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Higher Categories and Topological Quantum Field Theories

Shawn X. Cui

TL;DR

This work constructs a state-sum invariant Z of closed oriented 4-manifolds from a finite group G cross braided spherical fusion category (G-BSFC) and shows it extends to a (3+1)-dimensional TQFT. The invariant generalizes Crane-Yetter and Yetter-type invariants, and admits a ω-twisted variant when the G-grading is trivial or concentrated on the identity sector, connecting to twisted Dijkgraaf-Witten theory. The authors also develop a higher-categorical perspective by associating a monoidal 2-category and a spherical Gray category structure to a G-BSFC, clarifying the relation between 4D state-sum invariants and higher category axioms. The results provide a framework that enriches 4D topology with categorical data and opens multiple avenues for refining invariants and exploring their smooth-structure sensitivity and presentations via Kirby diagrams.

Abstract

We construct a state-sum type invariant of smooth closed oriented $4$-manifolds out of a $G$-crossed braided spherical fusion category ($G$-BSFC) for $G$ a finite group. The construction can be extended to obtain a $(3+1)$-dimensional topological quantum field theory (TQFT). The invariant of $4$-manifolds generalizes several known invariants in literature such as the Crane-Yetter invariant from a ribbon fusion category and Yetter's invariant from homotopy $2$-types. If the $G$-BSFC is concentrated only at the sector indexed by the trivial group element, a cohomology class in $H^4(G,U(1))$ can be introduced to produce a different invariant, which reduces to the twisted Dijkgraaf-Witten theory in a special case. Although not proven, it is believed that our invariants are strictly different from other known invariants. It remains to be seen if the invariants are sensitive to smooth structures. It is expected that the most general input to the state-sum type construction of $(3+1)$-TQFTs is a spherical fusion $2$-category. We show that a $G$-BSFC corresponds to a monoidal $2$-category with certain extra structure, but that structure does not satisfy all the axioms of a spherical fusion $2$-category given by M. Mackaay. Thus the question of what axioms properly define a spherical fusion $2$-category is open.

Higher Categories and Topological Quantum Field Theories

TL;DR

This work constructs a state-sum invariant Z of closed oriented 4-manifolds from a finite group G cross braided spherical fusion category (G-BSFC) and shows it extends to a (3+1)-dimensional TQFT. The invariant generalizes Crane-Yetter and Yetter-type invariants, and admits a ω-twisted variant when the G-grading is trivial or concentrated on the identity sector, connecting to twisted Dijkgraaf-Witten theory. The authors also develop a higher-categorical perspective by associating a monoidal 2-category and a spherical Gray category structure to a G-BSFC, clarifying the relation between 4D state-sum invariants and higher category axioms. The results provide a framework that enriches 4D topology with categorical data and opens multiple avenues for refining invariants and exploring their smooth-structure sensitivity and presentations via Kirby diagrams.

Abstract

We construct a state-sum type invariant of smooth closed oriented -manifolds out of a -crossed braided spherical fusion category (-BSFC) for a finite group. The construction can be extended to obtain a -dimensional topological quantum field theory (TQFT). The invariant of -manifolds generalizes several known invariants in literature such as the Crane-Yetter invariant from a ribbon fusion category and Yetter's invariant from homotopy -types. If the -BSFC is concentrated only at the sector indexed by the trivial group element, a cohomology class in can be introduced to produce a different invariant, which reduces to the twisted Dijkgraaf-Witten theory in a special case. Although not proven, it is believed that our invariants are strictly different from other known invariants. It remains to be seen if the invariants are sensitive to smooth structures. It is expected that the most general input to the state-sum type construction of -TQFTs is a spherical fusion -category. We show that a -BSFC corresponds to a monoidal -category with certain extra structure, but that structure does not satisfy all the axioms of a spherical fusion -category given by M. Mackaay. Thus the question of what axioms properly define a spherical fusion -category is open.

Paper Structure

This paper contains 23 sections, 28 theorems, 117 equations, 31 figures, 3 tables.

Table of Contents

  1. Introduction
  2. $G$-crossed Braided Spherical Fusion Categories ($G$-BSFCs)
  3. Definition of $G$-BSFCs
  4. Strictifying $G$-BSFCs
  5. Skeletonizing $G$-BSFCs
  6. Partition Function
  7. Definition of Partition Function I
  8. Definition of Partition Function II
  9. Definition of Partition Function III
  10. Examples and Variations
  11. Crane-Yetter Invariant
  12. Yetter's Invariant from Homotopy $2$-types
  13. Trivial $G$-grading with a Trivial $G$-action
  14. Variation: Trivial $G$-grading
  15. Proof of Theorem \ref{['thm:main']}
  16. ...and 8 more sections

Key Result

Theorem 1.1

The formula for $Z_{{\mathcal{C}}_{G}^{\times}}(M, \mathcal{T})$ is independent of the choice of the triangulation $\mathcal{T}$ and thus $Z_{{\mathcal{C}}_{G}^{\times}}(M):= Z_{{\mathcal{C}}_{G}^{\times}}(M, \mathcal{T})$ is an invariant of closed smooth oriented $4$-manifolds.

Figures (31)

  • Figure 1: Example of a graph diagram
  • Figure 2: Colorings of a graph diagram
  • Figure 3: Interpretations of a graph diagram
  • Figure 4: Regular isotopies of a graph diagram (I)
  • Figure 5: Regular isotopies of a graph diagram (II). Note that in $(b)$, $|A| = |B|$.
  • ...and 26 more figures

Theorems & Definitions (56)

  • Theorem 1.1: Main, informal
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 1.5: Informal
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • ...and 46 more