Pure spinor

In the domain of mathematics known as representation theory, pure spinors (or simple spinors) are spinors that are annihilated, under the Clifford algebra representation, by a maximal isotropic subspace of a vector space ${\displaystyle V}$ with respect to a scalar product ${\displaystyle Q}$. They were introduced by Élie Cartan^[1] in the 1930s and further developed by Claude Chevalley.^[2]

They are a key ingredient in the study of spin structures and higher dimensional generalizations of twistor theory,^[3] introduced by Roger Penrose in the 1960s. They have been applied to the study of

Clifford algebra and pure spinors

Consider a complex vector space ${\displaystyle V}$, with either even dimension ${\displaystyle 2n}$ or odd dimension ${\displaystyle 2n+1}$, and a nondegenerate complex

${\displaystyle Q}$

${\displaystyle Q(u,v)}$

${\displaystyle (u,v)}$

${\displaystyle Cl(V,Q)}$

${\displaystyle V}$

${\displaystyle u\otimes v+v\otimes u=2Q(u,v),\quad \forall \ u,v\in V.}$

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. In terms of stratification of spinor modules by orbits of the spin group ${\displaystyle Spin(V,Q)}$, pure spinors correspond to the smallest orbits, which are the Shilov boundary of the stratification by the orbit types of the spinor representation on the irreducible spinor (or half-spinor) modules.

Pure spinors, defined up to projectivization, are called projective pure spinors. For ${\displaystyle \,V\,}$ of even dimension ${\displaystyle 2n}$, the space of projective pure spinors is the homogeneous space ${\displaystyle SO(2n)/U(n)}$; for ${\displaystyle \,V\,}$ of odd dimension ${\displaystyle 2n+1}$, it is ${\displaystyle SO(2n+1)/U(n)}$.

Irreducible Clifford module, spinors, pure spinors and the Cartan map

The irreducible Clifford/spinor module

Following Cartan^[1] and Chevalley,^[2] we may view ${\displaystyle V}$ as a direct sum

${\displaystyle V=V_{n}\oplus V_{n}^{*}\ {\text{ or }}\ V=V_{n}\oplus V_{n}^{*}\oplus \mathbf {C} ,}$

where ${\displaystyle V_{n}\subset V}$ is a totally isotropic subspace of dimension ${\displaystyle n}$, and ${\displaystyle V_{n}^{*}}$ is its dual space, with scalar product defined as

${\displaystyle Q(v_{1}+w_{1},v_{2}+w_{2}):=w_{2}(v_{1})+w_{1}(v_{2}),\quad v_{1},v_{2}\in V_{n},\ w_{1},w_{2}\in V_{n}^{*},}$

or

${\displaystyle Q(v_{1}+w_{1}+a_{1},v_{2}+w_{2}+a_{2}):=w_{2}(v_{1})+w_{1}(v_{2})+a_{1}a_{2},\quad a_{1},a_{2}\in \mathbf {C} ,}$

respectively.

The Clifford algebra representation ${\displaystyle \Gamma _{X}\in \mathrm {End} (\Lambda (V_{n}))}$ as endomorphisms of the irreducible Clifford/spinor module ${\displaystyle \Lambda (V_{n})}$, is generated by the linear elements ${\displaystyle X\in V}$, which act as

${\displaystyle \Gamma _{v}(\psi )=v\wedge \psi \ {\text{ (wedge product) }}\ {\text{for }}v\in V_{n}\ {\text{ and }}\Gamma _{w}(\psi )=\iota (w)\psi \ {\text{ (inner product) }}{\text{for}}\ w\in V_{n}^{*},}$

for either ${\displaystyle V=V_{n}\oplus V_{n}^{*}}$ or ${\displaystyle V=V_{n}\oplus V_{n}^{*}\oplus \mathbf {C} }$, and

${\displaystyle \Gamma _{a}\psi =(-1)^{p}a\ \psi ,\quad a\in \mathbf {C} ,\ \psi \in \Lambda ^{p}(V_{n}),}$

for ${\displaystyle V=V_{n}\oplus V_{n}^{*}\oplus \mathbf {C} }$, when ${\displaystyle \psi }$ is homogeneous of degree ${\displaystyle p}$.

Pure spinors and the Cartan map

A pure spinor ${\displaystyle \psi }$ is defined to be any element ${\displaystyle \psi \in \Lambda (V_{n})}$ that is annihilated by a maximal isotropic subspace ${\displaystyle w\subset V}$ with respect to the scalar product ${\displaystyle \,Q\,}$. Conversely, given a maximal isotropic subspace it is possible to determine the pure spinor that annihilates it, up to multiplication by a complex number, as follows.

Denote the Grassmannian of maximal isotropic (${\displaystyle n}$-dimensional) subspaces of ${\displaystyle V}$ as ${\displaystyle \mathbf {Gr} _{n}^{0}(V,Q)}$. The Cartan map ^[1][12][13]

${\displaystyle \mathbf {Ca} :\mathbf {Gr} _{n}^{0}(V,Q)\rightarrow \mathbf {P} (\Lambda (V_{n}))}$

is defined, for any element ${\displaystyle w\in \mathbf {Gr} _{n}^{0}(V,Q)}$, with basis ${\displaystyle (X_{1},\dots ,X_{n})}$, to have value

${\displaystyle \mathbf {Ca} (w):=\mathrm {Im} (\Gamma _{X_{1}}\cdots \Gamma _{X_{n}});}$

i.e. the image of ${\displaystyle \Lambda (V_{n})}$ under the endomorphism formed from taking the product of the Clifford representation endomorphisms ${\displaystyle \{\Gamma _{X_{i}}\in \mathrm {End} (\Lambda (V_{n}))\}_{i=1,\dots ,n}}$, which is independent of the choice of basis ${\displaystyle (X_{1},\cdots ,X_{n})}$. This is a ${\displaystyle 1}$-dimensional subspace, due to the isotropy conditions,

${\displaystyle Q(X_{i},X_{j})=0,\quad 1\leq i,j\leq n,}$

which imply

${\displaystyle \Gamma _{X_{i}}\Gamma _{X_{j}}+\Gamma _{X_{j}}\Gamma _{X_{i}}=0,\quad 1\leq i,j\leq n,}$

and hence ${\displaystyle \mathbf {Ca} (w)}$ defines an element of the projectivization ${\displaystyle \mathbf {P} (\Lambda (V_{n}))}$ of the irreducible Clifford module ${\displaystyle \Lambda (V_{n})}$. It follows from the isotropy conditions that, if the projective class ${\displaystyle [\psi ]}$ of a spinor ${\displaystyle \psi \in \Lambda (V_{n})}$ is in the image ${\displaystyle \mathbf {Ca} (w)}$ and ${\displaystyle X\in w}$, then

${\displaystyle \Gamma _{X}(\psi )=0.}$

So any spinor ${\displaystyle \psi }$ with ${\displaystyle [\psi ]\in \mathbf {Ca} (w)}$ is annihilated, under the Clifford representation, by all elements of ${\displaystyle w}$. Conversely, if ${\displaystyle \psi }$ is annihilated by ${\displaystyle \Gamma _{X}}$ for all ${\displaystyle X\in w}$, then ${\displaystyle [\psi ]\in \mathbf {Ca} (w)}$.

If ${\displaystyle V=V_{n}\oplus V_{n}^{*}}$ is even dimensional, there are two connected components in the isotropic Grassmannian ${\displaystyle \mathbf {Gr} _{n}^{0}(V,Q)}$, which get mapped, under ${\displaystyle \mathbf {Ca} }$, into the two half-spinor subspaces ${\displaystyle \Lambda ^{+}(V_{n}),\Lambda ^{-}(V_{n})}$ in the direct sum decomposition

${\displaystyle \Lambda (V_{n})=\Lambda ^{+}(V_{n})\oplus \Lambda ^{-}(V_{n}),}$

where ${\displaystyle \Lambda ^{+}(V_{n})}$ and ${\displaystyle \Lambda ^{-}(V_{n})}$ consist, respectively, of the even and odd degree elements of ${\displaystyle \Lambda ^{(}V_{n})}$ .

The Cartan relations

Define a set of bilinear forms ${\displaystyle \{\beta _{m}\}_{m=0,\dots 2n}}$ on the spinor module ${\displaystyle \Lambda (V_{n})}$, with values in ${\displaystyle \Lambda ^{m}(V^{*})\sim \Lambda ^{m}(V)}$ (which are isomorphic via the scalar product ${\displaystyle Q}$), by

${\displaystyle \beta _{m}(\psi ,\phi )(X_{1},\dots ,X_{m}):=\beta _{0}(\psi ,\Gamma _{X_{1}}\cdots \Gamma _{X_{m}}\phi ),\quad {\text{for }}\psi ,\phi \in \Lambda (V_{n}),\ X_{1},\dots ,X_{m}\in V,}$

where, for homogeneous elements ${\displaystyle \psi \in \Lambda ^{p}(V_{n})}$, ${\displaystyle \phi \in \Lambda ^{q}(V_{n})}$ and volume form ${\displaystyle \Omega }$ on ${\displaystyle \Lambda (V_{n})}$,

${\displaystyle \beta _{0}(\psi ,\phi )\,\Omega ={\begin{cases}\psi \wedge \phi \quad {\text{if }}p+q=n\\0\quad {\text{otherwise. }}\end{cases}}}$

As shown by Cartan,^[1] pure spinors ${\displaystyle \psi \in \Lambda (V_{n})}$ are uniquely determined by the fact that they satisfy the following set of homogeneous quadratic equations, known as the Cartan relations:^[1][12][13]

${\displaystyle \beta _{m}(\psi ,\psi )=0\quad \forall \ m\equiv n\mod (4),\quad 0\leq m<n}$

on the standard irreducible spinor module.

These determine the image of the submanifold of maximal isotropic subspaces of the vector space ${\displaystyle V,}$ with respect to the scalar product ${\displaystyle Q}$, under the Cartan map, which defines an embedding of the Grassmannian of isotropic subspaces of ${\displaystyle V}$ in the projectivization of the spinor module (or half-spinor module, in the even dimensional case), realizing these as projective varieties.

There are therefore, in total,

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

Cartan relations, signifying the vanishing of the bilinear forms ${\displaystyle \beta _{m}}$ 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 ${\displaystyle \,{\tfrac {1}{2}}\,n(n-1)\,}$ when ${\displaystyle \,V\,}$ is of even dimension ${\displaystyle 2n}$ and ${\displaystyle \,{\tfrac {1}{2}}\,n(n+1)\,}$ when ${\displaystyle \,V\,}$ has odd dimension ${\displaystyle 2n+1}$, and the Cartan map is an embedding of the connected components of this in the projectivization of the half-spinor modules when ${\displaystyle \,V\,}$ is of even dimension and in the irreducible spinor module if it is of odd dimension, 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.

In 6 dimensions or fewer, all spinors are pure. In 7 or 8  dimensions, there is a single pure spinor constraint. In 10 dimensions, there are 10 constraints

${\displaystyle \psi \;\Gamma _{\mu }\,\psi =0~,\quad \mu =1,\dots ,10,}$

where ${\displaystyle \,\Gamma _{\mu }\,}$ are the Gamma matrices that represent the vectors in ${\displaystyle \,\mathbb {C} ^{10}\,}$ that generate the Clifford algebra. However, only ${\displaystyle 5}$ of these are independent, so the variety of projectivized pure spinors for ${\displaystyle V=\mathbb {C} ^{10}}$ is ${\displaystyle 10}$ (complex) dimensional.

Applications of pure spinors

Supersymmetric Yang Mills theory

For ${\displaystyle d=10}$ dimensional, ${\displaystyle N=1}$ supersymmetric Yang-Mills theory, the super-ambitwistor correspondence,^[4][5] consists of an equivalence between the supersymmetric field equations and the vanishing of supercurvature along super null lines, which are of dimension ${\displaystyle (1|16)}$, where the ${\displaystyle 16}$ Grassmannian dimensions correspond to a pure spinor. Dimensional reduction gives the corresponding results for ${\displaystyle d=6}$, ${\displaystyle N=2}$ and ${\displaystyle d=4}$, ${\displaystyle N=3}$ or ${\displaystyle 4}$.

String theory and generalized Calabi-Yau manifolds

Pure spinors were introduced in string quantization by Nathan Berkovits.^[6] Nigel Hitchin^[14] introduced

${\displaystyle \tau }$-functions