Weak Coupling Expansion of Yang-Mills Theory on Recursive Infinite Genus Surfaces

TL;DR

This paper investigates the weak coupling expansion of 2D Yang-Mills theory on recursively structured infinite genus surfaces, revealing insights into the analytic properties of the partition function and the moduli space of flat connections.

Contribution

It extends the analysis of Yang-Mills partition functions to infinite genus surfaces with recursive structures, providing a generalized Migdal formula and insights into moduli space regularization.

Findings

Weak coupling expansion is analytic in the Euler characteristic.

Moduli space of flat connections may be well-defined on infinite genus surfaces.

Generalized Migdal formula applies to recursive infinite genus surfaces.

Abstract

We analyze the partition function of two dimensional Yang-Mills theory on a family of surfaces of infinite genus. These surfaces have a recursive structure, which was used by one of us to compute the partition function that results in a generalized Migdal formula. In this paper we study the `small area' (weak coupling) expansion of the partition function, by exploiting the fact that the generalized Migdal formula is analytic in the (complexification of the) Euler characteristic. The structure of the perturbative part of the weak coupling expansion suggests that the moduli space of flat connections (of the SU(2) and SO(3) theories) on these infinite genus surfaces are well defined, perhaps in an appropriate regularization.