Abstract

We present all the symmetry superalgebras mathfrak{g} of all warped AdSk ×wMd − k, k > 2, flux backgrounds in d = 10, 11 dimensions preserving any number of supersymmetries. First we give the conditions for g to decompose into a direct sum of the isometry algebra of AdSk and that of the internal space Md − k. Assuming this decomposition, we identify all symmetry superalgebras of AdS3 backgrounds by showing that the isometry groups of internal spaces act transitively on spheres. We demonstrate that in type II and d = 11 theories the AdS3 symmetry superalgebras may not be simple and also present all symmetry superalgebras of heterotic AdS3 backgrounds. Furthermore, we explicitly give the symmetry superalgebras of AdSk, k > 3, backgrounds and prove that they are all classical.

Highlights

  • One way to find the symmetry superalgebra, g = g0 ⊕ g1, of a product AdSk × M d−k, k > 2, background in a supergravity theory is to assume that it is a classical superalgebra1 whose even subalgebra decomposes as g0 = so(k − 1, 2) ⊕ t0 and the dimension of the odd subspace g1 is the number of Killing spinors N, where so(k − 1, 2) is the Lie algebra of isometries of AdSk subspace

  • We present all the symmetry superalgebras g of all warped AdSk ×w M d−k, k > 2, flux backgrounds in d = 10, 11 dimensions preserving any number of supersymmetries

  • The conditions (2.9), (2.10) and (2.12) we have found on the Killing spinor bilinears in the previous section for g0 = so(k − 1, 2) ⊕ t0 can derived in an elegant way after imposing that the internal space Md−k is compact without boundary and the fields are smooth

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Summary

Introduction

In these computations, the geometry of the internal space M d−k is used in an essential way to determine all (anti-) commutators. The second ingredient in our proof is the closure of KSAs for superymmetric d = 11 and IIB backgrounds shown in [15, 16] We use this to demonstrate that in all cases the superJacobi identities and the explicit dependence of the Killing spinors on the AdS coordinates are sufficient to determine the commutator [t0, g1] from those of {g1, g1} and [so(k−1, 2), g1]. In appendix D, we present the construction of KSAs for AdS3 backgrounds with a low number of supersymmetries without the use of the results of [10]

Definition of KSAs
AdS Killing spinors
Global conditions for the decomposition of g0
Left and right superalgebra
Left or right superalgebra
Structure constants of KSAs
The AdS3 KSAs are direct sums of Left and Right KSAs
Structure theorems
Killing spinors
KSAs for heterotic backgrounds
Extended supersymmetry
Conclusions
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