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

  1. Oct 5, 2008 #1
    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 [tex]S=S^{+}\oplus S^{-}[/tex] where [tex]S^{\pm}[/tex] are dual to each other. The algebra of pure spinors in this case is given by [tex]u^{\alpha}\in S^{+}[/tex] such that [tex]u^{\alpha}\Gamma^{m}_{\alpha\beta}u^{\beta}=0[/tex].

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

    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?

  2. jcsd
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