# Reparameterization Invariant Model of a Supersymmetric System: BRST and Supervariable Approaches

Authors: A. Tripathi; B. Chauhan; A. K. Rao; R. P. Malik
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

We carry out the Becchi-Rouet-Stora-Tyutin (BRST) quantization of the one0+1-dimensional (1D) model of a free massive spinning relativistic particle (i.e., a supersymmetric system) by exploiting its classical infinitesimal and continuous reparameterization symmetry transformations. We use the modified Bonora-Tonin (BT) supervariable approach (MBTSA) to BRST formalism to obtain the nilpotent (anti-)BRST symmetry transformations of the target space variables and the (anti-)BRST invariant Curci-Ferrari- (CF-) type restriction for the 1D model of our supersymmetric (SUSY) system. The nilpotent (anti-)BRST symmetry transformations for other variables of our model are derived by using the (anti-)chiral supervariable approach (ACSA) to BRST formalism. Within the framework of the latter, we have shown the existence of the CF-type restriction by proving the (i) symmetry invariance of the coupled Lagrangians and (ii) the absolute anticommutativity property of the conserved (anti-)BRST charges. The application of the MBTSA to a physical SUSY system (i.e., a 1D model of a massive spinning particle) is a novel result in our present endeavor. In the application of ACSA, we have considered only the (anti-)chiral super expansions of the supervariables. Hence, the observation of the absolute anticommutativity of the (anti-)BRST charges is a novel result. The CF-type restriction is universal in nature as it turns out to be the same for the SUSY and non-SUSY reparameterization (i.e., 1D diffeomorphism) invariant models of the (non-)relativistic particles.


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## Body

## 1. Introduction

The Becchi-Rouet-Stora-Tyutin (BRST) quantization scheme is one of the most elegant approaches to quantize the locally gaugeand diffeomorphism invariant theories where the local classical transformation parameters are traded with the (anti-)ghost fields at the quantum level [1–4]. For the quantization of the classical supersymmetric gauge theories (with the bosonic and fermionic transformation parameters), the BRST quantization scheme requires the fermionic as well as the bosonic (anti-)ghost fields/variables (see, e.g., [5, 6]). Some of the key characteristic features of the BRST quantization scheme are (i) for a given local gauge and/or diffeomorphism symmetry, there exist two nilpotent symmetries which are christened as the BRST and anti-BRST symmetries, (ii) the (anti-)BRST symmetries sab are fermionic (i.e., nilpotent) and absolutely anticommuting (i.e., sbsab+sabsb=0) in nature, (iii) the quantum gauge (i.e., BRST) invariance and unitarity are respected together in the perturbative computations at any arbitrary order, (iv) there is appearance of the Curci-Ferrari- (CF-) type restriction(s) which are responsible for the absolute anticommutativity of the (anti-)BRST transformations and they lead to the existence of the coupled (but equivalent) Lagrangian (densities) for the BRST quantized theory which respect both (i.e., BRST and anti-BRST) nilpotent quantum symmetries, and (v) the (anti-)BRST symmetries transform a bosonic field/variable to a fermionic field/variable and vice-versa. Hence, these symmetries are supersymmetric type (i.e., SUSY type).Physically, the nilpotency property of the (anti-)BRST symmetry transformations corresponds to thefermionic nature of these quantum symmetries and the absolute anticommutativity property encodes the linear independence of the BRST and anti-BRST symmetry transformations. The former property encodes the SUSY-type transformations. As pointed out earlier, the absolute anticommutativity property owes its dependence on the existence of the CF-type restriction(s). The BRST approach to Abelian 1-form gauge theory is an exception where the CF-type restriction is trivial (but it turns out to be the limiting case of the non-Abelian 1-form theory which is endowed with the CF condition [7]). It is the usual superfield approach (USFA) to BRST formalism [8–15] which provides the interpretation and origin for the abstract mathematical properties (i.e., nilpotency and anticommutativity) that are associated with the (anti-)BRST symmetries. Furthermore, the USFA leads to the derivation of the CF condition [7] in the context of a 4D 1-form non-Abelian theory (see, e.g., [10–12]) which is found to be an (anti-)BRST invariant quantity. Hence, it is a physical restriction on the BRST quantized theory.The USFA to BRST formalism [8–15] leads to the derivation of only the off-shell nilpotent symmetries that are associated with the gauge and (anti-)ghost fields/variables. It does not shed any light on the (anti-)BRST transformations that are associated with the matter fields in an interacting gauge theory. There have been consistent extensions of the USFA (see, e.g., [16–18]) where additional quantum gauge invariant restrictions on the superfields have been imposed to derive the (anti-)BRST symmetry transformations for the gauge, (anti-)ghost, and matter fields together. This extended version of the superfield approach to BRST formalism has been called as the augmented version of superfield approach (AVSA). In our recent works (see, e.g., [19–21]), we have been able to develop a simpler off-shoot of AVSA where only the (anti-)chiral superfields/supervariables have been taken into account. The quantum gauge (i.e., (anti-)BRST) invariant restrictions on these (anti-)chiral superfields/supervariables have led to the deduction of (anti-)BRST symmetry transformations for all the fields/variables. This approach to BRST formalism has been named as the (anti-)chiral superfield/supervariable approach (ACSA) to BRST formalism where the existence of the CF-type restriction(s) has been shown by proving (i) the absolute anticommutativity of the (anti-)BRST charges and (ii) the invariance of the coupled (but equivalent) Lagrangian (densities) of the (anti-)BRST invariant theories.It has been a challenging problem to apply the superfield approach to BRST formalism in the context of (super)string and gravitational theories which are diffeomorphism invariant. In a recent paper [22], it has been proposed that the diffeomorphism symmetry can be taken into consideration within the framework of superfield approach to BRST formalism. This approach to BRST formalism has been called as the modified Bonora-Tonin superfield/supervariable approach (MBTSA) to BRST formalism which has been recently applied to the physical system of a 1D scalar (non-)relativistic particles [23, 24]. To be precise, judicious combination of MBTSA and ACSA has been very fruitful in our recent works [23, 24] where we have been able to derive the proper (anti-)BRST symmetries for all the quantum variables along with the CF-type restriction in a systematic fashion. In the proposal by Bonora and Malik [25], the diffeomorphism symmetry transformations have been incorporated into the superfields which are defined on a D,2-dimensional supermanifold on which a D-dimensional ordinary diffeomorphism invariant theory is generalized. We have exploited the mathematical rigor and beauty of the MBTSA in our present endeavor for the BRST analysis as well as quantization of a 1D diffeomorphism invariant SUSY system.To be precise, in our present investigation, we have applied the theoretical beauty and strength of MBTSA to derive the off-shell nilpotent (anti-)BRST transformations for the target space canonically conjugatebosonic variables (xμ and pμ) and fermionic variables ψμ,ψ5 along with the (anti-)BRST invariant CF-type restrictions: B+B¯+iC¯Ċ−C¯̇C=0 which is responsible for (i) the validity (cf. Equation (22)) of the absolute anticommutativity (i.e., sb,sab=sbsab+sabsb=0) of the off-shell nilpotent (sab2=0) (anti-)BRST symmetry transformations sab and (ii) the derivation of LB and LB¯ which are coupled (but equivalent) (cf. Equation (24)). The proper (anti-)BRST transformations of the rest of the variables have been derived by using the ACSA to BRST formalism. It is worth pointing out that, in the case of MBTSA, we have taken into account the full super expansions of the supervariables along all the possible Grassmannian directions of the general1,2-dimensional supermanifold. On the other hand, we have performed only the (anti-)chiral super expansions of the supervariables in the context of ACSA to BRST formalism. We have derived the exact expression for the CF-type restriction by demanding (i) the invariance of the coupled (but equivalent) Lagrangians and (ii) the validity of the absolute anticommutativity of the off-shell nilpotent (anti-)BRST symmetries as well as conserved (anti-)BRST charges in the ordinary space and in the superspace (within the framework of ACSA to BRST formalism). These derivations of the CF-type restrictions, by various theoretical methods, are novel results in our present investigation.The following key factors have been at the heart of our curiosity to pursue our present investigation. First, we have been able to apply the MBTSA to a reparameterization invariant model of thescalar relativistic particle to derive the (anti-)BRST symmetry transformations as well as the (anti-)BRST invariant CF type of restriction [24]. Thus, it has been very important for us to apply the same mathematical technique (i.e., MBTSA) to a physically interesting SUSY model of a reparameterization invariant theory where the fermionic as well as bosonic variables exist. Second, we have been very curious to verify the universality of the CF-type restriction in the context of BRST quantization of the 1D diffeomorphism (i.e., reparameterization) invariant theories. We find that the nature and form of the CF-type restriction are the same for the SUSY as well as non-SUSY theories. Third, it has been very interesting to note that the gauge-fixing and Faddeev-Popov ghost terms together are the same for the reparameterization invariant scalar and SUSY relativistic as well as a nonrelativistic particle [23]. Finally, our present investigation (as well as others [23, 24]) is our modest initial steps to apply the MBTSA as well as the ACSA to BRST formalism together to physically interesting 4D (and higher dimensional) diffeomorphism invariant theories which are important from the point of view of the modern developments in gravitational as well as (super)string theories (and related extended objects) of the theoretical high energy physics.The theoretical contents of our present endeavor are organised as follows. In Section2, we discuss a couple of continuous and infinitesimal symmetry transformations and establish their relationship with the infinitesimal and continuous 1D diffeomorphism (i.e., reparameterization) symmetry transformations. Our Section 3 is devoted to the upgradation of the classical infinitesimal reparameterization symmetry transformations to the quantum off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations. The latter property is satisfied due to the existence of an (anti-)BRST invariant CF-type restriction. This section also contains the coupled (but equivalent) Lagrangians that respect both the (anti-)BRST symmetry transformations on the submanifold (of the subspace of quantum variables) where the CF-type restriction is satisfied. In Section 4, we derive the (anti-)BRST transformations for the target space fermionic as well as bosonic variables. In addition, we deduce the CF type of restriction by exploiting the theoretical strength of MBTSA. We utilize the potential of ACSA to BRST formalism to obtain the nilpotent (anti-)BRST symmetry transformations for the other variables of our BRST-invariant theory in Section 5. We capture the (anti-)BRST invariance of the coupled (but equivalent) Lagrangians within the ambit of ACSA and establish, once again, the existence of our quantum (anti-)BRST invariant CF-type restriction in Section 6. Section 7 of our present endeavor contains theoretical proof of the nilpotency and absolute anticommutativity of the conserved and off-shell nilpotent (anti-)BRST charges. Finally, in Section 8, we summarize our key accomplishments and point out a few directions for further investigations.In our AppendicesA, B, and C, we collect some explicit computations that corroborate a few key claims and statements that have been made in the main body of the text of our present endeavor. Our Appendix D is devoted to provide an alternative proof of the existence of an (anti-)BRST invariant CF-type restriction on our theory.Convention and notations: we take the flat metric tensor of the D-dimensional target spacetime manifold as ημν=+1,−1,−1,−1⋯ so that the dot product between two nonnull vectors Pμ,Qμ is P·Q=ημνPμQν=P0Q0−PiQi where the Greek indices μ,ν,λ⋯=0,1,2⋯D−1 and Latin indices i,j,k⋯=1,2,3⋯D−1. We adopt the convention of the left-derivative w.r.t. all the fermionic variables. We always denote the nilpotent (anti-)BRST transformations by the notations sab. Our 1D model is generalized onto a 1,2-dimensional supermanifold which is parameterized by the superspace coordinates ZM=τ,θ,θ¯ where τ is the bosonic evolution parameter and a pair of Grassmannian variable θ,θ¯ satisfy θ2=θ¯2=0,θθ¯+θ¯θ=0. In our present investigation, we shall focus only on the 1,1-dimensional (anti-)chiral super submanifolds of our chosen general1,2-dimensional supermanifold in the context of ACSA.


## 2. Preliminaries: Continuous and Infinitesimal Reparameterization Symmetry Transformations

We divide our present section into two parts. We discuss, in our Section2.1, a few classical infinitesimal continuous symmetries and their relationships with the classical infinitesimal reparameterization symmetry transformations. Our Section 2.2 is devoted to a concise description of the quantum (anti-)BRST symmetry transformations.


### 2.1. Some Classical Infinitesimal and Continuous Symmetries

We begin with the followingfirst-order Lagrangian Lf for the free one 0+1-dimensional (1D) massive spinning (i.e., SUSY) relativistic particle (see, e.g., [5, 26])
(1)Lf=pμẋμ−e2p2−m2+i2ψμψ̇μ−ψ5ψ̇5+iχpμψμ−mψ5,where we have parametrized the trajectory of the particle by τ and defined the “generalized” velocities ẋμ=dxμ/dτ,ψ̇μ=dψμ/dτ w.r.t. to it. This 1D trajectory is embedded in the D-dimensional flat Minkowskian target spacetime manifold where xμ,pμ (with μ=0,1,2,⋯D−1) are the canonical conjugate pair of spacetime and momenta variables which are function of the evolution parameter τ. We have the fermionic variables χ,ψμ,ψ5 in our theory. The Lagrangian (1) also has the bosonic variable e and fermionic variable χ as the Lagrange multiplier variables. These latter variables behave like the “gauge” and “superaguge” variables due to their transformation properties under the gauge and supersymmetric gauge transformations. In fact, the fermionic variable ψμ is the superpartner of xμ and the other fermionic variable ψ5 has been introduced to incorporate the rest mass m into the Lagrangian Lf where the mass-shell condition p2−m2=0 is satisfied by the free (ṗμ=0) massive spinning relativistic particle. Our present 1D model is interesting because it provides a prototype example of a supersymmetric (SUSY) gauge theory and its generalization to the 4D theory provides an example of the supergravity theory where fermionic variable ψμ becomes the Rarita-Schwinger Lorentz vector spin 3/2 field and the einbein variable e turns itself into the vierbein field.The action integralS=∫−∞+∞dτLf respects the following infinitesimal and continuous reparameterization symmetry transformations δr:
(2)δrxμ=εẋμ,δrψμ=εψ̇μ,δrpμ=εṗμ,δrψ5=εψ̇5,δrχ=ddτεχ,δre=ddτεe,because the first-order Lagrangian Lf transforms, under the above infinitesimal reparameterization symmetry transformation δr, as
(3)δrLf=ddτεLf⟹δrS=0,where δrbasically corresponds to the infinitesimal 1D diffeomorphism/reparameterization transformation: τ⟶τ′=τ−ετ. Here, the transformation parameter ετ is infinitesimal. It is an elementary exercise to note that, if we set all the fermionic variables equal to zero (i.e., χ,ψμ,ψ5=0), we obtain an infinitesimal gauge symmetry transformation δg from the infinitesimal reparameterization symmetry transformation (2) as follows:
(4)δgxμ=ξpμ,δgpμ=0,δge=ddτξ=ξ̇,δgψμ=δgψ5=δgχ=0,where we have identified eε=ξ and used the Eular-Lagrange eqations of motions (EL-EOMs): ẋμ=epμ,ṗμ=0. In Equation (4), the bosonic infinitesimal transformation parameter ξτ is nothing but the gauge transformation parameter. It can be readily checked that we have the following transformation for Lf and S under δg:
(5)δgLf=ddτξ2p2+m2⟹δgS=0,S=∫−∞+∞dτLf.We have infinitesimal classical supergauge symmetry transformationsδsg in our theory which transforms the fermionic variables into their bosonic counterparts and vice-versa. These continuous and infinitesimal symmetry transformations are
(6)δsgxμ=κψμ,δsgpμ=0,δsgψμ=iκpμ,δsgψ5=iκm,δsgχ=iκ̇,δsge=2κχ,where the infinitesimal transformation parameter κτ is fermionic (i.e., κ2=0) in nature. It is straightforward to observe that we have the following:
(7)δsgLf=ddτκ2pμψμ+mψ5⟹δsgS=0.Under the combinedδ=δg+δsgclassical symmetry transformation δ, we have the following continuous and infinitesimal symmetry transformations δ, namely,
(8)δxμ=ξpμ+κψμ,δψμ=iκpμ,δpμ=0,δe=ξ̇+2κχ,δχ=iκ̇,δψ5=iκm,which lead to the transformation of the first-order Lagrangian Lf as
(9)δLf=ddτξ2p2+m2+κ2pμψμ+mψ5⟹δS=0.Thus, the continuous and infinitesimal transformationδ is indeed a symmetry transformation for the action integral S=∫−∞∞dτLf due to Gauss’s divergence theorem.As the gauge symmetry transformation (4) can be incorporated into the reparameterization symmetry transformations (2) with the help of some EL-EOMs and some identification of the transformation parameters, in exactly similar fashion, the combined (super)gauge symmetry transformations (8) can be incorporated into the reparameterization symmetry transformations (2) if we take the help of the following EL-EOMs, namely,
(10)ṗμ=0,ẋμ=epμ−iχψμ,ψ̇μ=χpμ,ψ̇5=mχ,and identify the transformation parameters as eε=ξ and −iεχ=κ (see, e.g., [5, 26] for details). Thus, we note that the classical infinitesimal reparameterization symmetry transformations (2) are a set of very general kind of symmetry transformations whose special cases are the continuous and infinitesimal symmetry transformations (4) and (8).


### 2.2. Quantum (Anti-)BRST Symmetries Corresponding to the Classical (Super)gauge Symmetry Transformations

Theclassical continuous and infinitesimal (super)gauge symmetry transformations (8) can be elevated to their counterpart quantum nilpotent sab2=0, infinitesimal, and continuous (anti-)BRST transformations sab as follows [5, 6, 26]:
(11)sabxμ=c¯pμ+β¯ψμ,sabψμ=iβ¯pμ,sabe=c¯̇+2β¯χ,sabc=ib¯,sabc¯=−iβ¯2,sabpμ=0,sabβ¯=0,sabβ=−iγ,sabγ=0,sabχ=iβ¯̇,sabb=2iβ¯γ,sabψ5=iβ¯m,sabb¯=0,(12)sbxμ=cpμ+βψμ,sbψμ=iβpμ,sbe=ċ+2βχ,sbpμ=0,sbc=−iβ2,sbβ=0,sbc¯=ib,sbβ¯=iγ,sbγ=0,sbχ=iβ̇,sbb¯=−2iβγ,sbb=0,sbψ5=iβm,where the fermionicc2=c¯2=cc¯+c¯c=0 (anti-)ghost variables c¯c and the bosonicβ2=β¯2≠0,ββ¯=β¯β (anti-)ghost variables β¯β correspond to the bosonic and fermionic gauge and supergauge transformation parameters ξ and κ of Equation (8), respectively. The variables b¯b are the Nakanishi-Lautrup-type auxiliary variables, and γ is an additional fermionic γ2=0 variable in our BRST-quantized (as well as invariant) theory.It is straightforward to note that the anticommutativity property of the off-shell nilpotentsab2=0 (anti-)BRST symmetry transformations (sab), namely,
(13)sb,sabxμ=ib+b¯+2ββ¯pμ,sb,sabe=iddτb+b¯+2ββ¯,is true if and only if the CF-type restriction b+b¯+2ββ¯=0 is invoked from outside. However, this restriction is a physical constraint because it is an (anti-)BRST invariant (i.e., sabb+b¯+2ββ¯=0) quantity. It can be readily checked that
(14)sb,sabΦ=0,Φ=pμ,ψμ,ψ5,b,b¯,β,β¯,c,c¯,γ.In other words, we observe that the off-shell nilpotentsab2=0 (anti-)BRST symmetry transformations sab are absolutely anticommuting (i.e., sb,sab=sbsab+sabsb=0) provided the entire theory is considered on the quantum submanifold where the CF-type restriction b+b¯+2ββ¯=0 is satisfied. It is the existence of this physical restriction that leads to the existence of coupled (but equivalent) Lagrangians
(15)Lb=Lf+sbsabi2e2+χψ5−12c¯c,Lb¯=Lf−sabsbi2e2+χψ5−12c¯c,which incorporate the gauge-fixing and Faddeev-Popov ghost terms in addition to the first-order Lagrangian Lf of Equation (1). In the full blaze of glory, the above Lagrangians (in terms of all the appropriate variables) are as follows [6, 26]:
(16)Lb=Lf+bė+2β¯β+b2−ic¯̇ċ+β¯2β2−2eβ¯β̇+γχ+2iχβc¯̇−β¯ċ+mβ¯β̇−β¯̇β+γχ+2γβc¯−β¯c−γ̇ψ5,(17)Lb¯=Lf−b¯ė−2β¯β+b¯2−ic¯̇ċ+β¯2β2+2iχβc¯̇−β¯ċ+2eβ¯̇β−γχ+mβ¯β̇−β¯̇β+γχ+2γβc¯−β¯c−γ̇ψ5,where the subscripts b and b¯ are appropriate because the Lagrangian Lb depends on the Nakanishi-Lautrup auxiliary variable bbut the Lagrangian Lb¯ contains the auxiliary variable b¯ in its full expression. It is straightforward to check that Lb and Lb¯ of our theory respect the perfect BRST and anti-BRST transformations because we have
(18)sbLb=ddτβ2pμψμ+mψ5+c2p2+m2+bċ+2βχ,(19)sabLb¯=ddτβ¯2pμψμ+mψ5+c¯2p2+m2−b¯c¯̇+2β¯χ.As a consequence, the action integralsS1=∫−∞∞dτLb and S2=∫−∞∞dτLb¯ remain invariant under the BRST and anti-BRST symmetry transformations (12) and (11), respectively. We define a perfect symmetry as the one under which the action integral remains invariant without any use of the CF-type restriction and/or EL-EOMs.The BRST quantization of the massive spinning particle can be performed using the (anti-)BRST transformations (11) and (12) which correspond to the classical (super)gauge symmetry transformations (8). In our recent publication [26], we have discussed all the details of this quantization scheme. However, we have not touched the continuous and infinitesimal reparameterization transformations (2). We focus on the latter classical symmetry transformations in the next section for the BRST analysis as it is our modest first step towards our main goal to discuss the diffeomorphism invariant SUSY theories in the physical 3+1-dimensional (4D) and higher dimensional D>4 spacetime.


## 2.1. Some Classical Infinitesimal and Continuous Symmetries

We begin with the followingfirst-order Lagrangian Lf for the free one 0+1-dimensional (1D) massive spinning (i.e., SUSY) relativistic particle (see, e.g., [5, 26])
(1)Lf=pμẋμ−e2p2−m2+i2ψμψ̇μ−ψ5ψ̇5+iχpμψμ−mψ5,where we have parametrized the trajectory of the particle by τ and defined the “generalized” velocities ẋμ=dxμ/dτ,ψ̇μ=dψμ/dτ w.r.t. to it. This 1D trajectory is embedded in the D-dimensional flat Minkowskian target spacetime manifold where xμ,pμ (with μ=0,1,2,⋯D−1) are the canonical conjugate pair of spacetime and momenta variables which are function of the evolution parameter τ. We have the fermionic variables χ,ψμ,ψ5 in our theory. The Lagrangian (1) also has the bosonic variable e and fermionic variable χ as the Lagrange multiplier variables. These latter variables behave like the “gauge” and “superaguge” variables due to their transformation properties under the gauge and supersymmetric gauge transformations. In fact, the fermionic variable ψμ is the superpartner of xμ and the other fermionic variable ψ5 has been introduced to incorporate the rest mass m into the Lagrangian Lf where the mass-shell condition p2−m2=0 is satisfied by the free (ṗμ=0) massive spinning relativistic particle. Our present 1D model is interesting because it provides a prototype example of a supersymmetric (SUSY) gauge theory and its generalization to the 4D theory provides an example of the supergravity theory where fermionic variable ψμ becomes the Rarita-Schwinger Lorentz vector spin 3/2 field and the einbein variable e turns itself into the vierbein field.The action integralS=∫−∞+∞dτLf respects the following infinitesimal and continuous reparameterization symmetry transformations δr:
(2)δrxμ=εẋμ,δrψμ=εψ̇μ,δrpμ=εṗμ,δrψ5=εψ̇5,δrχ=ddτεχ,δre=ddτεe,because the first-order Lagrangian Lf transforms, under the above infinitesimal reparameterization symmetry transformation δr, as
(3)δrLf=ddτεLf⟹δrS=0,where δrbasically corresponds to the infinitesimal 1D diffeomorphism/reparameterization transformation: τ⟶τ′=τ−ετ. Here, the transformation parameter ετ is infinitesimal. It is an elementary exercise to note that, if we set all the fermionic variables equal to zero (i.e., χ,ψμ,ψ5=0), we obtain an infinitesimal gauge symmetry transformation δg from the infinitesimal reparameterization symmetry transformation (2) as follows:
(4)δgxμ=ξpμ,δgpμ=0,δge=ddτξ=ξ̇,δgψμ=δgψ5=δgχ=0,where we have identified eε=ξ and used the Eular-Lagrange eqations of motions (EL-EOMs): ẋμ=epμ,ṗμ=0. In Equation (4), the bosonic infinitesimal transformation parameter ξτ is nothing but the gauge transformation parameter. It can be readily checked that we have the following transformation for Lf and S under δg:
(5)δgLf=ddτξ2p2+m2⟹δgS=0,S=∫−∞+∞dτLf.We have infinitesimal classical supergauge symmetry transformationsδsg in our theory which transforms the fermionic variables into their bosonic counterparts and vice-versa. These continuous and infinitesimal symmetry transformations are
(6)δsgxμ=κψμ,δsgpμ=0,δsgψμ=iκpμ,δsgψ5=iκm,δsgχ=iκ̇,δsge=2κχ,where the infinitesimal transformation parameter κτ is fermionic (i.e., κ2=0) in nature. It is straightforward to observe that we have the following:
(7)δsgLf=ddτκ2pμψμ+mψ5⟹δsgS=0.Under the combinedδ=δg+δsgclassical symmetry transformation δ, we have the following continuous and infinitesimal symmetry transformations δ, namely,
(8)δxμ=ξpμ+κψμ,δψμ=iκpμ,δpμ=0,δe=ξ̇+2κχ,δχ=iκ̇,δψ5=iκm,which lead to the transformation of the first-order Lagrangian Lf as
(9)δLf=ddτξ2p2+m2+κ2pμψμ+mψ5⟹δS=0.Thus, the continuous and infinitesimal transformationδ is indeed a symmetry transformation for the action integral S=∫−∞∞dτLf due to Gauss’s divergence theorem.As the gauge symmetry transformation (4) can be incorporated into the reparameterization symmetry transformations (2) with the help of some EL-EOMs and some identification of the transformation parameters, in exactly similar fashion, the combined (super)gauge symmetry transformations (8) can be incorporated into the reparameterization symmetry transformations (2) if we take the help of the following EL-EOMs, namely,
(10)ṗμ=0,ẋμ=epμ−iχψμ,ψ̇μ=χpμ,ψ̇5=mχ,and identify the transformation parameters as eε=ξ and −iεχ=κ (see, e.g., [5, 26] for details). Thus, we note that the classical infinitesimal reparameterization symmetry transformations (2) are a set of very general kind of symmetry transformations whose special cases are the continuous and infinitesimal symmetry transformations (4) and (8).


## 2.2. Quantum (Anti-)BRST Symmetries Corresponding to the Classical (Super)gauge Symmetry Transformations

Theclassical continuous and infinitesimal (super)gauge symmetry transformations (8) can be elevated to their counterpart quantum nilpotent sab2=0, infinitesimal, and continuous (anti-)BRST transformations sab as follows [5, 6, 26]:
(11)sabxμ=c¯pμ+β¯ψμ,sabψμ=iβ¯pμ,sabe=c¯̇+2β¯χ,sabc=ib¯,sabc¯=−iβ¯2,sabpμ=0,sabβ¯=0,sabβ=−iγ,sabγ=0,sabχ=iβ¯̇,sabb=2iβ¯γ,sabψ5=iβ¯m,sabb¯=0,(12)sbxμ=cpμ+βψμ,sbψμ=iβpμ,sbe=ċ+2βχ,sbpμ=0,sbc=−iβ2,sbβ=0,sbc¯=ib,sbβ¯=iγ,sbγ=0,sbχ=iβ̇,sbb¯=−2iβγ,sbb=0,sbψ5=iβm,where the fermionicc2=c¯2=cc¯+c¯c=0 (anti-)ghost variables c¯c and the bosonicβ2=β¯2≠0,ββ¯=β¯β (anti-)ghost variables β¯β correspond to the bosonic and fermionic gauge and supergauge transformation parameters ξ and κ of Equation (8), respectively. The variables b¯b are the Nakanishi-Lautrup-type auxiliary variables, and γ is an additional fermionic γ2=0 variable in our BRST-quantized (as well as invariant) theory.It is straightforward to note that the anticommutativity property of the off-shell nilpotentsab2=0 (anti-)BRST symmetry transformations (sab), namely,
(13)sb,sabxμ=ib+b¯+2ββ¯pμ,sb,sabe=iddτb+b¯+2ββ¯,is true if and only if the CF-type restriction b+b¯+2ββ¯=0 is invoked from outside. However, this restriction is a physical constraint because it is an (anti-)BRST invariant (i.e., sabb+b¯+2ββ¯=0) quantity. It can be readily checked that
(14)sb,sabΦ=0,Φ=pμ,ψμ,ψ5,b,b¯,β,β¯,c,c¯,γ.In other words, we observe that the off-shell nilpotentsab2=0 (anti-)BRST symmetry transformations sab are absolutely anticommuting (i.e., sb,sab=sbsab+sabsb=0) provided the entire theory is considered on the quantum submanifold where the CF-type restriction b+b¯+2ββ¯=0 is satisfied. It is the existence of this physical restriction that leads to the existence of coupled (but equivalent) Lagrangians
(15)Lb=Lf+sbsabi2e2+χψ5−12c¯c,Lb¯=Lf−sabsbi2e2+χψ5−12c¯c,which incorporate the gauge-fixing and Faddeev-Popov ghost terms in addition to the first-order Lagrangian Lf of Equation (1). In the full blaze of glory, the above Lagrangians (in terms of all the appropriate variables) are as follows [6, 26]:
(16)Lb=Lf+bė+2β¯β+b2−ic¯̇ċ+β¯2β2−2eβ¯β̇+γχ+2iχβc¯̇−β¯ċ+mβ¯β̇−β¯̇β+γχ+2γβc¯−β¯c−γ̇ψ5,(17)Lb¯=Lf−b¯ė−2β¯β+b¯2−ic¯̇ċ+β¯2β2+2iχβc¯̇−β¯ċ+2eβ¯̇β−γχ+mβ¯β̇−β¯̇β+γχ+2γβc¯−β¯c−γ̇ψ5,where the subscripts b and b¯ are appropriate because the Lagrangian Lb depends on the Nakanishi-Lautrup auxiliary variable bbut the Lagrangian Lb¯ contains the auxiliary variable b¯ in its full expression. It is straightforward to check that Lb and Lb¯ of our theory respect the perfect BRST and anti-BRST transformations because we have
(18)sbLb=ddτβ2pμψμ+mψ5+c2p2+m2+bċ+2βχ,(19)sabLb¯=ddτβ¯2pμψμ+mψ5+c¯2p2+m2−b¯c¯̇+2β¯χ.As a consequence, the action integralsS1=∫−∞∞dτLb and S2=∫−∞∞dτLb¯ remain invariant under the BRST and anti-BRST symmetry transformations (12) and (11), respectively. We define a perfect symmetry as the one under which the action integral remains invariant without any use of the CF-type restriction and/or EL-EOMs.The BRST quantization of the massive spinning particle can be performed using the (anti-)BRST transformations (11) and (12) which correspond to the classical (super)gauge symmetry transformations (8). In our recent publication [26], we have discussed all the details of this quantization scheme. However, we have not touched the continuous and infinitesimal reparameterization transformations (2). We focus on the latter classical symmetry transformations in the next section for the BRST analysis as it is our modest first step towards our main goal to discuss the diffeomorphism invariant SUSY theories in the physical 3+1-dimensional (4D) and higher dimensional D>4 spacetime.


## 3. Quantum (Anti-)BRST Symmetries Corresponding to the Classical Reparameterization Symmetry

In this section, we discuss thequantum (anti-)BRST symmetry transformations corresponding to the classical infinitesimal reparameterization symmetry transformations (2). This is essential and important because we wish to perform the BRST quantization of a 1D diffeomorphism (i.e., reparameterization) invariant SUSY theory. We exploit the standard techniques and tricks of the BRST formalism to elevate the above classical symmetry to its counterpart quantum symmetries. In fact, the off-shell nilpotent sab2=0 (anti-)BRST symmetry transformations (corresponding to the classical Equation (2)) are
(20)sabψμ=C¯ψ̇μ,sabpμ=C¯ṗμ,sabe=ddτC¯e,sabxμ=C¯ẋμ,sabC¯=C¯C¯̇,sabχ=ddτC¯χ,sabψ5=C¯ψ̇5,sabC=iB¯,sabB¯=0,sabB=ḂC¯−BC¯̇,(21)sbxμ=Cẋμ,sbpμ=Cṗμ,sbe=ddτCe,sbψμ=Cψ̇μ,sbψ5=Cψ̇5,sbχ=ddτCχ,sbC¯=iB,sbB=0,sbC=CĊ,sbB¯=B¯̇C−B¯Ċ,where B and B¯ are the Nakanishi-Lautrup auxiliary and C¯C are the (anti-)ghost variables of our theory. As far as the absolute anticommutativity property (i.e., sb,sab=0) of the above transformations is concerned, we note the following:
(22)sb,sabsμ=iB+B¯+iC¯Ċ−C¯̇Cṡμ,sb,sabΦ=iddτB+B¯+iC¯Ċ−C¯̇CΦ,sb,sabΨ=0,Ψ=B,B¯,C,C¯,where sμ=xμτ,pμτ,ψμτ,ψ5τ and Φ=eτ,χτ. Thus, we note that the absolute anticommutativity property of the (anti-)BRST symmetry transformations (cf. Equations (20) and (21)) is satisfied (i.e., sb,sab=sbsab+sabsb=0) if and only if we invoke the (anti-)BRST invariant (i.e., sabB+B¯+iC¯Ċ−C¯̇C=0) CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0. We note, therefore, that a CF-type constraint exists on our theory which is the root cause behind the absolute anticommutativity (i.e., sb,sab=0) of the (anti-)BRST symmetry transformations and it leads to the existence of the coupled (but equivalent) (anti-)BRST invariant Lagrangians as
(23)LB=Lf+sbsabi2e2+χψ5−12C¯C,LB¯=Lf−sabsbi2e2+χψ5−12C¯C,where the (anti-)BRST symmetry transformations sab are the quantum symmetries (cf. Equations (20) and (21)) corresponding to the classical infinitesimal reparameterization symmetry transformations (2). It will be noted that the quantities in the square brackets of (23) are the same as quoted in Equation (15). However, the (anti-)BRST symmetry transformations in (23) are different from (15) as are the notations for the Nakanishi-Lautrup-type auxiliary and (anti-)ghost variables (cf. Equation (11), (12), (20), and (21) for details).The above coupled (but equivalent) Lagrangians (23) can be expressed in terms of the auxiliary and basic variables in an explicit form as
(24)LB=Lf+Beė−i2C¯̇C+C¯Ċ+B22−ie2C¯̇Ċ−ieėC¯̇C−C¯̇C¯ĊC,LB¯=Lf−B¯eė−i2C¯Ċ+C¯̇C+B¯22−ie2C¯̇Ċ−ieėC¯Ċ−C¯̇C¯ĊC.We note that the pure Faddeev-Popov ghost part (i.e.,−C¯̇C¯ĊC) of the above coupled (but equivalent) Lagrangians is the same. It can be readily checked that the EL-EOMs from Lagrangians LB and LB¯, w.r.t. the auxiliary variables B and B¯, lead to the following:
(25)B=−eė+2iC¯̇C+iC¯Ċ,B¯=eė−2iC¯Ċ−iC¯̇C,which are responsible for the derivation of the CF-type restriction: B+B¯+iC¯Ċ−C¯̇C=0. The above Lagrangians LB and LB¯ of our theory respect the BRST and anti-BRST transformations in a precise and perfect manner because it is interesting to check that
(26)sbLB=ddτCLf+e2BĊ+eėBC−iBC¯ĊC+B2C,(27)sabLB¯=ddτC¯Lf−e2B¯C¯̇−eėB¯C¯−iB¯C¯̇C¯C+B¯2C¯.As a consequence, the action integralsS1=∫−∞∞dτLB and S2=∫−∞∞dτLB¯ respect the continuous and nilpotent symmetries sb and sab because of the Gauss’s divergence theorem (where all the physical variables of our theory vanish off as τ⟶±∞). We can also apply sb on LB¯ and sab on LB. The ensuing results are as follows:
(28)sbLB¯=ddτCLf−e2iC¯̇ĊC+B¯Ċ−eėiC¯ĊC+B¯C+i2B¯−BC¯ĊC+B¯2C+B+B¯+iC¯Ċ−C¯̇CiC¯C¨C+eėĊ−2B¯Ċ+2iC¯̇ĊC+ddτB+B¯+iC¯Ċ−C¯̇Ce2Ċ−B¯C,(29)sabLB=ddτC¯Lf+e2iC¯̇C¯Ċ+BC¯̇+eėiC¯̇C¯C+BC¯+i2B−B¯C¯̇C¯C+B2C¯+B+B¯+iC¯Ċ−C¯̇CiC¯¨C¯C−eėC¯̇−2BC¯̇+2iC¯̇C¯Ċ−ddτB+B¯+iC¯Ċ−C¯̇Ce2C¯̇+BC¯,which demonstrate that the coupled Lagrangians LB and LB¯ of our theory are equivalent in the sense that both of them respect both (i.e., BRST and anti-BRST) transformations due to the validity of the physical CF-type restriction: B+B¯+iC¯Ċ−C¯̇C=0.It is very interesting to point out that the contributions of the term “χψ5” in Equation (23) turn out to be total derivatives because we observe that
(30)sbsabχψ5=ddτiBχ−C¯Ċχ−C¯Cχ̇ψ5−C¯Cχψ̇5,−sabsbχψ5=−ddτiB¯χ+C¯̇Cχ+C¯Cχ̇ψ5+C¯Cχψ̇5.As a consequence, the gauge-fixing and Faddeev-Popov ghost terms of the LagrangiansLB and LB¯ of Equation (24) originate from the same terms as the ones derived in our earlier work [6] on the massless spinning relativistic particle. Thus, we note that the variable (i.e., ψ5), corresponding to the mass term for a massive spinning relativistic particle, does not contribute anything new to the gauge-fixing and Faddeev-Popov ghost terms. In other words, the dynamics of our theory (at the BRST quantized level) is unaffected by the presence of the “χψ5” term. This is a novel observation in our theory which is radically different from our earlier work [26] where the “χψ5” term contributes to the dynamics. The observations in Equation (30) also imply that the absolute anticommutativity property sb,sabχψ5=id/dτB+B¯+iC¯Ċ−C¯̇C of the (anti-)BRST symmetries sab is satisfied (i.e., sb,sab=0) only when the CF-type restriction is imposed from outside.According to Noether’s theorem, the continuous symmetry invariance of the action integrals, corresponding to the transformations (26) and (27), leads to the derivation of the conserved currents (i.e., conserved charges for our 1D system) as
(31)JB=iBC¯CĊ+B2C+Be2Ċ+BeėC+12eCp2−m2+iχCpμψμ−mψ5,(32)JB¯=iB¯C¯C¯̇C+B¯2C¯−B¯e2C¯̇−B¯eėC¯+12eC¯p2−m2+iχC¯pμψμ−mψ5.The conservation lawdJr/dτ=0 (with r=B,B¯) can be proven by using the EL-EOMs that derived from the Lagrangians LB and LB¯. For instance, we point out that the following EL-EOMs w.r.t. the variables (xμ,pμ,ψμ,ψ5,χ,e,B,B¯,C,C¯), namely,
(33)ṗμ=0,ẋμ=epμ−iχψμ,ψμ=χpμ,ψ̇5=mχ,pμψμ=mψ5,B−i2C¯̇C+ieė+C¯Ċ=0,eḂ+ieC¯̇Ċ−ieC¯¨C+12p2−m2=0,iḂC¯−iBC¯̇+ieėC¯̇+ie2C¯¨+C¯C¯¨C+2C¯C¯̇Ċ=0,−iBĊ−2iḂC−3ieėĊ−ie2C¨−iė2C−iee¨C+C¯CC¨+2C¯̇CĊ=0,are obtained from LB. The equations of motion that are different from (33) and emerge out from LB¯ (as the EL-EOMs) are as follows:
(34)B¯+i2C¯Ċ+ieė+C¯̇C=0,−eB¯̇+ieC¯̇Ċ−ieC¯C¨+12p2−m2=0,iB¯̇C−iB¯Ċ−ieėĊ−ie2C¨+C¯CC¨+2C¯̇CĊ=0,−iB¯C¯̇−2iB¯̇C¯+3ieėC¯̇+ie2C¯¨+iė2C¯+iee¨C¯+C¯C¯¨C+2C¯C¯̇Ċ=0.These conserved currentsJB,JB¯ lead to the definition of the conserved charges QB and QB¯ which are the same (i.e., JB=QB,JB¯=QB¯) as the conserved currents quoted in Equations (31) and (32). This is due to the fact that we are dealing with a 1D system.We close this section with the following key comments. First, we observe that the off-shell nilpotent (anti-)BRST symmetry transformations (20) and (21) are absolutely anticommuting in nature provided we invoke the (anti-)BRST invariant CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 from outside. Second, this restriction can be obtained from the coupled (but equivalent) Lagrangians LB and LB¯ if we use the EL-EOMs (cf. Equation (25)) w.r.t. the Nakanishi-Lautrup-type auxiliary variables B and B¯. Third, we observe that the term χψ5 in the square bracket of Equation (23) does not contribute anything to the dynamics as well as to the gauge-fixing and Faddeev-Popov ghost terms. Fourth, the coupled Lagrangians LB and LB¯ are equivalent in the sense that both of them respect both off-shell nilpotent (anti-)BRST symmetries on a submanifold of the quantum variables where the CF-type constraint B+B¯+iC¯Ċ−C¯̇C=0 is satisfied. This key observation is an alternative proof of the existence of CF-type restriction on our theory. Finally, we observe that the absolute anticommutativity (i.e., QB,QB¯=0) of the conserved (i.e., Q̇B¯B=0) and off-shell nilpotent (i.e., QB¯B2=0) (anti-)BRST charges QB¯B is satisfied only due to the validity of the existence of the CF-type restriction (cf. Section 6).


## 4. Off-Shell Nilpotent Symmetries of the Target Space Variables and CF-Type Restriction: MBTSA

In this section, we derive the off-shell nilpotent (anti-)BRST symmetry transformations for the target space variablesxμ,pμ,ψμ,ψ5 which are scalars w.r.t. the 1D space of the trajectory for the massive spinning relativistic particle that is embedded in the D-dimensional target space. For this purpose, we exploit the theoretical power and potential of MBTSA (see, e.g., [22, 25]). In this context, first of all, we promote the 1D diffeomorphism transformation τ⟶τ′=fτ≡τ−ετ (where ετ is the infinitesimal transformation parameter corresponding to the 1D diffeomorphism (i.e., reparameterization) symmetry transformation) to its counterpart on the 1,2-dimensional supermanifold as
(35)fτ⟶fτ,θ,θ¯=τ−θC¯−θ¯C+θθ¯hτ,where the general 1,2-dimensional supermanifold is parameterized by a bosonic (i.e., evolution) coordinate τ and a pair of Grassmannian variables θ,θ¯ that satisfy θ2=θ¯2=0,θθ¯+θ¯θ=0. In the above diffeomorphism transformation, the function fτ is any arbitrary function of the evolution parameter τ such that it is finite at τ=0 and vanishes off at τ=±∞. In other words, fτ is a physically well-defined function of τ. It is worth pointing out that the coefficients of the Grassmannian variables θ,θ¯, in Equation (35), are nothing but the fermionic (i.e., C2=C¯2=0,CC¯+C¯C=0) (anti-)ghost variables C¯C of the (anti-)BRST symmetry transformations (20) and (21) corresponding to the infinitesimal reparameterization symmetry transformations (2). In other words, the infinitesimal reparameterization bosonic transformation parameter ετ has been replaced by the fermionic (anti-)ghost variables C¯C of the (anti-)BRST symmetry transformations. This has been done purposely in view of the fact that in earlier works (see, e.g., [10–12]), it has been established that the translational generators ∂θ,∂θ¯, along the Grassmannian directions θ,θ¯, are intimately connected with the nilpotent (anti-)BRST transformations sab in the ordinary space. In other words, we have already taken into account sabτ=−C¯,sbτ=−C, which are the generalization of the classical infinitesimal 1D diffeomorphism symmetry transformation δrτ=−ετ to its quantum counterparts sab within the framework of BRST formalism. It is worthwhile to point out that the secondary variable hτ of the expansion (37) has to be determined from other consistency conditions which we elaborate in our forthcoming paragraphs.According to the basic tenets of MBTSA, we have togeneralize all the ordinary variables of the Lagrangians (24) onto the 1,2-dimensional supermanifold as their counterpart supervariables where the generalization of the 1D diffeomorphism transformation (cf. Equation (35)) has to be incorporated (in a suitable fashion) into the expressions for the supervariables. For instance, we shall have the following generalization as far as the generic target space variable sμτ (cf. Equation (22)) is concerned, namely,
(36)sμτ⟶S~μfτ,θ,θ¯,θ,θ¯≡S~μτ−θC¯−θ¯C+θθ¯h,θ,θ¯,where the pair of variables θ,θ¯, as pointed out earlier, is the Grassmannian variables (i.e., θ2=θ¯2=0,θθ¯+θ¯θ=0) of the superspace coordinates ZM=τ,θ,θ¯ that characterize the 1,2-dimensional supermanifold on which our 1D ordinary theory of the reparameterization invariant massive spinning particle is considered. Now, following the techniques of MBTSA, we have the following super expansion of (36) along all the possible directions of the Grassmannian variables of the 1,2-dimensional supermanifold, namely,
(37)S~μfτ,θ,θ¯,θ,θ¯=Sμfτ,θ,θ¯+θR¯μfτ,θ,θ¯+θ¯Rμfτ,θ,θ¯+θθ¯Pμfτ,θ,θ¯,where on the r.h.s., we have the secondary supervariables which are also function of fτ,θ,θ¯. As a consequence, we can have the following Taylor expansions (for those secondary variables that are present on the r.h.s.), namely,
(38)Sμτ−θC¯−θ¯C+θθ¯h=sμτ−θC¯ṡμτ−θ¯Cṡμτ+θθ¯hṡμ−C¯Cs¨μ,θR¯μτ−θC¯−θ¯C+θθ¯h≡θR¯μτ−θθ¯R¯̇μτ,θ¯Rμτ−θC¯−θ¯C+θθ¯h≡θ¯Rμτ+θθ¯Ṙμτ,θθ¯Pμτ−θC¯−θ¯C+θθ¯h≡θθ¯Pμτ.At this juncture, we would like to lay stress on the fact that, in the super expansion (37), all the supervariables on the r.h.s. have to be ordinary variables as all of them are Lorentz scalars w.r.t. the 1D trajectory of the particle (that is embedded in the D-dimensional flat Minkowskian target space). It is worthwhile to point out that a pure Lorentz (bosonic or fermionic) scalar is the one which does not transform at all under any kind of spacetime and/or internal transformations. As a result, the expansion (37) can be written as
(39)S~μfτ,θ,θ¯,θ,θ¯=sμτ+θR¯μτ+θ¯Rμτ+θθ¯Pμτ≡sμτ+θsabsμτ+θ¯sbsμτ+θθ¯sbsabsμτ,where sab are the (anti-)BRST symmetry transformations (20) and (21). This is due to fact that the (anti-)BRST symmetry transformations sab have been shown to be deeply connected with the translational generators ∂θ,∂θ¯ along the θ,θ¯-directions of the 1,2-dimensional supermanifold (see, e.g., [10–12] for details).It is evident that we have to compute the values ofRμ, R¯μ, and Pμ (in terms of the auxiliary and basic variables of the Lagrangians (24)) so that we can obtain the off-shell nilpotent (anti-)BRST symmetry transformations for the generic variable sμτ. At this stage, the so called “horizontality condition” (HC) comes to our help where we demand that S~μfτ,θ,θ¯,θ,θ¯=sμτ. This relationship can be explicitly expressed as
(40)sμτ+θR¯μ−C¯ṡμ+θ¯Rμ−Cṡμ+θθ¯hṡμ−C¯Cx¨μ−CR¯̇μ+C¯Ṙμ+Pμ≡sμτ,where we have collected all the terms from Equation (38) to express (36). Physically, the above requirement corresponds to the fact that a Lorentz-scalar does not transform under any kind of physically well-defined spacetime transformations. Needless to say, the relationship (40) implies that we have the following explicit relationships:
(41)Rμ=Cṡμ,R¯μ=C¯ṡμ,Pμ=CR¯̇μ−C¯Ṙμ+C¯Cṡμ−hs¨μ.It is straightforward to note that we have already obtainedsbsμ=Cṡμ and sabsμ=C¯ṡμ as is evident from Equation (39). The requirement of the absolute anticommutativity (that is one of the sacrosanct properties of the (anti-)BRST symmetry transformations) implies that we have the following equalities, namely,
(42)sbsabsμ=−sabsbsμ,sb,sabsμ=0.On the other hand, the requirement of the off-shell nilpotency (that isanother sacrosanct property of the (anti-)BRST symmetry transformations) leads to the following:
(43)sbC=CĊ,sabC¯=C¯C¯̇.On top of the already obtained off-shell nilpotent (anti-)BRST symmetry transformations:sbsμ=Cṡμ,sabsμ=C¯ṡμ,sbC=CĊ,sabC¯=C¯C¯̇, we take into account the standard (anti-)BRST symmetry transformations sbC¯=iB and sabC=iB¯ in terms of the Nakanishi-Lautrup auxiliary variables. These standard inputs lead to the determination of the l.h.s. and r.h.s. of the first equality in Equation (42) as [22, 25]
(44)sbsabsμ=iB−C¯Ċṡμ−C¯Cs¨μ≡Pμτ,−sabsbsμ=−iB¯−C¯̇Cṡμ−C¯Cs¨μ≡Pμτ,where Pμτ is present in the expansion (39). A close look at Equation (44) implies that we have B+B¯+iC¯Ċ−C¯̇C=0 which is nothing but the CF-type restriction. In addition, the observation of Equation (41) implies that there is an explicit expression for Pμ in terms of hτ (that is present in the expansion of fτ,θ,θ¯ in Equation (35)). Plugging in the values of Rμ=Cṡμ and R¯μ=C¯ṡμ into Equation (41) leads to [22, 25]
(45)Pμτ=−C¯̇C+C¯Ċ+hṡμ+C¯Cs¨μ.Comparison of (44) and (45) yields (see, e.g., [22, 25] for details)
(46)h=−iB−C¯̇C≡+iB¯−C¯ĊB+B¯+iC¯Ċ+C¯̇C=0.Thus, we note that the comparison of the values ofhτ (that is determined from the comparison between Equations (44) and (45)) leads to the deduction of the (anti-)BRST invariant (i.e., sabB+B¯+iC¯Ċ−C¯̇C=0) CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 which plays an important role in the proof: sb,sab=0.We wrap up this section with the following useful and important remarks. First, we have taken into account the standard choice in the BRST formalism which issbC¯=iB,sabC=iB¯. In other words, we have made the following (anti-)chiral super expansions for the (anti-)chiral supervariables (in view of ∂θ¯↔sab,∂θ↔sb,), namely,
(47)Cτ⟶Fcτ,θ=Cτ+θiB¯τ≡Cτ+θsabCτ,C¯τ⟶F¯acτ,θ¯=C¯τ+θ¯iBτ≡C¯τ+θ¯sbC¯τ,where Fcτ,θ and F¯acτ,θ¯ are the chiral and antichiral supervariables that have been defined on the 1,1-dimensional suitably chosen chiral and antichiral super submanifolds of the general1,2-dimensional supermanifold. Second, we have seen that (cf. Equation (22)) the absolute anticommutativity property (i.e., sb,sab=0) of the (anti-)BRST transformations sab is satisfied if and only if the CF-type restriction (46) is satisfied. Third, we point out that the requirement of the following
(48)sb,sabC=0,sbB¯=B¯̇C−B¯Ċ,sb,sabC¯=0,sabB=ḂC¯−BC¯̇,leads to the derivation of (anti-)BRST symmetry transformations on the Nakanishi-Lautrup auxiliary variables BB¯. Fourth, within the framework of MBTSA, the CF-type restriction (46) is derived from the expression for hτ due to the consistency condition (i.e., sbsabsμ=−sabsbsμ≡Pμ). Fifth, the (anti-)BRST symmetry transformations (sbsμ=Cṡμ,sabsμ=C¯ṡμ) on the generic variable sμ≡xμ,pμ,ψμ,ψ5 imply that we have already obtained the following (anti-)BRST symmetry transformations:
(49)sabxμ=C¯ẋμ,sabpμ=C¯ṗμ,sabψμ=C¯ψ̇μ,sabψ5=C¯ψ̇5,sbxμ=Cẋμ,sbpμ=Cṗμ,sbψμ=Cψ̇μ,sbψ5=Cψ̇5,for the target space variables xμ,pμ,ψμ,ψ5 of our theory that are present in the first-order Lagrangian Lf (cf. Equation (1)) for the massive spinning (i.e., SUSY) relativistic particle. Finally, the explicit form of Equation (39) can be written, after the application of HC, as follows:
(50)Xμhfτ,θ,θ¯,θ,θ¯=xμτ+θC¯ẋμ+θ¯Cẋμ+θθ¯iB−C¯Ċẋμ−C¯Cx¨μ≡xμτ+θsabxμ+θ¯sbxμ+θθ¯sbsabxμ,Pμhfτ,θ,θ¯,θ,θ¯=pμτ+θC¯ṗμ+θ¯Cṗμ+θθ¯iB−C¯Ċṗμ−C¯Cp¨μ≡pμτ+θsabpμ+θ¯sbpμ+θθ¯sbsabpμ,Ψμhfτ,θ,θ¯,θ,θ¯=ψμτ+θC¯ψ̇μ+θ¯Cψ̇μ+θθ¯iB−C¯Ċψ̇μ−C¯Cψ¨μ≡ψμτ+θsabψμ+θ¯sbψμ+θθ¯sbsabψμ,Ψ5hfτ,θ,θ¯,θ,θ¯=ψ5τ+θC¯ψ̇5+θ¯Cψ̇5+θθ¯iB−C¯Ċψ̇5−C¯Cψ¨5≡ψ5τ+θsabψ5+θ¯sbψ5+θθ¯sbsabψ5,where the superscript h denotes the full expansion of the supervariables after the application of HC. A straightforward comparison of (39) with (50) shows that we have already derived the (anti-)BRST symmetry transformations (49) as the coefficients of θθ¯ in the super expansions (50) along with sbsabxμ,sbsabψμ,sbsabpμ,sbsabψ5 which are the coefficients of θθ¯. We also note, from Equation (50), that we have a mapping: sb↔∂θ¯θ=0,sab↔∂θθ¯=0. This observation is consistent with results obtained in Refs. [10–12].


## 5. Off-Shell Nilpotent (Anti-)BRST Symmetries for Other Variables of Our Theory: ACSA

In this section, we exploit the basic principle behind ACSA to BRST formalism to obtain, first of all, the off-shell nilpotent BRST symmetry transformations (21) by generalizing the basic and auxiliary variables of the Lagrangian LB (cf. Equation (24)) on the antichiral1,1-dimensional super submanifold (of the most general1,2-dimensional supermanifold) as
(51)Bτ⟶B~τ,θ¯=Bτ+θ¯f1τ,eτ⟶Eτ,θ¯=eτ+θ¯f2τ,χτ⟶Kτ,θ¯=χτ+θ¯b1τ,Cτ⟶Fτ,θ¯=Cτ+θ¯b2τ,C¯τ⟶F¯τ,θ¯=C¯τ+θ¯b3τ,B¯τ⟶B¯~τ,θ¯=B¯τ+θ¯f3τ,where f1,f2,f3 are the fermionic and b1,b2,b3 are bosonic secondary variables. These variables have to be expressed in terms of the auxiliary and basic variables that are present in LB. For this purpose, we exploit the BRST-invariant restrictions. It will be noted that the antichiral 1,1-dimensional super submanifold is parameterized by the superspace coordinates ZM=τ,θ¯ where τ is the bosonic evolution parameter and Grassmannian variable θ¯ is fermionic (θ¯2=0) in nature. In addition to (51), we have the antichiral limit (i.e., θ=0) of the expansions (50) as follows:
(52)Xμhaτ,θ¯=xμτ+θ¯Cẋμ,Ψμhaτ,θ¯=ψμτ+θ¯Cψ̇μ,Pμhaτ,θ¯=pμτ+θ¯Cṗμ,Ψ5haτ,θ¯=ψ5τ+θ¯Cψ̇5,where the superscript ha denotes the antichiral limit of the super expansions (of the supervariables (cf. Equation (50))) that have been obtained after the application of HC. It is straightforward to note that the BRST invariance (sbB=0) of the variable B implies that we have the following (with f1τ=0), namely,
(53)B~τ,θ¯⟶B~bτ,θ¯=Bτ+θ¯0=Bτ+θ¯sbB,where the superscript b stands for the antichiral supervariable that has been obtained after the BRST invariant sbB=0 restriction. In other words, we have already obtained the BRST symmetry transformation sbB=0 as the coefficient of θ¯ in (53) due to our knowledge of sb↔∂θ¯ (i.e., ∂θ¯B~bτ,θ¯=sbB=0).The off-shell nilpotency of the BRST symmetry transformations (21) ensures that we have the following BRST invariant quantities:
(54)sbCẋμ=0,sbCṗμ=0,sbCĊ=0,sbCė+Ċe=0,sbĊχ+Cχ̇=0,sbCψ̇μ=0,sbCψ̇5=0,sbB¯̇C−B¯Ċ=0.The abovequantum gauge (i.e., BRST) invariant quantities must be independent of the Grassmannian variable θ¯ when they are generalized onto 1,1-dimensional antichiral super submanifold (of the most general1,2-dimensional supermanifold)) on which our 1D ordinary theory has been generalized. In other words, we have the validity of the following equalities in terms of the supervariables and ordinary variables, namely,
(55)Fτ,θ¯Ẋμhaτ,θ¯=Cτẋμτ,Fτ,θ¯Ḟτ,θ¯=CτĊτ,Fτ,θ¯Ψ̇μhaτ,θ¯=Cτψ̇μτ,Fτ,θ¯Ψ̇5haτ,θ¯=Cτψ̇μτ,Fτ,θ¯Ṗμhaτ,θ¯=Cτṗμτ,B~bτ,θ¯=Bτ,Ḟτ,θ¯Eτ,θ¯+Fτ,θ¯Ėτ,θ¯=Ċτeτ+Cτėτ,Ḟτ,θ¯Kτ,θ¯+Fτ,θ¯K̇τ,θ¯=Ċτχτ+Cτχ̇τ,B¯~̇τ,θ¯Fτ,θ¯−B¯~τ,θ¯Ḟτ,θ¯=B¯̇τCτ−B¯τĊτ,where the supervariables with superscripts ha and b have already been explained and derived in Equations (52) and (53). The substitutions of the expansions from (52) and (51) lead to the determination of the secondary variables of the latter equation (cf. Equation (51)) as
(56)f2τ=Ċe+Cė,b1τ=Ċχ+Cχ̇,b2τ=CĊ,b3τ=iB,f3τ=B¯̇C−B¯Ċ.The above relationships demonstrate that we have already obtained the secondary variables of the super expansion (51) in terms of the auxiliary and basic variables of LB (and the Nakanishi-Lautrup auxiliary variable B¯τ of the Lagrangian LB¯ (cf. Equation (24))).The substitutions of the above expressions for the secondary variables into the super expansions (51) (besides Equations (52) and (53)) are as follows:
(57)Ebτ,θ¯=eτ+θ¯eĊ+ėC≡eτ+θ¯sbe,Kbτ,θ¯=χτ+θ¯Cχ̇+Ċχ≡χτ+θ¯sbχ,Fbτ,θ¯=Cτ+θ¯CĊ≡Cτ+θ¯sbC,F¯bτ,θ¯=C¯τ+θ¯iB≡C¯τ+θ¯sbC¯,B¯~bτ,θ¯=B¯τ+θ¯B¯̇C−B¯Ċ≡B¯τ+θ¯sbB¯,where the superscript b on the supervariables denotes the antichiral supervariables that have been obtained after the imposition of the BRST (i.e., quantum gauge) invariant restrictions in Equation (55). It is clear from (57) that we have a mapping ∂θ¯↔sb which demonstrates that the off-shell nilpotency ∂θ¯2=0 of the translational generator ∂θ¯ along θ¯-direction of the 1,1-dimensional antichiral super submanifold and off-shell nilpotency sb2=0 of the BRST transformations (21) in the ordinary space are interrelated. A careful look at Equations (50) and (57) demonstrates that we have already derived the BRST symmetry transformations (21) for all the variables of LB as the coefficients of θ¯.Now, we dwell a bit on the derivation of the anti-BRST transformations (20) within the framework of ACSA. Towards this end in mind, we note that the following are the chiral (i.e., θ¯=0) limit of the full super expansions in (50), namely,
(58)Xμhcτ,θ=xμτ+θC¯ẋμ≡xμτ+θsabxμ,Pμhcτ,θ=pμτ+θC¯ṗμ≡pμτ+θsabpμ,Ψμhcτ,θ=ψμτ+θC¯ψ̇μ≡ψμτ+θsabψμ,Ψ5hcτ,θ=ψ5τ+θC¯ψ̇5≡ψ5τ+θsabψ5,where the superscript hc stands for the chiral limit of the supervariables that have been derived after the application of HC in Equation (50). The above expansions in (58) would be utilized in the anti-BRST invariant restrictions on the chiral supervariables which we are going to discuss as follows. It can be readily checked that we have the following interesting anti-BRST invariant quantities:
(59)sabC¯ẋμ=0,sabC¯ṗμ=0,sabC¯C¯̇=0,sabC¯̇e+C¯ė=0,sabC¯̇χ+C¯χ̇=0,sabC¯ψ̇μ=0,sabC¯ψ̇5=0,sabḂC¯−BC¯̇=0,where the anti-BRST symmetry transformations sab are the ones that have been listed in Equation (20). Keeping in our mind the mapping sab↔∂θ and the observation sabB¯=0,we have the following:
(60)B¯~τ,θ⟶B¯~abτ,θ=B¯τ+θ0=B¯τ+θsabB¯,where the superscript ab denotes the expansion of the supervariable that has been obtained after the application of the anti-BRST invariant restriction B¯~τ,θ=B¯τ that is obtained due to the anti-BRST invariance sabB¯=0. We also note that ∂θB¯~abτ,θ=sabB¯=0. Exploiting the basic principle of ACSA to BRST formalism, we obtain the following equalities in terms of chiral and ordinary variables, namely,
(61)F¯τ,θẊμhcτ,θ=C¯τẋμτ,F¯τ,θṖμhcτ,θ=C¯τṗμτ,F¯τ,θΨ̇μhcτ,θ=C¯τψ̇μτ,F¯τ,θΨ̇5hcτ,θ=C¯τψ̇5τ,F¯τ,θF¯̇τ,θ=C¯τC¯̇τ,B¯~abτ,θ=B¯τ,F¯̇τ,θEτ,θ+F¯τ,θĖτ,θ=C¯̇τeτ+C¯τėτ,F¯̇τ,θKτ,θ+F¯τ,θK̇τ,θ=C¯̇τχτ+C¯τχ̇τ,B~̇τ,θF¯τ,θ−B~τ,θF¯̇τ,θ=ḂτC¯τ−BτC¯̇τ,where the chiral supervariables with superscripts hc and ab have been discussed and explained in Equations (58) and (60). It is worth pointing out that the equalities in (61) are nothing but the generalization of our observations in (59) where the anti-BRST invariant quantities have been obtained (because of the off-shell nilpotency of the anti-BRST symmetry transformations (20)). At this stage, it is crucial to point out that, besides our chiral supervariables in (58) and (60), we have the following generalizations:
(62)Bτ⟶B~τ,θ=Bτ+θf¯1τ,eτ⟶Eτ,θ=eτ+θf¯2τ,χτ⟶Kτ,θ=χτ+θb¯1τ,Cτ⟶Fτ,θ=Cτ+θb¯2τ,C¯τ⟶F¯τ,θ=C¯τ+θb¯3τ.The abovechiral supervariables are defined, and their expansions have been carried out on a 1,1-dimensional super submanifold that is characterized by the superspace coordinates ZM=τ,θ where the Grassmannian coordinate θ is fermionic θ2=0 in nature. It is straightforward to draw the conclusion that the secondary variables b¯1,b¯2,b¯3 and f¯1,f¯2,f¯3, on the r.h.s. of Equation (62), are bosonic and fermionic sets, respectively.The stage is now set to utilize the equalities (61) where we have to plug in the chiral super expansions (58) as well as the chiral generalizations (62). This exercise leads to the following relationships between the secondary variables of (62) and the basic as well as auxiliary variables of Lagrangian LB¯ (and the Nakanishi-Lautrup auxiliary variable B of the perfectly BRST invariant Lagrangian LB), namely,
(63)f¯1=ḂC¯−BC¯̇,f¯2=C¯ė+C¯̇e,b¯1=C¯χ̇+C¯̇χ,b¯2=iB¯,b¯3=C¯C¯̇.In other words, we have already determined the secondary variables (i.e., the coefficients ofθ) of the super expansions (62). The substitutions of (63) into the chiral super expansion (62) on the 1,1-dimensional chiral super submanifold leads to the following:
(64)B~abτ,θ=Bτ+θḂC¯−BC¯̇≡Bτ+θsabB,Eabτ,θ=eτ+θeC¯̇+ėC¯≡eτ+θsabe,Kabτ,θ=χτ+θC¯̇χ+C¯χ̇≡χτ+θsabχ,Fabτ,θ=Cτ+θiB¯≡Cτ+θsabC,F¯abτ,θ=C¯τ+θC¯C¯̇≡C¯τ+θsabC¯,where the superscript ab denotes the chiral supervariables that have been obtained after the application of the anti-BRST (i.e., quantum gauge) invariant restrictions in Equation (61). It is evident, from Equation (64), that we have already obtained the anti-BRST symmetry transformations (cf. Equation (20)) of the variables B,e,χ,C,C¯ as the coefficients of θ in the chiral super expansions (62). We also observe, in the above chiral super expansions, that there is a mapping ∂θ↔sab which agrees with the result of Refs. [10–12]. For the sake of completeness, we perform the step-by-step computations of the mathematical relationships obtained in Equation (63) in our Appendix A.


## 6. Symmetry Invariance of the Lagrangians: ACSA

In this section, we use the results of the previous section to capture the (anti-)BRST invariance of the Lagrangians (cf. Equations (26)–(29)) within the framework of ACSA. To accomplish this goal, first of all, we generalize the Lagrangians (24) to their counterparts (anti-)chiral super Lagrangians as follows:
(65)LB⟶L~Bacτ,θ¯=L~facτ,θ¯+B~bτ,θ¯Ebτ,θ¯Ėbτ,θ¯−i2F¯̇bτ,θ¯Fbτ,θ¯+F¯bτ,θ¯Ḟbτ,θ¯+12B~bτ,θ¯B~bτ,θ¯−iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯Ḟbτ,θ¯−iEbτ,θ¯Ėbτ,θ¯F¯̇bτ,θ¯Fbτ,θ¯−F¯̇bτ,θ¯F¯bτ,θ¯Ḟbτ,θ¯Fbτ,θ¯,(66)LB¯⟶L~B¯cτ,θ=L~fcτ,θ−B¯~abτ,θEabτ,θĖabτ,θ−i2F¯abτ,θḞabτ,θ+F¯̇abτ,θFabτ,θ+12B¯~abτ,θB¯~abτ,θ−iEabτ,θEabτ,θF¯̇abτ,θḞabτ,θ−iEabτ,θĖabτ,θF¯abτ,θḞabτ,θ−F¯̇abτ,θF¯abτ,θḞabτ,θFabτ,θ,where the superscripts ac and c, on the r.h.s., denote the antichiral and chiral versions of the first-order Lagrangian Lf (cf. Equation (1)). In other words, we have the following:
(67)Lf⟶L~facτ,θ¯=Pμbτ,θ¯Ẋμbτ,θ¯−12Ebτ,θ¯Pμbτ,θ¯Pμbτ,θ¯−m2+i2Ψμbτ,θ¯Ψ̇μbτ,θ¯−Ψ5bτ,θ¯Ψ̇5bτ,θ¯+iχ~bτ,θ¯Pμbτ,θ¯Ψμbτ,θ¯−mΨ5bτ,θ¯,(68)Lf⟶L~fcτ,θ=Pμabτ,θẊμabτ,θ−12Eabτ,θPμabτ,θPμabτ,θ−m2+i2Ψμabτ,θΨ̇μabτ,θ−Ψ5abτ,θΨ̇5abτ,θ+iχ~abτ,θPμabτ,θΨμabτ,θ−mΨ5abτ,θ,where the superscripts b and ab on the supervariables of L~fac and L~fc have been already explained in the previous section. The superscripts ac and c on the super Lagrangians, on the l.h.s. of Equations (65) and (66), denote the antichiral and chiral generalizations of the ordinary Lagrangians (24). Keeping in our mind the mappings: sb↔∂θ¯,sab↔∂θ, we observe the (anti-)BRST invariance of the first-order Lagrangian in the language of ACSA to BRST formalism as follows:
(69)∂∂θ¯L~fac=ddτCLf≡sbLf,∂∂θL~fc=ddτC¯Lf≡sabLf.In other words, we have accomplished the objective of establishing a precise connection between the (anti-)BRST invariance ofLf in the ordinary space (cf. Equations (3), (27), and (26)) and superspace within the purview of ACSA. The results in (69) will be useful in the proof of the (anti-)BRST invariance (cf. Equations (27) and (26)) of the coupled (but equivalent) Lagrangians LB and LB¯. Geometrically, it is clear from our observation in (69) that super Lagrangians L~fac,c are the unique sum of (anti-)chiral supervariables (obtained after the (anti-)BRST invariant restrictions) such that their translations along θ,θ¯-directions in the superspace produce the total derivatives in the ordinary space.We now focus on the BRST and anti-BRST invariance ofLB and LB¯ within the purview of ACSA. In the explicit expressions of (65) and (66), we substitute the super expansions of (52), (57), (58), and (64) and apply the derivatives ∂θ¯,∂θ on them due to the mappings: ∂θ¯↔sb,∂θ↔sab. It is straightforward to check that we have the following explicit relationships between the invariances in the superspace and ordinary space:
(70)∂∂θ¯L~Bac=ddτCLf+e2BĊ+eėBC−iBC¯ĊC+B2C=sbLB,∂∂θL~B¯c=ddτC¯Lf−e2B¯C¯̇−eėB¯C¯−iB¯C¯̇C¯C+B¯2C¯=sabLB¯.We would like to emphasize that the super LagrangianL~Bac is a unique sum of antichiral supervariables (derived after the applications of the BRST invariant restrictions) such that its translation along θ¯-direction of the 1,1-dimensional antichiral super submanifold leads to a total derivative in the ordinary space. The latter is nothing but the BRST invariance of the ordinary Lagrangian LB (cf. Equation (26)). In exactly similar fashion, we can provide a geometrical interpretation for the anti-BRST invariance of LB¯ (cf. Equation (27)) in the terminology of the superspace translational generator ∂θ along the θ-direction of the suitably chosen chiral1,1-dimensional super submanifold.At this juncture, we concentrate on the deduction of the CF-type restrictionB+B¯+iC¯Ċ−C¯̇C=0 in the proof of the equivalence between the Lagrangians LB and LB¯ (cf. Equation (24)) within the ambit of ACSA to BRST formalism. In other words, we capture the transformations sabLB (cf. Equation (29)) and sbLB¯ (cf. Equation (28)) in the terminology of the ACSA. Towards this central goal in our mind, first of all, we generalize the ordinary Lagrangian LB to its counterpart chiral super Lagrangian as
(71)LB⟶L~Bcτ,θ=L~fcτ,θ+B~abτ,θEabτ,θĖabτ,θ−i2F¯̇abτ,θFabτ,θ+F¯abτ,θḞabτ,θ+12B~abτ,θB~abτ,θ−iEabτ,θEabτ,θF¯̇abτ,θḞabτ,θ−iEabτ,θĖabτ,θF¯̇abτ,θFabτ,θ−F¯̇abτ,θF¯abτ,θḞabτ,θFabτ,θ,where the superscripts c and ab have already been explained in our earlier discussions. It is very interesting to observe that we have the following:
(72)∂∂θL~Bcτ,θ=ddτC¯Lf+e2iC¯̇C¯Ċ+BC¯̇+eėiC¯̇C¯C+BC¯+i2B−B¯C¯̇C¯C+B2C¯+B+B¯+iC¯Ċ−C¯̇CiC¯¨C¯C−eėC¯̇−2BC¯̇+2iC¯̇C¯Ċ−ddτB+B¯+iC¯Ċ−C¯̇Ce2C¯̇+BC¯≡sabLB.The above equation shows that the LagrangianLB respects the anti-BRST symmetry transformations (20) only when the CF-type restriction is invoked from outside. In a subtle manner, we have obtained the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 within the ambit of ACSA while proving the anti-BRST invariance of the Lagrangian LB. We now demonstrate the BRST invariance of the Lagrangian LB¯and existence of the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 within the purview of theoretical tricks and techniques of ACSA to BRST formalism. Towards these aims in our mind, we generalize the Lagrangian LB¯ onto 1,1-dimensional antichiral super submanifold as follows:
(73)LB¯⟶L~B¯acτ,θ¯=L~facτ,θ¯−B¯~bτ,θ¯Ebτ,θ¯Ėbτ,θ¯−i2F¯bτ,θ¯Ḟbτ,θ¯+F¯̇bτ,θ¯Fbτ,θ¯+12B¯~bτ,θ¯B¯~bτ,θ¯−iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯Ḟbτ,θ¯−iEbτ,θ¯Ėbτ,θ¯F¯bτ,θ¯Ḟbτ,θ¯−F¯̇bτ,θ¯F¯bτ,θ¯Ḟbτ,θ¯Fbτ,θ¯,where the superscripts ac and b have been already explained in our earlier discussions. Keeping in our mind the mapping: ∂θ¯↔sb, we observe the following interesting relationship between sbLB¯ and its counterpart in superspace, namely,
(74)∂∂θ¯L~B¯acτ,θ¯=ddτCLf−e2iC¯̇ĊC+B¯Ċ−eėiC¯ĊC+B¯C+i2B¯−BC¯ĊC+B¯2C+B+B¯+iC¯Ċ−C¯̇CiC¯C¨C+eėĊ−2B¯Ċ+2iC¯̇ĊC+ddτB+B¯+iC¯Ċ−C¯̇Ce2Ċ−B¯C≡sbLB¯,where the r.h.s. is nothing but the operation of sb on the Lagrangian LB¯ (cf. Equation (28)). In other words, we have established an intimate relationship between the BRST symmetry transformation on LB¯and operation of the translational generator ∂θ¯ on the antichiral super Lagrangian L~B¯acτ,θ¯ (defined on the 1,1-dimensional (anti-)chiral super submanifold). A careful and close look on the r.h.s. of (74) demonstrates that we have derived the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 while proving the BRST invariance of LB¯ within the purview of ACSA.We end this section with the following concluding remarks. First, we have captured the (anti-)BRST invariance (cf. Equation (69)) of the first-order Lagrangian Lf within the ambit of ACSA. Second, we have been able to express the (anti-)BRST invariance (cf. Equations (26) and (27)) of the Lagrangians LB and LB¯ of our theory within the purview of ACSA (cf. Equation (70)). Third, we have been able to demonstrate that our observations in Equations (28) and (29) can also be expressed in superspace (cf. Equations (72) and (74)) within the ambit of ACSA. Finally, we have derived the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0, in a subtle manner, by expressing the transformations sbLB¯ and sabLB in the language of ACSA (cf. Equations (72) and (74) for details). In other words, in the ordinary space, whatever we have seen in the proof of the absolute anticommutativity of (anti-)BRST symmetry transformations (cf. Equation (22)), the same restriction appears when we discuss sbLB¯ and sabLB in the superspace by using the formal theoretical techniques of ACSA.


## 7. Off-Shell Nilpotency and Absolute Anticommutativity of the Conserved (Anti-)BRST Charges

In this section, we prove the off-shell nilpotencyQB¯B2=0 and absolute anticommutativity QB,QB¯=0 of the (anti-)BRST charges QB¯B which have already been derived in our Section 3 where JB=QB and JB¯=QB¯ (cf. Equations (31) and (32)). In Section 7.1, we discuss the above properties of the (anti-)BRST charges QB¯B in the ordinary space. Our Section 7.2 contains the theoretical material related with the techniques of capturing the nilpotency and anticommutativity properties of the above charges within the ambit of ACSA. It is quite interesting to state that the CF-type restriction appears when we prove the absolute anticommutativity property of the charges in the ordinary space as well as in the superspace (within the purview of ACSA). We describe explicitly the computation of sbLB¯ in our Appendix B where the algebra is a bit more involved, and ultimately, we demonstrate the existence of the CF-type restriction in the ordinary space (cf. Equation (28)) which has also appeared in Equation (74) within the ambit of ACSA.


### 7.1. Off-Shell Nilpotency and Absolute Anticommutativity Properties: Ordinary Space

In this subsection, we primarily exploit the theoretical potential of the well-known relationship between the continuous symmetry transformations and their generators. In other words, we can prove the off-shell nilpotencyQB2=QB¯2=0 of the (anti-)BRST charges QB¯B in the ordinary space by using the standard relationship between the infinitesimal continuous (anti-)BRST transformations sab and their generators (QB¯B) as
(75)sbQB=−iQB,QB=0,QB2=0,sabQB¯=−iQB¯,QB¯=0,QB¯2=0.The above proofs of the off-shell nilpotency of the conserved charges are nothing but the reflection of the off-shell nilpotencysab2=0 of the (anti-)BRST symmetry transformations (20) and (21) in the ordinary space. It would be worthwhile to point out the fact that, in the computation of the l.h.s. of (75), we have directly applied the (anti-)BRST symmetry transformations (20) and (21) on the appropriate form of the conserved (i.e., Q̇B¯B=0) (anti-)BRST charges (cf. Equations (31) and (32)). In other words, the straightforward application of sb on QB gives us a zero result. The same is the situation (i.e., sabQB¯=0) when we apply the anti-BRST symmetry transformations sab on the anti-BRST charge QB¯. Hence, the Noether conserved charges (cf. Equations (31) and (32)) are off-shell nilpotent of order two (i.e., QB¯B2=0) in the ordinary space due to the key relationship that is given in (75).The above explicit proof of the off-shell nilpotency of the (anti-)BRST charges ensures that they should be able to be written asan exact quantity w.r.t. the off-shell nilpotent sab2=0 (anti-)BRST symmetry transformations sab. Towards this goal in mind, we use the following EL-EOMs (derived from the Lagrangians LB and LB¯), namely,
(76)pμψμ=mψ5,B=−eė+2iC¯̇C+iC¯Ċ,eḂ+ieC¯̇Ċ−ieC¯¨C+12p2−m2=0,BĊ+2ḂC+3eėĊ+e2C¨+ė2C+ee¨C−iC¯C¨C−2iC¯̇ĊC=0,B¯=eė−2iC¯Ċ−iC¯̇C,eB¯̇−ieC¯̇Ċ+ieC¯C¨−12p2−m2=0,B¯C¯̇+2B¯̇C¯−3eėC¯̇−e2C¯¨−ė2C¯−ee¨C¯−iC¯¨C¯C−2iC¯̇C¯Ċ=0,to get rid of the constraints p2−m2≈0 and pμψμ−mψ5≈0 from the expressions for the (anti-)BRST charges QB¯B (cf. Equations (31) and (32)) to recast them as
(77)QB¯1=e2B¯̇C¯−B¯C¯̇+iC¯¨C¯C+C¯̇C¯Ċ+2ieėC¯̇C¯C,QB1=e2BĊ−ḂC−iC¯C¨C+C¯̇ĊC−2ieėC¯ĊC.We have discussed different forms of the (anti-)BRST charges in our AppendixC where the emphasis is laid on the derivation of the expressions for the (anti)BRST charges (cf. Equation (77)). The above expressions of the conserved (anti-)BRST charges can be mathematically expressed in the following exact forms w.r.t. the off-shell nilpotent sab2=0 (anti-)BRST symmetry transformations sab, namely,
(78)QB¯1=sabie2C¯Ċ−C¯̇C,QB1=sbie2C¯̇C−C¯Ċ.Now it is straightforward to note thatsabQB¯1=0 and sbQB1=0 due to the off-shell nilpotency [sab2=0] of the (anti-)BRST symmetry transformations sab. Thus, we conclude, from our observations in Equation (78), that the nilpotency of the (anti-)BRST transformations [sab] is deeply connected with the nilpotency of their generators (anti-)BRST charges [QB¯B] which becomes completely transparent from the direct observations of the following computations:
(79)QB¯12=0⇔sabQB¯1=−iQB¯1,QB¯1=0⇔sab2=0,QB12=0⇔sbQB1=−iQB1,QB1=0⇔sb2=0.The above equation completes our discussion on the proof of the off-shell nilpotency of the conserved (anti-)BRST charges in theordinary space.Now, we dwell on the proof of the absolute anticommutativity (i.e.,QB1QB¯1+QB¯1QB1=0) of the conserved off-shell nilpotent (anti-)BRST charges [QB¯B1]. Towards this central objective in mind, first of all, we assume the sanctity and validity of the CF-type restriction, B+B¯+iC¯Ċ−C¯̇C=0, right from the beginning. As a result, we can express the (anti-)BRST charges QB¯B1, in the alternative forms, as follows:
(80)QB¯1⟶QB¯2=e2BC¯̇−ḂC¯+2iC¯̇C¯Ċ+2ieėC¯̇C¯C,QB1⟶QB2=e2B¯̇C−B¯Ċ−2iC¯̇ĊC−2ieėC¯ĊC.We point out that it is because of the use of the CF-type restriction [B+B¯+iC¯Ċ−C¯̇C=0] that we have been able to express QB¯2 in terms of the Nakanishi-Lautrup auxiliary variable Bτ and QB2 in the language of other Nakanishi-Lautrup-type auxiliary variable B¯τ. At this crucial stage, we observe the following interesting relationships:
(81)QB2=sab−ie2ĊC,QB¯2=sbie2C¯̇C¯.In other words, we have been able to express the anti-BRST charge [QB¯2] as the BRST exact quantity. On the other hand, we have been able to write the BRST charge [QB2] as an exact quantity w.r.t. the nilpotent anti-BRST transformation sab. A close and careful observation of (81) leads to the following (due to the well-known relationship between the continuous (anti-)BRST symmetry transformations sab and their generators as conserved (anti-)BRST charges QB¯B2), namely,
(82)sabQB2=−iQB2,QB¯2=0⇔sab2=0,sbQB¯2=−iQB¯2,QB2=0⇔sb2=0.As a result, we observe that the absolute anticommutativity of the (anti-)BRST chargesQB¯B2 is related to the nilpotency sab2=0 of the (anti-)BRST symmetries.We would like to lay stress on the key results that have been seen in Equation (82). It is very interesting (due to the validity of the CF-type restriction on our theory) to pinpoint that (i) the anticommutativity of the BRST charge QB2with the anti-BRST charge QB¯2 is intimately connected with the nilpotency sb2=0 of the BRST transformations sb and (ii) the anticommutativity property of the anti-BRST charge QB¯2with the BRST charge QB2 owes its origin to the nilpotency sab2=0 of the anti-BRST transformations sab. We conclude this subsection with the following remarks. First, we have shown that the nilpotency of the (anti-)BRST charges QB¯B is deeply related with the nilpotency of the (anti-)BRST transformations sab. Second, we have been able to express the modified form of the BRST charge QB1 and anti-BRST charge QB¯1 as the exact expressions w.r.t. the BRST transformations sb and anti-BRST transformations sab (cf. Equation (78)), respectively. Third, it is due to the existence of the CF-type restriction on our theory that we have been able to express another modified form of the BRST charge QB2 as an exact expression w.r.t. the anti-BRST transformations saband the anti-BRST charge QB¯2 in the BRST-exact form. This exercise has enabled us to prove the absolute anticommutativity (i.e., QB2,QB¯2=0) of the nilpotent (anti-)BRST charges QB¯B2. Finally, the proof of the absolute anticommutativity property (cf. Equation (82)) crucially depends on the existence of the CF-type restriction. Thus, in a subtle manner, we have derived and corroborated the sanctity of the existence of the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 on our theory. We have provided an alternative proof for the appearance of the CF-type restriction (on our 1D reparameterization invariant SUSY theory) in our Appendix D. This completes our discussions on the absolute anticommutativity property of the conserved (anti-)BRST charges (in the ordinary Minkowski space).


### 7.2. Off-Shell Nilpotency and Absolute Anticommutativity Properties: ACSA to BRST Formalism in Superspace

In this subsection, we capture the properties of the nilpotency (i.e., fermionic nature) and absolute anticommutativity (i.e., linear independence) of the (anti-)BRST charges within the purview of ACSA where the superspace consideration on the1,1-dimensional (anti-)chiral super submanifolds has been taken into account. First of all, we focus on the off-shell nilpotency [QB¯B2=0] of the (anti-)BRST charges [QB¯B]. In this context, keeping in our knowledge the mappings: ∂θ↔sab, ∂θ¯↔sb, it can be readily seen that the expressions for the (anti-)BRST charges that have been quoted in Equation (78) can be translated into the superspace as follows:
(83)QB¯1=∂∂θiEabτ,θEabτ,θF¯abτ,θḞabτ,θ−F¯̇abτ,θFabτ,θ≡∫dθiEabτ,θEabτ,θF¯abτ,θḞabτ,θ−F¯̇abτ,θFabτ,θ,(84)QB1=∂∂θ¯iEbτ,θ¯Ebτ,θ¯F¯bτ,θ¯Ḟbτ,θ¯−F¯̇bτ,θ¯Fbτ,θ¯≡∫dθ¯iEbτ,θ¯Ebτ,θ¯F¯bτ,θ¯Ḟbτ,θ¯−F¯̇bτ,θ¯Fbτ,θ¯,where the supervariables with the superscripts ab and b have been obtained in Equations (64) and (57), respectively. At this stage, the off-shell nilpotency QB¯B12=0 of the conserved (anti-)BRST charges QB¯B1 can be written in the superspace (by using the theoretical techniques and tricks of ACSA to BRST formalism) as
(85)∂θQB¯1=0⇔∂θ2=0,∂θ¯QB1=0⇔∂θ¯2=0.Thus, we conclude that the off-shell nilpotency of the anti-BRST chargeQB¯1 is deeply related to the nilpotency ∂θ2=0 of the translational generator ∂θ along the θ-direction of the 1,1-dimensional chiral super submanifold of the general1,2-dimensional supermanifold. Similar type of comments can be made in the context of the off-shell nilpotency of the BRST charge QB1 and its intimate relationship with the nilpotency ∂θ¯2=0 of the translational generator ∂θ¯ on the antichiral super submanifold.We concentrate now on capturing the absolute anticommutativity of the (anti-)BRST charges within the purview of ACSA where the superspace of the1,1-dimensional (anti-)chiral super submanifolds is taken into consideration. Towards this central goal in our mind, we express the modified forms of the (anti-)BRST charges QB¯B2 of Equation (81) in the following mathematical expression within the framework of ACSA, namely,
(86)QB¯2=∂∂θ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯F¯bτ,θ¯≡∫dθ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯F¯bτ,θ¯,(87)QB2=∂∂θ−iEabτ,θEabτ,θḞabτ,θFabτ,θ≡∫dθ−iEabτ,θEabτ,θḞabτ,θFabτ,θ,where the supervariables with the superscripts ab and b have been quoted in Equations (64) and (57), respectively. At this crucial juncture, we note the following:
(88)∂θ¯QB¯2=0⇔∂θ¯2=0,∂θQB2=0⇔∂θ2=0.The above relations are nothing but the explicit proof of the absolute anticommutativity relations of the conserved (anti-)BRST chargesQB¯B2 (within the ambit of ACSA).We wrap up this subsection with the following comments. First, the off-shell nilpotencyQB¯B2=0 of the (anti-)BRST charges QB¯B is intimately connected with the nilpotency ∂θ2=0,∂θ¯2=0 of the translational generators ∂θ,∂θ¯ along the θ,θ¯-directions of the 1,1-dimensional chiral and antichiral super submanifolds. Second, in the ordinary space, the above statements of the off-shell nilpotency are captured in Equations (75) and (79). Third, the absolute anticommutativity of the BRST charge with the anti-BRST charge is related to the nilpotency ∂θ2=0 of the translational generator ∂θ along the θ-direction of the chiral super submanifold. The absolute anticommutativity of the anti-BRST charge with the BRST charge, on the other hand, is connected with the nilpotency ∂θ¯2=0 of the translational generator ∂θ¯ along the θ¯-direction of the antichiral super submanifold. Fourth, the above statements have been corroborated in the ordinary space by Equation (82) where the off-shell nilpotency sab2=0 of the (anti-)BRST transformations sab and the anticommutativity QB2,QB¯2=0 of the (anti-)BRST charges QB¯B2 are found to be interconnected in an intimate and beautiful manner.


## 7.1. Off-Shell Nilpotency and Absolute Anticommutativity Properties: Ordinary Space

In this subsection, we primarily exploit the theoretical potential of the well-known relationship between the continuous symmetry transformations and their generators. In other words, we can prove the off-shell nilpotencyQB2=QB¯2=0 of the (anti-)BRST charges QB¯B in the ordinary space by using the standard relationship between the infinitesimal continuous (anti-)BRST transformations sab and their generators (QB¯B) as
(75)sbQB=−iQB,QB=0,QB2=0,sabQB¯=−iQB¯,QB¯=0,QB¯2=0.The above proofs of the off-shell nilpotency of the conserved charges are nothing but the reflection of the off-shell nilpotencysab2=0 of the (anti-)BRST symmetry transformations (20) and (21) in the ordinary space. It would be worthwhile to point out the fact that, in the computation of the l.h.s. of (75), we have directly applied the (anti-)BRST symmetry transformations (20) and (21) on the appropriate form of the conserved (i.e., Q̇B¯B=0) (anti-)BRST charges (cf. Equations (31) and (32)). In other words, the straightforward application of sb on QB gives us a zero result. The same is the situation (i.e., sabQB¯=0) when we apply the anti-BRST symmetry transformations sab on the anti-BRST charge QB¯. Hence, the Noether conserved charges (cf. Equations (31) and (32)) are off-shell nilpotent of order two (i.e., QB¯B2=0) in the ordinary space due to the key relationship that is given in (75).The above explicit proof of the off-shell nilpotency of the (anti-)BRST charges ensures that they should be able to be written asan exact quantity w.r.t. the off-shell nilpotent sab2=0 (anti-)BRST symmetry transformations sab. Towards this goal in mind, we use the following EL-EOMs (derived from the Lagrangians LB and LB¯), namely,
(76)pμψμ=mψ5,B=−eė+2iC¯̇C+iC¯Ċ,eḂ+ieC¯̇Ċ−ieC¯¨C+12p2−m2=0,BĊ+2ḂC+3eėĊ+e2C¨+ė2C+ee¨C−iC¯C¨C−2iC¯̇ĊC=0,B¯=eė−2iC¯Ċ−iC¯̇C,eB¯̇−ieC¯̇Ċ+ieC¯C¨−12p2−m2=0,B¯C¯̇+2B¯̇C¯−3eėC¯̇−e2C¯¨−ė2C¯−ee¨C¯−iC¯¨C¯C−2iC¯̇C¯Ċ=0,to get rid of the constraints p2−m2≈0 and pμψμ−mψ5≈0 from the expressions for the (anti-)BRST charges QB¯B (cf. Equations (31) and (32)) to recast them as
(77)QB¯1=e2B¯̇C¯−B¯C¯̇+iC¯¨C¯C+C¯̇C¯Ċ+2ieėC¯̇C¯C,QB1=e2BĊ−ḂC−iC¯C¨C+C¯̇ĊC−2ieėC¯ĊC.We have discussed different forms of the (anti-)BRST charges in our AppendixC where the emphasis is laid on the derivation of the expressions for the (anti)BRST charges (cf. Equation (77)). The above expressions of the conserved (anti-)BRST charges can be mathematically expressed in the following exact forms w.r.t. the off-shell nilpotent sab2=0 (anti-)BRST symmetry transformations sab, namely,
(78)QB¯1=sabie2C¯Ċ−C¯̇C,QB1=sbie2C¯̇C−C¯Ċ.Now it is straightforward to note thatsabQB¯1=0 and sbQB1=0 due to the off-shell nilpotency [sab2=0] of the (anti-)BRST symmetry transformations sab. Thus, we conclude, from our observations in Equation (78), that the nilpotency of the (anti-)BRST transformations [sab] is deeply connected with the nilpotency of their generators (anti-)BRST charges [QB¯B] which becomes completely transparent from the direct observations of the following computations:
(79)QB¯12=0⇔sabQB¯1=−iQB¯1,QB¯1=0⇔sab2=0,QB12=0⇔sbQB1=−iQB1,QB1=0⇔sb2=0.The above equation completes our discussion on the proof of the off-shell nilpotency of the conserved (anti-)BRST charges in theordinary space.Now, we dwell on the proof of the absolute anticommutativity (i.e.,QB1QB¯1+QB¯1QB1=0) of the conserved off-shell nilpotent (anti-)BRST charges [QB¯B1]. Towards this central objective in mind, first of all, we assume the sanctity and validity of the CF-type restriction, B+B¯+iC¯Ċ−C¯̇C=0, right from the beginning. As a result, we can express the (anti-)BRST charges QB¯B1, in the alternative forms, as follows:
(80)QB¯1⟶QB¯2=e2BC¯̇−ḂC¯+2iC¯̇C¯Ċ+2ieėC¯̇C¯C,QB1⟶QB2=e2B¯̇C−B¯Ċ−2iC¯̇ĊC−2ieėC¯ĊC.We point out that it is because of the use of the CF-type restriction [B+B¯+iC¯Ċ−C¯̇C=0] that we have been able to express QB¯2 in terms of the Nakanishi-Lautrup auxiliary variable Bτ and QB2 in the language of other Nakanishi-Lautrup-type auxiliary variable B¯τ. At this crucial stage, we observe the following interesting relationships:
(81)QB2=sab−ie2ĊC,QB¯2=sbie2C¯̇C¯.In other words, we have been able to express the anti-BRST charge [QB¯2] as the BRST exact quantity. On the other hand, we have been able to write the BRST charge [QB2] as an exact quantity w.r.t. the nilpotent anti-BRST transformation sab. A close and careful observation of (81) leads to the following (due to the well-known relationship between the continuous (anti-)BRST symmetry transformations sab and their generators as conserved (anti-)BRST charges QB¯B2), namely,
(82)sabQB2=−iQB2,QB¯2=0⇔sab2=0,sbQB¯2=−iQB¯2,QB2=0⇔sb2=0.As a result, we observe that the absolute anticommutativity of the (anti-)BRST chargesQB¯B2 is related to the nilpotency sab2=0 of the (anti-)BRST symmetries.We would like to lay stress on the key results that have been seen in Equation (82). It is very interesting (due to the validity of the CF-type restriction on our theory) to pinpoint that (i) the anticommutativity of the BRST charge QB2with the anti-BRST charge QB¯2 is intimately connected with the nilpotency sb2=0 of the BRST transformations sb and (ii) the anticommutativity property of the anti-BRST charge QB¯2with the BRST charge QB2 owes its origin to the nilpotency sab2=0 of the anti-BRST transformations sab. We conclude this subsection with the following remarks. First, we have shown that the nilpotency of the (anti-)BRST charges QB¯B is deeply related with the nilpotency of the (anti-)BRST transformations sab. Second, we have been able to express the modified form of the BRST charge QB1 and anti-BRST charge QB¯1 as the exact expressions w.r.t. the BRST transformations sb and anti-BRST transformations sab (cf. Equation (78)), respectively. Third, it is due to the existence of the CF-type restriction on our theory that we have been able to express another modified form of the BRST charge QB2 as an exact expression w.r.t. the anti-BRST transformations saband the anti-BRST charge QB¯2 in the BRST-exact form. This exercise has enabled us to prove the absolute anticommutativity (i.e., QB2,QB¯2=0) of the nilpotent (anti-)BRST charges QB¯B2. Finally, the proof of the absolute anticommutativity property (cf. Equation (82)) crucially depends on the existence of the CF-type restriction. Thus, in a subtle manner, we have derived and corroborated the sanctity of the existence of the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 on our theory. We have provided an alternative proof for the appearance of the CF-type restriction (on our 1D reparameterization invariant SUSY theory) in our Appendix D. This completes our discussions on the absolute anticommutativity property of the conserved (anti-)BRST charges (in the ordinary Minkowski space).


## 7.2. Off-Shell Nilpotency and Absolute Anticommutativity Properties: ACSA to BRST Formalism in Superspace

In this subsection, we capture the properties of the nilpotency (i.e., fermionic nature) and absolute anticommutativity (i.e., linear independence) of the (anti-)BRST charges within the purview of ACSA where the superspace consideration on the1,1-dimensional (anti-)chiral super submanifolds has been taken into account. First of all, we focus on the off-shell nilpotency [QB¯B2=0] of the (anti-)BRST charges [QB¯B]. In this context, keeping in our knowledge the mappings: ∂θ↔sab, ∂θ¯↔sb, it can be readily seen that the expressions for the (anti-)BRST charges that have been quoted in Equation (78) can be translated into the superspace as follows:
(83)QB¯1=∂∂θiEabτ,θEabτ,θF¯abτ,θḞabτ,θ−F¯̇abτ,θFabτ,θ≡∫dθiEabτ,θEabτ,θF¯abτ,θḞabτ,θ−F¯̇abτ,θFabτ,θ,(84)QB1=∂∂θ¯iEbτ,θ¯Ebτ,θ¯F¯bτ,θ¯Ḟbτ,θ¯−F¯̇bτ,θ¯Fbτ,θ¯≡∫dθ¯iEbτ,θ¯Ebτ,θ¯F¯bτ,θ¯Ḟbτ,θ¯−F¯̇bτ,θ¯Fbτ,θ¯,where the supervariables with the superscripts ab and b have been obtained in Equations (64) and (57), respectively. At this stage, the off-shell nilpotency QB¯B12=0 of the conserved (anti-)BRST charges QB¯B1 can be written in the superspace (by using the theoretical techniques and tricks of ACSA to BRST formalism) as
(85)∂θQB¯1=0⇔∂θ2=0,∂θ¯QB1=0⇔∂θ¯2=0.Thus, we conclude that the off-shell nilpotency of the anti-BRST chargeQB¯1 is deeply related to the nilpotency ∂θ2=0 of the translational generator ∂θ along the θ-direction of the 1,1-dimensional chiral super submanifold of the general1,2-dimensional supermanifold. Similar type of comments can be made in the context of the off-shell nilpotency of the BRST charge QB1 and its intimate relationship with the nilpotency ∂θ¯2=0 of the translational generator ∂θ¯ on the antichiral super submanifold.We concentrate now on capturing the absolute anticommutativity of the (anti-)BRST charges within the purview of ACSA where the superspace of the1,1-dimensional (anti-)chiral super submanifolds is taken into consideration. Towards this central goal in our mind, we express the modified forms of the (anti-)BRST charges QB¯B2 of Equation (81) in the following mathematical expression within the framework of ACSA, namely,
(86)QB¯2=∂∂θ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯F¯bτ,θ¯≡∫dθ¯iEbτ,θ¯Ebτ,θ¯F¯̇bτ,θ¯F¯bτ,θ¯,(87)QB2=∂∂θ−iEabτ,θEabτ,θḞabτ,θFabτ,θ≡∫dθ−iEabτ,θEabτ,θḞabτ,θFabτ,θ,where the supervariables with the superscripts ab and b have been quoted in Equations (64) and (57), respectively. At this crucial juncture, we note the following:
(88)∂θ¯QB¯2=0⇔∂θ¯2=0,∂θQB2=0⇔∂θ2=0.The above relations are nothing but the explicit proof of the absolute anticommutativity relations of the conserved (anti-)BRST chargesQB¯B2 (within the ambit of ACSA).We wrap up this subsection with the following comments. First, the off-shell nilpotencyQB¯B2=0 of the (anti-)BRST charges QB¯B is intimately connected with the nilpotency ∂θ2=0,∂θ¯2=0 of the translational generators ∂θ,∂θ¯ along the θ,θ¯-directions of the 1,1-dimensional chiral and antichiral super submanifolds. Second, in the ordinary space, the above statements of the off-shell nilpotency are captured in Equations (75) and (79). Third, the absolute anticommutativity of the BRST charge with the anti-BRST charge is related to the nilpotency ∂θ2=0 of the translational generator ∂θ along the θ-direction of the chiral super submanifold. The absolute anticommutativity of the anti-BRST charge with the BRST charge, on the other hand, is connected with the nilpotency ∂θ¯2=0 of the translational generator ∂θ¯ along the θ¯-direction of the antichiral super submanifold. Fourth, the above statements have been corroborated in the ordinary space by Equation (82) where the off-shell nilpotency sab2=0 of the (anti-)BRST transformations sab and the anticommutativity QB2,QB¯2=0 of the (anti-)BRST charges QB¯B2 are found to be interconnected in an intimate and beautiful manner.


## 8. Conclusions

The USFA to BRST formalism (see, e.g., [10–12]) is useful in the context of the gauge theories where the spacetime coordinates do not change. Thus, it was a challenge to include the diffeomorphism (i.e., the general spacetime transformations) within the framework of Bonora-Tonin (BT) superfield approach to BRST formalism (see, e.g., [10–12]). This was achieved by Bonora in Ref. [22] which has been christened by us as the MBTSA where the generalization of the 1D diffeomorphism [i.e., τ⟶τ′=fτ≡τ−ετ] to the 1,2-dimensional supermanifold (cf. Equation (35)) has played an important role in the derivation of the (anti-)BRST symmetry transformations (cf. Equation (50)) for the target space variables (xμ,pμ,ψμ,ψ5). In addition, this approach has enabled us to deduce the (anti-)BRST invariant CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 that is responsible for the absolute anticommutativity of the (anti-)BRST symmetry transformations (cf. Equations (20) and (21)) and existence of the coupled (but equivalent) Lagrangians (24) for our theory.We have taken into account the standard (anti-)BRST symmetry transformations (sbC¯=iB,sabC=iB¯) for the (anti-)ghost variables C¯C which have, in a subtle manner, forced us to consider the (anti-)chiral super expansions (cf. Equation (47)). This has provided us the clue to adopt the ACSA to BRST formalism for the deduction of the proper (anti-)BRST transformations for the rest of the variables of our theory (cf. Section 5). Within the purview of ACSA, we have derived the CF-type restriction when we have proven the equivalence of the coupled (but equivalent) Lagrangians (cf. Section 6). Furthermore, it is the validity of the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 that has enabled us to write (i) the BRST charge as an exact quantity w.r.t. the nilpotent anti-BRST transformation and (ii) the anti-BRST charge as an exact expression w.r.t. the nilpotent BRST transformation. These observations have been responsible for the proof of the absolute anticommutativity of the (anti-)BRST charges (cf. Section 7). In other words, it is the proof of the anticommutativity of the conserved and nilpotent charges QB¯B2 which lead to the existence of the CF-type restriction on our SUSY theory (cf. Section 7).We would like to emphasize that theobservation of the absolute anticommutativity property, in the context of the conserved (anti-)BRST charges, is a novel observation because of the fact that only the (anti-)chiral super expansions have been considered within the ambit of ACSA. This observation of the absolute anticommutativity property is obvious when one takes into account the full super expansions of the supervariables along all the possible Grassmannian directions of the suitably chosen supermanifold on which the ordinary theory is generalized. Furthermore, the appearance of the CF-type restriction in the computations of sbLB¯ and sabLB (cf. Equations (28) and (29)) in the ordinary space and its analogue in the superspace are very interesting observations in our present endeavor (cf. Section 6). The other observation that merits a clear and special mention is the universality of the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 in the context of the reparameterization (i.e., 1D diffeomorphism) invariant non-SUSY theory of a scalar relativistic particle as well as a nonrelativistic particle [23, 24] and our present SUSY system of a spinning relativistic particle (where the nilpotent SUSY transformations exist between the specific set of bosonic and fermionic variables of our SUSY theory).It is worthwhile to mention that, for theD-dimensional diffeomorphism invariant theory [22, 25] where the infinitesimal diffeomorphism symmetry transformation is xμ⟶xμ′=xμ−εμx (with μ=0,1,2⋯D−1), the general form of the CF-type restriction has been obtained as Bμ+B¯μ+iC¯ρ∂ρCμ+Cρ∂ρC¯μ=0 where the (anti-)ghost fields C¯μCμ correspond to the infinitesimal transformation parameter εμx in the general coordinate transformation xμ′=xμ−εμxand the Nakanishi-Lautrup fields B¯μBμ appear in the (anti-)BRST symmetry transformations: sbC¯μ=iBμ, sabCμ=iB¯μ. It is straightforward to note that the CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 is the limiting case of the above generalD-dimensional CF-type restriction in the case of the BRST approach to D-dimensional diffeomorphism invariant theory [22, 25]. Thus, our theoretical treatments of the reparameterization (i.e., 1D diffeomorphism) invariant theories of the scalar and spinning relativistic particles are correct.One of the highlights of ACSA to BRST formalism is the observation that it distinguishes between the suitably chosen1,1-dimensional chiral and antichiral super submanifolds in the proof of the absolute anticommutativity of the conserved (anti-)BRST charges. For instance, we note that the anticommutativity of the BRST charge QB2with the anti-BRST charge QB¯2 is connected with the nilpotency ∂θ2=0 of the translational generator ∂θ along the θ-direction of the chiral super submanifold (cf. Equation (88)). On the other hand, the anticommutativity of the anti-BRST charge QB¯2with the BRST charge QB2 crucially depends on the nilpotency ∂θ¯2=0 of the translational generator ∂θ¯ (cf. Equation (88)) along the θ¯-direction of the antichiral super submanifold (cf. Section 7 for details). This observation is a reflection of our discussion on the absolute anticommutativity property of the (anti-)BRST charges in the ordinary space (cf. Section 7) where the off-shell nilpotency of the (anti-)BRST transformations (cf. Equation (82)) plays a decisive role.We plan to extend our present study to the physical3+1-dimensional (4D) theories of the gravitation and higher dimensional (super)string theories where there is existence of the diffeomorphism invariance. In other words, we plan to apply the ideas of MBTSA and ACSA together to find out the (anti-)BRST symmetries of the above mentioned theories. The mathematical elegance, rigor, and beauty of the MBTSA [22, 25] should find more applications to some physical systems of interest in theoretical high-energy physics. We envisage to take up these challenges in our future investigations. Before we end this section, it is worthwhile to point out that in our earlier works (see, e.g., [27, 28]), we have applied the techniques and tricks of ACSA to obtain the nilpotent symmetries of the N=2 SUSY quantum mechanical models of interest. However, we have found that the conserved charges are not absolutely anticommuting. Thus, our observation of the absolute anticommutativity property in the context of (anti-)BRST charges is novel and interesting.



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*Source: 2056629-2021-11-10.xml*