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

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.