Cohomological aspects of Abelian gauge theory

TL;DR

This paper explores the cohomological structure of two-dimensional free Abelian gauge theory within the BRST formalism, highlighting its topological features and invariants, and relating it to known topological field theories.

Contribution

It derives BRST and co-BRST charges, formulates the Hodge decomposition, and identifies topological invariants, connecting the theory to Witten- and Schwarz-type topological field theories.

Findings

Derived conserved, nilpotent BRST and co-BRST charges.

Expressed the Hodge decomposition in terms of these charges.

Identified topological invariants related by duality.

Abstract

We discuss some aspects of cohomological properties of a two-dimensional free Abelian gauge theory in the framework of BRST formalism. We derive the conserved and nilpotent BRST- and co-BRST charges and express the Hodge decomposition theorem in terms of these charges and a conserved bosonic charge corresponding to the Laplacian operator. It is because of the topological nature of free U(1) gauge theory that the Laplacian operator goes to zero when equations of motion are exploited. We derive two sets of topological invariants which are related to each-other by a certain kind of duality transformation and express the Lagrangian density of this theory as the sum of terms that are BRST- and co-BRST invariants. Mathematically, this theory captures together some of the key features of Witten- and Schwarz type of topological field theories.