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 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]

  1. ^ Cartan, Élie (1981) [1938]. The theory of spinors. New York: Dover Publications. ISBN 978-0-486-64070-9. MR 0631850.
  2. ^ 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. ^ 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. ^ 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. ^ Cite error: The named reference Berk was invoked but never defined (see the help page).
  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. ^ 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. ^ 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. ^ 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.