Turaev-Viro invariants as an extended TQFT

TL;DR

The paper develops an extended Turaev–Viro framework by extending TV invariants to 3-manifolds with corners using a spherical fusion category $\mathcal{C}$ and its Drinfeld center $Z(\mathcal{C})$. It builds a 3–2–1 extended TQFT: polytope-decomposition independence is established, surfaces with boundary are labeled by $Z(\mathcal{C})$, and extended 3-manifolds with tubes yield invariants living in centers, aligning with Reshetikhin–Turaev theory for $Z(\mathcal{C})$. A sequence of constructions—polytope decompositions, state spaces, and gluing laws—ensures a coherent extended theory and sets up the conjectured equivalence $Z_{TV,\mathcal{C}}(M)=Z_{RT, Z(\mathcal{C})}(M)$, with partial proofs and explicit computations supporting the framework. The work culminates in concrete computations (cylinder and sphere with holes) that illustrate how center data governs boundary and tube contributions, paving the way for full equivalence results in subsequent papers.

Abstract

In this paper we show how one can extend Turaev-Viro invariants, defined for an arbitrary spherical fusion category $C$, to 3-manifolds with corners. We demonstrate that this gives an extended TQFT which conjecturally coincides with the Reshetikhin-Turaev TQFT corresponding to the Drinfeld center $Z(C)$. In the present paper we give a partial proof of this statement.

Figures (23)

• Figure 1: Round coupons
• Figure 3: Graph on a sphere and its "flattening" to the plane
• Figure 4: Graphical presentation of the half-braiding $\varphi_Y\colon Y\otimes V\to V\otimes Y$, $Y\in \mathop{\mathrm{Obj}}\nolimits Z(\mathcal{C})$, $V\in \mathop{\mathrm{Obj}}\nolimits \mathcal{C}$
• Figure 5: Half-braiding $I(V)\otimes W\to W\otimes I(V)$.
• Figure 6: Diagram $\hat{\Gamma}$ on the sphere and its flattening to the plane. Arc $\gamma$ is shown by a double line.

Theorems & Definitions (47)

• Lemma 1.1
• Corollary 1.2
• Lemma 1.3
• Theorem 1.4
• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4