A naive matrix-model approach to two-dimensional quantum gravity coupled to matter of arbitrary central charge

TL;DR

This paper explores a matrix-model approach to two-dimensional quantum gravity coupled with matter of arbitrary central charge, successfully reproducing expected critical exponents for small n and extending to larger n without signs of instability.

Contribution

It introduces a simplified matrix-model method for analyzing 2D quantum gravity coupled to matter with any central charge, providing accurate critical exponents for small n and extending to larger n.

Findings

Accurately reproduces string susceptibility exponents for n=0,1,2

Obtains consistent exponents for n=3 and coupled Ising models

No signs of tachyonic instability at genus zero

Abstract

In the usual matrix-model approach to random discretized two-dimensional manifolds, one introduces n Ising spins on each cell, i.e. a discrete version of 2D quantum gravity coupled to matter with a central charge n/2. The matrix-model consists then of an integral over $2^{n}$ matrices, which we are unable to solve for $n>1$. However for a fixed genus we can expand in the cosmological constant g for arbitrary values of n, and a simple minded analysis of the series yields for n=0,1 and 2 the expected results for the exponent $\gamma_{string}$ with an amazing precision given the small number of terms that we considered. We then proceed to larger values of n. Simple tests of universality are successfully applied; for instance we obtain the same exponents for n=3 or for one Ising model coupled to a one dimensional target space. The calculations are easily extended to states Potts models, through an integration over $q^{n}$ matrices. We see no sign of the tachyonic instability of the theory, but we have only considered genus zero at this stage.