The Virasoro-Shapiro amplitude in AdS5 x S5 and level splitting of 10d conformal symmetry

The genus zero contribution to the four-point correlator < O-p1 O-p2 O-p3 O-p4 > of half-BPS single-particle operators O-p in N = 4 super Yang-Mills, at strong coupling, computes the Virasoro-Shapiro amplitude of closed superstrings in AdS(5) x 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(5) x S-5 up to a small number of ambiguities at each order. 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 which 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) <-> S-5 exchange, and it enjoys intriguing properties, which we explain and detail in various cases.