Critical Varieties in the Grassmannian

We introduce a family of spaces called critical varieties. Each critical variety is a subset of one of the positroid varieties in the Grassmannian. The combinatorics of positroid varieties is captured by the dimer model on a planar bipartite graph G, and the critical variety is obtained by restricting to Kenyon’s critical dimer model associated to a family of isoradial embeddings of G. This model is invariant under square/spider moves on G, and we give an explicit boundary measurement formula for critical varieties which does not depend on the choice of G. This extends our recent results for the critical Ising model, and simultaneously also includes the case of critical electrical networks. We systematically develop the basic properties of critical varieties. In particular, we study their real and totally positive parts, the combinatorics of the associated strand diagrams, and introduce a shift map motivated by the connection to zonotopal tilings and scattering amplitudes.