2d partition function in Ω-background and vortex/instanton correspondence

Abstract: We derive the exact vortex partition function in 2d N=(2,2) gauge theory on the Ω-background, applying the localization scheme in the Higgs phase. We show that the partition function at a finite Ω-deformation parameter ϵ 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 N=2 theory with two Ω-deformation parameters (ϵ₁, ϵ₂).