# Pure spinor

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{{Short description|Class of spinors constructed using Clifford algebras}}
In the domain of [mathematics](/source/mathematics) known as [representation theory](/source/representation_theory), '''pure spinors''' (or '''simple spinors''') are [spinor](/source/spinor)s that are annihilated, under the [Clifford algebra representation](/source/Clifford_algebra), by a [maximal isotropic subspace](/source/Quadric_(algebraic_geometry)) of a vector space <math> V </math> with respect to a scalar product <math>Q</math>.  
They were introduced by [Élie Cartan](/source/%C3%89lie_Cartan)<ref name="Car">{{cite book | last=Cartan | first=Élie |author-link= Élie Cartan  |title=The theory of spinors | orig-date=1938 | url=https://books.google.com/books?isbn=0486640701 | publisher=[Dover Publications](/source/Dover_Publications) | location=New York | isbn=978-0-486-64070-9 | mr=631850 | year=1981}}</ref> in the 1930s and further developed by [Claude Chevalley](/source/Claude_Chevalley).<ref name ="Ch">{{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}}</ref> 
 
They are a key ingredient in the study of [spin structure](/source/spin_structure)s and higher dimensional generalizations of [twistor theory](/source/twistor_theory),<ref name="PenRin">{{Cite book|title=Spinors and Space-Time|author-link= Roger Penrose | last1=Penrose|first1=Roger|last2=Rindler|author-link2=Wolfgang Rindler|first2=Wolfgang|publisher=Cambridge University Press|year=1986|isbn=9780521252676|pages=Appendix|language=en|doi=10.1017/cbo9780511524486}}</ref> introduced by [Roger Penrose](/source/Roger_Penrose) in the 1960s. 
They have been applied to the study of [supersymmetric Yang-Mills theory](/source/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory) in 10D,<ref name="W2">{{cite journal | first1=E. | last1=Witten  | author-link= Witten | title=Twistor-like transform in ten dimensions | journal=Nuclear Physics  |  volume=B266 | pages=245–264 | year=1986 | issue=2 | doi= 10.1016/0550-3213(86)90090-8 | bibcode=1986NuPhB.266..245W }}</ref><ref name="HS1">{{cite journal | first1=J. | last1=Harnad | first2=S. | last2=Shnider | author1-link= John Harnad |title=Constraints and Field Equations for Ten Dimensional Super Yang-Mills Theory | journal=Commun. Math. Phys.  | volume=106 | pages=183–199 | year=1986 | issue=2 | doi= 10.1007/BF01454971 | bibcode=1986CMaPh.106..183H | s2cid=122622189 | url=http://projecteuclid.org/euclid.cmp/1104115696 }}</ref>  [superstrings](/source/Superstring_theory),<ref name="Berk"/>  [generalized complex structures](/source/generalized_complex_structures)<ref name="Hi">{{cite journal |authorlink=Nigel Hitchin |last=Hitchin |first=Nigel |doi=10.1093/qmath/hag025 |title=Generalized Calabi-Yau manifolds |journal=[Quarterly Journal of Mathematics](/source/Quarterly_Journal_of_Mathematics) |volume=54 |year=2003 |issue=3 |pages=281–308 }}</ref>
<ref name="Gu">{{cite journal |last=Gualtieri |first=Marco |doi=10.4007/annals.2011.174.1.3 |title=Generalized complex geometry |journal=[Annals of Mathematics](/source/Annals_of_Mathematics) |series=(2) |volume=174 |year=2011 |issue=1 |pages=75–123 | doi-access=free |arxiv=0911.0993 }}</ref>  and  parametrizing solutions of [integrable hierarchies](/source/Integrable_system).<ref name="DJKM3">{{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=Transformation groups for soliton equations IV. A new hierarchy of soliton equations of KP type | journal=Physica | volume=4D | issue=11 | year=1982  |pages=343–365 }}</ref><ref name="DJKM4">{{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=Transformation groups for soliton equations | journal=In: Nonlinear Integrable Systems - Classical Theory and Quantum Theory | publisher=World Scientific (Singapore) |  year=1983 | editor= M. Jimbo and T. Miwa  | pages=943–1001 }}</ref><ref name="BHH">{{cite journal | last1=Balogh | first1=F. | last2=Harnad | first2=J.  | last3=Hurtubise | first3=J. |author2-link= John Harnad | author3-link= Jacques Hurtubise (mathematician)| title=Isotropic Grassmannians, Plücker and Cartan maps | journal=Journal of Mathematical Physics | volume=62 | year=2021| issue=2 | pages=121701| doi=10.1063/5.0021269 | arxiv=2007.03586 | s2cid=220381007 }}</ref>

==Clifford algebra and pure spinors==
Consider a [complex](/source/complex_number) [vector space](/source/vector_space) <math> V </math>, with either even dimension <math> 2n </math> or odd  dimension <math> 2n+1 </math>, and a  nondegenerate complex [scalar product](/source/scalar_product) 
<math> Q </math>, with values <math> Q(u,v) </math> on pairs of vectors <math> (u, v) </math>.  
The [Clifford algebra](/source/Clifford_algebra) <math> Cl(V, Q) </math> is the quotient of the full [tensor algebra](/source/tensor) 
on <math> V </math> by the ideal generated by the relations
::<math>u\otimes v + v \otimes u = 2 Q(u,v), \quad \forall \ u, v \in V. </math>

[Spinor](/source/Spinor)s are [modules](/source/module_(mathematics)) of the Clifford algebra, and so in particular there is an action of the
elements of <math> V </math>  on the space of spinors.  The complex subspace <math> V^0_\psi \subset V </math> that annihilates 
a given nonzero spinor <math> \psi </math> has dimension <math> m \le n </math>. If <math> m=n </math>  then <math> \psi </math> is said to be a '''pure spinor'''. In terms of stratification of spinor modules by orbits of the [spin group](/source/spin_group) <math>Spin(V,Q)</math>, pure spinors correspond to the smallest orbits, which are the [Shilov boundary](/source/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 <math>\,V\,</math> of even dimension <math>2n</math>, the space of projective pure spinors is the [homogeneous space](/source/homogeneous_space)
<math> SO(2n)/U(n)</math>; for <math>\,V\,</math>  of odd dimension <math>2n+1</math>, it is <math> SO(2n+1)/U(n)</math>.

==Irreducible Clifford module, spinors, pure spinors and the Cartan map ==
=== The irreducible Clifford/spinor module ===
Following Cartan<ref name="Car"/> and Chevalley,<ref name="Ch"/> 
we may view <math>V</math> as a direct sum 
::<math>V= V_n \oplus V_n^*\ \text{ or }\  V= V_n \oplus V_n^*\oplus\mathbf{C},</math>
where <math>V_n\subset V</math> is a totally isotropic subspace of dimension <math>n</math>, and <math>V^*_n</math> is its [dual space](/source/dual_space), with scalar product defined as
::<math> 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, </math>

or
::<math> 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}, </math>

respectively.

The Clifford algebra representation <math>\Gamma_X \in \mathrm{End}(\Lambda(V_n))</math> as endomorphisms of the irreducible Clifford/spinor module <math>\Lambda(V_n)</math>, is generated by the linear elements <math>X\in V</math>, which act as
::<math> \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, </math>

for  either <math>V= V_n \oplus V_n^*</math> or <math>V= V_n \oplus V_n^*\oplus\mathbf{C}</math>, and
:: <math> \Gamma_a \psi = (-1)^p a\ \psi, \quad a \in \mathbf{C}, \  \psi \in \Lambda^p(V_n), </math>

for  <math>V= V_n \oplus V_n^*\oplus\mathbf{C}</math>, when <math> \psi </math> is homogeneous of degree <math>p</math>.

===Pure spinors and the Cartan map===
A '''pure spinor''' <math>\psi</math> is defined to be any element <math>\psi\in \Lambda (V_n) </math> that is annihilated by a [maximal isotropic subspace](/source/Quadric_(algebraic_geometry)) <math>w\subset V</math> with respect to the scalar product <math>\,Q\,</math>. 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 (<math>n</math>-dimensional) subspaces of <math>V</math> as <math>\mathbf{Gr}^0_n(V, Q)</math>. The '''Cartan map''' <ref name ="Car"/><ref name="HS2"/><ref name="HS3"/>
:: <math> \mathbf{Ca}:  \mathbf{Gr}^0_n(V, Q)\rightarrow  \mathbf{P}(\Lambda (V_n)) </math>

is defined, for any element <math>w\in  \mathbf{Gr}^0_n(V, Q)</math>, with basis <math>(X_1, \dots, X_n)</math>, to have value 
:: <math>\mathbf{Ca}(w): = \mathrm{Im}(\Gamma_{X_1}\cdots \Gamma_{X_n});</math>

i.e. the image of <math>\Lambda (V_n) </math> under the endomorphism formed from taking the product of the Clifford representation endomorphisms
<math>\{\Gamma_{X_i} \in \mathrm{End}(\Lambda (V_n))\}_{i=1, \dots, n}</math>, which is  independent of the choice of basis <math>(X_1, \cdots , X_n)</math>.
This is a <math>1</math>-dimensional subspace, due to the isotropy conditions,
:: <math>Q(X_i, X_j) =0, \quad 1\le i, j \le n, </math>
which imply

:: <math>\Gamma_{X_i} \Gamma_{X_j} + \Gamma_{X_j} \Gamma_{X_i}=0, \quad 1\le i, j \le n, </math>

and hence <math>\mathbf{Ca}(w)</math> defines an element of the projectivization <math> \mathbf{P}(\Lambda (V_n))</math> of the irreducible Clifford module  <math>\Lambda (V_n)</math>.
It follows from the isotropy conditions that, if the projective class <math>[\psi]</math> of a spinor <math>\psi \in \Lambda(V_n)</math> is in the image <math>\mathbf{Ca}(w)</math> and <math>X\in w</math>, then
:: <math> \Gamma_X(\psi) =0. </math>
So any spinor <math>\psi</math> with <math>[\psi]\in \mathbf{Ca}(w)</math> is annihilated, under the Clifford representation, by all elements of <math>w</math>. Conversely, if <math>\psi</math> is annihilated by <math>\Gamma_X</math> for all <math>X \in w</math>, then <math>[\psi]\in \mathbf{Ca}(w)</math>.

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

where <math>\Lambda^+(V_n)</math> and  <math> \Lambda^-(V_n) </math>  consist, respectively, of the even and odd degree elements of <math>\Lambda^(V_n) </math> .

===The Cartan relations ===
Define a set of bilinear forms <math>\{\beta_m\}_{m=0, \dots 2n}</math> on the spinor module <math>\Lambda(V_n)</math>, 
with values in <math>\Lambda^m(V^*) \sim  \Lambda^m(V)</math> (which are isomorphic via the scalar product <math>Q</math>), by
::<math> \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, </math>

where, for homogeneous elements <math>\psi\in \Lambda^p(V_n)</math>, 
<math>\phi\in \Lambda^q(V_n)</math> and [volume form](/source/volume_form) <math>\Omega</math> on <math>\Lambda(V_n)</math>,
::<math> \beta_0(\psi, \phi)\,\Omega = \begin{cases} 
 \psi \wedge \phi  \quad \text{if }p+q = n \\
0 \quad \text{otherwise. }
\end{cases} </math>
As shown by Cartan,<ref name ="Car"/> pure spinors <math>\psi\in \Lambda(V_n)</math> are uniquely determined by the fact that they satisfy the following set of homogeneous [quadratic equation](/source/quadratic_equation)s, known as the '''Cartan relations''':<ref name ="Car"/><ref name="HS2">{{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 }}</ref><ref name="HS3">{{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 }}</ref>
::<math> \beta_m(\psi, \psi) =0 \quad \forall\  m \equiv n \mod(4), \quad 0\le m <  n </math>
on the standard irreducible spinor module.

These determine the image of the submanifold of maximal isotropic subspaces of the vector space <math>V,</math> with respect to the scalar product <math>Q</math>, under the '''Cartan map''', which defines an embedding of the Grassmannian of isotropic subspaces of <math>V</math> 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,
::<math> \sum_{0\le m \le n-1 \atop m \equiv n, \text{ mod } 4} {\text{dim}(V) \choose m} </math>

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

:<math> 2^{n-1} - \tfrac{1}{2}\,n(n-1) - 1 </math>

in the <math>\,2n\,</math> dimensional case, and

:<math> 2^n - \tfrac{1}{2}\,n(n+1) - 1 </math>

in the <math>\,2n + 1\,</math> dimensional case.

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

:<math>\psi \; \Gamma_\mu \, \psi = 0~, \quad \mu= 1, \dots, 10, </math>

where <math>\,\Gamma_\mu\,</math> are the [Gamma matrices](/source/Gamma_matrices) that represent the vectors 
in <math>\,\mathbb{C}^{10}\,</math> that generate the Clifford algebra. However, only <math>5</math> of these are independent, so the variety of projectivized pure spinors for <math> V =\mathbb{C}^{10} </math> is <math>10</math> (complex) dimensional.

== Applications of pure spinors ==

=== Supersymmetric Yang Mills theory ===

For <math>d=10 </math> dimensional, <math> N=1</math> [supersymmetric Yang-Mills theory](/source/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory), the '''super-ambitwistor''' correspondence,<ref name = "W2"/><ref name = "HS1"/>  consists of an equivalence between the [supersymmetric field equations](/source/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory) and the vanishing of supercurvature along '''super null lines''', which are of dimension <math>(1 | 16) </math>, where the <math>16</math> Grassmannian dimensions correspond to a pure spinor. Dimensional reduction gives the corresponding results for <math>d=6</math>, <math> N=2</math> and <math>d=4</math>, <math> N=3</math> or <math> 4</math>.

===String theory and generalized Calabi-Yau manifolds===

Pure spinors were introduced in string quantization by Nathan Berkovits.<ref name="Berk">{{cite journal |last1=Berkovits |first1=Nathan |year=2000 |title=Super-Poincare Covariant Quantization of the Superstring 
|journal=[Journal of High Energy Physics](/source/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 }}</ref> [Nigel Hitchin](/source/Nigel_Hitchin)<ref>{{cite journal |authorlink=Nigel Hitchin |last=Hitchin |first=Nigel |doi=10.1093/qmath/hag025 |title=Generalized Calabi-Yau manifolds |journal=[Quarterly Journal of Mathematics](/source/Quarterly_Journal_of_Mathematics) |volume=54 |year=2003 |issue=3 |pages=281&ndash;308 }}
</ref>
introduced [generalized Calabi–Yau manifold](/source/generalized_Calabi%E2%80%93Yau_manifold)s, where the [generalized complex structure](/source/generalized_complex_structure) is defined by a pure spinor.  These spaces describe the geometry of [flux compactification](/source/Compactification_(physics))s in string theory.

=== Integrable systems===

In the approach to [integrable hierarchies](/source/Integrable_systems) developed by [Mikio Sato](/source/Mikio_Sato),<ref name="Sa">{{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}}</ref>  and his students,<ref name="DJKM1">{{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 }}</ref><ref name="DJKM2">{{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 }}</ref> equations of the hierarchy are viewed as compatibility conditions for commuting flows on an infinite dimensional [Grassmannian](/source/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](/source/Hilbert_space) under a suitably defined complex scalar product. They therefore serve as moduli for solutions of the BKP integrable hierarchy,<ref name="DJKM3"/><ref name="DJKM4"/><ref name="BHH"/> parametrizing the associated BKP [<math>\tau</math>-functions](/source/Tau_function_(integrable_systems)), which are generating functions for the flows. Under the '''Cartan map''' correspondence, these may be expressed as infinite dimensional Fredholm [Pfaffians](/source/Pfaffians).<ref name="BHH"/>

==References==
{{Reflist}}

== Bibliography ==

* {{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

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Adapted from the Wikipedia article [Pure spinor](https://en.wikipedia.org/wiki/Pure_spinor) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Pure_spinor?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.