Counting the massive vacua of N=1^∗ super Yang-Mills theory

Abstract: We compute the number of massive vacua of N=4$$ \mathcal{N}=4 $$ supersymmetric Yang-Mills theory on ^ℝ4$$ {\mathrm{\mathbb{R}}}^4 $$ mass-deformed to preserve N=1$$ \mathcal{N}=1 $$ supersymmetry, for any gauge group G. We use semi-classical techniques and efficiently reproduce the known counting for A, B and C type gauge groups, present the generating function for both O(2n) and SO(2n), and compute the supersymmetric index for gauge groups of exceptional type. A crucial role is played by the classification of nilpotent orbits, as well as global properties of their centralizers. We give illustrative examples of new features of our analysis for the D-type algebras.