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

Benjamin Balsam|arXiv (Cornell University)|Dec 2, 2010

Geometric and Algebraic Topology4 citations

TL;DR

This paper establishes a natural isomorphism between Turaev-Viro and Reshetikhin-Turaev topological quantum field theories at the level of closed surfaces, showing that their state spaces are canonically isomorphic via surface parametrizations and cut systems. The key result extends the known equality of 3-manifold invariants to an equivalence of extended TQFTs, with explicit computation of state sums on manifolds with embedded ribbon graphs using cell decompositions and gluing axioms.

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.

Motivation & Objective

• To extend the known equality of Turaev-Viro and Reshetikhin-Turaev 3-manifold invariants to an equivalence of extended TQFTs at the level of surfaces.
• To resolve the dependence on decomposition choices in modular functor constructions by showing that TV and RT theories yield isomorphic state spaces via natural isomorphisms induced by surface cut systems.
• To provide a systematic method for computing Turaev-Viro state sums on 3-manifolds with embedded ribbon graphs, including those with uncolored strands incident to coupons.
• To demonstrate that the TQFTs are equivalent not only in dimension 3 but also in lower dimensions, particularly at the surface level, by constructing natural identifications between state spaces.

Proposed method

• Uses surface parametrizations from Bakalov-Kirillov to relate pairs-of-pants decompositions (RT) with cell decompositions (TV), enabling comparison between non-local and local constructions.
• Applies the Lego-Teichmüller game moves to relate different cut systems and constructs natural isomorphisms between TV state spaces that correspond to RT gluing maps.
• Employs the gluing axiom and projector $ H_{TV}(\Sigma) \to Z_{TV}(\Sigma) $ to show that the resulting state space is independent of decomposition choices.
• Uses handlebody cobordisms and cylinder maps to verify commutativity of diagrams, particularly for F- and S-moves, ensuring consistency across different parametrizations.
• Applies the state sum formula and surgery techniques to reduce general ribbon graphs to simpler cases with only colored strands, proving invariance under such reductions.
• Leverages the Drinfeld center construction $ Z(\mathcal{C}) $ of a spherical fusion category $ \mathcal{C} $ to relate TV invariants to modular RT invariants.

Experimental results

Research questions

• RQ1 Is there a natural isomorphism between the Turaev-Viro and Reshetikhin-Turaev TQFTs at the level of closed surfaces, independent of decomposition choices?
• RQ2 Can the Turaev-Viro state sum be consistently extended to 3-manifolds with embedded ribbon graphs, including those with uncolored strands?
• RQ3 Do the F- and S-moves in the Lego-Teichmüller game induce natural isomorphisms between TV state spaces that match the RT gluing maps?
• RQ4 Is the equivalence of TQFTs preserved when restricting to extended manifolds with corners and boundary components labeled by objects in $ Z(\mathcal{C}) $?
• RQ5 Can the state sum computation for TV invariants be reduced to a canonical form via cobordism and surgery techniques, preserving invariance?

Key findings

• For any closed surface $ \Sigma $, there exists a natural isomorphism $ Z_{TV,\mathcal{C}}(\Sigma) \cong Z_{RT,Z(\mathcal{C})}(\Sigma) $, establishing equivalence of the two TQFTs at the surface level.
• The dimension of the state space on a genus $ g $ surface is $ \mathcal{D}^{2g-2} \sum_{i \in \mathrm{Irr}(\mathcal{C})} d_i^{2-2g} $, matching both TV and RT theories.
• The F-move and S-move in the Lego-Teichmüller game induce natural isomorphisms between TV state spaces that are compatible with the RT gluing maps, ensuring consistency across decompositions.
• The Turaev-Viro state sum for a 3-manifold with a ribbon graph is invariant under the presence of uncolored strands incident to coupons, as such configurations can be reduced via cobordism to equivalent colored ones.
• The extended TQFTs are equivalent as 3-2-1 theories when restricted to manifolds with corners, provided mild conditions on the types of manifolds are imposed.
• The identity map on the state space of a 4-punctured sphere is realized via a handlebody cobordism $ \mathcal{H} $, showing that gluing such a cobordism does not alter the state sum value, thus preserving invariance.