Quantum intersection numbers and the GromovWitten invariants of $\mathbb{CP}^1$
Abstract
The notion of a quantum taufunction 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 taufunction was then described by the first author, the coefficients of the logarithm of this series are called the quantum intersection numbers. Because of the KontsevichWitten theorem, a part of the quantum intersection numbers coincides with the classical intersection numbers of psiclasses on the moduli spaces of stable algebraic curves. In this paper, we relate the quantum intersection numbers to the stationary relative GromovWitten invariants of $(\mathbb{CP}^1,0,\infty)$ with an insertion of a Hodge class. Using the OkounkovPandharipande 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 onepart double Hurwitz numbers.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.16464
 arXiv:
 arXiv:2402.16464
 Bibcode:
 2024arXiv240216464B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematical Physics
 EPrint:
 15 pages