,$$

The terms $\left.\delta_{\epsilon}\phi^{b} \right|_{\phi^{f}=0}\, ,\, \left.
  S_{,f} \right|_{\phi^{f}=0}\, ,\, \left.  (\delta_{\epsilon}\phi^{f}
  )_{,f_{1}} \right|_{\phi^{f}=0} $ vanish automatically because they are odd in fermion fields $\phi^{f}$ and so we are left with

$$\left.\left\{S_{,b}\, 
(\delta_{\epsilon}
\phi^{b})_{,f_{1}}
+S_{,ff_{1}}\, 
\delta_{\epsilon}\phi^{f}
\right\}\right|_{\phi^{f}=0}=0\, .$$

This is valid for any fields $\phi^{b}$ and any supersymmetry parameter $\epsilon$. For a supersymmetric field configuration $\epsilon$ is a Killing spinor $\left.  \delta_{\epsilon}\phi^{f} \right|_{\phi^{f}=0}$ and we obtain the KSIs

$$\label{eq:ksis}
\mathcal{E}(\phi^{b})
\left. (\delta_{\epsilon}
\phi^{b})_{,f_{1}}
\right|_{\phi^{f}=0}
=0\, .$$

These non-trivial identities are linear relations between the bosonic equations of motion and can be used to solve Tod’s problem, obtain BPS bounds etc. Let’s see some examples.

Example: $N=1,d=4$ Supergravity.
--------------------------------

This is the simplest supergravity theory. Its field content is $\{e^{a}{}_{\mu},\psi_{\mu}\}$. The bosonic action (Einstein-Hilbert’s) and the equations of motion (Einstein’s) are

$$\left. S \right|_{\psi_{\mu}=0}
=\int d^{4}x \sqrt{|g|}\, 
R\, , \,\,\, \Rightarrow\,\,\,
\mathcal{E}_{a}{}^{\mu}({e}) \sim G_{a}{}^{\mu}\, .$$

The supersymmetry transformations of the graviton and gravitino are

$$\delta_{\epsilon} e^{a}{}_{\mu} =
-i\bar{\epsilon}\gamma^{a}\psi_{\mu}\, ,
\hspace{1cm}
\delta_{\epsilon} \psi_{\mu} =
\nabla_{\mu}\epsilon
=\partial_{\mu}\epsilon 
-{\textstyle\frac{1}{4}}\omega_{\mu}{}^{ab}
\gamma_{ab}\epsilon\, .$$

The KSIs can be readily computed from the general formula Eq. (\[eq:ksis\]) and simplified

$$-i\bar{\epsilon} \gamma^{a}  
G_{a}{}^{\mu}=0\, ,\,\,\,\, 
\Rightarrow 
R=0\, ,\,\,\,
-i\bar{\epsilon} \gamma^{a}  
R_{a}{}^{\mu}=0\, .$$

On the other hand, in trying to solve 