Class of spinors constructed using Clifford algebras
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
V
{\displaystyle V}
with respect to a scalar product
Q
{\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 supersymmetric Yang-Mills theory in 10D,[ 4] [ 5] superstrings ,[ 6] generalized complex structures [ 7]
[ 8] and parametrizing solutions of integrable hierarchies .[ 9] [ 10] [ 11]
^ Cartan, Élie (1981) [1938]. The theory of spinors . New York: Dover Publications . ISBN 978-0-486-64070-9 . MR 0631850 .
^ 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 .
^ Penrose, Roger ; Rindler, Wolfgang (1986). Spinors and Space-Time . Cambridge University Press. pp. Appendix. doi :10.1017/cbo9780511524486 . ISBN 9780521252676 .
^ 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 .
^ 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 .
^ Cite error: The named reference Berk
was invoked but never defined (see the help page ).
^ Hitchin, Nigel (2003). "Generalized Calabi-Yau manifolds". Quarterly Journal of Mathematics . 54 (3): 281–308. doi :10.1093/qmath/hag025 .
^ 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 .
^ 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.
^ 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.
^ 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 .