Simplifications in Lagrangian BV quantization exemplified by the anomalies of chiral $W_3$ gravity

TL;DR

This paper simplifies the BV quantization process for gauge theories, especially in chiral W3 gravity, by introducing a new proof of key cohomological properties, leading to easier anomaly calculations and anomaly cancellation methods.

Contribution

It provides a new proof of the acyclicity of the Koszul--Tate differential, simplifying anomaly computations in BV formalism, exemplified through chiral W3 gravity.

Findings

One-loop anomaly computed without negative ghost number terms.

The full anomaly is determined up to local counterterms.

Background charges can be incorporated to cancel anomalies.

Abstract

The Batalin--Vilkovisky (BV) formalism is a useful framework to study gauge theories. We summarize a simple procedure to find a a gauge--fixed action in this language and a way to obtain one--loop anomalies. Calculations involving the antifields can be greatly simplified by using a theorem on the antibracket cohomology. The latter is based on properties of a `Koszul--Tate differential', namely its acyclicity and nilpotency. We present a new proof for this acyclicity, respecting locality and covariance of the theory. This theorem then implies that consistent higher ghost terms in various expressions exist, and it avoids tedious calculations. This is illustrated in chiral $W_3$ gravity. We compute the one--loop anomaly without terms of negative ghost number. Then the mentioned theorem and the consistency condition imply that the full anomaly is determined up to local counterterms. Finally we show how to implement background charges into the BV language in order to cancel the anomaly with the appropriate counterterms. Again we use the theorem to simplify the calculations, which agree with previous results.