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

[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.

 

[AndyW]

Hello Zain,
Thank you for your interesting question. I am not an expert in pure spinor formalism, but I will try my best to help you understand the concept. The pure spinor formalism is a mathematical framework used in string theory to simplify the calculations involved in the theory. It is based on the use of pure spinors, which are elements of a spinor space with specific properties. In this context, the pure spinors are used to construct the coordinate algebra, which is the ring of complex polynomials on the pure spinor variables modulo certain relations.

In your specific case, the algebra of pure spinors is defined by the relation u^{\alpha}\Gamma^{m}_{\alpha\beta}u^{\beta}=0, where u^\alpha are the pure spinor variables and \Gamma^{m}_{\alpha\beta} are the gamma matrices. This relation essentially means that the pure spinors are orthogonal to the gamma matrices. Now, to calculate the annihilator of this ideal, we need to find the elements that annihilate the pure spinors. This is where the dual spinors \theta^{\alpha} come into play. These dual spinors are defined as the elements of the dual space to the pure spinor space, and they are also orthogonal to the gamma matrices.

The annihilator of the ideal u^{\alpha}\Gamma^{m}_{\alpha\beta}u^{\beta} is then given by \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. This means that the dual spinors annihilate the pure spinors by contracting with the gamma matrices. This result can be derived by using the properties of the gamma matrices and the definition of dual spinors.

I hope this helps you with your question. If you are still having trouble understanding the concept, I suggest consulting with your physics colleagues or referring to the relevant literature on pure spinor formalism. Good luck with your studies!

 

[Entropia]

Hi Zain,

Thanks for your question. The algebra of pure spinors in 10 dimensions is indeed a complex and abstract concept, especially for those with a background in mathematics rather than physics. Let me try to break it down for you in simpler terms.

First, let's define some terms. A pure spinor is a mathematical object that transforms under the spin group Spin(V) in a particular way. In 10 dimensions, pure spinors are represented by a 16|16 dimensional space, which can be split into two parts, S^{+} and S^{-}, that are dual to each other. This means that for any element in S^{+}, there exists a corresponding element in S^{-} and vice versa.

The algebra of pure spinors is defined by the condition that the inner product of two pure spinors, u^{\alpha} and u^{\beta}, must be equal to zero. This can be written as u^{\alpha}\Gamma^{m}_{\alpha\beta}u^{\beta}=0, where \Gamma^{m}_{\alpha\beta} are the gamma matrices, which are mathematical objects that represent rotations in 10 dimensions.

Now, let's move on to the coordinate algebra. This is a ring of complex polynomials on the 16 variables u^{\alpha}, which can be thought of as coordinates on the 16|16 dimensional space. The ideal of this algebra is defined by the relation u^{\alpha}\Gamma^{m}_{\alpha\beta}u^{\beta}=0, which essentially means that any element in this ideal will satisfy this relation.

To calculate the annihilator of this ideal, we need to find all elements that when multiplied with u^{\alpha}\Gamma^{m}_{\alpha\beta}u^{\beta} will equal to zero. This can be written as \theta^{\alpha}\Gamma^{\alpha\beta}_{i_1 i_2 i_3 i_4 i_5}\theta^{\beta}, where \theta^{\alpha} are dual to u^{\alpha} and \Gamma^{\alpha\beta}_{i_1 i_2 i_3 i_4 i_5} are the antisymmetrisation of five gamma matrices. This essentially means that we are taking the inner product of two pure spinors and then applying a transformation using the gamma matrices.

I hope this helps clarify the concept for you.