Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 078, 30 pages      arXiv:0801.1445

Deligne-Beilinson Cohomology and Abelian Link Invariants

Enore Guadagnini a and Frank Thuillier b
a) Dipartimento di Fisica ''E. Fermi'' dell'Università di Pisa and Sezione di Pisa dell'INFN, Italy
b) LAPTH, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux cedex, France

Received July 14, 2008, in final form October 27, 2008; Published online November 11, 2008

For the Abelian Chern-Simons field theory, we consider the quantum functional integration over the Deligne-Beilinson cohomology classes and we derive the main properties of the observables in a generic closed orientable 3-manifold. We present an explicit path-integral non-perturbative computation of the Chern-Simons link invariants in the case of the torsion-free 3-manifolds S3, S1 × S2 and S1 × Σg.

Key words: Deligne-Beilinson cohomology; Abelian Chern-Simons; Abelian link invariants.

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  1. Schwarz A.S., The partition function of degenerate quadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2 (1978), 247-252.
    Schwarz A.S., The partition function of a degenerate functional, Comm. Math. Phys. 67 (1979), 1-16.
  2. Hagen C.R., A new gauge theory without an elementary photon, Ann. Physics 157 (1984), 342-359.
  3. Polyakov A.M., Fermi-Bose transmutations induced by gauge fields, Modern Phys. Lett. A 3 (1988), 325-328.
  4. Witten E., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-399.
  5. Jones V.F.R., A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 103-111.
    Jones V.F.R., Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), 335-388.
  6. Reshetikhin N.Y., Turaev V.G., Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1-26.
    Reshetikhin N.Y., Turaev V.G., Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547-597.
  7. Deligne P., Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5-58.
  8. Beilinson A.A., Higher regulators and values of L-functions, J. Soviet Math. 30 (1985), 2036-2070.
  9. Esnault H., Viehweg E., Deligne-Beilinson cohomology, in Beilinson's Conjectures on Special Values of L-Functions, Editors M. Rapaport, P. Schneider and N. Schappacher, Perspect. Math., Vol. 4, Academic Press, Boston, MA, 1988, 43-91.
  10. Jannsen U., Deligne homology, Hodge-D-conjecture, and motives, in Beilinson's Conjectures on Special Values of L-Functions, Editors M. Rapaport, P. Schneider and N. Schappacher, Perspect. Math., Vol. 4, Academic Press, Boston, MA, 1988, 305-372.
  11. Brylinski J.L., Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics, Vol. 107, Birkhäuser Boston, Inc., Boston, MA, 1993.
  12. Cheeger J., Simons J., Differential characters and geometric invariants, Stony Brook Preprint, 1973 (reprinted in Geometry and Topology Proc. (1983-84), Editors J. Alexander and J. Harer, Lecture Notes in Math., Vol. 1167, Springer, Berlin, 1985, 50-90).
  13. Koszul J.L., Travaux de S.S. Chern et J. Simons sur les classes caractéristiques, Seminaire Bourbaki, Vol. 1973/1974, Lecture Notes in Math., Vol. 431, Springer, Berlin, 1975, 69-88.
  14. Harvey R., Lawson B., Zweck J., The de Rham-Federer theory of differential characters and character duality, Amer. J. Math. 125 (2003), 791-847, math.DG/0512251.
  15. Alvarez M., Olive D.I., The Dirac quantization condition for fluxes on four-manifolds, Comm. Math. Phys. 210 (2000), 13-28, hep-th/9906093.
    Alvarez M., Olive D.I., Spin and Abelian electromagnetic duality on four-manifolds, Comm. Math. Phys. 217 (2001), 331-356, hep-th/0003155.
  16. Alvarez A., Olive D.I., Charges and fluxes in Maxwell theory on compact manifolds with boundary, Comm. Math. Phys. 267 (2006), 279-305, hep-th/0303229.
  17. Alvarez O., Topological quantization and cohomology, Comm. Math. Phys. 100 (1985), 279-309.
  18. Gawedzki K., Topological Actions in two-dimensional quantum field theories, in Nonperturbative Quantum Field Theory (Cargèse, 1987), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 185, Plenum, New York, 1988, 101-141.
  19. Witten E., Topological quantum field theory, Comm. Math. Phys. 117 (1988), 353-386.
  20. Witten E., Dynamics of quantum field theory, in Quantum Fields and Strings: A Course for Mathematicians (Princeton, NJ, 1996/1997), Editors P. Deligne et al., Amer. Math. Soc., Providence, RI, 1999, Vol. 2, 1119-1424.
  21. Freed D.S., Locality and integration in topological field theory, in Group Theoretical Methods in Physics, Vol. 2, Editors M.A. del Olmo, M. Santander and J.M. Guilarte, CIEMAT, 1993, 35-54, hep-th/9209048.
  22. Zucchini R., Relative topological integrals and relative Cheeger-Simons differential characters J. Geom. Phys. 46 (2003), 355-393, hep-th/0010110.
    Zucchini R., Abelian duality and Abelian Wilson loops, Comm. Math. Phys. 242 (2003), 473-500, hep-th/0210244.
  23. Hopkins M.J., Singer I.M., Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005), 329-452, math.AT/0211216.
  24. Woodhouse N.M.J., Geometric quantization, 2nd ed., Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992.
  25. Bauer M., Girardi G., Stora R., Thuillier F., A class of topological actions, J. High Energy Phys. 2005 (2005), no. 8, 027, 35 pages, hep-th/0406221.
  26. Mackaay M., Picken R., Holonomy and parallel transport for Abelian gerbes, Adv. Math. 170 (2002), 287-339, math.DG/0007053.
  27. Godement R., Topologie algébrique et théorie des faisceaux, Actualit'es Sci. Ind., no. 1252, Publ. Math. Univ. Strasbourg, no. 13, Hermann, Paris, 1958 (reprinted, 1998).
  28. Bott R., Tu L.W., Differential forms in algebraic topology, Graduate Texts in Mathematics, Vol. 82, Springer-Verlag, New York - Berlin, 1982.
  29. Rolfsen D., Knots and links, Mathematics Lecture Series, no. 7, Publish or Perish, Inc., Berkeley, Calif., 1976.
  30. Calugareanu G., Sur les classes d'isotopie des noeuds tridimensionnels et leurs invariants, Czechoslovak Math. J. 11 (1961), 588-625.
  31. Calugareanu G., L'intégrale de Gauss et l'Analyse des noeuds tridimensionnels, Rev. Math. Pures Appl. 4 (1959), 5-20.
    Pohl W.F., The self-linking number of a closed space curve, J. Math. Mech. 17 (1967/1968), 975-985.
  32. Guadagnini E., Martellini M., Mintchev M., Wilson lines in Chern-Simons theory and link invariants, Nuclear Phys. B 330 (1990), 575-607.
  33. Guadagnini E., The link invariants of the Chern-Simons field theory. New developments in topological quantum field theory, de Gruyter Expositions in Mathematics, Vol. 10, Walter de Gruyter & Co., Berlin, 1993.
  34. Feynman R.P., Hibbs A.R., Quantum mechanics and path integrals, McGraw-Hill, New York, 1965.
  35. Coleman S., Aspects of symmetry, Cambridge University Press, New York, 1985.
  36. Elworthy D. and Truman A., Feynman maps, Cameron-Martin formulae and anharmonic oscillators, Ann. Inst. Henri Poincaré Phys. Théor. 41 (1984), 115-142.
  37. Ashtekar A., Lewandowski J., Representation theory of analytic holonomy C*-algebras, in Knots and Quantum Gravity (Riverside, CA, 1993), Editors J. Baez, Oxford Lecture Ser. Math. Appl., Vol. 1, Oxford Univ. Press, New York, 21-61, gr-qc/9311010.
  38. Baez J.C., Link invariants, holonomy algebras, and functional integration, J. Funct. Anal. 127 (1995), 108-131, hep-th/9301063.
  39. Lickorish W.B.R., Invariants for 3-manifolds from the combinatorics of the Jones polynomial, Pacific J. Math. 149 (1991), 337-386.
  40. Morton H.R., Strickland P.M., Satellites and surgery invariants, in Knots 90 (Osaka, 1990), Editor A. Kawauchi, de Gruyter, Berlin, 1992.
  41. Kirby R., A calculus for framed links in S3, Invent. Math. 45 (1978), 35-56.

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