On topological charge carried by nexuses and center vortices

TL;DR

This paper investigates the topological charge in the center vortex-nexus framework of gauge theories, linking it to homotopy groups and revealing new types of nexus configurations that can carry topological charge.

Contribution

It explicitly relates global topological charge to linkage numbers and homotopy classes, introducing new nexus types and clarifying their topological and physical interpretations.

Findings

Global topological charge is a linkage number of vortex and nexus surfaces.

Homotopy $\Pi_3(S^3) ext{ and } \Pi_2(S^2)$ explain charge quantization.

New nexus configurations can carry topological charge without vortices.

Abstract

In this paper we further explore the question of topological charge in the center vortex-nexus picture of gauge theories. Generally, this charge is locally fractionalized in units of 1/N for gauge group SU(N), but globally quantized in integral units. We show explicitly that in d=4 global topological charge is a linkage number of the closed two-surface of a center vortex with a nexus world line, and relate this linkage to the Hopf fibration, with homotopy $\Pi_3(S^3)\simeq Z$; this homotopy insures integrality of the global topological charge. We show that a standard nexus form used earlier, when linked to a center vortex, gives rise naturally to a homotopy $\Pi_2(S^2)\simeq Z$, a homotopy usually associated with 't Hooft-Polyakov monopoles and similar objects which exist by virtue of the presence of an adjoint scalar field which gives rise to spontaneous symmetry breaking. We show that certain integrals related to monopole or topological charge in gauge theories with adjoint scalars also appear in the center vortex-nexus picture, but with a different physical interpretation. We find a new type of nexus which can carry topological charge by linking to vortices or carry d=3 Chern-Simons number without center vortices present; the Chern-Simons number is connected with twisting and writhing of field lines, as the author had suggested earlier. In general, no topological charge in d=4 arises from these specific static configurations, since the charge is the difference of two (equal) Chern-Simons number, but it can arise through dynamic reconnection processes. We complete earlier vortex-nexus work to show explicitly how to express globally-integral topological charge as composed of essentially independent units of charge 1/N.