Hidden Conformal Symmetry in AdS${}_2\times$S${}^2$ Beyond Tree Level

Abstract

Correlators of a certain one-dimensional superconformal field theory dual to hypermultiplets in AdS${}_2\times$S${}^2$ exhibit a hidden four-dimensional conformal symmetry which allows one to repackage all tree-level 4-point correlators into a single four-dimensional object corresponding to a contact diagram arising from a massless ${\varphi }^4$ theory in AdS${}_2\times$S${}^2$. This theory serves as a toy model for IIB string theory in AdS${}_5\times$S${}^5$ and is interesting in its own right because AdS${}_2\times$S${}^2$ describes the near-horizon limit of extremal black holes in four dimensions. We argue that after acting with an $SU(1,1)\times SU(2)$ Casimir, all one-loop correlators can similarly be encoded by a four-dimensional function which arises from a one-loop scalar bubble diagram in AdS${}_2\times$S${}^2$, explaining how the hidden conformal symmetry extends beyond tree level. Finally, we conjecture a scalar effective field theory with a derivative interaction in AdS${}_2\times$S${}^2$ whose Witten diagrams should directly reproduce 4-point correlators to all loops without acting with Casimirs.

5 pages plus appendices; v2: minor changes and added references, published version; v3: typos fixed