of an even number of fermionic fields) and $\phi^{f}$ for the fermionic fields (or products of an odd number of fermionic fields) and $\epsilon$ are the infinitesimal, local, parameters of the supersymmetry transformations, which are fermionic.

Then, a bosonic configuration (i.e. a configuration with vanishing fermionic fields $\phi^{f}=0$) will be invariant under the infinitesimal supersymmetry transformation generated by the parameter $\epsilon^{\alpha}(x)$ if it satisfies the [*Killing spinor equations*]{} (one equation for each $\phi^{f}$), which have the generic form

$$\label{eq:genericKillingspinorequation}
  \delta_{\epsilon} \phi^{f} \sim \partial \epsilon +\phi^{b} \epsilon=0\, .$$

The concept of unbroken supersymmetry is a generalization of the concept of isometry, an infinitesimal general coordinate transformation generated by $\xi^{\mu}(x)$ that leaves the metric $g_{\mu\nu}$ invariant because it satisfies the [*Killing (vector) equation*]{}

$$\label{eq:Killingvectorequation}
  \delta_{\xi} g_{\mu\nu}= 2\nabla_{(\mu}\xi_{\nu)}=0\, .$$

As it is well known, in this case, to each bosonic symmetry we associate a generator

$$\xi_{(I)}^{\mu}(x)\rightarrow P_{I}\, ,$$

of a symmetry algebra

$$[P_{I},P_{J}]  =f_{IJ}{}^{K}  P_{K}\, ,\,\,\,
\Leftrightarrow [\xi_{(I)},\xi_{(J)}]  =f_{IJ}{}^{K}  \xi_{(K)}\, ,$$

where the brackets in the right are Lie brackets of vector fields.

In our case, the unbroken supersymmetries are associated to the odd generators

$$\epsilon_{(n)}^{\alpha}(x)\rightarrow\mathcal{Q}_{n}\, ,$$

of a superalgebra

$$[\mathcal{Q}_{n},P_{I}]  =f_{nI}{}^{m}\mathcal{Q}_{m}  \, ,
\hspace{1cm}
\{\mathcal{Q}_{n},\mathcal{Q}_{m}\}  =f_{nm}{}^{I}P_{I}\, .$$

The calculation of these commutators 