Quantum intersection numbers and the Gromov–Witten invariants of CP^1

Letters in Mathematical Physics. 2024. Vol. 114. Article 131.

Blot X., Buryak A.

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 (CP1,0,∞) 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.

Keywords: Quantum integrable system, Квантовая интегрируемая система, риманова поверхность, Riemann surface

Publication based on the results of: Representation theory of vertex algebras and quantum groups with applications to knot theory, integrable systems, geometry of moduli spaces, and enumerative invariants (2024)