  Dimer models arise as effective descriptions in a variety of physical
contexts, and provide paradigmatic examples of systems subject to strong local
constraints. Here we present a quantum version of the venerable Kasteleyn
model, which has an unusual phase transition from a dimer solid to a U(1)
liquid. We show how the phase structure of the quantum model can be understood
in terms of the quantum mechanics of one-dimensional strings and determine the
exact value of the critical coupling. By constructing effective models to
describe the properties of these strings, we calculate properties such as the
dimer-dimer correlation function in the neighborhood of the transition. We also
discuss the full phase structure of the model, in the ground state and at
nonzero temperature.
