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

TL;DR

This paper classifies gapped phases in (2+1)d with non-invertible symmetries using SymTFTs, revealing an infinite variety of boundary conditions and phases, especially for finite abelian groups, through novel theta constructions.

Contribution

It introduces non-minimal boundary conditions via theta constructions, expanding the classification of gapped phases and non-invertible symmetries in (2+1)d theories.

Findings

Infinite non-invertible symmetries from non-minimal BCs

Rich structure of gapped phases for finite abelian groups

Explicit examples for G=Z2 and other finite abelian groups

Abstract

We use the Symmetry Topological Field Theory (SymTFT) to study and classify gapped phases in (2+1)d for a class of categorical symmetries, referred to as being of bosonic type. The SymTFTs for these symmetries are given by twisted and untwisted (3+1)d Dijkgraaf-Witten (DW) theories for finite groups G. A finite set of boundary conditions (BCs) of these DW theories is well-known: these simply involve imposing Dirichlet and Neumann conditions on the (3+1)d gauge fields. We refer to these as minimal BCs. The key new observation here is that for each DW theory, there exists an infinite number of other BCs, that we call non-minimal BCs. These non-minimal BCs are all obtained by a 'theta construction', which involves stacking the Dirichlet BC with 3d TFTs having G 0-form symmetry, and gauging the diagonal G symmetry. On the one hand, using the non-minimal BCs as symmetry BCs gives rise to an infinite number of non-invertible symmetries having the same SymTFT, while on the other hand, using the non-minimal BCs as physical BCs in the sandwich construction gives rise to an infinite number of (2+1)d gapped phases for each such non-invertible symmetry. Our analysis is thoroughly exemplified for G = $\mathbb{Z_2}$ and more generally any finite abelian group, for which the resulting non-invertible symmetries and their gapped phases already reveal an immensely rich structure.