Tropical Lagrangian coamoebae and free resolutions

TL;DR

This paper explores the relationship between the combinatorics of degenerations of Lagrangian submanifolds in complex tori and the homological algebra of mirror coherent sheaves, introducing tropical Lagrangian coamoebae as a new tool.

Contribution

It introduces tropical Lagrangian coamoebae associated to free resolutions, linking combinatorial data to mirror symmetry and extending spectral theory of dimer models to higher dimensions.

Findings

Reconstruction of algebraic data from tropical coamoebae

Mirror correspondence between modules and constructible sheaves

Generalization of dimer model spectral theory

Abstract

We study the coamoebae of Lagrangian submanifolds of $(\mathbb{C}^\times)^n$, specifically how the combinatorics of their degenerations encodes the homological algebra of mirror coherent sheaves. Concretely, to a minimal free resolution $F^\bullet$ of a module $M$ over $\mathbb{C}[z_1^{\pm 1}, \dotsc, z_n^{\pm 1}]$ we associate a simplicial complex $T(F^\bullet) \subset T^n$. We call $T(F^\bullet)$ a tropical Lagrangian coamoeba. We show that the discrete information in $F^\bullet$ can often be recovered from $T(F^\bullet)$, and that more generally $M$ is mirror to a certain constructible sheaf supported on $T(F^\bullet)$. The resulting interplay between coherent sheaves on $(\mathbb{C}^\times)^n$ and simplicial complexes in $T^n$ provides a higher-dimensional generalization of the spectral theory of dimer models in $T^2$, as well as a symplectic counterpart to the theory of brane brick models.