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 ${\mathbb{Z}}_N$ $1$-form symmetry and the $\theta$ periodicity for $4$d pure Yang-Mills theory, which was originally discussed by Gaiotto $\mathit{\text{et}}\mathit{\text{al.}}$ in the continuum description. For this purpose, we define the topological charge of the lattice $SU(N)$ gauge theory coupled with the background ${\mathbb{Z}}_N$ $2$-form gauge fields $B_p$ by generalizing Lüscher’s construction of the $SU(N)$ topological charge. We show that this lattice topological charge enjoys the fractional $1\mathrm{/}N$ shift completely characterized by the background gauge field $B_p$, and this rigorously proves the mixed 't Hooft anomaly with the finite lattice spacings. As a consequence, the Yang-Mills vacua at $\theta$ and $\theta +2\pi$ are distinct as the symmetry-protected topological states when the confinement is assumed.