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Groupoids and Lie Bigebras in Gauge and String Theories

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Differential Geometrical Methods in Theoretical Physics

Part of the book series: NATO ASI Series ((ASIC,volume 250))

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Abstract

This talk deals with two seemingly disparate mathematical concepts: Lie groupoids and algebroids, and Lie bigebras (bialgebras), also known as Lie-Hopf-algebras (LHA), which have recently found to have applications in quantum field theory, gauge theory, and string theory. Although many of the mathematical concepts have been around for some time, the interest in groupoids lies in potential application to string theories, ‘whereas Lie bigebras appear in recent work by Drinfel’d and others on “quantum groups” and their relation to the classical Yang-Baxter equation.

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References

  1. A. Connes, ‘Sur la théorie noncommutative de l’intégration,’ in LNM 725, Berlin-Heidelberg-New York: Springer Verlag, 1979, pp. 19–143. ‘Feuilletages et algèbres d’opérateurs,’ in Séminaire Bourbak. 79/80, LNM 842, Berlin-Heidelberg-New York: Springer Verlag, 1980, exp. 551. ‘A Survey of Foliations and Operator Algebras,’ in Ref. 2, pp. 521–632.

    Google Scholar 

  2. R. Kadison, Ed., Operator Algebras and Application., 2 vols. Proc. Symp. Pure Math. vol 38, Providence: AMS, 1982.

    MATH  Google Scholar 

  3. D. Kastler, ‘Lecture Notes on Connes’ Cyclic Cohomology, U. C. Irvine, 1984, and private communications. ‘Cyclic Cohomology within the Differential Envelope,’ Preprint CPT-84/P.1697 (To be published as a book by Hermann, Paris). ‘An Invitation to Alain Connes’ Cyclic Cohomology,’ ZiF Preprint 132b, Bielefeld, 1985. Private notes, received January 1988.

    Google Scholar 

  4. J. Bellissard, ‘K-Theory of C*-Algebras in Solid-State Physics,’ Lecture at the Conference on Statistical Mechanics and Field Theor.: Mathematical Aspect., Groningen, 26–30 August, 1985, Preprint CPT-85/P.1816, December 1985 (Contains an extensive Bibliography).

    Google Scholar 

  5. M. E. Mayer, ‘Quantization of Gauge Theories,’ vv Acta Phys. Austriac., Suppl. XXII., 481 (1981);

    Google Scholar 

  6. M. E. Mayer, ‘A Comment on “Ghosts and Geometry” ,’ Phys. Lett. B5., 355 (1983). ‘Quantum Theory in Vector Bundles,’ in: Fundamental Aspects of Quantum Theor., A. Frigerio and V. Gorini, eds., NATO ASI Series B: Physics Vol. 144, New York: Plenum Press, 1986, pp. 403–410.

    Google Scholar 

  7. M. E. Mayer, ‘Gauge Fields as Quantized Connection Forms,’ in: Differential Geometrical Methods in Mathematical Physics (Bonn 1975., K. Bleuler and A. Reetz, eds., LNM vol 570, Berlin-Heidelberg-New York: Springer Verlag, 1977 pp. 307–349.

    Chapter  Google Scholar 

  8. E. Witten, ‘Noncommutative Geometry and String Field Theory,’ Nucl. Phys. B26., 253–294 (1986).

    Article  MathSciNet  Google Scholar 

  9. E. Witten, ‘Interacting Field Theory of Open Superstrings,’ Nucl. Phys. B27., 291–324 (1986).

    Article  MathSciNet  Google Scholar 

  10. See, e. g., J. H. Schwarz, ‘Superstrings,’ Phys. Report. 89 (1982); Lectures at: Jerusalem Winter School, December 1984; Scottish Universities Summer School, June 1985; Ecole Normale Supérieure, August 1985; UC Irvine, 1986. M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theor., 2 vols., Cambridge Univ. Press, 1987. See also J. Schwarz (Ed.), Superstrings: The First Fifteen Year., 2 vols., World Scientific, Singapore, 1985.

    MATH  Google Scholar 

  11. R. Kugo and I. Ojima, ‘Local Covariant Operator Formalism of Non-Abelian Gauge Theories and Quark Confinement Problem,’ Prog. Theor. Phys. (Kyoto) Suppl. 66(1979). Cf. also Ref. 5 (the second reference contains an extensive bibliography).

    Google Scholar 

  12. J. Westman, ‘Harmonic Analysis on Groupoids,’ Pac. J. Math. 27, 621 (1968);

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Westman, ‘Cohomology For Ergodic Groupoids,’ Trans. AM., 146,465 (1969); Several Preprints, Univ. of Calif., Irvine, 1969–1975.

    Article  MathSciNet  Google Scholar 

  14. A. Ramsay, ‘Virtual Groups and Group Actions,’ Advances i. Math. 6, 253–322 (1971), and earlier work by G. W. Mackey quoted there.

    Article  MathSciNet  MATH  Google Scholar 

  15. B. L. Reinhart, The Differential Geometry of Foliation., Ergebnisse der Mathematik, Vol. 99 , Berlin-Heidelberg-New York: Springer Verlag, 1983. [Contains a detailed bibliography.] A. Kumpera and D. Spencer, Lie Equations, Vol.., Annals of Math. Studies, Nr. 73, Princeton University Press, Princeton, 1972. Appendix.

    Book  MATH  Google Scholar 

  16. J. N. Renault, ‘C*-Algebras of Groupoids and Foliations,’ In Ref. 2, Part 1, P 339; A Groupoid Approach to C*-Algebra., Lecture Notes In Mathematics No. 79., Springer Verlag, Berlin-Heidelberg-New York, 1980.

    Google Scholar 

  17. B. Lawson, Th. Quantitative Theory o. Foliations, AMS, Providence 1977;

    Google Scholar 

  18. R. Bott, Lectures on Characteristic Classes and Foliation., LNM 279, Berlin-Heidelberg-New York: Springer Verlag, 1972, Pp. 1 – 94. For a discussion of the fundamental groupoid, see E. H. Sp anier, A lgebraic Topo log., Sec. 7, New Yô rk: Mc Gr aw Hill, 1966; also Ref. 18.

    Google Scholar 

  19. L. Gross, ‘A Poincaré Lemma for Connection Forms,’ J. Funct. Anal., 63, 1–46 (1985); Private Communication, Nov. 1986.

    Article  MathSciNet  MATH  Google Scholar 

  20. H. G. Loos, ‘Internal Holonomy Groups of Yang-Mills Fields,’ J. Math. Phys. 8, 2114–2124 (1967).

    Article  MATH  Google Scholar 

  21. M. B. Menskii, ‘The Path Group and the Interaction of Quantized Strings,’ Zh. Eksp. Teor. Fiz. 90, 416 (1986); [Sov. Phys. JET. (1987)] and additional papers of the same author quoted there (in particular, in English: Lett. Math. Phys. 3, 513 (1979).

    MathSciNet  Google Scholar 

  22. G. Segal, this volume page.

    Google Scholar 

  23. F. Kamber and P. Tondeur, Foliated Bundles and Characteristic Classe., LNM 493, Berlin-Heidelberg-New York: Springer Verlag, 1975.

    MATH  Google Scholar 

  24. Yu. I. Manin, Gauge Theory and Complex Geometr., Nauka, Moscow, 1984 .

    Google Scholar 

  25. K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometr., LMS Lecture Notes 124, Cambridge University Press, 1987.

    Book  MATH  Google Scholar 

  26. M. F. Atiyah, ‘Complex-analytic Connections in Fibre Bundles,’ Trans. AM. 85, 181–207 (1957);

    Article  MathSciNet  MATH  Google Scholar 

  27. H. K. Nickerson, ‘On Differential Operators and Connections,’ Trans. AM. 99, 509–539 (1961) .

    Article  MathSciNet  MATH  Google Scholar 

  28. M. E. Mayer, Vector Bundles and Gauge Theor. (to be completed in 1988, Springer 1989?), in particular Chapter 4 Part II. This is a very incomplete bibliography on a vast subject, compiled in a hurry; it comes with my sincere apologies to authors who have not been cited.

    Google Scholar 

  29. V. G. Drinfel’d, ‘Kvantovye Gruppy (Quantum Groups),’ Zapiski Nauchnykh Seminarov LOM. T. 155, Leningrad, 1955, pp. 18–49. [English version: Proceedings of the International Congress of Mathematicians, Berkeley, 1986. A. M. Gleason, Ed., AMS, Providence, 1987, pp. 798–820.]

    Google Scholar 

  30. S. Doplicher, R. Haag, and J. Roberts, ‘Fields, Observables, and Gauge Transformations,’ Commun. Math. Phys. 13, 1; 15, 173 (1969).

    Article  Google Scholar 

  31. S. Doplicher, R. Haag, and J. Roberts, ‘Local Observables and Particle Statistics,’ Commun. Math. Phys. 23, 199 (1971); 35, 49 (1974).

    Article  MathSciNet  Google Scholar 

  32. S. Doplicher and J.E. Roberts, ‘Fields, Statistics and Nonabelian Gauge Groups,’ Commun. Math. Phys. 28, 331 (1972).

    Article  MathSciNet  Google Scholar 

  33. S. Doplicher and J.E. Roberts, ‘Compact Lie Groups Associated with Endomorphisms of C*-Algebras,’ Bull. AMS 1., 333 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  34. S. Doplicher and J. Roberts, ‘C*-Algebras and Duality for Compact Groups: Why there is a Compact Group of Symmetries in Particle Physics,’ Preprint, to appear in Proc. of fntern. Conf. o. Mathematical Physic., Marseille, July 16–25, 1986.

    Google Scholar 

  35. K. Fredenhagen and R. Haag, ‘Generally Covariant Quantum Field Theory and Scaling Limits,’ DESY Preprin. 86–066, June 1986.

    Google Scholar 

  36. R. Haag, H. Narnhofer, and U. Stein, ‘On Quantum Field Theory in Gravitational Background,’ Commun. Math. Phys. 94, 219–238 (1984).

    Article  MathSciNet  Google Scholar 

  37. A. Lichnerowicz, ‘Quantum Mechanics and Deformations of Geometrical Dynamics,’ in Quantum Theory, Groups, Fields, and Particle., A. O. Barut, ed., pp. 3–82, Dordrecht: Reidel, 1983.

    Chapter  Google Scholar 

  38. M. E. Mayer, ‘Automorphisms of C*-Algebras, Fell Bundles, W*-Bigebras, and the Description of Internal Symmetries in Algebraic Quantum Theory,’ Acta Phys. Austriac. Suppl. VIII, 177–226 (1971).

    Google Scholar 

  39. M. E. Mayer, ‘The Uses of Group—Theoretical Duality Theorems in Quantum Theory,’ in Proceedings of the Conference on Differential-Geometric Methods in Physic., K. Bleuler and A. Reetz, Eds., Bonn 1973.

    Google Scholar 

  40. J. E. Roberts, ‘Local Cohomology and Superselection Structure,’ Commun. Math. Phys. 51, 107 (1976); Lecture at the RCP No. 25, Strasbourg., December 1976.

    Article  MATH  Google Scholar 

  41. A. Weinstein, ‘The Local Structure of Poisson Manifolds,’ J. Diff. Geom. 18, 523–557 (1983).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, University of California, Irvine, CA, 92717, USA

    Meinhard E. Mayer

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  1. Institute for Theoretical Nuclear Physics, University of Bonn, Bonn, Germany

    K. Bleuler  & M. Werner  & 

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Mayer, M.E. (1988). Groupoids and Lie Bigebras in Gauge and String Theories. In: Bleuler, K., Werner, M. (eds) Differential Geometrical Methods in Theoretical Physics. NATO ASI Series, vol 250. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7809-7_8

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  • DOI: https://doi.org/10.1007/978-94-015-7809-7_8

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-8459-0

  • Online ISBN: 978-94-015-7809-7

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