Gromov-Witten theory, Hurwitz theory, and completed cycles

TL;DR

This paper establishes a precise connection between the stationary Gromov-Witten theory of curves and Hurwitz enumeration via completed cycles, providing explicit descriptions and deriving Toda equations for the projective line.

Contribution

It reveals an explicit equivalence between stationary Gromov-Witten invariants and Hurwitz coverings using completed cycles, advancing understanding of curve enumerations.

Findings

Complete description of stationary Gromov-Witten theory for the projective line and elliptic curve.

Derived Toda equations for the relative stationary theory of the projective line.

Established an explicit equivalence linking Gromov-Witten theory and Hurwitz enumeration.

Abstract

We establish an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles. The stationary sector is formed, by definition, by the descendents of the point class. Completed cycles arise naturally in the theory of shifted symmetric functions. Using this equivalence, we give a complete description of the stationary Gromov-Witten theory of the projective line and elliptic curve. Toda equations for the relative stationary theory of the projective line are derived.