(Spontaneously broken) Abelian Chern-Simons theories

Abstract

A detailed analysis of Chem-Simons (CS) theories in which a compact Abelian direct product gauge group U(1) k is spontaneously broken down to a direct product of cyclic groups H ⋍ Z N (1) × … x Z N (k) is presented. The spectrum features global H charges, vortices carrying magnetic flux labeled by the elements of H and dyonic combinations. Due to the Aharonov-Bohm effect these particles exhibit topological interactions. The remnant of the U(1) k CS term in the discrete H gauge theory describing the effective long distance physics of such a model is shown to be a 3-cocycle for H governing the non-trivial topological interactions for the magnetic fluxes implied by the U(1) k CS term. It is noted that there are in general three types of 3-cocycles for a finite Abelian gauge group H: one type describes topological interactions between vortices carrying flux with respect to the same cyclic group in the direct product H, another type gives rise to topological interactions among vortices carrying flux with respect to two different cyclic factors of H and a third type leading to topological interactions between vortices carrying flux with respect to three different cyclic factors. Among other things, it is demonstrated that only the first two types can be obtained from a spontaneously broken U(1) k CS theory. The 3-cocycles that cannot be reached in this way turn out to be the most interesting. They render the theory non-Abelian and in general lead to dualities with planar theories with a non-Abelian finite gauge group. In particular, the CS theory with finite gauge group H ⋍ Z 2 × Z 2 × Z 2 defined by such a 3-cocycle is shown to be dual to the planar discrete D 4 gauge theory with D 4 the dihedral group of order 8.