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

TL;DR

This paper computes the first curvature correction to four-point string amplitudes of KK modes on AdS3×S3, revealing new integral formulas involving polylogarithms and providing insights into operator dimensions in the dual D1-D5 CFT.

Contribution

It derives a novel integral representation of the AdS3×S3 Virasoro-Shapiro amplitude with KK modes, generalizing to arbitrary modes using Mellin formalism, and connects results to strong-coupling CFT data.

Findings

Derived amplitude in special and general cases using polylogarithms.

Provided operator anomalous dimensions and OPE data at strong coupling.

Confirmed consistency of scaling dimensions with classical string theory.

Abstract

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.