Noncommutative geometry and the BV formalism: application to a matrix model

TL;DR

This paper applies the Batalin-Vilkovisky (BV) formalism within noncommutative geometry to analyze a U(2)-matrix model, introducing spectral triples that geometrically encode gauge and ghost fields, and solving the classical master equation.

Contribution

It introduces a geometric framework using spectral triples to describe the BV formalism in noncommutative geometry, providing a new perspective on gauge theories.

Findings

Constructed BV and auxiliary spectral triples from gauge and ghost fields.

Derived a general solution to the classical master equation.

Demonstrated the sum of fermionic actions equals the BV action.

Abstract

We analyze a U(2)-matrix model derived from a finite spectral triple. By applying the BV formalism, we find a general solution to the classical master equation. To describe the BV formalism in the context of noncommutative geometry, we define two finite spectral triples: the BV spectral triple and the BV auxiliary spectral triple. These are constructed from the gauge fields, ghost fields and anti-fields that enter the BV construction. We show that their fermionic actions add up precisely to the BV action. This approach allows for a geometric description of the ghost fields and their properties in terms of the BV spectral triple.