M-theory on Complex P bundles and Calabi–Yau Relatives
نویسنده
چکیده
These are written up notes of the talk I gave at Simons Workshop 2004, based on preprints hep-th/0402153,0403002,0403038 co-authored by J. Gauntlett, J. Sparks, and D. Waldram. I review the construction of some supersymmetric solutions of 11d supergravity of the type AdS5×M6 where M6 are complex P bundles over Kähler four-manifolds, closely resembling twistor spaces. Then I discuss the dualization of some of these solutions yielding new Sasaki–Einstein metrics on S × S. In addition, I briefly review basic facts about Sasaki–Einstein geometry and discuss general features of the field theory duals of these geometries. 1 Supersymmetric AdS5 ×M6 solutions Our first goal is to construct supersymmetric solutions of eleven dimensional supergravity which contain a (warped) AdS5 factor in the metric. The strategy will be to consider the most general ansatz for the four-form flux G and Killing spinor, compatible with the AdS5 symmetry. Recall that the bosonic fields of 11d supergravity are a metric gMN and a fourform GMNPQ. A supersymmetric solution of this theory is a configuration obeying the condition δψM = ∇̂Mη − 1 288 ( GNPQRΓ̂ NPQR M − 8GMNPQΓ̂ ) η = 0 (1.1) which sets to zero the variation of the gravitino field, the G equation of motion and Bianchi identity d ∗̂G + 1 2 G ∧G = 0 dG = 0 (1.2) and the Einstein equations. Our metric ansatz is the following dŝ11 = e [ds(AdS5) + ds (M6)] (1.3) where the warp-factor λ is a function on M6. The G field has arbitrary components (to be determined) along M6, while the spinorial supersymmetry parameter is of the form η = ψ ⊗ ξ (1.4) where, crucially, ξ is a non-chiral spinor on M6 and ψ is a Killing spinor in AdS5, namely it obeys ∇μψ = i 2mγμψ . (1.5) Of course a non-chiral spinor in six dimensions can be always decomposed in its chiral components, which are irreducible representations of Spin(6)
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تاریخ انتشار 2004