Turaev-Viro invariants as an extended TQFT III

Abstract

In the third paper in this series, we examine the Reshetikhin-Turaev and Turaev-Viro TQFTs at the level of surfaces. In particular, we show that for a closed surface $Σ$, $Z_{TV, \mathcal{C}}(Σ) \cong Z_{RT, Z(\C)}(Σ)$, thus extending the equality of 3-manifold invariants proved in an earlier paper to an equivalence of TQFTs. We also describe how to compute Turaev-Viro state sums for 3-manifolds with embedded ribbon graphs.

19 pages, many figures. Added explanation of equivalence of theories for extended 3-manifolds and added description of TV theory for manifolds with embedded ribbon graphs