Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models

The Lieb-Schultz-Mattis theorem dictates that a trivial symmetric insulator in lattice models is prohibited if lattice translation symmetry and $U(1)$ charge conservation are both preserved. In this paper, we generalize the Lieb-Schultz-Mattis theorem to systems with higher-form symmetries, which act on extended objects of dimension $n>0$. The prototypical lattice system with higher-form symmetry is the pure abelian lattice gauge theory whose action consists only of the field strength. We first construct the higher-form generalization of the Lieb-Schultz-Mattis theorem with a proof. We then apply it to the $U(1)$ lattice gauge theory description of the quantum dimer model on bipartite lattices. Finally, using the continuum field theory description in the vicinity of the Rokhsar-Kivelson point of the quantum dimer model, we diagnose and compute the mixed 't Hooft anomaly corresponding to the higher-form Lieb-Schultz-Mattis theorem.