Kramers-Wannier-like duality defects in (3+1)d gauge theories

TL;DR

The paper extends Kramers-Wannier duality beyond (1+1)d by constructing non-invertible topological defects in (3+1)d gauge theories from mixed anomalies between zero- and one-form ${\mathbb{Z}_2}$ symmetries. By gauging the 1-form symmetry and dressing the remaining defect with a minimal ${\mathcal{A}}^{2,1}$ TQFT, the authors obtain a genuine KW-like defect ${\mathcal{N}}(M_3)$ with fusion rules ${\mathcal{N}}(M_3) \times {\mathcal{N}}(M_3) = {1\over |H^0(M_3,\mathbb{Z}_2)|} \sum_{\Sigma} (-1)^{Q(\Sigma)} L(\Sigma)$ and ${\mathcal{N}} \times L = (-1)^{Q} {\mathcal{N}}$, $L\times L=1$, signaling non-invertibility. The work shows that any theory with a self-duality under gauging admits such KW-like non-invertible defects, realized via a condensate mechanism that recasts the duality as a domain-wall action and yields self-duality under $TST$ (linear case) or $TKST$ (anti-linear case). The authors illustrate the construction in concrete (3+1)d theories—$SO(3)$ YM at $\theta=\pi$, ${\mathcal{N}}=1$ $SO(3)$ YM, and ${\mathcal{N}}=4$ $SU(2)$ YM at $\tau=i$—and discuss 2+1d analogs, highlighting the interplay between higher-form symmetries, anomalies, and dualities. These KW-like defects provide a continuum realization of higher-dimensional dualities, with potential implications for phase structure, S- and T-transformations, and non-invertible symmetry actions in quantum field theory.

Abstract

We introduce a class of non-invertible topological defects in (3+1)d gauge theories whose fusion rules are the higher-dimensional analogs of those of the Kramers-Wannier defect in the (1+1)d critical Ising model. As in the lower-dimensional case, the presence of such non-invertible defects implies self-duality under a particular gauging of their discrete (higher-form) symmetries. Examples of theories with such a defect include SO(3) Yang-Mills (YM) at $θ= π$, $\mathcal{N}=1$ SO(3) super YM, and $\mathcal{N}=4$ SU(2) super YM at $τ= i$. We also introduce an analogous construction in (2+1)d, and give a number of examples in Chern-Simons-matter theories.

Figures (3)

• Figure 1: Any theory ${\mathcal{X}}$ with a self-duality admits a non-invertible defect with Kramers-Wannier-type fusion rules. ${\mathcal{X}}^*$ is the $TST$ (or $TKST$) transform of ${\mathcal{X}}$.
• Figure 2: A theory with a Kramers-Wannier defect has a self-duality since the mesh of ${\mathcal{C}}$ can be replaced by a set of topologically trivial loops of ${\mathcal{N}}$.
• Figure 3: Phase diagram of $SO(3)$ YM as a function of $\theta$.