Supersymmetry of AdS and flat IIB backgrounds

TL;DR

This paper systematically classifies supersymmetric IIB supergravity backgrounds with warped AdS and flat spacetime factors, establishing relations between geometry, fluxes, and preserved supersymmetries, and deriving new mathematical theorems connecting Killing spinors to Dirac operators.

Contribution

It provides a comprehensive classification of supersymmetric warped AdS and flat IIB backgrounds, including explicit supersymmetry counts and new Lichnerowicz-type theorems linking Killing spinors to Dirac zero modes.

Findings

AdS backgrounds preserve N=2^{[n/2]} k for n≤4 and N=2^{[n/2]+1} k for 4<n≤6 supersymmetries.

Flat backgrounds preserve N=2^{[n/2]} k for 2<n≤4 and N=2^{[(n+1)/2]} k for 4<n≤7 supersymmetries.

Killing spinors do not factorize into AdS and internal space components.

Abstract

We present a systematic description of all warped $AdS_n\times_w M^{10-n}$ and ${\mathbb{R}}^{n-1,1}\times_w M^{10-n}$ IIB backgrounds and identify the a priori number of supersymmetries $N$ preserved by these solutions. In particular, we find that the $AdS_n$ backgrounds preserve $N=2^{[{n\over2}]} k$ for $n\leq 4$ and $N=2^{[{n\over2}]+1} k$ for $4<n\leq 6$ supersymmetries and for $k\in {\mathbb{N}}_{+}$ suitably restricted. In addition under some assumptions required for the applicability of the maximum principle, we demonstrate that the Killing spinors of $AdS_n$ backgrounds can be identified with the zero modes of Dirac-like operators on $M^{10-n}$ establishing a new class of Lichnerowicz type theorems. Furthermore, we adapt some of these results to ${\mathbb{R}}^{n-1,1}\times_w M^{10-n}$ backgrounds with fluxes by taking the AdS radius to infinity. We find that these backgrounds preserve $N=2^{[{n\over2}]} k$ for $2<n\leq 4$ and $N=2^{[{n+1\over2}]} k$ for $4<n\leq 7$ supersymmetries. We also demonstrate that the Killing spinors of $AdS_n\times_w M^{10-n}$ do not factorize into Killing spinors on $AdS_n$ and Killing spinors on $M^{10-n}$.