  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 $(\epsilon_1,\epsilon_2)$.
