Supersymmetric dS _n solutions for n ⩾ 5in D = 11 supergravity

Abstract

We determine the necessary and sufficient conditions for warped product $dS_n$ solutions, $5 \leq n \leq 10$, to preserve supersymmetry in $D=11$ supergravity, without assuming factorization of the Killing spinors. We prove that for $7 \leq n \leq 10$, all such solutions are flat, with vanishing 4-form. We also show that the only warped product $dS_6$ solutions are either the maximally supersymmetric $AdS_7 \times S^4$ solution, or $\mathbb{R}^{1,6} \times N_4$ where $N_4$ is hyperK\"ahler, with vanishing 4-form. Supersymmetric warped product $dS_5$ solutions are then classified; it is shown that all such solutions are generalized M5-brane configurations, for which the transverse space is $\mathbb{R} \times N_4$, and $N_4$ is a hyperK\"ahler manifold. If the 4-form is covariantly constant, then $N_4$ admits a hyperK\"ahler potential.

D = 11 dS 5 backgrounds with enhanced supersymmetry

“…If the Hessian of the warp factor satisfies the condition mentioned above, then we prove that for a N = 16 solution, there must be eight σ + spinors, and also eight σ − spinors. The resulting conditions on the flux and geometry imply that such N = 16 solutions correspond to the class of solutions obtained in [15] for which the four-form is parallel. If, however, the Hessian of the warp factor does not satisfy the condition, then for a N = 16 solution, it follows that there must be 16 linearly independent σ + spinors on the hyper-Kähler manifold, which implies that this manifold must be R 4 .…”

“…In this paper, we shall classify the warped product dS 5 × w M 6 solutions in D = 11 supergravity which exhibit enhanced supersymmetry. In [15] it was shown that supersymmetric warped product dS 5 × w M 6 solutions must preserve N = 8k supersymmetries for k = 1, 2, 3, 4. Minimal supersymmetry therefore corresponds to N = 8 supersymmetry-these were fully classified in [15].…”

“…In [15] it was shown that supersymmetric warped product dS 5 × w M 6 solutions must preserve N = 8k supersymmetries for k = 1, 2, 3, 4. Minimal supersymmetry therefore corresponds to N = 8 supersymmetry-these were fully classified in [15]. Furthermore, it was also noted that the only possible N = 32 warped-product dS 5 × w M 6 solutions are R 1,10 with vanishing four-form, or the maximally supersymmetric AdS 7 × S 4 solution.…”

“…We find two classes of N = 16 dS 5 × w M 6 backgrounds. The first is a special class of solutions constructed in [15] for which the four-form is parallel; the geometry is a generalized M5-brane configuration with transverse space R × N 4 , where N 4 is a hyper-Kähler manifold and the M5-brane harmonic function is determined by a hyper-Kähler potential on N 4 . The second class is a stack of M5 branes with transverse space R 5 [17].…”

“…In order to construct the classification of all D = 11 dS 5 solutions with enhanced supersymmetry, we begin by taking the necessary conditions obtained for a supersymmetric solution to preserve (at least) the minimal N = 8 possible supersymmetry, which were obtained in [15]. Such conditions imply the existence of two preferred co-ordinates s and t in the internal manifold, and also impose certain conditions on the four-form flux, when written in terms of these co-ordinates.…”