Toda theory in AdS2 and \U0001d4b2An-algebra structure of boundary correlators

Abstract

We consider the conformal An Toda theory in AdS2. Due to the bulk full Virasoro symmetry, this system provides an instance of a non-gravitational AdS2/CFT1 correspondence where the 1d boundary theory enjoys enhanced ``frac{1}{2}- Virasoro" symmetry. General boundary correlators are expected to be captured by the restriction of chiral correlators in a suitable WAn Virasoro extension. At next-to-leading order in weak coupling expansion they have been conjectured to match the subleading terms in the large central charge expansion of the dual \U0001d4b2An correlators. We explicitly test this conjecture on the boundary four point functions of the Toda scalar fields dual to \U0001d4b2An generators with next-to-minimal spin 3 and 4. Our analysis is valid in the generic rank case and extends previous results for specific rank-2 Toda theories. On the AdS side, the extension is straightforward and requires the computation of a finite set of tree Witten diagrams. This is due to simple rank dependence and selection rules of cubic and quartic couplings. On the boundary, we exploit crossing symmetry and specific meromorphic properties of the W-algebra correlators at large central charge. We present the required 4-point functions in closed form for any rank and verify the bulk-boundary correspondence in full details.