Resurgence in 2-dimensional Yang-Mills and a genus altering deformation

TL;DR

This paper explores the resurgence properties of 2D Yang-Mills theory on surfaces, introducing a deformation that analytically continues the genus, revealing new saddle solutions and mathematical structures.

Contribution

It introduces a genus-altering deformation in 2D Yang-Mills theory, analyzes new saddle solutions, and derives novel PDEs for the partition function.

Findings

Deformation leads to non-integer effective genus.

Identification of new saddle solutions and their properties.

Derivation of partial differential equations for the partition function.

Abstract

We study resurgence in the context of the partition function of 2-dimensional $SU(N)$ and $U(N)$ Yang-Mills theory on a surface of genus $h$. After discussing the properties of the transseries in the undeformed theory, we add a term to the action to deform the theory. The partition function can still be calculated exactly, and the deformation has the effect of analytically continuing the effective genus parameter in the exact answer to be non-integer. In the deformed theory we find new saddle solutions and study their properties. In this context each saddle contributes an asymptotic series to the transseries which can be analysed using Borel-\`Ecalle resummation. For specific values of the deformation parameter we find Cheshire cat points where the asymptotic series in the transseries truncate to a few terms. We also find new partial differential equations satisfied by the partition function, and a number of applications of these are explained, including low-order/low-order resurgence.