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 s
ubspace. 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.
We consider the superfield formulation of supersymmetric gauge and matter field theories on a three-dimensional sphere with rigid ${cal N}=2$ supersymmetry, as well as with ${cal N}> 2$. The construction is based on a supercoset $SU(2|1)/U(1)$ contai
ning $S^3$ as the bosonic subspace. We derive an explicit form of $SU(2|1)/U(1)$ 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.
We consider the type IIB Green-Schwarz superstring theory on AdS(2)xS(2)xT(6) supported by homogeneous Ramond-Ramond 5-form flux and its type IIA T-duals. One motivation is to understand the solution of this theory based on integrability. This backgr
ound is a limit of a 1/4 supersymmetric supergravity solution describing four intersecting D3-branes and represents a consistent embedding of AdS(2)xS(2) into critical superstring theory. Its AdS(2)xS(2) part with corresponding fermions can be described by a classically integrable PSU(1,1|2)/SO(1,1)xU(1) supercoset sigma-model. We point out that since the RR 5-form field has non-zero components along the 6-torus directions one cannot, in general, factorize the 10d superstring theory into the supercoset part plus 6 bosons and 6 additional massless fermions. Still, we demonstrate that the full superstring model (i) is classically integrable, at least to quadratic order in fermions, and (ii) admits a consistent classical truncation to the supercoset part. Following the analogy with other integrable backgrounds and starting with the finite-gap equations of the PSU(1,1|2)/SO(1,1)xU(1) supercoset we propose a set of asymptotic Bethe ansatz equations for a subset of the quantum string states.
