  For each positive rational number epsilon, the theory of epsilon-stable
quasimaps to certain GIT quotients W//G developed in arXiv:1106.3724[math.AG]
gives rise to a Cohomological Field Theory. Furthermore, there is an asymptotic
theory corresponding to epsilon --> 0. For epsilon >1 one obtains the usual
Gromov-Witten theory of W//G, while the other theories are new. However, they
are all expected to contain the same information and in particular the
numerical invariants should be related by wall-crossing formulas. In this paper
we analyze the genus zero picture and find that the wall-crossing in this case
significantly generalizes toric mirror symmetry (the toric cases correspond to
abelian groups G). In particular, we give a geometric interpretation of the
mirror map as a generating series of quasimap invariants. We prove our
wall-crossing formulas for all targets W//G which admit a torus action with
isolated fixed points, as well as for zero loci of sections of homogeneous
vector bundles on such W//G.

Algebraic Geometry