Applications of the theory of Floer to symmetric spaces

Abstract

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 $G_{\mathbb{C}}/P_{\sigma }$. We prove that the Pontryagin ring $H_{-\ast }(\Omega (G/K))$ of the based loop space $\Omega (G/K)$ is isomorphic to the Floer cohomology ring $H{F}^{\ast }(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$ 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 $T^{\ast }(G/K)$ to $G_{\mathbb{C}}/P_{\sigma }$ 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 $\Omega (G/K)$ to $C{F}^{\ast }(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$. 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.

Combined with arXiv:2102.03103, 36 pages, Comments welcome

References in corpus (3)

• Symplectic cohomology and duality for the wrapped Fukaya category
• The Lagrangian Floer-quantum-PSS package and canonical orientations in Floer theory
• Peterson conjecture via Lagrangian correspondences and wonderful compactifications

Cited by in corpus (1)

• Quantum characteristic classes, moment correspondences and the Hamiltonian groups of coadjoint orbits