has a clear (possibly warped) product structure, we can distinguish internal and spacetime (super-) symmetries and, if we choose this vacuum, our choice implies spontaneous compactification.

Tod’s problem
=============

This is the problem of finding *all* the supersymmetric bosonic field configurations, i.e. all the bosonic field configurations $\phi^{b}$ for which a SUGRA’s Killing spinor equations

$$\left.
\delta_{\epsilon}\phi^{f}
\right|_{\phi^{f}=0}\sim 
\partial \epsilon+\phi^{b}
\epsilon =0\, ,$$

have a solution $\epsilon$, which includes all the possible supersymmetric vacua and compactifications.

Observe that, as we announced, not all supersymmetric bosonic field configurations satisfy the classical bosonic equations of motion for which we use the notations $\left.\frac{\delta S}{\delta \phi^{b}}\right|_{\phi^{f}=0}
\equiv \left.  S_{,b}\right|_{\phi^{f}=0} \equiv \mathcal{E}(\phi^{b})$. Actually, the bosonic equations of motion of supersymmetric bosonic field configurations satisfy the so-called [*Killing spinor identities*]{} (KSIs) [@Kallosh:1993wx; @Bellorin:2005hy]that relate different equations of motion of a supersymmetric theory. These identities can be derived as follows: The supersymmetry invariance of the action implies, for arbitrary local supersymmetry parameters $\epsilon$

$$\delta_{\epsilon}S
=
\int d^{d}x\, (
S_{,b}\, 
\delta_{\epsilon}\phi^{b}   
+ S_{,f}\, 
\delta_{\epsilon}\phi^{f}
)
=0\, .$$

Taking the functional derivative w.r.t. the fermions and setting them to zero

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