In the domain of mathematics known as representation theory, pure spinors (or simple spinors) are spinors that are annihilated, under the Clifford algebra representation, by a maximal isotropic subspace of a vector space with respect to a scalar product . They were introduced by Élie Cartan in the 1930s and further developed by Claude Chevalley.
They are a key ingredient in the study of spin structures and higher dimensional generalizations of twistor theory, introduced by Roger Penrose in the 1960s. They have been applied to the study of supersymmetric Yang-Mills theory in 10D, superstrings, generalized complex structures and parametrizing solutions of integrable hierarchies.
Clifford algebra and pure spinors
Consider a complex vector space , with either even dimension or odd 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. In terms of stratification of spinor modules by orbits of the spin group , pure spinors correspond to the smallest orbits, which are the Shilov boundary of the stratification by the orbit types of the spinor representation on the irreducible spinor (or half-spinor) modules.
Pure spinors, defined up to projectivization, are called projective pure spinors. For of even dimension , the space of projective pure spinors is the homogeneous space ; for of odd dimension , it is .
Irreducible Clifford module, spinors, pure spinors and the Cartan map
The irreducible Clifford/spinor module
Following Cartan and Chevalley, we may view as a direct sum
where is a totally isotropic subspace of dimension , and is its dual space, with scalar product defined as
or
respectively.
The Clifford algebra representation as endomorphisms of the irreducible Clifford/spinor module , is generated by the linear elements , which act as
for either or , and
for , when is homogeneous of degree .
Pure spinors and the Cartan map
A pure spinor is defined to be any element that is annihilated by a maximal isotropic subspace with respect to the scalar product . Conversely, given a maximal isotropic subspace it is possible to determine the pure spinor that annihilates it, up to multiplication by a complex number, as follows.
Denote the Grassmannian of maximal isotropic (-dimensional) subspaces of as . The Cartan map
is defined, for any element , with basis , to have value
i.e. the image of under the endomorphism formed from taking the product of the Clifford representation endomorphisms , which is independent of the choice of basis . This is a -dimensional subspace, due to the isotropy conditions,
which imply
and hence defines an element of the projectivization of the irreducible Clifford module . It follows from the isotropy conditions that, if the projective class of a spinor is in the image and , then
So any spinor with is annihilated, under the Clifford representation, by all elements of . Conversely, if is annihilated by for all , then .
If is even dimensional, there are two connected components in the isotropic Grassmannian , which get mapped, under , into the two half-spinor subspaces in the direct sum decomposition
where and consist, respectively, of the even and odd degree elements of .
The Cartan relations
Define a set of bilinear forms on the spinor module , with values in (which are isomorphic via the scalar product ), by
where, for homogeneous elements , and volume form on ,
As shown by Cartan, pure spinors are uniquely determined by the fact that they satisfy the following set of homogeneous quadratic equations, known as the Cartan relations:
on the standard irreducible spinor module.
These determine the image of the submanifold of maximal isotropic subspaces of the vector space with respect to the scalar product , under the Cartan map, which defines an embedding of the Grassmannian of isotropic subspaces of in the projectivization of the spinor module (or half-spinor module, in the even dimensional case), realizing these as projective varieties.
There are therefore, in total,