Quark zero modes in intersecting center vortex gauge fields

TL;DR

This paper investigates the existence and localization of zero modes of the Dirac operator in the background of intersecting center vortex gauge fields in two and four dimensions, revealing their dependence on topological charge and flux.

Contribution

It demonstrates the conditions under which normalizable zero modes appear in vortex backgrounds on compact manifolds, linking zero modes to topological features and vortex intersections.

Findings

Zero modes exist when net flux exceeds 1 in 2D.

Zero modes are localized at vortex intersections in 4D.

On tori, zero modes correspond to non-zero topological charge.

Abstract

The zero modes of the Dirac operator in the background of center vortex gauge field configurations in $\R^2$ and $\R^4$ are examined. If the net flux in D=2 is larger than 1 we obtain normalizable zero modes which are mainly localized at the vortices. In D=4 quasi-normalizable zero modes exist for intersecting flat vortex sheets with the Pontryagin index equal to 2. These zero modes are mainly localized at the vortex intersection points, which carry a topological charge of $\pm 1/2$. To circumvent the problem of normalizability the space-time manifold is chosen to be the (compact) torus $\T^2$ and $\T^4$, respectively. According to the index theorem there are normalizable zero modes on $\T^2$ if the net flux is non-zero. These zero modes are localized at the vortices. On $\T^4$ zero modes exist for a non-vanishing Pontryagin index. As in $\R^4$ these zero modes are localized at the vortex intersection points.