Lagrangian pairs of pants

Abstract

We construct a Lagrangian submanifold, inside the cotangent bundle of a real torus, which we call a Lagrangian pair of pants. It is given as the graph of the differential of a smooth function defined on the real blow up of a Lagrangian coamoeba. Lagrangian pairs of pants are the main building blocks in a construction of smooth Lagrangian submanifolds of $({\mathbb{C}}^{\ast }{)}^n$ which lift tropical subvarieties in ${\mathbb{R}}^n$. As an example we explain how to lift tropical curves in ${\mathbb{R}}^2$ to Lagrangian submanifolds of $({\mathbb{C}}^{\ast }{)}^2$. We also give several new examples of Lagrangian submanifolds inside toric varieties, some of which are monotone.

50 pages, 18 Figures

References in corpus (5)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Lagrangian pairs of pants
• Tropically constructed Lagrangians in mirror quintic threefolds
• Lagrangian submanifolds from tropical hypersurfaces
• Lagrangian 3-torus fibrations

Cited by in corpus (9)

• Self-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry
• Lagrangian pairs of pants
• Tropical Lagrangian Hypersurfaces are Unobstructed
• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• Tropically constructed Lagrangians in mirror quintic threefolds
• Lagrangian submanifolds from tropical hypersurfaces
• Tropical Lagrangians in toric del-Pezzo surfaces
• Some statistics about Tropical Sandpile Model
• Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds