Coarse-Graining the Lin-Maldacena Geometries

TL;DR

This paper develops a coarse-grained, macroscopic description of the Lin-Maldacena geometries at large N, linking entropy to geometric data and identifying a typical state geometry as a zero-temperature dual of the field theory.

Contribution

It introduces a method to describe typical states via coarse-grained geometries and derives an entropy formula, connecting microstates to macroscopic gravitational descriptions.

Findings

Coarse-grained geometries are singular when entropy is non-zero.

The entropy is maximized for a typical state geometry.

The typical state geometry corresponds to the zero-temperature limit of the field theory.

Abstract

The Lin-Maldacena geometries are nonsingular gravity duals to degenerate vacuum states of a family of field theories with SU(2|4) supersymmetry. In this note, we show that at large N, where the number of vacuum states is large, there is a natural `macroscopic' description of typical states, giving rise to a set of coarse-grained geometries. For a given coarse-grained state, we can associate an entropy related to the number of underlying microstates. We find a simple formula for this entropy in terms of the data that specify the geometry. We see that this entropy function is zero for the original microstate geometries and maximized for a certain ``typical state'' geometry, which we argue is the gravity dual to the zero-temperature limit of the thermal state of the corresponding field theory. Finally, we note that the coarse-grained geometries are singular if and only if the entropy function is non-zero.