Killing spinor equations are equations that result from supersymmetric transformations. One example of those is for example
∇μϵ+i2Aμγ∗ϵ+i4ImNABγμνGμνA(ImLB−iγ∗ReLB)γμϵ=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?