Musings

Hagedorn

While I was doing those endless recompiles, I was very much enjoying reading Aharony et al’s paper on the phase structure of large-N gauge theories.

The idea is to consider a large-N gauge theory at finite temperature and on a finite-volume $S^3$. Working in finite volume introduces a dimensionless parameter, $\gamma=R \Lambda_{QCD}$ which, if it’s small, means that the theory can be studied in perturbation theory. Even though we are in finite volume, at infinite $N$, the theory still has sharp phase transitions.

The free theory is well-known to exhibit string-like behaviour at low temperatures, with a Hagedorn transition at a certain finite temperature, $T_H$. One wants to extend this analysis to the interacting theory and study the phase structure as a function of two parameters, $RT$ and $\gamma$.

The method (pioneered by Sundborg in the $\mathcal{N}=4$ supersymmetric case), is to work in the gauge

where

For small $\gamma$ (in the $\mathcal{N}=4$ case, at weak coupling), the remaining degrees of freedom can be integrated out, leaving a single matrix integral for $\alpha$ or, more properly, for $U=e^{i\alpha}$. The effective action for $U$ can be computed in perturbation theory.

The resulting large-N unitary matrix integral is dominated at low temperatures by a uniform distribution of eigenvalues on the circle. In the free theory, the Hagedorn transition is associated to the longest “wavelength” mode on the circle going unstable. In the high temperature phase, a sinusoidal eigenvalue distribution, vanishing at $\theta=\pi$, dominates. It has long been expected that this Hagedorn transition (at $\gamma=0$) would be connected by a line of phase transitions to the deconfinement transition (at $\gamma=\infty$)

Aharony et al study the corrections to the effective action for $U$ to order $\lambda^2$ ($\lambda=g_{YM}^2 N$ is the 't Hooft coupling). Depending on the sign of a certain coefficient in the effective action (which, in turn, depends on the matter content), the Hagedorn transition is 2^nd order and is followed at a yet-higher temperature by another “Gross-Witten”-type phase transition. Alternatively, with the opposite sign, the theory undergoes a 1^st order phase transition at a temperature somewhat below $T_H$.

Matching this behaviour onto what we believe to be true in infinite volume suggests an intricate structure to the aforementioned phase diagram.