  We investigate two-dimensional (2d) melting in the presence of a
one-dimensional (1d) periodic potential as, for example, realized in recent
experiments on 2d colloids subjected to two interfering laser beams. The
topology of the phase diagram is found to depend primarily on two factors: the
relative orientation of the 2d crystal and the periodic potential troughs,
which select a set of Bragg planes running parallel to the troughs, and the
commensurability ratio p= a'/d of the spacing a' between these Bragg planes to
the period d of the periodic potential. The complexity of the phase diagram
increases with the magnitude of the commensurabilty ratio p. Rich phase
diagram, with ``modulated liquid'', ``floating'' and ``locked floating'' solid
and smectic phases are found. Phase transitions between these phases fall into
two broad universality classes, roughening and melting, driven by the
proliferation of discommensuration walls and dislocations, respectively. We
discuss correlation functions and the static structure factor in these phases
and make detailed predictions of the universal features close to the phase
boundaries. We predict that for charged systems with highly screened
short-range interactions these melting transitions are generically reentrant as
a function of the strength of the periodic potential, prediction that is in
accord with recent 2d colloid experiments. Implications of our results for
future experiments are also discussed.
