Topological aspects in non-Abelian gauge theory

TL;DR

This paper explores the topological features of a two-dimensional free non-Abelian gauge theory through BRST cohomology, linking Hodge decomposition to topological invariants and showing the Lagrangian's invariance properties.

Contribution

It establishes a connection between BRST cohomology, Hodge decomposition, and topological invariants in a non-interacting 2D non-Abelian gauge theory, highlighting its topological nature.

Findings

Identification of two sets of topological invariants

Expression of the Lagrangian as a sum of BRST and co-BRST invariants

Connection between Laplacian vanishing and topological properties

Abstract

We discuss the BRST cohomology and exhibit a connection between the Hodge decomposition theorem and the topological properties of a two dimensional free non-Abelian gauge theory having no interaction with matter fields. The topological nature of this theory is encoded in the vanishing of the Laplacian operator when equations of motion are exploited. We obtain two sets of topological invariants with respect to BRST and co-BRST charges on the two dimensional manifold and show that the Lagrangian density of the theory can be expressed as the sum of terms that are BRST- and co-BRST invariants.