The Crepant Transformation Conjecture and Fourier--Mukai Transforms

Suppose that X and Y are Kahler manifolds or orbifolds which are related by a crepant resolution or flop F.  It is expected that the Gromov--Witten potentials of X and Y should be related by analytic continuation in Kahler parameters combined with a linear symplectomorphism between Givental's symplectic spaces for X and Y.  This linear symplectomorphism is expected to coincide, in a precise sense which I will explain, with the Fourier--Mukai transform on K-theory induced by F.  In this talk I will prove these conjectures, as well as their torus-equivariant generalizations, in the case where X and Y are toric.  

This is joint work with Hiroshi Iritani and Yunfeng Jian