Sphalerons, knots, and dynamical compactification in Yang-Mills-Chern-Simon theories

TL;DR

This paper investigates how Yang-Mills-Chern-Simons theories in three dimensions dynamically become compact, analyzing solitons like sphalerons and vortices, and revealing their topological and physical properties through a gauge-invariant framework.

Contribution

It demonstrates the dynamical compactification of YMCS theories and characterizes solitons as knots and domain walls, linking topological configurations to gauge invariance and physical stability.

Findings

YMCS theory dynamically compactifies, with non-compact states having higher vacuum energy.

Sphalerons form domain walls with half-integer CS number, confining knots in a compact domain.

Center vortices and nexuses exhibit similar compactification and topological properties.

Abstract

Euclidean d=3 SU(2) Yang-Mills-Chern-Simons (YMCS) theory, including Georgi-Glashow (GGCS) theory, may have solitons in the presence of appropriate mass terms. For integral CS level k and for solitons carrying integral CS number, YMCS is gauge-invariant and consistent. However, individual solitons such as sphalerons and linked center vortices with CS number of 1/2 and writhing center vortices with arbitrary CS number are non-compact; a condensate of them threatens compactness of the theory. We study various forms of the non-compact theory in the dilute-gas approximation, treating the parameters of non-compact large gauge transformations as collective coordinates. We conclude: 1) YMCS theory dynamically compactifies; non-compact YMCS have infinitely higher vacuum energy than compact YMCS. 2) An odd number of sphalerons is associated with a domain- wall sphaleron, a pure-gauge configuration on a closed surface enclosing them and with a half-integral CS number. 3) We interpret the domain-wall sphaleron in terms of fictitious closed Abelian magnetic field lines that express the links of the Hopf fibration. Sphalerons are over- and under-crossings of knots in the field lines; the domain-wall sphaleron is a superconducting wall confining these knots to a compact domain. 4) Analogous results hold for center vortices and nexuses. 5) For a CS term induced with an odd number of fermion doublets, domain-wall sphalerons are related to non-normalizable fermion modes. 6) GGCS with monopoles is compactified with center-vortex-like strings.