Wall-Crossings in Toric Gromov-Witten Theory I: Crepant Examples

Abstract

Let X be a Gorenstein orbifold and let Y be a crepant resolution of X. We state a conjecture relating the genus-zero Gromov--Witten invariants of X to those of Y, which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan--Graber, and prove our conjecture when X = P(1,1,2) and X = P(1,1,1,3). As a consequence, we see that the original form of the Bryan--Graber Conjecture holds for P(1,1,2) but is probably false for P(1,1,1,3). Our methods are based on mirror symmetry for toric orbifolds.

71 pages, v2: typos corrected and references modified, v3: corrected errors in Proposition 2.9 and in Summary, v4: major revision, exposition unified from a viewpoint of VSHS, many signpostings for the logical structure; the authorship has changed, with Alessio Corti withdrawing

References in corpus (6)

• Computing Genus-Zero Twisted Gromov-Witten Invariants
• Quantum Cohomology and Crepant Resolutions: A Conjecture
• Real and integral structures in quantum cohomology I: toric orbifolds
• On the Crepant Resolution Conjecture in the Local Case
• Topological Strings and (Almost) Modular Forms
• The Equivariant Gromov--Witten Theory of CP^1 and Integrable hierarchies

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