I will review the deep connection between the homology of certain algebraic (cluster) varieties defined within the Grassmannian and the “non-transcendental” functions that arise in perturbative quantum field theories. For gauge theories in the planar limit, these functions are related to boundaries of the totally positive Grassmannian, which until recently was thought to be entirely described by the combinatorics of the positroid stratification. I describe novel structures needed for physics, associated with certain Calabi-Yau geometries defined within positroids. I will also discuss the generalization of these ideas to a broader class of cluster varieties associated with non-planar physics.