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 ${\displaystyle V}$ 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 1960s.

Definition[]

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.

Projective pure spinors[]

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 dimensions, or fewer, 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 ${\displaystyle \,\mathbb {C} ^{10}\,}$ 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 case.

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[]

• 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).