Construction of topological field theories using BV

TL;DR

This paper details the construction of topological field theories using the BV formalism, highlighting the role of the energy momentum tensor, BRST invariance, and the treatment of symmetries in topological Yang--Mills theory.

Contribution

It introduces a comprehensive BV-based framework for topological field theories, including energy momentum tensor analysis and symmetry recovery methods.

Findings

Energy momentum tensor is BRST invariant and antifield dependent.

In topological theories, energy momentum is quantum BRST exact, with conditions at each  in .

Full energy momentum tensor is classically BRST exact in the antibracket sense.

Abstract

We discuss in detail the construction of topological field theories using the Batalin--Vilkovisky (BV) quantisation scheme. By carefully examining the dependence of the antibracket on an external metric, we show that differentiating with respect to the metric and the BRST charge do not commute in general. We introduce the energy momentum tensor in this scheme and show that it is BRST invariant, both for the classical and quantum BRST operators. It is antifield dependent, guaranteeing gauge independence. For topological field theories, this energy momentum has to be quantum BRST exact. This leads to conditions at each order in $\hbar$. As an example of this procedure, we consider topological Yang--Mills theory. We show how the reducible set of symmetries used in topological Yang--Mills can be recovered by means of trivial systems and canonical transformations. Self duality of the antighosts is properly treated by introducing an infinite tower of auxiliary fields. Finally, it is shown that the full energy momentum tensor is classically BRST exact in the antibracket sense.