Genus-one open string amplitudes on AdS$_5\times$S$^3$ from CFT

We bootstrap one-loop string corrections to the four-point function of half-BPS operators in a 4d $\mathcal{N}=2$ SCFT with flavour group $SO(8)$, dual to gluon scattering at genus one on AdS$_5\times$S$^3$. We identify an 8-dimensional organising principle which governs the spectrum of double-trace anomalous dimensions, valid to all orders in the string length. This has precise implications for the structure of one-loop Mellin amplitudes, which we explicitly compute for the first three orders beyond the field-theory limit. We also consider the corresponding position space representation, which is entirely determined by the square of a certain differential operator acting on a simpler "pre-correlator". Finally, we show that the flat-space limit of the Mellin amplitudes exactly matches the logarithmic terms of the genus-one amplitude in 8-dimensional flat space, which we compute via a partial-wave analysis.