Hagedorn vs. Hawking-Page transition in string theory
Alex Buchel, Leopoldo A. Pando Zayas
TL;DR
The paper addresses the confinement/deconfinement transition in $N=4$ SU$(N)$ SYM on $\mathbb{R}\times S^3$ with a chemical potential for a $U(1)\subset SO(6)_R$ by analyzing a single-charge AdS$_5$ black hole in $D=5$ ${\cal N}=8$ gauged supergravity. Using holographic renormalization, it derives finite expressions for the ADM mass $M$ and the generalized free energy $I_E$, confirming the relation $I_E=\beta(M-\mu_{\tilde{q}}\tilde{q})-S_{BH}$ with the physical chemical potential $\mu_{\tilde{q}}$. The work further discusses the connection between the Hawking-Page-type transition and the Hagedorn transition in a PP-wave background, highlighting how the high-$J$ sector maps to a finite free energy at the transition. It clarifies prior mass subtraction issues and provides a concrete framework for comparing large-$N$ gauge theory thermodynamics with gravity results, illuminating the phase structure at fixed $T$ and $\mu_J$.
Abstract
We study the supergravity dual to the confinement/deconfinement phase transition for the N=4 SU(N) SYM on R x S^3 with a chemical potential conjugate to a U(1)\subset SO(6)_R charge. The appropriate supergravity system is a single charge black hole in D=5 N=8 gauged supergravity. Application of the gauge/string theory holographic renormalization approach leads to new expressions for the black hole ADM mass and its generalized free energy. We comment on the relation of this phase transition to the Hagedorn transition for strings in the maximally supersymmetric plane wave background with null RR five form field strength.