Differential equations aspects of quantum cohomology

TL;DR

This paper explores the role of quantum differential equations in quantum cohomology, highlighting their universal properties and connections to integrable systems like KdV, and demonstrates their effectiveness in encoding geometric information through examples.

Contribution

It presents a framework linking quantum differential equations with integrable systems and illustrates their geometric applications in Gromov-Witten invariants and related conjectures.

Findings

Quantum differential equations have universal properties beyond quantum cohomology.

The framework connects quantum cohomology with integrable systems like KdV.

Effective encoding of geometric information via quantum differential equations.

Abstract

The quantum differential equations can be regarded as examples of equations with certain universal properties which are of wider interest beyond quantum cohomology itself. We present this point of view as part of a framework which accommodates the KdV equation and other well known integrable systems. In the case of quantum cohomology, the theory is remarkably effective in packaging geometric information; we illustrate this with reference to simple examples of Gromov-Witten invariants, variations of Hodge structure, the Reconstruction Theorem, and the Crepant Resolution Conjecture. Based on lectures given at the summer school "Geometric and Topological Methods for Quantum Field Theory", Villa de Leyva, 2007.