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Concept
Pure spinor
Summary
In the domain of mathematics known as representation theory, pure spinors (or simple spinors) are spinors that are annihilated under the Clifford action by a maximal isotropic subspace of the space
of vectors with respect to the scalar product determining the Clifford algebra.
They were introduced by Élie Cartan
in the 1930s to classify complex structures.
Pure spinors were a key ingredient in the study of spin geometry and twistor theory,
introduced by Roger Penrose in the 1960s.
Consider a complex vector space with either even complex dimension
or odd complex dimension and a nondegenerate complex scalar product
with values on pairs of vectors .
The Clifford algebra is the quotient of the full tensor algebra
on by the ideal generated by the relations
Spinors are modules of the Clifford algebra, and so in particular there is an action of the
elements of on the space of spinors. The complex subspace that annihilates
a given nonzero spinor has dimension . If then is said to be a pure spinor.
Every pure spinor is annihilated by a maximal isotropic subspace of with respect to the scalar product . Conversely, given a maximal isotropic subspace it is possible to determine the pure spinor that it annihilates it up to multiplication by a complex number. Pure spinors defined up to projectivization are called projective pure spinors. For of dimension , the space of projective pure spinors is the homogeneous space
As shown by Cartan, pure spinors are uniquely determined by the fact that they satisfy a set of homogeneous quadratic equations on the standard irreducible spinor module, the Cartan relations, which determine the image of maximal isotropic subspaces of the vector space under the Cartan map. In 7 dimensions, or fewer, all spinors are pure. In 8 dimensions there is a single pure spinor constraint. In 10 dimensions, there are 10 constraints
where are the Gamma matrices that represent the vectors
in that generate the Clifford algebra.
Official source
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