Note on refined topological vertex, Jack polynomials and instanton counting

TL;DR

This paper computes the refined topological vertex using Jack symmetric functions and derives the partition function for elliptic N=2 models, matching Nekrasov's instanton counting results for N=2* theories.

Contribution

It introduces a method to calculate the refined topological vertex with Jack functions and connects it to elliptic N=2 models and Nekrasov's instanton partition functions.

Findings

Refined topological vertex computed via Jack symmetric functions.

Partition function for elliptic N=2 models derived.

Results agree with Nekrasov instanton counting for N=2* theories.

Abstract

In this article, we calculated the refined topological vertex for the one parameter case using the Jack symmetric functions. Also, we obtain the partition function for elliptic N=2 models, the results coincide with those of Nekrasov instanton counting partition functions for the $N=2^{\ast}$ theories.