A bstract We derive a modular anomaly equation satisfied by the prepotential of the N = 2 ? $$ mathcal{N}={2}^{star } $$ 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(2 r + 1) theory is mapped to that of the Sp(2 r ) 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 N = 4 $$ mathcal{N}=4 $$ supersymmetric theories. These results extend the analysis for the simply laced groups presented in a companion paper.
展开▼
