Supersymmetric Yang--Mills Matrix Integrals Revisited

TL;DR

This paper computes the twisted partition function of 4D supersymmetric Yang--Mills theory reduced to a point, revealing connections to nilpotent orbits, Plancherel measures, and representations of p-adic groups.

Contribution

It provides a residue-based evaluation of the partition function for all simple gauge groups and uncovers unexpected links to representation theory and measure theory.

Findings

Partition function expressed as a sum of residues classified by nilpotent orbits

Residue multiplicities are proportional to their common value

Sum over residues matches the Plancherel measure for Yang's particle system

Abstract

We evaluate the twisted partition function of four-dimensional $\mathcal{N} = 1$ supersymmetric Yang--Mills theory reduced to a point for all simple gauge groups. The partition function is expressed as a sum of residues. The types of residues that appear are classified by distinguished nilpotent orbits. Surprisingly, the multiplicity of residues of each type is proportional to their common value. The sum over residues has the same form as the Plancherel measure for Yang's system of particles. Intriguingly, the precise constants appearing in the Plancherel measure also appear in the formal degrees of unipotent discrete series representations of $p$-adic groups.