The Crepant Transformation Conjecture for Toric Complete Intersections

TL;DR

This paper proves the Crepant Transformation Conjecture for toric complete intersections, establishing gauge equivalence of quantum connections after analytic continuation and linking it to Fourier-Mukai transformations, under weak geometric assumptions.

Contribution

It demonstrates the conjecture for a broad class of toric stacks and intersections, connecting quantum cohomology, K-theory, and mirror symmetry through explicit transformations.

Findings

Equivariant quantum connections are gauge-equivalent after analytic continuation.

The gauge transformation corresponds to a Fourier-Mukai transform.

Results hold for non-compact and non-Gorenstein toric stacks.

Abstract

Let X and Y be K-equivalent toric Deligne-Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y, with a Fourier-Mukai transformation between the K-groups of X and Y, via an equivariant version of the Gamma-integral structure on quantum cohomology. We prove similar results for toric complete intersections. We impose only very weak geometric hypotheses on X and Y: they can be non-compact, for example, and need not be weak Fano or have Gorenstein coarse moduli space. Our main tools are the Mirror Theorems for toric Deligne-Mumford stacks and toric complete intersections, and the Mellin-Barnes method for analytic continuation of hypergeometric functions.