# A simple question on the algebra of pure spinors in 10 dimensions

1. Oct 5, 2008

### zazzou

Hi.
I'm looking at Berkovits' pure spinor formulism of string theory. I am a PhD student studying mathematics and so am having trouble with some of the physics behind the mathematics.

Say we have V a 10 dimensional vector space and we pick an irreducible representation of Spin(V) - which is a 16|16 dimensional space $$S=S^{+}\oplus S^{-}$$ where $$S^{\pm}$$ are dual to each other. The algebra of pure spinors in this case is given by $$u^{\alpha}\in S^{+}$$ such that $$u^{\alpha}\Gamma^{m}_{\alpha\beta}u^{\beta}=0$$.

I am looking at the coordinate algebra that is defined by this: i.e. the ring of complex polynomials on the 16 variables $$u^\alpha$$ modulo the relation with the gamma matrices above. I want to calculate the annihilator of this ideal $$u^{\alpha}\Gamma^{m}_{\alpha\beta}u^{\beta}$$ which is supposed to be $$\theta^{\alpha}\Gamma^{\alpha\beta}_{i_1 i_2 i_3 i_4 i_5}\theta^{\beta}$$ where $$\Gamma^{\alpha\beta}_{i_1 i_2 i_3 i_4 i_5}$$ is the antisymmetrisation of five gamma matrices and $$\theta^{\alpha}$$ are dual to $$u^\alpha$$.

I cannot see how this is the case. Could anyone help me with this simple question. Essentially, I think that I'm not too confident in working with these gamma matrices?

Thanks.
Zain.

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