Quantum intersection numbers and the Gromov-Witten invariants of $\mathbb{CP}^1$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916932358242304 |
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| author | Blot, Xavier Buryak, Alexandr |
| author_facet | Blot, Xavier Buryak, Alexandr |
| contents | The notion of a quantum tau-function for a natural quantization of the KdV hierarchy was introduced in a work of Dubrovin, Guéré, Rossi, and the second author. A certain natural choice of a quantum tau-function was then described by the first author, the coefficients of the logarithm of this series are called the quantum intersection numbers. Because of the Kontsevich-Witten theorem, a part of the quantum intersection numbers coincides with the classical intersection numbers of psi-classes on the moduli spaces of stable algebraic curves. In this paper, we relate the quantum intersection numbers to the stationary relative Gromov-Witten invariants of $(\mathbb{CP}^1,0,\infty)$ with an insertion of a Hodge class. Using the Okounkov-Pandharipande approach to such invariants (with the trivial Hodge class) through the infinite wedge formalism, we then give a short proof of an explicit formula for the ``purely quantum'' part of the quantum intersection numbers, found by the first author, which in particular relates these numbers to the one-part double Hurwitz numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16464 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum intersection numbers and the Gromov-Witten invariants of $\mathbb{CP}^1$ Blot, Xavier Buryak, Alexandr Algebraic Geometry Mathematical Physics The notion of a quantum tau-function for a natural quantization of the KdV hierarchy was introduced in a work of Dubrovin, Guéré, Rossi, and the second author. A certain natural choice of a quantum tau-function was then described by the first author, the coefficients of the logarithm of this series are called the quantum intersection numbers. Because of the Kontsevich-Witten theorem, a part of the quantum intersection numbers coincides with the classical intersection numbers of psi-classes on the moduli spaces of stable algebraic curves. In this paper, we relate the quantum intersection numbers to the stationary relative Gromov-Witten invariants of $(\mathbb{CP}^1,0,\infty)$ with an insertion of a Hodge class. Using the Okounkov-Pandharipande approach to such invariants (with the trivial Hodge class) through the infinite wedge formalism, we then give a short proof of an explicit formula for the ``purely quantum'' part of the quantum intersection numbers, found by the first author, which in particular relates these numbers to the one-part double Hurwitz numbers. |
| title | Quantum intersection numbers and the Gromov-Witten invariants of $\mathbb{CP}^1$ |
| topic | Algebraic Geometry Mathematical Physics |
| url | https://arxiv.org/abs/2402.16464 |