Macdonald's identities and the large N limit of $YM_2$ on the cylinder

TL;DR

This paper rigorously computes the large N limit of the SU(N) 2D Yang-Mills partition function on a cylinder with special boundary conditions, revealing an unexpected asymptotic behavior of the free energy.

Contribution

It provides a rigorous calculation of the large N limit for specific boundary conditions using MacDonald's identities, challenging previous predictions.

Findings

Free energy asymptotic to N times a functional of eigenvalue densities

Contradicts earlier predictions based on the complex Burgers equation

Utilizes MacDonald's identity to factor the partition function

Abstract

We give a rigorous calculation of the large N limit of the partition function of SU(N) gauge theory on a 2D cylinder in the case where one boundary holomony is a so-called special element of type $\rho$. By MacDonald's identity, the partition function factors in this case as a product over positive roots and it is straightforward to calculate the large N asymptotics of the free energy. We obtain the unexpected result that the free energy in these cases is asymptotic to N times a functional of the limit densities of eigenvalues of the boundary holonomies. This appears to contradict the predictions of Gross-Matysin and Kazakov-Wynter that the free energy should have a limit governed by the complex Burgers equation.