#### Alladi Ramakrishnan Hall

#### S-duality and modular anomaly equations in N=2 SQCD

#### Madhusudhan Raman

##### IMSc

*We use equivariant localization techniques to study the non-perturbative data of N=2 superconformal gauge theories with gauge group SU(N) and 2N fundamental flavours. In order to identify and better understand the underlying modular structures, the instanton corrections to the prepotential, the dual periods, and the period matrix are calculated in a locus of special vacua possessing a Z_N symmetry. In a semi-classical expansion, we show that these observables are constrained by S-duality via a modular anomaly equation which takes the form of a recursion relation. These constraints allow us to resum the instanton contributions to these observables, making it possible to determine quasi-modular parts of these observables to arbitrarily high order in the effective coupling constant. For N = 2, 3, 4, and 6, the S-duality group contains a subgroup that is also a subgroup of the modular group. We show that the solutions to the modular anomaly equations are meromorphic modular functions of subgroups of SL(2,Z).*

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