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

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.

arXiv comment16 pages, LaTeX2e

References in corpus (4)

• Quantum Cohomology and Virasoro Algebra
• Virasoro constraints and the Chern classes of the Hodge bundle
• The Topological CP^1 Model and the Large-N Matrix Integral
• Topological recursion relations in genus 2

Cited by in corpus (7)

• Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
• An algebro-geometric proof of Witten's conjecture
• Toda hierarchies and their applications
• Generalized string equations for double Hurwitz numbers
• Laplacian growth in a channel and Hurwitz numbers
• Constellations and $\tau$-functions for rationally weighted Hurwitz numbers
• Asymptotics of quantum weighted Hurwitz numbers