Fracton models with crystalline symmetries in two dimensions

We construct dipole conserving models invariant under crystalline symmetries. Excitations of these models are fractons. We further show how to gauge the dipole symmetries. Upon gauging we introduce gauge fields that transform under irreducible representations of discrete rotations. As a concrete example, we discuss a D₄ symmetric model on a lattice, which can be coupled to a spin-2 gauge theory of point group C₄. The known traceless scalar charge theory is a special continuum limit of our model. Discussions about charge mobility and Wilson strips are included. Next, we generalize our model to wallpaper groups p4mm (symmorphic) and p4gm (non-symmorphic). Where in p4gm (non-symmorphic) case we show that fields with two flavors are necessary to implement fractional translations.