arXiv 2103.003822021-03math.SG

Floer理论在对称空间中的应用

Applications of the theory of Floer to symmetric spaces

Hanwool Bae, Chi Hong Chow, Naichung Conan Leung

备注：: Combined with arXiv:2102.03103, 36 pages, Comments welcome

摘要：

我们研究了Bott-Samelson将Morse理论应用于任何紧致对称空间$G/K$和相关的实旗流形$G_{\mathbb{R}}/B$（它是复偏旗簇$G_{\mathbb{C}}/P_σ$的实轨迹）所考虑的问题。证明了基环空间$Ω(G/K)$的Pontryagin环$H_{-*}(Ω(G/K))$与局部化后的Floer上同调环$HF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$同构.当$G/K$是李群时，这是彼得森的一个猜想，由拉姆-下佐野在复旗簇的量子上同调的背景下组合证明。我们的方法是几何的性质：我们构建了一个拉格朗日对应$T^*(G/K)$ $G_{\mathbb{C}}/P_σ$几何组成的余切纤维$G_{\mathbb{R}}/B$，并计算相关的Ma 'u-Wehrheim-Woodward的$A_{\infty}$同态从Floer模型的$Ω(G/K)$到$CF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$的线性部分。关键是利用$G/K$的几何构造特定的扰动数据，使我们能够减少计算的情况下，当$G/K$是一个环面。

英文摘要：

We quantize the problem considered by Bott-Samelson who applied Morse theory to any compact symmetric space $G/K$ and the associated real flag manifold $G_{\mathbb{R}}/B$ which is a real locus of a complex partial flag variety $G_{\mathbb{C}}/P_σ$. We prove that the Pontryagin ring $H_{-*}(Ω(G/K))$ of the based loop space $Ω(G/K)$ is isomorphic to the Floer cohomology ring $HF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$ after localization. When $G/K$ is a Lie group, this is a conjecture of Peterson, proved combinatorially by Lam-Shimozono, in the context of quantum cohomologies of complex flag varieties. Our approach is geometric in nature: we construct a Lagrangian correspondence from $T^*(G/K)$ to $G_{\mathbb{C}}/P_σ$ which geometrically composes with a cotangent fiber to $G_{\mathbb{R}}/B$, and compute the linear part of the associated Ma'u-Wehrheim-Woodward's $A_{\infty}$ homomorphism from a Floer model of $Ω(G/K)$ to $CF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$. The crux is to make use of the geometry of $G/K$ to construct specific perturbation data which enables us to reduce the computations to the case when $G/K$ is a torus.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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