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