  We introduce a lattice field theory that describes the transition between a
superfluid (SF) and a bosonic topological Mott Insulator (tMI) -- a $U(1)$
symmetry protected topological phase labeled by an integer level $k$ and
possessing an even integer $2k\frac{e^2}{h}$ quantized Hall conductance. Our
model differs from the usual $2+1$d XY model by a topological term that
vanishes on closed manifolds and in the absence of an applied gauge field,
which implies that the critical exponents of the SF-tMI transition are
identical to those of the well-studied $2+1$d XY transition. Our formalism
predicts a "level-shift" symmetry: in the absence of an applied gauge field,
the bulk correlation functions of all local operators are identical for models
differing by the topological term. %, for example, near the SF-MI and SF-tMI
transitions, and hence the extremely well-studied critical exponents of the
$2+1$d XY model apply to the SF-tMI transition. In the presence of a background
gauge field, the topological term leads to a quantized Hall response in the tMI
phase, and we argue that this quantized Hall effect persists in the vicinity of
the phase transition into the SF phase. Our formalism paves the way for other
exact lattice descriptions of symmetry-protected-topological (SPT) phases, and
mappings of critical exponents between transitions from symmetry-breaking to
trivial states and transitions from symmetry-breaking to SPT states. A similar
"level shift" symmetry should appear between all group cohomology SPT states
protected by the same symmetry.
