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Spinors in 2022

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Mathematics Going Forward

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2313))

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Abstract

In 1997 Springer published a second edition of the book The Algebraic Theory of Spinors by Claude Chevalley (cf. The algebraic theory of spinors, 1197), initially published in 1954 by Columbia University Press. On the Springer side, Catriona BYRNE was responsible for the project. (It was part of a more comprehensive endeavour supported by the CNRS to publish Claude Chevalley’s Complete Works, that also involved Michel Broué.) Because, in the interval between the two editions, so much had happened in Mathematics and in Physics involving spinors I was asked to provide a Postface to the second edition (cf. Bourguignon, Postface: spinors in 1995, 1997). This is one of the opportunities offered to me by Catriona over the years. At this time, spinors were indeed one of my main objects of study. I decided to entitle this contribution Spinors in 1995.

Since then, the topic of spinors continued to attract a lot of attention. This is why, for the special volume of the Lecture Notes in Mathematics dedicated to Catriona, I decided to contribute a follow up to the Postface. It was therefore natural to entitle it Spinors in 2022. I hope this short article will convince you of the appropriateness of the initiative.

Dedicated to Catriona BYRNE on the occasion of her retirement, in recognition of the great support provided over the years and of opportunities offered.

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References

  1. I. Agricola, S.G. Chiossi, T. Friedrich and J. Höll. Spinorial Description of SU3- and G2-Manifolds. J. Geom. Phys.98, 535–555 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Bär. Real Killing Spinors and Holonomy. Commun. Math. Phys.154, 509–521 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Bär, P. Gauduchon and A. Moroianu. Generalized Cylinders in Semi-Riemannian and Spin Geometry. Math. Z.249(3), 545–580 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Binz and R. Pferschy. The Dirac Operator and the Change of Metric. C.R. Math. Rep. Acad. Sci. Canada, V, 269–274 (1983).

    Google Scholar 

  5. J.-P. Bourguignon. Postface: Spinors in 1995, In: C. Chevalley The Algebraic Theory of Spinors (second edition), 199–210, Springer (1997).

    Google Scholar 

  6. J.-P. Bourguignon, Th. Branson, A. Chamseddine, O. Hijazi and R.J. Stanton (eds). Dirac Operators: Yesterday and Today. Proc. Summer School-Workshop held in Beirut in 2001, Int. Press. (2005).

    Google Scholar 

  7. J.-P. Bourguignon and P. Gauduchon. Spineurs, opérateurs de Dirac et variations de métriques. Commun. Math. Phys.144, 581–599 (1992).

    Article  MATH  Google Scholar 

  8. J.-P. Bourguignon, O. Hijazi, J.-L. Milhorat, A. Moroianu and S. Moroianu. A Spinorial Approach to Riemannian and Conformal Geometry. Monographs Math., European Math. Soc. (2015).

    Google Scholar 

  9. R. Brauer and H. Weyl. Spinors in n Dimensions. Amer. J. Math.57, 425–449 (1935).

    Article  MathSciNet  MATH  Google Scholar 

  10. É. Cartan. Les groupes projectifs qui ne laissent invariante aucune multiplicité plane. Bull. Soc. Math. France41, 53–96 (1913).

    Article  MathSciNet  MATH  Google Scholar 

  11. É. Cartan. Les groupes projectifs continus réels qui ne laissent invariante aucune multiplicité plane. J. Math. Pures Appl. (6) 10, 149–186 (1914).

    Google Scholar 

  12. É. Cartan. La théorie des spineurs, Gauthier-Villars, Paris (1937); second edition, The Theory of Spinors, Hermann, Paris (1966).

    Google Scholar 

  13. Q. Chen, J. Jost, J.Y. Liu and G.F. Wang. Dirac-Harmonic Maps. Math. Z.254, 409–432 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  14. Q. Chen, J. Jost, G. Wang and M. Zhu. The Boundary Value Problem for Dirac-Harmonic Maps. J. Eur. Math. Soc.15, 997–1031 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Chevalley. The Algebraic Theory of Spinors. Columbia Univ. Press (1954); second edition, Springer (1997).

    Google Scholar 

  16. P.A.M. Dirac. The Quantum Theory of the Electron. Proc. Roy. Soc.A117, 610–624 (1928).

    MATH  Google Scholar 

  17. Th. Friedrich. Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachr.97 (1980), 117–146.

    Article  MathSciNet  MATH  Google Scholar 

  18. Th. Friedrich. On the Spinor Representation of Surfaces in Euclidean 3-space. J. Geom. Phys.28, 143–157 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  19. Th. Friedrich, Dirac-Operatoren in der Riemannschen Geometrie. Adv. Lectures in Math., Fr. Viehweg & Sohn (1997); English edition: Dirac Operators in Riemannian Geometry, Grad. Stud, Math. 25, Amer. Math. Soc. (2000).

    Google Scholar 

  20. Th. Friedrich and E.C. Kim, Some Remarks on the Hijazi Inequality and Generalizations of the Killing Equation for Spinors. J. Geom. Phys.37, 1–14 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Gromov and H.B. Lawson Jr. Spin and Scalar Curvature in the Presence of a Fundamental Group. Ann. of Math.111(2), 209–230 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Gromov and H.B. Lawson Jr. The Classification of Simply Connected Manifolds of Positive Scalar Curvature. Ann. of Math.111(3), 423–434 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Herzlich and A. Moroianu. Generalized Killing Spinors and Conformal Eigenvalue Estimates for spinc Manifolds. Ann. Global Anal. Geom.17, 341–370 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  24. O. Hijazi. A Conformal Lower Bound for the Smallest Eigenvalue of the Dirac Operator and Killing Spinors. Commun. Math. Phys.104, 151–162 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  25. N. Hitchin. Harmonic Spinors, Adv. Math.14, 1–55 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  26. Y. Homma and U. Semmelmann. The Kernel of the Rarita-Schwinger Operator on Riemannian Spin Manifolds, Commun. Math. Phys.370, 853–871 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  27. B. Julia. Système linéaire associé aux équations d’Einstein. C. R. Acad. Sc. Paris295, 113–116 (1982).

    MATH  Google Scholar 

  28. J.L. Kazdan and F.W. Warner. Existence and Conformal Deformation of Metrics with Prescribed Gaussian and Scalar Curvatures. Ann. of Math.101(2), 317–331 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  29. R. Kusner and N. Schmitt. The Spinor Representation of Surfaces in Space. arXiv: dg-ga/961005, 1–52 (1996).

    Google Scholar 

  30. H.B. Lawzon Jr. and M.L. Michelsohn. Spin Geometry, Princeton Math. Series 38, Princeton Univ. Press (1989).

    Google Scholar 

  31. A. Lichnerowicz. Spineurs harmoniques. C.R. Acad. Sci. ParisA257, 7–9 (1963).

    MathSciNet  MATH  Google Scholar 

  32. S. Montiel. Dirac Operators and Hypersurfaces. Proc. 9thInt. Workshop Diff. Geom.9, 1–15 (2005).

    Google Scholar 

  33. B. Morel. Surfaces in S3 and H3, Actes Sém. théorie spectrale et géométrie Institut Fourier23, 131–144 (2004-2005).

    Article  Google Scholar 

  34. B. Morel. The Energy-Momentum Tensor as a Second Fundamental Form. arXiv: math/0302205, 1–13 (2003).

    Google Scholar 

  35. A. Moroianu. Parallel and Killing Spinors on spinc Manifolds. Commun. Math. Phys. 187, 417–428 (1997).

    Article  MATH  Google Scholar 

  36. A. Moroianu. Spinc Manifolds and Complex Contact Structures, Commun. Math. Phys.193, 661–673 (1998).

    Article  MATH  Google Scholar 

  37. R. Nakad. Lower Bounds for the Eigenvalues of the Dirac Operator on Spinc Manifolds. J. Geom. Phys.60, 1634–1642 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  38. W. Pauli. Zur Quantenmechanik des magnetischen Elektrons. Z. Physik43, 601–623 (1927).

    Article  MATH  Google Scholar 

  39. R. Penrose. A Spinor Approach to General Relativity. Ann. Phys. 10(2), 171–201 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  40. R. Penrose and R. Rindler. Spinors and Space-Time, Cambridge Univ. Press, Cambridge (1984).

    Book  MATH  Google Scholar 

  41. J. Rosenberg. C∗-Algebras, Positive Scalar Curvature, and the Novikov Conjecture. Publ. Math. Inst. Hautes Études Sci.58, 197–212 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Rosenberg and S. Stolz. A “Stable” Version of the Gromov-Lawson Conjecture. In: The Czech Centennial, 405-418, Contemp. Math. 181, American Mathematical Society (1995).

    Google Scholar 

  43. S. Stolz. Simply Connected Manifolds of Positive Scalar Curvature. Ann. of Math.136(3), 511–540 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  44. S. Stolz. Positive Scalar Curvature – Constructions and Obstructions. arXiv: math-DG/2202.05904, 1–34 (2022).

    Google Scholar 

  45. A. Trautmann. The Dirac Operator on Hypersurfaces. Acta Phys. Polon. B26, 1283–1310 (1995).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CNRS-Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France

    Jean-Pierre Bourguignon

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  1. Jean-Pierre Bourguignon
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Correspondence to Jean-Pierre Bourguignon .

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  1. CMLA, École Normale Supérieure, Cachan CX, France

    Jean-Michel Morel

  2. Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013, Paris, France

    Bernard Teissier

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Bourguignon, JP. (2023). Spinors in 2022. In: Morel, JM., Teissier, B. (eds) Mathematics Going Forward . Lecture Notes in Mathematics, vol 2313. Springer, Cham. https://doi.org/10.1007/978-3-031-12244-6_12

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