Pure spinor

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 [1] 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 1960's.

Definition

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.

Projective pure spinors

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 or less dimensions 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. It was shown by Cartan that there are, in general,

quadratic relations, signifying the vanishing of the quadratic forms with values in the exterior spaces for

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

in the dimensional case and

in the dimensional one.

Pure spinors in string theory

Pure spinors were introduced in string quantization by Nathan Berkovits.[2] 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 compactifications in string theory.

References

  1. Cartan, Élie (1981) [1938], The theory of spinors, New York: Dover Publications, ISBN 978-0-486-64070-9, MR 0631850
  2. Berkovits, Nathan (2000). "Super-Poincare Covariant Quantization of the Superstring". Journal of High Energy Physics. 2000 (4): 18–18. doi:10.1088/1126-6708/2000/04/018.
  • Cartan, Élie. Lecons sur la Theorie des Spineurs, Paris, Hermann (1937).
  • Chevalley, Claude. The algebraic theory of spinors and Clifford Algebras. Collected Works. Springer Verlag (1996).
  • Charlton, Philip. The geometry of pure spinors, with applications, PhD thesis (1997).
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.