BRST Formalism and Zero Locus Reduction

TL;DR

This paper explores the structure of the zero locus in BRST quantization, revealing deep connections between gauge symmetries, observables, and algebraic brackets, and relates these to classical equations like the Yang-Baxter equation.

Contribution

It establishes a correspondence between BRST symmetries, observables, and Hamiltonian vector fields on the zero locus, and generalizes to bi-QP-manifolds for BRST-anti-BRST symmetry.

Findings

Zero locus carries an (anti)bracket with opposite parity to the fundamental bracket.

On-shell gauge symmetries correspond to Hamiltonian vector fields on the zero locus.

The classical Yang-Baxter equation emerges as a condition for nilpotent vector fields.

Abstract

In the BRST quantization of gauge theories, the zero locus $Z_Q$ of the BRST differential $Q$ carries an (anti)bracket whose parity is opposite to that of the fundamental bracket. We show that the on-shell BFV/BV gauge symmetries are in a 1:1 correspondence with Hamiltonian vector fields on $Z_Q$, and observables of the BRST theory are in a 1:1 correspondence with characteristic functions of the bracket on $Z_Q$. By reduction to the zero locus, we obtain relations between bracket operations and differentials arising in different complexes (the Gerstenhaber, Schouten, Berezin-Kirillov, and Sklyanin brackets); the equation ensuring the existence of a nilpotent vector field on the reduced manifold can be the classical Yang-Baxter equation. We also generalize our constructions to the bi-QP-manifolds which from the BRST theory viewpoint corresponds to the BRST-anti-BRST-symmetric quantization.