Augmented Superfield Approach to Nilpotent Symmetries in the Modified Version of 2D Proca Theory

We derive the complete set of off-shell nilpotent and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST and (anti-)co-BRST symmetry transformations for all the fields of the modified version of two (1+1)-dimensional (2D) Proca theory by exploiting the"augmented"superfield formalism where the (dual-)horizontality conditions and (dual-)gauge-invariant restrictions are exploited together. We capture the (anti-)BRST and (anti-)co-BRST invariance of the Lagrangian density in the language of superfield formalism. We also express the nilpotency and absolute anticommutativity of the (anti-)BRST and (anti-)co-BRST charges within the framework of augmented superfield formalism. This exercise leads to some novel observations which have, hitherto, not been pointed out in the literature within the framework of superfield approach to BRST formalism. For the sake of completeness, we also mention, very briefly, a unique bosonic symmetry, the ghost-scale symmetry and discrete symmetries of the theory and show that the algebra of conserved charges captures the cohomological aspects of differential geometry. Thus, our present modified 2D Proca theory is a model for the Hodge Theory.


Introduction
One of the earliest known gauge theories (with U(1) gauge symmetry) is the Abelian 1-form (A (1) = dx µ A µ , µ = 0, 1, 2, ...D − 1) Maxwell theory which describes the massless vector boson (A µ ) with (D − 2) degrees of freedom in any arbitrary D-dimensions of spacetime. Thus, in the physical four dimensions of spacetime, A µ has two degrees of freedom. Its massive generalization is a Proca theory that describes a vector boson with three degrees of freedom in the physical four (3+1)-dimensions of spacetime. The central goal of our present investigation is to study the two (1+1)-dimensional (2D) Stueckelberg-modified version of Proca theory which also incorporates a pseudo-scalar field on physical and mathematical grounds [2,3]. This model is very special because it is endowed with mass together with various kinds of internal symmetries * . The existence of the above (continuous as well as discrete) symmetries renders the model to become an example for the Hodge theory [2,3].
Recently, in a set of papers [4][5][6], we have demonstrated that the N = 2 supersymmetric (SUSY) quantum mechanical models also provide physical examples of Hodge theory because of their specific continuous and discrete symmetry transformations which provide the physical realizations of the de Rham cohomological operators † and Hodge duality ( * ) operation of differential geometry [7][8][9][10][11][12]. However, these SUSY models are not gauge theories because they are not endowed with the first-class constraints in the terminology of Dirac's prescription for the classification scheme of constraints [13,14]. One of the characteristic features of these SUSY models is that they have mass but do not possess gauge symmetries that are primarily generated by the first-class constraints (see, e.g. [14,15]).
We have also provided the physical realizations of the de Rham cohomological operators of the differential geometry in the context of Abelian p-form (p = 1, 2, 3) gauge theories in D = 2p dimensions of spacetime within the framework of BRST formalism [16][17][18][19]. As a consequence, these theories are also the field theoretic models for the Hodge theory. One of the decisive features of these theories is the observation that they have gauge symmetry (generated by the first-class constraints) but they do not have mass. The modified version of 2D Proca theory is, thus, a special field theory which possesses mass as well as various kinds of internal symmetries and, as has turned out, it also provides a field theoretic model for the Hodge theory within the framework of BRST formalism [2,3].
One of the most intuitive approaches to understand the abstract mathematical properties associated with the (anti-)BRST symmetries is the geometrical superfield formalism (see, e.g. [20][21][22][23]) where the celebrated horizontality condition (HC) plays very important role as far as the derivation of (anti-)BRST symmetry transformations for the gauge fields and associated (anti-)ghost fields, for a given gauge theory, are concerned. In the augmented version [24][25][26][27] of the above superfield formalism, the HC blends together with the gauge-invariant restrictions (GIRs) in a beautiful fashion enabling us to derive the (anti-) BRST symmetry transformations for the gauge, (anti-)ghost and matter fields of a given interacting gauge theory. The central objective of our present paper is to apply extensively the above augmented version ‡ of the geometrical superfield formalism to discuss various aspects of the modified 2D Proca theory within the framework of BRST formalism.
In our present investigation, we derive the off-shell nilpotent and absolutely anticommuting (anti-)BRST and (anti-)co-BRST symmetry transformations by exploiting the theoretical power of augmented version of superfield formalism. In fact, we exploit the celebrated (dual-)horizontality conditions [(D)HCs] and (dual-)gauge invariant restrictions [(D)GIRs] to obtain the proper (anti-)BRST and (anti-)co-BRST symmetry transformations for all the fields of the modified version of 2D Proca theory. We provide the geometrical meaning for the above nilpotent symmetry transformations in the language of translational generators along the Grassmannian directions of the (2, 2)-dimensional supermanifold, on which, our ordinary modified version of 2D Proca theory is generalized.
Some of the key observations of our present investigation are contained in our subsections 3.3 and 4.4 where we have expressed the (anti-)BRST and (anti-)co-BRST charges in terms of the superfields (obtained after the applications of (D)HC and (D)GIR), Grassmannian derivatives and Grassmannian differentials. The off-shell nilpotency and absolute anticommutativity properties of the (anti-)BRST and (anti-)co-BRST symmetries (and their corresponding generators) emerge very naturally within the framework of our augmented version of the geometrical superfield formalism. We have also captured the (anti-) BRST and (anti-)co-BRST invariance of the Lagrangian density within the ambit of our augmented version of superfield formalism in a very simple and straightforward fashion.
From our fundamental off-shell nilpotent and absolutely anticommuting (anti-)BRST and (anti-)co-BRST symmetry transformations, we have also computed a unique bosonic symmetry. Furthermore, we have discussed the ghost-scale symmetry and a couple of discrete symmetries for our present theory. By exploiting the fundamental tricks of Noether theorem, we have computed the exact expressions for the (anti-)BRST, (anti-)co-BRST, a bosonic and ghost charges of our present 2D modified-Proca theory. We have also derived the exact extended BRST algebra amongst the above charges and demonstrated that this algebraic structure is analogous to the algebraic relationship obeyed by the de Rham cohomological operators of differential geometry. Ultimately, we have established that our present 2D modified Proca theory is a perfect field theoretic model for the Hodge theory.
The main motivating factors behind our present investigations are as follows. First, it is very important for us to put the basic ideas of our augmented version of superfield formalism on firmer footings by applying it to various interesting physical systems which are BRST invariant. Second, it is essential for us to establish the correctness of our earlier results [3] where we have discussed about the off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations for the 2D modified Proca theory. Finally, our present endeavor is our modest step towards our main goal of applying our ideas to find out the 4D massive models for the Hodge theory which might enforce the existence of fields that would turn out to be the candidates for the dark matter [28,29]. We have already shown the emergence and existence of the latter (as a pseudo-scalar field with a negative kinetic term) in our study of the modified version of 2D Proca theory (see, e.g. [2,3] for details). ‡ We have christened the extended version of the usual Bonora-Tonin superfield formalism [20,21] as the augmented superfield formalism where, in addition to the HC, other physically relevant restrictions (consistent with the HC) are also imposed on the superfields defined on the supermanifold.
The contents of our present paper are organized as follows. In Sec. 2, we recapitulate the bare essentials of the usual Proca theory and discuss the gauge symmetry transformations of the Stueckelberg-modified version of it in any arbitrary D-dimensions of spacetime. We also point out a key observation that plays an important role in the superfield formulation of our present theory.
Our Sec. 3 is devoted to the derivation of off-shell nilpotent (anti-)BRST symmetry transformations within the framework of augmented superfield formalism. We also capture the (anti-)BRST invariance of the Lagrangian density and provide geometrical meaning for the nilpotency and anticommutativity properties of the (anti-)BRST charges within the framework of augmented superfield formalism and point out some novel results.
In Sec. 4, we deal with the (anti-)co-BRST symmetry transformations for the 2D Stueckelberg-modified Proca theory by exploiting the augmented version of superfield formalism. Furthermore, we demonstrate the (anti-)co-BRST invariance of the Lagrangian density within the framework of superfield formalism. In addition, we capture the properties of nilpotency and anticommutativity of the (anti-)co-BRST charges in the language of superfield approach to BRST formalism and mention some new observations. Our Sec. 5 describes, very briefly, a unique bosonic symmetry, the ghost-scale symmetry and discrete symmetries of our present theory. We also compute the expressions for the bosonic and ghost charges and concisely mention the consequences that ensue, due to the operation of discrete symmetries of our theory, on all the conserved charges.
In Sec. 6, we present the algebraic structure of all the generators of the above continuous symmetries and establish its connection with the cohomological operators of differential geometry. We demonstrate that our present theory is a model for the Hodge theory.
Finally, we make some concluding remarks in Sec. 7.

Preliminaries: Gauge Symmetries in Proca Theory
Let us begin with the Lagrangian density (L 0 ) of a Proca theory (with a mass parameter m) in any arbitrary D-dimensions of spacetime. This can be expressed as follows: Here is the exterior derivative and the 1-form A (1) = dx µ A µ defines the vector boson A µ . In physical four (3+1)-dimensions of spacetime, the bosonic field A µ has three degree of freedom and m has the dimension of mass in natural units (where = c = 1). In the massless limit (i.e. m → 0), we obtain the 4D Maxwell Lagrangian density from (1) which respects the gauge invariance under the transformations: where Λ is the local gauge parameter. It is evident that, in the Proca theory, the gauge symmetry transformations (2) are lost because of the presence of mass term. In some sense, a Proca theory is a generalization of the Maxwell's theory as the latter is the massless (m → 0) limit of the former (where the usual U(1) gauge symmetry is respected). By exploiting the Stueckelberg's formalism, one can restore the gauge symmetry (2) for the original Lagrangian density (1) where the field A µ is replaced by A µ ∓ 1 m ∂ µ φ. As a consequence, we obtain the following Stueckelberg Lagrangian density which respects the following local, continuous and infinitesimal gauge transformations where φ is a real scalar field. The key points, at this stage, are as follows. First, by introduction of the Stueckelberg's field φ, we have converted the second-class constraints of the original Lagrangian density (1) into the first-class variety in the terminology of Dirac's prescription for the classification scheme [13,14]. Second, the whole of our above discussions is true in any arbitrary D-dimensions of spacetime where µ, ν, λ, ... = 0, 1, 2, ...D −1 and the gauge symmetry transformations (4) are also valid in this arbitrary dimension of spacetime. Third, the Lagrangian density (3) describes, in the physical four dimensions of spacetime, a theory where the mass and gauge invariance co-exist together in a beautiful fashion. We close this section with the following remarks. First, the gauge symmetry transformations (4) are valid in any arbitrary dimension of spacetime for the Stueckelberg-modified Lagrangian density (L s ) at the classical level. This symmetry, therefore, could be exploited for the (anti-)BRST symmetry transformations at the quantum level. Second, the quantity A µ ∓ 1 m ∂ µ φ is a gauge-invariant quantity because δ g [A µ ∓ 1 m ∂ µ φ] = 0 for δ g A µ = ∂ µ Λ and δ g φ = ± mΛ. These observations would play very important roles in our further discussions on the derivation of (anti-)BRST symmetries within the framework of superfield formalism.

(Anti-)BRST Symmetries: Superfield Formalism
In this section, we derive the full set of proper (anti-)BRST symmetry transformations by exploiting the strength of HC and GIR. Furthermore, we capture the (anti-)BRST invariance of the Lagrangian density and the nilpotency as well as absolute anticommutativity properties of the (anti-)BRST charges within the framework of superfield formalism
The central role, in the superfield approach [20][21][22][23], is played by the horizontality condition (HC) which requires that the gauge-invariant quantity F µν , owing its origin to the exterior derivative, must remain independent of the Grassmannian variables when it is generalized onto a (D, 2)-dimensional supermanifold. In other words, the ordinary curvature 2-form F (2) = d A (1) = (dx µ ∧ dx ν /2!) F µν must be equal (i.e. F (2) =F (2) ) to the super curvature 2-form (F (2) ) as illustrated below: In the above, the super exterior derivatived (withd 2 = 0) and super 1-form connectioñ A (1) are defined on the (D, 2)-dimensional supermanifold as We have taken ∂ M = (∂ µ , ∂ θ , ∂θ) as the superspace derivative on the (D, 2)-dimensional supermanifold. Physically, the equality dA (1) =dÃ (1) of the HC implies that the gaugeinvariant electric and magnetic fields of the ordinary theory should not be affected by the presence of the Grassmannian variables θ andθ of the supermanifold on which the ordinary theory has been generalized due the prescription laid down by the superfield formalism. The superfields B µ (x, θ,θ), F (x, θ,θ),F (x, θ,θ) of (6) are the generalizations of the gauge field (A µ ), ghost field (C) and anti-ghost field (C), respectively, of ordinary Ddimensional BRST-invariant theory because the above superfields can be expanded along the Grassmannian directions of the (D, 2)-dimensional supermanifold as (see, e.g. [20]) where (A µ , C,C) are the basic fields of an arbitrary D-dimensional (anti-)BRST invariant theory and rest of the fields, on the r.h.s. of (7), are the secondary fields of the theory which can be expressed in terms of the basic and auxiliary fields of the ordinary D-dimensional theory by exploiting HC. It is clear that (R µ , s,s) and (S µ , B 1 , B 2 , B 3 , B 4 ) are the fermionic and bosonic secondary fields, respectively, on the r.h.s. of (7).
One can expand the expressiondÃ (1) of (5) in the following explicit form using (6). This expansion, in its full blaze of glory, is as follows In a similar fashion, one can also expand the r.h.s. of (5) as: The HC requires that F (2) = [dx µ ∧ dx ν /2!] F µν should be equal toF (2) Written in an explicit form, we have the following connections from the comparison between the equations (8) and (9) due to the celebrated HC, namely; The substitution of the expansions (7) yields the following relationships amongst the secondary fields and the basic as well as auxiliary fields of the ordinary 2D theory, namely; The last entry in the above is nothing but the celebrated Curci-Ferrari condition [30] which turns out to be trivial in the case of Abelian 1-form modified Proca gauge theory. Taking the help of (11), we have the following expansions (if we choose B 4 = B, B 1 = −B) namely; which yields the following off-shell nilpotent (anti-)BRST symmetries for the gauge (A µ ) and (anti-)ghost fields (C)C of the theory A few noteworthy points, at this stage, are as follows. First, the superscript (h) on the superfields in (12) denotes the expansion of the superfields after the application of HC. Second, the transformations s (a)b B = 0 on the Nakanishi-Lautrup auxiliary fields B have been derived from the nilpotency condition. Third, it can be checked that the last entry of (10) is satisfied:F (12). Finally, we have the following mappings § (see, e.g., [24][25][26][27] for details) (14) § Truly, the exact relationship is: where M (x) is the 2D ordinary field andM (h) (x, θ,θ) is the corresponding superfield, obtained after the application of the HC. However, we shall continue with the mapping (14) but shall keep in mind that the precise connection between the (anti-)BRST transformations s (a)b and (∂ θ , ∂θ) is: s b ↔ ∂θ and s ab ↔ ∂ θ .
Thus, we note that the Grassmannian translation generators (∂ θ , ∂θ) are connected with the off-shell nilpotent (s 2 (a)b = 0) and absolutely anticommuting (s b s ab + s ab s b = 0) fermionic (anti-)BRST symmetry transformations s (a)b . These properties have their origin in the properties ∂ 2 θ = 0, ∂ 2 θ = 0, ∂ θ ∂θ + ∂θ∂ θ = 0 of the Grassmannian translation generators (∂ θ , ∂θ) when the above relations are taken in their operator form. Now we exploit the strength of the augmented version of superfield formalism [24][25][26][27] to derive the (anti-)BRST symmetry transformations for the real scalar field φ. To this end in mind, we recall that the quantity (A µ ∓ 1 m ∂ µ φ) is a gauge invariant quantity. Thus, this physical quantity should remain unaffected by the presence of the Grassmannian variables (θ,θ) when it is generalized onto a (D, 2)-dimensional supermanifold. In other words, in the language of differential geometry, the following is true: (12)] and the zero-form superfield Φ(x, θ,θ) has the following superexpansion along the Grassmannian directions (θ,θ) of the (D, 2)dimensional supermanifold, namely;.
In the above, it is evident that the pair of fields [f (x),f (x)] are fermionic secondary fields and (φ(x), b(x)) are bosonic in nature. In the limit (θ,θ) → 0, we retrieve back our real scalar Stueckelberg field φ(x) of the original D-dimensional ordinary theory. The gauge-invariant restriction (GIR) in (15) leads to the following relationships: The substitution of (17) into (16) leads to where the superscript (g) on the superfield Φ(x, θ,θ) corresponds to the superexpansion of this superfield after the application of GIR. It is clear, from the above equation, that: We note that, ultimately, it is the combination of HC and GIR which leads to the derivation of a full set of the correct off-shell nilpotent (s 2 (a)b = 0) and absolutely anticommuting (s b s ab + s ab s b = 0) (anti-)BRST transformations for all the fields of the D-dimensional modified Proca theory within the framework of augmented superfield formalism.

Lagrangian Densities: (Anti-)BRST Invariance
Exploiting the full set of (anti-)BRST symmetry transformations, we can derive the (anti-) BRST invariant Lagrangian densities (that incorporate the gauge-fixing and Faddeev-Popov ghost terms) by exploiting the standard techniques of BRST approach, namely; In explicit form, the total (anti-)BRST invariant Lagrangian densities (containing two signatures) look in the following form: Using the full set of (anti-)BRST symmetries (13) and (19), we can check that the above Lagrangian densities transform to the total spacetime derivatives as As a consequence, the action integral S = d D−1 x L B remains invariant for the physically well-defined fields of the theory. The above symmetry transformations, according to Noether's theorem, lead to the following expressions for the (anti-)BRST charges Q (a)b : which are found to be conserved (Q (a)b = 0) and nilpotent of order two (Q 2 (a)b = 0). These charges are the generators of transformations listed in (13) and (19) and they are derived from the following Noether conserved currents: In the proof of the conservation laws (∂ µ J µ (a)b = 0), we have to use the following Euler-Lagrange (E-L) equations of motion that emerge from the Lagrangian densities (21). The (anti-)BRST invariance of the Lagrangian density (21) can be also captured in the language of superfield formalism. To this end in mind, first of all, we note that the Stueckelberg Lagrangian density L s [cf. (3)] can be written as within the framework of superfield formalism where the superfields are obtained after the applications of HC and GIR [cf. (12), (18)]. It is straightforward to check that In view of the mappings (14), it is evident that Stueckelberg Lagrangian densities are Exploiting the techniques of superfield formalism, the full (anti-)BRST invariant Lagrangian densities (21) (that incorporate the gauge-fixing and Faddev-Popov ghost terms) can be expressed in three different ways (modulo a total spacetime derivative), namely; Taking into account the nilpotency (∂ 2 θ = ∂ 2 θ = 0) and anticommutativity (∂ θ ∂θ + ∂θ ∂ θ = 0) property of the generator (∂ θ , ∂θ), it is straightforward to prove that where the mappings (14) and results from (26) have been taken into considerations. We would like to lay emphasis on the fact that there is no contradiction amongst (20), (22), (28) and (29). This is due to the observation that, in reality, we have: However, we have thrown away the total spacetime derivative term from the Lagrangian density (21). If we keep this term in (21), then, we have s (a)b L B = 0 instead of the expressions in (22).

Conserved Charges: Superfield Approach
We can also express the (anti-)BRST charges in terms of superfields (obtained after the application of HC and GIR), the Grassmannian derivatives (∂ θ , ∂θ) and differentials (dθ, dθ). For instance, we note the following expression for the BRST charge: in the language of superfields (after the application of HC). It is clear, from the mappings (14), that the above expression implies: in the ordinary D-dimensions of spacetime where the (anti-)BRST transformations s (a)b and ordinary fields are defined. The proof of the nilpotency of the BRST charge becomes simple now due to the nilpotency (s 2 b = 0) of s b and that of the translation generator ∂θ (because ∂ 2 θ = 0). In exactly similar fashion, we can express the anti-BRST charge Q ab as: Once again, the proof of nilpotency of anti-BRST charge Q ab becomes pretty simple because of the fact that, in the ordinary D-dimensional spacetime, the expression (32) can be written in the following form by exploiting the mappings (14), namely; where s 2 ab = 0 implies that Q 2 ab = 0 (due to s ab Q ab = i {Q ab , Q ab } = 0). Within the framework of superfield formalism, the nilpotency of Q ab is encoded in the nilpotency of ∂ θ (because ∂ 2 θ = 0). In other words, we can explicitly verify the nilpotency of the conserved (anti-)BRST charges in terms of translational generators along the Grassmannian directions (θ,θ) as: There are other alternative forms of the conserved and nilpotent (anti-)BRST charges, within the framework of the superfield formalism, that are also interesting. For instance, it can checked that the anti-BRST charge can be expressed as: In exactly similar fashion, we can express the conserved BRST charge as: The nilpotency (Q 2 (a)b = 0) of the (anti-)BRST charges Q (a)b are encoded in the observation that the following is true, namely; where the nilpotency (∂ 2 θ = ∂ 2 θ = 0) of ∂ θ and ∂θ plays an important role. We close this subsection with the remark that the following observations in the context of expression for the (anti-)BRST charges: lead to the proof of absolute anticommutativity of (anti-)BRST charges because it can be readily checked that the following are true, namely; because of the nilpotency of the (anti-)BRST transformations s (a)b . These observations can also be captured in the language of the superfield formalism because imply the following: A close look at (38), (39) and (40) shows that the nilpotency and anticommutativity property are inter-related. In other words, the observations ∂ 2 θ = ∂ 2 θ = 0 and ∂ θ ∂θ + ∂θ ∂ θ = 0 are inter-connected. For instance, in the latter relation when we set ∂ θ = ∂θ, we obtain the former relation ∂ 2 θ = ∂ 2 θ = 0 which actually provides the connection between the properties of the anticommutativity and nilpotency associated with the (anti-)BRST transformations.
We wrap up this subsection with the remark that it is the strength of the superfield approach to BRST formalism that we have obtained various expressions for the (anti-) BRST charges in the language of (anti-)BRST symmetry transformations. Some of the results are completely novel as, to the best of our knowledge, these expressions have not been pointed out in the literature. In fact, these new expressions are responsible, with the help of mapping in (14), to establish the nilpotency and absolute anticommutativity of the (anti-)BRST symmetries (and corresponding charges) within the framework of superfield formalism. For instance, the relationship, given in (38), demonstrate that the nilpotency property and absolute anticommutativity property are intertwined.

(Anti-)co-BRST Symmetries: Superfield Approach
In this section, first of all, we discuss about the dual-gauge transformations for the gaugefixed Lagrangian densities and show that a particular kind of restriction must be imposed on the dual-gauge parameter if we wish to maintain the dual-gauge symmetry in the theory. Then, we derive the off-shell nilpotent and absolutely anticommuting (anti-)co-BRST symmetry transformations by exploiting the strength of dual-HC (DHC) and dual-GIR (DGIR). After this, we prove the (anti-)co-BRST invariance of the Lagrangian densities within the framework of superfield formalism. Finally, we capture the (anti-)co-BRST invariance of the conserved charges, their nilpotency and absolute anticommutativity within the ambit of the superfield approach to BRST formalism.

Dual-gauge Transformations for the Lagrangian Densities in Two-Dimensions of Spacetime
Analogous to the local gauge symmetry transformations (4), we wish to discuss, in this subsection, the dual-gauge transformations which would be, finally, generalized to the (anti-)co-BRST symmetry transformations. In the two (1+1)-dimensions of spacetime, a particular part of the Lagrangian density [i.
This is a pseudo-scalar field because it changes sign under parity. In their most general form, the 2D gauge-fixed Lagrangian densities for the modified Proca theory [without the fermionic (anti-)ghost fields] are as follows ¶ (see, e.g. [3]) where (B,B, B,B) are the Nakanishi-Lautrup type auxiliary fields andφ is a pseudo-scalar field that has been incorporated in the theory on mathematical as well as physical grounds [2,3]. It will be noted that the (pseudo-)scalar fields (φ)φ have been added/subtracted in a symmetrical fashion to the kinetic and gauge-fixing terms, respectively, so that we would have appropriate discrete symmetries in the the theory [cf. (89) below]. Let us discuss the dual-gauge transformations δ (1,2) dg : ¶ For the 2D theory, we adopt the convention and notations such that the background flat Minkowskian spacetime manifold is endowed with a metric η µν with signatures (+1, −1) so that P · Q = η µν P µ Q ν = P 0 Q 0 − P 1 Q 1 is the dot product between two non-null 2D vectors P µ and Q µ . We also choose the antisymmetric Levi-Civita tensor ε µν such that ε 01 = +1 = ε 10 , ε µν ε µν = −2!, ε µν ε νλ = δ λ µ , etc.
where = ∂ 2 0 − ∂ 2 1 is the d'Alembertian operator, Σ is the local and infinitesimal dualgauge parameter and the superscripts (1, 2) denote the dual-gauge transformations for the Lagrangian densities L (b 1 ) and L (b 2 ) , respectively. We note that the Lagrangian densities L (b 1 ,b 2 ) transform, under the above dual-gauge transformations δ (1,2) dg , as follows: Thus, it is clear that, to maintain the dual-gauge symmetries in the 2D theory, we have to impose the condition ( + m 2 ) Σ = 0, from outside, on the dual-gauge parameter Σ. A few noteworthy points, at this juncture, are as follows. First, we point out that the nomenclature of the dual-gauge symmetry is appropriate because δ In other words, the total gauge-fixing terms (∂ · A ± mφ), owing their fundamental origin to the dual-exterior derivative , remain invariant. Second, the dual-gauge parameter has to be restricted by ( + m 2 ) Σ = 0 to maintain the dual-gauge symmetry in the theory. One can take care of this restriction by introducing the (anti-)ghost fields (C)C within the framework of BRST formalism as we shall see in subsection 4.3. Third, the dual-gauge symmetry transformations exist only in the specific two (1 + 1)-dimensions of spacetime for the Abelian 1-form gauge theory whereas the local gauge as well as (anti-)BRST symmetries exist in any arbitrary dimension of spacetime. Fourth, the perfect analogue of the gauge symmetry [cf. (4)] does not exist for the dual-gauge symmetry (because we have to impose the restriction ( + m 2 ) Σ = 0 from outside). Finally, in the forthcoming sections, we shall see that one can have perfect (anti-)dual-BRST [or (anti-)co-BRST] symmetries in the theory where Σ will be replaced by the (anti-)ghost fields (C)C. We claim that there would not be any restrictions on anything (from outside) when we shall discuss the full (anti-)BRST and (anti-)co-BRST invariant Lagrangian densities of the theory.

Nilpotent (Anti-)co-BRST Symmetry Transformations: Geometrical Superfield Technique
As prescribed by the superfield formalism, first of all, we generalize the 2D arbitrary theory onto the (2, 2)-dimensional supermanifold and promote the ordinary co-exterior derivative δ = − * d * onto the (2, 2)-dimensional superfield, as where the (⋆) operator is the Hodge duality operation, defined on the (2, 2)-dimensional supermanifold. The working rule for the operation of (⋆) has been worked out in our earlier paper [31] and we exploit these results in the following dual-HC (DHC): We note that the operation of co-exterior derivative δ = − * d * on the connection 1-form (A (1) = dx µ A µ ) yields (∂ · A) which is a zero-form. We can add/subtract a scalar field φ to it as is the case with the gauge-fixing terms (∂ · A ± mφ) that have been incorporated in L (b1,b2) .
where the l.h.s. is (− ⋆d ⋆Ã (1) ) and r.h.s. is obviously equal to the Lorentz condition for the gauge-fixing [i.e. (∂ · A)]. The definition ofd andÃ (1) are quoted in (6) and the expression for the expansions of the superfields are listed in (7).
The explicit expression for the computation of the l.h.s. of the DHC, in equation (45), is as follows (see, e.g. [31] for details) where S θ θ and Sθθ coefficients, in the above, have turned up while taking the Hodge duality (⋆) operation on the following 4-forms (defined on the (2, 2)-dimensional supermanifold) while the computations of d ⋆ A (1) is performed, namely; It is to be noted that (d ⋆Ã (1) ) is a four-form on the (2, 2)-dimensional supermanifold and when we perform another (⋆) operation on it, the differentials of (47) appear. In the above, S θ θ and Sθθ are symmetric in θ andθ and all the other coefficients of the l.h.s. of (45) have been worked out in our earlier paper [31]. On the comparison of the l.h.s. and r.h.s. of (45), we obtain the following: It is clear that, unlike the HC where all the secondary fields of expansions (7) are exactly and uniquely determined, in the case of DHC, the secondary fields are not uniquely determined and there can be various (non-)local choices for the solution of (48) (see, e.g. [32] for details). Thus, we have the complete freedom to make the choices. Finally, we select the following local expressions * * for the solution of (48), namely; which, unambiguously, satisfy ∂ · R (1) = ∂ · R (2) = ∂ · S = 0 and B 2 + B 3 = 0. Ultimately, we obtain the following expansions for the superfields along the Grassmannian (θ,θ)-directions of the (2, 2)-dimensional supermanifold after the application of DHC: ≡C + θ (s adC ) +θ (s dC ) + θθ (s d s adC ), (50) * * From now on, we shall focus only on the Lagrangian density L (b1) of (41) and its generalization to the (anti-)co-BRST invariant Lagrangian density (57) (see below). However, it is straightforward to make the local choices for the Lagrangian density L (b2) , too. For instance, we can choose R where the superscript (dh) denotes the expansions of the superfields after the application of DHC. A close look at the above expansions demonstrates that we have already obtained the (anti-)co-BRST symmetry transformations for the gauge field (A µ ) and corresponding (anti-)ghost fields (C)C. Physically, the DHC states that the dual-gauge invariant quantity [i.e. δ (1,2) dg (∂ ·A = 0)], which is nothing but the Lorentz condition (∂ ·A) for the gauge-fixing, does not depend on the Grassmannian variables θ andθ in any form.
To obtain the (anti-)co-BRST symmetry transformations for theφ field, we exploit the strength of augmented superfield formalism where we demand that all the dual-gauge [or (anti-)co-BRST] invariant quantities should remain independent of the Grassmannian variables θ andθ. In this context, we observe that δ under the dual-gauge transformations (42). Thus, we demand the following dual-GIR on the superfields of the (2, 2)-dimensional supermanifold, namely; We note that the DGIR combines DHC and the dual-gauge invariance together in a fruitful fashion. Taking the help from (50) and using the following expansion for the superfield Φ(x, θ,θ) along the Grassmannian (θ,θ)-directions of the (2, 2)-dimensional supermanifold: we obtain the following results: It is obvious, from the above, that f 1 (x) and f 2 (x) are fermionic in nature and b 1 (x) is bosonic. Thus, we note that we have obtained the secondary fields of expansions (52) in terms of the basic as well as auxiliary fields of the ordinary 2D BRST-invariant theory. Plugging in the above values in (52), we deduce the following superexpansion: where the superscript (dg) denotes the superexpansion after the application of dual-GIR (DGIR) on the superfields of the (2, 2)-dimensional supermanifold. A careful observation of (50) and (54) leads to the derivation of the following fermionic (anti-)co-BRST symmetry transformations for the whole theory, namely; It is clear that the 2D (anti-)co-BRST symmetry transformations (55) are derived from the superexpansions (50) and (54) which are present on the (2, 2)-dimensional supermanifold. Hence, there should be some connection between the 2D (anti-)co-BRST symmetries and the superfield formalism on (2, 2)-dimensional superspace. A close observation, at the superexpansions in (50) and (54), leads to the following mappings: whereÑ (dh,dg) (x, θ,θ) is the superfield obtained after the application of DHC and DGIR on the (2, 2)-dimensional supermanifold and N(x) is the ordinary 2D field of our present (anti-)co-BRST invariant theory. It is evident that the transformations (55) would be automatically off-shell nilpotent and absolutely anticommuting because these are identified with the translational operators (∂ θ , ∂θ), along the Grassmannian directions (θ,θ) of the (2, 2)-dimensional supermanifold, which satisfy ∂ 2 θ = ∂ 2 θ = 0, ∂ θ ∂θ + ∂θ ∂ θ = 0 due to their inhernt properties. We close this subsection with the remark that Φ (x, θ,θ) = φ(x) because φ(x) is a dual-gauge invariant quantity as is clear from δ

Lagrangian Densities: (Anti-)co-BRST Invariance
The (anti-)co-BRST invariant version of the 2D Lagrangian density L (b 1 ) of (41), in its full blaze of glory, is the one that incorporates the FP-ghost terms, namely; It will be noted that the gauge-fixing and Faddeev-Popov ghost terms of the above Lagrangian density are same as that of the (anti-)BRST invariant Lagrangian density (21). Under the off-shell nilpotent and absolutely anticommuting (anti-)co-BRST symmetry transformations s (a)d [cf. (55)], the above Lagrangian density transforms to the total spacetime derivatives as illustrated below: Hence, the action integral S = dx L B of our theory remains invariant under s (a)d due to the Gauss divergence theorem applied for the physically meaningful fields of the theory. We note that the gauge-fixing and Faddeev-Popov ghost terms of the Lagrangian densities (21) have been derived by exploiting the off-shell nilpotent (anti-)BRST symmetry transformations [cf. (20)]. In exactly similar fashion, it is interesting to observe that The above expressions show that there are three different ways (modulo a total spacetime derivative term) to write the kinetic term plus the Faddeev-Popov (FP) ghost term in the language of the off-shell nilpotent (s 2 a(d) = 0) (anti-)co-BRST symmetry transformations s a(d) which also absolutely anticommute (i.e. s d s ad + s ad s d = 0) with each-other in their operator form.
We can express the above three relations in the language of superfield formalism because we observe that the following expressions: also lead to the derivation of the sum of a part of kinetic term and FP-ghost terms. Ultimately, this exercise implies that the sum of kinetic and FP-ghost terms: is always (anti-)dual-BRST invariant quantity (modulo a total spacetime derivative) because this is trivially true when we take into account the nilpotency and absolute anticommutativity of the (anti-)co-BRST symmetry transformations. In other words, we conclude that We have already seen that a part of the kinetic term and the total of FP-ghost terms can be expressed in terms of the superfields obtained after the application of DHC and DGIR [cf. (60)]. Furthermore, the kinetic term 1 2 ∂ µ φ ∂ µ φ for the φ field and the total gauge fixing term † † [B (∂ · A + m φ) + 1/2(B 2 )] would remain intact within the framework † † We note that B (∂ · A + m φ) → B (∂ µ B µ(dh) + mφ) in the superfield formalism and it is trivial to check that B (∂ · B (dh) ) = B (∂ · A) [cf. (50)] so that B (∂ · A + m φ) → B (∂ · A + m φ) without any change whatsoever when we generalize it onto the (2, 2)-dimensional supermanifold.
of superfield formalism as they are the dual-gauge [or (anti-)co-BRST] invariant quantities. The rest of the terms can be generalized onto the (2, 2)-dimensional supermanifold as: where the symbols have already been explained earlier and they are nothing but the superexpansions after the application of the DHC and DGIR [cf. (50), (54)].
It is interesting to note that the last term of (62) is always an (anti-)co-BRST invariant quantity because we observe the following: where we have taken the expansions of B µ (x, θ,θ) from (50). Taking the help of the mappings (56), we note the following: is always a total spacetime derivative. The rest of the terms in (62) are also (anti-)co-BRST invariant quantity because we check that Ultimately, we conclude that is always a total spacetime derivative. As a consequence, this specific part of the Lagrangian density (i.e. m Eφ − 1 2 ∂ µφ ∂ µφ + 1 2 m 2 A µ A µ ) is an (anti-)co-BRST invariant quantity. Finally, we have the total expression for the 2D Lagrangian density (57) in the superfield formalism, on the (2, 2)-dimensional supermanifold, as where all the symbols have been explained earlier. The (anti-)co-BRST invariance of the Lagrangian density, within the framework of superfield formalism, is Due to the above observations, it is clear that the action integral would remain invariant under the (anti-)co-BRST symmetry transformations. Finally, we would like to state that we have accomplished our goal of capturing the (anti-)co-BRST invariance of the action integral within the framework of superfield formalism where we have used the superfields that have been obtained after the application of DHC and DGIR. We further note that the expressions in (58) and (67) match very nicely. The appearance of the terms like B ∂ µC , B ∂ µ C, i B ∂ µ B in the parenthesis of above equation is due to the same kind of arguments as we offered at the end of equation (29) in the context of (anti-)BRST symmetries and corresponding Lagrangian invariance under these symmetries.

Nilpotency and Anticommutativity of the Conserved (Anti-) co-BRST Charges: Superfield Formulation
Exploiting the standard technique of Noether theorem and using the appropriate equations of motion, we obtain the following expressions for the conserved and off-shell nilpotent (anti-)co-BRST [or (anti-)dual BRST] charges: which have been derived from the Lagrangian density (57) that has led to the following conserved (i.e. ∂ µ J µ (a)d = 0) Noether currents The conservation law (i.e. ∂ µ J µ (a)d = 0) can be proven by exploiting the following equations of motion emerging from the Lagrangian density (57), namely; It is straightforward to check that the (anti-)co-BRST charges can be expressed in terms of the (anti-)co-BRST symmetry transformations as: Exploiting the mapping (56), it can be seen that the above expressions could be recast in the language of the superfields, obtained after the application of DHC and DGIR, as From the above expressions, too, one can prove the off-shell nilpotency (Q 2 (a)d = 0) of the charges Q (a)d by observing that the following is true, namely; The above observation of the nilpotency of Q (a)d is intimately connected with the nilpotency ∂ 2 θ = ∂ 2 θ = 0 of translational generators ( ∂ θ , ∂θ) along the Grassmannian directions. The nilpotency of Q (a)d can also be proven by the following expressions of Q (a)d in terms of the (anti-)co-BRST symmetry transformations s (a)d , namely; Thus, it is clear that the following will be true, namely; due to the nilpotency of s (a)d (i.e. s 2 (a)d = 0 ⇔ Q 2 (a)d = 0). In the language of superfield formalism, the expressions (74) can be written as which demonstrate trivially the following where the nilpotency of ∂ θ and ∂θ (i.e. ∂ 2 θ = ∂ 2 θ = 0) plays a decisive role. To prove the absolute anticommutativity of Q (a)d , we note the following interesting expressions for the conserved (anti-)co-BRST charges: The above expressions automatically imply the following: Thus, we point out a very interesting observation that the absolute anticommutativity property of the (anti-)co-BRST charges is deeply and clearly connected with the nilpotency of the (anti-)co-BRST symmetry transformations (i.e. s 2 (a)d = 0). These expressions (78) could be also written in terms of superfields, translational generators (∂ θ , ∂θ) and differentials (dθ, dθ) defined on the (2, 2)-dimensional supermanifold, as The above expressions capture the anticommutativity property of the (anti-)co-BRST charges in the language of superfield formalism, as where the properties ∂ 2 θ = 0, ∂ 2 θ = 0 play important roles when we use the expressions for Q (a)d from (80). The anticommutativity property is hidden in (81) in view of the mapping (56) which imply that (81) can be written as: s d Q ad = i {Q ad , Q d } = 0 and s ad Q d = i {Q d , Q ad } = 0 primarily due to ∂ 2 θ = 0, ∂ 2 θ = 0. We close this subsection with the remark that the nilpotency and absolute anticommutativity properties of the (anti-)co-BRST symmetry transformations (and their corresponding conserved charges) are related with the properties ∂ 2 θ = 0, ∂ 2 θ = 0 and ∂ θ ∂θ + ∂θ ∂ θ = 0. These relations are, in turn, inter-connected with each other because the limiting case of the latter (i.e. ∂ θ ∂θ + ∂θ ∂ θ = 0) leads to the derivation of the former (∂ 2 θ = 0, ∂ 2 θ = 0) when we set ∂ θ = ∂θ in the latter relationship of anticommutativity.

On a Unique Bosonic Symmetry, the Ghost-Scale Symmetry and the Discrete Symmetries
From the four nilpotent (s 2 (a)b = s 2 (a)d = 0) symmetries of the theory, we can construct a unique bosonic symmetry ‡ ‡ s ω = {s b , s d } ≡ −{s ab , s ad }, under which, the relevant fields of our present theory, described by the Lagrangian density (57), transform as modulo an overall factor of (− i). We note that {s d , s ad } = 0, {s d , s ab } = 0, {s b , s ad } = 0, {s b , s ab } = 0. One of the decisive features of the above bosonic symmetry is the observation that the ghost part of the Lagrangian density remains invariant. Under the above transformations (82), the Lagrangian density (57) transforms as As a consequence, the action integral S = dx L B remains invariant. The above symmetry transformation, according to Noether theorem, leads to the derivation of the following conserved charge (as the analogue of the Laplacian operator): which emerge from the Noether conserved (∂ µ J µ ω = 0) current The conserved charge (84) is the generator of the continuous and infinitesimal bosonic symmetry transformations (82) which can be checked by using the standard formula. Our theory, described by the Lagrangian density (57), is endowed with the following ghost-scale symmetry transformations (with a global scale parameter Ω), namely; where the numbers in the exponentials denote the ghost numbers of the fields. The infinitesimal version of the above scale transformations (s g ), for Ω = 1, are which are generated by the following ghost charge Q g [2,3]: The above charge has been derived from the conserved current J µ g = i(C ∂ µ C − ∂ µC C). The conservation law ∂ µ J µ g = 0 can be proven by using the Euler-Lagrange equations of motion ( + m 2 )C = 0 and ( + m 2 ) C = 0 which emerge from (57).
In addition to the above continuous symmetries, we have a set of suitable discrete symmetries in the theory. These symmetries are as follows: It is straightforward to check that the Lagrangian density (57) remains invariant under the above discrete symmetry transformations. Further, it can be readily checked that: where the operator ( * ) is nothing but the operation of the above discrete symmetry transformations on the conserved charges of the theory. We note that two successive operations of the discrete symmetry transformations leave the conserved charges intact. On the other hand, a single operation of the discrete symmetry transformations interchanges each of the pairs (Q b , Q d ) and (Q ab , Q ad ) such that: (Q b ↔ Q d , Q ab ↔ Q ad ) and Q g → − Q g .
There is a simpler way to check the sanctity of the extended BRST algebra listed in (91) where we use the well-known relationship between the continuous symmetry transformations and their generators. For instance, the above algebra can be obtained from the following transformations on the conserved charges, namely; where the l.h.s. can be calculated in a straightforward manner by exploiting the expressions for the six conserved charges and the corresponding continuous symmetry transformations that have been mentioned in the main body of our text. A comparison between (91) and (92) demonstrates that Q ω and ∆ are the Casimir operators for the above algebras. Furthermore, a close look at these algebras leads to the following clear-cut two-to-one mappings: between the conserved charges and the cohomological operators. Furthermore, we note that we have the following beautiful relationship [2,3]: which provides the physical realization of the relationship between the co-exterior derivative and exterior derivative (i.e. δ = − * d * ) defined on an even dimensional spacetime manifold.
In the above equation (95), we observe that it is the interplay between the continuous symmetries (i.e. s (a)b , s (a)d ) and the discrete symmetries (89) that provide the analogue of relationship δ = − * d * . In fact, the latter [i.e. equation (89)] leads to the physical realization of the Hodge duality ( * ) operation of the differential geometry. Thus, it is now clear that the ( * ) in (95) is nothing but the discrete symmetry transformations (89). The minus sign on the r.h.s of (95) is governed by two successive operations of the discrete symmetry transformations (89) on the generic field Ψ: * ( * Ψ) = − Ψ (see, e.g. [33]). One of the distinguishing features of the cohomological operators (d, δ, ∆) is the observation that when they operate on a differential form of a specific degree, the consequences turn out to be completely different. For instance, when the (co-)exterior derivatives operate on a form (f n ) of degree n, they (lower)raise the degree of the form by one (i.e. δ f n ∼ f n−1 , d f n ∼ f n+1 ). On the contrary, when ∆ acts on a form of degree n, it does not change the degree at all (i.e. ∆ f n ∼ f n ). We have to capture these properties in the language of the symmetry properties and conserved charges of our present theory.
The above algebraic features could be also captured in the language of conserved charges. To this end in mind, let us define a state |ψ n in the quantum Hilbert space of states, as: where the eigenvalue n is the ghost number because Q g is the ghost charge [cf. (88)]. Due to the algebra (91), respected by the various charges, it can be readily checked that i Q g Q b |ψ n = (n + 1) Q b |ψ n , i Q g Q ad |ψ n = (n + 1) Q ad |ψ n , i Q g Q d |ψ n = (n − 1) Q d |ψ n , i Q g Q ab |ψ n = (n − 1) Q ab |ψ n , i Q g Q ω |ψ n = n Q ω |ψ n .
Thus, we note that the ghost numbers for the states Q b |ψ n , Q d |ψ n and Q ω |ψ n are (n + 1), (n − 1) and n, respectively. In exactly similar fashion, the states Q ad |ψ n , Q ab |ψ n and Q ω |ψ n also carry the ghost numbers (n + 1), (n − 1) and n, respectively. These properties are exactly like the consequences that ensue due to the operations of the cohomological operators (d, δ, ∆) on a differential form of degree n We conclude that, if the degree of a form is identified with the ghost number, then, the operation of (d , δ, ∆) on this given form is exactly like the operation of the set (Q b , Q d , Q ω ) and/or (Q ad , Q ab , Q ω ) on the state with ghost number equal to the degree of the form. Thus, the mappings (94) are correct as far as the algebraic structures of (91) and (92) are concerned and we have two-to-one mapping from the conserved charges of the theory to the de Rham cohomological operators (d, δ, ∆) of differential geometry. A close look at equations (90) and (91) leads to the conclusion that the algebra (91) remains invariant under any number of operations of discrete (duality) symmetry transformations (89). This establishes that our present 2D theory is a perfect model for the Hodge theory where the continuous symmetry transformations (and corresponding generators) provide the physical realizations of the cohomological operators. On the other hand, it is the discrete symmetry transformations of the theory that are the physical analogue of the Hodge duality ( * ) operation of differential geometry. Finally, we observe that the ghost number of a specific state in the quantum Hilbert space provides the physical analogue of the degree of a form of differential geometry as far as its cohomological aspects are concerned.

Conclusions
In our present endeavor, we have applied the augmented version of superfield formalism to derive the off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations for the modified version of 2D Proca theory. We have exploited the theoretical strength of horizontality condition (HC) and gauge invariant restriction (GIR) to derive the (anti-) BRST symmetries for all the fields of our present 2D theory. In addition, we have made use of the dual-HC (DHC) and dual-GIR (DGIR) to obtain the complete set of (anti-)co-BRST symmetry transformations for all the fields of our present theory. The local gauge symmetry transformations [cf. (4)] are the perfect "classical" version of the (anti-)BRST symmetries which exist in any arbitrary dimensions of spacetime. However, there is no such perfect "classical" analogue (see, e.g., subsection 4.1) for the (anti-)co-BRST symmetries of our present theory. The latter symmetries exist only in specific dimensions of spacetime and they are always "quantum" in nature. For instance, for the Abelian 1-form gauge theory, these "quantum" symmetries exist only in two dimensions of spacetime.
In our subsections 3.3 and 4.4, we have expressed (anti-)BRST and (anti-)co-BRST charges in various forms due to our knowledge of the superfield approach to BRST formal-ism. In these subsections, we have been able to provide the meaning of nilpotency and absolute anticommutativity in the language of superfield formalism. We have been able to establish connections between the properties of nilpotency and absolute anticommutativity, too. In fact, it is the strength of the superfield formalism that we have expressed (anti-) BRST and (anti-)co-BRST charges in a completely novel fashions which have, hitherto, not been pointed out in the literature. Thus, there are completely novel results in our subsections 3.3 and 4.4 as far as our present investigation is concerned.
In addition to the above results, there are applications of dual-HC and dual-GIR (DGIR) in deducing the all set of (anti-)co-BRST symmetry transformations for all the fields of our present theory. These derivations are also novel results. In particular, the application of DGIR, in the derivation of the (anti-)co-BRST symmetry transformations for the pseudoscalar field (φ), is a completely new result which has not been discussed in the literature. The symmetries of the theory enforce the pseudo-scalar field to have a negative kinetic term. Since this field is massive [i.e. ( + m 2 )φ = 0], it is a very good candidature for the dark-matter [28,29]. We lay emphasis on the fact that the Stuckelberg's scalar field (φ) has always a positive kinetic term and, hence, it is an ordinary matter. Both these fields (i.e. φ, φ) have interactions with electric field (E) and gauge-fixing term(∂ · A), respectively, via a mass-coupling. Therefore, they experience only the gravitational interaction.
In our investigation, we have provided physical realizations of the de Rham cohomological operators in the language of the continuous symmetry transformations (and their corresponding charges). Further, we have shown that a set of discrete symmetry transformations provide the physical analogue of the Hodge duality ( * ) operation of differential geometry. Ultimately, we have shown that, at the algebraic level, the set of six conserved charges of our theory obey exactly the same algebra as that of the de Rham cohomological operators of differential geometry. This algebra remains invariant [cf. (90)] under the discrete symmetry transformations (89) which are the analogue of Hodge theory ( * ) operation. The degree of a form finds its physical analogue as the ghost number of a state in the quantum Hilbert space of states. Thus, our present 2D modified version of Proca theory turns out to be perfect model for the Hodge theory. The unique feature of our present theory is the co-existence of the mass and various kinds of internal symmetries together in a physically and mathematically meaningful manner.
It would be nice future endeavor to study the above kind of possibilities in the cases of 3D and 4D massive gauge theories [34,35] where the gauge invariance and mass would co-exist together. In other words, we would like to study whether Stueckelberg's type of technique would be able to modify the above theories in such a way that they could also become massive field theoretic models for the Hodge theory. We speculate that such kind of situation will exist and these models will provide candidates for the dark matter in more physical 3D and 4D of spacetime (analogous to the massive pseudo-scalarφ of our present 2D theory). Our speculation is based on the fact that we have already shown that the 4D free Abelian 2-form gauge theory is a model for the Hodge theory where a massless pseudo-scalar field does exist with a negative kinetic term (see, e.g. [17,18] for details). We are currently intensively busy with such kinds of problems and, hopefully, we shall be able to report about our progress in our future publications [36].