Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces of compact type

TL;DR

This paper computes the Lagrangian Floer homology for pairs of real forms in Hermitian symmetric spaces of compact type, generalizing known inequalities and demonstrating volume minimization of certain Lagrangian spheres.

Contribution

It extends Floer homology calculations to pairs of real forms in Hermitian symmetric spaces, generalizing the Arnold-Givental inequality and proving volume minimization of specific Lagrangian spheres.

Findings

Calculated Floer homology for real form pairs in Hermitian symmetric spaces.

Generalized Arnold-Givental inequality for irreducible cases.

Proved volume minimization of a Lagrangian sphere in complex hyperquadric.

Abstract

In this paper we calculate the Lagrangian Floer homology $HF(L_0, L_1 : {\mathbb Z}_2)$ of a pair of real forms $(L_0,L_1)$ in a monotone Hermitian symmetric space $M$ of compact type in the case where $L_0$ is not necessarily congruent to $L_1$. In particular, we have a generalization of the Arnold-Givental inequality in the case where $M$ is irreducible. As its application, we prove that the totally geodesic Lagrangian sphere in the complex hyperquadric is globally volume minimizing under Hamiltonian deformations.