Quantum groups in the Higgs phase

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 the quasi-Hopf algebra $D^{\omega}(H)$ by means of a 3-cocycle $\omega$ on $H$. 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.