Gromov–Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda hierarchies
Gromov–Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda hierarchies
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DOI：
10.2140/gt.2016.20.2135
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发表时间：
2014-01
影响因子：
2
通讯作者：
T. Milanov;Yefeng Shen;Hsian-Hua Tseng
中科院分区：
文献类型：
--
作者：
T. Milanov;Yefeng Shen;Hsian-Hua Tseng
我们构造了一个Hirota二次方程（HQE）形式的可积族，它控制Fano轨道射影曲线$\mathbb{P}^1_{a_1，a_2，a_3}$的Gromov-维滕（GW）不变量.我们构造的顶点算子是根据$\mathbb{P}^1_{a_1，a_2，a_3}$的$K$-理论，通过Iritani对Chern特征标映射的$\Gamma$-类修改给出的.我们还确定我们的HQE与适当的Kac-Wakimoto层次的ADE类型。特别地，我们得到了关于$\mathbb{P}^1 $的GW不变量的著名的户田猜想的推广.
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 $\Gamma$-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 $\mathbb{P}^1$ .