  The 2+1d continuum Lifshitz theory of a free compact scalar field plays a
prominent role in a variety of quantum systems in condensed matter physics and
high energy physics. It is known that in compact space, it has an infinite
ground state degeneracy. In order to understand this theory better, we consider
two candidate lattice regularizations of it using the modified Villain
formalism. We show that these two lattice theories have significantly different
global symmetries (including a dipole global symmetry), anomalies, ground state
degeneracies, and dualities. In particular, one of them is self-dual. Given
these theories and their global symmetries, we can couple them to corresponding
gauge theories. These are two different $U(1)$ tensor gauge theories. The
resulting models have excitations with restricted mobility, i.e., fractons.
Finally, we give an exact lattice realization of the fracton/lineon-elasticity
dualities for the Lifshitz theory, scalar and vector charge gauge theories.
