Instantons and the 5D U(1) gauge theory with extra adjoint

TL;DR

This paper computes the partition function of 5D supersymmetric U(1) gauge theory with adjoint matter in an Omega-background, revealing its connection to the topology of instanton moduli spaces and comparing methods.

Contribution

It provides a novel calculation of the partition function with adjoint matter and links it to the Poincare polynomial of instanton moduli spaces, clarifying discrepancies with existing methods.

Findings

Partition function encodes topological information.

With adjoint matter, the partition function reproduces the Poincare polynomial.

Comparison with refined topological vertex results highlights differences.

Abstract

In this paper we compute the partition function of 5D supersymmetric U(1) gauge theory with extra adjoint matter in general $\Omega$-background. It is well known that such partition functions encode very rich topological information. We show in particular that unlike the case with no extra matter, the partition function with extra adjoint at some special values of the parameters directly reproduces the generating function for the Poincare polynomial of the moduli space of instantons. Comparing our results with those recently obtained by Iqbal et. al., who used the refined topological vertex method, we present our comments on apparent discrepancies.