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
They were introduced by Élie Cartan[1] in the 1930s and further developed by Claude Chevalley.
[2] They are a key ingredient in the study of spin structures and higher dimensional generalizations of twistor theory,[3] introduced by Roger Penrose in the 1960s.
They have been applied to the study of supersymmetric Yang-Mills theory in 10D,[4][5] superstrings,[6] generalized complex structures[7] [8] and parametrizing solutions of integrable hierarchies.
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
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.
is its dual space, with scalar product defined as or respectively.
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.
-dimensional subspace, due to the isotropy conditions, which imply and hence
is even dimensional, there are two connected components in the isotropic Grassmannian
consist, respectively, of the even and odd degree elements of
are uniquely determined by the fact that they satisfy the following set of homogeneous quadratic equations, known as the Cartan relations:[1][12][13] on the standard irreducible spinor module.
These determine the image of the submanifold of maximal isotropic subspaces of the vector space
, under the Cartan map, which defines an embedding of the Grassmannian of isotropic subspaces of
There are therefore, in total, Cartan relations, signifying the vanishing of the bilinear forms
, corresponding to these skew symmetric elements of the Clifford algebra.
However, since the dimension of the Grassmannian of maximal isotropic subspaces of
, and the Cartan map is an embedding of the connected components of this in the projectivization of the half-spinor modules when
is of even dimension and in the irreducible spinor module if it is of odd dimension, the number of independent quadratic constraints is only in the
In 7 or 8 dimensions, there is a single pure spinor constraint.
of these are independent, so the variety of projectivized pure spinors for
supersymmetric Yang-Mills theory, the super-ambitwistor correspondence,[4][5] consists of an equivalence between the supersymmetric field equations and the vanishing of supercurvature along super null lines, which are of dimension
Grassmannian dimensions correspond to a pure spinor.
Pure spinors were introduced in string quantization by Nathan Berkovits.
[6] Nigel Hitchin[14] introduced generalized Calabi–Yau manifolds, where the generalized complex structure is defined by a pure spinor.
These spaces describe the geometry of flux compactifications in string theory.
In the approach to integrable hierarchies developed by Mikio Sato,[15] and his students,[16][17] equations of the hierarchy are viewed as compatibility conditions for commuting flows on an infinite dimensional Grassmannian.
Under the (infinite dimensional) Cartan map, projective pure spinors are equivalent to elements of the infinite dimensional Grassmannian consisting of maximal isotropic subspaces of a Hilbert space under a suitably defined complex scalar product.
Under the Cartan map correspondence, these may be expressed as infinite dimensional Fredholm Pfaffians.