Communications in Mathematical Physics

, Volume 66, Issue 3, pp 197–221 | Cite as

Graded manifold theory as the geometry of supersymmetry

  • John Dell
  • Lee Smolin


Building upon Kostant's graded manifold theory, we present a new way of introducing spinors into the spacetime manifold, by expanding the algebra of functions on spacetime to a graded algebra. The elements of differential geometry are generalized to accomodate the expanded algebra of functions and in this enriched geometry we find the elements of supersymmetry and of supergravity theory. The geometrical role of the supergravity fields is discussed and a derivation of their transformation rules is given.


Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics 
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  1. 1.
    Golfand, Yu. A., Likhtman, E.P.: JETP Letters13, 452 (1971)Google Scholar
  2. 2.
    Wess, J., Zumino, B.: Nucl. Phys. B70, 39 (1974); Phys. Lett. B49, 52 (1974)Google Scholar
  3. 3.
    Freedman, D.Z., Nieuwenhuizen, P. van, Ferrara, S.: Phys. Rev. D13, 3214 (1976); Freedman, D.Z., Nieuwenhuizen, P. van: Phys. Rev. D14, 912 (1976)Google Scholar
  4. 4.
    Deser, S., Zumino, B.: Phys. Lett. B62, 335 (1976)Google Scholar
  5. 5.
    Trautman, A.: In: Lectures on general relativity, Brandeis 1964, Summer Institute on Theoretical Physics, Vol. I. Englewood Cliffs, NJ: Prentice Hall 1965; Hawking, S.W., Ellis, G.F.R.: The large scale structure of spacetime. New York: Cambridge University Press 1973Google Scholar
  6. 6.
    Kostant, B.: Graded manifolds, graded Lie theory and prequantization. In: Differential geometric methods in mathematical physics. Lecture notes in mathematics, Vol. 570. Berlin, Heidelberg, New York: Springer 1977Google Scholar
  7. 7.
    Noether, E.: Gött. Nachr.235 (1918)Google Scholar
  8. 8.
    Batchelor, M.: Proc. Am. Math. Soc. (to be published)Google Scholar
  9. 9.
    Volkov, D.V., Akulov, A.D.: Phys. Lett. B46, 109 (1973); Salam, A., Strathdee, J.: Nucl. Phys. B76, 477 (1974); Ferrara, S., Wess, J., Zumino, B.: Phys. Lett. B51, 239 (1974)Google Scholar
  10. 10.
    Salam, A., Strathdee, J.: Phys. Rev. D11, 1521 (1975)Google Scholar
  11. 11.
    Batchelor, M.: Personal communicationGoogle Scholar
  12. 12.
    Nath, P., Arnowitt, R.: Phys. Lett. B56, 177 (1975); Nath, P.: Supersymmetry and gauged supersymmetry. In: Gauge theories and modern field theory. Arnowitt, R., Nath, P. (eds.). Cambridge, MA: MIT Press 1976Google Scholar
  13. 13.
    Siegal, W.: Preprints, Harvard University (Dec. 1977); Gates, S.J., Siegel, W.: Preprint, Harvard University (Aug. 1978); Nucl. Phys. B (submitted)Google Scholar
  14. 14.
    Wess, J., Zumino, B.: Phys. Lett. B74, 51 (1978); Zumino, B.: Cargese Summer School Lectures 1978. New York, London: Plenum Press (to be published)Google Scholar
  15. 15.
    Smolin, L.: Preprint HUTP-78/AO48, Harvard University (Nov. 1978)Google Scholar
  16. 16.
    Stella, K.S., West, P.C.: Phys. Lett. B74, 330 (1978); Ferrara, S., Nieuwenhuizen, P. van: Phys. Lett. B74, 333 (1978)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • John Dell
    • 1
  • Lee Smolin
    • 2
  1. 1.Department of Physics and AstronomyUniversity of MarylandCollege ParkUSA
  2. 2.Lyman Laboratory of PhysicsHarvard UniversityCambridgeUSA

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