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
$V$ of vectors with respect to the scalar product determining the Clifford algebra.
They were introduced by Élie Cartan
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 1960s.

Definition

Consider a complex vector space $V$ with either even complex dimension
$2n$ or odd complex dimension $2n+1$ and a nondegenerate complex scalar product
$Q$, with values $Q(u,v)$ on pairs of vectors $(u, v)$.
The Clifford algebra $Cl(V, Q)$ is the quotient of the full tensor algebra
on $V$ by the ideal generated by the relations

:$u\otimes v + v \otimes u = Q(u,v), \quad \forall \ u, v \in V.$

Spinors are modules of the Clifford algebra, and so in particular there is an action of the
elements of $V$ on the space of spinors. The complex subspace $V^0_\psi \subset V$ that annihilates
a given nonzero spinor $\psi$ has dimension $m \le n$. If $m=n$ then $\psi$ is said to be a ''pure spinor''.

Projective pure spinors

Every pure spinor is annihilated by a maximal isotropic subspace of $\,V\,$ with respect to the scalar product $\,Q\,$. 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 $\,V\,$ of dimension $\,2n\,$, the space of projective pure spinors is the homogeneous space

:$SO(2n)/U(n)~.$

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 $\,V\,$ under the Cartan map. In 7 dimensions, or fewer, all spinors are pure. In 8 dimensions there is a single pure spinor constraint. In 10 dimensions, there are 10 constraints

:$\psi \; \Gamma_\mu \, \psi = 0~,$

where $\,\Gamma_\mu\,$ are the Gamma matrices that represent the vectors
in $\,\mathbb^\,$ that generate the Clifford algebra. It was shown by Cartan that there are, in general,

:$\sum_$

quadratic relations, signifying the vanishing of the quadratic forms with values in the exterior spaces $\,\Lambda^m(V)\,$ for

:$m \equiv n, \text 4$

corresponding to these skew symmetric elements of the Clifford algebra. However, since the dimension of the Grassmannian of maximal isotropic subspaces of $\,V\,$ is $\,\tfrac\,n (n-1)\,$ and the Cartan map is an embedding of this in the projectivization of the half-spinor module when $\,V\,$ is of even dimension $2n$ and the irreducible spinor module if it is of odd dimension $\,2n+1\,$, the number of independent quadratic constraints is only

:$2^ - \tfrac\,n(n-1) - 1$

in the $\,2n\,$ dimensional case and

:$2^n - \tfrac\,n(n-1) - 1$

in the $\,2n + 1\,$ dimensional case.

Pure spinors in string theory

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 compactifications in string theory.

References

{{Reflist

* 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).

Spinors