The Witten Index Beyond the Adiabatic Approximation

TL;DR

This paper investigates the Witten index in supersymmetric gauge theories on a torus, addressing orbifold singularities and the breakdown of the adiabatic approximation by considering zero-momentum fields within a fundamental domain.

Contribution

It introduces a method to handle orbifold singularities and Gribov copies by restricting fields to a fundamental domain with specific boundary conditions.

Findings

Resolved the breakdown of the adiabatic approximation at singularities.

Developed a framework for defining zero-momentum fields globally.

Provided insights into the structure of the moduli space in supersymmetric theories.

Abstract

We attempt to deal with the orbifold singularities in the moduli space of flat connections for supersymmetric gauge theories on the torus. At these singularities the energy gap in the transverse fluctuations vanishes and the resulting breakdown of the adiabatic approximation is resolved by considering the full set of zero-momentum fields. These can not be defined globally, due to the problem of Gribov copies. For this reason we restrict the fields to the fundamental domain, containing no gauge copies, but requiring a boundary condition in field space.