Pure spinor

In a field 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. They were introduced by Élie Cartan in the 1930s 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 1960s, where they became among the basic objects of study in twistor theory.

Definition

Consider a complex vector space C²ⁿ with even complex dimension 2n and a quadratic form Q, which maps a vector v to complex number Q(v). The Clifford algebra Cℓ_2n(C) is the ring generated by products of vectors in C²ⁿ subject to the relation

${\displaystyle v^{2}=Q(v).\,}$

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 C²ⁿ 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 C²ⁿ. 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

${\displaystyle \psi \Gamma ^{\mu }\psi =0.\,}$

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

${\displaystyle {2n \choose n-4}}$

constraints.

Pure spinors in string theory

Recently pure spinors have attracted attention in string theory. In the year 2000 Nathan Berkovits, professor at Instituto de Fisica Teorica in São Paulo-Brazil introduced the pure spinor formalism in his paper Super-Poincare covariant quantization of the superstring. In 2002 Nigel Hitchin introduced generalized Calabi–Yau manifolds in his paper 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. The geometry of pure spinors, with applications, PhD thesis (1997).
• Pure spinor on arxiv.org