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Supersymmetric Liouville theory in AdS$_2$ and AdS/CFT

In a series of recent papers, a special kind of AdS$_2$/CFT$_1$ duality was observed: the boundary correlators of elementary fields that appear in the Lagrangian of a 2d conformal theory in rigid AdS$_2$ background are the same as the correlators of the corresponding primary operators in the chiral half of that 2d CFT in flat space restricted to the real line. The examples considered were: (i) the Liouville theory where the operator dual to the Liouville scalar in AdS$_2$ is the stress tensor; (ii) the abelian Toda theory where the operators dual to the Toda scalars are the $\mathcal W$-algebra generators; (iii) the non-abelian Toda theory where the Liouville field is dual to the stress tensor while the extra gauged WZW theory scalars are dual to non-abelian parafermionic operators. By direct Witten diagram computations in AdS$_2$ one can check that the structure of the boundary correlators is indeed consistent with the Virasoro (or higher) symmetry. Here we consider a supersymmetric generalization: the $\mathcal N=1$ superconformal Liouville theory in AdS$_2$. We start with the super Liouville theory coupled to 2d supergravity and show that a consistent restriction to rigid AdS$_2$ background requires a non-zero value of the supergravity auxiliary field and thus a modification of the Liouville potential from its familiar flat-space form. We show that the Liouville scalar and its fermionic partner are dual to the chiral half of the stress tensor and the supercurrent of the super Liouville theory on the plane. We perform tests supporting the duality by explicitly computing AdS$_2$ Witten diagrams with bosonic and fermionic loops.