I was reading Kiritsis' notes (http://arxiv.org/abs/hep-th/9709062), at page 105/106 (equation 10.1), where he has a covariant action of the superstring including the gravitino. I have problems showing that this is invariant under the given supersymmetry. The action is $$S=\int \sqrt{g}\left(g^{ab}\partial_a X^\mu \partial_b X_\mu+\frac{i}{2}\psi^\mu\gamma^a\partial_a\psi_\mu+\frac{i}{2}(\chi_a\gamma^b\gamma^a\psi^\mu)\left(\partial_bX^\mu-\frac{i}{4}\chi_b\psi^\mu\right)\right)$$
and the supersymmetry transformation is supposed to be
$$\delta g_{ab}=i\epsilon(\gamma_a\chi_b+\gamma_b\chi_a)\\ \delta\chi_a=2\nabla_a\epsilon\\ \delta X^\mu=i\epsilon\psi^\mu\\ \delta\psi^\mu=\gamma^a\left(\partial_aX^\mu-\frac{i}{2}\chi_a\psi^\mu\right)\epsilon$$
But when doing the variation I fail. For instance, trying to locate all terms that only have one $X$ and one $\psi$, I do not get that these sum to zero. However, I am not sure I do it correctly. If I understand it correctly (which is not explained well in the paper) the gamma matrices satisfy $\{\gamma^a,\gamma^b\}=-2g^{ab}$ since the metric is not gauge fixed yet. But under the supersymmetry variation, the metric also changes. Does this mean that we need a variation $\delta \gamma=\ldots$? If so what is this variation of these Dirac matrices? (I tried something like $\gamma\sim \epsilon (\chi^a)^T$ but did not make it work with the anti commutation relation)
If someone has some other reference about the covariant action of the superstring and BRST quantization, let me know, many books seem to only deal with the gauge fixed version.
