The Hagedorn Transition, Deconfinement and N=4 SYM Theory

TL;DR

The paper investigates how deconfinement and Hagedorn thermodynamics emerge in large $N$ gauge theories by studying zero-coupling ${\cal N}=4$ SYM on $S^3$ with a singlet constraint and a broad class of toy models. It develops a necklace-counting framework to enumerate singlet bound states, revealing an intrinsic Hagedorn spectrum that coexists with a deconfinement transition; crucially, the Hagedorn temperature $T_H$ coincides with the deconfinement temperature once next-to-leading $1/N$ fluctuations are included. The analysis provides explicit results for ${\cal N}=4$ SYM, including constituent partition sums and a computed $x_H \approx 0.072$, and shows how the large-$N$ saddle must be augmented by eigenvalue-density fluctuations to recover Hagedorn behaviour. AdS/CFT considerations then connect these weak-coupling thermodynamics to gravitational physics, suggesting a dual Hawking-Page–like transition to AdS black holes even at vanishing tension. Overall, the work identifies a general mechanism in which Hagedorn thermodynamics underlies deconfinement in large-$N$ gauge theories and highlights potential applicability to broader gauge theories and holographic duals.

Abstract

N=4 Super Yang-Mills theory supplies us with a non-Abelian 4D gauge theory with a meaningful perturbation expansion, both in the UV and in the IR. We calculate the free energy on a 3-sphere and observe a deconfinement transition for large N at zero coupling. The same thermodynamic behaviour is found for a wide class of toy models, possibly also including the case of non-zero coupling. Below the transition we also find Hagedorn behaviour, which is identified with fluctuations signalling the approach to the deconfined phase. The Hagedorn and the deconfinement temperatures are identical. Application of the AdS/CFT correspondence gives a connection between string Hagedorn behaviour and black holes.