Gromov–Witten Theory of Toric Birational Transformations

Abstract

Abstract We investigate the effect of a general toric wall crossing on genus zero Gromov–Witten theory. Given two complete toric orbifolds $X_{+}$ and $X_{-}$ related by wall crossing under variation of geometric invariant theory quotients, we prove that their respective $I$-functions are related by linear transformation and asymptotic expansion. We use this comparison to deduce a similar result for birational complete intersections in $X_{+}$ and $X_{-}$. This extends the work of the previous authors in [2] to the case of complete intersections in toric varieties and generalizes some of the results of Coates–Iritani–Jiang [15] on the crepant transformation conjecture to the setting of non-zero discrepancy.