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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 $\Psi^i_\alpha = Q^i_\alpha + c^{ik} S_{k\alpha}$, 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 $c^{(ij)}$, $\Lambda = -6 c^{ij}c_{ij}$. 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 Kahler sigma models. We find the proper redefinition of $\theta$ coordinates ensuring the AdS-covariant form of the analytic superfield component expansions.
E. Ivanov and N. Zaigraev, $\mathcal{N}=2$ AdS hypermultiplets in harmonic superspace, arXiv:2509.01406 [hep-th].
