Pure spinor

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

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

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

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

${\displaystyle SO(2n)/U(n).}$

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 ${\displaystyle V}$ 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

${\displaystyle \psi \Gamma _{\mu }\psi =0,}$

where ${\displaystyle \Gamma _{\mu }}$ are the Gamma matrices that represent the vectors in ${\displaystyle \mathbf {C} ^{10}}$ that generate the Clifford algebra. It was shown by Cartan that there are, in general,

${\displaystyle \sum _{0\leq m\leq n-1 \atop m\equiv n,\ {\text{mod }}4}{{\text{dim}}(V) \choose m}}$

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

${\displaystyle m\equiv n,\ {\text{mod }}4}$

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

${\displaystyle 2^{n-1}-{\tfrac {1}{2}}n(n-1)-1}$

in the ${\displaystyle 2n}$ dimensional case and

${\displaystyle 2^{n}-{\tfrac {1}{2}}n(n-1)-1}$

in the ${\displaystyle 2n+1}$ dimensional one.

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.

• 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).