B Cone Construction and Supersymmetry

It is well known that Sd and AdSd can be described as surfaces embedded in ℝd+1 and ℝ2,d−1. What is less known, especially in the physics literature, is that geometric Killing spinors on the latter are induced from parallel spinors on the former. This was proven by Bär [49] in the Riemannian case and by Kath [336] in the pseudo-Riemannian case. In this appendix, I briefly review this result.

Consider a Riemannian spin manifold (M, g ) having geometric Killing spinors ψ satisfying the differential equation

𝜖 ∇m ψ = − ---Γ m ψ. (555 ) 2R
R is related to the curvature of the manifold and 𝜖 is a sign, to be spelled out below. From a physics point of view, the right-hand side of this equation is the remnant of the gravitino supersymmetry transformation in the presence of non-trivial fluxes proportional to the volume form of the manifold (M, g ). Mathematically, it is a rather natural extension of the notion of covariantly constant Killing spinors. The statement that the manifold (M, g) allows an embedding in a higher-dimensional Riemannian space ^M corresponds, metrically, to considering the metric of a cone ^g in ^M with base space M. Thus,
^ + 2 ( r )2 M = ℝ × M and ^g = dr + R- g , (556 )
where R > 0 is the radius of curvature of (M, g). There exists a similar construction in the pseudo-Riemannian case in which the cone is now along a timelike direction. In the following, I will distinguish two different cases, though part of the analysis will be done simultaneously:

To establish an explicit map between Killing spinors in both manifolds, one needs to relate their spin connections. To do so, consider a local coframe 𝜃i for (M, g ) and ^𝜃a for (M^,g^), defined as

^r ^i -r i 𝜃 = dr and 𝜃 = R 𝜃 . (557 )
The connection coefficients i ω j and a ^ω b satisfy the corresponding structure equations
d𝜃i + ωij ∧ 𝜃j = 0 and d ^𝜃a + ω^ab ∧ ^𝜃b = 0 . (558 )
Given the relation between coframes, the connections are related as
ω^ij = ωij and ^ωir = 1-^𝜃i = 1𝜃i . (559 ) r R
Let ^∇ denote the spin connection on (M^ ,^g):
^∇ = d + 1^ωabΓ^ , (560 ) 4 ab
where ^γ a are the gamma-matrices for the relevant Clifford algebra. Plugging in the expression for the connection coefficients for the cone, one finds
1 ij 1 i ^∇ = d + --ω ^Γ ij +--𝜃 ^Γ ir. (561 ) 4 2R
To continue we have to discuss the embedding of Clifford algebras in order to recognise the above connection intrinsically on (M, g). This requires distinguishing two cases, according to the signature of (M, g ).

 B.1 (M, g) Riemannian
 B.2 (M, g) of signature (1,d − 1)

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