Hi.(adsbygoogle = window.adsbygoogle || []).push({});

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?

Thanks.

Zain.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**