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arXiv · 1401.5778

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

Todor Milanov·Yefeng Shen·Hsian-Hua Tseng
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Abstract

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 $\mathbb{P}^1_{a_1,a_2,a_3}$. 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 $Γ$-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 $\mathbb{P}^1$ .

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Todor Milanov, Yefeng Shen, Hsian-Hua Tseng. 2014-07-29. Gromov--Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies. https://doi.org/10.2140/gt.2016.20.2135

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