Skip to main content
SpringerLink
Log in

Strong connections on quantum principal bundles

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the two-sphere fibrationS 2→RP 2. A certain class of strongU q (2)-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq-dependent hermitian metric. A particular form of the Yang-Mills action on a trivialU q (2)-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent ofq.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abe, E.: Hopf Algebras. Cambridge: Cambridge University Press 1980

    MATH  Google Scholar 

  2. Aref'eva, I.Ya., Arutyunov, G.E.: Uniqueness ofU q (N) as a quantum gauge group and representations of its differential algebra.hep-th/9305176

  3. Aref'eva, I.Ya., Volovich, I.V.: Quantum group gauge fields. Mod. Phys. Lett. A6 (10), 893–907 (1991).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Booss, B., Bleecker, D.D.: Topology and Analysis. The Atiyah-Singer Formula and Gauge-Theoretic Physics. Berlin-Heidelberg-New York: Springer-Verlag, 1985

    MATH  Google Scholar 

  5. Bourbaki, N.: Algèbre Homologique. Paris: Masson, 1980

    MATH  Google Scholar 

  6. Brzeziński, T.: Differential Geometry of Quantum Groups and Quantum Fibre Bundles. Ph.D. thesis, Cambridge University, 1994

  7. Brzeziński, T.: Translation Map in Quantum Principal Bundles. To appear in J. Geom. Phys.hep-th/9407145

  8. Brzeziński, T., Majid, S.: Quantum Group Gauge Theory on Classical Spaces. Phys. Lett. B298, 339–43 (1993)hep-th/9210024

    Article  MathSciNet  ADS  Google Scholar 

  9. Brzeziński, T., Majid, S.: Quantum Group Gauge Theory on Quantum Spaces. Commun. Math. Phys.157, 591–638 (1993)hep-th/9208007; Erratum, Commun. Math. Phys.167, 235 (1995)

    Article  ADS  Google Scholar 

  10. Budzyński, R.J., Kondracki, W.: Quantum Principal Fiber Bundles: Topological Aspects.hep-th/9401019

  11. Connes, A.: The Action Functional in Non-Commutative Geometry. Commun. Math. Phys.117, 673–83 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Connes, A.: Non-Commutative Geometry. New York-London: Academic Press, 1994

    Google Scholar 

  13. Connes, A., Rieffel, M.: Yang-Mills for Non-Commutative Two-Tori. Contemp. Math.62, 237–66 (1987)

    MathSciNet  Google Scholar 

  14. Coquereaux, R.: Noncommutative Geometry and Theoretical Physics. J. Geom. Phys.6, (3) 425–90 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Doebner, H.D., Hennig, J.D., Lücke, W.: Mathematical Guide to Quantum Groups. In: Doebner, H.D., Hennig, J.D. (eds.) Quantum Groups. Proceedings of the 8th International Workshop on Mathematical Physics, Clausthal, West Germany, 19–26 July 1989, Berlin-Heidelberg-New York. Springer-Verlag 1990, pp. 29–63

    Google Scholar 

  16. Dubois-Violette, M.: Dérivations et calcul différentiel non commutatif. C.R. Acad. Sci. Paris307 (Série I), 403–8 (1988)

    MathSciNet  MATH  Google Scholar 

  17. Dubois-Violette, M., Kerner, R., Madore, J.: Noncommutative differential geometry of matrix algebras. J. Math. Phys.31 (2), 316–22 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Dubois-Violette, M., Kerner, R., Madore, J.: Noncommutative differential geometry and new models of gauge theory. J. Math. Phys.31 (2), 323–30 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Durdevic, M.: Geometry of Quantum Principal Bundles I. Commun. Math. Phys.175 (3), 457–521 (1996)q-alg/9507019

    Article  MathSciNet  ADS  Google Scholar 

  20. Durdevic, M.: Geometry of Quantum Principal Bundles II (Extended Version).q-alg/9412005

  21. Durdevic, M.: Quantum Principal Bundles and Corresponding Gauge Theories.q-alg/9507021

  22. Durdevic, M.: Quantum Principal Bundles. In: Proceedings of the XXII International Conference on Differential Geometric Methods in Theoretical Physics. Ixtapa, Mexico 1993.hep-th/9311029

  23. Durdevic, M.: On Framed Quantum Principal Bundles.q-alg/9507020

  24. Fadeev, L.D., Reshetikhin, N.Yu., Takhtadzhyan, L.A.: Quantization of Lie Groups and Lie Algebras. Leningrad Math. J.1 (1), 193–225 (1990)

    MathSciNet  Google Scholar 

  25. Hazewinkel, M.: Formal Groups and Applications. New York-London: Academic Press, 1978

    MATH  Google Scholar 

  26. Majid, S.: Introduction to Braided Geometry andq-Minkowski space. In: Castellani, L., Wess, J. (eds.) Proceedings of the International School of Physics “Enrico Fermi” CXXVII. IOS Press 1996hep-th/9410241

  27. Manin, Yu.I.: Quantum Groups and Noncommutative Geometry. Publications du CRM 1561, Univ. de Montreal, 1988

  28. Pflaum, M.J.: Quantum Groups on Fibre Bundles. Commun. Math. Phys.166 (2), 279–315 (1994)hep-th/9401085

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Podleś, P.: Quantum Spheres. Lett. Math. Phys.14, 521–31 (1987)

    Google Scholar 

  30. Rieffel, M.: Critical Points of Yang-Mills for Noncommutative Two-Tori. J. Differential Geom.31, 535–46 (1990)

    MathSciNet  MATH  Google Scholar 

  31. Schneider, H.J.: Principal Homogenous Spaces for Arbitrary Hopf Algebras. Isr. J. Math.72 (1–2), 167–95 (1990)

    Google Scholar 

  32. Schneider, H.J.: Hopf Galois Extensions, Crossed Products, and Clifford Theory. In: Bergen, J., Montgomery, S. (eds.), Advances in Hopf Algebras. Lecture Notes in Pure and Applied Mathematics.158, New York: Marcel Dekker, Inc. 1994

    Google Scholar 

  33. Shnider, S., Sternberg, S.: Quantum Groups. International Press Inc., 1993

  34. Spera, M.: A Symplectic Approach to Yang-Mills Theory for Non Commutative Tori. Can. J. Math.44 (2), 368–87 (1992)

    MathSciNet  MATH  Google Scholar 

  35. Sweedler, M.E.: Hopf Algebras. New York: Benjamin, 1969

    Google Scholar 

  36. Takhtadzhyan, L.A.: Lectures on Quantum Groups. In: Introduction to Quantum Groups and Integrable Massive Models of Quantum Field Theory. Singapore: World Scientific, 1990, pp 69–197

    Google Scholar 

  37. Trautman, A.: Differential Geometry for Physicists. Paris: Bibliopolis, 1984

    MATH  Google Scholar 

  38. Woronowicz, S.L.: Compact Matrix Pseudogroups. Commun. Math. Phys.111, 613–65 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. Woronowicz, S.L.: Differential Calculus on Compact Matrix Pseudogroups (Quantum Groups). Commun. Math. Phys.122, 125–70 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Wu, K., Zhang, R.-J.: Algebraic approach to gauge theory and its noncommutative extension. Commun. Theor. Phys.17 (2), 175–82 (1992)

    MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. Piotr M. Hajac

    Present address: Department of Mathematical Methods in Physics, Warsaw University, ul. Hoża 74, 00-682, Warsaw, Poland

Authors and Affiliations

  1. Mathematics Section, International Centre for Theoretical Physics, Strada Costiera 11, 34014, Trieste, Italy

    Piotr M. Hajac

Authors
  1. Piotr M. Hajac
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Connes

This work was in part supported by the NSF grant 1-443964-21858-2. Writing up the revised version was partially supported by the KBN grant 2 P301 020 07 and by a visiting fellowship at the International Centre for Theoretical Physics in Trieste.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hajac, P.M. Strong connections on quantum principal bundles. Commun.Math. Phys. 182, 579–617 (1996). https://doi.org/10.1007/BF02506418

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02506418

Keywords

Search

Navigation