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

Masanori Hanada, Jonathan Maltz

TL;DR

This work proposes a gauge-theory description for the small Schwarzschild black hole in AdS$_5\times$S$^5$ by interpreting a BH as a bound state of eigenvalues of the scalar fields in 4d ${\cal N}=4$ SYM on $\mathbb{R}^{1}\times$S$^3$ and its emission as a reduction in dynamical degrees of freedom. By analyzing the microcanonical ensemble and leveraging the AdS/CFT correspondence for the large BH, the authors derive the energy–temperature relation $E \sim 1/(G_{\rm 10,N} T^7)$ for the small BH, and show that Hagedorn behavior emerges in the strong-coupling regime via long closed strings or winding matrix degrees of freedom. The framework also yields consistent ABJM (11d) black-hole scalings and extends to generic theories with a single characteristic scale, suggesting a universal mechanism where small BHs are just smaller realizations of large BHs in gauge theory language. The results underscore the central role of eigenvalue dynamics in gauge/gravity duality and propose avenues for direct gauge-theory tests, including Monte Carlo simulations and matrix RG approaches, to probe horizon physics and information-related aspects. Overall, the paper provides a coherent gauge-theory narrative linking large and small BHs through dynamical truncation of matrix degrees of freedom and captures key gravity-side expectations in the strongly coupled regime.

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.

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

TL;DR

This work proposes a gauge-theory description for the small Schwarzschild black hole in AdSS by interpreting a BH as a bound state of eigenvalues of the scalar fields in 4d SYM on S and its emission as a reduction in dynamical degrees of freedom. By analyzing the microcanonical ensemble and leveraging the AdS/CFT correspondence for the large BH, the authors derive the energy–temperature relation for the small BH, and show that Hagedorn behavior emerges in the strong-coupling regime via long closed strings or winding matrix degrees of freedom. The framework also yields consistent ABJM (11d) black-hole scalings and extends to generic theories with a single characteristic scale, suggesting a universal mechanism where small BHs are just smaller realizations of large BHs in gauge theory language. The results underscore the central role of eigenvalue dynamics in gauge/gravity duality and propose avenues for direct gauge-theory tests, including Monte Carlo simulations and matrix RG approaches, to probe horizon physics and information-related aspects. Overall, the paper provides a coherent gauge-theory narrative linking large and small BHs through dynamical truncation of matrix degrees of freedom and captures key gravity-side expectations in the strongly coupled regime.

Abstract

Based on 4d SYM on S, a gauge theory description of a small black hole in AdSS 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 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.

Paper Structure

This paper contains 15 sections, 26 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Stringy interpretation of the field theory degrees of freedom
  3. Emission of D-branes and black hole evaporation
  4. The large black hole (large energy and $\lambda\gg 1$)
  5. Emission of eigenvalues
  6. Gauge fixing
  7. 10d Schwarzschild at $1/\lambda \ll N_{\rm BH}/N < 1$ ($\lambda\gg 1$)
  8. The phase transition from the big black hole to the small black hole
  9. The Hagedorn growth at $N_{\rm BH}/N \lesssim 1/\lambda$ ($\lambda\gg 1$)
  10. The case of weak coupling $\lambda\ll 1$
  11. SO(6) breaking
  12. Other cases
  13. ABJM theory and 11d black hole
  14. More generic theories
  15. Discussions

Figures (8)

  • Figure 1: The microcanonical $E$-vs-$T$ phase diagram of 4d SYM on $\mathbb{R}^{1}\times$S$^{3}$ at strong coupling ($\lambda=g_{\rm YM}^2N\gg 1$), obtained by assuming the validity of the AdS/CFT duality.
  • Figure 2: Interpretation of matrix components as sites and links of a nonlocal lattice.
  • Figure 3: $X_{M_1,ik}X_{M_2,kl}X_{M_3,lj}X_{M_4,ji}$ in ${\rm Tr}(X_{M_1}X_{M_2}X_{M_3}X_{M_4})|{\rm \it{Vac}}\rangle$.
  • Figure 4: $X_{M_1,ik}X_{M_2,kl}X_{M_3,ll}X_{M_4,li}$ in ${\rm Tr}(X_{M_1}X_{M_2}X_{M_3}X_{M_4})|{\rm \it{Vac}}\rangle$.
  • Figure 5: AdS black hole is obtained by rolling up black 3-brane.
  • ...and 3 more figures