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By J. Gilard
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Extra resources for Spinorial Geometry and Supergravity [thesis]
Example text
19). However, these are not all independent, as we shall see shortly. ) The independent equations are the conditions that we seek to solve and interpret. Let us first consider the time component D0 η1 = 0. 34) Also, there are three more conditions, but they are simply the complex conjugate equations to the ones above, and therefore provide no additional independent constraints. e. as to whether i = α or i = α ¯ . However, we find that the D = 11 supercovariant derivative is real, in the sense that the holomorphic and antiholomorphic components are related by complex conjugation and dualisation with respect to the antiholomorphic spinor basis.
4) so that the dimension of the space of metric deformations is the Hodge number h1,1 . 4). The machinery developed in this chapter can also be applied to known compactification solutions for the heterotic string, to determine the allowable α′ corrections to the backgrounds. For instance, in [35] the deformations were computed for the conifold and the U (n)-invariant Calabi-Yau metric. C HAPTER 4 S PINORIAL G EOMETRY IN D = 11 S UPERGRAVITY In this chapter, we describe a systematic method for solving the Killing spinor equations of supergravity, which was first proposed in [64].
4) at zeroth order in ′ α. 4), and collect the terms linear in α′ . However, let us first define the Lichnerowicz operator ∆L , which arises naturally in this calculation. For any Riemannian manifold (M,˚ g) ˚ we have with associated Levi-Civita connection ∇, Rij (˚ g + ǫh) = Rij + ǫ∆L hij + O(ǫ2 ) . 1) In other words, ∆L is the first-order deformation of the Ricci tensor under a small perturbation of the metric. One can show that ∆L hij = − − 1 ˚2 ˚ikjl hkl + 1 ∇ ˚i∇ ˚j ∇ ˚ k hkj + 1 ∇ ˚ k hki ∇ hij − R 2 2 2 1˚ ˚ k 1˚ k 1˚ k ∇i ∇j h k + R ki h j + Rkj h i .