Categorical Symmetries and Fiber Functors from Multiple Gaugeable Homomorphisms from 6D ${\cal N}=(2,0)$ SCFTs

TL;DR

The paper develops a higher-categorical framework for 6D ${\cal N}=(2,0)$ SCFTs and their 4D descendants, using SymTFT/TO and fiber-functor language to study intrinsic versus non-intrinsic non-invertible symmetries. By analyzing gauging-by-gauging procedures and relative QFTs, it identifies the total quantum dimension of a relative gaugeable algebra ${\cal A}_{\epsilon\rho}$ as a key diagnostic: intrinsic non-invertible configurations exhibit a larger quantum dimension than their non-intrinsic counterparts. This criterion is articulated in the context of class ${\cal S}$ theories descended from 6D theories, with a precise formula relating dimensions of relative algebras and their subalgebras. The results connect to broader mathematical structures, including modular tensor categories, fusion categories, and 7D CS/7D SymTFT constructions, and point toward further exploration via cobordism theory, factorisation homology, and the AGT correspondence.

Abstract

Exploiting the symmetry topological field theory/topological order correspondence (SymTFT/TO), together with the higher-categorical structure of 6D N =(2,0) SCFTs, we prove that the total quantum dimension of the relative gaugeable algebra leading to intrinsic non-invertible symmetries between class S theories is greater with respect to the non-intrinsic case. From a higher-categorical perspective, this supports the idea that multiplicity is allowed to exceed unity in some superselection sectors.

Figures (15)

• Figure 1: The S-matrix corresponds to the Hopf link, and encodes the information of mutual-statistics. As shown in these pictures, the linking between particle worldlines encodes information of mutual-statistics.
• Figure 2: The ribbon structure describing self-statistics is here drawn as the worldline of a particle excitation $x$ winding around itself. Such information is encoded in the T-matrix.
• Figure 3: The Freed-Moore-Teleman setup, with $\tilde{F}$ denoting a relative QFT. Specifying the topological data $(\sigma,\rho)$, the resulting theory, $\tilde{F}_{_{\rho}}$ is absolute.
• Figure 4: Gauging corresponds to gauging an algebra in a TFT. Idempotency ensures the resulting theory can be effectively thought of as featuring a unique defect, as shown on the RHS.
• Figure 5: gauging two different subalgebras, ${\cal A}_{_1}, {\cal A}_{_2}\ \subset\ {\cal A}$, the resulting theory corresponds to one with a changed phase with a gauging defect resulting from a relative gaugeable algebra, ${\cal A}_{_{12}}$ ending in the bulk. The defect at the endpoint is nontrivial, and can therefore be thought of as a Hom$(\mathbb{1}_{_{{\cal C}}},{\cal A}_{_{12}})$.