Refined Topological Vertex, Cylindric Partitions and the U(1) Adjoint Theory

TL;DR

This paper explores the partition function of a 5D U(1) gauge theory with an adjoint hypermultiplet, revealing its connection to cylindric partitions, Schur processes, and operator traces, with implications for (q,t) identities.

Contribution

It demonstrates that the partition function is a refined cylindric partition generating function and relates it to operator traces, advancing understanding of refined topological vertex applications.

Findings

Partition function is a periodic Schur process.

Partition function refines cylindric plane partition generating functions.

Connection established between partition functions and operator traces.

Abstract

We study the partition function of the compactified 5D U(1) gauge theory (in the Omega-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 Omega-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) identities using the refined topological vertex.