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Topological field theories and fractional statistics Bergeron, Mario


We examine the problem of determining which representations of the braid group on a Riemann surface are carried by the wave function of a quantized Abelian Chern-Simons theory interacting with non-dynamical matter. We generalize the quantization of Chern-Simons theory to the case where the coefficient of the Chern-Simons term, k, is rational, the Riemann surface has arbitrary genus and the total matter charge is non-vanishing. We find an explicit solution of the SchrOdinger equation. We find that the wave functions carry a representation of the braid group as well as a projective representation of the discrete group of large gauge transformations. We find a fundamental constraint that relates the charges of the particles, q, the coefficient k and the genus of the manifold, g. We study the non-linear sigma model with a Chern-Simons term. We find the canonical structures of the model using Dirac bracket, accounting for the non-trivial constraints of the sigma-model. We show that solutions to the field equation are represented by solitons. We also recover braid group representations for the low energy limit of soliton exchange.

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