Towards computational foundations of generalized symmetries

Topology and symmetry in non-Abelian gauge theories are considered with lattice regularization. Recently, the concept of symmetry has been generalized; the important ingredients are given by higher-form, higher-group, and non-invertible symmetries. At first, we start from an extension of Luescher’s construction of topology on the lattice. Thus we recover the SU(N)/Z_N principal bundle structure from lattice SU(N) gauge fields coupling to Z_N 2-form gauge fields. We then explicitly demonstrate the fractional topological charge. Our construction can be applied to analyzing the higher-group and non-invertible symmetries in the SU(N) gauge theory, and also provide a computational foundation of those in lattice simulations. Finally, we realize an axial U(1) non-invertible symmetry on lattice chiral gauge theory with an anomalous matter content.