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I'm reading about twistors from the book of Huggett and Tod: $\textit{ An introduction to twistor theory}$. I'm trying to understand everything and reproduce every equation that comes here. So, naturally, I came to a dead road to me.

In the book they state that

and I can't verify this. I only reach this:

$ \begin{eqnarray} \nabla^{A'(A}\beta^{B)}=\bar\varphi^{A'C'}\nabla^{(B}_{\hspace{.2cm}C'}\alpha^{A)}+{k\over 4}\nabla^{A'(A}\alpha^{B)}-\varphi^B_{\hspace{.2cm}C}\nabla^{A'(C}\alpha^{A)}-\varphi^A_{\hspace{.2cm}C}\nabla^{A'(C}\alpha^{B)}+{1\over 4}\alpha^{(A}\nabla^{B)A'}k+X^{CC'}\nabla^{A'(A}\nabla_{CC'}\alpha^{B)} \end{eqnarray}$

I really don't know how the last two terms are zero. This book is full of typos (for example, in equation 5.17, the last term it really is $-k\alpha^A/4$) so I was hopping that there was another typo... or maybe something about the properties of flat spaces that I'm not considering.

In advance I'm very grateful for any help.

P. S. $\alpha^A$ is any spinor, $k$ is such that $\mathcal L_{\textbf{X}}g=kg$, and naturally $\textbf X$ is a conformal Killing vector field.

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  • $\begingroup$ So I just solve my doubt. The first of the two last terms is a mistake, and the last is zero due to flat properties of the Minkowski space and the choose of an orthogonal basis for the set of vector fields, which translates in the fact that the covariant derivative commutes. $\endgroup$ – raul Oct 31 '15 at 23:37
  • $\begingroup$ and yes... that is still a typo. $\endgroup$ – raul Oct 31 '15 at 23:39

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