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

TL;DR

This paper explores non-invertible topological defects in 3+1d quantum field theories with one-form symmetries, revealing their fusion rules, algebraic structures, and implications for phases and dualities in gauge theories.

Contribution

It introduces a comprehensive framework for understanding non-invertible defects, their fusion rules involving 2+1d TQFTs, and their constraints on phases in 3+1d QFTs, including gauge theories.

Findings

Fusion rules involve 2+1d TQFTs like gauge theories and Chern-Simons theories.

Some non-invertible symmetries prevent trivial gapped phases.

Duality and triality defects appear in various gauge theories, including Maxwell and super Yang-Mills.

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, $\mathbb{Z}_N$ 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, ${\cal N}=1,$ and ${\cal N}=4$ super Yang-Mills theories.