The Equivariant Gromov--Witten Theory of CP¹ and Integrable hierarchies

2005/08/26 by Todor Milanov, Milanov, Todor E. · 2 citations
Mathematics · Physics and Astronomy · #53D45 #Algebraic structures and combinatorial models #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.math-ph/0508054

openalex publication_date 2005/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 P¹ satisfies the HQE. Our prove is based on the quantization formalism developed in \citeG1, \citeG2, and on the equivariant mirror model of \C P¹. 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.

Cited by

• Asymmetric rational reductions of 2D-Toda hierarchy and a generalized Frobenius manifold
• Generalized Frobenius Manifold Structures on the Orbit Spaces of Affine Weyl Groups I

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