Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Supersymmetry and Complex Geometry

  • Chapter
Several Complex Variables III

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 9))

  • 1015 Accesses

Abstract

In the past years supersymmetric theories have gained great importance in physics. By this one intends field theoretical models based on a new form of symmetry dubbed supersymmetry. Supersymmetry connects boson and fermion fields with each other [13], [46], [48], [26], [6]. The observed properties of particles cannot satisfy the demands of supersymmetry (for instance, supersymmetry would lead to the equality of mass for the boson and the corresponding fermion). However, an increasing number of physicists have arrived at the conviction that the action functional of interactions encountered in nature must be supersymmetrical (although for the ground state (the physical vacuum) and, consequently, for the observed spectra of particles supersymmetry is broken). Perhaps the most weighty foundation for such a belief is the mathematical beauty of the supersymmetric theories and the remarkable property of cancellation of the divergencies appearing in these theories. It is question of the circumstance that in quantum field theories one encounters divergencies arising from the integration over large momenta (ultraviolet divergencies). In supersymmetry the most dangerous of these divergencies cancel. Moreover, there exist models completely free of ultraviolet divergencies. Presently great hopes are put on such supersymmetric theories which take account of the presence of gravitational interactions. Thus and important constituent part of these theories is played by supergravity, a supersymmetric theory containing Einstein’s theory of gravity.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  1. Baranov, M.A., Roslyi, A.A., Schwarz, A.S.: Super light like geodesies in supergravitation. Yad. Fiz. 41, 285–287 (1985) [Russian]. English transl.: Sov. J. Nucl. Phys. 41, 180-181 (1985). Zbl. 592.58013.

    MathSciNet  Google Scholar 

  2. Berezin, F.A.: Introduction to the algebra and analysis of anticommuting variables. Moscow: MGU 1983 [Russian]. Zbl. 527.15020. English transl.: Introduction to superanalysis (Part I). Dordrecht: Reidel 1987.

    Google Scholar 

  3. Berezin, F.A., Kats, G.I.: Lie groups with commuting and anticommuting variables. Mat. Sb. Nov. Ser. 82 (124), 343–359 (1970) [Russian]. English transl.: Math. USSR, Sb. 11 (1970), 311-325 (1971).

    Google Scholar 

  4. Bernshtem, I.N., Leites, D.A.: How to integrate differential forms on supermanifolds. Funkts. Anal. Prilozh. 11, No. 3, 70–71 (1976) [Russian]. English transl.: Funct. Anal. Appl. 11, 219-221 (1978). Zbl. 364.58005.

    Google Scholar 

  5. Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 113 (1974) 219–271 (1975). Zbl. 302.32015. Russian transl.: Usp. Mat. Nauk 38, No. 2 (230) 149-193 (1983).

    MathSciNet  Google Scholar 

  6. Fayet, P., Ferrara, S.: Supersymmetry. Phys. Rep. 32, 250–334 (1977).

    MathSciNet  Google Scholar 

  7. Ferber, A.: Supertwistors and conformal supersymmetry. Nucl. Phys. B 132, 55–64 (1978)

    Article  MathSciNet  Google Scholar 

  8. Ferrara, S., Zumino, B.: Supergauge invariant Yang-Mills theories. Nucl. Phys. B 79, 413–421 (1974).

    Article  Google Scholar 

  9. Gaiduk, A.V., Khudaverdyan, O.M., Schwarz, A.S.: Integration over surfaces in superspace. Teor. Mat. Fiz. 52, 375–383 (1982) [Russian]. English transl.: Theor. Math. Phys. 52, 862-868 (1983). Zbl. 513.58015.

    MathSciNet  Google Scholar 

  10. Galperin, A., Ivanov, E., Kalitzin, S., Ogievetsky, V., Sokachev, E.: Unconstrained N = 2 matter, Yang-Mills and supergravity theories in harmonic superspace. Classical Quantum Gravity 1, 469–498 (1984).

    Google Scholar 

  11. Gates, S.J., Siegel, W.: Understanding constraints in superspace formulation of supergravity. Nucl Phys. B 163, 519–545 (1980).

    Article  MathSciNet  Google Scholar 

  12. Gayduk, A.V. (= Gaiduk, A.V.), Romanov, V.N., Schwarz, A.S.: Supergravity and field space democracy. Commun. Math. Phys. 79, 507–528 (1981).

    Article  MathSciNet  Google Scholar 

  13. Gol’fand, Yu.A., Likhtman, E.P.: Extension of the algebra of generators of the Poincare group and violation of P-invariance. Zh. Eksper. Teor. Fiz. 13, 452–451 (1971) [Russian].

    Google Scholar 

  14. Green, P.S., Isenberg, J., Yasskin, P.B.: Non-self-dual gauge field theories. Phys. Lett. B 78, 462–464 (1978).

    Article  Google Scholar 

  15. Grimm, R., Sohnius, M, Wess, J.: Extended supersymmetry and gauge theories. Nucl. Phys. B 133, 275–284 (1978).

    Article  MathSciNet  Google Scholar 

  16. Henkin, G.M.(= Khenkin, G.M.): Tangent Cauchy-Riemann equations and Yang-Mills, Higgs and Dirac fields. In: Proc. of the ICM, Warszawa, Aug. 16–24, 1983, 809–827. Amster-dam etc.: North Holland 1984. Zbl. 584.58050.

    Google Scholar 

  17. Howe, P.: Supergravity in superspace. Nucl. Phys. B 199, 309–364 (1982).

    Article  MathSciNet  Google Scholar 

  18. Khudaverdian, O.M. (= Khudaverdyan, O.M.), Schwarz, A.S. (= Shvarts, A.S.), Tyupkin, Yu.S.: Integral invariants for supercanonical transformations. Lett. Math. Phys. 5, 517–522 (1981) Zbl. 521.58054

    Article  MathSciNet  MATH  Google Scholar 

  19. Khudaverdyan, O.M., Schwarz, A.S.: Additive and multiplicative functionals. Preprint ITEF-3. Moscow: 1980 [Russian]

    Google Scholar 

  20. Khudaverdyan, O.M., Schwarz, A.S.: Multiplicative functionals and gauge fields. Teor. Mat. Fiz. 46, 187–198 (1981) [Russian]

    Google Scholar 

  21. Khudaverdyan, O.M., Schwarz A.S.: Normal gauging in supergravity. Teor. Mat. Fiz. 57, 354–362 (1983) [Russian]

    MathSciNet  Google Scholar 

  22. Lee, H.-C. : The universal integral invariants of Hamiltonian systems and application to the theory of canonical transformations. Proc. R. Soc. Edinb., Sect. A 62, 237–246 (1947) Zbl. 30, 55

    MATH  Google Scholar 

  23. Manin, Yu.I. : Flag superspaces and the supersymmetric Yang-Mills equations. In: Problems of high energy physics and quantum theory of fields. Tr. Mezhdunarodnogo Seminara, Pro- tvino, 46–73. Serdukhov: IFVE 1982 [Russian]

    Google Scholar 

  24. Manin, Yu.I.: Supersymmetry and supergravity in the space of null supergeodesics. In: Group- theoretic methods in physics. Tr. Mezdunarodnogo Seminara, Zvenigorod, November 1982, Vol. 1, 203–208. Moscow: Nauka 1983 [Russian]. Zbl. 599.58004

    Google Scholar 

  25. Manin, Yu.I.: Gauge fields and complex geometry. Moscow: Nauka 1984. [Russian]. Zbl. 576.53002. English transl.: Berlin etc.: Springer-Verlag 1988

    MATH  Google Scholar 

  26. Ogievetskii, V.I., Mezinchesku, L.: The symmetry between bosons and fermions and super- fields. Usp. Fiz. Nauk 117, No. 4, 637–683 (1975) [Russian]

    Article  Google Scholar 

  27. Ogievetskii, V.I., Sokachev, E.S.: The simplest group for Einstein supergravity. Yad. Fiz. 31, 264–279 (1980) [Russian]

    MathSciNet  Google Scholar 

  28. Ogievetskii, V.I., Sokachev, E.S.: Axial gravitational superfield and the formalism of differential geometry. Yad. Fiz. 31, 821–840 (1980) [Russian]. Zbl. 569.35039. English transl.: Sov. J. Nucl. Phys. 31, 424-433 (1980).

    MathSciNet  Google Scholar 

  29. Ogievetskii, V.I., Sokachev, E.S.: Normal gauge in supergravity. Yad. Fiz .32, 862–869 (1980) [Russian]

    MathSciNet  Google Scholar 

  30. Rosly, A.A.(= Roslyi, A.A.): Geometry of N = 1 Yang-Mills theory in curved superspace. J. Phys. A 15, 1663–1667 (1982).

    Article  MathSciNet  Google Scholar 

  31. Rosly, A.A.: Gauge fields in superspace and twistors. Classical Quantum Gravity 2, 693–699 (1985). Zbl. 576.35075

    Article  MathSciNet  MATH  Google Scholar 

  32. Rosly, A.A., Schwarz, A.S.: Geometry of N = 1 supergravity. Commun. Math. Phys. 95, 161–184 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  33. Rosly, A.A., Schwarz, A.S.: Geometry of N = 1 supergravity, II. Commun. Math. Phys. 96, 285–309 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  34. Roslyi, A.A.: Constraints in supersymmetric Yang-Mills theory as integrability conditions. In: Group-theoretic methods in physics. Tr. Mezhdunarodnogo Seminara, Zvenigorod, November, 1982, pp. 263–268. Moscow: Nauka 1982 [Russian]. Zbl. 599.58056

    Google Scholar 

  35. Roslyi, A.A., Schwarz, A. A.: The geometry of nonminimal and alternative minimal supergravity. Yad. Fiz. 37, 786–794 (1983) [Russian]. Zbl. 592.53067. English transl.: Sov. J. Nucl. Phys. 37, 466-471 (1983).

    MathSciNet  Google Scholar 

  36. Salam, A., Strathdee, J.: Super-gauge transformations. Nuclear Phys. B 76, 477–482 (1974).

    MathSciNet  Google Scholar 

  37. Salam, A., Strathdee, J.: Super-symmetry and non-Abelian gauges. Phys. Lett. B 51, 353–355 (1974).

    Article  MathSciNet  Google Scholar 

  38. Schwarz, A.S.: Are the field and space variables on equal footing? Nucl. Phys. B 171, 154–166 (1980).

    Article  Google Scholar 

  39. Schwarz, A.S.: Supergravity, complex geometry and G-structures. Commun. Math. Phys. 87, 37–63 (1982).

    Article  MATH  Google Scholar 

  40. Schwarz, A.S.: Supergravity and complex geometry. Yad. Fiz. 34, 1114–1149 (1981) [Russian]

    Google Scholar 

  41. Sohnius, M.F.: Bianchi identities for supersymmetric gauge theories. Nucl. Phys. B 136, 461–474 (1978)

    Article  MathSciNet  Google Scholar 

  42. Sternberg, S.: Lectures on differential geometry. Englewood Cliffs: Prentice Hall 1964. Zbl. 129, 131

    MATH  Google Scholar 

  43. Szczyrba, W.: A symplectic structure for the Einstein-Maxwell field. Rep. Math. Phys. 12, 169–191 (1977). Zbl. 396.53020

    Article  MathSciNet  MATH  Google Scholar 

  44. Van Nieuwenhuizen, P.: Supergravity. Phys. Rep. 68, 189–398 (1981).

    Article  Google Scholar 

  45. Volkov, D.V.: Phenomenological Langrangeans. In: The Physics of elementary particles and the atomic nucleus, 3–41. Moscow: Atomizdat 1973 [Russian]

    Google Scholar 

  46. Volkov, D.V., Akulov, V.P.: The Goldstine field with one half spin. Teor. Mat. Fiz. 18, 39–50 (1974) [Russian].

    MathSciNet  Google Scholar 

  47. Ward, R.S.: On self-a lal fields. Phys. Lett. A 61, 81–82 (1977).

    Article  MathSciNet  Google Scholar 

  48. Wess, J.: Supersymmetry - supergravity. In: Topics in quantum field theory and gauge theories. Proc. VIII Internat. Seminar on Teor. Phys., Salamanca, June 1977. Lect. Notes Phys. 77, 81–125. Berlin etc.: Springer 1978.

    Google Scholar 

  49. Wess, J., Zumino, B.: Supergauge transformations in four dimensions. Nucl. Phys. B 70, 39–50 (1974).

    Article  MathSciNet  Google Scholar 

  50. Wess, J., Zumino, B.: Superspace formulation of supergravity. Phys. Lett. B 66, 361–364 (1977).

    Article  MathSciNet  Google Scholar 

  51. Witten, E.: An interpretation of classical Yang-Mills theory. Phys. Lett. B 77, 394–398 (1978).

    Article  Google Scholar 

Download references

Authors
  1. A. A. Roslyľ
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. M. Khudaverdyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. S. Schwarz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Central Economic and Mathematical Institute, Academy of Sciences of the USSR, ul. Kravisoka 32, 117418, Moscow, USSR

    G. M. Khenkin

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Roslyľ, A.A., Khudaverdyan, O.M., Schwarz, A.S. (1989). Supersymmetry and Complex Geometry. In: Khenkin, G.M. (eds) Several Complex Variables III. Encyclopaedia of Mathematical Sciences, vol 9. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61308-1_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-61308-1_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64785-7

  • Online ISBN: 978-3-642-61308-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation