The Equivariant Gromov--Witten Theory of CP^1 and Integrable hierarchies

TL;DR

This paper constructs an integrable hierarchy related to the equivariant Gromov-Witten theory of CP^1, demonstrating its connection to the 2-Toda hierarchy and providing a new proof of the equivariant Toda conjecture.

Contribution

It develops a vertex operator-based integrable hierarchy for CP^1's equivariant Gromov-Witten theory and links it to the 2-Toda hierarchy through coordinate transformations.

Findings

The equivariant total descendant potential satisfies the Hirota Quadratic Equations.

The hierarchy transforms into the 2-Toda hierarchy under certain variable changes.

Provides a new proof of the equivariant Toda conjecture.

Abstract

We construct an integrable hierarchy in terms of vertex operators and Hirota Quadratic Equations (HQE shortly) and we show that the equivariant total descendant potential of $\C P^1$ satisfies the HQE. Our prove is based on the quantization formalism developed in \cite{G1}, \cite{G2}, and on the equivariant mirror model of $\C P^1.$ The vertex operators in our construction obey certain transformation law under change of coordinates, which might be important for generalizing the HQE to other manifolds. We also show that under certain change of the variables, which is due to E. Getzler, the HQE are transformed into the HQE of the 2-Toda hierarchy. Thus we obtain a new proof of the equivariant Toda conjecture.