A proposal of the gauge theory description of the small Schwarzschild black hole in AdS$_5\times$S$^5$

TL;DR

This paper proposes a gauge theory framework for understanding small black holes in AdS$_5\times$S$^5$, linking eigenvalue dynamics to black hole properties and deriving energy relations consistent with duality principles.

Contribution

It introduces a gauge theory description of small black holes in AdS$_5\times$S$^5$, connecting eigenvalue emission to black hole thermodynamics and deriving energy relations via the AdS/CFT correspondence.

Findings

Derivation of the energy of small ten-dimensional Schwarzschild black holes.

Identification of Hagedorn behavior of long strings at low energy.

Support for the gauge theory description through consistency with known black hole relations.

Abstract

Based on 4d ${\cal N}=4$ SYM on $\mathbb{R}^{1}\times$S$^3$, a gauge theory description of a small black hole in AdS$_5\times$S$^5$ is proposed. The change of the number of dynamical degrees of freedom associated with the emission of the scalar fields' eigenvalues plays a crucial role in this description. By analyzing the microcanonical ensemble, the Hagedorn behavior of long strings at low energy is obtained. Modulo an assumption based on the AdS/CFT duality for a large black hole, the energy of the small ten-dimensional Schwarzschild black hole $E\sim 1/(G_{\rm 10,N}T^7)$ is derived. A heuristic gauge theory argument supporting this assumption is also given. The same argument applied to the ABJM theory correctly reproduces the relation for the eleven-dimensional Schwarzschild black hole. One of the consequences of our proposal is that the small and large black holes are very similar when seen from the gauge theory point of view.