$\mathcal{N}=2$ AdS hypermultiplets in harmonic superspace

TL;DR

This work develops a harmonic superspace framework for ${ m N}=2$ supersymmetry in AdS$_4$ by embedding the AdS supergroup ${ m OSp}(2|4)$ as a subalgebra of the ${ m N}=2$ superconformal algebra ${rak su}(2,2|2)$ via a constant symmetric matrix $c^{ik}$. The authors construct an ${ m OSp}(2|4)$-invariant mass term for the AdS hypermultiplet, realized through an extra ${ m SO}(2)$ rotation and a background field, and demonstrate how this reduces to the central-charge extended Poincaré case in the flat limit. A superfield Weyl rescaling is explicitly derived to obtain an ${ m OSp}(2|4)$-invariant analytic integration measure, enabling unconstrained AdS$_4$-deformed ${ m N}=2$ hyper-Kähler sigma models, together with a redefinition of Grassmann coordinates to achieve AdS-covariant component expansions. The paper also presents the harmonic-action formulation of a massive ${ m N}=2$ AdS vector multiplet and discusses the broader implications for AdS supergravity and higher-spin theories within the harmonic superspace setting. Overall, this work lays foundational tools for consistent ${ m N}=2$ AdS holography and AdS higher-spin constructions in HSS, with future directions identified toward AdS supergravity, coset constructions, and AdS sigma models.

Abstract

We present the harmonic superspace formulation of $\mathcal{N}=2$ hypermultiplet in AdS$_4$ background, starting from the proper realization of $4D, \mathcal{N}=2$ superconformal group $SU(2,2|2)$ on the analytic subspace coordinates. The key observation is that $\mathcal{N}=2$ AdS$_4$ supergroup $OSp(2|4)$ can be embedded as a subgroup in the superconformal group through introducing a constant symmetric matrix $c^{(ij)}$ and identifying the AdS supercharge as $Ψ^i_α= Q^i_α+ c^{ik} S_{kα}$, with $Q$ and $S$ being generators of the standard and conformal $4D, {\cal N}=2$ supersymmetries. Respectively, the AdS cosmological constant is given by the square of $c^{(ij)}$, $Λ= -12 c^{ij}c_{ij}$. We construct the $OSp(2|4)$ invariant hypermultiplet mass term by adding, to the coordinate AdS transformations, a piece realized as an extra $SO(2)$ rotation of the hypermultiplet superfield. It is analogous to the central charge $x^5$ transformation of flat $\mathcal{N}=2$ supersymmetry and turns into the latter in the super Minkowski limit. As another new result, we explicitly construct the superfield Weyl transformation to the $OSp(2|4)$ invariant AdS integration measure over the analytic superspace, which provides, in particular, a basis for unconstrained superfield formulations of the AdS$_4$-deformed $\mathcal{N}=2$ hyper Kähler sigma models. We find the proper redefinition of $θ$ coordinates ensuring the AdS-covariant form of the analytic superfield component expansions.