Brane Tilings and Cluster Algebras

We discuss combinatorial aspects of connections between cluster algebras and string theory. In particular, physicists such as R. Eager, S. Franco, A. Hanany, K.D. Kennaway, R.-K. Seong, D. Vegh, B. Wecht, and others study certain families of quivers and construct duals for them given as tilings of a torus, known as a brane tiling. In joint work with University of Minnesota REU students I. Jeong, S. Zhang, M. Leoni, S. Neel, and P. Turner, we investigated several such examples, including a six-vertex quiver associated to the dP3 lattice and periodic quivers coming from Gale-Robinson sequences, and obtain combinatorial formulas for cluster varialbes with principal coefficients as subgraphs of these brane tilings. I will not assume prior background on cluster algebras nor string theory.