[Paper Review] Non-invertible Symmetries and Higher Representation Theory I

Thomas Bartsch, Mathew Bullimore+2 more|arXiv (Cornell University)|Aug 11, 2022

Black Holes and Theoretical PhysicsPhysics and Astronomy44 references34 citations

TL;DR

The paper develops the global categorical symmetry arising from gauging finite higher groups in three dimensions, showing the full symmetry category is the (D-1)-fusion category of (D-1)-representations, and analyzes both finite groups and split 2-groups in D=3.

ABSTRACT

The purpose of this paper is to investigate the global categorical symmetries that arise when gauging finite higher groups in three or more dimensions. The motivation is to provide a common perspective on constructions of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. This paper focusses on gauging finite groups and split 2-groups in three dimensions. In addition to topological Wilson lines, we show that this generates a rich spectrum of topological surface defects labelled by 2-representations and explain their connection to condensation defects for Wilson lines. We derive various properties of the topological defects and show that the associated symmetry category is the fusion 2-category of 2-representations. This allows us to determine the full symmetry categories of certain gauge theories with disconnected gauge groups. A subsequent paper will examine gauging more general higher groups in higher dimensions.

Motivation & Objective

• Motivate a systematic construction of finite non-invertible symmetries in dimensions D>2.
• Provide a precise mathematical description of symmetry categories arising from gauging finite higher groups.
• Focus on three-dimensional theories with finite groups and split 2-groups to elucidate higher representation structures.
• Connect condensation defects and SPT insertions to the resulting topological surface defects.
• Apply the framework to gauge theories with disconnected gauge groups and outline extensions to higher dimensions.

Proposed method

• Gauging a finite group G in three dimensions by analyzing networks of G symmetry defects in the parent theory.
• Labeling simple topological surfaces by data such as G-orbits and cohomology classes, identified with irreducible 2-representations of G.
• Extending the construction to split 2-groups G = A[1] ⋊ H with A abelian and vanishing Postnikov class.
• Describing symmetry categories as the fusion 2-category 2-Rep(G) of 2-representations.
• Providing equivalent characterizations via subgroups and SPT phases to give physical intuition for simple surfaces.
• Computing fusion, 1-morphisms, and composition of 1-morphisms to match the 2-representation structure.

Experimental results

Research questions

• RQ1 What is the full symmetry category that arises when gauging finite groups and finite split 2-groups in three dimensions?
• RQ2 How do simple topological surfaces in the gauged theory relate to higher representations (2-representations) of the gauged group?
• RQ3 How do condensation defects and SPT insertions organize and classify topological defects in D=3 gaugings?
• RQ4 What is the impact of gauging 1-form and 0-form components on theories with disconnected gauge groups?
• RQ5 How can the results for finite groups and split 2-groups inform general higher-group gauging in higher dimensions?

Key findings

• Simple topological surfaces in 3D gauged theories are labeled by a G-orbit, a cohomology class in H^2(G, U(1)^O), and (for 2-groups) a collection of characters χ_j: A → U(1).
• The spectrum of topological defects and their fusion/morphism structure reproduce the 2-representation category 2-Rep(G) for the gauged finite group G and its split 2-group.
• For split 2-groups G = A[1] ⋊ H, simple surfaces are labeled by an H-orbit, a class in H^2(G, U(1)^O), and a set of A-characters satisfying compatibility, yielding a 2-representation description.
• The symmetry category of gauge theories with disconnected gauge groups in three dimensions is given by 2-Rep(G), extending previous constructions by including condensation and SPT defects.
• A framework is established to compute the full symmetry category via gauging procedures,: objects, 1-morphisms, and fusion rules align with 2-representation theory.
• The results connect the physical construction of simple topological surfaces to mathematical descriptions of 2-representations and Mackey-type inductions.