The Equivariant Gromov–Witten Theory of $\mathbb{C}P^{1}$ and Integrable Hierarchies

We construct an integrable hierarchy in terms of vertex operators and Hirota quadratic equations (HQE, in short) and we show that the equivariant total descendent potential of $\mathbb{C}P^{1}$ satisfies the HQE. Our proof is based on the quantization formalism developed in Givental [5, 7] and on the equivariant mirror model of $\mathbb{C}P^{1}$ 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.