Applications of the theory of Floer to symmetric spaces

TL;DR

This paper establishes a geometric link between the Floer cohomology of real flag manifolds and the Pontryagin ring of loop spaces of symmetric spaces, confirming a conjecture for Lie groups.

Contribution

It constructs a geometric framework connecting Floer cohomology and loop space homology for symmetric spaces, proving a conjecture of Peterson in this context.

Findings

Pontryagin ring of loop space is isomorphic to Floer cohomology after localization

Constructed a Lagrangian correspondence linking cotangent bundles and flag varieties

Reduced complex computations to the case of tori using geometric perturbation data

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 $H_{-*}(\Omega(G/K))$ of the based loop space $\Omega(G/K)$ 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 $G_{\mathbb{C}}/P_{\sigma}$ 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 $CF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$. 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.