Topological aspects of Abelian gauge theory in superfield formalism

We discuss some of the topological features of a non-interacting two (1 + 1)dimensional Abelian gauge theory in the framework of superfield formulation. This theory is described by a BRST invariant Lagrangian density in the Feynman gauge. We express the local and continuous symmetries, Lagrangian density, topological invariants and symmetric energy momentum tensor of the theory in the language of superfields by exploiting the nilpotent (anti-)BRSTand (anti-)co-BRST symmetries. In fact, the above superfields are defined to deduce these nilpotent symmetries in the framework of superfield formalism. ∗ E-mail address: [email protected] There are many areas of research in the modern developments of theoretical high energy physics that have brought together mathematicians as well as theoretical physicists to share their key insights of those specific fields (of investigations) in a constructive and illuminating manner. The subject of topological field theories (TFTs) [1-3] is one such area that has provided a meeting-ground for both variety of researchers to enrich their understanding in a coherent and consistent fashion. Recently, two (1 + 1)-dimensional (2D) free Abelianand self-interacting non-Abelian gauge theories (having no interaction with matter fields) have been shown [4,5] to belong to a new class of TFTs that capture together some of the key features of Wittenand Schwarz type of TFTs [1,2]. Furthermore, in a set of papers [6-9], these 2D freeas well as interacting (non-)Abelian gauge theories have been shown to represent a class of field theoretical models for the Hodge theory where symmetries of the Lagrangian density (and corresponding generators) have been identified with the de Rham cohomology operators of differential geometry. In fact, these symmetries and corresponding generators have been exploited to establish the topological nature of the free Abelianand self-interacting non-Abelian gauge theories [4,5]. The analogues of the above cohomological operators, in terms of the symmetries (and corresponding generators), have also been found for physical four (3 + 1)-dimensional free Abelian two-form gauge theory [10]. The geometrical interpretation for the above local and conserved generators for the 2D theories have been established [11-13] in the framework of the superfield formalism [14-18] where it has been shown that these conserved charges correspond to the translation generators along the Grasmannian (odd)as well as bosonic (even) directions of a compact four (2 + 2)-dimensional supermanifold. In these endeavours, a generalized version of the so-called horizontality condition [14-16] has been exploited w.r.t. the three † super de Rham cohomology operators (d̃, δ̃, ∆̃ = d̃δ̃ + δ̃d̃) of differential geometry defined on the (2 + 2)-dimensional compact supermanifold (without a boundary). In all our previous attempts [11-13] to provide the geometrical interpretation for the generators of the (anti-)BRST symmetries, (anti-)co-BRST symmetries and a bosonic symmetry in the framework of superfield formulation, we have not found a way to capture the topological features of the 2D free Abelianand self-interacting non-Abelian gauge theories (without having any interaction with matter fields). The purpose of our present paper is to show that the nilpotent (sb = s̄ 2 b = s 2 d = s̄ 2 d = 0) (anti-)BRST symmetries (s̄b)sb and (anti-)co-BRST symmetries (s̄d)sd, Lagrangian density, topological invariants and symmetric energy momentum tensor for the free 2D Abelian gauge theory can be expressed in terms of the superfields alone and a possible geometrical interpretation can be provided for the above physical quantities in the framework of superfield formalism. We show, in particular, that the Lagrangian density and the symmetric energy momentum tensor can be written as the sum of quantities that can be expressed in terms of the Grassmannian derivatives † On an ordinary flat compact manifold without a boundary, a set (d, δ,∆) of three cohomological operators can be defined which obey the algebra: d = δ = 0,∆ = (d + δ) = dδ + δd ≡ {d, δ}, [∆, d] = [∆, δ] = 0 where d = dx∂μ and δ = ± ∗ d∗ (with ∗ as the Hodge duality operation) are the nilpotent (of order two) exteriorand co-exterior derivatives and ∆ is the Laplacian operator [19-22].