Abstract:

A dimer model is a bipartite graph embedded in a real 2-torus. Every such graph determines naturally a noncommutative algebra A whose centre is a semigroup algebra R. Under mild assumptions, it has been shown that the dimer model encodes the minimal projective resolution of A as an (A,A)-bimodule and, moreover, that A is a CY3-algebra (it is a `noncommutative crepant resolution' of R). I will describe work in preparation that constructs the resolution much more simply from a toric cell complex. This `noncommutative cellular resolution' provides an analogue of the cellular resolutions in commutative algebra constructed by Bayer-Sturmfels, and it makes possible a generalisation of the dimer model construction to arbitrary dimension. This is joint work in preparation with Alexander Quintero Velez.