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

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


Introduction
Gauge theories describe three (i.e. strong, weak, electromagnetic) out of four fundamental interactions of nature which are characterized by first-class constraints in the context of Dirac's prescription for the classification scheme of constraints [1,2]. The existence of the first-class constraints, in a given system, is the key signature of a gauge theory. Many interesting theories, in the domain of physics, are expressed by the suitable Lagrangians that are invariant under the gauge symmetry transformations. These symmetries are generated by the first-class constraints in a given gauge theory. For the covariant canonical quantization of the gauge theory, the Becchi-Rouet-Stora-Tyutin (BRST) quantization procedure plays a decisive role where we replace the infinitesimal local gauge parameter by ghost and antighost fields [3][4][5][6]. Thus, in this formalism, we have two fermionic type global BRST (s b ) and anti-BRST (s ab ) transformations at the quantum level (for a given local gauge symmetry transformation at the classical level). These symmetry transformations are endowed with two important properties: (i) nilpotency of order two (i.e. s 2 b = 0, s 2 ab = 0), and (ii) absolute anti-commutativity (i.e. s b s ab + s ab s b = 0). The first property signifies that theses quantum BRST and anti-BRST symmetry transformations are fermionic in nature whereas second property shows that both symmetry transformations are linearly independent of each other. Besides the (anti-)BRST symmetry transformations, we have two more fermionic and linearly independent symmetry transformations which are christened as the co-BRST (s d ) and anti-co-BRST (s ad ) symmetry transformations. The latter fermionic type symmetry transformations are valid for any D-dimensional p-form (p = 1, 2, 3, ...) gauge theories in D = 2p dimensions of spacetime. We point out that there are some specific systems such as rigid rotor and Christ-Lee (CL) model in one-dimension (1D) that respect the (anti-)co-BRST transformations along with to the (anti-)BRST transformations [7][8][9][10].
The CL model is one of the simplest examples of gauge-invariant system which is described by a singular Lagrangian [14]. Physically, the CL model represents the motion of a point particle moving in a plane under the influence of a central potential. The CL model has been studied at the classical and quantum levels in different prospectives [14][15][16][17][18][19][20]. This model is endowed with the first-class constraints in the Dirac's terminology for the classification scheme of constraints [1,2]. This model is also quantized by using the Faddeev-Jackiw quantization where all the primary and derived constraints are treated on equal footing without any type of further classification [20]. Within the framewok of BRST formalism, the CL model respects six independent continuous symmetries (i.e. BRST, anti-BRST, co-BRST, anti-co-BRST, ghost-scale and bosonic symmetries) (see, e.g. [9] for detail). The BT-superfield formalism has been applied to obtain the absolutely anti-commuting and offshell nilpotent (anti-)BRST as well as (anti-)co-BRST symmetry transformations where the technique of celebrated horizontality condition (HC) and dual horizontality condition have been used [10], respectively.
In our recent set of papers [21][22][23][24][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 supersubmanifolds of the most general (D, 2)-dimensional supermanifold. The proof of the absolute anti-commutativity property of Noether's conserved charges is obvious in the case of BTsuperfield formalism where the full super expansions of superfields/supervariables are taken into account. In the case of ACSA, we have also been able to show the nilpotency and absolute anti-commutativity properties of conserved charges despite the fact that we have taken only one Grassmannian variable into account. In our present endeavor, we derive the (anti-)BRST together with (anti-)co-BRST symmetry transformations where some specific sets of (anti-)BRST and (anti-)co-BRST invariant restrictions play very important role. We also show the absolute anti-commutativity as well as nilpotency properties of (anti-)BRST and (anti-) co-BRST conserved charges within the realm of ACSA to BRST formalism.
Against the background of the above paragraph, it has been found that the nilpotency of the N = 2 super charges are true for any N = 2 supersymmetric (SUSY) quantum mechanical models (see, e.g. [26][27][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 anti-commutativity of the N = 2 super conserved charges. Rather, it has been found that the anti-commutator 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 anti-commutativity of the charges for all N = 2 SUSY theories.
The different sections of our present paper are arranged as follows. In Sec. 2, we discuss the (anti-)BRST and (anti-)co-BRST symmetry transformations for the CL model and derive the conserved charges. Our Sec. 3 deals with the ACSA to BRST formalism where we derive the (anti-)BRST symmetry transformations. Sec. 4 is devoted to the derivation of (anti-)co-BRST symmetry transformations by using the ACSA to BRST formalism where the super expansions of (anti-)chiral supervariables are utilized in a fruitful manner. In Sec. 5, we express the conserved (anti-)BRST and (anti-)co-BRST charges on the (1, 1)-dimensional supersub-manifolds [of the most general (1, 2)-dimensional supermanifold] on which our theory is generalized and provide the proof of nilpotency and absolute anticommutativity properties of the (anti-)BRST along with (anti-)co-BRST charges within the ambit of ACSA to BRST formalism. In Sec. 6, we discuss the (anti-)BRST and (anti-)co-BRST invariances of the Lagrangian within the scope of ACSA. Finally, we point out our key results and discovery in Sec. 7 and mention a few future scopes for further investigation.

Preliminaries: Symmetries and their Corresponding
Generator for the Christ-Lee Model The first-order and gauge-invariant Lagrangian of the Christ-Lee (CL) model * in (0 + 1)dimension (1D) of spacetime in polar coordinates system is given by [14,16,19], whereṙ andφ are the generalized velocities, p r and p ϕ are their corresponding canonical momenta, respectively and z is a Lagrange multiplier which enforces a constraint p ϕ ≈ 0. This Lagrangian explains that a two dimensional particle moves under the influence of the central potential V (r) bounded from below.
The above system has a primary constraint as follow The time derivative of the primary constraint Φ 1 leads to the following secondary constraint It is clear that both Φ 1 and Φ 2 are first-class constraints. The gauge symmetry transformation generator can be written in terms of first-class constraints as where χ(τ ) is an infinitesimal and time dependent local gauge parameter andχ(τ ) = dχ/dτ . Using the definition of a generator where φ is the generic variable that is present in the first-order Lagrangian L f . We deduce the following local gauge transformations by exploiting Eq. (5), namely; It is elementary to check that, under the above local gauge symmetry transformations, the Lagrangian under consideration remains invariant (i.e. δL f = 0). The (anti-)BRST invariant Lagrangian for the (0 + 1)-dimensional CL model containing the gauge-fixing and Faddeev-Popov ghost terms is given by [15] L =ṙ p r +φ p ϕ − 1 2 where the Nakanishi-Lautrup type auxiliary 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 C 2 =C 2 = 0, CC +CC = 0) have ghost numbers (−1) + 1, respectively. The above Lagrangian respects the following off-shell nilpotent [i.e. s 2 (a)b = s 2 (a)d = 0] and absolutely anti-commuting (i.e. s b s ab + s ab s b = 0 and s d s ad + s ad s d = 0) (anti-)BRST along with (anti-)co-BRST symmetry transformations: It can be clearly checked that under the above (anti-)BRST [Eq. (8)] and (anti-)co-BRST [Eq. (9)] symmetry transformations the Lagrangian [Eq. (7)] remains quasi-invariant (i.e. modulo a total time derivative): where the equivalent forms of the above charges are written with the help of the equation of motion: can be easily proven by using the following interesting Euler-Lagrange equations of motion † (EOMs) derived from Lagrangian L of our theory [Eq. (7)], namely; † Besides these EOMs, we use a equationB + B = 0 derived from the EOMs (13) to prove the conservation law for the (anti-)BRST together with (anti-)co-BRST conserved charges [Eqs. (11), (12)].
The (anti-)co-BRST and (anti-)BRST conserved charges are the generators of the (anti-) co-BRST and (anti-)BRST symmetry transformations, respectively. As one can easily check that following relationships are true where ξ denotes any generic variable present in the Lagrangian L of our theory. The subscript (±) on the square brackets denotes the (anti)commutator which depend on the nature of generic variables ξ being (fermionic)bosonic in nature.
where b 1 , b 2 are the bosonic derived variables and f 1 , f 2 , f 3 , f 4 , f 5 , f 6 are the fermionic derived variables due to fermionic nature ofθ. We determine the precise value of these derived variables in terms of the auxiliary and basic variables present in the BRST invariant Lagrangian (7) by using the BRST invariant quantities/restrictions. According to the basic principles of ACSA, the BRST invariant quantities must remain independent of the Grassmannian variable (θ) when they are generalized onto the (1, 1)dimensional anti-chiral supersub-manifold. The BRST invariant quantities are the specific combinations of the variables present in Lagrangian (7). These are given as follows We generalize the above BRST invariant restrictions onto the (1, 1)-dimensional anti-chiral supersub-manifolds (of the suitably chosen most common (1, 2)-dimensional supermanifold) The above restrictions lead to the derivation of the derived variables in terms of the basic and auxiliary variables. To determine the value of these variables, we perform the step-by-step explicit calculations. For this purpose, first of all, we use the generalization of the trivial BRST invariant restrictions given in the first line of Eq. (17) as: After substituting the above value of derived variables from (18) to (15), we get the following expressions for the anti-chiral supervariables, namely; where the superscript (b) on the anti-chiral supervariables denotes that these supervariables have been obtained after the use of BRST invariant quantities. It is clear that the coefficients of Grassmannian variableθ are simply the quantum BRST symmetries (8). Now, in the case of non-trivial BRST invariant restrictions: s b (zĊ) = 0 and s b (ϕ C) = 0, the following generalizations onto (1, 1)-dimensional supersub-manifold, namely; lead to the following interesting results where κ 1 and κ 2 are the proportionality constants. To determine the value of these constants, we use the generalization of BRST invariant restriction s b (φ − z) = 0 aṡ Finally, to determine the value of constants, we generalize the BRST invariant restrictions s b (Ḃ z + iĊĊ) = 0 and s b (B ϕ + iC C) = 0 onto (1, 1)-dimensional supersub-manifold as: Using the results obtained in Eq. (22) and Eq. (23), it is clear that κ 1 = κ 2 = 1. Therefore, we obtain the value of derived variables as: f 1 (τ ) =Ċ(τ ), f 2 (τ ) = C(τ ), b 2 (τ ) = B(τ ), thus, we get the following expansions for anti-chiral supervariables: Thus, in view of the above Eq. (24), we have a connection between the BRST symmetry transformation (s b ) and partial derivative (∂θ) on the anti-chiral supersub-manifold defined by the mapping: s b ←→ ∂θ (see, e.g. [11][12][13] for details). To be more clear, the BRST transformation of any generic variable ψ(τ ) is equal to the translation of the corresponding anti-chiral supervariable Ψ (b) (τ,θ) along theθ-direction. Mathematically, it can be represented as In other words, one can say, the coefficient of ϑ in the expansion of an anti-chiral supervariable is simply the quantum BRST symmetry transformation of the corresponding variable.
We are now in the stage to derive the quantum anti-BRST symmetry transformations using chiral supervariable approach. In this context, we use the chiral super expansions of the chiral supervariables where we generalize (0 + 1)-dimensional variables onto the (1, 1)-dimensional supersub-manifold of the suitably chosen most common (1, 2)-dimensional supermanifold. The chiral super expansions of the ordinary variables are as follows As the physical quantities remain independent of the Grassmannian variable ϑ which imply that the anti-BRST invariant restrictions can be generalized onto the (1, 1)-dimensional supersub-manifold of the most common (1, 2)-dimensional supermanifold as follows R(τ, ϑ) = r(τ ), P r (τ, ϑ) = p r (τ ), P ϕ (τ, ϑ) = p ϕ (τ ),B(τ, ϑ) = B(τ ),F (τ, ϑ) =C(τ ), The above generalizations of the anti-BRST invariant restrictions [Eq. (26)] lead to the derivation of the chiral derived variables in terms of the auxiliary and basic variables present in the Lagrangian L, namely; The above values of chiral derived variables are derived in a similar fashion as the anti-chiral derived variables are derived. After the substitution of the above derived variables into the chiral super expansions (25), we get the following expressions for the chiral supervariables onto (1, 1)-dimensional supersub-manifold as Here, the coefficients of ϑ are simply the anti-BRST symmetry transformations (see, e.g. [11][12][13] for detail). In fact, the anti-BRST transformation of any generic variable ψ(τ ) is simply the translation of the corresponding chiral supervariable Ψ (ab) (τ, ϑ) along the ϑdirection. Mathematically, this statement can be corroborated as s ab ψ(τ ) = ∂ ∂ϑ Ψ (ab) (τ, ϑ) = ∂ ϑ Ψ (ab) (τ, ϑ). Thus, it is clear that there is a mapping between the quantum anti-BRST symmetry transformation (s ab ) and the Grassmannian partial derivative (∂ ϑ ) defined on the chiral supersub-manifold with the mapping: s ab ←→ ∂ ϑ .

Nilpotent (Anti-)co-BRST Symmetry Transformations: (Anti-)Chiral Supervariable Approach
In this section, we derive the nilpotent (anti-)co-BRST symmetry transformations using the (anti-)chiral supervariable approach (ACSA) where we use the expansions of the (anti-)chiral supervariables and the (anti-)co-BRST invariant restrictions. Toward this goal in our mind, first of all, we determine the co-BRST symmetries by exploiting the chiral super expansions given in Eq. (25) and the co-BRST invariant restrictions. The co-BRST invariant restrictions are given as: According to the basic rules of ACSA, the above co-BRST invariant restrictions can be generalized onto the (1, 1)-dimensional supersub-manifold [of the suitably chosen most common (1, 2)-dimensional supermanifold] as: At this stage, we determine the value of derived variables of Eq. (25) using the above generalizations of the co-BRST invariant restrictions. To derive the value of the derived variables, first of all, we use the first line entry of Eq. (31) where the trivial co-BRST invariant quantities are generalized which implies the following relationships: After substituting the above value of derived variables into the expressions of the chiral super expansions [Eq. (25)], we obtain the following chiral super expansions: where superscript (d) on the chiral supervariables denotes the supervariables obtained after the application of the co-BRST (i.e. dual-BRST) invariant restrictions. For the non-trivial case, first of all, we generalize the co-BRST invariant restriction s d (zC) = 0 and s d (ϕĊ) = 0 onto (1, 1)-dimensional supersub-manifold as which lead to the following relationships for the derived variables whereκ 1 andκ 2 are the proportionality constants. To determine the value of these constants, we further use the generalizations of the co-BRST invariant restrictions s d (ϕ +ż) = 0, s d (ϕṗ ϕ + iĊĊ) = 0 and s d (z p ϕ − iC C) = 0 as: The results of the above three relations in Eq. (36) imply that proportionality constant are equal (i.e.κ 1 =κ 2 ) and their values are equal to minus one (i.eκ 1 =κ 2 = − 1). Therefore, we get the value of the derived variables as:f 1 =C,f 2 = −Ċ,b 1 = p ϕ . As a result, we have the following chiral super expansions of the ordinary variables: where superscript (d) on the supervariables denotes the same meaning as in Eq. (33). From the above equations, it is clear that the translation of any generic chiral supervariable Ψ (d) (τ, ϑ) along the ϑ-direction generates co-BRST symmetry transformation (s d ) of the corresponding variable ψ(τ ). Mathematically, we can express this statements as s d ψ(τ ) = ∂ ϑ Ψ (d) (τ, ϑ). In other words, the coefficients of the ϑ are simply the co-BRST symmetry transformations. Thus, the co-BRST symmetry transformation (s d ) is connected with the Grassmannian derivative ∂ ϑ (i.e. s d ←→ ∂ ϑ ) [11][12][13]. Now for the derivation of anti-co-BRST symmetry transformations, we use the anti-chiral super expansions of the supervariables [Eq. (15)] and anti-co-BRST invariant restrictions. The anti-co-BRST invariant restrictions are listed as s ad (r, p r , p ϕ , B, C) = 0, s ad (z C) = 0, s ad (ϕĊ) = 0, The generalization of these anti-co-BRST invariant restrictions onto the (1, 1)-dimensional supersub-manifold (of the most common (1, 2)-dimensional supermanifold) are as follows: R(τ,θ) = r(τ ), P r (τ,θ) = p r (τ ), P ϕ (τ,θ) = p ϕ (τ ),B(τ,θ) = B(τ ), F (τ,θ) = C(τ ), The above generalizations of the anti-co-BRST invariant restrictions, finally, lead to the derivation of the derived variables as follows: Thus, we have determined all the derived variables using the same technique as we have used in the derivation of the co-BRST symmetry transformations. Finally, after substituting the value of derived variables into Eq. (15), we obtain the following expressions for the anti-chiral expansions of the supervariables, namely; where superscript (ad) on the anti-chiral supervariables denote the fact that anti-chiral supervariables are obtained after the application of quantum anti-co-BRST invariant restrictions [Eq. (38)]. Here, it is clear, the coefficient of Grassmannian variableθ is simply the quantum anti-co-BRST symmetry transformation (s ad ). To be more clear, the anti-co-BRST symmetry transformation (s ad ) of any generic variable ψ(τ ) is equal to the translation of the corresponding anti-chiral supervariable Ψ (ad) (τ,θ) along theθ-direction i.e. s ad ψ(τ ) = ∂θΨ (ad) (τ,θ). This implies that the anti-co-BRST symmetry (s ad ) is connected with the Grassmannian translation generator (∂θ) as s ad ←→ ∂θ [11][12][13].

Nilpotency and Absolute Anti-Commutativity of the Noether Conserved Charges: ACSA
In this section, we deduce the nilpotency and absolute anti-commutativity properties of the conserved (anti-)BRST along with (anti-)co-BRST charges in the language of ACSA.
For this, first of all, we show the nilpotency of the (anti-)BRST together with (anti-)co-BRST conserved charges. It is straightforward to express the expressions of the (anti-)BRST together with (anti-)co-BRST charges in terms of the (anti-)chiral supervariables and partial derivatives (∂θ, ∂ ϑ ) with an equivalent integral form as follows where the superscripts (b) and (ab) stand for the anti-chiral and chiral supervariables that have been obtained after the application of the BRST and anti-BRST invariant restrictions, respectively. The superscripts (d) and (ad) show the chiral and anti-chiral supervariables that is obtained after the application of co-BRST and anti-co-BRST invariant restrictions, respectively. It is clear that the nilpotency (∂ 2 ϑ = 0, ∂ 2 ϑ = 0) of the translational generators (∂θ, ∂ ϑ ) implies that 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; Now, we are in a stage to show the absolute anti-commutativity of the (anti-)BRST along with (anti-)co-BRST charges. For this purpose, we write the charges in terms of the (anti-) chiral supervariables and the derivatives (∂ ϑ , ∂θ) of the Grassmannian variables (θ, ϑ) where the superscripts (a)b and (a)d denote the same meaning as explained earlier. Here, it is straightforward to check that the nilpotency (∂ 2 ϑ = 0, ∂ 2 ϑ = 0) of the translational generators (∂θ, ∂ ϑ ) implies that the following relations which show the absolute anti-commutativity property of the (anti-)BRST as well as (anti-)co-BRST conserved charges. The property of absolute anti-commutativity of conserved charges can also be captured explicitly in an ordinary space by using the following (anti-)BRST exact and (anti-)co-BRST exact forms of the charges, namely;

Conclusions
In our present analysis, for the first time, we have derived the off-shell nilpotent quantum (anti-)BRST along with (anti-)co-BRST symmetry transformations within the scope of ACSA. We have also discussed the nilpotency along with absolute anti-commutativity properties of the corresponding (anti-)BRST along with (anti-)co-BRST conserved charges of the ordinary (0 + 1)-dimensional gauge invariant Christ-Lee model within the ambit of (anit-)chiral supervariable approach (ACSA) to BRST formalism.
The novel remarks of our present endeavor are the derivation of the off-shell nilpotent (anti-)BRST along with (anti-)co-BRST symmetry transformations (cf. Sec. 4) and the proof of nilpotency and the anti-commutativity properties of the (anti-)BRST and (anti-)co-BRST charges in spite of the fact that we have taken into account only the (anti-)chiral super expansions of the supervariables (cf. Sec. 5). The nilpotency and anti-commutativity properties of the above conserved charges and derivation of the corresponding (anti-)BRST and (anti-)co-BRST symmetry transformations are obvious when the full super expansions of the supervariables (i.e. BT-supervariable formalism [29][30][31][32]) is taken into account. However, for the present study, we have shown these properties with the help of only (anti-)chiral super expansions of the (anti-)chiral supervariables.
It is worthwhile to mention that the nilpotency of the BRST as well as anti-BRST conserved charges is connected with the nilpotency (∂ 2 ϑ = ∂ 2 ϑ = 0) of the translational generators ∂θ and ∂ ϑ , respectively. On the other hand, nilpotency of the co-BRST and antico-BRST charges is connected with nilpotency (∂ 2 ϑ = ∂ 2 ϑ = 0) of the translational generators ∂ ϑ and ∂θ, respectively. However, we have shown (cf. Sec. 5) that the absolute anticommutativity of the BRST charge with anti-BRST charge is connected with the nilpotency (∂ 2 ϑ = 0) of the translational generator (∂ ϑ ) and absolute anti-commutativity of anti-BRST charge with BRST charge is connected with the nilpotency (∂ 2 ϑ = 0) of the translational generator (∂θ). On the contrary, the absolute anti-commutativity of the co-BRST charge with anti-co-BRST charge is connected with the nilpotency of the translational generator (∂θ) and the absolute anti-commutativity of the anti-co-BRST charge with co-BRST charge is deeply related with the nilpotency of the translational generator (∂ 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. Sec. 6).
The above issues, within the scope of ACSA to BRST approach, would be discussed in our future investigations for the various gauge-invariant models/theories like ABJM theory [33][34][35], supersymmetric Chern-Simons theory [36], Jackiw-Pi model, Freedman-Townsend model, Abelian gauge theory with higher derivative matter fields. In fact, our standard techniques of ACSA to BRST formalism are applicable wherever gauge invariance is present in the theory. Furthermore, there is a interesting and important work [37] that would be discussed in the future for different prospects of the theoretical and physical point of view in the domain of theoretical high energy physics.

Data Availability
No data were used to support this study.

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