Lagrangian gauge theories on supermanifolds

  • Ugo Bruzzo
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1251)


We describe an approach to the formulation of superspace supersysmmetric field theories based on the theory of supermanifolds (in the sense of DeWitt-Rogers). As a first step, we set up a variational calculus on fibered supermanifolds. Suitable definitions of the properties of local gauge and general invariance of a supermanifold field theory are given and equivalence of these invariances to generalizations of Utiyama theorem is proved. We show that, under some conditions, the extremality of the action functional is locally equivalent to a set of differential equations on the supermanifold.

Another main point in this article is the generalization of Noether theorem to supermanifold field theory. It is indeed shown that the above mentioned invariances are equivalent to a pair of differential identities (strong conservation laws).

The paper ends with the discussion of an example, namely, superspace N=1 supergravity.


Local Injection General Invariance Variational Calculus Superspace Formulation Supersymmetric Field Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bruzzo U., and R.Cianci. Mathematical Foundations of Supermanifold Field Theories: Action Functional and Reduction to Spacetime. Boll. Un. Mat. Italiana in press.Google Scholar
  2. 2.
    Bruzzo U., and R. Cianci. Variational Calculus on Supermanifolds and Invariance Properties of Superspace Field Theories. Preprint, Dipartimento di Matematica, Genova (1986).zbMATHGoogle Scholar
  3. 3.
    Choquet-Bruhat Y. The Cauchy Problem in Classical Supergravity. Lett. Math. Phys. 7 (1983) 459–467.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Choquet-Bruhat Y. The Cauchy Problem in Extended Supergravity, N=1, d=11. Commun. Math. Phys. 97 (1985) 541–552.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bao D., Y. Choquet-Bruhat, J. Isenberg, and P.B. Yasskin. The Well-Posedness of (N=1) Classical Supergravity. J. Math. Phys. 24 (1985) 329–333.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dell J., and L. Smolin. Graded Manifold Theory as the Geometry of Supersymmetry. Commun. Math. Phys. 66 (1979) 197–221.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hernandez Ruiperez D., and J. Muñoz Masqué. Global Variational Calculus on Graded Manifolds, I: Graded Jet Bundles, Structure 1-Form and Graded Infinitesimal Contact Transformations. J. Math. Pures et Appl. 63 (1984) 283–309. Higher Order Jet Bundles for Graded Manifolds (Superjets). In M. Modugno (ed), Proceedings of the International Meeting on Geometry and Physics-Florence, October 1982. Pitagora Editrice, Bologna (1983).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hernandez Ruiperez D., and J. Muñoz Masqué. Global Variational Calculus on Greded Manifolds, II. J. Math. Pures et Appl. 64 (1985) 87–104.zbMATHGoogle Scholar
  9. 9.
    Perez-Rendon A., and D. Hernandez Ruiperez. Toward a Classical Field Theory on Graded Manifolds. In S. Benenti, M. Ferraris, M. Francaviglia (eds), Proceedings Journees Relativistes 1983, Pitagora Editrice, Bologna (1985).Google Scholar
  10. 10.
    Lopez Almorox A. Le Theoreme de Utiyama dans les varietés gradées de Kostant. Proceedings Journées Relativistes 1986, Toulouse, April 1986 (to appear).Google Scholar
  11. 11.
    Kerner R. Graded Gauge Theory. Commun. Math. Phys. 91 (1983) 213–234.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kerner R., and E.M. Da Silva Maia. Graded Gauge Theories over Supersymmetric Space. J. Math. Phys. 24 (1983) 361–368.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bruzzo U., and R. Cianci. Structure of Supermanifolds and Supersymmetry Transformations. Commun. Math. Phys. 95 (1984) 393–400.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    DeWitt B. Supermanifolds. Cambridge Univ. Press, London (1984).zbMATHGoogle Scholar
  15. 15.
    Rogers A. A Global Theory of Supermanifolds. J. Math. Phys. 21 (1980) 1352–1365.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jadczyk A., and K. Pilch. Superspaces and Supersymmetries. Commun. Math. Phys. 78 (1981) 373–390.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rogers A. Super Lie Groups: Global Topology and Local Structure. J. Math. Phys. 22 (1981) 939–945.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hoyos J., M. Quiros, J. Ramirez Mittelbrunn, and F. J. De Urries. Generalized Supermanifolds. I: Superspaces and Linear Operators. J. Math. Phys. 25 (1984) 833–840.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hoyos J., M. Quiros, J. Ramirez Mittelbrunn, and F. J. De Urries. Generalized Supermanifolds. II: Analysis on Superspaces. J. Math. Phys. 25 (1984) 841–846.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hoyos J., M. Quiros, J. Ramirez Mittelbrunn, and F. J. De Urries. Generalized Supermanifolds. III: P-Supermanifolds. J. Math. Phys. 25 (1984) 847–854.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rothstein M. On the Resolution of Some Difficulties in the Theory of G Manifolds. Preprint, University of Washington, Seattle, USA (1985).Google Scholar
  22. 22.
    Boyer C. P., and S. Gitler, The Theory of G-Supermanifolds. Trans. A.M.S. 285 (1984) 241.MathSciNetzbMATHGoogle Scholar
  23. 23.
    Bruzzo U., and R. Cianci. Mathematical Theory of Super Fibre Bundles. Class. Quantum Grav. 1 (1984) 213–226.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Bruzzo U., and R. Cianci. An Existence Result for Super Lie Groups. Lett. Math. Phys. 8 (1984) 279–288.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Bruzzo U., and R. Cianci. Differential Equations, Frobenius Theorem, and Local Flows on Supermanifolds. J. Phys. A: Math. Gen. 18 (1985) 417–423.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vladimirov V. S., and I. V. Volovich. Superanalysis. I. Differential Calculus. Theor. Math. Phys. 59 (1984) 317–335.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Vladimirov V. S., and I. V. Volovich. Superanalysis. II. Integral Calculus. Theor. Math. Phys. 60 (1985) 743–765.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rabin J. M., and L. Crane. Global Properties of Supermanifolds. Commun. Math. Phys. 100 (1985) 141–160.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Bruzzo U., and R. Cianci. On the Structure of Superfields in a Field Theory on a Supermanifold. Lett. Math. Phys. 11 (1986) 21–26.MathSciNetCrossRefGoogle Scholar
  30. 30.
    Jadczyk A., and K. Pilch. Classical Limit of CAR and Self-Duality of the Infinite Dimensional Grassmann Algebra. In Jancewicz and Lukierski (eds), Quantum Theory of Particles and Fields, World Scientific, Singapore (1983).Google Scholar
  31. 31.
    Garcia P. L. Gauge Algebras, Curvature and Symplectic Structures. J. Diff. Geom. 12 (1977) 209–227.zbMATHGoogle Scholar
  32. 32.
    Husemoller D. Fibre Bundles. McGraw-Hill, New York (1966).CrossRefzbMATHGoogle Scholar
  33. 33.
    Trautman A. Noether Equations and Conservation Laws. Commun. Math. Phys. 6 (1967) 248–261.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Garcia P. L., and A. Perez-Rendon. Symplectic Approach to the Theory of Quantized Fields. I. Commun. Math. Phys. 13 (1969) 24–44.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Catenacci R., C. Reina, and P. Teofilatto. On the Body of Supermanifolds. J. Math. Phys. 26 (1985) 671–676.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lang S. Differential Manifolds, Addison-Wesley, Reading MA (1972)zbMATHGoogle Scholar
  37. 37.
    Abraham R., J. E. Marsden, and T. Ratiu. Manifolds and Tensor Analysis and Applications. Addison-Wesley, Reading MA (1983).zbMATHGoogle Scholar
  38. 38.
    Garcia P. L., and A. Perez-Rendon. Symplectic Approach to the Theory of Quantized Fields. II Arch. Rat. Mech. Anal. 43 (1971) 101–124.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Goldschmidt H., and S. Sternberg. The Hamilton-Cartan Formalism in the Calculus of Variations. Ann. Inst. Fourier Grenoble 23 (1973) 203–267.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Bleecker D. Gauge Theory and Variational Principles. Addison-Wesley, Reading MA (1981).zbMATHGoogle Scholar
  41. 41.
    Rogers A. Consistent Superspace Integration. J. Math. Phys. 26 (1985) 385–392.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Rogers A. On the Existence of Global Integral Forms on Supermanifolds. J. Math. Phys 26 (1985) 2749–2753.MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Rogers A. Realising the Berezin Integral as a Superspace Contour Integral. J. Math. Phys. 27 (1986) 710–717.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Rabin J. M. Berezin Integration on General Fermionic Supermanifolds. Commun. Math. Phys. 103 (1986) 431–440.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Picken R. F., and K. Sundermeyer. Integration on Supermanifolds and a Generalized Cartan Calculus. Commun. Math. Phys. 102 (1985) 585–604.MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Bruzzo U. Geometry of Rigid Supersymmetry. Hadronic J. in press.Google Scholar
  47. 47.
    Hille E., and R. S. Phillips. Functional Analysis and Semi-Groups. AMS Coll. Publ. Vol. XXXI, Providence, RI (1957).Google Scholar
  48. 48.
    D'Auria R., P. Fre', P. K. Townsend, and P. Van Nieuwenhuizen. Invariance of Actions, Rheonomy, and the New Minimal N=1 Supergravity in the Group Manifold Approach. Ann. Phys. (NY) 155 (1984) 423–446.MathSciNetCrossRefGoogle Scholar
  49. 49.
    Bruzzo U. Supermanifolds and Geometry of Superspace. In preparation.Google Scholar
  50. 50.
    Wess J., and B. Zumino. Superspace Formulation of Supergravity. Phys. Lett. B 66 (1977) 361–364.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Ugo Bruzzo
    • 1
  1. 1.Dipartimento di MatematicaUniversita' di GenovaGenovaItaly

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