The Equivariant Gromov--Witten Theory of CP1 and Integrable hierarchies
Todor E. Milanov
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 P1 satisfies the HQE. Our prove is based on the quantization formalism developed in G1, G2, and on the equivariant mirror model of 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.
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