  We study four-dimensional $\mathcal{N}=2$ supersymmetric $U(N)$ gauge theory
with $2N$ fundamental hypermultiplets in the self-dual $\Omega$-background. The
partition function simplifies at special points of the parameter space and is
related to the partition function of two-dimensional Yang-Mills theory on
$S^{2}$. We also consider the insertion of a Wilson loop operator in
two-dimensional Yang-Mills theory and find the corresponding operator in the
four-dimensional $\mathcal{N}=2$ gauge theory.
