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On the topology of real Lagrangians in toric symplectic manifolds

Joé Brendel, Joontae Kim, Jiyeon Moon

TL;DR

The paper develops a real Delzant framework for toric symplectic manifolds, showing that the topology of real Lagrangians L = Fix(R^σ) arising from Δ-invariant polytope symmetries is completely determined by combinatorial data (Δ and σ). By constructing ν_R and K_R as real analogues of the Delzant maps, L is diffeomorphic to ν_R^{-1}(0)/K_R and enjoys convexity and tightness properties. The authors prove a real Delzant theorem and demonstrate its power by classifying and realizing all connected real Lagrangians in toric monotone del Pezzo surfaces, including CP^2 and various blow-ups, via explicit lifted antisymplectic involutions. They also establish obstructions (e.g., Arnold’s lemma) that restrict possible topologies, and provide concrete examples showing how different σ yield a range of real Lagrangian diffeomorphism types such as RP^2, T^2, S^2, and RP^2#RP^2. The work thereby connects combinatorial polytope data to Lagrangian topology, with implications for rigidity and classification of real Lagrangians in toric del Pezzo surfaces.

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.

On the topology of real Lagrangians in toric symplectic manifolds

TL;DR

The paper develops a real Delzant framework for toric symplectic manifolds, showing that the topology of real Lagrangians L = Fix(R^σ) arising from Δ-invariant polytope symmetries is completely determined by combinatorial data (Δ and σ). By constructing ν_R and K_R as real analogues of the Delzant maps, L is diffeomorphic to ν_R^{-1}(0)/K_R and enjoys convexity and tightness properties. The authors prove a real Delzant theorem and demonstrate its power by classifying and realizing all connected real Lagrangians in toric monotone del Pezzo surfaces, including CP^2 and various blow-ups, via explicit lifted antisymplectic involutions. They also establish obstructions (e.g., Arnold’s lemma) that restrict possible topologies, and provide concrete examples showing how different σ yield a range of real Lagrangian diffeomorphism types such as RP^2, T^2, S^2, and RP^2#RP^2. The work thereby connects combinatorial polytope data to Lagrangian topology, with implications for rigidity and classification of real Lagrangians in toric del Pezzo surfaces.

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.

Paper Structure

This paper contains 17 sections, 22 theorems, 133 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Basic geometry
  3. Basics
  4. Compatible maps
  5. Toric symplectic manifolds
  6. Basics
  7. The Delzant construction
  8. Lifting symmetries of the moment polytope
  9. Symmetries of the moment polytope
  10. Lifted symplectomorphisms
  11. Lifted antisymplectic involutions
  12. Classification of compatible maps
  13. Real Delzant construction
  14. Convexity and Tightness
  15. Real Lagrangians in toric symplectic del Pezzo surfaces
  16. ...and 2 more sections

Key Result

Theorem A

Let $(M,\omega)$ be a toric symplectic manifold with moment map $\mu$ and moment polytope $\Delta$. Furthermore, let $\sigma \in \mathcal{S}_{\Delta}$ be an involution of $\Delta$. Then $\sigma$ lifts to an antisymplectic involution $R^{\sigma}$ of $M$,

Figures (10)

  • Figure 1: The flip $\sigma$ on $\Delta$ and the corresponding antisymplectic involution $R^{\sigma}$ on $S^2$.
  • Figure 2: Moment polytope $\Delta$ and $\mathcal{S}_{\Delta}$ for toric symplectic del Pezzo surfaces.
  • Figure 3: $\mathop{\mathrm{Fix}}\nolimits(\sigma)$ in $\Delta$.
  • Figure 4: Examples in which convexity or tightness fail.
  • Figure 5: Involutions on the 2-simplex $\Delta$
  • ...and 5 more figures

Theorems & Definitions (58)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • ...and 48 more