2d Partition Function in Omega-background and Vortex/Instanton Correspondence

TL;DR

This paper derives the exact vortex partition function in 2d supersymmetric gauge theory on the Omega-background, revealing a differential equation system that links vortex and instanton partition functions through a quantized twisted F-term framework.

Contribution

It introduces a novel derivation of the vortex partition function in 2d $ ext{N}=(2,2)$ theories on the Omega-background and establishes a new correspondence with 4d Nekrasov partition functions.

Findings

Partition function satisfies a system of differential equations.

Differential equations are a quantized version of twisted F-term equations.

Established vortex-instanton correspondence at the Higgs branch root.

Abstract

We derive the exact vortex partition function in 2d $\mathcal{N}$ = (2,2) gauge theory on the Omega-background, applying the localization scheme in the Higgs phase. We show that the partition function at a finite Omega-deformation parameter $\epsilon$ satisfies a system of differential equations, which can be interpreted as a quantized version of the twisted F-term equations characterizing the SUSY vacua. Using the differential equations derived in this paper, we show the correspondence between the partition function of the two-dimensional vortex string worldsheet theory and the Nekrasov partition function at the root of Higgs branch of the four-dimensional $\mathcal{N}$ = 2 theory with two Omega-deformation parameters $(\epsilon_1,\epsilon_2)$.