# Modular and duality properties of surface operators in $\mathcal{N}={2}^{\star }$ gauge theories

Journal of High Energy Physics, Jul 2017

We calculate the instanton partition function of the four-dimensional $\mathcal{N}={2}^{\star }$ 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 $\mathcal{N}=2$ or to $\mathcal{N}={2}^{\star }$ gauge theories.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP07%282017%29068.pdf

S. K. Ashok, M. Billò, E. Dell’Aquila, M. Frau, R. R. John, A. Lerda. Modular and duality properties of surface operators in $\mathcal{N}={2}^{\star }$ gauge theories, Journal of High Energy Physics, 2017, 68, DOI: 10.1007/JHEP07(2017)068