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In this article, we study the localizaiton of the partition function of BPSvortices in $\mathcal{N}=(2,2)$ $U(N)$ super Yang-Mills theory with $N$-flavoron $\R^2$. The vortex partition function for $\mathcal{N}=(2,2)$ superYang-Mills theory is obtained from the one in $\mathcal{N}=(4,4)$ superYang-Mills theory by mass deformation. We show that the partition function canbe written as $Q$-exact form and integration in the partition functions islocalized to the fixed points which are related to $N$-tuple one dimensionalpartitions of positive integers.

Localization of Vortex Partition Functions in $\mathcal{N}=(2,2) $ Super Yang-Mills theory