Non-Invertible Higher-Categorical Symmetries

TL;DR

The paper develops a general framework to encode non-invertible higher-categorical symmetries of quantum field theories across dimensions by gauging 0-form sub-symmetries of invertible higher-form/higher-group structures. It introduces a higher-category (and localized-subcategory) formalism to capture topological defects and their multi-level fusion and condensation, providing practical procedures for constructing non-invertible symmetries and computing their fusion rules. The authors illustrate the approach with a wide array of 3d-4d-5d-6d examples, including Pin+, Spin(8), O(2), Sc(4N), and various 2- and 3-group constructions, and cross-check results against anomaly-based methods. The work offers a unifying, constructive path to non-invertible symmetries, linking higher-category theory, condensations, and higher-group anomalies, with potential implications for UV completions and string-theoretic realizations.

Abstract

We sketch a procedure to capture general non-invertible symmetries of a d-dimensional quantum field theory in the data of a higher-category, which captures the local properties of topological defects associated to the symmetries. We also discuss fusions of topological defects, which involve condensations/gaugings of higher-categorical symmetries localized on the worldvolumes of topological defects. Recently some fusions of topological defects were discussed in the literature where the dimension of topological defects seems to jump under fusion. This is not possible in the standard description of higher-categories. We explain that the dimension-changing fusions are understood as higher-morphisms of the higher-category describing the symmetry. We also discuss how a 0-form sub-symmetry of a higher-categorical symmetry can be gauged and describe the higher-categorical symmetry of the theory obtained after gauging. This provides a procedure for constructing non-invertible higher-categorical symmetries starting from invertible higher-form or higher-group symmetries and gauging a 0-form symmetry. We illustrate this procedure by constructing non-invertible 2-categorical symmetries in 4d gauge theories and non-invertible 3-categorical symmetries in 5d and 6d theories. We check some of the results obtained using our approach against the results obtained using a recently proposed approach based on 't Hooft anomalies.

Figures (28)

• Figure 1: Example of non-genuine defects arising at junctions of genuine defects. Here $D'_1$ and $D_2$ are genuine line and surface defects respectively. $D_1$ is a non-genuine line defect arising at the end of $D_2$ and $D_0$ is a non-genuine local operator that can arise at an end of $D'_1$ along $D_2$.
• Figure 2: An example of a non-genuine defect arising at the junctions of genuine and other non-genuine defects. Here $D_0$ is a non-genuine local operator that can arise at an end of $D'_1$ along $D_1$, where $D'_1$ is a genuine line defect, while $D_1$ itself is a non-genuine line defect that can arise at an end of the genuine surface defect $D_2$.
• Figure 3: Fusion of codimension-1 topological defects that describes a monoidal structure on the objects in the symmetry higher-category.
• Figure 4: A 1-morphism $D_{d-2}^{(1,2)}$ from $D_{d-1}^{(1)}$ to $D_{d-1}^{(2)}$ is a codimension-2 topological defect living between codimension-1 topological defects $D_{d-1}^{(1)}$ and $D_{d-1}^{(2)}$. To specify the direction of the morphism, we need to pick a "time" direction, which is taken to run from bottom to top of the figure.
• Figure 5: Fusing two codimension-2 defects $D_{d-2}^{(1,2)}$ and $D_{d-2}^{(2,3)}$ leads to the defect $D_{d-2}^{(2,3)} \circ D_{d-2}^{(1,2)}$. This is described in the higher-category as a composition of 1-morphisms, and to describe the direction of the morphisms and composition, we need to pick a "time" direction, which is taken to run from bottom to top of the figure.