Topological interactions in broken gauge theories

TL;DR

This thesis explores topological interactions in gauge theories with broken symmetry, focusing on the role of Chern-Simons terms, Hopf algebras, and their deformations, revealing new insights into the structure of vortices and gauge groups.

Contribution

It introduces a framework connecting Chern-Simons actions, 3-cocycles, and Hopf algebra deformations in broken gauge theories, including nonabelian cases.

Findings

Chern-Simons terms induce topological interactions governed by 3-cocycles.

Hopf algebra D(H) deforms into a quasi-Hopf algebra with CS terms.

Certain 3-cocycles, like Z_2 x Z_2 x Z_2, lead to dual nonabelian gauge theories.

Abstract

This thesis deals with planar gauge theories in which some gauge group G is spontaneously broken to a finite subgroup H. The spectrum consists of magnetic vortices, global H charges and dyonic combinations exhibiting topological Aharonov-Bohm interactions. Among other things, we review the Hopf algebra D(H) related to this residual discrete H gauge theory, which provides an unified description of the spin, braid and fusion properties of the aforementioned particles. The implications of adding a Chern-Simons (CS) term to these models are also addressed. We recall that the CS actions for a compact gauge group G are classified by the cohomology group H^4(BG,Z). For finite groups H this classification boils down to the cohomology group H^3(H,U(1)). Thus the different CS actions for a finite group H are given by the inequivalent 3-cocycles of H. It is argued that adding a CS action for the broken gauge group G leads to additional topological interactions for the vortices governed by a 3-cocycle for the residual finite gauge group H determined by a natural homomorphism from H^4(BG,Z) to H^3(H,U(1)). Accordingly, the related Hopf algebra D(H) is deformed into a quasi-Hopf algebra. These general considerations are illustrated by CS theories in which the direct product of some U(1) gauge groups is broken to a finite subgroup H. It turns out that not all conceivable 3-cocycles for finite abelian gauge groups H can be obtained in this way. Those that are not reached are the most interesting. A Z_2 x Z_2 x Z_2 CS theory given by such a 3-cocycle, for instance, is dual to an ordinary gauge theory with nonabelian gauge group the dihedral group of order eight. Finally, the CS theories with nonabelian finite gauge group a dihedral or double dihedral group are also discussed in full detail.