Renormalization Group Approach to the Continuum Limit of Matrix Models of Quantum Gravity with Preferred Foliation

TL;DR

This paper applies the Functional Renormalization Group Equation to a matrix model of 2D quantum gravity with foliation, exploring its continuum limits and fixed points, and suggesting the method's usefulness for higher-dimensional models.

Contribution

It demonstrates the application of FRGE to dually weighted matrix models of quantum gravity, revealing fixed points consistent with known results and highlighting the potential for higher-dimensional studies.

Findings

Identified fixed points compatible with 2D CDT results

Showed scheme dependence affects precision of fixed points

Supported FRGE as a useful tool for quantum gravity models

Abstract

This contribution is not intended as a review but, by suggestion of the editors, as a glimpse ahead into the realm of dually weighted tensor models for quantum gravity. This class of models allows one to consider a wider class of quantum gravity models, in particular one can formulate state sum models of spacetime with an intrinsic notion of foliation. The simplest one of these models is the one proposed by Benedetti and Henson, which is a matrix model formulation of two-dimensional Causal Dynamical Triangulations (CDT). In this paper we apply the Functional Renormalization Group Equation (FRGE) to the Benedetti-Henson model with the purpose of investigating the possible continuum limits of this class of models. Possible continuum limits appear in this FRGE approach as fixed points of the renormalization group flow where the size of the matrix acts as the renormalization scale. Considering very small truncations, we find fixed points that are compatible with analytically known results for CDT in two dimensions. By studying the scheme dependence of our results we find that precision results require larger truncations than the ones considered in the present work. We conclude that our work suggests that the FRGE is a useful exploratory tool for dually weighted matrix models. We thus expect that the FRGE will be a useful exploratory tool for the investigation of dually weighted tensor models for CDT in higher dimensions.