Touching Random Surfaces and Liouville Gravity

Abstract

Large $N$ matrix models modified by terms of the form $ g(\TrΦ^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 $γ$ to ${γ\overγ-1}$. We study this effect in Ł gravity and attribute it to a change of the interaction term from $O e^{α_+ ϕ}$ for $g<g_t$ to $O e^{α_- ϕ}$ for $g=g_t$ ($α_+$ and $α_-$ are the two roots of the conformal invariance condition for the Ł dressing of a matter operator $O$). Thus, the new critical behavior is explained by the unconventional branch of Ł dressing in the action.