Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions

Abstract

We discuss a variety of codimension-one, non-invertible topological defects in general 3+1d QFTs with a discrete one-form global symmetry. These include condensation defects from higher gauging of the one-form symmetries on a codimension-one manifold, each labeled by a discrete torsion class, and duality and triality defects from gauging in half of spacetime. The universal fusion rules between these non-invertible topological defects and the one-form symmetry surface defects are determined. Interestingly, the fusion coefficients are generally not numbers, but 2+1d TQFTs, such as invertible SPT phases, Z N gauge theories, and U (1) N Chern-Simons theories. The associativity of these algebras over TQFT coefficients relies on nontrivial facts about 2+1d TQFTs. We further prove that some of these non-invertible symmetries are intrinsically incompatible with a trivially gapped phase, leading to nontrivial constraints on renormalization group flows. Duality and triality defects are realized in many familiar gauge theories, including free Maxwell theory, non-abelian gauge theories with orthogonal gauge groups, N = 1, and N = 4 super Yang-Mills theories.

Non-invertible Global Symmetries in the Standard Model

“…(More precisely, the fusion coefficient is the partition function of the TQFT T evaluated on the three-manifold M .) Similar fusion algebras over TQFT coefficients have recently been explored in [32,35].…”

“…In the past year, non-invertible symmetries have been constructed in many familiar continuum and lattice gauge theories in higher than two spacetime dimensions [30,31,29,[32][33][34][35][36]. These non-invertible symmetries are realized by gauging a higher-form global symmetry in either half of the spacetime [31,29,34,35], or on a higher-codimensional submanifold [32]. The operators D p N in QED in this paper are realized from generalizations of such gauging constructions.…”

Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond

“…In 2+1d, the non-abelian anyons in a low-energy TQFT, described by the theory of modular tensor categories, can be viewed as the non-invertible generalization of one-form global symmetries (see [104][105][106][107]41] for recent discussions). Recently, the non-invertible Kramer-Wannier duality defects have been generalized to 3+1d gauge theories [108,109,102,110,47,111,112] via the gauging of higherform global symmetries. More generally, in any theory with a higher-form symmetry, the most fundamental non-invertible symmetries are condensation defects [42][43][44][45][46] arising from higher gauging [41].…”

“…Condensation defects in 2+1d were systematically analyzed in [41], generalizing the results of [85,113]. The non-invertible fusion "coefficients" of these n-dimensional topological defects are generally not numbers, but n-dimensional TQFTs [41,47]. More recently, infinitely many non-invertible global symmetries were identified in the Standard Model and axion models from ABJ anomalies [8,9].…”

“…There are also recent applications to 2d adjoint QCD, clarifying the infrared behavior [94]. In four-dimensional theories the dynamical implications of non-invertible symmetry are just beginning to emerge with recent results on anomalies of such symmetries [109,47] and applications to Yang-Mills theory [102].…”