Universal Superspace Unitary Operator and Nilpotent (Anti-)dual BRST Symmetries: Superfield Formalism

We exploit the key concepts of the augmented version of superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism to derive the superspace (SUSP) dual unitary operator (and its Hermitian conjugate) and demonstrate their utility in the derivation of the nilpotent and absolutely anticommuting (anti-)dual BRST symmetry transformations for a set of interesting models of the Abelian 1-form gauge theories. These models are the one (0+1)-dimensional (1D) rigid rotor, modified versions of the two (1+1)-dimensional (2D) Proca as well as anomalous gauge theories and 2D model of a self-dual bosonic field theory. We show the universality of the SUSP dual unitary operator and its Hermitian conjugate in the cases of all the Abelian models under consideration. These SUSP dual unitary operators, besides maintaining the explicit group structure, provide the alternatives to the dual-horizontality condition (DHC) and dual-gauge invariant restrictions (DGIRs) of the superfield formalism. The derivation of the dual unitary operators and corresponding (anti-)dual BRST symmetries are completely novel results in our present investigation.


Introduction
A classical gauge theory is endowed with the local gauge symmetries which are generated by the first-class constraints in the terminology of Dirac's prescription for the classification scheme [1,2]. Thus, one of the decisive features of a classical gauge theory is the existence of the first-class constraints on it. The above cited classical local gauge symmetries are traded with the quantum gauge [i.e. (anti-)BRST] symmetries within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. The existence of the Curci-Ferrari (CF) condition(s) [3] is one of the key signatures of a quantum gauge theory when it is BRST quantized. The geometrical superfield approach [4][5][6][7][8][9][10] to BRST formalism is one of the most elegant methods which leads to the derivation of the nilpotent and absolutely anticommuting (anti-)BRST transformations for a given D-dimensional gauge theory. In addition, this usual superfield formalism [6][7][8] also leads to the deduction of the (anti-)BRST invariant CF-conditions (which are the key signature of the quantum gauge theories). Thus, we observe that, in one stroke, the usual superfield formalism (USF) produces the CF-type condition(s) as well as the proper quantum gauge [i.e., (anti-)BRST] symmetries for a quantum gauge theory. It is, therefore, clear that the USF sheds light on various aspects of quantum gauge theories when they are discussed within the framework of BRST formalism.
The USF [4][5][6][7][8][9][10], however, leads to the derivation of nilpotent (anti-)BRST symmetry transformations only for the gauge field and associated (anti-)ghost fields of a given quantum gauge theory. It does not shed any light on the derivation of the (anti-)BRST symmetry transformations, associated with the matter fields, in a given interacting quantum gauge theory where there is a coupling between the gauge field and matter fields. In a set of papers [11][12][13][14][15], the above superfield formalism has been consistently extended so as to derive precisely the (anti-)BRST symmetry transfromations for the gauge, matter and (anti-)ghost fields together. Whereas the usual superfield formalism exploits the theoretical potential and power of the horizontality condition (HC), its extended version utilizes the theoretical strength of the HC as well as the gauge invariant restrictions (GIRs) together in a consistent manner. The extended version of the USF has been christened [11][12][13][14][15] as the augmented version of superfield formalism (AVSF). One of the key observations of the applications of USF and AVSF is the fact that the group structure of the (non-)Abelian gauge theories remains somewhat hidden but the geometry of these theories becomes quite explicit as we take the help of the cohomological operators of differential geometry.
The purpose of our present investigation is to exploit the theoretical strength of AVSF to derive the superspace dual unitary operators for the 1D and 2D interesting models of the Abelian 1-form gauge theories corresponding to the (anti-)dual BRST [i.e., (anti-)co-BRST] symmetry transformations which have been shown to exist for the above models. These models are the 1D rigid rotor, modified versions of the 2D Proca as well as anomalous gauge theories and 2D self-dual bosonic field theory. In fact, these models have been shown to provide the physical examples of Hodge theory within the framework of BRST formalsm where the (anti-)BRST as well as (anti-)co-BRST symmetries exist together with a unique bosonic symmetry and the ghost-scale symmetry [16][17][18][19][20]. The universal superspace (SUSP) unitary operators, corresponding to the nilpotent (anti-)BRST symmetry transformations, have already been shown to exist for the above models (see, e.g. [21] for details). The central theme of our present investigation is to derive the SUSP dual unitary operators (from the above universal unitary operators). The derivation of the unitary SUSP operators is important because the group structure of the theory is maintained and it remains explicit throughout the whole discussion within the framework of AVSF. The forms of this SUSP unitary operators were first suggested in an earlier work on the superfield approach to the non-Abelian 1-form gauge theory [7]. These expressions, however, were intuitively chosen but not derived theoretically. Moreover, the Hermitian conjugate unitary operator, corresponding to the chosen SUSP unitary operator, was derived after imposing some outside conditions on the fields and Grassmannian variables (of the SUSP unitary operator).
In our present investigation, we have derived the dual SUSP unitary operators (i.e. SUSP dual unitary operator and its Hermitian conjugate) which provide the alternatives to the dual horizontality condition (DHC) and dual gauge invariant restrictions (DGIRs). This derivation is a completely new result because it leads to the derivation of the nilpotent and absolutely anticommutating (anti-)co-BRST symmetry transformations which have been derived earlier within the framework of superfield approach where the DHC and DGIRs have played some decisive roles [22][23][24][25]. In fact, we have obtained the proper dual SUSP unitary operator (and its Hermitian conjugate) from the universal unitary operators that have been derived in our earlier works [26,27] on interacting gauge theories. To be specific, we have already derived the explicit form of the SUSP unitary operator (and its Hermitian conjugate) in the 4D interacting Abelian 1-form gauge theory with Dirac and complex scalar fields [26] as well as 4D non-Abelian gauge theory with Dirac fields [27]. In our present investigation, we have obtained the dual SUSP unitary operator (and its Hermitian conjugate) from the duality operation on the universal SUSP unitary operators (that have already been derived in our earlier work [26] for the 4D interacting Abelian theory). The form of the SUSP dual unitary operator (and its Hermitian conjugate) turns out to be universal for all the Abelian 1-form gauge models under consideration which are defined on the one and two dimensional Minkowskian flat spacetime manifold.
Our present investigation is essential on the following key considerations. First and foremost, as we have shown the universality of the SUSP unitary operator (and its Hermitian conjugate) in the context of the models under consideration for the derivations of the (anti-)BRST symmetries, similarly, we have to derive the universal SUSP dual unitary operator (and its Hermitian conjugate) for the (anti-)co-BRST transformations for the sake of completeness. We have accomplished this goal in present investigation. Second, the existence of the SUSP dual unitary operator (and its Hermitian conjugate) provides the alternatives to the DHC and DGIRs that have been invoked in the derivation of the nilpotent (anti-)co-BRST symmetry transformations within in the framework of AVSF. One of the highlights of our present investigation is the observation that the SUSP dual unitary operator and its Hermitian conjugate turn out to be universal for all the Abelian models that have been considered in our present endeavor. Third, the Abelian 1-form theories (that have been considered here) are intresting because these have been shown to provide the physical examples of Hodge theory. Fourth, we have found out the (anti-)co-BRST symmetry transformation for a new model which has not been considered in our earlier works on the superfield approach to BRST formalism [22][23][24][25]. We have obtained, for the first time, the (anti-)co-BRST symmetry transformations for the modified version of the 2D anomalous gauge theory. Thus, it is a novel result in our present endeavor. Finally, our present attempt is our modest first-step towards our central goal of establishing that these SUSP dual unitary operators are universal even in the case of non-Abelian theories.
The contents of our present investigation are organized as follows. In Sec. 2, we briefly discuss the (anti-)dual BRST symmetry transformations in the Lagrangian fromulation for the 1D rigid rotor, modified versions of the 2D Proca as well as anomalous gauge theories and 2D self-dual bosonic field theory. We exploit the theoretical strength of the DHC and DGIRs to derive the above nilpotent symmetries within the framework of superfield formalism in Sec. 3. Our Sec. 4 deals with the derivation of the above nilpotent symmetries by using the SUSP dual unitary operators. In Sec. 5, we summarize our key results and point out a few future directions for further investigation.

Preliminaries: (Anti-)dual-BRST Symmetries
To begin with, we discuss here the nilpotent (s 2 (a)d = 0) and absolutely anticommuting (s d s ad + s ad s d = 0) (anti-)dual BRST symmetries s (a)d in the Lagrangian formulation for the 1D rigid rotor which is described by the following first-order Lagrangian (see, e.g. [16]) where the pair (ṙ,θ) is the generalized velocities corresponding to the generalized polar coordinates (r, ϑ) of the rigid rotor. We have taken the unit mass (m = 1) while defining the pair (p r , p ϑ ) as the conjugate momenta corresponding to the coordinates (r, ϑ). Here λ(t) is the "gauge" variable of the theory (which is a 1-form λ (1) = dt λ(t) on a 1D manifold) and B(t) is the Nakanishi-Lautrup type auxiliary variable. The anticommuting (C(t)C(t) +C(t) C(t) = 0) fermionic (C 2 =C 2 = 0) (anti-)ghost variables (C)C are required to maintain the unitarity in the theory. All the variables are the function of an evolution parameter t and an overdot (ṙ,θ,λ,Ċ,Ċ, etc.) corresponds to a single derivative (i.e.ṙ = dr/dt,v = dv/dt, etc.) with respect to t. It can be readily checked [16] that, under the following (anti-)dual BRST symmetry transformations (s (a)d ) the Lagrangian (1) and gauge-fixing term remain invariant (s (a)d L B = 0, s (a)d (λ − p r ) = 0). We now focus our attention on the (anti-)dual BRST symmetry transformations for the modified version of 2D Proca theory (with mass parameter m) which is described by the following (anti-)BRST invariant Lagrangian density (see, e.g. [17,18] for details) where the 1-form A (1) = dx µ A µ defines the 2D gauge potential A µ and the corresponding curvature tensor F µν is defined from the 2-form is the exterior derivative. In 2D, the curvature tensor F µν contains only one independent component which is nothing but the electric field E. The latter turns out to be a pseudoscalar in two (1 + 1)-dimensions of spacetime. In the above, we have a pair (φ,φ) of fields which is constructed by a scalar Stueckelberg field φ and a pseudoscalar fieldφ. The latter has been introduced in the theory on the physical as well as mathematical grounds [17,18]. The fermionic (C 2 =C 2 = 0, CC +C C = 0) fields are the (anti-)ghost fields (C) C which are required to maintain the unitarity in the theory. It can be readily checked that the following nilpotent (s 2 (a)d = 0) and absolutely anticommuting (s d s ad + s ad s d = 0) (anti-)dual BRST symmetry transformations (s (a)d ) leave the action integral invariant because the Lagrangian density transforms to the total spacetime derivatives (see, e.g. [17], [18] for details). It is to be noted that the total gaugefixing term remains invariant under s (a)d [i.e. s (a)d (∂ · A + mφ) = 0]. Another modified version of the 2D Abelian 1-form model is the bosonized version of anomalous Abelian 1-form gauge theory which is described by the following (anti-)BRST invariant Lagrangian density (see, e.g., [19] for details) where, as explained earlier, the 2-form curvature F µν has only electric field as its existing component and a is the ambiguity parameter in the regularization of the fermionic determinant when the 2D chiral Schwinger model (with electric charge e = 1) is bosonized in terms of the scalar field φ. We have introduced an extra 2D bosonic field σ(x) in the theory to convert the second-class constraints of the original 2D chiral Schwinger model into the first-class system so that we could have the "classical" gauge and "quantum" (anti-)BRST symmetries in the theory (see, e.g. [19] for details). The other symbols (C)C and B(x) have already been explained earlier. The Lagrangian density (5) can be re-expressed as which is endowed with the following (anti-)co-BRST symmetries where we have introduced an auxiliary field B(x) to linearize the kinetic term (− 1 of our modified 2D anomalous Abelian 1-form gauge theory. The symmetry invariance can be explicitly checked, by using the above transformations, where the action integral S = d 2 x L (a) B remains invariant because the above Lagrangian density transforms to the total spacetime derivatives (see, e.g. [19] for details).
Finally, we concentrate on a theoretically interesting system of the Abelian 1-form model of the 2D self-dual bosonic field theory which is described by the following (anti-)BRST invariant Lagrangian density (see, e.g. [20] for details) where an overdot on fields (e.g.v = ∂v/∂t,φ = ∂φ/∂t) corresponds to the expression for the "generalized" velocities (where a derivative with respect to the evolution parameter t is taken into account) and the prime on the fields Here v(x) field is the Wess-Zumino (WZ) field and φ(x) field is the 2D self-dual bosonic field (of our present 2D self-dual field theory).
Rest of the symbols have already been explained earlier. The above Lagrangian density (8) is endowed with the following (anti-)dual-BRST symmetry transformations (s (a)d ) because the Lagrangian density (8) transforms to the total "time" derivatives as Thus, the action integral S = d 2 x L (s) B remains invariant under s (a)d for the physical fields that vanish off at t = ±∞.
The decisive features of the (anti-)dual BRST [i.e. (anti-)co-BRST] symmetry transformations are the observations that (i) they are nilpotent of order two (i.e. s 2 (a)d = 0) which demonstrates their fermionic nature, (ii) these nilpotent symmetries are also absolutely anticommuting (s d s ad + s ad s d = 0) in nature that shows the linear independent of s d and s ad , (iii) the gauge-fixing terms, owing their origin to the co-exterior derivative (see, e.g. [16][17][18][19][20]), remain invariant under the (anti-)dual BRST symmetry transformations (s (a)d ). Thus, the nomenclature (anti-)co-BRST symmetries is appropriate for these symmetries. This observation should be contrasted with the (anti-)BRST symmetries where the total kinetic term, owing its origin to the exterior derivative, remains invariant [16][17][18][19][20].
In this connection, we note that the gauge-fixing term (λ − p r ) remains invariant under s (a)d . Furthermore, we observe that this term has its geometrical origin in the co-exterior derivative (δ) because δλ (1) = * d * (dtλ(t)) ≡λ(t) where ( * ) is the Hodge duality operation on the 1D manifold. Here we have taken the 1-form as: λ (1) = dtλ(t). According to the basic tenets of AVSF, the invariance of the gauge-fixing term implies that this quantity should remain independent of the "soul" coordinates (θ,θ) when we generalize it onto the (1, 2)-dimensional supermanifold parameterized by the superspace coordinates (t, θ,θ) where the pair (θ,θ) is a set of Grassmannian variables (with θ 2 =θ 2 = 0, θθ +θθ = 0). In older literature [28], the latter coordinates have been christened as the "soul" coordinates and t has been called as the body coordinate. In other words, we have the following equality where ⋆ is the Hodge duality operation on the (1, 2)-dimensional supermanifold on which our 1D ordinary theory is generalized. The other quantities in the equation (11) are where the supervariables Λ(t, θ,θ), F (t, θ,θ),F (t, θ,θ) and P r (t, θ,θ) have the following expansions along the (θ,θ)-directions of (1, 2)-dimensional supermanifold [25]: We note, in the above, that the secondary variables (R,R, s,s, K,K) are fermionic and (S, B 1 ,B 1 , B 2 ,B 2 , L) are bosonic in nature. It is elementary to verify that, in the limit θ =θ = 0, we get back our 1D variables (λ, C,C, p r ) that are present in the Lagrangian (1). The dual horizontality condition (DHC) [cf. (11)] leads to the following [25] The above relationships prove that some of the secondary variables are zero and others are interconnected in a definite and precise manner. It is worthwhile to mention that the condition B 1 +B 2 = 0 is the trivial CF-type condition. This restriction is a physical condition in our theory because it is an (anti-)co-BRST invariant quantity. We resort to the additional restrictions on the supervariables that are motivated by the basic requirements of AVSF which state that the (anti-)co-BRST invariant quantities should be independent of the "soul" coordinates. In this connection, we observe the following which, ultimately, imply the following equalities due to DGIRs, namely; where the new notations (with R(t, θ,θ) = r(t)) are explicitly written as In the above, we have chosen B 1 (t) = −B 2 (t) = − B and taken the inputs from (14). The conditions (15) are now supported by the observations: s d (λC) = 0 and s ad (λC) = 0. These two conditions lead to the following restrictions on the supervariables: Finally, we obtain the expressions for the secondary variables in terms of the original variables of the Lagrangian (1) as (see, e.g. [25] for details): The substitution of these values into the expansions (13) and (17) leads to the following final expressions for the expansion of the supervariables (see, e.g. [25] for details) where the superscript (d) on the supervariables denotes the expansions that have been obtained after the application of DHC and DGIRs. A careful and close look at the above expansions demonstrate that we have already obtained the non-trival (anti-)co-BRST symmetry transformations for the variables (λ, C,C, p r ) of the 1D rigid rotor. The trivial nilpotent (anti-)co-BRST symmetry transformations s (a)d [r, p θ , θ] = 0 are self-evident. It is clear that there is a geometrical meaning of s (a)d in the language of translational operators (∂ θ , ∂θ) along the Grassmannian directions (θ,θ) of the (1, 2)-dimensional supermanifold. The nilpotency (∂ θ 2 = ∂θ 2 = 0) and absolute anticommutativity (∂ θ ∂θ + ∂θ ∂ θ ) of these generators provide the geometrical meaning to the nilpotency (s 2 (a)d = 0) and absolute anticommutativity (s d s ad + s ad s d = 0) of the (anti-)co-BRST symmetries.
We now focus on the derivation of the (anti-)co-BRST symmetry transformations (s (a)d ) in the context of the modified versions of the 2D Proca and anomalous Abelian 1-form gauge theories within the framework of AVSF. In this connection, first of all, we observe that the gauge-fixing term (∂ · A ± mφ) remains invariant [i.e. s (a)d (∂ · A ± mφ) = 0] under s (a)d (because, separately and independently, we have: s (a)d (∂ · A) = 0, s (a)d φ = 0). We note that (∂ · A) has its origin in the co-exterior derivative (δ) because δA (1) = − * d * (dx µ A µ ) = (∂·A). Thus, we have to generalize this relationship onto the (2, 2)-dimensional supermanifold parametrized by the superspace co-ordinates (x µ , θ,θ). Thus, according to the basic tenets of AVSF, we have the following equality (see, e.g. [18] for details) where ⋆ is the Hodge duality operation on the (2, 2)-dimensional supermanifold and other relevant symbols have already been explained earlier. In our earlier works [18], the l. h. s. of relation (21) has been already computed clearly by taking the help of the Hodge duality operation ⋆ defined on the (2, 2)-dimensional supermanifold [22]. At this stage, we would like to clarify some of the new symbols used in equation (21). We have the generalization of the ordinary exterior derivative d = dx µ ∂ µ and Abelian 1-form A (1) = dx µ A µ onto the (2, 2)-dimensional supermanifold as where the superfields B µ (x, θ,θ), F (x, θ,θ) andF (x, θ,θ) have the following expansions along (θ,θ)-directions of the (2, 2)-dimensional supermanifold where (A µ (x), C(x),C(x)) are the basic fields of the modified versions of the 2D Proca and anomalous gauge theories. The set of secondary fields (R µ , s,s) are fermionic and (S µ , B 1 , B 2 , B 3 , B 4 ) are bosonic in nature (because of the fermionic nature of the Grassmannian variable (θ,θ)). The dual horizontality condition (21) leads to the following very useful relationships (see, e.g. [18] for details) where the relation B 2 + B 3 = 0 is like the CF-type condition which turns out to be a trivial relationship. We would like to state that the details of the equation (24) have been worked out in our earlier work on the superfield approach to the modified version of 2D Proca theory [18]. The interesting point is that the above conditions are true in the AVSF approach to the modified version of 2D anomalous gauge theory, too.
The above relations do not lead to the exact form of R µ , S µ and (B 2 , B 3 ). The CF-type condition B 2 + B 3 = 0 allows us to choose B 2 = B so that B 3 = − B. Now, we exploit the virtue of the AVSF to derive the exact forms of the secondary fields and observe that the following (anti-)co-BRST invariant quantity permits us to demand that the superfield generalization of the above quantity on the (2, 2)-dimensional supermanifold must be independent of the soul co-ordinates (θ,θ). Thus, we have the following equality In the above, the expansions for the superfields because it is clear from (24) that s =s = 0 and B 1 = B 4 = 0. The substitution of the explicit expansion of F (d) (x, θ,θ),F (d) (x, θ,θ) and B µ (x, θ,θ) into (26) leads to the following relationships when we equate the coefficients of θ,θ and θθ equal to zero, namely; leading to the final determination of the secondary fields (with the help from (24)) as Thus, we have the following explicit expansions of the superfield: It is very clear that we have derived the following (anti-)co-BRST symmetry transformations for the fields (A µ (x), C(x),C(x)) due to superfield formalism: To determine the (anti-)dual-BRST symmetry transformations for theφ(x) field of the modified version of the 2D Proca theory (cf. Eq. (3)), we observe that s (a)d A µ − 1 m ε µν ∂ νφ = 0. Thus, according to the basic requirements of AVSF, we demand that this quantity should remain independent of the "soul" coordinates, namely; Now if we taken the expansion of the superfield we obtain, from (32), the relationships which show the fermionic nature of (f 4 , f 5 ) and bosonic nature of b 4 . Thus, the final expansion of (33), in terms of the (anti-)co-BRST symmetries s (a)d , is We have, therefore, derived all the non-trival (anti-)co-BRST symmetry transformations for the fields A µ , C,C andφ of the modified version of 2D Proca theory. The rest of the transformations are trivial (e.g. s (a)d φ = 0 and s (a)d B = 0) and they can be derived in a straightforward manner from the AVSF because φ(x) → Φ(x, θ,θ) = φ(x), B(x) → B(x, θ,θ) = B(x). We re-emphasize that the transformations (31) are common to the modified versions of 2D Proca and anomalous gauge theories. As far as the latter theory is concerned, we have to derive the (anti-)co-BRST symmetry transformations for the scalar fields φ(x) and σ(x). In this connection, we observe that the following useful quantities are (anti-)co-BRST invariant, namely; Thus, according to the basic requirement of AVSF, we have the following equality due to the restriction on the superfield where the expansions for B (d) µ (x, θ,θ) and Φ(x, θ,θ) are given in (30) and (42) (see below). Substitution of these values into (35) yields the following relationships:f 1 = − C, f 1 = −C, b 1 = − B which imply the following expansions for the scalar superfield Φ (d) (x, θ,θ) after the application of DGIRs, namely; We have to determine the (anti-)co-BRST transformations on the field σ(x). In this regards, we have the following equality due to AVSF where the expansion of Φ (d) (x, θ,θ) is given in (36) and we have taken the following general expansions of Σ(x, θ,θ) along the Grassmannian (θ,θ)-directions of the (2, 2)-dimensional supermanifold, namely; where the secondary fields (P (x),P (x)) are fermionic and Q is bosonic (due to the fermionic nature of θ andθ). It is straightforward to observe, from (37), that we have: .
The above values imply that the super expansions (38) is where −1) ). Thus, we have derived all the non-trivial (anti-)co-BRST symmetry transformations for the modified version of 2D anomalous gauge theory within the framework of AVSF.
We are now in the position to discuss the superfield approach to the derivation of the (anti-)co-BRST symmetries for the 2D self-dual chiral bosonic field theory. First of all, we generalize the relevant fields of the 2D theory onto the (2, 2)-dimensional superfield parametrized by the superspace co-ordinates (x µ , θ,θ) as which have the following expansions along the Grassmannian directions (i.e. (θ,θ)directions) of the (2, 2)-dimensional supermanifold [20] Φ(x, θ,θ) = φ(x) + i θf 1 where the set (S, B 1 ,B 1 , B 2 ,B 2 , b 1 , b 2 ) is made up of the bosonic secondary fields and the fermionic secondary fields are (R,R, s,s, f 1 ,f 1 , f 2 ,f 2 ). We obtain the basic fields (λ, φ, v, C,C) of the theory in the limit θ =θ = 0. We shall obtain the exact expressions for the secondary fields in terms of the basic and auxiliary fields of the theory by exploiting the physically motivated restrictions on the superfields. First of all, we take into account the appropriate generalizations of the exterior derivative and connection 1-form onto the (2, 2)-dimensional supermanifold, as [20]: It should be noted that even though we have generalized the ordinary theory onto the (2, 2)-dimensional supermanifold, the super exterior derivative (d) has been defined on the (1, 2)-dimensional super sub-manifold. This is due to the peculiarity of the gauge field in the case of 2D self-dual bosonic field theory where only one component of the 2D gauge field couples with the matter fields but the other component of the gauge field remains inert (see, e.g. [20] for details). The basic tenets of AVSF state that all the (anti-)co-BRST invariant quantities should be independent of the "soul" coordinates (θ,θ). In this connection, we note that the following are the (anti-)co-BRST invariant quantities (see, eg. [20] for details) Thus, the above quantities in the square brackets, when generalized on the (2, 2)dimensional supermanifold, should be independent of the "soul" coordinates (θ,θ). Plugging in the expansions from (42), we obtain the followinḡ We shall see that these relationships would be useful in our further discussions. For instance, we observe that the following are the invariant quantities: In the above expressions, it is elementary to note that δ λ (1) = + * d * λ (1) is nothing buṫ λ i.e. δ λ (1) = + * d * (dt λ(x)) =λ(x) . We have to generalize this relationship on the (2, 2)-dimensional supermanifold as whereδ = ⋆d ⋆. Here ⋆ is the Hodge duality operation on the (1, 2)-dimensional supersubmanifold of the general (2, 2)-dimensional supermanifold andδ is the super co-exterior derivative (withd = dt ∂ t + d θ ∂ θ + dθ ∂θ). It is to be noted that the gauge field λ is a function of x µ (µ = 0, 1) but the geometrical quantitiesd andδ as well as d = dt ∂ t and δ = * d * are defined in terms of t only. In other words, d and δ are defined on the 1D sub-manifold of the 2D ordinary Minkowskian spacetime manifold andd andδ are defined on the (1, 2)-dimensional super-submanifold of the (2, 2)-dimensional supermanifold on which our ordinary 2D theory is generalized. The l.h.s. of (47) has been worked out in our earlier work. The following relationship emerges from (47): This condition is nothing but the analogue of the CF-type restriction which is essential as far as the proof of the absolute anticommutativity property (i.e. s d s ad + s ad s d = 0) of the nilpotent (anti-)dual BRST symmetry transformations s (a)d is concerned. This condition is also (anti-)dual BRST invariant under the above symmetry transformations s (a)d . Thus, this restriction is a physical condition on the model under consideration within the realm of BRST formalism. In fact, the whole theory is defined on the constrained hypersurface (defined by the above trivial constrained condition) that is embedded in the 2D Minkowskian spacetime manifold on which the whole of our present theory is defined. Ultimately, we concentrate on the following (anti-)co-BRST invariance which imply the following restrictions on the supervariables where the expansions for F (d) andF (d) are as follows: Here the superscript (d) denotes the super-expansions obtained after the application of DHC given in (47). Plugging in the expressions from (43) and (51), we obtain which imply the following: At this stage, we are free to choose the auxiliary field B in such a manner that s (a)d B = 0. The latter condition is essential because of the requirements of nilpotency and absolute anticommutativity. We choose the following in terms of the basic fields as which serves our purpose. Finally, we have the following expansions (see, e.g. [20]) where the superscript (d) denotes the expansion of the superfields after the imposition of the DHC and DGIRs within the framework of AVSF. Thus, we note that we have derived all the (anti-)co-BRST symmetry transformations listed in Eq. (9). The nilpotency and absolute anticmmutativity of

SUSP Dual Unitary Operator: Universal Aspects
The precise expressions for the SUSP unitary operator and its Hermitian conjugate have been explicitly derived in our earlier work [26] on the 4D interacting Abelian 1-form gauge theory with Dirac and complex scalar fields where we have provided the alternatives to the HC and GIRs in the context of the derivation of the (anti-)BRST symmetries of this theory. These forms are expressed, in terms of the familiar symbols, as follows which satisfy UU † = U † U = 1. It is important to point out that the above explicit expressions have been derived by exploiting the theoretical strength behind the concept of covariant derivatives. The expressions (56) can be also written in the exponential forms as which very clearly demonstrate the validity of unitary condition: UU † = U † U = 1. The basic idea behind the covariant derivative also leads to the transformation property of the 1-form A (1) = dx µ A µ gauge connection under the (anti-)BRST symmetry transformations, in the language of SUSP unitary operator and its Hermitian conjugate, as [21,26] A (1) x, θ,θ). In this expression, the superfield B (h) µ (x, θ,θ) yields the (anti-)BRST symmetry transformations for the gauge field A µ (x) and the superfields (F (h) (x, θ,θ),F (h) (x, θ,θ)) yield the (anti-)BRST symmetry transformations for the ghost and anti-ghost fields, respectively. Here the superscript (h) denotes the expressions of the superfields after the application of the HC. The equation (58) provides an alternative to the HC in terms of the SUSP unitary operator U and its Hermitian conjugate U † (see, e.g. [21] for details). We shall see below that we can derive the proper (anti-)dual-BRST symmetry transformations for the relevant fields/variables from the equations like (56), (57) and (58) which would be obtained after the application of the duality transformations (A µ → − ε µν A ν , C →C,C → C).
We focus, first of all, on the derivation of the (anti-)co-BRST symmetry transformations for the 1D rigid rotor where the form of the unitary operator and its Hermitian conjugate is same as given in (56) and (57) with the replacement x → t (i.e. U (x, θ,θ)| x=t = U (t, θ,θ)), where all the fields are functions of t only (i.e. B(t), C(t),C(t)). There is a duality in the theory where λ → p r , C →C andC → C for the presence of the (anti-)dual-BRST symmetry transformations s (a)d . This is due to the fact that the role of λ, p r , C andC change in a symmetrical fashion when we go from the (anti-)BRST symmetries to the (anti-)dual-BRST symmetries. A careful and close look at Eqn. (2) shows that the role of B in the (anti-)BRST symmetry transformations is traded with (r − a) in the (anti-) co-BRST symmetries. Thus, we have the SUSP dual unitary operator and its Hermitian conjugate operator from the unitary operators (56) (with the replacement B → (r − a)) as which also satisfyŨŨ † =Ũ †Ũ = 1 and they can be exponentiated as so that we have the validity of unitary conditionŨŨ † =Ũ †Ũ = 1 in a straightforward manner. Now the DHC can be expressed in the following fashion t, θ,θ). It should be noted that we have already taken into account the dual transformations r , F (d) andF (d) have been already given in Eq. (20) and we have the expression for the super-exterior derivative asd = dt ∂ t + dθ ∂ θ + dθ ∂θ. Written in the explicit forms, the quantum dual gauge [i.e. (anti-)co-BRST] transformation (61) implies the following expressions for P (d) (r) (t, θ,θ), F (d) (t, θ,θ) andF (d) (t, θ,θ) in terms of the SUSP dual unitary operator and its Hermitian conjugate, namely; The explicit substitution ofŨ andŨ † from (59) into the above relationships yields exactly the same result as (20) for the expansions of P (d) In the above, we have constructed a 1-form p (1) (r) = dtp (r) (t) on the 1D manifold for the derivation of the (anti-)dual-BRST symmetries. This should be contrasted with the 1-form λ (1) = dtλ(t) that was taken into account in the context of the derivation of the (anti-)BRST symmetries [21]. We have done it because of the fact that there is a duality (i.e. λ → p r , C →C,C → C) in the theory when we go from s (a)b → s (a)d . Thus, the super 1-formλ (1) (h) (t, θ,θ) →P (d) (r) (t, θ,θ) such that the appropriate super 1-form P (1) (r) (t, θ,θ) is defined, for the derivation of the (anti-)co-BRST symmetry transformations. Thus, now we have P (1) (r) (t, θ,θ) = dt P (r) (t, θ,θ) + dθ F (t, θ,θ) + dθF (t, θ,θ). We note that the above 1-form is derived from the definition of super 1-formλ θ,θ) that has been used for the derivation of the (anti-)BRST symmetries [21]. From relationship (61), it can be checked thatdP = dp r (t) = 0 (where we have operated byd from the left onP (1)(d) (r) and taken into account the fact thatdp (1) (r) = dp (1) (r) = 0 anddŨ ∧dŨ † = 0). To be more precise, it can be checked that d p (1) r = dtp r (t) and ∂ θ p r (t) = ∂θ p r (t) = 0. The explicit form ofdŨ anddŨ † are as follows: The claimdŨ ∧dŨ † = 0 can be proven by collecting all the coefficients of (dt ∧ dθ), (dt ∧ dθ), (dθ ∧ dθ), (dθ ∧ dθ) and (dθ ∧ dθ) and showing that these are exactly zero. There is a simpler method to prove this statement by looking carefully at the exponential forms of U and U † [cf. (60)]. We note that the exponents are the same modulo a sign factor. Therefore, the quantitydŨ ∧dŨ † would imply the wedge product between the same quantities (i.e. exponents). Since the exponents are bosonic in nature, their wedge product would always be zero. Thus, we conclude thatdŨ ∧dŨ † = 0 which implies that a 2-form (d P r = 0) cannot be defined on a 1D manifold. Hence, the r.h.s. ofdŨ ∧dŨ † is zero.
We concentrate now on the modified version of 2D Proca theory as well as the anomalous gauge theory and express the DHC and DGIRs in terms of the SUSP dual unitary operator and its Hermitian conjugate. In these theories, there is a duality in the sense that the transformations:  [21]. Thus, we define the dual super 1-form connection, as an input for the derivation of the (anti-)co-BRST symmetries, as: It is to be noted that we have derivedÃ (1) (d) (x, θ,θ) from the usual super 1-formÃ (1) = dx µ B µ (x, θ,θ) + dθF (x, θ,θ) + dθF (x, θ,θ) by the replacements: B µ → − ε µν B ν , F →F andF → F due to the presence of duality in our theory. In the context of (anti-)co-BRST symmetries, it will be noted that the usual definition of the super 1-form (i.e.Ã (1) ) is taken into account in a subtle manner. Under the (anti-)dual BRST symmetry transformations, the above super 1-form transforms in the superspace as where, for the modified version of 2D Proca theory, the form of SUSP dual unitary operator U and its Hermitian conjugateŨ † are: The above can be explicitly written (in terms of coefficients of θ,θ and θθ) as: The substitution of (72) into (70) yields the followinḡ where we have equated the coefficients of dx µ , dθ and dθ from the l.h.s. and r.h.s. of (70). The last entry in the above equation leads to the following: Similarly, we have the following super expansions in an explicit form: Thus, we have derived the proper (anti-)co-BRST symmetry transformations for the basic fields A µ (x) C(x) andC(x) which are common for the modified versions of the 2D Proca and anomalous gauge theories. In the latter case, however, we have to replace (E − mφ) by B(x) in the definition of the SUSP dual unitary operator and its Hermitian conjugate. We shall now focus on the derivation of proper (anti-)co-BRST symmetries for the additional fields in these theories. Let us express the superfieldφ (d) (x, θ,θ) in the language of the SUSP dual unitary operator and its Hermitian conjugate. We have seen that s (a)d [E − mφ] = 0 due to the on-shell nilpotency (i.e. s 2 (a)d C = s 2 (a)dC = 0) in the theory because of the fact that the (anti-)ghost fields (C) C obey the on-shell conditions: ( + m 2 ) C = 0, ( + m 2 )C = 0. Thus, the combination [E − mφ] is an (anti-)co-BRST invariant quantity which can be generalized onto the (2, 2)-dimensional supermanifold due to AVSF. This can be expressed in the language of superfields, derived after the application of the DHCs and DGIRs, as: Using the expression for B This relation, finally, leads to the following expression forΦ (d) (x, θ,θ) in terms of the SUSP dual unitary operators (i.e.Ũ andŨ † ), namely; It is very interesting to check that the r.h.s. yields the expansions (34) when we use the on-shell conditions: ( + m 2 ) C = 0, ( + m 2 )C = 0 and ( + m 2 ) B = 0 . It is important to point out that the contributions, from the second term of (78), cancel out with the extra piece that emerges from the last term on the r.h.s. of (78). Thus, we have expressed all is a 1-form on the 1D sub-manifold of the general 2D ordinary spacetime manifold. The above equation, taking into account the definition (83), is as follows in the component form where we have taken into account the comparison of the coefficients of dx µ , dθ and dθ from r.h.s. and l.h.s. The substitution of the explicit form ofŨ andŨ † from (82) leads to the following expansions from the superfields (cf. Eq. (55)): A close look at the expansions shows that we have already derived the (anti-)co-BRST symmetry transformations s (a)d for the fields φ(x), C(x),C(x). We note that s (a)d [φ − v] = 0. This observation implies immediately, due to the basic tenets of AVSF, that we have the following expansion of the superfield corresponding to the WZ-field v(x), namely; where the (anti-)co-BRST symmetry transformations s (a)d are listed in (9). The above equation (87) can also be written in terms ofŨ andŨ † as: This is due to the fact that a super 1-form can be written exactly like (83) in terms of . It goes without saying that we can repeat the above exercise to obtain the superspace transformation like (87) and (88). We observe that s (a)d λ − 2 φ = s (a)d λ − 2 v = 0. Thus, we have the following restrictions (due to these invariances) on the superfields, defined on (2, 2)-dimensional supermanifold, according to to basic tenets of AVSF, namely; which implies that the superfield Λ (d) (x, θ,θ) can be expressed (from both the above relationships) in terms of the SUSP dual unitary operatorsŨ andŨ † as: The above expression finally leads to: Thus, we have provided the alternatives to the DHC and DGIRs used in Sec. 3, in the language ofŨ andŨ † and obtained all the non-trivial (anti-)co-BRST symmetries of the 2D self-dual bosonic field theory. We conclude this section with the remarks that SUSP dual unitary operatorsŨ andŨ † provide the alternatives to the DHC and DGIRs within the framework of AVSF where the explicit group structure is maintained.

Conclusions
For the Abelian 1-form U(1) gauge theories, it is important to have explicit existence and appearance of the group structure in any kind of computation. The SUSP dual unitary operator and its Hermitian conjugate do exactly the same job in our present endeavor and, that is why, their derivation is important. In our earlier works [26,27], we have explicitly derived the exact form of the SUSP unitary operator and its Hermitian conjugate for the cases of the interacting (i) 4D Abelian U(1) gauge theory with the Dirac and complex scalar fields, and (ii) 4D non-Abelian SU(N) gauge theory with Dirac fields, in the context of nilpotent (anti-)BRST symmetries. The universal nature of the SUSP unitary operator and its Hermitian conjugate has also been established in our recently published work [21] for the case of the 1D and 2D Abelian U(1) gauge theories. In fact, we have been able to derive the dual unitary operator and its Hermitian conjugate from the above universal unitary operator by exploiting the virtues of the duality symmetry in our theory where C →C,C → C and A µ → − ε µν A ν . As it turns out, we observe that the mathematical form of the SUSP dual unitary operator and its Hermitian conjugate is universal in exactly the same way as the SUSP unitary operator and its Hermitian conjugate are (see, e.g. [21]). We would like to dwell a bit on the duality aspects of our statement. In the case of 2D Abelian 1-form gauge theory, it can be seen that the self-duality condition: form potential corresponding to the Abelian 1-form potential A µ . Furthermore, we observe that when we go from the (anti-)BRST symmetries (particularly in the ghost sector of our theory), there is a transformation from C →C andC → C. Thus, for a 2D 1-form theory, the transformations A µ → − ε µν A ν , C →C,C → C are the duality transformations which have been exploited in the definition of super 1-forms (cf. (69), (83)). However, in the case of 1D Abelian 1-form theory (i.e. a rigid rotor), we observe that there is a duality: λ → p r , C →C,C → C. This observation has been exploited in the statements that have followed equations (61) and (63) in the definition of P (1) (d) (x, θ,θ). Similar kind of arguments have been exploited in the case of 2D self-dual field theory where we have expressed the DHC and DGIRs in the language ofŨ andŨ † .
In our present endeavor, we have applied the AVSF to derive the (anti-)co-BRST symmetry transformations for a new model in 2D. This model is nothing but the modified version of the 2D anomalous gauge theory which has already been proven to provide a tractable model for the Hodge theory [19]. Thus, it is a novel result in our present endeavor. The precise derivation of the (anti-)co-BRST symmetries establishes the sanctity and correctness of the working-rule that has been laid down for the Hodge duality (⋆) operation on the (1, 2) and (2, 2)-dimensional supermanifolds [22]. Thus, we conclude that the AVSF is a powerful theoretical technique that can be applied to interesting physical systems and one can derive the appropriate form of the BRST-type symmetries. The key concepts (that play important roles in the application of the AVSF) are the DHC and DGIRs. One of the key observations of our present endeavor is the fact that the geometrical meaning of the (anti-)co-BRST symmetries, in the language of the translational generators (∂ θ , ∂θ) along the Grassmannian directions of the appropriately chosen supermanifold, remains the same when we exploit the theoretical strength of the DHC and DGIRs.
We would like to lay emphasis on the fact that the models of the Abelian 1-form gauge theories in 1D and 2D (that have been considered in our present endeavor) are interesting because these models provide the tractable physical examples of Hodge theory within the framework of BRST formalism [16][17][18][19][20]. Such models are mathematically as well as physically very rich because there are many continuous symmetries in the theory which enable these theories to be quantized without the definition of the canonical conjugate momenta corresponding to the fields of these theories [29][30][31][32]. In the context of gauge theories, it has been shown, in our earlier works [29][30][31][32], that there exist six continuous internal symmetries for such theories which are so powerful that they lead to the canonical quantization of these theories at the level of creation and annihilation operators. The above symmetries have also played very important roles in the proof of 2D (non-)Abelian 1-form gauge theories (without any interaction with matter fields) to be a new class [33] of topological field theories (TFTs) that capture a few key aspects of the Witten-type TFTs and some salient features of the Schwartz-type TFTs.
We have succeeded in obtaining universal SUSP unitary operator and its Hermitian conjugate that are primarily connected with the (anti-)BRST symmetries in the cases of 4D interacting Abelian 1-form gauge theories with Dirac fields, 2D and 1D Abelian gauge theories. In our present endeavor, we have obtained the SUSP dual unitary operator and its Hermitian conjugate in the cases of 2D and 1D Abelian 1-form gauge theories that are connected with the (anti-)co-BRST symmetry transformations. One of the immediate goal for us is to extend our work to the 2D non-Abelian 1-form gauge theory (without any interaction with matter fields) so that we could derive the SUSP unitary operator and its Hermitian conjugate as well as the SUSP dual unitary operator and its Hermitian conjugate. This is essential because we have already shown that this 2D non-Abelian model is an example of the Hodge theory where the (anti-)BRST and (anti-)co-BRST symmetries exist along with other internal symmetries. We have already made some progress in this direction and our results would be reported in our future publication [34].