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

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 $\phi^4$ 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.