Pure spinor

In a field of mathematics known as representation theory pure spinors are spinor representations of the special orthogonal group that are annihilated by the largest possible subspace of the Clifford algebra. They were introduced by Elie Cartan in the 1930's to classify complex structures. Pure spinors were introduced into the realm of theoretical physics, and elevated in their importance in the study of spin geometry more generally, by Roger Penrose in the 1960's, where they became among the basic objects of study in twistor theory.

Definition

Consider a complex vector space C^2"n" with even complex dimension "2n" and a quadratic form "Q", which maps a vector "v" to complex number "Q(v)". The Clifford algebra Cliff_2n is the ring generated by products of vectors in C^2"n" subject to the relation

:$v^2=Q(v)$.

Spinors are modules of the Clifford algebra, and so in particular there is an action of C²ⁿ on the space of spinors. The subset of C^2"n" that annihilates a given spinor ψ is a complex subspace C^"m". If ψ is nonzero then "m" is less than or equal to "n". If "m" is equal to "n" then ψ is said to be a "pure spinor".

The set of pure spinors

Every pure spinor is annihilated by a half-dimensional subspace of "C"²ⁿ. Conversely given a half-dimensional subspace it is possible to determine the pure spinor that it annihilates up to multiplication by a complex number. Pure spinors defined up to complex multiplication are called projective pure spinors. The space of projective pure spinors is the homogeneous space

:SO("2n")/U("n").

Not all spinors are pure. In general pure spinors may be separated from impure spinors via a series of quadratic equations called pure spinor constraints. However in 6 or less real dimensions all spinors are pure. In 8 dimensions there is, projectively, a single pure spinor constraint. In 10 dimensions, the case relevant for superstring theory, there are 10 constraints

:$psiGamma^\{mu\}psi\; =\; 0.,$

where Γ^μ are the gamma matrices, which represent the vectors C²ⁿ that generate the Clifford algebra. In general there are

:$\{2n\; choose\; n\; -\; 4\}$

constraints.

Pure spinors in string theory

Recently pure spinors have attracted attention in string theory. In the year 2000 Nathan Berkovits, professor at [http://www.ift.unesp.br/ Instituto de Fisica Teorica] in São Paulo-Brazil introduced the pure spinor formalism in his paper [http://xxx.lanl.gov/abs/hep-th/0001035 Super-Poincare covariant quantization of the superstring] . This formalism is the only known quantization of the superstring which is manifestly covariant with respect to both spacetime and worldsheet supersymmetry. In 2002 Nigel Hitchin introduced generalized Calabi-Yau manifolds in his paper [http://xxx.lanl.gov/abs/math.DG/0209099 Generalized Calabi-Yau manifolds] , where the generalized complex structure is defined by a pure spinor. These spaces describe the geometries of flux compactifications in string theory.

References

* 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. [http://csusap.csu.edu.au/~pcharlto/charlton_thesis.pdf The geometry of pure spinors, with applications] , PhD thesis (1997).
* [http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=hep-th&level=2&index1=5751497 Pure spinor on arxiv.org]