pure spinor

Consider a complex vector space $V$, with either even dimension $2n$ or odd 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

::$u\backslash otimes\; v\; +\; v\; \backslash otimes\; u\; =\; 2\; Q(u,v),\; \backslash quad\; \backslash forall\; \backslash \; u,\; v\; \backslash in\; V.$

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 $V^0\_\backslash psi\; \backslash subset\; V$ that annihilates

a given nonzero spinor $\backslash psi$ has dimension $m\; \backslash le\; n$. If $m=n$ then $\backslash psi$ is said to be a pure spinor. In terms of stratification of spinor modules by orbits of the spin group $Spin(V,Q)$, 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 $\backslash ,V\backslash ,$ of even dimension $2n$, the space of projective pure spinors is the homogeneous space

$SO(2n)/U(n)$; for $\backslash ,V\backslash ,$ of odd dimension $2n+1$, it is $SO(2n+1)/U(n)$.

Following Cartan and Chevalley,

we may view $V$ as a direct sum

::$V=\; V\_n\; \backslash oplus\; V\_n^*\backslash \; \backslash text\{\; or\; \}\backslash \; V=\; V\_n\; \backslash oplus\; V\_n^*\backslash oplus\backslash mathbf\{C\},$

where $V\_n\backslash subset\; V$ is a totally isotropic subspace of dimension $n$, and $V^*\_n$ is its dual space, with scalar product defined as

::$Q(v\_1\; +\; w\_1,v\_2\; +\; w\_2)\; :=\; w\_2(v\_1)\; +\; w\_1(v\_2),\backslash quad\; v\_1,\; v\_2\; \backslash in\; V\_n,\; \backslash \; w\_1,\; w\_2\; \backslash in\; V^*\_n,$

or

::$Q(v\_1\; +\; w\_1\; +\; a\_1,v\_2\; +\; w\_2+a\_2)\; :=\; w\_2(v\_1)\; +\; w\_1(v\_2)\; +\; a\_1\; a\_2,\backslash quad\; a\_1,\; a\_2\; \backslash in\; \backslash mathbf\{C\},$

respectively.

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

::$\backslash Gamma\_v(\backslash psi)\; =\; v\; \backslash wedge\; \backslash psi\; \backslash \; \backslash text\{\; (wedge\; product)\; \}\; \backslash \; \backslash text\; \{for\; \}\; v\; \backslash in\; V\_n\; \backslash \; \backslash text\{\; and\; \}\; \backslash Gamma\_w(\backslash psi)\; =\; \backslash iota(w)\; \backslash psi\; \backslash \; \backslash text\{\; (inner\; product)\; \}\; \backslash text\{for\}\backslash \; w\; \backslash in\; V^*\_n,$

for either $V=\; V\_n\; \backslash oplus\; V\_n^*$ or $V=\; V\_n\; \backslash oplus\; V\_n^*\backslash oplus\backslash mathbf\{C\}$, and

:: $\backslash Gamma\_a\; \backslash psi\; =\; (-1)^p\; a\backslash \; \backslash psi,\; \backslash quad\; a\; \backslash in\; \backslash mathbf\{C\},\; \backslash \; \backslash psi\; \backslash in\; \backslash Lambda^p(V\_n),$

for $V=\; V\_n\; \backslash oplus\; V\_n^*\backslash oplus\backslash mathbf\{C\}$, when $\backslash psi$ is homogeneous of degree $p$.

A pure spinor $\backslash psi$ is defined to be any element $\backslash psi\backslash in\; \backslash Lambda\; (V\_n)$ that is annihilated by a maximal isotropic subspace $w\backslash subset\; V$ with respect to the scalar product $\backslash ,Q\backslash ,$. 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 ($n$-dimensional) subspaces of $V$ as $\backslash mathbf\{Gr\}^0\_n(V,\; Q)$. The Cartan map

:: $\backslash mathbf\{Ca\}:\; \backslash mathbf\{Gr\}^0\_n(V,\; Q)\backslash rightarrow\; \backslash mathbf\{P\}(\backslash Lambda\; (V\_n))$

is defined, for any element $w\backslash in\; \backslash mathbf\{Gr\}^0\_n(V,\; Q)$, with basis $(X\_1,\; \backslash dots,\; X\_n)$, to have value

:: $\backslash mathbf\{Ca\}(w):\; =\; \backslash mathrm\{Im\}(\backslash Gamma\_\{X\_1\}\backslash cdots\; \backslash Gamma\_\{X\_n\});$

i.e. the image of $\backslash Lambda\; (V\_n)$ under the endomorphism formed from taking the product of the Clifford representation endomorphisms

$\backslash \{\backslash Gamma\_\{X\_i\}\; \backslash in\; \backslash mathrm\{End\}(\backslash Lambda\; (V\_n))\backslash \}\_\{i=1,\; \backslash dots,\; n\}$, which is independent of the choice of basis $(X\_1,\; \backslash cdots\; ,\; X\_n)$.

This is a $1$-dimensional subspace, due to the isotropy conditions,

:: $Q(X\_i,\; X\_j)\; =0,\; \backslash quad\; 1\backslash le\; i,\; j\; \backslash le\; n,$

which imply

:: $\backslash Gamma\_\{X\_i\}\; \backslash Gamma\_\{X\_j\}\; +\; \backslash Gamma\_\{X\_j\}\; \backslash Gamma\_\{X\_i\}=0,\; \backslash quad\; 1\backslash le\; i,\; j\; \backslash le\; n,$

and hence $\backslash mathbf\{Ca\}(w)$ defines an element of the projectivization $\backslash mathbf\{P\}(\backslash Lambda\; (V\_n))$ of the irreducible Clifford module $\backslash Lambda\; (V\_n)$.

It follows from the isotropy conditions that, if the projective class $[\backslash psi]$ of a spinor $\backslash psi\; \backslash in\; \backslash Lambda(V\_n)$ is in the image $\backslash mathbf\{Ca\}(w)$ and $X\backslash in\; w$, then

:: $\backslash Gamma\_X(\backslash psi)\; =0.$

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

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

:: $\backslash Lambda(V\_n)\; =\; \backslash Lambda^+(V\_n)\; \backslash oplus\; \backslash Lambda^-(V\_n),$

where $\backslash Lambda^+(V\_n)$ and $\backslash Lambda^-(V\_n)$ consist, respectively, of the even and odd degree elements of $\backslash Lambda^(V\_n)$ .

Define a set of bilinear forms $\backslash \{\backslash beta\_m\backslash \}\_\{m=0,\; \backslash dots\; 2n\}$ on the spinor module $\backslash Lambda(V\_n)$,

with values in $\backslash Lambda^m(V^*)\; \backslash sim\; \backslash Lambda^m(V)$ (which are isomorphic via the scalar product $Q$), by

::$\backslash beta\_m(\backslash psi,\; \backslash phi)(X\_1,\; \backslash dots,\; X\_m)$

:=\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 $\backslash psi\backslash in\; \backslash Lambda^p(V\_n)$,

$\backslash phi\backslash in\; \backslash Lambda^q(V\_n)$ and volume form $\backslash Omega$ on $\backslash Lambda(V\_n)$,

::$\backslash beta\_0(\backslash psi,\; \backslash phi)\backslash ,\backslash Omega\; =\; \backslash begin\{cases\}$

\psi \wedge \phi \quad \text{if }p+q = n \\

0 \quad \text{otherwise. }

\end{cases}

As shown by Cartan, pure spinors $\backslash psi\backslash in\; \backslash Lambda(V\_n)$ are uniquely determined by the fact that they satisfy the following set of homogeneous quadratic equations, known as the Cartan relations:{{cite journal | last1=Harnad | first1=J. | last2=Shnider | first2=S. |author-link= John Harnad| title=Isotropic geometry and twistors in higher dimensions. I. The generalized Klein correspondence and spinor flags in even dimensions | journal=Journal of Mathematical Physics | volume=33 | issue=9 | year=1992 | doi=10.1063/1.529538 | pages=3197–3208 }}{{cite journal | last1=Harnad | first1=J. | last2=Shnider | first2=S. |author-link= John Harnad | title=Isotropic geometry and twistors in higher dimensions. II. Odd dimensions, reality conditions, and twistor superspaces | journal=Journal of Mathematical Physics | volume=36 | issue=9 | year=1995 | doi=10.1063/1.531096 | pages=1945–1970 | doi-access=free }}

::

on the standard irreducible spinor module.

These determine the image of the submanifold of maximal isotropic subspaces of the vector space $V,$ with respect to the scalar product $Q$, under the Cartan map, which defines an embedding of the Grassmannian of isotropic subspaces of $V$ 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,

::$\backslash sum\_\{0\backslash le\; m\; \backslash le\; n-1\; \backslash atop\; m\; \backslash equiv\; n,\; \backslash text\{\; mod\; \}\; 4\}\; \{\backslash text\{dim\}(V)\; \backslash choose\; m\}$

Cartan relations, signifying the vanishing of the bilinear forms $\backslash beta\_m$ with values in the exterior spaces $\backslash ,\backslash Lambda^m(V)\backslash ,$ for $m\; \backslash equiv\; n,\; \backslash text\{\; mod\; \}\; 4$, corresponding to these skew symmetric elements of the Clifford algebra. However, since the dimension of the Grassmannian of maximal isotropic subspaces of $\backslash ,V\backslash ,$ is $\backslash ,\backslash tfrac\{1\}\{2\}\backslash ,n\; (n-1)\backslash ,$ when $\backslash ,V\backslash ,$ is of even dimension $2n$ and $\backslash ,\backslash tfrac\{1\}\{2\}\backslash ,n\; (n+1)\backslash ,$ when $\backslash ,V\backslash ,$ has odd dimension $2n\; +1$, and the Cartan map is an embedding of the connected components of this in the projectivization of the half-spinor modules when $\backslash ,V\backslash ,$ is of even dimension and in the irreducible spinor module if it is of odd dimension, the number of independent quadratic constraints is only

:$2^\{n-1\}\; -\; \backslash tfrac\{1\}\{2\}\backslash ,n(n-1)\; -\; 1$

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

:$2^n\; -\; \backslash tfrac\{1\}\{2\}\backslash ,n(n+1)\; -\; 1$

in the $\backslash ,2n\; +\; 1\backslash ,$ 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

:$\backslash psi\; \backslash ;\; \backslash Gamma\_\backslash mu\; \backslash ,\; \backslash psi\; =\; 0~,\; \backslash quad\; \backslash mu=\; 1,\; \backslash dots,\; 10,$

where $\backslash ,\backslash Gamma\_\backslash mu\backslash ,$ are the Gamma matrices that represent the vectors

in $\backslash ,\backslash mathbb\{C\}^\{10\}\backslash ,$ that generate the Clifford algebra. However, only $5$ of these are independent, so the variety of projectivized pure spinors for $V\; =\backslash mathbb\{C\}^\{10\}$ is $10$ (complex) dimensional.

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

Pure spinors were introduced in string quantization by Nathan Berkovits.{{cite journal |last1=Berkovits |first1=Nathan |year=2000 |title=Super-Poincare Covariant Quantization of the Superstring

|journal=Journal of High Energy Physics |volume=2000 |issue=4 |pages=18 |doi = 10.1088/1126-6708/2000/04/018|doi-access=free |arxiv=hep-th/0001035 }} Nigel Hitchin{{cite journal |authorlink=Nigel Hitchin |last=Hitchin |first=Nigel |doi=10.1093/qmath/hag025 |title=Generalized Calabi-Yau manifolds |journal=Quarterly Journal of Mathematics |volume=54 |year=2003 |issue=3 |pages=281–308 }}

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.

In the approach to integrable hierarchies developed by Mikio Sato,{{cite journal|last= Sato|first = Mikio|author-link= Mikio Sato | title= Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds| journal= Kokyuroku, RIMS, Kyoto Univ.|pages= 30–46 | year =1981}} and his students,{{cite journal | last1=Date | first1=Etsuro | last2=Jimbo | first2=Michio | last3=Kashiwara | first3=Masaki | last4=Miwa | first4=Tetsuji | author2-link= Michio Jimbo | author3-link= Masaki Kashiwara | author4-link= Tetsuji Miwa |title=Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III– | journal=Journal of the Physical Society of Japan | publisher=Physical Society of Japan | volume=50 | issue=11 | year=1981 | issn=0031-9015 | doi=10.1143/jpsj.50.3806 | pages=3806–3812| bibcode=1981JPSJ...50.3806D }}{{cite journal | last1=Jimbo | first1=Michio | last2=Miwa | first2=Tetsuji |author1-link= Michio Jimbo | author2-link= Tetsuji Miwa | title=Solitons and infinite-dimensional Lie algebras | journal=Publications of the Research Institute for Mathematical Sciences | publisher=European Mathematical Society Publishing House | volume=19 | issue=3 | year=1983 | issn=0034-5318 | doi=10.2977/prims/1195182017 | pages=943–1001| doi-access=free }} equations of the hierarchy are viewed as compatibility conditions for commuting flows on an infinite dimensional Grassmannian. Under the (infinite dimensional) Cartan map, projective pure spinors are equivalent to elements of the infinite dimensional Grassmannian consisting of maximal isotropic subspaces of a Hilbert space under a suitably defined complex scalar product. They therefore serve as moduli for solutions of the BKP integrable hierarchy, parametrizing the associated BKP $\backslash tau$-functions, which are generating functions for the flows. Under the Cartan map correspondence, these may be expressed as infinite dimensional Fredholm Pfaffians.

{{Reflist}}

• {{cite book |last=Cartan |first=Élie |author-link=Élie Cartan |year=1981 |orig-year=1966 |title=The Theory of Spinors |place=Paris, FR |publisher=Hermann (1966)|series= Dover Publications |edition=reprint |isbn=978-0-486-64070-9}}
• {{cite book |last=Chevalley |first=Claude |author-link=Claude Chevalley |year=1996 |orig-year=1954 |title=The Algebraic Theory of Spinors and Clifford Algebras |publisher=Columbia University Press (1954); Springer (1996) |edition=reprint |isbn=978-3-540-57063-9}}
• Charlton, Philip. [http://csusap.csu.edu.au/~pcharlto/charlton_thesis.pdf The geometry of pure spinors, with applications], PhD thesis (1997).

Category:Spinors