2021/02/28 by Bae, Hanwool, Chow, Chi Hong, Leung, Naichung Conan
#57R58 #57T15 #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2103.00382
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 Gℝ/B which is a real locus of a complex partial flag variety Gℂ/Pσ. 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^*(Gℝ/B,Gℝ/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 Gℂ/Pσ which geometrically composes with a cotangent fiber to Gℝ/B, and compute the linear part of the associated Ma'u-Wehrheim-Woodward's A∞ homomorphism from a Floer model of Ω(G/K) to CF^*(Gℝ/B,Gℝ/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.