S-duality and the prepotential of $ \mathcal{N}={2}^{\star } $ theories (II): the non-simply laced algebras

We derive a modular anomaly equation satisfied by the prepotential of the \( \mathcal{N}={2}^{\star } \) supersymmetric theories with non-simply laced gauge algebras, including the classical B r and C r infinite series and the exceptional F 4 and G 2 cases. This equation determines the exact prepotential recursively in an expansion for small mass in terms of quasi-modular forms of the S-duality group. We also discuss the behaviour of these theories under S-duality and show that the prepotential of the SO(2r + 1) theory is mapped to that of the Sp(2r) theory and viceversa, while the exceptional F 4 and G 2 theories are mapped into themselves (up to a rotation of the roots) in analogy with what happens for the \( \mathcal{N}=4 \) supersymmetric theories. These results extend the analysis for the simply laced groups presented in a companion paper.