Toda theory in AdS$_{2}$ and $\mathcal WA_{n}$-algebra structure of boundary correlators

We consider the conformal $A_{n}$ Toda theory in AdS$_{2}$. Due to the bulk full Virasoro symmetry, this system provides an instance of a non-gravitational $\text{AdS}_{2}$/$\text{CFT}_{1}$ 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 $\mathcal WA_{n}$ 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 $\mathcal WA_{n}$ correlators. We explicitly test this conjecture on the boundary four point functions of the Toda scalar fields dual to $\mathcal WA_{n}$ 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, the CFT calculation is made feasible by exploiting crossing symmetry and specific meromorphic properties of the $\mathcal W$-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.