We introduce a family of spaces called critical varieties. Each critical variety is a subset of one of 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 $G$.
Special cases include critical electrical networks and Baxter's critical $Z$-invariant Ising model associated to rhombus tilings of polygons in the plane. In the case of regular polygons, our formula yields new simple expressions for response matrices of electrical networks and for correlation matrices of the Ising model.
We systematically develop the basic properties of critical varieties. In particular, we study their 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.
Speaker(s): Pavel Galashin (UCLA)