We made the following study on algebraic quantum foeld theory.
We first studied 3-dimensional topological quantum field theories arising from unitary 6j-symbols in a joint work [38] with N. Sato and M. Wakui. The Turaev-Viro-Ocneanu construction gives a numerical invariant of closed 3-manifolds from such 6j-symbols. Ocneanu has a quantum double construction for such 6j-symbols as an analogue of the quantum double construction of Drinfel'd, and it produces a modular tensor category. Then by applying the Reshetikhin-Turaev construction to this tensor category, we also obtain a numerical invariant of closed 3-manifolds. It is a natural problem to study a relation between these two invariants, and we have proved that they are identical. Ocneanu showed that such unitary 6j-symbols arise from subfactors and there have been constructed such 6j-symbols, not arising directly from quantum groups. Sato and Wakui have used this result for explicit computations of this invariant.
Next, we obtained a classification result [40] of 2-dimensional local conformal nets of factors with R. Longo. If we have diffeomorphism covariance, we can define a central charge, as a numerical invariant of such nets, and in the case this number is less than 1, we have obtained a complete classification of such nets. We had a similar classification result for 1-dimensional nets [36] in the previous year, also with Longo, and we again have pairs of A-D-E Dynkin diagrams as invariants, but now D_{2n+1} and E_7 do appear in the classification, though they did not appear in the classification of 1-dimensional nets. The new feature of the 2-dimensional classification theory is emergence of 2-cohomology groups for tensor categories. Our main technical tool is vanishing of such 2-cohomology groups for certain tensor categories related to the Virasoro algebra.