The structure of $\mathcal{N} = {2}$ supersymmetric nonlinear sigma models in AdS₄

Abstract

We present a detailed study of the most general $\mathcal{N} = {2}$ supersymmetric sigma models in four-dimensional anti-de Sitter space (AdS₄) formulated in terms of $\mathcal{N} = 1$ chiral superfields. The target space is demonstrated to be a non-compact hyperkähler manifold restricted to possess a special Killing vector field which generates an SO(2) group of rotations on the two-sphere of complex structures and necessarily leaves one of them invariant. All hyperkähler cones, that is the target spaces of $\mathcal{N} = {2}$ superconformal sigma models, prove to possess such a vector field that belongs to the Lie algebra of an isometry group SU(2) acting by rotations on the complex structures. A unique property of the $\mathcal{N} = {2}$ sigmamodelsconstructedisthatthealgebraofOSp(2|4)transformationsclosesoff the mass shell. We uncover the underlying $\mathcal{N} = {2}$ superfield formulation for the $\mathcal{N} = {2}$ sigma models constructed and compute the associated $\mathcal{N} = {2}$ supercurrent. We give a special analysis of the most general systems of self-interacting $\mathcal{N} = {2}$ tensor multiplets in AdS₄ and their dual sigma models realized in terms of $\mathcal{N} = 1$ chiral multiplets. We also briefly discuss the relationship between our results on $\mathcal{N} = {2}$ supersymmetric sigma models formulated in the $\mathcal{N} = 1$ AdS superspace and the off-shell sigma models constructed in the $\mathcal{N} = {2}$ AdS superspace in arXiv:0807.3368.