What are Killing Spinor Equations?

Question

Killing spinor equations are equations that result from supersymmetric transformations. One example of those is for example

$$\nabla_{\mu}\epsilon+\frac{i}{2}A_{\mu}\gamma_*\epsilon+\frac{i}{4}Im\mathcal{N}_{AB}\gamma_{\mu\nu}G^{\mu\nu A}(Im L^{B}-i\gamma_* Re L^{B})\gamma_{\mu}\epsilon=0$$ This is the usual one present in $N=2, d=4$ Supergravity theories.

As suggested by some books and papers on the web, this is the vanishing of the gravitini supersymmetry.

My related questions are:

1- Why do we have to solve these equations known as killing spinor equations? In other words, how are their solutions important or beneficial in any way?
2- I understand that there in supergravity there is a graviton, 2 gravitini and a 'so-called' graviphoton. So if fermions here should vanish in $N=2$ supergravity theories (those coupled to vector multiplets), why don't we see vanishing of the graviton supersymmetry or gravi-photon supersymmetry but instead we see vanishing of gaugino supersymmetry? Where am I confused about here?

Answer 1

The underlying point of the question is: how to define a supersymmetric background ? At the classical level, this is done by requiring the vanishing of the supersymmetric variations of the various fields present in the theory. The supersymmetric variation of a bosonic field is a fermionic field but if we want a Lorentz invariant background, the background values of the fermionic fields are set to zero and so the supersymmetric variations of the bosonic fields always vanish in the background and the non-trivial constraints on the possibly non trivial background values of the bosonic fields come from the requirement of vanishing of the supersymmetric variations of the fermionic fields. The Killing spinor equation is the constraint on the background coming from the vanishing of the supersymmetric variation of the gravitino.

Comments

Thanks a lot for your interesting answers @40227 . I would like to ask you about the second part of my question. I restate it in more details here.

I understand that there in $N=2$ $D=4$ supergravity coupled to vector multiplets there is a graviton, 2 gravitini and a 'so-called' graviphoton. This can be found here http://arxiv.org/pdf/math/0002122.pdf where the author says: "As we neglect here the hypermultiplets, we have to consider the basic supergravity multiplet and the vector multiplets.The supergravity sector contains the graviton, 2 gravitini and a so-called graviphoton" So if fermions here should vanish in $N=2$ supergravity theories (those coupled to vector multiplets), we should see vanishing of gravitini supersymmetry only in order to give us the first Killing spinor equation.My question is that I see the presence of both: the vanishing gravitini and gaugini supersymmetry in some papers in $N=2, D=4$ supergravity coupled to vector muliplets. Where does the gaugini supersymmetry come from?

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As indicated in my answer, to obtain a supersymmetric background one needs the vanishing of the supersymmetric variation of all the fermionic fields. In particular in a supergravity theory coupled with vector multiplets one needs the vanishing of the supersymmetric variation of both the gravitini present in the gravity multiplet and of the gaugini present in the vector multiplets. The Killing spinor equation is the equation associated to the gravitini and there is another equation associated to the gaugini.

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Ah thanks @40227 I got it now. Your answers are always very helpful. So, the gravitini is present in the gravity multiplet and gaugini is present in the vector multiplet. What confused me is the fact that Van Proeyen did not mention this in the attached paper, or I did not read it clearly, or even worse, I might have not understood it. He said, "The supergravity sector contains the graviton, 2 gravitini and a so-called graviphoton." So, he implied that the supergravity sector is the one containing the gravitini. So, does this mean that the supergravity sector the same thing as the gravity multiplet? 

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Yes.