Lagrangian Pairs of Pants

Abstract

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 an exact one form on the real blowup of a Lagrangian coamoeba. Lagrangian pairs of pants are the main building blocks in a construction of smooth Lagrangian submanifolds of $( {\mathbb{C}}^*)^n$ that 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.