Quantum cohomology of the Hilbert scheme of points on an elliptic surface

TL;DR

This paper computes the quantum multiplication with divisor classes on Hilbert schemes of points on elliptic surfaces, expressing results via explicit operators on Fock space and revealing meromorphic quasi-Jacobi form structure.

Contribution

It provides explicit formulas for quantum multiplication on Hilbert schemes of elliptic surfaces, extending previous work and establishing new GW/PT correspondences and computations.

Findings

Explicit formulas for quantum multiplication with divisors on Hilbert schemes.

Identification of structure constants as meromorphic quasi-Jacobi forms.

Extension of results to equivariant quantum multiplication on product surfaces.

Abstract

We determine the quantum multiplication with divisor classes on the Hilbert scheme of points on an elliptic surface $S \to \Sigma$ for all curve classes which are contracted by the induced fibration $S^{[n]} \to \Sigma^{[n]}$. The formula is expressed in terms of explicit operators on Fock space. The structure constants are meromorphic quasi-Jacobi forms of index $0$. Combining with work of Hu-Li-Qin, this determines the quantum multiplication with divisors on the Hilbert scheme of elliptic surfaces with $p_g(S)>0$. We also determine the equivariant quantum multiplication with divisor classes for the Hilbert scheme of points on the product $E \times \mathbb{C}$. The proof of our formula is based on Nesterov's Hilb/PT wall-crossing, a newly established GW/PT correspondence for the product of an elliptic surface times a curve, and new computations in the Gromov-Witten theory of an elliptic curve.