Non-Perturbative Solution of Matrix Models Modified by Trace-Squared Terms

TL;DR

This paper provides a non-perturbative analysis of large N matrix models with trace-squared modifications, revealing new universality classes and their relation to Liouville theory, including a double-scaling limit at a critical coupling.

Contribution

It introduces a non-perturbative solution for matrix models with trace-squared terms, identifying new universality classes and their connection to Liouville theory, and explores the effects of operator modifications on gravitational dimensions.

Findings

For g<g_t, the model shares universality with the g=0 case.

At g=g_t, a new universality class emerges with a different string susceptibility exponent.

A double-scaling limit is defined at the critical g=g_t.

Abstract

We present a non-perturbative solution of large $N$ matrix models modified by terms of the form $ g(\Tr\Phi^4)^2$, which add microscopic wormholes to the random surface geometry. For $g<g_t$ the sum over surfaces is in the same universality class as the $g=0$ theory, and the string susceptibility exponent is reproduced by the conventional Liouville interaction $\sim e^{\alpha_+ \phi}$. For $g=g_t$ we find a different universality class, and the string susceptibility exponent agrees for any genus with Liouville theory where the interaction term is dressed by the other branch, $e^{\alpha_- \phi}$. This allows us to define a double-scaling limit of the $g=g_t$ theory. We also consider matrix models modified by terms of the form $g O^2$, where $O$ is a scaling operator. A fine-tuning of $g$ produces a change in this operator's gravitational dimension which is, again, in accord with the change in the branch of the Liouville dressing.