The quantum differential equation of the Hilbert scheme of points in the plane

TL;DR

This paper studies the quantum differential equation associated with the Hilbert scheme of points in the plane, focusing on intertwining operators and providing an exact solution connecting Donaldson-Thomas and Gromov-Witten theories.

Contribution

It introduces a detailed analysis of the quantum differential equation for the Hilbert scheme, including explicit solutions and applications of intertwining operators.

Findings

Derived an exact solution to the connection problem between Donaldson-Thomas and Gromov-Witten points.

Analyzed properties of intertwining operators and their role in the quantum differential equation.

Expanded on previous work to deepen understanding of the quantum geometry of the Hilbert scheme.

Abstract

We discuss here basic properties of the quantum differential equation of the Hilbert scheme of points in the plane. Our emphasis is on intertwining operators (which shift equivariant parameters) and their applications. In particular, we obtain an exact solution to the connection problem from the Donaldson-Thomas point q=0 to the Gromov-Witten point q=-1. The paper is an expanded version of what was once the last Section of arXiv:math/0411210.