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

Todor Milanov IPMU Yefeng Shen IPMU Hsian-Hua Tseng Ohio State University

Algebraic Geometry Mathematical Physics Quantum Algebra Representation Theory Symplectic Geometry mathscidoc:1801.01004

Geometry and Topology, 20, 2135--2218, 2016
We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov--Witten (GW) invariants of the Fano orbifold projective curve P^1_{a1,a2,a3}. The vertex operators in our construction are given in terms of the K-theory of P^1_{a1,a2,a3} via Iritani's Γ-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 .
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@inproceedings{todor2016gromov--witten,
  title={Gromov--Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies},
  author={Todor Milanov, Yefeng Shen, and Hsian-Hua Tseng},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180104233632485987869},
  booktitle={Geometry and Topology},
  volume={20},
  pages={2135--2218},
  year={2016},
}
Todor Milanov, Yefeng Shen, and Hsian-Hua Tseng. Gromov--Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies. 2016. Vol. 20. In Geometry and Topology. pp.2135--2218. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180104233632485987869.
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