Global dipole symmetry, compact Lifshitz theory, tensor gauge theory, and fractons

We study field theories with global dipole symmetries and gauge symmetries. The famous Lifshitz theory is an example of a symmetry. in detail its 1+1d version compact field. When this symmetry promoted to $U(1)$ symmetry, the corresponding tensor This known lead fractons. In order resolve various subtleties precise meaning these or symmetries, we place on lattice then take continuum limit. Interestingly, limit not unique. Different limits different theories, whose operators, defects, etc. are different. also consider $\mathbb Z_N$ group. Surprisingly, several physical observables, such as ground state degeneracy mobility defects depend sensitively number sites lattice. Our analysis forces us think carefully about that do act standard Hilbert space theory, but only presence defects. refer them time-like discuss detail. These allow phrase restrictions (including those fractons) consequence