Double scaling limits of Dirac ensembles and Liouville quantum gravity

TL;DR

This paper explores the connection between noncommutative geometry, spectral triples, and quantum gravity by analyzing Dirac ensembles and their relation to Liouville quantum gravity, deriving critical exponents and differential equations through rigorous random matrix theory methods.

Contribution

It introduces a novel framework linking spectral triples with quantum gravity models and derives critical exponents and differential equations from double scaling limits.

Findings

Derives critical exponents of minimal models from Liouville quantum gravity.

Shows partition function asymptotics satisfy Painlevé I equation.

Establishes rigorous connection between noncommutative geometry and quantum gravity.

Abstract

In this paper we study ensembles of finite real spectral triples equipped with a path integral over the space of possible Dirac operators. In the noncommutative geometric setting of spectral triples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus, this path integral serves as a noncommutative analogue of integration over metrics, a key feature of a theory of quantum gravity. From these integrals in the so-called double scaling limit we derive critical exponents of minimal models from Liouville conformal field theory coupled with gravity. Additionally, the asymptotics of the partition function of these models satisfy differential equations such as Painlev\'e I, as a reduction of the KDV hierarchy, which is predicted by conformal field theory. This is all proven using well-established and rigorous techniques from random matrix theory.