Index of subfactors and statistics of quantum fields - II. Correspondences, braid group statistics and Jones polynomial

The endomorphism semigroup End(M) of an infinite factor M is endowed with a natural conjugation (modulo inner automorphisms) {Mathematical expression}, where γ is the canonical endomorphism of M into Ï(M). In Quantum Field Theory conjugate endomorphisms are shown to correspond to conjugate superselection sectors in the description of Doplicher, Haag and Roberts. On the other hand one easily sees that conjugate endomorphisms correspond to conjugate correspondences in the setting of A. Connes. In particular we identify the canonical tower associated with the inclusion[Figure not available: see fulltext.] relative to a sector Ï. As a corollary, making use of our previously established index-statistics correspondence, we completely describe, in low dimensional theories, the statistics of a selfconjugate superselection sector Ï with 3 or less channels, in particular of sectors with statistical dimension d(Ï)<2, by obtaining the braid group representations of V. Jones and Birman, Wenzl and Murakami. The statistics is thus described in these cases by the polynomial invariants for knots and links of Jones and Kauffman. Selconjugate sectors are subdivided into real and pseudoreal ones and the effect of this distinction on the statistics is analyzed. The FYHLMO polynomial describes arbitrary 2-channels sectors. © 1990 Springer-Verlag.