Gamma classes and quantum cohomology

Abstract

The $\hat{\Gamma }$-class is a characteristic class for complex manifolds with transcendental coefficients. It defines an integral structure of quantum cohomology, or more precisely, an integral lattice in the space of flat sections of the quantum connection. We present several conjectures (the $\hat{\Gamma }$-conjectures) about this structure, particularly focusing on the Riemann–Hilbert problem it poses. We also discuss a conjectural functoriality of quantum cohomology under birational transformations.