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

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.

arXiv comment22 pages, this is the second part of an earlier version, major revision of the exposition

References in corpus (6)

• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
• The Topological CP^1 Model and the Large-N Matrix Integral
• The equivariant Gromov-Witten theory of P^1
• Gromov-Witten theory, Hurwitz theory, and completed cycles
• The equivariant Toda lattice, I
• Hirota Quadratic Equations for the Extended Toda Hierarchy

Cited by in corpus (6)

• Quantum Cohomology and Crepant Resolutions: A Conjecture
• Gromov--Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies
• Additional symmetries of the extended bigraded Toda hierarchy
• Darboux transformations and Fay identities for the extended bigraded Toda hierarchy
• $N=1$ formal genus $0$ Gromov-Witten theories and Givental's formalism
• 4D limit of melting crystal model and its integrable structure