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 global fusions of topological defects, where some of the directions of the defects are wrapped along compact sub-manifolds in spacetime while some others are left non-compact, and argue that such a global fusion is described as a local fusion plus a gauging of a higher-categorical symmetry localized along the compact sub-manifold. Recently some fusions of topological defects were discussed in the literature where the dimension of topological defects seems to jump under fusion. We explain how such fusions can be understood in terms of fusion operations of a higher-category, where the dimension does not jump. 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.

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.

In this paper we present various 4d$$ \mathcal{N} $$ N = 1 dualities involving theories obtained by gluing two E[USp(2N)] blocks via the gauging of a common USp(2N) symmetry with the addition of 2L fundamental matter chiral fields. For L = 0 in particular the theory has a quantum deformed moduli space with chiral symmetry breaking and its index takes the form of a delta-function. We interpret it as the Identity wall which identifies the two surviving USp(2N) of each E[USp(2N)] block. All the dualities are derived from iterative applications of the Intriligator-Pouliot duality. This plays for us the role of the fundamental duality, from which we derive all others. We then focus on the 3d version of our 4d dualities, which now involve the $$ \mathcal{N} $$ N = 4 T[SU(N)] quiver theory that is known to correspond to the 3d S-wall. We show how these 3d dualities correspond to the relations S2 = −1, S−1S = 1 and STS = T−1S−1T−1 for the S and T generators of SL(2, ℤ). These observations lead us to conjecture that E[USp(2N)] can also be interpreted as a 4d S-wall.

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