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

2023/12/20 by Georg Oberdieck, Oberdieck, Georg, Aaron Pixton +1 · 2 citations
Mathematics · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2312.13188

Abstract

We determine the quantum multiplication with divisor classes on the Hilbert scheme of points on an elliptic surface S → Σ for all curve classes which are contracted by the induced fibration S^[n] → Σ^[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 × 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.

Cited by

• The quantum integrable hierarchy for the Gromov-Witten theory of elliptic curves
• Gromov-Witten theory of Hilbⁿ(ℂ²) and Noether-Lefschetz theory of A_g

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