Wall-crossing in genus zero quasimap theory and mirror maps

Abstract

For each positive rational number , the theory of -stable quasimaps to certain GIT quotients W==G developed in [CKM14] gives rise to a Cohomological Field Theory. Furthermore, there is an asymptotic theory corresponding to ! 0. For > 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 nd that the wall-crossing in this case signicantly 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 xed points, as well as for zero loci of sections of homogeneous vector bundles on such W==G.