Quantum noncommutative gravity in two dimensions

TL;DR

This paper investigates the quantization of noncommutative gravity in two dimensions, performing exact path integrals for noncommutative Jackiw-Teitelboim gravity and analyzing quantum fluctuations and anomalies.

Contribution

It provides an exact path integral formulation for noncommutative 2D gravity and explores quantum fluctuations and anomalies within this framework.

Findings

Exact path integral for noncommutative Jackiw-Teitelboim gravity

Calculation of the conformal anomaly and Polyakov action

Analysis of quantum fluctuations in the matter sector

Abstract

We study quantisation of noncommutative gravity theories in two dimensions (with noncommutativity defined by the Moyal star product). We show that in the case of noncommutative Jackiw-Teitelboim gravity the path integral over gravitational degrees of freedom can be performed exactly even in the presence of a matter field. In the matter sector, we study possible choices of the operators describing quantum fluctuations and define their basic properties (e.g., the Lichnerowicz formula). Then we evaluate two leading terms in the heat kernel expansion, calculate the conformal anomaly and the Polyakov action (as an expansion in the conformal field).