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In chapter $7$ of introduction to AdS/CFT correspondence Nastase states following:

... we generalize the first order action for the particle. The generalisation is done by introducing objects $\theta^A,A=1,2,...,N$ which are $N$ spacetime spinors and worldsheet scalars, i.e. $\theta^{A,\alpha}$, $\color{red}{\textrm{with}\ \alpha\ \textrm{a spacetime spinor index in}}$ ten dimension.

My confusion is with the highlighted text why do we need $\alpha$ indices? Spinors are represented by column vector and they are acted upon by $\Gamma$ matrices so I don't see from where these indices are popping up. If these indices were for worldsheet I could have assumed that we have done some sort of "pullback" on the spinors living in background spacetime to the worldsheet. But since these spinors are scalar that wouldn't have made any sense as well.

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  • $\begingroup$ Are these spinors of spin $\frac12$ or $\frac32$? The latter famously needs a spacetime index. $\endgroup$
    – J.G.
    Dec 24, 2021 at 8:51

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