  It was recently shown that IIB supergravity on AdS$_5\times$S$^5$ enjoys 10d
conformal symmetry and that superstring theory on this background can be
described using a 10d scalar effective field theory. In this paper we adapt
these two complementary approaches to correlators of hypermultiplets in
AdS$_2\times$S$^2$. In particular, we show that 4-point correlators of
$1/2$-BPS operators in the 1d boundary can be computed using 4d conformal
symmetry and a 4d effective action in the bulk. The 4d conformal symmetry is
realised by acting with Casimirs of $SU(1,1|2)$, and is generically broken by
higher derivative corrections. We point out similar structure underlying
$\alpha'$ corrections to IIB supergravity in AdS$_5\times$S$^5$. In particular,
while the $\alpha'^3$ corrections can be written in terms of a sixth order
Casimir acting on a 10d conformal block, similar structure does not appear in
higher-order corrections. We note however that a specific combination of higher
derivative corrections can give rise to Witten diagrams with higher dimensional
symmetry at the integrand level, with breaking then arising from the measure.
