Question about Green, Schwarz, Witten Appendix 4.A on N=1SYM

[petergreat]

It's really just a dumb question about spinors. The book says that for N=1 SYM, in 3 or 4D we use Majorana spinors. In 6D we use Weyl spinors. And in 10D we use Majorana-Weyl spinors, so that the number of fermionic states matches D-2.

My questions is, in 4D, Weyl spinors (chiral spinors) also have only 2 degrees of freedom. Why must we use Majorana spinors instead?

In which dimensions do Majorana spinors exist? And which dimensions allow Majorana-Weyl spinors? What are the gamma matrices in these representations? What's the best reference for these issues?

Thanks in advance.

[crackjack]

For a short review, I would suggest the appendix of the 2nd volume of Polchinski's book.

[Simon_Tyler]

Sorry for the late post.

It's got to do with the representation theory of Clifford Algebras and its modulo 8 (Bott) periodicity.
Some references are:
Clifford Algebras in Physics: http://arxiv.org/abs/hep-th/0506011
The Pin Groups in Physics: C, P, and T: http://arxiv.org/abs/math-ph/0012006
http://demonstrations.wolfram.com/TrigramsAndRealCliffordAlgebras/

The first one is a nice clear short discussion that should answer all of the questions you posed in your post. The second reference has some really interesting material by some of the people most knowledgeable about this stuff. The final reference is just some pretty pictures that I made.