Formula omitted]-dimensional topological quantum field theory from subfactors and Dehn surgery formula for 3-manifold invariants

Abstract

In this paper, we establish the general theory of ( 2 + 1 ) -dimensional topological quantum field theory (in short, TQFT) with a Verlinde basis. It is a consequence that we have a Dehn surgery formula for 3-manifold invariants for this kind of TQFT's. We will show that Turaev–Viro–Ocneanu unitary TQFT's obtained from subfactors satisfy the axioms of TQFT's with Verlinde bases. Hence, in a Turaev–Viro–Ocneanu TQFT, we have a Dehn surgery formula for 3-manifolds. It turns out that this Dehn surgery formula is nothing but the formula of the Reshetikhin–Turaev invariant constructed from a tube system, which is a modular category corresponding to the quantum double construction of a C * -tensor category. In Sato and Wakui (J. Knot Theory Ramif. 12 (2003) 543), we will exhibit computations of Turaev–Viro–Ocneanu invariants for several “basic 3-manifolds”. In the appendix, we discuss the relationship between the system of M ∞ - M ∞ bimodules arising from the asymptotic inclusion M ∨ M op ⊂ M ∞ constructed from N ⊂ M and the tube system obtained from a subfactor N ⊂ M .