Magnetic discrete gauge field in the confining vacua and the supersymmetric index

TL;DR

The paper provides evidence that magnetic $\mathbb{Z}_q$ discrete gauge fields arise in the confining vacua of Yang-Mills theory, by explicitly matching the supersymmetric index (Witten index) of pure $\mathcal{N}=1$ SYM across ultraviolet and infrared descriptions for several gauge groups. It conducts detailed checks for $G=SU(2)$, $SO(3)$, $SU(N)/\mathbb{Z}_N$, and $SO(N)$, using commuting holonomies, generalized Stiefel-Whitney classes, and discrete theta angles, and shows sector-by-sector index matching. A uniform argument for general connected gauge groups is presented, leveraging 1-form global symmetries and spectral flow to equate UV and IR counts in each charge sector. The results reinforce the view that confining vacua host topological magnetic degrees of freedom and provide a coherent framework tying together UV data, IR dynamics, and higher-form symmetries in supersymmetric gauge theories.

Abstract

It has recently been argued that the confining vacua of Yang-Mills theory in the far infrared can have topological degrees of freedom given by magnetic $\mathbb{Z}_q$ gauge field, both in the non-supersymmetric case and in the N=1 supersymmetric case. In this short note we give another piece of evidence by computing and matching the supersymmetric index of the pure super Yang-Mills theory both in the ultraviolet and in the infrared.