Duke Mathematical Journal

The geometry of degree-four characteristic classes and of line bundles on loop spaces I

J.-L. Brylinski and D. A. Mclaughlin

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Duke Math. J., Volume 75, Number 3 (1994), 603-638.

First available in Project Euclid: 20 February 2004

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Zentralblatt MATH identifier

Primary: 57R20: Characteristic classes and numbers
Secondary: 58B20: Riemannian, Finsler and other geometric structures [See also 53C20, 53C60] 58D15: Manifolds of mappings [See also 46T10, 54C35]


Brylinski, J.-L.; Mclaughlin, D. A. The geometry of degree-four characteristic classes and of line bundles on loop spaces I. Duke Math. J. 75 (1994), no. 3, 603--638. doi:10.1215/S0012-7094-94-07518-2. https://projecteuclid.org/euclid.dmj/1077287811

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  • [1] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.
  • [2] A. Beilinson, Higher regulators and values of $L$-functions, J.Soviet Math 30 (1985), 2036–2070.
  • [3] J. Benabou, Introduction to bicategories, Reports of the Midwest Category Seminar, Lecture Notes in Math., vol. 47, Springer, Berlin, 1967, pp. 1–77.
  • [4] L. Breen, Bitorseurs et cohomologie non abélienne, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 401–476.
  • [5] L. Breen, Théorie de Schreier supérieure, preprint, 1991.
  • [6] J. Brodski, A model of a classical field theory for $\mathrmMap(X,G)$, preprint, 1992.
  • [7] J.-L. Brylinski, The Kaehler geometry of the space of knots on a smooth threefold, preprint, 1990.
  • [8] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics, vol. 107, Birkhäuser Boston Inc., Boston, MA, 1993.
  • [9] J.-L. Brylinski and D. A. McLaughlin, A geometric construction of the first Pontryagin class, Quantum topology, Ser. Knots Everything, vol. 3, World Sci. Publishing, River Edge, NJ, 1993, pp. 209–220.
  • [10] J.-L. Brylinski and D. A. McLaughlin, Čech cocycles for characteristic classes, preprint, 1991.
  • [11] E. Cartan, La topologie des espaces représentatifs des groupes de Lie, Actualités Scientifiques et Industrielles, vol. 358, Hermann, Paris, 1936, and in Oeuvres Complètes, Part 1, Vol. 2, Gauthier-Villars, Paris, 1952, 1307–1330.
  • [12] P. Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. (1974), no. 44, 5–77.
  • [13] P. Deligne, Seminar on WZW theories, Inst. for Adv. Stud., Princeton, 1991, Spring.
  • [14] P. Deligne, J. S. Milne, A. Ogus, and K.-Y. Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin, 1982.
  • [15] R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), no. 2, 393–429.
  • [16] J. Duskin, An outline of a theory of higher-dimensional descent, Bull. Soc. Math. Belg. Sér. A 41 (1989), no. 2, 249–277.
  • [17] E. Friedlander, Étale homotopy of simplicial schemes, Annals of Mathematics Studies, vol. 104, Princeton University Press, Princeton, N.J., 1982.
  • [18] K. Gawedzki, Topological actions in two-dimensional quantum field theories, Nonperturbative quantum field theory (Cargèse, 1987) eds. G. t'Hooft and Jaffe and G. Mack and P. K. Mitter and R. Stora, NATO Adv. Sci. Inst. Ser. B Phys., vol. 185, Plenum, New York, 1988, pp. 101–141.
  • [19] J. Giraud, Cohomologie non abélienne, Springer-Verlag, Berlin, 1971.
  • [20] M. Gotay, R. Lashof, J. Śniatycki, and A. Weinstein, Closed forms on symplectic fibre bundles, Comment. Math. Helv. 58 (1983), no. 4, 617–621.
  • [21] A. Grothendieck, Compléments sur les biextensions: Propriétés générales des biextensions des schémas en groupes, Groupe de Monodromie en Géométrie Algébrique, Lecture Notes in Math. #7, vol. 288, Springer-Verlag, Berlin, 1972.
  • [22] R. Hain and R. MacPherson, Higher logarithms, Illinois J. Math. 34 (1990), no. 2, 392–475.
  • [23] T. P. Killingback, World-sheet anomalies and loop geometry, Nuclear Phys. B 288 (1987), no. 3-4, 578–588.
  • [24] B. Kostant, Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, 87–208. Lecture Notes in Math., Vol. 170.
  • [25] B. Mazur and W. Messing, Universal extensions and one dimensional crystalline cohomology, Lecture Notes in Mathematics, vol. 370, Springer-Verlag, Berlin, 1974.
  • [26] J. Mickelsson, Kac-Moody groups, topology of the Dirac determinant bundle, and fermionization, Comm. Math. Phys. 110 (1987), no. 2, 173–183.
  • [27] D. McLaughlin, Orientation and string structures on loop space, Pacific J. Math. 155 (1992), no. 1, 143–156.
  • [28] A. Pressley and G. Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1986.
  • [29] G. Segal, The definition of conformal field theory, preprint; condensed version in Differential Geometrical Methods in Theoretical Physics (Como 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 250, Kluwer, Dordrecht, 1988, 165–171.
  • [30] G. Segal, January 1990, Lecture at Math. Sci. Res. Inst., Berkeley.
  • [31] A. Weil, Introduction à l'étude des variétés kählériennes, Publications de l'Institut de Mathématique de l'Université de Nancago, VI. Actualités Sci. Ind. no. 1267, Hermann, Paris, 1958.
  • [32] A. Weinstein, The symplectic structure on moduli space, (in honor of Andreas Floer), preprint PAM-551, Berkeley, May 1992.
  • [33] E. Witten, The index of the Dirac operator in loop space, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 161–181.
  • [34] E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399.
  • [35] E. Witten, Free fermions on an algebraic curve, The mathematical heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 329–344.

See also

  • See also: J.-L. Brylinski, D. A. McLaughlin. The geometry of degree-$4$ characteristic classes and of line bundles on loop spaces II. Duke Math. J. Vol. 83, No. 1 (1996), pp. 105–139.