ON OCNEANUS THEORY OF ASYMPTOTIC INCLUSIONS FOR SUBFACTORS, TOPOLOGICAL QUANTUM-FIELD THEORIES AND QUANTUM DOUBLES

A fully detailed account of Ocneanu's theorem is given that the Hilbert space associated to the two-dimensional torus in a Turaev-Viro type (2 + 1)-dimensional topological quantum field theory arising from a finite depth subfactor N subset of M has a natural basis labeled by certain M infinity-M infinity bimodules of the asymptotic inclusion M boolean OR (M' boolean AND M infinity) subset of M infinity, and moreover that all these bimodules are given by the basic construction from MV(M' boolean AND M infinity) subset of M infinity if the fusion graph is connected. This Hilbert space is an analogue of the space of conformal blocks in conformal field theory. It is also shown that after passing to the asymptotic inclusions we have S- and T-matrices, analogues of the Verlinde identity and Vafa's result on roots of unity. It is explained that the asymptotic inclusions can be regarded as a subfactor analogue of the quantum double construction of Drinfel'd. These claims were announced by A. Ocneanu in several talks, but he has not published his proofs, so details are given here along the lines outlined in his talks.