Boundary correlators of Liouville and Toda theories on AdS2 and AdS/CFT

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We observe that boundary correlators of the elementary scalar field of the Liouville theory defined on rigid AdS2 background are the same 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: correlators of Toda scalars on AdS2 are directly related to those of the chiral 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 AdS2/CFT1 duality where the CFT1 is the chiral half of a 2d CFT. Similar relation applies also to a non-abelian Toda theory containing a Liouville scalar coupled to a 2d sigma-model originating from the SL(2,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± of the non-abelian Toda CFT. The duality is checked at the next-to-leading order in the large central charge expansion by matching the chiral CFT correlators of (T,V+,V−) with the boundary correlators of the three Toda scalars given by the tree-level and one-loop Witten diagrams in AdS2. Based on arXiv:1904.12753,1907.01357.