arXiv 1409.35872014-09math.AGmath.COmath.QA

旗流形和周期Toda格的量子上同调环的仿射变形

An affine deformation of the quantum cohomology ring of flag manifolds and periodic Toda lattice

Augustin-Liviu Mare, Leonardo C. Mihalcea

备注：: v3: we strengthen the main result so it holds across all Lie types
DOI：: 10.1112/plms.12077

摘要：

考虑广义旗流形$G/B$和相应的仿射旗流形$\mathcal{Fl}_G$。本文利用$\mathcal{Fl}_G$中Schubert簇的曲线邻域构造了$\mathcal{Fl}_G$的仿射Gromov-Witten不变量，得到了以$G$的仿射根系中的单根为指标的“仿射量子Chevalley”算子族$Λ_0, \ldots, Λ_n$.这些算子作用在上同调环$\mathrm{H}^*(\mathcal{Fl}_G)$上，系数在$\mathbb{Z}[q_0, \ldots,q_n]$中。通过分析这些算子的交换性和不变性，我们导出了两个量子上同调环的存在性，它们满足Guest和Otofuji对$G= \mathrm{SL}_n(\mathbb{C})$的性质.第一个量子环是由因子生成的$\mathrm{H}^*(\mathcal{Fl}_G)$的子代数的变形。第二个环，表示为$\mathrm{QH}^*_{\mathrm{af}}(G/B)$，通过添加仿射量子参数$q_0$使普通量子上同调环$\mathrm{QH}^*(G/B)$变形。我们证明了$\mathrm{QH}^*_{\mathrm{af}}(G/B)$是一个Frobenius代数，并且新的量子积决定了一个平坦的Dubrovin联络.此外，我们开发了一个类似的Givental和Kim形式主义，这个环，我们推导出的介绍$\mathrm{QH}^*_{\mathrm{af}}(G/B)$的生成元和关系。理想的关系是由运动积分的周期户田晶格相关联的$G$的扩展Dynkin图的对偶。

英文摘要：

Consider the generalized flag manifold $G/B$ and the corresponding affine flag manifold $\mathcal{Fl}_G$. In this paper we use curve neighborhoods for Schubert varieties in $\mathcal{Fl}_G$ to construct certain affine Gromov-Witten invariants of $\mathcal{Fl}_G$, and to obtain a family of "affine quantum Chevalley" operators $Λ_0, \ldots, Λ_n$ indexed by the simple roots in the affine root system of $G$. These operators act on the cohomology ring $\mathrm{H}^*(\mathcal{Fl}_G)$ with coefficients in $\mathbb{Z}[q_0, \ldots,q_n]$. By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for $G= \mathrm{SL}_n(\mathbb{C})$. The first quantum ring is a deformation of the subalgebra of $\mathrm{H}^*(\mathcal{Fl}_G)$ generated by divisors. The second ring, denoted $\mathrm{QH}^*_{\mathrm{af}}(G/B)$, deforms the ordinary quantum cohomology ring $\mathrm{QH}^*(G/B)$ by adding an affine quantum parameter $q_0$. We prove that $\mathrm{QH}^*_{\mathrm{af}}(G/B)$ is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. Further, we develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of $\mathrm{QH}^*_{\mathrm{af}}(G/B)$ by generators and relations. The ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of $G$.

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英文摘要：

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