Non-semisimple WRT at the boundary of Crane-Yetter

TL;DR

The paper formalizes the long-standing intuition that the Witten–Reshetikhin–Turaev 3-TQFT is a boundary condition for a Crane–Yetter 4-TQFT, extending the framework to non-semisimple settings. It proves Crane–Yetter can be realized as a once-extended TQFT and constructs a boundary condition using skein theory, enabling reconstruction of WRT and its non-semisimple analogues through anomaly-trivializing data. The authors develop a non-semisimple skein-theoretic 4-TQFT $ extsc{S}_{ ext{I}}$ supported by a chromatic, modified-trace input category, and show how the boundary condition yields a projective WRT/DGGPR theory via a resolved anomaly. This work bridges 3D and 4D TQFTs in both semisimple and non-semisimple contexts, offering a robust framework for anomalous and projective TQFTs with potential fully extended generalizations.

Abstract

We prove the slogan, promoted by Walker and Freed-Teleman twenty years ago, that "The Witten-Reshetikhin-Turaev 3-TQFT is a boundary condition for the Crane-Yetter 4-TQFT" and generalize it to the non-semisimple case following ideas of Jordan, Reutter and Walker. To achieve this, we prove that the Crane-Yetter 4-TQFT and its non-semisimple version arXiv:2306.03225 are once-extended TQFTs, using the main result of arXiv:2412.14649. We define a boundary condition, partially defined in the non-semisimple case, for this 4D theory. When the ribbon category used is modular, possibly non-semisimple, we check that the composition of this boundary condition with the values of the 4-TQFT on bounding manifolds reconstructs the Witten-Reshetikhin-Turaev 3-TQFTs and their non-semisimple versions arXiv:1912.02063, in a sense that we make precise.

Figures (14)

• Figure 1: Left: A framed vertex (with blackboard framing, coming out of the page) colored by a morphism $f \in \mathop{\mathrm{Hom}}\nolimits_\mathcal{A}({\mathrm{1}\mkern-4mu{}\mathrm{l}}, X^*\otimes Z \otimes Y) \simeq \mathop{\mathrm{Hom}}\nolimits_\mathcal{A}(X, Z \otimes Y) \simeq \mathop{\mathrm{Hom}}\nolimits_\mathcal{A}( X\otimes Y^*, Z) \simeq \cdots$ Right: A coupon representing the same morphism $f \in \mathop{\mathrm{Hom}}\nolimits_\mathcal{A}( X\otimes Y^*, Z)$.
• Figure 2: An $\mathcal{I}$-colored ribbon graph in a 3-manifold with collared boundary.
• Figure 3: The gluing of skeins.
• Figure 4: The action of the skein category on the skein module. At the top left is a skein $S \in \mathop{\mathrm{Sk}}\nolimits_\mathcal{I}(M;X\sqcup Y)$. At the top right is a morphism $T_+ \in \mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathrm{SkCat}}\nolimits_\mathcal{I}(\Sigma_+)}(X\sqcup Y, Z)$. At the bottom is the skein $S\cdot T_+ \in \mathop{\mathrm{Sk}}\nolimits_\mathcal{I}(M;Z)$.
• Figure 5: The isotopy $\varphi$ of $M_{12}\underset{\Sigma_2}{\cup}M_{23}$ realizing the coend relations. It intertwines the two maps $M_{12}\underset{\Sigma_2\times I}{\cup}\Sigma_2\times [-1,1]\underset{\Sigma_2\times I}{\cup}M_{23} \to M_{12}\underset{\Sigma_2\times I}{\cup}M_{23}$ induced by the unitor diffeomorphisms for $M_{12}$ and $M_{23}$.

Theorems & Definitions (35)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Theorem 3.5
• proof
• Example 4.1