Center vortices, nexuses, and fractional topological charge

Abstract

It has been remarked in several previous works that the combination of center vortices and nexuses (a nexus is a monopole-like soliton whose world line mediates certain allowed changes of field strengths on vortex surfaces) carries topological charge quantized in units of $1/N$ for gauge group $\mathrm{SU}(N).$ These fractional charges arise from the interpretation of the standard topological charge $\ensuremath{\int}G\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}$ as a sum of (integral) intersection numbers weighted by certain (fractional) traces. We show that without nexuses the sum of intersection numbers gives vanishing topological charge (since vortex surfaces are closed and compact). With nexuses residing as world lines on vortices, the contributions to the total intersection number are weighted by different trace factors, and yield a picture of the total topological charge as a linking of a closed nexus world line with a vortex surface; this linking gives rise to non-vanishing but integral total topological charge. This reflects the standard $2\ensuremath{\pi}$ periodicity of the theta angle. We argue that the Witten-Veneziano ${\ensuremath{\eta}}^{\ensuremath{'}}$ relation, naively violating $2\ensuremath{\pi}$ periodicity, scales properly with N at large N without requiring $\ensuremath{\theta}$ periodicity of $2\ensuremath{\pi}N.$ This reflects the underlying composition of localized fractional topological charges, which are in general widely separated. Some simple models are given of this behavior. In the intersection-number picture of topological charge, nexuses lead to non-standard surfaces for all $\mathrm{SU}(N)$ and intersections of surfaces which do not constitute manifolds for $N>2.$ We generalize previously introduced nexuses to all $\mathrm{SU}(N)$ in terms of a set of fundamental nexuses, which can be distorted into a configuration resembling the 't Hooft--Polyakov monopole with no strings. Nexuses can also be exhibited with thick non-singular strings, which generate vortices with nexus (and anti-nexus) world lines appearing as boundaries on the vortex surface. The existence of localized but widely separated fractional topological charges, adding to integers only on long distance scales, has important implications for fermion zero modes and the existence of standard chiral condensates in chiral symmetry breakdown, avoiding the usual difficulty when the number of flavors exceeds one, as we review.