摘要 : The prepotential of \( \mathcal{N}={2}^{\star } \) 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 m... 展开 The prepotential of \( \mathcal{N}={2}^{\star } \) 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 \( \mathcal{N}={2}^{\star } \) 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 \( \mathrm{S}\mathrm{L}\left(2,\ \mathbb{Z}\right) \). 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 \( \mathcal{N}=2 \) 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 收起