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

TL;DR

The paper extends the SymTFT approach to gapped phases in (2+1)d with non-invertible, all-bosonic fusion 2-category symmetries by employing 3+1d Dijkgraaf-Witten theories and their gapped boundaries. It develops a full classification of minimal and non-minimal gapped boundary conditions, and shows how to realize and analyze a wide zoo of beyond-Landau phases, including phases with 2-groups, 2-representations, and non-invertible 1-form symmetries, via SymTFT sandwiches. The work provides explicit constructions and phase maps for cases based on S_3 and D_8 (and their dihedral/d_2n generalizations), including superstar-type vacua with mixed topological orders across distinct symmetry-breaking vacua. It also outlines how generalized gauging and boundary data generate rich phase structures, paving the way toward gapless generalizations and lattice realizations. Overall, the framework yields a comprehensive, systematic method to catalog and understand gapped phases with categorical symmetries in 2+1 dimensions.

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.