Superfield approach to topological aspects of non - Abelian gauge theory

We discuss some of the key topological features of a two (1 + 1)-dimensional (2D) self-interacting non-Abelian gauge theory (having no interaction with matter fields) in the framework of superfield formalism. We provide the geometrical interpretation for the Lagrangian density, symmetric energy momentum tensor, topological invariants, etc., by exploiting the on-shell nilpotent BRST-and co-BRST symmetries that emerge after the application of the generalized versions of horizontality condition. We show that the above physical quantities geometrically correspond to the translations of some local (but composite) chiral superfields along one of the two independent Grassmannian directions of the four (2 + 2)-dimensional supermanifold. 1 Introduction The modern developments in the physically and mathematically rich subject of topological field theories (TFTs) have encompassed in their ever widening horizons a host of diverse and distinct areas of research in theoretical physics and mathematics. In this context, mention can be made of such interesting topics as Chern-Simon theories, topological string theories and matrix models, 2D topological gravity, Morse theory, Donaldson and Jones polynomials, etc. (see, e.g., Ref. [1] and references therein for details). Without going into the subtleties and intricacies, TFTs can be classified into two types. The Witten type TFTs [2] are the ones where the Lagrangian density turns out to be the Becchi-Rouet-Stora-Tyutin (BRST) (anti-)commutator. The conserved and nilpotent BRST charge for such TFTs generates a symmetry that is a combination of a topological shift symmetry and some type of local gauge symmetries. On the other hand, the Schwarz type of TFTs [3] are characterized by the existence of a conserved and nilpotent BRST charge that generates only some local gauge type of symmetries for a Lagrangian density that cannot be totally expressed as a BRST (anti-)commutator (see, e.g., Ref. [1] for details). For both types of TFT, there are no energy excitations in the physical sector of the theory because of the fact that energy-momentum tensor turns out to be a BRST (anti-)commutator and all the physical states (including the vacuum) of this theory are supposed to be BRST invariant (w.r.t. the conserved, nilpotent, metric independent and hermitian BRST charge). Recently, in a set of papers [4-7], the free 2D Abelian-and self-interacting non-Abelian gauge theories (without any kind of interaction with matter fields) have been shown to belong to a new class of TFTs because the Lagrangian density of the theory turns out to bear an appearance similar to the Witten type …