It is shown that two(1 + 1)-dimensional (2D) free Abelian-and self-interacting non-Abelian gauge theories (without any interaction with matter fields) belong to a new class of topological field theories. These new theories capture together some of the key features of Witten-and Schwarz type of topological field theories because they are endowed with symmetries that are reminiscent of the Schwarz type theories but their Lagrangian density has the appearance of the Witten type theories. The topological invariants for these theories are computed on a 2D compact manifold and their recursion relations are obtained. These new theories are shown to provide a class of tractable field theoretical models for the Hodge theory in two dimensions of flat (Minkowski) spacetime where there are no propagating degrees of freedom associated with the 2D gauge boson. *

We demonstrate that the four (3 + 1)-dimensional (4D) free Abelian 2-form gauge theory presents a tractable field theoretical model for the Hodge theory where the well-defined symmetry transformations correspond to the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above continuous symmetry transformations, obey an algebra that is reminiscent of the algebra obeyed by the cohomological operators. The discrete symmetry transformation of the theory represents the realization of the Hodge duality operation that exists in the relationship between the exterior and co-exterior derivatives of differential geometry. Thus, we provide the realizations of all the mathematical quantities, associated with the de Rham cohomological operators, in the language of the symmetries of the present 4D free Abelian 2-form gauge theory.PACS : 11.15.-q, 03.70.+k Keywords: Free 4D Abelian 2-form gauge theory, anticommuting (anti-)BRST symmetries, anticommuting (anti-)co-BRST symmetries, de Rham cohomological operators, analogues of the Curci-Ferrari conditions 1 The exterior derivative d = dx µ ∂ µ (with d 2 = 0), the co-exterior derivative δ = ± * d * (with δ 2 = 0) and the Laplacian operator ∆ = dδ + δd constitute the set of de Rham cohomological operators of differential geometry on a compact manifold without a boundary. These operators follow the algebra: d 2 = δ 2 = 0, ∆ = (d + δ) 2 ≡ {d, δ}, [d, ∆] = 0, [δ, ∆] = 0. The operator * is the Hodge duality operation on a given manifold.

Abstract:We discuss BRST and anti-BRST transformations for an Abelian antisymmetric gauge field in 4D and find that, in order for them to anticommute, we have to impose a condition on the auxiliary fields. This condition is similar to the Curci-Ferrari condition for the 4D non-Abelian 1-form gauge theories and represents a consistency requirement. We interpret it as a signal that our Abelian 2-form gauge field theory is based on gerbes. To support this interpretation we discuss, in particular, the case of the 1-gerbe for our present field theory and write the relevant equations and symmetry transformations for 2-gerbes.

We provide an alternative to the gauge covariant horizontality condition which is responsible for the derivation of the nilpotent (anti-)BRST symmetry transformations for the gauge and (anti-)ghost fields of a (3 + 1)dimensional (4D) interacting 1-form non-Abelian gauge theory in the framework of the usual superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism. The above covariant horizontality condition is replaced by a gauge invariant restriction on the (4, 2)-dimensional supermanifold, parameterized by a set of four spacetime coordinates x µ (µ = 0, 1, 2, 3) and a pair of Grassmannian variables θ andθ. The latter condition enables us to derive the nilpotent (anti-)BRST symmetry transformations for all the fields of an interacting 1-form 4D non-Abelian gauge theory where there is an explicit coupling between the gauge field and the Dirac fields. The key differences and striking similarities between the above two conditions are pointed out clearly.

The derivation of the nilpotent Becchi-Rouet-Stora-Tyutin (BRST)-and anti-BRST symmetries for the matter fields, present in any arbitrary interacting gauge theory, has been a long-standing problem in the framework of superfield approach to BRST formalism. These nilpotent (anti-)BRST symmetries for the Dirac fields are derived in the superfield formulation for the interacting Abelian gauge theory in four (3 + 1)-dimensions (4D) of spacetime. The same type of symmetries are deduced for the 4D complex scalar fields having a gauge invariant interaction with the U(1) gauge field. The above interacting theories are considered on a six (4 + 2)-dimensional supermanifold parametrized by four even spacetime coordinates and a couple of odd elements of the Grassmann algebra. The invariance of the conserved matter (super)currents and the horizontality condition on the (super)manifolds play very important roles in the above derivations. The geometrical origin and interpretation for all the above off-shell nilpotent symmetries are provided in the framework of superfield formalism. .in † The set (d, δ, ∆) of operators, defined on a compact manifold without a boundary, is called the set of de Rham cohomological operators where δ = ± * d * , d = dx µ ∂ µ , ∆ = (d + δ) 2 are called the (co-)exterior derivatives ((δ)d) and the Laplacian operator (∆) respectively. Here * is the Hodge duality operation on the manifold. These operators obey an algebra: d 2 = δ 2 = 0, ∆ = {d, δ}, [δ, ∆] = 0, [d, ∆] = 0 showing that the Laplacian operator ∆ is the Casimir operator for the whole algebra (see, eg, [17,18] for details).

It is shown that the four (3 + 1)-dimensional (4D) free Abelian 2-form gauge theory provides an example of (i) a class of field theoretical models for the Hodge theory, and (ii) a possible candidate for the quasi-topological field theory (q-TFT). Despite many striking similarities with some of the key topological features of the two (1 + 1)-dimensional (2D) free Abelian (and self-interacting non-Abelian) gauge theories, it turns out that the 4D free Abelian 2-form gauge theory is not an exact TFT. To corroborate this conclusion, some of the key issues are discussed. In particular, it is shown that the (anti-)BRST and (anti-)co-BRST invariant quantities of the 4D 2-form Abelian gauge theory obey the recursion relations that are reminiscent of the exact TFTs but the Lagrangian density of this theory is not found to be able to be expressed as the sum of (anti-)BRST and (anti-)co-BRST exact quantities as is the case with the topological 2D free Abelian (and self-interacting non-Abelian) gauge theories. *

We apply the superfield approach to the toy model of a rigid rotor and show the existence of the nilpotent and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations, under which, the kinetic term and the action remain invariant. Furthermore, we also derive the off-shell nilpotent and absolutely anticommuting (anti-) co-BRST symmetry transformations, under which, the gauge-fixing term and the Lagrangian remain invariant. The anticommutator of the above nilpotent symmetry transformations leads to the derivation of a bosonic symmetry transformation, under which, the ghost terms and the action remain invariant. Together, the above transformations (and their corresponding generators) respect an algebra that turns out to be a physical realization of the algebra obeyed by the de Rham cohomological operators of differential geometry. Thus, our present model is a toy model for the Hodge theory.

Abstract:We derive the off-shell nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for all the fields of a free Abelian 2-form gauge theory by exploiting the geometrical superfield approach to BRST formalism. The above four (3 + 1)-dimensional (4D) theory is considered on a (4, 2)-dimensional supermanifold parameterized by the four even spacetime variables x µ (with µ = 0, 1, 2, 3) and a pair of odd Grassmannian variables θ andθ (with θ 2 =θ 2 = 0, θθ +θθ = 0). One of the salient features of our present investigation is that the above nilpotent (anti-)BRST symmetry transformations turn out to be absolutely anticommuting due to the presence of a Curci-Ferrari (CF) type of restriction. The latter condition emerges due to the application of our present superfield formalism. The actual CF condition, as is well-known, is the hallmark of a 4D non-Abelian 1-form gauge theory. We demonstrate that our present 4D Abelian 2-form gauge theory imbibes some of the key signatures of the 4D non-Abelian 1-form gauge theory. We briefly comment on the generalization of our supperfield approach to the case of Abelian 3-form gauge theory in four (3 + 1)-dimensions of spacetime.