Flow of Hagedorn singularities and phase transitions in large $N$ gauge theories

TL;DR

This paper analyzes the singularity structure of the large-$N$ QCD partition function with massive adjoint fermions, revealing how phase transitions and Hagedorn behavior are connected to the flow and collision of singularities in the complex temperature plane.

Contribution

It generalizes Lee-Yang-Fisher analysis to large-$N$ gauge theories, identifying mechanisms for Hagedorn singularities and phase transitions via singularity flow.

Findings

Identified two mechanisms for Hagedorn singularities: inflow from zero and collision of conjugate pairs.

Connected singularity flow to phase transitions and center-symmetry changes.

Extended analysis to theories with massive adjoint fermions on small three-spheres.

Abstract

We investigate the singularity structure of the $(-1)^F$ graded partition function in QCD with $n_f \geq 1$ massive adjoint fermions in the large-$N$ limit. Here, $F$ is fermion number and $N$ is the number of colors. The large $N$ partition function is made reliably calculable by taking space to be a small three-sphere $S^3$. Singularites in the graded partition function are related to phase transitions and to Hagedorn behavior in the $(-1)^F$-graded density of states. We study the flow of the singularities in the complex "inverse temperature" $\beta$ plane as a function of the quark mass. This analysis is a generalization of the Lee-Yang-Fisher-type analysis for a theory which is always in the thermodynamic limit thanks to the large $N$ limit. We identify two distinct mechanisms for the appearance of physical Hagedorn singularities and center-symmetry changing phase transitions at real positive $\beta$, inflow of singularities from the $\beta=0$ point, and collisions of complex conjugate pairs of singularities.