Free Fermions and Thermal AdS/CFT

TL;DR

The paper provides an exact finite-$N$ solution for finite-temperature $U(N)$ gauge theories on $S^3$ in terms of a representation sum, and shows that in the large-$N$ limit the deconfinement transition has a Douglas–Kazakov–type character. By focusing on the Tr$U$ sector, it derives a continuum large-$N description in terms of a Young tableaux density $u(h)$ and a saddlepoint equation with a fermionic structure, yielding three dominant saddles corresponding to Thermal AdS, small, and large AdS black holes. A striking result is the emergence of a free-fermion phase-space description where $u(h)$ and the eigenvalue density $\sigma(\theta)$ are related as functional inverses, effectively embedding the gauge theory dynamics into a two-dimensional phase-space region with uniform density. This phase-space picture closely mirrors the Lin–Lunin–Maldacena (LLM) construction and provides a potential route to interpreting bulk geometry redundancies and bulk reconstruction from boundary data. The findings suggest the phase-space geometry of the gauge theory is robust to perturbative corrections and may encode essential features of the dual gravity dynamics across the Hawking–Page transition.

Abstract

The dynamics of finite temperature U(N) gauge theories on $S^3$ can be described, at weak coupling, by an effective unitary matrix model. Here we present an exact solution to these models, for any value of $N$, in terms of a sum over representations. Taking the large $N$ limit of this solution provides a new perspective on the deconfinement transition which is supposed to be dual to the Hawking-Page transition. The large $N$ phase transition manifests itself here in a manner similar to the Douglas-Kazakov phase transition in 2d Yang-Mills theory. We carry out a complete analysis of the saddle representation in the simplest case involving only the order parameter ${\rm Tr}U$. We find that the saddle points corresponding to thermal $AdS$, the small black hole and the large black hole can all be described in terms of free fermions. They all admit a simple phase space description {\it a la} the BPS geometries of Lin, Lunin and Maldacena.

Figures (9)

• Figure 1: Young Tableaux
• Figure 2: Contour for Solution class 1
• Figure 3: Plot of $u(h)$ vs. h for Solution Class 1. The value of $\xi$ increases from 0.5 as one goes from the leftmost graph to the right.
• Figure 4: Contour for Solution Class 2
• Figure 5: Plot of $\tilde{u}(h)$ vs. h for Solution Class 2. The value of $\xi$ decrease from 0.5 to 0 as one goes from the rightmost to the leftmost graph.