On the topology of real Lagrangians in toric symplectic manifolds

TL;DR

This paper investigates the topology of real Lagrangian submanifolds in toric symplectic manifolds, establishing a combinatorial classification and realizing all possible types in certain surfaces.

Contribution

It introduces a real analog of the Delzant construction for real Lagrangians, linking their topology to combinatorial data.

Findings

Diffeomorphism type of real Lagrangians is determined by combinatorial data.

All possible diffeomorphism types are realized in toric symplectic del Pezzo surfaces.

Established a classification framework for real Lagrangians in toric symplectic manifolds.

Abstract

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data. As an application, we realize all possible diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces.