# Hamiltonian analysis and Faddeev-Jackiw formalism for two Dimensional
Quadratic Gravity expressed as BF theory

Jaime Manuel Cabrera and Jorge Mauricio Paulin Fuentes  División Académica de
Ciencias Básicas, Universidad Juárez Autónoma de Tabasco,
Km 1 Carretera Cunduacán-Jalpa, Apartado Postal 24, 86690 Cunduacán, Tabasco,
México

###### Abstract

We examine the model of Two-Dimensional Quadratic Gravity as a consequence of
symmetry breaking within the framework of background field (BF) theory. This
theory is essentially an extension of BF theory, introducing an additional
polynomial term that operates on both the gauge and background fields. We
analyze the theory using the Dirac and Faddeev-Jackiw procedures, determining
the form of the gauge transformation, the full structure of the constraints,
the counting of degrees of freedom, and the generalized Faddeev-Jackiw
brackets. Additionally, we demonstrate the coincidence of the Faddeev-Jackiw
and Dirac’s brackets. Finally, we provide some remarks and discuss prospects.

###### keywords:

Hamilton dynamics , Faddeev-Jackiw symplectic formalism , Topological theories

###### MSC:

[2010], 00-01, 99-00

††journal: Journal of LaTeX Templates

##  1 INTRODUCTION

Throughout several decades, physicists have dedicated significant effort to
generalize and extend General Relativity (GR) in various ways, with the goal
of achieving a unified framework that encompasses both gravity and the other
fundamental interactions within the framework of gauge theory, specifically
Yang-Mills (YM) theory, as well as through the exploration of Topological
Field Theories (TFT) [1, 2]. As previously mentioned, the search for a unified
model of fundamental interactions has motivated the utilization of different
frameworks. For example, one approach involves introducing higher dimensions
in gravity models, where the metric remains the fundamental field and its
components can be associated with the fundamental interactions [3, 4]. On the
other hand, the literature extensively highlights that elementary interactions
can be comprehended by considering the connection associated with an internal
symmetry group [5], where space-time is not considered a dynamic source.
Numerous attempts have been made to construct Yang-Mills type theories of
gravity. In this context, various formulations known as pure connection
actions for gravity have been proposed [6, 7, 8, 9, 10]. The fundamental field
in this perspective is the gauge field, which is associated with a specific
symmetry group. Consequently, the metric no longer assumes the primary role in
describing the gravitational field; instead, it becomes an object that can be
derived from more fundamental elements. As a result, GR naturally emerges as a
consequence of the proposed gauge theory. Another relevant approach to the
unification of fundamental forces lies in the interplay between topological
field theories and gravity. Within this framework, it has been discovered that
GR can be formulated through a polynomial action principle that links it to
the topological background field (BF) theory. This fact is quite surprising
because, unlike BF theories, GR contains propagating degrees of freedom. In
recent years, there has been considerable interest in the scientific community
regarding the BF formulations of GR. These formulations have provided the
groundwork for the covariant approach to quantum gravity called spinfoam
models [11], which aim to quantize the gravitational field through a path
integral method. Montesinos et al. [12] have explored the different aspects of
BF formulation in D-dimensions, covering a range of related models within the
scope of GR. Despite the theoretical advances achieved through various
alternative formulations to GR, a satisfactory quantum theory of gravity has
not yet been established in the context of four-dimensional space-time.
Therefore, we can resort to studying gravitational models in lower dimensions
[13] in order to gain a better understanding of some quantum properties of the
gravitational field. In this regard, lower-dimensional BF models emerge as
suitable environments for the analysis of quantum gravity in reduced
dimensions (see [14] and references therein). However, in lower-dimensional
models, certain trivial features can be observed. For example, gravity in two
dimensions cannot be explained by the framework of GR since, in this case, the
theory becomes meaningless. However, Jackiw and Teitelboim [15, 16] suggested
a model that enables the resolution of this problem by introducing a dilatonic
field. On the other hand, the formulation of Jackiw-Teitelboim’s approach as a
BF gauge theory of $SO(2,1)$ has been developed in [17]. There is another
possibility to write down the action for 2-dimensional gravity in Polynomial
BF form [18], using a like construction of the Freidel-Starodubsetv [19, 20,
21]. These models consist of two main components: a topological term involving
assumed gauge symmetries, and a second quadratic term in the B fields that
breaks the symmetry $SO(3)$. The action of GR originates from the introduction
of symmetry breaking, accompanied by an additional term that is quadratic in
curvature. The combination described gives the theory a structure that
resembles modified gravity theories of the f(R) variety. These theories have
the potential to provide an explanation for the recent acceleration of the
universe, avoiding the need for the concept of dark energy [22, 23, 24].
Additionally, there exists a range of compelling motives to explore secondary
quadratic curvature Lagrangians within the domain of gravitational models.
These motivations encompass several critical areas, including the analysis of
quantum corrections [25, 26], the resolution of divergences via
renormalization techniques [27, 28, 29], the examination of effective actions
stemming from the framework of string theory [30], and the investigation of
inflationary mechanisms that operate independently of the presence of scalar
fields [31].

Within the framework of the BF gauge approach to gravity, the theory is
fundamentally defined by constraints and is characterized by the presence of
gauge symmetries. In the context of gravity theories, this symmetry not only
indicates that the physical content of the theory remains unchanged when the
coordinates change, but it is also intricately related to the dynamics of the
theory [32]. When considering such systems, the standard method to study these
theories is the Dirac-Bergmann method [33, 34, 35, 36, 37, 38]. In this
approach, the Hamiltonian is a linear combination of constraints, and the
dynamic behavior of the system is characterized by these constraints. This
formalism categorizes constraints into primary, secondary, and tertiary
classes, and distinguishes between first-class and second-class constraints.
The presence of first-class constraints results in the emergence of gauge
symmetries, whereas the existence of second-class constraints leads to a
reduction in the number of phase space variables.

In this regard, achieving a precise characterization of the dynamic structure
in Polynomial BF theories requires a thorough evaluation of the relevant
constraints. Therefore, conducting a comprehensive analysis and careful
consideration is necessary to identify these constraints. Until now, no
comprehensive study of the canonical description and constraint analysis of
the Polynomial BF theory proposed by Paszko-da Rocha [18] in the full phase
space has been conducted. However, an initial approach has been made in the
reduced phase space and presented in [39]. It is worth noting that
understanding the nature of constraints in the full phase space and their
implications for the canonical description of the Polynomial BF model is a
matter of great importance in both classical and quantum gravity. The
implementation of the Dirac-Bergmann formalism requires careful consideration,
as obtaining and classifying the constraints within the formalism is not an
easy task. Within this context, an additional and powerful approach that has
been developed for investigating canonical singular systems was introduced by
Faddeev and Jackiw [40]. In this formalism, all constraints are treated
equally, without the need to impose any additional classification. A central
requirement of this approach is to have a first-order Lagrangian that
describes the system. The constraints are derived from the zero modes of the
symplectic two-form matrix and are gradually introduced into the Lagrangian
until all the constraints in the theory are identified [41, 42]. This approach
has been extended through modifications to include the Dirac-Bergmann
algorithm, as detailed in [43, 44]. It is noteworthy that the symplectic
matrix holds crucial information within the theory. The inverse of this matrix
yields the FJ generalized brackets, aligning with the Dirac brackets [44, 45].

The Faddeev-Jackiw formalism has been successfully implemented in various
physical systems, including exotic actions for both Abelian and non-Abelian
gravity [46], four-dimensional BF theory [47], the Bonzom–Livine model for
gravity [48], the Christ-Lee model [49], a particle constrained to move in a
torus knot [50], and the Jackiw-Teitelboim model as BF theory in two
dimensions [51].

The primary goal of this article is to conduct a thorough analysis of the
Polynomial BF action in 1+1 dimensions, as proposed by Paszko-da Rocha [18].
We aim to achieve this by applying two classical methods commonly used in
dealing with constrained systems. The initial approach we will employ is
inspired by the contributions of Dirac and Bergmann. In this methodology, we
treat all the variables within the theory as dynamic variables. This approach
offers the distinct advantage of fully utilizing the phase space, rather than
restricting our attention solely to variables explicitly linked with
generalized velocities in the action. Furthermore, our analysis is conducted
without fixing the gauge, and we exclusively eliminate second-class
constraints by utilizing Dirac brackets. Consequently, we preserve the
integrity of the first-class constraints. These unique characteristics
differentiate our approach from the analysis presented in [39], where this
specific aspect was not taken into account. In the second description, the
modified FJ approach simplifies the need to classify constraints into first
and second classes, treating all constraints uniformly [43, 44]. Within the
framework of the modified Faddeev–Jackiw formalism (FJ), we explore the
structure of constraints and the derivation of generalized FJ brackets. As a
final step, we aim to obtain the gauge transformations.

The structure of this paper is organized as follows. In Section II, a
comprehensive analysis is conducted on the model proposed in [18]. This
analysis involves an in-depth exploration of the relationship between the two-
dimensional Polynomial BF action and quadratic gravity, employing a
methodology closely aligned with that presented in [39]. Section III is
dedicated to carry out a comprehensive Hamiltonian analysis, we will
demonstrate that a thorough examination of the dynamical structure reveals the
entire set of physical constraints, encompassing primary and secondary. In
Section IV, our focus turns to identifying the first and second-class
constraints within the Polynomial BF theory and determining the corresponding
physical degrees of freedom in our model. Besides, we derive the extended
action and Hamiltonian. Following this, we proceed to establish the
fundamental quantization brackets. Finally, we delve into the derivation of
gauge transformations and subsequently identify the generators associated with
these transformations. Section V is devoted to a meticulous presentation of
the Faddeev–Jackiw symplectic analysis of the Polynomial BF action. This
analysis serves as the foundation for the determination of all physical
constraints through the solution of zero-modes within the singular symplectic
matrix. Additionally, we establish the core quantization brackets, articulate
the functional measure in the path integral, and recognize the physical
degrees of freedom linked to our model through the introduction of a gauge-
fixing procedure.

##  2 2D quadratic gravity expressed as a Polynomial BF action

As mentioned in [18], the action of quadratic gravity can be represented as a
2D polynomial BF action. In this formulation, the process begins by
constructing a BF action that remains invariant under $SO(3)$. Subsequently, a
quadratic interaction term in A and B is introduced. As a result of
introducing this interaction term, the initial invariance is broken, leading
to a reduction to $SO(2)$. Rather than introducing the interaction term
directly through Lagrange multipliers, it was suggested to follow an approach
akin to that proposed by MacDowell-Mansouri [19].

The theory is formulated by means of 0-form field $B=\frac{1}{2}B^{IJ}T_{IJ}$,
and the curvature111Where $F\equiv dA+A\wedge
A=\frac{1}{4}(\partial_{\alpha}A^{IJ}_{\beta}-\partial_{\beta}A^{IJ}_{\alpha}+A^{IK}_{\alpha}A{{}_{\beta
K}}^{J}-A^{IK}_{\beta}A{{}_{\alpha K}}^{J})T_{IJ}dx^{\alpha}\wedge
dx^{\beta}$. $F=\frac{1}{4}F^{IJ}_{\alpha\beta}T_{IJ}dx^{\alpha}\wedge
dx^{\beta}$ written in terms of a 1-form field $A=A^{IJ}_{\mu}dx^{\mu}T_{IJ}$,
where $T_{IJ}$ (with $I=0,1,2$) are the generators of the $SO(3)$ group and
metric $\eta_{IJ}=diag(+,+,+)$.

The action of the BF model with interaction term is given by [18]

$\displaystyle S=\int_{\mathcal{M}}tr[-\frac{1}{2}B\wedge
F+k^{2}(PB)(PB)(QA)\wedge A],$ (1)

where $k$ is a constant parameter. To develop the polynomial BF action (1),
the pivotal insight in reference [18] was to treat P and Q as projection
operators defined by222$\epsilon^{IJK}$ is the Levi-Civita symbol and, by
convention we have $\epsilon^{012}=1$.

$\displaystyle P^{IJ,KL}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\epsilon^{IJ2}\epsilon^{KL2},$ (2) $\displaystyle
Q_{IJ,KL}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\epsilon{{}_{IJ}}^{M}\epsilon{{}_{KL}}^{N}\epsilon_{MN2}.$
(3)

Let P be a projection operator into the subalgebra of Lorentz generators
$T_{ab}$ (with a = 0,1) and translations $T_{a2}=P_{a}$. It is evident that
the operator (2) does not project any translations; in other words, it does
not affect the translations but does leave the Lorentz generators invariant

$\displaystyle(PT)_{ab}=T_{ab},\quad(PT)_{a2}=0,$ (4)

and the operator Q described in (3) does not project any Lorentz generators;
instead, it exchanges the translations

$\displaystyle(QT)_{ab}=\epsilon{{}_{a}}^{b}T_{b2},\quad(QT)_{a2}=0.$ (5)

Instead of employing differential forms, we work with the components of the B
and F fields, resulting in [18, 39]

$S=\int
d^{2}x\epsilon^{\mu\nu}\bigg{[}\frac{1}{4}B_{KL}F^{KL}_{\mu\nu}+\frac{k^{2}}{2}(B_{ab}B^{ab})\epsilon_{mn}A^{m}_{\nu}A^{n}_{\nu}\bigg{]}.$
(6)

where $\epsilon^{\mu\nu}$ represents the Levi-Civita Symbol for the spacetime.

It has been demonstrated in [18] that the action given in (6) is equivalent to
a quadratic gravity model. To see this one ﬁrst decomposes the connection
$A^{IJ}_{\mu}$ into zweiben $lA^{a2}_{\mu}=e^{a}_{\mu}$ and Lorentz connection
$A^{ab}_{\mu}=\omega^{ab}_{\mu}=\epsilon^{ab}\omega_{\mu}$ as follows:

$\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}A^{IJ}_{\mu}T_{IJ}=\frac{1}{2}A^{ab}_{\mu}T_{ab}+A^{a2}_{\mu}T_{a2},$
(7) $\displaystyle=$
$\displaystyle\frac{1}{2}\omega^{ab}_{\mu}T_{ab}+\frac{1}{l}e^{a}_{\mu}P_{a}.$

The parameter $l$ is introduced for dimensional considerations, given that the
zweibein is dimensionless. As we delve further, we will discover that this
parameter is connected to the cosmological constant. Additionally, the
components of the connection curvature $A^{IJ}$ are linked to the curvature of
the Lorentz connection $\omega$

$\displaystyle F^{ab}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\epsilon^{ab}(\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu})-\frac{1}{l^{2}}(e^{a}_{\mu}e^{b}_{\nu}-e^{a}_{\nu}e^{b}_{\mu})=R^{ab}_{\mu\nu}-\frac{1}{l^{2}}(e^{a}_{\mu}e^{b}_{\nu}-e^{a}_{\nu}e^{b}_{\mu}),$
(8)

and the torsion

$\displaystyle F^{a2}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{l}(\partial_{\mu}e^{a}_{\nu}-\partial_{\nu}e^{a}_{\mu}+\epsilon{{}^{a}}_{b}\omega_{\mu}e^{b}_{\nu}-\epsilon{{}^{a}}_{b}\omega_{\nu}e^{b}_{\mu})=\frac{1}{l}T^{a}_{\mu\nu}.$
(9)

By utilizing equations (7) and (8), as well as (9), we can express the action
of the theory in the polynomial BF action (6) in the following way 333The
following notations are used $e=dete^{a}_{\mu}$ is the determinant of the
zweiben, and $B_{a}=B_{a2}$, $B_{ab}=\epsilon_{ab}B$.

$\displaystyle S[e^{a}_{\mu},\omega_{\mu},B,B_{a}]=\int
d^{2}x\bigg{[}\frac{1}{2}B\bigg{(}\frac{1}{2}\epsilon_{ab}\epsilon^{\mu\nu}R^{ab}_{\mu\nu}-\frac{2}{l^{2}}e\bigg{)}+\frac{1}{2l}\epsilon^{\mu\nu}B_{a}T^{a}_{\mu\nu}+\frac{2k^{2}}{l^{2}}B^{2}e\bigg{]}.$
(10)

By varying the action (10) with respect to
$(e^{a}_{\mu},\omega_{\mu},B,B_{a})$, the following equations of motion are
derived

$\displaystyle\frac{\delta S}{\delta e^{a}_{\mu}}$ $\displaystyle:$
$\displaystyle\frac{1}{l}\epsilon^{\mu\nu}(D_{\nu}B_{a}-\epsilon_{ab}\frac{B}{l}e^{b}_{\nu}(1-2k^{2}B))=0,$
(11) $\displaystyle\frac{\delta S}{\delta\omega_{\mu}}$ $\displaystyle:$
$\displaystyle\epsilon^{\mu\nu}(\partial_{\nu}B+\frac{1}{l}\epsilon_{ab}B^{a}e^{b}_{\nu})=0,$
(12) $\displaystyle\frac{\delta S}{\delta B}$ $\displaystyle:$
$\displaystyle\frac{1}{4}\epsilon_{ab}\epsilon^{\mu\nu}R^{ab}_{\mu\nu}-\frac{e}{l^{2}}(1-4k^{2}B)=0,$
(13) $\displaystyle\frac{\delta S}{\delta B_{a}}$ $\displaystyle:$
$\displaystyle\epsilon^{\mu\nu}T^{a}_{\mu\nu}=0.$ (14)

Furthermore, as a special case of the Lagrangian (10), the Jackiw-Teitelboim
model emerges when we set $k=0$ in equation (13). This choice results in the
elimination of an interaction term. Notably, in this scenario, the B field
retains its arbitrariness, allowing for the establishment of an identification
with the dilatonic scalar field. Additionally, by selecting the cosmological
constant $\Lambda=1/l^{2}$ and defining $R=g^{\mu\nu}R_{\mu\nu}$ as the
curvature scalar, the action (10) seamlessly transforms into the Jackiw-
Teitelboim action [15, 16]

$\displaystyle S_{JT}=\frac{1}{2}\int\sqrt{g}B(R-2\Lambda),$ (15)

On the other hand, in the region where $k\neq 0$, we can use equation (13) to
solve for the field B, and we can establish a relationship between the B
field, curvature, and the parameter $l^{2}$ [18, 39]

$\displaystyle B=-\frac{l^{2}}{8k^{2}}(R-\frac{2}{l^{2}}).$ (16)

and then substitute it into action (10) in order to obtain444where
$G=k^{2}/8\pi$ and $2\Lambda=1/l^{2}$

$\displaystyle S=\frac{1}{16\pi G}\int
d^{2}x\sqrt{g}(R-2\Lambda)-\frac{1}{128\pi G\Lambda}\int d^{2}x\sqrt{g}R^{2}.$
(17)

Hence, the two-dimensional Polynomial BF action (1) can be regarded as
equivalent to a quadratic gravitational action, at least in the classical
sense.

##  3 Hamiltonian Constraint Analysis

Hence, in order to develop the Hamiltonian analysis, we assume for the space-
time a topological structure of the form $M=\Re\times\Sigma$, where the real
line $\Re$ represents an evolution parameter, and $\Sigma$ is a 1-dimensional
manifold arbitrary but fixed topology, representing the ”space”. By performing
the 1+1 decomposition, we can write the action (10) as555Covariant derivatives
$D_{1}B_{a}=\partial_{1}B_{a}+\epsilon{{}_{a}}^{b}\omega_{1}B_{b}$.

$\displaystyle S[e^{a}_{\mu},\omega_{\mu},B,B_{a}]=\int
d^{2}x\bigg{(}B\dot{\omega}_{1}+\frac{1}{l}B_{a}\dot{e}^{a}_{1}+\omega_{0}(\partial_{1}B+\frac{1}{l}\epsilon{{}^{a}}_{b}B_{a}e^{b}_{1})+\frac{e^{a}_{0}}{l}(D_{1}B_{a}-\frac{1}{l}\epsilon_{ab}Be^{b}_{1}(1-2k^{2}B))\bigg{)},$
(18)

The definition of the momenta
$\Pi^{Q_{L}}=(\Pi,\Pi^{a},\Pi^{\mu},\Pi^{\mu}_{a})$ canonically conjugate to
the configuration variables $Q_{L}=(B,B_{a},\omega_{\mu},e^{a}_{\mu})$ are
determined by

$\Pi=\frac{\delta{\mathcal{L}}}{\delta\dot{B}},\qquad\Pi^{a}=\frac{\delta{\mathcal{L}}}{\delta\dot{B}_{a}},\qquad\Pi^{\mu}=\frac{\delta{\mathcal{L}}}{\delta\dot{\omega}_{\mu}},\qquad\Pi^{\mu}_{a}=\frac{\delta{\mathcal{L}}}{\delta\dot{e}^{a}_{\mu}}.$
(19)

The matrix elements of the Hessian is given by

$H_{LM}=\frac{\partial^{2}{\mathcal{L}}}{\partial\dot{Q}_{L}\partial\dot{Q}_{M}}=0.$
(20)

Notice that the rank of the Hessian is zero, thus, we expect $9$ primary
constraints. From the definition of the momenta $(\ref{16})$ we identify to
the following primary constraints

$\displaystyle\phi:=\Pi\approx 0,\quad\phi^{a}:=\Pi^{a}\approx
0,\quad\phi^{0}:=\Pi^{0}\approx 0,\quad\phi^{1}:=\Pi^{1}-B\approx
0,\quad\phi^{0}_{a}:=\Pi^{0}_{a}\approx
0,\quad\phi^{1}_{a}:=\Pi^{1}_{a}-\frac{1}{l}B_{a}\approx 0.$ (21)

Due to the constraints (21) the motion is restricted to a subspace
$\Gamma_{1}$ of the full phase-sapce $\Gamma$. The Phase-space $\Gamma$ of
this system includes the dynamical fields $(Q_{L},\Pi^{Q_{L}})$.

The two remaining terms in the action (18) give the canonical Hamiltonian and
$H_{c}$ is only defined on $\Gamma_{1}$

$H_{c}=-\int
dx\bigg{(}\omega_{0}(\partial_{1}B+\frac{1}{l}\epsilon{{}^{a}}_{b}e^{b}_{1}B_{a})+\frac{e^{a}_{0}}{l}(D_{1}B_{a}-\frac{1}{l}\epsilon_{ab}e^{b}_{1}B(1-2k^{2}B))\bigg{)},$
(22)

this expression, independent of velocities, solely encompasses spatial
derivatives of coordinates and momenta. Additionally, $H_{c}$ can be
considered as the restriction to the hypersurface $\Gamma_{1}$ of a function
$H_{P}$, which we shall call primary Hamiltonian defined all over phase-space.
Upon incorporating the primary constraints (21), the primary Hamiltonian takes
on the following form

$H_{P}=H_{c}+\int
dx\left[\check{\lambda}\phi+\check{\lambda}_{a}\phi^{a}+\check{\lambda}_{\mu}\phi^{\mu}+\check{\lambda}^{a}_{\mu}\phi^{\mu}_{a}\right],$
(23)

where
$(\check{\lambda},\check{\lambda}_{a},\check{\lambda}_{\mu},\check{\lambda}^{a}_{\mu})$
are the corresponding Lagrange multipliers associated of these constraints
$(\phi,\phi^{a},\phi^{\mu},\phi^{\mu}_{a})$. The fundamental Poisson brackets
of the theory are determined by the commutation relations

$\displaystyle\\{B(x),\Pi(y)\\}$ $\displaystyle=$ $\displaystyle\delta(x-y),$
$\displaystyle\\{B_{a}(x),\Pi^{b}(y)\\}$ $\displaystyle=$
$\displaystyle\delta^{b}_{a}\delta(x-y),$
$\displaystyle\\{e^{a}_{\mu}(x),\Pi^{\nu}_{b}(y)\\}$ $\displaystyle=$
$\displaystyle\delta^{a}_{b}\delta^{\nu}_{\mu}\delta(x-y),$
$\displaystyle\\{\omega_{\mu}(x),\Pi^{\nu}(y)\\}$ $\displaystyle=$
$\displaystyle\delta^{\nu}_{\mu}\delta(x-y).$ (24)

The next step is to observe if there are more constraints, so that, we
calculate the following 9$\times$9 matrix whose entries are the Poisson
brackets among the constraints (21). The non-vanishing Poisson brackets are
given by

$\displaystyle\\{\phi(x),\phi^{1}(y)\\}$ $\displaystyle=$
$\displaystyle\delta(x-y),$ $\displaystyle\\{\phi^{a}(x),\phi^{1}_{b}(y)\\}$
$\displaystyle=$ $\displaystyle\frac{1}{l}\delta^{a}_{b}\delta(x-y),$ (25)

and expressed in matrix form, namely,
$W=\\{\widetilde{\Phi}^{A}(x),\widetilde{\Phi}^{B}(y)\\}$

$\displaystyle\left[\begin{array}[]{l|cccccc}&\phi(y)&\phi^{b}(y)&\phi^{0}(y)&\phi^{1}(y)&\phi^{0}_{b}(y)&\phi^{1}_{b}(y)\\\
\hline\cr\phi(x)&0&0&0&\\{\phi(x),\phi^{1}(y)\\}&0&0\\\
\phi^{a}(x)&0&0&0&0&0&\\{\phi^{a}(x),\phi^{1}_{b}(y)\\}\\\
\phi^{0}(x)&0&0&0&0&0&0\\\ \phi^{1}(x)&\\{\phi^{1}(x),\phi(y)\\}&0&0&0&0&0\\\
\phi^{0}_{a}(x)&0&0&0&0&0&0\\\
\phi^{1}_{a}(x)&0&\\{\phi^{1}_{a}(x),\phi^{b}(y)\\}&0&0&0&0\\\
\end{array}\right],$ (33)

where, $\widetilde{\Phi}^{A}=(\phi,\phi^{a},\phi^{\mu},\phi^{\mu}_{a})$. We
can see that this matrix has rank=6 and 3 null-vectors. By using these 3 null-
vectors and the evolution of $\phi^{0}_{a}$ and $\phi^{0}$ produces the
following $3$ secondary constraints

$\displaystyle\dot{\phi}^{0}$ $\displaystyle=$
$\displaystyle\\{\phi^{0}(x),{H}_{P}\\}\approx
0\quad\Rightarrow\quad\Psi:=\partial_{1}B+\frac{1}{l}\epsilon{{}^{a}}_{b}e^{b}_{1}B_{a}\approx
0,$ $\displaystyle\dot{\phi}^{0}_{a}$ $\displaystyle=$
$\displaystyle\\{\phi^{0}_{a}(x),{H}_{P}\\}\approx
0\quad\Rightarrow\quad\Psi_{a}:=D_{1}B_{a}-\frac{1}{l}\epsilon_{ab}e^{b}_{1}B(1-2Bk^{2})\approx
0,$ (34)

and the rank enables us to determine the specific values for the Lagrangian
multipliers

$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\\{\phi(x),H_{P}\\}\approx
0\quad\Rightarrow\check{\lambda}_{1}-\partial_{1}\omega_{0}-\frac{1}{l^{2}}\epsilon_{ab}e^{b}_{1}e^{a}_{0}(1-4k^{2}B)\approx
0,$ $\displaystyle\dot{\phi}^{a}$ $\displaystyle=$
$\displaystyle\\{\phi^{a}(x),H_{P}\\}\approx
0\quad\Rightarrow\check{\lambda}^{a}_{1}-\partial_{1}e^{a}_{0}-\epsilon{{}^{a}}_{b}\omega_{1}e^{b}_{0}+\epsilon{{}^{a}}_{b}e^{b}_{1}\omega_{0}\approx
0,$ $\displaystyle\dot{\phi}^{1}$ $\displaystyle=$
$\displaystyle\\{\phi^{1}(x),H_{P}\\}\approx
0\quad\Rightarrow\check{\lambda}-\frac{1}{l}\varepsilon{{}_{a}}^{b}B_{b}e^{a}_{0}\approx
0,$ $\displaystyle\dot{\phi}^{1}_{a}$ $\displaystyle=$
$\displaystyle\\{\phi^{1}_{a}(x),H_{P}\\}\approx
0\quad\Rightarrow\check{\lambda}_{a}+\epsilon{{}_{a}}^{b}B_{b}\omega_{0}-\frac{1}{l}\epsilon_{ab}e^{b}_{0}B(1-2k^{2}B)\approx
0.$ (35)

Consistency demands the preservation of the secondary constraints over time.
In this theory, there are no third-order constraints.

##  4 First- and second-class constraints

###  4.1 Separation of constraints

To categorize the full set of constraints, we must distinguish between first
and second-class constraints by identifying them among the primary and
secondary constraints. To achieve this goal, we must calculate the rank and
null-vectors of a 12x12 matrix. The matrix’s elements will consist of the
Poisson brackets between primary and secondary constraints. This process
involves the following steps

$\displaystyle\\{\phi(x),\phi^{1}(y)\\}$ $\displaystyle=$
$\displaystyle\delta(x-y),$ $\displaystyle\\{\phi(x),\Psi(y)\\}$
$\displaystyle=$ $\displaystyle-\partial_{1}\delta(x-y),$
$\displaystyle\\{\phi(x),\Psi_{b}(y)\\}$ $\displaystyle=$
$\displaystyle\frac{1}{l}\epsilon_{bc}e^{c}_{1}(1-4Bk^{2})\delta(x-y),$
$\displaystyle\\{\phi^{a}(x),\phi^{1}_{b}(y)\\}$ $\displaystyle=$
$\displaystyle\frac{1}{l}\delta^{a}_{b}\delta(x-y),$
$\displaystyle\\{\phi^{a}(x),\Psi(y)\\}$ $\displaystyle=$
$\displaystyle-\frac{1}{l}\epsilon{{}^{a}}_{b}e^{b}_{1}\delta(x-y),$
$\displaystyle\\{\phi^{a}(x),\Psi_{b}(y)\\}$ $\displaystyle=$
$\displaystyle-\delta^{a}_{b}\partial_{1}\delta(x-y)+\epsilon{{}^{a}}_{b}\omega_{1}\delta(x-y),$
$\displaystyle\\{\phi^{1}(x),\Psi_{b}\\}$ $\displaystyle=$
$\displaystyle-\epsilon{{}_{b}}^{c}B_{c},$
$\displaystyle\\{\phi^{1}_{a}(x),\Psi(y)\\}$ $\displaystyle=$
$\displaystyle-\frac{1}{l}\epsilon{{}^{b}}_{a}B_{b}\delta(x-y),$
$\displaystyle\\{\phi^{1}_{a}(x),\Psi_{b}(y)\\}$ $\displaystyle=$
$\displaystyle-\frac{1}{l}B\epsilon_{ab}(1-2Bk^{2})\delta(x-y).$ (36)

Furthermore, it can be represented in matrix form as follows

$\displaystyle\left[\begin{array}[]{l|cccccccc}&\phi(y)&\phi^{b}(y)&\phi^{0}(y)&\phi^{1}(y)&\phi^{0}_{b}(y)&\phi^{1}_{b}(y)&\Psi(y)&\Psi_{b}(y)\\\
\hline\cr\phi(x)&0&0&0&\\{\phi(x),\phi^{1}(y)\\}&0&0&\\{\phi(x),\Psi(y)\\}&\\{\phi(x),\Psi_{b}(y)\\}\\\
\phi^{a}(x)&0&0&0&0&0&\\{\phi^{a}(x),\phi^{1}_{b}(y)\\}&\\{\phi^{a}(x),\Psi(y)\\}&\\{\phi^{a}(x),\Psi_{b}(y)\\}\\\
\phi^{0}(x)&0&0&0&0&0&0&0&0\\\
\phi^{1}(x)&\\{\phi^{1}(x),\phi(y)\\}&0&0&0&0&0&0&\\{\phi^{1}(x),\Psi_{b}(y)\\}\\\
\phi^{0}_{a}(x)&0&0&0&0&0&0&0&0\\\
\phi^{1}_{a}(x)&0&\\{\phi^{1}_{a}(x),\phi^{b}(y)\\}&0&0&0&0&\\{\phi^{1}_{a}(x),\Psi(y)\\}&\\{\phi^{1}_{a}(x),\Psi_{b}(y)\\}\\\
\Psi(x)&\\{\Psi(x),\phi(y)\\}&\\{\Psi(x),\phi^{b}(y)\\}&0&0&0&\\{\Psi(x),\phi^{1}_{b}(y)\\}&0&0\\\
\Psi_{a}(x)&\\{\Psi_{a}(x),\phi(y)\\}&\\{\Psi_{a}(x),\phi^{b}(y)\\}&0&\\{\Psi_{a}(x),\phi^{1}(y)\\}&0&\\{\Psi_{a}(x),\phi^{1}_{b}(y)\\}&0&0\\\
\end{array}\right].$

This matrix has a vanishing determinant which tells us that there is at least
one first class constraints. After a long calculation, we found that this
matrix has a rank=6 and 6 null vectors, thus, the theory presents a set of 6
first class constraints and 6 second class constraints. The set of null
vectors of the matrix (36) are given by

$\displaystyle V^{1}$ $\displaystyle=$
$\displaystyle(-\epsilon{{}_{a}}^{c}B_{c}\delta(x-y),-\epsilon_{ba}B(1-2Bk^{2})\delta(x-y),0,-\frac{1}{l}\epsilon_{ab}e^{b}_{1}(1-4Bk^{2})\delta(x-y),0,$
$\displaystyle
l(\delta^{b}_{a}\partial_{1}\delta(x-y)-\epsilon{{}^{b}}_{a}\omega_{1}\delta(x-y)),\delta^{b}_{a}\delta(x-y)),$
$\displaystyle V^{2}$ $\displaystyle=$
$\displaystyle(0,-\epsilon{{}^{c}}_{b}B_{c}\delta(x-y),0,\partial_{1}\delta(x-y),0,\epsilon{{}^{b}}_{c}e^{c}_{1}\delta(x-y),\delta(x-y),0),$
$\displaystyle V^{3}$ $\displaystyle=$
$\displaystyle(0,0,\delta^{b}_{a}\delta(x-y),0,0,0,0,0),$ $\displaystyle
V^{4}$ $\displaystyle=$ $\displaystyle(0,0,0,0,\delta(x-y),0,0,0).$ (38)

In order to identified the following 6 first class constraints, we used the
contraction of these null vectors with the constraints (21) and (34)

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\Psi+\partial_{1}\chi^{1}+\epsilon{{}^{a}}_{b}e^{b}_{1}\chi^{1}_{a}+\epsilon{{}_{a}}^{b}B_{b}\chi^{a}\approx
0,$ $\displaystyle\gamma_{a}$ $\displaystyle=$
$\displaystyle\Psi_{a}+lD_{1}\chi^{1}_{a}-\frac{1}{l}\epsilon_{ab}e^{b}_{1}\chi^{1}(1-4k^{2}B)+\epsilon_{ab}(B\chi^{b}(1-2k^{2}B)-\chi
B^{b})\approx 0,$ $\displaystyle\gamma^{0}_{a}$ $\displaystyle=$
$\displaystyle\phi^{0}_{a}=\Pi^{0}_{a}\approx 0,$ $\displaystyle\gamma^{0}$
$\displaystyle=$ $\displaystyle\phi^{0}=\Pi^{0}\approx 0,$ (39)

and the following 6 second class constraints

$\displaystyle\chi$ $\displaystyle=$ $\displaystyle\phi=\Pi\approx 0,$
$\displaystyle\chi^{a}$ $\displaystyle=$ $\displaystyle\phi^{a}=\Pi^{a}\approx
0,$ $\displaystyle\chi^{1}$ $\displaystyle=$
$\displaystyle\phi^{1}=\Pi^{1}-B\approx 0,$ $\displaystyle\chi^{1}_{a}$
$\displaystyle=$
$\displaystyle\phi^{1}_{a}=\Pi^{1}_{a}-\frac{1}{l}B_{a}\approx 0.$ (40)

It is worth noting that these constraints have not been previously documented
in the literature. The comprehensive description of these constraints across
the entire phase space is essential for understanding the fundamental gauge
transformations and for establishing the Dirac brackets. Moreover, these
constraints will prove pivotal in advancing the process of quantization.

After identifying and categorizing all constraints as first or second-class,
it is crucial to analyze the Poisson algebra associated with these
constraints. This analysis is essential to validate that the separation has
been carried out accurately. Next, we will proceed to calculate the algebra of
the constraints, expressing it as follows:

$\displaystyle\\{\gamma(x),\gamma(y)\\}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\\{\gamma_{a}(x),\gamma(y)\\}$ $\displaystyle=$
$\displaystyle\epsilon{{}_{a}}^{b}\gamma_{b}\delta(x-y)\approx 0,$
$\displaystyle\\{\gamma_{a}(x),\gamma_{b}(y)\\}$ $\displaystyle=$
$\displaystyle-\epsilon_{ab}(1-4k^{2}B)\gamma\delta(x-y)+4k^{2}\epsilon_{ab}\chi^{1}\Psi\delta(x-y)\approx
0,$ $\displaystyle\\{\gamma(x),\chi(y)\\}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\\{\gamma(x),\chi^{b}(y)\\}$ $\displaystyle=$
$\displaystyle\epsilon{{}_{a}}^{b}\chi^{a}\delta(x-y)\approx 0,$
$\displaystyle\\{\gamma(x),\chi^{1}(y)\\}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\\{\gamma(x),\chi^{1}_{b}(y)\\}$ $\displaystyle=$
$\displaystyle-\epsilon{{}_{b}}^{a}\chi^{1}_{a}\approx 0,$
$\displaystyle\\{\gamma_{a}(x),\chi(y)\\}$ $\displaystyle=$
$\displaystyle\frac{4}{l}k^{2}\epsilon_{ab}e^{b}_{1}\chi^{1}\delta(x-y)+\epsilon_{ab}(1-4k^{2}B)\chi^{b}\approx
0,$ $\displaystyle\\{\gamma_{a}(x),\chi^{b}(y)\\}$ $\displaystyle=$
$\displaystyle-\epsilon{{}_{a}}^{b}\chi\delta(x-y)\approx 0,$
$\displaystyle\\{\gamma_{a}(x),\chi^{1}(y)\\}$ $\displaystyle=$ $\displaystyle
l\epsilon{{}_{a}}^{b}\chi^{1}_{b}\delta(x-y)\approx 0,$
$\displaystyle\\{\gamma_{a}(x),\chi^{1}_{b}(y)\\}$ $\displaystyle=$
$\displaystyle-\frac{1}{l}\epsilon_{ab}\chi^{1}(1-4Bk^{2})\delta(x-y)\approx
0,$ $\displaystyle\\{\chi(x),\chi^{1}(y)\\}$ $\displaystyle=$
$\displaystyle\delta(x-y),$ $\displaystyle\\{\chi^{a}(x),\chi^{1}_{b}(y)\\}$
$\displaystyle=$ $\displaystyle\frac{1}{l}\delta^{a}_{b}\delta(x-y).$ (41)

In the previous section, we noted that setting $k=0$ removes us from quadratic
gravity. However, as $k$ approaches zero, we move towards the $SO(3)$
algebra666By defining $T_{ab}=\epsilon_{ab}T$ and letting $T_{a2}$ be $P_{a}$,
we can reformulate the algebra: $[T,T]=0$, $[T,P_{a}]=\epsilon_{a}^{b}P_{b}$,
$[P_{a},P_{b}]=-\epsilon_{ab}T$. [39]. Additionally, the results in equation
(41) correspond well with the classification in references [33, 52, 53].

###  4.2 Physical degrees of freedom

With the correct identification of the constraints, we can carry out the
counting of degrees of freedom in the following form: there are $18$ canonical
variables
$(e^{a}_{\mu},\Pi^{\mu}_{a},\omega_{\mu},\Pi^{\mu},B,\Pi,B_{a},\Pi^{a})$, $6$
first class constraints $(\gamma,\gamma_{a},\gamma^{0}_{a},\gamma^{0})$ and
$6$ second class constraints $(\chi,\chi^{a},\chi^{1}_{a},\chi^{1})$ and one
concludes that the $S[e^{a}_{\mu},\omega_{\mu},B,B_{a}]$ action for gravity in
two dimensions is devoid of degrees of freedom, therefore, the theory is
topological. Certainly, when we treat the second class constraints given in
(40) as strong equations, we find that the aforementioned relationships
simplify to the conventional constraints described in [39].

###  4.3 The extended action and Hamiltonian

Moreover, the process of identifying Lagrange multipliers along with the
constraints will enable us to establish the extended action. Through the
utilization of the first-class constraints (39), the second-class constraints
(40), and the Lagrange multipliers (35), we determine that the extended action
can be expressed in the following form

$\displaystyle S_{E}[Q_{L},P^{L},\lambda_{A},u_{A}]$ $\displaystyle=$
$\displaystyle\int
d^{2}x(\dot{Q}_{L}P^{L}-\mathcal{H}-\tilde{\lambda}\gamma-\tilde{\lambda}^{a}\gamma_{a}-\check{\lambda}_{0}\gamma^{0}-\check{\lambda}^{a}_{0}\gamma^{0}_{a}-u\chi-
u_{a}\chi^{a}$ (42) $\displaystyle-u^{a}_{1}\chi^{1}_{a}-u_{1}\chi^{1}),$

and

$\displaystyle\mathcal{H}=-\omega_{0}\gamma-\frac{e^{a}_{0}}{l}\gamma_{a},$
(43)

where $Q_{L}=(e^{a}_{\mu},\omega_{\mu},B,B_{a})$,
$P^{L}=(\Pi^{\mu}_{a},\Pi^{\mu},\Pi,\Pi^{a})$ and
$\lambda_{A}=(\tilde{\lambda},\tilde{\lambda}^{a},\check{\lambda}_{0},\check{\lambda}^{a}_{0})$,
$u_{A}=(u,u_{a},u^{a}_{1},u_{1})$, are the Lagrange multipliers that enforce
the first and second class constraints. We are to observable, by considering
the second class constraints as strong equation, that the Hamiltonian (43) is
reduced to the usual expression defined on a reduced phase space context. From
the extend action we can identify the extend Hamiltonian, which is given by

$\displaystyle H_{E}=\int
dx(\mathcal{H}+\tilde{\lambda}\gamma+\tilde{\lambda}^{a}\gamma_{a}+\check{\lambda}^{a}_{0}\gamma^{0}_{a}+\check{\lambda}_{0}\gamma^{0}),$
(44)

thus, the extended Hamiltonian is a linear combination of first-class
constraints as expected.

###  4.4 Dirac brackets

Considering all the information we have gathered thus far, we are ready to
formulate the Dirac brackets. There are two approaches to constructing them.
The first method involves eliminating the second-class constraints and
retaining the first-class ones, while the second approach entails fixing the
gauge and transforming the first-class constraints into second-class ones. In
this section, we will focus on the former approach, which involves eliminating
the second-class constraints and preserving the first-class ones. To derive
the Dirac brackets, we will utilize a matrix whose elements exclusively
consist of Poisson brackets between the second-class constraints, denoted as
$C^{\alpha\beta}(x,y)=\\{\zeta^{\alpha}(x),\zeta^{\beta}(y)\\}$. A
straightforward yields

$\displaystyle[C^{\alpha\beta}(x,y)]=\left(\begin{array}[]{cccc}0&0&\delta(x-y)&0\\\
0&0&0&\frac{1}{l}\delta^{a}_{b}\delta(x-y)\\\ -\delta(x-y)&0&0&0\\\
0&-\frac{1}{l}\delta^{b}_{a}\delta(x-y)&0&0\\\ \end{array}\right),$ (49)

its inverse is given by

$\displaystyle[C_{\alpha\beta}(x,y)]=\left(\begin{array}[]{cccc}0&0&-\delta(x-y)&0\\\
0&0&0&-l\delta^{a}_{b}\delta(x-y)\\\ \delta(x-y)&0&0&0\\\
0&l\delta^{b}_{a}\delta(x-y)&0&0\\\ \end{array}\right).$ (54)

The Dirac brackets among two functionals $A$, $B$ are expressed by

$\displaystyle\\{A(x),B(y)\\}_{D}=\\{A(x),B(y)\\}_{P}-\int
dudv\\{A(x),\zeta^{\alpha}(u)\\}C_{\alpha\beta}(u,v)\\{\zeta^{\beta}(v),B(y)\\}.$
(55)

The standard Poisson bracket, denoted as $\\{A(x),B(y)\\}_{P}$, is applied to
the functionals $A$ and $B$. Meanwhile, the set of second class constraints,
represented as $\zeta^{\alpha}=(\chi,\chi^{a},\chi^{1},\chi^{1}_{a})$, is
involved. Consequently, with the utilization of equations (54) and (55), we
derive the Dirac brackets for the theory as follows

$\displaystyle\\{e^{a}_{\mu}(x),\Pi^{\nu}_{b}(y)\\}_{D}$ $\displaystyle=$
$\displaystyle\delta^{a}_{b}\delta^{\nu}_{\mu}\delta(x-y),$
$\displaystyle\\{e^{a}_{1}(x),B_{b}(y)\\}_{D}$ $\displaystyle=$ $\displaystyle
l\delta^{a}_{b}\delta(x-y),$ $\displaystyle\\{\omega_{1}(x),B(y)\\}_{D}$
$\displaystyle=$ $\displaystyle\delta(x-y),$
$\displaystyle\\{\omega_{\mu}(x),\Pi^{\nu}(y)\\}_{D}$ $\displaystyle=$
$\displaystyle\delta^{\nu}_{\mu}\delta(x-y).$ (56)

It is noteworthy to mention that in the Polynomial BF action, the fields
$(e^{a}_{1},B_{a},\omega_{1},B)$ have acquired a non-commutative nature, just
as it happens in the JT model [51] with the fields $(A^{i}_{x},\phi_{i})$.

By using the brackets given in (56), we calculate the Dirac brackets among the
first class and second class constraints given by

$\displaystyle\\{\gamma(x),\gamma(y)\\}_{D}$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle\\{\gamma_{a}(x),\gamma\\}_{D}$ $\displaystyle=$
$\displaystyle\epsilon{{}_{a}}^{b}(\gamma_{b}+\epsilon_{bc}(1-2k^{2}B)\chi^{c}\chi^{1}+l\epsilon{{}^{c}}_{b}\chi^{1}_{c}\chi)\delta(x-y),$
$\displaystyle\\{\gamma_{a}(x),\gamma_{b}(y)\\}_{D}$ $\displaystyle=$
$\displaystyle-\epsilon_{ab}[(1-4k^{2}B)\gamma+\epsilon{{}^{c}}_{d}\chi^{1}_{c}(l(1-4k^{2}B)\chi^{d}+4k^{2}e^{d}_{1}\chi^{1})]\delta(x-y).$
(57)

We can observe that only squares of second class constraints appear. In fact,
the Dirac brackets among first class constraints must be square of second
class constraints and linear of first class constraints [33, 52, 53].

###  4.5 Gauge generator

Currently, we are able to compute gauge transformations across the full phase
space. Achieving the accurate gauge symmetry involves employing Dirac’s
conjecture to build a gauge generator utilizing the first-class constraints.
The arrangement of constraints outlined in the full phase space will furnish
us with the fundamental gauge transformations. To accomplish this, we will
utilize Castellani’s algorithm for the generation of the gauge generator.

We establish the gauge transformation generator as

$\displaystyle
G=\int_{\sum}\left[D_{0}\tau^{a}\gamma^{0}_{a}+D_{0}\tau\gamma^{0}+\theta\gamma+\theta^{a}\gamma_{a}\right].$
(58)

Hence, we determine that the gauge transformations in the phase space are

$\displaystyle\delta{{}_{0}}B$ $\displaystyle=$
$\displaystyle-\epsilon{{}_{ab}}\theta^{a}B^{b},$
$\displaystyle\delta{{}_{0}}B_{a}$ $\displaystyle=$
$\displaystyle\epsilon_{ab}\theta B^{b}-\epsilon_{ab}\theta^{b}B(1-2k^{2}B),$
$\displaystyle\delta{{}_{0}}\omega_{0}$ $\displaystyle=$ $\displaystyle
D_{0}\tau,$ $\displaystyle\delta{{}_{0}}\omega_{1}$ $\displaystyle=$
$\displaystyle-\partial_{1}\theta-\frac{1}{l}\epsilon_{ab}\theta^{a}e^{b}_{1}(1-4k^{2}B),$
$\displaystyle\delta{{}_{0}}e^{a}_{0}$ $\displaystyle=$ $\displaystyle
D_{0}\tau^{a},$ $\displaystyle\delta{{}_{0}}e^{a}_{1}$ $\displaystyle=$
$\displaystyle-lD_{1}\theta^{a}+\epsilon{{}^{a}}_{b}\theta e^{b}_{1},$
$\displaystyle\delta{{}_{0}}\Pi$ $\displaystyle=$
$\displaystyle-\frac{4}{l}k^{2}\epsilon_{ab}\theta^{a}e^{b}_{1}\chi^{1}-\epsilon_{ab}(1-4k^{2}B)\theta^{a}\chi^{b},$
$\displaystyle\delta{{}_{0}}\Pi^{a}$ $\displaystyle=$
$\displaystyle\epsilon{{}^{a}}_{b}\theta\chi^{b}-\epsilon_{ab}\theta^{b}\chi,$
$\displaystyle\delta{{}_{0}}\Pi^{0}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\delta{{}_{0}}\Pi^{1}$ $\displaystyle=$
$\displaystyle-\epsilon{{}_{ab}}\theta^{a}B^{b}-l\epsilon{{}_{a}}^{b}\theta^{a}\chi^{1}_{b},$
$\displaystyle\delta{{}_{0}}\Pi^{0}_{a}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\delta{{}_{0}}\Pi^{1}_{a}$ $\displaystyle=$
$\displaystyle\epsilon{{}_{a}}^{b}\theta\Pi^{1}_{b}-\frac{1}{l}\epsilon_{ab}\theta^{b}B(1-2k^{2}B)-\frac{1}{l}\epsilon_{ab}\theta^{b}\chi^{1}(1-4k^{2}B).$
(59)

Although previously analyzed in the reduced phase space [39], the gauge
transformations in the full phase space (59) have not been fully reported.
Upon comparing our transformations with the earlier work, we observe a general
agreement in the configuration variables, except for the transformation of the
field $e^{a}_{1}$, where we identify an additional term
$\epsilon{{}^{a}}_{b}\theta e^{b}_{1}$ not present in the reduced phase space
analysis of [39].

On the other hand, by redefining the parameter as follows:
$\tau=-\theta=\zeta$, $\tau^{a}=-l\theta^{a}=\xi^{a}$, the gauge
transformations (59) can be expressed in a covariant form. This leads to the
following gauge symmetry

$\displaystyle B$ $\displaystyle\rightarrow$ $\displaystyle
B+\frac{1}{l}\epsilon_{ab}\xi^{a}B^{b},$ $\displaystyle B_{a}$
$\displaystyle\rightarrow$ $\displaystyle B_{a}-\epsilon{{}_{a}}^{b}\zeta
B_{b}+\frac{1}{l}\epsilon_{ab}\xi^{b}B(1-2k^{2}B),$ $\displaystyle
e^{a}_{\mu}$ $\displaystyle\rightarrow$ $\displaystyle
e^{a}_{\mu}+D_{\mu}\xi^{a}-\epsilon{{}^{a}}_{b}\zeta e^{b}_{\mu},$
$\displaystyle\omega_{\mu}$ $\displaystyle\rightarrow$
$\displaystyle\omega_{\mu}+\partial_{\mu}\zeta+\frac{1}{l^{2}}\epsilon_{ab}(1-4k^{2}B)\xi^{a}e^{b}_{\mu}.$
(60)

The results of (60) are sufficient for the construction of Poincaré
transformation. Starting from redefine the gauge parameters as
$\zeta=-\tilde{\zeta}+\omega_{\mu}V^{\mu}$ and $\xi^{a}=e^{a}_{\mu}V^{\mu}$
[54]

$\displaystyle\delta_{0}B$ $\displaystyle=$
$\displaystyle\delta_{PGT}B-\epsilon_{\mu\nu}\frac{\delta
S}{\delta\omega_{\mu}}V^{\nu},$ $\displaystyle\delta_{0}B_{a}$
$\displaystyle=$
$\displaystyle\delta_{PGT}B_{a}-\varepsilon_{\mu\nu}\frac{\delta S}{\delta
e^{a}_{\mu}}V^{\nu},$ $\displaystyle\delta_{0}e^{a}_{\mu}$ $\displaystyle=$
$\displaystyle\delta_{PGT}e^{a}_{\mu}+\varepsilon_{\mu\nu}\frac{\delta
S}{\delta B_{a}}V^{\nu},$ $\displaystyle\delta_{0}\omega_{\mu}$
$\displaystyle=$
$\displaystyle\delta_{PGT}\omega_{\mu}+\varepsilon_{\mu\nu}\frac{\delta
S}{\delta B}V^{\nu},$ (61)

where

$\displaystyle\delta_{PGT}B$ $\displaystyle=$ $\displaystyle
V^{\mu}\partial_{\mu}B,$ $\displaystyle\delta_{PGT}B_{a}$ $\displaystyle=$
$\displaystyle
V^{\mu}\partial_{\mu}B_{a}+\epsilon{{}_{a}}^{b}B_{b}\tilde{\zeta},$
$\displaystyle\delta_{PGT}e^{a}_{\mu}$ $\displaystyle=$ $\displaystyle
e^{a}_{\alpha}\partial_{\mu}V^{\alpha}+\partial_{\alpha}e^{a}_{\mu}V^{\alpha}+\epsilon{{}^{a}}_{b}e^{b}_{\mu}\tilde{\zeta},$
$\displaystyle\delta_{PGT}\omega_{\mu}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\tilde{\zeta}+\omega_{\alpha}\partial_{\mu}V^{\alpha}+\partial_{\alpha}\omega_{\mu}V^{\alpha}.$
(62)

We can see that the gauge symmetries (61) take back to the Poincaré symmetries
up to terms proportional to the equations of motion (14).

The fundamental gauge transformations of the model are expressed by (60) and
do not align with diffeomorphisms. However, in any theory characterized by
background independence, diffeomorphisms covariance is fundamental. This
symmetry should emerge from the foundational gauge transformation. Hence,
diffeomorphisms can be identified by redefining the gauge parameters as
$\xi^{a}=e^{a}_{\mu}V^{\mu}$ and $\zeta=\omega_{\mu}V^{\mu}$, where $V$
represents a vector field

$\displaystyle\delta{{}_{0}}B$ $\displaystyle=$
$\displaystyle\frac{1}{l}\epsilon{{}_{a}}^{b}e^{a}_{\mu}V^{\mu}B_{b},$
$\displaystyle\delta{{}_{0}}B_{a}$ $\displaystyle=$
$\displaystyle-\epsilon{{}_{a}}^{b}\omega_{\mu}V^{\mu}B_{b}+\frac{1}{l}\epsilon_{ab}B(1-2k^{2}B)e^{b}_{\mu}V^{\mu},$
$\displaystyle\delta{{}_{0}}e^{a}_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}(e^{a}_{\alpha}V^{\alpha})+\epsilon{{}^{a}}_{b}\omega_{\mu}(e^{b}_{\alpha}V^{\alpha})-\epsilon{{}^{a}}_{b}e^{b}_{\mu}\omega_{\alpha}V^{\alpha},$
$\displaystyle\delta{{}_{0}}\omega_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}(\omega_{\alpha}V^{\alpha})+\frac{1}{l^{2}}\epsilon_{ab}(1-4k^{2}B)e^{a}_{\alpha}e^{b}_{\mu}V^{\alpha},$
(63)

the gauge transformation (60) is given by the following expression

$\displaystyle B$ $\displaystyle\rightarrow$ $\displaystyle
B+\mathfrak{L}_{V}B-\epsilon_{\mu\alpha}\frac{\delta
S}{\delta\omega_{\mu}}V^{\alpha},$ $\displaystyle B_{a}$
$\displaystyle\rightarrow$ $\displaystyle
B_{a}+\mathfrak{L}_{V}B_{a}-\epsilon_{\mu\alpha}\frac{\delta S}{\delta
e^{a}_{\mu}}V^{\alpha},$ $\displaystyle e^{a}_{\mu}$
$\displaystyle\rightarrow$ $\displaystyle
e^{a}_{\mu}+\mathfrak{L}_{V}e^{a}_{\mu}+\epsilon_{\mu\alpha}\frac{\delta
S}{\delta B_{a}}V^{\alpha},$ $\displaystyle\omega_{\mu}$
$\displaystyle\rightarrow$
$\displaystyle\omega_{\mu}+\mathfrak{L}_{V}\omega_{\mu}+\epsilon_{\mu\alpha}\frac{\delta
S}{\delta B}V^{\alpha}.$ (64)

Where the symbol $\mathfrak{L}$ represents the Lie derivative. Therefore,
diffeomorphisms are obtained as an internal symmetry of the theory from the
fundamental gauge transformations (on shell).

In conclusion, we conducted a pure Hamiltonian analysis for the action (10).
This analysis yielded the extended action, extended Hamiltonian, the complete
structure of constraints on the full phase space, their algebra, the count of
degrees of freedom, and the fundamental gauge transformations. While working
on the complete phase space introduces a set of first and second-class
constraints, using the second-class constraints allows us to construct Dirac
brackets, which will be valuable in the quantization of the theory. It is
relevant to emphasize that when examining the scenario with k=0 in the model,
results associated with the Jackiw-Teitelboim model are obtained.

##  5 Faddeev-Jackiw Analysis

In this section, we examine the BF Polynomial action within the framework of
the modified Faddeev-Jackiw formalism [43, 44]. Throughout this analysis, we
not only address the constraint structure but also provide a detailed
description of all the fundamental brackets present in the system.
Furthermore, we carry out the determination of gauge transformations of the
symplectic variables.

###  5.1 Symplectic formalism

An essential prerequisite for the Faddeev–Jackiw formalism is the presence of
an initial Lagrangian of first order [40]

$\displaystyle\mathcal{L}(\xi,\dot{\xi})=K^{(0)}_{i}\dot{\xi}^{(0)i}-V^{(0)}(\xi),\qquad
i=1,...,N.$ (65)

Here, $\xi^{(0)i}$ represents a set of symplectic variables, with the
canonical 1-form denoted by $\mathrm{K}^{(0)}_{i}$, and $V^{(0)}(\xi^{(0)})$
represents the initial symplectic potential.

Within the Faddeev-Jackiw (FJ) framework, the Euler-Lagrange equations of
motion are [40, 41, 42]

$\int f^{(0)}_{ij}(x,y)\dot{\xi}^{j}(y)dy=\frac{\delta}{\delta\xi^{i}(x)}\int
V^{(0)}(y)dy.$ (66)

Additionally, the construction of a symplectic matrix $f^{(0)}_{ij}$ is
defined through the canonical 1-form $\mathrm{K}^{(0)}_{i}$

$f^{(0)}_{ij}(x,y)=\frac{\delta\mathrm{K}_{j}(y)}{\delta\xi^{i}(x)}-\frac{\delta\mathrm{K}_{i}(x)}{\delta\xi^{j}(y)}.$
(67)

If $f^{(0)}$ is regular, all symplectic variables can be determined using
(66). However, if $f^{(0)}$ is singular, it implies the presence of
constraints in the system. Suppose the rank of $f^{(0)}_{ij}$ is $r$, then
$N-r=M$, In this scenario, there are $M$ zero-mode vectors
$(v^{(0)}_{i})^{T}_{\alpha}$ of (67). These zero-mode vectors satisfy the
equation

$(v^{(0)}_{i})^{T}_{\alpha}f^{(0)}_{ij}(x,y)=0,\qquad\alpha=1,...,M.$ (68)

Hence, by utilizing the equation of motion (66), it is possible to formulate

$\displaystyle\Omega^{(0)}_{\alpha}$ $\displaystyle=$ $\displaystyle\int
dx(v^{(0)}_{i})^{T}_{\alpha}(x)\frac{\delta}{\delta\xi^{(0)i}(x)}\int
dyV^{(0)}(\xi).$ (69)

In the FJ symplectic formalism, the quantities $\Omega^{(0)}_{\alpha}$
represent restrictions.

From (18), the first-order symplectic Lagrangian density of Polynomial BF
action is given by

$\displaystyle\mathcal{L}^{(0)}=B\dot{\omega}_{1}+\frac{1}{l}B_{a}\dot{e}^{a}_{1}-V^{(0)},$
(70)

here the superscript (0) means initial Lagrangian and where
$V^{(0)}=-\omega_{0}(\partial_{1}B+\frac{1}{l}\epsilon{{}^{a}}_{b}B_{a}e^{b}_{1})-\frac{e^{a}_{0}}{l}(D_{1}B_{a}-\frac{1}{l}\epsilon_{ab}e^{b}_{1}B(1-2k^{2}B))$
is called the symplectic potential.

In the [FJ] framework, it should be emphasized that the choice of symplectic
variables is flexible, allowing us to opt for either the configuration
variables or the phase space variables. Previous sections involved the
construction of Dirac brackets through the elimination of second-class
constraints. Consequently, in order to derive these outcomes within the [FJ]
framework, we will opt for the configuration space as the symplectic variables
[40, 41, 42]. To achieve this, we specifically select the following symplectic
variables
$\xi{{}^{(0)i}}(x)=\\{B,B_{a},\omega_{0},\omega_{1},e^{a}_{0},e^{a}_{1}\\}$
from the symplectic Lagrangian $\mathcal{L}^{(0)}$, and the components of the
symplectic 1-forms are
$\mathrm{K}^{(0)}_{i}(x)=\\{0,0,0,B,0,\frac{1}{l}B_{a}\\}$. Therefore,
employing our chosen set of symplectic variables results in the symplectic
matrix (67) taking a specific form

$\displaystyle
f^{(0)}_{ij}(x,y)=\left[\begin{array}[]{l|cccccc}&B&B_{b}&\omega_{0}&\omega_{1}&e^{b}_{0}&e^{b}_{1}\\\
\hline\cr B&0&0&0&1&0&0\\\ B_{a}&0&0&0&0&0&\frac{1}{l}\delta^{a}_{b}\\\
\omega_{0}&0&0&0&0&0&0\\\ \omega_{1}&-1&0&0&0&0&0\\\ e^{a}_{0}&0&0&0&0&0&0\\\
e^{a}_{1}&0&-\frac{1}{l}\delta^{b}_{a}&0&0&0&0\end{array}\right]\delta(x-y).$
(78)

Clearly, the symplectic matrix (78) is singular and has dimensions of
$[9\times 9]$. Upon calculating the matrix rank, which is found to be 6, we
deduce that M=9-6=3. This fact results in the presence of zero-modes
$(v_{i}^{(0)})_{1}^{T}=(0,0,v^{\omega_{0}},0,0,0)$ and
$(v_{i}^{(0)})_{2}^{T}=(0,0,0,0,v^{e^{a}_{0}},0)$, where $v^{\omega_{0}}$ and
$v^{e^{a}_{0}}$ are arbitrary functions. Consequently, by using (69), the
constraints we compute are

$\displaystyle\Omega^{(0)}$ $\displaystyle=$ $\displaystyle\int
dx(v^{(0)}_{i})^{T}_{1}(x)\frac{\delta}{\delta\xi^{(0)i}(x)}\int
dyV^{(0)}(\xi),$ $\displaystyle=$ $\displaystyle-\int
dxv^{\omega_{0}}(x)(\partial_{1}B+\frac{1}{l}\epsilon{{}^{a}}_{b}B_{a}e^{b}_{1}),$
$\displaystyle\rightarrow$
$\displaystyle-(\partial_{1}B+\frac{1}{l}\epsilon{{}^{a}}_{b}B_{a}e^{b}_{1})=0,$
$\displaystyle\Omega^{(0)}_{a}$ $\displaystyle=$ $\displaystyle\int
dx(v^{(0)}_{i})^{T}_{2}(x)\frac{\delta}{\delta\xi^{(0)i}(x)}\int
dyV^{(0)}(\xi),$ $\displaystyle=$ $\displaystyle-\int
dxv^{e^{a}_{0}}(x)(D_{1}B_{a}-\frac{1}{l}\epsilon_{ab}e^{b}_{1}B(1-2k^{2}B)),$
$\displaystyle\rightarrow$
$\displaystyle-(D_{1}B_{a}-\frac{1}{l}\epsilon_{ab}e^{b}_{1}B(1-2k^{2}B))=0,$

thus we identify the following constraints

$\displaystyle\Omega^{(0)}$ $\displaystyle=$
$\displaystyle-(\partial_{1}B+\frac{1}{l}\epsilon{{}^{a}}_{b}B_{a}e^{b}_{1})=0,$
$\displaystyle\Omega^{(0)}_{a}$ $\displaystyle=$
$\displaystyle-(D_{1}B_{a}-\frac{1}{l}\epsilon_{ab}e^{b}_{1}B(1-2k^{2}B))=0,$
(80)

the constraints acquired in (80) are the secondary constraints identified
through the Dirac method in the previous section. To investigate the presence
of additional constraints, we will proceed with the implementation of the
modified FJ formalism [43, 44]. In the case where the symplectic matrix f
demonstrates singularity, it becomes possible to convert the equations of
motion (66) along with the consistency condition of the constraint (69), we
have

$\displaystyle f^{(0)}_{ij}\dot{\xi}^{j}$ $\displaystyle=$
$\displaystyle\frac{\partial V}{\partial\xi^{i}},$
$\displaystyle\frac{\partial\Omega^{(0)}_{\alpha}}{\partial\xi^{i}}\dot{\xi}^{i}$
$\displaystyle=$ $\displaystyle 0.$ (81)

We can reformulate (81) as

$\displaystyle F^{(1)}_{ij}\dot{\xi}^{j}=Z_{i}(\xi).$ (82)

At this stage, we have utilized the constraints identified (69) so far to
derive the expression (82), and then obtain

$\displaystyle F^{(1)}_{ij}=\left(\begin{array}[]{ccc}f^{(0)}_{ij}\\\
\frac{\partial\Omega^{(0)}}{\partial\xi^{j}}\\\
\frac{\partial\Omega^{(0)}_{a}}{\partial\xi^{j}}\end{array}\right),\qquad
Z_{i}(\xi)=\left(\begin{array}[]{cccc}\frac{\partial
V^{(0)}(\xi)}{\partial\xi^{i}}\\\ 0\\\ 0\\\ \end{array}\right).$ (89)

Thus the symplectic matrix $F^{(1)}_{ij}$ is deduced as follows

$\displaystyle F^{(1)}_{ij}(x,y)=\left(\begin{array}[]{ccc}f^{(0)}_{ij}\\\
\frac{\partial\Omega^{(0)}}{\partial\xi^{j}}\\\
\frac{\partial\Omega^{(0)}_{a}}{\partial\xi^{j}}\end{array}\right)=\left[\begin{array}[]{l|cccccc}&B&B_{b}&\omega_{0}&\omega_{1}&e^{b}_{0}&e^{b}_{1}\\\
\hline\cr B&0&0&0&1&0&0\\\ B_{a}&0&0&0&0&0&\frac{1}{l}\delta^{a}_{b}\\\
\omega_{0}&0&0&0&0&0&0\\\ \omega_{1}&-1&0&0&0&0&0\\\ e^{a}_{0}&0&0&0&0&0&0\\\
e^{a}_{1}&0&-\frac{1}{l}\delta^{b}_{a}&0&0&0&0\\\
\frac{\partial\Omega^{(0)}}{\partial\xi^{j}}&\frac{\partial\Omega^{(0)}}{\partial
B}&\frac{\partial\Omega^{(0)}}{\partial
B_{b}}&\frac{\partial\Omega^{(0)}}{\partial\omega_{0}}&\frac{\partial\Omega^{(0)}}{\partial\omega_{1}}&\frac{\partial\Omega^{(0)}}{\partial
e^{b}_{0}}&\frac{\partial\Omega^{(0)}}{\partial e^{b}_{1}}\\\
\frac{\partial\Omega^{(0)}_{a}}{\partial\xi^{j}}&\frac{\partial\Omega^{(0)}_{a}}{\partial
B}&\frac{\partial\Omega^{(0)}_{a}}{\partial
B_{b}}&\frac{\partial\Omega^{(0)}_{a}}{\partial\omega_{0}}&\frac{\partial\Omega^{(0)}_{a}}{\partial\omega_{1}}&\frac{\partial\Omega^{(0)}_{a}}{\partial
e^{b}_{0}}&\frac{\partial\Omega^{(0)}_{a}}{\partial e^{b}_{1}}\\\
\end{array}\right]\delta(x-y).$ (102)

The coefficient matrix (102) mentioned in (82) is non-square matrix. However,
despite this, it still exhibits linearly independent zero modes

$\displaystyle(v^{(1)}_{i})_{1}^{T}$ $\displaystyle=$
$\displaystyle(0,-\varepsilon{{}^{c}}_{a}B_{c}\delta(x-y),v^{\omega_{0}},\partial^{y}_{1}\delta(x-y),v^{e^{a}_{0}},\varepsilon{{}^{a}}_{c}e^{c}_{1}\delta(x-y),\delta(x-y),0),$
$\displaystyle(v^{(1)}_{i})_{2}^{T}$ $\displaystyle=$
$\displaystyle(-\varepsilon{{}_{ab}}B^{b}\delta(x-y),\varepsilon_{ab}B(1-2k^{2}B)\delta(x-y),v^{\omega_{0}},-\frac{1}{l}\varepsilon_{ab}e^{b}_{1}(1-4k^{2}B)\delta(x-y),v^{e^{a}_{0}},$
(103) $\displaystyle
l(\delta^{b}_{a}\partial^{y}_{1}\delta(x-y)+\varepsilon{{}_{a}}^{b}\omega_{1}\delta(x-y)),0,\delta^{b}_{a}\delta(x-y)).$

Multiplying it on the left side of $F^{(1)}_{ij}$ results in zero. Expanding
this process to both sides of equation (82), we obtain the corresponding
constraints [43, 44]

$(v^{(1)}_{i})_{c}^{T}Z_{i}=0,$ (104)

where $c=1,2$, we obtain that (104) is an identity given by

$(v^{(1)}_{i})_{1}^{T}Z_{i}=-\frac{e^{a}_{0}}{l}\varepsilon{{}_{a}}{{}^{b}}\Omega^{(0)}_{b}\biggm{|}_{\Omega^{(0)}_{b}=0}=0,$
(105)

and

$(v^{(1)}_{i})_{2}^{T}Z_{i}=\omega_{0}\varepsilon{{}_{a}}^{b}\Omega^{(0)}_{b}\biggm{|}_{\Omega^{(0)}_{b}=0}+\frac{e^{b}_{0}}{l}\varepsilon_{ba}(1-4k^{2}B)\Omega^{(0)}\biggm{|}_{\Omega^{(0)}=0}=0,$
(106)

thus, in [FJ] formalism there are not more constraints for the theory under
study.

Analogous to the conventional Faddeev–Jackiw approach, we have the ability to
incorporate Lagrange multipliers $e^{a}_{0}=\dot{\lambda}^{a}$ and
$\omega_{0}=\dot{\lambda}$ that align with (105) and (106) into the
Lagrangian, thereby formulating a novel Lagrangian

$\mathcal{L}^{(1)}=B\dot{\omega}_{1}+\frac{1}{l}B_{a}\dot{e}^{a}_{1}+\Omega^{(0)}_{a}\dot{\lambda}^{a}+\Omega^{(0)}\dot{\lambda}-V^{(1)}.$
(107)

In the scenario where
$V^{(1)}=V^{(0)}\mid_{\Omega^{(0)}{a}=0,\Omega^{(0)}=0}=0$, the symplectic
potential vanishes, highlighting the theory’s general covariance, similar to
the principle observed in General Relativity. Thus, referring to (107), we can
define a new set of symplectic variables
$\xi{{}^{(1)i}}(x)=\\{B,B_{a},\lambda,\omega_{1},\lambda^{a},e^{a}_{1}\\}$,
accompanied by the corresponding symplectic 1-forms
$\mathrm{K}^{(1)}_{i}(x)=\\{0,0,\Omega^{(0)},B,\Omega^{(0)}_{a},\frac{1}{l}B_{a}\\}$.
By employing these revised symplectic variables and 1-forms, we can proceed to
compute the symplectic matrix accordingly

$\displaystyle
f^{(1)}_{ij}(x,y)=\left[\begin{array}[]{l|cccccc}&B&B_{b}&\lambda&\omega_{1}&\lambda^{b}&e^{b}_{1}\\\
\hline\cr
B&0&0&\partial^{y}_{1}&1&-\frac{1}{l}\epsilon_{bc}e^{c}_{1}(1-4k^{2}B)&0\\\
B_{a}&0&0&\frac{1}{l}\epsilon{{}^{a}}_{c}e^{c}_{1}&0&\delta^{a}_{b}\partial^{y}_{1}-\epsilon{{}^{a}}_{b}\omega_{1}&\frac{1}{l}\delta^{a}_{b}\\\
\lambda&-\partial^{x}_{1}&-\frac{1}{l}\epsilon{{}^{b}}_{c}e^{c}_{1}&0&0&0&-\frac{1}{l}\epsilon{{}^{c}}_{b}B_{c}\\\
\omega_{1}&-1&0&0&0&\epsilon{{}_{b}}^{c}B_{c}&0\\\
\lambda^{a}&\frac{1}{l}\epsilon_{ab}e^{b}_{1}(1-4k^{2}B)&-\delta^{a}_{b}\partial^{x}_{1}-\epsilon{{}_{b}}^{a}\omega_{1}&0&-\epsilon{{}_{a}}^{c}B_{c}&0&\frac{1}{l}\epsilon_{ab}B(1-2k^{2}B)\\\
e^{a}_{1}&0&-\frac{1}{l}\delta^{b}_{a}&\frac{1}{l}\epsilon{{}^{c}}_{a}B_{c}&0&\frac{1}{l}\epsilon_{ab}B(1-2k^{2}B)&0\\\
\end{array}\right]$

$\displaystyle\times\delta(x-y).$ (109)

###  5.2 FJ brackets

The symplectic matrix $f^{(1)}_{ij}$ is a singular $[9\times 9]$ matrix.
Despite this singularity, we have established that there are no additional
constraints. The noninvertibility of (LABEL:51) signifies the existence of
gauge symmetry within the theory. To eliminate redundant degrees of freedom
and derive the desired Faddeev–Jackiw generalized brackets, additional gauge-
fixing conditions must be applied. Since there are two linearly independent
null eigenvectors, we opt for the following gauge constraints

$\displaystyle\omega_{0}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
e^{a}_{0}(x)$ $\displaystyle=$ $\displaystyle 0.$ (110)

Following that, we add the Lagrangian multipliers $\alpha_{a}$ and $\theta$
corresponding to the previously discussed gauge fixing, with the goal of
developing a new symplectic Lagrangian. The expression for the symplectic
Lagrangian is now

$\mathcal{L}^{(2)}=B\dot{\omega}_{1}+\frac{1}{l}B_{a}\dot{e}^{a}_{1}+(\alpha_{a}+\Omega^{(0)}_{a})\dot{\lambda}^{a}+(\theta+\Omega^{(0)})\dot{\lambda},$
(111)

therefore, we can characterize the following set of symplectic variables
$\xi{{}^{(2)i}}(x)=\\{B,B_{a},\lambda,\theta,\omega_{1},\lambda^{a},\alpha_{a},e^{a}_{1}\\}$
and the symplectic 1-forms
$\mathrm{K}^{(2)}_{i}(x)=\\{0,0,\Omega^{(0)}+\theta,0,B,\Omega^{(0)}_{a}+\alpha_{a},0,\frac{1}{l}B_{a}\\}$.
Additionally, by employing these symplectic variables, we establish the
expression for the symplectic matrix

$\displaystyle f^{(2)}_{ij}(x,y)=$

$\displaystyle\left[\begin{array}[]{l|cccccccc}&B&B_{b}&\lambda&\theta&\omega_{1}&\lambda^{b}&\alpha_{b}&e^{b}_{1}\\\
\hline\cr
B&0&0&\partial^{y}_{1}&0&1&-\frac{1}{l}\epsilon_{bc}e^{c}_{1}(1-4k^{2}B)&0&0\\\
B_{a}&0&0&\frac{1}{l}\epsilon{{}^{a}}_{c}e^{c}_{1}&0&0&\delta^{a}_{b}\partial^{y}_{1}-\epsilon{{}^{a}}_{b}\omega_{1}&0&\frac{1}{l}\delta^{a}_{b}\\\
\lambda&-\partial^{x}_{1}&-\frac{1}{l}\epsilon{{}^{b}}_{c}e^{c}_{1}&0&-1&0&0&0&-\frac{1}{l}\epsilon{{}^{c}}_{b}B_{c}\\\
\theta&0&0&1&0&0&0&0&0\\\
\omega_{1}&-1&0&0&0&0&\epsilon{{}_{b}}^{c}B_{c}&0&0\\\
\lambda^{a}&\frac{1}{l}\epsilon_{ab}e^{b}_{1}(1-4k^{2}B)&-\delta^{b}_{a}\partial^{x}_{1}-\epsilon{{}_{a}}^{b}\omega_{1}&0&0&-\epsilon{{}_{a}}^{c}B_{c}&0&-\delta^{b}_{a}&\frac{1}{l}\epsilon_{ab}B(1-2k^{2}B)\\\
\alpha_{a}&0&0&0&0&0&\delta^{b}_{a}&0&0\\\
e^{a}_{1}&0&-\frac{1}{l}\delta^{b}_{a}&\frac{1}{l}\epsilon{{}^{c}}_{a}B_{c}&0&0&\frac{1}{l}\epsilon_{ab}B(1-2k^{2}B)&0&0\\\
\end{array}\right]$

$\displaystyle\times\delta(x-y).$ (113)

Thus far, a non-singular matrix has been obtained, allowing for the deduction
of its inverse matrix

$\displaystyle[f^{(2)}_{ij}(x,y)]^{-1}=$

$\displaystyle\left[\begin{array}[]{l|cccccccc}&B&B_{b}&\lambda&\theta&\omega_{1}&\lambda^{b}&\alpha_{b}&e^{b}_{1}\\\
\hline\cr B&0&0&0&0&-1&0&-\epsilon{{}_{a}}^{c}B_{c}&0\\\
B_{a}&0&0&0&\epsilon{{}^{c}}_{b}B_{c}&0&0&\epsilon_{ba}B(1-2k^{2}B)&-l\delta^{a}_{b}\\\
\lambda&0&0&0&1&0&0&0&0\\\
\theta&0&-\epsilon{{}^{c}}_{a}B_{c}&-1&0&\partial^{x}_{1}&0&0&\epsilon{{}^{a}}_{c}e^{c}_{1}\\\
\omega_{1}&1&0&0&-\partial^{y}_{1}&0&0&\frac{1}{l}\epsilon_{ac}e^{c}_{1}(1-4k^{2}B)&0\\\
\lambda^{a}&0&0&0&0&0&0&\delta^{b}_{a}&0\\\
\alpha_{a}&\epsilon{{}_{b}}^{c}B_{c}&\epsilon_{ba}B(1-2k^{2}B)&0&0&-\frac{1}{l}\epsilon_{bc}e^{c}_{1}(1-4k^{2}B)&-\delta^{a}_{b}&0&l(\delta^{a}_{b}\partial^{x}_{1}+\varepsilon{{}_{b}}^{a}\omega_{1})\\\
e^{a}_{1}&0&l\delta^{b}_{a}&0&-\epsilon{{}^{b}}_{c}e^{c}_{1}&0&0&-l(\delta^{b}_{a}\partial^{y}_{1}+\varepsilon{{}^{b}}_{a}\omega_{1})&0\\\
\end{array}\right]$

$\displaystyle\times\delta(x-y).$ (115)

By calculating the inverse of this symplectic matrix(5.2), we can recognize
the standard Faddeev–Jackiw generalized brackets for the physical variables
through a defined procedure

$\displaystyle\\{\xi_{i}^{(2)}(x),\xi_{j}^{(2)}(y)\\}_{FJ}=[f^{(2)}_{ij}(x,y)]^{-1},$
(116)

thus, the following brackets are identified

$\displaystyle\\{\omega_{1}(x),B(y)\\}_{FJ}$ $\displaystyle=$
$\displaystyle\delta(x-y),$ $\displaystyle\\{e^{a}_{1}(x),B_{b}(y)\\}_{FJ}$
$\displaystyle=$ $\displaystyle l\delta^{a}_{b}\delta(x-y).$ (117)

It is noteworthy that the generalized (FJ) brackets correspond to those
derived using the Dirac method mentioned earlier. Specifically, a redefinition
of the fields, introducing the momenta as provided by

$\displaystyle\Pi$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle\Pi^{a}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\Pi^{1}_{a}$
$\displaystyle=$ $\displaystyle\frac{1}{l}B_{a},$ $\displaystyle\Pi^{1}$
$\displaystyle=$ $\displaystyle B.$ (118)

It allows us to derive the following generalized Faddeev–Jackiw (FJ) brackets
and demonstrate their equivalence with the Dirac brackets obtained earlier
(56)

$\displaystyle\\{e^{a}_{1}(x),\Pi^{1}_{b}(y)\\}_{FJ}$ $\displaystyle=$
$\displaystyle\delta^{a}_{b}\delta(x-y),$
$\displaystyle\\{e^{a}_{1}(x),B_{a}(y)\\}_{FJ}$ $\displaystyle=$
$\displaystyle l\delta^{a}_{b}\delta(x-y),$
$\displaystyle\\{\omega_{1}(x),\Pi^{1}(y)\\}_{FJ}$ $\displaystyle=$
$\displaystyle\delta(x-y),$ $\displaystyle\\{\omega_{1}(x),B(y)\\}_{FJ}$
$\displaystyle=$ $\displaystyle\delta(x-y).$ (119)

Now, it is essential to introduce a bracket for observables on $\sum$. This
operator must align with the commutator in the classical limit. In this
context, for any pair of observables $\mathcal{O}_{1}$, $\mathcal{O}_{2}$
defined in the phase space, possessing a symplectic structure such as
$\\{\xi^{final}_{i},\xi^{final}_{j}\\}$ , we can denote the following
relationship [55]

$\displaystyle\\{\mathcal{O}_{1}(\xi),\mathcal{O}_{2}(\xi)\\}_{FJ}=\sum_{ij}\int
d^{3}r\frac{\delta\mathcal{O}_{1}(\xi)}{\delta\xi_{i}(r)}(f^{(final)}_{ij})^{-1}\frac{\delta\mathcal{O}_{2}(\xi)}{\delta\xi_{j}(r)}.$
(120)

The transition of the Faddeev-Jackiw bracket by operator commutattion
relations according to

$\displaystyle\\{\mathcal{O}_{1}(\xi),\mathcal{O}_{2}(\xi)\\}_{FJ}\quad\rightarrow\frac{1}{i\hbar}[\mathcal{O}_{1}(\xi),\mathcal{O}_{2}(\xi)].$
(121)

###  5.3 Path integral

In this section, we will determine the transition amplitude [56, 57]. To
accomplish this, we will rewrite the matrix (113) as follows in order

$\displaystyle\left[\begin{array}[]{l|cccccccc}&B&B_{b}&\omega_{1}&e^{b}_{1}&\lambda&\theta&\lambda^{b}&\alpha_{b}\\\
\hline\cr
B&0&0&1&0&\partial^{y}_{1}&0-&\frac{1}{l}\epsilon_{bc}e^{c}_{1}(1-4k^{2}B)&0\\\
B_{a}&0&0&0&\frac{1}{l}\delta^{a}_{b}&\frac{1}{l}\epsilon{{}^{a}}_{c}e^{c}_{1}&0&\delta^{a}_{b}\partial^{y}_{1}-\epsilon{{}^{a}}_{b}\omega_{1}&0\\\
\omega_{1}&-1&0&0&0&0&0&\epsilon{{}_{b}}^{c}B_{c}&0\\\
e^{a}_{1}&0&-\frac{1}{l}\delta^{b}_{a}&0&0&\frac{1}{l}\epsilon{{}^{c}}_{a}B_{c}&0&\frac{1}{l}\epsilon_{ab}B(1-2k^{2}B)&0\\\
\lambda&-\partial^{x}_{1}&-\frac{1}{l}\epsilon{{}^{b}}_{c}e^{c}_{1}&0&-\frac{1}{l}\epsilon{{}^{c}}_{b}B_{c}&0&-1&0&0\\\
\theta&0&0&0&0&1&0&0&0\\\
\lambda^{a}&\frac{1}{l}\epsilon_{ab}e^{b}_{1}(1-4k^{2}B)&-\delta^{b}_{a}\partial^{x}_{1}-\epsilon{{}_{a}}^{b}\omega_{1}&-\epsilon{{}_{a}}^{c}B_{c}&\frac{1}{l}\epsilon_{ab}B(1-2k^{2}B)&0&0&0&-\delta^{b}_{a}\\\
\alpha_{a}&0&0&0&0&0&0&\delta^{b}_{a}&0\\\ \end{array}\right]$

$\displaystyle\times\delta(x-y).$ (123)

To calculate the determinant of the symplectic two-form in (LABEL:58a), we
will examine its general block form structure

$\displaystyle
f^{(2)}_{ij}=\left(\begin{array}[]{cc}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&\mathcal{D}\\\ \end{array}\right),\qquad
with\qquad\mathcal{C}=-\mathcal{B}^{T}.$ (126)

In this context, the specific expression for each sub-matrix $\mathcal{A}$,
$\mathcal{B}$, and $\mathcal{D}$ in equation (126) is revealed to be

$\displaystyle\small{\mathcal{A}=\left[\begin{array}[]{l|cccc}&B&B_{b}&\omega_{1}&e^{b}_{1}\\\
\hline\cr B&0&0&1&0\\\ B_{a}&0&0&0&\frac{1}{l}\delta^{a}_{b}\\\
\omega_{1}&-1&0&0&0\\\ e^{a}_{1}&0&-\frac{1}{l}\delta^{b}_{a}&0&0\\\
\end{array}\right],\quad\mathcal{B}=\left[\begin{array}[]{l|cccc}&\lambda&\theta&\lambda^{b}&\alpha_{b}\\\
\hline\cr
B&\partial^{y}_{1}&0&-\frac{1}{l}\epsilon_{bc}e^{c}_{1}(1-4k^{2}B)&0\\\
B_{a}&\frac{1}{l}\epsilon{{}^{a}}_{c}e^{c}_{1}&0&\delta^{a}_{b}\partial^{y}_{1}-\epsilon{{}^{a}}_{b}\omega_{1}&0\\\
\omega_{1}&0&0&\epsilon{{}_{b}}^{c}B_{c}&0\\\
e^{a}_{1}&\frac{1}{l}\epsilon{{}^{c}}_{a}B_{c}&0&\frac{1}{l}\epsilon_{ab}B(1-2k^{2}B)&0\\\
\end{array}\right],\quad\mathcal{D}=\left[\begin{array}[]{l|cccc}&\lambda&\theta&\lambda^{b}&\alpha_{b}\\\
\hline\cr\lambda&0&-1&0&0\\\ \theta&1&0&0&0\\\
\lambda^{a}&0&0&0&-\delta^{b}_{a}\\\ \alpha_{a}&0&0&\delta^{b}_{a}&0\\\
\end{array}\right]}.$

Using the standard identity for any matrix

$\displaystyle\left(\begin{array}[]{cc}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&\mathcal{D}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\mathcal{A}&0\\\
\mathcal{C}&\mathcal{I}\\\
\end{array}\right)\left(\begin{array}[]{cc}\mathcal{I}&\mathcal{A}^{-1}\mathcal{B}\\\
0&\mathcal{D}-\mathcal{C}\mathcal{A}^{-1}\mathcal{B}\\\ \end{array}\right),$
(134)

assuming that $\mathcal{A}^{-1}$ exists, to see that

$\displaystyle det\left(\begin{array}[]{cc}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&\mathcal{D}\\\
\end{array}\right)=(det\mathcal{A})[det(\mathcal{D}-\mathcal{C}\mathcal{A}^{-1}\mathcal{B})],$
(137)

Therefore, utilizing equation (137) and performing algebraic manipulations, it
becomes feasible to demonstrate the determinant of $f^{(2)}_{ij}$ is evaluated
in terms of the symplectic two-form from and is equal

$\displaystyle det[f^{(2)}_{ij}]=\frac{1}{l^{4}}.$ (138)

Presently, it is noteworthy to observe that according to [56, 57], the
functional measure within the path integral linked to our model, under the
time gauge, is

$\displaystyle
d\mu=(\prod_{i}[D\xi^{(2)i}])(det[f^{(2)}_{ij}(x,y)])^{1/2}=(\prod_{i}[D\xi^{(2)i}])[\frac{1}{l^{2}}]=(\prod_{i}[D\xi^{(2)i}])\Lambda.$
(139)

It is evident that the path integral measure in (139) is associated with the
cosmological constant. The approximation proposed by Batalin, Fradkin, and
Vilkovisky (BFV) is utilized in [39], incorporating the concept of constraints
in Path Integral Quantization during the calculation of the transition
amplitude. It is observed that the functional measure in the time gauge equals
1, a distinction from the measure identified in the current study.

The partition function incorporates the symplectic two-form straightforwardly;
since this element is crucial for the Faddeev-Jackiw method, no additional
effort is needed, except for calculating its determinant. In accordance with
(139) and the Lagrangian (111), the partition function takes the following
form

$\displaystyle Z$ $\displaystyle=$ $\displaystyle\int d\mu
exp[\frac{i}{\hbar}\int d^{3}x\mathcal{L}^{(2)}],$ (140) $\displaystyle=$
$\displaystyle\int DBDB_{a}D\omega_{1}De^{a}_{1}D\lambda D\theta
D\lambda^{a}D\alpha_{a}[\frac{1}{l^{2}}]exp[\frac{i}{\hbar}\int
d^{3}x(B\dot{\omega}_{1}+\frac{1}{l}B_{a}\dot{e}^{a}_{1}+(\alpha_{a}+\Omega^{(0)}_{a})\dot{\lambda}^{a}+(\theta+\Omega^{(0)})\dot{\lambda})].$

###  5.4 FJ physical degrees of freedom

As mentioned earlier, in the Faddeev–Jackiw (FJ) approach, there’s no need to
categorize constraints as first or second class, as all constraints hold the
same significance. Consequently, the count of physical degrees of freedom can
be performed as follows: with $6$ dynamical variables represented by
$(B,\omega_{1},B_{a},e^{a}_{1},)$, and $6$ constraints
$(\Omega^{(0)}_{a},\Omega^{(0)},e^{a}_{0},\omega_{0})$, the theory exhibits no
physical degrees of freedom.

###  5.5 FJ gauge generator

We finish this section by calculating the gauge transformations of the theory,
for this aim we calculate the modes of the matrix (109), this mode are given
by

$\displaystyle(w^{(1)}_{i})_{1}^{T}$ $\displaystyle=$
$\displaystyle(0,\epsilon{{}_{b}}^{c}B_{c}\delta(x-y),-\delta(x-y),\partial^{y}_{1}\delta(x-y),0,\epsilon{{}^{b}}_{c}e^{c}_{1}\delta(x-y)),$
$\displaystyle(w^{(1)}_{i})_{2}^{T}$ $\displaystyle=$
$\displaystyle(-\epsilon{{}_{a}}^{c}B_{c}\delta(x-y),-\epsilon_{ab}B(1-2k^{2}B)\delta(x-y),0,-\frac{1}{l}\epsilon_{ac}e^{c}_{1}(1-4k^{2}B)\delta(x-y),-\delta^{b}_{a}\delta(x-y),l(\delta^{b}_{a}\partial^{y}_{1}\delta(x-y)$
(141) $\displaystyle+\epsilon{{}_{b}}^{a}\omega_{1}\delta(x-y))).$

Consistent with the [FJ] symplectic formalism, the zero modes
$(w^{(1)}_{i})_{1}^{T}$ and $(w^{(1)}_{i})_{2}^{T}$ act as the generator of
the infinitesimal gauge transformation of the action (70). Specificall we have
that

$\displaystyle\delta_{G}\xi^{(1)i}=\int
dx(w^{(1)}_{i})^{T}_{c}\varepsilon^{c}$ (142)

where $"\varepsilon^{c}"$ is a set of infinitesimal arbitrary parameters.

From the above, we can see that the gauge transformation is therefore given by
777Where $\epsilon^{1}=\tilde{\theta}=-\tau$ and
$\epsilon^{2}=\tilde{\theta}^{a}=-\tau^{a}$.

$\displaystyle\delta_{G}\xi^{(1)a}=(\delta_{G}B,\delta_{G}B_{a},\delta_{G}\lambda,\delta_{G}\omega_{1},\delta_{G}\lambda^{a},\delta_{G}e^{a}_{1})=\int
dx[(w^{(1)}_{i})^{T}_{1}\varepsilon^{1}+(w^{(2)}_{i})^{T}_{1}\varepsilon^{2}],$
(143)

we obtain 888coming back to the original variables,
$\partial_{t}\delta{{}_{G}}\lambda=\delta_{G}\omega_{0}$ and
$\partial_{t}\delta{{}_{G}}\lambda^{a}=\delta_{G}e^{a}_{0}$.

$\displaystyle\delta{{}_{G}}B$ $\displaystyle=$
$\displaystyle-\epsilon_{ab}\tilde{\theta}^{a}B^{b},$
$\displaystyle\delta{{}_{G}}B_{a}$ $\displaystyle=$
$\displaystyle\epsilon_{ab}\tilde{\theta}B^{b}-\epsilon_{ab}\tilde{\theta}^{b}B(1-2k^{2}B),$
$\displaystyle\delta{{}_{G}}\omega_{0}$ $\displaystyle=$
$\displaystyle-\partial_{0}\tilde{\theta}=\partial_{0}\tau,$
$\displaystyle\delta{{}_{G}}\omega_{1}$ $\displaystyle=$
$\displaystyle-\partial_{1}\tilde{\theta}-\frac{1}{l}\epsilon_{ab}\tilde{\theta}^{a}e^{b}_{1}(1-4k^{2}B),$
$\displaystyle\delta{{}_{G}}e^{a}_{0}$ $\displaystyle=$
$\displaystyle-\partial_{0}\tilde{\theta^{a}}=\partial_{0}\tau^{a},$
$\displaystyle\delta{{}_{G}}e^{a}_{1}$ $\displaystyle=$ $\displaystyle-
lD_{1}\tilde{\theta}^{a}+\epsilon{{}^{a}}_{b}\tilde{\theta}e^{b}_{1}.$ (144)

Using the FJ symplectic framework, we have successfully reproduced most
components of the gauge transformations described in the Dirac approach.
However, a discrepancy is observed in the transformations of the fields
$e^{a}_{0}$ between the FJ and Dirac approaches. Montani and Wotzasek (MW)
[58] propose a method to obtain complete transformations across the entire
configuration space. According to the MW method, one should write the
functional variation of the corresponding Lagrangians999In term of matrix form
$(\frac{\delta\mathcal{L}^{(1)}}{\delta\xi^{(1)i}})^{T}=\left(\begin{array}[]{cccccc}\frac{\delta\mathcal{L}^{(1)}}{\delta\xi^{(1)1}}&\frac{\delta\mathcal{L}^{(1)}}{\delta\xi^{(1)2}}&\frac{\delta\mathcal{L}^{(1)}}{\delta\xi^{(1)3}}&\frac{\delta\mathcal{L}^{(1)}}{\delta\xi^{(1)4}}&\frac{\delta\mathcal{L}^{(1)}}{\delta\xi^{(1)5}}&\frac{\delta\mathcal{L}^{(1)}}{\delta\xi^{(1)6}}\end{array}\right)$.
to zero. Therefore,

$\displaystyle\frac{\delta\mathcal{L}^{(1)}}{\delta\xi^{(1)i}}=f^{(1)}_{ij}\dot{\xi}^{(1)j}-\frac{\partial
V^{(1)}}{\partial\xi^{(1)i}}.$ (145)

So, according to MW, on multiplying (145) by the zero modes (141), we have

$\displaystyle\epsilon^{c}(w^{(1)}_{i})^{T}_{c}\left(\frac{\delta\mathcal{L}^{(1)}}{\delta\xi^{(1)i}}=f^{(1)}_{ij}\dot{\xi}^{(1)j}-\frac{\partial
V^{(1)}}{\partial\xi^{(1)i}}\right),\quad c=1,2.$ (146)

from (146), one finds the GT over all the configuration space. Using this
equation, the GT can be written as

$\displaystyle(\epsilon^{1}(w^{(1)}_{i})^{T}_{1}+\epsilon^{2}(w^{(1)}_{i})^{T}_{2})\frac{\delta\mathcal{L}^{(1)}}{\delta\xi^{(1)i}}=(\epsilon^{1}(w^{(1)}_{i})^{T}_{1}+\epsilon^{2}(w^{(1)}_{i})^{T}_{2})f^{(1)}_{ij}\dot{\xi}^{(1)j}$
(147)

Specifically we have that

$\displaystyle\mathcal{A}_{1}\frac{\delta\mathcal{L}^{(1)}}{\delta
B}+\mathcal{A}_{2}\frac{\delta\mathcal{L}^{(1)}}{\delta
B_{a}}+\mathcal{A}_{3}\frac{\delta\mathcal{L}^{(1)}}{\delta\lambda}+\mathcal{A}_{4}\frac{\delta\mathcal{L}^{(1)}}{\delta\omega_{1}}+\mathcal{A}_{5}\frac{\delta\mathcal{L}^{(1)}}{\delta\lambda^{a}}+\mathcal{A}_{6}\frac{\delta\mathcal{L}^{(1)}}{\delta
e^{a}_{1}}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{1}(\dot{\omega}_{1}-\frac{1}{l}\epsilon_{cb}e^{b}_{1}(1-4k^{2}B)\dot{\lambda}^{c}-\partial_{1}\dot{\lambda})+$
$\displaystyle\mathcal{A}_{2}(\frac{1}{l}\dot{e}^{a}_{1}-\partial_{1}\dot{\lambda}^{a}+\epsilon{{}_{b}}^{a}\omega_{1}\dot{\lambda}^{b}+\frac{1}{l}\epsilon{{}^{a}}_{b}e^{b}_{1}\dot{\lambda})+$
$\displaystyle\mathcal{A}_{3}(-\partial_{t}\Omega^{(0)})+\mathcal{A}_{4}(-\dot{B}+\epsilon{{}_{a}}^{b}B_{b}\dot{\lambda}^{a})+$
$\displaystyle\mathcal{A}_{5}(-\partial_{t}\Omega^{(0)}_{a})+\mathcal{A}_{6}(-\frac{1}{l}\dot{B}_{a}-\frac{1}{l}\epsilon_{ba}B(1-2k^{2}B)\dot{\lambda}^{b}$
$\displaystyle+\frac{1}{l}\epsilon{{}^{b}}_{a}B_{b}\dot{\lambda})).$

where
$\mathcal{A}_{1}=-\epsilon{{}_{a}}^{c}B_{c}\epsilon^{2},\mathcal{A}_{2}=\epsilon{{}_{a}}^{b}B_{b}\epsilon^{1}-\epsilon_{ab}B(1-2k^{2}B)\epsilon^{2},\mathcal{A}_{3}=-\epsilon^{1},\mathcal{A}_{4}=[-\partial_{1}\epsilon^{1}-\frac{1}{l}\epsilon_{ab}e^{b}_{1}(1-4k^{2}B)\epsilon^{2}],\mathcal{A}_{5}=-\epsilon^{2},\mathcal{A}_{6}=\epsilon{{}^{a}}_{b}e^{b}_{1}\epsilon^{1}-lD_{1}\epsilon^{2}$.

Now, coming back to the original variables

$\displaystyle\dot{\lambda}$ $\displaystyle\rightarrow$
$\displaystyle\omega_{0},$ $\displaystyle\dot{\lambda}^{a}$
$\displaystyle\rightarrow$ $\displaystyle e^{a}_{0},$
$\displaystyle\frac{\delta\mathcal{L}^{(1)}}{\delta\lambda}$
$\displaystyle\rightarrow$
$\displaystyle\frac{\delta\mathcal{L}^{(1)}}{\delta\omega_{0}}\partial_{t},$
$\displaystyle\frac{\delta\mathcal{L}^{(1)}}{\delta\lambda^{a}}$
$\displaystyle\rightarrow$ $\displaystyle\frac{\delta\mathcal{L}^{(1)}}{\delta
e^{a}_{0}}\partial_{t},$ $\displaystyle\frac{\delta\mathcal{L}^{(1)}}{\delta
B}$ $\displaystyle=$ $\displaystyle\frac{\delta\mathcal{L}^{(0)}}{\delta B},$
$\displaystyle\frac{\delta\mathcal{L}^{(1)}}{\delta B_{a}}$ $\displaystyle=$
$\displaystyle\frac{\delta\mathcal{L}^{(0)}}{\delta B_{a}},$
$\displaystyle\frac{\delta\mathcal{L}^{(1)}}{\delta\omega_{1}}$
$\displaystyle=$
$\displaystyle\frac{\delta\mathcal{L}^{(0)}}{\delta\omega_{1}},$
$\displaystyle\frac{\delta\mathcal{L}^{(1)}}{\delta e^{a}_{1}}$
$\displaystyle=$ $\displaystyle\frac{\delta\mathcal{L}^{(0)}}{\delta
e^{a}_{1}},$ (148)

and implementing the equations of motion for the gauge fields

$\displaystyle\frac{\delta\mathcal{L}^{(0)}}{\delta\omega_{0}}$
$\displaystyle=$ $\displaystyle\Omega^{(0)},$
$\displaystyle\frac{\delta\mathcal{L}^{(0)}}{\delta e^{a}_{0}}$
$\displaystyle=$ $\displaystyle\Omega^{(0)}_{a},$ (149)

we obtain

$\displaystyle(-1)D_{0}\tilde{\theta}\frac{\partial\mathcal{L}^{(0)}}{\partial\omega_{0}}+(-1)D_{0}\tilde{\theta}^{a}\frac{\partial\mathcal{L}^{(0)}}{\partial
e^{a}_{0}}+(-1)\epsilon_{ab}\tilde{\theta}^{a}B^{b}\frac{\partial\mathcal{L}^{(0)}}{\partial
B}+(\epsilon_{ab}\tilde{\theta}B^{b}-\epsilon_{ab}\tilde{\theta}^{b}B(1-2k^{2}B))\frac{\partial\mathcal{L}^{(0)}}{\partial
B_{a}}+$
$\displaystyle(-1)(lD_{1}\tilde{\theta}^{a}-\epsilon{{}^{a}}_{b}\tilde{\theta}e^{b}_{1})\bigg{(}\frac{\partial\mathcal{L}^{(0)}}{\partial
e^{a}_{1}}-\partial_{t}\frac{\partial\mathcal{L}^{(0)}}{\partial\dot{e}^{a}_{1}}\bigg{)}+(-1)(\partial_{1}\tilde{\theta}+\frac{1}{l}\epsilon_{ab}\tilde{\theta}^{a}e^{b}_{1}(1-4k^{2}B))\bigg{(}\frac{\partial\mathcal{L}^{(0)}}{\partial\omega_{1}}-\partial_{t}\frac{\partial\mathcal{L}^{(0)}}{\partial\dot{\omega}_{1}}\bigg{)}=0.$
(150)

Then, we can find that the gauge field transformations are given by

$\displaystyle\delta{{}_{G}}B$ $\displaystyle=$
$\displaystyle-\epsilon_{ab}\tilde{\theta}^{a}B^{b},$
$\displaystyle\delta{{}_{G}}B_{a}$ $\displaystyle=$
$\displaystyle\epsilon_{ab}\tilde{\theta}B^{b}-\epsilon_{ab}\tilde{\theta}^{b}B(1-2k^{2}B),$
$\displaystyle\delta{{}_{G}}e^{a}_{0}$ $\displaystyle=$ $\displaystyle
D_{0}\tau^{a},$ $\displaystyle\delta{{}_{G}}e^{a}_{1}$ $\displaystyle=$
$\displaystyle-
lD_{1}\tilde{\theta}^{a}+\epsilon{{}^{a}}_{b}\tilde{\theta}e^{b}_{1},$
$\displaystyle\delta{{}_{G}}\omega_{0}$ $\displaystyle=$ $\displaystyle
D_{0}\tau,$ $\displaystyle\delta{{}_{G}}\omega_{1}$ $\displaystyle=$
$\displaystyle-\partial_{1}\tilde{\theta}-\frac{1}{l}\epsilon_{ab}\tilde{\theta}^{a}e^{b}_{1}(1-4k^{2}B).$
(151)

It can be appreciated that the transformations obtained in equation
$(\ref{61})$ coincide with those derived through the Dirac method in equation
$(\ref{38})$.

##  6  Conclusions and prospects

In this paper, we conduct a Hamiltonian canonical analysis on the action
describing the Polynomial BF model system. Within the full phase space, we
identify and distinguish all the constraints of the theory, classifying them
into first and second classes. Additionally, we determine the algebra of the
first-class constraints, as shown in (41) and count the degrees of freedom.
The total degrees of freedom are calculated as $18$ (canonical variables) -
$2\times 6$ (first-class constraints) - $6$ (second-class constraints) = $0$,
indicating a system with zero degrees of freedom. Moreover, we have removed
the second-class constraints by introducing Dirac brackets, as we see from
(57). Castellani’s approach enables us to deduce the fundamental gauge
structure corresponding to deformed $SO(3)$ transformations, as shown in (59).
Specifying the gauge parameters allows us to derive diffeomorphisms and
Poincaré symmetries originating from the fundamental gauge symmetry. On the
contrary, treating the second-class constraints as strong equalities leads to
outcomes consistent with the conventional expressions in the literature.

In comparison, we derive the structures of all the physical constraints in the
Polynomial BF model using the modified FJ formalism. When we plug all obtained
constraints into the Lagrangian through Lagrangian multipliers, we observe
that the resulting two-form symplectic matrix remains singular, as shown in
(109). This singularity implies that the investigated theory exhibits
characteristics of a gauge system with additional symmetry. For effective
quantization, we require suitable gauge conditions to modify the two-form
symplectic matrix and render it non-singular. The modification enables us to
derive basic quantum brackets from the inverse of the non-singular matrix. The
transition amplitude can then be determined by assessing the determinant of
the nonzero matrix. Additionally, we calculate the count of degrees of freedom
in the theory, finding that it possesses zero degrees of freedom. Moreover, by
employing the MW algorithm in the FJ framework, we successfully obtain
generators for gauge transformations and shifts throughout the entire
configuration space, eliminating the need for Castellani’s algorithm.

Upon completing our calculations, we confirm the classical equivalence between
Dirac brackets and FJ brackets, as obtained through the symplectic approach.
In the FJ formulation, the process of identifying constraints is less
intricate and involves fewer limitations compared to the Dirac method,
rendering the FJ approach more elegant and efficient.

Recently, it was demonstrated [18] that the Polynomial BF model for gravity in
two dimensions has employed a mechanism similar to that found in the
MacDowell-Mansouri theory. Additionally, the Polynomial BF model, presented in
[18], can be related to dilaton theories in two dimensions, as described in
[39]. These theories play a fundamental role in the investigation of gravity
in two dimensions. By incorporating a suitable dilaton field, it becomes
possible to derive various gravitational models, including notable examples
like the one proposed by Jackiw–Teitelboim and the quadratic gravitational
formulation. The establishment of a connection between the Polynomial BF model
and dilaton theory provides valuable insights for formulating an interaction
term, facilitates the dynamic integration of torsion, and allows for an
extension to four dimensions.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgements

J.M.C. welcomes the support of the Universidad Juárez Autónoma de Tabasco for
providing a suitable work environment during the research period. J.M.C. also
thanks CONACyT for their support through a grant for postdoctoral studies
under grant No. 3873825.

## References

  * [1] Peter Peldan 1990 Gravity coupled to matter without the metric. Phys. Lett. B, 248:62-66.
  * [2] K. Krasnov and R. Percacci 2018 Gravity and Unification: A review, Class. Quant. Grav. 35 143001 [arXiv:1712.03061]
  * [3] D.B. Fairlie 1979 Higgs’ Fields and the Determination of the Weinberg Angle. Phys. Lett. B, 82:97$-$100\.
  * [4] Yuval Ne’eman, S. Sternberg, and D. Fairlie. Superconnections for electroweak su(2/1) and extensions, and the mass of the Higgs. Phys. Rept., 406:303$-$377, 2005.
  * [5] Weinberg S 2005 The Quantum Theory of Fields vol IIII (Cambridge: Cambridge University Press)
  * [6] R. Capovilla, T. Jacobson and J. Dell 1991 A Pure spin connection formulation of gravity, Class. Quant. Grav. 8, 59-73.
  * [7] K. Krasnov, Pure Connection Action Principle for General Relativity 2011 Phys. Rev. Lett. 106, 251103.
  * [8] J. E. Rosales-Quintero 2016 Anti-self-dual gravity and supergravity from a pure connection formulation, Int. J. Mod. Phys. A 31 no.12, 1650064.
  * [9] E. Mitsou 2019 Spin connection formulations of real Lorentzian General Relativity, Class. Quant. Grav. 36, 04500
  * [10] J. Chagoya, M. Sabido and A. Silva-García 2023 Vector-tensor gravity from a broken gauge symmetry, Class. Quant. Grav. 40 095007 [arXiv:2207.06599]
  * [11] Sergei Alexandrov, Marc Geiller, and Karim Noui 2012 Spin Foams and Canonical Quantization, SIGMA 8, 055, arXiv:1112.1961. A. Perez 2013 The Spin Foam Approach to Quantum Gravity, Living Rev.Rel. 16, 3, arXiv:1205.2019 [gr-qc]. Rovelli C and Vidotto F 2014 Covariant Loop Quantum Gravity (Cambridge Monographs on Mathematical Physics) (Cambridge: Cambridge University Press)
  * [12] Mariano Celada, Diego Gonzlez, and Merced Montesinos 2016 BF gravity. Class. Quant. Grav., 33(21):213001, arXiv:1610.02020.
  * [13] John David Brown. Lower Dimensional Gravity, World Scientific, Singapore, (1988).
  * [14] S. Carlip, Quantum gravity in 2+1 dimensions, Cambridge University Press UK, 276p, (1998).
  * [15] R. Jackiw, Liouville field theory: A two-dimensional model for gravity, in Quantum Theory Of Gravity, S. Christensen, ed., pp. 403-420. Adam Hilger, Bristol, (1984).
  * [16] C. Teitelboim, The Hamiltonian structure of two-dimensional space-time and its relation with the conformal anomaly, in Quantum Theory Of Gravity, S. Christensen, ed., pp. 327-344. Adam Hilger, Bristol, 1984.
  * [17] T. Fukuyama and K. Kamimura, Gauge Theory of Two-Dimensional Gravities, Phys. Lett. B169 (1985) 259. K. Isler and C. A. Trugenberger,Gauge Theory of Two-Dimensional Quantum Gravity, Phys. Rev. Lett. 63 (1989) 834, A. H. Chamseddine and D. Wyler, Topological gravity in 1 + 1 dimensions, Nucl. Phys. B340 (1990) 595.
  * [18] Paszko, R., da Rocha, R 2015 Quadratic gravity from BF theory in two and three dimensions. Gen. Relativ. Gravit. 47, 94.
  * [19] MacDowell S W and Mansouri F 1977 Unified geometric theory of gravity and supergravity Phys. Rev. Lett. 38 739$-$42
  * [20] Smolin L and Starodubtsev A 2003 General relativity with a topological phase: an action principle arXiv:hep-th/0311163
  * [21] Freidel L and Starodubtsev A 2005 Quantum gravity in terms of topological observables arXiv:hep-th/0501191
  * [22] De Felice, A., Tsujikawa, S 2010 f(R) Theories. Living Rev. Relativ. 13, 3. https://doi.org/10.12942/lrr-2010-3
  * [23] S. Nojiri and S. D. Odintsov 2011 Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models, Phys. Rept. 505, 59 [arXiv:1011.0544 [gr-qc]].
  * [24] Min-Xing Xu, T. Harko and Shi-Dong Liang 2016 Quantum Cosmology of f(R, T) gravity, Eur. Phys. J. C 76 449. https://doi.org/10.1140/epjc/s10052-016-4303-6
  * [25] C. Lanczos, Ann. Math. 39, 842 (1938).
  * [26] A. Pais and G. E. Uhlenbeck, Phys. Rev. 79, 145 (1950).
  * [27] B. S. DeWitt, Phys. Rep. 19C, 295 (1975).
  * [28] G. ’t Hooft and M. Veltman, Ann. Inst. Henri Poincaré 20, 69 (1974).
  * [29] K. S. Stelle, Phys. Rev. D 16, 953 (1977).
  * [30] A. L. Maroto and I. L. Shapiro, Phys.Lett. B 414, 34 (1997).
  * [31] A. A. Starobinsky, Phys. Lett. B 91, 99 (1980).
  * [32] N. Bodendorfer, J. Lewandowski, and J. Swiezewski, General relativity in the radial gauge: Reduced phase space and canonical structure, Physical Review D 92 (2015) 084041, arXiv:1506.09164 [gr-qc].
  * [33] Dirac, P. Generalized Hamiltonian dynamics. Can. J. Math. 1950, 2, 129.
  * [34] Dirac, P. The Hamiltonian form of field dynamics. Can. J. Math. 1951, 3, 1.
  * [35] Dirac, P. The theory of gravitation in Hamiltonian form. Proc. R. Soc. (Lond.) 1958, 246, 333.
  * [36] Bergmann, P. Non Linear Field Theories. Phys. Rev. 1949, 75, 680.
  * [37] Bergmann, P. Brunings, J. Non Linear Field Theories II. Canonical Equations and Quantization. Rev. Mod. Phys. 1949, 21, 480.
  * [38] Bergmann, P. Penfield, R. Schiller, R.; Zatzkis, H. The Hamiltonian of the General Theory of Relativity with Electromagnetic Field. Phys. Rev. 1950, 80, 81.
  * [39] C. E. Valcárcel 2017 Constraint Analysis of Two-Dimensional Quadratic Gravity from BF Theory, Gen. Rel. Grav. 49:11.
  * [40] Faddeev L Dand Jackiw R 1988 Hamiltonian reduction of unconstrained and constrained systems Phys. Rev. Lett. 60 1692
  * [41] Barcelos-Neto J and Wotzasek C 1992 Symplectic quantization of constrained systems Mod. Phys. Lett A 7 1737–42
  * [42] Barcelos-Neto J and Wotzasek C 1992 Faddeev-Jackiw quantization and constraints Int. J. Mod. Phys. A 7 4981–5003
  * [43] Long Z Wand Liu B 2002 A new method to deal with constraints in the context of the Faddeev-Jackiw method to treat constrained systems Europhys. Lett. 58 1
  * [44] Liao L and Huang Y C 2007 Non-equivalence of Faddeev-Jackiw method and Dirac-Bergmann algorithm and the modiﬁcation of Faddeev-Jackiw method for keeping the equivalence Ann. Phys. 322 2469
  * [45] García J A and Pons J M 1997 Equivalence of Faddeev-Jackiw and dirac approaches for gauge theories Int. J. Mod. Phys. A 12 451
  * [46] Escalante A and Manuel J 2015 Faddeev-Jackiw quantization of an Abelian and non-Abelian exotic action for gravity in three dimensions Annals Phys. 361 585–604
  * [47] Escalante A and Cavildo-Sánchez P 2016 Faddeev–Jackiw quantization of four dimensional BF theory Annals Phys. 374 375-394
  * [48] Escalante A and Manuel J 2017 Hamiltonian dynamics and Faddeev-Jackiw quantization of 3D gravity with a Barbero-Immirzi like parameter Eur. Phys. J. C77 303
  * [49] Anjali S, S. Gupta 2020 Faddeev–Jackiw quantization of Christ–Lee model Mod. Phys. Lett. A 35, 2050072
  * [50] Anjali S, Gupta, S. 2022 Particle on a torus knot: symplectic analysis. Eur. Phys. J. Plus 137, 511
  * [51] Manuel J and Mauricio Paulin J M. 2023 Canonical analysis and modified Faddeev-Jackiw approach for the Jackiw-Teitelboim model in two dimensions. Physica Scripta 98.10 : 105207.
  * [52] Amorim R 1995 On the BFT-BFV quantization of gauge invariant systems with linear second class constraints Z. Phys C67 695
  * [53] Henneaux Mand Teitelboim C 1991 Quantization of Gauge Systems University of Chicago Press)
  * [54] Blagojevic M 2002 Gravitation and Gauge Symmetries (IoP Publishing)(https://doi.org/10.1887/0750307676)
  * [55] Rodríguez-Tzompantzi O 2021 Symplectic realization of two interacting spin-two fields in three dimensions. Journal of High Energy Physics, 1-23.
  * [56] L. Liao, Y.C. Huang 2007 Path integral quantization corresponding to Faddeev-Jackiw canonical quantization, Phys. Rev. D 75 025025.
  * [57] D.J. Toms 2015 Faddeev-Jackiw quantization and the path integral, Phys.Rev.D 92 105026, 10.
  * [58] Montani H and Wotzasek C 1993 Faddeev-Jackiw quantization of non-Abelian system Mod. Phys. Lett. A 8 3387