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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
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