(2+1)-dimensional topological quantum field theory from subfactors and Dehn surgery formula for 3-manifold invariants

TL;DR

This paper develops a (2+1)-dimensional topological quantum field theory framework using subfactors, establishing a Dehn surgery formula for 3-manifold invariants, and connecting it to quantum doubles and modular categories.

Contribution

It introduces a TQFT with Verlinde basis derived from subfactors and proves a Dehn surgery formula for 3-manifold invariants within this framework.

Findings

Established a general theory of (2+1)-dimensional TQFT with Verlinde basis.

Derived a Dehn surgery formula for 3-manifold invariants in this TQFT.

Connected the Dehn surgery formula to Reshetikhin-Turaev invariants from modular categories.

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 the forthcoming paper, we will exbit computations of Turaev-Viro-Ocneanu invariants for several ``basic 3-manifolds ''. In Appendix, we discuss the relationship between the system of M_{infinity}-M_{infinity} bimodules arising from the asymptotic inclusion M V M^{op} subset M_{infinity} constructed from N subset M and the tube system obtained from a subfactor N subset M.