Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913623817846784 |
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| author | Oberdieck, Georg |
| author_facet | Oberdieck, Georg |
| contents | We conjecture that the generating series of Gromov-Witten invariants of the Hilbert schemes of $n$ points on a K3 surface are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture in genus $0$ and for at most $3$ markings - for all Hilbert schemes and for arbitrary curve classes. In particular, for fixed $n$, the reduced quantum cohomologies of all hyperkähler varieties of $K3^{[n]}$-type are determined up to finitely many coefficients.
As an application we show that the generating series of $2$-point Gromov-Witten classes are vector-valued Jacobi forms of weight $-10$, and that the fiberwise Donaldson-Thomas partition functions of an order two CHL Calabi-Yau threefold are Jacobi forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_03361 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface Oberdieck, Georg Algebraic Geometry High Energy Physics - Theory We conjecture that the generating series of Gromov-Witten invariants of the Hilbert schemes of $n$ points on a K3 surface are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture in genus $0$ and for at most $3$ markings - for all Hilbert schemes and for arbitrary curve classes. In particular, for fixed $n$, the reduced quantum cohomologies of all hyperkähler varieties of $K3^{[n]}$-type are determined up to finitely many coefficients. As an application we show that the generating series of $2$-point Gromov-Witten classes are vector-valued Jacobi forms of weight $-10$, and that the fiberwise Donaldson-Thomas partition functions of an order two CHL Calabi-Yau threefold are Jacobi forms. |
| title | Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface |
| topic | Algebraic Geometry High Energy Physics - Theory |
| url | https://arxiv.org/abs/2202.03361 |