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

Milanov, Todor¹ ; Shen, Yefeng² ; Tseng, Hsian-Hua³

¹ Kavli IPMU, University of Tokyo (WPI), 5-1-5 Kashiwanoha, Kashiwa 2778583, Japan
² Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305, United States
³ Department of Mathematics, Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210, United States

Geometry & topology, Tome 20 (2016) no. 4, pp. 2135-2218.

Résumé

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 ${\mathbb{P}}_{a_1,a_2,a_3}^1$. The vertex operators in our construction are given in terms of the $K$–theory of ${\mathbb{P}}_{a_1,a_2,a_3}^1$ 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.

Détail

DOI : 10.2140/gt.2016.20.2135

Classification : 14N35, 17B69
Keywords: Gromov–Witten theory, Fano orbifold curves, ADE-Toda hierarchies

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