Topology and higher dimensional representations

TL;DR

This paper investigates topological charges in SU(3) gauge theory using different representations, demonstrating that physical configurations have integer topological charge and fractional charges are lattice artifacts, impacting understanding of gaugino condensates.

Contribution

It shows that the sextet and fundamental index theorems yield consistent integer topological charges in the continuum, clarifying the nature of topological configurations in lattice gauge theory.

Findings

Topological charge from sextet index always integer in the continuum.

Fractional topological charges are lattice artifacts, not physical.

SU(2) adjoint index theorem also yields integer charge.

Abstract

SU(3) gauge theory in the 2-index symmetric (sextet) and fundamental representations is considered in symmetric and periodic boxes. Using the overlap formulation in the quenched approximation it is shown that the topological charge obtained from the sextet index theorem always leads to an integer value and agrees with the charge obtained from the fundamental index theorem in the continuum. At larger lattice spacing configurations exist with fractional topological charge if the sextet index is used but these are lattice artifacts and the probability of finding such a configuration rapidly approaches zero. By considering the decomposition of the sextet representation with respect to an SU(2) subgroup it is shown that the SU(2) adjoint index theorem leads to integer charge as well. We conclude that the non-zero value of the bilinear gaugino condensate in N=1 super-Yang-Mills theory cannot be attributed to configurations with fractional topological charge once periodic boundary conditions are imposed.