Black Hole Thermodynamics from Calculations in Strongly-Coupled Gauge Theory

TL;DR

The paper probes the gauge/gravity duality between ten-dimensional black holes with 0-brane charge and strongly coupled $SU(N)$ quantum mechanics at finite temperature by building a Gaussian (mean-field) approximation that preserves key symmetries and resums planar diagrams via Schwinger-Dyson gap equations. It computes the finite-temperature partition function and finds a power-law behavior of the free energy, $\beta F$, that closely matches the gravity prediction, supporting the duality beyond BPS arguments. The results yield a concrete density-matrix model for the black hole in the strongly coupled gauge theory and reveal a Gross-Witten transition and regime boundaries that illuminate where the supergravity description applies. The work provides a framework for exploring horizon physics and spacetime locality from the gauge theory and highlights the limitations and extensions needed to access deeper strong-coupling regimes.

Abstract

We develop an approximation scheme for the quantum mechanics of N D0-branes at finite temperature in the 't Hooft large-N limit. The entropy of the quantum mechanics calculated using this approximation agrees well with the Bekenstein-Hawking entropy of a ten-dimensional non-extremal black hole with 0-brane charge. This result is in accord with the duality conjectured by Itzhaki, Maldacena, Sonnenschein and Yankielowicz. Our approximation scheme provides a model for the density matrix which describes a black hole in the strongly-coupled quantum mechanics.

Figures (5)

• Figure 1: The solid curve is the power law fit (\ref{['results:fit']}) for $\beta F$. The data points are calculated from numerical solutions to the gap equations.
• Figure 2: Energy vs. $\beta$. For $\beta >2.5$ fixing $\lambda$ by fitting $\beta F$ to a power law leads to the solid middle line, while the Schwinger-Dyson gap equation for lambda leads to the lower dashed line. The upper dot-dashed line is the semiclassical energy of the black hole.
• Figure 3: Energy vs. $\beta$. Same as Fig. 2, but plotted on a $\log$--$\log$ scale.
• Figure 4: The one-plaquette coupling $\lambda$ vs. $\beta$. The Gross-Witten transition occurs when $\lambda = 2$. For $\beta < 2.5$ we use the Schwinger-Dyson gap equation to determine $\lambda$. For $\beta > 2.5$ the Schwinger-Dyson gap equation gives the dashed line, while fitting $\beta F$ to a power law gives the solid line.
• Figure 5: Range of eigenvalues (radius of the Wigner semi-circle) vs. $\beta$. The upper solid red curve is for the scalar fields in the scalar multiplets; the lower solid blue curve is for the scalar fields in the gauge multiplet. The dashed black curve is the Schwarzschild radius of the black hole. These results were calculated with $\beta F$ fit to a power law for $\beta >2.5$.