Non-invertible higher-categorical symmetries

• DOI: 10.21468/scipostphys.14.1.007
• Publication Date: 2023-01-26T17:34:43Z
• Authors: Lakshya Bhardwaj; Lea E. Bottini; Sakura Schäfer-Na...; Apoorv Tiwari
• Keywords: High Energy Physics - Theory; Strongly Correlated Electrons (cond-mat.str-el); Physics; QC1-999; FOS: Physical sciences; Mathematics - Category Theory; 01 natural sciences; Condensed Matter - Strongly Correlated Electrons; High Energy Physics - Theory (hep-th); 0103 physical sciences; FOS: Mathematics; Category Theory (math.CT)

We sketch a procedure to capture general non-invertible symmetries of a dd-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.