Negative specific heat from non-planar interactions and small black holes in AdS/CFT

Abstract

The gravity side of the gauge/gravity duality predicts the existence of small black holes with negative specific heat. A free theory of strings has a Hagedorn behavior, but it does not lead to negative specific heat. To understand such states one needs to consider a theory of interacting strings. In the dual gauge theory, the string interactions are related to non-planar diagrams. In this paper the simplest gauged matrix model of two free matrices, that has Hagedorn behavior is analyzed in detail. A simple double trace deformation of the Hamiltonian, proportional to the square of the free Hamiltonian is studied. If the interaction has a negative sign, mimicking a gravitational attraction, one produces states with negative specific heat perturbatively and one can still compute the equation of state relating the entropy and the energy. A more general argument based on non-planar interactions that are random and that grow faster in strength than the planar diagrams as a function of the planar energy suggests that states with negative specific heat appear generically.