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 $(\mathbb{C}^*)^n$ 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 $(\mathbb{C}^*)^2$. We also give several new examples of Lagrangian submanifolds inside toric varieties, some of which are monotone.