Touching Random Surfaces and Liouville Gravity

TL;DR

This paper explores how modifications in matrix models and Liouville gravity lead to a sudden change in string susceptibility, explained by a shift in the interaction term's exponential dressing, revealing new critical behavior.

Contribution

It demonstrates the connection between matrix model modifications and Liouville gravity, explaining a sudden change in critical behavior via a shift in the exponential dressing of the interaction term.

Findings

String susceptibility exponent jumps at a critical point g_t.

Change in the interaction term from e^{α_+ φ} to e^{α_- φ}.

New critical behavior explained by the unconventional branch of Liouville dressing.

Abstract

Large $N$ matrix models modified by terms of the form $ g(\Tr\Phi^n)^2$ generate random surfaces which touch at isolated points. Matrix model results indicate that, as $g$ is increased to a special value $g_t$, the string susceptibility exponent suddenly jumps from its conventional value $\gamma$ to ${\gamma\over\gamma-1}$. We study this effect in \L\ gravity and attribute it to a change of the interaction term from $O e^{\alpha_+ \phi}$ for $g<g_t$ to $O e^{\alpha_- \phi}$ for $g=g_t$ ($\alpha_+$ and $\alpha_-$ are the two roots of the conformal invariance condition for the \L\ dressing of a matter operator $O$). Thus, the new critical behavior is explained by the unconventional branch of \L\ dressing in the action.