Critical analysis of topological charge determination in the background of center vortices in SU(2) lattice gauge theory

TL;DR

This paper critically examines various methods of determining topological charge in SU(2) lattice gauge theory with center vortices, revealing complexities and discrepancies among definitions and interpretations.

Contribution

It compares multiple topological charge definitions in vortex backgrounds and investigates Dirac zeromodes, highlighting challenges and inconsistencies.

Findings

Different topological charge definitions yield inconsistent results.

Discrepancies between Dirac operator-based and geometric definitions.

Interpretation of topological charge in vortex backgrounds is complex and subtle.

Abstract

We analyze topological charge contributions from classical SU(2) center vortices with shapes of planes and spheres using different topological charge definitions, namely the center vortex picture of topological charge, a discrete version of F\~{F} in the plaquette and hypercube definitions and the lattice index theorem. For the latter the zeromodes of the Dirac operator in the fundamental and adjoint representations using both the overlap and asqtad staggered fermion formulations are investigated. We find several problems for the individual definitions and discuss the discrepancies between the different topological charge definitions. Our results show that the interpretation of topological charge in the background of center vortices is rather subtle.