S-duality and the prepotential in N = 2 ⋆ theories (I): the ADE algebras

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 SL(2, Z). The 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 N = 2 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.