The three avtars of the Chern-Simons term in 3 space-time dimensions

Abstract

I discuss in detail the three "avtars" of the Chern-Simons (C-S) term in 2 + 1 dimensions i.e. (i) C-S term as gauge field mass term (ii) C-S term as a purely kinematic term (iii) gauge theories with purely C-S action. In the first case we find that because of the C-S term one has massive gauge quanta and still the theory is gauge invariant. Such a C-S term can be generated either by spontaneous symmetry breaking or by radiative corrections. The dramatic effect of this term is that the vortices of the abelian (or nonabelian) Higgs model now have finite, quantized charge and angular momentum. In the second case the C-S term is not really independent but can be expressed in terms of the basic quanta of the 0(3) nonlinear σ-model or CP¹ model. In either case one finds that due to this term the soliton of the model has fractional spin and statistic interpolating between fermions and bosons. The relevance of this in the context of high-T_c, superconductivity is discussed in some detail. Finally the third avtar has to do with some recent work of Witten where he has shown that the Hilbert spaces of the quantum Yang-Mills theory with pure C-S action can be interpreted as the spaces of the conformal blocks in 1 + 1 dimensional conformal theories.