# Modular properties of surface operators in N=2 SQCD

## TL;DR

This paper investigates the properties of half-BPS surface operators in N=2 SQCD with SU(2) gauge group, computing their effective superpotential and exploring duality constraints to connect field theory and conformal field theory descriptions.

## Contribution

It provides a novel computation of the twisted superpotential for surface operators in N=2 SQCD using localization and Seiberg-Witten data, and relates it to conformal blocks in Liouville CFT.

## Key findings

- Resummed instanton contributions into elliptic functions and modular forms.
- Matching of superpotential results with Liouville CFT conformal blocks.
- Insights into the modular properties of surface operators in supersymmetric gauge theories.

## Abstract

We study half-BPS surface operators in N=2 supersymmetric QCD in four dimensions with gauge group SU(2) and four fundamental flavours. We compute the twisted chiral superpotential that describes the effective theory on the surface operator using equivariant localization as well as the Seiberg-Witten data. We then use the constraints imposed by S-duality to resum the instanton contributions to the twisted superpotential into elliptic functions and (quasi-) modular forms. The resummed results match what one would obtain from the description of surface operators as the insertion of a degenerate operator in a spherical conformal block in Liouville CFT.