On continuous 2-category symmetries and Yang-Mills theory

A bstract We study a 4d gauge theory U(1) N− 1 ⋊ 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 categorical 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 ZN⁽¹⁾ 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 ⋊ 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.