Non-abelian Toda theory on AdS$_2$ and AdS$_2$/CFT$_2^{1/2}$ duality

It was recently observed that boundary correlators of the elementary scalar field of the Liouville theory on AdS$_2$ background are the same (up to a non-trivial proportionality coefficient) as the correlators of the chiral stress tensor of the Liouville CFT on the complex plane restricted to the real line. The same relation generalizes to the conformal abelian Toda theory: boundary correlators of Toda scalars on AdS$_2$ are directly related to the correlation functions of the chiral $\mathcal W$-symmetry generators in the Toda CFT and thus are essentially controlled by the underlying infinite-dimensional symmetry. These may be viewed as examples of AdS$_2$/CFT$_1$ duality where the CFT$_1$ is the chiral half of a 2d CFT; we shall refer to this as AdS$_{2}$/CFT$^{1/2}_{2}$. In this paper we demonstrate that this duality applies also to the non-abelian Toda theory containing a Liouville scalar coupled to a 2d $\sigma$-model originating from the $SL(2, \mathbb R)/U(1)$ gauged WZW model. Here the Liouville scalar is again dual to the chiral stress tensor $T$ while the other two scalars are dual to the parafermionic operators $V^\pm$ of the non-abelian Toda CFT. We explicitly check the duality at the next-to-leading order in the large central charge expansion by matching the chiral CFT correlators of $(T, V^+,V^-)$ (computed using a free field representation) with the boundary correlators of the three Toda scalars given by the tree-level and one-loop Witten diagrams in AdS$_{2}$.