AdS Virasoro-Shapiro amplitude with KK modes

A bstract We determine the first curvature correction for the string amplitude of two supergravity states and two Kaluza-Klein modes on AdS 5 ×S 5 , which is dual to the correlator $$ \left\langle {\mathcal{O}}_2{\mathcal{O}}_2{\mathcal{O}}_p{\mathcal{O}}_p\right\rangle $$ O 2 O 2 O p O p of half-BPS operators in $$ \mathcal{N} $$ N = 4 SYM theory. The result has the form of an integral over the Riemann sphere as for the usual Virasoro-Shapiro amplitude, with the insertion of single-valued multiple polylogarithms of weight three. The result fixes OPE data of single-trace operators in $$ \mathcal{N} $$ N = 4 SYM theory at strong coupling, including operators with non-zero R -charge and odd spin. We successfully check our results by comparing to data available from integrability, localisation and consistency with a 10 d effective action.