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

TL;DR

This paper uses bootstrap methods to analyze four-point correlators in 6D supergravity on AdS3×S3, uniquely determining the spectrum parameter n=21, matching string theory on K3, and confirming results with flat-space amplitude calculations.

Contribution

It introduces a bootstrap approach that constrains the spectrum parameter n to be 21, matching IIB string theory on K3, and analyzes the spectrum of double-trace operators at one-loop order.

Findings

Bootstrap equations fix n=21, corresponding to K3 compactification.

Spectrum of unprotected double-trace operators is analyzed and anomalous dimensions are rationalized.

Flat-space limit matches recent one-loop amplitude calculations.

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].