The Virasoro-Shapiro amplitude in AdS$_5times$S$^5$ and level splitting of 10d conformal symmetry

The genus zero contribution to the four-point correlator $langle {cal O}_{p_1}{cal O}_{p_2}{cal O}_{p_3}{cal O}_{p_4}rangle$ of half-BPS single-particle operators ${cal O}_p$ in $mathcal{N}=4$ super Yang-Mills at strong coupling computes the Virasoro-Shapiro amplitude of closed superstrings on $AdS_5times S^5$. Combining Mellin space techniques, the large $p$ limit, and data about the spectrum of two-particle operators at tree level in supergravity, we design a bootstrap algorithm which heavily constrains its $alpha$ expansion. We use crossing symmetry, polynomiality in the Mellin variables and the large $p$ limit to stratify the Virasoro-Shapiro amplitude away from the ten-dimensional flat space limit. Then we analyse the spectrum of exchanged two-particle operators at fixed order in the $alpha$ expansion. We impose that the ten-dimensional spin of the spectrum visible at that order is bounded above in the same way as in the flat space amplitude. This constraint determines the Virasoro-Shapiro amplitude in $AdS_5times S^5$ up to a small number of ambiguities at each order and we compute it explicitly for $(alpha)^{5,6,7,8,9}$. As the order of $alpha$ grows, the ten dimensional spin grows, and the set of visible two-particle operators opens up. Operators illuminated for the first time receive a string correction to their anomalous dimensions. This correction is uniquely determined and lifts the residual degeneracy of tree level supergravity due to ten-dimensional conformal symmetry. We encode the lifting of the residual degeneracy in a characteristic polynomial. This object carries information about all orders in $alpha$.It is analytic in the quantum numbers, symmetric under an $AdS_5 leftrightarrow S^5$ exchange, and it enjoys intriguing properties, which we explain and detail in various cases.