#### Volume 20, issue 4 (2016)

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

### Todor Milanov, Yefeng Shen and Hsian-Hua Tseng

Geometry & Topology 20 (2016) 2135–2218
##### Abstract

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 ${ℙ}_{{a}_{1},{a}_{2},{a}_{3}}^{1}$. The vertex operators in our construction are given in terms of the $K$–theory of ${ℙ}_{{a}_{1},{a}_{2},{a}_{3}}^{1}$ 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 ${ℙ}^{1}$ to all Fano orbifold curves.

##### Keywords
Gromov–Witten theory, Fano orbifold curves, ADE-Toda hierarchies
##### Mathematical Subject Classification 2010
Primary: 14N35, 17B69
##### Publication
Accepted: 5 November 2015
Published: 15 September 2016
Proposed: Yasha Eliashberg
Seconded: Gang Tian, Jim Bryan
##### Authors
 Todor Milanov Kavli IPMU University of Tokyo (WPI) 5-1-5 Kashiwanoha Kashiwa 2778583 Japan Yefeng Shen Department of Mathematics Stanford University 450 Serra Mall, Building 380 Stanford, CA 94305 United States Hsian-Hua Tseng Department of Mathematics Ohio State University 100 Math Tower 231 West 18th Avenue Columbus, OH 43210 United States