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

Abstract

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,

$${\mathbb {Z}}_N$$

Z N

gauge theories, and

$$U(1)_N$$

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,

$${{{\mathcal {N}}}}=1,$$

N = 1 ,

and

$${{{\mathcal {N}}}}=4$$

N = 4

super Yang-Mills theories.