# Holomorphic anomaly of 2d Yang-Mills theory on a torus revisited

## TL;DR

This paper revisits the holomorphic anomaly in 2d Yang-Mills theory on a torus, proposing an anti-holomorphic deformation of the partition function that satisfies a traditional anomaly equation and analyzing its behavior across different coupling regimes.

## Contribution

It introduces a natural anti-holomorphic deformation of the 2d Yang-Mills partition function that obeys a holomorphic anomaly equation, clarifying its topological string interpretation.

## Key findings

- Proposes an anti-holomorphic deformation satisfying a holomorphic anomaly equation.
- Provides a closed-form expression for the deformed partition function.
- Shows simplification of the partition function in the weak coupling limit.

## Abstract

We study the large $N$ 't Hooft expansion of the chiral partition function of 2d $U(N)$ Yang-Mills theory on a torus. There is a long-standing puzzle that no explicit holomorphic anomaly equation is known for the partition function, although it admits a topological string interpretation. Based on the chiral boson interpretation we clarify how holomorphic anomaly arises and propose a natural anti-holomorphic deformation of the partition function. Our deformed partition function obeys a fairly traditional holomorphic anomaly equation. Moreover, we find a closed analytic expression for the deformed partition function. We also study the behavior of the deformed partition function both in the strong coupling/large area limit and in the weak coupling/small area limit. In particular, we observe that drastic simplification occurs in the weak coupling/small area limit, giving another nontrivial support for our anti-holomorphic deformation.