The Arithmetic of Supersymmetric Vacua

TL;DR

This paper derives explicit formulas for counting supersymmetric vacua in 4D N=1 super Yang-Mills theories on a circle, incorporating gauge algebra, line operators, and duality invariance, with applications to N=1* theories.

Contribution

It provides a novel arithmetic and duality-invariant method to count and classify vacua of supersymmetric gauge theories with detailed formulas.

Findings

Explicit formulas for vacua count in pure N=1 super Yang-Mills

SL(2,Z) duality invariance in vacuum enumeration

Refined index distinguishing vacua by unbroken gauge groups

Abstract

We provide explicit formulas for the number of vacua of four-dimensional pure N=1 super Yang-Mills theories on a circle, with any simple gauge algebra and any choice of center and spectrum of line operators. These form a key ingredient in the semi-classical calculation of the number of massive vacua of N=1* gauge theories with gauge algebra su(n) compactified on a circle. Using arithmetic, we express that number in an SL(2,Z) duality invariant manner. We confirm our tally of massive vacua of the N=1* theories by a count of inequivalent extrema of the exact superpotential. Furthermore, we compute a formula for a refined index that distinguishes massive vacua according to their unbroken discrete gauge group.