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R-charged AdS_{5} black holes and large N unitary matrix models

Pallab Basu, Spenta R. Wadia

TL;DR

The paper establishes a nonperturbative gauge/gravity link by mapping the R-charged AdS$_5$ black hole phases to the R-charge sector of large-$N$ ${\cal N}=4$ SYM via an effective unitary matrix model of the thermal Polyakov loop. A crucial result is the logarithmic term in the fixed-charge sector, which decisively shapes the phase structure and enables precise matching of saddle points and critical exponents with the supergravity description, including a cusp catastrophe and a third-order transition. Through analyses at zero, weak, and strong coupling, the work demonstrates consistent multi-saddle dynamics (I, II, III) and universal critical behavior, with explicit double-scaling limits yielding tractable partition-function forms that mirror gravitational thermodynamics. The approach provides a powerful framework for understanding black-hole thermodynamics in AdS via matrix models, and suggests extensions to other charged AdS backgrounds and Kerr-AdS black holes.

Abstract

Using the AdS/CFT, we establish a correspondence between the intricate thermal phases of R-charged AdS_{5} blackholes and the R-charge sector of the N=4 gauge theory, in the large N limit. Integrating out all fields in the gauge theory except the thermal Polyakov line, leads to an effective unitary matrix model. In the canonical ensemble, a logarithmic term is generated in the non-zero charge sector of the matrix model. This term is important to discuss various supergravity properties like i) the non-existence of thermal AdS as a solution, ii) the existence of a point of cusp catastrophe in the phase diagram and iii) the matching of saddle points and the critical exponents of supergravity and those of the effective matrix model.

R-charged AdS_{5} black holes and large N unitary matrix models

TL;DR

The paper establishes a nonperturbative gauge/gravity link by mapping the R-charged AdS black hole phases to the R-charge sector of large- SYM via an effective unitary matrix model of the thermal Polyakov loop. A crucial result is the logarithmic term in the fixed-charge sector, which decisively shapes the phase structure and enables precise matching of saddle points and critical exponents with the supergravity description, including a cusp catastrophe and a third-order transition. Through analyses at zero, weak, and strong coupling, the work demonstrates consistent multi-saddle dynamics (I, II, III) and universal critical behavior, with explicit double-scaling limits yielding tractable partition-function forms that mirror gravitational thermodynamics. The approach provides a powerful framework for understanding black-hole thermodynamics in AdS via matrix models, and suggests extensions to other charged AdS backgrounds and Kerr-AdS black holes.

Abstract

Using the AdS/CFT, we establish a correspondence between the intricate thermal phases of R-charged AdS_{5} blackholes and the R-charge sector of the N=4 gauge theory, in the large N limit. Integrating out all fields in the gauge theory except the thermal Polyakov line, leads to an effective unitary matrix model. In the canonical ensemble, a logarithmic term is generated in the non-zero charge sector of the matrix model. This term is important to discuss various supergravity properties like i) the non-existence of thermal AdS as a solution, ii) the existence of a point of cusp catastrophe in the phase diagram and iii) the matching of saddle points and the critical exponents of supergravity and those of the effective matrix model.

Paper Structure

This paper contains 20 sections, 68 equations, 5 figures.

Table of Contents

  1. Introduction
  2. R-charged blackholes in $AdS_{5}$ and critical phenomena
  3. Equation of state
  4. Critical Phenomena
  5. Free YM theory
  6. Effective action with chemical potential
  7. Canonical Ensemble
  8. Phase Structure
  9. Small coupling model
  10. Effective action and phase diagram at strong coupling
  11. Finite temperature effective actions in the gauge theory
  12. Non-zero charge sector
  13. Model effective action at strong coupling
  14. Universal neighborhood of critical point and the critical exponents
  15. Partition function near the critical point
  16. ...and 5 more sections

Figures (5)

  • Figure 1: Plot of $\beta(r_{+})$ for q increasing from the left. The third graph from the left is for $q_{crit}$
  • Figure 2: Plots of $D(\rho)$ with increasing $a=c$ from the above and with fixed $b$ and $q<q_{crit}$
  • Figure 3: Top three graphs of Fig $2$ showing emergence of two saddle points in the region $\rho > {1\over2}$
  • Figure 4: Plots of $D(T,\rho)$ for $T=0$, $T=T_{crit}$,$T=T_{01}$ from below.
  • Figure 5: Plots of $D(T,\rho)$ with with fixed $a$ and increasing $b$ from the top, showing a critical transition in the region $\rho>{1\over2}$ (from the top, the $3rd$ graph has a point of inflection)