Gromov--Witten Theory of CP¹ and Integrable Hierarchies

2006/04/29 by Todor Milanov, Milanov, Todor E. · 1 citation
Mathematics · #34S30 #Algebraic Geometry and Number Theory #FOS: Physical sciences #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.math-ph/0605001

openalex publication_date 2006/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 P¹. The idea is to adopt the formalism developed in \citeG1,GM to the mirror model of \C P¹. 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 \citeG3, we obtain a new proof of the following version of the Toda conjecture: the total descendant potential of \C P¹ (known also as the partition function of the \C P¹ topological sigma model) is a tau-function of the Extended Toda Hierarchy.

Citations

• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants

Cited by

• Quantum Cohomology and Crepant Resolutions: A Conjecture

Related

• The Equivariant Gromov--Witten Theory of CP¹ and Integrable hierarchies
• A_n-1 singularities and nKdV hierarchies
• Descendant Gromov-Witten Invariants, Simple Hurwitz Numbers, and the Virasoro Conjecture for P¹
• The Toda equations and the Gromov-Witten theory of the Riemann sphere
• Witten's D₄ Integrable Hierarchies Conjecture