New symmetries for Abelian gauge theory in superfield formulation

We show the existence of some new local, covariant and continuous symmetries for the BRST invariant Lagrangian density of a free two ($1 + 1$)-dimensional (2D) Abelian U(1) gauge theory in the framework of superfield formalism. The Noether conserved charges corresponding to the above local continuous symmetries find their geometrical origin as the translation generators along the odd (Grassmannian)- and even (bosonic) directions of the four ($2 + 2)$-dimensional compact supermanifold. Some new discrete symmetries are shown to exist in the superfield formulation. The logical origin for the existence of BRST- and co-BRST symmetries is shown to be encoded in the Hodge decomposed versions (of the 2D fermionic vector fields) that are consistent with the discrete symmetries of the theory.


Introduction
The superfield approach [1][2][3][4][5] to Becchi-Rouet-Stora-Tyutin (BRST) formalism is a wellestablished technique which provides the geometrical origin for the existence of (anti-)BRST charges as the generators of translation along the Grassmannian directions of the compact supermanifold that is parametrized by the spacetime coordinates and two extra anticommuting (Grassmannian) variables. In fact, in this scheme, the (p + 1)-form super curvature tensor for a p-form (p = 1, 2, 3, ...) gauge theory is restricted to be flat along the Grassmannian directions of the supermanifold. This restriction, popularly known as the horizontality condition † , leads to the derivation of the (anti-)BRST symmetry transformations for the Lagrangian density of a p-form gauge theory. In this derivation, the mathematical power of the super exterior derivatived alone (which is only one of the three de Rham cohomology operators ‡ of differential geometry) is exploited when it operates on the super p-form potential of a p-form gauge theory to make it a (p + 1)-form curvature tensor through Maurer-Cartan equation. Thus, it is an interesting endeavour to explore the possibility of the existence of some new local symmetries by exploiting the other two super de Rham cohomology operators (δ: co-exterior derivative;∆: Laplacian operator) of differential geometry and find out their geometrical interpretation in the language of some kind of translation generators on an appropriately chosen compact supermanifold.
The purpose of the present paper is to show the existence of some new local, covariant and continuous symmetries for the free 2D Abelian gauge theory that emerge due to the operation of super co-exterior derivativeδ (δ = − * d * ,δ 2 = 0, * = Hodge duality operation) and super Laplacian operator∆ (∆ =dδ +δd) on the super one-form connectionÃ together with the analogue of the horizontality conditions w.r.t. these super de Rham cohomology operators. In fact, we demonstrate that (anti-)co-BRST symmetry-and a bosonic symmetry transformations emerge when we exploit these super cohomological operators on a four (2 + 2)-dimensional compact supermanifold and they turn out to be exactly same as the new local symmetries obtained recently in a set of papers in the Lagrangian formalism alone [11][12][13][14][15]. It has been established in these works that the 2D free-as well as interacting (non-)Abelian (one-form) gauge theories provide the field theoretical models for the Hodge theory where all the de Rham cohomology operators find their interpretation as the local Noether charges that generate these new local, continuous and covariant symmetries. Such symmetries and corresponding generators (conserved Noether charges) have also been shown to exist for the four (3 + 1)-dimensional free two-form Abelian gauge theory [16]. In these attempts, the local Noether charges have also been shown to refine the BRST coho-mology [12] and define the analogue of the Hodge decomposition theorem(HDT) § in the quantum Hilbert space of states [11][12][13][14][15][16][17]. Exploiting these ideas, it has been shown that 2D free (non-)Abelian gauge theories belong to a new class of topological field theories (TFTs) [17]. However, in all the above attempts, the geometrical origin for the existence of these charges has not yet been discussed. In the present work, we show that the (anti-)BRSTand (anti-)co-BRST symmetry generators (conserved and nilpotent Noether charges (Q b )Q b and (Q d )Q d respectively) are the translation generators along the Grassmannian (odd) directions of the (2 + 2)-dimensional compact supermanifold and they owe their origin to the super cohomological operatorsd andδ. A bosonic symmetry, generated by the Casimir operator, turns out to be the translation generator along the bosonic (even) direction of the supermanifold and its origin is encoded in the super operator∆. This even (bosonic) direction on the supermanifold is equivalent to a couple of intertwined Grassmannian directions. The local conserved charges in the theory provide an analogue of the set (d, δ, ∆).
The outline of our present paper is as follows. In Sec. 2, we set up the notations and recapitulate some of the salient features of our earlier works [11][12][13][14][15][16][17]. Section 3 is devoted to the derivation of (anti-)BRST symmetry transformations through horizontality condition [3,4]. In Sec. 4, we exploit the super co-exterior derivative and derive the (anti-)co-BRST symmetry transformations exploiting the analogue of the horizontality condition w.r.t.δ. We discuss some interesting discrete symmetries and the Hodge decomposed versions of 2D vectors in Sec. 5. A local bosonic symmetry is obtained in Sec. 6 using the super Laplacian operator∆. Finally, we make some concluding remarks in Sec. 7.

BRST-and dual BRST symmetries: A brief sketch
Let us start off with the BRST invariant Lagrangian density L b for the free two (1 + 1)dimensional ¶ Abelian gauge theory in the Feynman gauge [6,[18][19][20] where F µν = ∂ µ A ν −∂ ν A µ is the field strength tensor (curvature two-form) derived from oneform A = dx µ A µ (with A µ = vector potential) by the application of the exterior derivative d (i.e. F = dA = 1 2 dx µ ∧dx ν F µν ). The gauge-fixing term (zero-form) is derived from one-form A by the application of the co-exterior derivative δ (i.e. (∂ · A) = δA; δ = − * d * ; * = Hodge duality operation). Thus, in some sense, F = dA and (∂·A) = δA are "Hodge dual" to eachother. The (anti-)ghost fields (C)C are anti-commuting (C 2 =C 2 = 0, CC +CC = 0) in nature. Under the following on-shell (✷C = ✷C = 0) nilpotent (s 2 b = 0,s 2 b = 0, s bsb +s b s b = § This theorem states that, on a compact manifold without a boundary, any arbitrary n-form f n , (n = 0, 1, 2..) can be uniquely written as the sum of a harmonic form h n (∆h n = dh n = δh n = 0), an exact form de n−1 and a co-exact form δc n+1 as: f n = h n + de n−1 + δc n+1 [7][8][9][10]. ¶ We follow here the conventions and notations such that the 2D flat Minkowski metric is: the Lagrangian density (2.1) remains invariant. The same Lagrangian density is also invariant under the following on-shell (✷C = ✷C = 0) nilpotent (s 2 d =s 2 d = 0, s dsd +s d s d = 0) (anti-)dual BRST transformations (s d )s d on the basic fields [11,12,17] We christen the above new continuous, covariant and nilpotent symmetry as the (anti-) dual BRST symmetry because it is the gauge-fixing term (∂ · A) = δA (Hodge dual to the curvature two-form F = dA) that remains invariant. In contrast, it is the curvature two-form F = dA that remains invariant under the (anti-)BRST symmetry transformations (2.2). The anti-commutator of the above two symmetries leads to yet another new bosonic type symmetry transformation s w (s under which the ghost fields remain invariant. The Noether conserved charges (Q r ) corresponding to the above continuous symmetries are the generators for the above transformations [11][12][13][14][15][16][17]. This statement can be concisely expressed as where brackets [ , ] ± stand for the (anti-)commutators for any arbitrary generic field Ψ being (fermionic)bosonic in nature. Here the ghost charge Q g (Q g = −i dx[CĊ +CĊ]) generates the continuous scale transformations: C → e −Σ C,C → e ΣC , A µ → A µ (where Σ is a global parameter) for the invariance of the Lagrangian density (2.1). Now we wish to discuss some of the discrete symmetries present in the theory. It is interesting to note that s b ↔s b under the discrete symmetry transformations : On the other hand, s d ↔s d when we take: Under yet another discrete symmetry transformations [17,15] the Lagrangian density (2.1) remains form-invariant and the symmetry transformations (2.2) and (2.3) are related to one-another. Furthermore, this discrete symmetry turns out to be the analogue of the Hodge * duality operations of the differential geometry as one of the key relationships:

exists for any arbitrary
Here the notations, followed in Ref. [20], are adopted. In fact, in its totality, a BRST transformation δ B is the product of an anti-commuting spacetime independent parameter η and s b (i.e. δ B = η s b ).
generic field Ψ of the theory [15,17]. The (±) sign in this relationship is dictated by the existence of the corresponding sign in the operation: * ( * Ψ) = ± Ψ where * is nothing but the discrete transformations (2.6). The Lagrangian density (2.1) and the corresponding symmetric energy-momentum tensor T µν can be expressed, modulo some total derivatives, as [11,12,17] (2.8) The expressions in (2.7) establish the topological nature of 2D free Abelian gauge theory as topological invariants and their recursion relations have been obtained in Ref. [17]. The algebra amongst the conserved charges of the theory are reminiscent of the algebra obeyed by the de Rham cohomology operators of differential geometry. Thus, the present theory is a field theoretic model for the Hodge theory and it represents a new class of topological field theory which captures some of the key features of Witten-and Schwarz type TFTs.

(3.4)
Here the signs in the expansion are chosen for the later convenience. It is straightforward to see that the local fields R µ (x),R µ (x), C(x),C(x), s(x),s(x) are fermionic (anti-commuting) in nature and the bosonic (commuting) local fields are: A µ (x), S µ (x), ±E(x), ±(∂ · A)(x) in the above expansion so that bosonic-and fermionic degrees of freedom can match. It is interesting to note that the above expansion is such that: (Φ(x, θ,θ)) 2 = 0, (Φ(x, θ,θ)) 2 = 0, Φ(x, θ,θ)Φ(x, θ,θ) +Φ(x, θ,θ) Φ(x, θ,θ) = 0 and [B µ (x, θ,θ), B ν (x, θ,θ)] = 0. As a consequence, it is straightforward to verify thatÃ ∧Ã = 1 2 [Ã,Ã] = 0. The super curvature tensor (two-formF ) for the gauge theory can be constructed by exploiting (3.1) (i.e.F =dÃ +Ã ∧Ã). For the U(1) gauge theorỹ where use has been made of the fact that the nilpotency of the super exterior derivative (d 2 = 0) implies the following relations for the wedge products on the supermanifold etc. Now the soul-flatness (or horizontality) condition imposes the following restrictioñ In the language of the component superfields of (3.3), the above condition implies and the following conditions on the local component fields of the superfield B µ (x, θ,θ) The conditions (3.7) lead to the following solutions It will be noticed that the signs in the expansion (3.4) are chosen such that the condition: ∂ θ Φ+∂θΦ = 0 is satisfied trivially. Furthermore, the solutions in (3.9) automatically satisfy the conditions in (3.8). Now the expansion in (3.4) can be expressed as (3.10) We conclude that the horizontality condition on the super two-form curvature tensor for the U(1) Abelian gauge theory leads to the derivation of BRST-and anti-BRST symmetries for the Lagrangian density (2.1). The corresponding conserved and nilpotent charges find their geometrical origin as the translation generators along the Grassmannian directions of the supermanifold. In other words, it is the power ofd that provides the geometrical interpretation for Q b andQ b as translation generators (cf. (2.5)). Thus, the mapping is: d ⇔ (Q b ,Q b ) but the ordinary exterior derivative d is identified with Q b alone because the latter increases the ghost number of a state by one [11,12,17] as d increases the degree of a form by one on which it operates [7][8][9][10].

Super co-exterior derivative and (anti-)co-BRST symmetry transformations
We operate the super co-exterior derivativeδ = − * d * on the super one-form connectioñ A of (3.1), with the Hodge duality operation * defined on the differentials and their wedge products (for the case of (2 + 2)-dimensional compact supermanifold), as * (dx µ ) = ε µν (dx ν ), * (dθ) = (dθ), * (dθ) = (dθ), where ε µθ = −ε θµ , ε µθ = −εθ µ and s θθ = sθ θ etc. It is obvious that the operation (δÃ) would result in a superscalar (zero-form) superfield (asδ reduces the degree of a super form by one on which it operates). The explicit expression for this superfield is The analogue of the horizontality condition with the super co-exterior derivativeδ is to equate Eqn. (4.2) to the gauge-fixing term δA = (∂ · A) (i.e.,δÃ = δA). This restriction leads to the following conditions on the superfields and an additional restriction on the local field components of the expansion (3.4) for the bosonic superfield B µ (x, θ,θ). The latter conditions are The solutions for the restriction (4.3) are listed below which automatically satisfy the restrictions (4.4). It will be noticed that the choice of the signs in the expansion (3.4) are such that the restriction ∂ θΦ + ∂θΦ = 0 is satisfied trivially. In terms of solutions (4.5), the expansion (3.4) can be re-expressed as

(4.6)
It is worth pointing out that the above expansion can be directly obtained from the definition of * operation in Section 2 (cf. Eqn. (2.6)). Now exploiting dual-and anti-dual BRST symmetries (discussed in Sec. 2), we can rewrite Eqn. (4.6) as which is the analogue of Eqn. (v) A single restrictionδÃ = δA produces co-BRST-and anti-co-BRST symmetry transformations for the Lagrangian density (2.1). Thus, the mapping is:δ ⇔ (Q d ,Q d ) but the ordinary co-exterior derivative δ is identified with Q d alone because it decreases the ghost number of a state by one [11,12,17] as δ reduces the degree of a given form by one on which it operates [7][8][9][10].

Discrete symmetries
We have discussed a few discrete symmetries at the fag end of Sec. 2. Now we exploit these discrete symmetries vis-a-vis our superfield expansion (3.4). We emphasize the fact that, for the BRST-and dual BRST symmetries, we have shown that: s(x) = 0,s(x) = 0 in the expansion (3.4). Thus, we shall now be concentrating on (3.4) only for this case. First of all, it is straightforward to verify that under the following discrete transformations the superfields in (3.4) undergo the following change Furthermore, it can be trivially checked that the above transformations still satisfy: Φ 2 = 0,Φ 2 = 0, ΦΦ +ΦΦ = 0 and [B µ , B ν ] = 0. Yet another interesting point is to see that in the limit: θ → 0,θ → 0, the above transformations reduce to : C → ±iC,C → ±iC, A µ → A µ . Thus, transformations (5.2) are the generalization of the discrete symmetry (2.6). A close look at the expressions for R µ ,R µ , S µ in equations (3.9) and (4.5) allows us to write down the Hodge decomposed versions for these 2D fermionic (R µ ,R µ )-and bosonic (S µ ) vectors (appearing in the expansion of the bosonic superfield B µ (x, θ,θ)) as which are solutions to the transformations: R µ → −R µ ,R µ → −R µ , S µ → S µ under the discrete transformations (2.6). However, it is interesting to note that the r.h.s. of the expression for S µ is the equation of motion for the 2D photon: ∂ µ F µν + ∂ ν (∂ · A) = 0 (with F 10 = E). Thus, S µ turns out to be zero on the on-shell. It can be checked that S µ = −ε µν ✷A ν also transforms as S µ → S µ under (2.6) because ✷ → ✷ under ∂ µ → ±iε µν ∂ ν . Now let us concentrate on the discrete symmetries: that connect BRST-to anti-BRST-as well as co-BRST-to anti-co-BRST symmetry transformations. The generalized version of these symmetries, vis-a-vis our superfield expansion (3.4), is: under which the superfields transform as It will be noticed that in the limit θ → 0,θ → 0, we get back our original discrete symmetries: It is interesting to point out that the solutions (5.3) are no longer the appropriate solutions for the present case. In fact, taking the help of (3.9) and (4.5), now the solutions for the 2D fermionic vectors are which are nothing but the orthogonal Hodge decomposed version of the corresponding solution in (5.3). Now, for the present case where (∂ · A) ↔ −(∂ · A), E ↔ −E, it is clear that any arbitrary linear combination: S µ = P ∂ µ (∂ · A) + Q ε µν ∂ ν E (where P and Q are some c-number constants) would lead to S µ = 0 for the requirement S µ → S µ (cf. (5.4)) to be satisfied. The origin for the existence of the (anti-)BRST-and (anti-)co-BRST symmetries in the theory is encoded in the orthogonal relations (5.3) and (5.6) for the Hodge decomposed versions of R µ andR µ . In fact, these relations show that ∂ µ C(∂ µC ) and ε µν ∂ νC (ε µν ∂ ν C) are the separate and independent symmetry transformations for the Lagrangian density (2.1). In the language of the BRST cohomology and HDT, this is the logical explanation for the existence of (anti-)BRST-and (anti-)co-BRST symmetries for the Lagrangian density (2.1) of a free 2D Abelian gauge theory.

Conclusions
We have demonstrated the existence of some new local symmetries by exploiting the mathematical power of the super de Rham cohomology operators of differential geometry defined on a (2 + 2)-dimensional compact supermanifold. As conserved and nilpotent (anti-)BRST charges (Q b )Q b are connected with the super exterior derivatived [3,4], in a similar fashion (anti-)co-BRST charges (Q d )Q d are connected with the super co-exterior derivativeδ. These nilpotent charges turn out to be the translation generators along the Grassmannian directions of the supermanifold. A bosonic charge Q w is shown to be related with the super Laplacian operator∆. This charge turns out to be the translation generator along the bosonic direction (which is equivalent to a couple of intertwined Grassmannian directions) of the supermanifold. The mapping between super operators (d,δ,∆) and the local conserved charges is: The analogy between the ghost number of a state in the quantum Hilbert space and the degree of a differential form allows one to relate the ordinary de Rham cohomology operators (d, δ, ∆) with the conserved charges as: In the setting of the superfield formulation, the above mappings find their geometrical interpretation. The interplay between the discrete-and continuous symmetries of the theory allows one to write down the Hodge decomposed versions for the 2D fermionic vectors which provide an unambiguous explanation for the existence of (anti-)BRST-and (anti-)co-BRST symmetries in the theory. It would be nice to extend these ideas to the interacting case [14,15].