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3), but that h′ij = hij + ∇ ′ vector v. We can show that such a vector exists in the following way. 2), then v is determined by ˚i∇ ˚ i hj j . 4) ∇ 2 4 The left-hand side is an elliptic operator acting on v, and can be inverted if its kernel is orthogonal to the right-hand side of the equation [46], [22]. Now, suppose that X3 is an irreducible Calabi-Yau manifold and that w is in the kernel of the operator. Then, we have 0= ˚i∇ ˚ k wk dvol = − ˚k∇ ˚ k wi + 1 ∇ wi ∇ 2 X3 X3 ˚ k wi ∇ ˚ k wi + 1 (∇ ˚ k wk )2 dvol .

45) gives the condition ¯ Ω0,0γ = Ωα,βγ gαβ − Ωγ,β β . 51) This is another constraint on the geometry of spacetime which will be investigated shortly. 40), we have Gαβγ = −2iΩα,βγ + 2igα[β| ¯ ¯ ¯ Ω0,0|γ] . 37) imply that Ω0,α¯ β¯ = Ωα,0 ¯ β¯ . 53) This is a geometric constraint which comes from the torsion-free condition of the Levi-Civita connection, which will be considered in the next subsection. 39). Together they imply the following two constraints: Gα¯ 1 α¯ 2 α¯ 3 Fαβ ¯ 1 β2 β3 = 6iΩ[α¯ 1 ,α¯ 2 α¯ 3 ] 1 γ ¯1 γ ¯2 Ωα,¯ = ¯ γ1 γ ¯2 + 3Ω[α,¯ β1 β2 β3 + 6iΩ[β1 |,0|β2 gβ3 ]α ¯ γ1 γ ¯2 ] ǫ ¯ .

In other words, we may simplify the spinor without making the Killing spinor equations any more complicated. The Spin(1, 10)-invariance of D means that any two spinors lying in the same orbit of Spin(1, 10) yield identical solutions to the Killing spinor equations, up to local Lorentz transformations of the fields. This means that we can use the canonical form for the orbit to make the solution of the Killing spinor equations relatively straightforward. 5. We may expand the Killing spinor equations in terms of this basis and after some straightforward computation, the equations reduce to a set of algebraic and differential constraints on the fields of the theory.

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