$AdS_3 \times S^3$ Virasoro-Shapiro amplitude with KK modes

We study the first curvature correction to the string amplitude of four Kaluza--Klein (KK) modes on $AdS_3 \times S^3 \times M_4$, with $M_4=K3$ or $T^4$, in type IIB string theory, which is holographically dual to the four--point correlator $\langle \mathcal{O}_{p_1} \mathcal{O}_{p_2} \mathcal{O}_{p_3} \mathcal{O}_{p_4} \rangle$ of certain half--BPS operators in the boundary D1--D5 CFT. The result takes the form of an integral over the Riemann sphere, analogous to the flat-space Virasoro--Shapiro amplitude, but with insertions of single-valued multiple polylogarithms of weight three. Our results are obtained in two steps. First, we derive the $AdS_3 \times S^3$ Virasoro--Shapiro amplitude in the special case $\langle \mathcal{O}_{p} \mathcal{O}_{p} \mathcal{O}_{1} \mathcal{O}_{1} \rangle$, by matching the CFT block expansion with an ansatz based on single-valued multiple polylogarithms. We then employ the $AdS \times S$ Mellin formalism to generalize the result to the general case of four arbitrary KK modes $\langle \mathcal{O}_{p_1} \mathcal{O}_{p_2} \mathcal{O}_{p_3} \mathcal{O}_{p_4} \rangle$. Our analysis yields an infinite set of results for operator anomalous dimensions and OPE data in D1--D5 CFT at strong coupling. In particular, the resulting scaling dimensions of certain operators are shown to be consistent with classical string theory computations.