The BV construction for finite spectral triples

TL;DR

This paper integrates the Batalin-Vilkovisky (BV) formalism into noncommutative geometry for finite spectral triples, revealing a geometric interpretation of gauge theory quantization steps and connecting BV complexes with Hochschild cohomology.

Contribution

It demonstrates that the BV and BRST complexes can be expressed using noncommutative geometric objects and shows their equivalence to Hochschild complexes in this setting.

Findings

BV formalism naturally fits into noncommutative geometry

BV and BRST complexes coincide with Hochschild complexes

Detailed construction for $U(n)$-gauge theories on matrix algebras

Abstract

This article presents how the BV formalism naturally inserts in the framework of noncommutative geometry for gauge theories induced by finite spectral triples. Reaching this goal entails that not only all the steps of the BV construction, from the introduction of ghost/anti-ghost fields to the construction of the BRST complex, can be expressed using noncommutative geometric objects, but also that the method to go from one step in the construction to the next one has an intrinsically noncommutative geometric nature. Moreover, we prove that both the classical BV and BRST complexes coincide with another cohomological theory, naturally appearing in noncommutative geometry: the Hochschild complex of a coalgebra. The construction is presented in detail for $U(n)$-gauge theories induced by spectral triples on the algebra $M_n(\mathbb{C})$.