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

TL;DR

The work analyzes a broad class of non-invertible codimension-one defects in 3+1d QFTs with a discrete one-form symmetry, unveiling condensation, duality, and triality defects whose fusion coefficients are 2+1d TQFTs or SPTs. By formulating higher gauging on submanifolds and half-spacetime, the authors derive universal fusion rules, reveal the necessity of 2+1d topological data for associativity, and demonstrate that certain non-invertible symmetries constrain RG flows, prohibiting trivially gapped IR phases under precise arithmetic conditions. They provide explicit realizations in classic theories, notably Maxwell theory and several gauge theories—including ${ m SU}(N)$, ${ m SO}(8)$, and ${ m N}=1,4$ super Yang–Mills—thereby illustrating how duality and triality defects manifest in known settings. The results offer a robust framework for classifying non-invertible symmetries in higher dimensions, with implications for phase structure, domain walls, and the interplay between $0$- and higher-form symmetries in quantum field theory.

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.

Figures (9)

• Figure 1: Definition of the condensation defect $\mathcal{C}_{\frac{N \ell}{2}}(M)$. The vertical red line corresponds to the codimension-one manifold $M$ on which the condensation defect is supported ($x=0$), and the arrow indicates the orientation of $M$. Here $\ell = 0,1$ for even $N$, and $\ell = 0$ for odd $N$.
• Figure 2: Lattice regularization for the condensation defect. The $b_1$ gauge field lives on the right of the red vertical line, and the $b_2$ gauge field lives on the right of the blue vertical line. Integrating out $b_2$ sets $b_1=0$ almost everywhere except for the interior of the green shaded region.
• Figure 3: The duality defect $\mathcal{D}_2(M)$ defined by gauging the $\mathbb{Z}_N^{(1)}$ symmetry only for $x>0$. Again, the red arrow indicates the orientation of $M$.
• Figure 4: The charge conjugation defect $\mathcal{U}(M)$, which does not depend on the orientation of $M$.
• Figure 5: The orientation reversal $\overline{\mathcal{D}}_2(\overline{M})$ of the duality defect. The arrow shows that the orientation of $\overline{M}$ is opposite to that of $M$.