Superfield theories on $S^3$ and their localization

We consider the superfield formulation of supersymmetric gauge and matter field theories on a three-dimensional sphere with rigid $\mathcal{N}=2$ supersymmetry, as well as with $\mathcal{N}>2$. The construction is based on a supercoset $SU(2\mathrm{\mid }1)\mathrm{/}U(1)$ containing $S^3$ as the bosonic subspace. We derive an explicit form of $SU(2\mathrm{\mid }1)\mathrm{/}U(1)$ supervielbein and covariant derivatives, and use them to construct classical superfield actions for gauge and matter supermultiplets in this superbackground. We then apply superfield methods for computing one-loop partition functions of these theories and demonstrate how the localization technique works directly in the superspace.