The Toda equations and the Gromov-Witten theory of the Riemann sphere

TL;DR

This paper explores the implications of Toda equations on the Gromov-Witten theory of the Riemann sphere, providing new formulas, predictions, and proofs related to enumerative invariants and Hodge integrals.

Contribution

It introduces new series formulas for Gromov-Witten invariants, investigates degree 1 invariants with applications to Hodge integrals, and derives a differential equation for Hurwitz numbers, all based on conjectural Toda equations.

Findings

Closed series forms for all 1-point invariants across genera and degrees

Degree 1 invariants with new applications to Hodge integrals

A differential equation for classical simple Hurwitz numbers

Abstract

Consequences of the Toda equations arising from the conjectural matrix model for the Riemann sphere are investigated. The Toda equations determine the Gromov-Witten descendent potential (including all genera) of the Riemann sphere from the degree 0 part. Degree 0 series computations via Hodge integrals then lead to higher degree predictions by the Toda equations. First, closed series forms for all 1-point invariants of all genera and degrees are given. Second, degree 1 invariants are investigated with new applications to Hodge integrals. Third, a differential equation for the generating function of the classical simple Hurwitz numbers (in all genera and degrees) is found -- the first such equation. All these results depend upon the conjectural Toda equations. Finally, proofs of the Toda equations in genus 0 and 1 are given.