Crystal melting on toric surfaces

We study the relationship between the statistical mechanics of crystal melting and instanton counting in N=4N=4 supersymmetric U(1)U(1) gauge theory on toric surfaces. We argue that, in contrast to their six-dimensional cousins, the two problems are related but not identical. We develop a vertex formalism for the crystal partition function, which calculates a generating function for the dimension 0 and 1 subschemes of the toric surface, and describe the modifications required to obtain the corresponding gauge theory partition function.

► Study of Hilbert scheme of points and curves on toric surfaces.
► Vertex formalism for crystal melting partition function on such surfaces.
► Comparison to instanton partition function of D=4N=4U(1)D=4N=4U(1) SYM.