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

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