[Paper Review] Turaev-Viro invariants as an extended TQFT II

Benjamin Balsam|arXiv (Cornell University)|Oct 6, 2010

Geometric and Algebraic Topology4 citations

TL;DR

This paper establishes that Turaev-Viro TQFT invariants for closed 3-manifolds with links are equivalent to Reshetikhin-Turaev invariants when the input category is the Drinfeld center of a spherical fusion category. Using an extended TQFT framework, the authors prove equality via decomposition into building blocks, mapping class group actions, and surgery formulas, confirming the TV and RT theories compute identical invariants for closed 3-manifolds.

ABSTRACT

In this paper, we present the next step in the proof that $Z_{TV,\C} = Z_{RT, Z(\C)}$, namely that the theories give the same 3-manifold invariants. In future papers we will show that this equality extends to an equivalence of TQFTs.

Motivation & Objective

• To prove that Turaev-Viro invariants for closed 3-manifolds coincide with Reshetikhin-Turaev invariants when the input category is the Drinfeld center of a spherical fusion category.
• To extend the TV TQFT to an extended 3-2-1 TQFT structure compatible with Lurie’s framework.
• To show that the extended TV theory computes the same invariants as the RT theory on closed 3-manifolds, including those with embedded links.
• To demonstrate that the mapping class group actions on the torus are realized by the twist and S-matrices, matching the RT theory.
• To establish a surgery formula for the extended TV theory, which implies the main invariance result as a corollary.

Proposed method

• Decompose the 3-sphere with a link into finite building blocks and verify that the TV and RT theories agree on each block using state sum computations.
• Use the Drinfeld center $ Z( cal C) $ of a spherical fusion category $ cal C $ as the input for the RT theory, leveraging its modularity and braided structure.
• Compute the Hilbert space assigned to the torus via a direct sum decomposition involving $ \bigoplus_Z \langle Z, Z^* \rangle_{\ncal C} $, showing compatibility with the RT theory.
• Apply the gluing axiom to extend agreement from blocks to the full closed 3-manifold, ensuring invariance under topological gluing.
• Verify that the generators $ T $ and $ S $ of the mapping class group act via multiplication by the twist and S-matrices, respectively, in the TV theory.
• Derive a surgery formula for the extended TV theory, which reduces the main invariance statement to a computation on standard surgery presentations.

Experimental results

Research questions

• RQ1 Do the extended Turaev-Viro TQFT and the Reshetikhin-Turaev TQFT assign the same invariant to every closed 3-manifold with an embedded link when the input category is the Drinfeld center of a spherical fusion category?
• RQ2 How do the mapping class group actions on the torus in the TV theory compare to those in the RT theory, particularly in terms of the S and T matrices?
• RQ3 Can the extended TV TQFT be shown to satisfy a surgery formula that matches the RT theory’s surgery invariance?
• RQ4 Is the Hilbert space assigned to the torus in the TV theory isomorphic to the space of invariants $ \bigoplus_Z \operatorname{Hom}_{Z(\ncal C)}(\mathbf{1}, Z \otimes Z^*) $, as in the RT theory?
• RQ5 Does the state sum computation on decomposed 3-manifolds yield the same result as the RT invariant, via gluing and composition maps?

Key findings

• The extended Turaev-Viro TQFT assigns the same invariant to $ S^3 $ with a link as the Reshetikhin-Turaev TQFT when the input category is $ Z(\ncal C) $, the Drinfeld center of a spherical fusion category $ \ncal C $.
• The Hilbert space of the torus in the TV theory is isomorphic to $ \bigoplus_{Z \in \operatorname{Irr}(Z(\ncal C))} \langle Z, Z^* \rangle_{\ncal C} $, matching the RT theory’s assignment.
• The mapping class group generators $ T $ and $ S $ act on the torus Hilbert space via multiplication by the twist and S-matrices, respectively, as in the RT theory.
• The extended TV theory satisfies a surgery formula that reduces the invariant of any closed 3-manifold to a state sum computation on a surgery presentation.
• The state sum computation on $ \mathbb{T}^2 \times I $ yields a projector onto $ \bigoplus_Z \operatorname{Hom}_{Z(\ncal C)}(\mathbf{1}, Z \otimes Z^*) $, confirming consistency with the RT theory.
• The invariant of $ S^2 \times S^1 $ with two unlinked tubes labeled $ Z $ and $ W $ is $ \delta_{Z,W} $, matching the RT theory’s result and confirming orthogonality of distinct anyons.