Multi-spin systems on a randomly triangulated surface

TL;DR

This paper studies spin systems on randomly triangulated surfaces as models for conformal matter coupled to 2D gravity, revealing universality and critical behavior constraints.

Contribution

It introduces a framework connecting spin systems on random surfaces to conformal matter fields, establishing universality assumptions and critical behavior classifications.

Findings

At critical points, models show mean field or conformal matter behavior.

String susceptibility exponent values are limited to 1/n, with n≥2.

Unitary case restricts the exponent to specific discrete values.

Abstract

We consider a class of spin systems on randomly triangulated surfaces as discrete approximations to conformal matter fields coupled to 2d gravity. On the basis of certain universality assumptions we argue that at critical points with diverging string susceptibility the model either exhibits mean field behaviour or it can effectively be described by a conformal matter system with central charge less than or equal to 1 coupled to 2d gravity. As a particular consequence we conclude in the unitary case that the string susceptibility exponent is limited to possible values of the form 1/n, n=2,3,4,..., where n=2 corresponds to mean field behaviour.