Interacting 2D Field-Theoretic Model for Hodge Theory

A. Tripathi(a), A. K. Rao(a), R. P. Malik(a,b)
(a) Physics Department, Institute of Science,
Banaras Hindu University, Varanasi - 221 005, (U.P.), India

(b) DST Centre for Interdisciplinary Mathematical Sciences,
Institute of Science, Banaras Hindu University, Varanasi - 221 005, India
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract: We take up the St${\ddot{u}}$ckelberg-modified version of the two
(1+1)-dimensional (2D) Proca theory, in interaction with the Dirac fields, to
study its various continuous and discrete symmetry transformations and show
that this specific interacting 2D field-theoretic model provides a tractable
example for the Hodge theory because its symmetries (and corresponding
conserved charges) provide the physical realizations of the de Rham
cohomological operators of differential geometry at the algebraic level. The
physical state of this theory is chosen to be the harmonic state (of the Hodge
decomposed state) in the quantum Hilbert space which is annihilated by the
conserved and nilpotent (anti-)BRST as well as (anti-)co-BRST charges. A
physical consequence of this study is an observation that the 2D anomaly, at
the quantum level, does not lead to any problem as far as the consistency and
unitarity of our present 2D theory is concerned. In other words, our present
2D field-theoretic model is amenable to particle interpretation despite the
presence of the local chiral symmetry (which is associated with the nilpotent
(anti-)co-BRST symmetry transformations) besides the presence of the nilpotent
(anti-)BRST symmetries (which are connected with the local gauge symmetry).
The physicality condition with the (anti-)co-BRST charges implies that the 2D
anomaly term is trivial in our present theory. Hence, our 2D theory is
consistent, unitary and amenable to particle interpretation.

PACS numbers: 11.15.-q; 03.70.+k; 11.30.-j; 11.30.Pb; 11.30.Qc.

Keywords: St${\ddot{u}}$ckelberg modified 2D Proca theory; interacting Dirac
fields; (anti-)BRST symmetries; (anti-)co-BRST symmetries; de Rahm
cohomological operators; Hodge decomposition theorem; harmonic state as the
physical state; physicality criteria; 2D anomaly and (anti-)co-BRST
symmetries; consistency and unitarity of our 2D theory

##  1 Introduction

Mathematics is the language of all the key branches of modern-day scientific
pursuits. Its creative usefulness and glorious presence are undeniable truths
in the domain of theoretical physics. The applications and realizations of the
mathematical ideas and concepts in the realm of theoretical physics have
played very important and decisive roles in the development of the latter. In
this context, it is pertinent to point out that it has been possible to show,
within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism [1-4], that
the Abelian $p$-form $(p=1,2,3...)$ gauge theories in $D=2p$ dimensions of
spacetime provide the field-theoretic examples of Hodge theory (see, e.g.
[5-7] and references therein) where the discrete and continuous symmetries
(and the conserved charges) have been able to provide the physical
realizations of the de Rham cohomological operators of differential geometry
[8-12] at the algebraic level. Furthermore, it has been found that some of the
${\cal N}=2$ supersymmetric (SUSY) quantum mechanical models are also examples
of Hodge theory (see, e.g. [13-17]) at the algebraic level (without the
central extension(s)). Whereas the above Abelian $p$-form $(p=1,2,3...)$ gauge
theories are massless theories due to the presence of gauge symmetries, the
${\cal N}=2$ SUSY quantum mechanical models are massive and they respect two
nilpotent SUSY transformations which are not absolutely anticommuting in
nature. However, the (anti-)BRST symmetries (corresponding to the gauge
symmetry transformations of the Abelian $p$-form gauge theory) are nilpotent
of order two but absolutely anticommuting in nature. These properties are very
sacrosanct.

Against the backdrop of the above, we have shown that the
St$\ddot{u}$ckelberg-modified 2D Proca as well as the 4D Abelian 2-form gauge
theories (without any interaction with matter fields) provide the massive
models of Hodge theories where the mass and the quantum (anti-)BRST as well as
(anti-)co-BRST symmetries co-exist together [18-20]. The central purpose of
our present investigation is to establish that the 2D modified Proca theory,
in interaction with the Dirac fields, provides a field-theoretic example of
Hodge theory where the continuous symmetry transformations (and corresponding
conserved Noether charges) provide the physical realizations of the
cohomological operators of differential geometry and the existence of a couple
of discrete symmetry transformations is at the heart of the physical
realizations of the Hodge duality $*$ operation of differential geometry.
Thus, our present 2D theory is a novel example of a massive and interacting
model of Hodge theory within the framework of BRST formalism. The existence of
the nilpotent (anti-)BRST and (anti-)co-BRST symmetries provides the physical
realizations of the nilpotent (co-) exterior derivatives of differential
geometry. The (anti-)BRST symmetry transformations correspond to the local
gauge symmetry in the theory. On the other hand, it is the local chiral
symmetry transformation (in the massless limit of the Dirac fields) that leads
to the existence of a proper set of (anti-)co-BRST transformations in the
theory.

The study of the field-theoretic models of Hodge theory, within the ambit of
BRST formalism, is physically important on the following grounds. First, it
has been possible to show that the (non-)Abelian 1-form gauge theories, in two
$(1+1)$\- dimensions of spacetime, provide a new [21] type of topological
field theory (TFT) that captures a few key aspects of the Witten-type TFT [22]
and some salient features of the Schwarz-type TFT [23]. Second, we have been
able to establish that the 4D Abelian 2-form and 6D Abelian 3-form gauge
theories are tractable field-theoretic examples of Hodge theory and they
provide the models of quasi-TFTs where the topological invariants and their
perfect recursion relations exist [5, 24, 25]. Third, it has been established
that the massive models of Hodge theory (without any interaction with matter
fields) allow the existence of fields with negative kinetic terms [18-20]
which have been called as the “phantom” fields in the domain of cosmology.
Such kinds of exotic fields have played important roles in the context of
cyclic, bouncing and self-accelerated models of Universe (see, e.g. [26-30]).
In our earlier works [18-20], we have laid the emphasis on the relevance of
such fields (with negative kinetic terms) in the context of providing a set of
possible candidates for the dark matter and dark energy within the framework
of quantum field theory. Finally, in our present endeavor, we establish that
the massive and interacting 2D modified version of the Proca theory with the
Dirac fields shows that the vector and axial-vector currents (and their
corresponding charges) are conserved together because the 2D anomaly term is
trivial when we choose the physical state as the harmonic state (of the Hodge
decomposed state in the total quantum Hilbert space) which is annihilated by
the conserved and nilpotent BRST and co-BRST charges together.

The following key factors have been at the heart of our present investigation.
First, in the physical four $(3+1)$-dimensions of QED with the Dirac fields,
it has been well-established that the vector and axial-vector currents can not
be conserved together due to the presence of the anomaly term
$(\sim\vec{E}\cdot\vec{B})$ which is a pseudo-scalar constructed with the
polar-vector electric field $(\vec{E})$ and the axial-vector magnetic field
$(\vec{B})$ (see, e.g. [31] for details). However, in the case of 2D, there is
presence of only the electric field $(E)$ which becomes a pseudo-scalar and
the anomaly term. We have been motivated to check the status of this anomaly
term within the framework of BRST formalism. Our present endeavor is an
attempt in that direction. Second, we have been working on the idea of dual-
BRST symmetry for quite sometime and we have shown [32, 33] that the 2D QED
with the Dirac fields (interacting with a massless photon) leads to the
definition of a conserved co-BRST charge which contains the pseudo-scalar
electric field $(E)$. We have been curious to establish a connection between
the co-BRST charge and the anomaly term $(E)$ in 2D. We have accomplished this
goal in our present endeavor. Finally, this is, for the first time, we have
proven a massive as well as an interacting gauge theory to be an example of
the Hodge theory. We have already established the existence of the field-
theoretic models of Hodge theory with mass and massless gauge fields in our
earlier works [18-20, 5-7]. Our present 2D theory is also a perfect model of a
duality invariant theory [34] because of the existence of the discrete
symmetry transformations [see, Eq. (10) below].

Our present interacting theory respects six continuous and a couple of
discrete symmetry transformations cf. Eq. (10) below]. Out of these, at the
most economical level, three symmetries are very fundamental. These are the
BRST, co-BRST and discrete symmetries. The conserved and nilpotent BRST charge
annihilates the physical (i.e. harmonic) state which implies that the operator
forms of the first-class constraints must annihilate the physical state. The
decisive presence of the discrete symmetry transformations [cf. Eq. (10)
below] ensures that our theory is a perfect model (cf. Secs. 5, 6 below) of
the duality invariant theory [34]. Hence, the dual versions of the above
first-class constraints must also annihilate the physical state. Thus, the
annihilation of the physical state by the first-class constraints and their
dual is very sacrosanct and it must be respected at the tree as well as at the
loop-level diagrams. These dual versions of constraints [cf. Eq. (47) below]
emerge when we demand that the physical state must be annihilated by the co-
BRST charge which is nilpotent but conserved only at the tree-level diagrams
because it is derived from the co-BRST current [cf. Eq. (13) below] which is
also conserved at the tree-level diagrams. However, it is the sanctity of the
requirements of a perfect duality-invariant theory that (i) the 2D anomaly
term, and (ii) its invariance under the time-evolution, must also annihilate
the physical state of our theory [cf. Eq. (48) below]. This implies,
primarily, that the vector and axial-vector currents are found to be conserved
together for our 2D interacting theory with the Dirac fields. Hence, the 2D
anomaly term becomes trivial.

The theoretical contents of our present endeavor are organized as follows.
First, to set the notations and convention, we recapitulate the bare
essentials of the (anti-)BRST symmetries of the D-dimensional Lagrangian
density for the modified version of Proca (i.e. massive Abelian 1-form
$A^{(1)}=d\,x^{\mu}\,A_{\mu}$) gauge theory where there is a coupling between
the gauge field $A_{\mu}$ (with $\mu=0,1,2...D-1)$ and the conserved current
constructed with the fermionic Dirac fields in Sec. 2. Our Sec. 3 is devoted
to the discussion on the nilpotent (anti-)co-BRST symmetry transformations for
the same [i.e. (anti-)BRST invariant] Lagrangian density in two
(1+1)-dimensions of spacetime. We derive the bosonic and ghost-scale symmetry
transformations in our Sec. 4. Our Sec. 5 deals with the derivation of the
algebras obeyed by the above (anti-)BRST, (anti-)co-BRST, a unique bosonic and
the ghost-scale continuous symmetries and corresponding conserved charges. In
our Sec. 6, we elaborate on the connection between the algebraic structures of
the above symmetries (and corresponding conserved charges) and the de Rham
cohomological operators of differential geometry. In Sec. 7, we dwell a bit on
the irrelevance of the 2D anomaly term as far as our discussion on the
importance of the (anti-)dual-BRST [i.e. (anti-)co-BRST] transformations (nd
corresponding charges) is concerned. Finally, in Sec. 8, we summarize our
results and lay emphasis on the physical importance of our present work.

In our Appendices A, B and C, we perform some explicit computations which
corroborate some of the claims that have been made in the main body of our
text.

Convention and Notations: We adopt the background 2D Minkowskian flat
spacetime manifold that is endowed with the metric tensor $\eta_{\mu\nu}=$
diag $(+1,-1)$ so that the dot product $(P\cdot Q)$ between the two non-null
vectors $P_{\mu}$ and $Q_{\mu}$ is defined as: $P\cdot
Q=\eta_{\mu\nu}\,P^{\mu}\,Q^{\nu}=P_{0}\,Q_{0}-P_{i}\,Q_{i}$ where the Greek
indices $\mu,\,\nu,\,\lambda...=0,1$ correspond to the spacetime directions
and $i,\,j,\,k,...=1$ stand for the space direction. The antisymmetric
$(\varepsilon_{\mu\nu}=-\varepsilon_{\nu\mu})$ Levi-Civita tensor
$\varepsilon_{\mu\nu}$ is chosen such that
$\varepsilon_{01}=+\varepsilon^{10}=+1$ and
$\varepsilon_{\mu\nu}\,\varepsilon^{\mu\nu}=-\,2!,\,$
$\varepsilon_{\mu\lambda}\,\varepsilon^{\mu\nu}=-\,1!\,\delta^{\nu}_{\lambda}$,
etc. We denote the (anti-)BRST symmetry transformations by $s_{(a)b}$ and
(anti-)co-BRST symmetry transformations by $s_{(a)d}$ in the whole body of our
text. We take the convention of the left-derivative w.r.t. the fermionic
fields ($\bar{\psi},\,\psi,\,\bar{C},\,C$) which anticommute among themselves
but commute with all the bosonic fields of our theory. We choose the $2\times
2$ Dirac $\gamma$-matrices in terms of the traceless and hermitian $2\times 2$
Pauli-matrices as: $\gamma_{0}=\sigma_{1},\,\gamma_{1}=i\,\sigma_{2}$ which
satisfy $\\{\gamma_{\mu},\,\gamma_{\nu}\\}=2\,\eta_{\mu\nu}$ and lead to the
definition of $\gamma_{5}=\gamma_{0}\,\gamma_{1}\equiv-\,\sigma_{3}$ which is
the diagonal matrix [like the $\gamma_{5}$ matrix in the Weyl representation
(see, e.g. [35] for details) of Dirac’s matrices] and hermitian (i.e.
$\gamma_{5}^{\dagger}=\gamma_{5}$). We also have:
$(\gamma_{5})^{2}=I,\,\gamma_{5}\,\gamma_{\mu}=\varepsilon_{\mu\nu}\,\gamma^{\nu},\,\gamma_{5}=-(1/2!)\,\varepsilon_{\mu\nu}\,\gamma^{\mu}\,\gamma^{\nu}$
and $\\{\gamma_{5},\gamma_{\mu}\\}=0$ where $\gamma_{0}=\gamma^{0}$ and
$\gamma^{1}=-\,\gamma_{1}$ due to the choice of the signatures of our 2D flat
metric tensor.

Standard Definitions: We recapitulate the following definitions of the
differential geometry [8-12] which will be useful in our discussions (as far
as the connection between the physical aspects and mathematical ideas of our
present endeavor are concerned).

(i) The de Rham Cohomological Operators: The set of three operators
$(d,\,\delta,\,\Delta)$ constitute the de Rham cohomological operators
(defined the D-dimensional compact manifold without a boundary) where
$d=d\,x^{\mu}\,\partial_{\mu}$ (with $\mu=0,\,1,\,2...D-1)$ is the exterior
derivative, $\delta=\mp*\,d\,*$ is the co-exterior (or dual-exterior)
derivative and $\Delta=(d+\delta)^{2}$ is the Laplacian operator. Here $*$ is
the Hodge duality operation on the above-mentioned compact manifold without a
boundary. For the even dimensions of spacetime $\delta=-\,*\,d\,*$. These
operators obey the algebra:
$d^{2}=\delta^{2}=0,\,\Delta=(d+\delta)^{2}=d\,\delta+\delta\,d=\\{d,\,\delta\\}$,
$[\Delta,\,d]=0,\,[\Delta,\,\delta]=0$ which is popularly known as the Hodge
algebra (that is not a Lie algebra).

(ii) The Hodge Decomposition Theorem: On a compact manifold without a
boundary, any arbitrary form $f_{n}$ (of non-zero degree $n$) can be uniquely
written as the sum of the harmonic form $(h_{n})$, an exact form
$(d\,g_{n-1})$ and a co-exact form $(\delta\,k_{n+1})$ as:
$f_{n}=h_{n}+d\,g_{n-1}+\delta\,k_{n+1}$ where the harmonic form $h_{n}$
satisfies: $d\,h_{n}=\delta\,h_{n}=0$. In other words, we have
$\Delta\,h_{n}=0$. Thus, the harmonic form $(h_{n})$ is closed and co-closed
together.

##  2 Preliminaries: (Anti-)BRST Symmetry Invariant Lagrangian Density in Any
Arbitrary Dimension

We begin with the following D-dimensional (anti-)BRST invariant interacting
Lagrangian density in the ’t Hooft gauge (see, e.g. [36] for details)

$\displaystyle{\cal L}_{B}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}+\frac{m^{2}}{2}\,{A_{\mu}}{A^{\mu}}+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi-m\,A_{\mu}\,\partial^{\mu}\,\phi+\bar{\psi}\,(i\,\gamma^{\mu}\,\partial_{\mu}-m^{\prime})\,\psi$
(1) $\displaystyle-$ $\displaystyle
e\,\bar{\psi}\,\gamma^{\mu}\,A_{\mu}\,\psi+B\,(\partial\cdot
A+m\,\phi)+\frac{1}{2}\,B^{2}-i\,\partial_{\mu}\,\bar{C}\,\partial^{\mu}\,C+i\,m^{2}\,\bar{C}\,C.$

where the kinetic term for the massive photon has its origin in the exterior
derivative $(d=d\,x^{\mu}\,\partial_{\mu}$ with $d^{2}=0)$ of the differential
geometry [8-12] because we note that
$d\,A^{(1)}=\frac{1}{2!}\,(d\,x^{\mu}\wedge d\,x^{\nu})\,F_{\mu\nu}$. Here
$A^{(1)}=d\,x^{\mu}\,A_{\mu}$ (with $\mu=0,\,1,\,2...D-1)$ defines the massive
vector field $(A_{\mu})$ with rest mass $m$ and
$F_{\mu\nu}=\partial_{\mu}\,A_{\nu}-\partial_{\nu}\,A_{\mu}$ is the field
strength tensor. The pure scalar field $\phi$ has been invoked because of
St${\ddot{u}}$ckelberg’s formalism and it has positive kinetic term. The
fermionic $(\psi^{2}=\bar{\psi}^{2}=0,\,\psi\,\bar{\psi}+\bar{\psi}\,\psi=0)$
Dirac fields (with the rest mass $m^{\prime}$) have interaction with the
massive gauge field $A_{\mu}$ through minimal interaction
$(-\,e\,\bar{\psi}\,\gamma^{\mu}\,\psi\,A_{\mu})$ coupling. The gauge-fixing
term for the gauge field owes its origin to the exterior derivative
$(\delta=-*\,d\,*)$ because $\delta\,A^{(1)}=(\partial\cdot A)$ and we have
added the $m\,\phi$ term to it on the dimensional ground. In our theory, $B$
is the Nakanishi-Lautrup auxiliary field and the fermionic
$(C^{2}=\bar{C}^{2}=0,\,C\,\bar{C}+\bar{C}\,C=0)$ (anti-)ghost fields
$(\bar{C})\,C$ have been invoked to preserve the unitarity in the theory at
any order of perturbative computations. The above Lagrangian density $({\cal
L}_{B})$ respects the following off-shell nilpotent $[s_{(a)b}^{2}=0]$ and
absolutely anticommuting $(s_{b}\,s_{ab}+s_{ab}\,s_{b}=0)$ (anti-)BRST
transformations $[s_{(a)b}]$, namely;

$\displaystyle s_{ab}\,A_{\mu}=\partial_{\mu}\,\bar{C},\qquad
s_{ab}\,\phi=m\,\bar{C},\qquad s_{ab}\,\psi=-i\,e\,\bar{C}\,\psi,\qquad
s_{ab}\,\bar{\psi}=-i\,e\,\bar{\psi}\,\bar{C},$ $\displaystyle
s_{ab}\,\bar{C}=0,\qquad s_{ab}\,C=-i\,B,\qquad s_{ab}\,B=0,$ $\displaystyle
s_{b}\,A_{\mu}=\partial_{\mu}\,C,\qquad s_{b}\,\phi=m\,C,\qquad
s_{b}\,\psi=-i\,e\,C\,\psi,\qquad s_{b}\,\bar{\psi}=-i\,e\,\bar{\psi}\,C,$
$\displaystyle s_{b}\,C=0,\qquad s_{b}\,\bar{C}=i\,B,\qquad s_{b}\,B=0,$ (2)

because the Lagrangian density $({\cal L}_{B})$ transforms to

$\displaystyle s_{ab}\,{\cal
L}_{B}=\partial_{\mu}\,[B\,\partial^{\mu}\,\bar{C}],\qquad s_{b}\,{\cal
L}_{B}=\partial_{\mu}\,[B\,\partial^{\mu}\,C],$ (3)

thereby rendering the action integral $S=\int d^{D}\,x\,{\cal L}_{B}$
invariant [i.e. $s_{(a)b}\,S=0$] for the physical fields that vanish-off as
$x\rightarrow\pm\,\infty$ due to Gauss’s divergence theorem. As a consequence
of the celebrated Noether’s theorem, we have the conserved currents
$[J^{\mu}_{(a)b}]$ corresponding to the off-shell nilpotent, absolutely
anticommuting, infinitesimal and continuous (anti-)BRST symmetry
transformations $s_{(a)b}$ as:

$\displaystyle J^{\mu}_{(ab)}$ $\displaystyle=$ $\displaystyle
B\,\partial^{\mu}\,\bar{C}+m\,\bar{C}\,\partial^{\mu}\,\phi-F^{\mu\nu}\,\partial_{\nu}\,\bar{C}-m^{2}\,A^{\mu}\,\bar{C}-e\,\bar{\psi}\,\gamma^{\mu}\,\bar{C}\,\psi,$
$\displaystyle J^{\mu}_{(b)}$ $\displaystyle=$ $\displaystyle
B\,\partial^{\mu}\,C+m\,C\,\partial^{\mu}\,\phi-F^{\mu\nu}\,\partial_{\nu}\,C-m^{2}\,A^{\mu}\,C-e\,\bar{\psi}\,\gamma^{\mu}\,C\,\psi.$
(4)

The conservation law $[\partial_{\mu}\,J^{\mu}_{(a)b}=0]$ can be proven
readily by using the following Euler-Lagrange (EL) equations of motion (EOMs)
from the Lagrangian density $({\cal L}_{B})$, namely;

$\displaystyle\partial_{\mu}\,F^{\mu\nu}+m^{2}\,A^{\nu}-\partial^{\nu}\,B-m\,\partial^{\nu}\,\phi-e\,\bar{\psi}\,\gamma^{\nu}\,\psi=0,\qquad\Box\,\phi-m\,(\partial\cdot
A)-m\,B=0,$ $\displaystyle
i\,(\partial_{\mu}\,\bar{\psi})\,\gamma^{\mu}+m^{\prime}\,\bar{\psi}+e\,\bar{\psi}\,\gamma^{\mu}\,A_{\mu}=0,\qquad
i\,\gamma^{\mu}\,\partial_{\mu}\,\psi-m^{\prime}\,\psi-e\,\gamma^{\mu}\,A_{\mu}\,\psi=0,$
$\displaystyle(\Box+m^{2})\,\bar{C}=0,\qquad(\Box+m^{2})\,C=0,\qquad
B+\partial\cdot A+m\,\phi=0.$ (5)

The conserved (anti-)BRST charges $(Q_{ab}=\int
d^{D-1}\,x\,J^{0}_{(ab)},\,Q_{b}=\int d^{D-1}\,x\,J^{0}_{(b)})$ are

$\displaystyle Q_{ab}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}\,x\,\Big{[}B\,\dot{\bar{C}}+m\,\dot{\phi}\,\bar{C}-F^{0i}\,\partial_{i}\,\bar{C}-m^{2}\,A^{0}\,\bar{C}-e\,\bar{\psi}\,\gamma^{0}\,\bar{C}\,\psi\Big{]}$
$\displaystyle\equiv$ $\displaystyle\int
d^{D-1}\,x\,\big{(}B\,\dot{\bar{C}}-\dot{B}\,\bar{C}\big{)},$ $\displaystyle
Q_{b}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}\,x\,\Big{[}B\,\dot{C}+m\,\dot{\phi}\,C-F^{0i}\,\partial_{i}\,C-m^{2}\,A^{0}\,C-e\,\bar{\psi}\,\gamma^{0}\,C\,\psi\Big{]}$
(6) $\displaystyle\equiv$ $\displaystyle\int
d^{D-1}\,x\,\big{(}B\,\dot{C}-\dot{B}\,C\big{)},$

where the final expressions for the concise form of the conserved charges have
been obtained due to the following EL-EOM that has been chosen from (5),
namely;

$\displaystyle\partial_{i}\,F^{0i}=m^{2}\,A^{0}-\dot{B}-m\,\dot{\phi}-e\,\bar{\psi}\,\gamma^{0}\,\psi.$
(7)

In other words, first of all, we have expressed
$(-F^{0i}\,\partial_{i}\,\bar{C})$ and $(-F^{0i}\,\partial_{i}\,C)$ as
$[(\partial_{i}\,F^{0i})\,\bar{C}-\partial_{i}\,(F^{0i}\,\bar{C})]$ and
$[(\partial_{i}\,F^{0i})\,C-\partial_{i}\,(F^{0i}\,C)]$, respectively, and
applied the Gauss divergence theorem to drop the total space derivative terms.
It is straightforward to note that the conserved (anti-)BRST charges are off-
shell nilpotent $[Q_{(a)b}^{2}=0]$ of order two and absolutely anticommuting
$(Q_{b}\,Q_{ab}+Q_{ab}\,Q_{b}=0)$ in nature due to the following observations
in terms of the conserved charges (6) and the nilpotent (anti-)BRST
transformations (2), namely;

$\displaystyle s_{b}\,Q_{b}=-i\,\\{Q_{b},Q_{b}\\}=0\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad Q_{b}^{2}=0,$ $\displaystyle
s_{ab}\,Q_{ab}=-i\,\\{Q_{ab},Q_{ab}\\}=0\qquad$ $\displaystyle\Rightarrow$
$\displaystyle\qquad Q_{ab}^{2}=0,$ $\displaystyle
s_{b}\,Q_{ab}=-i\,\\{Q_{ab},Q_{b}\\}=0\qquad$ $\displaystyle\Rightarrow$
$\displaystyle\qquad\\{Q_{ab},Q_{b}\\}=0,$ $\displaystyle
s_{ab}\,Q_{b}=-i\,\\{Q_{b},Q_{ab}\\}=0\qquad$ $\displaystyle\Rightarrow$
$\displaystyle\qquad\\{Q_{b},Q_{ab}\\}=0,$ (8)

where the l.h.s. of the above relationships have been computed by directly
applying the (anti-)BRST symmetry transformations [cf. Eq. (2)] on the
expressions for the nilpotent (anti-)BRST charges [cf. Eq. (6)] where the
basic principle behind the continuous symmetry transformations and their
generators (as the conserved Noether charges) has been exploited to establish
the off-shell nilpotency $([Q_{(a)b}]^{2}=0)$ and absolute anticommutativity
$(\\{Q_{b},Q_{ab}\\}=0)$ of the conserved ${\dot{Q}}_{(a)b}=0$ (anti-)BRST
charges $[Q_{(a)b}]$.

We wrap-up this section with the following crucial and decisive remarks.
First, we note that the kinetic term of the gauge field
$(-\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu})$ remains invariant (i.e.
$s_{(a)b}\,F_{\mu\nu}=0$) under the (anti-)BRST symmetry transformations
$[s_{(a)b}]$. Second, as pointed out earlier, the kinetic term has its origin
in the exterior derivative of differential geometry. Third, we observe that
the ghost number is increased by one by the BRST symmetry transformation when
it applies on a generic field (e.g.
$s_{b}\,A_{\mu}=\partial_{\mu}\,C,\,s_{b}\,\bar{C}=i\,B$, etc.) in contrast to
the anti-BRST symmetry transformations that decreases (e.g.
$s_{ab}\,A_{\mu}=\partial_{\mu}\,\bar{C},\,s_{ab}\,C=-i\,B$, etc.) the ghost
number by one. Fourth, the absolute anticommutativity (i.e.
$Q_{b}\,Q_{ab}+Q_{ab}\,Q_{b}=0,\,s_{b}\,s_{ab}+s_{ab}\,s_{b}=0$) of the
(anti-)BRST charges and the continuous fermionic $[s_{(a)b}^{2}=0]$ symmetry
transformations ensure that the (anti-)BRST symmetries (and their
corresponding conserved charges) can not be together identified with the
exterior derivative of differential geometry. Fifth, the absolute
anticommutativity property ensures that the nilpotent (anti-)BRST symmetry
transformations are different from the ${\cal N}=2$ SUSY symmetry
transformations which are nilpotent but not absolutely anticommuting in
nature. Sixth, it is evident that, for the identification of the co-exterior
derivative, we have to invoke and discuss another kind of nilpotent symmetries
for our theory.

##  3 (Anti-)Dual-BRST Symmetries: Modified Interacting 2D Proca Theory with
Dirac Fields

As we have modified the mass term $(\frac{m^{2}}{2}\,A_{\mu}\,A^{\mu})$ by
exploiting the theoretical tricks of the St$\ddot{u}$ckelberg formalism [36],
in exactly similar fashion, we can modify the 2D kinetic term
$(-\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}=\frac{1}{2}\,E^{2})$ of the gauge field
by invoking a pseudo-scalar field $\tilde{\phi}$ along with a linearizing
Nakanishi-Lautrup type auxiliary field ${\cal B}$ as (see, e.g. [19] for
details):

$\displaystyle{\cal L}_{\cal B}$ $\displaystyle=$ $\displaystyle{\cal
B}\,(E-m\,\tilde{\phi})-\frac{1}{2}\,{\cal
B}^{2}+m\,E\,\tilde{\phi}-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\partial^{\mu}\,\tilde{\phi}+\frac{m^{2}}{2}\,{A_{\mu}}{A^{\mu}}+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi$
(9) $\displaystyle-$ $\displaystyle
m\,A_{\mu}\,\partial^{\mu}\,\phi+\bar{\psi}\,(i\,\gamma^{\mu}\,\partial_{\mu}-m^{\prime})\,\psi-e\,\bar{\psi}\,\gamma^{\mu}\,A_{\mu}\,\psi+B\,(\partial\cdot
A+m\,\phi)+\frac{1}{2}\,B^{2}$ $\displaystyle-$ $\displaystyle
i\,\partial_{\mu}\,\bar{C}\,\partial^{\mu}\,C+i\,m^{2}\,\bar{C}\,C.$

It should be noted that the kinetic term
$-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\partial^{\mu}\,\tilde{\phi}$ for
the pseudo-scalar field $\tilde{\phi}$ carries a negative sign with it. This
is due to the fact that we have the following discrete symmetry in our theory
(modulo total derivatives) provided we have the positive and negative signs
for the kinetic terms for the pure scalar $(\phi)$ and pseudo-scalar
$(\tilde{\phi})$ fields, respectively, namely;

$\displaystyle
A_{\mu}\to\mp\,i\,\varepsilon_{\mu\nu}A^{\nu},\quad\phi\to\mp\,i\,\tilde{\phi},\quad\tilde{\phi}\to\mp\,i\,\phi,\quad
C\to\pm\,i\,\bar{C},\quad\bar{C}\to\pm\,i\,C,$ $\displaystyle{\cal
B}\to\mp\;i\,B,\quad B\to\mp\;i\,{\cal
B},\quad\psi\to\psi,\quad\bar{\psi}\to\bar{\psi},\quad
e\to\pm\;i\,e\,\gamma_{5},$ $\displaystyle(\partial\cdot A)\to\pm\;i\,E,\quad
E\to\pm\;i\,(\partial\cdot
A),\quad-m\,A_{\mu}\,\partial^{\mu}\,\phi\to+m\,E\,\tilde{\phi}.$ (10)

We would like to mention that the mass term
$(\frac{m^{2}}{2}\,A_{\mu}\,A^{\mu})$ for the gauge field $(A_{\mu})$ remains
invariant under $(A_{\mu}\to\mp\,i\,\varepsilon_{\mu\nu}A^{\nu})$. Hence, the
masses associated with $\phi$ and $\tilde{\phi}$ fields are the same
[otherwise the discrete symmetry (10) will be violated]. We shall see later
that the above two discrete symmetry transformations (10) are very important
for our whole discussion because they provide the physical realizations of the
Hodge duality $*$ operation of differential geometry [8-12] as the analogue of
relationship: $\delta=\pm\,*\,d\,*$ in terms of the off-shell nilpotent
(anti-)BRST and (anti-)co-BRST transformations (cf. Sec. 5).

It is very interesting to point out that the 2D Lagrangian density (9)
respects the following (anti-)co-BRST symmetry transformations $[s_{(a)d}]$,
namely;

$\displaystyle s_{ad}\,A_{\mu}=-\varepsilon_{\mu\nu}\,\partial^{\nu}\,C,\quad
s_{ad}\,\tilde{\phi}=-m\,C,\quad s_{ad}\,\psi=i\,e\,C\,\gamma_{5}\,\psi,\quad
s_{ad}\,\bar{\psi}=-i\,e\,\bar{\psi}\,\gamma_{5}\,C,$ $\displaystyle
s_{ad}\,\bar{C}=i\,{\cal B},\quad s_{ad}\,C=0,\quad s_{ad}\,B=s_{ad}\,{\cal
B}=s_{ad}\,\phi=0,\quad s_{ad}\,E=\Box C,$ $\displaystyle
s_{d}\,A_{\mu}=-\varepsilon_{\mu\nu}\,\partial^{\nu}\,\bar{C},\quad
s_{d}\,\tilde{\phi}=-m\,\bar{C},\quad
s_{d}\,\psi=i\,e\,\bar{C}\,\gamma_{5}\,\psi,\quad
s_{d}\,\bar{\psi}=-i\,e\,\bar{\psi}\,\gamma_{5}\,\bar{C},$ $\displaystyle
s_{d}\,\bar{C}=0,\quad s_{d}\,C=-i\,{\cal B},\quad s_{d}\,B=s_{d}\,{\cal
B}=s_{d}\,\phi=0,\quad s_{d}\,E=\Box\bar{C},$ (11)

because [for the massless $(m^{\prime}=0)$ fermions], we have the following

$\displaystyle s_{ad}\,{\cal L}_{\cal B}=\partial_{\mu}\,\Big{[}{\cal
B}\,\partial^{\mu}\,C+m\,\varepsilon^{\mu\nu}\,(m\,C\,A_{\nu}+\phi\,\partial_{\nu}\,C)+m\,\tilde{\phi}\,\partial^{\mu}\,C\Big{]},$
$\displaystyle s_{d}\,{\cal L}_{\cal B}=\partial_{\mu}\,\Big{[}{\cal
B}\,\partial^{\mu}\,\bar{C}+m\,\varepsilon^{\mu\nu}\,(m\,\bar{C}\,A_{\nu}+\phi\,\partial_{\nu}\,\bar{C})+m\,\tilde{\phi}\,\partial^{\mu}\,\bar{C}\Big{]},$
(12)

which render the action integral $S=\int d^{2}\,x\,{\cal L}_{\cal B}$
invariant for the physical fields that vanish-off as $x\rightarrow\pm\,\infty$
due to Gauss’s divergence theorem. According to Noether’s theorem, the above
continuous symmetry transformations lead to the derivation of the Noether
conserved currents $[J^{\mu}_{(r)},r=d,ad]$ at the classical (i.e. tree
Feynman diagram) level as:

$\displaystyle J^{\mu}_{(ad)}$ $\displaystyle=$ $\displaystyle{\cal
B}\,\partial^{\mu}\,C+m\,C\,\partial^{\mu}\,\tilde{\phi}$ $\displaystyle-$
$\displaystyle\varepsilon^{\mu\nu}\big{[}B\,(\partial_{\nu}\,C)+m^{2}\,C\,A_{\nu}+m\,\phi\,(\partial_{\nu}\,C)\big{]}+e\,\bar{\psi}\,\gamma^{\mu}\gamma_{5}\,C\,\psi,$
$\displaystyle J^{\mu}_{(d)}$ $\displaystyle=$ $\displaystyle{\cal
B}\,\partial^{\mu}\,\bar{C}+m\,\bar{C}\,\partial^{\mu}\,\tilde{\phi}$ (13)
$\displaystyle-$
$\displaystyle\varepsilon^{\mu\nu}\big{[}B\,(\partial_{\nu}\,\bar{C})+m^{2}\,\bar{C}\,A_{\nu}+m\,\phi\,(\partial_{\nu}\,\bar{C})\big{]}+e\,\bar{\psi}\,\gamma^{\mu}\gamma_{5}\,\bar{C}\,\psi.$

The conservation law for the above currents requires that
$\partial_{\mu}J^{\mu}_{(5)}=\partial_{\mu}(\bar{\psi}\,\gamma^{\mu}\,\gamma_{5}\,\psi)=0$
which is true only at the tree-level Feynman diagram (when the massless limit
of the Dirac fields are taken into account). These tree-level conserved
currents lead to the definition of the conserved $(\dot{Q}_{(r)}=0$ with
$r=d,ad)$ charges $Q_{(r)}=\int d\,x\,J_{r}^{0}$ (with $r=d,ad$) as:

$\displaystyle Q_{ad}$ $\displaystyle=$ $\displaystyle\int d\,x\,\Big{[}{\cal
B}\,\dot{C}+m\,\dot{\tilde{\phi}}\,C+B\,\partial_{1}\,C+m^{2}\,A_{1}\,C+m\,\phi\,\partial_{1}\,C+e\,\bar{\psi}\,\gamma^{0}\,\gamma_{5}\,C\,\psi\Big{]},$
$\displaystyle Q_{d}$ $\displaystyle=$ $\displaystyle\int d\,x\,\Big{[}{\cal
B}\,\dot{\bar{C}}+m\,\dot{\tilde{\phi}}\,\bar{C}+B\,\partial_{1}\,\bar{C}+m^{2}\,A_{1}\,\bar{C}+m\,\phi\,\partial_{1}\,\bar{C}$
(14) $\displaystyle+$ $\displaystyle
e\,\bar{\psi}\,\gamma^{0}\,\gamma_{5}\,\bar{C}\,\psi\Big{]}.$

It is worthwhile to mention here that the conservation law
$(\partial_{\mu}\,J^{\mu}_{(r)}=0,\,r=d,ad)$ for the Noether conserved
currents (13) and conserved $({\dot{Q}}_{(r)}=0,\,r=d,ad)$ charges (14) can be
readily proven, at the classical (i.e. tree Feynman diagram) level, by using
the following EL-EOMs that are derived from the Lagrangian density (9):

$\displaystyle\varepsilon^{\mu\nu}\,\big{[}\partial_{\nu}\,{\cal
B}+m\,\partial_{\nu}\,\tilde{\phi}\big{]}=m^{2}\,A^{\mu}-\partial^{\mu}\,B-m\,\partial^{\mu}\,\phi-e\,\bar{\psi}\,\gamma^{\mu}\,\psi,$
$\displaystyle\Box\,\tilde{\phi}+m\,E-m\,{\cal
B}=0,\qquad\Box\,\phi-m\,(\partial\cdot A)-m\,B=0,$ $\displaystyle
i\,(\partial_{\mu}\,\bar{\psi})\,\gamma^{\mu}+m^{\prime}\,\bar{\psi}+e\,\bar{\psi}\,\gamma^{\mu}\,A_{\mu}=0,\qquad
i\,\gamma^{\mu}\,\partial_{\mu}\,\psi-m^{\prime}\,\psi-e\,\gamma^{\mu}\,A_{\mu}\,\psi=0,$
$\displaystyle(\Box+m^{2})\,\bar{C}=0,\qquad(\Box+m^{2})\,C=0,\qquad
B=-\,(\partial\cdot A+m\,\phi)$ $\displaystyle{\cal
B}=(E-m\,\tilde{\phi})\equiv-(\varepsilon^{\mu\nu}\,\partial_{\mu}\,A_{\nu}+m\,\tilde{\phi}).$
(15)

A close look and keen observation of these equations imply that we have:
$(\Box+m^{2})\,\phi=0,\,(\Box+m^{2})\,\tilde{\phi}=0,\,(\Box+m^{2})\,B=0,\,(\Box+m^{2})\,{\cal
B}=0$. The last entry [i.e. $(\Box+m^{2})\,{\cal B}=0$] is true only at the
classical level (i.e. tree-level Feynman diagrams) in the massless limit
$(m^{\prime}=0)$ of the Dirac fields. This is due to the fact that we use:
$\partial_{\mu}\,(\bar{\psi}\,\gamma^{\mu}\,\gamma_{5}\,\psi)=0$. We shall
offer more comments on it in Sec. 7 where the 2D anomaly term will be
discussed. It is also worthwhile to mention that in the derivation of
$(\Box+m^{2})\,B=0$, we have used the conservation of vector current (i.e.
$\partial_{\mu}\,J^{\mu}_{(m)}=e\,\partial_{\mu}\,[\bar{\psi}\,\gamma^{\mu}\,\psi]=0$)
that is true at the classical as well as quantum level. Furthermore, we can
derive the simpler forms of the conserved charges $Q_{r}\,(r=d,ad)$ using the
Gauss’s divergence theorem and the following EL-EOM that has been taken from
the top entry of Eq. (15), namely;

$\displaystyle\dot{\cal
B}=\partial_{1}\,B+m\,(\partial_{1}\,\phi)+e\,\bar{\psi}\,\gamma_{1}\,\psi-m^{2}\,A_{1}-m\,\dot{\tilde{\phi}},$
(16)

which leads to the following concise expressions

$\displaystyle Q_{ad}^{(1)}$ $\displaystyle=$ $\displaystyle\int
d\,x\,\big{(}{\cal B}\,\dot{C}-\dot{\cal B}\,C\big{)},$ $\displaystyle
Q_{d}^{(1)}$ $\displaystyle=$ $\displaystyle\int d\,x\,\big{(}{\cal
B}\,\dot{\bar{C}}-\dot{\cal B}\,\bar{C}\big{)}.$ (17)

It is now crystal clear, from the (anti-)co-BRST symmetry transformations (11)
and the above concise expressions for the conserved (anti-)co-BRST charges
(17), that we have the following explicit relationships, namely;

$\displaystyle s_{d}\,Q_{d}^{(1)}=-i\,\\{Q_{d}^{(1)},Q_{d}^{(1)}\\}=0\quad$
$\displaystyle\Rightarrow$ $\displaystyle\quad[Q_{d}^{(1)}]^{2}=0,$
$\displaystyle
s_{ad}\,Q_{ad}^{(1)}=-i\,\\{Q_{ad}^{(1)},Q_{ad}^{(1)}\\}=0\quad$
$\displaystyle\Rightarrow$ $\displaystyle\quad[Q_{ad}^{(1)}]^{2}=0,$
$\displaystyle s_{d}\,Q_{ad}^{(1)}=-i\,\\{Q_{ad}^{(1)},Q_{d}^{(1)}\\}=0\quad$
$\displaystyle\Rightarrow$ $\displaystyle\quad
Q_{ad}^{(1)}\,Q_{d}^{(1)}+Q_{d}^{(1)}\,Q_{ad}^{(1)}=0,$ $\displaystyle
s_{ad}\,Q_{d}^{(1)}=-i\,\\{Q_{d}^{(1)},Q_{ad}^{(1)}\\}=0\quad$
$\displaystyle\Rightarrow$ $\displaystyle\quad
Q_{d}^{(1)}\,Q_{ad}^{(1)}+Q_{ad}^{(1)}\,Q_{d}^{(1)}=0.$ (18)

The above equations capture the off-shell nilpotency (i.e.
$[Q_{(r)}^{(1)}]^{2}=0,\,r=d,ad$) and absolute anticommutativity (i.e.
$Q_{d}^{(1)}\,Q_{ad}^{(1)}+Q_{ad}^{(1)}\,Q_{d}^{(1)}=0$) of the (anti-)co-BRST
charges where we have exploited the theoretical strength of the continuous
symmetry transformations and their generators as the conserved Noether
charges. It is straightforward to compute the l.h.s. of Eq. (18) by directly
applying the (anti-)co-BRST symmetry transformations (11) on the concise
expressions for the conserved $({\dot{Q}}_{(a)d}^{(1)}=0)$ (anti-)co-BRST
charges that have been quoted in Eq. (17) to check the sanctity of Eq. (18).

We wrap-up this section with the following key and decisive remarks. First of
all, we note that the (anti-)co-BRST symmetry transformations (11) are off-
shell nilpotent (i.e. $s_{(a)d}^{2}=0$) of order two and they are absolutely
anticommuting (i.e. $s_{d}\,s_{ad}+s_{ad}\,s_{d}=0$) in nature. Second, we
observe that the total gauge-fixing term [i.e. $(\partial.A+m\,\phi)$] remains
invariant under the (anti-)co-BRST symmetry transformations [i.e.
$s_{(a)d}\,(\partial.A)=0,\,s_{(a)d}\,\phi=0$]. Third, as already mentioned
earlier, the gauge-fixing term $(\partial.A)$ of the gauge field $(A_{\mu})$
owes its origin to the co-exterior derivative because
$\delta\,A^{(1)}=-\,*\,d\,*\,(d\,x^{\mu}\,A_{\mu})=(\partial.A)$. In the total
gauge-fixing term $(\partial.A+m\,\phi)$, it is evident that the term
$m\,\phi$ has been added on the dimensional ground (in the natural units where
$\hbar=c=1$). Fourth, to be precise, we have four forms of Lagrangian
densities for our 2D theory as discussed thoroughly in our earlier work [19].
However, we have chosen only one for the sake of bravity in Eq. (9). We have
provided a synopsis of the varieties of Lagrangian densities in our Appendix
A. Fifth, the absolute anticommutativity $(s_{d}\,s_{ad}+s_{ad}\,s_{d}=0)$ of
the (anti-)co-BRST symmetry transformations (11) (and their intimate
connection with the invariance of the gauge-fixing term owing its origin to
the co-exterior derivative $\delta=-\,*\,d\,*$) demonstrates that only one of
them can be identified with $\delta$. Sixth, we note that the co-BRST symmetry
transformations decrease the ghost number by one (i.e.
$s_{d}\,A_{\mu}=-\varepsilon_{\mu\nu}\,\partial^{\nu}\,\bar{C},\,s_{d}\,C=-i\,{\cal
B}$, etc.). On the contrary, the anti-co-BRST symmetry transformations
increase (i.e.
$s_{ad}\,A_{\mu}=-\varepsilon_{\mu\nu}\,\partial^{\nu}\,C,\,s_{ad}\,\bar{C}=i\,{\cal
B}$, etc.) the ghost number by one for the fields on which they act. Finally,
we note that the conservation of the currents in (13) and charges [cf. Eqs.
(14), (17)] is valid only at the classical (i.e. tree Feynman diagram) level.
We shall discuss more on this issue in our Sec. 7 to show that the
conservation law persists at the loop-level, too.

##  4 Unique Bosonic and Ghost-Scale Continuous Symmetry Transformations:
Conserved Charges

It is evident that, so far, we have discussed four fermionic [i.e. (anti-)BRST
and (anti-)co-BRST] symmetry transformations for our theory. We have also
noted that $s_{b}\,s_{ab}+s_{ab}\,s_{b}=0$ and $s_{d}\,s_{ad}+s_{ad}\,s_{d}=0$
due to the absolute anticommutativity properties of the (anti-)BRST and
(anti-)co-BRST symmetry transformations. With the above observations as
inputs, we note that we have the following bosonic symmetry transformations
$(s_{w},s_{\bar{w}})$, namely;

$\displaystyle s_{w}=\\{s_{b},s_{d}\\},\qquad
s_{\bar{w}}=\\{s_{ad},s_{ab}\\},$ (19)

that are constructed with the existing four fermionic symmetry transformations
$s_{(a)b}$ and $s_{(a)d}$ of our theory. These explicit symmetry
transformations are

$\displaystyle
s_{w}\,A_{\mu}=-i\,\varepsilon_{\mu\nu}\,\partial^{\nu}\,B-i\,\partial_{\mu}\,{\cal
B},\qquad s_{w}\,\phi=-i\,m\,{\cal B},\qquad s_{w}\,\tilde{\phi}=-i\,m\,{B},$
$\displaystyle s_{w}\,\psi=e\,B\,\gamma_{5}\,\psi-e\,{\cal B}\,\psi,\qquad
s_{w}\,\bar{\psi}=e\,B\,\bar{\psi}\,\gamma_{5}+e\,{\cal B}\,\bar{\psi},$
$\displaystyle s_{w}\,C=s_{w}\,\bar{C}=s_{w}\,B=s_{w}\,{\cal B}=0,\qquad
s_{w}^{2}\neq 0,$ (20) $\displaystyle
s_{\bar{w}}\,A_{\mu}=i\,\varepsilon_{\mu\nu}\,\partial^{\nu}\,B+i\,\partial_{\mu}\,{\cal
B},\qquad s_{\bar{w}}\,\phi=i\,m\,{\cal B},\qquad
s_{\bar{w}}\,\tilde{\phi}=i\,m\,{B},$ $\displaystyle
s_{\bar{w}}\,\psi=-(e\,B\,\gamma_{5}\,\psi-e\,{\cal B}\,\psi),\qquad
s_{\bar{w}}\,\bar{\psi}=-(e\,B\,\bar{\psi}\,\gamma_{5}+e\,{\cal
B}\,\bar{\psi}),$ $\displaystyle
s_{\bar{w}}\,C=s_{\bar{w}}\,\bar{C}=s_{\bar{w}}\,B=s_{\bar{w}}\,{\cal
B}=0,\qquad s_{\bar{w}}^{2}\neq 0,$ (21)

where we have considered all the basic and auxiliary fields of our chosen
Lagrangian density in Eq. (9). A close look at (20) and (21) demonstrates that
the bosonic symmetry transformations $s_{w}$ and $s_{\bar{w}}$ are not
independent of each-other as they satisfy the following

$\displaystyle(s_{w}+s_{\bar{w}})\,\Sigma=0,\qquad\Sigma=A_{\mu},\phi,\tilde{\phi},C,\bar{C},B,{\cal
B},\psi,\bar{\psi},$ (22)

where $\Sigma$ is the generic field of the Lagrangian density (9). In other
words, we have a unique bosonic symmetry in the thoery. The sanctity of the
above statement can be verified by the following transformation properties of
the Lagrangian density (9):

$\displaystyle s_{w}\,{\cal L}_{\cal B}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\Big{[}i\,m\,\varepsilon^{\mu\nu}\,\big{\\{}\phi\,(\partial_{\nu}\,B)+m\,B\,A_{\nu}\big{\\}}+i\,m\,\tilde{\phi}\,\partial^{\mu}\,B+i\,{\cal
B}\,\partial^{\mu}\,B-i\,B\,\partial^{\mu}\,{\cal B}\Big{]},$ $\displaystyle
s_{\bar{w}}\,{\cal L}_{\cal B}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\,\Big{[}i\,m\,\varepsilon^{\mu\nu}\,\big{\\{}\phi\,(\partial_{\nu}\,B)+m\,B\,A_{\nu}\big{\\}}+i\,m\,\tilde{\phi}\,\partial^{\mu}\,B+i\,{\cal
B}\,\partial^{\mu}\,B$ (23) $\displaystyle-$ $\displaystyle
i\,B\,\partial^{\mu}\,{\cal B}\Big{]}.$

This observation, once again, corroborates the fact that there is an existence
of a unique [i.e. $(s_{w}+s_{\bar{w}})\,{\cal L}_{\cal B}=0$] bosonic symmetry
$(s_{w})$ in our theory.

According to the Noether theorem, the above continuous bosonic symmetry leads
to the derivation of the following conserved
$(\partial_{\mu}\,J^{\mu}_{(w)}=0)$ Noether current $(J^{\mu}_{(w)})$:

$\displaystyle J^{\mu}_{(w)}$ $\displaystyle=$ $\displaystyle
i\,\varepsilon^{\mu\nu}\,\Big{[}{\cal B}\,\partial_{\nu}\,{\cal
B}-B\,\partial_{\nu}\,B+m\,\tilde{\phi}\,\partial_{\nu}\,{\cal
B}-m\,\phi\,\partial_{\nu}\,B-m^{2}\,B\,A_{\nu}\Big{]}$ (24) $\displaystyle+$
$\displaystyle i\,m\,B\,\partial^{\mu}\,\tilde{\phi}-i\,m\,{\cal
B}\,\partial^{\mu}\,\phi+i\,m^{2}\,{\cal
B}\,A^{\mu}+i\,e\,B\,\bar{\psi}\,\gamma^{\mu}\,\gamma_{5}\,\psi-i\,e\,{\cal
B}\,\bar{\psi}\,\gamma^{\mu}\,\psi.$

The conservation law $(\partial_{\mu}\,J^{\mu}_{(w)}=0)$ can be proven by
using the beauty and strength of the EL-EOMs that have been derived in Eq.
(15) from the Lagrangian density (9). The bosonic conserved charge $Q_{w}=\int
d\,x\,J^{0}_{(w)}$ (with $\gamma_{0}\,\gamma_{5}=\gamma_{1}$) is as follows:

$\displaystyle Q_{w}$ $\displaystyle=$ $\displaystyle\int
d\,x\,\Big{[}i\,B\,\partial_{1}\,B-i\,{\cal B}\,\partial_{1}\,{\cal
B}+i\,m\,B\,\dot{\tilde{\phi}}-i\,m\,{\cal
B}\,\dot{\phi}+i\,m\,\phi\,\partial_{1}\,B-i\,m\,\tilde{\phi}\,\partial_{1}\,{\cal
B}$ (25) $\displaystyle+$ $\displaystyle i\,m^{2}\,B\,A_{1}+i\,m^{2}\,{\cal
B}\,A_{0}+i\,e\,B\,\bar{\psi}\,\gamma_{1}\,\psi-i\,e\,{\cal
B}\,\bar{\psi}\,\gamma_{0}\,\psi\Big{]}.$

It is straightforward to check that the above conserved charge $(Q_{w})$ is
the generator of the continuous and infinitesimal bosonic symmetry
transformations in Eq. (20).

We have a ghost-scale symmetry in our theory where only the fermionic
$(C^{2}={\bar{C}}^{2}=0,\,C\,\bar{C}+\bar{C}\,C=0)$ (anti-)ghost fields
$(\bar{C})\,C$ transform continuously by a scale factor:

$\displaystyle C\rightarrow
C=e^{\Omega}\,C,\qquad\bar{C}\rightarrow\bar{C}=e^{-\Omega}\,\bar{C},\qquad\Sigma\rightarrow
e^{0}\,\Sigma,$ (26)

In the above, the field $\Sigma=A_{\mu},\phi,\tilde{\phi},B,{\cal
B},\psi,\bar{\psi}$ stands for the generic field of ${\cal L}_{\cal B}$
[without the fermionic (anti-)ghost fields] and $\Omega$ in the exponentials
of Eq. (26) is the global (i.e. spacetime independent) scale factor. The signs
in the exponents of $C$ and $\bar{C}$ denote the ghost numbers $(\pm 1)$,
respectively. The infinitesimal versions of (26) are

$\displaystyle s_{g}\,C=+C,\qquad s_{g}\,\bar{C}=-\bar{C},\qquad
s_{g}\,\Sigma=0,$ (27)

where, for the sake of brevity, we have taken $\Omega=1$. The invariance of
${\cal L}_{\cal B}$ under the continuous and infinitesimal global ghost-scale
symmetry transformations (27) leads to the existence of the following
Noether’s conserved current:

$\displaystyle
J^{\mu}_{(g)}=i\,\Big{[}\bar{C}\,(\partial^{\mu}\,C)-(\partial^{\mu}\,\bar{C})\,C\Big{]}.$
(28)

The conservation law $(\partial_{\mu}\,J^{\mu}_{(g)}=0)$ can be proven by
using the EL-EOMs that have been quoted in Eq. (15). The conserved charge
$Q_{g}=\int d\,x\,J^{0}_{(g)}$ is as follows:

$\displaystyle Q_{g}=i\,\int
d\,x\,\big{(}\bar{C}\,\dot{C}-\dot{\bar{C}}\,C\big{)}.$ (29)

The subscripts $g$ in Eqs. (27), (28) and (29) denote the infinitesimal
transformations, conserved current and conserved charge corresponding to the
presence of an infinitesimal and continuous global ghost-scale symmetry
transformation in our theory.

We end this section with the following key comments. First of all, we note
that the (anti-)ghost fields $(\bar{C})\,C$ do not transform under the bosonic
symmetry transformations [cf. Eq. (20)]. Second, out of the four fermionic
symmetries [i.e. $s_{(a)b}$ and $s_{(a)d}$] of our theory, it can be checked
that the anticommutators:
$\\{s_{b},s_{ab}\\},\,\\{s_{d},s_{ad}\\},\,\\{s_{b},s_{ad}\\},\,\\{s_{d},s_{ab}\\}$
turn out to be absolutely zero. Third, only two non-trivial anticommutators
(i.e. $s_{w}=\\{s_{b},s_{d}\\},\,s_{\bar{w}}=\\{s_{ad},s_{ab}\\}$) exist.
However, only one of them turns out to be independent which defines a unique
bosonic symmetry transformation $(s_{w})$ for our theory [cf. Eq. (20)]
because of our observation: $s_{w}+s_{\bar{w}}=0$ [cf. Eqs. (20), (21), (23)].
Fourth, under the ghost-scale symmetry transformations, only the (anti-)ghost
fields transform by a global scale factor. However, all the rest of the fields
do not transform at all. Finally, we observe the following interesting
relationships, namely;

$\displaystyle s_{w}\,Q_{r}$ $\displaystyle=$
$\displaystyle-i\,\big{[}Q_{r},Q_{w}\big{]}=0,\qquad\qquad\quad
r=b,ab,d,ad,g,w,$ $\displaystyle s_{g}\,Q_{s}$ $\displaystyle=$
$\displaystyle-i\,\big{[}Q_{s},Q_{g}\big{]}=+Q_{s},\qquad\qquad s=b,ad,$
$\displaystyle s_{g}\,Q_{t}$ $\displaystyle=$
$\displaystyle-i\,\big{[}Q_{t},Q_{g}\big{]}=-Q_{t},\qquad\qquad t=d,ab,$ (30)

which demonstrate that the conserved charge $Q_{w}$ is the Casimir operator
for the whole algebra constructed with the conserved charges
$Q_{r}(r=b,ab,d,ad,g,w)$. Furthermore, we note that the ghost number for the
conserved charges $Q_{b}$ and $Q_{ad}$ is $+1$ and the same for the conserved
charges $Q_{d}$ and $Q_{ab}$ is $(-1)$. We shall corroborate these statements
very clearly in our forthcoming sections 5 and 6, respectively, where these
observations will be shown to be important in the proof of our present theory
to be a model for Hodge theory.

##  5 Algebraic Structures: Symmetries and Charges

In this section, we assimilate all the continuous symmetry transformations for
our 2D theory and point out their algebraic structure where they act like
operators. In other words, when we say that $\\{s_{b},s_{ab}\\}=0$, it implies
that we have: $\\{s_{b},s_{ab}\\}\,\Sigma=0$ where
$\Sigma=A_{\mu},C,\bar{C},B,{\cal B},\psi,\bar{\psi},\phi,\tilde{\phi}$ is the
generic field. It is straightforward to note that we obtain the following
extended BRST-algebra that is obeyed by the infinitesimal and continuous
symmetry transformation operators $s_{r}\,(r=b,ab,d,ad,g,w)$ of our theory,
namely:

$\displaystyle\\{s_{b},s_{ab}\\}=0,\qquad\\{s_{d},s_{ad}\\}=0,\qquad
s_{(a)b}^{2}=0,\qquad s_{(a)d}^{2}=0,$
$\displaystyle\\{s_{b},s_{ad}\\}=0,\qquad\\{s_{d},s_{ab}\\}=0,\qquad\\{s_{d},s_{b}\\}=-\\{s_{ab},s_{ad}\\}=s_{w},$
$\displaystyle[s_{w},s_{r}]=0,\qquad r=b,ab,d,ad,g,w,$
$\displaystyle[s_{g},s_{b}]=+s_{b},\qquad[s_{g},s_{ad}]=+s_{ad},\qquad[s_{g},s_{ab}]=-s_{ab},\qquad[s_{g},s_{d}]=-s_{d},$
(31)

which shows that the bosonic symmetry transformation $(s_{w})$ commute with
all the other continuous and infinitesimal symmetry transformations
$s_{r}\,(r=b,ab,d,ad,g)$ of our 2D theory. On the other hand, the commutator
of $s_{g}$ with $s_{r}\,(r=b,ad)$ produces the transformations $s_{r}$ with
positive sign (which is just the opposite of the commutators of $s_{g}$ with
$s_{r}\,(r=d,ab)$ where there is a negative sign on the r.h.s.).

There is a very beautiful relationship between the fermionic symmetry
transformations and the discrete symmetry transformations (10) of our 2D
theory. We very clearly observe the validity of the following relationships:

$\displaystyle s_{d}=\pm\,*\,s_{b}\,*,\qquad s_{ab}=\pm\,*\,s_{ad}\,*,$ (32)

where $*$, in the above, stands for the discrete symmetry transformations (10)
of our theory. A close look at (32) demonstrates that we have captured the
mathematical relationship: $\delta=\pm\,*\,d\,*$ of differential geometry in
the terminology of the symmetry transformations where the continuous and
discrete symmetry transformations are intertwined together and play a very
important and decisive role. Following the basic concepts behind the duality
invariant theories (see, e.g. [34] for details), we observe that the $(\pm)$
signs in the relationship: $s_{d}=\pm\,*\,s_{b}\,*$ are governed by the double
operations of $*$ operator [i.e. the discrete symmetry transformations] on a
specific field. In other words, we note that for the generic field $\Psi$:

$\displaystyle*\,(*\,\Psi)\equiv-\Psi,\qquad\Psi=C,\bar{C},A_{\mu},B,{\cal
B},\phi,\tilde{\phi},$
$\displaystyle*\,(*\,\Psi)\equiv+\Psi,\qquad\Psi=\psi,\bar{\psi}.$ (33)

The above observations show that, only for the Dirac fields
$(\psi,\bar{\psi})$, we have the relationships:
$s_{d}\,\psi=+\,*\,s_{b}\,*\,\psi,\,s_{d}\,\bar{\psi}=+\,*\,s_{b}\,*\,\bar{\psi}$.
On the contrary, for the rest of the fields of our theory, we have:
$s_{d}=-*\,s_{b}\,*$ where there is a negative sign on the r.h.s. We have
provided more discussions on it in our Appendix B.

According to the basic principle behind the Noether theorem, we know that the
continuous symmetry transformations and the Noether conserved charges (i.e.
symmetry generators) are deeply connected with one-another. Hence, it turns
out that the algebra (31), obeyed by the symmetry transformation operators, is
also respected and replicated by the conserved charges
$Q_{r}\,(r=b,ab,d,ad,g,w)$. In other words, the algebra satisfied by the
conserved charges (which are responsible for the existence of the continuous
and infinitesimal symmetry transformations in our theory) respect the
following:

$\displaystyle Q_{(a)b}^{2}=0,\qquad Q_{(a)d}^{2}=0,\qquad
Q_{w}=\\{Q_{b},Q_{d}\\}=-\,\\{Q_{ad},Q_{ab}\\},$
$\displaystyle[Q_{w},Q_{r}]=0,\qquad r=b,ab,d,ad,g,w,$
$\displaystyle\\{Q_{b},Q_{ab}\\}=\\{Q_{d},Q_{ad}\\}=\\{Q_{b},Q_{ad}\\}=\\{Q_{ab},Q_{d}\\}=0,$
$\displaystyle i\,[Q_{g},Q_{b}]=+\,Q_{b},\qquad i\,[Q_{g},Q_{ad}]=+\,Q_{ad},$
$\displaystyle i\,[Q_{g},Q_{d}]=-\,Q_{d},\qquad i\,[Q_{g},Q_{ab}]=-\,Q_{ab}.$
(34)

It is crystal clear that the bosonic conserved charge $Q_{w}$, constructed
from the off-shell nilpotent (i.e. fermionic) conserved charges, is the
Casimir operator for the whole algebra. A close and clear look at (34) also
shows, in a subtle manner, that the ghost numbers for the set of charges
$(Q_{b},Q_{d},Q_{w})$ are $(+1,-1,0)$, respectively. On the other hand, the
set of charges $(Q_{ad},Q_{ab},Q_{w})$ carries the ghost numbers equal to
$(+1,-1,0)$, respectively, too. Hence, we observe that there are two sets of
conserved charges, in the algebra (34), which carry the ghost numbers
$(+1,-1,0)$ in our theory. This observation would play a very important role
in the next section where we shall establish the connection of the algebra
(obeyed by the physical conserved charges and continuous as well as discrete
symmetries of our theory) with that of the algebra respected by the de Rham
cohomological operators of the differential geometry [8-12] which are purely
mathematical in nature.

##  6 Physical Symmetries, Conserved Charges and Cohomological Operators:
Algebraic Connection

In this section, we establish the precise connection between the discrete and
continuous physical symmetries (and conserved charges) of our field-theoretic
model and the de Rham cohomological operators of differential geometry [8-12]
at the level of algebra. In this context, first of all, we recall that the
following explicit algebra [8-12]

$\displaystyle
d^{2}=0,\qquad\delta^{2}=0,\qquad\Delta=d\,\delta+\delta\,d=\\{d,\delta\\},$
$\displaystyle[\Delta,d]=0,\qquad[\Delta,\delta]=0,\qquad\Delta^{2}\neq 0,$
(35)

is satisfied by the de Rham cohomological operators $(d,\delta,\Delta)$ where
the (co-)exterior derivatives $(\delta)\,d$ obey: $\delta=\pm\,*\,d\,*$. Here
the symbol $*$ stands for the Hodge duality operator on the compact manifold
without a boundary. Furthermore, we note that when $d$ operates on an
arbitrary form $(f_{n})$ of degree $n$, it raises the degree by one (i.e.
$d\,f_{n}\sim f_{n+1}$). On the other hand, we have: $\delta\,f_{n}\sim
f_{n-1}$ which demonstrates that the operation of $\delta$ on a form decreases
the degree of the form by one. It is worthwhile to mention that here (i.e.
$\delta\,f_{n}\sim f_{n-1}$) the degree of the form is non-zero (i.e.
$n=1,2,3...$). The degree of a given form $(f_{n})$ remains unchanged when it
is operated upon by the Laplacian operator $\Delta$.

Against the backdrop of the above paragraph and equation (35), it is very
important for us to provide the physical realizations of (35) and capture the
key properties of the operations of the cohomological operators on a given
differential form $(f_{n})$ of a non-zero degree $n$ in the language of the
physical symmetries and conserved charges (in the quantum Hilbert space). In
this context, a clear and close look at the algebra (34) demonstrates that
there exists a deep connection between the cohomological operators and
conserved charges of our theory at the algebraic level. In fact, we find that
the following two-to-one mapping exists between the conserved charges and the
cohomological operators:

$\displaystyle(Q_{b},Q_{ad})\rightarrow
d,\qquad(Q_{d},Q_{ab})\rightarrow\delta,$ $\displaystyle
Q_{w}=\\{Q_{b},Q_{d}\\}\equiv-\\{Q_{ad},Q_{ab}\\}\rightarrow\Delta.$ (36)

As pointed out earlier, the absolute anticommutativity (i.e.
$\\{s_{b},s_{ab}\\}=0,\,\\{s_{d},s_{ad}\\}=0$) properties of the (anti-)BRST
and (anti-)co-BRST symmetry transformations (and their corresponding conserved
charges) imply that only one of these symmetries (and corresponding conserved
charge) can be identified with $d$ and $\delta$, respectively. It turns out
that, when $Q_{b}$ is identified with the exterior derivative $d$, the
corresponding dual-BRST charge $Q_{d}$ is identified with $\delta$. As a
consequence, we find that the Laplacian operator $\Delta$ is identified with
$Q_{w}=\\{Q_{b},Q_{d}\\}$ due to the fact that
$\Delta=\\{d,\delta\\}=(d+\delta)^{2}$. In exactly similar fashion, we find
that when $Q_{ad}$ provides the physical realization of $d$, then, the co-
exterior derivative is identified with the anti-BRST charge $Q_{ab}$. Hence,
the Laplacian operator turns out to be $Q_{w}=-\\{Q_{ad},Q_{ab}\\}$. The
negative sign appears here due to the fact that $Q_{w}$ is also defined as:
$Q_{w}=\\{Q_{d},Q_{b}\\}$ where there is a plus sign on the r.h.s. (in view of
$s_{w}+s_{\bar{w}}=0$).

As far as the change in the degree of the form, due to the operations of the
cohomological operators is concerned, we point out that this property can be
realized in the quantum Hilbert space of states where we define (in terms of
the conserved ghost charge $Q_{g}$) any arbitrary quantum state with a non-
zero ghost number $n$ as:

$\displaystyle i\,Q_{g}\,\mid\psi>_{n}=n\,\mid\psi>_{n}.$ (37)

The algebra (34) now immediately leads to the following interesting and
important observations as far as the ghost numbers of some specific states
[constructed with $(Q_{w},\,Q_{b},\,Q_{d})$ and $(Q_{w},\,Q_{ad},\,Q_{ab})$]
in the quantum Hilbert space are concerned, namely;

$\displaystyle i\,Q_{g}\,Q_{b}\,\mid\psi>_{n}$ $\displaystyle=$
$\displaystyle(n+1)\,Q_{b}\,\mid\psi>_{n},$ $\displaystyle
i\,Q_{g}\,Q_{d}\,\mid\psi>_{n}$ $\displaystyle=$
$\displaystyle(n-1)\,Q_{d}\,\mid\psi>_{n},$ $\displaystyle
i\,Q_{g}\,Q_{w}\,\mid\psi>_{n}$ $\displaystyle=$ $\displaystyle
n\,Q_{w}\,\mid\psi>_{n}.$ (38)

The above equations imply that the quantum states
$Q_{b}\,\mid\psi>_{n},\,Q_{d}\,\mid\psi>_{n}$ and $Q_{w}\,\mid\psi>_{n}$ have
the ghost numbers $(n+1),\,(n-1),\,n$, respectively. In other words, the ghost
number increases by one due to the operation of $Q_{b}$ but decreases by one
due to the operation of $Q_{d}$. On the contrary, the ghost number of a state
remains intact due to the operation of $Q_{w}$ on it. In exactly similar
fashion, given equation (37) as an input, we have the following mathematical
relationships, namely;

$\displaystyle i\,Q_{g}\,Q_{ad}\,\mid\psi>_{n}$ $\displaystyle=$
$\displaystyle(n+1)\,Q_{ad}\,\mid\psi>_{n},$ $\displaystyle
i\,Q_{g}\,Q_{ab}\,\mid\psi>_{n}$ $\displaystyle=$
$\displaystyle(n-1)\,Q_{ab}\,\mid\psi>_{n},$ $\displaystyle
i\,Q_{g}\,Q_{w}\,\mid\psi>_{n}$ $\displaystyle=$ $\displaystyle
n\,Q_{w}\,\mid\psi>_{n},$ (39)

which demonstrates that the ghost numbers are $(n+1),\,(n-1)$ and $n$ for the
states $Q_{ad}\,\mid\psi>_{n},\,Q_{ab}\,\mid\psi>_{n}$ and
$Q_{w}\,\mid\psi>_{n}$, respectively. Hence, the operations of the set of
charges $(Q_{ad},\,Q_{ab},\,Q_{w})$, on a quantum state of the ghost number
$n$, are exactly like the operations of the set of de Rham cohomological
operators $(d,\,\delta,\,\Delta)$ on a form of degree $n$. We can capture
mathematically, the observations made in Eqs. (38) and (39), as follows:

$\displaystyle Q_{b}\,\mid\psi>_{n}\quad\sim\quad\mid\psi>_{n+1},\qquad
Q_{ad}\,\mid\psi>_{n}\quad\sim\quad\mid\psi>_{n+1},$ $\displaystyle
Q_{d}\,\mid\psi>_{n}\quad\sim\quad\mid\psi>_{n-1},\qquad
Q_{ab}\,\mid\psi>_{n}\quad\sim\quad\mid\psi>_{n-1},$ $\displaystyle
Q_{w}\,\mid\psi>_{n}\quad\sim\quad\mid\psi>_{n},\qquad\quad
Q_{w}\,\mid\psi>_{n}\quad\sim\quad\mid\psi>_{n}.$ (40)

Thus, we observe that the changes in the degree of a given form, due to the
operations of $(d,\,\delta,\,\Delta)$ are exactly like the changes in the
ghost numbers of a suitably chosen quantum state due to the operations of
$(Q_{b},\,Q_{d},\,Q_{w})$ and/or $(Q_{ad},\,Q_{ab},\,Q_{w})$. In equations
(37) to (40), we have denoted the ghost numbers of the states by the
subscripts on these states.

At this crucial juncture, we are in the position to capture the Hodge
decomposition theorem of differential geometry [8-12] in the language of the
conserved charges of our 2D interacting theory in the quantum Hilbert space of
states, namely;

$\displaystyle\mid\psi>_{n}$ $\displaystyle=$
$\displaystyle\mid\omega>_{n}\,+\,Q_{b}\,\mid\chi>_{n-1}\,+\,Q_{d}\,\mid\lambda>_{n+1}$
(41) $\displaystyle\equiv$
$\displaystyle\mid\omega>_{n}\,+\,Q_{ad}\,\mid\sigma>_{n-1}\,+\,Q_{ab}\,\mid\kappa>_{n+1},$

where $\mid\psi>_{n}$ is any arbitrary quantum state in the Hilbert space with
the ghost number $n$. For the sake of generality, we have chosen the non-null
states $\mid\chi>_{n-1},\,\mid\sigma>_{n-1},\,\mid\lambda>_{n+1}$ and
$\mid\kappa>_{n+1}$ in the Hodge decomposed state $\mid\psi>_{n}$ in terms of
the two sets of charges $(Q_{w},\,Q_{b},\,Q_{d})$ and
$(Q_{w},\,Q_{ad},\,Q_{ab})$, respectively. We note that the states
$\mid\chi>_{n-1}$ and $\mid\sigma>_{n-1}$ are the BRST-exact and anti-co-BRST-
exact, respectively. On the other hand, we have the states
$\mid\lambda>_{n+1}$ and $\mid\kappa>_{n+1}$ which are co-BRST-exact and anti-
BRST-exact, respectively. In the Hodge-decomposed state (41), we have the
quantum state $\mid\omega>_{n}$ as the harmonic state which is the most
symmetric state in the whole theory because it is annihilated by all the
fermionic conserved charges. In other words, we have the following:

$\displaystyle Q_{(a)b}\,\mid\omega>_{n}=0,\qquad
Q_{(a)d}\,\mid\omega>_{n}=0.$ (42)

We choose the harmonic state as the physical state of the theory which is
annihilated by the conserved and nilpotent BRST charge as well as the co-BRST
charge. We have, purposely, not written the anti-BRST as well as the anti-co-
BRST charges because these do not lead to any new restrictions on the physical
state of the theory. To be economical and precise, we demand that the physical
state (i.e. $\mid phys>$) is the one which is annihilated by, at least, the
conserved BRST and co-BRST charges. Thus, the physical space of our 2D theory
is a subspace of the total quantum Hilbert space which satisfies the
physicality criteria:

$\displaystyle Q_{b}\,\mid phys>=0,\qquad Q_{d}\,\mid phys>=0.$ (43)

In the next section, we shall discuss the physical consequences of (43), in
detail, to demonstrate that our present study is physically interesting and
useful.

##  7 Harmonic State as Physical State: Consequences

As pointed out in the previous section, it is the harmonic state in the Hodge
decomposed state (of the total quantum Hilbert space) which is the most
symmetric state of our entire theory. This state has to be annihilated by the
conserved and nilpotent BRST and co-BRST charges (which is the most economical
and precise requirement). In this context, we have the following (from the
condition: $Q_{b}\mid phys>=0$), namely;

$\displaystyle\Pi^{0}\equiv B\mid phys>=0\quad$ $\displaystyle\Rightarrow$
$\displaystyle\quad-(\partial\cdot A+m\,\phi)\mid phys>=0,$
$\displaystyle\dot{\Pi}^{0}\equiv\dot{B}\mid phys>=0\quad$
$\displaystyle\Rightarrow$ $\displaystyle\quad-\partial_{0}\,(\partial\cdot
A+m\,\phi)\mid phys>=0,$ (44) $\displaystyle\equiv$
$\displaystyle(\vec{\bigtriangledown}\cdot\vec{E}-m\,\Pi_{\phi}-e\,\psi^{\dagger}\,\psi)\mid
phys>=0,$

where we have used the concise expressions for charge from Eq. (6) and assumed
that the (anti-)ghost fields are not physical fields. The above restrictions
on the physical state are consistent with Dirac’s quantization condition where
the physical state of a quantum gauge theory must be annihilated by the
operator form of the first-class constraints (of the corresponding classical
gauge theory). It is clear from the Lagrangian density (1) [and/or (9)] that
we have: $\Pi^{0}=-(\partial\cdot A+m\,\phi)\equiv B$ and
$\dot{\Pi}^{0}=\dot{B}\equiv\vec{\bigtriangledown}\cdot\vec{E}-m\,\Pi_{\phi}-e\,\psi^{\dagger}\,\psi$
where $\Pi_{\phi}=\dot{\phi}-m\,A_{0}$ is the canonical conjugate momentum
w.r.t. the pure scalar field $\phi$ (which is nothing but the
St$\ddot{u}$ckelberg-field in the context of our 2D Proca theory). We would
like to point out that $\Pi^{0}$ is the conjugate momentum w.r.t. $A_{0}$
field and
$\dot{\Pi}^{0}=\dot{B}=\vec{\bigtriangledown}\cdot\vec{E}-m\,\Pi_{\phi}-e\,\psi^{\dagger}\,\psi$
emerges out from the EL-EOMs (5) and/or (15). In the 2D case, we have
$\vec{\bigtriangledown}\cdot\vec{E}=\partial_{1}\,E$ and, therefore, the
Gauss-law of divergence is:
$\dot{B}=\partial_{1}\,E-m\,\Pi_{\phi}-e\,\psi^{\dagger}\,\psi$ which is
derived from the top entry of Eq. (15) with the inputs: $B=-(\partial\cdot
A+m\,\phi)$ and ${\cal B}=E-m\,\tilde{\phi}$. In more precise language, within
the ambit of BRST formalism, we have the restrictions on the physical state
[cf. Eq. (44)] as the quantum generalizations of the first-class constraints
($\Pi^{0}\approx 0$ and
$\vec{\bigtriangledown}\cdot\vec{E}-m\,\Pi_{\phi}-e\,\psi^{\dagger}\,\psi\approx
0$) on the classical gauge theory which become operators at the quantum level
and they annihilate the physical state.

Let us focus on the proof of the conservation law $(\dot{Q}_{b}=0)$ for the
concise form of the nilpotent BRST charge $(Q_{b})$ that is quoted in Eq. (6).
In this context, we note that the conserved current $J^{\mu}_{(b)}$ [cf. Eq.
(4)] corresponding to the infinitesimal and continuous BRST symmetry
transformations (2) defines the BRST charge $(Q_{b})$ in a straightforward
manner. A close look at the expression of $J^{\mu}_{(b)}$ in Eq. (4) shows
that, for the proof of the conservation law $\partial_{\mu}\,J^{\mu}_{(b)}=0$,
we invoke the validity of the conservation law
$\partial_{\mu}\,J^{\mu}_{(m)}=\partial_{\mu}\,[\bar{\psi}\,\gamma^{\mu}\,\psi]=0$
where $J^{\mu}_{(m)}$ is the polar-vector current constructed with the Dirac
fields (which are the interacting matter fields in our theory). There is no
problem for the proof of $\partial_{\mu}\,J^{\mu}_{(m)}=0$ as the EL-EOM of
Eq. (15) lead to it. It is straightforward to note that we have the following
when we take a direct time derivative on (6), namely;

$\displaystyle\dot{Q}_{b}=\int
d^{D-1}\,x\,\big{[}B\,\ddot{C}-\ddot{B}\,C\big{]}.$ (45)

Using the EL-EOMs from (5) and/or (15), we note that $(\Box+m^{2})\,C=0$ and
$(\Box+m^{2})\,B=0$. In the derivation of the latter, we have used
$(\Box+m^{2})\,\phi=0$ and the conservation of the matter current***The
conservation of the matter current
$(J^{\mu}_{(m)}=e\,\bar{\psi}\,\gamma^{\mu}\,\psi)$, constructed with the
Dirac fields $(\bar{\psi},\,\psi)$, is sacrosanct because this leads to the
conservation of the electric charge which is universal and true at the
classical as well as quantum level (in any physically allowed process). [i.e.
$\partial_{\mu}\,J^{\mu}_{(m)}\equiv
e\,\partial_{\mu}\,(\bar{\psi}\,\gamma^{\mu}\,\psi)=0$] which is valid due to
the local gauge and/or BRST symmetry invariance in our theory. As a
consequence of the above EL-EOMs, we have:
$\ddot{C}=\vec{\bigtriangledown}^{2}\,C-m^{2}\,C$ and
$\ddot{B}=\vec{\bigtriangledown}^{2}\,B-m^{2}\,B$. Substitutions of these into
(45) lead to the following explicit form of the conservation law, namely;

$\displaystyle\dot{Q}_{b}=\int
d^{D-1}\,x\,\big{[}\vec{\bigtriangledown}\cdot(B\,\vec{\bigtriangledown}\,C-C\,\vec{\bigtriangledown}\,B)\big{]}\quad\longrightarrow\quad
0.$ (46)

Thus, we find that the conservation of BRST charge is true in any arbitrary
dimension of spacetime because we see that $\dot{Q}_{b}\rightarrow 0$ due to
Gauss’s divergence theorem. For 2D case, we have $\dot{Q}_{b}=\int
d\,x\,\big{[}\partial_{1}\,(B\,\partial_{1}\,C-C\,\partial_{1}\,B)\big{]}\rightarrow
0$. We conclude that the physicality criterion with the nilpotent and
conserved BRST charge: $Q_{b}\,\mid phys>=0$ leads to the annihilation of the
physical state by the operator forms of the primary constraint
$(\Pi^{0}\approx 0)$ and the secondary constraint:
$(\vec{\bigtriangledown}\cdot\vec{E}-m\,\Pi_{\phi}-e\,\psi^{\dagger}\,\psi\approx
0)$ on our 2D St$\ddot{u}$ckelberg’s modified Proca theory. It is clear that
both these constraints are first-class.

Due to the presence of a set of discrete symmetry transformations (10) and our
observations in Secs. 5 and 6, it is obvious that our present 2D theory is (i)
a perfect model of a duality-invariant theory [34], and (ii) an example of
Hodge theory. As far as the duality property is concerned, it will be noted
that the discrete symmetry transformation on the gauge field (i.e.
$A_{\mu}\rightarrow\mp\,i\,\varepsilon_{\mu\nu}\,A^{\nu}$) owes its origin to
the self-duality (see, e.g. [5, 7] for details) condition [i.e.
$*\,A^{(1)}=*\,(d\,x^{\mu}\,A_{\mu})$] on the 1-form
$(A^{(1)}=d\,x^{\mu}\,A_{\mu})$ gauge field $A_{\mu}$ where the $*$ is the
Hodge duality operation. Rest of the transformations in (10) are consistent
with this self-duality requirement. For a perfect duality invariant gauge
theory, it is very sacrosanct requirement that the dual to the first-class
constraints [cf. Eq. (44)] must also annihilate the physical state of the
theory. In this context, it is interesting to point out that
$(E-m\,\tilde{\phi})$ is the dual of the gauge-fixing term $(\partial\cdot
A+m\,\phi)$ due to the discrete symmetry transformations (10). Similarly, we
note that $\partial_{0}\,(E-m\,\tilde{\phi})$ is also dual to
$[\partial_{0}\,(\partial\cdot A+m\,\phi)]$. The dual restrictions emerge out
from the co-BRST charge $Q_{d}^{(1)}$ as discussed below. We lay emphasis on
the fact that operator forms of the first-class constraints and their dual
must annihilate the physical state at the level of tree and loop diagrams. In
other words, these constraints/restrictions on the physical state are
universal for a perfect duality invariant gauge theory and they must be
respected at the classical as well as quantum level.

Against the backdrop of the above, the physicality criterion:
$Q_{d}^{(1)}\,\mid phys>=0$ imposes exactly the conditions on the physical
state due to the duality considerations, namely;

$\displaystyle{\cal B}\,\mid phys>=0\quad$ $\displaystyle\Rightarrow$
$\displaystyle\quad(E-m\,\tilde{\phi})\,\mid phys>=0,$ $\displaystyle\dot{\cal
B}\,\mid phys>=0\quad$ $\displaystyle\Rightarrow$
$\displaystyle\quad\partial_{0}\,(E-m\,\tilde{\phi})\,\mid phys>=0$ (47)
$\displaystyle\equiv$
$\displaystyle(\partial_{1}\,\Pi^{0}+m\,\Pi_{\tilde{\phi}}+e\,\psi^{\dagger}\,\gamma_{5}\,\psi+m\,\partial_{1}\,\phi-m^{2}\,A_{1})\,\mid
phys>=0,$

where we have used the concise expression for the conserved co-BRST charge
$Q_{d}^{(1)}$ [cf. Eq. (17)] and EL-EOMs from Eq. (15) in the derivation of
$\partial_{0}\,(E-m\,\tilde{\phi})$. We have also used the expressions:
$\Pi^{0}=B,\,\Pi_{\tilde{\phi}}=-\,\dot{\tilde{\phi}},\,{\cal
B}=E-m\,\tilde{\phi}$. Let us now concentrate on the physical consequence that
emerges out from the top entry of (47). A close and careful look at the 2D
(anti-)BRST and (anti-)co-BRST invariant Lagrangian density (9) demonstrates
that the pseudo-scalar field $\tilde{\phi}$ has appeared in the theory with a
negative kinetic term. Such fields have been christened as the “phantom”
fields in the realm of cosmology and they have been found to be useful in the
context of cyclic, bouncing and self-accelerated models of Universe (see e.g.
[26-30] for details). In a nut-shell, such exotic fields (with negative
kinetic terms) are unphysical in the sense that they have not yet been
detected by the experiments. Thus, just like the (anti-)ghost fields, such
kinds of fields (e.g. the pseudo-scalar field $\tilde{\phi}$) do not impose
any restriction on the physical state of a quantum gauge theory. This leads us
to draw the conclusion that physically, the restrictions in (47), imply the
following (from the physicality requirement $Q_{d}^{(1)}\,\mid phys>=0$),
namely;

$\displaystyle{\cal B}\,\mid phys>=0\quad$ $\displaystyle\Rightarrow$
$\displaystyle\quad(E-m\,\tilde{\phi})\,\mid phys>=0\qquad\Rightarrow\quad
E\,\mid phys>=0,$ $\displaystyle\dot{\cal B}\,\mid phys>=0\quad$
$\displaystyle\Rightarrow$
$\displaystyle\quad\partial_{0}\,(E-m\,\tilde{\phi})\,\mid
phys>=0\quad\Rightarrow\quad\dot{E}\,\mid phys>=0.$ (48)

At this crucial juncture, we recall that the electric field
$E=-\frac{1}{2}\,\varepsilon^{\mu\nu}\,F_{\mu\nu}\equiv-\varepsilon^{\mu\nu}\,\partial_{\mu}\,A_{\nu}$
is also the anomaly term in 2D because
$\partial_{\mu}\,(\bar{\psi}\,\gamma^{\mu}\,\gamma_{5}\,\psi)=-\frac{\alpha}{2}\,\varepsilon^{\mu\nu}\,F_{\mu\nu}=\alpha\,E$
where $\alpha$ is a constant factor (that is not very important for our
present discussion). Thus, the anomaly term and the requirement of its time-
evolution invariance must annihilate the physical state.

We have the conservation $({\dot{Q}}_{b}=0)$ of the BRST charge due to the
conservation $(\partial_{\mu}\,J^{\mu}_{(b)}=0)$ of the BRST invariant
$(s_{b}\,J^{\mu}_{(b)}=0)$ Noether current $(J^{\mu}_{(b)})$ which, in a
precise language, implies the following in terms of the physical state (see,
e.g. [36] for details)

$\displaystyle<phys\mid\partial_{\mu}\,J^{\mu}_{(b)}\mid
phys>=0\quad\Rightarrow\quad<phys\mid{\dot{Q}}_{(b)}\mid phys>=0.$ (49)

Against this as a backdrop, we now concentrate on the proof of the
conservation $({\dot{Q}}_{d}^{(1)}=0)$ of the concise form of the co-BRST
charge in Eq. (17). It is straightforward to note that we have the following
explicit expression for $({\dot{Q}}_{d}^{(1)}\equiv d\,Q_{d}^{(1)}/d\,t)$,
namely;

$\displaystyle{\dot{Q}}_{d}^{(1)}=\int d\,x\,\big{[}{\cal
B}\,\ddot{\bar{C}}-\ddot{\cal B}\,\bar{C}\big{]},$ (50)

where we have applied directly the time derivative on (17). We have the EL-
EOMs: $(\Box+m^{2})\,\bar{C}=0$ and $(\Box+m^{2})\,{\cal B}=0$. The latter is
true at the classical level (where there are no loop-diagrams and the massless
limit of the Dirac fermions is taken into account). To be precise, we lay
emphasis on the fact that we find that $(\Box+m^{2})\,{\cal
B}=e\,\partial_{\mu}\,\big{[}\bar{\psi}\,\gamma^{\mu}\,\gamma_{5}\,\psi\big{]}$
where the r.h.s. is precisely equal to zero in the classical limit as stated
above. When we consider the triangle Feynman diagram, we obtain the ABJ
anomaly term on the r.h.s. where
$\partial_{\mu}\,(\bar{\psi}\,\gamma^{\mu}\,\gamma_{5}\,\psi)=\alpha\,E$ for
the 2D theory (with $\alpha$ as a constant factor). Thus, we find that Eq.
(50) reduces to the following explicit form:

$\displaystyle{\dot{Q}}_{d}^{(1)}$ $\displaystyle=$ $\displaystyle\int
d\,x\,\big{[}{\cal
B}\,(\partial_{1}^{2}\,\bar{C}-m^{2}\,\bar{C})-(\partial_{1}^{2}\,{\cal
B}-m^{2}\,{\cal B}+\alpha\,E)\,\bar{C}\big{]}$ (51) $\displaystyle\equiv$
$\displaystyle\int d\,x\,\Big{[}\partial_{1}\big{\\{}{\cal
B}\,\partial_{1}\,\bar{C}-(\partial_{1}\,{\cal
B})\,\bar{C}\big{\\}}\Big{]}-\alpha\,\int d\,x\,E\,\bar{C},$

where we have used
$\ddot{\bar{C}}=\partial_{1}\,\bar{C}-m^{2}\,\bar{C},\,\ddot{\cal
B}=\partial_{1}^{2}\,{\cal B}-m^{2}\,{\cal B}+\alpha\,E$. It is
straightforward to note that the first term of (51) goes to zero as
$x\rightarrow\pm\,\infty$ due to Gauss’s divergence theorem. However, the
second term (due to 2D anomaly) remains intact. In view of (49), we note that:
$<phys\mid\,\partial_{\mu}\,J^{\mu}_{(d)}\,\mid
phys>\,\Rightarrow\,<phys\mid\,{\dot{Q}}_{d}^{(1)}\,\mid phys>$. However, as
pointed out earlier, our theory is a perfect model of duality invariant theory
[34] because $A_{\mu}\rightarrow\mp\,i\,\varepsilon_{\mu\nu}\,A^{\nu}$ [cf.
Eq. (10)] arises due to the self-duality restriction (see, e.g. [5, 7] for
details). Thus, the restrictions (48) are valid at the tree-level as well as
loop-level Feynman diagrams†††It is the very elegant interplay between the co-
BRST symmetry transformations (11) and the discrete symmetry transformations
(10) that ensures the conservation of the co-BRST charge at the tree as well
as the loop-level diagrams (when we choose the physical state as the harmonic
state).. At this crucial stage, we find the following explicit expressions
(see, e.g. [36] for details)

$\displaystyle<phys\mid\partial_{\mu}\,J^{\mu}_{(d)}\mid phys>$
$\displaystyle=$ $\displaystyle\alpha\,<phys\mid E\mid
phys>=0\quad\Rightarrow$ $\displaystyle<phys\mid{\dot{Q}}_{d}^{(1)}\mid phys>$
$\displaystyle=$ $\displaystyle\alpha\,\int<phys\mid E\,\bar{C}\mid
phys>\,dx=0.$ (52)

The r.h.s. of the above expressions are zero due to the fact that, in view of
Eq. (48), we have: $E\,\mid phys>=0$. In the proof of the conservation of co-
BRST charge, we note that the anti-ghost field $\bar{C}$ does not impose any
restriction on the physical state. Hence, in 2D, the anomaly term is trivial
in the sense that the vector and axial-vector currents (and corresponding
Noether charges) are conserved together as far as the physical state of our 2D
interacting theory is concerned. This is the basic reason behind the
consistency and unitarity of the 2D anomalous Abelian 1-form gauge theories
(see, e.g. [37-39] for details).

##  8 Summary and Outlook

In our present investigation, we have demonstrated that an interacting 2D
modified version of Proca theory (in interaction with the Dirac fields)
represents a tractable field-theoretic model for the Hodge theory where the
discrete and continuous symmetries of this theory provide the physical
realizations of the de Rham cohomological operators of differential geometry
at the algebraic level. Whereas the infinitesimal and continuous symmetries
(and corresponding conserved charges) of our present 2D theory provide the
physical realizations of the cohomological operators, the discrete symmetry
transformations correspond to the Hodge duality $*$ operation of differential
geometry‡‡‡Under the discrete symmetry transformations (10), the kinetic term
$(\bar{\psi}\,i\,\gamma^{\mu}\,\partial_{\mu}\,\psi)$ and the mass term
$(-m\,\bar{\psi}\,\psi)$ of the Dirac fields remain trivially invariant. The
interaction term $(-\,e\,\bar{\psi}\,\gamma^{\mu}\,A_{\mu}\,\psi)$ is also
found to respect the discrete symmetry transformations (10) [cf. Appendix C
for details].. The ghost number considerations of the quantum states in the
total Hilbert space, in terms of the conserved charges, have been able to
provide the physical analogue of the operations of cohomological operators on
the differential form of a given degree and the changes (in the degree) that
ensue.

For our 2D interacting theory, we have been able to demonstrate that (i) the
nilpotent (anti-)BRST symmetry transformations (and conserved charges) exist
[cf. Eqs. (2), (6)] corresponding to the local gauge symmetry in the theory
which is generated by the first-class constraints, and (ii) the nilpotent
(anti-)co-BRST symmetries (and corresponding conserved charges) are present
[cf. Eqs. (11), (17)] in the theory due to the existence of the local chiral
symmetry (in the massless limit of the Dirac fields). The latter symmetry
exists due to the presence of a duality symmetry in the theory [cf. Eq. (10)].
As is well-known, there is presence of the ABJ anomaly whenever the local
gauge and chiral symmetries co-exist together in a theory with Dirac fields
(in interaction with a gauge field). However, at the classical level (in the
massless limit of the Dirac fields), this anomaly term is zero (at the tree
level Feynman diagrams). For the QED in 2D, the anomaly term is nothing but
the electric field (i.e.
$\frac{\alpha}{2}\,\varepsilon^{\mu\nu}\,F_{\mu\nu}=\alpha\,E$ with $\alpha$
as a constant factor) which appears in the conservation law of the chiral
current (i.e.
$\partial_{\mu}\,J^{\mu}_{(5)}=\partial_{\mu}\,(\bar{\psi}\,\gamma^{\mu}\,\gamma_{5}\,\psi)=\alpha\,E$).

As per Dirac’s quantization condition, a physical state of the total quantum
Hilbert space must be annihilated by the operator form of the constraints of a
given classical theory. In our present 2D theory, this key feature is
incorporated in the requirement that the physical state must be annihilated by
the conserved and nilpotent BRST charge [cf. Eq. (44)]. However, if a theory
is perfectly duality invariant, the dual version of these constraints must
also annihilate the physical state at the quantum level [cf. Eq. (47)]. Our
present 2D theory is not only a perfect field-theoretic example of Hodge
theory, it is also a beautiful model of the duality invariant theory [34] due
to the presence of the discrete symmetry transformations (10). This is the
reason that we have the restrictions (44) and (47) on the physical state of
our theory which are very sacrosanct and they must be respected at the
classical as well as the quantum level. In other words, the restrictions (44)
and (47) are true at the tree as well as loop-level Feynman diagrams for our
2D interacting theory. This is precisely the reason that the 2D anomaly term
becomes trivial and, as a consequence, the vector ($J^{\mu}_{(b)}$) and axial-
vector ($J^{\mu}_{(d)}$) currents are conserved together leading to the
conservation of the BRST and co-BRST charges at the tree as well as at the
loop-level diagrams.

We end this section with a few crucial remarks. First of all, the simultaneous
conservation of the BRST and co-BRST charges (and corresponding Noether
currents) are true only in the case of 2D theory of QED with Dirac fields.
Second, in our earlier works [32, 33] on the dual-BRST symmetries (without
mass term for the photon), we have shown that the physical state is
annihilated by the electric field $(E)$ and its time-evolution $(\dot{E})$
invariance (i.e. $E\,\mid phys>=0,\,\dot{E}\,\mid phys>=0$). However, in such
theories, the infrared divergence problem exists. In contrast, in our present
theory of massive photon, such divergences do not exist (see, e.g. [36]).
Third, if QED with massive photon (along with a pseudo-scalar field) is taken
into account, our conjecture is the observation that the anomaly term will be:
$(E-m\,\tilde{\phi})$. Fourth, our present analysis provides the basic reason
behind the consistency and unitarity of the 2D anomalous Abelian gauge theory
with Dirac fields (see, e.g. [37-39]). Finally, the existence of the negative
kinetic term for the fields is not a problem in the modern-day cosmology where
the cyclic, bouncing and self- accelerated models of Universe require the
existence of such kind of “phantom” fields and particles (see, e.g. [26-30]).
This pseudo-scalar field and an axial-vector field (with negative kinectic
terms) have also been shown to provide a possible set of candidates for dark
matter and dark energy [18-20]. Within the framework of (SUSY) quantum field
theories, the St${\ddot{u}}$ckelberg-boson has been able to shed light on the
infrared problem in QED and it has been shown to play an important role in
providing a possible candidate for the ultralight dark matter. These issues
have been recently discussed in a set of very interesting papers [40-42]. It
will be nice future endeavor to apply BRST approach to these (SUSY) field
theoretic models.

Acknowledgments

Two (AT, AKR) of us gratefully acknowledge the financial support from Banaras
Hindu University (BHU) under its BHU-Fellowship program. One (RPM) of us has
learnt the basics of gauge theory, constraints, anomalous gauge theory, etc.,
from Prof. R. Rajaraman who happens to be one of the most influential mentors
of the QFT group at BHU. All the authors, very humbly and respectfully,
dedicate their present work to him.

Appendix A: On Different Forms of Lagrangian Density

The central theme of our present Appendix is to highlight that we have
discussed a few different forms of the (anti-)BRST and (anti-)co-BRST
invariant Lagrangian densities (without the Dirac fields) in our earlier work
[19] besides the Lagrangian density (9) that has been taken for discussion in
our present endeavor. For instance, we have taken into account the following
Lagrangian densities (without Dirac fields), namely;

${\cal
L}_{(B_{1})}=\frac{1}{2}\,(E-m\,\tilde{\phi})+m\,E\,\tilde{\phi}-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\partial^{\mu}\,\tilde{\phi}+\frac{m^{2}}{2}\,A_{\mu}\,A^{\mu}+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi$
$-m\,A_{\mu}\,\partial^{\mu}\,\phi-\frac{1}{2}\,(\partial\cdot
A+m\,\phi)^{2}-i\,\partial_{\mu}\,\bar{C}\,\partial^{\mu}\,C+i\,m^{2}\,\bar{C}\,C,$
$None$ ${\cal
L}_{(B_{2})}=\frac{1}{2}\,(E+m\,\tilde{\phi})-m\,E\,\tilde{\phi}-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\partial^{\mu}\,\tilde{\phi}+\frac{m^{2}}{2}\,A_{\mu}\,A^{\mu}+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi$
$+m\,A_{\mu}\,\partial^{\mu}\,\phi-\frac{1}{2}\,(\partial\cdot
A-m\,\phi)^{2}-i\,\partial_{\mu}\,\bar{C}\,\partial^{\mu}\,C+i\,m^{2}\,\bar{C}\,C,$
$None$

which are intimately connected with each-other by the discrete symmetry
transformations: $A_{\mu}\rightarrow A_{\mu},\,C\rightarrow
C,\,\bar{C}\rightarrow\bar{C},\,\phi\rightarrow-\phi,\,\tilde{\phi}\rightarrow-\tilde{\phi}$.
Thus, they do not describe two different physical systems. It is
straightforward to check that the following on-shell nilpotent and absolutely
anticommuting (anti-)BRST symmetry transformations $[s_{(a)b}]$

$s_{ab}\,A_{\mu}=\partial_{\mu}\,\bar{C},\qquad s_{ab}\,C=i\,(\partial\cdot
A+m\,\phi),\qquad s_{ab}\,\phi=m\,\bar{C},$
$s_{ab}\,(\bar{C},E,\tilde{\phi})=0,\qquad s_{ab}\,(\partial\cdot
A+m\,\phi)=(\Box+m^{2})\,\bar{C},$ $s_{b}\,A_{\mu}=\partial_{\mu}\,C,\qquad
s_{b}\,\bar{C}=-i\,(\partial\cdot A+m\,\phi),\qquad s_{b}\,\phi=m\,C,$
$s_{b}\,(C,E,\tilde{\phi})=0,\qquad s_{b}\,(\partial\cdot
A+m\,\phi)=(\Box+m^{2})\,C,$ $None$

transform the Lagrangian density ${\cal L}_{B_{1}}$ to a total spacetime
derivative thereby rendering the action integral $S=\int d\,x\,{\cal
L}_{B_{1}}$ invariant. In exactly similar fashion, we can write the on-shell
nilpotent and absolutely anticommuting (anti-)BRST transformations for ${\cal
L}_{B_{2}}$ from (A.3) by the replacements: $A_{\mu}\rightarrow
A_{\mu},\,C\rightarrow
C,\,\bar{C}\rightarrow\bar{C},\,\phi\rightarrow-\phi,\,\tilde{\phi}\rightarrow-\tilde{\phi}$.

The Lagrangian density (A.1) also respects an on-shell nilpotent and
absolutely anticommuting set of (anti-)co-BRST symmetry transformations
$s_{(a)d}$, namely;

$s_{ad}\,A_{\mu}=-\varepsilon_{\mu\nu}\,\partial^{\nu}\,C,\qquad
s_{ad}\,\bar{C}=i\,(E-m\,\tilde{\phi}),\qquad s_{ad}\,\tilde{\phi}=-m\,C,$
$s_{ad}\,(C,\partial\cdot A+m\,\phi,\phi)=0,\qquad
s_{ad}\,(E-m\,\tilde{\phi})=(\Box+m^{2})\,C,$
$s_{d}\,A_{\mu}=-\varepsilon_{\mu\nu}\,\partial^{\nu}\,\bar{C},\qquad
s_{d}\,C=-i\,(E-m\,\tilde{\phi}),\qquad s_{d}\,\tilde{\phi}=-m\,\bar{C},$
$s_{d}\,(\bar{C},\partial\cdot A+m\,\phi,\phi)=0,\qquad
s_{d}\,(E-m\,\tilde{\phi})=(\Box+m^{2})\,\bar{C}.$ $None$

It is straightforward to note that the Lagrangian density (A.1) transforms to
a total spacetime derivative under the on-shell nilpotent (anti-)dual BRST
symmetry transformations $s_{(a)d}$. As a consequence, the action integral
$S=\int d\,x\,{\cal L}_{B_{1}}$ remains invariant under $s_{(a)d}$. It goes
without saying that the (anti-)co-BRST symmetries for the Lagrangian density
${\cal L}_{B_{2}}$ can be obtained from (A.4) by the replacements:
$A_{\mu}\rightarrow A_{\mu},\,C\rightarrow
C,\,\bar{C}\rightarrow\bar{C},\,\phi\rightarrow-\phi,\,\tilde{\phi}\rightarrow-\tilde{\phi}$.
As far as the off-shell nilpotent (anti-)BRST and (anti-)co-BRST invariant
Lagrangian densities (without the Dirac fields) are concerned, we have the
following [19]:

${\cal L}_{\cal B}={\cal B}\,(E-m\,\tilde{\phi})-\frac{1}{2}\,{\cal
B}^{2}+m\,E\,\tilde{\phi}-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\partial^{\mu}\,\tilde{\phi}+\frac{m^{2}}{2}\,{A_{\mu}}{A^{\mu}}+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi$
$-m\,A_{\mu}\,\partial^{\mu}\,\phi+B\,(\partial\cdot
A+m\,\phi)+\frac{1}{2}\,B^{2}-i\,\partial_{\mu}\,\bar{C}\,\partial^{\mu}\,C+i\,m^{2}\,\bar{C}\,C,$
$None$ ${\cal
L}_{\cal{\bar{B}}}={\cal{\bar{B}}}\,(E+m\,\tilde{\phi})-\frac{1}{2}\,{\cal{\bar{B}}}^{2}-m\,E\,\tilde{\phi}-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\partial^{\mu}\,\tilde{\phi}+\frac{m^{2}}{2}\,{A_{\mu}}{A^{\mu}}+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi$
$+m\,A_{\mu}\,\partial^{\mu}\,\phi+{\bar{B}}\,(\partial\cdot
A-m\,\phi)+\frac{1}{2}\,\bar{B}^{2}-i\,\partial_{\mu}\,\bar{C}\,\partial^{\mu}\,C+i\,m^{2}\,\bar{C}\,C.$
$None$

We point out that we have invoked new type of Nakanishi-Lautrup type auxiliary
fields $({\cal{\bar{B}}},\bar{B})$ in the Lagrangian density (A.6) for the
linearization of the kinetic and gauge-fixing terms. In our present endeavor,
we have concentrated only on the Lagrangian density (A.5) with the Dirac
fields $(\bar{\psi},\psi)$ in interaction with the massive gauge field
$(A_{\mu})$ for the sake of brevity. However, one can also focus on the
Lagrangian density (A.6) for which the (anti-)BRST as well as the (anti-)co-
BRST symmetry transformations have been written in our earlier work [19] on
the modified 2D Proca theory without Dirac fields. It can be trivially checked
that (A.6) can also be generalized with Dirac fields $(\bar{\psi},\psi)$ in
interaction with massive gauge field $(A_{\mu})$ as we have done in Eq. (9).

Appendix B: On the Physical Analogue of $\delta=\pm*\,d\,*$

As discussed in Sec. 5, it is a well-known fact that the (co-)exterior
derivatives $(\delta)\,d$ of the differential geometry are connected to each-
other by the precise relationship: $\delta=\pm\,*\,d\,*$ where ($\pm$) signs
are dictated by the dimensionality of the compact manifold without a boundary
and the degree of the form that is invoked in the inner product [8-12]. The
central purpose of our present Appendix is to provide the physical realization
of ${\it this}$ relationship in the terminology of the discrete and continuous
symmetry transformations of our present 2D theory. In this context, we take a
few explicit examples to demonstrate that all the key ingredients of the
algebraic relationship $\delta=\pm\,*\,d\,*$ can be physically realized. We
note that, for the Dirac fields $(\bar{\psi},\psi)$ and the gauge field
$A_{\mu}$, we have the following [34]

$*\,(*\,\psi)=+\psi,\qquad*\,(*\,\bar{\psi})=+\bar{\psi},\qquad*\,(*\,A_{\mu})=-\,A_{\mu},$
$None$

where $*$ stands for the discrete symmetry transformations (10). First of all,
we can verify that the following relationships are correct, namely;

$s_{d}\,\psi=+*\,s_{b}\,*\,\psi,\qquad
s_{d}\,\bar{\psi}=+*\,s_{b}\,*\,\bar{\psi},\qquad
s_{d}\,A_{\mu}=-\,*s_{b}\,*\,A_{\mu},$ $None$

where the signs on the r.h.s. of (B.2) have been dictated by the signs in
(B.1) as per the principles behind a duality invariant theory (see, e.g. [34]
for details). In Eq. (B.2), the symbol $*$ stands, once again, for the
discrete symmetry transformations (10). The nilpotent BRST symmetry
transformations $s_{b}$ are given in Eq. (2) and the co-BRST symmetry
transformations $s_{d}$ are quoted in Eq. (11). It is crystal clear that (B.2)
provides the physical realization of $\delta=\pm*\,d\,*$ where there is a
mapping: $\delta\leftrightarrow s_{d}$ and $d\leftrightarrow s_{b}$.
Furthermore, the Hodge duality $*$ operation of differential geometry is
realized in terms of the discrete symmetry transformations (10). We note that,
exactly like Eq. (B.2), we have the validity of the following relationships
between $s_{ab}$ and $s_{ad}$, namely;

$s_{ab}\,\psi=+*\,s_{ad}\,*\,\psi,\qquad
s_{ab}\,\bar{\psi}=+*\,s_{ad}\,*\,\bar{\psi},\qquad
s_{ab}\,A_{\mu}=-\,*s_{ad}\,*\,A_{\mu},$ $None$

which demonstrate that we have the mappings between the mathematical
quantities $(\delta)\,d$ and the physical symmetry transformations
$(s_{ab})\,s_{ad}$ as: $\delta\leftrightarrow s_{ab},\,d\leftrightarrow
s_{ad}$. In addition, the notation $*$ in (B.3) denotes, once again, the
discrete symmetry transformations (10).

We end this Appendix with the remarks that, it is the peculiarity of the
specific two $(1+1)$-dimensional (2D) theory that we have also the validity of
the inverse relationships corresponding to (B.2) and (B.3) as:

$s_{b}\,\psi=+*\,s_{d}\,*\,\psi,\qquad
s_{b}\,\bar{\psi}=+*\,s_{d}\,*\,\bar{\psi},\qquad
s_{b}\,A_{\mu}=-\,*s_{d}\,*\,A_{\mu},$
$s_{ad}\,\psi=+*\,s_{ab}\,*\,\psi,\qquad
s_{ad}\,\bar{\psi}=+*\,s_{ab}\,*\,\bar{\psi},\qquad
s_{ad}\,A_{\mu}=-\,*s_{ab}\,*\,A_{\mu}.$ $None$

Thus, it appears that we have also the sanctity of the mappings:
$d\leftrightarrow(s_{d},\,s_{ab})$ and
$\delta\leftrightarrow(s_{b},\,s_{ad})$. However, this is not true as is
evident from our discussions on the ghost number considerations (cf. Sec. 6)
in the total quantum Hilbert space in terms of conserved charges corresponding
to the continuous symmetry transformations generated by
$s_{b},\,s_{ab},\,s_{d},\,s_{ad},\,s_{w}$ and $s_{g}$. It turns out that the
ghost number increases by one when we apply the charges $(Q_{b},\,Q_{ad})$ on
a state with the ghost number $n$. On the contrary, the ghost number decreases
by one due to the applications of $(Q_{d},\,Q_{ab})$ on a state with the ghost
number $n$ in the total quantum Hilbert space of states. Hence, the following
identifications, namely;

$d\leftrightarrow(s_{b},s_{ad}),\qquad\qquad\delta\leftrightarrow(s_{d},s_{ab}),$
$None$

are true as the conserved charges corresponding to the continuous symmetry
transformations $(s_{b},s_{ad})$ and $(s_{d},s_{ab})$ capture the real and
sacrosanct properties of $d$ and $\delta$. Furthermore, we note that we have
not taken other fields: $\phi,\,\tilde{\phi},\,C,\,\bar{C},\,B,\,{\cal B}$ of
our theory in our present discussion. However, as pointed out earlier, these
fields will obey exactly the same kind of relationships as has been respected
by the $A_{\mu}$ field [cf. Eq. (33)].

Appendix C: On the Discrete Symmetries in Component Form

We demonstrate here the invariance of the Dirac fields in interaction with a
massive gauge field (i.e. $-e\,\bar{\psi}\,\gamma^{\mu}\,A_{\mu}\,\psi$) under
the discrete symmetry transformations (10) where we take into account the
component forms of the Dirac fields as well as the gauge field ($A_{\mu}$).
With the choices: $\gamma_{0}=\sigma_{1},\,\gamma^{0}=\gamma_{0}$, we note
that we have the following for our 2D theory:

$\bar{\psi}=\psi^{\dagger}\,\gamma^{0}=(\psi_{1}^{*},\,\psi_{2}^{*})\,\sigma_{1}\equiv(\psi_{2}^{*},\,\psi_{1}^{*}).$
$None$

It is elementary to verify that the kinetic term for the fermionic Dirac
Lagrangian density (i.e. $\bar{\psi}\,i\,\gamma^{\mu}\,\partial_{\mu}\,\psi$)
remains invariant under (10) because:
$\psi\rightarrow\psi,\,\bar{\psi}\rightarrow\bar{\psi}$. Written in the
component form, the interaction term
($-\,e\,\bar{\psi}\,\gamma^{\mu}\,A_{\mu}\,\psi$) in 2D is as follows:

$-\,e\,\bar{\psi}\,\gamma^{\mu}\,A_{\mu}\,\psi=-\,e\,\bar{\psi}\,\gamma^{0}\,A_{0}\,\psi-\,e\,\bar{\psi}\,\gamma^{1}\,A_{1}\,\psi$
$\equiv-\,e\,\psi_{2}^{*}\,A_{0}\,\psi_{2}-\,e\,\psi_{1}^{*}\,A_{0}\,\psi_{1}+\,e\,\psi_{2}^{*}\,A_{1}\,\psi_{2}-\,e\,\psi_{1}^{*}\,A_{1}\,\psi_{1}.$
$None$

In the above, we have used (C.1) and taken into account
$\gamma^{0}=\gamma_{0}=\sigma_{1},\,\gamma^{1}=-\,\gamma_{1}=-\,i\,\sigma_{2}$.
We have also considered the column vectors: $\psi=(\psi_{1},\psi_{2})^{T}$ and
$A_{\mu}=(A_{1},A_{2})^{T}$. It is straightforward to check that the latter
satisfies: $A_{\mu}\rightarrow\mp\,i\,\varepsilon_{\mu\nu}\,A^{\nu}$ correctly
in the matrix form, too. In other words, we obtain:
$A_{0}\rightarrow\pm\,i\,A_{1},\,A_{1}\rightarrow\pm\,i\,A_{0}$ from the
matrix forms of $\varepsilon_{\mu\nu}$ and $A_{\mu}$ where
$\varepsilon_{\mu\nu}$ is a $2\times 2$ matrix and $A_{\mu}$ is a $2\times 1$
matrix.

To check the invariance of the interaction term
($-\,e\,\bar{\psi}\,\gamma^{\mu}\,A_{\mu}\,\psi$) under (10), first of all, we
note that we have the following transformations under (10), namely;

$-\,e\,\bar{\psi}\,\gamma^{\mu}\,A_{\mu}\,\psi\quad\rightarrow\quad-(\pm\,i\,e\,\gamma_{5})\,\bar{\psi}\,\gamma^{\mu}\,(\mp\,i\,\varepsilon_{\mu\nu}\,A^{\nu})\,\psi$
$=e\,\bar{\psi}\,\gamma_{5}\gamma^{\mu}\,\varepsilon_{\mu\nu}\,A^{\nu}\,\psi.$
$None$

In the above, we have used: $\gamma_{5}\,\bar{\psi}=-\,\bar{\psi}\,\gamma_{5}$
(as $\gamma_{5}\,\gamma_{0}=-\gamma_{0}\,\gamma_{5}$). Using the
straightforward relationships:
$\gamma^{\mu}\,\varepsilon_{\mu\nu}=-\gamma_{5}\,\gamma_{\nu},\,\gamma_{5}^{2}=I$,
we observe that the explicit form of (C.3) is nothing but the l.h.s. of (C.2)
which proves the point. However, we take into account the explicit expression
(C.3) in the component form (with
$\gamma^{0}=\gamma_{0},\,\gamma^{1}=-\gamma_{1},\,A^{0}=A_{0},\,A^{1}=-A_{1}$)
as

$\gamma^{\mu}\,\varepsilon_{\mu\nu}\,A^{\nu}=\gamma^{0}\,\varepsilon_{01}\,A^{1}+\gamma^{1}\,\varepsilon_{10}\,A^{0}=-\gamma_{0}\,A_{1}+\gamma_{1}\,A_{0},$
$None$

to obtain the following form of the interaction term [cf. Eq. (C.3)]

$-e\,\bar{\psi}\,\gamma_{5}\,\gamma_{0}\,A_{1}\,\psi+e\,\bar{\psi}\,\gamma_{5}\,\gamma_{1}\,A_{0}\,\psi.$
$None$

Substitutions of the component forms:
$\bar{\psi}=(\psi_{2}^{*},\,\psi_{1}^{*}),\,\psi=(\psi_{1},\,\psi_{2})^{T},\,\gamma_{5}=-\sigma_{3},\,\gamma_{0}=\sigma_{1},\,\gamma_{1}=i\,\sigma_{2}$
leads to the derivation of the following from (C.5):

$-e\,\bar{\psi}\,\gamma_{5}\,\gamma_{0}\,A_{1}\,\psi=e\,\psi_{2}^{*}\,A_{1}\,\psi_{2}-e\,\psi_{1}^{*}\,A_{1}\,\psi_{1},$
$e\,\bar{\psi}\,\gamma_{5}\,\gamma_{1}\,A_{0}\,\psi=-e\,\psi_{2}^{*}\,A_{0}\,\psi_{2}-e\,\psi_{1}^{*}\,A_{0}\,\psi_{1}.\
$ $None$

The sum of the above two terms [cf. Eqs. (C.5), (C.6)] is nothing but the Eq.
(C.2). Thus, even in the component form, the interaction term remains
invariant under (10). There is a simpler way to verify the same thing. If we
substitute $\gamma_{5}=\gamma_{0}\,\gamma_{1}$ into (C.5) and use
$\gamma^{0}=\gamma_{0},\,\gamma^{1}=-\gamma_{1},\,\gamma_{0}^{2}=I,\,\gamma_{1}^{2}=-I$
and $\gamma_{0}\,\gamma_{1}=-\gamma_{1}\,\gamma_{0}$, we observe that Eq.
(C.5) reduces to:
$(-e\,\bar{\psi}\,\gamma^{0}\,A_{0}\,\psi-e\,\bar{\psi}\,\gamma^{1}\,A_{1}\,\psi)$
which was the starting point in Eq. (C.2).

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