Conformal Dimensions On Causal Random Geometry

TL;DR

This paper explores how matter fields, specifically the Ising model, interact with two-dimensional causal dynamical triangulations (CDT) in quantum gravity, revealing that conformal dimensions on CDT match those on fixed lattices, using analytical and lattice methods.

Contribution

It introduces an analytical approach to determine conformal dimensions on CDT, adapting the Duplantier-Sheffield framework to the one-dimensional nature of CDT.

Findings

Conformal dimensions on CDT match those on fixed lattices.

Analytical methods confirm the quenched model results.

Adaptation of lattice and stochastic methods to CDT context.

Abstract

We investigate the interaction between matter and causal dynamical triangulations (CDT) in the context of two-dimensional quantum gravity. We focus on the Ising model coupled to CDT, contrasting this with Liouville gravity and the relation to the Knizhnik-Polyakov-Zamolodchikov (KPZ) formula. We demonstrate analytically for the quenched model that the conformal dimensions of fields on CDT align with those on a fixed lattice. We do this using a combination of lattice methods and adapting the Duplantier-Sheffield framework to CDT, emphasizing the one-dimensional nature of CDT and its description via a stochastic differential equation.