Monotone Lagrangians in flag varieties

TL;DR

This paper derives a Maslov index formula for gradient holomorphic discs and uses it to classify all monotone Lagrangian non-toric fibers in Gelfand-Cetlin systems on partial flag manifolds.

Contribution

It introduces a new Maslov index formula for gradient holomorphic discs and applies it to classify monotone Lagrangian fibers in partial flag manifolds.

Findings

Derived a Maslov index formula for gradient holomorphic discs

Classified all monotone Lagrangian non-toric fibers in Gelfand-Cetlin systems

Provided tools for understanding Lagrangian submanifolds in flag varieties

Abstract

In this paper, we give a formula for the Maslov index of a gradient holomorphic disc, which is a relative version of the Chern number formula of a gradient holomorphic sphere for a Hamiltonian $S^1$-action. Using the formula, we classify all monotone Lagrangian non-toric fibers of Gelfand-Cetlin systems on partial flag manifolds.