The question is about the spinor representation decomposed under subgroups. It's a common technique in string theory when parts of dimensions are compactified and ignored, and we are only interested in the remaining sub-symmetry. I'm learning it from the appendix B in Polchinski's big book volume II. For a particular decomposition, SO(2k+1,1) → SO(2l+1,1) × SO(2k-2l) (B.1.43), the Weyl spinors decompose as the formula (B.1.44), 2(adsbygoogle = window.adsbygoogle || []).push({}); ^{k}→ (2^{l}, 2^{k-l-1})+(2'^{l},2'^{k-l-1}) and 2'^{k}→ (2'^{l}, 2^{k-l-1})+(2^{l},2'^{k-l-1}), where 2^{k}and 2'^{k}are the Weyl representations of Lorentz group SO(2k+1,1) with chirality +1 and -1 respectively.

Specifically on the case SO(9,1)→ SO(5,1)× SO(4) with decomposetions 16 → (4,2)+(4',2'), which appears at (B.6.3). My question is **the contradiction with minimum representations**. By checking Majorana and Weyl conditions, the minimum spinors for d=6 and d=4 have 8 and 4 components respectively. (Ref. table B.1 Polchinski) So How can you find (4,2) representation for SO(5,1)× SO(4)?

Also, I'm very interested in the prove of (B.1.44) under (B.1.43)? How is it proved by comparing the eigenvalues of $\Gamma^{+}\Gamma^{-}-\frac{1}{2}$ as claimed by Polchinski?

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# Spinor representations decomposed under subgroups in Joe's big book

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