Superfield Approaches to a Model of
Bosonic String: Curci-Ferrari Type Restrictions

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: Exploiting the theoretical potential of the modified Bonora-Tonin
superfield approach (MBTSA) as well as the (anti-)chiral superfield approach
(ACSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the complete
set of off-shell nilpotent (anti-) BRST symmetry transformations corresponding
to the classical two $(1+1)$-dimensional (2D) diffeomorphism symmetry
transformations on the world-sheet (that is traced out by the motion of a
model of bosonic string). Only the BRST symmetry transformations for this
model have been discussed in the earlier literature. We derive the (anti-)BRST
invariant Curci-Ferrari (CF) type restrictions (using MBTSA) which turn out to
be the root-cause behind the absolute anticommutativity of the above
(anti-)BRST symmetry transformations. We capture the symmetry invariance of
the (anti-)BRST invariant Lagrangian densities within the ambit of ACSA. The
derivation of the proper anti-BRST transformations (corresponding to the
already known BRST transformations) and the (anti-)BRST invariant CF-type
restrictions are the novel results in our present endeavor.

PACS numbers: 04.60.Cf.; 11.25.Sq; 11.30.-j

Keywords: A model of bosonic string; 2D diffeomorphism symmetry; modified
Bonora-Tonin superfield approach; (anti-)chiral superfield approach;
(anti-)BRST symmetries; CF-type restrictions; off-shell nilpotency; absolute
anticommutativity

##  1 Introduction

Superfield approaches (see, e.g. [1-12]) to Becchi-Rouet-Stora-Tyutin (BRST)
formalism are geometrically elegant, mathematically rich and physically very
intuitive as they provide the geometrical basis for the off-shell nilpotency
and absolute anticommutativity of the quantum (anti-)BRST symmetry
transformations that are associated with a given classical local gauge
symmetry transformation for a classically gauge invariant theory. In the above
usual superfield approaches [1-12], only the $p$-form ($p=1,2,3...$) gauge
theories have been considered which are characterized by the existence of the
first-class constraints on them in the terminology of Dirac’s prescription for
the classification scheme of constraints (see, e.g. [13, 14]). It has been a
challenging problem to incorporate the diffeomorphism invariant theories in
the domain of the superfield approaches to BRST formalism. An attempt has been
made by Delbourgo, etal., (see, e.g. [15]) in this direction where a
diffeomorphism invariant gravitational theory has been considered. However, in
our present endeavor, we shall not discuss anything connected with the
superfield approach developed in [15] for the BRST-analysis of our present
two-dimensional (2D) diffeomorphism invariant theory.

A very successful application of the superfield approach [4-6] to BRST
formalism (in the context of D-dimensional non-Abelian 1-form gauge theory)
has been performed by Bonora and Tonin (BT). We have exploited the theoretical
techniques and tricks of this approach in the context of BRST-analysis of the
higher $p$-form ($p=2,3$) Abelian gauge theories [16, 17]. It has been a very
exciting problem to incorporate the diffeomorphism symmetry transformations
within the framework of BT-superfield formalism. A breakthrough, in this
direction, has been made by Bonora in a very recent paper [18] where the
D-dimensional diffeomorphism invariant theory has been discussed within the
ambit of BT-superfield approach [4-6]. We have christened this theoretical
technique as the modified version of BT-superfield approach (MBTSA) to BRST
formalism [18] and applied its theoretical potential in the context of 1D
diffeomorphism (i.e. reparameterization) invariant models of (i) a free non-
SUSY (NSUSY) non-relativistic particle [19], (ii) a free scalar NSUSY
relativistic particle [20], and (iii) a free SUSY (i.e. spinning) relativistic
particle [21] and established that the Curci-Ferrari (CF) type of restriction
as well as the gauge-fixing and Faddeev-Popov ghost terms are the same for all
these 1D diffeomorphism (i.e. reparametrization) invariant models. In
particular, it has been gratifying to show the universal nature of the CF-type
restriction in all the above 1D diffeomorphism (i.e. reparameterization)
invariant NSUSY as well as SUSY relativistic free particles and NSUSY and non-
relativistic free particle.

In the applications of MBTSA [18], it turns out that we have to take into
account the full super expansions of the superfields defined on the (D,
2)-dimensional supermanifold. In other words, we perform the super expansion
of the above superfields along all the possible Grassmannian directions of the
(D, 2)-dimensional supermanifold on which a D-dimensional ordinary
diffeomorphism invariant theory is generalized. The idea of horizontality
condition (HC) enables us to derive the (anti-)BRST symmetry transformations
for the scalars, vectors, tensors, etc. However, we invoke the Nakanishi-
Lautrup type auxiliary fields $(\bar{B}_{\mu})B_{\mu}$ (with
$\mu=0,\,1,\,2...D-1$) in the standard nilpotent (anti-)BRST symmetry
transformations:
$s_{b}\,\bar{C}_{\mu}=i\,B_{\mu},\,s_{b}\,B_{\mu}=0,\,s_{ab}\,C_{\mu}=i\,\bar{B}_{\mu},\,s_{ab}\,\bar{B}_{\mu}=0$
of the (anti-)ghost fields $(\bar{C}_{\mu})C\mu$ in the case of the
D-dimensional diffeomorphism invariant theory. This forces us to consider the
(anti-)chiral super expansions of the superfields [cf. Eq. (27) below]. At
this juncture, it becomes essential for us to take into account the
(anti-)chiral superfield approach (ACSA) to BRST formalism (see e.g. [22-26])
which has been developed by us.

The fundamental ideas behind the ACSA have emerged from its application to the
description of a set of ${\cal N}=2$ SUSY quantum mechanical models (see, e.g.
[27-31]) where the nilpotency property of the two fermionic symmetries exists
but the absolute anticommutativity property (of these symmetries) does not.
The presence of the two conserved fermionic charges and two nilpotent
symmetries of the ${\cal N}=2$ SUSY models are similar to the nilpotent
(anti-)BRST symmetries and corresponding charges of the BRST-quantized gauge
theories. However, there is a distinct difference at the level of absolute
anticommutativity property. Whereas the BRST and anti-BRST symmetries
absolutely anticommute, the fermionic ${\cal N}=2$ SUSY symmetries do not. If
we take the full super expansions of the superfields [1-12,18], the absolute
anticommutativity is guaranteed. Hence, we have taken the (anti-)chiral super
expansions of the superfields so that we could avoid the absolute
anticommutativity property (but retain the nilpotency property). This is why
we have adopted ACSA in the context of ${\cal N}=2$ SUSY quantum mechanical
models. In our recent publications [22-26], however, we have applied ACSA in
the context of BRST invariant theories and shown that, despite the
(anti-)chiral super expansions within the framework of ACSA, the conserved
BRST and anti-BRST charges absolutely anticommute (unlike the nilpotent ${\cal
N}=2$ SUSY charges which do not). We have supplemented MBTSA with the
theoretical strength of ACSA in our present endeavor so that we can compute
all the (anti-)BRST transformations for all the fields of our diffeomorphism
invariant theory.

The central theme of our present investigation is to apply the ideas of MBTSA
and ACSA to BRST formalism in the realm of a 2D diffeomorphism invariant
theory of a model of bosonic string and derive (i) all the (anti-)BRST
symmetries of this theory in a consistent and clear fashion, and (ii) the CF-
type restrictions which are responsible for the absolute anticommutativity of
the (anti-)BRST symmetry transformations. We have also derived the BRST and
anti-BRST invariant Lagrangian densities and captured their symmetry
invariance(s) in the language of ACSA to BRST formalism. We would like to lay
emphasis on the fact that the theoretical potential of MBTSA has been
responsible for the derivation of (i) the (anti-)BRST symmetry transformations
for the pure Lorentz scalars, and (ii) the (anti-)BRST invariant CF-type
restrictions. However, we have been able to derive all the proper (anti-)BRST
transformations for all the other fields by using ACSA.

The following motivating factors have been at the heart of our present
investigation. First, we have already used the beautiful blend of theoretical
ideas behind MBTSA and ACSA in the cases of some 1D diffeomorphism (i.e.
reparameterization) invariant theories of SUSY (i.e. spinning) relativistic
particle, NSUSY (i.e. scalar) relativistic particles and NSUSY and non-
relativistic system of a free particle for the discussion of BRST analysis.
However, these models are also endowed with the gauge symmetry transformations
which are a kind of subset of the reparameterization symmetry transformations
(under specific limits). Thus, it is a challenging problem for us to use the
theoretical strength of MBTSA and ACSA in the context of a 2D diffeomorphism
invariant theory which does not respect a gauge symmetry transformation. We
have discussed, in our present endeavor, a model of bosonic string which has
the 2D diffeomorphism symmetry invariance but it does not respect a gauge
symmetry transformation. Second, one of the sacrosanct aspects of BRST
formalism is the existence of the quantum BRST and anti-BRST symmetries
together for a given classical gauge/diffeomorphism symmetry transformation.
For our present bosonic string, only the BRST symmetries are known in
literature [32]. Thus, it is a challenge for us to derive the proper anti-BRST
symmetry transformations corresponding to the above BRST symmetry
transformations. We have accomplished this goal in our present endeavor.
Finally, the hallmark of a BRST-quantized theory is the existence of the CF-
type restrictions [33,34] which provide the independent identity to the BRST
and anti-BRST symmetries at the quantum level. We have derived these
restrictions, too.

The theoretical contents of our present endeavor are organized as follows. In
Sec. 2, we concisely discuss the (anti-)BRST symmetry transformations for the
gauge-fixed Lagrangian densities of the bosonic string theory. Our Sec. 3 is
devoted to the derivation of the Curci-Ferrari (CF) type restrictions for our
BRST invariant theory within the framework of MBTSA. In addition, we also
derive the (anti-)BRST symmetry transformations for the target space
coordinates and the determinant of the modified version of the inverse of the
2D metric tensor. Sec. 4 contains the derivation of the nilpotent (anti-)BRST
symmetries for the other fields of our theory by exploiting the theoretical
potential of ACSA. We capture the (anti-)BRST invariances of the Lagrangian
densities using ACSA in Sec. 5. Finally, we make some concluding remarks in
our Sec. 6.

##  2 Preliminary: (Anti-)BRST Symmetries

We begin with the following (anti-)BRST invariant Lagrangian densities [${\cal
L}_{(a)b}$] for the model of the bosonic string of our theory (see, e.g. [35]
for details)

$\displaystyle{\cal L}_{ab}$ $\displaystyle=$ $\displaystyle{\cal
L}_{0}-\bar{B}_{1}A_{1}-{\bar{B}}_{0}A_{0}+i\,\Bigl{[}C_{1}\,(\partial_{0}\,\bar{C}^{1}+\partial_{1}\,\bar{C}^{0})+C_{0}(\partial_{a}\,\bar{C}^{a})+(\partial_{a}C_{0})\,\bar{C}^{a}\Bigr{]}\,A_{0}$
(1) $\displaystyle+$ $\displaystyle
i\,\Bigl{[}C_{0}\,(\partial_{0}\,\bar{C}^{1}+\partial_{1}\,\bar{C}^{0})+C_{1}\,(\partial_{a}\,\bar{C}^{a})+(\partial_{a}\,C_{1})\,\bar{C}^{a}\Bigr{]}\,A_{1}$
$\displaystyle+$ $\displaystyle
i\,\Bigl{[}C_{1}\,(\partial_{0}\,\bar{C}^{0}-\partial_{1}\,\bar{C}^{1})+C_{0}\,(\partial_{0}\,\bar{C}^{1}-\partial_{1}\,\bar{C}^{0})\Bigr{]}\,A_{2},$
$\displaystyle{\cal L}_{b}$ $\displaystyle=$ $\displaystyle{\cal
L}_{0}+B_{1}A_{1}+B_{0}A_{0}-i\,\big{[}\bar{C}_{1}(\partial_{0}C^{1}+\partial_{1}C^{0})+\bar{C}_{0}(\partial_{a}C^{a})-C^{a}\partial_{a}\bar{C}_{0}\big{]}\,A_{0}$
(2) $\displaystyle-$ $\displaystyle
i\,\big{[}\bar{C}_{0}(\partial_{0}C^{1}+\partial_{1}C^{0})-C^{a}\,\partial_{a}\,\bar{C}_{1}+\bar{C}_{1}\,(\partial_{a}\,C^{a})\big{]}\,A_{1}$
$\displaystyle-$ $\displaystyle
i\,\big{[}\bar{C}_{1}\,(\partial_{0}C^{0}\,-\partial_{1}C^{1})+\bar{C}_{0}\,(\partial_{0}\,C^{1}\,-\partial_{1}\,C^{0})\big{]}\,A_{2}.$

where the 2D diffeomorphism invariant classical action integral $(S_{0})$
w.r.t. ${\cal L}_{0}$ is:

$\displaystyle S_{0}=\int d^{2}\,\xi\,{\cal
L}_{0}\equiv\int_{-\infty}^{+\infty}d\,\tau\,\int_{\sigma=0}^{\sigma=\pi}d\,\sigma\,\Big{[}-\frac{1}{2\kappa}\,{\tilde{g}}^{mn}\,\partial_{a}\,X^{\mu}\,\partial_{b}\,X_{\mu}+E\,\big{(}\mbox{det}\,\tilde{g}+1\big{)}\Big{]}.$
(3)

In the above, we have taken the notation
$\xi^{a}=(\xi^{0},\,\xi^{1})=(\tau,\,\sigma)$ where $\tau$ is the evolution
parameter (with $-\infty<\tau<+\infty$) and $\sigma$ denotes the length of the
bosonic string (with $0\leq\sigma\leq\pi$). The modified version of the
inverse of the 2D metric tensor is: ${\tilde{g}}^{mn}={\sqrt{-g}}\,g^{mn}$
where $g^{mn}$ is the inverse of the 2D metric tensor $g_{mn}$ and $g=\,\,$det
$(g_{mn})$. The coordinates $X^{\mu}(\xi)\equiv X^{\mu}(\tau,\sigma)$ (where
$\mu=0,1,2...D-1$) correspond to the D-dimensional flat Minkowskian target
space and $a,b,c,...l,m,n=0,1$ are the “time” and space directions on the
world-sheet. The symbol $\kappa$ denotes the string tension parameter and $E$
is the Lagrange multiplier density which ensures that det $\tilde{g}=-1$ so
that we can have two degrees of freedom for the metric field tensor which,
being symmetric, has only three degrees of freedom on a 2D flat spacetime
manifold. In other words, we have (see, e.g. [32] for details) the following
diffeomorphism of ${\tilde{g}}^{mn}$, namely;

$\displaystyle{\tilde{g}}^{mn}=\begin{pmatrix}A_{1}+A_{2}&A_{0}\\\
A_{0}&A_{1}-A_{2}\end{pmatrix}.$ (4)

The flat limit (i.e. ${\tilde{g}}^{mn}\rightarrow\eta^{mn})$ can be obtained
by the gauge-fixing conditions: $A_{0}=A_{1}=0$. The latter choices imply that
we have $A_{2}^{2}=1$ when we demand $\mbox{det}\,\tilde{g}=-\,1$. This input
leads to ${\tilde{g}}^{mn}\rightarrow\eta^{mn}=\mbox{diag}\,(+1,\,-1)$ for the
choice $A_{2}=+1$ where $\eta_{mn}=\eta^{mn}=\mbox{diag}\,(+1,-1)$ are the
flat metric tensor $(\eta_{mn})$ and its inverse $(\eta^{mn})$ on the 2D
Minkowskian spacetime manifold. In the derivation of the gauge-fixing and
Faddeev-Popov ghost terms, we have taken the standard prescription of the BRST
formalism (see, e.g. [32]), namely;

$\displaystyle{\cal L}_{ab}$ $\displaystyle=$ $\displaystyle{\cal
L}_{0}+s_{ab}\,\big{[}\,i\,C_{0}A_{0}+i\,C_{1}A_{1}\big{]},$
$\displaystyle{\cal L}_{b}$ $\displaystyle=$ $\displaystyle{\cal
L}_{0}+s_{b}\,\big{[}-\,i\,\bar{C}_{0}A_{0}-i\,\bar{C}_{1}A_{1}\big{]},$ (5)

where the full set of nilpotent $[(s_{(a)b})^{2}=0]$ (anti-)BRST
transformations $[s_{(a)b}]$ are

$\displaystyle s_{ab}X^{\mu}=\bar{C}^{a}\,\partial_{a}\,X^{\mu},\qquad
s_{ab}C^{a}=i\bar{B}^{a},\qquad
s_{ab}\bar{C}^{a}=\bar{C}^{m}\,\partial_{m}\,\bar{C}^{a},\,\,$ $\displaystyle
s_{ab}\,E=\partial_{a}\,(\bar{C}^{a}E),\qquad s_{ab}\,\bar{B}^{a}=0,\qquad
s_{ab}(\mbox{det}\,{\tilde{g}})=\bar{C}^{m}\,\partial_{m}\,(\mbox{det}\,{\tilde{g}}),\qquad$
$\displaystyle
s_{ab}\,{\tilde{g}}^{mn}=\partial_{a}(\bar{C}^{a}\,{\tilde{g}}^{mn})\,-\,(\partial_{a}\,\bar{C}^{m})\,{\tilde{g}}^{an}\,-\,(\partial_{a}\,\bar{C}^{n})\,{\tilde{g}}^{ma},$
$\displaystyle
s_{ab}\,B^{a}={\bar{C}}^{m}\,\partial_{m}\,B^{a}-B^{m}\,\partial_{m}\,{\bar{C}}^{a},$
(6) $\displaystyle s_{b}X^{\mu}=C^{a}\,\partial_{a}\,X^{\mu},\qquad
s_{b}\bar{C}^{a}=iB^{a},\qquad s_{b}\,B^{a}=0,\qquad
s_{b}C^{a}=C^{b}\,\partial_{b}\,C^{a},$ $\displaystyle
s_{b}\,{\tilde{g}}^{mn}=\partial_{a}(C^{a}\,{\tilde{g}}^{mn})\,-\,(\partial_{a}\,C^{m})\,{\tilde{g}}^{an}\,-\,(\partial_{a}\,C^{n})\,{\tilde{g}}^{ma},\qquad
s_{b}\,E=\partial_{a}\,(C^{a}E),$ $\displaystyle
s_{b}\,{\bar{B}}^{a}={C}^{m}\,\partial_{m}\,{\bar{B}}^{a}-{\bar{B}}^{m}\,\partial_{m}\,{C}^{a},\qquad
s_{b}(\mbox{det}\,{\tilde{g}})=C^{a}\,\partial_{a}\,(\mbox{det}\,{\tilde{g}}).$
(7)

Here the fermionic
$[(C^{a})^{2}=(\bar{C}^{a})^{2}=0,\,C^{a}\,C^{b}+C^{b}\,C^{a}=0,\,C^{a}\,\bar{C}^{b}+\bar{C}^{b}\,C^{a}=0,\,\bar{C}^{a}\,\bar{C}^{b}+\bar{C}^{b}\,\bar{C}^{a}=0$,
etc.] (anti-)ghost fields are $(\bar{C}^{a})C^{a}$ and the bosonic Nakanishi-
Lautrup auxilary fields are $(\bar{B}^{a})B^{a}$. From the above, we can
derive the (anti-)BRST symmetry transformations for the component gauge fields
$A_{0},\,A_{1}$ and $A_{2}$ as follows:

$\displaystyle
s_{ab}A_{0}=\bar{C}^{m}\,\partial_{m}\,A_{0}-(\partial_{0}\,\bar{C}^{1}-\partial_{1}\,\bar{C}^{0})\,A_{2}-(\partial_{0}\,\bar{C}^{1}+\partial_{1}\,\bar{C}^{0})\,A_{1},$
$\displaystyle
s_{ab}A_{1}=\bar{C}^{m}\,\partial_{m}\,A_{1}-(\partial_{1}\,\bar{C}^{0}+\partial_{0}\,\bar{C}^{1})\,A_{0}-(\partial_{0}\,\bar{C}^{0}-\partial_{1}\,\bar{C}^{1})\,A_{2},$
$\displaystyle
s_{ab}A_{2}=\bar{C}^{m}\,\partial_{m}\,A_{2}-(\partial_{1}\,\bar{C}^{0}-\partial_{0}\,\bar{C}^{1})A_{0}-(\partial_{0}\,\bar{C}^{0}-\partial_{1}\,\bar{C}^{1})A_{1},$
(8) $\displaystyle
s_{b}A_{0}=C^{m}\,\partial_{m}\,A_{0}-(\partial_{0}\,C^{1}-\partial_{1}\,C^{0})\,A_{2}-(\partial_{0}\,C^{1}+\partial_{1}\,C^{0})\,A_{1},$
$\displaystyle
s_{b}A_{1}=C^{m}\,\partial_{m}\,A_{1}-(\partial_{1}\,C^{0}+\partial_{0}\,C^{1})\,A_{0}-(\partial_{0}\,C^{0}-\partial_{1}\,C^{1})\,A_{2},$
$\displaystyle
s_{b}A_{2}=C^{m}\,\partial_{m}\,A_{2}-(\partial_{1}\,C^{0}-\partial_{0}\,C^{1})A_{0}-(\partial_{0}\,C^{0}-\partial_{1}\,C^{1})A_{1}.$
(9)

It is interesting to note that this CF-type restrictions:
$B^{a}+\bar{B}^{a}+i\,(C^{m}\,\partial_{m}\,\bar{C}^{a}+\bar{C}^{m}\,\partial_{m}\,C^{a})=0$
appear in the following simple cases of the proof of absolute
anticommutativity property:

$\displaystyle\\{s_{b},s_{ab}\\}\,X^{\mu}$ $\displaystyle=$ $\displaystyle
i\,\Big{[}B^{a}+\bar{B}^{a}+i\,(C^{m}\,\partial_{m}\,\bar{C}^{a}+\bar{C}^{m}\,\partial_{m}\,C^{a})\Big{]}\,\big{(}\partial_{a}\,X^{\mu}\big{)},$
$\displaystyle\\{s_{b},s_{ab}\\}\,E$ $\displaystyle=$ $\displaystyle
i\,\partial_{a}\,\Big{[}\\{B^{a}+\bar{B}^{a}+i\,(C^{m}\,\partial_{m}\,\bar{C}^{a}+\bar{C}^{m}\,\partial_{m}\,C^{a})\\}\,E\Big{]},$
$\displaystyle\\{s_{b},s_{ab}\\}\,\tilde{g}^{mn}$ $\displaystyle=$
$\displaystyle
i\,\partial_{k}\,\Big{[}\big{\\{}B^{k}+\bar{B}^{k}+i\,(C^{l}\,\partial_{l}\,\bar{C}^{k}+\bar{C}^{l}\,\partial_{l}\,C^{k})\big{\\}}\,\tilde{g}^{mn}\Big{]}$
(10) $\displaystyle-$ $\displaystyle
i\,\partial_{k}\,\Big{[}B^{m}+\bar{B}^{m}+i\,(C^{l}\,\partial_{l}\,\bar{C}^{m}+\bar{C}^{l}\,\partial_{l}\,C^{m})\Big{]}\,\tilde{g}^{kn}$
$\displaystyle-$ $\displaystyle
i\,\partial_{k}\,\Big{[}B^{n}+\bar{B}^{n}+i\,(C^{l}\,\partial_{l}\,\bar{C}^{n}+\bar{C}^{l}\,\partial_{l}\,C^{n})\Big{]}\,\tilde{g}^{km}.$

Thus, the off-shell nilpotent $[(s_{(a)b})^{2}=0]$ (anti-)BRST symmetry
transformations [cf. Eqs. (7), (6)] are the proper set of quantum symmetry
transformations.

We end this section with the following remarks. First, the off-shell nilpotent
$[s_{(a)b}^{2}=0]$ (anti-)BRST symmetry transformations (7) and (6) correspond
to the classical 2D diffeomorphism symmetry transformations:
$\xi^{a}\,\rightarrow\,g^{a}(\xi)=\xi^{a}-\varepsilon^{a}(\xi)$ where
$g^{a}(\xi)$ is a physically well-defined function of $\xi^{a}$ on the 2D
world-sheet such that it is finite at $\tau=0$ and $\sigma=0$ but vanishes off
at $\tau\rightarrow\pm\infty$ and $\sigma=\pi$. The infinitesimal version of
these transformations are: $g^{a}(\xi)=\xi^{a}-\varepsilon^{a}(\xi)$ where
$\varepsilon^{a}(\xi)$ (with $a=0,1$) are the 2D infinitesimal diffeomorphism
transformation parameters. Second, according to the basic tenets of BRST
formalism, the parameters $\varepsilon^{a}(\xi)$ have been replaced by the
fermionic (anti-)ghost fields $(\bar{C}^{a})C^{a}$ in the (anti-)BRST symmetry
transformations (7) and (6). Third, it is crystal clear, from Eq. (10), that
the (anti-)BRST symmetry transformations $s_{(a)b}$ are absolutely
anticommuting (i.e. $\\{s_{b},\,s_{ab}\\}=0$) in nature only on the
submanifold of the quantum Hilbert space of fields where the CF-type
restrictions:
$B^{a}+\bar{B}^{a}+i\,(C^{m}\,\partial_{m}\,\bar{C}^{a}+\bar{C}^{m}\,\partial_{m}\,C^{a})=0$
are satisfied. Finally, we note that the target space coordinates
$X^{\mu}({\xi})$ and $[\mbox{det}\,\tilde{g}(\xi)]$ transform as pure Lorentz
scalars [i.e. $X^{\mu^{\prime}}(\xi^{{}^{\prime}})=X^{\mu}(\xi)$,
$\,\mbox{det}\,\tilde{g}^{\prime}(\xi^{\prime})=\mbox{det}\,\tilde{g}(\xi)$]
under the infinitesimal and continuous diffeomorphism symmetry
transformations:
$\xi^{a}\,\rightarrow\,g^{a}(\xi)=\xi^{a}-\varepsilon^{a}(\xi)$.

##  3 CF-Type Restrictions: MBTSA

According to the basic tenets of MBTSA to BRST formalism, first of all, we
generalize the 2D infinitesimal diffeomorphism transformations:
$\xi^{a}\longrightarrow\xi^{{}^{\prime}a}=g^{a}(\xi)=\xi^{a}-\varepsilon^{a}(\xi)$
to its counterpart onto the $(2,2)$-dimensional supermanifold as (see, e.g.
[18] for details)

$\displaystyle
g^{a}(\xi)\quad\longrightarrow\quad\tilde{g}^{a}(\xi,\theta,\bar{\theta})=\xi^{a}-\theta\,{\bar{C}}^{a}(\xi)-\bar{\theta}\,C^{a}(\xi)+\theta\,\bar{\theta}\,f^{a}(\xi),$
(11)

where the $(2,2)$-dimensional supermanifold is parameterized by the superspace
coordinates $Z^{M}=(\xi^{a},\theta,\bar{\theta})$. Here
$\xi^{a}=(\xi^{0},\xi^{1})\equiv(\tau,\sigma)$ are the bosonic world-sheet
coordinates and a pair of Grassmannian variables $(\theta,\bar{\theta})$
satisfy: $\theta^{2}={\bar{\theta}}^{2}=0$,
$\theta\,\bar{\theta}+\bar{\theta}\,\theta=0$. In Eq. (11), the fermionic
(anti-)ghost $({\bar{C}}^{a})C^{a}$ fields are the ones that are present in
the (anti-)BRST transformations (7) and (6). In view of the mappings
$(s_{b}\leftrightarrow\partial_{\bar{\theta}}\mid_{\theta=0},\,s_{ab}\leftrightarrow\partial_{\theta}\mid_{\bar{\theta}=0})$
established by Bonora and Tonin [4,5], the coefficients of $\theta$ and
$\bar{\theta}$ in (11) have been taken to be the (anti-)ghost fields because,
according to the standard BRST prescription, the classical infinitesimal
diffeomorphism symmetry transformations:
$\delta\,\xi^{a}=-\,\varepsilon^{a}(\xi)$ have been promoted to the quantum
level by the (anti-)BRST symmetry transformations:
$s_{ab}\,\xi^{a}=-\,{\bar{C}}^{a},\,s_{b}\,\xi^{a}=-\,C^{a}$. The coefficients
of $\theta\,\bar{\theta}$ in (11) [i.e. $f^{a}(\xi)$] have to be determined
from other consistency conditions of the BRST formalism which we elaborate
below.

To derive the CF-type restrictions and the (anti-)BRST symmetry
transformations
$s_{ab}\,X^{\mu}={\bar{C}}^{a}\,\partial_{a}\,X^{\mu},\,s_{b}\,X^{\mu}=C^{a}\,\partial_{a}\,X^{\mu}$,
we generalize the target space ordinary coordinate fields $X^{\mu}(\xi)$ onto
the $(2,2)$-dimensional supermanifold as

$\displaystyle
X^{\mu}(\xi)\quad\longrightarrow\quad{\tilde{X}}^{\mu}[\tilde{g}(\xi,\theta,\bar{\theta}),\theta,\bar{\theta}]$
$\displaystyle=$ $\displaystyle{\cal
X}^{\mu}[\tilde{g}(\xi,\theta,\bar{\theta})]+\theta\,{\bar{R}}^{\mu}[\tilde{g}(\xi,\theta,\bar{\theta})]$
(12) $\displaystyle+$
$\displaystyle\bar{\theta}\,R^{\mu}[\tilde{g}(\xi,\theta,\bar{\theta})]+\theta\,\bar{\theta}\,S^{\mu}[\tilde{g}(\xi,\theta,\bar{\theta})],$

where
${\tilde{X}}^{\mu}\,[\tilde{g}(\xi,\theta,\bar{\theta}),\theta,\bar{\theta}]$
are the superfields whose arguments incorporate the super diffeomorphism
transformations (11) and, on the r.h.s, we have the secondary superfields
which have the following super expansions [as their arguments are
transformations (11)], namely;

$\displaystyle\theta\,\bar{\theta}\,S^{\mu}\,\big{[}\xi^{a}-\theta\,\bar{C}^{a}-\bar{\theta}\,C^{a}+\theta\,\bar{\theta}\,f^{a}\big{]}$
$\displaystyle\equiv$
$\displaystyle\theta\,\bar{\theta}\,S^{\mu}(\xi^{a})\equiv\theta\,\bar{\theta}\,S^{\mu}(\xi),$
$\displaystyle\bar{\theta}\,R^{\mu}\,\big{[}\xi^{a}-\theta\,\bar{C}^{a}-\bar{\theta}\,C^{a}+\theta\,\bar{\theta}\,f^{a}\big{]}$
$\displaystyle\equiv$
$\displaystyle\bar{\theta}\,R^{\mu}(\xi)+\theta\,\bar{\theta}\,\bar{C}^{a}\,\partial_{a}\,R^{\mu}(\xi),$
$\displaystyle\theta\,\bar{R}^{\mu}\,\big{[}\xi^{a}-\theta\,\bar{C}^{a}-\bar{\theta}\,C^{a}+\theta\,\bar{\theta}\,f^{a}\big{]}$
$\displaystyle\equiv$
$\displaystyle\theta\,\bar{R}^{\mu}(\xi)-\theta\,\bar{\theta}\,C^{a}\,\partial_{a}\,\bar{R}^{\mu}(\xi),$
$\displaystyle{\cal
X}^{\mu}\,\big{[}\xi^{a}-\theta\,\bar{C}^{a}-\bar{\theta}\,C^{a}+\theta\,\bar{\theta}\,f^{a}\big{]}$
$\displaystyle\equiv$ $\displaystyle
X^{\mu}(\xi)-\theta\,\bar{C}^{a}\,\partial_{a}\,X^{\mu}-\bar{\theta}\,C^{a}\,\partial_{a}\,X^{\mu}$
(13) $\displaystyle+$
$\displaystyle\theta\,\bar{\theta}\,[f^{a}\,\partial_{a}\,X^{\mu}-\bar{C}^{a}\,C^{m}\partial_{a}\,\partial_{m}\,X^{\mu}],$

where ${\cal
X}^{\mu}\,\big{(}\xi^{a}-\theta\,\bar{C}^{a}-\bar{\theta}\,C^{a}+\theta\,\bar{\theta}\,f^{a}\big{)}|_{\theta\,=\,\bar{\theta}\,=\,0}=X^{\mu}(\xi)$
and the Taylor expansions have been taken around
$\theta\,=\,\bar{\theta}\,=\,0$. Collecting the coefficients of $\theta$,
$\bar{\theta}$ and $\theta\,\bar{\theta}$, from the r.h.s. of the above
equation, we obtain the following:

$\displaystyle\tilde{X^{\mu}}\,[\tilde{g}(\xi,\,\theta,\,\bar{\theta}),\,\theta,\,\bar{\theta}]=X^{\mu}({\xi})+\theta\,[\bar{R}^{\mu}-\bar{C}^{a}\,\partial_{a}\,X^{\mu}]+\bar{\theta}\,[R^{\mu}-C^{a}\,\partial_{a}\,X^{\mu}]$
$\displaystyle+\theta\,\bar{\theta}\,[f^{a}\,\partial_{a}\,X^{\mu}-\bar{C}^{a}\,C^{m}\,\partial_{a}\,\partial_{m}\,X^{\mu}-C^{a}\,\partial_{a}\,\bar{R}^{\mu}+\bar{C}^{a}\,\partial_{a}\,R^{\mu}+S^{\mu}].$
(14)

We note that the target space coordinate fields $X^{\mu}(\xi)$ are the pure
scalars with respect to the 2D world-sheet on which we have taken the
diffeomorphism symmetry transformations
$\xi^{a}\rightarrow\xi^{a}{{}^{\prime}}=g^{a}(\xi)$. Thus, physically, it is
evident that, ultimately, the restrictions on the (2, 2)-dimensional
superfield
$\tilde{X^{\mu}}\,[\tilde{g}(\xi,\,\theta,\,\bar{\theta}),\,\theta,\,\bar{\theta}]$
is the following

$\displaystyle
X^{\mu}(\xi)\rightarrow\tilde{X^{\mu}}\,[\tilde{g}(\xi,\,\theta,\,\bar{\theta}),\,\theta,\,\bar{\theta}]=X^{\mu}(\xi).$
(15)

This is what has been called as the horizontality condition (HC) in [18]. This
HC [cf. Eq. (15)] amounts to setting the coefficients of $\theta$,
$\bar{\theta}$ and $\theta\,\bar{\theta}$ in the expression (14) equal to
zero. In other words, we have the following

$\displaystyle
R^{\mu}=C^{a}\,\partial_{a}\,X^{\mu},\qquad\bar{R}^{\mu}=\bar{C}^{a}\,\partial_{a}\,X^{\mu},$
$\displaystyle
S^{\mu}=C^{a}\,\partial_{a}\,\bar{R}^{\mu}-\bar{C}^{a}\,\partial_{a}\,R^{\mu}+\bar{C}^{a}\,C^{m}\,\partial_{a}\,\partial_{m}\,X^{\mu}-f^{a}\,\partial_{a}\,X^{\mu}.$
(16)

The last entry can be explicitly written by plugging in the values of
$R^{\mu}$ and $\bar{R}^{\mu}$ as

$\displaystyle
S^{\mu}=C^{a}\,\partial_{a}\,[\bar{C}_{m}\,\partial_{m}\,X^{\mu}]-\bar{C}^{a}\,\partial_{a}\,[C^{m}\,\partial_{m}\,X^{\mu}]+\bar{C}^{a}\,C^{m}\,\partial_{a}\,\partial_{m}\,X^{\mu}-f^{a}\,\partial_{a}\,X^{\mu}.$
(17)

Now it is straightforward to check that we have the following:

$\displaystyle
S^{\mu}=\big{[}C^{a}\,\partial_{a}\,\bar{C}^{m}-\bar{C}^{a}\,\partial_{a}\,C^{m}-f^{m}\big{]}\,(\partial_{m}\,X^{\mu})-\bar{C}^{m}\,C^{a}\,\partial_{m}\,\partial_{a}\,X^{\mu}.$
(18)

As pointed out earlier, the coefficients of $\theta\,\bar{\theta}$ [i.e.
$f^{a}(\xi)$] in Eq. (11) and their presence in (18) can be computed by the
requirements of the consistency conditions of BRST formalism.

One of the sacrosanct properties of a pure scalar field/superfield is the
observation that it should not transform under any kind of internal,
spacetime, supersymmetric, etc., transformations. As a consequence, the
secondary superfields of the r.h.s. of (12) are

$\displaystyle{\cal
X}^{\mu}[\tilde{g}\,(\xi,\,\theta,\,\bar{\theta})]=X^{\mu}(\xi),\qquad\bar{R}^{\mu}[\tilde{g}\,(\xi,\,\theta,\,\bar{\theta})]=\bar{R}^{\mu}(\xi),$
$\displaystyle
R^{\mu}[\tilde{g}\,(\xi,\,\theta,\,\bar{\theta})]=R^{\mu}(\xi),\qquad
S^{\mu}[\tilde{g}\,(\xi,\,\theta,\,\bar{\theta})]=S^{\mu}(\xi).$ (19)

Similarly, the l.h.s. is:
$\tilde{X}^{\mu}[\tilde{g}(\xi,\theta,\bar{\theta}),\theta,\bar{\theta}]=\tilde{X}^{\mu}(\xi,\theta,\bar{\theta})$.
Substitutions of these equalities into (12) yield the following expressions in
terms of $s_{(a)b}$, namely;

$\displaystyle{\tilde{X}}^{\mu}(\xi,\theta,\bar{\theta})$ $\displaystyle=$
$\displaystyle
X^{\mu}(\xi)+\theta\,{\bar{R}}^{\mu}(\xi)+\bar{\theta}\,R^{\mu}(\xi)+\theta\,\bar{\theta}\,S^{\mu}(\xi)$
(20) $\displaystyle\equiv$ $\displaystyle
X^{\mu}(\xi)+\theta\,(s_{ab}\,X^{\mu})+\bar{\theta}\,(s_{b}\,X^{\mu})+\theta\,\bar{\theta}\,(s_{b}\,s_{ab}\,X^{\mu}),$

in a view of the Bonora-Tonin (BT) mappings:
$s_{b}\leftrightarrow\partial_{\bar{\theta}}\mid_{\theta=0}$, and
$s_{ab}\leftrightarrow\partial_{\theta}\mid_{\bar{\theta}=0}$ which was
established in the realm of D-dimensional non-Abelian 1-form gauge theory
[4,5]. In fact, a close look at (20) demonstrates that this expansion is
exactly like the BT-superfield approach to BRST formalism in the context of
gauge theories. Thus, it is clear from (16) and (18), that we have obtained
the following [in terms of the (anti-)BRST symmetry transformations
$(s_{(a)b})$ of (7) and (6)], namely;

$\displaystyle
R^{\mu}=C^{a}\,\partial_{a}\,X^{\mu}=s_{b}\,X^{\mu},\qquad\bar{R}^{\mu}=\bar{C}^{a}\,\partial_{a}\,X^{\mu}=s_{ab}\,X^{\mu},$
$\displaystyle
S^{\mu}=[C^{a}\,\partial_{a}\,\bar{C}^{m}-\bar{C}^{a}\,\partial_{a}\,C^{m}-f^{m}]\,(\partial_{m}\,X^{\mu})-\bar{C}^{a}\,C^{m}\,\partial_{a}\,\partial_{m}\,X^{\mu}\equiv
s_{b}\,s_{ab}\,X^{\mu}.$ (21)

The absolute anticommutativity requirement (i.e.
$\\{s_{b},s_{ab}\\}\,X^{\mu}=0$) implies that the following equality is true,
namely;

$\displaystyle
s_{b}\,\bar{R}^{\mu}=-\,s_{ab}\,R^{\mu}\quad\Longleftrightarrow\quad
s_{b}\,s_{ab}\,X^{\mu}=-\,s_{ab}\,s_{b}\,X^{\mu}.$ (22)

The explicit computations of $s_{b}\,\bar{R}^{\mu}$ and ($-\,s_{ab}\,R^{\mu}$)
yield

$\displaystyle s_{b}\,\bar{R}^{\mu}$ $\displaystyle=$ $\displaystyle
i\,B^{m}\,\partial_{m}\,X^{\mu}-\bar{C}^{a}\,C^{m}\,\partial_{a}\,\partial_{m}\,X^{\mu}-\bar{C}^{a}\,(\partial_{a}\,C^{m})\,(\partial_{m}\,X^{\mu}),$
$\displaystyle-\,s_{ab}\,R^{\mu}$ $\displaystyle=$
$\displaystyle-\,i\,\bar{B}^{m}\,\partial_{m}\,X^{\mu}-\bar{C}^{a}\,C^{m}\,\partial_{a}\,\partial_{m}\,X^{\mu}+C^{a}\,(\partial_{a}\,\bar{C}^{m})\,(\partial_{m}\,X^{\mu}),$
(23)

where we have used $s_{b}\,\bar{C}^{a}=i\,B^{a}$ and
$s_{ab}\,C^{a}=i\,\bar{B}^{a}$. In addition, we have taken
$s_{b}\,C^{a}=C^{m}\,\partial_{m}\,C^{a}$ and
$s_{ab}\,\bar{C}^{a}=\bar{C}^{m}\,\partial_{m}\,\bar{C}^{a}$ which are derived
from the nilpotency requirements: $s_{b}^{2}\,X^{\mu}=0$ and
$s_{ab}^{2}\,X^{\mu}=0$. The above equality (22) implies [from (23)] that we
have

$\displaystyle
B^{m}+\bar{B}^{m}+i\,(C^{a}\,\partial_{a}\,\bar{C}^{m}+\bar{C}^{a}\,\partial_{a}\,C^{m})=0,$
(24)

which is nothing but the CF-type restrictions that have been obtained [cf. Eq.
(10)] from the requirement of the absolute anticommutativity property (i.e.
$\\{s_{b},s_{ab}\\}=0$) of the (anti-)BRST symmetry transformations (7) and
(6).

At this crucial juncture, we are in the position to determine the explicit
expression for $f^{a}(\xi)$ that is present in Eqs. (11) and (18) by demanding
the equality of each of the equations present in (23) with the expression for
$S^{\mu}$ in (21). In other words, we find that

$\displaystyle
S^{\mu}=s_{b}\,\bar{R}^{\mu}\equiv-\,s_{ab}\,R^{\mu}\Longrightarrow$
$\displaystyle\big{[}C^{a}\,\partial_{a}\,\bar{C}^{m}-\bar{C}^{a}\,\partial_{a}\,C^{m}-f^{m}(\xi)\big{]}\,\partial_{m}\,X^{\mu}-\bar{C}^{a}\,C^{m}\,\partial_{a}\,\partial_{m}\,X^{\mu}$
$\displaystyle=\big{(}i\,B^{m}-\bar{C}^{a}\,\partial_{a}\,C^{m}\big{)}\,\big{(}\partial_{m}\,X^{\mu}\big{)}-\bar{C}^{a}\,C^{m}\,\partial_{a}\,\partial_{m}\,X^{\mu}$
$\displaystyle\equiv\big{(}-\,i\,\bar{B}^{m}+C^{a}\,\partial_{a}\,\bar{C}^{m}\big{)}\,\big{(}\partial_{m}\,X^{\mu}\big{)}-\bar{C}^{a}\,C^{m}\,\partial_{a}\,\partial_{m}\,X^{\mu}.$
(25)

A close look at (25) implies that there are two ways to equate the l.h.s
[containing $f^{m}(\xi)$] with the r.h.s. of the above equation, namely;

$\displaystyle f^{m}(\xi)=-\,i\,B^{m}+\bar{C}^{a}\,\partial_{a}\,C^{m}\equiv
i\,\bar{B}^{m}-C^{a}\,\partial_{a}\,\bar{C}^{m},$ (26)

which lead to the derivation of the CF-type restrictions (24). Thus, we
conclude that the CF-type restrictions are hidden in the determination of
$f^{a}(\xi)$ of equation (11) by exploiting the absolute anticommutativity
property (i.e. $\\{s_{b},s_{ab}\\}\,X^{\mu}=0$) within the ambit of MBTSA to
BRST formalism. Ultimately, we observe that the above logistics can be
repeated in the case of a pure scalar $(\mbox{det}\,{\tilde{g}})$ to derive
the CF-type restrictions (24) and the (anti-)BRST transformations:
$s_{ab}\,(\mbox{det}\,\tilde{g})=\bar{C}^{a}\,\partial_{a}\,(\mbox{det}\,\tilde{g})$
and
$s_{b}\,(\mbox{det}\,\tilde{g})=C^{a}\,\partial_{a}\,(\mbox{det}\,\tilde{g})$,
too.

We wrap-up this section with the following remarks. First of all, we have
taken the standard (anti-)BRST symmetry transformations:
$s_{ab}\,C^{a}=i\,\bar{B}^{a},\,s_{b}\,\bar{C}^{a}=i\,B^{a},\,s_{ab}\,\bar{B}^{a}=0,\,s_{b}\,B^{a}=0$
which imply the following [in the terminology of the (anti-)chiral superfield
approach (ACSA) to BRST formalism [22-26]], namely;

$\displaystyle C^{m}(\xi)\quad$ $\displaystyle\rightarrow$ $\displaystyle\quad
F^{m\,(c)}_{(ab)}(\xi,\,\theta)=C^{m}(\xi)+\theta\,(i\,\bar{B}^{m})\equiv
C^{m}(\xi)+\theta\,(s_{ab}\,C^{m}),$ $\displaystyle\bar{C}^{m}(\xi)\quad$
$\displaystyle\rightarrow$
$\displaystyle\quad\bar{F}^{m\,(ac)}_{(b)}(\xi,\,\bar{\theta})=\bar{C}^{m}(\xi)+\bar{\theta}\,(i\,B^{m})\equiv\bar{C}^{m}(\xi)+\bar{\theta}\,(s_{b}\,\bar{C}^{m}),$
$\displaystyle B^{m}(\xi)\quad$ $\displaystyle\rightarrow$
$\displaystyle\quad\tilde{B}^{m\,(ac)}_{(b)}(\xi,\,\bar{\theta})=B^{m}(\xi)+\bar{\theta}\,(0)\equiv
B^{m}(\xi)+\bar{\theta}\,(s_{b}\,B^{m}),$ $\displaystyle\bar{B}^{m}(\xi)\quad$
$\displaystyle\rightarrow$
$\displaystyle\quad\tilde{\bar{B}}^{m\,(c)}_{(ab)}(\xi,\,\theta)=\bar{B}^{m}(\xi)+\theta\,(0)\equiv\bar{B}^{m}(\xi)+\theta\,(s_{ab}\,\bar{B}^{m}),$
(27)

where the superscripts $(c)$ and $(ac)$ on the superfields [cf. the l.h.s. of
(27)] denote the chiral and anti-chiral versions of the full super expansions
and the subscripts $(b)$ and $(ab)$ denote the fact that the coefficients of
$(\bar{\theta})\,\theta$ in the above expansions lead to the determination of
BRST and anti-BRST symmetry transformations. In other words, we are sure about
the nilpotent (anti-)BRST symmetry transformations:
$s_{ab}\,C^{a}=i\,\bar{B}^{a},\,s_{ab}\,\bar{B}^{a}=0,\,s_{b}\,\bar{C}^{a}=i\,B^{a},\,s_{b}\,B^{a}=0$
in terms of the (anti-)chiral superfield expansions in Eq. (27). Second, it is
the off-shell nilpotency requirements $s_{(a)b}^{2}\,X^{\mu}=0$ which lead to
$s_{b}\,C^{a}=C^{m}\,\partial_{m}\,C^{a}$ and
$s_{ab}\,\bar{C}^{a}=\bar{C}^{m}\,\partial_{m}\,\bar{C}^{a}$. However, we have
to obtain these transformations within the realm of superfield approach.
Furthermore, it is the requirement of the absolute anticommutativity
properties: $\\{s_{b},s_{ab}\\}\,C^{a}=0,\,\\{s_{b},s_{ab}\\}\,\bar{C}^{a}=0$
which yield
$s_{b}\,\bar{B}^{a}=C^{m}\,\partial_{m}\,\bar{B}^{a}-\bar{B}^{m}\,\partial_{m}\,C^{a}$
and
$s_{ab}\,B^{a}=\bar{C}^{m}\,\partial_{m}\,B^{a}-B^{m}\,\partial_{m}\,\bar{C}^{a}$.
We have to obtain, however, these symmetry transformations too, by using the
techniques of the superfield approach to BRST formalism which we accomplish in
our next section. Third, we note that the HC condition (15) has led to the
following full super expansion of the target space coordinate superfield,
namely;

$\displaystyle\tilde{X}^{\mu\,(h)}(\xi,\,\theta,\,\bar{\theta})$
$\displaystyle=$ $\displaystyle
X^{\mu}(\xi)+\theta\,(\bar{C}^{a}\,\partial_{a}\,X^{\mu})+\bar{\theta}\,(C^{a}\,\partial_{a}\,X^{\mu})$
(28) $\displaystyle+$
$\displaystyle\theta\,\bar{\theta}\,\big{[}(i\,B^{a}-\bar{C}^{m}\,\partial_{m}\,C^{a})\,\partial_{a}\,X^{\mu}-\bar{C}^{m}\,C^{a}\,\partial_{m}\,\partial_{a}\,X^{\mu}\big{]}$
$\displaystyle\equiv$ $\displaystyle
X^{\mu}(\xi)+\theta\,(s_{ab}\,X^{\mu})+\bar{\theta}\,(s_{b}\,X^{\mu})+\theta\,\bar{\theta}\,(s_{b}\,s_{ab}\,X^{\mu}),$

where the superscript $(h)$ denotes the target space coordinate superfield
that has been obtained after the application of HC which, ultimately, leads to
(20). Here the coefficients of $\theta$ and $\bar{\theta}$ are the (anti-)BRST
symmetry transformations $[s_{(a)b}]$ that are listed in Eqs. (7) and (6).
Finally, we comment that an expansion like (28) can be also written for the
derivation of the (anti-)BRST symmetry transformations for the scalar
$(\mbox{det}\,\tilde{g})$.

##  4 (Anti-)BRST Symmetries of Other Fields: ACSA

In this section, we exploit the theoretical strength of ACSA to BRST formalism
[22-26] to derive all the (anti-)BRST symmetry transformations (7) and (6)
except such transformations for the target space coordinates $X^{\mu}$ and
$(\mbox{det}\,{\tilde{g}})$ which have already been derived in the previous
section by using MBTSA to BRST formalism. We are inspired to use, in our
present section, ACSA to BRST formalism because of our observations in Eq.
(27). First of all, we focus on the derivation of the BRST symmetry
transformations (7) which have not been derived in the previous section. Thus,
we wish to obtain:
$s_{b}\,C^{a}=C^{m}\,\partial_{m}\,C^{a},\,s_{b}\,\bar{B}^{a}=C^{m}\,\partial_{m}\,\bar{B}^{a}-\bar{B}^{m}\,\partial_{m}\,C^{a},\,s_{b}\,{\tilde{g}}^{mn}=\partial_{a}\,(C^{a}\,\tilde{g}^{mn})-(\partial_{a}\,C^{m})\,\tilde{g}^{an}-(\partial_{a}\,C^{n})\,{\tilde{g}}^{ma},\,s_{b}\,E=(\partial_{a}\,C^{a})\,E+C^{a}\,(\partial_{a}\,E)$.
In this context, first of all, we generalize the ordinary 2D fields
$C^{a}(\xi),\,\bar{B}^{a}(\xi),\,E(\xi)$ and ${\tilde{g}}^{mn}(\xi)$ onto a
$(2,1)$-dimensional anti-chiral super sub-manifold of the general
$(2,2)$-dimensional supermanifold as

$\displaystyle C^{m}(\xi)\quad$ $\displaystyle\rightarrow$ $\displaystyle\quad
F^{m\,(ac)}(\xi,\bar{\theta})=C^{m}(\xi)+\bar{\theta}\,b_{1}^{m}(\xi),$
$\displaystyle\bar{B}^{m}(\xi)\quad$ $\displaystyle\rightarrow$
$\displaystyle\quad\bar{\cal
B}^{m\,(ac)}(\xi,\bar{\theta})=\bar{B}^{m}(\xi)+\bar{\theta}\,f_{1}^{m}(\xi),$
$\displaystyle E(\xi)\quad$ $\displaystyle\rightarrow$
$\displaystyle\quad{\cal
E}^{(ac)}(\xi,\bar{\theta})=E(\xi)+\bar{\theta}\,f_{2}(\xi),$
$\displaystyle\tilde{g}^{mn}(\xi)\quad$ $\displaystyle\rightarrow$
$\displaystyle\quad\tilde{G}^{mn\,(ac)}(\xi,\bar{\theta})=\tilde{g}^{mn}(\xi)+\bar{\theta}\,\tilde{R}^{mn}(\xi),$
(29)

where the 2D fields $(f_{1}^{m},\,f_{2},\,\tilde{R}^{ab})$ are fermionic
secondary fields and $b_{1}^{m}(\xi)$ is a bosonic secondary field due to the
fermionic $(\bar{\theta}^{2}=0)$ nature of the Grassmannian variable
$\bar{\theta}$. The above $(2,1)$-dimensional anti-chiral super sub-manifold
is parameterized by $(\xi^{a},\bar{\theta})$ where
$\xi^{a}\equiv(\tau,\sigma)$ are the bosonic coordinates and $\bar{\theta}$ is
the fermionic $(\bar{\theta}^{2}=0)$ Grassmannian variable. The superscript
$(ac)$ on the superfields denotes the anti-chiral super expansions of the
above anti-chiral superfields along $\bar{\theta}$-direction of the above
super submanifold.

The basic tenets of ACSA to BRST formalism require that the BRST-invariant
(i.e. quantum gauge invariant) quantities should be independent of the
Grassmannian variables as the latter are only the mathematical artifacts that
are useful in the context of theoretical techniques of SUSY theories. In this
connection, we note that the following BRST (i.e. quantum gauge) invariant
quantities are useful and important for us, namely;

$\displaystyle s_{b}\,\big{[}C^{a}\,\partial_{a}\,X^{\mu}\big{]}=0,\qquad
s_{b}\big{[}C^{a}\,\partial_{a}\,\bar{B}^{m}-\bar{B}^{a}\,\partial_{a}\,C^{m}\big{]}=0,$
$\displaystyle
s_{b}\,\big{[}C^{a}\,\partial_{a}\,E+(\partial_{a}\,C^{a})\,E\big{]}=0,$
$\displaystyle
s_{b}\,\big{[}C^{a}\,\partial_{a}\,\tilde{g}^{mn}+(\partial_{a}\,C^{a})\,\tilde{g}^{mn}-(\partial_{a}\,C^{m})\,\tilde{g}^{an}-(\partial_{a}\,C^{n})\,\tilde{g}^{ma}\big{]}=0.$
(30)

The above invariant quantities are obtained by a close observation of the
transformations (7) where an off-shell nilpotency property $(s_{b}^{2}=0)$
exists for the BRST-symmetry transformations. We focus on
$s_{b}\,[C^{a}\,\partial_{a}\,X^{\mu}]=0$ which implies the following
restriction

$\displaystyle
F^{m\,(ac)}(\xi,\bar{\theta})\,\partial_{m}\,X^{\mu\,(h,ac)}(\xi,\bar{\theta})=C^{m}(\xi)\,\partial_{m}\,X^{\mu}(\xi),$
(31)

where $X^{\mu\,(h,ac)}(\xi,\bar{\theta})$ is the anti-chiral limit of the full
super expansion containing the nilpotent (anti-)BRST symmetries as the
coefficients of $\theta$ and $\bar{\theta}$. In other words, we have:

$\displaystyle
X^{\mu\,(h,ac)}(\xi,\bar{\theta})=X^{\mu}(\xi)+\bar{\theta}\,(C^{a}\,\partial_{a}\,X^{\mu}).$
(32)

Plugging in the appropriate super expansions for $F^{a}(\xi,\bar{\theta})$
from (29) as well as the super expansion for
$X^{\mu\,(h,ac)}(\xi,\bar{\theta})$ from (32), we obtain the explicit
expression for the secondary fields as:
$b_{1}^{m}(\xi)=C^{a}\,\partial_{a}\,C^{m}$. As a consequence, we have the
following final expansion

$\displaystyle
F^{m\,(ac)}_{(b)}(\xi,\bar{\theta})=C^{m}(\xi)+\bar{\theta}\,(C^{a}\,\partial_{a}\,C^{m})\equiv
C^{m}(\xi)+\bar{\theta}\,(s_{b}\,C^{m}),$ (33)

where the subscript $(b)$ on the superfield (on the l.h.s.) denotes that the
above anti-chiral superfield has been obtained after the application of the
BRST invariant restrictions (31) and the coefficient of $\bar{\theta}$ is
nothing but the BRST symmetry transformation for the field $C^{m}(\xi)$ which
also encodes the relationships:
$\partial_{\bar{\theta}}\,F^{m\,(ac)}(\xi,\bar{\theta})=s_{b}\,C^{m}(\xi)$ and
$\partial_{\bar{\theta}}^{2}=0\Leftrightarrow s_{b}^{2}=0$. The latter
establishes the connection between the nilpotency properties of
$\partial_{\bar{\theta}}$ and $s_{b}$.

At this juncture, we now concentrate on the derivation of $f_{2}(\xi)$ in the
expansion of ${\cal E}^{ac}(\xi,\bar{\theta})$ in Eq. (29). For this purpose,
we note that $s_{b}\,[C^{m}\,\partial_{m}\,E+(\partial_{m}\,C^{m})\,E]=0$.
Following the basic principle of ACSA, the expressions in the square bracket
have to be generalized onto the $(2,1)$-dimensional anti-chiral super sub-
manifold with the following BRST (i.e. quantum gauge) symmetry invariant
restriction

$\displaystyle F_{(b)}^{m\,(ac)}(\xi,\bar{\theta})\,\partial_{m}\,{\cal
E}^{(ac)}(\xi,\bar{\theta})+\big{[}\partial_{m}\,F_{(b)}^{m\,(ac)}(\xi,\bar{\theta})\big{]}\,{\cal
E}^{(ac)}(\xi,\bar{\theta})$
$\displaystyle=C^{m}(\xi)\,\big{[}\partial_{m}\,E(\xi)\big{]}+\big{[}\partial_{m}\,C^{m}(\xi)\big{]}\,E(\xi),$
(34)

where the expansions of $F_{(b)}^{m\,(ac)}(\xi,\bar{\theta})$ and ${\cal
E}^{(ac)}(\xi,\bar{\theta})$ have been quoted in Eqs. (33) and (29),
respectively. Substitutions of these super expansions into the l.h.s. and
comparison with the r.h.s. of the restriction (34), lead to the following
condition

$\displaystyle(\partial_{m}\,C^{a})\,(\partial_{a}\,C^{m})\,E+C^{a}\,(\partial_{a}\,\partial_{m}\,C^{m})\,E+C^{a}\,(\partial_{a}\,C^{m})\,(\partial_{m}\,E)$
$\displaystyle-(\partial_{m}\,C^{m})\,f_{2}-C^{m}\,(\partial_{m}\,f_{2})=0.$
(35)

In other words, the restriction (34) implies that BRST invariant quantity must
be independent of $\bar{\theta}$. A careful and close look at the above
equation leads to:

$\displaystyle\partial_{m}\,\big{[}C^{a}\,(\partial_{a}\,C^{m})\,E-C^{m}\,f_{2}\big{]}=0.$
(36)

Substituting for
$C^{a}\,(\partial_{a}\,C^{m})\,E=\partial_{a}\,[C^{a}\,C^{m}\,E]-\,(\partial_{a}\,C^{a})\,C^{m}\,E-\,C^{a}\,C^{m}\,(\partial_{a}\,E),$
we obtain the following from the above equation:

$\displaystyle\partial_{m}\,[\partial_{a}\,\\{C^{a}\,C^{m}\,E\\}-\,(\partial_{a}\,C^{a})\,C^{m}\,E-C^{a}\,C^{m}\,(\partial_{a}\,E)-C^{m}\,f_{2}]=0.$
(37)

It is clear that the first term in the square-bracket will be zero if we
operate the derivative $(\partial_{m})$ from outside. Thus, the final
expression is as follows:

$\displaystyle\partial_{m}\,[C^{m}\,\\{\partial_{a}\,(C^{a}\,E)-f_{2}\\}]=0.$
(38)

Integrating over $d^{2}\,\xi=d\,\sigma\,d\,\tau$ and taking the physicality
condition that all the fields must vanish off as $\tau\rightarrow\pm\,\infty$
and at $\sigma=0$, $\sigma=\pi$, we obtain the precise value of $f_{2}(\xi)$
as

$\displaystyle f_{2}=\partial_{a}\,(C^{a}\,E)\qquad[\mbox{for}\,\,C^{m}\neq
0].$ (39)

Hence, we have the following final expansion for the superfield ${\cal
E}^{(ac)}(\xi,\bar{\theta})$

$\displaystyle{\cal
E}_{(b)}^{(ac)}(\xi,\bar{\theta})=E(\xi)+\bar{\theta}\,\big{[}\partial_{n}(C^{n}\,E)\big{]}\equiv
E(\xi)+\bar{\theta}\,(s_{b}\,E),$ (40)

which leads to the derivation of the BRST symmetry transformation
$s_{b}\,E=\partial_{a}\,(C^{a}\,E)$ as the coefficient of $\bar{\theta}$ in
the above equation implying, once again, that $\partial_{\bar{\theta}}\,{\cal
E}_{(b)}^{(ac)}(\xi,\bar{\theta})=s_{b}\,E(\xi)$. This relationship
establishes the connection between $s_{b}$ and translational generator
$\partial_{\bar{\theta}}$ along the $\bar{\theta}$-direction of the (2,
1)-dimensional anti-chiral super sub-manifold and it also demonstrates that
$s_{b}^{2}=0\Leftrightarrow\partial_{\bar{\theta}}^{2}=0$ (which is the
connection between the nilpotency properties). It goes without saying that the
subscript $(b)$ on the l.h.s. denotes that the super expansion (40) has been
obtained after the application of the BRST invariant restriction (34).

We now focus on the BRST invariance:
$s_{b}\,[C^{n}\,\partial_{n}\,\bar{B}^{m}-\bar{B}^{n}\,\partial_{n}\,C^{m}]=0$.
This observation can be generalized onto the (2, 1)-dimensional anti-chiral
super sub-manifold with the following restriction on the anti-chiral
superfields, namely;

$\displaystyle F_{(b)}^{m\,(ac)}(\xi,\bar{\theta})\,\partial_{m}\,{\bar{\cal
B}}^{n\,(ac)}(\xi,\bar{\theta})-{\bar{\cal
B}}^{m\,(ac)}(\xi,\bar{\theta})\,\partial_{m}\,F_{(b)}^{n\,(ac)}(\xi,\bar{\theta})$
$\displaystyle=C^{m}(\xi)\,\partial_{m}\,\bar{B}^{n}(\xi)-\bar{B}^{m}(\xi)\,\partial_{m}\,C^{n}(\xi).$
(41)

The substitutions of expansions from (29) and (33) lead to the following
equality:

$\displaystyle
C^{n}\,\big{[}\partial_{n}\,f^{m}_{1}+{\bar{B}}^{a}\,(\partial_{a}\,\partial_{n}\,C^{m})-(\partial_{n}\,C^{a})(\partial_{a}\,{\bar{B}}^{m})\big{]}+\big{[}f^{a}_{1}+{\bar{B}}^{n}\,(\partial_{n}\,C^{a})\big{]}(\partial_{a}\,C^{m})=0.$
(42)

In the above, the term:
$-\,(\partial_{n}\,C^{a})(\partial_{a}\,{\bar{B}}^{m})$ can be written as:
$-\,\partial_{n}\,[C^{a}\,\partial_{a}\,{\bar{B}}^{m}]+C^{a}\,\partial_{n}\,\partial_{a}\,{\bar{B}}^{m}$.
It is elementary to note that the second term will vanish off when we shall
multiply by $C^{n}$ from the left (i.e.
$C^{n}\,C^{a}\,\partial_{a}\,\partial_{n}\,\bar{B}^{m}=0$). The substitution
of the leftover term (i.e.
$-\,\partial_{n}\,[C^{a}\,\partial_{a}\,{\bar{B}}^{m}]$) into (42) leads to:

$\displaystyle
C^{n}\,\partial_{n}\,\big{[}f^{m}_{1}+{\bar{B}}^{a}\,(\partial_{a}\,C^{m})-C^{a}\,\partial_{a}\,{\bar{B}}^{m}\big{]}+\big{[}f^{a}_{1}+{\bar{B}}^{n}\,(\partial_{n}\,C^{a})-C^{n}\,\partial_{n}\,{\bar{B}}^{a}\big{]}(\partial_{a}\,C^{m})=0.$
(43)

It is straightforward to note that
$f^{m}_{1}=C^{a}\,\partial_{a}\,{\bar{B}}^{m}-{\bar{B}}^{a}\,(\partial_{a}\,C^{m})$
satisfies the above equation very beautifully. Thus, we have, ultimately, the
following expansion [cf. Eq. (29)]:

$\displaystyle\bar{\cal
B}^{m\,(ac)}_{(b)}(\xi,\bar{\theta})=\bar{B}^{m}(\xi)+\bar{\theta}\,[C^{a}\,\partial_{a}\,{\bar{B}}^{m}-{\bar{B}}^{a}\,\partial_{a}\,C^{m}]\equiv\bar{B}^{m}(\xi)+\bar{\theta}\,[s_{b}\,\bar{B}^{m}(\xi)].$
(44)

Hence , we have derived the BRST transformations:
$s_{b}\,\bar{B}^{m}=C^{a}\,\partial_{a}\,{\bar{B}}^{m}-{\bar{B}}^{a}\,\partial_{a}\,C^{m}$
as the coefficient of $\bar{\theta}$ in the above super expansion. It should
be noted that the subscript $(b)$ on the superfield [cf. l.h.s. of Eq. (44)]
denotes that $\bar{\cal B}^{m\,(ac)}_{(b)}(\xi,\bar{\theta})$ has been derived
after the imposition of the BRST invariant restriction (41).

At this stage, we now wish to derive the BRST symmetry transformation
$[s_{b}\,{\tilde{g}}^{mn}=\partial_{k}(C^{k}\,{\tilde{g}}^{mn})\,-\,(\partial_{k}\,C^{m})\,{\tilde{g}}^{kn}\,-\,(\partial_{k}\,C^{n})\,{\tilde{g}}^{mk}]$
using the theoretical strength of ACSA to BRST formalism. Towards this goal in
mind, we have the following restriction on the anti-chiral superfields which
have their super expansions in (29) and (33), namely;

$\displaystyle
F^{k(ac)}_{(b)}(\xi,\bar{\theta})\,\partial_{k}\,\tilde{G}^{mn(ac)}(\xi,\bar{\theta})+\big{[}\partial_{k}\,F^{k(ac)}_{(b)}(\xi,\bar{\theta})\big{]}\,\tilde{G}^{mn(ac)}(\xi,\bar{\theta})$
$\displaystyle-\big{[}\partial_{k}\,F^{m(ac)}_{(b)}(\xi,\bar{\theta})\big{]}\,\tilde{G}^{kn(ac)}(\xi,\bar{\theta})-\big{[}\partial_{k}\,F^{n(ac)}_{(b)}(\xi,\bar{\theta})\big{]}\,\tilde{G}^{km(ac)}(\xi,\bar{\theta})$
$\displaystyle=C^{k}(\xi)\,\big{[}\partial_{k}\,\tilde{g}^{mn}(\xi)\big{]}+\big{[}\partial_{k}\,C^{k}(\xi)\big{]}\,\tilde{g}^{mn}(\xi)$
$\displaystyle-\big{[}\partial_{k}\,C^{m}(\xi)\big{]}\,\tilde{g}^{kn}(\xi)-\big{[}\partial_{k}\,C^{n}(\xi)\big{]}\,\tilde{g}^{mk}(\xi).$
(45)

The above restriction has been obtained by a close look at the off-shell
nilpotency property $(s_{b}^{2}\,\tilde{g}^{mn}=0)$ of the BRST symmetry
transformations (7). This restriction on the anti-chiral superfields leads to
the following condition on the basic and secondary fields

$\displaystyle
C^{k}\,(\partial_{k}\,\tilde{R}^{mn})+(\partial_{k}\,C^{k})\,\tilde{R}^{mn}-(\partial_{k}\,C^{l})\,(\partial_{l}\,C^{k})\,\tilde{g}^{mn}-C^{l}\,(\partial_{k}\,\partial_{l}\,C^{k})\,\tilde{g}^{mn}$
$\displaystyle-C^{l}\,(\partial_{l}\,C^{k})\,(\partial_{k}\,\tilde{g}^{mn})-(\partial_{k}\,C^{m})\,\tilde{R}^{kn}+(\partial_{k}\,C^{l})\,(\partial_{l}\,C^{m})\,\tilde{g}^{kn}+C^{l}\,(\partial_{k}\,\partial_{l}\,C^{m})\,\tilde{g}^{kn}$
$\displaystyle-(\partial_{k}\,C^{n})\,\tilde{R}^{mk}+(\partial_{k}\,C^{l})\,(\partial_{l}\,C^{n})\,\tilde{g}^{mk}+C^{l}\,(\partial_{k}\,\partial_{l}\,C^{n})\,\tilde{g}^{mk}=0,$
(46)

where we have used the super expansions from (29) and (33). It is
straightforward to note that the first five terms, in the above, lead to the
following total derivative, namely;

$\displaystyle\partial_{k}\,\big{[}C^{k}\,\tilde{R}^{mn}-C^{l}\,(\partial_{l}\,C^{k})\,\tilde{g}^{mn}\big{]}\equiv\partial_{k}\,\big{[}C^{k}\,\\{\tilde{R}^{mn}-\partial_{l}\,(C^{l}\,\tilde{g}^{mn})\\}\big{]},$
(47)

where we have used:
$-\,C^{l}\,(\partial_{l}\,C^{k})\,\tilde{g}^{mn}=-\,\partial_{l}\,\big{[}C^{l}\,C^{k}\,\tilde{g}^{mn}\big{]}+(\partial_{l}\,C^{l})\,\tilde{g}^{mn}+C^{l}\,C^{k}\,(\partial_{l}\,\tilde{g}^{mn})$
and $\partial_{k}\,\partial_{l}\,(C^{l}\,C^{k}\,\tilde{g}^{mn})=0$. Adding and
substracting:
$\partial_{k}\,\big{[}C^{k}\,(\partial_{l}\,C^{m})\,\tilde{g}^{ln}+C^{k}\,(\partial_{l}\,C^{n})\,\tilde{g}^{ml}\big{]}$
we obtain the following equation from (46):

$\displaystyle\partial_{k}\,\big{[}C^{k}\,\\{\tilde{R}^{mn}-\partial_{l}\,(C^{l}\,\tilde{g}^{mn})+(\partial_{l}\,C^{m})\,\tilde{g}^{ln}+(\partial_{l}\,C^{n})\,\tilde{g}^{ml}\\}\big{]}$
$\displaystyle-\partial_{k}\,\big{[}C^{k}\,(\partial_{l}\,C^{m})\,\tilde{g}^{ln}+C^{k}\,(\partial_{l}\,C^{n})\,\tilde{g}^{ml}\big{]}=0.$
(48)

Expanding the total derivative in the second entry of the above equation and
rearranging these, we obtain the following interesting equation, namely;

$\displaystyle\partial_{k}\,\big{[}C^{k}\,\\{\tilde{R}^{mn}-\partial_{l}\,(C^{l}\,\tilde{g}^{mn})+(\partial_{l}\,C^{m})\,\tilde{g}^{ln}+(\partial_{l}\,C^{n})\,\tilde{g}^{ml}\\}\big{]}$
$\displaystyle-(\partial_{k}\,C^{m})\,\big{[}\tilde{R}^{nk}-\partial_{l}\,(C^{l}\,\tilde{g}^{nk})+(\partial_{l}\,C^{k})\,\tilde{g}^{ln}\big{]}$
$\displaystyle-(\partial_{k}\,C^{n})\,\big{[}\tilde{R}^{mk}-\partial_{l}\,(C^{l}\,\tilde{g}^{mk})+(\partial_{l}\,C^{k})\,\tilde{g}^{lm}\big{]}=0.$
(49)

Adding and subtracting
$(\partial_{k}\,C^{m})\,(\partial_{l}\,C^{n})\,\tilde{g}^{lk}+(\partial_{k}\,C^{n})\,(\partial_{l}\,C^{m})\,\tilde{g}^{lk}$
in the above, we finally obtain the following very nice looking equation:

$\displaystyle\partial_{k}\,\big{[}C^{k}\,\\{\tilde{R}^{mn}-\partial_{l}\,(C^{l}\,\tilde{g}^{mn})+(\partial_{l}\,C^{m})\,\tilde{g}^{ln}+(\partial_{l}\,C^{n})\,\tilde{g}^{ml}\\}\big{]}$
$\displaystyle-(\partial_{k}\,C^{m})\,\big{[}\tilde{R}^{nk}-\partial_{l}\,(C^{l}\,\tilde{g}^{nk})+(\partial_{l}\,C^{k})\,\tilde{g}^{ln}+(\partial_{l}\,C^{n})\,\tilde{g}^{lk}\big{]}$
$\displaystyle-(\partial_{k}\,C^{n})\,\big{[}\tilde{R}^{mk}-\partial_{l}\,(C^{l}\,\tilde{g}^{mk})+(\partial_{l}\,C^{k})\,\tilde{g}^{lm}+(\partial_{l}\,C^{m})\,\tilde{g}^{lk}\big{]}=0.$
(50)

It should be noted that what we have added and subtracted in (49) is basically
equal to zero on its own because we make the following observation:

$\displaystyle\tilde{g}^{lk}\,\big{[}(\partial_{k}\,C^{m})\,(\partial_{l}\,C^{n})+(\partial_{k}\,C^{n})\,(\partial_{l}\,C^{m})\big{]}=0.$
(51)

In other words, the last entries in the second and third lines of Eq. (50) are
zero on their own. We note that the symmetric indices in $(\tilde{g}^{lk})$
and anti-symmetric indices $(l,k)$ in the square-bracket are sum-up to yield
zero. It is straightforward now to point out that

$\displaystyle\tilde{R}^{mn}=\partial_{k}\,(C^{k}\,\tilde{g}^{mn})-(\partial_{k}\,C^{m})\,\tilde{g}^{kn}-(\partial_{k}\,C^{n})\,\tilde{g}^{mk},$
(52)

satisfies the above equation (50). As a consequence, we have the following

$\displaystyle\tilde{G}^{mn(ac)}_{(b)}(\xi,\bar{\theta})$ $\displaystyle=$
$\displaystyle\tilde{g}^{mn}(\xi)+\bar{\theta}\,\big{[}\partial_{k}\,(C^{k}\,\tilde{g}^{mn})-(\partial_{k}\,C^{m})\,\tilde{g}^{kn}-(\partial_{k}\,C^{n})\,\tilde{g}^{mk}\big{]}$
(53) $\displaystyle\equiv$
$\displaystyle\tilde{g}^{mn}(\xi)+\bar{\theta}\,[s_{b}\,\tilde{g}^{mn}(\xi)],$

where the coefficient of $\bar{\theta}$ is nothing but the BRST symmetry
transformation for $\tilde{g}^{mn}(\xi)$ that has been quoted in (7). The
subscript $(b)$ on the l.h.s. of the above equation denotes that the anti-
chiral superfield $\tilde{G}^{mn}_{(b)}(\xi,\bar{\theta})$ has been obtained
after the application of the BRST invariant restriction on a specific
combination of superfields [cf. Eq. (45)].

We set out now to derive the anti-BRST symmetry transformations (6) by using
ACSA to BRST formalism where, first of all, we generalize the following basic
and auxiliary fields of our theory onto a $(2,1)$-dimensional chiral super
submanifold

$\displaystyle B^{m}(\xi)\quad$ $\displaystyle\rightarrow$
$\displaystyle\quad{\cal
B}^{m(c)}(\xi,\theta)=B^{m}(\xi)+\theta\,\bar{f}_{1}^{m}(\xi),$ $\displaystyle
E(\xi)\quad$ $\displaystyle\rightarrow$ $\displaystyle\quad{\cal
E}^{(c)}(\xi,\theta)=E(\xi)+\theta\,\bar{f}_{2}(\xi),$
$\displaystyle\bar{C}^{m}(\xi)\quad$ $\displaystyle\rightarrow$
$\displaystyle\quad{\bar{F}}^{m(c)}(\xi,\theta)=\bar{C}^{m}(\xi)+\theta\,\bar{b}_{1}^{m}(\xi),$
$\displaystyle\tilde{g}^{mn}(\xi)\quad$ $\displaystyle\rightarrow$
$\displaystyle\quad\tilde{G}^{mn(c)}(\xi,\theta)=\tilde{g}^{mn}(\xi)+\theta\,\tilde{\bar{R}}^{mn}(\xi),$
(54)

where $(\bar{f}_{1}^{m},\,\bar{f}_{2}^{m},\,\tilde{\bar{R}}^{mn})$ are the
fermionic and $\bar{b}^{m}_{1}$ is the bosonic secondary fields that are to be
determined in terms of the basic and auxiliary fields of the (anti-)BRST
invariant Lagrangian densities ${\cal L}_{(a)b}$ [cf. Eqs. (1), (2)]. It is
elementary to note that, in the limit $\theta=0$, we retrieve the bosonic and
auxiliary fields of ${\cal L}_{(a)b}$. We point out that
$s_{ab}\,\bar{B}^{m}(\xi)=0$ implies that we have $\bar{\cal
B}^{m}_{(ab)}(\xi,\,\theta)=\bar{B}^{m}(\xi)$ where $\bar{\cal
B}^{m}_{(ab)}(\xi,\,\theta)$ is the superfield that has been obtained after
the restriction on the chiral superfield $\bar{\cal B}^{m}(\xi,\,\theta)$ that
is obtained in the generalization $\bar{B}^{m}(\xi)\rightarrow\bar{\cal
B}^{m}(\xi,\,\theta)$ on the chiral super submanifold [which is parameterized
by $(\xi^{a},\,\theta)$ where $\xi^{a}$ characterize the 2D world-sheet and
$\theta$ is the fermionic $(\theta^{2}=0)$ Grassmannian variable]. The
superscript $(ab)$ denotes the chiral superfield which leads to the derivation
of $[s_{ab}\,\bar{B}(\xi)=0]$ as the coefficient of $\theta$ in its expansion:
$\bar{\cal
B}^{m}_{(ab)}(\xi,\,\theta)=\bar{B}^{m}(\xi)+\theta\,(0)\,\equiv\,\bar{B}^{m}(\xi)+\theta\,(s_{ab}\,\bar{B}^{m})$.
It should be further noted that we have not devoted time on the derivation of
the (anti-)BRST symmetries that have already been derived and mentioned in
Sec. 3 where the theoretical strength of MBTSA has been exploited.

A close and careful observation of the anti-BRST symmetry transformations (6)
demonstrates that we have the following very useful and interesting
combinations of fields

$\displaystyle
s_{ab}\,\big{[}\bar{C}^{a}\,\partial_{a}\,X^{\mu}\big{]}=0,\qquad
s_{ab}\big{[}\bar{C}^{a}\,\partial_{a}\,B^{m}-B^{a}\,\partial_{a}\,\bar{C}^{m}\big{]}=0,$
$\displaystyle
s_{ab}\,\big{[}\bar{C}^{a}\,\partial_{a}\,E+(\partial_{a}\,\bar{C}^{a})\,E\big{]}=0,$
$\displaystyle
s_{ab}\,\big{[}\bar{C}^{a}\,\partial_{a}\,\tilde{g}^{mn}+(\partial_{a}\,\bar{C}^{a})\,\tilde{g}^{mn}-(\partial_{a}\,\bar{C}^{m})\,\tilde{g}^{an}-(\partial_{a}\,\bar{C}^{n})\,\tilde{g}^{ma}\big{]}=0,$
(55)

as the anti-BRST invariant quantities. The fundamental requirement of ACSA is
that the generalizations of the quantities [present in the square bracket of
(55)] onto a suitably chosen $(2,1)$-dimensional chiral super submanifold
should be independent of the Grassmannian variable $\theta$. As a consequence,
we have the following restrictions

$\displaystyle\bar{F}^{a(c)}(\xi,\theta)\,\partial_{a}\,X^{\mu(h,c)}(\xi,\theta)=\bar{C}^{a}(\xi)\,\partial_{a}\,X^{\mu}(\xi),$
$\displaystyle\bar{F}^{a\,(c)}(\xi,\theta)\,\partial_{a}\,{{\cal
B}}^{m\,(c)}(\xi,\theta)-{{\cal
B}}^{a\,(c)}(\xi,\theta)\,\partial_{a}\,\bar{F}^{m\,(c)}(\xi,\theta)=\bar{C}^{a}(\xi)\,\partial_{a}\,B^{m}(\xi)-B^{a}(\xi)\,\partial_{a}\,\bar{C}^{m}(\xi),$
$\displaystyle\bar{F}^{a(c)}(\xi,\theta)\,\partial_{a}\,{\cal
E}^{(c)}(\xi,\theta)+\big{[}\partial_{a}\,\bar{F}^{a(c)}(\xi,\theta)\big{]}\,{\cal
E}^{(c)}(\xi,\theta)=\bar{C}^{a}(\xi)\,\partial_{a}\,E(\xi)+\big{[}\partial_{a}\,\bar{C}^{a}(\xi)\big{]}\,E(\xi),$
$\displaystyle\bar{F}^{a(c)}(\xi,\theta)\,\partial_{a}\,\tilde{G}^{mn(c)}(\xi,\theta)+\big{[}\partial_{a}\,\bar{F}^{a(c)}(\xi,\theta)\big{]}\,\tilde{G}^{mn(c)}(\xi,\theta)$
$\displaystyle-\big{[}\partial_{a}\,\bar{F}^{m(c)}(\xi,\theta)\big{]}\,\tilde{G}^{an(c)}(\xi,\theta)-\big{[}\partial_{a}\,\bar{F}^{n(c)}(\xi,\theta)\big{]}\,\tilde{G}^{ma(c)}(\xi,\theta)$
$\displaystyle=\bar{C}^{a}(\xi)\,\big{[}\partial_{a}\,\tilde{g}^{mn}(\xi)\big{]}+\big{[}\partial_{a}\,\bar{C}^{a}(\xi)\big{]}\,\tilde{g}^{mn}(\xi)-\big{[}\partial_{a}\,\bar{C}^{m}(\xi)\big{]}\,\tilde{g}^{an}(\xi)-\big{[}\partial_{a}\,\bar{C}^{n}(\xi)\big{]}\,\tilde{g}^{ma}(\xi),$
(56)

where we have taken the super expansions from (54) and
$X^{\mu(h,c)}(\xi,\theta)$ is the chiral limit $(\bar{\theta}=0)$ of the full
expansion [cf. (28)]. In other words, we have the following

$\displaystyle
X^{\mu(h,c)}(\xi,\theta)=X^{\mu}(\xi)+\theta\,[\bar{C}^{a}\,\partial_{a}\,X^{\mu}(\xi)],$
(57)

where the superscript $(h,c)$ denotes the chiral version of the full expansion
of $X^{\mu(h)}(\xi,\theta)$ that has been obtained in the previous section
[cf. Eq. (28)].

We would like to lay emphasis on the fact that all the secondary fields
$(\bar{f}_{1}^{m},\,f_{2},\,\tilde{\bar{R}}^{mn})$ and $\bar{b}_{1}^{m}$ can
be computed in exactly similar manner as we have done in the case of
determination of the BRST symmetry transformations $(s_{b})$ for the super
expansions in Eq. (29). It turns out that, adopting this logic, we obtain the
following:

$\displaystyle\bar{f}^{m}_{1}=\bar{B}^{a}\,\partial_{a}\,\bar{C}^{m}-(\partial_{a}\,\bar{B}^{a})\,\bar{C}^{m},\qquad
f_{2}=(\partial_{a}\,\bar{C}^{a})\,E+\bar{C}^{a}\,(\partial_{a}\,E),$
$\displaystyle\bar{b}_{1}^{m}=\bar{C}^{a}\,\partial_{a}\,\bar{C}^{m},\qquad\tilde{\bar{R}}^{mn}=\partial_{a}\,(\bar{C}^{a}\,\tilde{g}^{mn})-(\partial_{a}\,\bar{C}^{m})\,\tilde{g}^{an}-(\partial_{a}\,\bar{C}^{n})\,\tilde{g}^{ma}.$
(58)

Substitutions of the above secondary fields into the chiral super expansions
of Eq. (54), we obtain the following final super expansions

$\displaystyle{\cal B}^{m(c)}_{(ab)}(\xi,\theta)$ $\displaystyle=$
$\displaystyle
B^{m}(\xi)+\theta\,\big{[}\bar{C}^{a}\,\partial_{a}\,B^{m}-B^{a}\,\partial_{a}\,\bar{C}^{m}\big{]}\equiv
B^{m}(\xi)+\theta\,\big{[}s_{ab}\,B^{m}(\xi)\big{]},$ $\displaystyle{\cal
E}^{(c)}_{(ab)}(\xi,\theta)$ $\displaystyle=$ $\displaystyle
E(\xi)+\theta\,\big{[}\partial_{a}\,(\bar{C}^{a}\,E)\big{]}\equiv
E(\xi)+\theta\,\big{[}s_{ab}\,E(\xi)\big{]},$
$\displaystyle{\bar{F}}^{m(c)}_{(ab)}(\xi,\theta)$ $\displaystyle=$
$\displaystyle\bar{C}^{m}(\xi)+\theta\,\big{[}\bar{C}^{a}\,\partial_{a}\,\bar{C}^{m}\big{]}\equiv\bar{C}^{m}(\xi)+\theta\,\big{[}s_{ab}\,\bar{C}^{m}(\xi)\big{]},$
$\displaystyle{\tilde{G}}^{mn(c)}_{(ab)}(\xi,\theta)$ $\displaystyle=$
$\displaystyle\tilde{g}^{mn}(\xi)+\theta\,\big{[}\partial_{a}\,(\bar{C}^{a}\,\tilde{g}^{mn})-(\partial_{a}\,\bar{C}^{m})\,\tilde{g}^{an}-(\partial_{a}\,\bar{C}^{n})\,\tilde{g}^{ma}\big{]}$
(59) $\displaystyle\equiv$
$\displaystyle\tilde{g}^{mn}(\xi)+\theta\,\big{[}s_{ab}\,\tilde{g}^{mn}(\xi)\big{]},$

where the subscript $(ab)$ on the chiral superfields on the l.h.s. of the
above equation (59) denotes that the above superfields have been obtained
after the quantum gauge (i.e. anti-BRST) invariant restrictions on the chiral
superfields [cf. Eq. (56)] have been imposed. It can be readily checked that
we have obtained the anti-BRST symmetry transformations:
$s_{ab}\,B^{m}=\bar{C}^{a}\,\partial_{a}\,B^{m}-B^{a}\,\partial_{a}\,\bar{C}^{m},\,s_{ab}\,E=\partial_{a}\,[\bar{C}^{a}\,E],\,s_{ab}\,\bar{C}^{m}=\bar{C}^{a}\,\partial_{a}\,\bar{C}^{m},\,s_{ab}\,\tilde{g}^{mn}=\partial_{a}\,(\bar{C}^{a}\,\tilde{g}^{mn})-(\partial_{a}\,\bar{C}^{m})\,\tilde{g}^{an}-(\partial_{a}\,\bar{C}^{n})\,\tilde{g}^{ma}$
as the coefficients of the chiral super expansions in (59). It is nice to note
that $\partial_{\theta}\,\Omega_{(ab)}(\xi,\theta)=s_{ab}\,\omega(\xi)$ where
the generic chiral superfield $\Omega_{(ab)}(\xi,\theta)$ stands for the
l.h.s. of (59) and $\omega=B^{m},\,E,\,\bar{C}^{m},\,\tilde{g}^{mn}$ generic
ordinary field.

We end this section with the following remarks. First, we have derived the
(anti-)BRST symmetry transformations for the fields by exploiting the
theoretical tricks of ACSA to BRST formalism. These fields are the ones for
which the MBTSA has not been able to derive the (anti-)BRST symmetry
transformations. Second, a careful and close observation of the theoretical
contents of Secs. 3 and 4 demonstrate that we have derived all the nilpotent
(anti-)BRST symmetry transformations for our theory by exploiting the
theoretical strength of MBTSA and ACSA. Finally, the (anti-)BRST symmetry
transformations for the component fields $A_{0},\,A_{1}$ and $A_{2}$ of
$\tilde{g}^{mn}$ [cf. Eq. (4)] can be obtained from the exact expressions for
$s_{b}\,\tilde{g}^{mn}(\xi)$ and $s_{ab}\,\tilde{g}^{mn}(\xi)$ that have been
quoted in (7) and (6). To be more transparent, we find the following anti-
chiral super expansions:

$\displaystyle A_{0}(\xi)\quad\rightarrow\quad{\cal
A}^{(ac)}_{0(b)}(\xi,\bar{\theta})$ $\displaystyle=$ $\displaystyle
A_{0}(\xi)+\bar{\theta}\,\big{[}C^{m}\,\partial_{m}\,A_{0}-(\partial_{0}\,C^{1}-\partial_{1}\,C^{0})\,A_{2}$
$\displaystyle-$
$\displaystyle(\partial_{0}\,C^{1}+\partial_{1}\,C^{0})\,A_{1}\big{]}\equiv
A_{0}(\xi)+\bar{\theta}\,\big{[}s_{b}\,A_{0}(\xi)\big{]},$ $\displaystyle
A_{1}(\xi)\quad\rightarrow\quad{\cal A}^{(ac)}_{1(b)}(\xi,\bar{\theta})$
$\displaystyle=$ $\displaystyle
A_{1}(\xi)+\bar{\theta}\,\big{[}C^{m}\,\partial_{m}\,A_{1}-(\partial_{1}\,C^{0}+\partial_{0}\,C^{1})\,A_{0}$
$\displaystyle-$
$\displaystyle(\partial_{0}\,C^{0}-\partial_{1}\,C^{1})\,A_{2}\big{]}\equiv
A_{1}(\xi)+\bar{\theta}\,\big{[}s_{b}\,A_{1}(\xi)\big{]},$ $\displaystyle
A_{2}(\xi)\quad\rightarrow\quad{\cal A}^{(ac)}_{2(b)}(\xi,\bar{\theta})$
$\displaystyle=$ $\displaystyle
A_{2}(\xi)+\bar{\theta}\,\big{[}C^{m}\,\partial_{m}\,A_{2}-(\partial_{1}\,C^{0}-\partial_{0}\,C^{1})A_{0}$
(60) $\displaystyle-$
$\displaystyle(\partial_{0}\,C^{0}-\partial_{1}\,C^{1})A_{1}\big{]}\equiv
A_{2}(\xi)+\bar{\theta}\,\big{[}s_{b}\,A_{2}(\xi)\big{]},$

where the coefficients of $\bar{\theta}$ are nothing but the BRST symmetry
transformations [cf. Eq. (9)] on $A_{0}(\xi),\,A_{1}(\xi)$ and $A_{2}(\xi)$.
In exactly similar fashion, we can obtain the anti-BRST symmetry
transformations on $A_{0},\,A_{1}$ and $A_{2}$ from the following chiral super
expansions:

$\displaystyle A_{0}(\xi)\quad\rightarrow\quad{\cal
A}^{(c)}_{0(ab)}(\xi,\theta)$ $\displaystyle=$ $\displaystyle
A_{0}(\xi)+\theta\,\big{[}\bar{C}^{m}\,\partial_{m}\,A_{0}-(\partial_{0}\,\bar{C}^{1}-\partial_{1}\,\bar{C}^{0})\,A_{2}$
$\displaystyle-$
$\displaystyle(\partial_{0}\,\bar{C}^{1}+\partial_{1}\,\bar{C}^{0})\,A_{1}\big{]}\equiv
A_{0}(\xi)+\theta\,\big{[}s_{ab}\,A_{0}(\xi)\big{]},$ $\displaystyle
A_{1}(\xi)\quad\rightarrow\quad{\cal A}^{(c)}_{1(ab)}(\xi,\theta)$
$\displaystyle=$ $\displaystyle
A_{1}(\xi)+\theta\,\big{[}\bar{C}^{m}\,\partial_{m}\,A_{1}-(\partial_{1}\,\bar{C}^{0}+\partial_{0}\,\bar{C}^{1})\,A_{0}$
$\displaystyle-$
$\displaystyle(\partial_{0}\,\bar{C}^{0}-\partial_{1}\,\bar{C}^{1})\,A_{2}\big{]}\equiv
A_{1}(\xi)+\theta\,\big{[}s_{ab}\,A_{1}(\xi)\big{]},$ $\displaystyle
A_{2}(\xi)\quad\rightarrow\quad{\cal A}^{(c)}_{2(ab)}(\xi,\theta)$
$\displaystyle=$ $\displaystyle
A_{2}(\xi)+\theta\,\big{[}\bar{C}^{m}\,\partial_{m}\,A_{2}-(\partial_{1}\,\bar{C}^{0}-\partial_{0}\,\bar{C}^{1})A_{0}$
(61) $\displaystyle-$
$\displaystyle(\partial_{0}\,\bar{C}^{0}-\partial_{1}\,\bar{C}^{1})A_{1}\big{]}\equiv
A_{2}(\xi)+\theta\,\big{[}s_{ab}\,A_{2}(\xi)\big{]}.$

In the above, the coefficients of $\theta$ are nothing but the anti-BRST
symmetry transformations for the component fields $A_{0},\,A_{1}$ and $A_{2}$
[cf. Eq. (8)]. We lay emphasis on the fact that the super expansions in (60)
and (61) are very crucial and important as will be clear in the next section
where we shall discuss the symmetry invariances.

##  5 Invariance of the Lagrangian Densities: ACSA

In this section, we capture the (anti-)BRST invariance of the Lagrangian
densities (1) and (2) in terms of the (anti-)chiral superfields that have been
obtained after the imposition of the (anti-)BRST invariant restrictions. In
this connection, it is worth pointing out that we have already computed the
BRST invariance of ${\cal L}_{b}$ and anti-BRST invariance of ${\cal L}_{ab}$
in the ordinary space in our earlier work [35]. To be precise, the action
integrals $S_{1}=\int d^{2}\,\xi\,{\cal L}_{b}$ and $S_{2}=\int
d^{2}\,\xi\,{\cal L}_{ab}$ remain invariant under the continuous,
infinitesimal and nilpotent transformations in (7) and (6). In this
connection, first of all, we note that the following are true for the
classical Lagrangian density $({\cal L}_{0})$, namely;

$\displaystyle s_{b}\,{\cal L}_{0}=\partial_{a}\,\big{[}C^{a}\,{\cal
L}_{0}\big{]},\qquad\qquad s_{ab}\,{\cal
L}_{0}=\partial_{a}\,\big{[}\bar{C}^{a}\,{\cal L}_{0}\big{]},$ (62)

and the total Lagrangian densities ${\cal L}_{b}$ and ${\cal L}_{ab}$
transform as [35]:

$\displaystyle s_{b}\,{\cal L}_{b}$ $\displaystyle=$
$\displaystyle\partial_{a}\,\Bigl{[}C^{a}\big{(}{\cal
L}_{0}+B_{0}\,A_{0}+B_{1}\,A_{1}\big{)}+i\,\bar{C}_{1}C^{b}\,\partial_{b}\,(C^{a}\,A_{1})+i\,\bar{C}_{1}\,C^{a}(\partial_{0}\,C^{1}+\partial_{1}\,C^{0})\,A_{0}$
(63) $\displaystyle+$ $\displaystyle
i\,\bar{C}_{0}C^{b}\,\partial_{b}\,(C^{a}\,A_{0})+i\,\bar{C}_{0}\,C^{a}\,(\partial_{0}\,C^{1}+\partial_{1}\,C^{0})\,A_{1}+i\,\bar{C}_{0}\,C^{a}\,(\partial_{0}\,C^{1}-\partial_{1}\,C^{0})\,A_{2}$
$\displaystyle+$ $\displaystyle
i\,\bar{C}_{1}\,C^{a}\,(\partial_{0}\,C^{0}-\partial_{1}\,C^{1})\,A_{2}\Bigr{]},$
$\displaystyle s_{ab}\,{\cal L}_{ab}$ $\displaystyle=$
$\displaystyle\partial_{a}\,\Bigl{[}\bar{C}^{a}\big{(}{\cal
L}_{0}-\bar{B}_{0}\,A_{0}-\bar{B}_{1}\,A_{1}\big{)}-i\,C_{1}\,\bar{C}^{b}\,\partial_{b}\,(\bar{C}^{a}\,A_{1})-i\,C_{1}\,\bar{C}^{a}(\partial_{0}\,\bar{C}^{1}+\partial_{1}\,\bar{C}^{0})\,A_{0}$
(64) $\displaystyle-$ $\displaystyle
i\,C_{0}\,\bar{C}^{b}\,\partial_{b}\,(\bar{C}^{a}\,A_{0})-i\,C_{0}\,\bar{C}^{a}\,(\partial_{0}\,\bar{C}^{1}+\partial_{1}\,\bar{C}^{0})\,A_{1}-i\,C_{0}\,\bar{C}^{a}\,(\partial_{0}\,\bar{C}^{1}-\partial_{1}\,\bar{C}^{0})\,A_{2}$
$\displaystyle-$ $\displaystyle
i\,C_{1}\,\bar{C}^{a}\,(\partial_{0}\,\bar{C}^{0}-\partial_{1}\,\bar{C}^{1})\,A_{2}\Bigr{]}.$

The above observations demonstrate that $s_{b}\,S_{1}=0$ and $s_{ab}\,S_{2}=0$
for the physical fields of the (anti-)BRST invariant theories which vanish off
at $\sigma=0,\,\pi$ and $\tau\rightarrow\pm\,\infty$ due to Gauss’s divergence
theorem. We mention, in passing, that $s_{b}\,S_{0}=0$ and $s_{ab}\,S_{0}=0$
(where $S_{0}=\int d^{2}\,\xi\,{\cal L}_{0}$) due to the (anti-)BRST
transformations for ${\cal L}_{0}$ in (62).

First of all, we capture the (anti-)BRST invariance of the action integral
$S_{0}=\int d^{2}\,\xi\,{\cal L}_{0}$ within the realm of ACSA. In this
regard, we note the following (anti-)chiral generalizations of ${\cal L}_{0}$
to its counterpart super Lagrangians (i.e. ${\cal L}_{0}\rightarrow\tilde{\cal
L}_{0}$) on the $(2,1)$-dimensional (anti-) chiral super submanifolds, namely;

$\displaystyle{\cal L}_{0}\quad\rightarrow\quad\tilde{\cal
L}_{0}^{(ac)}(\xi,\bar{\theta})$ $\displaystyle=$
$\displaystyle-\frac{1}{2k}\,{\tilde{G}}^{mn(ac)}_{(b)}(\xi,\bar{\theta})\,\partial_{m}\,{\tilde{X}}^{\mu(h,ac)}(\xi,\bar{\theta})\,\partial_{n}\,{\tilde{X}}_{\mu}^{(h,ac)}(\xi,\bar{\theta})$
$\displaystyle+$ $\displaystyle{\cal
E}^{(ac)}_{(b)}(\xi,\bar{\theta})\,\big{[}\mbox{det}\,\tilde{G}^{(ac)}_{(b)}(\xi,\bar{\theta})+1\big{]},$
$\displaystyle{\cal L}_{0}\quad\rightarrow\quad\tilde{\cal
L}_{0}^{(c)}(\xi,\theta)$ $\displaystyle=$
$\displaystyle-\frac{1}{2k}\,{\tilde{G}}^{mn(c)}_{(ab)}(\xi,\theta)\,\partial_{m}\,{\tilde{X}}^{\mu(h,c)}(\xi,\theta)\,\partial_{n}\,{\tilde{X}}_{\mu}^{(h,c)}(\xi,\theta)$
(65) $\displaystyle+$ $\displaystyle{\cal
E}^{(c)}_{(ab)}(\xi,\theta)\,\big{[}\mbox{det}\,\tilde{G}^{(c)}_{(ab)}(\xi,\theta)+1\big{]},$

where the super Lagrangian densities (on the l.h.s.) carry superscripts $(ac)$
and $(c)$ to denote that these have been defined on the (2, 1)-dimensional
(anti-)chiral super submanifolds of the (2, 2)-dimensional general
supermanifold (that has been chosen for our discussion). The superfields with
subscripts $(b)$ and $(ab)$ as well as with superscripts $(ac)$, $(c)$,
$(h,\,c)$ and $(h,\,ac)$ have already been explained in our previous and
present sections. The equation (62) can be captured in the superspace (where
ACSA plays an important role). The mappings:
$s_{b}\leftrightarrow\partial_{\bar{\theta}}$,
$s_{ab}\leftrightarrow\partial_{\theta}$ lead to the following observations:

$\displaystyle\frac{\partial}{\partial\,\bar{\theta}}\,\tilde{\cal
L}_{0}^{(ac)}(\xi,\bar{\theta})=\partial_{a}\,\big{[}C^{a}\,{\cal
L}_{0}\big{]}\equiv s_{b}\,{\cal
L}_{0},\qquad\;\;\frac{\partial}{\partial\,\theta}\,\tilde{\cal
L}_{0}^{(c)}(\xi,\theta)=\partial_{a}\,\big{[}\bar{C}^{a}\,{\cal
L}_{0}\big{]}\equiv s_{ab}\,{\cal L}_{0}.$ (66)

Thus, the (anti-)BRST symmetry invariances of ${\cal L}_{0}$ have been
expressed in the language of ACSA to BRST formalism. We have performed this
exercise separately because, on its own, the original classical Lagrangian
density ${\cal L}_{0}$ transforms to the total derivatives [cf. Eq. (62)]
under the (anti-)BRST symmetry transformations.

We would like to express the symmetry transformations (63) and (64) in the
realm of ACSA where the super expansions in (27), (32), (33), (40), (44), (53)
and (60) will be playing decisive role for the BRST invariance [cf. Eq. (63)].
On the other hand, the super expansions (27), (57), (59) and (61) will be very
useful in capturing the anti-BRST invariance [cf. Eq. (64)]. With these inputs
at our disposal, we set out to capture the BRST invariance in terms of
$\partial_{\bar{\theta}}$ and ${\tilde{\cal L}^{ac}_{b}}(\xi,\,\bar{\theta})$.
Here the latter is given by the following expression in the language of the
anti-chiral superfields that have been derived after the imposition of the
BRST-invariant restrictions. These anti-chiral superfields might also be the
limiting cases of the full super expansions that have been derived in Sec. 3,
namely;.

$\displaystyle\tilde{\cal L}_{b}^{(ac)}(\xi,\bar{\theta})$ $\displaystyle=$
$\displaystyle\tilde{\cal L}_{0}^{(ac)}(\xi,\bar{\theta})+B_{0}(\xi)\,{\cal
A}_{0(b)}^{(ac)}(\xi,\bar{\theta})+B_{1}(\xi)\,{\cal
A}_{1(b)}^{(ac)}(\xi,\bar{\theta})$ (67) $\displaystyle-$ $\displaystyle
i\,\Big{[}{\bar{F}}_{1(b)}^{(ac)}(\xi,\bar{\theta})\,\Big{\\{}\partial_{0}\,{F}^{1(ac)}_{(b)}(\xi,\bar{\theta})+\partial_{1}\,{F}^{0(ac)}_{(b)}(\xi,\bar{\theta})\Big{\\}}$
$\displaystyle+$
$\displaystyle{\bar{F}}^{(ac)}_{0(b)}(\xi,\bar{\theta})\,\Big{\\{}\partial_{a}\,{F}^{a(ac)}_{(b)}(\xi,\bar{\theta})\Big{\\}}-{F}^{a(ac)}_{(b)}(\xi,\bar{\theta})\,\Big{\\{}\partial_{a}\,{\bar{F}}^{(ac)}_{0(b)}(\xi,\bar{\theta})\Big{\\}}\Big{]}\,{\cal
A}_{0(b)}^{(ac)}(\xi,\bar{\theta})$ $\displaystyle-$ $\displaystyle
i\,\Big{[}{\bar{F}}_{0(b)}^{(ac)}(\xi,\bar{\theta})\,\Big{\\{}\partial_{0}\,{F}^{1(ac)}_{(b)}(\xi,\bar{\theta})+\partial_{1}\,{F}^{0(ac)}_{(b)}(\xi,\bar{\theta})\Big{\\}}$
$\displaystyle-$
$\displaystyle{F}^{a(ac)}_{(b)}(\xi,\bar{\theta})\,\Big{\\{}\partial_{a}\,{\bar{F}}^{(ac)}_{1(b)}(\xi,\bar{\theta})\Big{\\}}+{\bar{F}}^{(ac)}_{1(b)}(\xi,\bar{\theta})\,\Big{\\{}\partial_{a}\,{F}^{a(ac)}_{(b)}(\xi,\bar{\theta})\Big{\\}}\Big{]}\,{\cal
A}_{1(b)}^{(ac)}(\xi,\bar{\theta})$ $\displaystyle-$ $\displaystyle
i\,\Big{[}{\bar{F}}_{1(b)}^{(ac)}(\xi,\bar{\theta})\,\Big{\\{}\partial_{0}\,{F}^{0(ac)}_{(b)}(\xi,\bar{\theta})-\partial_{1}\,{F}^{1(ac)}_{(b)}(\xi,\bar{\theta})\Big{\\}}$
$\displaystyle+$
$\displaystyle{\bar{F}}_{0(b)}^{(ac)}(\xi,\bar{\theta})\,\Big{\\{}\partial_{0}\,{F}^{1(ac)}_{(b)}(\xi,\bar{\theta})-\partial_{1}\,{F}^{0(ac)}_{(b)}(\xi,\bar{\theta})\Big{\\}}\Big{]}\,{\cal
A}_{2(b)}^{(ac)}(\xi,\bar{\theta}),$

where we have taken the ordinary fields $B_{0}(\xi)$ and $B_{1}(\xi)$ because
we know that $B^{m}(\xi)\rightarrow{\cal
B}^{m}_{(b)}(\xi,\,\bar{\theta})=B^{m}(\xi)$ due to the BRST invariance
$[s_{b}\,B^{m}(\xi)=0]$ of $B^{m}(\xi)]$. Ultimately, it turns out that we
obtain the following due to operation of $\partial_{\bar{\theta}}$ on
$\tilde{\cal L}_{b}^{(ac)}(\xi,\bar{\theta})$:

$\displaystyle\frac{\partial}{\partial\,\bar{\theta}}\,\tilde{\cal
L}_{b}^{(ac)}(\xi,\bar{\theta})$ $\displaystyle=$
$\displaystyle\partial_{a}\,\Bigl{[}C^{a}\big{(}{\cal
L}_{0}+B_{0}\,A_{0}+B_{1}\,A_{1}\big{)}+i\,\bar{C}_{1}C^{b}\,\partial_{b}\,(C^{a}\,A_{1})$
(68) $\displaystyle+$ $\displaystyle
i\,\bar{C}_{1}\,C^{a}(\partial_{0}\,C^{1}+\partial_{1}\,C^{0})\,A_{0}+i\,\bar{C}_{0}C^{b}\,\partial_{b}\,(C^{a}\,A_{0})$
$\displaystyle+$ $\displaystyle
i\,\bar{C}_{0}\,C^{a}\,(\partial_{0}\,C^{1}+\partial_{1}\,C^{0})\,A_{1}+i\,\bar{C}_{0}\,C^{a}\,(\partial_{0}\,C^{1}-\partial_{1}\,C^{0})\,A_{2}$
$\displaystyle+$ $\displaystyle
i\,\bar{C}_{1}\,C^{a}\,(\partial_{0}\,C^{0}-\partial_{1}\,C^{1})\,A_{2}\Bigr{]}\equiv
s_{b}\,{\cal L}_{b},$

It is evident that the above equation captures the BRST invariance of the
Lagrangian density ${\cal L}_{b}$ in the superspace (as is clear from our
observation on the r.h.s.).

We can repeat the same exercise for the anti-BRST invariance. For this
purpose, first of all, we generalize ${\cal L}_{ab}$ to its counterpart chiral
super Lagrangian density on the (2, 1)-dimensional chiral super submanifold as

$\displaystyle\tilde{\cal L}_{ab}^{(c)}(\xi,\theta)$ $\displaystyle=$
$\displaystyle\tilde{\cal L}_{0}^{(c)}(\xi,\theta)-\bar{B}_{0}(\xi)\,{\cal
A}_{0(ab)}^{(c)}(\xi,\theta)-\bar{B}_{1}(\xi)\,{\cal
A}_{1(ab)}^{(c)}(\xi,\theta)$ $\displaystyle+$ $\displaystyle
i\,\Big{[}{F}_{1(ab)}^{(c)}(\xi,\theta)\,\Big{\\{}\partial_{0}\,{\bar{F}}^{1(c)}_{(ab)}(\xi,\theta)+\partial_{1}\,{\bar{F}}^{0(c)}_{(ab)}(\xi,\theta)\Big{\\}}$
$\displaystyle+$
$\displaystyle{F}^{(c)}_{0(ab)}(\xi,\theta)\,\Big{\\{}\partial_{a}\,{\bar{F}}^{a(c)}_{(ab)}(\xi,\theta)\Big{\\}}+\Big{\\{}\partial_{a}\,{F}^{(c)}_{0(ab)}(\xi,\theta)\Big{\\}}\,{\bar{F}}^{a(c)}_{(ab)}(\xi,\theta)\Big{]}\,{\cal
A}_{0(ab)}^{(c)}(\xi,\theta)$ (69) $\displaystyle+$ $\displaystyle
i\,\Big{[}{F}_{0(ab)}^{(c)}(\xi,\theta)\,\Big{\\{}\partial_{0}\,{\bar{F}}^{1(c)}_{(ab)}(\xi,\theta)+\partial_{1}\,{\bar{F}}^{0(c)}_{(ab)}(\xi,\theta)\Big{\\}}$
$\displaystyle+$
$\displaystyle{F}^{(c)}_{1(ab)}(\xi,\theta)\,\Big{\\{}\partial_{a}\,{\bar{F}}^{a(c)}_{(ab)}(\xi,\theta)\Big{\\}}+\Big{\\{}\partial_{a}\,{F}^{(c)}_{1(ab)}(\xi,\bar{\theta})\Big{\\}}\,{\bar{F}}^{a(c)}_{(ab)}(\xi,\theta)\Big{]}\,{\cal
A}_{1(ab)}^{(c)}(\xi,\theta)$ $\displaystyle+$ $\displaystyle
i\,\Big{[}{F}_{1(ab)}^{(c)}(\xi,\theta)\,\Big{\\{}\partial_{0}\,{\bar{F}}^{0(c)}_{(ab)}(\xi,\theta)-\partial_{1}\,{\bar{F}}^{1(c)}_{(ab)}(\xi,\theta)\Big{\\}}$
$\displaystyle+$
$\displaystyle{F}_{0(ab)}^{(c)}(\xi,\theta)\,\Big{\\{}\partial_{0}\,{\bar{F}}^{1(c)}_{(ab)}(\xi,\theta)-\partial_{1}\,{\bar{F}}^{0(c)}_{(ab)}(\xi,\theta)\Big{\\}}\Big{]}\,{\cal
A}_{2(ab)}^{(c)}(\xi,\theta),$

where the ordinary fields ${\bar{B}}_{0}(\xi)$ and ${\bar{B}}_{1}(\xi)$ are
present in the above super Lagrangian density because
$s_{ab}\,{\bar{B}}^{a}=0$ which implies that
${\bar{B}}^{a}(\xi)\rightarrow\bar{\cal
B}_{(ab)}^{a(c)}(\xi,\theta)={\bar{B}}^{a}(\xi)+\theta\,(0)\equiv{\bar{B}}^{a}(\xi)$.
In other words, there is no chiral $\theta$-dependence on the r.h.s. of the
super expansion of the superfield $\bar{\cal B}_{(ab)}^{a(c)}(\xi,\theta)$.
The rest of the notations for the chiral superfields have already been
explained earlier. At this juncture, in view of the mapping:
$s_{ab}\leftrightarrow\partial_{\theta}$, we can capture the anti-BRST
invariance (64) by applying a derivative $\partial_{\theta}$ on (69) which
yields the following

$\displaystyle\frac{\partial}{\partial\,\theta}\,\tilde{\cal
L}_{ab}^{(c)}(\xi,\theta)$ $\displaystyle=$
$\displaystyle\partial_{a}\,\Bigl{[}\bar{C}^{a}\big{(}{\cal
L}_{0}-\bar{B}_{0}\,A_{0}-\bar{B}_{1}\,A_{1}\big{)}-i\,C_{1}\,\bar{C}^{b}\,\partial_{b}\,(\bar{C}^{a}\,A_{1})$
(70) $\displaystyle-$ $\displaystyle
i\,C_{1}\,\bar{C}^{a}(\partial_{0}\,\bar{C}^{1}+\partial_{1}\,\bar{C}^{0})\,A_{0}-i\,C_{0}\,\bar{C}^{b}\,\partial_{b}\,(\bar{C}^{a}\,A_{0})$
$\displaystyle-$ $\displaystyle
i\,C_{0}\,\bar{C}^{a}\,(\partial_{0}\,\bar{C}^{1}+\partial_{1}\,\bar{C}^{0})\,A_{1}-i\,C_{0}\,\bar{C}^{a}\,(\partial_{0}\,\bar{C}^{1}-\partial_{1}\,\bar{C}^{0})\,A_{2}$
$\displaystyle-$ $\displaystyle
i\,C_{1}\,\bar{C}^{a}\,(\partial_{0}\,\bar{C}^{0}-\partial_{1}\,\bar{C}^{1})\,A_{2}\Bigr{]}\equiv
s_{ab}\,{\cal L}_{ab}.$

Hence we have captured the anti-BRST symmetry invariance (64) in the language
of ACSA to BRST formalism [as is evident from the r.h.s. of (70)].

We close this section with the following remark. We can capture the basic
ideas behind the derivations of ${\cal L}_{b}$ and ${\cal L}_{ab}$ which have
been explained in Eq. (5). In view of the mappings:
$s_{b}\leftrightarrow\partial_{\bar{\theta}},\,s_{ab}\leftrightarrow\partial_{\theta}$,
we can express the super (anti-)BRST invariant Lagrangian densities
corresponding to the ordinary Lagrangian densities [cf. Eq. (5)] as

$\displaystyle\tilde{\cal L}_{ab}^{(c)}(\xi,\theta)$ $\displaystyle=$
$\displaystyle{\cal
L}_{0}^{(c)}(\xi,\theta)+\frac{\partial}{\partial\,\theta}\,\Big{[}i\,F^{(c)}_{0(ab)}(\xi,\theta)\,{\cal
A}^{(c)}_{0(ab)}(\xi,\theta)+i\,F^{(c)}_{1(ab)}(\xi,\theta)\,{\cal
A}^{(c)}_{1(ab)}(\xi,\theta)\Big{]},$ $\displaystyle\tilde{\cal
L}_{b}^{(ac)}(\xi,\bar{\theta})$ $\displaystyle=$ $\displaystyle{\cal
L}_{0}^{(ac)}(\xi,\bar{\theta})+\frac{\partial}{\partial\,\bar{\theta}}\,\Big{[}-i\,\bar{F}^{(ac)}_{0(b)}(\xi,\bar{\theta})\,{\cal
A}^{(ac)}_{0(b)}(\xi,\bar{\theta})-i\,\bar{F}^{(ac)}_{1(b)}(\xi,\bar{\theta})\,{\cal
A}^{(ac)}_{1(b)}(\xi,\bar{\theta})\Big{]},$ (71)

where all the symbols have been explained in our earlier discussion. It is
crystal clear, from the above expression, that the (anti-)BRST invariance of
the action integrals $S_{1}=\int d^{2}\,\xi\,{\cal L}_{b}$ and $S_{2}=\int
d^{2}\,\xi\,{\cal L}_{ab}$ can be captured in the terminology of ACSA to BRST
formalism because $s_{b}\,S_{1}$ and $s_{ab}\,S_{2}$ will be zero in the
ordinary space. Furthermore, we note that $\partial_{\theta}\,\tilde{\cal
L}_{ab}^{(c)}(\xi,\theta)$ and $\partial_{\bar{\theta}}\,\tilde{\cal
L}_{b}^{(ac)}(\xi,\bar{\theta})$ will always produce the total derivatives in
the ordinary space thereby rendering the action integrals (i.e. $S_{1}$ and
$S_{2}$) equal to zero [cf. Eq. (71)]. To be precise, the nilpotency
$(\partial_{\bar{\theta}}^{2}=0,\,\partial_{\theta}^{2}=0)$ property of the
translational generators $(\partial_{\theta},\,\partial_{\bar{\theta}})$ will
ensure that $\partial_{\theta}\,\tilde{\cal L}_{ab}^{(c)}(\xi,\theta)$ and
$\partial_{\bar{\theta}}\,\tilde{\cal L}_{b}^{(ac)}(\xi,\bar{\theta})$ will be
always the total derivatives in the ordinary space. Hence, we are able to
capture the symmetry invariance(s) of the action integrals (corresponding to
the Lagrangian densities ${\cal L}_{b}$ and ${\cal L}_{ab}$) using ACSA.

##  6 Conclusions

In our present endeavor, we have exploited the theoretical potential of MBTSA
and ACSA to derive all the (anti-)BRST symmetry transformations for the 2D
diffeomorphism symmetry invariant model of a bosonic string theory. These
symmetry transformations are proper because they are off-shell nilpotent
$[s_{(a)b}^{2}=0]$ of order two and absolutely anticommuting (i.e.
$s_{b}\,s_{ab}+s_{ab}\,s_{b}=0$) in nature [cf. Eqs. (10),(7),(6)]. The latter
property of the (anti-) BRST symmetry transformations [$s_{(a)b}$] is
satisfied if and only if we invoke the sanctity of the CF-type restrictions:
$B^{a}+\bar{B}^{a}+i\,(C^{m}\,\partial_{m}\,\bar{C}^{a}+\bar{C}^{m}\,\partial_{m}\,C^{a})=0$
(with $a,m=0,1$) which define a submanifold in the quantum Hilbert space of
fields where the Nakanishi-Lautrup type auxiliary fields as well as
(anti-)ghost fields are present algebraically in a specific manner [cf. Eq.
(24)]. These restrictions are physical in some sense because they are
(anti-)BRST symmetry invariant [cf. Eqs. (7),(6)] on the submanifold (that is
defined above). Hence, their imposition on our BRST-quantized theory is
logical.

By applying the theoretical strength of MBTSA, we have been able to derive, in
one stroke, the (anti-)BRST symmetry transformations together for the Lorentz
pure scalar fields [e.g. $X^{\mu}(\xi),\,(\mbox{det}\,\tilde{g})$] and the 2D
version of the universal CF-type restrictions:
$B^{a}+\bar{B}^{a}+i\,(C^{m}\,\partial_{m}\,\bar{C}^{a}+\bar{C}^{m}\,\partial_{m}\,C^{a})=0$.
These 2D restrictions are the limiting case of the D-dimensional
diffeomorphism invarant theory where the superfield approach (developed by
Bonora [18]) leads to the existence of the D-dimensional CF-type restrictions:
$B_{\mu}+\bar{B}_{\mu}+i\,(C^{\rho}\,\partial_{\rho}\,\bar{C}_{\mu}+\bar{C}^{\rho}\,\partial_{\rho}\,C_{\mu})=0$
(with $\mu=0,\,1,\,2...D-1$) where the fermionic (anti-)ghost fields
$(\bar{C}_{\mu})C_{\mu}$ correspond to the D-dimensional infinitesimal and
continuous diffeomorphism symmetry transformations: $x_{\mu}\rightarrow
x_{\mu}^{{}^{\prime}}=x_{\mu}-\epsilon_{\mu}(x)$. In these infinitesimal
transformations, the parameters $\epsilon_{\mu}(x)$ are the diffeomorphism
transformation parameters. The symbols $(\bar{B}_{\mu})B_{\mu}$ are nothing
but the Nakanishi-Lautrup type auxiliary fields in the D-dimensional theory.
The existence of the D-dimensional CF-type restrictions:
$B^{\mu}+\bar{B}^{\mu}+i\,(C^{\rho}\,\partial_{\rho}\,\bar{C}^{\mu}+\bar{C}^{\rho}\,\partial_{\rho}\,C^{\mu})=0$
are universal and, so far, their presence have been shown explicitly in the
cases of 2D and 1D diffeomorphism invariant theories (see, e.g. [19-21] for
details).

Within the ambit of MBTSA, it becomes evident that we have to take, at least,
the helps of the (anti-)chiral superfield expansions [cf. Eq. (27)] so that we
can obtain $s_{b}\,\bar{C}_{\mu}=i\,B_{\mu}$ and
$s_{ab}\,C_{\mu}=i\,\bar{B}_{\mu}$ for the D-dimensional diffeomorphism
invariant theory (see e.g. [18] for details) in addition to the validity of
off-shell nilpotency property so that we can obtain:
$s_{b}\,C_{\mu}=C^{\rho}\,\partial_{\rho}\,C_{\mu}$ and
$s_{ab}\,\bar{C}_{\mu}=\bar{C}^{\rho}\,\partial_{\rho}\,\bar{C}_{\mu}$. The
above two inputs are essential for the completeness of MBTSA. Hence, we have
exploited the theoretical potential of the ACSA to BRST formalism [22-26] so
that both the above inputs can be taken care of. As a consequence, it becomes
important to blend together the ideas from the MBTSA and ACSA so that we can
derive all the (anti-)BRST symmetry transformations for the all the fields of
a diffeomorphism invariant theory along with the derivation of appropriate
(anti-)BRST invariant CF-type restrictions. This is what we have precisely
done in our present investigation. Our earlier works [19-21] on the 1D
diffeomorphism invariant models of the relativistic and non-relativistic
particles (of SUSY and non-SUSY varieties) have also exploited the ideas
behind MBTSA and ACSA together to obtain the 1D version
$[B+\bar{B}+i\,(\bar{C}\,\dot{C}-\dot{\bar{C}}\,C)=0]$ of the universal
D-dimensional CF-type restrictions.

In our earlier work [35] on our present bosonic string, we have computed the
expressions for the BRST and anti-BRST charges in the flat space. In the paper
by Kato and Ogawa [32], the nilpotency of BRST charge has been proven to
demonstrate that quantum version of the theory is valid only when $D=26$ and
$\alpha_{0}=1$. It will be very nice future endeavor for us to take the
expression for the anti-BRST charge and plug in the normal mode expansions of
the fields (with creation and annihilation operators in it) so that the
quantum version of it can be obtained. With appropriate boundary conditions on
the target space coordinate fields and (anti-)ghost fields, it will be
challenging to derive $D=26$ and $\alpha_{0}=1$ from the requirement of the
nilpotency of the anti-BRST charge in the flat limit. We are presently
involved with this problem and our results/observations will be reported
elsewhere.

Acknowledgments

Two of us (AT and AKR) would like to express their deep sense of gratefulness
towards Banaras Hindu University (BHU) for its financial support through the
BHU-fellowship. Prof G. Rajasekaran is one of the very prominent mentors of
our group at BHU and all three of us would like to dedicate this work very
respectfully to him on the occasion of his $85^{th}$ birth-anniversary which
will be celebrated on the forthcoming $22^{nd}$ February 2021.

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