Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part II

TL;DR

This paper uses SymTFT to classify and analyze gapped phases with categorical symmetries in 2+1 dimensions, focusing on non-Abelian finite groups and their boundary conditions, revealing new symmetry-breaking patterns and topological orders.

Contribution

It extends the SymTFT framework to non-Abelian groups, detailing boundary conditions and symmetry structures, and explores rich gapped phases with novel symmetry-breaking patterns.

Findings

Classification of boundary conditions into minimal and non-minimal types.

Identification of symmetry structures including 2-groups and fusion 2-categories.

Discovery of phases with unique symmetry-breaking and topological order patterns.

Abstract

We use the Symmetry Topological Field Theory (SymTFT) to systematically characterize gapped phases in 2+1 dimensions with categorical symmetries. The SymTFTs that we consider are (3+1)d Dijkgraaf-Witten (DW) theories for finite groups $G$, whose gapped boundaries realize all so-called ``All Bosonic type" fusion 2-category symmetries. In arXiv:2408.05266 we provided the general framework and studied the case where $G$ is an abelian group. In this work we focus on the case of non-Abelian $G$. Gapped boundary conditions play a central role in the SymTFT construction of symmetric gapped phases. These fall into two broad families: minimal and non-minimal boundary conditions, respectively. The first kind corresponds to boundaries on which all line operators are obtainable as boundary projections of bulk line operators. The symmetries on such boundaries include (anomalous) 2-groups and 2-representation categories of 2-groups. Conversely non-minimal boundaries contain line operators that are intrinsic to the boundaries. The symmetries on such boundaries correspond to fusion 2-categories where modular tensor categories intertwine non-trivially with the above symmetry types. We discuss in detail the generalized charges of these symmetries and their condensation patterns that give rise to a zoo of rich beyond-Landau gapped phases. Among these are phases that exhibit novel patterns of symmetry breaking wherein the symmetry broken vacua carry distinct kinds of topological order. We exemplify this framework for the case where $G$ is a dihedral group.