Gromov–Witten theory of orbifold projective lines and integrable hierarchies

This is a joint work with H.-H. Tseng and Y. Schen. Let $X$ be the sphere with 3 orbifold points and a positive (orbifold) Euler characteristic. There are finitely many such orbifolds and they are in 1-to-1 correspondence with the Dynkin diagrams of type ADE. Our goal is to construct an integrable hierarchy in the Hirota bilinear form that governs the Gromov–Witten invariants of $X$. Following my earlier work with Givental and Frenkel, we obtain a realization of the basic representation of the corresponding affine Lie algebra in terms of the solutions of the quantum differential equations. Once the representation is constructed we can identify our hierarchy with a particular class of the so called Kac–Wakimoto hierarchies.