FROM SUBFACTORS TO 3-DIMENSIONAL TOPOLOGICAL QUANTUM-FIELD THEORIES AND BACK - A DETAILED ACCOUNT OF OCNEANUS THEORY

A full proof of Ocneanu's theorem is given that one can produce a rational unitary polyhedral 3-dimensional topological quantum held theory of Turaev-Viro type from a subfactor with finite index and finite depth, and vice versa. The key argument is an equivalence between flatness of a connection in paragroup theory and invariance of a state sum under one of the three local moves of tetrahedra. This was announced by A. Ocneanu and he gave a proof of Frobenius reciprocity and the pentagon relation, which produces a 3-dimensional TQFT via the Turaev-Viro machinery, but he has not published a proof of the converse direction of the equivalence. Details are given here along the lines suggested by him.