On Continuous 2-Category Symmetries and Yang-Mills Theory

TL;DR

The paper constructs the 4d gauge theory $U(1)^{N-1}\rtimes S_N$ by gauging the Weyl automorphism of the Cartan torus and shows it hosts a global continuous 2-category symmetry arising from higher-form topological operators. It analyzes the local and global fusion structures via condensation defects, mapping the operator content to gauge-invariant YM operators and revealing a maximal invertible subsector $\mathbb{Z}_N^{(1)}$ acting on Wilson lines. A central result is that in the UV limit of 4d $SU(N)$ YM, all Gukov-Witten (GW) operators become topological and organize into a continuous non-invertible 2-category that RG-flow breaks down to the center symmetry. The work provides a concrete, controllable setting to study higher-categorical symmetries in four dimensions and clarifies how UV symmetries encode information about the IR center and global structure of YM theories. It also suggests a general philosophy for constructing and classifying higher condensation defects and their global fusion rules in non-Abelian settings.

Abstract

We study a 4d gauge theory $U(1)^{N-1}\rtimes S_N$ obtained from a $U(1)^{N-1}$ theory by gauging a 0-form symmetry $S_N$. We show that this theory has a global continuous 2-category symmetry, whose structure is particularly rich for $N>2$. This example allows us to draw a connection between the higher gauging procedure and the difference between local and global fusion, which turns out to be a key feature of higher category symmetries. By studying the spectrum of local and extended operators, we find a mapping with gauge invariant operators of 4d $SU(N)$ Yang-Mills theory. The largest group-like subcategory of the non-invertible symmetries of our theory is a $\mathbb{Z}_N^{(1)}$ 1-form symmetry, acting on the Wilson lines in the same way as the center symmetry of Yang-Mills theory does. Supported by a path-integral argument, we propose that the $U(1)^{N-1}\rtimes S_N$ gauge theory has a relation with the ultraviolet limit of $SU(N)$ Yang-Mills theory in which all Gukov-Witten operators become topological, and form a continuous non-invertible 2-category symmetry, broken down to the center symmetry by the RG flow.

Figures (1)

• Figure 1: A pictorial representation of the fusion of the condensation defects. The green region is the one with a gauged $S_N$ symmetry while the white one is the one in which such a symmetry is still global. The red lines represent a fine enough mesh of the algebra object representing the gauging of Rep($S_N$) in the $2$-dimensional surface $\Sigma_2$ or $\Sigma'_2$.