On flat connections with non-diagonalizable holonomies

TL;DR

This paper provides an explicit description of non-diagonalizable flat connections with commuting holonomies in the context of supersymmetric gauge theories, addressing a long-standing puzzle about the Witten index.

Contribution

It offers the first explicit construction of such flat connections for the Spin(7) gauge group, expanding understanding of moduli spaces in supersymmetric theories.

Findings

Explicit description of non-diagonalizable flat connections for Spin(7)

Addresses the existence of flat connections with non-Cartan holonomies

Contributes to the understanding of Witten index calculations in supersymmetric gauge theories

Abstract

Recently the long-standing puzzle about counting the Witten index in N=1 supersymmetric gauge theories was resolved. The resolution was based on existence (for higher orthogonal $SO(N), N \geq 7$ and exceptional gauge groups) of flat connections on $T^{3}$ which have commuting holonomies but cannot be gauged to a Cartan torus. A number of papers has been published which studied moduli spaces and some topological characteristics of those flat connections. In the present letter an explicit description of such flat connection for the basic case of $Spin(7)$ is given.