Unitary, anomalous Master Ward Identity and its connections to the Wess-Zumino condition, BV formalism and $L_\infty$-algebras

TL;DR

This paper explores the mathematical structure of anomalies in quantum field theory, connecting the Master Ward identity with the Wess-Zumino condition, BV formalism, and $L_$-algebras, using perturbation theory.

Contribution

It establishes a link between the cocycle condition of the anomalous Master Ward identity and the Wess-Zumino and BV consistency relations, highlighting the role of $L_$-algebras.

Findings

The cocycle condition relates to the Wess-Zumino consistency relation.

Connections are made between the Master Ward identity and BV formalism.

The generalized Jacobi identity for $L_$-algebras underpins the anomaly structure.

Abstract

The C*-algebraic construction of QFT by Buchholz and one of us relies on the causal structure of spacetime and a classical Lagrangian. In one of our previous papers we have introduced additional structure into this construction, namely an action of symmetries, which is related to fixing renormalisation conditions. This action characterizes anomalies and satisfies a cocycle condition which is summarized in the unitary anomalous Master Ward identity. Here (using perturbation theory) we show how this cocycle condition is related to the Wess-Zumino consistency relation and the consistency relation for the anomaly in the BV formalism, where the latter is the generalized Jacobi identity for the associated $L_\infty$-algebra.