Critical Lin-Lunin-Maldacena geometries

Abstract

We study the critical behavior of the Lin-Lunin-Maldacena (LLM) geometry in the case when a droplet in the LLM base space develops a cusp. This cusp is a generic feature of the density of complex eigenvalues in the dual complex matrix model (CMM) computing the correlation functions of huge 1/2-BPS operators in $\mathcal{N}=4$ SYM theory. It is also related to the criticality in CMM describing the pure $2D$ quantum gravity behavior. The supergravity dual -- LLM metric in the vicinity of the tip of the cusp -- acquires a universal $ISO(1,3)\times SO(5)$ symmetric form, with a naked singularity along a half-infinite line. Both massless and massive particles get trapped by this line singularity for almost any impact parameter. Generic trajectories ending on the singular line reach it in finite affine time, while the corresponding observer time diverges. An explicit analytic solution for a large class of massless trajectories together with the absence of stochastic behavior in the vicinity of the cusp hint on a certain integrability of the problem.