Generalized Semilocal Theories and Higher Hopf Maps

TL;DR

This paper extends semilocal theories to higher Hopf bundles, classifying new defects as constrained instantons and generalized sphalerons, while exploring their geometric structures and limitations.

Contribution

It generalizes semilocal theories to realize higher Hopf bundles and classifies associated defects, expanding the understanding of gauge theories and topological configurations.

Findings

Classified semilocal defects via higher Hopf bundles.

Constructed semilocal spaces with Stiefel bundles over Grassmannians.

Failed to realize the final Hopf bundle octo f, indicating limitations.

Abstract

\def\mon{S^3\stackrel{S^1}{\rightarrow}S^2} \def\inst{S^7\stackrel{S^3}{\rightarrow}S^4} \def\octo{S^{15}\stackrel{S^7}{\rightarrow}S^8} In semilocal theories, the vacuum manifold is fibered in a non-trivial way by the action of the gauge group. Here we generalize the original semilocal theory (which was based on the Hopf bundle $\mon$) to realize the next Hopf bundle $\inst$, and its extensions $S^{2n+1}\stackrel{S^3}\rightarrow \H P^n$. The semilocal defects in this class of theories are classified by $\pi_3(S^3)$, and are interpreted as constrained instantons or generalized sphaleron configurations. We fail to find a field theoretic realization of the final Hopf bundle $\octo$, but are able to construct other semilocal spaces realizing Stiefel bundles over Grassmanian spaces.