Quantumgroups in the Higgs Phase

TL;DR

This paper explores how residual finite symmetry groups in the Higgs phase lead to discrete gauge theories described by quantum groups, and how adding a Chern-Simons term deforms these structures, affecting topological interactions.

Contribution

It introduces the deformation of quantum groups D(H) into quasi-Hopf algebras via a 3-cocycle, classifying inequivalent theories by H^3(H,U(1)), and discusses Coulomb screening in the Higgs phase.

Findings

Residual symmetry groups lead to discrete gauge theories.

Chern-Simons terms deform quantum groups into quasi-Hopf algebras.

Screening mechanisms affect Coulomb but not Aharonov-Bohm interactions.

Abstract

In the Higgs phase we may be left with a residual finite symmetry group H of the condensate. The topological interactions between the magnetic- and electric excitations in these so-called discrete H gauge theories are completely described by the Hopf algebra or quantumgroup D(H). In 2+1 dimensional space time we may add a Chern-Simons term to such a model. This deforms the underlying Hopf algebra D(H) into a quasi-Hopf algebra by means of a 3-cocycle H. Consequently, the finite number of physically inequivalent discrete H gauge theories obtained in this way are labelled by the elements of the cohomology group H^3(H,U(1)). We briefly review the above results in these notes. Special attention is given to the Coulomb screening mechanism operational in the Higgs phase. This mechanism screens the Coulomb interactions, but not the Aharonov-Bohm interactions. (Invited talk given by Mark de Wild Propitius at `The III International Conference on Mathematical Physics, String Theory and Quantum Gravity', Alushta, Ukraine, June 13-24, 1993. To be published in Theor. Math. Phys.)