Batalin-Vilkovisky quantization of fuzzy field theories

TL;DR

This paper extends Batalin-Vilkovisky quantization methods to fuzzy noncommutative field theories, including equivariant theories with Hopf algebra symmetry, and demonstrates calculations on fuzzy spheres and tori.

Contribution

It develops a generalized BV quantization framework for fuzzy noncommutative theories with Hopf algebra symmetry, including braided $L_$-algebras, and applies it to specific models.

Findings

Computed perturbative correlation functions for scalar and Chern-Simons theories on fuzzy spheres.

Extended BV techniques to theories with Hopf algebra symmetry, including braided scalar fields.

Provided explicit examples demonstrating the quantization process on fuzzy geometries.

Abstract

We apply the modern Batalin-Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogs of the field theories proposed recently through the notion of `braided $L_\infty$-algebras'. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern-Simons theories on the fuzzy $2$-sphere, as well as for braided scalar field theories on the fuzzy $2$-torus.