# Modular and duality properties of surface operators in N=2* gauge theories

## TL;DR

This paper computes the instanton partition function for N=2* SU(N) gauge theories with surface operators, revealing modular properties and linking four-dimensional defects to two-dimensional sigma models.

## Contribution

It introduces a novel calculation of the instanton partition function with surface operators and demonstrates its modular structure, connecting 4D gauge theories to 2D sigma models.

## Key findings

- Partition function expressed in elliptic and quasi-modular forms.
- Effective twisted superpotential satisfies a modular anomaly equation.
- Results match 2D sigma model descriptions of surface defects.

## Abstract

We calculate the instanton partition function of the four-dimensional N=2* SU(N) gauge theory in the presence of a generic surface operator, using equivariant localization. By analyzing the constraints that arise from S-duality, we show that the effective twisted superpotential, which governs the infrared dynamics of the two-dimensional theory on the surface operator, satisfies a modular anomaly equation. Exploiting the localization results, we solve this equation in terms of elliptic and quasi-modular forms which resum all non-perturbative corrections. We also show that our results, derived for monodromy defects in the four-dimensional theory, match the effective twisted superpotential describing the infrared properties of certain two-dimensional sigma models coupled either to pure N=2 or to N=2* gauge theories.