AdS3 × S3 Virasoro-Shapiro amplitude with KK modes

A bstract We study the first curvature correction to the string amplitude of four Kaluza-Klein (KK) modes on AdS 3 × S 3 × M 4 , with M 4 = K 3 or T 4 , in type IIB string theory, which is holographically dual to the four-point correlator $$ \left\langle {\mathcal{O}}_{p1}{\mathcal{O}}_{p2}{\mathcal{O}}_{p3}{\mathcal{O}}_{p4}\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:math> 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 × S 3 Virasoro-Shapiro amplitude in the special case $$ \left\langle {\mathcal{O}}_p{\mathcal{O}}_p{\mathcal{O}}_1{\mathcal{O}}_1\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:msub> <mml:mi>O</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>O</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:mfenced> </mml:math> , by matching the CFT block expansion with an ansatz based on single-valued multiple polylogarithms. We then employ the AdS × S Mellin formalism to generalize the result to the general case of four arbitrary KK modes $$ \left\langle {\mathcal{O}}_{p1}{\mathcal{O}}_{p2}{\mathcal{O}}_{p3}{\mathcal{O}}_{p4}\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:math> . 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.