A bstract The prepotential of N = 2 ? $$ mathcal{N}={2}^{star } $$ 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 ? $$ mathcal{N}={2}^{star } $$ 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 S L 2 , ? $$ mathrm{S}mathrm{L}left(2, mathbb{Z}ight) $$ . 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 $$ mathcal{N}=2 $$ 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.
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