathcal{H}$. Also, $(\ast,\ast)$ is an
invariant quadratic form on $\mathfrak{h}$ which, by the duality induced by it
between $\mathfrak{h}$ and $\mathfrak{h}^{*}$, allows to define the action $S$
in terms of $\mu\in\mathfrak{h}$. The coupling constant of the theory is
$\epsilon$.

The purpose of this work is to initiate a study of the relation between the
original theory (1.1), the quadratic action (1.2) and the symplectic integral
(1.3). We will show below that such a relation exist but it is not a direct
one and involves instead a generalization of the action functional (1.1). To
do this, we follow the strategy used in NA loc CS  for computing the moment
map $\mu$ in the case of the conventional 3-dimensional CS theories and make
the necessary modifications in order to locate the 1-form $\omega$ and the
time direction $\mathbb{R}$ within the general construction. As a
complementary and more direct approach, we shall recover the quadratic action
(1.2) from a different perspective using known field-theoretic manipulations
that are usually employed to dualize a theory, e.g. like in T-duality. The
outcome is that the partition function of the theory takes the desired form
(1.3), where $X=\mathcal{A}/\mathcal{S}$ is a symplectic quotient space,

$\left(\mu,\mu\right)=\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge
CS\left(\mathbb{A}\right)-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(\Phi^{2}\right)+\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right)$
(1.4)

is a quadratic action functional333The reader interested in a fast deduction
of (1.4), may take a quick glance to section (5) first.  generalizing (1.1)
and

$\text{exp}\;\hat{\underline{\Omega}}=\mathcal{D}\mathbb{A}|_{X},\text{ \ \
}\hat{\underline{\Omega}}=-\frac{i}{2}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\hat{\delta}\mathbb{A}\wedge\hat{\delta}\mathbb{A}\right)$
(1.5)

is the path integral symplectic measure, where $\hat{\underline{\Omega}}$ is a
symplectic form defined on the quotient $X$. The Lie algebra valued function
$\Phi$ and the vector field $\mathcal{R}$ on $\mathbb{M}$ will be introduced
below. In this work, we will construct (1.3) primarily for integrable field
theories of the principal chiral model (PCM) type, as the main illustrative
example.

The paper is organized as follows. In section (2) we construct, from general
results in symplectic geometry, an action functional that is, by definition,
of the quadratic form (1.2). The construction is rather general and relies on
a variant of the symplectic form originally used in NA loc CS . In section
(3), we show how this new action functional is related to the original 4d CS
theory action (1.1). The quadratic action generalizing (1.1) depends on the
usual 1-form $\omega$, a contact 1-form $\alpha$ naturally linked to the non-
triviality of the circle bundle M over $C$, and a pair of parameters. We also
comment on a strategy devised to recover the theory (1.1), which is based on a
partial gauge fixing and involves a degenerate limit in one of the parameters.
In section (4), we specialize the general construction to a particular example
concerning one of the simplest known Seifert manifolds $\text{M}=S^{3}$, i.e.
the 3-sphere. This case covers integrable field theories of the principal
chiral model type, where the underlying circle bundle structure is that of the
Hopf fibration of $S^{3}$ over $C=\mathbb{CP}^{1}$. We study, respectively,
the Riemannian and Kähler metrics on the space of gauge connections
$\mathcal{A}$ and on an important quotient space $\overline{\mathcal{A}}$,
which is identified with the symplectic manifold $X$ in the path integral
expression (1.3). Then, we work out the Hamiltonian approach in order to
implement the necessary partial gauge fixing and degenerate limit mentioned in
section (3). After recovering the 4d CS theory for these theories, as an
example, we end up by re-deriving the Lax connection for the
$\lambda$-deformed PCM. In section (5), we deduce the generalized 4d CS theory
quadratic action (1.2) from simple duality manipulations. In section (6) we
address, at the formal level, the path integral measure and its relation to
the symplectic form $\hat{\underline{\Omega}}$ defined on
$\overline{\mathcal{A}}$, hence completing the construction of the canonical
integral expression (1.3). Finally, in section (7) we make some comments and
provide further explanations concerning the results presented along the text.

##  2 Moment map and the quadratic action

The goal of this relatively long section is to introduce an action functional
$S$, on a 4-dimensional manifold $\mathbb{M}$, that is proportional to the
square of a moment map $\mu$, associated to the Hamiltonian action of a
symmetry group $\mathcal{H}$ on a symplectic space $\overline{\mathcal{A}}$,
constructed out of the space $\mathcal{A}$ of Chern-Simons gauge connections
defined on $\mathbb{M}$. In order to do this, we use known facts from
symplectic geometry to correctly identify $\mathcal{H}$ and
$\overline{\mathcal{A}}$, to construct $\mu$ and study its symmetry properties
and to introduce the main result of this section, which is materialized in the
expression (2.81) (or equivalently (1.4)) defining the generalization of the
4d CS theory (1.1). We follow NA loc CS  closely and proceed formally.

Warning. We are not specifying reality conditions over some quantities. As a
consequence, some objects that are usually expected to be real by definition
could emerge as purely imaginary.

###  2.1 Moment map and its properties

Consider a principal $G$-bundle $P$, where $G$ is a Lie group with Lie algebra
$\mathfrak{g}$, over a 4-dimensional orientable manifold $\mathbb{M}$, such
that $\partial\mathbb{M}=0$. Denote by $\mathcal{A}$ the space of connections
$\mathbb{A}$ on P and identify it with the vector space
$\Omega_{\mathbb{M}}^{1}\otimes\mathfrak{g}$ of 1-forms on $\mathbb{M}$ taking
values in the Lie algebra $\mathfrak{g}$. Denote by $\mathcal{G}$, the group
of gauge transformations acting on $\mathcal{A}$. The Lie algebra of
$\mathcal{G}$, denoted by $\mathcal{G}_{\text{Lie}}$, consists of elements in
the vector space $\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$ of Lie algebra
valued functions on $\mathbb{M}$.

Introduce a pre-symplectic form $\hat{\Omega}\in\Omega_{\mathcal{A}}^{2}$ on
the space of connections $\mathcal{A}$, defined by

$\hat{\Omega}=-\frac{1}{2}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\hat{\delta}\mathbb{A}\wedge\hat{\delta}\mathbb{A}\right),$
(2.1)

where $\kappa,\Omega\in\Omega_{\mathbb{M}}^{1}$ are globally defined and
everywhere non-zero 1-forms on $\mathbb{M}$ that satisfy, respectively, the
following set of conditions:

$i_{\mathcal{R}}\kappa=1,\text{ \ \
}i_{\mathcal{R}}\left(d_{\mathbb{M}}\kappa\right)=0$ (2.2)

and

$i_{\mathcal{R}}\Omega=0,\text{ \ \ }\pounds_{\mathcal{R}}\Omega=0.$ (2.3)

The symbol $\hat{\delta}$ denotes the exterior derivative on $\mathcal{A}$,
$d_{\mathbb{M}}$ denotes the exterior derivative on $\mathbb{M}$,
$\mathcal{R}\in\mathfrak{X}_{\mathbb{M}}$ is a vector field on $\mathbb{M}$
defined by the first condition (normalization condition) in (2.2) once
$\kappa$ is specified, $i_{\mathcal{R}}$ is the interior product or
contraction with $\mathcal{R}$ and $\pounds_{\mathcal{R}}=d_{\mathbb{M}}\circ
i_{\mathcal{R}}+i_{\mathcal{R}}\circ d_{\mathbb{M}}$ is the Lie derivative
along $\mathcal{R}$. The symbol Tr, denotes an invariant quadratic form
defined on the Lie algebra $\mathfrak{g}$. As a consequence of (2.2) and
(2.3), we also have that

$\pounds_{\mathcal{R}}\kappa=0,\text{ \ \
}i_{\mathcal{R}}\left(d_{\mathbb{M}}\Omega\right)=0.$ (2.4)

The 2-form $\hat{\Omega}$ is closed and invariant under the action of
$\mathcal{G}$ and, in particular, under the action of the group
$\mathcal{S}=U(1)\times U(1)$ generated by the following set of independent
shifts

${}^{\kappa}\mathbb{A}=\mathbb{A}+s\kappa,\text{ \ \
}^{\Omega}\mathbb{A}=\mathbb{A}+s^{\prime}\Omega,$ (2.5)

where $s,s^{\prime}\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$ are
arbitrary and, as a consequence of this, the 2-form $\hat{\Omega}$ is
degenerate along elements $\mathbb{A}$ of the form $\mathbb{A}=s\kappa$,
$\mathbb{A}=s^{\prime}\Omega$. Thus, we take the quotient of $\mathcal{A}$ by
the action of the group $\mathcal{S}$ and define the symplectic space
$\overline{\mathcal{A}}=\mathcal{A}/\mathcal{S}$. Under the quotient, the pre-
symplectic form $\hat{\Omega}$ on $\mathcal{A}$ descends to a symplectic form
on $\overline{\mathcal{A}}$, which becomes a symplectic space naturally
associated to a generalization of the 4d CS theory. The action of the gauge
group $\mathcal{G}$ on $\mathcal{A}$ also descends, under the quotient, to a
well-defined action on $\overline{\mathcal{A}}$ and the 2-form $\hat{\Omega}$
on $\overline{\mathcal{A}}$ is invariant under the action of $\mathcal{G}$.
The action of $\mathcal{G}$ is non-linear, contrary to the linear action of
$\mathcal{S}$.

The gauge group $\mathcal{G}$ and the group $\mathcal{H}$ acting on
$\overline{\mathcal{A}}$ in a Hamiltonian way are related, but are not the
same. Thus, in order to identify $\mathcal{H}$ properly, we first consider the
action of $\mathcal{G}$ on $\overline{\mathcal{A}}$ and subsequently extend
the algebraic structure of $\mathcal{G}_{\text{Lie}}$, until the defining
condition of a Hamiltonian action on $\overline{\mathcal{A}}$ is fulfilled.
Because of we are interested in computing the square of the moment map $\mu$
associated to the action of $\mathcal{H}$ on $\overline{\mathcal{A}}$, the Lie
algebra of $\mathcal{H}$, denoted by $\mathfrak{h}$, must be endowed with a
well-defined invariant inner product $(*,*)$. If such an inner product exists,
the action functional we are seeking is defined to be proportional to the
square $(\mu,\mu)$. However, before we continue, let us review some facts from
symplectic geometry that are necessary for accomplishing this task.

Consider a Lie group $\mathcal{H}$ with Lie algebra $\mathfrak{h}$, a
symplectic manifold $X$ and assume that the action of $\mathcal{H}$ on $X$
preserves the symplectic form $\hat{\Omega}$ on $X$. The action
$\mathcal{H}\circlearrowright X$ of $\mathcal{H}$ on $X$ is said to be
Hamiltonian, when there exists an algebra homomorphism from $\mathfrak{h}$ to
the algebra of functions on $X$ under the Poisson bracket. In terms of the
moment map $\mu:X\rightarrow\mathfrak{h}^{*}$ and the elements
$\eta,\lambda\in\mathfrak{h}$, this statement is equivalent to the condition
that $\mu$ satisfy

$\left\\{\left\langle\mu,\eta\right\rangle,\left\langle\mu,\lambda\right\rangle\right\\}=\left\langle\mu,\left[\eta,\lambda\right]\right\rangle,$
(2.6)

where $\left\langle\ast,\ast\right\rangle$ is the dual pairing between
$\mathfrak{h}$ and $\mathfrak{h}^{*}$, $\\{*,*\\}$ is the Poisson bracket and
$[*,*]$ is the Lie bracket. The homomorphism from $\mathfrak{h}$ to the
algebra of functions on $X$ is given by $f=\left\langle\mu,\eta\right\rangle$
and the moment map $\mu$, by definition, satisfy the relation

$d\left\langle\mu,\eta\right\rangle=-i_{V(\eta)}\hat{\Omega},$ (2.7)

where $V(\eta)$ is the induced vector field on $X$ generated by the action of
$\eta$. The symbol $d$ denotes the exterior derivative on $X$. The equation
(2.6) also reflects, infinitesimally, the condition that the map $\mu$ commute
with the action of $\mathcal{H}$ on $X$ and the co-adjoint action of
$\mathcal{H}$ on $\mathfrak{h}^{*}$. The Poisson bracket of two functions on
$X$, is given by the expression

$\left\\{f_{1},f_{2}\right\\}=\hat{\delta}f_{2}(V_{f_{1}})=-\hat{\delta}f_{1}(V_{f_{2}}),$
(2.8)

where $V_{f_{1}},V_{f_{2}}\in\mathfrak{X}_{X}$ are the Hamiltonian vector
fields associated to the functions $f_{1},f_{2}$ on $X$, respectively. The
relation between $V_{f}$ and $f$ is of the form (2.7), i.e. we have that
$df=-i_{V_{f}}\hat{\Omega}$.

Let us return to the case of interest and construct $\mu$ in two stages. The
first one dealing with the action of the gauge group and the second one with
the action of the $\mathcal{R}$ vector field.

Consider the action of the gauge group $\mathcal{G}$ on the elements of
$\mathcal{A}$, given by the known relation

${}^{g}\mathbb{A}=g^{-1}\mathbb{A}g+g^{-1}d_{\mathbb{M}}g.$ (2.9)

Infinitesimally, we write $g=\exp\eta$ with
$\eta\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$. This action induces, on
the space $\mathcal{A}$, the following vector field

$V(0,\eta,0)=d_{\mathbb{A}}\eta,$ (2.10)

where $d_{\mathbb{A}}=d_{\mathbb{M}}+\left[\mathbb{A},\cdot\;\right]$. The
notation $(*,*,*)$ with three entries will become clearer as we proceed.

Now, we compute the moment map $\mu$ associated to the action of (2.10). Start
by calculating the contraction on the left hand side (lhs) of the defining
expression

$-i_{V(0,\eta,0)}\hat{\Omega}=\hat{\delta}\left\langle\mu,(0,\eta,0)\right\rangle.$
(2.11)

We find that

$-i_{V(0,\eta,0)}\hat{\Omega}=\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(d_{\mathbb{A}}\eta\wedge\hat{\delta}\mathbb{A}\right).$
(2.12)

After integrating by parts, we obtain

$\left\langle\mu,(0,\eta,0)\right\rangle=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\big{(}\eta
F_{\mathbb{A}}\big{)}-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\left(\Omega\wedge\kappa\right)\wedge\text{Tr}\Big{(}\eta(\mathbb{A}-\mathbb{A}_{0})\Bigl{)},$
(2.13)

where $F_{\mathbb{A}}=d_{\mathbb{M}}\mathbb{A}+\mathbb{A}\wedge\mathbb{A}$ is
the curvature of the connection $\mathbb{A}$ and $\mathbb{A}_{0}$ is a
constant connection with respect to $\hat{\delta}$. In what follows we take a
vanishing basepoint, i.e. $\mathbb{A}_{0}=0$.

Using the general identity

$F_{\mathbb{A}-\theta s}=F_{\mathbb{A}}+\theta\wedge
d_{\mathbb{A}}s-d_{\mathbb{M}}\theta s,$ (2.14)

where $\theta\in\Omega_{\mathbb{M}}^{1}$ and
$s\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$, it is not difficult to show
that (2.13) descends to a functional on $\overline{\mathcal{A}}$, as it is
invariant under both transformations generated by the shift group
$\mathcal{S}$.

In order to find the Poisson algebra for two functionals of the form (2.13),
we use the general result (2.8). We find that

$\big{\\{}\left\langle\mu,(0,\eta,0)\right\rangle,\left\langle\mu,(0,\lambda,0)\right\rangle\big{\\}}=\hat{\delta}\left\langle\mu,(0,\lambda,0)\right\rangle\left(V(0,\eta,0)\right)=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(d_{\mathbb{A}}\eta\wedge
d_{\mathbb{A}}\lambda\right).$ (2.15)

Using the identity

$\text{Tr}\left(d_{\mathbb{A}}\epsilon\wedge
d_{\mathbb{A}}\epsilon^{\prime}\right)=\text{Tr}\left(\left[\epsilon,\epsilon^{\prime}\right]F_{\mathbb{A}}\right)+d_{\mathbb{M}}\text{Tr}\left(\epsilon
d_{\mathbb{A}}\epsilon^{\prime}\right),$ (2.16)

for arbitrary
$\epsilon,\epsilon^{\prime}\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$, we
obtain

$\big{\\{}\left\langle\mu,(0,\eta,0)\right\rangle,\left\langle\mu,(0,\lambda,0)\right\rangle\big{\\}}=\left\langle\mu,(0,\left[\eta,\lambda\right],0)\right\rangle-c\left(\eta,\lambda\right),$
(2.17)

where we notice the presence of a Lie algebra cocycle defined by the integral

$c\left(\eta,\lambda\right)=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\left(\Omega\wedge\kappa\right)\wedge\text{Tr}\left(\eta
d_{\mathbb{M}}\lambda\right).$ (2.18)

If the cohomology class of this cocycle is not zero, something we assume from
now on, the action of the gauge group $\mathcal{G}$ on the symplectic space
$\overline{\mathcal{A}}$ is not Hamiltonian, as it violates the homomorphism
condition (2.6). The cocycle (2.18) then determines a central
extension444There are two contributions in (2.18) but, as we shall see, one of
them is required to vanish. $\mathbb{R}$ of the gauge algebra
$\mathcal{G}_{\text{Lie}}$ which, as a vector space, is given by
$\tilde{\mathcal{G}}_{\text{Lie}}=\mathcal{G}_{\text{Lie}}\oplus\mathbb{R}$
and comes equipped with the bracket

$\Big{[}(0,\eta,a),(0,\lambda,b)\Big{]}=\Big{(}0,\left[\eta,\lambda\right],c(\eta,\lambda)\Big{)},$
(2.19)

where $\eta,\lambda\in\mathcal{G}_{\text{Lie}}$ and $a,b\in\mathbb{R}$. The
pairing between an element $\mu\in\tilde{\mathcal{G}}_{\text{Lie}}^{*}$ and an
element $(0,\eta,a)\in\tilde{\mathcal{G}}_{\text{Lie}}$ is defined by

$\left\langle\mu,(0,\eta,a)\right\rangle=\left\langle\mu,(0,\eta,0)\right\rangle-a.$
(2.20)

We assume that the central subgroup of $\tilde{\mathcal{G}}$ acts trivially on
$\overline{\mathcal{A}}$, so that the moment map for the central generator
$(0,0,a)$ of $\tilde{\mathcal{G}}_{\text{Lie}}$ is constant. Then, by
construction, we realize that the new moment map for the action of
$\tilde{\mathcal{G}}$ on $\overline{\mathcal{A}}$, which is now given by

$\left\langle\mu,(0,\eta,a)\right\rangle=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\big{(}\eta
F_{\mathbb{A}}\big{)}-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\left(\Omega\wedge\kappa\right)\wedge\text{Tr}\big{(}\eta\mathbb{A}\big{)}-a,$
(2.21)

satisfies the Hamiltonian action condition, i.e.

$\big{\\{}\left\langle\mu,(0,\eta,a)\right\rangle,\left\langle\mu,(0,\lambda,b)\right\rangle\big{\\}}=\left\langle\mu,\left[(0,\eta,a),(0,\lambda,b)\right]\right\rangle.$
(2.22)

Once the moment map is properly identified, the next step is to introduce a
non-degenerate, invariant inner product $(*,*)$ on
$\tilde{\mathcal{G}}_{\text{Lie}}$ in order to dualize
$\tilde{\mathcal{G}}_{\text{Lie}}^{*}$ and subsequently compute the square
$(\mu,\mu)$. However, the Lie algebra $\tilde{\mathcal{G}}_{\text{Lie}}$ does
not provide the algebraic structure we are looking for and this is because we
still have not taken into account the induced action of the vector field
$\mathcal{R}\in\mathfrak{X}_{\mathbb{M}}$ on the spaces
$\mathcal{G}_{\text{Lie}}$ and $\mathcal{A}$.

In order to show how the vector field $\mathcal{R}$ acts on the gauge algebra,
let us write the cocycle (2.18) in a slightly different form. Take

$c\left(\eta,\lambda\right)=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\eta
d_{\mathbb{M}}\lambda\right)+\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge
d_{\mathbb{M}}\kappa\wedge\text{Tr}\left(\eta d_{\mathbb{M}}\lambda\right)$
(2.23)

and consider the contracted 5-form

$0=i_{\mathcal{R}}\Big{(}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\wedge\text{Tr}\left(\eta
d_{\mathbb{M}}\lambda\right)\Big{)}=-\Omega\wedge
d_{\mathbb{M}}\kappa\wedge\text{Tr}\left(\eta
d_{\mathbb{M}}\lambda\right)+\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\Big{(}\eta
i_{\mathcal{R}}\left(d_{\mathbb{M}}\lambda\right)\Big{)}.$ (2.24)

Using this result above gives

$c\left(\eta,\lambda\right)=\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(\eta\pounds_{\mathcal{R}}\lambda\right)+d\left(\eta,\lambda\right),$
(2.25)

where, for further reference, we have defined

$d\left(\eta,\lambda\right)=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\eta
d_{\mathbb{M}}\lambda\right).$ (2.26)

The first contribution in (2.25) is antisymmetric and exhibits a $U(1)$ action
generated by the vector field $\mathcal{R}$ on the Lie algebra elements of
$\mathcal{G}_{\text{Lie}}$. Hence, the rigid action of $\mathcal{R}$ on
$\mathbb{M}$ induces a natural $U(1)$ group action on the gauge algebra
$\mathcal{G}_{\text{Lie}}$, as well as on the space $\mathcal{A}$, something
we shall see right below. Notice that (2.26) is antisymmetric as well if
$d_{\mathbb{M}}\Omega\wedge d_{\mathbb{M}}\kappa=0$ holds.

Let us understand now, from the symplectic geometry point of view, what are
the implications of the action of the vector field $\mathcal{R}$ on the space
of gauge connections $\mathcal{A}$.

The action of $\mathcal{R}$ on $\mathbb{M}$, induces the following vector
field on $\mathcal{A}$, given by

$V(p,0,0)=p\pounds_{\mathcal{R}}\mathbb{A},$ (2.27)

where $p\in\mathbb{R}$. Notice that, because $\kappa,\Omega$ are invariant,
i.e. $\pounds_{\mathcal{R}}\kappa=\pounds_{\mathcal{R}}\Omega=0$, the $U(1)$
action of $\mathcal{R}$ on $\mathcal{A}$, also descends to a corresponding
action on the quotient space $\overline{\mathcal{A}}$.

As done before with the action of the gauge group, now we compute the moment
map $\mu$ associated to the action of the vector field (2.27). Compute then,
the contraction on the lhs of the defining relation

$-i_{V(p,0,0)}\hat{\Omega}=\hat{\delta}\left\langle\mu,(p,0,0)\right\rangle.$
(2.28)

Consider the expression

$-i_{V(p,0,0)}\hat{\Omega}=\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(p\pounds_{\mathcal{R}}\mathbb{A}\wedge\hat{\delta}\mathbb{A}\right)$
(2.29)

and notice the following result

$\pounds_{\mathcal{R}}\left(\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}\wedge\hat{\delta}\mathbb{A}\right)\right)=\Omega\wedge\kappa\wedge\text{Tr}\left(\pounds_{\mathcal{R}}\mathbb{A}\wedge\hat{\delta}\mathbb{A+A}\wedge\pounds_{\mathcal{R}}\left(\hat{\delta}\mathbb{A}\right)\right).$
(2.30)

Then, inside integrals, we can replace

$\Omega\wedge\kappa\wedge\text{Tr}\left(\pounds_{\mathcal{R}}\mathbb{A}\wedge\hat{\delta}\mathbb{A}\right)=\Omega\wedge\kappa\wedge\text{Tr}\left(\pounds_{\mathcal{R}}\left(\hat{\delta}\mathbb{A}\right)\wedge\mathbb{A}\right).$
(2.31)

Using this fact above, we obtain that

$\left\langle\mu,(p,0,0)\right\rangle=\frac{p}{2}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\pounds_{\mathcal{R}}\mathbb{A}\wedge\mathbb{A}\right).$
(2.32)

The latter expression is manifestly invariant under the action of
$\mathcal{S}$ and also descends to $\overline{\mathcal{A}}$.

The actions of $\mathcal{R}$ on $\mathcal{G}_{\text{Lie}}$ and $\mathcal{A}$
are intertwined. To see this, we compute the Poisson algebra for two
functionals of the form (2.13) and (2.32). Thus,

$\big{\\{}\left\langle\mu,(p,0,0)\right\rangle,\left\langle\mu,(0,\lambda,0)\right\rangle\big{\\}}=\hat{\delta}\left\langle\mu,(0,\lambda,0)\right\rangle\left(V(p,0,0)\right)=-p\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\pounds_{\mathcal{R}}\mathbb{A}\wedge
d_{\mathbb{A}}\lambda\right).$ (2.33)

To simplify this expression, consider the following result

$d_{\mathbb{M}}\Big{(}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}\pounds_{\mathcal{R}}\lambda\right)\Big{)}=d_{\mathbb{M}}\left(\Omega\wedge\kappa\right)\wedge\text{Tr}\left(\mathbb{A}\pounds_{\mathcal{R}}\lambda\right)+\Omega\wedge\kappa\wedge\text{Tr}\Big{(}d_{\mathbb{M}}\mathbb{A}\pounds_{\mathcal{R}}\lambda-\mathbb{A}\wedge
d_{\mathbb{M}}\left(\pounds_{\mathcal{R}}\lambda\right)\Big{)}.$ (2.34)

Then, inside the integral, we can write

$\Omega\wedge\kappa\wedge\text{Tr}\Big{(}\mathbb{A}\wedge
d_{\mathbb{M}}\left(\pounds_{\mathcal{R}}\lambda\right)\Big{)}=d_{\mathbb{M}}\left(\Omega\wedge\kappa\right)\wedge\text{Tr}\left(\mathbb{A}\pounds_{\mathcal{R}}\lambda\right)+\Omega\wedge\kappa\wedge\text{Tr}\left(d_{\mathbb{M}}\mathbb{A}\pounds_{\mathcal{R}}\lambda\right).$
(2.35)

After acting with $\pounds_{\mathcal{R}}$ on $d_{\mathbb{A}}\lambda$ and
simplifying, we find that

$\big{\\{}\left\langle\mu,(p,0,0)\right\rangle,\left\langle\mu,(0,\lambda,0)\right\rangle\big{\\}}=\left\langle\mu,\left(0,-p\pounds_{\mathcal{R}}\lambda,0\right)\right\rangle.$
(2.36)

The last result follows because $\partial\mathbb{M}=0$, imply that

$\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\Big{(}\mathbb{A}\wedge\mathbb{A}\pounds_{\mathcal{R}}\lambda-\mathbb{A\wedge}\pounds_{\mathcal{R}}\left[\mathbb{A},\lambda\right]\Big{)}=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\pounds_{\mathcal{R}}\Big{(}\Omega\wedge\kappa\wedge\text{Tr}\big{(}\mathbb{A\wedge
A\lambda}\big{)}\Big{)}=0.$ (2.37)

From these results, it turns out that the algebraic structure extending
(2.19), is given by the semi-direct product $U(1)\ltimes\tilde{\mathcal{G}}$
of the $U(1)$ group generated by the rigid action of $\mathcal{R}$ and the
central extension of the gauge group $\tilde{\mathcal{G}}$ . As a vector
space, the Lie algebra of $U(1)\ltimes\tilde{\mathcal{G}}$ is identified with
$\mathbb{R}\oplus\tilde{\mathcal{G}}_{\text{Lie}}=\mathbb{R}\oplus\mathcal{G}_{\text{Lie}}\oplus\mathbb{R}$
and comes equipped with the following bracket Loop

$\Big{[}\left(p,\eta,a\right),\left(q,\lambda,b\right)\Big{]}=\Big{(}0,\left[\eta,\lambda\right]-p\pounds_{\mathcal{R}}\lambda+q\pounds_{\mathcal{R}}\eta,c(\eta,\lambda)\Big{)}.$
(2.38)

Summarizing, the complete moment map associated to the action of the vector
field

$V(p,\eta,a)=d_{\mathbb{A}}\eta+p\pounds_{\mathcal{R}}\mathbb{A}$ (2.39)

on the quotient space $\overline{\mathcal{A}}$, takes the form

$\left\langle\mu,\left(p,\eta,a\right)\right\rangle=\frac{p}{2}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\pounds_{\mathcal{R}}\mathbb{A}\wedge\mathbb{A}\right)-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\eta
F_{\mathbb{A}}\right)-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\left(\Omega\wedge\kappa\right)\wedge\text{Tr}\left(\eta\mathbb{A}\right)-a.$
(2.40)

The Poisson algebra for two functionals (2.40) is given by the desired
expression, see (2.6),

$\big{\\{}\left\langle\mu,\left(p,\eta,a\right)\right\rangle,\left\langle\mu,\left(q,\lambda,b\right)\right\rangle\big{\\}}=\left\langle\mu,\left[\left(p,\eta,a\right),\left(q,\lambda,b\right)\right]\right\rangle.$
(2.41)

Thus, the action of the group $U(1)\ltimes\tilde{\mathcal{G}}$ on
$\overline{\mathcal{A}}$, is Hamiltonian. Its Lie algebra
$\mathbb{R}\oplus\tilde{\mathcal{G}}_{\text{Lie}}$ admits Loop  a hyperbolic,
non-degenerate and invariant inner product defined by

$\Big{(}\left(p,\eta,a\right),\left(q,\lambda,b\right)\Big{)}=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(\eta\lambda\right)-pb-qa,$ (2.42)

provided (2.26) vanish and where the top-form given by555Below, we will see
from duality arguments, why we have defined (2.42) in terms of the 4-form
(2.43).

$\Omega\wedge\kappa\wedge d_{\mathbb{M}}\kappa\in\Omega_{\mathbb{M}}^{4}$
(2.43)

is globally defined and everywhere non-zero over $\mathbb{M}$. Notice that
$\mathcal{H}=U(1)\ltimes\tilde{\mathcal{G}}$ and
$\mathfrak{h}=\mathbb{R}\oplus\tilde{\mathcal{G}}_{\text{Lie}}$ because of
$\mathcal{H}$ acts on the quotient space $\overline{\mathcal{A}}$ in a
Hamiltonian fashion and $\mathfrak{h}$ is endowed with a well-defined
invariant inner product.

A comment is in order. The condition that (2.26) vanishes is absent in NA loc
CS ; Loop , because of the cocycle used there is, roughly speaking, recovered
from (2.25) by setting $\Omega\rightarrow 1$ and
$\mathbb{M}\rightarrow\text{M}$. To show how such a condition emerges, we
verify the invariance property of the inner product (2.42) under the adjoint
action of $\mathfrak{h}$, which is given by

$\Big{(}\left[\left(p,\eta,a\right),\left(q,\lambda,b\right)\right],\left(r,\phi,c\right)\Big{)}=\Big{(}\left(p,\eta,a\right),\left[\left(q,\lambda,b\right),\left(r,\phi,c\right)\right]\Big{)}.$
(2.44)

Then, (2.44) boils down to

$rd(\eta,\lambda)=pd(\lambda,\phi)$ (2.45)

and because $r,p,\eta,\lambda,\phi$ are all arbitrary, we end up by enforcing
that $d(\ast,\ast)=0$. We will interpret this as a condition to be imposed
over the gauge parameters $\eta\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$.
Although an expression generalizing (2.45) will be considered below in (7.13),
once we notice the existence of a second vector field $\mathcal{R}^{\prime}$,
besides $\mathcal{R}$, acting on $\mathcal{G}_{\text{Lie}}$ and $\mathcal{A}$
as well. More on this below.

Once $\mathfrak{h}$ is equipped with an appropriate inner product, one is
encouraged to use the definition (2.42) in order to dualize the moment map
$\mu\in\mathfrak{h}^{*}$ and by this we mean solving the following equation

$\left\langle\mu,\left(q,\lambda,b\right)\right\rangle=\left(\mu,\left(q,\lambda,b\right)\right),$
(2.46)

for an element $\mu=\left(p,\eta,a\right)\in\mathfrak{h}$ on the right hand
side (rhs). We quickly find that

$\mu=\left(1,\frac{\Omega\wedge\kappa\wedge
F_{\mathbb{A}}+d_{\mathbb{M}}\left(\Omega\wedge\kappa\right)\wedge\mathbb{A}}{\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa},-\frac{1}{2}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\pounds_{\mathcal{R}}\mathbb{A}\wedge\mathbb{A}\right)\right).$
(2.47)

A few words concerning the latter result. Any 4-form
$\gamma\in\Omega_{\mathbb{M}}^{4}\otimes\mathfrak{g}$ is proportional to
(2.43) and can be written as

$\gamma=\phi\,\Omega\wedge\kappa\wedge d_{\mathbb{M}}\kappa,$ (2.48)

for some $\phi\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$. Thus, ‘dividing’
by (2.43) actually means picking the term $\phi$, so that

$\frac{\gamma}{\Omega\wedge\kappa\wedge d_{\mathbb{M}}\kappa}=\phi.$ (2.49)

This unusual notation, originally introduced in NA loc CS , turns out to be
very useful for performing algebraic calculations.

Let us simplify the expression (2.47) further. Consider now the contracted
5-form

$0=i_{\mathcal{R}}\Big{(}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\wedge\mathbb{A}\Big{)}=-\Omega\wedge
d_{\mathbb{M}}\kappa\wedge\mathbb{A+}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa i_{\mathcal{R}}\mathbb{A}\text{.}$ (2.50)

From this follows that we can express the contraction
$i_{\mathcal{R}}\mathbb{A}$ in an equivalent way

$i_{\mathcal{R}}\mathbb{A=}\frac{\Omega\wedge
d_{\mathbb{M}}\kappa\wedge\mathbb{A}}{\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa}.$ (2.51)

Also, introduce the quantities

$\Phi=\frac{\Omega\wedge\kappa\wedge
F_{\mathbb{A}}+d_{\mathbb{M}}\Omega\wedge\kappa\wedge\mathbb{A}}{\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa},\text{ \ \
}B=\frac{d_{\mathbb{M}}\Omega\wedge\kappa\wedge\mathbb{A}}{\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa}.$ (2.52)

In this way, we get a more compact expression for the moment map (2.47), i.e.

$\mu=\left(1,\Phi-i_{\mathcal{R}}\mathbb{A},a\right),$ (2.53)

where

$a=-\frac{1}{2}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\pounds_{\mathcal{R}}\mathbb{A}\wedge\mathbb{A}\right).$
(2.54)

Under the shift ${}^{\kappa}\mathbb{A}=\mathbb{A+\kappa}s$, we have that

${}^{\kappa}\Phi=\Phi+s,\text{ \ \
}i_{\mathcal{R}}\left({}^{\kappa}\mathbb{A}\right)=i_{\mathcal{R}}\mathbb{A}+s,\text{
\ \ }^{\kappa}a=a,$ (2.55)

while under ${}^{\Omega}\mathbb{A}=\mathbb{A}+\Omega s^{\prime}$, we get

${}^{\Omega}\Phi=\Phi,\text{ \ \
}i_{\mathcal{R}}(^{\Omega}\mathbb{A})=i_{\mathcal{R}}\mathbb{A},\text{ \ \
}^{\Omega}a=a.$ (2.56)

To show this, we use the identity (2.14). Thus, the moment map
$\mu\in\mathfrak{h}$ is, as expected, invariant under the action of the shift
group $\mathcal{S}$

$\mu(^{\kappa}\mathbb{A})=\mu(\mathbb{A}),\text{ \ \
}\mu(^{\Omega}\mathbb{A})=\mu(\mathbb{A})$ (2.57)

and it is a well-defined functional on the quotient space
$\overline{\mathcal{A}}$. In particular, any object defined in terms of it is
invariant too.

Now we verify the equivariance property of $\mu$ under the action of
$\mathfrak{h}$ on $\overline{\mathcal{A}}$. The co-adjoint action of an
element $(q,\lambda,b)\in\mathfrak{h}$ on the moment map
$\mu\in\mathfrak{h}^{*}$, is defined by

$ad^{*}_{(p,\eta,a)}\mu\left(\mathbb{A}\right)=\mu\big{(}V(p,\eta,a)\big{)},$
(2.58)

where we have emphasized that $\mu$ depends on $\mathbb{A}$. Recall that
$V(p,\eta,a)$ represents the action of $(p,\eta,a)\in\mathfrak{h}$ on
$\mathbb{A}$. By pairing this expression against an element
$(q,\lambda,b)\in\mathfrak{h}$ and working out (2.40) to first order we find,
after some algebra, that

$\left\langle
ad^{*}_{(p,\eta,a)}\mu\left(\mathbb{A}\right),(q,\lambda,b)\right\rangle=\left\langle\mu(\mathbb{A}),ad_{(p,\eta,a)}(q,\lambda,b)\right\rangle,$
(2.59)

where

$ad_{(p,\eta,a)}(q,\lambda,b)=[(p,\eta,a),(q,\lambda,b)]$ (2.60)

is the adjoint action of $(p,\eta,a)$ on $(q,\lambda,b)$, which is given by
the bracket (2.38).

For completeness, it is important to study how $\mu$ behaves under the action
of $\tilde{\mathcal{G}}\subset\mathcal{H}$, i.e. under finite gauge
transformations. After some algebra, we obtain cf. Loop

$\displaystyle\left\langle\mu\left({}^{g}\mathbb{A}\right),\left(q,\lambda,b\right)\right\rangle=\left\langle
Ad^{\ast}_{(0,g,a)}\mu(\mathbb{A}),(q,\lambda,b)\right\rangle=\left\langle\mu(\mathbb{A}),Ad_{(0,g,a)}(q,\lambda,b)\right\rangle$
(2.61) $\displaystyle=\left\langle\mu\left(\mathbb{A}\right),\left(q,g\lambda
g^{-1}+q\pounds_{\mathcal{R}}gg^{-1},b+\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\left(\Omega\wedge\kappa\right)\wedge\text{Tr}\left(\mathbb{I\lambda}\right)+\frac{q}{2}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{I\wedge}\pounds_{\mathcal{R}}\mathbb{I}\right)\right)\right\rangle,$

where $\mathbb{I}=g^{-1}d_{\mathbb{M}}g$. To first order in $\eta$ with
$g=e^{\eta}$, we have

$\left\langle\mu\left({}^{g}\mathbb{A}\right),\left(q,\lambda,b\right)\right\rangle=\left\langle\mu\left(\mathbb{A}\right),\left(q,\lambda,b\right)\right\rangle+\left\langle\mu\left(\mathbb{A}\right),\left(0,\left[\eta,\lambda\right]+q\pounds_{\mathcal{R}}\eta,c(\eta,\lambda)\right)\right\rangle,$
(2.62)

or

$\left\langle
ad_{(0,\eta,a)}^{\ast}\mu\left(\mathbb{A}\right),\left(q,\lambda,b\right)\right\rangle=\left\langle\mu\left(\mathbb{A}\right),\left[\left(0,\eta,a\right),\left(q,\lambda,b\right)\right]\right\rangle,$
(2.63)

where

$ad_{(0,\eta,a)}^{\ast}\mu\left(\mathbb{A}\right)=\mu\left({}^{g}\mathbb{A}\right)-\mu\left(\mathbb{A}\right),$
(2.64)

in agreement with (2.58) for $p=0$.

Furthermore, by setting
$\nu^{\prime}=(q^{\prime},\lambda^{\prime},b^{\prime})\equiv
Ad_{(0,g,a)}(q,\lambda,b)$ we may ask if the norm of $\nu=(q,\lambda,b)$ is
preserved under the Adjoint action of $\tilde{\mathcal{G}}$. From the inner
product (2.42), we find

$\displaystyle\big{(}\nu^{\prime},\nu^{\prime}\big{)}$
$\displaystyle=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\big{(}\lambda^{\prime
2}\big{)}-2q^{\prime}b^{\prime}$ (2.65)
$\displaystyle=\big{(}\nu,\nu\big{)}-2q\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{I}\big{(}\lambda+\frac{q}{2}g^{-1}\pounds_{\mathcal{R}}g\big{)}\right)-q^{2}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(g^{-1}\pounds_{\mathcal{R}}g\mathbb{I\wedge
I}\right),$

where we have used
$\pounds_{\mathcal{R}}\mathbb{I}=g^{-1}d_{\mathbb{M}}\left(\pounds_{\mathcal{R}}gg^{-1}\right)g$
and integrated by parts with $\partial\mathbb{M}=0$. Now, using the contracted
5-form

$0=i_{\mathcal{R}}\Big{(}\Omega\wedge\kappa\wedge\text{Tr}\big{(}\mathbb{I\wedge
I\wedge I}\big{)}\Big{)}=-\Omega\wedge\text{Tr}\big{(}\mathbb{I\wedge I\wedge
I}\big{)}+3\Omega\wedge\kappa\wedge\text{Tr}\left(g^{-1}\pounds_{\mathcal{R}}g\mathbb{I\wedge
I}\right),$ (2.66)

we get

$\Omega\wedge\kappa\wedge\text{Tr}\left(g^{-1}\pounds_{\mathcal{R}}g\mathbb{I\wedge
I}\right)=-\Omega\wedge\chi(g),$ (2.67)

where we have introduced a 3-form $\chi(g)\in\Omega_{\mathbb{M}}^{3}$, defined
by

$\chi(g)=-\frac{1}{3}\text{Tr}\left(\mathbb{I\wedge I\wedge I}\right).$ (2.68)

Then, we obtain the equivalent form

$\big{(}\nu^{\prime},\nu^{\prime}\big{)}=\big{(}\nu,\nu\big{)}-2q\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{I}\big{(}\lambda+\frac{q}{2}g^{-1}\pounds_{\mathcal{R}}g\big{)}\right)+q^{2}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\chi(g).$
(2.69)

The second term on the rhs right above is related to the ‘obstruction’ (2.26),
while the third term resembles the behavior of a Chern-Simons theory under the
action of finite gauge transformations. It is a modified Wess-Zumino (WZ)
term. This will be verified later on when we consider the quadratic expression
$(\mu,\mu)$ for the dualized moment map $\mu$. To first order in $\eta$ the
inner product is then invariant if (2.26) vanishes.

After this digression on symmetry properties, we now proceed to simplify the
quantity $a$ defined in (2.54). Let us write

$\displaystyle-2a=\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\pounds_{\mathcal{R}}\mathbb{A}\wedge\mathbb{A}\right)=\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\Big{(}d_{\mathbb{M}}\left(i_{\mathcal{R}}\mathbb{A}\right)\wedge\mathbb{A+}i_{\mathcal{R}}\left(d_{\mathbb{M}}\mathbb{A}\right)\wedge\mathbb{A}\Big{)}.$
(2.70)

Using

$\displaystyle
d_{\mathbb{M}}\Big{(}\Omega\wedge\kappa\wedge\text{Tr}\left(i_{\mathcal{R}}\mathbb{AA}\right)\Big{)}=d_{\mathbb{M}}\left(\Omega\wedge\kappa\right)\wedge\text{Tr}\left(i_{\mathcal{R}}\mathbb{AA}\right)+\Omega\wedge\kappa\wedge\text{Tr}\Big{(}d_{\mathbb{M}}\left(i_{\mathcal{R}}\mathbb{A}\right)\wedge\mathbb{A+}i_{\mathcal{R}}\mathbb{A}d_{\mathbb{M}}\mathbb{A}\Big{)},$
(2.71)

we can write inside the integral

$\Omega\wedge\kappa\wedge\text{Tr}\Big{(}d_{\mathbb{M}}\left(i_{\mathcal{R}}\mathbb{A}\right)\wedge\mathbb{A}\Big{)}=-d_{\mathbb{M}}\left(\Omega\wedge\kappa\right)\wedge\text{Tr}\left(i_{\mathcal{R}}\mathbb{AA}\right)-\Omega\wedge\kappa\wedge\text{Tr}\left(i_{\mathcal{R}}\mathbb{A}d_{\mathbb{M}}\mathbb{A}\right).$
(2.72)

Now, consider the contracted 5-form

$\displaystyle
0=i_{\mathcal{R}}\Big{(}\Omega\wedge\kappa\wedge\text{Tr}\big{(}d_{\mathbb{M}}\mathbb{A}\wedge\mathbb{A}\big{)}\Big{)}=-\Omega\wedge\text{Tr}\left(d_{\mathbb{M}}\mathbb{A}\wedge\mathbb{A}\right)+\Omega\wedge\kappa\wedge\text{Tr}\Big{(}i_{\mathcal{R}}\left(d_{\mathbb{M}}\mathbb{A}\right)\wedge\mathbb{A}+d_{\mathbb{M}}\mathbb{A}i_{\mathcal{R}}\mathbb{A}\Big{)}.$
(2.73)

From this follows that

$\Omega\wedge\kappa\wedge\text{Tr}\Big{(}i_{\mathcal{R}}\left(d_{\mathbb{M}}\mathbb{A}\right)\wedge\mathbb{A}\Big{)}=\Omega\wedge\text{Tr}\left(d_{\mathbb{M}}\mathbb{A}\wedge\mathbb{A}\right)-\Omega\wedge\kappa\wedge\text{Tr}\left(d_{\mathbb{M}}\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right).$
(2.74)

Now, inserting both results above into their respective positions, gives

$\displaystyle-2a=\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Bigl{\\{}\Omega\wedge\text{Tr}\left(\mathbb{A\wedge}d_{\mathbb{M}}\mathbb{A}\right)+2\Omega\wedge\kappa\wedge$
$\displaystyle\text{Tr}\left(i_{\mathcal{R}}\mathbb{AA}\wedge\mathbb{A}\right)$
(2.75)
$\displaystyle-2\Omega\wedge\kappa\wedge\text{Tr}\left(i_{\mathcal{R}}\mathbb{A}F_{\mathbb{A}}\right)-d_{\mathbb{M}}\left(\Omega\wedge\kappa\right)\wedge\text{Tr}\left(i_{\mathcal{R}}\mathbb{AA}\right)\Bigr{\\}}.$

We can simplify this expression even further by considering the contracted
5-form

$\displaystyle
0=i_{\mathcal{R}}\Big{(}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A\wedge
A}\wedge\mathbb{A}\right)\Big{)}=-\Omega\wedge\text{Tr}\left(\mathbb{A\wedge
A}\wedge\mathbb{A}\right)+3\Omega\wedge\kappa\wedge\text{Tr}\left(i_{\mathcal{R}}\mathbb{AA}\wedge\mathbb{A}\right).$
(2.76)

Using this result and the definitions introduced above, allows to write

$-2a=\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge
CS\left(\mathbb{A}\right)-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\Big{(}\left(2\Phi-B-
i_{\mathcal{R}}\mathbb{A}\right)i_{\mathcal{R}}\mathbb{A}\Big{)},$ (2.77)

where

$CS(\mathbb{A})=\text{Tr}\left(\mathbb{A}\wedge
d_{\mathbb{M}}\mathbb{A+}\frac{2}{3}\mathbb{A\wedge A\wedge A}\right),$ (2.78)

being an element of $\Omega_{\mathbb{M}}^{3}$, is the well known Chern-Simons
3-form.

###  2.2 Generalized 4d CS theory action functional

Armed with the previous results, now we are able to compute the square of the
moment map (2.47) with respect to the inner product (2.42). As an element of
$\mathfrak{h}$, the moment map is of the form $\mu=(p,\eta,a)$, thus

$\left(\mu,\mu\right)=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(\eta^{2}\right)-2a,$ (2.79)

equals

$\left(\mu,\mu\right)=\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge
CS\left(\mathbb{A}\right)-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(\Phi^{2}-Bi_{\mathcal{R}}\mathbb{A}\right),$
(2.80)

by virtue of (2.53) and (2.77). Equivalently, we have the final form (see
(1.4))

$\left(\mu,\mu\right)=\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge
CS\left(\mathbb{A}\right)-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(\Phi^{2}\right)+\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right).$
(2.81)

At this point, we see that the 1-form $\Omega$ corresponds to a generalization
of the ‘twist’ form $\omega$ entering the definition of the conventional 4d CS
theory (1.1). The action (2.81) is also invariant under the following changes
(rescalings)

$\kappa\rightarrow t\kappa,\text{ \ \ }\Omega\rightarrow\Omega,$ (2.82)

for any non-zero function $t\in\Omega_{\mathbb{M}}^{0}$. The normalization
condition $i_{\mathcal{R}}\kappa=1$, requires $\mathcal{R}\rightarrow
t^{-1}\mathcal{R}$. In particular, we have that $\Phi\rightarrow t^{-1}\Phi$
and $\Omega\wedge\kappa\wedge d_{\mathbb{M}}\kappa\rightarrow
t^{2}\Omega\wedge\kappa\wedge d_{\mathbb{M}}\kappa$ and the sign of (2.43) is
preserved under the arbitrary rescalings (2.82), reflecting the orientability
of $\mathbb{M}$.

The $\mathcal{S}$-invariant action functional on the quotient space
$\overline{\mathcal{A}}$, is then defined by

$S=ic\left(\mu,\mu\right).$ (2.83)

The action (2.83) can be interpreted as a quadratic completion of the 4d CS
theory defined in terms of the Lagrangian density $L\sim\omega\wedge
CS(\mathbb{A})$. To find its functional variation, we use the following
results

$\displaystyle\Omega\wedge\delta CS\left(\mathbb{A}\right)$
$\displaystyle=2\Omega\wedge\text{Tr}\left(\delta\mathbb{A}\wedge
F_{\mathbb{A}}\right)+d_{\mathbb{M}}\Omega\wedge\text{Tr}\left(\delta\mathbb{A}\wedge\mathbb{A}\right),$
(2.84) $\displaystyle\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\delta\text{Tr}\left(\Phi^{2}\right)$
$\displaystyle=-2\Omega\wedge\text{Tr}\Big{(}\delta\mathbb{A}\wedge\left(\kappa\wedge
d_{\mathbb{A}}\Phi-d_{\mathbb{M}}\kappa\Phi\right)\Big{)}.$

Then,

$\delta(\mu,\mu)=2\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\text{Tr}\left(\delta\mathbb{A\wedge}F_{\mathbb{A}-\kappa\Phi}\right)+\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\text{Tr}\left(\delta\mathbb{A\wedge
A}\right)+\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\delta\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right),$
(2.85)

where we have used (2.14). To simplify this expression further, consider the
contracted 5-form

$0=i_{\mathcal{R}}\Big{(}\
d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A\wedge\delta
A}\right)\Big{)}=d_{\mathbb{M}}\Omega\wedge\text{Tr}(\mathbb{A}\wedge\delta\mathbb{A})-d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}(i_{\mathcal{R}}\mathbb{A}\delta\mathbb{A}-\mathbb{A}i_{\mathcal{R}}\delta\mathbb{A}).$
(2.86)

From this result, it follows that

$d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\delta\mathbb{A}\right)=d_{\mathbb{M}}\Omega\wedge\text{Tr}\Big{(}\delta\mathbb{A\wedge}\left(\mathbb{A-\kappa}i_{\mathcal{R}}\mathbb{A}\right)\Big{)}.$
(2.87)

Then, the general variation takes the form

$\delta(\mu,\mu)=2\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\text{Tr}\left(\delta\mathbb{A\wedge}F_{\mathbb{A}-\kappa\Phi}\right)+2\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\text{Tr}\Big{(}\delta\mathbb{A\wedge}\left(\mathbb{A-\kappa}i_{\mathcal{R}}\mathbb{A}\right)\Big{)}$
(2.88)

and from this we get the equations of motion (eom) of the theory, which are
given by

$\Omega\wedge
F_{\mathbb{A}-\kappa\Phi}=d_{\mathbb{M}}\Omega\wedge\left(\mathbb{A-\kappa}i_{\mathcal{R}}\mathbb{A}\right).$
(2.89)

To understand better the rôle played by the object $\Phi$ introduced before in
(2.52), we rewrite such a definition in the equivalent form

$\kappa\wedge\Big{(}\Omega\wedge
F_{\mathbb{A}-\kappa\Phi}-d_{\mathbb{M}}\Omega\wedge\mathbb{A}\Big{)}=0.$
(2.90)

This means that $\Phi$ is actually determined by wedging the eom of the theory
against the 1-form $\kappa$. Later on, we shall see how (2.90) raises as a
genuine eom from a well-defined variational problem applied to an extended
action functional $S(\mathbb{A},\Phi)$.

Now, let us we analyze the behavior of the quadratic expression (2.80) under
the action of the gauge group $\mathcal{G}$. Using (2.9), we get the
transformation properties of the CS 3-form under the action of finite gauge
transformations, namely,

$CS\left({}^{g}\mathbb{A}\right)=CS(\mathbb{A})+d_{\mathbb{M}}\Big{(}\text{Tr}\left(\mathbb{A}\wedge\mathbb{J}\right)\Big{)}+\chi(g),$
(2.91)

where $\mathbb{J}=d_{\mathbb{M}}gg^{-1}$. Using the contracted 5-form

$0=i_{\mathcal{R}}\Big{(}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A\wedge
J}\right)\Big{)}=d_{\mathbb{M}}\Omega\wedge\text{Tr}\left(\mathbb{A\wedge
J}\right)-d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(i_{\mathcal{R}}\mathbb{AJ-A}i_{\mathcal{R}}\mathbb{J}\right),$
(2.92)

we find that

$\displaystyle\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left({}^{g}X^{2}\right)=\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(X^{2}\right)+d_{\mathbb{M}}\Omega\wedge\text{Tr}\left(\mathbb{A\wedge
J}\right)+d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\big{(}\mathbb{J}Y\big{)},$
(2.93)

where

$\begin{aligned} X^{2}=\Phi^{2}-Bi_{\mathcal{R}}\mathbb{A},\text{ \ \
}Y=-2\Phi+2i_{\mathcal{R}}\mathbb{A}-B(\mathbb{J})+i_{\mathcal{R}}\mathbb{J}\end{aligned},$
(2.94)

with $B(\mathbb{J})$ given by the second relation of (2.52) with $\mathbb{J}$
in the place of $\mathbb{A}$. Putting all together gives, cf. (2.69)

${}^{g}(\mu,\mu)=(\mu,\mu)+\int_{\mathbb{M}}\Omega\wedge\chi(g)+\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\big{(}Y\mathbb{J}\big{)}.$
(2.95)

The last contribution on the rhs right above is again related to the
obstruction (2.26), while the second term on the rhs is a generalization of
the usual WZ shift $\chi(g)$ common to CS theories. Thus, for a path integral
formulation of the theory it is reasonable to consider that further analytic
constraints over the gauge group elements $g$ must, in principle, be imposed
in order to obtain an honest gauge theory. In this paper we will adopt this
point of view.

The eom (2.89) also reflects a lack of gauge covariance. If we demand that the
eom (2.89) are preserved under gauge transformations, i.e. if

$\Omega\wedge
F_{{}^{g}\mathbb{A}-\kappa^{g}\Phi}=d_{\mathbb{M}}\Omega\wedge\left({}^{g}\mathbb{A}-\kappa
i_{\mathcal{R}}(^{g}\mathbb{A})\right),$ (2.96)

then an equation for $g$ emerge. Indeed, using

${}^{g}\Phi=g^{-1}\left(\Phi+B(\mathbb{J})\right)g$ (2.97)

and (2.89), we find that (2.96) holds, provided $g\in\mathcal{G}$ satisfy the
following equation

$\Omega\wedge\kappa\wedge d_{\mathbb{A}}B(\mathbb{J})=0.$ (2.98)

This equation can be solved $\forall\mathbb{A}$, if we choose $g$ to be such
that $B(\mathbb{J})=0$. Notice that (2.26) can be written alternatively as

$d(\eta,\lambda)=-\int_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(\eta B(d_{\mathbb{M}}\lambda)\right).$
(2.99)

Then, if $B(j)=0$ for $j=\mathbb{I}=\mathbb{J}=0$, we have that (2.99)
vanishes as well. In this case, (2.97) reduces to

${}^{g}\Phi=g^{-1}\Phi g$ (2.100)

and the field $\Phi$ then transforms in the adjoint. Below, we will see how
(2.26) extends the usual ‘boundary’ condition imposed over the gauge
parameters $\eta$ in conventional 4d CS theories, that they must vanish at the
set of poles of the twist 1-form $\omega$.

We now proceed to verify if the quadratic action (2.83) is a real number. To
show it, we invoke the same reality conditions used in unifying , when applied
to the conventional 4d CS theories. In the present case, these conditions
impose suitable properties to be obeyed by the 1-forms $\Omega,\kappa$ and the
gauge connection $\mathbb{A}$. Notice that all manipulations done above have
been at the formal level and, as a consequence of this, it should not come as
a surprise to find that some objects are purely imaginary instead of real.
This is because we have not specified any reality conditions on the Lie
algebra $\mathfrak{g}$, the gauge connection $\mathbb{A}$, the central
extensions, the manifold $\mathbb{M}$ and so on.

Let us first declare that the Lie algebra $\mathfrak{g}$ is actually a complex
Lie algebra. Let $\tau:\mathfrak{g}\rightarrow\mathfrak{g}$ be an anti-linear
involutive automorphism. It provides $\mathfrak{g}$ with an action of the
cyclic group $\mathbb{Z}_{2}$. Its fixed point subset is a real Lie subalgebra
$\mathfrak{g}_{\mathbb{R}}$ of $\mathfrak{g}$, regarded itself as a real Lie
algebra. The anti-linear involution $\tau$ is compatible with the bilinear
form on $\mathfrak{g}$, in the sense that

$\overline{\text{Tr}(ab)}=\text{Tr}(\tau(a)\tau(b)),$ (2.101)

for any $a,b$ in the Lie algebra $\mathfrak{g}$ or by extension in
$\mathcal{G}_{\text{Lie}}$. Denote by $x$ a set of local coordinates on the
manifold $\mathbb{M}$ and endow it with a complex structure. The complex
conjugation $x\rightarrow\overline{x}$ defines an involution
$\nu:\mathbb{M}\rightarrow\mathbb{M}$, which also provides $\mathbb{M}$ with a
$\mathbb{Z}_{2}$ action. We then require $\gamma\in\Omega_{\mathbb{M}}^{p}$
and $\rho\in\Omega_{\mathbb{M}}^{p}\otimes\mathfrak{g}$ to be equivariant
under this action of $\mathbb{Z}_{2}$, i.e.

$\overline{\gamma}=\nu^{\ast}\gamma,\text{ \ \ }\tau(\rho)=\nu^{*}\rho.$
(2.102)

For example, $\gamma$ represents any of the differential forms $\Omega,\kappa$
and their exterior derivatives and $\rho$ represents the gauge connection
$\mathbb{A}$ the Lie algebra valued functions $\Phi$ and the contraction
$i_{\mathcal{R}}\mathbb{A}$. The rest of the proof goes exactly like in
unifying  (see §2.5) after noticing the fact that $\nu$ has the effect of
conjugating the complex structure on $\mathbb{M}$ and thus also of reversing
its orientation. Let us perform a sample computation to see how this works.
Consider the first term in (2.81) and conjugate it, then

$\displaystyle\int_{\mathbb{M}}\overline{\Omega}\wedge\overline{CS(\mathbb{A})}$
$\displaystyle=\int_{\mathbb{M}}\overline{\Omega}\wedge
CS(\tau(\mathbb{A)})=\int_{\mathbb{M}}\nu^{\ast}\Omega\wedge
CS(\nu^{\ast}\mathbb{A})$ (2.103)
$\displaystyle=\int_{\mathbb{M}}\nu^{\ast}\Big{(}\Omega\wedge
CS(\mathbb{A})\Big{)}=\int_{\nu\left(\mathbb{M}\right)}\Omega\wedge
CS(\mathbb{A})=-\int_{\mathbb{M}}\Omega\wedge CS(\mathbb{A}).$

After repeating for each contribution in (2.81) (or in the form (3.23) below),
we end up with

$\overline{(\mu,\mu)}=-(\mu,\mu).$ (2.104)

As a consequence, the action (2.83) obeys $\overline{S}=S$ and
$S\in\mathbb{R}$. The gauge transformations are also required to preserve the
conditions (2.102) for $\rho=\mathbb{A}$, which is equivalent to restricting
the gauge elements to the ones satisfying the condition $\tau(g)=\nu^{\ast}g$.
In the latter expression, $\tau$ also denotes the lift of the antilinear
automorphism $\tau$ acting on $\mathcal{G}_{\text{Lie}}$ to the group
$\mathcal{G}$. With this, the last contribution to (2.95) also flips sign.
After repeating the same procedure for the pre-symplectic form (2.1), we
conclude that $\underline{\hat{\Omega}}=i\hat{\Omega}$ is real.

For future reference, now we introduce a second vector field
$\mathcal{R}^{\prime}\in\mathfrak{X}_{\mathbb{M}}$ besides $\mathcal{R}$,
defined by the following set of conditions

$i_{\mathcal{R}^{\prime}}\kappa=0,\text{ \ \
}i_{\mathcal{R}^{\prime}}\left(d_{\mathbb{M}}\kappa\right)=0$ (2.105)

and

$i_{\mathcal{R}^{\prime}}\Omega=1,\text{ \ \
}\pounds_{\mathcal{R}^{\prime}}\Omega=0.$ (2.106)

As a consequence of (2.105) and (2.106), we also have that

$\pounds_{\mathcal{R}^{\prime}}\kappa=0,\text{ \ \
}i_{\mathcal{R}^{\prime}}\left(d_{\mathbb{M}}\Omega\right)=0.$ (2.107)

The obstruction (2.26) can be written in terms of the vector field
$\mathcal{R}^{\prime}$ as well. In fact, by using the contracted 5-form

$0=i_{\mathcal{R}^{\prime}}\Big{(}d_{\mathbb{M}}\Omega\wedge\Omega\wedge\kappa\wedge\text{Tr}\left(\eta
d_{\mathbb{M}}\lambda\right)\Big{)}=d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\eta
d_{\mathbb{M}}\lambda\right)+d_{\mathbb{M}}\Omega\wedge\Omega\wedge\kappa\text{Tr}\left(\eta\pounds_{\mathcal{R}^{\prime}}\lambda\right),$
(2.108)

we find that

$d(\eta,\lambda)=\int\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\Omega\wedge\kappa\text{Tr}\left(\eta\pounds_{\mathcal{R}^{\prime}}\lambda\right).$
(2.109)

A possible solution to the condition $B(d_{\mathbb{M}}g)=0$, may be taken to
be of the form

$d_{\mathbb{M}}g=\kappa\pounds_{\mathcal{R}}g+\gamma(g),$ (2.110)

where666Not to be confused with the $\gamma$ used above.
$\gamma\in\text{ker}_{i_{\mathcal{R}}}\cap\text{ker}_{i_{\mathcal{R}^{\prime}}}$
is such that $d_{\mathbb{M}}\Omega\wedge\gamma=0$. Notice that
$\pounds_{\mathcal{R}^{\prime}}g=0$.

Now we have formally constructed the generalized 4d CS theory action
functional and studied some of its properties, the next natural step is to
explore how it is related to the conventional 4d CS theory, i.e. we would like
to know what the actions (2.83) and (1.1) have in common. Fortunately,
answering this question is precisely the content of the next section.

##  3 Relation to the 4d Chern-Simons theory

We now solve explicitly the conditions (2.2) and (2.3) imposed over $\kappa$
and $\Omega$ in a simple way. To do so, we take the 4-dimensional manifold
$\mathbb{M}$ to be of the form $\mathbb{M}=\mathbb{R}\times\text{M}$, where
the $\mathbb{R}$ factor is identified with the time direction and where M is
taken to be the total space of a non-trivial circle bundle of degree $n\neq 0$
over a Riemann surface $C$. The time direction $\mathbb{R}$ and the typical
fiber $S^{1}$ of M being provided by the cylinder $\Sigma=\mathbb{R}\times
S^{1}$. Thus, we have that M is defined by

$S^{1}\overset{n}{\longrightarrow}\text{M}\overset{\underline{\pi}}{\longrightarrow}C.$
(3.1)

This is a natural generalization of the space $\mathbb{M}=\Sigma\times C$
originally used to define the 4d CS theories, in which $\text{M}=S^{1}\times
C$ is a trivial $S^{1}$ fiber bundle over the base manifold $C$. In the
present case, the non-trivial bundle structure in (3.1) provides extra room
for introducing some geometric structures that can be exploited in order to
generalize the 4d CS theories. In particular, the circle bundle (3.1) admits
the existence of invariant contact structures defined on the total space M.
Notice that locally $\mathbb{R}\times\text{M}$ and $\Sigma\times C$ coincide.
We follow NA loc CS  closely.

Consider the following solutions $\kappa,\Omega\in\Omega_{\mathbb{M}}^{1}$ to
the conditions (2.2) and (2.3), given by

$\kappa=\kappa_{\tau}d\tau+\tilde{\kappa},\text{ \ \
}\Omega=\Omega_{\tau}d\tau+\tilde{\Omega},$ (3.2)

where

$\kappa_{\tau}=\frac{1}{2}\alpha_{\tau},\text{ \ \
}\tilde{\kappa}=\frac{1}{2}\alpha\text{ \ \ }\Omega_{\tau}=2\zeta,\text{ \ \
}\tilde{\Omega}=\omega-\frac{\Omega_{\tau}}{\alpha_{\tau}}\alpha.$ (3.3)

We have chosen $\alpha_{\tau}\neq 0\in\mathbb{R}$ and $\zeta>0\in\mathbb{R}$
to be non-zero constants, $\alpha\in\Omega_{\text{M}}^{1}$ to be a
contact777Any compact, orientable 3-manifold possesses a contact structure
Martinet . Recall that $\mathbb{M}=\mathbb{R}\times\text{M}$ is assumed to be
orientable and such that $\partial\mathbb{M}=0$. 1-form on the 3-dimensional
manifold M and
$\omega=\underline{\pi}^{\ast}\omega_{C}\in\Omega_{\text{M}}^{1}$, with
$\omega_{C}\in\Omega_{C}^{1,0}$ to be a $(1,0)$-form on the Riemann surface
$C$. The ‘twist’ 1-form $\omega_{C}$ is a meromorphic differential with a set
of zeroes and poles on $C$ denoted, respectively, by $\mathfrak{z}$ and
$\mathfrak{p}$. Notice that, in order for the solutions (3.2) to exists, the
manifold M must admit a contact structure, hence $(\text{M},\alpha)$ is
required to be a contact manifold888For a comprehensive description of contact
manifolds, the reader is referred to the references Etnyre ; Blair ; Geiges  .
Furthermore, over (3.2) we have imposed the conditions
$\partial_{\tau}\alpha=0$ and $\partial_{\tau}\omega=0$. Concerning the
rescalings (2.82), we notice that $\kappa\rightarrow t\kappa$, implies

$\alpha_{\tau}\rightarrow t\alpha_{\tau},\text{ \ \ }\alpha\rightarrow
t\alpha$ (3.4)

and from this follows that $\Omega$ is invariant, as initially assumed in
(2.82).

Split now the exterior derivative

$d_{\mathbb{M}}=d\tau\wedge\partial_{\tau}+d_{\text{M}},$ (3.5)

where $d_{\text{M}}$ denotes the exterior differential on M.

The total space M admits a free $U(1)$ action arising from the rotations of
the fibers $S^{1}$ and also admits a natural contact structure, which is
invariant under the action of $U(1)$. Such a contact 1-form $\alpha$ is
introduced via the Boothby-Wang construction Boothby-Wang , see also Blair ;
Geiges . This means that $\alpha$ is defined by the relation

$d_{\text{M}}\alpha=n\underline{\pi}^{\ast}\sigma_{C},$ (3.6)

where $\sigma_{C}\in\Omega_{C}^{1,1}$ is a symplectic 2-form on $C$,
normalized to one, i.e.

$\mathop{\displaystyle\int}\nolimits_{C}\sigma_{C}=1$ (3.7)

and where $n>0$ (after a suitable choice of orientation) is the degree of the
bundle M. The geometric meaning of the relation (3.6) is that $\alpha$ defines
a $U(1)$-connection on M regarded now as the total space of a $U(1)$ principal
bundle over $C$, that is induced by the symplectic form $\sigma_{C}$ and has a
non-trivial curvature given by $d_{\text{M}}\alpha$. The contact 1-form
$\alpha$ constructed in this way is $U(1)$ invariant, globally defined on M
and satisfies the contact condition999In general, if $(\text{M},\alpha)$ is a
$2n+1$ dimensional contact manifold, the contact condition is
$\alpha\wedge(d_{\text{M}}\alpha)^{n}\neq 0$. Compare this against its
analogue on the $2n$ dimensional symplectic manifold $(\text{M},\sigma)$,
given by $\sigma^{n}\neq 0$. that the top-form $\alpha\wedge
d_{\text{M}}\alpha\in\Omega_{\text{M}}^{3}$ is nowhere vanishing all over M.

The action of $U(1)$ along the fibers is generated by the Reeb vector field
$R\in\mathfrak{X}_{\text{M}}$. It is a non-vanishing vector field, globally
defined over M and canonically associated to $\alpha$ by the normalization
condition $\alpha(R)=i_{R}\alpha=1$. The 1-form $\alpha$ is invariant because,
as a connection 1-form on M, it separates any tangent space $T_{p}\text{M}$
over $p\in\text{M}$ into vertical and horizontal spaces. R is vertical and
$d_{\text{M}}\alpha$ is a horizontal 2-form, thus $\pounds_{R}\alpha=0$.
Furthermore, the integral of $\alpha$ over any fiber $S^{1}$ is normalized to
one, i.e.

$\mathop{\displaystyle\int}\nolimits_{S^{1}}\alpha=1.$ (3.8)

Hence, after a fiber integration, we get that

$\mathop{\displaystyle\int}\nolimits_{\text{M}}\alpha\wedge
d_{\text{M}}\alpha=n\mathop{\displaystyle\int}\nolimits_{\text{M}}\alpha\wedge\underline{\pi}^{\ast}\sigma_{C}=n\mathop{\displaystyle\int}\nolimits_{C}\sigma_{C}=n$
(3.9)

is never zero because, by assumption, the bundle M is non-trivial.

Returning to the solutions (3.2), for the vector field
$\mathcal{R}\in\mathfrak{X}_{\mathbb{M}}$, we have taken

$\mathcal{R=}\frac{1}{\alpha_{\tau}}\partial_{\tau}+R.$ (3.10)

All conditions written in (2.2) and (2.3) are satisfied after noticing that
$\omega$ is a horizontal 1-form, i.e. it is obtained by pulling $\omega_{C}$
back from $C$ to $\mathbb{M}$, with vanishing $d\tau$ component. Actually,
both conditions are solved in general by first considering

$\kappa=\frac{1}{2}\Big{(}\alpha_{\tau}d\tau+\alpha\Big{)},\text{ \ \
}\mathcal{R=}f_{\tau}\partial_{\tau}+R,$ (3.11)

for $\alpha_{\tau},f_{\tau}\in\Omega_{\mathbb{M}}^{0}$ arbitrary non-zero
functions. The normalization condition $i_{\mathcal{R}}\kappa=1$, requires
that $f_{\tau}=1/\alpha_{\tau}$ and
$i_{\mathcal{R}}\left(d_{\mathbb{M}}\kappa\right)=0$ is equivalent to
$\partial_{\tau}\alpha-d\alpha_{\tau}=0$. The condition
$i_{\mathcal{R}}\Omega=0$ can be solved by taking

$\Omega=\omega+\Omega_{\tau}\Big{(}d\tau-\frac{\alpha}{\alpha_{\tau}}\Big{)},$
(3.12)

such that $i_{\mathcal{R}}\omega=0$ and with
$\Omega_{\tau}\in\Omega_{\mathbb{M}}^{0}$ being an arbitrary non-zero
function. Finally, the invariance condition $\pounds_{\mathcal{R}}\Omega=0$
implies $i_{\mathcal{R}}\left(d_{\mathbb{M}}\Omega\right)=0$, which is
equivalent to

$i_{\mathcal{R}}\left(d_{\mathbb{M}}\omega+d_{\mathbb{M}}\Omega_{\tau}\wedge\Big{(}d\tau-\frac{\alpha}{\alpha_{\tau}}\Big{)}-\Omega_{\tau}\Big{(}\frac{d_{\mathbb{M}}\alpha}{\alpha_{\tau}}-\frac{d_{\mathbb{M}}\alpha_{\tau}\wedge\alpha}{\alpha_{\tau}^{2}}\Big{)}\right)=0.$
(3.13)

Taking $\alpha$ and $\omega$ as defined above and restricting $\Omega_{\tau}$
to be a constant, requires $\alpha_{\tau}$ to be a constant too. Thus, in
(3.3) we have chosen a simple non-trivial solution. The explicit form of the
vector field $\mathcal{R}^{\prime}\in\mathfrak{X}_{\mathbb{M}}$ introduced
above in the last section, will be given in due course.

Now, we consider a key result concerning the 4-form (2.43) introduced above.
Notice that

$\omega\wedge
d_{\text{M}}\alpha=n\underline{\pi}^{\ast}\omega_{C}\wedge\underline{\pi}^{\ast}\sigma_{C}=n\underline{\pi}^{\ast}\left(\omega_{C}\wedge\sigma_{C}\right)=0,$
(3.14)

because of $\omega_{C}\wedge\sigma_{C}\in\Omega_{C}^{2,1}$ is a $(2,1)$-form
on $C$, hence it vanishes by dimensionality reasons. Then, the result (3.14)
imply that the 4-dimensional top-form (2.43) takes the more explicit form

$\Omega\wedge\kappa\wedge d_{\mathbb{M}}\kappa=\zeta d\tau\wedge\alpha\wedge
d_{\text{M}}\alpha.$ (3.15)

This expression is globally defined on $\mathbb{M}$ and nowhere vanishing. It
is basically proportional to the volume form $d\text{Vol}_{\mathbb{M}}$ of the
manifold $\mathbb{M}$. If we integrate (3.15) over $\mathbb{M}$, we get

$\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa=n\zeta\Delta\tau,$ (3.16)

where we have integrated the $\tau$ direction over a finite interval of size
$\Delta\tau$. At this point, it is interesting to compare (3.15) with the top-
form obtained from the symplectisation Geiges
$\left(\mathbb{R}\times\text{M},\sigma=d_{\mathbb{M}}\left(e^{\zeta\tau/2}\alpha\right)\right)$
of the contact manifold $(\text{M},\alpha)$, which is given by

$\sigma\wedge\sigma=\zeta e^{\zeta\tau}d\tau\wedge\alpha\wedge
d_{\text{M}}\alpha.$ (3.17)

The factor $e^{\zeta\tau}$ can be absorbed into $\sigma$ but this spoils the
closedness of the symplectic form $\sigma$.
In what follows, we denote $d_{\text{M}}=d$ in order to avoid clutter.

To see more explicitly how the action (2.83) generalize the conventional 4d
Chern-Simons theories, we proceed by interpreting $\zeta$ as a deformation
parameter. Exhibiting $\zeta$ in the expressions

$\displaystyle\Omega\wedge\kappa$ $\displaystyle=\omega\wedge\kappa+2\zeta
d\tau\wedge\alpha,$ (3.18) $\displaystyle d_{\mathbb{M}}\Omega\wedge\kappa$
$\displaystyle=d\omega\wedge\kappa-\frac{2\zeta}{\alpha_{\tau}}\kappa\wedge
d\alpha,$ $\displaystyle\Omega\wedge d_{\mathbb{M}}\kappa$
$\displaystyle=\zeta\Big{(}d\tau-\frac{\alpha}{\alpha_{\tau}}\Big{)}\wedge
d\alpha,$

we find that

$\Phi=\frac{\Phi_{-1}}{\zeta}+\Phi_{0},\text{ \ \
}i_{\mathcal{R}}\mathbb{A=}\left(i_{\mathcal{R}}\mathbb{A}\right)_{0}.$ (3.19)

Right above, we get

$\Phi_{-1}=\frac{\omega\wedge\kappa\wedge
F_{\mathbb{A}}+d\omega\wedge\kappa\wedge\mathbb{A}}{d\tau\wedge\alpha\wedge
d\alpha},\text{ \ \ }\Phi_{0}=2\frac{d\tau\wedge\alpha\wedge
F_{\mathbb{A}}-\alpha_{\tau}^{-1}\kappa\wedge
d\alpha\wedge\mathbb{A}}{d\tau\wedge\alpha\wedge d\alpha},$ (3.20)

and

$i_{\mathcal{R}}\mathbb{A=}\frac{\left(d\tau-\alpha/\alpha_{\tau}\right)\wedge
d\alpha\wedge\mathbb{A}}{d\tau\wedge\alpha\wedge d\alpha}.$ (3.21)

Now, inserting these results into the action (2.81), we obtain the following
$\zeta$ expansion

$S=S_{0}+\zeta S_{1}+\zeta^{-1}S_{-1},$ (3.22)

where

$\displaystyle S_{0}$
$\displaystyle=ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\omega\wedge
CS\left(\mathbb{A}\right)-2ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d\tau\wedge\alpha\wedge
d\alpha\text{Tr}\left(\Phi_{0}\Phi_{-1}\right)+ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d\omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right),$
(3.23) $\displaystyle S_{1}$
$\displaystyle=2ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\left(d\tau-\frac{\alpha}{\alpha_{\tau}}\right)\wedge
CS\left(\mathbb{A}\right)-ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d\tau\wedge\alpha\wedge
d\alpha\text{Tr}\left(\Phi_{0}^{2}\right)-2i\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\frac{\kappa\wedge
d\alpha}{\alpha_{\tau}}\wedge\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right),$
$\displaystyle S_{-1}$
$\displaystyle=-ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d\tau\wedge\alpha\wedge
d\alpha\text{Tr}\left(\Phi_{-1}^{2}\right).$

The first term in the rhs of the first line above matches perfectly with the
4d Chern-Simons theory (1.1). Thus, if we are interested in recovering the 4d
CS theories it is desirable to find a way to do so. Fortunately, inspired by
(3.22) we can implement the following two-step strategy:

  * •

Step I, we gauge fix the $\kappa$-shift symmetry by imposing the gauge fixing
condition $\Phi\approx 0$. This step simplifies drastically the expressions
(3.23) and we end up with an expansion (3.22) involving only the powers
$\zeta^{0}=1$ and $\zeta^{1}=\zeta$ in the deformation parameter $\zeta$.
Because of ${}^{\kappa}\Phi=\Phi+s$, this gauge fixing condition is
accessible. Yet, we still need to verify if it is a good gauge fixing
condition. This requires running the Dirac algorithm.
The partially gauged fixed theory is still invariant under the
$\Omega$-shifts.

  * •

Step II, we take $\zeta\rightarrow 0$ at the end. This is a degenerate limit
rendering several expressions ill-defined. For instance, from (3.18) we
realize that in this limit, the 2-form $\Omega\wedge\kappa$ vanishes at the
set of zeroes $\mathfrak{z}^{\prime}$ of the twist 1-form $\omega$ and the
term $d_{\mathbb{M}}\Omega\wedge\kappa$ localizes at the set of poles
$\mathfrak{p}^{\prime}$ of $\omega$. In the first case, the pre-symplectic
form (2.1) vanishes at the set $\mathfrak{z}^{\prime}$ and from (3.15), we
have that the first contribution to the inner product (2.42) is absent. Thus,
$\zeta$ can be interpreted also as a regularizing parameter.

Let us notice that these steps make no sense if performed in reverse order.
However, by assuming everything is fine we get, after setting $\Phi=\zeta=0$,
the partially gauge fixed action

$S=ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\omega\wedge
CS\left(\mathbb{A}\right)+ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d\omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right).$
(3.24)

When $\zeta\rightarrow 0$, $\Omega=\omega$ and the $\Omega$-shift symmetry is
reduced to ${}^{\omega}\mathbb{A}=\mathbb{A}+s\omega$. There is a new
‘boundary’ term in (3.24) that is not present in the original
theory101010Boundary terms are understood as those contributions to the action
that localize at the set of poles $\mathfrak{p}\in C$. In the present case,
this can be seen by using local bundle coordinates.. However, by demanding
that this contribution vanish, we can fix part of the analytic structure of
the connection $\mathbb{A}$ that later on will define the Lax connection
$\mathscr{L}$ of an integrable field theory associated to the 4d CS theory. We
will work out this explicitly in the next section. Then, if the field
$\mathbb{A}$ satisfies the condition

$\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d\omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right)=0,$
(3.25)

the action functional (3.24) formally reduces, to that of the 4d Chern-Simons
theory (1.1)

$S=ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\omega\wedge
CS\left(\mathbb{A}\right).$ (3.26)

The only trace of the non-triviality of the circle bundle M over $C$ lies in
the integration domain $\mathbb{M}$ and locally, the generalized and the
original theory (1.1) coincide. Notice the important rôle played by the
deformation parameter $\zeta$ in achieving the final expression (3.26).

Now we have clarified how the generalized and the usual 4d CS theory are
related, we proceed now to make a choice for the 3-dimensional contact
manifold $(\text{M},\alpha)$ in order to work out explicitly the generalized
4d CS theory in a well known situation. In the next section, we choose
$\text{M}=S^{3}$. Within our construction, this manifold is naturally
associated to integrable field theories of the PCM type.

##  4 Principal Chiral Model type theories

In this section, we specialize the construction introduced above to a
particular case and consider an example corresponding to the description of
integrable field theories of the Principal Chiral Model (PCM) type, i.e. we
take $\text{M}=S^{3}$, $C=S^{2}$. We first gather some basic results
concerning the Hopf fibration of $S^{3}$. Then, following NA loc CS  we show
that the induced metric on the space $\overline{\mathcal{A}}$ is Kähler with
respect to the symplectic form $\hat{\Omega}$ and a complex structure $J$ to
be defined below. This is an essential result required to prove (1.5). We also
perform the Hamiltonian analysis in this case, where we implement a partial
gauge fixing for the action of the shift group $\mathcal{S}$ by means of the
condition $\Phi\approx 0$ and subsequently, take the limit $\zeta\rightarrow
0$, where we recover the known 4d CS theory action, making contact with the
discussion made in section (3). Finally, as an example of a solution to the
condition (3.25), we re-derive the Lax connection for the lambda deformed PCM.

###  4.1 A contact form and the Hopf fibration

This case corresponds to the Riemann surface $C=\mathbb{CP}^{1}$, which is the
spectral space associated to integrable field theories of the principal chiral
model type CY . Thus, we have a $S^{1}$ bundle over $S^{2}$ and

$S^{1}\longrightarrow S^{3}\overset{\underline{\pi}}{\longrightarrow}S^{2}$
(4.1)

is the Hopf fibration. The degree of this bundle is $n=1$ Bott-Tu .
Furthermore, it is known that $S^{3}$ is one of the simplest Seifert manifolds
Orlik .

In what follows we gather some basic facts concerning the Hopf fibration of
$S^{3}$ and then move to the explicit construction of the contact 1-form
$\alpha$ and its associated Reeb vector field $R$. Here, for sake of
completeness, we try to be as self-contained as possible.

Consider $S^{3}$ as the unit sphere in $\mathbb{C}^{2}$ with coordinates
$\left(z_{0},z_{1}\right)$ and $\mathbb{CP}^{1}$ as the quotient space of
$S^{3}$ under the equivalence relation
$\left(z_{0},z_{1}\right)\sim\lambda\left(z_{0},z_{1}\right)$, for any
$\lambda\in S^{1}$. Define the projection
$\underline{\pi}:S^{3}\rightarrow\mathbb{CP}^{1}$ by the natural map

$\underline{\pi}:(z_{0},z_{1})\longrightarrow\left[z_{0},z_{1}\right],$ (4.2)

where $\left[z_{0},z_{1}\right]$ are the homogeneous coordinates of
$\mathbb{CP}^{1}$. Let

$\mathcal{U}_{i}=\left\\{\left[z_{0},z_{1}\right]:z_{i}\neq 0\right\\},\text{
\ \ }i=0,1$ (4.3)

be the coordinate charts of $\mathbb{CP}^{1}$. From (4.3), we have that the
local coordinates on $\mathcal{U}_{0}$ and $\mathcal{U}_{1}$ are,
respectively, given by $w=z_{1}/z_{0}$ and $z=z_{0}/z_{1}$, so that $w=1/z$ .

The bundle structure is introduced via the local trivializations

$\Phi_{i}:\underline{\pi}^{-1}\left(\mathcal{U}_{i}\right)\longrightarrow\mathcal{U}_{i}\times
S^{1},\text{ \ \
}\Phi_{i}(z_{0},z_{1})=\left(\left[z_{0},z_{1}\right],\frac{z_{i}}{\left|z_{i}\right|}\right),$
(4.4)

with inverses given by

$\Phi_{i}^{-1}:\mathcal{U}_{i}\times
S^{1}\longrightarrow\underline{\pi}^{-1}\left(\mathcal{U}_{i}\right),\text{ \
\
}\Phi_{i}^{-1}\left(\left[z_{0},z_{1}\right],e^{i\sigma}\right)=\frac{e^{i\sigma}\left|z_{i}\right|}{\sqrt{\left|z_{0}\right|^{2}+\left|z_{1}\right|^{2}}z_{i}}\left(z_{0},z_{1}\right).$
(4.5)

For $j\neq i$, one has that

$\Phi_{j}\Phi_{i}^{-1}:\mathcal{U}_{i}\times
S^{1}\longrightarrow\mathcal{U}_{j}\times S^{1},\text{ \ \
}\Phi_{j}\Phi_{i}^{-1}\left(\left[z_{0},z_{1}\right],e^{i\sigma}\right)=\left(\left[z_{0},z_{1}\right],e^{i\sigma}\frac{z_{j}\left|z_{i}\right|}{z_{i}\left|z_{j}\right|}\right),$
(4.6)

so the transition map is given by

$t_{ji}:\mathcal{U}_{j}\cap\mathcal{U}_{i}\longrightarrow S^{1},\text{ \ \
}\left[z_{0},z_{1}\right]\rightarrow\frac{z_{j}\left|z_{i}\right|}{z_{i}\left|z_{j}\right|}.$
(4.7)

Let us write some expressions in a more explicit way. Define $S^{2}$ and
$S^{3}$ by the elements $(u_{1},u_{2},u_{3})\in\mathbb{R}^{3}$ and
$(x_{1},x_{2},x_{3},x_{4})\in\mathbb{R}^{4}$ obeying
$u_{1}^{2}+u_{2}^{2}+u_{3}^{2}=1$ and
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1,$ respectively. In terms of the
complex coordinates of $\mathbb{C}^{2}$ introduced above, we set
$z_{0}=x_{1}+ix_{2}$ and $z_{1}=x_{3}+ix_{4}$. The latter defining the complex
structures to be considered here.

Let $(x,y)$ be the stereographic projection coordinates of a point in the
southern hemisphere $\mathcal{U}_{1}$ of $S^{2}$ from the north pole. We have
that

$(x,y)=\left(\frac{u_{1}}{1-u_{3}},\frac{u_{2}}{1-u_{3}}\right),$ (4.8)

and

$z=x+iy=\frac{u_{1}+iu_{2}}{1-u_{3}}=\frac{x_{1}+ix_{2}}{x_{3}+ix_{4}}=\frac{z_{0}}{z_{1}}.$
(4.9)

In a similar way, the stereographic coordinates $(u,v)$ of the northern
hemisphere $\mathcal{U}_{0}$ projected from the south pole are

$(u,v)=\left(\frac{u_{1}}{1+u_{3}},\frac{u_{2}}{1+u_{3}}\right).$ (4.10)

Then,

$w=u-iv=\frac{u_{1}-iu_{2}}{1+u_{3}}=\frac{x_{3}+ix_{4}}{x_{1}+ix_{2}}=\frac{z_{1}}{z_{0}}.$
(4.11)

On the equator of $S^{2}$, $u_{3}=0$,
$\left|z_{0}\right|=\left|z_{1}\right|=1/\sqrt{2}$ and the transition function
$t_{01}$ become $t_{01}=z_{0}/z_{1}=u_{1}+iu_{2}$.

From (4.4) and (4.5), we write for $i=1$, the bundle coordinate relations

$\left(z,e^{i\sigma}\right)=\left(\frac{z_{0}}{z_{1}},\frac{z_{1}}{\left|z_{1}\right|}\right),\text{
\ \
}\left(z_{0},z_{1}\right)=\frac{e^{i\sigma}}{\sqrt{1+\left|z\right|^{2}}}\left(z,1\right).$
(4.12)

We now use the local trivialization coordinates (4.12) over the chart
$\mathcal{U}_{1}\subset\mathbb{CP}^{1}$ to compute the contact form $\alpha$
and mainly to run the Hamiltonian analysis of the theory below111111We will be
mostly working on the chart $\mathcal{U}_{1}$, covering all $\mathbb{CP}^{1}$
but the north pole. If something special occurs at the point $z=\infty$, we
will properly comment on it when necessary..

Notice that $\underline{\pi}:(z_{0},z_{1})\rightarrow z=z_{0}/z_{1}$. Consider
the symplectic form on $\mathbb{CP}^{1}$, given by the Kähler form

$\sigma_{C}=\frac{i}{2\pi}\frac{dz\wedge
d\overline{z}}{\big{(}1+\left|z\right|^{2}\big{)}^{2}}.$ (4.13)

To find the pull-back $\underline{\pi}^{\ast}\sigma_{C},$ simply take
$z=z_{0}/z_{1}$ in order to obtain a local expression on M, which is given by

$\underline{\pi}^{\ast}\sigma_{C}=\frac{i}{2\pi}\Big{(}dz_{0}\wedge
d\overline{z}_{0}+dz_{1}\wedge d\overline{z}_{1}\Big{)},$ (4.14)

where we have used $\left|z_{0}\right|^{2}+\left|z_{1}\right|^{2}=1$ and
$d\big{(}\left|z_{0}\right|^{2}+\left|z_{1}\right|^{2}\big{)}=0$ in order to
reach the final form. The contact 1-form is then given by the defining
relation (3.6), implying in

$\alpha=\frac{i}{4\pi}\Big{(}z_{0}d\overline{z}_{0}-\overline{z}_{0}dz_{0}+z_{1}d\overline{z}_{1}-\overline{z}_{1}dz_{1}\Big{)}.$
(4.15)

Now, the Reeb vector field satisfying the condition $\alpha(R)=1$, is

$R=2\pi
i\Big{(}z_{0}\partial_{z_{0}}-\overline{z}_{0}\partial_{\overline{z}_{0}}+z_{1}\partial_{z_{1}}-\overline{z}_{1}\partial_{\overline{z}_{1}}\Big{)}$
(4.16)

and its integral curves, given by

$\big{(}z_{0}(t),z_{1}(t)\big{)}=\lambda(t)(z_{0},z_{1}),\text{ \ \
}\lambda(t)=e^{2\pi it}\in S^{1},$ (4.17)

with $t\in\mathbb{R}$ are, not surprisingly, the $S^{1}$ fibers of the Hopf
bundle. Then, we get

$\left(\frac{z_{0}(t)}{z_{1}(t)},\frac{z_{1}(t)}{\left|z_{1}(t)\right|}\right)=\left(z,e^{i(\sigma+2\pi
t)}\right).$ (4.18)

In real coordinates, we alternatively have

$\displaystyle\alpha=\frac{1}{2\pi}\Big{(}x_{1}dx_{2}-x_{2}dx_{1}+x_{3}dx_{4}-x_{4}dx_{3}\Big{)},\text{
\ \ }d\alpha=\frac{1}{\pi}\Big{(}dx_{1}\wedge dx_{2}+dx_{3}\wedge
dx_{4}\Big{)}$ (4.19)

and from this follows that

$\alpha\wedge d\alpha=\frac{1}{2\pi^{2}}d\text{Vol}_{S^{3}},$ (4.20)

where

$d\text{Vol}_{S^{3}}=i^{\ast}\left(i_{u}d\text{Vol}_{\mathbb{R}^{4}}\right)=\sum_{i=1}^{4}(-1)^{i-1}x_{i}dx_{1}\wedge...\wedge\widehat{dx_{i}}\wedge...\wedge
dx_{4}.$ (4.21)

Right above, $u=\left(x_{1},x_{2},x_{3},x_{4}\right)$ is a unit vector normal
to $S^{3},$ $d\text{Vol}_{\mathbb{R}^{4}}=dx_{1}\wedge dx_{2}\wedge
dx_{3}\wedge dx_{4}$ is the volume form of $\mathbb{R}^{4}$ and $i$ is the
inclusion map $S^{3}\hookrightarrow\mathbb{R}^{4}$. The $2\pi^{2}$ is the
3-dimensional surface volume of a 3-sphere of unit radius. The integral of
$\alpha\wedge d\alpha$ over M is one, cf. (3.9).

In the local bundle coordinates $(z,\sigma)$, which are the ones that we will
use later on to run the Dirac algorithm, we have

$\displaystyle\alpha=\alpha_{z}dz+\alpha_{\overline{z}}d\overline{z}+\alpha_{\sigma}d\sigma=-\frac{i}{4\pi}\frac{\overline{z}dz-
zd\overline{z}}{1+\left|z\right|^{2}}+\frac{d\sigma}{2\pi},\text{ \ \
}d\alpha=ig_{z\overline{z}}dz\wedge d\overline{z}=\frac{i}{2\pi}\frac{dz\wedge
d\overline{z}}{\big{(}1+\left|z\right|^{2}\big{)}^{2}}.$ (4.22)

The contraction (2.51), is equivalent to

$i_{\mathcal{R}}\mathbb{A=}\frac{A_{\tau}}{\alpha_{\tau}}+i_{R}A=\frac{A_{\tau}}{\alpha_{\tau}}+\frac{d\alpha\wedge
A}{\alpha\wedge d\alpha},$ (4.23)

where we have used the decomposition

$\mathbb{A}=A_{\tau}d\tau+A,\text{ \ \
}A=A_{z}dz+A_{\overline{z}}d\overline{z}+A_{\sigma}d\sigma.$ (4.24)

Notice that

$i_{R}A=\frac{A_{\sigma}}{\alpha_{\sigma}},\text{ \ \
}\alpha_{\sigma}=\frac{1}{2\pi}.$ (4.25)

Then, in these coordinates the Reeb vector field takes the simple form

$R=\frac{1}{\alpha_{\sigma}}\partial_{\sigma}$ (4.26)

and from (3.10), we get that

$\mathcal{R}=\frac{1}{\alpha_{\tau}}\partial_{\tau}+\frac{1}{\alpha_{\sigma}}\partial_{\sigma}.$
(4.27)

Alternatively, the expression (4.26) can be obtained from (4.16) and (4.12).

In the rest of this work we will set $\alpha_{\tau}=\alpha_{\sigma}$. This
particular choice ensures that in local coordinates, the vector fields
$\mathcal{R}$ and $\mathcal{R}^{\prime}$ are proportional to the usual light-
cone expressions $\partial_{\pm}$, respectively. Indeed, locally
$\Sigma=\mathbb{R}\times S^{1}$ is a Minkowskian cylinder and the integral
curves of the vector field $\mathcal{R}$ are spirals drawn on $\Sigma$. After
introducing light-cone coordinates via the definitions
$\sigma^{\pm}=\tau\pm\sigma$,
$\partial_{\pm}=\frac{1}{2}(\partial_{\tau}\pm\partial_{\sigma})$, we have
that $\mathcal{R}=4\pi\partial_{+}$. Then, (3.10) can be understood as a
global definition of the light-cone vector $\partial_{+}\in T\Sigma$. The
global counterpart of $\partial_{-}$, denoted by $\mathcal{R}^{\prime}$ and
announced before, will be introduced later on when needed.

Now, we verify explicitly the conditions (2.102) for $\kappa,\Omega$ with
$x=(\tau,\sigma,z)$ and $\overline{x}=(\tau,\sigma,\overline{z})$. The reality
conditions for $\kappa$ are trivially satisfied, while for $\Omega$ they imply
that $\overline{\varphi(z)}=\varphi(\overline{z})$. The latter condition is
equivalent to the statement that the zeroes and the poles in the sets
$\mathfrak{z}$, $\mathfrak{p}$, are either real of coming in complex conjugate
pairs unifying ; Lacroix . In what follows, we will assume that this is the
case.

Let us consider the expression (2.26) in more detail. After writing

$d_{\mathbb{M}}\Omega=d\tilde{\Omega}=d\omega-4\pi\zeta
d\alpha=\Omega_{z\overline{z}}dz\wedge d\overline{z},$ (4.28)

we get that

$d(\eta,\lambda)=\frac{1}{2\pi}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d\tau\wedge
d\sigma\wedge dz\wedge
d\overline{z}\Omega_{z\overline{z}}\text{Tr}\left(\eta\partial_{-}\lambda\right).$
(4.29)

Then, the condition for the inner product (2.42) to be invariant under the
adjoint action of $\mathfrak{h}$, requires restricting the elements
$\eta\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$ of the gauge algebra to
depend on the $\Sigma$ coordinates $\tau,\sigma$ only through the light-cone
coordinate $\sigma^{+}$. In general they may be chosen to satisfy the global
condition $\pounds_{\mathcal{R}^{\prime}}\eta=0$, see (2.109). In the
degenerate limit $\zeta\rightarrow 0$, it suffices instead to impose the
condition $\eta|_{\mathfrak{p}}=0$, that the gauge parameters vanish at the
set of poles $\mathfrak{p}$ of the twist 1-form $\omega$. This is already a
well known fact in the literature. We also verify explicitly that
$d_{\mathbb{M}}\Omega\wedge d_{\mathbb{M}}\kappa=0$ in both situations, so the
expression (2.26) is anti-symmetric.

###  4.2 Riemannian and Kähler metrics on $\mathcal{A}$ and
$\overline{\mathcal{A}}$

We work now in the Hopf coordinates, $0\leq\eta\leq\pi/2,$
$0\leq\xi_{1},\xi_{2}\leq 2\pi$, which are defined by

$z_{0}=e^{i\xi_{1}}\sin\eta,\text{ \ \ }z_{1}=e^{i\xi_{2}}\cos\eta.$ (4.30)

Some useful expressions to be used in what follows are

$\alpha=\frac{1}{2\pi}\Big{(}\sin^{2}\eta d\xi_{1}+\cos^{2}\eta
d\xi_{2}\Big{)},\text{ \ \ \ }d\alpha=\frac{1}{\pi}\sin\eta\cos\eta
d\eta\wedge\left(d\xi_{1}-d\xi_{2}\right).$ (4.31)

In these coordinates, the round metric of $\text{M}=S^{3}$ takes the form

$g_{\text{M}}=d\eta\otimes d\eta+\sin^{2}\eta d\xi_{1}\otimes
d\xi_{1}+\cos^{2}\eta d\xi_{2}\otimes d\xi_{2}.$ (4.32)

We want to relate $g_{\text{M}}$ to the pull-back of the Kähler metric $g_{C}$
on $C$ associated to the Kähler form (4.13) given by

$g_{C}=\frac{1}{2\pi}\frac{dz\otimes d\overline{z}+d\overline{z}\otimes
dz}{\big{(}1+\left|z\right|^{2}\big{)}^{2}}$ (4.33)

and to the contact form $\alpha$. In order to compute
$\underline{\pi}^{\ast}g_{C}$, we set $z=z_{0}/z_{1}$ as before and obtain

$\pi\left(\underline{\pi}^{\ast}g_{C}\right)=d\eta\otimes
d\eta+\sin^{2}\eta\cos^{2}\eta\left(d\xi_{1}-d\xi_{2}\right)\otimes\left(d\xi_{1}-d\xi_{2}\right).$
(4.34)

The metric $g_{\text{M}}$, then takes the compact form

$g_{\text{M}}=\pi\left(\underline{\pi}^{\ast}g_{C}\right)+\left(2\pi\right)^{2}\alpha\otimes\alpha.$
(4.35)

In this guise, we see that the action of the vector field $R$ is an isometry
of the metric $g_{\text{M}}$ and this follows from

$i_{R}(d\eta)=0,\text{ \ \ }i_{R}\left(d\xi_{1}-d\xi_{2}\right)=0$ (4.36)

as can be seen from the expression for $d\alpha$ written in (4.31).

Now, we introduce a pseudo-Riemannian metric $g_{\mathbb{M}}$ on the
4-dimensional manifold $\mathbb{M}=\mathbb{R}\times\text{M}$. It is defined by
$g_{\mathbb{M}}=\sqrt{\rho}\left(d\tau\otimes d\tau-g_{\text{M}}\right)$,
where $\rho$ is a real positive constant to be fixed below. In matrix form, we
have that

$\left[g_{\mathbb{M}}\right]=\sqrt{\rho}\left(\begin{array}[]{cccc}1&0&0&0\\\
0&-1&0&0\\\ 0&0&-\sin^{2}\eta&0\\\ 0&0&0&-\cos^{2}\eta\end{array}\right).$
(4.37)

In order to proceed, we need an explicit expression for the second vector
field $\mathcal{R}^{\prime}\in\mathfrak{X}_{\mathbb{M}}$, satisfying the
conditions (2.105) and (2.106). We quickly find that

$\mathcal{R}^{\prime}=\frac{\kappa_{\tau}}{\Omega_{\tau}}\Big{(}\frac{1}{\alpha_{\tau}}\partial_{\tau}-R\Big{)}.$
(4.38)

Let us write now

$\Omega=\omega+\Omega^{\prime},\text{ \ \
}\Omega^{\prime}=\Omega_{\tau}\left(d\tau-\frac{\alpha}{\alpha_{\tau}}\right).$
(4.39)

Thus $i_{\mathcal{R}^{\prime}}\Omega^{\prime}=1$, because of
$i_{\mathcal{R}^{\prime}}\omega=0$. In the local bundle coordinates
$(z,\sigma)$, the vector field (4.38) becomes
$\mathcal{R^{\prime}}=(1/\Omega_{\tau})\partial_{-}$ and we see that (4.38)
corresponds, up to a constant, to a global definition of the tangent vector
$\partial_{-}\in T\Sigma$. Its integral curves are also spirals drawn on
$\Sigma$. Thus, the vectors $\partial_{\pm}$ span $T\Sigma$ locally, as
expected.

Consider now the contracted 5-form

$0=i_{\mathcal{R}^{\prime}}\Big{(}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\wedge\gamma\Big{)}=\kappa\wedge
d_{\mathbb{M}}\kappa\wedge\gamma+\Omega\wedge\kappa\wedge d_{\mathbb{M}}\kappa
i_{\mathcal{R}^{\prime}}\gamma,$ (4.40)

for any $\gamma\in\Omega_{\mathbb{M}}^{1}.$ This result allows to write the
contraction with $\mathcal{R}^{\prime}$ as

$i_{\mathcal{R}^{\prime}}\gamma=-\frac{\kappa\wedge
d_{\mathbb{M}}\kappa\wedge\gamma}{\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa}.$ (4.41)

Using a similar result for $i_{\mathcal{R}}\gamma$, see (2.51), we get the
basic contractions

$\displaystyle\begin{aligned}
i_{\mathcal{R}}(d\tau)&=\frac{1}{\alpha_{\tau}},\text{ \ \ \ \,
}i_{\mathcal{R}}(d\eta)=0,\text{ \ \,
}i_{\mathcal{R}}(d\xi_{1})=\frac{1}{\alpha_{\tau}},\text{ \ \ \ \ \ \ \
}i_{\mathcal{R}}(d\xi_{2})=\frac{1}{\alpha_{\tau}},\\\
i_{\mathcal{R}^{\prime}}(d\tau)&=\frac{1}{2\Omega_{\tau}},\text{ \ \
}i_{\mathcal{R}^{\prime}}(d\eta)=0,\text{ \ \
}i_{\mathcal{R}^{\prime}}(d\xi_{1})=-\frac{1}{2\Omega_{\tau}},\text{ \ \
}i_{\mathcal{R}^{\prime}}(d\xi_{2})=-\frac{1}{2\Omega_{\tau}}.\end{aligned}$
(4.42)

From these results, we find that

$g_{\mathbb{M}}\left(\mathcal{R},\mathcal{R}\right)=g_{\mathbb{M}}\left(\mathcal{R}^{\prime},\mathcal{R}^{\prime}\right)=0,\text{
\ \
}\pounds_{\mathcal{R}}g_{\mathbb{M}}=\pounds_{\mathcal{R}^{\prime}}g_{\mathbb{M}}=0.$
(4.43)

Thus, $\mathcal{R}$ and $\mathcal{R}^{\prime}$ are light-like Killing vectors.
They are not orthogonal under $g_{\mathbb{M}}$ and the explicit value for
$g_{\mathbb{M}}\left(\mathcal{R},\mathcal{R}^{\prime}\right)$ will not be
important in what follows.

Once $g_{\mathbb{M}}$ has been defined, we consider now a Riemannian inner
product on the space $\mathcal{A}$ of gauge connections defined by the
expression

$\left(\mathbb{A},\mathbb{A}^{\prime}\right)_{\mathbb{M}}=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\text{Tr}\left(\mathbb{A\wedge\ast
A^{\prime}}\right),$ (4.44)

for $\mathbb{A},\mathbb{A}^{\prime}\in\mathcal{A}$ and use it to implement an
orthogonal decomposition of $\mathcal{A}$ in terms of the quotient space
$\overline{\mathcal{A}}=\mathcal{A}/\mathcal{S}$ and space $\mathcal{S}$ of
Abelian shifts. The Hodge duality operator $*$ to be used in what follows is
defined in terms of the metric introduced in (4.37), the coordinates
$x^{\mu}=(\tau,\eta,\xi_{1},\xi_{2})$, with $\mu=1,2,3,4$ and the orientation
$d\tau\wedge d\eta\wedge d\xi_{1}\wedge d\xi_{2}$. It gives

$\displaystyle\ast(d\tau)$ $\displaystyle=\sqrt{\rho}\sin\eta\cos\eta
d\eta\wedge d\xi_{1}\wedge d\xi_{2},\text{ \ \
}\ast(d\eta)=\sqrt{\rho}\sin\eta\cos\eta d\tau\wedge d\xi_{1}\wedge d\xi_{2},$
(4.45) $\displaystyle\ast(d\xi_{1})$ $\displaystyle=-\sqrt{\rho}\cot\eta
d\tau\wedge d\eta\wedge d\xi_{2},\text{ \ \ \ \ \ \
}\ast(d\xi_{2})=\sqrt{\rho}\tan\eta d\tau\wedge d\eta\wedge d\xi_{1}.$

We also have the useful expressions

$\ast 1=2\pi^{2}\rho d\tau\wedge\alpha\wedge d\alpha,\text{ \ \
}\ast(d\tau)=2\pi^{2}\sqrt{\rho}\alpha\wedge d\alpha,\text{ \ \
}\ast\alpha=\frac{1}{2}\sqrt{\rho}d\tau\wedge d\alpha.$ (4.46)

First, we consider the shift group space $\mathcal{S}$. Elements in this space
are of the form $s\kappa$, $s^{\prime}\Omega$ and satisfy the relations

$\left(s\kappa,s^{\prime}\kappa\right)_{\mathbb{M}}=0,\text{ \ \
}\left(s\Omega,s^{\prime}\Omega\right)_{\mathbb{M}}=0,\text{ \ \
}\left(s\Omega,s^{\prime}\kappa\right)_{\mathbb{M}}=-\frac{\sqrt{\rho}}{\alpha_{\tau}}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(ss^{\prime}\right),$ (4.47)

where we have used

$\ast\kappa=\frac{\sqrt{\rho}}{\alpha_{\tau}}\kappa\wedge
d_{\mathbb{M}}\kappa,\text{ \ \
}\ast\Omega=\ast\omega-\frac{\sqrt{\rho}}{\alpha_{\tau}}\Omega\wedge
d_{\mathbb{M}}\kappa,$ (4.48)

with

$\omega=m\Big{(}d\eta+i\sin\eta\cos\eta\left(d\xi_{1}-d\xi_{2}\right)\Big{)},\text{
\ \ }\ast\omega=m\sqrt{\rho}\Big{(}\sin\eta\cos\eta d\tau\wedge d\xi_{1}\wedge
d\xi_{2}-2\pi id\tau\wedge d\eta\wedge\alpha\Big{)}$ (4.49)

and $m=\varphi(z_{0}/z_{1})e^{i(\xi_{1}+\xi_{2})}/z_{1}^{2}$. The norm of any
element belonging to $\mathcal{S}$ is then

$\left(s\kappa+s^{\prime}\Omega,s\kappa+s^{\prime}\Omega\right)_{\mathbb{M}}=2\left(s\Omega,s^{\prime}\kappa\right)_{\mathbb{M}}.$
(4.50)

Notice that the last term in (4.47) is proportional to the first contribution
on the rhs of the inner product defined before in (2.42) and we can consider
(4.50) as being defined independently of any metric in $\mathbb{M}$. Some
useful results used are

$\kappa\wedge\ast\omega=\omega\wedge\ast\kappa=\omega\wedge\ast\omega=\Omega\wedge\ast\omega=\omega\wedge\ast\Omega=0.$
(4.51)

Second, we consider the quotient space $\overline{\mathcal{A}}$. Elements in
this space are taken to be of the form
$\mathbb{A}^{\perp}=\mathbb{A}-p\kappa-q\Omega$, where
$p,q\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$ are to be determined by the
orthogonality conditions

$\left(\mathbb{A}^{\perp},s\kappa\right)_{\mathbb{M}}=0,\text{ \ \
}\left(\mathbb{A}^{\perp},s\Omega\right)_{\mathbb{M}}=0.$ (4.52)

We find that

$p=i_{\mathcal{R}}\mathbb{A}+X,\text{ \ \
}q=i_{\mathcal{R}^{\prime}}\mathbb{A},$ (4.53)

where

$X=\frac{\alpha_{\tau}}{\sqrt{\rho}}\frac{\mathbb{A}\wedge\ast\omega}{\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa}.$ (4.54)

As a consequence, the norm of any element $\mathbb{A}$ can be decomposed in
the form

$\left(\mathbb{A},\mathbb{A}\right)_{\mathbb{M}}=\left(\mathbb{A}^{\perp},\mathbb{A}^{\perp}\right)_{\mathbb{M}}+2\left(p\kappa,q\Omega\right)_{\mathbb{M}}.$
(4.55)

Now, we proceed to relate the first term on the rhs in (4.55) to the pre-
symplectic form (2.1) in order to define a Kähler metric structure on
$\overline{\mathcal{A}}$. Using

$i_{\mathcal{R}^{\prime}}\mathbb{A}^{\perp}=i_{\mathcal{R}^{\prime}}\mathbb{A}-q=0,\text{
\ \
}i_{\mathcal{R}^{\prime}}\left(\Omega\wedge\mathbb{A}^{\perp}\right)=\mathbb{A}^{\perp}$
(4.56)

and the contracted 5-form

$0=i_{\mathcal{R}^{\prime}}\left(\Omega\wedge\mathbb{A}^{\perp}\wedge\ast\mathbb{A}^{\perp}\right)=\mathbb{A}^{\perp}\wedge\ast\mathbb{A}^{\perp}+\Omega\wedge\mathbb{A}^{\perp}\wedge\left(i_{\mathcal{R}^{\prime}}\circ\ast\right)\mathbb{A}^{\perp},$
(4.57)

we have that

$\mathbb{A}^{\perp}\wedge\ast\mathbb{A}^{\perp}=-\Omega\wedge\mathbb{A}^{\perp}\wedge\left(i_{\mathcal{R}^{\prime}}\circ\ast\right)\mathbb{A}^{\perp}.$
(4.58)

In a similar way, we use

$i_{\mathcal{R}}\mathbb{A}^{\perp}=i_{\mathcal{R}}\mathbb{A}-p=-X,\text{ \ \
}i_{\mathcal{R}}\left(\Omega\wedge\kappa\wedge\mathbb{A}^{\perp}\right)=-\Omega\wedge\mathbb{A}^{\perp}+\Omega\wedge\kappa
i_{\mathcal{R}}\mathbb{A}^{\perp}$ (4.59)

and the contracted 5-form

$\displaystyle 0=i_{\mathcal{R}}\Big{(}\Omega\wedge\kappa\wedge$
$\displaystyle\mathbb{A}^{\perp}\wedge\left(i_{\mathcal{R}^{\prime}}\circ\ast\right)\mathbb{A}^{\perp}\Big{)}$
(4.60)
$\displaystyle=i_{\mathcal{R}}\left(\Omega\wedge\kappa\wedge\mathbb{A}^{\perp}\right)\wedge\left(i_{\mathcal{R}^{\prime}}\circ\ast\right)\mathbb{A}^{\perp}-\Omega\wedge\kappa\wedge\mathbb{A}^{\perp}\wedge\left(i_{\mathcal{R}}\circ
i_{\mathcal{R}^{\prime}}\circ\ast\right)\mathbb{A}^{\perp}$

to find that

$\mathbb{A}^{\perp}\wedge\ast\mathbb{A}^{\perp}=\Omega\wedge\kappa\wedge\mathbb{A}^{\perp}\wedge\ast_{2}\mathbb{A}^{\perp}-\Omega\wedge\kappa
X\wedge\ast_{3}\mathbb{A}^{\perp},$ (4.61)

where we have introduced two ‘effective’ Hodge duality operators defined by

$\ast_{2}=i_{\mathcal{R}}\circ i_{\mathcal{R}^{\prime}}\circ\ast\text{ \ \
}\ast_{3}=-i_{\mathcal{R}^{\prime}}\circ\ast.$ (4.62)

Then, (4.55) becomes

$\left(\mathbb{A},\mathbb{A}\right)_{\mathbb{M}}=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}^{\perp}\mathbb{\wedge\ast}_{2}\mathbb{A}^{\perp}\right)+\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(X\mathbb{\ast}_{3}\mathbb{A}^{\perp}\right)+2\left(p\kappa,q\Omega\right)_{\mathbb{M}}.$
(4.63)

The expression (4.63) can be simplified a bit more if now we introduce the
quantity

$\Pi\left(\mathbb{A}\right)=\mathbb{A}-(i_{\mathcal{R}}\mathbb{A})\kappa-(i_{\mathcal{R}^{\prime}}\mathbb{A})\Omega$
(4.64)

and write $\mathbb{A}^{\perp}=\Pi\left(\mathbb{A}\right)-X\kappa.$ Notice that
$i_{\mathcal{R}}\circ\Pi=i_{\mathcal{R}^{\prime}}\circ\Pi=0$, hence
$\Pi\left(\mathbb{A}\right)\subset\text{ker}_{i_{\mathcal{R}}}\cap\text{ker}_{i_{\mathcal{R}^{\prime}}}$.
Thus, after using the fact that $\ast_{2}\kappa=$ $\ast_{3}\kappa=0$, we get

$\displaystyle\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}^{\perp}\mathbb{\wedge\ast}_{2}\mathbb{A}^{\perp}\right)$
$\displaystyle=\Omega\wedge\kappa\wedge\text{Tr}\big{(}\Pi\left(\mathbb{A}\right)\mathbb{\wedge\ast}_{2}\Pi\left(\mathbb{A}\right)\big{)},$
(4.65)
$\displaystyle\Omega\wedge\kappa\wedge\text{Tr}\left(X\mathbb{\ast}_{3}\mathbb{A}^{\perp}\right)$
$\displaystyle=\Omega\wedge\kappa\wedge\text{Tr}\big{(}X\mathcal{\ast}_{3}\Pi\left(\mathbb{A}\right)\big{)}.$

Explicitly, we find that

$\Pi\left(\mathbb{A}\right)=A_{\eta}d\eta+\Big{(}\frac{\alpha_{2}}{\alpha_{\tau}}A_{1}-\frac{\alpha_{1}}{\alpha_{\tau}}A_{2}\Big{)}\left(d\xi_{1}-d\xi_{2}\right)-\frac{1}{4\zeta}\Big{(}A_{\tau}-A_{1}-A_{2}\Big{)}\omega,$
(4.66)

where we have taken

$\mathbb{A}=A_{\tau}d\tau+A_{\eta}d\eta+A_{1}d\xi_{1}+A_{2}d\xi_{2}$ (4.67)

and used (4.31) with $\alpha=\alpha_{1}d\xi_{1}+\alpha_{2}d\xi_{2}$. Notice
that the 1-forms $d\eta$ and $\left(d\xi_{1}-d\xi_{2}\right)$ span the space
$\text{ker}_{i_{R}}=\Omega_{\text{M,Hor}}^{1}$ of horizontal 1-forms defined
by the vector field $R$, cf. (4.36) above.

On the one hand, we obtain that

$\Omega\wedge\kappa\wedge\mathcal{\ast}_{3}(d\eta)=0,\text{ \ \
}\Omega\wedge\kappa\wedge\mathcal{\ast}_{3}(d\xi_{1}-d\xi_{2})=0,$ (4.68)

showing that the second contribution on the rhs in (4.63) is absent. On the
other hand, from the explicit expressions

$\displaystyle\mathcal{\ast}_{2}(d\tau)$ $\displaystyle=0,\text{ \ \
}\mathcal{\ast}_{2}(d\eta)=-\frac{\pi\sqrt{\rho}}{\zeta}\sin\eta\cos\eta\left(d\xi_{1}-d\xi_{2}\right),$
(4.69) $\displaystyle\mathcal{\ast}_{2}(d\xi_{1})$
$\displaystyle=\frac{\pi\sqrt{\rho}}{\zeta}\cot\eta d\eta,\text{ \ \
}\mathcal{\ast}_{2}(d\xi_{2})=-\frac{\pi\sqrt{\rho}}{\zeta}\tan\eta d\eta,$

we find that the ‘effective’ 2-dimensional Hodge duality operator $\ast_{2}$
actually defines a complex structure on the space
$\Omega_{\text{M},\text{Hor}}^{1}$, i.e. it obeys $\ast_{2}^{2}=-1$, provided
we fix the constant $\rho$ to be

$\sqrt{\rho}=\alpha_{\tau}\Omega_{\tau}=\frac{\zeta}{\pi}.$ (4.70)

It satisfies the basic relation

$\ast_{2}1=\zeta d\alpha=\underline{\pi}^{\ast}(\zeta\sigma_{C}),$ (4.71)

where we have used the defining relation (3.6) with $n=1$.

As a consequence of all these results, we get the desired relation

$\left(\mathbb{A},\mathbb{A}\right)_{\mathbb{M}}=\hat{\Omega}\big{(}\Pi\left(\mathbb{A}\right),\mathbb{\ast}_{2}\Pi\left(\mathbb{A}\right)\big{)}+2\left(p\kappa,q\Omega\right)_{\mathbb{M}},$
(4.72)

where we have

$\hat{\Omega}\big{(}\Pi\left(\mathbb{A}\right),\mathbb{\ast}_{2}\Pi\left(\mathbb{A}\right)\big{)}=-\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\Big{(}\Pi\left(\mathbb{A}\right)\mathbb{\wedge\ast}_{2}\Pi\left(\mathbb{A}\right)\Big{)},$
(4.73)

with $\hat{\Omega}$ being the pre-symplectic form defined in (2.1). Recall
that now $\hat{\Omega}|_{\overline{\mathcal{A}}}$ is symplectic.

Let us now consider the gauge group $\mathcal{G}$ and use (2.110) to write

$\mathbb{I}=\kappa g^{-1}\pounds_{\mathcal{R}}g+g^{-1}\gamma(g).$ (4.74)

Using ${}^{g}\mathbb{A}=g^{-1}\mathbb{A}g+\mathbb{I}$ and (4.74) in (4.64),
gives

$\Pi(^{g}\mathbb{A})=g^{-1}\Pi(\mathbb{A})g+g^{-1}\gamma(g).$ (4.75)

Then
$\Pi\left({}^{g}\mathbb{A}\right)\subset\text{ker}_{i_{\mathcal{R}}}\cap\text{ker}_{i_{\mathcal{R}^{\prime}}}$
and the quotient space $\overline{\mathcal{A}}$ is preserved by the action of
a ‘restricted’ gauge group defined by the elements $g$ satisfying (2.110).

Notice that $d_{\mathbb{M}}\Omega\sim d\eta\wedge(d\xi_{1}-d\xi_{2})$. In this
case, in order to get $d_{\mathbb{M}}\Omega\wedge\gamma=0$, we propose a
linear combination of the form

$g^{-1}\gamma(g)=Xd\eta+Y(d\xi_{1}-d\xi_{2}),$ (4.76)

with

$X=g^{-1}\partial_{\eta}g,\text{ \ \
}Y=\frac{\alpha_{2}}{\alpha_{\tau}}g^{-1}\partial_{1}g-\frac{\alpha_{1}}{\alpha_{\tau}}g^{-1}\partial_{2}g.$
(4.77)

This choice in (4.75) provide the usual gauge field transformations

${}^{g}A_{i}=g^{-1}A_{i}g+g^{-1}\partial_{i}g,\text{ \ \ }i=\tau,\eta,1,2,$
(4.78)

with $g$ obeying $\pounds_{\mathcal{R}^{\prime}}g=0$, namely

$\partial_{\tau}g-\partial_{1}g-\partial_{2}g=0.$ (4.79)

Furthermore, the WZ-type term (2.67) becomes

$\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\chi(g)=-\frac{\zeta}{\pi}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d\tau\wedge
d\eta\wedge d\xi_{1}\wedge
d\xi_{2}\text{Tr}\Big{(}g^{-1}\pounds_{\mathcal{R}}g[X,Y]\Big{)}.$ (4.80)

The expression (4.74) exhibits an orthogonal decomposition, under the inner
product (4.44), of the current
$\mathbb{I}\in\Omega_{\mathbb{M}}^{1}\otimes\mathfrak{g}$ along
$\text{im}_{i_{\mathcal{R}}}$ and
$\text{ker}_{i_{R}}\subset\text{ker}_{i_{\mathcal{R}}}$ while keeping
$\mathbb{I}\in\text{ker}_{i_{\mathcal{R}^{\prime}}}$. This follows from the
fact that $\text{im}_{i_{\mathcal{R}}}^{\perp}=\text{ker}_{i_{R}}$, because of

$\kappa\wedge\ast d\eta=0,\text{ \ \ }\kappa\wedge\ast d(\xi_{1}-\xi_{2})=0.$
(4.81)

Alternatively, in the local bundle coordinates $(z,\sigma)$ over
$\mathcal{U}_{1}$, we have that $d_{\mathbb{M}}\Omega\sim dz\wedge
d\overline{z}$ and this time we propose instead

$g^{-1}\gamma(g)=Xdz+Yd\overline{z},$ (4.82)

with

$X=g^{-1}\partial_{z}g-\frac{\alpha_{z}}{\alpha_{\tau}}g^{-1}\partial_{\sigma}g,\text{
\ \
}Y=g^{-1}\partial_{\overline{z}}g-\frac{\alpha_{\overline{z}}}{\alpha_{\tau}}g^{-1}\partial_{\sigma}g.$
(4.83)

Using this in

$\Pi(\mathbb{A})=\Big{(}A_{z}-\frac{\alpha_{z}}{\alpha_{\tau}}A_{\sigma}\Big{)}dz+\Big{(}A_{\overline{z}}-\frac{\alpha_{\overline{z}}}{\alpha_{\tau}}A_{\sigma}\Big{)}d\overline{z}-\frac{1}{4\zeta}\Big{(}A_{\tau}-A_{\sigma}\Big{)}\omega,$
(4.84)

gives

${}^{g}A_{i}=g^{-1}A_{i}g+g^{-1}\partial_{i}g,\text{ \ \
}i=\tau,\sigma,z,\overline{z},$ (4.85)

for the gauge field transformations. The condition
$\pounds_{\mathcal{R}^{\prime}}g=0$, now takes the form

$\partial_{\tau}g-\partial_{\sigma}g=0.$ (4.86)

This is precisely the condition $\partial_{-}g=0$ we found before. The WZ-type
term takes the same form as in (4.80), but with the volume form $d\tau\wedge
d\sigma\wedge dz\wedge d\overline{z}$.

Now, we consider the WZ-type term. Using (4.74) in (4.80), allows to
write121212This can be shown in an explicit way by using the results (4.79) or
(4.86).

$\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\chi(g)=2\Omega_{\tau}\mathop{\displaystyle\int}\nolimits_{\mathbb{R}\times\text{M}}d\tau\wedge\chi(g)^{\prime},$
(4.87)

where we have defined, cf. (2.68),

$\chi(g)^{\prime}=-\frac{1}{3}\text{Tr}(I\wedge I\wedge I),$ (4.88)

with $I=g^{-1}dg$ being the contribution along M in the decomposition
$\mathbb{I}=I_{\tau}d\tau+I$. In principle, the WZ-type term contribution can
be set to zero if further analytic restrictions are imposed over the elements
$g\in\mathcal{G}$. For example, by enforcing the holomorphicity condition
$\partial_{\overline{z}}g=0$ when using the local bundle coordinates
$(z,\sigma)$. Also notice the possibility of taking the limit
$\zeta\rightarrow 0$ as well. It is important to emphasize that at this level
of analysis it is not clear if (4.87) could, alternatively, be related to some
quantization condition as occurs in conventional CS theories.

Finally, the result (4.73) shows that the metric induced by the Riemannian
metric (4.44) on the quotient space $\overline{\mathcal{A}}$ is Kähler, with
respect to the symplectic form $\hat{\Omega}|_{\overline{\mathcal{A}}}$ and
complex structure

$J=\ast_{2}=i_{\mathcal{R}}\circ i_{\mathcal{R}^{\prime}}\circ\ast.$ (4.89)

This is an important result and we will come back to it later on when
considering the symplectic measure of the generalized 4d CS theory path
integral, see (1.5).

###  4.3 Hamiltonian analysis and 4d Chern-Simons theory

In order to perform the Hamiltonian analysis and implement explicitly the
steps I and II mentioned before in section (3), we first need to isolate the
time differential $d\tau$ from all expressions. The exterior derivative
$d_{\mathbb{M}}$ was already introduced in (3.5) and now we write it in terms
of the local bundle coordinates $(z,\sigma)$ over $\mathcal{U}_{1}$. Thus,

$d_{\mathbb{M}}=d\tau\wedge\partial_{\tau}+d,\text{ \ \
}d=dz\wedge\partial_{z}+d\overline{z}\wedge\partial_{\overline{z}}+d\sigma\wedge\partial_{\sigma}.$
(4.90)

Some quantities of interest to be used below are:
i) The curvature of the connection $\mathbb{A}$ under the decomposition
(4.24), namely,

$F_{\mathbb{A}}=F_{A}+d\tau\wedge F_{\tau},\text{ \ \
}F_{\tau}=\partial_{\tau}A-d_{A}A_{\tau},$ (4.91)

where $F_{A}=dA+A\wedge A$ is the curvature of $A$ and $d_{A}=d+[A,*]$.
ii) The Lie algebra value field $\Phi$ defined previously in (2.52), which now
takes the form

$\Phi=2\frac{\big{(}\Omega_{\tau}\tilde{\kappa}-\kappa_{\tau}\tilde{\Omega}\big{)}\wedge
F_{A}+\tilde{\Omega}\wedge\tilde{\kappa}\wedge
F_{\tau}+d\tilde{\Omega}\wedge\left(\kappa_{\tau}A-\tilde{\kappa}A_{\tau}\right)}{\Omega_{\tau}\alpha\wedge
d\alpha}.$ (4.92)

iii) The Chern-Simons 3-form

$CS(\mathbb{A})=-d\tau\wedge\text{Tr}\left(A\wedge\partial_{\tau}A-2A_{\tau}F_{A}\right)-d\tau\wedge
d\text{Tr}\left(A_{\tau}A\right)+CS(A).$ (4.93)

Also, introduce the following variables

$A^{\prime}_{\tau}=A_{\tau}-\kappa_{\tau}\Phi,\text{ \ \
}A^{\prime}=A-\tilde{\kappa}\Phi.$ (4.94)

They are invariant under $\kappa$-shifts and will be useful for writing
results coming from the Hamiltonian analysis in a more compact way.

From the expression (2.81), we quickly obtain the Lagrangian of the theory

$\displaystyle
L=ic\bigg{\\{}\mathop{\displaystyle\int}\nolimits_{\text{M}}\tilde{\Omega}\wedge\text{Tr}\big{(}A\wedge\partial_{\tau}A$
$\displaystyle-2A_{\tau}F_{A}\big{)}-\frac{1}{2}\Omega_{\tau}\mathop{\displaystyle\int}\nolimits_{\text{M}}\alpha\wedge
d\alpha\text{Tr}\left(\Phi^{2}\right)$ (4.95)
$\displaystyle+\Omega_{\tau}\mathop{\displaystyle\int}\nolimits_{\text{M}}CS(A)+\mathop{\displaystyle\int}\nolimits_{\text{M}}d\tilde{\Omega}\wedge\text{Tr}\Big{(}A_{\tau}A+\left(\kappa_{\tau}A-\tilde{\kappa}A_{\tau}\right)i_{\mathcal{R}}\mathbb{A}\Big{)}\bigg{\\}}.$

Because of the Lagrangian is quadratic in $\Phi$, we see from (4.91), that the
theory is not only linear in the velocities $\partial_{\tau}A$ but also
quadratic on them.

To find the canonical momenta, we compute the variation of the Lagrangian
(4.95) with respect to all field components in
$\delta(\partial_{\tau}\mathbb{A})$. Using the following result

$\Omega_{\tau}\alpha\wedge
d\alpha\delta_{\partial_{\tau}A}\Phi=2\tilde{\Omega}\wedge\tilde{\kappa}\wedge\delta(\partial_{\tau}A),$
(4.96)

we find that

$\delta_{\partial_{\tau}A}L=\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\Big{(}\delta(\partial_{\tau}A)\wedge
P\Big{)},$ (4.97)

where the 2-form $P\in\Omega_{\text{M}}^{2}\otimes\mathfrak{g}$ is given by

$P=ic\left(\tilde{\Omega}\wedge
A-2\tilde{\Omega}\wedge\tilde{\kappa}\Phi\right).$ (4.98)

The components in the expansion

$P=P_{z}d\overline{z}\wedge d\sigma+P_{\overline{z}}d\sigma\wedge
dz+P_{\sigma}dz\wedge d\overline{z}$ (4.99)

are, actually, the usual canonical momenta defined by

$\displaystyle P_{z}$ $\displaystyle=\frac{\delta
L}{\delta\left(\partial_{\tau}A_{z}\right)}=ic\Big{(}\Omega_{\overline{z}}\left(A_{\sigma}-2\kappa_{\sigma}\Phi\right)-\Omega_{\sigma}\left(A_{\overline{z}}-2\kappa_{\overline{z}}\Phi\right)\Big{)},$
(4.100) $\displaystyle P_{\overline{z}}$ $\displaystyle=\frac{\delta
L}{\delta\left(\partial_{\tau}A_{\overline{z}}\right)}=ic\Big{(}\Omega_{\sigma}\left(A_{z}-2\kappa_{z}\Phi\right)-\Omega_{z}\left(A_{\sigma}-2\kappa_{\sigma}\Phi\right)\Big{)},$
$\displaystyle P_{\sigma}$ $\displaystyle=\frac{\delta
L}{\delta\left(\partial_{\tau}A_{\sigma}\right)}=ic\Big{(}\Omega_{z}\left(A_{\overline{z}}-2\kappa_{\overline{z}}\Phi\right)-\Omega_{\overline{z}}\left(A_{z}-2\kappa_{z}\Phi\right)\Big{)},$

where we have used the local decompositions

$\tilde{\Omega}=\Omega_{z}dz+\Omega_{\overline{z}}d\overline{z}+\Omega_{\sigma}d\sigma,\text{
\ \
}\tilde{\kappa}=\kappa_{z}dz+\kappa_{\overline{z}}d\overline{z}+\kappa_{\sigma}d\sigma,\text{
\ \ }A=A_{z}dz+A_{\overline{z}}d\overline{z}+A_{\sigma}d\sigma.$ (4.101)

Also, notice that

$P_{\tau}=\frac{\delta L}{\delta\left(\partial_{\tau}A_{\tau}\right)}=0.$
(4.102)

From the absence of the term $\partial_{\tau}A_{\tau}$ in the Lagrangian and
the very form of $P$, we easily detect the presence of three primary
constraints given by

$P_{\tau}\approx 0,\text{ \ \ }\tilde{\kappa}\wedge
P+ic\tilde{\Omega}\wedge\tilde{\kappa}\wedge A\approx 0,\text{ \ \
}\tilde{\Omega}\wedge P\approx 0.$ (4.103)

In components, the last two constraints can be written in the form

$\tilde{\kappa}\wedge P+ic\tilde{\Omega}\wedge\tilde{\kappa}\wedge
A=\phi_{\tilde{\kappa}}dz\wedge d\overline{z}\wedge d\sigma\approx 0,\text{ \
\ }\tilde{\Omega}\wedge P=\phi_{\tilde{\Omega}}dz\wedge d\overline{z}\wedge
d\sigma\approx 0,$ (4.104)

with

$\displaystyle\phi_{\tilde{\kappa}}=\kappa_{z}P_{z}+\kappa$
$\displaystyle{}_{\overline{z}}P_{\overline{z}}+\kappa_{\sigma}P_{\sigma}$
(4.105)
$\displaystyle+ic\Big{(}\left(\Omega_{\overline{z}}\kappa_{\sigma}-\Omega_{\sigma}\kappa_{\overline{z}}\right)A_{z}+\left(\Omega_{\sigma}\kappa_{z}-\Omega_{z}\kappa_{\sigma}\right)A_{\overline{z}}+\left(\Omega_{z}\kappa_{\overline{z}}-\Omega_{\overline{z}}\kappa_{z}\right)A_{\sigma}\Big{)}\approx
0,$

and

$\phi_{\tilde{\Omega}}=\Omega_{z}P_{z}+\Omega_{\overline{z}}P_{\overline{z}}+\Omega_{\sigma}P_{\sigma}\approx
0.$ (4.106)

Then, the three primary constraints are

$P_{\tau}\approx 0,\text{ \ \ }\phi_{\tilde{\kappa}}\approx 0,\text{ \ \
}\phi_{\tilde{\Omega}}\approx 0.$ (4.107)

Let us now introduce the compact notation

$\text{Tr}\left(A,B\right)_{\left(z,\sigma\right)}=\mathop{\displaystyle\int}\nolimits_{\text{M}}d\text{Vol}\text{Tr}\left(AB\right),$
(4.108)

where $d\text{Vol}=dz\wedge d\overline{z}\wedge d\sigma$.

The canonical Poisson bracket of the theory is defined by

$\left\\{f,g\right\\}=\text{Tr}\left(\frac{\delta f}{\delta
A_{i}(\sigma,z)},\frac{\delta g}{\delta P_{i}(\sigma,z)}-\frac{\delta
f}{\delta P_{i}(\sigma,z)},\frac{\delta g}{\delta
A_{i}(\sigma,z)}\right)_{\left(z,\sigma\right)},$ (4.109)

where $i,j=\tau,\sigma,z,\overline{z}$. Then, the phase space coordinates
satisfy the usual relations

$\left\\{A_{i}(\sigma,z)_{\mathbf{1}},P_{j}(\sigma^{\prime},z^{\prime})_{\mathbf{2}}\right\\}=C_{\mathbf{12}}\delta_{ij}\delta_{\sigma\sigma^{\prime}}\delta_{zz^{\prime}},$
(4.110)

where $C_{\mathbf{12}}=\eta^{AB}T_{A}\otimes T_{B}$ is the tensor Casimir of
the Lie algebra $\mathfrak{g}$,
$\delta_{\sigma\sigma^{\prime}}=\delta(\sigma-\sigma^{\prime})$ and
$\delta_{zz^{\prime}}=\delta(z-z^{\prime})$ are Dirac delta distributions. The
latter can be written locally in the form

$\delta_{zz^{\prime}}=-\frac{1}{2\pi
i}\partial_{\overline{z}}\left(\frac{1}{z-z^{\prime}}\right).$ (4.111)

Define now the quantities

$\mathcal{S}_{\kappa}(s)=\text{Tr}\left(s,\phi_{\kappa}\right)_{\left(z,\sigma\right)},\text{
\ \
}\mathcal{S}_{\Omega}(s^{\prime})=\text{Tr}\left(s^{\prime},\phi_{\Omega}\right)_{\left(z,\sigma\right)},$
(4.112)

where $s,s\textasciiacute\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$ are
arbitrary and

$\phi_{\kappa}=\kappa_{\tau}P_{\tau}+\phi_{\tilde{\kappa}},\text{ \ \
}\phi_{\Omega}=\Omega_{\tau}P_{\tau}+\phi_{\tilde{\Omega}}.$ (4.113)

They generate both Abelian $U(1)\times U(1)$ shifts in $\mathcal{S}$

$\delta_{\kappa}A_{i}=\left\\{A_{i},\mathcal{S}_{\kappa}(s)\right\\}=\kappa_{i}s,\text{
\ \
}\delta_{\Omega}A_{i}=\left\\{A_{i},\mathcal{S}_{\Omega}(s^{\prime})\right\\}=\Omega_{i}s^{\prime},$
(4.114)

under the bracket (4.110) and Poisson commute between.

The canonical Hamiltonian is given by the Legendre transformation

$H=\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\left(P\wedge\partial_{\tau}A\right)+\mathop{\displaystyle\int}\nolimits_{\text{M}}d\text{Vol}\text{Tr}\left(P_{\tau}\partial_{\tau}A_{\tau}\right)-L$
(4.115)

and the total Hamiltonian is given by

$H_{T}=H+\text{Tr}\left(u_{\tau},P_{\tau}\right)_{(z,\sigma)}+\mathcal{S}_{\kappa}(u)+\mathcal{S}_{\Omega}(u^{\prime}),$
(4.116)

where $u_{\tau},u,u^{\prime}\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$ are
arbitrary Lagrange multipliers.

The variation $\delta H$ must depend Teitel  only on the variations $(\delta
A$, $\delta P)$ and $(\delta A_{\tau}$, $\delta P_{\tau})$. In order to find
it, we first compute the variation of the $P$-dependent contribution to $H$,
then use (4.98), (4.102) and finally subtract $\delta L$. We find that

$\delta H=\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\left(\delta
P\wedge\partial_{\tau}A\right)+\mathop{\displaystyle\int}\nolimits_{\text{M}}d\text{Vol}\text{Tr}\left(\delta
P_{\tau}\partial_{\tau}A_{\tau}\right)+ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\left[\delta(\partial_{\tau}A)\wedge\left(\tilde{\Omega}\wedge
A-2\tilde{\Omega}\wedge\tilde{\kappa}\Phi\right)\right]-\delta L.$ (4.117)

To compute $\delta L$ correctly, we calculate the variation of $L$ without
performing any integration by parts involving the derivative $\partial_{\tau}$
along the $\tau$ direction. Thus, by using the identity

$\displaystyle\Omega_{\tau}\alpha\wedge d\alpha\text{Tr}\left(\delta\Phi
X\right)$ $\displaystyle=2\text{Tr}\left[\delta
A\wedge\left(-(\Omega_{\tau}\tilde{\kappa}-\kappa_{\tau}\tilde{\Omega})\wedge
d_{A}X+\Omega_{\tau}d\tilde{\kappa}X-\tilde{\Omega}\wedge\tilde{\kappa}\left[A_{\tau},X\right]\right)\right]$
(4.118) $\displaystyle+2\text{Tr}\left[\delta
A_{\tau}\left(-\tilde{\Omega}\wedge
d\tilde{\kappa}X+\tilde{\Omega}\wedge\tilde{\kappa}\wedge
d_{A}X\right)\right]+2\text{Tr}\left[\delta(\partial_{\tau}A)\wedge\left(\tilde{\Omega}\wedge\tilde{\kappa}X\right)\right],$

with $X=\Phi$ being an element of
$\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$, we find from (4.95), that

$\displaystyle\delta L$
$\displaystyle=ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\left[\delta
A\wedge\left(-\tilde{\Omega}\wedge\partial_{\tau}A+2\tilde{\Omega}\wedge
d_{A^{\prime}}A_{\tau}^{\prime}+2\Omega_{\tau}F_{A^{\prime}}\right)\right]$
(4.119)
$\displaystyle+ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\left[\delta
A_{\tau}\left(-2\tilde{\Omega}\wedge
F_{A^{\prime}}\right)\right]+ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\left[\delta(\partial_{\tau}A)\wedge\left(\tilde{\Omega}\wedge
A-2\tilde{\Omega}\wedge\tilde{\kappa}\Phi\right)\right]$
$\displaystyle+2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}d\tilde{\Omega}\wedge
d\sigma\wedge\text{Tr}\Big{[}\delta
A_{\tau}\left(A_{\sigma}-\kappa_{\sigma}i_{\mathcal{R}}\mathbb{A}\right)-\delta
A_{\sigma}\left(A_{\tau}-\kappa_{\tau}i_{\mathcal{R}}\mathbb{A}\right)\Big{]}.$

Then,

$\displaystyle\delta H$
$\displaystyle=\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\left(\delta
P\wedge\partial_{\tau}A\right)+ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\left[\delta
A\wedge\left(\tilde{\Omega}\wedge\partial_{\tau}A-2\tilde{\Omega}\wedge
d_{A^{\prime}}A_{\tau}^{\prime}-2\Omega_{\tau}F_{A^{\prime}}\right)\right]$
(4.120)
$\displaystyle+\mathop{\displaystyle\int}\nolimits_{\text{M}}d\text{Vol}\text{Tr}\left(\delta
P_{\tau}\partial_{\tau}A_{\tau}\right)+ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\left[\delta
A_{\tau}\left(2\tilde{\Omega}\wedge F_{A^{\prime}}\right)\right]$
$\displaystyle-2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}d\tilde{\Omega}\wedge
d\sigma\wedge\text{Tr}\Big{[}\delta
A_{\tau}\left(A_{\sigma}-\kappa_{\sigma}i_{\mathcal{R}}\mathbb{A}\right)-\delta
A_{\sigma}\left(A_{\tau}-\kappa_{\tau}i_{\mathcal{R}}\mathbb{A}\right)\Big{]},$

only depends on the phase space coordinates variations, as required.

Before we continue, as a consistency test, let us find the Lagrangian eom of
the theory. After integrating by parts the term $\partial_{\tau}(\delta A)$
and simplifying, we obtain

$\displaystyle\delta L$
$\displaystyle=-2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\left[\delta
A\left(\tilde{\Omega}\wedge
F_{\tau}^{\prime}-\Omega_{\tau}F_{A^{\prime}}\right)+\delta
A_{\tau}\left(\tilde{\Omega}\wedge F_{A^{\prime}}\right)\right]$ (4.121)
$\displaystyle+2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}d\tilde{\Omega}\wedge
d\sigma\wedge\text{Tr}\Big{[}\delta
A_{\tau}\left(A_{\sigma}-\kappa_{\sigma}i_{\mathcal{R}}\mathbb{A}\right)-\delta
A_{\sigma}\left(A_{\tau}-\kappa_{\tau}i_{\mathcal{R}}\mathbb{A}\right)\Big{]},$

where
$F_{\tau}^{\prime}=\partial_{\tau}A^{\prime}-d_{A^{\prime}}A_{\tau}^{\prime}$.
The eom extracted from (4.121) also derive from (2.89) and both are perfectly
equivalent.

Now, we return to the expression (4.120) and find the functional derivatives
of $H$ with respect to the phase space coordinate fields. We get that

$\displaystyle\frac{\delta H}{\delta A_{\tau}}$
$\displaystyle=2ic\Big{(}\Omega_{z}F_{\overline{z}\sigma}^{\prime}+\Omega_{\sigma}F_{z\overline{z}}^{\prime}+\Omega_{\overline{z}}F_{\sigma
z}^{\prime}-\Omega_{z\overline{z}}\left(A_{\sigma}-\kappa_{\sigma}i_{\mathcal{R}}\mathbb{A}\right)\Big{)},$
(4.122) $\displaystyle\frac{\delta H}{\delta A_{z}}$
$\displaystyle=icQ_{\overline{z}\sigma},\text{ \ \ }\frac{\delta H}{\delta
A_{\overline{z}}}=icQ_{\sigma z},\text{ \ \ }\frac{\delta H}{\delta
A_{\sigma}}=icQ_{z\overline{z}}+2ic\Omega_{z\overline{z}}\left(A_{\tau}-\kappa_{\tau}i_{\mathcal{R}}\mathbb{A}\right)$

and

$\frac{\delta H}{\delta P_{\tau}}=\partial_{\tau}A_{\tau},\text{ \ \
}\frac{\delta H}{\delta P_{z}}=\partial_{\tau}A_{z},\text{ \ \ }\frac{\delta
H}{\delta P_{\overline{z}}}=\partial_{\tau}A_{\overline{z}},\text{ \ \
}\frac{\delta H}{\delta P_{\sigma}}=\partial_{\tau}A_{\sigma},$ (4.123)

where we have used

$\displaystyle Q$
$\displaystyle=\tilde{\Omega}\wedge\partial_{\tau}A-2\tilde{\Omega}\wedge
d_{A^{\prime}}A_{\tau}^{\prime}-2\Omega_{\tau}F_{A^{\prime}},$ (4.124)
$\displaystyle=Q_{\overline{z}\sigma}d\overline{z}\wedge d\sigma+Q_{\sigma
z}d\sigma\wedge dz+Q_{z\overline{z}}dz\wedge d\overline{z},$ $\displaystyle
F_{A^{\prime}}$
$\displaystyle=F_{\overline{z}\sigma}^{\prime}d\overline{z}\wedge
d\sigma+F_{\sigma z}^{\prime}d\sigma\wedge
dz+F_{z\overline{z}}^{\prime}dz\wedge d\overline{z},$ $\displaystyle
d\tilde{\Omega}$ $\displaystyle=\Omega_{z\overline{z}}dz\wedge d\overline{z}.$

Armed with these expressions, now we are able to verify the time preservation
of the primary constraints. Starting with $P_{\tau}$, we find that

$\big{\\{}\text{Tr}\left(\eta,P_{\tau}\right)_{(z,\sigma)},H_{T}\big{\\}}=\gamma(\eta)\approx
0$ (4.125)

and from this we obtain a secondary constraint given by

$\gamma(\eta)=-2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\tilde{\Omega}\wedge\text{Tr}\left(\eta
F_{A^{\prime}}\right)+2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}d\tilde{\Omega}\wedge\text{Tr}\Big{(}\eta\left(A-\tilde{\kappa}i_{\mathcal{R}}\mathbb{A}\right)\Big{)}.$
(4.126)

Now, we consider $\mathcal{S}_{\kappa}(s)$ and obtain

$\displaystyle\left\\{H_{T},\mathcal{S}_{\kappa}(s)\right\\}=$
$\displaystyle-2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\text{Tr}\left[s\left((\Omega_{\tau}\tilde{\kappa}-\kappa_{\tau}\tilde{\Omega})\wedge
F_{A^{\prime}}+\tilde{\Omega}\wedge\tilde{\kappa}\wedge
F_{\tau}^{\prime}\right)\right]$ (4.127)
$\displaystyle+2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}d\tilde{\Omega}\wedge\text{Tr}\Big{[}s\Big{(}\tilde{\kappa}(A_{\tau}-\kappa_{\tau}i_{\mathcal{R}}\mathbb{A})-\kappa_{\tau}(A-\tilde{\kappa}i_{\mathcal{R}}\mathbb{A})\Big{)}\Big{]}.$

By using (4.92), (3.2), we get that

$(\Omega_{\tau}\tilde{\kappa}-\kappa_{\tau}\tilde{\Omega})\wedge
F_{A^{\prime}}+\tilde{\Omega}\wedge\tilde{\kappa}\wedge
F_{\tau}^{\prime}=d\tilde{\Omega}\wedge\left(\tilde{\kappa}A_{\tau}-\kappa_{\tau}A\right)$
(4.128)

and from (4.23), (4.25), we have

$A_{\tau}-\kappa_{\tau}i_{\mathcal{R}}\mathbb{A=}\frac{1}{2}\left(A_{\tau}-A_{\sigma}\right),\text{
\ \
}A_{\sigma}-\kappa_{\sigma}i_{\mathcal{R}}\mathbb{A}=-\frac{1}{2}\left(A_{\tau}-A_{\sigma}\right).$
(4.129)

Thus,

$\left\\{H_{T},\mathcal{S}_{\kappa}(s)\right\\}=0.$ (4.130)

For $\mathcal{S}_{\Omega}(s^{\prime})$, we follow a similar calculation to
obtain

$\left\\{H_{T},\mathcal{S}_{\Omega}(s^{\prime})\right\\}=2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}d\tilde{\Omega}\wedge\text{Tr}\left[s^{\prime}\left(\tilde{\Omega}(A_{\tau}-\kappa_{\tau}i_{\mathcal{R}}\mathbb{A})-\Omega_{\tau}(A-\tilde{\kappa}i_{\mathcal{R}}\mathbb{A})\right)\right]$
(4.131)

and after using (3.2), we get that

$\left\\{H_{T},\mathcal{S}_{\Omega}(s^{\prime})\right\\}=0.$ (4.132)

Not surprisingly, the canonical Hamiltonian of the theory is invariant under
the two independent shifts in $\mathcal{S}$.

There is a single secondary constraint $\gamma(\eta)$. From the expression
(4.126), it is not difficult to verify that it is invariant under both shift
transformations as well, hence

$\left\\{\gamma(\eta),\mathcal{S}_{\kappa}(s)\right\\}=0,\text{ \ \
}\left\\{\gamma(\eta),\mathcal{S}_{\Omega}(s^{\prime})\right\\}=0.$ (4.133)

It remains to verify if its time preservation introduce a tertiary constraint
or if instead it determines some of the Lagrange multipliers. From the
expression

$\left\\{H_{T},\gamma(\eta)\right\\}=\left\\{H,\gamma(\eta)\right\\}-\text{Tr}\left(u_{\tau},\frac{\delta\gamma(\eta)}{\delta
A_{\tau}}\right)_{(z,\sigma)}\approx 0,$ (4.134)

we realize that this is actually a condition over the Lagrange multiplier
$u_{\tau}$, provided that $\frac{\delta\gamma(\eta)}{\delta A_{\tau}}\neq 0$
and it is not difficult to check that this is indeed the case. The explicit
form for $u_{\tau}$ is not required in what follows, hence we do not need to
perform the calculation explicitly.

It is clear from (4.126) that the computation of the Poisson algebra for two
$\gamma^{\prime}s$ is rather tedious. Fortunately, at this stage, we are
mainly interested in implementing the step I mentioned above131313We will
consider step II below as well., see section (3), in the Hamiltonian
formulation. Thus, we gauge fix the first class constraint that generates the
$\kappa$-shift symmetry by choosing the following gauge fixing condition

$\Phi\approx 0.$ (4.135)

Now, the pair $\phi_{\kappa}\approx 0$ $,\Phi\approx 0$ of constraints become
a second class set and we impose them strongly by means of a Dirac bracket.
Notice that

$\displaystyle\left\\{\Phi(\sigma,z)_{\mathbf{1}},\phi_{\kappa}(\sigma^{\prime},z^{\prime})_{\mathbf{2}}\right\\}=C_{\mathbf{12}}\delta_{\sigma\sigma^{\prime}}\delta_{zz^{\prime}},\text{
\ \
}\left\\{\phi_{\kappa}(\sigma,z)_{\mathbf{1}},\Phi(\sigma^{\prime},z^{\prime})_{\mathbf{2}}\right\\}^{-1}=C_{\mathbf{12}}\delta_{\sigma\sigma^{\prime}}\delta_{zz^{\prime}}.$
(4.136)

Then, (4.135) is also a good gauge fixing condition.

The Dirac bracket is given by

$\displaystyle\big{\\{}f,g\big{\\}}^{\ast}$
$\displaystyle=\left\\{f,g\right\\}$ (4.137)