FRG: Collaborative Research: Dimers in Combinatorics and Physics

Statistical mechanics is the mathematical study of matter at small scales. Its primary goals are to analyze phase transitions: for example liquid-to-solid transitions where the physical properties of a substance change abruptly. The dimer model was originally conceived as a simplified model of two-dimensional matter in which phase transitions can be studied. Recent work, however, has linked the model to many other areas of mathematics, from combinatorics to string theory, where ''brane dimers'' are proposed as fundamental descriptions of spacetime at small scales. The PIs propose to jointly investigate a number of interrelated topics in mathematics and physics, each of which has the dimer model as its underlying combinatorial structure. This project will lead to the organization of workshops and regular meetings of the PIs and their graduate students and postdoctoral fellows, continuing the PIs' efforts to get young mathematicians and physicists involved in these topics. The PIs will contribute to the mathematical community through their mentorship of young scholars, research talks in conferences and workshops, papers published in peer-reviewed journals, and books on a selection of these topics.

The dimer model studies the set of all dimers, or perfect matchings, on a planar bipartite graph G on a disk or Riemann surface. Despite the simple definition, there are many open problems about the dimer model, as well as applications to geometry, algebra, and physics. There is a fundamental connection between the dimer model on the disk and the Grassmannian, via the fact that generating functions of dimers satisfy Plucker relations. This fact leads to the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian, and the beautiful combinatorics of the positive Grassmannian. This project will explore a myriad of generalizations of the objects mentioned above, and will significantly improve our understanding of: the dimer model on non-planar graphs; limiting behaviors of the dimer model on a torus and other surfaces; the connection between dimers on a torus and brane tilings in string theory; soliton solutions to the KP equation and the bipartite graphs realizable as soliton graphs; the relationship between convex polygon tilings and the corresponding bipartite planar graphs with Kasteleyn weightings; the connection between the dimer model and triangulations of the amplituhedron, an object whose volume computes scattering amplitudes; and higher-dimensional dimer models, colored quivers and a generalized notion of cluster mutation, exciting new objects motivated by dualities in supersymmetric quantum field theory.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.