Tropical Lagrangian coamoebae and free resolutions

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 $\mathbb{C}[z_1^{\pm 1},\dots ,z_n^{\pm 1}]$ 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.