I am trying to understand spinors in 5+1 dimensions but I have troubles when going through dimensional reduction to 4+1 dimensions. The dimension of six dimensional Weyl representation is 4 and the algebra is $SO(1,5) \approx SU^*(4)$. I am using pseudoMajorana-Weyl spinors so that an extra index is present and the pseudo-Majorana condition is given by $\overline{\Psi_i^A} = \Psi ^{Ai}$ where $\bar \cdot$ is some conjugation squaring to $-1$.

I understand that the supersymmetry algebra is given by $$ \{Q_{A i} ,Q_{B i} \} = 2 \epsilon_{ij} \Sigma^M_{AB} P_M $$ where $\Sigma^M$ are Weyl matrices, $P$ is momentum and indices run $M=0,1,2,3,4,5$ and $A=1,2,3,4$. . $\Sigma^M_{AB}$ matrices are antisymmetric in their $AB$ indices.

I have problems when trying to dimensionally reduce a spinor action like $$ S=\int i \Psi^i \ \Sigma^M \ \partial_M \Psi_i = \int i \Psi^{Ai} \ \Sigma^M_{AB} \ \partial_M \Psi^B_i . $$ Take the conjugation to be $\Psi^{A} =(\psi^\alpha , \;\bar \lambda^{\dot \alpha}) \rightarrow \overline{\Psi^A} = (- \lambda^{ \alpha}, \;\bar \psi^{\dot \alpha} ) $.

When dimensionally reducing the algebra from SU*(4) to SO(1,3) seems to be convenient to split the spinor index $A$ into a couple $(\alpha, \dot \alpha)$ so that $\Psi^{1A} =(\psi^\alpha , \;\bar \lambda^{\dot \alpha}) $ where $\psi$ and $\lambda$ are SL(2,C) spinors.

It seems also to be convenient to choose the Weyl matrices so that $$ \Sigma^\mu_{\alpha \dot \beta} = \sigma^\mu_{\alpha \dot \beta}; \qquad \Sigma^\mu_{\alpha \beta} = 0 = \Sigma^\mu_{\dot\alpha \dot \beta} $$ being as usual $\sigma^\mu = (1,\sigma^i)$. ($\Sigma^4$ and $\Sigma^5$ are not a problem and however do not contribute since $\partial_5=0=\partial_4$).

This is reasonable since it gives a term like $-\psi \sigma^\mu \partial_\mu \bar\psi$ for the first spinor, but I do not understand how this works for the second one: by antisymmetry of $\Sigma$ it should be $$ \Sigma^\mu_{\dot \alpha \beta} = -\Sigma^\mu_{\beta \dot \alpha } = - \sigma^\mu_{\beta \dot \alpha } $$ that has unpleasant indices comparing to the expected index structure for $\bar \lambda {\bar \sigma}^\mu \lambda$, where $ {\bar\sigma^\mu}^{\dot \alpha \beta} = (1,-\sigma^i)$ ; I have tired to write $\Sigma^\mu$ explicitly but I do not quite manage to get a nice expression like $-{\bar\sigma}^\mu$ for the lower-left block; I guess I am missing something on the notation.

ADDED Ok it feels a bit stupid to add this so close after I sent the question, but reviewing Ramond's great Field theory I realized it could be all related to the identity $$ \sigma^2 \sigma^i \sigma^2 = - {\sigma^i }^* = - { \sigma^i }^T $$ that would ideed provide the desired bars and so on, but I am not quite able to write down in an explicit and elegant way the indices.


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