  A quantum many-body system with a conserved electric charge can have a DC
resistivity that is either exactly zero (implying it supports dissipationless
current) or nonzero. Exactly zero resistivity is related to conservation laws
that prevent the current from degrading. In this paper, we carefully examine
the situations in which such a circumstance can occur. We find that exactly
zero resistivity requires either continuous translation symmetry, or an
internal symmetry that has a certain kind of "mixed anomaly" with the electric
charge. (The symmetry could be a generalized global symmetry associated with
the emergence of unbreakable loop or higher dimensional excitations.) However,
even if one of these is satisfied, we show that there is still a mechanism to
get nonzero resistivity, through critical fluctuations that drive the
susceptibility of the conserved quantity to infinity; we call this mechanism
"critical drag". Critical drag is thus a mechanism for resistivity that, unlike
conventional mechanisms, is unrelated to broken symmetries. We furthermore
argue that an emergent symmetry that has the appropriate mixed anomaly with
electric charge is in fact an inevitable consequence of compressibility in
systems with lattice translation symmetry. Critical drag therefore seems to be
the only way (other than through irrelevant perturbations breaking the emergent
symmetry, that disappear at the renormalization group fixed point) to get
nonzero resistivity in such systems. Finally, we present a very simple and
concrete model -- the "Quantum Lifshitz Model" -- that illustrates the critical
drag mechanism as well as the other considerations of the paper.
