Counting the Massive Vacua of N=1* Super Yang-Mills Theory

TL;DR

This paper calculates the number of massive vacua in N=1* super Yang-Mills theory for various gauge groups, using semi-classical methods and algebraic classifications, providing new insights especially for D-type algebras.

Contribution

It introduces a comprehensive counting method for vacua across all gauge groups, including exceptional types, and presents generating functions and indices for these cases.

Findings

Reproduces known vacuum counts for A, B, C gauge groups

Provides generating functions for O(2n) and SO(2n)

Computes supersymmetric index for exceptional groups

Abstract

We compute the number of massive vacua of N=4 supersymmetric Yang-Mills theory mass-deformed to preserve 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.