Gromov–Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda hierarchies

Abstract

We construct an integrable hierarchy in the form of Hirota quadratic equations (HQEs) that governs the Gromov–Witten invariants of the Fano orbifold projective curve ${\mathbb{P}}_{a_1,a_2,a_3}^1$. The vertex operators in our construction are given in terms of the $K$–theory of ${\mathbb{P}}_{a_1,a_2,a_3}^1$ via Iritani’s $\Gamma$–class modification of the Chern character map. We also identify our HQEs with an appropriate Kac–Wakimoto hierarchy of ADE type. In particular, we obtain a generalization of the famous Toda conjecture about the GW invariants of ${\mathbb{P}}^1$ to all Fano orbifold curves.