Philip Engel (Harvard), Tilings and Hurwitz Theory

Consider the tilings of an oriented surface by triangles, or squares, or hexagons, up to combinatorial equivalence. The combinatorial curvature of a vertex is 6, 4, or 3 minus the number of adjacent polygons, respectively. Tilings are naturally stratified into all such having the same set of non-zero curvatures. We outline a proof that for squares and hexagons, the generating function for the number of tilings in a fixed stratum lies in a ring of quasi-modular forms of specified level and weight. First, we rephrase the problem in terms of Hurwitz theory of an elliptic orbifold---a quotient of the plane by an orientation-preserving wallpaper group. In turn, we produce a formula for the number of tilings in terms of characters of the symmetric group. Generalizing techniques pioneered by Eskin and Okounkov, who studied the pillowcase orbifold, we express the generating function for a stratum in terms of the q-trace of an operator acting on Fock space. The key step is to compute the trace in a different basis to express it as an infinite product, then apply the Jacobi triple product formula to conclude quasi-modularity.