Lattice construction of mixed 't Hooft anomaly with higher-form symmetry

In this talk, we give the lattice regularized formulation of the mixed 't Hooft anomaly between the ZN $1$-form symmetry and the θ periodicity for $4$d pure Yang-Mills theory, which was originally discussed by Gaiotto et al. in the continuum description. For this purpose, we define the topological charge of the lattice $SU(N)$ gauge theory coupled with the background ZN $2$-form gauge fields Bₚ by generalizing Luscher's construction of the $SU(N)$ topological charge. We show that this lattice topological charge enjoys the fractional $1/N$ shift completely characterized by the background gauge field Bₚ, and this rigorously proves the mixed 't Hooft anomaly with the finite lattice spacings. As a consequence, the Yang-Mills vacua at θ and θ+2π are distinct as the symmetry-protected topological states when the confinement is assumed.