Supersymmetry of AdS and flat IIB backgrounds

Abstract

We present a systematic description of all warped $Ad{S}_n{\times }_w{M}^{10-n}$ 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 $Ad{S}_n$ backgrounds preserve $N=2^{[\frac{n}{2}]}k$ for $n\le 4$ and $N=2^{[\frac{n}{2}]+1}k$ for $4<n\le 6$ 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 $Ad{S}_n$ 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 $N=2^{[\frac{n}{2}]}k$ for $2<n\le 4$ and $N=2^{[\frac{n+1}{2}]}k$ for $4<n\le 7$ supersymmetries. We also demonstrate that the Killing spinors of $Ad{S}_n{\times }_w{M}^{10-n}$ do not factorize into Killing spinors on $Ad{S}_n$ and Killing spinors on $M^{10-n}$.

arXiv comment46 pages, minor typos corrected

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