Quantum Gravity on AdS$_3\times$S$^3$ from CFT: Bootstrapping $n=21$

TL;DR

This work performs a precise one-loop bootstrap for the four-point correlator of $SO(n)$ tensor multiplets in AdS$_3\times S^3$ with $(2,0)$ supergravity, reconstructing the correlator at order $1/c^2$ in both position and Mellin space. By enforcing crossing symmetry and the leading-log constraints from the OPE, the authors show that the strong bootstrap uniquely fixes the flavour count to $n=21$, corresponding to IIB on K3, while tree-level data leaves $n$ unconstrained. They analyze mixing among double-trace operators, derive new CFT data (notably anomalous dimensions in the singlet/tensor/graviton sectors), and demonstrate that in the $n=21$ theory several quantities become rational, with one anomalous dimension vanishing. The flat-space limit reproduces the known one-loop six-dimensional amplitude, providing a nontrivial consistency check and connecting CFT bootstrap data to string theory and UV finiteness, with implications for swampland considerations and future extensions to more general AdS$_3\times S^3$ setups.

Abstract

We consider the simplest four-point scattering amplitude of $SO(n)$ tensor multiplets in six-dimensional (2,0) supergravity on AdS$_3\times$S$^3$. Using crossing symmetry and the consistency of the operator product expansion in the dual CFT, we explicitly construct the one-loop contribution to the correlator at order $1/c^2$, both in position space and in Mellin space. We show that a strong form of the bootstrap equations imposes constraints on the value of $n$. Remarkably, we find that our bootstrap approach uniquely determines $n=21$, which corresponds to the spectrum of IIB string theory compactified on K3. This stands in sharp contrast to the tree-level correlator for which $n$ is unconstrained. We also analyse the spectrum of unprotected double-trace operators and solve the mixing problem in the first case that involves both tensor and graviton correlators. When $n=21$, the anomalous dimensions rationalise and one of them vanishes. Lastly, we study the flat-space limit of the correlator and find perfect agreement with the one-loop amplitude recently obtained in [arXiv:2510.24558].