Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories

TL;DR

This work elevates group-theoretical fusion categories to the 2-categorical setting by defining group-theoretical fusion 2-categories as bimodules over twisted group-graded 2-vector spaces and proving a characterizing criterion via $\Omega\mathfrak{C}\simeq \mathbf{Rep}(H)$. It develops Morita theory, explicit underlying 2-category decompositions, and partial fusion rules to classify these objects, and then studies fiber 2-functors and Tambara-Yamagami defects, obtaining a full classification of fusion 2-categories admitting fiber 2-functors and a 2-categorical Tambara-Yamagami theory. The results connect higher categorical symmetry, gauging, and duality in $(2+1)$-dimensional physics, providing both broad structural theorems and concrete examples. The framework lays the mathematical groundwork for analyzing gapped boundaries, dualities, and TY-type defects in topological phases using higher representation theory.

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.

Figures (3)

• Figure 1: Starting with some fusion 2-category $\mathfrak{C}$, taking bimodules with respect to some algebras $\mathcal{A}$ and $\mathcal{B}$ moves between different bulks, namely $\mathbf{Bimod}_{\mathfrak{C}}(\mathcal{A})$ and $\mathbf{Bimod}_{\mathfrak{C}}(\mathcal{B})$. The solid lines denote boundaries. In particular, $\mathbf{Bimod}_{\mathfrak{C}}(\mathcal{A},\mathcal{B})$ is a boundary between $\mathbf{Bimod}_{\mathfrak{C}}(\mathcal{A})$ and $\mathbf{Bimod}_{\mathfrak{C}}(\mathcal{B})$.
• Figure 2: Morita equivalences between fusion 2-categories constructed from an arbitrary finite 2-group $\mathcal{G}={A}[1]\boldsymbol{\cdot}H[0]$. Recall that there is an equivalence $\mathbf{2Rep}(A[1])\simeq \mathbf{2Vect}_{\widehat{A}[0]}$ of finite semisimple 2-categories.
• Figure 3: Morita equivalences between fusion 2-categories constructed from a split finite 2-group $\mathcal{G}={A}[1]\rtimes H[0]$. The arrow going to the bottom left of the square represents gauging $\mathbf{2Vect}_{H[0]}$, and the arrow going to the bottom right of the square represents gauging $\mathbf{2Rep}(H[0])$. Recall that there is an equivalence $\mathbf{2Rep}(\widehat{A}[0])\simeq \mathbf{2Vect}_{A[1]}$ of finite semisimple 2-categories.

Theorems & Definitions (109)

• Theorem 1: Theorem \ref{['thm:grouptheoreticalrecognition']}
• Theorem 2: Theorem \ref{['thm:fiber2functor']}
• Theorem 3: Proposition \ref{['prop:2TYclassification']}, Proposition \ref{['prop:fusionD']}
• Definition 2.2.1
• Definition 2.3.1
• Remark 2.3.2
• Example 2.3.3
• Definition 2.3.4
• Definition 2.3.5
• Definition 2.3.6