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

Lakshya Bhardwaj, Daniel Pajer+4 more|arXiv (Cornell University)|Aug 9, 2024

Liquid Crystal Research Advancements6 citations

TL;DR

This paper uses Symmetry Topological Field Theory (SymTFT) to classify gapped phases in (2+1)d with bosonic-type fusion 2-category symmetries, revealing an infinite family of non-minimal boundary conditions and corresponding gapped phases, exemplified for G = Z2 and general 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.

Motivation & Objective

• Characterize and classify gapped phases in (2+1)d with bosonic-type fusion 2-category symmetries.
• Show that (3+1)d Dijkgraaf–Witten theories admit infinitely many gapped boundary conditions (non-minimal BCs).
• Demonstrate how non-minimal BCs, via theta constructions, generate infinite families of non-invertible symmetries sharing the same SymTFT.
• Provide explicit analysis for G = Z2 and general abelian groups to illustrate the structure of non-invertible symmetries and associated phases.

Proposed method

• Apply the SymTFT framework to (3+1)d DW theories for finite groups G.
• Classify gapped boundary conditions as minimal (Dirichlet/Neumann) and non-minimal (Dirichlet/Neumann stacked with 3d TFTs and gauging).
• Use theta-constructions to generate non-minimal BCs by coupling stacked 3d TFTs with diagonal gauging.
• Describe the Drinfeld center Z( S ) for bosonic-type symmetries as Z(2Vec_G^τ) and organize topological defects accordingly.
• Analyze boundary conditions and resulting (2+1)d gapped phases via the sandwich (interval compactification) approach.

Experimental results

Research questions

• RQ1 How can fusion 2-category bosonic-type symmetries be realized as boundary data of (3+1)d DW theories?
• RQ2 What is the full set of gapped boundary conditions for (3+1)d Z2 DW theory, including non-minimal ones?
• RQ3 How do non-minimal BCs give rise to non-invertible symmetries and an infinite landscape of (2+1)d gapped phases?
• RQ4 What is the structure of gapped phases for abelian groups G, and how does theta-construction affect them?

Key findings

• There exists an infinite number of non-minimal gapped boundary conditions for each (3+1)d DW theory, beyond the familiar Dirichlet/Neumann minimal BCs.
• Non-minimal BCs are obtained by stacking Dirichlet BC with a 3d TFT and gauging the diagonal G symmetry (theta construction).
• The same SymTFT (DW theory for G) yields an infinite family of non-invertible symmetries realized on symmetry boundaries.
• For G = Z2 and general abelian G, the paper explicitly demonstrates rich structures of non-invertible symmetries and corresponding (2+1)d gapped phases.
• The analysis uses the Drinfeld center Z( S ) = Z(2Vec_G^τ) to organize topological defects and boundary data in the SymTFT framework.
• The results support a higher-categorical Landau paradigm (SymTFT) for classifying symmetric phases in (2+1)d.