Boundary correlators in WZW model on AdS$_2$

Boundary correlators of elementary fields in some 2d conformal field theories defined on AdS$_2$ have a particularly simple structure. For example, the correlators of the Liouville scalar happen to be the same as the correlators of the chiral component of the stress tensor on a plane restricted to the real line. Here we show that an analogous relation is true also in the WZW model: boundary correlators of the WZW scalars have the same structure as the correlators of chiral Ka\v{c}-Moody currents. This is checked at the level of the tree and one-loop Witten diagrams in AdS$_2$. We also compute some tree-level correlators in a generic $\sigma$-model defined on AdS$_2$ and show that they simplify only in the WZW case where an extra Ka\v{c}-Moody symmetry appears. In particular, the terms in 4-point correlators having logarithmic dependence on 1d cross-ratio cancel only at the WZW point. One motivation behind this work is to learn how to compute AdS$_2$ loop corrections in 2d models with derivative interactions related to the study of correlators of operators on Wilson loops in string theory in AdS.