## Gromov-Witten theory of Fano orbifold curves, gamma integral structures and ADE-Toda hierarchies.(English)Zbl 1394.14037

Summary: 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 $$\mathbb{P}^1_{a_{1},a_{2},a_{3}}$$. 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 $$\mathbb{P}^1$$ to all Fano orbifold curves.

### MSC:

 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 17B69 Vertex operators; vertex operator algebras and related structures 14J45 Fano varieties 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Full Text: