We construct new compactifications with good properties of moduli spaces of maps from nonsingular marked curves to a large class of GIT quotients. This generalizes from a unified perspective many particular examples considered earlier in the literature.

For each positive rational number ε, the theory of ε-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 ε → 0. For ε > 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.

In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds F (n 1 , . . . , n l , n). This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of F (n 1 , . . . , n l , n) to a certain Gorenstein toric Fano variety P (n 1 , . . . , n l , n) which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of P (n 1 , . . . , n l , n) and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of P (n 1 , . . . , n l , n).

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