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. 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, and a nondegenerate complex scalar product. The Clifford algebra is the quotient of the full tensor algebra by the ideal generated by the relations

Spinors are modules 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, the space of projective pure spinors is the homogeneous space.

Irreducible Clifford module, spinors, pure spinors and the Cartan map

The irreducible Clifford/spinor module

Following Cartan and Chevalley, as a direct sum

is its dual space, with scalar product defined as

or

respectively.

, which act as

, and

.

Pure spinors and the Cartan map

A pure spinor that is annihilated by a maximal isotropic subspace. 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.

. The Cartan map

, to have value

-dimensional subspace, due to the isotropy conditions,

which imply

, then

.

in the direct sum decomposition

.

The Cartan relations

), by

and volume form,

As shown by Cartan, 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.

, under the Cartan map 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,

, corresponding to these skew symmetric elements of the Clifford algebra. However, since the dimension of the Grassmannian of maximal isotropic subspaces, and the Cartan map is of even dimension and in the irreducible spinor module if it is of odd dimension, the number of independent quadratic constraints is only

dimensional case, and

dimensional case.

In 6dimensions or fewer, all spinors are pure. In 7or 8 dimensions, there is a single pure spinor constraint. In 10dimensions, there are 10constraints

are the Gamma matrices (complex) dimensional.

Applications of pure spinors

Supersymmetric Yang Mills theory

supersymmetric Yang-Mills theory, the super-ambitwistor correspondence, consists of an equivalence between the supersymmetric field equations and the vanishing of supercurvature along super null lines.

String theory and generalized Calabi-Yau manifolds

Pure spinors were introduced in string quantization by Nathan Berkovits. Nigel Hitchin introduced generalized Calabi–Yau manifolds, where the generalized complex structure is defined by a pure spinor. These spaces describe the geometry of flux compactificationss in string theory.

Integrable systems

In the approach to integrable hierarchies developed by Mikio Sato, and his students, 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. They therefore serve as moduli for solutions of the BKP integrable hierarchy, parametrizing the associated BKP \tau-functions, which are generating functions for the flows. Under the Cartan map correspondence, these may be expressed as infinite dimensional Fredholm Pfaffians.

References

Bibliography

• Charlton, Philip. The geometry of pure spinors, with applications, PhD thesis (1997).