[数学] 射影线上的等变轨道结构与可积 层次结构 [推广有奖]

摘要翻译：
设$\cp^1_{k,m}为$\cp^1_上的orbifold结构，分别由$z\mapsto z^k_和$w\mapsto w^m将0和$\infty_邻域均匀化得到。$环面在射影线上的对角作用自然地引起$\cp^1_{k,m}上的orbifold作用。$本文证明了如果k和m是余素数，则Givental对$\cp^1_{k,m}的等变总下降Gromov-Witten势的预测满足某些Hirota二次方程（简称HQE）。我们还证明了在适当改变变量后，类似于Getzler在等变Gromov-Witten理论中对$\cp^1$的改变，HQE转化为2-Toda族的HQE,即$\cp^1_{k,m}$的Gromov-Witten势是2-Toda族的tau函数。更准确地说，我们从下位势通过一些平移得到了2-Toda族的tau-函数序列。后一个条件是序列中的所有tau函数都是由一个tau函数通过平移得到的，这对2-Toda层次的解施加了严重的限制。我们的定理导致了一个新的可积族（我们建议称之为等变双分次Toda族）的发现。我们猜想，这个新的层次结构支配，即唯一确定$\cp^1_{k,m}的等变Gromov-Witten不变量。$\cp^1_{k,m}的等变Gromov-Witten不变量
---
英文标题：
《Equivariant orbifold structures on the projective line and integrable
  hierarchies》
---
作者：
Todor E. Milanov and Hsian-Hua Tseng
---
最新提交年份：
2007
---
分类信息：

一级分类：Mathematics        数学
二级分类：Symplectic Geometry        辛几何
分类描述：Hamiltonian systems, symplectic flows, classical integrable systems
哈密顿系统，辛流，经典可积系统
--
一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
--

---
英文摘要：
  Let $\CP^1_{k,m}$ be the orbifold structure on $\CP^1$ obtained via uniformizing the neighborhoods of 0 and $\infty$ respectively by $z\mapsto z^k$ and $w\mapsto w^m.$ The diagonal action of the torus on the projective line induces naturally an orbifold action on $\CP^1_{k,m}.$ In this paper we prove that if k and m are co-prime then Givental's prediction of the equivariant total descendent Gromov-Witten potential of $\CP^1_{k,m}$ satisfies certain Hirota Quadratic Equations (HQE for short). We also show that after an appropriate change of the variables, similar to Getzler's change in the equivariant Gromov-Witten theory of $\CP^1$, the HQE turn into the HQE of the 2-Toda hierarchy, i.e., the Gromov-Witten potential of $\CP^1_{k,m}$ is a tau-function of the 2-Toda hierarchy. More precisely, we obtain a sequence of tau-functions of the 2-Toda hierarchy from the descendent potential via some translations. The later condition, that all tau-functions in the sequence are obtained from a single one via translations, imposes a serious constraint on the solution of the 2-Toda hierarchy. Our theorem leads to the discovery of a new integrable hierarchy (we suggest to be called the Equivariant Bi-graded Toda Hierarchy). We conjecture that this new hierarchy governs, i.e., uniquely determines, the equivariant Gromov-Witten invariants of $\CP^1_{k,m}.$
---
PDF链接：
https://arxiv.org/pdf/0707.3172