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

We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov–Witten (GW) invariants of the Fano orbifold projective curve P1a1,a2,a3 with positive orbifold Euler characteristic. 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 P1. Contents