Wall-crossing in genus zero quasimap theory and mirror maps

Abstract: 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.