The equivariant Gromov-Witten theory of P^1

TL;DR

This paper expresses equivariant Gromov-Witten invariants of P^1 as matrix elements in Fock space and shows they are governed by the 2-Toda hierarchy, advancing the understanding of curve Gromov-Witten theory.

Contribution

It provides an explicit operator framework for equivariant Gromov-Witten invariants of P^1 and links them to integrable hierarchies, specifically the 2-Toda hierarchy.

Findings

Equivariant Gromov-Witten invariants are expressed as matrix elements in Fock space.

The theory is governed by the 2-Toda hierarchy.

This approach advances the algebraic understanding of Gromov-Witten invariants for P^1.

Abstract

We express all equivariant Gromov-Witten invariants of the projective line as matrix elements of explicit operators acting in the Fock space. As a consequence, we prove the equivariant theory is governed by the 2-Toda hierarchy of Ueno and Takasaki. This is the second in a sequence of three papers devoted to the Gromov-Witten theory of nonsingular target curves (the first paper of the series is math.AG/0204305).