Quantum cohomology of the Hilbert scheme of points in the plane
The Hilbert scheme of points Hilb\(_n(\mathbb{C}^2)\) is the crepant resolution of the orbifold quotient \((\mathbb{C}^2)^n/S_n\) containing the configurations of \(n\) distinct points in \(\mathbb{C}^2\). Its geometry and algebraic invariants have been a subject of intense study in the last decade due to connections with the string theory. In particular, the ring structure of its torus equivariant quantum cohomology has been established (the natural action of the complex torus on \((\mathbb{C}^2)^n/S_n\) extends to the Hilbert scheme). In the paper under review, the authors give an explicit formula for \(M_D\), the operator of multiplication by the first Chern class \(D\) of the Hilbert scheme in small quantum cohomology. The matrix elements of \(M_D\) give counts of rational curves meeting three given subvarieties of the scheme, and an associated differential equation is an integrable non-stationary deformation of the Calogero-Sutherland equation for quantum particles on a torus. The formula implies that \(D\) generates the small quantum cohomology ring, and completes establishing the four-way correspondence between the quantum cohomology of Hilb\(_n(\mathbb{C}^2)\), the equivariant orbifold cohomology of \((\mathbb{C}^2)^n/S_n\), and the Gromov-Witten and Donaldson-Thomas invariants of \(\mathbb{P}^1\times\mathbb{C}^2\).
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