BV quantization of dynamical fuzzy spectral triples

TL;DR

This paper systematically studies gauge symmetries in dynamical fuzzy spectral triples for quantum gravity, developing classical and quantum BV formalisms, and analyzing how background Dirac operators influence correlation functions.

Contribution

It introduces a homological construction of quantum correlation functions in fuzzy spectral triples and examines the impact of gauge symmetry breaking by background Dirac operators.

Findings

Ghost and antifield contributions depend on the background Dirac operator.

Explicit BV formalism for perturbative quantum correlation functions.

Analysis of gauge symmetry effects in specific models.

Abstract

This paper provides a systematic study of gauge symmetries in the dynamical fuzzy spectral triple models for quantum gravity that have been proposed by Barrett and collaborators. We develop both the classical and the perturbative quantum BV formalism for these models, which in particular leads to an explicit homological construction of the perturbative quantum correlation functions. We show that the relevance of ghost and antifield contributions to such correlation functions depends strongly on the background Dirac operator $D_0$ around which one perturbs, and in particular on the amount of gauge symmetry that it breaks. This will be illustrated by studying quantum perturbations around 1.) the gauge-invariant zero Dirac operator $D_0=0$ in a general $(p,q)$-model, and 2.) a simple example of a non-trivial $D_0$ in the quartic $(0,1)$-model.