Adjoint fermion zero-modes for SU(N) calorons

TL;DR

This paper addresses the analytic construction of adjoint Dirac zero-modes in the background of $Q=1$ SU($N$) calorons, extending previous SU(2) results. It develops a comprehensive framework combining ADHM, Nahm-ADHM, and a replica trick to obtain zero-modes with both periodic and anti-periodic time boundary conditions, and provides explicit expressions and density profiles. The work analyzes normalisability, periodicity, and the decomposition of zero-modes into contributions associated with constituent monopoles, validating results against index-theorem expectations. These results supply tools for semiclassical analyses of finite-temperature SUSY Yang-Mills and related gauge theories, with potential implications for supersymmetry breaking and confinement mechanisms at finite temperature.

Abstract

We derive analytic formulas for the zero-modes of the Dirac equation in the adjoint representation in the background field of Q=1 SU(N) calorons. Solutions with various boundary conditions are obtained, including the physically most relevant cases of periodic and antiperiodic ones. The latter are essential ingredients in a semiclassical treatment of finite temperature supersymmetric Yang-Mills theory. A detailed discussion of adjoint zero-modes in several other contexts is also presented.