# Superfield Approach to Nilpotency and Absolute Anticommutativity of Conserved Charges: 2D Non-Abelian 1-Form Gauge Theory

Authors: S. Kumar; B. Chauhan; R. P. Malik
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

We exploit the theoretical strength of augmented version of superfield approach (AVSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism to express the nilpotency and absolute anticommutativity properties of the (anti-)BRST and (anti-)co-BRST conserved charges for the two(1+1)-dimensional (2D) non-Abelian 1-form gauge theory (without any interaction with matter fields) in the language of superspace variables, their derivatives, and suitable superfields. In the proof of absolute anticommutativity property, we invoke the strength of Curci-Ferrari (CF) condition for the (anti-)BRST charges. No suchoutside condition/restriction is required in the proof of absolute anticommutativity of the (anti-)co-BRST conserved charges. The latter observation (as well as other observations) connected with (anti-)co-BRST symmetries and corresponding conserved charges arenovel results of our present investigation. We also discuss the (anti-)BRST and (anti-)co-BRST symmetry invariance of the appropriate Lagrangian densities within the framework of AVSA. In addition, we dwell a bit on the derivation of the above fermionic (nilpotent) symmetries by applying the AVSA to BRST formalism, whereonly the (anti)chiral superfields are used.


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## Body

## 1. Introduction

The principle oflocal gauge invariance is at the heart of standard model of particle physics, where there is a stunning degree of agreement between theory and experiment. One of the most elegant approaches to covariantly quantize the above kinds of gauge theories (based on the principle oflocal gauge invariance) is Becchi-Rouet-Stora-Tyutin (BRST) formalism, whereeach local gauge symmetry is traded withtwo nilpotent symmetries that are christened as the BRST and anti-BRST symmetries. The latter symmetries are thequantum version of the gauge symmetries and their very existence ensures the covariant canonical quantization of a given gauge theory. The decisive features of the above quantum (anti-)BRST symmetries are the observations that (i) they are nilpotent of order two and (ii) they are absolutely anticommuting in nature. In the language of theoretical physics, the nilpotency property ensures the fermionic (supersymmetric-type) nature of the (anti-)BRST symmetries and the linear independence of BRST and anti-BRST symmetries is encoded in the property of absolute anticommutativity of the above (anti-)BRST symmetries.The superfield approach to BRST formalism [1–8] provides the geometrical basis for the properties of nilpotency and absolute anticommutativity which are associated with the (anti-)BRST symmetries. In the aboveusual superfield approach [1–8], the celebrated horizontality condition (HC) plays a key and decisive role. The HC leads, however, to the derivation (as well as geometrical interpretation) of the (anti-)BRST symmetries that are associated with the gauge and corresponding (anti)ghost fieldsonly. It doesnot shed any light on the (anti-)BRST symmetries that are associated with thematterfields in a giveninteractinggauge theory. In a set of papers [9–12], the aboveusualsuperfield formalism has been systematically generalized so as to derive the (anti-)BRST symmetries for the gauge,matter, and (anti)ghost fieldstogether. The latter superfield approach [9–12] has been christened as the augmented version of superfield approach to BRST formalism, where, consistent with the HC, additional restrictions (i.e., gauge invariant conditions) are also invoked. We shall exploit the latter superfield approach [9–12] to discuss a few key features of the 2D non-Abelian 1-form gauge theory (without any interaction with matter fields) which have already been discussed within the framework of BRST formalism [13–16].To be more specific, in the above works [13–16], we have shown the existence of the nilpotent (anti-)BRST as well as (anti-)co-BRST symmetry transformations for the 2D non-Abelian 1-form gauge theory. The central theme of our present investigation is to capture the nilpotency and absolute anticommutativity of the (anti-)BRST and (anti-)co-BRST conserved charges [13–16] within the framework of augmented version of superfield approach (AVSA) to BRST formalism. In the proof of the absolute anticommutativity of the (anti-)BRST charges (within the framework of AVSA), we invoke the CF condition to recast the expressions for these charges in an appropriate form and, then only, the superfield formalism is applied. However, in the case of theabove proof of the (anti-)co-BRST charges, we donot invoke any CF-type restrictions. In our present investigation, we have proven the nilpotency and absolute anticommutativity of the conserved (anti-)BRST and (anti-)co-BRST charges that have been derived fromtwo sets of coupled Lagrangian densities (cf. (1) and (39) below) for our 2D non-Abelian 1-form theory.In the BRST approach to a given gauge theory, the existence of the (anti-)BRST symmetries and their conserved charges is well known. However, we have been able to establish the existence of (anti-)co-BRST symmetry transformations (in addition to the nilpotent (anti-)BRST symmetry transformations) in the case of a toy model of a rigid rotor in one (0+1) dimension of spacetime [17]. Furthermore, we have demonstrated the existence of such (i.e., (anti-)co-BRST) symmetries in the cases of Abelian p-form (p=1,2,3) gauge theories in the two (1+1) dimensions, four (3+1) dimensions, and six (5+1) dimensions of spacetime (see, e.g., [18] and references therein). In other words, we have established that the nilpotent (anti-)co-BRST symmetries exist for any arbitrary Abelian p-form (p=1,2,3,…) gauge theory in D=2p dimensions of spacetime [18]. One of the decisive features of the (anti-)co-BRST symmetries is the observation that it is the gauge-fixing term that remains invariant under these transformations (unlike the kinetic term that remains invariant under the (anti-)BRST transformations). The geometrical origin for these observations has been provided in our review article (see, e.g., [18] and references therein).We concentrate on the 2D non-Abelian theory (without any interaction with matter fields) because this theory has been shown [13] to be a perfect model of Hodge theory as well as anew model of topological field theory (TFT) which captures a few aspects of Witten-type TFTs [19] and some salient features of Schwarz-type TFTs [20]. The equivalence of the coupled Lagrangian densities of this 2D theory with respect to the (anti-)co-BRST symmetries has been established in our recent publication [14]. We have also discussed the CF-type restrictions for this theory within the framework of superfield approach [15], where we have demonstrated the existence of a tower of CF-type restrictions. This happens forthis theory because it is a TFT where there areno physical propagating degrees of freedom for the 2D gauge field. In another work [16], we have derived all the conserved currents and charges for this 2D theory and shown their algebraic structure that is found to be reminiscent of the Hodge algebra [21–24]. In other words, we have provided the physical realizations of the de Rham cohomological operators of differential geometry (and their algebra) in the language of the continuous (as well as discrete) symmetries, corresponding conserved charges, and their algebra in operator form.We have exploited the key ideas of AVSA to BRST formalism to derive the (anti-)BRST and (anti-)co-BRST symmetry transformations by using HC and dual-HC (DHC) as well as the (anti)chiral superfield approach to BRST formalism (see AppendicesA and B below) in the context of our present 2D non-Abelian 1-form gauge theory. In our earlier works [9–12], we havenever been able to capture the nilpotency as well as absolute anticommutativity properties of the (anti-)BRST and (anti-)co-BRST charges. The central objective of our present paper is to achieve this goal in the case of 2D non-Abelian 1-form gauge theory. To the best of our knowledge, this issue is being pursued for the first time in our present endeavor. Thus, thenovelty in our present investigation is the observation that the nilpotency of the fermionic symmetry transformations and CF-type restrictions play a decisive role in capturing the nilpotency and absolute anticommutativity properties of the conserved (anti-)BRST and (anti-)co-BRST charges in the ordinary 2D spacetime (see Section 5 below). However, it is the nilpotency of the translational generators (along the Grassmannian directions) that plays a crucial role for thesame purpose within the framework of AVSA to BRST formalism on the supermanifold (see Section 6 below).The following key factors have spurred our curiosity to pursue our present investigation. First, to add somenew ideas to the existing technique(s) of the superfield formalism is a challenging problem. In this context, we have expressed the fermionic charges (i.e., nilpotent (anti-)BRST and (anti-)co-BRST) in the language of the superfields and derivatives defined on the (2, 2)-dimensional supermanifold. Second, in our earlier works [14, 15], we have derived the expressions for the conserved fermionic charges in the ordinary 2D space. It is a challenging problem to express their nilpotency and absolute anticommutativity properties in terms of the quantities that are defined on the (2,2)-dimensional supermanifold. Third, it is also an interesting as well asnovel idea to discuss various aspects of the (anti-)co-BRST charges within the framework of AVSA to BRST formalism. Finally, the insights and understandings, gained in our present investigation, would turn out to be useful when we shall discuss the 4D Abelian 2-form and 6D Abelian 3-form gauge theories within the framework of AVSA to BRST formalism. In fact, we have already shown, in our earlier works [25, 26], that the above 4D and 6D Abelian 2-form and 3-form gauge theories are the models for the Hodge theory and theydo support the existence of the (anti-)BRST and (anti-)co-BRST symmetries (as well as their corresponding conserved charges) in addition to theother continuous symmetries (and corresponding charges). There exist discrete symmetries, too, in these theories [25, 26]. All these symmetries (and corresponding conserved charges) are required for the proof that the above models are the tractable field theoretic examples of Hodge theory.Our present paper is organized as follows. In Section2, we discuss the nilpotent (fermionic) (anti-)BRST and (anti-)co-BRST symmetries in the Lagrangian formulation. Section 3 is devoted to the discussion of horizontality condition (HC) that leads to the derivation of (anti-)BRST symmetries for the gauge field and corresponding fermionic (anti)ghost fields along with the CF condition. Section 4 deals with the dual-HC (DHC) which enables us to derive the (anti-)co-BRST symmetries that exist for the 2D non-Abelian 1-form gauge theory. The subject matter of Section 5 concerns itself with the discussion of nilpotency and absolute anticommutativity properties of the fermionic charges within the framework of BRST formalism in 2D ordinary spacetime. In Section 6, we discuss the nilpotency and absolute anticommutativity of the fermionic charges within the framework of AVSA to BRST formalism on a (2,2)-dimensional supermanifold, where the CF condition plays an important role for (anti-)BRST charges. Finally, we discuss the key results of our present investigation in Section 7, where we point out a few possible theoretical directions that might be pursued for future investigations.In AppendicesA and B, we derive the (anti-)BRST and (anti-)co-BRST symmetry transformations by exploiting the ideas of (anti)chiral superfield approach to BRST formalism which match with theones derived in the main body of the text. We express the (anti-)BRST and (anti-)co-BRST invariance of the Lagrangian densities in the language of the AVSA to BRST formalism in Appendix C.We note that the theoretical materials, contained in Sections5 and 6, are deeply interrelated. In fact, sometimes, it is due to our observations in Section 5 that we have been able to express the nilpotency and anticommutativity properties of the charges in Section 6 within the framework of AVSA to BRST formalism. On the other hand, at times, it is our knowledge of the AVSA to BRST formalism (cf. Section 6) that has turned out to be handy for our derivations of the above properties in 2D ordinary space (cf. Section 5).Convention and Notations. We take the 2D ordinary Minkowskian background spacetime to beflat with a metric tensor ημν=diag(+1,-1), where the Greek indices μ,ν,λ,…=0,1 correspond to the time and space directions, respectively. We choose 2D Levi-Civita tensor εμν to obey the properties: εμνεμν=-2!, εμνενλ=δμλ, ε01=+1=ε10, and so forth. In 2D, the curvature tensor (i.e., field strength tensor) Fμν has only one existing component F01=E=-εμν[∂μAν+(i/2)(Aμ×Aν)] because Fμν=∂μAν-∂νAμ+i(Aμ×Aν). Here, in the SU(N) Lie algebraic space, we have adopted the notations A·B=AaBa and (A×B)a=fabcAbBc for the nonnull vectors Aa and Ba, where a,b,c=1,2,…,N2-1 and fabc are the structure constants in the SU(N) Lie algebra [Ta,Tb]=fabcTc for the generators Ta, which are present in the definition of 1-form potential Aμ=Aμ·T=AμaTa and curvature 2-form field strength tensor Fμν=Fμν·T=FμνaTa and so forth. Throughout the whole body of our text, we denote the (anti-)BRST and (anti-)co-BRST fermionic (s(a)b2=s(a)d2=0) symmetry transformations by s(a)b and s(a)d, respectively.


## 2. Preliminaries: Nilpotent (Fermionic) Symmetries

We discuss here the (anti-)BRST and (anti-)co-BRST symmetries (and derive their corresponding conserved charges) in the Lagrangian formulation of the 2D non-Abelian1-form (A(1)=dxμAμ=dxμAμ·T) gauge theory within the framework of BRST formalism. The starting coupled Lagrangian densities, in the Curci-Ferrari gauge [27, 28], are(1)LB=B·E-12B·B+B·∂μAμ+12B·B+B¯·B¯-i∂μC¯·DμC,LB¯=B·E-12B·B-B¯·∂μAμ+12B·B+B¯·B¯-iDμC¯·∂μC, where B, B, and B¯ are the Nakanishi-Lautrup type auxiliary fields that have been invoked for various purposes. For instance, B is introduced in the theory to linearize the kinetic term (-(1/4)Fμν·Fμν=(1/2)E·E≡B·E-(1/2)B·B) and auxiliary fields B and B¯ satisfy the Curci-Ferrari restriction: B+B¯+(C×C¯)=0, where the (anti)ghost fields C¯ and C are fermionic (i.e., (Ca)2=(C-a)2=0, CaC¯b+C¯bCa=0, CaCb+CbCa=0, C¯aC¯b+C¯bC¯a=0, C¯aCb+CbC¯a=0, etc.) in nature and they are required in the theory for the validity of unitarity. In the above, we have the covariant derivatives [DμC=∂μC+i(Aμ×C) and DμC¯=∂μC¯+i(Aμ×C¯)] on the (anti)ghost fields in the adjoint representation.The Lagrangian densities in (1) respect the following off-shell nilpotent (s(a)b2=0) (anti-)BRST symmetries transformations (s(a)b):(2)sbAμ=DμC,sbC=-i2C×C,sbC¯=iB,sbB=0,sbB·B=0,sbB¯=iB¯×C,sbE=iE×C,sbB=iB×C,sbB·E=0,sabAμ=DμC¯,sabC¯=-i2C¯×C¯,sabC=iB¯,sabB¯=0,sabB·B=0,sabE=iE×C¯,sabB=iB×C¯,sabB=iB×C¯,sabB·E=0,because the Lagrangian densities LB and LB¯ transform under s(a)b as(3)sbLB=∂μB·DμC,sabLB¯=-∂μB¯·DμC¯,sabLB=-∂μB¯+C×C¯·∂μC¯+B+B¯+C×C¯·Dμ∂μC¯,sbLB¯=∂μB+C×C¯·∂μC-B+B¯+C×C¯·Dμ∂μC.It should be noted thatboth the Lagrangian densities in (1) respectboth (i.e., BRST and anti-BRST) symmetries on the constrained hypersurface, where the CF condition (B+B¯+(C×C¯)=0) is satisfied. In other words, we note that sbLB¯=-∂μ[B¯·∂μC] and sabLB=∂μ[B·∂μC¯] because of the validity of CF condition. As a consequence, the action integrals S=∫d2xLB and S=∫d2xLB¯ remain invariant under the (anti-)BRST symmetries on the above hypersurface located in the 2D Minkowskian spacetime manifold. It is interesting to point out that the absolute anticommutativity {sb,sab}=0 isalso satisfied on the above hypersurface, which is defined by the field equation: B+B¯+(C×C¯)=0.According to the celebrated Noether’s theorem, the above continuous symmetries lead to the derivations of conserved currents and charges. These (anti-)BRST charges, corresponding to the above continuous symmetriess(a)b, are (see, e.g., [14] for details)(4)Qab=∫dxB¯˙·C¯-B¯·D0C¯+12C¯×C¯·C˙,Qb=∫dxB·D0C-B˙·C-12C¯˙·C×C,where a single dot on a field denotes the ordinary time derivative (e.g., C˙=∂C/∂t).The above conserved chargesQ(a)b are nilpotent (Qb2=Qab2=0) of order two and they obey absolute anticommutativity property (i.e., QbQab+QabQb=0). These properties can be mathematically expressed as follows:(5)sbQb=-iQb,Qb=0⟹Qb2=0,sabQab=-iQab,Qab=0⟹Qab2=0,sabQb=-iQb,Qab=0⟹Qb,Qab=0⟺sbQab=-iQab,Qb=0.The preciseness of the above expressions can be verified by taking into account the nilpotent (anti-)BRST symmetry transformation s(a)b (cf. (2)) and expressions for the nilpotent (anti-)BRST charges from (4). It should be noted that the property of absolute anticommutativity of the (anti-)BRST charges (i.e., {Qb,Qab}=0) istrue only when we use the CF condition (i.e., B+B¯+(C×C¯)=0).The Lagrangian densities (1)also respect the following off-shell nilpotent (s(a)d2=0) and absolutely anticommuting (sdsad+sadsd=0) (anti-)co-BRST [i.e., (anti)dual BRST] symmetry transformations (s(a)d) (see, e.g., [13, 14]):(6)sadAμ=-εμν∂νC,sadC=0,sadC¯=iB,sadB=0,sadB¯=0,sadE=Dμ∂μC,sad∂μAμ=0,sadB=0,sdAμ=-εμν∂νC¯,sdC¯=0,sdC=-iB,sdB=0,sdB¯=0,sdE=Dμ∂μC¯,sd∂μAμ=0,sdB=0,because the above Lagrangian densities transform, under s(a)d, as follows:(7)sadLB¯=∂μB·∂μC,sdLB=∂μB·∂μC¯,sadLB=∂μB·DμC+εμνC¯·∂νC×C+i∂μAμ·B×C,sdLB¯=∂μB·DμC¯-εμνC·∂νC¯×C¯+i∂μAμ·B×C¯.It is clear thatboth the Lagrangian densities respectboth (i.e., co-BRST and anti-co-BRST) fermionic symmetry transformations on a hypersurface, where the CF-type restrictions B×C=0 and B×C¯=0 are satisfied. We lay emphasis on the observation that absolute anticommutativity {sd,sad}=0 is satisfiedwithout any use of CF-type restrictions B×C=0 and B×C¯=0. More elaborate discussions about these CF-type restrictions (and other related restrictions) can be found in our earlier works (see, e.g., [14, 16] for details).The Noether conserved (Q˙(a)d=0) charges Q(a)d, corresponding to the continuous and nilpotent symmetry transformations (6), are(8)Qd=∫dxB·C¯˙+B·∂1C¯≡∫dxB·C¯˙-D0B·C¯+∂1C¯×C·C¯,Qad=∫dxB·C˙-B¯·∂1C≡∫dxB·C˙-D0B·C-C¯×∂1C·C.The above charges are found to be nilpotent (Q(a)d2=0) and absolutely anticommuting (QdQad+QadQd=0) in nature. These claims can be verified in a straightforward fashion by taking the help of symmetries (6) and expressions of the charges (8) as follows:(9)sdQd=-iQd,Qd=0⟹Qd2=0,sadQad=-iQad,Qad=0⟹Qad2=0,sdQad=-iQad,Qd=0⟺sadQd=-iQd,Qad=0⟹Qd,Qad=0.In fact, in this simple proof, one has to verify the left-hand side of the above equations. In the forthcoming sections, we shall exploit the beauty and strength of the AVSA to BRST formalism to capture the above properties in a cogent and consistent manner.


## 3. Horizontality Condition: Off-Shell Nilpotent (Anti-)BRST Symmetry Transformations

We concisely mention here the key points associated with the geometrical origin of the nilpotent (anti-)BRST symmetries and existence of the CF condition within the framework of Banora-Tonin (BT) superfield formalism [4, 5]. In this connection, first of all, we generalize the 2D ordinary theory onto (2,2)-dimensional supermanifold, where the non-Abelian 1-form gauge field Aμ(x) and (anti)ghost fields (C¯)C are generalized onto their corresponding superfields with the following expansions (incorporating the secondary fields Rμ,R¯μ,Sμ,B1,B2,B¯1,B¯2,s,s¯) on the (2,2)-dimensional supermanifolds [4, 5]:(10)Aμx⟶Bμx,θ,θ¯=Aμx+θR¯μx+θ¯Rμx+iθθ¯Sμx,Cx⟶Fx,θ,θ¯=Cx+iθB¯1+iθ¯B1+iθθ¯sx,C¯x⟶F¯x,θ,θ¯=C¯x+iθB¯2+iθ¯B2+iθθ¯s¯x,where the supermanifold is characterized by the superspace coordinates ZM=(xμ,θ,θ¯). The 2D ordinary bosonic coordinates xμ(μ=0,1) and the Grassmannian coordinates (θ,θ¯) (with θ2=θ¯2=θθ¯+θ¯θ=0) specify the superspace coordinate ZM and all the superfields, defined on the supermanifold, are function of them. The 2-form super curvature is(11)F~2=dZM∧dZN2!F~MNx,θ,θ¯≡d~A~1+iA~1∧A~1,where the super curvature tensor F~MN=(F~μν,F~μθ,F~μθ¯,F~θθ,F~θθ¯,F~θ¯θ¯). In the above equation, the ordinary exterior derivative d=dxμ∂μ and non-Abelian 1-form (A(1)=dxμAμ) gauge connection have been generalized onto the (2,2)-dimensional supermanifold as(12)d=dxμ∂μ⟶d~=dxμ∂μ+dθ∂θ+dθ¯∂θ¯,d~2=0,A1=dxμAμ⟶A~1=dxμBμx,θ,θ¯+dθF¯x,θ,θ¯+dθ¯Fx,θ,θ¯,where (∂μ,∂θ,∂θ¯) are the superspace derivatives (with ∂μ=∂/∂xμ, ∂θ=∂/∂θ, and ∂θ¯=∂/∂θ¯).We have observed earlier that the kinetic term(-(1/4)Fμν·Fμν=B·E-(B·B)/2) of the Lagrangian densities (1) remains invariant under the (anti-)BRST symmetries (2) and it has its origin in the exterior derivative d (i.e., F(2)=dA(1)+iA(1)∧A(1)). This gauge invariant quantity should remain independent of the Grassmannian variables (θ,θ¯) as the latter are onlymathematical artifacts and they cannot be physically realized. Thus, we have the following equality due to the gauge invariant restriction (GIR):(13)-14F~MNx,θ,θ¯·F~MNx,θ,θ¯=-14Fμνx·Fμνx.The celebrated horizontality condition (HC) requires that the Grassmannian components of F~MN(x,θ,θ¯)=(F~μν,F~μθ,F~μθ¯,F~θθ,F~θθ¯,F~θ¯θ¯) should be set equal to zero so that, ultimately, we should have the following equality:(14)-14F~μνx,θ,θ¯·F~μνx,θ,θ¯=-14Fμνx·Fμνx.The requirement of HC leads to the following [4, 5, 11, 12]:(15)Rμx=DμC,R¯μx=DμC¯,s=iB¯×C,s¯=-iB×C¯,Sμ=DμB+DμC×C¯≡-DμB¯+C×DμC¯,B1=-12C×C,B¯2=-12C¯×C¯,B1+B2+C×C¯=0,where the last entry is nothing but the celebrated CF condition (B+B¯+(C×C¯)=0) if we identify B¯1=B¯ and B2=B. It is crystal clear that the HC leads to the derivation of the secondary fields in terms of the auxiliary and basic fields of the starting Lagrangian densities (1). The substitution of the above expressions for the secondary fields into the super expansion (10) leads to the following [4, 5, 11, 12]:(16)Bμhx,θ,θ¯=Aμx+θDμC¯+θ¯DμC+iθθ¯DμB+DμC×C¯≡Aμx+θsabAμ+θ¯sbAμ+θθ¯sbsabAμ,Fhx,θ,θ¯=Cx+θiB¯+θ¯-i2C×C+θθ¯-B¯×C≡Cx+θsabC+θ¯sbC+θθ¯sbsabC,F¯hx,θ,θ¯=C¯x+θ-i2C¯×C¯+θ¯iB+θθ¯B×C¯≡C¯x+θsabC¯+θ¯sbC¯+θθ¯sbsabC¯,where the superscript (h) on the superfields denotes the fact that these superfields have been obtained after the application of HC. A close look at the above expressions demonstrates that the coefficients of (θ,θ¯) are nothing but the anti-BRST and BRST transformations (2), respectively, which have been listed for the Lagrangian densities (1).Due to application of HC, ultimately, we obtain the following expression for the super curvature tensor (as we have already setF~μθ=F~μθ¯=F~θθ¯=F~θθ=F~θ¯θ¯=0):(17)F~μνhx,θ,θ¯=∂μBνh-∂νBμh+iBμh×Bνh.Substitution of the expression for Bμ(h)(x,θ,θ¯), from (16), yields(18)F~μνhx,θ,θ¯=Fμνx+θiFμν×C¯+θ¯iFμν×C+θθ¯-Fμν×C×C¯-Fμν×B≡Fμνh+θsabFμν+θ¯sbFμν+θθ¯sbsabFμν,which leads to the derivation of the (anti-)BRST symmetry transformations for Fμν (cf. (2)). It is now crystal clear that the requirements of gauge invariant restrictions in (14) and (13) are satisfied due to HC and, in this process, we have obtained the (anti-)BRST symmetry transformations forall the fields (as well as the CF condition) for our theory. We have derived these (anti-)BRST symmetry transformations by exploiting the potential of (anti)chiral superfields approach to BRST formalism in Appendix A.


## 4. Dual Horizontality Condition: Nilpotent (Anti-)Co-BRST Symmetry Transformations

We exploit here the dual-HC (DHC) to derive the (anti-)co-BRST symmetry transformations for the (anti)ghost fields and basic tenets of AVSA to obtain the precise form of the (anti-)co-BRST symmetry transformations associated with the gauge field(Aμ=Aμ·T) of our 2D non-Abelian theory. In this context, first of all, we note that the gauge-fixing term (∂μAμ) has its origin in the coexterior derivative (δ=-∗d∗) of the differential geometry in the following sense (see, e.g., [21–24] for details):(19)δA1=-∗d∗dxμAμ=∂μAμ,δ2=0,where δ=-∗d∗ is the coexterior derivative and ∗ is the Hodge duality operator on 2D Minkowskianflat spacetime manifold. It is clear that the Lorentz gauge-fixing term (∂μAμ) is a 0-form which emerges out from the 1-form (A(1)=dxμAμ) due to application of the coexterior derivative (δ=-∗d∗) which reduces the degree of a form by one.We have seen that the gauge-fixing term(∂μAμ) remains invariant under the (anti-)co-BRST symmetry transformations (cf. (6)). We generalize this observation onto our chosen (2,2)-dimensional supermanifold as follows:(20)δ~A~1=δA1,δ~=-⋆d~⋆,δ~2=0d~2=0,where δ~ is the super coexterior derivative defined on the (2,2)-dimensional supermanifold and ⋆ is the Hodge duality operator on the (2,2)-dimensional supermanifold (see, e.g., [29] for details). The left-hand side of (20) has already been computed in our previous work [29]. We quote here the result of operation of δ~ on A~(1) as 0-form; namely,(21)∂μBμ+∂θF¯+∂θ¯F+sθ¯θ¯∂θ¯F¯+sθθ∂θF=∂μAμ,where sθθ and sθ¯θ¯ appear in the following Hodge duality ⋆ operation:(22)⋆dxμ∧dxν∧dθ¯∧dθ¯=εμνsθ¯θ¯,⋆dxμ∧dxν∧dθ∧dθ=εμνsθθ.These factors (i.e., sθθ and sθ¯θ¯) are essential to get back the 4-forms ((dxμ∧dxν∧dθ∧dθ) and (dxμ∧dxν∧dθ¯∧dθ¯)) if we applyanother⋆ on (22). In other words, we have super Hodge duality ⋆ on the 0-form as follows:(23)⋆εμνsθθ=±dxμ∧dxν∧dθ∧dθ,⋆εμνsθ¯θ¯=±dxμ∧dxν∧dθ¯∧dθ¯.The equality in (21) ultimately leads to(24)∂θF=0,∂θ¯F¯=0,∂μBμ+∂θF¯+∂θ¯F=∂μAμbecause of the fact that there areno terms carrying the factors sθθ and sθ¯θ¯ on the right-hand side.At this stage, we substitute the expressions ofBμ(x,θ,θ¯), F(x,θ,θ¯), and F¯(x,θ,θ¯) into (24) to derive the following important relationships:(25)∂μRμ=0,∂μR¯μ=0,∂μSμ=0,s=0,B¯1=0,B2=0,s¯=0,B1+B¯2=0.The last entry, in the above, is just like the CF-type restriction which istrivial. With the choices B1=-B and B¯2=B, we obtain the following expansions:(26)Fdhx,θ,θ¯=Cx+θ¯-iB≡Cx+θ¯sdC,F¯dhx,θ,θ¯=C¯x+θiB≡C¯x+θ¯sadC¯,where the superscript (dh) denotes the expansions of the superfields after the application of DHC. It is self-evident that we have already obtained the (anti-)co-BRST symmetry transformation (4) for the (anti)ghost fields (C¯)C of our theory as(27)sdC=-iB,sadC=0,sdC¯=0,sadC=iB.Thus, the DHC leads to the derivation of (anti-)co-BRST symmetry transformations for the (anti)ghost fields and very useful restrictions on the secondary fields in (25).We are now in the position to derive the (anti-)co-BRST symmetry transformationss(a)d for the gauge field Aμ. We exploit here the idea of AVSA to BRST formalism, which states that the (anti-)co-BRST invariant quantities should be independent of the “soul” coordinates   (θ,θ¯). During the early days of the developments of superspace technique, the bosonic coordinates xμ of the superspace coordinates ZM=(xμ,θ,θ¯) were called the “body” coordinates and the Grassmannian variables (θ,θ¯) were christened as the “soul” coordinates. In this context, we observe that the following is true:(28)sadεμνAν·∂μB-i∂μC¯·∂μC=0.Thus, we have the following equality due to AVSA to BRST formalism:(29)εμνBνx,θ,θ¯·∂μBx-i∂μF¯dhx,θ,θ¯·∂μFdhx,θ,θ¯≡εμνAνx·∂μBx-i∂μC¯x·∂μCx.The substitution of the expansions from (26) yields the following:(30)εμνR¯ν+∂μC=0,εμνRν+∂μC¯=0,εμνSν-∂μB=0.It is worthwhile to point out that we havenot taken any super expansion of B(x) on the left-hand side in (29) because of the fact that s(a)dB(x)=0. In other words, we have taken B(x)→B~(x,θ,θ¯)=B(x). Ultimately, the relation in (30) produces the following:(31)Rμ=-εμν∂νC¯,R¯μ=-εμν∂νC,Sμ=εμν∂νB.The substitution of these expressions into the super expansions of Bμ(x,θ,θ¯) leads to the following (in terms of the (anti-)co-BRST symmetry transformations (6)):(32)Bμdgx,θ,θ¯=Aμx+θεμν∂νC+θ¯-εμν∂νC¯+θθ¯+εμν∂νB≡Aμx+θsadAμ+θ¯sdAμ+θθ¯sdsadAμ.Here the superscript (dg) on Bμ(x,θ,θ¯) denotes the expansion that has been obtained after the application of (anti-)co-BRST (i.e., dual gauge) invariant restriction (29). We end this section with the remark that we have obtainedall the (anti-)co-BRST symmetry transformations for our 2D non-Abelian 1-form gauge theory by exploiting the theoretical strength of DHC and basic tenets of AVSA to BRST formalism.


## 5. Nilpotency and Absolute Anticommutativity of the Fermionic Charges: Ordinary 2D Spacetime

We, first of all, capture the nilpotency and absolute anticommutativity of the (anti-)BRST and (anti-)co-BRST charges in theordinary space where the concepts/ideas behind the continuous symmetry and their generators (as well as the nilpotency of the (anti-)BRST and (anti-)co-BRST symmetry transformations) play very important roles. We would like to lay stress on the fact that some of the key results of our present section have been obtained due to our knowledge of the AVSA to BRST formalism that is contained in Section 6. Towards this goal in mind, we observe a few aspects of the conserved charges (listed in (4) and (8)) corresponding to the (anti-)BRST and (anti-)co-BRST symmetries of the Lagrangian densities (1). Using the (anti-)BRST and (anti-)co-BRST symmetry transformations of (2) and (6), we observe that the following are true:(33)Qb=sb∫dxB·A0+iC¯˙·C,Qab=sab∫dxiC¯·C˙-B¯·A0,Qd=sd∫dxB·A1+B·A0,Qad=sad∫dxB·A1-B¯·A0.It should be noted that we have expressed the conserved charges in (4) and (8) in terms of the transformations in (2) and (6). It is elementary now to check that the nilpotency of the charges is satisfied:(34)sbQb=-iQb,Qb=0,sabQab=-iQab,Qab=0,sdQd=-iQd,Qd=0,sadQad=-iQad,Qad=0,due to the nilpotency properties of (anti-)BRST and (anti-)co-BRST symmetry transformations (i.e., s(a)b2=0,s(a)d2=0). In the above, we have used the basic principles behind the continuous symmetries and symmetry generators (as the conserved charges of the theory). Wealso point out that we have taken into accountone of the expressions for Q(a)b and Q(a)d from (4) and (8) which have been explicitly derived in Section 2.To prove the absolute anticommutativity properties of the (anti-)BRST and (anti-) co-BRST conserved charges, we note the following useful relationships:(35)Qd=sad∫dx-iC¯·C¯˙+C¯2·A0×C¯,Qad=sd∫dxiC·C˙-C2·A0×C,Qb=sab∫dxiC·C˙-C2·A0×C,Qab=sb∫dx-iC¯·C¯˙+C¯2·A0×C¯, which establish the absolute anticommutativity properties of the (anti-)co-BRST and nilpotent (anti-)BRST charges as follows:(36)sadQd=-iQd,Qad=0,sdQad=-iQad,Qd=0,sabQb=-iQb,Qab=0,sbQab=-iQab,Qb=0,due to, once again, the nilpotency (s(a)b2=0,s(a)d2=0) properties of the (anti-)BRST and (anti-)co-BRST symmetry transformations. It is interesting to point out that the expressions in the square brackets for the pair (Qb,Qad) and the pair (Qd,Qab) areexactly thesame (as is evident from (35)). We would like to make a few remarks at this stage. A close look at (35) and (36) establishes one of the key observations that the nilpotency of symmetriesand absolute anticommutativity properties of the conserved (anti-)BRST and (anti-)co-BRST charges are interrelated. Furthermore, we would like to mention that, in the expressions for Q(a)d in (35), we have dropped total space derivative terms in our computations. It is very important to emphasize here that, in the expressions for Q(a)b (cf. (4)), we have utilized the strength of CF condition (B+B¯+(C×C¯)=0) to recast these expressions in asuitable form before expressing them in the form in (35). To elaborate on it, we take a simple example where the expression for the BRST charges Qb (cf. (4)), emerging from the Noether conserved current, is(37)Qb=∫dxB·D0C-B˙·C-C¯˙2·C×C. Using the CF condition B+B¯+(C×C¯)=0 (associated with the (anti-)BRST symmetries), we can recast the above expression in the following suitable form:(38)Qb=∫dxB¯˙·C-B¯·D0C-C×C¯·D0C+C¯˙2·C×C+C˙×C¯·C≡∫dxB¯˙·C-B¯·D0C+C¯˙2·C×C-iC×C¯·A0×C. The above form of the BRST charge has been expressed in the anti-BRSTexact form as given in (35). A similar kind of argument has gone into the expression for the anti-BRST charge Qab (cf. (35)), where we have been able to express it as the BRSTexact form. No such kinds of arguments have been invoked in the cases of the (anti-)co-BRST charges (cf. (35)) which have been expressed as the co-BRSTexact and anti-co-BRSTexact forms.We have modified the Lagrangian densities (1) in our earlier works [14, 16] by incorporating a couple of fermionic Lagrange multiplier fields (λ and λ¯ with λ2=λ¯2=0 and λλ¯+λ¯λ=0) in such a manner that the modified Lagrangian densities [14, 16],(39)LBλ¯=B·E-12B·B+B·∂μAμ+12B·B+B¯·B¯-i∂μC¯·DμC+λ¯·B×C,LB¯λ=B·E-12B·B-B¯·∂μAμ+12B·B+B¯·B¯-iDμC¯·∂μC+λ·B×C¯, respect the followingperfect (anti-)co-BRST symmetries transformations:(40)sadAμ=-εμν∂νC,sadC=0,sadC¯=iB,sadB=0,sadE=Dμ∂μC,sad∂μAμ=0,sadλ=-i∂μAμ,sadλ¯=0,sdAμ=-εμν∂νC¯,sdC¯=0,sdC=-iB,sdB=0,sdE=Dμ∂μC¯,sd∂μAμ=0,sdλ¯=-i∂μAμ,sdλ=0. It can be checked that the above (anti-)co-BRST symmetry transformations are off-shell nilpotent and absolutely anticommuting in nature (where we do not invoke any kinds of CF-type restrictions for its validity). Wealso note that the superscripts (λ) and (λ¯) on the Lagrangian densities are logically correct because the Lagrange multipliers λ and λ¯ characterize these Lagrangian densities. Furthermore, we observe that these Lagrange multiplier fields carry the ghost numbers equal to (+1) and (−1), respectively. Finally, it can be explicitly checked that the following are true:(41)sdLBλ¯=∂μB·∂μC¯,sadLB¯λ=∂μB·∂μC,sdLB¯λ=∂μB·DμC¯-εμν∂νC¯×C¯·C,sadLBλ¯=∂μB·DμC+εμνC¯·∂νC×C,which demonstrate that the action integrals S=∫d2xLB(λ¯) and S=∫d2xLB¯(λ) remain invariant under the (anti-)co-BRST symmetry transformations. We would like to lay emphasis on the fact thatboth the Lagrangian densities LB(λ¯) and LB¯(λ) respectboththe co-BRST and anti-co-BRST symmetries (cf. (40)) separately and independently.A close look at the transformations (41) and (8) (cf. Section 2) demonstrates that the expressions for the charges Qd(λ¯)=Qd and Qad(λ)=Qad (cf. (8)) remain thesame as far as the Lagrangian densities in (1)andLB(λ¯) as well as LB¯(λ) are concerned. However, we note that the anti-co-BRST charge Qad(λ¯) (derived from the Lagrangian density LB(λ¯)) and co-BRST charge Qd(λ) (derived from the Lagrangian density LB¯(λ)) would be different from (8). These conserved charges and their expressions have been derived in our earlier work (see, e.g., [16] for details). We quote here these expressions explicitly:(42)Qadλ¯=∫dxB·C˙-∂1B·C+C¯·∂1C×C≡∫dxB·C˙-D0B·C+∂1C¯×C·C+C¯·∂1C×C≡∫dxB·C˙-D0B·C-C¯·∂1C×C,Qdλ=∫dxB·C¯˙+∂1B¯·C¯-∂1C¯×C¯·C≡∫dxB·C¯˙-D0B·C¯-C¯×∂1C·C¯-∂1C¯×C¯·C≡∫dxB·C¯˙-D0B·C¯+C¯×∂1C¯·C.In the above equivalent expressions, we have utilized the equations of motion (derived from the Lagrangian densities in (39)) and we havealso dropped the total space derivative terms. To prove the nilpotency [(Qad(λ¯))2=0,(Qd(λ))2=0] of the above charges, we note that they can be expressed in terms of the (anti-)co-BRST transformations as(43)Qadλ¯=sad∫dx-iC¯·D0C+iC¯˙·C,Qdλ=sd∫dxiC¯·C˙-iD0C¯·C. The above expressions for the (anti-)co-BRST charges produce the last entry in the expressions for the charges Qad(λ¯) and Qd(λ) in (42). It can be now trivially checked that(44)sadQadλ¯=-iQadλ¯,Qadλ¯=0⟺sad2=0,sdQdλ=-iQdλ,Qdλ=0⟺sd2=0.Thus, we observe that the nilpotency of the charges Qad(λ¯) and Qd(λ) is deeply connected with the nilpotency of the (anti-)-co-BRST symmetries (i.e., s(a)d2=0) when we exploit the beauty and strength of the connection between the continuous symmetries and their corresponding generators. We would like to state that the nilpotency of the charges Qd(λ¯)=Qd (cf. (8) and (9)) and Qad(λ)=Qad has already been proven in (9). This happens because of the fact that the expressions for Qad(λ) and Qd(λ¯) are the same as given in (8) for the Lagrangian densities (1). Thus, we have proven the nilpotency ofall the charges derived from the modified Lagrangian densities (39), where λ and λ¯ are present.We now focus on the proof of the property of absolute anticommutativity of the chargesQad(λ¯) and Qd(λ), which are nontrivial (cf. (42)). In this connection, we would like to point out that the absolute anticommutativity of the charges Qd(λ¯)=Qd and Qad(λ)=Qad has already been proven in our present section itself. We note that the following are true:(45)Qdλ=sad∫dx-iC¯·C¯˙+C¯2·A0×C¯,Qadλ¯=sd∫dxiC·C˙-C2·A0×C. The above expressions demonstrate that the absolute anticommutativity property of the (anti-)co-BRST charges (i.e., {Qd(λ),Qad(λ¯)}=0) istrue and this property is primarily connected with the off-shell nilpotency (s(a)d2=0) of the (anti-)co-BRST symmetry transformations (s(a)d) that are present in our 2D non-Abelian theory (cf. (40)). To corroborate the above statements, it is straightforward to note that(46)sadQdλ=-iQdλ,Qadλ¯=0⟺sad2=0,sdQadλ¯=-iQadλ¯,Qdλ=0⟺sd2=0. From the above relationships, it is crystal clear that the absolute anticommutativity (i.e., {Qd(λ),Qad(λ¯)}=0) for the (anti-)co-BRST charges is deeply connected with the nilpotency (s(a)d2=0) property of the (anti-)co-BRST symmetry transformations (s(a)d) for the Lagrangian densities (39). We wrap up this section with the remark that we have proven the nilpotency and absolute anticommutativity properties of the (anti-)co-BRST charges for the Lagrangian densities (1) as well as (39) where we donot invoke any kinds of CF-type restrictions. This observation isnovel and drastically different from the proof of the absolute anticommutativity property of the conserved and nilpotent (anti-)BRST charges where it is mandatory for us to invoke the CF condition.


## 6. Nilpotency and Absolute Anticommutativity of the Fermionic Charges: Superfield Approach

We express here the properties of nilpotency and absolute anticommutativity by exploiting the geometrical AVSA to BRST formalism. In this connection, first of all, we recall that the (anti-)BRST symmetry transformationss(a)b have been shown to be connected with the translational generators (∂θ,∂θ¯) along (θ,θ¯)-directions of the (2,2)-dimensional supermanifold through the following mappings:(47)sb⟷∂∂θ¯θ=0,sab⟷∂∂θθ¯=0.We can very well choose the Grassmannian variables to be (θ1,θ2) and identify the nilpotent symmetries, sb↔∂θ1|θ2=0 and sab↔∂θ2|θ1=0, because there are other nilpotent (s(a)d2=0) symmetries s(a)d in our theory, too. Thelatter nilpotent symmetries could be identified with translational generators as sd↔∂θ3|θ4=0 and sad↔∂θ4|θ3=0, where we shall have another set of a pair of Grassmannian variables (θ3,θ4). However, for the sake of brevity, we have chosen only (θ,θ¯) as the Grassmannian variables so that we could discuss the (anti-)BRST and (anti-)co-BRST symmetries separately and independently. The above mappings imply that the nilpotency of the (anti-)BRST symmetries (i.e., s(a)b2=0) is intimately connected with the nilpotency (∂θ2=∂θ¯2=0) of the translational generators (∂θ,∂θ¯). This observation is utilized in expressing the expressions for the conserved and nilpotent (anti-)BRST charges in (33) as follows:(48)Qab=∂∂θ∫dxiF¯hx,θ,θ¯·F˙hx,θ,θ¯-B¯x·B0hx,θ,θ¯θ¯=0≡∫dθ∫dxiF¯hx,θ,θ¯·F˙hx,θ,θ¯-B¯x·B0hx,θ,θ¯θ¯=0,Qb=∂∂θ¯∫dxBx·B0hx,θ,θ¯+iF¯˙hx,θ,θ¯·Fhx,θ,θ¯θ=0≡∫dθ¯∫dxBx·B0hx,θ,θ¯+iF¯˙hx,θ,θ¯·Fhx,θ,θ¯θ=0.The above expressions establish the nilpotency of the (anti-)BRST charges Q(a)b because(49)∂θQab=0⟺∂θ2=0⟺sabQab=-iQab,Qab=0,∂θ¯Qb=0⟺∂θ¯2=0⟺sbQb=-iQb,Qb=0.It should be noted that we donot invoke any kinds of CF-type restrictions for the proof of off-shell nilpotency of the above (anti-)BRST charges.We capture now the absolute anticommutativity property of the (anti-)BRST symmetry generatorsQ(a)b in the language of the AVSA to BRST formalism. In this context, we concentrate on the expressions for (anti-)BRST charges that have been quoted in (35). It can be checked that we have the following expressions for these charges in the language of the AVSA to BRST formalism:(50)Qab=∂∂θ¯∫dx-iF¯hx,θ,θ¯·F¯˙hx,θ,θ¯+12F¯hx,θ,θ¯·B0hx,θ,θ¯×F¯hx,θ,θ¯θ=0=∫dθ¯∫dx-iF¯hx,θ,θ¯·F¯˙hx,θ,θ¯+12F¯hx,θ,θ¯·B0hx,θ,θ¯×F¯hx,θ,θ¯θ=0,Qb=∂∂θ∫dxiFhx,θ,θ¯·F˙hx,θ,θ¯-12Fhx,θ,θ¯·B0hx,θ,θ¯×Fhx,θ,θ¯θ¯=0=∫dθ∫dxiFhx,θ,θ¯·F˙hx,θ,θ¯-12Fhx,θ,θ¯·B0hx,θ,θ¯×Fhx,θ,θ¯θ¯=0.It is straightforward to note that the nilpotency properties of the translational generators (∂θ,∂θ¯) along the Grassmannian directions imply that(51)∂θ¯Qab=0⟺∂θ¯2=0,∂θQb=0⟺∂θ2=0.The above observations lead us to draw the conclusion that the absolute anticommutativity (QbQab+QabQb=0) of the (anti-)BRST charges (cf. (36)) in the ordinary space can be captured in the language of the superfield approach to BRST formalism.We briefly comment here on the expressions for the (anti-)co-BRST chargesQ(a)d that have been expressed intwo different ways in (33) and (35). We have established earlier that the following mappings are true in the cases of sd and sad:(52)sd⟷limθ=0∂∂θ¯,sad⟷limθ¯=0∂∂θ.We can very well repeat here the previous footnote written in our manuscript. However, this would beonly an academic exercise. The main issue is the fact that we discuss the (anti-)BRST and (anti-)co-BRST symmetries, within the framework of AVSA to BRST formalism, separately and independently. Thus, when we focus on (θ1,θ2), we do not bother about (θ3,θ4) and vice versa. This is precisely the reason why we have taken, for the sake of brevity,only the (2,2)-dimensional supermanifold for our discussion, where, at a time, only a pair of Grassmannian variables are taken into account. Thus, the nilpotency of the (anti-)co-BRST charges can be expressed in terms of the quantities on the (2,2)-dimensional supermanifold as follows:(53)Qd=∂∂θ¯∫dxBx·B1dgx,θ,θ¯+Bx·B0dgx,θ,θ¯θ=0≡∫dθ¯∫dxBx·B1dgx,θ,θ¯+Bx·B0dgx,θ,θ¯θ=0,where the superscript (dg) denotes the superfields (cf. (32)) that have been obtained after the application of (anti-)co-BRST invariant restriction in (29). Similarly, we note that the following is correct:(54)Qad=∂∂θ∫dxBx·B1dgx,θ,θ¯-B¯x·B0dgx,θ,θ¯θ¯=0≡∫dθ∫dxBx·B1dgx,θ,θ¯-B¯x·B0dgx,θ,θ¯θ¯=0. It is crystal clear, from (53) and (54), that the following are true:(55)∂θ¯Qd=0⟺∂θ¯2=0,∂θQad=0⟺∂θ2=0.The above relationships, in the ordinary 2D space, correspond to the following explicit expressions in the language of anticommutators:(56)sdQd=-iQd,Qd=0⟺Qd2=0⟺sd2=0,sadQad=-iQad,Qad=0⟺Qad2=0⟺sad2=0.Thus, we have captured the nilpotency property of the (anti-)co-BRST charges in the language of the quantities that are defined on the (2,2)-dimensional supermanifold. In fact, the nilpotency (Q(a)d2=0) of the (anti-)co-BRST charges is deeply connected with the nilpotency (∂θ2=0,∂θ¯2=0) of the translational generators (∂θ,∂θ¯) along the Grassmannian directions (θ,θ¯) of the (2,2)-dimensional supermanifold.Now we dwell a bit on the absolute anticommutativity property of the (anti-)co-BRST chargesQ(a)d that have been expressed in (35). Taking the inputs from (52), (32), and (26), we have the following:(57)Qad=∂∂θ¯∫dxiFdhx,θ,θ¯·F˙dhx,θ,θ¯-12Fdhx,θ,θ¯·B0dgx,θ,θ¯×Fdhx,θ,θ¯θ=0≡∫dθ¯∫dxiFdhx,θ,θ¯·F˙dhx,θ,θ¯-12Fdhx,θ,θ¯·B0dgx,θ,θ¯×Fdhx,θ,θ¯θ=0,Qd=∂∂θ∫dx-iF¯dhx,θ,θ¯·F¯˙dhx,θ,θ¯+12F¯dhx,θ,θ¯·B0dgx,θ,θ¯×F¯dhx,θ,θ¯θ¯=0≡∫dθ∫dx∫dx-iF¯dhx,θ,θ¯·F¯˙dhx,θ,θ¯+12F¯dhx,θ,θ¯·B0dgx,θ,θ¯×F¯dhx,θ,θ¯θ¯=0.Thus, the expressions for the (anti-)co-BRST charges (cf. (57)) imply that(58)∂θQd=0⟺∂θ2=0⟺sadQd=-iQd,Qad=0,∂θ¯Qad=0⟺∂θ¯2=0⟺sdQad=-iQad,Qd=0.The above expressions capture the absolute anticommutativity property of the (anti-)co-BRST charge (i.e., {Qd,Qad}=0) in the language of AVSA to BRST formalism. We observe, once again, that it is the nilpotency (∂θ2=∂θ¯2=0) of the translational generators (∂θ,∂θ¯) that plays a decisive role in capturing the nilpotency as well as absolute anticommutativity properties of the (anti-)co-BRST charges in the terminology of AVSA to BRST formalism.Finally, we would like to comment briefly on the nilpotency and absolute anticommutativity properties of the (anti-)co-BRST charges(Qd(λ),Qad(λ¯)) that have been derived from the Lagrangian densities (39) and listed in (42) in different forms. We would like to lay emphasis on the fact that the Lagrangian densities (39) are veryspecial in the sense that these Lagrangian densities respect proper (anti-)co-BRST symmetry transformations (listed in (40)) separately and independently (cf. (41)), where we donot invoke any kinds of CF-type restrictions from outside. Thus, as far as symmetry considerations are concerned, these Lagrangian densities are really beautiful from the point of view of the proper (anti-)co-BRST symmetry transformations (41). Within the framework of AVSA to BRST formalism, it can be checked that the expressions in (43) are(59)Qadλ¯=∂∂θ∫dx-iF¯dhx,θ,θ¯·F˙dhx,θ,θ¯+F¯dhx,θ,θ¯·B0dgx,θ,θ¯×Fdhx,θ,θ¯+iF¯dhx,θ,θ¯·Fdhx,θ,θ¯θ¯=0≡∫dθ∫dx-iF¯dhx,θ,θ¯·F˙dhx,θ,θ¯+F¯dhx,θ,θ¯·B0dgx,θ,θ¯×Fdhx,θ,θ¯+iF¯dhx,θ,θ¯·Fdhx,θ,θ¯θ¯=0,Qdλ=∂∂θ¯∫dxiF¯dhx,θ,θ¯·F˙dhx,θ,θ¯-F¯˙dhx,θ,θ¯·Fdhx,θ,θ¯+B0dgx,θ,θ¯×F¯dhx,θ,θ¯·Fdhx,θ,θ¯θ=0≡∫dθ¯∫dxiF¯dhx,θ,θ¯·F˙dhx,θ,θ¯-F¯˙dhx,θ,θ¯·Fdhx,θ,θ¯+B0dgx,θ,θ¯×F¯dhx,θ,θ¯·Fdhx,θ,θ¯θ=0,where the superfields with superscripts (dh) and (dg) have been explained in Section 4. It is clear, from the above expressions, that we have the following:(60)∂θQadλ¯=0⟺∂θ2=0⟺sadQadλ¯=-iQadλ¯,Qadλ¯=0,∂θ¯Qdλ=0⟺∂θ¯2=0⟺sdQdλ¯=-iQdλ,Qdλ=0.The above relations prove the nilpotency of Qd(λ) and Qad(λ¯), which is also connected with the nilpotency of the translational generators (∂θ,∂θ¯) along the Grassmannian directions (θ,θ¯) of the (2,2)-dimensional supermanifold on which our 2Dordinary theory is considered. To be more precise, the nilpotency of the above (anti-)co-BRST charges (which have been derived from the Lagrangian densities (39)) becomes very transparent when concentrating on thethird andsixth lines in (60). In particular, the anticommutator of the conserved charges withthemselves being zero immediately implies the nilpotency property (of these conserved charges). Let us now concentrate on the forms of the (anti-)co-BRST charges that have been written in (45). As is evident from (46), the absolute anticommutativity property of the (anti-)co-BRST charges is primarily hidden in (45) and is deeply connected with the nilpotency property of the (anti-)co-BRST symmetry transformations (s(a)d). Thus, we express the forms of the (anti-)co-BRST charges (45) in the language of AVSA to BRST formalism as follows:(61)Qadλ¯=∂∂θ¯∫dxiFdhx,θ,θ¯·F˙dhx,θ,θ¯-12Fdhx,θ,θ¯·B0dgx,θ,θ¯×Fdhx,θ,θ¯θ=0≡∫dθ¯∫dxiFdhx,θ,θ¯·F˙dhx,θ,θ¯-12Fdhx,θ,θ¯·B0dgx,θ,θ¯×Fdhx,θ,θ¯θ=0,Qdλ=∂∂θ∫dx-iF¯dhx,θ,θ¯·F¯˙dhx,θ,θ¯+12F¯dhx,θ,θ¯·B0dgx,θ,θ¯×F¯dhx,θ,θ¯θ¯=0≡∫dθ∫dx-iF¯dhx,θ,θ¯·F¯˙dhx,θ,θ¯+12F¯dhx,θ,θ¯·B0dgx,θ,θ¯×F¯dhx,θ,θ¯θ¯=0,where the superfields with superscripts (dh) and (dg) have already been explained in Section 4. It is straightforward to note, from the above equation, that(62)∂θQdλ=0⟺∂θ2=0⟺sadQdλ=-iQdλ,Qadλ¯=0,∂θ¯Qadλ¯=0⟺∂θ¯2=0⟺sdQadλ¯=-iQadλ¯,Qdλ=0.We end this section with the remark that the absolute anticommutativity property of the (anti-)co-BRST charges is deeply connected with the nilpotency property (∂θ2=∂θ¯2=0) of the translational generators (∂θ,∂θ¯) along the Grassmannian directions (θ,θ¯) of the (2, 2)-dimensional supermanifold on which our 2D ordinary non-Abelian theory is generalized.


## 7. Conclusions

We have exploited the theoretical strength of the AVSA to BRST formalism to express the properties of the nilpotency and absolute anticommutativity of the fermionic conserved charges (i.e., (anti-)BRST and (anti-)co-BRST charges) of our self-interacting 2D non-Abelian theory (without any interaction with matter fields). We havenot achieved this goal in our earlier works [9–12] on the AVSA to BRST formalism. Thus, the results in our present investigation are achieved for the first time. It is straightforward to express the nilpotency property of the fermionic (i.e., conserved (anti-)BRST and (anti-)co-BRST) charges in the language of the AVSA to BRST formalism. However, the property of absolute anticommutativity is captured, within the framework of AVSA to BRST formalism, by applying specific mathematical trick, where the CF condition plays a decisive role.We would like to lay emphasis on the contents of Section5, where we have been able to exploit the virtues of symmetry principles to express the (anti-)BRST and (anti-)co-BRST charges in variousexact forms. These theoretical expressions have been exploited, in turn, to capture the nilpotency and absolute anticommutativity properties in the language of AVSA to BRST formalism in Section 6. We observe that the CF condition (B+B¯+(C×C¯)=0) enables us to express the BRST charge as an anti-BRSTexact form and anti-BRST charge as a BRSTexact form. We would like to lay emphasis on the fact that the contents of Sections 5 and 6 are intertwined in an elegant manner. Although it appears, from our statements in this paragraph, that the contents of Section 5 have influenced our results in Section 6, we would like to stress that, many times, our understandings of the contents of Section 6 have influenced our results in Section 5. Thus, to be precise, the key results of our present endeavor are influenced by our knowledge ofboth the above sections which are interrelated. These results play an important role in establishing the absolute anticommutativity properties of the above fermionic charges. Thus, first of all, we have proven the nilpotency and absolute anticommutativity of the (anti-)BRST and (anti-)co-BRST charges in the language of symmetry properties (cf. Section 5). In particular, we have shown that it is the nilpotency of the (anti-)BRST and (anti-)co-BRST symmetry transformations that has played a decisive role in the proof of the aboveproperties in theordinary2D space of our non-Abelian 1-form gauge theory. In fact, the results of Section 5 have been translated into the language of AVSA to BRST formalism in Section 6.The proof of the nilpotency and absolute anticommutativity properties in the language of AVSA to BRST formalism for the fermionic (anti-)BRST and (anti-)co-BRST charges is anovel result because, in our earlier works on AVSA to BRST formalism [9–12], we have not achieved this goal. In our very recent works [30–32], we have captured the property of the absolute anticommutativity of nilpotent charges within the framework of (anti)chiral superfield approach to BRST formalism. However, we havenot doneso within the framework of AVSA to BRST formalism, where thefull expansions of the superfields are taken into account. We plan to exploit our present idea to consolidate it by applying it to the cases of 1D toy model of the rigid rotor, 2D self-dual bosonic theory, modified versions of 2D Proca and anomalous gauge theory, and 4D Abelian 2-form and 6D Abelian 3-form gauge theories, where we have demonstrated the existence of (anti-)BRST and (anti-)co-BRST charges, as these theories are the models for the Hodge theory [18, 30–32].



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