BRST cohomology and Hodge decomposition theorem in Abelian gauge theory

TL;DR

This paper explores the BRST cohomology and Hodge decomposition in 2D free U(1) gauge theory, revealing its topological features and invariants through conserved charges and Laplacian operators.

Contribution

It introduces a dual-BRST charge, relates the Hodge decomposition to gauge invariance, and demonstrates the topological nature of the theory using cohomology and invariants.

Findings

Identification of a dual-BRST charge that preserves gauge-fixing

Expression of the Lagrangian as BRST and co-BRST invariants

Derivation of topological invariants on a 2D compact manifold

Abstract

We discuss the Becchi-Rouet-Stora-Tyutin (BRST) cohomology and Hodge decomposition theorem for the two dimensional free U(1) gauge theory. In addition to the usual BRST charge, we derive a local, conserved and nilpotent co(dual)-BRST charge under which the gauge-fixing term remains invariant. We express the Hodge decomposition theorem in terms of these charges and the Laplacian operator. We take a single photon state in the quantum Hilbert space and demonstrate the notion of gauge invariance, no-(anti)ghost theorem, transversality of photon and establish the topological nature of this theory by exploiting the concepts of BRST cohomology and Hodge decomposition theorem. In fact, the topological nature of this theory is encoded in the vanishing of the Laplacian operator when equations of motion are exploited. On the two dimensional compact manifold, we derive two sets of topological invariants with respect to the conserved and nilpotent BRST- and co-BRST charges and express the Lagrangian density of the theory as the sum of terms that are BRST- and co-BRST invariants. Mathematically, this theory captures together some of the key features of both Witten- and Schwarz type of topological field theories.