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 with respect to a scalar product . 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 supersymmetric Yang-Mills theory in 10D,[4][5] superstrings,[6] generalized complex structures[7] [8] and parametrizing solutions of integrable hierarchies.[9][10][11]

Clifford algebra and pure spinors

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Consider a complex vector space  , with either even dimension   or odd dimension  , and a nondegenerate complex scalar product  , with values   on pairs of vectors  . The Clifford algebra   is the quotient of the full tensor algebra on   by the ideal generated by the relations

 

Spinors are modules of the Clifford algebra, and so in particular there is an action of the elements of   on the space of spinors. The complex subspace   that annihilates a given nonzero spinor   has dimension  . If   then   is said to be a pure spinor. In terms of stratification of spinor modules by orbits of the spin group  , 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   of even dimension  , the space of projective pure spinors is the homogeneous space  ; for   of odd dimension  , it is  .

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

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The irreducible Clifford/spinor module

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Following Cartan[1] and Chevalley,[2] we may view   as a direct sum

 

where   is a totally isotropic subspace of dimension  , and   is its dual space, with scalar product defined as

 

or

 

respectively.

The Clifford algebra representation   as endomorphisms of the irreducible Clifford/spinor module  , is generated by the linear elements  , which act as

 

for either   or  , and

 

for  , when   is homogeneous of degree  .

Pure spinors and the Cartan map

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A pure spinor   is defined to be any element   that is annihilated by a maximal isotropic subspace   with respect to the scalar product  . 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 ( -dimensional) subspaces of   as  . The Cartan map [1][12][13]

 

is defined, for any element  , with basis  , to have value

 

i.e. the image of   under the endomorphism formed from taking the product of the Clifford representation endomorphisms  , which is independent of the choice of basis  . This is a  -dimensional subspace, due to the isotropy conditions,

 

which imply

 

and hence   defines an element of the projectivization   of the irreducible Clifford module  . It follows from the isotropy conditions that, if the projective class   of a spinor   is in the image   and  , then

 

So any spinor   with   is annihilated, under the Clifford representation, by all elements of  . Conversely, if   is annihilated by   for all  , then  .

If   is even dimensional, there are two connected components in the isotropic Grassmannian  , which get mapped, under  , into the two half-spinor subspaces   in the direct sum decomposition

 

where   and   consist, respectively, of the even and odd degree elements of   .

The Cartan relations

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Define a set of bilinear forms   on the spinor module  , with values in   (which are isomorphic via the scalar product  ), by

 

where, for homogeneous elements  ,   and volume form   on  ,

 

As shown by Cartan,[1] pure spinors   are uniquely determined by the fact that they satisfy the following set of homogeneous quadratic equations, known as the Cartan relations:[1][12][13]

 

on the standard irreducible spinor module.

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

 

Cartan relations, signifying the vanishing of the bilinear forms   with values in the exterior spaces   for  , corresponding to these skew symmetric elements of the Clifford algebra. However, since the dimension of the Grassmannian of maximal isotropic subspaces of   is   when   is of even dimension   and   when   has odd dimension  , and the Cartan map is an embedding of the connected components of this in the projectivization of the half-spinor modules when   is of even dimension and in the irreducible spinor module if it is of odd dimension, the number of independent quadratic constraints is only

 

in the   dimensional case, and

 

in the   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

 

where   are the Gamma matrices that represent the vectors in   that generate the Clifford algebra. However, only   of these are independent, so the variety of projectivized pure spinors for   is   (complex) dimensional.

Applications of pure spinors

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Supersymmetric Yang Mills theory

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For   dimensional,   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  , where the   Grassmannian dimensions correspond to a pure spinor. Dimensional reduction gives the corresponding results for  ,   and  ,   or  .

String theory and generalized Calabi-Yau manifolds

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

Integrable systems

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In the approach to integrable hierarchies developed by Sato,[15] and his students,[16][17] 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,[9][10][11] parametrizing the associated BKP  -functions, which are generating functions for the flows. Under the Cartan map correspondence, these may be expressed as infinite dimensional Fredholm Pfaffians.[11]

References

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  1. ^ a b c d e Cartan, Élie (1981) [1938]. The theory of spinors. New York: Dover Publications. ISBN 978-0-486-64070-9. MR 0631850.
  2. ^ a b Chevalley, Claude (1996) [1954]. The Algebraic Theory of Spinors and Clifford Algebras (reprint ed.). Columbia University Press (1954); Springer (1996). ISBN 978-3-540-57063-9.
  3. ^ Penrose, Roger; Rindler, Wolfgang (1986). Spinors and Space-Time. Cambridge University Press. pp. Appendix. doi:10.1017/cbo9780511524486. ISBN 9780521252676.
  4. ^ a b Witten, E. (1986). "Twistor-like transform in ten dimensions". Nuclear Physics. B266 (2): 245–264. Bibcode:1986NuPhB.266..245W. doi:10.1016/0550-3213(86)90090-8.
  5. ^ a b Harnad, J.; Shnider, S. (1986). "Constraints and Field Equations for Ten Dimensional Super Yang-Mills Theory". Commun. Math. Phys. 106 (2): 183–199. Bibcode:1986CMaPh.106..183H. doi:10.1007/BF01454971. S2CID 122622189.
  6. ^ a b Berkovits, Nathan (2000). "Super-Poincare Covariant Quantization of the Superstring". Journal of High Energy Physics. 2000 (4): 18. arXiv:hep-th/0001035. doi:10.1088/1126-6708/2000/04/018.
  7. ^ Hitchin, Nigel (2003). "Generalized Calabi-Yau manifolds". Quarterly Journal of Mathematics. 54 (3): 281–308. doi:10.1093/qmath/hag025.
  8. ^ Gualtieri, Marco (2011). "Generalized complex geometry". Annals of Mathematics. (2). 174 (1): 75–123. arXiv:0911.0993. doi:10.4007/annals.2011.174.1.3.
  9. ^ a b Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1982). "Transformation groups for soliton equations IV. A new hierarchy of soliton equations of KP type". Physica. 4D (11): 343–365.
  10. ^ a b Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1983). M. Jimbo and T. Miwa (ed.). "Transformation groups for soliton equations". In: Nonlinear Integrable Systems - Classical Theory and Quantum Theory. World Scientific (Singapore): 943–1001.
  11. ^ a b c Balogh, F.; Harnad, J.; Hurtubise, J. (2021). "Isotropic Grassmannians, Plücker and Cartan maps". Journal of Mathematical Physics. 62 (2): 121701. arXiv:2007.03586. doi:10.1063/5.0021269. S2CID 220381007.
  12. ^ a b Harnad, J.; Shnider, S. (1992). "Isotropic geometry and twistors in higher dimensions. I. The generalized Klein correspondence and spinor flags in even dimensions". Journal of Mathematical Physics. 33 (9): 3197–3208. doi:10.1063/1.529538.
  13. ^ a b Harnad, J.; Shnider, S. (1995). "Isotropic geometry and twistors in higher dimensions. II. Odd dimensions, reality conditions, and twistor superspaces". Journal of Mathematical Physics. 36 (9): 1945–1970. doi:10.1063/1.531096.
  14. ^ Hitchin, Nigel (2003). "Generalized Calabi-Yau manifolds". Quarterly Journal of Mathematics. 54 (3): 281–308. doi:10.1093/qmath/hag025.
  15. ^ Sato, Mikio (1981). "Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds". Kokyuroku, RIMS, Kyoto Univ.: 30–46.
  16. ^ Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1981). "Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III–". Journal of the Physical Society of Japan. 50 (11). Physical Society of Japan: 3806–3812. Bibcode:1981JPSJ...50.3806D. doi:10.1143/jpsj.50.3806. ISSN 0031-9015.
  17. ^ Jimbo, Michio; Miwa, Tetsuji (1983). "Solitons and infinite-dimensional Lie algebras". Publications of the Research Institute for Mathematical Sciences. 19 (3). European Mathematical Society Publishing House: 943–1001. doi:10.2977/prims/1195182017. ISSN 0034-5318.

Bibliography

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