Supersymmetry of AdS and flat IIB backgrounds

Abstract

We present a systematic description of all warped AdS n × w M 10−n and R n−1,1 × 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 ]+1 k for 4 < n ≤ 6 supersymmetries and for k ∈ 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 R n−1,1 × w M 10−n backgrounds with fluxes by taking the AdS radius to infinity. We find that these backgrounds preserve N = 2 ] k for 2 < n ≤ 4 and N = 2 ] k for 4 < n ≤ 7supersymmetries. We also demonstrate that the Killing spinors of AdS n × w M 10−n do not factorize into Killing spinors on AdS n and Killing spinors on M 10−n .

Supersymmetry of IIA warped flux AdS and flat backgrounds

“…This result is similar to that already established for the D=11 and IIB backgrounds in [12,13] where it was shown that such a factorization leads to an incorrect counting of Killing spinors. We also examine the factorization of Killing spinors for R n−1,1 × w M 10−n backgrounds.…”

“…A first step towards the classification of AdS backgrounds is to identify the fractions of supersymmetry that are preserved. This has been established for D=11 and IIB AdS backgrounds in [12] and [13], respectively. The novelty of this approach is that no restrictions have been put on either the form of the fields or that of the Killing spinors.…”

Geometry and supersymmetry of heterotic warped flux AdS backgrounds

“…This generalizes the theorem we have shown for dH = 0 configurations in [23] and [24]. We find that after choosing an appropriate Dirac operator that couples to fluxes on M 7 , all the zero modes of this Dirac operator are Killing spinors of the gravitino, dilatino and, in the perturbation theory, the gaugino KSEs.…”

“…Despite many developments, the geometry of all such backgrounds has not been specified and their classification remains an open problem, for some selected publications see [5]- [21]. Only recently, the fractions of supersymmetry preserved by flux, warped AdS backgrounds of 11-dimensional and type II supergravities have been specified in [22][23][24] without making any assumptions on the form of the fields and that of the Killing spinors apart from imposing the symmetries of AdS space on the former. The computation of these fractions has also uncovered a new class of Lichnerowicz type theorems.…”

“…The only differences are that B 4 must be a hyper-Kähler manifold with respect to the ds 2 (B 4 ) metric, and that F sd = 0. The metric and 3-form G of M 7 are given as in (2.35) but now we have that∇ [22][23][24]. However, there are some differences.…”