The BV formalism: theory and application to a matrix model

TL;DR

This paper reviews the BV formalism for 0-dimensional gauge theories, providing an algorithm to construct extended theories with ghost fields and applying it to a specific matrix model with U(2) symmetry.

Contribution

It introduces a systematic method to extend gauge theories within the BV formalism and applies it explicitly to a matrix model with gauge symmetry.

Findings

Constructed extended BV theories for gauge models

Explicit solution of the classical master equation for a matrix model

Demonstrated the application of BV formalism to matrix models

Abstract

We review the BV formalism in the context of $0$-dimensional gauge theories. For a gauge theory $(X_{0}, S_{0})$ with an affine configuration space $X_{0}$, we describe an algorithm to construct a corresponding extended theory $(\tilde{X}, \tilde{S})$, obtained by introducing ghost and anti-ghost fields, with $\tilde{S}$ a solution of the classical master equation in $\mathcal{O}_{\tilde{X}}$. This construction is the first step to define the (gauge-fixed) BRST cohomology complex associated to $(\tilde{X}, \tilde{S})$, which encodes many interesting information on the initial gauge theory $(X_{0}, S_{0})$. The second part of this article is devoted to the application of this method to a matrix model endowed with a $U(2)$-gauge symmetry, explicitly determining the corresponding $\tilde{X}$ and the general solution $\tilde{S}$ of the classical master equation for the model.