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

Authors: B. Chauhan; S. Kumar
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

We derive the off-shell nilpotent of order two and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST, and (anti-)co-BRST symmetry transformations for the Christ–Lee (CL) model in one0+1-dimension (1D) of spacetime by exploiting the (anti-)chiral supervariable approach (ACSA) to BRST formalism where the quantum symmetry (i.e., (anti-)BRST along with (anti-)co-BRST) invariant quantities play a crucial role. We prove the nilpotency and absolute anticommutativity properties of the (anti-)BRST along with (anti-)co-BRST conserved charges within the scope of ACSA to BRST formalism where we take only one Grassmannian variable into account. We also show the (anti-)BRST and (anti-)co-BRST invariances of the Lagrangian within the scope of ACSA.


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## Body

## 1. Introduction

Gauge theories describe three (i.e., strong, weak, and electromagnetic) out of four fundamental interactions of nature which are characterized by first-class constraints in the context of Dirac’s prescription for the classification scheme of constraints [1, 2]. The existence of the first-class constraints, in a given system, is the key signature of a gauge theory. Many interesting theories, in the domain of physics, are expressed by the suitable Lagrangians that are invariant under the gauge symmetry transformations. These symmetries are generated by the first-class constraints in a given gauge theory. For the covariant canonical quantization of the gauge theory, the Becchi-Rouet-Stora-Tyutin (BRST) quantization procedure plays a decisive role where we replace the infinitesimal local gauge parameter by ghost and antighost fields [3–6]. Thus, in this formalism, we have two fermionic-type global BRST sb and anti-BRST sab transformations at the quantum level (for a given local gauge symmetry transformation at the classical level). These symmetry transformations are endowed with two important properties: (i) nilpotency of order two (i.e., sb2=0,sab2=0) and (ii) absolute anticommutativity (i.e., sbsab+sabsb=0). The first property signifies that these quantum BRST and anti-BRST symmetry transformations are fermionic in nature whereas the second property shows that both symmetry transformations are linearly independent of each other. Besides the (anti-)BRST symmetry transformations, we have two more fermionic and linearly independent symmetry transformations which are christened as the co-BRST (sd) and anti-co-BRST (sad) symmetry transformations. The latter fermionic-type symmetry transformations are valid for any D-dimensional p-form (p=1,2,3,⋯) gauge theories in D=2p dimensions of spacetime. We point out that there are some specific systems such as rigid rotor and Christ–Lee (CL) model in one dimension (1D) that respect the (anti-)co-BRST transformations along with the (anti-)BRST transformations [7–10].The geometrical interpretation and the emergence of nilpotent (anti-)BRST symmetry transformations have been shown within the ambit of Bonora-Tonin (BT) superfield formalism [11–13] where the Grassmannian variables ϑ,ϑ¯ and their corresponding derivatives ∂ϑ,∂ϑ¯ (with properties ϑ2=ϑ¯2=0,∂ϑ2=∂ϑ¯2=0 and ϑϑ¯+ϑ¯ϑ=0,∂ϑ∂ϑ¯+∂ϑ¯∂ϑ=0) play a very important role. In BT-superfield approach, we see the connections between the (anti-)BRST symmetry transformations (sab) (with properties sb2=sab2=0 and sbsab+sabsb=0) and Grassmannian translational generators ∂ϑ,∂ϑ¯ because of the fact that both have the same algebraic structure. In this formalism, any D-dimensional Minkowskian manifold is generalized onto the D,2-dimensional supermanifold. This suitably chosen supermanifold is denoted by the superspace coordinates xμ,ϑ,ϑ¯ where xμμ=0,1,2,⋯,D−1 are the spacetime coordinates and ϑ,ϑ¯ are a pair of Grassmannian variables.The CL model is one of the simplest examples of gauge-invariant system which is described by a singular Lagrangian [14]. Physically, the CL model represents the motion of a point particle moving in a plane under the influence of a central potential. The CL model has been studied at the classical and quantum levels in different perspectives [14–20]. This model is endowed with the first-class constraints in the Dirac’s terminology for the classification scheme of constraints [1, 2]. This model is also quantized by using the Faddeev-Jackiw quantization where all the primary and derived constraints are treated on equal footing without any type of further classification [20]. Within the framework of BRST formalism, the CL model respects six independent continuous symmetries (i.e., BRST, anti-BRST, co-BRST, anti-co-BRST, ghost-scale, and bosonic symmetries) (see, e.g., [9] for detail). The BT-superfield formalism has been applied to obtain the absolutely anticommuting and off-shell nilpotent (anti-)BRST as well as (anti-)co-BRST symmetry transformations where the techniques of celebrated horizontality condition (HC) and dual horizontality condition have been used [10], respectively.In our recent set of papers [21–25], we have used a newly proposed formalism which has been called by us as the (anti-)chiral superfield/supervariable approach (ACSA) to BRST formalism. In this approach, we take into account only one Grassmannian variable in the expression for the superfield/supervariable. Thus, the resulting superfield/supervariable turns into (anti-)chiral version of the superfield/supervariable. In other words, in this formalism, any D-dimensional Minkowskian manifold is generalized onto D,1-dimensional super-submanifolds of the most generalD,2-dimensional supermanifold. The proof of the absolute anticommutativity property of Noether’s conserved charges is obvious in the case of BT-superfield formalism where the full super expansions of superfields/supervariables are taken into account. In the case of ACSA, we have also been able to show the nilpotency and absolute anticommutativity properties of conserved charges despite the fact that we have taken only one Grassmannian variable into account. In our present endeavor, we derive the (anti-)BRST together with (anti-)co-BRST symmetry transformations where some specific sets of (anti-)BRST and (anti-)co-BRST invariant restrictions play a very important role. We also show the absolute anticommutativity as well as nilpotency properties of (anti-)BRST and (anti-)co-BRST conserved charges within the realm of ACSA to BRST formalism.Against the background of the above paragraph, it has been found that the nilpotency of theN=2 super charges is true for any N=2 supersymmetric (SUSY) quantum mechanical models (see, e.g., [26–28]) within the ambit of (anti-)chiral supervariable approach to BRST formalism. However, the application of ACSA, in the realm of N=2 SUSY quantum mechanical model, does not lead to the absolute anticommutativity of the N=2 super conserved charges. Rather, it has been found that the anticommutator of the above N=2 SUSY conserved charges leads to the time translation of the variable on which it acts. Thus, it is crystal clear that ACSA to BRST formalism does not lead to the derivation of absolute anticommutativity of the charges for all N=2 SUSY theories.The different sections of our present paper are arranged as follows. In Section2, we discuss the (anti-)BRST and (anti-)co-BRST symmetry transformations for the CL model and derive the conserved charges. Our Section 3 deals with the ACSA to BRST formalism where we derive the (anti-)BRST symmetry transformations. Section 4 is devoted to the derivation of (anti-)co-BRST symmetry transformations by using the ACSA to BRST formalism where the super expansions of (anti-)chiral supervariables are utilized in a fruitful manner. In Section 5, we express the conserved (anti-)BRST and (anti-)co-BRST charges on the 1,1-dimensional super-submanifolds (of the most general1,2-dimensional supermanifold) on which our theory is generalized and provide the proof of nilpotency and absolute anticommutativity properties of the (anti-)BRST along with (anti-)co-BRST charges within the ambit of ACSA to BRST formalism. In Section 6, we discuss the (anti-)BRST and (anti-)co-BRST invariances of the Lagrangian within the scope of ACSA. Finally, we point out our key results and discovery in Section 7 and mention a few future scopes for further investigation.


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

The first-order and gauge-invariant Lagrangian of the Christ–Lee (CL) model [the other equivalent second-order Lagrangian (Ls) [14] associated with the Christ–Lee model is Ls=1/2ṙ2+1/2ṙ2φ̇−z2−Vr; but we choose only the first-order Lagrangian Lf in our present work because it respects maximum symmetry transformations] in 0+1-dimension (1D) of spacetime in polar coordinates system is given by [14, 16, 19]
(1)Lf=ṙpr+φ̇pφ−12pr2−12r2pφ2−zpφ−Vr,whereṙ and φ̇ are the generalized velocities, pr and pφ are their corresponding canonical momenta, respectively, and z is a Lagrange multiplier which enforces a constraint pφ≈0. This Lagrangian explains that a two-dimensional particle moves under the influence of the central potential Vr bounded from below.The above system has a primary constraint as follows:(2)Φ1=∂Lf∂ż=pz≈0.The time derivative of the primary constraintΦ1 leads to the following secondary constraint:
(3)dΦ1dτ=ddτ∂Lf∂ż≈0⇒Φ2=pφ≈0.It is clear that bothΦ1 and Φ2 are first-class constraints. The gauge symmetry transformation generator can be written in terms of first-class constraints as
(4)G=χ̇τΦ1+χτΦ2,whereχτ is an infinitesimal and time-dependent local gauge parameter and χ̇τ=dχ/dτ. Using the definition of a generator
(5)δϕτ=−iϕτ,G,ϕ=r,pr,φ,pφ,z,whereϕ is the generic variable that is present in the first-order Lagrangian Lf. We deduce the following local gauge transformations by exploiting Equation (5), namely,
(6)δzτ=χ̇τ,δφτ=χτ,δrτ,prτ,pφτ,Vr=0.It is elementary to check that, under the above local gauge symmetry transformations, the Lagrangian under consideration remains invariant (i.e.,δLf=0).The (anti-)BRST invariant Lagrangian for the0+1-dimensional CL model containing the gauge-fixing and Faddeev–Popov ghost terms is given by [15]
(7)L=ṙpr+φ̇pφ−12pr2−12r2pφ2−zpφ−Vr+12B2+Bż+φ+iC¯C−iC¯̇Ċ,where the Nakanishi–Lautrup-typeauxiliary variable B is used to linearize the gauge-fixing term and the Faddeev–Popov (anti-)ghost variables C¯C are used to make the Lagrangian BRST invariant. These fermionic variables C¯C (with C2=C¯2=0,CC¯+C¯C=0) have ghost numbers −1+1, respectively. The above Lagrangian respects the following off-shell nilpotent (i.e., sab2=sad2=0) and absolutely anticommuting (i.e., sbsab+sabsb=0 and sdsad+sadsd=0) (anti-)BRST along with (anti-)co-BRST symmetry transformations:
(8)sabz=C¯̇,sabz=C¯̇,sabφ=C¯,sabC=−iB,sabr,pr,pφ,B,C¯=0,sbz=Ċ,sbφ=C,sbC¯=iB,sbr,pr,pφ,B,C=0,(9)sadz=C,sadφ=−Ċ,sadC¯=−ipφ,sadr,pr,pφ,B,C=0,sdz=C¯,sdφ=−C¯̇,sdC=ipφ,sdr,pr,pφ,B,C¯=0.It can be clearly checked that under the above (anti-)BRST (Equation (8)) and (anti-)co-BRST (Equation (9)) symmetry transformations the Lagrangian (Equation (7)) remains quasi-invariant (i.e., modulo a total time derivative):
(10)sbL=ddτBĊ,sabL=ddτBC¯̇,sdL=−ddτpφC¯̇,sadL=−ddτpφĊ.As a result, the action integralS=∫dτL remains invariant under the (anti-)BRST as well as (anti-)co-BRST symmetry transformations (i.e., sabS=0,sadS=0). According to Noether’s theorem, the invariance of the above Lagrangian under the nilpotent (anti-)BRST together with (anti-)co-BRST symmetry transformations leads to the following (anti-)BRST charges Qab and (anti-)co-BRST charges Qad, namely,
(11)Qab=BC¯̇+pφC¯≡BC¯̇−BC¯,Qb=BĊ+pφC≡BĊ−BC,(12)Qad=BC−pφĊ≡BC+BĊ,Qd=BC¯−pφC¯̇≡BC¯+BC¯̇,where the equivalent forms of the above charges are written with the help of the equation of motion:pφ=−B. The above charges are nilpotent of order two (i.e., Qab2=Qad2=0) and anticommuting in nature (i.e., QbQab+QabQb=0 and QdQad+QadQd=0). The conservation law for these charges (i.e., d/dτQab=0 and d/dτQad=0) can be easily proven by using the following interesting Euler-Lagrange equations of motion [besides these EOMs, we use an equation B¨+B=0 derived from the EOMs (13) to prove the conservation law for the (anti-)BRST together with (anti-)co-BRST conserved charges (Equations (11) and (12))] (EOMs) derived from Lagrangian L of our theory (Equation (7)), namely,
(13)B+pφ=0,B=ṗφ,B=−ż+φ,ṗr−pφ2r3+V′r=0,ṙ=pr,φ̇−z−pφr2=0,C¯+C¯¨=0,C+C¨=0.The (anti-)co-BRST and (anti-)BRST conserved charges are the generators of the (anti-)co-BRST and (anti-)BRST symmetry transformations, respectively. As one can easily check that the following relationships are true:(14)sdξ=−iξ,Qd±,sadξ=−iξ,Qad±,sbξ=−iξ,Qb±,sabξ=−iξ,Qab±,whereξ denotes any generic variable present in the Lagrangian L of our theory. The subscript ± on the square brackets denotes the (anti)commutator which depends on the nature of generic variables ξ being (fermionic) bosonic in nature.


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

In this section, we determine the nilpotent (anti-)BRST symmetry transformations (cf. Equation (8)) by using (anti-)chiral supervariable approach (ACSA) to BRST formalism where we shall use the (anti-)chiral super expansions of supervariables. Towards this goal, first of all, we generalize the ordinary variables of the Lagrangian (7) onto 1,1-dimensional antichiral super-submanifold (of the most common1,2-dimensional supermanifold) as follows:
(15)zτ⟶Zτ,ϑ¯=zτ+ϑ¯f1τ,(16)φτ⟶Θτ,ϑ¯=φτ+ϑ¯f2τ,(17)Cτ⟶Fτ,ϑ¯=Cτ+iϑ¯b1τ,(18)C¯τ⟶F¯τ,ϑ¯=C¯τ+iϑ¯b2τ,(19)rτ⟶Rτ,ϑ¯=rτ+ϑ¯f3τ,(20)prτ⟶Prτ,ϑ¯=prτ+ϑ¯f4τ,(21)pφτ⟶Pφτ,ϑ¯=pφτ+ϑ¯f5τ,(22)Bτ⟶B~τ,ϑ¯=Bτ+ϑ¯f6τ,whereb1,b2 are the bosonic derived variables and f1,f2,f3,f4,f5,f6 are the fermionic derived variables due to fermionic nature of ϑ¯. We determine the precise value of these derived variables in terms of the auxiliary and basic variables present in the BRST invariant Lagrangian (7) by using the BRST invariant quantities/restrictions.According to the basic principles of ACSA, the BRST invariant quantitiesmust remain independent of the Grassmannian variable (ϑ¯) when they are generalized onto the 1,1-dimensional antichiral super-submanifold. The BRST invariant quantities are the specific combinations of the variables present in Lagrangian (7). These are given as follows:
(23)sbr,pr,pφ,B,C=0,sbzĊ=0,sbφC=0,sbBz+iC¯̇Ċ=0,sbφ̇−z=0,sbBφ+iC¯C=0.We generalize the above BRST invariant restrictions onto the1,1-dimensional antichiral super-submanifolds (of the suitably chosen most common1,2-dimensional supermanifold)
(24)Rτ,ϑ¯=rτ,Prτ,ϑ¯=prτ,Pφτ,ϑ¯=pφτ,B~τ,ϑ¯=Bτ,Fτ,ϑ¯=Cτ,(25)Zτ,ϑ¯Ḟτ,ϑ¯=zτĊτ,Θτ,ϑ¯Fτ,ϑ¯=φτCτ,(26)B~̇τ,ϑ¯Zτ,ϑ¯+iF¯̇τ,ϑ¯Ċτ,ϑ¯=Ḃτzτ+iC¯̇τĊτ,(27)Θ̇τ,ϑ¯−Zτ,ϑ¯=φ̇τ−zτ,(28)B~τ,ϑ¯Θτ,ϑ¯+iF¯τ,ϑ¯Fτ,ϑ¯=Bτφτ+iC¯τCτ.The above restrictions lead to the derivation of the derived variables in terms of thebasic and auxiliary variables. To determine the value of these variables, we perform the step-by-step explicit calculations. For this purpose, first of all, we use the generalization of the trivial BRST invariant restrictions given in the first line of Equation (24) as
(29)Pφτ,ϑ¯=pφτ⇒f5=0,B~τ,ϑ¯=Bτ⇒f6=0,Rτ,ϑ¯=rτ⇒f3=0,Fτ,ϑ¯=Cτ⇒b1=0,Prτ,ϑ¯=prτ⇒f4=0.After substituting the above value of derived variables from (29) to (15), we get the following expressions for the antichiral supervariables, namely,
(30)Cτ⟶Fbτ,ϑ¯=Cτ+ϑ¯0≡Cτ+ϑ¯sbCτ,rτ⟶Rbτ,ϑ¯=rτ+ϑ¯0≡rτ+ϑ¯sbrτ,prτ⟶Prbτ,ϑ¯=prτ+ϑ¯0≡prτ+ϑ¯sbprτ,pφτ⟶Pφbτ,ϑ¯=pφτ+ϑ¯0≡pφτ+ϑ¯sbpφτ,Bτ⟶B~bτ,ϑ¯=Bτ+ϑ¯0≡Bτ+ϑ¯sbBτ,where the superscriptb on the antichiral supervariables denotes that these supervariables have been obtained after the use of BRST invariant quantities. It is clear that the coefficients of Grassmannian variable ϑ¯ are simply the quantum BRST symmetries (8). Now, in the case of nontrivial BRST invariant restrictions: sbzĊ=0 and sbφC=0, the following generalizations onto 1,1-dimensional super-submanifold, namely,
(31)Zτ,ϑ¯Ḟbτ,ϑ¯=zτĊτ,Θτ,ϑ¯Fbτ,ϑ¯=φτCτ,lead to the following interesting results:
(32)f1τĊτ=0⇒f1τ∝Ċτ,⇒f1τ=κ1Ċτ,f2τCτ=0⇒f2τ∝Cτ,⇒f2τ=κ2Cτ,whereκ1 and κ2 are the proportionality constants. To determine the value of these constants, we use the generalization of BRST invariant restriction sbφ̇−z=0 as
(33)Θ̇τ,ϑ¯−Zτ,ϑ¯=φ̇τ−zτ⇒κ1=κ2.Finally, to determine the value of constants, we generalize the BRST invariant restrictionssbḂz+iC¯̇Ċ=0 and sbḂφ+iC¯C=0 onto 1,1-dimensional super-submanifold as
(34)B~̇bτ,ϑ¯Zτ,ϑ¯+iF¯̇τ,ϑ¯Ḟbτ,ϑ¯=Ḃτzτ+iC¯̇τĊτ⇒ḃ2τ=κ1Ḃτ,(35)B~bτ,ϑ¯Θτ,ϑ¯+iF¯τ,ϑ¯Fbτ,ϑ¯=Bτφτ+iC¯τCτ⇒b2τ=κ2Bτ.Using the results obtained in Equations (33) and (34), it is clear that κ1=κ2=1. Therefore, we obtain the value of derived variables as f1τ=Ċτ,f2τ=Cτ,b2τ=Bτ; thus, we get the following expansions for antichiral supervariables:
(36)zτ⟶Zbτ,ϑ¯=zτ+ϑ¯Ċτ≡zτ+ϑ¯sbzτ,(37)φτ⟶Θbτ,ϑ¯=φτ+ϑ¯Cτ≡φτ+ϑ¯sbφτ,(38)C¯τ⟶F¯bτ,ϑ¯=C¯τ+ϑ¯iBτ≡C¯τ+ϑ¯sbC¯τ.Thus, in view of Equation (36), we have a connection between the BRST symmetry transformation sb and partial derivative ∂ϑ¯ on the antichiral super-submanifold defined by the mapping: sb↔∂ϑ¯ (see, e.g., [11–13] for details). To be more clear, the BRST transformation of any generic variable ψτ is equal to the translation of the corresponding antichiral supervariable Ψbτ,ϑ¯ along the ϑ¯-direction. Mathematically, it can be represented as sbψτ=∂/∂ϑ¯Ψbτ,ϑ¯=∂ϑ¯Ψbτ,ϑ¯. In other words, one can say that the coefficient of ϑ¯ in the expansion of an antichiral supervariable is simply the quantum BRST symmetry transformation of the corresponding variable.We are now in the stage to derive the quantum anti-BRST symmetry transformations usingchiral supervariable approach. In this context, we use the chiral super expansions of the chiral supervariables where we generalize 0+1-dimensional variables onto the 1,1-dimensional super-submanifold of the suitably chosen most common1,2-dimensional supermanifold. The chiral super expansions of the ordinary variables are as follows:
(39)zτ⟶Zτ,ϑ=zτ+ϑf¯1τ,(40)φτ⟶Θτ,ϑ=φτ+ϑf¯2τ,(41)Cτ⟶Fτ,ϑ=Cτ+iϑb¯1τ,(42)C¯τ⟶F¯τ,ϑ=C¯τ+iϑb¯2τ,(43)rτ⟶Rτ,ϑ=rτ+ϑf¯3τ,(44)prτ⟶Prτ,ϑ=prτ+ϑf¯4τ,(45)pφτ⟶Pφτ,ϑ=pφτ+ϑf¯5τ,(46)Bτ⟶B~τ,ϑ=Bτ+ϑf¯6τ,where derived variablesb¯1,b¯2 are the bosonic and derived variables f¯1,f¯2,f¯3,f¯4,f¯5,f¯6 are fermionic in nature. The anti-BRST invariant restrictions also must remain independent of the Grassmannian variable (ϑ) when they are generalized onto the 1,1-dimensional chiral super-submanifold. The anti-BRST invariant restrictions are given as
(47)sabr,pr,pφ,B,C¯=0,sabzC¯̇=0,sabφC¯=0,sabḂz+iC¯̇Ċ=0,sabφ̇−z=0,sabBφ+iC¯C=0.As the physical quantities remain independent of the Grassmannian variableϑ which imply that the anti-BRST invariant restrictions can be generalized onto the 1,1-dimensional super-submanifold of the most common1,2-dimensional supermanifold as follows:
(48)Rτ,ϑ=rτ,Prτ,ϑ=prτ,Pφτ,ϑ=pφτ,B~τ,ϑ=Bτ,F¯τ,ϑ=C¯τ,Zτ,ϑF¯̇τ,ϑ=zτC¯̇τ,Θτ,ϑF¯τ,ϑ=φτC¯τ,B~̇τ,ϑZτ,ϑ+iF¯̇τ,ϑḞτ,ϑ=Ḃτzτ+iC¯̇τĊτ,Θ̇τ,ϑ−Zτ,ϑ=φ̇τ−zτ,B~τ,ϑΘτ,ϑ+iF¯τ,ϑFτ,ϑ=Bτφτ+iC¯τCτ.The above generalizations of the anti-BRST invariant restrictions (Equation (47)) lead to the derivation of the chiral derived variables in terms of the auxiliary and basic variables present in the Lagrangian L, namely,
(49)b¯2=0,f¯3=0,f¯4=0,f¯5=0,f¯6=0,f¯1=C¯̇,f¯2=C¯,b¯1=−B.The above values ofchiral derived variables are obtained in a similar fashion as the antichiral derived variables are obtained. After the substitution of the above derived variables into the chiral super expansions (39), we get the following expressions for the chiral supervariables onto 1,1-dimensional super-submanifold as
(50)zτ⟶Zabτ,ϑ=zτ+ϑC¯̇≡zτ+ϑsabzτ,φτ⟶Θabτ,ϑ=φτ+ϑC¯≡φτ+ϑsabφτ,Cτ⟶Fabτ,ϑ=Cτ+ϑ−iB≡Cτ+ϑsabCτ,C¯τ⟶F¯abτ,ϑ=C¯τ+ϑ0≡C¯τ+ϑsabC¯τ,rτ⟶Rabτ,ϑ=rτ+ϑ0≡rτ+ϑsabrτ,prτ⟶Prabτ,ϑ=prτ+ϑ0≡prτ+ϑsabprτ,pφτ⟶Pφabτ,ϑ=pφτ+ϑ0≡pφτ+ϑsabpφτ,Bτ⟶B~abτ,ϑ=Bτ+ϑ0≡Bτ+ϑsabBτ.Here, the coefficients ofϑ are simply the anti-BRST symmetry transformations (see, e.g., [11–13] for detail). In fact, the anti-BRST transformation of any generic variable ψτ is simply the translation of the corresponding chiral supervariable Ψabτ,ϑ along the ϑ-direction. Mathematically, this statement can be corroborated as sabψτ=∂/∂ϑΨabτ,ϑ=∂ϑΨabτ,ϑ. Thus, it is clear that there is a mapping between the quantum anti-BRST symmetry transformation sab and the Grassmannian partial derivative ∂ϑ defined on the chiral super-submanifold with the mapping: sab↔∂ϑ.


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

In this section, we derive the nilpotent (anti-)co-BRST symmetry transformations using the (anti-)chiral supervariable approach (ACSA) where we use the expansions of the (anti-)chiral supervariables and the (anti-)co-BRST invariant restrictions. Toward this goal in our mind, first of all, we determine the co-BRST symmetries by exploiting thechiral super expansions given in Equation (39) and the co-BRST invariant restrictions. The co-BRST invariant restrictions are given as
(51)sdr,pr,pφ,B,C¯=0,sdzC¯=0,sdφC¯̇=0,sdφṗφ+iC¯̇Ċ=0,sdzpφ−iC¯C=0,sdφ+ż=0.According to thebasic rules of ACSA, the above co-BRST invariant restrictions can be generalized onto the 1,1-dimensional super-submanifold (of the suitably chosen most common1,2-dimensional supermanifold) as
(52)Rτ,ϑ=rτ,Prτ,ϑ=prτ,Pφτ,ϑ=pφτ,B~τ,ϑ=Bτ,F¯τ,ϑ=C¯τ,(53)Zτ,ϑF¯τ,ϑ=zτC¯τ,Θτ,ϑF¯̇τ,ϑ=φτC¯̇τ,(54)Θτ,ϑṖφτ,ϑ+iF¯̇τ,ϑḞτ,ϑ=φτṗφτ+iC¯̇τĊτ,(55)Zτ,ϑPφτ,ϑ−iF¯τ,ϑFτ,ϑ=zτpφτ−iC¯τCτ,(56)Θτ,ϑ+Żτ,ϑ=φτ+żτ.At this stage, we determine the value of derived variables of Equation (39) using the above generalizations of the co-BRST invariant restrictions. To derive the value of the derived variables, first of all, we use the first line entry of Equation (52) where the trivial co-BRST invariant quantities are generalized which implies the following relationships:
(57)Pφτ,ϑ=pφτ⇒f¯5=0,B~τ,ϑ=Bτ⇒f¯6=0,Rτ,ϑ=rτ⇒f¯3=0,F¯τ,ϑ=C¯τ⇒b¯2=0,Prτ,ϑ=prτ⇒f¯4=0.After substituting the above value of derived variables into the expressions of thechiral super expansions (Equation (39)), we obtain the following chiral super expansions:
(58)C¯τ⟶F¯dτ,ϑ=C¯τ+ϑ0≡C¯τ+ϑsdC¯τ,(59)rτ⟶Rdτ,ϑ=rτ+ϑ0≡rτ+ϑsdrτ,(60)prτ⟶Prdτ,ϑ=prτ+ϑ0≡prτ+ϑsdprτ,(61)pφτ⟶Pφdτ,ϑ=pφτ+ϑ0≡pφτ+ϑsdpφτ,(62)Bτ⟶B~dτ,ϑ=Bτ+ϑ0≡Bτ+ϑsdBτ,where superscriptd on the chiral supervariables denotes the supervariables obtained after the application of the co-BRST (i.e., dual-BRST) invariant restrictions. For the nontrivial case, first of all, we generalize the co-BRST invariant restriction sdzC¯=0 and sdφC¯̇=0 onto 1,1-dimensional super-submanifold as
(63)Zτ,ϑF¯dτ,ϑ=zτC¯τ,Θτ,ϑF¯̇dτ,ϑ=φτC¯̇τ,which lead to the following relationships for the derived variables:
(64)f¯1τC¯τ=0⇒f¯1τ∝C¯τ,⇒f¯1τ=−κ¯1C¯τ,f¯2τC¯̇τ=0⇒f¯2τ∝C¯̇τ,⇒f¯2τ=κ¯2C¯̇τ,whereκ¯1 and κ¯2 are the proportionality constants. To determine the value of these constants, we further use the generalizations of the co-BRST invariant restrictions sdφ+ż=0,sdφṗφ+iC¯̇Ċ=0 and sdzpφ−iC¯C=0 as
(65)Θτ,ϑ+Żτ,ϑ=φτ+żτ⇒κ¯1=κ¯2,(66)Θτ,ϑṖφdτ,ϑ+iF¯̇dτ,ϑḞτ,ϑ=φτṗφτ+iC¯̇τĊτ⇒b¯̇1τ=−κ¯2ṗφτ,(67)Zτ,ϑPφdτ,ϑ−iF¯dτ,ϑFτ,ϑ=zτpφτ−iC¯τCτ⇒b¯1τ=−κ¯1pφτ.The results of the above three relations in Equation (65) imply that proportionality constant is equal (i.e., κ¯1=κ¯2) and their values are equal to minus one (i.e., κ¯1=κ¯2=−1). Therefore, we get the value of the derived variables as f¯1=C¯,f¯2=−C¯̇,b¯1=pφ. As a result, we have the following chiral super expansions of the ordinary variables:
(68)zτ⟶Zdτ,ϑ=zτ+ϑC¯τ≡zτ+ϑsdzτ,φτ⟶Θdτ,ϑ=φτ+ϑ−C¯̇τ≡φτ+ϑsdφτ,Cτ⟶Fdτ,ϑ=Cτ+ϑipφτ≡Cτ+ϑsdCτ,where superscriptd on the supervariables denotes the same meaning as in Equation (58). From the above equations, it is clear that the translation of any generic chiral supervariable Ψdτ,ϑ along the ϑ-direction generates co-BRST symmetry transformation (sd) of the corresponding variable ψτ. Mathematically, we can express these statements as sdψτ=∂ϑΨdτ,ϑ. In other words, the coefficients of ϑ are simply the co-BRST symmetry transformations. Thus, the co-BRST symmetry transformation sd is connected with the Grassmannian derivative ∂ϑ (i.e., sd↔∂ϑ) [11–13].Now, for the derivation of anti-co-BRST symmetry transformations, we use theantichiral super expansions of the supervariables (Equation (15)) and anti-co-BRST invariant restrictions. The anti-co-BRST invariant restrictions are listed as
(69)sadr,pr,pφ,B,C=0,sadzC=0,sadφĊ=0,sadzpφ−iC¯C=0,sadφṗφ+iC¯̇Ċ=0,sadż+φ=0.The generalization of these anti-co-BRST invariant restrictions onto the1,1-dimensional super-submanifold (of the most common1,2-dimensional supermanifold) is as follows:
(70)Rτ,ϑ¯=rτ,Prτ,ϑ¯=prτ,Pφτ,ϑ¯=pφτ,B~τ,ϑ¯=Bτ,Fτ,ϑ¯=Cτ,Zτ,ϑ¯Fτ,ϑ¯=zτCτ,Θτ,ϑ¯Ḟτ,ϑ¯=φτĊτ,Zτ,ϑ¯Pφτ,ϑ¯−iF¯τ,ϑ¯Fτ,ϑ¯=zτpφτ−iC¯τCτ,Θτ,ϑ¯Ṗφτ,ϑ¯+iF¯̇τ,ϑ¯Ḟτ,ϑ¯=φτpφτ+iC¯̇τĊτ,Żτ,ϑ¯+Θτ,ϑ¯=żτ+φτ.The above generalizations of the anti-co-BRST invariant restrictions, finally, lead to the derivation of the derived variables as follows:(71)b1=0,b2=−pφ,f1=C,f2=−Ċ,f3=0,f4=0,f5=0,f6=0.Thus, we have determined all the derived variables using the same technique as we have used in the derivation of the co-BRST symmetry transformations. Finally, after substituting the value of derived variables into Equation (15), we obtain the following expressions for the antichiral expansions of the supervariables, namely,
(72)zτ⟶Zadτ,ϑ¯=zτ+ϑ¯C≡zτ+ϑ¯sadzτ,φτ⟶Θadτ,ϑ¯=φτ+ϑ¯−Ċ≡φτ+ϑ¯sadφτ,Cτ⟶Fadτ,ϑ¯=Cτ+ϑ¯0≡Cτ+ϑ¯sadCτ,C¯τ⟶F¯adτ,ϑ¯=C¯τ+ϑ¯−ipφ≡C¯τ+ϑ¯sadC¯τ,rτ⟶Radτ,ϑ¯=rτ+ϑ¯0≡rτ+ϑ¯sadrτ,prτ⟶Pradτ,ϑ¯=prτ+ϑ¯0≡prτ+ϑ¯sadprτ,pφτ⟶Pφadτ,ϑ¯=pφτ+ϑ¯0≡pφτ+ϑ¯sadpφτ,Bτ⟶B~adτ,ϑ¯=Bτ+ϑ¯0≡Bτ+ϑ¯sadBτ,where superscriptad on the antichiral supervariables denotes the fact that antichiral supervariables are obtained after the application of quantum anti-co-BRST invariant restrictions (Equation (69)). Here, it is clear that the coefficient of Grassmannian variable ϑ¯ is simply the quantum anti-co-BRST symmetry transformation (sad). To be more clear, the anti-co-BRST symmetry transformation sad of any generic variable ψτ is equal to the translation of the corresponding antichiral supervariable Ψadτ,ϑ¯ along the ϑ¯-direction, i.e., sadψτ=∂ϑ¯Ψadτ,ϑ¯. This implies that the anti-co-BRST symmetry (sad) is connected with the Grassmannian translation generator (∂ϑ¯) as sad↔∂ϑ¯ [11–13].


## 5. Nilpotency and Absolute Anticommutativity of the Noether Conserved Charges: ACSA

In this section, we deduce the nilpotency and absolute anticommutativity properties of the conserved (anti-)BRST along with (anti-)co-BRST charges in the language of ACSA. For this, first of all, we show the nilpotency of the (anti-)BRST together with (anti-)co-BRST conserved charges. It is straightforward to express the expressions of the (anti-)BRST together with (anti-)co-BRST charges in terms of the (anti-)chiral supervariables and partial derivatives∂ϑ¯,∂ϑ with an equivalent integral form as follows:
(73)Qb=∂∂ϑ¯iF¯̇bτ,ϑ¯Fbτ,ϑ¯−iF¯bτ,ϑ¯Ḟbτ,ϑ¯≡∫dϑ¯iF¯̇bτ,ϑ¯Fbτ,ϑ¯−iF¯bτ,ϑ¯Ḟbτ,ϑ¯,Qab=∂∂ϑiF¯abτ,ϑḞabτ,ϑ−iF¯̇abτ,ϑFabτ,ϑ≡∫dϑiF¯abτ,ϑḞabτ,ϑ−iF¯̇abτ,ϑFabτ,ϑ,Qd=∂∂ϑiF¯dτ,ϑḞdτ,ϑ−iF¯̇dτ,ϑFdτ,ϑ≡∫dϑiF¯dτ,ϑḞdτ,ϑ−iF¯̇dτ,ϑFdτ,ϑ,Qad=∂∂ϑ¯iF¯̇adτ,ϑ¯Fadτ,ϑ¯−iF¯adτ,ϑ¯Ḟadτ,ϑ¯≡∫dϑ¯iF¯̇adτ,ϑ¯Fadτ,ϑ¯−iF¯adτ,ϑ¯Ḟadτ,ϑ¯,where the superscriptsb and ab stand for the antichiral and chiral supervariables that have been obtained after the application of the BRST and anti-BRST invariant restrictions, respectively. The superscripts d and ad show the chiral and antichiral supervariables that are obtained after the application of co-BRST and anti-co-BRST invariant restrictions, respectively. It is clear that the nilpotency (∂ϑ¯2=0,∂ϑ2=0) of the translational generators ∂ϑ¯,∂ϑ implies that
(74)∂ϑ¯Qb=0⇔sbQb=−iQb,Qb=0,(75)∂ϑQab=0⇔sabQab=−iQab,Qab=0,(76)∂ϑQd=0⇔sdQd=−iQd,Qd=0,(77)∂ϑ¯Qad=0⇔sadQad=−iQad,Qad=0,which show the nilpotency [Qab2=Qad2=0] of the conserved charges within the ambit of ACSA to BRST formalism. Thus, we have shown that there is a deep connection between the nilpotency (∂ϑ¯2=0,∂ϑ2=0) of the translational generator ∂ϑ¯,∂ϑ and the nilpotency (i.e., Qab2=Qad2=0) of the (anti-)BRST and (anti-)co-BRST charges [Qab,Qad]. The above nilpotency property can be also captured in an ordinary space where we use the (anti-)BRST exact as well as (anti-)co-BRST exact forms of the charges, namely,
(78)Qb=−isbC¯Ċ−C¯̇C,Qab=+isabC¯Ċ−C¯̇C,Qd=isdC¯Ċ−C¯̇C,Qad=−isadC¯Ċ−C¯̇C.The above expressions show the nilpotency property of the (anti-)BRST along with (anti-)co-BRST conserved charges, in a simpler way, in an ordinary space (cf. (74)).Now, we are in a stage to show the absolute anticommutativity of the (anti-)BRST along with (anti-)co-BRST charges. For this purpose, we write the charges in terms of the (anti-)chiral supervariables and the derivatives∂ϑ,∂ϑ¯ of the Grassmannian variables ϑ¯,ϑ(79)Qb=−i∂∂ϑḞabτ,ϑFabτ,ϑ≡−i∫dϑḞabτ,ϑFabτ,ϑ,Qab=i∂∂ϑ¯F¯̇bτ,ϑ¯F¯bτ,ϑ¯≡i∫dϑ¯F¯̇bτ,ϑ¯F¯bτ,ϑ¯,Qd=i∂∂ϑ¯F¯̇adτ,ϑ¯F¯adτ,ϑ¯≡i∫dϑ¯F¯̇adτ,ϑ¯F¯adτ,ϑ¯,Qad=−i∂∂ϑḞdτ,ϑFdτ,ϑ≡−i∫dϑḞdτ,ϑFdτ,ϑ,where the superscriptsab and ad denote the same meaning as explained earlier. Here, it is straightforward to check that the nilpotency (∂ϑ¯2=0,∂ϑ2=0) of the translational generators ∂ϑ¯,∂ϑ implies the following relations:
(80)∂ϑQb=0⇔sabQb=−iQb,Qab=0,∂ϑ¯Qab=0⇔sbQab=−iQab,Qb=0,∂ϑ¯Qd=0⇔sadQd=−iQd,Qad=0,∂ϑQad=0⇔sdQad=−iQad,Qd=0,which show the absolute anticommutativity property of the (anti-)BRST as well as (anti-)co-BRST conserved charges. The property of absolute anticommutativity of conserved charges can also be captured explicitly in an ordinary space by using the following (anti-)BRST exact and (anti-)co-BRST exact forms of the charges, namely,
(81)Qb=−isabĊC,Qab=+isbC¯̇C¯,Qd=isadC¯̇C¯,Qad=−isdĊC.


## 6. Invariances of Lagrangian in ACSA

In this section, we discuss the (anti-)BRST along with (anti-)co-BRST invariances of the Lagrangian (7) within the scope of ACSA to BRST formalism. For this purpose, foremost, we generalize the ordinary Lagrangian of 0+1-dimensional onto the suitably chosen 1,1-dimensional (anti-)chiral super-submanifold of the most common1,2-dimensional supermanifold. The expressions of the (anti-)chiral super Lagrangian are
(82)Lτ⟶L~acτ,ϑ¯=ṙτprτ+Θ̇bτ,ϑ¯pφτ−12pr2τ−12r2pφ2τ−Zbτ,ϑ¯pφτ−Vr+12B2τ+BτŻbτ,ϑ¯+Θbτ,ϑ¯−iF¯̇bτ,ϑ¯Ċτ+iF¯bτ,ϑ¯Cτ,Lτ⟶L~cτ,ϑ=ṙτprτ+Θ̇abτ,ϑpφτ−12pr2τ−12r2pφ2τ−Zabτ,ϑpφτ−Vr+12B2τ+BτŻabτ,ϑ+Θabτ,ϑ−iC¯̇τḞabτ,ϑ+iC¯τFabτ,ϑ,where the superscriptsac and c on the super Lagrangians denote the antichiral and chiral super Lagrangians (containing antichiral and chiral supervariables), respectively. It is evident that under the application of translational generators ∂ϑ¯,∂ϑ, we get the (anti-)BRST invariance of Lagrangian (L) with the following results:
(83)∂∂ϑ¯L~acτ,ϑ¯=ddτBτĊτ,∂∂ϑL~cτ,ϑ=ddτBτC¯̇τ,which imply that the generalized version of super Lagrangians remains quasi-invariant (i.e., up to a total time derivative) under the translational generators∂ϑ¯,∂ϑ within the scope of ACSA which are consistent with Equation (10).Now, we capture the (anti-)co-BRST invariance of the Lagrangian (7) within the scope of (anti-)chiral supervariable approach. For this, we generalize the ordinary Lagrangian into (anti-)co-BRST super Lagrangian where 0+1-dimensional theory is generalized onto the 1,1-dimensional (anti-)chiral super-submanifold of the common1,2-dimensional supermanifold as follows:
(84)Lτ⟶L~c,dτ,ϑ=ṙτprτ+Θ̇dτ,ϑpφτ−12pr2τ−12r2pφ2τ−Zdτ,ϑpφτ−Vr+12B2τ+BτŻdτ,ϑ+Θdτ,ϑ−iC¯̇τḞdτ,ϑ+iC¯τFdτ,ϑ,Lτ⟶L~ac,adτ,ϑ¯=ṙτprτ+Θ̇adτ,ϑ¯pφτ−12pr2τ−12r2pφ2τ−Zadτ,ϑ¯pφτ−Vr+12B2τ+BτŻadτ,ϑ¯+Θadτ,ϑ¯−iF¯̇adτ,ϑ¯Ċτ+iF¯adτ,ϑ¯Cτ,where the superscriptsc,d and ac,ad denote the super Lagrangians (containing the chiral and antichiral supervariables) obtained after the application of the co-BRST and anti-co-BRST invariant restrictions, respectively. It is straightforward to check that
(85)∂∂ϑL~c,dτ,ϑ=−ddτpφτC¯̇τ,∂∂ϑ¯L~ac,adτ,ϑ¯=−ddτpφτĊτ,which show the (anti-)co-BRST invariance of the LagrangianL within the ambit of ACSA to BRST formalism. At the end of this section, we have the following concluding remarks. There are deep connections between the (anti-)BRST symmetries sab and derivatives ∂ϑ¯,∂ϑ of the Grassmannian variables ϑ¯,ϑ with the following mappings: sb↔∂ϑ¯ and sab↔∂ϑ. Similarly, in the case of (anti-)co-BRST symmetry transformations, it is clear that these symmetry transformations are also connected with the derivatives ∂ϑ¯,∂ϑ of Grassmannian variables with the mappings: sd↔∂ϑ and sad↔∂ϑ¯ (cf. Sections 4 and 5).


## 7. Conclusions

In our present analysis, for the first time, we have derived the off-shell nilpotentquantum (anti-)BRST along with (anti-)co-BRST symmetry transformations within the scope of ACSA. We have also discussed the nilpotency along with absolute anticommutativity properties of the corresponding (anti-)BRST along with (anti-)co-BRST conserved charges of the ordinary0+1-dimensional gauge invariant Christ–Lee model within the ambit of (anti-)chiral supervariable approach (ACSA) to BRST formalism.Thenovel remarks of our present endeavor are the derivation of the off-shell nilpotent (anti-)BRST along with (anti-)co-BRST symmetry transformations (cf. Section 4) and the proof of nilpotency and the anticommutativity properties of the (anti-)BRST and (anti-)co-BRST charges in spite of the fact that we have taken into account only the (anti-)chiral super expansions of the supervariables (cf. Section 5). The nilpotency and anticommutativity properties of the above conserved charges and derivation of the corresponding (anti-)BRST and (anti-)co-BRST symmetry transformations are obvious when the full super expansions of the supervariables (i.e., BT-supervariable formalism [29–32]) are taken into account. However, for the present study, we have shown these properties with the help of only (anti-)chiral super expansions of the (anti-)chiral supervariables.It is worthwhile to mention that the nilpotency of the BRST as well as anti-BRST conserved charges is connected with the nilpotency∂ϑ¯2=∂ϑ2=0 of the translational generators ∂ϑ¯ and ∂ϑ, respectively. On the other hand, nilpotency of the co-BRST and anti-co-BRST charges is connected with nilpotency (∂ϑ2=∂ϑ¯2=0) of the translational generators ∂ϑ and ∂ϑ¯, respectively. However, we have shown (cf. Section 5) that the absolute anticommutativity of the BRST charge with anti-BRST charge is connected with the nilpotency (∂ϑ2=0) of the translational generator (∂ϑ) and absolute anticommutativity of anti-BRST charge with BRST charge is connected with the nilpotency (∂ϑ¯2=0) of the translational generator (∂ϑ¯). On the contrary, the absolute anticommutativity of the co-BRST charge with anti-co-BRST charge is connected with the nilpotency of the translational generator (∂ϑ¯) and the absolute anticommutativity of the anti-co-BRST charge with co-BRST charge is deeply related with the nilpotency of the translational generator (∂ϑ). These statements are completely novel for the present model. We have also captured the (anti-)BRST along with (anti-)co-BRST invariances of the Lagrangian within the scope of ACSA. In fact, the action corresponding to the (anti-)chiral super Lagrangian is independent of Grassmannian variables ϑ,ϑ¯ which is completely novel for the present CL model (cf. Section 6).The above issues, within the scope of ACSA to BRST approach, would be discussed in our future investigations for the various gauge-invariant models/theories like ABJM theory [33–35], supersymmetric Chern-Simons theory [36], Jackiw-Pi model, Freedman-Townsend model, and Abelian gauge theory with higher derivative matter fields. In fact, our standard techniques of ACSA to BRST formalism are applicable wherever gauge invariance is present in the theory. Furthermore, there is an interesting and important work [37] that would be discussed in the future for different prospects of the theoretical and physical point of view in the domain of theoretical high-energy physics.



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*Source: 5518304-2021-10-28.xml*