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Gromov--Witten Theory of CP^1 and Integrable Hierarchies

Todor E. Milanov
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Abstract

The ancestor Gromov--Witten invariants of a compact {\Kahler} manifold $X$ can be organized in a generating function called the total ancestor potential of $X$. In this paper, we construct Hirota Quadratic Equations (HQE shortly) for the total ancestor potential of $\C P^1$. The idea is to adopt the formalism developed in \cite{G1,GM} to the mirror model of $\C P^1$. We hope that the ideas presented here can be generalized to other manifolds as well. As a corollary, using the twisted loop group formalism from \cite{G3}, we obtain a new proof of the following version of the Toda conjecture: the total descendant potential of $\C P^1$ (known also as the partition function of the $\C P^1$ topological sigma model) is a tau-function of the Extended Toda Hierarchy.

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Todor E. Milanov. 2006-04-29. Gromov--Witten Theory of CP^1 and Integrable Hierarchies. https://arxiv.org/abs/math-ph/0605001

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