Gromov-Witten Theory of Toric Birational Transformations

TL;DR

This paper studies how genus zero Gromov-Witten invariants of toric orbifolds change under wall crossing, establishing relations between their $I$-functions and extending results to complete intersections and non-zero discrepancy cases.

Contribution

It proves that $I$-functions of toric orbifolds related by wall crossing are linearly related and extends these results to complete intersections and non-zero discrepancy scenarios.

Findings

$I$-functions are related by linear transformation and asymptotic expansion.

Results extend to complete intersections in toric varieties.

Generalizes crepant transformation conjecture to non-zero discrepancy cases.

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 GIT, 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 Acosta-Shoemaker to the case of complete intersections in toric varieties, and generalizes some of the results of Coates-Iritani-Jiang on the crepant transformation conjecture to the setting of non-zero discrepancy.