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 $\mathbb{ P}^1_{ a_1,a_2,a_3}$. The vertex operators in our construction are given in terms of the $K$-theory of $\mathbb{ P}^1_{ a_1,a_2,a_3}$ 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 $\mathbb{ P}^1$ .
Categories: math.AG math-ph math.MP math.QA math.RT math.SG