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arXiv · 2312.13188

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

Georg Oberdieck·Aaron Pixton
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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.

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Georg Oberdieck, Aaron Pixton. 2023-12-20. Quantum cohomology of the Hilbert scheme of points on an elliptic surface. https://arxiv.org/abs/2312.13188

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