Fusion 2-categories with no line operators are grouplike

TL;DR

The paper addresses when indecomposable objects in fusion 2-categories with endomorphism category $\, ext{Vec}$ or $\, ext{SVec}$ form a finite group under fusion. It develops a state-operator calculus and condensation framework to prove that in the strongly fusion (bosonic) case the indecomposable objects form a finite group, and in the fermionic case with $\, ext{SVec}$, they still form a finite group realized as a central double cover of the component group $\\pi_0 \\mathcal{C}$. It further shows that the components themselves form a group and discusses connections to extended (super)group cohomology classifications. This advances the classification of higher-dimensional topological orders by reducing the problem to algebraic data in fusion 2-categories.

Abstract

We show that if $\mathcal{C}$ is a fusion $2$-category in which the endomorphism category of the unit object is $\rm{Vec}$ or $\rm{SVec}$, then the indecomposable objects of $\mathcal{C}$ form a finite group.

Figures (5)

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• Figure 3: $\eta^*_X$ is by definition the universal map such that the composition with $\eta_X$ can be filled. The resulting framing of $\eta^*_X \circ \eta_X$ is homotopic to the blackboard framing of Figure \ref{['AnnulusFraming']}.
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Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Remark
• Definition 2.1
• Remark
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6: douglas2018fusion