Lagrangian fibers of Gelfand-Cetlin systems

TL;DR

This paper provides a detailed topological analysis of Gelfand-Cetlin fibers in integrable systems, revealing their structure and relation to vanishing cycles, and introduces an algorithm to determine Lagrangian fibers from combinatorics.

Contribution

It offers a comprehensive description of the topology of Gelfand-Cetlin fibers, including their diffeomorphism types and connection to vanishing cycles, along with a new combinatorial algorithm.

Findings

Fibers over interior points of faces are isotropic and diffeomorphic to $(S^1)^k imes N$.

The manifolds $N$ are exactly the vanishing cycles in toric degenerations.

An algorithm to read off Lagrangian fibers from ladder diagram combinatorics.

Abstract

A Gelfand-Cetlin system is a completely integrable system defined on a partial flag manifold whose image is a rational convex polytope called a Gelfand-Cetlin polytope. Motivated by the study of Nishinou-Nohara-Ueda on the Floer theory of Gelfand-Cetlin systems, we provide a detailed description of topology of Gelfand-Cetlin fibers. In particular, we prove that any fiber over an interior point of a k-dimensional face of the Gelfand-Cetlin polytope is an isotropic submanifold and is diffeomorphic to $(S^1)^k \times N$ for some smooth manifold $N$. We also prove that such $N$'s are exactly the vanishing cycles shrinking to points in the associated toric variety via the toric degeneration. We also devise an algorithm of reading off Lagrangian fibers from the combinatorics of the ladder diagram.