Six-Dimensional Supergravity on S^3 X AdS_3 and 2d Conformal Field Theory

TL;DR

The paper establishes a precise correspondence between the Kaluza-Klein spectrum of six-dimensional supergravity on S^3×AdS_3 and the spectrum of dual two-dimensional conformal field theories, computed via representation theory and boundary chiral algebras. It shows that KK states organize into short multiplets of the AdS supergroup SU(1,1|2)×SU(1,1|2) and that all chiral primaries of the boundary CFT (for the D1-D5/K3 system) are accounted for, including a resolution to Vafa's missing-states puzzle by incorporating multi-particle descendants and boundary degrees of freedom. The analysis extends to various 6d/5d supergravity compactifications, revealing that the KK spectrum encodes the cohomology of the target space in the dual CFT’s sigma-model, and providing detailed matches for multiple supersymmetry cases (e.g., (2,2), (1,1), (3,0)/(4,0)). Overall, the work demonstrates that KK spectra, constrained by AdS symmetries and boundary chiral algebras, offer a robust, largely representation-theory–driven bridge between gravity in AdS_3 and 2d CFTs, with implications for elliptic genera, singletons, and holographic duality across string/M-theory compactifications.

Abstract

In this paper we study the relation between six-dimensional supergravity compactified on S^3 X AdS_3 and certain two-dimensional conformal field theories. We compute the Kaluza-Klein spectrum of supergravity using representation theory; these methods are quite general and can also be applied to other compactifications involving anti-de Sitter spaces. A detailed comparison between the spectrum of the two-dimensional conformal field theory and supergravity is made, and we find complete agreement. This applies even at the level of certain non-chiral primaries, and we propose a resolution to the puzzle of the missing states recently raised by Vafa. As a further illustration of the method the Kaluza-Klein spectra of F-theory on M^6 X S^3 X AdS_3 and of M-theory on M^6 X S^2 X AdS_3 are computed, with M^6 some Calabi-Yau manifold.