STRANGE STATISTICS, BRAID GROUP REPRESENTATIONS AND MULTIPOINT FUNCTIONS IN THE N-COMPONENT MODEL

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Abstract
The statistics of fields in low dimensions is studied from the point of view of the braid group Bn of n strings. Explicit representations MR for the N-component model, N=2 to 5, are derived by solving the Yang-Baxter-like braid group relations for the statistical matrix R, which describes the transformation of the bilinear product of two N-component fields under the transposition of coordinates. When R2≠1 the statistics is neither Bose-Einstein nor Fermi-Dirac; it is strange. It is shown that for each N, the N+1 parameter family of solutions obtained is the most general one under a given set of constraints including “charge” conservation. Extended Nth order (N>2) Alexander-Conway relations for link polynomials are derived. They depend nonhomogeneously only on one of the N+1 parameters. The N=3 and 4 ones agree with those previously derived by Akutsu et al. Flat connections ω defining integrable systems of the N-component model are derived from the representations. The monodromy of the solution of such a system also carries a representation Mω of Pn⊂Bn. For N=2, Mω=MR, but the equality may not hold in general. The connections also lead directly to solutions of the classical Yang-Baxter equation. A generalization of Kohno’s monodromy representation of Bn associated with the algebra sl(N,C) is given. Applications of the braid group representations to statistical models, conformal field theory and many-body systems of extended objects are briefly discussed.

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