Refined topological vertex, cylindric partitions and U(1)U(1) adjoint theory

We study the partition function of the compactified 5D U(1)U(1) gauge theory (in the Ω-background) with a single adjoint hypermultiplet, calculated using the refined topological vertex. We show that this partition function is an example a periodic Schur process and is a refinement of the generating function of cylindric plane partitions. The size of the cylinder is given by the mass of adjoint hypermultiplet and the parameters of the Ω  -background. We also show that this partition function can be written as a trace of operators which are generalizations of vertex operators studied by Carlsson and Okounkov. In the last part of the paper we describe a way to obtain (q,t)(q,t) identities using the refined topological vertex.