Gamma classes and quantum cohomology

TL;DR

This paper explores the Gamma-class as a characteristic class for complex manifolds, its role in quantum cohomology, and presents conjectures related to its structure and functoriality under birational transformations.

Contribution

It introduces the Gamma-conjectures, linking Gamma-classes to quantum cohomology and proposes conjectural functoriality properties under birational maps.

Findings

Formulation of Gamma-conjectures about the integral structure in quantum cohomology.

Discussion of the Riemann-Hilbert problem related to Gamma-structures.

Conjectural framework for functoriality of quantum cohomology under birational transformations.

Abstract

The 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 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.