S-duality and the prepotential in (FORMULA PRESENTED.)

Abstract: The prepotential of N=2⋆supersymmetric theories with unitary gauge groups in an Ω background satisfies a modular anomaly equation that can be recursively solved order by order in an expansion for small mass. By requiring that S-duality acts on the prepotential as a Fourier transform we generalise this result to N=2⋆ theories with gauge algebras of the D and E type and show that their prepotentials can be written in terms of quasi-modular forms of SL2,ℤThe results are checked against microscopic multi-instanton calculus based on localization for the A and D series and reproduce the known 1-instanton prepotential of the pure theories for any gauge group of ADE type. Our results can also be used to obtain the multi-instanton terms in the exceptional theories for which the microscopic instanton calculus and the ADHM construction are not available.