Higher Categories and Topological Quantum Field Theories

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.

The published version renamed as "Four Dimensional Topological Quantum Field Theories from $G$-crossed Braided Categories" for accuracy

References in corpus (5)

• On Gauging Symmetry of Modular Categories
• On $G$--equivariant modular categories
• Dichromatic state sum models for four-manifolds from pivotal functors
• Four-manifolds up to connected sum with complex projective planes
• Evaluating TQFT invariants from G-crossed braided spherical fusion categories via Kirby diagrams with 3-handles

Cited by in corpus (17)

• Non-Invertible Higher-Categorical Symmetries
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• Symmetry-protected self-correcting quantum memories
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTs
• TQFT, Symmetry Breaking, and Finite Gauge Theory in 3+1D
• A 3-categorical perspective on G-crossed braided categories
• (3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases
• Semisimple 4-dimensional topological field theories cannot detect exotic smooth structure
• Anomaly of $(2+1)$-Dimensional Symmetry-Enriched Topological Order from $(3+1)$-Dimensional Topological Quantum Field Theory
• Subsystem Symmetry Fractionalization and Foliated Field Theory
• A relation between chiral central charge and ground state degeneracy in 2+1-dimensional topological orders
• Lieb-Schultz-Mattis Theorem and the Filling Constraint
• Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)
• $G$-crossed braided zesting
• Condensation Completion and Defects in 2+1D Topological Orders
• Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization