[Paper Review] Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories

Thibault D. Décoppet, Matthew Yu|arXiv (Cornell University)|Jun 13, 2023

Carbohydrate Chemistry and Synthesis11 citations

TL;DR

This paper defines and classifies group-theoretical fusion 2-categories, characterizes fusion 2-categories with fiber 2-functors, and classifies Tambara-Yamagami (TY) defects and TY fusion 2-categories, using G-graded 2-vector space constructions and 4-cocycle twisting.

ABSTRACT

We introduce group-theoretical fusion 2-categories, a strong categorification of the notion of a group-theoretical fusion 1-category. Physically speaking, such fusion 2-categories arise by gauging subgroups of a global symmetry. We show that group-theoretical fusion 2-categories are completely characterized by the property that the braided fusion 1-category of endomorphisms of the monoidal unit is Tannakian. Then, we describe the underlying finite semisimple 2-category of group-theoretical fusion 2-categories, and, more generally, of certain 2-categories of bimodules. We also partially describe the fusion rules of group-theoretical fusion 2-categories, and investigate the group gradings of such fusion 2-categories. Using our previous results, we classify fusion 2-categories admitting a fiber 2-functor. Next, we study fusion 2-categories with a Tambara-Yamagami defect, that is $\mathbb{Z}/2$-graded fusion 2-categories whose non-trivially graded factor is $\mathbf{2Vect}$. We classify these fusion 2-categories, and examine more closely the more restrictive notion of Tambara-Yamagami fusion 2-categories. Throughout, we give many examples to illustrate our various results.

Motivation & Objective

• Introduce group-theoretical fusion 2-categories as a categorification of group-theoretical fusion 1-categories and relate them to gauging subgroups of global symmetry.
• Characterize fusion 2-categories that admit fiber 2-functors via Tannakian endomorphism categories of the monoidal unit.
• Classify fusion 2-categories with Tambara-Yamagami defects and Tambara-Yamagami fusion 2-categories through finite group data and cocycles.
• Describe the underlying 2-categories and partial fusion rules for group-theoretical fusion 2-categories and their bimodule 2-categories.

Proposed method

• Construct and study 2VectG and 2VectπG to model G-graded and π-twisted structures.
• Use rigid algebras and bimodule 2-categories to realize Morita equivalences and to define modules, bimodules, and duals in fusion 2-categories.
• Prove that a fusion 2-category is group-theoretical iff its ΩC is braided equivalent to Rep(H) for some finite group H.
• Classify fiber 2-functors by showing C is equivalent to C(G,H,π,ψ) with exact group factorizations and cocycle data.
• Develop a 2-categorical analogue of TY classification by constructing TY 2-categories from A ≀ Z/2 with a 4-cocycle π, and determine when they admit fiber 2-functors.
• Relate TY defects to group-graded fusion 2-categories and describe their fusion rules via condensation picture.

Experimental results

Research questions

• RQ1 When is a fusion 2-category group-theoretical in the 2-categorical sense?
• RQ2 What are the necessary and sufficient conditions for a fusion 2-category to admit a fiber 2-functor?
• RQ3 How can Tambara-Yamagami defects be modeled and classified in the fusion 2-category setting?
• RQ4 What are the explicit data (groups, cocycles) that classify TY 2-categories and their equivalences?
• RQ5 How do duality and condensation behave in the presence of fiber 2-functors and TY defects in (2+1)-dimensional theories?

Key findings

• A fusion 2-category C is group-theoretical iff its ΩC is Tannakian, i.e., braided equivalent to Rep(H) for some finite group H.
• A fusion 2-category admits a fiber 2-functor if and only if it is equivalent to C(G,H,π,ψ) for a finite group G with an exact factorization by subgroups H and K, together with cocycle data π and ψ subject to specified constraints.
• Every Tambara-Yamagami 2-category can be constructed from data of a finite abelian group A and a class π in H4(A ≀ Z/2; k×) with restriction to A⊕A trivial; equivalence of TY 2-categories reduces to a group isomorphism respecting the wreath product structure and trivializing π/π′.
• A Tambara-Yamagami 2-category TY(A,π) is monoidally classified up to equivalence by A and π; the trivially graded factor is a direct sum of |A| copies of Mod(VectA).
• For odd-order A, TY(A,triv) admits a fiber 2-functor but is not equivalent to 2-representations of any finite 2-group when A is nontrivial; specific cases with A=Z/2 yield instances that are equivalent to 2-representations of a finite 2-group.