Wall-crossing in genus zero quasimap theory and mirror maps

Ionuţ Ciocan-Fontanine, Bumsig Kim

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47 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)400-448
Number of pages49
JournalAlgebraic Geometry
Volume1
Issue number4
DOIs
StatePublished - Oct 1 2014

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