Christ–Lee Model: (Anti-)Chiral Supervariable Approach to BRST Formalism

B. Chauhan(a), S. Kumar(a)

(a)Physics Department, Centre of Advance Studies, Institute of Science,
Banaras Hindu University, Varanasi - 221 005, (U.P.), India

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract: We derive the off-shell nilpotent of order two and absolutely anti-
commuting Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST and (anti-)co-BRST
symmetry transformations for the Christ–Lee (CL) model in one (0 +
1)-dimension (1D) of spacetime by exploiting the (anti-)chiral supervariable
approach (ACSA) to BRST formalism where the quantum symmetry [i.e. (anti-)BRST
along with (anti-)co-BRST] invariant quantities play a crucial role. We prove
the nilpotency and absolute anti-commutativity properties of the (anti-) BRST
along with (anti-)co-BRST conserved charges within the scope of ACSA to BRST
formalism where we take only one Grassmannian variable into account. We also
show the (anti-)BRST and (anti-)co-BRST invariances of the Lagrangian within
the scope of ACSA.

PACS numbers: 12.90.+B, 03.70.+k, 11.10Kk, 11.15.-q

Keywords: Christ–Lee model; (anti-)BRST and (anti-)co-BRST symmetry
transformations; (anti-)chiral supervariable approach; (anti-)BRST invariant
restrictions; (anti-)co-BRST invariant restrictions; nilpotency and absolute
anti-commutativity properties

##  1 Introduction

Gauge theories describe three (i.e. strong, weak, electromagnetic) out of four
fundamental interactions of nature which are characterized by first-class
constraints in the context of Dirac’s prescription for the classification
scheme of constraints [1, 2]. The existence of the first-class constraints, in
a given system, is the key signature of a gauge theory. Many interesting
theories, in the domain of physics, are expressed by the suitable Lagrangians
that are invariant under the gauge symmetry transformations. These symmetries
are generated by the first-class constraints in a given gauge theory. For the
covariant canonical quantization of the gauge theory, the Becchi-Rouet-Stora-
Tyutin (BRST) quantization procedure plays a decisive role where we replace
the infinitesimal local gauge parameter by ghost and anti-ghost fields [3-6].
Thus, in this formalism, we have two fermionic type global BRST $(s_{b})$ and
anti-BRST $(s_{ab})$ transformations at the quantum level (for a given local
gauge symmetry transformation at the classical level). These symmetry
transformations are endowed with two important properties: (i) nilpotency of
order two (i.e. $s_{b}^{2}=0,\;s_{ab}^{2}=0$), and (ii) absolute anti-
commutativity (i.e. $s_{b}\,s_{ab}+s_{ab}\,s_{b}=0$). The first property
signifies that theses quantum BRST and anti-BRST symmetry transformations are
fermionic in nature whereas second property shows that both symmetry
transformations are linearly independent of each other. Besides the
(anti-)BRST symmetry transformations, we have two more fermionic and linearly
independent symmetry transformations which are christened as the co-BRST
($s_{d}$) and anti-co-BRST ($s_{ad}$) symmetry transformations. The latter
fermionic type symmetry transformations are valid for any D-dimensional
$p$-form ($p=1,2,3,...$) gauge theories in $D=2p$ dimensions of spacetime. We
point out that there are some specific systems such as rigid rotor and
Christ–Lee (CL) model in one-dimension (1D) that respect the (anti-)co-BRST
transformations along with to the (anti-)BRST transformations [7-10].

The geometrical interpretation and the emergence of nilpotent (anti-)BRST
symmetry transformations have been shown within the ambit of Bonora-Tonin (BT)
superfield formalism [11-13] where the Grassmannian variables
$(\vartheta,\bar{\vartheta})$ and their corresponding derivatives
$(\partial_{\vartheta},\partial_{\bar{\vartheta}})$ (with properties
$\vartheta^{2}=\bar{\vartheta}^{2}=0,\;\partial_{\vartheta}^{2}=\partial_{\bar{\vartheta}}^{2}=0$
and
$\vartheta\,\bar{\vartheta}+\bar{\vartheta}\,\vartheta=0,\;\partial_{\vartheta}\,\partial_{\bar{\vartheta}}+\partial_{\bar{\vartheta}}\,\partial_{\vartheta}=0$)
play very important role. In BT-superfield approach, we see the connections
between the (anti-) BRST symmetry transformations [$s_{(a)b}$] (with
properties $s_{b}^{2}=s_{ab}^{2}=0$ and $s_{b}\,s_{ab}+s_{ab}\,s_{b}=0)$ and
Grassmannian translational generators
$(\partial_{\vartheta},\partial_{\bar{\vartheta}})$ because of the fact that
both have same algebraic structure. In this formalism, any D-dimensional
Minkowskian manifold is generalized onto the (D, 2)-dimensional supermanifold.
This suitably chosen supermanifold is denoted by the superspace coordinates
$(x^{\mu},\vartheta,\bar{\vartheta})$ where $x^{\mu}(\mu=0,1,2,...,D-1)$ are
the spacetime coordinates and $(\vartheta,\bar{\vartheta})$ are a pair of
Grassmannian variables.

The CL model is one of the simplest examples of gauge-invariant system which
is described by a singular Lagrangian [14]. Physically, the CL model
represents the motion of a point particle moving in a plane under the
influence of a central potential. The CL model has been studied at the
classical and quantum levels in different prospectives [14-20]. This model is
endowed with the first-class constraints in the Dirac’s terminology for the
classification scheme of constraints [1, 2]. This model is also quantized by
using the Faddeev-Jackiw quantization where all the primary and derived
constraints are treated on equal footing without any type of further
classification [20]. Within the framewok of BRST formalism, the CL model
respects six independent continuous symmetries (i.e. BRST, anti-BRST, co-BRST,
anti-co-BRST, ghost-scale and bosonic symmetries) (see, e.g. [9] for detail).
The BT-superfield formalism has been applied to obtain the absolutely anti-
commuting and off-shell nilpotent (anti-)BRST as well as (anti-)co-BRST
symmetry transformations where the technique of celebrated horizontality
condition (HC) and dual horizontality condition have been used [10],
respectively.

In our recent set of papers [21-25], we have used a newly proposed formalism
which has been called by us as the (anti-)chiral superfield/supervariable
approach (ACSA) to BRST formalism. In this approach, we take into account only
one Grassmannian variable in the expression for the superfield/supervariable.
Thus, the resulting superfield/supervariable turns into (anti-)chiral version
of the superfield/supervariable. In other words, in this formalism, any
D-dimensional Minkowskian manifold is generalized onto (D, 1)-dimensional
supersub-manifolds of the most general (D, 2)-dimensional supermanifold. The
proof of the absolute anti-commutativity property of Noether’s conserved
charges is obvious in the case of BT-superfield formalism where the full super
expansions of superfields/supervariables are taken into account. In the case
of ACSA, we have also been able to show the nilpotency and absolute anti-
commutativity properties of conserved charges despite the fact that we have
taken only one Grassmannian variable into account. In our present endeavor, we
derive the (anti-)BRST together with (anti-)co-BRST symmetry transformations
where some specific sets of (anti-)BRST and (anti-)co-BRST invariant
restrictions play very important role. We also show the absolute anti-
commutativity as well as nilpotency properties of (anti-)BRST and (anti-) co-
BRST conserved charges within the realm of ACSA to BRST formalism.

Against the background of the above paragraph, it has been found that the
nilpotency of the ${\cal N}=2$ super charges are true for any ${\cal N}=2$
supersymmetric (SUSY) quantum mechanical models (see, e.g. [26-28]) within the
ambit of (anti-)chiral supervariable approach to BRST formalism. However, the
application of ACSA, in the realm of ${\cal N}=2$ SUSY quantum mechanical
model, does not lead to the absolute anti-commutativity of the ${\cal N}=2$
super conserved charges. Rather, it has been found that the anti-commutator of
the above ${\cal N}=2$ SUSY conserved charges leads to the time translation of
the variable on which it acts. Thus, it is crystal clear that, ACSA to BRST
formalism does not lead to the derivation of absolute anti-commutativity of
the charges for all ${\cal N}=2$ SUSY theories.

The different sections of our present paper are arranged as follows. In Sec.
2, we discuss the (anti-)BRST and (anti-)co-BRST symmetry transformations for
the CL model and derive the conserved charges. Our Sec. 3 deals with the ACSA
to BRST formalism where we derive the (anti-)BRST symmetry transformations.
Sec. 4 is devoted to the derivation of (anti-)co-BRST symmetry transformations
by using the ACSA to BRST formalism where the super expansions of
(anti-)chiral supervariables are utilized in a fruitful manner. In Sec. 5, we
express the conserved (anti-)BRST and (anti-)co-BRST charges on the (1,
1)-dimensional supersub-manifolds [of the most general (1, 2)-dimensional
supermanifold] on which our theory is generalized and provide the proof of
nilpotency and absolute anti-commutativity properties of the (anti-)BRST along
with (anti-)co-BRST charges within the ambit of ACSA to BRST formalism. In
Sec. 6, we discuss the (anti-)BRST and (anti-)co-BRST invariances of the
Lagrangian within the scope of ACSA. Finally, we point out our key results and
discovery in Sec. 7 and mention a few future scopes for further investigation.

##  2 Preliminaries: Symmetries and their Corresponding Generator for the
Christ–Lee Model

The first-order and gauge-invariant Lagrangian of the Christ–Lee (CL)
model***The other equivalent second-order Lagrangian ($L_{s}$) [14] associated
with the Christ–Lee model is
$L_{s}=\frac{1}{2}\,\dot{r}^{2}+\frac{1}{2}\,\dot{r}^{2}\,(\dot{\varphi}-z)^{2}-V(r)$.
But we choose only the first-order Lagrangian $L_{f}$ in our present work
because it respects maximum symmetry transformations. in $(0+1)$-dimension
(1D) of spacetime in polar coordinates system is given by [14, 16, 19],

$\displaystyle
L_{f}=\dot{r}\,p_{r}+\dot{\varphi}\,p_{\varphi}-\frac{1}{2}\,p^{2}_{r}-\frac{1}{2r^{2}}\,p^{2}_{\varphi}-z\,p_{\varphi}-V(r),$
(1)

where $\dot{r}$ and $\dot{\varphi}$ are the generalized velocities, $p_{r}$
and $p_{\varphi}$ are their corresponding canonical momenta, respectively and
$z$ is a Lagrange multiplier which enforces a constraint $p_{\varphi}\approx
0$. This Lagrangian explains that a two dimensional particle moves under the
influence of the central potential $V(r)$ bounded from below.

The above system has a primary constraint as follow

$\displaystyle\Phi_{1}=\frac{\partial L_{f}}{\partial{\dot{z}}}=p_{z}\approx
0.$ (2)

The time derivative of the primary constraint $\Phi_{1}$ leads to the
following secondary constraint

$\displaystyle\frac{d\Phi_{1}}{d\tau}=\frac{d}{d\tau}\Big{(}\frac{\partial
L_{f}}{\partial{\dot{z}}}\Big{)}\approx
0\Rightarrow\Phi_{2}=p_{\varphi}\approx 0.$ (3)

It is clear that both $\Phi_{1}$ and $\Phi_{2}$ are first-class constraints.
The gauge symmetry transformation generator can be written in terms of first-
class constraints as

$\displaystyle G={\dot{\chi}}(\tau)\,\Phi_{1}+\chi(\tau)\,\Phi_{2},$ (4)

where $\chi(\tau)$ is an infinitesimal and time dependent local gauge
parameter and ${\dot{\chi}}(\tau)={d\chi}/{d\tau}$. Using the definition of a
generator

$\displaystyle\delta\;\phi(\tau)=-i\,[\phi(\tau),\;G],\qquad\phi=r,p_{r},\varphi,p_{\varphi},z,$
(5)

where $\phi$ is the generic variable that is present in the first-order
Lagrangian $L_{f}$. We deduce the following local gauge transformations by
exploiting Eq. (5), namely;

$\displaystyle\delta\,z(\tau)=\dot{\chi}(\tau),\qquad\delta\,\varphi(\tau)=\chi(\tau),\qquad\delta[r(\tau),\,p_{r}(\tau),\,p_{\varphi}(\tau),\;V(r)]=0.$
(6)

It is elementary to check that, under the above local gauge symmetry
transformations, the Lagrangian under consideration remains invariant (i.e.
$\delta L_{f}=0$).

The (anti-)BRST invariant Lagrangian for the (0 + 1)-dimensional CL model
containing the gauge-fixing and Faddeev-Popov ghost terms is given by [15]

$\displaystyle
L=\dot{r}\,p_{r}+\dot{\varphi}\,p_{\varphi}-\frac{1}{2}\,p^{2}_{r}-\frac{1}{2r^{2}}\,p^{2}_{\varphi}-z\,p_{\varphi}-V(r)+\frac{1}{2}\,{\cal
B}^{2}+{\cal B}\,(\dot{z}+\varphi)+i\,\bar{C}\,C-i\,\dot{\bar{C}}\,\dot{C},$
(7)

where the Nakanishi–Lautrup type auxiliary variable ${\cal B}$ is used to
linearize the gauge-fixing term and the Faddeev–Popov (anti-)ghost variables
$(\bar{C})C$ are used to make the Lagrangian BRST invariant. These fermionic
variables $(\bar{C})C$ (with $C^{2}=\bar{C}^{2}=0,C\bar{C}+\bar{C}C=0$) have
ghost numbers $(-1)+1$, respectively. The above Lagrangian respects the
following off-shell nilpotent [i.e. $s_{(a)b}^{2}=s_{(a)d}^{2}=0$] and
absolutely anti-commuting (i.e. $s_{b}\,s_{ab}+s_{ab}\,s_{b}=0$ and
$s_{d}\,s_{ad}+s_{ad}\,s_{d}=0$) (anti-)BRST along with (anti-)co-BRST
symmetry transformations:

$\displaystyle
s_{ab}z=\dot{\bar{C}},\quad\;s_{ab}\varphi=\bar{C},\quad\;s_{ab}\,C=-\,i\,{\cal
B},\quad s_{ab}[r,\,p_{r},\,p_{\varphi},{\cal B},\,\bar{C}]=0,$ $\displaystyle
s_{b}z=\dot{C},\;\;\quad s_{b}\varphi=C,\;\quad\;s_{b}\bar{C}=i\,{\cal
B},\qquad\;\;s_{b}[r,\,p_{r},\,p_{\varphi},\,{\cal B},\,C]=0,$ (8)
$\displaystyle~{}~{}~{}~{}s_{ad}\,z=C,\quad s_{ad}\,\varphi=-\dot{C},\quad
s_{ad}\,\bar{C}=-i\,p_{\varphi},\quad s_{ad}\,[r,\,p_{r},\,p_{\varphi},\,{\cal
B},\,C]=0,$ $\displaystyle~{}~{}~{}~{}s_{d}\,z=\bar{C},\;\quad
s_{d}\,\varphi=-\dot{\bar{C}},\;\;\quad s_{d}\,C=i\,p_{\varphi},\qquad
s_{d}\,[r,\,p_{r},\,p_{\varphi},\,{\cal B},\,\bar{C}]=0.$ (9)

It can be clearly checked that under the above (anti-)BRST [Eq. (8)] and
(anti-)co-BRST [Eq. (9)] symmetry transformations the Lagrangian [Eq. (7)]
remains quasi-invariant (i.e. modulo a total time derivative):

$\displaystyle s_{b}\,L=\frac{d}{d\tau}\big{(}{\cal
B}\,\dot{C}\big{)},\;\,\qquad\qquad s_{ab}\,L=\frac{d}{d\tau}\big{(}{\cal
B}\,\dot{\bar{C}}\big{)},$ $\displaystyle
s_{d}\,L=-\frac{d}{d\tau}\big{(}p_{\varphi}\,\dot{\bar{C}}\big{)},\quad\qquad
s_{ad}\,L=-\frac{d}{d\tau}\big{(}p_{\varphi}\,\dot{C}\big{)}.$ (10)

As a result, the action integral $S=\int d\tau\,L$ remains invariant under the
(anti-)BRST as well as (anti-)co-BRST symmetry transformations [i.e.
$s_{(a)b}\,S=0,\;s_{(a)d}\,S=0$]. According to Noether’s theorem, the
invariance of the above Lagrangian under the nilpotent (anti-)BRST together
with (anti-)co-BRST symmetry transformations leads to the following
(anti-)BRST charges $[{\cal Q}_{(a)b}]$ and (anti-)co-BRST charges $[{\cal
Q}_{(a)d}]$, namely;

$\displaystyle{\cal Q}_{ab}=\;{\cal
B}\,\dot{\bar{C}}+p_{\varphi}\,\bar{C}\,\;\equiv\,\;{\cal
B}\,\dot{\bar{C}}-\dot{\cal B}\,{\bar{C}},$ $\displaystyle{\cal
Q}_{b}\;\,=\;{\cal B}\,\dot{C}+p_{\varphi}\,C\,\;\equiv\,\;{\cal
B}\,\dot{C}-\dot{\cal B}\,C,$ (11) $\displaystyle{\cal Q}_{ad}=\;{\cal
B}\,C-p_{\varphi}\,\dot{C}\,\;\equiv\,\;{\cal B}\,C+\dot{\cal B}\,\dot{C},$
$\displaystyle{\cal Q}_{d}\;\,=\;{\cal
B}\,\bar{C}-p_{\varphi}\,\dot{\bar{C}}\,\;\equiv\,\;{\cal
B}\,\bar{C}+\dot{\cal B}\,\dot{\bar{C}},$ (12)

where the equivalent forms of the above charges are written with the help of
the equation of motion: $p_{\varphi}=-\,\dot{\cal B}$. The above charges are
nilpotent of order two [i.e. ${\cal Q}_{(a)b}^{2}={\cal Q}_{(a)d}^{2}=0$] and
anti-commuting in nature (i.e. ${\cal Q}_{b}\,{\cal Q}_{ab}+{\cal
Q}_{ab}\,{\cal Q}_{b}=0$ and ${\cal Q}_{d}\,{\cal Q}_{ad}+{\cal Q}_{ad}\,{\cal
Q}_{d}=0$). The conservation law for these charges [i.e. $\frac{d}{d\tau}{\cal
Q}_{(a)b}=0$ and $\frac{d}{d\tau}{\cal Q}_{(a)d}=0$] can be easily proven by
using the following interesting Euler-Lagrange equations of motion†††Besides
these EOMs, we use a equation $\ddot{\cal B}+{\cal B}=0$ derived from the EOMs
(13) to prove the conservation law for the (anti-)BRST together with
(anti-)co-BRST conserved charges [Eqs. (11), (12)]. (EOMs) derived from
Lagrangian $L$ of our theory [Eq. (7)], namely;

$\displaystyle\dot{\cal B}+p_{\varphi}=0,\quad{\cal
B}=\dot{p}_{\varphi},\quad{\cal
B}=-(\dot{z}+\varphi),\quad\dot{p}_{r}-\frac{p^{2}_{\varphi}}{r^{3}}+V^{\prime}(r)=0,$
$\displaystyle\dot{r}=p_{r},\quad\dot{\varphi}-z-\frac{p_{\varphi}}{r^{2}}=0,\quad\bar{C}+\ddot{\bar{C}}=0,\qquad
C+\ddot{C}=0.$ (13)

The (anti-)co-BRST and (anti-)BRST conserved charges are the generators of the
(anti-) co-BRST and (anti-)BRST symmetry transformations, respectively. As one
can easily check that following relationships are true

$\displaystyle s_{d}\,\xi=-i\,\big{[}\xi,\;{\cal Q}_{d}\big{]}_{\pm},\qquad
s_{ad}\,\xi=-i\,\big{[}\xi,\;{\cal Q}_{ad}\big{]}_{\pm},$ $\displaystyle
s_{b}\,\xi=-i\,\big{[}\xi,\;{\cal Q}_{b}\big{]}_{\pm},\qquad
s_{ab}\,\xi=-i\,\big{[}\xi,\;{\cal Q}_{ab}\big{]}_{\pm},$ (14)

where $\xi$ denotes any generic variable present in the Lagrangian $L$ of our
theory. The subscript $(\pm)$ on the square brackets denotes the
(anti)commutator which depend on the nature of generic variables $\xi$ being
(fermionic)bosonic in nature.

##  3 Nilpotent Quantum (Anti-)BRST Symmetry Transformations: (Anti-)Chiral
Supervariable Approach

In this section, we determine the nilpotent (anti-)BRST symmetry
transformations [cf. Eq. (8)] by using (anti-)chiral supervariable approach
(ACSA) to BRST formalism where we shall use the (anti-)chiral super expansions
of supervariables. Towards this goal, first of all, we generalize the ordinary
variables of the Lagrangian (7) onto (1, 1)-dimensional anti-chiral supersub-
manifold [of the most common (1, 2)-dimensional supermanifold] as follows,

$\displaystyle z(\tau)\,\longrightarrow\;{\cal
Z}(\tau,\bar{\vartheta})\;=\;z(\tau)+\bar{\vartheta}\,f_{1}(\tau),$
$\displaystyle\varphi(\tau)\,\longrightarrow\;\Theta(\tau,\bar{\vartheta})\;=\;\varphi(\tau)+\bar{\vartheta}\,f_{2}(\tau),$
$\displaystyle
C(\tau)\longrightarrow\;F(\tau,\bar{\vartheta})\;=\;C(\tau)+i\,\bar{\vartheta}\,b_{1}(\tau),$
$\displaystyle\bar{C}(\tau)\longrightarrow\;\bar{F}(\tau,\bar{\vartheta})\;=\;\bar{C}(\tau)+i\,\bar{\vartheta}\,b_{2}(\tau),$
$\displaystyle
r(\tau)\;\longrightarrow\;R(\tau,\bar{\vartheta})\;=\;r(\tau)+\bar{\vartheta}\,f_{3}(\tau),$
$\displaystyle
p_{r}(\tau)\longrightarrow\;P_{r}(\tau,\bar{\vartheta})=\;p_{r}(\tau)+\bar{\vartheta}\,f_{4}(\tau),$
$\displaystyle
p_{\varphi}(\tau)\longrightarrow\;P_{\varphi}(\tau,\bar{\vartheta})=\;p_{\varphi}(\tau)+\bar{\vartheta}\,f_{5}(\tau),$
$\displaystyle{\cal B}(\tau)\;\,\longrightarrow\;{\tilde{\cal
B}}(\tau,\bar{\vartheta})\,=\;{\cal B}(\tau)+\bar{\vartheta}\,f_{6}(\tau),$
(15)

where $b_{1},b_{2}$ are the bosonic derived variables and
$f_{1},f_{2},f_{3},f_{4},f_{5},f_{6}$ are the fermionic derived variables due
to fermionic nature of $\bar{\vartheta}$. We determine the precise value of
these derived variables in terms of the auxiliary and basic variables present
in the BRST invariant Lagrangian (7) by using the BRST invariant
quantities/restrictions.

According to the basic principles of ACSA, the BRST invariant quantities must
remain independent of the Grassmannian variable ($\bar{\vartheta}$) when they
are generalized onto the (1, 1)-dimensional anti-chiral supersub-manifold. The
BRST invariant quantities are the specific combinations of the variables
present in Lagrangian (7). These are given as follows

$\displaystyle s_{b}(r,p_{r},p_{\varphi},{\cal B},C)=0,\quad
s_{b}(z\,\dot{C})=0,\quad s_{b}(\varphi\,C)=0,$ $\displaystyle s_{b}(\dot{\cal
B}\,z+i\,\dot{\bar{C}}\,\dot{C})=0,\quad s_{b}(\dot{\varphi}-z)=0,\quad
s_{b}({\cal B}\,\varphi+i\,\bar{C}\,C)=0.$ (16)

We generalize the above BRST invariant restrictions onto the (1,
1)-dimensional anti-chiral supersub-manifolds (of the suitably chosen most
common (1, 2)-dimensional supermanifold)

$\displaystyle
R(\tau,\bar{\vartheta})=r(\tau),\;P_{r}(\tau,\bar{\vartheta})=p_{r}(\tau),\;P_{\varphi}(\tau,\bar{\vartheta})=p_{\varphi}(\tau),\;{\tilde{\cal
B}}(\tau,\bar{\vartheta})={\cal B}(\tau),\;F(\tau,\bar{\vartheta})=C(\tau),$
$\displaystyle{\cal
Z}(\tau,\bar{\vartheta})\,\dot{F}(\tau,\bar{\vartheta})=z(\tau)\,\dot{C}(\tau),\quad\Theta(\tau,\bar{\vartheta})\,F(\tau,\bar{\vartheta})=\varphi(\tau)\,C(\tau),$
$\displaystyle{\dot{\tilde{\cal B}}}(\tau,\bar{\vartheta})\,{\cal
Z}(\tau,\bar{\vartheta})+i\,\dot{\bar{F}}(\tau,\bar{\vartheta})\,\dot{C}(\tau,\bar{\vartheta})=\dot{\cal
B}(\tau)\,z(\tau)+i\,\dot{\bar{C}}(\tau)\,\dot{C}(\tau),$
$\displaystyle\dot{\Theta}(\tau,\bar{\vartheta})-{\cal
Z}(\tau,\bar{\vartheta})=\dot{\varphi}(\tau)-z(\tau),$
$\displaystyle{\tilde{\cal
B}}(\tau,\bar{\vartheta})\,\Theta(\tau,\bar{\vartheta})+i\,{\bar{F}}(\tau,\bar{\vartheta})\,F(\tau,\bar{\vartheta})={\cal
B}(\tau)\,\varphi(\tau)+i\,{\bar{C}}(\tau)\,C(\tau).$ (17)

The above restrictions lead to the derivation of the derived variables in
terms of the basic and auxiliary variables. To determine the value of these
variables, we perform the step-by-step explicit calculations. For this
purpose, first of all, we use the generalization of the trivial BRST invariant
restrictions given in the first line of Eq. (17) as:

$\displaystyle
P_{\varphi}(\tau,\bar{\vartheta})=p_{\varphi}(\tau)\Longrightarrow
f_{5}=0,\qquad\;\,{\tilde{\cal B}}(\tau,\bar{\vartheta})={\cal
B}(\tau)\Longrightarrow f_{6}=0,$ $\displaystyle
R(\tau,\bar{\vartheta})\;=r(\tau)\;\;\Longrightarrow
f_{3}=0,\qquad\;\,F(\tau,\bar{\vartheta})=C(\tau)\Longrightarrow b_{1}=0,$
$\displaystyle P_{r}(\tau,\bar{\vartheta})=p_{r}(\tau)\;\Longrightarrow
f_{4}=0.$ (18)

After substituting the above value of derived variables from (18) to (15), we
get the following expressions for the anti-chiral supervariables, namely;

$\displaystyle
C(\tau)\longrightarrow\;F^{(b)}(\tau,\bar{\vartheta})=C(\tau)+\bar{\vartheta}\,(0)\;\equiv
C(\tau)+\bar{\vartheta}\,[s_{b}\,C(\tau)],$ $\displaystyle
r(\tau)\;\longrightarrow\;R^{(b)}(\tau,\bar{\vartheta})=r(\tau)+\bar{\vartheta}\,(0)\;\;\equiv
r(\tau)+\bar{\vartheta}\,[s_{b}\,r(\tau)],$ $\displaystyle
p_{r}(\tau)\longrightarrow\;P_{r}^{(b)}(\tau,\bar{\vartheta})=p_{r}(\tau)+\bar{\vartheta}\,(0)\equiv
p_{r}(\tau)+\bar{\vartheta}\,[s_{b}\,p_{r}(\tau)],$ $\displaystyle
p_{\varphi}(\tau)\longrightarrow\;P_{\varphi}^{(b)}(\tau,\bar{\vartheta})=p_{\varphi}(\tau)+\bar{\vartheta}\,(0)\equiv
p_{\varphi}(\tau)+\bar{\vartheta}\,[s_{b}\,p_{\varphi}(\tau)],$
$\displaystyle{\cal B}(\tau)\;\longrightarrow\;{\tilde{\cal
B}}^{(b)}(\tau,\bar{\vartheta})={\cal
B}(\tau)+\bar{\vartheta}\,(0)\;\equiv{\cal
B}(\tau)+\bar{\vartheta}\,[s_{b}\,{\cal B}(\tau)],$ (19)

where the superscript $(b)$ on the anti-chiral supervariables denotes that
these supervariables have been obtained after the use of BRST invariant
quantities. It is clear that the coefficients of Grassmannian variable
$\bar{\vartheta}$ are simply the quantum BRST symmetries (8). Now, in the case
of non-trivial BRST invariant restrictions: $s_{b}(z\,\dot{C})=0$ and
$s_{b}(\varphi\,C)=0$, the following generalizations onto (1, 1)-dimensional
supersub-manifold, namely;

$\displaystyle{\cal
Z}(\tau,\bar{\vartheta})\,\dot{F}^{(b)}(\tau,\bar{\vartheta})=z(\tau)\,\dot{C}(\tau),\qquad\Theta(\tau,\bar{\vartheta})\,F^{(b)}(\tau,\bar{\vartheta})=\varphi(\tau)\,C(\tau),$
(20)

lead to the following interesting results

$\displaystyle f_{1}(\tau)\,\dot{C}(\tau)=0\quad$
$\displaystyle\Longrightarrow$ $\displaystyle\quad
f_{1}(\tau)\propto\dot{C}(\tau),$ $\displaystyle\Longrightarrow$
$\displaystyle\quad f_{1}(\tau)=\kappa_{1}\,\dot{C}(\tau),$ $\displaystyle
f_{2}(\tau)\,C(\tau)=0\quad$ $\displaystyle\Longrightarrow$
$\displaystyle\quad f_{2}(\tau)\propto C(\tau),$ (21)
$\displaystyle\Longrightarrow$ $\displaystyle\quad
f_{2}(\tau)=\kappa_{2}\,C(\tau),$

where $\kappa_{1}$ and $\kappa_{2}$ are the proportionality constants. To
determine the value of these constants, we use the generalization of BRST
invariant restriction $s_{b}(\dot{\varphi}-z)=0$ as

$\displaystyle\dot{\Theta}(\tau,\bar{\vartheta})-{\cal
Z}(\tau,\bar{\vartheta})=\dot{\varphi}(\tau)-z(\tau)\quad\Longrightarrow\quad\kappa_{1}=\kappa_{2}.$
(22)

Finally, to determine the value of constants, we generalize the BRST invariant
restrictions $s_{b}(\dot{\cal B}\,z+i\,\dot{\bar{C}}\,\dot{C})=0$ and
$s_{b}({\cal B}\,\varphi+i\,\bar{C}\,C)=0$ onto $(1,1)$-dimensional supersub-
manifold as:

$\displaystyle{\dot{\tilde{\cal B}}}^{(b)}(\tau,\bar{\vartheta})\,{\cal
Z}(\tau,\bar{\vartheta})+i\,\dot{\bar{F}}(\tau,\bar{\vartheta})\,\dot{F}^{(b)}(\tau,\bar{\vartheta})=\dot{\cal
B}(\tau)\,z(\tau)+i\,\dot{\bar{C}}(\tau)\,\dot{C}(\tau)\;\;\Longrightarrow\;\dot{b}_{2}(\tau)=\kappa_{1}\dot{\cal
B}(\tau),$ $\displaystyle{\tilde{\cal
B}}^{(b)}(\tau,\bar{\vartheta})\,\Theta(\tau,\bar{\vartheta})+i\,{\bar{F}}(\tau,\bar{\vartheta})\,F^{(b)}(\tau,\bar{\vartheta})={\cal
B}(\tau)\,\varphi(\tau)+i\,{\bar{C}}(\tau)\,C(\tau)\;\Longrightarrow{b}_{2}(\tau)=\kappa_{2}\,{\cal
B}(\tau).$ (23)

Using the results obtained in Eq. (22) and Eq. (23), it is clear that
$\kappa_{1}=\kappa_{2}=1$. Therefore, we obtain the value of derived variables
as: $f_{1}(\tau)=\dot{C}(\tau),\;f_{2}(\tau)=C(\tau),\;b_{2}(\tau)={\cal
B}(\tau)$, thus, we get the following expansions for anti-chiral
supervariables:

$\displaystyle z(\tau)\;\longrightarrow{\cal
Z}^{(b)}(\tau,\bar{\vartheta})=z(\tau)+\bar{\vartheta}\,[\dot{C}(\tau)]\;\;\,\;\equiv\;z(\tau)+\bar{\vartheta}\,[s_{b}\,z(\tau)],$
$\displaystyle\varphi(\tau)\;\longrightarrow\Theta^{(b)}(\tau,\bar{\vartheta})=\varphi(\tau)+\bar{\vartheta}\,[C(\tau)]\;\;\,\equiv\;\varphi(\tau)+\bar{\vartheta}\,[s_{b}\,\varphi(\tau)],$
$\displaystyle\bar{C}(\tau)\longrightarrow\bar{F}^{(b)}(\tau,\bar{\vartheta})=\bar{C}(\tau)+\bar{\vartheta}\,[i\,B(\tau)]\;\equiv\;\bar{C}(\tau)+\bar{\vartheta}\,[s_{b}\,\bar{C}(\tau)].$
(24)

Thus, in view of the above Eq. (24), we have a connection between the BRST
symmetry transformation $(s_{b})$ and partial derivative
$(\partial_{\bar{\vartheta}})$ on the anti-chiral supersub-manifold defined by
the mapping: $s_{b}\longleftrightarrow\partial_{\bar{\vartheta}}$ (see, e.g.
[11-13] for details). To be more clear, the BRST transformation of any generic
variable $\psi(\tau)$ is equal to the translation of the corresponding anti-
chiral supervariable $\Psi^{(b)}(\tau,\bar{\vartheta})$ along the
$\bar{\vartheta}$-direction. Mathematically, it can be represented as
$s_{b}\psi(\tau)=\frac{\partial}{\partial{\bar{\vartheta}}}\Psi^{(b)}(\tau,\bar{\vartheta})=\partial_{\bar{\vartheta}}\Psi^{(b)}(\tau,\bar{\vartheta})$.
In other words, one can say, the coefficient of $\bar{\vartheta}$ in the
expansion of an anti-chiral supervariable is simply the quantum BRST symmetry
transformation of the corresponding variable.

We are now in the stage to derive the quantum anti-BRST symmetry
transformations using chiral supervariable approach. In this context, we use
the chiral super expansions of the chiral supervariables where we generalize
(0 + 1)-dimensional variables onto the (1, 1)-dimensional supersub-manifold of
the suitably chosen most common (1, 2)-dimensional supermanifold. The chiral
super expansions of the ordinary variables are as follows

$\displaystyle z(\tau)\;\longrightarrow\;{\cal
Z}(\tau,\vartheta)=z(\tau)+\vartheta\,\bar{f}_{1}(\tau),$
$\displaystyle\varphi(\tau)\;\longrightarrow\;\Theta(\tau,\vartheta)=\varphi(\tau)+\vartheta\,\bar{f}_{2}(\tau),$
$\displaystyle
C(\tau)\longrightarrow\;F(\tau,\vartheta)=C(\tau)+i\,\vartheta\,\bar{b}_{1}(\tau),$
$\displaystyle\bar{C}(\tau)\longrightarrow\;\bar{F}(\tau,\vartheta)=\bar{C}(\tau)+i\,\vartheta\,\bar{b}_{2}(\tau),$
$\displaystyle
r(\tau)\;\longrightarrow\;R(\tau,\vartheta)=r(\tau)+\vartheta\,\bar{f}_{3}(\tau),$
$\displaystyle
p_{r}(\tau)\longrightarrow\;P_{r}(\tau,\vartheta)=p_{r}(\tau)+\vartheta\,\bar{f}_{4}(\tau),$
$\displaystyle
p_{\varphi}(\tau)\longrightarrow\;P_{\varphi}(\tau,\vartheta)=p_{\varphi}(\tau)+\vartheta\,\bar{f}_{5}(\tau),$
$\displaystyle{\cal B}(\tau)\;\longrightarrow\;{\tilde{\cal
B}}(\tau,\vartheta)={\cal B}(\tau)+\vartheta\,\bar{f}_{6}(\tau),$ (25)

where derived variables $\bar{b}_{1},\bar{b}_{2}$ are the bosonic and derived
variables
$\bar{f}_{1},\bar{f}_{2},\bar{f}_{3},\bar{f}_{4},\bar{f}_{5},\bar{f}_{6}$ are
fermionic in nature. The anti-BRST invariant restrictions also must remain
independent of the Grassmannian variable ($\vartheta$) when they are
generalized onto the (1, 1)-dimensional chiral supersub-manifold. The anti-
BRST invariant restrictions are given as

$\displaystyle s_{ab}(r,p_{r},p_{\varphi},{\cal B},\bar{C})=0,\qquad
s_{ab}(z\,\dot{\bar{C}})=0,\qquad s_{ab}(\varphi\,\bar{C})=0,$ $\displaystyle
s_{ab}(\dot{\cal B}\,z+i\,\dot{\bar{C}}\,\dot{C})=0,\quad
s_{ab}(\dot{\varphi}-z)=0,\quad s_{ab}({\cal B}\,\varphi+i\,\bar{C}\,C)=0.$
(26)

As the physical quantities remain independent of the Grassmannian variable
$\vartheta$ which imply that the anti-BRST invariant restrictions can be
generalized onto the (1, 1)-dimensional supersub-manifold of the most common
(1, 2)-dimensional supermanifold as follows

$\displaystyle
R(\tau,\vartheta)=r(\tau),\;P_{r}(\tau,\vartheta)=p_{r}(\tau),\;P_{\varphi}(\tau,\vartheta)=p_{\varphi}(\tau),\;{\tilde{\cal
B}}(\tau,\vartheta)={\cal B}(\tau),\;\bar{F}(\tau,\vartheta)=\bar{C}(\tau),$
$\displaystyle{\cal
Z}(\tau,\vartheta)\,\dot{\bar{F}}(\tau,\vartheta)=z(\tau)\,\dot{\bar{C}}(\tau),\quad\Theta(\tau,\vartheta)\,\bar{F}(\tau,\vartheta)=\varphi(\tau)\,\bar{C}(\tau),$
$\displaystyle{\dot{\tilde{\cal B}}}(\tau,\vartheta)\,{\cal
Z}(\tau,\vartheta)+i\,\dot{\bar{F}}(\tau,\vartheta)\,\dot{F}(\tau,\vartheta)=\dot{\cal
B}(\tau)\,z(\tau)+i\,\dot{\bar{C}}(\tau)\,\dot{C}(\tau),$
$\displaystyle\dot{\Theta}(\tau,\vartheta)-{\cal
Z}(\tau,\vartheta)=\dot{\varphi}(\tau)-z(\tau),$ $\displaystyle{\tilde{\cal
B}}(\tau,\vartheta)\,\Theta(\tau,\vartheta)+i\,{\bar{F}}(\tau,\vartheta)\,F(\tau,\vartheta)={\cal
B}(\tau)\,\varphi(\tau)+i\,{\bar{C}}(\tau)\,C(\tau).$ (27)

The above generalizations of the anti-BRST invariant restrictions [Eq. (26)]
lead to the derivation of the chiral derived variables in terms of the
auxiliary and basic variables present in the Lagrangian $L$, namely;

$\displaystyle\bar{b}_{2}=0,\;\;\bar{f}_{3}=0,\;\;\bar{f}_{4}=0,\;\;\bar{f}_{5}=0,\;\;\bar{f}_{6}=0,\;\;\bar{f}_{1}=\dot{\bar{C}},\;\;\bar{f}_{2}=\bar{C},\;\;\bar{b}_{1}=-\,{\cal
B}.$ (28)

The above values of chiral derived variables are derived in a similar fashion
as the anti-chiral derived variables are derived. After the substitution of
the above derived variables into the chiral super expansions (25), we get the
following expressions for the chiral supervariables onto (1, 1)-dimensional
supersub-manifold as

$\displaystyle z(\tau)\;\,\longrightarrow\;{\cal
Z}^{(ab)}(\tau,\vartheta)=z(\tau)+\vartheta\,(\dot{\bar{C}})\,\;\quad\equiv\;\;z(\tau)+\vartheta\,[s_{ab}\,z(\tau)],$
$\displaystyle\varphi(\tau)\;\,\longrightarrow\;\Theta^{(ab)}(\tau,\vartheta)=\varphi(\tau)+\vartheta\,(\bar{C})\,\quad\equiv\;\;\varphi(\tau)+\vartheta\,[s_{ab}\varphi(\tau)],$
$\displaystyle
C(\tau)\longrightarrow\;F^{(ab)}(\tau,\vartheta)=C(\tau)+\vartheta\,(-i{\cal
B})\;\equiv\;\;C(\tau)+\vartheta\,[s_{ab}\,C(\tau)],$
$\displaystyle\bar{C}(\tau)\longrightarrow\;\bar{F}^{(ab)}(\tau,\vartheta)=\bar{C}(\tau)+\vartheta\,(0)\;\quad\;\equiv\;\;\bar{C}(\tau)+\vartheta\,[s_{ab}\bar{C}(\tau)],$
$\displaystyle
r(\tau)\;\,\longrightarrow\;R^{(ab)}(\tau,\vartheta)=r(\tau)+\vartheta\,(0)\;\,\quad\;\equiv\;\;r(\tau)+\vartheta\,[s_{ab}\,r(\tau)],$
$\displaystyle
p_{r}(\tau)\longrightarrow\;P_{r}^{(ab)}(\tau,\vartheta)=p_{r}(\tau)+\vartheta\,(0)\;\quad\equiv\;\;p_{r}(\tau)+\vartheta\,[s_{ab}\,p_{r}(\tau)],$
$\displaystyle
p_{\varphi}(\tau)\longrightarrow\;P_{\varphi}^{(ab)}(\tau,\vartheta)=p_{\varphi}(\tau)+\vartheta\,(0)\,\quad\equiv\;\;p_{\varphi}(\tau)+\vartheta\,[s_{ab}\,p_{\varphi}(\tau)],$
$\displaystyle{\cal B}(\tau)\;\longrightarrow\;{\tilde{\cal
B}}^{(ab)}(\tau,\vartheta)={\cal
B}(\tau)+\vartheta\,(0)\;\;\quad\equiv\;\;{\cal
B}(\tau)+\vartheta\,[s_{ab}\,{\cal B}(\tau)].$ (29)

Here, the coefficients of $\vartheta$ are simply the anti-BRST symmetry
transformations (see, e.g. [11-13] for detail). In fact, the anti-BRST
transformation of any generic variable $\psi(\tau)$ is simply the translation
of the corresponding chiral supervariable $\Psi^{(ab)}(\tau,\vartheta)$ along
the $\vartheta$-direction. Mathematically, this statement can be corroborated
as
$s_{ab}\psi(\tau)=\frac{\partial}{\partial\vartheta}\Psi^{(ab)}(\tau,\vartheta)=\partial_{\vartheta}\Psi^{(ab)}(\tau,\vartheta)$.
Thus, it is clear that there is a mapping between the quantum anti-BRST
symmetry transformation $(s_{ab})$ and the Grassmannian partial derivative
$(\partial_{\vartheta})$ defined on the chiral supersub-manifold with the
mapping: $s_{ab}\longleftrightarrow\partial_{\vartheta}$.

##  4 Nilpotent (Anti-)co-BRST Symmetry Transformations: (Anti-)Chiral
Supervariable Approach

In this section, we derive the nilpotent (anti-)co-BRST symmetry
transformations using the (anti-)chiral supervariable approach (ACSA) where we
use the expansions of the (anti-)chiral supervariables and the (anti-)co-BRST
invariant restrictions. Toward this goal in our mind, first of all, we
determine the co-BRST symmetries by exploiting the chiral super expansions
given in Eq. (25) and the co-BRST invariant restrictions. The co-BRST
invariant restrictions are given as:

$\displaystyle s_{d}(r,p_{r},p_{\varphi},{\cal
B},\bar{C})=0,\quad\;s_{d}(z\,{\bar{C}})=0,\quad\;s_{d}(\varphi\,\dot{\bar{C}})=0,$
$\displaystyle
s_{d}(\varphi\,\dot{p}_{\varphi}+i\,\dot{\bar{C}}\,\dot{C})=0,\quad
s_{d}(z\,p_{\varphi}-i\,\bar{C}\,C)=0,\quad s_{d}(\varphi+\dot{z})=0.$ (30)

According to the basic rules of ACSA, the above co-BRST invariant restrictions
can be generalized onto the (1, 1)-dimensional supersub-manifold [of the
suitably chosen most common (1, 2)-dimensional supermanifold] as:

$\displaystyle
R(\tau,\vartheta)=r(\tau),\;P_{r}(\tau,\vartheta)=p_{r}(\tau),\;P_{\varphi}(\tau,\vartheta)=p_{\varphi}(\tau),\;{\tilde{\cal
B}}(\tau,\vartheta)={\cal B}(\tau),\;\bar{F}(\tau,\vartheta)=\bar{C}(\tau),$
$\displaystyle{\cal
Z}(\tau,\vartheta)\,{\bar{F}}(\tau,\vartheta)=z(\tau)\,{\bar{C}}(\tau),\quad\Theta(\tau,\vartheta)\,\dot{\bar{F}}(\tau,\vartheta)=\varphi(\tau)\,\dot{\bar{C}}(\tau),$
$\displaystyle\Theta(\tau,\vartheta)\,\dot{P}_{\varphi}(\tau,\vartheta)\,+i\,\dot{\bar{F}}(\tau,\vartheta)\,\dot{F}(\tau,\vartheta)=\varphi(\tau)\,\dot{p}_{\varphi}(\tau)+i\,\dot{\bar{C}}(\tau)\,\dot{C}(\tau),$
$\displaystyle{\cal
Z}(\tau,\vartheta)\,P_{\varphi}(\tau,\vartheta)-i\,{\bar{F}}(\tau,\vartheta)\,F(\tau,\vartheta)=z(\tau)\,p_{\varphi}(\tau)-i\,{\bar{C}}(\tau)\,C(\tau),$
$\displaystyle\Theta(\tau,\vartheta)+\dot{\cal
Z}(\tau,\vartheta)=\varphi(\tau)+\dot{z}(\tau).$ (31)

At this stage, we determine the value of derived variables of Eq. (25) using
the above generalizations of the co-BRST invariant restrictions. To derive the
value of the derived variables, first of all, we use the first line entry of
Eq. (31) where the trivial co-BRST invariant quantities are generalized which
implies the following relationships:

$\displaystyle
P_{\varphi}(\tau,\vartheta)=p_{\varphi}(\tau)\Longrightarrow\;\bar{f}_{5}=0,\qquad\,\;\;\;{\tilde{\cal
B}}(\tau,\vartheta)={\cal B}(\tau)\Longrightarrow\bar{f}_{6}=0,$
$\displaystyle
R(\tau,\vartheta)\;=r(\tau)\;\Longrightarrow\;\bar{f}_{3}=0,\qquad\quad\bar{F}(\tau,\vartheta)=\bar{C}(\tau)\Longrightarrow\bar{b}_{2}=0,$
$\displaystyle
P_{r}(\tau,\vartheta)=p_{r}(\tau)\Longrightarrow\;\bar{f}_{4}=0.$ (32)

After substituting the above value of derived variables into the expressions
of the chiral super expansions [Eq. (25)], we obtain the following chiral
super expansions:

$\displaystyle\bar{C}(\tau)\,\longrightarrow\;\bar{F}^{(d)}(\tau,\vartheta)=\bar{C}(\tau)+\vartheta\,(0)\;\equiv\;\bar{C}(\tau)+\vartheta\,[s_{d}\,\bar{C}(\tau)],$
$\displaystyle
r(\tau)\;\,\longrightarrow\;R^{(d)}(\tau,\vartheta)=r(\tau)+\vartheta\,(0)\;\;\equiv\;r(\tau)+\vartheta\,[s_{d}\,r(\tau)],$
$\displaystyle
p_{r}(\tau)\longrightarrow\;P_{r}^{(d)}(\tau,\vartheta)=p_{r}(\tau)+\vartheta\,(0)\;\equiv\;p_{r}(\tau)+\vartheta\,[s_{d}\,p_{r}(\tau)],$
$\displaystyle
p_{\varphi}(\tau)\longrightarrow\;P_{\varphi}^{(d)}(\tau,\vartheta)=p_{\varphi}(\tau)+\vartheta\,(0)\equiv\;p_{\varphi}(\tau)+\vartheta\,[s_{d}\,p_{\varphi}(\tau)],$
$\displaystyle{\cal B}(\tau)\;\longrightarrow\;{\tilde{\cal
B}}^{(d)}(\tau,\vartheta)={\cal B}(\tau)+\vartheta\,(0)\,\;\equiv\;{\cal
B}(\tau)+\vartheta\,[s_{d}\,{\cal B}(\tau)],$ (33)

where superscript $(d)$ on the chiral supervariables denotes the
supervariables obtained after the application of the co-BRST (i.e. dual-BRST)
invariant restrictions. For the non-trivial case, first of all, we generalize
the co-BRST invariant restriction $s_{d}(z\,{\bar{C}})=0$ and
$s_{d}(\varphi\,\dot{\bar{C}})=0$ onto (1, 1)-dimensional supersub-manifold as

$\displaystyle{\cal
Z}(\tau,\vartheta)\,{\bar{F}}^{(d)}(\tau,\vartheta)=z(\tau)\,{\bar{C}}(\tau),\qquad\Theta(\tau,\vartheta)\,\dot{\bar{F}}^{(d)}(\tau,\vartheta)=\varphi(\tau)\,\dot{\bar{C}}(\tau),$
(34)

which lead to the following relationships for the derived variables

$\displaystyle\bar{f}_{1}(\tau)\,\bar{C}(\tau)=0\quad$
$\displaystyle\Longrightarrow$
$\displaystyle\quad\bar{f}_{1}(\tau)\propto\bar{C}(\tau),$
$\displaystyle\Longrightarrow$
$\displaystyle\quad\bar{f}_{1}(\tau)=-\,\bar{\kappa}_{1}\bar{C}(\tau),$
$\displaystyle\bar{f}_{2}(\tau)\,\dot{\bar{C}}(\tau)=0\quad$
$\displaystyle\Longrightarrow$
$\displaystyle\quad\bar{f}_{2}(\tau)\propto\dot{\bar{C}}(\tau),$ (35)
$\displaystyle\Longrightarrow$
$\displaystyle\quad\bar{f}_{2}(\tau)=\bar{\kappa}_{2}\dot{\bar{C}}(\tau),$

where $\bar{\kappa}_{1}$ and $\bar{\kappa}_{2}$ are the proportionality
constants. To determine the value of these constants, we further use the
generalizations of the co-BRST invariant restrictions
$s_{d}(\varphi+\dot{z})=0,\;s_{d}(\varphi\,\dot{p}_{\varphi}+i\,\dot{\bar{C}}\,\dot{C})=0$
and $s_{d}(z\,p_{\varphi}-i\,\bar{C}\,C)=0$ as:

$\displaystyle\Theta(\tau,\vartheta)+\dot{\cal
Z}(\tau,\vartheta)=\varphi(\tau)+\dot{z}(\tau)\quad$
$\displaystyle\Longrightarrow$
$\displaystyle\bar{\kappa}_{1}=\bar{\kappa}_{2},$
$\displaystyle\Theta(\tau,\vartheta)\,\dot{P}_{\varphi}^{(d)}(\tau,\vartheta)+i\,\dot{\bar{F}}^{(d)}(\tau,\vartheta)\,\dot{F}(\tau,\vartheta)$
$\displaystyle=$
$\displaystyle\varphi(\tau)\,\dot{p}_{\varphi}(\tau)\,+i\,\dot{\bar{C}}(\tau)\,\dot{C}(\tau)$
$\displaystyle\Longrightarrow$
$\displaystyle\;\;\dot{\bar{b}}_{1}(\tau)=-\,\bar{\kappa}_{2}\,\dot{p}_{\varphi}(\tau),$
$\displaystyle{\cal
Z}(\tau,\vartheta)\,P_{\varphi}^{(d)}(\tau,\vartheta)-i\,{\bar{F}}^{(d)}(\tau,\vartheta)\,F(\tau,\vartheta)$
$\displaystyle=$ $\displaystyle
z(\tau)\,p_{\varphi}(\tau)\,-i\,{\bar{C}}(\tau)\,C(\tau)$ (36)
$\displaystyle\Longrightarrow$
$\displaystyle\bar{b}_{1}(\tau)=-\,\bar{\kappa}_{1}\,p_{\varphi}(\tau).$

The results of the above three relations in Eq. (36) imply that
proportionality constant are equal (i.e. $\bar{\kappa}_{1}=\bar{\kappa}_{2}$)
and their values are equal to minus one (i.e
$\bar{\kappa}_{1}=\;\bar{\kappa}_{2}=-\,1$). Therefore, we get the value of
the derived variables as:
$\bar{f}_{1}=\bar{C},\;\bar{f}_{2}=-\,\dot{\bar{C}},\;\bar{b}_{1}=p_{\varphi}$.
As a result, we have the following chiral super expansions of the ordinary
variables:

$\displaystyle z(\tau)\;\longrightarrow\;{\cal
Z}^{(d)}(\tau,\vartheta)=z(\tau)+\vartheta\,[\bar{C}(\tau)]\;\;\;\;\equiv
z(\tau)+\vartheta\,[s_{d}\,z(\tau)],$
$\displaystyle\varphi(\tau)\;\longrightarrow\;\Theta^{(d)}(\tau,\vartheta)=\varphi(\tau)+\vartheta\,[-\,\dot{\bar{C}}(\tau)]\equiv\varphi(\tau)+\vartheta\,[s_{d}\,\varphi(\tau)],$
$\displaystyle
C(\tau)\longrightarrow\;\bar{F}^{(d)}(\tau,\vartheta)=C(\tau)+\vartheta\,[i\,p_{\varphi}(\tau)]\;\equiv
C(\tau)+\vartheta\,[s_{d}\,C(\tau)].$ (37)

where superscript $(d)$ on the supervariables denotes the same meaning as in
Eq. (33). From the above equations, it is clear that the translation of any
generic chiral supervariable $\Psi^{(d)}(\tau,\vartheta)$ along the
$\vartheta$-direction generates co-BRST symmetry transformation ($s_{d}$) of
the corresponding variable $\psi(\tau)$. Mathematically, we can express this
statements as
$s_{d}\psi(\tau)=\partial_{\vartheta}\Psi^{(d)}(\tau,\vartheta)$. In other
words, the coefficients of the $\vartheta$ are simply the co-BRST symmetry
transformations. Thus, the co-BRST symmetry transformation $(s_{d})$ is
connected with the Grassmannian derivative $\partial_{\vartheta}$ (i.e.
$s_{d}\longleftrightarrow\partial_{\vartheta}$) [11-13].

Now for the derivation of anti-co-BRST symmetry transformations, we use the
anti-chiral super expansions of the supervariables [Eq. (15)] and anti-co-BRST
invariant restrictions. The anti-co-BRST invariant restrictions are listed as

$\displaystyle s_{ad}(r,p_{r},p_{\varphi},{\cal
B},C)=0,\qquad\;s_{ad}(z\,{C})=0,\qquad\;s_{ad}(\varphi\,\dot{C})=0,$
$\displaystyle s_{ad}(z\,p_{\varphi}-i\,\bar{C}\,C)=0,\quad
s_{ad}(\varphi\,\dot{p}_{\varphi}+i\,\dot{\bar{C}}\,\dot{C})=0,\quad
s_{ad}(\dot{z}+\varphi)=0.$ (38)

The generalization of these anti-co-BRST invariant restrictions onto the (1,
1)-dimensional supersub-manifold (of the most common (1, 2)-dimensional
supermanifold) are as follows:

$\displaystyle
R(\tau,\bar{\vartheta})=r(\tau),\;P_{r}(\tau,\bar{\vartheta})=p_{r}(\tau),\;P_{\varphi}(\tau,\bar{\vartheta})=p_{\varphi}(\tau),\;{\tilde{\cal
B}}(\tau,\bar{\vartheta})={\cal B}(\tau),\;F(\tau,\bar{\vartheta})=C(\tau),$
$\displaystyle{\cal
Z}(\tau,\bar{\vartheta})\,F(\tau,\bar{\vartheta})=z(\tau)\,C(\tau),\quad\Theta(\tau,\bar{\vartheta})\,\dot{F}(\tau,\bar{\vartheta})=\varphi(\tau)\,\dot{C}(\tau),$
$\displaystyle{\cal
Z}(\tau,\bar{\vartheta})\,P_{\varphi}(\tau,\bar{\vartheta})-i\,{\bar{F}}(\tau,\bar{\vartheta})\,F(\tau,\bar{\vartheta})=z(\tau)\,p_{\varphi}(\tau)-i\,{\bar{C}}(\tau)\,C(\tau),$
$\displaystyle\Theta(\tau,\bar{\vartheta})\,\dot{P}_{\varphi}(\tau,\bar{\vartheta})\,+i\,\dot{\bar{F}}(\tau,\bar{\vartheta})\,\dot{F}(\tau,\bar{\vartheta})=\varphi(\tau)\,p_{\varphi}(\tau)+i\,\dot{\bar{C}}(\tau)\,\dot{C}(\tau),$
$\displaystyle\dot{\cal
Z}(\tau,\bar{\vartheta})+\Theta(\tau,\bar{\vartheta})=\dot{z}(\tau)+\varphi(\tau).$
(39)

The above generalizations of the anti-co-BRST invariant restrictions, finally,
lead to the derivation of the derived variables as follows:

$\displaystyle
b_{1}=0,\;\;b_{2}=-\,p_{\varphi},\;\;f_{1}={C},\;\;f_{2}=-\,\dot{C},\;\;f_{3}=0,\;\;f_{4}=0,\;\;f_{5}=0,\;\;f_{6}=0.$
(40)

Thus, we have determined all the derived variables using the same technique as
we have used in the derivation of the co-BRST symmetry transformations.
Finally, after substituting the value of derived variables into Eq. (15), we
obtain the following expressions for the anti-chiral expansions of the
supervariables, namely;

$\displaystyle z(\tau)\;\longrightarrow\;{\cal
Z}^{(ad)}(\tau,\bar{\vartheta})=z(\tau)+\bar{\vartheta}\,(C)\quad\;\;\;\equiv\;\;z(\tau)+\bar{\vartheta}\,[s_{ad}\,z(\tau)],$
$\displaystyle\varphi(\tau)\;\longrightarrow\;\Theta^{(ad)}(\tau,\bar{\vartheta})=\varphi(\tau)+\bar{\vartheta}\,(-\,\dot{C})\;\;\equiv\;\;\varphi(\tau)+\bar{\vartheta}\,[s_{ad}\,\varphi(\tau)],$
$\displaystyle
C(\tau)\longrightarrow\;F^{(ad)}(\tau,\bar{\vartheta})=C(\tau)+\bar{\vartheta}\,(0)\qquad\equiv\;\;C(\tau)+\bar{\vartheta}\,[s_{ad}\,C(\tau)],$
$\displaystyle\bar{C}(\tau)\longrightarrow\;\bar{F}^{(ad)}(\tau,\bar{\vartheta})=\bar{C}(\tau)+\bar{\vartheta}\,(-\,i\,p_{\varphi})\equiv\;\;\bar{C}(\tau)+\bar{\vartheta}\,[s_{ad}\,\bar{C}(\tau)],$
$\displaystyle
r(\tau)\;\longrightarrow\;R^{(ad)}(\tau,\bar{\vartheta})=r(\tau)+\bar{\vartheta}\,(0)\qquad\;\equiv\;\;r(\tau)+\bar{\vartheta}\,[s_{ad}\,r(\tau)],$
$\displaystyle
p_{r}(\tau)\longrightarrow\;P_{r}^{(ad)}(\tau,\bar{\vartheta})=p_{r}(\tau)+\bar{\vartheta}\,(0)\;\;\,\quad\equiv\;\;p_{r}(\tau)+\bar{\vartheta}\,[s_{ad}\,p_{r}(\tau)],$
$\displaystyle
p_{\varphi}(\tau)\longrightarrow\;P_{\varphi}^{(ad)}(\tau,\bar{\vartheta})=p_{\varphi}(\tau)+\bar{\vartheta}\,(0)\quad\;\;\equiv\;\;p_{\varphi}(\tau)+\bar{\vartheta}\,[s_{ad}\,p_{\varphi}(\tau)],$
$\displaystyle{\cal B}(\tau)\,\longrightarrow\;{\tilde{\cal
B}}^{(ad)}(\tau,\bar{\vartheta})={\cal
B}(\tau)+\bar{\vartheta}\,(0)\qquad\equiv\;\;{\cal
B}(\tau)+\bar{\vartheta}\,[s_{ad}\,{\cal B}(\tau)],$ (41)

where superscript $(ad)$ on the anti-chiral supervariables denote the fact
that anti-chiral supervariables are obtained after the application of quantum
anti-co-BRST invariant restrictions [Eq. (38)]. Here, it is clear, the
coefficient of Grassmannian variable $\bar{\vartheta}$ is simply the quantum
anti-co-BRST symmetry transformation ($s_{ad}$). To be more clear, the anti-
co-BRST symmetry transformation $(s_{ad})$ of any generic variable
$\psi(\tau)$ is equal to the translation of the corresponding anti-chiral
supervariable $\Psi^{(ad)}(\tau,\bar{\vartheta})$ along the
$\bar{\vartheta}$-direction i.e.
$s_{ad}\psi(\tau)=\partial_{\bar{\vartheta}}\Psi^{(ad)}(\tau,\bar{\vartheta})$.
This implies that the anti-co-BRST symmetry ($s_{ad}$) is connected with the
Grassmannian translation generator ($\partial_{\bar{\vartheta}}$) as
$s_{ad}\longleftrightarrow\partial_{\bar{\vartheta}}$ [11-13].

##  5 Nilpotency and Absolute Anti-Commutativity of the Noether Conserved
Charges: ACSA

In this section, we deduce the nilpotency and absolute anti-commutativity
properties of the conserved (anti-)BRST along with (anti-)co-BRST charges in
the language of ACSA. For this, first of all, we show the nilpotency of the
(anti-)BRST together with (anti-)co-BRST conserved charges. It is
straightforward to express the expressions of the (anti-)BRST together with
(anti-)co-BRST charges in terms of the (anti-)chiral supervariables and
partial derivatives $(\partial_{\bar{\vartheta}},\partial_{\vartheta})$ with
an equivalent integral form as follows

$\displaystyle{\cal Q}_{b}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\bar{\vartheta}}\Big{[}i\,\dot{\bar{F}}^{(b)}(\tau,\bar{\vartheta})\,F^{(b)}(\tau,\bar{\vartheta})-i\,\bar{F}^{(b)}(\tau,\bar{\vartheta})\,\dot{F}^{(b)}(\tau,\bar{\vartheta})\Big{]}$
$\displaystyle\equiv$
$\displaystyle\int\,d\bar{\vartheta}\Big{[}i\,\dot{\bar{F}}^{(b)}(\tau,\bar{\vartheta})\,F^{(b)}(\tau,\bar{\vartheta})-i\,\bar{F}^{(b)}(\tau,\bar{\vartheta})\,\dot{F}^{(b)}(\tau,\bar{\vartheta})\Big{]},$
$\displaystyle{\cal Q}_{ab}$ $\displaystyle=$
$\displaystyle\,\frac{\partial}{\partial\vartheta}\,\Big{[}i\,\bar{F}^{(ab)}(\tau,\vartheta)\,\dot{F}^{(ab)}(\tau,\vartheta)-i\,\dot{\bar{F}}^{(ab)}(\tau,\vartheta)\,F^{(ab)}(\tau,\vartheta)\Big{]}$
(42) $\displaystyle\equiv$
$\displaystyle\,\int\,d\vartheta\,\Big{[}i\,\bar{F}^{(ab)}(\tau,\vartheta)\,\dot{F}^{(ab)}(\tau,\vartheta)-i\,\dot{\bar{F}}^{(ab)}(\tau,\vartheta)\,F^{(ab)}(\tau,\vartheta)\Big{]},$
$\displaystyle{\cal Q}_{d}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\vartheta}\,\Big{[}i\,\bar{F}^{(d)}(\tau,\vartheta)\,\dot{F}^{(d)}(\tau,\vartheta)-i\,\dot{\bar{F}}^{(d)}(\tau,\vartheta)\,F^{(d)}(\tau,\vartheta)\Big{]}$
$\displaystyle\equiv$
$\displaystyle\int\,d\vartheta\,\Big{[}i\,\bar{F}^{(d)}(\tau,\vartheta)\,\dot{F}^{(d)}(\tau,\vartheta)-i\,\dot{\bar{F}}^{(d)}(\tau,\vartheta)\,F^{(d)}(\tau,\vartheta)\Big{]},$
$\displaystyle{\cal Q}_{ad}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\bar{\vartheta}}\,\Big{[}i\,\dot{\bar{F}}^{(ad)}(\tau,\bar{\vartheta})\,F^{(ad)}(\tau,\bar{\vartheta})-i\,\bar{F}^{(ad)}(\tau,\bar{\vartheta})\,\dot{F}^{(ad)}(\tau,\bar{\vartheta})\Big{]}$
(43) $\displaystyle\equiv$
$\displaystyle\int\,d\bar{\vartheta}\,\Big{[}i\,\dot{\bar{F}}^{(ad)}(\tau,\bar{\vartheta})\,F^{(ad)}(\tau,\bar{\vartheta})-i\,\bar{F}^{(ad)}(\tau,\bar{\vartheta})\,\dot{F}^{(ad)}(\tau,\bar{\vartheta})\Big{]},$

where the superscripts $(b)$ and $(ab)$ stand for the anti-chiral and chiral
supervariables that have been obtained after the application of the BRST and
anti-BRST invariant restrictions, respectively. The superscripts $(d)$ and
$(ad)$ show the chiral and anti-chiral supervariables that is obtained after
the application of co-BRST and anti-co-BRST invariant restrictions,
respectively. It is clear that the nilpotency
($\partial_{\bar{\vartheta}}^{2}=0,\;\partial_{\vartheta}^{2}=0$) of the
translational generators $(\partial_{\bar{\vartheta}},\;\partial_{\vartheta})$
implies that

$\displaystyle\partial_{\bar{\vartheta}}\;{\cal
Q}_{b}=0\quad\;\,\Longleftrightarrow\quad s_{b}\;{\cal
Q}_{b}\quad\;=\;-\;i\;{\\{{\cal Q}_{b},{\cal Q}_{b}}\\}=0,$
$\displaystyle\partial_{\vartheta}\;{\cal
Q}_{ab}=0\quad\Longleftrightarrow\quad s_{ab}\;{\cal
Q}_{ab}\;\,\,=\;-\;i\;{\\{{\cal Q}_{ab},{\cal Q}_{ab}}\\}=0,$
$\displaystyle\partial_{\vartheta}\;{\cal
Q}_{d}=0\quad\;\Longleftrightarrow\quad s_{d}\;{\cal
Q}_{d}\;\quad=\;-\;i\;{\\{{\cal Q}_{d},{\cal Q}_{d}}\\}=0,$
$\displaystyle\partial_{\bar{\vartheta}}\;{\cal
Q}_{ad}=0\quad\Longleftrightarrow\quad s_{ad}\;{\cal
Q}_{ad}\;\;=\;-\;i\;{\\{{\cal Q}_{ad},{\cal Q}_{ad}}\\}=0,$ (44)

which show the nilpotency [${\cal Q}_{(a)b}^{2}={\cal Q}_{(a)d}^{2}=0$] of the
conserved charges within the ambit of ACSA to BRST formalism. Thus, we have
shown that there is a deep connection between the nilpotency
($\partial_{\bar{\vartheta}}^{2}=0,\;\partial_{\vartheta}^{2}=0$) of the
translational generator $(\partial_{\bar{\vartheta}},\;\partial_{\vartheta})$
and the nilpotency [i.e. ${\cal Q}_{(a)b}^{2}={\cal Q}_{(a)d}^{2}=0$] of the
(anti-)BRST and (anti-)co-BRST charges [${\cal Q}_{(a)b},\;{\cal Q}_{(a)d}$].
The above nilpotency property can be also captured in an ordinary space where
we use the (anti-)BRST exact as well as (anti-)co-BRST exact forms of the
charges, namely;

$\displaystyle{\cal Q}_{b}$ $\displaystyle=$
$\displaystyle-\,i\,s_{b}\,\big{(}\bar{C}\,\dot{C}-\dot{\bar{C}}\,C\big{)},\qquad{\cal
Q}_{ab}=+\,i\,s_{ab}\,\big{(}\bar{C}\,\dot{C}-\dot{\bar{C}}\,C\big{)},$
$\displaystyle{\cal Q}_{d}$ $\displaystyle=$ $\displaystyle
i\,s_{d}\,\big{(}\bar{C}\,\dot{C}-\dot{\bar{C}}\,C\big{)},\;\;\;\qquad{\cal
Q}_{ad}=-\,i\,s_{ad}\,\big{(}\bar{C}\,\dot{C}-\dot{\bar{C}}\,C\big{)},$ (45)

the above expressions show the nilpotency property of the (anti-)BRST along
with (anti-) co-BRST conserved charges, in a simpler way, in an ordinary space
[cf. (44)].

Now, we are in a stage to show the absolute anti-commutativity of the
(anti-)BRST along with (anti-)co-BRST charges. For this purpose, we write the
charges in terms of the (anti-) chiral supervariables and the derivatives
$(\partial_{\vartheta},\;\partial_{\bar{\vartheta}})$ of the Grassmannian
variables $(\bar{\vartheta},\vartheta)$

$\displaystyle{\cal Q}_{b}$ $\displaystyle=$
$\displaystyle-\,i\,\frac{\partial}{\partial\vartheta}\,\Big{[}\dot{F}^{(ab)}(\tau,\vartheta)\,F^{(ab)}(\tau,\vartheta)\Big{]}\,\equiv-\,i\,\int\,d\vartheta\,\Big{[}\dot{F}^{(ab)}(\tau,\vartheta)\,F^{(ab)}(\tau,\vartheta)\Big{]},$
$\displaystyle{\cal Q}_{ab}$ $\displaystyle=$
$\displaystyle\;\;\;i\,\frac{\partial}{\partial\bar{\vartheta}}\,\Big{[}\dot{\bar{F}}^{(b)}(\tau,\bar{\vartheta})\,\bar{F}^{(b)}(\tau,\bar{\vartheta})\Big{]}\quad\equiv\;\;\;i\,\int\,d\bar{\vartheta}\,\Big{[}\dot{\bar{F}}^{(b)}(\tau,\bar{\vartheta})\,\bar{F}^{(b)}(\tau,\bar{\vartheta})\Big{]},$
$\displaystyle{\cal Q}_{d}$ $\displaystyle=$
$\displaystyle\;\;\;i\,\frac{\partial}{\partial\bar{\vartheta}}\,\Big{[}\dot{\bar{F}}^{(ad)}(\tau,\bar{\vartheta})\,\bar{F}^{(ad)}(\tau,\bar{\vartheta})\Big{]}\,\equiv\;\;\;i\,\int
d\bar{\vartheta}\,\Big{[}\dot{\bar{F}}^{(ad)}(\tau,\bar{\vartheta})\,\bar{F}^{(ad)}(\tau,\bar{\vartheta})\Big{]},$
$\displaystyle{\cal Q}_{ad}$ $\displaystyle=$
$\displaystyle-\,i\,\frac{\partial}{\partial\vartheta}\,\Big{[}\dot{F}^{(d)}(\tau,\vartheta)\,F^{(d)}(\tau,\vartheta)\Big{]}\quad\equiv-\,i\,\int
d\vartheta\,\Big{[}\dot{F}^{(d)}(\tau,\vartheta)\,F^{(d)}(\tau,\vartheta)\Big{]},$
(46)

where the superscripts $(a)b$ and $(a)d$ denote the same meaning as explained
earlier. Here, it is straightforward to check that the nilpotency
($\partial_{\bar{\vartheta}}^{2}=0,\;\partial_{\vartheta}^{2}=0$) of the
translational generators $(\partial_{\bar{\vartheta}},\;\partial_{\vartheta})$
implies that the following relations

$\displaystyle\partial_{\vartheta}\;{\cal
Q}_{b}=0\;\quad\,\Longleftrightarrow\quad s_{ab}\;{\cal Q}_{b}=-\,i\;{\\{{\cal
Q}_{b},{\cal Q}_{ab}}\\}=0,$ $\displaystyle\partial_{\bar{\vartheta}}\;{\cal
Q}_{ab}=0\quad\Longleftrightarrow\quad s_{b}\;{\cal Q}_{ab}=-\,i\;{\\{{\cal
Q}_{ab},{\cal Q}_{b}}\\}=0,$ $\displaystyle\partial_{\bar{\vartheta}}\;{\cal
Q}_{d}=0\;\quad\Longleftrightarrow\quad s_{ad}\;{\cal Q}_{d}=-\,i\;{\\{{\cal
Q}_{d},{\cal Q}_{ad}}\\}=0,$ $\displaystyle\partial_{\vartheta}\;{\cal
Q}_{ad}=0\quad\Longleftrightarrow\quad s_{d}\;{\cal Q}_{ad}=-\,i\;{\\{{\cal
Q}_{ad},{\cal Q}_{d}}\\}=0,$ (47)

which show the absolute anti-commutativity property of the (anti-)BRST as well
as (anti-)co-BRST conserved charges. The property of absolute anti-
commutativity of conserved charges can also be captured explicitly in an
ordinary space by using the following (anti-)BRST exact and (anti-)co-BRST
exact forms of the charges, namely;

$\displaystyle{\cal Q}_{b}=-\,i\,s_{ab}\,\big{(}\dot{C}\,C\big{)},\qquad{\cal
Q}_{ab}=+\,i\,s_{b}\,\big{(}\dot{\bar{C}}\,\bar{C}\big{)},$
$\displaystyle{\cal
Q}_{d}=i\,s_{ad}\,\big{(}\dot{\bar{C}}\,\bar{C}\big{)},\;\;\;\qquad{\cal
Q}_{ad}=-\,i\,s_{d}\,\big{(}\dot{C}\,C\big{)}.$ (48)

##  6 Invariances of Lagrangian in ACSA

In this section, we discus the (anti-)BRST along with (anti-)co-BRST
invariances of the Lagrangian (7) within the scope of ACSA to BRST formalism.
For this purpose, foremost, we generalize the ordinary Lagrangian of (0 +
1)-dimensional onto the suitably chosen (1, 1)-dimensional (anti-)chiral
supersub-manifold of the most common (1, 2)-dimensional supermanifold. The
expressions of the (anti-)chiral super Lagrangian are

$\displaystyle L(\tau)\longrightarrow\tilde{L}^{(ac)}(\tau,\bar{\vartheta})$
$\displaystyle=$
$\displaystyle\dot{r}(\tau)\,p_{r}(\tau)+\dot{\Theta}^{(b)}(\tau,\bar{\vartheta})\,p_{\varphi}(\tau)-\frac{1}{2}\,p^{2}_{r}(\tau)-\frac{1}{2r^{2}}\,p^{2}_{\varphi}(\tau)$
$\displaystyle-$ $\displaystyle{\cal
Z}^{(b)}(\tau,\bar{\vartheta})\,p_{\varphi}(\tau)-V(r)+\frac{1}{2}\,{\cal
B}^{2}(\tau)+{\cal B}(\tau)\,\big{[}\dot{\cal Z}^{(b)}(\tau,\bar{\vartheta})$
$\displaystyle+$
$\displaystyle\Theta^{(b)}(\tau,\bar{\vartheta})\big{]}-i\,\dot{\bar{F}}^{(b)}(\tau,\bar{\vartheta})\,\dot{C}(\tau)+i\,\bar{F}^{(b)}(\tau,\bar{\vartheta})\,C(\tau),$
$\displaystyle L(\tau)\longrightarrow\tilde{L}^{(c)}(\tau,\vartheta)$
$\displaystyle=$
$\displaystyle\dot{r}(\tau)\,p_{r}(\tau)+\dot{\Theta}^{(ab)}(\tau,\vartheta)\,p_{\varphi}(\tau)-\frac{1}{2}\,p^{2}_{r}(\tau)-\frac{1}{2r^{2}}\,p^{2}_{\varphi}(\tau)$
(49) $\displaystyle-$ $\displaystyle{\cal
Z}^{(ab)}(\tau,\vartheta)\,p_{\varphi}(\tau)-V(r)+\frac{1}{2}\,{\cal
B}^{2}(\tau)+{\cal B}(\tau)\,\big{[}\dot{\cal Z}^{(ab)}(\tau,\vartheta)$
$\displaystyle+$
$\displaystyle\Theta^{(ab)}(\tau,\vartheta)\big{]}-i\,\dot{\bar{C}}(\tau)\,\dot{F}^{(ab)}(\tau,\vartheta)+i\,\bar{C}(\tau)\,F^{(ab)}(\tau,\vartheta),$

where the superscripts $(ac)$ and $(c)$ on the super Lagrangians denote the
anti-chiral and chiral super Lagrangians (containing anti-chiral and chiral
supervariables), respectively. It is evident that under the application of
translational generators
$(\partial_{\bar{\vartheta}},\;\partial_{\vartheta})$, we get the (anti-) BRST
invariance of Lagrangian ($L$) with the following results

$\displaystyle\frac{\partial}{\partial{\bar{\vartheta}}}\Big{[}\tilde{L}^{(ac)}(\tau,\bar{\vartheta})\Big{]}=\frac{d}{d\,\tau}[{\cal
B}(\tau)\,\dot{C}(\tau)],$
$\displaystyle\frac{\partial}{\partial{\vartheta}}\Big{[}\tilde{L}^{(c)}(\tau,\vartheta)\Big{]}\;=\frac{d}{d\,\tau}[{\cal
B}(\tau)\,\dot{\bar{C}}(\tau)],$ (50)

which imply that generalized version of super Lagrangians remain quasi-
invariant (i.e. up to a total time derivative) under the translational
generators $(\partial_{\bar{\vartheta}},\;\partial_{\vartheta})$ within the
scope of ACSA which are consistent with Eq. (10).

Now, we capture the (anti-)co-BRST invariance of the Lagrangian (7) within the
scope of (anti-)chiral supervariable approach. For this, we generalize the
ordinary Lagrangian into (anti-)co-BRST super Lagrangian where (0 +
1)-dimensional theory is generalized onto the (1, 1)-dimensional (anti-)chiral
supersub-manifold of the common (1, 2)-dimensional supermanifold as follows:

$\displaystyle L(\tau)\longrightarrow\tilde{L}^{(c,\,d)}(\tau,\vartheta)$
$\displaystyle=$
$\displaystyle\dot{r}(\tau)\,p_{r}(\tau)+\dot{\Theta}^{(d)}(\tau,\vartheta)\,p_{\varphi}(\tau)-\frac{1}{2}\,p^{2}_{r}(\tau)-\frac{1}{2r^{2}}\,p^{2}_{\varphi}(\tau)$
$\displaystyle-$ $\displaystyle{\cal
Z}^{(d)}(\tau,\vartheta)\,p_{\varphi}(\tau)-V(r)+\frac{1}{2}\,{\cal
B}^{2}(\tau)+{\cal B}(\tau)\,\big{[}\dot{\cal Z}^{(d)}(\tau,\vartheta)$
$\displaystyle+$
$\displaystyle\Theta^{(d)}(\tau,\vartheta)\big{]}-i\,\dot{\bar{C}}(\tau)\,\dot{F}^{(d)}(\tau,\vartheta)+i\,\bar{C}(\tau)\,F^{(d)}(\tau,\vartheta),$
$\displaystyle
L(\tau)\longrightarrow\tilde{L}^{(ac,\,ad)}(\tau,\bar{\vartheta})$
$\displaystyle=$
$\displaystyle\dot{r}(\tau)\,p_{r}(\tau)+\dot{\Theta}^{(ad)}(\tau,\bar{\vartheta})\,p_{\varphi}(\tau)-\frac{1}{2}\,p^{2}_{r}(\tau)-\frac{1}{2r^{2}}\,p^{2}_{\varphi}(\tau)$
(51) $\displaystyle-$ $\displaystyle{\cal
Z}^{(ad)}(\tau,\bar{\vartheta})\,p_{\varphi}(\tau)-V(r)+\frac{1}{2}\,{\cal
B}^{2}(\tau)+{\cal B}(\tau)\,\big{[}\dot{\cal Z}^{(ad)}(\tau,\bar{\vartheta})$
$\displaystyle+$
$\displaystyle\Theta^{(ad)}(\tau,\bar{\vartheta})\big{]}-i\,\dot{\bar{F}}^{(ad)}(\tau,\bar{\vartheta})\,\dot{C}(\tau)+i\,\bar{F}^{(ad)}(\tau,\bar{\vartheta})\,C(\tau),$

where the superscripts $(c,\,d)$ and $(ac,\,ad)$ denote that the super
Lagrangians (containing the chiral and anti-chiral supervariables) obtained
after the application of the co-BRST and anti-co-BRST invariant restrictions,
respectively. It is straightforward to check that

$\displaystyle\frac{\partial}{\partial{\vartheta}}\Big{[}\tilde{L}^{(c,d)}(\tau,\vartheta)\Big{]}\;\;\,=\;-\,\frac{d}{d\,\tau}\big{[}p_{\varphi}(\tau)\,\dot{\bar{C}}(\tau)\big{]},$
$\displaystyle\frac{\partial}{\partial{\bar{\vartheta}}}\Big{[}\tilde{L}^{(ac,ad)}(\tau,\bar{\vartheta})\Big{]}=\;-\,\frac{d}{d\,\tau}\big{[}p_{\varphi}(\tau)\,\dot{C}(\tau)\big{]},$
(52)

which show the (anti-)co-BRST invariance of the Lagrangian $L$ within the
ambit of ACSA to BRST formalism. At the end of this section, we have the
following concluding remarks. There are deep connections between the
(anti-)BRST symmetries $(s_{(a)b})$ and derivatives
$(\partial_{\bar{\vartheta}},\;\partial_{\vartheta})$ of the Grassmannian
variables $(\bar{\vartheta},\vartheta)$ with the following mappings:
$s_{b}\longleftrightarrow\partial_{\bar{\vartheta}}$ and
$s_{ab}\longleftrightarrow\partial_{\vartheta}$. Similarly, in the case of
(anti-)co-BRST symmetry transformations, it is clear that these symmetry
transformations are also connected with the derivatives
$(\partial_{\bar{\vartheta}},\;\partial_{\vartheta})$ of Grassmannian
variables with the mappings: $s_{d}\longleftrightarrow\partial_{\vartheta}$
and $s_{ad}\longleftrightarrow\partial_{\bar{\vartheta}}$ [cf. Secs. 4, 5].

##  7 Conclusions

In our present analysis, for the first time, we have derived the off-shell
nilpotent quantum (anti-)BRST along with (anti-)co-BRST symmetry
transformations within the scope of ACSA. We have also discussed the
nilpotency along with absolute anti-commutativity properties of the
corresponding (anti-)BRST along with (anti-)co-BRST conserved charges of the
ordinary (0 + 1)-dimensional gauge invariant Christ–Lee model within the ambit
of (anit-)chiral supervariable approach (ACSA) to BRST formalism.

The novel remarks of our present endeavor are the derivation of the off-shell
nilpotent (anti-)BRST along with (anti-)co-BRST symmetry transformations (cf.
Sec. 4) and the proof of nilpotency and the anti-commutativity properties of
the (anti-)BRST and (anti-)co-BRST charges in spite of the fact that we have
taken into account only the (anti-)chiral super expansions of the
supervariables (cf. Sec. 5). The nilpotency and anti-commutativity properties
of the above conserved charges and derivation of the corresponding (anti-)BRST
and (anti-)co-BRST symmetry transformations are obvious when the full super
expansions of the supervariables (i.e. BT-supervariable formalism [29-32]) is
taken into account. However, for the present study, we have shown these
properties with the help of only (anti-)chiral super expansions of the
(anti-)chiral supervariables.

It is worthwhile to mention that the nilpotency of the BRST as well as anti-
BRST conserved charges is connected with the nilpotency
($\partial_{\bar{\vartheta}}^{2}=\partial_{\vartheta}^{2}=0)$ of the
translational generators $\partial_{\bar{\vartheta}}$ and
$\partial_{\vartheta}$, respectively. On the other hand, nilpotency of the co-
BRST and anti-co-BRST charges is connected with nilpotency
($\partial_{\vartheta}^{2}=\partial_{\bar{\vartheta}}^{2}=0$) of the
translational generators $\partial_{\vartheta}$ and
$\partial_{\bar{\vartheta}}$, respectively. However, we have shown (cf. Sec.
5) that the absolute anti-commutativity of the BRST charge with anti-BRST
charge is connected with the nilpotency ($\partial_{\vartheta}^{2}=0$) of the
translational generator ($\partial_{\vartheta}$) and absolute anti-
commutativity of anti-BRST charge with BRST charge is connected with the
nilpotency ($\partial_{\bar{\vartheta}}^{2}=0$) of the translational generator
($\partial_{\bar{\vartheta}}$). On the contrary, the absolute anti-
commutativity of the co-BRST charge with anti-co-BRST charge is connected with
the nilpotency of the translational generator ($\partial_{\bar{\vartheta}}$)
and the absolute anti-commutativity of the anti-co-BRST charge with co-BRST
charge is deeply related with the nilpotency of the translational generator
($\partial_{\vartheta}$). These statements are completely novel for the
present model. We have also captured the (anti-)BRST along with (anti-)co-BRST
invariances of the Lagrangian within the scope of ACSA. In fact, the action
corresponding to the (anti-)chiral super Lagrangian is independent of
Grassmannian variables $(\vartheta,\bar{\vartheta})$ which is completely novel
for the present CL model (cf. Sec. 6).

The above issues, within the scope of ACSA to BRST approach, would be
discussed in our future investigations for the various gauge-invariant
models/theories like ABJM theory [33-35], supersymmetric Chern-Simons theory
[36], Jackiw-Pi model, Freedman-Townsend model, Abelian gauge theory with
higher derivative matter fields. In fact, our standard techniques of ACSA to
BRST formalism are applicable wherever gauge invariance is present in the
theory. Furthermore, there is a interesting and important work [37] that would
be discussed in the future for different prospects of the theoretical and
physical point of view in the domain of theoretical high energy physics.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors of this study declare that there is no conflicts of interest.

Acknowledgments: B. Chauhan and S. Kumar are thankful to the DST-INSPIRE and
BHU fellowships for financial support, respectively. The authors also thank
Dr. R. Kumar for a careful reading of the manuscript and for important as well
as significant suggestions.

arXiv identifier

arXiv: 2102.03845 [hep-th]

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