Gauge and matter superfield theories on $S^2$

We develop a superfield formulation of gauge and matter field theories on a two-dimensional sphere with rigid N=(2,2) as well as extended supersymmetry. The construction is based on a supercoset SU(2|1)/[U(1) x U(1)] containing $S^2$ as the bosonic subspace. We derive an explicit form of supervielbein and covariant derivatives on this coset, 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.