Christ-Lee Model: (Anti-)Chiral Supervariable Approach to BRST Formalism
aa r X i v : . [ h e p - t h ] F e b Christ–Lee Model: (Anti-)Chiral SupervariableApproach to BRST Formalism
B. Chauhan ( a ) , S. Kumar ( a )( a ) Physics Department, Centre of Advance Studies, Institute of Science,Banaras Hindu University, Varanasi - 221 005, (U.P.), India
E-mails: [email protected]; [email protected]
Abstract:
We derive the off-shell nilpotent and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST and (anti-)co-BRST symmetry transformations for theChrist–Lee (CL) model in one (0 + 1)-dimension of spacetime by exploiting the (anti-)chiralsupervariable approach (ACSA) to BRST formalism where a few specific and appropriatesets of invariant quantities play a decisive role. We prove the nilpotency and absolute anti-commutativity properties of the (anti-)BRST and (anti-)co-BRST conserved charges withinthe framework of ACSA to BRST formalism where we take only one
Grassmannian vari-able into account. We also show the (anti-)BRST and (anti-)co-BRST invariances of theLagrangian within the framework of ACSA.PACS numbers: 11.15.-q, 03.70.+k, 11.10Kk, 12.90.+b
Keywords:
Christ-Lee model; (anti-)BRST and (anti-)co-BRST symmetry transformations;(anti-)chiral supervariable approach; (anti-)BRST invariant restrictions; (anti-)co-BRST in-variant restrictions; nilpotency and absolute anticommutativity properties
Introduction
Gauge theories describe three (i.e. strong, weak, electromagnetic) out of four fundamentalinteractions of nature which are characterized by first-class constraints in the language ofDirac’s prescription for the classification scheme of constraints [1, 2]. The existence of thefirst-class constraints, in a given system, is the key signature of the gauge theory. Manypowerful theories in physics are described by the Lagrangians that are invariant under thegauge symmetry transformations. These symmetry transformations are generated by thefirst-class constraints in a given theory. To quantize a given gauge theory, Becchi-Rouet-Stora-Tyutin (BRST) formalism plays an important role where we replace the infinitesimallocal gauge parameter by ghost and anti-ghost fields [3-6]. Thus, in this formalism, wehave two supersymmetric type global BRST and anti-BRST symmetry transformations atthe quantum level for a given local gauge transformation at classical level. These symmetrytransformations have two important properties: (i) nilpotency of order two, and (ii) absoluteanticommutativity. The first property signifies that theses symmetry transformations arefermionic in nature whereas second property shows that both symmetry transformations arelinearly independent of each other. Besides (anti-)BRST symmetry transformations, we havetwo more linearly independent symmetry transformations which are christened as the co-BRST and anti-co-BRST symmetry transformations. The latter symmetry transformationsare true for any D-dimensional p -form ( p = 1 , , , ... ) gauge theories in D = 2 p dimensionsof spacetime. We point out that there are some specific systems such as rigid rotor andChrist–Lee (CL) model in one-dimension that respect the (anti-)co-BRST transformationsin addition to the (anti-)BRST transformations [7-10].The geometrical interpretation and origin of nilpotent (anti-)BRST symmetry transfor-mations have been shown within the framework of Bonora-Tonin (BT) superfield formalism[7-9] where the Grassmannian variables ( θ, ¯ θ ) and their corresponding derivatives ( ∂ θ , ∂ ¯ θ )(with properties θ = ¯ θ = 0 , θ ¯ θ + ¯ θ θ = 0 and ∂ θ = ∂ θ = 0 , ∂ θ ∂ ¯ θ + ∂ ¯ θ ∂ θ = 0) play veryimportant role. In BT-superfield approach, we see the connections between the (anti-)BRSTsymmetry transformations and Grassmannian translational generators ( ∂ θ , ∂ ¯ θ ) because ofthe fact that both have same algebraic structure. In this formalism, any D-dimensionalMinkowskian manifold is generalized onto the (D, 2)-dimensional supermanifold. The latteris parametrized by the superspace coordinates ( x µ , θ, ¯ θ ) where x µ ( µ = 0 , , , ..., D −
1) arethe spacetime coordinates and ( θ, ¯ θ ) are a pair of Grassmannian coordinates.The CL model is one of the simplest examples of gauge-invariant system that is describedby a singular Lagrangian [14]. Physically, CL model represents a point particle moving ina plane under the influence of a central potential. The CL model has been studied atthe classical and quantum levels in different prospectives [14-19]. This model is endowedwith the first-class constraints in the Dirac’s terminology for the classification scheme ofconstraints [1, 2]. Basically, the CL model respects six independent continuous symmetries(i.e. BRST, anti-BRST, co-BRST, anti-co-BRST, bosonic and ghost-scale) at the quantumlevel (see, e.g. [9] for detail). The BT-superfield formalism has been applied to obtain the off-shell nilpotent and absolutely anticommuting (anti-)BRST and (anti-)co-BRST symmetrytransformations where the technique of celebrated horizontality condition (HC) and dualhorizontality condition have been used [10].In our recent set of papers [20-24], we have used a newly proposed formalism which is2hristened as the (anti-)chiral superfield/supervariable approach (ACSA) to BRST formal-ism. In this formalism, we set one of the Grassmannian variables equal to zero in the ex-pression of superfield/supervariable which, in turn, implies that the superfield/supervariableconverts into (anti-)chiral version of the superfield/supervariable. In other words, in thisformalism, any D-dimensional Minkowskian manifold is generalized onto (D, 1)-dimensionalsuper submanifolds of the general (D, 2)-supermanifold. The proof of the absolute anticom-mutativity property of Noether’s conserved charges is obvious in the case of BT-superfieldformalism where 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 oneGrassmannian variable into account. In our present endeavor, we derive the (anti-)BRSTand (anti-)co-BRST symmetry transformations where some specific sets of (anti-)BRST and(anti-)co-BRST invariant restrictions play very important role and also show the nilpotencyand absolute anticommutativity properties of (anti-)BRST and (anti-)co-BRST conservedcharges within the framework of ACSA to BRST formalism.The different sections of our present paper are organized as follows. In Sec. 2, we discussthe (anti-)BRST and (anti-)co-BRST symmetry transformations for the CL model and derivethe conserved charges. Our Sec. 3 deals with the ACSA to BRST formalism where we derivethe (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 superexpansions of (anti-)chiral supervariables are utilized in a fruitful manner. In Sec. 5, weexpress the conserved (anti-)BRST and (anti-)co-BRST charges on the (1, 1)-dimensionalsuper submanifolds [of the general (1, 2)-dimensional supermanifold] on which our theory isgeneralized and provide the proof of nilpotency and absolute anticommutativity propertiesof the (anti-)BRST and (anti-)co-BRST charges within the framework of ACSA to BRSTformalism. In Sec. 6, we discuss the (anti-)BRST and (anti-)co-BRST invariances of theLagrangian within the ambit of ACSA to BRST formalism. Finally, we point out our keyresults in Sec. 7 and mention a few future directions for further investigation. The first-order and gauge-invariant Lagrangian of the Christ–Lee (CL) model ∗ in (0 + 1)-dimension of spacetime in polar coordinates is given by [14, 16, 19], L f = ˙ r p r + ˙ ϑ p ϑ − p r − r p ϑ − z p ϑ − V ( r ) , (1)where ˙ r and ˙ ϑ are the generalized velocities, p r and p ϑ are their corresponding canonicalmomenta, respectively and z is a Lagrange multiplier which enforces the constraint p ϑ ≈ ∗ The other equivalent Lagrangians [14] associated with the Christ–Lee model are given as: L = ˙ r − V ( r ) and L s = ˙ r + ˙ r ( ˙ ϑ − z ) − V ( r ). However, we choose only the first-order Lagrangian L f in ourpresent work because of presence of maximum number of variables in it which is theoretically more appealing. V ( r ) bounded from below.The above system has a primary constraint as followΦ = ∂L f ∂ ˙ z = p z ≈ . (2)The time evolution of the primary constraint Φ leads to the following secondary constraint d Φ dt = ddt (cid:16) ∂L f ∂ ˙ z (cid:17) ≈ ⇒ Φ = p ϑ ≈ . (3)It is clear that both Φ and Φ are first-class constraints. The gauge symmetry transforma-tion generator can be written in terms of first-class constraints as G = ˙ χ ( t ) Φ + χ ( t ) Φ , (4)where χ ( t ) is an infinitesimal and time dependent local gauge parameter. Using the definitionof a generator δ φ ( t ) = − i [ φ ( t ) , G ] , φ = r, p r , ϑ, p ϑ , z, (5)where φ is generic variables present in Lagrangian L f . We obtain the following local (timedependent) gauge transformations using Eq. (5) as: δ z = ˙ χ ( t ) , δ ϑ = χ ( t ) , δ [ r, p r , p ϑ , V ( r )] = 0 . (6)It is straightforward to check that under the above gauge symmetry transformations, thefirst-order Lagrangian ( L f ) remains invariant (i.e. δL f = 0).The (anti-)BRST invariant Lagrangian for the (0 + 1)-dimensional CL model containingthe guage-fixing and Faddeev-Popov (anti-)ghost variables is given by [15] L = ˙ r p r + ˙ ϑ p ϑ − p r − r p ϑ − z p ϑ − V ( r ) + 12 B + B ( ˙ z + ϑ ) + i ¯ C C − i ˙¯ C ˙ C, (7)where the Nakanishi–Lautrup type auxiliary variable B is used to linearize the gauge-fixingterm and ( ¯ C ) C are the Faddeev–Popov (anti-)ghost variables used to make the LagrangianBRST invariant. These fermionic variables ( ¯ C ) C (with C = ¯ C = 0 , C ¯ C + ¯ CC = 0)have ghost numbers ( −
1) + 1, respectively. The above Lagrangian respects the followingoff-shell nilpotent ( s a ) b = s a ) d = 0) as well as absolutely anticommuting (i.e. s b s ab + s ab s b = 0 , s d s ad + s ad s d = 0) (anti-)BRST ( s ( a ) b ) and (anti-)co-BRST ( s ( a ) d ) symmetrytransformations: s ab z = ˙¯ C, s ab ϑ = ¯ C, s ab C = − i B, s ab [ r, p r , p ϑ , B, ¯ C ] = 0 ,s b z = ˙ C, s b ϑ = C, s b ¯ C = i B, s b [ r, p r , p ϑ , B, C ] = 0 , (8) s ad z = C, s ad ϑ = − ˙ C, s ad ¯ C = − i p ϑ , s ad [ r, p r , p ϑ , B, C ] = 0 ,s d z = ¯ C, s d ϑ = − ˙¯ C, s d C = i p ϑ , s d [ r, p r , p ϑ , B, ¯ C ] = 0 . (9)4t can be clearly checked that under the above off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations the Lagrangian remains quasi-invariant (modulo a totaltime derivative): s b L = ddt (cid:0) B ˙ C (cid:1) , s ab L = ddt (cid:0) B ˙¯ C (cid:1) ,s d L = − ddt (cid:0) p ϑ ˙¯ C (cid:1) , s ad L = − ddt (cid:0) p ϑ ˙ C (cid:1) . (10)As a consequence, the action integral S = R dt L remains invariant under the (anti-)BRSTand (anti-)co-BRST symmetry transformations [i.e. s ( a ) b S = 0 , s ( a ) d S = 0]. According tothe Noether’s theorem, the invariances of the Lagrangian under the above continuous (anti-)BRST and (anti-)co-BRST symmetry transformations leads to the following (anti-)BRSTcharges ( Q ( a ) b ) and (anti-)co-BRST charges ( Q ( a ) d ), namely; Q b = B ˙ C + p ϑ C ≡ B ˙ C − ˙ B C,Q ab = B ˙¯ C + p ϑ ¯ C ≡ B ˙¯ C − ˙ B ¯ C, (11) Q d = B ¯ C − p ϑ ˙¯ C ≡ B ¯ C + ˙ B ˙¯ C,Q ad = B C − p ϑ ˙ C ≡ B C + ˙ B ˙ C, (12)where the equivalent forms of the (anti-)BRST and (anti-)co-BRST charges are writtenwith the help of equation of motion: p ϑ = − ˙ B . These charges are nilpotent of ordertwo (i.e. Q a ) b = 0 , Q a ) d = 0) and anticommuting in nature (i.e. Q b Q ab + Q ab Q b = 0and Q d Q ad + Q ad Q d = 0). The conservation law for these charges (i.e. ddt Q ( a ) b = 0 and ddt Q ( a ) d = 0) can be proven by using the following Euler-Lagrange equations of motion † (EOMs) derived from Lagrangian L , namely;˙ B + p ϑ = 0 , B = ˙ p ϑ , B = − ( ˙ z + ϑ ) , ˙ p r − p ϑ r + V ′ ( r ) = 0 , ˙ r = p r , ˙ ϑ − z − p ϑ r = 0 , ¨¯ C + ¯ C = 0 , ¨ C + C = 0 . (13)The (anti-)BRST and (anti-)co-BRST charges are the generators of the (anti-)BRST and(anti-)co-BRST symmetry transformations, respectively. As one can check that followingrelationships are true s b ψ = − i (cid:2) ψ, Q b (cid:3) ± , s ab ψ = − i (cid:2) ψ, Q ab (cid:3) ± ,s d ψ = − i (cid:2) ψ, Q d (cid:3) ± , s ad ψ = − i (cid:2) ψ, Q ad (cid:3) ± , (14)where ψ denotes any generic variable present in the Lagrangian L . The subscripts ( ± ) onthe square brackets denote the (anti)commutator which depend on the nature of genericvariables ψ being (fermionic)bosonic in nature. † Besides these EOMs, we use a equation ¨ B + B = 0 derived from the EOMs (13) to prove the conservationlaw for (anti-)BRST and (anti-)co-BRST charges [Eqs. (11), (12)]. Off-Shell Nilpotent (Anti-)BRST Symmetry Trans-formations: (Anti-)Chiral Supervariable Approach
In this section, we derive the off-shell nilpotent (anti-)BRST symmetry transformations [cf.Eq. (8)] by using ACSA to BRST formalism where we shall use the expressions of the (anti-)chiral expansions of supervariables. Towards this goal, first of all, we generalize the ordinaryvariables of the Lagrangian (7) onto (1, 1)-dimensional anti-chiral super submanifold (of thegeneral (1, 2)-dimensional supermanifold) as follows, z ( t ) −→ Z ( t, ¯ θ ) = z ( t ) + ¯ θ f ( t ) ,ϑ ( t ) −→ Θ( t, ¯ θ ) = ϑ + ¯ θ f ( t ) ,C ( t ) −→ F ( t, ¯ θ ) = C ( t ) + i ¯ θ b ( t ) , ¯ C ( t ) −→ ¯ F ( t, ¯ θ ) = ¯ C ( t ) + i ¯ θ b ( t ) ,r ( t ) −→ R ( t, ¯ θ ) = r ( t ) + ¯ θ f ( t ) ,p r ( t ) −→ P r ( t, ¯ θ ) = p r + ¯ θ f ( t ) ,p ϑ ( t ) −→ P ϑ ( t, ¯ θ ) = p ϑ ( t ) + ¯ θ f ( t ) ,B ( t ) −→ ˜ B ( t, ¯ θ ) = B ( t ) + ¯ θ f ( t ) , (15)where b , b are the bosonic secondary variables and f , f , f , f , f , f are the fermionicsecondary variables. We determine the precise value of these secondary variables in termsof the basic and auxiliary variables present in the BRST invariant Lagrangian (7) by usingthe BRST invariant quantities/restrictions.According to the basic tenets of ACSA to BRST formalism, the BRST invariant quantitiesmust remain independent of the Grassmannian variable (¯ θ ) when they are generalized ontothe (1, 1)-dimensional anti-chiral super submanifold. The BRST invariant quantities aresome specific combinations of the basic and auxiliary variables. These are given as follows s b ( r, p r , p ϑ , B, C ) = 0 , s b ( z ˙ C ) = 0 , s b ( ϑ C ) = 0 ,s b ( ˙ B z + i ˙¯ C ˙ C ) = 0 , s b ( ˙ ϑ − z ) = 0 , s b ( B ϑ + i ¯ C C ) = 0 . (16)We generalize the above BRST invariant quantities onto the (1, 1)-dimensional anti-chiralsuper submanifold as R ( t, ¯ θ ) = r ( t ) , P r ( t, ¯ θ ) = p r ( t ) , P ϑ ( t, ¯ θ ) = p ϑ ( t ) , ˜ B ( t, ¯ θ ) = B ( t ) , F ( t, ¯ θ ) = C ( t ) , Z ( t, ¯ θ ) ˙ F ( t, ¯ θ ) = z ( t ) ˙ C ( t ) , Θ( t, ¯ θ ) F ( t, ¯ θ ) = ϑ ( t ) C ( t ) , ˙˜ B ( t, ¯ θ ) Z ( t, ¯ θ ) + i ˙¯ F ( t, ¯ θ ) ˙ C ( t, ¯ θ ) = ˙ B ( t ) z ( t ) + i ˙¯ C ( t ) ˙ C ( t ) , ˙Θ( t, ¯ θ ) − Z ( t, ¯ θ ) = ˙ ϑ ( t ) − z ( t ) , ˜ B ( t, ¯ θ ) Θ( t, ¯ θ ) + i ¯ F ( t, ¯ θ ) F ( t, ¯ θ ) = B ( t ) ϑ ( t ) + i ¯ C ( t ) C ( t ) . (17)The above restrictions lead to the derivation of the secondary variables in terms of the basicand auxiliary variables. To determine the value of secondary variables, we perform the step-by-step explicit calculations. For this purpose, first of all, we use the generalization of the6rivial BRST invariant restrictions given in the first line of Eq. (17) as: P ϑ ( t, ¯ θ ) = p ϑ ( t ) = ⇒ f = 0 , ˜ B ( t, ¯ θ ) = B ( t ) = ⇒ f = 0 ,R ( t, ¯ θ ) = r ( t ) = ⇒ f = 0 , F ( t, ¯ θ ) = C ( t ) = ⇒ b = 0 ,P r ( t, ¯ θ ) = p r ( t ) = ⇒ f = 0 . (18)After substituting the value of secondary variables (18) into (15), we get the following ex-pressions for the anti-chiral super expansions, namely; C ( t ) −→ F ( b ) ( t, ¯ θ ) = C ( t ) + ¯ θ (0) ≡ C ( t ) + ¯ θ [ s b C ( t )] ,r ( t ) −→ R ( b ) ( t, ¯ θ ) = r ( t ) + ¯ θ (0) ≡ r ( t ) + ¯ θ [ s b r ( t )] ,p r ( t ) −→ P ( b ) r ( t, ¯ θ ) = p r ( t ) + ¯ θ (0) ≡ p r ( t ) + ¯ θ [ s b p r ( t )] ,p ϑ ( t ) −→ P ( b ) ϑ ( t, ¯ θ ) = p ϑ ( t ) + ¯ θ (0) ≡ p ϑ ( t ) + ¯ θ [ s b p ϑ ( t )] ,B ( t ) −→ ˜ B ( b ) ( t, ¯ θ ) = B ( t ) + ¯ θ (0) ≡ B ( t ) + ¯ θ [ s b B ( t )] , (19)where the superscript ( b ) on the anti-chiral supervariables denotes that these supervariableshave been obtained after the use of BRST invariant quantities. It is clear that the coeffi-cients of Grassmannian variable ¯ θ are nothing but the BRST symmetry transformations (8).Now, in the case of non-trivial BRST invariant restrictions, the generalization of the BRSTinvariant restrictions s b ( z ˙ C ) = 0 and s b ( ϑ C ) = 0 Z ( t, ¯ θ ) ˙ F ( b ) ( t, ¯ θ ) = z ( t ) ˙ C ( t ) , Θ( t, ¯ θ ) F ( b ) ( t, ¯ θ ) = ϑ ( t ) C ( t ) , (20)lead to the following interesting relationships f ( t ) ˙ C ( t ) = 0 = ⇒ f ( t ) ∝ ˙ C ( t ) , = ⇒ f ( t ) = κ ˙ C ( t ) ,f ( t ) C ( t ) = 0 = ⇒ f ( t ) ∝ C ( t ) , = ⇒ f ( t ) = κ C ( t ) , (21)where κ and κ are the proportionality constants. Now, we use the generalization of BRSTinvariant restriction s b ( ˙ ϑ − z ) = 0 as˙Θ( t, ¯ θ ) − Z ( t, ¯ θ ) = ˙ ϑ ( t ) − z ( t ) = ⇒ κ = κ . (22)Finally, in order to determine the value of constants, we generalize the BRST invariantrestrictions s b ( ˙ B z + i ˙¯ C ˙ C ) = 0 and s b ( B ϑ + i ¯ C C ) = 0 onto (1 ,
1) super submanifold as:˙˜ B ( b ) ( t, ¯ θ ) Z ( t, ¯ θ ) + i ˙¯ F ( t, ¯ θ ) ˙ F ( b ) ( t, ¯ θ ) = ˙ B ( t ) z ( t ) + i ˙¯ C ( t ) ˙ C ( t ) = ⇒ ˙ b ( t ) = κ ˙ B ( t ) , ˜ B ( b ) ( t, ¯ θ ) Θ( t, ¯ θ ) + i ¯ F ( t, ¯ θ ) F ( b ) ( t, ¯ θ ) = B ( t ) ϑ ( t ) + i ¯ C ( t ) C ( t ) = ⇒ b ( t ) = κ B ( t ) . (23)Using the results obtained in Eq. (22) and Eq. (23), it is clear that κ = κ = 1. Therefore,we get the value of secondary variables as: f ( t ) = ˙ C ( t ) , f ( t ) = C ( t ) , b ( t ) = B ( t ) and weobtain the following expansions of anti-chiral supervariables: z ( t ) −→ Z ( b ) ( t, ¯ θ ) = z ( t ) + ¯ θ [ ˙ C ( t )] ≡ z ( t ) + ¯ θ [ s b z ( t )] ,ϑ ( t ) −→ Θ ( b ) ( t, ¯ θ ) = ϑ ( t ) + ¯ θ [ C ( t )] ≡ ϑ ( t ) + ¯ θ [ s b ϑ ( t )] , ¯ C ( t ) −→ ¯ F ( b ) ( t, ¯ θ ) = ¯ C ( t ) + ¯ θ [ i B ( t )] ≡ ¯ C ( t ) + ¯ θ [ s b ¯ C ( t )] . (24)7hus, in view of the above Eq. (24), we have a connection between the BRST symmetrytransformation ( s b ) and partial derivative ( ∂ ¯ θ ) on the anti-chiral super submanifold definedby the mapping: s b ←→ ∂ ¯ θ .We are now in the position to derive the anti-BRST symmetry transformations usingchiral supervariable approach. In this context, we use the chiral super expansions of thechiral supervariables where we generalize (0 + 1)-dimensional variables onto the (1, 1)-dimensional super submanifold of the general (1, 2)-dimensional supermanifold. The chiralsuper expansions of the variables are z ( t ) −→ Z ( t, θ ) = z ( t ) + θ ¯ f ( t ) ,ϑ ( t ) −→ Θ( t, θ ) = ϑ ( t ) + θ ¯ f ( t ) ,C ( t ) −→ F ( t, θ ) = C ( t ) + i θ ¯ b ( t ) , ¯ C ( t ) −→ ¯ F ( t, θ ) = ¯ C ( t ) + i θ ¯ b ( t ) ,r ( t ) −→ R ( t, θ ) = r ( t ) + θ ¯ f ( t ) ,p r ( t ) −→ P r ( t, θ ) = p r ( t ) + θ ¯ f ( t ) ,p ϑ ( t ) −→ P ϑ ( t, θ ) = p ϑ ( t ) + θ ¯ f ( t ) ,B ( t ) −→ ˜ B ( t, θ ) = B ( t ) + θ ¯ f ( t ) , (25)where ¯ b , ¯ b are the bosonic secondary variables and ¯ f , ¯ f , ¯ f , ¯ f , ¯ f , ¯ f are the fermionicsecondary variables. The anti-BRST invariant quantities also must remain independent ofthe Grassmannian variable ( θ ) when they are generalized onto the (1 + 1)-dimensional chiralsuper submanifold. The anti-BRST invariant restrictions are given as follows s ab ( r, p r , p ϑ , B, ¯ C ) = 0 , s ab ( z ˙¯ C ) = 0 , s ab ( ϑ ¯ C ) = 0 ,s ab ( ˙ B z + i ˙¯ C ˙ C ) = 0 , s ab ( ˙ ϑ − z ) = 0 , s ab ( B ϑ + i ¯ C C ) = 0 . (26)As the physical quantities remain independent of the Grassmannian variable θ which implythat the anti-BRST invariant restrictions can be generalized onto the (1, 1)-dimensionalsuper submanifold of the general (1, 2)-dimensional supermanifold as follows R ( t, θ ) = r ( t ) , P r ( t, θ ) = p r ( t ) , P ϑ ( t, θ ) = p ϑ ( t ) , ˜ B ( t, θ ) = B ( t ) , ¯ F ( t, θ ) = ¯ C ( t ) , Z ( t, θ ) ˙¯ F ( t, θ ) = z ( t ) ˙¯ C ( t ) , Θ( t, θ ) ¯ F ( t, θ ) = ϑ ( t ) ¯ C ( t ) , ˙˜ B ( t, θ ) Z ( t, θ ) + i ˙¯ F ( t, θ ) ˙ F ( t, θ ) = ˙ B ( t ) z ( t ) + i ˙¯ C ( t ) ˙ C ( t ) , ˙Θ( t, θ ) − Z ( t, θ ) = ˙ ϑ ( t ) − z ( t ) , ˜ B ( t, θ ) Θ( t, θ ) + i ¯ F ( t, θ ) F ( t, θ ) = B ( t ) ϑ ( t ) + i ¯ C ( t ) C ( t ) . (27)The generalizations of the above anti-BRST invariant restrictions [Eq. (26)] lead to thederivation of the secondary variables in terms of the auxiliary and basic variables present inthe Lagrangian L , namely;¯ b = 0 , ¯ f = 0 , ¯ f = 0 , ¯ f = 0 , ¯ f = 0 , ¯ f = ˙¯ C, ¯ f = ¯ C, ¯ b = − B. (28)The above values of chiral secondary variables are derived in a similar fashion as the anti-chiral secondary variables are derived. After the substitution of the above secondary variables8nto the chiral super expansions (25), we get the following expressions for the chiral supervariables generalized onto the (1, 1)-dimensional super submanifold z ( t ) −→ Z ( ab ) ( t, θ ) = z ( t ) + θ ( ˙¯ C ) ≡ z ( t ) + θ [ s ab z ( t )] ,ϑ ( t ) −→ Θ ( ab ) ( t, θ ) = ϑ ( t ) + θ ( ¯ C ) ≡ ϑ ( t ) + θ [ s ab ϑ ( t )] ,C ( t ) −→ F ( ab ) ( t, θ ) = C ( t ) + θ ( − i B ) ≡ C ( t ) + θ [ s ab C ( t )] , ¯ C ( t ) −→ ¯ F ( ab ) ( t, θ ) = ¯ C ( t ) + θ (0) ≡ ¯ C ( t ) + θ [ s ab ¯ C ( t )] ,r ( t ) −→ R ( ab ) ( t, θ ) = r ( t ) + θ (0) ≡ r ( t ) + θ [ s ab r ( t )] ,p r ( t ) −→ P ( ab ) r ( t, θ ) = p r ( t ) + θ (0) ≡ p r ( t ) + θ [ s ab p r ( t )] ,p ϑ ( t ) −→ P ( ab ) ϑ ( t, θ ) = p ϑ ( t ) + θ (0) ≡ p ϑ ( t ) + θ [ s ab p ϑ ( t )] ,B ( t ) −→ ˜ B ( ab ) ( t, θ ) = B ( t ) + θ (0) ≡ B ( t ) + θ [ s ab B ( t )] . (29)Here, the coefficients of θ are nothing but the anti-BRST symmetry transformations. Thus,it is clear that there is a connection between the anti-BRST symmetry transformation ( s ab )and the partial derivative ( ∂ θ ) defined on the chiral super submanifold as: s ab ←→ ∂ θ . In this section, we derive the off-shell nilpotent (anti-)co-BRST symmetry transformationsusing 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 goalin our mind, first of all, we derive the co-BRST symmetry transformations by exploiting thechiral super expansions given in Eq. (25) and co-BRST invariant restrictions. The co-BRSTinvariant restrictions are given as: s d ( r, p r , p ϑ , B, ¯ C ) = 0 , s d ( z ¯ C ) = 0 , s d ( ϑ ˙¯ C ) = 0 ,s d ( ϑ ˙ p ϑ + i ˙¯ C ˙ C ) = 0 , s d ( z p ϑ − i ¯ C C ) = 0 , s d ( ϑ + ˙ z ) = 0 . (30)According to the basic tenets of ACSA to BRST formalism the above co-BRST invariantrestrictions can be generalized onto the (1, 1)-dimensional super submanifold (of the general(1, 2)-dimensional supermanifold) as: R ( t, θ ) = r ( t ) , P r ( t, θ ) = p r ( t ) , P ϑ ( t, θ ) = p ϑ ( t ) , ˜ B ( t, θ ) = B ( t ) , ¯ F ( t, θ ) = ¯ C ( t ) , Z ( t, θ ) ¯ F ( t, θ ) = z ( t ) ¯ C ( t ) , Θ( t, θ ) ˙¯ F ( t, θ ) = ϑ ( t ) ˙¯ C ( t ) , Θ( t, θ ) ˙ P ϑ ( t, θ ) + i ˙¯ F ( t, θ ) ˙ F ( t, θ ) = ϑ ( t ) ˙ p ϑ ( t ) + i ˙¯ C ( t ) ˙ C ( t ) , Z ( t, θ ) P ϑ ( t, θ ) − i ¯ F ( t, θ ) F ( t, θ ) = z ( t ) p ϑ ( t ) − i ¯ C ( t ) C ( t ) , Θ( t, θ ) + ˙ Z ( t, θ ) = ϑ ( t ) + ˙ z ( t ) . (31)At this stage, we derive the value of secondary variables of Eq. (25) using the above general-izations of the co-BRST invariant restrictions. To derive the value of the secondary variables,9e use the first line entry of Eq. (31) where only trivial co-BRST invariant quantities aregeneralized as follows: P ϑ ( t, θ ) = p ϑ ( t ) = ⇒ ¯ f = 0 , ˜ B ( t, θ ) = B ( t ) = ⇒ ¯ f = 0 ,R ( t, θ ) = r ( t ) = ⇒ ¯ f = 0 , ¯ F ( t, θ ) = ¯ C ( t ) = ⇒ ¯ b = 0 ,P r ( t, θ ) = p r ( t ) = ⇒ ¯ f = 0 . (32)After substituting the above value of secondary variables into the expressions of the chiralsuper expansions [Eq. (25)], we obtain chiral super expansions as;¯ C ( t ) −→ ¯ F ( d ) ( t, θ ) = ¯ C ( t ) + θ (0) ≡ ¯ C ( t ) + θ [ s d ¯ C ( t )] ,r ( t ) −→ R ( d ) ( t, θ ) = r ( t ) + θ (0) ≡ r ( t ) + θ [ s d r ( t )] ,p r ( t ) −→ P ( d ) r ( t, θ ) = p r ( t ) + θ (0) ≡ p r ( t ) + θ [ s d p r ( t )] ,p ϑ ( t ) −→ P ( d ) ϑ ( t, θ ) = p ϑ ( t ) + θ (0) ≡ p ϑ ( t ) + θ [ s d p ϑ ( t )] ,B ( t ) −→ ˜ B ( d ) ( t, θ ) = B ( t ) + θ (0) ≡ B ( t ) + θ [ s d B ( t )] , (33)where superscript ( d ) denotes the supervariables obtained after the application of the co-BRST (dual-BRST) invariant restrictions. For the non-trivial case, first of all, we generalizethe co-BRST invariant restriction s d ( z ¯ C ) = 0 and s d ( ϑ ˙¯ C ) = 0 as, Z ( t, θ ) ¯ F ( d ) ( t, θ ) = z ( t ) ¯ C ( t ) , Θ( t, θ ) ˙¯ F ( d ) ( t, θ ) = ϑ ( t ) ˙¯ C ( t ) , (34)which lead to the following relationships¯ f ( t ) ¯ C ( t ) = 0 = ⇒ ¯ f ( t ) ∝ ¯ C ( t ) , = ⇒ ¯ f ( t ) = − ¯ κ ¯ C ( t ) , ¯ f ( t ) ˙¯ C ( t ) = 0 = ⇒ ¯ f ( t ) ∝ ˙¯ C ( t ) , = ⇒ ¯ f ( t ) = ¯ κ ˙¯ C ( t ) , (35)where ¯ κ and ¯ κ are the proportionality constants. To determine the value of these con-stants, we further use the generalization of the co-BRST invariant restrictions s d ( ϑ + ˙ z ) =0 , s d ( ϑ ˙ p ϑ + i ˙¯ C ˙ C ) = 0 and s d ( z p ϑ − i ¯ C C ) = 0 as:Θ( t, θ ) + ˙ Z ( t, θ ) = ϑ ( t ) + ˙ z ( t ) = ⇒ ¯ κ = ¯ κ , Θ( t, θ ) ˙ P ( d ) ϑ ( t, θ ) + i ˙¯ F ( d ) ( t, θ ) ˙ F ( t, θ ) = ϑ ( t ) ˙ p ϑ ( t ) + i ˙¯ C ( t ) ˙ C ( t ) = ⇒ ˙¯ b ( t ) = − ¯ κ ˙ p ϑ ( t ) , Z ( t, θ ) P ( d ) ϑ ( t, θ ) − i ¯ F ( d ) ( t, θ ) F ( t, θ ) = z ( t ) p ϑ ( t ) − i ¯ C ( t ) C ( t ) = ⇒ ¯ b ( t ) = − ¯ κ p ϑ ( t ) . (36)The results of the above three relations in Eq. (36) imply that proportionality constant areequal (i.e. ¯ κ = ¯ κ ) and their values are: ¯ κ = ¯ κ = −
1. Therefore, we get the value of thesecondary variables as: ¯ f = ¯ C, ¯ f = − ˙¯ C, ¯ b = p ϑ . As a result, we have the following chiralexpansions of the ordinary variables: z ( t ) −→ Z ( d ) ( t, θ ) = z ( t ) + θ [ ¯ C ( t )] ≡ z ( t ) + θ [ s d z ( t )] ,ϑ ( t ) −→ Θ ( d ) ( t, θ ) = ϑ ( t ) + θ [ − ˙¯ C ( t )] ≡ ϑ ( t ) + θ [ s d ϑ ( t )] ,C ( t ) −→ ¯ F ( d ) ( t, θ ) = C ( t ) + θ [ i p ϑ ( t )] ≡ C ( t ) + θ [ s d C ( t )] . (37)10here superscript ( d ) on the supervariables denotes the same meaning as in Eq. (33). Thecoefficients of the θ are nothing but the co-BRST symmetry transformations. It is clear thatco-BRST symmetry ( s d ) is connected with the Grassmannian derivative ∂ θ (i.e. s d ←→ ∂ θ ).Now for the derivation of anti-co-BRST symmetry transformations, we use the anti-chiralsuper expansions of the supervariables [Eq. (15)] and anti-co-BRST invariant restrictionswhich are the combinations of some specific set of ordinary variables. These restrictions are: s ad ( r, p r , p ϑ , B, C ) = 0 , s ad ( z C ) = 0 , s ad ( ϑ ˙ C ) = 0 ,s ad ( z p ϑ − i ¯ C C ) = 0 , s ad ( ϑ ˙ p ϑ + i ˙¯ C ˙ C ) = 0 , s ad ( ˙ z + ϑ ) = 0 . (38)The generalization of these anti-co-BRST invariant restrictions onto the (1, 1)-dimensionalsuper submanifold (of the general (1, 2)-dimensional supermanifold) are as follows: R ( t, ¯ θ ) = r ( t ) , P r ( t, ¯ θ ) = p r ( t ) , P ϑ ( t, ¯ θ ) = p ϑ ( t ) , ˜ B ( t, ¯ θ ) = B ( t ) , F ( t, ¯ θ ) = C ( t ) , Z ( t, ¯ θ ) F ( t, ¯ θ ) = z ( t ) C ( t ) , Θ( t, ¯ θ ) ˙ F ( t, ¯ θ ) = ϑ ( t ) ˙ C ( t ) , Z ( t, ¯ θ ) P ϑ ( t, ¯ θ ) − i ¯ F ( t, ¯ θ ) F ( t, ¯ θ ) = z ( t ) p ϑ ( t ) − i ¯ C ( t ) C ( t ) , Θ( t, ¯ θ ) ˙ P ϑ ( t, ¯ θ ) + i ˙¯ F ( t, ¯ θ ) ˙ F ( t, ¯ θ ) = ϑ ( t ) p ϑ ( t ) + i ˙¯ C ( t ) ˙ C ( t ) , ˙ Z ( t, ¯ θ ) + Θ( t, ¯ θ ) = ˙ z ( t ) + ϑ ( t ) . (39)The above generalizations of the anti-co-BRST invariant restrictions, finally, lead to thederivation of secondary variables as follows: b = 0 , f = 0 , f = 0 , f = 0 , f = 0 , f = C, f = − ˙ C, b = − p ϑ . (40)Thus, we have determined all the secondary variables using the same technique as in the caseof co-BRST symmetry transformations. Finally, after substituting the value of secondaryvariables into Eq. (15), we obtain the following expressions of the anti-chiral expansions: z ( t ) −→ Z ( ad ) ( t, ¯ θ ) = z ( t ) + ¯ θ ( C ) ≡ z ( t ) + ¯ θ [ s ad z ( t )] ,ϑ ( t ) −→ ¯ θ ( ad ) ( t, ¯ θ ) = ϑ ( t ) + ¯ θ ( − ˙ C ) ≡ ϑ ( t ) + ¯ θ [ s ad ϑ ( t )] ,C ( t ) −→ F ( ad ) ( t, ¯ θ ) = C ( t ) + ¯ θ (0) ≡ C ( t ) + ¯ θ [ s ad C ( t )] , ¯ C ( t ) −→ ¯ F ( ad ) ( t, ¯ θ ) = ¯ C ( t ) + ¯ θ ( − i p ϑ ) ≡ ¯ C ( t ) + ¯ θ [ s ad ¯ C ( t )] ,r ( t ) −→ R ( ad ) ( t, ¯ θ ) = r ( t ) + ¯ θ (0) ≡ r ( t ) + ¯ θ [ s ad r ( t )] ,p r ( t ) −→ P ( ad ) r ( t, ¯ θ ) = p r ( t ) + ¯ θ (0) ≡ p r ( t ) + ¯ θ [ s ad p r ( t )] ,p ϑ ( t ) −→ P ( ad ) ϑ ( t, ¯ θ ) = p ϑ ( t ) + ¯ θ (0) ≡ p ϑ ( t ) + ¯ θ [ s ad p ϑ ( t )] ,B ( t ) −→ ˜ B ( ad ) ( t, ¯ θ ) = B ( t ) + ¯ θ (0) ≡ B ( t ) + ¯ θ [ s ad B ( t )] , (41)where superscript ( ad ) on the anti-chiral supervariables denote the fact that supervariablesobtained after the application of anti-co-BRST invariant restrictions [Eq. (38)]. Here, it isclear, the coefficient of Grassmannian variable ¯ θ is nothing but the anti-co-BRST symmetrytransformations. This implies that the anti-co-BRST symmetry ( s ad ) is connected with thederivative ( ∂ ¯ θ ) of Grassmannian variable ¯ θ as: s ad ←→ ∂ ¯ θ .11 ACSA to Nilpotency and Absolute Anticommutativ-ity of the Conserved Charges
In this section, we prove the nilpotency and absolute anticommutativity properties of theconserved (anti-)BRST and (anti-)co-BRST charges in the language of ACSA to BRSTformalism. First of all, we show the nilpotency of the (anti-)BRST and (anti-)co-BRSTcharges. It is straightforward to express the (anti-)BRST and (anti-)co-BRST charges interms of the (anti-)chiral supervariables and partial derivatives ( ∂ ¯ θ , ∂ θ ) with an equivalentintegral form as follows Q b = ∂∂ ¯ θ h i ˙¯ F ( b ) ( t, ¯ θ ) F ( b ) ( t, ¯ θ ) − i ¯ F ( b ) ( t, ¯ θ ) ˙ F ( b ) ( t, ¯ θ ) i ≡ Z d ¯ θ h i ˙¯ F ( b ) ( t, ¯ θ ) F ( b ) ( t, ¯ θ ) − i ¯ F ( b ) ( t, ¯ θ ) ˙ F ( b ) ( t, ¯ θ ) i ,Q ab = ∂∂θ h i ¯ F ( ab ) ( t, θ ) ˙ F ( ab ) ( t, θ ) − i ˙¯ F ( ab ) ( t, θ ) F ( ab ) ( t, θ ) i ≡ Z dθ h i ¯ F ( ab ) ( t, θ ) ˙ F ( ab ) ( t, θ ) − i ˙¯ F ( ab ) ( t, θ ) F ( ab ) ( t, θ ) i , (42) Q d = ∂∂θ h i ¯ F ( d ) ( t, θ ) ˙ F ( d ) ( t, θ ) − i ˙¯ F ( d ) ( t, θ ) F ( d ) ( t, θ ) i ≡ Z dθ h i ¯ F ( d ) ( t, θ ) ˙ F ( d ) ( t, θ ) − i ˙¯ F ( d ) ( t, θ ) F ( d ) ( t, θ ) i ,Q ad = ∂∂ ¯ θ h i ˙¯ F ( ad ) ( t, ¯ θ ) F ( ad ) ( t, ¯ θ ) − i ¯ F ( ad ) ( t, ¯ θ ) ˙ F ( ad ) ( t, ¯ θ ) i ≡ Z d ¯ θ h i ˙¯ F ( ad ) ( t, ¯ θ ) F ( ad ) ( t, ¯ θ ) − i ¯ F ( ad ) ( t, ¯ θ ) ˙ F ( ad ) ( t, ¯ θ ) i , (43)where the superscripts ( b ) and ( ab ) stand for the anti-chiral and chiral supervariables thathave been obtained after the application of the (anti-)BRST invariant restrictions, respec-tively. The superscripts ( d ) and ( ad ) denote the chiral and anti-chiral superfields obtainedafter the application of (anti-)co-BRST invariant restrictions, respectively. It is crystal clearthat the nilpotency ( ∂ θ = 0 , ∂ θ = 0) of the translational generators ( ∂ ¯ θ , ∂ θ ) implies that ∂ ¯ θ Q b = 0 ⇐⇒ ∂ θ = 0 ⇐⇒ s b Q b = − i { Q b , Q b } = 0 ,∂ θ Q ab = 0 ⇐⇒ ∂ θ = 0 ⇐⇒ s ab Q ab = − i { Q ab , Q ab } = 0 ,∂ θ Q d = 0 ⇐⇒ ∂ θ = 0 ⇐⇒ s d Q d = − i { Q d , Q d } = 0 ,∂ ¯ θ Q ad = 0 ⇐⇒ ∂ θ = 0 ⇐⇒ s ad Q ad = − i { Q ad , Q ad } = 0 , (44)which show the nilpotency ( Q a ) b = 0 , Q a ) d = 0) of the conserved charges within the ambitof ACSA to BRST formalism. Thus, we have shown that there is a deep connection betweenthe nilpotency ( ∂ θ = 0 , ∂ θ = 0) of the translational generator ( ∂ ¯ θ , ∂ θ ) and the nilpotency(i.e. Q a ) b = 0 , Q a ) d = 0) of the (anti-)BRST and (anti-)co-BRST charges ( Q ( a ) b , Q ( a ) d ).The above nilpotency property can be also captured in an ordinary space where we use the12anti-)BRST-exact and (anti-)co-BRST-exact forms of the charges, namely; Q b = − i s b (cid:0) ¯ C ˙ C − ˙¯ C C (cid:1) , Q ab = + i s ab (cid:0) ¯ C ˙ C − ˙¯ C C (cid:1) ,Q d = i s d (cid:0) ¯ C ˙ C − ˙¯ C C (cid:1) , Q ad = − i s ad (cid:0) ¯ C ˙ C − ˙¯ C C (cid:1) , (45)which show the nilpotency property of the (anti-)BRST and (anti-)co-BRST charges, in asimpler way, in an ordinary space.Now, we are in a stage to show the absolute anticommutativity of the (anti-)BRST and(anti-)co-BRST charges. For this purpose, we write the charges in terms of the (anti-)chiralsupervariables and the derivatives ( ∂ θ , ∂ ¯ θ ) of the Grassmannian variables (¯ θ, θ ) Q b = − i ∂∂θ h ˙ F ( ab ) ( t, θ ) F ( ab ) ( t, θ ) i ≡ − i Z dθ h ˙ F ( ab ) ( t, θ ) F ( ab ) ( t, θ ) i ,Q ab = i ∂∂ ¯ θ h ˙¯ F ( b ) ( t, ¯ θ ) ¯ F ( b ) ( t, ¯ θ ) i ≡ i Z d ¯ θ h ˙¯ F ( b ) ( t, ¯ θ ) ¯ F ( b ) ( t, ¯ θ ) i ,Q d = i ∂∂ ¯ θ h ˙¯ F ( ad ) ( t, ¯ θ ) ¯ F ( ad ) ( t, ¯ θ ) i ≡ i Z d ¯ θ h ˙¯ F ( ad ) ( t, ¯ θ ) ¯ F ( ad ) ( t, ¯ θ ) i Q ad = − i ∂∂θ h ˙ F ( d ) ( t, θ ) F ( d ) ( t, θ ) i ≡ − i Z dθ h ˙ F ( d ) ( t, θ ) F ( d ) ( t, θ ) i , (46)where the superscripts ( a ) b and ( a ) d denote the same meaning as explained earlier. Here,it is straightforward to check that the nilpotency ( ∂ θ = 0 , ∂ θ = 0) of the translationalgenerators ( ∂ ¯ θ , ∂ θ ) implies that the following relations ∂ θ Q b = 0 ⇐⇒ ∂ θ = 0 ⇐⇒ s ab Q b = − i { Q b , Q ab } = 0 ,∂ ¯ θ Q ab = 0 ⇐⇒ ∂ θ = 0 ⇐⇒ s b Q ab = − i { Q ab , Q b } = 0 ,∂ ¯ θ Q d = 0 ⇐⇒ ∂ θ = 0 ⇐⇒ s ad Q d = − i { Q d , Q ad } = 0 ,∂ θ Q ad = 0 ⇐⇒ ∂ θ = 0 ⇐⇒ s d Q ad = − i { Q ad , Q d } = 0 , (47)which show the absolute anticommutativity property of the (anti-)BRST and (anti-)co-BRSTcharges. The property of absolute anticommutativity of conserved charges can also be cap-tured explicitly in an ordinary space by using the following (anti-)BRST exact and (anti-)co-BRST exact forms of the charges, namely; Q b = − i s ab (cid:0) ˙ C C (cid:1) , Q ab = + i s b (cid:0) ˙¯ C ¯ C (cid:1) ,Q d = i s ad (cid:0) ˙¯ C ¯ C (cid:1) , Q ad = − i s d (cid:0) ˙ C C (cid:1) . (48) In this section, we discus the (anti-)BRST and (anti-)co-BRST invariances of the Lagrangian(7) within the framework of ACSA to BRST formalism. For this purpose, first of all, wegeneralize the ordinary Lagrangian of (0+1)-dimensional onto the suitably chosen (1, 1)-dimensional (anti-)chiral super submanifold of the general (1, 2)-dimensional supermanifold.13he expressions of the (anti-)chiral super Lagrangian are L ( t ) −→ ˜ L ( ac ) ( t, ¯ θ ) = ˙ r ( t ) p r ( t ) + ˙Θ ( b ) ( t, ¯ θ ) p ϑ ( t ) − p r ( t ) − r p ϑ ( t ) − Z ( b ) ( t, ¯ θ ) p ϑ ( t ) − V ( r ) + 12 B ( t ) + B ( t ) [ ˙ Z ( b ) ( t, ¯ θ ) + Θ ( b ) ( t, ¯ θ )] − i ˙¯ F ( b ) ( t, ¯ θ ) ˙ C ( t )+ i ¯ F ( b ) ( t, ¯ θ ) C ( t ) ,L ( t ) −→ ˜ L ( c ) ( t, θ ) = ˙ r ( t ) p r ( t ) + ˙Θ ( ab ) ( t, θ ) p ϑ ( t ) − p r ( t ) − r p ϑ ( t ) − Z ( ab ) ( t, θ ) p ϑ ( t ) − V ( r ) + 12 B ( t ) + B ( t ) [ ˙ Z ( ab ) ( t, θ ) + Θ ( ab ) ( t, θ )] − i ˙¯ C ( t ) ˙ F ( ab ) ( t, θ )+ i ¯ C ( t ) F ( ab ) ( t, θ ) , (49)where the superscripts ( ac ) and ( c ) on the super Lagrangians denote the anti-chiral and chiralLagrangians (containing anti-chiral and chiral supervariables), respectively that have beenobtained after the application of the (anti-)BRST invariant restrictions. Under the applica-tion of translational generators ( ∂ ¯ θ , ∂ θ ), we get the (anti-)BRST invariance of Lagrangian( L ) with the following results ∂∂ ¯ θ h ˜ L ( ac ) ( t, ¯ θ ) i = dd t [ B ( t ) ˙ C ( t )] , ∂∂θ h ˜ L ( c ) ( t, θ ) i = dd t [ B ( t ) ˙¯ C ( t )] , (50)which imply that generalized Lagrangians remain quasi-invariant (or up to a total timederivative) under the translational generators ( ∂ ¯ θ , ∂ θ ) within the framework of ACSA toBRST formalism which have been captured in the ordinary space [cf. Eq. (10)].Now, we would like to capture the (anti-)co-BRST invariance of the Lagrangian ( L )within the framework of (anti-)chiral supervariable approach to BRST formalism. 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 submanifoldof the (1, 2)-dimensional supermanifold as follows: L ( t ) −→ ˜ L ( c, d ) ( t, θ ) = ˙ r ( t ) p r ( t ) + ˙Θ ( d ) ( t, θ ) p ϑ ( t ) − p r ( t ) − r p ϑ ( t ) − Z ( d ) ( t, θ ) p ϑ ( t ) − V ( r ) + 12 B ( t ) + B ( t ) [ ˙ Z ( d ) ( t, θ ) + Θ ( d ) ( t, θ )] − i ˙¯ C ( t ) ˙ F ( d ) ( t, θ )+ i ¯ C ( t ) F ( d ) ( t, θ ) ,L ( t ) −→ ˜ L ( ac, ad ) ( t, ¯ θ ) = ˙ r ( t ) p r ( t ) + ˙Θ ( ad ) ( t, ¯ θ ) p ϑ ( t ) − p r ( t ) − r p ϑ ( t ) − Z ( ad ) ( t, ¯ θ ) p ϑ ( t ) − V ( r ) + 12 B ( t ) + B ( t ) [ ˙ Z ( ad ) ( t, ¯ θ ) + Θ ( ad ) ( t, ¯ θ )] − i ˙¯ F ( ad ) ( t, ¯ θ ) ˙ C ( t )+ i ¯ F ( ad ) ( t, ¯ θ ) C ( t ) , (51)where the superscripts ( c, d ) and ( ac, ad ) denote that the super Lagrangians (containingthe chiral and anti-chiral supervariables) obtained after the application of the co-BRST andanti-co-BRST invariant restrictions, respectively. It is straightforward to check that ∂∂θ h ˜ L ( c,d ) ( t, θ ) i = − dd t (cid:2) p ϑ ( t ) ˙¯ C ( t ) (cid:3) , ∂∂ ¯ θ h ˜ L ( ac,ad ) ( t, ¯ θ ) i = − dd t (cid:2) p ϑ ( t ) ˙ C ( t ) (cid:3) , (52)14hich show the (anti-)co-BRST invariance of the Lagrangian L within the ambit of ACSA toBRST formalism. At the end of this section, we have the following concluding remarks. Thereare deep connections between the (anti-)BRST symmetry transformations ( s ( a ) b ) and deriva-tives ( ∂ ¯ θ , ∂ θ ) of the Grassmannian variables (¯ θ, θ ) with the following mappings: s b ←→ ∂ ¯ θ and s ab ←→ ∂ θ . Similarly, in the case of (anti-)co-BRST symmetry transformations, it isclear that these symmetry transformations are also connected with the derivatives ( ∂ ¯ θ , ∂ θ )of Grassmannian variables with the mappings: s d ←→ ∂ θ and s ad ←→ ∂ ¯ θ [cf. Secs. 4, 5]. In our present investigation, we have derived the off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations. We have also discussed the nilpotency and absoluteanticommutativity properties of the corresponding conserved (anti-)BRST and (anti-)co-BRST charges of the ordinary (0 + 1)-dimensional gauge invariant Christ–Lee model withinthe framework of (anit-)chiral supervariable approach (ACSA) to BRST formalism.The novel observations of our present endeavor are the derivation of the off-shell nilpo-tent (anti-)BRST, (anti-)co-BRST symmetry transformations (cf. Sec. 4) and the proof ofnilpotency and the anticommutativity properties of the (anti-)BRST and (anti-)co-BRSTcharges in spite of the fact that we have taken into account only the (anti-)chiral super ex-pansions of the supervariables (cf. Sec. 5). The nilpotency and anticommutativity propertiesof the above charges and derivation of the corresponding (anti-)BRST and (anti-)co-BRSTsymmetry transformations are obvious when the full super expansions of the BT-superfieldformalism [25-28] is taken into account.It is worthwhile to mention that the nilpotency of the BRST and anti-BRST charges isconnected with the nilpotency of the translational generators ∂ ¯ θ and ∂ θ , respectively. Onthe other hand, nilpotency of the co-BRST and anti-co-BRST charges is connected withnilpotency of the translational generators ∂ θ and ∂ ¯ θ , respectively. However, we have estab-lished (cf. Sec. 5) that the absolute anticommutativity of the BRST charge with anti-BRSTcharge is connected with the nilpotency of the translational generator ( ∂ θ ) and absolute an-ticommutativity of anti-BRST charge with BRST charge is connected with the nilpotencyof the translational generator ( ∂ ¯ θ ). On the contrary, the absolute anticommutativity of theco-BRST charge with anti-co-BRST charge is connected with the nilpotency of the transla-tional generator ( ∂ ¯ θ ) and the absolute anticommutativity of the anti-co-BRST charge withco-BRST charge is connected with the nilpotency of the translational generator ( ∂ θ ). Theseobservations are completely novel for the present model.We have also captured the (anti-)BRST and (anti-)co-BRST invariances of the La-grangian within the framework of ACSA to BRST formalism which are completely novelfor the present CL model. These are the issues that would be discussed in our future in-vestigations for the various model like ABJM theory [29-31], supersymmetric Chern-Simonstheory [32], Jackiw-Pi model, Freedman-Townsend model and Abelian gauge theory withhigher derivative matter fields within the framework of ACSA to BRST formalism.15 ata Availability No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflicts of interest.
Acknowledgments:
B. Chauhan and S. Kumar are grateful to the DST-INSPIRE andBHU fellowships for financial support, respectively. The authors also thank Dr. R. Kumarfor a careful reading of the manuscript and for important as well as significant suggestions.
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