Symmetric gauge theories and Lieb-Schultz-Mattis-type constraints

It is known that certain symmetric gauge theories can only exist on the surface of a higher dimensional symmetry protected topological (SPT) state, in which case the symmetric gauge theory is anomalous. The anomaly class of such a gauge theory is represented by the bulk SPT index. Given an anomaly class, what kind of the symmetric gauge theories are allowed? Based on physical arguments and tensor-network constructions, we point out a sharp mathematical relationship between the symmetry properties of Abelian gauge theories and the anomaly class: the cup product. When the physical system hosts Lieb-Schultz-Mattis-type constraints, which is a specific type of anomaly class, our result sharply determines the physically allowed symmetric gauge theories via the preimage of the cup product. As applications, we algebraically compute the physically allowed skyrmion quantum numbers in various Neel states in 2+1D, and gauge charge/monopole projective representations in U(1) quantum spin liquids in 3+1D. We also compute the allowed Z2 gauge theories in several 2+1D quantum spin systems.